diff --git "a/process_9/tokenized_finally.jsonl" "b/process_9/tokenized_finally.jsonl"
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@@ -0,0 +1,8236 @@
+{"id": "3385.png", "formula": "\\begin{align*} 0 = \\sum _ { j = 0 } ^ m r _ j s _ { n - j } . \\end{align*}"}
+{"id": "897.png", "formula": "\\begin{align*} T _ \\alpha = \\partial _ { \\alpha } + \\sum _ { \\beta < n } B _ { \\alpha \\beta } ( x _ { \\beta } \\partial _ { n } - x _ { n } \\partial _ { \\beta } ) , \\ ; \\ ; \\mbox { f o r } \\alpha < n , \\end{align*}"}
+{"id": "5110.png", "formula": "\\begin{align*} c _ { \\sigma } : = \\sqrt { \\frac { \\sin ( \\pi \\sigma ) } { \\pi } } . \\end{align*}"}
+{"id": "7814.png", "formula": "\\begin{align*} J ^ { w _ 1 } ( X ) & = - \\frac { 2 } { m + 3 } \\\\ J ^ { w _ 1 } ( Y ) & = - 2 \\left ( \\frac { 1 } { m + 1 } - \\frac { 2 } { m + 2 } + \\frac { 1 } { m + 3 } \\right ) \\end{align*}"}
+{"id": "2327.png", "formula": "\\begin{align*} \\zeta ( 1 , 2 ) = \\zeta ( 3 ) \\end{align*}"}
+{"id": "1279.png", "formula": "\\begin{align*} u _ { a , b } = ( b ^ \\ast \\otimes a ^ \\ast \\otimes \\varepsilon _ { a \\otimes b } ) \\circ ( b ^ \\ast \\otimes \\eta _ a \\otimes b \\otimes ( a \\otimes b ) ^ \\ast ) \\circ ( \\eta _ b \\otimes ( a \\otimes b ) ^ \\ast ) \\end{align*}"}
+{"id": "7926.png", "formula": "\\begin{align*} \\mu _ 1 ( a \\cdot b , c ) + \\mu _ 1 ( a , b ) \\cdot c = ~ & \\mu _ 1 ( a , b \\cdot c ) + a \\cdot \\mu _ 1 ( b , c ) , \\\\ R _ 1 ( a ) \\cdot R ( b ) + R ( a ) \\cdot R _ 1 ( b ) + \\mu _ 1 ( R ( a ) , R ( b ) ) = ~ & R _ 1 \\big ( R ( a ) \\cdot b + a \\cdot R ( b ) \\big ) + R \\big ( \\mu _ 1 ( R ( a ) , b ) ~ + ~ \\mu _ 1 ( a , R ( b ) ) \\big ) \\\\ ~ & + R \\big ( R _ 1 ( a ) \\cdot b + a \\cdot R _ 1 ( b ) \\big ) + \\kappa ~ \\mu _ n ( a , b ) , \\end{align*}"}
+{"id": "7334.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\sup _ { t \\in \\lbrack 0 , T ] } \\left \\vert \\int _ { 0 } ^ { t } \\Pi _ { \\tau \\varepsilon } ( s ) d s \\right \\vert ^ { 2 } + \\sup _ { t \\in \\lbrack 0 , T ] } \\left \\vert \\sum \\limits _ { i = 1 } ^ { d } \\int _ { 0 } ^ { t } \\Lambda _ { \\tau \\varepsilon } ^ { i } ( s ) d W _ { i } ( s ) \\right \\vert ^ { 2 } \\right ] \\leq C \\varepsilon ^ { 3 } . \\end{align*}"}
+{"id": "2357.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in \\mathfrak { S } _ { n + 1 } } { \\rm s g n } ( \\sigma ) x _ { a _ { \\sigma ( 1 ) } } x _ { b _ { 1 } } x _ { a _ { \\sigma ( 2 ) } } x _ { b _ { 2 } } \\cdots x _ { a _ { \\sigma ( n ) } } x _ { b _ { n } } x _ { a _ { \\sigma ( n + 1 ) } } & \\equiv - \\sum _ { \\sigma \\in \\mathfrak { S } _ { n + 1 } } \\mathrm { s g n } ( \\sigma ) f _ { b _ { 1 } } ( x _ { a _ { \\sigma ( 1 ) } } , \\ , x _ { a _ { \\sigma ( 2 ) } } x _ { b _ { 2 } } \\cdots x _ { a _ { \\sigma ( n ) } } x _ { b _ { n } } x _ { a _ { \\sigma ( n + 1 ) } } ) \\\\ & = 0 , \\end{align*}"}
+{"id": "6461.png", "formula": "\\begin{align*} \\| f \\| _ { C ^ \\tau _ * } = \\sup _ j 2 ^ { j \\tau } \\| \\psi _ j ( D ) f \\| _ { L ^ \\infty } < \\infty \\end{align*}"}
+{"id": "5287.png", "formula": "\\begin{align*} & \\chi _ B ( j ) = \\begin{cases} & 1 \\ \\ j \\in B , \\\\ & 0 \\ \\ j \\notin B , \\end{cases} \\\\ & \\chi _ { [ i ] } ( j ) = \\begin{cases} & 1 \\ \\ j \\in [ i ] = \\{ i \\ * ( n + 1 ) + 1 , \\ldots , ( i + 1 ) \\ * ( n + 1 ) \\} , \\\\ & 0 \\ \\ j \\notin [ i ] . \\end{cases} \\end{align*}"}
+{"id": "7868.png", "formula": "\\begin{align*} n _ 1 ( t , x ) = \\sup _ { g \\in C ^ 2 ( J ) } \\# F ( x ) \\cap B ( g , t ) , \\end{align*}"}
+{"id": "1823.png", "formula": "\\begin{align*} [ T ( a ) + a , T ( b ) + b ] _ { \\rho } = & [ T ( a ) , T ( b ) ] + \\rho ( T ( a ) ) b - \\rho ( T ( b ) ) a + \\lambda [ a , b ] _ V , \\end{align*}"}
+{"id": "8167.png", "formula": "\\begin{align*} \\| \\sum _ j A _ j X _ j \\| ^ 2 = \\| \\sum _ j X _ j ^ * P _ j X _ j \\| \\end{align*}"}
+{"id": "3265.png", "formula": "\\begin{align*} \\widehat { P } _ n ( m ) = \\frac { \\langle \\phi _ m , \\psi _ n \\rangle } { \\langle \\phi _ m , \\psi _ 0 \\rangle } , n , m = 0 , \\ldots , N . \\end{align*}"}
+{"id": "4999.png", "formula": "\\begin{align*} \\# \\bigg \\{ \\vec { v } = ( v _ 1 , v _ 2 , \\ldots , v _ N ) \\in \\{ 1 , 2 , \\ldots , n - 1 \\} ^ N : \\Re \\bigg ( \\prod _ { j = 1 } ^ N h \\big ( e _ n ( v _ j ) \\big ) \\bigg ) \\geq 1 \\bigg \\} \\leq \\big ( c + o ( 1 ) \\big ) \\big ( n - 1 \\big ) ^ N . \\end{align*}"}
+{"id": "5155.png", "formula": "\\begin{align*} D ( A ' , \\Omega , x ) = 1 D ( \\widetilde A , x ) = 1 , \\end{align*}"}
+{"id": "7688.png", "formula": "\\begin{align*} \\lim _ { | x | \\to 1 } | f ( x ) | ^ p \\Phi _ n ( x ) = 0 . \\end{align*}"}
+{"id": "2253.png", "formula": "\\begin{align*} \\sigma ^ m ( b ) = \\sigma ( a ) \\sigma ^ 2 ( a ) \\cdots \\sigma ^ m ( a ) = \\sigma ^ m ( a ) \\sigma ^ { m - 1 } ( a ) \\cdots \\sigma ( a ) = x ^ m y ^ m . \\end{align*}"}
+{"id": "6533.png", "formula": "\\begin{align*} A ( G ) = \\begin{pmatrix} \\mathbf { 0 } & C \\\\ C ^ T & \\mathbf { 0 } \\end{pmatrix} \\end{align*}"}
+{"id": "6545.png", "formula": "\\begin{align*} a = v , \\end{align*}"}
+{"id": "3212.png", "formula": "\\begin{align*} 0 & = h ' _ { g _ { u _ { n } } } ( 1 ) \\\\ & = - 2 \\beta A ( u _ { n } ) + \\left ( ( 1 - \\frac { 3 } { 2 } \\beta ) p + 3 \\beta - 2 \\right ) C ( u _ { n } ) + ( 2 - \\beta ) B ( u _ { n } ) \\\\ & \\ \\ \\ + \\frac { \\beta } { 4 } \\int _ { \\mathbb { R } ^ { 3 } } \\int _ { \\mathbb { R } ^ { 3 } } e ^ { - | x - y | } u _ { n } ^ { 2 } ( x ) u _ { n } ^ { 2 } ( y ) d x d y . \\end{align*}"}
+{"id": "5399.png", "formula": "\\begin{align*} \\begin{cases} \\Lambda ^ { - 1 } \\mathbb I _ n \\leq A \\leq \\Lambda \\mathbb I _ n , \\ \\Lambda \\geq 1 \\\\ | A ( x , t ) - A ( y , s ) | \\leq K ( | x - y | + | t - s | ^ { 1 / 2 } ) . \\end{cases} \\end{align*}"}
+{"id": "625.png", "formula": "\\begin{align*} Z G \\left ( X , Y \\right ) = G \\left ( \\nabla _ { Z } X , Y \\right ) + G \\left ( X , \\nabla _ { Z } ^ { \\dag } Y \\right ) , \\end{align*}"}
+{"id": "8115.png", "formula": "\\begin{gather*} P _ { d + 1 , x } = \\{ ( t _ 0 , \\ldots , t _ d ) \\in \\mathbb R _ { \\geqslant 0 } ^ { d + 1 } \\ , | \\ , t _ 0 + \\cdots + t _ d \\leqslant x \\} , \\\\ \\Delta _ { d , x } = \\{ ( t _ 0 , \\ldots , t _ d ) \\in \\mathbb R _ { \\geqslant 0 } ^ { d + 1 } \\ , | \\ , t _ 0 + \\cdots + t _ d = x \\} . \\end{gather*}"}
+{"id": "7710.png", "formula": "\\begin{align*} \\rho > 0 \\ \\ i n \\ \\ X , \\ \\ \\ \\ \\rho = 0 \\ \\ o n \\ \\ \\partial X , \\ \\ \\ \\ d \\rho \\neq 0 \\ \\ o n \\ \\ \\partial X . \\end{align*}"}
+{"id": "2930.png", "formula": "\\begin{align*} | \\ | \\lambda q ( \\lambda ) | - 1 \\ | \\le \\min _ { U \\in \\mathcal U } \\| A q ( A ) - U \\| | = \\max _ { \\| x \\| = 1 } | \\ \\| A q ( A ) x \\| - 1 \\ | . \\end{align*}"}
+{"id": "4316.png", "formula": "\\begin{align*} m _ j ( h ) = \\frac { 1 } { b _ j \\pi _ j } \\sum _ { k = 0 } ^ j \\widehat { h } ( k ) \\pi _ k = - \\frac { 1 } { b _ j \\pi _ j } \\sum _ { k = j + 1 } ^ \\infty \\widehat { h } ( k ) \\pi _ k ; \\end{align*}"}
+{"id": "1054.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma _ { - 2 m } ( V W \\Delta ^ { - m } ) & = - \\frac { m ( m + 1 ) } { 3 | | \\xi | | ^ { 2 m + 4 } } V ^ { a } W ^ { b } \\mathrm { R i c } _ { c d } \\xi _ { a } \\xi _ { b } \\xi _ { c } \\xi _ { d } \\\\ & + \\frac { 2 m } { 3 | | \\xi | | ^ { 2 m + 2 } } V ^ { a } W ^ { b } \\xi _ { c } ( \\xi _ { a } \\mathrm { R i c } _ { b c } + \\xi _ { b } \\mathrm { R i c } _ { a c } - R _ { c a d b } \\xi _ { d } ) + o ( \\mathbf { 1 } ) . \\end{aligned} \\end{align*}"}
+{"id": "1020.png", "formula": "\\begin{align*} \\ell _ t = 2 \\ , \\ell _ b \\qquad ( \\beta = \\gamma ) \\Psi = \\frac { 3 } { 2 } . \\end{align*}"}
+{"id": "6704.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & \\chi _ { 0 } ( v ) = \\frac { 1 } { \\sqrt { \\rho } } M , \\chi _ { i } ( v ) = \\frac { v _ { i } - u _ { i } } { \\sqrt { R \\rho \\theta } } M , \\mbox { f o r $ i = 1 , 2 , 3 $ , } \\\\ & \\chi _ { 4 } ( v ) = \\frac { 1 } { \\sqrt { 6 \\rho } } ( \\frac { | v - u | ^ { 2 } } { R \\theta } - 3 ) M , \\\\ & \\langle \\chi _ { i } , \\frac { \\chi _ { j } } { M } \\rangle = \\delta _ { i j } , \\mbox { f o r ~ ~ $ i , j = 0 , 1 , 2 , 3 , 4 $ } , \\end{array} \\right . \\end{align*}"}
+{"id": "7904.png", "formula": "\\begin{align*} ( a , u ) \\cdot _ \\ltimes ( b , v ) = ( a \\cdot b , a \\cdot v + u \\cdot b ) , ( a , u ) , ( b , v ) \\in A \\oplus M . \\end{align*}"}
+{"id": "6622.png", "formula": "\\begin{gather*} \\Pi _ \\pm : L ^ { 2 } ( \\mathbb { R } ^ { 2 } ) \\to L ^ { 2 } ( \\mathbb { R } ^ { 2 } _ { + } ) , \\Pi _ \\pm u ( x , y ) : = \\frac { u ( x , y ) \\pm u ( x , - y ) } { \\sqrt { 2 } } , \\end{gather*}"}
+{"id": "3134.png", "formula": "\\begin{align*} \\sigma | _ x ( y ) = \\begin{cases} \\sigma ( y ) & \\\\ y & \\end{cases} \\end{align*}"}
+{"id": "7139.png", "formula": "\\begin{align*} r ( \\theta ) = _ \\lambda \\frac { \\langle \\lambda , \\theta \\rangle } { \\Big \\langle \\lambda , \\left ( ( V , V ) ^ { \\oplus 3 } \\right ) ^ { \\lambda > 0 } \\Big \\rangle } , \\end{align*}"}
+{"id": "2372.png", "formula": "\\begin{align*} S = \\{ a _ { 1 } , \\dots , a _ { n } , 1 - a _ { 1 } , \\dots , 1 - a _ { n } , 0 , 1 \\} . \\end{align*}"}
+{"id": "4892.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty & \\sum _ { n = 1 } ^ \\infty \\frac { ( 2 n ) ^ { k - 1 } ( 2 x ) ^ { 2 n } } { { 2 n \\choose n } } \\frac { t ^ k } { k ! } = \\frac { x } { ( 1 - x ^ 2 ) ^ { 1 / 2 } } \\left ( x \\sqrt { 1 - x ^ 2 } P \\left ( x ^ 2 , \\frac { t } { 1 - x ^ 2 } \\right ) + \\arcsin ( x ) Q \\left ( x ^ 2 , \\frac { t } { 1 - x ^ 2 } \\right ) \\right ) . \\end{align*}"}
+{"id": "2469.png", "formula": "\\begin{align*} g _ { i j } = \\delta _ { i j } + \\frac { 1 } { 2 } \\sum _ { k , l = 1 } ^ n \\frac { \\partial ^ 2 g _ { i j } } { \\partial x ^ k \\partial x _ l } ( x _ 0 ) x ^ k x ^ l + \\mathrm { o } ( \\abs { x } ^ 2 ) \\ , , \\end{align*}"}
+{"id": "4931.png", "formula": "\\begin{align*} \\binom { n } { k - 2 } = \\sum _ { i = 3 } ^ { k - 1 } { \\left ( \\frac { j } { 2 } \\right ) } ^ { i - 2 } \\binom { n } { k - i } + { \\left ( \\frac { j } { 2 } \\right ) } ^ { k - 3 } \\binom { n } { 1 } + \\sum _ { i = 3 } ^ { k - 1 } { \\left ( \\frac { j } { 2 } \\right ) } ^ { i - 3 } \\frac { { \\left ( j + 1 \\right ) } { \\left ( i - 1 \\right ) } } { k - { \\left ( i - 1 \\right ) } } \\binom { n } { k - i } . \\end{align*}"}
+{"id": "3421.png", "formula": "\\begin{align*} U ^ * _ \\epsilon = P ^ U | _ { ( U ^ * , \\rho ^ * , S ^ * ) } , \\rho ^ * _ \\epsilon = P ^ \\rho | _ { ( U ^ * , \\rho ^ * , S ^ * ) } , S ^ * _ \\epsilon = P ^ S | _ { ( U ^ * , \\rho ^ * , S ^ * ) } \\end{align*}"}
+{"id": "4467.png", "formula": "\\begin{align*} \\langle [ Y , E ] , Z \\rangle - \\langle [ Z , Y ] , E \\rangle + \\langle [ Z , E ] , Y \\rangle = 0 \\end{align*}"}
+{"id": "6819.png", "formula": "\\begin{align*} \\partial _ { t t } u _ 1 - \\kappa \\Delta \\partial _ t u _ 1 - \\kappa _ 1 \\Delta u _ 1 = 0 . \\end{align*}"}
+{"id": "2199.png", "formula": "\\begin{align*} & \\varphi ( x _ 1 , x _ 2 , x _ 3 ) \\\\ & \\ ; \\ ; \\ ; = \\left ( 2 \\sqrt { A G ( x _ 1 ) } \\ , \\cos \\frac { x _ 2 } { 2 \\sqrt { A } } , \\ , 2 \\sqrt { A G ( x _ 1 ) } \\ , \\sin \\frac { x _ 2 } { 2 \\sqrt { A } } , \\ , f ( x _ 1 ) \\cos x _ 3 , \\ , f ( x _ 1 ) \\sin x _ 3 \\right ) . \\end{align*}"}
+{"id": "5772.png", "formula": "\\begin{align*} - \\frac { 2 4 1 2 0 } { 1 3 0 9 } < a \\leq - \\frac { 8 0 4 } { 7 1 } , - \\frac { 4 } { 5 } a < b < \\frac { 4 8 2 4 - 3 6 7 9 a } { 4 9 2 6 } , \\frac { - 3 5 a - 6 b } { 7 2 } < c \\leq - \\frac { 5 8 ( 5 a + 6 b ) - 3 3 5 } { 7 1 } . \\end{align*}"}
+{"id": "1507.png", "formula": "\\begin{align*} h ( p ) p = \\bigl ( 1 - \\tfrac 1 p \\bigr ) ^ { - 1 } \\bigl ( 1 - \\tfrac { \\chi ( p ) } { p } \\bigr ) ^ { - 1 } \\bigl ( 1 + \\chi ( p ) ( 1 - \\tfrac { 1 } { p } ) \\bigr ) \\end{align*}"}
+{"id": "1859.png", "formula": "\\begin{align*} y _ { 2 i - 1 , 2 n ' _ 0 - 1 } & = n ( i - 1 ) + n ' _ 0 \\mbox { \\ f o r a l l } 1 \\leqslant i \\leqslant m ^ { + } _ 0 , \\mbox { a n d } \\\\ y _ { 2 i , 2 n ' _ 0 - 1 } & = M N - n i + n ' _ 0 \\mbox { \\ f o r a l l } 1 \\leqslant i \\leqslant m _ 0 . \\end{align*}"}
+{"id": "5758.png", "formula": "\\begin{align*} h _ 1 ( x _ 0 , x ) = \\begin{pmatrix} x ^ 2 & x x _ 0 \\\\ x _ 0 x & x _ 0 ^ 2 \\end{pmatrix} \\quad h _ 2 ( x _ 0 , x ) = \\begin{pmatrix} x ^ 2 & x _ 0 x \\\\ x x _ 0 & x _ 0 ^ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "4650.png", "formula": "\\begin{align*} 3 - \\pi ^ 2 \\max \\{ 0 , 2 - \\gamma \\} ^ 2 > 0 \\iff \\gamma > 2 - \\frac { \\sqrt { 3 } } { \\pi } = : \\gamma _ c ' . \\end{align*}"}
+{"id": "507.png", "formula": "\\begin{align*} f ( x ) & = \\sum _ { i = 1 } ^ n V ( x _ i ) + \\sum _ { 1 \\le i < j \\le n } J _ { i j } K ( x _ i - x _ j ) , \\end{align*}"}
+{"id": "1391.png", "formula": "\\begin{align*} \\partial _ t \\Phi _ { \\beta , \\varepsilon } ( x , t ; t _ 0 ) = - \\beta \\Phi _ { \\beta + 1 , \\varepsilon } ( x , t ; t _ 0 ) . \\end{align*}"}
+{"id": "3237.png", "formula": "\\begin{align*} \\theta ^ { n } _ { \\varphi _ { j } } \\geq e ^ { \\varphi _ { j } } \\mu _ { k } \\theta ^ { n } _ { \\varphi _ { k } } = e ^ { \\varphi _ { k } } \\mu _ { k } . \\end{align*}"}
+{"id": "5731.png", "formula": "\\begin{align*} \\phi ( M ) = \\begin{pmatrix} \\langle J _ { 1 1 } , M \\rangle & \\cdots & \\langle J _ { 1 t } , M \\rangle \\\\ \\vdots & \\ddots & \\vdots \\\\ \\langle J _ { t 1 } , M \\rangle & \\cdots & \\langle J _ { t t } , M \\rangle \\end{pmatrix} . \\end{align*}"}
+{"id": "2521.png", "formula": "\\begin{align*} F _ 1 x & = ( x _ { 0 0 0 } , x _ { 0 0 1 } , x _ { 0 1 0 } , x _ { 0 1 1 } ) \\\\ F _ 2 x & = ( x _ { 0 0 0 } , x _ { 0 0 1 } , x _ { 1 0 0 } , x _ { 1 0 1 } ) \\\\ F _ 3 x & = ( x _ { 0 0 0 } , x _ { 0 1 0 } , x _ { 1 0 0 } , x _ { 1 1 0 } ) \\end{align*}"}
+{"id": "3137.png", "formula": "\\begin{align*} a _ { k + m } = F P \\left ( \\sigma ^ { m + k } \\right ) = F P \\left ( \\sigma ^ k \\right ) = a _ k \\end{align*}"}
+{"id": "1472.png", "formula": "\\begin{align*} \\tilde { Q } _ n = \\begin{bmatrix} \\frac { 1 } { f _ { 1 } - \\alpha _ n } e _ M ( \\theta ) 1 _ { \\{ 1 \\in J _ w \\} } + \\hat { Z } _ { 1 } \\\\ \\vdots \\\\ \\frac { 1 } { f _ { y } - \\alpha _ n } e _ M ( \\theta ) 1 _ { \\{ y \\in J _ w \\} } + \\hat { Z } _ { y } \\end{bmatrix} , \\end{align*}"}
+{"id": "2734.png", "formula": "\\begin{align*} b ( 1 2 p ) = \\left \\{ \\begin{array} { l l l } 2 4 & p \\equiv 1 1 & \\pmod { 1 2 } , \\\\ 2 8 & p \\equiv 5 & \\pmod { 1 2 } , \\\\ 3 4 & p \\equiv 7 & \\pmod { 1 2 } , \\\\ 4 0 & p \\equiv 1 & \\pmod { 1 2 } . \\end{array} \\right . \\end{align*}"}
+{"id": "8081.png", "formula": "\\begin{align*} \\tfrac { 1 } { \\rho _ 1 } + \\tfrac { 1 } { \\rho _ 2 - \\rho _ 1 } = \\tfrac { \\rho _ 1 + \\rho _ 2 } { \\rho _ 1 ( \\rho _ 2 - \\rho _ 1 ) } . \\end{align*}"}
+{"id": "4023.png", "formula": "\\begin{align*} w ( y , s ) = ( T - t ) ^ { - \\frac { 1 } { p - 1 } } u ( x , t ) , \\ ; y = \\frac { x } { ( T - t ) ^ { \\frac { 1 } { 2 k } } } , \\ ; \\ ; s = - \\ln ( T - t ) . \\end{align*}"}
+{"id": "1363.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t ^ 2 u - \\Delta u + \\partial _ t u = 0 , & t > 0 , x \\in \\mathbb { R } ^ n , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\ \\partial _ t u ( 0 , x ) = u _ 1 ( x ) , & x \\in \\mathbb { R } ^ n . \\end{array} \\right . \\end{align*}"}
+{"id": "7435.png", "formula": "\\begin{align*} [ H , \\ , T _ \\eta ] = \\sum _ { \\mu = 1 } ^ { n } T _ \\mu \\alpha _ { \\mu \\eta } , \\alpha _ { \\mu \\eta } ^ { \\dag } = \\alpha _ { \\mu \\eta } , [ \\alpha _ { \\mu \\eta } , H ] = 0 . \\end{align*}"}
+{"id": "4896.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty a _ n \\frac { t ^ { n + 1 } } { ( n + 1 ) ! } = \\frac { 3 } { 2 } P \\left ( \\frac { 1 } { 4 } , \\frac { 2 } { 3 } t \\right ) = \\frac { 6 e ^ { t / 2 } \\left ( \\arcsin ( e ^ { t / 2 } / 2 ) - \\arcsin ( 1 / 2 ) \\right ) } { ( 4 - e ^ t ) ^ { 1 / 2 } } . \\end{align*}"}
+{"id": "5445.png", "formula": "\\begin{align*} \\begin{matrix} P ^ * & = & ( \\ , p _ 1 ^ * , p _ 2 ^ * , p _ 3 ^ * , p _ 4 ^ * \\ , \\mid \\ , p _ { 1 2 } ^ * , p _ { 1 3 } ^ * , p _ { 2 3 } ^ * , p _ { 1 4 } ^ * , p _ { 2 4 } ^ * , p _ { 3 4 } ^ * \\ , \\mid \\ , p _ { 1 2 3 } ^ * , p _ { 1 2 4 } ^ * , p _ { 1 3 4 } ^ * , p _ { 2 3 4 } ^ * \\ , ) \\\\ & = & ( p _ { 2 3 4 } , - p _ { 1 3 4 } , p _ { 1 2 4 } , - p _ { 1 2 3 } \\ , \\mid \\ , p _ { 3 4 } , - p _ { 2 4 } , p _ { 1 4 } , p _ { 2 3 } , - p _ { 1 3 } , p _ { 1 2 } \\ , \\mid \\ , p _ 4 , - p _ 3 , p _ 2 , - p _ 1 ) . \\end{matrix} \\end{align*}"}
+{"id": "6216.png", "formula": "\\begin{align*} E ^ * _ \\nu \\mathcal { W } _ { ( \\mu , d ) } = E ^ * _ \\nu E _ { \\mu } P ( E _ { \\mu - 1 } ) \\mathcal { L } _ \\nu , \\end{align*}"}
+{"id": "7217.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\pm \\infty } \\| e ^ { i t \\triangle } \\vec { u } ^ { \\pm } - \\vec { u } ( t ) \\| _ { L _ x ^ 2 h ^ 1 } = 0 . \\end{align*}"}
+{"id": "2977.png", "formula": "\\begin{align*} [ X _ 3 ] & = [ Z _ 2 ] + [ Y _ 1 ] \\\\ [ Z _ { n - 1 } ] & = [ Y _ n ] + [ X _ { n - 1 } ] \\end{align*}"}
+{"id": "450.png", "formula": "\\begin{align*} B _ n ( - x ) = ( - 1 ) ^ n \\big ( B _ n ( x ) + n x ^ { n - 1 } \\big ) , \\end{align*}"}
+{"id": "4571.png", "formula": "\\begin{gather*} | T ( f ) ( q _ { l _ n } ) | = | \\int f d \\mu _ n | = | \\int f _ n d \\mu _ n + \\int \\sum _ { m \\in B \\backslash \\{ n \\} } f _ m | \\geq \\\\ | \\int f _ n d \\mu _ n | - | \\int \\sum _ { m \\in B \\backslash \\{ n \\} } f _ m | \\geq \\varepsilon - \\varepsilon / 3 = 2 \\varepsilon / 3 . \\end{gather*}"}
+{"id": "6797.png", "formula": "\\begin{align*} \\alpha ( x _ { [ 4 ] } ) + \\alpha \\circ ( 3 \\ , \\ , 4 ) ( x _ { [ 4 ] } ) = \\sum _ { i \\in [ r ] } \\beta _ { i , 1 } ( x _ { I _ { i , 1 } } ) \\beta _ { i , 2 } ( x _ { I _ { i , 2 } } ) \\dots \\beta _ { i , d _ i } ( x _ { I _ { i , d _ i } } ) \\end{align*}"}
+{"id": "2473.png", "formula": "\\begin{align*} F _ A = F _ 0 + \\d A . \\end{align*}"}
+{"id": "3651.png", "formula": "\\begin{align*} & \\hat { p } ( X _ 1 , \\ldots , X _ i , X _ i ' , \\ldots , X _ j , X _ j ' , \\ldots , X _ m ) \\\\ & = p _ 1 + p _ 2 ( X _ i + X ' _ i + X _ j + X ' _ j ) + p _ { 4 } ( X _ i + X ' _ i ) ( X _ j + X ' _ j ) + p _ { 5 } ( X _ i + X _ j ) ( X ' _ i + X ' _ j ) \\end{align*}"}
+{"id": "2745.png", "formula": "\\begin{align*} c _ 2 ( \\Omega _ Z ) = - 3 \\pi ^ * K _ C \\cdot H - 2 \\pi ^ * c _ 1 ( E ) \\cdot H + 3 H ^ 2 . \\end{align*}"}
+{"id": "5008.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | g \\zeta _ { h n } - \\bar g \\zeta _ { h n } | = \\infty \\quad g , \\bar g \\in G \\bar g g ^ { - 1 } \\notin K _ h , \\end{align*}"}
+{"id": "3041.png", "formula": "\\begin{align*} \\sigma _ 0 = ( 1 ) ( 2 , 3 ) ( 4 , 5 , 6 ) ( 7 , 8 , 9 , 1 0 ) ( 1 1 , 1 2 , 1 3 , 1 4 , 1 5 ) \\cdots \\end{align*}"}
+{"id": "5194.png", "formula": "\\begin{align*} \\frac { X - 1 } { ( X + 1 ) ( 2 X - 1 ) } = \\frac { A ( 2 X + 1 ) + B ( X + 1 ) } { ( X + 1 ) ( 2 X - 1 ) } = \\frac { A ( 2 X + 1 ) } { X + 1 } + \\frac { B ( X + 1 ) } { 2 x + 1 } \\enspace . \\end{align*}"}
+{"id": "521.png", "formula": "\\begin{align*} \\alpha _ i ( t , x _ 1 , \\ldots , x _ n ) = \\widehat \\alpha _ i ( t , x _ i ) , \\end{align*}"}
+{"id": "1811.png", "formula": "\\begin{align*} \\langle a _ { 2 } ^ * \\otimes a _ { 1 } , T \\rangle = \\langle a _ { 2 } ^ { * } , T ( a _ { 1 } ) \\rangle , \\forall a _ 1 \\in V _ 1 , a ^ { * } _ { 2 } \\in V ^ { * } _ { 2 } . \\end{align*}"}
+{"id": "6351.png", "formula": "\\begin{align*} \\mathcal { A } _ { P } ^ { ( 1 ) } ( F , G ) & = m a x _ { i = 1 } ^ 3 \\frac { q _ i \\bigg \\{ | \\langle f _ i , g _ i \\rangle | + \\| f _ i \\| \\| g _ i \\| \\bigg \\} } { 2 } \\\\ & = m a x _ { i = 1 } ^ 3 \\bigg \\{ 1 , \\frac { | 1 + \\beta | + \\left | \\sqrt { \\alpha ^ 2 + ( 1 + \\beta ) ^ 2 } \\right | } { 2 } , \\frac { | 1 - \\beta | + \\left | \\sqrt { \\alpha ^ 2 + ( 1 - \\beta ) ^ 2 } \\right | } { 2 } \\bigg \\} . \\end{align*}"}
+{"id": "1976.png", "formula": "\\begin{align*} \\sum _ { j = j _ 0 - N } ^ { j _ 0 + N } \\sum _ { k \\in { E } } P _ { R _ { j , k } } = P _ { I _ { j _ 0 + N } ^ 1 } \\otimes P _ { I _ { j _ 0 + N } ^ 2 } \\otimes & \\cdots \\otimes P _ { I _ { j _ 0 + N } ^ d } \\\\ & - P _ { I _ { j _ 0 - N } ^ 1 } \\otimes P _ { I _ { j _ 0 - N } ^ 2 } \\otimes \\cdots \\otimes P _ { I _ { j _ 0 - N } ^ d } . \\end{align*}"}
+{"id": "3981.png", "formula": "\\begin{align*} q ( n , t ) = \\sum _ { \\Omega ^ { n } _ { k } } \\prod _ { j = 1 } ^ { k } \\frac { \\theta ^ { x _ { j } } } { x _ { j } ! } \\frac { ( \\alpha t ) ^ { k } } { k ! } e ^ { - \\alpha t \\left ( e ^ { \\theta } - 1 \\right ) } , \\ n \\ge 0 , \\end{align*}"}
+{"id": "2538.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x ^ i _ t = & ~ [ A x ^ i _ t + B \\alpha ^ i _ t + f ( \\nu ^ { N , i } _ { \\boldsymbol { x } _ t } ) + b ( \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } _ t } ) ] d t + \\sigma d W ^ i _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x ^ i _ 0 = & ~ \\xi ^ i , \\end{aligned} \\right . \\end{align*}"}
+{"id": "6294.png", "formula": "\\begin{align*} \\langle x , y \\rangle + \\langle x , - y \\rangle = \\begin{cases} ( q - 1 ) \\langle x \\rangle _ { } , & \\\\ 0 , & \\end{cases} \\end{align*}"}
+{"id": "5686.png", "formula": "\\begin{gather*} g _ { a , b } ( x _ b ) = x _ a x _ b x _ a ^ { - 1 } , g _ { a , b } ( x _ d ) = x _ d d \\neq b , \\\\ f _ { a , b , c } ( x _ c ) = x _ c [ x _ a , x _ b ] , f _ { a , b , c } ( x _ d ) = x _ d d \\neq c . \\end{gather*}"}
+{"id": "2571.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { t _ 0 , x _ 0 , \\xi , \\alpha } = & ~ [ A x _ t ^ { t _ 0 , x _ 0 , \\xi , \\alpha } + B \\alpha _ t + f ( \\nu _ t ^ { * , t _ 0 , \\xi } ) + b ( \\mu _ t ^ { * , t _ 0 , \\xi } ) ] d t + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ { t _ 0 } ^ { t _ 0 , x _ 0 , \\xi , \\alpha } = & ~ x _ 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "4621.png", "formula": "\\begin{align*} \\frac { 1 - L _ p r ^ p } { 1 - r } = \\frac { 1 / 2 } { 1 - ( 2 L _ p ) ^ { - 1 / p } } = \\frac { p } { 2 p ( 1 - ( 2 L _ p ) ^ { - 1 / p } ) } . \\end{align*}"}
+{"id": "4876.png", "formula": "\\begin{align*} I _ n ^ { ( \\mathbf { s } ) } = \\{ ( e _ 1 , \\ldots , e _ n ) \\in \\mathbb { Z } ^ n \\mid 0 \\leq e _ i < s _ i 1 \\leq i \\leq n \\} . \\end{align*}"}
+{"id": "4473.png", "formula": "\\begin{align*} \\| f \\| _ s ^ 2 = \\int _ { \\mathbb { R } ^ 3 } \\Big ( f ^ 2 + \\sum _ { i = 1 } ^ 3 \\left ( \\partial _ i f \\right ) ^ 2 + \\sum _ { i , j = 1 } ^ 3 \\left ( \\partial _ i \\partial _ j f \\right ) ^ 2 \\Big ) \\ , \\end{align*}"}
+{"id": "2059.png", "formula": "\\begin{align*} f ( m ; z ) = \\sum _ { n = 0 } ^ \\infty b _ 5 ( n ) q ^ { \\frac { 2 4 n + m ( 5 a + b ) + 4 } { 2 4 } } \\cdot \\prod _ { n = 1 } ^ \\infty ( 1 - q ^ { 5 m n } ) ^ a ( 1 - q ^ { m n } ) ^ b . \\end{align*}"}
+{"id": "4837.png", "formula": "\\begin{align*} \\int _ S \\int _ { \\R ^ d } ( \\nabla f \\cdot \\nabla \\phi + K \\phi ) \\ , \\dd \\rho ( x , s ) = K \\int _ S \\int _ { \\R ^ d } \\phi \\ , \\dd \\bar \\rho ( x ) \\dd \\mu ( s ) , \\end{align*}"}
+{"id": "2553.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\varphi _ t ^ { * , \\xi } = & - [ ( A - B ^ 2 R ^ { - 1 } P _ t ) \\varphi _ t ^ { * , \\xi } + P _ t f ( \\nu _ t ^ { * , \\xi } ) + P _ t b ( \\mu _ t ^ { * , \\xi } ) \\\\ & + Q l ( \\nu _ t ^ { * , \\xi } ) - P _ t B h ( \\mu _ t ^ { * , \\xi } ) ] d t + \\Lambda _ t ^ { 0 , * , \\xi } d W ^ 0 _ t , \\\\ \\varphi _ T ^ { * , \\xi } = & ~ G g ( \\nu _ T ^ { * , \\xi } ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "3358.png", "formula": "\\begin{align*} m ' \\mathrm { R e s } _ { \\tau _ \\alpha : L \\hookrightarrow L _ { \\alpha } } ( x _ 1 ) = m ' x _ \\alpha \\end{align*}"}
+{"id": "7207.png", "formula": "\\begin{align*} K ^ { \\mathrm { t o p } } _ 0 ( \\mathbb { S } _ A ) \\cong \\bigotimes _ { i = 1 } ^ k K ^ { \\mathrm { t o p } } _ 0 \\left ( \\mathbb { S } ( d _ i ) _ { w _ i } \\right ) . \\end{align*}"}
+{"id": "5519.png", "formula": "\\begin{gather*} a = \\frac 1 2 \\cdot \\# \\{ i \\mid \\varsigma ( i ) \\neq i , \\quad \\dim \\phi _ { \\pi _ i } = 1 \\} , b = \\frac 1 2 \\cdot \\# \\{ i \\mid \\varsigma ( i ) \\neq i , \\dim \\phi _ { \\pi _ i } = 2 \\} , \\\\ c = \\# \\{ i \\mid \\varsigma ( i ) = i \\} . \\end{gather*}"}
+{"id": "4131.png", "formula": "\\begin{align*} \\int _ t ^ { \\varepsilon _ n ^ \\alpha } a ' ( s ) \\frac { ( U ) ^ 2 ( s ) } { 2 } d s = a ( \\varepsilon _ n ^ \\alpha ) \\frac { ( U ) ^ 2 ( \\varepsilon _ n ^ \\alpha ) } { 2 } - a ( t ) \\frac { ( U ) ^ 2 ( t ) } { 2 } - \\int _ t ^ { \\varepsilon _ n ^ \\alpha } a ( s ) U ( s ) U ' ( s ) d s \\end{align*}"}
+{"id": "3371.png", "formula": "\\begin{align*} \\frac { 1 + z + \\dots + z ^ { \\ell - 1 } } { 1 + z + \\dots + z ^ { q - 1 } } \\ , h _ P ^ \\ast ( z ) = a ( z ) + z ^ \\ell \\ , b ( z ) \\ , , \\end{align*}"}
+{"id": "2878.png", "formula": "\\begin{align*} \\# \\{ \\tau _ { s _ k } \\in { \\bf { S } } _ { s _ k } : f ^ 1 _ { \\tau _ { s _ k } } \\not \\equiv 0 \\} \\prod _ { i = 1 } ^ { k - 1 } \\max _ { \\tau _ { s _ { i + 1 } } \\in { \\bf { S } } _ { s _ { i + 1 } } } \\# \\{ \\tau _ { s _ i } \\in { \\bf { S } } _ { s _ i } : \\tau _ { s _ i } \\subset \\tau _ { s _ { i + 1 } } , f _ { \\tau _ { s _ i } } ^ 1 \\not \\equiv 0 \\} . \\end{align*}"}
+{"id": "3109.png", "formula": "\\begin{align*} | Y _ i | = \\ell _ i q _ { \\ell _ i } \\end{align*}"}
+{"id": "2813.png", "formula": "\\begin{align*} d \\nu _ s = \\nu _ s d \\left ( R _ s + [ R ] _ s \\right ) , s \\in [ t , T ] , \\nu _ t = 1 . \\end{align*}"}
+{"id": "6825.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\alpha _ n ^ i \\partial _ t ^ { i + n } u _ { n - 1 } + \\partial _ t ^ { n } u _ { n - 1 } - \\sum _ { i = 0 } ^ { 2 n - 1 } \\beta _ n ^ { i + 1 } \\Delta \\partial _ t ^ { i } u _ { n - 1 } = 0 . \\end{align*}"}
+{"id": "1891.png", "formula": "\\begin{align*} S _ { n , q } : = \\{ x _ { i _ 1 } \\otimes \\cdots \\otimes x _ { i _ q } , 1 \\leq i _ 1 \\leq \\ldots \\leq i _ q \\leq n \\} . \\end{align*}"}
+{"id": "1887.png", "formula": "\\begin{align*} f ( 2 x + y ) + f ( x + 2 y ) = \\frac { f ( x ) f ( y ) \\sum _ { k = 0 } ^ { n } ( 2 ^ { n - k } + 2 ^ k ) \\binom { n } { k } f ( x ) ^ { \\frac { k } { n } } f ( y ) ^ { \\frac { n - k } { n } } } { \\left ( 2 f ( x ) ^ { \\frac { 2 } { n } } + 5 f ( x ) ^ { \\frac { 1 } { n } } f ( y ) ^ { \\frac { 1 } { n } } + 2 f ( y ) ^ { \\frac { 2 } { n } } \\right ) ^ n } , . \\end{align*}"}
+{"id": "951.png", "formula": "\\begin{align*} - \\epsilon ^ 2 \\delta ^ 2 b _ { \\alpha } u _ n ( x ) \\leq u ( 0 ) - u ( x ) + \\sum _ { \\beta = 1 } ^ { n - 1 } x _ \\beta u _ \\beta ( x ) \\leq C b _ \\alpha ^ 2 \\end{align*}"}
+{"id": "943.png", "formula": "\\begin{align*} \\omega _ \\epsilon : = \\{ x \\in \\Omega : | x _ \\beta | < \\epsilon \\delta \\frac { b _ \\alpha } { \\sqrt { b _ \\beta } } , \\beta = \\alpha + 1 , \\ldots , n - 1 , x _ n < \\epsilon ^ 2 \\delta ^ 2 b _ \\alpha \\} \\end{align*}"}
+{"id": "589.png", "formula": "\\begin{align*} & \\big ( \\mu _ n ^ { ( 1 2 ) } - \\mu _ { n + 1 } ^ { ( 1 2 ) } - 1 \\big ) Y _ { n + 1 , p } - \\big ( \\mu _ { n - 1 } ^ { ( 1 2 ) } - \\mu _ n ^ { ( 1 2 ) } + 1 \\big ) Y _ { n , p } = \\\\ & \\big ( \\psi ^ { 0 0 } _ { n , p } \\big ) ^ 2 + ( 2 \\mu _ n ^ { ( 1 2 ) } - \\mu ^ { ( 1 ) } - \\mu ^ { ( 2 ) } - \\mu ^ { ( 3 ) } - \\mu _ p ^ { ( 1 2 3 ) } ) \\ , \\psi ^ { 0 0 } _ { n , p } - ( \\mu ^ { ( 1 ) } - \\mu _ p ^ { ( 1 2 3 ) } ) ( \\mu ^ { ( 3 ) } - \\mu ^ { ( 2 ) } ) \\ , , \\end{align*}"}
+{"id": "1590.png", "formula": "\\begin{align*} \\rho ( s ) T _ { b _ { 1 } } + T _ { s _ { 1 } } = \\rho ( s ' ) T _ { b _ { 1 } } + T _ { s _ { 1 } ' } = T _ { b _ { 2 } } \\end{align*}"}
+{"id": "1161.png", "formula": "\\begin{align*} \\Gamma ( k - 3 / 2 ) S ( D _ 1 , D _ 2 ) = 2 \\pi ^ { k - \\frac { 3 } { 2 } } D _ 2 ^ { k - \\frac { 3 } { 2 } } p _ { D _ 2 } ( D _ 1 ) . \\end{align*}"}
+{"id": "654.png", "formula": "\\begin{align*} V \\cdot \\psi _ 1 = \\psi _ 2 . \\end{align*}"}
+{"id": "4065.png", "formula": "\\begin{align*} \\sigma ' _ { \\mid I } & = \\sigma _ 1 = \\sigma _ { \\mid I } , \\\\ \\sigma ' _ { \\mid P \\setminus I } & = \\sigma _ 2 = \\sigma _ { \\mid P \\setminus I } , \\end{align*}"}
+{"id": "2512.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } T ^ { c _ 2 ( m ) } x _ { 0 0 } = x _ { 0 1 } \\quad \\textup { a n d } \\lim _ { n \\to \\infty } T ^ { c _ 1 ( n ) } x _ { 0 1 } = x _ { 1 1 } , \\end{align*}"}
+{"id": "6475.png", "formula": "\\begin{align*} W F ^ { - \\frac 1 2 - \\epsilon + \\tau } ( \\omega ^ { ( 2 ) } _ { G } | _ { \\cal { Q } } ) \\subset ( N _ + \\times N _ { - } ) \\cap ( N _ - \\times N _ { + } ) = \\emptyset . \\end{align*}"}
+{"id": "8048.png", "formula": "\\begin{align*} & s _ { { \\textrm { i n d } , m , 2 } _ { n _ r } } ( t ) = \\sum _ { n _ t = 1 } ^ { N _ t } \\alpha _ { m , 2 } \\sum _ { n _ m = 1 } ^ { N _ m } x _ { n _ t } \\left ( t - \\tau _ { \\textrm { i n d } , m , 2 , n _ m , n _ t , n _ r } \\right ) \\\\ & \\times e ^ { \\mathrm { j } 2 \\pi { f _ c } \\left ( t - \\tau _ { \\textrm { i n d } , m , p , q , 2 , n _ t } \\right ) } e ^ { \\mathrm { j } 2 \\pi { f _ { D , 2 } } \\left ( t - \\tau _ { \\textrm { i n d } , m , 2 , n _ m , n _ t , n _ r } \\right ) } e ^ { \\mathrm { j } { \\phi _ { m , n _ m } } } , \\end{align*}"}
+{"id": "3855.png", "formula": "\\begin{align*} - \\frac { 1 } { { m } ^ z ( w ) } = w + { m } ^ z ( w ) - \\frac { | z | ^ 2 } { w + { m } ^ z ( w ) } , \\mbox { w i t h } \\mathrm { I m } [ m ^ z ( w ) ] \\mathrm { I m } w > 0 , \\end{align*}"}
+{"id": "3867.png", "formula": "\\begin{align*} \\# \\{ i : x _ i \\equiv { v _ j } \\} + \\# \\{ i : x _ i \\equiv \\overline { v _ j } \\} = \\# \\{ i : y _ i \\equiv { v _ j } \\} + \\# \\{ i : y _ i \\equiv \\overline { v _ j } \\} . \\end{align*}"}
+{"id": "568.png", "formula": "\\begin{align*} T _ { W _ { n J } , \\psi } ( \\mu _ n ( Q ) ) & = \\frac { 1 } { n } \\sum _ { i , j = 1 } ^ n J _ { i j } \\int _ \\R \\int _ \\R \\psi ( x , y ) Q _ i ( d x ) Q _ j ( d y ) , \\\\ I ( \\mu _ n ( Q ) ) & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n H ( Q _ i | \\rho ) , \\end{align*}"}
+{"id": "7100.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } ( - 1 ) ^ i _ { \\mathbb { Q } } H ^ i ( ( X , d ) , \\varphi _ d ) = _ { X , d } . \\end{align*}"}
+{"id": "3280.png", "formula": "\\begin{gather*} x _ 2 - c _ 1 x _ 2 ( x _ 3 ) ^ 2 = d _ 1 , \\\\ x _ 3 - c _ 2 ( x _ 2 ) ^ 2 x _ 3 = d _ 2 , \\end{gather*}"}
+{"id": "1622.png", "formula": "\\begin{align*} g ( u _ i ) - g ( v _ i ) & = \\inf _ { x , y \\in M \\setminus A _ i } \\big ( g ( x ) + d ( x , u _ i ) - g ( y ) + d ( y , v _ i ) \\big ) \\\\ & \\ge \\inf _ { x , y \\in M \\setminus A _ i } \\big ( d ( x , u _ i ) + d ( y , v _ i ) - ( 1 - \\varepsilon ) d ( x , y ) \\big ) \\\\ & \\ge ( 1 - \\varepsilon ) d ( u _ i , v _ i ) , \\end{align*}"}
+{"id": "4838.png", "formula": "\\begin{align*} \\partial _ t \\Pi = \\nabla _ x \\cdot ( \\Pi \\nabla _ s f ( x , s _ 1 ) ) + \\nabla _ y \\cdot ( \\Pi \\nabla _ y f ( y , s _ 2 ) ) + K \\int _ { S ^ 2 } ( \\Gamma ( s _ 1 , s _ 2 ) \\Pi ( x , y , s _ 1 ' , s _ 2 ' ) - \\Pi ) \\dd s _ 1 ' \\dd s _ 2 ' \\end{align*}"}
+{"id": "5592.png", "formula": "\\begin{align*} g _ y ( X , Y ) = g _ { i j } ( y ) X ^ i Y ^ j \\end{align*}"}
+{"id": "2436.png", "formula": "\\begin{align*} I _ { \\gamma } ^ { \\mathfrak { m } } ( 0 ' ; a _ { 1 } , \\dots , a _ { k } ; 0 ' ) & = \\sum _ { 0 \\leq i \\leq j \\leq k } I _ { \\mathrm { d c h } } ^ { \\mathfrak { m } } ( 0 ' ; a _ { 1 } , \\dots , a _ { i } ; 1 ' ) I _ { C _ { 1 } } ^ { \\mathfrak { m } } ( 1 ' ; a _ { i + 1 } , \\dots , a _ { j } ; 1 ' ) I _ { \\mathrm { d c h } ^ { - 1 } } ^ { \\mathfrak { m } } ( 1 ' ; a _ { j + 1 } , \\dots , a _ { k } ; 0 ' ) . \\end{align*}"}
+{"id": "36.png", "formula": "\\begin{align*} q ' _ j & = \\sum _ { i \\in A _ j } q _ i = \\sum _ { i \\in [ k ] \\setminus [ m ] : \\tilde { \\delta } _ { i - m } \\in ( \\nu _ { j + 1 } , \\nu _ { j } ] } q _ i = \\sum _ { i \\in [ k ] \\setminus [ m ] : \\tilde { \\delta } _ { i - m } \\in ( \\nu _ { j + 1 } , \\nu _ { j } ] } 0 . 5 \\tilde { q } _ { i - m } = 0 . 5 \\P \\{ \\tilde { X } \\in ( \\nu _ { j + 1 } , \\nu _ { j } ] \\} . \\end{align*}"}
+{"id": "306.png", "formula": "\\begin{align*} - F ^ { s - 1 } d b & = - \\sum _ { i = 0 } ^ { s - 1 } ( a _ { i } ( 1 - T ) ^ { p ^ { s - i } } ) ^ { p ^ { i } - 1 } d ( a _ { i } ( 1 - T ) ^ { p ^ { s - i } } ) - \\sum _ { i = 0 } ^ { s - 1 } ( a _ { i } ' T ^ { p ^ { s - i } } ) ^ { p ^ { i } - 1 } d ( a _ { i } ' T ^ { p ^ { s - i } } ) \\\\ & = - ( 1 - T ) ^ { p ^ { s } } \\sum _ { i = 0 } ^ { s - 1 } a _ { i } ^ { p ^ { i } - 1 } d a _ { i } - T ^ { p ^ { s } } \\sum _ { i = 0 } ^ { s - 1 } a _ { i } '^ { p ^ { i } - 1 } d a _ { i } ' \\\\ & = ( - F ^ { s - 1 } d a ) ( 1 - T ) ^ { p ^ { s } } + ( - F ^ { s - 1 } d a ' ) T ^ { p ^ { s } } \\end{align*}"}
+{"id": "1918.png", "formula": "\\begin{align*} P _ 0 u _ 1 + \\lambda u _ 1 = g . \\end{align*}"}
+{"id": "2690.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { m i n ( m , n ) } \\binom { a } { m - k } \\binom { b } { n - k } \\binom { a + b + k } { k } = \\binom { a + n } { m } \\binom { b + m } { n } \\end{align*}"}
+{"id": "6175.png", "formula": "\\begin{align*} \\sum _ { p \\ , \\leq \\ , x } ( \\log g _ { u } ( p - 1 ) ) ^ { \\lambda } = \\sum _ { \\substack { d \\ , \\le \\ , x \\\\ ( d , \\ , a _ { 2 } ) \\ , = \\ , 1 } } \\rho _ { \\lambda } ( d ) \\pi ( x ; \\l ( d ) , 1 ) \\end{align*}"}
+{"id": "3744.png", "formula": "\\begin{align*} m ( y ) & = \\sum _ { k = 0 } ^ { \\deg m } \\left ( D ^ k m ( t ) - t D ^ { k + 1 } m ( t ) \\right ) y ^ k \\\\ m ( y ) - m ( t ) & = ( y - t ) \\sum _ { k = 1 } ^ { \\deg m } D ^ k m ( t ) \\cdot y ^ { k - 1 } . \\end{align*}"}
+{"id": "2845.png", "formula": "\\begin{align*} B _ { s } : = R \\otimes _ { R ^ s } R ( 1 ) . \\end{align*}"}
+{"id": "6037.png", "formula": "\\begin{align*} X _ 4 + X _ 6 = X _ 5 + X _ 6 = X _ 1 + X _ 2 + X _ 3 + ( \\gamma + \\gamma ^ q ) X _ 6 = 0 . \\end{align*}"}
+{"id": "809.png", "formula": "\\begin{align*} \\lambda ^ 2 - 2 a ' \\lambda - 2 \\beta = 0 . \\end{align*}"}
+{"id": "7117.png", "formula": "\\begin{align*} D ^ b ( \\mathrm { H i l b } ( \\mathbb { C } ^ 3 , 2 ) ) = \\langle i _ { \\ast } ( p ^ { \\ast } D ^ b ( \\Delta ) \\otimes \\mathcal { O } ( - 1 ) ) , D ^ b ( ( \\mathbb { C } ^ 3 ) ^ { \\times 2 } / \\mathfrak { S } _ 2 ) ) . \\end{align*}"}
+{"id": "2042.png", "formula": "\\begin{align*} \\lambda _ 1 = \\mu _ 0 , \\lambda _ 2 = 6 \\mu _ 0 - \\mu _ 1 , \\lambda _ 3 = 1 9 \\mu _ 0 - 7 \\mu _ 1 + \\frac { 1 } { 2 } \\mu _ 2 \\end{align*}"}
+{"id": "1964.png", "formula": "\\begin{align*} \\rho ( \\xi ) = \\left \\{ \\begin{array} { l l } 0 & \\mbox { f o r } \\ ; \\xi \\leq - 1 , \\\\ 1 & \\mbox { f o r } \\ ; \\xi \\geq 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "7224.png", "formula": "\\begin{align*} \\big \\| \\int ^ t _ { t _ 0 } e ^ { i ( t - \\tau ) \\triangle } \\vec { F } ( \\tau ) d \\tau \\big \\| _ { U ^ 2 _ { \\triangle } ( h ^ { 1 } ; J \\times \\mathbb { R } ^ 2 ) } & \\lesssim \\sum ^ { k } _ { m = 1 } \\big \\| \\int _ { J ^ m } e ^ { - i \\tau \\triangle } \\vec { F } ( \\tau ) d \\tau \\big \\| _ { L ^ 2 _ x h ^ { 1 } } + \\big [ \\sum ^ { k } _ { m = 1 } ( \\| \\vec { F } \\| _ { D U ^ { 2 } _ { \\triangle } ( h ^ { 1 } ; J ^ m \\times \\mathbb { R } ^ 2 ) } ) ^ 2 \\big ] ^ { 1 / 2 } . \\end{align*}"}
+{"id": "6653.png", "formula": "\\begin{align*} c _ { n } & = \\prod _ { k = - \\kappa + 1 } ^ { - \\kappa + n } \\lambda _ { k } ^ { 2 } , n \\in \\mathbb { N } \\cap [ 0 , \\kappa ] , \\\\ c _ { n } & = \\prod _ { k = 0 } ^ { \\kappa - 1 } \\lambda _ { k } ^ { 2 } \\sum _ { i = 1 } ^ { \\eta } \\prod _ { j = 1 } ^ { n - \\kappa } \\lambda _ { i , j } ^ { 2 } , n \\in \\mathbb { N } \\cap [ \\kappa + 1 , \\kappa + p ] \\end{align*}"}
+{"id": "116.png", "formula": "\\begin{align*} \\mathbf G ^ { [ 2 ] } ( \\kappa h ; \\alpha ) = \\int _ { - \\pi } ^ { \\pi } \\frac { \\sin ^ 2 ( \\theta ) \\ , | \\widehat { \\alpha } ( \\theta ) | ^ 2 } { \\sinh ^ 2 ( 2 \\kappa h ) + \\sin ^ 2 ( \\theta ) } \\ , \\frac { d \\theta } { 2 \\pi } . \\end{align*}"}
+{"id": "4044.png", "formula": "\\begin{align*} y H _ { n - 1 } ( y , s ) = H _ n ( y , s ) + I ^ { - 2 } ( s ) 2 ( n - 1 ) H _ { n - 2 } ( y , s ) . \\end{align*}"}
+{"id": "5691.png", "formula": "\\begin{gather*} k _ j = \\begin{cases} \\sigma ( j ) & ( 1 \\le j \\le i + 1 ) \\\\ \\min \\{ k _ { 2 i + 2 - j } , \\cdots , k _ { i + 1 } \\} & ( i + 2 \\le j \\le 2 i ) \\\\ \\min \\{ k _ { j - 2 i } , \\cdots , k _ { i + 1 } \\} & ( 2 i + 1 \\le j \\le 3 i ) . \\end{cases} \\end{gather*}"}
+{"id": "910.png", "formula": "\\begin{align*} ( B \\Psi - \\Phi ) _ n = - B + 2 B K x _ n - \\Phi _ n = 0 , \\end{align*}"}
+{"id": "2635.png", "formula": "\\begin{align*} \\lceil x \\rceil = & \\lambda ( e , b ) \\lambda ( b , b ^ 2 ) . . . \\lambda ( b ^ { m - 1 } , b ^ m ) \\\\ \\lceil x \\rceil = & \\lambda ( e , a ) \\lambda ( a , a ^ 2 ) . . . \\lambda ( a ^ { n - 1 } , a ^ n ) \\\\ \\lceil x \\rceil = & \\lambda ( e , a ) \\lambda ( a , a ^ 2 ) . . . \\lambda ( a ^ n , a ^ n b ) . . . \\lambda ( b ^ { m - 1 } , b ^ m ) \\end{align*}"}
+{"id": "318.png", "formula": "\\begin{align*} F ( t ) - t = \\left ( \\frac { t _ { 1 } } { t _ { 2 } } , \\frac { t _ { 2 } } { t _ { 1 } } , \\frac { t _ { 3 } } { t _ { 1 } ^ { p } t _ { 2 } ^ { p ^ { 2 } } } \\right ) , \\end{align*}"}
+{"id": "6161.png", "formula": "\\begin{align*} ( M _ { ( P ^ \\perp + z P ) U } R _ q , R _ { \\overline q } M _ { U ^ * ( P + z P ^ \\perp ) } ) = ( V _ 1 ' \\tau _ { \\rm B C L } \\mathfrak r _ q \\tau _ { \\rm B C L } ^ * , \\tau _ { \\rm B C L } \\mathfrak r _ { \\overline q } \\tau _ { \\rm B C L } ^ * V _ 2 ' ) = \\tau _ { \\rm B C L } ( V _ 1 \\mathfrak r _ q , \\mathfrak r _ { \\overline q } V _ 2 ) \\tau _ { \\rm B C L } ^ * . \\end{align*}"}
+{"id": "776.png", "formula": "\\begin{align*} \\left \\| \\left ( T _ { \\varphi } ( x _ { j } ) \\right ) _ { j = 1 } ^ { n } \\right \\| _ { Y ^ { \\rm d u a l } ( F ^ * ) } = \\sup _ { ( y _ { j } ) _ { j = 1 } ^ { \\infty } \\in B _ { Y ( F ) } } \\left | \\sum _ { j = 1 } ^ { n } \\varphi ( x _ { j } \\otimes y _ { j } ) \\right | \\leq \\left \\| \\varphi \\cdot \\right \\| \\left \\| ( x _ { j } ) _ { j = 1 } ^ { n } \\right \\| _ { X ( E ) } . \\end{align*}"}
+{"id": "688.png", "formula": "\\begin{align*} G = i \\ \\overline { v } \\ \\widehat { v } . \\end{align*}"}
+{"id": "6940.png", "formula": "\\begin{align*} \\left [ q ^ d \\right ] \\sum _ { m = 0 } ^ N ( - y ) ^ m s _ { ( \\ell , 1 ^ { N - m } ) } ( z _ 1 , \\ldots , z _ { N + 1 } ) = & ( - 1 ) ^ N \\left [ t ^ \\ell \\right ] \\frac { y t ^ { N ( d - 1 ) + 1 } } { ( 1 - t ) ^ { N d } ( 1 - y t ) ^ { d + 1 } } . \\end{align*}"}
+{"id": "1996.png", "formula": "\\begin{align*} \\sum _ { w \\in N ( u ) } z _ w = ( q _ n - d ( u ) ) z _ u > \\Omega ( \\sqrt { n } ) . \\end{align*}"}
+{"id": "758.png", "formula": "\\begin{align*} V _ { \\alpha , \\beta , \\gamma } = \\mathbb { C } [ \\partial ] v , L _ \\lambda v = ( \\partial + \\alpha \\lambda + \\beta ) v , H _ \\lambda v = \\gamma v . \\end{align*}"}
+{"id": "5230.png", "formula": "\\begin{align*} s _ \\tau + t _ \\tau = a _ \\tau + e , \\operatorname { m a x } \\lbrace s _ \\tau , t _ \\tau \\rbrace \\geq a _ \\tau + 1 \\end{align*}"}
+{"id": "4118.png", "formula": "\\begin{align*} \\mathcal { K } _ c ^ { \\alpha } ( \\varphi , \\varphi ) = \\norm { \\left ( - \\mathcal { L } _ c ^ { \\alpha } \\right ) ^ { 1 / 2 } ( \\varphi ) } ^ 2 _ { L ^ 2 ( 0 , 1 ) } \\end{align*}"}
+{"id": "4060.png", "formula": "\\begin{align*} ( x \\rhd y ) \\rhd z - x \\rhd ( y \\rhd z ) = ( y \\rhd x ) \\rhd z - y \\rhd ( x \\rhd z ) , \\end{align*}"}
+{"id": "5849.png", "formula": "\\begin{align*} { } N \\langle f ( A ) x , x \\rangle ^ p + { N } ^ p \\langle f ( A ) x , x \\rangle ^ p \\Big \\{ \\displaystyle \\sum _ { n = N + 1 } ^ { \\infty } \\frac { 1 } { n ^ p } \\Big \\} & < N \\Big ( \\frac { p } { p - 1 } \\Big ) ^ { p } \\langle f ( A ) x , x \\rangle ^ p . \\end{align*}"}
+{"id": "1868.png", "formula": "\\begin{align*} y _ { ( 2 r ' _ 0 - 1 ) m + m ^ { + } _ 0 , j } & = M N - x _ { m ^ { + } _ 0 , j } + ( r ' _ 0 - \\tfrac { 1 } { 2 } ) ( m n ) + \\tfrac { 3 } { 2 } \\mbox { \\ i f } m ^ { + } _ 0 + j \\mbox { i s o d d a n d } \\\\ y _ { ( 2 r ' _ 0 - 1 ) m + m ^ { + } _ 0 , j } & = M N - x _ { m ^ { + } _ 0 , j } - ( r ' _ 0 - \\tfrac { 1 } { 2 } ) ( m n ) + \\tfrac { 3 } { 2 } \\mbox { \\ i f } m ^ { + } _ 0 + j \\mbox { i s e v e n . } \\end{align*}"}
+{"id": "6505.png", "formula": "\\begin{align*} \\binom { n - 1 } { r - 1 } < 2 \\binom { n - r - 1 } { r - 1 } . \\end{align*}"}
+{"id": "2846.png", "formula": "\\begin{align*} \\mathsf { N } _ { V _ 1 \\otimes V _ 2 } = \\mathsf { N } _ { V _ 1 } \\otimes \\mathsf { N } _ { V _ 2 } , \\end{align*}"}
+{"id": "5342.png", "formula": "\\begin{align*} V = [ J , F ( G ) ] = O _ 2 ( G ) \\times U , \\end{align*}"}
+{"id": "7821.png", "formula": "\\begin{align*} Q _ n & = n ^ n \\prod _ { i = 1 } ^ { n } c _ { i , n } , \\\\ c _ { i , n } & = \\frac { \\binom { 2 i - 2 } { i - 1 } \\binom { 2 n - 2 i } { n - i } } { \\binom { 2 n - 1 } { n - 1 } } \\end{align*}"}
+{"id": "7547.png", "formula": "\\begin{align*} A _ x ( z ) = \\begin{pmatrix} T _ x ( y , y ' ) \\end{pmatrix} _ { y , y ' = 0 } ^ { q - 1 } + z \\begin{pmatrix} T _ x ( y , y ' + q ) \\end{pmatrix} _ { y , y ' = 0 } ^ { q - 1 } , z \\in \\mathbb C . \\end{align*}"}
+{"id": "4751.png", "formula": "\\begin{align*} u \\succ v : = l ( T ( u ) ) v u \\prec v : = r ( T ( v ) ) u , \\forall u , v \\in V . \\end{align*}"}
+{"id": "577.png", "formula": "\\begin{align*} \\Phi _ 2 ( \\mu ^ t ) = \\int _ 0 ^ 1 H \\big ( \\mu ^ 0 _ u \\circ ( ( 1 - t ) \\mathrm { I d } + t T _ u ) ^ { - 1 } \\big ) \\ , d u \\end{align*}"}
+{"id": "5766.png", "formula": "\\begin{align*} - \\frac { 5 8 3 ( 4 a + 7 b ) } { 1 4 5 8 } < c \\leq - \\frac { 5 8 3 ( 4 a + 7 b ) - 5 8 3 2 } { 1 4 5 8 } . \\end{align*}"}
+{"id": "5225.png", "formula": "\\begin{align*} \\Psi _ 0 : l [ [ v ] ] \\rightarrow L ^ \\times \\otimes _ { \\mathbb { Z } } \\mathbb { F } _ p = H ^ 1 ( G _ L , \\mu _ p ( L ) ) \\end{align*}"}
+{"id": "5576.png", "formula": "\\begin{align*} \\int _ { S ^ { m n - 1 } } \\omega _ k \\ , d \\sigma \\sim 2 ^ { k n ( ( | J | - 1 / p _ J ) - ( | J | - 1 ) ) } = 2 ^ { - k n ( 1 / p _ J - 1 ) } . \\end{align*}"}
+{"id": "6762.png", "formula": "\\begin{align*} \\norm { \\psi _ t } _ { L ^ 2 ( \\mathbb { R } ^ 3 ) } = \\norm { \\psi } _ { L ^ 2 ( \\mathbb { R } ^ 3 ) } \\mathcal { E } [ \\psi _ t , \\varphi _ t ] = \\mathcal { E } [ \\psi , \\varphi ] \\ ; t \\in \\mathbb { R } . \\end{align*}"}
+{"id": "7147.png", "formula": "\\begin{align*} \\mathbb { E } ( d ; \\delta ) = \\langle a ^ { \\ast } \\mathbb { D } ( d ; \\delta ) _ { p } , a ^ { \\ast } \\mathbb { D } ( d ; \\delta ) _ { p + 1 } , \\cdots , a ^ { \\ast } \\mathbb { D } ( d ; \\delta ) _ q \\rangle \\end{align*}"}
+{"id": "864.png", "formula": "\\begin{align*} f _ { + } ( z , t , s ) & : = \\exp \\left ( - \\sum _ { n = 1 } ^ { \\infty } \\frac { z ^ { n } } { n } \\int _ { \\{ S _ n > 0 \\} } e ^ { i t S _ n } e ^ { i { s } \\cdot { C } _ n } d \\mathbb { P } \\right ) , \\\\ f _ { - } ( z , t , s ) & : = \\exp \\left ( + \\sum _ { n = 1 } ^ { \\infty } \\frac { z ^ { n } } { n } \\int _ { \\{ S _ n \\le 0 \\} } e ^ { i t S _ n } e ^ { i { s } \\cdot { C } _ n } d \\mathbb { P } \\right ) . \\end{align*}"}
+{"id": "3316.png", "formula": "\\begin{gather*} P _ { n _ 1 ( k _ 1 ) , n _ 2 ( k _ 2 ) } ( \\mu _ { m _ 1 ( j _ 1 ) } , \\mu _ { m _ 2 ( j _ 2 ) } ) = R _ { k _ 1 , k _ 2 } \\big ( y _ { j _ 1 } , y _ { j _ 2 } ; \\alpha _ 0 , \\alpha _ 1 , \\alpha _ 2 , N _ 1 , N _ 2 ; q ^ 2 \\big ) , \\end{gather*}"}
+{"id": "5294.png", "formula": "\\begin{align*} E _ s ( A ) : = \\# \\{ ( a _ 1 , . . . , a _ s , b _ 1 , . . . , b _ s ) \\in A ^ { 2 s } : a _ 1 + . . . + a _ s = b _ 1 + . . . + b _ s \\} . \\end{align*}"}
+{"id": "7339.png", "formula": "\\begin{align*} \\mathcal { I } _ { ( m , m ) } = \\big \\{ ( i , j , t , p ) \\mid \\varrho ( y , z ) = ( i , j , t , p ) , \\ \\ ( y , z ) \\in X _ { ( m , m ) } \\big \\} . \\end{align*}"}
+{"id": "5984.png", "formula": "\\begin{align*} T _ b ^ + ( 1 ) : = \\inf \\lbrace T ( i ) \\in \\mathcal { T } _ r : X ( T ( i ) ) > b \\rbrace . \\end{align*}"}
+{"id": "5283.png", "formula": "\\begin{align*} \\dim L _ { \\pi } = m \\ * n + m / 2 - | \\pi | . \\end{align*}"}
+{"id": "65.png", "formula": "\\begin{align*} g ( \\nabla \\Psi , \\vec { \\nu } ) = c o n s t \\ \\ \\textnormal { o n e a c h l e a f } \\Sigma _ { t } \\end{align*}"}
+{"id": "2375.png", "formula": "\\begin{align*} d L _ { \\gamma } ( u ) = \\sum _ { 1 \\leq i < j \\leq n } L _ { \\gamma } ( \\partial _ { a _ { i } , a _ { j } } u ) \\cdot d \\log ( a _ { i } - a _ { j } ) + \\sum _ { 1 \\leq i \\leq j \\leq n } L _ { \\gamma } ( \\partial _ { 1 - a _ { i } , a _ { j } } u ) \\cdot d \\log ( 1 - a _ { i } - a _ { j } ) \\end{align*}"}
+{"id": "547.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\mathcal { W } _ 2 ^ 2 ( P _ i , Q _ i ) \\le \\mathcal { W } _ 2 ^ 2 ( P , Q ) . \\end{align*}"}
+{"id": "5301.png", "formula": "\\begin{align*} M _ S f ( x ) : = \\sup _ { r > 0 } \\frac { 1 } { | S ^ { n - 1 } | } \\int _ { S ^ { n - 1 } } | f ( x + r \\theta ) | d \\theta . \\end{align*}"}
+{"id": "1251.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle - \\Delta _ p u _ p = f & \\Omega , \\\\ \\displaystyle | \\nabla u _ p | ^ { p - 2 } \\nabla u _ p \\cdot \\nu + \\lambda | u _ p | ^ { p - 2 } u _ p = g & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "5674.png", "formula": "\\begin{align*} \\left ( \\frac { r + 1 } { r + 1 - r \\alpha _ p } \\right ) ^ { - 1 } = 1 - ( 1 - \\frac { 1 } { r + 1 } ) \\alpha _ p = \\frac { 1 } { r + 1 } + ( 1 - \\alpha _ p ) - \\frac { 1 - \\alpha _ p } { r + 1 } \\asymp ( p - p _ c ) \\vee \\frac { 1 } { r } \\end{align*}"}
+{"id": "2468.png", "formula": "\\begin{align*} - \\Delta F _ * + F _ * = 2 \\pi J _ * \\end{align*}"}
+{"id": "1368.png", "formula": "\\begin{align*} E _ L ( t ) & = \\begin{cases} O ( t ^ { - a _ 0 } ) & ( 1 < a _ 0 < n ) , \\\\ O ( t ^ { - n + \\delta } ) & ( a _ 0 \\ge n ) \\end{cases} \\end{align*}"}
+{"id": "7818.png", "formula": "\\begin{align*} \\frac { J ^ { w _ 1 } ( X _ { i : n } ) } { J ^ { w _ 1 } ( X _ { i + 1 : n } ) } & = \\frac { c _ { i , n } } { c _ { i + 1 , n } } \\left ( \\frac { 2 i - 1 } { 2 i + 1 } \\right ) \\frac { E \\left ( M ( B _ { 2 i : 2 n } ) \\right ) } { E \\left ( M ( B _ { 2 i + 2 : 2 n } ) \\right ) } \\\\ & = \\frac { i ( 2 n - 2 i - 1 ) } { ( 2 i + 1 ) ( n - i ) } \\frac { E \\left ( M ( B _ { 2 i : 2 n } ) \\right ) } { E \\left ( M ( B _ { 2 i + 2 : 2 n } ) \\right ) } \\\\ & \\le \\frac { E \\left ( M ( B _ { 2 i : 2 n } ) \\right ) } { E \\left ( M ( B _ { 2 i + 2 : 2 n } ) \\right ) } . \\end{align*}"}
+{"id": "6507.png", "formula": "\\begin{align*} ( 2 R - I ) ^ 2 = 4 R ^ 2 - 4 R + I = 4 R - 4 R + I = I \\end{align*}"}
+{"id": "3189.png", "formula": "\\begin{align*} \\hat { v } _ { n } ( x ) : = \\int _ { \\mathbb { R } ^ { 3 } } \\frac { 1 - e ^ { - | x - y | } } { | x - y | } u _ { n } ( y ) u ( y ) d y . \\end{align*}"}
+{"id": "2905.png", "formula": "\\begin{align*} Y = \\left ( \\frac { \\eta } { m + 1 } x _ { n + 1 } ^ { m + 1 } + \\lambda x _ { n + 1 } \\right ) e _ { n + 1 } . \\end{align*}"}
+{"id": "5033.png", "formula": "\\begin{align*} \\gamma ( x ) = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } } e ^ { - \\frac { | x | ^ 2 } { 2 } } \\ , . \\end{align*}"}
+{"id": "3221.png", "formula": "\\begin{align*} \\mu _ 2 ( G ) > \\frac { 1 } { \\delta + 1 } \\left ( 2 + \\frac { 2 } { \\binom { d + 1 } { 2 } - 1 } \\right ) , \\end{align*}"}
+{"id": "3941.png", "formula": "\\begin{align*} H _ { \\Gamma _ n } ( p _ n ( x ) ) = \\operatorname { T r } \\nabla ^ 2 b _ { \\Omega _ n } ( p _ n ( x ) ) = \\operatorname { T r } \\nabla ^ 2 b _ { \\Omega _ n } ( x ) + \\operatorname { O } ( h ) . \\end{align*}"}
+{"id": "2429.png", "formula": "\\begin{align*} f ( w _ { 1 } \\cdot w _ { 2 } ) = f ( w _ { 1 } ) w _ { 2 } + w _ { 1 } f ( w _ { 2 } ) \\quad ( w _ { 1 } , w _ { 2 } \\in \\mathcal { R } ) \\end{align*}"}
+{"id": "5677.png", "formula": "\\begin{align*} \\zeta ( p ) \\geq \\Phi ( G ) \\log \\frac { \\Phi ( G ) } { 1 - p } + ( 1 - \\Phi ( G ) ) \\log \\frac { \\Phi ( G ) } { p } \\end{align*}"}
+{"id": "2150.png", "formula": "\\begin{align*} - \\mathrm { d i v } _ { \\vec { R } } \\ ; \\sigma \\left ( \\vec { x } , \\vec { y } , \\nabla u ( \\vec { x } ) + { \\bf R } ^ T \\nabla _ { \\vec { y } } u _ 1 ( \\vec { x } , \\vec { y } ) \\right ) = 0 \\vec { x } \\in \\Omega \\end{align*}"}
+{"id": "2085.png", "formula": "\\begin{align*} h ( x _ 1 , \\ldots , x _ n ) = ( \\psi ( x ) , x _ 2 , \\ldots , x _ n ) . \\end{align*}"}
+{"id": "4557.png", "formula": "\\begin{align*} H _ f ( G ) = \\frac { 1 } { 3 } \\Big ( 4 f ( 1 ) - f ( 4 ) \\Big ) n + \\frac { 2 } { 3 } \\Big ( f ( 4 ) - f ( 1 ) \\Big ) m + \\Gamma _ { \\ ! \\ ! f } ( G ) \\ , . \\end{align*}"}
+{"id": "6320.png", "formula": "\\begin{align*} x . \\phi : = D ( \\phi ) ( x ) . \\end{align*}"}
+{"id": "995.png", "formula": "\\begin{align*} 0 \\ = \\ - { \\rm c u r l } \\left ( { \\rm c u r l } \\ , u \\right ) + { \\rm D } ( { \\rm d i v } \\ , u ) \\ , - \\ , \\Delta u \\end{align*}"}
+{"id": "1833.png", "formula": "\\begin{align*} & \\mathcal { R } ( e _ 1 ) = \\frac { 1 } { 2 } e _ 1 + 2 \\alpha e _ 2 + 2 \\beta e _ 5 + 2 \\gamma e _ 6 , \\ \\ \\mathcal { R } ( e _ 2 ) = 0 , \\ \\ \\mathcal { R } ( e _ 3 ) = e _ 3 - \\alpha e _ 1 + \\delta e _ 5 - 2 \\beta e _ 6 , \\\\ & \\mathcal { R } ( e _ 4 ) = e _ 4 - \\beta e _ 1 - \\delta e _ 2 + \\mu e _ 6 , \\ \\ \\mathcal { R } ( e _ 5 ) = \\mathcal { R } ( e _ 6 ) = 0 , \\ \\ \\mathcal { R } ( e _ 7 ) = e _ 7 - \\gamma e _ 1 + 2 \\beta e _ 2 - \\mu e _ 5 . \\end{align*}"}
+{"id": "7848.png", "formula": "\\begin{align*} \\textup { S h i f t } _ \\mu : T ^ * \\widetilde Q & \\to T ^ * \\widetilde Q \\\\ \\alpha & \\mapsto \\alpha - \\alpha _ \\mu , \\end{align*}"}
+{"id": "3569.png", "formula": "\\begin{align*} a _ 2 a _ 3 = z _ 1 a _ 1 ^ 2 + z _ 2 a _ 1 a _ 2 + z _ 3 a _ 2 ^ 2 . \\end{align*}"}
+{"id": "8064.png", "formula": "\\begin{align*} \\max _ { p } ~ \\sum _ { \\widetilde { n } = 1 } ^ { K } | { \\bf w } ^ H _ p { \\bf y } _ { R _ 0 } [ \\widetilde { n } ] | ^ 2 = \\widetilde { \\beta } _ 0 , \\end{align*}"}
+{"id": "7928.png", "formula": "\\begin{align*} \\delta _ \\mathrm { m R B A } ( \\mu _ 1 , R _ 1 ) = ( \\delta _ \\mathrm { H o c h } ( \\mu _ 1 ) , ~ - \\widetilde { \\delta } _ \\mathrm { H o c h } ( R _ 1 ) - \\Psi ^ 2 ( \\mu _ 1 ) ) = 0 . \\end{align*}"}
+{"id": "3802.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 n + h ^ 2 - 3 h - 3 + d } { 2 } . \\end{align*}"}
+{"id": "3683.png", "formula": "\\begin{align*} \\mathcal { H } [ U \\cup \\{ u _ 1 \\} ] = \\mathcal { G } [ U \\cup \\{ u _ 1 \\} ] \\cong K _ { t + 2 } ^ { 3 } \\quad \\mathcal { H } [ U \\cup \\{ v \\} ] = \\mathcal { G } [ U \\cup \\{ v \\} ] \\cong K _ { t + 2 } ^ { 3 } . \\end{align*}"}
+{"id": "2740.png", "formula": "\\begin{align*} ( \\Lambda - \\alpha _ X \\pi ^ * H ) \\cdot \\bar { C } = - \\alpha _ X H \\cdot C < 0 . \\end{align*}"}
+{"id": "2697.png", "formula": "\\begin{align*} \\binom { a + j } { j } \\sum _ { k = j } ^ { n - j } \\binom { n - j } { k - j } \\bigl { ( } \\sum _ { l = 0 } ^ { a } \\binom { n - k } { a + j - l } \\binom { k - j } { a - l } \\binom { n - j + l } { l } \\bigr { ) } G ( n , k , a ) \\end{align*}"}
+{"id": "1807.png", "formula": "\\begin{align*} \\{ x , y \\} = x \\vartriangleright y - y \\vartriangleright x + [ x , y ] . \\end{align*}"}
+{"id": "5609.png", "formula": "\\begin{align*} R ^ i _ { j k l } = \\frac { { \\delta } \\hat { \\mathbb { G } } ^ i _ { j l } } { { \\delta } x ^ k } - \\frac { { \\delta } \\hat { \\mathbb { G } } ^ i _ { j k } } { { \\delta } x ^ l } + \\hat { \\mathbb { G } } ^ i _ { k s } \\hat { \\mathbb { G } } ^ s _ { j l } - \\hat { \\mathbb { G } } ^ s _ { j k } \\hat { \\mathbb { G } } ^ i _ { l s } \\end{align*}"}
+{"id": "13.png", "formula": "\\begin{align*} \\alpha _ m = \\max \\{ t \\ , | \\ , Q - t J - Y \\succeq 0 , Y \\in \\R ^ { Z _ m \\times Z _ m } _ { \\geq 0 } \\} . \\end{align*}"}
+{"id": "121.png", "formula": "\\begin{align*} 2 \\int _ 0 ^ { t } \\int \\bigl ( | \\widetilde \\psi ^ h | ^ 2 \\widetilde \\psi ^ h \\bigr ) ( s , x ) \\overline { \\bigl [ e ^ { - i ( t - s ) \\Delta } f \\bigr ] } ( x ) \\ , d x \\ , d s = 2 \\int _ 0 ^ { t } \\bigl \\langle e ^ { i ( t - s ) \\Delta } \\bigl ( | \\widetilde \\psi ^ h | ^ 2 \\widetilde \\psi ^ h \\bigr ) ( s ) , f \\bigr \\rangle \\ , d s . \\end{align*}"}
+{"id": "4080.png", "formula": "\\begin{align*} V = F ^ { - 1 } V \\supset F ^ { - 2 } V \\supset \\ldots \\supset 0 . \\end{align*}"}
+{"id": "3800.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 n - 3 + d } { 2 } . \\end{align*}"}
+{"id": "1.png", "formula": "\\begin{align*} S _ M ( X ; \\pi , \\hat { \\phi } , W ) = \\sum _ { p } \\frac { a _ { \\pi } ( p ) \\log p } { \\sqrt { p } } \\hat { \\phi } \\left ( \\frac { \\log p } { M \\log X } \\right ) \\left ( \\frac { 2 } { p } \\right ) \\sum _ { ( d , 2 ) = 1 } M _ Z ( d ) \\left ( \\frac { d } { p } \\right ) W \\left ( \\frac { d } { X } \\right ) . \\end{align*}"}
+{"id": "7847.png", "formula": "\\begin{align*} \\widetilde B _ 1 = B _ { \\mbox { \\tiny { $ \\ ! \\langle \\ ! J \\mathcal { K } \\ ! \\rangle $ } } } + \\widetilde J _ i d \\widetilde { \\mathcal { Y } } ^ i , \\end{align*}"}
+{"id": "6953.png", "formula": "\\begin{align*} \\mathsf G ( z ) = \\sum _ { d = 1 } ^ { \\infty } z ^ { d } \\cdot \\frac { N } { d } \\binom { ( N + 1 ) ( d - 1 ) } { d - 1 } = 1 - \\frac { 1 } { ( 1 + t ) ^ N } , \\end{align*}"}
+{"id": "62.png", "formula": "\\begin{align*} f ( x _ 1 , . . . , x _ n ) = x _ n \\end{align*}"}
+{"id": "629.png", "formula": "\\begin{align*} \\Gamma ^ { \\dag } \\left ( X , Y , Z \\right ) \\equiv \\left ( \\nabla _ { Z } ^ { \\dag } G \\right ) \\left ( X , Y \\right ) = - \\Gamma \\left ( X , Y , Z \\right ) . \\end{align*}"}
+{"id": "5920.png", "formula": "\\begin{align*} \\pi _ X ^ 4 = [ \\Delta _ X ] - \\pi _ X ^ 0 - \\pi _ X ^ 2 - \\pi _ X ^ 6 - \\pi _ X ^ 8 \\ , . \\end{align*}"}
+{"id": "1484.png", "formula": "\\begin{align*} J = \\sum _ { d \\le \\Delta , \\ , d | P ( z ) } h ( d ) \\end{align*}"}
+{"id": "2658.png", "formula": "\\begin{align*} d s ^ { 2 } = ( d x _ { 1 } ) ^ { 2 } + x _ 1 ( d x _ { 2 } ) ^ { 2 } + x _ 4 ( d x _ { 3 } ) ^ { 2 } + x _ 3 ( d x _ { 4 } ) ^ { 2 } . \\end{align*}"}
+{"id": "6522.png", "formula": "\\begin{align*} U = ( 2 P - I ) ( 2 Q - I ) . \\end{align*}"}
+{"id": "834.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { p ( x ) } { x ^ { k _ 1 } } - \\frac { p ( x ^ { * } ) } { ( x ^ { * } ) ^ { k _ 1 } } = - \\frac { 1 } { l _ 2 } \\frac { ( 1 - \\theta ) } { \\left ( \\frac { K k _ 1 } { k _ 1 p ( x ^ { * } ) - p ' ( x ^ { * } ) x ^ { * } } + \\theta \\right ) } - \\frac { l _ 1 } { l _ 2 } . \\end{aligned} \\end{align*}"}
+{"id": "3300.png", "formula": "\\begin{align*} \\phi _ { m _ 1 } : = \\phi _ { m _ 1 ( j ) } \\mu _ { m _ 1 } : = \\mu _ { m _ 1 ( j ) } \\end{align*}"}
+{"id": "6897.png", "formula": "\\begin{align*} \\phi _ M ( x ) : = \\phi ( x / M ) - \\phi ( 2 x / M ) \\end{align*}"}
+{"id": "5176.png", "formula": "\\begin{align*} E ^ 2 u ( x ) = \\begin{cases} E ^ { j } _ { n , i } ( u | _ { U ^ { j } _ { n , i } \\setminus T ^ { j } _ { n , i } } ) ( x ) , & x \\in U ^ { j } _ { n , i } \\ ; \\ ; j \\ ; , \\\\ u ( x ) , & , \\end{cases} \\end{align*}"}
+{"id": "1666.png", "formula": "\\begin{align*} \\left [ \\mathbf { 1 } + \\lambda \\mathtt { G } + \\lambda ^ 2 \\mathtt { H } + \\lambda ^ 3 \\mathtt { I } \\right ] ^ { - 1 } = \\mathbf { 1 } - \\lambda \\mathtt { G } - \\lambda ^ 2 \\left [ \\mathtt { H } - \\mathtt { G } ^ 2 \\right ] + \\lambda ^ 3 \\mathtt { R } \\ , , \\end{align*}"}
+{"id": "4308.png", "formula": "\\begin{gather*} L ^ m _ { j } = \\alpha _ 1 \\ell _ { j + m } + \\alpha _ 2 \\ell _ { j + m - 2 } + \\cdots + \\alpha _ { m + 1 } \\ell _ { j - m } , \\end{gather*}"}
+{"id": "593.png", "formula": "\\begin{align*} { \\rho } ^ { 0 0 } _ { n , p } = - \\widehat { \\mathcal { Q } } _ n ( j _ { p + 1 } ^ { ( 1 2 3 ) } ) - \\widehat { \\mathcal { Q } } _ n ( - j _ p ^ { ( 1 2 3 ) } ) + ( j ^ { ( 3 ) } - j ^ { ( 4 ) } - 1 ) ( j ^ { ( 3 ) } - j ^ { ( 4 ) } ) \\ , . \\end{align*}"}
+{"id": "4286.png", "formula": "\\begin{align*} ( q ) _ { \\infty } \\sum _ { k = 1 } ^ { \\infty } \\frac { q ^ { k ( k + 1 ) } } { ( q ) _ k ^ { 2 } ( 1 - q ^ k ) } F ( 0 ; q ^ k ; - q ^ k ) = \\frac { 1 } { 2 } \\frac { ( q ) _ { \\infty } } { ( - q ) _ { \\infty } } \\sum _ { k = 1 } ^ { \\infty } \\frac { q ^ { k ( k + 1 ) / 2 } } { ( q ) _ k ( 1 - q ^ k ) } \\left ( \\frac { ( - q ) _ k } { ( q ) _ k } - 1 \\right ) . \\end{align*}"}
+{"id": "3493.png", "formula": "\\begin{align*} \\textbf { e r r } ^ { ( 5 ) } = \\mathcal { O } \\Bigg ( \\varepsilon ^ 6 \\sqrt { \\mathcal { K } ^ { ( \\mathrm { T _ 0 } ) } _ { \\mathrm { r } } } \\frac { 1 } { | \\xi - z | ^ { 4 - 2 \\mathrm { r } } } \\Big \\Vert | \\mathrm { E } | ^ { 2 } \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\Omega ) } \\Bigg ) . \\end{align*}"}
+{"id": "5407.png", "formula": "\\begin{align*} m _ { s c } ^ 2 ( z ) + z m _ { s c } ( z ) + 1 = 0 . \\end{align*}"}
+{"id": "353.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } + \\gamma u _ t - \\Delta u + f ( u ) = g , ( x , t ) \\in \\Omega \\times \\mathbb { R } ^ + , \\\\ u | _ { \\partial \\Omega } = 0 , t \\in \\mathbb { R } ^ + , \\\\ u | _ { t = 0 } = u _ 0 , \\ u _ t | _ { t = 0 } = u _ 1 , x \\in \\Omega . \\end{cases} \\end{align*}"}
+{"id": "5067.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 f _ { k - 1 } ( y ) g ( y ) \\ , d y = 0 \\quad k \\ge 4 k \\equiv \\epsilon \\ ; ( 2 ) . \\end{align*}"}
+{"id": "461.png", "formula": "\\begin{align*} \\frac { F ( z ) L ( z ) } { \\sinh ^ 2 \\ ! \\Big ( \\frac { \\sqrt { 5 } F _ j } { 2 } z \\Big ) } = \\frac { 4 } { \\sqrt { 5 } } e ^ { L _ j z } \\coth \\ ! \\Big ( \\frac { \\sqrt { 5 } F _ j } { 2 } z \\Big ) . \\end{align*}"}
+{"id": "711.png", "formula": "\\begin{align*} { { { \\bf { m } } } ^ { \\left ( { e q } \\right ) } } = { \\bf { M } } { { \\bf { g } } ^ { \\left ( { e q } \\right ) } } = { \\left [ { T , 0 , 0 , 0 , \\overline w T , 0 , 0 } \\right ] ^ { \\rm T } } . \\end{align*}"}
+{"id": "6215.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\nu } = \\{ \\xi \\in V \\mid E ^ * _ { \\nu - 1 } A _ 1 E ^ * _ { \\nu } \\xi = 0 , \\ \\xi _ { x } = 0 \\ \\ \\partial ( x , x _ 0 ) \\neq \\nu \\} . \\end{align*}"}
+{"id": "5478.png", "formula": "\\begin{align*} F _ N ( a , b ; t ) & = \\frac { ( t q ) _ N } { ( t ) _ N } { } _ 3 \\phi _ 2 \\begin{bmatrix} q ^ { - N } , & b / a , & q & ; q , a t q ^ { N + 1 } \\\\ b q , & t q \\end{bmatrix} \\\\ & = \\frac { ( 1 - t q ^ N ) } { ( 1 - t ) } \\sum _ { n = 0 } ^ { N } ( - 1 ) ^ n \\begin{bmatrix} N \\\\ n \\end{bmatrix} \\frac { ( b / a ) _ n ( q ) _ n ( a t ) ^ n q ^ { n ( n + 1 ) / 2 } } { ( b q ) _ n ( t q ) _ n } . \\end{align*}"}
+{"id": "1626.png", "formula": "\\begin{align*} g ( 1 ) - g ( n ) & < g ( 1 ) - g ( k ) + g ( l ) - g ( n ) - ( 1 - \\alpha ) \\Big ( 3 - \\Big | \\frac { 1 } { k } - \\frac { 1 } { l } \\Big | \\Big ) \\\\ & < 2 + \\frac { 1 } { k } + 3 - \\frac { 1 } { n } + \\frac { 1 } { l } - 3 + \\Big | \\frac { 1 } { k } - \\frac { 1 } { l } \\Big | + 3 \\alpha \\\\ & = 2 - \\frac { 1 } { n } + \\frac { 1 } { k } + \\frac { 1 } { l } + \\Big | \\frac { 1 } { k } - \\frac { 1 } { l } \\Big | + \\frac { 2 } { n } - \\frac { 2 } { n + 1 } - 3 \\alpha \\\\ & \\le 2 + \\frac { 1 } { n } - 3 \\alpha \\\\ & \\le ( 1 - \\alpha ) d ( 1 , n ) . \\end{align*}"}
+{"id": "3457.png", "formula": "\\begin{align*} \\Big \\Vert u \\Big \\Vert _ { \\mathrm { H } ^ { 1 , \\frac { 1 } { 2 } } \\Big ( \\partial \\Omega \\times \\mathbb { R } \\Big ) } ^ 2 = \\int _ \\mathbb { R } \\int _ { \\partial \\Omega } \\Big [ | \\nabla _ { \\textbf { t a n } } u ( \\mathrm { x } , t ) | ^ 2 + u ^ 2 ( \\mathrm { x } , t ) + \\Big ( \\partial _ { t } ^ { \\frac { 1 } { 2 } } u ( \\mathrm { x } , t ) \\Big ) ^ 2 \\Big ] d \\sigma _ { \\mathrm { x } } d \\mathrm { t } , \\end{align*}"}
+{"id": "1390.png", "formula": "\\begin{align*} \\Psi ( x , t ; t _ 0 ) : = t _ 0 + t + A _ { \\varepsilon } ( x ) . \\end{align*}"}
+{"id": "8210.png", "formula": "\\begin{align*} \\begin{aligned} \\Vert \\Lambda ^ { \\alpha } ( u v ) - u \\Lambda ^ { \\alpha } v \\Vert _ { \\dot { B } ^ { s } _ { 2 , 1 } } \\lesssim _ { \\alpha , N , s , s _ 1 , s _ 2 } \\Vert u \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - s _ 1 } _ { 2 , 1 } } \\Vert v \\Vert _ { \\dot { B } ^ { s + s _ 1 + \\alpha - 1 } _ { 2 , 1 } } + \\Vert u \\Vert _ { \\dot { B } ^ { s + s _ 2 + \\alpha } _ { 2 , 1 } } \\Vert v \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } - s _ 2 } _ { 2 , 1 } } . \\end{aligned} \\end{align*}"}
+{"id": "977.png", "formula": "\\begin{align*} T ( \\delta _ \\xi ) = \\frac { \\delta _ \\xi } { \\sqrt { 1 + \\gamma ( \\xi ) ^ 2 } } . \\end{align*}"}
+{"id": "1913.png", "formula": "\\begin{align*} w = w ( t , v ) = w _ 0 ( t ) w _ 1 ( v _ 1 ) \\cdots w _ d ( v _ d ) , \\end{align*}"}
+{"id": "1628.png", "formula": "\\begin{align*} | a - g ( p ) | & \\le \\frac { 1 } { 2 } \\Big ( \\max \\big \\{ g ( p ) \\colon p \\in \\cup _ { i = 1 } ^ n \\{ x _ i , y _ i , u _ i , v _ i \\} \\big \\} \\\\ & \\qquad - \\min \\big \\{ g ( p ) \\colon p \\in \\cup _ { i = 1 } ^ n \\{ x _ i , y _ i , u _ i , v _ i \\} \\big \\} \\Big ) \\le 1 \\end{align*}"}
+{"id": "2876.png", "formula": "\\begin{align*} w _ { R / W _ k } ( x ) = \\frac { c ' } { ( 1 + | x | ^ 2 / ( R / W _ k ) ^ 2 ) ^ { 1 0 } } , \\qquad \\| w _ { R / W _ k } \\| _ 1 = 1 . \\end{align*}"}
+{"id": "7022.png", "formula": "\\begin{align*} \\begin{cases} v _ { t t } + \\Delta ^ 2 v + \\Delta \\theta = 0 , \\\\ \\theta _ t - \\Delta \\theta - \\Delta v _ t = 0 , \\\\ ( v , v _ t , \\theta ) ( 0 , x ) = ( v _ 0 , v _ 1 , \\theta _ 0 ) ( x ) . \\end{cases} \\end{align*}"}
+{"id": "1844.png", "formula": "\\begin{align*} V \\left ( \\mathcal { P } _ { m , n } \\right ) = \\left \\{ \\left ( i , j \\right ) : 1 \\leqslant i \\leqslant m , 1 \\leqslant j \\leqslant n \\right \\} , \\end{align*}"}
+{"id": "3896.png", "formula": "\\begin{align*} ( - \\Delta ) ^ \\gamma : \\ ; \\mathbb H ^ { \\gamma } & \\to \\mathbb H ^ { - \\gamma } ; \\\\ ( - \\Delta ) ^ \\gamma \\varphi & = \\sum _ { n = 1 } ^ \\infty \\lambda _ n ^ \\gamma \\left ( \\int \\limits _ \\Omega \\varphi ( x ) e _ n ( x ) d x \\right ) e _ n . \\end{align*}"}
+{"id": "3354.png", "formula": "\\begin{gather*} n ^ { ( w ) } : = [ M ^ { ( w ) } : K _ w ] , \\\\ m ^ { ( w ) } : = \\prod _ { w ' \\in T \\setminus \\{ w \\} } n ^ { ( w ' ) } , \\end{gather*}"}
+{"id": "7577.png", "formula": "\\begin{align*} \\iint G ( p , q ) d \\mu ( p ) d \\mu ( q ) & \\leq \\iint \\log \\frac { 1 } { | z _ 0 ( p ) - z _ 0 ( q ) | } d \\mu ( p ) d \\mu ( q ) + C _ 1 \\\\ & = \\iint \\log \\frac { 1 } { | z - w | } d \\omega ( z ) d \\omega ( w ) + C _ 1 \\\\ & \\leq - \\log \\delta + \\log 4 + C _ 1 . \\end{align*}"}
+{"id": "4099.png", "formula": "\\begin{align*} J ( d _ 1 , \\ldots , d _ n ; e _ 1 , \\ldots , e _ n ) & \\leq \\sum _ { 0 \\leq j \\leq n } \\frac { \\beta ^ { n + 1 } - \\beta ^ j } { \\beta - 1 } \\sum _ { 0 \\leq k \\leq j } r ^ k \\sum _ { ( m _ \\ell ) \\in S ^ { ( 2 ) } _ { j - k } } \\prod _ { \\ell = 2 } ^ n d _ \\ell ^ { m _ \\ell } \\\\ & \\leq \\sum _ { 0 \\leq j \\leq n } \\frac { \\beta ^ { n + 1 } - \\beta ^ j } { \\beta - 1 } \\sum _ { 0 \\leq k \\leq j } r ^ k \\cdot \\alpha ^ { j - k } \\cdot \\# S ^ { ( 2 ) } _ { j - k } . \\end{align*}"}
+{"id": "7769.png", "formula": "\\begin{align*} f : [ t , + \\infty ) \\rightarrow [ 0 , 1 ] , \\ f ( t ) = 1 , f ( + \\infty ) = \\lim \\limits _ { s \\rightarrow + \\infty } f ( s ) = 0 \\end{align*}"}
+{"id": "7556.png", "formula": "\\begin{align*} X : \\det ( \\lambda I - A ( z ) ) = 0 \\end{align*}"}
+{"id": "6717.png", "formula": "\\begin{align*} A _ { j } ( v ) = L ^ { - 1 } _ { M } [ \\hat { A } _ { j } ( v ) M ] \\mbox { a n d } B _ { i j } ( v ) = L ^ { - 1 } _ { M } [ \\hat { B } _ { i j } ( v ) M ] . \\end{align*}"}
+{"id": "3699.png", "formula": "\\begin{align*} q ^ b _ t = X _ 0 + \\int _ 0 ^ t b ( q ^ b _ s , s ) d s + W _ t \\end{align*}"}
+{"id": "6106.png", "formula": "\\begin{align*} \\begin{aligned} | \\mathcal H ( k , d , l ) | = \\binom { n - d + 1 } { k - d + 1 } - \\binom { n - d - l + 1 } { k - d + 1 } + ( d - 1 ) \\binom { n - d - l + 1 } { k - d - l + 2 } \\sim c \\binom { n } { k - d } , \\end{aligned} \\end{align*}"}
+{"id": "7.png", "formula": "\\begin{align*} \\Phi \\ , : \\ , \\left ( \\C ^ { Z _ m \\times Z _ m } \\right ) ^ { G _ m } \\to \\bigoplus _ { i = 1 } ^ k \\C ^ { m _ i \\times m _ i } , \\end{align*}"}
+{"id": "4951.png", "formula": "\\begin{align*} [ x , y ] = \\mathfrak { D } ( x ) \\cdot y - x \\cdot \\mathfrak { D } ( y ) . \\end{align*}"}
+{"id": "4746.png", "formula": "\\begin{gather*} r ^ * ( \\eth _ k ( a ) ) v ^ * = r ^ * ( a ) \\beta _ k ^ * ( v ^ * ) + \\alpha _ k ^ * ( r ^ * ( a ) v ^ * ) , \\\\ l ^ * ( \\eth _ k ( a ) ) v ^ * = l ^ * ( a ) \\beta _ k ^ * ( v ^ * ) + \\alpha _ k ^ * ( l ^ * ( a ) v ^ * ) . \\end{gather*}"}
+{"id": "890.png", "formula": "\\begin{align*} \\begin{aligned} 0 < 2 \\sigma _ 2 ( \\lambda ( D ^ 2 u ) ) = \\ , & 2 \\sum _ { 1 \\leq i < j \\leq n } ( u _ { i i } u _ { j j } - u _ { i j } ^ 2 ) \\\\ = \\ , & ( \\Delta u ) ^ 2 - \\sum _ i u _ { i i } ^ 2 - \\sum _ { i \\neq j } u _ { i j } ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "2976.png", "formula": "\\begin{align*} [ X _ i ] & = [ X _ { i + 1 } ] + [ Y _ i ] \\end{align*}"}
+{"id": "73.png", "formula": "\\begin{align*} \\lambda _ 1 ^ { - \\mathcal { L } } ( D ) \\leq Q ( v ) = Q _ { 1 } ( v ) . \\end{align*}"}
+{"id": "1326.png", "formula": "\\begin{align*} \\int _ { B _ r ( x _ 0 ) } \\Big ( \\left | \\nabla u ( x ) \\right | ^ p - \\left | \\nabla v ( x ) \\right | ^ p \\Big ) \\ , d x & = \\int _ { B _ r ( x _ 0 ) } \\left ( \\int _ 0 ^ 1 \\frac { d } { d s } \\left | \\nabla u _ s ( x ) \\right | ^ p \\ , d s \\right ) \\ , d x \\\\ & = \\int _ { B _ r ( x _ 0 ) } \\left ( \\int _ 0 ^ 1 p \\left | \\nabla u _ s ( x ) \\right | ^ { p - 2 } \\nabla u _ s ( x ) \\cdot \\nabla ( u - v ) ( x ) \\ , d s \\right ) \\ , d x . \\end{align*}"}
+{"id": "6405.png", "formula": "\\begin{align*} x _ { 0 } ^ { ( k + 4 ) } x _ { 0 } ^ { ( k ) } = v ^ { \\bullet } \\bigg [ e _ { 0 , 2 } e _ { 0 , 1 } e _ { 1 , 2 } \\left ( \\prod _ { j = 2 } ^ { k + 1 } ( e _ { j , j + 1 } ) ^ 2 \\right ) y _ { k + 2 } ^ { ( 3 ) } \\bigg ] + v ^ { \\bullet } ( x _ { 2 } ^ { ( k + 2 ) } ) ^ 2 . \\end{align*}"}
+{"id": "441.png", "formula": "\\begin{align*} \\frac { \\mathbf { E } _ \\rho ^ t ( S ' ) - \\mathbf { E } _ \\rho ^ t ( S ) } { t } & = \\frac { \\sum _ { i = 1 } ^ n \\mathcal { E } _ \\rho ^ t ( S ' , h _ i ^ t \\circ f _ i ^ t ) - \\mathbf { E } _ \\rho ^ t ( S ) } { t } \\\\ & > \\frac { \\sum _ { i = 1 } ^ n \\mathcal { E } _ \\rho ^ s ( S ' , h _ i ^ s \\circ f _ i ^ t ) - \\mathbf { E } _ \\rho ^ s ( S ) } { s } \\\\ & > \\sum _ { i = 1 } ^ n \\mathcal { E } _ \\rho ( S ' , \\pi \\circ \\tilde { f } _ i ^ t ) - \\mathbf { E } _ \\rho ( S ) . \\end{align*}"}
+{"id": "7140.png", "formula": "\\begin{align*} \\rho _ A : = _ a \\left ( \\frac { \\sum _ { i > a } w _ i } { \\sum _ { i > a } d _ i } - \\frac { \\sum _ { i \\leq a } w _ i } { \\sum _ { i \\leq a } d _ i } \\right ) . \\end{align*}"}
+{"id": "1851.png", "formula": "\\begin{align*} x _ { i , j } + x _ { i , j + 1 } & = S - a _ j , & x _ { i , n + 1 - j } + x _ { i , n - j } & = a _ j , \\\\ x _ { m + 1 - i , j } + x _ { m + 1 - i , j + 1 } & = S - a _ j , \\mbox { a n d } & x _ { m + 1 - i , n + 1 - j } + x _ { m + 1 - i , n - j } & = a _ j . \\end{align*}"}
+{"id": "2921.png", "formula": "\\begin{align*} T _ \\infty ( r , F ) = m _ \\infty ( r , F ) + N _ \\infty ( r , F ) . \\end{align*}"}
+{"id": "7781.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ X | \\nabla ^ { g ^ + } u | _ { g ^ + } ^ p d V _ { g ^ + } & = \\int _ { X \\varepsilon } x ^ { p - n - 1 } | \\nabla ^ g u | _ g ^ p d V _ g \\\\ & = \\int _ { X \\varepsilon } x ^ { p - n - 1 } | f ' ( x ) | ^ p \\cdot | \\nabla ^ g x | _ g ^ p d V _ g \\\\ & = \\int _ 0 ^ \\varepsilon x ^ { p - n - 1 } | f ' ( x ) | ^ p \\cdot V o l ( \\Sigma _ x ) d x \\end{aligned} \\end{align*}"}
+{"id": "6361.png", "formula": "\\begin{align*} X : = \\left [ \\begin{array} { c c c c | c c c c } A _ { d _ A } & & & & & & & \\\\ & I _ n & & & & & & \\\\ & & \\ddots & & & & & \\\\ & & & I _ n & & & & \\\\ \\hline & & & & D _ { d _ D } & & & \\\\ & & & & & I _ m & & \\\\ & & & & & & \\ddots & \\\\ & & & & & & & I _ m \\\\ \\end{array} \\right ] , \\end{align*}"}
+{"id": "1781.png", "formula": "\\begin{align*} p _ { \\tilde { A } , a } \\left ( C \\right ) \\left \\langle Y , B , a \\right \\rangle = 0 . \\end{align*}"}
+{"id": "989.png", "formula": "\\begin{align*} u = \\left ( \\begin{array} { c } u _ 1 \\\\ u _ 2 \\\\ u _ 3 \\\\ \\end{array} \\right ) \\ , , u _ i \\in { \\rm H } ^ { 1 } ( \\Omega ) , \\ P = \\left ( \\begin{array} { c c c } P _ { 1 1 } & P _ { 1 2 } & P _ { 1 3 } \\\\ P _ { 2 1 } & P _ { 2 2 } & P _ { 2 3 } \\\\ P _ { 3 1 } & P _ { 3 2 } & P _ { 3 3 } \\end{array} \\right ) \\ , P ^ T e _ i \\in { \\rm H } ( { \\rm c u r l } \\ , ; \\Omega ) \\ , \\ P ^ T e _ i \\in { \\rm H } ( { \\rm d i v } \\ , ; \\Omega ) \\end{align*}"}
+{"id": "3203.png", "formula": "\\begin{align*} I ( g _ { u } ( \\theta ) ) = \\frac { \\theta ^ { 6 } } { 2 } A ( u ) + \\frac { \\theta ^ { 6 } } { 4 } \\int _ { \\mathbb { R } ^ { 3 } } \\int _ { \\mathbb { R } ^ { 3 } } \\frac { 1 - e ^ { - \\theta ^ { - 2 } | x - y | } } { | x - y | } u ( x ) ^ { 2 } u ( y ) ^ { 2 } d x d y + \\frac { \\theta ^ { 4 p - 6 } } { p } C ( u ) . \\end{align*}"}
+{"id": "706.png", "formula": "\\begin{align*} { \\partial _ t } = \\sum \\limits _ { n = 1 } ^ { + \\infty } { { \\varepsilon ^ n } \\partial { t _ n } } , \\ ; \\ ; \\ ; \\nabla = \\varepsilon { \\nabla _ 1 } , \\ ; \\ ; \\ ; { g _ i } = \\sum \\limits _ { n = 1 } ^ { + \\infty } { { \\varepsilon ^ n } g _ i ^ { \\left ( n \\right ) } } , \\ ; \\ ; \\ ; \\bar F = \\varepsilon { \\bar F ^ { \\left ( 1 \\right ) } } , \\end{align*}"}
+{"id": "2526.png", "formula": "\\begin{align*} B _ 1 + \\cdots + B _ k \\subset \\{ n \\in \\N : T ^ n x _ { \\vec 0 } \\in E \\} = A \\end{align*}"}
+{"id": "2259.png", "formula": "\\begin{align*} \\epsilon _ 0 d _ 0 + \\dots + \\epsilon _ { l - 1 } d _ { l - 1 } + d _ l = \\epsilon ' _ 0 d _ 0 + \\dots + \\epsilon ' _ { l - 1 } d _ { l - 1 } . \\end{align*}"}
+{"id": "3459.png", "formula": "\\begin{align*} H ^ { 1 , \\frac { 1 } { 2 } } \\Big ( \\Omega \\times \\mathbb { R } , \\mathcal { L } \\Big ) : = \\Big \\{ \\mathrm { u } \\in H ^ { 1 , \\frac { 1 } { 2 } } \\Big ( \\Omega \\times \\mathbb { R } \\Big ) : \\mathcal { L } u \\in \\mathrm { L } ^ 2 \\Big ( \\Omega \\times \\mathbb { R } \\Big ) \\Big \\} , \\end{align*}"}
+{"id": "6913.png", "formula": "\\begin{align*} \\left [ q ^ d \\right ] \\mathsf { u } \\prod _ { i } \\bigg ( \\frac { z _ i ( \\alpha _ i + y ) } { \\alpha _ i ( z _ i + y ) } \\bigg ) ^ { b _ i } \\bigg ( \\frac { z _ i + y } { z _ i } \\bigg ) ^ { - 1 } \\frac { d h _ i } { d q } z _ i ^ { d + 1 } \\bigg ( \\frac { R ( z _ i ) } { \\prod _ { j } ( z _ j - \\alpha _ i ) } \\bigg ) ^ { b _ i } \\cdot \\prod _ { i < j } ( z _ i - z _ j ) ^ 2 \\bigg { | } _ { \\epsilon = 0 } . \\end{align*}"}
+{"id": "5120.png", "formula": "\\begin{align*} \\frac { 1 } { ( p + 1 ) \\tilde { q _ { \\theta } } ' } = \\frac { \\theta } { q _ { \\theta } } , \\frac { 1 } { ( p + 1 ) \\tilde { r _ { \\theta } } ' } = \\frac { \\theta } { r _ { \\theta } } + \\frac { 2 ( 1 - \\theta ) } { p d } , \\textmd { ( t h e H \\ \" o l d e r i n e q u a l i t y , o r s a y , i n t e r p o l a t i o n ) } \\end{align*}"}
+{"id": "531.png", "formula": "\\begin{align*} H ( Q _ k \\ , | \\ , P ) = \\frac { 1 } { Q ( B _ k ) } \\int _ { B _ k } \\log \\frac { d Q } { d P } \\ , d Q - \\log Q ( B _ k ) \\end{align*}"}
+{"id": "4369.png", "formula": "\\begin{align*} D _ H H ^ \\beta ( ( \\sigma _ 1 ^ 0 , . . . , \\sigma _ l ^ 0 , x _ 0 ) ( T ) ) = D _ H h ^ \\beta ( x _ 0 ( T ) ) \\circ w ^ 2 . \\end{align*}"}
+{"id": "3216.png", "formula": "\\begin{align*} | V | \\leq 1 + k \\sum _ { i = 0 } ^ { m - 1 } ( k - 1 ) ^ i . \\end{align*}"}
+{"id": "4544.png", "formula": "\\begin{align*} \\alpha ( \\hat { E } ^ i ) = \\left ( \\frac { \\alpha \\wedge E ^ i } { v } \\right ) = \\imath _ { \\hat { E } ^ i } \\alpha \\ , , \\alpha ( \\tilde { E } ^ i ) = \\left ( \\frac { \\alpha \\wedge E ^ i } { w } \\right ) = \\imath _ { \\tilde { E } ^ i } \\alpha \\ , . \\end{align*}"}
+{"id": "3838.png", "formula": "\\begin{align*} \\rho = \\left ( 3 + i _ { 1 } , 4 + i _ { 2 } , \\dots , h - 2 + i _ { h - 4 } , k + \\frac { - h ^ 2 + 3 h + 4 } { 2 } - i \\right ) , \\end{align*}"}
+{"id": "58.png", "formula": "\\begin{align*} X : p \\mapsto \\left . \\frac { d } { d t } \\right \\vert _ { t = 0 } \\Big ( \\exp ( t \\frak { X } ) ( p ) \\Big ) . \\end{align*}"}
+{"id": "8021.png", "formula": "\\begin{align*} A \\leq S \\left ( \\begin{bmatrix} Q _ 0 & r o w ( Q _ j ^ * ) _ { j \\geq m + 1 } \\\\ \\ : c o l ( Q _ j ) _ { j \\geq m + 1 } & T _ Q \\otimes I _ d ^ { m + 1 } \\end{bmatrix} ; 0 \\right ) : = \\Tilde { A } . \\end{align*}"}
+{"id": "6187.png", "formula": "\\begin{align*} E ^ * _ { m } A _ 1 E ^ * _ m = M ^ { 0 , 0 } _ { \\lfloor \\frac { m } { 2 } \\rfloor , \\lfloor \\frac { m } { 2 } \\rfloor } . \\end{align*}"}
+{"id": "4558.png", "formula": "\\begin{align*} \\xi _ { 1 } = f ( 2 ) - \\frac { 2 } { 3 } f ( 1 ) - \\frac { 1 } { 3 } f ( 4 ) \\xi _ { 2 } = f ( 3 ) - \\frac { 1 } { 3 } f ( 1 ) - \\frac { 2 } { 3 } f ( 4 ) \\end{align*}"}
+{"id": "5595.png", "formula": "\\begin{align*} G _ k u ^ k = 0 \\end{align*}"}
+{"id": "1018.png", "formula": "\\begin{align*} W _ { \\rm { m p } } ( \\overline { \\mathbf { U } } ) = 0 \\Rightarrow \\boldsymbol { \\alpha } = 0 ( \\boldsymbol { \\mathfrak { K } } = 0 ) . \\end{align*}"}
+{"id": "2520.png", "formula": "\\begin{align*} F ^ * x = ( x _ \\epsilon : \\epsilon \\ne \\vec 0 ) \\end{align*}"}
+{"id": "6620.png", "formula": "\\begin{align*} \\int _ \\Gamma \\big | [ u ] \\big | ^ 2 \\dd \\sigma & = \\sum _ { j = 1 } ^ M \\int _ { \\Gamma _ j } | u _ j - u _ { j - 1 } | ^ 2 \\dd \\sigma \\le 2 \\sum _ { j = 1 } ^ M \\int _ { \\Gamma _ j } \\big ( | u _ j | ^ 2 + | u _ { j - 1 } | ^ 2 \\big ) \\dd \\sigma . \\end{align*}"}
+{"id": "8224.png", "formula": "\\begin{align*} \\begin{aligned} X ( T ) \\le C X _ 0 , \\end{aligned} \\end{align*}"}
+{"id": "5797.png", "formula": "\\begin{align*} \\widetilde { A } ( z ) = \\langle A \\hat { k } _ z , \\hat { k } _ z \\rangle , ( z \\in \\Omega ) . \\end{align*}"}
+{"id": "2504.png", "formula": "\\begin{align*} N \\mapsto \\mu _ N = \\frac { 1 } { | \\Phi _ N | } \\sum _ { n \\in \\Phi _ N } \\delta _ { T ^ n a } \\end{align*}"}
+{"id": "6460.png", "formula": "\\begin{align*} \\| f \\| _ { C ^ \\tau } : = \\displaystyle \\sum _ { | \\alpha | \\le [ \\tau ] } \\| \\partial ^ \\alpha _ { x } f \\| _ { L ^ \\infty ( \\mathbb { R } ^ n ) } + \\displaystyle \\sum _ { | \\alpha | = [ \\tau ] } \\sup _ { x \\neq y } \\frac { | \\partial ^ \\alpha _ { x } f ( x ) - \\partial ^ \\alpha _ { x } f ( y ) | } { | x - y | ^ { \\tau - [ \\tau ] } } < \\infty . \\end{align*}"}
+{"id": "1879.png", "formula": "\\begin{align*} \\left | f ( x ) - \\frac { 1 } { 3 ^ { m l } } f \\left ( \\frac { x } { 3 ^ m } \\right ) \\right | \\leq \\sum _ { s = 0 } ^ { m - 1 } \\frac { 1 } { 3 ^ { l s } } Q \\left ( \\frac { x } { 3 ^ { s + 1 } } , \\frac { x } { 3 ^ { s + 1 } } \\right ) \\end{align*}"}
+{"id": "2964.png", "formula": "\\begin{align*} \\{ \\mu _ k \\} _ { k \\geq 0 } \\ ; , { \\sum } _ { k = 0 } ^ { \\infty } \\ \\mu _ k = \\infty , a , q \\geq 1 , \\ ; \\ ; a \\geq b , \\ ; \\ ; p _ 1 , p _ 2 \\geq q . \\end{align*}"}
+{"id": "1214.png", "formula": "\\begin{align*} P _ m ( M ) \\geq \\binom { E } { m } \\prod _ { k = 0 } ^ { m - 1 } \\binom { M - 2 k } { 2 } \\prod _ { k = 0 } ^ { M - 2 m - 1 } \\frac { E - i - m } { E } . \\end{align*}"}
+{"id": "3540.png", "formula": "\\begin{align*} \\gamma ^ { \\textbf { e x t } } _ { 1 } \\mathrm { U } _ { \\mathrm { e } } = - \\mathbb { H } _ { \\alpha _ \\mathrm { m } } \\Big [ \\mathrm { U } _ { \\mathrm { e } } \\Big ] ( \\mathrm { x } , t ) - \\Big ( \\frac { 1 } { 2 } I _ { d } - \\mathcal { K } _ { \\alpha _ \\mathrm { m } } ^ { * } \\Big ) \\mathcal { S } _ { \\alpha _ \\mathrm { m } } ^ { - 1 } \\Big ( \\frac { 1 } { 2 } I _ { d } - \\mathcal { K } _ { \\alpha _ \\mathrm { m } } \\Big ) \\Big [ \\gamma ^ { \\textbf { e x t } } _ { 0 } \\mathrm { U } _ { \\mathrm { e } } \\Big ] . \\end{align*}"}
+{"id": "2347.png", "formula": "\\begin{align*} \\zeta ^ { \\mathfrak { m } } ( \\Bbbk ' , \\Bbbk ^ { \\dagger } ) = \\zeta ^ { \\mathfrak { m } } ( \\Bbbk \\star \\Bbbk ' ) \\end{align*}"}
+{"id": "397.png", "formula": "\\begin{align*} f = f ^ 0 + x y f ^ { x y } \\end{align*}"}
+{"id": "8.png", "formula": "\\begin{align*} \\Phi \\colon ( \\C ^ { Z \\times Z } ) ^ { G } \\to \\bigoplus _ { i = 1 } ^ k \\C ^ { m _ i \\times m _ i } \\ , \\ , \\ , \\ , A \\mapsto \\bigoplus _ { i = 1 } ^ k U _ i ^ * A U _ i . \\end{align*}"}
+{"id": "6548.png", "formula": "\\begin{align*} U = \\exp \\left ( \\gamma ( U ^ T - U ) \\right ) . \\end{align*}"}
+{"id": "5717.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t \\nu _ t ( \\dd x ) & = \\kappa _ { \\nu _ t } ^ + ( \\dd x ) - \\kappa _ { \\nu _ t } ^ - ( \\dd x ) \\\\ \\end{aligned} \\end{align*}"}
+{"id": "6278.png", "formula": "\\begin{align*} \\varphi ( t ) = \\left ( \\norm { a _ { J _ { 1 / \\sqrt { j } } } } _ 1 - \\lambda \\right ) t + \\sqrt { 1 - j t ^ 2 } \\norm { a _ { I \\backslash J _ { 1 / \\sqrt { j } } } } , 1 / \\sqrt { j + 1 } < t \\leq 1 / \\sqrt { j } , \\end{align*}"}
+{"id": "1741.png", "formula": "\\begin{align*} M \\left \\| \\sum _ { | k | \\sim M } \\frac { \\omega _ k } { \\sqrt { \\lambda _ k } } e ^ { 2 \\pi i k x } \\right \\| _ { L ^ p } & = M \\left \\| \\sum _ { | k | \\sim M } \\Phi _ M ( k ) \\frac { \\omega _ k } { \\sqrt { \\lambda _ k } } e ^ { 2 \\pi i k x } \\right \\| _ { L ^ p } \\\\ & = \\left \\| \\sum _ { | k | \\sim M } \\Psi _ M ( k ) \\omega _ k e ^ { 2 \\pi i k x } \\right \\| _ { L ^ p } \\\\ & \\lesssim \\left \\| \\sum _ { | k | \\sim M } \\omega _ k e ^ { 2 \\pi i k x } \\right \\| _ { L ^ p } \\ , . \\end{align*}"}
+{"id": "8070.png", "formula": "\\begin{align*} p | _ 1 = \\rho _ 1 g \\left ( \\sum _ { j = 1 } ^ N D _ j + b - z \\right ) . \\end{align*}"}
+{"id": "3829.png", "formula": "\\begin{align*} a _ 1 + a _ 2 + \\cdots + a _ { h - 2 } = \\frac { ( h - 1 ) ( 2 n - 5 ) - ( a _ { h - 1 } + a _ h ) + d } { 2 } . \\end{align*}"}
+{"id": "95.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ \\Psi u = f ( u ) & A ( r _ { 1 } , r _ { 2 } ) \\\\ u \\equiv c _ { 1 } & \\{ r _ { 1 } \\} \\times N \\\\ u \\equiv c _ { 2 } & \\{ r _ { 2 } \\} \\times N . \\end{cases} \\end{align*}"}
+{"id": "5278.png", "formula": "\\begin{align*} \\begin{cases} & \\sum _ { i = 1 } ^ { n + 1 } t _ i = 0 , \\\\ & \\sum _ { i = 1 } ^ { n + 1 + i } t _ i = 0 , \\\\ & \\sum _ { i \\in B } t _ i = 0 , \\ \\ \\forall B \\in \\pi . \\end{cases} \\end{align*}"}
+{"id": "1254.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle - \\Delta _ p u _ p = \\frac { A } { \\left | x \\right | } & \\ \\Omega , \\\\ | \\nabla u _ p | ^ { p - 2 } \\nabla u _ p \\cdot \\nu + \\lambda u ^ { p - 1 } _ p = \\gamma & \\ \\partial \\Omega \\ , , \\end{cases} \\end{align*}"}
+{"id": "1299.png", "formula": "\\begin{align*} \\alpha _ { a \\otimes b , c , d } \\circ \\alpha _ { a , b , c \\otimes d } = ( \\alpha _ { a , b , c } \\otimes d ) \\circ \\alpha _ { a , b \\otimes c , d } \\circ ( a \\otimes \\alpha _ { b , c , d } ) , \\end{align*}"}
+{"id": "771.png", "formula": "\\begin{align*} e m + \\gamma = \\widehat { \\iota } ( D ) c \\geq \\widehat { \\iota } ( D ) ( b _ 1 d + 1 ) . \\end{align*}"}
+{"id": "4762.png", "formula": "\\begin{align*} \\delta = ( \\eth _ 1 \\otimes \\eth _ 2 - \\eth _ 2 \\otimes \\eth _ 1 ) \\Delta . \\end{align*}"}
+{"id": "7596.png", "formula": "\\begin{align*} \\tilde { G } ( p , q ) = - \\frac { 1 } { 2 \\pi } \\log | z ( p ) - z ( q ) | + O ( 1 ) \\end{align*}"}
+{"id": "1435.png", "formula": "\\begin{align*} \\mathcal { E } ( t ; t _ 0 , x _ 0 ) & : = \\frac { 1 } { 2 } \\int _ { B _ { t _ 0 - t } ( x _ 0 ) \\cap \\Omega } ( | \\partial _ t u ( t , x ) | ^ 2 + | \\nabla u ( t , x ) | ^ 2 + | u ( t , x ) | ^ 2 ) \\ , d x \\end{align*}"}
+{"id": "2173.png", "formula": "\\begin{align*} \\gamma ^ 2 _ { i j } & = \\frac { 1 } { 2 | A _ 0 | } \\ , \\alpha _ { i j } ^ 0 \\ , ( \\overline { \\alpha } _ 1 ^ { \\ , 0 } T _ { 2 3 } - \\overline { \\alpha } _ 2 ^ { \\ , 0 } T _ { 1 3 } + \\overline { \\alpha } _ 3 ^ { \\ , 0 } T _ { 1 2 } ) ^ 2 \\end{align*}"}
+{"id": "4835.png", "formula": "\\begin{align*} \\int _ S \\int _ { B _ R } \\rho _ \\epsilon \\nabla g \\cdot \\nabla \\phi - \\epsilon \\ , \\rho _ \\epsilon \\cdot \\Delta \\phi + \\rho _ \\epsilon \\phi \\ , \\dd x \\dd \\mu ( s ) = \\int _ S \\int _ { B _ R } \\left ( \\int _ S \\rho _ \\epsilon ( x , s ' ) \\ , \\dd \\mu ( s ' ) \\right ) \\phi \\ , \\dd x \\dd \\mu ( s ) , \\end{align*}"}
+{"id": "3392.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty u ( h _ 1 - i ^ 2 , h _ 2 + i ) = 0 \\end{align*}"}
+{"id": "7525.png", "formula": "\\begin{align*} C ( p , q ) & = - 2 \\partial _ p \\left ( \\log \\left | \\frac { \\theta _ 1 ( p ) \\theta _ 1 ( q ) } { \\theta _ 1 ( p - q ) } \\right | - \\frac { 2 \\pi } { \\Im \\tau } \\left ( \\Im p \\right ) \\left ( \\Im q \\right ) \\right ) d p \\\\ & = - \\left ( \\frac { \\theta _ 1 ' ( p ) } { \\theta _ 1 ( p ) } - \\frac { \\theta _ 1 ' ( p - q ) } { \\theta _ 1 ( p - q ) } + \\frac { 2 \\pi i } { \\Im \\tau } \\Im q \\right ) d p . \\end{align*}"}
+{"id": "561.png", "formula": "\\begin{align*} H ( \\overline { Q } ) \\le \\frac 1 n \\sum _ { i = 1 } ^ n H ( Q _ i ) . \\end{align*}"}
+{"id": "1551.png", "formula": "\\begin{align*} X _ { s _ { k } ^ { \\prime \\prime } } \\left ( \\omega ^ { \\prime } \\right ) = X _ { s _ { k } ^ { \\prime \\prime } } \\left ( \\omega \\right ) \\end{align*}"}
+{"id": "4801.png", "formula": "\\begin{align*} \\mathcal F _ { 0 , q + 1 } \\mathcal B ( U _ { n , k } ) & = \\phi _ { q + 1 } ( n , k ) ( b _ 1 + k - 1 ) \\dotsm ( b _ q + k - 1 ) k U _ { n , k } , \\\\ \\mathcal A ( U _ { n , k - 1 } ) & = \\quad \\psi _ p ( n , k ) ( b _ 1 + k - 1 ) \\dotsm ( b _ q + k - 1 ) k U _ { n , k } , \\\\ \\mathcal B ( k U _ { n , k } ) & = \\quad \\phi _ { q } ( n , k ) ( b _ 1 + k - 1 ) \\dotsm ( b _ q + k - 1 ) k U _ { n , k } . \\end{align*}"}
+{"id": "1664.png", "formula": "\\begin{align*} e ^ { \\lambda \\mathtt { P } } = \\mathbf { 1 } + \\lambda \\mathtt { P } + \\mbox { \\small $ \\frac { 1 } { 2 } $ } \\lambda ^ 2 \\mathtt { P } ^ 2 + \\lambda ^ 3 \\mathtt { T } ^ { ( \\lambda ) } ( \\mathtt { P } ) \\ , , \\end{align*}"}
+{"id": "2438.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 \\leq i \\leq k } } I _ { \\mathrm { d c h } } ^ { \\mathfrak { m } } ( 0 ' ; a _ { 1 } , a _ { 2 } , \\dots , a _ { i } ; 1 ' ) I _ { \\mathrm { d c h } ^ { - 1 } } ^ { \\mathfrak { m } } ( 1 ' ; a _ { i + 1 } , \\dots , a _ { k } ; 0 ' ) = I _ { \\mathrm { d c h } \\circ \\mathrm { d c h } ^ { - 1 } } ^ { \\mathfrak { m } } ( 0 ' ; a _ { 1 } , \\dots , a _ { k } ; 0 ' ) = \\delta _ { k , 0 } , \\end{align*}"}
+{"id": "677.png", "formula": "\\begin{align*} [ X _ 1 ] = ( \\frac { \\mu } { 2 } d \\overline { z } - \\frac { 2 } { \\mu } h _ z ^ 2 d z + \\frac { \\tau _ 0 } { 2 \\mu } d \\overline { z } ) J - \\frac { 1 } { \\mu } d h \\sqrt { \\tau _ 0 } I . \\end{align*}"}
+{"id": "380.png", "formula": "\\begin{align*} \\Omega ( M ) _ N = \\frac { \\Omega ( M ) _ { [ N ] } } { I _ \\Omega ( N ) } \\end{align*}"}
+{"id": "4411.png", "formula": "\\begin{align*} \\upsilon = r _ 0 + \\upsilon _ 1 q , \\upsilon _ 1 = d ^ { ( 0 ) } _ 1 + \\cdots + d ^ { ( 0 ) } _ k q ^ { k - 1 } , \\end{align*}"}
+{"id": "192.png", "formula": "\\begin{align*} Q ^ + _ { A _ 0 } \\psi + \\pi ^ - ( a \\cdot \\psi ) = 0 , \\ , \\ , \\ , d ^ * a = 0 , \\ , \\ , \\ , d ^ + a + F ^ + _ { A _ 0 } = \\rho ^ { - 1 } ( \\mu ( \\psi ) ) . \\end{align*}"}
+{"id": "504.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\hat f _ i ( s _ 0 , X ) \\bar f _ i ( t _ 0 ) \\not \\equiv . \\end{align*}"}
+{"id": "4295.png", "formula": "\\begin{align*} C _ 1 ( q ) & = \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n + 1 } q ^ { n ( n + 1 ) / 2 } } { 1 - q ^ n } = \\sum _ { k = 0 } ^ { \\infty } \\frac { k q ^ { k ^ 2 } } { ( q ) _ k ^ { 2 } } , \\\\ R _ 1 ( q ) & = \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n + 1 } q ^ { n ( 3 n + 1 ) / 2 } } { 1 - q ^ n } . \\end{align*}"}
+{"id": "1025.png", "formula": "\\begin{align*} \\dim A _ i = 4 i + 1 - m _ i - n _ i . \\end{align*}"}
+{"id": "4499.png", "formula": "\\begin{align*} V ( u + h ) = \\frac { 1 } { 2 } \\int _ { [ 0 , 1 ] } \\big ( ( u ' ) ^ 2 + 2 u ' h ' + ( h ' ) ^ 2 \\big ) \\ , , \\end{align*}"}
+{"id": "3579.png", "formula": "\\begin{align*} \\lambda = n ^ { - \\tfrac { b } { t + b + 2 b \\nu } } . \\end{align*}"}
+{"id": "6752.png", "formula": "\\begin{align*} a \\in [ 0 , \\frac { 1 } { 2 } ) , \\mbox { a n d } t \\leq T _ { m a x } = \\frac { 1 } { 4 C _ 1 } \\frac { 1 } { \\eta _ { 0 } \\varepsilon ^ a + \\varepsilon ^ { \\frac { 1 } { 2 } - a } } . \\end{align*}"}
+{"id": "4524.png", "formula": "\\begin{align*} ^ 2 E _ S [ f ] = \\int _ S f ^ i \\ , \\imath _ { \\hat { \\mathfrak { E } } _ i } \\ ! v \\ , , \\end{align*}"}
+{"id": "1941.png", "formula": "\\begin{align*} \\begin{aligned} & \\| \\lambda ^ { 1 / 2 } | u _ { \\varepsilon } | + | D _ v u _ { \\varepsilon } | + | ( - \\Delta _ x ) ^ { 1 / 6 } u _ { \\varepsilon } | \\| \\\\ & \\le N \\| | \\vec f _ { \\varepsilon } | + \\lambda ^ { - 1 / 2 } | g _ { \\varepsilon } | \\| \\le N \\| | \\vec f | + \\lambda ^ { - 1 / 2 } | g | \\| . \\end{aligned} \\end{align*}"}
+{"id": "1159.png", "formula": "\\begin{align*} S = \\frac { ( 4 \\pi ) ^ { 2 k - 3 } } { \\Gamma ( 2 k - 3 ) } \\left ( 1 + C ( p ) + O ( k ^ { - \\alpha } D ^ { - \\alpha } + p ^ { - 5 / 4 + \\epsilon } k ^ { - 1 3 / 1 2 } D ^ { 7 / 8 + \\epsilon } ) \\right ) . \\end{align*}"}
+{"id": "1980.png", "formula": "\\begin{align*} \\lambda _ 1 ^ 2 x _ u ^ 2 = \\bigg ( \\sum _ { u w \\in E ( G ) } x _ w \\bigg ) ^ 2 \\leq d ( u ) \\sum _ { u w \\in E ( G ) } x _ w ^ 2 \\leq d ( u ) , \\end{align*}"}
+{"id": "6144.png", "formula": "\\begin{align*} ( R _ { \\overline q } \\otimes U ^ * P + R _ { \\overline q } M _ z \\otimes U ^ * P ^ \\perp ) ( z ^ n \\otimes \\xi ) = \\overline q ^ n z ^ n \\otimes U ^ * P \\xi + \\overline q ^ { n + 1 } z ^ { n + 1 } \\otimes U ^ * P ^ \\perp \\xi \\end{align*}"}
+{"id": "2810.png", "formula": "\\begin{align*} d R _ s = \\rho _ s d s + \\eta _ s d W ^ R _ s , s \\in [ 0 , T ] , R _ 0 = 0 , \\end{align*}"}
+{"id": "5908.png", "formula": "\\begin{align*} A = \\left ( \\begin{smallmatrix} B _ 1 & & & \\\\ & B _ 2 & & \\\\ & & \\ddots & \\\\ & & & B _ r \\end{smallmatrix} \\right ) \\ , , \\end{align*}"}
+{"id": "3401.png", "formula": "\\begin{align*} ( x _ 2 y ) x _ 1 = x _ 1 ( y x _ 2 ) . \\end{align*}"}
+{"id": "573.png", "formula": "\\begin{align*} I ( \\mu _ m ) & = H ( \\mu ^ m \\ , | \\ , \\overline { \\mu } ) = \\int _ { [ 0 , 1 ] \\times \\R } \\log \\frac { d \\mu ^ m } { d \\overline { \\mu } } \\ , d \\mu ^ m \\\\ & = \\int _ { [ 0 , 1 ] \\times \\R } \\log \\frac { d \\mu ^ m } { d \\mu } \\ , d \\mu ^ m + \\int _ { [ 0 , 1 ] \\times \\R } \\log \\frac { d \\mu } { d \\overline { \\mu } } \\ , d \\mu ^ m . \\end{align*}"}
+{"id": "1250.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\phi _ 1 | D u | + \\int _ { \\partial \\Omega } \\psi | u | \\ , d \\mathcal H ^ { N - 1 } = \\int _ { \\Omega ' } | \\mu | = \\sup \\left \\{ L ( F ) \\ > : \\ > F \\in C _ 0 ( \\Omega ' ) ^ N \\ \\ \\| F \\| _ \\infty \\le 1 \\right \\} \\ , . \\end{align*}"}
+{"id": "2988.png", "formula": "\\begin{align*} g ( x , y ) & = \\frac { 1 } { \\sqrt { ( - y + b ) ^ 2 + ( x - a ) ^ 2 } } \\cdot ( - y + b , x - a ) \\ ; , \\\\ h ( x , y ) & = \\left ( \\tfrac { r } { \\sqrt { ( x - a ) ^ 2 + ( y - b ) ^ 2 } } - 1 \\right ) \\cdot ( x - a , y - b ) \\ ; . \\end{align*}"}
+{"id": "2457.png", "formula": "\\begin{align*} J ( u , \\ , A ) : = \\frac { 1 } { 2 } \\d j ( u , \\ , A ) + \\frac { 1 } { 2 } F _ A \\end{align*}"}
+{"id": "1296.png", "formula": "\\begin{align*} \\pi ^ i _ b \\circ f \\circ \\iota ^ j _ a = \\pi ^ i _ b \\circ g \\circ \\iota ^ j _ a . \\end{align*}"}
+{"id": "3793.png", "formula": "\\begin{align*} n : = k + k ' + \\frac { ( l + l ' ) } { 2 } + \\sum _ { r = 1 } ^ k i _ r + \\sum _ { r = 1 } ^ { k ' } i ' _ r + \\sum _ { r = 1 } ^ l j _ r + \\sum _ { r = 1 } ^ { l ' } j ' _ r . \\end{align*}"}
+{"id": "819.png", "formula": "\\begin{align*} d x ( t ) = \\alpha x ( t ) d t + \\sigma x ( t ) d W ( t ) , \\ \\ x ( 0 ) = x . \\end{align*}"}
+{"id": "3554.png", "formula": "\\begin{align*} 1 - \\alpha \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } = \\mathcal { O } ( \\delta ^ \\mathrm { h } ) \\iff \\omega ^ 2 = \\omega _ 0 ^ 2 - \\omega _ \\mathrm { p } ^ 2 \\dfrac { \\varepsilon _ \\mathrm { m } + \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } } { \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } ( \\varepsilon _ \\infty ^ { - 1 } \\varepsilon _ \\mathrm { m } - 1 ) - \\varepsilon _ \\mathrm { m } } + \\mathcal { O } ( \\delta ^ \\mathrm { h } ) \\end{align*}"}
+{"id": "4.png", "formula": "\\begin{align*} S _ 1 = & S _ { 0 , 1 } + S _ { 0 , - 1 } + S _ { 1 , 1 } - S _ { 1 , - 1 } , \\end{align*}"}
+{"id": "6615.png", "formula": "\\begin{align*} f _ { n } ^ { + } ( x , y ) : = e ^ { i k x - 2 y } \\chi _ { n } ( x ) \\widetilde { \\chi } _ { n } ( y ) , f _ { n } ^ { - } ( x , y ) : = - e ^ { i k x + 2 y } \\chi _ { n } ( x ) \\widetilde { \\chi } _ { n } ( y ) . \\end{align*}"}
+{"id": "3996.png", "formula": "\\begin{align*} \\hat { \\mathcal { M } } _ { \\beta } ( t ) \\overset { d } { = } \\hat { \\mathcal { M } } ( T _ { 2 \\beta } ( t ) ) , \\ t > 0 , \\end{align*}"}
+{"id": "2086.png", "formula": "\\begin{align*} \\int _ X h \\ , | \\alpha | : = \\int _ { U } h \\ , | f | \\ , d \\mu _ n , \\end{align*}"}
+{"id": "2560.png", "formula": "\\begin{align*} \\eta _ t : = \\exp \\int _ t ^ T ( A - B ^ 2 R ^ { - 1 } P _ s ) d s . \\end{align*}"}
+{"id": "5441.png", "formula": "\\begin{align*} \\langle \\ , p _ { 1 3 } \\ , , \\ , p _ { 1 4 } \\ , , \\ , p _ { 1 2 } p _ { 3 4 } \\ , \\rangle = \\langle \\ , p _ { 1 3 } , p _ { 1 4 } , p _ { 3 4 } \\ , \\rangle \\ , \\ , \\cap \\ , \\ , \\langle \\ , p _ { 1 2 } , p _ { 1 3 } , p _ { 1 4 } \\ , \\rangle . \\end{align*}"}
+{"id": "650.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = \\frac { 1 } { 2 } \\sqrt { c _ 1 } \\ X _ 1 \\cdot \\nu _ 1 \\cdot \\varphi - \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi \\end{align*}"}
+{"id": "6439.png", "formula": "\\begin{align*} h _ n ( x , y ) = \\sum _ { k = 0 } ^ n \\begin{bmatrix} n \\\\ k \\end{bmatrix} x ^ k y ^ { n - k } . \\end{align*}"}
+{"id": "7774.png", "formula": "\\begin{align*} \\begin{aligned} ( \\int _ t ^ { + \\infty } [ A ( s ) ] ^ { \\frac { 1 } { 1 - p } } d s ) ^ { 1 - p } & \\leq \\int _ t ^ { + \\infty } [ ( \\mathcal { A } ( q ) + \\varepsilon ) ( \\frac { e ^ s } { 2 } ) ^ n ] ^ { \\frac { 1 } { 1 - p } } d s ) ^ { 1 - p } \\\\ & = ( \\mathcal { A } ( q ) + \\varepsilon ) \\frac { 1 } { 2 ^ n } ( \\int _ t ^ { + \\infty } e ^ { \\frac { n s } { 1 - p } } d s ) ^ { 1 - p } \\\\ & = \\frac { 1 } { 2 ^ n } ( \\frac { n } { p - 1 } ) ^ { p - 1 } ( \\mathcal { A } ( q ) + \\varepsilon ) e ^ { n t } \\end{aligned} \\end{align*}"}
+{"id": "8134.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac { 1 } { n } \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ^ { \\otimes n } \\otimes \\overline A ) = \\lim _ { n \\rightarrow + \\infty } \\frac { 1 } { n } \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ^ { \\otimes n } \\otimes \\overline B ) \\end{align*}"}
+{"id": "559.png", "formula": "\\begin{align*} M _ n = n \\sup _ { Q \\in \\P ( \\R ) } \\left ( \\int _ { \\R } V \\ , d Q + \\frac 1 2 \\int _ { \\R } \\int _ { \\R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \\right ) , \\end{align*}"}
+{"id": "2205.png", "formula": "\\begin{align*} & r _ 1 = - 4 b ( a d - 4 b ^ 2 ) ^ 3 , \\\\ & r _ 2 = 2 ( a - d ) ( a d - 4 b ^ 2 ) ^ 3 , \\\\ & r _ 4 = 4 b ( a d - 4 b ^ 2 ) ^ 3 . \\end{align*}"}
+{"id": "3269.png", "formula": "\\begin{gather*} \\widehat { P } _ n ( m ) = \\frac { 1 } { \\sqrt { h _ n } } R _ n \\big ( y _ m ; \\alpha , \\beta , \\gamma , \\delta ; q ^ 2 \\big ) , \\end{gather*}"}
+{"id": "777.png", "formula": "\\begin{align*} { \\cal I } ( E ; F ) \\stackrel { 1 } { = } ( E \\otimes _ { \\alpha } F ^ * ) ^ * \\cap \\mathcal { L } ( E , F ) \\end{align*}"}
+{"id": "2083.png", "formula": "\\begin{align*} h ( x ) = ( \\varphi _ 1 ( x ) , x _ 2 , \\ldots , x _ n ) . \\end{align*}"}
+{"id": "1476.png", "formula": "\\begin{align*} \\tilde { Q } _ n ( s ) = \\begin{bmatrix} \\frac { 1 } { f _ { g ( ( s - 1 ) \\ell _ w ^ * + 1 ) } - \\alpha _ n } e _ M ( \\theta ) 1 _ { \\{ 1 \\in J _ w ^ { [ s ] } \\} } + \\hat { Z } _ { s , 1 } \\\\ \\vdots \\\\ \\frac { 1 } { f _ { g ( s \\ell _ w ^ * ) } - \\alpha _ n } e _ M ( \\theta ) 1 _ { \\{ \\ell _ w ^ * \\in J _ w ^ { [ s ] } \\} } + \\hat { Z } _ { s , \\ell _ w ^ * } \\end{bmatrix} , \\end{align*}"}
+{"id": "3435.png", "formula": "\\begin{align*} D _ z = \\partial _ z + \\sum _ { j \\geq 0 } \\partial _ z ^ { j + 1 } v \\partial _ { \\partial _ z ^ j v } . \\end{align*}"}
+{"id": "2477.png", "formula": "\\begin{align*} \\norm { \\rho _ i \\nu _ k } _ { W ^ { - 1 , p } ( U _ i ) } \\lesssim \\norm { \\rho _ i \\nu _ k } _ { W ^ { - 1 , 1 } ( U _ i ) } ^ \\alpha \\abs { \\rho _ i \\nu _ k } ( U _ i ) ^ { 1 - \\alpha } ( k = 1 , 2 , \\dots d ) , \\end{align*}"}
+{"id": "1801.png", "formula": "\\begin{align*} { \\rm s i g n } \\left \\langle i _ { 1 } , i _ { 1 } + 1 , \\dots , \\widehat { i _ { j } + \\epsilon } , \\dots , i _ { r } , i _ { r } + 1 \\right \\rangle = \\left ( - 1 \\right ) ^ { s + i _ { j } + \\epsilon } . \\end{align*}"}
+{"id": "7741.png", "formula": "\\begin{align*} 2 \\cdot ( \\frac { V o l ( \\partial X , \\hat { g } ) } { \\mathcal { A } ( p ) } ) ^ { \\frac { 1 } { n } } \\leq \\delta e ^ { \\coth ^ { - 1 } ( 1 + a ( \\delta ) ) } = \\delta e ^ { \\frac { 1 } { 2 } \\ln \\frac { a ( \\delta ) + 2 } { a ( \\delta ) } } = \\delta ( 1 + \\frac { 2 } { a ( \\delta ) } ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
+{"id": "7164.png", "formula": "\\begin{align*} \\mathcal { C } ( 1 ) ^ { \\times d } = \\mathcal { C } ( d ) ^ { \\lambda } \\stackrel { q } { \\leftarrow } \\mathcal { C } ( d ) ^ { \\lambda \\geq 0 } \\stackrel { p } { \\to } \\mathcal { C } ( d ) , \\end{align*}"}
+{"id": "4757.png", "formula": "\\begin{align*} \\beta _ 2 ( \\partial _ 1 ( a ) ) \\cdot b = \\beta _ 1 ( \\partial _ 2 ( a ) ) \\cdot b , \\forall a , b \\in A . \\end{align*}"}
+{"id": "1073.png", "formula": "\\begin{align*} \\int \\limits _ { | | \\xi | | = 1 } \\sigma _ { - 4 } ( D _ { k } ^ { - 3 } ) d \\xi = \\frac { i } { 2 } \\gamma ^ { a } \\biggl ( k ^ { - 2 } \\bigl [ k ^ { - 2 } , \\{ k ^ { - 1 } , \\delta _ { a } k \\} \\bigr ] k ^ { - 2 } \\biggr ) . \\end{align*}"}
+{"id": "3966.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\mathrm { d } } { \\mathrm { d } t } \\bar { q } ( 0 , t ) & = - \\lambda \\bar { q } ( 0 , t ) , \\\\ \\frac { \\mathrm { d } } { \\mathrm { d } t } \\bar { q } ( n , t ) & = - \\lambda \\bar { q } ( n , t ) + \\frac { \\lambda } { ( 1 - \\rho ) ^ { - r } - 1 } \\sum _ { j = 1 } ^ { n } \\binom { r + j - 1 } { j } \\rho ^ { j } \\bar { q } ( n - j , t ) , \\ n \\ge 1 , \\end{aligned} \\end{align*}"}
+{"id": "4526.png", "formula": "\\begin{align*} \\big \\{ ^ 2 E _ { S _ 1 } [ f _ 1 ] , ^ 2 E _ { S _ 2 } [ f _ 2 ] \\big \\} = 0 \\ , , \\big \\{ \\mathrm { T r } \\ , h _ { \\gamma _ 1 } , \\mathrm { T r } \\ , h _ { \\gamma _ 2 } \\big \\} = 0 \\ , , \\end{align*}"}
+{"id": "4067.png", "formula": "\\begin{align*} \\langle \\delta ( \\mathcal { T } ) , \\mathcal { T } ' \\otimes \\mathcal { T } '' \\rangle & = \\frac { 1 } { | \\min ( \\mathcal { T } ) | } \\sum _ { \\substack { Y \\circledcirc \\mathcal { T } \\\\ \\mathcal { T } _ { | Y } \\approx \\mathcal { T } ' , \\\\ \\mathcal { T } _ { | X \\setminus Y } \\approx \\mathcal { T } '' } } \\sigma ( \\mathcal { T } ' ) \\sigma ( \\mathcal { T } '' ) = \\frac { 1 } { | \\min ( \\mathcal { T } ) | } | A | \\end{align*}"}
+{"id": "4956.png", "formula": "\\begin{align*} [ \\varphi ( x ) , [ y , z ] ] + [ \\varphi ( y ) , [ z , y ] ] + [ \\varphi ( z ) , [ x , y ] ] = 0 . \\end{align*}"}
+{"id": "668.png", "formula": "\\begin{align*} d g _ 1 ' = ( \\log \\sqrt { \\tau _ 0 } ) _ z d z \\ g ' _ 1 + \\frac { 1 } { \\sqrt { \\tau _ 0 } } \\left ( Q _ 0 d z + \\frac { \\tau _ 0 } { 4 } d \\overline { z } \\right ) \\ J \\widehat { g _ 1 ' } I , \\end{align*}"}
+{"id": "4541.png", "formula": "\\begin{align*} \\mathrm { e s s } \\sup _ { x \\in U } \\chi = \\mathrm { i n f } \\{ a \\in \\mathbb { R } : \\mu \\big ( f ^ { - 1 } ( a , \\infty ) \\big ) = 0 \\} \\end{align*}"}
+{"id": "249.png", "formula": "\\begin{align*} \\Q [ \\mathsf { f } ] ( y ) = \\left ( \\begin{array} { c c c c c } { \\bf R } [ f _ 1 ( y ) ] & 0 & \\dots & 0 & 0 \\\\ 0 & { \\bf R } [ f _ 2 ( y ) ] & \\dots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & { \\bf R } [ f _ { d - 1 } ( y ) ] & 0 \\\\ 0 & 0 & \\dots & 0 & { \\bf R } [ f _ d ( y ) ] \\end{array} \\right ) , \\end{align*}"}
+{"id": "4831.png", "formula": "\\begin{align*} \\int _ S \\int _ { B _ R } \\epsilon e ^ { - g / \\epsilon \\ , } \\nabla A ( h ) \\cdot \\nabla \\phi + e ^ { - g / \\epsilon \\ , } A ( h ) \\phi \\ , \\dd x \\dd \\mu ( s ) = \\int _ S \\int _ { B _ R } \\left ( \\int _ S e ^ { - g / \\epsilon \\ , } h ( x , s ' ) \\dd \\mu ( s ' ) \\right ) \\phi \\dd x \\dd \\mu ( s ) , \\end{align*}"}
+{"id": "1346.png", "formula": "\\begin{align*} u _ r ( x ) : = \\frac { u ( r x ) } { r } . \\end{align*}"}
+{"id": "2073.png", "formula": "\\begin{align*} P M Q = \\begin{bmatrix} I _ k \\\\ 0 \\\\ D \\end{bmatrix} , \\end{align*}"}
+{"id": "523.png", "formula": "\\begin{align*} V _ { \\mathrm { d e t } } = \\sup _ { y \\in \\R ^ n } \\left ( \\int _ { \\R ^ n } g \\ , d \\gamma _ { y , T } - \\frac { 1 } { n } H ( \\gamma _ { y , T } \\ , | \\ , \\gamma _ T ) \\right ) = \\sup _ { y \\in \\R ^ n } \\left ( \\int _ { \\R ^ n } g \\ , d \\gamma _ { y , T } - \\frac { | y | ^ 2 } { 2 n T } \\right ) , \\end{align*}"}
+{"id": "482.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\binom n k \\frac { F _ { j k } } { L _ j ^ k } B _ { n - k } = 0 , \\qquad \\mbox { $ n $ e v e n } , \\end{align*}"}
+{"id": "4168.png", "formula": "\\begin{align*} \\sum _ { | j | < \\kappa ^ r } | \\gamma _ j | ^ 2 I _ 2 = \\kappa ^ { 2 \\beta - 1 } i _ 2 ( 2 \\beta - 1 ) \\sum _ { | j | < \\kappa ^ r } | \\gamma _ j | ^ 2 \\Bigl [ 2 \\pi | j _ h | ^ 2 U _ e ^ 2 + \\Bigl ( \\frac { 2 \\pi } 3 | j | ^ 2 + 2 \\pi j _ z ^ 2 \\Bigr ) U _ f ^ 2 \\Bigr ] . \\end{align*}"}
+{"id": "8009.png", "formula": "\\begin{align*} S _ { M \\otimes I _ d } ( I ) \\oplus S _ { M \\otimes I _ d } ( I ) = S _ { M \\otimes I _ d } ( S _ { M \\otimes I _ d } ( J ) ; J ) \\end{align*}"}
+{"id": "7515.png", "formula": "\\begin{align*} \\theta _ 1 ( z ) = \\theta _ 1 ( z ; \\tau ) = - i \\sum _ { k = - \\infty } ^ { \\infty } ( - 1 ) ^ k e ^ { \\pi i \\tau ( k + \\frac { 1 } { 2 } ) ^ 2 + ( 2 k + 1 ) \\pi i z } . \\end{align*}"}
+{"id": "1049.png", "formula": "\\begin{align*} \\sqrt { ( g ) } = 1 - \\frac { 1 } { 6 } \\mathrm { R i c } _ { a b } x ^ { a } x ^ { b } + o ( { \\mathbf { x ^ { 2 } } } ) , \\end{align*}"}
+{"id": "8096.png", "formula": "\\begin{align*} T _ 2 ^ * T _ 1 = [ M _ z ^ { \\alpha } ] ^ * A _ r \\otimes M _ z ^ { \\alpha } = r A _ r [ M _ z ^ { \\alpha } ] ^ * \\otimes M _ z ^ { \\alpha } = r ( A _ r [ M _ z ^ { \\alpha } ] ^ * \\otimes M _ z ^ { \\alpha } ) = r T _ 1 T _ 2 ^ * . \\end{align*}"}
+{"id": "1150.png", "formula": "\\begin{align*} \\theta _ { S , \\mu } \\left ( \\gamma ( \\tau , z ) \\right ) ( c \\tau + d ) ^ { - \\frac { g } { 2 } } e \\left ( - c S [ z ] ( c \\tau + d ) ^ { - 1 } \\right ) = \\sum _ { \\eta \\in ( 2 S ) ^ { - 1 } \\mathbb Z ^ g / \\mathbb Z ^ g } \\varepsilon _ { S } ( \\eta , \\mu ; \\gamma ) \\theta _ { S , \\eta } ( \\tau , z ) , \\end{align*}"}
+{"id": "774.png", "formula": "\\begin{align*} \\alpha _ { X , Y } ^ { } ( u ; M \\otimes N ) & \\leq \\left \\| ( x _ { j } ) _ { j = 1 } ^ { n } \\right \\| _ { X ( M ) } \\cdot \\left \\| ( y _ { j } ) _ { j = 1 } ^ { n } \\right \\| _ { Y ( N ) } \\\\ & \\leq \\left \\| ( x _ { j } ) _ { j = 1 } ^ { n } \\right \\| _ { X ( E ) } \\cdot \\left \\| ( y _ { j } ) _ { j = 1 } ^ { n } \\right \\| _ { Y ( F ) } \\leq \\alpha _ { X , Y } ^ { } ( u ; E \\otimes F ) + \\eta . \\end{align*}"}
+{"id": "5530.png", "formula": "\\begin{align*} ( 1 , 2 , \\ldots , 2 n ) = \\prod _ { i = 1 } ^ r \\prod _ { j = 1 } ^ r I _ { i , j } = I _ { 1 , 1 } \\cdot I _ { 1 , 2 } \\cdot I _ { 1 , 3 } \\cdots I _ { r , r } , \\end{align*}"}
+{"id": "4633.png", "formula": "\\begin{align*} \\phi _ \\lambda ( x , t ) : = \\int _ 0 ^ t \\tfrac p \\lambda \\tau ^ { p - 1 } + \\min \\{ \\phi _ - ' ( x , \\tau ) , p \\lambda \\tau ^ { p - 1 } \\} \\ , d \\tau = \\tfrac 1 \\lambda t ^ p + \\int _ 0 ^ t \\min \\{ \\phi _ - ' ( x , \\tau ) , p \\lambda \\tau ^ { p - 1 } \\} \\ , d \\tau . \\end{align*}"}
+{"id": "4181.png", "formula": "\\begin{align*} p _ n < n , \\lim _ { n \\rightarrow \\infty } p _ n / n = 1 \\quad \\lim _ { n \\rightarrow \\infty } n - p _ n - \\lambda \\log ( n ) = + \\infty . \\end{align*}"}
+{"id": "748.png", "formula": "\\begin{align*} q _ { i j } ( x ) & = 0 | i - j | > m , \\\\ q _ { i j } ( x ) & \\in ( 0 , M ] 0 < | i - j | \\leq m , \\end{align*}"}
+{"id": "1261.png", "formula": "\\begin{align*} \\min _ { u \\in B V ( \\Omega ) } \\tilde J ( u ) = \\min _ { u \\in B V ( \\Omega ) } J ( u ) \\ , . \\end{align*}"}
+{"id": "935.png", "formula": "\\begin{align*} \\begin{aligned} u ( x _ 0 ) & = \\varphi ( x _ 0 ) = \\frac { 1 } { 2 } \\sum _ { \\beta = \\alpha + 1 } ^ { n - 1 } b _ \\beta x _ \\beta ^ 2 + \\frac { 1 } { 6 } \\sum _ { \\xi , \\beta , \\gamma \\geq \\alpha + 1 } \\varphi _ { \\xi \\beta \\gamma } ( 0 ) x _ \\xi x _ \\beta x _ \\gamma + O ( | \\tilde { x } | ^ 4 ) \\\\ & \\geq \\frac { 1 } { 2 } \\delta ^ 2 b _ \\alpha ^ 2 - C \\delta ^ 3 b _ \\alpha ^ 2 \\geq \\frac { 7 } { 1 6 } \\delta ^ 2 b _ \\alpha ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "7723.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { A } ( p ) & = 2 ^ n \\cdot V o l ( \\partial X , \\hat { g } ^ p ) = 2 ^ n \\int _ { \\partial X } d V _ { \\hat { g } ^ p } \\\\ & = 2 ^ n \\int _ { \\partial X } e ^ { - n ( s _ p - s _ q ) } d V _ { \\hat { g } ^ q } \\end{aligned} \\end{align*}"}
+{"id": "5461.png", "formula": "\\begin{align*} ( n , k , d ) = & ( 2 7 , 6 , 1 5 ) , ( 2 7 , 8 , 1 3 ) , ( 2 7 , 9 , 1 2 ) , ( 2 7 , 1 1 , 1 0 ) , ( 2 8 , 1 1 , 1 1 ) , \\\\ & ( 2 9 , 1 1 , 1 1 ) , ( 3 0 , 7 , 1 6 ) , ( 3 0 , 1 2 , 1 1 ) . \\end{align*}"}
+{"id": "1997.png", "formula": "\\begin{align*} B : = \\Big \\{ v \\in L : z _ v > 0 \\Big \\} , ~ ~ C : = \\Big \\{ v \\in L : z _ v < 0 \\Big \\} \\end{align*}"}
+{"id": "5150.png", "formula": "\\begin{align*} F _ { \\delta } = \\left \\{ x \\in \\partial \\Omega : \\ , \\exists r ^ x _ i \\searrow 0 \\ ; \\ ; \\ ; \\dfrac { | A ' \\cap B ( x , r ^ x _ i ) | } { | B ( x , r ^ x _ i ) \\cap \\Omega | } \\in [ \\delta , 1 - \\delta ] \\right \\} . \\end{align*}"}
+{"id": "3475.png", "formula": "\\begin{align*} \\textbf { e r r } ^ { ( 1 ) } = \\mathcal { O } \\Bigg ( \\delta \\Big \\Vert \\gamma ^ { \\textbf { i n t } } _ { 1 } \\mathrm { U } _ { \\mathrm { i } } \\Big \\Vert _ { \\mathrm { H } ^ { - \\frac { 1 } { 2 } , - \\frac { 1 } { 4 } } \\Big ( \\partial \\Omega \\times \\mathbb { R } \\Big ) } \\Big \\Vert \\nabla \\Phi ^ { \\textbf { e } } ( \\xi , t ; \\mathrm { z } , \\cdot ) \\Big \\Vert _ { \\mathrm { H } ^ { \\frac { 1 } { 2 } , \\frac { 1 } { 4 } } \\Big ( \\partial \\Omega \\times ( 0 , \\mathrm { t } ) \\Big ) } \\Bigg ) . \\end{align*}"}
+{"id": "5417.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { b = 1 } ^ N | G _ { a b } G _ { b c } ( t , z ) | ^ 2 \\prec \\frac { \\Im m _ N ( t , z ) } { N \\eta } . \\end{align*}"}
+{"id": "7745.png", "formula": "\\begin{align*} ( \\mathbb { R } ^ n \\times \\mathbb { S } ^ 1 ( \\lambda ) , g ^ + = d t ^ 2 + \\sinh ^ 2 t g _ { \\mathbb { S } ^ { n - 1 } } + \\cosh ^ 2 t g _ { \\mathbb { S } ^ 1 ( \\lambda ) } ) . \\end{align*}"}
+{"id": "1223.png", "formula": "\\begin{align*} \\int _ { X ^ - } \\phi \\dd \\mu ^ - = 0 . \\end{align*}"}
+{"id": "4637.png", "formula": "\\begin{align*} \\int _ \\Omega \\phi ( x , | \\nabla u _ \\infty | ) \\ , d x - \\int _ \\Omega \\phi ( x , | \\nabla u | ) \\ , d x = : \\epsilon > 0 . \\end{align*}"}
+{"id": "6403.png", "formula": "\\begin{align*} \\mu _ 1 \\colon e _ 1 e _ 1 ' & = [ e _ 6 e _ 7 e _ 9 ] + v ^ { - 1 } [ e _ 3 e _ 4 ] , & \\mu _ 3 \\colon e _ 3 e _ 3 ' & = [ e _ 6 e _ 9 e _ { 1 3 } ] + v ^ { - 1 } [ e _ 1 e _ 5 ] , \\\\ \\mu _ 4 \\colon e _ 4 e _ 4 ' & = v ^ { - 1 } [ e _ { 1 } ^ 2 e _ { 2 } ] + v [ e _ { 6 } e _ { 7 } ^ 2 e _ { 9 } ^ 2 e _ { 1 2 } ] , & \\mu _ 6 \\colon e _ 6 e _ 6 ' & = v ^ { - 1 } [ e _ 3 e _ 4 e _ { 1 1 } e _ { 1 3 } ] + v [ e _ 1 ^ 2 e _ { 5 } ^ 2 ] . \\end{align*}"}
+{"id": "487.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom n k \\frac { 2 ^ k L _ { j k } } { L _ j ^ { k } } B _ { n - k } = n \\Big ( \\frac { \\sqrt 5 F _ j } { L _ j } \\Big ) ^ { n - 1 } , \\qquad \\mbox { $ n $ o d d } . \\end{align*}"}
+{"id": "5751.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } a _ { i j } x _ i x _ j + \\sum _ { i = 1 } ^ { m } a _ i x _ i + a _ 0 , \\end{align*}"}
+{"id": "3228.png", "formula": "\\begin{align*} \\tilde { I } ( u , v ) = \\int _ { X } ( u - v ) ( \\theta ^ { n } _ { u } - \\theta ^ { n } _ { V } ) \\end{align*}"}
+{"id": "856.png", "formula": "\\begin{align*} \\begin{aligned} \\mu = \\left ( - \\frac { 1 } { a ' k _ 2 \\sigma } \\right ) \\ln \\left ( \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) , \\varrho = \\left [ \\frac { 1 } { k _ 2 \\sigma } \\ln \\frac { K k _ 2 } { k _ 2 - 1 } \\right ] ^ { 2 } . \\end{aligned} \\end{align*}"}
+{"id": "4709.png", "formula": "\\begin{align*} L ( f ) = \\int _ { - \\pi } ^ { \\pi } f ( e ^ { i \\theta } ) h ( e ^ { i \\theta } ) d m _ { \\lambda } ( \\theta ) \\end{align*}"}
+{"id": "4593.png", "formula": "\\begin{align*} \\ln \\frac { m _ { 2 l + 2 } N + n _ 0 - b } { n _ 0 - b } = \\ln \\frac { m _ { 2 l + 2 } N + n _ 0 - b } { m _ 0 N + n _ 0 - b } + \\ln \\frac { m _ { 0 } N + n _ 0 - b } { n _ 0 - b } \\leq { \\frac { l + 2 } { 9 8 C _ k } } . \\end{align*}"}
+{"id": "925.png", "formula": "\\begin{align*} \\epsilon _ 1 b _ { n - 1 } u _ n ( x ) \\leq u ( y ) - u ( x ) + \\sum _ { \\alpha = 1 } ^ { n - 1 } ( y _ \\alpha - x _ \\alpha ) u _ \\alpha ( x ) \\leq C b _ { n - 1 } ^ 2 \\end{align*}"}
+{"id": "6472.png", "formula": "\\begin{align*} \\| \\psi \\omega ^ { ( 2 ) } _ { G } \\| ^ 2 _ { H ^ { - \\frac { 1 } { 2 } - \\epsilon } ( M \\times M ) } & \\le \\displaystyle { \\sum ^ \\infty _ { j = 1 } } \\frac { C } { \\lambda _ j ^ { 3 + 2 \\epsilon } } \\le \\displaystyle { \\sum ^ \\infty _ { j = 1 } } \\frac { C ' } { j ^ { 1 + \\epsilon } } < \\infty \\end{align*}"}
+{"id": "4792.png", "formula": "\\begin{align*} & \\left ( \\frac { n + a _ { p + 1 } + p + 1 } { n + 1 } S _ n - 1 \\right ) R _ { p , b } ( n ) \\\\ & \\quad = \\frac { n + a _ { p + 1 } + p + 1 } { n + 1 } R _ { p , b } ( n + 1 ) S _ n - R _ { p , b } ( n ) \\\\ & \\quad = \\frac { n + a _ { p + 1 } + p + 1 } { n + 1 } R _ { p , b } ( n + 1 ) L _ b + \\left ( \\frac { n + a _ { p + 1 } + p + 1 } { n + b + 1 } R _ { p , b } ( n + 1 ) - R _ { p , b } ( n ) \\right ) . \\end{align*}"}
+{"id": "2525.png", "formula": "\\begin{align*} H _ \\epsilon = \\bigcap _ { b \\in B _ 1 \\times \\cdots \\times B _ k } T ^ { - \\overline { \\eta ( \\epsilon ) } \\cdot b } E \\end{align*}"}
+{"id": "1491.png", "formula": "\\begin{align*} y _ d = \\frac { \\mu ( d ) } { h ( d ) } \\sum _ { m \\equiv 0 ( \\mod d ) } g ( m ) \\rho _ { m } . \\end{align*}"}
+{"id": "7991.png", "formula": "\\begin{align*} w ( e ^ { i t } ) = \\sum _ { j = - m } ^ m c _ j e ^ { i j t } \\end{align*}"}
+{"id": "5915.png", "formula": "\\begin{align*} h ^ { - 1 , 1 } = 1 \\ , , h ^ { 0 , 0 } = 2 0 \\ , , h ^ { 1 , - 1 } = 1 \\ , . \\end{align*}"}
+{"id": "1961.png", "formula": "\\begin{align*} s = \\beta - \\sum _ { j = 1 } ^ d \\frac { a _ j } { p _ j } + \\frac { \\nu } { p _ { } } , \\end{align*}"}
+{"id": "72.png", "formula": "\\begin{align*} v = 0 \\Omega _ { 1 } \\setminus D \\end{align*}"}
+{"id": "6922.png", "formula": "\\begin{align*} q ( z _ i + y ) \\prod _ { p = 1 } ^ { r } ( 1 + z _ i x _ p ) = z _ i R ( z _ i ) \\end{align*}"}
+{"id": "7895.png", "formula": "\\begin{align*} a * _ R b = R ( a ) \\cdot b + a \\cdot R ( b ) , ~ a , b \\in A . \\end{align*}"}
+{"id": "60.png", "formula": "\\begin{align*} \\Sigma _ { t + a } = r ^ { - 1 } ( t ) = \\{ t + a \\} \\times N , \\end{align*}"}
+{"id": "7289.png", "formula": "\\begin{align*} \\frac { d } { d t } X ^ * ( t , \\omega ) = A ( t ) X ^ * ( t , \\omega ) + B ( t ) R ^ { - 1 } ( t ) B ^ T ( t ) \\Upsilon ( t , \\omega ) . \\end{align*}"}
+{"id": "8120.png", "formula": "\\begin{align*} ( ( L , \\varphi ) ^ { d + 1 } ) \\geqslant - \\operatorname { \\widehat { \\deg } } _ { \\xi ' } ( R ) - \\frac 1 2 \\nu ( \\Omega _ \\infty ) ( d + 1 ) \\ln \\binom { r + \\delta - 1 } { \\delta } , \\end{align*}"}
+{"id": "7542.png", "formula": "\\begin{align*} C ^ { ( 2 , - 1 ) } ( u , v ; a ) = \\frac { \\theta _ 1 ' ( 0 ) \\theta _ 1 ( u - a ) ^ 2 \\theta _ 1 ( v ) ^ 2 \\theta _ 1 ( u - v + 2 a ) } { \\theta _ 1 ( u - v ) \\theta _ 1 ( v - a ) ^ 2 \\theta _ 1 ( u ) ^ 2 \\theta _ 1 ( 2 a ) } \\frac { d u ^ 2 } { d v } \\end{align*}"}
+{"id": "861.png", "formula": "\\begin{align*} ( a + b ) \\cdot c + c = a \\cdot c + b \\cdot c \\end{align*}"}
+{"id": "3625.png", "formula": "\\begin{align*} ( { 1 \\over 2 } \\sigma ^ 2 \\theta ^ 2 - \\lambda - 1 ) g ( x ) + \\sigma ^ 2 \\theta g ' ( x ) + { 1 \\over 2 } \\sigma ^ 2 g '' ( x ) + e ^ { - \\theta x } ( x - \\rho ) ^ 2 + \\theta \\int _ x ^ b g ( z ) d z + \\zeta = 0 . \\end{align*}"}
+{"id": "6982.png", "formula": "\\begin{align*} \\Delta \\phi ( f ) = \\phi ' ( f ) \\Delta f + \\phi '' ( f ) | \\nabla f | ^ 2 . \\end{align*}"}
+{"id": "2160.png", "formula": "\\begin{align*} \\begin{vmatrix} R _ { 1 2 1 2 } - c & R _ { 1 2 1 3 } & R _ { 1 2 2 3 } \\\\ R _ { 1 2 1 3 } & R _ { 1 3 1 3 } - c & R _ { 1 3 2 3 } \\\\ R _ { 1 2 2 3 } & R _ { 1 3 2 3 } & R _ { 2 3 2 3 } - c \\end{vmatrix} \\geq 0 , \\end{align*}"}
+{"id": "3863.png", "formula": "\\begin{align*} \\frac { \\partial G ^ { z } _ { \\mathfrak { u v } } } { \\partial w _ { a B } } = - G ^ { z } _ { \\mathfrak { u } a } G ^ { z } _ { B \\mathfrak { v } } , \\frac { \\partial G ^ { z } _ { \\mathfrak { u v } } } { \\partial \\overline { w _ { a B } } } = - G ^ { z } _ { \\mathfrak { u } B } G ^ { z } _ { a \\mathfrak { v } } , \\end{align*}"}
+{"id": "4974.png", "formula": "\\begin{align*} E _ { v \\rightarrow c } ^ { ( 0 ) } \\left ( \\mathcal { X } _ { v } [ m ] \\right ) = \\begin{cases} \\frac { 1 } { 1 + \\exp \\left ( l _ { \\mathrm { B P } } \\left ( a _ { v } \\right ) \\right ) } , & m = 0 \\\\ \\frac { 1 } { M } \\left ( 1 - \\frac { 1 } { 1 + \\exp \\left ( l _ { \\mathrm { B P } } \\left ( a _ { v } \\right ) \\right ) } \\right ) . & 1 \\leq m \\leq M \\end{cases} \\end{align*}"}
+{"id": "5380.png", "formula": "\\begin{align*} \\Pi _ D = \\int _ \\Sigma d x d y \\ , H ( x , y ) ^ i { } _ j \\frac { \\delta } { \\delta \\hat E ^ i ( x ) } \\wedge \\frac { \\delta } { \\delta A _ j ( y ) } . \\end{align*}"}
+{"id": "2684.png", "formula": "\\begin{align*} M _ { S _ 1 } ( 2 n , j , 0 ) & = \\begin{cases} 0 , & 0 \\leq j < n ; \\\\ ( - 1 ) ^ n , & j = n . \\end{cases} \\\\ M _ { S _ 1 } ( 2 n , j , 1 ) & = ( - 1 ) ^ n \\binom { 2 n } { n } \\binom { n } { j } \\end{align*}"}
+{"id": "1253.png", "formula": "\\begin{align*} \\Omega = B _ R : = \\{ x \\in \\mathbb { R } ^ N : | x | < R \\} . \\end{align*}"}
+{"id": "5024.png", "formula": "\\begin{align*} \\mathcal { E } _ k ( i ) : = \\mathcal { E } ( i ) \\cap \\Omega _ k , \\end{align*}"}
+{"id": "841.png", "formula": "\\begin{align*} J _ { i } ( \\tau _ i ^ { * } ) = \\mathbb { E } \\{ \\bar { \\theta } _ 1 y _ i ^ { * } - e ^ { - \\beta \\tau _ i ^ { * } } K \\} \\end{align*}"}
+{"id": "1893.png", "formula": "\\begin{align*} & \\theta \\circ ( \\theta _ 1 \\circ ( \\theta _ { 1 , 1 } , \\ldots , \\theta _ { 1 , k _ 1 } ) , \\ldots , \\theta _ n \\circ ( \\theta _ { n , 1 } , \\ldots , \\theta _ { n , k _ n } ) ) \\\\ = { } & ( \\theta \\circ ( \\theta _ 1 , \\ldots , \\theta _ n ) ) \\circ ( \\theta _ { 1 , 1 } , \\ldots , \\theta _ { 1 , k _ 1 } , \\ldots , \\theta _ { n , 1 } , \\ldots , \\theta _ { n , k _ n } ) , \\end{align*}"}
+{"id": "89.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { \\Psi } v = f ' ( u ) \\ , v & \\O \\\\ v = 0 & \\partial \\O . \\end{cases} \\end{align*}"}
+{"id": "2663.png", "formula": "\\begin{align*} ( x , & y , z , \\zeta ) ( x ' , y ' , z ' , \\zeta ' ) \\\\ & = ( x + x ' , y + y ' , z + z ' + x y ' - x ' y , \\zeta + \\zeta ' + \\xi ( x y ' - x ' y ) ) . \\end{align*}"}
+{"id": "5796.png", "formula": "\\begin{align*} k ( z , w ) = k _ w ( z ) = \\langle k _ w , k _ z \\rangle \\end{align*}"}
+{"id": "442.png", "formula": "\\begin{align*} Q ( W ) = \\int _ S | ( d W ) ^ N | ^ 2 - | \\langle k , W \\rangle | ^ 2 \\end{align*}"}
+{"id": "3647.png", "formula": "\\begin{align*} \\Delta _ { m - 1 } = \\left \\{ ( x _ 1 , \\ldots , x _ m ) \\in [ 0 , 1 ] ^ m \\colon x _ 1 + \\cdots + x _ m = 1 \\right \\} . \\end{align*}"}
+{"id": "3779.png", "formula": "\\begin{align*} A _ { j _ 1 , j _ 2 } = 2 \\sum _ { r = 0 } ^ { j _ 1 - 1 } C _ r A _ { j _ 1 - 1 - r , j _ 2 } + ( 2 j _ 2 + 1 ) C _ { j _ 1 + j _ 2 } . \\end{align*}"}
+{"id": "7785.png", "formula": "\\begin{align*} \\frac { \\det g _ x } { \\det \\hat { g } } = \\det ( \\hat { g } ^ { - 1 } g _ x ) = \\det ( I _ n + O ( x ) ) = 1 + O ( x ) \\end{align*}"}
+{"id": "5331.png", "formula": "\\begin{align*} p ^ f + ( - 1 ) ^ f = 2 r ^ 2 . \\end{align*}"}
+{"id": "3786.png", "formula": "\\begin{align*} C _ i ( 1 ) = \\sum _ { r = 0 } ^ { i - 1 } C _ { r , i - 1 - r } ( 1 ) + \\sum _ { r = 0 } ^ { i - 2 } A _ { r , i - 2 - r } \\end{align*}"}
+{"id": "2675.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { 2 n } ( - 1 ) ^ k \\binom { 2 n } { k } ^ m \\binom { k + r - 1 } { k } \\binom { 2 n - k + r - 1 } { 2 n - k } \\frac { 1 } { 2 } S ( k , r ) \\frac { 1 } { 2 } S ( 2 n - k , r ) \\end{align*}"}
+{"id": "1632.png", "formula": "\\begin{align*} w _ r ( c ( a ) ) = n \\ ; , \\end{align*}"}
+{"id": "2899.png", "formula": "\\begin{align*} \\left ( \\alpha + 1 \\right ) \\kappa \\dot { P } - \\varpi \\dot { P } = \\alpha P . \\end{align*}"}
+{"id": "2744.png", "formula": "\\begin{align*} c _ 1 ( \\Omega _ Z ) = \\pi ^ * c _ 1 ( E ) + \\pi ^ * K _ C - 3 H \\end{align*}"}
+{"id": "5611.png", "formula": "\\begin{align*} \\mathbf { K } ( y , V ) = \\frac { R _ { i k } V ^ i V ^ k } { \\left ( g _ { i j } g _ { k l } - g _ { i k } g _ { j l } \\right ) y ^ i y ^ j V ^ k V ^ l } \\end{align*}"}
+{"id": "4482.png", "formula": "\\begin{align*} \\Omega ( \\mathbb { X } , \\mathbb { Y } ) = \\langle Y _ 2 , X _ 1 \\rangle _ { H } - \\langle X _ 2 , Y _ 1 \\rangle _ { H } \\ , . \\end{align*}"}
+{"id": "7923.png", "formula": "\\begin{align*} & \\mu _ t = \\sum _ { i = 0 } ^ \\infty \\mu _ i t ^ i , \\mu _ i \\in \\mathrm { H o m } ( A ^ { \\otimes 2 } , A ) \\mu _ 0 = \\mu , \\\\ & R _ t = \\sum _ { i = 0 } ^ \\infty R _ i t ^ i , R _ i \\in \\mathrm { H o m } ( A , A ) R _ 0 = R , \\end{align*}"}
+{"id": "5296.png", "formula": "\\begin{align*} f _ { k , N } ( x ) = \\sum _ { a = 1 } ^ N e ^ { 2 \\pi i a ^ k x } , \\end{align*}"}
+{"id": "6273.png", "formula": "\\begin{align*} \\hat { x } = \\hat { x } ( t ) = t \\norm { H _ r ( - t p ) } ^ { - 1 } H _ r ( - t p ) = - t \\norm { H _ r ( p ) } ^ { - 1 } H _ r ( p ) . \\end{align*}"}
+{"id": "4989.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 \\Phi ^ { i j } ( p , q ) \\left ( \\frac { q _ i } { q ^ 0 } - \\frac { p _ i } { p ^ 0 } \\right ) = \\sum _ { j = 1 } ^ 3 \\Phi ^ { i j } ( p , q ) \\left ( \\frac { q _ j } { q ^ 0 } - \\frac { p _ j } { p ^ 0 } \\right ) = 0 . \\end{align*}"}
+{"id": "377.png", "formula": "\\begin{align*} \\{ ( u , \\alpha ) , ( v , \\beta ) \\} = ( [ u , v ] , \\L _ u \\beta ) . \\end{align*}"}
+{"id": "8092.png", "formula": "\\begin{align*} \\langle D _ T ^ 2 h , h \\rangle = 0 \\iff D _ T h = 0 \\iff \\| T h \\| = \\| h \\| . \\end{align*}"}
+{"id": "7354.png", "formula": "\\begin{align*} | \\mathcal { I } ' _ { k - 1 } | = { k + 3 \\choose 4 } . \\end{align*}"}
+{"id": "7190.png", "formula": "\\begin{align*} w ( x ) : = \\frac { ( 1 - q _ 1 ^ { - 1 } x ) ( 1 - q _ 2 ^ { - 1 } x ) } { ( 1 - x ) ( 1 - q ^ { - 1 } x ) } . \\end{align*}"}
+{"id": "4334.png", "formula": "\\begin{align*} \\eta _ { \\Delta } = \\sum _ i ( - 1 ) ^ { \\deg \\psi _ i ^ * } \\pi _ 1 ^ * \\psi _ i \\wedge \\pi _ 2 ^ * \\psi _ i ^ * . \\end{align*}"}
+{"id": "6870.png", "formula": "\\begin{align*} \\varepsilon u _ i = v _ i ^ { 2 ^ t } . \\end{align*}"}
+{"id": "6194.png", "formula": "\\begin{align*} & \\mathcal { B } _ { i , i } = \\{ ( i , i , t , p ) \\mid ( i , i , t , p ) \\in \\mathcal { I } ' _ { k } \\} , \\\\ & \\mathcal { B } _ { i , i - l } = \\{ ( i , i - l , t , p ) \\mid ( i , i - l , t , p ) \\in \\mathcal { I } ' _ { k } \\} , \\\\ & \\mathcal { B } _ { i - l , i } = \\{ ( i - l , i , t , p ) \\mid ( i - l , i , t , p ) \\in \\mathcal { I } ' _ { k } \\} . \\end{align*}"}
+{"id": "5212.png", "formula": "\\begin{align*} C _ j = \\sum _ { i = 1 } ^ { s - 1 } H _ i + \\sum _ { k = j _ s } ^ j p _ k + L + p _ j . \\end{align*}"}
+{"id": "2763.png", "formula": "\\begin{align*} 2 a \\cdot ( B _ n / P _ k ) - 2 b \\cdot \\dim ( B _ n / P _ k ) & \\geq 2 ( 2 n - k ) ^ 2 - k ( 4 n - 3 k + 1 ) \\\\ & = 8 n ^ 2 - 1 2 n k + 5 k ^ 2 - k \\\\ & = ( 2 n - 2 k ) ( 4 n - 2 k ) + k ^ 2 - k > 0 . \\end{align*}"}
+{"id": "680.png", "formula": "\\begin{align*} d G = \\sqrt { \\tau _ 0 } / 4 \\left \\{ \\left ( \\left ( 1 + 4 Q _ 0 / \\tau _ 0 \\right ) d z + \\left ( 1 + 4 \\overline { Q _ 0 } / \\tau _ 0 \\right ) d \\overline { z } \\right ) u _ 1 \\right . \\\\ \\left . - i \\left ( \\left ( 1 - 4 Q _ 0 / \\tau _ 0 \\right ) d z - \\left ( 1 - 4 \\overline { Q _ 0 } / \\tau _ 0 \\right ) d \\overline { z } \\right ) u _ 2 \\right \\} \\end{align*}"}
+{"id": "5104.png", "formula": "\\begin{align*} ( - \\Delta _ x ) ^ { \\sigma } = \\frac { \\sin ( \\pi \\sigma ) } { \\pi } \\int _ 0 ^ { \\infty } m ^ { \\sigma - 1 } \\frac { - \\Delta _ x } { - \\Delta _ x + m } \\ , d m . \\end{align*}"}
+{"id": "2371.png", "formula": "\\begin{align*} \\int _ { 0 < t _ { 1 } < \\cdots < t _ { n } < 1 } \\prod _ { j = 1 } ^ { n } \\frac { d \\gamma ( t _ { j } ) } { \\gamma ( t _ { j } ) - z _ { j } } . \\end{align*}"}
+{"id": "6268.png", "formula": "\\begin{align*} ( \\forall i \\in \\{ 1 , \\ldots , n \\} ) ~ \\big ( H _ r ( x ) \\big ) _ i = \\begin{cases} x _ i , & , \\\\ 0 , & . \\end{cases} \\end{align*}"}
+{"id": "5313.png", "formula": "\\begin{align*} U _ 0 : = \\int _ 1 ^ { \\sqrt { x } } , ~ U _ 1 : = \\int _ { \\sqrt { x } } ^ q , ~ U _ 2 : = \\int _ q ^ x , ~ U _ 3 : = \\int _ x ^ { \\infty } \\textrm { w i t h } q : = \\exp ( M - m ) . \\end{align*}"}
+{"id": "2741.png", "formula": "\\begin{align*} \\alpha _ X = \\frac { b ( \\Gamma ) } { a ( \\Gamma ) } \\end{align*}"}
+{"id": "5221.png", "formula": "\\begin{align*} D _ 1 = \\begin{pmatrix} - \\mu _ 1 - \\frac { 1 } { 1 2 } S & & \\\\ & - \\mu _ 2 - \\frac { 1 } { 1 2 } S & \\\\ & & - \\mu _ 3 - \\frac { 1 } { 1 2 } S \\end{pmatrix} , \\end{align*}"}
+{"id": "7376.png", "formula": "\\begin{align*} \\frac { k _ 0 } { q _ 0 } + \\frac { k _ 1 } { q _ 1 } + \\dots + \\frac { k _ l } { q _ l } = \\frac { 1 } { p } + \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "3549.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } ^ 2 P } { \\mathrm { d } \\mathrm { t } ^ 2 } + \\gamma \\frac { \\mathrm { d } P } { \\mathrm { d } \\mathrm { t } } + \\omega _ 0 ^ 2 P = \\varepsilon _ \\infty \\omega ^ 2 _ p u ^ \\mathrm { i } \\end{align*}"}
+{"id": "244.png", "formula": "\\begin{align*} g [ \\xi ] ( \\lambda ) = g ( \\lambda ) \\stackrel { d e f } { = } { \\bf E } \\exp ( \\lambda , \\xi ) , \\end{align*}"}
+{"id": "471.png", "formula": "\\begin{align*} B _ n ( x ) - B _ n ( y ) = n y ^ { n - 1 } , \\end{align*}"}
+{"id": "5774.png", "formula": "\\begin{align*} - \\frac { 1 1 ( 3 a + 8 b ) } { 3 1 } < c \\leq - \\frac { 1 1 ( 3 a + 8 b ) - 9 3 } { 3 1 } . \\end{align*}"}
+{"id": "5703.png", "formula": "\\begin{gather*} \\beta _ { a , b , c } = e _ { a } ^ { b , c } \\wedge \\beta , \\gamma _ { a , b , c } = e _ { a } ^ { b , c } \\wedge \\gamma . \\end{gather*}"}
+{"id": "1621.png", "formula": "\\begin{align*} F ( g ) = ( 1 - \\varepsilon ) \\int _ { \\widetilde { M } \\setminus ( \\Gamma _ { 1 , A _ i } \\cup \\Gamma _ { 2 , A _ i } ) } \\tilde { f } d \\mu + \\int _ { \\Gamma _ { 1 , A _ i } \\cup \\Gamma _ { 2 , A _ i } } \\tilde { g } d \\mu > ( 1 - \\varepsilon ) ( 1 - 3 \\gamma ) - 2 \\gamma > 1 - \\varepsilon - 5 \\gamma \\end{align*}"}
+{"id": "6856.png", "formula": "\\begin{align*} & ( \\alpha ^ { p ^ { e ' } + 1 } ( v _ 1 ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ 1 ) , \\dots , \\alpha ^ { p ^ { e ' } + 1 } ( v _ s ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ s ) , ( v _ { s + 1 } ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ { s + 1 } ) , \\dots , ( v _ n ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ n ) ) \\\\ = & ( u _ 1 g ( a _ 1 ) , \\dots , u _ s g ( a _ s ) , u _ { s + 1 } g ( a _ { s + 1 } ) , \\dots , u _ n g ( a _ n ) ) . \\end{align*}"}
+{"id": "3199.png", "formula": "\\begin{align*} & f ' ( \\theta , u ) \\mid _ { \\theta = 1 } = 4 A ( u ) + 4 H ( u ) + ( 4 p - 8 ) C ( u ) < k < 0 , \\\\ & f '' ( \\theta , u ) = 1 2 \\theta ^ { 2 } A ( u ) + ( 4 p - 8 ) ( 4 p - 9 ) \\theta ^ { 4 p - 1 0 } C ( u ) + 1 2 \\theta ^ { 2 } H ( u ) < k < 0 . \\end{align*}"}
+{"id": "5605.png", "formula": "\\begin{align*} \\frac { \\delta } { \\delta x ^ i } = \\frac { \\partial } { \\partial x ^ i } - \\hat { \\mathbb { G } } ^ k _ i \\frac { \\partial } { \\partial y ^ k } , \\ \\ \\delta y ^ j = d y ^ j + \\hat { \\mathbb { G } } ^ j _ k d x ^ k . \\end{align*}"}
+{"id": "3790.png", "formula": "\\begin{align*} \\widetilde { C } _ { i , j } ( 1 ) = 2 j \\sum _ { r = 0 } ^ { i - 1 } ( r + 1 ) C _ r C _ { i + j - r - 1 } = \\frac { 2 j } { 2 j + 1 } A _ { i - 1 , j } \\end{align*}"}
+{"id": "830.png", "formula": "\\begin{align*} A x ^ { k _ 1 } + p ( x ) + \\frac { K } { \\bar { \\theta } _ 2 } \\left ( \\frac { x ^ { * } } { x } \\right ) ^ { - k _ 1 } = \\frac { 1 } { l _ 2 } \\left [ \\frac { 1 - \\theta } { \\bar { \\theta } _ 2 - \\theta } - l _ 1 \\right ] . \\end{align*}"}
+{"id": "5993.png", "formula": "\\begin{align*} \\tilde { C } ^ { ( q , r ) } ( b ; h ) = \\dfrac { \\Xi ^ { ( q , r ) } ( b ; h ) - \\dfrac { \\rho ^ { ( q ) } _ { b } ( b ; h ) } { W ^ { ( q ) } ( b ) } Z ^ { ( q ) } ( b ; \\Phi ( q + r ) ) } { \\dfrac { Z ^ { ( q ) } ( b ; \\Phi ( q + r ) ) } { W ^ { ( q ) } ( b ) } - \\dfrac { r } { \\Phi ( q + r ) } } . \\end{align*}"}
+{"id": "1933.png", "formula": "\\begin{align*} \\begin{aligned} & ( | A u | ^ p ) ^ { 1 / p } _ { Q _ { \\nu r , c \\nu r } ( z _ 0 ) } = ( \\nu r ) ^ { - 1 } ( | A \\widetilde u | ^ p ) ^ { 1 / p } _ { Q _ { 1 , c } } , \\\\ & \\bigg ( | A u - ( A u ) _ { Q _ r ( z _ 0 ) } | ^ p \\bigg ) ^ { 1 / p } _ { Q _ r ( z _ 0 ) } = ( \\nu r ) ^ { - 1 } \\bigg ( | A \\widetilde u - ( A \\widetilde u ) _ { Q _ { 1 / \\nu } } | ^ p \\bigg ) _ { Q _ { 1 / \\nu } } ^ { 1 / p } . \\end{aligned} \\end{align*}"}
+{"id": "7865.png", "formula": "\\begin{align*} | h ( \\theta ) | & \\ge \\left | \\int _ { \\theta _ 0 } ^ { \\theta } h ' ( s ) d s \\right | - C \\Delta = \\int _ { \\theta _ 0 } ^ { \\theta } \\left | h ' ( s ) \\right | d s - C \\Delta \\\\ & = \\int _ { \\theta _ 0 } ^ \\theta \\int _ { \\theta _ 0 } ^ s h '' ( u ) d u d s - C \\Delta \\geq c t ( \\theta - \\theta _ 0 ) ^ 2 - C \\Delta . \\end{align*}"}
+{"id": "2730.png", "formula": "\\begin{align*} & A ( I ) = n u l l , k ( I ) = 1 , p r o d ( I ) = c a r t e s i a n , \\\\ & A ( I * p t ) = n u l l , k ( I * p t ) = 3 , p r o d ( I * p t ) = j o i n . \\end{align*}"}
+{"id": "6321.png", "formula": "\\begin{align*} \\frac { \\max \\limits _ { 1 \\le j \\le n } \\sum \\limits _ { i = 1 } ^ j ( X _ j - \\bar { X } ) - \\min \\limits _ { 1 \\le j \\le n } \\sum \\limits _ { i = 1 } ^ j ( X _ j - \\bar { X } ) } { \\Big \\{ \\frac 1 { n } \\sum \\limits _ { i = 1 } ^ n ( X _ i - \\bar { X } ) ^ 2 \\Big \\} ^ { 1 / 2 } } \\end{align*}"}
+{"id": "1043.png", "formula": "\\begin{align*} \\mathcal { W } \\ , ( \\Delta ^ { - m + 1 } ) = \\frac { n - 2 } { 1 2 } v _ { n - 1 } \\int _ { M } R ( g ) ~ v o l _ { g } , \\end{align*}"}
+{"id": "7075.png", "formula": "\\begin{align*} \\operatorname { A p } ( H , n ) : = \\{ s \\in H \\ , | \\ , s - n \\not \\in H \\} . \\end{align*}"}
+{"id": "7796.png", "formula": "\\begin{align*} I _ { \\theta } = \\int _ { 0 } ^ { \\csc ( \\theta ) } \\frac { 2 r } { r ^ 2 + \\tau ^ { 2 } } d r . \\end{align*}"}
+{"id": "6149.png", "formula": "\\begin{align*} \\tau _ { \\rm B C L } ( V _ 1 , V _ 2 ) = ( M _ { ( P ^ \\perp + z P ) U } , M _ { U ^ * ( P + z P ^ \\perp ) } ) \\tau _ { \\rm B C L } \\end{align*}"}
+{"id": "918.png", "formula": "\\begin{align*} F ^ { i j } v _ { i j } \\geq \\delta _ 1 \\left ( \\frac { 1 } { b _ { n - 1 } ^ { 1 / k } } + \\sum _ { i = 1 } ^ n F ^ { i i } \\right ) \\mbox { i n } \\omega . \\end{align*}"}
+{"id": "1463.png", "formula": "\\begin{align*} D _ { w } = \\frac { 1 } { L } \\sum _ { i = 1 } ^ { L } 1 _ { \\Delta _ { \\theta , i } \\neq \\hat { \\Delta } _ { \\theta , i } } \\end{align*}"}
+{"id": "337.png", "formula": "\\begin{align*} p ( X ( t ) | Y _ { t } ) & = p [ X ( t ) | ( y ( t ) , Y _ { t - 1 } ) ] = \\frac { p ( y ( t ) | X ( t ) | Y _ { t - 1 } ) p ( X ( t ) | Y _ { t - 1 } ) } { p ( y ( t ) | Y _ { t - 1 } ) } \\\\ & = \\frac { p ( y ( t ) | X ( t ) ) } { p ( y ( t ) | Y _ { t - 1 } ) } p ( X ( t ) | Y _ { t - 1 } ) \\end{align*}"}
+{"id": "152.png", "formula": "\\begin{align*} \\begin{aligned} b ( \\omega , v _ 1 - v ) = \\beta ( \\theta ) | v _ 1 - v | ^ \\gamma \\ , , \\ \\cos \\theta = \\frac { ( v _ 1 - v ) \\cdot \\omega } { | v _ 1 - v | } \\ , , \\ - 3 < \\gamma \\leq 1 \\ , , \\end{aligned} \\end{align*}"}
+{"id": "3898.png", "formula": "\\begin{align*} & s ( t ) + \\theta \\int _ 0 ^ t a ( t - \\tau ) s ( \\tau ) d \\tau = 1 , \\ ; t \\ge 0 , \\\\ & r ( t ) + \\theta \\int _ 0 ^ t a ( t - \\tau ) r ( \\tau ) d \\tau = a ( t ) , \\ ; t > 0 , \\end{align*}"}
+{"id": "4829.png", "formula": "\\begin{align*} \\nabla \\cdot ( \\rho \\nabla g ) + \\epsilon \\Delta \\rho - \\rho = - \\int _ S \\rho ( x , s ' ) \\dd \\mu ( s ' ) \\end{align*}"}
+{"id": "3378.png", "formula": "\\begin{align*} \\delta _ p = \\delta _ p ( f ) : = \\limsup _ { n \\to \\infty } \\sup _ X \\| ( f ^ n ) _ * [ X ] \\| _ W ^ { 1 / n } \\end{align*}"}
+{"id": "563.png", "formula": "\\begin{align*} \\phi ( x , y ) = \\sum _ { i = 1 } ^ L c _ i a _ i ( x ) b _ i ( y ) , \\end{align*}"}
+{"id": "5347.png", "formula": "\\begin{align*} f _ 1 & = i , \\\\ f _ 2 & = - \\Gamma \\mu i ^ { \\otimes 2 } , \\\\ f _ k & = 0 , & & . \\end{align*}"}
+{"id": "1623.png", "formula": "\\begin{align*} d ( u , x ) + d ( v , y ) & \\ge ( 1 - \\varepsilon ) \\big ( d ( u , v ) - d ( x , y ) \\big ) + \\varepsilon \\big ( d ( u , x ) + d ( v , y ) \\big ) \\\\ & \\ge ( 1 - \\varepsilon ) \\big ( d ( u , v ) - d ( x , y ) \\big ) + 2 ( 1 - \\varepsilon ) d ( x , y ) \\\\ & = ( 1 - \\varepsilon ) \\big ( d ( u , v ) + d ( x , y ) \\big ) . \\end{align*}"}
+{"id": "1456.png", "formula": "\\begin{align*} \\gamma _ { \\varepsilon } \\varphi _ { \\beta , \\varepsilon } ' ( s ) = - \\beta e ^ { - s } M ( \\gamma _ { \\varepsilon } - \\beta , \\gamma _ { \\varepsilon } + 1 ; s ) . \\end{align*}"}
+{"id": "2079.png", "formula": "\\begin{align*} \\begin{bmatrix} T _ { 1 1 } & 0 \\\\ 0 & U \\end{bmatrix} S M T = \\begin{bmatrix} T _ { 1 1 } X _ 1 T \\\\ U Y _ 1 T \\end{bmatrix} = \\begin{bmatrix} T \\\\ 0 \\\\ D \\end{bmatrix} . \\end{align*}"}
+{"id": "8082.png", "formula": "\\begin{align*} \\hat { \\mathrm { C } } = \\mathrm { M } ^ V + F r ^ 2 k ^ 2 \\mathrm { E } , \\end{align*}"}
+{"id": "29.png", "formula": "\\begin{align*} q _ i = \\begin{cases} 0 . 5 \\tilde { q } _ i , & i \\in [ m ] \\\\ 0 . 5 \\tilde { q } _ { i - m } , & i \\in [ k ] \\setminus [ m ] . \\end{cases} \\end{align*}"}
+{"id": "2747.png", "formula": "\\begin{align*} d ^ * \\geq 2 B + 2 ( g ( S ) - q ( S ) ) \\geq 2 + 2 ( N - 1 ) = 2 N . \\end{align*}"}
+{"id": "844.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d x _ i ( t ) = \\alpha x _ i ( t ) d t + \\sigma x _ i ( t ) d W _ i ( t ) , \\\\ & x _ i ( 0 ) = x . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2143.png", "formula": "\\begin{align*} { \\mathcal { W } ^ p } ( \\mathrm { d i v } , \\Omega ) = \\displaystyle { \\Bigl \\{ \\vec { u } \\in { L ^ p ( \\Omega ; \\R ^ n ) } \\ ; \\mid \\ ; \\mathrm { d i v } \\ ; \\vec { u } \\in { L ^ p ( \\Omega } \\Bigr \\} } \\end{align*}"}
+{"id": "7802.png", "formula": "\\begin{align*} s _ { 1 } ( \\theta ) = \\arcsin ( \\frac { M \\sin ^ 2 ( \\theta ) - 1 } { \\cos ( \\theta ) } ) , \\ s _ { 2 } ( \\theta ) = \\pi - s _ { 1 } ( \\theta ) \\end{align*}"}
+{"id": "4672.png", "formula": "\\begin{align*} f _ { \\ell , n _ q } ( A _ 0 , \\dots , A _ { n _ q - 1 } ) = 1 + \\sum _ { m = 0 } ^ { \\ell - 1 } \\sum _ { \\substack { 0 \\leq s _ k \\\\ \\ \\sum _ { k = 0 } ^ { n _ q - 1 } s _ k = \\ell - m \\\\ \\sum _ { k = 0 } ^ { n _ q - 1 } q ^ k s _ k \\equiv 0 \\bmod { \\ell } } } b _ { s _ 0 , \\dots , s _ { n _ q - 1 } } A _ 0 ^ { s _ 0 } \\cdots A _ { n _ q - 1 } ^ { s _ { n _ q - 1 } } . \\end{align*}"}
+{"id": "5065.png", "formula": "\\begin{align*} g ( y ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\Re ( u ) = - \\frac 5 2 } \\widetilde { g } ( u ) y ^ { - u } \\ , d u \\quad y > 0 . \\end{align*}"}
+{"id": "7657.png", "formula": "\\begin{align*} \\frac { P ( B _ r ) } { | B _ r | ^ p } = 2 \\pi ^ { 1 - p } r ^ { 1 - 2 p } , \\end{align*}"}
+{"id": "2681.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ^ m H ( n , k , a ) \\end{align*}"}
+{"id": "1156.png", "formula": "\\begin{align*} \\Delta ^ * _ p ( 1 , n ) = \\sum _ { L M = p } \\frac { \\mu ( L ) } { \\nu ( L ) } \\sum _ { \\ell | L ^ \\infty } \\frac { \\ell } { \\nu ( \\ell ) ^ 2 } \\sum _ { d _ 1 , d _ 2 | \\ell } c _ \\ell ( d _ 1 ) c _ \\ell ( d _ 1 ) \\sum _ { v | ( n , L ) } \\frac { v \\ \\mu ( v ) } { \\nu ( v ) } \\sum _ { b | ( \\frac { n } { v } , v ) } \\sum _ { e | ( d _ 2 , \\frac { n } { b ^ 2 } ) } \\Delta _ M ( d _ 1 , \\tfrac { n d _ 2 } { e ^ 2 b ^ 2 } ) , \\end{align*}"}
+{"id": "2568.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { * , t _ 0 , x _ 0 , \\xi } = & ~ [ A x _ t ^ { * , t _ 0 , x _ 0 , \\xi } - B ^ 2 R ^ { - 1 } U ( t , x _ t ^ { * , t _ 0 , x _ 0 , \\xi } , \\nu _ t ^ { * , t _ 0 , \\xi } ) - B h ( \\mu _ t ^ { * , t _ 0 , \\xi } ) + f ( \\nu _ t ^ { * , t _ 0 , \\xi } ) + b ( \\mu _ t ^ { * , t _ 0 , \\xi } ) ] d t \\\\ & + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ { t _ 0 } ^ { * , t _ 0 , x _ 0 , \\xi } = & ~ x _ 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "181.png", "formula": "\\begin{align*} \\begin{aligned} p _ n ( \\varsigma ) = 1 + \\varsigma + \\cdots + \\varsigma ^ n \\ , , \\ \\varsigma \\in \\R \\end{aligned} \\end{align*}"}
+{"id": "5587.png", "formula": "\\begin{align*} v ^ \\alpha \\frac { \\partial } { \\partial v ^ \\alpha } = \\frac 1 2 \\left ( y ^ i \\frac { \\partial } { \\partial y ^ i } - \\sqrt { - 1 } u ^ k \\frac { \\partial } { \\partial y ^ k } \\right ) . \\end{align*}"}
+{"id": "86.png", "formula": "\\begin{align*} v = X ( u ) = g ( \\nabla u , X ) . \\end{align*}"}
+{"id": "3886.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } u _ k ( x ) | x | ^ { - \\tau _ - } = k . \\end{align*}"}
+{"id": "2182.png", "formula": "\\begin{align*} & R _ { i j k l } = R ^ B _ { i j k l } , \\\\ & R _ { i j k p } = R _ { i j p q } = R _ { i p q r } = 0 , \\\\ & R _ { i p j q } = - f H ^ f _ { i j } h _ { p q } , \\\\ & R _ { p q r s } = f ^ 2 \\{ R ^ F _ { p q r s } - | | { \\rm g r a d } \\ : f | | ^ 2 ( h _ { p r } h _ { q s } - h _ { p s } h _ { q r } ) \\} , \\end{align*}"}
+{"id": "435.png", "formula": "\\begin{align*} \\phi ( h ) ( z ) = h ^ * \\sigma \\Big ( \\frac { \\partial } { \\partial z } , \\frac { \\partial } { \\partial z } \\Big ) ( z ) d z ^ 2 , \\end{align*}"}
+{"id": "3092.png", "formula": "\\begin{align*} \\sigma _ 1 & = \\prod _ { n = 0 } ^ { \\infty } \\alpha _ { 2 n + 1 } = ( 1 , 2 ) ( 3 , 4 ) ( 5 , 6 ) ( 7 , 8 ) \\cdots \\\\ \\sigma _ 2 & = \\gamma _ 1 \\prod _ { n = 2 } ^ { \\infty } \\alpha _ { 2 n + 1 } = ( 1 , 2 , 3 , 4 ) ( 5 , 6 ) ( 7 , 8 ) \\cdots \\\\ \\sigma _ 3 & = \\prod _ { n = 0 } ^ { \\infty } \\gamma ^ 2 _ { 8 n + 1 } \\gamma _ { 8 n + 5 } = \\prod _ { n = 0 } ^ { \\infty } \\alpha ' _ { 8 n + 1 } \\alpha ' _ { 8 n + 2 } \\gamma _ { 8 n + 5 } \\\\ & = ( 1 , 3 ) ( 2 , 4 ) ( 5 , 6 , 7 , 8 ) ( 9 , 1 1 ) ( 1 0 , 1 2 ) ( 1 3 , 1 4 , 1 5 , 1 6 ) \\cdots \\end{align*}"}
+{"id": "7734.png", "formula": "\\begin{align*} g ^ { E ' } = e ^ { - 2 s _ { E ' } } g ^ + = e ^ { - 2 r } g ^ + = \\frac { x ^ 2 } { ( \\delta ' ) ^ 2 } g ^ + = \\frac { 1 } { ( \\delta ' ) ^ 2 } g \\end{align*}"}
+{"id": "7322.png", "formula": "\\begin{align*} J ( u ( \\cdot ) ) : = \\mathbb { E } \\left [ \\Phi ( X ^ { u } ( T ) ) + \\int _ { 0 } ^ { T } f ( t , X ^ { u } ( t ) , u ( t ) ) d t \\right ] \\end{align*}"}
+{"id": "4604.png", "formula": "\\begin{align*} \\sin 2 \\pi \\theta ( n + l ) & = \\sin 2 \\pi \\left ( \\theta ( n ) + k l + \\frac { O ( 1 ) } { 1 + n - b } \\right ) \\\\ & = \\sin ( 2 \\pi \\theta ( n ) + 2 \\pi k l ) + \\frac { O ( 1 ) } { 1 + n - b } , \\end{align*}"}
+{"id": "6300.png", "formula": "\\begin{align*} ( \\gamma _ 1 * \\gamma _ 2 ) ( t ) = \\begin{cases} \\gamma _ 1 ( t ) & \\hbox { f o r } t \\in [ 0 , \\ell _ 1 ] \\\\ \\gamma _ 2 ( t - \\ell _ 1 ) & \\hbox { f o r } t \\in [ \\ell _ 1 , \\ell _ 1 + \\ell _ 2 ] . \\end{cases} \\end{align*}"}
+{"id": "5375.png", "formula": "\\begin{align*} V _ \\mathfrak { J } \\subsetneq V _ \\circ \\subset T \\C ^ \\omega = V _ \\circ ^ { \\omega \\omega } = V _ \\mathfrak { J } ^ { \\omega \\omega } . \\end{align*}"}
+{"id": "3783.png", "formula": "\\begin{align*} C _ { i _ 1 , \\ldots , i _ k } ( 1 ) = \\sum _ { r = 1 } ^ k C _ { i _ r } ( 1 ) C _ { i _ 1 , \\ldots , \\widehat { i _ r } , \\ldots , i _ k } + \\sum _ { 1 \\leq r < s \\leq k } \\widetilde { C } _ { i _ r , i _ s } ( 1 ) C _ { i _ 1 , \\ldots , \\widehat { i _ r } , \\ldots , \\widehat { i _ s } , \\ldots , i _ k } . \\end{align*}"}
+{"id": "768.png", "formula": "\\begin{align*} \\nu ^ * A = \\mu ^ * h ^ * D + E . \\end{align*}"}
+{"id": "6068.png", "formula": "\\begin{align*} \\vert \\vert \\eta _ \\epsilon - \\eta \\vert \\vert _ { H ^ 1 ( \\Omega ) } = O ( \\epsilon ^ 2 ) . \\end{align*}"}
+{"id": "8105.png", "formula": "\\begin{align*} T _ 1 ' T _ 2 ' = U T _ 2 ' T _ 1 ' , T _ 1 ' U = U T _ 1 ' , T _ 2 ' U = U T _ 2 ' \\end{align*}"}
+{"id": "6767.png", "formula": "\\begin{align*} \\widehat { \\Phi } ( x ) & = \\int d k \\ , \\abs { k } ^ { - 1 } \\left ( e ^ { 2 \\pi i k x } a _ k + e ^ { - 2 \\pi i k x } a ^ * _ k \\right ) \\end{align*}"}
+{"id": "7406.png", "formula": "\\begin{align*} B ^ { ( 1 ) } = { \\sigma ^ { ( 1 ) } } ^ * B - 2 E . \\end{align*}"}
+{"id": "7013.png", "formula": "\\begin{align*} | \\nabla f _ { \\varepsilon } | = | \\phi ' _ { \\varepsilon } ( f ) | | \\nabla f | = \\frac { | f | } { \\sqrt { f ^ 2 + \\varepsilon ^ 2 } } | \\nabla f | \\leq | \\nabla f | . \\end{align*}"}
+{"id": "2021.png", "formula": "\\begin{align*} \\aligned \\langle \\phi _ 1 , \\phi _ 2 \\rangle _ { G _ g , a } & = \\sum _ { \\gamma } \\frac { \\widehat { \\phi _ 1 } ( - \\gamma ) - \\widehat { \\phi _ 1 } ( 0 ) } { \\gamma } \\cdot \\frac { \\widehat { \\overline { \\phi _ 2 } } ( \\gamma ) - \\widehat { \\overline { \\phi _ 2 } } ( 0 ) } { \\gamma } \\endaligned \\end{align*}"}
+{"id": "6802.png", "formula": "\\begin{align*} \\Pi ( x _ { [ 5 ] } ) = \\pi ^ { a , \\ell _ a } _ \\ell ( x _ B | x _ 4 | x _ { P _ a } ) \\beta _ b ( x _ { A _ b } , x _ { P _ b } ) , \\end{align*}"}
+{"id": "7883.png", "formula": "\\begin{align*} \\# \\mathcal G ' \\gtrsim | \\log \\delta | ^ { - 1 } \\# \\mathcal G = | \\log \\delta | ^ { - 1 } \\sum _ { \\tilde R \\in \\tilde { \\mathcal R } } \\sum _ { R \\in \\mathcal { R } ( \\tilde R ) } \\# \\mathcal { G } ( R ) \\sim | \\log \\delta | ^ { - 1 } \\mu _ 2 M \\# \\tilde { \\mathcal { R } } , \\end{align*}"}
+{"id": "994.png", "formula": "\\begin{align*} \\varrho \\ , { u } _ { i , t t } & = \\sigma _ { j i , j } , \\ \\ , i = 1 , 2 , 3 , \\\\ \\varrho \\ , j \\ , { \\vartheta } _ { i , t t } & = m _ { j i , j } ^ * + \\epsilon _ { i j k } \\ , \\sigma _ { j k } , i = 1 , 2 , 3 , \\end{align*}"}
+{"id": "2089.png", "formula": "\\begin{align*} \\ker A = \\{ ( - B ^ { - 1 } C y , y ) \\mid y \\in K ^ { n - m } \\} . \\end{align*}"}
+{"id": "2134.png", "formula": "\\begin{align*} \\mathrm { c o r a n k } _ { \\Omega ( G ) } ( \\displaystyle \\bigoplus _ { v \\in S } { } ^ { \\pm } \\widetilde { K } _ v ( E _ { p } / L _ \\infty ) ) = \\mathrm { c o r a n k } _ { \\Omega ( G ) } ( \\displaystyle \\bigoplus _ { v \\in S _ p } { } ^ { \\pm } \\widetilde { K } _ v ( E _ { p } / L _ \\infty ) ) = 2 . \\end{align*}"}
+{"id": "7115.png", "formula": "\\begin{align*} ( \\mathbb { C } ^ { \\ast } ) ^ 2 \\stackrel { \\cong } { \\to } T : = \\{ ( t _ 1 , t _ 2 , t _ 3 ) \\in ( \\mathbb { C } ^ { \\ast } ) ^ { \\times 3 } \\mid t _ 1 t _ 2 t _ 3 = 1 \\} . \\end{align*}"}
+{"id": "6502.png", "formula": "\\begin{align*} \\frac { r ^ 2 } { n } < \\frac { 1 } { 1 8 c r } \\le e ^ { - 1 } \\le e ^ { - ( r - 1 ) ( 8 r - 1 4 ) / ( 4 r - 6 ) ^ 2 } = e ^ { - ( r - 1 ) 2 k / ( k + 1 ) ^ 2 } . \\end{align*}"}
+{"id": "7235.png", "formula": "\\begin{align*} \\mathcal { R } _ j = \\left \\{ \\left ( j _ { 1 } , j _ { 2 } , j _ { 3 } \\right ) \\in \\left ( \\mathbb { Z } ^ { 2 } \\right ) ^ { 3 } : j _ { 1 } - j _ { 2 } + j _ { 3 } = j \\left | j _ { 1 } \\right | ^ { 2 } - \\left | j _ { 2 } \\right | ^ { 2 } + \\left | j _ { 3 } \\right | ^ { 2 } = | j | ^ { 2 } \\right \\} , \\end{align*}"}
+{"id": "2454.png", "formula": "\\begin{align*} - \\Delta F _ * + F _ * = 2 \\pi J _ * . \\end{align*}"}
+{"id": "762.png", "formula": "\\begin{align*} \\phi _ { \\lambda } ( [ a _ { \\mu } b ] ) = [ \\phi _ { \\lambda } ( a ) _ { \\lambda + \\mu } b ] + [ a _ { \\mu } \\phi _ { \\lambda } ( b ) ] , \\forall \\ a , b \\in \\mathcal { A } . \\end{align*}"}
+{"id": "6186.png", "formula": "\\begin{align*} ( M ^ { t , p } _ { i , j } ) _ { x y } = \\left \\{ \\begin{array} { l l } 1 & \\ ( x _ 0 , x , y ) \\in X ^ { x _ 0 } _ { ( i , j , t , p ) } , \\\\ 0 & \\end{array} \\right . \\ \\ ( x , y \\in X ) . \\end{align*}"}
+{"id": "8203.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { j \\le j _ 0 } \\psi _ j 2 ^ { j r } \\Vert \\dot { \\Delta } _ j R ( f , g ) \\Vert _ { L ^ 2 } \\le C \\Big ( \\sup _ { j ' \\in \\mathbb { Z } } 2 ^ { j ' ( r _ 3 + \\frac { N } { 2 } - \\beta ) } \\Vert \\dot \\Delta _ { j ' } g \\Vert _ { L ^ 2 } \\Big ) \\\\ \\times \\sum _ { j \\le j _ 0 } \\sum _ { j ' > j - 4 } 2 ^ { ( j - j ' ) ( r + \\frac { N } { 2 } - \\beta ) } & 2 ^ { j ' ( r - r _ 3 ) } \\Vert \\dot \\Delta _ { j ' } f \\Vert _ { L ^ 2 } \\psi _ j . \\end{aligned} \\end{align*}"}
+{"id": "3991.png", "formula": "\\begin{align*} \\hat { q } ( n , t ) = \\sum _ { \\Omega ^ { n } _ { k } } \\prod _ { j = 1 } ^ { k } \\frac { ( 1 - p ) ^ { x _ { j } } } { x _ { j } k ! } \\left ( - \\frac { \\lambda t } { \\ln p } \\right ) ^ { k } e ^ { - \\lambda t } , \\end{align*}"}
+{"id": "6427.png", "formula": "\\begin{align*} \\mathcal { L } ^ { ( \\alpha ) } _ n ( x ) = \\frac { ( q ^ { \\alpha + 1 } ; q ) _ n } { ( q ; q ) _ n } { } _ 1 \\phi _ 1 \\left ( \\begin{gathered} q ^ { - n } \\\\ q ^ { \\alpha + 1 } \\end{gathered} ; \\ , q , - q ^ { n + \\alpha + 1 } x \\right ) , \\alpha > - 1 . \\end{align*}"}
+{"id": "4266.png", "formula": "\\begin{align*} T _ 2 = \\frac { d } { ( c - d ) ( q ) _ N } \\left ( 1 - \\frac { ( d q ) _ N } { ( c q ) _ N } \\right ) + \\frac { 1 } { ( c q ) _ N } \\sum _ { k = 1 } ^ N \\frac { \\left ( \\frac { c q } { d } \\right ) _ k ( d q ) ^ k ( d q ) _ { N - k } } { ( q ) _ k ( 1 - q ^ k ) ( q ) _ { N - k } } . \\end{align*}"}
+{"id": "187.png", "formula": "\\begin{align*} c h ( \\mathfrak { s } ^ + _ { 3 / 2 } \\otimes L ) - c h ( \\mathfrak { s } ^ - _ { 3 / 2 } \\otimes L ) = ( c h ( \\mathfrak { s } ^ { + } _ { 1 / 2 } \\otimes L ) - c h ( \\mathfrak { s } ^ { - } _ { 1 / 2 } \\otimes L ) ) ( c h ( T X ) + 1 ) . \\end{align*}"}
+{"id": "7609.png", "formula": "\\begin{align*} C ( S ) = \\sum _ { i \\in S , j \\in V \\setminus S , ( i , j ) \\in E } w _ { i j } \\end{align*}"}
+{"id": "5346.png", "formula": "\\begin{align*} m _ 1 & = d i , \\\\ m _ 2 & = \\pi \\mu i ^ { \\otimes 2 } , \\\\ m _ 3 & = \\pi \\mu ( \\Gamma \\mu \\otimes 1 - 1 \\otimes \\Gamma \\mu ) i ^ { \\otimes 3 } , \\\\ m _ k & = 0 , & & . \\end{align*}"}
+{"id": "1277.png", "formula": "\\begin{align*} E = 3 H + ( C _ 1 + 2 C _ 2 ) + ( C _ 1 ' + 2 C _ 2 ' ) + 2 ( C _ 1 '' + 2 C _ 2 '' ) . \\end{align*}"}
+{"id": "4280.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } \\frac { q ^ { j ^ 2 } } { ( q ) _ { j } ^ { 2 } } \\sum _ { n = 1 } ^ { j } \\frac { q ^ n } { ( 1 - q ^ n ) ^ 2 } = \\frac { 1 } { 2 } \\left . \\frac { d ^ 2 } { d z ^ 2 } \\sum _ { j = 0 } ^ { \\infty } \\frac { q ^ { j ^ 2 } } { ( z q ) _ j ( z ^ { - 1 } q ) _ j } \\right | _ { z = 1 } . \\end{align*}"}
+{"id": "5276.png", "formula": "\\begin{align*} \\dim L _ { \\pi } = 2 n + 2 - ( | \\pi | + 1 ) = 2 n + 1 - | \\pi | . \\end{align*}"}
+{"id": "2844.png", "formula": "\\begin{align*} ( \\omega , s _ 1 , \\cdots , s _ i ) \\star ( \\omega ' , s ' _ 1 , \\cdots , s ' _ j ) = ( \\omega \\omega ' , ( \\omega ' ) ^ { - 1 } s _ 1 \\omega ' , \\cdots , ( \\omega ' ) ^ { - 1 } s _ i \\omega ' , s ' _ 1 , \\cdots , s ' _ j ) , \\end{align*}"}
+{"id": "946.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\varphi } { \\partial x _ \\beta ^ 2 } ( x ' , \\rho ( x ' ) ) \\geq b _ \\beta - C \\delta \\tilde { b } _ \\beta , \\ \\beta = 1 , \\ldots , n - 1 \\end{align*}"}
+{"id": "5999.png", "formula": "\\begin{align*} v _ { b ^ * } ' ( b ^ * ) & = - \\theta C ^ 1 _ { b ^ { * } } W ^ { ( \\theta ) } ( b ^ * ) + \\beta Z ^ { ( \\theta ) } ( b ^ * ) \\\\ & \\quad + \\lambda \\left [ \\left ( \\theta \\dfrac { { C ^ { ( \\theta , r ) } ( b ^ * ; w ) } + \\rho _ { b ^ { * } } ^ { ( \\theta ) } ( b ^ * ; w ) } { Z ^ { ( \\theta ) } ( b ^ * ) } - w ( 0 ) \\right ) W ^ { ( \\theta ) } ( b ^ * ) - \\rho ^ { ( \\theta ) } _ { b ^ * } ( b ^ * ; w ' _ { + } ) \\right ] = 1 . \\end{align*}"}
+{"id": "6256.png", "formula": "\\begin{align*} \\delta ( A ^ { [ n ] } b ) & = \\delta ( A ) \\sum _ { j _ 2 , \\dots , j _ \\ell \\geq 0 } \\prod _ { r = 2 } ^ \\ell \\delta _ { j _ r } ( A ) + \\sum _ { j _ 1 = 1 } ^ m \\sum _ { j _ 2 \\cdots j _ \\ell = 0 } \\prod _ { r = 1 } ^ \\ell \\delta _ { j _ r } ( A ) . \\end{align*}"}
+{"id": "2113.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & - D & 0 \\\\ 0 & 0 & I _ { \\ell - k } \\\\ 0 & 0 & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "2996.png", "formula": "\\begin{align*} x v ' & = f ' - v = \\frac { 2 } { v ^ { - 2 } + 1 } - v = \\frac { - v ( v - 1 ) ^ 2 } { 1 + v ^ 2 } \\ ; , \\\\ \\frac { v ' } { v } & = - \\frac { v ^ 2 - 2 v + 1 } { x ( 1 + v ^ 2 ) } = - \\frac { 1 } { x } + \\frac { 2 v } { x ( 1 + v ^ 2 ) } \\ ; . \\end{align*}"}
+{"id": "2643.png", "formula": "\\begin{align*} & \\lambda _ x | _ { E _ { d ^ * ( y ' f ' ) } } = \\lambda _ { y ' f ' } \\\\ & \\lambda _ x | _ { E _ { d ^ * ( y ) } } = \\lambda _ y \\\\ & \\lambda _ x | _ { E _ { [ w b ^ { - 1 } , w a ] } } = \\phi ' \\end{align*}"}
+{"id": "1009.png", "formula": "\\begin{align*} \\kappa _ { i [ j k ] } = \\mathbf { A } _ { [ j k ] , i } = \\frac { 1 } { 2 } \\left ( A _ { j k , i } - A _ { k j , i } \\right ) = \\frac { 1 } { 2 } \\left ( - \\epsilon _ { j k l } \\vartheta _ { l , i } + \\epsilon _ { k j l } \\vartheta _ { l , i } \\right ) = \\frac { 1 } { 2 } \\left ( \\epsilon _ { k j l } + \\epsilon _ { k j l } \\right ) \\vartheta _ { l , i } = \\epsilon _ { k j l } \\ , \\vartheta _ { l , i } . \\end{align*}"}
+{"id": "5923.png", "formula": "\\begin{align*} [ \\Delta _ X ] = \\sum _ { i = 0 } ^ { 2 d } \\ ; \\pi _ X ^ i \\end{align*}"}
+{"id": "7795.png", "formula": "\\begin{align*} I _ { \\theta } = \\int _ { 0 } ^ { \\csc ( \\theta ) } \\frac { 1 } { 2 \\pi i } \\int _ { \\vert z \\vert = 1 } \\frac { 2 r } { ( r + \\tau z ) ( \\tau - r z ) } d z d r , \\end{align*}"}
+{"id": "5649.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 n - 2 } I _ \\gamma ( W _ i , W _ i ) & = \\sum _ { i = 1 } ^ { 2 n - 2 } \\int _ 0 ^ d [ g _ T ( D _ T ^ T W _ i , D _ T ^ T W _ i ) - g _ T ( W _ i , W _ i ) \\mathbf { K } ( T , W _ i ) ] d t \\\\ & = \\sum _ { i = 1 } ^ { 2 n - 2 } \\int _ 0 ^ d [ ( s ' _ \\lambda ( t ) ) ^ 2 - ( s _ \\lambda ( t ) ) ^ 2 \\mathbf { K } ( T , E _ i ) ] d t \\\\ & = \\int _ 0 ^ d [ ( 2 n - 2 ) ( s ' _ \\lambda ( t ) ) ^ 2 - \\mathbf { R i c } ^ \\perp ( T ) ( s _ \\lambda ( t ) ) ^ 2 ] d t \\\\ & \\leq ( 2 n - 2 ) \\int _ 0 ^ d [ ( s ' _ \\lambda ( t ) ) ^ 2 - \\lambda ( s _ \\lambda ( t ) ) ^ 2 ] d t = 0 . \\end{align*}"}
+{"id": "3307.png", "formula": "\\begin{align*} A v = w . \\end{align*}"}
+{"id": "5566.png", "formula": "\\begin{align*} \\Tilde { \\Phi } _ 2 ( \\beta _ 0 , \\eta ) = \\bigg [ \\tfrac { 2 } { L _ { 2 m } } \\bigg [ \\frac { \\tilde { N _ 2 } } { L _ { 2 m } } \\bigg ( \\frac { \\eta ^ { 3 - 2 \\mu - \\nu } - \\beta _ 0 ^ { 3 - 2 \\mu - \\nu } } { ( 2 - \\mu ) ( 3 - 2 \\mu - \\nu ) } - \\frac { \\beta _ 0 ^ { 2 - \\mu } \\eta ^ { 1 - \\mu - \\nu } - \\beta _ 0 ^ { 3 - 2 \\mu - \\nu } } { ( 2 - \\mu ) ( 1 - \\mu - \\nu ) } \\bigg ) \\end{align*}"}
+{"id": "4035.png", "formula": "\\begin{align*} L ^ { \\infty } _ M ( \\R ) = \\{ g ( 1 + | y | ^ M ) ^ { - 1 } g \\in L ^ \\infty ( \\R ) \\} , \\end{align*}"}
+{"id": "1966.png", "formula": "\\begin{align*} w _ { n , I } ( x ) = \\sqrt { \\frac { | I | } { 2 } } e ^ { i \\alpha _ I x } \\bigl \\{ g _ I \\bigl ( | I | ( x + e _ { n , I } ) \\bigr ) + g _ I \\bigl ( | I | ( x - e _ { n , I } ) \\bigr ) \\bigr \\} . \\end{align*}"}
+{"id": "7025.png", "formula": "\\begin{align*} \\lambda ^ 3 + ( \\sigma + | \\xi | ^ 2 ) \\lambda ^ 2 + 2 | \\xi | ^ 4 \\lambda + ( \\sigma + | \\xi | ^ 2 ) | \\xi | ^ 4 = 0 . \\end{align*}"}
+{"id": "7778.png", "formula": "\\begin{align*} \\begin{aligned} \\liminf \\limits _ { t \\rightarrow + \\infty } \\frac { c a p _ p ( B ^ { g ^ + } _ q ( t ) ) } { e ^ { n t } } & \\geq \\liminf \\limits _ { t \\rightarrow + \\infty } \\frac { n ^ p } { ( p - 1 ) ^ { p - 1 } } \\frac { V o l ( B ^ { g ^ + } _ q ( t ) ) } { e ^ { n t } } \\\\ & = \\frac { n ^ p } { ( p - 1 ) ^ { p - 1 } } \\lim \\limits _ { t \\rightarrow + \\infty } \\frac { \\frac { d } { d t } [ V o l ( B ^ { g ^ + } _ q ( t ) ) ] } { n e ^ { n t } } \\\\ & = \\frac { 1 } { 2 ^ n } ( \\frac { n } { p - 1 } ) ^ { p - 1 } \\mathcal { A } ( q ) \\end{aligned} \\end{align*}"}
+{"id": "6045.png", "formula": "\\begin{align*} D J ( \\Omega , V ) = \\lim _ { t \\rightarrow 0 ^ + } \\frac { J ( \\Omega _ t ) - J ( \\Omega ) } { t } . \\end{align*}"}
+{"id": "5153.png", "formula": "\\begin{align*} G = \\left \\{ x \\in \\partial \\Omega : \\ , D ( A ' , \\Omega , x ) = 1 D ( A , \\Omega , x ) \\ne 1 \\right \\} \\end{align*}"}
+{"id": "3535.png", "formula": "\\begin{align*} \\Big \\Vert \\mathrm { E } \\Big \\Vert _ { \\mathrm { L } ^ 2 \\Big ( \\Omega \\Big ) } = \\mathcal { O } \\Big ( \\delta | \\log { \\delta } | ^ \\mathrm { h } \\Big ) . \\end{align*}"}
+{"id": "3809.png", "formula": "\\begin{align*} a _ 1 = \\frac { 3 n + h ^ 2 - 3 h - 1 3 - d } { 2 } , \\end{align*}"}
+{"id": "2242.png", "formula": "\\begin{align*} x d = \\sigma ( d ) x , \\ y d = \\sigma ^ { - 1 } ( d ) y , \\ y x = a , \\ x y = \\sigma ( a ) , \\ d \\in D . \\end{align*}"}
+{"id": "6148.png", "formula": "\\begin{align*} \\mathfrak r _ q h = D _ { V ^ * } h + q V D _ { V ^ * } V ^ * h + \\cdots q ^ n V ^ n D _ { V ^ * } V ^ { * n } h + \\cdots . \\end{align*}"}
+{"id": "5140.png", "formula": "\\begin{align*} \\mathcal N ( Q _ i ) = \\left \\{ Q _ j \\in \\mathcal W \\setminus \\{ Q _ i \\} \\ , : \\ , ( Q _ i \\cup Q _ j ) \\right \\} . \\end{align*}"}
+{"id": "2947.png", "formula": "\\begin{align*} \\widehat N ^ p _ S ( \\bar z ; w ) = N _ S ( \\bar z ; w ) = N _ S ( \\bar z ) \\cap \\{ w \\} ^ \\perp = N _ { T _ S ( \\bar z ) } ( w ) \\end{align*}"}
+{"id": "2402.png", "formula": "\\begin{align*} { \\rm e v } _ { 1 / 2 } ^ { \\mathfrak { m } } ( e _ { a _ { 1 } } \\cdots e _ { a _ { k } } ) = 0 . \\end{align*}"}
+{"id": "3325.png", "formula": "\\begin{gather*} \\psi _ { n _ 2 ( k _ 2 ' ) } = \\sum _ { j _ 2 = 0 } ^ { N _ 2 } \\big \\langle \\psi _ { n _ 2 ( k _ 2 ' ) } , \\phi _ { m _ 2 ( j _ 2 ) } ^ { m _ 1 ( j _ 1 ) } \\big \\rangle \\phi _ { m _ 2 ( j _ 2 ) } ^ { m _ 1 ( j _ 1 ) } , \\end{gather*}"}
+{"id": "6287.png", "formula": "\\begin{align*} \\langle y \\rangle _ { } + \\langle - y \\rangle _ { } = \\begin{cases} 1 & \\\\ 0 & \\end{cases} \\quad \\sum _ { i = 0 } ^ { N - 1 } \\langle y + \\frac { i } { N } \\rangle _ { } = \\langle N y \\rangle _ { } + \\frac { N - 1 } { 2 } . \\end{align*}"}
+{"id": "7737.png", "formula": "\\begin{align*} H = \\Delta _ { g ^ + } r = \\Delta _ { g ^ + } ( - \\ln x ) = n - x \\Delta _ g x . \\end{align*}"}
+{"id": "1228.png", "formula": "\\begin{align*} Q = \\alpha \\otimes Q ^ x = Q _ y \\otimes \\beta \\end{align*}"}
+{"id": "4646.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\ , + \\infty } \\Phi ^ \\lambda [ \\eta _ n ] = - \\infty . \\end{align*}"}
+{"id": "1325.png", "formula": "\\begin{align*} & \\int _ { B _ r ( x _ 0 ) } \\left | \\nabla u ( x ) - \\nabla v ( x ) \\right | ^ p \\ , d x \\\\ & \\qquad \\le C \\left ( \\int _ { B _ r ( x _ 0 ) } \\left ( \\left | \\nabla u ( x ) \\right | ^ p - \\left | \\nabla v ( x ) \\right | ^ p \\right ) \\ , d x \\right ) ^ { \\frac { p } { 2 } } \\left ( \\int _ { B _ r ( x _ 0 ) } \\big ( \\left | \\nabla u ( x ) \\right | + \\left | \\nabla v ( x ) \\right | \\big ) ^ p \\ , d x \\right ) ^ { 1 - \\frac { p } { 2 } } , \\end{align*}"}
+{"id": "3888.png", "formula": "\\begin{align*} \\int _ { \\Omega } u \\mathcal { L } _ \\mu ^ * ( \\xi ) \\ , | x | ^ { \\tau _ + ( \\mu ) } d x = \\int _ { \\Omega } f \\xi \\ , | x | ^ { \\tau _ + ( \\mu ) } d x + c _ \\mu k \\xi ( 0 ) , \\quad \\forall \\ , \\xi \\in C ^ { 1 . 1 } _ 0 ( \\Omega ) , \\end{align*}"}
+{"id": "4134.png", "formula": "\\begin{align*} T _ n ^ { \\alpha } ( x ) = \\sqrt { 2 ( 2 n + \\alpha + 1 ) } x ^ { \\alpha + 1 / 2 } P _ n ^ { ( \\alpha , 0 ) } ( 1 - 2 x ^ 2 ) , \\end{align*}"}
+{"id": "4561.png", "formula": "\\begin{align*} 2 \\xi _ 2 = \\frac { 2 \\big ( 9 ( \\ln 3 ) ^ a - 8 ( \\ln 4 ) ^ a \\big ) } { 3 } & < \\xi _ 1 = \\frac { 2 \\big ( 3 ( \\ln 2 ) ^ a - 2 ( \\ln 4 ) ^ a \\big ) } { 3 } < \\frac { \\xi _ 2 } { 2 } = \\frac { 9 ( \\ln 3 ) ^ a - 8 ( \\ln 4 ) ^ a } { 6 } . \\end{align*}"}
+{"id": "5177.png", "formula": "\\begin{align*} A _ 0 = \\left \\{ x \\in Q \\ , : \\ , u ( x ) \\le 0 \\right \\} A _ 1 = \\left \\{ x \\in Q \\ , : \\ , u ( x ) \\ge 1 \\right \\} . \\end{align*}"}
+{"id": "773.png", "formula": "\\begin{align*} b - b _ 1 = l ( p - q ) + ( p - q r ) + 1 \\leq l ( p - q ) \\leq ( l q - 1 ) ( r - 1 ) = b _ 1 ( r - 1 ) . \\end{align*}"}
+{"id": "4230.png", "formula": "\\begin{align*} \\varepsilon _ \\Phi ( X ) = - \\frac { q ^ { 1 - g } h } { \\zeta ( 2 ) ( 1 - q ^ { - 1 } ) ( q ^ 2 - 1 ) } + \\sum _ { j = 1 } ^ { 2 g } \\frac { Z ( q \\gamma _ j ^ { - 1 } ) } { Z ' ( \\gamma _ j ^ { - 1 } ) } \\frac { \\gamma _ j } { \\gamma _ j - 1 } + \\frac { 1 } { 2 \\pi i } \\sum _ { N = 0 } ^ { X - 1 } \\oint _ { C _ \\rho } \\frac { 1 } { u ^ { N + 1 } } \\frac { Z ( q u ) } { Z ( u ) } \\ , d u . \\end{align*}"}
+{"id": "7636.png", "formula": "\\begin{align*} Z _ i = ( P _ i , w ( P _ i ) ) , R _ j = ( S _ j , w ( S _ j ) ) , ( 0 \\le i \\le 1 1 , \\ j = 1 , 2 ) \\end{align*}"}
+{"id": "7808.png", "formula": "\\begin{align*} E ( z , w ) = \\exp - \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } \\frac { g ( u ) } { \\overline { u - w } ( u - z ) } d a ( u ) = 1 - ( T _ { w } ^ { * - 1 } \\varphi , T _ { z } ^ { * - 1 } \\varphi ) \\ \\ z , w \\in \\mathbb { C } . \\end{align*}"}
+{"id": "5665.png", "formula": "\\begin{align*} T _ o ^ \\alpha T _ o ^ \\beta & = ( T ^ \\alpha + \\sqrt { - 1 } T ^ { \\alpha + n } ) ( T ^ \\beta + \\sqrt { - 1 } T ^ { \\beta + n } ) \\\\ & = T ^ \\alpha T ^ \\beta - T ^ { \\alpha + n } T ^ { \\beta + n } + \\sqrt { - 1 } ( T ^ { \\alpha + n } T ^ \\beta + T ^ \\alpha T ^ { \\beta + n } ) \\end{align*}"}
+{"id": "4360.png", "formula": "\\begin{align*} \\underline { d } x ( t ) : = \\left \\{ \\begin{array} { c c l } x ' ( t ) & { \\rm i f } & t \\in [ 0 , T ] \\setminus \\{ \\tau _ i : i \\in \\{ 0 , . . . , k + 1 \\} \\} \\\\ x ' ( \\tau _ i + ) & { \\rm i f } & t = \\tau _ i , i \\in \\{ 0 , . . . , k \\} \\\\ x ' ( T - ) & { \\rm i f } & t = T . \\end{array} \\right . \\end{align*}"}
+{"id": "846.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\tau = \\inf \\{ t : \\ W ( t ) \\geq a ' t + b ' \\} , \\\\ & a ' = \\frac { \\sigma } { 2 } - \\frac { \\alpha } { \\sigma } , b ' = \\frac { 1 } { \\sigma } \\ln \\left ( \\frac { K } { \\bar { \\theta } _ 1 x } \\cdot \\frac { k _ 2 } { k _ 2 - 1 } \\right ) = \\frac { 1 } { \\sigma } \\ln \\left ( \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) ^ { \\frac { 1 } { k _ 2 } } > 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3208.png", "formula": "\\begin{align*} 2 A ( u ) + 4 B ( u ) + p C ( u ) + \\omega \\int _ { \\mathbb { R } ^ { 3 } } | u | ^ { 2 } d x = 0 . \\end{align*}"}
+{"id": "5408.png", "formula": "\\begin{align*} | { \\Im } m _ { s c } ( z ) | \\sim \\begin{cases} \\sqrt { \\kappa + \\eta } , & \\mbox { i f } E \\in [ - 2 , 2 ] , \\\\ \\frac { \\eta } { \\sqrt { \\kappa + \\eta } } , & \\mbox { o t h e r w i s e } \\ , , \\end{cases} \\end{align*}"}
+{"id": "1573.png", "formula": "\\begin{align*} \\left ( z _ { 4 } s z _ { 1 } s \\right ) ^ { - 1 } z _ { 4 } z _ { 1 } & = s ^ { - 1 } z _ { 1 } ^ { - 1 } s ^ { - 1 } z _ { 1 } \\mbox { a n d } \\\\ z _ { 4 } \\left ( z _ { 4 } s z _ { 1 } s \\right ) ^ { - 1 } z _ { 1 } & = z _ { 4 } \\left ( s ^ { - 1 } z _ { 1 } ^ { - 1 } s ^ { - 1 } \\right ) z _ { 4 } ^ { - 1 } z _ { 1 } . \\end{align*}"}
+{"id": "2640.png", "formula": "\\begin{align*} q ( c ( z ' ) ) = & q ( c ( z _ 1 . . . z _ { n - 2 } ) c ( z _ { n - 1 } ' z _ n ' z _ { n + 1 } ' ) \\\\ = & q ( c ( z _ 1 . . . z _ { n - 2 } ) c _ a c _ b c _ b ) \\\\ = & q ( c ( z _ 1 . . . z _ { n - 2 } ) ) a b ^ 2 \\\\ = & q ( c ( z _ 1 . . . z _ { n - 2 } ) ) b a \\\\ = & q ( c ( z ) ) \\end{align*}"}
+{"id": "1995.png", "formula": "\\begin{align*} ( q _ n - d ( u ) ) z _ u = \\sum _ { w \\in N ( u ) } z _ w . \\end{align*}"}
+{"id": "7130.png", "formula": "\\begin{align*} \\chi + \\mathfrak { g } ^ { \\lambda > 0 } = \\sum _ { i = 1 } ^ k v _ i \\tau _ { d _ i } + \\sum _ { i = 1 } ^ k ( \\psi _ i - \\rho _ i ) . \\end{align*}"}
+{"id": "2147.png", "formula": "\\begin{align*} X _ p = \\left \\{ u \\in W ^ { 1 , p } _ \\sharp ( Y ^ m ) | \\left ( { \\bf I } _ m - { \\bf R } { \\bf R } ^ T \\right ) \\ ; \\nabla _ y u = \\vec { 0 } \\right \\} \\end{align*}"}
+{"id": "3072.png", "formula": "\\begin{align*} \\sigma ^ { m _ r } ( x ) = x \\end{align*}"}
+{"id": "3746.png", "formula": "\\begin{align*} ( x I - A ) ^ { - 1 } = \\frac { ( x - 2 2 ) ( x + 8 ) \\left ( A + ( x + 4 ) I _ { 6 0 } \\right ) + ( 9 x + 8 2 ) J _ { 6 0 } + ( x - 2 2 ) I _ 3 \\otimes J _ { 2 0 } } { ( x - 2 2 ) ( x - 2 ) ( x + 6 ) ( x + 8 ) } . \\end{align*}"}
+{"id": "7415.png", "formula": "\\begin{align*} f ( q ) = q - 2 q ^ 2 - 3 q ^ 3 + 4 q ^ 4 + 6 q ^ 5 + 6 q ^ 6 - 1 6 q ^ 7 - 8 q ^ 8 + 9 q ^ 9 - 1 2 q ^ { 1 0 } + 1 2 q ^ { 1 1 } + O ( q ^ { 1 2 } ) , \\end{align*}"}
+{"id": "403.png", "formula": "\\begin{align*} f = f ^ 0 ( \\sigma _ i , p ^ i , p ) + q p ^ 1 f ^ 1 + q p ^ 2 f ^ 2 , \\end{align*}"}
+{"id": "5262.png", "formula": "\\begin{align*} \\kappa _ p ^ { ( N ) } ( k _ 1 , \\ldots , k _ p ) = 0 \\ \\ { \\text i f } \\ \\ p > 2 \\ \\ { \\text a n d } \\ \\ \\sum _ { i = 1 } ^ p | k _ i | \\leq 2 \\ * N . \\end{align*}"}
+{"id": "6235.png", "formula": "\\begin{align*} H ( p ) = - \\partial _ x ^ 2 + ( p + B x ) ^ 2 - \\partial _ z ^ 2 \\end{align*}"}
+{"id": "2811.png", "formula": "\\begin{align*} d \\gamma _ s = \\gamma _ s ( \\mu _ s d s + \\sigma _ s d W ^ 1 _ s ) , s \\in [ 0 , T ] , \\end{align*}"}
+{"id": "3179.png", "formula": "\\begin{align*} [ \\tau ( \\rho _ i ) ] : = \\tau ( a _ i ) \\cdot \\tau ( e _ i ' ) \\cdot h _ i \\tau ( e _ i ' ) \\cdot h _ i ^ 2 \\tau ( e _ i ' ) \\dots \\end{align*}"}
+{"id": "6105.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal H ( k , d , l ) = & \\left \\{ F \\in \\binom { [ n ] } { k } : [ d - 1 ] \\subseteq F , \\ , F \\cap [ d , d + l - 1 ] \\neq \\emptyset \\right \\} \\\\ & \\cup \\left \\{ F \\in \\binom { [ n ] } { k } : | [ d - 1 ] \\cap F | = d - 2 , \\ , [ d , d + l - 1 ] \\subseteq F \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "2015.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } t \\ , e ^ { i z t } \\ , d t = - \\frac { 1 } { z ^ 2 } ~ \\Im ( z ) > 0 , \\end{align*}"}
+{"id": "3987.png", "formula": "\\begin{align*} \\Lambda _ { n } ^ { k } = \\left \\{ ( x _ { 1 } , x _ { 2 } , \\dots , x _ { n - k + 1 } ) : \\sum _ { j = 1 } ^ { n - k + 1 } x _ { j } = k , \\ \\ \\sum _ { j = 1 } ^ { n - k + 1 } j x _ { j } = n , \\ x _ { j } \\in \\mathbb { N } \\cup \\{ 0 \\} \\right \\} . \\end{align*}"}
+{"id": "7195.png", "formula": "\\begin{align*} K _ T ( \\mathcal { D T } ( d ) ) \\otimes _ { \\mathbb { K } } \\mathbb { F } = \\bigoplus _ { \\begin{subarray} { c } d _ 1 + \\cdots + d _ k = d \\\\ 0 \\leq v _ 1 / d _ 1 \\leq \\cdots \\leq v _ k / d _ k < 1 \\end{subarray} } \\mathbb { F } \\cdot [ \\mathcal { F } _ { d _ 1 , w _ 1 } ] \\ast \\cdots \\ast [ \\mathcal { F } _ { d _ k , w _ k } ] \\end{align*}"}
+{"id": "5349.png", "formula": "\\begin{align*} \\theta \\wedge \\omega \\wedge d \\theta ^ { n + 1 - k } = \\zeta _ \\theta \\wedge d \\theta ^ { n + 2 - k } . \\end{align*}"}
+{"id": "5044.png", "formula": "\\begin{align*} \\Lambda _ f \\bigl ( s , \\tfrac { a } { q } \\bigr ) = i ^ k \\chi ( \\bar { a } ) q ^ { 1 - 2 s } \\Lambda _ f \\bigl ( 1 - s , - \\tfrac { \\bar { a } } { q } \\bigr ) . \\end{align*}"}
+{"id": "782.png", "formula": "\\begin{align*} \\max _ { \\tau _ i \\in \\mathcal { S } ^ { i } } J _ i ( \\tau _ i ) = \\max _ { \\tau _ i \\in \\mathcal { S } ^ { i } } \\mathbb { E } \\left \\{ e ^ { - \\beta \\tau _ i } ( x _ i ( \\tau _ i ) - K ) \\right \\} . \\end{align*}"}
+{"id": "656.png", "formula": "\\begin{align*} \\nabla _ X \\psi _ 1 = \\frac { 1 } { 2 } X _ 1 \\cdot \\psi _ 1 - \\frac { 1 } { 2 } B ( X ) \\cdot \\psi _ 1 \\end{align*}"}
+{"id": "7978.png", "formula": "\\begin{align*} & \\eta ( \\phi ) = \\eta \\left ( | \\xi ^ H | ^ { 2 n - 2 } \\varphi _ h \\right ) \\\\ & = ( 2 n - 2 ) | \\xi ^ H | ^ { 2 n - 4 } \\left \\langle \\eta , \\xi ^ H \\right \\rangle \\varphi _ h + | \\xi ^ H | ^ { 2 n - 2 } \\eta ( \\varphi _ h ) \\\\ & = \\frac { 2 n - 2 } { 2 n + 1 } | \\xi ^ H | ^ { 2 n - 2 } \\left \\langle \\eta , \\xi ^ H \\right \\rangle ( l - 3 k ) + \\frac { 2 n - 2 } { 2 n + 1 } | \\xi ^ H | ^ { 2 n - 2 } \\left \\langle \\eta , \\xi ^ H \\right \\rangle \\left ( 3 k - l \\right ) = 0 . \\end{align*}"}
+{"id": "5917.png", "formula": "\\begin{align*} X : = \\bigcap _ { v \\in V _ 6 } Q ( v ) \\ , , \\end{align*}"}
+{"id": "569.png", "formula": "\\begin{align*} M _ n ^ { \\psi } ( Q ) / n = T _ { W _ { n J } , \\psi } ( \\mu _ n ( Q ) ) - I ( \\mu _ n ( Q ) ) . \\end{align*}"}
+{"id": "1320.png", "formula": "\\begin{align*} k n c & \\le \\lambda ^ 2 c = \\sum _ { u \\sim v } \\sum _ { w \\sim u } \\mathrm { x } _ w = d ( v ) c + \\sum _ { u \\sim v } \\sum _ { \\substack { w \\sim u , \\\\ w \\neq v } } \\mathrm { x } _ w \\\\ & \\le d ( v ) c + \\sum _ { u \\in S _ 1 } \\sum _ { \\substack { w \\sim u \\\\ w \\in L _ 1 \\cup L _ 2 } } \\mathrm { x } _ w + 2 e ( S _ 1 ) \\alpha + 2 e ( L ) + e ( L _ 1 , S _ 1 ) \\alpha + e ( N _ 1 , S _ 2 ) \\alpha . \\end{align*}"}
+{"id": "414.png", "formula": "\\begin{align*} g _ H ( D , M , Q , S ) : = \\vert S _ H ( D , M , Q , S ) \\vert . \\end{align*}"}
+{"id": "5066.png", "formula": "\\begin{align*} \\frac 1 { 2 \\pi i } \\int _ { \\Re ( u ) = \\sigma } \\widetilde { f } ( 1 - u ) \\widetilde { g } ( u ) \\ , d u = \\int _ 0 ^ \\infty f ( y ) g ( y ) \\ , d y . \\end{align*}"}
+{"id": "4764.png", "formula": "\\begin{align*} [ r _ { 1 2 } , r _ { 1 3 } ] + [ r _ { 1 3 } , r _ { 2 3 } ] + [ r _ { 1 2 } , r _ { 2 3 } ] = 0 , \\end{align*}"}
+{"id": "2701.png", "formula": "\\begin{align*} \\binom { n - a - 2 j + l } { k - j } \\binom { k - j } { a - l } = \\binom { n - a - 2 j + l } { a - l } \\binom { n - 2 a - 2 j + 2 l } { k - j - a + l } \\end{align*}"}
+{"id": "1221.png", "formula": "\\begin{align*} t = \\sum _ { i \\in Q } 2 ^ i , \\end{align*}"}
+{"id": "793.png", "formula": "\\begin{align*} \\mathcal { J } _ i ( \\tau _ i , \\tau _ { - i } ) = \\mathbb { E } \\Bigg \\{ e ^ { - \\beta \\tau _ i } \\Big ( x _ i ( \\tau _ i ) - K \\Big ) \\Bigg \\} \\end{align*}"}
+{"id": "1706.png", "formula": "\\begin{align*} H _ { \\tau , 0 } : = \\Theta _ \\tau ( h ) = \\int d x \\ , d y \\ , \\varphi _ \\tau ^ * ( x ) h ( x ; y ) \\varphi _ \\tau ( y ) \\ , , \\end{align*}"}
+{"id": "3494.png", "formula": "\\begin{align*} \\textbf { e r r } ^ { ( 1 ) } = \\mathcal { O } \\Bigg ( \\delta ^ { 7 } \\sqrt { \\mathcal { K } ^ { ( \\mathrm { T _ 0 } ) } _ { \\mathrm { r } } } \\frac { 1 } { | \\xi - z | ^ { \\frac { 7 } { 2 } - 2 \\mathrm { r } } } \\Big \\Vert | \\mathrm { E } | ^ { 2 } \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\Omega ) } \\Bigg ) \\end{align*}"}
+{"id": "438.png", "formula": "\\begin{align*} \\mathcal { E } ( S ' , h \\circ f ^ { - 1 } ) - \\mathcal { E } ( S , h ) = - 4 \\textrm { R e } \\int _ S \\phi \\cdot \\frac { \\mu } { 1 - | \\mu | ^ 2 } + 2 \\int _ S e ( h ) \\cdot \\frac { | \\mu | ^ 2 } { 1 - | \\mu | ^ 2 } d A . \\end{align*}"}
+{"id": "8225.png", "formula": "\\begin{align*} \\partial _ t u _ L + \\mu \\Lambda ^ \\alpha u _ L = 0 , u _ L | _ { t = 0 } = u _ 0 . \\end{align*}"}
+{"id": "2430.png", "formula": "\\begin{align*} f ( p ( X _ { 1 } ) ) = \\frac { d } { d X _ { 1 } } p ( X _ { 1 } ) \\quad ( p \\in \\mathcal { H } [ [ t ] ] ) . \\end{align*}"}
+{"id": "3932.png", "formula": "\\begin{align*} d _ { ( t _ 0 , x _ 0 ) } T ( s , y ) = y + s \\nabla b _ { \\Omega } ( x _ 0 ) + t _ 0 d _ { x _ 0 } \\nabla b _ { \\Omega } ( y ) , \\forall ( s , y ) \\in \\R \\times T _ { x _ 0 } \\partial \\Omega . \\end{align*}"}
+{"id": "7656.png", "formula": "\\begin{align*} \\kappa _ { E _ p } = p H ( p ) | E _ p | ^ { p - 1 } . \\end{align*}"}
+{"id": "6120.png", "formula": "\\begin{align*} \\mathcal F _ 2 ^ { ( 2 ) } = \\{ F _ { 2 , 1 } , F _ { 2 , 2 } , F _ { 2 , 3 } , F _ { 2 , 4 } \\} , \\end{align*}"}
+{"id": "7437.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} T _ 1 & T _ 2 & \\ldots & T _ n \\end{matrix} \\right ) \\left ( A - P \\right ) \\left ( \\begin{matrix} \\sigma _ 1 \\\\ \\sigma _ 2 \\\\ \\ldots \\\\ \\sigma _ n \\\\ \\end{matrix} \\right ) = 0 , \\end{align*}"}
+{"id": "1682.png", "formula": "\\begin{align*} E _ 0 ^ { ( \\lambda ) } = E _ 0 + \\lambda \\langle \\psi _ 0 | \\mathtt { H } _ 1 \\psi _ 0 \\rangle + \\lambda ^ 2 \\langle \\psi _ 0 | \\mathtt { H } _ 2 \\psi _ 0 \\rangle - \\lambda ^ 2 \\sum _ { k = 1 } ^ { \\mathtt { w } } \\frac { | \\langle \\psi _ 0 | \\mathtt { H } _ 1 \\psi _ k \\rangle | ^ 2 } { E _ k - E _ 0 } + \\lambda ^ 3 E ^ { ( \\lambda ) } \\end{align*}"}
+{"id": "694.png", "formula": "\\begin{align*} \\rho = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { f _ i } } , \\ ; \\ ; \\ ; \\ ; \\ ; \\rho { { \\bf { u } } ^ { e q } } = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { { \\bf { c } } _ i } { f _ i } } . \\end{align*}"}
+{"id": "4997.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } < \\prod _ { j = 1 } ^ m \\bigg ( 1 - \\frac { 1 } { p _ j } \\bigg ) < 1 . \\end{align*}"}
+{"id": "1874.png", "formula": "\\begin{align*} q ( 2 x + y ) + q ( 2 x - y ) = \\frac { 2 q ( x ) q ( y ) [ q ( x ) + 1 6 q ( y ) + 2 4 \\sqrt { q ( x ) q ( y ) } ] } { [ 4 \\sqrt { q ( y ) } - \\sqrt { q ( x ) } ] ^ { 4 } } \\end{align*}"}
+{"id": "7934.png", "formula": "\\begin{align*} ( \\mu _ 1 , R _ 1 ) - ( \\mu _ 1 ' , R _ 1 ' ) = \\big ( \\delta _ \\mathrm { H o c h } ( \\varphi _ 1 ) , ~ - \\Psi ^ 1 ( \\varphi _ 1 ) \\big ) = \\delta _ \\mathrm { m R B A } ( \\varphi _ 1 , 0 ) . \\end{align*}"}
+{"id": "4333.png", "formula": "\\begin{align*} L ( f ) = \\sum _ { \\ p } \\iota _ { \\Gamma } ( p ) . \\end{align*}"}
+{"id": "6683.png", "formula": "\\begin{align*} Q ( t ) = \\det \\begin{bmatrix} s _ { - r } & s _ { - r + 1 } & \\ldots & s _ { - r + K - 1 } & 1 \\\\ s _ { - r + 1 } & s _ { - r + 2 } & \\ldots & s _ { - r + K } & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ s _ { - r + K } & s _ { - r + K + 1 } & \\ldots & s _ { - r + 2 K - 1 } & t ^ { K } \\end{bmatrix} . \\end{align*}"}
+{"id": "2678.png", "formula": "\\begin{align*} S ( n , m , a ) = M _ S ( n , 0 , m - 1 ; a ) \\end{align*}"}
+{"id": "4767.png", "formula": "\\begin{gather*} \\partial _ 1 ( a ) \\cdot \\partial _ 2 ( b ) + \\eth _ 1 ( a \\cdot \\partial _ 2 ( b ) ) - \\partial _ 2 ( a ) \\cdot \\partial _ 1 ( b ) - \\eth _ 2 ( a \\cdot \\partial _ 1 ( b ) ) \\\\ \\qquad { } \\overset { \\eqref { e q : q a d m 2 } } { = } a \\cdot \\eth _ 1 ( \\partial _ 2 ( b ) ) - a \\cdot \\eth _ 2 ( \\partial _ 1 ( b ) ) \\overset { \\eqref { e q : v i p 1 } } { = } 0 . \\end{gather*}"}
+{"id": "6285.png", "formula": "\\begin{align*} \\tilde { f } ( \\mathbf { m } ) = \\big ( \\Psi \\cdot U \\cdot ( \\Psi ' ) ^ { - 1 } \\big ) \\mathbf { m ' } , \\end{align*}"}
+{"id": "1264.png", "formula": "\\begin{align*} y ^ 2 + a _ 1 x y + a _ 3 y = x ^ 3 + a _ 2 x ^ 2 + a _ 4 x + a _ 6 . \\end{align*}"}
+{"id": "837.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & y _ i ( \\tau _ i ) : = e ^ { - \\beta \\tau _ i } x _ i ( \\tau _ i ) , y _ { - i } ( \\tau _ { - i } ) : = \\frac { 1 } { N } \\sum \\limits _ { j = 1 , j \\neq i } ^ N e ^ { - \\beta \\tau _ j } x _ j ( \\tau _ j ) , \\\\ & y _ i ^ { * } ( \\tau _ i ^ { * } ) : = e ^ { - \\beta \\tau _ i ^ { * } } x _ i ( \\tau _ i ^ { * } ) , y _ { - i } ^ { * } ( \\tau _ { - i } ^ { * } ) : = \\frac { 1 } { N } \\sum \\limits _ { j = 1 , j \\neq i } ^ N e ^ { - \\beta \\tau _ j ^ { * } } x _ j ( \\tau _ j ^ { * } ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2749.png", "formula": "\\begin{align*} B = h ^ { 1 , 1 } ( S ) = b _ 2 ( S ) = 1 { \\rm a n d } A = q ( S ) = 0 . \\end{align*}"}
+{"id": "990.png", "formula": "\\begin{align*} \\vartheta : = \\ , \\mathbf { A } \\ , , \\qquad \\textbf { A n t i } \\ , \\vartheta = \\mathbf { A } \\ , , \\end{align*}"}
+{"id": "6451.png", "formula": "\\begin{align*} \\Delta _ { x , \\alpha } ^ n \\{ 1 \\} = L _ n ^ { ( \\alpha ) } ( x ) . \\end{align*}"}
+{"id": "6814.png", "formula": "\\begin{align*} \\varepsilon \\partial _ { t t } u + \\partial _ t u - \\kappa \\Delta u = 0 , \\end{align*}"}
+{"id": "7471.png", "formula": "\\begin{align*} S _ m ^ { ( 1 ) } ( n ) = \\sum _ { \\nu = 1 } ^ n \\nu ^ m = \\frac { 1 } { m + 1 } \\sum _ { k = 0 } ^ m { m + 1 \\choose k } B _ k n ^ { m - k + 1 } \\ , , \\end{align*}"}
+{"id": "6631.png", "formula": "\\begin{align*} s _ { \\theta ^ { ' } } ( v , v ) \\ge s _ { \\theta } ( v , v ) - c \\varepsilon \\| v \\| ^ 2 _ s = ( 1 - c \\varepsilon ) s _ \\theta ( v , v ) - c \\varepsilon \\big ( 1 - \\Lambda _ 1 ( S _ \\theta ) \\big ) \\| v \\| ^ 2 _ { L ^ 2 ( \\Omega _ { \\frac { \\pi } { 4 } } ) } . \\end{align*}"}
+{"id": "5019.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ { N + 2 } B _ 1 ( ( 4 i , 0 ) ) \\cup \\Big ( ( 4 , 4 ( N + 2 ) ) \\times ( - \\varepsilon , + \\varepsilon ) \\Big ) . \\end{align*}"}
+{"id": "1839.png", "formula": "\\begin{align*} R ( f , 0 ) ( p ) & = \\sum _ { k = 0 } ^ \\infty \\pi ( A ^ k u ) ( p ) = \\pi \\left ( \\left ( \\sum _ { k = 0 } ^ \\infty A ^ k ( p ) \\right ) u ( p ) \\right ) = \\pi ( ( I _ n - A ) ^ { - 1 } u ) ( p ) . \\end{align*}"}
+{"id": "8157.png", "formula": "\\begin{align*} \\bigoplus _ { \\begin{subarray} { c } \\boldsymbol { b } = ( b _ 1 , \\ldots , b _ r ) \\in \\mathbb N ^ r \\\\ | \\boldsymbol { b } | = b _ 1 + \\cdots + b _ r = \\delta \\\\ b _ 1 \\mu _ 1 + \\cdots + b _ r \\mu _ r = t \\end{subarray} } S ^ { b _ 1 } ( V _ 1 / V _ 0 ) \\otimes \\cdots \\otimes S ^ { b _ r } ( V _ r / V _ { r - 1 } ) . \\end{align*}"}
+{"id": "6584.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left | \\frac { e ^ { q t } } { t ^ { \\ell - 1 } } \\exp ( t Q ) y - \\sum _ { k = 1 } ^ { m } e ^ { i t \\theta _ k } v _ k \\right | = 0 , \\end{align*}"}
+{"id": "6692.png", "formula": "\\begin{align*} t _ { \\infty } ( ( s _ { i , - 1 } , 1 , \\lambda _ { i , 2 } ^ { 2 } ) ) & = \\lim _ { b \\to \\infty } \\frac { b ^ { 2 } s _ { i , - 1 } ^ { 2 } - b s _ { i , - 1 } + s _ { i , - 1 } + s _ { i , - 1 } \\lambda _ { i , 2 } ^ { 2 } } { b ^ { 2 } - b \\lambda _ { i , 2 } ^ { 2 } } \\\\ & = s _ { i , - 1 } ^ { 2 } > 0 , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] . \\end{align*}"}
+{"id": "4643.png", "formula": "\\begin{align*} \\Phi ^ \\lambda [ \\xi ] = \\langle \\xi , \\ , \\Gamma ^ \\lambda \\xi \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 3 ) } , \\qquad \\forall \\ , \\xi \\in D \\end{align*}"}
+{"id": "7712.png", "formula": "\\begin{align*} K [ g ^ + ] + 1 = O ( \\rho ^ 2 ) \\end{align*}"}
+{"id": "158.png", "formula": "\\begin{align*} \\begin{aligned} L [ \\hat { f } ] = \\nu \\hat { f } - K [ \\hat { f } ] = \\hat { \\textrm { g } } \\ , , \\ | \\hat { f } | = \\tfrac { 1 } { \\nu } | \\hat { \\textrm { g } } + K [ \\hat { f } ] | < \\tfrac { 1 } { \\nu _ 0 } ( | \\hat { \\textrm { g } } | + | K [ \\hat { f } ] | ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "69.png", "formula": "\\begin{align*} ( \\nabla u ( x _ { 0 } ) ) ^ { \\top } = 0 . \\end{align*}"}
+{"id": "5285.png", "formula": "\\begin{align*} \\dim L _ { \\pi } = \\sum \\dim L _ { \\pi _ { i , j } } , \\end{align*}"}
+{"id": "5744.png", "formula": "\\begin{align*} \\begin{aligned} ( { \\rm I d } _ q \\otimes P ^ M ) g ( X ^ M ) ( { \\rm I d } _ q \\otimes P ^ M ) & = \\sum _ { i = 1 , j = 1 } ^ { m } B _ { i j } \\otimes P ^ M { X ^ M _ i } { X ^ M _ j } P ^ M \\\\ & = \\sum _ { i = 1 , j = 1 } ^ { m } B _ { i j } \\otimes M ^ s _ { i j } \\succeq 0 . \\end{aligned} \\end{align*}"}
+{"id": "6012.png", "formula": "\\begin{align*} \\tilde { g } ^ { ( q ) } ( x ; h ) = \\dfrac { W ^ { ( q ) } ( x ) } { W ^ { ( q ) } ( b ) } \\left ( \\rho ^ { ( q ) } _ { b } ( b ; h ) + \\tilde { g } ^ { ( q ) } ( b ; h ) \\right ) - \\rho ^ { ( q ) } _ { b } ( x ; h ) . \\end{align*}"}
+{"id": "974.png", "formula": "\\begin{align*} [ T ( f ) ] ( \\xi ) : = \\frac { f ( \\xi ) } { \\sqrt { ( 1 + \\gamma ( \\xi ) ^ 2 ) } } . \\end{align*}"}
+{"id": "609.png", "formula": "\\begin{align*} \\mu _ m ^ { ( 1 2 ) } \\big ( \\mathfrak { s } \\boldsymbol { J } \\big ) = \\mu _ m ^ { ( 1 2 3 ) } \\big ( \\boldsymbol { J } \\big ) \\mu _ m ^ { ( 1 2 3 ) } \\big ( \\mathfrak { s } \\boldsymbol { J } \\big ) = \\mu _ m ^ { ( 1 2 ) } \\big ( \\boldsymbol { J } \\big ) . \\end{align*}"}
+{"id": "3817.png", "formula": "\\begin{align*} a _ 1 + a _ 2 + \\cdots + a _ { h - 3 } = \\frac { ( h - 2 ) ( 2 n - 4 ) - ( a _ { h - 2 } + a _ { h - 1 } + a _ h ) + d } { 2 } , \\end{align*}"}
+{"id": "7165.png", "formula": "\\begin{align*} m _ i : = \\left \\lceil \\frac { v i } { d } \\right \\rceil - \\left \\lceil \\frac { v ( i - 1 ) } { d } \\right \\rceil + \\delta _ i ^ d - \\delta _ i ^ 1 \\in \\mathbb { Z } , \\end{align*}"}
+{"id": "159.png", "formula": "\\begin{align*} \\begin{aligned} N _ 1 [ K [ \\hat { f } ] ] < k N _ 2 [ \\hat { f } ] \\ , , \\ N _ 3 ^ { ( r + 1 / 2 ) } [ K [ \\hat { f } ] ] < k N _ 3 ^ { ( r ) } [ \\hat { f } ] \\end{aligned} \\end{align*}"}
+{"id": "2279.png", "formula": "\\begin{align*} p = c _ 1 T _ 1 + c _ 2 T _ 2 + \\dots + c _ s T _ s , c _ i \\in k ^ * , T _ i , i = 1 , 2 , \\dots , n , \\end{align*}"}
+{"id": "753.png", "formula": "\\begin{align*} [ a _ { ( m ) } , b _ { ( n ) } ] = \\sum _ { j \\in \\mathbb { Z _ { + } } } \\left ( \\begin{array} { c c c } m \\\\ j \\end{array} \\right ) ( a _ { ( j ) } b ) _ { ( m + n - j ) } . \\end{align*}"}
+{"id": "5919.png", "formula": "\\begin{align*} A \\cap \\wedge ^ 3 U = 0 \\iff A ^ \\perp + F _ u = \\wedge ^ 3 V _ 6 ^ \\vee \\iff A ^ \\perp \\cap F _ u = 0 \\ , , \\end{align*}"}
+{"id": "8022.png", "formula": "\\begin{align*} \\begin{bmatrix} Q _ 0 - \\Tilde { A } & X & r o w ( Q _ j ^ * ) _ { j \\geq m + 1 } \\\\ X ^ * & ( Q _ { i - j } \\otimes I _ d ) _ { i = 1 , j = 1 } ^ m - S ( m - 1 ) \\otimes I _ d & ( Q _ { i - j } \\otimes I _ d ) _ { i = 1 , j = m + 1 } ^ { m + 1 , \\infty } \\\\ c o l ( Q _ j ) _ { j \\geq m + 1 } & ( Q _ { i - j } \\otimes I _ d ) _ { i = m + 1 , j = 1 } ^ { \\infty , m } & T _ Q \\otimes I _ d ^ { m + 1 } \\end{bmatrix} \\end{align*}"}
+{"id": "6943.png", "formula": "\\begin{align*} h _ i & = \\left [ t ^ i \\right ] \\frac { 1 } { ( 1 - z _ 1 t ) \\cdots ( 1 - z _ { N + 1 } t ) } = \\left [ t ^ i \\right ] \\frac { 1 } { t ^ { N + 1 } P ( 1 / t ) } . \\end{align*}"}
+{"id": "7125.png", "formula": "\\begin{align*} \\rho - \\sum _ { i = 1 } ^ k \\rho _ i = \\frac { 1 } { 2 } \\mathfrak { g } ^ { \\lambda < 0 } = - \\frac { 1 } { 2 } \\mathfrak { g } ^ { \\lambda > 0 } , \\end{align*}"}
+{"id": "6380.png", "formula": "\\begin{align*} & 2 \\pi = \\sum _ { n \\geq 4 } v _ { n } ^ { \\mathrm { i n t } } \\left ( 2 \\pi - \\frac { \\pi } { 2 } n \\right ) + \\sum _ { n _ 1 , n _ 2 } v _ { n _ 1 , n _ 2 } ^ { \\mathrm { e x } } \\left ( \\frac { 3 \\pi } { 2 } - \\frac { \\pi } { 4 } ( n + n _ 1 ) \\right ) + \\sum _ { n _ 1 , n _ 2 } v _ { n _ 1 , n _ 2 } ^ { \\mathrm { c l } } \\left ( \\frac { 3 \\pi } { 2 } - \\frac { \\pi } { 4 } ( n + n _ 1 ) \\right ) , \\\\ & \\sum _ { n } v _ n ^ { \\mathrm { e x } } = 3 , \\end{align*}"}
+{"id": "102.png", "formula": "\\begin{align*} \\varphi '' + \\Big ( ( m - 1 ) \\frac { \\sigma ' } { \\sigma } - \\Phi ' \\Big ) \\varphi ' - f ' ( u ) \\varphi \\leq \\varphi '' + \\Big ( ( m - 1 ) \\frac { \\sigma ' } { \\sigma } - \\Phi ' \\Big ) \\varphi ' + B = 0 . \\end{align*}"}
+{"id": "3640.png", "formula": "\\begin{align*} e ^ { k , l } : = e _ k ^ * e _ l \\in M _ n \\beta _ e : = ( e _ 1 , \\dots , e _ n ) \\in M _ { 1 , n } ( M _ { 1 , n } ) . \\end{align*}"}
+{"id": "641.png", "formula": "\\begin{align*} \\mathcal { A } = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ p e _ j \\cdot \\big ( ( \\widetilde { \\nabla } _ Y B ) ( X , e _ j ) - ( \\widetilde { \\nabla } _ X B ) ( Y , e _ j ) \\big ) , \\end{align*}"}
+{"id": "5193.png", "formula": "\\begin{align*} G ( X ) = \\frac { 1 - X } { 1 - X - 2 X ^ 2 } = \\frac { X - 1 } { 2 X ^ 2 + X - 1 } = \\frac { X - 1 } { ( X + 1 ) ( 2 X - 1 ) } \\enspace . \\end{align*}"}
+{"id": "3234.png", "formula": "\\begin{align*} \\mu _ { j } = e ^ { - u _ { j } } \\theta ^ { n } _ { u } + e ^ { - v _ { j } } \\theta ^ { n } _ { v } . \\end{align*}"}
+{"id": "1139.png", "formula": "\\begin{align*} M _ p = \\max \\{ d ( x _ 0 , p ) , d ( u , p ) \\} . \\end{align*}"}
+{"id": "6073.png", "formula": "\\begin{align*} V ( \\vec { K } ' _ 1 ) & = \\{ x _ 1 , x _ 2 , y _ 1 , y _ 2 , y _ 3 , y _ 4 \\} \\\\ A ( \\vec { K } ' _ 1 ) & = \\{ x _ 1 x _ 2 , y _ 1 y _ 2 , y _ 2 y _ 3 , y _ 3 y _ 4 , y _ 4 y _ 1 , y _ 1 x _ 1 , y _ 2 x _ 1 , y _ 3 x _ 1 , y _ 4 x _ 1 , x _ 2 y _ 1 , x _ 2 y _ 2 , x _ 2 y _ 3 , x _ 2 y _ 4 \\} \\\\ \\end{align*}"}
+{"id": "3718.png", "formula": "\\begin{align*} \\partial _ t S ( x , t ) + \\frac 1 2 ( \\nabla S ( x , t ) ) ^ 2 - \\frac 1 2 \\frac { \\nabla ^ 2 \\sqrt { \\rho ( x , t ) } } { \\sqrt { \\rho ( x , t ) } } = 0 \\end{align*}"}
+{"id": "7256.png", "formula": "\\begin{align*} \\over - \\int f d \\mu _ 0 = \\int g d \\mu _ 0 \\geq \\inf _ { f \\in \\mathcal { A } } \\int f d \\mu _ 0 . \\end{align*}"}
+{"id": "6886.png", "formula": "\\begin{align*} f ( \\mathbf w ) = f _ C ( \\mathbf w ) . \\end{align*}"}
+{"id": "3924.png", "formula": "\\begin{align*} \\partial _ t u - ( 1 + D _ t ^ { \\{ m \\} } ) \\Delta u & = g ( x ) p ( t ) + f _ 1 ( u ) \\Omega , t \\in ( 0 , T ) , \\\\ u & = 0 \\partial \\Omega , \\ ; t \\ge 0 , \\\\ u ( 0 ) & = \\xi \\Omega , \\\\ \\int _ \\Omega \\kappa ( x ) u ( t , x ) d x & = \\psi ( t ) , \\ ; t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "473.png", "formula": "\\begin{align*} \\sum _ { k = c _ 1 } ^ { c _ 2 } { F _ { i w _ k + m } v _ k z ^ { w _ k } } = \\frac { 1 } { \\sqrt 5 } \\big ( \\alpha ^ m h ( \\alpha ^ i z ) + \\beta ^ m h ( \\beta ^ i z ) \\big ) \\ , , \\end{align*}"}
+{"id": "6302.png", "formula": "\\begin{align*} d _ \\infty ( \\underline { x } , \\underline { x } ' ) = \\max _ { 1 \\leq i \\leq n } | x _ i - x ' _ i | . \\end{align*}"}
+{"id": "2213.png", "formula": "\\begin{gather*} \\dot { y } = - a - v y \\end{gather*}"}
+{"id": "655.png", "formula": "\\begin{align*} \\nabla _ X \\psi = \\frac { 1 } { 2 } X _ 1 \\cdot V \\cdot \\psi - \\frac { 1 } { 2 } S ( X ) \\cdot N \\cdot \\psi . \\end{align*}"}
+{"id": "1095.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ Z _ { i , j } \\right ] = \\frac { \\pi } { 4 } \\| \\mathbf { x } \\| . \\end{align*}"}
+{"id": "1483.png", "formula": "\\begin{align*} R ( y ) = \\sum _ { d < y } \\tau _ 3 ( d ) | r _ d | \\end{align*}"}
+{"id": "4609.png", "formula": "\\begin{align*} \\theta ( n + 2 ) = \\theta ( n ) + 1 + \\frac { K _ 1 } { \\pi ( n - b ) } \\left ( 1 0 0 - \\frac { 1 } { 2 } \\sin 4 \\pi \\theta ( n ) + \\frac { O ( 1 ) } { n - b } \\right ) . \\end{align*}"}
+{"id": "7794.png", "formula": "\\begin{align*} I _ { \\theta } = - \\frac { 1 } { \\pi } \\int _ { D _ { \\theta } } \\frac { 1 } { \\overline { u + 1 } ( u - 1 ) } d a ( u ) \\end{align*}"}
+{"id": "5624.png", "formula": "\\begin{align*} \\hat { \\mathbb { G } } ^ i _ k u ^ k = 2 J ^ i _ k \\hat { \\mathbb { G } } ^ k . \\end{align*}"}
+{"id": "4538.png", "formula": "\\begin{align*} \\sup _ { x \\in U } f & : = \\inf \\{ a \\in \\mathbb { R } : f ^ { - 1 } ( a , \\infty ) = \\varnothing \\} \\ , , \\\\ \\mathrm { e s s } \\ , \\sup _ { x \\in U } f & : = \\inf \\{ a \\in \\mathbb { R } : \\mu ( f ^ { - 1 } ( a , \\infty ) ) = 0 \\} \\ , . \\end{align*}"}
+{"id": "3629.png", "formula": "\\begin{align*} f ' ( \\alpha ) = - c \\end{align*}"}
+{"id": "546.png", "formula": "\\begin{align*} \\log \\int _ { \\R ^ n } e ^ { f } \\ , d \\gamma _ t - \\sup _ { y \\in \\R ^ n } \\left ( \\int _ { \\R ^ n } f \\ , d \\gamma _ { y , t } - H ( \\gamma _ { y , t } \\ , | \\ , \\gamma _ t ) \\right ) = \\inf _ { y \\in \\R ^ n } H ( \\gamma _ { y , t } \\ , | \\ , P ) . \\end{align*}"}
+{"id": "8037.png", "formula": "\\begin{align*} ( - D _ r \\widehat h _ r - \\widehat h _ r D _ r ) ( x , y , z ) = ( 0 , y , d _ r y ) + ( 0 , 0 , - d _ r y - x + z ) = ( x , y , z ) - ( x , 0 , x ) . \\end{align*}"}
+{"id": "5467.png", "formula": "\\begin{align*} R _ { 1 , N } ( a , b , t ) = \\sum _ { j = 0 } ^ { N } \\frac { ( a q ) _ j } { ( b q ) _ j } t ^ j \\left ( 1 - \\frac { ( b - a t q ) q ^ j } { 1 - t } \\right ) + \\frac { b - 1 } { 1 - t } . \\end{align*}"}
+{"id": "981.png", "formula": "\\begin{align*} ( k - j - s - 1 ) a _ { s + 1 , t } & = ( - w _ 2 + j + s + t ) a _ { s , t } , \\\\ ( 2 w _ 2 + n - 2 t - 2 ) a _ { s , t + 1 } & = 2 ( - w _ 2 + j + s + t ) a _ { s , t } . \\end{align*}"}
+{"id": "2903.png", "formula": "\\begin{align*} 0 = \\int _ \\Sigma \\Delta x _ { n + 1 } \\ , d A = \\int _ \\Sigma \\left ( \\eta x _ { n + 1 } ^ m + \\lambda \\right ) \\nu _ { n + 1 } \\ , d A = \\int _ \\Sigma \\eta x _ { n + 1 } ^ m \\nu _ { n + 1 } \\ , d A , \\end{align*}"}
+{"id": "3473.png", "formula": "\\begin{align*} \\mathbb { A } ^ { \\textbf { e x t } } : = \\mathbb { H } + \\Big ( \\frac { 1 } { 2 } I _ { d } - \\mathcal { K } ^ { * } \\Big ) \\mathcal { S } ^ { - 1 } \\Big ( \\frac { 1 } { 2 } I _ { d } - \\mathcal { K } \\Big ) . \\end{align*}"}
+{"id": "352.png", "formula": "\\begin{align*} ( d ( S ( T ) y _ 1 , S ( T ) y _ 2 ) ) ^ 2 \\leq & C [ d ( y _ 1 , y _ 2 ) ^ 2 - ( d ( S ( T ) y _ 1 , S ( T ) y _ 2 ) ) ^ 2 \\\\ & + g _ 1 ( \\varrho _ T ^ 1 ( y _ 1 , y _ 2 ) , \\varrho _ T ^ 2 ( y _ 1 , y _ 2 ) , \\cdots , \\varrho _ T ^ m ( y _ 1 , y _ 2 ) ) ] ^ \\beta \\\\ & + g _ 2 ( \\varrho _ T ^ 1 ( y _ 1 , y _ 2 ) , \\varrho _ T ^ 2 ( y _ 1 , y _ 2 ) , \\cdots , \\varrho _ T ^ m ( y _ 1 , y _ 2 ) ) . \\end{align*}"}
+{"id": "1452.png", "formula": "\\begin{align*} \\frac { d ^ m } { d s ^ m } M ( b , c ; s ) = \\frac { ( b ) _ m } { ( c ) _ m } M ( b + m , c + m ; s ) , \\end{align*}"}
+{"id": "5606.png", "formula": "\\begin{align*} \\hat { \\omega } ^ i _ j = \\hat { \\mathbb { G } } ^ i _ { j k } d x ^ k \\end{align*}"}
+{"id": "3361.png", "formula": "\\begin{gather*} \\mu ( \\mathrm { R e s } _ { \\tau _ \\alpha : L \\hookrightarrow L _ { \\alpha } } ( x _ 1 ) ) _ { \\alpha } = \\mu ( x _ \\alpha ) _ \\alpha , \\\\ ( y _ i ) _ i = \\left ( \\mathrm { R e s } _ { L K _ i / K _ i } ( \\tilde { y } _ i ) \\right ) _ i , \\\\ \\mu \\mathrm { R e s } _ { L K _ { \\mathcal { I } } / L } ( x _ 1 ) = \\mu z . \\end{gather*}"}
+{"id": "2332.png", "formula": "\\begin{align*} Z ( m _ { 0 } , \\dots , m _ { 2 n } ) : = \\zeta ( \\{ 2 \\} ^ { m _ { 0 } } , 1 , \\{ 2 \\} ^ { m _ { 1 } } , 3 , \\{ 2 \\} ^ { m _ { 2 } } , \\dots , 1 , \\{ 2 \\} ^ { m _ { 2 n - 1 } } , 3 , \\{ 2 \\} ^ { m _ { 2 n } } ) \\end{align*}"}
+{"id": "2772.png", "formula": "\\begin{align*} U _ A ^ T A G = C , U _ B ^ T B G = S , \\end{align*}"}
+{"id": "2592.png", "formula": "\\begin{align*} \\mu ^ i + h ( \\mu ^ i ) = - R ^ { - 1 } B \\Big [ P _ t \\nu ^ { N , i } _ { \\boldsymbol { x } } + \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } ) \\Big ] . \\end{align*}"}
+{"id": "5689.png", "formula": "\\begin{gather*} \\tau _ * ( \\alpha _ { ( 0 , i ) } ) = e _ { 2 , 1 } ^ { 1 } \\wedge ( e _ { 3 , 1 } ^ { 1 } + e _ { 3 , 2 } ^ 2 ) \\wedge \\cdots \\wedge ( \\sum _ { j = 1 } ^ i e _ { i + 1 , j } ^ { j } ) \\in \\bigwedge ^ { i } U . \\end{gather*}"}
+{"id": "5541.png", "formula": "\\begin{align*} \\alpha ( 0 ) = 0 \\beta ( 0 ) = 0 \\end{align*}"}
+{"id": "5718.png", "formula": "\\begin{align*} \\dd \\theta _ { \\nu _ t } = c _ { \\nu _ t } \\sqrt { u _ t } \\ , \\dd \\gamma , \\end{align*}"}
+{"id": "5049.png", "formula": "\\begin{align*} \\frac { \\Gamma ( k - 1 ) } { ( 4 \\pi ) ^ { k - 1 } } \\sum _ { g \\in H _ k ( N , \\chi ) } \\rho _ g ( n ) \\overline { \\rho _ g ( m ) } = \\delta _ { n = m } + 2 \\pi i ^ { - k } \\sum _ { \\substack { q \\geq 1 \\\\ N \\mid q } } \\frac { S _ { \\chi } ( m , n ; q ) } { q } J _ { k - 1 } \\ ! \\left ( \\frac { 4 \\pi \\sqrt { m n } } { q } \\right ) , \\end{align*}"}
+{"id": "1001.png", "formula": "\\begin{align*} { \\rm D } ^ 2 \\widetilde { W } ( { \\rm D } \\phi ) . ( \\xi \\otimes \\eta , \\xi \\otimes \\eta ) > 0 \\forall \\ \\eta , \\xi \\in \\mathbb { R } ^ 3 , \\ \\ \\lVert \\eta \\rVert = \\lVert \\xi \\rVert = 1 , \\end{align*}"}
+{"id": "4992.png", "formula": "\\begin{align*} & | F _ { n + 1 } | \\lesssim \\mathbf { M } ^ { 1 - } , | \\nabla _ p F _ { n + 1 } | \\lesssim \\mathbf { M } ^ { 1 - } , \\\\ & | \\nabla _ x F _ { n + 1 } | \\lesssim \\mathbf { M } ^ { 1 - } , | \\nabla _ x \\nabla _ p F _ { n + 1 } | \\lesssim \\mathbf { M } ^ { 1 - } , \\\\ & | \\nabla _ x ^ 2 F _ { n + 1 } | \\lesssim \\mathbf { M } ^ { 1 - } , | \\nabla _ x ^ 2 \\nabla _ p F _ { n + 1 } | \\lesssim \\mathbf { M } ^ { 1 - } . \\end{align*}"}
+{"id": "4247.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( \\frac { b } { a } ) _ n a ^ n } { ( 1 - q ^ n ) ( b ) _ n } = \\sum _ { n = 1 } ^ { \\infty } \\frac { a ^ n - b ^ n } { 1 - q ^ n } . \\end{align*}"}
+{"id": "6514.png", "formula": "\\begin{align*} A ( G ) = \\begin{pmatrix} \\mathbf { 0 } & C \\\\ C ^ T & \\mathbf { 0 } \\end{pmatrix} . \\end{align*}"}
+{"id": "3754.png", "formula": "\\begin{align*} \\tau _ i ( - 6 ) & = \\frac { 1 } { q _ i ( 2 ) } \\sum _ { k = 1 } ^ { \\deg m _ i } D ^ k m _ i ( 2 ) A [ \\mathcal C _ i ] ^ { k - 1 } . \\end{align*}"}
+{"id": "770.png", "formula": "\\begin{align*} \\widehat { \\iota } ( D ) = \\frac { \\beta - e } { d } \\end{align*}"}
+{"id": "1466.png", "formula": "\\begin{align*} I ( W _ { 1 : M } ^ { [ t ] } ; S _ n ^ { [ t ] } ) = 0 , \\end{align*}"}
+{"id": "7788.png", "formula": "\\begin{align*} \\begin{gathered} E ( z , w ) = \\exp - \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } \\frac { u - w } { u - z } \\frac { g ( u ) } { \\vert u - w \\vert ^ 2 } d a ( u ) = \\\\ \\exp - \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } \\frac { g ( u ) } { \\overline { u - w } ( u - z ) } d a ( u ) \\end{gathered} \\end{align*}"}
+{"id": "6442.png", "formula": "\\begin{align*} \\lambda _ n = \\int _ { - \\infty } ^ { + \\infty } \\frac { e ^ { - \\theta ^ 2 + 2 m \\theta } } { ( z q ^ { 1 / 2 } e ^ { - 2 i k \\theta } ; q ) _ { \\infty } } \\frac { h _ n ( - q e ^ { 2 m k i } , q ^ { 1 / 2 } e ^ { 2 k i \\theta } ) } { ( q ; q ) _ n } d \\theta . \\end{align*}"}
+{"id": "5682.png", "formula": "\\begin{align*} y _ j = \\alpha ^ { q ^ j } x _ 0 + \\alpha ^ { q ^ { j + 1 } } x _ 1 + \\alpha ^ { q ^ { j + 2 } } x _ 2 + . . . + \\alpha ^ { q ^ { j + i } } x _ i + . . . + \\alpha ^ { q ^ { j + n } } x _ n \\end{align*}"}
+{"id": "157.png", "formula": "\\begin{align*} \\begin{aligned} | \\L ^ { - 1 } g ( v ) | \\lesssim ( 1 + | v | ) ^ { - m } \\| ( 1 + | v | ) ^ m g \\| _ { L ^ \\infty } \\ ( \\forall v \\in \\R ^ 3 ) \\end{aligned} \\end{align*}"}
+{"id": "7179.png", "formula": "\\begin{align*} \\mathcal { F } _ { d , w } : = \\Phi ( \\mathcal { E } _ { d , w } ) \\in \\mathbb { S } ^ { \\mathrm { g r } } ( d ) _ w . \\end{align*}"}
+{"id": "6374.png", "formula": "\\begin{align*} \\tilde { X } _ \\cdot = \\mathcal { G } _ \\mu ( B _ \\cdot ^ H ) . \\end{align*}"}
+{"id": "4508.png", "formula": "\\begin{align*} \\dd _ { ( \\phi , \\pi ) } \\Phi _ x ( h , 0 ) = \\langle D _ 1 \\Phi _ x ( \\phi , \\pi ) , h \\rangle _ { H ^ 1 } \\ , . \\end{align*}"}
+{"id": "724.png", "formula": "\\begin{align*} { u _ z } = \\frac { 1 } { \\rho } \\left [ { { f _ 0 } + { f _ 1 } + { f _ 2 } + { f _ 3 } + { f _ 4 } + { f _ 7 } + { f _ 8 } + { f _ 9 } + { f _ { 1 0 } } + 2 \\left ( { { f _ 5 } + { f _ { 1 1 } } + { f _ { 1 4 } } + { f _ { 1 5 } } + { f _ { 1 8 } } } \\right ) + \\frac { 1 } { 2 } { F _ z } } \\right ] - 1 , \\end{align*}"}
+{"id": "8059.png", "formula": "\\begin{align*} \\mathop { { \\mathrm { m i n i m i z e } } } \\limits _ { { \\bf w } _ p } ~ { \\bf w } _ p ^ H \\widetilde { \\bf \\Upsilon } _ { c } { \\bf w } _ p \\ ; \\ ; \\textrm { s u b j e c t t o } ~ { \\bf w } _ p ^ H { \\bf \\Upsilon } _ { p , t } { \\bf w } _ p = 1 . \\end{align*}"}
+{"id": "3525.png", "formula": "\\begin{align*} \\Big ( \\mathbb { G } ^ { ( \\mathrm { k } ) } - \\mathbb { G } ^ { ( 0 ) } \\Big ) ( \\mathrm { x } , \\mathrm { y } ) = a | \\mathrm { x } - \\mathrm { y } | ^ 2 \\log { | \\mathrm { x } - \\mathrm { y } | } + b | \\mathrm { x } - \\mathrm { y } | ^ 2 + \\mathcal { O } ( | \\mathrm { x } - \\mathrm { y } | ^ 3 ) , \\end{align*}"}
+{"id": "2456.png", "formula": "\\begin{align*} \\star \\omega : = ( * \\omega ) ^ \\# , \\star v : = * ( v ^ \\flat ) \\end{align*}"}
+{"id": "7650.png", "formula": "\\begin{align*} \\bar \\kappa : = \\inf \\left \\{ \\ , \\kappa > R ^ { - 1 } \\ , : \\ , \\ , \\right \\} . \\end{align*}"}
+{"id": "8055.png", "formula": "\\begin{align*} & { \\bf h } _ { u , m , k } = \\sum _ { l _ m = 1 } ^ { L _ m } \\sum _ { k = 1 } ^ { K } { \\alpha _ { l _ m } } e ^ { - \\mathrm { j } \\frac { 2 \\pi k d } { K } } { \\bf a } _ t ( \\phi _ { l _ m } , f _ k ) r ( k T _ s - \\tau _ { l _ m } ) , \\ ; \\in \\mathbb { C } ^ { { N _ m } \\times 1 } , \\end{align*}"}
+{"id": "2649.png", "formula": "\\begin{align*} x _ { \\lambda } x _ { \\mu } x _ { \\nu } = x _ { \\lambda } ( x _ { \\mu } x _ { \\nu } ) \\lambda ( \\mu \\nu ) \\end{align*}"}
+{"id": "686.png", "formula": "\\begin{align*} d v ' = \\left \\{ ( \\log \\sqrt { \\tau _ 0 } ) _ z \\ d z + \\frac { i } { \\sqrt { \\tau _ 0 } } \\left ( Q _ 0 d z + \\frac { \\tau _ 0 } { 4 } d \\overline { z } \\right ) J \\right \\} v ' . \\end{align*}"}
+{"id": "1525.png", "formula": "\\begin{align*} \\left \\Vert r - \\frac { 1 } { c } \\tilde { r } \\right \\Vert _ { \\infty } = \\max _ { S \\subseteq E } \\left | r \\left ( S \\right ) - \\frac { 1 } { c } \\tilde { r } \\left ( S \\right ) \\right | < \\varepsilon . \\end{align*}"}
+{"id": "7264.png", "formula": "\\begin{align*} \\overline { { m d i m } } _ M ( T , s f , d ) = 0 ~ ~ ~ ~ s = - \\frac { F ( \\mu _ s ) } { \\int f d \\mu _ s } . \\end{align*}"}
+{"id": "463.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 0 \\\\ n - k \\equiv 0 \\ , ( \\mathrm { m o d } \\ , 2 ) } } ^ n \\binom { n } { k } 2 ^ k F _ { k } ( \\sqrt { 5 } ) ^ { n - k } & \\frac { B _ { n - k + 2 } } { n - k + 2 } + \\frac { 2 } { \\sqrt { 5 } } \\sum _ { \\substack { k = 0 \\\\ n - k \\equiv 1 \\ , ( \\mathrm { m o d } \\ , 2 ) } } ^ { n - 1 } \\binom { n } { k } ( \\sqrt { 5 } ) ^ { n - k } \\frac { B _ { n - k + 1 } } { n - k + 1 } \\\\ & = \\frac { 2 } { 5 ( n + 1 ) } \\Big ( \\frac { 2 ^ { n + 1 } F _ { n + 2 } } { n + 2 } - 1 \\Big ) . \\end{align*}"}
+{"id": "6274.png", "formula": "\\begin{align*} T _ \\lambda ( u ) = - t ^ * S _ \\lambda \\left ( p _ \\lambda ( u ) \\right ) , \\end{align*}"}
+{"id": "7168.png", "formula": "\\begin{align*} i _ { \\ast } \\mathcal { E } _ { d , v } & = i _ { \\ast } p _ { \\ast } ( \\mathcal { O } _ { \\mathcal { Z } } \\otimes \\mathbb { C } ( m _ 1 , \\ldots , m _ d ) ) \\\\ & \\cong p _ { \\ast } ( i _ { \\ast } \\mathcal { O } _ { \\mathcal { Z } } \\otimes q ^ { \\ast } \\mathcal { O } _ { \\mathcal { Y } ( d ) ^ { \\lambda } } ( m _ 1 , \\ldots , m _ d ) ) . \\end{align*}"}
+{"id": "2555.png", "formula": "\\begin{align*} \\mu _ t ^ { * , \\xi } = \\rho ( - R ^ { - 1 } B P _ t \\nu _ t ^ { * , \\xi } - R ^ { - 1 } B \\varphi _ t ^ { * , \\xi } ) \\end{align*}"}
+{"id": "498.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } { n \\choose k } 6 ^ k ( \\sqrt { 5 } ) ^ { n - k } \\big ( { 1 - 3 ^ { n - k - 1 } } \\big ) F _ { k } B _ { n - k } = n 2 ^ { n - 1 } L _ { 2 n - 2 } + \\sum _ { m = 1 } ^ { n } { n \\choose m } m 4 ^ { m - 1 } L _ { m - 1 } . \\end{align*}"}
+{"id": "8004.png", "formula": "\\begin{align*} \\langle S f , f \\rangle = i n f \\Big \\{ \\Big \\langle \\begin{pmatrix} A & B \\\\ B ^ * & C \\end{pmatrix} \\begin{pmatrix} f \\\\ g \\end{pmatrix} , \\begin{pmatrix} f \\\\ g \\end{pmatrix} \\Big \\rangle : \\ : g \\in \\mathcal { H } _ 2 \\Big \\} \\end{align*}"}
+{"id": "3737.png", "formula": "\\begin{align*} ( A ^ 2 ) _ { i j } & = \\frac 1 3 ( M _ { i j } + 2 8 ) - 4 A _ { i j } \\in \\{ 9 - 4 A _ { i j } , 1 0 - 4 A _ { i j } \\} . \\end{align*}"}
+{"id": "2016.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\frac { 1 } { 4 } \\left ( C - e ^ { - t / 2 } \\Phi ( e ^ { - 2 t } , 2 , 1 / 4 ) \\right ) e ^ { i z t } \\ , d t = - \\frac { 1 } { 2 z ^ 2 } \\left ( \\frac { \\Gamma ' } { \\Gamma } \\left ( \\frac { s } { 2 } \\right ) - \\frac { \\Gamma ' } { \\Gamma } \\left ( \\frac { 1 } { 4 } \\right ) \\right ) \\end{align*}"}
+{"id": "2566.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { * , t _ 0 , \\xi } = & ~ [ A x _ t ^ { * , t _ 0 , \\xi } - B ^ 2 R ^ { - 1 } U ( t , x _ t ^ { * , t _ 0 , \\xi } , \\nu _ t ^ { * , t _ 0 , \\xi } ) - B h ( \\mu _ t ^ { * , t _ 0 , \\xi } ) + f ( \\nu _ t ^ { * , t _ 0 , \\xi } ) + b ( \\mu _ t ^ { * , t _ 0 , \\xi } ) ] d t \\\\ & + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ { t _ 0 } ^ { * , t _ 0 , \\xi } = & ~ \\xi , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7213.png", "formula": "\\begin{align*} F ( v _ i ) = \\sum _ { j = i } ^ d b _ { j + 1 - i } v _ j \\end{align*}"}
+{"id": "848.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\lambda _ { + } ( t ) = \\frac { \\sigma } { 2 } - \\frac { \\alpha } { \\sigma } + \\sqrt { \\Big ( \\frac { \\sigma } { 2 } - \\frac { \\alpha } { \\sigma } \\Big ) ^ 2 - 2 t } , \\\\ & \\lambda _ { - } ( t ) = \\frac { \\sigma } { 2 } - \\frac { \\alpha } { \\sigma } - \\sqrt { \\Big ( \\frac { \\sigma } { 2 } - \\frac { \\alpha } { \\sigma } \\Big ) ^ 2 - 2 t } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3903.png", "formula": "\\begin{align*} u ( t ) = \\sum _ { n = 1 } ^ \\infty u _ n ( t ) e _ n , \\ ; F ( t ) = \\sum _ { n = 1 } ^ \\infty F _ n ( t ) e _ n . \\end{align*}"}
+{"id": "6293.png", "formula": "\\begin{align*} \\langle x , \\frac { 1 } { q - 1 } \\rangle = \\sum _ { N = 0 } ^ \\infty \\langle x \\rangle _ N = \\langle x \\rangle _ { } . \\end{align*}"}
+{"id": "4914.png", "formula": "\\begin{align*} ( A _ \\mathcal { L } ) ^ T \\vec { x } = \\vec { b } _ \\mathcal { L } . \\end{align*}"}
+{"id": "3601.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\frac { 1 } { ( 1 + y ^ \\beta ) ^ { \\alpha / \\beta } } \\ , d y & = 2 ^ { - \\alpha / \\beta } \\int _ 0 ^ \\infty { \\left ( \\frac { 1 } { 2 } + \\frac { y ^ \\beta } { 2 } \\right ) ^ { - \\alpha / \\beta } } \\ , d y \\leq 2 ^ { - \\alpha / \\beta } \\int _ 0 ^ \\infty { \\left ( \\frac { 1 } { 2 } + \\frac { y ^ \\alpha } { 2 } \\right ) ^ { - 1 } } \\ , d y , \\end{align*}"}
+{"id": "6957.png", "formula": "\\begin{align*} \\binom { - \\chi - N + d ( N + 1 ) } { k } = \\binom { - \\chi + d ( N + 1 ) } { k } - \\sum _ { \\ell = 1 } ^ k \\frac { N } { \\ell } \\binom { ( N + 1 ) ( \\ell - 1 ) } { \\ell - 1 } \\binom { - \\chi + ( d - \\ell ) ( N + 1 ) } { k - \\ell } . \\end{align*}"}
+{"id": "2857.png", "formula": "\\begin{align*} ( t _ \\lambda x ) \\cdot _ \\ell \\mu = x ( \\mu + \\rho ) - \\rho + \\ell \\lambda \\end{align*}"}
+{"id": "2224.png", "formula": "\\begin{align*} W _ { k + 1 } ^ D = W _ { k } ^ D * \\boldsymbol { \\mathcal { A } } - \\boldsymbol { \\alpha } _ { k } \\times W _ { k } ^ D - \\boldsymbol { \\beta } _ { k } \\times W _ { k - 1 } ^ D , \\\\ V _ { k + 1 } \\times \\boldsymbol { \\beta } _ { k + 1 } = \\boldsymbol { \\mathcal { A } } * V _ { k } - V _ { k } \\times \\boldsymbol { \\alpha } _ { k } - V _ { k - 1 } , \\end{align*}"}
+{"id": "305.png", "formula": "\\begin{align*} b = ( a _ { s - 1 } ( 1 - T ) ^ { p } , \\ldots , a _ { 1 } ( 1 - T ) ^ { p ^ { s - 1 } } , a _ { 0 } ( 1 - T ) ^ { p ^ { s } } ) + ( a _ { s - 1 } ' T ^ { p } , \\ldots , a _ { 1 } ' T ^ { p ^ { s - 1 } } , a _ { 0 } T ^ { p ^ { s } } ) . \\end{align*}"}
+{"id": "5486.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\frac { ( a t q ) _ \\infty ( q ) _ \\infty } { ( t ) _ \\infty ( b q ) _ \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( b ) _ n ( t ) _ n q ^ n } { ( a t q ) _ n ( q ) _ n } . \\end{align*}"}
+{"id": "2262.png", "formula": "\\begin{align*} \\tau _ i = ( \\alpha _ i z _ 1 + \\beta _ i ( z _ 2 ) , u _ i z _ 2 + v _ i ) , \\ \\alpha _ i , u _ i \\in k ^ * , v _ i \\in k , \\ \\beta _ i ( z _ 2 ) \\in k [ z _ 2 ] , d _ i : = \\deg ( \\beta _ i ) > 1 . \\end{align*}"}
+{"id": "1019.png", "formula": "\\begin{align*} \\alpha _ 2 = 0 \\ , \\Leftrightarrow \\ , a _ 2 = 0 \\ , \\Leftrightarrow \\ , \\ell _ t = 2 \\ , \\ell _ b \\ , \\Leftrightarrow \\ , \\beta = \\gamma 2 \\ , \\alpha _ 1 + 3 \\ , \\alpha _ 3 = 0 \\ , \\Leftrightarrow \\ , a _ 3 = 0 \\ , \\Leftrightarrow \\ , \\Psi = \\frac { 3 } { 2 } , \\end{align*}"}
+{"id": "5280.png", "formula": "\\begin{align*} L _ { \\pi } : = \\{ t \\in \\R ^ { m \\ * ( n + 1 ) } : \\sum _ { j = 1 } ^ { n + 1 } t _ { i ( n + 1 ) + j } = 0 , \\ \\forall \\ 0 \\leq i \\leq m - 1 ; \\ \\ \\sum _ { j \\in B } t _ j = 0 \\ \\forall B \\in \\pi \\} . \\end{align*}"}
+{"id": "2670.png", "formula": "\\begin{align*} \\binom { n } { k } \\sup _ { g \\in \\Pi _ 1 ^ { - 1 } ( U ) } ( \\phi ( g ) ) \\geq \\# \\Pi _ 1 ^ { - 1 } ( U ) \\sup _ { g \\in \\Pi _ 1 ^ { - 1 } ( U ) } ( \\phi ( g ) ) \\geq \\sum _ { g \\in \\Pi _ 1 ^ { - 1 } ( U ) } \\phi ( g ) . \\end{align*}"}
+{"id": "6130.png", "formula": "\\begin{align*} \\begin{aligned} | \\mathcal H ( d + 2 , d , 3 ) | - \\frac { 3 } { 2 } \\binom { n - d } { 2 } & = \\sum _ { i = d } ^ { d + 2 } \\binom { n - i } { 2 } + ( d - 1 ) ( n - k ) - \\frac { 3 } { 2 } \\binom { n - d } { 2 } \\\\ & = \\frac { ( n - d - 1 ) ( n - d - 4 ) } { 4 } + \\binom { n - d - 2 } { 2 } + ( d - 1 ) ( n - k ) > 0 , \\end{aligned} \\end{align*}"}
+{"id": "1362.png", "formula": "\\begin{align*} E [ u ] ( t ) + \\int _ 0 ^ t \\int _ { \\Omega } a ( x ) | \\partial _ t u ( s , x ) | ^ 2 \\ , d x d s & = E [ u ] ( 0 ) . \\end{align*}"}
+{"id": "2855.png", "formula": "\\begin{align*} X = ` ` \\varprojlim _ { i \\in I } \" X _ i , \\end{align*}"}
+{"id": "6508.png", "formula": "\\begin{align*} U : = ( 2 P - I ) ( 2 Q - I ) . \\end{align*}"}
+{"id": "5573.png", "formula": "\\begin{align*} \\dfrac { 1 } { \\Phi _ 1 [ \\alpha _ 0 ^ { * } ( \\beta _ 0 ) , \\beta _ 0 , u _ 1 ] } - \\dfrac { 2 l _ b \\gamma _ b ( \\alpha ^ { * } _ 0 ( \\beta _ 0 ) ) ^ { \\nu + 1 } } { \\theta _ b - \\theta _ m } = 0 \\end{align*}"}
+{"id": "7228.png", "formula": "\\begin{align*} v ( t , x ) : = v _ { a } ( t , x ) = \\sum _ { j } a _ { j } e ^ { i j x } e ^ { i j ^ { 2 } t } . \\end{align*}"}
+{"id": "8133.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac { 1 } { n } \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ^ { \\otimes n } \\otimes \\overline A ) = \\lim _ { n \\rightarrow + \\infty } \\frac { 1 } { n } \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ^ { \\otimes n } \\otimes \\overline A ^ { \\otimes p } ) . \\end{align*}"}
+{"id": "3344.png", "formula": "\\begin{align*} \\Lambda _ { C } = \\alpha [ \\Lambda _ { A } , \\Lambda _ { B } ] _ q + \\beta \\big ( \\Lambda _ { A \\cap B } \\Lambda _ { A \\cup B } + \\Lambda _ { A \\backslash B } \\Lambda _ { B \\backslash A } \\big ) , \\end{align*}"}
+{"id": "5028.png", "formula": "\\begin{align*} \\Lambda _ N ( \\Omega ) + \\varepsilon \\ge N \\Lambda _ 1 ( \\Omega ) = N h _ 1 ( \\Omega ) . \\end{align*}"}
+{"id": "108.png", "formula": "\\begin{align*} \\theta _ { \\Psi } ( r , \\xi ) = \\theta _ { \\Psi } ( r ) \\end{align*}"}
+{"id": "2112.png", "formula": "\\begin{align*} \\begin{bmatrix} A _ { 1 1 } & A _ { 1 2 } & 0 \\\\ A _ { 2 1 } & A _ { 2 2 } & 0 \\\\ A _ { 3 1 } & A _ { 3 2 } & 0 \\end{bmatrix} + \\begin{bmatrix} 0 & - D & 0 \\\\ 0 & 0 & I _ { \\ell - k } \\\\ 0 & 0 & 0 \\end{bmatrix} \\in Z ^ 0 , \\end{align*}"}
+{"id": "3233.png", "formula": "\\begin{align*} \\theta ^ { n } _ { \\varphi } = e ^ { \\varphi - u } \\theta ^ { n } _ { u } + e ^ { \\varphi - v } \\theta ^ { n } _ { v } . \\end{align*}"}
+{"id": "5516.png", "formula": "\\begin{align*} \\dim H _ l ( H , \\Pi _ 0 \\cdot \\chi _ H ^ { - 1 } ) \\leq \\sum _ { j = 1 } ^ { j _ 0 } \\sum _ { k = 0 } ^ \\infty d _ { l , j , k } . \\end{align*}"}
+{"id": "3686.png", "formula": "\\begin{align*} V _ i ' = V _ i \\cap N ( v ) \\cap \\left ( \\bigcap _ { u \\in U _ 1 \\setminus V _ i } N ( u ) \\right ) \\quad i \\in [ t ] \\cup \\{ t + 1 , t + 4 \\} . \\end{align*}"}
+{"id": "3638.png", "formula": "\\begin{align*} K ' ( x ) = e ^ { \\theta ( x - a ) } \\int _ 0 ^ \\infty [ h ' ( a + z ) + c ] \\nu ( d z ) + \\lambda c + 2 ( x - \\rho ) , x < a \\end{align*}"}
+{"id": "5419.png", "formula": "\\begin{align*} T : = S - \\Pi , ( \\Pi ) _ { a b } : = \\frac { 1 } { N } , 1 \\leq a , b \\leq N . \\end{align*}"}
+{"id": "1617.png", "formula": "\\begin{align*} S ( x ^ * , \\alpha ) = \\{ y \\in B _ X \\colon x ^ * ( y ) > 1 - \\alpha \\} \\end{align*}"}
+{"id": "7263.png", "formula": "\\begin{align*} \\int n d \\mu \\leq \\overline { { m d i m } } _ M ( T , f + n , d ) - \\over = n , \\end{align*}"}
+{"id": "4378.png", "formula": "\\begin{align*} ( \\mathcal { P } _ \\theta ) \\left \\{ \\begin{array} { c l } { \\rm M a x i m i z e } & \\sum _ { i = 1 } ^ { l } \\theta _ i J _ i ( x , u ) \\\\ { \\rm s u b j e c t \\ ; t o } & ( x , u ) \\in A d m ( { \\mathcal B } ) . \\end{array} \\right . \\end{align*}"}
+{"id": "3239.png", "formula": "\\begin{align*} d _ { p } ( x , y ) = \\inf \\left \\{ \\sum _ { i = 0 } \\rho ( x _ { n } , x _ { n + 1 } ) ^ { p } : x _ { 0 } , \\dots , x _ { n + 1 } \\in X x = x _ { 0 } , y = x _ { n + 1 } \\right \\} \\end{align*}"}
+{"id": "4777.png", "formula": "\\begin{align*} u \\diamond v : = \\rho ( T ( u ) ) v , u \\ast v : = \\mu ( T ( u ) ) v , \\forall u , v \\in V . \\end{align*}"}
+{"id": "3953.png", "formula": "\\begin{align*} \\mathrm { P r } \\{ \\xi ( t ) = n \\} = \\sum _ { \\Omega _ { n } } \\prod _ { j = 1 } ^ { n } \\frac { ( t \\lambda _ j ) ^ { x _ j } } { x _ j ! } e ^ { - \\sum _ { j = 1 } ^ { \\infty } t \\lambda _ j } . \\end{align*}"}
+{"id": "8195.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { Z } } \\varphi _ j ( \\xi ) = 1 , \\ \\ \\ \\ \\forall \\xi \\in \\mathbb { R } ^ N \\setminus \\{ 0 \\} , \\end{align*}"}
+{"id": "5386.png", "formula": "\\begin{align*} \\omega _ ( ( v _ 1 , \\eta _ 1 ) , ( v _ 2 , \\eta _ 2 ) ) = \\eta _ 2 ( v _ 1 ) - \\eta _ 1 ( v _ 2 ) . \\end{align*}"}
+{"id": "5651.png", "formula": "\\begin{align*} \\mathcal { L } ( X ) = \\left \\{ \\begin{array} { c c } g _ X ( X , \\cdot ) , & X \\not = 0 , \\\\ 0 , & X = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "1171.png", "formula": "\\begin{align*} j = j _ k = \\Big \\lfloor \\frac { \\omega _ k ( k + 1 ) } { 1 + \\omega _ k } \\Big \\rfloor . \\end{align*}"}
+{"id": "7714.png", "formula": "\\begin{align*} R i c [ g ^ + ] \\geq - n g ^ + \\ \\ a n d \\ \\ R [ g ^ + ] + n ( n + 1 ) = o ( e ^ { - 2 t } ) \\end{align*}"}
+{"id": "7927.png", "formula": "\\begin{align*} \\widetilde { \\delta } _ \\mathrm { H o c h } ( R _ 1 ) + \\Psi ^ 2 ( \\mu _ 1 ) = 0 . \\end{align*}"}
+{"id": "4287.png", "formula": "\\begin{align*} \\frac { 1 } { ( d q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { n ( - 1 ) ^ { n - 1 } d ^ n q ^ { n ( n + 1 ) / 2 } } { ( q ) _ n } + \\sum _ { n = 1 } ^ { \\infty } \\frac { d ^ n q ^ { n ( n + 1 ) } } { ( q ) _ n ( d q ) _ n ( 1 - q ^ n ) } = \\sum _ { n = 1 } ^ { \\infty } \\frac { ( d q ) ^ n } { ( q ) _ n ( 1 - q ^ n ) } . \\end{align*}"}
+{"id": "4514.png", "formula": "\\begin{align*} \\Omega ( \\mathbb { X } _ { \\Phi _ { \\ ! x } } , \\mathbb { Y } ) = \\dd \\Phi _ x ( \\mathbb { Y } ) \\ , , \\forall \\mathbb { Y } \\in H ^ 1 ( 0 , 1 ) \\times H ^ 1 ( 0 , 1 ) \\ , . \\end{align*}"}
+{"id": "831.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & A ( x ^ * ) ^ { k _ 1 } + p ( x ^ * ) = - \\frac { K } { \\bar { \\theta } _ 2 } , \\\\ & k _ 1 A ( x ^ * ) ^ { k _ 1 - 1 } + p ' ( x ^ * ) = 0 , \\\\ & A x ^ { k _ 1 } + p ( x ) + \\frac { K } { \\bar { \\theta } _ 2 } \\left ( \\frac { x ^ { * } } { x } \\right ) ^ { - k _ 1 } = \\frac { 1 } { l _ 2 } \\left [ \\frac { 1 - \\theta } { \\bar { \\theta } _ 2 - \\theta } - l _ 1 \\right ] \\end{aligned} \\right . \\end{align*}"}
+{"id": "5778.png", "formula": "\\begin{align*} a _ { s , t } \\le ( r ^ 2 + r + 1 ) s + \\left \\lfloor \\frac { ( r ^ 2 + r + 1 ) t } { r + 1 } \\right \\rfloor = ( r ^ 2 + r + 1 ) s + r t . \\end{align*}"}
+{"id": "5358.png", "formula": "\\begin{align*} \\nabla _ { [ i } W _ { j k ] } { } ^ { l m } = - 2 C _ { [ i j } { } ^ { [ l } g _ { k ] } { } ^ { m ] } \\end{align*}"}
+{"id": "3727.png", "formula": "\\begin{align*} h _ j ( x , t ) = \\begin{cases} 1 ( x , t ) \\in D _ j \\\\ 0 ( x , t ) \\in \\underset { m = j + 2 } { U ^ \\infty } ( D _ m \\setminus D _ { m - 1 } ) \\end{cases} \\end{align*}"}
+{"id": "4466.png", "formula": "\\begin{align*} \\langle [ E , Y ] , Y \\rangle = 0 , \\end{align*}"}
+{"id": "5323.png", "formula": "\\begin{align*} w _ { 0 } T _ { 2 k - 1 } = - q ^ { - 1 } w _ { 0 } . \\end{align*}"}
+{"id": "6598.png", "formula": "\\begin{align*} \\Gamma : = \\bigcup _ { j = 1 } ^ { M } \\Gamma _ { j } \\Gamma _ { j } = \\{ ( r , \\theta _ { j } ) : r \\geq 0 \\} , \\end{align*}"}
+{"id": "721.png", "formula": "\\begin{align*} \\rho { u _ x } = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { c _ { i x } } { f _ i } + \\frac { { \\Delta t } } { 2 } { F _ x } } , \\end{align*}"}
+{"id": "2351.png", "formula": "\\begin{align*} g _ { i } = f _ { a _ { i } } ( S ( x _ { a _ { 1 } } \\cdots x _ { a _ { i - 1 } } ) , x _ { a _ { i + 1 } } \\cdots x _ { a _ { n } } v ) \\end{align*}"}
+{"id": "7245.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u _ j + \\Delta _ { \\mathbb { R } ^ 2 } u _ j = \\sum \\limits _ { \\mathcal { R } ( j ) } u _ { j _ 1 } \\bar { u } _ { j _ 2 } u _ { j _ 3 } , \\\\ u _ j ( 0 ) = u _ { 0 , j } , \\end{cases} \\end{align*}"}
+{"id": "6525.png", "formula": "\\begin{align*} U = \\sum _ r e ^ { i \\theta _ r } E _ r = \\exp ( i H ) , \\end{align*}"}
+{"id": "4264.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { N - k } \\frac { \\left ( \\frac { d } { c } \\right ) _ j q ^ j \\left ( q ^ { - ( N - k ) } \\right ) _ j } { ( q ) _ j \\left ( q ^ { - ( N - k ) } / c \\right ) _ j } = \\frac { ( d q ) _ { N - k } } { ( c q ) _ { N - k } } . \\end{align*}"}
+{"id": "3482.png", "formula": "\\begin{align*} \\varphi ( \\mathrm { v } , \\mathrm { y } , \\mathrm { t } , \\tau ) : = \\displaystyle \\int _ { 0 } ^ { \\tau } \\dfrac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 ( \\mathrm { s } - \\tau ) ^ 2 } \\textbf { e x p } \\Big ( - \\dfrac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 ( \\mathrm { s } - \\tau ) } \\Big ) \\Phi ^ { \\textbf { e } } ( \\xi , \\mathrm { t } ; \\mathrm { z } , \\mathrm { s } ) d \\mathrm { s } . \\end{align*}"}
+{"id": "460.png", "formula": "\\begin{align*} - \\frac { d } { d z } \\coth z = \\frac { 1 } { \\sinh ^ 2 z } = \\frac { 1 } { z ^ 2 } - \\sum _ { n = 1 } ^ \\infty ( 2 n - 1 ) 2 ^ { 2 n } B _ { 2 n } \\frac { z ^ { 2 n - 2 } } { ( 2 n ) ! } \\ , . \\end{align*}"}
+{"id": "2028.png", "formula": "\\begin{align*} \\Psi ( t ) = W ( \\Delta _ t ) = W ( R _ t \\ast \\widetilde { R _ t } ) . \\end{align*}"}
+{"id": "6212.png", "formula": "\\begin{align*} & E ^ * _ { i + k - 1 } A _ 1 E ^ * _ { i + k - 2 } A _ 1 E ^ * _ { i + k - 3 } \\cdots E ^ * _ { i + 1 } A _ 1 E ^ * _ { i } = \\big ( ( \\frac { k - 2 } { 2 } ) ! \\big ) ^ 2 \\frac { k } { 2 } M ^ { \\frac { k - 2 } { 2 } , 0 } _ { \\frac { 2 m - i - k + 1 } { 2 } , \\frac { i - 1 } { 2 } } \\\\ & M ^ { 0 , 0 } _ { \\frac { i + k - 1 } { 2 } , \\frac { 2 m - i - k + 1 } { 2 } } M ^ { \\frac { k - 2 } { 2 } , 0 } _ { \\frac { 2 m - i - k + 1 } { 2 } , \\frac { i - 1 } { 2 } } = \\frac { k } { 2 } M ^ { \\frac { 2 m - k } { 2 } , \\frac { i - 1 } { 2 } } _ { \\frac { i + k - 1 } { 2 } , \\frac { i - 1 } { 2 } } \\end{align*}"}
+{"id": "3806.png", "formula": "\\begin{align*} 3 n - 9 & = 4 n - 8 - d , \\\\ 3 n - 7 & = 4 n - 8 - d , \\\\ 3 n - 5 & = 4 n - 8 - d , \\\\ 3 n - 3 & = 4 n - 8 - d . \\end{align*}"}
+{"id": "6787.png", "formula": "\\begin{align*} \\mathcal { E } ^ { ( 3 ) } [ \\psi _ t , \\varphi _ t ] & = \\norm { \\left ( - \\Delta + \\Phi _ { \\varphi _ t } + M \\right ) ^ { 3 / 2 } \\psi _ t } _ { L ^ 2 ( \\mathbb { R } ^ 3 ) } ^ 2 , \\end{align*}"}
+{"id": "4545.png", "formula": "\\begin{align*} \\alpha ( \\tilde { E } _ i ) = \\left ( \\frac { \\alpha \\wedge E ^ i } { w } \\right ) = \\left ( \\frac { \\alpha \\wedge E ^ i } { v } \\right ) \\left ( \\frac { v } { w } \\right ) = \\alpha ( \\hat { E } ^ i ) \\left ( \\frac { v } { w } \\right ) \\ , , \\end{align*}"}
+{"id": "1756.png", "formula": "\\begin{align*} w _ { n = k + m , k , m } \\left ( C \\mathcal { Z } , \\mathcal { Z } \\right ) = { \\left \\lfloor \\frac { k + m - 1 } { 2 } \\right \\rfloor \\choose \\frac { m } { 2 } } . \\end{align*}"}
+{"id": "7668.png", "formula": "\\begin{align*} \\bar p : = \\inf \\{ \\ , p \\ , : \\ , | E _ p | = | \\Omega | \\ , \\} , \\end{align*}"}
+{"id": "1400.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } E [ u ^ { ( j ) } ] ( t ) = E [ u ] ( t ) \\end{align*}"}
+{"id": "6852.png", "formula": "\\begin{align*} & ( \\beta \\lambda u _ 1 h ( a _ 1 ) f ^ { p ^ { e ' } } ( a _ 1 ) , \\dots , \\beta \\lambda u _ s h ( a _ s ) f ^ { p ^ { e ' } } ( a _ s ) , \\lambda u _ { s + 1 } h ( a _ { s + 1 } ) f ^ { p ^ { e ' } } ( a _ { s + 1 } ) , \\dots , \\lambda u _ n h ( a _ n ) f ^ { p ^ { e ' } } ( a _ n ) ) \\\\ = & ( u _ 1 g ( a _ 1 ) , \\dots , u _ s g ( a _ s ) , u _ { s + 1 } g ( a _ { s + 1 } ) , \\dots , u _ n g ( a _ n ) ) . \\end{align*}"}
+{"id": "2832.png", "formula": "\\begin{align*} d X _ s ^ n = \\gamma _ s ^ { - \\frac 1 2 } d ( u _ s ^ n - H _ s ^ n ) + ( u _ s ^ n - H _ s ^ n ) d \\gamma _ s ^ { - \\frac 1 2 } + d [ \\gamma ^ { - \\frac 1 2 } , u ^ n - H ^ n ] _ s . \\end{align*}"}
+{"id": "3347.png", "formula": "\\begin{gather*} \\Lambda _ { \\{ 1 , 4 \\} } = \\alpha \\big [ \\Lambda _ { \\{ 1 , 3 \\} } , \\Lambda _ { \\{ 3 , 4 \\} } \\big ] _ q + \\beta \\big ( \\Lambda _ { \\{ 3 \\} } \\Lambda _ { \\{ 1 , 3 , 4 \\} } + \\Lambda _ { \\{ 1 \\} } \\Lambda _ { \\{ 4 \\} } \\big ) \\end{gather*}"}
+{"id": "6515.png", "formula": "\\begin{align*} C = \\begin{pmatrix} 1 & 0 & 1 & 0 \\\\ 1 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 1 \\end{pmatrix} , \\hat { C } = \\begin{pmatrix} \\frac { 1 } { \\sqrt { 6 } } & 0 & \\frac { 1 } { 2 } & 0 \\\\ [ 3 m m ] \\frac { 1 } { \\sqrt { 6 } } & \\frac { 1 } { \\sqrt { 2 } } & 0 & 0 \\\\ [ 3 m m ] \\frac { 1 } { \\sqrt { 3 } } & 0 & 0 & 0 \\\\ [ 3 m m ] 0 & 0 & \\frac { 1 } { 2 } & \\frac { 1 } { \\sqrt { 2 } } \\end{pmatrix} . \\end{align*}"}
+{"id": "1289.png", "formula": "\\begin{align*} \\pi _ a ^ j \\circ \\iota _ a ^ i & = ( \\pi _ 1 ^ { \\lfloor \\frac { j } { n _ 2 } \\rfloor } \\otimes \\pi _ 2 ^ { j \\ ; \\mathrm { m o d } \\ ; n _ 2 } ) \\circ ( \\iota _ 1 ^ { \\lfloor \\frac { i } { n _ 2 } \\rfloor } \\otimes \\iota _ 2 ^ { i \\ ; \\mathrm { m o d } \\ ; n _ 2 } ) \\\\ & = ( \\pi _ 1 ^ { \\lfloor \\frac { j } { n _ 2 } \\rfloor } \\circ \\iota _ 1 ^ { \\lfloor \\frac { i } { n _ 2 } \\rfloor } ) \\otimes ( \\pi _ 2 ^ { j \\ ; \\mathrm { m o d } \\ ; n _ 2 } \\circ \\iota _ 2 ^ { i \\ ; \\mathrm { m o d } \\ ; n _ 2 } ) , \\end{align*}"}
+{"id": "3077.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ { k } ) = F _ X ( \\sigma ^ { 2 k } ) = \\infty \\end{align*}"}
+{"id": "400.png", "formula": "\\begin{align*} \\Lambda _ 1 ^ n T ^ * E = \\{ \\eta \\in \\Lambda ^ n T ^ * E ~ | ~ \\iota _ u \\iota _ v \\eta = 0 \\ ; \\ ; \\forall u , v \\in T _ { } E \\} ~ . \\end{align*}"}
+{"id": "7576.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } D _ { V , h } ( \\mu ^ { \\varepsilon } ) = D _ { V , h } ( \\mu ) \\end{align*}"}
+{"id": "2993.png", "formula": "\\begin{align*} & \\alpha = \\beta a ( t ) = b ( t ) t \\ge 0 \\ ; , \\\\ & \\alpha \\ne \\beta a ( t ) \\ne b ( t ) t \\ge 0 \\ ; , \\end{align*}"}
+{"id": "2728.png", "formula": "\\begin{align*} A u t ( I ^ { \\times k } ) \\cong \\left ( \\prod _ { i = 1 } ^ k A u t ( I ) \\right ) \\rtimes S y m ( k ) \\cong ( \\mathbb { Z } / 2 \\mathbb { Z } ) ^ k \\rtimes S y m ( k ) . \\end{align*}"}
+{"id": "5803.png", "formula": "\\begin{align*} \\omega ( | A | ) = \\| | A | \\| = \\| A \\| = \\| | A ^ { * } | \\| = \\omega ( | A ^ { * } | ) . \\end{align*}"}
+{"id": "3158.png", "formula": "\\begin{align*} m ^ { 1 } z ^ { 1 } \\ , + \\ , m ^ { 2 } z ^ { 2 } \\ , + \\ , m ^ { 3 } z ^ { 3 } \\ , + \\ , m ^ { 4 } z ^ { 4 } \\ , \\ , = \\ , \\ , 0 \\end{align*}"}
+{"id": "1588.png", "formula": "\\begin{align*} \\tilde { T } : W _ S & \\to V = W _ { b _ 1 } \\oplus W _ { b _ 2 } \\oplus W _ { b _ 3 } \\\\ w = ( w _ e ) _ { e \\in S } & \\mapsto ( \\tilde { T } _ 1 ( w ) , \\tilde { T } _ 2 ( w ) , \\tilde { T } _ 2 ( w ) ) \\end{align*}"}
+{"id": "4769.png", "formula": "\\begin{gather*} R ( a ) \\cdot R ( b ) = R ( R ( a ) \\cdot b + a \\cdot R ( b ) ) , \\\\ [ R ( a ) , R ( b ) ] = R ( [ R ( a ) , b ] + [ a , R ( b ) ] ) , \\forall a , b \\in A . \\end{gather*}"}
+{"id": "7379.png", "formula": "\\begin{align*} ( \\nu _ 1 + i \\nu _ 2 ) \\big ( f _ { + } - f _ { - } \\big ) = - \\alpha \\big ( \\partial _ { \\overline { z } } f _ { + } + \\partial _ { \\overline { z } } f _ { - } \\big ) \\partial _ { \\overline { z } } f _ { + } = \\partial _ { \\overline { z } } f _ { - } \\Sigma , \\end{align*}"}
+{"id": "6206.png", "formula": "\\begin{align*} E ^ * _ { i + k } A _ 1 E ^ * _ { i + k - 1 } A _ 1 E ^ * _ { i + k - 2 } \\cdots E ^ * _ { i + 1 } A _ 1 E ^ * _ { i } = \\big ( ( \\frac { k } { 2 } ) ! \\big ) ^ 2 M ^ { \\frac { 2 m - k } { 2 } , \\frac { i - 1 } { 2 } } _ { \\frac { i + k - 1 } { 2 } , \\frac { i - 1 } { 2 } } . \\end{align*}"}
+{"id": "5981.png", "formula": "\\begin{align*} W _ { 0 } ^ { ( q , r ) } ( x ) = W ^ { ( q + r ) } ( x ) , Z _ { 0 } ^ { ( q , r ) } ( x ) = Z ^ { ( q + r ) } ( x ) . \\end{align*}"}
+{"id": "4679.png", "formula": "\\begin{align*} \\phi _ { n } ^ { \\lambda } ( z ) & = \\epsilon _ n r ^ { n } \\left [ \\frac { n + 2 \\lambda } { 2 \\lambda } P _ { n } ^ { \\lambda } ( \\cos \\theta ) + i \\sin \\theta P _ { n - 1 } ^ { \\lambda + 1 } ( \\cos \\theta ) \\right ] , \\\\ \\bar { z } \\overline { \\phi _ { n - 1 } ^ { \\lambda } ( z ) } & = \\epsilon _ { n - 1 } r ^ { n } \\left [ \\frac { n } { 2 \\lambda } P _ { n } ^ { \\lambda } ( \\cos \\theta ) - i \\sin \\theta P _ { n - 1 } ^ { \\lambda + 1 } ( \\cos \\theta ) \\right ] , \\end{align*}"}
+{"id": "4446.png", "formula": "\\begin{align*} \\Gamma - \\lim _ { n \\to \\infty } J _ n & = \\Gamma - \\lim _ { n \\to \\infty } J . \\end{align*}"}
+{"id": "7591.png", "formula": "\\begin{align*} \\Delta f = \\frac { 4 } { \\rho _ U ( z ) ^ 2 } \\frac { \\partial } { \\partial z } \\frac { \\partial } { \\partial \\overline { z } } f , \\end{align*}"}
+{"id": "2376.png", "formula": "\\begin{align*} d I ( z _ { 0 } ; z _ { 1 } , \\dots , z _ { n } ; z _ { n + 1 } ) = \\sum _ { i = 1 } ^ { n } d \\log \\left ( \\frac { z _ { i + 1 } - z _ { i } } { z _ { i } - z _ { i - 1 } } \\right ) I ( z _ { 0 } ; z _ { 1 } , \\dots , \\widehat { z _ { i } } , \\dots , z _ { n } ; z _ { n + 1 } ) . \\end{align*}"}
+{"id": "6108.png", "formula": "\\begin{align*} | \\mathcal G ( k , d ) | & = \\binom { n - d + 1 } { k - d + 1 } - \\binom { n - k + 1 } { k - d + 1 } + \\binom { n - k - 1 } { k - d - 1 } + 2 ( n - k ) + d - 3 \\sim ( k - d ) \\binom { n } { k - d } . \\end{align*}"}
+{"id": "2407.png", "formula": "\\begin{align*} { \\rm C u t } _ { ( \\emptyset ; 1 ) } ( \\psi \\mid _ { z \\to 0 } ) = ( \\psi \\mid _ { z \\to 0 } ) \\cdot ( u + v ) . \\end{align*}"}
+{"id": "2559.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\nu _ t ^ { * , \\xi } = & [ ( A - B ^ 2 R ^ { - 1 } P _ t ) \\nu _ t ^ { * , \\xi } - B ^ 2 R ^ { - 1 } \\varphi _ t ^ { * , \\xi } - B h ( \\mu _ t ^ { * , \\xi } ) + f ( \\nu _ t ^ { * , \\xi } ) + b ( \\mu _ t ^ { * , \\xi } ) ] d t + \\sigma _ 0 d W ^ 0 _ t , \\\\ \\nu _ 0 ^ { * , \\xi } = & ~ \\mathbb { E } [ \\xi ] . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2595.png", "formula": "\\begin{align*} \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) ) } ^ { N , i } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } h \\Big ( \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) ) } ^ { N , j } \\Big ) = - R ^ { - 1 } B \\Big [ P _ t \\nu ^ { N , i } _ { \\boldsymbol { x } } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } \\phi ^ { N , j } ( t , \\boldsymbol { x } ) \\Big ] . \\end{align*}"}
+{"id": "6416.png", "formula": "\\begin{align*} j _ F ^ p ( x ) = ( j _ \\rho ^ p ( x _ 1 ) , \\ldots , j _ \\rho ^ p ( x _ n ) ) . \\end{align*}"}
+{"id": "1300.png", "formula": "\\begin{align*} \\lambda _ { a \\otimes b } = ( \\lambda _ a \\otimes b ) \\circ \\alpha _ { I , a , b } , \\end{align*}"}
+{"id": "1535.png", "formula": "\\begin{align*} g _ { s ^ { - 1 } , k , i } \\circ g _ { e , j , k } \\circ g _ { s , i , j } = \\mathrm { i d } _ { b _ i } = g _ { s ^ { - 1 } , k , i } \\circ g _ { s , j , k } \\circ g _ { e , i , j } \\end{align*}"}
+{"id": "4078.png", "formula": "\\begin{align*} x _ 0 ^ * \\mathcal { V } \\stackrel { \\simeq } { \\longrightarrow } ( ] x [ ^ * ( \\mathcal { V } ) ^ { \\nabla = 0 } \\stackrel { \\simeq } { \\longrightarrow } x _ 1 ^ * \\mathcal { V } \\end{align*}"}
+{"id": "5050.png", "formula": "\\begin{align*} S _ { \\chi } ( m , n ; q ) = \\sum _ { \\substack { a \\bmod { q } \\\\ \\gcd ( a , q ) = 1 } } \\chi ( a ) e \\ ! \\left ( \\frac { m a + n \\bar { a } } { q } \\right ) \\end{align*}"}
+{"id": "8147.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac 1 n d ( \\xi _ n ^ Y , \\xi _ { n \\varphi ^ Y } ) = \\lim _ { n \\rightarrow + \\infty } \\frac 1 n \\int _ { \\omega \\in \\Omega } d _ \\omega ( \\xi _ n ^ Y , \\xi _ { n \\varphi ^ Y } ) \\ , \\nu ( \\mathrm { d } \\omega ) . \\end{align*}"}
+{"id": "6437.png", "formula": "\\begin{align*} f ( 0 , y _ 1 , x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) = \\sum _ { n _ 1 = 0 } ^ { \\infty } c _ { n _ 1 } ( x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) y _ 1 ^ { n _ 1 } . \\end{align*}"}
+{"id": "1332.png", "formula": "\\begin{align*} \\int _ { B _ { 9 / 1 0 } } \\big ( \\left | \\nabla u ( x ) \\right | ^ p - \\left | \\nabla v ( x ) \\right | ^ p \\big ) \\ , d x \\le \\left | \\left \\{ u = 0 \\right \\} \\cap B _ { 9 / 1 0 } \\right | + \\sigma J _ p ( v , B _ 1 ) . \\end{align*}"}
+{"id": "4815.png", "formula": "\\begin{align*} \\frac { \\partial \\rho } { \\partial t } = \\nabla \\cdot ( \\rho \\nabla E _ s [ \\rho ] ) + K ( \\bar { \\rho } \\otimes \\mu - \\rho ) \\end{align*}"}
+{"id": "5857.png", "formula": "\\begin{align*} | \\langle A \\hat { k } _ { \\lambda } , \\hat { k } _ { \\lambda } \\rangle | ^ p { ( 1 + 2 ^ { p - 1 } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } ) } < \\frac { 1 } { 2 } ( \\frac { p } { p - 1 } ) ^ { p } \\langle ( | A | ^ { 2 p \\alpha } + | A ^ * | ^ { 2 p ( 1 - \\alpha ) } ) \\hat { k } _ { \\lambda } , \\hat { k } _ { \\lambda } \\rangle \\end{align*}"}
+{"id": "5760.png", "formula": "\\begin{align*} [ \\Omega _ A ] \\cdot [ \\omega ' _ A ] = 0 . \\end{align*}"}
+{"id": "5402.png", "formula": "\\begin{align*} M _ 1 = N ^ 4 \\operatorname { l o g } ( 2 N ( 1 + | | V | | _ { \\infty } ) ) + N ( | | V | | _ 1 ^ { 1 / 2 } + [ V ] _ { 1 / 2 } ^ { 1 / 2 } + 1 ) . \\end{align*}"}
+{"id": "7010.png", "formula": "\\begin{align*} L _ w f ( g ) : = - \\int \\langle \\nabla f , \\nabla g \\rangle d m _ w , g \\in W ^ { 1 , 2 } \\cap L ^ { \\infty } . \\end{align*}"}
+{"id": "480.png", "formula": "\\begin{align*} B _ n \\Big ( \\frac { \\beta ^ j } { L _ j } \\Big ) = ( - 1 ) ^ n B _ n \\Big ( \\frac { \\alpha ^ j } { L _ j } \\Big ) . \\end{align*}"}
+{"id": "370.png", "formula": "\\begin{align*} \\mathcal { N } ( I _ \\mu ) = \\left \\lbrace f \\in C ^ \\infty ( M ) ~ ~ \\Big \\vert ~ ~ \\lbrace f , I _ \\mu \\rbrace \\subset I _ \\mu \\right \\rbrace . \\end{align*}"}
+{"id": "7813.png", "formula": "\\begin{align*} J ^ { w _ 1 } ( X ) & = - \\frac { 1 } { 2 } \\int _ { - \\infty } ^ { \\infty } w _ 1 ( x ) f ^ 2 ( x ) d x \\\\ & = - \\frac { 1 } { 2 } E ( \\Lambda _ X ^ { w _ 1 } ( U ) ) , \\end{align*}"}
+{"id": "381.png", "formula": "\\begin{align*} \\L _ \\xi ( \\mu - \\phi ) _ \\zeta = ( \\mu - \\phi ) _ { [ \\xi , \\zeta ] } \\end{align*}"}
+{"id": "7585.png", "formula": "\\begin{align*} h ( s ) = \\frac { 1 } { z - s } \\end{align*}"}
+{"id": "2005.png", "formula": "\\begin{align*} \\lim _ { y \\to + \\infty } \\frac { Q ( i y ) } { y } = 0 . \\end{align*}"}
+{"id": "2814.png", "formula": "\\begin{align*} d \\nu _ s ^ { - 1 } = - \\nu _ s ^ { - 1 } d R _ s , s \\in [ t , T ] , \\nu _ t ^ { - 1 } = 1 . \\end{align*}"}
+{"id": "6109.png", "formula": "\\begin{align*} \\mathcal S ( k , 3 ) = & \\left \\{ F \\in \\binom { [ n ] } { k } : [ 2 ] \\subseteq F , \\ , F \\cap [ 3 , k - 1 ] \\neq \\emptyset \\right \\} \\cup \\left \\{ F \\in \\binom { [ n ] } { k } : [ 2 ] \\cup [ k , k + 2 ] \\subseteq F \\right \\} \\\\ & \\cup \\left ( \\bigcup _ { i = 1 } ^ { 2 } \\left \\{ F \\in \\binom { [ n ] } { k } : \\{ i \\} \\cup [ 3 , k - 1 ] \\subseteq F , \\ , | [ k , k + 2 ] \\cap F | = 2 \\right \\} \\right ) , \\end{align*}"}
+{"id": "5053.png", "formula": "\\begin{align*} L ( s , f \\times \\bar { g } ) = \\zeta ^ { ( N ) } ( 2 s ) \\sum _ { m = 1 } ^ \\infty \\frac { f _ m \\overline { \\rho _ g ( m ) } } { m ^ s } \\quad \\quad \\zeta ^ { ( N ) } ( s ) = \\zeta ( s ) \\prod _ { p \\mid N } \\bigl ( 1 - p ^ { - s } \\bigr ) . \\end{align*}"}
+{"id": "2856.png", "formula": "\\begin{align*} Y = ` ` \\varinjlim _ { j \\in J } \" Y _ j \\end{align*}"}
+{"id": "2829.png", "formula": "\\begin{align*} d ( \\alpha _ s \\beta _ s ) = - \\alpha _ s X _ s d ( \\nu _ s \\gamma _ s ) + \\beta _ s d ( \\gamma _ s ^ { - \\frac 1 2 } \\nu _ s ^ { - 1 } ) - X _ s d [ \\gamma ^ { - \\frac 1 2 } \\nu ^ { - 1 } , \\nu \\gamma ] _ s , s \\in [ t , T ] . \\end{align*}"}
+{"id": "6713.png", "formula": "\\begin{align*} D _ { i j } = \\partial _ { x _ { j } } u _ { i } + \\partial _ { x _ { i } } u _ { j } - \\frac { 2 } { 3 } \\delta _ { i j } \\nabla _ { x } \\cdot u , \\end{align*}"}
+{"id": "7531.png", "formula": "\\begin{align*} \\mathbb V H ( x , y ) = \\frac { 1 } { 2 } \\left ( ( a + i b ) \\frac { \\partial f ( z ) } { \\partial z } + ( a - i b ) \\frac { \\partial \\overline { f ( z ) } } { \\partial \\overline z } \\right ) = \\Re \\left ( \\mathbb V ^ { 1 , 0 } f \\right ) . \\end{align*}"}
+{"id": "156.png", "formula": "\\begin{align*} \\begin{aligned} \\tfrac { \\partial \\nu ( v ) } { \\partial | v | } \\geq 0 \\ \\textrm { f o r } 0 \\leq \\gamma \\leq 1 \\ , , \\textrm { a n d } \\tfrac { \\partial \\nu ( v ) } { \\partial | v | } < 0 \\ \\textrm { f o r } - 3 < \\gamma < 0 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "2482.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = v ^ { p } & & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta v & = f ( | \\nabla u | ) & & \\quad \\mbox { i n } \\Omega . \\end{aligned} \\right . \\end{align*}"}
+{"id": "39.png", "formula": "\\begin{align*} \\P \\left \\{ Y = 2 ^ { - i } \\right \\} = r 2 ^ i , \\end{align*}"}
+{"id": "4402.png", "formula": "\\begin{align*} = r ^ r \\int _ { ( 0 , \\frac { 1 } { r } ) ^ r } \\sum _ { k = 1 } ^ r ( - 1 ) ^ { k + 1 } \\binom { r } { k } u ( x , k t ( \\sum _ { i = 1 } ^ r z _ i ) ) d z _ 1 \\dots d z _ r . \\end{align*}"}
+{"id": "4966.png", "formula": "\\begin{align*} \\begin{aligned} \\min \\limits _ { r } & r \\\\ s . t . & { \\rm C _ { 2 } } , \\\\ & R _ { \\rm F A } ( \\lambda ^ { \\star } , 2 , r ) \\leq \\tau \\end{aligned} \\end{align*}"}
+{"id": "3738.png", "formula": "\\begin{align*} A ^ 2 + 4 A - 1 2 I _ { 6 0 } = 9 J _ { 6 0 } + I _ 3 \\otimes J _ { 2 0 } . \\end{align*}"}
+{"id": "8243.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi ( t , \\tau ) & : = \\sum _ { j \\le j _ 0 } \\sum _ { j ' \\le j + 4 } 2 ^ { j ' ( \\frac { N } { 2 } + 1 ) } \\Vert \\dot \\Delta _ { j ' } u ( \\tau ) \\Vert _ { L ^ 2 } \\psi _ j ( t , \\tau ) \\\\ & + \\sum _ { j \\le j _ 0 } \\sum _ { j ' > j - 4 } 2 ^ { ( j - j ' ) ( \\frac { N } { 2 } + \\bar { s } + \\delta s \\alpha ) } 2 ^ { j ' ( \\frac { N } { 2 } + 1 ) } \\Vert \\dot \\Delta _ { j ' } u ( \\tau ) \\Vert _ { L ^ 2 } \\psi _ j ( t , \\tau ) , \\end{aligned} \\end{align*}"}
+{"id": "8215.png", "formula": "\\begin{align*} T _ v \\cdot \\nabla g : = \\sum _ { i = 1 } ^ N T _ { v _ i } \\partial _ { x _ i } g = \\sum _ { i = 1 } ^ N \\sum _ { j \\in \\mathbb { Z } } \\dot { S } _ { j - 1 } v _ i \\dot \\Delta _ j \\partial _ { x _ i } g . \\end{align*}"}
+{"id": "3851.png", "formula": "\\begin{align*} \\Omega _ 0 : = \\left [ 1 - \\frac { n ^ { \\tau } } { \\sqrt { n } } , ~ 1 + \\frac { n ^ { \\tau } } { \\sqrt { n } } \\right ] \\times \\left [ - \\frac { n ^ { \\tau / 2 } } { n ^ { 1 / 4 } } , ~ \\frac { n ^ { \\tau / 2 } } { n ^ { 1 / 4 } } \\right ] , \\end{align*}"}
+{"id": "2480.png", "formula": "\\begin{align*} - \\Delta F _ * + F _ * = J _ * \\end{align*}"}
+{"id": "2901.png", "formula": "\\begin{align*} - \\frac { 1 } { \\sqrt { 1 + { f ' } ^ 2 } } = \\frac { \\eta } { m + 1 } f ^ { m + 1 } + \\lambda f + c , \\end{align*}"}
+{"id": "6269.png", "formula": "\\begin{align*} T _ \\lambda ( u ) = \\begin{cases} 0 , & \\nabla h ( u ) = \\lambda v , \\\\ - t ^ * \\norm { H _ r \\big ( p _ \\lambda ( u ) \\big ) } ^ { - 1 } H _ r \\big ( p _ \\lambda ( u ) \\big ) , & \\nabla h ( u ) \\neq \\lambda v , \\end{cases} \\end{align*}"}
+{"id": "2192.png", "formula": "\\begin{align*} & H ^ f _ { 1 1 1 } = \\frac { \\partial H ^ f _ { 1 1 } } { \\partial x _ 1 } - 2 \\Gamma ^ 1 _ { 1 1 } H ^ f _ { 1 1 } - 2 \\Gamma ^ 2 _ { 1 1 } H ^ f _ { 1 2 } , \\\\ & H ^ f _ { 1 2 1 } = \\frac { \\partial H ^ f _ { 1 2 } } { \\partial x _ 1 } - \\Gamma ^ 1 _ { 1 1 } H ^ f _ { 1 2 } - \\Gamma ^ 2 _ { 1 1 } H ^ f _ { 2 2 } - \\Gamma ^ 1 _ { 1 2 } H ^ f _ { 1 1 } - \\Gamma ^ 2 _ { 1 2 } H ^ f _ { 1 2 } , \\\\ & H ^ f _ { 2 2 1 } = \\frac { \\partial H ^ f _ { 2 2 } } { \\partial x _ 1 } - 2 \\Gamma ^ 1 _ { 1 2 } H ^ f _ { 1 2 } - 2 \\Gamma ^ 2 _ { 1 2 } H ^ f _ { 2 2 } . \\end{align*}"}
+{"id": "2660.png", "formula": "\\begin{align*} \\Gamma = \\bigoplus _ { n = 1 } ^ \\infty \\Gamma _ n , F = \\prod _ { n = 1 } ^ \\infty F _ n \\quad G = \\Gamma \\rtimes F \\end{align*}"}
+{"id": "7872.png", "formula": "\\begin{align*} \\# \\{ f \\in F _ B ( x ) : \\Delta ( f , k _ x ) \\le \\Delta _ x \\} = n _ 3 ( \\Delta _ x , x ) , \\end{align*}"}
+{"id": "5944.png", "formula": "\\begin{align*} T _ { 0 , K } & = \\{ x \\in \\Q _ q ^ k \\colon | x \\cdot \\gamma ^ { ( j ) } ( a ) | \\leq \\delta ^ { - j } \\} \\\\ & = \\{ x \\in \\Q _ q ^ k \\colon M _ a ^ { T } x \\in T _ { \\delta } \\} = M _ { a } ^ { - T } T _ { \\delta } \\end{align*}"}
+{"id": "1727.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to \\infty } \\sup _ { \\xi \\in \\mathcal { C } _ p } | \\beta _ { \\tau , m } ^ \\xi ( \\zeta ) | = 0 \\ , \\ , . \\end{align*}"}
+{"id": "2591.png", "formula": "\\begin{align*} \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) ) } ^ { N , i } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } h \\Big ( \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) ) } ^ { N , j } \\Big ) = - R ^ { - 1 } B \\Big [ P _ t \\nu ^ { N , i } _ { \\boldsymbol { x } } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } \\Phi ( t , \\nu ^ { N , j } _ { \\boldsymbol { x } } ) \\Big ] \\end{align*}"}
+{"id": "1729.png", "formula": "\\begin{align*} \\zeta ( t ) = \\begin{cases} 1 & \\ , | t | \\leq 1 \\\\ 0 & \\ , | t | > 2 . \\end{cases} \\end{align*}"}
+{"id": "1430.png", "formula": "\\begin{align*} \\mathcal { U } _ { N L } ( t ) & : = \\int _ 0 ^ t U ( t - s ) \\begin{pmatrix} 0 \\\\ - | u ( s ) | ^ { p - 1 } u ( s ) \\end{pmatrix} \\ , d s \\\\ & \\in C ( [ 0 , T ] ; D ( \\mathcal { A } ) ) \\cap C ^ 1 ( [ 0 , T ] ; \\mathcal { H } ) . \\end{align*}"}
+{"id": "6282.png", "formula": "\\begin{align*} \\varphi \\big ( \\varrho _ 1 \\cdot ( \\varrho _ 2 \\varrho _ \\infty \\varrho _ 2 ^ { - 1 } \\varrho _ \\infty ^ { - 1 } ) \\big ) = \\varphi ( \\varrho _ 1 ) ; \\end{align*}"}
+{"id": "3624.png", "formula": "\\begin{align*} { 1 \\over 2 } \\sigma ^ 2 e ^ { - \\theta x } f '' ( x ) + e ^ { - \\theta x } ( x - \\rho ) ^ 2 - ( 1 + \\lambda ) e ^ { - \\theta x } f ( x ) + \\theta \\int _ x ^ b f ( z ) e ^ { - \\theta z } d z + \\zeta = 0 , \\end{align*}"}
+{"id": "2271.png", "formula": "\\begin{align*} \\alpha \\beta = ( \\alpha _ 1 \\beta _ 1 , \\alpha _ 2 \\beta _ 2 , \\dots , \\alpha _ n \\beta _ n ) , \\ \\forall \\alpha , \\beta \\in ( k ^ * ) ^ n \\end{align*}"}
+{"id": "5557.png", "formula": "\\begin{align*} L _ 1 ( u _ 1 ( \\alpha _ 0 ) ) u _ 1 ' ( \\alpha _ 0 ) = - 2 l _ b \\gamma _ b \\alpha _ 0 / ( \\theta _ b - \\theta _ m ) , \\end{align*}"}
+{"id": "7686.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { | a _ n | ^ 2 + | b _ n | ^ 2 } { c _ { 2 / p } ( n ) } \\le \\frac { 1 } { 1 - | \\cos { \\pi } / { p } | } \\| f \\| ^ 2 _ { p } , \\end{align*}"}
+{"id": "5154.png", "formula": "\\begin{align*} G _ { \\delta } = & \\left \\{ x \\in \\partial \\Omega \\ , : \\ , \\exists r ^ x _ i \\searrow 0 \\ ; \\ ; \\ ; \\dfrac { | A \\cap B ( x , r ^ x _ i ) | } { | B ( x , r ^ x _ i ) \\cap \\Omega | } < 1 - \\delta \\right . \\\\ & \\left . \\dfrac { | A ' \\cap B ( x , r ) | } { | B ( x , r ) \\cap \\Omega | } > \\frac 1 2 0 < r < \\delta \\right \\} . \\end{align*}"}
+{"id": "1900.png", "formula": "\\begin{align*} r = \\frac { 2 q } { 2 q - 3 } . \\end{align*}"}
+{"id": "8240.png", "formula": "\\begin{align*} Z _ s ^ h ( t ) : = t ^ s \\Big ( \\Vert \\sigma ^ h ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } } _ { 2 , 1 } } + \\Vert u ^ h ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } } \\Big ) = t ^ s \\sum _ { j > j _ 0 } \\Big ( 2 ^ { j \\frac { N } { 2 } } \\| \\dot \\Delta _ j \\sigma \\| _ { L ^ 2 } + 2 ^ { j ( \\frac { N } { 2 } + 1 - \\alpha ) } \\| \\dot \\Delta _ j u \\| _ { L ^ 2 } \\Big ) , \\end{align*}"}
+{"id": "3583.png", "formula": "\\begin{align*} \\mathcal { H } = \\left \\{ f ( x ) = \\sum _ { i \\geq 1 } f _ i \\phi _ i ( x ) , x \\in [ 0 , 1 ] : \\sum _ { i } \\frac { f _ i ^ 2 } { a _ i } < \\infty \\right \\} , \\end{align*}"}
+{"id": "322.png", "formula": "\\begin{align*} p ( Y _ t | \\theta , h ) = & \\int p ( Y _ t | X _ t ) p ( X _ t | \\theta , h ) d X _ t \\\\ = & \\int p ( Y _ t | X _ t ) \\prod _ { t = 1 } ^ { T } p ( x _ t | x _ { t - 1 } , \\theta , h ) p ( x _ 0 ) d x _ 0 \\cdots d x _ t \\\\ = & \\int p ( Y _ t | X _ t ) \\prod _ { t = 1 } ^ { T } p ( x _ t | x _ { t - 1 } , \\theta ) p ( \\theta | h ) p ( x _ 0 ) d x _ 0 \\cdots d x _ t \\end{align*}"}
+{"id": "4870.png", "formula": "\\begin{align*} \\zeta _ { C B } ( - k ) = \\frac { 1 } { 3 } \\left ( \\frac { 2 } { 3 } \\right ) ^ k p _ k \\left ( \\frac { 1 } { 4 } \\right ) + \\frac { 1 } { 3 } \\left ( \\frac { 2 } { 3 } \\right ) ^ { k + 1 } q _ k \\left ( \\frac { 1 } { 4 } \\right ) \\frac { \\pi } { \\sqrt { 3 } } \\in \\Q + \\Q \\frac { \\pi } { \\sqrt { 3 } } . \\end{align*}"}
+{"id": "6036.png", "formula": "\\begin{align*} & \\ell _ 1 ^ \\perp : X _ 4 = X _ 5 = X _ 6 . \\end{align*}"}
+{"id": "6978.png", "formula": "\\begin{align*} \\rho _ U ( \\epsilon ( \\gamma - 1 ) ) & = \\rho _ U ( \\epsilon \\gamma - 1 - ( \\epsilon - 1 ) ) = p _ J ( \\psi _ U ( \\epsilon \\gamma ) ) - p _ J ( \\psi _ U ( \\epsilon ) ) \\\\ & = p _ { \\Gamma } ( \\psi _ U ( \\epsilon ) ) p _ { J } ( \\psi _ U ( \\gamma ) ) + p _ J ( \\psi _ U ( \\epsilon ) ) - p _ J ( \\psi _ U ( \\epsilon ) ) \\\\ & = \\epsilon p _ J ( \\psi _ U ( \\gamma ) ) = \\epsilon \\rho _ U ( \\gamma ) . \\end{align*}"}
+{"id": "5942.png", "formula": "\\begin{align*} M _ a = M _ b \\left ( \\begin{array} { c c c c } 1 & 0 & \\dots & 0 \\\\ ( 1 ! ) ^ { - 1 } ( b - a ) & 1 & \\dots & 0 \\\\ ( 2 ! ) ^ { - 1 } ( b - a ) ^ 2 & ( 1 ! ) ^ { - 1 } ( b - a ) & \\dots & 0 \\\\ \\vdots & & \\ddots & \\\\ ( ( k - 1 ) ! ) ^ { - 1 } ( b - a ) ^ { k - 1 } & ( ( k - 2 ) ! ) ^ { - 1 } ( b - a ) ^ { k - 2 } & \\dots & 1 \\end{array} \\right ) \\end{align*}"}
+{"id": "4935.png", "formula": "\\begin{align*} x + y = \\binom { n - 1 } { ( k - 1 ) / 2 + t } + \\cdots + \\binom { n - 1 } { k - 1 } . \\end{align*}"}
+{"id": "1196.png", "formula": "\\begin{align*} e _ { 2 , i k } \\ ! + \\ ! \\gamma _ { i k } \\frac { \\sum _ { j \\neq i } | e ^ { ( n ) } _ { 1 , j i k } | ^ { 2 } } { ( 1 + \\nu _ { i k } \\gamma _ { i k } ) ^ { 2 } } \\ ! - \\ ! 2 \\Re { \\{ e _ { 3 , i k } e ^ { ( n ) } _ { 1 , i i k } \\} } \\ ! - \\ ! 2 \\nu _ { i k } | e _ { 3 , i k } | ^ { 2 } \\ ! = \\ ! 0 . \\end{align*}"}
+{"id": "4765.png", "formula": "\\begin{gather*} [ r _ { 1 2 } , r _ { 1 3 } ] = \\sum _ { i , j } [ a _ i , a _ j ] \\otimes b _ i \\otimes b _ j , [ r _ { 1 3 } , r _ { 2 3 } ] = \\sum _ { i , j } a _ i \\otimes a _ j \\otimes [ b _ i , b _ j ] , \\\\ [ r _ { 1 2 } , r _ { 2 3 } ] = \\sum _ { i , j } a _ i \\otimes [ b _ i , a _ j ] \\otimes b _ j . \\end{gather*}"}
+{"id": "4130.png", "formula": "\\begin{align*} \\int _ t ^ { \\varepsilon _ n ^ \\alpha } a ( s ) U ( s ) U ' ( s ) d s = \\frac { ( U ' ) ^ 2 ( t ) } { 2 } - \\frac { ( U ' ) ^ 2 ( \\varepsilon _ n ^ \\alpha ) } { 2 } \\end{align*}"}
+{"id": "5477.png", "formula": "\\begin{align*} { } _ 3 \\phi _ 2 \\begin{bmatrix} q ^ { - N } , & a , & b & ; q , q \\\\ c , & q ^ { 1 - N } / t \\end{bmatrix} = \\frac { ( a t ) _ n } { ( t ) _ n } { } _ 3 \\phi _ 2 \\begin{bmatrix} q ^ { - N } , & c / b , & a & ; q , b t q ^ { N } \\\\ c , & a t \\end{bmatrix} \\end{align*}"}
+{"id": "665.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = - \\frac { 1 } { 2 } \\sqrt { | c _ 1 | } \\ X _ 1 \\cdot \\nu _ 1 \\cdot \\varphi - \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi \\end{align*}"}
+{"id": "8191.png", "formula": "\\begin{align*} \\rho - 1 \\in C _ b ( [ 0 , + \\infty ) ; \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } ) , u \\in C _ { b } ( [ 0 , + \\infty ) ; \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } ) \\cap L ^ 1 ( \\mathbb { R } ^ + ; { \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } } ) . \\end{align*}"}
+{"id": "676.png", "formula": "\\begin{align*} 2 / \\mu \\ | h _ z | ^ 2 = \\mu / 2 \\ | T | ^ 2 = \\mu / 2 \\ ( 1 - \\nu ^ 2 ) = \\mu / 2 - \\tau _ 0 / { 2 \\mu } \\end{align*}"}
+{"id": "1042.png", "formula": "\\begin{align*} { \\mathcal { v } } ( f ) : = \\mathcal { W } \\ , ( f \\Delta ^ { - m } ) = v _ { n - 1 } \\int _ { M } f ~ v o l _ { g } , \\end{align*}"}
+{"id": "1545.png", "formula": "\\begin{align*} & \\varphi _ { s , i , j } : \\Omega _ { b _ { i } } \\times \\Omega \\rightarrow \\Omega _ { b _ { j } } \\times \\Omega \\\\ & \\varphi _ { s , i , j } \\left ( \\omega _ { i } , \\omega \\right ) = \\left ( f _ { s , i , j } \\left ( \\omega _ { i , } X _ { s _ { i } } \\left ( \\omega \\right ) \\right ) , \\omega \\right ) \\end{align*}"}
+{"id": "7014.png", "formula": "\\begin{align*} L ^ { a c } _ w f \\cdot m _ w = e ^ { w } \\left ( L ^ { a c } f + \\langle \\nabla w , \\nabla f \\rangle \\right ) \\cdot m \\geq e ^ { - \\| w \\| _ { L ^ \\infty } } \\varepsilon _ 0 \\cdot m > 0 \\end{align*}"}
+{"id": "1503.png", "formula": "\\begin{align*} G ( w ) = \\sum _ { p \\le w } g ( p ) ( \\log p ) ^ 2 . \\end{align*}"}
+{"id": "2440.png", "formula": "\\begin{align*} \\mathfrak { r } \\bigl ( \\Phi ^ { \\circ } \\bigr ) & = \\mu ^ { 2 } \\Phi ^ { 2 } . \\end{align*}"}
+{"id": "1209.png", "formula": "\\begin{align*} | \\mathcal { A } | = 2 ^ { \\frac { 8 ( 8 + 1 ) } { 2 } } \\cdot \\prod _ { i = 1 } ^ 5 \\left ( \\prod _ { j = 2 } ^ { s _ i } \\left ( 2 ^ j - 1 \\right ) \\right ) = 2 ^ { 3 6 } \\cdot ( 3 ) \\cdot ( 3 \\cdot 7 ) = 6 3 \\cdot 2 ^ { 3 6 } \\end{align*}"}
+{"id": "5365.png", "formula": "\\begin{align*} A B C : = I _ { 0 1 2 } . \\end{align*}"}
+{"id": "6774.png", "formula": "\\begin{align*} G _ { x } ( k ) & = e ^ { - 2 i \\pi k x } \\abs { k } ^ { - 1 } , \\end{align*}"}
+{"id": "3050.png", "formula": "\\begin{align*} F _ { \\hat { X } _ { \\infty } } \\left ( \\hat { \\sigma } _ { \\infty } ^ k \\right ) = 0 . \\end{align*}"}
+{"id": "81.png", "formula": "\\begin{align*} [ \\Delta _ { \\Psi } , X ] = 0 , \\O \\end{align*}"}
+{"id": "5253.png", "formula": "\\begin{align*} f ( N \\ * x ) = f ( N \\ * x _ 1 , \\ldots , N \\ * x _ n ) = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } \\ * N ^ { n } } \\ * \\sum _ { k \\in \\mathbb { Z } ^ n } \\ * \\hat { f } ( k _ 1 \\ * N ^ { - 1 } , \\ldots , k _ n \\ * N ^ { - 1 } ) e ^ { i \\ * k \\cdot x } , \\end{align*}"}
+{"id": "1972.png", "formula": "\\begin{align*} | w _ { n , R } ( x ) | & \\leq 2 ^ { - d } | R | ^ { 1 / 2 } \\prod _ { i = 1 } ^ d \\bigg ( | g _ { I _ i } \\bigl ( | I _ i | ( x _ i + e _ { n _ i , I _ i } ) \\bigr ) | + | g _ { I _ i } \\bigl ( | I _ i | ( x _ i - e _ { n _ i , I _ { i } } ) | \\bigr ) \\bigg ) \\\\ & \\leq 2 ^ { - d } | R | ^ { 1 / 2 } \\sum _ { m = 1 } ^ { 2 ^ d } | G _ R ( \\Delta ( x + U _ m e _ { n , R } ) ) | , \\end{align*}"}
+{"id": "728.png", "formula": "\\begin{align*} { f _ { 1 6 } } = { f _ { 1 5 } } - f _ { 1 5 } ^ { \\left ( { e q } \\right ) } + f _ { 1 6 } ^ { \\left ( { e q } \\right ) } - \\delta y - \\delta z , \\end{align*}"}
+{"id": "4990.png", "formula": "\\begin{align*} K _ j ( \\gamma ) : = \\frac { ( 2 ^ j ) j ! } { ( 2 j ) ! } \\frac { 1 } { \\gamma ^ j } \\int _ { \\gamma } ^ { \\infty } e ^ { - \\lambda } ( \\lambda ^ 2 - \\gamma ^ 2 ) ^ { j - 1 / 2 } d \\lambda , \\quad ( j \\geq 0 ) . \\end{align*}"}
+{"id": "1813.png", "formula": "\\begin{align*} [ r _ { 1 2 } , r _ { 1 3 } ] + [ r _ { 1 2 } , r _ { 2 3 } ] + [ r _ { 1 3 } , r _ { 2 3 } ] = 0 , \\end{align*}"}
+{"id": "5241.png", "formula": "\\begin{align*} f ( a , b ) = \\bigg { \\{ } \\begin{array} { l r } \\chi _ { m a x \\{ a , b \\} } & \\ a \\neq b \\\\ \\chi _ { [ 0 , a ] } & \\ a = b \\end{array} \\end{align*}"}
+{"id": "1613.png", "formula": "\\begin{align*} \\left \\{ n \\in \\mathbb { N } \\mid x _ { n } = x _ { n } ^ { \\prime } \\right \\} \\in \\mathcal { U } . \\end{align*}"}
+{"id": "6054.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta p - k ^ 2 p = 2 ( \\eta - \\eta _ 0 ) - 2 \\Delta \\eta + 2 \\nabla \\cdot A \\ , \\ , \\ , \\ , \\Omega \\backslash K \\\\ [ 0 . 3 c m ] \\frac { \\partial p } { \\partial n } = 0 \\ , \\ , \\ , \\ , \\partial K \\\\ [ 0 . 3 c m ] p = 0 \\ , \\ , \\ , \\ , \\partial \\Omega \\end{cases} \\end{align*}"}
+{"id": "1051.png", "formula": "\\begin{align*} \\begin{aligned} { \\mathfrak a } _ { 2 } = & \\bigl ( \\delta _ { a b } + \\frac { 1 } { 3 } R _ { a c b d } x ^ { c } x ^ { d } \\bigr ) \\xi _ { a } \\xi _ { b } + o ( { \\mathbf { x ^ { 2 } } } ) , \\\\ { \\mathfrak a } _ { 1 } = & \\frac { 2 i } { 3 } \\mathrm { R i c } _ { a b } x ^ { a } \\xi _ { b } + o ( { \\mathbf { x ^ { 2 } } } ) . \\end{aligned} \\end{align*}"}
+{"id": "6049.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\eta ' = ( k ^ 2 ) ' \\eta + k ^ 2 \\eta ' \\ , \\ , \\ , \\ , \\Omega \\\\ [ 0 . 3 c m ] \\frac { \\partial \\eta ' } { \\partial n } = \\left ( - \\frac { \\partial ^ 2 \\eta } { \\partial n ^ 2 } \\right ) V \\cdot n + \\nabla \\eta \\cdot \\nabla _ \\Gamma ( V \\cdot n ) \\ , \\ , \\ , \\ , \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "5013.png", "formula": "\\begin{align*} \\mathrm { e } ^ { \\mathrm { i } \\vartheta } ( z _ 1 , z _ 2 , y ) & : = ( \\mathrm { e } ^ { \\mathrm { i } \\vartheta } z _ 1 , \\mathrm { e } ^ { \\mathrm { i } \\vartheta } z _ 2 , y ) , \\qquad \\qquad \\tau ( z _ { 1 } , z _ { 2 } , y ) : = ( z _ { 2 } , z _ { 1 } , y ) , \\\\ \\alpha ( z _ 1 , z _ 2 , y ) & : = ( z _ 1 , z _ 2 , \\alpha y ) \\quad \\alpha \\in O ( N - 4 ) . \\end{align*}"}
+{"id": "1244.png", "formula": "\\begin{align*} T _ k ( s ) : = & \\max ( - k , \\min ( s , k ) ) . \\end{align*}"}
+{"id": "3961.png", "formula": "\\begin{align*} \\mathcal { L } \\left ( t ^ { \\beta - 1 } E ^ { \\gamma } _ { \\alpha , \\beta } \\left ( x t ^ { \\alpha } \\right ) ; s \\right ) = \\frac { s ^ { \\alpha \\gamma - \\beta } } { ( s ^ { \\alpha } - x ) ^ { \\gamma } } , \\ s > | x | ^ { 1 / \\alpha } . \\end{align*}"}
+{"id": "5840.png", "formula": "\\begin{align*} \\| S \\| _ { b e r } = \\sup \\{ \\| S e _ i \\| : i \\in \\mathbb { Z } _ { + } \\} = 1 = \\| S \\| \\end{align*}"}
+{"id": "6980.png", "formula": "\\begin{align*} - \\int _ { X } \\langle \\nabla f , \\nabla g \\rangle d m = \\int _ { X } h g d m , g \\in W ^ { 1 , 2 } . \\end{align*}"}
+{"id": "1970.png", "formula": "\\begin{align*} P _ R = b _ R ( D ) \\Bigl [ \\bigotimes _ { j = 1 } ^ d ( \\operatorname { I d } + R _ { \\alpha _ { I _ j } } - R _ { \\alpha _ { I _ j } ' } ) \\Bigr ] b _ R ( D ) , \\end{align*}"}
+{"id": "1921.png", "formula": "\\begin{align*} P _ 0 u _ { k , j } + \\lambda u _ { k , j } = u _ k P _ 0 \\zeta _ j + \\div ( \\vec f _ k \\zeta _ j ) - \\vec f _ k \\cdot D _ v \\zeta _ j + g _ k \\zeta _ j - 2 ( a D _ v \\zeta _ j ) \\cdot D _ v u _ k . \\end{align*}"}
+{"id": "2952.png", "formula": "\\begin{align*} J ( \\bar x ; u ) : = \\{ i \\in J ( F ( \\bar x ) ) \\ , | \\ , \\nabla F ( \\bar x ) u \\in T _ { P _ i } ( F ( \\bar x ) ) \\} . \\end{align*}"}
+{"id": "6366.png", "formula": "\\begin{align*} R _ H ( t , s ) = \\int _ 0 ^ { t \\wedge s } K _ H ( t , r ) K _ H ( s , r ) \\d r , \\end{align*}"}
+{"id": "2787.png", "formula": "\\begin{align*} \\| r _ i ^ \\mathrm { S V D } \\| _ 2 = \\beta _ k | e _ k ^ T x _ i | . \\end{align*}"}
+{"id": "4641.png", "formula": "\\begin{align*} \\pi _ k : = \\left \\{ ( \\vec { r } _ k , \\vec { \\rho } _ k ) \\ ! \\in \\mathbb { R } ^ 6 \\ , \\big | \\ ; \\vec { r } _ k \\ ! = \\vec { 0 } \\right \\} , \\pi : = \\textstyle { \\bigcup _ { k = 1 } ^ 3 \\pi _ k } \\end{align*}"}
+{"id": "3360.png", "formula": "\\begin{align*} \\mu ( ( x _ \\alpha ) _ \\alpha , ( y _ i ) _ i , z , ( t _ { i j } ) _ { i , j } ) = \\mu f \\left ( x _ 1 , ( \\tilde { y } _ i ) _ i \\right ) . \\end{align*}"}
+{"id": "5827.png", "formula": "\\begin{align*} \\Pi T \\Pi ^ * = M _ z . \\end{align*}"}
+{"id": "2852.png", "formula": "\\begin{align*} \\Xi _ ! : = \\pi _ \\dag ( \\Xi _ ! ^ \\wedge ) , \\Xi _ * : = \\pi _ \\dag ( \\Xi _ * ^ \\wedge ) , \\end{align*}"}
+{"id": "6527.png", "formula": "\\begin{align*} \\log ( e ^ { i \\theta _ r } ) = \\log ( e ^ { i \\theta _ r + 2 k _ r \\pi } ) \\end{align*}"}
+{"id": "4049.png", "formula": "\\begin{align*} \\mathcal { R } _ s ( q ) = I ^ { - 2 } ( s ) y ^ { 2 k - 2 } \\left ( \\alpha _ 1 + \\alpha _ 2 y ^ { 2 k } e _ b + ( \\alpha _ 3 + \\alpha _ 4 y ^ { 2 k } e _ b ) q \\right ) , \\end{align*}"}
+{"id": "7021.png", "formula": "\\begin{align*} \\| w ( t , \\cdot ) \\| _ { L ^ 2 } ^ 2 \\simeq \\begin{cases} t ^ { 2 - \\frac { n } { 2 } } & \\mbox { i f } \\ \\ n \\leqslant 3 , \\\\ \\ln t & \\mbox { i f } \\ \\ n = 4 , \\end{cases} \\end{align*}"}
+{"id": "667.png", "formula": "\\begin{align*} S = \\left ( \\begin{array} { c c } 1 / 2 + \\alpha & \\beta \\\\ \\beta & 1 / 2 - \\alpha \\end{array} \\right ) \\end{align*}"}
+{"id": "383.png", "formula": "\\begin{align*} \\tilde { l } _ k ( \\sigma _ N ^ 1 , \\dots , \\sigma _ N ^ k ) = \\tilde { l } _ k ( \\sigma ^ 1 , \\dots , \\sigma ^ k ) _ N \\end{align*}"}
+{"id": "5207.png", "formula": "\\begin{align*} & ( 2 M + C + 1 ) + \\left ( ( M - 1 ) ( M + C + 2 ) + \\frac { ( M - 1 ) ( M - 2 ) } { 2 } \\right ) + ( n M ) \\\\ & = M ^ 2 + M + C M + \\frac { M ^ 2 + M } { 2 } + n M > z , \\end{align*}"}
+{"id": "4699.png", "formula": "\\begin{align*} \\Phi _ { r , \\theta } ( s , \\varphi ) = \\frac { \\left ( 1 - r s + | \\sin \\theta | + | \\sin \\varphi | \\right ) ^ { - 2 \\lambda } } { \\left ( 1 - r s + \\left | \\sin ( \\theta - \\varphi ) / 2 \\right | \\right ) ^ { 2 } } . \\end{align*}"}
+{"id": "8230.png", "formula": "\\begin{align*} \\sigma = \\bar { \\sigma } + \\tilde { \\sigma } \\in \\widetilde { L } ^ { \\infty } _ { T } ( \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } ) \\cap L ^ 1 _ T ( \\widetilde { B } ^ { \\frac { N } { 2 } + 1 , \\frac { N } { 2 } + 2 - \\alpha } ) , \\ u = \\bar { u } + u _ L \\in \\widetilde { L } ^ { \\infty } _ { T } ( \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } ) \\cap L ^ 1 _ { T } ( \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } ) . \\end{align*}"}
+{"id": "5370.png", "formula": "\\begin{align*} \\varphi = \\varphi ( \\mathsf { t } E ) . \\end{align*}"}
+{"id": "1753.png", "formula": "\\begin{align*} Q = \\sum \\partial \\sigma _ { i _ { 1 } , i _ { 1 } + 1 , \\dots , i _ { \\frac { m } { 2 } - 1 } , i _ { \\frac { m } { 2 } - 1 } + 1 , n , n + 1 , 1 } , \\end{align*}"}
+{"id": "4390.png", "formula": "\\begin{align*} w ( x ) : = 2 ^ { 2 \\varrho } \\sinh ^ { 2 \\alpha + 1 } x \\cosh ^ { 2 \\beta + 1 } x . \\end{align*}"}
+{"id": "1611.png", "formula": "\\begin{align*} \\ker T _ { S } = \\bigcap _ { e \\in S } W _ { e } ^ { 0 } = \\left ( \\sum _ { e \\in S } W _ { e } \\right ) ^ { 0 } \\end{align*}"}
+{"id": "7209.png", "formula": "\\begin{align*} \\left [ \\mathcal { O } _ { \\mathcal { Z } } ( \\chi ) \\right ] & = \\left [ \\mathcal { O } _ { \\mathcal { Z } } \\right ] _ 2 G _ 0 ^ { \\mathrm { t o p } } ( \\mathcal { Z } ) , \\\\ S \\left ( \\left [ \\mathcal { O } _ { \\mathcal { Z } } \\right ] \\right ) & = 1 _ { \\mathcal { Z } } H ^ { \\mathrm { B M } } _ 2 ( \\mathcal { Z } ) . \\end{align*}"}
+{"id": "6510.png", "formula": "\\begin{align*} U = \\exp ( i H ) . \\end{align*}"}
+{"id": "7951.png", "formula": "\\begin{align*} Z ( t ) = - 2 \\left \\langle J ( Z ) , \\xi ^ H \\right \\rangle \\qquad \\mbox { f o r a n y } Z \\in \\mathcal { H } , \\end{align*}"}
+{"id": "8006.png", "formula": "\\begin{align*} G : \\ : \\overline { r a n } ( C \\otimes I _ d ) \\rightarrow \\overline { r a n } ( A \\otimes I _ d ) , \\mbox { s u c h t h a t } G & = ( A \\otimes I _ d ) ^ { 1 / 2 } ( B \\otimes I _ d ) ( C \\otimes I _ d ) ^ { 1 / 2 } \\\\ & = ( A ^ { 1 / 2 } \\otimes I _ d ) ( B \\otimes I _ d ) ( C ^ { 1 / 2 } \\otimes I _ d ) \\\\ & = ( ( A ^ { 1 / 2 } B ) \\otimes I _ d ) ( C ^ { 1 / 2 } \\otimes I _ d ) \\\\ & = ( A ^ { 1 / 2 } B C ^ { 1 / 2 } ) \\otimes I _ d \\\\ & = G _ M \\otimes I _ d \\end{align*}"}
+{"id": "845.png", "formula": "\\begin{align*} \\mathcal { J } _ i ( \\tau _ i , \\tau _ { - i } ) = \\mathbb { E } \\left \\{ \\frac { e ^ { - \\beta \\tau _ i } x _ i ( \\tau _ i ) } { \\frac { 1 } { N } \\sum \\limits _ { j = 1 , j \\neq i } ^ N e ^ { - \\beta \\tau _ j } x _ j ( \\tau _ j ) } - e ^ { - \\beta \\tau _ i } K \\right \\} . \\end{align*}"}
+{"id": "3707.png", "formula": "\\begin{align*} F _ n ( \\beta ) : = n \\mathbb E \\sum _ { i = 0 } ^ { n - 1 } ( \\triangle q _ i ^ \\beta ) ^ 2 \\end{align*}"}
+{"id": "1164.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k } \\dbinom { k } { j } ^ 2 a ^ { p _ k ( k - j ) / k } b ^ { p _ k j / k } \\leq ( a + b ) ^ { p _ k } , \\end{align*}"}
+{"id": "6921.png", "formula": "\\begin{align*} [ q ^ d ] s _ { \\lambda _ k } = \\left [ q ^ d \\right ] e _ { N + 1 } ^ { k } e _ { N } ^ { d - k } = ( - 1 ) ^ { k N + ( d - k ) ( N - 1 ) } y ^ k . \\end{align*}"}
+{"id": "6640.png", "formula": "\\begin{align*} \\Lambda _ n ( Q ^ { 1 + \\varepsilon } _ \\theta ) & = ( 1 + \\varepsilon ) ^ 2 \\Lambda _ n ( Q ^ { 1 } _ \\theta ) = ( 1 + \\varepsilon ) ^ 2 \\Big [ - \\dfrac { 1 } { ( 2 n - 1 ) ^ 2 \\theta ^ 2 } + O ( 1 ) \\Big ] , \\end{align*}"}
+{"id": "5393.png", "formula": "\\begin{align*} \\mathcal { U } _ 0 - A _ 0 = \\Im d _ A \\oplus ( \\ker d _ { A _ 0 } ^ \\star ) _ N = \\Im ( d _ A \\vert _ D ) \\oplus \\ker ( d _ { A _ 0 } ) . \\end{align*}"}
+{"id": "1137.png", "formula": "\\begin{align*} \\begin{cases} x _ { 2 n + 1 } & = ( 1 - \\alpha _ n ) T x _ { 2 n } + \\alpha _ n u , \\\\ x _ { 2 n + 2 } & = ( 1 - \\beta _ n ) U x _ { 2 n + 1 } + \\beta _ n x _ { 2 n + 1 } . \\end{cases} \\end{align*}"}
+{"id": "3904.png", "formula": "\\begin{align*} & u _ n ' ( t ) + \\lambda _ n ( 1 + D _ t ^ { \\{ m \\} } ) u _ n ( t ) = F _ n ( t ) , \\\\ & u _ n ( 0 ) = \\xi _ n : = ( \\xi , e _ n ) . \\end{align*}"}
+{"id": "1670.png", "formula": "\\begin{align*} \\kappa ^ 2 _ { \\mathsf { A } } - \\kappa ^ 2 _ { \\mathsf { J } _ { \\mathsf { f } - 1 } } = \\sum \\limits _ { \\mathsf { g } = 1 } ^ { \\mathsf { f } - 1 } \\big [ \\kappa _ { \\mathsf { J } _ { \\mathsf { g } - 1 } } ^ 2 - \\kappa _ { { \\mathsf { J } _ { \\mathsf { g } } } } ^ 2 \\big ] \\leq \\frac { \\mathsf { f } - 1 } { \\mathsf { F } } \\ , \\big [ \\kappa ^ { 2 } _ { \\mathsf { A } } - \\kappa ^ { 2 } _ { \\mathsf { B } } \\big ] \\ , . \\end{align*}"}
+{"id": "7243.png", "formula": "\\begin{align*} \\| \\vec { u } \\| ^ 2 _ { \\tilde { X } _ { k _ 0 } ( [ 0 , T ] ) } : = \\sup _ { 0 \\leq j \\leq k _ 0 } \\sup _ { G _ k ^ { j } \\subset [ 0 , T ] } \\| \\vec { u } \\| ^ 2 _ { X ( G _ k ^ { j } ) } . \\end{align*}"}
+{"id": "357.png", "formula": "\\begin{align*} \\sup _ { t \\geq 0 } \\| v _ { \\delta _ 0 } ( t ) \\| _ { L ^ 2 } < \\varepsilon / 2 , \\ ; \\delta _ 0 = \\frac { 1 } { 2 } \\min _ { 1 \\leq i \\leq 4 } \\delta _ i . \\end{align*}"}
+{"id": "2274.png", "formula": "\\begin{align*} \\| M \\| : = \\max _ { 1 \\leq i , j \\leq n } | M [ i , j ] | . \\end{align*}"}
+{"id": "1536.png", "formula": "\\begin{align*} F ' ( f ) \\coloneqq \\begin{cases} F ( g _ { e , j , 1 } \\circ f \\circ g _ { e , 1 , i } ) & \\mbox { i f } i \\neq 1 \\mbox { a n d } j \\neq 1 , \\\\ F ( g _ { e , j , 1 } \\circ f ) & \\mbox { i f } i = 1 , \\\\ F ( f \\circ g _ { e , 1 , i } ) & \\mbox { i f } j = 1 . \\end{cases} \\end{align*}"}
+{"id": "7438.png", "formula": "\\begin{align*} \\left ( A - P \\right ) \\left ( \\begin{matrix} \\sigma _ 1 \\\\ \\sigma _ 2 \\\\ \\ldots \\\\ \\sigma _ n \\\\ \\end{matrix} \\right ) = 0 . \\end{align*}"}
+{"id": "3580.png", "formula": "\\begin{align*} \\sup _ { i , l } \\frac { 1 } { \\mu _ i \\mu _ l } \\left | \\sum _ j \\mu _ j \\langle \\phi _ i , \\psi _ j \\rangle \\langle \\phi _ l , \\psi _ j \\rangle \\right | ^ 2 & = \\sup _ { i , l } \\frac { 1 } { \\mu _ i \\mu _ l } \\left | \\sum _ j \\mu _ j \\langle \\phi _ i , \\phi _ j \\rangle \\langle \\phi _ l , \\phi _ j \\rangle \\right | ^ 2 \\\\ & = \\sup _ { i } \\frac { 1 } { \\mu _ i ^ 2 } \\left | \\sum _ j \\mu _ j \\langle \\phi _ i , \\phi _ j \\rangle ^ 2 \\right | ^ 2 = 1 . \\end{align*}"}
+{"id": "2686.png", "formula": "\\begin{align*} M _ { S _ 3 } ( 2 n , j , 0 ) = ( - 1 ) ^ j \\binom { 2 n } { n } \\binom { 2 j } { j } \\binom { 2 ( n - j ) } { n - j } \\end{align*}"}
+{"id": "5961.png", "formula": "\\begin{align*} \\int _ { Q } & | f | ^ p \\leq C ( \\log \\delta ^ { - 1 } ) ^ { 3 p } \\times \\\\ & \\left [ D _ p ( \\delta ^ { 1 - \\varepsilon } ) ^ p + q ^ { \\frac { p } { 2 } + \\frac { k ^ 2 + 7 k - 4 } { 2 } } \\delta ^ { - ( k ^ 2 + 4 k - 2 ) \\varepsilon } \\delta ^ { - \\frac { 1 } { k } ( \\frac { p } { 2 } + \\frac { k ( k - 3 ) } { 2 } ) } D _ { p - 2 k } ( \\delta ^ { 1 - \\frac { 1 } { k } } ) ^ { p - 2 k } \\right ] \\Big ( \\sum _ { K \\in P _ { \\delta _ 0 } } \\| f _ K \\| _ { L ^ p ( Q ) } ^ 2 \\Big ) ^ { p / 2 } . \\end{align*}"}
+{"id": "7938.png", "formula": "\\begin{align*} \\overline { \\mu } _ t = \\mu + \\overline { \\mu } _ 2 t ^ 2 + \\cdots ~ ~ ~ ~ ~ ~ ~ ~ \\overline { R } _ t = R + \\overline { R } _ 2 t ^ 2 + \\cdots . \\end{align*}"}
+{"id": "2362.png", "formula": "\\begin{align*} U ( b ) : = \\overbrace { e _ { 0 } e _ { 1 } e _ { 0 } e _ { 1 } \\cdots e _ { \\iota ^ { b } ( 1 ) } } ^ { b } + \\overbrace { e _ { 1 } e _ { 0 } e _ { 1 } e _ { 0 } \\cdots e _ { \\iota ^ { b } ( 0 ) } } ^ { b } \\in \\mathcal { A } . \\end{align*}"}
+{"id": "2668.png", "formula": "\\begin{align*} 1 = \\langle \\varpi \\otimes \\varpi ( \\cdot ) \\xi ' _ i , \\xi ' _ i \\rangle i . \\end{align*}"}
+{"id": "5231.png", "formula": "\\begin{align*} D = \\begin{pmatrix} d _ 1 & d d _ 1 \\\\ 0 & d _ 2 \\end{pmatrix} \\in \\operatorname { M a t } ( C ^ \\flat \\otimes _ { \\mathbb { F } _ p } \\mathbb { F } ) \\end{align*}"}
+{"id": "5295.png", "formula": "\\begin{align*} f _ A ( x ) = \\sum _ { a \\in A } e ^ { 2 \\pi i a \\cdot x } . \\end{align*}"}
+{"id": "7090.png", "formula": "\\begin{align*} p _ { \\varpi } ( P , Q ) = \\begin{cases} 1 & P = Q , \\\\ 0 & P \\neq Q \\varpi _ x ( P ) = \\varpi _ x ( Q ) , \\end{cases} \\end{align*}"}
+{"id": "254.png", "formula": "\\begin{align*} \\mathsf { f } = \\mathsf { f } ( y , \\mathsf { m } ) = 2 \\mathsf { m } \\pi \\frac { \\mathsf { H } ( \\| y \\| ) } { \\mathsf { H } ( b ) } , \\mathsf { H } ( r ) = \\int _ a ^ r \\frac { d z } { z ^ { n + 1 } H ( z , z ^ 2 ) } . \\end{align*}"}
+{"id": "1281.png", "formula": "\\begin{align*} f _ { \\ast \\ast } = f , ( f _ \\ast ) ^ \\ast = ( f ^ \\ast ) _ \\ast . \\end{align*}"}
+{"id": "627.png", "formula": "\\begin{align*} \\Gamma \\left ( X , Y , Z \\right ) \\equiv \\left ( \\nabla _ { Z } G \\right ) \\left ( X , Y \\right ) = Z G \\left ( X , Y \\right ) - G \\left ( \\nabla _ { Z } X , Y \\right ) - G \\left ( X , \\nabla _ { Z } Y \\right ) . \\end{align*}"}
+{"id": "7708.png", "formula": "\\begin{gather*} \\int _ { B _ r } | \\tilde { \\omega } _ { i _ 1 \\dots i _ k } ( x ^ \\prime , x ^ n ) | d x = \\int _ { B _ r \\cap \\mathbb { H } ^ n } | \\omega _ { i _ 1 \\dots i _ k } ( x ^ \\prime , x ^ n ) | d x + \\int _ { B _ r \\cap \\mathbb { H } ^ n _ { - } } | \\omega _ { i _ 1 \\dots i _ k } ( x ^ \\prime , - x ^ n ) | d x \\\\ = 2 \\int _ { B _ r \\cap \\mathbb { H } ^ n } | \\omega _ { i _ 1 \\dots i _ k } ( x ) | d x \\end{gather*}"}
+{"id": "3155.png", "formula": "\\begin{align*} S _ { 1 , \\lambda } : = \\inf _ { u \\in H ^ { 1 } \\left ( \\mathbb { B } ^ { N } \\right ) \\backslash \\{ 0 \\} } J _ { \\infty } ( u ) , S _ { m , \\lambda } : = m ^ { \\frac { p - 1 } { p + 1 } } S _ { 1 , \\lambda } , m = 2 , 3 , 4 , \\cdots \\end{align*}"}
+{"id": "3799.png", "formula": "\\begin{align*} a _ 1 + n - 1 + n = 2 n - 2 - a _ 1 + d + 2 n - 2 , \\end{align*}"}
+{"id": "7880.png", "formula": "\\begin{align*} \\mathcal G ( R ) = \\{ f \\in F _ B : ( f , R ) \\in \\mathcal G \\} . \\end{align*}"}
+{"id": "7652.png", "formula": "\\begin{align*} I ( V ) : = \\inf \\{ \\ , P ( F ) \\ , : \\ , F \\subset \\Omega \\ , , | F | = V \\ , \\} . \\end{align*}"}
+{"id": "757.png", "formula": "\\begin{align*} V _ { \\alpha , \\beta } = \\mathbb { C } [ \\partial ] v , L _ \\lambda v = ( \\partial + \\alpha \\lambda + \\beta ) v . \\end{align*}"}
+{"id": "5330.png", "formula": "\\begin{align*} G = T \\times C _ { p _ 1 } \\times \\cdots \\times C _ { p _ k } , \\end{align*}"}
+{"id": "90.png", "formula": "\\begin{align*} X _ { j } ( u ) ( x _ { 0 } ) = 0 , \\forall j = 1 , \\cdots , k , \\ \\ \\ \\forall x _ 0 \\in \\bar \\O . \\end{align*}"}
+{"id": "5376.png", "formula": "\\begin{align*} \\omega ( \\mathbb X _ 1 , \\mathbb X _ 2 ) = \\langle \\langle \\mathbb { X } _ 1 , j \\mathbb { X } _ 2 \\rangle \\rangle . \\end{align*}"}
+{"id": "2579.png", "formula": "\\begin{align*} \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } ^ * _ t } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } h ( \\mu ^ { N , j } _ { \\boldsymbol { \\alpha } ^ * _ t } ) = \\Delta _ { * , t } ^ { N , i } : = - R ^ { - 1 } B \\lambda ^ { N , i } _ { \\boldsymbol { y } _ t ^ { * } } , \\end{align*}"}
+{"id": "7855.png", "formula": "\\begin{align*} \\Gamma _ z = \\{ ( \\theta , z \\cdot \\gamma ( \\theta ) : \\theta \\in I \\} . \\end{align*}"}
+{"id": "4872.png", "formula": "\\begin{align*} A _ n ( x ) = \\sum _ { \\pi \\in \\mathfrak { S } _ n } x ^ { \\mathrm { e x c } ( \\pi ) } = \\sum _ { k = 0 } ^ { n - 1 } A ( n , k ) x ^ k . \\end{align*}"}
+{"id": "2465.png", "formula": "\\begin{align*} J ( u _ { | M \\setminus K } ) = J ( u ) _ { | M \\setminus V } = 0 \\end{align*}"}
+{"id": "6843.png", "formula": "\\begin{align*} ( v _ 1 ^ { p ^ e + 1 } f ^ { p ^ e } ( a _ 1 ) , v _ 2 ^ { p ^ e + 1 } f ^ { p ^ e } ( a _ 2 ) , \\dots , v _ n ^ { p ^ e + 1 } f ^ { p ^ e } ( a _ n ) , f _ { k - 1 } ^ { p ^ e } ) = ( u _ 1 g ( a _ 1 ) , u _ 2 g ( a _ 2 ) , \\dots , u _ n g ( a _ n ) , - g _ { n - k } ) , \\end{align*}"}
+{"id": "1749.png", "formula": "\\begin{align*} \\left \\langle C \\mathcal { Z } , i _ { 1 } , \\dots , i _ { m } \\right \\rangle = \\det \\left ( \\left ( \\begin{array} { c } C \\\\ I _ { i _ { 1 } , \\dots , i _ { m } } \\end{array} \\right ) \\mathcal { Z } \\right ) , \\end{align*}"}
+{"id": "495.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } { n \\choose k } 2 ^ k F _ { j k } & ( \\pm \\sqrt { 5 } F _ j ) ^ { n - k } \\big ( q ^ { 1 - ( n - k ) } - 1 \\big ) B _ { n - k } \\\\ = n F _ j & q ^ { 1 - n } \\sum _ { r = 1 } ^ { q - 1 } \\Big ( ( \\pm \\sqrt { 5 } F _ j r + q L _ j ) ^ { n - 1 } + ( \\pm \\sqrt { 5 } F _ j ( r - q ) + q L _ j ) ^ { n - 1 } \\Big ) . \\end{align*}"}
+{"id": "268.png", "formula": "\\begin{align*} h _ t ( z ) = h _ 0 ( e ^ { - t } z ) + H _ t ( \\phi _ 0 ) ( e ^ { - t } z ) , \\end{align*}"}
+{"id": "1582.png", "formula": "\\begin{align*} w _ r ^ { - 1 } w _ r x _ { a } t _ { r } ^ { 5 } x _ { b } t _ { r } ^ { 5 } x _ { c } \\left ( t _ { a } ^ { - 1 } t _ { r } ^ { - 1 } t _ { b } ^ { - 1 } t _ { r } ^ { - 1 } t _ { c } ^ { - 1 } \\right ) ^ { 5 } = e . \\end{align*}"}
+{"id": "4207.png", "formula": "\\begin{align*} \\sum _ { X = X _ 0 } ^ \\infty A ( X ) v ^ { X - 1 } \\end{align*}"}
+{"id": "971.png", "formula": "\\begin{align*} \\widehat { \\partial _ j f ( 0 , \\xi ) } = 2 \\pi i \\ ; y _ j ( \\xi ) \\widehat { f ( 0 , \\xi ) } . \\end{align*}"}
+{"id": "4465.png", "formula": "\\begin{align*} { d x \\over d t } = v , \\nabla _ v v = E - { g ( E , v ) \\over g ( v , v ) } v , \\end{align*}"}
+{"id": "1015.png", "formula": "\\begin{align*} a ^ { \\rm M } _ { 1 , 2 , 3 , 5 , 8 , 1 1 , 1 4 , 1 5 } & = 0 \\ , , a ^ { \\rm M } _ { 4 } = \\mu \\ , L _ c ^ 2 \\frac { 2 a _ 3 - a _ 1 } { 3 } \\ , , a ^ { \\rm M } _ { 1 0 } = \\mu \\ , L _ c ^ 2 \\frac { a _ 1 + a _ 2 } { 2 } \\ , , a ^ { \\rm M } _ { 1 3 } = \\mu \\ , L _ c ^ 2 \\frac { a _ 1 - a _ 2 } { 2 } \\ , . \\end{align*}"}
+{"id": "6253.png", "formula": "\\begin{align*} \\delta _ k ( A ^ { [ n ] } b ) = \\sum _ { j _ 1 \\cdots j _ \\ell = k } \\prod _ { r = 1 } ^ \\ell \\delta _ { j _ r } ( A ) ; \\end{align*}"}
+{"id": "3566.png", "formula": "\\begin{align*} \\binom { g - 1 } { 2 } + \\ell ( A / \\tilde { I } ) \\ge \\binom { \\mu ( I ) - 1 } { 2 } + \\ell ( A / I ) . \\end{align*}"}
+{"id": "2531.png", "formula": "\\begin{align*} X _ \\beta = \\{ ( x , y , w , z ) \\in X : w - x = \\beta \\} \\end{align*}"}
+{"id": "5121.png", "formula": "\\begin{align*} \\frac { 2 \\sigma } { l } + \\frac { d } { m } = \\frac { d } { 2 } , \\frac { 1 } { m ' } = \\frac { 1 } { m } + \\frac { p } { r _ { \\theta } } , \\frac { 1 } { l ' } = \\frac { 1 } { l } + \\frac { p } { q _ { \\theta } } \\textmd { ( S t r i c h a r t z e x p o n e n t r e l a t i o n s a n d t h e H \\ \" o l d e r i n e q u a l i t y ) } . \\end{align*}"}
+{"id": "4626.png", "formula": "\\begin{align*} \\int _ E \\frac { \\phi ' _ h ( x , | \\nabla u | ) } { | \\nabla u | } \\nabla u \\cdot \\nabla h \\ , d x = \\lim _ { \\epsilon \\to 0 ^ + } \\int _ E \\frac { \\phi ( x , g ( \\epsilon ) ) - \\phi ( x , g ( 0 ) ) } { \\epsilon } \\ , d x . \\end{align*}"}
+{"id": "2170.png", "formula": "\\begin{align*} R _ { i j k l } & = R _ { i j } R _ { k l } \\\\ & = \\alpha _ { i k } \\alpha _ { j l } - \\alpha _ { i l } \\alpha _ { j k } . \\end{align*}"}
+{"id": "4921.png", "formula": "\\begin{align*} \\binom { n } { 1 } + \\sum _ { i = 1 } ^ { k - 1 } \\frac { r + 1 + 3 ( i - 1 ) } { k - ( i - 1 ) } \\binom { n } { k - i } > 2 \\binom { n } { r } + \\sum _ { i = k - r + 1 } ^ { k - 1 } \\frac { 1 } { 2 } \\binom { n } { k - i } . \\end{align*}"}
+{"id": "2340.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in \\mathfrak { S } _ { n + 1 } } { \\rm s g n } ( \\sigma ) I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a _ { \\sigma ( 1 ) } , b _ { 1 } , a _ { \\sigma ( 2 ) } , b _ { 2 } , \\dots , a _ { \\sigma ( n ) } , b _ { n } , a _ { \\sigma ( n + 1 ) } ) = 0 . \\end{align*}"}
+{"id": "3100.png", "formula": "\\begin{align*} t _ n = 1 + 2 + 3 + \\cdots + n = \\frac { n ( n + 1 ) } { 2 } . \\end{align*}"}
+{"id": "2036.png", "formula": "\\begin{align*} \\Re \\left [ \\frac { \\Gamma ' } { \\Gamma } ( \\sigma + i t ) \\right ] = \\log | \\sigma + i t | + O ( | \\sigma + i t | ^ { - 1 } ) \\end{align*}"}
+{"id": "8008.png", "formula": "\\begin{align*} S _ { M \\otimes I _ d } ( I ) \\oplus S _ { M \\otimes I _ d } ( I ' ) = S _ { M \\otimes I _ d } ( S _ { M \\otimes I _ d } ( J ) ; I \\cup I ' ) \\end{align*}"}
+{"id": "5476.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\frac { 1 - b } { 1 - t } F ( \\frac { a t } { b } , t ; b ) . \\end{align*}"}
+{"id": "7410.png", "formula": "\\begin{align*} f ( q ) = \\sum _ { k = 1 } ^ \\infty b _ k q ^ k \\in S _ 4 ^ { \\textrm { n e w } } ( \\Gamma _ 0 ( N ) ) \\end{align*}"}
+{"id": "7526.png", "formula": "\\begin{align*} \\int f d \\mu _ { \\varepsilon , h } = \\int f ( \\Phi ( \\varepsilon , p ) ) d \\mu ( p ) , \\varepsilon \\in \\mathbb R , \\end{align*}"}
+{"id": "103.png", "formula": "\\begin{align*} v ( r , \\xi ) = \\Delta ^ { N } _ \\Gamma u ( r , \\xi ) . \\end{align*}"}
+{"id": "3529.png", "formula": "\\begin{align*} \\Big \\Vert \\nabla \\mathrm { H } \\Big \\Vert _ { \\mathbb { L } ^ 4 ( \\Omega ) } & = \\delta ^ { - \\frac { 1 } { 2 } } \\Big \\Vert \\nabla \\Tilde { \\mathrm { H } } \\Big \\Vert _ { \\mathbb { L } ^ 4 ( \\mathrm { B } ) } \\\\ & = \\mathcal { O } \\Big ( \\delta ^ { - \\frac { 1 } { 2 } } \\Big ) . \\end{align*}"}
+{"id": "3007.png", "formula": "\\begin{align*} f ^ 0 = f ^ 0 ( t , x ) = & - ( \\phi ^ 0 _ { b c , S } + \\phi ^ 0 _ { b c , T } ) | _ { y = 1 } , \\\\ f ^ 1 = f ^ 1 ( t , x ) = & - ( \\phi ^ 1 _ { b c , S } + \\phi ^ 1 _ { b c , R } ) | _ { y = 0 } . \\end{align*}"}
+{"id": "7461.png", "formula": "\\begin{align*} A _ { K , \\max } u ^ i = \\lambda u ^ i , B _ K u ^ i = e _ i , \\end{align*}"}
+{"id": "27.png", "formula": "\\begin{align*} A _ { \\gamma _ j , \\gamma _ { j + 1 } } = \\{ i : p _ i / q _ i \\in [ \\gamma _ j , \\gamma _ { j + 1 } ) \\} = \\{ i : \\delta _ i \\in [ \\nu _ j , \\nu _ { j + 1 } ) \\} . \\end{align*}"}
+{"id": "371.png", "formula": "\\begin{align*} F _ N = \\left \\lbrace f \\in C ^ \\infty ( M ) ~ ~ \\Big \\vert ~ ~ \\lbrace f , I _ N \\rbrace \\subset I _ N \\right \\rbrace . \\end{align*}"}
+{"id": "1247.png", "formula": "\\begin{align*} M = \\sup _ { u \\in W _ { 0 } ^ { 1 , 1 } ( \\Omega ) \\backslash \\{ 0 \\} } \\displaystyle \\frac { \\displaystyle \\int _ \\Omega f u \\ , d x } { \\displaystyle \\int _ { \\Omega } | \\nabla u | d x } . \\end{align*}"}
+{"id": "3485.png", "formula": "\\begin{align*} \\textbf { e r r } ^ { ( 4 ) } : = \\mathcal { O } \\Bigg ( \\alpha _ \\mathrm { p } ^ { \\frac { 1 } { 4 } } \\delta ^ { \\frac { 7 } { 4 } } \\Big \\Vert \\gamma ^ { \\textbf { i n t } } _ { 1 } \\mathrm { U } _ { \\mathrm { i } } \\Big \\Vert _ { \\mathrm { H } ^ { - \\frac { 1 } { 2 } , - \\frac { 1 } { 4 } } \\Big ( \\partial \\Omega \\times \\mathbb { R } \\Big ) } \\sqrt [ 4 ] { \\mathbb { V } _ 1 \\cdot \\mathbb { V } _ 2 } \\Bigg ) . \\end{align*}"}
+{"id": "8098.png", "formula": "\\begin{align*} T _ 1 T _ 2 ( z _ 1 ^ { m _ 1 } z _ 2 ^ { m _ 2 } \\eta ) & = M _ { z _ 1 } ^ { \\alpha _ 1 } ( \\alpha _ 2 z _ 1 ^ { m _ 1 } z _ 2 ^ { m _ 2 + 1 } U ^ { m _ 1 } \\eta ) \\\\ & = \\alpha _ 1 \\alpha _ 2 z _ 1 ^ { m _ 1 + 1 } z _ 2 ^ { m _ 2 + 1 } U ^ { m _ 1 } \\eta \\\\ & = ( I _ { H ^ 2 ( { \\mathbb { D } } ) } \\otimes U ^ * ) ( \\alpha _ 1 \\alpha _ 2 z _ 1 ^ { m _ 1 + 1 } z _ 2 ^ { m _ 2 + 1 } U ^ { m _ 1 + 1 } \\eta ) \\\\ & = \\widetilde { U } T _ 2 T _ 1 ( z _ 1 ^ { m _ 1 } z _ 2 ^ { m _ 2 } \\eta ) \\end{align*}"}
+{"id": "1489.png", "formula": "\\begin{align*} \\begin{cases} g ( p ) p = 1 + \\chi ( p ) \\bigl ( 1 - \\frac { 1 } { p } \\bigr ) { \\rm i f } \\ , \\ , p \\nmid q \\\\ g ( p ) = 0 { \\rm i f } \\ , \\ , p \\mid q , \\end{cases} \\end{align*}"}
+{"id": "7002.png", "formula": "\\begin{align*} v ( \\gamma _ { s _ 1 } ) = \\max _ { s \\in [ 0 , 1 ] } v ( \\gamma _ s ) \\quad v ( \\gamma _ { s _ 2 } ) = \\min _ { s \\in [ 0 , 1 ] } v ( \\gamma _ s ) . \\end{align*}"}
+{"id": "2835.png", "formula": "\\begin{align*} \\int \\frac { f ( x ) \\bar { f } ( y ) } { \\abs { x - y } } = 4 \\pi \\int \\frac { \\abs { \\hat { f } ( p ) } } { \\abs { p } ^ { 2 } } \\geq 0 \\end{align*}"}
+{"id": "6207.png", "formula": "\\begin{align*} E ^ * _ { i + k - 2 } A _ 1 E ^ * _ { i + k - 3 } A _ 1 E ^ * _ { i + k - 4 } \\cdots E ^ * _ { i + 1 } A _ 1 E ^ * _ { i } = \\big ( ( \\frac { k - 3 } { 2 } ) ! \\big ) ^ 2 \\frac { k - 1 } { 2 } M ^ { \\frac { k - 3 } { 2 } , \\frac { k - 3 } { 2 } } _ { \\frac { i + k - 3 } { 2 } , \\frac { 2 m - i } { 2 } } . \\end{align*}"}
+{"id": "8142.png", "formula": "\\begin{align*} ( \\overline L _ 0 \\cdots \\overline L _ d ) _ S \\geqslant \\delta _ 0 \\operatorname { \\widehat { \\mu } _ { \\max } ^ { \\mathrm { a s y } } } ( \\overline L _ 0 ) + \\sum _ { i = 1 } ^ d \\delta _ i \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L _ i ) . \\end{align*}"}
+{"id": "7506.png", "formula": "\\begin{align*} g _ + ( z ) + g _ - ( z ) - V ( z ) = \\ell _ j \\Sigma _ j \\end{align*}"}
+{"id": "7869.png", "formula": "\\begin{align*} \\# F _ B ( x ) = \\# F ( x ) \\cap ( 3 B ) \\cap B _ x = \\# F ( x ) \\cap B _ x \\sim \\mu _ 1 . \\end{align*}"}
+{"id": "1528.png", "formula": "\\begin{align*} B = \\left [ \\begin{matrix} 0 & 0 & 0 & 0 & 1 \\\\ 1 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & \\ddots & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 0 \\end{matrix} \\right ] \\in \\mathrm { G L } _ { k } \\mathbb { F } . \\end{align*}"}
+{"id": "4143.png", "formula": "\\begin{align*} B ( z ) = \\eta \\prod \\limits _ { j = 1 } ^ n \\frac { z _ j - z } { 1 - \\overline { z _ j } z } \\ , , | \\eta | = 1 \\ , , \\end{align*}"}
+{"id": "5524.png", "formula": "\\begin{align*} \\dim H _ l ( H , \\Pi \\otimes \\chi ^ { \\vee } ) \\le \\sum _ { j = 1 } ^ { j _ 0 } \\sum _ { k = 0 } ^ { \\infty } d _ { l , j , k } . \\end{align*}"}
+{"id": "5426.png", "formula": "\\begin{align*} | B _ 3 | \\prec \\frac { 1 } { N ^ { d - m _ 0 } } \\Big ( \\frac { C _ 0 } { N } \\Big ) ^ { m _ 0 - 1 } \\Psi ^ { m _ 2 } \\sum _ { j _ 2 , \\ldots , j _ d } \\big | G _ { j _ q y _ 2 } \\big | \\prod _ { i = 3 } ^ { m _ 1 } \\big | G _ { x _ i y _ i } \\big | . \\end{align*}"}
+{"id": "211.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { N } } u _ 0 ( \\varphi ( 0 , x ) - k \\Delta \\varphi ( 0 , x ) ) d x = \\nu ( 0 ) \\int _ { \\mathbb { R } ^ { N } } u _ 0 ( \\phi ( x ) - k \\Delta \\phi ( x ) ) d x = 0 . \\end{align*}"}
+{"id": "3608.png", "formula": "\\begin{align*} E \\left [ M \\right ] _ t = t \\int _ \\Re y ^ 2 1 _ { \\{ | y | < 1 \\} } \\nu ( d y ) < \\infty . \\end{align*}"}
+{"id": "6081.png", "formula": "\\begin{align*} \\pi _ I ( \\gamma ^ n ( x ) ) = V \\pi _ I ( x ) V ^ * x \\in A . \\end{align*}"}
+{"id": "3053.png", "formula": "\\begin{align*} g ( \\ell ) = \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } f ( k ) \\end{align*}"}
+{"id": "6976.png", "formula": "\\begin{align*} \\| Q ^ \\frac { 1 } { 2 } X h \\| ^ 2 = \\langle X ^ * Q X h , h \\rangle = \\lim _ n \\langle P ^ { * n } X ^ * X P ^ n h , h \\rangle \\leq \\| X \\| ^ 2 \\langle Q h , h \\rangle . \\end{align*}"}
+{"id": "4910.png", "formula": "\\begin{align*} ( \\frak F \\circ g ^ * ) ( [ V \\xrightarrow h X ; E ) & = \\frak F ( [ V ' \\xrightarrow h ' X ' ; ( g '' ) ^ * E ] ) \\\\ & = [ X ' \\xleftarrow h ' V ' \\xrightarrow { f ' \\circ h ' } Y ' ; ( g '' ) ^ * E ] ) . \\end{align*}"}
+{"id": "3355.png", "formula": "\\begin{align*} L \\otimes _ K K _ { i j } & \\rightarrow \\prod _ { j ' } L _ { i j j ' } \\\\ x \\otimes y & \\mapsto ( \\sigma _ { i j j ' } ( x ) y ) _ { j ' } , \\end{align*}"}
+{"id": "7897.png", "formula": "\\begin{align*} ( a , b ) \\bullet ( a ' , b ' ) = ( a \\cdot a ' , b \\cdot b ' ) , ( a , b ) , ( a ' , b ' ) \\in A \\oplus A . \\end{align*}"}
+{"id": "874.png", "formula": "\\begin{align*} \\binom { r } { r \\alpha } \\sim \\sqrt { \\frac { 1 } { 2 \\pi \\alpha ( 1 - \\alpha ) r } } \\ , \\ , \\Bigl ( \\frac { 1 } { \\alpha } \\Bigr ) ^ { \\alpha r } \\ , \\Bigl ( \\frac { 1 } { 1 - \\alpha } \\Bigr ) ^ { ( 1 - \\alpha ) r } . \\end{align*}"}
+{"id": "2778.png", "formula": "\\begin{align*} A ^ T A v _ i = \\sigma _ i ^ 2 v _ i , \\end{align*}"}
+{"id": "2545.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { \\xi } = & ~ [ A x _ t ^ { \\xi } - B ^ 2 R ^ { - 1 } y _ t ^ { \\xi } - B h ( \\mu _ t ) + f ( \\nu _ t ) + b ( \\mu _ t ) ] d t \\\\ & + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ d y _ t ^ { \\xi } = & - [ A y _ t ^ { \\xi } + Q x _ t ^ { \\xi } + Q l ( \\nu _ t ) ] d t + z _ t ^ { \\xi } d W _ t + z ^ { 0 , \\xi } _ t d W ^ 0 _ t , \\\\ x _ 0 ^ { \\xi } = & ~ \\xi , ~ y _ T ^ { \\xi } = G ( x _ T ^ \\xi + g ( \\nu _ T ) ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7997.png", "formula": "\\begin{align*} T _ w = c _ 0 I + \\sum _ { k = 1 } ^ m c _ k S ^ k + \\sum _ { k = 1 } ^ m c _ { - k } S ^ { * k } , \\end{align*}"}
+{"id": "6537.png", "formula": "\\begin{align*} H = \\sum _ { r } \\theta _ r \\left ( E _ { \\theta _ r } - \\overline { E _ { \\theta _ r } } . \\right ) \\end{align*}"}
+{"id": "3684.png", "formula": "\\begin{align*} V _ { i } ' = \\begin{cases} V _ { i } \\cap N ( u _ { t + 1 } ) \\cap N ( u _ { t + 2 } ) & i = 1 , \\\\ V _ { i } \\cap N ( v ) \\cap N ( u _ { t + 2 } ) & i = t + 1 , \\\\ V _ { i } \\cap N ( v ) \\cap N ( u _ { t + 1 } ) \\cap N ( u _ { t + 2 } ) & i \\in [ 2 , t ] \\cup \\{ t + 4 \\} . \\end{cases} \\end{align*}"}
+{"id": "4422.png", "formula": "\\begin{align*} I ^ { \\ast } ( x ) = \\inf _ { x _ n \\to x } \\liminf _ { n \\to \\infty } I ( x _ n ) . \\end{align*}"}
+{"id": "3674.png", "formula": "\\begin{align*} x _ { t + 1 } = \\frac { \\alpha } { t + 2 } \\pm 1 0 t \\delta ^ { 1 / 2 } , \\quad x _ { t + 2 } = \\frac { 1 - \\alpha } { t + 2 } \\pm 1 0 t \\delta ^ { 1 / 2 } . \\end{align*}"}
+{"id": "5610.png", "formula": "\\begin{align*} B ^ i _ { j k l } = \\hat { \\mathbb { G } } ^ i _ { j k l } = \\frac { \\partial ^ 3 \\hat { \\mathbb { G } } ^ i } { \\partial y ^ j \\partial y ^ k \\partial y ^ l } \\end{align*}"}
+{"id": "6385.png", "formula": "\\begin{align*} M ( \\alpha ) M ( \\beta ) = q ^ { \\Pi ( \\alpha , \\beta ) / 2 } M ( \\alpha + \\beta ) \\end{align*}"}
+{"id": "6022.png", "formula": "\\begin{align*} L ^ { \\pi ^ { ( x , y , i ) } } _ { r } ( t ) : = L ^ { \\pi ^ { ( y , i ) } } _ { r } ( t ) 1 _ { \\lbrace t > 0 \\rbrace } , R ^ { \\pi ^ { ( x , y , i ) } } _ { r } ( t ) : = ( y - x ) 1 _ { \\lbrace t = 0 \\rbrace } + R ^ { \\pi ^ { ( y , i ) } } _ { r } ( t ) 1 _ { \\lbrace t > 0 \\rbrace } , t \\geq 0 . \\end{align*}"}
+{"id": "5412.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N | G _ { i j } | ^ 2 = \\frac { 1 } { \\eta } \\Im G _ { i i } , \\end{align*}"}
+{"id": "6087.png", "formula": "\\begin{align*} d ( a _ k , b _ k ) = d ( a _ { k + 1 } ^ 2 , b _ { k + 1 } ^ 2 ) \\leq 2 d ( a _ { k + 1 } , b _ { k + 1 } ) k \\geq 0 . \\end{align*}"}
+{"id": "7180.png", "formula": "\\begin{align*} \\det ( ( \\mathfrak { g } ^ { \\nu > 0 } ) ^ { \\vee } ( 2 ) ) [ - \\dim \\mathfrak { g } ^ { \\nu > 0 } ] = ( \\det V _ 1 ) ^ { - d _ 2 } \\otimes ( \\det V _ 2 ) ^ { d _ 1 } [ d _ 1 d _ 2 ] . \\end{align*}"}
+{"id": "1052.png", "formula": "\\begin{align*} \\sigma ( V W ) = \\mathfrak v _ { 2 } + \\mathfrak v _ { 1 } = - V ^ { a } W ^ { b } \\xi _ { a } \\xi _ { b } + i V ^ { a } \\delta _ { a } ( W ^ { b } ) \\xi _ { b } . \\end{align*}"}
+{"id": "6974.png", "formula": "\\begin{align*} \\pi ( X ) = \\pi ( \\Phi ( X ) ) X \\in C ^ * ( I , \\mathcal T ( P ) ) . \\end{align*}"}
+{"id": "3081.png", "formula": "\\begin{align*} a _ { k , 2 } = a _ { k , 1 } - 1 \\end{align*}"}
+{"id": "7691.png", "formula": "\\begin{align*} \\| f \\| ^ p _ { p } = \\lim _ { r \\to 1 } \\int _ { \\mathbb { S } } | f ( r \\zeta ) | ^ p d \\sigma ( \\zeta ) . \\end{align*}"}
+{"id": "3486.png", "formula": "\\begin{align*} \\textbf { e r r } ^ { ( 5 ) } : = \\mathcal { O } \\Bigg ( \\delta ^ { \\frac { 3 } { 2 } } \\Big \\Vert \\gamma ^ { \\textbf { i n t } } _ { 0 } \\mathrm { U } _ { \\mathrm { i } } \\Big \\Vert _ { \\mathrm { H } ^ { \\frac { 1 } { 2 } , \\frac { 1 } { 4 } } \\Big ( \\partial \\Omega \\times \\mathbb { R } \\Big ) } \\sqrt { \\int _ { 0 } ^ { \\mathrm { T } _ 0 } \\Big | \\partial _ { t } \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) \\Big | ^ 2 d \\tau } \\Bigg ) . \\end{align*}"}
+{"id": "2732.png", "formula": "\\begin{align*} \\sigma _ i \\bigl ( e ^ { \\pm } _ { i j } \\bigr ) = - e ^ { \\mp } _ { i j } \\ , , \\sigma ^ { \\pm } _ { i j } ( e _ j ) = \\mp e _ i \\ , , \\end{align*}"}
+{"id": "7725.png", "formula": "\\begin{align*} \\begin{aligned} V o l ( B ( t ) \\setminus B ( \\frac { t } { 2 } + d ) , g ^ p ) & = \\int _ { B ( t ) \\setminus B ( \\frac { t } { 2 } + d ) } d V _ { g ^ p } \\leq e ^ { ( n + 1 ) d } \\int _ { B ( t ) \\setminus B ( \\frac { t } { 2 } + d ) } d V _ { g ^ q } \\\\ & = e ^ { ( n + 1 ) d } \\cdot V o l ( B ( t ) \\setminus B ( \\frac { t } { 2 } + d ) , g ^ q ) \\\\ & \\leq e ^ { ( n + 1 ) d } \\cdot V o l ( D ( t + d ) \\setminus D ( \\frac { t } { 2 } ) , g ^ q ) . \\end{aligned} \\end{align*}"}
+{"id": "6933.png", "formula": "\\begin{align*} ( - 1 ) ^ { j + 1 } e _ { N + 1 } ^ j e _ { N - j } & = ( - 1 ) ^ { N ( j + 1 ) } y ^ { j + 1 } \\left [ z ^ { j + 1 } \\right ] \\left ( \\left ( 1 + \\frac { z } { y } \\right ) ( 1 + z x _ 1 ) \\cdots ( 1 + z x _ r ) \\right ) \\\\ & = \\left [ t ^ { j + 1 } \\right ] \\left ( ( 1 - ( - 1 ) ^ { N - 1 } t ) ( 1 - ( - 1 ) ^ { N - 1 } x _ 1 y t ) \\cdots ( 1 - ( - 1 ) ^ { N - 1 } x _ r y t ) \\right ) , \\end{align*}"}
+{"id": "5243.png", "formula": "\\begin{align*} Z _ { N , \\beta } = \\int _ { \\mathbb { T } ^ N } \\prod _ { 1 \\leq j < k \\leq N } \\left | e ^ { i \\theta _ j } - e ^ { i \\theta _ k } \\right | ^ \\beta \\ * d \\theta = ( 2 \\pi ) ^ N \\ * \\frac { \\Gamma \\left ( 1 + \\frac { \\beta N } { 2 } \\right ) } { \\Gamma \\left ( 1 + \\frac { \\beta } { 2 } \\right ) ^ N } . \\end{align*}"}
+{"id": "709.png", "formula": "\\begin{align*} { \\rm O } ( \\epsilon ^ 0 ) : { { { \\bf { m } } } } ^ { ( 0 ) } = { { { \\bf { m } } } } ^ { ( e q ) } , \\end{align*}"}
+{"id": "1861.png", "formula": "\\begin{align*} y _ { ( 2 k ) m + 2 i - 1 , j } + y _ { ( 2 k ) m + 2 i - 1 , j + 1 } & = x _ { 2 i - 1 , j } + x _ { 2 i - 1 , j + 1 } \\mbox { a n d } \\\\ y _ { ( 2 k ) m + 2 i , j } + y _ { ( 2 k ) m + 2 i , j + 1 } & = x _ { 2 i , j } + x _ { 2 i , j + 1 } . \\end{align*}"}
+{"id": "4100.png", "formula": "\\begin{align*} n = 2 + \\frac { \\log ( r ) + \\log \\log ( r ) + \\log ( 2 ) / \\log ( r ) } { \\log ( 2 ) } , \\end{align*}"}
+{"id": "8017.png", "formula": "\\begin{align*} T _ Q - \\begin{bmatrix} S ( m ) & 0 \\\\ 0 & 0 \\end{bmatrix} \\geq 0 . \\end{align*}"}
+{"id": "6854.png", "formula": "\\begin{align*} \\beta \\lambda u _ i h ( a _ i ) f ^ { p ^ { e ' } } ( a _ i ) = u _ i g ( a _ i ) = \\lambda u _ i h ( a _ i ) f ^ { p ^ { e ' } } ( a _ i ) , \\ 1 \\leq i \\leq s . \\end{align*}"}
+{"id": "3568.png", "formula": "\\begin{align*} a _ 1 a _ 3 = y _ 1 a _ 1 ^ 2 + y _ 2 a _ 1 a _ 2 + y _ 3 a _ 2 ^ 2 , \\end{align*}"}
+{"id": "5084.png", "formula": "\\begin{align*} ( i \\partial _ t + ( - \\Delta _ { x } ) ^ { \\sigma } ) u = F , u ( 0 ) = u _ 0 \\in H ^ { \\sigma } ( \\mathbb { R } ^ d ) . \\end{align*}"}
+{"id": "3794.png", "formula": "\\begin{align*} n _ f : = k + \\frac { l } { 2 } + \\sum _ { r = 1 } ^ k i _ r + \\sum _ { r = 1 } ^ l j _ r \\quad n _ g : = k ' + \\frac { l ' } { 2 } + \\sum _ { r = 1 } ^ { k ' } i ' _ r + \\sum _ { r = 1 } ^ { l ' } j ' _ r \\end{align*}"}
+{"id": "1962.png", "formula": "\\begin{align*} t - s = \\frac { \\nu } { \\tau _ { } } - \\frac { \\nu } { p _ { } } . \\end{align*}"}
+{"id": "1828.png", "formula": "\\begin{align*} \\{ x , y \\} = x \\rhd y - y \\rhd x + [ x , y ] \\end{align*}"}
+{"id": "5642.png", "formula": "\\begin{align*} l \\leq d ( \\gamma ( t ) , f ( \\gamma ( t ) ) ) & \\leq d ( \\gamma ( t ) , f ( p ) ) + d ( f ( p ) , f ( \\gamma ( t ) ) ) . \\\\ & = d ( \\gamma ( t ) , f ( p ) ) + d ( p , \\gamma ( t ) ) = d ( p , f ( p ) ) = l . \\end{align*}"}
+{"id": "3780.png", "formula": "\\begin{align*} a _ { j _ 2 } ( t ) = ( 2 j _ 2 + 1 ) ( 1 - 2 t c ( t ) ) ^ { - 1 } \\vec { c } _ { j _ 2 } ( t ) . \\end{align*}"}
+{"id": "3048.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ k ) = \\sum _ { i \\in I } F _ { X _ i } ( \\sigma _ i ^ k ) \\end{align*}"}
+{"id": "2051.png", "formula": "\\begin{align*} \\Psi _ \\omega ( t ) : = e ^ { - \\omega t } \\Psi ( t ) + 2 \\omega \\int _ { 0 } ^ { t } e ^ { - \\omega u } \\Psi ( u ) \\ , d u + \\omega ^ 2 \\int _ { 0 } ^ { t } ( t - u ) e ^ { - \\omega u } \\Psi ( u ) \\ , d u \\end{align*}"}
+{"id": "1172.png", "formula": "\\begin{align*} a _ k ( x ) f _ k '' ( x ) + b _ k ( x ) f _ k ' ( x ) + p _ k c _ k ( x ) f _ k ( x ) = 0 , \\end{align*}"}
+{"id": "7777.png", "formula": "\\begin{align*} c a p _ p ( B ^ { g ^ + } _ q ( t ) ) \\geq ( \\int _ { V o l ( B ^ { g ^ + } _ q ( t ) ) } ^ { + \\infty } ( n \\tau ) ^ { \\frac { p } { 1 - p } } d \\tau ) ^ { 1 - p } = \\frac { n ^ p } { ( p - 1 ) ^ { p - 1 } } V o l ( B ^ { g ^ + } _ q ( t ) ) \\end{align*}"}
+{"id": "2349.png", "formula": "\\begin{align*} f _ { c } ( u , 1 ) = f _ { c } ( 1 , u ) = x _ { c } u \\quad ( u \\in \\mathfrak { X } ) \\end{align*}"}
+{"id": "7569.png", "formula": "\\begin{align*} \\left | \\int \\Re V d \\mu _ { \\varepsilon , h } - \\int \\Re V d \\mu - \\varepsilon \\int h d V d \\mu \\right | = o ( \\varepsilon ) . \\end{align*}"}
+{"id": "7622.png", "formula": "\\begin{align*} h ^ q ( \\Omega ^ 1 _ { \\widehat { W } } ) = h ^ { d - r - q } ( \\Omega ^ 1 _ V ) 0 \\le q \\le d - r . \\end{align*}"}
+{"id": "6847.png", "formula": "\\begin{align*} \\lambda u _ i h ( a _ i ) = v _ i ^ { p ^ e + 1 } \\neq 0 , \\ 1 \\leq i \\leq n , \\end{align*}"}
+{"id": "4789.png", "formula": "\\begin{align*} { } _ p F _ q \\left ( \\left . \\begin{matrix} a _ 1 , \\dots , a _ p \\\\ b _ 1 , \\dots , b _ q \\end{matrix} \\right | x \\right ) : = \\sum _ { k \\ge 0 } \\frac { ( a _ 1 ) _ k \\dotsm ( a _ p ) _ k } { ( b _ 1 ) _ k \\dotsm ( b _ q ) _ k } \\frac { x ^ k } { k ! } , \\end{align*}"}
+{"id": "2970.png", "formula": "\\begin{align*} { } ^ { \\perp } X = \\{ y \\in \\mathcal { T } \\ ; \\vline \\ ; \\mathrm { H o m } ( y , x ) = 0 \\ ; \\ ; x \\in X \\} \\end{align*}"}
+{"id": "490.png", "formula": "\\begin{align*} B _ n \\Big ( \\frac { 2 \\beta ^ j } { L _ j } \\Big ) + B _ n \\Big ( \\frac { F _ j \\sqrt 5 } { L _ j } \\Big ) = 0 , \\qquad \\mbox { $ n $ o d d } . \\end{align*}"}
+{"id": "937.png", "formula": "\\begin{align*} \\xi _ 2 = \\kappa _ 0 \\xi _ 1 ^ 2 + O ( | \\xi _ 1 | ^ 3 ) \\mbox { a s } \\xi _ 1 \\rightarrow 0 \\end{align*}"}
+{"id": "6681.png", "formula": "\\begin{align*} ( t _ { \\infty } ( \\mathbf { s } ) , \\infty ) = \\bigcup _ { 0 < a < b } ( t _ { a , b } ( \\mathbf { s } ) , T _ { a , b } ( \\mathbf { s } ) ) . \\end{align*}"}
+{"id": "2131.png", "formula": "\\begin{align*} \\mathrm { c o r a n k } _ { \\Lambda ( G ) } ( \\displaystyle \\bigoplus _ { v \\in S } J _ v ^ { \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) ) = \\mathrm { c o r a n k } _ { \\Lambda ( G ) } ( \\displaystyle \\bigoplus _ { v \\in S _ p } J _ v ^ { \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) ) . \\end{align*}"}
+{"id": "3323.png", "formula": "\\begin{gather*} \\delta _ { k _ 1 , k _ 1 ' } \\delta _ { k _ 2 , k _ 2 ' } = \\delta _ { n _ 1 ( k _ 1 ) , n _ 1 ( k _ 1 ' ) } \\delta _ { n _ 2 ( k _ 2 ) , n _ 2 ( k _ 2 ' ) } = \\big \\langle \\psi _ { n _ 1 ( k _ 1 ' ) } ^ { n _ 2 ( k _ 2 ' ) } , \\psi _ { n _ 1 ( k _ 1 ) } ^ { n _ 2 ( k _ 2 ) } \\big \\rangle _ { V _ 1 } \\big \\langle \\psi _ { n _ 2 ( k _ 2 ' ) } , \\psi _ { n _ 2 ( k _ 2 ) } \\big \\rangle _ { V _ 2 } . \\end{gather*}"}
+{"id": "3513.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\mathrm { E } | ^ 2 ( \\mathrm { y } ) d \\mathrm { y } = \\dfrac { 1 } { | 1 - \\alpha \\lambda ^ { ( 3 ) } _ { \\mathrm { n } _ { 0 } } | ^ 2 } \\Big | \\Big \\langle \\widehat { \\nabla \\mathrm { H } } ^ { \\textbf { i n } } ; e ^ { ( 3 ) } _ { \\mathrm { n } _ { 0 } } \\Big \\rangle _ { \\mathbb { L } ^ { 2 } ( \\mathrm { B } ) } \\Big | ^ 2 \\delta ^ 2 + \\mathcal { O } \\Big ( \\delta ^ { 2 } \\Big ) . \\end{align*}"}
+{"id": "1154.png", "formula": "\\begin{align*} \\epsilon _ { f \\otimes \\chi _ D } = \\chi _ D ( - r ) \\mu ( q ' ) q '^ { 1 / 2 } \\lambda _ f ( q ' ) \\epsilon _ f . \\end{align*}"}
+{"id": "8163.png", "formula": "\\begin{align*} \\| \\sum \\alpha _ j \\otimes C _ j \\| ^ 2 = \\| \\sum \\alpha _ j ^ * \\alpha _ j \\| \\end{align*}"}
+{"id": "243.png", "formula": "\\begin{align*} S [ \\xi ] ( u ) \\stackrel { d e f } { = } { \\bf P } ( \\ \\min _ { i \\in D } \\xi _ i > u \\ ) , \\ u \\ge 1 . \\end{align*}"}
+{"id": "5698.png", "formula": "\\begin{gather*} F _ { ( \\mu , \\nu ) } ( \\tau _ * ( \\alpha _ { ( \\xi , \\eta ) } ) ) = 0 ( \\xi , \\eta ) \\ngeq ( \\mu , \\nu ) . \\end{gather*}"}
+{"id": "4774.png", "formula": "\\begin{align*} a \\cdot b : = a \\ast b + b \\ast a { \\rm a n d } [ a , b ] : = a \\diamond b - b \\diamond a , \\forall a , b \\in A . \\end{align*}"}
+{"id": "5100.png", "formula": "\\begin{align*} f ^ { \\delta } ( \\xi _ 1 , \\xi _ 2 ) = \\sum _ { m \\in \\Lambda _ N } c _ d \\delta ^ { - d } 1 _ { \\{ | \\xi _ 2 - m | \\leq \\delta \\} } f ( \\xi _ 1 , m ) , \\delta < \\frac { 1 } { N } , \\end{align*}"}
+{"id": "5382.png", "formula": "\\begin{align*} [ \\Pi _ { F ^ { ( j ) } } , \\Pi _ { G ^ { ( k ) } } ] _ ( g _ 1 , \\cdots , g _ { j + k - 1 } ) & = \\cdots = \\{ \\{ \\{ \\{ F ^ { ( j ) } , G ^ { ( k ) } \\} ^ { [ 1 ] } , g _ 1 \\} ^ { [ 1 ] } , \\cdots \\} ^ { [ 1 ] } , g _ { j + k - 1 } \\} ^ { [ 1 ] } \\\\ & = \\Pi _ { \\{ F ^ { ( j ) } , G ^ { ( k ) } \\} ^ { [ 1 ] } } ( g _ 1 , \\cdots , g _ { j + k - 1 } ) \\end{align*}"}
+{"id": "2679.png", "formula": "\\begin{align*} M _ S ( n , j , t + 1 ; a ) = \\binom { n } { j } \\sum _ { u = 0 } ^ { \\lfloor \\frac { n - 2 j } { 2 } \\rfloor } \\binom { n - j } { u } M _ S ( n , j + u , t ; a ) \\end{align*}"}
+{"id": "4002.png", "formula": "\\begin{align*} \\hat { q } _ { \\beta } ( n , t ) = \\sum _ { k = 1 } ^ { n } \\underset { m _ j \\in \\mathbb { N } } { \\underset { m _ { 1 } + m _ { 2 } + \\dots + m _ { k } = n } { \\sum } } \\prod _ { j = 1 } ^ { k } \\frac { ( 1 - p ) ^ { m _ { j } } } { m _ { j } ! } \\left ( \\frac { - \\lambda t ^ { \\beta } } { \\ln p } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } ( - \\lambda t ^ { \\beta } ) . \\end{align*}"}
+{"id": "2027.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } \\Delta _ t ( x ) \\ , e ^ { i z x } d x = \\frac { 1 - \\cos ( z t ) } { z ^ 2 } \\end{align*}"}
+{"id": "1515.png", "formula": "\\begin{align*} p \\equiv a ( \\mod q ) { \\rm w i t h } \\ , \\ , \\ , \\chi ( a ) = 1 . \\end{align*}"}
+{"id": "1205.png", "formula": "\\begin{align*} \\pi _ { ( A , b ) } = \\pi _ { L , b _ 0 } \\circ \\pi _ U \\circ \\pi _ P , \\end{align*}"}
+{"id": "3059.png", "formula": "\\begin{align*} \\ell = \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } \\mu \\left ( \\frac { \\ell } { k } \\right ) s ( k ) . \\end{align*}"}
+{"id": "4948.png", "formula": "\\begin{align*} y = \\sum _ { i = 0 } ^ { ( k - 5 ) / 2 - t } \\frac { k - 2 + 2 t + i ( t + ( k - 1 ) / 2 ) } { n } \\binom { n } { k - 2 - i } \\end{align*}"}
+{"id": "2158.png", "formula": "\\begin{align*} & R _ { i j k l } = \\alpha _ { i k } \\alpha _ { j l } - \\alpha _ { i l } \\alpha _ { j k } . \\end{align*}"}
+{"id": "124.png", "formula": "\\begin{align*} T x : = [ ( f _ n ( x ) ) _ n ] \\ \\ \\ \\ ( \\forall x \\in E ) . \\end{align*}"}
+{"id": "4736.png", "formula": "\\begin{gather*} r _ { 1 2 } r _ { 1 3 } = \\sum _ { i , j } a _ i \\cdot a _ j \\otimes b _ i \\otimes b _ j , r _ { 1 3 } r _ { 2 3 } = \\sum _ { i , j } a _ i \\otimes a _ j \\otimes b _ i \\cdot b _ j , \\\\ r _ { 2 3 } r _ { 1 2 } = \\sum _ { i , j } a _ j \\otimes a _ i \\cdot b _ j \\otimes b _ i . \\end{gather*}"}
+{"id": "6753.png", "formula": "\\begin{align*} \\mathcal { H } ^ { ( N ) } = \\left ( L ^ 2 \\left ( \\mathbb { R } ^ { 3 } \\right ) \\right ) ^ { \\otimes _ s N } \\otimes \\mathcal { F } , \\end{align*}"}
+{"id": "1246.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle - \\Delta _ p u _ p = f & \\Omega , \\\\ \\displaystyle | \\nabla u _ p | ^ { p - 2 } \\nabla u _ p \\cdot \\nu + \\lambda | u _ p | ^ { p - 2 } u _ p = g & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "4532.png", "formula": "\\begin{align*} & s _ 0 ( t ) = \\frac { 1 } { \\sqrt { \\sinh 1 } } \\sinh \\left ( t - \\frac { 1 } { 2 } \\right ) \\ , , \\\\ & c _ 0 ( t ) = 1 \\ , , \\\\ & s _ k ( t ) = \\sqrt { \\frac { 2 } { 1 + 4 \\pi ^ 2 k ^ 2 } } \\sin ( 2 \\pi k t ) \\ , , k \\in \\mathbb { N } \\ , , \\\\ & c _ k ( t ) = \\sqrt { \\frac { 2 } { 1 + 4 \\pi ^ 2 k ^ 2 } } \\cos ( 2 \\pi k t ) \\ , , k \\in \\mathbb { N } \\ , , \\end{align*}"}
+{"id": "6192.png", "formula": "\\begin{align*} | \\mathcal { I } ' _ m | = { m + 4 \\choose 4 } \\ \\ \\ \\ m \\geq 1 \\end{align*}"}
+{"id": "7588.png", "formula": "\\begin{align*} \\left [ \\int C ( u , q ) d \\mu ( q ) \\right ] ^ 2 = \\int C ^ { ( 2 , - 1 ) } ( u , q ; a ) d V ( q ) d \\mu ( q ) , \\end{align*}"}
+{"id": "5841.png", "formula": "\\begin{align*} \\omega ( S ) = 1 , ~ ~ c ( S ) = 0 . \\end{align*}"}
+{"id": "2167.png", "formula": "\\begin{align*} & \\begin{vmatrix} R _ { a b } & \\overline { R } _ { a b } \\\\ R _ { c d } & \\overline { R } _ { c d } \\end{vmatrix} \\begin{vmatrix} R _ { p q } & \\overline { R } _ { p q } \\\\ R _ { r s } & \\overline { R } _ { r s } \\end{vmatrix} \\\\ = & ( R _ { a b } \\overline { R } _ { c d } - R _ { c d } \\overline { R } _ { a b } ) ( R _ { p q } \\overline { R } _ { r s } - R _ { r s } \\overline { R } _ { p q } ) \\\\ = & 2 ( R _ { a b p q } R _ { c d r s } - R _ { a b r s } R _ { c d p q } ) . \\end{align*}"}
+{"id": "42.png", "formula": "\\begin{align*} p & : = \\left ( 1 / 2 - 2 \\epsilon - \\epsilon ^ { 1 + \\alpha } + \\epsilon ^ { 1 + \\beta } - \\epsilon ^ { 1 + \\delta } , 1 / 2 + 2 \\epsilon , \\epsilon ^ { 1 + \\alpha } - \\epsilon ^ { 1 + \\beta } , \\epsilon ^ { 1 + \\delta } \\right ) , \\\\ q & : = \\left ( 1 / 2 , 1 / 2 - \\epsilon ^ { 1 + \\alpha } , \\epsilon ^ { 1 + \\alpha } , 0 \\right ) , \\end{align*}"}
+{"id": "37.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { D - 1 } \\left ( \\nu _ { j } - \\nu _ { j - 1 } \\right ) \\P \\left \\{ Y \\geq \\nu _ j \\right \\} , \\end{align*}"}
+{"id": "6362.png", "formula": "\\begin{align*} L _ { \\sigma } ( \\lambda ) : = \\lambda M _ { d _ A } - M _ { \\sigma ^ { - 1 } ( 1 ) } \\cdots M _ { \\sigma ^ { - 1 } ( d _ A ) } = \\lambda M _ { d _ A } - M _ { \\sigma } . \\end{align*}"}
+{"id": "4089.png", "formula": "\\begin{align*} \\alpha _ { \\pm } : = g \\pm \\sqrt { g ^ 2 - 1 } . \\end{align*}"}
+{"id": "4939.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k - 1 } \\frac { j } { 2 } \\binom { n } { i } + \\frac { j } { 2 } \\cdot \\binom { n } { r } < \\binom { n } { k } . \\end{align*}"}
+{"id": "3862.png", "formula": "\\begin{align*} W \\equiv W _ t = H ^ z _ t + Z = \\begin{pmatrix} 0 & X _ t \\\\ X _ t ^ * & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "362.png", "formula": "\\begin{align*} \\{ f _ 1 , f _ 2 \\} = \\omega ( v _ { f _ 1 } , v _ { f _ 2 } ) = \\L _ { v _ { f _ 1 } } ( f _ 2 ) \\end{align*}"}
+{"id": "8241.png", "formula": "\\begin{align*} Z _ { s , \\bar { s } } ( t ) : = Z _ { s , \\bar { s } } ^ \\ell ( t ) + Z _ s ^ h ( t ) . \\end{align*}"}
+{"id": "6929.png", "formula": "\\begin{align*} [ \\epsilon ^ 0 q ^ d ] \\frac { ( - 1 ) ^ { ( N - 1 ) d } } { \\det ( z _ i ^ { N - j + 1 } ) } \\ , \\begin{vmatrix} z _ 1 ^ { d + N } & z _ 1 ^ { d + N - 1 } & \\cdots & z _ 1 ^ { d + 1 } & f ( z _ { 1 } ) \\\\ z _ 2 ^ { d + N } & z _ 2 ^ { d + N - 1 } & \\cdots & z _ 2 ^ { d + 1 } & f ( z _ { 2 } ) \\\\ \\vdots & \\vdots & \\cdots & \\vdots & \\vdots \\\\ z _ { N + 1 } ^ { d + N } & z _ { N + 1 } ^ { d + N - 1 } & \\cdots & z _ { N + 1 } ^ { d + 1 } & f ( z _ { N + 1 } ) \\end{vmatrix} . \\end{align*}"}
+{"id": "7098.png", "formula": "\\begin{align*} \\sum _ { d \\geq 0 } \\mathrm { D T } _ { X , d } q ^ d = M ( - q ) ^ { \\chi ( X ) } , \\end{align*}"}
+{"id": "6836.png", "formula": "\\begin{align*} \\sum _ { v \\in M _ { K } } N _ { v } \\max _ { 1 \\le i \\le n } ( \\abs { \\log ( \\abs { u _ { i } } ) } _ { v } ) & = \\sum _ { v \\in M _ { K } } N _ { v } \\max _ { 1 \\le i \\le n } ( \\max ( M _ { v , i } , - m _ { v , i } ) ) \\\\ & = \\sum _ { v \\in M _ { K } } N _ { v } \\max ( M _ { v } , - m _ { v } ) \\\\ & \\le \\sum _ { v \\in M _ { K } } N _ { v } ( M _ { v } - m _ { v } ) \\\\ & \\le ( n + 1 ) \\sum _ { v \\in M _ { K } } N _ { v } M _ { v } . \\end{align*}"}
+{"id": "7350.png", "formula": "\\begin{align*} \\bigoplus ^ { m } _ { d = 0 } ( m - d - \\lceil \\frac { m - d } { 2 } \\rceil + 1 ) \\odot \\mathbb { C } ^ { ( 2 d + 2 ) \\times ( 2 d + 2 ) } . \\end{align*}"}
+{"id": "6703.png", "formula": "\\begin{align*} M \\equiv M _ { [ \\rho , u , \\theta ] } ( t , x , v ) : = \\frac { \\rho ( t , x ) } { \\sqrt { ( 2 \\pi R \\theta ( t , x ) ) ^ { 3 } } } \\exp \\big \\{ - \\frac { | v - u ( t , x ) | ^ { 2 } } { 2 R \\theta ( t , x ) } \\big \\} . \\end{align*}"}
+{"id": "5690.png", "formula": "\\begin{gather*} \\iota _ { i } \\tau _ * ( \\alpha _ { ( 0 , i ) } ) = \\sum _ { K \\in [ n ] ^ { 3 i } } a _ K e _ K \\in M _ i , \\end{gather*}"}
+{"id": "2052.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\Psi _ \\omega ( t ) \\ , e ^ { i z t } \\ , d t = - \\frac { 1 } { z ^ 2 } \\frac { \\xi ' } { \\xi } ( \\tfrac { 1 } { 2 } + \\omega - i z ) , \\Im ( z ) > 1 / 2 - \\omega \\end{align*}"}
+{"id": "5327.png", "formula": "\\begin{align*} X _ n = X _ { n - 1 } - \\alpha _ n U _ n , \\ , \\ , \\ , n \\geq 1 \\end{align*}"}
+{"id": "3484.png", "formula": "\\begin{align*} \\Big \\Vert \\mathcal { K } _ { } \\Big [ \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) \\Big ] \\Big \\Vert _ { \\mathrm { H } ^ { 1 , \\frac { 1 } { 2 } } \\Big ( \\partial \\Omega \\times \\mathbb { R } \\Big ) } ^ 2 = \\frac { 1 } { 4 } \\Big \\Vert \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) \\Big \\Vert _ { \\mathrm { H } ^ { 1 , \\frac { 1 } { 2 } } \\Big ( \\partial \\Omega \\times \\mathbb { R } \\Big ) } ^ 2 . \\end{align*}"}
+{"id": "6529.png", "formula": "\\begin{align*} H = H _ 0 + \\sum _ r 2 k _ r \\pi E _ { \\theta _ r } \\end{align*}"}
+{"id": "1126.png", "formula": "\\begin{align*} E X ^ { n } = \\int _ { 0 } ^ { \\infty } x ^ { n } f _ { g } ( x | \\beta ) d x = \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } . \\end{align*}"}
+{"id": "8076.png", "formula": "\\begin{align*} A x = b \\end{align*}"}
+{"id": "6191.png", "formula": "\\begin{align*} & i - p = | x \\cap y - z | \\leq | x - z | = m - j , \\\\ & i - p = | x \\cap y - z | \\leq | y - z | = m - t , \\\\ & j - p = | x \\cap z - y | \\leq | z - y | = m - t , \\end{align*}"}
+{"id": "1273.png", "formula": "\\begin{align*} M ' = ( M ^ \\perp ) ^ \\perp \\supseteq D _ 6 \\oplus A _ 1 ^ { m - 3 } \\supset M . \\end{align*}"}
+{"id": "5733.png", "formula": "\\begin{align*} \\mathcal { A } ^ { ' } = \\begin{pmatrix} \\mathcal { A } ^ { ' } _ { 1 1 } & \\cdots & \\mathcal { A } ^ { ' } _ { 1 q } \\\\ \\vdots & \\ddots & \\vdots \\\\ \\mathcal { A } ^ { ' } _ { q 1 } & \\cdots & \\mathcal { A } ^ { ' } _ { q q } \\end{pmatrix} = \\end{align*}"}
+{"id": "6942.png", "formula": "\\begin{align*} A _ i = \\sum _ { m = 0 } ^ { N } ( - y ) ^ m e _ { N + i - m } & = \\left [ t ^ { N + i } \\right ] ( 1 - y t + \\cdots + ( - 1 ) ^ N y ^ N t ^ N ) ( 1 + e _ 1 t + \\cdots + e _ { N + 1 } t ^ { N + 1 } ) \\\\ & = ( - 1 ) ^ { N + i } \\left [ t ^ { N + i } \\right ] \\frac { ( 1 - ( y t ) ^ { N + 1 } ) } { 1 - y t } \\cdot t ^ { N + 1 } P ( 1 / t ) . \\end{align*}"}
+{"id": "4481.png", "formula": "\\begin{align*} \\dd _ \\psi F ( h ) = \\langle D F ( \\psi ) , h \\rangle _ { H } \\ , , \\end{align*}"}
+{"id": "8209.png", "formula": "\\begin{align*} \\begin{aligned} \\Vert \\Lambda ^ { \\alpha } ( u v ) - u \\Lambda ^ { \\alpha } v \\Vert _ { L ^ p } \\lesssim _ { \\alpha , N } \\Vert \\nabla u \\Vert _ { L ^ { p _ 1 } } \\Vert \\Lambda ^ { \\alpha - 1 } v \\Vert _ { L ^ { p _ 2 } } + \\Vert \\Lambda ^ { \\alpha } u \\Vert _ { L ^ { p _ 1 } } \\Vert v \\Vert _ { L ^ { p _ 2 } } . \\end{aligned} \\end{align*}"}
+{"id": "7170.png", "formula": "\\begin{align*} \\mathfrak { g } _ l : = \\{ ( x _ { i , j } ) \\mid 1 \\leq i , j \\leq d , x _ { i , j } = 0 \\mbox { f o r } j \\leq i + l \\} . \\end{align*}"}
+{"id": "4032.png", "formula": "\\begin{align*} \\displaystyle \\rho _ s = \\frac { I ( s ) } { \\sqrt { 4 \\pi } } e ^ { - \\frac { I _ { s } ^ { 2 } y ^ 2 } { 4 } } , \\end{align*}"}
+{"id": "6042.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } - \\Delta \\eta - \\lambda \\eta = f \\ , \\ , \\ , \\ , \\Omega \\\\ \\eta = 0 \\ , \\ , \\ , \\ , \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "6340.png", "formula": "\\begin{align*} I ( u , a , b ) \\le I ( z , a , b ) = ( 1 + \\delta ) \\ , I ( z , a / ( 1 + \\delta ) , b / ( 1 + \\delta ) ) - \\delta \\norm { z } ^ 2 \\le - \\delta \\norm { z } ^ 2 . \\end{align*}"}
+{"id": "1817.png", "formula": "\\begin{align*} [ T ( a ) , T ( b ) ] = T \\big ( \\rho ( T ( a ) ) b - \\rho ( T ( b ) ) a \\big ) . \\end{align*}"}
+{"id": "4226.png", "formula": "\\begin{align*} \\widehat { \\mu } _ k ( y ) = \\frac { 1 } { k } \\sum _ { t = 0 } ^ { k - 1 } \\prod _ { j = 1 } ^ g J _ 0 \\left ( 2 \\left | \\sum _ { \\ell = 0 } ^ { k - 1 } \\frac { Z ( \\gamma _ { j , \\ell } ^ { - 1 } ) } { k \\gamma _ { j , \\ell } ^ { 1 - k } Z ' ( \\gamma _ j ^ { - 1 } ) } \\frac { \\gamma _ { j , \\ell } } { \\gamma _ { j , \\ell } - 1 } e ^ { 2 \\pi i [ \\ell t ] / k } \\right | y \\right ) , \\end{align*}"}
+{"id": "4979.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\rm a p p r o x } = \\mathcal { O } ( K L ( p _ { w _ { 1 } } ( M + 1 ) ^ { w _ { 1 } } + p _ { w _ { 2 } } ( M + 1 ) ^ { w _ { 2 } } ) ) , \\end{align*}"}
+{"id": "2536.png", "formula": "\\begin{align*} \\phi _ m ( x ) = \\begin{cases} 1 & x \\in A _ m ^ { ( 1 ) } \\\\ 0 & x \\notin A _ m ^ { ( 2 ) } . \\end{cases} \\end{align*}"}
+{"id": "7533.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { E _ { \\varphi } ( F _ { \\varepsilon , h } ) - E _ { \\varphi } ( F ) } { \\varepsilon } = 0 \\end{align*}"}
+{"id": "4409.png", "formula": "\\begin{align*} c = | c | \\left ( q ^ 2 - ( 2 a - 1 ) q + a ^ 2 - a + \\frac { c } { | c | } \\right ) . \\end{align*}"}
+{"id": "5209.png", "formula": "\\begin{align*} ( s - 1 ) L < \\sum _ { i = 1 } ^ { j _ { s } - 1 } ( a _ i + b _ i ) + a _ { j _ s } . \\end{align*}"}
+{"id": "6232.png", "formula": "\\begin{align*} H = - \\partial _ x ^ 2 + ( - i \\partial _ y + B x ) ^ 2 - \\partial _ z ^ 2 \\end{align*}"}
+{"id": "5897.png", "formula": "\\begin{align*} \\norm { \\varphi _ n ^ \\tau } = \\mathcal { O } ( h ^ 2 ) , \\ \\norm { \\varphi _ n ^ x } = \\mathcal { O } ( h ^ 3 / \\epsilon ) , \\ \\norm { \\varphi _ n ^ v } = \\mathcal { O } ( h ^ 3 / \\epsilon ) . \\end{align*}"}
+{"id": "997.png", "formula": "\\begin{align*} \\rho \\ , u _ { , t t } & = \\ , \\boldsymbol { \\sigma } \\ , , \\\\ \\rho \\eta \\ , \\tau _ { \\rm c } ^ 2 \\ , \\mathbf { A } _ { , t t } & = - \\ , \\ , \\boldsymbol { m } + \\ , \\boldsymbol { \\sigma } \\ , , \\end{align*}"}
+{"id": "4331.png", "formula": "\\begin{align*} d _ { T V } ( X , \\pi ) \\leq \\sup _ { h \\in \\mathcal { H } } \\sup _ { l \\in \\mathbb { Z } ^ + } | m _ l ( h ) | \\mathbb { P } ( X = n ) \\left ( n + 1 - \\mathbb { E } \\nu \\right ) P _ { n , n + 1 } . \\end{align*}"}
+{"id": "7097.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { Q } } K \\left ( \\mathbb { T } ( n d ) _ { n v } \\right ) \\geq \\dim _ { \\mathbb { Q } } K _ 0 ^ { \\mathrm { t o p } } \\left ( \\mathbb { T } ( n d ) _ { n v } \\right ) = p _ 2 ( n ) . \\end{align*}"}
+{"id": "4183.png", "formula": "\\begin{align*} \\mu _ t = \\left ( \\frac { i - t } { i + t } \\right ) \\cdot \\frac { \\bar { \\omega } } { | \\omega | } \\end{align*}"}
+{"id": "3915.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\tau ^ { - \\delta } ( t - \\tau ) ^ { - 2 \\gamma } d \\tau = B ( 1 - \\delta , 1 - 2 \\gamma ) t ^ { 1 - \\delta - 2 \\gamma } . \\end{align*}"}
+{"id": "6487.png", "formula": "\\begin{align*} \\delta u ( x , y ) : \\ , = u ( x + y ) + u ( x - y ) - 2 u ( x ) , \\end{align*}"}
+{"id": "407.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq a , b _ 1 , b _ 2 , \\ldots , b _ s \\leq n \\\\ ( a , n ) = 1 } } ( a - 1 , b _ 1 , \\ldots , b _ s , n ) = \\varphi ( n ) \\sigma _ s ( n ) , \\ ; \\ ; \\sigma _ s ( n ) = \\displaystyle \\sum _ { d \\mid n } d ^ s . \\end{align*}"}
+{"id": "6970.png", "formula": "\\begin{align*} S _ i ^ * Q P - Q S _ { d - i } = \\lim _ j ( S _ i ^ * P ^ { * j } P ^ j P - P ^ { * j } P ^ j S _ { d - i } ) = \\lim _ j P ^ { * j } ( S _ i ^ * P - S _ { d - i } ) P ^ j . \\end{align*}"}
+{"id": "4392.png", "formula": "\\begin{align*} \\le ( 2 \\pi t ) ^ { - \\frac { d } { 2 } } e ^ { - \\frac { | x - y | ^ 2 } { 4 t } } = K _ t ( x - y ) . \\end{align*}"}
+{"id": "2720.png", "formula": "\\begin{align*} \\binom { 2 n } { 2 n - j - u } \\binom { 2 n - j - u } { n } ( - 1 ) ^ { j + u + r - 1 } \\binom { j + u + r - 1 } { j + u } \\frac { 1 } { 2 } S ( n , r ) \\cdot \\\\ \\cdot \\sum _ { l = 0 } ^ { r - 1 } ( - 1 ) ^ l \\binom { 2 n - j - u + l } { l } \\frac { \\binom { 2 ( j + u + r - 1 - l ) } { j + u + r - 1 - l } \\binom { 2 ( n - j - u + l + 1 ) } { n - j - u + l + 1 } \\binom { n - j - u } { j + u + r - l - 1 } } { 2 \\binom { 2 n - j - u + l + 1 } { n } } \\end{align*}"}
+{"id": "5763.png", "formula": "\\begin{align*} 0 \\to L _ 1 \\to E \\to L _ 2 \\to 0 \\end{align*}"}
+{"id": "2097.png", "formula": "\\begin{align*} \\int _ X h \\ , | \\omega _ 2 \\wedge \\varphi ^ * \\eta | = \\int _ { y \\in Y } \\Big ( \\int _ { \\varphi ^ { - 1 } ( y ) } h \\ , | \\omega _ 2 | \\Big ) \\eta ( y ) , \\end{align*}"}
+{"id": "6103.png", "formula": "\\begin{align*} \\mathcal B ( k , d ) = \\left \\{ A \\in \\binom { [ n ] } { k } : [ d - 1 ] \\subseteq A , A \\cap [ d , k + 1 ] \\neq \\emptyset \\right \\} \\cup \\binom { [ k + 1 ] } { k } . \\end{align*}"}
+{"id": "3951.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } ^ { \\beta } } { \\mathrm { d } t ^ { \\beta } } f ( t ) : = \\left \\{ \\begin{array} { l l } \\dfrac { 1 } { \\Gamma { ( 1 - \\beta ) } } \\displaystyle \\int ^ t _ { 0 } ( t - s ) ^ { - \\beta } f ' ( s ) \\ , \\mathrm { d } s , \\ 0 < \\beta < 1 , \\\\ \\\\ f ' ( t ) , \\ \\beta = 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "7282.png", "formula": "\\begin{align*} \\frac { d } { d t } X ( t , \\omega ) = A ( t ) X ( t , \\omega ) + B ( t ) u ( t , \\omega ) , \\end{align*}"}
+{"id": "3438.png", "formula": "\\begin{align*} E _ v ( a f ( b _ { + 1 } ) ) = \\sum _ { i \\geq 0 } E ^ { ( i ) } _ v ( b ) ( { - D _ z } ) ^ { i + 1 } f ' ( b _ { + 1 } ) + E ^ { ( i ) } _ v ( a ) ( { - D _ z } ) ^ i ( f ( b _ { + 1 } ) - b _ { + 1 } f ' ( b _ { + 1 } ) ) \\end{align*}"}
+{"id": "4760.png", "formula": "\\begin{gather*} [ ( a + u ) , ( b + v ) ] : = [ a , b ] + ( \\rho ( a ) v - \\rho ( b ) u ) , \\\\ ( a + u ) \\cdot ( b + v ) : = a \\cdot b + ( \\mu ( a ) v + \\mu ( b ) u ) , \\forall a , b \\in A , u , v \\in V . \\end{gather*}"}
+{"id": "5082.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": "2515.png", "formula": "\\begin{align*} F ^ * ( x _ { 0 0 } , x _ { 0 1 } , x _ { 1 0 } , x _ { 1 1 } ) = ( x _ { 0 1 } , x _ { 1 0 } , x _ { 1 1 } ) \\end{align*}"}
+{"id": "2020.png", "formula": "\\begin{align*} \\ll \\frac { 1 } { X ^ 2 } \\sum _ { n } \\frac { \\Lambda ( n ) } { \\sqrt { n } } e ^ { c ( t - \\log n ) } = \\frac { e ^ { c t } } { X ^ 2 } \\left [ - \\frac { \\zeta ' } { \\zeta } \\left ( \\frac { 1 } { 2 } + c \\right ) \\right ] \\ll \\frac { t \\cdot e ^ { t / 2 } } { X ^ 2 } , \\end{align*}"}
+{"id": "7455.png", "formula": "\\begin{align*} [ \\hat { j } , \\ , p _ 0 ^ \\dagger ] = ( p _ 0 ^ \\dagger + 4 p _ 1 ^ \\dagger ) \\frac { 1 } { 2 \\hat { j } + 1 } , [ \\hat { j } , \\ , p _ 1 ^ \\dagger ] = ( p _ 0 ^ \\dagger J ^ 2 - p _ 1 ^ \\dagger ) \\frac { 1 } { 2 \\hat { j } + 1 } . \\end{align*}"}
+{"id": "246.png", "formula": "\\begin{align*} { \\bf P } ( \\forall i \\in D \\ \\Rightarrow \\ \\xi _ i \\ge x _ i ) = { \\bf P } \\left [ \\ \\cap _ { i \\in D } \\{ \\xi _ i \\ge x _ i \\} \\ \\right ] \\le \\end{align*}"}
+{"id": "1633.png", "formula": "\\begin{align*} | S | & \\leq ( k - 1 ) ( ( k - 1 ) ( \\ell + 3 ) - 1 ) + 1 \\\\ & = ( k - 1 ) ^ 2 ( \\ell + 3 ) - k + 2 \\\\ & < ( k - 1 ) ^ 2 ( \\ell + 3 ) , \\end{align*}"}
+{"id": "1295.png", "formula": "\\begin{align*} ( \\{ f _ j \\colon a \\to b \\mid j \\in J \\} ) ^ \\dagger = \\{ f _ j ^ \\dagger \\colon b \\to a \\mid j \\in J \\} . \\end{align*}"}
+{"id": "5272.png", "formula": "\\begin{align*} L _ { n _ 1 , \\ldots , n _ m } : = \\{ t = ( t _ 1 , \\ldots , t _ { n + 1 } ) \\in \\R ^ { n + 1 } : \\ \\sum _ { i = 1 } ^ { n + 1 } t _ i = 0 ; \\ \\ \\sum _ { M _ { j - 1 } < i \\leq M _ j } t _ i = 0 , \\ \\ 1 \\leq j \\leq m \\} , \\end{align*}"}
+{"id": "4825.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial \\Pi _ t } { \\partial t } = & \\ , \\nabla _ { x _ 1 } \\cdot ( \\Pi _ t \\nabla _ { x _ 1 } f ( x _ 1 , s _ 1 ) ) + \\nabla _ { x _ 2 } \\cdot ( \\Pi _ t \\nabla _ { x _ 2 } f ( x _ 2 , s _ 2 ) ) + \\\\ & + K ~ \\int _ S \\int _ S ( \\Gamma ( s _ 1 , s _ 2 ) \\Pi _ t ( x _ 1 , \\dd s _ 1 ' , x _ 2 , \\dd s _ 2 ' ) - \\Pi _ t ( x _ 1 , s _ 1 , x _ 2 , s _ 2 ) ) \\end{aligned} \\end{align*}"}
+{"id": "6613.png", "formula": "\\begin{align*} E _ n ( Q ^ \\gamma _ \\theta ) = - \\frac { \\gamma ^ { 2 } } { ( 2 n - 1 ) ^ { 2 } \\theta ^ { 2 } } + O ( 1 ) \\theta \\to 0 ^ + . \\end{align*}"}
+{"id": "4716.png", "formula": "\\begin{align*} D _ { z } \\left ( z \\widetilde { K } _ { \\lambda , 1 } ( z , w ) \\right ) = K _ { \\lambda , 2 } ( z , w ) , \\end{align*}"}
+{"id": "4652.png", "formula": "\\begin{align*} \\chi _ P ( a ) = \\Omega \\left ( a ^ { \\frac { q ^ { \\deg ( P ) } - 1 } { \\ell } } \\right ) = \\Omega \\left ( a ^ { \\frac { \\deg ( P ) ( q ^ { n _ q } - 1 ) } { n _ q \\ell } } \\right ) . \\end{align*}"}
+{"id": "4178.png", "formula": "\\begin{align*} h ( x + i \\alpha ) = g ( x + i \\alpha ) + \\eta _ i . \\end{align*}"}
+{"id": "2209.png", "formula": "\\begin{align*} & r _ 1 = - R _ { 1 2 1 2 } ( R _ { 2 3 2 3 } S _ { 1 2 1 3 1 } + R _ { 1 3 1 3 } S _ { 1 2 2 3 2 } ) = 0 , \\\\ & r _ 4 = R _ { 1 3 1 3 } ( R _ { 2 3 2 3 } S _ { 1 2 1 3 1 } - R _ { 1 2 1 2 } S _ { 1 3 2 3 3 } ) = 0 , \\\\ & r _ 6 = R _ { 2 3 2 3 } ( R _ { 1 3 1 3 } S _ { 1 2 2 3 2 } + R _ { 1 2 1 2 } S _ { 1 3 2 3 3 } ) = 0 . \\end{align*}"}
+{"id": "3377.png", "formula": "\\begin{align*} d ^ * _ p = d ^ * _ p ( f ) : = \\limsup _ { n \\to \\infty } \\sup _ S \\| ( f ^ n ) _ * ( S ) \\| _ W ^ { 1 / n } \\end{align*}"}
+{"id": "7440.png", "formula": "\\begin{align*} \\hat { j } = \\frac { 1 } { 2 } ( \\sqrt { \\hat { I } + 4 J ^ 2 } - \\hat { I } ) . \\end{align*}"}
+{"id": "5894.png", "formula": "\\begin{align*} \\xi ^ { x } _ { n } ( h ) = \\mathcal { O } ( h ^ 3 / \\epsilon ) . \\end{align*}"}
+{"id": "1639.png", "formula": "\\begin{align*} \\eta ( \\mathsf { I } , \\mathsf { J } ) \\ ; : = \\ ; 1 \\ , - \\ , \\frac { \\kappa _ { \\mathsf { J } } ^ 2 } { \\kappa _ { \\mathsf { I } } ^ 2 } \\ ; \\in \\ ; [ 0 , 1 ] \\ , , 1 \\leq \\mathsf { I } < \\mathsf { J } \\leq \\mathsf { L } \\ ; . \\end{align*}"}
+{"id": "1841.png", "formula": "\\begin{align*} u ' & : = ( A \\otimes I ) ( A \\otimes I - I \\otimes B ) ^ { - 1 } ( u \\otimes v ) \\\\ v ' & : = - ( I \\otimes B ) ( A \\otimes I - I \\otimes B ) ^ { - 1 } ( u \\otimes v ) . \\end{align*}"}
+{"id": "193.png", "formula": "\\begin{align*} Q ^ + \\psi + \\pi ^ { - } ( a \\cdot \\psi ) = 0 , \\ , \\ , \\ , d ^ * a = 0 , \\ , \\ , \\ , d ^ + a = \\rho ^ { - 1 } ( \\mu ( \\psi ) ) . \\end{align*}"}
+{"id": "1697.png", "formula": "\\begin{align*} \\mathrm { T r } ( h ^ { - 1 } ) = \\sum _ { k \\in \\N } \\frac { 1 } { 4 \\pi ^ 2 | k | ^ 2 + \\kappa } < \\infty , \\end{align*}"}
+{"id": "7028.png", "formula": "\\begin{align*} \\widehat { J } _ 0 : = \\frac { \\sin ( | \\xi | ^ 2 t ) } { | \\xi | ^ 2 } \\mathrm { e } ^ { - \\frac { 1 } { 2 \\sigma } | \\xi | ^ 4 t } \\ \\ \\mbox { a n d } \\ \\ \\widehat { J } _ 1 : = \\cos ( | \\xi | ^ 2 t ) \\mathrm { e } ^ { - \\frac { 1 } { 2 \\sigma } | \\xi | ^ 4 t } . \\end{align*}"}
+{"id": "3882.png", "formula": "\\begin{align*} ( 1 + \\delta ) \\| T ^ \\ast \\omega \\| _ { \\varphi _ 1 } ^ 2 + \\| S \\omega \\| _ { \\varphi _ 3 } ^ 2 \\ge \\int _ { \\Omega _ \\varepsilon } \\left ( \\sum _ { j , k = 1 } ^ n \\frac { \\partial ^ 2 \\varphi } { \\partial z _ j \\partial \\bar { z } _ k } \\omega _ j \\bar { \\omega } _ k - ( 1 + 1 / \\delta ) | \\partial \\psi | ^ 2 | \\omega | ^ 2 \\right ) e ^ { - \\varphi } \\end{align*}"}
+{"id": "3712.png", "formula": "\\begin{align*} \\hat F _ \\infty ( \\beta ) : = \\begin{cases} \\underset { n \\to \\infty } { l i m } ( F _ n ( \\beta ) - n d ) & \\forall \\beta \\in \\Sigma \\\\ + \\infty \\end{cases} \\end{align*}"}
+{"id": "6996.png", "formula": "\\begin{align*} \\int \\langle \\nabla ( \\eta ( - u - k ) _ + ) , \\nabla u \\rangle d m = \\int \\eta ( - u - k ) _ { + } \\left ( \\alpha _ 2 u \\log u + ( \\lambda - \\alpha _ 1 ) u \\right ) d m . \\end{align*}"}
+{"id": "813.png", "formula": "\\begin{align*} l _ 1 + l _ 2 \\Big ( \\frac { K } { \\bar { \\theta } _ 1 } \\cdot \\frac { k _ 2 } { k _ 2 - 1 } \\Big ) ^ { 1 - k _ 2 } x ^ { k _ 2 } = \\frac { 1 - \\theta } { \\bar { \\theta } _ 1 - \\theta } . \\end{align*}"}
+{"id": "4182.png", "formula": "\\begin{align*} \\mu _ t = \\left ( \\frac { i - t } { i + t } \\right ) \\cdot \\frac { \\bar { q } } { | q | } . \\end{align*}"}
+{"id": "3807.png", "formula": "\\begin{align*} \\l = ( 1 , 2 , \\ldots , n - 6 , n - 4 , n - 3 , n + 1 , 2 n - 4 ) \\end{align*}"}
+{"id": "8165.png", "formula": "\\begin{align*} \\langle A _ j \\eta , \\xi _ j \\rangle = 1 \\ \\ \\mbox { a n d } \\ \\ \\langle A _ j \\eta , \\xi _ k \\rangle = 0 , \\ \\mbox { i f } \\ k \\ne j . \\end{align*}"}
+{"id": "6995.png", "formula": "\\begin{align*} \\frac { 1 } { r _ 2 } = \\frac { 1 } { r _ 1 } - \\frac { 2 } { N } + \\frac { 1 } { \\delta } = \\frac { 1 } { 2 ^ * } - \\frac { 2 } { N } + \\frac { 2 } { \\delta } . \\end{align*}"}
+{"id": "6208.png", "formula": "\\begin{align*} M ^ { 0 , 0 } _ { \\frac { 2 m - i - k + 1 } { 2 } , \\frac { i + k - 3 } { 2 } } M ^ { \\frac { k - 3 } { 2 } , \\frac { k - 3 } { 2 } } _ { \\frac { i + k - 3 } { 2 } , \\frac { 2 m - i } { 2 } } & = \\frac { k - 1 } { 2 } M ^ { \\frac { 2 m - k + 1 } { 2 } , \\frac { 2 m - i - k + 1 } { 2 } } _ { \\frac { 2 m - i - k + 1 } { 2 } , \\frac { 2 m - i } { 2 } } \\end{align*}"}
+{"id": "2333.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 2 n } Z ( m _ { 2 n - j + 1 } , \\dots , m _ { 2 n } , m _ { 0 } , \\dots , m _ { 2 n - j } ) = \\zeta ( \\{ 2 \\} ^ { 2 n + \\sum _ { j = 0 } ^ { 2 n } m _ { j } } ) . \\end{align*}"}
+{"id": "7758.png", "formula": "\\begin{align*} 1 = \\lim \\limits _ { t \\rightarrow + \\infty } \\frac { V o l ( \\partial B ^ { g ^ + } _ p ( t ) ) } { V o l ( \\partial B ^ \\mathbb { H } _ o ( t ) ) } \\leq \\lim \\limits _ { t \\rightarrow 0 ^ + } \\frac { V o l ( \\partial B ^ { g ^ + } _ p ( t ) ) } { V o l ( \\partial B ^ \\mathbb { H } _ o ( t ) ) } = 1 \\end{align*}"}
+{"id": "4426.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { W ^ 1 _ p } \\leq C \\Vert \\epsilon \\Vert _ { L _ p } , 1 < p < \\infty \\epsilon = \\tfrac { 1 } { 2 } \\left ( \\nabla u + \\nabla u ^ T \\right ) . \\end{align*}"}
+{"id": "293.png", "formula": "\\begin{align*} f : X ^ { \\prime } = X _ { s } \\xrightarrow { f _ { s } } X _ { s - 1 } \\xrightarrow { f _ { s - 1 } } \\cdots \\xrightarrow { f _ { 1 } } X _ { 0 } = X \\end{align*}"}
+{"id": "185.png", "formula": "\\begin{align*} Q ^ + \\psi + \\pi ^ - ( a \\cdot \\psi ) = 0 , \\ , \\ , \\ , \\ , d ^ + a = \\rho ^ { - 1 } ( \\mu ( \\psi ) ) \\end{align*}"}
+{"id": "212.png", "formula": "\\begin{align*} & \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ { N } } \\omega \\varphi d x d t \\leq C ( p ) \\biggr ( \\mathcal { J } _ 1 + k \\mathcal { J } _ 2 + \\mathcal { J } _ 3 \\biggr ) , \\end{align*}"}
+{"id": "1057.png", "formula": "\\begin{align*} \\mathcal { G } ^ { \\Delta _ { T } } ( V , W ) = { \\frac { v _ { n - 1 } } { 6 } } ( V ) \\ , \\int _ { M } G ( V , W ) \\ , v o l _ { g } + { \\frac { v _ { n - 1 } } { 2 } } \\int _ { M } F ( V , W ) \\ , v o l _ { g } , \\end{align*}"}
+{"id": "7454.png", "formula": "\\begin{align*} \\begin{matrix} [ \\hat { j } , \\ , \\tau ^ \\dagger _ { - 1 } ] = - \\tau ^ \\dagger _ { - 1 } , [ \\hat { j } , \\ , \\tau ^ \\dagger _ { 1 } ] = \\tau ^ \\dagger _ { 1 } \\\\ \\end{matrix} \\end{align*}"}
+{"id": "5636.png", "formula": "\\begin{align*} W ( \\gamma ( t ) ) = s _ \\lambda ( t ) V ( \\gamma ( t ) ) . \\end{align*}"}
+{"id": "4098.png", "formula": "\\begin{align*} J ( d _ 1 , \\ldots , d _ n ; e _ 1 , \\ldots , e _ n ) \\leq \\begin{cases} \\frac { \\beta ^ { n + 1 } r ^ { n + 2 } } { ( r - 2 e \\alpha ) ( \\beta - 1 ) ( r - 1 ) } + \\frac { ( 2 e \\alpha ) ^ { n + 2 } \\beta ^ { n + 1 } } { ( \\beta - 1 ) ( r - 2 e \\alpha ) ( 2 e \\alpha \\beta - 1 ) } , r > 2 e \\alpha \\\\ \\frac { \\beta ^ { n + 1 } ( 2 e \\alpha ) ^ { n + 2 } } { ( 2 e \\alpha - r ) ( \\beta - 1 ) ( 2 e \\alpha - 1 ) } + \\frac { \\beta ^ { n + 1 } r ^ { n + 2 } } { ( \\beta - 1 ) ( 2 e \\alpha - r ) ( \\beta r - 1 ) } . , 2 e \\alpha > r . \\end{cases} \\end{align*}"}
+{"id": "4474.png", "formula": "\\begin{align*} F ( v ) = \\langle \\psi , v \\rangle _ { \\mathcal { H } } \\ , , \\forall v \\in \\mathcal { H } \\ , . \\end{align*}"}
+{"id": "2989.png", "formula": "\\begin{align*} Q : = \\left \\{ ( x , y ) \\in \\Omega ^ 2 : ( 1 - S ( x , y ) ) \\int _ { z \\in \\Omega } S ( x , z ) S ( y , z ) > 0 \\right \\} . \\end{align*}"}
+{"id": "7269.png", "formula": "\\begin{align*} \\psi ( T , \\gamma ) \\sharp \\eta ^ * = - \\nabla _ x \\sigma ( \\gamma ( T ) , e _ T \\sharp \\eta ^ * ) - \\int _ { \\Gamma } \\nabla _ m \\sigma ( \\gamma ' ( t ) , e _ T \\sharp \\eta ^ * , \\gamma ( T ) ) \\eta ^ * ( d \\gamma ' ) \\end{align*}"}
+{"id": "6736.png", "formula": "\\begin{align*} \\partial _ { x _ i } M = M \\big \\{ \\frac { \\partial _ { x _ i } \\rho } { \\rho } + \\frac { ( v - u ) \\cdot \\partial _ { x _ i } u } { R \\theta } + ( \\frac { | v - u | ^ { 2 } } { 2 R \\theta } - \\frac { 3 } { 2 } ) \\frac { \\partial _ { x _ i } \\theta } { \\theta } \\big \\} . \\end{align*}"}
+{"id": "4356.png", "formula": "\\begin{align*} \\int _ \\Omega \\Phi ( f _ n ( t ) ) \\ , \\mathrm { d } x - \\int _ \\Omega \\Phi ( f _ n ( t _ 0 ) ) \\ , \\mathrm { d } x = \\int _ { t _ 0 } ^ t \\Big \\langle \\partial _ t f _ n ( \\tau ) , \\Phi ' ( f _ n ( \\tau ) ) \\Big \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } \\ , \\mathrm { d } \\tau \\ , . \\end{align*}"}
+{"id": "6652.png", "formula": "\\begin{align*} \\lambda _ { i , 2 } = \\sqrt { 1 + \\tau _ i ( [ 0 , 1 ] ) } = \\sqrt { 1 + \\tau _ 1 ( [ 0 , 1 ] ) } = \\lambda _ { 1 , 2 } \\end{align*}"}
+{"id": "2941.png", "formula": "\\begin{align*} N _ \\Omega ( \\bar z ) : = \\{ z ^ * \\in \\R ^ n \\ , | \\ , \\exists z _ k \\to \\bar z , \\ , \\exists z _ k ^ * \\to z ^ * , \\ , \\forall k \\in \\N \\colon \\ , z _ k ^ * \\in \\widehat N _ \\Omega ( z _ k ) \\} . \\end{align*}"}
+{"id": "6844.png", "formula": "\\begin{align*} \\gcd ( p ^ r + 1 , p ^ s - 1 ) = \\left \\{ \\begin{array} { r c l } 1 & & { i f \\ \\frac { s } { \\gcd ( r , s ) } \\ i s \\ o d d \\ a n d \\ p \\ i s \\ e v e n , } \\\\ 2 & & { i f \\ \\frac { s } { \\gcd ( r , s ) } \\ i s \\ o d d \\ a n d \\ p \\ i s \\ o d d , } \\\\ p ^ { \\gcd ( r , s ) } + 1 & & { i f \\ \\frac { s } { \\gcd ( r , s ) } \\ i s \\ e v e n . } \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "3058.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ { \\ell } ) = \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } k C _ { \\sigma } ( k ) = \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } k = s ( k ) \\end{align*}"}
+{"id": "2979.png", "formula": "\\begin{align*} [ W _ { i + 1 } ] = [ W _ i ] + ( - 1 ) ^ { i } [ W _ 1 ] \\end{align*}"}
+{"id": "790.png", "formula": "\\begin{align*} { C } _ { \\theta } ( \\tau _ i , \\tau _ { - i } ) = \\theta \\underbrace { e ^ { - \\beta \\tau _ i } x _ i ( \\tau _ i ) } _ { } + ( 1 - \\theta ) \\underbrace { \\frac { e ^ { - \\beta \\tau _ i } x _ i ( \\tau _ i ) } { l _ 1 + l _ 2 \\left ( \\frac { 1 } { N } \\sum \\limits _ { j = 1 } ^ N e ^ { - \\beta \\tau _ j } x _ j ( \\tau _ j ) \\right ) } } _ { } . \\end{align*}"}
+{"id": "3331.png", "formula": "\\begin{gather*} a _ 1 = - q ^ { 2 \\alpha _ 0 + 2 \\alpha _ 2 + 2 } , a _ 2 = q ^ { - 2 N _ 2 } , a _ 3 = q ^ { - 2 N _ 1 } , b = - q ^ { 2 \\alpha _ 0 } , \\\\ 2 N = - \\hat { \\alpha } + N _ 1 + N _ 2 - 1 , \\end{gather*}"}
+{"id": "4376.png", "formula": "\\begin{align*} \\begin{array} { l } \\sum _ { i = 1 } ^ { l } \\theta _ i J _ i ( x , u ) = \\sum _ { i = 1 } ^ { l } \\theta _ i g _ i ^ 0 ( x ( T ) ) + \\int _ { 0 } ^ { T } \\sum _ { i = 1 } ^ { l } \\theta _ i f ^ 0 _ i ( t , x ( t ) , u ( t ) ) d t \\\\ \\le \\sum _ { i = 1 } ^ { l } \\theta _ i g _ i ^ 0 ( \\overline { x } ( T ) ) + \\sum _ { i = 1 } ^ { l } \\theta _ i D _ H g _ i ( \\overline { x } ( T ) ) \\cdot ( x ( T ) - \\overline { x } ( T ) ) + \\\\ \\int _ { 0 } ^ { T } \\sum _ { i = 1 } ^ { l } \\theta _ i f ^ 0 _ i ( t , x ( t ) , u ( t ) ) d t . \\end{array} \\end{align*}"}
+{"id": "1370.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t ^ 2 u - \\Delta u + \\partial _ t u = f ( u ) , & t > 0 , x \\in \\Omega , \\\\ u ( t , x ) = 0 , & t > 0 , x \\in \\partial \\Omega , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\ \\partial _ t u ( 0 , x ) = u _ 1 ( x ) , & x \\in \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "6820.png", "formula": "\\begin{align*} \\partial _ { t t } u _ 1 - \\omega \\kappa \\Delta \\partial _ t u _ 1 - ( 1 - \\omega ) \\kappa \\ , \\int _ 0 ^ \\infty g ( s ) \\Delta \\partial _ t u _ 1 ( t - s ) d s - \\kappa _ 1 \\Delta u _ 1 = 0 . \\end{align*}"}
+{"id": "6050.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } \\eta ^ 2 ( V \\cdot n ) d \\sigma + 2 \\int _ \\Omega \\eta \\eta ' d x = 0 . \\end{align*}"}
+{"id": "6234.png", "formula": "\\begin{align*} H = \\int ^ \\oplus _ \\mathbb { R } H ( p ) \\ , \\mathrm { d } p , \\end{align*}"}
+{"id": "2644.png", "formula": "\\begin{align*} \\lambda _ x | _ { E _ w } = \\lambda _ y | _ { E _ w } , \\lambda _ x ( w , w b ) = f \\lambda _ x ( w b ) = s ( f ) \\end{align*}"}
+{"id": "7703.png", "formula": "\\begin{gather*} \\mathcal { H } ^ k ( \\bar { M } ) : = \\{ \\gamma \\in H ^ 1 _ { \\operatorname { l o c } } \\Omega ^ k ( \\bar { M } ) | d \\omega = 0 = \\delta \\omega \\} , \\\\ \\mathcal { H } ^ k _ N ( \\bar { M } ) : = \\{ \\gamma \\in \\mathcal { H } ^ k ( \\bar { M } ) | n ( \\gamma ) = 0 \\} , \\\\ \\mathcal { H } ^ k _ D ( \\bar { M } ) : = \\{ \\gamma \\in \\mathcal { H } ^ k ( \\bar { M } ) | t ( \\gamma ) = 0 \\} , \\end{gather*}"}
+{"id": "944.png", "formula": "\\begin{align*} \\bar { y } _ n : = y _ n - \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( x ' _ 0 ) y _ \\beta . \\end{align*}"}
+{"id": "6344.png", "formula": "\\begin{align*} \\mathcal { A } _ { P } ^ { ( m ) } ( F , G ) = \\textit { m a x } \\left \\{ \\frac { \\left \\| E _ { \\Lambda , ( F , G ) } \\right \\| + \\rho { ( E _ { \\Lambda , ( F , G ) } ) } } { 2 } : D _ { p } \\in \\mathcal { D } ^ { P } _ { m } \\right \\} , \\end{align*}"}
+{"id": "1023.png", "formula": "\\begin{align*} f = x _ 0 p _ 0 ( u , v ) + x _ 1 p _ 1 ( u , v ) + x _ 2 p _ 2 ( u , v ) + g ( u , v ) \\in R _ d . \\end{align*}"}
+{"id": "5466.png", "formula": "\\begin{align*} F ( a , b , t , N ) = \\frac { 1 - a t q } { 1 - t } + \\frac { ( 1 - a q ) ( b - a t q ) } { ( 1 - b q ) ( 1 - t ) } t q F ( a , b , t , N ) + R _ { 1 , N } ( a , b , t ) , \\end{align*}"}
+{"id": "7757.png", "formula": "\\begin{align*} \\begin{aligned} d ( p _ 0 , y ) & \\leq d ( y , ( t _ 0 , w , \\theta ) ) + d ( ( t _ 0 , w , \\theta ) , ( t _ 0 , w _ 0 , \\theta ) ) + d ( ( t _ 0 , w _ 0 , \\theta ) , p _ 0 ) \\\\ & \\leq t - t _ 0 + \\sinh t _ 0 | w - w _ 0 | + \\lambda \\cosh t _ 0 | \\theta - \\theta _ 0 | \\end{aligned} \\end{align*}"}
+{"id": "6630.png", "formula": "\\begin{align*} \\big | s _ { \\theta ' } ( v , v ) - s _ { \\theta } ( v , v ) \\big | & = \\big | \\tan ^ 2 \\theta ' - \\tan ^ 2 \\theta \\big | \\int _ { \\Omega _ { \\frac { \\pi } { 4 } } \\setminus O y ^ + } | \\partial _ y v | ^ 2 \\dd x \\dd y \\\\ & \\le \\varepsilon \\int _ { \\Omega _ { \\frac { \\pi } { 4 } } \\setminus O y ^ + } | \\partial _ y v | ^ 2 \\dd x \\dd y \\le c \\varepsilon \\| v \\| ^ 2 _ s . \\end{align*}"}
+{"id": "4472.png", "formula": "\\begin{align*} \\langle u , v \\rangle : = \\int _ { \\mathbb { R } ^ 3 } \\big ( u v + \\sum _ { i = 1 } ^ 3 \\left ( \\partial _ i u \\right ) \\ , \\left ( \\partial _ i v \\right ) \\big ) \\ . \\end{align*}"}
+{"id": "1530.png", "formula": "\\begin{align*} \\tilde { p } \\left ( x _ { n + 1 } \\right ) = \\sum _ { i = 0 } ^ { r } q _ { i } \\left ( z _ { 1 } , \\ldots , z _ { n } \\right ) x _ { n + 1 } ^ { i } \\neq 0 \\end{align*}"}
+{"id": "3905.png", "formula": "\\begin{align*} u ( t ) = S ( t ) \\xi + \\int _ 0 ^ t S ( t - \\tau ) F ( \\tau ) d \\tau , \\end{align*}"}
+{"id": "7361.png", "formula": "\\begin{align*} ( d R _ x ) _ { 0 _ x } ( \\xi ) = \\xi \\end{align*}"}
+{"id": "8257.png", "formula": "\\begin{align*} \\begin{aligned} Z _ { s , \\bar { s } } ( t ) \\le C \\exp \\Big \\{ C \\int _ 0 ^ t \\Big ( \\Phi ( t , \\tau ) + \\Vert u ( \\tau ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } } \\Big ) \\dd \\tau \\Big \\} . \\end{aligned} \\end{align*}"}
+{"id": "6313.png", "formula": "\\begin{align*} & | f ( t , x ) - f _ n ( t , x ) | \\\\ & = \\int _ { [ - \\delta , \\delta ] } \\delta _ n ( y ) \\big | f ( t , x ) - f ( t , x - y ) \\big | \\dd y + \\int _ { [ - 1 , 1 ] \\setminus [ - \\delta , \\delta ] } \\delta _ n ( y ) \\big | f ( t , x ) - f ( t , x - y ) \\big | \\dd y . \\end{align*}"}
+{"id": "1084.png", "formula": "\\begin{align*} b _ { 1 } u _ { 1 } ^ { 2 } l ^ { 2 ( k _ { 1 } - k _ { 3 } ) } - b _ { 1 } b _ { 2 } u _ { 3 } ^ { 2 } = - p l ^ { - 2 k _ { 3 } } , \\end{align*}"}
+{"id": "1993.png", "formula": "\\begin{align*} q _ 1 x _ v & = d ( v ) x _ v + \\sum _ { u \\in N ( v ) } x _ u \\\\ & \\leq ( d ( v ) - 1 ) x _ v + \\sqrt { n } \\\\ & \\leq ( n - 2 ) x _ v + \\sqrt { n } , \\end{align*}"}
+{"id": "1359.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t ^ 2 u - \\Delta u + a ( x ) \\partial _ t u + | u | ^ { p - 1 } u = 0 , & t > 0 , x \\in \\Omega , \\\\ u ( t , x ) = 0 , & t > 0 , x \\in \\partial \\Omega , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\ \\partial _ t u ( 0 , x ) = u _ 1 ( x ) , & x \\in \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "4273.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\frac { d ^ { k - 1 } ( q / d ) _ { k - 1 } q ^ k } { ( q ) _ k } = \\frac { 1 } { 1 - d } \\left ( 1 - \\frac { ( q ) _ { \\infty } } { ( d q ) _ \\infty } \\right ) , \\end{align*}"}
+{"id": "5192.png", "formula": "\\begin{align*} \\ell _ k = \\frac { 2 ^ k + 2 \\cdot ( - 1 ) ^ k } { 3 } \\enspace . \\end{align*}"}
+{"id": "2400.png", "formula": "\\begin{align*} { \\rm e v } _ { 0 } ^ { \\mathfrak { m } } ( e _ { a _ { 1 } } \\cdots e _ { a _ { k } } ) = I ^ { \\mathfrak { m } } ( 1 , a _ { 1 } ' , \\dots , a _ { k } ' , \\infty ' ) \\end{align*}"}
+{"id": "3175.png", "formula": "\\begin{align*} \\frac { \\| g \\| _ { T _ t } } { \\| g \\| _ Z } \\geq C ^ { - 2 } \\lambda ^ { \\frac { t - t _ 0 ( g ) } { \\log ( \\lambda ) } - \\frac { t _ 1 - t _ 0 ( g ) } { \\log ( \\lambda ) } - 1 } = C ^ { - 2 } \\lambda ^ { - 1 } e ^ { t - t _ 1 } \\end{align*}"}
+{"id": "4480.png", "formula": "\\begin{align*} \\delta S = \\int _ M \\frac { \\delta S } { \\delta \\phi } \\delta \\phi \\ , , \\end{align*}"}
+{"id": "1592.png", "formula": "\\begin{align*} & - \\rho ( s ) ^ { - 1 } T _ { s _ 1 } - \\rho ( s ) ^ { - 1 } \\rho ( s ' ) ^ { - 1 } T _ { s _ 2 ' } \\\\ = & - \\rho ( s ) ^ { - 1 } \\left [ T _ { b _ { 2 } } - \\rho \\left ( s \\right ) T _ { b _ { 1 } } \\right ] - \\rho \\left ( s \\right ) ^ { - 1 } \\rho \\left ( s ' \\right ) ^ { - 1 } \\left [ T _ { b _ { 3 } } - \\rho \\left ( s ' \\right ) T _ { b _ { 2 } } \\right ] \\\\ = & T _ { b _ { 1 } } - \\rho \\left ( s \\right ) ^ { - 1 } \\rho \\left ( s ' \\right ) ^ { - 1 } T _ { b _ { 3 } } \\end{align*}"}
+{"id": "7369.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\norm { g _ k } _ { x _ k } ^ 2 < \\infty , \\end{align*}"}
+{"id": "5492.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } \\frac { q ^ n } { ( b q ) _ { n } ( q ) _ { n } } = \\frac { 1 } { ( b q ) _ { N } ( q ) _ { N } } \\sum _ { n = 0 } ^ { N } \\begin{bmatrix} N \\\\ n \\end{bmatrix} ( q ) _ n ( - b ) ^ n q ^ { n ( n + 1 ) / 2 } . \\end{align*}"}
+{"id": "3408.png", "formula": "\\begin{align*} a = a ( \\rho , p ) > 0 \\end{align*}"}
+{"id": "5694.png", "formula": "\\begin{gather*} c _ { i } \\iota _ { i } \\tau _ * ( \\alpha _ { ( 0 , i ) } ) = ( - 1 ) ^ i ( i + 1 ) ! \\ : e _ 1 \\wedge e _ 2 \\wedge \\cdots \\wedge e _ { i + 1 } \\otimes e _ { 1 } ^ * . \\end{gather*}"}
+{"id": "2452.png", "formula": "\\begin{align*} & 3 \\sum _ { \\substack { \\substack { ( a , b , c ) \\equiv ( 1 , 1 , 0 ) \\bmod { 2 } \\\\ a + b + c = w + 2 , \\ , b > 1 } } } ( p _ { a , 1 , b } + \\delta _ { a , 1 } \\delta _ { b , 3 } ) \\cdot I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a , b , c ) - \\sum _ { \\substack { \\substack { ( a , b , c ) \\equiv ( 0 , 0 , 0 ) \\bmod { 2 } \\\\ a + b + c = w + 2 } } } p _ { a , b , c } \\cdot I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a , b , c ) \\\\ & \\overset { ? } { = } \\frac { w ^ { 2 } ( w - 2 ) ( 5 w - 1 1 ) } { 2 4 } I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( w + 2 ) . \\end{align*}"}
+{"id": "5090.png", "formula": "\\begin{align*} \\| u \\| _ { Y _ T ^ 1 } = \\sum _ { k = 0 , 1 } \\| | \\nabla _ x | ^ { k \\sigma } u \\| _ { L ^ l _ t L ^ m _ x L ^ 2 _ y ( [ - T , T ] \\times \\mathbb { R } ^ d \\times \\mathbb { T } ) } , \\end{align*}"}
+{"id": "4209.png", "formula": "\\begin{align*} \\sum _ { X = 1 } ^ \\infty Q _ k ( X ) u ^ { X - 1 } = \\frac { Z ( u ) } { Z ( u ^ k ) ( 1 - u ) } . \\end{align*}"}
+{"id": "6452.png", "formula": "\\begin{align*} ( x + y ) ^ n = \\sum _ { k = 0 } ^ n \\begin{bmatrix} n \\\\ k \\end{bmatrix} y ^ k x ^ { n - k } . \\end{align*}"}
+{"id": "257.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathbb { E } [ u _ t ] \\bigg | _ { t = 0 } = \\frac { d } { d t } \\int _ \\Omega W ( x , u _ t ( x ) , \\nabla u _ t ( x ) ) \\ , d x \\bigg | _ { t = 0 } = 0 . \\end{align*}"}
+{"id": "3885.png", "formula": "\\begin{align*} p ^ \\# _ { \\mu _ 1 , \\mu _ 2 , \\theta } = 1 + \\frac { 2 + \\theta } { - \\tau _ + } { \\rm f o r } \\ ; \\ - \\frac { ( N - 2 - \\mu _ 1 ) ^ 2 } { 4 } \\leq \\mu _ 2 < 0 . \\end{align*}"}
+{"id": "2980.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { B ( t - t _ 0 ) } { B ( t ) - B ( t _ 0 ) } = 1 , t _ 0 \\geq 0 . \\end{align*}"}
+{"id": "4436.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\sup _ { n \\in \\N } d _ X ( x _ { n , m } , x _ n ) = 0 \\lim _ { n \\to \\infty } d _ X ( x _ { n , m } , x ) = 0 m \\in \\N . \\end{align*}"}
+{"id": "2817.png", "formula": "\\begin{align*} D ^ X _ s = \\gamma _ s X _ s + \\nu _ s ^ { - 1 } \\left ( d - \\gamma _ t x - \\int _ t ^ s X _ r d ( \\nu _ r \\gamma _ r ) \\right ) , s \\in [ t , T ] , D ^ X _ { t - } = d \\end{align*}"}
+{"id": "1523.png", "formula": "\\begin{align*} r ( A ) = \\lambda H ( X _ A ) . \\end{align*}"}
+{"id": "6030.png", "formula": "\\begin{align*} \\ell ^ { \\perp _ \\mu } = \\Delta . \\end{align*}"}
+{"id": "6566.png", "formula": "\\begin{align*} U = \\begin{pmatrix} 0 & - \\frac { 1 } { 3 } & 0 & \\frac { 2 } { 3 } & \\frac { 2 } { 3 } & 0 & 0 \\\\ [ 2 . 5 m m ] 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ [ 2 . 5 m m ] 0 & \\frac { 2 } { 3 } & 0 & \\frac { 2 } { 3 } & \\frac { 1 } { 3 } & 0 & 0 \\\\ [ 2 . 5 m m ] 0 & \\frac { 2 } { 3 } & 0 & - \\frac { 1 } { 3 } & \\frac { 2 } { 3 } & 0 & 0 \\\\ [ 2 . 5 m m ] 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ [ 2 . 5 m m ] 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ [ 2 . 5 m m ] 0 & 0 & 0 & 0 & 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "6110.png", "formula": "\\begin{align*} \\mathcal S _ 1 ( k , 3 ) = & \\left \\{ F \\in \\binom { [ n ] } { k } : [ 2 ] \\subseteq F , \\ , F \\cap [ 3 , k - 1 ] \\neq \\emptyset \\right \\} \\cup \\left \\{ F \\in \\binom { [ n ] } { k } : [ 2 ] \\cup [ k , k + 2 ] \\subseteq F \\right \\} \\\\ & \\cup \\left \\{ [ 2 , k ] \\cup \\{ i \\} : i \\in [ k + 1 , n ] \\right \\} \\ , \\cup \\ , \\left \\{ [ 2 , k + 2 ] \\setminus \\{ k \\} , \\ , [ k + 1 ] \\setminus \\{ 2 \\} , \\ , [ k + 2 ] \\setminus \\{ 2 , k + 1 \\} \\right \\} . \\end{align*}"}
+{"id": "3729.png", "formula": "\\begin{align*} \\begin{cases} d \\ , \\Delta u + u ( m ( x ) - u ) = 0 , \\ ; \\ ; u > 0 & \\ ; \\ ; \\Omega , \\\\ \\ ; u = 0 & \\ ; \\ ; \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "1897.png", "formula": "\\begin{align*} ( 1 + \\Delta _ { \\gamma _ 1 } ) = [ \\gamma _ 2 ^ { - 1 } ( 1 + \\Delta _ { \\gamma _ 1 \\gamma _ 2 ^ { - 1 } } ) \\gamma _ 2 ] \\cdot ( 1 + \\Delta _ { \\gamma _ 2 } ) . \\end{align*}"}
+{"id": "2359.png", "formula": "\\begin{align*} \\iota ( \\Bbbk , \\{ 2 \\} ^ { c } , 3 ) & = ( - 1 ) ^ { c } x _ { 2 c + 3 } \\cdot \\iota ( \\Bbbk ) \\quad ( \\Bbbk \\in W _ { 2 , 3 } ' ) , \\\\ \\theta ( \\Bbbk , 3 , \\{ 2 \\} ^ { c } ) & = ( - 1 ) ^ { c } x _ { 2 c + 2 } \\cdot s _ { 1 } ( \\theta ( \\Bbbk ) ) \\quad ( \\Bbbk \\in W _ { 2 , 3 } ) . \\end{align*}"}
+{"id": "2572.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\bar x _ t ^ { * , t _ 0 , \\xi } = & ~ [ A \\bar x _ t ^ { * , t _ 0 , \\xi } - B ^ 2 R ^ { - 1 } \\bar U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) - B h ( \\bar \\mu _ t ^ { * , t _ 0 , \\xi } ) + f ( \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) + b ( \\bar \\mu _ t ^ { * , t _ 0 , \\xi } ) ] d t \\\\ & + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ \\bar x _ { t _ 0 } ^ { * , t _ 0 , \\xi } = & ~ \\xi \\end{aligned} \\right . \\end{align*}"}
+{"id": "2900.png", "formula": "\\begin{align*} \\kappa ( s ) = \\eta x _ { n + 1 } ^ m ( s ) + \\lambda . \\end{align*}"}
+{"id": "4136.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left [ x ^ { 2 \\alpha + 1 } ( 1 - x ^ 2 ) \\frac { d } { d x } [ V ] \\right ] + \\gamma _ n ^ { \\alpha } ( 0 ) x ^ { 2 \\alpha + 1 } V = g ( x ) x ^ { 2 \\alpha + 1 } V , \\end{align*}"}
+{"id": "2384.png", "formula": "\\begin{align*} F _ { 1 } & = \\sum _ { s = 0 } ^ { 1 } \\sum _ { k = 1 } ^ { m } \\sum _ { t = 0 } ^ { 1 } \\left ( \\Delta _ { a _ { 1 } , \\iota ( t ) } ^ { \\alpha , \\beta } - \\Delta _ { \\iota ^ { c _ { k } } ( a _ { s + 1 } ) , \\iota ^ { c _ { k } + k + s } ( t ) } ^ { \\alpha , \\beta } \\right ) e _ { a _ { 1 } } W ( t ; c _ { k } + k + s - 2 ) g _ { k , s } \\\\ & = \\sum _ { s = 0 } ^ { 1 } \\sum _ { k = 1 } ^ { m } \\sum _ { t = 0 } ^ { 1 } \\left ( \\Delta _ { a _ { 1 } , \\iota ( t ) } ^ { \\alpha , \\beta } - \\Delta _ { a _ { s + 1 } , \\iota ^ { k + s } ( t ) } ^ { \\alpha , \\beta } \\right ) Y _ { k , t } ( s ) . \\end{align*}"}
+{"id": "2610.png", "formula": "\\begin{align*} f _ n ( t , z , x ) : = \\sup _ { \\| q \\| \\le n } \\Big ( q \\cdot z - g ( t , q , x ) \\Big ) . \\end{align*}"}
+{"id": "812.png", "formula": "\\begin{align*} l _ 1 + l _ 2 \\Big ( \\frac { K } { \\bar { \\theta } _ 1 } \\cdot \\frac { k _ 2 } { k _ 2 - 1 } \\Big ) ^ { 1 - \\frac { \\lambda _ + } { \\sigma } } x ^ { \\frac { \\lambda _ { + } } { \\sigma } } = \\frac { 1 - \\theta } { \\bar { \\theta } _ 1 - \\theta } \\end{align*}"}
+{"id": "6250.png", "formula": "\\begin{align*} \\delta ( G , [ n ] ) = \\frac { ( n - t ) ! } { n ( n - 2 ) ! } + \\sum _ { k = 2 } ^ { t - 1 } \\frac { 1 } { k ( k - 2 ) ! } \\delta ( G _ { ( [ k ] ) } , [ n ] \\setminus [ k ] ) , \\end{align*}"}
+{"id": "5866.png", "formula": "\\begin{align*} \\lim \\limits _ { q \\rightarrow \\infty } S ( q ) = 0 . \\end{align*}"}
+{"id": "5971.png", "formula": "\\begin{align*} b ' ( p ) \\geq ( 1 - \\frac { 1 } { k } ) ^ { - \\frac { p } { 2 k } } [ 1 + \\frac { 2 - k - k ^ 2 } { 2 k ( k - 1 ) } ] = ( 1 - \\frac { 1 } { k } ) ^ { - \\frac { p } { 2 k } } [ 1 - \\frac { k + 2 } { 2 k } ] \\geq 0 \\end{align*}"}
+{"id": "2443.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { d - 1 } ( \\{ i + 1 \\} , \\dots , \\{ d \\} , \\{ 1 \\} , \\dots , \\{ i \\} ) - \\begin{cases} ( \\{ 1 , \\dots , d \\} ) & d : \\ , { \\rm o d d } \\\\ 0 & d : \\ , { \\rm e v e n } \\end{cases} \\end{align*}"}
+{"id": "2100.png", "formula": "\\begin{align*} \\psi _ x : = \\psi _ { x ; k , \\ell , n } \\colon U _ { \\ell , n } \\to G ( k , n ) , \\end{align*}"}
+{"id": "2802.png", "formula": "\\begin{align*} Q _ A \\tilde { w } _ i & = \\tilde { c } _ i \\tilde { u } _ i ^ A , & Q _ A ^ T \\tilde { u } _ i ^ A & = \\tilde { c } _ i \\tilde { w } _ i + \\alpha _ { k + 1 } v _ { k + 1 } e _ { k + 1 } ^ T x _ i , \\\\ Q _ B \\tilde { w } _ i & = \\tilde { s } _ i \\tilde { u } _ i ^ B , & Q _ B ^ T \\tilde { u } _ i ^ B & = \\tilde { s } _ i \\tilde { w } _ i + \\check { \\beta } _ { k } v _ { k + 1 } e _ { k } ^ T \\hat { x } _ i , \\end{align*}"}
+{"id": "6840.png", "formula": "\\begin{align*} \\sum _ { w \\in M _ { K } } N _ { w } \\log ( r _ { w , \\alpha } ^ { - 1 } ) & = \\sum _ { w \\in M _ { K } \\setminus M ' _ { K , \\alpha } } N _ { w } \\log ( d ( P _ { 1 , w } , P _ { 2 , w } ) ^ { - 1 } ) + \\sum _ { w \\in M ' _ { K , \\alpha } } N _ { w } \\log ( \\rho _ { \\alpha } ^ { - 1 } ) \\\\ & \\le \\sum _ { w \\in M _ { K } } N _ { w } \\ , \\max ( 0 , \\log ( d ( P _ { 1 , w } , P _ { 2 , w } ) ^ { - 1 } ) ) + \\sharp M _ { \\alpha } \\ , \\log ( \\rho _ { \\alpha } ^ { - 1 } ) . \\end{align*}"}
+{"id": "704.png", "formula": "\\begin{align*} S _ i = { { w } _ i } \\rho c _ v \\frac { \\Delta t } { 2 } \\partial _ t ^ 2 T , \\end{align*}"}
+{"id": "1059.png", "formula": "\\begin{align*} \\nabla _ { e _ { i } } e _ { j } = \\alpha _ { i j k } e _ { k } , \\end{align*}"}
+{"id": "1872.png", "formula": "\\begin{align*} r ( x + 2 y ) + r ( 2 x + y ) = \\frac { r ( x ) r ( y ) [ 5 r ( x ) + 5 r ( y ) + 8 \\sqrt { r ( x ) r ( y ) } ] } { [ 2 r ( x ) + 2 r ( y ) + 5 \\sqrt { r ( x ) r ( y ) } ] ^ { 2 } } \\end{align*}"}
+{"id": "399.png", "formula": "\\begin{align*} \\theta = { \\textstyle \\sum } \\ , p ^ I \\ , \\d e _ I = { \\textstyle \\sum } \\ , p ^ I \\ , \\d e _ { i _ 1 } \\wedge \\cdots \\wedge \\d e _ { i _ n } . \\end{align*}"}
+{"id": "5538.png", "formula": "\\begin{align*} \\lambda _ 1 ( \\theta _ b ) \\dfrac { \\partial \\theta _ 1 } { \\partial r } ( \\alpha ( t ) , t ) = - l _ b \\gamma _ b \\dfrac { d \\alpha } { d t } , \\end{align*}"}
+{"id": "339.png", "formula": "\\begin{align*} p ( z ) \\approx \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } g ( z - z _ { i } ) \\end{align*}"}
+{"id": "3091.png", "formula": "\\begin{align*} \\gamma _ n ^ 2 = ( n , n + 2 ) ( n + 1 , n + 3 ) = \\alpha ' _ n \\alpha ' _ { n + 1 } . \\end{align*}"}
+{"id": "2012.png", "formula": "\\begin{align*} \\lambda _ { n + 1 } = \\frac { 1 } { n ! } \\frac { d ^ n } { d w ^ n } \\left [ \\frac { 1 } { ( 1 - w ) ^ 2 } \\ , \\frac { \\xi ' } { \\xi } \\left ( \\frac { 1 } { 1 - w } \\right ) \\right ] _ { w = 0 } , \\end{align*}"}
+{"id": "6447.png", "formula": "\\begin{align*} \\int _ u ^ v \\frac { x ^ n ( q x / u , q x / v ; q ) _ { \\infty } } { ( d x , c x ; q ) _ { \\infty } } d _ q x = \\frac { ( 1 - q ) v ( q , u / v , q v / u , d c u v ; q ) _ { \\infty } } { ( d u , d v , c u , c v ; q ) _ { \\infty } } W _ n ( d , c , u , v ) . \\end{align*}"}
+{"id": "1092.png", "formula": "\\begin{align*} z _ { 1 } ^ { 2 } - 5 z _ { 2 } ^ { 2 } = 4 p \\equiv 2 8 \\equiv 4 \\pmod 8 , \\\\ z _ { 1 } ^ { 2 } - 5 z _ { 3 } ^ { 2 } = - p \\equiv 1 \\pmod 8 . \\end{align*}"}
+{"id": "978.png", "formula": "\\begin{align*} \\lim _ { \\ell \\to \\infty } \\alpha _ \\ell = + \\infty . \\end{align*}"}
+{"id": "5274.png", "formula": "\\begin{align*} L _ { \\pi } : = \\{ t = ( t _ 1 , \\ldots , t _ { 2 n + 2 } ) \\in \\R ^ { 2 n + 2 } : \\sum _ { i = 1 } ^ { n + 1 } t _ i = 0 , \\ \\sum _ { i = 1 } ^ { n + 1 } t _ { n + 1 + i } = 0 ; \\ \\ \\sum _ { i \\in B } t _ i = 0 \\ \\forall B \\in \\pi \\} . \\end{align*}"}
+{"id": "528.png", "formula": "\\begin{align*} - H ( R \\ , | \\ , P ) + \\log Z & = \\int _ { \\R ^ n } f \\ , d R - H ( R ) \\le \\int _ { \\R ^ n } f \\ , d Q - H ( Q ) = - H ( Q \\ , | \\ , P ) + \\log Z . \\end{align*}"}
+{"id": "5802.png", "formula": "\\begin{align*} \\| | A | \\| = \\| A \\| = \\| A ^ { * } \\| = \\| | A ^ { * } | \\| . \\end{align*}"}
+{"id": "5369.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ A \\varphi = - \\star d _ A E & \\Sigma , \\\\ \\mathsf { n } d _ A \\varphi = ( - 1 ) ^ { \\Sigma - 1 } \\mathsf { t } E & \\partial \\Sigma . \\end{cases} \\end{align*}"}
+{"id": "6664.png", "formula": "\\begin{align*} V _ { \\eta , \\kappa } & = \\{ - k \\ ! : k \\in \\mathbb { N } \\cap [ 0 , \\kappa ] \\} \\sqcup \\{ ( i , j ) \\ ! : j \\in \\mathbb { N } _ { 1 } , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] \\} , \\\\ E _ { \\kappa } & = \\{ ( - k , - k + 1 ) \\ ! : k \\in \\mathbb { N } \\cap [ 1 , \\kappa ] \\} \\sqcup \\{ ( 0 , ( i , 1 ) ) \\ ! : i \\in \\mathbb { N } \\cap [ 1 , \\eta ] \\} , \\\\ E _ { \\eta , \\kappa } & = E _ { \\kappa } \\sqcup \\{ ( ( i , j ) , ( i , j + 1 ) ) \\ ! : i \\in \\mathbb { N } \\cap [ 1 , \\eta ] , j \\in \\mathbb { N } _ { 1 } \\} , \\end{align*}"}
+{"id": "2231.png", "formula": "\\begin{align*} H ( t ) = \\textrm { d i a g } \\left [ \\{ f _ k ( t ) \\} _ { k = 1 } ^ { 1 6 } \\right ] , f _ k ( t ) = \\alpha _ k + \\beta _ k \\cos ( 2 \\pi \\nu t ) + \\gamma _ k \\cos ( 4 \\pi \\nu t ) , \\end{align*}"}
+{"id": "447.png", "formula": "\\begin{align*} \\phi _ 1 & = z ( z - 1 ) \\frac { d z ^ 2 } { w ^ 2 } \\\\ \\phi _ 2 & = \\mu ( a , b ) ( z - a ) \\frac { d z ^ 2 } { w ^ 2 } , \\end{align*}"}
+{"id": "5161.png", "formula": "\\begin{align*} u ( z ) = \\chi _ { \\Omega _ 1 } ( z ) \\max \\left \\{ \\min \\left \\{ \\frac { | \\varphi ( c ) - z | - r } { r } , 1 \\right \\} , 0 \\right \\} . \\end{align*}"}
+{"id": "2381.png", "formula": "\\begin{align*} F _ { 1 } & = \\sum _ { l } \\partial _ { \\alpha , \\beta } ( w _ { l } e _ { p _ { l } } ) w ' _ { l } \\\\ F _ { 2 } & = \\sum _ { l } w _ { l } \\partial _ { \\alpha , \\beta } ( e _ { p _ { l } } w ' _ { l } ) \\\\ F _ { 3 } & = \\sum _ { l } D _ { \\alpha , \\beta } ( w _ { l } , p _ { l } , w ' _ { l } ) . \\end{align*}"}
+{"id": "7349.png", "formula": "\\begin{align*} V = \\sum _ { ( \\mu , d ) \\in \\Upsilon } \\mathcal { W } _ { ( \\mu , d ) } \\ \\ \\ \\ \\ ( ) . \\end{align*}"}
+{"id": "3983.png", "formula": "\\begin{align*} q _ { \\beta } ( n , t ) & = \\sum _ { k = 1 } ^ { n } \\mathrm { P r } \\{ X _ { 1 } + X _ { 2 } + \\dots + X _ { k } = n \\} \\mathrm { P r } \\{ N _ { \\beta } ( t ) = k \\} \\\\ & = \\sum _ { k = 1 } ^ { n } \\sum _ { \\Theta _ { n } ^ { k } } k ! \\prod _ { j = 1 } ^ { n } \\frac { ( \\theta ^ { j } / j ! ) ^ { x _ { j } } } { x _ { j } ! } \\left ( \\alpha t ^ { \\beta } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } \\left ( - \\alpha ( e ^ { \\theta } - 1 ) t ^ { \\beta } \\right ) , \\end{align*}"}
+{"id": "2827.png", "formula": "\\begin{align*} d \\gamma _ s ^ { - \\frac 1 2 } & = \\gamma _ s ^ { - \\frac 1 2 } \\left ( - \\frac 1 2 \\mu _ s + \\frac { 3 } { 8 } \\sigma _ s ^ 2 \\right ) d s - \\frac 1 2 \\gamma _ s ^ { - \\frac 1 2 } \\sigma _ s d W ^ 1 _ s , s \\in [ 0 , T ] . \\end{align*}"}
+{"id": "2539.png", "formula": "\\begin{align*} \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } } : = \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } \\alpha ^ j , \\nu ^ { N , i } _ { \\boldsymbol { x } } : = \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } x ^ j . \\end{align*}"}
+{"id": "6604.png", "formula": "\\begin{align*} T _ { \\Gamma , \\alpha } \\simeq \\alpha ^ 2 T _ { \\Gamma } , T _ \\Gamma : = T _ { \\Gamma , - 1 } , \\end{align*}"}
+{"id": "4564.png", "formula": "\\begin{gather*} \\Phi _ T ( M _ { f _ 1 f _ 2 } ) = \\Phi _ T ( M _ { f _ 1 } M _ { f _ 2 } ) = \\Phi _ T ( M _ { f _ 1 } ) \\Phi _ T ( M _ { f _ 2 } ) = \\\\ = ( M _ { g _ 1 } + R _ 1 ) ( M _ { g _ 2 } + R _ 2 ) = M _ { g _ 1 g _ 2 } + R _ 1 M _ { g _ 2 } + M _ { g _ 1 } R _ 2 + R _ 1 R _ 2 . \\end{gather*}"}
+{"id": "3334.png", "formula": "\\begin{gather*} \\psi _ s = ( \\beta L + \\gamma [ K , L ] _ q + \\delta B + \\varepsilon D _ 1 ) \\psi _ n \\end{gather*}"}
+{"id": "2612.png", "formula": "\\begin{align*} \\ \\begin{cases} \\partial _ t v _ n + \\partial _ x v _ n \\cdot B + \\frac { 1 } { 2 } \\mathrm { T r } \\big [ \\partial _ { x x } v _ n \\Sigma \\Sigma ^ \\top \\big ] + f _ n ( t , \\partial _ { x } v _ n \\Sigma , x ) [ 0 , 1 ) \\times \\R ^ \\ell = 0 , \\\\ v _ n ( 1 , x ) = F ( x ) x \\in \\R ^ \\ell \\end{cases} \\end{align*}"}
+{"id": "5995.png", "formula": "\\begin{align*} \\tau _ { 0 } ^ - ( r ) : = \\inf \\{ t > 0 : U ^ { b } _ { r } ( t ) < 0 \\} . \\end{align*}"}
+{"id": "7564.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon \\to 0 } E [ \\mu _ { \\varepsilon , h } - \\mu ] & = \\limsup _ { \\varepsilon \\to 0 } \\left ( E [ \\mu _ { \\varepsilon , h } ] + E [ \\mu ] - 2 \\iint G ( p , q ) d \\mu ( p ) d \\mu _ { \\varepsilon , h } ( q ) \\right ) \\\\ & \\leq E [ \\mu ] + E [ \\mu ] - 2 E [ \\mu ] = 0 , \\end{align*}"}
+{"id": "7836.png", "formula": "\\begin{align*} { \\bf i } _ { ( \\eta _ i ) _ { T ^ * \\widetilde Q } } ( \\widetilde \\Omega + \\widetilde { \\bf B } ) = d \\widetilde J _ { \\eta _ i } , \\end{align*}"}
+{"id": "840.png", "formula": "\\begin{align*} \\mathcal { J } _ i ( \\tau _ i ^ { * } , \\tau _ { - i } ^ { * } ) = \\mathbb { E } \\left \\{ y _ i ^ { * } \\widetilde { \\theta } _ 1 ( y _ i ^ { * } , y _ { - i } ^ { * } ) - e ^ { - \\beta \\tau _ i ^ { * } } K \\right \\} . \\end{align*}"}
+{"id": "2211.png", "formula": "\\begin{align*} f ^ { \\prime } { } ^ 2 = \\frac { 1 - k ^ 2 x ^ 2 - \\alpha ^ 2 x ^ { 2 \\alpha } } { x ^ 2 } \\end{align*}"}
+{"id": "5994.png", "formula": "\\begin{align*} v _ b ^ { w \\ , \\prime } ( b + ) = \\left ( \\theta \\dfrac { { C ^ { ( \\theta , r ) } ( b ; w ) } + \\rho _ b ^ { ( \\theta ) } ( b ; w ) } { Z ^ { ( \\theta ) } ( b ) } - w ( 0 ) \\right ) W ^ { ( \\theta ) } ( b ) - \\rho _ b ^ { ( \\theta ) } ( b ; w ' ) = v _ b ^ { w \\ , \\prime } ( b - ) . \\end{align*}"}
+{"id": "1718.png", "formula": "\\begin{align*} \\Theta _ \\tau ( \\xi ) \\big | _ { \\mathfrak { h } ^ { ( n ) } } = \\begin{cases} \\frac { p ! } { \\tau ^ p } { n \\choose p } P _ + \\left ( \\xi \\otimes \\mathbf { 1 } ^ { ( n - p ) } \\right ) P _ + & n \\geq p \\\\ 0 & \\ , , \\end{cases} \\end{align*}"}
+{"id": "1249.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ \\Omega | \\nabla T _ { k } ( u _ p ) | \\ , d x + \\int _ { \\partial \\Omega } ( \\lambda | T _ { k } ( u _ p ) | - g T _ { k } ( u _ p ) ) d \\mathcal H ^ { N - 1 } \\\\ & \\le \\int _ \\Omega f T _ { k } ( u _ p ) \\ , d x + \\frac { p - 1 } { p } | \\Omega | + \\frac { p - 1 } { p } \\int _ { \\partial \\Omega } \\lambda d \\mathcal H ^ { N - 1 } . \\end{aligned} \\end{align*}"}
+{"id": "527.png", "formula": "\\begin{align*} H ( Q _ k \\ , | \\ , P ) = - \\log P ( B _ k ) \\to 0 . \\end{align*}"}
+{"id": "5871.png", "formula": "\\begin{align*} \\R ^ n = \\bigsqcup _ { \\xi \\in \\Xi } \\xi , \\end{align*}"}
+{"id": "1342.png", "formula": "\\begin{align*} | \\nabla v ( \\overline { x } ) - q | = \\max _ { x \\in \\overline { B _ { 1 / 2 } } } | \\nabla v ( x ) - q | > C ( \\varepsilon a ) ^ { \\nu } . \\end{align*}"}
+{"id": "8206.png", "formula": "\\begin{align*} { \\Lambda } ^ \\alpha u ( x ) = ( - \\Delta ) ^ { \\frac { \\alpha } { 2 } } u ( x ) = c _ { \\alpha , N } \\ , \\mathrm { p . v . } \\int _ { \\mathbb { R } ^ N } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { N + \\alpha } } d y , \\end{align*}"}
+{"id": "2300.png", "formula": "\\begin{align*} ( x ^ 2 - \\mu _ n y ^ 2 ) ( \\mu _ { n + 1 } y ^ 2 - x ^ 2 ) & \\ge ( x ^ 2 - \\mu _ n y ^ 2 ) + ( \\mu _ { n + 1 } y ^ 2 - x ^ 2 ) - 1 \\\\ & = ( \\mu _ { n + 1 } - \\mu _ n ) y ^ 2 - 1 \\ge y ^ 2 - 1 \\ge \\frac 3 4 y ^ 2 . \\end{align*}"}
+{"id": "3697.png", "formula": "\\begin{align*} \\partial _ t S ( x , t ) + \\frac 1 2 ( \\nabla S ( x , t ) ) ^ 2 - \\frac 1 2 \\frac { \\nabla ^ 2 \\sqrt { \\rho ( x , t ) } } { \\sqrt { \\rho ( x , t ) } } = 0 \\end{align*}"}
+{"id": "3570.png", "formula": "\\begin{align*} e _ 1 ( I ) \\le \\lfloor \\frac { 1 } { 3 } \\binom { e _ 0 ( I ) - b + 1 } { 2 } \\rfloor - \\binom { \\mu ( m ) - d } { 2 } . \\end{align*}"}
+{"id": "1302.png", "formula": "\\begin{align*} ( a ^ \\ast \\otimes \\varepsilon ) \\circ \\alpha ^ { - 1 } _ { a ^ \\ast , a , a ^ \\ast } \\circ ( \\eta \\otimes a ^ \\ast ) = \\sigma _ { I , a ^ \\ast } , ( \\varepsilon \\otimes a ) \\circ \\alpha _ { a , a ^ \\ast , a } \\circ ( a \\otimes \\eta ) = \\sigma _ { a , I } , \\end{align*}"}
+{"id": "579.png", "formula": "\\begin{align*} V & = \\Big ( Z ( T ' , M _ n ) + e _ 1 M _ n ( F ) e _ 3 + e _ 3 M _ n ( F ) e _ 2 \\Big ) \\cap \\Lambda ^ \\perp \\\\ & = \\left ( Z ( T ' , M _ n ) \\cap \\Lambda ^ \\perp \\right ) + e _ 1 M _ n ( F ) e _ 3 + e _ 3 M _ n ( F ) e _ 2 \\end{align*}"}
+{"id": "4154.png", "formula": "\\begin{align*} \\lambda _ 1 = 0 , \\lambda _ { 2 } = a _ { 1 2 } + a _ { 1 3 } + a _ { 2 3 } + \\Delta , \\lambda _ { 3 } = a _ { 1 2 } + a _ { 1 3 } + a _ { 2 3 } - \\Delta \\end{align*}"}
+{"id": "2039.png", "formula": "\\begin{align*} | \\langle u _ 0 , \\eta _ j \\rangle | = | \\langle \\chi _ 0 , \\eta _ j \\rangle | \\ll \\frac { ( \\log j ) ^ 2 } { j } \\end{align*}"}
+{"id": "7859.png", "formula": "\\begin{align*} K ( J ) = \\int _ { G _ { \\theta ( J ) } ' } \\sum _ { z \\in Z _ \\delta } { \\bf 1 } _ { \\Gamma ^ { \\delta } _ z } ( \\theta ( J ) , y ) d y , \\end{align*}"}
+{"id": "179.png", "formula": "\\begin{align*} \\begin{aligned} \\tfrac { \\chi ( \\sqrt { | V _ \\shortparallel | ^ 2 + | V _ \\perp ' - \\zeta _ \\perp | ^ 2 } ) } { ( | V _ \\shortparallel | ^ 2 + | V _ \\perp ' - \\zeta _ \\perp | ^ 2 ) ^ { \\frac { 1 - \\gamma } { 2 } } } \\leq \\tfrac { 2 ^ { 1 - \\gamma } } { ( 1 + | V _ \\shortparallel | ^ 2 + | V _ \\perp ' - \\zeta _ \\perp | ^ 2 ) ^ { \\frac { 1 - \\gamma } { 2 } } } \\ , . \\end{aligned} \\end{align*}"}
+{"id": "7517.png", "formula": "\\begin{align*} G ( p , q ) - \\log \\frac { 1 } { | z _ { \\infty } ( p ) ^ { - 1 } - z _ { \\infty } ( q ) ^ { - 1 } | } = O ( 1 ) \\end{align*}"}
+{"id": "8229.png", "formula": "\\begin{align*} \\begin{aligned} \\Vert \\sigma ^ n \\Vert _ { \\widetilde { L } ^ \\infty _ T ( \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } ) } + \\Vert u ^ n \\Vert _ { \\widetilde { L } ^ \\infty _ T ( \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } ) } + \\Vert \\sigma ^ n \\Vert _ { L ^ 1 _ T ( \\widetilde { B } ^ { \\frac { N } { 2 } + 1 , \\frac { N } { 2 } + 2 - \\alpha } ) } + \\Vert u ^ n \\Vert _ { L ^ { 1 } _ { T } ( \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } ) } \\le C X _ 0 , \\end{aligned} \\end{align*}"}
+{"id": "247.png", "formula": "\\begin{align*} \\zeta [ \\sigma _ 1 , \\sigma _ 2 , \\rho ] ( u ) \\stackrel { d e f } { = } \\exp \\left \\{ \\ - \\frac { u ^ 2 } { 2 ( 1 - \\rho ^ 2 ) } \\ \\frac { \\sigma _ 1 ^ 2 + \\sigma _ 2 ^ 2 - 2 \\rho \\ \\sigma _ 1 \\ \\sigma _ 2 } { \\sigma _ 1 ^ 2 \\sigma _ 2 ^ 2 } \\ \\right \\} . \\end{align*}"}
+{"id": "5644.png", "formula": "\\begin{align*} d f ( \\tilde \\gamma _ s ( 0 , 0 ) ) = d f ( V ( 0 ) ) = V ( l ) = \\tilde \\gamma _ s ( l , 0 ) . \\end{align*}"}
+{"id": "2620.png", "formula": "\\begin{align*} & E _ { k , [ p , q ] } ^ 0 = \\{ n \\in \\mathbb { N } ^ k : p \\le n \\le q \\} \\\\ & E _ { k , [ p , q ] } ^ 1 = \\{ x \\in E _ { k , q } ^ 1 : s ( x ) , r ( x ) \\in E _ { k , [ p , q ] } ^ 0 \\} \\end{align*}"}
+{"id": "2297.png", "formula": "\\begin{align*} O _ { F _ 1 , F _ 2 , \\varepsilon } \\left ( B ^ { d \\eta _ d + \\varepsilon } \\right ) = O _ { F _ 1 , F _ 2 , \\varepsilon } \\left ( N ^ { \\eta _ d ( 1 + d \\tau ) + \\varepsilon } \\right ) = O _ { F _ 1 , F _ 2 , \\varepsilon } \\left ( N ^ { \\eta ' _ { d , F _ 1 , F _ 2 } + \\varepsilon } \\right ) . \\end{align*}"}
+{"id": "3481.png", "formula": "\\begin{align*} \\mathbb { E } ( \\mathrm { y } , t ) : = \\int _ { \\partial \\Omega } \\frac { ( \\mathrm { y } - \\mathrm { v } ) \\cdot \\nu _ { \\mathrm { v } } } { 2 \\pi | \\mathrm { y } - \\mathrm { v } | ^ 2 } \\Big [ \\varphi ( \\mathrm { v } , \\mathrm { y } , \\mathrm { t } , \\tau ) - \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) \\Big ] d \\sigma _ \\mathrm { v } . \\end{align*}"}
+{"id": "3348.png", "formula": "\\begin{align*} \\mu ( a _ 0 \\nearrow b _ 1 \\searrow a _ 1 ) = - \\iota _ { a _ 0 , b _ 1 } \\cdot \\iota _ { a _ 1 , b _ 1 } \\mbox { \\ \\ a n d \\ \\ } \\mu ( c _ 0 \\searrow d _ 1 \\nearrow c _ 1 ) = - \\iota _ { d _ 1 , c _ 0 } \\cdot \\iota _ { d _ 1 , c _ 1 } , \\end{align*}"}
+{"id": "7892.png", "formula": "\\begin{align*} P ( a ) \\cdot P ( b ) = P \\big ( P ( a ) \\cdot b + a \\cdot P ( b ) \\big ) + \\lambda ~ \\ ! P ( a \\cdot b ) , a , b \\in A . \\end{align*}"}
+{"id": "2445.png", "formula": "\\begin{align*} I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a , b ) + I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( b , a ) = 0 \\quad \\quad ( a + b = w + 2 , \\ , a , b \\geq 2 ) . \\end{align*}"}
+{"id": "7313.png", "formula": "\\begin{align*} a = & ( \\frac { q - 3 } { 2 } ) q ^ { m - 1 } + ( \\frac { q - 1 } { 2 } ) q ^ { m - 2 } + \\cdots + ( \\frac { q - 1 } { 2 } ) q ^ { m - j _ 0 - 1 } \\\\ & - ( \\frac { q - 3 } { 2 } ) q ^ { m - j _ 0 - 2 } - i _ { m - 1 } q ^ { m - j _ 0 - 3 } - \\cdots - i _ { j _ 0 + 2 } q - i _ { j _ 0 + 1 } . \\end{align*}"}
+{"id": "2269.png", "formula": "\\begin{align*} H _ m = H _ m ( \\alpha _ 1 , \\alpha _ 2 ) : = k [ z _ 1 ^ { \\pm 1 } , z _ 2 ^ { \\pm 1 } ] ( x , y ; \\sigma _ m , a ) \\end{align*}"}
+{"id": "2451.png", "formula": "\\begin{align*} q _ { a , b , c } & = \\frac { 1 } { 1 2 } ( a - b ) \\Big ( 1 7 ( a ^ { 3 } + b ^ { 3 } ) + 1 9 7 a b ( a + b ) + 1 2 ( a ^ { 2 } + b ^ { 2 } - 1 6 a b ) c - 2 2 2 ( a + b ) c ^ { 2 } - 1 6 8 c ^ { 3 } \\\\ & \\quad - 3 6 2 ( a ^ { 2 } + b ^ { 2 } ) - 1 2 3 8 a b + 7 9 8 ( a + b ) c + 1 6 7 4 c ^ { 2 } + 1 0 7 5 ( a + b ) - 3 7 7 4 c + 5 9 0 \\Big ) , \\end{align*}"}
+{"id": "6353.png", "formula": "\\begin{align*} \\psi ( g ) \\cdot \\psi ( g ) = \\sum _ { k \\in G } f ( \\psi ( g ) ^ { - 1 } \\psi ( k ) ) \\psi ( k ) . \\end{align*}"}
+{"id": "7409.png", "formula": "\\begin{align*} a _ p ( Y ) = 1 + ( p + p ^ 2 ) k _ p ( Y ) + p ^ 3 - \\# Y _ p . \\end{align*}"}
+{"id": "4968.png", "formula": "\\begin{align*} E _ { c \\rightarrow v } ^ { ( i ) } = \\log ( \\frac { 1 } { 1 - \\prod \\limits _ { v ' \\in \\mathcal { N } ( c ) \\backslash \\{ v \\} } ( 1 + \\exp ( E _ { v ' \\rightarrow c } ^ { ( i - 1 ) } ) ) ^ { - 1 } } ) , \\end{align*}"}
+{"id": "7343.png", "formula": "\\begin{align*} & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ X ^ { ( i , j , t , p ) } _ { ( m + 1 , m + 1 ) } = \\{ ( y , z ) \\in X _ { ( m + 1 , m + 1 ) } \\mid \\varrho ( y , z ) = ( i , j , t , p ) \\} . \\end{align*}"}
+{"id": "4991.png", "formula": "\\begin{align*} & \\sum _ { i , j = 1 } ^ 3 \\sigma ^ { i j } \\big ( u ^ 0 \\hat { p } _ i - u _ i \\big ) \\big ( u ^ 0 \\hat { p } _ j - u _ j \\big ) \\geq \\sum _ { i , j = 1 } ^ 3 \\frac 1 2 \\big ( u ^ 0 \\big ) ^ 2 \\sigma ^ { i j } \\hat { p } _ i \\hat { p } _ j - \\sum _ { i , j = 1 } ^ 3 \\sigma ^ { i j } u _ i u _ j . \\end{align*}"}
+{"id": "1442.png", "formula": "\\begin{align*} \\frac { b _ 2 ( x ) } { a ( x ) } = \\frac { 1 } { \\langle x \\rangle ^ { \\alpha } a ( x ) } \\left ( \\langle x \\rangle ^ { \\alpha } a ( x ) - a _ 0 - \\frac { a _ 0 \\alpha } { n - \\alpha } \\langle x \\rangle ^ { - 2 } \\right ) \\end{align*}"}
+{"id": "7057.png", "formula": "\\begin{align*} e ( G ) \\geq & f ( n , \\ , \\tilde { q } ( H ) ) + { \\rm { e x } } ( \\tilde { q } ( H ) - 1 , \\ , \\tilde { \\mathcal { B } } ( H ) ) + ( k ( H ) - 1 ) ^ 2 \\\\ = & e ( T _ 2 ( n - \\tilde { q } ( H ) + 1 ) ) + ( \\tilde { q } ( H ) - 1 ) ( n - \\tilde { q } ( H ) + 1 ) + { \\rm { e x } } ( \\tilde { q } ( H ) - 1 , \\ , \\tilde { \\mathcal { B } } ( H ) ) + ( k ( H ) - 1 ) ^ 2 . \\end{align*}"}
+{"id": "1784.png", "formula": "\\begin{align*} p _ { A \\backslash j } \\left ( W \\right ) \\left \\langle Y , B , j \\right \\rangle = 0 , \\end{align*}"}
+{"id": "1450.png", "formula": "\\begin{align*} s M ( b , c ; s ) & = s M ' ( b , c ; s ) + ( c - b ) M ( b , c ; s ) - ( c - b ) M ( b - 1 , c ; s ) , \\\\ c M ' ( b , c ; s ) & = c M ( b , c ; s ) - ( c - b ) M ( b , c + 1 ; s ) . \\end{align*}"}
+{"id": "1820.png", "formula": "\\begin{align*} a s _ A ( x , y , z ) + a s _ A ( y , x , z ) & = 0 , \\\\ a s _ A ( z , x , y ) + a s _ A ( z , y , x ) & = 0 . \\end{align*}"}
+{"id": "171.png", "formula": "\\begin{align*} \\begin{aligned} | v _ 1 | ^ 2 - | v | ^ 2 = | v _ 1 - v | ^ 2 + 2 v \\cdot ( v _ 1 - v ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "6379.png", "formula": "\\begin{align*} G _ { 1 } G _ { 2 } = v ^ { \\Pi ( G _ { 1 } , G _ { 2 } ) } G _ { 2 } G _ { 1 } . \\end{align*}"}
+{"id": "6158.png", "formula": "\\begin{align*} V _ 1 V _ 2 ^ * & = ( I _ { H ^ 2 } \\otimes P ^ \\perp U + M _ z \\otimes P U ) ( I _ { H ^ 2 } \\otimes P U + M _ z ^ * \\otimes P ^ \\perp U ) \\\\ & = I _ { H ^ 2 } \\otimes { P ^ \\perp } U P U + M _ z \\otimes P U P U + M _ z ^ * \\otimes { P ^ \\perp } U { P ^ \\perp } U + M _ z M _ z ^ * \\otimes P U { P ^ \\perp } U . \\end{align*}"}
+{"id": "6406.png", "formula": "\\begin{align*} \\mathrm { d e g } _ \\infty = ( 2 \\varpi _ 1 , 2 \\varpi _ 1 , 2 \\varpi _ 1 , 2 \\varpi _ 1 ) \\in \\mathsf { P } _ + ^ { \\oplus 4 } . \\end{align*}"}
+{"id": "3418.png", "formula": "\\begin{align*} E _ U ( \\tilde \\Phi ) = 0 , J _ 1 E _ S ( \\tilde \\Phi ) = D _ r E _ { \\tilde \\rho } ( \\tilde \\Phi ) \\end{align*}"}
+{"id": "1854.png", "formula": "\\begin{align*} z _ { i , j } & = x _ { \\eta ( i ) , j } , \\\\ z _ { m _ 0 ^ { + } , j } & = x _ { m _ 0 ^ { + } , j } , \\mbox { a n d } \\\\ z _ { m + 1 - i , j } & = x _ { m + 1 - \\eta ( i ) , j } . \\end{align*}"}
+{"id": "5937.png", "formula": "\\begin{align*} \\theta _ { \\delta } : = \\delta \\O \\times \\delta ^ 2 \\O \\times \\dots \\times \\delta ^ k \\O \\end{align*}"}
+{"id": "5052.png", "formula": "\\begin{align*} K _ n ( s , f , \\chi ) = \\zeta ^ { ( N ) } ( 2 s ) \\sum _ { m = 1 } ^ \\infty \\frac { f _ m } { m ^ s } \\frac { \\Gamma ( k - 1 ) } { ( 4 \\pi ) ^ { k - 1 } } \\sum _ { g \\in H _ k ( N , \\chi ) } \\rho _ g ( n ) \\overline { \\rho _ g ( m ) } = \\frac { \\Gamma ( k - 1 ) } { ( 4 \\pi ) ^ { k - 1 } } \\sum _ { g \\in H _ k ( N , \\chi ) } \\rho _ g ( n ) L ( s , f \\times \\bar { g } ) , \\end{align*}"}
+{"id": "1715.png", "formula": "\\begin{align*} \\mathfrak { B } _ p : = \\{ \\xi \\in \\mathfrak { S } ^ 2 ( \\mathfrak { h } ^ { ( p ) } ) : \\| \\xi \\| _ { \\mathfrak { S } ^ 2 ( \\mathfrak { h } ^ { ( p ) } ) } \\leq 1 \\} \\ , . \\end{align*}"}
+{"id": "2035.png", "formula": "\\begin{align*} \\aligned g _ 0 ( t ) & : = - 4 ( e ^ { t / 2 } + e ^ { - t / 2 } - 2 ) , g _ 1 ( t ) : = \\sum _ { n \\leq e ^ { | t | } } \\frac { \\Lambda ( n ) } { \\sqrt { n } } ( | t | - \\log n ) , \\\\ g _ \\infty ( t ) & : = - \\frac { | t | } { 2 } \\left [ \\frac { \\Gamma ' } { \\Gamma } \\left ( \\frac { 1 } { 4 } \\right ) - \\log \\pi \\right ] - \\frac { 1 } { 4 } \\left ( C - e ^ { - | t | / 2 } \\Phi ( e ^ { - 2 | t | } , 2 , 1 / 4 ) \\right ) \\endaligned \\end{align*}"}
+{"id": "1695.png", "formula": "\\begin{align*} \\lambda _ k : = 4 \\pi ^ 2 | k | ^ 2 + \\kappa \\end{align*}"}
+{"id": "7218.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta _ { \\mathbb { R } ^ 2 } ) u = u | u | ^ 2 , \\end{align*}"}
+{"id": "5320.png", "formula": "\\begin{align*} \\eta ( C u _ { 1 } ) = \\eta ( C u _ { 2 } ) = \\dotsc = \\eta ( C u _ { p } ) . \\end{align*}"}
+{"id": "5692.png", "formula": "\\begin{gather*} c _ { i } ( a _ K e _ K ) = a _ K ( e _ { \\sigma ( 1 ) } \\wedge e _ { \\sigma ( 2 ) } \\wedge \\cdots \\wedge e _ { \\sigma ( i + 1 ) } ) \\otimes e _ 1 ^ { * } . \\end{gather*}"}
+{"id": "4058.png", "formula": "\\begin{align*} ( I d \\otimes \\delta ) \\delta = \\tau ^ { 1 2 } ( I d \\otimes \\delta ) \\delta . \\end{align*}"}
+{"id": "2335.png", "formula": "\\begin{align*} I _ { \\mathrm { b l } } ^ { \\mathfrak { m } } ( l _ { 1 } , \\ldots , l _ { d } ) : = I ^ { \\mathfrak { m } } ( a _ { 0 } , \\ldots , a _ { n + 1 } ) \\end{align*}"}
+{"id": "3016.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { y } & = ( \\varphi _ 1 , \\varphi _ 2 + \\tfrac { \\mathrm { d } ^ \\beta } { \\mathrm { d } t ^ \\beta } \\varphi _ 1 , \\varphi _ 3 , \\ldots , \\varphi _ m ) \\end{aligned} \\end{align*}"}
+{"id": "283.png", "formula": "\\begin{align*} \\varphi '^ { ( m ) } _ { s } ( \\bar { a } ) = \\begin{cases} \\overline { - F ^ { s - 1 } d a } \\otimes \\overline { 1 } & ( ( m , p ) \\neq ( 2 , 2 ) ) , \\\\ \\overline { - F ^ { s - 1 } d a } \\otimes \\overline { 1 } + \\overline { d \\pi / \\pi ^ { 2 } } \\otimes \\sqrt { \\overline { a _ { 0 } \\pi ^ { 2 } } } & ( ( m , p ) = ( 2 , 2 ) ) \\end{cases} \\end{align*}"}
+{"id": "5769.png", "formula": "\\begin{align*} - \\frac { 4 } { 5 } a < b < - \\frac { 5 } { 6 } a . \\end{align*}"}
+{"id": "607.png", "formula": "\\begin{align*} \\varphi ^ { + + } _ { n , p } = \\varphi ^ { - - } _ { n , p } = : \\varphi ^ { D } _ { n , p } , \\varphi ^ { + - } _ { n , p } = \\varphi ^ { - + } _ { n , p } = : \\varphi ^ { A } _ { n , p } , \\varphi ^ { 0 + } _ { n , p } = \\varphi ^ { 0 - } _ { n , p } = : \\varphi ^ { V } _ { n , p } , \\varphi ^ { + 0 } _ { n , p } = \\varphi ^ { - 0 } _ { n , p } = : \\varphi ^ { H } _ { n , p } . \\end{align*}"}
+{"id": "4732.png", "formula": "\\begin{align*} \\partial ( a \\cdot b ) & = \\cdot \\sigma \\Delta ( a \\cdot b ) \\overset { \\eqref { e q : a s i i f s } } { = } \\cdot \\sigma \\bigg ( \\sum _ i a _ i ^ 1 \\cdot b \\otimes a _ i ^ 2 + \\sum _ i b _ i ^ 1 \\otimes a \\cdot b _ i ^ 2 \\bigg ) \\\\ & = \\sum _ i a _ i ^ 2 \\cdot a _ i ^ 1 \\cdot b + \\sum _ i a \\cdot b _ i ^ 2 \\cdot b _ i ^ 1 = \\partial ( a ) \\cdot b + a \\cdot \\partial ( b ) , \\end{align*}"}
+{"id": "4371.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\sum _ { \\alpha = 1 } ^ { m } \\lambda _ \\alpha D _ H g ^ \\alpha ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) \\\\ + \\sum _ { \\beta = 1 } ^ { q } \\mu _ \\beta D _ H h ^ \\beta ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) = 0 . \\end{array} \\right \\} \\end{align*}"}
+{"id": "6393.png", "formula": "\\begin{align*} \\sigma _ { i j } : = d _ i ^ { - 1 } \\varepsilon _ { i j } \\gcd ( d _ i , d _ j ) . \\end{align*}"}
+{"id": "2194.png", "formula": "\\begin{align*} & 2 ( \\Gamma ^ 1 _ { 1 1 } G - \\Gamma ^ 1 _ { 1 2 } F ) \\Delta + G _ 1 \\Delta - G \\Delta _ 1 \\\\ = & G ( G E _ 1 - 2 F F _ 1 + F E _ 2 ) + F ( F G _ 1 - G E _ 2 ) - ( F ^ 2 G _ 1 - 2 F G F _ 1 + G ^ 2 E _ 1 ) \\\\ = & 0 . \\end{align*}"}
+{"id": "7812.png", "formula": "\\begin{align*} J ^ x ( X ) = - \\frac { 1 } { 2 } \\int _ { - \\infty } ^ { \\infty } x f ^ 2 ( x ) d x \\end{align*}"}
+{"id": "2888.png", "formula": "\\begin{align*} 2 H = \\alpha \\frac { \\nu _ 3 } { z } + \\varpi , \\end{align*}"}
+{"id": "2687.png", "formula": "\\begin{align*} M _ { \\varPsi } ( 2 n , j , 0 ; l - 1 ) = ( - 1 ) ^ j \\frac { \\binom { 2 l } { l } \\binom { 2 n } { n } \\binom { 2 j } { j } \\binom { 2 ( n + l - j ) } { n + l - j } \\binom { 2 n - j } { n } } { \\binom { n + l } { n } \\binom { 2 n + l - j } { n } } \\end{align*}"}
+{"id": "737.png", "formula": "\\begin{align*} { p _ { E O S } } = \\frac { { \\rho R T } } { { 1 - b \\rho } } - \\frac { { a \\xi \\left ( T \\right ) { \\rho ^ 2 } } } { { 1 + 2 b \\rho - { b ^ 2 } { \\rho ^ 2 } } } , \\end{align*}"}
+{"id": "5867.png", "formula": "\\begin{align*} \\langle ( s \\cdot a ) \\hat { } , \\omega \\rangle & = \\langle \\omega , s \\cdot a \\rangle = \\langle s ^ { - 1 } \\omega , a \\rangle = \\langle \\hat a , s ^ { - 1 } \\omega \\rangle = \\langle s \\cdot \\hat a , \\omega \\rangle , \\end{align*}"}
+{"id": "6999.png", "formula": "\\begin{align*} - \\Delta v = | \\nabla v | ^ 2 + \\alpha _ 2 v + ( \\lambda - \\alpha _ 1 ) . \\end{align*}"}
+{"id": "4125.png", "formula": "\\begin{align*} U '' ( S ( x ) ) + \\left ( \\chi _ n ^ { \\alpha } ( c ) + \\theta _ { \\alpha } ( S ( x ) ) \\right ) U ( S ( x ) ) = 0 , \\ , \\ , x \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "7982.png", "formula": "\\begin{align*} S _ M = M \\cap L _ v \\not = \\emptyset . \\end{align*}"}
+{"id": "853.png", "formula": "\\begin{align*} \\Theta ( K ) = - \\gamma _ 2 \\left ( 1 - \\frac { 1 } { \\rho } \\right ) \\frac { k _ 2 - 1 } { k _ 2 } K ^ { \\rho - 1 } + \\frac { 2 \\gamma _ 1 } { ( k _ 2 \\sigma a ' ) ^ { 2 } } \\ln \\frac { K k _ 2 } { k _ 2 - 1 } \\end{align*}"}
+{"id": "6572.png", "formula": "\\begin{align*} \\begin{pmatrix} B & C \\\\ \\mathbf { 0 } & D \\end{pmatrix} , \\end{align*}"}
+{"id": "6251.png", "formula": "\\begin{align*} | { \\rm F i x } _ k ( C ) | & = \\sum _ { i _ 1 \\cdots i _ r = k } \\prod _ { s = 1 } ^ r | { \\rm F i x } _ { i _ s } ( C _ s ) | \\end{align*}"}
+{"id": "6471.png", "formula": "\\begin{align*} & \\Big | { \\mathcal F _ { ( t , s ) \\to ( \\xi _ 0 , \\eta _ 0 ) } } \\Big ( \\frac { \\psi ( t , s ) } { \\lambda _ j } e ^ { i { \\lambda _ j ( t - s ) } } \\Big ) ( \\xi _ 0 , \\eta _ 0 ) \\Big | ^ 2 \\\\ & = \\frac { 1 } { \\lambda _ j ^ 2 } \\left | { \\mathcal F } ( \\psi ) ( \\xi _ 0 - \\lambda _ j , \\eta _ 0 + \\lambda _ j ) \\right | ^ 2 \\\\ \\end{align*}"}
+{"id": "2488.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = | x | ^ { a } v ^ { p } & & \\quad \\mbox { i n } B _ R , \\\\ \\Delta v & = | x | ^ { b } v ^ { q } f ( | \\nabla u | ) & & \\quad \\mbox { i n } B _ R , \\end{aligned} \\right . \\end{align*}"}
+{"id": "6925.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { N } \\left ( \\bigg ( \\frac { z _ i ( \\alpha _ i + y ) } { \\alpha _ i ( z _ i + y ) } \\ , \\frac { R ( z _ i ) } { \\prod _ { j = 1 } ^ { N } ( z _ j - \\alpha _ i ) } \\bigg ) ^ { b _ i } \\prod _ { p = 1 } ^ { r } \\bigg ( \\frac { 1 + \\alpha _ i x _ p } { 1 + z _ i x _ p } \\bigg ) ^ { a _ i + m _ p + 1 } \\right ) = ( - 1 ) ^ { ( N - 1 ) \\sum b _ i } f ( z _ { N + 1 } ) . \\end{align*}"}
+{"id": "386.png", "formula": "\\begin{align*} \\{ \\{ f , h \\} , C ^ \\infty ( M ) \\} = \\{ \\{ h , C ^ \\infty ( M ) \\} , f \\} - \\{ \\{ f , C ^ \\infty ( M ) \\} , h \\} \\end{align*}"}
+{"id": "8161.png", "formula": "\\begin{align*} A = \\left [ \\begin{array} { l l } 0 & 0 \\\\ 1 & 0 \\end{array} \\right ] , \\ \\ \\ B = \\left [ \\begin{array} { l l } 1 & 0 \\\\ 0 & 0 \\end{array} \\right ] . \\end{align*}"}
+{"id": "6443.png", "formula": "\\begin{align*} \\lim _ { q \\to 1 } \\int _ a ^ b f ( x ) d _ q x = \\int _ a ^ b f ( x ) d x . \\end{align*}"}
+{"id": "324.png", "formula": "\\begin{align*} \\theta ( t + 1 ) = \\Phi ( \\theta ( t ) , h ( t + 1 ) , h ( t ) , \\eta ( t ) ) , \\end{align*}"}
+{"id": "2187.png", "formula": "\\begin{align*} & R _ { 1 2 1 2 } = K \\Delta , R _ { 1 2 1 3 } = R _ { 1 2 2 3 } = 0 , \\\\ & R _ { 1 3 1 3 } = - f H ^ f _ { 1 1 } , R _ { 1 3 2 3 } = - f H ^ f _ { 1 2 } , R _ { 2 3 2 3 } = - f H ^ f _ { 2 2 } . \\end{align*}"}
+{"id": "7992.png", "formula": "\\begin{align*} w ( e ^ { i t } ) = | p ( e ^ { i t } ) | ^ 2 \\end{align*}"}
+{"id": "3070.png", "formula": "\\begin{align*} C _ { \\sigma } ( \\ell _ j ) = q _ j \\ell _ j \\qquad \\end{align*}"}
+{"id": "6832.png", "formula": "\\begin{align*} A & = \\ell _ { b } ( m _ { a } ^ 2 - u _ { a } ^ 2 ) + \\ell _ { a } ( m _ { b } ^ 2 - u _ { b } ^ 2 ) + \\ell _ { a b } \\big ( ( m _ { a } + u _ { a } ) ( m _ { b } + u _ { b } ) + ( m _ { a } - u _ { a } ) ( m _ { b } - u _ { b } ) \\big ) \\\\ & = \\ell _ { b } m _ { a } ^ 2 + \\ell _ { a } m _ { b } ^ 2 + 2 \\ell _ { a b } m _ { a } m _ { b } + 2 \\ell _ { a b } u _ { a } u _ { b } - \\ell _ { b } u _ { a } ^ 2 - \\ell _ { a } u _ { b } ^ 2 . \\end{align*}"}
+{"id": "3005.png", "formula": "\\begin{align*} \\| f \\| _ { X ^ r _ i } = \\| f _ \\Phi \\| _ { L ^ 2 _ y ( I _ i ; H ^ r _ x ) } , \\end{align*}"}
+{"id": "7980.png", "formula": "\\begin{align*} \\theta ' ( s ) & = \\frac { 2 n - 1 } { r ^ 2 ( s ) } \\left ( r ( s ) \\cos { \\left ( \\theta ( s ) \\right ) } - c r ^ 3 ( s ) \\right ) \\\\ & = \\frac { 2 n - 1 } { r ^ 2 ( s ) } \\left ( - \\phi _ 0 r ^ { 2 - 2 n } ( s ) - \\frac { 2 c } { 2 n + 1 } r ^ 3 ( s ) \\right ) . \\end{align*}"}
+{"id": "4599.png", "formula": "\\begin{align*} \\begin{cases} \\cot ( \\pi \\theta ( n + 1 ) - \\pi k ) = \\cot \\pi \\theta ( n ) - \\frac { V ( n ) } { \\sin \\pi k } , \\\\ \\cot ( \\pi \\tilde { \\theta } ( n + 1 ) - \\pi \\tilde { k } ) = \\cot \\pi \\tilde { \\theta } ( n ) - \\frac { V ( n ) } { \\sin \\pi \\tilde { k } } , \\end{cases} \\end{align*}"}
+{"id": "7740.png", "formula": "\\begin{align*} a ( \\delta ) = \\frac { 1 } { n } \\min \\{ - x \\Delta _ g x | _ { \\partial X _ \\delta } \\} > 0 . \\end{align*}"}
+{"id": "4711.png", "formula": "\\begin{align*} ( \\lambda + 1 ) K _ { \\lambda , 1 } ( z , w ) = \\left [ \\left ( r \\frac { d } { d r } \\right ) ^ 2 + ( 2 \\lambda + 3 ) r \\frac { d } { d r } + ( \\lambda + 2 ) ( \\lambda + 1 ) \\right ] C ( z , w ) . \\end{align*}"}
+{"id": "4734.png", "formula": "\\begin{align*} [ ( S _ 1 , S _ 2 ) , ( T _ 1 , T _ 2 ) ] : = ( - [ S _ 1 , T _ 1 ] , [ S _ 2 , T _ 2 ] ) = ( T _ 1 S _ 1 - S _ 1 T _ 1 , S _ 2 T _ 2 - T _ 2 S _ 2 ) , \\end{align*}"}
+{"id": "1061.png", "formula": "\\begin{align*} \\nabla ^ { ( s ) } _ { e _ { i } } = e _ { i } - \\frac { 1 } { 4 } \\alpha _ { i j k } \\gamma ^ { j } \\gamma ^ { k } , \\end{align*}"}
+{"id": "7553.png", "formula": "\\begin{align*} R _ N ( w , z ) = \\frac { 1 } { w - z } \\begin{pmatrix} 0 & I _ q \\end{pmatrix} Y ^ { - 1 } ( w ) Y ( z ) \\begin{pmatrix} I _ q \\\\ 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "5310.png", "formula": "\\begin{align*} W _ a ( x ) : = \\sup _ { \\zeta ( \\rho ) = 0 , \\rho = \\beta + i \\gamma \\atop a < \\Re \\rho = \\beta , | \\gamma | < x } \\frac { x ^ \\beta } { | \\rho | } , W ( x ) : = \\sup _ { \\zeta ( \\rho ) = 0 , \\rho = \\beta + i \\gamma \\atop \\theta < \\Re \\rho = \\beta , | \\gamma | < x } \\frac { x ^ \\beta } { | \\rho | } . \\end{align*}"}
+{"id": "3537.png", "formula": "\\begin{align*} \\Big \\Vert | \\mathrm { E } | ^ 2 \\Big \\Vert _ { \\mathrm { L } ^ 2 \\Big ( \\Omega \\Big ) } = \\mathcal { O } \\Big ( \\delta | \\log \\delta | ^ { \\frac { 3 \\mathrm { h } } { 2 } } \\Big ) . \\end{align*}"}
+{"id": "6549.png", "formula": "\\begin{align*} ( 2 Q - I ) e _ i = e _ j . \\end{align*}"}
+{"id": "4082.png", "formula": "\\begin{align*} D _ i : = \\sum _ { ( m _ j ) \\in S _ i } \\prod _ { j = 1 } ^ n \\binom { d _ j + m _ j - 1 } { m _ j } \\end{align*}"}
+{"id": "2848.png", "formula": "\\begin{align*} \\mu _ { \\Delta ^ \\wedge _ w } ( f \\otimes 1 ) = \\mu _ { \\Delta ^ \\wedge _ w } ( 1 \\otimes v ^ { - 1 } ( f ) ) , \\end{align*}"}
+{"id": "5891.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { \\tilde { x } } _ n ( s ) = \\dot { \\tilde { v } } _ n ( s ) , \\ \\ \\dot { \\tilde { v } } _ n ( s ) = \\frac { 1 } { \\epsilon } \\tilde { B } ( x _ { n + \\frac { 1 } { 2 } } ) \\tilde { v } _ n ( s ) + F ( { \\tilde { x } } _ n ( s ) ) , \\ \\ { \\tilde { x } } _ n ( 0 ) = x ( t _ n ) , \\ \\ { \\tilde { v } } _ n ( 0 ) = v ( t _ n ) , \\ \\ \\ 0 < s \\leq h , \\end{aligned} \\end{align*}"}
+{"id": "6464.png", "formula": "\\begin{align*} p ^ { \\# } ( x , \\xi ) = \\displaystyle \\sum _ { j = 0 } ^ { \\infty } J _ { \\epsilon _ { j } } p ( x , \\xi ) \\psi _ j ( \\xi ) . \\end{align*}"}
+{"id": "6614.png", "formula": "\\begin{align*} \\chi _ { n } ( x ) & = 0 x \\in ( - \\infty , n ) \\cup ( 2 n , \\infty ) , \\\\ \\chi _ { n } ( x ) & = 1 x \\in ( n + 1 , 2 n - 1 ) , \\\\ \\widetilde { \\chi } _ { n } ( y ) & = 0 y \\in ( - \\infty , - a n ) \\cup ( a n , \\infty ) , \\\\ \\widetilde { \\chi } _ { n } ( y ) & = 1 y \\in ( - a n + 1 , a n - 1 ) . \\end{align*}"}
+{"id": "669.png", "formula": "\\begin{align*} [ T ] + \\nu [ N ] = \\frac { 2 } { \\mu } h _ z J + \\nu I , \\end{align*}"}
+{"id": "123.png", "formula": "\\begin{align*} T x = T ( x _ 1 , x _ 2 , \\dots ) : = ( \\alpha _ 1 T _ 1 x _ 1 , \\alpha _ 2 T _ 2 x _ 2 , \\dots ) . \\end{align*}"}
+{"id": "5290.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p } \\le \\| \\hat f \\| _ { L ^ q } \\textrm { i f $ 1 \\le p \\le 2 $ a n d $ \\frac { 1 } { q } = 1 - \\frac { 1 } { p } $ } \\end{align*}"}
+{"id": "1915.png", "formula": "\\begin{align*} P _ 0 u + \\lambda u = \\div \\vec f + g , \\end{align*}"}
+{"id": "2794.png", "formula": "\\begin{align*} J = X C Y ^ T , \\widehat { J } = \\widehat { X } S Y ^ T . \\end{align*}"}
+{"id": "4045.png", "formula": "\\begin{align*} \\partial _ s H _ n ( y , s ) = n ( n - 1 ) \\left ( 1 - \\frac { 1 } { k } \\right ) I ^ { - 2 } ( s ) H _ { n - 2 } ( y , s ) . \\end{align*}"}
+{"id": "5780.png", "formula": "\\begin{align*} a _ { s , t } \\le r ^ 3 + r ( r + 1 ) s + r t + 1 \\le s + r ( r + 1 ) s + r t = ( r ^ 2 + r + 1 ) s + r t . \\end{align*}"}
+{"id": "4811.png", "formula": "\\begin{align*} ( \\mathcal F _ \\pi ( \\phi _ n ) ) ' = ( \\pi ( x ) \\phi _ n ) ' = \\pi ' ( x ) \\phi _ n + \\pi ( x ) \\phi _ n ' = ( \\mathcal F _ { \\pi ' } + \\mathcal P ) ( \\phi _ n ) . \\end{align*}"}
+{"id": "1701.png", "formula": "\\begin{align*} H _ 0 : = \\Theta ( h ) = \\int d x \\ , d y \\ , \\overline { \\varphi } ( x ) h ( x ; y ) \\varphi ( y ) \\ , . \\end{align*}"}
+{"id": "5060.png", "formula": "\\begin{align*} \\sum _ { \\substack { a \\bmod { q } \\\\ a \\bar { a } \\equiv 1 \\bmod { q } } } e ( n \\bar { a } / q ) L _ f \\bigl ( 1 - s - u , - \\tfrac { \\bar { a } } { q } \\bigr ) = \\sum _ { \\substack { a \\bmod { q } \\\\ a \\bar { a } \\equiv 1 \\bmod { q } } } e ( n \\bar { a } / q ) \\sum _ { m = 1 } ^ \\infty \\frac { f _ m e ( - m \\bar { a } / q ) } { m ^ { 1 - s - u } } = \\sum _ { m = 1 } ^ \\infty \\frac { f _ m r _ q ( n - m ) } { m ^ { 1 - s - u } } , \\end{align*}"}
+{"id": "3069.png", "formula": "\\begin{align*} f _ { j } ( d \\ell _ j ) - f _ { j } ( \\ell _ j ) = f _ { j } ( d \\ell _ j ) - q _ j \\ell _ j \\geq 0 \\end{align*}"}
+{"id": "6551.png", "formula": "\\begin{align*} U e _ i = \\begin{cases} e _ { i + 2 } , \\quad i i \\neq n - 3 ; \\\\ e _ { i - 2 } , \\quad i i \\neq 0 ; \\\\ e _ { 1 } , \\quad i = 0 ; \\\\ e _ { n - 2 } , \\quad i = n - 3 . \\end{cases} \\end{align*}"}
+{"id": "5600.png", "formula": "\\begin{align*} C _ { i j s } J ^ i _ p J ^ j _ q = C _ { p q s } = C _ { s q p } = C _ { i j p } J ^ i _ s J ^ j _ q . \\end{align*}"}
+{"id": "7963.png", "formula": "\\begin{align*} \\theta ' ( s ) = \\frac { - \\frac { 2 c } { 3 } r ^ 3 ( s ) - \\varphi _ 0 } { r ^ 2 ( s ) } . \\end{align*}"}
+{"id": "4077.png", "formula": "\\begin{align*} ( ] x [ ^ * ( \\mathcal { V } ) ) ^ { \\nabla = 0 } \\stackrel { \\simeq } { \\longrightarrow } x _ 0 ^ * \\mathcal { V } \\end{align*}"}
+{"id": "5572.png", "formula": "\\begin{align*} \\alpha _ 0 ^ { * } = \\alpha _ 0 ^ { * } ( \\beta _ 0 ) = \\bigg ( \\frac { \\theta _ b - \\theta _ m } { 2 l _ b \\gamma _ b } \\bigg [ \\dfrac { 2 l _ m \\gamma _ m ( \\beta _ 0 ) ^ { \\nu + 1 } } { \\theta _ b - \\theta _ m } - \\dfrac { u _ c } { \\Phi _ 2 [ \\beta _ 0 , + \\infty , u _ 2 ] } \\bigg ] \\bigg ) ^ { \\frac { 1 } { { \\nu + 1 } } } \\end{align*}"}
+{"id": "3500.png", "formula": "\\begin{align*} \\mathbb { M } \\Big [ \\nabla \\mathrm { H } \\Big ] ( \\mathrm { x } ) = \\nabla \\int _ { \\Omega } \\nabla \\mathbb { G } ^ { ( 0 ) } ( \\mathrm { x } , \\mathrm { y } ) \\cdot \\nabla \\mathrm { H } ( \\mathrm { y } ) d \\mathrm { y } . \\end{align*}"}
+{"id": "115.png", "formula": "\\begin{gather*} \\Lambda ( z ; \\alpha ) : = \\alpha ( S - z ^ { - 1 } ) ^ { - 1 } , \\Gamma ( z ; \\alpha ) : = \\beta ( z - S ) ^ { - 1 } , \\end{gather*}"}
+{"id": "799.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow + \\infty } \\frac { 1 } { N } \\sum \\limits _ { j = 1 } ^ N e ^ { - \\beta \\tau _ j } x _ j ( \\tau _ j ) = \\mathbb { E } \\Big ( e ^ { - \\beta \\tau } x ( \\tau ) \\Big ) \\end{align*}"}
+{"id": "5333.png", "formula": "\\begin{align*} F ^ * ( G ) = T \\times C _ { p _ 1 } \\times \\cdots \\times C _ { p _ k } , \\end{align*}"}
+{"id": "5789.png", "formula": "\\begin{align*} f ( U ) = \\sum _ { V \\le U } g ( V ) \\mu ( V , U ) \\end{align*}"}
+{"id": "6796.png", "formula": "\\begin{align*} 0 = \\phi + \\phi \\circ ( 1 \\ , \\ , 3 ) + \\phi \\circ ( 1 \\ , \\ , 4 ) = \\phi \\circ ( 1 \\ , \\ , 3 ) = & \\alpha \\circ ( 1 \\ , \\ , 3 ) + \\alpha \\circ ( 3 \\ , \\ , 4 ) \\circ ( 1 \\ , \\ , 3 ) \\\\ = & \\alpha + \\alpha \\circ ( 1 \\ , \\ , 4 ) . \\end{align*}"}
+{"id": "1505.png", "formula": "\\begin{align*} \\nu = 1 - \\sum _ { w < p \\le z } g ( p ) - \\tfrac 9 2 \\alpha ^ 2 . \\end{align*}"}
+{"id": "5531.png", "formula": "\\begin{align*} ( g _ S ^ { - 1 } ( 1 ) , \\ldots , g _ S ^ { - 1 } ( 2 n ) ) = \\left ( \\prod _ { i = 1 } ^ r \\prod _ { j = i } ^ r I ^ + _ { i , j } \\right ) \\cdot \\left ( \\prod _ { j = 1 } ^ r \\prod _ { i = 1 } ^ j I ^ - _ { r + 1 - i , r + 1 - j } \\right ) . \\end{align*}"}
+{"id": "3164.png", "formula": "\\begin{align*} p ( x ) = \\big ( \\sum _ { i = 1 } ^ { l } \\max ^ { ~ ~ ~ ~ 2 } \\{ x _ i , 0 \\} + \\sum _ { j = l + 1 } ^ { m } x _ j ^ 2 \\big ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "64.png", "formula": "\\begin{align*} | \\nabla f | = \\alpha ( f ) \\Delta _ { \\Psi } f = \\beta ( f ) . \\end{align*}"}
+{"id": "4267.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { N } \\frac { ( - 1 ) ^ { n - 1 } \\left ( \\frac { c } { d } \\right ) _ { n } d ^ n q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( q ) _ n ( q ) _ { N - n } ( c q ) _ { n } } \\left ( n + \\sum _ { k = 1 } ^ n \\frac { q ^ k } { 1 - q ^ k } \\right ) \\\\ & = \\frac { c } { ( c - d ) ( q ) _ N } \\left ( 1 - \\frac { ( d q ) _ N } { ( c q ) _ N } \\right ) + \\frac { 1 } { ( c q ) _ N } \\sum _ { k = 1 } ^ N \\frac { \\left ( \\frac { c q } { d } \\right ) _ k ( d q ) ^ k ( d q ) _ { N - k } } { ( q ) _ k ( 1 - q ^ k ) ( q ) _ { N - k } } \\end{align*}"}
+{"id": "6955.png", "formula": "\\begin{align*} \\mathsf W _ { \\ell + 1 } = \\mathsf W _ { \\ell } \\cdot \\mathsf B ^ { N } . \\end{align*}"}
+{"id": "1890.png", "formula": "\\begin{align*} B _ { n , q } : = \\{ x _ { i _ 1 } \\otimes \\cdots \\otimes x _ { i _ q } , 1 \\leq i _ 1 , \\ldots , i _ q \\leq n \\} . \\end{align*}"}
+{"id": "2859.png", "formula": "\\begin{align*} \\mathsf { D } _ { \\mathsf { B } } = \\{ X \\in \\mathsf { D } \\mid \\forall n \\in \\Z , \\ , H ^ n ( X ) \\in \\mathsf { B } \\} . \\end{align*}"}
+{"id": "7318.png", "formula": "\\begin{align*} \\left [ \\frac { q ^ m - 1 } { 2 } - q ^ { m - 1 } - q + 3 , \\ \\frac { q ^ m - 1 } { 2 } - q ^ { m - 1 } - 2 \\right ] . \\end{align*}"}
+{"id": "2533.png", "formula": "\\begin{align*} \\overline { \\{ T ^ n \\pi ( g ) : n \\in \\Z \\} } = g Y . \\end{align*}"}
+{"id": "1153.png", "formula": "\\begin{align*} \\lambda _ { f } ( n ) = \\lambda _ t ( n ) = \\lambda _ s ( n ) \\end{align*}"}
+{"id": "3810.png", "formula": "\\begin{align*} a _ 1 + a _ 2 + \\cdots + a _ { h - 2 } = \\frac { ( h - 1 ) ( 2 n - 4 ) - ( a _ { h - 1 } + a _ h ) } { 2 } . \\end{align*}"}
+{"id": "5747.png", "formula": "\\begin{align*} ( { \\rm I d } _ 4 \\otimes P ) f ( X ^ 0 ) ( { \\rm I d } _ 4 \\otimes P ) & = \\begin{pmatrix} P ( X _ 1 ^ 0 X ^ 0 _ 2 + X ^ 0 _ 2 X ^ 0 _ 1 - X ^ 0 _ 2 X ^ 0 _ 2 ) P & 0 \\\\ 0 & 0 \\end{pmatrix} \\oplus \\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "7363.png", "formula": "\\begin{align*} \\eta _ { k + 1 } = - g _ { k + 1 } + \\beta _ { k + 1 } ( d \\exp _ { x _ k } ) _ { \\alpha _ k \\eta _ k } ( \\eta _ k ) . \\end{align*}"}
+{"id": "1096.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m U _ i \\geq \\sum _ { i = 1 } ^ m V _ i . \\end{align*}"}
+{"id": "7215.png", "formula": "\\begin{align*} \\bigcup _ { \\substack { a + b = d \\\\ a , b \\geq 1 } } B ( d ) \\times _ { B ( a ) \\times B ( b ) } \\left ( Z ( a ) ^ { \\mathrm { c l } } \\times _ { \\mathbb { C } ^ 2 } Z ( b ) ^ { \\mathrm { c l } } \\right ) \\end{align*}"}
+{"id": "6773.png", "formula": "\\begin{align*} T _ b & = - \\int d x \\ , b _ x ^ * \\Delta b _ x . \\end{align*}"}
+{"id": "5021.png", "formula": "\\begin{align*} \\mathcal { E } _ J : = \\{ \\mathcal { E } ( j ) \\} _ { j \\in J } . \\end{align*}"}
+{"id": "734.png", "formula": "\\begin{align*} { f _ { 1 3 } } = { f _ { 1 4 } } - \\frac { 1 } { 2 } \\left ( { { f _ 1 } - { f _ 2 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) - \\frac { 1 } { 8 } \\left ( { 2 { F _ x } - { F _ z } } \\right ) - \\frac { 1 } { 6 } \\rho { u _ z } , \\end{align*}"}
+{"id": "3704.png", "formula": "\\begin{align*} \\partial _ t \\rho + \\nabla ( \\rho v ) = 0 \\end{align*}"}
+{"id": "4882.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty B _ n ^ { ( k ) } \\frac { t ^ n } { n ! } = \\frac { \\mathrm { L i } _ k ( 1 - e ^ { - t } ) } { 1 - e ^ { - t } } . \\end{align*}"}
+{"id": "3719.png", "formula": "\\begin{align*} = \\int _ 0 ^ 1 \\int _ { \\R ^ d } [ ( - \\partial _ t \\lambda - \\nabla \\lambda ) v + v ^ 2 + \\frac { \\nabla ^ 2 \\sqrt \\rho } { \\sqrt \\rho } ] \\delta \\rho d x d t + [ \\lambda \\delta \\rho ] _ 0 ^ 1 + [ \\lambda v \\delta \\rho ] _ { - \\infty } ^ { + \\infty } \\end{align*}"}
+{"id": "6729.png", "formula": "\\begin{align*} 4 \\pi E ( t , x ) = - \\int _ { | y - x | \\leq t } \\int _ { { \\mathbb R } ^ 3 } \\frac { ( \\omega + \\hat { v } ) ( 1 - | \\hat { v } | ^ 2 ) } { ( 1 + \\hat { v } \\cdot \\omega ) ^ 2 } F ( t - | y - x | , y , v ) \\ , d v \\frac { d y } { | y - x | ^ 2 } + \\mbox { o t h e r t e r m s } , \\end{align*}"}
+{"id": "6357.png", "formula": "\\begin{align*} \\begin{aligned} P \\left ( \\frac { d } { d t } \\right ) x ( t ) & = B u ( t ) , \\\\ y ( t ) & = C x ( t ) + D u ( t ) , \\end{aligned} \\end{align*}"}
+{"id": "4692.png", "formula": "\\begin{align*} { } _ 2 \\ ! F _ { 1 } ( a , b ; a + b ; t ) & = 1 + \\frac { a b } { a + b } \\int _ 0 ^ t ( 1 - s ) ^ { - 1 } { } _ 2 \\ ! F _ { 1 } ( a , b ; a + b + 1 ; s ) d t \\\\ & \\asymp 1 + \\ln \\frac { 1 } { 1 - t } \\asymp \\ln \\left ( \\frac { 1 } { 1 - t } + 2 \\right ) , - 1 \\le t < 1 . \\end{align*}"}
+{"id": "646.png", "formula": "\\begin{align*} \\nabla _ Y \\mathcal { P } ( X ) = \\mathcal { P } ( B ( Y , X ) ) - B ( Y , f ( X ) ) + B ^ * ( Y , h ( X ) ) . \\end{align*}"}
+{"id": "3098.png", "formula": "\\begin{align*} F _ { \\Z } ( \\sigma _ 4 ^ k ) = \\begin{cases} | \\Z | & \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "587.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\big ( \\mu _ n ^ { ( 1 2 ) } - \\mu _ { n + 1 } ^ { ( 1 2 ) } \\big ) ^ 2 = \\big ( \\mu _ n ^ { ( 1 2 ) } + \\mu _ { n + 1 } ^ { ( 1 2 ) } \\big ) \\ , . \\end{align*}"}
+{"id": "2601.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } ^ { \\boldsymbol { \\xi } } [ | r ^ { N , i } ( t , \\boldsymbol { x } ^ * _ t ) | ^ 2 ] \\leq & \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } \\mathbb { E } ^ { \\boldsymbol { \\xi } } [ | x ^ { * , i } _ t - x ^ { * , j } _ t | ^ 2 ] \\Big ) \\\\ \\leq & \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 ] \\Big ) . \\end{aligned} \\end{align*}"}
+{"id": "6778.png", "formula": "\\begin{align*} \\norm { \\psi _ t } _ { H ^ 1 ( \\mathbb { R } ^ 3 ) } ^ 2 + \\norm { \\varphi _ t } _ { L ^ 2 ( \\mathbb { R } ^ 3 ) } ^ 2 \\leq 2 \\left ( \\mathcal { E } [ \\psi _ t , \\varphi _ t ] + C \\right ) = 2 \\left ( \\mathcal { E } [ \\psi , \\varphi ] + C \\right ) & \\leq C \\big ( \\norm { \\psi } _ { H ^ 1 ( \\mathbb { R } ^ 3 ) } ^ 2 + \\norm { \\varphi } _ { L ^ 2 ( \\mathbb { R } ^ 3 ) } ^ 2 \\big ) \\end{align*}"}
+{"id": "3095.png", "formula": "\\begin{align*} \\delta _ { d , r } ^ k = \\prod _ { h = 0 } ^ { k - 1 } \\delta _ { k d , r + h d } . \\end{align*}"}
+{"id": "5969.png", "formula": "\\begin{align*} b ( p ) : = ( p - k ( k + 1 ) ) ( 1 - \\frac { 1 } { k } ) ^ { - \\frac { p } { 2 k } } + 2 c ( p _ 0 ) , \\end{align*}"}
+{"id": "5521.png", "formula": "\\begin{align*} H \\cap P = ( H \\cap L ) \\ltimes U ^ { \\sigma } \\end{align*}"}
+{"id": "5819.png", "formula": "\\begin{align*} \\omega ( V ) = \\sup _ { \\| x \\| = 1 } | \\langle V x , x \\rangle | \\leq \\| V \\| = 1 . \\end{align*}"}
+{"id": "1712.png", "formula": "\\begin{align*} \\mathcal { W } ^ { \\mathrm { W i c k } } : = \\frac { 1 } { 2 } \\int d x \\ , d y \\ , \\bigl ( | \\varphi ( x ) | ^ 2 - \\mathbb { E } _ { \\mu } [ | \\varphi ( x ) | ^ 2 ] \\bigr ) \\ , w ( x - y ) \\ , \\bigl ( | \\varphi ( y ) | ^ 2 - \\mathbb { E } _ { \\mu } [ | \\varphi ( x ) | ^ 2 ] \\bigr ) \\ , , \\end{align*}"}
+{"id": "8110.png", "formula": "\\begin{align*} H _ L ( n ) = \\frac { ( L ^ d ) } { d ! } n ^ d + o ( n ^ d ) . \\end{align*}"}
+{"id": "3264.png", "formula": "\\begin{align*} \\langle \\psi _ n , \\psi _ m \\rangle = \\delta _ { m n } . \\end{align*}"}
+{"id": "1427.png", "formula": "\\begin{align*} \\lim _ { t \\to T _ { \\mathrm { m a x } } - 0 } \\| \\mathcal { U } ( t ) \\| _ { \\mathcal { H } } = \\infty . \\end{align*}"}
+{"id": "2764.png", "formula": "\\begin{align*} 2 a \\cdot ( C _ n / P _ k ) - 2 b \\cdot \\dim ( C _ n / P _ k ) & \\geq 2 ( 2 k - 2 ) ( 2 n - k + 1 ) - k ( 4 n - 3 k + 1 ) \\\\ & = 4 n k - k ^ 2 + 7 k - 8 n - 4 \\\\ & = \\frac { 2 n k } { 3 } - k ^ 2 + \\frac { 1 0 n k } { 3 } - 8 n + 7 k - 4 > 0 . \\end{align*}"}
+{"id": "4686.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ { \\pi } f ( e ^ { i \\varphi } ) e ^ { i \\varphi } \\phi _ { n - 1 } ( e ^ { i \\varphi } ) d m _ { \\lambda } ( \\varphi ) = 0 , n = 1 , 2 , \\dots , \\end{align*}"}
+{"id": "3837.png", "formula": "\\begin{align*} \\phi ( \\eta ) = \\left ( 3 + i _ { h - 4 } , 4 + i _ { h - 5 } , \\dots , h - 2 + i _ 1 , k + \\frac { - h ^ 2 + 3 h + 4 } { 2 } - i \\right ) . \\end{align*}"}
+{"id": "1810.png", "formula": "\\begin{align*} \\langle T ^ { * } ( a ^ { * } _ { 2 } ) , a _ 1 \\rangle = \\langle a ^ { * } _ { 2 } , T ( a _ 1 ) \\rangle . \\end{align*}"}
+{"id": "6418.png", "formula": "\\begin{align*} F _ { \\ast } \\left ( j _ { F } ^ { p } ( x ) - j _ { F } ^ { p } ( y ) \\right ) & \\le \\left ( F ( x ) + F ( y ) \\right ) ^ { p - \\tau } \\left ( \\sum _ { i } \\rho ( x _ { i } - y _ { i } ) ^ { p \\left ( \\frac { \\tau - 1 } { p - 1 } \\right ) } \\right ) ^ { \\frac { p - 1 } { p } } \\\\ & \\lesssim \\left ( F ( x ) + F ( y ) \\right ) ^ { p - \\tau } \\left ( \\sum _ { i } \\rho ( x _ { i } - y _ { i } ) ^ { p } \\right ) ^ { \\frac { \\tau - 1 } { p } } \\\\ & = ( F ( x ) + F ( y ) ) ^ { p - \\tau } F ( x - y ) ^ { \\tau - 1 } . \\end{align*}"}
+{"id": "8139.png", "formula": "\\begin{align*} ( \\overline L _ 0 | _ { Y } \\cdots \\overline L _ r | _ { Y } ) _ S \\geqslant \\sum _ { i = 0 } ^ r \\delta _ i \\ , \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L _ i | _ Y ) . \\end{align*}"}
+{"id": "7242.png", "formula": "\\begin{align*} \\| \\vec { u } \\| ^ 2 _ { X ( G _ k ^ { j } ) } : = \\sum _ { 0 \\leq i < j } 2 ^ { i - j } \\sum _ { G _ { \\alpha } ^ { i } \\subset G _ k ^ { j } } \\| P _ { \\xi ( G _ { \\alpha } ^ { i } ) , i - 2 \\leq \\cdot \\leq i + 2 } \\vec { u } \\| ^ 2 _ { U ^ 2 _ { \\triangle } ( h ^ { 1 } ; G _ { \\alpha } ^ { i } \\times \\mathbb { R } ^ 2 ) } + \\sum _ { i \\geq j } \\| P _ { \\xi ( G _ { k } ^ { j } ) , i - 2 \\leq \\cdot \\leq i + 2 } \\vec { u } \\| ^ 2 _ { U ^ 2 _ { \\triangle } ( h ^ { 1 } ; G _ { k } ^ { j } \\times \\mathbb { R } ^ 2 ) } . \\end{align*}"}
+{"id": "2099.png", "formula": "\\begin{align*} \\sigma ( V _ 1 , \\ldots , V _ m ) : = \\sigma \\big ( g _ 1 ^ * V _ 1 , \\ldots , g _ m ^ * V _ m \\big ) . \\end{align*}"}
+{"id": "4278.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\textup { s p t } ( n ) q ^ n = \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { n ( - 1 ) ^ { n - 1 } q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( q ) _ n ( 1 - q ^ n ) } , \\end{align*}"}
+{"id": "5070.png", "formula": "\\begin{align*} { } f \\ ! \\left ( \\frac { \\frac { - 1 } { H ^ 2 z } + a } { q } \\right ) = \\omega \\chi ( \\bar { a } ) ( - i H z ) ^ \\ell f \\ ! \\left ( \\frac { z - \\bar { a } } { q } \\right ) \\end{align*}"}
+{"id": "6591.png", "formula": "\\begin{align*} [ B ^ 2 , B C ] = 2 B [ B , B C ] = 2 B ( [ B , B ] C + B [ B , C ] ) = O . \\end{align*}"}
+{"id": "350.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\Omega } F ( u ) d x = ( f ( u ) , u _ t ) . \\end{align*}"}
+{"id": "6324.png", "formula": "\\begin{align*} \\inf _ { w \\in A } \\ , E ( w ) \\le c : = \\inf _ { h \\in \\widetilde { H } } \\ , \\sup _ { u \\in h ( Q ) } \\ , E ( u ) \\le \\sup _ { u \\in Q } \\ , E ( u ) , \\end{align*}"}
+{"id": "7743.png", "formula": "\\begin{align*} ( \\mathbb { H } ^ { n + 1 } , g _ \\mathbb { H } = d t ^ 2 + \\sinh ^ 2 t g _ { \\mathbb { S } ^ n } ) \\end{align*}"}
+{"id": "1752.png", "formula": "\\begin{align*} \\partial P \\left ( Y , \\mathcal { Z } \\right ) = \\partial P \\left ( Y , \\mathcal { Z } ^ { ' } \\right ) = 0 . \\end{align*}"}
+{"id": "660.png", "formula": "\\begin{align*} \\nabla _ X \\Psi _ 2 = - \\frac { 1 } { 2 } \\left ( X - \\langle X , T _ 1 \\rangle ( T _ 1 + f _ 1 ) - \\langle X , T _ 2 \\rangle ( T _ 2 + f _ 2 ) \\right ) \\cdot \\Psi _ 1 + \\frac { 1 } { 2 } S ( X ) \\cdot \\Psi _ 2 , \\end{align*}"}
+{"id": "4772.png", "formula": "\\begin{align*} a \\cdot b = a \\ast b + b \\ast a , \\forall a , b \\in A . \\end{align*}"}
+{"id": "2997.png", "formula": "\\begin{align*} \\ln \\frac { f ( x ) } { x } - \\frac { 2 } { f ( x ) / x - 1 } = \\ln \\frac { \\beta } { x } + \\frac { 2 } { 1 - \\beta / \\alpha } . \\end{align*}"}
+{"id": "98.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ \\Psi u = f ( u ) & A ( r _ { 1 } , r _ { 2 } ) \\\\ u \\equiv c _ { 1 } & \\{ r _ { 1 } \\} \\times N \\\\ u \\equiv c _ { 2 } & \\{ r _ { 2 } \\} \\times N . \\end{cases} \\end{align*}"}
+{"id": "540.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\partial _ i f ( x ) \\big ( T _ i ( x _ i ) - x _ i \\big ) Q ( x ) d x & = \\int _ { \\R } \\hat { f } ' _ i ( x _ i ) \\big ( T _ i ( x _ i ) - x _ i \\big ) Q _ i ( x _ i ) d x _ i , \\end{align*}"}
+{"id": "588.png", "formula": "\\begin{align*} \\mu _ { n + 2 } ^ { ( 1 2 ) } - 2 \\mu _ { n + 1 } ^ { ( 1 2 ) } + \\mu _ n ^ { ( 1 2 ) } = 2 \\ , , \\end{align*}"}
+{"id": "2821.png", "formula": "\\begin{align*} d \\psi _ s & = \\rho \\left ( \\frac { \\lambda } { \\lambda + \\rho } + \\theta _ s \\right ) \\psi _ s d s + \\frac { \\rho \\lambda } { \\lambda + \\rho } \\sqrt { \\gamma _ 0 } \\zeta _ s ( 1 - K _ s ) d s + \\sum _ { j = 1 } ^ m \\phi _ s ^ j d W _ s ^ j , s \\in [ 0 , T ] , \\\\ \\psi _ T & = - \\frac 1 2 \\sqrt { \\gamma _ 0 } \\xi . \\end{align*}"}
+{"id": "5734.png", "formula": "\\begin{align*} X ^ 0 _ i : = \\bordermatrix { & & & ( i + 1 ) & & \\cr & 0 & \\cdots & 1 & \\cdots & 0 \\cr & \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\cr ( i + 1 ) & 1 & \\cdots & 0 & \\cdots & 0 \\cr & \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\cr & 0 & \\cdots & 0 & \\cdots & 0 } . \\end{align*}"}
+{"id": "7748.png", "formula": "\\begin{align*} d ( p _ 0 , y ) = d ( ( t _ 0 , w , \\theta ) , ( t , w _ 0 , \\theta _ 0 ) ) \\end{align*}"}
+{"id": "6568.png", "formula": "\\begin{align*} \\langle \\delta _ { \\omega ' } , \\hat { E } _ i \\delta _ \\omega \\rangle = 0 \\end{align*}"}
+{"id": "1909.png", "formula": "\\begin{align*} | P _ { k } - P _ { k - 1 } | ( x , t ) \\le | u - P _ k | ( x , t ) + | u - P _ { k - 1 } | ( x , t ) = \\mathcal O ( t ^ { a _ { k - 1 } } ) . \\end{align*}"}
+{"id": "7672.png", "formula": "\\begin{align*} \\mathrm { d i m } _ \\mathcal { H } ( \\partial E \\cap \\partial \\Omega ) = \\alpha . \\end{align*}"}
+{"id": "5504.png", "formula": "\\begin{align*} F _ N ( a , b ; t ) & = \\frac { 1 - t q ^ N } { 1 - t } + \\frac { ( 1 - t q ^ N ) } { ( 1 - t ) } \\sum _ { n = 1 } ^ { N } ( - 1 ) ^ n \\begin{bmatrix} N \\\\ n \\end{bmatrix} \\frac { ( b / a ) _ n ( q ) _ n ( a t ) ^ n q ^ { n ( n + 1 ) / 2 } } { ( b q ) _ n ( t q ) _ n } \\\\ & = \\frac { 1 - t q ^ N } { 1 - t } + \\frac { ( 1 - t q ^ N ) ( b - a ) t } { ( 1 - t ) ( 1 - b q ) ( 1 - t q ) } \\sum _ { n = 0 } ^ { N - 1 } ( - 1 ) ^ { n } \\frac { ( q ) _ { N } ( b q / a ) _ { n } ( a t ) ^ { n } q ^ { ( n + 2 ) ( n + 1 ) / 2 } } { ( q ) _ { N - n - 1 } ( b q ^ 2 ) _ { n } ( t q ^ 2 ) _ { n } } . \\end{align*}"}
+{"id": "3270.png", "formula": "\\begin{gather*} \\alpha = - q ^ { p _ 0 + p _ 2 } , \\beta = q ^ { p _ 0 - p _ 2 } , \\gamma = q ^ { p _ 0 - p _ 1 } , \\delta = - q ^ { p _ 2 + p _ 3 } . \\end{gather*}"}
+{"id": "3835.png", "formula": "\\begin{align*} a _ 1 = \\frac { n + ( h ^ 2 - 3 h - 9 ) + d } { 2 } \\end{align*}"}
+{"id": "3365.png", "formula": "\\begin{align*} M _ N ( x , E , \\omega ) = \\prod _ { k = N } ^ 1 \\begin{pmatrix} E - \\lambda v ( x + k \\omega ) - v _ 1 ( k ) & - 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "2043.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } \\ , \\lambda _ { n } = \\sum _ { k = 0 } ^ { n - 1 } \\frac { 1 } { k ! ( k + 3 ) ! ( n - k - 1 ) ! } \\left ( \\Bigl ( k - \\frac { 4 n - 1 } { 2 } \\Bigr ) ^ 2 + 2 n + \\frac { 7 } { 4 } \\right ) ( - 1 ) ^ k \\mu _ k \\end{align*}"}
+{"id": "6560.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { ( n - 2 ) / 2 } \\frac { 2 k \\pi } { n - 1 } \\sin \\left ( \\frac { 2 k \\pi } { n - 1 } \\cdot b \\right ) = \\frac { \\pi \\csc \\left ( \\frac { b \\pi } { 2 a + 1 } \\right ) \\left ( 2 ( a + 1 ) + \\sin \\left ( \\frac { 2 b \\pi ( a + 1 ) } { 2 a + 1 } \\right ) \\csc \\left ( \\frac { b \\cdot \\pi } { 2 a + 1 } \\right ) \\right ) } { 4 a + 2 } \\end{align*}"}
+{"id": "408.png", "formula": "\\begin{align*} \\sum _ { \\substack { a = 1 \\\\ ( a , n ) = 1 } } ^ n ( a - 1 , n ) \\chi ( a ) = \\varphi ( n ) \\sigma _ 0 \\left ( \\frac { n } { d } \\right ) , \\end{align*}"}
+{"id": "3735.png", "formula": "\\begin{align*} \\mathfrak { f } _ 1 ( x ) & = ( x + 5 ) ^ { 4 1 } ( x - 6 ) ( x - 9 ) ^ 2 ( x - 1 1 ) ^ 7 ( x - 1 3 ) ^ 8 , \\\\ \\mathfrak { f } _ 2 ( x ) & = ( x + 5 ) ^ { 4 1 } ( x - 9 ) ^ 3 ( x - 1 1 ) ^ 5 ( x - 1 3 ) ^ 8 ( x ^ 2 - 1 9 x + 8 2 ) . \\end{align*}"}
+{"id": "215.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ { N } } \\omega \\phi d x \\leq C ( p ) \\biggr ( C T ^ { - \\frac { N } { 2 } } R ^ N + k C T ^ { - \\frac { N } { 2 } } ( \\ln R ) ^ \\frac { 2 - N } { 2 } + C ( \\ln R ) ^ \\frac { 2 - N } { 2 } \\biggr ) . \\end{align*}"}
+{"id": "266.png", "formula": "\\begin{align*} U _ t : = \\tilde { U } _ { 2 t } , \\end{align*}"}
+{"id": "1473.png", "formula": "\\begin{align*} U _ n = \\sum _ { i \\in J _ w } \\tilde { \\Delta } _ { \\theta , i } \\prod _ { j \\in J _ w , j \\neq i } ( f _ j - \\alpha _ n ) + \\prod _ { j \\in J _ w } ( f _ j - \\alpha _ n ) z , \\end{align*}"}
+{"id": "4880.png", "formula": "\\begin{align*} Q ( x , t ) : = \\sum _ { n = 0 } ^ \\infty q _ { n - 1 } ( x ) \\frac { t ^ n } { n ! } \\end{align*}"}
+{"id": "2058.png", "formula": "\\begin{align*} f ( m ; z ) : = \\frac { \\eta ( 5 z ) } { \\eta ( z ) } \\eta ^ a ( 5 m z ) \\eta ^ b ( m z ) , \\end{align*}"}
+{"id": "4458.png", "formula": "\\begin{align*} \\Theta ( u _ n ) = u _ n \\otimes u _ n \\end{align*}"}
+{"id": "2126.png", "formula": "\\begin{align*} \\mathrm { c o r a n k } _ { \\Lambda ( G ) } H ^ { 1 } ( L _ { \\infty , w } , E _ { p ^ { \\infty } } ) = \\mathrm { c o r a n k } _ { \\Omega ( G ) } H ^ { 1 } ( L _ { \\infty , w } , E _ { p } ) . \\end{align*}"}
+{"id": "2499.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\dfrac { | ( \\Phi _ N - t ) \\cap \\Phi _ N | } { | \\Phi _ N | } = 1 \\end{align*}"}
+{"id": "2831.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } E \\left [ \\int _ t ^ T ( D _ s ^ n - D _ s ) ^ 2 \\gamma _ s ^ { - 1 } d s \\right ] = 0 \\end{align*}"}
+{"id": "6906.png", "formula": "\\begin{align*} \\frac { 1 } { e _ { \\mathbb { C } ^ * } ( \\textrm { N } _ { \\vec d } ) } = \\prod _ { i , j \\in [ N ] , \\ , \\ , i \\ne j } \\bigg ( e _ { \\mathbb { C } ^ * } \\left ( \\pi _ { \\star } \\left ( \\mathcal { K } _ i ^ \\vee ( a _ j - a _ { i } ) \\right ) \\right ) \\bigg ) ^ { - 1 } \\prod _ { i , j \\in [ N ] , \\ , i \\ne j } e _ { \\mathbb { C } ^ * } \\left ( \\pi _ { \\star } \\left ( \\mathcal { K } ^ { \\vee } _ i \\otimes \\mathcal { K } _ j ( a _ { j } - a _ { i } ) \\right ) \\right ) . \\end{align*}"}
+{"id": "5670.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } t f ( t ) = \\frac { 1 } { 4 } \\end{align*}"}
+{"id": "7281.png", "formula": "\\begin{align*} H ( s , \\gamma ( s ) , \\Psi _ L ( s , \\gamma ) , m ^ * ( s ) , u ^ * _ E ( s , \\gamma ( s ) ) ) = \\max _ { u \\in U } H ( s , \\gamma ( s ) , \\Psi _ L ( s , \\gamma ) , m ^ * ( s ) , u ) . \\end{align*}"}
+{"id": "8168.png", "formula": "\\begin{align*} ( | \\lambda | ^ 2 + 1 ) I - ( A + \\lambda B ) ^ * ( A + \\lambda B ) = \\left [ \\begin{array} { c c } 1 - \\| b \\| ^ 2 & - \\overline { \\lambda } a ^ * - b ^ * C \\\\ - \\lambda a - C ^ * b & ( | \\lambda | ^ 2 + 1 ) I - a a ^ * - C ^ * C \\end{array} \\right ] . \\end{align*}"}
+{"id": "1800.png", "formula": "\\begin{align*} \\det \\left ( \\begin{matrix} C \\\\ I _ { i _ { 1 } , i _ { 1 } + 1 , \\ldots , \\widehat { i _ { j } + \\epsilon } , \\ldots , i _ { r } , i _ { r } + 1 } \\end{matrix} \\right ) = \\left ( - 1 \\right ) ^ { k + i _ { j } - \\epsilon } \\det \\left ( C _ { [ n ] \\setminus \\{ i _ { 1 } , i _ { 1 } + 1 , \\ldots , \\widehat { i _ { j } + \\epsilon } , \\ldots , i _ { r } , i _ { r } + 1 \\} } \\right ) . \\end{align*}"}
+{"id": "6931.png", "formula": "\\begin{align*} e _ j = ( - 1 ) ^ { j - 1 } \\left [ z ^ { N + 1 - j } \\right ] ( y + z ) ( 1 + z x _ 1 ) \\cdots ( 1 + z x _ r ) . \\end{align*}"}
+{"id": "271.png", "formula": "\\begin{align*} \\tau = \\inf \\left \\{ u \\in [ 0 , t ] : | m _ { u } ( z _ { t - u } ) - 1 | \\geq \\frac { \\varphi ^ { 3 / 2 } } { N \\eta _ { z _ { t - u } } } \\right \\} \\wedge t . \\end{align*}"}
+{"id": "3893.png", "formula": "\\begin{align*} ~ \\textbf { x } _ { \\kappa _ l } = ( x _ { i _ { t + 1 } } , \\ldots , x _ { i _ { t + 1 } + n - 2 } , x _ { i _ { t + 1 } + n - 2 } + l d ) , \\ ; 1 \\leq l \\leq j \\end{align*}"}
+{"id": "4094.png", "formula": "\\begin{align*} \\sum _ { k | n } \\frac { 1 } { k } \\chi ^ { ( k ) } _ { n / k } = 0 , \\end{align*}"}
+{"id": "2278.png", "formula": "\\begin{align*} V _ T : = k \\{ z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\dots z _ n ^ { i _ n } : - N ' \\leq i _ j \\leq N ' , 1 \\leq j \\leq n \\} \\end{align*}"}
+{"id": "1265.png", "formula": "\\begin{align*} \\Delta = ( t + \\alpha ) ^ { n _ 1 } ( t + \\beta ) ^ { n _ 2 } ( \\hdots ) ^ 2 = t ^ { 2 4 } + ( \\alpha + \\beta ) t ^ { 2 3 } + \\hdots . \\end{align*}"}
+{"id": "4969.png", "formula": "\\begin{align*} E _ { v \\rightarrow c } ^ { ( i ) } = l _ { \\rm M P A } ( a _ { v } ) + \\sum _ { c ' \\in \\mathcal { N } ( v ) \\backslash \\{ c \\} } E _ { c ' \\rightarrow v } ^ { ( i - 1 ) } , \\end{align*}"}
+{"id": "7913.png", "formula": "\\begin{align*} ( \\delta _ \\mathrm { m R B A } ) ^ 2 ( u ) = ~ & \\delta _ \\mathrm { m R B A } ( \\delta _ \\mathrm { H o c h } ( u ) , - u ) \\\\ = ~ & \\big ( ( \\delta _ \\mathrm { H o c h } ) ^ 2 ( u ) , ~ \\widetilde { \\delta } _ \\mathrm { H o c h } ( u ) - \\Psi ^ 1 \\circ \\delta _ \\mathrm { H o c h } ( u ) \\big ) = 0 \\end{align*}"}
+{"id": "1014.png", "formula": "\\begin{align*} \\mu _ { \\rm e } & > 0 , \\quad \\ , 3 \\ , \\lambda _ { \\rm e } + 2 \\ , \\mu _ { \\rm e } > 0 , \\qquad \\mu _ { \\rm c } \\geq 0 , \\\\ \\mu _ { \\rm m i c r o } & > 0 , 3 \\ , \\lambda _ { \\rm m i c r o } + 2 \\ , \\mu _ { \\rm m i c r o } = 0 , \\\\ a _ 1 & > 0 , \\qquad \\ \\ a _ 2 > 0 , a _ 3 > 0 . \\end{align*}"}
+{"id": "3274.png", "formula": "\\begin{gather*} \\tilde { w } ( m , n ) : = \\tilde { w } ( m , n , p _ 0 , p _ 1 , p _ 2 , p _ 3 ; q ) = \\frac { \\rho \\big ( m , \\alpha , \\beta , \\gamma , \\delta ; q ^ 2 \\big ) } { h _ n ( \\alpha , \\beta , \\gamma , \\delta ; q ^ 2 ) } , \\end{gather*}"}
+{"id": "2423.png", "formula": "\\begin{align*} \\tilde { L } ( { \\rm C u t } _ { ( \\{ 0 \\} ^ { N - 1 } ; \\emptyset ) } ( \\tau ( w ) ) ) = ( - 1 ) ^ { N - 1 } \\tilde { L } \\left ( \\tau \\left ( { \\rm C u t } _ { ( \\emptyset ; \\{ 1 \\} ^ { N - 1 } ) } ( w ) \\right ) \\right ) , \\end{align*}"}
+{"id": "2115.png", "formula": "\\begin{align*} f _ i ( x ) : = \\sum _ { a \\in A } c _ { i a } x ^ a , 1 \\le i \\le n , \\end{align*}"}
+{"id": "1825.png", "formula": "\\begin{align*} [ a , b ] _ T : = \\rho ( T ( a ) ) b - \\rho ( T ( b ) ) a + \\lambda [ a , b ] _ V , ~ a , b \\in V . \\end{align*}"}
+{"id": "796.png", "formula": "\\begin{align*} \\mathcal { J } _ i ( \\tau _ i , \\tau _ { - i } ) = \\mathbb { E } \\left \\{ { C } _ { \\theta } ( \\tau _ i , \\tau _ { - i } ) - e ^ { - \\beta \\tau _ i } K \\right \\} \\end{align*}"}
+{"id": "8113.png", "formula": "\\begin{gather*} \\widehat { \\deg } _ + ( \\overline V ) : = \\sup _ { W \\subset V } \\widehat { \\deg } ( \\overline W ) = \\int _ 0 ^ { + \\infty } \\dim _ { k } ( \\mathcal F ^ t ( \\overline V ) ) \\ , \\mathrm { d } t , \\\\ \\widehat { \\deg } ( \\overline V ) = - \\int _ { \\mathbb R } \\ , t \\ , \\mathrm { d } \\dim _ k ( \\mathcal F ^ t ( \\overline V ) ) \\ , \\mathrm { d } t . \\end{gather*}"}
+{"id": "3747.png", "formula": "\\begin{align*} D ^ 1 m ( x ) & = x ^ 3 - 1 0 x ^ 2 - 2 4 4 x - 5 3 6 , \\\\ D ^ 2 m ( x ) & = x ^ 2 - 1 0 x - 2 4 4 , \\\\ D ^ 3 m ( x ) & = x - 1 0 , \\\\ D ^ 4 m ( x ) & = 1 . \\end{align*}"}
+{"id": "3962.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( e ^ { - s Y _ { \\beta } ( t ) } \\right ) = E _ { \\beta , 1 } \\left ( - s t ^ { \\beta } \\right ) , \\ s > 0 . \\end{align*}"}
+{"id": "6440.png", "formula": "\\begin{align*} \\Psi _ n ^ { ( a ) } ( x , y ) = \\sum _ { k = 0 } ^ n \\begin{bmatrix} n \\\\ k \\end{bmatrix} ( a ; q ) _ k x ^ k y ^ { n - k } \\end{align*}"}
+{"id": "5159.png", "formula": "\\begin{align*} c \\widetilde Q _ 1 \\cap c \\widetilde Q _ 2 = \\emptyset . \\end{align*}"}
+{"id": "7360.png", "formula": "\\begin{align*} T M \\oplus T M : = \\{ ( \\xi , \\eta ) : \\xi , \\eta \\in T _ x M , x \\in M \\} . \\end{align*}"}
+{"id": "5762.png", "formula": "\\begin{align*} \\mu _ \\sigma ( E ) : = \\begin{cases} - \\frac { \\Re Z ( E ) } { \\Im Z ( E ) } , & ~ \\Im Z ( E ) > 0 \\\\ + \\infty , & . \\end{cases} \\end{align*}"}
+{"id": "5164.png", "formula": "\\begin{align*} T _ n = \\bigcup ^ { 8 ^ n } _ { i = 1 } T _ { n , i } \\end{align*}"}
+{"id": "8202.png", "formula": "\\begin{align*} \\sum _ { j \\le j _ 0 } \\psi _ j 2 ^ { j r } \\Vert \\dot { \\Delta } _ j ( T _ f g ) \\Vert _ { L ^ 2 } \\le C \\Big ( \\sup _ { j ' \\le j _ 0 + 4 } 2 ^ { j ' ( r - r _ 2 ) } \\Vert \\dot \\Delta _ { j ' } g \\Vert _ { L ^ 2 } \\Big ) \\sum _ { j \\le j _ 0 } \\sum _ { j ' \\le j + 4 } 2 ^ { j r _ 2 } \\Vert \\dot \\Delta _ { j ' } f \\Vert _ { L ^ \\infty } \\psi _ j . \\end{align*}"}
+{"id": "2557.png", "formula": "\\begin{align*} k ( t , x , y ) : = \\rho ( - R ^ { - 1 } B P _ t x - R ^ { - 1 } B y ) \\quad \\end{align*}"}
+{"id": "1266.png", "formula": "\\begin{align*} y ^ 2 + t ^ 2 x y + t a _ 3 ' y = x ^ 3 + t a _ 2 ' x ^ 2 + t a _ 4 ' x + t ^ 2 a _ 6 ' . \\end{align*}"}
+{"id": "6227.png", "formula": "\\begin{align*} Q ( H _ a ( p ) ) = \\{ \\psi \\in H ^ 1 ( \\Sigma ) : \\ , h _ a [ \\psi ; p ] < \\infty \\ , , \\ ; \\psi ( x , 0 ) = 0 , \\ ; \\ , x \\in \\partial \\Sigma \\setminus W _ 0 ( a ) \\} , \\end{align*}"}
+{"id": "3722.png", "formula": "\\begin{align*} i \\partial _ t \\psi + \\frac 1 2 \\nabla ^ 2 \\psi = 0 \\quad , | \\psi _ 0 | ^ 2 = \\rho _ 0 , | \\psi _ 1 | ^ 2 = \\rho _ 1 , \\end{align*}"}
+{"id": "7990.png", "formula": "\\begin{align*} \\binom { n } { t - a } \\leq \\left ( \\frac { e n } { t - a } \\right ) ^ { t - a } \\leq \\left ( \\frac { e n } { t - t / \\rho } \\right ) ^ { t - a } \\leq \\left ( \\frac { e } { 1 - 1 / \\rho } \\right ) ^ t \\cdot \\left ( \\frac { n } { t } \\right ) ^ { t - a } . \\end{align*}"}
+{"id": "1369.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u - \\Delta u + b ( t ) \\partial _ t u = 0 , \\end{align*}"}
+{"id": "6288.png", "formula": "\\begin{align*} \\langle y \\rangle _ { } = \\sum _ { i = 0 } ^ { \\ell - 1 } y _ i \\langle \\frac { q ^ i } { q ^ \\ell - 1 } \\rangle _ { } . \\end{align*}"}
+{"id": "135.png", "formula": "\\begin{align*} d _ i ( x ) = \\left \\{ \\begin{array} { l l l l l } d _ i x \\in X _ { k - 1 } \\subseteq ( \\Sigma X ) _ { m - 1 } & \\mbox { i f } 0 \\leq i \\leq k , \\\\ x \\in X _ { k } \\subseteq ( \\Sigma X ) _ { m } & \\mbox { i f } k + 1 \\leq i \\leq m , \\end{array} \\right . \\end{align*}"}
+{"id": "2038.png", "formula": "\\begin{align*} | \\langle u _ n , \\eta _ j \\rangle | = | \\langle \\chi _ n , \\eta _ j \\rangle | \\ll \\frac { \\log j } { j } ~ ( 0 < | n | \\leq N ) , | \\langle u _ { \\pm ( N + 1 ) } , \\eta _ j \\rangle | \\ll \\frac { \\log j } { j } \\end{align*}"}
+{"id": "4379.png", "formula": "\\begin{align*} T _ x ^ t f : = T ^ t f ( x ) = u ( x , t ) , \\end{align*}"}
+{"id": "4683.png", "formula": "\\begin{align*} c _ n & = \\lim _ { r \\rightarrow 1 - } \\int _ { - \\pi } ^ { \\pi } f ( r e ^ { i \\varphi } ) \\overline { \\phi _ { n } ( e ^ { i \\varphi } ) } d m _ { \\lambda } ( \\varphi ) , \\\\ \\tilde { c } _ n & = \\lim _ { r \\rightarrow 1 - } \\int _ { - \\pi } ^ { \\pi } f ( r e ^ { i \\varphi } ) e ^ { i \\varphi } \\phi _ { n - 1 } ( e ^ { i \\varphi } ) d m _ { \\lambda } ( \\varphi ) , \\end{align*}"}
+{"id": "1301.png", "formula": "\\begin{align*} \\alpha _ { c , a , b } \\circ \\sigma _ { a \\otimes b , c } \\circ \\alpha _ { a , b , c } = ( \\sigma _ { a , c } \\otimes b ) \\circ \\alpha _ { a , c , b } \\circ ( a \\otimes \\sigma _ { b , c } ) , \\end{align*}"}
+{"id": "8131.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac { 1 } { n } \\operatorname { \\widehat { \\mu } _ { \\min } } ( \\overline L ^ { \\otimes n } \\otimes \\overline A ) = \\operatorname { \\widehat { \\mu } _ { \\min } } ( \\overline L ) . \\end{align*}"}
+{"id": "146.png", "formula": "\\begin{align*} \\widetilde { x } ( [ 0 , \\ldots , \\widehat { k } , k + 1 ] \\otimes [ 0 , \\ldots , \\ell ] ) = 0 \\end{align*}"}
+{"id": "2427.png", "formula": "\\begin{align*} { \\rm c u t } _ { ( u ; v ) } ( w ) \\coloneqq \\begin{cases} w ' & { \\rm i f } \\ w = u w ' v \\\\ 0 & { \\rm o t h e r w i s e } \\end{cases} \\end{align*}"}
+{"id": "1658.png", "formula": "\\begin{align*} \\frac { \\phi } { \\mathsf { M } } \\eta ( \\mathsf { A } _ 1 , \\mathsf { A } _ { \\mathsf { M } + 1 } ) \\geq \\frac { 2 ^ { - 6 } \\ , \\mathsf { q } ^ { - 1 } \\ , \\pmb { \\eta } } { \\lfloor 2 ^ { - 1 0 } \\ , \\pmb { \\eta } \\ , \\mathsf { q } ^ { - 1 } \\lambda ^ { - 1 } \\rfloor } \\geq \\frac { 2 ^ { - 6 } \\ , \\mathsf { q } ^ { - 1 } \\ , \\pmb { \\eta } } { 2 ^ { - 1 0 } \\ , \\pmb { \\eta } \\ , \\mathsf { q } ^ { - 1 } \\lambda ^ { - 1 } } = 2 ^ 4 \\lambda \\ , . \\end{align*}"}
+{"id": "5079.png", "formula": "\\begin{align*} \\mathcal { F } ( P _ { \\leq N } f ) ( \\xi ) : = \\eta ^ d \\left ( \\frac { \\xi } { N } \\right ) \\mathcal { F } ( f ) ( \\xi ) , \\xi \\in \\mathbb { R } ^ m \\times \\mathbb { Z } ^ n , \\end{align*}"}
+{"id": "6680.png", "formula": "\\begin{align*} \\frac { \\mathcal { D } \\left ( ( s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { 2 m - 1 } , ( s _ k ) _ { k = 0 } ^ { m } \\right ) } { a \\mathcal { D } \\left ( ( s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { 2 m - 1 } , ( 1 , 0 , \\ldots , 0 ) \\right ) } \\stackrel { a \\to 0 + } { \\longrightarrow } \\infty . \\end{align*}"}
+{"id": "6410.png", "formula": "\\begin{align*} \\mathcal { J } ^ { w } _ { \\theta _ { 0 } } \\mathcal { J } ^ { u } _ { \\theta _ { 0 } } \\mathcal { H } ( \\theta ) = \\mathcal { J } ^ { w + u } _ { \\theta _ { 0 } } \\mathcal { H } ( \\theta ) . \\end{align*}"}
+{"id": "950.png", "formula": "\\begin{align*} y _ n ^ 2 = \\left ( \\bar { y } _ n + \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( x ' _ 0 ) y _ \\beta \\right ) ^ 2 \\leq 2 \\bar { y } _ n ^ 2 + C \\epsilon ^ 2 \\delta ^ 2 b _ \\alpha | y ' | ^ 2 . \\end{align*}"}
+{"id": "2184.png", "formula": "\\begin{align*} & R _ { 1 2 1 2 } = - f f ^ { \\prime \\prime } E , R _ { 1 2 1 3 } = - f f ^ { \\prime \\prime } F , R _ { 1 3 1 3 } = - f f ^ { \\prime \\prime } G , \\\\ & R _ { 1 2 2 3 } = R _ { 1 3 2 3 } = 0 , \\\\ & R _ { 2 3 2 3 } = f ^ 2 ( K - f ^ { \\prime } { } ^ 2 ) \\Delta , \\end{align*}"}
+{"id": "476.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { \\binom n k L _ { j k + m } B _ { n - k } ( x ) z ^ k } = \\alpha ^ m B _ n ( x + \\alpha ^ j z ) + \\beta ^ m B _ n ( x + \\beta ^ j z ) . \\end{align*}"}
+{"id": "7946.png", "formula": "\\begin{align*} \\Phi ^ { s ' } ( a ) = ~ & U s ' ( a ) - s ' R ( a ) \\\\ = ~ & U \\varphi s ( a ) - \\varphi s R ( a ) \\\\ = ~ & \\varphi U s ( a ) - \\varphi s R ( a ) = \\varphi ( \\Phi ^ s ( a ) ) = \\Phi ^ s ( a ) . \\end{align*}"}
+{"id": "4074.png", "formula": "\\begin{align*} \\nabla : = d - \\left ( \\begin{array} { c c } 0 & 0 \\\\ \\omega & 0 \\end{array} \\right ) , \\end{align*}"}
+{"id": "7552.png", "formula": "\\begin{align*} Y ( z ) = ( I _ { 2 q } + O ( z ^ { - 1 } ) ) \\begin{pmatrix} z ^ N I _ q & 0 \\\\ 0 & z ^ { - N } I _ q \\end{pmatrix} z \\to \\infty . \\end{align*}"}
+{"id": "6592.png", "formula": "\\begin{align*} [ B B ^ * , B C ] & = [ B B ^ * , B ] C + B [ B B ^ * , C ] = - [ B , B B ^ * ] C - B [ C , B B ^ * ] \\\\ & = - ( [ B , B ] B ^ * + B [ B , B ^ * ] ) C - B \\big ( [ C , B ] B ^ * + B [ C , B ^ * ] \\big ) = O . \\end{align*}"}
+{"id": "7722.png", "formula": "\\begin{align*} g ^ p = e ^ { - 2 s _ p } g ^ + \\ \\ a n d \\ \\ g ^ q = e ^ { - 2 s _ q } g ^ + \\end{align*}"}
+{"id": "7765.png", "formula": "\\begin{align*} \\lim \\limits _ { m \\rightarrow 0 ^ + } \\mathcal { A } ( r _ m , \\cdot , \\cdot ) = 0 . \\end{align*}"}
+{"id": "2486.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = f ( u ) & & \\quad \\mbox { i n } \\Omega , \\\\ u & = \\infty & & \\quad \\mbox { o n } \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1605.png", "formula": "\\begin{align*} \\lambda H ( X _ { C \\setminus \\{ c \\} } ) = r ( C \\setminus \\{ c \\} ) = r ( C ) = \\lambda H ( X _ C ) . \\end{align*}"}
+{"id": "3073.png", "formula": "\\begin{align*} x \\in \\bigcup _ { j = 1 } ^ r X _ { i _ j } . \\end{align*}"}
+{"id": "1357.png", "formula": "\\begin{align*} \\{ u = 0 \\} \\cap B _ { s / 5 0 } \\ne \\varnothing . \\end{align*}"}
+{"id": "2251.png", "formula": "\\begin{align*} b = \\sigma ^ { - ( m - 1 ) } ( a ) \\sigma ^ { - ( m - 2 ) } ( a ) \\cdots \\sigma ^ { - 1 } ( a ) a \\in D . \\end{align*}"}
+{"id": "1141.png", "formula": "\\begin{align*} \\mathbb { B } _ { k , S ; ( n , g ) } ( \\tau , z ) = \\mathbb { B } _ { k , S [ u ] ; ( n , g ) } ( \\tau , ( u ^ t ) ^ { - 1 } z ) , \\end{align*}"}
+{"id": "1766.png", "formula": "\\begin{align*} { \\frac { n - 1 } { 2 } \\choose \\frac { m } { 2 } } = { \\frac { k + m - 1 } { 2 } \\choose \\frac { m } { 2 } } \\end{align*}"}
+{"id": "7653.png", "formula": "\\begin{align*} H ( p ) : = \\inf \\left \\{ \\ , \\frac { P ( F ) } { | F | ^ p } \\ , : \\ , F \\subset \\Omega \\ , , | F | > 0 \\ , \\right \\} . \\end{align*}"}
+{"id": "5697.png", "formula": "\\begin{gather*} \\eta _ j = \\sum _ { k \\in L _ { l + j } } \\nu _ k ( 1 \\le j \\le l ( \\eta ) ) . \\end{gather*}"}
+{"id": "3215.png", "formula": "\\begin{align*} \\mu _ 2 ( G ) = \\min _ { u \\in [ 1 ~ \\ldots ~ 1 ] ^ \\perp } \\frac { u ^ T L ( G ) u } { u ^ T u } \\leq \\frac { \\tilde { x } ^ T L ( G ) \\tilde { x } } { \\tilde { x } ^ T \\tilde { x } } = \\frac { \\mu _ 2 ( H ) } { k } . \\end{align*}"}
+{"id": "1657.png", "formula": "\\begin{align*} \\mathsf { A } = \\mathsf { A } _ 1 \\ , , \\quad \\quad \\qquad \\mathsf { B } = \\mathsf { A } _ { \\mathsf { M } + 1 } \\ , , \\quad \\quad \\qquad \\mathsf { F } = \\mathsf { M } : = \\lfloor 2 ^ { - 1 0 } \\pmb { \\eta } \\ , \\mathsf { q } ^ { - 1 } \\lambda ^ { - 1 } \\rfloor \\ , , \\quad \\quad \\qquad \\phi = \\frac { 1 } { 2 } \\ , . \\end{align*}"}
+{"id": "6685.png", "formula": "\\begin{align*} t _ { 1 } ( \\mathbf { s } ) & = \\inf _ { 0 < a < b \\le 1 } t _ { a , b } ( \\mathbf { s } ) \\\\ T _ { 1 } ( \\mathbf { s } ) & = \\sup _ { 0 < a < b \\le 1 } T _ { a , b } ( \\mathbf { s } ) , \\end{align*}"}
+{"id": "7473.png", "formula": "\\begin{align*} \\psi _ m ^ { ( a ) } ( n ) = \\psi _ m ^ { ( a ) } ( n - 1 ) + \\psi ^ { ( a - 1 ) } _ m ( n ) \\ , . \\end{align*}"}
+{"id": "5010.png", "formula": "\\begin{align*} & \\Big \\| \\phi _ i ( g _ j ) \\ , ( w _ { i n } \\circ g _ j ^ { - 1 } ) - \\sum _ { s = j + 1 } ^ { m } \\phi _ i ( g _ s ) \\ , ( w _ { i } \\circ g _ s ^ { - 1 } ) ( \\ , \\cdot \\ , - g _ s \\xi _ { r n } + g _ j \\xi _ { r n } ) \\Big \\| ^ 2 \\\\ & = \\Big \\| \\phi _ i ( g _ j ) \\ , ( w _ { i n } \\circ g _ j ^ { - 1 } ) - \\sum _ { s = j } ^ { m } \\phi _ i ( g _ s ) \\ , ( w _ { i } \\circ g _ s ^ { - 1 } ) ( \\ , \\cdot \\ , - g _ s \\xi _ { r n } + g _ j \\xi _ { r n } ) \\Big \\| ^ 2 + \\Big \\| \\phi _ i ( g _ j ) \\ , ( w _ i \\circ g _ j ) \\Big \\| ^ 2 + o _ n ( 1 ) . \\end{align*}"}
+{"id": "119.png", "formula": "\\begin{align*} \\chi ( x ) = \\begin{cases} 1 & : \\abs { x } \\leq 1 \\\\ 0 & : \\abs { x } \\geq 2 \\end{cases} \\end{align*}"}
+{"id": "5179.png", "formula": "\\begin{align*} \\mathcal E _ 1 ( O , \\gamma ) = \\inf _ \\Gamma \\mathcal { H } ^ { n - 1 } ( \\Gamma ) , \\end{align*}"}
+{"id": "5055.png", "formula": "\\begin{align*} 2 \\pi J _ { k - 1 } ( 4 \\pi y ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\Re ( u ) = \\sigma _ 0 } \\frac { \\Gamma _ \\C ( u + \\frac { k - 1 } { 2 } ) } { \\Gamma _ \\C ( - u + \\frac { k + 1 } { 2 } ) } y ^ { - 2 u } \\ , d u , \\end{align*}"}
+{"id": "1554.png", "formula": "\\begin{align*} \\left ( X _ { b _ { i } } , X _ { s _ { i } } , X _ { b _ { j } } \\right ) \\left ( \\tilde { \\omega } \\right ) & = \\left ( \\omega _ { i } , X _ { s _ { i } } \\left ( \\omega \\right ) , \\omega _ { j } \\right ) \\quad \\\\ \\left ( X _ { b _ { i } } , X _ { s _ { i } } , X _ { b _ { j } } \\right ) \\left ( \\tilde { \\omega } ^ { \\prime } \\right ) & = \\left ( \\omega _ { i } , X _ { s _ { i } } \\left ( \\omega ^ { \\prime } \\right ) , \\omega _ { j } \\right ) . \\end{align*}"}
+{"id": "6365.png", "formula": "\\begin{align*} \\mathcal { S } ^ { \\mathbb { B } } : = \\left [ \\begin{array} { c | c } A ( \\lambda ) ^ { \\mathcal { B } } & - ( e _ { \\ell } e _ k ^ { T } ) \\otimes B \\\\ \\hline ( e _ j e _ i ^ { T } ) \\otimes C & D ( \\lambda ) ^ { \\mathcal { B } } \\\\ \\end{array} \\right ] . \\end{align*}"}
+{"id": "7965.png", "formula": "\\begin{align*} \\cos { \\left ( \\theta ( s ) \\right ) } = \\frac { c } { 3 } r ^ 2 ( s ) > 0 \\qquad \\mbox { a n d } \\theta ' ( s ) = - \\frac { 2 c } { 3 } r ( s ) < 0 . \\end{align*}"}
+{"id": "8024.png", "formula": "\\begin{align*} T _ Q = \\begin{bmatrix} Q _ 0 & Q _ { - 1 } & Q _ { - 2 } & \\cdots & \\cdots \\\\ Q _ { 1 } & Q _ 0 \\otimes I _ d & Q _ { - 1 } \\otimes I _ d & \\cdots & \\cdots \\\\ Q _ { 2 } & Q _ { 1 } \\otimes I _ d & Q _ 0 \\otimes I _ d \\otimes I _ d & \\cdots & \\cdots \\\\ \\vdots & \\vdots & \\vdots & \\ddots & & \\\\ \\end{bmatrix} \\end{align*}"}
+{"id": "4000.png", "formula": "\\begin{align*} \\hat { q } _ { \\beta } ( n , t ) & = \\sum _ { k = 1 } ^ { n } \\mathrm { P r } \\{ X _ { 1 } + X _ { 2 } + \\dots + X _ { k } = n \\} \\mathrm { P r } \\{ N _ { \\beta } ( t ) = k \\} \\\\ & = \\sum _ { k = 1 } ^ { n } \\sum _ { \\Theta _ { n } ^ { k } } k ! \\prod _ { j = 1 } ^ { n } \\frac { ( ( 1 - p ) ^ { j } / j ) ^ { x _ { j } } } { x _ { j } ! } \\left ( \\frac { - \\lambda t ^ { \\beta } } { \\ln p } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } ( - \\lambda t ^ { \\beta } ) , \\end{align*}"}
+{"id": "6290.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\ell - 1 } y _ i \\cdot \\frac { q ^ i } { q ^ \\ell - 1 } = y = \\sum _ { i ' = 0 } ^ { \\ell ' - 1 } y _ { i ' } ' \\cdot \\frac { q ^ { i ' } } { q ^ { \\ell ' } - 1 } . \\end{align*}"}
+{"id": "2781.png", "formula": "\\begin{align*} s _ i ^ 2 A ^ T A g _ i = c _ i ^ 2 B ^ T B g _ i , \\end{align*}"}
+{"id": "44.png", "formula": "\\begin{align*} \\frac { q _ 0 } { q _ 1 } = e ^ { \\sqrt \\beta } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\frac { 1 - q _ 0 } { 1 - q _ 1 } \\le \\frac { 1 } { 2 - e ^ { - \\sqrt { \\beta } } } , \\end{align*}"}
+{"id": "3488.png", "formula": "\\begin{align*} \\Big | \\partial _ { \\mathrm { x } _ { i } } \\Phi ( \\mathrm { x } , t ; \\mathrm { y } , \\tau ) \\Big | \\lesssim \\frac { \\alpha ^ \\mathrm { r } } { ( t - \\tau ) ^ \\mathrm { r } } \\frac { 1 } { | \\mathrm { x } - \\mathrm { y } | ^ { 3 - 2 \\mathrm { r } } } , \\ \\ \\ \\mathrm { r } < 2 , \\ i = 1 , 2 , \\end{align*}"}
+{"id": "1896.png", "formula": "\\begin{align*} \\Theta _ { K / F , S _ 0 } ^ E ( s ) : = \\prod _ { v \\not \\in S _ 0 } P _ { v } ^ { \\ast , G } ( N v ^ { - s } ) ^ { - 1 } , \\end{align*}"}
+{"id": "2587.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } _ t ^ * : = ( \\alpha _ t ^ { * , i } ) _ { 1 \\leq i \\leq N } = \\left ( - R ^ { - 1 } B ( P _ t x _ t ^ { * , i } + \\varphi _ t ^ { * , i } ) - h \\Big ( k _ i ^ N ( t , \\nu ^ { N , 1 } _ { \\boldsymbol { x } ^ * _ t } , \\ldots , \\nu ^ { N , N } _ { \\boldsymbol { x } ^ * _ t } , \\lambda ^ { N , 1 } _ { \\boldsymbol { \\varphi } ^ * _ t } , \\ldots , \\lambda ^ { N , N } _ { \\boldsymbol { \\varphi } ^ * _ t } ) \\Big ) \\right ) _ { 1 \\leq i \\leq N } . \\end{align*}"}
+{"id": "332.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial h } L ( h , Y _ 1 , \\cdots , Y _ n , \\Theta _ m ) | _ { h = \\hat { h } _ { n , m } } = 0 . \\end{align*}"}
+{"id": "7944.png", "formula": "\\begin{align*} \\widetilde { \\delta } _ \\mathrm { H o c h } ( \\Phi ^ s ) + \\Psi ^ 2 ( \\chi ^ s ) = 0 . \\end{align*}"}
+{"id": "4444.png", "formula": "\\begin{align*} v _ { n _ k } ( x ) : = \\begin{cases} 0 , & x \\notin \\Sigma _ { n _ k } \\\\ z _ { n _ k } , & x \\in \\Sigma _ { n _ k } . \\end{cases} \\end{align*}"}
+{"id": "4691.png", "formula": "\\begin{align*} r \\frac { d } { d r } \\left [ { } _ 2 \\ ! F _ { 1 } \\left ( \\lambda , \\lambda ; 2 \\lambda + 1 ; A \\right ) \\right ] = \\frac { \\lambda ^ 2 } { 2 \\lambda + 1 } { } _ 2 \\ ! F _ { 1 } \\left ( \\lambda + 1 , \\lambda + 1 ; 2 \\lambda + 2 ; A \\right ) r \\frac { d } { d r } A , \\end{align*}"}
+{"id": "402.png", "formula": "\\begin{align*} \\iota _ u \\iota _ { u ' } ( \\Psi ^ * \\alpha ) = \\Psi ^ * ( \\iota _ { \\Psi _ * u } \\iota _ { \\Psi _ * u ' } \\alpha ) = 0 . \\end{align*}"}
+{"id": "3161.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } \\ ; \\ ; \\varphi ( x ) \\mbox { s u b j e c t t o } \\ ; \\ ; g _ i ( x ) \\leq 0 ~ i = 1 , . . . , l , h _ j ( x ) = 0 ~ j = l + 1 , . . . , m \\end{align*}"}
+{"id": "5873.png", "formula": "\\begin{align*} c ( t ) = \\underset { r \\in R } { \\sum } c _ r t ^ r , \\end{align*}"}
+{"id": "4542.png", "formula": "\\begin{align*} \\hat { E } ^ i = \\left ( \\frac { \\cdot \\wedge E ^ i } { v } \\right ) \\ , , \\tilde { E } ^ i = \\left ( \\frac { \\cdot \\wedge E ^ i } { w } \\right ) \\ , , \\end{align*}"}
+{"id": "2748.png", "formula": "\\begin{align*} d = \\bar { L } ^ 2 \\leq K _ S ^ 2 + 4 g ( S ) - 4 , \\end{align*}"}
+{"id": "1116.png", "formula": "\\begin{gather*} \\sum _ { m = 0 } ^ { n } ( - 1 ) ^ { n - m } \\binom { n } { m } 2 ^ { - m } \\sum _ { j = 0 } ^ { m } \\binom { m } { j } \\left ( \\beta \\right ) ^ { \\left ( n - j \\right ) } \\left ( \\beta + m - j \\right ) ^ { \\left ( j \\right ) } \\\\ = \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ \\frac { n ! } { 2 ^ { n / 2 } ( n / 2 ) ! } \\left ( \\beta \\right ) ^ { \\left ( n / 2 \\right ) } & & n \\end{array} \\right . . \\end{gather*}"}
+{"id": "8213.png", "formula": "\\begin{align*} \\sigma : = \\begin{cases} \\frac { \\sqrt { \\kappa \\gamma } } { \\gamma - 1 } ( \\rho ^ { \\gamma - 1 } - 1 ) , \\ & \\gamma > 1 , \\\\ \\sqrt { \\kappa } \\ln \\rho , \\ & \\gamma = 1 . \\end{cases} \\end{align*}"}
+{"id": "4087.png", "formula": "\\begin{align*} 1 - 2 g t + t ^ 2 = \\prod _ { n \\geq 1 } ( 1 - t ^ n ) ^ { e _ n } \\end{align*}"}
+{"id": "5232.png", "formula": "\\begin{align*} \\sigma ( \\alpha ) = \\alpha D ^ { - 1 } \\sigma ( D ) = \\alpha \\begin{pmatrix} \\frac { \\sigma ( d _ 1 ) } { d _ 1 } & \\sigma ( d ) \\frac { \\sigma ( d _ 1 ) } { d _ 1 } - d \\frac { \\sigma ( d _ 2 ) } { d _ 2 } \\\\ 0 & \\frac { \\sigma ( d _ 2 ) } { d _ 2 } \\end{pmatrix} \\end{align*}"}
+{"id": "2179.png", "formula": "\\begin{align*} & R _ { i j k l } - c ( \\delta _ { i k } \\delta _ { j l } - \\delta _ { i l } \\delta _ { j k } ) = \\alpha _ { i k } \\alpha _ { j l } - \\alpha _ { i l } \\alpha _ { j k } , \\\\ & \\beta _ { i j k } = \\beta _ { i k j } , \\end{align*}"}
+{"id": "382.png", "formula": "\\begin{align*} \\pi ^ * \\ , \\tilde { l } _ k ( \\sigma _ N ^ 1 , \\dots , \\sigma _ N ^ k ) & = \\varsigma ( k ) ~ \\iota _ { v _ N ^ k } \\dots \\iota _ { v _ N ^ 1 } ~ \\omega _ N \\\\ & = \\varsigma ( k ) ~ \\iota _ { v ^ k | _ N } \\dots \\iota _ { v ^ 1 | _ N } ~ j ^ * \\omega \\\\ & = j ^ * \\big ( \\varsigma ( k ) \\ , \\iota _ { v ^ k } \\dots \\iota _ { v ^ 1 } \\omega \\big ) \\\\ & = \\pi ^ * \\ , \\tilde { l } _ k ( \\sigma ^ 1 , \\dots , \\sigma ^ k ) _ N , \\end{align*}"}
+{"id": "7631.png", "formula": "\\begin{align*} t _ 5 ^ 3 = t _ 3 ^ 3 = t _ 4 ^ 3 . \\end{align*}"}
+{"id": "2508.png", "formula": "\\begin{align*} x _ { 0 0 } & = a \\\\ x _ { 0 1 } & = \\lim _ { m \\to \\infty } T ^ { c _ 2 ( m ) } a \\\\ x _ { 1 0 } & = \\lim _ { j \\to \\infty } T ^ { c _ 1 ( j ) } a \\\\ x _ { 1 1 } & = \\lim _ { j \\to \\infty } \\lim _ { m \\to \\infty } T ^ { c _ 1 ( j ) + c _ 2 ( m ) } a = \\lim _ { m \\to \\infty } \\lim _ { j \\to \\infty } T ^ { c _ 1 ( j ) + c _ 2 ( m ) } a . \\end{align*}"}
+{"id": "3502.png", "formula": "\\begin{align*} \\mathbb { T } \\Big [ \\nabla \\mathrm { H } \\Big ] ( \\mathrm { x } ) = \\int _ { \\Omega } \\nabla \\nabla \\cdot \\Big ( \\mathbb { G } ^ { ( \\mathrm { k } ) } - \\mathbb { G } ^ { ( 0 ) } \\Big ) ( \\mathrm { x } , \\mathrm { y } ) \\nabla \\mathrm { H } ( \\mathrm { y } ) d \\mathrm { y } . \\end{align*}"}
+{"id": "8181.png", "formula": "\\begin{align*} | a _ 1 | ^ 2 + | a _ 2 | ^ 2 + | b _ 1 | ^ 2 + | b _ 2 | ^ 2 = \\| a \\| ^ 2 + \\| b \\| ^ 2 = 1 \\end{align*}"}
+{"id": "722.png", "formula": "\\begin{align*} \\rho { u _ y } = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { c _ { i y } } { f _ i } + \\frac { { \\Delta t } } { 2 } { F _ y } } , \\end{align*}"}
+{"id": "5555.png", "formula": "\\begin{align*} u _ 1 ( \\beta _ 0 ) = 0 , \\end{align*}"}
+{"id": "6171.png", "formula": "\\begin{align*} \\| h \\| ^ 2 - \\| V _ 1 ^ * V _ 2 ^ * \\cdots V _ { i - 1 } ^ * V _ { i + 1 } ^ * \\cdots V _ d ^ * h \\| ^ 2 = \\| D _ { V _ { ( i ) } ^ * } h \\| ^ 2 . \\end{align*}"}
+{"id": "6738.png", "formula": "\\begin{align*} | I _ 4 | = | \\int _ { { \\mathbb R } ^ 3 } \\int _ { { \\mathbb R } ^ 3 } ( E + v \\times B ) \\cdot v \\frac { ( \\partial ^ { \\alpha } F ) ^ 2 } { 2 \\mu } \\ , d v \\ , d x | \\leq C \\| E \\| _ { L ^ { \\infty } } \\| \\langle v \\rangle ^ { \\frac { 1 } { 2 } } \\frac { \\partial ^ { \\alpha } F } { \\sqrt { \\mu } } \\| ^ 2 , \\end{align*}"}
+{"id": "7674.png", "formula": "\\begin{align*} \\| f \\| _ p ^ p = \\sup _ { 0 < r < 1 } \\int _ { \\mathbb { S } } | f ( r \\zeta ) | ^ p d \\sigma ( \\zeta ) < \\infty . \\end{align*}"}
+{"id": "4274.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( q ) _ { n - 1 } z ^ n q ^ n } { ( 1 - d q ^ n ) ( z q ) _ n } & = z \\sum _ { n = 1 } ^ { \\infty } \\frac { ( z q / d ) _ { n - 1 } } { ( z q ) _ n } \\frac { d ^ { n - 1 } q ^ n } { 1 - z q ^ { n } } \\end{align*}"}
+{"id": "7615.png", "formula": "\\begin{align*} p _ { 1 , \\lambda } = x _ 0 ^ 3 x _ 6 ^ 3 + x _ 1 ^ 3 x _ 7 ^ 3 + x _ 2 ^ 3 x _ 8 ^ 3 - 3 \\lambda x _ 3 x _ 4 x _ 5 x _ 6 x _ 7 x _ 8 , \\end{align*}"}
+{"id": "1180.png", "formula": "\\begin{align*} \\left ( \\mathfrak { M } ^ \\prime ( f ) \\eta \\right ) ( x , m ) = f \\left ( O ^ m x \\right ) \\eta ( x , m ) , \\ ; \\left ( \\mathcal { F } ' ( O ^ k ) \\eta \\right ) ( x , m ) = \\eta ( x , m - k ) . \\end{align*}"}
+{"id": "7359.png", "formula": "\\begin{align*} b _ i = \\begin{cases} \\frac { 1 } { 2 } ( i + 1 ) p - 1 & \\\\ \\frac { 1 } { 2 } i p + 1 & \\\\ \\frac { 1 } { 2 } p ( p - 1 ) & \\end{cases} \\end{align*}"}
+{"id": "5706.png", "formula": "\\begin{gather*} H _ i ( \\pi _ 1 ( U \\Sigma _ g ) , \\Q ) = \\Q [ 0 ] \\oplus H [ 1 ] \\oplus H [ 2 ] \\oplus \\Q [ 3 ] . \\end{gather*}"}
+{"id": "6754.png", "formula": "\\begin{align*} i \\partial _ t \\Psi _ { N , t } = H ^ { \\rm F } _ { N , \\alpha } \\Psi _ { N , t } \\end{align*}"}
+{"id": "7353.png", "formula": "\\begin{align*} | \\mathcal { I } ' _ m | = { m + 4 \\choose 4 } \\ \\ \\ \\ m \\geq 1 \\end{align*}"}
+{"id": "5029.png", "formula": "\\begin{align*} \\tilde { \\Lambda } _ 1 ( \\Omega ) = h _ 1 ( \\Omega ) . \\end{align*}"}
+{"id": "7387.png", "formula": "\\begin{align*} \\sigma \\cdot x = \\sigma _ 1 x _ 1 + \\sigma _ 2 x _ 2 \\sigma \\cdot \\nabla = \\sigma _ 1 \\partial _ 1 + \\sigma _ 2 \\partial _ 2 . \\end{align*}"}
+{"id": "1360.png", "formula": "\\begin{align*} E [ u ] ( t ) & : = \\frac { 1 } { 2 } \\int _ { \\Omega } ( | \\partial _ t u ( t , x ) | ^ 2 + | \\nabla u ( t , x ) | ^ 2 ) \\ , d x + \\frac { 1 } { p + 1 } \\int _ { \\Omega } | u ( t , x ) | ^ { p + 1 } \\ , d x \\end{align*}"}
+{"id": "4312.png", "formula": "\\begin{align*} m _ j ( h ) = \\frac { 1 } { P _ { j , j + 1 } } \\left ( \\widehat { h } ( j ) + \\sum _ { k = 0 } ^ { j - 1 } m _ k ( h ) \\sum _ { l = 0 } ^ k P _ { j , l } \\right ) , \\end{align*}"}
+{"id": "4042.png", "formula": "\\begin{align*} e _ { b _ 0 } ( y ) = ( p - 1 ) ^ { - 1 } \\left ( \\sum _ { l = 0 } ^ L \\left ( - \\frac { b y ^ { 2 k } } { p - 1 } \\right ) ^ l + \\left ( - \\frac { b y ^ { 2 k } } { p - 1 } \\right ) ^ { L + 1 } \\right ) , \\end{align*}"}
+{"id": "1129.png", "formula": "\\begin{align*} H _ { j , n } = ( - 1 ) ^ { n } \\binom { j } { n } ( \\beta ) ^ { ( j ) } \\sqrt { \\frac { n ! } { \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } } } . \\end{align*}"}
+{"id": "4324.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\left | \\mathbb { P } ( X > j ) - \\sum _ { k = j + 1 } ^ \\infty \\sum _ { i = 0 } ^ \\infty \\mathbb { P } ( X = i ) P _ { i , k } \\right | \\leq \\sum _ { j = 0 } ^ \\infty \\sum _ { i = 0 } ^ \\infty \\mathbb { P } ( X = i ) \\left | \\sum _ { k = j + 1 } ^ \\infty \\left ( Q _ { i , k } - P _ { i , k } \\right ) \\right | . \\end{align*}"}
+{"id": "5998.png", "formula": "\\begin{align*} b ^ * : = \\inf \\{ b \\geq 0 : G ( b ) \\leq 0 \\} . \\end{align*}"}
+{"id": "1411.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t ^ 2 u - \\Delta u + a ( x ) \\partial _ t u = 0 , & t > 0 , x \\in \\Omega , \\\\ u ( x , t ) = 0 , & t > 0 , x \\in \\partial \\Omega , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\ \\partial _ t u ( 0 , x ) = u _ 1 ( x ) , & x \\in \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "8036.png", "formula": "\\begin{align*} \\begin{cases} ( - 1 ) ^ { m + r + 1 } d _ m ^ B \\widehat h _ m - \\widehat h _ m d _ m ^ A = 0 , & 0 \\leq m < r , \\\\ - d _ r ^ B \\widehat h _ r - \\widehat h _ r d _ r ^ A = f _ r - g _ r , & \\end{cases} \\end{align*}"}
+{"id": "7916.png", "formula": "\\begin{align*} \\chi = \\delta _ \\mathrm { H o c h } ( \\varphi _ 1 ) ~ ~ ~ ~ ~ ~ \\Phi = - \\widetilde { \\delta } _ \\mathrm { H o c h } ( u ) - \\Psi ^ 1 ( \\varphi _ 1 ) . \\end{align*}"}
+{"id": "7101.png", "formula": "\\begin{align*} \\sum _ { d \\geq 0 } \\mathrm { D T } _ { d } q ^ d = \\exp \\left ( \\sum _ { n \\geq 1 } ( - 1 ) ^ { n - 1 } n N _ n q ^ n \\right ) = \\prod _ { d \\geq 1 } ( 1 - ( - q ) ^ d ) ^ { d \\Omega _ d } , \\end{align*}"}
+{"id": "6066.png", "formula": "\\begin{align*} a ( \\eta , v ) = \\int _ { \\Omega } \\nabla \\eta \\nabla v d x - \\int _ { \\Omega } k ^ 2 \\eta v d x . \\end{align*}"}
+{"id": "2181.png", "formula": "\\begin{align*} & R ( X _ i , X _ j ) X _ k = R ^ B ( X _ i , X _ j ) X _ k , \\\\ & R ( X _ i , X _ j ) Y _ p = 0 , \\\\ & R ( X _ i , Y _ p ) X _ j = f ^ { - 1 } H ^ f ( X _ i , X _ j ) Y _ p , \\\\ & R ( X _ i , Y _ p ) Y _ q = - f \\ , h _ { p q } ( \\nabla ^ B _ { X _ i } \\ , { \\rm g r a d } \\ : f ) , \\\\ & R ( Y _ p , Y _ q ) X _ i = 0 , \\\\ & R ( Y _ p , Y _ q ) Y _ r = R ^ F ( Y _ p , Y _ q ) Y _ r - | | { \\rm g r a d } \\ : f | | ^ 2 ( h _ { q r } Y _ p - h _ { p r } Y _ q ) , \\end{align*}"}
+{"id": "6467.png", "formula": "\\begin{align*} & \\omega ^ { ( 2 ) } _ { G } ( u , v ) = \\int _ { M } \\left ( \\frac { e ^ { i A ^ \\frac { 1 } { 2 } ( t - s ) } } { A ^ { \\frac { 1 } { 2 } } } u \\right ) ( s , y ) v ( s , y ) d s d y , \\end{align*}"}
+{"id": "6304.png", "formula": "\\begin{align*} [ e _ \\alpha , e _ \\beta ] = e _ { \\alpha + \\beta } [ e _ \\alpha , e _ \\beta ] = 0 . \\end{align*}"}
+{"id": "4105.png", "formula": "\\begin{align*} \\beta ( y _ 1 , y _ 2 ; t ) = ( \\int ^ { \\gamma _ t ( y ) } _ { \\gamma _ t ( y _ 2 ) } \\omega _ i ) . \\end{align*}"}
+{"id": "4908.png", "formula": "\\begin{align*} \\frak F ( [ V \\xrightarrow h X ; E ] ) \\bullet \\frak F ( [ W \\xrightarrow k Y ; F ] ) & = [ X \\xleftarrow h V \\xrightarrow { f \\circ h } Y ; E ] \\bullet [ Y \\xleftarrow k W \\xrightarrow { g \\circ k } Z ; F ] \\\\ & = [ X \\xleftarrow { h \\circ { k } '' } V ' \\xrightarrow { ( g \\circ k ) \\circ ( f ' \\circ f ' ) } Z ; { { k } '' } ^ * E \\oplus ( f ' \\circ { h } ' ) ^ * F ] \\end{align*}"}
+{"id": "1521.png", "formula": "\\begin{align*} X _ { S } \\left ( \\omega \\right ) = \\left ( X _ { s _ { i } } \\left ( \\omega \\right ) \\right ) _ { i = 1 } ^ { n } . \\end{align*}"}
+{"id": "7677.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 1 + } c ( \\alpha ) = 0 . \\end{align*}"}
+{"id": "494.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { n \\choose k } 2 ^ k F _ { j k } ( - \\sqrt { 5 } F _ j ) ^ { n - k } & B _ { n - k } ( \\alpha ) \\\\ = n F _ j 2 ^ { 1 - n } & \\Big ( ( \\sqrt { 5 } F _ j - L _ { j - 3 } ) ^ { n - 1 } + ( - \\sqrt { 5 } F _ j - L _ { j - 3 } ) ^ { n - 1 } \\Big ) . \\end{align*}"}
+{"id": "5186.png", "formula": "\\begin{align*} A _ n = \\{ ( f , g ) \\in S _ n ^ 2 : \\gcd ( f , g ) = 1 \\} \\enspace , \\enspace B _ n = \\{ ( f , g ) \\in S _ n ^ 2 : \\gcd ( f , g ) \\neq 1 \\} \\enspace . \\end{align*}"}
+{"id": "7203.png", "formula": "\\begin{align*} K _ i ^ { \\mathrm { t o p } } ( \\mathrm { P e r f } ^ { \\mathrm { g r } } ( \\mathcal { X } \\times \\mathbb { C } ^ { \\ast } ) ) = K _ i ^ { \\mathrm { t o p } } ( \\mathrm { P e r f } ( \\mathcal { X } ) ) = K _ i ^ { \\mathrm { t o p } } ( B G ) = \\begin{cases} K ( B G ) _ { \\mathbb { Q } } , & i = 0 , \\\\ 0 , & i = 1 , \\end{cases} \\end{align*}"}
+{"id": "739.png", "formula": "\\begin{align*} { \\left ( { { { { D \\mathord { \\left / { \\vphantom { D D } } \\right . \\kern - \\nulldelimiterspace } D } } _ 0 } } \\right ) ^ 2 } = 1 - K t , \\end{align*}"}
+{"id": "4960.png", "formula": "\\begin{align*} \\mathbf { I } = \\mathop { \\vee } \\limits _ { u \\in \\mathcal { U } _ { a c } ^ { \\mathbf { a } } } \\mathbf { s } _ { u } , \\end{align*}"}
+{"id": "2040.png", "formula": "\\begin{align*} \\sum _ { i > M } | \\langle u _ n , \\psi _ i \\rangle | ^ 2 = \\left \\Vert \\xi \\begin{bmatrix} { } ^ { t } X _ 1 ^ { - 1 } X _ 3 \\\\ 1 \\end{bmatrix} { } ^ { t } X _ 4 ^ { - 1 } \\right \\Vert _ { \\ell ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "751.png", "formula": "\\begin{align*} C u r \\mathcal { G } = \\mathbb { C } [ \\partial ] \\otimes \\mathcal { G } , [ a _ \\lambda b ] = [ a , b ] , \\end{align*}"}
+{"id": "7015.png", "formula": "\\begin{align*} L _ w g = L _ w \\left ( f - ( M - \\varepsilon _ 0 ) \\right ) _ + \\geq \\chi _ { \\{ f > M - \\varepsilon _ 0 \\} } L ^ { a c } _ { w } f \\cdot m _ w \\geq 0 . \\end{align*}"}
+{"id": "7469.png", "formula": "\\begin{align*} N _ { \\gamma } + \\widetilde { N } _ { \\gamma } \\cdot F _ { \\gamma } = N _ { \\gamma } \\cdot \\left ( \\mathrm { I d } + N _ { \\gamma } ^ { - 1 } \\cdot \\widetilde { N } _ { \\gamma } \\cdot F _ { \\gamma } \\right ) \\end{align*}"}
+{"id": "1798.png", "formula": "\\begin{align*} c _ { n = k + m , k , m } \\left ( C \\mathcal { Z } , \\mathcal { Z } \\right ) = \\begin{cases} \\frac { 2 k + m - 1 } { m + 1 } { \\frac { k + m - 2 } { 2 } \\choose \\frac { m - 1 } { 2 } } & { \\rm f o r \\ ; } k \\ ; { \\rm o d d } , \\\\ 2 { \\frac { k + m - 1 } { 2 } \\choose \\frac { m + 1 } { 2 } } & { \\rm f o r \\ ; } k \\ ; { \\rm e v e n . } \\end{cases} \\end{align*}"}
+{"id": "2935.png", "formula": "\\begin{align*} \\begin{pmatrix} I & X \\\\ & I \\end{pmatrix} \\begin{pmatrix} A & C \\\\ & B \\end{pmatrix} \\begin{pmatrix} I & - X \\\\ & I \\end{pmatrix} = \\begin{pmatrix} A \\\\ & B \\end{pmatrix} \\end{align*}"}
+{"id": "5197.png", "formula": "\\begin{align*} D _ { n , k } = \\binom { n - 1 } { k - 1 } \\enspace . \\end{align*}"}
+{"id": "1581.png", "formula": "\\begin{align*} w _ r ^ { - 1 } w _ r x _ { a } t _ { r } ^ { 5 } x _ { b } t _ { r } ^ { 5 } x _ { c } \\left ( t _ { a } ^ { - 1 } t _ { r } ^ { - 1 } t _ { b } ^ { - 1 } t _ { r } ^ { - 1 } t _ { c } ^ { - 1 } \\right ) ^ { 5 } = e . \\end{align*}"}
+{"id": "5621.png", "formula": "\\begin{align*} \\mathbb { G } ^ { \\beta } = \\hat { \\mathbb { G } } ^ { \\beta } + \\sqrt { - 1 } \\hat { \\mathbb { G } } ^ { \\beta + n } \\end{align*}"}
+{"id": "7937.png", "formula": "\\begin{align*} \\big ( \\overline { \\mu } _ t = \\varphi _ t \\circ \\mu _ t \\circ ( \\varphi _ t ^ { - 1 } \\otimes \\varphi _ t ^ { - 1 } ) , ~ \\overline { R } _ t = \\varphi _ t \\circ R _ t \\circ \\varphi _ t ^ { - 1 } \\big ) . \\end{align*}"}
+{"id": "4268.png", "formula": "\\begin{align*} \\textup { s p t } ( n ) = n p ( n ) - \\frac { 1 } { 2 } N _ 2 ( n ) . \\end{align*}"}
+{"id": "2505.png", "formula": "\\begin{align*} \\mu _ N ( E ) = \\dfrac { | A \\cap \\Phi _ N | } { | \\Phi _ N | } \\end{align*}"}
+{"id": "7648.png", "formula": "\\begin{align*} m ( \\Omega ) & : = \\inf \\{ \\ , | E _ 1 | \\ , : \\ , \\ , \\} , \\\\ M ( \\Omega ) & : = \\sup \\{ \\ , | E _ 1 | \\ , : \\ , \\ , \\} , \\end{align*}"}
+{"id": "536.png", "formula": "\\begin{align*} G _ 1 ( M ( t ) ) & = \\int _ { \\R ^ n } f \\Big ( ( 1 - t ) x _ 1 + t T _ 1 ( x _ 1 ) , \\dots , ( 1 - t ) x _ n + t T _ n ( x _ n ) \\Big ) \\prod _ { i = 1 } ^ { n } Q ^ * _ i ( d x _ i ) , \\end{align*}"}
+{"id": "7877.png", "formula": "\\begin{align*} Y _ 2 ( R ) = \\{ x \\in C _ s R \\cap E _ 2 : R _ x R \\} , \\end{align*}"}
+{"id": "5965.png", "formula": "\\begin{align*} f ^ { ( H , \\alpha , \\beta ) } _ K : = f ^ { ( H , \\alpha ) } _ K \\end{align*}"}
+{"id": "1071.png", "formula": "\\begin{align*} \\Delta = \\sum _ { a = 1 , 2 } \\delta _ { a } ^ { 2 } . \\end{align*}"}
+{"id": "1603.png", "formula": "\\begin{align*} f _ { n } \\left ( g _ { s ^ { \\prime \\prime } , k , i } \\right ) \\circ f _ { n } \\left ( g _ { s ^ { \\prime } , j , k } \\right ) \\circ f _ { n } \\left ( g _ { s , i , j } \\right ) \\restriction _ { T _ { b _ { i } } \\left ( V ^ { \\prime \\prime } \\right ) } = \\mathrm { i d } _ { T _ { b _ { i } } \\left ( V ^ { \\prime \\prime } \\right ) } , \\end{align*}"}
+{"id": "3616.png", "formula": "\\begin{align*} f ( x ) = M f ( x ) = f ( \\beta ) + D + d ( x - \\beta ) , x \\ge b . \\end{align*}"}
+{"id": "4518.png", "formula": "\\begin{align*} \\{ \\Pi _ y , \\mathcal { K } \\} = - \\int _ 0 ^ 1 \\ ! \\ ! \\mathcal { E } _ y ( x ) \\Phi _ x \\ , \\mathrm { d } x \\ , . \\end{align*}"}
+{"id": "3181.png", "formula": "\\begin{align*} \\| \\phi ^ k ( g ) \\| _ T = \\| \\phi ^ { - m } \\circ \\phi ^ { m } \\circ \\phi ^ k ( g ) \\| _ T > \\| \\phi ^ { m } \\circ \\phi ^ k ( g ) \\| _ T > \\| \\phi ^ k ( g ) \\| _ T \\end{align*}"}
+{"id": "149.png", "formula": "\\begin{align*} \\partial _ i ( e ( T ) \\otimes 1 ) : = \\sum _ { j = 1 } ^ { i + 3 } \\sum _ { \\sigma \\in H } ( - 1 ) ^ { j - 1 } e ( \\sigma ( T _ j ) ) \\otimes x _ { a _ j } \\in V _ { ( d - 1 , d - i + 2 , 1 ^ { i - 1 } ) } \\otimes _ K R ( - d - i ) = F _ { i - 1 } , \\end{align*}"}
+{"id": "174.png", "formula": "\\begin{align*} \\begin{aligned} \\Xi = \\{ \\gamma | \\ , \\Phi _ 1 ( \\gamma ) \\cap \\Phi _ 2 ( \\gamma ) \\neq \\emptyset , - 3 < \\gamma \\leq 1 \\} \\ , , \\end{aligned} \\end{align*}"}
+{"id": "6458.png", "formula": "\\begin{align*} \\lambda ( F _ 1 , F _ 2 ) = \\mu ( F _ 1 , F _ 2 ) + \\frac { i } { 2 } \\sigma ( F _ 1 , F _ 2 ) \\end{align*}"}
+{"id": "5602.png", "formula": "\\begin{align*} - C _ { i p b } J ^ i _ s J ^ p _ a = C _ { a b s } . \\end{align*}"}
+{"id": "6180.png", "formula": "\\begin{align*} - K _ T + L _ i - L _ j i , j = 1 , 2 , 3 . \\end{align*}"}
+{"id": "4362.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\sum _ { i \\neq j } \\theta _ i D _ H g _ i ^ 0 ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) \\\\ + \\sum _ { \\alpha = 1 } ^ { m } \\lambda _ \\alpha D _ H g ^ \\alpha ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) \\\\ + \\sum _ { \\beta = 1 } ^ { q } \\mu _ \\beta D _ H h ^ \\beta ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) = 0 . \\end{array} \\right \\} \\end{align*}"}
+{"id": "4624.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 ^ + } \\frac { \\phi ( x , g ( \\epsilon ) ) - \\phi ( x , g ( 0 ) ) } { \\epsilon } = \\phi ' _ h ( x , g ( 0 ) ) g ' ( 0 ) = \\frac { \\phi ' _ h ( x , | \\nabla u | ) } { | \\nabla u | } \\nabla u \\cdot \\nabla h \\end{align*}"}
+{"id": "5167.png", "formula": "\\begin{align*} \\bigcap _ { n \\geq 1 } U _ n = C . \\end{align*}"}
+{"id": "7383.png", "formula": "\\begin{align*} L _ { \\lambda } ( x ) = \\frac { \\sqrt { \\lambda } } { 2 \\pi } K _ 1 \\bigl ( - i \\sqrt { \\lambda } | x | \\bigr ) \\frac { x _ 1 - i x _ 2 } { | x | } , x = ( x _ 1 , x _ 2 ) \\in \\mathbb { R } ^ 2 \\setminus \\{ 0 \\} , \\end{align*}"}
+{"id": "5482.png", "formula": "\\begin{align*} ( 1 - t ) F _ N ( 0 , t ; t ) = ( 1 - t q ^ N ) \\sum _ { n = 0 } ^ { N } \\begin{bmatrix} N \\\\ n \\end{bmatrix} \\frac { ( q ) _ n ( t ) ^ { 2 n } q ^ { n ^ 2 } } { ( t q ) _ n ( t q ) _ n } . \\end{align*}"}
+{"id": "7887.png", "formula": "\\begin{align*} d _ y \\Big ( \\begin{array} { c } \\partial _ { x _ 1 } \\Phi ( x , y ) \\\\ \\partial ^ 2 _ { x _ 1 } \\Phi ( x , y ) / | \\nabla _ x \\Phi ( x , y ) | \\end{array} \\Big ) \\Big | _ { ( x , y ) = ( 0 , 0 ) } \\quad \\textrm { i s i n v e r t i b l e } . \\end{align*}"}
+{"id": "196.png", "formula": "\\begin{align*} & \\frac { ( \\psi _ t \\psi ^ * _ t ) _ 0 - ( \\psi \\psi ^ * ) _ 0 } { t } = \\frac { \\psi _ t \\psi ^ * _ t - \\psi \\psi ^ * } { t } - \\frac { 1 } { 2 } \\left ( \\frac { \\psi _ t \\psi ^ * _ t - \\psi \\psi ^ * } { t } \\right ) 1 \\\\ & = \\frac { \\psi _ t ( \\psi ^ * _ t - \\psi ^ * ) } { t } + \\frac { ( \\psi _ t - \\psi ) \\psi ^ * } { t } - \\frac { 1 } { 2 } \\Big \\{ \\frac { \\psi _ t ( \\psi ^ * _ t - \\psi ^ * ) } { t } + \\frac { ( \\psi _ t - \\psi ) \\psi ^ * } { t } \\Big \\} 1 . \\end{align*}"}
+{"id": "7986.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { \\lfloor c _ s n d ^ { - \\frac { s } { s - 2 } } \\rfloor } \\binom { n } { r } ( e r p ) ^ { \\frac { r s } { 2 } } \\leq \\sum _ { r = 1 } ^ { \\lfloor c _ s n d ^ { - \\frac { s } { s - 2 } } \\rfloor } \\left ( \\frac { e n } { r } \\cdot ( e r p ) ^ { \\frac { s } { 2 } } \\right ) ^ r = \\sum _ { r = 1 } ^ { \\lfloor c _ s n d ^ { - \\frac { s } { s - 2 } } \\rfloor } ( e ^ { \\frac { s } { 2 } + 1 } r ^ { \\frac { s - 2 } { 2 } } n p ^ { \\frac { s } { 2 } } ) ^ r . \\end{align*}"}
+{"id": "7041.png", "formula": "\\begin{align*} R _ { \\ , \\mathfrak { t r } ^ p , \\mathfrak u ^ p } = 0 \\end{align*}"}
+{"id": "1226.png", "formula": "\\begin{align*} \\tilde L _ 2 ( h , k ) ( y ) = k ( y ) + \\frac { \\displaystyle { \\int _ { X ^ - } e ^ { - c ( x , y ) + \\phi ( x ) } h ( x ) \\dd \\mu ^ - ( x ) } } { \\displaystyle { \\int _ { X ^ - } e ^ { - c ( x , y ) + \\phi ( x ) } \\dd \\mu ^ - ( x ) } } , L _ 2 ( h , k ) = G ( \\phi , \\psi ) \\tilde L _ 2 ( h , k ) . \\end{align*}"}
+{"id": "4675.png", "formula": "\\begin{align*} \\chi ( A ) = \\chi _ { { \\mathfrak { P } } _ { 1 , 1 } } ( A ) \\prod _ { j = 2 } ^ k \\chi _ { { \\mathfrak { P } _ { j , 1 } } } ^ { a _ j } ( A ) , \\end{align*}"}
+{"id": "7142.png", "formula": "\\begin{align*} \\mathbb { D } ( d ; \\delta ) = \\Big \\langle \\boxtimes _ { i = 1 } ^ k \\mathbb { M } ( d _ i ) _ { w _ i } \\Big \\rangle . \\end{align*}"}
+{"id": "3614.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\tau _ n ( \\omega ) = \\infty , a . s . \\end{align*}"}
+{"id": "3152.png", "formula": "\\begin{align*} - \\Delta _ { \\mathbb { B } ^ { N } } v - \\lambda v = | v | ^ { p - 1 } v H ^ { - 1 } \\left ( \\mathbb { B } ^ { N } \\right ) , v \\in H ^ { 1 } \\left ( \\mathbb { B } ^ { N } \\right ) . \\end{align*}"}
+{"id": "2393.png", "formula": "\\begin{align*} \\partial _ { 0 , 1 } ( w _ { 1 } U ( t _ { 1 } ) \\cdots w _ { k } U ( t _ { k } ) w _ { k + 1 } ) = \\sum _ { i = 1 } ^ { k } w _ { 1 } U ( t _ { 1 } ) \\cdots w _ { i } V ( t _ { i } ) w _ { i + 1 } \\cdots U ( t _ { k } ) w _ { k + 1 } \\end{align*}"}
+{"id": "7043.png", "formula": "\\begin{align*} \\mathfrak u ^ p _ { 0 } : = \\{ Z \\in \\mathcal K ^ G ( M ) : \\nabla _ { \\nu _ p } Z = 0 \\} \\end{align*}"}
+{"id": "6583.png", "formula": "\\begin{align*} U ( C _ 6 ) = \\begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 \\end{pmatrix} , U ( P _ 7 ) = \\begin{pmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 1 & 0 & 0 \\end{pmatrix} \\end{align*}"}
+{"id": "2183.png", "formula": "\\begin{align*} & \\frac { 1 } { \\ , 2 \\ , } \\frac { \\partial \\ ; \\ ; } { \\partial x _ i } | | { \\rm g r a d } \\ : f | | ^ 2 = H ^ f ( X _ i , { \\rm g r a d } \\ : f ) = \\sum _ { j , k } H ^ f _ { i j } g ^ { j k } f _ k , \\\\ & \\sum _ { l , m } f _ l \\ , g ^ { l m } R ^ B _ { i j k m } = - ( R ^ B ( X _ i , X _ j ) X _ k ) f . \\end{align*}"}
+{"id": "4258.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac { ( - 1 ) ^ { n - 1 } \\left ( \\frac { c } { d } \\right ) _ { n } d ^ n q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( q ) _ n ( q ) _ { N - n } ( c q ) _ { n } } \\left ( n + \\sum _ { k = 1 } ^ n \\frac { q ^ k } { 1 - q ^ k } \\right ) = : T _ 1 + T _ 2 , \\end{align*}"}
+{"id": "744.png", "formula": "\\begin{align*} X ^ { ( i ) } ( t ) & = x + \\int _ 0 ^ t A ( s , \\overline { X ^ { ( i ) } } ( s ) , i ) d s + \\int _ 0 ^ t B ( s , \\overline { X ^ { ( i ) } } ( s ) , i ) d W ( s ) \\\\ & \\quad \\ + \\int _ 0 ^ t \\int _ { \\{ | z | < 1 \\} } H ( s , \\overline { X ^ { ( i ) } } ( s ) , i , z ) \\widetilde { N } ( d s , d z ) \\\\ & \\quad \\ + \\int _ 0 ^ t \\int _ { \\{ | z | \\ge 1 \\} } J ( s , \\overline { X ^ { ( i ) } } ( s ) , i , z ) N ( d s , d z ) , \\end{align*}"}
+{"id": "1809.png", "formula": "\\begin{align*} \\langle a ^ * , r ( b ^ * ) \\rangle = \\langle a ^ * \\otimes b ^ * , r \\rangle , \\end{align*}"}
+{"id": "2692.png", "formula": "\\begin{align*} \\binom { n - j } { j } \\binom { n - 2 j } { k - j } = \\binom { n - j } { k - j } \\binom { n - k } { n - k - j } \\end{align*}"}
+{"id": "7999.png", "formula": "\\begin{align*} T _ Q \\ : : = \\ : Q _ 0 \\otimes I _ { \\mathcal { F } ^ 2 _ d } \\ , + \\ , \\sum _ { 0 < | v | \\leq n } Q _ v \\otimes L _ v \\ , + \\ , \\sum _ { 0 < | v | \\leq n } Q _ v ^ * \\otimes L _ v ^ * . \\end{align*}"}
+{"id": "1324.png", "formula": "\\begin{align*} \\int _ { B _ r ( x _ 0 ) } \\left | \\nabla v ( x ) \\right | ^ { p - 2 } \\nabla v ( x ) \\cdot \\nabla \\varphi ( x ) \\ , d x = 0 \\end{align*}"}
+{"id": "3853.png", "formula": "\\begin{align*} \\Omega _ 2 : = \\left [ 1 + \\sqrt { \\frac { \\gamma _ n } { 4 n } } + \\frac { C _ n } { \\sqrt { 4 n \\gamma _ n } } , ~ 1 + \\frac { n ^ { \\tau } } { \\sqrt { n } } \\right ] \\times \\left [ - \\frac { n ^ { \\tau / 2 } } { n ^ { 1 / 4 } } , ~ \\frac { n ^ { \\tau / 2 } } { n ^ { 1 / 4 } } \\right ] , \\end{align*}"}
+{"id": "229.png", "formula": "\\begin{align*} \\partial _ t m _ t = \\div \\left ( \\nabla a ( m _ t , \\cdot ) m _ t \\right ) - a ( m _ t , x ) m _ t \\ , . \\end{align*}"}
+{"id": "5687.png", "formula": "\\begin{gather*} \\{ e _ { a , b } ^ { b } \\mid 1 \\le a , b \\le n , \\ ; a \\ne b \\} \\cup \\{ e _ { a , b } ^ { c } \\mid 1 \\le a , b , c \\le n , \\ ; a < b , \\ ; a \\ne c \\ne b \\} \\\\ = \\{ e _ { a , b } ^ { c } \\mid 1 \\le a , b , c \\le n , \\ ; a < b \\} . \\end{gather*}"}
+{"id": "2417.png", "formula": "\\begin{align*} { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { n } ( w _ { 1 } * w _ { 2 } ) ) = \\sum _ { i + j = n } { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { i } w _ { 1 } ) * { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { j } w _ { 2 } ) \\end{align*}"}
+{"id": "7177.png", "formula": "\\begin{align*} \\Phi ( \\mathcal { E } ) = ( \\mathcal { E } \\otimes _ { \\mathbb { C } } \\mathrm { S y m } ( \\mathfrak { g } ) , d _ { \\Phi ( \\mathcal { E } ) } ) , \\ d _ { \\Phi ( \\mathcal { E } ) } ( u \\otimes v ) = d _ { \\mathcal { E } } ( u ) \\otimes v + \\sum _ { i = 1 } ^ k ( u \\cdot e _ i ^ { \\vee } ) \\otimes ( e _ i \\cdot v ) \\end{align*}"}
+{"id": "597.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { 2 } \\Big ( \\psi ^ { - 0 } _ { n - 1 , p } \\psi ^ { - 0 } _ { n , p } \\rho ^ { 0 + } _ { n , p } - 2 \\psi ^ { - 0 } _ { n - 1 , p } \\rho ^ { 0 + } _ { n - 1 , p } \\psi ^ { - 0 } _ { n , p - 1 } + \\rho ^ { 0 + } _ { n - 2 , p } \\psi ^ { - 0 } _ { n - 1 , p - 1 } \\psi ^ { - 0 } _ { n , p - 1 } \\Big ) & = & - \\varphi ^ { - + } _ { n - 1 , p } \\psi ^ { - 0 } _ { n , p - 1 } \\\\ & & = & - \\psi ^ { - 0 } _ { n - 1 , p } \\varphi ^ { - + } _ { n , p } \\ , . \\end{aligned} \\end{align*}"}
+{"id": "4756.png", "formula": "\\begin{gather*} \\beta _ 2 ( \\mu ( \\partial _ 1 ( a ) ) v ) - \\beta _ 1 ( \\mu ( \\partial _ 2 ( a ) ) v ) + \\mu ( \\partial _ 1 ( a ) ) \\alpha _ 2 ( v ) - \\mu ( \\partial _ 2 ( a ) ) \\alpha _ 1 ( v ) = 0 , \\end{gather*}"}
+{"id": "66.png", "formula": "\\begin{align*} \\Sigma _ t = \\{ ( x ' , x _ n ) \\in \\mathbb { R } ^ n \\ | \\ x ' \\in \\mathbb { S } _ t ^ { n - 1 } , \\ x _ n \\in \\mathbb { R } \\} \\end{align*}"}
+{"id": "4712.png", "formula": "\\begin{align*} \\widetilde { K } _ { \\lambda , 1 } ( z , w ) & = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( n + \\lambda + 3 ) ( n + \\lambda + 2 ) } { 2 \\lambda + 2 } \\phi _ { n } ( z ) \\overline { \\phi _ { n } ( w ) } , \\\\ \\widetilde { K } _ { \\lambda , 2 } ( z , w ) & = K _ { \\lambda , 2 } ( z , w ) ( 1 - | z | ^ 2 ) ( 1 - | w | ^ 2 ) . \\end{align*}"}
+{"id": "1849.png", "formula": "\\begin{align*} x _ { i - 1 , j - 1 } + x _ { m + 2 - i , n + 2 - j } & = S ( i - 1 , j - 1 ) , \\\\ x _ { i - 1 , j } + x _ { m + 2 - i , n + 1 - j } & = S ( i - 1 , j ) , \\mbox { a n d } \\\\ x _ { i , j - 1 } + x _ { m + 1 - i , n + 2 - j } & = S ( i , j - 1 ) , \\end{align*}"}
+{"id": "1531.png", "formula": "\\begin{align*} A _ { k } = \\left ( z _ { k , i , j } \\right ) _ { 1 \\le i , j \\le n } . \\end{align*}"}
+{"id": "962.png", "formula": "\\begin{align*} \\mathcal { H } _ \\Phi : = { \\big \\{ F ( z , \\omega ) = f ( \\Phi ^ { - 1 } ( z , \\omega ) , \\omega ) ; \\ ; f \\in H ^ 1 _ { \\rm l o c } ( \\mathbb { R } ^ n ; L ^ 2 ( \\Omega ) ) \\ ; \\ ; \\big \\} } \\end{align*}"}
+{"id": "83.png", "formula": "\\begin{align*} \\Delta _ { \\Psi } u = \\Delta u - g ( \\nabla \\Psi , \\nabla u ) \\end{align*}"}
+{"id": "6687.png", "formula": "\\begin{align*} Q ( t ) = ( 1 - t ) \\det \\begin{bmatrix} s ' _ { 0 } & s ' _ { 1 } & \\ldots & s ' _ { m - 1 } & 1 \\\\ s ' _ { 1 } & s ' _ { 2 } & \\ldots & s ' _ { m } & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ s ' _ { m } & s ' _ { m + 1 } & \\ldots & s ' _ { 2 m - 1 } & t ^ { m } \\end{bmatrix} , \\end{align*}"}
+{"id": "6547.png", "formula": "\\begin{align*} ( a , b ) = ( a ' , b ' ) . \\end{align*}"}
+{"id": "3676.png", "formula": "\\begin{align*} | L _ { \\mathcal { G } } ( u ) | / n ^ 2 & \\le \\binom { t } { 2 } \\left ( \\frac { 1 } { t + 2 } + \\epsilon \\right ) ^ 2 + t \\left ( \\frac { 1 } { t + 2 } + \\epsilon \\right ) \\left ( \\frac { 1 } { t + 2 } + 2 \\epsilon \\right ) \\\\ & \\le \\frac { t ( t + 1 ) } { 2 ( t + 2 ) ^ 2 } + t \\epsilon + \\frac { t ( t + 3 ) } { 2 } \\epsilon ^ 2 \\le \\frac { t ( t + 1 ) } { 2 ( t + 2 ) ^ 2 } + 2 t \\epsilon . \\end{align*}"}
+{"id": "6060.png", "formula": "\\begin{align*} \\begin{cases} \\ , \\ , p \\in H ^ 1 _ 0 ( \\Omega ) \\ , \\ , \\ , \\ , \\\\ [ 0 . 3 c m ] \\int _ { \\Omega } \\nabla v \\nabla p d x - \\int _ { \\Omega } k ^ 2 v p d x = - 2 \\int _ { \\Omega } ( \\nabla \\eta - A ) \\nabla v d x - 2 \\int _ { \\Omega } ( \\eta - \\eta _ 0 ) v d x \\ ; v \\in H ^ 1 _ 0 ( \\Omega ) . \\end{cases} \\end{align*}"}
+{"id": "2569.png", "formula": "\\begin{align*} \\begin{aligned} & V ( t _ 0 , x _ 0 , \\nu _ 0 ) \\\\ : = & \\frac { 1 } { 2 } \\mathbb { E } \\Big \\{ \\int _ { t _ 0 } ^ T \\Big [ Q \\left ( x _ t ^ { * , t _ 0 , x _ 0 , \\xi } + l ( \\nu _ t ^ { * , t _ 0 , \\xi } ) \\right ) ^ 2 + R \\left ( \\alpha _ t ^ { * , t _ 0 , x _ 0 , \\xi } + h ( \\mu _ t ^ { * , t _ 0 , \\xi } ) \\right ) ^ 2 \\Big ] d t + G \\left ( x _ T ^ { * , t _ 0 , x _ 0 , \\xi } + g ( \\mu _ T ^ { * , t _ 0 , \\xi } ) \\right ) ^ 2 \\Big \\} \\end{aligned} \\end{align*}"}
+{"id": "7608.png", "formula": "\\begin{align*} B _ { \\varepsilon , m } = \\max _ { x ( \\theta ) i n L _ { \\varepsilon , m } } \\min _ { y _ m ( x ( \\theta ) ) \\in \\Omega ^ * _ m } | | x ( \\theta ) - y ( \\theta ) | | _ \\pi = \\max _ { x \\in L _ { \\varepsilon , m } } | | x ( \\theta ) - y ^ * _ m ( x ( \\theta ) ) | | _ \\pi . \\end{align*}"}
+{"id": "7754.png", "formula": "\\begin{align*} \\begin{aligned} b _ { \\gamma _ + } ( 0 _ n , \\theta ) & = \\lim \\limits _ { t \\rightarrow \\infty } [ s _ E ( y ) - s _ { p _ 0 } ( y ) ] \\\\ & = \\lim \\limits _ { t \\rightarrow \\infty } [ t - \\cosh ^ { - 1 } ( \\cosh t \\cdot \\cosh ( \\lambda | \\theta - \\theta _ 0 | ) ) ] \\\\ & = - \\ln \\cosh ( \\lambda | \\theta - \\theta _ 0 | ) . \\end{aligned} \\end{align*}"}
+{"id": "4244.png", "formula": "\\begin{align*} \\frac { ( - a q ) _ { \\infty } } { ( b q ) _ { \\infty } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - b / a ) _ n a ^ n q ^ { n ( n + 1 ) / 2 } } { ( q ) _ n ( b q ) _ n } . \\end{align*}"}
+{"id": "21.png", "formula": "\\begin{align*} ( M N T _ { 2 } ) _ { i j } = \\left \\{ \\begin{array} { c c } { i + 2 j \\choose 3 j + 1 } & i \\geq j \\geq 0 ; \\\\ 0 & i < j , \\end{array} \\right . \\end{align*}"}
+{"id": "7680.png", "formula": "\\begin{align*} \\Upsilon ( v ) = \\frac { v } { P ^ 2 _ { S ( v ) } } \\end{align*}"}
+{"id": "7684.png", "formula": "\\begin{align*} \\| f \\| _ { r , p r } ^ { p r } = c _ r \\int _ 0 ^ { t _ \\circ } \\mu ( t ) t ^ { r - 1 } d t , \\end{align*}"}
+{"id": "2176.png", "formula": "\\begin{align*} & r _ 1 = - | A _ 0 | h _ { 1 2 1 2 } ^ 0 , r _ 2 = - 2 | A _ 0 | h _ { 1 2 1 3 } ^ 0 , r _ 3 = - 2 | A _ 0 | h _ { 1 2 2 3 } ^ 0 , \\\\ & r _ 4 = - | A _ 0 | h _ { 1 3 1 3 } ^ 0 , r _ 5 = - 2 | A _ 0 | h _ { 1 3 2 3 } ^ 0 , r _ 6 = - | A _ 0 | h _ { 2 3 2 3 } ^ 0 . \\end{align*}"}
+{"id": "800.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\bar { \\theta } _ 1 : = \\lim _ { N \\rightarrow + \\infty } \\widetilde { \\theta } _ 1 = \\theta + \\frac { 1 - \\theta } { l _ 1 + l _ 2 \\mathbb { E } \\Big ( e ^ { - \\beta \\tau } x ( \\tau ) \\Big ) } , \\\\ & \\bar { K } _ 1 : = \\lim _ { N \\rightarrow + \\infty } K ' _ 1 = \\frac { K } { \\bar { \\theta } _ 1 } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3918.png", "formula": "\\begin{align*} u ( t + h ) - u ( t ) & = [ S ( t + h ) - S ( t ) ] \\xi + \\int _ 0 ^ t S ( \\tau ) D _ h f ( u ) ( t - \\tau ) d \\tau \\\\ & + \\int _ t ^ { t + h } S ( \\tau ) f ( u ( t + h - \\tau ) , \\mathcal H u ( t + h - \\tau ) ) d \\tau . \\end{align*}"}
+{"id": "1835.png", "formula": "\\begin{align*} & [ T ( a ) , T ( b ) ] - T \\big ( \\rho ( T ( a ) ) b - \\rho ( T ( b ) ) a + \\lambda [ a , b ] _ V \\big ) \\\\ & = T ( a ) \\cdot T ( b ) - T ( b ) \\cdot T ( a ) - T \\big ( ( \\mathfrak { l } - \\mathfrak { r } ) ( T ( a ) ) b - ( \\mathfrak { l } - \\mathfrak { r } ) ( T ( b ) ) a + \\lambda ( a \\cdot _ V b - b \\cdot _ V a \\big ) = 0 . \\end{align*}"}
+{"id": "5978.png", "formula": "\\begin{align*} { W ^ { ( q ) } } ( 0 ) & = \\left \\{ \\begin{array} { l l } 0 & \\textrm { i f $ X $ i s o f u n b o u n d e d v a r i a t i o n , } \\\\ \\dfrac 1 { c } & \\textrm { i f $ X $ i s o f b o u n d e d v a r i a t i o n , } \\end{array} \\right . \\end{align*}"}
+{"id": "2984.png", "formula": "\\begin{align*} t ( K , u ) & = t ^ { \\mathbf { A } } ( F , W ) \\cdot \\frac { r ^ { - r } } { \\prod _ { i \\in R } \\pi ( A _ i ) } \\ ; \\mbox { a n d } \\\\ t ( K , u ' ) & = \\prod _ { e \\in E } d _ e \\cdot r ^ { - r } \\ ; . \\end{align*}"}
+{"id": "1708.png", "formula": "\\begin{align*} \\gamma _ { \\tau , p } ( x _ 1 , \\ldots , x _ p ; y _ 1 , \\ldots , y _ p ) : = \\rho _ { \\tau } ( \\varphi ^ * _ \\tau ( y _ 1 ) \\ldots \\varphi _ \\tau ^ * ( y _ p ) \\varphi _ \\tau ( x _ 1 ) \\ldots \\varphi _ \\tau ( x _ p ) ) \\ , . \\end{align*}"}
+{"id": "6860.png", "formula": "\\begin{align*} f ( x ) = r ( x ) \\prod _ { i = 1 } ^ { s } ( x - a _ i ) , \\end{align*}"}
+{"id": "2352.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } f _ { c } ( S ( x _ { a _ { i + 1 } } \\cdots x _ { a _ { n } } ) , x _ { a _ { 1 } } \\cdots x _ { a _ { i } } ) = \\begin{cases} x _ { c + a _ { 1 } + \\cdots + a _ { n } } & n : { \\rm e v e n } \\\\ 0 & n : { \\rm o d d } . \\end{cases} \\end{align*}"}
+{"id": "4749.png", "formula": "\\begin{gather*} \\partial ( a \\succ b ) = \\partial ( a ) \\succ b + a \\succ \\partial ( b ) , \\\\ \\partial ( a \\prec b ) = \\partial ( a ) \\prec b + a \\prec \\partial ( b ) , \\forall a , b \\in A . \\end{gather*}"}
+{"id": "2228.png", "formula": "\\begin{align*} \\tau _ i = h ( i - 1 ) + a , i = 1 , \\dots , M , h = \\frac { b - a } { M - 1 } . \\end{align*}"}
+{"id": "5835.png", "formula": "\\begin{align*} U ( e _ 1 ) & = e _ 2 \\\\ U ( e _ 2 ) & = e _ 1 . \\end{align*}"}
+{"id": "7947.png", "formula": "\\begin{align*} [ R ( x ) , R ( y ) ] = R \\big ( [ R ( x ) , y ] + [ x , R ( y ) ] \\big ) + \\kappa ~ \\ ! [ x , y ] , x , y \\in \\mathfrak { g } . \\end{align*}"}
+{"id": "8252.png", "formula": "\\begin{align*} \\begin{aligned} Z _ s ^ h ( t ) \\le C \\Big ( \\Vert \\sigma _ 0 \\Vert _ { \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } _ { 2 , 1 } } + \\Vert u _ 0 \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } } \\Big ) + C \\int _ 0 ^ t \\Vert u ( \\tau ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } } Z _ { s , \\bar { s } } ( \\tau ) \\dd \\tau . \\end{aligned} \\end{align*}"}
+{"id": "2119.png", "formula": "\\begin{align*} P = \\{ \\ell _ 1 , \\ell _ 1 + 1 , \\ldots , r _ 1 \\} \\times \\ldots \\times \\{ \\ell _ n , \\ell _ n + 1 , \\ldots , r _ n \\} , \\end{align*}"}
+{"id": "992.png", "formula": "\\begin{align*} W = \\frac { 1 } { 2 } \\Big [ & ( \\mu ^ * + \\varkappa ) \\ , { \\rm e } ^ * _ { j i } { \\rm e } ^ * _ { j i } + \\mu ^ * \\ , { \\rm e } ^ * _ { j i } { \\rm e } ^ * _ { i j } + \\lambda \\ , { \\rm e } ^ * _ { i i } { \\rm e } ^ * _ { j j } + \\gamma \\ , \\mathfrak { K } _ { j i } \\mathfrak { K } _ { j i } + \\beta \\ , \\mathfrak { K } _ { j i } \\mathfrak { K } _ { i j } + \\alpha \\ , \\mathfrak { K } _ { i i } \\mathfrak { K } _ { j j } \\Big ] . \\end{align*}"}
+{"id": "6769.png", "formula": "\\begin{align*} K ( t , k , x ) & = \\int d y \\ , q ( t , x , y ) \\abs { k } ^ { - 1 } e ^ { - 2 \\pi i k y } \\psi _ t ( y ) . \\end{align*}"}
+{"id": "5581.png", "formula": "\\begin{align*} J ^ i _ k = \\left \\{ \\begin{array} { l l } \\delta ^ i _ { k + n } , & 1 \\leq k \\leq n , \\\\ - \\delta ^ i _ { k - n } , & n + 1 \\leq k \\leq 2 n . \\end{array} \\right . \\end{align*}"}
+{"id": "1509.png", "formula": "\\begin{align*} \\delta ( w , z ) = \\sum _ { w < p \\le z } g ( p ) \\end{align*}"}
+{"id": "3519.png", "formula": "\\begin{align*} \\Big \\Vert \\nabla \\mathrm { H } \\Big \\Vert _ { \\mathbb { L } ^ { 2 } \\Big ( \\Omega \\Big ) } ^ { 2 } = \\mathcal { O } \\Big ( \\delta ^ { 2 - 2 h } \\Big ) \\ \\ h < 1 . \\end{align*}"}
+{"id": "4611.png", "formula": "\\begin{align*} \\varrho _ \\phi ( u ) & : = \\int _ \\Omega \\phi ( x , | u ( x ) | ) \\ , d x . \\end{align*}"}
+{"id": "365.png", "formula": "\\begin{align*} C ^ \\infty ( N ) = C ^ \\infty ( M ) / I _ { N } \\ , . \\end{align*}"}
+{"id": "876.png", "formula": "\\begin{align*} | | \\xi _ 1 | | _ { \\mathfrak { Y } } & = \\sup \\bigg \\{ | | S ( \\xi _ 1 , \\widehat { D } ) | | ~ : ~ \\widehat { D } \\in \\mathfrak { Y } \\bigg \\} \\\\ & = \\sup \\bigg \\{ | | \\sum _ { i = 1 } ^ { n } \\xi _ 1 ( d _ i ) \\overline { \\mu } ( D _ i ) | | ~ : ~ \\widehat { D } = ( D _ i , d _ i ) , ~ i = 1 , 2 , . . \\bigg \\} \\\\ & = \\sum _ { i = 1 } ^ { n } \\xi _ 1 ( d _ i ) \\overline { \\mu } ( D _ i ) \\\\ & = 0 \\end{align*}"}
+{"id": "6871.png", "formula": "\\begin{gather*} m ( j ) _ { i ( 1 ) } = m _ { i ( 1 ) } + 1 , \\dots , m ( j ) _ { i ( j ) } = m _ { i ( j ) } + 1 , \\\\ m ( j ) _ { i } = m _ { i } i \\ne i ( 1 ) , \\dots , i ( j ) . \\end{gather*}"}
+{"id": "3176.png", "formula": "\\begin{align*} \\rho _ i : = e _ i \\cdot h _ i \\bar e _ i \\cdot h _ i h _ i ' e _ i \\cdot h _ i h _ i ' h _ i \\bar e _ 1 \\dots \\end{align*}"}
+{"id": "7324.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l l } \\psi ^ { u } ( t ) = \\psi ( t , X ^ { u } ( t ) , u ( t ) ) , & \\psi _ { x } ^ { u } ( t ) = \\psi _ { x } ( t , X ^ { u } ( t ) , u ( t ) ) , & \\psi _ { x x } ^ { u } ( t ) = \\psi _ { x x } ( t , X ^ { u } ( t ) , u ( t ) ) . \\end{array} \\end{align*}"}
+{"id": "1720.png", "formula": "\\begin{align*} A ^ { \\xi } _ \\tau ( \\zeta ) & : = \\tilde { \\rho } _ { \\tau , \\zeta } ( \\Theta _ \\tau ( \\xi ) ) \\ , . \\end{align*}"}
+{"id": "1776.png", "formula": "\\begin{align*} s _ { i } = \\min \\left ( s \\in \\left [ n \\right ] \\mid s > s _ { i - 1 } \\ { \\rm a n d } \\ { \\rm s i g n } \\left ( s \\right ) = - { \\rm s i g n } \\left ( s _ { i - 1 } \\right ) \\right ) . \\end{align*}"}
+{"id": "1016.png", "formula": "\\begin{align*} I ( \\varphi , \\overline { \\mathbf { R } } ) = \\dd \\int _ { \\Omega } \\left [ W _ { \\rm { m p } } ( \\overline { \\mathbf { U } } ) + W _ { \\rm { c u r v } } ( \\boldsymbol { \\alpha } ) \\right ] d V { \\to } \\textrm { \\ \\ m i n . } { \\rm w . r . t . } ( \\varphi , \\overline { \\mathbf { R } } ) \\ , , \\end{align*}"}
+{"id": "666.png", "formula": "\\begin{align*} F _ 1 = \\frac { 1 } { \\sqrt { | c _ 1 | } } \\ \\langle \\langle \\nu _ 1 \\cdot \\varphi , \\varphi \\rangle \\rangle . \\end{align*}"}
+{"id": "2695.png", "formula": "\\begin{align*} M _ Q ( n , j , 0 ; a ) = \\binom { a + j } { j } \\sum _ { k = j } ^ { n - j } \\binom { n - j } { k - j } \\bigl { ( } \\binom { a + n - k } { a + j } \\binom { a + k } { a } \\bigr { ) } G ( n , k , a ) \\end{align*}"}
+{"id": "6396.png", "formula": "\\begin{align*} \\pi ^ { ( \\omega ) } _ { i j } : = \\Pi ( e ^ { ( \\omega ) } _ i , e ^ { ( \\omega ) } _ j ) . \\end{align*}"}
+{"id": "6181.png", "formula": "\\begin{align*} ( b \\phi - \\phi b ) W = 0 \\ \\ . \\end{align*}"}
+{"id": "5814.png", "formula": "\\begin{align*} c ( M _ z ) = c ( \\Pi V \\Pi ^ * ) = c ( V ) \\leq \\| V \\| = 1 . \\end{align*}"}
+{"id": "7031.png", "formula": "\\begin{align*} J _ 0 ( t , | D | ) u _ 1 ( x ) - J _ 0 ( t , x ) P _ { u _ 1 } & = \\int _ { | y | \\leqslant t ^ { \\alpha _ 0 } } \\big ( J _ 0 ( t , x - y ) - J _ 0 ( t , x ) \\big ) u _ 1 ( y ) \\mathrm { d } y \\\\ & \\quad + \\int _ { | y | \\geqslant t ^ { \\alpha _ 0 } } J _ 0 ( t , x - y ) u _ 1 ( y ) \\mathrm { d } y - J _ 0 ( t , x ) \\int _ { | y | \\geqslant t ^ { \\alpha _ 0 } } u _ 1 ( y ) \\mathrm { d } y \\end{align*}"}
+{"id": "5825.png", "formula": "\\begin{align*} \\omega ( R T ) = \\omega ( R ) \\omega ( T ) . \\end{align*}"}
+{"id": "2736.png", "formula": "\\begin{align*} V ( X , E ) : = \\{ D \\in N ^ 1 ( X ) \\ , | \\ , \\Lambda + \\pi ^ * D \\ \\} . \\end{align*}"}
+{"id": "7550.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\oint _ { \\gamma } P _ n ( z ) \\frac { A ( z ) ^ L } { z ^ { M + N } } z ^ k d z = H _ n \\delta _ { n , k } 0 \\leq k \\leq n , \\end{align*}"}
+{"id": "5063.png", "formula": "\\begin{align*} \\widetilde { f } _ n ( s ) = \\int _ 0 ^ \\infty f _ n ( y ) y ^ { s - 1 } \\ , d y = \\frac { ( - 1 ) ^ { \\lfloor { n / 2 } \\rfloor } \\Gamma _ \\C ( s ) } { 2 ^ s \\Gamma _ \\C ( \\frac { s + n + 1 } { 2 } ) \\Gamma _ \\C ( \\frac { s - n + 1 } { 2 } ) } \\quad \\Re ( s ) > 0 . \\end{align*}"}
+{"id": "8129.png", "formula": "\\begin{align*} ( \\overline L _ 0 \\cdots \\overline L _ d ) _ S \\geqslant \\sum _ { i = 0 } ^ d \\delta _ i \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L _ i ) . \\end{align*}"}
+{"id": "46.png", "formula": "\\begin{align*} \\chi _ P ( P _ a , P _ a ) & = \\sum _ { i = 1 } ^ k k P _ a ^ 2 ( i ) - 1 = \\sum _ { i = 1 } ^ k k \\frac { 1 } { k ^ 2 } \\left ( 1 + \\epsilon v _ { a , i } \\right ) ^ 2 - 1 = \\sum _ { i = 1 } ^ k \\frac { 1 } { k } \\left ( 1 + 2 \\epsilon v _ { a , i } + \\epsilon ^ 2 v _ { a , i } ^ 2 \\right ) - 1 = \\epsilon ^ 2 , \\end{align*}"}
+{"id": "7377.png", "formula": "\\begin{align*} f _ { + } = f _ { - } - \\frac { \\alpha } { 2 } \\big ( f _ { + } + f _ { - } \\big ) = \\big ( \\partial _ { \\nu } f _ { + } - \\partial _ { \\nu } f _ { - } \\big ) \\Sigma , \\end{align*}"}
+{"id": "153.png", "formula": "\\begin{align*} \\begin{aligned} \\beta ( \\theta ) \\sin \\theta = \\beta ( \\tfrac { \\pi } { 2 } - \\theta ) \\sin ( \\tfrac { \\pi } { 2 } - \\theta ) \\ , , \\ \\beta ( \\theta ) \\sin \\theta = \\beta ( \\pi - \\theta ) \\sin ( \\pi - \\theta ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "4385.png", "formula": "\\begin{align*} T ^ t f ( x ) : = \\frac { 1 } { \\pi } \\int _ \\mathbb { R } f ( \\xi ) \\frac { t } { ( \\xi - x ) ^ 2 + t ^ 2 } d t = f * K _ { t , e } ( x ) , \\end{align*}"}
+{"id": "8258.png", "formula": "\\begin{align*} \\sup _ { j \\in \\mathbb { Z } } \\int _ 0 ^ t \\psi ^ { r ' } _ j ( t , \\tau ) 2 ^ { j \\alpha } \\dd \\tau = \\sup _ { j \\in \\mathbb { Z } } 2 ^ { j \\alpha } \\int _ 0 ^ t e ^ { - r ' \\bar { \\mu } 2 ^ { j \\alpha } ( t - \\tau ) } t ^ { s r ' } \\tau ^ { - s r ' } \\dd \\tau \\le C . \\end{align*}"}
+{"id": "7254.png", "formula": "\\begin{align*} \\over = \\sup _ { \\mu \\in M ( X ) } F ( \\mu ) + \\int f d \\mu , \\end{align*}"}
+{"id": "3188.png", "formula": "\\begin{align*} \\left \\| v _ { n } \\right \\| _ { 6 } = C \\left \\| v _ { n } \\right \\| _ { D } \\le C \\left \\| u _ { n } \\right \\| _ { \\frac { 1 2 } { 5 } } ^ { 2 } \\le C . \\end{align*}"}
+{"id": "781.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d x _ i ( t ) = \\alpha x _ i ( t ) d t + \\sigma x _ i ( t ) d W _ i ( t ) , \\\\ & x _ i ( 0 ) = x \\end{aligned} \\right . \\end{align*}"}
+{"id": "6004.png", "formula": "\\begin{align*} V ( x , i ) = \\lim _ { n \\rightarrow \\infty } v ^ - _ n ( x , i ) = \\lim _ { n \\rightarrow \\infty } v ^ + _ n ( x , i ) , \\end{align*}"}
+{"id": "7418.png", "formula": "\\begin{align*} \\left [ \\hat { N } , a ^ \\dagger \\right ] = a ^ \\dagger , \\left [ \\hat { N } , a \\right ] = - a . \\end{align*}"}
+{"id": "6503.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r a _ i = m , \\ a _ 1 \\ge 1 , \\ a _ i \\ge 2 \\ \\mbox { f o r } \\ 1 < i < r , \\ \\mbox { a n d } \\ a _ r = m - a _ { r - 1 } \\ge 0 . \\end{align*}"}
+{"id": "2316.png", "formula": "\\begin{align*} \\d X _ t = b _ 0 ( X _ t ) \\d t + \\sigma _ 0 ( X _ t ) \\d W _ t , \\end{align*}"}
+{"id": "7366.png", "formula": "\\begin{align*} ~ \\lim _ { k \\to \\infty } \\norm { g _ k } _ { x _ k } = 0 . \\end{align*}"}
+{"id": "6610.png", "formula": "\\begin{align*} U _ \\theta : = \\big \\{ ( x , y ) : | \\arg ( x + i y ) | < \\theta \\big \\} , \\end{align*}"}
+{"id": "7601.png", "formula": "\\begin{align*} R : = \\left \\{ \\begin{array} { l l } j ^ * Q ^ X & \\mbox { i f } R ^ X = \\eta _ 1 ^ X Q ^ X , \\\\ R : = j ^ * R ^ X & \\mbox { o t h e r w i s e } . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "4416.png", "formula": "\\begin{align*} z = \\vartheta _ n + \\varsigma _ n + \\theta _ n , \\end{align*}"}
+{"id": "4468.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { H ^ 1 } : = \\int _ I \\big ( u v + u ' v ' \\big ) \\ , , \\end{align*}"}
+{"id": "590.png", "formula": "\\begin{align*} \\xi _ { 1 2 } & = ( \\mu ^ { ( 3 ) } - \\mu ^ { ( 2 ) } ) ( \\mu ^ { ( 1 ) } - \\mu _ p ^ { ( 1 2 3 ) } ) \\ , , \\\\ \\xi _ { 2 3 } & = ( \\mu ^ { ( 1 ) } - \\mu ^ { ( 2 ) } ) ( \\mu ^ { ( 3 ) } - \\mu _ p ^ { ( 1 2 3 ) } ) \\ , , \\\\ \\xi _ 0 & = ( \\mu ^ { ( 1 ) } \\mu ^ { ( 3 ) } - \\mu ^ { ( 2 ) } \\mu _ p ^ { ( 1 2 3 ) } ) ( \\mu _ p ^ { ( 1 2 3 ) } - \\mu ^ { ( 1 ) } + \\mu ^ { ( 2 ) } - \\mu ^ { ( 3 ) } ) + ( \\mu ^ { ( 2 ) } + \\mu ^ { ( 3 ) } ) ( \\mu ^ { ( 1 ) } + \\mu ^ { ( 1 2 3 ) } ) \\ , . \\end{align*}"}
+{"id": "754.png", "formula": "\\begin{align*} [ L _ \\lambda L ] = ( \\partial + 2 \\lambda ) L , [ L _ \\lambda H ] = ( \\partial + \\lambda ) H , [ H _ \\lambda L ] = \\lambda H , [ H _ \\lambda H ] = 0 . \\end{align*}"}
+{"id": "3528.png", "formula": "\\begin{align*} \\Big \\Vert \\nabla \\Tilde { \\mathrm { H } } \\Big \\Vert _ { \\mathbb { L } ^ 4 ( \\mathrm { B } ) } \\lesssim \\Big \\Vert \\Tilde { \\mathrm { H } } \\Big \\Vert _ { H ^ { \\frac { 3 } { 2 } } ( \\mathrm { B } ) } & \\lesssim \\Big \\Vert \\partial _ { \\nu } \\Tilde { \\mathrm { H } } \\Big \\Vert _ { H ^ { \\frac { 1 } { 2 } } ( \\partial \\mathrm { B } ) } + \\delta ^ 2 \\Big \\Vert \\Tilde { \\mathrm { H } } \\Big \\Vert _ { \\mathbb { L } ^ 2 ( \\partial \\mathrm { B } ) } = \\mathcal { O } \\Big ( 1 \\Big ) . \\end{align*}"}
+{"id": "7987.png", "formula": "\\begin{align*} \\mathbb { E } ( Y ) = & \\sum _ { R \\in \\mathcal { R } } \\mathbb { P } ( R \\mbox { i s b a d } ) \\leq \\sum _ { r = 1 } ^ { t } \\binom { n } { r } \\binom { \\binom { r } { 2 } } { \\lceil r s / 2 \\rceil } p ^ { \\lceil \\frac { r s } { 2 } \\rceil } \\\\ < & \\sum _ { r = 1 } ^ { t } n ^ { r } ( e r p ) ^ { \\frac { r s } { 2 } } < t \\big ( e ^ { \\frac { s } { 2 } } t ^ { \\frac { s } { 2 } } n p ^ { \\frac { s } { 2 } } \\big ) ^ { t } = t ^ { \\frac { t s } { 2 } + 1 } e ^ { \\frac { t s } { 2 } } n . \\end{align*}"}
+{"id": "5739.png", "formula": "\\begin{align*} \\langle \\mathcal { A } , M ^ s \\rangle = \\sum _ { i = 1 , j = 1 } ^ { m } \\langle A _ { i j } , M ^ s _ { i j } \\rangle = E ^ T \\left ( \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } \\otimes M ^ s _ { i j } \\right ) E < 0 . \\end{align*}"}
+{"id": "2661.png", "formula": "\\begin{align*} r s [ R , G ] = s r [ R , G ] r s r ' s ' [ R , G ] = r r ' s s ' [ R , G ] . \\end{align*}"}
+{"id": "6104.png", "formula": "\\begin{align*} \\mathcal C ( k , d ) = & \\left \\{ A \\in \\binom { [ n ] } { k } : [ d - 1 ] \\subseteq A , A \\cap [ d , k ] \\neq \\emptyset \\right \\} \\\\ & \\cup \\left \\{ A \\in \\binom { [ n ] } { k } : | A \\cap [ d - 1 ] | = d - 2 , [ d , k ] \\subseteq A \\right \\} , \\end{align*}"}
+{"id": "746.png", "formula": "\\begin{align*} P ^ X & ( s , x ; t , { \\cal X } ) \\\\ & = e ^ { - \\nu ( \\{ | z | \\ge 1 \\} ) ( t - s ) } P ^ Z ( s , x ; t , { \\cal X } ) \\\\ & \\ \\ \\ + \\int _ { s } ^ t \\int _ { \\{ | x _ 2 | \\ge 1 \\} } e ^ { - \\nu ( \\{ | z | \\ge 1 \\} ) ( t _ 1 - s ) } P ^ X ( t _ 1 , x _ 1 + J ( t _ 1 , x _ 1 , x _ 2 ) ; t , { \\cal X } ) P ^ Z ( s , x ; t _ 1 , d x _ 1 ) \\nu ( d x _ 2 ) d t _ 1 . \\end{align*}"}
+{"id": "186.png", "formula": "\\begin{align*} c h ( \\mathfrak { s } ^ { \\pm } _ { 1 / 2 } \\otimes L ) \\ , c h ( T X ) = c h ( \\mathfrak { s } ^ { \\pm } \\otimes L \\otimes T X ) = c h ( \\mathfrak { s } ^ { \\pm } _ { 3 / 2 } \\otimes L ) + c h ( \\mathfrak { s } ^ { \\mp } _ { 1 / 2 } \\otimes L ) . \\end{align*}"}
+{"id": "960.png", "formula": "\\begin{align*} \\sum _ { k , \\ell = 1 } ^ n A _ { k \\ell } ( y , \\omega ) \\ , \\xi _ k \\ , \\xi _ \\ell \\geqslant a _ 0 { \\vert \\xi \\vert } ^ 2 . \\end{align*}"}
+{"id": "7044.png", "formula": "\\begin{align*} \\tau _ t = \\mathrm { d } \\phi _ t \\circ g _ t \\ , , \\end{align*}"}
+{"id": "3085.png", "formula": "\\begin{align*} s ( k ) = \\sum _ { \\substack { d | k \\\\ d \\geq 1 } } d \\end{align*}"}
+{"id": "3187.png", "formula": "\\begin{align*} B ( u _ { n } - u ) - ( B ( u _ { n } ) - B ( u ) ) = 2 I _ { n } ^ { ( 1 ) } + 4 I _ { n } ^ { ( 2 ) } - 4 I _ { n } ^ { ( 3 ) } - 4 I _ { n } ^ { ( 4 ) } + 2 A . \\end{align*}"}
+{"id": "4210.png", "formula": "\\begin{align*} \\sum _ { X = 1 } ^ Y Q _ k ( X ) u ^ { X - 1 } ( 1 - u ) & = \\sum _ { X = 0 } ^ { Y - 1 } ( Q _ k ( X + 1 ) - Q _ k ( X ) ) u ^ X - Q _ k ( Y ) u ^ Y . \\end{align*}"}
+{"id": "4090.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n ( e _ i - d _ i ) > 0 . \\end{align*}"}
+{"id": "6842.png", "formula": "\\begin{align*} ( v _ 1 ^ { p ^ e + 1 } f ^ { p ^ e } ( a _ 1 ) , v _ 2 ^ { p ^ e + 1 } f ^ { p ^ e } ( a _ 2 ) , \\dots , v _ n ^ { p ^ e + 1 } f ^ { p ^ e } ( a _ n ) ) = ( u _ 1 g ( a _ 1 ) , u _ 2 g ( a _ 2 ) , \\dots , u _ n g ( a _ n ) ) . \\end{align*}"}
+{"id": "1602.png", "formula": "\\begin{align*} \\left ( f _ { n } \\left ( g _ { s ^ { \\prime } , j , k } \\right ) \\right ) \\left ( T _ { b _ { j } } \\left ( v \\right ) \\right ) = T _ { b _ { k } } \\left ( v \\right ) \\quad \\quad \\left ( f _ { n } \\left ( g _ { s ^ { \\prime \\prime } , k , i } \\right ) \\right ) \\left ( T _ { b _ { k } } \\left ( v \\right ) \\right ) = T _ { b _ { i } } \\left ( v \\right ) . \\end{align*}"}
+{"id": "5527.png", "formula": "\\begin{align*} \\sigma _ j : = \\Psi ( g _ j ) ^ { - 1 } \\circ \\sigma \\circ \\Psi ( g _ j ) . \\end{align*}"}
+{"id": "4432.png", "formula": "\\begin{align*} \\lim _ { a \\to \\infty } \\sup _ { u \\in X } \\Vert W ( u - \\tau _ a \\circ u ) \\Vert _ { L _ p } \\leq C \\lim _ { a \\to \\infty } \\sup _ { u \\in X } \\Vert u - \\tau _ a \\circ u \\Vert _ { L _ p } = 0 . \\end{align*}"}
+{"id": "3954.png", "formula": "\\begin{align*} \\Omega _ n = \\left \\{ ( x _ { 1 } , x _ { 2 } , \\dots , x _ { n } ) : \\sum _ { j = 1 } ^ { n } j x _ { j } = n , \\ x _ { j } \\in \\mathbb { N } \\cup \\{ 0 \\} \\right \\} . \\end{align*}"}
+{"id": "1806.png", "formula": "\\begin{align*} x \\triangleright \\lbrack y , z ] & = [ x \\triangleright y , z ] + [ y , x \\triangleright z ] , \\\\ \\lbrack x , y ] \\triangleright z & = a s _ { \\triangleright } ( x , y , z ) - a s _ { \\triangleright } ( y , x , z ) . \\end{align*}"}
+{"id": "6542.png", "formula": "\\begin{align*} \\pi _ u = \\{ v + g ^ 0 , v + g ^ 1 , \\cdots , v + g ^ { n - 2 } \\} . \\end{align*}"}
+{"id": "3080.png", "formula": "\\begin{align*} a _ k = \\sum _ { i \\in I } a _ { i , k } = \\sum _ { \\substack { i \\in I \\\\ \\ell _ i \\leq k } } a _ { i , k } \\end{align*}"}
+{"id": "5227.png", "formula": "\\begin{align*} g ( v ^ { \\Omega _ { \\tau , \\sigma , J , x } } ) _ \\tau = ( v ^ { \\Omega _ { \\tau , \\sigma , J , x } } \\omega _ \\tau ^ { \\Omega _ { \\tau , \\sigma , J , x } } ( g ) ) _ \\tau = \\omega _ { \\sigma , J , x } ( g ) ( v ^ { \\Omega _ { \\tau , \\sigma , J , x } } ) _ \\tau , \\end{align*}"}
+{"id": "5870.png", "formula": "\\begin{align*} \\begin{array} { r l } F _ 1 : = & \\max ( b , ~ - 5 + 2 b , ~ - 3 + a + 2 b , ~ 2 a + b , ~ 4 a + 2 b ) , \\\\ F _ 2 : = & \\max ( b , ~ - 2 + 2 a + b , ~ - 2 + 3 a + 2 b , ~ - 4 + 4 a + 2 b ) , \\end{array} \\end{align*}"}
+{"id": "5242.png", "formula": "\\begin{align*} p _ N ^ \\beta ( \\theta _ 1 , \\ldots , \\theta _ N ) = \\frac { 1 } { Z _ { N , \\beta } } \\prod _ { 1 \\leq j < k \\leq N } \\left | e ^ { i \\theta _ j } - e ^ { i \\theta _ k } \\right | ^ \\beta , \\ \\ \\ \\end{align*}"}
+{"id": "8222.png", "formula": "\\begin{align*} X ( T ) : = \\Vert \\sigma \\Vert _ { \\widetilde { L } ^ { \\infty } _ { T } ( \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } ) } + \\Vert u \\Vert _ { \\widetilde { L } ^ { \\infty } _ { T } ( \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } ) } + \\Vert \\sigma \\Vert _ { L ^ { 1 } _ { T } ( \\widetilde { B } ^ { \\frac { N } { 2 } + 1 , \\frac { N } { 2 } + 2 - \\alpha } ) } + \\Vert u \\Vert _ { L ^ { 1 } _ { T } ( \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } ) } \\end{align*}"}
+{"id": "3468.png", "formula": "\\begin{align*} \\mathbb { I } _ { ( \\Omega ) } \\Big [ \\psi \\Big ] ( \\mathrm { x } , t ) = \\Big ( \\overline { \\mathbb { I } } _ { ( B ) } \\Big [ \\hat { \\psi } \\Big ] \\Big ) ^ \\vee , \\end{align*}"}
+{"id": "1675.png", "formula": "\\begin{align*} \\| \\mathtt { y } \\| = \\| \\mathtt { y } ^ * \\mathtt { y } \\| ^ { \\frac { 1 } { 2 } } = \\mbox { \\small $ \\frac { 1 } { 2 } $ } \\left \\| \\left [ Q ^ { \\perp } - Q \\right ] \\mathtt { P } ^ * Q ^ { \\perp } \\mathtt { P } \\left [ Q ^ { \\perp } - Q \\right ] + Q ^ { \\perp } \\mathtt { P } ^ * Q \\mathtt { P } Q ^ { \\perp } \\right \\| ^ { \\frac { 1 } { 2 } } \\leq \\mbox { \\small $ \\frac { 1 } { 2 } $ } \\ , ( 1 + 1 ) ^ { \\frac { 1 } { 2 } } = 2 ^ { - \\frac { 1 } { 2 } } \\ , . \\end{align*}"}
+{"id": "982.png", "formula": "\\begin{align*} a _ { k - j , 0 } & = \\frac { \\Gamma \\bigl ( - w _ 2 + k \\bigr ) \\Gamma \\bigl ( 0 \\bigr ) \\Gamma \\bigl ( w _ 2 + \\frac { n } { 2 } \\bigr ) } { \\Gamma \\bigl ( - w _ 2 + j \\bigr ) \\Gamma \\bigl ( - j \\bigr ) \\Gamma \\bigl ( w _ 2 + \\frac { n - 2 k } { 2 } \\bigr ) } \\\\ & = ( - 1 ) ^ j j ! \\frac { \\Gamma \\bigl ( - w _ 2 + k \\bigr ) \\Gamma \\bigl ( w _ 2 + \\frac { n } { 2 } \\bigr ) } { \\Gamma \\bigl ( - w _ 2 + j \\bigr ) \\Gamma \\bigl ( w _ 2 + \\frac { n - 2 k } { 2 } \\bigr ) } . \\end{align*}"}
+{"id": "2627.png", "formula": "\\begin{align*} \\mu | _ { 2 , [ m - e _ j , m + e _ 1 ] } ^ * = \\phi ^ j = \\lambda _ x | _ { 2 , [ m - e _ j , m + e _ 1 ] } ^ * \\end{align*}"}
+{"id": "4596.png", "formula": "\\begin{align*} \\abs { \\sum _ { l = n _ 0 } ^ { n } \\frac { e ^ { 2 \\pi i \\theta ( l ) } } { l - b } } \\leq \\frac { C ( E , K ) } { n _ 0 - b } . \\end{align*}"}
+{"id": "2355.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in \\mathfrak { S } _ { n + 1 } } \\mathrm { s g n } ( \\sigma ) f _ { b _ { 1 } } ( x _ { a _ { \\sigma ( 1 ) } } , \\ , x _ { a _ { \\sigma ( 2 ) } } x _ { b _ { 2 } } u _ { \\sigma } ) = \\sum _ { \\sigma \\in \\mathfrak { S } _ { n + 1 } } \\mathrm { s g n } ( \\sigma ) x _ { b _ { 1 } } x _ { a _ { \\sigma ( 2 ) } } f _ { b _ { 2 } } ( x _ { a _ { \\sigma ( 1 ) } } , u _ { \\sigma } ) = 0 , \\end{align*}"}
+{"id": "6760.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t \\psi _ t ( x ) & = h ( t ) \\psi _ t ( x ) , \\\\ [ 2 m m ] i \\partial _ t \\varphi _ t ( k ) & = \\ \\ \\varphi _ t ( k ) + \\sqrt { \\alpha } \\abs { k } ^ { - 1 } \\int d x \\ , e ^ { - 2 \\pi i k x } \\abs { \\psi _ t ( x ) } ^ 2 \\end{cases} \\end{align*}"}
+{"id": "5415.png", "formula": "\\begin{align*} \\max _ { i , j } | G _ { i j } ( t , z ) - \\delta _ { i j } m _ { s c } ( z ) | \\prec \\Psi : = \\frac { 1 } { N \\eta } , N ^ { - \\frac { 1 } { 3 } + \\epsilon } \\leq \\Psi \\leq N ^ { - \\epsilon } , \\end{align*}"}
+{"id": "3379.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { a \\in \\sigma ( f ^ { - n } ( V ) ) } \\mathrm { V o l } ( \\Gamma _ n ( a ) ) d a & \\ge \\int _ { a \\in \\sigma ( f ^ { - n } ( V ) ) } \\mathrm { C a r d } \\left ( \\Psi ^ j _ n \\cap \\sigma ^ { - 1 } ( a ) \\right ) d a \\\\ & = \\mathrm { V o l } ( \\sigma ( \\Psi ^ j _ n ) ) \\ge e ^ { n ( h _ \\nu - 2 \\varepsilon ) } e ^ { - n ( 2 m _ j \\chi ^ + _ j + \\ldots + 2 m _ q \\chi ^ + _ q ) } e ^ { - 8 k n \\varepsilon } . \\end{aligned} \\end{align*}"}
+{"id": "7027.png", "formula": "\\begin{align*} \\alpha _ { \\pm } : = \\sqrt [ 3 ] { \\frac { 1 } { 2 } ( 3 \\sqrt { 6 9 } + 1 1 ) } \\pm \\sqrt [ 3 ] { \\frac { 1 } { 2 } ( 3 \\sqrt { 6 9 } - 1 1 ) } , \\end{align*}"}
+{"id": "1208.png", "formula": "\\begin{align*} A & = \\left [ \\begin{smallmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 1 & 1 \\end{smallmatrix} \\right ] , & b & = \\left [ \\begin{smallmatrix} 0 \\\\ 1 \\\\ 0 \\end{smallmatrix} \\right ] . \\end{align*}"}
+{"id": "6202.png", "formula": "\\begin{align*} \\bigoplus ^ { m } _ { d = 0 } ( m - d - \\lfloor \\frac { m - d + 1 } { 2 } \\rfloor + 1 ) \\odot \\mathbb { C } ^ { ( d + 1 ) \\times ( d + 1 ) } . \\end{align*}"}
+{"id": "2329.png", "formula": "\\begin{align*} \\zeta ( \\overbrace { 2 , 2 \\ldots , 2 } ^ { n } ) = \\frac { \\pi ^ { 2 n } } { ( 2 n + 1 ) ! } \\end{align*}"}
+{"id": "7690.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 1 + } \\| f \\| _ { \\alpha , \\alpha p } = \\| f \\| _ { p } . \\end{align*}"}
+{"id": "5563.png", "formula": "\\begin{align*} \\dfrac { 1 } { \\Phi _ 1 [ \\alpha _ 0 , \\beta _ 0 , L _ 1 ( 0 ) , N _ 1 ( 0 ) ] } = \\dfrac { 2 l _ b \\gamma _ b ( \\alpha _ 0 ) ^ { \\nu + 1 } } { \\theta _ b - \\theta _ m } \\end{align*}"}
+{"id": "7460.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\dot { u } ( t ) & = & A u ( t ) , & t > 0 , \\\\ u ( 0 ) & = & u _ 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "6564.png", "formula": "\\begin{align*} \\alpha = \\begin{cases} \\frac { t - s } { 2 } , \\quad s , t \\\\ [ 2 m m ] \\frac { s + t + 1 } { 2 } \\quad s t \\\\ [ 2 m m ] \\frac { - t - s - 1 } { 2 } , \\quad s t \\\\ [ 2 m m ] \\frac { s - t } { 2 } , \\quad s , t \\\\ [ 2 m m ] \\end{cases} . \\end{align*}"}
+{"id": "2590.png", "formula": "\\begin{align*} \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) ) } ^ { N , i } : = k _ i ^ N ( t , \\nu ^ { N , 1 } _ { \\boldsymbol { x } } , \\ldots , \\nu ^ { N , N } _ { \\boldsymbol { x } } , \\lambda ^ { N , 1 } _ { \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) } , \\ldots , \\lambda ^ { N , N } _ { \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) } ) , \\quad \\mu ^ i : = k \\big ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } , \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } ) \\big ) , \\end{align*}"}
+{"id": "6170.png", "formula": "\\begin{align*} V _ i V _ { ( i ) } = V _ i V _ 1 V _ 2 \\cdots V _ { i - 1 } V _ { i + 1 } \\cdots V _ d = q ( i , 1 ) q ( i , 2 ) \\cdots q ( i , i - 1 ) V . \\end{align*}"}
+{"id": "3386.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\ell r _ j s _ { n - j } = 0 . \\end{align*}"}
+{"id": "1559.png", "formula": "\\begin{align*} \\tilde { \\rho } ( f _ r ) = ( y _ { r , i , j } ) _ { 1 \\le i , j \\le n } . \\end{align*}"}
+{"id": "6530.png", "formula": "\\begin{align*} H = i S + \\pi E _ { - 1 } , \\end{align*}"}
+{"id": "567.png", "formula": "\\begin{align*} M _ n ^ \\psi ( Q ) = & \\sum _ { i = 1 } ^ n \\int _ { \\R } V ( x ) \\tilde { Q } _ { i } ( d x ) + \\sum _ { i , j = 1 } ^ n J _ { \\pi _ n ( i ) \\pi _ n ( j ) } \\int _ { \\R } \\int _ { \\R } \\psi ( x , y ) \\tilde { Q } _ { i } ( d x ) \\tilde { Q } _ { j } ( d y ) - \\sum _ { i = 1 } ^ n H ( \\tilde { Q } _ { i } ) , \\end{align*}"}
+{"id": "596.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { 2 } \\Big ( \\psi ^ { + 0 } _ { n , p } \\psi ^ { + 0 } _ { n - 1 , p } \\rho ^ { 0 + } _ { n - 2 , p } - 2 \\psi ^ { + 0 } _ { n , p } \\rho ^ { 0 + } _ { n - 1 , p } \\psi ^ { + 0 } _ { n - 1 , p - 1 } + \\rho ^ { 0 + } _ { n , p } \\psi ^ { + 0 } _ { n , p - 1 } \\psi ^ { + 0 } _ { n - 1 , p - 1 } \\Big ) & = & - \\varphi ^ { + + } _ { n , p } \\psi ^ { + 0 } _ { n - 1 , p - 1 } \\\\ & & = & - \\psi ^ { + 0 } _ { n , p } \\varphi ^ { + + } _ { n - 1 , p } \\ , . \\end{aligned} \\end{align*}"}
+{"id": "325.png", "formula": "\\begin{align*} C _ i \\dot { V } _ i = - g _ { L , i } ( V _ i - V _ L ) + \\sum _ u I _ { u , i } + I _ { e x t , i } , V _ i < V _ { t h , i } \\end{align*}"}
+{"id": "3156.png", "formula": "\\begin{align*} \\left \\langle u _ { 1 } , u _ { 2 } \\right \\rangle _ { H _ { \\lambda } } = \\int _ { \\mathbb { B } ^ { N } } u _ { 1 } ^ { p } u _ { 2 } \\mathrm { ~ d } V _ { \\mathbb { B } ^ { N } } = \\int _ { \\mathbb { B } ^ { N } } u _ { 2 } ^ { p } u _ { 1 } \\mathrm { ~ d } V _ { \\mathbb { B } ^ { N } } , \\ ; \\left \\| u _ { i } \\right \\| _ { H _ { \\lambda } } = \\int _ { \\mathbb { B } ^ { N } } u _ { i } ^ { p + 1 } \\mathrm { ~ d } V _ { \\mathbb { B } ^ { N } } , \\ ; i = 1 , 2 . \\end{align*}"}
+{"id": "4963.png", "formula": "\\begin{align*} \\begin{aligned} \\min \\limits _ { w _ { c } , r } & [ w _ { c } , r ] \\\\ s . t . & { \\rm C _ { 1 } : } \\ w _ { c } \\geq 2 \\\\ & { \\rm C _ { 2 } : } \\ 0 \\textless r \\textless 1 \\\\ & { \\rm C _ { 3 } : } \\ R _ { \\rm F A } ( \\lambda ^ { \\star } , w _ { c } , r ) \\leq \\tau \\end{aligned} \\end{align*}"}
+{"id": "8049.png", "formula": "\\begin{align*} s _ { { \\textrm { i n d } , m } _ { n _ r } } ( t ) = s _ { { \\textrm { i n d } , m , 2 } _ { n _ r } } ( t ) + s _ { { \\textrm { i n d } , m , 3 } _ { n _ r } } ( t ) + s _ { { \\textrm { i n d } , m , 4 } _ { n _ r } } ( t ) . \\end{align*}"}
+{"id": "5001.png", "formula": "\\begin{align*} & \\Pr \\bigg ( S _ N \\geq \\frac { n N } { 2 } + 2 C \\sqrt { N } n \\bigg ) = \\Pr \\bigg ( \\frac { S _ N - n N / 2 } { \\sigma \\sqrt { N } } \\geq \\frac { 2 C \\sqrt { N } n } { \\sigma \\sqrt { N } } \\bigg ) \\\\ & \\leq \\Pr \\bigg ( \\frac { S _ N - n N / 2 } { \\sigma \\sqrt { N } } \\geq 4 \\sqrt { 3 } C \\bigg ) = 1 - \\Phi ( 4 \\sqrt { 3 } C ) + O \\bigg ( \\frac { 1 } { \\sqrt { N } } \\bigg ) . \\end{align*}"}
+{"id": "1470.png", "formula": "\\begin{align*} g ( x ) = \\begin{cases} x \\ ! \\ ! \\ ! \\ ! \\mod y , & \\\\ y , & \\end{cases} \\end{align*}"}
+{"id": "6884.png", "formula": "\\begin{align*} \\kappa _ p ^ { \\alpha _ g \\alpha _ h } = \\kappa ^ \\alpha _ { \\psi , p } - \\kappa ^ \\alpha _ { \\xi , p } , \\kappa _ p ^ { \\beta _ g \\beta _ h } = \\kappa ^ \\alpha _ { \\psi , p } + \\kappa ^ \\alpha _ { \\xi , p } . \\end{align*}"}
+{"id": "5963.png", "formula": "\\begin{align*} f ^ { ( H , \\alpha ) } _ K : = f ^ { ( H ) } _ K \\end{align*}"}
+{"id": "6732.png", "formula": "\\begin{align*} \\mathcal { E } _ { N } ( t ) \\mid _ { t = 0 } \\leq C \\eta ^ { 2 } _ { 0 } \\varepsilon ^ { 2 } . \\end{align*}"}
+{"id": "7998.png", "formula": "\\begin{align*} T _ w = T _ p ^ * T _ p \\end{align*}"}
+{"id": "5653.png", "formula": "\\begin{align*} H ( f ) ( X , Y ) = X Y ( f ) - D ^ { \\nabla f } _ X Y ( f ) = \\left ( \\frac { \\partial ^ 2 f } { \\partial x ^ i \\partial x ^ j } - \\hat { \\mathbb { G } } ^ k _ { i j } ( \\nabla f ) \\frac { \\partial f } { \\partial x ^ k } \\right ) X ^ i Y ^ j . \\end{align*}"}
+{"id": "1207.png", "formula": "\\begin{align*} f = \\bar { V } _ 2 \\oplus \\bar { V } _ 0 \\bar { V } _ 1 \\oplus \\bar { V } _ 0 \\bar { V } _ 2 \\oplus \\bar { V } _ 1 \\bar { V } _ 2 = m _ { 3 } + m _ { 4 } + m _ { 2 } + m _ { 1 } \\end{align*}"}
+{"id": "2049.png", "formula": "\\begin{align*} ( - 1 ) ^ { n } \\mu _ n = ( n + 1 ) ! \\ , a _ { n + 1 } - \\sum _ { k = 0 } ^ { n - 1 } ( - 1 ) ^ { k } b _ { n , k } \\ , \\mu _ k \\end{align*}"}
+{"id": "4729.png", "formula": "\\begin{align*} \\mathfrak { B } _ d ( ( \\partial _ k + \\eth _ k ^ * ) ( a + a ^ * ) , b + b ^ * ) & = \\mathfrak { B } _ d ( \\partial _ k ( a ) + \\eth _ k ^ * ( a ^ * ) , b + b ^ * ) = \\langle \\partial _ k ( a ) , b ^ * \\rangle + \\langle \\eth _ k ^ * ( a ^ * ) , b \\rangle \\\\ & = \\langle a , \\partial _ k ^ * ( b ^ * ) \\rangle + \\langle a ^ * , \\eth _ k ( b ) \\rangle = \\mathfrak { B } _ d ( a + a ^ * , ( \\eth _ k + \\partial _ k ^ * ) ( b + b ^ * ) ) . \\end{align*}"}
+{"id": "1358.png", "formula": "\\begin{align*} { \\mathcal { K } } = \\N . \\end{align*}"}
+{"id": "5139.png", "formula": "\\begin{align*} \\psi _ i ( x ) = 1 - \\sum _ { j \\ne i } \\psi _ j ( x ) = 1 x \\in \\frac 1 2 Q _ i . \\end{align*}"}
+{"id": "4217.png", "formula": "\\begin{align*} \\cos ( \\phi - ( X - 1 ) \\theta ) u ^ { X - 1 } = \\frac { e ^ { i \\phi } } { 2 } ( e ^ { - i \\theta } u ) ^ { X - 1 } + \\frac { e ^ { - i \\phi } } { 2 } ( e ^ { i \\theta } u ) ^ { X - 1 } , \\end{align*}"}
+{"id": "6371.png", "formula": "\\begin{align*} \\mathcal { A } _ M : = \\{ \\phi \\in \\mathcal { A } : \\phi ( \\omega ) \\in S _ M , \\ \\mathbb { P } \\textit { - } \\mathrm { a . s . } \\} . \\end{align*}"}
+{"id": "7832.png", "formula": "\\begin{gather*} f ^ { ( d ) } \\ ; : = \\ ; \\big ( \\partial _ s ^ d \\ , f _ s \\big ) _ { s = 0 } \\end{gather*}"}
+{"id": "8042.png", "formula": "\\begin{align*} s _ { \\textrm { d i r } _ { n _ r } } ( t ) & = \\sum _ { n _ t = 1 } ^ { N _ t } \\alpha _ { 1 } x _ { n _ t } ( t - \\tau _ { \\textrm { d i r } , n _ r , n _ t } ) e ^ { \\mathrm { j } 2 \\pi { f _ c } ( t - \\tau _ { \\textrm { d i r } , n _ r , n _ t } ) } e ^ { \\mathrm { j } 2 \\pi { f _ { D , 1 } } ( t - \\tau _ { \\textrm { d i r } , n _ r , n _ t } ) } . \\end{align*}"}
+{"id": "1202.png", "formula": "\\begin{align*} \\pi _ { ( D , b ) } & = \\pi _ { ( d i a g ( D _ 1 , \\ldots , D _ l ) , b ) } = \\\\ & = \\pi _ { ( d i a g ( P _ 1 U _ 1 L _ 1 , \\ldots , P _ l U _ l L _ l ) , b ) } = \\\\ & = \\pi _ { ( P U L , b ) } = \\\\ & = \\pi _ { ( L , b ) } \\circ \\pi _ { ( U , 0 ) } \\circ \\pi _ { ( P , 0 ) } , \\end{align*}"}
+{"id": "5324.png", "formula": "\\begin{align*} \\varphi ( v _ { t } ) = \\varphi ( v _ { 0 } T _ { i _ { 1 } } \\dotsb T _ { i _ { r } } ) = \\varphi ( v _ { 0 } ) T _ { i _ { 1 } } \\dotsb T _ { i _ { r } } = w _ { 0 } \\beta _ { i _ { 1 } } \\dotsb \\beta _ { i _ { r } } . \\end{align*}"}
+{"id": "4903.png", "formula": "\\begin{align*} f _ j ( q , x , y ) = \\frac { j ! ( q + 1 ) _ j } { ( x ) _ j ( y ) _ j } . \\end{align*}"}
+{"id": "303.png", "formula": "\\begin{align*} \\tau _ { D ' } ^ { - 1 } ( T ^ { * } _ { X } X ( \\log D ' ) ) & = \\bigcup _ { I '' \\subset I ' - \\{ r \\} } T ^ { * } _ { D _ { I '' } } X \\cup \\bigcup _ { \\substack { I '' \\subset I ' \\\\ r \\in I '' } } T ^ { * } _ { D _ { I '' } } X \\\\ & = \\tau _ { E ' } ^ { - 1 } ( T ^ { * } _ { X } X ( \\log E ' ) ) \\cup i _ { r \\circ } \\tau _ { i _ { r } ^ { * } E ' } ^ { - 1 } ( T ^ { * } _ { D _ { r } } D _ { r } ( \\log i _ { r } ^ { * } E ' ) ) . \\end{align*}"}
+{"id": "1674.png", "formula": "\\begin{align*} \\eta ( \\mathsf { I } _ { \\mathsf { F } - 1 } , \\mathsf { I } _ { \\mathsf { F } } ) = \\eta ( \\mathsf { J } _ { \\mathsf { F } - 1 } , \\mathsf { B } ) = 1 - \\kappa ^ 2 _ { \\mathsf { B } } \\kappa ^ { - 2 } _ { \\mathsf { J } _ { \\mathsf { F } - 1 } } \\geq 1 - \\kappa ^ 2 _ { \\mathsf { J } _ { \\mathsf { F } } } \\kappa ^ { - 2 } _ { \\mathsf { J } _ { \\mathsf { F } - 1 } } = \\eta ( \\mathsf { J } _ { \\mathsf { F } - 1 } , \\mathsf { J } _ { \\mathsf { F } } ) \\end{align*}"}
+{"id": "4046.png", "formula": "\\begin{align*} P _ n ( \\mathcal { L } _ s q ) = \\left ( 1 - \\frac { n } { 2 k } \\right ) q _ n + ( 1 - \\frac { 1 } { k } ) ( n + 1 ) ( n + 2 ) I ^ { - 2 } q _ { n + 2 } . \\end{align*}"}
+{"id": "7335.png", "formula": "\\begin{align*} \\partial ( x , y ) & = | x \\cup y - x \\cap y | = | x | + | y | - 2 | x \\cap y | \\ \\ \\ \\ \\ \\ \\ \\ . \\end{align*}"}
+{"id": "5265.png", "formula": "\\begin{align*} & j ( p _ 1 , \\ldots , p _ m ; t _ 1 , \\ldots , t _ p ) : = \\\\ & \\min \\left ( 1 , \\ \\max \\left ( 0 , \\sum _ { i = 1 } ^ { p _ 1 } t _ i , \\sum _ { i = 1 } ^ { p _ 1 + p _ 2 } t _ i , \\ldots , \\sum _ { i = 1 } ^ { p _ 1 + \\ldots + p _ { m - 1 } } t _ i \\right ) + \\max \\left ( 0 , \\sum _ { i = 1 } ^ { p _ 1 } ( - t _ i ) , \\ldots , \\sum _ { i = 1 } ^ { p _ 1 + \\ldots + p _ { m - 1 } } ( - t _ i ) \\right ) \\right ) . \\end{align*}"}
+{"id": "1070.png", "formula": "\\begin{align*} \\Delta _ { h } = h ^ { - 1 } \\Delta \\ , h ^ { - 1 } , \\end{align*}"}
+{"id": "3380.png", "formula": "\\begin{align*} \\mathcal { J } : = \\{ \\gamma \\in \\mathcal { O } ( M , \\mathbb { C } ^ k ) : \\gamma ( \\lambda ) \\in J _ \\lambda \\ , \\forall \\lambda \\in M \\} \\end{align*}"}
+{"id": "2948.png", "formula": "\\begin{align*} & y _ k \\in \\Upsilon ( \\bar x + t _ k u _ k , h ( \\bar x ) + t _ k \\mu _ k ) ) = \\Psi ( \\bar x + t _ k u _ k ) , \\\\ & \\norm { y _ k - y } \\leq t _ k \\kappa \\norm { ( u _ k , \\mu _ k ) } \\end{align*}"}
+{"id": "3227.png", "formula": "\\begin{align*} I _ { \\psi } ( u , v ) : = \\int _ { X } \\psi ( u - v ) ( \\theta ^ { n } _ { u } + \\theta ^ { n } _ { v } ) \\end{align*}"}
+{"id": "16.png", "formula": "\\begin{align*} \\overline { T } _ { n } [ { \\bf \\Lambda } ] = \\lambda S _ n [ 1 ] , \\end{align*}"}
+{"id": "4038.png", "formula": "\\begin{align*} \\psi ( d _ 0 , . . . , d _ { 2 k - 1 } , y , s _ 0 ) = \\sum _ { i = 0 } ^ { 2 k - 1 } d _ i I ^ { - \\delta } ( s _ 0 ) y ^ i , \\end{align*}"}
+{"id": "6041.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u - k ^ 2 u = g \\ , \\ , \\ , \\ , \\Omega \\backslash K \\\\ [ 0 . 4 c m ] u = 0 \\ , \\ , \\ , \\ , \\partial K \\\\ [ 0 . 4 c m ] u = 0 \\ , \\ , \\ , \\ , \\partial \\Omega \\end{cases} \\end{align*}"}
+{"id": "3106.png", "formula": "\\begin{align*} \\sigma _ 2 = \\prod _ { n = 1 } ^ { \\infty } \\alpha _ { 6 n - 5 } \\prod _ { n = 1 } ^ { \\infty } \\gamma _ { 6 n - 3 } = ( 1 , 2 ) ( 3 , 4 , 5 , 6 ) ( 7 , 8 ) ( 9 , 1 0 , 1 1 , 1 2 ) \\cdots . \\end{align*}"}
+{"id": "4200.png", "formula": "\\begin{align*} \\Phi ( D ) : = \\mathcal { N } D \\prod _ { \\substack { P \\mid D \\\\ P } } \\left ( 1 - \\frac { 1 } { \\mathcal { N } P } \\right ) \\end{align*}"}
+{"id": "3466.png", "formula": "\\begin{align*} \\gamma ^ { \\textbf { i n t } } _ { 1 , \\mathrm { x } } \\mathcal { V } _ { \\Omega \\times \\mathbb { R } } \\Big [ \\psi \\Big ] ( \\mathrm { x } , t ) = \\delta \\Big ( \\gamma ^ { \\textbf { i n t } } _ { 1 , \\xi } \\overline { \\mathcal { V } } _ { B \\times \\mathbb { R } } \\Big [ \\hat { \\psi } \\Big ] \\Big ) ^ \\vee \\end{align*}"}
+{"id": "2437.png", "formula": "\\begin{align*} I _ { C _ { 1 } } ^ { \\mathfrak { m } } ( 1 ' ; a _ { i + 1 } , \\dots , a _ { j } ; 1 ' ) = \\begin{cases} \\frac { \\mu ^ { j - i } } { ( j - i ) ! } & a _ { i + 1 } = \\cdots = a _ { j } = 1 , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "617.png", "formula": "\\begin{align*} \\widetilde { \\varphi } ^ { H } _ { n , p } & = \\varphi ^ { H } _ { n , p } - \\psi _ { n , p } \\ , , \\\\ \\widetilde { \\varphi } ^ { V } _ { n , p } & = \\varphi ^ { V } _ { n , p } - \\rho _ { n , p } \\ , . \\end{align*}"}
+{"id": "5753.png", "formula": "\\begin{align*} \\tilde f ( X _ 0 , X ) = X _ 0 ^ 2 f ( X / X _ 0 ) , \\quad X \\in \\mathbb { R } ^ { m } , \\ X _ 0 \\neq 0 \\in \\mathbb { R } . \\end{align*}"}
+{"id": "3876.png", "formula": "\\begin{align*} \\rho ( X ) \\stackrel { } { = } 1 + \\sqrt { \\frac { \\alpha _ n } { 4 n } } + \\frac { 1 } { \\sqrt { 4 n \\alpha _ n } } G _ n , \\alpha _ n : = \\log n - 2 \\log \\log n - \\log ( 2 \\pi ) , \\end{align*}"}
+{"id": "5860.png", "formula": "\\begin{align*} { } f ^ p ( b e r A ) \\Big ( 1 + 2 ^ { p - 1 } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } \\Big ) & < \\Big ( \\frac { p } { p - 1 } \\Big ) ^ { p } b e r ( f ^ p ( A ) ) . \\end{align*}"}
+{"id": "4625.png", "formula": "\\begin{align*} \\Big | \\frac { g ( \\epsilon ) - g ( 0 ) } { \\epsilon } \\Big | = \\Big | \\frac { | \\nabla u + \\epsilon \\nabla h | - | \\nabla u | } { \\epsilon } \\Big | \\le | \\nabla h | \\le | \\nabla u | + | \\nabla h | . \\end{align*}"}
+{"id": "1401.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } I _ 0 [ u _ 0 ^ { ( j ) } , u _ 1 ^ { ( j ) } ] = I _ 0 [ u _ 0 , u _ 1 ] . \\end{align*}"}
+{"id": "3367.png", "formula": "\\begin{align*} M _ N ( z , E ) = \\prod _ { k = N } ^ 1 \\begin{pmatrix} E - \\lambda v ( z + k \\omega ) - v _ 1 ( k ) & - 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "302.png", "formula": "\\begin{align*} \\tau _ { E ' / D ' } ^ { - 1 } ( \\tau _ { E ' / D ' } ( C ) ) = C \\cup { i _ { r } } _ { \\circ } i _ { r } ^ { \\circ } C , \\end{align*}"}
+{"id": "270.png", "formula": "\\begin{align*} f _ j ( z ) = | z - E _ j | ^ \\gamma \\ , \\chi ( \\frac { z - E _ j } { \\theta } ) + | 2 \\theta | ^ \\gamma \\ , \\left ( 1 - \\chi ( \\frac { z - E _ j } { \\theta } ) \\right ) , \\theta = \\lambda / N , \\end{align*}"}
+{"id": "6712.png", "formula": "\\begin{align*} \\Theta : = \\varepsilon \\partial _ { t } G + \\varepsilon P _ { 1 } ( v \\cdot \\nabla _ { x } G ) - \\varepsilon ( E + v \\times B ) \\cdot \\nabla _ { v } G - Q ( G , G ) . \\end{align*}"}
+{"id": "8118.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { - \\ln \\| s _ 1 \\wedge \\cdots \\wedge s _ { r _ n } \\| _ { n \\varphi , \\det } } { n ^ 2 / 2 } = ( ( L , \\varphi ) \\cdot ( L , \\varphi ) ) ' , \\end{align*}"}
+{"id": "5472.png", "formula": "\\begin{align*} { } _ 3 \\phi _ 2 \\begin{bmatrix} q ^ { - N } , & q , & a q & ; q , q \\\\ b q , & q ^ { 1 - N } / t \\end{bmatrix} = F _ N ( a , b ; t ) , \\end{align*}"}
+{"id": "3860.png", "formula": "\\begin{align*} \\frac { 1 } { n ^ 2 } \\sum _ { a = 1 } ^ { n } \\sum _ { B = n + 1 } ^ { 2 n } G _ { a B } G _ { B \\bar a } = \\frac { 1 } { n ^ 2 } \\sum _ { a = 1 } ^ { n } \\sum _ { B \\neq \\bar { a } } G _ { a B } G _ { B \\bar a } + \\frac { 1 } { n ^ 2 } \\sum _ { a = 1 } ^ { n } G _ { a \\bar { a } } G _ { \\bar a \\bar a } , \\end{align*}"}
+{"id": "1044.png", "formula": "\\begin{align*} { \\mathcal R } ( f ) : = \\mathcal { W } \\ , ( f \\Delta ^ { - m + 1 } ) = \\frac { n - 2 } { 1 2 } v _ { n - 1 } \\int _ { M } f R ( g ) v o l _ { g } . \\end{align*}"}
+{"id": "6812.png", "formula": "\\begin{align*} \\tilde { \\alpha } ( d , d , a _ 3 , \\dots , a _ k ) = & \\alpha ' ( d , d , a _ 3 , \\dots , a _ k ) + \\tilde { \\sigma } ( d , d , a _ 3 , \\dots , a _ k ) \\\\ = & \\sigma ' ( d , a _ 3 , \\dots , a _ k ) + \\delta ' ( d , a _ 3 , \\dots , a _ k ) + \\sigma ' ( d , a _ 3 , \\dots , a _ k ) = \\delta ' ( d , a _ 3 , \\dots , a _ k ) \\end{align*}"}
+{"id": "2314.png", "formula": "\\begin{align*} \\mathbb { G } _ { n , \\Delta } ( f ) = \\mathbb { G } _ { n \\Delta } ( f ) + \\frac { 1 } { \\sqrt { n \\Delta } } \\mathbb { A } _ { n , \\Delta } , \\end{align*}"}
+{"id": "7960.png", "formula": "\\begin{align*} \\varphi _ h ( \\xi ) = \\frac { c } { 3 } | \\xi ^ H | ^ 3 - \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle , \\end{align*}"}
+{"id": "7270.png", "formula": "\\begin{align*} H ( s , \\gamma ( s ) , \\psi ( s , \\gamma ) , e _ s \\sharp \\eta ^ * , u ^ * ( s , \\gamma ) ) = \\max _ { u \\in U } H ( s , \\gamma ( s ) , \\psi ( s , \\gamma ) , e _ s \\sharp \\eta ^ * , u ) . \\end{align*}"}
+{"id": "7149.png", "formula": "\\begin{align*} \\eta _ { \\lambda } : = \\langle \\lambda , ( \\mathcal { X } ' ( d ) ^ { \\vee } ) ^ { \\lambda > 0 } \\rangle \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "5842.png", "formula": "\\begin{align*} b e r ( D ) = \\sup \\{ | \\langle D e _ i , e _ i \\rangle | : i \\in \\mathbb { Z } _ { + } \\} = \\sup \\{ | \\langle \\lambda _ i e _ { i } , e _ i \\rangle | : i \\in \\mathbb { Z } _ { + } \\} = \\sup _ { i } { | \\lambda _ i | } , \\end{align*}"}
+{"id": "4510.png", "formula": "\\begin{align*} \\dd _ { ( \\phi , \\pi ) } \\Phi _ x ( 0 , h ) = \\langle D _ 2 \\Phi _ x ( \\phi , \\pi ) , h \\rangle _ { H ^ 1 } \\ , . \\end{align*}"}
+{"id": "2460.png", "formula": "\\begin{align*} \\bigcup _ { B \\in \\mathcal { F } ^ \\prime } B \\subset V , \\mu \\left ( V \\setminus \\bigcup _ { B \\in \\mathcal { F } ^ \\prime } B \\right ) = 0 \\end{align*}"}
+{"id": "6136.png", "formula": "\\begin{align*} V _ 2 V _ 1 ^ * = q V _ 1 ^ * V _ 2 . \\end{align*}"}
+{"id": "2576.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x ^ { * , i } _ t = & ~ [ A x ^ { * , i } _ t + B \\alpha ^ { * , i } _ t ] d t + \\sigma d W ^ i _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x ^ { * , i } _ 0 = & ~ \\xi ^ i . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3389.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\ell r _ j s _ { h - g _ j } = 0 . \\end{align*}"}
+{"id": "4552.png", "formula": "\\begin{align*} H _ f ( G ) = \\sum _ { i = 1 } ^ 4 n _ { i } \\ , f ( i ) , \\end{align*}"}
+{"id": "8091.png", "formula": "\\begin{align*} { D _ T } = ( I - T ^ * T ) ^ { 1 / 2 } \\ a n d \\ { D _ { T ^ * } } = ( I - T T ^ * ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "4084.png", "formula": "\\begin{align*} H ^ 1 _ f ( G _ { \\Q } , Z ( U _ n ( x ) ) ) = W _ n \\supset W _ { n + 1 } = 0 \\end{align*}"}
+{"id": "4653.png", "formula": "\\begin{align*} \\delta _ \\chi : = \\begin{cases} 0 & \\mbox { w h e n $ \\chi $ i s e v e n , } \\\\ 1 & \\mbox { w h e n $ \\chi $ i s o d d , } \\end{cases} \\end{align*}"}
+{"id": "5206.png", "formula": "\\begin{align*} z : = n ( M + C ) + M ^ 2 + \\frac { M ( M + 1 ) } { 2 } + M C . \\end{align*}"}
+{"id": "2602.png", "formula": "\\begin{align*} \\underset { 0 \\leq t \\leq T } { \\sup } \\Bigg [ \\frac { 1 } { N } \\underset { i = 1 } { \\overset { N } { \\sum } } \\mathbb { E } ^ { \\boldsymbol { \\xi } } [ | \\phi ^ { N , i } ( t , \\boldsymbol { x } ^ * _ t ) - \\Psi ^ { N , i } ( t , \\boldsymbol { x } ^ * _ t ) | ^ 2 ] \\Bigg ] \\leq \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N ^ 2 } \\underset { i , j = 1 } { \\overset { N } { \\sum } } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 \\Big ) \\end{align*}"}
+{"id": "7443.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\tau ^ \\dagger _ \\theta = \\sum _ { k = 0 } ^ s p _ k ^ \\dagger \\sigma _ { k } ^ \\theta , & \\theta $ s $ , \\\\ \\displaystyle \\tau ^ \\dagger _ \\theta = \\sum _ { k = 1 } ^ s m _ k ^ \\dagger \\sigma _ { k } ^ \\theta , & \\end{cases} \\end{align*}"}
+{"id": "4443.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } J _ n ( v ) = J ( v ) \\end{align*}"}
+{"id": "5452.png", "formula": "\\begin{align*} G ^ i \\overline { G ^ i } = \\begin{pmatrix} I _ { n - p - q - r } & O & O & O \\\\ O & J - I _ p & \\omega ^ 2 J & \\omega J \\\\ O & \\omega J & J - I _ q & \\omega ^ 2 J \\\\ O & \\omega ^ 2 J & \\omega J & J - I _ r \\\\ \\end{pmatrix} , \\end{align*}"}
+{"id": "2863.png", "formula": "\\begin{align*} ( d / d t ) v + A v = 0 , v \\in \\R ^ D , \\end{align*}"}
+{"id": "3241.png", "formula": "\\begin{align*} I _ { \\psi } ( u , v ) = \\int _ { X } \\psi ( u - v ) ( \\theta ^ { n } _ { u } + \\theta ^ { n } _ { v } ) \\end{align*}"}
+{"id": "5973.png", "formula": "\\begin{align*} | B _ { \\delta ^ { - 1 } } | ^ { - ( k - 1 ) } \\prod _ { j = 1 } ^ k \\int _ { B _ { \\delta ^ { - 1 } } } \\Big ( \\sum _ { K _ j \\in P _ { \\delta } ( I _ j ) } | g _ { K _ j } | ^ 2 \\Big ) = \\int _ { B _ { \\delta ^ { - 1 } } } \\prod _ { j = 1 } ^ k \\Big ( \\sum _ { K _ j \\in P _ { \\delta } ( I _ j ) } | g _ { K _ j } | ^ 2 \\Big ) . \\end{align*}"}
+{"id": "3978.png", "formula": "\\begin{align*} u _ { 2 \\beta } ( x , t ) = \\frac { 1 } { 2 t ^ { \\beta } } W _ { - \\beta , 1 - \\beta } \\left ( - \\frac { | x | } { t ^ { \\beta } } \\right ) , \\ t > 0 , \\ x \\in \\mathbb { R } , \\end{align*}"}
+{"id": "7494.png", "formula": "\\begin{align*} c _ { m l } & = \\begin{cases} 1 / a _ { m m } = ( m - 2 ) ! & l = m \\ , , \\\\ - ( m - 2 ) ! \\sum _ { k = 2 } ^ { m - 1 } a _ { m k } c _ { k l } & 2 \\le l \\le m - 1 \\ , , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "7434.png", "formula": "\\begin{align*} [ H , \\ , p ^ \\dag A ] = p ^ \\dag A P . \\end{align*}"}
+{"id": "6658.png", "formula": "\\begin{align*} \\underline { Q } ( t ) = \\det \\begin{bmatrix} s _ 0 & s _ 1 & \\ldots & s _ { m - 1 } & 1 \\\\ s _ 1 & s _ 2 & \\ldots & s _ { m } & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ s _ { m } & s _ { m + 1 } & \\ldots & s _ { 2 m - 1 } & t ^ { m } \\end{bmatrix} ; \\end{align*}"}
+{"id": "3909.png", "formula": "\\begin{align*} \\| S * g ( t ) \\| ^ 2 _ { \\mathbb H ^ \\mu } = \\sum _ { n = 1 } ^ \\infty \\lambda _ n ^ \\mu \\left ( \\int _ 0 ^ t \\omega ( t - \\tau , \\lambda _ n ) g _ n ( \\tau ) d \\tau \\right ) ^ 2 , \\ ; g _ n ( \\tau ) = ( g ( \\tau ) , e _ n ) . \\end{align*}"}
+{"id": "640.png", "formula": "\\begin{align*} \\Phi ( \\mathcal { P } ( Z ) ) = \\mathcal { P } ' ( \\Phi ( Z ) ) \\end{align*}"}
+{"id": "4690.png", "formula": "\\begin{align*} r \\frac { d } { d r } \\left [ \\frac { 1 } { 1 - z \\bar { w } } \\right ] & = \\frac { z \\bar { w } } { ( 1 - z \\bar { w } ) ^ 2 } , z = r e ^ { i \\theta } , \\\\ r \\frac { d } { d r } \\left [ \\frac { 1 } { | 1 - z w | ^ { 2 \\lambda } } \\right ] & = 2 \\lambda \\frac { { \\mathrm R e } \\ , ( z w ) - | z w | ^ 2 } { | 1 - z w | ^ { 2 \\lambda + 2 } } , z = r e ^ { i \\theta } , \\\\ r \\frac { d } { d r } A & = A \\left ( 1 + 2 \\frac { { \\mathrm R e } \\ , ( z w ) - | z w | ^ 2 } { | 1 - z w | ^ { 2 } } \\right ) = A \\frac { 1 - | z w | ^ 2 } { | 1 - z w | ^ { 2 } } , z = r e ^ { i \\theta } . \\end{align*}"}
+{"id": "5537.png", "formula": "\\begin{align*} \\theta _ 2 ( \\beta ( t ) , t ) = \\theta _ m . \\end{align*}"}
+{"id": "2951.png", "formula": "\\begin{align*} \\Lambda ^ \\alpha ( \\bar x ) : = \\{ \\lambda \\in \\R ^ m \\ , | \\ , \\nabla _ x L ^ \\alpha ( \\bar x , \\lambda ) = 0 \\} . \\end{align*}"}
+{"id": "7223.png", "formula": "\\begin{align*} H ^ { s _ 2 } h ^ { s _ 1 } : = \\{ \\vec { \\phi } = \\{ \\phi _ { p } \\} : | | \\vec { \\phi } | | _ { H ^ { s _ 2 } h ^ { s _ 1 } } ^ 2 = \\sum \\limits _ { p \\in \\mathbb { Z } ^ 2 } \\langle p \\rangle ^ { 2 s _ 1 } | | \\phi _ p | | _ { H ^ { s _ 2 } } ^ { 2 } < + \\infty \\} . \\end{align*}"}
+{"id": "2908.png", "formula": "\\begin{align*} \\varphi ( A ) = \\sum _ { j = 1 } ^ d \\delta _ j ( A ) f _ j ( p ( A ) ) . \\end{align*}"}
+{"id": "5126.png", "formula": "\\begin{align*} \\ell ( \\gamma ) \\le C | x - y | , \\mathcal H ^ 1 ( \\gamma \\cap \\partial \\Omega ) = 0 . \\end{align*}"}
+{"id": "2077.png", "formula": "\\begin{align*} S M = \\begin{bmatrix} X _ 1 \\\\ Y _ 1 \\end{bmatrix} . \\end{align*}"}
+{"id": "3375.png", "formula": "\\begin{align*} a _ 1 p _ 1 + \\dots + a _ d p _ d & = a _ 1 ( \\alpha _ 1 q v _ { 1 , 1 } + \\dots + \\alpha _ d q v _ { d , 1 } ) + \\dots + a _ d ( \\alpha _ 1 q v _ { d , 1 } + \\dots + \\alpha _ d q v _ { d , d } ) \\\\ & = q \\alpha _ 1 ( a _ 1 v _ { 1 , 1 } + \\dots + a _ d v _ { 1 , d } ) + \\dots + q \\alpha _ d ( a _ 1 v _ { d , 1 } + \\dots + v _ { d , d } ) \\ , . \\end{align*}"}
+{"id": "3297.png", "formula": "\\begin{gather*} \\Lambda _ { \\{ 1 \\} } = A _ 1 , \\Lambda _ { \\{ 2 \\} } = A _ { N _ 1 } , \\Lambda _ { \\{ 3 \\} } = A _ { N _ 2 } , \\Lambda _ { \\{ 4 \\} } = A _ 0 , \\\\ \\Lambda _ { \\{ 1 , 2 \\} } = L _ 1 \\otimes 1 , \\Lambda _ { \\{ 2 , 3 \\} } = M _ 2 , \\Lambda _ { \\{ 3 , 4 \\} } = 1 \\otimes K _ 1 , \\\\ \\Lambda _ { \\{ 1 , 2 , 3 \\} } = L _ 2 , \\Lambda _ { \\{ 1 , 2 , 3 , 4 \\} } = A _ 2 , \\Lambda _ { \\{ 2 , 3 , 4 \\} } = K _ 2 . \\end{gather*}"}
+{"id": "3652.png", "formula": "\\begin{align*} \\vec { y } & = \\left ( x _ 1 , \\ldots , x _ { i - 1 } , ( x _ i + x _ j ) / 2 , 0 , x _ { i + 1 } , \\ldots , x _ { j - 1 } , 0 , ( x _ i + x _ j ) / 2 , x _ { j + 1 } , \\ldots , x _ m \\right ) , \\\\ \\vec { z } & = \\left ( x _ 1 , \\ldots , x _ { i - 1 } , 0 , ( x _ i + x _ j ) / 2 , x _ { i + 1 } , \\ldots , x _ { j - 1 } , ( x _ i + x _ j ) / 2 , 0 , x _ { j + 1 } , \\ldots , x _ m \\right ) . \\end{align*}"}
+{"id": "162.png", "formula": "\\begin{align*} \\begin{aligned} F _ \\kappa = \\sum _ { n = 0 } ^ \\infty \\kappa ^ n F _ n \\ , , \\end{aligned} \\end{align*}"}
+{"id": "5394.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d X ^ \\epsilon ( t ) = L \\Psi ( X ^ \\epsilon ( t ) ) d t + \\epsilon \\int _ { Z } f ( t , X ^ \\epsilon ( t - ) , z ) \\widetilde { N } ^ { \\epsilon ^ { - 1 } } ( d z , d t ) , \\ t \\in [ 0 , T ] , \\\\ & X ^ \\epsilon ( 0 ) = x \\in L ^ 2 ( \\mu ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "6013.png", "formula": "\\begin{align*} \\tilde { g } ^ { ( q ) } ( b ; h ) = \\tilde { \\delta } _ 1 + \\tilde { g } ^ { ( q ) } ( b ; h ) \\tilde { \\delta } _ 2 + \\tilde { \\delta } _ 3 , \\end{align*}"}
+{"id": "3792.png", "formula": "\\begin{align*} A _ { j _ 1 , \\ldots , j _ { 2 l } } = \\mu _ { j _ 1 , \\ldots , j _ { 2 l } } \\prod _ { k = 1 } ^ { 2 l } \\left [ \\frac { ( 2 j _ k + 1 ) ! } { ( j _ k ! ) ^ 2 } \\right ] . \\end{align*}"}
+{"id": "7958.png", "formula": "\\begin{align*} & { \\rm { s p a n } } \\{ Z , [ Z _ 1 , Z _ 2 ] \\ , : \\ , \\mbox { w i t h } Z , Z _ 1 , Z _ 2 \\in \\mathcal { H } ^ 0 _ \\xi \\} = \\\\ & = { \\rm { s p a n } } \\left \\{ V _ 1 , W _ 1 , \\ldots , V _ { n - 1 } , W _ { n - 1 } , \\tau - \\frac { k ( \\xi ) | \\mathcal { P } _ H ( \\nu ) | } { 2 } \\ , \\eta \\right \\} . \\end{align*}"}
+{"id": "395.png", "formula": "\\begin{align*} \\d f = \\big ( f ^ x ( x ) + x ( f ^ x ) ' ( x ) + y f ^ { x y | x } ( x ) + x y ( f ^ { x y | x } ) ' ( x ) + y f ^ { x y | y } ( y ) \\big ) \\ , \\d x \\\\ + \\big ( f ^ y ( y ) + y ( f ^ y ) ' ( y ) + x f ^ { x y | y } ( y ) + x y ( f ^ { x y | y } ) ' ( y ) + x f ^ { x y | x } ( x ) \\big ) \\ , \\d y . \\end{align*}"}
+{"id": "822.png", "formula": "\\begin{align*} \\beta v ( x ) - \\alpha x v ' ( x ) - \\frac { 1 } { 2 } \\sigma ^ 2 x ^ 2 v '' ( x ) - f ( x ) = 0 \\end{align*}"}
+{"id": "6539.png", "formula": "\\begin{align*} f _ 0 & = \\{ ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 0 ) \\} \\\\ f _ 1 & = \\{ ( 1 , 3 ) , ( 3 , 2 ) , ( 2 , 1 ) \\} \\\\ f _ 2 & = \\{ ( 0 , 2 ) , ( 2 , 3 ) , ( 3 , 0 ) \\} \\\\ f _ 3 & = \\{ ( 0 , 3 ) , ( 3 , 1 ) , ( 1 , 0 ) \\} \\end{align*}"}
+{"id": "6492.png", "formula": "\\begin{align*} | A | & = \\sum _ { Q \\in \\mathcal Q _ B } | A \\cap Q | + \\sum _ { Q \\in \\mathcal G _ L } | A \\cap Q | \\\\ & \\leq \\sum _ { Q \\in \\mathcal Q _ B } \\delta _ 1 | Q | + \\sum _ { Q \\in \\mathcal G _ L } \\delta _ 2 | Q | \\\\ & \\leq \\delta _ 1 | B | + \\delta _ 2 . \\end{align*}"}
+{"id": "4382.png", "formula": "\\begin{align*} W _ t ( x , y ) = \\sum _ { n = 0 } ^ \\infty e ^ { - ( 2 n + d ) t } \\sum _ { | \\nu | = n } \\tilde { h } _ \\nu ( x ) \\tilde { h } _ \\nu ( y ) . \\end{align*}"}
+{"id": "8250.png", "formula": "\\begin{align*} \\begin{aligned} Z ^ \\ell _ { s , \\bar { s } } ( t ) \\le C Z _ { s - 1 , \\bar { s } } ^ \\ell ( t ) + C \\int _ 0 ^ t Z _ { s , \\bar { s } } ( \\tau ) \\Psi ( t , \\tau ) \\dd \\tau , \\end{aligned} \\end{align*}"}
+{"id": "1571.png", "formula": "\\begin{align*} \\left ( z _ { 4 } s z _ { 1 } s z _ { 2 } ^ { i } \\right ) ^ { - 1 } z _ { 2 } \\left ( z _ { 4 } s z _ { 1 } s z _ { 2 } ^ { i - 1 } \\right ) & = \\left [ \\left ( z _ { 4 } s z _ { 1 } s z _ { 2 } ^ { i } \\right ) ^ { - 1 } , z _ { 2 } \\right ] \\mbox { o r } \\\\ z _ { 2 } \\left ( z _ { 4 } s z _ { 1 } s z _ { 2 } ^ { i } \\right ) ^ { - 1 } \\left ( z _ { 4 } s z _ { 1 } s z _ { 2 } ^ { i - 1 } \\right ) & = e . \\end{align*}"}
+{"id": "4162.png", "formula": "\\begin{align*} \\Phi _ k ( t ) = \\Phi ( \\chi _ k ^ { } | t | ) \\end{align*}"}
+{"id": "4415.png", "formula": "\\begin{align*} z = \\eta _ 0 + \\cdots + \\eta _ k q ^ k + \\displaystyle \\sum _ { i = 1 } ^ { \\infty } \\frac { \\mu _ i } { q ^ i } \\eta _ j , \\mu i \\in \\mathfrak { N } . \\end{align*}"}
+{"id": "2688.png", "formula": "\\begin{align*} M _ { \\varPsi } ( 2 n , j , 1 ; l - 1 ) = ( - 1 ) ^ j S ( n , l ) \\binom { n } { j } \\sum _ { v = 0 } ^ { n - j } ( - 1 ) ^ v S ( n + l - j - v , n ) \\binom { 2 ( j + v ) } { j + v } \\binom { n - j } { v } \\end{align*}"}
+{"id": "4967.png", "formula": "\\begin{align*} E _ { l } = \\sum _ { k = 1 } ^ { K } | \\mathbf { y } [ k + K ( l - 1 ) ] | ^ { 2 } = \\sum _ { u = 1 } ^ { N } a _ { u } \\vert \\vert \\mathbf { x } _ { u } \\vert \\vert _ { 2 } + K \\delta ^ { 2 } . \\end{align*}"}
+{"id": "1337.png", "formula": "\\begin{align*} \\left | B _ { 9 / 1 0 } \\cap \\left \\{ u = 0 \\right \\} \\right | \\le C \\varepsilon ^ { p ^ * } . \\end{align*}"}
+{"id": "7042.png", "formula": "\\begin{align*} R _ { \\hat { \\nu } _ p , \\hat { \\nu } _ p } = 0 . \\end{align*}"}
+{"id": "7697.png", "formula": "\\begin{align*} \\int _ { \\partial \\mathbb { D } } | \\hat f ( z ) | | d z | = 2 \\int _ { - 1 } ^ { 1 } \\frac { f ( x ) d x } { \\sqrt { 1 - x ^ 2 } } . \\end{align*}"}
+{"id": "1309.png", "formula": "\\begin{align*} ( f \\otimes g ) ^ \\dagger = f ^ \\dagger \\otimes g ^ \\dagger , \\end{align*}"}
+{"id": "6576.png", "formula": "\\begin{align*} \\sum _ i \\begin{pmatrix} E _ i & \\mathbf { 0 } \\\\ \\mathbf { 0 } & \\mathbf { 0 } \\end{pmatrix} + \\sum _ j \\begin{pmatrix} \\mathbf { 0 } & \\mathbf { 0 } \\\\ \\mathbf { 0 } & F _ j \\end{pmatrix} = I \\end{align*}"}
+{"id": "5475.png", "formula": "\\begin{align*} F _ N ( a , b ; t ) = \\frac { ( 1 - b ) ( 1 - t q ^ N ) } { ( 1 - t ) ( 1 - b q ^ N ) } \\sum _ { n = 0 } ^ { N } \\begin{bmatrix} N \\\\ n \\end{bmatrix} \\frac { ( a t q / b ) _ n ( q ) _ n ( b ) _ { N - n } b ^ n } { ( t q ) _ n ( b ) _ N } . \\end{align*}"}
+{"id": "1191.png", "formula": "\\begin{align*} \\tilde p : = p _ \\beta ( \\beta ) , \\end{align*}"}
+{"id": "2137.png", "formula": "\\begin{align*} 0 = \\mathrm { r a n k } _ { \\Lambda ( G ) } ( \\mathfrak { X } ^ { \\pm / \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) ) = \\mathrm { r a n k } _ { \\Omega ( G ) } ( \\mathcal { Y } ^ { \\pm / \\pm } ( E _ { p } / L _ \\infty ) ) \\end{align*}"}
+{"id": "3604.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ \\infty \\frac { ( \\ell + 1 ) ^ { 1 - \\delta } } { \\ell ^ 2 } \\leq \\sum _ { \\ell = 1 } ^ \\infty \\frac { \\ell ^ { 1 - \\delta } + 1 } { \\ell ^ 2 } = \\frac { \\pi ^ 2 } { 6 } + \\sum _ { \\ell = 1 } ^ \\infty \\frac { 1 } { \\ell ^ { 1 + \\delta } } < \\infty , \\end{align*}"}
+{"id": "143.png", "formula": "\\begin{align*} ( d _ i y ) ( [ \\mathbf { a } ] \\otimes [ \\mathbf { b } ] ) = 0 , \\end{align*}"}
+{"id": "1404.png", "formula": "\\begin{align*} \\Theta ( x , t ; t _ 0 ) : = t _ 0 + t + \\langle x \\rangle ^ { 2 - \\alpha } \\end{align*}"}
+{"id": "3581.png", "formula": "\\begin{align*} \\lambda = n ^ { - ~ \\tfrac { t + c } { 2 t + c + 2 ( t + c ) \\nu } } , \\end{align*}"}
+{"id": "7080.png", "formula": "\\begin{align*} t = \\frac { x ^ { i _ r } } { y ^ { i _ s } } . \\end{align*}"}
+{"id": "354.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } + \\gamma u _ t - \\Delta u = G ( t ) , ( x , t ) \\in \\Omega \\times \\mathbb { R } ^ + , \\\\ u | _ { \\partial \\Omega } = 0 , t \\in \\mathbb { R } ^ + , \\\\ \\xi _ u ( 0 ) = \\xi _ 0 , x \\in \\Omega . \\end{cases} \\end{align*}"}
+{"id": "6160.png", "formula": "\\begin{align*} \\tau _ { \\rm B C L } ( V _ 1 , V _ 2 ) = ( M _ { ( P ^ \\perp + z P ) U } , M _ { U ^ * ( P + z P ^ \\perp ) } ) \\tau _ { \\rm B C L } . \\end{align*}"}
+{"id": "4445.png", "formula": "\\begin{align*} v _ { n _ k } ( x ) : = \\begin{cases} 0 , & x \\notin \\Sigma _ { n _ k } \\\\ z _ { n _ k } , & x \\in \\Sigma _ { n _ k } . \\end{cases} \\end{align*}"}
+{"id": "3679.png", "formula": "\\begin{align*} I _ { s m a l l } = \\left \\{ i \\in [ t + 4 ] \\colon | N ( v ) \\cap V _ i | \\le \\epsilon ^ { 1 / 4 } n \\right \\} , \\quad I _ { \\emptyset } = \\left \\{ i \\in [ t + 4 ] \\colon N ( v ) \\cap V _ i = \\emptyset \\right \\} . \\end{align*}"}
+{"id": "78.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { \\Psi } w = A w & \\O \\\\ w = 0 & \\partial \\O . \\end{cases} \\end{align*}"}
+{"id": "7616.png", "formula": "\\begin{align*} p _ { 2 , \\lambda } = x _ 3 ^ 3 x _ 9 ^ 3 + x _ 4 ^ 3 x _ { 1 0 } ^ 3 + x _ 5 ^ 3 x _ { 1 1 } ^ 3 - 3 \\lambda x _ 0 x _ 1 x _ 2 x _ 9 x _ { 1 0 } x _ { 1 1 } . \\end{align*}"}
+{"id": "867.png", "formula": "\\begin{align*} \\mu ( B _ d ( x , r ) ) = c _ d r ^ Q \\end{align*}"}
+{"id": "2673.png", "formula": "\\begin{align*} \\varphi ( 2 n , m , r - 1 ) = \\sum _ { k = 0 } ^ { 2 n } ( - 1 ) ^ k \\binom { 2 n } { k } ^ m P ( k , r ) P ( 2 n - k , r ) \\end{align*}"}
+{"id": "6715.png", "formula": "\\begin{align*} \\mu ( \\theta ) = & - R \\theta \\int _ { \\mathbb { R } ^ { 3 } } \\hat { B } _ { i j } ( \\frac { v - u } { \\sqrt { R \\theta } } ) B _ { i j } ( \\frac { v - u } { \\sqrt { R \\theta } } ) \\ , d v > 0 , i \\neq j , \\\\ \\kappa ( \\theta ) = & - R ^ { 2 } \\theta \\int _ { \\mathbb { R } ^ { 3 } } \\hat { A } _ { j } ( \\frac { v - u } { \\sqrt { R \\theta } } ) A _ { j } ( \\frac { v - u } { \\sqrt { R \\theta } } ) \\ , d v > 0 , \\end{align*}"}
+{"id": "150.png", "formula": "\\begin{align*} v = [ v ] _ { \\rm e s s } + [ v ] _ { \\rm r e s } , \\end{align*}"}
+{"id": "5643.png", "formula": "\\begin{align*} V ( l ) = J ( T ( l ) ) = J ( d f ( T ( 0 ) ) ) = d f ( J ( T ( 0 ) ) ) = d f ( V ( 0 ) ) . \\end{align*}"}
+{"id": "2800.png", "formula": "\\begin{align*} J _ k = X _ { k + 1 } \\begin{bmatrix} C _ k \\\\ 0 \\end{bmatrix} Y _ k ^ T , \\check { J } _ k = \\widehat { X } _ k S _ k Y _ k ^ T , \\end{align*}"}
+{"id": "53.png", "formula": "\\begin{align*} | \\nabla f | = \\alpha ( f ) \\Delta f = \\beta ( f ) . \\end{align*}"}
+{"id": "5219.png", "formula": "\\begin{align*} \\mathring { R } = \\begin{pmatrix} D _ 1 & O _ 1 & O _ 2 \\\\ - O _ 1 & D _ 2 & O _ 3 \\\\ - O _ 2 & - O _ 3 & D _ 3 \\end{pmatrix} , \\end{align*}"}
+{"id": "919.png", "formula": "\\begin{align*} x _ n - y _ n \\geq ( 2 - \\epsilon ) b _ { n - 1 } \\mbox { o n } \\partial \\omega \\cap \\{ x _ n = 2 b _ { n - 1 } \\} . \\end{align*}"}
+{"id": "3295.png", "formula": "\\begin{gather*} [ A _ k , A _ l ] = [ A _ k , K ] = [ A _ k , L ] = 0 k , l \\in \\{ 0 , 1 , 2 , 3 , 4 , 5 \\} , \\end{gather*}"}
+{"id": "5960.png", "formula": "\\begin{align*} D _ p ( \\delta ) : = \\sup _ { \\delta _ 0 \\in q ^ { - \\N } \\cap [ \\delta , 1 ] } \\mathfrak { D } _ p ( \\delta _ 0 ) \\end{align*}"}
+{"id": "5075.png", "formula": "\\begin{align*} f ( z ) = c \\sum _ { ( \\xi _ { m + 1 } , . . . , \\xi _ { d } ) \\in \\mathbb { Z } ^ n } \\int _ { ( \\xi _ 1 , . . . , \\xi _ { m } ) \\in \\mathbb { R } ^ m } ( \\mathcal { F } f ) ( \\xi ) e ^ { i z \\cdot \\xi } \\ , d \\xi _ 1 . . . d \\xi _ m . \\end{align*}"}
+{"id": "4231.png", "formula": "\\begin{align*} \\sum _ { X = 1 } ^ \\infty F _ \\Phi ( X ) u ^ { X - 1 } = \\frac { Z ( q u ) } { Z ( u ) ( 1 - u ) } . \\end{align*}"}
+{"id": "2707.png", "formula": "\\begin{align*} \\phi ( 2 n , m , r - 1 ) = \\frac { 1 } { 4 } \\varPsi ( 2 n , m , r - 1 ) \\end{align*}"}
+{"id": "1787.png", "formula": "\\begin{align*} p _ { A \\backslash j } \\left ( W \\right ) \\left \\langle Y , B , j \\right \\rangle = 0 . \\end{align*}"}
+{"id": "1220.png", "formula": "\\begin{align*} \\mathcal { I } = \\bigcup _ { j \\in \\mathcal { I } _ { m i n } } \\{ i \\in [ N ] , j \\preceq i \\} . \\end{align*}"}
+{"id": "1075.png", "formula": "\\begin{align*} \\sigma ( P Q ) ( x , \\xi ) = \\sum \\limits _ { \\beta } \\frac { ( - i ) ^ { | \\beta | } } { | \\beta | ! } \\partial ^ { \\xi } _ { \\beta } \\sigma ( P ) ( x , \\xi ) \\partial _ { \\beta } \\sigma ( Q ) ( x , \\xi ) , \\end{align*}"}
+{"id": "803.png", "formula": "\\begin{align*} v ( x ) = A x ^ { k _ 1 } + B x ^ { k _ 2 } \\end{align*}"}
+{"id": "586.png", "formula": "\\begin{align*} j _ n ^ { ( 1 2 ) } = n + a _ { 1 2 } \\ , , \\end{align*}"}
+{"id": "5293.png", "formula": "\\begin{align*} \\partial _ t u = i \\sum _ { j = 1 } ^ d \\partial ^ 2 _ { x _ j } u \\end{align*}"}
+{"id": "508.png", "formula": "\\begin{align*} \\log \\int _ { \\R ^ n } e ^ { f ( x ) } \\ , d x - \\int _ { \\R ^ n } f \\ , d Q + H ( Q ) = H ( Q \\ , | \\ , P ) \\end{align*}"}
+{"id": "5456.png", "formula": "\\begin{align*} \\det M = \\det \\begin{pmatrix} J - A & \\omega ^ 2 J - B & \\omega J - C \\\\ \\omega J - C & J - A & \\omega ^ 2 J - B \\\\ \\omega ^ 2 J - B & \\omega J - C & J - A \\end{pmatrix} . \\end{align*}"}
+{"id": "5289.png", "formula": "\\begin{align*} U _ \\lambda ( f ) : = \\{ x : | f ( x ) | > \\lambda \\} . \\end{align*}"}
+{"id": "543.png", "formula": "\\begin{align*} \\int _ \\R ( | T _ i ( x _ i ) | + | x _ i | ) Q _ i ( x _ i ) d x _ i = \\int _ \\R | x _ i | Q ^ * _ i ( x _ i ) d x _ i + \\int _ \\R | x _ i | Q _ i ( x _ i ) d x _ i < \\infty . \\end{align*}"}
+{"id": "4665.png", "formula": "\\begin{align*} \\deg ( u ( t ) / v ( t ) ) : = \\max \\{ \\deg u ( t ) , \\deg v ( t ) \\} \\leq \\frac { g + \\ell - 1 } { g _ 0 + \\ell - 1 } , \\end{align*}"}
+{"id": "7372.png", "formula": "\\begin{align*} P _ { s } ( \\tau ) : = \\frac { \\abs { \\{ p \\in \\mathcal { P } : r _ { p , s } \\leq \\tau \\} } } { \\abs { \\mathcal { P } } } , \\end{align*}"}
+{"id": "8154.png", "formula": "\\begin{align*} \\bigoplus _ { \\begin{subarray} { c } \\boldsymbol { b } = ( b _ 1 , \\ldots , b _ r ) \\in \\mathbb N ^ r \\\\ | \\boldsymbol { b } | = b _ 1 + \\cdots + b _ r = \\delta \\end{subarray} } S ^ { b _ 1 } ( V ^ { ( 1 ) } ) \\otimes \\cdots \\otimes S ^ { b _ r } ( V ^ { ( r ) } ) \\longrightarrow S ^ { \\delta } ( V ) , \\end{align*}"}
+{"id": "1415.png", "formula": "\\begin{align*} \\left ( \\lambda - \\mathcal { A } \\right ) \\begin{pmatrix} u \\\\ v \\end{pmatrix} = \\begin{pmatrix} f \\\\ g \\end{pmatrix} . \\end{align*}"}
+{"id": "7634.png", "formula": "\\begin{align*} \\{ \\overline { \\rho _ i } \\ , \\ , \\vert \\ , \\ , i \\in I , I \\subseteq \\{ 0 , \\dots , 5 \\} , | I | = 5 \\} \\cup \\{ \\tau _ 1 , \\tau _ 2 \\} . \\end{align*}"}
+{"id": "7284.png", "formula": "\\begin{align*} \\frac { d } { d t } P _ 1 ( t ) = - P _ 1 ( t ) A ( t ) - A ^ T ( t ) P _ 1 ( t ) + P _ 1 ( t ) B ( t ) R ^ { - 1 } ( t ) B ( t ) P _ 1 ( t ) - ( Q _ x ( t ) + Q _ m ( t ) ) \\end{align*}"}
+{"id": "1024.png", "formula": "\\begin{align*} \\begin{array} { r c l } p _ 0 ( u , v ) & = & \\sum _ { i = 0 } ^ { d - 1 } { \\binom { d - 1 } { i } } a _ i u ^ { d - 1 - i } v ^ { i } , \\\\ p _ 1 ( u , v ) & = & \\sum _ { i = 0 } ^ { d - 1 } { \\binom { d - 1 } i } b _ i u ^ { d - 1 - i } v ^ { i } , \\\\ p _ 2 ( u , v ) & = & \\sum _ { i = 0 } ^ { d - 1 } { \\binom { d - 1 } i } c _ i u ^ { d - 1 - i } v ^ { i } , \\\\ g ( u , v ) & = & \\sum _ { i = 0 } ^ { d } { \\binom { d } i } g _ i u ^ { d - i } v ^ { i } . \\end{array} \\end{align*}"}
+{"id": "2950.png", "formula": "\\begin{align*} \\Upsilon ( x , \\alpha ) = \\begin{cases} \\{ y \\in Q \\ , | \\ , G ( y ) = x \\} & \\alpha \\geq 0 , \\\\ \\varnothing & \\alpha < 0 \\end{cases} \\forall x \\in \\R ^ n , \\ , \\forall \\alpha \\in \\R . \\end{align*}"}
+{"id": "6878.png", "formula": "\\begin{align*} \\psi _ 0 / \\psi ' _ 0 = \\xi / \\psi . \\end{align*}"}
+{"id": "5996.png", "formula": "\\begin{align*} \\mathbb { E } _ { x } \\Big [ e ^ { - q \\tau _ { 0 } ^ - ( r ) } \\Big ] & = Z _ { b } ^ { ( q , r ) } ( x ) - r Z ^ { ( q ) } ( b ) \\overline { W } ^ { ( q + r ) } ( x - b ) \\\\ & \\quad - q \\dfrac { Z ^ { ( q ) } ( b , \\Phi ( q + r ) ) } { Z ^ { ( q ) \\prime } ( b , \\Phi ( q + r ) ) } \\left ( W _ { b } ^ { ( q , r ) } ( x ) - r W ^ { ( q ) } ( b ) \\overline { W } ^ { ( q + r ) } ( x - b ) \\right ) , \\end{align*}"}
+{"id": "5205.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n + M } \\hat { C } _ j & = \\sum _ { j = 1 } ^ n \\hat { C } _ j + M C _ { \\max } + h M ( M + 1 ) / 2 \\\\ & > M ( C + 1 ) + h M ( M + 1 ) / 2 > z , \\end{align*}"}
+{"id": "151.png", "formula": "\\begin{align*} \\begin{aligned} v _ 1 ' = v _ 1 - [ ( v _ 1 - v ) \\cdot \\omega ] \\omega \\ , , \\ v ' = v + [ ( v _ 1 - v ) \\cdot \\omega ] \\omega \\ , , \\end{aligned} \\end{align*}"}
+{"id": "5321.png", "formula": "\\begin{align*} \\eta ( D _ { 1 } ) = \\frac { 1 } { \\kappa } \\eta \\left ( C u _ { 1 } \\right ) = \\frac { 1 } { \\kappa } \\eta \\left ( C u _ { p } \\right ) = \\eta ( D _ { 2 } ) . \\end{align*}"}
+{"id": "3427.png", "formula": "\\begin{align*} \\begin{gathered} E _ U ( r ^ { n - 1 } \\rho J _ { 1 , 1 } ) = 2 r ^ { n - 1 } \\rho U , E _ \\rho ( r ^ { n - 1 } \\rho J _ { 1 , 1 } ) = 2 r ^ { n - 1 } U ^ 2 , \\\\ E _ S ( r ^ { n - 1 } \\rho J _ { 1 , 1 } ) = - 2 r ^ { n - 1 } p ' , E ^ { ( 1 ) } _ S ( r ^ { n - 1 } \\rho J _ { 1 , 1 } ) = \\tfrac { 2 } { n } r ^ n p ' \\end{gathered} \\end{align*}"}
+{"id": "4323.png", "formula": "\\begin{align*} \\mathbb { P } ( T > t + 1 ) = \\frac { \\rho } { 1 + \\rho } \\mathbb { P } ( N > t ) \\leq \\frac { \\rho \\mathbb { E } r ^ N } { ( 1 + \\rho ) r ^ { t + 1 } } = \\frac { 1 + \\rho - [ ( 1 + \\rho ) ^ 2 - 4 \\rho r ] ^ { 1 / 2 } } { 2 ( 1 + \\rho ) r ^ { t + 1 } } , \\end{align*}"}
+{"id": "7131.png", "formula": "\\begin{align*} C = \\{ ( i , j ) \\mid \\ , 1 \\leq i , j \\leq d \\} , \\ D = \\{ ( i , j ) \\mid \\ , 1 \\leq j < i \\leq d \\} , \\ E = \\{ 1 , \\ldots , d \\} . \\end{align*}"}
+{"id": "7395.png", "formula": "\\begin{align*} \\frac { \\alpha } { 4 } + \\frac { | \\mu _ n ( H ) | + 1 } { \\alpha } \\leq E _ n ( \\alpha ) \\mu _ n ( S ( E _ n ( \\alpha ) ) ) = - \\frac { E _ n ( \\alpha ) } { \\alpha } \\leq \\frac { \\alpha } { 4 } - \\frac { | \\mu _ n ( H ) | + 1 } { \\alpha } . \\end{align*}"}
+{"id": "255.png", "formula": "\\begin{align*} \\nabla \\mathsf { f } = \\left [ \\frac { \\partial f _ i } { \\partial y _ j } = 2 m _ i \\pi \\frac { \\dot { \\mathsf { H } } ( z ) } { \\mathsf { H } ( b ) } \\frac { y _ j } { z } : 1 \\le i \\le d , 1 \\le j \\le N \\right ] = \\frac { 2 \\pi } { \\mathsf { H } ( b ) } \\frac { \\mathsf { m } \\otimes y } { z ^ { n + 2 } H ( z , z ^ 2 ) } . \\end{align*}"}
+{"id": "3415.png", "formula": "\\begin{align*} P ^ \\rho = \\eta ^ \\rho - \\tau \\rho _ t - \\xi \\rho _ r , P ^ U = \\eta ^ U - \\tau U _ t - \\xi U _ r , P ^ S = \\eta ^ S - \\tau S _ t - \\xi S _ r . \\end{align*}"}
+{"id": "1597.png", "formula": "\\begin{align*} \\left ( f _ { n } \\left ( g _ { s , i , j } \\right ) \\right ) \\left ( w \\right ) = \\varphi _ { s , i , j } \\left ( w , 0 \\right ) = \\varphi _ { s , i , j } \\left ( T _ { ( b _ { i } , s _ { i } ) } \\left ( v \\right ) \\right ) = T _ { b _ { j } } \\left ( v \\right ) \\end{align*}"}
+{"id": "7655.png", "formula": "\\begin{align*} \\mathfrak { V } ( p ) : = \\{ \\ , V \\ , : \\ , \\ , \\} . \\end{align*}"}
+{"id": "1420.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { U } ( t ) = \\mathcal { A } \\mathcal { U } ( t ) , t > 0 . \\end{align*}"}
+{"id": "945.png", "formula": "\\begin{align*} l ( y ' ) : = u ( x _ 0 ) + \\sum _ { \\beta = 1 } ^ { n - 1 } \\nabla _ \\beta u ( x _ 0 ) y _ \\beta . \\end{align*}"}
+{"id": "4484.png", "formula": "\\begin{align*} & \\dd _ { ( \\psi _ 1 , \\psi _ 2 ) } g ( h , 0 ) = \\langle D _ 1 g ( \\psi _ 1 , \\psi _ 2 ) , h \\rangle _ { H } \\ , , \\\\ & \\dd _ { ( \\psi _ 1 , \\psi _ 2 ) } g ( 0 , h ) = \\langle D _ 2 g ( \\psi _ 1 , \\psi _ 2 ) , h \\rangle _ { H } \\ , , \\end{align*}"}
+{"id": "4750.png", "formula": "\\begin{align*} a \\succ b = R ( a ) \\cdot b , a \\prec b = a \\cdot R ( b ) , \\forall a , b \\in A . \\end{align*}"}
+{"id": "4608.png", "formula": "\\begin{align*} \\ln R ( n + 2 ) ^ 2 = & \\ln R ( n ) ^ 2 - \\sum _ { j = 0 } ^ 1 V ( n + j ) \\sin 2 \\pi \\theta ( n + j ) + \\frac { O ( 1 ) } { ( n - b ) ^ 2 } \\\\ = & \\ln R ( n ) ^ 2 - \\frac { K _ 1 } { n - b } ( 1 - \\cos 4 \\pi \\theta ( n ) ) + \\frac { O ( 1 ) } { ( n - b ) ^ 2 } , \\\\ = & \\ln R ( n ) ^ 2 - \\frac { K _ 1 } { n - b } ( 1 - \\cos 4 \\pi \\theta ( n ) + \\frac { O ( 1 ) } { n - b } ) \\end{align*}"}
+{"id": "5252.png", "formula": "\\begin{align*} S _ N ( f ( N \\cdot ) ) = \\sum _ { 1 \\leq i , j \\leq N } f ( N \\ * ( \\theta _ i - \\theta _ j ) _ c ) . \\end{align*}"}
+{"id": "4073.png", "formula": "\\begin{align*} P ^ X _ { x , y } ( f _ x ) = f _ y . \\end{align*}"}
+{"id": "671.png", "formula": "\\begin{align*} \\nu _ z = - \\frac { 1 } { 2 } h _ z - ( \\alpha - I \\beta ) h _ { \\overline { z } } . \\end{align*}"}
+{"id": "1929.png", "formula": "\\begin{align*} U = D _ v ^ { \\alpha } D _ x ^ { \\beta } u \\end{align*}"}
+{"id": "7222.png", "formula": "\\begin{align*} | | u | | _ { L ^ p _ t L ^ q _ z ( I _ t \\times \\mathbb { R } ^ m \\times \\mathbb { T } ^ n ) } = \\left ( \\int _ { I _ t } \\left ( \\int _ { \\mathbb { R } ^ m \\times \\mathbb { T } ^ n } | u ( t , z ) | ^ q \\ , d z \\right ) ^ { \\frac { p } { q } } \\ , d t \\right ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
+{"id": "7719.png", "formula": "\\begin{align*} g ^ p = e ^ { - 2 s _ p } g ^ + = e ^ { 2 ( r - s _ p ) } g \\end{align*}"}
+{"id": "1162.png", "formula": "\\begin{align*} \\frac { D } { a ^ 2 } = \\frac { f ^ 2 D _ 0 } { a ^ 2 } = \\frac { ( f _ o / a _ o ) ^ 2 f _ 2 ^ 2 D _ 0 } { a _ 2 ^ 2 } = \\frac { ( f _ o / a _ o ) ^ 2 ( f _ 2 / 2 ^ { \\beta - 1 } ) ^ 2 D _ 0 } { 4 } = A ^ 2 \\cdot m , \\end{align*}"}
+{"id": "976.png", "formula": "\\begin{align*} \\langle \\delta _ \\chi , \\delta _ \\pi \\rangle _ { L ^ 2 ( G ^ \\wedge ) } = \\int _ { G ^ \\wedge } \\delta _ \\chi ( \\xi ) \\ ; \\delta _ \\pi ( \\xi ) \\ , d \\nu ( \\xi ) = \\left \\{ \\begin{array} { c c l } 1 , & & \\chi = \\pi , \\\\ 0 , & & \\chi \\not = \\pi . \\end{array} \\right . \\end{align*}"}
+{"id": "5534.png", "formula": "\\begin{align*} c _ 1 ( \\theta _ 2 ) \\gamma _ 2 ( \\theta _ 2 ) \\dfrac { \\partial \\theta _ 2 } { \\partial t } = \\dfrac { 1 } { r ^ { \\nu } } \\dfrac { \\partial } { \\partial r } \\bigg ( \\lambda _ 2 ( \\theta _ 2 ) r ^ { \\nu } \\dfrac { \\partial \\theta _ 2 } { \\partial r } \\bigg ) , \\ ; \\ ; \\ ; \\beta ( t ) < r < \\infty , \\ ; \\ ; \\ ; 0 < \\nu < 1 , \\end{align*}"}
+{"id": "2795.png", "formula": "\\begin{align*} Q _ A V _ k & = U _ { k + 1 } J _ k , & Q _ A ^ T U _ { k + 1 } & = V _ k J _ k ^ T + \\alpha _ { k + 1 } v _ { k + 1 } e _ { k + 1 } ^ T , \\\\ Q _ B \\widehat { V } _ k & = \\widehat { U } _ { k } \\widehat { J } _ k , & Q _ B ^ T \\widehat { U } _ { k } & = \\widehat { V } _ k \\widehat { J } _ k ^ T + \\hat { \\beta } _ { k } \\hat { v } _ { k + 1 } e _ { k } ^ T , \\end{align*}"}
+{"id": "1462.png", "formula": "\\begin{align*} D _ { r } = \\frac { 1 } { L } \\sum _ { i = 1 } ^ { L } 1 _ { W _ { \\theta , i } \\neq \\hat { W } _ { \\theta , i } } \\end{align*}"}
+{"id": "516.png", "formula": "\\begin{align*} y = \\mathbf { X } \\beta + \\varepsilon , \\varepsilon \\sim \\gamma _ { \\sigma ^ 2 } , \\end{align*}"}
+{"id": "4701.png", "formula": "\\begin{align*} C _ p ( u ) : = \\sup _ { 0 \\le r < 1 } \\int _ { - \\pi } ^ { \\pi } | u ( r \\cos \\theta , r \\sin \\theta ) | ^ p d m _ { \\lambda } ( \\theta ) < \\infty . \\end{align*}"}
+{"id": "3626.png", "formula": "\\begin{align*} { 1 \\over 2 } \\sigma ^ 2 g ''' + \\sigma ^ 2 \\theta g '' + \\left ( { 1 \\over 2 } \\sigma ^ 2 \\theta ^ 2 - \\lambda - 1 \\right ) g ' - \\theta g + 2 e ^ { - \\theta x } ( x - \\rho ) - \\theta e ^ { - \\theta x } ( x - \\rho ) ^ 2 = 0 . \\end{align*}"}
+{"id": "1564.png", "formula": "\\begin{align*} \\mathbb { K } = \\mathbb { L } \\left ( \\left \\{ \\xi _ { k , i , j } \\right \\} _ { \\substack { 1 \\le k \\le 4 , \\\\ 1 \\le i , j \\le n } } \\right ) , \\end{align*}"}
+{"id": "870.png", "formula": "\\begin{align*} \\partial _ \\mu E = \\left \\{ x \\in G : \\ , \\min \\left \\{ \\frac { \\mu ( E \\cap B _ d ( x , r ) ) } { \\mu ( B _ d ( x , r ) ) } , \\frac { \\mu ( E ^ c \\cap B _ d ( x , r ) ) } { \\mu ( B _ d ( x , r ) ) } \\right \\} > \\epsilon \\ , \\ , \\forall r > 0 \\right \\} . \\end{align*}"}
+{"id": "6972.png", "formula": "\\begin{align*} \\Phi ( \\Phi ( X ) Y ) = \\Phi ( X \\Phi ( Y ) ) = \\Phi ( \\Phi ( X ) \\Phi ( Y ) ) . \\end{align*}"}
+{"id": "6201.png", "formula": "\\begin{align*} { \\rm d i m } ( T ) = \\sum _ { ( \\mu , d ) \\in \\Psi } ( d + 1 ) ^ 2 < \\sum _ { ( \\mu , d ) \\in \\Upsilon } ( d + 1 ) ^ 2 & = \\sum ^ m _ { d = 0 } ( m - d - \\lfloor \\frac { m - d + 1 } { 2 } \\rfloor + 1 ) ( d + 1 ) ^ 2 \\\\ & = { m + 4 \\choose 4 } . \\end{align*}"}
+{"id": "6763.png", "formula": "\\begin{align*} \\mathcal { F } _ { b , \\psi _ t } = \\bigoplus _ { k = 0 } ^ \\infty \\mathcal F _ { b , \\psi _ t } ^ { ( k ) } , \\mathcal { F } _ { b } ^ { ( k ) } = \\left ( L ^ 2 _ { \\perp \\psi _ t } ( \\mathbb { R } ^ 3 ) \\right ) ^ { \\otimes _ s k } \\end{align*}"}
+{"id": "7510.png", "formula": "\\begin{align*} E _ \\varphi ( F ) = \\max _ { F ' \\in \\mathcal F } E _ \\varphi ( F ' ) . \\end{align*}"}
+{"id": "3076.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ k ) = F _ { \\hat { X } _ 0 } ( \\hat { \\sigma } _ 0 ^ k ) . \\end{align*}"}
+{"id": "5004.png", "formula": "\\begin{align*} z = l a + \\sum \\limits _ { i = 1 } ^ { a - 1 } c _ i ( g + i ) \\end{align*}"}
+{"id": "2446.png", "formula": "\\begin{align*} I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a , 1 , b ) = - I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a + b + 1 ) - \\sum _ { j = 2 } ^ { a - 1 } I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( j , a + 1 - j , b ) - \\sum _ { j = 2 } ^ { b - 1 } I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a , b + 1 - j , j ) \\end{align*}"}
+{"id": "2879.png", "formula": "\\begin{align*} \\widetilde { W } _ { B _ { r _ 1 } } ( x ) = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { ( 2 ^ k ) ^ { 1 0 } } \\int H ( x - y ) \\chi _ { A _ k } ( y ) d y . \\end{align*}"}
+{"id": "114.png", "formula": "\\begin{align*} H _ 2 ^ { [ 2 ] } \\bigl ( \\alpha ( t ) \\bigr ) = \\pm \\int _ { - \\pi } ^ \\pi 4 \\sin ^ 2 ( \\theta ) | \\widehat { \\alpha } ( t , \\theta ) | ^ 2 \\ , \\tfrac { d \\theta } { 2 \\pi } , \\end{align*}"}
+{"id": "2060.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty a ( n ) q ^ n = \\eta ^ { 4 - a } ( 3 0 z ) \\eta ^ { 4 - b } ( 6 z ) g ( m ; 6 z ) \\in S _ { 2 m - 2 } ( \\Gamma _ 0 ( 1 8 0 ) , \\chi _ 5 ) . \\end{align*}"}
+{"id": "6217.png", "formula": "\\begin{align*} E ^ * _ \\nu \\xi \\neq 0 , \\ E ^ * _ i \\xi = 0 \\ ( i \\neq \\nu ) \\ \\ E ^ * _ { \\nu - 1 } A _ 1 E ^ * _ \\nu \\xi = 0 \\ \\ ( 0 \\leq \\nu \\leq m ) , \\end{align*}"}
+{"id": "988.png", "formula": "\\begin{align*} ( { \\rm D } \\mathbf { A } ) _ { i j l } = A _ { i j , l } = - \\epsilon _ { i j k } \\ , { \\rm a x l } ( \\mathbf { A } ) _ { k , l } = - \\epsilon _ { i j k } ( { \\rm D } { \\rm a x l } ( \\mathbf { A } ) ) _ { k l } . \\end{align*}"}
+{"id": "2564.png", "formula": "\\begin{align*} U ( t , x , \\nu ) : = P _ t x + \\Phi ( t , \\nu ) \\end{align*}"}
+{"id": "1952.png", "formula": "\\begin{align*} \\sum \\limits _ { j \\in \\mathbb { Z } } \\overline { \\hat { \\varphi } ( 2 ^ { - j \\vec { a } } \\xi ) } \\hat { \\psi } ( 2 ^ { - j \\vec { a } } \\xi ) = 1 \\ \\ \\ \\ \\ \\xi \\neq 0 . \\end{align*}"}
+{"id": "3082.png", "formula": "\\begin{align*} A _ 2 & = ( \\overline { 1 , 5 , 4 , 5 , 1 , 8 } ) - ( \\overline { 1 , 1 , 1 , 1 , 1 , 1 } ) \\\\ & = ( \\overline { 0 , 4 , 3 , 4 , 0 , 7 } ) . \\end{align*}"}
+{"id": "3770.png", "formula": "\\begin{align*} B _ c ( \\sigma \\cdot x ) = B _ { \\sigma ^ { - 1 } \\cdot c } ( x ) \\end{align*}"}
+{"id": "2214.png", "formula": "\\begin{align*} \\dot { x } _ 1 & = - x _ 1 ^ 2 + 4 x _ 1 x _ 2 - x _ 1 + 2 x _ 2 \\\\ \\dot { x } _ 2 & = x _ 1 ^ 2 + 2 x _ 2 ^ 2 + x _ 1 \\end{align*}"}
+{"id": "3944.png", "formula": "\\begin{align*} \\int _ { \\Gamma _ \\infty } | \\nabla _ { \\Gamma _ \\infty } w _ \\infty ( x ) | ^ 2 \\ , d \\mu _ { \\Gamma _ \\infty } & \\leq \\liminf _ { n \\to + \\infty } \\int _ { \\Gamma _ \\infty } | \\nabla _ { \\Gamma _ \\infty } w _ n ( x ) | ^ 2 \\ , d \\mu _ { \\Gamma _ \\infty } , \\\\ \\lim _ { n \\to \\infty } \\int _ { \\Gamma _ \\infty } w _ n ( x ) f ( \\tau _ n ( x ) ) \\ , d \\mu _ { \\Gamma _ \\infty } & = \\int _ { \\Gamma _ \\infty } w _ \\infty ( x ) f ( x ) \\ , d \\mu _ { \\Gamma _ \\infty } . \\end{align*}"}
+{"id": "6124.png", "formula": "\\begin{align*} \\begin{aligned} \\pi ( \\mathcal F _ 1 ) & = \\{ F \\in \\pi ( \\mathcal F ) : [ d - 1 ] \\subseteq F , F \\cap \\{ d , d + 1 \\} \\neq \\emptyset , d + 2 \\in F \\} \\\\ & \\cup \\{ F \\in \\pi ( \\mathcal F ) : 1 \\notin F , [ 2 , d - 1 ] \\subseteq F , \\{ d , d + 1 \\} \\subseteq F , d + 2 \\in F \\} . \\end{aligned} \\end{align*}"}
+{"id": "7459.png", "formula": "\\begin{align*} \\begin{matrix} \\bar { \\tau } _ 1 = [ a _ 0 , \\hat { j } ] + a _ 0 , & \\bar { \\tau } _ { - 1 } = - [ a _ 0 , \\hat { j } ] + a _ 0 , \\\\ \\bar { \\tau } ^ \\dagger _ 1 = [ \\hat { j } , a _ 0 ^ \\dagger ] + a _ 0 ^ \\dagger , & \\bar { \\tau } _ { - 1 } = - [ \\hat { j } , a _ 0 ^ \\dagger ] + a _ 0 ^ \\dagger , \\\\ \\end{matrix} \\end{align*}"}
+{"id": "4962.png", "formula": "\\begin{align*} R _ { \\rm F A } ( \\lambda , w _ { c } , r ) = \\frac { 1 - \\lambda } { \\lambda } ( 1 - ( 1 - \\lambda ) ^ { \\frac { w _ { c } } { r } - 1 } ) ^ { w _ { c } } , \\end{align*}"}
+{"id": "5396.png", "formula": "\\begin{align*} d X ^ { g } _ { \\varepsilon } ( t ) + ( \\varepsilon - L ) \\Psi ( X ^ { g } _ { \\varepsilon } ( t ) ) d t = \\int _ { Z } f ( t , X ^ { g } _ { \\varepsilon } ( t ) , z ) ( g ( t , z ) - 1 ) \\nu ( d z ) d t , \\ \\ [ 0 , T ] , \\end{align*}"}
+{"id": "6805.png", "formula": "\\begin{align*} \\alpha ( d , d , x _ { [ 3 , m ] } , y _ { [ m + 1 , 5 ] } ) = \\sum _ { i \\in [ r ] } \\beta _ { i , 0 } ( d , x _ { I _ { i , 0 } } , y _ { J _ { i , 0 } } ) \\beta _ { i , 1 } ( x _ { I _ { i , 1 } } , y _ { J _ { i , 1 } } ) \\end{align*}"}
+{"id": "766.png", "formula": "\\begin{align*} N + E _ N ^ + - E _ N ^ - = f ^ * ( f _ * N ) . \\end{align*}"}
+{"id": "4770.png", "formula": "\\begin{gather*} ( - \\alpha _ 2 ) ( \\mu ( \\partial _ 1 ( a ) ) v ) - ( - \\alpha _ 1 ) ( \\mu ( \\partial _ 2 ( a ) ) v ) + \\mu ( \\partial _ 1 ( a ) ) \\alpha _ 2 ( v ) - \\mu ( \\partial _ 2 ( a ) ) \\alpha _ 1 ( v ) \\\\ \\qquad { } = ( \\mu ( \\partial _ 1 ( a ) ) \\alpha _ 2 ( v ) - \\alpha _ 2 ( \\mu ( \\partial _ 1 ( a ) ) v ) ) - ( \\mu ( \\partial _ 2 ( a ) ) \\alpha _ 1 ( v ) - \\alpha _ 1 ( \\mu ( \\partial _ 2 ( a ) ) v ) ) \\\\ \\qquad { } = - \\mu ( \\partial _ 2 ( \\partial _ 1 ( a ) ) ) v + \\mu ( \\partial _ 1 ( \\partial _ 2 ( a ) ) ) v = 0 . \\end{gather*}"}
+{"id": "2304.png", "formula": "\\begin{align*} \\mathfrak h ( 0 ) = p \\mathfrak h ( p ) = 0 , \\end{align*}"}
+{"id": "3946.png", "formula": "\\begin{align*} E _ \\Omega ( u ) | _ { \\Omega } = u , \\| E _ \\Omega \\| _ { \\mathcal { L } ( H ^ 2 ( \\Omega ) , H ^ 2 ( D ) ) } \\leq C . \\end{align*}"}
+{"id": "2034.png", "formula": "\\begin{align*} I _ \\infty = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\Re \\left [ \\frac { 1 } { 2 } \\frac { \\Gamma ' } { \\Gamma } \\left ( \\frac { s } { 2 } \\right ) - \\frac { 1 } { 2 } \\log \\pi \\right ] \\Bigl ( | \\Phi _ 1 ( \\phi ; z ) | ^ 2 + | \\Phi _ 1 ( \\phi ; - z ) | ^ 2 \\Bigr ) \\ , d z , \\end{align*}"}
+{"id": "8121.png", "formula": "\\begin{align*} ( ( L , \\varphi ) ^ { d + 1 } ) = - \\operatorname { \\widehat { \\deg } } _ { \\xi ' } ( R ) \\end{align*}"}
+{"id": "2428.png", "formula": "\\begin{align*} \\psi _ { 1 } ( t ) \\coloneqq \\frac { d } { d t } \\log \\Gamma _ { 1 } ( t ) = - \\sum _ { n = 2 } ^ { \\infty } \\zeta ^ { \\mathfrak { m } } ( n ) ( - t ) ^ { n - 1 } . \\end{align*}"}
+{"id": "7066.png", "formula": "\\begin{align*} I _ { S _ 1 } \\cap ( I _ { S _ 2 } I _ { S _ 0 } ^ { m - 1 } ) = I _ { S _ 1 } I _ { S _ 0 } ^ { m - 1 } . \\end{align*}"}
+{"id": "8158.png", "formula": "\\begin{align*} \\operatorname { s q } ^ t ( S ^ \\delta ( V ) ) \\cong \\bigoplus _ { \\begin{subarray} { c } \\boldsymbol { b } = ( b _ 1 , \\ldots , b _ r ) \\in \\mathbb N ^ r \\\\ | \\boldsymbol { b } | = b _ 1 + \\cdots + b _ r = \\delta \\\\ b _ 1 \\mu _ 1 + \\cdots + b _ r \\mu _ r = t \\end{subarray} } S ^ { b _ 1 } ( V _ 1 / V _ 0 ) \\otimes \\cdots \\otimes S ^ { b _ r } ( V _ r / V _ { r - 1 } ) \\end{align*}"}
+{"id": "2818.png", "formula": "\\begin{align*} X ^ { ( 1 ) } \\sim X ^ { ( 2 ) } & X ^ { ( 1 ) } _ . = X ^ { ( 2 ) } _ . \\ ; \\ ; d P \\times d s \\Omega \\times [ t , T ] , \\\\ & X ^ { ( 1 ) } _ { t - } = X ^ { ( 2 ) } _ { t - } \\ , ( = x ) X ^ { ( 1 ) } _ { T } = X ^ { ( 2 ) } _ { T } \\ , ( = \\xi ) . \\end{align*}"}
+{"id": "6059.png", "formula": "\\begin{align*} \\begin{cases} \\ , \\ , p _ \\epsilon \\in H ^ 1 _ 0 ( \\Omega ) \\ , \\ , \\ , \\\\ [ 0 . 3 c m ] \\int _ { \\Omega } \\nabla v \\nabla p _ \\epsilon d x - \\int _ { \\Omega } k ^ 2 v p _ \\epsilon d x = - 2 \\int _ { \\Omega } ( \\nabla \\eta _ \\epsilon - A ) \\nabla v d x - 2 \\int _ { \\Omega } ( \\eta _ \\epsilon - \\eta _ 0 ) v d x \\ , \\ , \\forall \\ , v \\in H ^ 1 _ 0 ( \\Omega ) , \\end{cases} \\end{align*}"}
+{"id": "5091.png", "formula": "\\begin{align*} \\| u \\| _ { Y _ T ^ 2 } = \\sum _ { k = 0 , 1 } \\| | \\partial _ y | ^ { k \\sigma } u \\| _ { L ^ l _ t L ^ m _ x L ^ 2 _ y ( [ - T , T ] \\times \\mathbb { R } ^ d \\times \\mathbb { T } ) } , \\end{align*}"}
+{"id": "1304.png", "formula": "\\begin{align*} ( g _ 1 + g _ 2 ) \\circ f = g _ 1 \\circ f + g _ 2 \\circ f , g \\circ ( f _ 1 + f _ 2 ) = g \\circ f _ 1 + g \\circ f _ 2 , \\end{align*}"}
+{"id": "6454.png", "formula": "\\begin{align*} \\int _ u ^ v \\frac { ( q x / u , q x / v , a b r x ; q ) _ { \\infty } } { ( a x , b x , c x ; q ) _ { \\infty } } d _ q x & = \\frac { ( 1 - q ) v ( q , u / v , q v / u ; q ) _ { \\infty } ( a c u v , b c u v , a b r / c ; q ) _ { \\infty } } { ( a u , a v , b u , b v , c u , c v ; q ) _ { \\infty } } \\\\ & \\times { } _ 3 \\phi _ 2 \\left ( \\begin{gathered} c u , c v , c u v / r \\\\ a c u v , b c u v \\end{gathered} ; \\ , q , \\frac { a b r } { c } \\right ) . \\end{align*}"}
+{"id": "3103.png", "formula": "\\begin{align*} \\limsup _ { x \\rightarrow \\infty } F _ { \\N } ( \\sigma ^ k ) = \\infty \\end{align*}"}
+{"id": "4850.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\R ^ { 2 d } \\times S ^ 2 } & | x - y | ^ 2 \\ , \\dd \\Pi _ t ( x , y , s _ 1 , s _ 2 ) \\leq \\\\ \\leq & \\left ( \\int _ { \\R ^ { 2 d } \\times S ^ 2 } | x - y | ^ 2 \\ , \\dd \\Pi _ 0 ( x , y , s _ 1 , s _ 2 ) \\right ) e ^ { - \\frac m 2 t } + c e ^ { - \\frac m 2 t } \\int _ 0 ^ t e ^ { \\frac m 2 s - \\int _ 0 ^ s K ( u ) \\ , \\dd u } \\ , \\dd s \\end{aligned} \\end{align*}"}
+{"id": "2615.png", "formula": "\\begin{align*} \\partial _ t \\mu ^ \\phi = - \\partial _ x \\Big \\{ \\Big [ b ( t , \\cdot , \\mu ^ \\phi ) + \\sigma \\phi ( t , \\cdot , \\mu ^ \\phi ) \\Big ] \\mu ^ \\phi \\Big \\} . \\end{align*}"}
+{"id": "5424.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ { d - m _ 0 } } \\sum _ { j _ 1 , j _ 2 , \\ldots , j _ { d } } \\prod _ { p = 1 } ^ { m _ 0 - 1 } ( T ^ { k _ p } ) _ { j _ { p } j _ { p + 1 } } \\Big ( \\prod _ { i = 1 } ^ { m _ 1 } G _ { x _ i y _ i } \\prod _ { l = 1 } ^ { m _ 2 } \\big ( G _ { w _ l w _ l } - m _ { s c } \\big ) ( t , z ) \\Big ) , \\end{align*}"}
+{"id": "3856.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n f ^ { - } _ k ( \\sigma _ i ) \\leq \\# \\{ \\sigma _ i \\in \\Omega _ k \\} \\leq \\sum _ { i = 1 } ^ n f ^ { + } _ k ( \\sigma _ i ) , k = 1 , 2 . \\end{align*}"}
+{"id": "6095.png", "formula": "\\begin{align*} \\psi _ n = P _ { k _ n } \\pi _ n P _ { k _ n } . \\end{align*}"}
+{"id": "4511.png", "formula": "\\begin{align*} \\{ \\Phi _ x , \\Pi _ y \\} & = \\langle D _ 1 \\Phi _ x , D _ 2 \\Pi _ y \\rangle _ { H ^ 1 } - \\langle D _ 1 \\Pi _ x , D _ 2 \\Phi _ y \\rangle _ { H ^ 1 } \\\\ & = \\langle \\mathcal { E } _ x , \\mathcal { E } _ y \\rangle _ { H ^ 1 } = \\mathcal { E } _ x ( y ) = \\mathcal { E } _ y ( x ) \\ , . \\end{align*}"}
+{"id": "2238.png", "formula": "\\begin{align*} q _ { k + 1 } ( \\lambda ) & = \\lambda * q _ { k } ( \\lambda ) - q _ { k } ( \\lambda ) \\times \\boldsymbol { \\alpha } _ { k } ^ H - q _ { k - 1 } ( \\lambda ) \\times \\boldsymbol { \\beta } _ { k } ^ H \\\\ p _ { k + 1 } ( \\lambda ) \\times \\boldsymbol { \\beta } _ { k + 1 } & = \\lambda * p _ { k } ( \\lambda ) - p _ { k } ( \\lambda ) \\times \\boldsymbol { \\alpha } _ { k } - p _ { k - 1 } ( \\lambda ) , \\end{align*}"}
+{"id": "4248.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( q ) _ n \\left ( \\frac { b } { a } \\right ) _ n ( a ) _ { N - n } a ^ n } { ( b ) _ n ( 1 - q ^ n ) ( a ) _ { N } } = \\displaystyle \\sum _ { m = 1 } ^ { \\infty } \\frac { ( a ^ m - b ^ m ) ( 1 - q ^ { m N } ) } { 1 - q ^ m } . \\end{align*}"}
+{"id": "2647.png", "formula": "\\begin{align*} & s ( \\mu \\nu ) = \\mu \\nu ( w _ 2 w _ 3 ) = \\mu \\nu | ^ * _ { E _ [ w _ 2 , w _ 2 w _ 3 ] } ( w _ 3 ) = \\nu ( w _ 3 ) = s ( \\nu ) , \\\\ & r ( \\mu \\nu ) = \\mu \\nu ( e ) = \\mu \\nu | _ { E _ { w _ 2 } } ( e ) = \\mu ( e ) = r ( \\mu ) \\end{align*}"}
+{"id": "6910.png", "formula": "\\begin{align*} \\mathsf u \\sum _ { | \\vec { d } | = d } \\left [ h _ 1 ^ { d _ 1 } \\ldots h _ N ^ { d _ N } \\right ] & \\bigg \\{ \\prod _ { i } \\bigg ( \\frac { z _ i ( \\alpha _ i + y ) } { \\alpha _ i ( z _ i + y ) } \\bigg ) ^ { b _ i } \\bigg ( \\frac { z _ i + y } { z _ i } \\bigg ) ^ { d _ i } \\bigg ( \\frac { h _ i } { R ( z _ i ) } \\bigg ) ^ { d _ i + 1 } z _ i ^ { d + 1 } \\bigg ( \\frac { R ( z _ i ) } { \\prod _ { j } ( z _ j - \\alpha _ i ) } \\bigg ) ^ { b _ i } \\\\ & \\cdot \\prod _ { i < j } ( z _ i - z _ j ) ^ 2 \\bigg \\} \\bigg { | } _ { \\epsilon = 0 } . \\end{align*}"}
+{"id": "549.png", "formula": "\\begin{align*} \\frac { 1 } { \\binom { n } { k } } \\sum _ { S \\subset [ n ] , \\ , | S | = k } \\mathcal { W } _ 2 ^ 2 ( P _ { S } , Q _ { S } ) = \\frac { 1 } { | \\Pi | } \\sum _ { ( S _ 1 , \\ldots , S _ m ) \\in \\Pi } \\frac { 1 } { m } \\sum _ { i = 1 } ^ m \\mathcal { W } _ 2 ^ 2 ( P _ { S _ i } , Q _ { S _ i } ) . \\end{align*}"}
+{"id": "4868.png", "formula": "\\begin{align*} p _ { k + 1 } ( x ) & = 2 ( k x + 1 ) p _ k ( x ) + 2 x ( 1 - x ) p ' _ k ( x ) + q _ k ( x ) , \\\\ q _ { k + 1 } ( x ) & = \\left ( 2 ( k + 1 ) x + 1 \\right ) q _ k ( x ) + 2 x ( 1 - x ) q ' _ k ( x ) . \\end{align*}"}
+{"id": "8069.png", "formula": "\\begin{align*} f ( \\{ E _ l \\} ) \\propto \\prod _ { 1 \\leq i < j \\leq N } ^ { } | E _ i - E _ j | ^ { \\beta } \\exp \\left ( - A \\sum _ { i = 1 } ^ { N } E _ i ^ 2 \\right ) , \\end{align*}"}
+{"id": "7929.png", "formula": "\\begin{align*} \\varphi _ t = \\sum _ { i = 0 } ^ \\infty \\varphi _ i t ^ i : \\big ( A [ [ t ] ] = ( A [ [ t ] ] , \\mu _ t ) , R _ t \\big ) \\rightarrow \\big ( A [ [ t ] ] ' = ( A [ [ t ] ] , \\mu _ t ' ) , R _ t ' \\big ) ~ ~ \\varphi _ 0 = \\mathrm { i d } _ A \\end{align*}"}
+{"id": "4577.png", "formula": "\\begin{align*} Y ( n ) = \\frac { 1 } { \\sin \\pi k } \\begin{pmatrix} \\sin \\pi k & 0 \\\\ - \\cos \\pi k & 1 \\end{pmatrix} \\begin{pmatrix} u ( n - 1 ) \\\\ u ( n ) \\end{pmatrix} . \\end{align*}"}
+{"id": "145.png", "formula": "\\begin{align*} x ( [ \\mathbf { a ' } , \\mathbf { a } ] \\otimes [ 0 , \\mathbf { b } ] ) = 0 \\end{align*}"}
+{"id": "1011.png", "formula": "\\begin{align*} \\mu ^ { \\rm N } & = \\mu ^ * + \\frac { \\varkappa } { 2 } , \\varkappa ^ { \\rm N } = \\frac { \\varkappa } { 2 } , \\qquad \\ , \\lambda ^ { \\rm N } = \\lambda , \\\\ \\gamma ^ { \\rm N } & = \\frac { \\beta + \\gamma } { 2 } , \\qquad \\ \\ \\beta ^ { \\rm N } = \\frac { \\gamma - \\beta } { 2 } , \\ \\ \\ \\ \\ \\ \\ \\alpha ^ { \\rm N } = \\alpha \\ , . \\end{align*}"}
+{"id": "2709.png", "formula": "\\begin{align*} \\binom { j + r - 1 } { j } \\sum _ { l = 0 } ^ { r - 1 } \\binom { 2 n - j + l } { l } \\binom { 2 n - j } { r - 1 - l } M _ { \\phi } ( 2 n , j + r - 1 - l , 0 ; r - 1 ) \\end{align*}"}
+{"id": "2766.png", "formula": "\\begin{align*} \\tau : = { \\rm m i n } \\{ t \\in \\R \\ , | \\ , K _ X + t H \\ \\} . \\end{align*}"}
+{"id": "2235.png", "formula": "\\begin{align*} [ q \\times \\boldsymbol { \\alpha } , p \\times \\boldsymbol { \\beta } ] & = \\boldsymbol { \\alpha } ^ H \\times [ q , p ] \\times \\boldsymbol { \\beta } , \\\\ [ q + s , p + r ] & = [ q , p ] + [ s , p ] + [ q , r ] + [ s , r ] . \\end{align*}"}
+{"id": "7889.png", "formula": "\\begin{align*} | \\dot f _ { y , r } ( x _ 1 ) - \\dot f _ { y ' , r ' } ( x _ 1 ) | & = \\Big | \\frac { \\Phi _ 1 ( x , y ) } { \\Phi _ 2 ( x , y ) } - \\frac { \\Phi _ 1 ( x ' , y ' ) } { \\Phi _ 2 ( x ' , y ' ) } \\Big | \\\\ & \\geq \\Big | \\frac { \\Phi _ 1 ( x ' , y ) } { \\Phi _ 2 ( x ' , y ) } - \\frac { \\Phi _ 1 ( x ' , y ' ) } { \\Phi _ 2 ( x ' , y ' ) } \\Big | - \\Big | \\frac { \\Phi _ 1 ( x , y ) } { \\Phi _ 2 ( x , y ) } - \\frac { \\Phi _ 1 ( x ' , y ) } { \\Phi _ 2 ( x ' , y ) } \\Big | . \\end{align*}"}
+{"id": "7001.png", "formula": "\\begin{align*} C _ 1 ( \\beta ) : = \\alpha - \\beta K \\quad C _ 2 ( \\beta ) : = \\frac { N \\alpha ( 1 - \\beta ) } { 4 \\beta } \\left ( 1 - \\frac { \\beta ( ( 2 - \\beta ) K - \\alpha ) } { 2 \\alpha ( 1 - \\beta ) } \\right ) ^ 2 , \\end{align*}"}
+{"id": "4752.png", "formula": "\\begin{gather*} ( a + u ) \\succ ( b + v ) : = \\hat T ( a + u ) \\cdot ( b + v ) = T ( u ) \\cdot ( b + v ) = T ( u ) \\cdot b + l ( T ( u ) ) v , \\\\ ( a + u ) \\prec ( b + v ) : = ( a + u ) \\cdot \\hat T ( b + v ) = ( a + u ) \\cdot T ( v ) = a \\cdot T ( v ) + r ( T ( v ) ) u , \\end{gather*}"}
+{"id": "1090.png", "formula": "\\begin{align*} 4 z _ { 3 } ^ { 2 } - 2 z _ { 2 } ^ { 2 } = 5 p . \\end{align*}"}
+{"id": "2467.png", "formula": "\\begin{align*} - \\Delta F _ { B } + F _ { B } = 2 J ( u , B ) , \\end{align*}"}
+{"id": "3278.png", "formula": "\\begin{gather*} x _ 1 - a _ 1 x _ 2 x _ 3 = b _ 1 , \\\\ x _ 2 - a _ 2 x _ 1 x _ 3 = b _ 2 , \\\\ x _ 3 - a _ 3 x _ 1 x _ 2 = b _ 3 , \\end{gather*}"}
+{"id": "2605.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\bar { x } ^ i _ t = & ~ \\Big [ ( A - B ^ 2 R ^ { - 1 } P _ t ) \\bar { x } ^ i _ t - B ^ 2 R ^ { - 1 } \\Phi ( t , \\bar { \\nu } _ t ^ i ) - B h \\Big ( k \\big ( t , \\bar { \\nu } _ t ^ i , \\Phi ( t , \\bar { \\nu } _ t ^ i ) \\big ) \\Big ) \\Big ] d t \\\\ & + \\sigma d W _ t ^ i + \\sigma _ 0 d W _ t ^ 0 , \\\\ \\bar { x } ^ i _ { t _ 0 } = & ~ \\xi ^ i , \\end{aligned} \\right . \\end{align*}"}
+{"id": "514.png", "formula": "\\begin{align*} \\delta _ \\square ( W _ 1 , W _ 2 ) : = \\inf _ { \\varphi } d _ \\square ( W _ 1 , W _ 2 ^ \\varphi ) , \\end{align*}"}
+{"id": "3262.png", "formula": "\\begin{gather*} K \\psi _ n = \\lambda _ n \\psi _ n , \\\\ L \\psi _ n = a _ n \\psi _ { n - 1 } + b _ n \\psi _ n + a _ { n + 1 } \\psi _ { n + 1 } . \\end{gather*}"}
+{"id": "7861.png", "formula": "\\begin{align*} E = \\bigcup _ { J \\in \\mathcal { J } } J \\times G _ { \\theta ( J ) } ' . \\end{align*}"}
+{"id": "3646.png", "formula": "\\begin{align*} | a _ s | & = \\left | \\frac { \\widehat v \\cdot \\nabla \\eta } { \\eta } \\right | \\le C ( 1 + | \\widehat v | ) \\end{align*}"}
+{"id": "4179.png", "formula": "\\begin{align*} h _ { m f _ 2 ^ - } ( x ) & = h _ { m f _ 1 ^ - } ( x ) + g _ { m ( f _ 2 ^ - - f _ 1 ^ - ) } ( x + m f _ 1 ^ - \\alpha ) - \\sum _ { i = m f _ 2 ^ - } ^ { m f _ 1 ^ - - 1 } \\eta _ i \\\\ h _ { m f _ 2 ^ + } ( x ) & = h _ { m f _ 1 ^ + } ( x ) + g _ { m ( f _ 2 ^ + - f _ 1 ^ + ) } ( x + m f _ 1 ^ + \\alpha ) + \\sum _ { i = m f _ 1 ^ + } ^ { m f _ 2 ^ + - 1 } \\eta _ i \\end{align*}"}
+{"id": "4167.png", "formula": "\\begin{align*} \\sum _ { \\kappa \\le | k | < 2 \\kappa } | k | ^ { - 4 } S _ k ^ { \\ll } = \\sum _ { 1 \\le | j | < \\kappa ^ r } | \\gamma _ j | ^ 2 \\sum _ { \\kappa \\le | k | < 2 \\kappa } | k | ^ { - 4 } | k - j | ^ { 2 \\beta } ( U _ e ^ 2 \\xi _ { k j } ^ 2 + U _ f ^ 2 \\upsilon _ { k j } ^ 2 ) . \\end{align*}"}
+{"id": "1931.png", "formula": "\\begin{align*} \\begin{aligned} & \\| \\partial _ t ^ j D _ x ^ l D _ v ^ { m } u \\| _ { L _ p ( Q _ { 1 / 2 } ) } \\\\ & \\le N ( d , \\delta , R , j , l , m ) ( \\| | D _ x u | + | D _ v u | + \\lambda ^ { 1 / 2 } | u | \\| _ { L _ p ( Q _ { R } ) } ) , R \\in ( 1 / 2 , 1 ] , \\end{aligned} \\end{align*}"}
+{"id": "5047.png", "formula": "\\begin{align*} \\sum _ { \\substack { q \\ge 1 \\\\ N \\mid q } } \\frac { r _ q ( n ) } { q ^ { 2 s } } = \\begin{cases} \\frac { \\sigma _ { 1 - 2 s } ( n ; N ) } { \\zeta ^ { ( N ) } ( 2 s ) } & n \\neq 0 , \\\\ N ^ { 1 - 2 s } \\prod _ { p \\mid N } ( 1 - p ^ { - 1 } ) \\frac { \\zeta ( 2 s - 1 ) } { \\zeta ^ { ( N ) } ( 2 s ) } & n = 0 . \\end{cases} \\end{align*}"}
+{"id": "4329.png", "formula": "\\begin{align*} d _ { T V } ( \\pi , X ) \\leq \\left | 1 - \\frac { p } { 2 p - 1 } \\mathbb { P } ( X = 0 ) \\right | , \\end{align*}"}
+{"id": "5601.png", "formula": "\\begin{align*} C _ { i j p } J ^ i _ s J ^ j _ q J ^ q _ b J ^ p _ a = C _ { i j s } J ^ i _ p J ^ p _ a J ^ j _ q J ^ q _ b \\end{align*}"}
+{"id": "5122.png", "formula": "\\begin{align*} \\lim _ { t _ 1 , t _ 2 \\rightarrow \\infty } \\norm { \\int _ { t _ 1 } ^ { t _ 2 } e ^ { - i s ( ( - \\Delta _ { x } ) ^ { \\sigma } - \\partial _ { y } ^ { 2 \\sigma } ) } ( | u | ^ p u ) \\ , d s } _ { H ^ { \\sigma } _ { x , y } ( \\R ^ d \\times \\mathbb { T } ) } = 0 . \\end{align*}"}
+{"id": "8047.png", "formula": "\\begin{align*} { \\widetilde { \\bf s } } _ { \\textrm { d i r } } [ f _ k ] = { \\bf A } _ { \\textrm { d i r } , k } { \\bf F } _ k { \\bf s } _ k , \\end{align*}"}
+{"id": "2677.png", "formula": "\\begin{align*} M _ S ( n , j , t ; a ) = \\binom { n - j } { j } \\sum _ { v = 0 } ^ { n - 2 j } \\binom { n - 2 j } { v } \\binom { n } { j + v } ^ t F ( n , j + v , a ) \\end{align*}"}
+{"id": "1045.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { W } \\ , ( f | D | ^ { - n } ) = 2 ^ { m } v _ { n - 1 } \\int _ { M } f ~ v o l _ { g } , \\\\ \\mathcal { W } \\ , ( f | D | ^ { - n + 2 } ) = - 2 ^ { m } \\frac { n - 2 } { 2 4 } v _ { n - 1 } \\int _ { M } f R ( g ) ~ v o l _ { g } . \\end{aligned} \\end{align*}"}
+{"id": "4196.png", "formula": "\\begin{align*} \\mathcal { G } ( X ) = 1 + \\underset { \\alpha \\in { S _ g } _ { X } } { \\prod } \\frac { 1 } { \\sqrt { \\mathrm { E x t } _ X ( \\alpha ) } } . \\end{align*}"}
+{"id": "8140.png", "formula": "\\begin{align*} ( \\overline L _ 0 \\cdots \\overline L _ d ) _ S \\geqslant \\sum _ { i = 1 } ^ d & \\delta _ i \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L _ i ) \\\\ & - \\int _ { \\Omega } \\int _ { X _ \\omega ^ { \\mathrm { a n } } } \\ln \\| s \\| _ { \\varphi _ { 0 , \\omega } } c _ 1 ( L _ { 1 , \\omega } , \\varphi _ { 1 , \\omega } ) \\cdots c _ 1 ( L _ { d , \\omega } , \\varphi _ { d , \\omega } ) \\ , \\nu ( \\mathrm { d } \\omega ) . \\end{align*}"}
+{"id": "5007.png", "formula": "\\begin{align*} A = \\{ a , 2 g + 1 \\} = \\{ 2 , 7 \\} . \\end{align*}"}
+{"id": "5699.png", "formula": "\\begin{gather*} \\tau _ * ( \\alpha _ { ( \\xi , \\eta ) } ) = \\tau ( \\xi , 1 ) \\wedge \\cdots \\wedge \\tau ( \\xi , l ) \\wedge \\tau ( \\eta , 1 ) \\wedge \\cdots \\wedge \\tau ( \\eta , l ( \\eta ) ) \\in \\bigwedge ^ i U , \\end{gather*}"}
+{"id": "379.png", "formula": "\\begin{align*} L _ \\infty ( M , \\omega ) _ N = \\frac { L _ \\infty ( M , \\omega ) _ { [ N ] } } { I _ { L _ \\infty } ( N ) } ~ . \\end{align*}"}
+{"id": "1737.png", "formula": "\\begin{align*} F & : = \\left \\| \\sum _ { k \\in \\Z } \\ \\frac { \\omega _ k } { \\sqrt { \\lambda _ k } } e ^ { 2 \\pi i k x } \\right \\| _ { L ^ p } ^ p \\chi _ { \\left ( \\sum _ { k \\in \\Z } \\frac { | \\omega _ k | ^ 2 } { \\lambda _ k } \\right ) ^ { 1 / 2 } \\leq B } \\\\ G & : = e ^ { \\frac { 2 } { p } \\| \\sum _ { k \\in \\Z } \\ \\frac { \\omega _ k } { \\sqrt { \\lambda _ k } } e ^ { 2 \\pi i k x } \\| _ { L ^ p } ^ p } \\ , \\chi _ { \\left ( \\sum _ { k \\in \\Z } \\frac { | \\omega _ k | ^ 2 } { \\lambda _ k } \\right ) ^ { 1 / 2 } \\leq B } \\ , . \\end{align*}"}
+{"id": "818.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow + \\infty } \\frac { 1 } { N } \\sum \\limits _ { j = 1 } ^ N \\int _ 0 ^ { \\tau _ j } e ^ { - \\beta t } f ( x _ j ( t ) ) d t = \\mathbb { E } \\int _ 0 ^ { \\tau } e ^ { - \\beta t } f ( x ( t ) ) d t \\end{align*}"}
+{"id": "4365.png", "formula": "\\begin{align*} \\begin{array} { l l } G _ i ^ 0 ( X ( T ) ) & = \\int _ { 0 } ^ { T } f _ i ^ 0 ( s , x ( s ) , u ( s ) ) d s + g _ i ^ 0 ( x ( T ) ) \\\\ \\null & \\ge G _ i ^ 0 ( X _ 0 ( T ) ) = \\int _ { 0 } ^ { T } f _ i ^ 0 ( s , x _ 0 ( s ) , u _ 0 ( s ) ) d s + g _ i ^ 0 ( x _ 0 ( T ) ) \\end{array} \\end{align*}"}
+{"id": "7081.png", "formula": "\\begin{gather*} d _ { > n } ( t _ 1 , \\dots , t _ \\ell ) \\in \\Big \\{ \\sum a _ { i _ 1 , \\dots , i _ \\ell } t _ 1 ^ { i _ 1 } \\cdots t _ \\ell ^ { i _ \\ell } \\ , | \\ , a _ { i _ 1 , \\dots , i _ \\ell } = 0 i _ 1 + \\cdots + i _ \\ell \\leq n \\Big \\} , \\\\ d _ { \\ge n } ( t _ 1 , \\dots , t _ \\ell ) \\in \\Big \\{ \\sum a _ { i _ 1 , \\dots , i _ \\ell } t _ 1 ^ { i _ 1 } \\cdots t _ \\ell ^ { i _ \\ell } \\ , | \\ , a _ { i _ 1 , \\dots , i _ \\ell } = 0 i _ 1 + \\cdots + i _ \\ell < n \\Big \\} . \\end{gather*}"}
+{"id": "4993.png", "formula": "\\begin{align*} ( z - r ) ( \\hat { f } ( z ) - \\hat { g } ( z ) ) = f ( z ) - g ( z ) - \\hat { g } ( z ) ( s - r ) \\end{align*}"}
+{"id": "6226.png", "formula": "\\begin{align*} h _ a [ \\psi ; p ] = \\int _ \\Sigma \\big ( | \\nabla \\psi ( x , z ) | ^ 2 + ( p + B x ) ^ 2 | \\psi ( x , z ) | ^ 2 \\big ) \\ , \\mathrm { d } x \\mathrm { d } z \\end{align*}"}
+{"id": "3696.png", "formula": "\\begin{align*} \\partial _ t \\rho ( x , t ) + \\nabla ( \\rho ( x , t ) \\nabla S ( x , t ) ) = 0 \\end{align*}"}
+{"id": "1702.png", "formula": "\\begin{align*} \\gamma _ p ( x _ 1 , \\ldots , x _ p ; y _ 1 , \\ldots , y _ p ) : = \\rho ( \\overline { \\varphi } ( y _ 1 ) \\ , \\ldots \\ , \\overline { \\varphi } ( y _ p ) \\ , \\varphi ( x _ 1 ) \\ , \\ldots \\ , \\varphi ( x _ p ) ) \\ , . \\end{align*}"}
+{"id": "8075.png", "formula": "\\begin{align*} \\mathrm { M } ^ V = d i a g ( M ^ { V , \\mu _ 1 } , \\dots , M ^ { V , \\mu _ N } ) . \\end{align*}"}
+{"id": "8205.png", "formula": "\\begin{align*} & [ \\Lambda ^ { \\alpha - 1 } , \\dot S _ { j - 1 } v \\cdot \\nabla ] \\dot { \\Delta } _ j \\sigma = \\sum _ { \\vert j - j ' \\vert \\le 4 } [ \\Lambda ^ { \\alpha - 1 } \\dot { \\Delta } _ { j ' } , \\dot { S } _ { j - 1 } v \\cdot \\nabla ] \\dot { \\Delta } _ j \\sigma \\\\ & = \\sum _ { | j ' - j | \\leq 4 } 2 ^ { j ' ( N + \\alpha - 1 ) } \\int _ { \\R ^ N } \\widetilde { h } ( 2 ^ { j ' } y ) \\big ( \\dot S _ { j - 1 } v ( x - y ) - \\dot S _ { j - 1 } v ( x ) \\big ) \\cdot \\nabla \\dot \\Delta _ j \\sigma ( x - y ) \\dd y , \\end{align*}"}
+{"id": "2026.png", "formula": "\\begin{align*} \\Delta _ t ( x ) = \\begin{cases} \\displaystyle { \\frac { 1 } { 2 } ( t - | x | ) } , & | x | \\leq t ; \\\\ [ 5 p t ] 0 , & | x | > t . \\end{cases} \\end{align*}"}
+{"id": "8073.png", "formula": "\\begin{align*} p | _ i & = \\rho _ i g \\left ( \\sum _ { j = i } ^ N D _ j + b - z + \\sum _ { j = 1 } ^ { i - 1 } \\frac { \\rho _ j } { \\rho _ i } D _ j \\right ) \\\\ & = \\rho _ i g \\left ( \\sum _ { j = 1 } ^ N D _ j + b - z + \\sum _ { j = 1 } ^ { i - 1 } \\frac { \\rho _ j - \\rho _ i } { \\rho _ i } D _ j \\right ) , \\end{align*}"}
+{"id": "1797.png", "formula": "\\begin{align*} { \\rm s i g n } \\left \\langle Y _ { 2 } , i _ { 1 } , i _ { 1 } + 1 , \\dots , i _ { r - 1 } + 1 , \\widehat { n } , n + 1 \\right \\rangle = - { \\rm s i g n } \\left \\langle Y _ { 2 } , i _ { 1 } , i _ { 1 } + 1 , \\dots , i _ { r - 1 } + 1 , n , \\widehat { n + 1 } \\right \\rangle \\end{align*}"}
+{"id": "292.png", "formula": "\\begin{align*} n _ { i ' } ' = \\begin{cases} n _ { i ' } & ( i ' \\neq i ) , \\\\ \\sum _ { i '' = 1 } ^ { r '' } n _ { i '' } & ( i ' = i ) \\end{cases} \\end{align*}"}
+{"id": "7302.png", "formula": "\\begin{align*} y : = x - t w , \\d y = t ^ { d } \\d w , \\end{align*}"}
+{"id": "1519.png", "formula": "\\begin{align*} \\begin{array} { c c } ( k , b , c ) = \\left ( \\frac { 2 m } { m ^ 2 - 1 } , - x _ i ^ 2 \\frac { ( m ^ 2 + 1 ) } { m ( m ^ 2 - 1 ) } , - \\frac { \\tau ^ 6 + 2 \\tau ^ 5 + 4 \\tau ^ 4 + 8 \\tau ^ 3 + 9 \\tau ^ 2 + 4 \\tau + 1 } { 4 \\tau ^ 2 ( \\tau + 1 ) ^ 2 } \\right ) , m \\in \\mathbb { Q } \\setminus \\{ 0 , \\pm 1 \\} , i = 1 , 2 , 3 & \\textrm { i f } n = 4 \\\\ \\end{array} \\end{align*}"}
+{"id": "6434.png", "formula": "\\begin{align*} \\mathcal { P } _ n ^ { ( \\alpha , \\beta ) } ( x ) = { } _ 2 \\phi _ 1 \\left ( \\begin{gathered} q ^ { - n } , \\alpha \\beta q ^ { n + 1 } \\\\ \\alpha q \\end{gathered} ; \\ , q , q x \\right ) . \\end{align*}"}
+{"id": "5767.png", "formula": "\\begin{align*} & - \\frac { 1 1 0 7 7 3 0 0 8 } { 1 1 2 8 1 0 6 3 } < a \\leq - \\frac { 6 6 4 7 9 } { 9 5 5 8 } , - \\frac { a } { 2 } < b < - \\frac { 4 ( 2 6 5 4 5 0 1 a - 1 3 8 4 6 6 2 6 ) } { 3 2 5 1 7 0 7 1 } , \\\\ & - \\frac { 2 1 3 5 0 0 a + 1 1 5 5 5 9 b } { 2 5 6 4 1 9 } < c \\leq - \\frac { 5 8 3 ( 4 a + 7 b ) - 5 8 3 2 } { 1 4 5 8 } \\end{align*}"}
+{"id": "1898.png", "formula": "\\begin{align*} \\Theta _ { K / F , S } ^ { E ( m ) } ( 0 ) = \\Theta _ { K / F , S } ^ E ( m ) . \\end{align*}"}
+{"id": "7315.png", "formula": "\\begin{align*} \\frac { q ^ m - 1 } { 2 } - q ^ { m - 1 } = l q ^ { m - j } + h . \\end{align*}"}
+{"id": "5925.png", "formula": "\\begin{align*} n \\cdot \\alpha = j _ * ( Z _ * ( \\alpha ) ) \\end{align*}"}
+{"id": "6088.png", "formula": "\\begin{align*} \\hat \\alpha ^ { - 1 } ( a _ 0 , a _ 1 , a _ 2 , \\cdots ) = ( a _ 1 , a _ 2 , a _ 3 , \\cdots ) . \\end{align*}"}
+{"id": "478.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { \\binom n k L _ { j k + m } B _ { n - k } z ^ k } = \\alpha ^ m B _ n ( \\alpha ^ j z ) + \\beta ^ m B _ n ( \\beta ^ j z ) . \\end{align*}"}
+{"id": "4497.png", "formula": "\\begin{align*} \\delta V = \\int _ 0 ^ 1 u ' ( \\delta u ) ' = u ' \\delta u \\big | _ 0 ^ 1 - \\int _ 0 ^ 1 u '' \\delta u = u ' ( 1 ) \\delta u ( 1 ) - u ' ( 0 ) \\delta u ( 0 ) - \\int _ 0 ^ 1 u '' \\delta u \\ , , \\end{align*}"}
+{"id": "154.png", "formula": "\\begin{align*} \\begin{aligned} K f ( v ) = K _ 1 f ( v ) - K _ 2 f ( v ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "4678.png", "formula": "\\begin{align*} D _ { z } f ( z ) & = \\partial _ z f + \\lambda \\frac { f ( z ) - f ( \\bar { z } ) } { z - \\bar { z } } , \\\\ D _ { \\bar { z } } f ( z ) & = \\partial _ { \\bar { z } } f - \\lambda \\frac { f ( z ) - f ( \\bar { z } ) } { z - \\bar { z } } . \\end{align*}"}
+{"id": "8184.png", "formula": "\\begin{align*} \\alpha ^ 4 ( | a _ 1 | ^ 2 + | b _ 1 | ^ 2 ) = \\alpha ^ 2 [ ( | a _ 1 | ^ 2 + | a _ 2 | ^ 2 ) | b _ 1 | ^ 2 + ( | b _ 1 | ^ 2 + | b _ 2 | ^ 2 ) | a _ 1 | ^ 2 ] . \\end{align*}"}
+{"id": "1954.png", "formula": "\\begin{align*} \\| f \\| _ { \\dot { F } ^ s _ { \\vec { p } q } ( \\vec { a } ) } : = \\Big \\| \\Big ( \\sum _ { j \\in \\mathbb { Z } } ( 2 ^ { s j } | \\varphi _ { j } \\ast f | ) ^ q \\Big ) ^ { 1 / q } \\Big \\| _ { \\vec { p } } < \\infty , \\end{align*}"}
+{"id": "3749.png", "formula": "\\begin{align*} A ^ 3 = - 4 A ^ 2 + 1 2 A + 2 0 8 J _ { 6 0 } - 8 I _ 3 \\otimes J _ { 2 0 } = 2 8 A - 4 8 I _ { 6 0 } + 1 7 2 J _ { 6 0 } - 1 2 I _ 3 \\otimes J _ { 2 0 } . \\end{align*}"}
+{"id": "4223.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\left ( \\exp \\left ( i \\left ( k X _ s ^ { ( \\Theta ) } + t \\right ) \\left ( \\frac { \\theta ( \\gamma _ j ) + 2 \\pi \\ell } { k } \\right ) \\right ) \\right ) _ { j = 1 } ^ g = \\left ( \\exp \\left ( i ( \\theta _ j + 2 \\pi [ \\ell t ] / k ) \\right ) \\right ) _ { j = 1 } ^ g , \\end{align*}"}
+{"id": "111.png", "formula": "\\begin{align*} h \\leq h _ 0 : = \\min \\Bigl \\{ 1 , \\tfrac 1 { 1 0 0 } \\bigl [ \\| \\psi _ 0 \\| _ { L ^ 2 } ^ 2 + \\| \\phi _ 0 \\| _ { L ^ 2 } ^ 2 \\bigr ] ^ { - 1 } \\Bigr \\} . \\end{align*}"}
+{"id": "7416.png", "formula": "\\begin{align*} \\left [ a , \\ , a ^ \\dagger \\right ] = \\hat { I } , [ a , \\ , \\hat { I } ] = 0 = [ a ^ \\dagger , \\ , \\hat { I } ] . \\end{align*}"}
+{"id": "6756.png", "formula": "\\begin{align*} [ a _ k , a ^ * _ l ] & = \\delta ( k - l ) , [ a _ k , a _ l ] = [ a ^ * _ k , a ^ * _ l ] = 0 \\end{align*}"}
+{"id": "4507.png", "formula": "\\begin{align*} \\{ \\Phi _ x , \\Pi _ y \\} = \\langle D _ 1 \\Phi _ x , D _ 2 \\Pi _ y \\rangle _ { H ^ 1 } - \\langle D _ 1 \\Pi _ x , D _ 2 \\Phi _ y \\rangle _ { H ^ 1 } \\ , . \\end{align*}"}
+{"id": "1383.png", "formula": "\\begin{align*} M ( b , c ; s ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( b ) _ n } { ( c ) _ n } \\frac { s ^ n } { n ! } , s \\in [ 0 , \\infty ) , \\end{align*}"}
+{"id": "6808.png", "formula": "\\begin{align*} \\alpha ( d , d , x _ 3 , x _ 4 , x _ 5 ) = \\sum _ { i \\in [ q ] , j \\in [ t ] } \\lambda _ { i j } \\tau _ i ( d , d , x _ 3 ) \\tilde { \\gamma } _ j ( x _ 4 , x _ 5 ) + & \\sum _ { i \\in [ q ] , j \\in [ t ] } \\lambda ' _ { i j } \\tau _ i ( d , d , x _ 4 ) \\tilde { \\gamma } _ j ( x _ 3 , x _ 5 ) \\\\ + & \\sum _ { i \\in [ q ] , j \\in [ t ] } \\lambda '' _ { i j } \\tau _ i ( d , d , x _ 5 ) \\tilde { \\gamma } _ j ( x _ 3 , x _ 4 ) \\end{align*}"}
+{"id": "879.png", "formula": "\\begin{align*} & \\inf \\bigg \\{ k > 0 : ~ ( L ) \\int _ { \\mathcal { J } } \\theta \\bigg ( t , ( \\xi _ n - \\xi ) ( t ) \\bigg ) d \\overline { \\mu } \\leq 1 \\bigg \\} \\to 0 \\\\ & \\implies ~ ( L ) \\int _ { \\mathcal { J } } \\theta \\big ( t , ( \\xi _ n - \\xi ) ( t ) \\big ) \\to 0 \\\\ & ~ i . e . , ~ \\theta ( t , ( \\xi _ n - \\xi ) ( t ) = 0 ~ \\overline { \\mu } - a . e . ~ a s ~ n \\to \\infty \\\\ & \\xi _ n - \\xi = 0 ~ a . e . ~ a s ~ n \\to \\infty \\end{align*}"}
+{"id": "6793.png", "formula": "\\begin{align*} \\alpha ' ( x _ 1 , x _ 2 , x _ 3 , \\dots , x _ k ) + \\alpha ' ( x _ 1 , x _ 2 , x _ 3 , \\dots , x _ k ) = \\sum _ { i \\in [ r _ 2 ] } \\gamma _ i ( x _ 1 , x _ { J _ { i , 1 } } ) \\gamma ' _ i ( x _ 2 , x _ { J _ { i , 2 } } ) \\gamma '' _ { i , 1 } ( x _ { J ' _ { i , 1 } } ) \\cdots \\gamma '' _ { i , d ' _ i } ( x _ { J ' _ { i , d ' _ i } } ) . \\end{align*}"}
+{"id": "6275.png", "formula": "\\begin{align*} & 0 \\in \\lambda \\partial \\norm { \\cdot } _ 1 ( x ^ * ) + \\nabla h ( x ^ * ) + p = \\lambda \\partial \\norm { \\cdot } _ 1 ( x ^ * ) + \\left [ \\alpha \\left ( 1 + \\norm { x ^ * } ^ 2 \\right ) ^ { - 1 / 2 } + \\beta \\right ] x ^ * + p \\\\ \\Leftrightarrow ~ ~ & ( \\forall i \\in \\{ 1 , \\ldots , n \\} ) \\left ( \\exists u _ i \\in \\partial | \\cdot | ( x ^ * _ i ) \\right ) ~ 0 = v _ i + \\left [ \\alpha \\left ( 1 + \\norm { x ^ * } ^ 2 \\right ) ^ { - 1 / 2 } + \\beta \\right ] x ^ * _ i , \\end{align*}"}
+{"id": "6303.png", "formula": "\\begin{align*} \\square [ \\underline { n } ] = \\square [ n _ 1 ] * \\dots * \\square [ n _ p ] \\end{align*}"}
+{"id": "779.png", "formula": "\\begin{align*} \\left ( M \\otimes _ { \\alpha _ { X , Y } ^ { } } \\left ( F / L \\right ) ^ * \\right ) ^ * \\stackrel { 1 } { = } \\mathcal { L } _ { X ; Y ^ { \\rm d u a l } } \\left ( M ; \\left ( F / L \\right ) ^ { * * } \\right ) \\stackrel { 1 } { = } \\mathcal { L } _ { X ; Y ^ { \\rm d u a l } } \\left ( M ; F / L \\right ) . \\end{align*}"}
+{"id": "7132.png", "formula": "\\begin{align*} D : \\chi = \\sum _ { 1 \\leq j < i \\leq d } c _ { i j } ( \\beta _ i - \\beta _ j ) + \\sum _ { 1 \\leq i \\leq d } d _ i \\beta _ i , \\end{align*}"}
+{"id": "4827.png", "formula": "\\begin{align*} \\nabla \\cdot ( \\rho \\nabla g ) - \\rho = - \\bar \\rho \\otimes \\mu , \\end{align*}"}
+{"id": "3986.png", "formula": "\\begin{align*} q _ { \\beta } ( n , t ) = \\sum _ { k = 1 } ^ { n } \\sum _ { \\Lambda _ { n } ^ { k } } k ! \\prod _ { j = 1 } ^ { n - k + 1 } \\frac { ( \\theta ^ { j } / j ! ) ^ { x _ { j } } } { x _ { j } ! } \\left ( \\alpha t ^ { \\beta } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } \\left ( - \\alpha ( e ^ { \\theta } - 1 ) t ^ { \\beta } \\right ) . \\end{align*}"}
+{"id": "7358.png", "formula": "\\begin{align*} | \\mathcal { I } ' _ { k } | & = \\sum ^ k _ { i = 0 } | \\mathcal { B } _ { i , i } | + | \\{ ( i , i , t , p ) \\mid ( i , i , t , p ) \\in \\mathcal { I } ' _ { k - 1 } \\} | + | \\mathcal { I } ' _ { k - 1 } | \\\\ [ 0 . 1 c m ] & = \\sum ^ k _ { i = 0 } | \\mathcal { B } _ { i , i } | + \\sum ^ { k - 1 } _ { i = 0 } | \\mathcal { B } _ { i , i } | + | \\mathcal { I } ' _ { k - 1 } | \\\\ [ 0 . 1 c m ] & = \\frac { ( k + 1 ) ( k + 2 ) ( k + 3 ) } { 6 } + { k + 3 \\choose 4 } = { k + 4 \\choose 4 } , \\ \\end{align*}"}
+{"id": "178.png", "formula": "\\begin{align*} \\begin{aligned} | K _ 2 ^ \\chi f ( v ) | \\leq W _ 1 ( v ) + W _ 2 ( v ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "4623.png", "formula": "\\begin{align*} \\lim _ { | y | \\to 0 } \\frac { \\phi _ + ' ( x , | y | ) } { | y | } y \\cdot z = \\Big ( \\lim _ { | y | \\to 0 ^ + } \\phi _ + ' ( x , | y | ) \\Big ) \\Big ( \\lim _ { | y | \\to 0 ^ + } \\frac { y } { | y | } \\cdot z \\Big ) = 0 \\end{align*}"}
+{"id": "2708.png", "formula": "\\begin{align*} \\phi ( 2 n , m , r - 1 ) = \\sum _ { k = 0 } ^ { 2 n } ( - 1 ) ^ k \\binom { 2 n } { k } ^ m ( \\frac { 1 } { 2 } S ( k , r ) ) ( \\frac { 1 } { 2 } S ( 2 n - k , r ) ) \\end{align*}"}
+{"id": "7278.png", "formula": "\\begin{align*} \\frac { d } { d t } X ( t , \\omega ) = v _ E ( t , X ( t , \\omega ) ) . \\end{align*}"}
+{"id": "3507.png", "formula": "\\begin{align*} \\Big \\Vert \\nabla \\mathrm { H } \\Big \\Vert _ { \\mathbb { L } ^ 2 ( \\Omega ) } ^ 2 = \\sum _ { \\substack { \\mathrm { n } \\in \\mathbb { N } \\\\ \\mathrm { i } = 1 , 2 , 3 } } \\Big | \\Big \\langle \\nabla \\mathrm { H } ; \\mathrm { e } ^ { ( \\mathrm { i } ) } _ { \\mathrm { n } } \\Big \\rangle _ { \\mathbb { L } ^ { 2 } ( \\Omega ) } \\Big | ^ 2 . \\end{align*}"}
+{"id": "272.png", "formula": "\\begin{align*} \\frac { { \\rm R e } \\ m _ u ( z ) } { | z | ^ 2 - 1 } = \\frac { 1 } { N } \\sum _ j \\frac { 1 } { | z - z _ j ( u ) | ^ 2 } , \\end{align*}"}
+{"id": "5341.png", "formula": "\\begin{align*} V = [ y , F ( G ) ] = V _ 1 \\times \\cdots \\times V _ k \\end{align*}"}
+{"id": "808.png", "formula": "\\begin{align*} \\mathbb { E } \\Big ( e ^ { - \\beta \\tau ^ { * } } \\Big ) = e ^ { - \\lambda b ' } \\end{align*}"}
+{"id": "2999.png", "formula": "\\begin{align*} F ( y ) = c _ 0 - \\ln x . \\end{align*}"}
+{"id": "794.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d x _ i ( t ) = \\alpha x _ i ( t ) d t + \\sigma x _ i ( t ) d W _ i ( t ) , \\\\ & x _ i ( 0 ) = x . \\end{aligned} \\right . \\end{align*}"}
+{"id": "1939.png", "formula": "\\begin{align*} \\phi _ n ( z ) = \\phi ( t / n ^ 2 , x / n ^ 3 , v / n ) , u _ n = u \\phi _ n , \\vec f _ n = \\vec f \\phi _ n . \\end{align*}"}
+{"id": "6026.png", "formula": "\\begin{align*} { L } ^ { \\pi ^ \\prime } ( T ( n ) ) = & x + X ( T ( 1 ) ) - \\inf _ { s \\in [ 0 , T ( 1 ) ] } X ( s ) + \\sum _ { k = 2 } ^ n ( X ( T ( k ) ) - \\inf _ { s \\in [ T ( k - 1 ) , T ( k ) ] } X ( s ) ) \\\\ \\geq & X ( T ( 1 ) ) + \\sum _ { k = 2 } ^ n ( X ( T ( k ) ) - X ( T ( k - 1 ) ) ) = X ( T ( n ) ) . \\end{align*}"}
+{"id": "2441.png", "formula": "\\begin{align*} \\bigoplus _ { d = 1 } ^ { \\infty } \\bigoplus _ { \\substack { l _ { 1 } , \\ldots , l _ { d } \\geq 2 \\\\ \\# \\{ l _ { i } : { \\rm e v e n } \\} = 1 } } \\mathbb { Q } \\cdot x _ { l _ { 1 } } \\cdots x _ { l _ { d } } . \\end{align*}"}
+{"id": "427.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } | \\Lambda _ { n } ( m ) | t ^ { n } = \\frac { m ! t ^ { m } } { ( 1 - t ) ^ { m } } \\prod _ { n = 2 } ^ { \\infty } \\frac { 1 } { 1 - t ^ { n } } , \\end{align*}"}
+{"id": "6398.png", "formula": "\\begin{align*} M ^ { ( \\omega ) } ( \\alpha ) M ^ { ( \\omega ) } ( \\beta ) = q ^ { \\Pi ^ { ( \\omega ) } ( \\alpha , \\beta ) / 2 } M ^ { ( \\omega ) } ( \\alpha + \\beta ) \\end{align*}"}
+{"id": "5352.png", "formula": "\\begin{align*} ( S \\wedge S ) \\circ ( S \\wedge S ) & = 2 S ^ 2 \\wedge S ^ 2 , \\\\ ( S \\wedge S ) \\circ ( S \\wedge T ) & = 0 , \\\\ ( S \\wedge S ) \\circ ( T \\wedge T ) & = 0 , \\\\ ( S \\wedge T ) \\circ ( S \\wedge T ) & = S ^ 2 \\wedge T ^ 2 . \\end{align*}"}
+{"id": "578.png", "formula": "\\begin{align*} \\pi _ 0 ( e ) = \\pi _ 0 ( e ^ 2 ) = \\pi _ 0 ( e ) ^ 2 + \\sum _ { \\omega \\in W _ f ^ + } \\pi _ { - \\omega } ( e ) \\pi _ { \\omega } ( e ) . \\end{align*}"}
+{"id": "5765.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } a < b < - \\frac { 4 } { 7 } a . \\end{align*}"}
+{"id": "2252.png", "formula": "\\begin{align*} \\sigma ^ i ( a ) d = \\sigma ^ i ( a \\sigma ^ { - i } ( d ) ) = \\sigma ^ i ( \\sigma ^ { - i } ( d ) a ) = d \\sigma ^ i ( a ) , \\end{align*}"}
+{"id": "5822.png", "formula": "\\begin{align*} \\omega ( V ) = \\omega ( \\Pi V \\Pi ^ * ) = \\omega ( M _ z \\oplus U ) = 1 . \\end{align*}"}
+{"id": "3894.png", "formula": "\\begin{align*} \\partial _ t u - ( 1 + D _ t ^ { \\{ m \\} } ) \\Delta u & = f ( u , \\mathcal H u ) \\Omega , t \\in ( 0 , T ) , \\\\ u & = 0 \\partial \\Omega , \\ ; t \\ge 0 , \\\\ u ( 0 ) & = \\xi \\Omega , \\end{align*}"}
+{"id": "7610.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 ) = 2 | x _ 1 - x _ 2 | + \\theta x _ 1 + 3 | 1 - x 2 | \\end{align*}"}
+{"id": "1274.png", "formula": "\\begin{align*} w _ j = 2 v _ j + m _ j F \\in \\tilde M . \\end{align*}"}
+{"id": "7530.png", "formula": "\\begin{align*} \\mathbb V = ( a + i b ) \\frac { \\partial } { \\partial z } + ( a - i b ) \\frac { \\partial } { \\partial \\overline z } = \\mathbb V ^ { 1 , 0 } + \\mathbb V ^ { 0 , 1 } . \\end{align*}"}
+{"id": "1755.png", "formula": "\\begin{align*} j _ { 1 } = 1 \\ , \\ , { \\rm a n d } \\ , \\ , I _ { 1 } = \\left \\{ j _ { 1 } + 1 , j _ { 2 } , , j _ { 2 } + 1 \\dots , j _ { l _ { 1 } } + 1 \\right \\} , \\dots , I _ { r } = \\left \\{ j _ { l _ { r - 1 } } , j _ { l _ { r - 1 } } + 1 , \\dots , j _ { \\frac { m } { 2 } - 1 } , j _ { \\frac { m } { 2 } - 1 } + 1 \\right \\} . \\end{align*}"}
+{"id": "8155.png", "formula": "\\begin{gather*} V ^ { \\otimes \\lambda } : = V ^ { \\otimes \\lambda _ 1 } \\otimes \\cdots \\otimes V ^ { \\otimes \\lambda _ p } \\\\ \\Lambda ^ { \\lambda } ( V ) : = \\Lambda ^ { \\lambda _ 1 } ( V ) \\otimes \\cdots \\otimes \\Lambda ^ { \\lambda _ p } ( V ) , \\\\ S ^ { \\lambda } ( V ) : = S ^ { \\lambda _ 1 } ( V ) \\otimes \\cdots \\otimes S ^ { \\lambda _ p } ( V ) . \\end{gather*}"}
+{"id": "4214.png", "formula": "\\begin{align*} & \\sum _ { X = 1 } ^ \\infty ( Q _ k ( X ) - \\mathrm { M T } _ k ( X ) + c X ^ { r - 1 } q ^ { X / 2 k } ) u ^ { X - 1 } \\\\ & = \\frac { Z ( u ) } { Z ( u ^ k ) ( 1 - u ) } - \\frac { d q } { 1 - q u } + \\frac { c } { u ( 1 - q ^ { 1 / 2 k } u ) ^ r } \\sum _ { \\ell = 0 } ^ { r - 1 } A ( r - 1 , \\ell ) q ^ { ( \\ell + 1 ) / 2 k } u ^ { \\ell + 1 } . \\end{align*}"}
+{"id": "5619.png", "formula": "\\begin{align*} \\Omega ^ { \\alpha } _ { \\beta } = R ^ { \\alpha } _ { \\beta ; \\mu \\bar { \\nu } } d z ^ { \\mu } \\wedge d z ^ { \\bar { \\nu } } + S ^ { \\alpha } _ { \\beta \\mu ; \\bar { \\nu } } \\delta v ^ { \\mu } \\wedge d z ^ { \\bar { \\nu } } + P ^ { \\alpha } _ { \\beta \\bar { \\nu } ; \\mu } d z ^ { \\mu } \\wedge \\delta v ^ { \\bar { \\nu } } + Q ^ { \\alpha } _ { \\beta \\mu \\bar { \\nu } } \\delta v ^ { \\mu } \\wedge \\delta v ^ { \\bar { \\nu } } . \\end{align*}"}
+{"id": "4841.png", "formula": "\\begin{align*} \\partial _ t \\eta = \\nabla \\cdot ( \\eta \\nabla F ( x ) ) \\end{align*}"}
+{"id": "1227.png", "formula": "\\begin{align*} \\tilde L _ 1 ( h , k ) ( x ) = h ( x ) + \\int _ { X ^ + } k ( y ) \\dd Q ^ x ( y ) , \\tilde L _ 2 ( h , k ) ( y ) = k ( y ) + \\int _ { X ^ - } h ( x ) \\dd Q _ y ( x ) , \\end{align*}"}
+{"id": "2710.png", "formula": "\\begin{align*} M _ { \\phi } ( 2 n , j , t ; r - 1 ) = \\frac { 1 } { 4 } M _ { \\varPsi } ( 2 n , j , t ; r - 1 ) \\end{align*}"}
+{"id": "3464.png", "formula": "\\begin{align*} \\Big \\Vert \\psi \\Big \\Vert _ { H ^ { - 1 , - \\frac { 1 } { 2 } } \\Big ( \\Omega \\times \\mathbb { R _ + } \\Big ) } = \\sup _ { 0 \\neq u \\in H ^ { 1 , \\frac { 1 } { 2 } } \\Big ( \\Omega \\times \\mathbb { R _ + } \\Big ) } \\dfrac { \\Big | \\Big \\langle \\psi , u \\Big \\rangle _ { \\Omega \\times \\mathbb { R _ + } } \\Big | } { \\Big \\Vert u \\Big \\Vert _ { \\mathrm { H } ^ { 1 , \\frac { 1 } { 2 } } \\Big ( \\Omega \\times \\mathbb { R _ + } \\Big ) } } . \\end{align*}"}
+{"id": "2956.png", "formula": "\\begin{align*} \\widehat N ^ p _ C ( F ( \\bar x ) ; \\nabla F ( \\bar x ) u ) & = N _ C ( F ( \\bar x ) ; \\nabla F ( \\bar x ) u ) = N _ C ( F ( \\bar x ) ) \\cap \\{ \\nabla F ( \\bar x ) u \\} ^ \\perp \\\\ & = \\left ( \\prod _ { i = 1 } ^ \\ell N _ { \\mathcal Q _ { m _ i } } ( F _ i ( \\bar x ) ) \\right ) \\cap \\{ \\nabla F ( \\bar x ) u \\} ^ \\perp \\supset \\prod _ { i = 1 } ^ \\ell N _ { \\mathcal Q _ { m _ i } } ( F _ i ( \\bar x ) ) \\cap \\{ \\nabla F _ i ( \\bar x ) u \\} ^ \\perp . \\end{align*}"}
+{"id": "8135.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac 1 n \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ^ { \\otimes n } \\otimes \\overline M ) = \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ) \\end{align*}"}
+{"id": "6446.png", "formula": "\\begin{align*} \\int _ u ^ v \\frac { x ^ n ( q x / u , q x / v ; q ) _ { \\infty } } { ( b x , c x ; q ) _ { \\infty } } d _ q x = \\frac { ( 1 - q ) v ( q , u / v , q v / u , b c u v ; q ) _ { \\infty } } { ( b u , b v , c u , c v ; q ) _ { \\infty } } W _ n ( b , c , u , v ) . \\end{align*}"}
+{"id": "6044.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } - \\Delta \\eta - k ^ 2 \\eta = 0 \\ , \\ , \\ , \\ , \\Omega \\\\ \\eta = 0 \\ , \\ , \\ , \\ , \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "92.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ \\Psi v \\geq 0 & \\O \\\\ \\partial _ { \\vec \\nu } v = - 2 \\mathrm { I I } ( X , X ) \\leq 0 & \\partial \\O . \\end{cases} \\end{align*}"}
+{"id": "7620.png", "formula": "\\begin{align*} \\Delta ^ \\vee : = \\{ n \\in N _ { \\R } \\ \\vert \\ \\langle m , n \\rangle \\geq - 1 \\ \\forall m \\in \\Delta \\} \\end{align*}"}
+{"id": "6341.png", "formula": "\\begin{align*} \\displaystyle { \\sum _ { i = 1 } ^ N p _ i = 1 } , \\ ; \\ ; \\ ; 0 \\leq p _ i \\leq 1 , \\ ; \\ ; i = 1 , 2 , . . . , N . \\end{align*}"}
+{"id": "6390.png", "formula": "\\begin{align*} b ' _ { i j } = b _ { \\sigma ^ { - 1 } ( i ) , \\sigma ^ { - 1 } ( j ) } , \\pi ' _ { i j } = \\pi _ { \\sigma ^ { - 1 } ( i ) , \\sigma ^ { - 1 } ( j ) } , A ' _ i = A _ { \\sigma ^ { - 1 } ( i ) } . \\end{align*}"}
+{"id": "7501.png", "formula": "\\begin{align*} \\iint \\frac { h ( s ) - h ( t ) } { s - t } d \\mu ( s ) d \\mu ( t ) = \\int h ( s ) V ' ( s ) d \\mu ( s ) \\end{align*}"}
+{"id": "5463.png", "formula": "\\begin{align*} ( A ) _ 0 & : = ( A ; q ) _ 0 = 1 , \\\\ ( A ) _ n & : = ( A ; q ) _ n = ( 1 - A ) ( 1 - A q ) \\cdots ( 1 - A q ^ { n - 1 } ) , n \\geq 1 , \\\\ ( A ) _ { \\infty } & : = ( A ; q ) _ { \\i } = \\lim _ { n \\to \\i } ( A ; q ) _ n , | q | < 1 , \\\\ ( A _ 1 , A _ 2 , \\cdots , A _ m ; q ) _ n & : = ( A _ 1 ; q ) _ n ( A _ 2 ; q ) _ n \\cdots ( A _ m ; q ) _ n . \\end{align*}"}
+{"id": "7321.png", "formula": "\\begin{align*} \\mathcal { U } [ 0 , T ] : = \\{ u ( \\cdot ) | u ( \\cdot ) \\mathbb { F } u ( t ) \\in U , \\mathbb { P } - a . s . , \\forall t \\in \\lbrack 0 , T ] \\} , \\end{align*}"}
+{"id": "5956.png", "formula": "\\begin{align*} ( \\xi _ { A _ 1 } - \\xi _ { A _ 1 } ) \\cdots ( \\xi _ { A _ 1 } - \\xi _ { A _ k } ) = \\sum _ { j = 0 } ^ { k } ( - 1 ) ^ { j } e _ { j } ( A ) \\xi _ { A _ 1 } ^ { k - j } . \\end{align*}"}
+{"id": "7662.png", "formula": "\\begin{align*} F ( \\kappa _ { E _ p } ) = P ( E _ p ) - \\kappa _ { E _ p } | E _ p | \\end{align*}"}
+{"id": "5040.png", "formula": "\\begin{align*} \\mathbb { P } ( \\mathcal { F } ) : = \\textrm { P r o j } ( \\oplus _ { m \\geqslant 0 } \\textrm { S y m } ^ { m } \\mathcal { F } ) \\end{align*}"}
+{"id": "741.png", "formula": "\\begin{align*} J a = \\frac { { { c _ p } \\left ( { { T _ w } - { T _ { s a t } } } \\right ) } } { { { h _ { l v } } } } , \\end{align*}"}
+{"id": "2693.png", "formula": "\\begin{align*} M _ Q ( n , j , 0 ; a ) & = \\sum _ { k = j } ^ { n - j } \\binom { n - j } { k - j } \\binom { n - k } { n - k - j } \\binom { a + k } { a } \\binom { a + n - k } { a } G ( n , k , a ) \\\\ & = \\sum _ { k = j } ^ { n - j } \\binom { n - j } { k - j } \\bigl { ( } \\binom { a + n - k } { a } \\binom { n - k } { n - k - j } \\bigr { ) } \\binom { a + k } { a } G ( n , k , a ) \\end{align*}"}
+{"id": "2255.png", "formula": "\\begin{align*} u y ^ m u ^ { - 1 } x ^ m = u \\sigma ^ { - m } ( u ^ { - 1 } ) y ^ m x ^ m = y ^ m x ^ m = b \\end{align*}"}
+{"id": "4898.png", "formula": "\\begin{align*} \\left ( \\frac { 2 } { 3 } \\right ) ^ n p _ n \\left ( \\frac { 1 } { 4 } \\right ) = \\sum _ { k = 0 } ^ n B _ { n - k } ^ { ( - k ) } . \\end{align*}"}
+{"id": "6267.png", "formula": "\\begin{align*} M _ { 1 , k } = \\begin{cases*} p _ { - 1 } - \\frac { \\alpha _ { k } } { 2 \\lambda } - \\frac { b c } { 2 } - \\frac { a \\lambda } { 2 } , & \\\\ \\frac { b - \\alpha _ k } { 2 \\lambda } - \\frac { ( b + 1 ) c } { 2 } - p _ { - 1 } , & \\end{cases*} \\end{align*}"}
+{"id": "5579.png", "formula": "\\begin{align*} \\overline { ( I + J ) ^ k } = \\sum _ { \\ell = 0 } ^ k \\overline { I ^ \\ell } \\cdot \\overline { J ^ { k - \\ell } } . \\end{align*}"}
+{"id": "2190.png", "formula": "\\begin{align*} H ^ f _ { 1 1 } H ^ f _ { 2 2 } - H ^ f _ { 1 2 } { } ^ 2 + K ( f _ 1 { } ^ 2 G - 2 f _ 1 f _ 2 F + f _ 2 { } ^ 2 E ) = c K \\Delta \\end{align*}"}
+{"id": "3630.png", "formula": "\\begin{align*} f ' ( b ) = d \\end{align*}"}
+{"id": "6656.png", "formula": "\\begin{align*} Q ( t ) = ( 1 - t ) \\det \\begin{bmatrix} s ' _ { 0 } & s ' _ { 1 } & \\ldots & s ' _ { m - 1 } & 1 \\\\ s ' _ { 1 } & s ' _ { 2 } & \\ldots & s ' _ { m } & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ s ' _ { m } & s ' _ { m + 1 } & \\ldots & s ' _ { 2 m - 1 } & t ^ { m } \\end{bmatrix} , \\end{align*}"}
+{"id": "5959.png", "formula": "\\begin{align*} | e _ { J - i } ( B ) p _ { i + 1 } ( B ) - e _ { J - i } ( A ) p _ { i + 1 } ( A ) | & = | e _ { J - i } ( B ) ( p _ { i + 1 } ( B ) - p _ { i + 1 } ( A ) ) + p _ { i + 1 } ( A ) ( e _ { J - i } ( B ) - e _ { J - i } ( A ) ) | \\\\ & \\leq \\max ( | p _ { i + 1 } ( B ) - p _ { i + 1 } ( A ) | , | e _ { J - i } ( B ) - e _ { J - i } ( A ) | ) \\leq \\delta \\end{align*}"}
+{"id": "4123.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 x J _ { \\alpha } ( \\lambda _ i ^ { ( \\alpha ) } x ) J _ { \\alpha } ( \\lambda _ j ^ { ( \\alpha ) } x ) d x = \\frac { | J _ { \\alpha + 1 } ( \\lambda _ j ^ { ( \\alpha ) } ) | ^ 2 } { 2 } \\delta _ { i , j } . \\end{align*}"}
+{"id": "6239.png", "formula": "\\begin{align*} H = - \\partial _ x ^ 2 + ( - i \\partial _ y + B x ) ^ 2 - \\partial _ z ^ 2 \\end{align*}"}
+{"id": "5983.png", "formula": "\\begin{align*} U _ r ^ b ( t ) = X ( t ) , t \\in [ 0 , T _ 0 ^ + ( 1 ) ) , \\end{align*}"}
+{"id": "936.png", "formula": "\\begin{align*} \\begin{aligned} x _ n ^ * - x _ n ^ 0 = \\ , & x _ n - \\sum _ { \\beta = 1 } ^ \\alpha x _ \\beta \\rho _ \\beta ( 0 , \\tilde { x } ) - \\rho ( 0 , \\tilde { x } ) \\\\ > \\ , & \\rho ( \\hat { x } , \\tilde { x } ) - \\sum _ { \\beta = 1 } ^ \\alpha x _ \\beta \\rho _ \\beta ( 0 , \\tilde { x } ) - \\rho ( 0 , \\tilde { x } ) \\geq 0 \\end{aligned} \\end{align*}"}
+{"id": "4037.png", "formula": "\\begin{align*} \\left | q _ { m } \\right | \\le I ^ { - \\delta } ( s ) , \\forall \\ ; 0 \\leq m \\leq [ M ] , \\ ; \\ ; m \\not = 2 k , \\end{align*}"}
+{"id": "3213.png", "formula": "\\begin{align*} 2 B ( u _ { n } ) + ( p - 2 ) C ( u _ { n } ) = 0 . \\end{align*}"}
+{"id": "88.png", "formula": "\\begin{align*} \\Delta _ { \\Psi } v = X ( \\Delta _ { \\Psi } u ) = X ( f ( u ) ) = f ' ( u ) X ( u ) = f ' ( u ) v . \\end{align*}"}
+{"id": "1187.png", "formula": "\\begin{align*} \\chi ( g ) = \\lim _ { l \\to \\infty } \\chi _ { _ { ^ { k ( l ) } \\ ! { \\scriptstyle \\lambda } } } ( g ) ~ ~ g \\in \\mathfrak { S } _ { \\widehat { \\mathbf { n } } } . \\end{align*}"}
+{"id": "1233.png", "formula": "\\begin{align*} \\theta _ 1 , \\theta _ 2 \\in \\mathbb { R } , \\cos ( \\theta _ 1 ) + \\cos ( \\theta _ 2 ) = 2 \\cos \\Big ( \\frac { \\theta _ 1 + \\theta _ 2 } { 2 } \\Big ) \\cos \\Big ( \\frac { \\theta _ 1 - \\theta _ 2 } { 2 } \\Big ) \\end{align*}"}
+{"id": "6281.png", "formula": "\\begin{align*} \\mu & = \\sup \\{ \\varphi ( t ) : 1 / \\sqrt { n } \\leq t \\leq 1 \\} = \\max \\{ \\varphi ( t _ { j + 1 } + ) : j = 1 , \\ldots , n - 1 \\} \\\\ & = \\left ( \\norm { a _ { J ^ * } } _ 1 - \\lambda \\right ) / \\sqrt { j ^ * + 1 } + \\norm { a _ { I \\backslash J ^ * } } / \\sqrt { 1 + j ^ * } . \\end{align*}"}
+{"id": "5951.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } \\tau _ { A _ i } - \\sum _ { i = 1 } ^ { k } \\tau _ { B _ i } = B ( 0 , \\delta ) . \\end{align*}"}
+{"id": "6243.png", "formula": "\\begin{align*} D ( H ( p ) ) = \\{ \\psi \\in H ^ 2 ( \\Sigma ) \\cap H ^ 1 _ 0 ( \\Sigma ) : \\ : H ( p ) \\psi \\in L ^ 2 ( \\Sigma ) \\} . \\end{align*}"}
+{"id": "3700.png", "formula": "\\begin{align*} n \\sum _ { i = 0 } ^ { n - 1 } ( \\Delta q ^ b _ i ( \\omega ) ) ^ 2 \\end{align*}"}
+{"id": "717.png", "formula": "\\begin{align*} { f _ { \\bar i } } \\left ( { { { \\bf { x } } _ b } , t + \\Delta t } \\right ) = f _ i ^ * \\left ( { { { \\bf { x } } _ b } , t } \\right ) , \\end{align*}"}
+{"id": "7941.png", "formula": "\\begin{align*} ( \\chi , \\Phi ) - ( \\chi ' , \\Phi ' ) = \\delta _ \\mathrm { m R B A } ( \\theta , 0 ) . \\end{align*}"}
+{"id": "3844.png", "formula": "\\begin{align*} \\rho = \\left ( 4 + i _ 1 , 5 + i _ 2 , \\dots , h - 1 + i _ { h - 4 } , k + \\frac { - h ^ 2 + 2 h + 1 1 } { 2 } - i \\right ) \\end{align*}"}
+{"id": "2223.png", "formula": "\\begin{align*} \\langle V _ 1 , \\dots , V _ n \\rangle = \\mathcal { K } _ { n } ( \\boldsymbol { \\mathcal { A } } , A ) , \\langle W _ 1 ^ D , \\dots , W _ n ^ D \\rangle = \\mathcal { K } _ n ^ { D } ( B ^ D , \\boldsymbol { \\mathcal { A } } ) . \\end{align*}"}
+{"id": "2459.png", "formula": "\\begin{align*} \\inf \\left \\{ r > 0 \\colon \\bar { B } _ r ( x _ 0 ) \\in \\mathcal { F } \\right \\} = 0 \\textrm { f o r a n y } x _ 0 \\in M . \\end{align*}"}
+{"id": "6841.png", "formula": "\\begin{align*} u _ i = \\prod _ { 1 \\leq j \\leq n , i \\neq j } ( a _ i - a _ j ) ^ { - 1 } , \\end{align*}"}
+{"id": "4725.png", "formula": "\\begin{gather*} \\eth _ k ( a ) \\cdot b = a \\cdot \\partial _ k ( b ) + \\eth _ k ( a \\cdot b ) , \\\\ a \\cdot \\eth _ k ( b ) = \\partial _ k ( a ) \\cdot b + \\eth _ k ( a \\cdot b ) , \\forall a , b \\in A . \\end{gather*}"}
+{"id": "6025.png", "formula": "\\begin{align*} { L } ^ { \\pi ^ \\prime } ( 0 ) - L ^ \\pi ( 0 ) + R ^ \\pi ( 0 ) = R ^ \\pi ( 0 ) \\geq 0 . \\end{align*}"}
+{"id": "7074.png", "formula": "\\begin{align*} \\alpha ( \\langle 3 , 7 , 8 \\rangle ) = \\big \\{ 0 ^ 2 , 1 ^ 2 \\big \\} \\qquad \\alpha ( \\langle 6 , 1 3 , 1 4 , 1 5 , 1 6 \\rangle ) = \\big \\{ 0 ^ 5 , 1 ^ 5 , 6 , 1 1 \\big \\} . \\end{align*}"}
+{"id": "6349.png", "formula": "\\begin{align*} q _ i \\| f _ i \\| \\ ; \\left \\| \\tfrac { 1 } { A } f _ i + \\epsilon _ { i } h _ i \\right \\| = \\sqrt { \\tfrac { q _ i ^ 2 } { A ^ 2 } \\| f _ i \\| ^ 4 + q _ i ^ 2 \\epsilon ^ 2 _ { i } \\| f _ i \\| ^ 2 \\| h _ i \\| ^ 2 + 2 \\tfrac { q _ i ^ 2 \\epsilon _ { i } } { A } \\| f _ i \\| ^ 2 R e ( \\langle f _ i , h _ i \\rangle ) } < \\frac { q _ i } { A } \\| f _ i \\| ^ 2 = c \\end{align*}"}
+{"id": "4494.png", "formula": "\\begin{align*} \\dd _ u K ( h ) = \\int _ { [ 0 , 1 ] } u h \\ , . \\end{align*}"}
+{"id": "2268.png", "formula": "\\begin{align*} W ^ m \\subseteq \\sum _ { i = 1 } ^ m V ^ { m } x ^ i + \\sum _ { i = 1 } ^ m V ^ { m } y ^ i + V ^ { m } , \\ m \\geq 1 . \\end{align*}"}
+{"id": "1834.png", "formula": "\\begin{align*} \\{ a , b \\} = a \\circ b - b \\circ a , a \\rhd b = a \\succ b - b \\prec a . \\end{align*}"}
+{"id": "3460.png", "formula": "\\begin{align*} \\mathbb { I } [ f ] ( \\mathrm { x } , \\mathrm { t } ) = \\int _ { \\Omega } \\Phi ( \\mathrm { x } , \\mathrm { t } ; \\mathrm { y } ) f ( \\mathrm { y } ) d \\mathrm { y } , \\end{align*}"}
+{"id": "3019.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } _ 1 & = u _ 1 \\\\ \\dot { x } _ 2 & = u _ 2 \\\\ \\dot { x } _ 3 & = x _ 1 + \\tfrac { u _ 2 ^ 2 } { 2 u _ 1 } \\ , , \\end{aligned} \\end{align*}"}
+{"id": "5223.png", "formula": "\\begin{align*} \\lbrace \\kappa \\in \\operatorname { H o m } _ { \\mathbb { Q } _ p } ( K , \\overline { \\mathbb { Q } } _ p ) \\mid \\kappa | _ k = \\tau \\rbrace = \\lbrace \\tau _ 0 , \\ldots , \\tau _ { e - 1 } \\rbrace \\end{align*}"}
+{"id": "3536.png", "formula": "\\begin{align*} \\Big \\Vert \\Tilde { \\mathrm { E } } \\Big \\Vert _ { \\mathrm { H } ^ 2 \\Big ( B \\Big ) } = \\mathcal { O } \\Big ( 1 \\Big ) . \\end{align*}"}
+{"id": "6670.png", "formula": "\\begin{align*} \\frac { \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( a s ^ { ( 1 ) } _ { k - 1 } ) _ { k = 0 } ^ { m - 1 } \\right ) } { a b \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( 1 , 0 , \\ldots , 0 ) \\right ) } & = \\frac { \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( s ^ { ( 1 ) } _ { k - 1 } ) _ { k = 0 } ^ { m - 1 } \\right ) } { b \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( 1 , 0 , \\ldots , 0 ) \\right ) } . \\end{align*}"}
+{"id": "4681.png", "formula": "\\begin{align*} C ( z , w ) & = \\sum _ { n = 0 } ^ { \\infty } \\phi _ { n } ( z ) \\overline { \\phi _ { n } ( w ) } , \\\\ P ( z , w ) & = C ( z , w ) + \\bar { z } w C ( w , z ) . \\end{align*}"}
+{"id": "1213.png", "formula": "\\begin{align*} P _ { \\geq 1 } ( M ) = 1 - P _ 0 ( M ) = 1 - \\prod _ { i = 0 } ^ { M - 1 } \\frac { E - i } { E } , \\end{align*}"}
+{"id": "5203.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n - 1 } \\binom { n - 1 } { j } \\cdot \\left ( - \\frac { 1 } { 2 } \\right ) ^ j \\cdot 1 ^ { n - 1 - j } = \\left ( 1 - \\frac { 1 } { 2 } \\right ) ^ { n - 1 } - 1 = \\left ( \\frac { 1 } { 2 } \\right ) ^ { n - 1 } - 1 \\enspace . \\end{align*}"}
+{"id": "801.png", "formula": "\\begin{align*} J _ i ( \\tau _ i ) = \\mathbb { E } \\Bigg \\{ \\bar { \\theta } _ 1 e ^ { - \\beta \\tau _ i } \\Big ( x _ i ( \\tau _ i ) - \\bar { K } _ 1 \\Big ) \\Bigg \\} = \\bar { \\theta } _ 1 \\mathbb { E } \\Bigg \\{ e ^ { - \\beta \\tau _ i } \\Big ( x _ i ( \\tau _ i ) - \\bar { K } _ 1 \\Big ) \\Bigg \\} \\end{align*}"}
+{"id": "100.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\varphi '' + \\Big ( ( m - 1 ) \\frac { \\sigma ' } { \\sigma } - \\Phi ' \\Big ) \\varphi ' + B = 0 & \\textnormal { i n } \\ I \\\\ \\varphi ( r _ 1 ) = 1 \\\\ \\varphi ' ( r _ 1 ) = b < 0 \\end{array} \\right . \\end{align*}"}
+{"id": "5037.png", "formula": "\\begin{align*} f ( \\varepsilon ) = \\frac { P _ { K ^ * } ( B _ \\varepsilon ) } { \\varepsilon } , \\varepsilon > 0 \\ , , \\end{align*}"}
+{"id": "1705.png", "formula": "\\begin{align*} \\varphi ^ * _ \\tau ( g ) : = \\tau ^ { - 1 / 2 } \\ , b ^ * ( g ) \\ , , \\varphi _ \\tau ( g ) : = \\tau ^ { - 1 / 2 } \\ , b ( g ) \\ , , \\end{align*}"}
+{"id": "1316.png", "formula": "\\begin{align*} f \\otimes 0 _ { b , b ' } = 0 _ { a \\otimes b , a ' \\otimes b ' } = 0 _ { a , a ' } \\otimes g . \\end{align*}"}
+{"id": "4220.png", "formula": "\\begin{align*} \\varphi ( n ) = ( \\exp ( i n ( \\theta ( \\gamma _ j ) + 2 \\pi \\ell ) / k ) ) _ { \\ell = 1 , j = 1 } ^ { g , k } . \\end{align*}"}
+{"id": "7971.png", "formula": "\\begin{align*} \\alpha ( \\xi ) \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle = k ( \\xi ) \\left \\langle \\eta , \\xi ^ H \\right \\rangle \\quad \\mbox { f o r } \\xi \\in M \\smallsetminus S _ M . \\end{align*}"}
+{"id": "823.png", "formula": "\\begin{align*} \\tau _ i ^ { * } = \\inf \\{ t : \\ x _ i ( t ) \\leq x ^ * \\} . \\end{align*}"}
+{"id": "6624.png", "formula": "\\begin{align*} D ( h ^ { N } _ { \\theta } ) & = \\Pi _ { + } D ( h _ { \\theta } ) = H ^ { 1 } ( \\mathbb { R } ^ { 2 } _ { + } \\setminus \\Gamma _ { \\theta } ^ { + } ) , \\\\ D ( h ^ { D } _ { \\theta } ) & = \\Pi _ { - } D ( h _ { \\theta } ) = \\{ v \\in H ^ { 1 } ( \\mathbb { R } ^ { 2 } _ { + } \\setminus \\Gamma _ { \\theta } ^ { + } ) : v ( \\cdot , 0 ) = 0 \\} , \\end{align*}"}
+{"id": "924.png", "formula": "\\begin{align*} - \\epsilon _ 1 b _ { n - 1 } u _ n ( x ) \\leq u ( 0 ) - u ( x ) + \\sum _ { \\alpha = 1 } ^ { n - 1 } x _ \\alpha u _ \\alpha ( x ) \\leq C b _ { n - 1 } ^ 2 . \\end{align*}"}
+{"id": "1447.png", "formula": "\\begin{align*} s u '' ( s ) + ( c - s ) u ' ( s ) - b u ( s ) = 0 . \\end{align*}"}
+{"id": "6138.png", "formula": "\\begin{align*} V ^ { * j } V _ 1 = q ^ { j - 1 } V _ 2 ^ * V ^ { * j - 1 } V ^ { * j } V _ 2 = \\overline q ^ j V _ 1 ^ * V ^ { * j - 1 } . \\end{align*}"}
+{"id": "4263.png", "formula": "\\begin{align*} T _ 2 ^ { * } : = \\frac { 1 } { ( c q ) _ N } \\sum _ { n = 1 } ^ N \\frac { \\left ( c q \\right ) _ { N - n } ( c q ) ^ n } { ( q ) _ n ( q ) _ { N - n } } \\sum _ { k = 1 } ^ n \\left [ \\begin{matrix} n \\\\ k \\end{matrix} \\right ] \\frac { \\left ( \\frac { q c } { d } \\right ) _ k \\left ( \\frac { d } { c } \\right ) _ { n - k } \\left ( \\frac { d } { c } \\right ) ^ { k } } { 1 - q ^ k } , \\end{align*}"}
+{"id": "7456.png", "formula": "\\begin{align*} A ^ \\dagger = \\tau ^ \\dagger _ 1 \\frac { 1 } { 2 \\sqrt { \\hat { j } + 1 } } \\frac { 1 } { \\sqrt { ( N + 1 ) + \\hat { j } + 1 + 1 } } \\frac { \\sqrt { 2 ( \\hat { j } + 1 ) + 1 } } { \\sqrt { 2 ( \\hat { j } + 1 ) - 1 } } , \\end{align*}"}
+{"id": "2992.png", "formula": "\\begin{align*} g ' ( t ) = - ( m _ F - 1 ) f ( t ) ^ { - m _ F } f ' ( t ) \\ge m _ F - 1 , \\end{align*}"}
+{"id": "6415.png", "formula": "\\begin{align*} F ( x + y ) + F ( x - y ) - 2 = & F ( x + y ) - F ( x ) + F ( x - y ) - F ( x ) \\\\ = & \\int _ 0 ^ 1 \\langle \\mathrm { D } F ( x + t y ) - \\mathrm { D } F ( x - t y ) , y \\rangle \\mathrm { d } t \\\\ = & 2 \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\langle \\mathrm { D } ^ 2 F ( x - t y + 2 s y ) y , y \\rangle s \\mathrm { d } s \\mathrm { d } t \\\\ \\lesssim & \\| D ^ { 2 } F \\| _ { L ^ { \\infty } ( B _ { 3 / 2 } \\setminus B _ { 1 / 2 } ) } F ( y ) ^ { 2 } . \\end{align*}"}
+{"id": "4971.png", "formula": "\\begin{align*} E _ { v \\rightarrow c } ^ { ( i ) } ( { \\mathcal { X } } _ { v } [ m ] ) = \\prod \\limits _ { \\mathclap { c ' \\in \\mathcal { N } ( v ) \\backslash \\{ c \\} } } E _ { c ' \\rightarrow v } ^ { ( i - 1 ) } ( { \\mathcal { X } } _ { v } [ m ] ) , \\end{align*}"}
+{"id": "2659.png", "formula": "\\begin{align*} q _ \\pi ( G _ 0 ) = q _ \\pi ( V \\times K ) = q _ \\pi ( V ) q _ \\pi ( K ) . \\end{align*}"}
+{"id": "497.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } { n \\choose k } 6 ^ k \\big ( 1 - 3 ^ { n - k - 1 } \\big ) F _ { j k } ( \\sqrt { 5 } F _ j ) ^ { n - k } B _ { n - k } = n F _ j \\sum _ { m = 0 } ^ { n - 1 } x { n - 1 \\choose m } \\big ( 2 ^ { n - 1 } + 4 ^ m \\big ) L _ j ^ { n - 1 - m } L _ { j m } . \\end{align*}"}
+{"id": "2516.png", "formula": "\\begin{align*} a \\in F \\textup { a n d } \\pi ( b ) = \\pi ( a ) \\implies b \\in F \\end{align*}"}
+{"id": "424.png", "formula": "\\begin{align*} \\hat { H } _ t ( y ) : = h \\vee \\bigg ( ( 1 - \\alpha ) ^ { - \\frac { 1 } { ( 1 - \\alpha ) \\gamma _ 1 - 1 } } \\big ( \\inf _ { s \\leq t } Y _ s ( y ) \\big ) ^ { \\frac { 1 } { ( 1 - \\alpha ) \\gamma _ 1 - 1 } } \\bigg ) , \\end{align*}"}
+{"id": "7797.png", "formula": "\\begin{align*} I _ { \\theta } = - \\frac { 1 } { \\pi } \\int _ { D _ { \\theta } } \\frac { 1 } { \\overline { u + 1 } ( u - 1 ) } d a ( u ) = \\ln ( 2 \\sin \\theta ) + i ( \\frac { \\pi } { 2 } - \\theta ) . \\end{align*}"}
+{"id": "7323.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X ^ { u } ( t ) = & b ( t , X ^ { u } ( t ) , u ( t ) ) d t + \\sigma ( t , X ^ { u } ( t ) , u ( t ) ) d W ( t ) , \\\\ X ^ { u } ( 0 ) = & x _ { 0 } , \\end{array} \\right . \\end{align*}"}
+{"id": "2509.png", "formula": "\\begin{align*} \\begin{gathered} \\lim _ { j \\to \\infty } ( T \\times T ) ^ { c _ 1 ( j ) } ( x _ { 0 0 } , x _ { 0 1 } ) = ( x _ { 1 0 } , x _ { 1 1 } ) \\\\ \\lim _ { m \\to \\infty } ( T \\times T ) ^ { c _ 2 ( m ) } ( x _ { 0 0 } , x _ { 1 0 } ) = ( x _ { 0 1 } , x _ { 1 1 } ) \\end{gathered} \\end{align*}"}
+{"id": "7079.png", "formula": "\\begin{align*} f ^ H _ i = \\bigg ( \\prod _ { j = 2 } Z _ { r _ j } ^ { h _ j ^ { ( i + ) } } \\bigg ) - \\bigg ( \\prod _ { j = 1 } Z _ { r _ j } ^ { h _ j ^ { ( i - ) } } \\bigg ) , \\end{align*}"}
+{"id": "1053.png", "formula": "\\begin{align*} \\mathcal { G } ^ { \\Delta } ( V , W ) : = \\mathcal { W } \\left ( V W \\Delta ^ { - m } \\right ) , \\end{align*}"}
+{"id": "8220.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Vert \\dot { \\Delta } _ j d \\Vert _ { L ^ 2 } + ( \\frac { 3 } { 4 } ) ^ \\alpha \\mu 2 ^ { j \\alpha } \\Vert \\dot { \\Delta } _ j d \\Vert _ { L ^ 2 } \\le \\frac { 8 } { 3 } \\lambda 2 ^ j \\Vert \\dot { \\Delta } _ j \\sigma \\Vert _ { L ^ 2 } + \\Vert g _ j \\Vert _ { L ^ 2 } + \\Vert \\nabla v \\Vert _ { L ^ { \\infty } } \\Vert \\dot { \\Delta } _ j d \\Vert _ { L ^ 2 } , \\end{align*}"}
+{"id": "545.png", "formula": "\\begin{align*} \\log \\int _ { \\R ^ n } e ^ { f } \\ , d \\gamma _ t - \\left ( \\int _ { \\R ^ n } f \\ , d \\gamma _ { y , t } - H ( \\gamma _ { y , t } \\ , | \\ , \\gamma _ t ) \\right ) = H ( \\gamma _ { y , t } \\ , | \\ , P ) , \\end{align*}"}
+{"id": "2680.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ^ m G ( n , k , a ) \\end{align*}"}
+{"id": "1145.png", "formula": "\\begin{align*} h _ { k , S , ( n , g ) } ( \\tau , z ; \\tau _ 0 , z _ 0 ) : = \\det ( \\tau - \\bar { \\tau } _ 0 ) ^ { - k } e \\left ( - ( \\tau - \\bar { \\tau } _ 0 ) ^ { - 1 } S [ z - \\overline { z } _ 0 ] \\right ) . \\end{align*}"}
+{"id": "4885.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty b _ n x ^ n & = \\sum _ { j = 0 } ^ \\infty \\frac { ( j ! ) ^ 2 x ^ { 2 j } } { ( 1 - x ) ^ 2 ( 1 - 2 x ) ^ 2 \\cdots ( 1 - ( j + 1 ) x ) ^ 2 } = \\frac { 1 } { ( 1 - x ) ^ 2 } \\sum _ { j = 0 } ^ \\infty f _ j \\left ( 2 - \\frac { 1 } { x } , 2 - \\frac { 1 } { x } \\right ) , \\end{align*}"}
+{"id": "3465.png", "formula": "\\begin{align*} \\mathcal { V } _ { \\Omega \\times \\mathbb { R } } \\Big [ \\psi \\Big ] ( \\mathrm { x } , t ) = \\delta ^ 2 \\Big ( \\overline { \\mathcal { V } } _ { B \\times \\mathbb { R } } \\Big [ \\hat { \\psi } \\Big ] \\Big ) ^ \\vee \\end{align*}"}
+{"id": "3655.png", "formula": "\\begin{align*} \\partial \\mathcal { H } = \\left \\{ A \\in \\binom { V ( \\mathcal { H } ) } { r - 1 } \\colon B \\in \\mathcal { H } A \\subseteq B \\right \\} . \\end{align*}"}
+{"id": "6017.png", "formula": "\\begin{align*} \\tilde { g } ^ { ( q ) } ( b ; h ) & = W ^ { ( q + r ) } ( b ) \\Bigg [ - \\dfrac { \\rho ^ { ( q ) } _ { b } ( b ; h ) + \\tilde { g } ^ { { ( q ) } } ( b ; h ) } { W ^ { ( q ) } ( b ) } r \\int _ b ^ { \\infty } e ^ { - \\Phi ( q + r ) y } W ^ { ( q ) } ( y ) d y + \\int _ b ^ { \\infty } e ^ { - \\Phi ( q + r ) u } h ( u ) d u \\\\ & + r \\int _ { b } ^ { \\infty } e ^ { - \\Phi ( q + r ) y } \\rho _ b ^ { ( q ) } ( y ; h ) d y \\Bigg ] + \\tilde { g } ^ { ( q ) } ( b ; h ) + \\dfrac { r } { \\Phi ( r + q ) } W ^ { ( q + r ) } ( b ) e ^ { - \\Phi ( q + r ) b } \\tilde { g } ^ { ( q ) } ( b ; h ) . \\end{align*}"}
+{"id": "7188.png", "formula": "\\begin{align*} & i _ { \\ast } [ \\mathcal { E } _ { d , v } ] = \\left ( 1 - q _ 1 ^ { - 1 } \\right ) ^ { d - 1 } \\left ( 1 - q _ 2 ^ { - 1 } \\right ) ^ { d - 1 } \\cdot \\\\ & h _ { \\ast } \\left ( \\mathcal { O } \\left ( \\chi \\right ) \\prod _ { j \\geq i + 2 } ( 1 - q ^ { - 1 } \\mathcal { O } ( \\beta _ j - \\beta _ i ) ) \\prod _ { j > i } ( 1 - q _ 1 ^ { - 1 } \\mathcal { O } ( \\beta _ i - \\beta _ j ) ) ( 1 - q _ 2 ^ { - 1 } \\mathcal { O } ( \\beta _ i - \\beta _ j ) ) \\right ) , \\end{align*}"}
+{"id": "3225.png", "formula": "\\begin{align*} \\mu _ 2 ( G ) > \\frac { 1 } { \\delta + 1 } \\left ( 2 + \\frac { 2 } { \\binom { d + 1 } { 2 } - 1 } \\right ) = \\frac { 1 } { \\delta + 1 } \\left ( 2 + \\frac { 2 } { D - 1 } \\right ) = \\frac { 2 D } { ( D - 1 ) ( \\delta + 1 ) } . \\end{align*}"}
+{"id": "6129.png", "formula": "\\begin{align*} \\begin{aligned} | \\mathcal H ( k , d , 2 ) | - ( k - d - \\frac { 1 } { 2 } ) \\binom { n - d } { k - d } & = ( d + \\frac { 3 } { 2 } - k ) \\binom { n - d } { k - d } + d \\binom { n - d - 1 } { k - d } \\\\ & = ( 2 d + \\frac { 3 } { 2 } - k ) \\binom { n - d } { k - d } - d \\binom { n - d - 1 } { k - d - 1 } . \\end{aligned} \\end{align*}"}
+{"id": "7801.png", "formula": "\\begin{align*} I _ { g } = \\frac { 1 } { 2 \\pi } \\int _ { \\mathcal { U } } \\ ( - \\cot ( \\theta ) + i ) g ( u ( t , \\theta ) ) d t d \\theta + \\frac { 1 } { 2 \\pi } \\int _ { \\mathcal { L } } ( \\cot ( \\theta ) - i ) g ( u ( t , \\theta ) ) d t d \\theta . \\end{align*}"}
+{"id": "652.png", "formula": "\\begin{align*} \\nabla _ X \\psi _ 1 = \\frac { 1 } { 2 } X _ 1 \\cdot \\psi _ 1 - \\frac { 1 } { 2 } S ( X ) \\cdot N \\cdot \\psi _ 1 \\end{align*}"}
+{"id": "4277.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { n ( - 1 ) ^ { n - 1 } q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( q ) _ n ( 1 - q ^ n ) } & = \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( 1 - q ^ n ) ^ 2 } - \\sum _ { j = 1 } ^ { \\infty } \\frac { q ^ { j ^ 2 } } { ( q ) _ { j } ^ { 2 } } \\sum _ { n = 1 } ^ { j } \\frac { q ^ n } { ( 1 - q ^ n ) ^ 2 } . \\end{align*}"}
+{"id": "74.png", "formula": "\\begin{align*} \\sup _ { M } v = \\sup _ { \\partial M } v . \\end{align*}"}
+{"id": "198.png", "formula": "\\begin{align*} \\lambda _ W = ( 2 - h ) ^ { 2 r } \\ , 2 ^ { m - 1 } ( 1 - d ) = 2 ^ { m - 1 } \\ , 2 ^ { 2 r } ( 1 - d ) = 2 ^ { 2 r + s + m - 1 } ( 1 - d ) . \\end{align*}"}
+{"id": "1526.png", "formula": "\\begin{align*} d _ \\mathrm { h a m m } ( \\sigma , \\tau ) = \\frac { 1 } { n } | \\{ i \\in \\left [ n \\right ] \\mid \\sigma ( i ) \\neq \\tau ( i ) \\} | \\end{align*}"}
+{"id": "3732.png", "formula": "\\begin{align*} & T _ 2 : = T \\cap \\{ c _ 1 < 1 / 6 \\} & \\ ; n = 2 , \\\\ & T _ n : = T , & \\ ; n \\ge 3 . \\end{align*}"}
+{"id": "4627.png", "formula": "\\begin{align*} g ( \\theta t + ( 1 - \\theta ) s ) & = | \\theta \\nabla u + \\theta t \\nabla h + ( 1 - \\theta ) \\nabla u + ( 1 - \\theta ) s \\nabla h | \\\\ & \\le | \\theta \\nabla u + \\theta t \\nabla h | + | ( 1 - \\theta ) \\nabla u + ( 1 - \\theta ) s \\nabla h | = \\theta g ( t ) + ( 1 - \\theta ) g ( s ) , \\end{align*}"}
+{"id": "3927.png", "formula": "\\begin{gather*} \\alpha \\circ \\beta = \\beta \\circ \\alpha , \\\\ \\alpha ( x y ) = \\alpha ( x ) \\alpha ( y ) \\beta ( x y ) = \\beta ( x ) \\beta ( y ) , \\\\ \\alpha ( x ) ( y z ) = ( x y ) \\beta ( z ) . \\end{gather*}"}
+{"id": "7692.png", "formula": "\\begin{align*} \\| f \\| _ { \\alpha , p } ^ p & = c ( \\alpha ) \\int _ { \\mathbb { B } } | f ( x ) | ^ p \\Phi ^ \\alpha _ n ( | x | ) ( 1 - | x | ^ 2 ) ^ { - n } \\frac { d V ( x ) } { { \\omega _ n } } \\\\ & = { c ( \\alpha ) } \\int _ 0 ^ 1 \\Psi ( r ) d r \\int _ { \\mathbb { S } } | f ( r \\zeta ) | ^ p d \\sigma ( \\zeta ) \\\\ & \\le \\| f \\| ^ p _ { p } , \\end{align*}"}
+{"id": "364.png", "formula": "\\begin{align*} I _ N = \\left \\{ f \\in C ^ \\infty ( M ) ~ \\Big | ~ f | _ N = 0 \\right \\} ~ . \\end{align*}"}
+{"id": "228.png", "formula": "\\begin{align*} \\partial _ t m _ t = \\div \\left ( \\nabla \\bar { a } ( m _ t , \\cdot ) \\pi \\right ) \\ , , \\end{align*}"}
+{"id": "1243.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle - \\Delta _ p u _ p = f & \\Omega , \\\\ \\displaystyle | \\nabla u _ p | ^ { p - 2 } \\nabla u _ p \\cdot \\nu + \\lambda | u _ p | ^ { p - 2 } u _ p = g & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "6627.png", "formula": "\\begin{align*} \\Tilde h ^ { N / D } _ { \\theta } ( v , v ) & = \\int _ { \\Omega _ \\theta \\setminus O y ^ + } | \\nabla v | ^ 2 \\dd x \\dd y - \\int _ 0 ^ \\infty | v ( 0 ^ + , y ) - v ( 0 ^ - , y ) | ^ 2 \\dd y , \\\\ D ( \\Tilde h ^ { N } _ { \\theta } ) & = H ^ 1 ( \\Omega _ \\theta \\setminus O y ^ + ) , D ( \\Tilde h ^ { D } _ { \\theta } ) = \\{ v \\in H ^ 1 ( \\Omega _ \\theta \\setminus O y ^ + ) : \\ , v = 0 \\partial \\Omega _ \\theta \\} . \\end{align*}"}
+{"id": "6375.png", "formula": "\\begin{align*} \\tilde { B } _ \\cdot ^ H : = B _ \\cdot ^ H + ( R _ H h ) ( \\cdot ) = \\int _ 0 ^ \\cdot K _ H ( \\cdot , s ) ( \\d W _ s + ( K _ H ^ * h ) ( s ) \\d s ) . \\end{align*}"}
+{"id": "2116.png", "formula": "\\begin{align*} \\int _ U J ( \\psi _ A ) \\ , d \\mu _ n = \\int _ { y \\in Y } \\# \\big ( U \\cap \\psi _ A ^ { - 1 } ( y ) \\big ) \\Omega _ Y ( y ) = \\sum _ { d \\in \\N } d | Y _ d | , \\end{align*}"}
+{"id": "6691.png", "formula": "\\begin{align*} \\overline { Q } _ { i } ^ { a , b } ( t ) & = ( b - t ) \\det \\begin{bmatrix} b s _ { i , - 1 } - 1 & 1 \\\\ b - \\lambda _ { i , 2 } ^ { 2 } & t \\end{bmatrix} , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] , \\ a , b \\in ( 0 , \\infty ) , \\ a < b \\\\ \\overline { P } _ { i } ^ { a , b } ( 0 ) & = \\sigma \\left ( \\frac { \\overline { Q } _ { i } ^ { a , b } ( t ) - \\overline { Q } _ { i } ^ { a , b } ( 0 ) } { t } \\right ) , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] , \\ a , b \\in ( 0 , \\infty ) , \\ a < b . \\end{align*}"}
+{"id": "6589.png", "formula": "\\begin{align*} ( \\widehat { B } \\widehat { C } ) ^ * = \\widehat { C } ^ * \\widehat { B } ^ * = \\widehat { C } \\widehat { B } = \\widehat { B } \\widehat { C } . \\end{align*}"}
+{"id": "5926.png", "formula": "\\begin{align*} \\pi _ X ^ 3 = [ \\Delta _ X ] - \\pi _ X ^ 0 - \\pi _ X ^ 2 - \\pi _ X ^ 4 - \\pi _ X ^ 6 \\ , . \\end{align*}"}
+{"id": "3317.png", "formula": "\\begin{align*} P _ { n _ 1 ( 0 ) } ^ { V _ 1 } ( m _ 1 ; n _ 2 ) = 1 = P _ { n _ 1 } ^ { V _ 1 } ( m _ 1 ( 0 ) ; n _ 2 ) P _ { n _ 2 ( 0 ) } ^ { V _ 2 } ( m _ 2 ; m _ 1 ) = 1 = P _ { n _ 2 } ^ { V _ 2 } ( m _ 2 ( 0 ) ; m _ 1 ) . \\end{align*}"}
+{"id": "996.png", "formula": "\\begin{align*} \\mathcal { A } = & \\int _ { 0 } ^ { T } \\int _ \\Omega \\bigg ( \\frac { 1 } { 2 } \\ , \\rho \\ , \\| u _ { , t } \\| ^ 2 + \\frac { 1 } { 2 } \\ , \\rho \\ , \\eta \\ , \\tau _ { \\rm c } ^ 2 \\ , \\ , \\ , \\| \\ , \\mathbf { A } _ { , t } \\| ^ 2 - W \\bigg ) \\ , d v \\ , d t \\end{align*}"}
+{"id": "5962.png", "formula": "\\begin{align*} f ^ { ( H ) } _ K = f _ K 1 _ { H / 2 < | f _ K | \\leq H } . \\end{align*}"}
+{"id": "6823.png", "formula": "\\begin{align*} \\alpha _ n ^ i = \\begin{cases} \\varepsilon _ n + \\alpha _ { n - 1 } ^ 1 & i = 1 , \\\\ \\varepsilon _ n \\alpha _ { n - 1 } ^ { i - 1 } + \\alpha _ { n - 1 } ^ i & n \\geq 3 i = 2 , \\ldots , n - 1 , \\\\ \\varepsilon _ n \\alpha _ { n - 1 } ^ { n - 1 } & i = n , \\end{cases} \\end{align*}"}
+{"id": "2596.png", "formula": "\\begin{align*} | r ^ { N , i } ( t , \\boldsymbol { x } ) | \\leq \\frac { C } { N } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } | x ^ i - x ^ j | \\Big ) . \\end{align*}"}
+{"id": "5263.png", "formula": "\\begin{align*} t _ i = \\frac { k _ i } { N } , \\ \\ 1 \\leq i \\leq p . \\end{align*}"}
+{"id": "6334.png", "formula": "\\begin{align*} C _ \\rho \\cap T ( N ) = \\emptyset . \\end{align*}"}
+{"id": "226.png", "formula": "\\begin{align*} \\partial _ t m _ t = \\div \\left ( \\nabla a ( m _ t , \\cdot ) m _ t \\right ) \\ , , \\ , \\ , \\ , \\ , \\ , a ( m , x ) : = \\frac { \\delta F } { \\delta m } ( m , x ) + \\frac { \\sigma ^ 2 } { 2 } \\log \\left ( \\frac { m ( x ) } { \\pi ( x ) } \\right ) - \\frac { \\sigma ^ 2 } { 2 } \\operatorname { K L } ( m | \\pi ) . \\end{align*}"}
+{"id": "2685.png", "formula": "\\begin{align*} M _ { S _ 2 } ( 2 n , j , 0 ; a ) & = ( - 1 ) ^ { n - j } \\binom { a + n } { a } \\binom { a + j } { j } \\binom { a } { n - j } , \\\\ M _ { S _ 2 } ( 2 n , j , 1 ; a ) & = ( - 1 ) ^ { n - j } \\binom { 2 n } { n } \\sum _ { u = 0 } ^ { n - j } ( - 1 ) ^ u \\binom { n } { j + u } \\binom { j + u } { u } \\binom { a + j + u } { j + u } \\binom { a + n } { 2 n - j - u } \\end{align*}"}
+{"id": "7695.png", "formula": "\\begin{align*} \\| F - F _ r \\| _ { p } = \\| \\hat { f } - f _ r \\| _ p \\le \\epsilon . \\end{align*}"}
+{"id": "6328.png", "formula": "\\begin{align*} \\inf _ { u \\in A } \\ , E ( u ) \\le c : = \\inf _ { h \\in \\widetilde { H } } \\ , \\sup _ { u \\in h ( Q ) } \\ , E ( u ) \\le \\sup _ { u \\in Q } \\ , E ( u ) , \\end{align*}"}
+{"id": "5669.png", "formula": "\\begin{align*} V ^ i V ^ j \\hat { \\mathbb { G } } ^ { k } _ { i j } ( T ) & = 2 J ^ k _ i \\hat { \\mathbb { G } } ^ { i } _ { j } ( T ) V ^ j - \\hat { \\mathbb { G } } ^ { k } _ { i } ( T ) J ^ i _ j V ^ j \\\\ & = 4 J ^ k _ i \\hat { \\mathbb { G } } ^ { j } ( T ) J _ j ^ i + \\hat { \\mathbb { G } } ^ { k } _ { i } ( T ) T ^ i \\\\ & = - 4 \\hat { \\mathbb { G } } ^ { k } ( T ) + 2 \\hat { \\mathbb { G } } ^ { k } ( T ) = - 2 \\hat { \\mathbb { G } } ^ { k } ( T ) = - T ^ i T ^ j \\hat { \\mathbb { G } } ^ { k } _ { i j } ( T ) . \\end{align*}"}
+{"id": "6354.png", "formula": "\\begin{align*} \\begin{aligned} A \\left ( \\frac { d } { d t } \\right ) x ( t ) & = B u ( t ) , \\\\ y ( t ) & = C x ( t ) + D \\left ( \\frac { d } { d t } \\right ) u ( t ) \\ , \\ , \\ , \\ , t \\geq 0 , \\end{aligned} \\end{align*}"}
+{"id": "1853.png", "formula": "\\begin{align*} x _ { i , j } + x _ { i , j + 1 } = x _ { i + 2 , j } + x _ { i + 2 , j + 1 } . \\end{align*}"}
+{"id": "6214.png", "formula": "\\begin{align*} V = \\textsf { W } _ 0 + \\textsf { W } _ 1 + \\cdots + \\textsf { W } _ m \\ \\ ( ) . \\end{align*}"}
+{"id": "7112.png", "formula": "\\begin{align*} \\textbf { V } ( d ) : = \\frac { 3 } { 2 } [ 0 , \\beta _ i - \\beta _ j ] + [ 0 , \\beta _ k ] \\subset M ( d ) _ { \\mathbb { R } } , \\end{align*}"}
+{"id": "938.png", "formula": "\\begin{align*} E _ n = \\frac { - \\varepsilon _ 1 } { \\sqrt { \\varepsilon _ 1 ^ 2 + | \\hat { x } | ^ 2 } } e _ 1 + \\frac { | \\hat { x } | } { \\sqrt { \\varepsilon _ 1 ^ 2 + | \\hat { x } | ^ 2 } } e _ 2 . \\end{align*}"}
+{"id": "1148.png", "formula": "\\begin{align*} \\lambda _ { k , S , D } = \\Gamma ( \\ell ) ( \\det 2 S ) ^ { \\ell - 1 / 2 } 2 ^ { - g / 2 } ( 2 \\pi D ) ^ { - \\ell } \\ , \\ell = k - g / 2 - 1 . \\end{align*}"}
+{"id": "6092.png", "formula": "\\begin{align*} x = \\sum _ { g \\in F } f _ g u _ g . \\end{align*}"}
+{"id": "248.png", "formula": "\\begin{align*} \\mathbb { E } [ u ] = \\int _ \\Omega W ( x , u ( x ) , \\nabla u ( x ) ) \\ , d x . \\end{align*}"}
+{"id": "4744.png", "formula": "\\begin{gather*} l ( \\eth _ k ( a ) ) v = l ( a ) \\alpha _ k ( v ) + \\beta _ k ( l ( a ) v ) , \\\\ r ( \\eth _ k ( a ) ) v = r ( a ) \\alpha _ k ( v ) + \\beta _ k ( r ( a ) v ) . \\end{gather*}"}
+{"id": "772.png", "formula": "\\begin{align*} b - b _ 1 = \\sum _ { i = 2 } ^ r b _ i = l ( p - q ) + ( p - q r ) + 1 . \\end{align*}"}
+{"id": "7499.png", "formula": "\\begin{align*} \\int f d \\mu _ { \\varepsilon , h } = \\int f \\left ( s + \\varepsilon h ( s ) \\right ) d \\mu ( s ) , \\varepsilon \\in \\mathbb R . \\end{align*}"}
+{"id": "7356.png", "formula": "\\begin{align*} & | \\mathcal { B } _ { i , i - l } | = | \\{ ( i - 1 , i - l , t , p ) \\mid ( i - 1 , i - l , t , p ) \\in \\mathcal { I } ' _ { k - 1 } \\} | , \\ \\ \\ \\ \\ \\\\ & | \\mathcal { B } _ { i - l , i } | = | \\{ ( i - l , i - 1 , t , p ) \\mid ( i - l , i - 1 , t , p ) \\in \\mathcal { I } ' _ { k - 1 } \\} | . \\end{align*}"}
+{"id": "1282.png", "formula": "\\begin{align*} a \\otimes ( s \\bullet f ) = s \\bullet ( a \\otimes f ) , ( s \\bullet f ) \\otimes a = s \\bullet ( f \\otimes a ) , \\end{align*}"}
+{"id": "7241.png", "formula": "\\begin{align*} \\sum _ { J _ l \\subset G _ k ^ j } N ( J _ l ) \\lesssim \\sum _ { J _ l \\subset G _ k ^ j } \\int _ { J _ l } N ( t ) ^ 3 d t \\lesssim \\int _ { G _ k ^ j } N ( t ) ^ 3 d t \\lesssim \\sum _ { \\alpha = k 2 ^ j } ^ { ( k + 1 ) 2 ^ j - 1 } \\int _ { J ^ { \\alpha } } N ( t ) ^ 3 d t \\lesssim 2 ^ j \\epsilon _ 3 . \\end{align*}"}
+{"id": "6322.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\le \\frac { \\sum \\limits _ { i = 1 } ^ n ( x _ i - \\bar { x } ) ( y _ i - \\bar { y } ) } { ( x _ n - x _ 1 ) ( y _ n - y _ 1 ) } \\le \\left \\{ \\begin{array} { l l } \\frac { n } { 4 } , & \\mbox { i f $ n $ i s e v e n } \\\\ & \\\\ \\frac { n ^ 2 - 1 } { 4 n } , & \\mbox { i f $ n $ i s o d d } . \\end{array} \\right . \\end{align*}"}
+{"id": "4517.png", "formula": "\\begin{align*} \\{ \\Pi _ y , \\mathcal { K } \\} ( \\phi , \\pi ) & = - \\dd _ { ( \\phi , \\pi ) } \\mathcal { K } ( \\mathbb { X } _ { \\Pi _ { \\ ! y } } \\ ! ) = - \\big ( \\dd _ \\phi K \\ , \\circ \\ , \\mathrm { p r o j } _ 1 \\big ) ( \\mathbb { X } _ { \\Pi _ { \\ ! y } } ) \\\\ & = - \\dd _ \\phi K ( X _ 1 ) = - \\dd _ \\phi K ( \\mathcal { E } _ y ) = - \\int _ { [ 0 , 1 ] } \\ ! \\ ! \\mathcal { E } _ y \\phi \\ , . \\end{align*}"}
+{"id": "6238.png", "formula": "\\begin{align*} \\phi _ { n , m } ( x , z ) = \\sqrt { \\textstyle { \\frac { 2 } { d } } } \\ , h _ n \\big ( x + \\textstyle { \\frac { p } { B } } \\big ) \\ , \\sin \\textstyle { \\frac { \\pi m z } { d _ j } } , ( - 1 ) ^ { j - 1 } z \\in ( 0 , d _ j ) , \\ ; j = 1 , 2 , \\end{align*}"}
+{"id": "4018.png", "formula": "\\begin{align*} \\bar { q } _ { \\beta } ( n , t ) = \\sum _ { k = 1 } ^ { n } \\underset { m _ j \\in \\mathbb { N } } { \\underset { m _ { 1 } + m _ { 2 } + \\dots + m _ { k } = n } { \\sum } } \\left ( \\frac { \\lambda t ^ { \\beta } ( 1 - \\rho ) ^ { r } } { 1 - ( 1 - \\rho ) ^ { r } } \\right ) ^ { k } \\prod _ { j = 1 } ^ { k } \\rho ^ { m _ { j } } \\binom { r + m _ { j } - 1 } { m _ { j } } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } ( - \\lambda t ^ { \\beta } ) . \\end{align*}"}
+{"id": "7828.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\Delta ( u ) d u = \\int _ { 0 } ^ { 1 } \\left [ w _ 1 ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) - w _ 2 ( G ^ { - 1 } ( u ) ) g ( G ^ { - 1 } ( u ) ) \\right ] d u \\geq 0 . \\end{align*}"}
+{"id": "384.png", "formula": "\\begin{align*} \\mathcal { Q } = \\left \\lbrace f \\in C ^ \\infty ( M ) ~ ~ \\Big \\vert ~ ~ \\{ f , C ^ \\infty ( M ) \\} \\subset I _ N \\right \\rbrace ~ . \\end{align*}"}
+{"id": "6646.png", "formula": "\\begin{align*} s _ { i , k } = t _ { 1 } ( ( s _ { i , k + 1 } , \\ldots , s _ { k + N _ i + 1 } ) ) , k \\in \\mathbb { Z } \\cap [ - \\kappa - 1 , p - N _ i - 3 ] , \\end{align*}"}
+{"id": "4048.png", "formula": "\\begin{align*} P _ { n } \\left ( \\mathcal { M } ( q ) ( s ) \\right ) = \\left \\{ \\begin{array} { r c l } \\frac { p } { p - 1 } + O ( I ^ { - \\delta } ( s ) ) & & n = 2 k \\\\ O ( I ^ { - \\delta } ( s ) ) & & n \\ne 2 k , n \\in \\{ 0 , 1 , . . . , [ M ] \\} \\end{array} \\right . . \\end{align*}"}
+{"id": "3178.png", "formula": "\\begin{align*} [ \\tau ( \\rho _ i ) ] : = \\tau ( a _ i ) \\cdot \\tau ( e _ i ' ) \\cdot h _ i \\tau ( \\bar e _ i ' ) \\cdot h _ i h _ i ' \\tau ( e _ i ' ) \\dots \\end{align*}"}
+{"id": "1524.png", "formula": "\\begin{align*} r \\left ( S \\right ) = \\dim \\left ( \\sum _ { e \\in S } W _ { e } \\right ) , \\end{align*}"}
+{"id": "1177.png", "formula": "\\begin{align*} \\mathfrak { W } _ 2 ^ { [ k , k ] } ( 1 + \\varepsilon , 1 - \\varepsilon ) & = 1 - \\frac { k - 1 } { 2 ( 2 k - 1 ) } \\varepsilon ^ 2 + O ( \\varepsilon ^ 3 ) , \\\\ \\mathfrak { M } _ 2 ^ { [ r ] } ( 1 + \\varepsilon , 1 - \\varepsilon ) & = 1 - \\frac { 1 - r } { 2 } \\varepsilon ^ 2 + O ( \\varepsilon ^ 3 ) \\end{align*}"}
+{"id": "148.png", "formula": "\\begin{align*} \\begin{array} { l l l l } 0 & = & \\partial x ( [ a ' , a - 1 ] \\otimes [ 0 , b ] ) \\\\ & = & x ( [ a - 1 ] \\otimes [ 0 , b ] ) \\\\ & & - x ( [ a ' ] \\otimes [ 0 , b ] ) & \\\\ & & + x ( [ a ' , a - 1 ] \\otimes [ b ] ) - x ( [ a ' , a - 1 ] \\otimes [ 0 ] ) . & \\end{array} \\end{align*}"}
+{"id": "4313.png", "formula": "\\begin{align*} d _ { T V } ( X , \\pi ) = \\sup _ { h \\in \\mathcal { H } } | \\mathbb { E } h ( X ) - \\mathbb { E } h ( \\pi ) | . \\end{align*}"}
+{"id": "1034.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } \\frac { 1 } { ( 1 + \\varepsilon ) ^ { j - 1 } } \\frac { 1 } { E ( t _ { i _ { j } } ) } ( i _ { j + 1 } - i _ j ) = \\infty . \\end{align*}"}
+{"id": "7704.png", "formula": "\\begin{gather*} \\tilde { g } _ { i j } ( x ^ \\prime , x ^ n ) : = \\begin{cases} g _ { i j } ( x ^ \\prime , | x ^ n | ) & i \\neq n j \\neq n i = j = n \\\\ \\operatorname { s g n } ( x ^ n ) g _ { i j } ( x ^ \\prime , | x ^ n | ) & , \\end{cases} \\end{gather*}"}
+{"id": "7291.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Upsilon ( t , \\omega ) = ( Q _ x ( t ) + Q _ m ( t ) ) X ^ * ( t , \\omega ) - A ^ T ( t ) \\Upsilon ( t , \\omega ) - Q _ m ( t ) \\overline { X } ^ * ( t ) \\end{align*}"}
+{"id": "2588.png", "formula": "\\begin{align*} \\phi ^ { N , i } ( t , \\boldsymbol { x } ) = \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } ) ~ ~ ~ ~ \\nu ^ { N , i } _ { \\boldsymbol { x } } = \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } x ^ j , \\end{align*}"}
+{"id": "2886.png", "formula": "\\begin{align*} \\begin{cases} x _ 1 \\mapsto \\frac { 4 a ^ 2 } { s ^ 2 } ( x _ 1 - \\tilde { \\nu _ 1 } ' ( l s ) x _ 2 - \\tilde { \\nu _ 1 } ( l s ) x _ 3 + \\tilde { \\nu _ 1 } ' ( l s ) l s x _ 3 ) \\\\ x _ 2 \\mapsto \\frac { 2 a } { s } ( x _ 2 - l s x _ 3 ) \\\\ x _ 3 \\mapsto x _ 3 \\end{cases} \\end{align*}"}
+{"id": "4101.png", "formula": "\\begin{align*} \\int ^ x _ y f ^ * \\omega = \\int ^ { f ( x ) } _ { f ( y ) } \\omega . \\end{align*}"}
+{"id": "2342.png", "formula": "\\begin{align*} \\mathfrak { X } _ { { \\rm e v } } ' \\coloneqq \\bigoplus _ { d = 1 } ^ { \\infty } \\bigoplus _ { \\substack { ( l _ { 1 } , \\dots , l _ { d } ) : \\ , { \\rm e v e n } \\\\ ( l _ { 1 } , \\dots , l _ { d } ) \\neq ( 1 , \\dots , 1 ) } } \\mathbb { Q } x _ { l _ { 1 } } \\cdots x _ { l _ { d } } . \\end{align*}"}
+{"id": "2637.png", "formula": "\\begin{align*} d ^ * ( \\lceil x \\rceil ) & = q ( c ( \\lceil x \\rceil ) ) \\\\ & = q ( c _ a c _ a c _ b c _ b c _ b c _ b c _ b c _ b c _ b c _ b ) \\\\ & = a ^ 2 b ^ 8 \\end{align*}"}
+{"id": "297.png", "formula": "\\begin{align*} R ' = R + D ' - D = \\sum _ { i ' \\in I ' } n _ { i ' } D _ { i ' } + \\sum _ { i ' \\in I - I ' } ( n _ { i ' } - 1 ) D _ { i ' } . \\end{align*}"}
+{"id": "3763.png", "formula": "\\begin{align*} \\Delta ( a b ) = ( \\Delta a ) b + ( - 1 ) ^ a a ( \\Delta b ) + \\{ a , b \\} \\end{align*}"}
+{"id": "6244.png", "formula": "\\begin{align*} \\lim _ { p \\to - \\infty } \\lambda _ k ( p ) = \\lambda _ { n ' , m ' } ^ \\mathrm { f r e e } \\end{align*}"}
+{"id": "1013.png", "formula": "\\begin{align*} \\lambda _ { } & = \\lambda ^ { \\rm N } \\ , , \\ \\ , \\ , \\ , \\mu _ { } = \\mu ^ { \\rm N } \\ , , \\ ! \\mu _ { } = \\varkappa ^ { \\rm N } \\ , , \\\\ \\alpha _ 1 & = \\frac { 2 } { L _ { \\rm c } ^ 2 \\ , \\mu ^ { \\rm N } } \\gamma ^ { \\rm N } \\ , , \\alpha _ 2 = \\frac { 2 } { L _ { \\rm c } ^ 2 \\ , \\mu ^ { \\rm N } } \\beta ^ { \\rm N } , \\qquad \\ , \\alpha _ 3 = \\frac { 2 } { L _ { \\rm c } ^ 2 \\ , \\mu ^ { \\rm N } } \\alpha ^ { \\rm N } \\ , . \\end{align*}"}
+{"id": "4359.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { t _ 0 } ^ t \\Big \\langle \\partial _ t f _ n ( \\tau ) , \\Phi ' ( f _ n ( \\tau ) ) \\Big \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } \\ , \\mathrm { d } \\tau = \\int _ { t _ 0 } ^ t \\Big \\langle \\partial _ t f ( \\tau ) , \\Phi ( f ( \\tau ) ) \\Big \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } \\ , \\mathrm { d } \\tau = 0 \\ , . \\end{align*}"}
+{"id": "7109.png", "formula": "\\begin{align*} \\textbf { W } ( d ) _ w : = \\frac { 3 } { 2 } [ 0 , \\beta _ i - \\beta _ j ] + w \\tau _ d \\subset \\textbf { W } ( d ) . \\end{align*}"}
+{"id": "7759.png", "formula": "\\begin{align*} ( X = \\mathbb { R } ^ 2 \\times \\mathbb { S } ^ 2 , \\ \\ g ^ m = \\lambda ^ 2 V g _ { \\mathbb { S } ^ 1 } + V ^ { - 1 } d r ^ 2 + r ^ 2 g _ { \\mathbb { S } ^ 2 } ) \\end{align*}"}
+{"id": "3229.png", "formula": "\\begin{align*} \\int _ { X } \\psi ( u - \\phi ) \\theta ^ { n } _ { v } & = \\int _ { X } \\psi ( \\tilde { u } - \\phi + C ) \\theta ^ { n } _ { \\tilde { v } } \\\\ & \\leq \\int _ { X } \\psi ( \\tilde { u } - \\phi ) \\theta ^ { n } _ { \\tilde { v } } + \\int _ { X } \\psi ( C ) \\theta ^ { n } _ { \\tilde { v } } < \\infty \\end{align*}"}
+{"id": "3615.png", "formula": "\\begin{align*} f ( x ) = M f ( x ) = f ( \\alpha ) + C + c ( \\alpha - x ) , x \\le a \\end{align*}"}
+{"id": "6944.png", "formula": "\\begin{align*} \\left [ t ^ { N - j } \\right ] \\frac { t ^ { N + 1 } P ( 1 / t ) } { 1 - y t } = \\left [ t ^ { N - j } \\right ] \\left ( ( 1 - t ) ^ { N } - \\frac { q t ^ { N } } { 1 - y t } \\right ) = \\begin{cases} ( - 1 ) ^ { N - j } \\binom { N } { N - j } & j > 0 \\\\ ( - 1 ) ^ N - q & j = 0 \\end{cases} . \\end{align*}"}
+{"id": "3706.png", "formula": "\\begin{align*} L ^ 2 _ { [ 0 , 1 ] } ( \\mathbb { P } ) : = \\{ \\beta : \\Omega \\times [ 0 , 1 ] \\rightarrow \\mathbb { R } ^ d \\ ; s . t . \\int _ \\Omega \\int _ 0 ^ 1 \\beta ^ 2 ( \\omega , s ) d s \\mathbb P ( d \\omega ) < \\infty \\} \\end{align*}"}
+{"id": "3009.png", "formula": "\\begin{align*} \\widetilde { \\phi } _ { b c , R } ^ 1 = \\phi _ { b c , R } ^ 1 + g ^ 1 , g ^ 1 = ( 1 - y ) ( \\phi ^ 1 _ { b c , S } + \\phi ^ 1 _ { b c , T } ) . \\end{align*}"}
+{"id": "4669.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 \\leq s _ k \\\\ \\ \\sum _ { k = 0 } ^ { n _ q - 1 } s _ k = \\ell - m \\\\ \\sum _ { k = 0 } ^ { n _ q - 1 } q ^ k s _ k \\equiv 0 \\bmod { \\ell } } } b _ { s _ 0 , \\dots , s _ { n _ q - 1 } } = a _ m , \\end{align*}"}
+{"id": "5771.png", "formula": "\\begin{align*} - \\frac { 8 0 4 } { 7 1 } < a < 0 , - \\frac { 4 } { 5 } a < b < - \\frac { 5 } { 6 } a , \\frac { - 3 5 a - 6 b } { 7 2 } < c \\leq - \\frac { 5 8 ( 5 a + 6 b ) - 3 3 5 } { 7 1 } \\end{align*}"}
+{"id": "2540.png", "formula": "\\begin{align*} \\mathcal { J } ^ i ( \\alpha ^ i , \\boldsymbol { \\alpha } ^ { - i } ) : = \\frac { 1 } { 2 } \\mathbb { E } \\Big \\{ \\int _ 0 ^ T \\Big [ Q \\left ( x ^ i _ t + l ( \\nu ^ { N , i } _ { \\boldsymbol { x } _ t } ) \\right ) ^ 2 + R \\left ( \\alpha ^ i _ t + h ( \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } _ t } ) \\right ) ^ 2 \\Big ] d t + G \\left ( x ^ i _ T + g ( \\nu ^ { N , i } _ { \\boldsymbol { x } _ T } ) \\right ) ^ 2 \\Big \\} , \\end{align*}"}
+{"id": "2013.png", "formula": "\\begin{align*} \\mu _ n = \\frac { d ^ n } { d X ^ n } \\left [ \\frac { 1 } { ( 1 - 2 X ) ^ 2 } \\frac { \\xi ' } { \\xi } ( 1 - X ) \\right ] _ { X = 0 } . \\end{align*}"}
+{"id": "3567.png", "formula": "\\begin{align*} \\widetilde { I ^ 2 } \\cap I = I ^ 2 . \\end{align*}"}
+{"id": "1058.png", "formula": "\\begin{align*} \\mathcal { G } ^ { \\Delta _ { T , E } } ( V , W ) = \\mathcal { G } ^ { \\Delta _ { T } } ( V , W ) + \\frac { 1 } { 2 } \\int _ { M } ( E ) \\ , g ( V , W ) ~ v o l _ { g } . \\end{align*}"}
+{"id": "1517.png", "formula": "\\begin{align*} X = 2 L ( 1 , \\chi ) x q ^ { - 1 } . \\end{align*}"}
+{"id": "1792.png", "formula": "\\begin{align*} \\left \\langle Y , B , i _ { 0 } \\right \\rangle = 0 . \\end{align*}"}
+{"id": "7970.png", "formula": "\\begin{align*} \\left \\langle V _ j , \\xi ^ H \\right \\rangle = \\left \\langle W _ j , \\xi ^ H \\right \\rangle = 0 \\quad \\mbox { f o r } j \\in \\{ 1 , \\ldots , n - 1 \\} \\mbox { a n d } \\xi \\in M \\smallsetminus S _ M . \\end{align*}"}
+{"id": "6901.png", "formula": "\\begin{align*} R ( I ) : = W ( I ) \\cap \\dot { B } ^ { \\frac { 4 } { d + 2 } } _ { d + 2 , \\frac { 2 ( d + 2 ) } { d } } ( I ) \\end{align*}"}
+{"id": "8093.png", "formula": "\\begin{align*} T ( m ) = \\begin{cases} T ^ m & \\ m \\geq 1 , \\\\ I & \\ m = 0 , \\\\ T ^ { * | m | } & ~ ~ m \\leq - 1 . \\end{cases} \\end{align*}"}
+{"id": "1396.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\| ( u ^ { ( j ) } ( t ) , \\partial _ t u ^ { ( j ) } ( t ) ) - ( u ( t ) , \\partial _ t u ( t ) ) \\| _ { \\mathcal { H } _ D } = 0 . \\end{align*}"}
+{"id": "1501.png", "formula": "\\begin{align*} K ( w ^ 2 ) = \\sum _ { d \\le w ^ 2 } h ( d ) ( \\log \\Delta / d ) ^ 2 . \\end{align*}"}
+{"id": "4403.png", "formula": "\\begin{align*} D _ { ( 2 ) , x } ^ r g _ { r , t } ( x ) = \\frac { r ^ r } { t ^ r } \\sum _ { k = 1 } ^ r ( - 1 ) ^ { k + 1 } \\binom { r } { k } \\frac { 1 } { k ^ r } \\Delta _ { \\frac { k t } { r } } ^ r f ( x ) . \\end{align*}"}
+{"id": "3308.png", "formula": "\\begin{align*} \\big \\{ \\psi _ { n _ 1 ( k ) } ^ { n _ 2 } \\big \\} _ { k = 0 } ^ { N _ 1 } , \\end{align*}"}
+{"id": "5455.png", "formula": "\\begin{align*} \\det M = \\det \\begin{pmatrix} A & O & O \\\\ O & A & O \\\\ O & O & A \\end{pmatrix} . \\end{align*}"}
+{"id": "7192.png", "formula": "\\begin{align*} \\mathcal { S } ' = \\bigoplus _ { v _ 1 / d _ 1 \\leq \\cdots \\leq v _ k / d _ k } \\mathbb { F } \\cdot A _ { d _ 1 , v _ 1 } \\ast \\cdots \\ast A _ { d _ k , v _ k } , \\end{align*}"}
+{"id": "3731.png", "formula": "\\begin{align*} \\begin{cases} d \\ , u '' + u ( m ( x ) - u ) = 0 , \\ ; \\ ; u > 0 & ( - 1 < x < 1 ) , \\\\ \\ ; u ' ( - 1 ) = u ' ( 1 ) = 0 . \\end{cases} \\end{align*}"}
+{"id": "7890.png", "formula": "\\begin{align*} F ( a , b ) = \\frac { \\Phi _ { 1 1 } ( a , b ) \\Phi _ 2 ( a , b ) ^ 2 - 2 \\Phi _ { 1 2 } ( a , b ) \\Phi _ 1 ( a , b ) \\Phi _ 2 ( a , b ) + \\Phi _ { 2 2 } ( a , b ) \\Phi _ 1 ( a , b ) ^ 2 } { \\Phi _ 2 ( a , b ) ^ 3 } \\end{align*}"}
+{"id": "4033.png", "formula": "\\begin{align*} \\mathcal { L } _ s H _ \\alpha ( y ) = \\left \\{ \\begin{array} { r c l } \\left ( 1 - \\frac { | \\alpha | } { 2 k } \\right ) H _ \\alpha ( y ) + \\end{array} \\right . \\end{align*}"}
+{"id": "7463.png", "formula": "\\begin{align*} z _ C \\coloneqq \\begin{pmatrix} z _ 1 \\\\ \\vdots \\\\ z _ { 2 m - n } \\end{pmatrix} , z _ K \\coloneqq \\begin{pmatrix} z _ { 2 m - n + 1 } \\\\ \\vdots \\\\ z _ { 2 m } \\end{pmatrix} \\end{align*}"}
+{"id": "6905.png", "formula": "\\begin{align*} f _ i ( z ) = \\begin{cases} z ^ { \\ell + d + N - i + 1 } & 1 \\le i \\le N - r \\\\ z ^ { N - i } ( z + y ) ^ { \\ell + 1 } & N - r + 1 \\le i \\le N \\end{cases} \\end{align*}"}
+{"id": "2706.png", "formula": "\\begin{align*} M _ Q ( n , j , 0 ; a ) = \\binom { a + j } { j } \\sum _ { l = 0 } ^ { a } \\binom { n - j + l } { l } \\binom { n - j } { a - l } M _ R ( n , j + a - l , 0 ; a ) \\end{align*}"}
+{"id": "5410.png", "formula": "\\begin{align*} \\frac { j } { N } = \\int _ { - \\infty } ^ { \\gamma _ j } \\dd \\mu _ { \\mathrm { s c } } ( x ) . \\end{align*}"}
+{"id": "6744.png", "formula": "\\begin{align*} - ( \\partial ^ { \\alpha } \\widetilde { E } , \\nabla _ { x } \\partial ^ { \\alpha } \\widetilde { \\rho } ) = ( \\nabla _ { x } \\cdot \\partial ^ { \\alpha } \\widetilde { E } , \\partial ^ { \\alpha } \\widetilde { \\rho } ) = - ( \\nabla _ { x } \\cdot \\partial ^ { \\alpha } \\widetilde { E } , \\nabla _ { x } \\cdot \\partial ^ { \\alpha } \\widetilde { E } ) . \\end{align*}"}
+{"id": "4260.png", "formula": "\\begin{align*} T _ 1 = \\frac { 1 } { ( q ) _ N } \\left ( 1 - \\frac { ( d q ) _ N } { ( c q ) _ N } \\right ) . \\end{align*}"}
+{"id": "8254.png", "formula": "\\begin{align*} t ^ s X _ j ( t ) \\le t _ 0 ^ s X _ j ( t _ 0 ) + C \\int _ 0 ^ t \\tau ^ s \\widetilde { R } _ j ( \\tau ) \\dd \\tau , \\ \\forall t \\ge t _ 0 . \\end{align*}"}
+{"id": "2518.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } T ^ { c ( n ) } ( x _ { 0 0 } , x _ { 0 1 } ) = ( x _ { 1 0 } , x _ { 1 1 } ) \\end{align*}"}
+{"id": "7566.png", "formula": "\\begin{align*} ( p , q ) \\mapsto H ( p , q ) = h ( p ) C ( p , q ) + h ( q ) C ( q , p ) \\end{align*}"}
+{"id": "1199.png", "formula": "\\begin{align*} { \\rm G a l } ( L ' / K ' ) = { \\rm G a l } ( L ' / N _ 1 ) \\times { \\rm G a l } ( L ' / N _ 2 ) . \\end{align*}"}
+{"id": "3857.png", "formula": "\\begin{align*} g ^ { - } _ 1 ( x ) = \\begin{cases} 1 , & | x - L | \\leq 4 l _ n / 5 , \\\\ 0 , & | x - L | \\geq l _ n , \\end{cases} , h _ 1 ^ - ( y ) = \\begin{cases} 1 , & | y | \\leq 4 h _ n / 5 , \\\\ 0 , & | y | \\geq h _ n . \\end{cases} \\end{align*}"}
+{"id": "1852.png", "formula": "\\begin{align*} x _ { i , j } + x _ { i , j + 1 } + x _ { i + 1 , j } + x _ { i + 1 , j + 1 } & = S \\mbox { \\ a n d } \\\\ x _ { i + 1 , j } + x _ { i + 1 , j + 1 } + x _ { i + 2 , j } + x _ { i + 2 , j + 1 } & = S , \\end{align*}"}
+{"id": "4225.png", "formula": "\\begin{align*} \\delta ( \\mathcal { S } _ k ( \\beta ) ) = \\lim _ { n \\to \\infty } \\lambda _ n ( [ - \\beta , \\beta ] ) = \\nu _ k ( [ - \\beta , \\beta ] ) , \\end{align*}"}
+{"id": "5273.png", "formula": "\\begin{align*} \\lambda = \\prod _ { j : n _ j > 1 } d t _ { M _ { j - 1 } + 1 } \\cdots d t _ { M _ j - 1 } , \\end{align*}"}
+{"id": "4325.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\sum _ { i = 0 } ^ \\infty \\mathbb { P } ( X = i ) \\left | \\sum _ { k = j + 1 } ^ \\infty \\left ( Q _ { i , k } - P _ { i , k } \\right ) \\right | = \\left | \\mathbb { E } \\sum _ { j = 0 } ^ \\infty \\sum _ { k = j + 1 } ^ \\infty \\left ( Q _ { X , k } - P _ { X , k } \\right ) \\right | = \\left | \\mathbb { E } X - \\mathbb { E } \\sum _ { j = 0 } ^ \\infty \\sum _ { k = j + 1 } ^ \\infty P _ { X , k } \\right | . \\end{align*}"}
+{"id": "5720.png", "formula": "\\begin{align*} \\Pi _ n : = ( L _ n ) _ { \\# } \\pi . \\end{align*}"}
+{"id": "633.png", "formula": "\\begin{align*} \\langle \\langle Z \\cdot \\varphi , \\varphi ' \\rangle \\rangle = \\langle \\langle \\varphi , Z \\cdot \\varphi ' \\rangle \\rangle . \\end{align*}"}
+{"id": "7776.png", "formula": "\\begin{align*} \\mathcal { C } h ( X , g ^ + ) = n \\Leftrightarrow Y ( \\partial X , [ \\hat { g } ] ) \\geq 0 . \\end{align*}"}
+{"id": "7735.png", "formula": "\\begin{align*} \\mathcal { A } ( E ' ) = 2 ^ n \\cdot V o l ( \\partial X , \\hat { g } ^ E ) = ( \\frac { 2 } { \\delta ' } ) ^ n \\cdot V o l ( \\partial X , \\hat { g } ) \\end{align*}"}
+{"id": "7388.png", "formula": "\\begin{align*} \\big ( A _ { - \\alpha c ^ 2 / 2 , \\alpha c ^ 2 / 2 } - & ( \\lambda + c ^ 2 / 2 ) \\big ) ^ { - 1 } = \\left ( A _ { 0 } - ( \\lambda + c ^ 2 / 2 ) \\right ) ^ { - 1 } \\\\ & + \\Phi _ { \\lambda + c ^ 2 / 2 } \\left ( I - \\alpha c ^ 2 M _ 3 \\mathcal { C } _ { \\lambda + c ^ 2 / 2 } \\right ) ^ { - 1 } \\alpha c ^ 2 M _ 3 \\Phi _ { \\overline { \\lambda } + c ^ 2 / 2 } ^ { \\ast } . \\end{align*}"}
+{"id": "2339.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { d - 1 } I _ { \\mathrm { b l } } ^ { \\mathfrak { m } } ( l _ { i + 1 } , \\dots , l _ { d } , l _ { 1 } , \\dots , l _ { i } ) = \\begin{cases} I _ { \\mathrm { b l } } ^ { \\mathfrak { m } } ( l _ { 1 } + \\cdots + l _ { d } ) & d : \\ , { \\rm o d d } \\\\ 0 & d : \\ , { \\rm e v e n } . \\end{cases} \\end{align*}"}
+{"id": "479.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { \\binom n k \\frac { F _ { j k + m } } { L _ j ^ { k } } B _ { n - k } } = \\begin{cases} F _ m B _ n \\Big ( \\frac { \\alpha ^ j } { L _ j } \\Big ) , & \\\\ [ 6 p t ] \\frac { { L _ m } } { { \\sqrt 5 } } B _ n \\Big ( \\frac { \\alpha ^ j } { L _ j } \\Big ) , & \\end{cases} \\end{align*}"}
+{"id": "7514.png", "formula": "\\begin{align*} G ( p , q ) = \\log \\left | \\frac { \\theta _ 1 ( p ) \\theta _ 1 ( q ) } { \\theta _ 1 ( p - q ) } \\right | - \\frac { 2 \\pi } { \\Im \\tau } \\left ( \\Im p \\right ) \\left ( \\Im q \\right ) \\end{align*}"}
+{"id": "2078.png", "formula": "\\begin{align*} T _ { 1 1 } X _ 1 = \\begin{bmatrix} I _ k \\\\ 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "7625.png", "formula": "\\begin{align*} ( \\C ^ r ) _ s : = \\{ p \\in \\C ^ r \\ \\vert \\ s ( p ) \\neq 0 \\} \\end{align*}"}
+{"id": "5152.png", "formula": "\\begin{align*} D ( A ' , \\Omega , x ) = 0 D ( A , \\Omega , x ) = 0 . \\end{align*}"}
+{"id": "1504.png", "formula": "\\begin{align*} K ( w ^ 2 , \\Delta ) = \\sum _ { w ^ 2 < d \\le \\Delta } h ( d ) ( \\log \\Delta / d ) ^ 2 . \\end{align*}"}
+{"id": "5673.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sup _ { v \\in V } T _ p ^ 2 P ^ k T _ p ( v , v ) = 0 , \\end{align*}"}
+{"id": "3049.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ k ) = F _ { \\hat { X } _ 0 } \\left ( \\hat { \\sigma } _ 0 ^ k \\right ) \\end{align*}"}
+{"id": "868.png", "formula": "\\begin{align*} \\mu ( E \\triangle E _ 1 ) = 0 \\mu ( E ^ c \\triangle E _ 0 ) = 0 . \\end{align*}"}
+{"id": "7037.png", "formula": "\\begin{align*} \\nabla _ { X _ p } Z = [ X , Z ] _ p \\ , . \\end{align*}"}
+{"id": "2528.png", "formula": "\\begin{align*} \\rho ( x ) = ( \\rho _ k ( x ) : k \\in \\N ) \\end{align*}"}
+{"id": "3939.png", "formula": "\\begin{align*} F _ 1 ( \\Omega _ n ) = & \\frac { 1 } { 2 h } \\int _ { U _ { h - t } ( \\Gamma _ \\infty ) } j _ 1 ( p _ n ( y ) , \\nabla b _ { \\Omega _ n } ( p _ n ( y ) ) , H _ { \\Gamma _ n } ( p _ n ( y ) ) ) \\ , \\det ( d T _ n ) \\ , d y \\\\ & + \\frac { 1 } { 2 h } \\int _ { U _ { h } ( \\Gamma _ n ) \\setminus { U _ { h - t } ( \\Gamma _ \\infty ) } } j _ 1 ( p _ n ( y ) , \\nabla b _ { \\Omega _ n } ( p _ n ( y ) ) , H _ { \\Gamma _ n } ( p _ n ( y ) ) ) \\ , \\det ( d T _ n ) \\ , d y . \\end{align*}"}
+{"id": "4479.png", "formula": "\\begin{align*} \\lim _ { h \\rightarrow 0 } \\frac { \\| f ( x + h ) - f ( x ) - \\dd _ x f ( h ) \\| _ 2 } { \\| h \\| _ 1 } = 0 \\ , . \\end{align*}"}
+{"id": "6755.png", "formula": "\\begin{align*} H ^ { \\rm F } _ { N , \\alpha } & = \\sum _ { j = 1 } ^ N \\left [ - \\Delta _ j + \\sqrt { \\frac { \\alpha } { N } } \\int d k \\ , \\abs { k } ^ { - 1 } \\left ( e ^ { 2 \\pi i k x _ j } a _ k + e ^ { - 2 \\pi i k x _ j } a ^ * _ k \\right ) \\right ] + \\mathcal { N } _ a . \\end{align*}"}
+{"id": "1928.png", "formula": "\\begin{align*} \\begin{aligned} & \\lambda \\| D ^ { m } _ v u \\| _ { L _ p ( Q _ r ) } + \\| D ^ { m + 1 } _ v u \\| _ { L _ p ( Q _ r ) } \\\\ & \\leq N \\| D ^ { m - 1 } _ v D _ x u \\| _ { L _ p ( Q _ { r _ 1 } ) } + N \\| D ^ { m } _ v u \\| _ { L _ p ( Q _ { r _ 1 } ) } . \\end{aligned} \\end{align*}"}
+{"id": "5776.png", "formula": "\\begin{align*} - \\frac { 2 9 7 6 } { 2 6 5 } < a \\leq - 8 , - \\frac { 1 } { 3 } a < b < \\frac { 9 9 2 - 1 9 7 a } { 8 5 6 } , \\frac { - 1 5 a - 8 b } { 3 2 } < c \\leq \\frac { - 3 3 a - 8 8 b + 9 3 } { 3 1 } \\end{align*}"}
+{"id": "5305.png", "formula": "\\begin{align*} I : = \\left [ Y , Y ^ { A _ 2 \\frac { \\log ( \\gamma _ 0 + 5 ) } { ( \\beta _ 0 - \\theta ) ^ 2 } } \\right ] , \\end{align*}"}
+{"id": "2542.png", "formula": "\\begin{align*} \\mathcal { J } ( \\mu , \\nu ; \\alpha ) : = \\frac { 1 } { 2 } \\mathbb { E } \\Big \\{ \\int _ 0 ^ T \\Big [ Q \\left ( x _ t ^ { \\xi , \\alpha } + l ( \\nu _ t ) \\right ) ^ 2 + R \\left ( \\alpha _ t + h ( \\mu _ t ) \\right ) ^ 2 \\Big ] d t + G \\left ( x _ T ^ { \\xi , \\alpha } + g ( \\nu _ T ) \\right ) ^ 2 \\Big \\} , \\end{align*}"}
+{"id": "2575.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t c ( t , \\nu ) + \\partial _ { \\nu } c ( t , \\nu ) A \\nu + \\frac { 1 } { 2 } \\partial _ { \\nu \\nu } c ( t , \\nu ) \\sigma _ 0 ^ 2 = 0 , \\\\ & c ( T , \\nu ) = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5536.png", "formula": "\\begin{align*} \\theta _ 1 ( \\beta ( t ) , t ) = \\theta _ m , \\end{align*}"}
+{"id": "2676.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { 2 n } ( - 1 ) ^ k \\binom { 2 n } { k } ^ m S ( k , r ) S ( 2 n - k , r ) \\end{align*}"}
+{"id": "3190.png", "formula": "\\begin{align*} B ( u _ { n } - u ) + B ( u ) = B ( u _ { n } ) + o ( 1 ) . \\end{align*}"}
+{"id": "2165.png", "formula": "\\begin{align*} & 2 S _ { 1 2 } - 3 S _ { 1 2 } + S _ { 1 2 } = 0 , \\\\ & ( 2 S _ { 1 2 } - 3 S _ { 1 2 } + S _ { 1 2 } ) S _ { 1 3 } = 0 \\end{align*}"}
+{"id": "1406.png", "formula": "\\begin{align*} \\Omega _ 1 ( s ) & = \\left \\{ x \\in \\Omega ; \\ , \\langle x \\rangle ^ { 2 - \\alpha } \\le t _ 0 + s \\right \\} , \\\\ \\Omega _ 2 ( s ) & = \\Omega \\setminus \\Omega _ 1 ( s ) = \\left \\{ x \\in \\Omega ; \\ , \\langle x \\rangle ^ { 2 - \\alpha } > t _ 0 + s \\right \\} . \\end{align*}"}
+{"id": "1819.png", "formula": "\\begin{align*} a s _ { A } ( x , x , y ) = a s _ { A } ( y , x , x ) = 0 , \\end{align*}"}
+{"id": "1514.png", "formula": "\\begin{align*} V ( q ) = \\prod _ { p < q ^ 2 } \\bigl ( 1 - \\frac 1 p \\bigr ) \\bigl ( 1 - \\frac { \\chi ( p ) } { p } \\bigr ) . \\end{align*}"}
+{"id": "2142.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle { \\mathcal { W } } ^ p _ { \\sharp } ( \\mathrm { d i v } _ { \\bf R } , Y ^ m ) = & \\displaystyle { \\Bigl \\{ \\vec { u } \\in { L ^ p _ \\sharp ( Y ^ m ; \\R ^ n ) } \\ ; \\mid \\ ; \\mathrm { d i v } _ { \\bf R } \\ ; \\vec { u } \\in { L ^ p _ \\sharp ( Y ^ m ) } \\Bigr \\} } \\end{array} \\end{align*}"}
+{"id": "2275.png", "formula": "\\begin{align*} \\beta _ m = \\alpha ^ { I + M + M ^ 2 + \\dots + M ^ { m - 1 } } , m \\geq 1 . \\end{align*}"}
+{"id": "5437.png", "formula": "\\begin{align*} \\log \\frac { \\Gamma \\left ( K + \\frac { L ^ 2 } { 2 } \\right ) } { \\Gamma \\left ( K - \\frac { L ^ 2 } { 2 } - 1 \\right ) } = ( L ^ 2 + 1 ) \\log K + \\cdots + O ( K ^ { - 1 0 0 0 } ) . \\end{align*}"}
+{"id": "1663.png", "formula": "\\begin{align*} N = \\left \\lceil 3 \\cdot 2 ^ { 1 2 } \\beta ^ { - 1 } \\pmb { \\eta } ^ { - 1 } \\mathsf { q } ^ 2 \\vartheta \\lambda ^ { - 1 } \\right \\rceil \\ , , \\end{align*}"}
+{"id": "392.png", "formula": "\\begin{align*} I _ { L _ \\infty } ( N ) = L _ \\infty ( M , \\omega ) _ { [ N ] } \\cap ( I _ \\mu + I _ N \\cap \\mathcal { Q } ) ~ . \\end{align*}"}
+{"id": "7988.png", "formula": "\\begin{align*} \\frac { a \\rho } { t - 1 } & \\leq \\frac { t } { \\rho } \\cdot \\left ( 1 - \\frac { \\varepsilon } { 2 } \\right ) \\cdot \\frac { \\rho } { t - 1 } = \\frac { t } { t - 1 } \\left ( 1 - \\frac { \\varepsilon } { 2 } \\right ) \\\\ & \\leq \\left ( 1 + \\frac { 1 } { 1 0 \\log d } \\right ) \\left ( 1 - \\frac { \\varepsilon } { 2 } \\right ) \\leq 1 - \\frac { \\varepsilon } { 4 } \\leq 1 - \\frac { t } { r - t } , \\end{align*}"}
+{"id": "2567.png", "formula": "\\begin{align*} \\mu _ t ^ { * , t _ 0 , \\xi } : = \\rho \\big ( - R ^ { - 1 } B \\mathbb E \\big [ U ( t , x _ t ^ { * , t _ 0 , \\xi } , \\nu _ t ^ { * , t _ 0 , \\xi } ) | \\mathcal { F } _ t ^ { W ^ 0 } \\big ] \\big ) \\quad \\nu _ t ^ { * , t _ 0 , \\xi } : = \\mathbb E \\big [ x _ t ^ { * , t _ 0 , \\xi } | \\mathcal { F } _ t ^ { W ^ 0 } \\big ] . \\end{align*}"}
+{"id": "2726.png", "formula": "\\begin{align*} F ' = H ^ 0 , H ^ 1 , H ^ 2 , \\dots , H ^ k = G ' \\end{align*}"}
+{"id": "6635.png", "formula": "\\begin{align*} E _ n ( Q _ \\theta ) \\equiv \\Lambda _ n ( Q _ \\theta ) = - \\dfrac { 1 } { ( 2 n - 1 ) ^ 2 \\theta ^ 2 } + O ( 1 ) \\theta \\to 0 ^ + , \\end{align*}"}
+{"id": "1832.png", "formula": "\\begin{align*} x \\blacktriangleright y = x \\rhd y + [ x , y ] . \\end{align*}"}
+{"id": "2125.png", "formula": "\\begin{align*} \\ker ( \\varphi _ { v } ^ { \\pm } ) = \\mathrm { I n d } ^ { G } _ { G _ { w } } \\ker ( \\psi _ { w , \\infty } ) \\cong \\ker ( \\psi _ { w , \\infty } ) \\widehat { \\otimes } _ { \\Omega ( G _ { w } ) } \\Omega ( G ) , \\end{align*}"}
+{"id": "6205.png", "formula": "\\begin{align*} E ^ * _ { i + k } A _ 1 E ^ * _ { i + k - 1 } A _ 1 E ^ * _ { i + k - 2 } \\cdots E ^ * _ { i + 1 } A _ 1 E ^ * _ { i } = \\big ( ( \\frac { k - 1 } { 2 } ) ! \\big ) ^ 2 \\frac { k + 1 } { 2 } M ^ { \\frac { k - 1 } { 2 } , 0 } _ { \\frac { 2 m - i - k } { 2 } , \\frac { i - 1 } { 2 } } . \\end{align*}"}
+{"id": "5903.png", "formula": "\\begin{align*} H ^ i ( w \\bullet \\mu ) \\cong \\begin{cases} H ^ 0 ( \\mu ) & \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "4397.png", "formula": "\\begin{align*} \\mathcal { J } ^ { - 1 } ( f ) ( x ) = f ( x ) = \\int _ 0 ^ \\infty \\hat { f } ( \\lambda ) \\varphi _ \\lambda ^ { ( \\alpha , \\beta ) } ( x ) d \\nu ( \\lambda ) , \\end{align*}"}
+{"id": "3356.png", "formula": "\\begin{align*} \\psi _ 1 ( ( x _ { \\alpha } ) _ { \\alpha } , ( y _ i ) _ i , z , ( t _ { i j } ) _ { i , j } ) = f _ 0 ( a ) . \\end{align*}"}
+{"id": "5518.png", "formula": "\\begin{align*} \\begin{cases} \\dim \\phi = N & \\\\ \\dim \\phi = 2 N , r = N \\dim \\phi _ j = 2 & \\end{cases} \\end{align*}"}
+{"id": "5580.png", "formula": "\\begin{align*} J = J ^ i _ k d x ^ k \\otimes \\frac { \\partial } { \\partial x ^ i } \\end{align*}"}
+{"id": "7994.png", "formula": "\\begin{align*} w ( e ^ { i t } ) = | p ( e ^ { i t } ) | ^ 2 \\end{align*}"}
+{"id": "3715.png", "formula": "\\begin{align*} \\tilde D f ( \\rho , v ) ( \\delta \\rho , \\delta v ) : = \\underset { \\epsilon \\to 0 } { l i m } \\frac 1 \\epsilon [ f ( \\rho + \\epsilon \\delta \\rho , v + \\epsilon \\delta v ) - f ( \\rho , v ) ] \\end{align*}"}
+{"id": "2385.png", "formula": "\\begin{align*} F _ { 2 } ^ { ( 1 ) } & = - \\sum _ { s = 0 } ^ { 1 } \\sum _ { k = 1 } ^ { m } e _ { a _ { 1 } } U ( c _ { k } + k + s - 1 ) \\partial _ { \\alpha , \\beta } ( g _ { k , s } ) , \\\\ F _ { 2 } ^ { ( 2 ) } & = - e _ { a _ { 1 } } \\partial _ { \\alpha , \\beta } ( g _ { 0 , 1 } ) . \\end{align*}"}
+{"id": "4251.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( q ) _ n ( q ) _ { n - 1 } ( a ) _ { N - n } a ^ n } { ( a ) _ n ( 1 - q ^ n ) ( a ) _ { N } } = \\displaystyle \\sum _ { n = 1 } ^ { N } \\frac { a q ^ { n - 1 } } { ( 1 - a q ^ { n - 1 } ) ^ 2 } . \\end{align*}"}
+{"id": "1064.png", "formula": "\\begin{align*} \\alpha _ { i j k } = - \\frac { 1 } { 2 } R _ { \\ell i j k } x ^ { \\ell } + \\mathbf { o ( } \\mathbf { x ) } , \\mathbf { } \\end{align*}"}
+{"id": "6169.png", "formula": "\\begin{align*} q _ i : = \\prod _ { j = 1 } ^ d q ( i , j ) . \\end{align*}"}
+{"id": "1065.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta ^ { ( s ) } = - \\partial _ { i } \\partial _ { i } + \\frac { 1 } { 3 } R _ { i j k \\ell } \\ , x ^ { j } x ^ { k } \\partial _ { i } \\partial _ { \\ell } + o ( \\mathbf { x ^ { 2 } ) } \\\\ + \\frac { 2 } { 3 } R _ { i j } \\ , x ^ { i } \\partial _ { j } + \\frac { 1 } { 4 } R _ { i \\ell j k } \\ , x ^ { \\ell } \\gamma ^ { j } \\gamma ^ { k } \\partial _ { i } + o ( \\mathbf { x ) } \\\\ + o ( \\mathbf { 1 ) } , \\end{aligned} \\end{align*}"}
+{"id": "2338.png", "formula": "\\begin{align*} Z ^ { \\mathfrak { m } } ( m _ { 0 } , \\dots , m _ { 2 n } ) = ( - 1 ) ^ { m _ { 0 } + \\cdots + m _ { 2 n } } I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( 2 m _ { 0 } + 2 , \\dots , 2 m _ { 2 n } + 2 ) \\end{align*}"}
+{"id": "4022.png", "formula": "\\begin{align*} f _ { b ( t ) } ( y ) = \\left ( p - 1 + b ( t ) y ^ { 2 k } \\right ) ^ { - \\frac { 1 } { p - 1 } } \\ ; \\ ; . \\end{align*}"}
+{"id": "1822.png", "formula": "\\begin{align*} & [ T ( a ) , T ( b ) ] = T \\big ( \\rho ( T ( a ) ) b - \\rho ( T ( b ) ) a + \\lambda [ a , b ] _ V \\big ) . \\end{align*}"}
+{"id": "6263.png", "formula": "\\begin{align*} \\Delta _ { k , k + 1 } & = \\tilde { \\varphi } \\left ( H _ { p _ k } ( z _ k ) - H ( z ^ * ) \\right ) - \\tilde { \\varphi } \\left ( H _ { p _ { k + 1 } } ( z _ { k + 1 } ) - H ( z ^ * ) \\right ) \\\\ & \\geq ( \\tilde { \\varphi } ) _ - ^ \\prime \\left ( H _ { p _ k } ( z _ k ) - H ( z ^ * ) \\right ) \\cdot [ H _ { p _ k } ( z _ k ) - H _ { p _ { k + 1 } } ( z _ { k + 1 } ) ] \\\\ & \\geq \\frac { H _ { p _ k } ( z _ k ) - H _ { p _ { k + 1 } } ( z _ { k + 1 } ) } { M _ { 2 , k } \\norm { z _ k - z _ { k - 1 } } } , \\end{align*}"}
+{"id": "5298.png", "formula": "\\begin{align*} D _ p ( L \\Omega = \\sqcup L \\theta ) = D _ p ( \\Omega = \\sqcup \\theta ) . \\end{align*}"}
+{"id": "6596.png", "formula": "\\begin{align*} \\exp ( t \\alpha ) \\exp ( - t ^ 2 \\beta ) \\exp ( ( t ^ 3 - p _ \\Gamma t ) \\Gamma ) x & = \\tilde U ^ * \\mbox { d i a g } ( e ^ { - a _ j t - b _ j t ^ 2 - \\gamma _ j ( t ^ 3 - p _ \\Gamma t ) } ) \\tilde U x \\\\ & = \\sum _ { j = 1 } ^ d e ^ { - a _ j t - b _ j t ^ 2 - \\gamma _ j ( t ^ 3 - p _ \\Gamma t ) } \\langle x , v _ { j } \\rangle v _ { j } , \\end{align*}"}
+{"id": "1799.png", "formula": "\\begin{align*} \\left \\langle i _ { 1 } , i _ { 1 } + 1 , \\dots , \\widehat { i _ { j } + \\epsilon } , \\dots , i _ { r } , i _ { r } + 1 \\right \\rangle = \\det \\left ( \\begin{matrix} C \\\\ I _ { i _ { 1 } , i _ { 1 } + 1 , \\ldots , \\widehat { i _ { j } + \\epsilon } , \\ldots , i _ { \\frac { m + 1 } { 2 } } , i _ { \\frac { m + 1 } { 2 } } + 1 } \\end{matrix} \\right ) \\det ( \\mathcal { Z } ) , \\end{align*}"}
+{"id": "3759.png", "formula": "\\begin{align*} f ( x ) & = x ^ 5 + 6 x ^ 4 - ( \\alpha + 9 4 ) x ^ 3 + ( 2 \\alpha + \\beta - 9 5 0 ) x ^ 2 + ( 1 0 4 \\alpha - 4 \\beta - 2 7 0 4 ) x + 1 9 6 \\alpha - 9 6 \\beta - 1 2 4 8 \\\\ g ( x ) & = x ^ 3 + 1 0 x ^ 2 + ( \\alpha + 2 2 ) x + 2 \\alpha - \\beta + 1 8 . \\end{align*}"}
+{"id": "349.png", "formula": "\\begin{align*} \\frac { d } { d t } E ( u , u _ t ) + ( \\eta , u _ t ) = 0 , \\ a . e . \\ t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "2129.png", "formula": "\\begin{align*} \\mathrm { c o r a n k } _ { \\Omega ( G ) } ( H ^ 1 ( G _ { \\infty } ^ { S } , E _ { p } ) ) = \\mathrm { c o r a n k } _ { \\Omega ( G ) } ( \\displaystyle \\bigoplus _ { v \\in S } { ^ { \\pm } \\widetilde { K } _ v ( E _ { p } / L _ \\infty ) } ) . \\end{align*}"}
+{"id": "6043.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } - \\Delta \\eta - \\lambda \\eta = 0 \\ , \\ , \\ , \\ , \\Omega \\\\ \\eta = 0 \\ , \\ , \\ , \\ , \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "5388.png", "formula": "\\begin{align*} \\sigma ( g h ) = \\sigma ( g ) + \\rho ( g ) \\sigma ( h ) . \\end{align*}"}
+{"id": "7094.png", "formula": "\\begin{align*} \\mathrm { N H i l b } ( d ) : = \\left ( V \\oplus ( V , V ) ^ { \\oplus 3 } \\right ) ^ { } / G L ( V ) , \\end{align*}"}
+{"id": "6156.png", "formula": "\\begin{align*} V _ 1 ^ * V _ 2 ( z ^ n \\otimes \\xi ) & = \\overline q ^ { 2 n } z ^ n \\otimes U ^ * P ^ \\perp U ^ * P \\xi + \\overline q ^ { 2 n + 2 } z ^ { n + 1 } \\otimes U ^ * P ^ \\perp U ^ * P ^ \\perp \\xi \\\\ & \\quad + \\overline q ^ { 2 n - 1 } z ^ { n - 1 } \\otimes U ^ * P U ^ * P \\xi + \\overline q ^ { 2 n + 1 } z ^ n \\otimes U ^ * P U ^ * P ^ \\perp \\xi . \\end{align*}"}
+{"id": "3238.png", "formula": "\\begin{align*} \\theta ^ { n } _ { \\varphi } = e ^ { \\varphi - u } \\theta ^ { n } _ { u } + e ^ { \\varphi - v } \\theta ^ { n } _ { v } . \\end{align*}"}
+{"id": "3972.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\mathrm { d } ^ { \\beta } } { \\mathrm { d } t ^ { \\beta } } q _ { \\beta } ( 0 , t ) & = - \\alpha \\left ( e ^ { \\theta } - 1 \\right ) q _ { \\beta } ( 0 , t ) , \\\\ \\frac { \\mathrm { d } ^ { \\beta } } { \\mathrm { d } t ^ { \\beta } } q _ { \\beta } ( n , t ) & = - \\alpha \\left ( e ^ { \\theta } - 1 \\right ) q _ { \\beta } ( n , t ) + \\alpha \\sum _ { j = 1 } ^ { n } \\frac { \\theta ^ { j } } { j ! } q _ { \\beta } ( n - j , t ) , \\ n \\ge 1 , \\end{aligned} \\end{align*}"}
+{"id": "4408.png", "formula": "\\begin{align*} \\upsilon = u _ 0 + u _ 1 q + u _ 2 q ^ 2 + u _ 3 q ^ 3 , u _ i \\in \\N . \\end{align*}"}
+{"id": "2321.png", "formula": "\\begin{align*} \\widehat \\theta _ T = \\operatorname { a r g \\ m i n } _ { \\theta \\in \\R ^ N } \\left \\{ \\mathcal L _ T ( \\theta ) + \\lambda \\| \\theta \\| _ 1 \\right \\} , \\end{align*}"}
+{"id": "3822.png", "formula": "\\begin{align*} a _ 1 = \\frac { 3 n + h ^ 2 - 2 h - 2 0 - d } { 2 } , \\end{align*}"}
+{"id": "6172.png", "formula": "\\begin{align*} 0 = \\langle \\oplus _ { i \\neq j = 1 } ^ d f _ j , D _ { V _ 1 ^ * } V _ 2 ^ * \\cdots V _ { i - 1 } ^ * V _ { i + 1 } ^ * \\cdots V _ d ^ * h & \\oplus D _ { V _ 2 ^ * } V _ 3 ^ * \\cdots V _ { i - 1 } ^ * V _ { i + 1 } ^ * \\cdots V _ d ^ * h \\\\ & \\oplus \\cdots \\oplus D _ { V _ { d - 1 } ^ * } V _ { d } ^ * h \\oplus D _ { V _ d ^ * } h \\rangle . \\end{align*}"}
+{"id": "1575.png", "formula": "\\begin{align*} \\varphi _ { 0 } \\left ( x _ { s } \\right ) = \\left ( s , 5 b _ { s } \\right ) \\varphi _ { 0 } \\left ( w _ { r } \\right ) = \\left ( f _ r , 0 \\right ) \\end{align*}"}
+{"id": "2490.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } \\frac { d s } { \\left ( \\int _ { 0 } ^ { s } F ( t ) d t \\right ) ^ { \\frac { p } { 2 p - q + 1 } } } < \\infty \\quad \\mbox { a n d } \\int _ { 1 } ^ { \\infty } \\frac { s \\ d s } { \\left ( \\int _ { 0 } ^ { s } F ( t ) d t \\right ) ^ { \\frac { p } { 2 p - q + 1 } } } = \\infty . \\end{align*}"}
+{"id": "6661.png", "formula": "\\begin{align*} \\overline { Q } ( t ) = ( b - t ) \\det \\begin{bmatrix} \\tilde { s } _ 0 ' & \\tilde { s } _ 1 ' & \\ldots & \\tilde { s } _ { m - 1 } ' & 1 \\\\ \\tilde { s } _ 1 ' & \\tilde { s } _ 2 ' & \\ldots & \\tilde { s } _ { m } ' & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ \\tilde { s } _ { m } ' & \\tilde { s } _ { m + 1 } ' & \\ldots & \\tilde { s } _ { 2 m - 1 } ' & t ^ { m } \\end{bmatrix} , \\end{align*}"}
+{"id": "6096.png", "formula": "\\begin{align*} F h ^ { m } = h \\sum \\limits _ { j = 1 } ^ { r } G _ j \\left ( x _ j g _ j - f _ j \\textbf { z } ^ { \\beta _ j } \\right ) + \\textbf { z } ^ { { \\alpha } } G , \\end{align*}"}
+{"id": "2159.png", "formula": "\\begin{align*} & S _ { i j k l m } = - S _ { j i k l m } = - S _ { i j l k m } = S _ { k l i j m } . \\end{align*}"}
+{"id": "2636.png", "formula": "\\begin{align*} d ^ * ( \\lfloor x \\rfloor ) & = q ( c ( \\lfloor x \\rfloor ) ) \\\\ & = q ( c _ b c _ b c _ a c _ a ) \\\\ & = b ^ 2 a ^ 2 \\end{align*}"}
+{"id": "560.png", "formula": "\\begin{align*} M _ n & \\ge \\sup _ { Q \\in \\P ( \\R ) } \\bigg ( n \\int _ { \\R } V ( x ) \\ , Q ( d x ) + \\frac 1 2 \\sum _ { i , j = 1 } ^ n J _ { i j } \\int _ { \\R } \\int _ { \\R } K ( x - y ) \\ , Q ( d x ) Q ( d y ) - n H ( Q ) \\bigg ) \\\\ & = n \\sup _ { Q \\in \\P ( \\R ) } \\bigg ( \\int _ { \\R } V \\ , d Q + \\frac 1 2 \\int _ { \\R } \\int _ { \\R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \\bigg ) . \\end{align*}"}
+{"id": "7685.png", "formula": "\\begin{align*} \\| f \\| _ { r , p r } ^ { p r } \\ge c _ r \\int _ 0 ^ { t _ 1 } ( \\mu _ 1 ( r / c ) ) t ^ { r - 1 } d t = c _ r c \\int _ 0 ^ { t _ 1 / c } ( \\mu _ 1 ( s ) ) s ^ { r - 1 } d s . \\end{align*}"}
+{"id": "1676.png", "formula": "\\begin{align*} \\mathbb { E } \\ , \\left [ \\mathcal { P } _ n \\ , | \\ , \\mathfrak { V } _ n \\wedge \\mathfrak { O } _ { 0 , n - 1 } \\right ] = \\mathbf { 0 } = \\mathbb { E } \\ , \\left [ \\mathcal { P } _ n \\ , | \\ , \\neg \\ , \\mathfrak { V } _ n \\wedge \\mathfrak { O } _ { 0 , n - 1 } \\right ] \\ , . \\end{align*}"}
+{"id": "338.png", "formula": "\\begin{align*} p ( y ( t ) | Y _ { t - 1 } ) = \\int p ( y ( t ) | X ( t ) ) p ( X ( t ) | Y _ { t - 1 } ) d X ( t ) \\end{align*}"}
+{"id": "5149.png", "formula": "\\begin{align*} F = \\left \\{ x \\in \\partial \\Omega : \\ , D ( A ' , \\Omega , x ) \\notin \\{ 0 , 1 \\} \\right \\} \\end{align*}"}
+{"id": "2093.png", "formula": "\\begin{align*} \\int _ D d y _ 1 \\wedge \\ldots \\wedge d y _ d = \\int _ D \\psi ^ * \\omega = \\int _ U \\omega . \\end{align*}"}
+{"id": "6005.png", "formula": "\\begin{align*} v ^ - _ n = \\Gamma v _ { n - 1 } ^ - \\leq \\Gamma V \\leq \\Gamma v _ { n - 1 } ^ + \\leq v ^ + _ n . \\end{align*}"}
+{"id": "5508.png", "formula": "\\begin{align*} F _ { N } ( a , b ; t ) = \\frac { 1 - t q ^ { N + 1 } } { 1 - t } - \\frac { ( b - a ) ( 1 - t q ^ { N + 1 } ) t } { ( 1 - t ) ( b - a t ) } + \\frac { ( b - a ) ( 1 - b q ^ { N + 2 } ) t } { ( 1 - b q ) ( b - a t ) } F _ { N } ( a , b q ; t ) . \\end{align*}"}
+{"id": "3919.png", "formula": "\\begin{align*} \\| u ( t + h ) - u ( t ) \\| ^ 2 _ { \\mathbb H ^ \\mu } & \\le 3 \\| [ S ( t + h ) - S ( t ) ] \\xi \\| ^ 2 _ { \\mathbb H ^ \\mu } \\\\ & + 3 \\int _ 0 ^ t \\| D _ h f ( u ) ( t - \\tau ) \\| ^ 2 _ { \\mathbb H ^ { \\mu - 1 } } d \\tau \\\\ & + 3 \\int _ t ^ { t + h } \\| f ( u ( t + h - \\tau ) , \\mathcal H u ( t + h - \\tau ) ) \\| _ { \\mathbb H ^ { \\mu - 1 } } ^ 2 d \\tau \\\\ & = F _ 1 ( t ) + F _ 2 ( t ) + F _ 3 ( t ) , \\end{align*}"}
+{"id": "1687.png", "formula": "\\begin{align*} H ( \\varphi ) : = \\int d x \\ , \\left ( | \\nabla \\varphi ( x ) | ^ 2 + \\kappa | \\varphi ( x ) | ^ 2 \\right ) + \\frac { 1 } { 2 } \\int d x \\ , d y \\ , | \\varphi ( x ) | ^ 2 \\ , w ( x - y ) | \\varphi ( y ) | ^ 2 \\ , . \\end{align*}"}
+{"id": "2185.png", "formula": "\\begin{align*} & r _ 3 = - f ^ 4 f ^ { \\prime \\prime } { } ^ 2 \\ , S ^ F _ { 2 3 2 3 2 } \\ , \\Delta , \\\\ & r _ 5 = - f ^ 4 f ^ { \\prime \\prime } { } ^ 2 \\ , S ^ F _ { 2 3 2 3 3 } \\ , \\Delta , \\end{align*}"}
+{"id": "1740.png", "formula": "\\begin{align*} \\Psi _ M ( \\xi ) : = \\frac { 1 } { \\sqrt { \\frac { 4 \\pi ^ 2 | \\xi | ^ 2 } { M ^ 2 } + \\frac { \\kappa } { M ^ 2 } } } \\Phi \\left ( \\frac { \\xi } { M } \\right ) \\ , . \\end{align*}"}
+{"id": "1464.png", "formula": "\\begin{align*} I ( \\theta ^ { [ t ] } ; Q _ n ^ { [ t ] } , U _ n ^ { [ t ] } | Q _ n ^ { [ 1 : t - 1 ] } , S _ n ^ { [ 1 : t - 1 ] } , U _ n ^ { [ 1 : t - 1 ] } ) = 0 , \\end{align*}"}
+{"id": "7238.png", "formula": "\\begin{align*} \\vec { u } _ { n } ^ { l } ( t ) : = \\sum _ { j = 1 } ^ { l } T _ { g _ { n } ^ { ( j ) } } \\left [ \\vec { v } ^ { j } \\left ( \\cdot + t _ { n } ^ { j } \\right ) \\right ] ( t ) + e ^ { i t \\Delta } \\vec { w } _ { n } ^ { l } . \\end{align*}"}
+{"id": "630.png", "formula": "\\begin{align*} \\Gamma \\left ( X , Y , Z \\right ) = C \\left ( X , J Y , Z \\right ) + g \\left ( X , \\left ( \\nabla _ { Z } J \\right ) Y \\right ) . \\end{align*}"}
+{"id": "2232.png", "formula": "\\begin{align*} U ( t ) & = \\textrm { d i a g } \\left [ \\left \\{ \\exp \\left ( - \\imath \\alpha _ k t - \\imath \\frac { \\beta _ k } { 2 \\pi \\nu } \\sin ( 2 \\pi \\nu t ) - \\imath \\frac { \\gamma _ k } { 4 \\pi \\nu } \\sin ( 4 \\pi \\nu t ) \\right ) \\right \\} _ { k = 1 } ^ { 1 6 } \\right ] . \\end{align*}"}
+{"id": "8028.png", "formula": "\\begin{align*} S ( m ) = \\begin{bmatrix} Q _ 0 & r o w ( Q _ i * ) _ { i = 1 } ^ m \\\\ \\ : c o l ( Q _ i ) _ { i = 1 } ^ { m } & S ( m - 1 ) \\otimes I _ d \\ : \\end{bmatrix} \\end{align*}"}
+{"id": "1046.png", "formula": "\\begin{align*} \\Delta = - \\frac { 1 } { \\sqrt { ( g ) } } \\partial _ { a } \\bigl ( \\sqrt { ( g ) } g ^ { a b } \\partial _ { b } \\bigr ) , \\end{align*}"}
+{"id": "6504.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r b _ i = m - 2 r + 3 = n - 2 r , \\ \\mbox { w i t h } \\ b _ i \\ge 0 , \\ \\mbox { f o r a l l } \\ 1 \\le i \\le r . \\end{align*}"}
+{"id": "139.png", "formula": "\\begin{align*} ( d _ i y ) ( [ \\mathbf { a ' } , \\mathbf { a } ] \\otimes [ 0 , \\mathbf { b } ] ) = y ( d ^ i [ \\mathbf { a ' } , \\mathbf { a } ] \\otimes [ 0 , \\mathbf { b } ] ) = 0 , \\end{align*}"}
+{"id": "75.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta _ \\Psi \\omega = f ( u _ 2 ) - f ( u _ 1 ) & \\textnormal { i n i n t } M \\\\ \\omega = 0 & \\textnormal { o n } \\ \\partial M , \\end{array} \\right . \\end{align*}"}
+{"id": "1375.png", "formula": "\\begin{align*} p _ { s u b c } ( n , \\alpha ) : = 1 + \\frac { 2 \\alpha } { n - \\alpha } . \\end{align*}"}
+{"id": "2286.png", "formula": "\\begin{align*} G _ 2 ( X _ 1 , X _ 2 ) : = F _ 2 \\left ( D a _ 1 X _ 1 + D a _ 2 X _ 2 , D a _ 3 X _ 1 + D a _ 4 X _ 2 \\right ) \\end{align*}"}
+{"id": "5992.png", "formula": "\\begin{align*} C ^ 1 _ b = \\dfrac { r ( \\beta Z ^ { ( \\theta ) } ( b ) - 1 ) } { \\theta \\Phi ( r + \\theta ) Z ^ { ( \\theta ) } ( b , \\Phi ( r + \\theta ) ) } + \\dfrac { \\beta } { \\Phi ( r + \\theta ) } . \\end{align*}"}
+{"id": "3650.png", "formula": "\\begin{align*} L ( \\vec { x } , \\vec { y } ) = \\left \\{ \\alpha \\cdot \\vec { x } + ( 1 - \\alpha ) \\cdot \\vec { y } \\colon \\alpha \\in [ 0 , 1 ] \\right \\} . \\end{align*}"}
+{"id": "3066.png", "formula": "\\begin{align*} F _ Y ( \\tau ^ { d \\ell } ) = F _ X ( \\sigma ^ { d \\ell } ) - q \\ell d \\in \\N \\end{align*}"}
+{"id": "8088.png", "formula": "\\begin{align*} \\alpha = \\frac { q ^ 2 \\left ( 1 - \\lambda _ N \\right ) } { q ^ 2 + h ^ 2 } , \\end{align*}"}
+{"id": "5338.png", "formula": "\\begin{align*} H _ f \\cap ( H _ f ) ^ t = \\{ ( h , f ( h ^ t ) ) \\ , : \\ , h \\in H \\cap H ^ t , \\ , f ( h ) = f ( h ^ t ) \\} \\end{align*}"}
+{"id": "2422.png", "formula": "\\begin{align*} \\tilde { L } ( { \\rm D } ( \\tau ( w ) ) ) = - \\tilde { L } ( \\tau ( D ( w ) ) ) = - \\tilde { L } ( D ( w ) ) , \\end{align*}"}
+{"id": "3122.png", "formula": "\\begin{align*} X _ 2 = X _ 1 \\setminus Y _ 1 = X \\setminus ( Y _ 1 \\cup Y _ 2 ) \\end{align*}"}
+{"id": "1097.png", "formula": "\\begin{align*} \\P \\left ( \\frac { 1 } { m } \\sum _ { i = 1 } ^ m V _ i \\leq \\left ( \\frac { \\pi } { 4 } + \\frac { 1 9 9 \\pi } { 3 2 0 0 k } \\right ) \\| \\mathbf { x } \\| \\right ) \\leq 2 \\exp \\left ( - \\frac { c _ 5 m } { k } \\right ) . \\end{align*}"}
+{"id": "3550.png", "formula": "\\begin{align*} \\varepsilon _ \\mathrm { p } ( \\omega ) = \\varepsilon _ \\infty \\Bigg [ 1 + \\dfrac { \\omega _ \\mathrm { p } ^ 2 } { \\omega _ 0 ^ 2 - \\omega ^ 2 - i \\gamma \\omega } \\Bigg ] . \\end{align*}"}
+{"id": "2646.png", "formula": "\\begin{align*} \\lambda _ y ( w b ^ { - n } , w b ^ { - n + 1 } ) = \\lambda _ x ( w b ^ { - n } , w b ^ { - n + 1 } ) = \\mu ( w b ^ { - n } , w b ^ { - n + 1 } ) \\end{align*}"}
+{"id": "2261.png", "formula": "\\begin{align*} \\sigma ' = \\tau _ 1 \\pi \\tau _ 2 \\cdots \\pi \\tau _ s \\pi , \\ s \\geq 1 , \\end{align*}"}
+{"id": "4877.png", "formula": "\\begin{align*} ( e ) = \\left | \\left \\{ 0 \\leq i < n : \\frac { e _ i } { s _ i } < \\frac { e _ { i + 1 } } { s _ { i + 1 } } \\right \\} \\right | , \\end{align*}"}
+{"id": "6711.png", "formula": "\\begin{align*} G = \\varepsilon L ^ { - 1 } _ { M } [ P _ { 1 } ( v \\cdot \\nabla _ { x } M ) ] + L ^ { - 1 } _ { M } \\Theta , \\end{align*}"}
+{"id": "4372.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\sum _ { i \\neq j } \\theta _ i ( D _ H g _ i ^ 0 ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) + D _ { G , 3 } f ^ 0 _ i ( T , x _ 0 ( T ) , u _ 0 ( T ) ) ) \\\\ + \\sum _ { \\alpha = 1 } ^ { m } \\lambda _ \\alpha D _ H g ^ \\alpha ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) \\\\ + \\sum _ { \\beta = 1 } ^ { q } \\mu _ \\beta D _ H h ^ \\beta ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) = 0 . \\end{array} \\right \\} \\end{align*}"}
+{"id": "4843.png", "formula": "\\begin{align*} \\begin{aligned} ( & \\nabla f ( x , s _ 1 ) - \\nabla F ( y ) ) \\cdot ( x - y ) = ( \\nabla f ( x , s _ 1 ) - \\nabla F ( x ) + \\nabla F ( x ) - \\nabla F ( y ) ) \\cdot ( x - y ) \\geq \\\\ & ( \\nabla f ( x , s _ 1 ) - \\nabla F ( x ) ) \\cdot ( x - y ) + m | x - y | ^ 2 \\geq \\frac m 2 | x - y | ^ 2 - \\frac 1 { 2 m } | \\nabla f ( x , s _ 1 ) - \\nabla F ( x ) | ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "1965.png", "formula": "\\begin{align*} \\rho ( \\xi ) ^ 2 + \\rho ( - \\xi ) ^ 2 = 1 \\ ; \\xi \\in \\R . \\end{align*}"}
+{"id": "4907.png", "formula": "\\begin{align*} c _ 1 ( L ) \\bullet c _ 1 ( L ' ) & = [ X \\xleftarrow { i _ { V ( s ) } } V ( s ) \\xrightarrow { i _ { V ( s ) } } X ; { \\bf 0 } ] \\bullet [ X \\xleftarrow { i _ { V ( s ' ) } } V ( s ' ) \\xrightarrow { i _ { V ( s ' ) } } X ; { \\bf 0 } ] \\\\ & = [ X \\xleftarrow { i _ { V ( ( s , s ' ) ) } } V ( ( s , s ' ) ) \\xrightarrow { i _ { V ( ( s , s ' ) ) } } X ; { \\bf 0 } ] \\\\ & = c _ 2 ( L \\oplus L ' ) = c _ 2 ( L ' \\oplus L ) = c _ 1 ( L ' ) \\bullet c _ 1 ( L ) . \\end{align*}"}
+{"id": "3496.png", "formula": "\\begin{align*} \\Big \\Vert | \\mathrm { E } | ^ { 2 } \\Big \\Vert _ { \\mathrm { L } ^ 2 \\Big ( \\Omega \\Big ) } = \\mathcal { O } \\Big ( \\delta ^ { - 1 } \\Big ) , \\end{align*}"}
+{"id": "685.png", "formula": "\\begin{align*} \\nabla _ X \\psi = - \\frac { 1 } { 2 } S ( X ) \\cdot \\nu _ 1 \\cdot \\psi \\end{align*}"}
+{"id": "6131.png", "formula": "\\begin{align*} \\sum _ { i = d } ^ { k - 1 } \\binom { n - i } { k - d } - ( k - d - \\frac { 1 } { 2 } ) \\binom { n - d } { k - d } & = \\sum _ { i = d } ^ { k - 1 } \\left ( \\binom { n - i } { k - d } - \\binom { n - d } { k - d } \\right ) + \\frac { 1 } { 2 } \\binom { n - d } { k - d } \\\\ & = \\frac { 1 } { 2 } \\binom { n - d } { k - d } - \\sum _ { i = d + 1 } ^ { k - 1 } \\sum _ { j = d + 1 } ^ { i } \\binom { n - j } { k - d - 1 } \\\\ & \\geq \\frac { 1 } { 2 } \\binom { n - d } { k - d } - \\frac { 1 } { 2 } ( k - d ) ( k - d - 1 ) \\binom { n - d - 1 } { k - d - 1 } > 0 . \\end{align*}"}
+{"id": "6118.png", "formula": "\\begin{align*} \\begin{aligned} | F \\cap [ d - 1 ] | = | F \\cap B | - | F \\cap ( B \\setminus [ d - 1 ] ) | \\geq d - 1 - | F \\cap ( B \\setminus [ d - 1 ] ) | \\geq d - 2 . \\end{aligned} \\end{align*}"}
+{"id": "7866.png", "formula": "\\begin{align*} | h ' ( \\theta ) | = | h ' ( \\theta ) - h ' ( \\theta _ 0 ) | \\gtrsim t | \\theta - \\theta _ 0 | \\gtrsim \\sqrt { t \\Delta } . \\end{align*}"}
+{"id": "206.png", "formula": "\\begin{align*} & \\mathcal { I } _ 1 \\leq C T ^ { \\frac { - \\gamma - 1 } { p - 1 } } R ^ N , \\\\ & \\mathcal { I } _ 2 \\leq C T ^ { \\frac { - \\gamma - 1 } { p - 1 } } R ^ { N - \\frac { 2 p } { p - 1 } } , \\\\ & \\mathcal { I } _ 3 \\leq C T ^ { 1 - \\frac { \\gamma } { p - 1 } } R ^ { N - \\frac { 2 p } { p - 1 } } . \\end{align*}"}
+{"id": "202.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\large \\displaystyle u _ t - k \\Delta u _ t - \\Delta u = I ^ \\gamma _ { 0 + } ( | u | ^ { p } ) + \\omega ( x ) , \\ , \\ ( t , x ) \\in ( 0 , \\infty ) \\times \\mathbb { R } ^ N , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\ , x \\in \\mathbb { R } ^ N , \\end{array} \\right . \\end{align*}"}
+{"id": "3508.png", "formula": "\\begin{align*} \\big | 1 - \\alpha \\lambda ^ { ( 3 ) } _ { \\mathrm { n } } \\Big | = \\begin{cases} \\delta ^ \\mathrm { h } & \\mathrm { n } = \\mathrm { n } _ 0 \\\\ 1 & \\mathrm { n } \\ne \\mathrm { n } _ 0 . \\end{cases} \\end{align*}"}
+{"id": "2188.png", "formula": "\\begin{align*} H ^ f _ { 1 1 } & = f _ { 1 1 } - \\Gamma ^ 1 _ { 1 1 } f _ 1 - \\Gamma _ { 1 1 } ^ 2 f _ 2 , \\\\ H ^ f _ { 1 2 } & = f _ { 1 2 } - \\Gamma _ { 1 2 } ^ 1 f _ 1 - \\Gamma _ { 1 2 } ^ 2 f _ 2 , \\\\ H ^ f _ { 2 2 } & = f _ { 2 2 } - \\Gamma _ { 2 2 } ^ 1 f _ 1 - \\Gamma _ { 2 2 } ^ 2 f _ 2 , \\end{align*}"}
+{"id": "1027.png", "formula": "\\begin{align*} M _ 2 = \\begin{pmatrix} 1 & 0 & b _ 0 & b _ 1 & c _ 0 & c _ 1 \\\\ 0 & 0 & b _ 1 & b _ 2 & c _ 1 & c _ 2 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ 0 & 1 & b _ { d - 2 } & b _ { d - 1 } & c _ { d - 2 } & c _ { d - 1 } \\end{pmatrix} . \\end{align*}"}
+{"id": "5868.png", "formula": "\\begin{align*} \\big | \\langle \\omega _ 0 , \\psi ( \\phi ( a ) \\pi _ s ) \\rangle \\big | & = \\big | \\tau ( \\phi ( a ) \\pi _ s ) \\big | = \\big | \\tau ( \\pi _ s ^ * \\pi _ s \\phi ( a ) \\pi _ s ) \\big | \\\\ & = \\big | \\tau ( \\pi _ s ^ * \\phi ( a ) \\pi _ s \\pi _ s ^ * ) \\big | \\leq \\big | \\tau ( \\pi _ s ^ * \\phi ( a ) ^ 2 \\pi _ s ) \\big | = 0 , \\end{align*}"}
+{"id": "7092.png", "formula": "\\begin{align*} \\begin{pmatrix} - y & 0 & 1 \\\\ k _ 2 & - y & 0 \\\\ 0 & 0 & - y \\end{pmatrix} \\begin{pmatrix} w \\\\ y ^ 2 \\\\ w y \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\\\ - k _ 2 ^ 2 k _ 3 \\end{pmatrix} . \\end{align*}"}
+{"id": "6402.png", "formula": "\\begin{align*} \\mathcal { C } _ { ( \\Delta ; E ) } : = \\bigcup _ { T \\in t ( \\Delta ) , ~ T \\nsupseteq E } \\mathcal { C } _ T \\cup \\mathcal { C } _ { Q _ E } , \\end{align*}"}
+{"id": "126.png", "formula": "\\begin{align*} \\int _ { W } g _ { u _ j } ^ p \\ , d \\mu & \\le \\int _ { U } g _ { u _ j } ^ p \\eta ^ p \\ , d \\mu = \\lim _ { k \\to \\infty } \\int _ { V _ k } g _ { u _ j } ^ p \\eta _ k ^ p \\ , d \\mu \\\\ & \\le \\lim _ { k \\to \\infty } \\int _ { V _ k } ( 2 ^ p \\eta ^ p _ k g _ f ^ p + ( 4 p M ) ^ p g _ { \\eta _ k } ^ p ) \\ , d \\mu \\le \\int _ { U } ( 2 ^ p g _ f ^ p + ( 4 p M ) ^ p g _ { \\eta } ^ p ) \\ , d \\mu < \\infty , \\end{align*}"}
+{"id": "5940.png", "formula": "\\begin{align*} \\theta _ K = \\gamma ( a ) + M _ a \\theta _ { \\delta } \\end{align*}"}
+{"id": "6058.png", "formula": "\\begin{align*} D _ T J ( x _ 0 ) = \\lim _ { \\epsilon \\rightarrow 0 } \\frac { 1 } { f ' ( \\epsilon ) } \\frac { d } { d \\epsilon } \\psi ( \\chi _ \\epsilon ( x _ 0 ) ) , \\end{align*}"}
+{"id": "2367.png", "formula": "\\begin{align*} a _ { i + 1 } = \\begin{cases} a _ { i } & i \\in \\{ b _ { 1 } , b _ { 1 } + b _ { 2 } , \\ldots , b _ { 1 } + \\cdots + b _ { m - 1 } \\} \\\\ 1 - a _ { i } & \\mathrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "5168.png", "formula": "\\begin{align*} \\textrm { s p t } ( h _ { n , i } ) : = \\{ x : \\ , h _ { n , i } ( x ) \\neq x \\} \\subset U _ n . \\end{align*}"}
+{"id": "2933.png", "formula": "\\begin{align*} ( \\lambda - M _ C ) ^ { - 1 } = \\begin{pmatrix} ( \\lambda - A ) ^ { - 1 } & ( \\lambda - A ) ^ { - 1 } C ( \\lambda - B ) ^ { - 1 } \\\\ & ( \\lambda - B ) ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
+{"id": "3879.png", "formula": "\\begin{align*} \\frac { \\abs { S _ n ( z , w ) } ^ 2 } { ( \\gamma n ) ^ { 3 / 2 } } \\lesssim \\frac { \\gamma e ^ { - x _ 1 - x _ 2 - y _ 1 ^ 2 - y _ 2 ^ 2 } } { \\gamma + \\sqrt { n / \\gamma } ( y _ 1 - y _ 2 ) ^ 2 } \\frac { ( x _ 1 - x _ 2 ) ^ 2 / \\sqrt { \\gamma } + \\sqrt { n } ( y _ 1 + y _ 2 ) ^ 2 } { ( \\gamma ^ { 1 / 4 } \\vee n ^ { 1 / 4 } y _ 1 ) ( \\gamma ^ { 1 / 4 } \\vee n ^ { 1 / 4 } y _ 2 ) } \\end{align*}"}
+{"id": "5832.png", "formula": "\\begin{align*} \\omega ( T ) = \\omega ( T _ 1 T _ 2 ) \\leq \\| T _ 1 \\| \\| T _ 2 \\| = \\omega ( T _ 1 ) \\omega ( T _ 2 ) . \\end{align*}"}
+{"id": "76.png", "formula": "\\begin{align*} \\Big ( f ( u _ 2 ) - f ( u _ 1 ) - f ' ( u _ 2 ) \\omega _ + \\Big ) \\omega _ + = 0 \\end{align*}"}
+{"id": "835.png", "formula": "\\begin{align*} \\begin{aligned} & - \\left ( \\frac { p ( x ) } { x ^ { k _ 1 } } + \\frac { l _ 1 } { l _ 2 } \\right ) + \\frac { p ( x ^ { * } ) } { ( x ^ { * } ) ^ { k _ 1 } } = \\frac { 1 } { l _ 2 } \\frac { ( 1 - \\theta ) } { \\left ( \\frac { K k _ 1 } { k _ 1 p ( x ^ { * } ) - p ' ( x ^ { * } ) x ^ { * } } + \\theta \\right ) } . \\end{aligned} \\end{align*}"}
+{"id": "6047.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\eta ' - k ^ 2 \\eta ' = 0 \\ , \\ , \\ , \\ , \\Omega \\\\ [ 0 . 3 c m ] \\frac { \\partial \\eta ' } { \\partial n } = \\left ( - \\frac { \\partial ^ 2 \\eta } { \\partial n ^ 2 } \\right ) V \\cdot n + \\nabla \\eta \\cdot \\nabla _ \\Gamma ( V \\cdot n ) \\ , \\ , \\ , \\ , \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "5069.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ \\pi \\frac { \\sin ( \\frac { k - 1 } { 2 } \\theta ) } { \\sin ( \\theta / 2 ) } | \\sin ( \\theta / 2 ) | g ( | \\sin ( \\theta / 2 ) | ) \\ , d \\theta = 0 \\quad k \\ge 4 . \\end{align*}"}
+{"id": "2202.png", "formula": "\\begin{align*} r _ 6 & = R _ { 1 3 1 3 } R _ { 2 3 2 3 } S _ { 1 2 2 3 2 } - R _ { 1 2 1 3 } R _ { 2 3 2 3 } S _ { 1 2 2 3 3 } - R _ { 1 2 1 3 } R _ { 2 3 2 3 } S _ { 1 3 2 3 2 } + R _ { 1 2 1 2 } R _ { 2 3 2 3 } S _ { 1 3 2 3 3 } \\\\ & = \\{ ( b + c ) ( R _ { 1 2 1 2 } - R _ { 1 3 1 3 } ) - ( a - d ) R _ { 1 2 1 3 } \\} R _ { 2 3 2 3 } { } ^ 2 \\\\ & = - ( b - c ) \\{ ( a - d ) ^ 2 + 4 ( b + c ) ^ 2 \\} \\{ ( b + c ) ^ 2 - a d \\} ^ 2 . \\end{align*}"}
+{"id": "258.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathbb { E } [ u _ t ] \\bigg | _ { t = 0 } & = \\int _ U \\left ( \\langle W _ u ( u ^ { - 1 } ( y ) ) , v ( y ) \\rangle + W _ \\xi ( u ^ { - 1 } ( y ) ) : [ \\nabla _ y v ] ( y ) [ \\nabla _ x u ] ( u ^ { - 1 } ( y ) ) \\right ) d y \\\\ & = \\int _ U \\left ( \\langle W _ u ( u ^ { - 1 } ( y ) ) , v ( y ) \\rangle + W _ \\xi ( u ^ { - 1 } ( y ) ) [ \\nabla _ x u ] ^ t ( u ^ { - 1 } ( y ) ) : [ \\nabla _ y v ] ( y ) \\right ) d y . \\end{align*}"}
+{"id": "7922.png", "formula": "\\begin{align*} \\Theta _ 0 ( u ) = ~ & \\frac { u } { 2 } , ~ u \\in C ^ 0 _ \\mathrm { R B A } ( ( A , P ) , ( M , Q ) ) = M , \\\\ \\Theta _ n ( f , g ) = ~ & ( f , ~ 2 ^ { n - 2 } g ) , ~ ( f , g ) \\in C ^ { n \\geq 1 } _ \\mathrm { R B A } ( ( A , P ) , ( M , Q ) ) . \\end{align*}"}
+{"id": "6883.png", "formula": "\\begin{align*} \\kappa _ p ( f , g _ \\zeta , h _ { \\alpha \\zeta ^ { - 1 } } ) ^ { \\beta _ g \\beta _ h } = \\kappa _ p ^ f ( f , g _ \\zeta , h _ { \\alpha \\zeta ^ { - 1 } } ) . \\end{align*}"}
+{"id": "2394.png", "formula": "\\begin{align*} V ( t _ { i } ) = \\widehat { e _ { 0 } } \\overbrace { e _ { 1 } e _ { 0 } e _ { 1 } \\cdots } ^ { t _ { i } - 1 } + \\widehat { e _ { 1 } } \\overbrace { e _ { 0 } e _ { 1 } e _ { 0 } \\cdots } ^ { t _ { i } - 1 } - \\overbrace { \\cdots e _ { 1 } e _ { 0 } e _ { 1 } } ^ { t _ { i } - 1 } \\widehat { e _ { 0 } } - \\overbrace { \\cdots e _ { 0 } e _ { 1 } e _ { 0 } } ^ { t _ { i } - 1 } \\widehat { e _ { 1 } } . \\end{align*}"}
+{"id": "7155.png", "formula": "\\begin{align*} \\mathcal { O } _ { s ^ { - 1 } ( 0 ) } = \\mathcal { O } _ { \\mathfrak { g } ^ { \\oplus 2 } } \\left [ \\mathfrak { g } ^ { \\vee } \\otimes \\mathcal { O } _ { \\mathfrak { g } ^ { \\oplus 2 } } [ 1 ] ; d _ s \\right ] , \\end{align*}"}
+{"id": "4305.png", "formula": "\\begin{gather*} \\alpha _ { i j } ^ k = \\alpha _ { i j } ^ { k + 1 } , i , j = 1 , \\dots , r , \\end{gather*}"}
+{"id": "220.png", "formula": "\\begin{align*} \\partial _ t m _ t ( x ) = - a ( m _ t , x ) m _ t ( x ) \\ , , \\ , \\ , \\ , a ( m , x ) : = \\frac { \\delta F } { \\delta m } ( m , x ) + \\frac { \\sigma ^ 2 } { 2 } \\log \\left ( \\frac { m ( x ) } { \\pi ( x ) } \\right ) - \\frac { \\sigma ^ 2 } { 2 } \\operatorname { K L } ( m | \\pi ) . \\end{align*}"}
+{"id": "2537.png", "formula": "\\begin{gather*} \\lim _ { N \\to \\infty } \\dfrac { | \\Phi _ N g \\cap \\Phi _ N | } { | \\Phi _ N | } = 1 \\\\ \\lim _ { N \\to \\infty } \\dfrac { | \\Phi _ N \\cap g \\Phi _ N | } { | \\Phi _ N | } = 1 \\end{gather*}"}
+{"id": "8060.png", "formula": "\\begin{align*} \\mathop { { \\mathrm { m i n i m i z e } } } \\limits _ { { \\bf W } _ p } & \\mathrm { T r } ( \\widetilde { \\bf \\Upsilon } _ { c } { \\bf W } _ p ) \\\\ \\textrm { s u b j e c t t o } & \\mathrm { T r } ( { \\bf \\Upsilon } _ { p , t } { \\bf W } _ p ) = 1 , \\ ; { \\bf W } _ p \\succeq { \\bf 0 } . \\end{align*}"}
+{"id": "3705.png", "formula": "\\begin{align*} \\partial _ t \\rho + \\nabla ( \\rho v ) = 0 \\end{align*}"}
+{"id": "7717.png", "formula": "\\begin{align*} c a p _ p ( B ^ { g ^ + } _ q ( t ) ) = O ( e ^ { n t } ) , \\ \\ t \\rightarrow + \\infty . \\end{align*}"}
+{"id": "8078.png", "formula": "\\begin{align*} N \\rho _ 1 \\leq \\lambda _ 1 \\leq \\sum _ { j = 1 } ^ N \\rho _ j \\end{align*}"}
+{"id": "5000.png", "formula": "\\begin{align*} & \\Pr \\bigg ( \\frac { n N } { 2 } \\leq S _ N \\leq \\frac { n N } { 2 } + \\frac { n - 1 } { 2 } \\bigg ) = \\Pr \\bigg ( 0 \\leq \\frac { S _ N - n N / 2 } { \\sigma \\sqrt { N } } \\leq \\frac { n - 1 } { 2 \\sigma \\sqrt { N } } \\bigg ) \\\\ & \\leq \\Pr \\bigg ( 0 \\leq \\frac { S _ N - n N / 2 } { \\sigma \\sqrt { N } } \\leq \\sqrt { \\frac { 3 } { N } } \\bigg ) = \\Phi \\bigg ( \\sqrt { \\frac { 3 } { N } } \\bigg ) - \\frac { 1 } { 2 } + O \\bigg ( \\frac { 1 } { \\sqrt { N } } \\bigg ) = o ( 1 ) . \\end{align*}"}
+{"id": "6891.png", "formula": "\\begin{align*} \\mathbb P _ { \\sf C } ( \\lbrace c \\rbrace ) = \\frac { u ^ { c ^ 2 / 2 } } { \\sum _ { n \\in \\mathbb Z } u ^ { n ^ 2 / 2 } } , c \\in \\Z . \\end{align*}"}
+{"id": "2410.png", "formula": "\\begin{align*} q _ { { \\bf b } } = \\begin{cases} 1 & { \\bf b } = \\emptyset \\\\ - u & { \\bf b } = ( 1 ) \\\\ - u v \\cdot q _ { { \\bf b } ' } & { \\bf b } = ( { \\bf b } ' , 1 - z , 1 ) \\\\ 0 & { \\bf b } = ( { \\bf b } ' , 1 , 1 ) \\\\ ( 2 u + v ) \\cdot q _ { { \\bf b } ' } & { \\bf b } = ( { \\bf b } ' , 1 - z ) . \\end{cases} \\end{align*}"}
+{"id": "5038.png", "formula": "\\begin{align*} P ^ s ( E , \\Omega ) = L _ s ( E \\cap \\Omega , E ^ c \\cap \\Omega ) + L _ s ( E \\cap \\Omega , E ^ c \\cap \\Omega ^ c ) + L _ s ( E \\cap \\Omega ^ c , E ^ c \\cap \\Omega ) . \\end{align*}"}
+{"id": "5198.png", "formula": "\\begin{align*} a _ n = \\sum _ { k = 2 } ^ n 2 ^ { n - k } \\cdot \\binom { n - 1 } { k - 1 } \\cdot \\frac { 2 ^ k + 2 \\cdot ( - 1 ) ^ k } { 3 } = 2 \\cdot \\frac { 4 ^ { n - 1 } - 1 } { 3 } \\enspace . \\end{align*}"}
+{"id": "188.png", "formula": "\\begin{align*} Q ^ { + } _ { A } \\psi = 0 , \\ , \\ , \\ , F ^ + _ { A } = \\rho ^ { - 1 } ( \\mu ( \\psi ) ) . \\end{align*}"}
+{"id": "502.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n | f _ i ( s ) | ^ 2 = \\mu \\log K ( G ( s ) , \\overline { G ( s ) } ) ~ ~ s \\in U . \\end{align*}"}
+{"id": "5512.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\large \\left ( \\frac { 1 - b } { 1 - t } \\large \\right ) \\large \\left ( 1 - \\frac { ( b - a t q ) } { ( b - a q ) } a q \\large \\right ) + \\frac { ( 1 - a q ) ( b - a t q ) ( b - a t q ^ 2 ) } { ( 1 - t ) ( b - a q ) } F ( a q , b ; t q ) , \\end{align*}"}
+{"id": "7995.png", "formula": "\\begin{align*} T _ w = c _ 0 I + \\sum _ { k = 1 } ^ m c _ k S ^ k + \\sum _ { k = 1 } ^ m c _ { - k } S ^ { * k } , \\end{align*}"}
+{"id": "177.png", "formula": "\\begin{align*} \\begin{aligned} \\Theta _ \\gamma ( v ) = 1 \\textrm { i f } 0 \\leq \\gamma \\leq 1 \\ , , \\textrm { a n d } \\Theta _ \\gamma ( v ) = \\nu ^ { - 1 } ( v ) \\textrm { i f } - 3 < \\gamma < 0 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "4601.png", "formula": "\\begin{align*} \\cot ( \\pi \\theta ( n + 1 ) - \\pi k ) = \\cot \\pi \\theta ( n ) - \\frac { V ( n ) } { \\sin \\pi k } \\end{align*}"}
+{"id": "6254.png", "formula": "\\begin{align*} \\omega ( y _ i b ^ j ) \\alpha ( y _ i b ^ j ) = \\omega ( y _ i b ^ { j + 1 } ) ; \\end{align*}"}
+{"id": "5480.png", "formula": "\\begin{align*} ( 1 - t ) F _ N ( 0 , b ; t ) = ( 1 - t q ^ N ) \\sum _ { n = 0 } ^ { N } \\begin{bmatrix} N \\\\ n \\end{bmatrix} \\frac { ( q ) _ n ( b t ) ^ n q ^ { n ^ 2 } } { ( b q ) _ n ( t q ) _ n } . \\end{align*}"}
+{"id": "3219.png", "formula": "\\begin{align*} D + 2 - 2 \\sqrt { D + 1 } > \\frac { 2 \\left ( \\frac { d ( d + 1 ) } { 2 } + 1 \\right ) - 1 } { 3 } = \\frac { D + 1 } { 3 } , \\end{align*}"}
+{"id": "4453.png", "formula": "\\begin{align*} ( A x + b ) \\cdot \\nu ( x ) = 0 \\Gamma _ R . \\end{align*}"}
+{"id": "4661.png", "formula": "\\begin{align*} C _ F \\ ; : \\ ; \\prod _ { j = 0 } ^ { \\ell - 1 } \\left ( y - \\sum _ { k = 0 } ^ { n _ q - 1 } \\zeta _ \\ell ^ { j q ^ k } \\sqrt [ \\ell ] { F _ { \\mathbf { v } _ k } } \\right ) = 0 . \\end{align*}"}
+{"id": "5685.png", "formula": "\\begin{align*} F _ 0 & = x ^ { 3 } + x ^ { 2 } y - x y ^ { 2 } + y ^ { 3 } + x ^ { 2 } z + x y z + y ^ { 2 } z - x z ^ { 2 } + z ^ { 3 } \\\\ F _ 1 & = x ^ { 3 } + x ^ { 2 } y - x ^ { 2 } z - x y z + y ^ { 2 } z + z ^ { 3 } \\\\ F _ 2 & = x ^ { 3 } - x ^ { 2 } y + x y ^ { 2 } + y ^ { 3 } + x ^ { 2 } z + x y z + y ^ { 2 } z - y z ^ { 2 } \\end{align*}"}
+{"id": "552.png", "formula": "\\begin{align*} z ^ \\top \\nabla ^ 2 f ( x ) z & = \\sum _ { i , j = 1 } ^ n z _ i z _ j \\partial _ { i j } f ( x ) = \\sum _ { i = 1 } ^ n z _ i ^ 2 V '' ( x _ i ) + \\sum _ { i , j = 1 } ^ n \\big ( z _ i ^ 2 - z _ i z _ j \\big ) J _ { i j } K '' ( x _ i - x _ j ) . \\end{align*}"}
+{"id": "1338.png", "formula": "\\begin{align*} \\delta : = \\frac { p ^ 2 } { n - p } > 0 , \\end{align*}"}
+{"id": "3263.png", "formula": "\\begin{align*} L \\phi _ m = \\mu _ m \\phi _ m , K \\phi _ m = \\tilde { a } _ m \\phi _ { m - 1 } + \\tilde { b } _ n \\phi _ m + \\tilde { a } _ { m + 1 } \\phi _ { m + 1 } , \\end{align*}"}
+{"id": "485.png", "formula": "\\begin{align*} B _ n \\Big ( \\frac { \\alpha ^ j } { L _ j } \\Big ) & = \\sum _ { k = 0 } ^ n { \\binom n k \\frac { { B _ k \\alpha ^ { j ( n - k ) } } } { { L _ j ^ { n - k } } } } \\\\ & = \\alpha \\sum _ { k = 0 } ^ n \\binom n k \\frac { B _ k F _ { j ( n - k ) } } { L _ j ^ { n - k } } + \\sum _ { k = 0 } ^ n \\binom n k \\frac { B _ k F _ { j ( n - k ) - 1 } } { L _ j ^ { n - k } } , \\end{align*}"}
+{"id": "8171.png", "formula": "\\begin{align*} ( a ^ * + \\overline { \\lambda } b ^ * C ) ( I + | \\lambda | ^ 2 T ) ^ { - 1 } ( a + \\lambda C ^ * b ) = \\| a \\| ^ 2 , \\ \\mbox { w h e r e } \\ T = I - a a ^ * - C ^ * C . \\end{align*}"}
+{"id": "3463.png", "formula": "\\begin{align*} \\Big \\Vert \\varphi \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\partial \\Omega ) } ^ 2 = \\int _ { \\partial \\Omega } | \\varphi | ^ 2 ( \\mathrm { x } , t ) d \\sigma ( \\mathrm { x } ) = \\delta \\int _ { \\partial B } | \\varphi | ^ 2 ( \\delta \\eta + \\mathrm { z } , \\alpha \\delta ^ 2 \\Tilde { \\tau } ) d \\sigma _ \\eta = \\delta \\Big \\Vert \\hat { \\varphi } \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\partial B ) } ^ 2 . \\end{align*}"}
+{"id": "6882.png", "formula": "\\begin{align*} \\alpha _ g \\alpha _ h = \\beta _ g \\beta _ h = \\alpha , \\alpha _ g \\beta _ h = \\beta _ g \\alpha _ h = - \\alpha . \\end{align*}"}
+{"id": "8198.png", "formula": "\\begin{align*} \\| u \\| _ { \\widetilde { L } ^ q ( \\widetilde { B } ^ { s _ 1 , s _ 2 } ) } : = \\sum _ { j = - \\infty } ^ { j _ 0 } 2 ^ { j s _ 1 } \\Vert \\dot { \\Delta } _ j u \\Vert _ { L ^ q _ T ( L ^ 2 ) } + \\sum _ { j = j _ 0 + 1 } ^ { + \\infty } 2 ^ { j s _ 2 } \\Vert \\dot { \\Delta } _ j u \\Vert _ { L ^ q _ T ( L ^ 2 ) } . \\end{align*}"}
+{"id": "7612.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( | | x ^ { k + 1 } ( \\theta ) - x ^ * ( \\theta ) | | _ \\pi ^ 2 | \\mathcal { F } _ k \\right ) & \\leq \\mathbb { E } | | x ^ 1 ( \\theta ) - x ^ * ( \\theta ) | | _ \\pi ^ 2 \\\\ & - \\sum _ { i = 1 } ^ k 2 \\alpha _ i \\mathbb { E } \\left ( \\mathbb { E } _ \\pi \\left ( f ( x _ i ( \\theta ) , \\theta ) - f ( x ^ * ( \\theta ) , \\theta ) | \\mathcal { F } _ i \\right ) \\right ) \\\\ & + \\sum _ { i = 1 } ^ k \\alpha _ i ^ 2 \\left ( G ^ 2 + V ^ 2 \\right ) \\end{align*}"}
+{"id": "3917.png", "formula": "\\begin{align*} \\int _ 0 ^ h ( t + h - \\tau ) ^ { - \\delta } d \\tau & = ( 1 - \\delta ) ^ { - 1 } [ ( t + h ) ^ { 1 - \\delta } - t ^ { 1 - \\delta } ] \\\\ & \\le h t ^ { - \\delta } \\le h ^ { 2 \\gamma } t ^ { - 2 \\gamma } h ^ { 1 - 2 \\gamma } t ^ { 2 \\gamma - \\delta } \\le h ^ { 2 \\gamma } t ^ { - 2 \\gamma } T ^ { 1 - \\delta } , \\end{align*}"}
+{"id": "459.png", "formula": "\\begin{align*} L ^ 2 ( z ) = 4 e ^ { L _ j z } \\cosh ^ 2 \\Big ( \\frac { \\sqrt { 5 } F _ j } { 2 } z \\Big ) . \\end{align*}"}
+{"id": "4025.png", "formula": "\\begin{align*} I ( s ) = e ^ { \\frac { s } { 2 } \\left ( 1 - \\frac { 1 } { k } \\right ) } . \\end{align*}"}
+{"id": "6223.png", "formula": "\\begin{align*} D ( H ( p ) ) = \\{ & \\psi : \\ : H ( p ) \\psi \\in L ^ 2 ( \\Sigma ) : \\ , \\psi ( x , 0 ) = 0 , \\ ; \\ , x \\in \\mathbb { R } , \\ ; \\ ; \\psi ( x , d ) = 0 \\ ; \\ ; \\ ; \\ ; | x | \\ge a \\ ; \\ ; \\\\ & \\& \\ ; \\ ; \\partial _ z \\psi ( x , d ) = 0 \\ ; \\ ; \\ ; \\ ; | x | < a , \\ ; \\ ; x \\in \\mathbb { R } \\} , \\end{align*}"}
+{"id": "2307.png", "formula": "\\begin{align*} \\lambda _ 0 ^ - ( 1 / 2 ) = \\lambda _ 0 ^ - ( - 1 / 2 ) = \\lambda _ { d , d - 2 } ^ + ( d - 3 / 2 ) = \\lambda _ { d , d - 2 } ^ + ( d - 5 / 2 ) = 1 . \\end{align*}"}
+{"id": "749.png", "formula": "\\begin{align*} [ \\partial a _ \\lambda b ] & = - \\lambda [ a _ \\lambda b ] , \\ \\ [ a _ \\lambda \\partial b ] = ( \\partial + \\lambda ) [ a _ \\lambda b ] \\ \\ \\mbox { ( c o n f o r m a l \\ s e s q u i l i n e a r i t y ) } , \\\\ { [ a _ \\lambda b ] } & = - [ b _ { - \\lambda - \\partial } a ] \\ \\ \\mbox { ( s k e w - s y m m e t r y ) } , \\\\ { [ a _ \\lambda [ b _ \\mu c ] ] } & = [ [ a _ \\lambda b ] _ { \\lambda + \\mu } c ] + [ b _ \\mu [ a _ \\lambda c ] ] \\ \\ \\mbox { ( J a c o b i \\ i d e n t i t y ) } . \\end{align*}"}
+{"id": "1502.png", "formula": "\\begin{align*} J ( w , \\Delta ) = \\sum _ { w < d \\le \\Delta } h ( d ) . \\end{align*}"}
+{"id": "4212.png", "formula": "\\begin{align*} A ( r - 1 , k ) = \\sum _ { j = 0 } ^ k \\binom { r } { j } ( - 1 ) ^ j ( k + 1 - j ) ^ { r - 1 } \\end{align*}"}
+{"id": "2671.png", "formula": "\\begin{align*} \\phi _ \\mu ( g ) = \\frac { J _ \\lambda ^ { ( 2 ) } ( a _ 1 , \\ldots , a _ N ) } { J _ \\lambda ^ { ( 2 ) } ( 1 , \\ldots , 1 ) } , \\mu = 2 \\lambda \\end{align*}"}
+{"id": "2808.png", "formula": "\\begin{align*} A ( \\omega _ i g _ i ) = ( \\omega _ i c _ i ) u _ i ^ A , \\gamma B ( \\omega _ i g _ i ) = ( \\gamma \\omega _ i s _ i ) u _ i ^ B , \\end{align*}"}
+{"id": "5671.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\frac { 1 } { r } \\log | \\partial B _ \\mathrm { i n t } ( v , r ) | = \\lim _ { r \\to \\infty } \\frac { 1 } { r } \\log | B _ \\mathrm { i n t } ( v , r ) | = \\gamma _ \\mathrm { i n t } ( p ) \\end{align*}"}
+{"id": "3798.png", "formula": "\\begin{align*} ( n - 2 ) + ( n - 1 ) & = d + 2 n - 2 \\\\ ( n - 1 ) + n & = d + 2 n - 2 . \\end{align*}"}
+{"id": "2797.png", "formula": "\\begin{align*} Q _ A V _ k & = U _ { k + 1 } J _ k , & Q _ A ^ T U _ { k + 1 } & = V _ k J _ k ^ T + \\alpha _ { k + 1 } v _ { k + 1 } e _ { k + 1 } ^ T , \\\\ Q _ B V _ k & = \\widehat { U } _ { k } \\check { J } _ k , & Q _ B ^ T \\widehat { U } _ { k } & = V _ k \\check { J } _ k ^ T + \\check { \\beta } _ { k } v _ { k + 1 } e _ { k } ^ T , \\end{align*}"}
+{"id": "4171.png", "formula": "\\begin{align*} H ^ { \\wedge Q ! ^ { ( i + 1 ) ( i + 2 ) / 2 } } = \\big ( H ^ { \\wedge Q ! ^ { i ( i + 1 ) / 2 } } \\big ) ^ { \\wedge Q ! ^ { i + 1 } } & \\subseteq \\big ( H _ { i + 1 } ( H \\cap \\Gamma ) \\big ) ^ { \\wedge Q ! ^ { i + 1 } } \\\\ & \\subseteq H _ { i + 1 } ^ { \\wedge Q ! ^ { i + 1 } } ( H \\cap \\Gamma ) ^ { \\wedge Q ! ^ { i + 1 } } H _ { i + 2 } \\subseteq H _ { i + 2 } ( H _ 1 \\cap \\Gamma ) . \\end{align*}"}
+{"id": "3309.png", "formula": "\\begin{gather*} K _ 2 \\big ( \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } \\big ) = \\lambda _ { n _ 1 } \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } . \\end{gather*}"}
+{"id": "5869.png", "formula": "\\begin{align*} \\phi ( x ) = \\phi \\big ( ( \\theta \\rtimes \\pi ) ( b ) \\big ) & = \\sum _ s \\phi ( \\theta ( b _ s ) \\pi _ s ) = \\phi ( \\theta ( b _ e ) ) \\\\ & = \\phi ( \\theta \\circ \\mathbb E ( b ) ) = \\phi \\big ( \\mathbb E _ \\theta \\circ ( \\theta \\rtimes \\pi ) ( b ) \\big ) = \\phi ( \\mathbb E _ \\theta ( x ) ) , \\end{align*}"}
+{"id": "4292.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } \\frac { ( d q ) _ { j } ( c q ^ { k } / d ) ^ { j } } { ( d q ^ { k + 1 } ) _ j ( q ) _ j } = \\frac { ( c q ^ { k + 1 } ) _ { \\infty } } { ( c q ^ { k } / d ) _ { \\infty } } \\sum _ { j = 0 } ^ { \\infty } \\frac { ( d q ) _ { j } q ^ { j ^ 2 + 2 j k } c ^ j } { ( d q ^ { k + 1 } ) _ j ( c q ^ { k + 1 } ) _ j ( q ) _ j } . \\end{align*}"}
+{"id": "4008.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\mathrm { d } ^ { \\beta } } { \\mathrm { d } t ^ { \\beta } } \\bar { q } _ { \\beta } ( 0 , t ) & = - \\lambda \\bar { q } _ { \\beta } ( 0 , t ) , \\\\ \\frac { \\mathrm { d } ^ { \\beta } } { \\mathrm { d } t ^ { \\beta } } \\bar { q } _ { \\beta } ( n , t ) & = - \\lambda \\bar { q } _ { \\beta } ( n , t ) + \\frac { \\lambda } { ( 1 - \\rho ) ^ { - r } - 1 } \\sum _ { j = 1 } ^ { n } \\binom { r + j - 1 } { j } \\rho ^ { j } \\bar { q } _ { \\beta } ( n - j , t ) , \\ n \\ge 1 , \\end{aligned} \\end{align*}"}
+{"id": "2022.png", "formula": "\\begin{align*} \\psi ~ \\mapsto ~ W ( \\phi ) : = \\sum _ { \\gamma } \\widehat { \\psi } ( \\gamma ) \\end{align*}"}
+{"id": "6524.png", "formula": "\\begin{align*} U = \\exp ( i H ) . \\end{align*}"}
+{"id": "1744.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to \\infty } \\frac { \\mathrm { T r } \\left ( \\Theta _ \\tau ( \\xi ) e ^ { - H ^ { \\prime , c } _ \\tau } \\right ) } { \\mathrm { T r } \\left ( e ^ { - H ^ { \\prime , c } _ \\tau } \\right ) } = \\frac { \\int d \\mu \\ , \\Theta ( \\xi ) e ^ { - H ^ c } } { \\int d \\mu \\ , e ^ { - H ^ c } } \\ , . \\end{align*}"}
+{"id": "1914.png", "formula": "\\begin{align*} u = \\div \\vec f + g . \\end{align*}"}
+{"id": "3726.png", "formula": "\\begin{align*} \\hat F _ \\infty ( \\beta ^ { b [ \\rho ' _ y , v ' _ y ] ) } = A ^ Q ( \\rho ' _ y , v ' _ y ) \\end{align*}"}
+{"id": "312.png", "formula": "\\begin{align*} - F ^ { s - 1 } d a = \\frac { \\alpha _ { 1 } d \\log t _ { 1 } + \\beta _ { 2 } t _ { 2 } d \\log t _ { 2 } + t _ { 1 } t _ { 2 } \\gamma } { t _ { 1 } ^ { n _ { 1 } } t _ { 2 } ^ { n _ { 2 } } } , \\end{align*}"}
+{"id": "670.png", "formula": "\\begin{align*} d ( h _ z ) = \\frac { 2 } { \\mu } \\mu _ z h _ z d z + \\frac { \\sqrt { \\tau _ 0 } } { 2 } \\left ( \\frac { 1 } { 2 } d \\overline { z } + ( \\alpha - I \\beta ) d z \\right ) \\end{align*}"}
+{"id": "4122.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { H } } ^ { 2 m + 1 } _ { \\alpha } \\left ( I \\right ) = \\{ \\varphi \\in L ^ 2 ( 0 , 1 ) ; ~ ~ ~ \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\left ( \\chi _ n ^ { \\alpha } ( c ) \\right ) ^ { 2 m + 1 } | a _ n ( \\varphi ) | ^ 2 < \\infty \\} . \\end{align*}"}
+{"id": "5922.png", "formula": "\\begin{align*} \\pi _ X ^ 6 = [ \\Delta _ X ] - \\pi _ X ^ 0 - \\pi _ X ^ 2 - \\pi _ X ^ 4 - \\pi _ X ^ 8 - \\pi _ X ^ { 1 0 } - \\pi _ X ^ { 1 2 } \\ , . \\end{align*}"}
+{"id": "5680.png", "formula": "\\begin{align*} \\frac { \\partial { F } } { \\partial x _ i } ( P ) = c _ { i - 1 } u _ { i - 1 } ^ { d - 1 } - c _ i u _ i ^ { d - 2 } u _ { i + 1 } = 0 \\end{align*}"}
+{"id": "6083.png", "formula": "\\begin{align*} _ H ^ G \\omega ( x ) = \\frac { 1 } { [ G : C _ G ( x ) ] } \\sum _ { a \\in A _ x } \\tilde \\omega ( a x a ^ { - 1 } ) . \\end{align*}"}
+{"id": "2742.png", "formula": "\\begin{align*} g ^ * \\Gamma \\cdot \\left ( g ^ * ( \\Lambda - \\beta _ X \\pi ^ * H ) \\right ) ^ { 2 n - 2 } = \\Gamma ' \\cdot \\left ( g ^ * ( \\Lambda - \\beta _ X \\pi ^ * H ) \\right ) ^ { 2 n - 2 } = 0 . \\end{align*}"}
+{"id": "6928.png", "formula": "\\begin{align*} \\left [ \\epsilon ^ 0 q ^ { d } \\right ] \\frac { ( - 1 ) ^ { ( N - 1 ) d } } { \\det ( z _ i ^ { N - j + 1 } ) } \\ , \\begin{vmatrix} z _ 1 ^ { d + N } & z _ 1 ^ { d + N - 1 } & \\cdots & z _ 1 ^ { d + 1 } \\\\ z _ 2 ^ { d + N } & z _ 2 ^ { d + N - 1 } & \\cdots & z _ 2 ^ { d + 1 } \\\\ \\vdots & \\vdots & \\cdots & \\vdots \\\\ z _ N ^ { d + N } & z _ N ^ { d + N - 1 } & \\cdots & z _ N ^ { d + 1 } \\end{vmatrix} f ( z _ { N + 1 } ) . \\end{align*}"}
+{"id": "3585.png", "formula": "\\begin{align*} \\Vert f \\Vert ^ 2 _ { \\mathcal { H } ' } = \\sum _ { i } \\frac { a _ i ^ { 2 \\nu + 1 } b _ i ^ { 2 \\nu } h _ i ^ 2 } { a _ i ^ { 2 \\nu + 1 } b _ i ^ { 2 \\nu } } = \\sum _ i h _ i ^ 2 = \\| h \\| ^ 2 < \\infty \\end{align*}"}
+{"id": "4784.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 ) , \\end{align*}"}
+{"id": "5031.png", "formula": "\\begin{align*} h _ 1 ( \\Omega ) \\le \\frac { | D v | ( X ) } { \\| v \\| _ 1 } \\le \\frac { p \\| u \\| _ p ^ { 1 - \\frac 1 p } \\| | \\nabla u | \\| _ p } { \\| u \\| _ p ^ p } = \\frac { p \\| | \\nabla u | \\| _ p } { \\| u \\| _ p } \\end{align*}"}
+{"id": "3662.png", "formula": "\\begin{align*} p _ { 1 } & = \\sum _ { v _ i v _ j v _ k \\in \\mathcal { G } [ W ] } X _ i X _ j X _ k , \\\\ p _ { 2 } & = \\sum _ { v _ i v _ j \\in L _ { \\mathcal { G } } ( v _ 1 ) [ W ] } X _ i X _ j = \\sum _ { v _ i v _ j \\in L _ { \\mathcal { G } } ( v _ 2 ) [ W ] } X _ i X _ j , \\\\ p _ { 3 } & = X _ { i _ 1 } + \\cdots + X _ { i _ k } , p _ { 4 } = X _ { i _ 1 } + \\cdots + X _ { i _ { k - 1 } } , \\quad p _ { 5 } = X _ { k } . \\end{align*}"}
+{"id": "4586.png", "formula": "\\begin{align*} \\abs { \\frac { 1 } { N } \\sum _ { l = 0 } ^ { N - 1 } \\cos ( \\theta + 2 \\pi k l ) } = \\abs { \\frac { 1 } { N } \\sum _ { l = 0 } ^ { N - 1 } \\cos ( \\theta + 2 \\pi k l ) - \\int _ { 0 } ^ { 2 \\pi } \\cos t d t } \\leq \\varepsilon . \\end{align*}"}
+{"id": "5361.png", "formula": "\\begin{align*} g = g _ 0 \\cdots g _ p & = ( h _ 0 t _ 0 ) ( h _ 1 t _ 1 ) \\cdots ( h _ p t _ p ) \\\\ & = ( h _ 0 h _ 1 ' \\cdots h _ p ' ) ( t _ 0 \\cdots t _ p ) \\end{align*}"}
+{"id": "5558.png", "formula": "\\begin{align*} L _ 1 ( u _ 1 ( \\beta _ 0 ) ) u _ 1 ' ( \\beta _ 0 ) = L _ 2 ( u _ 2 ( \\beta _ 0 ) ) u _ 2 ' ( \\beta _ 0 ) - 2 l _ m \\gamma _ m \\beta _ 0 / ( \\theta _ b - \\theta _ m ) , \\end{align*}"}
+{"id": "2358.png", "formula": "\\begin{align*} \\iota ( \\{ 2 \\} ^ { k _ { d } } , 3 , \\dots \\{ 2 \\} ^ { k _ { 1 } } , 3 ) & = ( - 1 ) ^ { k _ { 1 } + \\cdots + k _ { d } } x _ { 2 k _ { 1 } + 3 } \\cdots x _ { 2 k _ { d } + 3 } , \\\\ \\theta ( \\{ 2 \\} ^ { k _ { d } } , 3 , \\dots \\{ 2 \\} ^ { k _ { 1 } } , 3 , \\{ 2 \\} ^ { k _ { 0 } } ) & = ( - 1 ) ^ { k _ { 0 } + \\cdots + k _ { d } } x _ { 2 k _ { 0 } + 2 } x _ { 2 k _ { 1 } + 3 } \\cdots x _ { 2 k _ { d } + 3 } . \\end{align*}"}
+{"id": "7561.png", "formula": "\\begin{align*} \\left . \\frac { d } { d \\varepsilon } \\right | _ { \\varepsilon = 0 } V ( \\Phi ( \\varepsilon , p ) ) = h ( p ) d V ( p ) \\end{align*}"}
+{"id": "4288.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] n ( - 1 ) ^ { n - 1 } d ^ n q ^ { n ( n + 1 ) / 2 } + ( d q ) _ N \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { d ^ n q ^ { n ( n + 1 ) } } { ( d q ) _ n ( 1 - q ^ n ) } = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( d q ) ^ n ( d q ) _ { N - n } } { 1 - q ^ n } . \\end{align*}"}
+{"id": "3785.png", "formula": "\\begin{align*} C _ { i _ 1 , \\ldots , i _ k } ( 1 ) = \\sum _ { r = 0 } ^ { i _ 1 - 1 } C _ { r , i _ 1 - 1 - r , i _ 2 , \\ldots , i _ k } ( 1 ) + \\sum _ { r = 0 } ^ { i _ 1 - 2 } A _ { r , i _ 1 - 2 - r } C _ { i _ 2 , \\ldots , i _ k } + 2 \\sum _ { r = 2 } ^ k i _ r C _ { i _ 1 + i _ r - 1 , i _ 2 , \\ldots , \\widehat { i _ r } , \\ldots , i _ k } \\end{align*}"}
+{"id": "6866.png", "formula": "\\begin{align*} f ( x ) = r ( x ) \\prod _ { i = 1 } ^ { s } ( x - a _ i ) \\end{align*}"}
+{"id": "611.png", "formula": "\\begin{align*} \\begin{aligned} & ( C _ { a j } - C _ j - C _ a ) \\ , [ C _ { i k } , C _ { k \\ell } ] + ( C _ { a i } - C _ i - C _ a ) [ C _ { k \\ell } , C _ { j k } ] + ( C _ { a \\ell } - C _ \\ell - C _ a ) [ C _ { i j } , C _ { j k } ] \\\\ & \\qquad + ( C _ { a k } - C _ k - C _ a ) [ C _ { i \\ell } , C _ { j \\ell } ] = 0 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "2723.png", "formula": "\\begin{align*} \\binom { 2 n } { 2 n - j - u } M _ { \\varphi } ( 2 n , j + u , 0 ; r - 1 ) = \\frac { 1 } { 2 } S ( n , r ) \\cdot c ( n , j + u , r - 1 ) \\end{align*}"}
+{"id": "5332.png", "formula": "\\begin{align*} | T | = \\frac { 1 } { d } q ^ { n ( n - 1 ) / 2 } \\prod _ { i = 2 } ^ n ( q ^ i - 1 ) > \\frac { 1 } { 2 d } q ^ { n ^ 2 - 1 } , \\end{align*}"}
+{"id": "5222.png", "formula": "\\begin{align*} D _ 2 = D _ 3 = \\begin{pmatrix} - \\mu _ 1 + \\frac { 1 } { 6 } S & & \\\\ & - \\mu _ 2 + \\frac { 1 } { 6 } S & \\\\ & & - \\mu _ 3 + \\frac { 1 } { 6 } S \\end{pmatrix} . \\end{align*}"}
+{"id": "4505.png", "formula": "\\begin{align*} \\mathcal { E } ^ { ( 3 ) } _ { \\ ; \\mathbf { x _ 0 } } ( \\mathbf { x } ) = \\frac { 1 } { 8 \\pi } e ^ { - \\| \\mathbf { x } - \\mathbf { x _ 0 } \\| } \\end{align*}"}
+{"id": "5635.png", "formula": "\\begin{align*} s _ \\lambda ( t ) = \\sin ( \\sqrt { \\lambda } t ) \\end{align*}"}
+{"id": "418.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { \\substack { U \\pmod { D } \\\\ ( U , D ) = 1 } } \\psi ( U ) N _ 2 ( H , M , N , D , S , U ) = \\frac { \\varphi ( H ) ^ 2 \\psi ( S ) } { \\varphi ( M ) \\varphi ( N ) } \\sum _ { \\substack { J \\mid M \\\\ D \\mid J } } \\frac { \\mu ( J ) } { \\varphi ( D ) } . \\end{align*}"}
+{"id": "7477.png", "formula": "\\begin{align*} n ^ m = \\sum _ { k } c _ { m k } \\psi _ k ( n ) \\ , , \\end{align*}"}
+{"id": "290.png", "formula": "\\begin{align*} D _ { I '' } = \\bigcap _ { i \\in I '' } D _ { i } , \\end{align*}"}
+{"id": "554.png", "formula": "\\begin{align*} z ^ \\top \\nabla ^ 2 f ( x ) z & \\le \\sum _ { i = 1 } ^ n z _ i ^ 2 V '' ( x _ i ) \\le - \\kappa | z | ^ 2 , \\end{align*}"}
+{"id": "1731.png", "formula": "\\begin{align*} | I | \\leq \\sum _ k \\sum _ { k _ 1 + k _ 2 + k _ 3 = k } \\int _ { \\eta _ 1 + \\eta _ 2 + \\eta _ 3 = \\eta } d \\eta _ 1 \\ , & d \\eta _ 2 \\ , d \\eta _ 3 \\ , d \\eta \\ , \\frac { | c ( k , \\eta ) | } { \\left ( 1 + | \\eta + k ^ 2 | \\right ) ^ { 1 - b } } \\\\ & \\times | \\tilde { v } _ \\delta ( k _ 1 , \\eta _ 1 ) | | \\tilde { v } _ \\delta ( - k _ 2 , - \\eta _ 2 ) | | \\tilde { v } _ \\delta ( k _ 3 , \\eta _ 3 ) | . \\end{align*}"}
+{"id": "4489.png", "formula": "\\begin{align*} X _ { f \\ , 1 } = D _ 2 f \\ , , X _ { f \\ , 2 } = - D _ 1 f \\ , , \\end{align*}"}
+{"id": "238.png", "formula": "\\begin{align*} Q [ S ] ( u ) \\stackrel { d e f } { = } { \\bf P } \\left [ \\max _ { i \\in S } \\xi _ i > u \\ \\right ] , \\end{align*}"}
+{"id": "4275.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( 1 - d q ^ n ) ( 1 - q ^ n ) } = \\sum _ { n = 1 } ^ { \\infty } \\frac { n q ^ n ( q ^ { n + 1 } ) _ { \\infty } } { ( d q ^ n ) _ { \\infty } } . \\end{align*}"}
+{"id": "2790.png", "formula": "\\begin{align*} A \\tilde { V } _ r & = \\tilde { U } _ r \\tilde { \\Sigma } _ r , \\\\ A ^ T \\tilde { U } _ r & = \\tilde { V } _ r \\tilde { \\Sigma } _ r + q _ { k + 1 } b _ r ^ T , \\end{align*}"}
+{"id": "3287.png", "formula": "\\begin{gather*} p _ 0 = \\alpha _ 0 + \\alpha _ 3 , p _ 1 = \\alpha _ 0 - \\alpha _ 3 , p _ 2 = \\alpha _ 1 + \\alpha _ 2 , p _ 3 = \\alpha _ 1 - \\alpha _ 2 , \\end{gather*}"}
+{"id": "1652.png", "formula": "\\begin{align*} \\eta ( \\mathsf { I } _ { \\mathsf { w } - 1 } , \\mathsf { I } _ { \\mathsf { w } } ) \\ ; \\geq \\ ; 2 ^ { - 3 } \\ , \\mathsf { q } ^ { - 1 } \\ , \\pmb { \\eta } \\ , , \\qquad \\qquad \\mathsf { w } = 1 , \\dots , \\mathsf { q } \\ , , \\end{align*}"}
+{"id": "8122.png", "formula": "\\begin{align*} ( ( L , \\varphi _ 0 ) ^ { d + 1 } ) = - \\operatorname { \\widehat { \\deg } } _ { \\xi _ 0 ' } ( R ) . \\end{align*}"}
+{"id": "3663.png", "formula": "\\begin{align*} & p _ { \\mathcal { G } \\boxplus \\{ v _ 1 , v _ 2 \\} } ( X _ 1 , X _ 1 ' , X _ 2 , X _ 2 ' , X _ 3 \\ldots , X _ { m } ) \\\\ & = p _ 1 + p _ 2 ( X _ 1 + X _ 2 + X _ 1 ' + X _ 2 ' ) + p _ { 4 } ( X _ 1 + X _ 1 ' ) ( X _ 2 + X _ 2 ' ) + p _ 5 ( X _ 1 + X _ 2 ) ( X _ 1 ' + X _ 2 ' ) . \\end{align*}"}
+{"id": "5564.png", "formula": "\\begin{align*} \\dfrac { 1 } { \\Phi _ 1 [ \\alpha _ 0 , \\beta _ 0 , L _ 1 ( 0 ) , N _ 1 ( 0 ) ] } + \\dfrac { u _ c } { \\Phi _ 2 [ \\beta _ 0 , \\infty , L _ 2 ( u _ c ) , N _ 2 ( u _ c ) ] } = \\dfrac { 2 l _ m \\gamma _ m ( \\beta _ 0 ) ^ { \\nu + 1 } } { \\theta _ b - \\theta _ m } \\end{align*}"}
+{"id": "1394.png", "formula": "\\begin{align*} \\begin{pmatrix} u _ 0 ^ { ( j ) } \\\\ u _ 1 ^ { ( j ) } \\end{pmatrix} : = J _ { \\lambda _ j } \\begin{pmatrix} u _ 0 \\\\ u _ 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "6969.png", "formula": "\\begin{align*} Q = \\operatorname { S O T - } \\lim P ^ { * n } P ^ n . \\end{align*}"}
+{"id": "7502.png", "formula": "\\begin{align*} Q ( z ) = \\left ( \\frac { V ' ( z ) } { 2 } \\right ) ^ 2 - \\int \\frac { V ' ( z ) - V ' ( s ) } { z - s } d \\mu ( s ) . \\end{align*}"}
+{"id": "4556.png", "formula": "\\begin{align*} \\Gamma _ { \\ ! \\ ! f } ( G ) & = \\left ( f ( 2 ) - \\frac { 2 } { 3 } f ( 1 ) - \\frac { 1 } { 3 } f ( 4 ) \\right ) n _ 2 + \\left ( f ( 3 ) - \\frac { 1 } { 3 } f ( 1 ) - \\frac { 2 } { 3 } f ( 4 ) \\right ) n _ 3 \\ , . \\end{align*}"}
+{"id": "3015.png", "formula": "\\begin{align*} & \\begin{aligned} x = F _ x ( y , \\dot { y } , \\ldots , y ^ { ( r - 1 ) } ) \\end{aligned} \\\\ & \\begin{aligned} u = F _ u ( y , \\dot { y } , \\ldots , y ^ { ( r ) } ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "6673.png", "formula": "\\begin{align*} \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( 1 , 0 , \\ldots , 0 ) \\right ) & \\stackrel { a \\to 0 + } { \\longrightarrow } \\mathcal { D } \\left ( ( - s ^ { ( 1 ) } _ { k + 1 } ) _ { k = 0 } ^ { 2 m - 3 } , ( 1 , 0 , \\ldots , 0 ) \\right ) \\\\ & = ( - 1 ) ^ { m } \\mathcal { E } \\left ( ( - s ^ { ( 1 ) } _ { k + 1 } ) _ { k = 0 } ^ { 2 m - 3 } , ( 1 , 0 , \\ldots , 0 ) \\right ) , \\end{align*}"}
+{"id": "2406.png", "formula": "\\begin{align*} { \\rm C u t } _ { ( 0 ; \\emptyset ) } ( \\psi \\mid _ { z \\to 0 } ) = ( u + v ) \\cdot ( \\psi \\mid _ { z \\to 0 } ) . \\end{align*}"}
+{"id": "218.png", "formula": "\\begin{align*} V ^ { \\sigma } ( m ) = F ( m ) + \\frac { \\sigma ^ 2 } { 2 } \\operatorname { K L } ( m | \\pi ) \\ , , m \\in \\mathcal { P } ( \\mathbb { R } ^ d ) \\ , , \\end{align*}"}
+{"id": "142.png", "formula": "\\begin{align*} ( d _ { i } y ) ( [ \\mathbf { a } ] \\otimes [ \\mathbf { b } ] ) = y ( d ^ i [ \\mathbf { a } ] \\otimes [ \\mathbf { b } ] ) = 0 , \\end{align*}"}
+{"id": "4121.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { H } } ^ { 2 m } _ { \\alpha } \\left ( I \\right ) = \\{ \\varphi \\in L ^ 2 ( 0 , 1 ) ; ~ ~ ~ \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\left ( \\chi _ n ^ { \\alpha } ( c ) \\right ) ^ { 2 m } | a _ n ( \\varphi ) | ^ 2 < \\infty \\} \\end{align*}"}
+{"id": "4830.png", "formula": "\\begin{align*} - \\nabla \\cdot ( \\epsilon e ^ { - g / \\epsilon \\ , } \\nabla w ) + e ^ { - g / \\epsilon \\ , } w = \\int _ S e ^ { - g / \\epsilon \\ , } w ( x , s ' ) \\dd \\mu ( s ' ) . \\end{align*}"}
+{"id": "2672.png", "formula": "\\begin{align*} P ( n , r ) = \\binom { n + r - 1 } { n } \\frac { 1 } { 2 } S ( n , r ) \\end{align*}"}
+{"id": "5093.png", "formula": "\\begin{align*} \\frac { 1 } { \\tilde { r } ' } = \\frac { p + 1 } { r } , \\frac { 1 } { \\tilde { q } ' } > \\frac { p + 1 } { q } . \\end{align*}"}
+{"id": "798.png", "formula": "\\begin{align*} \\mathcal { J } _ i ( \\tau _ i , \\tau _ { - i } ) = \\mathbb { E } \\Bigg \\{ \\widetilde { \\theta } _ 1 e ^ { - \\beta \\tau _ i } \\Big ( x _ i ( \\tau _ i ) - K ' _ 1 \\Big ) \\Bigg \\} . \\end{align*}"}
+{"id": "4866.png", "formula": "\\begin{align*} \\zeta _ { C B } ( 1 ) = \\frac { 1 } { 3 } \\frac { \\pi } { \\sqrt { 3 } } , \\zeta _ { C B } ( 0 ) = \\frac { 1 } { 3 } + \\frac { 2 } { 9 } \\frac { \\pi } { \\sqrt { 3 } } , \\zeta _ { C B } ( - 1 ) = \\frac { 2 } { 3 } + \\frac { 2 } { 9 } \\frac { \\pi } { \\sqrt { 3 } } , \\zeta _ { C B } ( - 2 ) = \\frac { 4 } { 3 } + \\frac { 1 0 } { 2 7 } \\frac { \\pi } { \\sqrt { 3 } } . \\end{align*}"}
+{"id": "5828.png", "formula": "\\begin{align*} ( \\Pi R \\Pi ^ * ) M _ z = M _ z ( \\Pi R \\Pi ^ * ) . \\end{align*}"}
+{"id": "1678.png", "formula": "\\begin{align*} \\mathfrak { O } _ { 0 , n - 1 } = \\mathfrak { O } _ { 0 , n } \\sqcup \\big ( \\mathfrak { O } _ { 0 , n - 1 } \\setminus \\mathfrak { O } _ { n - 1 , n } \\big ) \\ , , \\qquad \\neg \\mathfrak { O } _ { 0 , N } = \\bigsqcup \\limits _ { k = 1 } ^ { N } \\ , \\left ( \\mathfrak { O } _ { 0 , k - 1 } \\setminus \\mathfrak { O } _ { k - 1 , k } \\right ) \\ , . \\end{align*}"}
+{"id": "7508.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { E _ \\varphi ( F _ { \\varepsilon , h } ) - E _ \\varphi ( F ) } { \\varepsilon } = 0 \\end{align*}"}
+{"id": "3490.png", "formula": "\\begin{align*} \\sqrt [ 4 ] { \\int _ { 0 } ^ \\mathrm { T _ 0 } \\Big [ | \\nabla _ { \\textbf { t a n } } \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) | ^ 2 + \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) ^ 2 + \\Big ( \\partial _ { t } ^ { \\frac { 1 } { 2 } } \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) \\Big ) ^ 2 \\Big ] d \\tau } = \\mathcal { O } \\Bigg ( \\sqrt [ 4 ] { \\mathcal { K } ^ { ( \\mathrm { T _ 0 } ) } _ { \\mathrm { r } } } \\frac { 1 } { | \\xi - z | ^ { \\frac { 3 } { 2 } - \\mathrm { r } } } \\Bigg ) . \\end{align*}"}
+{"id": "3721.png", "formula": "\\begin{align*} i \\partial _ t \\psi + \\frac 1 2 \\nabla ^ 2 \\psi = 0 , | \\psi _ o | ^ 2 = \\rho _ o , | \\psi _ 1 | ^ 2 = \\rho _ 1 \\end{align*}"}
+{"id": "8011.png", "formula": "\\begin{align*} S _ { M } \\otimes I _ 2 = S _ { M \\otimes I _ d } ( J ) . \\end{align*}"}
+{"id": "3268.png", "formula": "\\begin{align*} y _ j = q ^ { - j } + \\gamma \\delta q ^ { j + 1 } . \\end{align*}"}
+{"id": "4475.png", "formula": "\\begin{align*} v = \\sum _ { n = 1 } ^ { \\infty } \\langle v , e _ n \\rangle _ { \\mathcal { H } } \\ , e _ n \\ , , \\end{align*}"}
+{"id": "4150.png", "formula": "\\begin{align*} \\sup _ { M \\in \\mathbb { N } } \\ , \\sup _ { j = 1 , 2 , . . . , M } \\ , \\frac { 1 } { M } \\sum _ { k = 1 } ^ { M } B ^ M _ { i j } \\leq C , \\end{align*}"}
+{"id": "7133.png", "formula": "\\begin{align*} c ' _ { i j } = c _ { i j } + \\varepsilon , \\ c ' _ { l i } = c _ { l i } + \\varepsilon , \\ c ' _ { l j } = c _ { l j } - \\varepsilon . \\end{align*}"}
+{"id": "2302.png", "formula": "\\begin{align*} [ x _ 1 , x _ 2 , x _ 3 , x _ 4 ] = [ \\mathfrak h ( x _ 1 ) , \\mathfrak h ( x _ 2 ) , \\mathfrak h ( x _ 3 ) , \\mathfrak h ( x _ 4 ) ] . \\end{align*}"}
+{"id": "1518.png", "formula": "\\begin{align*} \\begin{array} { c c } ( k , b , c ) = \\left ( 1 - q , q x _ i ^ 2 , - \\frac { \\tau ^ 6 + 2 \\tau ^ 5 + 4 \\tau ^ 4 + 8 \\tau ^ 3 + 9 \\tau ^ 2 + 4 \\tau + 1 } { 4 \\tau ^ 2 ( \\tau + 1 ) ^ 2 } \\right ) & \\textrm { i f } n = 1 \\\\ ( k , b , c ) = \\left ( q - 1 , - q x _ i ^ 2 , - \\frac { \\tau ^ 6 + 2 \\tau ^ 5 + 4 \\tau ^ 4 + 8 \\tau ^ 3 + 9 \\tau ^ 2 + 4 \\tau + 1 } { 4 \\tau ^ 2 ( \\tau + 1 ) ^ 2 } \\right ) & \\textrm { i f } n = 2 \\\\ \\end{array} \\end{align*}"}
+{"id": "613.png", "formula": "\\begin{align*} & C _ { 1 3 } = - C _ { 2 3 } + C _ { 1 2 3 } - C _ { 1 2 } + C _ 1 + C _ 2 + C _ 3 \\\\ & C _ { 2 4 } = C _ { 2 3 4 } - C _ { 2 3 } - C _ { 3 4 } + C _ 2 + C _ 3 + C _ 4 \\ , , \\\\ & C _ { 1 4 } = C _ { 1 2 3 4 } - C _ { 1 2 3 } - C _ { 2 3 4 } + C _ { 2 3 } + C _ 1 + C _ 4 \\ , , \\\\ & C _ { 1 2 4 } = C _ { 1 2 3 4 } - C _ { 1 2 3 } - C _ { 3 4 } + C _ { 1 2 } + C _ { 3 } + C _ { 4 } \\ , , \\\\ & C _ { 1 3 4 } = C _ { 1 2 3 4 } - C _ { 2 3 4 } - C _ { 1 2 } + C _ { 3 4 } + C _ 1 + C _ 2 \\ , . \\end{align*}"}
+{"id": "2095.png", "formula": "\\begin{align*} | X | : = \\int _ X \\Omega _ X . \\end{align*}"}
+{"id": "2758.png", "formula": "\\begin{align*} \\frac { b ( \\Gamma ) } { a ( \\Gamma ) } = \\frac { \\Lambda ^ { 2 n - 1 } } { \\Lambda ^ { 2 n - 2 } \\cdot \\pi ^ * H } . \\end{align*}"}
+{"id": "1481.png", "formula": "\\begin{align*} A _ d = \\sum _ { n \\equiv 0 ( \\mod d ) } a _ n \\end{align*}"}
+{"id": "2413.png", "formula": "\\begin{align*} { \\rm e v } _ { 0 } ^ { \\mathfrak { m } } ( \\psi ) = 0 . \\end{align*}"}
+{"id": "3441.png", "formula": "\\begin{align*} \\varepsilon _ \\mathrm { p } ( \\omega , \\gamma ) = \\varepsilon _ \\infty \\Bigg [ 1 + \\dfrac { \\omega _ \\mathrm { p } ^ 2 } { \\omega _ 0 ^ 2 - \\omega ^ 2 - i \\gamma \\omega } \\Bigg ] \\end{align*}"}
+{"id": "7444.png", "formula": "\\begin{align*} [ J ^ 2 , \\ , \\tau ^ \\dagger _ \\theta ] = \\tau ^ \\dagger _ \\theta \\theta ( \\theta + 2 \\hat { j } + 1 ) . \\end{align*}"}
+{"id": "6992.png", "formula": "\\begin{align*} 0 = \\frac { 1 } { 2 } \\frac { d G ( t , \\beta ) } { d t } \\Big | _ { t = 0 } = \\int \\left ( \\langle \\nabla u , \\nabla \\phi \\rangle + \\alpha _ 1 u \\phi - \\frac { \\alpha _ 2 } { 2 } u \\phi \\log u ^ 2 - \\alpha _ 2 u \\phi + \\beta u \\phi \\right ) d m . \\end{align*}"}
+{"id": "880.png", "formula": "\\begin{align*} ( d _ 1 : = d _ 1 ' , \\ldots , d _ m ' : = d _ 2 ) , \\end{align*}"}
+{"id": "4619.png", "formula": "\\begin{align*} \\phi ( t ) = \\int _ 0 ^ t \\phi ' _ + ( \\tau ) \\ , d \\tau = \\int _ 0 ^ t \\phi ' _ - ( \\tau ) \\ , d \\tau , \\end{align*}"}
+{"id": "5742.png", "formula": "\\begin{align*} X ^ M _ i = \\begin{pmatrix} & & & ( v _ 1 ^ i ) ^ T \\\\ & 0 & & \\vdots \\\\ & & & ( v _ { r } ^ i ) ^ T \\\\ & & & \\\\ v _ 1 ^ i & \\cdots & v _ { r } ^ i & 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "3173.png", "formula": "\\begin{align*} f ^ n ( \\alpha \\setminus \\beta ' ) \\cap f ^ n ( \\beta ' ) = \\varnothing f ^ n ( \\beta ' ) \\subset [ f ^ n ( \\alpha ) ] \\end{align*}"}
+{"id": "7732.png", "formula": "\\begin{align*} d i s t _ { g ^ + } ( q , q ' ) = | r ( q ' ) - r ( q ) | = r ( q ) \\end{align*}"}
+{"id": "6994.png", "formula": "\\begin{align*} u = \\left ( \\beta I - \\Delta \\right ) ^ { - 1 } \\left ( \\alpha _ 2 u \\log u + ( \\beta + \\lambda - \\alpha _ 1 ) u \\right ) = R _ { \\beta } \\left ( \\alpha _ 2 u \\log u + \\alpha _ 3 u \\right ) , \\end{align*}"}
+{"id": "419.png", "formula": "\\begin{align*} d S _ t = S _ t \\mu d t + S _ t \\sigma d W _ t , t \\geq 0 , \\end{align*}"}
+{"id": "5746.png", "formula": "\\begin{align*} ( { \\rm I d } _ 4 \\otimes P ) g ( X ^ 0 ) ( { \\rm I d } _ 4 \\otimes P ) & = \\begin{pmatrix} P ( X ^ 0 _ 1 X ^ 0 _ 1 - X ^ 0 _ 2 X ^ 0 _ 2 ) P & 0 \\\\ 0 & P ( X ^ 0 _ 1 X ^ 0 _ 2 + X ^ 0 _ 2 X ^ 0 _ 1 ) P \\end{pmatrix} \\\\ & \\oplus \\begin{pmatrix} 0 & P ( X ^ 0 _ 1 X ^ 0 _ 2 - X ^ 0 _ 2 X ^ 0 _ 1 ) P \\\\ P ( X ^ 0 _ 2 X ^ 0 _ 1 - X ^ 0 _ 1 X ^ 0 _ 2 ) P & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "4132.png", "formula": "\\begin{align*} \\left ( \\mathcal { L } _ { c } ^ { \\alpha } + \\chi _ { n } ^ { \\alpha } ( c ) I d \\right ) \\left ( \\partial _ c ( \\psi _ { n , c } ^ { ( \\alpha ) } ) \\right ) + \\left ( \\partial _ c \\chi _ { n } ^ { \\alpha } ( c ) - 2 c x ^ 2 \\right ) \\psi _ { n , c } ^ { ( \\alpha ) } = 0 . \\end{align*}"}
+{"id": "5404.png", "formula": "\\begin{align*} \\int _ { B _ { 5 \\rho / 4 } } u ^ 2 ( x , 0 ) d x & \\le n _ 1 e ^ { M _ 1 ( 1 - \\beta ) + M _ 1 \\beta ( 1 - \\beta ) } \\left ( \\int _ { B _ { \\rho / 2 } } u ^ 2 ( x , 0 ) d x \\right ) ^ { \\beta ^ 2 } \\left ( N \\int _ { Q _ { 4 } } u ^ 2 ( x , t ) \\right ) ^ { 1 - \\beta + \\beta ( 1 - \\beta ) } \\\\ & = n _ 1 e ^ { M _ 1 ( 1 - \\beta ^ 2 ) } \\left ( \\int _ { B _ { \\rho / 2 } } u ^ 2 ( x , 0 ) d x \\right ) ^ { \\beta ^ 2 } \\left ( N \\int _ { Q _ { 4 } } u ^ 2 ( x , t ) \\right ) ^ { 1 - \\beta ^ 2 } . \\end{align*}"}
+{"id": "1385.png", "formula": "\\begin{align*} \\varphi _ { \\beta , \\varepsilon } ( s ) = e ^ { - s } M \\left ( \\gamma _ { \\varepsilon } - \\beta , \\gamma _ { \\varepsilon } ; s \\right ) , s \\ge 0 . \\end{align*}"}
+{"id": "3260.png", "formula": "\\begin{align*} [ X , Y ] _ q = q X Y - q ^ { - 1 } Y X . \\end{align*}"}
+{"id": "5949.png", "formula": "\\begin{align*} 0 \\in ( \\sum _ { i = 1 } ^ { k } \\tau _ { A _ i } - \\sum _ { i = 1 } ^ { k } \\tau _ { \\bar { K } _ i } + \\Box ) \\cap ( \\sum _ { i = 1 } ^ { k } \\tau _ { B _ i } - \\sum _ { i = 1 } ^ { k } \\tau _ { \\bar { K } _ i } + \\Box ) \\end{align*}"}
+{"id": "216.png", "formula": "\\begin{align*} \\lambda _ i - \\left \\lfloor \\frac { k ( m - ( 2 i - 1 ) ) } { 2 } \\right \\rfloor & \\geq 1 + k ( m - i ) - \\left \\lfloor \\frac { k ( m - ( 2 i - 1 ) ) } { 2 } \\right \\rfloor \\\\ & = 1 + k m - \\left \\lfloor \\frac { k ( m + 1 ) ) } { 2 } \\right \\rfloor \\\\ & = 1 + k ( m - 1 ) - \\left \\lfloor \\frac { k ( m - 1 ) ) } { 2 } \\right \\rfloor \\\\ & = 1 + \\left \\lceil \\frac { k ( m - 1 ) } { 2 } \\right \\rceil . \\end{align*}"}
+{"id": "8137.png", "formula": "\\begin{align*} ( \\overline L _ 0 \\cdots \\overline L _ d ) _ S \\geqslant \\sum _ { i = 0 } ^ d \\delta _ i \\ , \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } ( \\overline L _ i ) . \\end{align*}"}
+{"id": "1636.png", "formula": "\\begin{align*} \\psi ( x , t ) = \\left ( \\phi _ { ( I R ) } \\boxdot \\psi \\right ) ( x , t ) + \\left ( \\phi _ { ( I S ) } \\boxdot \\left ( x + t + \\left ( \\phi _ { ( I R ) } \\boxdot \\psi \\right ) \\right ) \\right ) ( x , t ) . \\end{align*}"}
+{"id": "2973.png", "formula": "\\begin{align*} \\operatorname { H o m } ( M ^ { \\ast } , X \\oplus Y ) \\cong \\operatorname { E x t } ^ 1 ( M ^ { \\ast } , \\Sigma ^ { - 1 } X \\oplus \\Sigma ^ { - 1 } Y ) = 0 \\end{align*}"}
+{"id": "7857.png", "formula": "\\begin{align*} \\int _ I \\sum _ { k \\geq k _ 0 } \\mathbb { P } \\big ( Z \\cap \\rho _ \\theta ^ { - 1 } ( G _ { \\theta , k } \\big ) d \\nu ( \\theta ) \\geq \\nu ( \\Theta ) = 1 . \\end{align*}"}
+{"id": "5.png", "formula": "\\begin{align*} S _ { i , j } = & X \\sum _ { \\substack { \\alpha \\leq Z \\\\ ( \\alpha , 2 ) = 1 } } \\frac { \\mu ( \\alpha ) } { 2 \\alpha ^ 2 } \\sum _ { m \\geq 1 } \\sum _ { \\substack { ( p , 2 \\alpha ) = 1 } } \\frac { a _ { \\pi } ( p ) \\log p } { p } \\hat { \\phi } \\left ( \\frac { \\log p } { M \\log X } \\right ) \\left ( \\frac { 2 j m } { p } \\right ) \\widetilde { W } _ { i } \\left ( \\frac { m X } { \\alpha ^ 2 p } \\right ) . \\end{align*}"}
+{"id": "7841.png", "formula": "\\begin{align*} \\widetilde A = \\widetilde Y ^ i \\otimes \\eta _ i , \\end{align*}"}
+{"id": "3677.png", "formula": "\\begin{align*} | V _ j \\setminus N ( u ) | \\le \\frac { 4 t \\epsilon n ^ 2 } { n / 3 } = 1 2 t \\epsilon n . \\end{align*}"}
+{"id": "7145.png", "formula": "\\begin{align*} - \\frac { 3 } { 2 } \\mathfrak { g } ^ { \\lambda > 0 } = \\frac { 3 } { 2 } \\sum _ { D ' } ( \\beta _ i - \\beta _ j ) , \\end{align*}"}
+{"id": "7750.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { A } ( p _ 0 ) & = 2 ^ n \\int _ { \\partial X } d V _ { \\hat { g } ^ { p _ 0 } } = 2 ^ n \\int _ { \\partial X } e ^ { - n ( s _ { p _ 0 } - s _ E ) | _ { \\partial X } } d V _ { \\hat { g } ^ E } \\\\ & = \\int _ { \\mathbb { S } ^ { n - 1 } \\times \\mathbb { S } ^ 1 ( \\lambda ) } e ^ { n b _ { \\gamma _ + } ( t _ 0 , w , \\theta ) } d \\Theta _ { n - 1 } d \\Theta _ \\lambda . \\end{aligned} \\end{align*}"}
+{"id": "1263.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c c c c } 1 & 2 & 2 ^ 2 & \\ldots & 2 ^ { m - 2 } \\\\ 1 & 3 & 3 ^ 2 & \\ldots & 3 ^ { m - 2 } \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 1 & m & m ^ 2 & \\ldots & m ^ { m - 2 } \\end{array} \\right ] \\left [ \\begin{array} { c } a _ 0 \\\\ a _ 1 \\\\ \\vdots \\\\ a _ { m - 2 } \\end{array} \\right ] = \\left [ \\begin{array} { c } 2 \\cdot 2 ^ 1 - m \\cdot 2 ^ 2 \\\\ 3 \\cdot 2 ^ 2 - m \\cdot 2 ^ 3 \\\\ \\vdots \\\\ m \\cdot 2 ^ { m - 1 } - m \\cdot 2 ^ m \\end{array} \\right ] . \\end{align*}"}
+{"id": "175.png", "formula": "\\begin{align*} \\begin{aligned} \\chi ( s ) \\equiv 0 \\textrm { f o r } 0 < s \\leq r \\ , , \\ \\chi ( s ) \\equiv 1 \\textrm { f o r } s \\geq 2 r \\ , , \\ 0 \\leq \\chi ( s ) \\leq 1 \\textrm { f o r a l l } s > 0 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "4961.png", "formula": "\\begin{align*} R _ { \\rm F A } = \\frac { \\sum \\limits _ { \\mathbf { a } \\in \\mathbb { B } ^ { N \\times 1 } \\atop | | \\mathbf { a } | | _ { 0 } = \\lambda N } \\vert \\hat { \\mathcal { U } } _ { a c } ^ { \\mathbf { a } } ( { \\mathbf { I } } ) - \\mathcal { U } _ { a c } ^ { \\mathbf { a } } \\vert } { \\sum \\limits _ { \\mathbf { a } \\in \\mathbb { B } ^ { N \\times 1 } \\atop | | \\mathbf { a } | | _ { 0 } = \\lambda N } | \\mathcal { U } ^ { \\mathbf { a } } _ { { a c } } | } \\leq { \\tau . } \\end{align*}"}
+{"id": "1347.png", "formula": "\\begin{align*} w ( x ) : = r v \\left ( \\frac { x } { r } \\right ) . \\end{align*}"}
+{"id": "4984.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { { \\rm d } h ( t ) } { { \\rm d } t } & = \\frac { - 2 } { t ^ { 2 } } + e ^ { t } ( \\frac { 2 } { t ^ 2 } - \\frac { 2 } { t } + 1 ) \\\\ & = \\frac { - 2 } { t ^ { 2 } } + ( 1 + t + \\frac { t ^ 2 } { 2 } + \\mathcal { O } ( t ^ 2 ) ) ( \\frac { 2 } { t ^ 2 } - \\frac { 2 } { t } + 1 ) \\\\ & \\geq \\frac { - 2 } { t ^ { 2 } } + ( 1 + t + \\frac { t ^ 2 } { 2 } ) ( \\frac { 2 } { t ^ 2 } - \\frac { 2 } { t } + 1 ) \\\\ & = \\frac { t ^ { 2 } } { 2 } > 0 , \\end{aligned} \\end{align*}"}
+{"id": "7371.png", "formula": "\\begin{align*} r _ { p , s } : = \\frac { t _ { p , s } } { \\min _ { s ^ \\prime \\in \\mathcal { S } } t _ { p , s ^ \\prime } } \\end{align*}"}
+{"id": "3051.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ k ) = F _ { \\hat { X } _ { 0 } } ( \\hat { \\sigma } _ { 0 } ^ k ) + F _ { \\hat { X } _ { \\infty } } ( \\hat { \\sigma } _ { \\infty } ^ k ) = F _ { \\hat { X } _ { 0 } } ( \\hat { \\sigma } _ { 0 } ^ k ) \\end{align*}"}
+{"id": "1856.png", "formula": "\\begin{align*} z _ { i , j } & = x _ { i , \\kappa ( j ) } , \\\\ z _ { i , n _ 0 ^ { + } } & = x _ { i , n _ 0 ^ { + } } , \\mbox { a n d } \\\\ z _ { i , n + 1 - j } & = x _ { i , n + 1 - \\kappa ( j ) } . \\end{align*}"}
+{"id": "7166.png", "formula": "\\begin{align*} G L ( V ) ^ { \\lambda \\geq 0 } \\to G L ( V ) ^ { \\lambda } = T ( d ) \\stackrel { \\chi } { \\to } \\mathbb { C } ^ { \\ast } , \\end{align*}"}
+{"id": "5068.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ { \\pi } \\cos ( \\tfrac { k - 1 } { 2 } \\theta ) g ( | \\sin ( \\theta / 2 ) | ) \\ , d \\theta = 0 \\quad k \\ge 5 . \\end{align*}"}
+{"id": "6164.png", "formula": "\\begin{align*} & V _ 1 V _ 2 f ( z ) = R _ q M _ z ^ 2 f ( z ) = q ^ 2 z ^ 2 f ( q z ) , \\\\ & V _ 2 V _ 1 f ( z ) = M _ z R _ q M _ z f ( z ) = M _ z q z f ( q z ) = q z ^ 2 f ( q z ) , \\end{align*}"}
+{"id": "7699.png", "formula": "\\begin{align*} R ^ j \\pi _ * \\omega _ F = 0 \\quad j \\geq 1 \\end{align*}"}
+{"id": "6084.png", "formula": "\\begin{align*} \\bigcap _ { n = 1 } ^ \\infty ( Z ( G ) \\cap \\textup { k e r } ( \\omega _ n ) ) \\subseteq Z ( G ) \\cap \\textup { k e r } ( \\omega ) = \\{ e \\} . \\end{align*}"}
+{"id": "6867.png", "formula": "\\begin{align*} ( v _ i ' ) ^ { p ^ { e ' } + 1 } = \\lambda u _ i h ( a _ i ) , \\ 1 \\leq i \\leq n . \\end{align*}"}
+{"id": "3692.png", "formula": "\\begin{align*} N _ { K ( \\gamma ) / K } \\left ( \\frac { 1 } { c } - \\gamma \\right ) & = \\prod _ { \\alpha ~ ~ ~ ~ g ( f ^ { n - 1 } ( z ) ) } \\left ( \\frac { 1 } { c } - \\alpha \\right ) = g \\left ( f ^ { n - 1 } \\left ( \\frac { 1 } { c } \\right ) \\right ) = g ( f ^ { ( n ) } ( 0 ) ) . \\end{align*}"}
+{"id": "6008.png", "formula": "\\begin{align*} g ^ { ( q ) } ( x ; h ) = \\dfrac { Z ^ { ( q ) } ( x ) } { Z ^ { ( q ) } ( b ) } \\left ( \\rho ^ { ( q ) } _ { b } ( b ; h ) + g ^ { ( q ) } ( b ; h ) \\right ) - \\rho ^ { ( q ) } _ { b } ( x ; h ) . \\end{align*}"}
+{"id": "6053.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\eta _ t - k ^ 2 \\eta _ t = 0 \\ , \\ , \\ , \\ , \\Omega \\backslash K _ t \\\\ [ 0 . 3 c m ] \\frac { \\partial \\eta _ t } { \\partial n } = 0 \\ , \\ , \\ , \\ , \\partial K _ t \\\\ [ 0 . 3 c m ] \\eta _ t = 0 \\ , \\ , \\ , \\ , \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "8126.png", "formula": "\\begin{align*} & \\quad \\ ; H ^ 1 ( Y \\times _ k X , I \\otimes p ^ * ( L ^ { \\otimes n } ) \\otimes q ^ * ( M ^ { \\otimes m } ) ) \\\\ & = H ^ 1 ( Y \\times _ k X , I \\otimes A ^ { \\otimes N } \\otimes p ^ * ( L ^ { \\otimes ( n - N ) } ) \\otimes q ^ * ( M ^ { \\otimes ( m - N ) } ) ) = \\boldsymbol { 0 } . \\end{align*}"}
+{"id": "79.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { \\Psi } u = f ( u ) & \\O \\\\ u = c _ { j } & ( \\partial \\O ) _ { j } \\end{cases} \\end{align*}"}
+{"id": "3366.png", "formula": "\\begin{align*} L ( E ) = \\liminf _ { N \\to \\infty } L _ N ( E ) . \\end{align*}"}
+{"id": "5014.png", "formula": "\\begin{align*} C _ { \\bf \\phi } : = \\left ( \\frac { p d _ { \\bf \\phi } } { ( p - 1 ) ( S ^ { \\bf \\phi } ) ^ \\frac { p } { p - 1 } } \\right ) ^ p . \\end{align*}"}
+{"id": "3117.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ { \\ell _ 1 } ) = \\ell _ 1 q _ 1 . \\end{align*}"}
+{"id": "7268.png", "formula": "\\begin{align*} u _ L ( t , \\omega ) = u _ K ( t , \\widehat { X } ( \\omega ) ) . \\end{align*}"}
+{"id": "6500.png", "formula": "\\begin{align*} ( k + 3 ) + \\left ( \\frac { 3 k + 1 } { k ^ 2 } \\right ) \\le k + 7 = 4 r , \\end{align*}"}
+{"id": "222.png", "formula": "\\begin{align*} \\frac { \\log \\bar { \\mu } _ { n + 1 } ( x ) - \\log \\bar { \\mu } _ n ( x ) } { \\tau } = - a ( \\bar { \\mu } _ { n } , x ) \\ , . \\end{align*}"}
+{"id": "7839.png", "formula": "\\begin{align*} \\widetilde { \\mathfrak { B } } _ { \\mbox { \\tiny { H G S } } } = \\{ \\eta _ 1 , . . . , \\eta _ k \\} , \\end{align*}"}
+{"id": "5336.png", "formula": "\\begin{align*} F ^ * ( G ) = T \\times ( C _ { p _ 1 } ) ^ { a _ 1 } \\times \\cdots \\times ( C _ { p _ t } ) ^ { a _ t } , \\end{align*}"}
+{"id": "984.png", "formula": "\\begin{align*} ( k - i - s - 1 ) a _ { s + 1 , t } & = \\left ( \\frac { n - 2 k } { 2 } + i + j + s + t \\right ) a _ { s , t } , \\\\ ( k - j - t - 1 ) a _ { s , t + 1 } & = \\left ( \\frac { n - 2 k } { 2 } + i + j + s + t \\right ) a _ { s , t } . \\end{align*}"}
+{"id": "2004.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ { n } G _ g ( t _ i , t _ j ) \\ , \\xi _ i \\overline { \\xi _ j } \\ , \\geq \\ , 0 \\end{align*}"}
+{"id": "6139.png", "formula": "\\begin{align*} D _ { V ^ { * } } ^ 2 = I - V V ^ * & = I - V _ 1 V _ 1 ^ * + V _ 1 V _ 1 ^ * - V _ 1 V _ 2 V _ 2 ^ * V _ 1 ^ * = D _ { V _ 1 ^ * } ^ 2 + V _ 1 D _ { V _ 2 ^ * } ^ 2 V _ 1 ^ * \\\\ & = I - V _ 2 V _ 2 ^ * + V _ 2 V _ 2 ^ * - V _ 2 V _ 1 V _ 1 ^ * V _ 2 ^ * = V _ 2 D _ { V _ 1 ^ * } ^ 2 V _ 2 ^ * + D _ { V _ 2 ^ * } ^ 2 . \\end{align*}"}
+{"id": "3891.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 ^ + } \\int _ { \\Omega \\setminus B _ r ( 0 ) } f ( x ) \\ , | x | ^ { \\tau _ + ( \\mu ) } d x = + \\infty , \\end{align*}"}
+{"id": "35.png", "formula": "\\begin{align*} \\left ( \\sqrt { q ' _ j } - \\sqrt { p ' _ j } \\right ) ^ 2 = q ' _ j \\left ( 1 - \\sqrt { 1 - \\frac { q ' _ j - p ' _ i } { q ' _ j } } \\right ) ^ 2 \\leq \\frac { ( q ' _ j - p ' _ i ) ^ 2 } { q ' _ j } . \\end{align*}"}
+{"id": "5134.png", "formula": "\\begin{align*} \\overline { A } ^ M = \\{ x \\in \\mathbb R ^ n \\ , : \\ , \\overline { D } ( A , x ) > 0 \\} , \\end{align*}"}
+{"id": "94.png", "formula": "\\begin{align*} [ \\Delta ^ M _ \\Psi , \\Delta ^ N _ \\Gamma ] = 0 . \\end{align*}"}
+{"id": "2939.png", "formula": "\\begin{align*} P = \\frac { 1 } { 2 \\pi i } \\int _ \\gamma ( \\lambda - M _ C ) ^ { - 1 } d \\lambda \\end{align*}"}
+{"id": "4804.png", "formula": "\\begin{align*} L ( k , m , \\lambda ) & = \\frac { ( n + \\lambda ) _ { m } ( n + \\lambda + \\epsilon _ 2 k m + ( 1 - \\delta _ { \\epsilon _ 2 , 1 } ) m ) _ m ^ { \\epsilon _ 2 } } { ( n + \\lambda + \\epsilon _ 2 m k ) _ m } , \\\\ M ( k , m , \\mu ) & = \\frac { ( n + \\epsilon ( \\mu + \\epsilon _ 1 k m ) ) _ m } { { ( n + \\epsilon \\mu ) _ m ( n + \\mu + \\epsilon _ 1 k m + ( 1 - \\delta _ { \\epsilon _ 1 , 1 } ) m ) _ m ^ { \\epsilon \\epsilon _ 1 } } } . \\end{align*}"}
+{"id": "5247.png", "formula": "\\begin{align*} F ( \\alpha ) : = \\lim _ { T \\to \\infty } F _ T ( \\alpha ) = \\begin{cases} | \\alpha | , & \\ 0 < | \\alpha | < 1 \\\\ 1 , & \\ | \\alpha | \\geq 1 . \\end{cases} . \\end{align*}"}
+{"id": "2603.png", "formula": "\\begin{align*} | \\phi ^ { N , i } ( 0 , \\boldsymbol { \\xi } ) - \\Psi ^ { N , i } ( 0 , \\boldsymbol { \\xi } ) | \\leq \\frac { C } { N } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 + \\frac { 1 } { N ^ 2 } \\underset { i , j = 1 } { \\overset { N } { \\sum } } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "4488.png", "formula": "\\begin{align*} \\dd f ( \\mathbb { Y } ) = \\langle D _ 1 f , Y _ 1 \\rangle _ { H } + \\langle D _ 2 f , Y _ 2 \\rangle _ { H } \\ , . \\end{align*}"}
+{"id": "2911.png", "formula": "\\begin{align*} \\mathcal A \\mathcal A + \\mathcal F = \\mathcal A \\mathcal A , \\ \\ \\mathcal P \\mathcal A \\mathcal A + \\mathcal F = \\mathcal P \\mathcal A \\mathcal A \\ \\ \\ \\ \\mathcal A \\mathcal A + \\mathcal K \\not \\subset \\mathcal P \\mathcal A \\mathcal A \\end{align*}"}
+{"id": "556.png", "formula": "\\begin{align*} H ( \\widetilde { Q } ) \\le \\sum _ { i = 1 } ^ { n } \\tilde { t } _ i H ( Q _ i ) , \\end{align*}"}
+{"id": "8170.png", "formula": "\\begin{align*} ( \\lambda a + C ^ * b ) ^ * E ( \\lambda ) ^ { - 1 } ( \\lambda a + C ^ * b ) = \\frac { 1 } { \\gamma } = 1 - \\| b \\| ^ 2 . \\end{align*}"}
+{"id": "7103.png", "formula": "\\begin{align*} \\mathcal { W } = \\{ ( \\beta _ i - \\beta _ j ) ^ { \\times 3 } \\mid 1 \\leq i , j \\leq d \\} , \\ \\mathcal { W } ^ f = \\mathcal { W } \\cup \\{ \\beta _ i \\mid 1 \\leq i \\leq d \\} . \\end{align*}"}
+{"id": "6335.png", "formula": "\\begin{align*} \\pi ( C _ \\rho ) \\cap T ( N ) = \\emptyset . \\end{align*}"}
+{"id": "1451.png", "formula": "\\begin{align*} M ( b , c ; s ) = \\frac { \\Gamma ( c ) } { \\Gamma ( b ) \\Gamma ( c - b ) } \\int _ 0 ^ 1 t ^ { b - 1 } ( 1 - t ) ^ { c - b - 1 } e ^ { t s } \\ , d t , \\end{align*}"}
+{"id": "1510.png", "formula": "\\begin{align*} \\delta = ( w , \\sqrt x ) \\le \\tfrac 1 4 . \\end{align*}"}
+{"id": "3017.png", "formula": "\\begin{align*} \\begin{aligned} x _ 1 ^ { ( \\alpha _ 1 ) } & = u _ 1 \\end{aligned} & \\begin{aligned} x _ 2 ^ { ( \\alpha _ 2 ) } & = u _ 2 \\end{aligned} \\begin{aligned} \\dot { x } _ 3 & = u _ 1 u _ 2 \\end{aligned} \\end{align*}"}
+{"id": "6474.png", "formula": "\\begin{align*} W F ^ { - \\frac 1 2 - \\epsilon + \\tau } ( \\omega ^ { + } | _ { \\cal { Q } } ) = W F ^ { - \\frac 1 2 - \\epsilon + \\tau } ( \\rho ^ * \\omega ^ { + } | _ { \\cal { Q } } ) = \\rho ^ * W F ^ { - \\frac 1 2 - \\epsilon + \\tau } ( \\omega ^ { + } | _ { \\cal { Q } } ) \\end{align*}"}
+{"id": "3339.png", "formula": "\\begin{gather*} L _ 2 \\big ( \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } \\big ) = v _ 1 \\otimes \\psi _ { n _ 2 - 1 } + v _ 2 \\otimes \\psi _ { n _ 2 } + v _ 3 \\otimes \\psi _ { n _ 2 + 1 } . \\end{gather*}"}
+{"id": "7528.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { E _ \\varphi [ \\mu _ { \\varepsilon , h } ] - E _ \\varphi [ \\mu ] } { \\varepsilon } = \\Re D _ { V , h } ( \\mu ) , \\end{align*}"}
+{"id": "7341.png", "formula": "\\begin{align*} & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ X ^ { ( i , j , t , p ) } _ { ( m , m + 1 ) } = \\{ ( y , z ) \\in X _ { ( m , m + 1 ) } \\mid \\varrho ( y , z ) = ( i , j , t , p ) \\} ; \\end{align*}"}
+{"id": "6962.png", "formula": "\\begin{align*} s _ { \\lambda } = ( - 1 ) ^ { s d } e _ { s + 1 } ^ { d } + e _ { s + 1 } . \\end{align*}"}
+{"id": "3143.png", "formula": "\\begin{align*} \\frac { k _ s ( \\overline { G _ n ' } ) } { \\binom { n } { s } } = \\frac { k _ s ( \\overline { G _ n } ) } { \\binom { n + \\lceil \\beta n \\rceil } { s } } \\ , . \\end{align*}"}
+{"id": "3003.png", "formula": "\\begin{align*} \\| f \\| _ { X ^ r _ { \\tau } } = \\| f _ \\Phi \\| _ { H ^ { r , 0 } } . \\end{align*}"}
+{"id": "5850.png", "formula": "\\begin{align*} \\langle f ( A ) x , x \\rangle ^ p \\Big ( 1 + N ^ { p - 1 } \\displaystyle \\sum _ { n = N + 1 } ^ { \\infty } \\frac { 1 } { n ^ p } \\Big ) & < \\Big ( \\frac { p } { p - 1 } \\Big ) ^ { p } \\langle f ( A ) ^ p x , x \\rangle . \\end{align*}"}
+{"id": "860.png", "formula": "\\begin{align*} a \\cdot ( b + c ) = a \\cdot b + a \\cdot c - a \\end{align*}"}
+{"id": "1911.png", "formula": "\\begin{align*} \\rho ( z , z _ 0 ) = \\max \\{ | t - t _ 0 | ^ { 1 / 2 } , | x - x _ 0 + ( t - t _ 0 ) v _ 0 | ^ { 1 / 3 } , | v - v _ 0 | \\} , \\end{align*}"}
+{"id": "3340.png", "formula": "\\begin{gather*} M _ 2 \\big ( \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } \\big ) = c _ { n _ 2 - 1 } \\psi _ { n _ 1 } ^ { n _ 2 - 1 } \\otimes \\psi _ { n _ 2 - 1 } + c _ { n _ 2 } \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } + c _ { n _ 2 + 1 } \\psi _ { n _ 1 } ^ { n _ 2 + 1 } \\otimes \\psi _ { n _ 2 + 1 } . \\end{gather*}"}
+{"id": "6893.png", "formula": "\\begin{align*} C ( z ) = \\begin{pmatrix} 0 & 1 \\\\ - \\mathfrak a ^ 2 ( L , s ) & \\frac { z + s + 1 } L \\end{pmatrix} + \\frac { 1 } { z + \\frac { 1 } 2 } \\begin{pmatrix} \\mathfrak a ( L , s ) \\varphi _ + ( L , s - 1 ) \\varphi _ - ( L , s ) & - \\varphi _ + ( L , s - 1 ) \\varphi _ - ( L , s - 1 ) \\\\ \\mathfrak a ^ 2 ( L , s ) \\varphi _ + ( L , s ) \\varphi _ - ( L , s ) & - \\mathfrak a ( L , s ) \\varphi _ + ( L , s ) \\varphi _ - ( L , s - 1 ) \\end{pmatrix} \\end{align*}"}
+{"id": "541.png", "formula": "\\begin{align*} - \\int _ { \\R } \\left ( T _ i ' ( x _ i ) - 1 \\right ) Q _ i ( x _ i ) d x _ i = \\int _ \\R \\left ( T _ i ( x _ i ) - x _ i \\right ) Q _ i ' ( x _ i ) \\ , d x _ i . \\end{align*}"}
+{"id": "4250.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n - 1 } a ^ n q ^ \\frac { n ( n + 1 ) } { 2 } } { ( 1 - q ^ n ) ( a q ) _ n } = \\sum _ { n = 1 } ^ { \\infty } \\frac { a ^ n q ^ n } { 1 - q ^ n } . \\end{align*}"}
+{"id": "1790.png", "formula": "\\begin{align*} \\bar { i } : = \\max \\left ( \\left \\{ 1 , \\dots , i _ { a } - 1 \\right \\} \\cap I ^ { c } \\right ) \\end{align*}"}
+{"id": "6382.png", "formula": "\\begin{align*} T ( X _ i , X _ j , \\emptyset ) = T ( Y _ i , Y _ j , \\emptyset ) = 0 \\end{align*}"}
+{"id": "6961.png", "formula": "\\begin{align*} s _ \\lambda ( z _ 1 , \\dots , z _ N ) = \\begin{vmatrix} e _ { s } & e _ { s + 1 } & e _ { s + 2 } & \\cdots & e _ { s + d - 1 } & e _ { s + d } \\\\ e _ { s - 1 } & e _ { s } & e _ { s + 1 } & \\cdots & e _ { s + d - 2 } & e _ { s + d - 1 } \\\\ \\vdots & \\vdots & \\vdots & \\cdots & \\vdots & \\vdots \\\\ e _ { s - d + 1 } & e _ { s - d + 2 } & e _ { s - d + 3 } & \\cdots & e _ s & e _ { s + 1 } \\\\ e _ { s - d } & e _ { s - d + 1 } & e _ { s - d + 2 } & \\cdots & e _ { s - 1 } & e _ s \\\\ \\end{vmatrix} . \\end{align*}"}
+{"id": "1625.png", "formula": "\\begin{align*} h _ i ( p ) - h _ i ( u _ i ) & = h _ i ( p ) - h _ i ( v ) + h _ i ( v ) - h _ i ( u _ i ) \\\\ & > - d ( p , v ) + d ( v , u _ i ) - \\delta ^ 2 / 2 \\\\ & > d ( p , u _ i ) - \\delta ^ 2 \\\\ & > ( 1 - \\delta ) d ( p , u _ i ) . \\end{align*}"}
+{"id": "582.png", "formula": "\\begin{align*} 2 C _ j \\ , [ C _ { i k } , C _ { k \\ell } ] + ( C _ { i j } - C _ i - C _ j ) [ C _ { k \\ell } , C _ { j k } ] + ( C _ { j \\ell } - C _ j - C _ \\ell ) [ C _ { j k } , C _ { i k } ] - ( C _ { j k } - C _ j - C _ k ) [ C _ { i j } , C _ { j \\ell } ] = 0 \\ , . \\end{align*}"}
+{"id": "6848.png", "formula": "\\begin{align*} \\gcd ( p ^ { e ' } + 1 , p ^ h - 1 ) = p ^ { \\gcd ( e ' , h ) } + 1 = p ^ e + 1 . \\end{align*}"}
+{"id": "1926.png", "formula": "\\begin{align*} \\begin{aligned} & \\ , \\| \\partial _ t ^ j D _ x ^ l D _ v ^ { m } u \\| _ { L _ p ( Q _ { 1 / 2 } ) } \\\\ & \\le N ( d , \\delta , p , j , l , m ) \\big ( \\| | D _ v u | + \\lambda ^ { 1 / 2 } | u | \\| _ { L _ p ( Q _ 1 ) } + \\sum _ { k = 0 } ^ { \\infty } 2 ^ { - 2 k } \\big ( | ( - \\Delta _ x ) ^ { 1 / 6 } u | ^ p ) ^ { 1 / p } _ { Q _ { 1 , 2 ^ k } } \\big ) . \\end{aligned} \\end{align*}"}
+{"id": "7411.png", "formula": "\\begin{align*} b _ p = a _ p ( Y ) \\end{align*}"}
+{"id": "7503.png", "formula": "\\begin{align*} \\left [ \\int \\frac { d \\mu ( s ) } { z - s } - \\frac { V ' ( z ) } { 2 } \\right ] ^ 2 = Q ( z ) , m _ 2 , \\end{align*}"}
+{"id": "8117.png", "formula": "\\begin{align*} { \\small \\begin{cases} { \\displaystyle ( ( L _ 0 , \\varphi _ 0 ) \\cdot ( L _ 1 , \\varphi _ 1 ) ) = \\frac { ( ( ( L _ 0 , \\varphi _ 0 ) + ( L _ 1 , \\varphi _ 1 ) ) ^ 2 ) - ( ( L _ 0 , \\varphi _ 0 ) ^ 2 ) - ( ( L _ 1 , \\varphi _ 1 ) ^ 2 ) } { 2 } } , \\\\ [ 2 e x ] { \\displaystyle ( ( L _ 0 , \\varphi _ 0 ) \\cdot ( L _ 1 , \\varphi _ 1 ) ) ' = \\frac { ( ( ( L _ 0 , \\varphi _ 0 ) + ( L _ 1 , \\varphi _ 1 ) ) ^ 2 ) ' - ( ( L _ 0 , \\varphi _ 0 ) ^ 2 ) ' - ( ( L _ 1 , \\varphi _ 1 ) ^ 2 ) ' } { 2 } } , \\end{cases} } \\end{align*}"}
+{"id": "8192.png", "formula": "\\begin{align*} \\begin{aligned} \\rho - 1 \\in { \\widetilde { L } ^ \\infty ( \\mathbb { R } ^ + ; \\widetilde { B } ^ { s , s + \\alpha - 1 } ) } , u \\in { \\widetilde { L } ^ \\infty ( \\mathbb { R } ^ + ; \\dot { B } ^ s _ { 2 , 1 } ) } \\cap { L ^ 1 ( \\mathbb { R } ^ + ; \\dot { B } ^ { s + \\alpha } _ { 2 , 1 } } ) . \\end{aligned} \\end{align*}"}
+{"id": "7038.png", "formula": "\\begin{align*} \\hat { \\hat { \\nu } } _ p = \\{ { [ X , Z ] _ p } : Z \\in \\mathcal { K } ^ G ( M ) , X p \\} . \\end{align*}"}
+{"id": "335.png", "formula": "\\begin{align*} m _ { j + 1 } & = \\arg \\min _ v I _ { K F } ( v ) \\\\ I _ { K F } ( v ) & = \\frac { 1 } { 2 } | y _ { j + 1 } - H v | _ { \\Gamma } ^ 2 + \\frac { 1 } { 2 } | v - F m _ j | _ { \\hat { C } _ { j + 1 } } ^ 2 , \\\\ \\end{align*}"}
+{"id": "4069.png", "formula": "\\begin{align*} x + y = 1 , \\end{align*}"}
+{"id": "687.png", "formula": "\\begin{align*} d v v ^ { - 1 } = \\frac { 1 } { 2 } a I + \\frac { 1 } { \\sqrt { \\tau _ 0 } } \\left ( \\frac { \\tau _ 0 } { 4 } d \\overline { z } + Q _ 0 d z \\right ) i J . \\end{align*}"}
+{"id": "2057.png", "formula": "\\begin{align*} \\chi ( n ) = \\left ( \\frac { ( - 1 ) ^ k \\prod _ { \\delta | N } \\delta ^ { r _ \\delta } } { n } \\right ) , \\end{align*}"}
+{"id": "3235.png", "formula": "\\begin{align*} \\theta ^ { n } _ { \\varphi _ { j } } = e ^ { \\varphi _ { j } } \\mu _ { j } . \\end{align*}"}
+{"id": "3634.png", "formula": "\\begin{align*} { 1 \\over 2 } \\sigma ^ 2 f '' ( b ) + ( b - \\rho ) ^ 2 - ( 1 + \\lambda ) f ( b ) + e ^ { \\theta b } \\zeta = 0 . \\end{align*}"}
+{"id": "2812.png", "formula": "\\begin{align*} d D ^ X _ s = - D ^ X _ s d R _ s + \\gamma _ s d X _ s , s \\in [ t , T ] , D ^ X _ { t - } = d . \\end{align*}"}
+{"id": "3472.png", "formula": "\\begin{align*} \\mathcal { S } \\mathbb { H } \\Big [ \\varphi \\Big ] = ( \\frac { 1 } { 2 } I _ { d } - \\mathcal { K } ) ( \\frac { 1 } { 2 } I _ { d } + \\mathcal { K } ) \\Big [ \\varphi \\Big ] . \\end{align*}"}
+{"id": "3324.png", "formula": "\\begin{gather*} \\psi _ { n _ 1 ( k _ 1 ' ) } ^ { n _ 2 ( k _ 2 ' ) } = \\sum _ { j _ 1 = 0 } ^ { N _ 1 } \\big \\langle \\psi _ { n _ 1 ( k _ 1 ' ) } ^ { n _ 2 ( k _ 2 ' ) } , \\phi _ { m _ 1 ( j _ 1 ) } \\big \\rangle \\phi _ { m _ 1 ( j _ 1 ) } \\end{gather*}"}
+{"id": "1707.png", "formula": "\\begin{align*} \\mathcal { Z } _ { \\tau } : = \\frac { Z _ { \\tau } } { Z _ { \\tau , 0 } } \\ , . \\end{align*}"}
+{"id": "5696.png", "formula": "\\begin{gather*} \\xi _ j - \\mu _ { \\sigma ( j ) } = \\sum _ { k \\in L _ j } \\nu _ k \\quad ( 1 \\le j \\le l ) , \\end{gather*}"}
+{"id": "5210.png", "formula": "\\begin{align*} H _ s = \\sum _ { k = j _ s } ^ { j _ { s + 1 } - 1 } p _ k + L + p _ { j _ { s + 1 } - 1 } . \\end{align*}"}
+{"id": "6903.png", "formula": "\\begin{align*} \\begin{cases} ( i \\dd _ t + \\Delta ) v _ n = g _ n ( F _ n + v _ n ) \\\\ v _ n ( 0 ) = v _ { n , 0 } \\in H ^ 2 ( \\R ^ d ) \\end{cases} \\end{align*}"}
+{"id": "3858.png", "formula": "\\begin{align*} L : = 1 + \\sqrt { \\frac { \\gamma _ n } { 4 n } } , l _ n : = \\frac { C _ n } { \\sqrt { 4 n \\gamma _ n } } , h _ n : = n ^ { - 1 / 4 + \\tau / 2 } , \\end{align*}"}
+{"id": "7527.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { E _ \\varphi [ \\mu _ { \\varepsilon , h } ] - E _ \\varphi [ \\mu ] } { \\varepsilon } = 0 \\end{align*}"}
+{"id": "4256.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { \\left ( \\frac { c } { d } \\right ) _ n ( d ) ^ n ( - 1 ) ^ { n - 1 } q ^ \\frac { n ( n + 1 ) } { 2 } } { ( c q ) _ n } = - \\sum _ { n = 1 } ^ { N } \\frac { \\left ( \\frac { d } { c } \\right ) _ { n } ( q ^ { - N } ) _ n q ^ n } { ( q ) _ n \\left ( \\frac { q ^ { - N } } { c } \\right ) _ n } , \\end{align*}"}
+{"id": "6296.png", "formula": "\\begin{align*} \\Gamma _ { } ( x , y ) = ( - \\theta ) ^ { \\frac { - q } { q - 1 } } \\cdot \\tilde { \\pi } \\cdot \\Pi ( - x , - y ) = ( - \\theta ) ^ { \\frac { - q } { q - 1 } } \\cdot \\tilde { \\pi } \\cdot ( - x ) \\cdot \\Gamma ( - x , 1 - y ) . \\end{align*}"}
+{"id": "6116.png", "formula": "\\begin{align*} \\begin{aligned} | B _ 2 | \\geq | B _ 1 \\cap B _ 2 | + \\sum _ { j = 1 } ^ { k ^ { | B _ 1 | - d + 1 } } | ( F _ j \\setminus B _ 1 ) \\cap B _ 2 | \\geq | B _ 1 \\cap B _ 2 | + k ( d - 1 - | B _ 1 \\cap B _ 2 | ) > k , \\end{aligned} \\end{align*}"}
+{"id": "6185.png", "formula": "\\begin{align*} X ^ { x _ 0 } _ { ( i , j , t , p ) } = \\{ ( x _ 0 , x , y ) \\in x _ 0 \\times X \\times X \\mid \\partial ( x _ 0 , x , y ) = ( i , j , t , p ) \\} . \\end{align*}"}
+{"id": "1671.png", "formula": "\\begin{align*} \\kappa ^ 2 _ { \\mathsf { B } - \\mathsf { F } + \\mathsf { f } } - \\kappa _ { \\mathsf { B } } ^ 2 = \\sum \\limits _ { \\mathsf { g } = 1 } ^ { \\mathsf { F } - \\mathsf { f } } \\big [ \\kappa _ { \\mathsf { B } - \\mathsf { g } } ^ 2 - \\kappa ^ 2 _ { \\mathsf { B } - \\mathsf { g } + 1 } \\big ] \\leq \\frac { \\phi ( \\mathsf { F } - \\mathsf { f } ) } { \\mathsf { F } } \\ , \\big [ \\kappa ^ { 2 } _ { \\mathsf { A } } - \\kappa ^ { 2 } _ { \\mathsf { B } } \\big ] \\ , . \\end{align*}"}
+{"id": "3114.png", "formula": "\\begin{align*} A = A _ 0 , A _ 1 , A _ 2 , \\ldots , A _ n , A _ { n + 1 } , \\ldots . \\end{align*}"}
+{"id": "5675.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k ( i ) } \\exp \\left [ \\gamma _ \\mathrm { i n t } ( p ) r _ i \\right ] \\leq \\exp \\left [ \\gamma _ \\mathrm { i n t } ( p ) r \\right ] \\sum _ { i = 0 } ^ { k ( i ) } e ^ { - c i } \\preceq \\exp \\left [ \\gamma _ \\mathrm { i n t } ( p ) r \\right ] . \\end{align*}"}
+{"id": "7654.png", "formula": "\\begin{align*} \\gamma ( 0 ) = x _ 0 , \\gamma ( 1 ) = x _ 1 , B _ r ( \\gamma ( t ) ) \\subset \\Omega , \\ , \\forall t \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "5119.png", "formula": "\\begin{align*} \\frac { 2 \\sigma } { q _ { \\theta } } + \\frac { d } { r _ { \\theta } } = \\frac { d } { 2 } - s , \\frac { 2 \\sigma } { q _ { \\theta } } + \\frac { d } { \\tilde { r } _ { \\theta } } + \\frac { 2 \\sigma } { \\tilde { q } _ { \\theta } } + \\frac { d } { r _ { \\theta } } = d , \\textmd { ( S t r i c h a r t z e x p o n e n t r e l a t i o n s ) } \\end{align*}"}
+{"id": "1487.png", "formula": "\\begin{align*} V ( z ) = \\prod _ { p | P ( z ) } ( 1 - g ( p ) ) . \\end{align*}"}
+{"id": "6881.png", "formula": "\\begin{align*} H ^ 1 ( \\Q _ p , V _ p ( E ) \\otimes V _ { g h } ) = \\bigoplus _ { ( a , b ) } H ^ 1 ( \\Q _ p , V _ p ( E ) \\otimes V _ { g h } ^ { a b } ) \\end{align*}"}
+{"id": "4926.png", "formula": "\\begin{align*} & ~ ~ \\left ( \\frac { k - 1 } { 2 k - 1 } + \\left ( \\frac { k - 1 } { 2 k - 1 } \\right ) ^ 2 + \\cdots \\right ) \\binom { n } { ( k + r + 1 ) / 2 } + \\binom { n } { r } \\\\ & < \\frac { k - 1 } { k } \\binom { n } { ( k + r + 1 ) / 2 } + \\frac { k - 1 } { 2 k - 1 } \\binom { n } { ( k + r + 1 ) / 2 } < \\binom { n } { k } . \\end{align*}"}
+{"id": "2833.png", "formula": "\\begin{align*} \\gamma _ s ^ { - \\frac 1 2 } d ( u _ s ^ n - H _ s ^ n ) & = \\gamma _ s ^ { - \\frac 1 2 } \\left ( \\rho _ s + \\mu _ s + \\frac 1 2 \\eta _ s ^ 2 - \\frac 3 8 \\sigma _ s ^ 2 \\right ) u _ s ^ n d s - \\gamma _ s ^ { - \\frac 1 2 } \\left ( \\frac 1 2 \\mu _ s - \\frac 1 8 \\sigma _ s ^ 2 \\right ) H _ s ^ n d s \\\\ & + \\gamma _ s ^ { - \\frac 1 2 } \\frac 1 2 \\sigma _ s ( u _ s ^ n - H _ s ^ n ) d W _ s ^ 1 + \\gamma _ s ^ { - \\frac 1 2 } Z _ s d v _ s ^ n . \\end{align*}"}
+{"id": "2388.png", "formula": "\\begin{align*} F _ { 3 } ^ { ( 1 ) } = F _ { 3 } ^ { ( 1 , 1 ) } + F _ { 3 } ^ { ( 1 , 2 ) } + F _ { 3 } ^ { ( 1 , 3 ) } \\end{align*}"}
+{"id": "6517.png", "formula": "\\begin{align*} \\deg _ { G ' } ( \\alpha ) = 2 , \\quad \\deg _ { G ' } ( a ) = \\deg _ { G } ( a ) . \\end{align*}"}
+{"id": "2847.png", "formula": "\\begin{align*} \\mathbf { 1 } = \\mathbf { 1 } \\odot \\mathbf { 1 } \\xrightarrow { f \\odot g } X \\odot X \\to X , \\end{align*}"}
+{"id": "3336.png", "formula": "\\begin{align*} ( \\delta ( \\lambda _ n - \\lambda _ s ) - q \\gamma \\lambda _ n ) B + ( \\varepsilon ( \\lambda _ n - \\lambda _ s ) - q \\gamma ) D _ 1 = 0 , \\end{align*}"}
+{"id": "455.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } { n \\choose k } F _ { j k } L _ { j ( n - k ) } = 2 ^ n F _ { j n } . \\end{align*}"}
+{"id": "4649.png", "formula": "\\begin{align*} f ^ \\lambda _ { \\lvert \\vec { x } \\rvert } \\ ! \\left ( \\vec { y } + \\frac { \\vec { x } } { 2 } \\right ) = \\sqrt { \\frac { 8 } { 3 \\pi } } \\ , \\lambda \\ , \\frac { \\mathrm { K } _ 2 \\Big ( \\sqrt { \\frac { 4 \\lambda } { 3 } } \\ , \\textstyle { \\sqrt { y ^ 2 + x ^ 2 + \\vec { x } \\ ! \\cdot \\ ! \\vec { y } } } \\ , \\Big ) } { y ^ 2 + x ^ 2 + \\vec { x } \\ ! \\cdot \\ ! \\vec { y } } . \\end{align*}"}
+{"id": "7408.png", "formula": "\\begin{align*} L ( Y , s ) : = \\sum _ { k = 1 } ^ \\infty \\frac { a _ k ( Y ) } { k ^ s } , \\end{align*}"}
+{"id": "2860.png", "formula": "\\begin{align*} \\mathsf { D } ' : = \\mathsf { D } / \\mathsf { D } _ { \\mathsf { B } } , \\end{align*}"}
+{"id": "5087.png", "formula": "\\begin{align*} ( - \\Delta _ x - \\Delta _ y ) \\phi _ j = \\lambda _ j ^ 2 \\phi _ j , \\lambda _ j > 0 . \\end{align*}"}
+{"id": "3921.png", "formula": "\\begin{align*} C _ 0 & = 3 \\gamma ^ { - 2 } \\| \\xi \\| ^ 2 _ { \\mathbb H ^ \\mu } + 6 T ( { L _ f ^ * } ^ 2 + { K _ f ^ * } ^ 2 \\| \\ell \\| ^ 2 _ { L ^ 1 } + \\epsilon ) { \\rho ^ * } ^ 2 , \\\\ C _ 1 & = 6 ( { L _ f ^ * } ^ 2 + { K _ f ^ * } ^ 2 \\| \\ell \\| ^ 2 _ { L ^ 1 } + \\epsilon ) . \\end{align*}"}
+{"id": "7852.png", "formula": "\\begin{align*} \\widetilde { J } ^ { - 1 } ( \\mu ) = \\widetilde { J } _ 1 ^ { - 1 } ( c _ 1 ) \\cap \\widetilde { J } _ 2 ^ { - 1 } ( c _ 2 ) = \\{ ( \\theta , \\phi , \\psi , p _ \\theta , p _ \\phi , p _ \\psi ) \\in T ^ * \\widetilde { Q } : p _ \\theta = \\tfrac { c _ 1 } { E ( \\phi ) } + c _ 2 \\ \\ p _ \\psi = c _ 2 \\} . \\end{align*}"}
+{"id": "8138.png", "formula": "\\begin{gather*} \\overline { L } _ { i , n } = \\overline L _ i ^ { \\otimes n } \\otimes \\overline A _ i , \\\\ \\delta _ { i , n } = ( L _ { 0 , n } \\cdots L _ { i - 1 , n } L _ { i + 1 , n } \\cdots L _ { d , n } ) . \\end{gather*}"}
+{"id": "3980.png", "formula": "\\begin{align*} q _ { \\beta } ( n , t ) = \\begin{cases*} E _ { \\beta , 1 } \\left ( - \\alpha ( e ^ { \\theta } - 1 ) t ^ { \\beta } \\right ) , \\ n = 0 , \\\\ \\displaystyle \\sum _ { \\Omega _ { n } } \\prod _ { j = 1 } ^ { n } \\frac { \\left ( \\alpha \\theta ^ { j } / j ! \\right ) ^ { x _ { j } } } { x _ { j } ! } z _ { n } ! t ^ { z _ { n } \\beta } E _ { \\beta , z _ { n } \\beta + 1 } ^ { z _ { n } + 1 } \\left ( - \\alpha ( e ^ { \\theta } - 1 ) t ^ { \\beta } \\right ) , \\ n \\ge 1 , \\end{cases*} \\end{align*}"}
+{"id": "4190.png", "formula": "\\begin{align*} l ( w ^ j ) = \\frac { 1 } { \\mathrm { s i n } ( \\theta _ j ' ) } \\underset { l } \\sum d _ l i ( w ^ j , w _ 1 ) \\mathrm { h e i g h t } ( C _ 1 ^ l ) \\end{align*}"}
+{"id": "2265.png", "formula": "\\begin{align*} \\sigma ' ( z _ 2 ) = \\lambda ' z _ 2 ^ { d _ 1 d _ 2 \\cdots d _ { s + 1 } } + \\end{align*}"}
+{"id": "2153.png", "formula": "\\begin{align*} & d s ^ 2 = d x _ 1 { } ^ 2 + f ( x _ 1 ) ^ 2 ( E d x _ 2 { } ^ 2 + 2 F d x _ 2 d x _ 3 + G d x _ 3 { } ^ 2 ) , \\\\ & d s ^ 2 = E d x _ 1 { } ^ 2 + 2 F d x _ 1 d x _ 2 + G d x _ 2 { } ^ 2 + f ( x _ 1 , x _ 2 ) ^ 2 d x _ 3 { } ^ 2 . \\end{align*}"}
+{"id": "172.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi _ 1 ( \\gamma ) : = \\{ b _ 0 | \\ , b _ 0 \\in [ 0 , 1 - \\gamma ] \\textrm { i f } - 1 < \\gamma \\leq 1 , \\textrm { a n d } b _ 0 \\in ( - \\gamma - 1 , 2 ) \\textrm { i f } - 3 < \\gamma \\leq - 1 \\} \\ , . \\end{aligned} \\end{align*}"}
+{"id": "1583.png", "formula": "\\begin{align*} w _ r ^ { - 1 } w _ r x _ { a } t _ { r } ^ { 5 } x _ { b } t _ { r } ^ { 5 } x _ { c } \\left ( t _ { a } ^ { - 1 } t _ { r } ^ { - 1 } t _ { b } ^ { - 1 } t _ { r } ^ { - 1 } t _ { c } ^ { - 1 } \\right ) ^ { 5 } = e \\end{align*}"}
+{"id": "6069.png", "formula": "\\begin{align*} \\begin{cases} \\Delta \\eta _ 1 ( x ) + \\lambda \\eta _ 1 ( x ) + \\lambda _ 1 \\eta ( x ) = \\lambda \\eta ( x _ 0 ) c a p ( \\omega ) \\lvert x - x _ 0 \\rvert ^ { - 1 } , \\ ; \\ ; \\ ; x \\in \\Omega \\\\ [ 0 . 3 c m ] \\eta _ 1 ( x ) = \\eta ( x _ 0 ) c a p ( \\omega ) \\lvert x - x _ 0 \\rvert ^ { - 1 } , \\ ; \\ ; \\ ; x \\in \\partial \\Omega \\end{cases} \\end{align*}"}
+{"id": "3656.png", "formula": "\\begin{align*} \\partial \\mathcal { H } = \\left \\{ A \\in \\binom { V ( \\mathcal { H } ) } { r - 1 } \\colon B \\in \\mathcal { H } A \\subseteq B \\right \\} . \\end{align*}"}
+{"id": "3995.png", "formula": "\\begin{align*} \\hat { \\mathcal { M } } _ { \\beta } ( t ) \\stackrel { d } { = } \\hat { \\mathcal { M } } ( Y _ { \\beta } ( t ) ) , \\ t \\ge 0 . \\end{align*}"}
+{"id": "6067.png", "formula": "\\begin{align*} \\lambda _ \\epsilon = \\lambda + 4 \\pi \\epsilon \\eta ( x _ 0 ) c a p ( \\omega ) + O ( \\epsilon ^ 2 ) \\end{align*}"}
+{"id": "1791.png", "formula": "\\begin{align*} \\left \\langle Y , B , i _ { r } \\right \\rangle = 0 = \\left \\langle Y , B , i _ { r } + 1 \\right \\rangle . \\end{align*}"}
+{"id": "6389.png", "formula": "\\begin{align*} A ' _ i A ' _ j = q ^ { \\pi ' _ { i j } } A ' _ j A ' _ i \\end{align*}"}
+{"id": "1418.png", "formula": "\\begin{align*} \\mathfrak { a } ( z , w ) & = ( ( \\lambda ^ 2 + \\lambda a ( x ) ) z , w ) _ { L ^ 2 } + ( \\nabla z , \\nabla w ) _ { L ^ 2 } \\end{align*}"}
+{"id": "7432.png", "formula": "\\begin{align*} [ H ^ n , \\ , p ^ \\dagger ] = [ H , \\ , p ^ \\dagger ] H ^ { n - 1 } + H [ H ^ { n - 1 } , p ^ \\dagger ] = p ^ \\dagger P H ^ { n - 1 } + [ H ^ { n - 1 } , p ^ \\dagger ] ( H + P ) , \\end{align*}"}
+{"id": "6674.png", "formula": "\\begin{align*} \\frac { \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( - s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { m - 1 } \\right ) } { b \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( 1 , 0 , \\ldots , 0 ) \\right ) } \\stackrel { a \\to 0 + } { \\longrightarrow } \\frac { \\mathcal { D } \\left ( ( - s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { 2 m - 3 } , ( - s ^ { ( 1 ) } _ { k } ) _ { k = m } ^ { 2 m - 1 } \\right ) } { b \\mathcal { E } \\left ( ( - s ^ { ( 1 ) } _ { k + 1 } ) _ { k = 0 } ^ { 2 m - 3 } , ( 1 , 0 , \\ldots , 0 ) \\right ) } > 0 . \\end{align*}"}
+{"id": "3925.png", "formula": "\\begin{align*} \\partial _ t u - ( 1 + m _ 0 ) \\Delta u - m _ 1 * \\Delta u & = g ( x ) p ( t ) + f _ 1 ( u ) , \\ ; m _ 0 = m ( 0 ) , m _ 1 = m ' . \\end{align*}"}
+{"id": "5599.png", "formula": "\\begin{align*} C _ { i j s } J ^ i _ p J ^ j _ q = C _ { p q s } , \\end{align*}"}
+{"id": "488.png", "formula": "\\begin{align*} B _ n \\Big ( \\frac { 2 \\alpha ^ j } { L _ j } \\Big ) - B _ n \\Big ( \\frac { F _ j \\sqrt 5 } { L _ j } \\Big ) = n \\Bigl ( { \\frac { { F _ j \\sqrt 5 } } { { L _ j } } } \\Bigr ) ^ { n - 1 } . \\end{align*}"}
+{"id": "4310.png", "formula": "\\begin{align*} \\langle v , w \\rangle _ { \\cal Q } \\ , = \\ , \\int _ 0 ^ T ( v ( t ) , w ( t ) ) \\ , d t \\quad \\mbox { f o r a l l \\ , $ w \\in { \\cal Q } $ \\ , a n d $ \\ , v \\in L ^ 2 ( 0 , T ; H ) $ } . \\end{align*}"}
+{"id": "7221.png", "formula": "\\begin{align*} \\Im ( \\sum _ { ( j , j _ 1 , j _ 2 , j _ 3 ) \\in \\mathcal { R } } u _ j \\bar { u } _ { j _ 1 } u _ { j _ 2 } \\bar { u } _ { j _ 3 } ) = 0 . \\end{align*}"}
+{"id": "2289.png", "formula": "\\begin{align*} \\sup _ { n \\ge 1 } \\frac { q _ n } n = \\frac { 3 8 1 } { 2 3 0 } \\cdotp \\end{align*}"}
+{"id": "1108.png", "formula": "\\begin{align*} \\min _ { \\xi , \\eta } p \\left ( \\xi , \\eta \\right ) = \\min \\left \\{ \\min _ { \\xi } p \\left ( \\xi , 1 \\right ) , \\min _ { \\xi } p \\left ( \\xi , \\frac { 9 9 } { 1 0 0 } \\right ) \\right \\} . \\end{align*}"}
+{"id": "7834.png", "formula": "\\begin{gather*} ( r + k ) c _ { r + 1 } = ( m - r ) c _ r \\end{gather*}"}
+{"id": "5011.png", "formula": "\\begin{align*} \\Big \\| u _ { i n } - \\sum _ { s = j + 1 } ^ { m } \\phi _ i ( g _ s ) \\ , ( w _ { i } \\circ g _ s ^ { - 1 } ) ( \\ , \\cdot \\ , - g _ s \\xi _ { r n } ) \\Big \\| ^ 2 = \\Big \\| u _ { i n } - \\sum _ { s = j } ^ { m } \\phi _ i ( g _ s ) \\ , ( w _ { i } \\circ g _ s ^ { - 1 } ) ( \\ , \\cdot \\ , - g _ s \\xi _ { r n } ) \\Big \\| ^ 2 + \\| w _ { i } \\| ^ 2 + o _ n ( 1 ) , \\end{align*}"}
+{"id": "2889.png", "formula": "\\begin{align*} 2 H = \\eta z ^ m + \\lambda . \\end{align*}"}
+{"id": "6536.png", "formula": "\\begin{align*} E _ { \\theta _ r } = \\overline { E _ { - \\theta _ r } } . \\end{align*}"}
+{"id": "5902.png", "formula": "\\begin{align*} \\rho = [ 2 , 1 , 0 , - 1 , - 2 ] \\ , . \\end{align*}"}
+{"id": "7560.png", "formula": "\\begin{align*} \\varphi ( z ) = \\Re V ( z ) = - \\frac { b } { 2 } \\log | \\lambda | + \\frac { 1 + c } { 2 } \\log | z | \\end{align*}"}
+{"id": "4486.png", "formula": "\\begin{align*} \\{ f , g \\} : = \\Omega ( \\mathbb { X } _ f , \\mathbb { X } _ g ) = \\dd f ( \\mathbb { X } _ g ) = - \\dd g ( \\mathbb { X } _ f ) \\ , , \\end{align*}"}
+{"id": "6824.png", "formula": "\\begin{align*} \\beta _ n ^ i = \\begin{cases} \\varepsilon _ n \\kappa _ n + \\kappa _ { n - 1 } & i = 1 , \\\\ \\varepsilon _ n \\omega _ n \\kappa _ { n - 1 } + \\beta _ { n - 1 } ^ 1 & i = 2 , \\\\ \\varepsilon _ n \\beta _ { n - 1 } ^ { i - 2 } + \\beta _ { n - 1 } ^ { i - 1 } & i = 3 , \\ldots , 2 n - 1 , \\\\ \\varepsilon _ n \\beta _ { n - 1 } ^ { 2 n - 2 } & i = 2 n . \\end{cases} \\end{align*}"}
+{"id": "5054.png", "formula": "\\begin{align*} K _ { 0 , n } ( s , f , \\chi ) = K _ n ( s , f , \\chi ) - f _ n n ^ { - s } \\zeta ^ { ( N ) } ( 2 s ) . \\end{align*}"}
+{"id": "336.png", "formula": "\\begin{align*} m _ { j + 1 } & = \\arg \\min _ v I _ { E n K F } ( v ) \\\\ \\hat { \\mu } _ { j + 1 } & = F ( m _ j ) \\\\ I _ { E n K F } ( v ) & = \\frac { 1 } { 2 } | y _ { j + 1 } - H ( v ) | _ { \\Gamma } ^ 2 + \\frac { 1 } { 2 } | v - \\hat { \\mu } _ { j + 1 } | _ { \\hat { C } _ { j + 1 } } ^ 2 . \\end{align*}"}
+{"id": "1035.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 + \\varepsilon ) ^ { j - 1 } } \\frac { 1 } { E ( t _ { i _ { j } } ) } ( i _ { j + 1 } - i _ j ) = \\Omega \\left ( \\frac { 1 } { j ^ 2 } \\right ) . \\end{align*}"}
+{"id": "4024.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial s } = I ^ { - 2 } ( s ) \\Delta w - \\frac { 1 } { 2 k } y \\cdot \\nabla w - \\frac { 1 } { p - 1 } w + | w | ^ { p - 1 } w , \\end{align*}"}
+{"id": "2683.png", "formula": "\\begin{align*} M _ S ( n , j , t ; a ) = \\binom { n - j } { j } \\sum _ { k = j } ^ { n - j } \\binom { n - 2 j } { k - j } \\binom { n } { k } ^ t F ( n , k , a ) \\end{align*}"}
+{"id": "5552.png", "formula": "\\begin{align*} [ L _ 1 ( u _ 1 ) \\eta ^ { \\nu } u _ 1 ' ] ' + 2 \\eta ^ { \\nu + 1 } N _ 1 ( u _ 1 ) u _ 1 ' = 0 , \\ ; \\ ; \\ ; \\alpha _ 0 < \\eta < \\beta _ 0 ; \\end{align*}"}
+{"id": "1262.png", "formula": "\\begin{align*} M ( 1 , 0 , \\lambda ) = M ( 1 , 0 , \\lambda , \\Omega ) = \\sup _ { u \\in W ^ { 1 , 1 } ( \\Omega ) \\setminus \\{ 0 \\} } \\left \\{ \\frac { \\displaystyle \\int _ { \\Omega } \\left | u \\right | d x } { \\displaystyle \\int _ { \\Omega } \\left | \\nabla u \\right | d x + \\lambda \\int _ { \\partial \\Omega } \\left | u \\right | d \\mathcal H ^ { N - 1 } } \\right \\} . \\end{align*}"}
+{"id": "2207.png", "formula": "\\begin{align*} & r _ 1 = ( b - c ) \\{ a ^ 2 + ( b + c ) ^ 2 \\} ( R _ { 1 2 1 2 } R _ { 1 3 1 3 } - R _ { 1 2 1 3 } { } ^ 2 ) , \\\\ & r _ 2 = 2 ( a + d ) ( b ^ 2 - c ^ 2 ) ( R _ { 1 2 1 2 } R _ { 1 3 1 3 } - R _ { 1 2 1 3 } { } ^ 2 ) , \\\\ & r _ 4 = ( b - c ) \\{ ( b + c ) ^ 2 + d ^ 2 \\} ( R _ { 1 2 1 2 } R _ { 1 3 1 3 } - R _ { 1 2 1 3 } { } ^ 2 ) . \\end{align*}"}
+{"id": "1252.png", "formula": "\\begin{align*} Q ( u ) = \\frac { 1 } { p } \\int _ { \\Omega } \\left | \\nabla u \\right | ^ { p } d x + \\int _ { \\partial \\Omega } \\frac { \\lambda } { p } \\left | u \\right | ^ { p } \\ d \\mathcal H ^ { N - 1 } - \\int _ { \\partial \\Omega } g u \\ d \\mathcal H ^ { N - 1 } - \\int _ { \\Omega } f u \\ d x . \\end{align*}"}
+{"id": "250.png", "formula": "\\begin{align*} \\Q [ \\mathsf { f } ] ( y ) = \\left ( \\begin{array} { c c c c c c } { \\bf R } [ f _ 1 ( y ) ] & 0 & \\dots & 0 & 0 & 0 \\\\ 0 & { \\bf R } [ f _ 2 ( y ) ] & \\dots & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & { \\bf R } [ f _ { d - 1 } ( y ) ] & 0 & 0 \\\\ 0 & 0 & \\dots & 0 & { \\bf R } [ f _ d ( y ) ] & 0 \\\\ 0 & 0 & \\dots & 0 & 0 & 1 \\end{array} \\right ) , \\end{align*}"}
+{"id": "3369.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 u _ N ( x + i y _ 0 ) d x & \\leq \\left ( 1 - \\frac { 2 y _ 0 } { \\rho } \\right ) \\int _ 0 ^ 1 u _ N ( x ) d x \\\\ & + \\frac { 2 y _ 0 } { \\rho } \\frac { k } { N } \\ln ( C _ v | \\lambda | ) \\\\ & + \\frac { 2 y _ 0 } { \\rho } \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N - k } \\int _ 0 ^ 1 \\ln \\left ( | E - \\lambda v ( x + i \\rho / 2 + j _ n \\omega ) - v _ 1 ( j _ n ) | + 1 \\right ) d x . \\end{align*}"}
+{"id": "1818.png", "formula": "\\begin{align*} T & = \\sum _ { i = 1 } ^ m T ( v _ i ) \\otimes v _ i ^ * = \\sum _ { i = 1 } ^ m \\sum _ { j = 1 } ^ n a _ { i j } e _ j \\otimes v _ i ^ * \\\\ & \\in A \\otimes V ^ * \\subset ( A \\ltimes _ { \\rho ^ * } V ^ * ) \\otimes ( A \\ltimes _ { \\rho ^ * } V ^ * ) . \\end{align*}"}
+{"id": "7184.png", "formula": "\\begin{align*} \\xi ( x ) : = \\frac { ( 1 - q _ 1 ^ { - 1 } x ) ( 1 - q _ 2 ^ { - 1 } x ) ( 1 - q ^ { - 1 } x ^ { - 1 } ) } { 1 - x } , \\end{align*}"}
+{"id": "4576.png", "formula": "\\begin{align*} \\frac { u ( 1 ) } { u ( 0 ) } = \\tan \\theta _ j . \\end{align*}"}
+{"id": "7011.png", "formula": "\\begin{align*} L \\left ( \\phi ( f ) \\right ) = \\phi ' ( f ) \\cdot L f + \\phi '' ( f ) | \\nabla f | ^ 2 \\cdot m . \\end{align*}"}
+{"id": "3368.png", "formula": "\\begin{align*} u _ N ( a + i y _ 0 ) & \\leq \\int _ { i y = 0 } u _ N ( x ) d \\mu _ { a + i y _ 0 } ( x ) + \\int _ { i y = i \\rho / 2 } u _ N ( x + i y ) d \\mu _ { a + i y _ 0 } ( x ) \\\\ & = \\int _ { i y = 0 } u _ N ( x + a ) d \\mu _ { i y _ 0 } ( x ) + \\int _ { i y = i \\rho / 2 } u _ N ( x + a + i y ) d \\mu _ { i y _ 0 } ( x ) . \\end{align*}"}
+{"id": "7673.png", "formula": "\\begin{align*} \\Delta _ { h } u ( x ) = ( 1 - | x | ^ 2 ) ^ 2 \\Delta u ( x ) + 2 ( n - 2 ) ( 1 - | x | ^ 2 ) \\sum _ { i = 1 } ^ { n } x _ { i } \\frac { \\partial u } { \\partial x _ { i } } ( x ) = 0 , \\end{align*}"}
+{"id": "2047.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 - 2 X ) ^ 2 ( 1 - X ) ^ 2 } \\sum _ { n = 0 } ^ { \\infty } \\lambda _ { n + 1 } \\left ( \\frac { X } { X - 1 } \\right ) ^ n = \\sum _ { n = 0 } ^ { \\infty } \\left [ \\sum _ { k = 1 } ^ { n + 1 } k \\ , 2 ^ { k - 1 } \\sum _ \\rho \\rho ^ { - ( n - k + 2 ) } \\right ] X ^ n . \\end{align*}"}
+{"id": "6799.png", "formula": "\\begin{align*} \\lambda _ { A \\ , \\ , B } + \\lambda _ { A \\ , \\ , C } + \\lambda _ { B \\ , \\ , C } = 0 \\end{align*}"}
+{"id": "3913.png", "formula": "\\begin{align*} D _ h f ( u ) ( t ) = f ( u ( t + h ) , \\mathcal H u ( t + h ) ) - f ( u ( t ) , \\mathcal H u ( t ) ) . \\end{align*}"}
+{"id": "2024.png", "formula": "\\begin{align*} C ( a ) = \\left \\{ \\phi \\in C _ c ^ \\infty ( \\R ) \\ , \\left | ~ { \\rm s u p p } \\ , \\psi \\subset [ - a , a ] \\right . \\right \\} . \\end{align*}"}
+{"id": "4582.png", "formula": "\\begin{align*} \\abs { \\frac { 1 } { N } \\sum _ { l = 0 } ^ { N - 1 } \\cos ( \\theta \\pm \\nu \\pi k l ) } \\leq \\varepsilon , \\end{align*}"}
+{"id": "1607.png", "formula": "\\begin{align*} P _ L ( \\omega _ L ) P _ { e ' } ( \\omega _ { e } ) = P _ C ( \\omega _ L , \\omega _ e , \\omega _ { e ' } ) = P _ L ( \\omega _ { L } ) P _ { e ' } ( \\omega _ { e ' } ) . \\end{align*}"}
+{"id": "1640.png", "formula": "\\begin{align*} v _ n \\ ; = \\ ; \\mathcal { T } _ n \\circ v _ { n - 1 } \\ ; , \\end{align*}"}
+{"id": "4285.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { \\infty } \\frac { n ( - 1 ) ^ { n - 1 } q ^ { n ( n + 1 ) / 2 } } { ( q ) _ n ( 1 + q ^ n ) } + ( q ) _ { \\infty } \\sum _ { k = 1 } ^ { \\infty } \\frac { q ^ { k ( k + 1 ) } } { ( q ) _ k ^ { 2 } ( 1 - q ^ k ) } F ( 0 ; q ^ k ; - q ^ k ) \\\\ & = \\frac { 1 } { 4 } \\left ( 1 - \\frac { ( q ) _ { \\infty } } { ( - q ) _ { \\infty } } \\right ) + \\frac { 1 } { 2 } \\frac { ( q ) _ { \\infty } } { ( - q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - q ) _ n } { ( q ) _ n } \\frac { q ^ n } { 1 - q ^ n } . \\end{align*}"}
+{"id": "3011.png", "formula": "\\begin{align*} R _ { b c } [ h ^ 0 , h ^ 1 ] = \\left ( \\begin{array} { c c } R _ { b c } ^ { 0 0 } & R _ { b c } ^ { 0 1 } \\\\ R _ { b c } ^ { 1 0 } & R _ { b c } ^ { 1 1 } \\end{array} \\right ) \\end{align*}"}
+{"id": "1770.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } p _ { A , i } \\left ( C \\right ) \\left \\langle Y , B , i \\right \\rangle = 0 , \\end{align*}"}
+{"id": "5851.png", "formula": "\\begin{align*} \\Big ( 1 + 2 ^ { p - 1 } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } \\Big ) b e r ^ p ( f ( A ) ) < \\Big ( \\frac { p } { p - 1 } \\Big ) ^ { p } ~ b e r ( f ( A ) ^ p ) ~ h o l d s ~ t r u e . \\end{align*}"}
+{"id": "2301.png", "formula": "\\begin{align*} X \\prod _ { i = 1 } ^ { d - 1 } \\left ( X - a _ i Y \\right ) ( X - p Y ) X \\prod _ { j = 1 } ^ { d - 2 } \\left ( X - b _ j Y \\right ) \\end{align*}"}
+{"id": "4299.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( q ) _ n q ^ { n ^ 2 } } { ( z q ) _ n ( z ^ { - 1 } q ) _ n } = \\frac { 1 } { ( q ) _ N } + ( 1 - z ) \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ n ( q ) _ n q ^ { n ( 3 n + 1 ) / 2 } } { ( q ) _ { n + N } } \\left ( \\frac { 1 } { 1 - z q ^ n } - \\frac { 1 } { z - q ^ n } \\right ) , \\end{align*}"}
+{"id": "8072.png", "formula": "\\begin{align*} p | _ 2 & = \\rho _ 2 g \\left ( \\sum _ { j = 2 } ^ N D _ j + b - z + \\frac { \\rho _ 1 } { \\rho _ 2 } \\ D _ 1 \\right ) = \\rho _ 2 g \\left ( \\sum _ { j = 1 } ^ N D _ j + b - z + \\frac { \\rho _ 1 - \\rho _ 2 } { \\rho _ 2 } \\ D _ 1 \\right ) . \\end{align*}"}
+{"id": "0.png", "formula": "\\begin{align*} S _ R ( X ; \\pi , \\hat { \\phi } , W ) & = \\sum _ { \\substack { l > Z \\\\ ( l , 2 ) = 1 } } \\mu ( l ) \\sum _ { ( m , 2 ) = 1 } W \\left ( \\frac { l ^ 2 m } { X } \\right ) E ( Y ; \\chi _ { 8 l ^ 2 m } , \\hat { \\phi } ) \\ll \\sum _ { l > Z } \\sum _ { X / l ^ 2 \\leq m \\leq 2 X / l ^ 2 } \\log ^ { 3 } ( X ) \\ll \\frac { X \\log ^ { 3 } X } { Z } . \\end{align*}"}
+{"id": "2102.png", "formula": "\\begin{align*} A ^ { 1 1 } + B ^ { 1 1 } D _ x & = H \\\\ A ^ { 2 1 } + B ^ { 2 1 } D _ x & = 0 \\\\ C ^ { 1 1 } D _ x & = D _ x H \\\\ C ^ { 2 1 } D _ x & = 0 . \\end{align*}"}
+{"id": "7799.png", "formula": "\\begin{align*} d a ( u ) = \\csc ^ 3 ( \\theta ) \\vert \\sin t + \\cos \\theta \\vert d t d \\theta \\end{align*}"}
+{"id": "164.png", "formula": "\\begin{align*} \\begin{aligned} \\Theta _ \\gamma ( v ) = 1 \\textrm { i f } 0 \\leq \\gamma \\leq 1 \\ , , \\textrm { a n d } \\Theta _ \\gamma ( v ) = \\nu ^ { - 1 } ( v ) \\textrm { i f } - \\tfrac { 3 } { 2 } < \\gamma < 0 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "4534.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { H ^ 1 } = \\sum _ { k = 0 } ^ \\infty ( a _ k ^ 2 + b _ k ^ 2 ) < + \\infty \\ , . \\end{align*}"}
+{"id": "2442.png", "formula": "\\begin{align*} M _ { d } ( ( S _ { 1 } , \\dots , S _ { r } ) ) = \\left ( I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( \\sum _ { j \\in S _ { 1 } } l _ { j } , \\dots , \\sum _ { j \\in S _ { r } } l _ { j } ) \\right ) _ { l _ { 1 } , \\dots , l _ { d } \\geq 2 } , \\end{align*}"}
+{"id": "7462.png", "formula": "\\begin{align*} B u = z . \\end{align*}"}
+{"id": "7786.png", "formula": "\\begin{align*} V o l ( \\Sigma _ x ) = \\int _ { \\Sigma _ x } d V _ { g _ x } = \\int _ { \\partial X } \\sqrt { \\frac { \\det g _ x } { \\det \\hat { g } } } d V _ { \\hat { g } } = V o l ( \\partial X , \\hat { g } ) + O ( x ) \\end{align*}"}
+{"id": "5383.png", "formula": "\\begin{align*} \\{ F ^ { ( 1 ) } , g \\} ^ { [ 1 ] } = \\Pi ^ { ( 1 ) } ( g ) . \\end{align*}"}
+{"id": "3620.png", "formula": "\\begin{align*} P _ x [ X _ { e _ q } \\in { \\cal A } ] = E _ x [ 1 \\{ X _ { e _ q } \\in { \\cal A } \\} 1 \\{ e _ q < \\tau ^ * _ 1 \\} ] + E _ x [ 1 \\{ X _ { e _ q } \\in { \\cal A } \\} 1 \\{ e _ q \\geq \\tau ^ * _ 1 \\} ] . \\end{align*}"}
+{"id": "334.png", "formula": "\\begin{align*} \\varepsilon _ { B O L D } & = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\frac { | B O L D _ { s i m u l a t i o n } ( t ) - B O L D _ { r e a l } ( t ) | ^ 2 } { | B O L D _ { r e a l } ( t ) | ^ 2 } . \\\\ \\varepsilon _ { h } & = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\frac { | h _ { s i m u l a t i o n } ( t ) - h _ { r e a l } ( t ) | ^ 2 } { | h _ { r e a l } ( t ) | ^ 2 } . \\end{align*}"}
+{"id": "3163.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } \\ ; \\ ; f _ { \\rho } ( x ) : = \\varphi ( x ) + \\rho \\sum _ { i = 1 } ^ { l } \\max \\{ \\ g _ i ( x ) , 0 \\} + \\rho \\sum _ { j = l + 1 } ^ { m } | h _ j ( x ) | \\mbox { s u b j e c t t o } \\ ; \\ ; x \\in \\R ^ n \\end{align*}"}
+{"id": "5303.png", "formula": "\\begin{align*} \\zeta ( s ) \\ne 0 ~ ~ ~ s = \\sigma + i t , ~ ~ \\sigma > 1 - \\frac { c } { \\log t } , \\end{align*}"}
+{"id": "6411.png", "formula": "\\begin{align*} ^ { C } D ^ { w } p ( \\theta ) = \\mathcal { L } \\big ( \\theta , p ( \\theta ) , ^ { C } D ^ { w } p ( \\theta ) \\big ) , p ( \\theta ) | _ { \\theta = 0 } = \\phi ( p ) \\end{align*}"}
+{"id": "4030.png", "formula": "\\begin{align*} 0 = - \\frac k 2 \\nabla f _ b - \\frac { 1 } { p - 1 } f _ b + | f _ b | ^ { p - 1 } f _ b . \\end{align*}"}
+{"id": "6319.png", "formula": "\\begin{align*} \\hat { X } ^ k _ t = x _ 0 ( t ) + \\int _ 0 ^ t K _ \\mu ( s , t ) \\mu _ { n _ k } ( s , \\hat { X } ^ k _ s ) \\dd s + K _ \\sigma ( t , t ) \\hat { M } ^ k _ t + \\int _ 0 ^ t \\hat { M } ^ k _ s \\partial _ 1 K _ \\sigma ( s , t ) \\dd s . \\end{align*}"}
+{"id": "5112.png", "formula": "\\begin{align*} \\nabla _ x \\phi & = \\frac { x } { \\abs { x } } , \\\\ \\Delta _ x \\phi & = \\frac { d - 1 } { \\abs { x } } , \\\\ \\partial _ { x _ k x _ l } \\phi & = \\frac { \\delta _ { x _ k x _ l } } { \\abs { x } } - \\frac { x _ k x _ l } { \\abs { x } ^ 3 } \\\\ \\Delta _ x ^ 2 \\phi & = \\begin{cases} - \\pi \\delta ( x ) , & d = 3 , \\\\ - ( d - 1 ) ( d - 3 ) \\abs { x } ^ { - 3 } , & d \\geq 4 . \\end{cases} \\end{align*}"}
+{"id": "5297.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 | f _ { k , N } ( x ) | ^ { 2 s } d x = E _ s ( A _ { k , N } ) . \\end{align*}"}
+{"id": "4413.png", "formula": "\\begin{align*} | s - r | = | \\kappa q | \\leq a ^ 2 - a - 1 . \\end{align*}"}
+{"id": "4834.png", "formula": "\\begin{align*} \\int _ S \\int _ { B _ R } \\rho \\nabla g \\cdot \\nabla \\phi + \\epsilon \\nabla \\rho \\cdot \\nabla \\phi + \\rho \\phi \\ , \\dd x \\dd \\mu ( s ) = \\int _ S \\int _ { B _ R } \\left ( \\int _ S \\rho ( x , s ' ) \\ , \\dd \\mu ( s ' ) \\right ) \\phi \\ , \\dd x \\dd \\mu ( s ) , \\end{align*}"}
+{"id": "5658.png", "formula": "\\begin{align*} s _ \\lambda ( t ) = \\left \\{ \\begin{array} { l c } \\sin ( \\sqrt { \\lambda } t ) , & \\lambda > 0 , \\\\ t , & \\lambda = 0 , \\\\ \\sinh ( \\sqrt { - \\lambda } t ) , & \\lambda < 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "5811.png", "formula": "\\begin{align*} C = M _ { \\Theta } , \\end{align*}"}
+{"id": "4151.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial u } { \\partial t } ( x , t ) = \\int _ { [ 0 , 1 ] } W ( x , y ) ( u ( y , t ) - u ( x , t ) ) d y \\\\ u ( x , 0 ) = u _ 0 ( x ) \\end{cases} \\end{align*}"}
+{"id": "2495.png", "formula": "\\begin{align*} \\lim \\limits _ { | x | \\nearrow R } u ( x ) = \\lim \\limits _ { | x | \\nearrow R } v ( x ) = \\infty \\end{align*}"}
+{"id": "5957.png", "formula": "\\begin{align*} ( q \\kappa ) ^ { k - 1 } | \\xi _ { A _ 1 } - \\xi _ { B _ 1 } | \\leq | \\sum _ { j = 0 } ^ { k } ( - 1 ) ^ { j } ( e _ { j } ( B ) - e _ { j } ( A ) ) \\xi _ { A _ 1 } ^ { k - j } | \\leq \\max _ { j } | e _ { j } ( B ) - e _ { j } ( A ) | . \\end{align*}"}
+{"id": "695.png", "formula": "\\begin{align*} { { \\bf { F } } _ i } \\left ( { { \\bf { x } } , t } \\right ) = f _ i ^ { \\left ( { e q } \\right ) } \\left ( { { { \\bf { u } } ^ { e q } } + \\Delta { \\bf { u } } } \\right ) - f _ i ^ { \\left ( { e q } \\right ) } \\left ( { { { \\bf { u } } ^ { e q } } } \\right ) , \\end{align*}"}
+{"id": "5451.png", "formula": "\\begin{align*} l _ i = ( 0 , \\dots , 0 , 1 , \\dots , 1 , \\omega , \\dots , \\omega , \\omega ^ 2 , \\dots , \\omega ^ 2 ) ^ T . \\end{align*}"}
+{"id": "4370.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } D _ { G , 2 } F ( t , ( \\sigma , \\xi ) , \\zeta ) \\\\ = ( D _ { G , 2 } f _ 1 ^ 0 ( t , \\xi , \\zeta ) \\circ w ^ 2 , . . . , D _ { G , 2 } f _ l ^ 0 ( t , \\xi , \\zeta ) \\circ w ^ 2 , D _ { G , 2 } f ( t , \\xi , \\zeta ) \\circ w ^ 2 ) . \\end{array} \\right \\} \\end{align*}"}
+{"id": "3320.png", "formula": "\\begin{gather*} \\frac { 1 } { h ( n _ 1 , n _ 2 ) } = \\frac { \\big \\langle \\phi _ { m _ 1 ( 0 ) } , \\psi _ { n _ 1 ( 0 ) } ^ { n _ 2 } \\big \\rangle _ { V _ 1 } } { \\big \\langle \\phi _ { m _ 1 ( 0 ) } , \\psi _ { n _ 1 } ^ { n _ 2 } \\big \\rangle _ { V _ 1 } } . \\end{gather*}"}
+{"id": "7952.png", "formula": "\\begin{align*} \\eta = - J \\nu ^ H \\end{align*}"}
+{"id": "3973.png", "formula": "\\begin{align*} G _ { \\beta } ( u , t ) = E _ { \\beta , 1 } \\left ( - \\alpha \\left ( e ^ { \\theta } - e ^ { \\theta u } \\right ) t ^ { \\beta } \\right ) . \\end{align*}"}
+{"id": "3129.png", "formula": "\\begin{align*} \\sigma ( x ) = x + 2 \\end{align*}"}
+{"id": "7119.png", "formula": "\\begin{align*} \\chi _ A : = \\sum _ { i = 1 } ^ k w _ i \\tau _ { d _ i } , \\ \\chi ' _ A : = \\chi _ A + \\mathfrak { g } ^ { \\lambda > 0 } . \\end{align*}"}
+{"id": "8251.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ t \\tau ^ s \\sum _ { j \\le j _ 0 } 2 ^ { j ( \\bar { s } + s \\alpha ) } R _ j ( \\tau ) e ^ { - \\bar { \\mu } 2 ^ { j \\alpha } ( t - \\tau ) } \\dd \\tau \\le C \\int _ 0 ^ t Z _ { s , \\bar { s } } ( \\tau ) \\Psi ( t , \\tau ) \\dd \\tau . \\end{aligned} \\end{align*}"}
+{"id": "1762.png", "formula": "\\begin{align*} \\mu ^ { n - M } \\underset { = + 1 } { \\underbrace { \\left ( - 1 \\right ) ^ { k + 1 + 1 + M } } } \\det \\left ( C _ { [ n ] \\setminus \\{ i _ { 1 } , i _ { 1 } + 1 , \\ldots , \\widehat { i _ { j } + \\epsilon } , \\dots , i _ { \\frac { m } { 2 } } , i _ { \\frac { m } { 2 } } + 1 \\} \\cup \\left \\{ M \\right \\} } \\right ) + o \\left ( \\mu ^ { n - M } \\right ) , \\end{align*}"}
+{"id": "3345.png", "formula": "\\begin{align*} \\Lambda _ { \\{ 1 , 3 \\} } = \\alpha [ \\Lambda _ { \\{ 1 , 2 \\} } , \\Lambda _ { \\{ 2 , 3 \\} } ] _ q + \\beta \\left ( \\Lambda _ { \\{ 2 \\} } \\Lambda _ { \\{ 1 , 2 , 3 \\} } + \\Lambda _ { \\{ 1 \\} } \\Lambda _ { \\{ 3 \\} } \\right ) . \\end{align*}"}
+{"id": "7967.png", "formula": "\\begin{align*} \\begin{cases} \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle = | \\xi ^ H | ^ 2 k ( \\xi ) , \\\\ \\left \\langle \\eta , \\xi ^ H \\right \\rangle = 2 | \\xi ^ H | ^ 2 \\frac { \\left \\langle \\nu _ \\xi , T \\right \\rangle } { | \\mathcal { P } _ H ( \\nu _ \\xi ) | } , \\\\ \\left \\langle V _ j , \\xi ^ H \\right \\rangle = \\left \\langle W _ j , \\xi ^ H \\right \\rangle = 0 \\quad \\mbox { f o r } j \\in \\{ 1 , \\ldots , n - 1 \\} . \\end{cases} \\end{align*}"}
+{"id": "6541.png", "formula": "\\begin{align*} U ^ 2 = \\exp \\left ( \\gamma ( U - U ^ T ) \\right ) \\end{align*}"}
+{"id": "3668.png", "formula": "\\begin{align*} d _ { \\Gamma _ { t + 2 } } ( i , j ) = \\begin{cases} t + 2 & \\{ i , j \\} \\in \\binom { [ t ] } { 2 } , \\\\ t + 1 & i \\in [ t ] j \\in [ t + 1 , t + 4 ] , \\\\ t & \\{ i , j \\} \\in \\{ \\{ t + 1 , t + 4 \\} , \\{ t + 2 , t + 3 \\} \\} , \\\\ t - 1 & \\{ i , j \\} \\in \\{ \\{ t + 1 , t + 3 \\} , \\{ t + 2 , t + 4 \\} \\} , \\\\ 1 & \\{ i , j \\} \\in \\{ \\{ t + 1 , t + 2 \\} , \\{ t + 3 , t + 4 \\} \\} . \\end{cases} \\end{align*}"}
+{"id": "7087.png", "formula": "\\begin{gather*} h _ { { R _ Z ^ H } } ( Z , Z _ \\bullet , Z _ \\bullet ' ) = Z _ { 1 3 } Z _ { 1 6 } + Z _ { 1 3 } Z _ { 1 6 } ' + Z _ { 1 4 } Z _ { 1 5 } ' + Z _ { 1 5 } Z _ { 1 4 } ' + Z _ { 1 6 } Z _ { 1 3 } ' + Z _ { 1 3 } ' Z _ { 1 4 } ' , \\\\ h _ H ( Z ) = 6 Z _ { 1 3 } Z _ { 1 6 } . \\end{gather*}"}
+{"id": "5471.png", "formula": "\\begin{align*} F _ { N } ( a , b ; t ) = ( 1 - t q ^ { N } ) \\sum _ { n = 0 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( a q ) _ n ( q ) _ n \\left ( \\frac { a t q } { b } \\right ) _ n ( a t q ^ 2 ) _ { N - 1 } ( t b ) ^ n q ^ { n ^ 2 } ( 1 - a t q ^ { 2 n + 1 } ) } { ( b q ) _ n ( t ) _ { n + 1 } ( a t q ^ 2 ) _ { N + n } } . \\end{align*}"}
+{"id": "1683.png", "formula": "\\begin{align*} \\frac { - 1 } { 2 \\pi i } \\oint _ { \\Gamma } ( \\mathtt { H } _ 0 - z ) ^ { - 1 } d z = \\frac { - 1 } { 2 \\pi i } \\oint _ { \\Gamma } \\mathtt { R } _ 0 ( z ) d z = \\psi _ 0 \\psi _ 0 ^ * \\ , , \\end{align*}"}
+{"id": "4206.png", "formula": "\\begin{align*} \\varepsilon _ { k } ( X ) = - \\frac { q ^ { 1 - g } h } { \\zeta ( k ) ( q - 1 ) ^ 2 } + \\sum _ { j = 1 } ^ { 2 g } \\sum _ { \\ell = 0 } ^ { k - 1 } \\frac { Z ( \\gamma _ { j , \\ell } ^ { - 1 } ) } { k \\gamma _ { j , \\ell } ^ { 1 - k } Z ' ( \\gamma _ j ^ { - 1 } ) } \\frac { \\gamma _ { j , \\ell } } { \\gamma _ { j , \\ell } - 1 } + \\frac { 1 } { 2 \\pi i } \\sum _ { N = 0 } ^ { X - 1 } \\oint _ { C _ \\rho } \\frac { 1 } { u ^ { N + 1 } } \\frac { Z ( u ) } { Z ( u ^ k ) } \\ , d u . \\end{align*}"}
+{"id": "7276.png", "formula": "\\begin{align*} H ( s , x , \\psi , m ^ * ( s ) , u ^ * _ E ( s , x ) ) = \\max _ { u \\in U } H ( s , x , \\psi , m ^ * ( s ) , u ) . \\end{align*}"}
+{"id": "3598.png", "formula": "\\begin{align*} \\tilde { f } = \\sum _ { i j } a _ j \\sqrt { a _ i } \\eta _ { i j } h _ i \\phi _ j = \\sum _ i a _ j \\beta _ j \\phi _ j . \\end{align*}"}
+{"id": "3207.png", "formula": "\\begin{align*} J ( u ) = A ( u ) + B ( u ) + C ( u ) + \\frac { \\omega } { 2 } \\int _ { \\mathbb { R } ^ { 2 } } u ^ { 2 } d x . \\end{align*}"}
+{"id": "7056.png", "formula": "\\begin{align*} I ( M ) = I ( E ) \\times I ( M ' ) . \\end{align*}"}
+{"id": "5234.png", "formula": "\\begin{align*} D _ \\sigma \\equiv \\begin{pmatrix} 1 & \\sigma ( d ) - d \\\\ 0 & 1 \\end{pmatrix} \\mathfrak { m } _ { C ^ \\flat } \\otimes _ { \\mathbb { F } _ p } \\overline { \\mathbb { F } } _ p . \\end{align*}"}
+{"id": "4386.png", "formula": "\\begin{align*} T ^ t f ( x ) : = \\int _ \\mathbb { R } f ( \\xi ) \\frac { e ^ { - \\frac { ( x - \\xi ) ^ 2 } { 4 b ^ 2 t } } } { 2 b \\sqrt { \\pi } \\sqrt { t } } d t = f * K _ { t , p } ( x ) . \\end{align*}"}
+{"id": "2850.png", "formula": "\\begin{align*} \\mu _ { \\nabla ^ \\wedge _ w } ( f \\otimes 1 ) = \\mu _ { \\nabla ^ \\wedge _ w } ( 1 \\otimes v ^ { - 1 } ( f ) ) , \\end{align*}"}
+{"id": "3439.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { i \\geq 0 } E ^ { ( i ) } _ v ( b _ { + 1 } ) ( { - D _ z } ) ^ { i + 1 } f ( b _ { + 1 } ) = & \\sum _ { i \\geq 0 } E ^ { ( i + 1 ) } _ v ( b ) ( { - D _ z } ) ^ { i + 1 } ( ( D _ z f ' ( b { + 1 } ) ) / a ) \\\\ & - \\sum _ { i \\geq 0 } \\sum _ { j \\geq 0 } { \\textstyle \\binom { i + j } { j } } E ^ { ( i + j ) } _ v ( a ) ( ( { - D _ z } ) ^ j b _ { + 1 } ) ( { - D _ z } ) ^ { i } ( ( D _ z f ' ( b _ { + 1 } ) ) / a ) \\end{aligned} \\end{align*}"}
+{"id": "3018.png", "formula": "\\begin{align*} \\begin{aligned} y _ 1 & = x _ 3 + \\sum _ { i = 1 } ^ { \\alpha _ 1 } ( - 1 ) ^ i x _ 1 ^ { ( \\alpha _ 1 - i ) } u _ 2 ^ { ( i - 1 ) } \\\\ y _ 2 & = x _ 2 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "7957.png", "formula": "\\begin{align*} H _ M ( \\xi ) = \\frac { 2 n + 1 } { 2 n - 1 } \\frac { r } { R ^ 2 } . \\end{align*}"}
+{"id": "2906.png", "formula": "\\begin{align*} \\int _ \\Omega \\left ( \\eta x _ { n + 1 } ^ m + \\lambda \\right ) d V = \\int _ \\Sigma \\langle Y , \\nu \\rangle \\ , d A . \\end{align*}"}
+{"id": "3901.png", "formula": "\\begin{align*} v ' ( t ) + \\lambda ( 1 + D ^ { \\{ m \\} } _ t ) v ( t ) & = g ( t ) , \\\\ v ( 0 ) & = v _ 0 . \\end{align*}"}
+{"id": "712.png", "formula": "\\begin{align*} \\rho c _ v \\partial _ { t _ 1 } T = { \\bar F } ^ { ( 1 ) } , \\end{align*}"}
+{"id": "4219.png", "formula": "\\begin{align*} E _ { M , k } ( X ) : = - \\sum _ { j = 1 } ^ { 2 g } \\sum _ { \\ell = 1 } ^ { k } \\frac { Z ( \\gamma _ { j , \\ell } ^ { - 1 } ) } { k \\gamma _ { j , \\ell } ^ { 1 - k } Z ' ( \\gamma _ j ^ { - 1 } ) } \\frac { \\gamma _ { j , \\ell } } { \\gamma _ { j , \\ell } - 1 } e ^ { i X ( \\theta ( \\gamma _ j ) + 2 \\pi \\ell ) / k } . \\end{align*}"}
+{"id": "738.png", "formula": "\\begin{align*} a = \\frac { { 0 . 4 5 7 2 4 { R ^ 2 } T _ c ^ 2 } } { { { p _ c } } } , b = \\frac { { 0 . 0 7 7 8 R { T _ c } } } { { { p _ c } } } , \\end{align*}"}
+{"id": "129.png", "formula": "\\begin{align*} c = \\sum _ { [ \\mathbf a ] \\in B [ m ] _ q } c _ { [ \\mathbf a ] } \\cdot [ \\mathbf a ] \\end{align*}"}
+{"id": "3671.png", "formula": "\\begin{align*} p _ { K _ { t + 2 } ^ 3 } ( y _ 1 , \\ldots , y _ { t + 2 } ) \\le \\frac { t ( t + 1 ) } { 2 ( t + 2 ) ^ 2 } - \\frac { t } { 6 ( t + 2 ) } \\sum _ { i = 1 } ^ { t + 2 } \\left ( y _ i - \\frac { 1 } { t + 2 } \\right ) ^ 2 \\end{align*}"}
+{"id": "1118.png", "formula": "\\begin{align*} f _ { N } ( x , y | \\rho ) = f _ { N } ( x ) f _ { N } ( x ) \\sum _ { n \\geq 0 } \\rho ^ { n } h _ { n } ( x ) h _ { n } ( y ) , \\end{align*}"}
+{"id": "6248.png", "formula": "\\begin{align*} \\delta _ k ( G ) = \\frac { 1 } { k ! } \\sum _ { j = 0 } ^ { t - k - 1 } \\frac { ( - 1 ) ^ j } { j ! } + \\frac { ( n - t ) ! } { k ! } \\sum _ { j = { t - k } } ^ { n - k } \\frac { ( - 1 ) ^ { j } } { ( n - k - j ) ! j ! } \\qquad \\end{align*}"}
+{"id": "5701.png", "formula": "\\begin{gather*} R ^ O _ 2 = \\ker \\tau ^ * _ O \\subset R _ 2 = \\ker \\tau ^ * . \\end{gather*}"}
+{"id": "1910.png", "formula": "\\begin{align*} \\partial _ t u - v \\cdot D _ x u - D _ { v _ i } ( a ^ { i j } ( z ) D _ { v _ j } u ) + \\div ( \\overline { b } u ) + b \\cdot D _ { v } u + c u + \\lambda u = \\div \\vec f + g . \\end{align*}"}
+{"id": "3890.png", "formula": "\\begin{align*} \\lim _ { | x | \\to 0 ^ + } u _ k ( x ) | x | ^ { - \\tau _ - ( \\mu ) } ( x ) = k . \\end{align*}"}
+{"id": "6465.png", "formula": "\\begin{align*} p ( x , \\xi ) = p ^ { \\# } ( x , \\xi ) + p ^ { b } ( x , \\xi ) , \\end{align*}"}
+{"id": "4500.png", "formula": "\\begin{align*} \\dd _ u V ( h ) = \\int _ { [ 0 , 1 ] } u ' h ' \\ , . \\end{align*}"}
+{"id": "6834.png", "formula": "\\begin{align*} 3 \\ell _ { a } \\ell _ { b } E _ { 0 } & = \\ell _ { b } ( \\ell _ { a } - \\ell _ { a b } ) ^ 2 + \\ell _ { a } ( \\ell _ { b } - \\ell _ { a b } ) ^ 2 - 2 \\ell _ { a b } ( \\ell _ { a } - \\ell _ { a b } ) ( \\ell _ { b } - \\ell _ { a b } ) \\\\ & = - 2 \\ell _ { a b } ^ 3 + 3 ( \\ell _ { a } + \\ell _ { b } ) \\ell _ { a b } ^ 2 - 6 \\ell _ { a } \\ell _ { b } \\ell _ { a b } + \\ell _ { a } \\ell _ { b } ( \\ell _ { a } + \\ell _ { b } ) . \\end{align*}"}
+{"id": "1688.png", "formula": "\\begin{align*} \\{ \\varphi ( x ) , \\overline { \\varphi } ( y ) \\} = i \\delta ( x - y ) , \\{ \\varphi ( x ) , \\varphi ( y ) \\} = \\{ \\overline { \\varphi } ( x ) , \\overline { \\varphi } ( y ) \\} = 0 \\ , . \\end{align*}"}
+{"id": "3518.png", "formula": "\\begin{align*} \\Big \\Vert \\nabla \\mathrm { H } ^ { } \\Big \\Vert ^ { 2 } _ { \\mathbb { L } ^ { 2 } \\Big ( \\Omega \\Big ) } = \\mathcal { O } ( \\delta ^ { 2 } ) . \\end{align*}"}
+{"id": "2157.png", "formula": "\\begin{align*} | \\widetilde { R } | = \\begin{vmatrix} R _ { 1 2 1 2 } & R _ { 1 2 1 3 } & R _ { 1 2 2 3 } \\\\ R _ { 1 2 1 3 } & R _ { 1 3 1 3 } & R _ { 1 3 2 3 } \\\\ R _ { 1 2 2 3 } & R _ { 1 3 2 3 } & R _ { 2 3 2 3 } \\end{vmatrix} \\geq 0 . \\end{align*}"}
+{"id": "4027.png", "formula": "\\begin{align*} f _ b ( y ) = \\left ( p - 1 + b y ^ { 2 k } \\right ) ^ { - \\frac { 1 } { p - 1 } } , \\end{align*}"}
+{"id": "7536.png", "formula": "\\begin{align*} \\Re V ( p ) = r _ 0 \\log | z _ 0 ( p ) | + O ( 1 ) , p \\to p _ 0 . \\end{align*}"}
+{"id": "1210.png", "formula": "\\begin{align*} U = \\begin{bmatrix} U _ 1 & & & & 0 \\\\ & 1 & & & \\\\ & & 1 & & \\\\ & & & 1 & \\\\ 0 & & & & U _ 5 \\\\ \\end{bmatrix} , \\end{align*}"}
+{"id": "4856.png", "formula": "\\begin{align*} \\partial _ t \\bar \\rho = \\nabla \\cdot ( \\bar \\rho \\nabla f ) + \\frac { 1 } K \\nabla \\cdot \\nabla \\cdot ( \\bar \\rho \\Sigma ) - \\frac 1 { 2 K } \\nabla \\cdot ( \\bar \\rho \\nabla V ) . \\end{align*}"}
+{"id": "4363.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\sum _ { i \\neq j } ( \\theta ^ 1 _ i - \\theta ^ 2 _ i ) D _ H g _ i ^ 0 ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) \\\\ + \\sum _ { \\alpha = 1 } ^ { m } ( \\lambda ^ 1 _ \\alpha - \\lambda ^ 2 _ \\alpha ) D _ H g ^ \\alpha ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) \\\\ + \\sum _ { \\beta = 1 } ^ { q } ( \\mu ^ 1 _ \\beta - \\mu ^ 2 _ \\beta ) D _ H h ^ \\beta ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) = 0 . \\end{array} \\right \\} \\end{align*}"}
+{"id": "6990.png", "formula": "\\begin{align*} \\int f ^ 2 \\log f ^ 2 d m = \\frac { 1 } { \\delta } \\int \\log | f | ^ { 2 \\delta } ( f ^ 2 d m ) \\leq \\frac { 1 + \\delta } { \\delta } \\log \\| f \\| ^ 2 _ { 2 + 2 \\delta } . \\end{align*}"}
+{"id": "3964.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\mathrm { d } } { \\mathrm { d } t } \\hat { q } ( 0 , t ) & = - \\lambda \\hat { q } ( 0 , t ) , \\\\ \\frac { \\mathrm { d } } { \\mathrm { d } t } \\hat { q } ( n , t ) & = - \\lambda \\hat { q } ( n , t ) - \\frac { \\lambda } { \\ln p } \\sum _ { j = 1 } ^ { n } \\frac { ( 1 - p ) ^ { j } } { j } \\hat { q } ( n - j , t ) , \\ n \\ge 1 , \\end{aligned} \\end{align*}"}
+{"id": "7726.png", "formula": "\\begin{align*} \\forall q \\in X , \\ \\ s _ E ( q ) = d i s t _ { g ^ + } ( E , q ) = \\inf \\limits _ { y \\in X } d i s t _ { g ^ + } ( y , q ) . \\end{align*}"}
+{"id": "1723.png", "formula": "\\begin{align*} \\mathcal { C } _ p : = \\mathfrak { B } _ p \\cup \\{ \\mathbf { 1 } _ p \\} \\ , . \\end{align*}"}
+{"id": "5768.png", "formula": "\\begin{align*} - \\frac { 6 6 4 7 9 } { 9 5 5 8 } < a < 0 , - \\frac { a } { 2 } < b < - \\frac { 4 } { 7 } a , - \\frac { 2 1 3 5 0 0 a + 1 1 5 5 5 9 b } { 2 5 6 4 1 9 } < c \\leq - \\frac { 5 8 3 ( 4 a + 7 b ) - 5 8 3 2 } { 1 4 5 8 } . \\end{align*}"}
+{"id": "7637.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { 5 } \\nu _ i P _ i + \\mu _ 1 S _ 1 + \\mu _ 2 S _ 2 = \\sum _ { i = 0 } ^ { 1 1 } \\lambda _ i P _ i \\end{align*}"}
+{"id": "5255.png", "formula": "\\begin{align*} S _ N ( f ) & = & \\sum _ { 1 \\leq j _ 1 , j _ 2 , \\ldots , j _ { n + 1 } \\leq N } f ( N \\ * ( \\theta _ { j _ 2 } - \\theta _ { j _ 1 } ) _ c , \\ldots , N \\ * ( \\theta _ { j _ { n + 1 } } - \\theta _ { j _ 1 } ) _ c ) \\\\ & = & \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } \\ * N ^ { n } } \\ * \\sum _ { k \\in \\mathbb { Z } ^ n } \\hat { f } ( k _ 1 \\ * N ^ { - 1 } , \\ldots , k _ n \\ * N ^ { - 1 } ) \\ * \\prod _ { j = 1 } ^ { n + 1 } T _ { N , k _ j } , \\end{align*}"}
+{"id": "3338.png", "formula": "\\begin{align*} L \\psi _ n = a _ n \\psi _ { n - 1 } + b _ n \\psi _ { n + 1 } + c _ n B \\psi _ n + d _ n D _ 1 \\psi _ n , \\end{align*}"}
+{"id": "3514.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\mathrm { E } | ^ 2 ( \\mathrm { y } ) d \\mathrm { y } = \\dfrac { 1 } { | 1 - \\alpha \\lambda ^ { ( 3 ) } _ { \\mathrm { n } _ { 0 } } | ^ 2 } \\Bigg [ | \\nabla \\mathrm { H } ^ { \\textbf { i n } } | ^ 2 ( \\mathrm { z } ) \\Big ( \\int _ \\mathrm { B } \\Tilde { \\mathrm { e } } ^ { ( 3 ) } _ { \\mathrm { n } _ { 0 } } ( \\mathrm { x } ) d \\mathrm { x } \\Big ) ^ 2 \\delta ^ 2 + \\mathcal { O } \\Big ( \\delta ^ { 3 } \\Big ) \\Bigg ] \\mathrm { h } < 1 . \\end{align*}"}
+{"id": "5657.png", "formula": "\\begin{align*} H ( r ) ( T , T ) = g _ T ( D ^ T _ T T , T ) = 0 , \\ \\ H ( r ) ( T , V ) = g _ T ( D ^ T _ T T , V ) = 0 . \\end{align*}"}
+{"id": "1388.png", "formula": "\\begin{align*} \\varphi _ { \\beta , \\varepsilon } ' ( s ) & = - \\frac { \\beta } { \\gamma _ { \\varepsilon } } e ^ { - s } M ( \\gamma _ { \\varepsilon } - \\beta , \\gamma _ { \\varepsilon } + 1 ; s ) , \\\\ \\varphi _ { \\beta , \\varepsilon } '' ( s ) & = \\frac { \\beta ( \\beta + 1 ) } { \\gamma _ { \\varepsilon } ( \\gamma _ { \\varepsilon } + 1 ) } e ^ { - s } M ( \\gamma _ { \\varepsilon } - \\beta , \\gamma _ { \\varepsilon } + 2 ; s ) . \\end{align*}"}
+{"id": "3878.png", "formula": "\\begin{align*} \\mu ( \\abs { z } ^ 2 ) ^ 2 = \\abs { z } ^ 2 - 2 \\log \\abs { z } - 1 \\ge \\frac { \\gamma + x + y ^ 2 } { 2 n } \\Bigl ( 1 - \\sqrt { \\frac { \\gamma } { n } } \\Bigr ) \\ge \\frac { \\gamma } { 2 n } \\Bigl ( 1 - \\sqrt { \\frac { \\gamma } { n } } \\Bigr ) + \\frac { x + y ^ 2 } { 3 n } \\end{align*}"}
+{"id": "2414.png", "formula": "\\begin{align*} { \\rm D } ( e _ { a } ) = e _ { a } e _ { a } - e _ { a } e _ { 0 } - e _ { 1 } e _ { a } = - e _ { 1 } e _ { 0 } \\quad ( a \\in \\{ 0 , 1 \\} ) . \\end{align*}"}
+{"id": "5709.png", "formula": "\\begin{align*} \\int _ { E } | f - f _ { Q } | = \\int _ { F } | f - f _ { Q } | . \\end{align*}"}
+{"id": "6837.png", "formula": "\\begin{align*} \\abs { P ( x ) } _ { v } & \\le \\max _ { h = ( h _ { 1 } , \\dotsc , h _ { n } ) \\in E } \\Big ( \\abs { a _ { h } } _ { v } \\prod _ { i = 1 } ^ n \\abs { x _ { i } } _ { v } ^ { h _ { i } } \\Big ) \\\\ & \\le \\max _ { h \\in H } ( \\abs { a _ { h } } _ { v } M _ { x } ^ { \\abs { h } } ) , \\end{align*}"}
+{"id": "5351.png", "formula": "\\begin{align*} ( S \\wedge T ) _ { i j k l } & : = S _ { i k } T _ { j l } + S _ { j l } T _ { i k } - S _ { i l } T _ { j k } - S _ { j k } T _ { i l } , \\\\ ( S \\circ T ) _ { i j } & : = S _ i { } ^ u T _ { u j } , \\end{align*}"}
+{"id": "2138.png", "formula": "\\begin{align*} F _ \\eta ( \\vec { x } ) = \\int _ \\Omega f \\left ( \\frac { \\vec { x } } { \\eta } , \\nabla u ( \\vec { x } ) \\right ) \\ , d \\vec { x } \\ ; , \\ ; u \\in W ^ { 1 , p } ( \\Omega ) \\end{align*}"}
+{"id": "2064.png", "formula": "\\begin{align*} \\mu ( X ) = \\lim _ { k \\to \\infty } \\mu ( X _ k ) = \\lim _ { k \\to \\infty } \\# \\pi _ k ( X ) q ^ { - k } . \\end{align*}"}
+{"id": "2966.png", "formula": "\\begin{align*} \\Lambda : = \\{ \\sigma : \\sigma \\{ 1 , 2 , \\ldots , N \\} \\} . \\end{align*}"}
+{"id": "5085.png", "formula": "\\begin{align*} \\frac { 2 } { \\tilde { p } } + \\frac { d } { \\tilde { q } } = \\frac { d } { 2 } - s + d ( 1 - \\sigma ) ( \\frac { 1 } { 2 } - \\frac { 1 } { q } ) , \\end{align*}"}
+{"id": "4975.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { m } & = \\underset { 0 \\leq m \\leq M } { \\arg \\max } \\ \\mathbf { P } _ { \\mathrm { M P A } } ^ { k } [ m , v ] , \\\\ \\hat { \\mathbf { x } } _ { v } [ k ] & = \\mathcal { X } _ { v } [ \\hat { m } ] , \\end{aligned} \\end{align*}"}
+{"id": "3261.png", "formula": "\\begin{gather*} Q = \\big ( q ^ { - 1 } - q ^ 3 \\big ) K L [ K , L ] _ q + q ^ 2 ( [ K , L ] _ q ) ^ 2 + B ( K L + L K ) + C _ 0 q ^ 2 K ^ 2 + C _ 1 q ^ { - 2 } L ^ 2 \\\\ \\hphantom { Q = } { } + D _ 0 \\big ( 1 + q ^ 2 \\big ) K + D _ 1 \\big ( 1 + q ^ { - 2 } \\big ) L . \\end{gather*}"}
+{"id": "878.png", "formula": "\\begin{align*} & \\overline { \\mu } ( \\mathbb { J } ) = \\sum \\limits _ { ( \\mathbb { J } ^ { ' } , t ^ { ' } ) \\in \\pi ^ { ' } } \\overline { \\mu } ( \\mathbb { J } \\cap \\mathbb { J } ^ { ' } ) . \\end{align*}"}
+{"id": "3474.png", "formula": "\\begin{align*} \\mathrm { U } _ { \\mathrm { e } } ( \\xi , t ) & = - \\mathcal { S } \\Big [ \\gamma ^ { \\textbf { e x t } } _ { 1 } \\mathrm { U } _ { \\mathrm { e } } \\Big ] ( \\xi , t ) + \\mathcal { D } \\Big [ \\gamma ^ { \\textbf { e x t } } _ { 0 } \\mathrm { U } _ { \\mathrm { e } } \\Big ] ( \\xi , t ) , \\ ( \\xi , t ) \\in \\mathbb { R } ^ 2 \\setminus \\overline { \\Omega } \\times ( 0 , \\mathrm { T } ) , \\end{align*}"}
+{"id": "5175.png", "formula": "\\begin{align*} E ^ 1 u ( x ) = \\begin{cases} E ^ { j } _ { n , i } ( u | _ { U ^ { j } _ { n , i } \\setminus T ^ { j } _ { n , i } } ) ( x ) , & x \\in U ^ { j } _ { n , i } \\ ; \\ ; j \\ ; , \\\\ u ( x ) , & , \\end{cases} \\end{align*}"}
+{"id": "448.png", "formula": "\\begin{align*} B _ n ( 1 + x ) - B _ n ( x ) = n x ^ { n - 1 } , \\end{align*}"}
+{"id": "3131.png", "formula": "\\begin{align*} F P ( \\sigma ^ k ) = 0 \\end{align*}"}
+{"id": "8177.png", "formula": "\\begin{align*} \\langle ( C ^ { 2 n } a , C b \\rangle = 0 \\ \\ \\ ( n = 0 , 1 , 2 , \\ldots ) . \\end{align*}"}
+{"id": "1662.png", "formula": "\\begin{align*} 1 - \\overline { \\sigma } - \\frac { 2 } { \\overline { \\sigma } } \\frac { \\lambda } { \\overline { \\tau } } = 1 - 2 ^ { - \\frac { 1 } { 2 } } \\ , . \\end{align*}"}
+{"id": "1971.png", "formula": "\\begin{align*} \\widehat { b _ R ( D ) f } : = b _ R \\hat { f } , \\end{align*}"}
+{"id": "2472.png", "formula": "\\begin{align*} A _ U = \\widetilde { A } _ U + g _ U ^ { - 1 } \\d g _ U \\end{align*}"}
+{"id": "8248.png", "formula": "\\begin{align*} \\begin{aligned} \\tau ^ s \\sum _ { j \\le j _ 0 } 2 ^ { j ( \\bar { s } + s \\alpha ) } \\psi _ j \\Vert \\nabla \\dot S _ { j - 1 } u ( \\tau ) \\Vert _ { L ^ { \\infty } } X _ j ( \\tau ) \\le C Z _ { s , \\bar { s } } ( \\tau ) \\Big ( \\sup _ { j \\le j _ 0 } \\| \\nabla \\dot S _ { j - 1 } u ( \\tau ) \\| _ { L ^ \\infty } \\psi _ j \\Big ) . \\end{aligned} \\end{align*}"}
+{"id": "3641.png", "formula": "\\begin{align*} \\alpha \\otimes x [ \\alpha _ { i , j } x ] _ { i , j } , \\alpha = [ \\alpha _ { i , j } ] _ { i , j } \\in M _ n x \\in X . \\end{align*}"}
+{"id": "7909.png", "formula": "\\begin{align*} a ~ \\widetilde { \\cdot } ~ u : = R ( a ) \\cdot u - S ( a \\cdot u ) u ~ \\widetilde { \\cdot } ~ a : = u \\cdot R ( a ) - S ( u \\cdot a ) , \\end{align*}"}
+{"id": "2503.png", "formula": "\\begin{align*} A = \\{ n \\in \\N : T ^ n a \\in E \\} . \\end{align*}"}
+{"id": "5565.png", "formula": "\\begin{align*} \\Tilde { \\Phi } _ 1 ( \\alpha _ 0 , \\eta ) = \\dfrac { 1 } { L _ m ^ 2 } \\bigg ( \\bigg ( \\tilde { N _ 1 } + \\frac { N _ { 1 M } \\tilde { L _ 1 } } { L _ { 1 m } } \\bigg ) \\bigg [ \\frac { \\eta ^ { 3 - \\nu } } { 3 - \\nu } - \\alpha _ 0 ^ 2 \\dfrac { \\eta ^ { 1 - \\nu } } { 1 - \\nu } + \\frac { 2 \\alpha _ 0 ^ 2 } { ( 3 - \\nu ) ( 1 - \\nu ) } \\bigg ] + \\tilde { L _ 1 } \\frac { \\eta ^ { 1 - \\nu } } { 1 - \\nu } \\bigg ) , \\end{align*}"}
+{"id": "4464.png", "formula": "\\begin{align*} & \\sum ( - 1 ) ^ { f _ \\sigma } \\cdots E _ { i j } E _ { j k } \\cdots - \\sum ( - 1 ) ^ { f _ { \\sigma ' } } \\cdots E _ { j k } E _ { i j } \\cdots \\\\ = & \\sum ( - 1 ) ^ { \\bar j + f _ { \\tilde \\sigma } } \\cdots E _ { i k } \\cdots - \\sum ( - 1 ) ^ { f _ { \\tilde \\sigma ' } } \\cdots E _ { j j } \\cdots \\\\ = & ( m - n ) \\sum ( - 1 ) ^ { f _ { \\tilde \\sigma } } \\cdots E _ { i k } \\cdots - C _ 1 \\sum ( - 1 ) ^ { f _ { \\tilde \\sigma ' } } \\cdots 1 \\cdots , \\end{align*}"}
+{"id": "1615.png", "formula": "\\begin{align*} & d \\left ( f _ { 1 } \\circ g _ { 1 } , f _ { 2 } \\circ g _ { 2 } \\right ) = \\lim _ { n \\rightarrow \\mathcal { U } } d \\left ( f _ { 1 , n } \\circ g _ { 1 , n } , f _ { 2 , n } \\circ g _ { 2 , n } \\right ) \\\\ \\le & \\lim _ { n \\rightarrow \\mathcal { U } } \\left [ d \\left ( f _ { 1 , n } , f _ { 2 , n } \\right ) + d \\left ( g _ { 1 , n } , g _ { 2 , n } \\right ) \\right ] = d \\left ( f _ { 1 } , f _ { 2 } \\right ) + d \\left ( g _ { 1 } , g _ { 2 } \\right ) \\end{align*}"}
+{"id": "1555.png", "formula": "\\begin{align*} X _ { s _ { i } } \\left ( \\omega \\right ) = X _ { s _ { i } } \\left ( \\tilde { \\omega } \\right ) = X _ { s _ { i } } \\left ( \\tilde { \\omega } ^ { \\prime } \\right ) = X _ { s _ { i } } \\left ( \\omega ^ { \\prime } \\right ) , \\end{align*}"}
+{"id": "663.png", "formula": "\\begin{align*} ( T _ 1 + f _ 1 ) \\cdot ( T _ 2 - f _ 2 ) \\cdot \\Psi _ 2 = i \\Psi _ 2 \\end{align*}"}
+{"id": "500.png", "formula": "\\begin{align*} K ( ( z , \\xi ) , \\overline { ( w , \\eta ) } ) = \\frac { K _ { D } ( z , \\overline { w } ) ^ { d _ { 0 } s + 1 } } { \\pi ^ { d _ { 0 } } } \\sum _ { j = 0 } ^ { d } \\frac { c ( s , j ) ( j + d _ { 0 } ) ! } { ( 1 - t ) ^ { j + d _ { 0 } + 1 } } , \\end{align*}"}
+{"id": "2633.png", "formula": "\\begin{align*} E _ { [ w _ 1 , w _ 2 ] } ^ 0 = & \\{ w \\in B S ( 2 , 1 ) ^ + : w _ 1 w , w w _ 2 \\} \\\\ E _ { [ w _ 1 , w _ 2 ] } ^ * = & \\{ x \\in E ^ 1 _ { w _ 2 } : s ( x ) , r ( x ) \\in E _ { [ w _ 1 , w _ 2 ] } ^ 0 \\} \\end{align*}"}
+{"id": "3911.png", "formula": "\\begin{align*} & \\ell ^ * _ 1 = \\sup _ { \\substack { h > 0 \\\\ t \\in ( 0 , T - h ] } } \\left ( \\frac t h \\right ) ^ \\gamma \\int _ t ^ { t + h } | \\ell ( \\tau ) | d \\tau < \\infty , \\\\ & 1 6 B ( 1 - \\delta , 1 - 2 \\gamma ) T ^ { 1 - \\delta } ( { L _ f ^ * } ^ 2 + { K _ f ^ * } ^ 2 { \\ell ^ * _ 2 } ^ 2 ) < 1 , \\end{align*}"}
+{"id": "1399.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } ( u ^ { ( j ) } ( t ) , \\partial _ t u ^ { ( j ) } ( t ) ) = ( u ( t ) , \\partial _ t u ( t ) ) C ( [ 0 , T ] ; H _ 0 ^ 1 ( \\Omega ) ) \\cap C ^ 1 ( [ 0 , T ] ; L ^ 2 ( \\Omega ) ) \\end{align*}"}
+{"id": "2959.png", "formula": "\\begin{align*} ( M _ 2 \\circ M _ 1 ) ( y _ 1 ) : = \\bigcup \\limits _ { y _ 2 \\in M _ 1 ( y _ 1 ) } M _ 2 ( y _ 2 ) \\qquad \\forall y _ 1 \\in \\R ^ { m _ 1 } . \\end{align*}"}
+{"id": "6597.png", "formula": "\\begin{align*} \\mathbb { E } [ | X _ t ( x ) | ^ 2 ] & = | \\exp ( t \\tilde A ) \\exp ( \\frac { t } { 2 } \\alpha ) \\exp ( - \\frac { t ^ 2 } { 2 } \\beta ) \\exp ( \\frac { p ( t ) } { 2 } \\Gamma ) x | ^ 2 \\\\ & = | \\sum _ { j = 1 } ^ d e ^ { - \\frac { a _ { j } } { 2 } t - \\frac { b _ { j } } { 2 } t ^ 2 - \\frac { \\gamma _ { j } } { 2 } ( t ^ 3 - p _ \\Gamma t ) } \\langle x , v _ { j } \\rangle \\exp ( t \\tilde A ) v _ { j } | ^ 2 . \\end{align*}"}
+{"id": "6031.png", "formula": "\\begin{align*} & q ^ 2 \\times \\left ( q ^ { n - 1 } + q ^ { n - 2 } \\right ) \\prod _ { i = 1 } ^ { n - 2 } ( q ^ { n - i } + 1 ) = \\left ( q ^ { n + 1 } + q ^ n \\right ) \\prod _ { i = 1 } ^ { n - 2 } ( q ^ { n - i } + 1 ) \\end{align*}"}
+{"id": "7040.png", "formula": "\\begin{align*} \\mathfrak { t r } ^ p : = \\{ X \\in \\mathcal K ^ G ( M ) : X \\textrm { i s a t r a n s v e c t i o n a t $ p $ w i t h } X _ p \\in \\nu ^ { ( 1 ) } _ p \\} \\subset \\mathfrak u ^ p . \\end{align*}"}
+{"id": "6759.png", "formula": "\\begin{align*} W ^ * ( f ) a _ k W ( f ) = a _ k + f ( k ) . \\end{align*}"}
+{"id": "529.png", "formula": "\\begin{align*} H ( Q \\ , | \\ , P ) \\le \\inf \\Big \\{ H ( R \\ , | \\ , P ) : R \\in \\P ( \\R ^ n ) , \\ H ( R ) < \\infty \\Big \\} = 0 , \\end{align*}"}
+{"id": "6709.png", "formula": "\\begin{align*} L _ { M } h : = Q ( h , M ) + Q ( M , h ) . \\end{align*}"}
+{"id": "7445.png", "formula": "\\begin{align*} [ j , \\ , \\tau ^ \\dagger _ \\theta ] = \\theta \\tau ^ \\dagger _ \\theta . \\end{align*}"}
+{"id": "7089.png", "formula": "\\begin{align*} p _ { R _ X } ( P , Q ) = \\begin{cases} 1 & P = Q , \\\\ 0 & P \\neq Q . \\end{cases} \\end{align*}"}
+{"id": "7191.png", "formula": "\\begin{align*} A _ { k _ { \\bullet } } : = \\mathrm { S y m } \\left ( \\frac { z _ 1 ^ { k _ 1 } \\cdots z _ d ^ { k _ d } } { ( 1 - q ^ { - 1 } z _ 1 ^ { - 1 } z _ 2 ) \\cdots ( 1 - q ^ { - 1 } z _ { d - 1 } ^ { - 1 } z _ d ) } \\cdot \\prod _ { j > i } w ( z _ i z _ j ^ { - 1 } ) \\right ) \\end{align*}"}
+{"id": "4950.png", "formula": "\\begin{align*} \\frac { k + 1 } { 2 } a _ 0 + \\sum _ { i = 1 } ^ { ( k - 5 ) / 2 - t } \\left ( \\frac { k - 1 } { 2 } - i \\right ) a _ i + \\sum _ { i = 1 } ^ { ( k - 5 ) / 2 - t } \\left ( \\frac { k - 3 } { 2 } - i \\right ) b _ i = \\binom { n } { k - 2 } . \\end{align*}"}
+{"id": "6077.png", "formula": "\\begin{align*} \\rho ( H ) & \\leq 7 n ( H ' ) - 3 ( 2 n ( H ' ) + 2 ) - 2 \\frac { n ( H ' ) - 7 } { 2 } \\\\ & = - 3 \\times 2 + 7 = 1 \\end{align*}"}
+{"id": "1237.png", "formula": "\\begin{align*} & \\le \\int _ { T } ^ { 2 T } \\prod _ { 1 \\le i \\le j } ( \\exp ( k \\Re G _ { i , j } ( t ) ) ^ 2 ( \\beta _ { j + 1 } ^ { 3 / 4 } | \\Re G _ { j + 1 , \\ell } ( t ) | ) ^ { M } d t \\\\ & \\ll ( \\beta _ { j + 1 } ^ { 3 / 2 } ) ^ { [ 1 / ( 1 0 \\beta _ { j + 1 } ) ] } \\int _ { T } ^ { 2 T } \\prod _ { 1 \\le i \\le j } \\left ( \\sum _ { 0 \\le n \\le 1 0 0 k \\beta _ { i } ^ { - 3 / 4 } } \\frac { ( k \\Re G _ { i , j } ( t ) ) ^ n } { n ! } \\right ) ^ 2 ( \\Re G _ { j + 1 , \\ell } ( t ) ) ^ { M } d t , \\end{align*}"}
+{"id": "7694.png", "formula": "\\begin{align*} F ( z ) = \\exp \\left ( \\int _ { - \\pi } ^ { \\pi } \\frac { e ^ { i t } + z } { e ^ { i t } - z } \\log f ( e ^ { i t } ) \\frac { d t } { 2 \\pi } \\right ) , \\end{align*}"}
+{"id": "705.png", "formula": "\\begin{align*} \\rho c _ v \\left [ g _ i + \\Delta t D _ i g _ i + \\frac { \\Delta t ^ 2 } { 2 } D _ i ^ 2 g _ i \\right ] - g _ i + ( 1 - \\rho c _ v ) \\left [ g _ i + \\Delta t d _ i g _ i + \\frac { \\Delta t ^ 2 } { 2 } d _ i ^ 2 g _ i \\right ] = - { \\left ( { { { \\bf { M } } ^ { - 1 } } \\Lambda { \\bf { M } } } \\right ) _ { i j } } \\left [ { g _ i - g _ i ^ { ( e q ) } } \\right ] + \\Delta t { \\bar F _ i } + \\vartheta \\Delta t { S _ i } , \\end{align*}"}
+{"id": "5447.png", "formula": "\\begin{align*} \\mathcal { B } _ r \\ , \\ , \\ , = \\ , \\ , \\ , \\bigcup _ { \\ell = 0 } ^ { d r } \\ , \\ , \\mathrm { G r } ( \\ell , n ) ^ { T _ \\lambda } , \\hbox { w h e r e } \\ , \\ , \\ , \\lambda \\ , = \\ , ( r , r , \\ldots , r ) . \\end{align*}"}
+{"id": "1323.png", "formula": "\\begin{align*} J _ p ( u , \\Omega ) : = \\int _ { \\Omega } \\Big ( | \\nabla u ( x ) | ^ p + \\chi _ { \\{ u > 0 \\} } ( x ) \\Big ) \\ , d x \\end{align*}"}
+{"id": "5528.png", "formula": "\\begin{align*} \\Psi ( l _ c ) ^ { - 1 } ( L ^ j \\ltimes ( L \\cap \\sigma _ j N ) ) = L ^ { j , c } \\ltimes ( L \\cap \\sigma _ { j , c } N ) , ( L ^ { j , c } \\ltimes ( L \\cap \\sigma _ { j , c } N ) ) \\cap P _ L = L _ L ^ { j , c } \\ltimes ( N _ L ^ { j , c } ( N _ L \\cap \\sigma _ { j , c } N ) ) . \\end{align*}"}
+{"id": "333.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial } { \\partial h } L ( h , Y _ 1 , \\cdots , Y _ n , \\Theta _ m ) & = \\frac { \\partial } { \\partial h } L ( h , Y _ 1 , \\cdots , Y _ n , \\Theta _ m ) | _ { h = \\tilde { h } } \\\\ & + ( h - \\tilde { h } ) \\frac { \\partial ^ 2 } { \\partial h ^ 2 } L ( h , Y _ 1 , \\cdots , Y _ n , \\Theta _ m ) | _ { h = \\tilde { h } } \\\\ & + \\frac { 1 } { 2 } \\zeta ( h - \\tilde { h } ) ^ 2 H ( Y _ 1 , \\cdots , Y _ n , \\tilde { h } ) , \\end{aligned} \\end{align*}"}
+{"id": "1376.png", "formula": "\\begin{align*} \\mathcal { A } = \\begin{pmatrix} 0 & 1 \\\\ \\Delta & - a ( x ) \\end{pmatrix} \\end{align*}"}
+{"id": "425.png", "formula": "\\begin{align*} K _ { n } ( \\lambda ) K _ { n } ( \\mu ) = \\sum _ { \\nu \\in \\mathcal { P } _ { n } ^ { } } a _ { \\lambda , \\mu } ^ { \\nu } ( n ) K _ { n } ( \\nu ) , \\end{align*}"}
+{"id": "6113.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal S _ 2 ( k , 3 ) = & \\left \\{ F \\in \\binom { [ n ] } { k } : [ 2 ] \\subseteq F , F \\cap [ 3 , k - 1 ] \\neq \\emptyset \\right \\} \\cup \\left \\{ F \\in \\binom { [ n ] } { k } : [ 2 ] \\cup [ k , k + 3 ] \\subseteq F \\right \\} \\\\ & \\cup \\left \\{ [ 2 , k + 1 ] , \\ , [ 2 , k - 1 ] \\cup \\{ k + 2 , k + 3 \\} , \\ , [ 2 , k ] \\cup \\{ k + 3 \\} , \\ , [ 2 , k + 2 ] \\setminus \\{ k \\} \\right \\} \\\\ & \\cup \\left \\{ [ 3 , k ] \\cup \\{ 1 , k + 2 \\} , \\ , [ 3 , k - 1 ] \\cup \\{ 1 , k + 1 , k + 3 \\} \\right \\} , \\end{aligned} \\end{align*}"}
+{"id": "7214.png", "formula": "\\begin{align*} \\dim U = 1 + d + \\dim B - \\dim = 1 + \\frac { d ( d + 1 ) } { 2 } . \\end{align*}"}
+{"id": "469.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\lfloor { n } / { 2 } \\rfloor } \\binom { n } { 2 k } \\frac { ( 5 F ^ 2 _ j ) ^ k } { 2 k + 1 } \\left ( \\frac { 4 ^ { k + 1 } - 1 } { k + 1 } L _ { j ( n - 2 k ) } { B _ { 2 k + 2 } } - \\frac { L _ j ^ { n - 2 k } } { 2 ^ n } \\right ) = 0 \\ , . \\end{align*}"}
+{"id": "91.png", "formula": "\\begin{align*} \\partial _ { \\vec \\nu } | X | ^ { 2 } & = - 2 \\mathrm { I I } ( X , X ) , \\partial \\O . \\end{align*}"}
+{"id": "5446.png", "formula": "\\begin{align*} B = B _ { \\lambda } \\ , = \\ , \\begin{small} \\begin{pmatrix} 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} . \\end{small} \\end{align*}"}
+{"id": "33.png", "formula": "\\begin{align*} \\left ( \\sqrt { p ' _ j } - \\sqrt { q ' _ j } \\right ) ^ 2 & = q ' _ j \\left ( \\sqrt { 1 + \\frac { p ' _ \\gamma - q ' _ j } { q ' _ j } } - 1 \\right ) ^ 2 \\leq \\frac { ( p ' _ i - q ' _ j ) ^ 2 } { q ' _ j } . \\end{align*}"}
+{"id": "4673.png", "formula": "\\begin{align*} ( - 1 ) ^ \\ell \\omega _ \\ell ^ { 0 + \\cdots + ( \\ell - 1 ) } ( A _ 0 ^ \\ell + A _ 1 ^ \\ell ) = - ( A _ 0 ^ \\ell + A _ 1 ^ \\ell ) . \\end{align*}"}
+{"id": "2839.png", "formula": "\\begin{align*} ( \\omega , s _ 1 , \\cdots , s _ i ) \\circledast ( \\omega ' , s ' _ 1 , \\cdots , s ' _ j ) = ( \\omega \\omega ' , ( \\omega ' ) ^ { - 1 } s _ 1 \\omega ' , \\cdots , ( \\omega ' ) ^ { - 1 } s _ i \\omega ' , s ' _ 1 , \\cdots , s ' _ j ) \\end{align*}"}
+{"id": "2398.png", "formula": "\\begin{align*} \\partial _ { c } ( e _ { a _ { 0 } } e _ { a _ { 1 } } \\cdots e _ { a _ { k } } e _ { a _ { k + 1 } } ) = \\sum _ { r \\in \\{ \\pm 1 \\} } r \\sum _ { i = 1 } ^ { k } { \\rm o r d } _ { z - c } ( a _ { i } - a _ { i + r } ) e _ { a _ { 0 } } \\cdots \\widehat { e _ { a _ { i } } } \\cdots e _ { a _ { k + 1 } } . \\end{align*}"}
+{"id": "3450.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) d \\tau = \\frac { 1 } { 2 } \\Large { \\Gamma } \\Big ( 0 , \\frac { | \\xi - \\mathrm { z } | ^ 2 } { 4 t } \\Big ) . \\end{align*}"}
+{"id": "5540.png", "formula": "\\begin{align*} \\theta _ 2 ( \\infty , t ) = 0 \\end{align*}"}
+{"id": "1397.png", "formula": "\\begin{align*} 0 \\le \\chi ( x ) \\le 1 \\ ( x \\in \\mathbb { R } ^ n ) , \\chi ( x ) = \\begin{cases} 1 & ( | x | \\le 1 ) , \\\\ 0 & ( | x | \\ge 2 ) . \\end{cases} \\end{align*}"}
+{"id": "5977.png", "formula": "\\begin{align*} \\overline { W } ^ { ( q ) } ( x ) = 0 , \\overline { \\overline { W } } ^ { ( q ) } ( x ) = 0 , Z ^ { ( q ) } ( x ) = 1 , \\textrm { a n d } \\overline { Z } ^ { ( q ) } ( x ) = x , x \\leq 0 . \\end{align*}"}
+{"id": "6523.png", "formula": "\\begin{align*} \\cos \\theta = 2 \\mu - 1 . \\end{align*}"}
+{"id": "1488.png", "formula": "\\begin{align*} \\sum _ { p \\le x } g ( p ) \\log p = \\kappa \\log x + O ( 1 ) \\end{align*}"}
+{"id": "2317.png", "formula": "\\begin{align*} \\mathcal L _ T ( b ) = \\frac 1 T \\int _ 0 ^ T \\left ( b ^ \\top ( X _ t ) a _ 0 ^ { - 1 } ( X _ t ) b ( X _ t ) \\d t - 2 b ^ \\top ( X _ t ) a _ 0 ^ { - 1 } ( X _ t ) \\d X _ t \\right ) . \\end{align*}"}
+{"id": "6785.png", "formula": "\\begin{align*} \\big ( 1 + \\abs { k } ^ 2 \\big ) G _ x ( k ) = G _ x ( k ) + ( 2 \\pi ) ^ { - 1 } \\left [ k \\cdot i \\nabla _ x , G _ x ( k ) \\right ] \\end{align*}"}
+{"id": "4688.png", "formula": "\\begin{align*} ( \\lambda + 1 ) K _ { \\lambda } ( z , w ) = r \\frac { d } { d r } \\left [ C ( z , w ) \\right ] + ( \\lambda + 1 ) C ( z , w ) . \\end{align*}"}
+{"id": "6123.png", "formula": "\\begin{align*} \\begin{aligned} \\pi ( \\mathcal F _ 2 ^ { ( 1 ) } ) & = \\{ F \\in \\pi ( \\mathcal F ) : [ d - 1 ] \\subseteq F , F \\cap \\{ d , d + 1 \\} \\neq \\emptyset , d + 2 \\notin F \\} \\\\ & \\cup \\{ F \\in \\pi ( \\mathcal F ) : 1 \\notin F , [ 2 , d - 1 ] \\subseteq F , \\{ d , d + 1 \\} \\subseteq F , d + 2 \\notin F \\} \\end{aligned} \\end{align*}"}
+{"id": "5645.png", "formula": "\\begin{align*} g _ T ( D ^ T _ { \\tilde \\gamma _ s } { \\tilde \\gamma _ s } , T ) | _ { t = 0 } ^ l = 0 . \\end{align*}"}
+{"id": "1176.png", "formula": "\\begin{align*} a = \\Big ( \\frac { 1 - q } { 2 } \\Big ) ^ { k / p _ k } , b = \\Big ( \\frac { q } { 2 } \\Big ) ^ { k / p _ k } . \\end{align*}"}
+{"id": "3906.png", "formula": "\\begin{align*} S ( t ) \\xi = \\sum _ { n = 1 } ^ \\infty \\omega ( t , \\lambda _ n ) \\xi _ n e _ n , \\ ; \\xi \\in L ^ 2 ( \\Omega ) . \\end{align*}"}
+{"id": "6826.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\alpha _ n ^ i \\partial _ t ^ { i + n + 1 } u _ { n } + \\partial _ t ^ { n + 1 } u _ { n } - \\sum _ { i = 1 } ^ { 2 n } \\beta _ n ^ i \\Delta \\partial _ t ^ i u _ n - \\kappa _ n \\Delta u _ n = 0 . \\end{align*}"}
+{"id": "1475.png", "formula": "\\begin{align*} C _ R ^ { [ 1 ] } & = \\frac { \\gamma _ r \\times N } { \\gamma _ r \\times \\ell _ r ^ * } = \\frac { N } { \\ell _ r ^ * } , C _ W ^ { [ 1 ] } = \\frac { N } { \\ell _ w ^ * } . \\end{align*}"}
+{"id": "4404.png", "formula": "\\begin{align*} P ^ \\beta _ \\alpha = \\{ ( x , y ) ; x \\geq 0 , \\alpha x \\leq y \\leq \\beta x \\} , ~ \\alpha , \\beta \\in \\R \\cup \\{ - \\infty , \\infty \\} , \\alpha < \\beta . \\end{align*}"}
+{"id": "4798.png", "formula": "\\begin{align*} \\frac { a ( n , k ) } { \\alpha ( n , k ) } - \\frac { a ( n , - c ) \\alpha ( n + i - 1 , - c ) } { \\alpha ( n , - c ) \\alpha ( n + i - 1 , k ) } = ( k + c ) { B _ i ( n , c ) } , \\end{align*}"}
+{"id": "7029.png", "formula": "\\begin{align*} \\widehat { H } _ 0 : = \\frac { 2 \\sigma + 1 } { 8 \\sigma ^ 2 } | \\xi | ^ 4 t \\cos ( | \\xi | ^ 2 t ) \\mathrm { e } ^ { - \\frac { 1 } { 2 \\sigma } | \\xi | ^ 4 t } \\ \\ \\mbox { a n d } \\ \\ \\widehat { H } _ 1 : = \\frac { 1 } { 2 \\sigma ^ 2 } | \\xi | ^ 4 t \\sin ( | \\xi | ^ 2 t ) \\mathrm { e } ^ { - \\frac { 1 } { 2 \\sigma } | \\xi | ^ 4 t } . \\end{align*}"}
+{"id": "3124.png", "formula": "\\begin{align*} F _ { X _ 2 } ( \\sigma _ 2 ^ k ) = 0 \\end{align*}"}
+{"id": "7430.png", "formula": "\\begin{align*} [ p , \\ , H ] = P p . \\end{align*}"}
+{"id": "4160.png", "formula": "\\begin{align*} R c ^ { \\pm \\phi } = R c \\mp \\dfrac { 1 } { 2 } d ^ * \\phi - \\dfrac { 1 } { 4 } \\phi ^ 2 \\end{align*}"}
+{"id": "5945.png", "formula": "\\begin{align*} M _ a ^ { - T } = M _ b ^ { - T } \\left ( \\begin{array} { c c c c c } 1 & O ( \\delta ) & O ( \\delta ^ 2 ) & \\dots & O ( \\delta ^ { k - 1 } ) \\\\ 0 & 1 & O ( \\delta ) & \\dots & O ( \\delta ^ { k - 2 } ) \\\\ 0 & 0 & 1 & \\dots & O ( \\delta ^ { k - 3 } ) \\\\ \\vdots & & & \\ddots & \\\\ 0 & 0 & 0 & \\dots & 1 \\end{array} \\right ) , \\end{align*}"}
+{"id": "6115.png", "formula": "\\begin{align*} | \\mathcal S _ 2 ( k , 3 ) | = | \\mathcal S _ 3 ( k , 3 ) | & = \\binom { n - 2 } { k - 2 } - \\binom { n - k + 1 } { k - 2 } + \\binom { n - k - 3 } { k - 6 } + 6 \\sim ( k - 3 ) \\binom { n } { k - 3 } . \\end{align*}"}
+{"id": "6998.png", "formula": "\\begin{align*} \\Delta ( \\phi ( u ) ) = \\phi ' ( u ) \\Delta u + \\phi '' ( u ) | \\nabla u | ^ 2 = \\frac { 1 } { u } \\Delta u - \\frac { 1 } { u ^ 2 } | \\nabla u | ^ 2 . \\end{align*}"}
+{"id": "4342.png", "formula": "\\begin{align*} H _ \\eta ( u _ 1 ( t ) | u _ 2 ( t ) ) & : = \\int _ \\Omega \\bigg [ f _ 1 ( t ) \\ln \\Big ( \\frac { f _ 1 ( t ) + \\eta } { f _ 2 ( t ) } \\Big ) - ( f _ 1 ( t ) - f _ 2 ( t ) ) \\bigg ] \\ , \\mathrm { d } x \\\\ [ 1 e x ] & \\quad + \\cfrac { b } { c } \\int _ \\Omega \\bigg [ g _ 1 ( t ) \\ln \\Big ( \\frac { g _ 1 ( t ) + \\eta } { g _ 2 ( t ) } \\Big ) - ( g _ 1 ( t ) - g _ 2 ( t ) ) \\bigg ] \\ , \\mathrm { d } x , t \\in [ 0 , T ] \\ , . \\end{align*}"}
+{"id": "1179.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\deg _ p ( N _ k ' ) = \\lim _ { k \\to \\infty } \\deg _ p ( N _ k '' ) . \\end{align*}"}
+{"id": "2779.png", "formula": "\\begin{align*} A A ^ T u _ i = \\sigma _ i ^ 2 u _ i , \\end{align*}"}
+{"id": "6775.png", "formula": "\\begin{align*} i \\left [ \\mathcal { N } , H ^ { \\rm { B } } ( t ) \\right ] = 2 i \\int d x \\int d k \\ , K ( t , k , x ) a _ k ^ * b _ x ^ * + . \\end{align*}"}
+{"id": "6445.png", "formula": "\\begin{align*} W _ n ( a , b , u , v ) = \\sum _ { k = 0 } ^ n \\begin{bmatrix} n \\\\ k \\end{bmatrix} \\frac { ( a v , b v ; q ) _ k } { ( a b u v ; q ) _ k } u ^ k v ^ { n - k } . \\end{align*}"}
+{"id": "5282.png", "formula": "\\begin{align*} [ 1 ] : = \\{ 1 , \\ldots , n + 1 \\} , \\ [ 2 ] : = \\{ n + 2 , \\ldots , 2 n + 2 \\} , \\ldots , [ m ] = \\{ ( m - 1 ) \\ * ( n + 1 ) + 1 , \\ldots , m \\ * ( n + 1 ) \\} . \\end{align*}"}
+{"id": "7820.png", "formula": "\\begin{align*} J ^ { w _ 1 } ( \\textbf { X } _ { S R S } ^ { ( n ) } ) & = \\frac { - 1 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { - \\infty } ^ { \\infty } w _ 1 ( x _ i ) f ^ 2 ( x _ i ) d x _ i \\right ) \\\\ & = \\frac { - 1 } { 2 } \\left ( - 2 J ^ { w _ 1 } ( X ) \\right ) ^ n \\\\ & = \\frac { - 1 } { 2 } \\left ( E ( \\Lambda _ X ^ { w _ 1 } ( U ) ) \\right ) ^ n \\end{align*}"}
+{"id": "8231.png", "formula": "\\begin{align*} I _ 2 ^ n = \\big \\langle [ \\Lambda ^ \\alpha , \\chi _ { k + 2 } ( u ^ n - u ) ] \\ , \\big ( \\chi _ k h ( \\chi _ { k + 2 } \\sigma ^ n ) \\big ) , \\phi \\big \\rangle + \\big \\langle [ \\Lambda ^ \\alpha , \\chi _ { k + 2 } u ] \\ , \\big ( \\chi _ k ( h ( \\chi _ { k + 2 } \\sigma ^ n ) - h ( \\chi _ { k + 2 } \\sigma ) ) \\big ) , \\phi \\rangle . \\end{align*}"}
+{"id": "41.png", "formula": "\\begin{align*} f ( n , \\phi , \\tilde { p } , \\tilde { q } ) : = \\P _ { ( x _ 1 , \\dots , x _ n ) \\sim \\tilde { p } ^ { \\otimes n } } ( \\phi ( y _ 1 , \\dots , y _ n ) \\ne p ) + \\P _ { ( x _ 1 , \\dots , x _ n ) \\sim \\tilde { q } ^ { \\otimes n } } ( \\phi ( y _ 1 , \\dots , y _ n ) \\ne q ) . \\end{align*}"}
+{"id": "6188.png", "formula": "\\begin{align*} A _ 1 = \\sum ^ m _ { \\stackrel { i = 0 } { i \\ e v e n } } ( M ^ { 0 , 0 } _ { \\frac { i } { 2 } , \\frac { 2 m - i } { 2 } } + M ^ { 0 , 0 } _ { \\frac { 2 m - i } { 2 } , \\frac { i } { 2 } } ) + \\sum ^ m _ { \\stackrel { i = 0 } { i \\ o d d } } ( M ^ { 0 , 0 } _ { \\frac { 2 m - i - 1 } { 2 } , \\frac { i - 1 } { 2 } } + M ^ { 0 , 0 } _ { \\frac { i - 1 } { 2 } , \\frac { 2 m - i - 1 } { 2 } } ) + M ^ { 0 , 0 } _ { \\lfloor \\frac { m } { 2 } \\rfloor , \\lfloor \\frac { m } { 2 } \\rfloor } \\end{align*}"}
+{"id": "6789.png", "formula": "\\begin{align*} \\phi ( x _ { [ k ] } ) = \\sum _ { i \\in [ r ' ] } \\beta ' _ { i , 1 } ( x _ { I ' _ { i , 1 } } ) \\dots \\beta ' _ { i , d ' _ i } ( x _ { I ' _ { i , d ' _ i } } ) \\end{align*}"}
+{"id": "3104.png", "formula": "\\begin{align*} \\sigma _ 1 ( x ) = \\begin{cases} x + 1 & \\\\ x - 1 & \\end{cases} \\end{align*}"}
+{"id": "7298.png", "formula": "\\begin{align*} \\Upsilon ( t , \\omega ) - \\overline { \\Upsilon } ( t ) = - P _ 1 ( t ) [ X ^ * ( t , \\omega ) - \\overline { X } ^ * ( t ) ] , \\end{align*}"}
+{"id": "4137.png", "formula": "\\begin{align*} \\mathcal { W } ( K _ n ^ { \\alpha } , H ^ { \\alpha } _ n ) ( y ) = \\frac { - ( 2 n + \\alpha + 1 ) } { y ^ { 2 \\alpha + 1 } ( 1 - y ^ 2 ) } . \\end{align*}"}
+{"id": "8052.png", "formula": "\\begin{align*} { \\widetilde { \\bf s } } _ { \\textrm { i n d } } [ f _ k ] = \\sum _ { m = 1 } ^ { M } { \\widetilde { \\bf s } } _ { \\textrm { i n d } , m } [ f _ k ] = { \\bf A } _ { \\textrm { i n d } , k } ( { \\bf \\Phi } ) { \\bf F } _ k { \\bf s } _ k , \\end{align*}"}
+{"id": "2272.png", "formula": "\\begin{align*} \\sigma ( z _ i ) & = \\alpha _ i z _ 1 ^ { M [ 1 , i ] } z _ 2 ^ { M [ 2 , i ] } \\cdots z _ n ^ { M [ n , i ] } = \\alpha _ i z ^ { M [ - , i ] } , \\ \\ i = 1 , 2 , \\dots , n \\end{align*}"}
+{"id": "1038.png", "formula": "\\begin{align*} \\dot { x } = - \\nabla f ( x ) = - x ^ 3 , x ( 0 ) = 1 . \\end{align*}"}
+{"id": "4527.png", "formula": "\\begin{align*} \\big \\{ h _ \\gamma ( \\lambda _ 2 , \\lambda _ 1 ) , ^ 2 \\ ! E _ S [ f ] \\big \\} \\ ! = \\ ! \\ ! \\int _ { x \\in S } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\mathrm { d } S ^ { b c } ( \\mathbf { x } ) f ^ k ( \\mathbf { x } ) v _ { a b c } ( \\mathbf { x } ) \\ ! \\int _ { \\lambda _ 1 } ^ { \\lambda _ 2 } \\ ! \\ ! \\ ! \\mathrm { d } t \\dot { \\gamma } ^ a ( t ) \\mathcal { E } ^ { ( 3 ) } _ { \\ ; \\ , \\gamma ( t ) } ( \\mathbf { x } ) h _ \\gamma ( \\lambda _ 2 , t ) \\frac { \\tau _ k } { 2 } h _ \\gamma ( t , \\lambda _ 1 ) \\ , . \\end{align*}"}
+{"id": "1340.png", "formula": "\\begin{align*} \\int _ { B _ 1 } \\left | u ( x ) - \\ell ( x ) \\right | ^ { \\frac { n ^ 2 } { n - 1 } } \\ , d x \\le C \\left \\| \\nabla u - q \\right \\| _ { L ^ n ( B _ 1 ) } ^ { \\frac { n ^ 2 } { n - 1 } } \\le C \\varepsilon ^ { \\frac { n ^ 2 } { n - 1 } } a ^ { \\frac { n ^ 2 } { n - 1 } } , \\end{align*}"}
+{"id": "4156.png", "formula": "\\begin{align*} \\varphi _ { ( x _ 1 , \\ldots , x _ k ) } : H ^ 0 ( E ) & \\longrightarrow \\bigoplus \\limits _ { i = 1 } ^ k E _ { x _ i } \\\\ s & \\longmapsto ( s ( x _ 1 ) , \\ldots , s ( x _ k ) ) \\end{align*}"}
+{"id": "3445.png", "formula": "\\begin{align*} \\mathbb { M } \\Big [ \\nabla \\mathrm { H } \\Big ] ( \\mathrm { x } ) = \\nabla \\int _ { \\Omega } \\nabla \\mathbb { G } ^ { ( 0 ) } ( \\mathrm { x } , \\mathrm { y } ) \\cdot \\nabla \\mathrm { H } ( \\mathrm { y } ) d \\mathrm { y } . \\end{align*}"}
+{"id": "4917.png", "formula": "\\begin{align*} 2 j \\binom { n } { r } + \\sum _ { i = r + 1 } ^ { k - 1 } \\frac { j - 1 } { 2 } \\binom { n } { i } < \\binom { n } { k } . \\end{align*}"}
+{"id": "732.png", "formula": "\\begin{align*} \\delta z = - \\frac { 1 } { 8 } { F _ z } . \\end{align*}"}
+{"id": "2045.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\lambda _ { n + 1 } w ^ n = \\frac { 1 } { ( 1 - w ) ^ 2 } \\ , \\frac { \\xi ' } { \\xi } \\left ( \\frac { 1 } { 1 - w } \\right ) . \\end{align*}"}
+{"id": "1459.png", "formula": "\\begin{align*} \\partial _ t \\Phi _ { \\beta , \\varepsilon } ( t , x ; t _ 0 ) & = - ( t _ 0 + t ) ^ { - \\beta - 1 } \\left [ \\beta \\varphi _ { \\beta , \\varepsilon } ( z ) + z \\varphi _ { \\beta , \\varepsilon } ' ( z ) \\right ] \\\\ & = - ( t _ 0 + t ) ^ { - \\beta - 1 } \\beta \\varphi _ { \\beta + 1 , \\varepsilon } ( z ) \\\\ & = - \\beta \\Phi _ { \\beta + 1 , \\varepsilon } ( t , x ; t _ 0 ) , \\end{align*}"}
+{"id": "5700.png", "formula": "\\begin{gather*} \\psi _ 1 \\iota ^ * ( \\beta ^ O _ { 1 , 3 , 4 } ) = - 2 ( n + 1 ) e _ n \\otimes ( e _ 1 ^ * \\wedge e _ 2 ^ * ) \\otimes e _ 1 ^ * \\otimes e _ 1 ^ * \\in V _ { 1 , 1 ^ 4 } . \\end{gather*}"}
+{"id": "3868.png", "formula": "\\begin{align*} - \\frac { m ^ z } { n ^ 2 } \\sum _ { p + q + 1 \\geq 3 } \\left ( \\cdots \\right ) - \\frac { \\mathfrak { m } ^ { z } } { n ^ 2 } \\sum _ { p + q + 1 \\geq 3 } \\left ( \\cdots \\right ) = O _ \\prec ( n ^ { - 1 / 2 } \\Psi ^ 3 + n ^ { - 3 / 2 } ) . \\end{align*}"}
+{"id": "5418.png", "formula": "\\begin{align*} \\frac { \\partial G _ { i j } } { \\partial h _ { a b } } = - \\frac { G _ { i a } G _ { b j } + G _ { i b } G _ { a j } } { 1 + \\delta _ { a b } } . \\end{align*}"}
+{"id": "4657.png", "formula": "\\begin{align*} \\Theta _ { f , D } = \\Theta _ { w _ N f , D } ^ t N . \\end{align*}"}
+{"id": "3808.png", "formula": "\\begin{align*} a _ 1 = \\frac { 3 n - 9 - d } { 2 } . \\end{align*}"}
+{"id": "1837.png", "formula": "\\begin{align*} L ( f , s ) = \\sum _ { n = 1 } ^ \\infty \\frac { f ( n ) } { n ^ s } . \\end{align*}"}
+{"id": "4428.png", "formula": "\\begin{align*} W ( v ) ( x ) = \\sum _ { \\xi \\in \\Z ^ d } \\mathbb { W } ( \\xi ) ( \\hat { v } ( \\xi ) ) e ^ { - 2 \\pi i \\xi \\cdot x } , \\end{align*}"}
+{"id": "417.png", "formula": "\\begin{align*} N _ l ( H , M , N , D , S , U ) = \\frac { \\varphi ( H ) } { \\varphi ( M ) \\varphi ( N ) } \\displaystyle \\sum _ { J | M } \\mu ( J ) \\displaystyle \\sum _ { I | N } \\mu ( I ) N _ { l - 1 } ( H , J , I , D , S , U ) . \\end{align*}"}
+{"id": "4889.png", "formula": "\\begin{align*} f _ j \\left ( 2 - \\frac { 1 } { x } , 2 - \\frac { 1 } { x } \\right ) & = \\frac { ( 1 - x ) ^ 2 } { ( 1 - 2 x ) ( 2 - x ) } f _ j \\left ( 3 - \\frac { 1 } { x } , 2 - \\frac { 1 } { x } \\right ) \\\\ & - \\frac { 1 - x } { 2 - x } \\left ( f _ { j + 1 } \\left ( 1 - \\frac { 1 } { x } , 2 - \\frac { 1 } { x } \\right ) - f _ j \\left ( 1 - \\frac { 1 } { x } , 2 - \\frac { 1 } { x } \\right ) \\right ) . \\end{align*}"}
+{"id": "2628.png", "formula": "\\begin{align*} \\mu | _ { E _ { 2 , m + e _ i - e _ j } } = \\lambda ^ j = \\lambda _ x | _ { E _ { 2 , m + e _ i - e _ j } } \\end{align*}"}
+{"id": "5726.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } x _ i x _ j \\quad g ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } B _ { i j } x _ i x _ j , \\end{align*}"}
+{"id": "4985.png", "formula": "\\begin{align*} \\frac { \\partial R _ { \\rm F A } } { \\partial r } = \\frac { w _ { c } ^ { 2 } ( 1 - \\lambda ) ^ { w _ { c } / r } \\log ( 1 - \\lambda ) } { \\lambda r ^ { 2 } } ( 1 - ( 1 - \\lambda ) ^ { w _ { c } / r - 1 } ) ^ { w _ { c } - 1 } . \\end{align*}"}
+{"id": "6295.png", "formula": "\\begin{align*} \\psi _ { y } ( \\xi _ { 1 , 0 } ) = \\prod _ { j = 0 } ^ \\infty \\left ( \\prod _ { i = 0 } ^ { c - 1 } ( 1 - \\frac { \\theta ^ { q ^ i } } { \\theta ^ { q ^ { c + \\ell j } } } ) \\cdot \\prod _ { i = c } ^ { \\ell - 1 } ( 1 - \\frac { \\theta ^ { q ^ i } } { \\theta ^ { q ^ { c + \\ell + \\ell j } } } ) \\right ) ^ { - 1 } . \\end{align*}"}
+{"id": "7924.png", "formula": "\\begin{align*} \\mu _ t \\big ( \\mu _ t ( a , b ) , c \\big ) = ~ & \\mu _ t \\big ( a , \\mu _ t ( b , c ) \\big ) , \\\\ \\mu _ t \\big ( R _ t ( a ) , R _ t ( b ) \\big ) = ~ & R _ t \\big ( \\mu _ t ( R _ t ( a ) , b ) + \\mu _ t ( a , R _ t ( b ) ) \\big ) + \\kappa ~ \\mu _ t ( a , b ) , \\end{align*}"}
+{"id": "2312.png", "formula": "\\begin{align*} \\mathbb { G } _ { n , \\Delta } ( f ) \\coloneqq \\frac { 1 } { \\sqrt { n \\Delta } } \\sum _ { k = 1 } ^ n f ( X _ { k \\Delta } ) \\Delta , \\end{align*}"}
+{"id": "3573.png", "formula": "\\begin{align*} P _ { I , N } ( z ) = \\frac { \\ell ( N / I N ) + ( 1 + b - \\ell ( N / I N ) ) z + \\sum _ { i = 2 } ^ { e _ 0 - b } z ^ i } { 1 - z } . \\end{align*}"}
+{"id": "7511.png", "formula": "\\begin{align*} E _ \\varphi ( F _ { \\varepsilon , h } ) - E _ \\varphi [ \\mu ^ F _ { \\varepsilon , h } ] = o ( \\varepsilon ) \\varepsilon \\to 0 , \\end{align*}"}
+{"id": "6314.png", "formula": "\\begin{align*} \\P \\big ( \\| X \\| _ { \\infty } \\vee \\| X ^ k \\| _ { \\infty } \\geq n \\big ) & \\leq \\P \\Big ( \\lbrace \\| X \\| _ { \\infty } \\geq n \\rbrace \\cup \\lbrace \\| X ^ k - X \\| _ { \\infty } + \\| X \\| _ { \\infty } \\geq n \\rbrace \\Big ) \\\\ & \\leq \\P \\big ( \\| X \\| _ { \\infty } \\geq n / 2 \\big ) + \\P \\big ( \\| X ^ k - X \\| _ { \\infty } \\geq n / 2 \\big ) \\\\ & \\leq \\delta / 4 + \\delta / 4 = \\delta / 2 . \\end{align*}"}
+{"id": "5663.png", "formula": "\\begin{align*} T ^ \\alpha = V ^ { \\alpha + n } , \\ \\ T ^ { \\alpha + n } = - V ^ \\alpha , \\ \\ T _ o ^ \\alpha = T ^ \\alpha + \\sqrt { - 1 } T ^ { \\alpha + n } . \\end{align*}"}
+{"id": "2698.png", "formula": "\\begin{align*} \\binom { a + j } { j } \\sum _ { l = 0 } ^ { a } \\binom { n - j + l } { l } \\sum _ { k = a + j - l } ^ { n - ( a + j - l ) } \\binom { n - j } { k - j } \\binom { n - k } { a + j - l } \\binom { k - j } { a - l } G ( n , k , a ) \\end{align*}"}
+{"id": "5023.png", "formula": "\\begin{align*} h _ M ( \\Omega ) = \\lim _ k h _ M ( \\Omega _ k ) . \\end{align*}"}
+{"id": "954.png", "formula": "\\begin{align*} m : = b _ { n - 1 } \\cdots b _ { \\alpha + 1 } b _ \\alpha ^ { k + \\alpha - n } . \\end{align*}"}
+{"id": "7271.png", "formula": "\\begin{align*} X ( t , \\omega ) = X ( 0 , \\omega ) + \\int _ { 0 } ^ t f ( \\tau , X ( \\tau , \\omega ) , e _ \\tau \\sharp \\eta , u _ L ( \\tau , \\omega ) ) d \\tau . \\end{align*}"}
+{"id": "7200.png", "formula": "\\begin{align*} ( \\alpha \\colon \\mathcal { F } \\rightleftarrows \\mathcal { G } \\colon \\beta ) , \\ \\alpha \\circ \\beta = \\beta \\circ \\alpha = \\times f , \\end{align*}"}
+{"id": "6072.png", "formula": "\\begin{align*} \\rho _ D ( R ) = 7 | R | - 3 m ( D [ R ] ) - 2 \\pi ( D [ R ] ) \\end{align*}"}
+{"id": "7568.png", "formula": "\\begin{align*} \\sup _ { | t | \\leq \\varepsilon } \\sup _ { p , q \\in K } | H ( \\Phi ( t , p ) , \\Phi ( t , q ) ) - H ( p , q ) | = o ( 1 ) \\end{align*}"}
+{"id": "4671.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n _ q - 1 } q ^ k s _ k \\equiv 0 \\bmod { \\ell } . \\end{align*}"}
+{"id": "4174.png", "formula": "\\begin{align*} \\partial ' _ { N } P ( n ) & = \\partial ' _ { N _ 0 N _ 1 } P \\big ( ( Q ' ! - 1 ) N _ 0 N _ 1 + n \\big ) \\cdots \\partial ' _ { N _ 0 N _ 1 } P ( n ) \\\\ & = \\rho \\big ( ( Q ' ! - 1 ) N _ 0 N _ 1 + n \\big ) \\cdots \\rho ( n ) \\cdot q \\big ( ( Q ' ! - 1 ) N _ 0 N _ 1 + n \\big ) \\cdots q ( n ) \\\\ & = \\rho \\big ( ( Q ' ! - 1 ) N _ 0 N _ 1 + n \\big ) \\cdots \\rho ( n ) \\cdot q ( n ) ^ { Q ' ! } \\gamma . \\end{align*}"}
+{"id": "6333.png", "formula": "\\begin{align*} \\widetilde { \\Gamma } ( ( M \\cap S _ \\rho ) \\times [ 0 , 1 ] ) = C _ \\rho \\cup \\Gamma ( A \\times ( 0 , 1 ] ) , \\end{align*}"}
+{"id": "7557.png", "formula": "\\begin{align*} ( \\psi _ j ) _ + = ( \\psi _ j ) _ - + f _ j \\frac { \\lambda ^ L } { z ^ { M + N } } \\gamma _ X . \\end{align*}"}
+{"id": "6033.png", "formula": "\\begin{align*} X _ 3 ^ 2 + X _ 1 X _ 5 + X _ 2 X _ 4 = 0 . \\end{align*}"}
+{"id": "2204.png", "formula": "\\begin{align*} & r _ 1 = - ( R _ { 1 2 1 2 } R _ { 1 3 1 3 } - R _ { 1 2 1 3 } { } ^ 2 ) S _ { 1 2 2 3 2 } , \\\\ & r _ 2 = - ( R _ { 1 2 1 2 } R _ { 1 3 1 3 } - R _ { 1 2 1 3 } { } ^ 2 ) ( S _ { 1 2 2 3 3 } + S _ { 1 3 2 3 2 } ) , \\\\ & r _ 4 = - ( R _ { 1 2 1 2 } R _ { 1 3 1 3 } - R _ { 1 2 1 3 } { } ^ 2 ) S _ { 1 3 2 3 3 } . \\end{align*}"}
+{"id": "8144.png", "formula": "\\begin{align*} \\widehat { \\mu } ^ { \\mathrm { a s y } } ( \\overline L ) = \\frac { ( \\overline L ^ { d + 1 } ) _ S } { ( d + 1 ) ( L ^ d ) } \\leqslant \\widehat { \\mu } _ { \\max } ^ { \\mathrm { a s y } } ( \\overline L ) . \\end{align*}"}
+{"id": "5345.png", "formula": "\\begin{align*} \\pi \\omega : = \\omega - d \\Gamma \\omega - \\Gamma d \\omega \\end{align*}"}
+{"id": "6154.png", "formula": "\\begin{align*} V _ 1 ^ * V _ 2 & = ( R _ { \\overline q } \\otimes U ^ * P ^ \\perp + R _ { \\overline q } M _ z ^ * \\otimes U ^ * P ) ( R _ { \\overline q } \\otimes U ^ * P + R _ { \\overline q } M _ z \\otimes U ^ * P ^ \\perp ) \\\\ & = R _ { \\overline q } ^ 2 \\otimes U ^ * P ^ \\perp U ^ * P + R _ { \\overline q } ^ 2 M _ z \\otimes U ^ * P ^ \\perp U ^ * P ^ \\perp + R _ { \\overline q } M _ z ^ * R _ { \\overline q } \\otimes U ^ * P U ^ * P \\\\ & + R _ { \\overline q } M _ z ^ * R _ { \\overline q } M _ z \\otimes U ^ * P U ^ * P ^ \\perp . \\end{align*}"}
+{"id": "1520.png", "formula": "\\begin{align*} y ^ 2 = \\tau ^ 4 + 6 \\tau ^ 3 + 7 \\tau ^ 2 + 2 \\tau + 1 . \\end{align*}"}
+{"id": "3969.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\mathrm { d } } { \\mathrm { d } t } q ( 0 , t ) & = - \\alpha \\left ( e ^ { \\theta } - 1 \\right ) q ( 0 , t ) , \\\\ \\frac { \\mathrm { d } } { \\mathrm { d } t } q ( n , t ) & = - \\alpha \\left ( e ^ { \\theta } - 1 \\right ) q ( n , t ) + \\alpha \\sum _ { j = 1 } ^ { n } \\frac { \\theta ^ { j } } { j ! } q ( n - j , t ) , \\ n \\ge 1 , \\end{aligned} \\end{align*}"}
+{"id": "6373.png", "formula": "\\begin{align*} \\tilde { X } _ \\cdot ^ h : = \\mathcal { G } _ \\mu \\left ( B _ \\cdot ^ H + ( R _ H h ) ( \\cdot ) \\right ) , \\end{align*}"}
+{"id": "7077.png", "formula": "\\begin{align*} f _ X ( x , y ) : = y ^ r + A _ { 1 } ( x ) y ^ { r - 1 } + A _ { 2 } ( x ) y ^ { r - 2 } + \\cdots + A _ { r - 1 } ( x ) y + A _ { r } ( x ) , \\end{align*}"}
+{"id": "7365.png", "formula": "\\begin{align*} ~ \\hat { \\beta } _ { k + 1 } ^ \\mathrm { H Z } : = \\max \\{ \\beta _ { k + 1 } ^ \\mathrm { H Z } , \\zeta _ { k + 1 } \\} , \\zeta _ { k + 1 } : = - \\frac { 1 } { \\norm { \\eta _ { k + 1 } } _ { x _ { k + 1 } } \\min \\{ \\zeta , \\norm { g _ { k + 1 } } _ { x _ { k + 1 } } \\} } , \\end{align*}"}
+{"id": "3714.png", "formula": "\\begin{align*} I ( b [ \\rho , v ] ) = \\hat F _ \\infty ( \\beta ^ { b [ \\rho , v ] } ) \\end{align*}"}
+{"id": "7816.png", "formula": "\\begin{align*} J ^ { w _ 1 } ( X ) & = - \\frac { 1 } { 2 } E \\left ( \\Lambda _ X ^ { w _ 1 } ( U ) \\right ) \\\\ & \\le - \\frac { 1 } { 2 } E \\left ( \\Lambda _ Y ^ { w _ 2 } ( U ) \\right ) \\\\ & = J ^ { w _ 2 } ( Y ) , \\end{align*}"}
+{"id": "4720.png", "formula": "\\begin{gather*} l ( a ) l ( b ) v = l ( a \\cdot b ) v , \\qquad \\ ! \\ ! \\ ! r ( b ) r ( a ) v = r ( a \\cdot b ) v , \\qquad \\ ! \\ ! \\ ! r ( b ) l ( a ) v = l ( a ) r ( b ) v , \\qquad \\ ! \\ ! \\ ! \\forall a , b \\in A , \\ , v \\in V . \\ ! \\end{gather*}"}
+{"id": "3765.png", "formula": "\\begin{align*} b ( \\Gamma / e ) = b ( \\Gamma ) - 1 . \\end{align*}"}
+{"id": "3526.png", "formula": "\\begin{align*} \\nabla \\Big ( \\mathbb { G } ^ { ( \\mathrm { k } ) } - \\mathbb { G } ^ { ( 0 ) } \\Big ) ( \\mathrm { x } , \\mathrm { y } ) = \\mathrm { C } | \\mathrm { x } - \\mathrm { y } | \\log { | \\mathrm { x } - \\mathrm { y } | } + \\mathrm { D } | \\mathrm { x } - \\mathrm { y } | + \\mathcal { O } ( | \\mathrm { x } - \\mathrm { y } | ^ 2 ) , \\end{align*}"}
+{"id": "304.png", "formula": "\\begin{align*} f \\colon X ' = X _ { s } \\xrightarrow { f _ { s } } X _ { s - 1 } \\xrightarrow { f _ { s - 1 } } \\cdots \\xrightarrow { f _ { 1 } } X _ { 0 } = X \\end{align*}"}
+{"id": "4291.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { n c ^ n q ^ { n ^ 2 } } { ( q ) _ n ( c q ) _ n } - \\sum _ { k = 1 } ^ { \\infty } \\frac { ( - c ) ^ k q ^ { \\frac { k ( k + 1 ) } { 2 } } } { ( q ) _ k ( 1 - q ^ k ) } \\sum _ { j = 0 } ^ { \\infty } \\frac { c ^ j q ^ { ( j + k ) ^ 2 } } { ( c q ) _ { j + k } ( q ) _ j } = \\frac { 1 } { ( c q ) _ { \\infty } } - 1 - \\frac { 1 } { ( c q ) _ { \\infty } } \\sum _ { k = 1 } ^ { \\infty } \\frac { ( - c ) ^ k q ^ { \\frac { k ( k + 3 ) } { 2 } } } { ( q ) _ k ( 1 - q ^ k ) } . \\end{align*}"}
+{"id": "1121.png", "formula": "\\begin{align*} L _ { n } ( x | \\beta ) = \\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \\frac { ( \\beta ) ^ { ( n ) } } { ( n - k ) ! ( \\beta ) ^ { ( k ) } } x ^ { k } / k ! . \\end{align*}"}
+{"id": "691.png", "formula": "\\begin{align*} \\underline { u } = \\frac { 1 } { 4 } \\left ( \\sum _ { j = 1 } ^ p e _ j \\cdot u ( e _ j ) + \\sum _ { j = p + 1 } ^ n e _ j \\cdot ( - u ^ * ( e _ j ) ) \\right ) . \\end{align*}"}
+{"id": "3582.png", "formula": "\\begin{align*} k ( x , y ) = \\sum _ { i \\geq 1 } a _ i \\phi _ i ( x ) \\phi _ i ( y ) , c ( x , y ) = \\sum _ { m \\geq 1 } b _ m \\psi _ m ( x ) \\psi _ m ( x ) , \\end{align*}"}
+{"id": "6710.png", "formula": "\\begin{align*} \\partial _ { t } G + P _ { 1 } ( v \\cdot \\nabla _ { x } G ) + P _ { 1 } ( v \\cdot \\nabla _ { x } M ) - ( E + v \\times B ) \\cdot \\nabla _ { v } G = \\frac { 1 } { \\varepsilon } L _ { M } G + \\frac { 1 } { \\varepsilon } Q ( G , G ) . \\end{align*}"}
+{"id": "6892.png", "formula": "\\begin{align*} C ( z ) & = \\begin{pmatrix} 0 & 1 \\\\ - \\mathfrak a ^ 2 ( L , s ) & \\frac { z + s + 1 } L \\end{pmatrix} \\\\ & \\qquad + \\sum _ { l \\in \\Z ' } \\frac { \\Delta \\sigma ( l ) } { z - l } \\begin{pmatrix} \\mathfrak a ( L , s ) \\varphi ( l + 1 ; L , s - 1 ) \\varphi ( l ; L , s ) & - \\varphi ( l + 1 ; L , s - 1 ) \\varphi ( l ; L , s - 1 ) \\\\ \\mathfrak a ^ 2 ( L , s ) \\varphi ( l + 1 ; L , s ) \\varphi ( l ; L , s ) & - \\mathfrak a ( L , s ) \\varphi ( l + 1 ; L , s ) \\varphi ( l ; L , s - 1 ) \\end{pmatrix} \\end{align*}"}
+{"id": "7181.png", "formula": "\\begin{align*} \\chi _ 1 + \\chi _ 2 + \\rho & = ( \\chi _ 1 + \\rho _ 1 ) + ( \\chi _ 2 + \\rho _ 2 ) + ( \\rho - \\rho _ 1 - \\rho _ 2 ) \\\\ & \\in \\textbf { W } ( d _ 1 ) _ { w _ 1 } + \\textbf { W } ( d _ 2 ) _ { w _ 2 } + \\frac { 1 } { 2 } \\mathfrak { g } ^ { \\nu < 0 } \\\\ & = \\textbf { W } ( d _ 1 ) _ { v _ 1 } + \\textbf { W } ( d _ 2 ) _ { v _ 2 } + \\frac { 3 } { 2 } \\mathfrak { g } ^ { \\nu < 0 } . \\end{align*}"}
+{"id": "310.png", "formula": "\\begin{align*} \\lambda _ { x } = \\sum _ { i = 1 } ^ { s } \\mu _ { i } , \\end{align*}"}
+{"id": "891.png", "formula": "\\begin{align*} S ^ { p q , r s } _ k = \\frac { \\partial ^ 2 S _ k [ D ^ 2 u ] } { \\partial r _ { p q } \\partial r _ { r s } } . \\end{align*}"}
+{"id": "5098.png", "formula": "\\begin{align*} E f = \\int _ { B _ 1 ^ { d - n } } \\sum _ { m \\in N ^ { - 1 } \\mathbb { Z } ^ n \\cap B _ 1 ^ { n } } f ( \\xi , m ) e ^ { i ( x \\cdot \\xi + y \\cdot m + t ( \\frac { \\mathfrak { m } ( N \\xi ) } { N ^ 2 \\mathfrak { m } ( N ) } + \\frac { \\mathfrak { m } ( N m ) } { N ^ 2 \\mathfrak { m } ( N ) } ) ) } \\ , d \\xi , \\end{align*}"}
+{"id": "1240.png", "formula": "\\begin{align*} & \\ll e ^ { O ( k 2 ^ { m / 2 } ) } \\left ( 2 ^ { m / 5 } \\cdot 2 ^ { 3 m / 4 } \\cdot 2 ^ { - m } \\right ) ^ { [ 2 ^ { 3 m / 4 } ] } T ( \\log T ) ^ { \\frac { k ^ 2 } { 2 } } \\mathcal { F } ( T , \\alpha _ 1 , \\alpha _ 2 ) ^ { \\frac { k ^ 2 } { 2 } } \\\\ & \\ll e ^ { O ( k 2 ^ { m / 2 } ) - 2 ^ { 3 m / 4 } } T ( \\log T ) ^ { \\frac { k ^ 2 } { 2 } } \\mathcal { F } ( T , \\alpha _ 1 , \\alpha _ 2 ) ^ { \\frac { k ^ 2 } { 2 } } . \\end{align*}"}
+{"id": "6720.png", "formula": "\\begin{align*} \\mu \\equiv M _ { [ 1 , 0 , \\frac { 3 } { 2 } ] } ( v ) : = ( 2 \\pi ) ^ { - \\frac { 3 } { 2 } } \\exp \\big ( - \\frac { | v | ^ { 2 } } { 2 } \\big ) . \\end{align*}"}
+{"id": "5781.png", "formula": "\\begin{align*} \\forall x , y \\in \\R \\exists \\ , t , a , b \\in \\R \\ ; \\left \\{ \\begin{array} { r c l } ( 1 - t ) \\ , x & = & a \\ , y \\\\ b \\ , x & = & t \\ , y \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "643.png", "formula": "\\begin{align*} \\mathcal { P } ' \\left ( \\partial _ Y \\Phi ( X ) \\right ) = \\partial _ Y \\left ( \\Phi \\left ( \\mathcal { P } ( X ) \\right ) \\right ) . \\end{align*}"}
+{"id": "3206.png", "formula": "\\begin{align*} 0 = A ( u ) + 5 B ( u ) + 3 C ( u ) + \\frac { 1 } { 4 } \\int _ { \\mathbb { R } ^ { 3 } } \\int _ { \\mathbb { R } ^ { 3 } } e ^ { - | x - y | } u ^ { 2 } ( x ) u ^ { 2 } ( y ) d x d y + \\frac { 3 \\omega } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } u ^ { 2 } d x . \\end{align*}"}
+{"id": "4595.png", "formula": "\\begin{align*} \\abs { \\sum _ { l = n _ 0 } ^ { n } \\frac { \\sin 2 \\pi \\theta ( l ) \\sin 2 \\pi \\theta _ j ( l ) } { l - b } } \\leq \\frac { C ( E , A , K ) } { n _ 0 - b } . \\end{align*}"}
+{"id": "6657.png", "formula": "\\begin{align*} Q ( t ) = \\det \\begin{bmatrix} s _ 0 & s _ 1 & \\ldots & s _ { m - 1 } & 1 \\\\ s _ 1 & s _ 2 & \\ldots & s _ { m } & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ s _ { m } & s _ { m + 1 } & \\ldots & s _ { 2 m - 1 } & t ^ { m } \\end{bmatrix} , \\end{align*}"}
+{"id": "1174.png", "formula": "\\begin{align*} & \\bigg ( \\sum _ { j = 0 } ^ { k } \\mathbb { P } ( Y _ 1 + \\cdots + Y _ k = j , \\ , Z _ 1 + \\cdots + Z _ k = j ) \\bigg ) ^ { 1 / p _ k } \\\\ & \\leq \\mathbb { P } ( Y _ 1 = \\cdots = Y _ k = 0 , \\ , Z _ 1 = \\cdots = Z _ k = 1 ) ^ { 1 / p _ k } \\\\ & + \\mathbb { P } ( Y _ 1 = \\cdots = Y _ k = 1 , \\ , Z _ 1 = \\cdots = Z _ k = 0 ) ^ { 1 / p _ k } . \\end{align*}"}
+{"id": "6404.png", "formula": "\\begin{align*} \\eta _ { i } ^ { ( k ) } \\colon x _ { i } ^ { ( k + 4 ) } x _ { i } ^ { ( k ) } = v ^ { m _ k } \\bigg [ e _ { i , i + 2 } e _ { i , i + 1 } e _ { i + 1 , i + 2 } \\bigg ( \\prod _ { j = i } ^ { k - 1 } ( e _ { j + 2 , j + 3 } ) ^ { 2 } \\bigg ) y _ { i + k + 2 } ^ { ( 3 ) } \\bigg ] + v ^ { n _ k } ( x _ { i + 2 } ^ { ( k + 2 ) } ) ^ 2 , \\end{align*}"}
+{"id": "5574.png", "formula": "\\begin{align*} J _ 1 ( \\beta _ 0 ) : = \\frac { 1 } { i _ 2 ( \\beta _ 0 ) } - \\bigg ( B ^ { * } \\beta _ 0 ^ { \\nu + 1 } - \\frac { u _ c } { M ( \\beta _ 0 ) } \\bigg ) ^ { \\tfrac { 1 } { \\nu + 1 } } , J _ 2 ( \\beta _ 0 ) : = \\frac { 1 } { i _ 1 ( \\beta _ 0 ) } - B ^ { * } \\beta _ 0 ^ { \\nu + 1 } . \\end{align*}"}
+{"id": "3831.png", "formula": "\\begin{align*} \\left ( a _ { h - 2 } , a _ { h - 1 } , a _ h \\right ) \\in \\left \\{ \\begin{aligned} & \\left ( n - 5 , n - 4 , n - 3 \\right ) , \\left ( n - 5 , n - 4 , n - 2 \\right ) , \\\\ & \\left ( n - 5 , n - 3 , n - 1 \\right ) , \\left ( n - 5 , n - 2 , n - 1 \\right ) , \\\\ & \\left ( n - 4 , n - 3 , n \\right ) , \\left ( n - 4 , n - 2 , n \\right ) , \\\\ & \\left ( n - 3 , n - 1 , n \\right ) , \\left ( n - 2 , n - 1 , n \\right ) \\end{aligned} \\right \\} . \\end{align*}"}
+{"id": "4003.png", "formula": "\\begin{align*} \\hat { q } _ { \\beta } ( n , t ) = \\sum _ { k = 1 } ^ { n } \\sum _ { \\Lambda _ { n } ^ { k } } k ! \\prod _ { j = 1 } ^ { n - k + 1 } \\frac { ( ( 1 - p ) ^ { j } / j ) ^ { x _ { j } } } { x _ { j } ! } \\left ( \\frac { - \\lambda t ^ { \\beta } } { \\ln p } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } ( - \\lambda t ^ { \\beta } ) , \\end{align*}"}
+{"id": "1212.png", "formula": "\\begin{align*} p ( y _ i | x _ i = 0 ) = p ( - y _ i | x _ i = 1 ) \\end{align*}"}
+{"id": "1361.png", "formula": "\\begin{align*} \\frac { d } { d t } E [ u ] ( t ) & = - \\int _ { \\Omega } a ( x ) | \\partial _ t u ( t , x ) | ^ 2 \\ , d x , \\end{align*}"}
+{"id": "1407.png", "formula": "\\begin{align*} \\int _ { \\Omega } a ( x ) ^ { \\frac { p + 1 } { p - 1 } } \\Theta ( s , x ; t _ 0 ) ^ { \\lambda - \\frac { p + 1 } { p - 1 } } \\ , d x & = \\int _ { \\Omega _ 1 ( s ) } a ( x ) ^ { \\frac { p + 1 } { p - 1 } } \\Theta ( s , x ; t _ 0 ) ^ { \\lambda - \\frac { p + 1 } { p - 1 } } \\ , d x \\\\ & + \\int _ { \\Omega _ 2 ( s ) } a ( x ) ^ { \\frac { p + 1 } { p - 1 } } \\Theta ( s , x ; t _ 0 ) ^ { \\lambda - \\frac { p + 1 } { p - 1 } } \\ , d x \\\\ & = : I ( s ) + I \\ ! I ( s ) . \\end{align*}"}
+{"id": "3497.png", "formula": "\\begin{align*} \\Big \\Vert | \\mathrm { E } | ^ 2 \\Big \\Vert _ { \\mathrm { L } ^ 2 \\Big ( \\Omega \\Big ) } = \\mathcal { O } \\Big ( \\delta | \\log \\delta | ^ { \\frac { 3 \\mathrm { h } } { 2 } } \\Big ) , \\end{align*}"}
+{"id": "4297.png", "formula": "\\begin{align*} z \\frac { \\partial } { \\partial z } \\frac { ( q ) _ { N } } { ( z q ) _ { N } ( z ^ { - 1 } q ) _ { N } } = z \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n + 1 } ( q ) _ n q ^ { n ( n + 1 ) / 2 } } { ( q ) _ { n + N } } \\left ( \\frac { 1 - q ^ n } { ( 1 - z q ^ n ) ^ 2 } - \\frac { 1 - q ^ n } { ( z - q ^ n ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "5258.png", "formula": "\\begin{align*} \\kappa _ p ^ { ( N ) } ( k _ 1 , \\ldots , k _ p ) = 0 . \\end{align*}"}
+{"id": "8045.png", "formula": "\\begin{align*} { \\mathbf s } _ { \\textrm { d i r } } [ \\widetilde { n } ] & = [ s _ { \\textrm { d i r } _ 1 } [ \\widetilde { n } ] , \\cdots , s _ { \\textrm { d i r } _ { N _ r } } [ \\widetilde { n } ] ] ^ T \\\\ & = \\sum _ { k = 1 } ^ { K } \\widetilde { \\alpha } _ { 1 , k } { \\bf a } _ r ( \\theta _ t , f _ k ) { \\bf a } _ t ^ T ( \\theta _ t , f _ k ) { \\bf F } _ k { \\bf s } _ k e ^ { \\mathrm { j } 2 \\pi { f _ k } \\frac { \\widetilde { n } \\triangle t } { K } } , \\end{align*}"}
+{"id": "5257.png", "formula": "\\begin{align*} \\kappa _ m ^ { ( N ) } ( k _ 1 , \\ldots , k _ m ) = \\kappa ( T _ { N , k _ 1 } , \\ldots , T _ { N , k _ m } ) . \\end{align*}"}
+{"id": "3200.png", "formula": "\\begin{align*} & \\bullet \\ \\ \\mu ^ { 2 } < \\rho ^ { 2 } - \\mu ^ { 2 } < \\rho _ { 1 } ^ { 2 } \\\\ & \\bullet \\ \\ \\mu ^ { 2 } < \\rho _ { 1 } ^ { 2 } < \\rho ^ { 2 } - \\mu ^ { 2 } \\\\ & \\bullet \\ \\ \\rho _ { 1 } ^ { 2 } < \\mu ^ { 2 } < \\rho ^ { 2 } - \\mu ^ { 2 } . \\end{align*}"}
+{"id": "4004.png", "formula": "\\begin{align*} \\bar { \\mathcal { M } } ( t ) \\stackrel { d } { = } \\sum _ { j = 1 } ^ { \\infty } j N _ { j } ( t ) , \\end{align*}"}
+{"id": "1565.png", "formula": "\\begin{align*} & \\deg _ { z } : G ^ { \\prime \\prime } \\rightarrow \\mathbb { Z } ^ { 4 } , \\\\ & \\deg _ { z } \\left ( x \\right ) = \\left ( \\deg _ { z _ { i } } \\left ( x \\right ) \\right ) _ { i = 1 } ^ { 4 } . \\end{align*}"}
+{"id": "526.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } f \\ , d Q - H ( Q ) = - H ( Q \\ , | \\ , P ) + \\log Z . \\end{align*}"}
+{"id": "4064.png", "formula": "\\begin{align*} \\langle \\delta ( P ) , Q \\otimes R \\rangle & = \\frac { 1 } { | \\min ( P ) | } \\sum _ { \\substack { I \\circledcirc P \\\\ I \\approx Q \\\\ P \\setminus I \\approx R } } \\sigma ( Q ) \\sigma ( R ) = \\frac { 1 } { | \\min ( P ) | } | A | \\end{align*}"}
+{"id": "7216.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u _ j + \\Delta u _ j = \\sum \\limits _ { \\mathcal { R } ( j ) } u _ { j _ 1 } \\bar { u } _ { j _ 2 } u _ { j _ 3 } , \\\\ u _ j ( 0 ) = u _ { 0 , j } , \\end{cases} \\end{align*}"}
+{"id": "3230.png", "formula": "\\begin{align*} I _ { \\psi } ( u , v ) & = \\int _ { X } \\psi ( u - v ) ( \\theta ^ { n } _ { u } + \\theta ^ { n } _ { v } ) \\\\ & \\leq \\int _ { X } ( \\psi ( u - \\phi ) + \\psi ( v - \\phi ) ) ( \\theta ^ { n } _ { u } + \\theta ^ { n } _ { v } ) . \\end{align*}"}
+{"id": "6369.png", "formula": "\\begin{align*} ( R _ H \\psi ) ( t ) = \\int _ 0 ^ t K _ H ( t , s ) ( K _ H ^ * \\psi ) ( s ) \\d s . \\end{align*}"}
+{"id": "7458.png", "formula": "\\begin{gather*} L _ z \\left | n , \\ , j \\ , j _ z \\right > = \\left ( \\frac { n - j } { 2 } - \\frac { n + j } { 4 } \\right ) \\left | n , \\ , j \\ , j _ z \\right > , \\\\ L ^ 2 \\left | n , \\ , j \\ , j _ z \\right > = \\frac { n + j } { 4 } \\left ( \\frac { n + j } { 4 } + 1 \\right ) \\left | n , \\ , j \\ , j _ z \\right > . \\end{gather*}"}
+{"id": "2938.png", "formula": "\\begin{align*} ( \\lambda - M _ C ) ^ { - 1 } = \\begin{pmatrix} ( \\lambda - A ) ^ { - 1 } & ( \\lambda - A ) ^ { - 1 } C ( \\lambda - B ) ^ { - 1 } \\\\ & ( \\lambda - B ) ^ { - 1 } \\end{pmatrix} \\end{align*}"}
+{"id": "6270.png", "formula": "\\begin{align*} \\mu ^ * = \\mu ^ * ( t ) = - t \\norm { H _ r ( p ) } , x ^ * = x ^ * ( t ) = - t \\frac { H _ r ( p ) } { \\norm { H _ r ( p ) } } . \\end{align*}"}
+{"id": "7050.png", "formula": "\\begin{align*} \\tilde { \\mathfrak { t r } } ^ p = [ \\mathfrak g ' , \\tilde { \\mathfrak { t r } ^ p _ 0 } ] + \\tilde { \\mathfrak { t r } } ^ p _ 0 = [ \\tilde { \\mathfrak g } , \\tilde { \\mathfrak { t r } ^ p _ 0 } ] + \\tilde { \\mathfrak { t r } } ^ p _ 0 . \\end{align*}"}
+{"id": "5638.png", "formula": "\\begin{align*} \\mathbf { K } ( T , W ) = \\mathbf { K } ( T , V ) = \\mathbf { H } ( V _ o ) . \\end{align*}"}
+{"id": "2404.png", "formula": "\\begin{align*} { \\rm e v } _ { x } ^ { \\mathfrak { m } } ( e _ { a _ { 1 } } \\cdots e _ { a _ { k } } ) = \\widetilde { { \\rm e v } _ { x } ^ { \\mathfrak { m } } } \\left ( \\mathbb { I } _ { { \\rm d c h } } ( \\overrightarrow { 1 } _ { z } ; a _ { 1 } , \\dots , a _ { k } ; - \\overrightarrow { 1 } _ { 1 - z } ) \\right ) , \\end{align*}"}
+{"id": "359.png", "formula": "\\begin{align*} \\mathcal { G } ^ m ( \\R ^ n \\times J ) & : = \\bigcap _ { j = 0 } ^ m C ^ j ( \\overline { J } , \\mathcal { H } ^ { m - j } ( \\R ^ n ) ) , \\ \\norm { u } _ { \\mathcal { G } ^ m ( \\R ^ n \\times J ) } : = \\max _ { 0 \\leq j \\leq m } \\norm { \\partial _ t ^ j u } _ { L ^ \\infty ( J , \\mathcal { H } ^ { m - j } ( \\R ^ n ) ) } . \\end{align*}"}
+{"id": "1930.png", "formula": "\\begin{align*} \\begin{aligned} & ( 1 + \\lambda ) \\| \\partial _ t U \\| _ { L _ p ( Q _ r ) } \\le ( 1 + \\lambda ) \\big ( \\| \\partial _ t U - v \\cdot D _ x U \\| _ { L _ p ( Q _ r ) } + r \\| D _ x U \\| _ { L _ p ( Q _ r ) } \\big ) \\\\ & \\le N ( 1 + \\lambda ) ( \\| U \\| _ { L _ p ( Q _ { r _ 1 } ) } + \\| D ^ { | \\alpha | - 1 } _ v D ^ { 1 + | \\beta | } _ x u \\| _ { L _ p ( Q _ { r _ 1 } ) } \\\\ & + \\| D _ x U \\| _ { L _ p ( Q _ { r } ) } ) \\le N \\| u \\| _ { L _ p ( Q _ R ) } , \\end{aligned} \\end{align*}"}
+{"id": "4375.png", "formula": "\\begin{align*} ( \\mathcal { P } _ \\theta ) \\left \\{ \\begin{array} { c l } { \\rm M a x i m i z e } & \\sum _ { i = 1 } ^ { l } \\theta _ i J _ i ( x , u ) \\\\ { \\rm s u b j e c t \\ ; t o } & ( x , u ) \\in A d m ( { \\mathcal M } ) . \\end{array} \\right . \\end{align*}"}
+{"id": "4980.png", "formula": "\\begin{align*} \\frac { \\sum \\limits _ { \\mathbf { a } \\in \\mathbb { B } ^ { N \\times 1 } \\atop | | \\mathbf { a } | | _ { 0 } = \\lambda N } | \\hat { \\mathcal { U } } _ { \\rm { a c } } ^ { \\mathbf { a } } ( \\mathbf { I } ) - \\mathcal { U } ^ { \\mathbf { a } } _ { \\rm { a c } } | } { \\binom { N } { \\lambda N } } = ( 1 - \\lambda ) \\sum _ { u = 1 } ^ { N } \\prod _ { l = 1 } ^ { L } ( 1 - ( 1 - \\lambda ) ^ { \\frac { w _ { c } } { r } - 1 } ) ^ { \\mathbf { S } [ l , u ] } . \\end{align*}"}
+{"id": "7507.png", "formula": "\\begin{align*} E _ \\varphi ( F ) = E _ \\varphi [ \\mu ^ F ] . \\end{align*}"}
+{"id": "3333.png", "formula": "\\begin{align*} L \\Psi ( n ) = \\{ L v \\colon v \\in \\Psi ( n ) \\} . \\end{align*}"}
+{"id": "4658.png", "formula": "\\begin{align*} \\tau _ \\chi L _ f ( \\chi , s ) = \\omega _ E \\tau _ { \\chi ^ { - 1 } } \\chi ( N _ E ) q ^ { ( 1 - s ) ( \\deg ( N _ E ) + 2 \\deg ( D ) - 4 ) } L _ { f } ( \\chi ^ { - 1 } , 2 - s ) , \\end{align*}"}
+{"id": "4742.png", "formula": "\\begin{gather*} T ( u ) \\cdot T ( v ) = T ( l ( T ( u ) ) v + r ( T ( v ) ) u ) , \\forall u , v \\in V , \\\\ \\partial _ k T = T \\alpha _ k , \\forall k = 1 , \\dots , m . \\end{gather*}"}
+{"id": "2594.png", "formula": "\\begin{align*} \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\Psi } ( t , \\boldsymbol { x } ) ) } ^ { N , i } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } h \\Big ( \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\Psi } ( t , \\boldsymbol { x } ) ) } ^ { N , j } \\Big ) = - R ^ { - 1 } B \\Big [ P _ t \\nu ^ { N , i } _ { \\boldsymbol { x } } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } \\Psi ^ { N , j } ( t , \\boldsymbol { x } ) \\Big ] . \\end{align*}"}
+{"id": "8175.png", "formula": "\\begin{align*} A = \\left [ \\begin{array} { c c } 0 & a ^ * \\\\ b & C \\end{array} \\right ] , \\end{align*}"}
+{"id": "6588.png", "formula": "\\begin{align*} [ t A , [ t A , J _ t ] ] = - [ t A , t W _ t C ] = - t ^ 2 W _ t [ A , C ] = O \\mbox { a n d } \\quad [ J _ t , t W _ t C ] = O . \\end{align*}"}
+{"id": "6155.png", "formula": "\\begin{align*} V _ 2 V _ 1 ^ * ( z ^ n \\otimes \\xi ) & = \\overline q ^ { 2 n } z ^ n \\otimes U ^ * P U ^ * P ^ \\perp \\xi + \\overline q ^ { 2 n - 2 } z ^ { n - 1 } \\otimes U ^ * P U ^ * P \\xi \\\\ & \\quad + \\overline q ^ { 2 n + 1 } z ^ { n + 1 } U ^ * P ^ \\perp U ^ * P ^ \\perp \\xi + \\overline q ^ { 2 n - 1 } z ^ n \\otimes U ^ * P ^ \\perp U ^ * P \\xi \\end{align*}"}
+{"id": "3267.png", "formula": "\\begin{gather*} \\mu _ m \\widehat { P } _ { n } ( m ) = a _ n \\widehat { P } _ { n - 1 } ( m ) + b _ n \\widehat { P } _ { n } ( m ) + a _ { n + 1 } \\widehat { P } _ { n + 1 } ( m ) . \\end{gather*}"}
+{"id": "3452.png", "formula": "\\begin{align*} \\mathcal { E } _ 1 \\Big ( \\frac { | \\xi - \\mathrm { z } | ^ 2 } { 4 \\mathrm { t } } \\Big ) = - \\gamma - \\ln \\frac { | \\xi - \\mathrm { z } | ^ 2 } { 4 \\mathrm { t } } + \\mathrm { e } _ { 1 } \\Big ( \\frac { | \\xi - \\mathrm { z } | ^ 2 } { 4 \\mathrm { t } } \\Big ) , \\end{align*}"}
+{"id": "1389.png", "formula": "\\begin{align*} \\Phi _ { \\beta , \\varepsilon } ( x , t ; t _ 0 ) = ( t _ 0 + t ) ^ { - \\beta } \\varphi _ { \\beta , \\varepsilon } ( z ) , z = \\frac { \\widetilde { \\gamma } _ \\varepsilon A _ \\varepsilon ( x ) } { t _ 0 + t } , \\end{align*}"}
+{"id": "930.png", "formula": "\\begin{align*} \\omega : = \\{ x \\in \\Omega : | x _ \\beta | < \\delta \\frac { b _ \\alpha } { \\sqrt { b _ \\beta } } , \\beta = \\alpha + 1 , \\ldots , n - 1 , x _ n < \\delta ^ 2 b _ \\alpha \\} , \\end{align*}"}
+{"id": "838.png", "formula": "\\begin{align*} \\begin{aligned} & \\widetilde { \\theta } _ 1 : = \\theta + \\frac { ( 1 - \\theta ) } { l _ 1 + l _ 2 \\left ( \\frac { 1 } { N } \\sum \\limits _ { j = 1 } ^ N e ^ { - \\beta \\tau _ j ^ { * } } x _ j ( \\tau _ j ^ { * } ) \\right ) } \\\\ & = \\theta + \\frac { ( 1 - \\theta ) } { l _ 1 + \\frac { l _ 2 } { N } e ^ { - \\beta \\tau _ i ^ { * } } x _ i ( \\tau _ i ^ { * } ) + \\frac { l _ 2 } { N } \\sum \\limits _ { j = 1 , j \\neq i } ^ N e ^ { - \\beta \\tau _ j ^ { * } } x _ j ( \\tau _ j ^ { * } ) } \\end{aligned} \\end{align*}"}
+{"id": "1257.png", "formula": "\\begin{align*} u _ p ( r ) = u _ p ( R ) + \\left ( \\frac { A } { N - 1 } \\right ) ^ { \\frac { 1 } { p - 1 } } \\left ( R - r \\right ) , \\end{align*}"}
+{"id": "7433.png", "formula": "\\begin{align*} [ H p ^ \\dag A ] = p ^ \\dag [ H + P , A ] + p ^ \\dag A P , \\end{align*}"}
+{"id": "2080.png", "formula": "\\begin{align*} \\begin{bmatrix} T ^ { - 1 } & 0 & 0 \\\\ 0 & I _ { \\ell - k } & 0 \\\\ 0 & 0 & I _ { n - \\ell } \\end{bmatrix} \\begin{bmatrix} T _ { 1 1 } & 0 \\\\ 0 & U \\end{bmatrix} S M T = \\begin{bmatrix} I _ k \\\\ 0 \\\\ D \\end{bmatrix} , \\end{align*}"}
+{"id": "4261.png", "formula": "\\begin{align*} & \\sum _ { k = 0 \\atop k \\neq m } ^ { n } \\left [ \\begin{matrix} n \\\\ k \\end{matrix} \\right ] \\frac { \\left ( \\frac { q } { x } \\right ) _ k ( x ) _ { n - k } } { 1 - q ^ { k - m } } x ^ k = ( - 1 ) ^ m q ^ { \\frac { m ( m + 1 ) } { 2 } } \\left [ \\begin{matrix} n \\\\ m \\end{matrix} \\right ] ( x q ^ { - m } ) _ n \\left ( \\sum _ { k = 0 } ^ { n - 1 } \\frac { x q ^ { k - m } } { 1 - x q ^ { k - m } } - \\sum _ { k = 0 \\atop k \\neq m } ^ { n } \\frac { q ^ { k - m } } { 1 - q ^ { k - m } } \\right ) . \\end{align*}"}
+{"id": "939.png", "formula": "\\begin{align*} \\bar { x } _ n - x _ n ^ 0 = \\frac { - \\varepsilon _ 1 \\bar { \\xi } _ 1 + | \\hat { x } | \\bar { \\xi } _ 2 } { \\sqrt { \\varepsilon _ 1 ^ 2 + | \\hat { x } | ^ 2 } } \\leq | D \\rho ( 0 , \\tilde { x } ) | | \\bar { \\xi } _ 1 | + | \\bar { \\xi } _ 2 | \\leq C \\epsilon _ 0 \\delta ^ 2 b _ \\alpha \\end{align*}"}
+{"id": "6818.png", "formula": "\\begin{align*} \\varepsilon \\partial _ { t t } u + \\partial _ t u - \\varepsilon \\omega \\kappa \\Delta \\partial _ t u - \\kappa \\Delta u = 0 . \\end{align*}"}
+{"id": "6662.png", "formula": "\\begin{align*} P ( 0 ) = \\sigma \\left ( \\frac { Q ( t ) - Q ( 0 ) } { t } \\right ) . \\end{align*}"}
+{"id": "1651.png", "formula": "\\begin{align*} \\mathsf { A } \\ ; = \\ ; \\mathsf { I } _ 0 \\ ; < \\ ; \\mathsf { I } _ 1 \\ ; < \\ ; \\dots \\ ; < \\ ; \\mathsf { I } _ { \\mathsf { F } } \\ ; = \\ ; \\mathsf { B } \\end{align*}"}
+{"id": "3108.png", "formula": "\\begin{align*} \\sigma = \\delta _ { 1 , 0 } ^ 2 = \\delta _ { 2 , 0 } \\delta _ { 2 , 1 } = ( \\ldots , - 4 , - 2 , 0 , 2 , 4 , 6 , \\ldots ) ( \\ldots , - 5 , - 3 , - 1 , 1 , 3 , 5 , \\ldots ) , \\end{align*}"}
+{"id": "2523.png", "formula": "\\begin{align*} ( \\phi x ) _ \\epsilon = x _ { \\phi ( \\epsilon ) } \\end{align*}"}
+{"id": "1579.png", "formula": "\\begin{align*} w _ r x _ a y _ { r , 1 } ^ { - 1 } = e , y _ { r , 1 } t _ { r } y _ { r , 2 } ^ { - 1 } = e , \\quad \\ldots \\quad , y _ { r , 3 5 } t _ { b } ^ { - 1 } y _ { r , 3 6 } = e . \\end{align*}"}
+{"id": "7247.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta _ { \\mathbb { R } ^ { m } \\times \\mathbb { T } ^ { n } } ) u & = F ( u ) = | u | ^ { p - 1 } u , \\\\ u ( 0 , x ) & = u _ { 0 } \\in H ^ { 1 } ( \\mathbb { R } ^ { m } \\times \\mathbb { T } ^ { n } ) , \\end{align*}"}
+{"id": "7573.png", "formula": "\\begin{align*} \\begin{aligned} E _ { \\varphi } [ \\mu _ \\varepsilon ] & = E _ { \\varphi } [ \\mu ] + \\varepsilon \\Re D _ { V , h } ( \\mu ) + o ( \\varepsilon ) , \\\\ E _ { \\varphi } [ \\mu ^ { \\varepsilon } ] & = E _ { \\varphi } [ \\mu ^ { \\varepsilon } _ { - \\varepsilon } ] + \\varepsilon \\Re D _ { V , h } \\left ( \\mu ^ { \\varepsilon } \\right ) + o ( \\varepsilon ) . \\end{aligned} \\end{align*}"}
+{"id": "2768.png", "formula": "\\begin{align*} G = \\sqrt { \\hat { A } ( X ) } \\omega \\end{align*}"}
+{"id": "2432.png", "formula": "\\begin{align*} f ( \\Phi ) = 0 , \\end{align*}"}
+{"id": "1924.png", "formula": "\\begin{align*} & ( z ) = \\zeta \\mathcal { A } h ( z ) - \\mathcal { A } ( \\zeta h ) ( z ) \\\\ & = \\int h ( t , x - y , v ) \\big ( \\zeta ( t , x , v ) - \\zeta ( t , x - y , v ) \\big ) \\ , \\frac { y } { | y | ^ { d + 5 / 3 } } \\ , d y \\\\ & = \\bigg ( \\int _ { | y | \\le 1 } \\ldots + \\int _ { | y | > 1 } \\ldots \\bigg ) = : I _ 1 ( z ) + I _ 2 ( z ) . \\end{align*}"}
+{"id": "6949.png", "formula": "\\begin{align*} \\mathsf W = \\mathsf A ^ { \\chi ( \\mathcal O _ C ) } \\cdot \\mathsf B ^ { \\chi ( E \\otimes L ) } . \\end{align*}"}
+{"id": "2967.png", "formula": "\\begin{align*} \\mathrm { E x t } ^ 1 ( X , Y ) = \\begin{cases*} k & X , Y \\\\ 0 & \\end{cases*} \\end{align*}"}
+{"id": "5757.png", "formula": "\\begin{align*} f ( x ) = \\begin{pmatrix} x ^ 2 & x \\\\ x & 1 \\end{pmatrix} , \\end{align*}"}
+{"id": "7288.png", "formula": "\\begin{align*} u ^ * ( t , \\omega ) = R ^ { - 1 } ( t ) B ^ T ( t ) \\Upsilon ( t , \\omega ) . \\end{align*}"}
+{"id": "130.png", "formula": "\\begin{align*} \\begin{array} { l l l l l l } \\partial _ q [ \\mathbf a ] & = \\partial _ q [ a _ 0 , \\dots , a _ q , a _ { q + 1 } ] \\\\ & = \\sum \\limits _ { i = 0 } ^ { q + 1 } ( - 1 ) ^ i \\cdot [ a _ 0 , \\dots , \\widehat { a _ i } , \\dots , a _ { q + 1 } ] \\in O [ m ] _ { q } \\\\ \\end{array} \\end{align*}"}
+{"id": "7236.png", "formula": "\\begin{align*} \\mathcal { R } = \\left \\{ \\left ( j , j _ { 1 } , j _ { 2 } , j _ { 3 } \\right ) \\in \\left ( \\mathbb { Z } ^ { 2 } \\right ) ^ { 4 } : j _ { 1 } - j _ { 2 } + j _ { 3 } = j \\left | j _ { 1 } \\right | ^ { 2 } - \\left | j _ { 2 } \\right | ^ { 2 } + \\left | j _ { 3 } \\right | ^ { 2 } = | j | ^ { 2 } \\right \\} . \\end{align*}"}
+{"id": "5368.png", "formula": "\\begin{align*} \\mathcal { V } _ A & \\doteq \\{ E \\in \\mathcal { E } _ A \\ | \\ \\exists \\varphi : E = \\star d _ A \\varphi \\} , \\\\ \\mathcal { H } _ A & \\doteq \\{ E \\in \\mathcal { E } _ A \\ | \\ d _ A E = 0 \\mathsf { t } E = 0 \\} . \\end{align*}"}
+{"id": "6937.png", "formula": "\\begin{align*} ( - 1 ) ^ { ( N - 1 ) d } \\left [ q ^ d \\right ] \\prod _ { i \\in [ N ] } \\bigg ( \\frac { \\alpha _ i - z _ { N + 1 } } { \\alpha _ i - y } \\bigg ) ^ { b _ i } \\bigg ( \\frac { z _ i } { z _ i - y } \\bigg ) ^ { - 1 } \\frac { z _ i ^ { d + 1 } } { P ' ( z _ i ) } \\prod _ { i , j \\in [ N ] , \\ , \\ , \\ , i \\neq j } ( z _ i - z _ j ) \\bigg { | } _ { \\epsilon = 0 } . \\end{align*}"}
+{"id": "3589.png", "formula": "\\begin{align*} \\tau _ j = \\sum _ i a _ i \\eta _ { i j } ^ 2 = \\frac { 2 5 6 } { j ^ 2 \\pi ^ 4 } \\left ( \\sum _ m b _ m m \\right ) ^ 2 \\sum _ i \\frac { a _ i } { i ^ 2 } \\lesssim \\frac { 1 } { j ^ 2 } \\end{align*}"}
+{"id": "7382.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\eta , \\tau } = A _ 0 + \\left ( \\eta I _ 2 + \\tau \\sigma _ 3 \\right ) \\delta _ { \\Sigma } ; \\end{align*}"}
+{"id": "3889.png", "formula": "\\begin{align*} \\lim _ { | x | \\to 0 ^ + } f ( x ) | x | ^ { 2 - \\tau _ - ( \\mu ) } = 0 , \\end{align*}"}
+{"id": "6659.png", "formula": "\\begin{align*} \\overline { Q } ( t ) = ( t - a ) ( b - t ) \\det \\begin{bmatrix} s _ 0 ' & s _ 1 ' & \\ldots & s _ { m - 2 } ' & 1 \\\\ s _ 1 ' & s _ 2 ' & \\ldots & s _ { m - 1 } ' & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ s _ { m - 1 } ' & s _ { m } ' & \\ldots & s _ { 2 m - 3 } ' & t ^ { m - 1 } \\end{bmatrix} , \\end{align*}"}
+{"id": "5634.png", "formula": "\\begin{align*} g _ T ( T , V ) = 0 , \\ \\ g _ T ( V , V ) = 1 , \\ \\ D ^ T _ T V = 0 . \\end{align*}"}
+{"id": "969.png", "formula": "\\begin{align*} \\xi _ y ( x ) = e ^ { 2 \\pi i \\ , y \\cdot x } , \\end{align*}"}
+{"id": "4197.png", "formula": "\\begin{align*} 1 \\cdot u = \\underset { w \\in U _ 0 } { \\sum } q _ { w } w \\ + \\underset { R \\in \\langle { \\mathcal { R } ^ { o r b } , \\mathcal { R } \\rangle } } { \\sum } c _ R R . \\end{align*}"}
+{"id": "5667.png", "formula": "\\begin{align*} 4 T _ o ^ \\alpha T _ o ^ \\beta \\frac { \\partial ^ 2 r } { \\partial z ^ \\alpha \\partial z ^ \\beta } & = T ^ i T ^ j \\frac { \\partial ^ 2 r } { \\partial x ^ i \\partial x ^ j } - V ^ i V ^ j \\frac { \\partial ^ 2 r } { \\partial x ^ i \\partial x ^ j } - 2 \\sqrt { - 1 } V ^ i T ^ i \\frac { \\partial ^ 2 r } { \\partial x ^ i \\partial x ^ j } . \\end{align*}"}
+{"id": "7885.png", "formula": "\\begin{align*} \\Vert \\Phi \\Vert _ { C ^ 3 ( U ) } = C _ 0 . \\end{align*}"}
+{"id": "6939.png", "formula": "\\begin{align*} \\chi ( \\mathsf { S y m } _ y L ^ { [ d ] } ) = \\frac { 1 } { ( 1 - y ) ^ \\chi } \\bigg [ 1 + ( - 1 ) ^ { ( N - 1 ) d + \\chi } y ^ { d + 1 } _ { t = 0 } \\ , \\ , \\omega \\bigg ] . \\end{align*}"}
+{"id": "1699.png", "formula": "\\begin{align*} \\varphi : = \\sum _ { k \\in \\N } \\frac { \\omega _ k } { \\sqrt { \\lambda _ k } } u _ k . \\end{align*}"}
+{"id": "7529.png", "formula": "\\begin{align*} D _ { V , h } ( \\mu ) = - \\iint \\left ( h ( p ) C ( p , q ) + h ( q ) C ( q , p ) \\right ) d \\mu ( p ) d \\mu ( q ) + \\int h d V d \\mu \\end{align*}"}
+{"id": "5172.png", "formula": "\\begin{align*} f _ { n , i } ^ j ( U _ { n , i } ^ j ) = U _ { } f _ { n , i } ^ j ( T _ { n , i } ^ j ) = T , \\end{align*}"}
+{"id": "4529.png", "formula": "\\begin{align*} \\big \\{ \\mathrm { T r } \\ , h _ { \\gamma } , ^ 2 \\ ! E _ S [ f ] \\big \\} & \\ ! = \\ ! \\int _ { \\mathbf { x } \\in S } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\mathrm { d } S ^ { b c } ( \\mathbf { x } ) \\ , f ^ i ( \\mathbf { x } ) v _ { a b c } ( \\mathbf { x } ) \\int _ 0 ^ 1 \\ ! \\ ! \\ ! \\mathrm { d } t \\ , \\dot { \\gamma } ^ { a } ( t ) \\mathcal { E } ^ { ( 3 ) } _ { \\ ; \\ , \\gamma ( t ) } ( \\mathbf { x } ) \\mathrm { T r } \\ , \\Big ( h _ { \\gamma } ( 1 , t ) \\frac { \\tau _ i } { 2 } h _ { \\gamma } ( t , 0 ) \\Big ) \\ , . \\end{align*}"}
+{"id": "5948.png", "formula": "\\begin{align*} \\sim _ { q } ( \\delta ^ { - \\frac { 1 } { k } } ) ^ { \\frac { p } { 2 } + k - \\frac { k ( k - 1 ) } { 2 } } ( \\delta ^ { - 1 + \\frac { 1 } { k } } ) ^ { \\frac { p - 2 k } { 2 } - \\frac { k ( k + 1 ) } { 2 } } = \\delta ^ { - ( \\frac { p } { 2 } - \\frac { k ( k + 1 ) } { 2 } ) } \\end{align*}"}
+{"id": "6453.png", "formula": "\\begin{align*} \\Delta _ { x , \\alpha } ^ n = \\left ( \\eta _ x - \\frac { x q ^ { \\alpha + 1 } } { 1 - q ^ { \\alpha + 1 } } \\lambda _ { \\alpha } \\eta _ x \\right ) ^ n = \\sum _ { k = 0 } ^ n \\begin{bmatrix} n \\\\ k \\end{bmatrix} \\frac { q ^ { k ^ 2 + k \\alpha } ( - x ) ^ k } { ( q ^ { \\alpha + 1 } ; q ) _ k } \\lambda _ { \\alpha } ^ k \\eta _ x ^ n . \\end{align*}"}
+{"id": "6850.png", "formula": "\\begin{align*} \\lambda u _ i h ( a _ i ) = ( v _ i ' ) ^ { p ^ { e ' } + 1 } , \\ 1 \\leq i \\leq n . \\end{align*}"}
+{"id": "1373.png", "formula": "\\begin{align*} 2 + ( n + 1 ) p - ( n - 1 ) p ^ 2 = 0 \\end{align*}"}
+{"id": "1085.png", "formula": "\\begin{align*} b _ { 1 } u _ { 1 } ^ { 2 } - b _ { 2 } u _ { 2 } ^ { 2 } = 4 ^ { m } p \\cdot 2 ^ { 2 k } \\implies b _ { 2 } \\equiv 0 \\pmod 2 , \\end{align*}"}
+{"id": "3198.png", "formula": "\\begin{align*} & 0 < k _ { 1 } < A ( u _ { n } ) < k _ { 2 } , \\\\ & 0 < \\eta _ { 1 } < | C ( u _ { n } ) | < \\eta _ { 2 } . \\end{align*}"}
+{"id": "3335.png", "formula": "\\begin{align*} - q \\gamma ( \\lambda _ n B + D _ 1 ) + ( \\lambda _ n - \\lambda _ s ) \\delta B + \\varepsilon ( \\lambda _ n - \\lambda _ s ) D _ 1 = 0 . \\end{align*}"}
+{"id": "7296.png", "formula": "\\begin{align*} & \\frac { d } { d t } [ X ^ * ( t , \\omega ) - \\overline { X } ^ * ( t ) ] = A ( t ) [ X ^ * ( t , \\omega ) - \\overline { X } ^ * ( t ) ] + B ( t ) R ^ { - 1 } ( t ) B ^ T ( t ) [ \\Upsilon ( t , \\omega ) - \\overline { \\Upsilon } ( t ) ] , \\\\ & \\frac { d } { d t } [ \\Upsilon ( t , \\omega ) - \\overline { \\Upsilon } ( t ) ] = ( Q _ x ( t ) + Q _ m ( t ) ) [ X ^ * ( t , \\omega ) - \\overline { X } ^ * ( t ) ] - A ^ T ( t ) [ \\Upsilon ( t , \\omega ) - \\overline { \\Upsilon } ( t ) ] . \\end{align*}"}
+{"id": "2141.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle { \\mathcal { W } } ^ p _ { \\sharp } ( \\mathrm { g r a d } _ { \\bf R } , Y ^ m ) = & \\displaystyle { \\Bigl \\{ u \\in { L ^ p _ \\sharp ( Y ^ m ) } \\ ; \\mid \\ ; \\mathrm { g r a d } _ { \\bf R } \\ ; u \\in { L ^ p _ \\sharp ( Y ^ m ; \\R ^ n ) } \\Bigr \\} } \\end{array} \\end{align*}"}
+{"id": "3283.png", "formula": "\\begin{align*} \\pm c _ 2 ( d _ 1 ) ^ 2 ( c _ 1 ) ^ { - 1 / 2 } = 0 . \\end{align*}"}
+{"id": "1574.png", "formula": "\\begin{align*} S _ { 0 } = & \\left \\{ x _ { s } , x _ { s } ^ { - 1 } \\right \\} _ { s \\in S } \\cup \\left \\{ w _ { r } , w _ { r } ^ { - 1 } \\right \\} _ { r \\in R } \\cup \\left \\{ e \\right \\} , \\\\ R _ { 0 } = & \\left \\{ x _ { s } x _ { s } ^ { - 1 } e = e \\right \\} _ { s \\in S } \\cup \\left \\{ w _ { s } w _ { s } ^ { - 1 } e = e \\right \\} _ { r \\in R } , \\end{align*}"}
+{"id": "1310.png", "formula": "\\begin{align*} \\alpha _ { a , b , c } ^ \\dagger = \\alpha ^ { - 1 } _ { a , b , c } , \\lambda _ a ^ \\dagger = \\lambda ^ { - 1 } _ a , \\sigma _ { a , b } ^ \\dagger = \\sigma _ { b , a } , \\end{align*}"}
+{"id": "3796.png", "formula": "\\begin{align*} \\l _ t \\leq \\begin{cases} 2 n - 2 & d \\in \\l , \\\\ 2 n - 4 & d \\not \\in \\l . \\end{cases} \\end{align*}"}
+{"id": "1745.png", "formula": "\\begin{align*} P \\left ( Y , \\mathcal { Z } \\right ) : = \\bigcup _ { I } S \\left ( I \\right ) \\bigcup _ { J } S \\left ( J , n , 1 \\right ) \\subset V _ { Y } , \\end{align*}"}
+{"id": "1087.png", "formula": "\\begin{align*} b _ { 1 } u _ { 1 } ^ { 2 } - b _ { 1 } b _ { 2 } u _ { 3 } ^ { 2 } = - p q ^ { 2 k } \\implies b _ { 1 } \\equiv 0 \\pmod { q ^ 2 } , \\end{align*}"}
+{"id": "2108.png", "formula": "\\begin{align*} v _ j = \\sum _ { i = 1 } ^ \\ell c _ { i j } e ' _ i , \\mbox { $ j = 1 , \\ldots , m $ } , \\end{align*}"}
+{"id": "653.png", "formula": "\\begin{align*} \\nabla _ X \\psi _ 2 = - \\frac { 1 } { 2 } X _ 1 \\cdot \\psi _ 2 - \\frac { 1 } { 2 } S ( X ) \\cdot N \\cdot \\psi _ 2 \\end{align*}"}
+{"id": "1724.png", "formula": "\\begin{align*} \\xi ( x _ 1 , \\ldots , x _ p ; y _ 1 , \\ldots , y _ p ) = \\prod _ { j = 1 } ^ p \\delta ( x _ j - y _ j ) \\ , . \\end{align*}"}
+{"id": "4863.png", "formula": "\\begin{align*} W = X \\# W _ { ( t b ( K ' _ 1 ) - 1 , \\dots , t b ( K ' _ n ) - 1 ) } ( L ' ) , \\end{align*}"}
+{"id": "7151.png", "formula": "\\begin{align*} \\lambda _ i ( t ) = ( \\overbrace { t ^ { - 1 } , \\ldots , t ^ { - 1 } } ^ k , 1 , \\ldots , 1 ) \\end{align*}"}
+{"id": "1467.png", "formula": "\\begin{align*} ( C _ R , \\ C _ W ) & = \\left ( \\frac { 2 } { 1 - \\frac { 2 } { N } } ( 1 - \\tilde { D } _ r ) , \\ \\frac { 2 } { 1 - \\frac { 2 } { N } } ( 1 - \\tilde { D } _ w ) \\right ) . \\end{align*}"}
+{"id": "4463.png", "formula": "\\begin{align*} & \\sum ( - 1 ) ^ { f _ \\sigma } \\cdots E _ { i j } E _ { j k } \\cdots - \\sum ( - 1 ) ^ { f _ \\sigma + ( \\bar i + \\bar j ) ( \\bar j + \\bar k ) } \\cdots E _ { j k } E _ { i j } \\cdots \\\\ = & \\sum \\delta _ { j j } ( - 1 ) ^ { f _ \\sigma } \\cdots E _ { i k } \\cdots - \\sum \\delta _ { i k } ( - 1 ) ^ { f _ \\sigma + ( \\bar i + \\bar j ) ( \\bar j + \\bar k ) } \\cdots E _ { j j } \\cdots \\end{align*}"}
+{"id": "6441.png", "formula": "\\begin{align*} & \\int _ { - \\infty } ^ { + \\infty } \\frac { e ^ { - \\theta ^ 2 + 2 m \\theta } } { ( y q ^ { 1 / 2 } e ^ { 2 k i \\theta } ; q ) _ { \\infty } ( z q ^ { 1 / 2 } e ^ { - 2 i k \\theta } ; q ) _ { \\infty } } d \\theta \\\\ & = \\sqrt { \\pi } e ^ { m ^ 2 } \\frac { ( - y q e ^ { 2 m k i } ; q ) _ { \\infty } ( - z q e ^ { - 2 m k i } ; q ) _ { \\infty } } { ( y z q ; q ) _ { \\infty } } . \\end{align*}"}
+{"id": "3273.png", "formula": "\\begin{gather*} P _ n ( m ) = R _ n \\big ( y _ m ; \\alpha , \\beta , \\gamma , \\delta ; q ^ 2 \\big ) . \\end{gather*}"}
+{"id": "1842.png", "formula": "\\begin{align*} f & = \\prod _ { i = 0 } ^ { n - 1 } f _ i ^ { j _ i } , & s & = s ( f ) + k , \\min { } _ s \\leqslant k \\leqslant \\max { } _ s & \\max { } _ { \\mbox { { \\scriptsize s c o r e } } } & \\geqslant \\sum _ { i = 0 } ^ { n - 1 } j _ i \\end{align*}"}
+{"id": "866.png", "formula": "\\begin{align*} \\partial _ \\mu E = \\left \\{ x \\in G : \\ , \\min \\left \\{ \\frac { \\mu ( E \\cap B _ d ( x , r ) ) } { \\mu ( B _ d ( x , r ) ) } , \\frac { \\mu ( E ^ c \\cap B _ d ( x , r ) ) } { \\mu ( B _ d ( x , r ) ) } \\right \\} > \\epsilon \\ , \\ , \\forall r > 0 \\right \\} , \\end{align*}"}
+{"id": "2753.png", "formula": "\\begin{align*} a ( H ) = \\pi _ 2 ^ * H \\cdot \\mu _ * ( \\ell _ 1 ) { \\rm a n d } a ( H ) - b ( H ) \\pi _ 1 ^ * H \\cdot \\mu ^ { - 1 } _ * ( \\ell _ 2 ) = 0 . \\end{align*}"}
+{"id": "716.png", "formula": "\\begin{align*} \\rho { c _ v } { \\partial _ { { t } } } T = \\nabla \\cdot \\left ( { \\lambda \\nabla T } \\right ) + { \\bar F } + \\left ( { \\vartheta - 1 } \\right ) \\frac { { \\Delta t } } { 2 } \\rho { c _ v } \\partial _ { { t } } ^ 2 T , \\end{align*}"}
+{"id": "6531.png", "formula": "\\begin{align*} U = \\sum _ r e ^ { i \\theta _ r } E _ r = \\exp ( i H ) , \\end{align*}"}
+{"id": "4647.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\ , 0 } S ( x ) = \\tfrac { \\sqrt { 3 } } { 2 } - \\frac { 2 \\pi } { 3 } + \\frac { \\pi } { 2 } \\gamma = \\frac { \\pi } { 2 } ( \\gamma - \\gamma _ c ) < 0 . \\end{align*}"}
+{"id": "1441.png", "formula": "\\begin{align*} b _ 1 ( x ) = \\Delta \\left ( \\frac { a _ 0 } { ( n - \\alpha ) ( 2 - \\alpha ) } \\langle x \\rangle ^ { 2 - \\alpha } \\right ) = a _ 0 \\langle x \\rangle ^ { - \\alpha } + \\frac { a _ 0 \\alpha } { n - \\alpha } \\langle x \\rangle ^ { - \\alpha - 2 } \\end{align*}"}
+{"id": "6167.png", "formula": "\\begin{align*} V _ j = R _ { q ^ { d - j } } M _ { z _ j } M _ { z _ j } R _ { q ^ { d - j } } \\end{align*}"}
+{"id": "8000.png", "formula": "\\begin{align*} T _ Q \\ : : = \\ : T _ F ^ * T _ F \\end{align*}"}
+{"id": "6372.png", "formula": "\\begin{align*} \\tilde { X } _ \\cdot = \\mathcal { G } _ \\mu ( B _ \\cdot ^ H ) . \\end{align*}"}
+{"id": "6423.png", "formula": "\\begin{align*} V _ { \\psi } \\overline { u } ( x , \\xi ) = \\overline { V _ { \\overline \\psi } u ( x , - \\xi ) } . \\end{align*}"}
+{"id": "4110.png", "formula": "\\begin{align*} d - N _ 0 \\frac { d u } { u } - \\sum _ { i = 1 } ^ n N _ i \\frac { d t _ i } { t _ i } . \\end{align*}"}
+{"id": "5043.png", "formula": "\\begin{align*} \\Lambda _ f ( s , \\alpha ) = \\Gamma _ \\C \\bigl ( s + \\tfrac { k - 1 } { 2 } \\bigr ) \\sum _ { n = 1 } ^ \\infty \\frac { f _ n e ( n \\alpha ) } { n ^ s } \\quad \\alpha \\in \\Q , \\end{align*}"}
+{"id": "7095.png", "formula": "\\begin{align*} \\boxtimes _ { i = 1 } ^ k \\mathbb { S } ( d _ i ) _ { v _ i + d _ i \\left ( \\sum _ { i > j } d _ j - \\sum _ { j > i } d _ j \\right ) } \\to \\mathcal { D T } ( d ) \\end{align*}"}
+{"id": "7942.png", "formula": "\\begin{align*} \\varphi ( a , u ) = ( a , u - \\theta ( a ) ) . \\end{align*}"}
+{"id": "6490.png", "formula": "\\begin{align*} \\mathcal V = \\{ \\pi \\cap B _ \\Lambda : \\} \\end{align*}"}
+{"id": "7295.png", "formula": "\\begin{align*} \\overline { X } ^ * ( 0 ) = \\overline { X _ 0 } , \\ \\ \\overline { \\Upsilon } ( T ) = - K _ x \\overline { X } ^ * ( T ) . \\end{align*}"}
+{"id": "1973.png", "formula": "\\begin{align*} U _ m : = ( v _ m ) , \\end{align*}"}
+{"id": "5464.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\frac { 1 - a t q } { 1 - t } + \\frac { ( 1 - a q ) ( b - a t q ) } { ( 1 - b q ) ( 1 - t ) } t q F ( a q , b q ; t q ) . \\end{align*}"}
+{"id": "373.png", "formula": "\\begin{align*} \\tilde { l } _ 2 ( ( v _ 1 , \\alpha _ 1 ) , \\dots , ( v _ 2 , \\alpha _ 2 ) ) = ( [ v _ 1 , v _ 2 ] , \\varsigma ( k ) \\iota ( v _ 1 \\wedge v _ 2 ) ~ \\omega ) \\end{align*}"}
+{"id": "4142.png", "formula": "\\begin{align*} \\mathcal { H } ^ { p , r } _ { \\alpha } \\left ( I \\right ) = \\{ u \\in L ^ p ( I ) ; ~ ~ ~ x ^ { ( \\alpha + k + 1 / 2 ) } \\left ( \\frac { 1 } { x } \\frac { d } { d x } \\right ) ^ { ( k ) } \\left ( x ^ { - ( \\alpha + 1 / 2 ) } u \\right ) \\in L ^ p ( I ) ~ \\ , , \\forall \\ , 0 \\leq k \\leq r \\} . \\end{align*}"}
+{"id": "6299.png", "formula": "\\begin{align*} F ( D ^ 2 u _ \\tau ) = f ( u _ \\tau ) \\quad \\Sigma _ \\tau \\cap \\{ u _ \\tau > 0 \\} , \\end{align*}"}
+{"id": "1883.png", "formula": "\\begin{align*} \\lim _ { j \\rightarrow \\infty } \\lim _ { m \\rightarrow \\infty } \\max \\{ \\left | \\frac { 1 } { 3 ^ l } \\right | ^ k G \\left ( \\frac { x } { 3 ^ { k + 1 } } , \\frac { x } { 3 ^ { k + 1 } } \\right ) , j \\leq k < m + j \\} & = \\lim _ { j \\to \\infty } \\left | \\frac { 1 } { 3 ^ l } \\right | ^ j G \\left ( \\frac { x } { 3 ^ { j + 1 } } , \\frac { x } { 3 ^ { j + 1 } } \\right ) \\\\ & = 0 . \\end{align*}"}
+{"id": "5800.png", "formula": "\\begin{align*} f ( A ) = \\displaystyle \\int _ { \\sigma ( A ) } f d \\mathcal { P } . \\end{align*}"}
+{"id": "804.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & k _ 1 = \\frac { 1 } { 2 } - \\frac { \\alpha } { \\sigma ^ 2 } - \\sqrt { \\Big ( \\frac { 1 } { 2 } - \\frac { \\alpha } { \\sigma ^ 2 } \\Big ) ^ 2 + \\frac { 2 \\beta } { \\sigma ^ 2 } } < 0 , \\\\ & k _ 2 = \\frac { 1 } { 2 } - \\frac { \\alpha } { \\sigma ^ 2 } + \\sqrt { \\Big ( \\frac { 1 } { 2 } - \\frac { \\alpha } { \\sigma ^ 2 } \\Big ) ^ 2 + \\frac { 2 \\beta } { \\sigma ^ 2 } } > 1 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6599.png", "formula": "\\begin{align*} t _ { \\Gamma , \\alpha } ( u , u ) = \\int _ { \\mathbb { R } ^ { 2 } \\setminus \\Gamma } | \\nabla u | ^ { 2 } \\dd x + \\alpha \\int _ \\Gamma \\big | [ u ] \\big | ^ 2 \\dd \\sigma \\end{align*}"}
+{"id": "7327.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } & \\mathcal { H } ( t , x , p , q , P , v , u ) \\\\ = & H ( t , x , p , q , v ) + \\dfrac { 1 } { 2 } \\sum \\limits _ { i = 1 } ^ { d } \\left ( \\sigma ^ { i } ( t , x , v ) - \\sigma ^ { i } ( t , x , u ) \\right ) ^ { \\intercal } P \\left ( \\sigma ^ { i } ( t , x , v ) - \\sigma ^ { i } ( t , x , u ) \\right ) \\\\ & - \\dfrac { 1 } { 2 } \\sum \\limits _ { i = 1 } ^ { d } \\left ( \\sigma ^ { i } ( t , x , u ) \\right ) ^ { \\intercal } P \\left ( \\sigma ^ { i } ( t , x , u ) \\right ) . \\end{array} \\end{align*}"}
+{"id": "941.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\sum _ { \\beta = 1 } ^ { \\alpha } b _ \\beta \\bar { x } _ \\beta ^ 2 \\leq C b _ \\alpha \\bar { x } _ n + b _ \\alpha O ( | \\bar { x } ' | ^ 3 ) \\leq C b _ \\alpha \\bar { x } _ n + C \\delta ^ 3 b _ \\alpha ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "8217.png", "formula": "\\begin{align*} j _ 0 : = \\frac { 1 } { \\alpha - 1 } \\log _ 2 \\frac { 4 \\lambda } { \\mu } , \\qquad 2 ^ { j _ 0 ( \\alpha - 1 ) } = \\frac { 4 \\lambda } { \\mu } . \\end{align*}"}
+{"id": "5502.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\frac { 1 } { 1 - t } + \\frac { ( b - a ) t q } { ( 1 - t ) ( 1 - b q ) } F ( a , b q ; t q ) . \\end{align*}"}
+{"id": "5450.png", "formula": "\\begin{align*} { \\rm d i m } ( W _ \\mu ) \\ , = \\ , d \\ell - b ( a + 1 ) ^ 2 - ( r - b ) a ^ 2 \\ , = \\ , ( d - a ) \\ell - ( a + 1 ) b . \\end{align*}"}
+{"id": "6163.png", "formula": "\\begin{align*} \\tau _ S ( V _ 1 \\mathfrak r _ { \\overline q } , \\mathfrak r _ { q } V _ 2 ) = ( M _ { z _ 1 } , M _ { z _ 2 } ) \\tau _ S . \\end{align*}"}
+{"id": "1563.png", "formula": "\\begin{align*} z _ { 3 } s ^ { \\prime } v _ { 1 } ^ { - 1 } = e , v _ { 1 } z _ { 1 } v _ { 2 } ^ { - 1 } = e , v _ { 2 } s ^ { \\prime } v _ { 3 } ^ { - 1 } = e , z _ { 3 } ^ { - 1 } v _ { 3 } v _ { 4 } ^ { - 1 } = e . \\end{align*}"}
+{"id": "8201.png", "formula": "\\begin{align*} \\sum _ { j \\le j _ 0 } \\psi _ j 2 ^ { j r } \\Vert \\dot { \\Delta } _ j ( T _ { f } g ) \\Vert _ { L ^ 2 } \\le C \\Big ( \\sum _ { j ' \\le j _ 0 + 4 } 2 ^ { j ' r _ 1 } \\Vert \\dot \\Delta _ { j ' } f \\Vert _ { L ^ \\infty } \\Big ) \\sum _ { j \\le j _ 0 } \\sum _ { \\vert j - j ' \\vert \\le 4 } 2 ^ { j ' ( r - r _ 1 ) } \\Vert \\dot \\Delta _ { j ' } g \\Vert _ { L ^ 2 } \\psi _ j . \\end{align*}"}
+{"id": "3102.png", "formula": "\\begin{align*} F _ { \\N } ( \\sigma ^ k ) = \\sum _ { d | k } d = s ( k ) \\end{align*}"}
+{"id": "2133.png", "formula": "\\begin{align*} \\mathrm { c o r a n k } _ { \\Lambda ( G ) } ( J _ v ^ { \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) ) & = \\mathrm { c o r a n k } _ { \\Omega ( G ) } ( ( J _ v ^ { \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) ) _ p ) \\\\ & = \\mathrm { c o r a n k } _ { \\Omega ( G ) } ( { ^ { \\pm } \\widetilde { K } _ v ( E _ { p } / L _ \\infty ) } ) . \\end{align*}"}
+{"id": "3372.png", "formula": "\\begin{align*} h _ d \\ , x ^ d + h _ { d - 1 } \\ , x ^ { d - 1 } + \\dots + h _ 0 = \\sum _ { k = - 1 } ^ { d - 1 } f _ k \\ , x ^ { k + 1 } ( 1 - x ) ^ { d - 1 - k } \\end{align*}"}
+{"id": "7698.png", "formula": "\\begin{align*} \\Delta \\log f ( x ) = - \\frac { f ' ( x ) ^ 2 } { f ( x ) ^ 2 } + \\frac { f '' ( x ) } { f ( x ) } \\ge 0 . \\end{align*}"}
+{"id": "2529.png", "formula": "\\begin{align*} x ( i ) = \\begin{cases} 1 _ { 2 \\Z } ( i ) & i \\in [ a ( n ) , a ( n ) + n ] n \\\\ 1 _ { 3 \\Z } ( i ) & i \\in [ b ( n ) , b ( n ) + n ] n \\\\ 1 _ { 5 \\Z } ( i ) & i \\in [ a ( n ) + b ( n ) , a ( n ) + b ( n ) + n ] n \\\\ 0 & \\\\ \\end{cases} \\end{align*}"}
+{"id": "7479.png", "formula": "\\begin{align*} B _ { a , b } = B _ { a - 1 , b } + B _ { b , a - 1 } \\ , . \\end{align*}"}
+{"id": "5654.png", "formula": "\\begin{align*} H ( f ) ( X , Y ) = g _ { \\nabla f } ( D ^ { \\nabla f } _ X \\nabla f , Y ) , \\ \\ \\ \\forall X , Y \\in T _ { \\mathbb { R } } M | _ { \\mathcal { U } _ f } . \\end{align*}"}
+{"id": "4391.png", "formula": "\\begin{align*} D _ { ( 2 ) } T ^ t g = T ^ t D _ { ( 2 ) } g . \\end{align*}"}
+{"id": "4717.png", "formula": "\\begin{align*} \\widetilde { K } _ { \\lambda } ( z , w ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { n + \\lambda + 2 } { \\lambda + 1 } \\phi _ { n } ( z ) \\overline { \\phi _ { n } ( w ) } . \\end{align*}"}
+{"id": "2724.png", "formula": "\\begin{align*} M _ { \\varphi } ( 2 n , j , 1 ; r - 1 ) = \\frac { 1 } { 2 } S ( n , r ) \\sum _ { u = 0 } ^ { n - j } \\binom { j + u } { u } c ( n , j + u , r - 1 ) \\end{align*}"}
+{"id": "1285.png", "formula": "\\begin{align*} \\gamma ^ \\dagger = \\gamma ^ { - 1 } . \\end{align*}"}
+{"id": "6946.png", "formula": "\\begin{align*} _ { t = \\frac { 1 } { y } } \\ , \\omega & = - _ { s = 0 } \\ , ( 1 - s ) ^ { - \\chi + ( N + 1 ) d } ( s + y - 1 ) ^ { \\chi - N d } y ^ { - d - 1 } \\frac { d s } { s ^ { d + 1 } } \\\\ & = - y ^ { - d - 1 } \\left [ s ^ { d } \\right ] ( 1 - s ) ^ { - \\chi + ( N + 1 ) d } ( s + y - 1 ) ^ { \\chi - N d } \\\\ & = - y ^ { - d - 1 } \\sum _ { k = 0 } ^ { d } ( - 1 ) ^ k \\binom { - \\chi + ( N + 1 ) d } { k } \\binom { \\chi - N d } { d - k } ( y - 1 ) ^ { \\chi - N d - d + k } . \\end{align*}"}
+{"id": "3776.png", "formula": "\\begin{align*} A _ { j _ 1 , \\ldots , j _ { 2 l } } = 2 \\sum _ { r = 1 } ^ { j _ 1 } C _ { r - 1 } A _ { j _ 1 - r , j _ 2 , \\ldots , j _ { 2 l } } + \\sum _ { r = 2 } ^ { 2 l } ( 2 j _ r + 1 ) C _ { j _ 1 + j _ r } A _ { j _ 2 , \\ldots , \\widehat { j _ r } , \\ldots , j _ { 2 l } } . \\end{align*}"}
+{"id": "6023.png", "formula": "\\begin{align*} L ^ { \\tilde { \\pi } } _ { r } ( t ) = L ^ { \\pi ^ { i , j ^ * } } _ { r } ( t ) , R ^ { \\tilde { \\pi } } _ { r } ( t ) = \\beta ( x - x _ { j ^ * } ) 1 _ { \\lbrace t = 0 \\rbrace } + R ^ { \\pi ^ { i , j ^ * } } _ { r } ( t ) 1 _ { \\lbrace t > 0 \\rbrace } , \\end{align*}"}
+{"id": "4573.png", "formula": "\\begin{align*} H u ( n ) = u ( n + 1 ) + u ( n - 1 ) + V ( n ) u ( n ) = E u ( n ) , \\ \\ n \\geq 1 , \\end{align*}"}
+{"id": "422.png", "formula": "\\begin{align*} v _ h ( y , h ) = 0 ~ ~ \\mathrm { a s } ~ ~ y = ( 1 - \\alpha ) h ^ { ( 1 - \\alpha ) \\gamma _ 1 - 1 } . \\end{align*}"}
+{"id": "7894.png", "formula": "\\begin{align*} R ( a ) \\cdot R ( b ) = R \\big ( R ( a ) \\cdot b + a \\cdot R ( b ) \\big ) + \\kappa ~ \\ ! a \\cdot b , ~ a , b \\in A . \\end{align*}"}
+{"id": "6027.png", "formula": "\\begin{align*} V ( x , i ) = \\sup _ { \\pi \\in \\mathcal { A } } V _ { \\pi } ( x , i ) \\leq V _ + ( x , i ) , ( x , i ) \\in [ 0 , \\infty ) \\times E . \\end{align*}"}
+{"id": "2076.png", "formula": "\\begin{align*} S \\begin{bmatrix} A _ { 1 1 } \\\\ A _ { 2 1 } \\end{bmatrix} = : \\begin{bmatrix} X _ 1 & X _ 2 \\\\ Y _ 1 & Y _ 2 \\end{bmatrix} , \\end{align*}"}
+{"id": "1969.png", "formula": "\\begin{align*} \\mathcal { P } _ I + \\mathcal { P } _ J = \\mathcal { P } _ { I \\cup J } \\end{align*}"}
+{"id": "713.png", "formula": "\\begin{align*} c _ s ^ 2 \\partial _ { x 1 } T = - \\frac { 1 } { \\Delta t } { \\varsigma } _ 1 { m } _ 1 ^ { ( 1 ) } , \\end{align*}"}
+{"id": "7492.png", "formula": "\\begin{align*} \\sum _ { \\kappa = 2 } ^ m a _ { \\mu \\kappa } c _ { \\kappa \\mu ' } = \\delta _ { \\mu \\mu ' } , ( 2 \\le \\mu , \\mu ' \\le m ) \\ , . \\end{align*}"}
+{"id": "7606.png", "formula": "\\begin{align*} x ^ * ( \\theta ) = \\arg \\min _ { x } \\pi ( f ) = \\int _ { V } F ( x ( \\theta ) , \\theta , v ) \\mu ( \\theta , d v ) . \\end{align*}"}
+{"id": "7246.png", "formula": "\\begin{align*} M ( t ) = \\sum _ { j , j ' \\in \\mathbb { Z } ^ 2 } \\langle j \\rangle ^ a \\langle j ' \\rangle ^ a \\big ( \\int _ { \\mathbb { R } ^ 2 } \\int _ { \\mathbb { R } ^ 2 } | w _ { j ' } ( t , y ) | ^ 2 \\frac { ( x - y ) } { | x - y | } \\cdot I m [ \\bar { w } _ j \\nabla w _ j ] ( t , x ) d x d y \\big ) . \\end{align*}"}
+{"id": "892.png", "formula": "\\begin{align*} H = \\Delta u + \\frac { B } { 2 } | x | ^ 2 , \\end{align*}"}
+{"id": "2981.png", "formula": "\\begin{align*} & \\lim _ { p \\downarrow p _ 0 } \\frac { ( d + 2 ) ( p - 1 ) } { 2 p } = \\frac { d + 2 - 6 \\alpha } { 6 } , \\\\ & \\lim _ { \\theta \\uparrow 3 } \\frac { ( d + 2 ) [ \\theta - ( \\theta - 1 ) p ] } { 2 \\alpha \\theta p } = \\frac { ( d + 2 ) ( 3 - 2 p ) } { 6 \\alpha p } \\in ( 0 , 1 ) , p \\in ( p _ 0 , 3 / 2 ) . \\end{align*}"}
+{"id": "276.png", "formula": "\\begin{align*} ( d _ { n - 1 } ( Y ) , d _ { n - 2 } ( Y ) , \\ldots , d _ 0 ( Y ) ) & = ( 1 , 2 , \\ldots , n ) , \\\\ ( d _ 0 ( Y ) , d _ { - 1 } ( Y ) , \\ldots , d _ { - n + 1 } ( Y ) ) & = ( n , n - 1 , \\ldots , 1 ) . \\end{align*}"}
+{"id": "464.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom { n } { k } \\frac { 2 ^ k } { k + 1 } \\left ( ( - 1 ) ^ k \\frac { F _ { j ( k + 1 ) } } { L ^ k _ j } \\Bigl ( \\frac { L _ j } { \\sqrt 5 F _ j } \\Bigr ) ^ n - ( 1 + ( - 1 ) ^ n ) \\frac { 2 ^ { k + 3 } - 2 } { k + 2 } F _ j B _ { k + 2 } \\right ) = 0 \\ , . \\end{align*}"}
+{"id": "1912.png", "formula": "\\begin{align*} \\begin{aligned} & _ { x , v } ' ( a , r ) : = \\sup _ { t , x , v } r ^ { - 8 d } \\\\ & \\times \\int _ { x _ 1 , x _ 2 \\in B _ { r ^ 3 } ( x ) } \\int _ { v _ 1 , v _ 2 \\in B _ r ( v ) } | a ( t , x _ 1 , v _ 1 ) - a ( t , x _ 2 , v _ 2 ) | \\ , d x _ 1 d x _ 2 \\ , d v _ 1 d v _ 2 \\le \\omega ( r ) . \\end{aligned} \\end{align*}"}
+{"id": "628.png", "formula": "\\begin{align*} C ^ { \\ast } \\left ( X , Y , Z \\right ) \\equiv \\left ( \\nabla _ { Z } ^ { \\ast } g \\right ) \\left ( X , Y \\right ) = - C \\left ( X , Y , Z \\right ) \\end{align*}"}
+{"id": "7085.png", "formula": "\\begin{gather*} h _ { { R _ Z ^ H } } ( Z , Z _ \\bullet , Z _ \\bullet ' ) = Z _ 4 Z _ 9 + Z _ { 7 } Z _ { 6 } ' + Z _ 6 Z _ { 7 } ' + Z _ 4 Z _ 9 ' , h _ H ( Z ) = 4 Z _ 4 Z _ 9 . \\end{gather*}"}
+{"id": "3023.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { \\bar { x } } _ { 1 , 1 } & = \\bar { x } _ { 1 , 2 } \\\\ \\dot { \\bar { x } } _ { 1 , 2 } & = \\bar { x } _ { 1 , 3 } \\\\ & \\vdotswithin { = } \\\\ \\dot { \\bar { x } } _ { 1 , \\rho _ 1 } & = \\bar { u } _ 1 \\end{aligned} \\begin{aligned} \\dot { \\bar { x } } _ { 2 , 1 } & = \\bar { x } _ { 2 , 2 } \\\\ \\dot { \\bar { x } } _ { 2 , 2 } & = \\bar { x } _ { 2 , 3 } \\\\ & \\vdotswithin { = } \\\\ \\dot { \\bar { x } } _ { 2 , \\rho _ 2 } & = \\bar { u } _ 2 \\end{aligned} \\end{align*}"}
+{"id": "3818.png", "formula": "\\begin{align*} \\left ( a _ { h - 2 } , a _ { h - 1 } , a _ h \\right ) = \\left \\{ \\begin{aligned} & \\left ( n - 5 , n - 4 , n - 3 \\right ) , \\left ( n - 5 , n - 4 , n - 2 \\right ) , \\\\ & \\left ( n - 5 , n - 3 , n - 1 \\right ) , \\left ( n - 5 , n - 2 , n - 1 \\right ) , \\\\ & \\left ( n - 4 , n - 3 , n \\right ) , \\left ( n - 4 , n - 2 , n \\right ) , \\\\ & \\left ( n - 3 , n - 1 , n \\right ) , \\left ( n - 2 , n - 1 , n \\right ) \\end{aligned} \\right \\} . \\end{align*}"}
+{"id": "7398.png", "formula": "\\begin{align*} a = \\frac { \\alpha _ { \\pm } } { 4 } - \\frac { k _ { \\pm } } { \\alpha _ { \\pm } } \\end{align*}"}
+{"id": "6593.png", "formula": "\\begin{align*} 4 8 [ \\alpha , \\Gamma ] = [ B ^ 2 + B B ^ * + B ^ * B + ( B ^ * ) ^ 2 , C ^ 2 + C C ^ * + C ^ * C + ( C ^ * ) ^ 2 ] . \\end{align*}"}
+{"id": "2148.png", "formula": "\\begin{align*} - \\mathrm { d i v } _ { \\bf R } \\ ; a \\left ( \\mathrm { g r a d } _ { \\bf R } \\ ; \\theta ( \\vec { x } , \\cdot ) \\right ) = f ( \\vec { x } , \\cdot ) \\ ; , a . e . \\ ; \\vec { x } \\in \\Omega \\end{align*}"}
+{"id": "3293.png", "formula": "\\begin{align*} \\Lambda _ 1 = A _ 1 , \\Lambda _ 2 = A _ 3 , \\Lambda _ 3 = A _ 0 , \\Lambda _ { 1 2 } = L , \\Lambda _ { 2 3 } = K , \\Lambda _ { 1 2 3 } = - A _ 2 , \\end{align*}"}
+{"id": "5812.png", "formula": "\\begin{align*} \\Pi C \\Pi ^ * = M _ { \\Theta } \\end{align*}"}
+{"id": "5141.png", "formula": "\\begin{align*} \\| \\nabla ( S _ { \\mathcal W } v ) \\| _ { L ^ 1 ( \\Omega ) } & = \\sum _ { i = 1 } ^ \\infty \\| \\nabla ( S _ { \\mathcal W } v ) \\| _ { L ^ 1 ( Q _ i ) } \\\\ & \\le C \\sum _ { i = 1 } ^ \\infty \\sum _ { Q _ j \\cap Q _ i \\ne \\emptyset } ( \\| D v \\| ( Q _ i \\cup Q _ j ) ) \\\\ & \\leq C \\| D v \\| ( \\Omega ) . \\end{align*}"}
+{"id": "1758.png", "formula": "\\begin{align*} { \\rm s i g n } \\left \\langle Z _ { * } , I \\backslash \\left \\{ j { \\rm t h } \\ , { \\rm e l e m e n t } \\right \\} \\right \\rangle = \\left ( - 1 \\right ) ^ { j } { \\rm s i g n } \\left \\langle I \\right \\rangle \\end{align*}"}
+{"id": "6134.png", "formula": "\\begin{align*} ( V _ 1 , V _ 2 ) = ( R _ { q } M _ { z _ 1 } , M _ { z _ 2 } ) ( M _ { z _ 1 } R _ { q } , M _ { z _ 2 } ) . \\end{align*}"}
+{"id": "1855.png", "formula": "\\begin{align*} z _ { i , j } & + z _ { i , j + 1 } + z _ { i + 1 , j } + z _ { i + 1 , j + 1 } \\\\ & = x _ { i , j } + x _ { i , j + 1 } + x _ { i + 1 , j } + x _ { i + 1 , j + 1 } = S . \\end{align*}"}
+{"id": "3266.png", "formula": "\\begin{align*} \\mu _ m \\langle \\phi _ m , \\psi _ n \\rangle & = \\langle L \\phi _ m , \\psi _ n \\rangle = \\langle \\phi _ m , L \\psi _ n \\rangle \\\\ & = a _ n \\langle \\phi _ m , \\psi _ { n - 1 } \\rangle + b _ n \\langle \\phi _ m , \\psi _ { n } \\rangle + a _ { n + 1 } \\langle \\phi _ m , \\psi _ { n + 1 } \\rangle , \\end{align*}"}
+{"id": "2895.png", "formula": "\\begin{align*} \\phi ( s , t ) = \\gamma ( s ) + \\sum _ { i = 1 } ^ { n - 1 } t _ i w _ i , t = ( t _ 1 , \\ldots , t _ { n - 1 } ) , \\end{align*}"}
+{"id": "472.png", "formula": "\\begin{align*} B _ n ( x ) - ( - 1 ) ^ n B _ n ( y ) = 0 , \\end{align*}"}
+{"id": "1460.png", "formula": "\\begin{align*} | \\Phi _ { \\beta , \\varepsilon } ( t , x ; t _ 0 ) | & \\le K _ { \\beta , \\varepsilon } ( t _ 0 + t ) ^ { - \\beta } \\left ( 1 + \\frac { \\widetilde { \\gamma } _ { \\varepsilon } A _ { \\varepsilon } ( x ) } { t _ 0 + t } \\right ) ^ { - \\beta } \\\\ & \\le C \\left ( t _ 0 + t + A _ { \\varepsilon } ( x ) \\right ) ^ { - \\beta } \\\\ & = C \\Psi ( t , x ; t _ 0 ) ^ { - \\beta } \\end{align*}"}
+{"id": "5548.png", "formula": "\\begin{align*} - \\Tilde { \\lambda _ 1 } ( 0 ) \\dfrac { \\partial T _ 1 } { \\partial r } \\bigg | _ { r = \\beta ( t ) } = - \\Tilde { \\lambda _ 2 } ( 0 ) \\dfrac { \\partial T _ 2 } { \\partial r } \\bigg | _ { r = \\beta ( t ) } + l _ m \\gamma _ m \\beta ' ( t ) / ( \\theta _ b - \\theta _ m ) , \\end{align*}"}
+{"id": "2754.png", "formula": "\\begin{align*} a ( H ) = H \\cdot \\pi _ { 2 * } \\mu _ * ( \\ell _ 1 ) . \\end{align*}"}
+{"id": "1827.png", "formula": "\\begin{align*} & [ N _ T ( x + a ) , N _ T ( y + b ) ] _ { \\rho } = [ x + T ( a ) , y + T ( b ) ] _ { \\rho } = [ x , y ] + [ x , T ( b ) ] - [ T ( a ) , y ] + [ T ( a ) , T ( b ) ] . \\end{align*}"}
+{"id": "5391.png", "formula": "\\begin{align*} ( \\ker d _ A ) _ { k = 0 } \\equiv 0 , \\end{align*}"}
+{"id": "2334.png", "formula": "\\begin{align*} & Z ( a _ { 1 } , b _ { 1 } , a _ { 2 } , b _ { 2 } , a _ { 3 } ) + Z ( a _ { 2 } , b _ { 1 } , a _ { 3 } , b _ { 2 } , a _ { 1 } ) + Z ( a _ { 3 } , b _ { 1 } , a _ { 1 } , b _ { 2 } , a _ { 2 } ) \\\\ = & Z ( a _ { 1 } , b _ { 2 } , a _ { 2 } , b _ { 1 } , a _ { 3 } ) + Z ( a _ { 2 } , b _ { 2 } , a _ { 3 } , b _ { 1 } , a _ { 1 } ) + Z ( a _ { 3 } , b _ { 2 } , a _ { 1 } , b _ { 1 } , a _ { 2 } ) . \\end{align*}"}
+{"id": "6979.png", "formula": "\\begin{align*} \\lambda ( \\alpha _ 1 , \\alpha _ 2 ) : = \\inf _ { f \\in \\mathcal { F } } \\int _ { X } \\left ( | \\nabla f | ^ 2 + \\alpha _ 1 f ^ 2 - \\frac { \\alpha _ 2 } { 2 } f ^ 2 \\log f ^ 2 \\right ) d m \\end{align*}"}
+{"id": "3242.png", "formula": "\\begin{align*} \\int _ { X } \\psi ( u - v ) \\theta ^ { n } _ { u } + \\int _ { X } \\psi ( u - v ) \\theta ^ { n } _ { v } = 0 . \\end{align*}"}
+{"id": "1297.png", "formula": "\\begin{align*} \\{ H f _ k \\mid 1 \\leq k \\leq n \\} = H \\sum _ { k = 1 } ^ n f _ k = H \\sum _ { k = 1 } ^ m g _ k = \\{ H g _ k \\mid 1 \\leq k \\leq m \\} , \\end{align*}"}
+{"id": "4820.png", "formula": "\\begin{align*} \\frac \\dd { \\dd t } \\eta _ t ^ n = A ( t , \\eta _ t ^ n ) = ( \\Delta f ^ n ( X ( t ) , S ( t ) ) - K ) \\eta _ t ^ n + K \\bar \\eta _ t ^ n \\otimes \\mu . \\end{align*}"}
+{"id": "5106.png", "formula": "\\begin{align*} [ ( - \\Delta _ x ) ^ { \\sigma } , B ] = \\frac { \\sin ( \\pi \\sigma ) } { \\pi } \\int _ 0 ^ { \\infty } m ^ { \\sigma } \\frac { 1 } { - \\Delta _ x + m } [ - \\Delta _ x , B ] \\frac { 1 } { - \\Delta _ x + m } \\ , d m . \\end{align*}"}
+{"id": "6345.png", "formula": "\\begin{align*} \\mathcal { A } _ { P } ^ { ( m ) } ( F ) : = i n f \\left \\{ \\mathcal { A } _ { P } ^ { ( m ) } ( F , G ) : \\right \\} . \\end{align*}"}
+{"id": "7007.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & b = 2 a + d , \\\\ & b - \\varepsilon = a ^ 2 + \\frac { N } { N - 1 } a d + \\frac { N } { 2 ( N - 1 ) } d ^ 2 , \\\\ & b + 1 = 2 a + \\frac { 3 N } { 2 ( N - 1 ) } d . \\end{aligned} \\right . \\end{align*}"}
+{"id": "1587.png", "formula": "\\begin{align*} T _ { s _ { i } } : V = W _ { b _ { 1 } } \\oplus W _ { b _ { 1 } } \\oplus W _ { b _ { 3 } } \\rightarrow W _ { s _ { i } } \\\\ T _ { s _ { i } } \\left ( v _ { 1 } , v _ { 2 } , v _ { 3 } \\right ) = v _ { j } - \\rho ( s ) \\left ( v _ { i } \\right ) , \\end{align*}"}
+{"id": "517.png", "formula": "\\begin{align*} f _ { y , \\mathbf { X } } ( \\beta ) : = \\sum _ { i = 1 } ^ p V ( \\beta _ i ) - \\frac { 1 } { 2 \\sigma ^ 2 } \\left | y - \\mathbf { X } \\beta \\right | ^ 2 . \\end{align*}"}
+{"id": "6466.png", "formula": "\\begin{align*} D _ x ^ \\beta p ^ { \\# } ( x , \\xi ) \\in \\begin{cases} S ^ { m } _ { 1 , \\delta } , & | \\beta | \\le \\tau \\\\ S ^ { m + \\delta ( | \\beta | - r ) } _ { 1 , \\delta } , & | \\beta | > \\tau \\end{cases} \\end{align*}"}
+{"id": "6650.png", "formula": "\\begin{align*} 1 + \\prod _ { j = 0 } ^ { \\kappa - 1 } \\lambda _ { - j } ^ { 2 } \\sum _ { i = 1 } ^ { \\eta } \\lambda _ { i , 1 } ^ { 2 } s _ { i , - \\kappa - 1 } \\le \\lambda _ { - \\kappa + 1 } ^ { 2 } , \\end{align*}"}
+{"id": "6718.png", "formula": "\\begin{align*} ( \\bar { \\rho } , \\bar { u } , \\bar { E } , \\bar { B } ) ( 0 , x ) = ( \\bar { \\rho } _ { 0 } , \\bar { u } _ { 0 } , \\bar { E } _ { 0 } , \\bar { B } _ { 0 } ) ( x ) . \\end{align*}"}
+{"id": "446.png", "formula": "\\begin{align*} | Q ( \\eta _ \\delta W ) - Q ( W ) | & \\leq \\int _ { \\delta ^ 2 \\leq | z | \\leq \\delta } | \\eta _ \\delta ( d W ) ^ N | ^ 2 + | ( W d \\eta _ \\delta ) ^ N | ^ 2 + 2 | \\eta _ \\delta ( d W ) ^ N | ^ 2 | ( W d \\eta _ \\delta ) ^ N | ^ 2 + O ( \\delta ^ 2 ) \\\\ & = O ( \\delta ^ 2 ) + O \\Big ( \\frac { 1 } { \\log \\delta ^ { - 1 } } \\Big ) = O \\Big ( \\frac { 1 } { \\log \\delta ^ { - 1 } } \\Big ) . \\end{align*}"}
+{"id": "3096.png", "formula": "\\begin{align*} \\sigma _ 4 & = \\delta _ { 3 , 0 } \\prod _ { n \\in \\Z } \\alpha _ { 3 n + 1 } \\\\ & = ( \\ldots , - 6 , - 3 , 0 , 3 , 6 , \\ldots ) \\cdots ( - 5 , - 4 ) ( - 2 , - 1 ) ( 1 , 2 ) ( 4 , 5 ) ( 7 , 8 ) \\cdots \\end{align*}"}
+{"id": "7138.png", "formula": "\\begin{align*} \\left \\langle \\nu ' , \\frac { 3 } { 2 } \\mathfrak { g } ^ { \\lambda < 0 } + \\sum _ { i = 1 } ^ k v _ i \\tau _ { d _ i } + d \\mu \\tau _ d \\right \\rangle & = \\left \\langle \\nu ' , \\frac { 3 } { 2 } \\mathfrak { g } ^ { \\lambda < 0 } + v _ k \\tau _ { d _ k } + d _ k \\mu \\tau _ { d _ k } \\right \\rangle \\\\ & \\leq \\left \\langle \\nu ' , \\frac { 3 } { 2 } \\mathfrak { g } ^ { \\lambda < 0 } + d _ k \\tau _ { d _ k } \\right \\rangle , \\end{align*}"}
+{"id": "6885.png", "formula": "\\begin{align*} | f | _ K ( \\mathbf w ) = \\sum _ { j = 0 } ^ k x _ j f ( \\mathbf v _ j ) . \\end{align*}"}
+{"id": "323.png", "formula": "\\begin{align*} h ( t + 1 ) = h ( t ) + \\zeta ( t ) \\end{align*}"}
+{"id": "12.png", "formula": "\\begin{align*} p _ { T _ 1 , T _ 2 } ( X ) = \\left ( \\prod _ { j = 1 } ^ { m - 1 } \\prod _ { s = j + 1 } ^ m \\frac { 1 } { r ( s , j ) ! \\ , u ( s , j ) ! } ( d _ { s \\to j } ) ^ { r ( s , j ) } ( d _ { j \\to s } ^ * ) ^ { u ( s , j ) } \\right ) \\cdot P _ { \\lambda } ( Z ) . \\end{align*}"}
+{"id": "3596.png", "formula": "\\begin{align*} C T ^ { 1 / 2 } = \\sum _ { m , i } b _ m \\sqrt { a _ i } \\langle \\psi _ m , \\phi _ i \\rangle \\psi _ m \\otimes \\phi _ i = \\sum _ { m , i } b _ m \\sqrt { a _ i } \\theta _ { m i } \\psi _ m \\otimes \\phi _ i \\end{align*}"}
+{"id": "1193.png", "formula": "\\begin{align*} A ^ U _ \\xi : = I ^ * ( { \\ge } \\xi , { < } \\nu ) \\end{align*}"}
+{"id": "6723.png", "formula": "\\begin{align*} P _ { 1 } ( v \\cdot \\nabla _ { x } M ) = P _ { 1 } \\big \\{ v \\cdot ( \\frac { | v - u | ^ { 2 } \\nabla _ { x } \\widetilde { \\theta } } { 2 R \\theta ^ { 2 } } + \\frac { ( v - u ) \\cdot \\nabla _ { x } \\widetilde { u } } { R \\theta } ) M \\big \\} + \\frac { 1 } { \\varepsilon } L _ { M } \\overline { G } . \\end{align*}"}
+{"id": "7537.png", "formula": "\\begin{align*} \\Re V ( p ) = r _ { \\infty } \\log | z _ { \\infty } ( p ) | + O ( 1 ) , p \\to p _ { \\infty } . \\end{align*}"}
+{"id": "4810.png", "formula": "\\begin{align*} \\mathcal D _ { i _ 0 } \\mathcal C _ { i _ 0 - 1 } \\cdots \\mathcal C _ { p } \\mathcal A U _ { n , k - 1 } = 0 , \\end{align*}"}
+{"id": "6938.png", "formula": "\\begin{align*} \\left [ \\epsilon ^ 0 q ^ { d } \\right ] \\frac { ( - 1 ) ^ { ( N - 1 ) d } } { \\det ( z _ i ^ { N - j + 1 } ) } \\begin{vmatrix} ( z _ 1 - y ) z _ 1 ^ { d + N - 1 } & \\cdots & ( z _ 1 - y ) z _ 1 ^ { d } \\\\ ( z _ 2 - y ) z _ 2 ^ { d + N - 1 } & \\cdots & ( z _ 2 - y ) z _ 2 ^ { d } \\\\ \\vdots & \\cdots & \\vdots \\\\ ( z _ N - y ) z _ N ^ { d + N - 1 } & \\cdots & ( z _ N - y ) z _ N ^ { d } \\end{vmatrix} \\prod _ { i \\in [ N ] } \\bigg ( \\frac { \\alpha _ i - z _ { N + 1 } } { \\alpha _ i - y } \\bigg ) ^ { b _ i } . \\end{align*}"}
+{"id": "5967.png", "formula": "\\begin{align*} f ^ { ( H ) } _ K = \\sum _ { \\alpha \\in 2 ^ { \\N } \\cap [ 1 , \\delta _ 0 ^ { - k ( k - 1 ) / 2 } ] } \\sum _ { \\beta \\in 2 ^ { \\N } \\cap [ 1 , \\nu / \\delta _ 0 ] } f ^ { ( H , \\alpha , \\beta ) } _ K \\end{align*}"}
+{"id": "2507.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\lim _ { m \\to \\infty } T ^ { c _ 1 ( j ) + c _ 2 ( m ) } a = \\lim _ { m \\to \\infty } \\lim _ { j \\to \\infty } T ^ { c _ 1 ( j ) + c _ 2 ( m ) } a . \\end{align*}"}
+{"id": "1416.png", "formula": "\\begin{align*} \\begin{dcases} \\lambda u - v = f , \\\\ \\lambda v - \\Delta u + a ( x ) v = g . \\end{dcases} \\end{align*}"}
+{"id": "4970.png", "formula": "\\begin{align*} l _ { \\rm B P } ( a _ { v } ) = \\log \\left ( \\frac { P ( a _ { v } = 1 ) } { P ( a _ { v } = 0 ) } \\right ) = l _ { \\rm M P A } ( a _ { v } ) + \\sum _ { c \\in \\mathcal { N } ( v ) } E _ { c \\rightarrow v } ^ { ( T ) } . \\end{align*}"}
+{"id": "927.png", "formula": "\\begin{align*} | F ^ { i j } ( \\nabla _ \\alpha \\varphi ) _ { i j } | \\leq C \\sqrt { b _ { n - 1 } } \\sum _ { i = 1 } ^ n F ^ { i i } \\mbox { i n } \\omega _ 1 . \\end{align*}"}
+{"id": "5291.png", "formula": "\\begin{align*} f ( x ) = \\int _ S a ( \\omega ) e ^ { 2 \\pi i \\omega x } d \\mu _ S ( \\omega ) , \\end{align*}"}
+{"id": "6633.png", "formula": "\\begin{align*} h _ \\theta ( u , u ) = \\int _ { \\Omega _ + } | \\nabla u _ + | ^ 2 \\dd x + \\int _ { \\Omega _ - } | \\nabla u _ - | ^ 2 \\dd x - \\int _ { \\Gamma _ \\theta } | u _ + - u _ - | ^ 2 \\dd \\sigma . \\end{align*}"}
+{"id": "4793.png", "formula": "\\begin{gather*} \\frac { n + a _ { p + 1 } + p + 1 } { n + 1 } \\frac { c _ { m - 1 } ( n + 1 ) } { ( p - m + 1 ) ! } \\frac { \\prod _ { i = 1 } ^ p ( n + 1 + a _ i + m - 1 ) } { \\prod _ { i = 1 } ^ { m - 1 } { ( n + 1 + i ) } } - \\frac { c _ m ( n ) } { ( p - m ) ! } \\frac { \\prod _ { i = 1 } ^ p ( n + a _ i + m ) } { \\prod _ { i = 1 } ^ m { ( n + i ) } } \\\\ = \\frac { \\prod _ { i = 1 } ^ { p + 1 } { ( n + a _ i + m ) } } { ( p + 1 - m ) ! \\prod _ { i = 1 } ^ m ( n + i ) } \\frac { ( n + a _ { p + 1 } + p + 1 ) c _ { m - 1 } ( n + 1 ) - ( p + 1 - m ) c _ m ( n ) } { n + a _ { p + 1 } + m } . \\end{gather*}"}
+{"id": "6332.png", "formula": "\\begin{align*} u = P _ N z + w = z + ( w - P _ M z ) \\in T ( N ) \\oplus M . \\end{align*}"}
+{"id": "5479.png", "formula": "\\begin{align*} F _ N ( a , 0 ; t ) = \\frac { ( 1 - t q ^ N ) } { ( 1 - t ) } \\sum _ { n = 0 } ^ { N } ( - 1 ) ^ n \\begin{bmatrix} N \\\\ n \\end{bmatrix} \\frac { ( q ) _ n ( a t ) ^ n q ^ { n ( n + 1 ) / 2 } } { ( t q ) _ n } . \\end{align*}"}
+{"id": "3148.png", "formula": "\\begin{align*} - \\Delta _ { \\mathbb { R } ^ N } u \\ , + \\ , u \\ , = \\ , u ^ p \\mbox { i n } \\ \\mathbb { R } ^ N , u \\in H ^ { 1 } ( \\mathbb { R } ^ N ) , u > 0 \\mbox { i n } \\ \\mathbb { R } ^ N , \\end{align*}"}
+{"id": "5859.png", "formula": "\\begin{align*} b e r ^ p ( A ) { ( 2 + 2 ^ { p } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } ) } < ( \\frac { p } { p - 1 } ) ^ { p } \\left ( b e r ( | A | ^ { 2 p \\alpha } ) + b e r ( | A ^ * | ^ { 2 p ( 1 - \\alpha ) } ) \\right ) ~ \\mbox { h o l d s } \\end{align*}"}
+{"id": "1884.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | 2 | ^ { n } \\mu ( 2 ^ { n + 1 } x , 2 ^ { n + 1 } y ) = 0 \\end{align*}"}
+{"id": "6063.png", "formula": "\\begin{align*} \\begin{cases} \\ , \\ , \\eta _ \\epsilon \\in H ^ 1 _ 0 ( \\Omega ) \\ , \\ , \\ , \\\\ [ 0 . 3 c m ] \\int _ { \\Omega _ \\epsilon } \\nabla \\eta _ \\epsilon \\nabla v d x - \\int _ { \\Omega } \\lambda _ \\epsilon \\eta _ \\epsilon v d x = 0 \\ , \\ , \\forall \\ , v \\in H ^ 1 _ 0 ( \\Omega ) \\end{cases} \\end{align*}"}
+{"id": "3123.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ k ) = F _ { Y _ 1 } ( \\tau _ 1 ^ k ) + F _ { Y _ 2 } ( \\tau _ 2 ^ k ) + F _ { X _ 2 } ( \\sigma _ 2 ^ k ) \\end{align*}"}
+{"id": "7404.png", "formula": "\\begin{align*} B _ 1 \\colon x = 0 , \\ ; \\ ; B _ 3 \\colon x - z = 0 , \\ ; \\ ; B _ 6 \\colon x z = u ^ 2 , \\ ; \\ ; B _ { 1 , 6 } \\colon x = u = 0 , \\ ; \\ ; P = ( 0 : 0 : 0 : 1 ) \\end{align*}"}
+{"id": "692.png", "formula": "\\begin{align*} { f _ i } \\left ( { { \\bf { x } } + { { \\bf { c } } _ i } \\Delta t , t + \\Delta t } \\right ) - { f _ i } \\left ( { { \\bf { x } } , t } \\right ) = { \\left ( { { { { \\bf { \\hat M } } } ^ { - 1 } } { \\bf { \\hat \\Lambda \\hat M } } } \\right ) _ { i j } } \\left [ { { f _ i } \\left ( { { \\bf { x } } , t } \\right ) - f _ i ^ { \\left ( { e q } \\right ) } \\left ( { { \\bf { x } } , t } \\right ) } \\right ] + { { \\bf { F } } _ i } \\left ( { { \\bf { x } } , t } \\right ) , \\end{align*}"}
+{"id": "4496.png", "formula": "\\begin{align*} \\lim _ { h \\rightarrow 0 } \\frac { 1 } { \\| h \\| } _ { \\ ! H ^ 1 } \\ ! & \\bigg | K ( u + h ) - K ( u ) - \\int _ { [ 0 , 1 ] } u h \\bigg | \\\\ & = \\lim _ { h \\rightarrow 0 } \\frac { 1 } { 2 \\| h \\| } _ { \\ ! H ^ 1 } \\ ! \\bigg | \\int _ { [ 0 , 1 ] } h ^ 2 \\bigg | \\leq \\lim _ { h \\rightarrow 0 } \\frac { 1 } { 2 } \\frac { \\| h \\| _ { H ^ 1 } ^ 2 } { \\| h \\| _ { H ^ 1 } } = 0 \\ , , \\end{align*}"}
+{"id": "477.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { \\binom n k F _ { j k + m } B _ { n - k } z ^ k } = \\frac { 1 } { \\sqrt 5 } \\big ( \\alpha ^ m B _ n ( \\alpha ^ j z ) - \\beta ^ m B _ n ( \\beta ^ j z ) \\big ) \\ , , \\end{align*}"}
+{"id": "7605.png", "formula": "\\begin{align*} f ( x ( \\theta ) , \\theta ) = \\mathbb { E } \\left ( F ( x ( \\theta ) , \\theta , v ) \\right ) \\end{align*}"}
+{"id": "1782.png", "formula": "\\begin{align*} \\sum _ { i \\in A } \\left ( - 1 \\right ) ^ { \\# i } p _ { A \\backslash i } \\left ( W \\right ) \\langle Y , B , i \\rangle = 0 . \\end{align*}"}
+{"id": "6193.png", "formula": "\\begin{align*} | \\mathcal { I } ' _ { k - 1 } | = { k + 3 \\choose 4 } . \\end{align*}"}
+{"id": "3154.png", "formula": "\\begin{align*} J ( u ) : = \\frac { \\| u \\| _ { \\lambda } ^ { 2 } } { \\left ( \\int _ { \\mathbb { B } ^ { N } } a ( x ) | u ( x ) | ^ { p + 1 } \\mathrm { ~ d } V _ { \\mathbb { B } ^ { N } } ( x ) \\right ) ^ { \\frac { 2 } { p + 1 } } } , J _ { \\infty } ( u ) : = \\frac { \\| u \\| _ { \\lambda } ^ { 2 } } { \\left ( \\int _ { \\mathbb { B } ^ { N } } | u ( x ) | ^ { p + 1 } \\mathrm { ~ d } V _ { \\mathbb { B } ^ { N } } ( x ) \\right ) ^ { \\frac { 2 } { p + 1 } } } \\\\ \\end{align*}"}
+{"id": "7623.png", "formula": "\\begin{align*} \\widehat { G } = \\{ \\chi : G \\rightarrow \\C ^ \\ast \\vert \\chi \\} . \\end{align*}"}
+{"id": "7402.png", "formula": "\\begin{align*} X \\colon \\ ; w ^ 2 = x y ( x - z ) ( y - z ) ( y z - ( x - t ) ^ 2 ) ( x z - ( y - t ) ^ 2 ) \\ ; . \\end{align*}"}
+{"id": "319.png", "formula": "\\begin{align*} F ( t ) - t = \\left ( \\frac { 1 } { t _ { 2 } ' } , t _ { 2 } ' , \\frac { t _ { 3 } ' } { t _ { 1 } '^ { p + p ^ { 2 } } t _ { 2 } '^ { p ^ { 2 } } } \\right ) \\end{align*}"}
+{"id": "1235.png", "formula": "\\begin{align*} \\int _ T ^ { 2 T } \\prod _ { 1 \\le i \\le r } ( \\cos ( t \\log p _ i ) ) ^ { a _ i } \\prod _ { r + 1 \\le i \\le s } ( \\cos ( 2 t \\log p _ i ) ) ^ { a _ i } d t = T g ( n ) + \\sideset { } { ' } \\sum _ { \\ell _ 1 , \\ldots , \\ell _ s } \\prod _ { 1 \\le i \\le s } \\frac { 1 } { 2 ^ { a _ i } } { a _ i \\choose \\ell _ i } \\int _ T ^ { 2 T } ( * ) d t . \\end{align*}"}
+{"id": "3660.png", "formula": "\\begin{align*} p ( ( x _ 1 + x _ 2 ) / 2 , ( x _ 1 + x _ 2 ) / 2 , x _ 3 , \\ldots , x _ m ) = p ( x _ 1 , x _ 2 , x _ 3 , \\ldots , x _ m ) = \\lambda ( p ) . \\end{align*}"}
+{"id": "8156.png", "formula": "\\begin{align*} \\alpha ^ { \\delta } \\prod _ { i = 1 } ^ d \\| e _ i \\| ^ { a _ i } \\leqslant \\| \\boldsymbol { e } ^ { \\boldsymbol { a } } \\| \\leqslant \\prod _ { i = 1 } ^ d \\| e _ i \\| ^ { a _ i } . \\end{align*}"}
+{"id": "3816.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 ( 2 n - 4 ) - ( a _ 2 + a _ 3 + a _ 4 ) + d } { 2 } . \\end{align*}"}
+{"id": "3056.png", "formula": "\\begin{align*} C _ { \\sigma } ( \\ell ) = \\frac { 1 } { \\ell } \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } \\mu \\left ( \\frac { \\ell } { k } \\right ) F _ X ( \\sigma ^ k ) \\end{align*}"}
+{"id": "4543.png", "formula": "\\begin{align*} \\beta = \\left ( \\frac { \\beta } { v } \\right ) v . \\end{align*}"}
+{"id": "5583.png", "formula": "\\begin{align*} \\left ( G _ { \\alpha \\bar { \\beta } } \\right ) = \\left ( \\frac { \\partial ^ 2 G } { \\partial v ^ { \\alpha } \\partial v ^ { \\bar { \\beta } } } \\right ) . \\end{align*}"}
+{"id": "3717.png", "formula": "\\begin{align*} \\partial _ t \\rho ( x , t ) + \\nabla ( \\rho ( x , t ) \\nabla S ( x , t ) ) = 0 \\end{align*}"}
+{"id": "6663.png", "formula": "\\begin{align*} \\underline { P } ( 0 ) & = \\sigma \\left ( \\frac { \\underline { Q } ( t ) - \\underline { Q } ( 0 ) } { t } \\right ) \\\\ \\overline { P } ( 0 ) & = \\sigma \\left ( \\frac { \\overline { Q } ( t ) - \\overline { Q } ( 0 ) } { t } \\right ) \\end{align*}"}
+{"id": "1876.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 0 \\\\ } } ^ { l } 2 ^ { l - k } \\binom { l } { k } = \\frac { 3 ^ { l } + 1 } { 2 } , l \\geq 0 . \\end{align*}"}
+{"id": "261.png", "formula": "\\begin{align*} L ( v ) & = \\int _ U \\left ( \\langle W _ u ( u ^ { - 1 } ( y ) ) , \\phi ( u ^ { - 1 } ( y ) ) \\rangle + W _ \\xi ( u ^ { - 1 } ( y ) ) : [ \\nabla _ x \\phi ] ( u ^ { - 1 } ( y ) ) \\right ) d y \\\\ & = \\int _ \\Omega \\left ( \\langle W _ u ( x ) , \\phi ( x ) \\rangle + W _ \\xi ( x ) : [ \\nabla _ x \\phi ] ( x ) \\right ) d x . \\end{align*}"}
+{"id": "4794.png", "formula": "\\begin{align*} \\begin{cases} \\frac { x - 1 } 2 P _ { n + 1 } = \\frac { 1 } { 2 ( 2 n + 3 ) } ( ( n + 2 ) P _ { n + 2 } - ( 2 n + 3 ) P _ { n + 1 } + ( n + 1 ) P _ n ) , \\\\ \\frac 1 { 2 n + 3 } ( P _ { n + 2 } ' - P _ n ' ) = P _ { n + 1 } . \\end{cases} \\end{align*}"}
+{"id": "7311.png", "formula": "\\begin{align*} d \\geq \\begin{cases} 2 \\delta + 1 & { \\rm i f ~ } \\delta \\equiv \\frac { q + 1 } 2 \\pmod { q } , \\\\ 2 \\delta - 1 & { \\rm o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "7154.png", "formula": "\\begin{align*} \\mathcal { Y } ( d ) : = \\mathfrak { g } ^ { \\oplus 2 } / G L ( d ) , \\end{align*}"}
+{"id": "4242.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( q ) _ n ( z c ) ^ n q ^ { n ^ 2 } } { ( z q ) _ n ( c q ) _ n } = z \\displaystyle \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( q ) _ n ( c q ) _ { N - n } ( c q ) ^ n } { ( z q ) _ n ( c q ) _ N } . \\end{align*}"}
+{"id": "4996.png", "formula": "\\begin{align*} h ( t ) = \\Phi _ { 1 0 5 } ( t ) & = t ^ { 4 8 } + t ^ { 4 7 } + t ^ { 4 6 } - t ^ { 4 3 } - t ^ { 4 2 } - 2 t ^ { 4 1 } - t ^ { 4 0 } - t ^ { 3 9 } + t ^ { 3 6 } + t ^ { 3 5 } + t ^ { 3 4 } + t ^ { 3 3 } \\\\ & + t ^ { 3 2 } + t ^ { 3 1 } - t ^ { 2 8 } - t ^ { 2 6 } - t ^ { 2 4 } - t ^ { 2 2 } - t ^ { 2 0 } + t ^ { 1 7 } + t ^ { 1 6 } + t ^ { 1 5 } + t ^ { 1 4 } + t ^ { 1 3 } \\\\ & + t ^ { 1 2 } - t ^ { 9 } - t ^ { 8 } - 2 t ^ { 7 } - t ^ { 6 } - t ^ { 5 } + t ^ { 2 } + t + 1 . \\end{align*}"}
+{"id": "6601.png", "formula": "\\begin{align*} \\partial ^ + _ N u = \\partial ^ - _ N u = : \\partial _ N u , \\alpha [ u ] = \\partial _ N u , \\partial ^ \\pm _ N u ( s ) : = \\lim _ { t \\to 0 ^ + } N ( s ) \\cdot \\nabla u \\big ( s \\pm t N ( s ) \\big ) , \\end{align*}"}
+{"id": "3965.png", "formula": "\\begin{align*} \\hat { G } ( u , t ) = \\exp \\left ( - \\lambda t \\left ( 1 - \\frac { \\ln ( 1 - ( 1 - p ) u ) } { \\ln p } \\right ) \\right ) . \\end{align*}"}
+{"id": "3101.png", "formula": "\\begin{align*} \\sigma = ( 1 ) ( 2 , 3 ) ( 4 , 5 , 6 ) ( 7 , 8 , 9 , 1 0 ) \\cdots \\left ( t _ { n - 1 } + 1 , t _ { n - 1 } + 2 , \\ldots , t _ n \\right ) \\cdots \\end{align*}"}
+{"id": "679.png", "formula": "\\begin{align*} g ' _ 2 = - \\frac { 1 } { \\nu } I \\widehat { g _ 1 ' } I + \\frac { 2 i } { \\mu \\nu } h _ z J g _ 1 ' \\end{align*}"}
+{"id": "6394.png", "formula": "\\begin{align*} \\sigma _ { i j } ' = \\begin{cases} - \\sigma _ { i j } & i = k j = k , \\\\ \\sigma _ { i j } + ( [ \\sigma _ { i k } ] _ + [ \\sigma _ { k j } ] _ + - [ - \\sigma _ { i k } ] _ + [ - \\sigma _ { k j } ] _ + ) \\alpha _ { i j } ^ k & , \\end{cases} \\end{align*}"}
+{"id": "7476.png", "formula": "\\begin{align*} S _ m ^ { ( a ) } ( n ) & = \\sum _ { \\nu _ a = 1 } ^ { n } \\sum _ { \\nu _ { a - 1 } = 1 } ^ { \\nu _ a } \\cdots \\sum _ { \\nu _ 1 = 1 } ^ { \\nu _ 2 } \\nu ^ m \\ , . \\end{align*}"}
+{"id": "1260.png", "formula": "\\begin{align*} J _ { p } ( u ) = \\frac { 1 } { p } \\int _ { \\Omega } \\left | \\nabla u \\right | ^ { p } \\ d x + \\frac { \\lambda } { p } \\int _ { \\partial \\Omega } \\left | u \\right | ^ { p } \\ d \\mathcal H ^ { N - 1 } - \\int _ { \\Omega } f u \\ d x \\end{align*}"}
+{"id": "207.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\psi ( t ) d t = \\int _ 0 ^ T \\biggl ( 1 - \\frac { t } { T } \\biggr ) ^ m d t = C ( m ) T . \\end{align*}"}
+{"id": "7724.png", "formula": "\\begin{align*} \\det g ^ p = e ^ { - 2 ( n + 1 ) s _ p } \\det g ^ + = e ^ { - 2 ( n + 1 ) ( s _ p - s _ q ) } \\det g ^ q \\leq e ^ { 2 ( n + 1 ) d } \\det g ^ q \\end{align*}"}
+{"id": "263.png", "formula": "\\begin{align*} \\sigma _ I \\cup \\sigma _ J \\cup \\sigma _ K = [ p t ] \\mbox { a n d } \\frac 1 r ( | \\lambda _ I | + | \\mu _ J | + | \\nu _ K | ) = \\frac 1 n ( | \\lambda | + | \\mu | + | \\nu | ) . \\end{align*}"}
+{"id": "1661.png", "formula": "\\begin{align*} \\mathfrak { S } _ { \\mathsf { m } } ^ { \\sigma , \\tau } : = \\mathfrak { C } ^ { \\sigma } _ { \\mathsf { m } } \\cap \\mathfrak { A } ^ { \\sigma , \\tau } _ { \\mathsf { m } } \\ ; . \\end{align*}"}
+{"id": "2210.png", "formula": "\\begin{align*} & R _ { 2 3 2 3 } S _ { 1 2 1 3 1 } = - R _ { 1 3 1 3 } S _ { 1 2 2 3 2 } = R _ { 1 2 1 2 } S _ { 1 3 2 3 3 } , \\end{align*}"}
+{"id": "842.png", "formula": "\\begin{align*} \\widetilde { \\theta } ( \\hat { y } _ i , y ^ { * } _ { - i } ) = \\theta + \\frac { ( 1 - \\theta ) } { l _ 1 + \\frac { l _ 2 } { N } e ^ { - \\beta \\hat { \\tau } _ i } x _ i ( \\hat { \\tau } _ i ) + \\frac { l _ 2 } { N } \\sum \\limits _ { j = 1 , j \\neq i } ^ N e ^ { - \\beta \\tau _ j ^ { * } } x _ j ( \\tau _ j ^ { * } ) } . \\end{align*}"}
+{"id": "5688.png", "formula": "\\begin{gather*} c _ { i } \\iota _ { i } \\tau _ * ( \\alpha _ { ( 0 , i ) } ) = ( - 1 ) ^ i ( i + 1 ) ! \\ : e _ 1 \\wedge e _ 2 \\wedge \\cdots \\wedge e _ { i + 1 } \\otimes e _ { 1 } ^ * . \\end{gather*}"}
+{"id": "1547.png", "formula": "\\begin{align*} \\varphi _ { s ^ { \\prime \\prime } , k , i } \\circ \\varphi _ { s ^ { \\prime } , j , k } \\circ \\varphi _ { s , i , j } = \\mathrm { i d } _ { \\Omega _ { b _ { i } } \\times \\Omega } . \\end{align*}"}
+{"id": "4513.png", "formula": "\\begin{align*} \\{ \\Phi _ x , \\mathcal { K } \\} = - \\dd \\mathcal { K } ( \\mathbb { X } _ { \\Phi _ { \\ ! x } } \\ ! ) \\ , , \\end{align*}"}
+{"id": "2177.png", "formula": "\\begin{align*} & \\{ R \\in S ^ 2 ( \\wedge ^ 2 V ^ * ) \\ : | \\ : \\mathfrak { S } _ { Y , Z , W } R ( X , Y , Z , W ) = 0 \\} \\\\ & \\{ S \\in S ^ 2 ( \\wedge ^ 2 V ^ * ) \\otimes V ^ * \\ : | \\ : \\mathfrak { S } _ { Z , W , U } S ( X , Y , Z , W , U ) = 0 \\} , \\end{align*}"}
+{"id": "3168.png", "formula": "\\begin{align*} d _ { T ' } ( u ( y ) , v ( y ) ) = d _ { T ' } ( g u ( x ) , g v ( x ) ) = d _ { T ' } ( u ( x ) , v ( x ) ) \\leq C . \\end{align*}"}
+{"id": "4919.png", "formula": "\\begin{align*} \\binom { n } { k } = \\sum _ { i = 1 } ^ { k - 1 } \\left ( \\frac { j - 1 } { 2 } \\right ) ^ i \\binom { n } { k - i } + \\left ( \\frac { j - 1 } { 2 } \\right ) ^ { k - 1 } \\binom { n } { 1 } + \\sum _ { i = 1 } ^ { k - 1 } \\left ( \\frac { j - 1 } { 2 } \\right ) ^ { i - 1 } \\frac { r + 1 + j ( i - 1 ) } { k - ( i - 1 ) } \\binom { n } { k - i } . \\end{align*}"}
+{"id": "3311.png", "formula": "\\begin{align*} ( 1 \\otimes K _ 1 ) \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } = \\lambda _ { n _ 2 } \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } , \\end{align*}"}
+{"id": "1584.png", "formula": "\\begin{align*} x _ c w _ { r _ j } x _ a x _ b x _ c ^ { - 1 } x _ b ^ { - 1 } x _ a ^ { - 1 } w _ { r _ j } ^ { - 1 } = \\left [ x _ c , w _ { r _ j } x _ a x _ b \\right ] . \\end{align*}"}
+{"id": "470.png", "formula": "\\begin{align*} B _ n ( - x ) \\pm B _ n ( - y ) = ( - 1 ) ^ n \\big ( B _ n ( x ) \\pm B _ n ( y ) + n ( x ^ { n - 1 } \\pm y ^ { n - 1 } ) \\big ) . \\end{align*}"}
+{"id": "697.png", "formula": "\\begin{align*} { { \\bf { F } } _ { { \\mathop { \\rm i n t } } } } = - G \\psi \\left ( { \\bf { x } } \\right ) \\sum \\limits _ { i = 1 } ^ { 1 8 } { \\varpi \\left ( { { { \\left | { { { \\bf { c } } _ i } } \\right | } ^ 2 } } \\right ) } \\psi \\left ( { { \\bf { x } } + { { \\bf { c } } _ i } \\Delta t } \\right ) { { \\bf { c } } _ i } , \\end{align*}"}
+{"id": "4017.png", "formula": "\\begin{align*} \\mathrm { P r } \\{ X _ { 1 } + X _ { 2 } + \\dots + X _ { k } = n \\} & = \\underset { m _ j \\in \\mathbb { N } } { \\underset { m _ { 1 } + m _ { 2 } + \\dots + m _ { k } = n } { \\sum } } \\prod _ { j = 1 } ^ { k } \\mathrm { P r } \\{ X _ { j } = m _ j \\} \\\\ & = \\underset { m _ j \\in \\mathbb { N } } { \\underset { m _ { 1 } + m _ { 2 } + \\dots + m _ { k } = n } { \\sum } } \\left ( \\frac { ( 1 - \\rho ) ^ { r } } { 1 - ( 1 - \\rho ) ^ { r } } \\right ) ^ { k } \\prod _ { j = 1 } ^ { k } \\rho ^ { m _ { j } } \\binom { r + m _ { j } - 1 } { m _ { j } } , \\end{align*}"}
+{"id": "4431.png", "formula": "\\begin{align*} \\lim _ { a \\to \\infty } \\sup _ { u \\in X } \\Vert u - \\tau _ { a } \\circ u \\Vert _ { L _ p } ^ p & \\leq \\lim _ { a \\to \\infty } \\int _ { \\{ u \\geq a \\} } \\vert u \\vert ^ p \\dd x \\leq \\lim _ { a \\to \\infty } \\sup _ { E \\colon \\vert E \\vert < R a ^ { - p } } \\int _ E \\vert u \\vert ^ p \\dd x = 0 . \\end{align*}"}
+{"id": "564.png", "formula": "\\begin{align*} T _ { W _ \\ell , \\phi } ( \\mu ) = \\sum _ { i = 1 } ^ L c _ i \\int _ { [ 0 , 1 ] ^ 2 } \\bar a _ i ( u ) \\bar b _ i ( v ) W _ \\ell ( u , v ) d u d v , \\end{align*}"}
+{"id": "40.png", "formula": "\\begin{align*} \\P ( X \\geq \\delta _ i ) & = \\P ( Y \\geq \\delta _ i ' ) = \\P ( Y \\geq 2 ^ { - j } ) = \\sum _ { i \\in [ j ] } \\P \\left \\{ Y = 2 ^ { - i } \\right \\} = \\sum _ { i \\in [ j ] } r 2 ^ i = 2 r \\left ( 2 ^ { j } - 1 \\right ) \\leq 2 r 2 ^ j . \\end{align*}"}
+{"id": "1982.png", "formula": "\\begin{align*} \\big | ( \\lambda _ 1 x _ u ^ 2 + \\overline { \\lambda } _ 1 \\overline { x } _ u ^ 2 ) - ( \\lambda _ 1 x _ v ^ 2 + \\overline { \\lambda } _ 1 \\overline { x } _ v ^ 2 ) \\big | = O \\Big ( \\frac { 1 } { n } \\Big ) . \\end{align*}"}
+{"id": "4560.png", "formula": "\\begin{align*} 2 \\xi _ 2 = - \\frac { 2 a ( a - 1 ) ( 8 a ^ 2 - a - 1 ) } { 3 } & < \\xi _ 1 = - \\frac { 2 a ( a - 1 ) ( 2 a ^ 2 + 2 a - 1 ) } { 3 } \\\\ [ 3 m m ] & < \\frac { \\xi _ 2 } { 2 } = - \\frac { a ( a - 1 ) ( 8 a ^ 2 - a - 1 ) } { 6 } \\end{align*}"}
+{"id": "8256.png", "formula": "\\begin{align*} \\begin{cases} s \\in [ 0 , 1 - \\frac { 1 } { \\alpha } ) , & \\textrm { i f } \\ ; \\ ; \\bar { s } = \\frac { N } { 2 } + 1 - \\alpha , \\\\ s \\geq 0 , \\ ; \\bar { s } + s \\alpha \\leq \\frac { N } { 2 } + 1 - \\alpha , & \\textrm { i f } \\ ; \\ ; \\bar { s } = - s _ 0 \\in ( - \\frac { N } { 2 } , \\frac { N } { 2 } + 1 - \\alpha ) . \\end{cases} \\end{align*}"}
+{"id": "4354.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| f _ n - f \\| _ { L _ 2 ( ( 0 , T ) , H ^ 1 ( \\Omega ) ) } = \\lim _ { n \\to \\infty } \\left \\| \\partial _ t f _ n - \\partial _ t f \\right \\| _ { L _ 2 ( ( 0 , T ) , H ^ 1 ( \\Omega ) ' ) } = 0 \\ , . \\end{align*}"}
+{"id": "7141.png", "formula": "\\begin{align*} O : = \\bigcup _ { w , w ' \\in \\mathbb { Z } } O _ { w , w ' } . \\end{align*}"}
+{"id": "2700.png", "formula": "\\begin{align*} \\binom { a + j } { j } \\sum _ { l = 0 } ^ { a } \\binom { n - j + l } { l } \\binom { n - j } { a + j - l } \\sum _ { k = a + j - l } ^ { n - ( a + j - l ) } \\binom { n - a - 2 j + l } { k - j } \\binom { k - j } { a - l } G ( n , k , a ) \\end{align*}"}
+{"id": "6714.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } \\partial _ { t } E - \\nabla _ { x } \\times B = \\rho u , \\partial _ { t } B + \\nabla _ { x } \\times E & = 0 , \\\\ \\nabla _ { x } \\cdot E = 1 - \\rho , \\nabla _ { x } \\cdot B & = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "4412.png", "formula": "\\begin{align*} \\sigma ( \\upsilon ) = r _ 0 + \\sigma ( \\upsilon _ 1 ) . \\end{align*}"}
+{"id": "132.png", "formula": "\\begin{align*} \\begin{array} { l l l l l } \\partial \\phi ( [ \\mathbf { a } , \\mathbf { a ' } ] ) & = & & \\partial \\left ( [ \\mathbf { a } ] \\otimes ( s ^ 0 ) ^ { k + 1 } [ \\mathbf { a ' } ] \\right ) \\\\ & = & & \\partial [ \\mathbf { a } ] \\otimes ( s ^ 0 ) ^ { k + 1 } [ \\mathbf { a ' } ] + ( - 1 ) ^ { | \\mathbf a | + 1 } [ \\mathbf { a } ] \\otimes \\partial ( s ^ 0 ) ^ { k + 1 } [ \\mathbf { a ' } ] \\\\ & = & & \\phi ( \\partial ( [ \\mathbf { a } , \\mathbf { a ' } ] ) ) \\\\ \\end{array} \\end{align*}"}
+{"id": "345.png", "formula": "\\begin{align*} A \\subset \\bigcup _ { j = 0 } ^ { N _ j } B _ { b L ^ r \\alpha } ( v _ j ) \\end{align*}"}
+{"id": "3971.png", "formula": "\\begin{align*} E ( \\mathcal { M } ( t ) ) & = \\alpha \\theta e ^ { \\theta } t , \\\\ \\operatorname { V a r } ( \\mathcal { M } ( t ) ) & = \\alpha \\theta ( \\theta + 1 ) e ^ { \\theta } t , \\\\ \\operatorname { C o v } ( \\mathcal { M } ( s ) , \\mathcal { M } ( t ) ) & = \\alpha \\theta ( \\theta + 1 ) e ^ { \\theta } \\min \\{ s , t \\} , \\end{align*}"}
+{"id": "4050.png", "formula": "\\begin{align*} P _ - ( \\partial _ s q ) = \\partial _ s q _ - - I ^ { - 2 } ( 1 - \\frac { 1 } { k } ) \\sum _ { n = [ M ] - 1 } ^ { [ M ] } ( n + 1 ) ( n + 2 ) q _ { n + 2 } ( s ) H _ n . \\end{align*}"}
+{"id": "3151.png", "formula": "\\begin{align*} \\int _ { \\mathbb { B } ^ { N } } a ( x ) \\left | u _ { n } \\right | ^ { p + 1 } \\mathrm { ~ d } V _ { \\mathbb { B } ^ { N } } ( x ) & - \\int _ { \\mathbb { B } ^ { N } } a ( x ) | u | ^ { p + 1 } \\mathrm { ~ d } V _ { \\mathbb { B } ^ { N } } ( x ) \\\\ & = \\int _ { \\mathbb { B } ^ { N } } a ( x ) \\left | u _ { n } - u \\right | ^ { p + 1 } \\mathrm { ~ d } V _ { \\mathbb { B } ^ { N } } ( x ) + o ( 1 ) . \\end{align*}"}
+{"id": "5764.png", "formula": "\\begin{align*} ( 5 0 a + 1 2 5 b + 1 3 c ) ( 2 a + 5 b ) < 0 . \\end{align*}"}
+{"id": "2246.png", "formula": "\\begin{align*} V _ n = k + k z _ 1 + \\cdots + k z _ n , \\ c _ n = \\frac { 1 } { 2 ^ n \\sqrt 5 ^ { n ( n + 1 ) } n ! } . \\end{align*}"}
+{"id": "2383.png", "formula": "\\begin{align*} \\partial _ { \\alpha , \\beta } ( e _ { a _ { 1 } } U ( c _ { k } + k + s - 1 ) e _ { \\iota ^ { c _ { k } } ( a _ { s + 1 } ) } ) = \\sum _ { t = 0 } ^ { 1 } \\bigg ( \\Delta _ { \\iota ^ { c _ { k } } ( a _ { s + 1 } ) , \\iota ^ { c _ { k } + k + s - 2 } ( t ) } ^ { \\alpha , \\beta } - \\sum _ { t = 0 } ^ { 1 } \\Delta _ { a _ { 1 } , \\iota ( t ) } ^ { \\alpha , \\beta } \\bigg ) e _ { a _ { 1 } } W ( t ; c _ { k } + k + s - 2 ) e _ { \\iota ^ { c _ { k } } ( a _ { s + 1 } ) } \\end{align*}"}
+{"id": "4730.png", "formula": "\\begin{gather*} ( \\eth _ k + \\partial _ k ^ * ) ( a + a ^ * ) \\star ( b + b ^ * ) = ( a + a ^ * ) \\star ( \\partial _ k + \\eth _ k ^ * ) ( b + b ^ * ) + ( \\eth _ k + \\partial _ k ^ * ) ( ( a + a ^ * ) \\star ( b + b ^ * ) ) , \\\\ ( a + a ^ * ) \\star ( \\eth _ k + \\partial _ k ^ * ) ( b + b ^ * ) = ( \\partial _ k + \\eth _ k ^ * ) ( b + b ^ * ) \\star ( b + b ^ * ) + ( \\eth _ k + \\partial _ k ^ * ) ( ( a + a ^ * ) \\star ( b + b ^ * ) ) . \\end{gather*}"}
+{"id": "1345.png", "formula": "\\begin{align*} \\left \\| u \\right \\| _ { \\mathcal { L } ^ { p , \\lambda } ( \\Omega ) } : = \\left \\| u \\right \\| _ { L ^ p ( \\Omega ) } + [ u ] _ { \\mathcal { L } ^ { p , \\lambda } ( \\Omega ) } . \\end{align*}"}
+{"id": "5904.png", "formula": "\\begin{align*} \\lambda = e _ 1 + e _ 4 \\ , , e _ 2 + e _ 4 \\ , , e _ 3 + e _ 4 \\ , , e _ 1 + e _ 5 \\ , , e _ 2 + e _ 5 \\ , , e _ 3 + e _ 5 \\ , . \\end{align*}"}
+{"id": "854.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Delta _ 1 = \\left ( \\frac { 2 \\gamma _ 1 } { \\gamma _ 2 } \\frac { 1 } { ( k _ 2 \\sigma a ' ) ^ { 2 } } \\frac { \\rho } { ( 1 - \\rho ) ^ { 2 } } \\right ) ^ { \\frac { 1 } { 2 - \\rho } } \\\\ & \\Delta _ 2 = \\gamma _ 1 \\left ( \\frac { 2 t _ 0 } { k _ 2 \\sigma a ' } - \\frac { 1 } { k _ 2 \\sigma ( a ' ) ^ { 3 } } \\right ) + \\gamma _ 2 \\left ( \\frac { 1 } { \\rho } - 1 \\right ) \\left ( \\frac { k _ 2 - 1 } { k _ 2 } \\right ) ^ { \\rho } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2365.png", "formula": "\\begin{align*} S _ { 2 , 2 } = \\left \\{ \\begin{array} { c } A _ { 1 } B _ { 1 } B _ { 2 } A _ { 2 } , \\ , A _ { 1 } B _ { 1 } A _ { 1 } B _ { 2 } A _ { 2 } , \\ , A _ { 1 } B _ { 1 } B _ { 2 } A _ { 1 } A _ { 2 } , \\ , A _ { 1 } B _ { 1 } A _ { 2 } B _ { 2 } A _ { 2 } , \\ , A _ { 1 } A _ { 2 } B _ { 1 } B _ { 2 } A _ { 2 } , \\\\ A _ { 1 } B _ { 1 } A _ { 1 } B _ { 2 } A _ { 1 } A _ { 2 } , \\ , A _ { 1 } B _ { 1 } A _ { 1 } A _ { 2 } B _ { 2 } A _ { 2 } , \\ , A _ { 1 } A _ { 2 } B _ { 1 } A _ { 2 } B _ { 2 } A _ { 2 } \\end{array} \\right \\} . \\end{align*}"}
+{"id": "7716.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow + \\infty } \\frac { c a p _ p ( B ^ { g ^ + } _ q ( t ) ) } { e ^ { n t } } = \\frac { 1 } { 2 ^ n } ( \\frac { n } { p - 1 } ) ^ { p - 1 } \\mathcal { A } ( q ) . \\end{align*}"}
+{"id": "7061.png", "formula": "\\begin{align*} Y ^ { [ h ] } : = \\bigcup _ { N : \\dim N = h } Y _ N \\end{align*}"}
+{"id": "6159.png", "formula": "\\begin{align*} & V _ 2 ^ * V _ 1 - V _ 1 V _ 2 ^ * = ( I - M _ z M _ z ^ * ) \\otimes P U { P ^ \\perp } U . \\end{align*}"}
+{"id": "4164.png", "formula": "\\begin{align*} ( 1 + x ) ^ { - 4 } \\ge 1 - 4 x \\ ; . \\end{align*}"}
+{"id": "6112.png", "formula": "\\begin{align*} \\begin{aligned} | \\mathcal S _ 1 ( k , 3 ) | = \\binom { n - 2 } { k - 2 } - \\binom { n - k + 1 } { k - 2 } + \\binom { n - k - 2 } { k - 5 } + n - k + 3 \\sim ( k - 3 ) \\binom { n } { k - 3 } . \\end{aligned} \\end{align*}"}
+{"id": "6483.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ \\infty ( \\Omega ) } = \\sup _ \\Omega | f | . \\end{align*}"}
+{"id": "1417.png", "formula": "\\begin{align*} ( \\lambda ^ 2 + \\lambda a ( x ) ) u - \\Delta u = h , \\end{align*}"}
+{"id": "6495.png", "formula": "\\begin{align*} \\tilde u ( x ) = \\frac { M - u ( 2 R x ) } { M - m } + \\gamma \\end{align*}"}
+{"id": "511.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\int _ { \\R ^ n } e ^ { f ( x ) } d x = \\sup _ { Q \\in \\P ( \\R ) } \\left ( \\int _ { \\R } V \\ , d Q + \\frac 1 2 \\int _ { \\R } \\int _ { \\R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \\right ) . \\end{align*}"}
+{"id": "869.png", "formula": "\\begin{align*} h _ d ( x , r ) = \\min \\left \\{ \\frac { \\mu ( E \\cap B _ d ( x , r ) ) } { \\mu ( B _ d ( x , r ) ) } , \\frac { \\mu ( E ^ c \\cap B _ d ( x , r ) ) } { \\mu ( B _ d ( x , r ) ) } \\right \\} . \\end{align*}"}
+{"id": "4796.png", "formula": "\\begin{align*} \\psi _ 0 ( n + 1 , k ) a ( n , k ) = \\psi _ 0 ( n , k ) a ( n , k - 1 ) . \\end{align*}"}
+{"id": "4698.png", "formula": "\\begin{align*} \\left | r \\frac { d } { d r } \\left [ C ( z , w ) \\right ] \\right | \\lesssim & \\frac { 1 } { | 1 - z \\overline { w } | ^ { 2 \\lambda + 2 } } \\ln \\left ( \\frac { | 1 - z \\bar { w } | ^ 2 } { | 1 - z w | ^ 2 } + 2 \\right ) + \\frac { 1 } { | 1 - z \\overline { w } | ^ { 2 \\lambda + 1 } } \\frac { 1 } { | 1 - z w | } \\\\ \\lesssim & \\frac { | 1 - z \\overline { w } | ^ { - 1 } | 1 - z w | ^ { - 1 } } { ( | 1 - z \\overline { w } | + | 1 - z w | ) ^ { 2 \\lambda } } , z = r e ^ { i \\theta } . \\end{align*}"}
+{"id": "7917.png", "formula": "\\begin{align*} H ^ n _ \\mathrm { m R B A } ( ( A , R ) , ( M , S ) ) : = \\frac { Z ^ n _ \\mathrm { m R B A } ( ( A , R ) , ( M , S ) ) } { B ^ n _ \\mathrm { m R B A } ( ( A , R ) , ( M , S ) ) } , n \\geq 0 \\end{align*}"}
+{"id": "7761.png", "formula": "\\begin{align*} g ^ m = d t ^ 2 + \\lambda ^ 2 V g _ { \\mathbb { S } ^ 1 } + r ^ 2 g _ { \\mathbb { S } ^ 2 } \\end{align*}"}
+{"id": "807.png", "formula": "\\begin{align*} \\tau ^ { * } = \\inf \\{ t : \\ W ( t ) \\geq a ' t + b ' \\} . \\end{align*}"}
+{"id": "5879.png", "formula": "\\begin{align*} H ( x , p ) = \\frac { 1 } { 2 } \\norm { p + \\frac { \\tilde { B } ( x ) } { 2 \\epsilon } x } ^ 2 + U ( x ) , \\end{align*}"}
+{"id": "4904.png", "formula": "\\begin{align*} \\frac { q x + 2 - x } { ( 1 - x ) ^ 2 } \\sum _ { j = 0 } ^ \\infty f _ j \\left ( q , 2 - \\frac { 1 } { x } , 2 - \\frac { 1 } { x } \\right ) = \\frac { 2 } { 1 - x } - \\frac { 1 } { q x + 2 - 2 x } + \\frac { ( q x + 1 - x ) ^ 2 } { ( 1 - 2 x ) ^ 2 ( q x + 2 - 2 x ) } \\sum _ { j = 0 } ^ \\infty f _ j \\left ( q , 3 - \\frac { 1 } { x } , 3 - \\frac { 1 } { x } \\right ) . \\end{align*}"}
+{"id": "7121.png", "formula": "\\begin{align*} v _ i = w _ i + d _ i \\left ( \\sum _ { j > i } d _ j - \\sum _ { j < i } d _ j \\right ) . \\end{align*}"}
+{"id": "673.png", "formula": "\\begin{align*} [ T ] = \\frac { 1 } { \\mu } J ( \\partial _ x h + I \\partial _ y h ) = \\frac { 2 } { \\mu } h _ z J \\end{align*}"}
+{"id": "5340.png", "formula": "\\begin{align*} G = ( O _ 2 ( G ) { : } K ) \\times G _ 0 = S _ 4 \\times G _ 0 \\end{align*}"}
+{"id": "2292.png", "formula": "\\begin{align*} \\eta ' _ { d , F _ 1 , F _ 2 } = \\begin{cases} \\eta _ d & \\\\ & \\\\ \\vartheta _ d & \\end{cases} \\end{align*}"}
+{"id": "510.png", "formula": "\\begin{align*} \\log \\int _ { \\R ^ n } e ^ f \\ , d \\gamma _ t \\le \\sup _ { y \\in \\R ^ n } \\left ( \\int _ { \\R ^ n } f \\ , d \\gamma _ { y , t } - H ( \\gamma _ { y , t } \\ , | \\ , \\gamma _ t ) \\right ) + \\frac { t ^ 2 } { 2 } \\sum _ { i , j = 1 } ^ n \\int _ { \\R ^ n } | \\partial _ { i j } f | ^ 2 \\ , d \\gamma _ { y ^ * , t } . \\end{align*}"}
+{"id": "6252.png", "formula": "\\begin{align*} \\delta _ k ( A \\wr _ I B ) = \\sum _ { \\ell = 0 } ^ n \\delta _ \\ell ( B ) \\sum _ { j _ 1 + \\dots + j _ \\ell = k } \\prod _ { r = 1 } ^ { \\ell } \\delta _ { j _ r } ( A ) \\quad \\textup { a n d h e n c e } \\delta ( A \\wr _ I B ) = \\sum _ { \\ell = 0 } ^ n \\delta _ \\ell ( B ) \\delta ( A ) ^ \\ell . \\end{align*}"}
+{"id": "6810.png", "formula": "\\begin{align*} \\alpha ' ( d , d , a _ 3 , \\dots , a _ k ) = \\sigma ' ( d , a _ 3 , \\dots , a _ k ) + \\delta ' ( d , a _ 3 , \\dots , a _ k ) \\end{align*}"}
+{"id": "4145.png", "formula": "\\begin{align*} \\frac { 1 - | B ( z ) | ^ 2 } { 1 - | z | ^ 2 } = \\sum \\limits _ { k = 1 } ^ n \\left ( \\prod \\limits _ { j = 1 } ^ { k - 1 } \\left | \\frac { z - z _ j } { 1 - \\overline { z _ j } z } \\right | ^ 2 \\right ) \\frac { 1 - | z _ k | ^ 2 } { \\left | 1 - \\overline { z _ k } z \\right | ^ 2 } \\ , , | z | \\not = 1 \\ , , \\end{align*}"}
+{"id": "2705.png", "formula": "\\begin{align*} M _ R ( n , j + a - l , 0 ; a ) = \\binom { n - ( j + a - l ) } { j + a - l } \\sum _ { k = a + j - l } ^ { n - ( a + j - l ) } \\binom { n - 2 ( a + j - l ) } { k - ( j + a - l ) } G ( n , k , a ) \\end{align*}"}
+{"id": "7104.png", "formula": "\\begin{align*} \\rho = \\frac { 1 } { 2 } \\mathfrak { g } ^ { \\lambda < 0 } = \\frac { 1 } { 2 } \\sum _ { j < i } ( \\beta _ i - \\beta _ j ) , \\end{align*}"}
+{"id": "2552.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { * , \\xi } = & ~ [ ( A - B ^ 2 R ^ { - 1 } P _ t ) x _ t ^ { * , \\xi } - B ^ 2 R ^ { - 1 } \\varphi _ t ^ { * , \\xi } - B h ( \\mu _ t ^ { * , \\xi } ) \\\\ & + f ( \\nu _ t ^ { * , \\xi } ) + b ( \\mu _ t ^ { * , \\xi } ) ] d t + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ 0 ^ { * , \\xi } = & ~ \\xi , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4051.png", "formula": "\\begin{align*} A = \\chi \\sum _ { j = 2 } ^ { K } d _ { j } ( e _ b q _ + ) ^ j S = \\chi \\sum _ { j = 2 } ^ K \\sum _ { \\ell = 0 } ^ { j - 1 } \\tilde d _ { j , \\ell } e _ { b } ^ j ( q _ + ) ^ \\ell ( q _ - ) ^ { j - \\ell } , d _ j , \\tilde d _ { j , \\ell } \\in \\mathbb { R } . \\end{align*}"}
+{"id": "7964.png", "formula": "\\begin{align*} t ' ( s ) = - \\frac { 2 c } { 3 } r ^ 3 ( s ) , \\end{align*}"}
+{"id": "4193.png", "formula": "\\begin{align*} w ^ i & = \\underset { j } \\sum c ^ j w ^ j \\\\ \\implies l ( w ^ i ) \\mathrm { s i n } ( \\theta ) & = \\underset { j } \\sum c ^ j l ( w ^ j ) \\mathrm { s i n } ( \\theta _ j ' ) \\end{align*}"}
+{"id": "3373.png", "formula": "\\begin{align*} h _ 1 \\le h _ j j = 2 , \\dots , d - 1 . \\end{align*}"}
+{"id": "225.png", "formula": "\\begin{align*} \\mathcal W _ 2 ( \\mu _ 0 , \\mu _ 1 ) = \\inf \\left \\{ \\int _ 0 ^ 1 \\int _ { \\mathbb { R } ^ d } | \\nu _ s | ^ 2 m _ s ( d x ) d s : \\ , \\ , \\ , \\partial _ s m _ s + \\div ( \\nu _ s m _ s ) = 0 \\ , , \\ , m _ { i } = \\mu _ i \\ , , \\ , \\ , i = 0 , 1 \\right \\} , \\end{align*}"}
+{"id": "8261.png", "formula": "\\begin{align*} ( t + 1 ) ^ s \\Big ( \\Vert ( \\sigma , u ) ^ \\ell ( t ) \\Vert _ { \\dot { B } ^ { \\bar { s } + s \\alpha } _ { 2 , 1 } } + \\Vert \\sigma ^ h ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } } _ { 2 , 1 } } + \\Vert u ^ h ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } } \\Big ) \\le 2 ^ s Z _ { s , \\bar { s } } ( t ) \\le C . \\end{align*}"}
+{"id": "8242.png", "formula": "\\begin{align*} \\begin{aligned} Z _ { s , \\bar { s } } ^ \\ell ( t ) \\le C \\Vert ( \\sigma _ 0 , u _ 0 ) \\Vert ^ \\ell _ { \\dot { B } ^ { \\bar { s } } _ { 2 , 1 } } + C \\int _ 0 ^ t \\Phi ( t , \\tau ) Z _ { s , \\bar { s } } ( \\tau ) \\dd \\tau , \\end{aligned} \\end{align*}"}
+{"id": "4319.png", "formula": "\\begin{align*} m _ i ( h ) = \\left ( \\frac { 1 + \\rho } { \\rho } \\right ) \\widehat { h } ( i ) + \\frac { 1 } { \\rho ^ { i + 1 } } \\sum _ { j = 0 } ^ { i - 1 } \\widehat { h } ( j ) \\rho ^ j = \\left ( \\frac { 1 + \\rho } { \\rho } \\right ) \\widehat { h } ( i ) - \\frac { 1 } { \\rho ^ { i + 1 } } \\sum _ { j = i } ^ { \\infty } \\widehat { h } ( j ) \\rho ^ j , \\end{align*}"}
+{"id": "3667.png", "formula": "\\begin{align*} \\Gamma _ { t + 2 } = \\begin{cases} \\{ 1 3 4 , 2 3 4 \\} \\boxplus \\{ 3 , 4 \\} & t = 1 , \\\\ K _ { t + 2 } ^ { 3 } \\boxplus \\{ t + 1 , t + 2 \\} & t \\geq 2 . \\end{cases} \\end{align*}"}
+{"id": "636.png", "formula": "\\begin{align*} \\partial _ X \\varphi = \\nabla _ X \\varphi - \\frac { 1 } { 2 } ( \\sqrt { c _ 1 } \\ X _ 1 \\cdot \\nu _ 1 + \\sqrt { c _ 2 } \\ X _ 2 \\cdot \\nu _ 2 ) \\cdot \\varphi + \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi \\end{align*}"}
+{"id": "7000.png", "formula": "\\begin{align*} \\int _ X | \\nabla v | ^ 2 \\Delta \\phi d m \\geq 2 \\int _ X \\phi \\left ( \\frac { ( \\Delta v ) ^ 2 } { N } + \\langle \\nabla v , \\nabla \\Delta v \\rangle + K | \\nabla v | ^ 2 \\right ) d m = \\int _ X \\phi d \\mu , \\end{align*}"}
+{"id": "6260.png", "formula": "\\begin{align*} b _ i ( \\alpha \\tau ) = \\alpha _ 1 \\tau _ 1 \\alpha _ 2 \\tau _ 2 \\cdots \\alpha _ { k _ i } \\tau _ { k _ i } = \\alpha _ 1 \\left ( \\alpha _ 2 ^ { \\tau _ 1 ^ { - 1 } } \\alpha _ 3 ^ { ( \\tau _ 1 \\tau _ 2 ) ^ { - 1 } } \\cdots \\alpha _ { k _ i } ^ { ( \\tau _ 1 \\cdots \\tau _ { k _ i } ) ^ { - 1 } } \\right ) \\tau _ 1 \\tau _ 2 \\cdots \\tau _ { k _ i } , \\end{align*}"}
+{"id": "5470.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\frac { ( a q ) _ { \\infty } } { ( b q ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( b / a ) _ n } { ( q ) _ n } \\frac { ( a q ) ^ n } { 1 - t q ^ n } . \\end{align*}"}
+{"id": "6647.png", "formula": "\\begin{align*} 1 + \\sum _ { i = 1 } ^ { \\eta } \\lambda _ { i , 1 } ^ { 2 } s _ { i , - 1 } \\le \\sum _ { i = 1 } ^ { \\eta } \\lambda _ { i , 1 } ^ { 2 } , \\end{align*}"}
+{"id": "2360.png", "formula": "\\begin{align*} & \\zeta ^ { \\mathfrak { m } } ( \\{ 2 \\} ^ { k _ { d } } , 3 , \\dots \\{ 2 \\} ^ { k _ { 1 } } , 3 , \\{ 2 \\} ^ { k _ { 0 } } ) \\\\ & = ( - 1 ) ^ { k _ { 0 } + \\cdots + k _ { d } + d } I ^ { \\mathfrak { m } } ( \\overbrace { 0 , 1 , \\dots , 0 , 1 , 0 } ^ { 2 k _ { d } + 3 } , \\overbrace { 0 , 1 , \\dots , 0 , 1 , 0 } ^ { 2 k _ { d - 1 } + 3 } , \\dots , \\overbrace { 0 , 1 , \\dots , 0 , 1 , 0 } ^ { 2 k _ { 1 } + 3 } , \\overbrace { 0 , 1 , \\dots , 0 , 1 } ^ { 2 k _ { 0 } + 2 } ) \\\\ & = ( - 1 ) ^ { k _ { 0 } + \\cdots + k _ { d } } L _ { B } ( x _ { 2 k _ { 0 } + 2 } x _ { 2 k _ { 1 } + 3 } \\cdots x _ { 2 k _ { d } + 3 } ) , \\end{align*}"}
+{"id": "7232.png", "formula": "\\begin{align*} g ( t , x ) : = g _ { b } ( t , x ) = \\sum _ { j } b _ { j } e ^ { i j x } e ^ { i j ^ { 2 } t } . \\end{align*}"}
+{"id": "5251.png", "formula": "\\begin{align*} S _ N ( f ) = \\sum _ { 1 \\leq j _ 1 , j _ 2 , \\ldots , j _ { n + 1 } \\leq N } f ( N \\ * ( \\theta _ { j _ 2 } - \\theta _ { j _ 1 } ) _ c , \\ldots , N \\ * ( \\theta _ { j _ { n + 1 } } - \\theta _ { j _ 1 } ) _ c ) , \\end{align*}"}
+{"id": "2974.png", "formula": "\\begin{align*} [ Z _ 3 ] + [ Y _ 2 ] = [ Z _ 1 ] \\\\ [ Z _ 1 ] + [ Y _ 3 ] = [ Z _ 2 ] \\end{align*}"}
+{"id": "6672.png", "formula": "\\begin{align*} \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( - s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { m - 1 } \\right ) & \\stackrel { a \\to 0 + } { \\longrightarrow } \\mathcal { D } \\left ( ( - s ^ { ( 1 ) } _ { k + 1 } ) _ { k = 0 } ^ { 2 m - 3 } , ( - s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { m - 1 } \\right ) \\\\ & = ( - 1 ) ^ { m } \\mathcal { D } \\left ( ( - s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { 2 m - 3 } , ( - s ^ { ( 1 ) } _ { k } ) _ { k = m } ^ { 2 m - 1 } \\right ) \\end{align*}"}
+{"id": "484.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom n k \\frac { L _ { j k } } { L _ j ^ k } B _ { n - k } = 0 , \\qquad \\mbox { $ n $ o d d } . \\end{align*}"}
+{"id": "6518.png", "formula": "\\begin{align*} G _ a = \\frac { 1 } { \\deg ( a ) } J - I \\end{align*}"}
+{"id": "4471.png", "formula": "\\begin{align*} \\int _ a ^ b u ' v = \\tilde { u } ( b ) \\tilde { v } ( b ) - \\tilde { u } ( a ) \\tilde { v } ( a ) - \\int _ a ^ b u v ' \\ , , \\quad \\forall a , b \\in \\overline { I } \\ , . \\end{align*}"}
+{"id": "1578.png", "formula": "\\begin{align*} w _ r ^ { - 1 } ( w _ r x _ { a } t _ { r } ^ { 5 } x _ { b } t _ { r } ^ { 5 } x _ { c } \\left ( t _ { a } ^ { - 1 } t _ { r } ^ { - 1 } t _ { b } ^ { - 1 } t _ { r } ^ { - 1 } t _ { c } ^ { - 1 } \\right ) ^ { 5 } ) = e \\end{align*}"}
+{"id": "7308.png", "formula": "\\begin{align*} \\overline { M } _ 3 = ( \\mathop { q - 2 } \\limits _ { m - 1 } , \\underbrace { q - 1 , \\ldots , q - 1 , } _ { m - 2 - i } \\mathop { q - 2 } _ { i } , \\underbrace { q - 1 , \\ldots , q - 1 , } _ { i - j - 1 } \\mathop { q - 2 } \\limits _ { j } , \\underbrace { q - 1 , \\ldots , q - 1 } _ { j } ) , \\end{align*}"}
+{"id": "7798.png", "formula": "\\begin{align*} u ( t , \\theta ) = \\csc ( \\theta ) e ^ { i t } + i \\cot ( \\theta ) , ( t , \\theta ) \\in ( - \\pi , \\pi ] \\times ( 0 , \\pi ) \\end{align*}"}
+{"id": "285.png", "formula": "\\begin{align*} \\varphi _ { s } ^ { ( D ' \\subset D , R ) } ( \\bar { a } ) = \\begin{cases} \\overline { - F ^ { s - 1 } d a } \\otimes \\overline { 1 } & ( p \\neq 2 ) , \\\\ \\overline { - F ^ { s - 1 } d a } \\otimes \\overline { 1 } + \\sum _ { \\substack { i \\in I - I ' \\\\ n _ { i } = 2 } } \\left ( \\overline { d t _ { i } / t _ { i } ^ { 2 } } \\otimes \\sqrt { \\overline { a _ { 0 } t _ { i } ^ { 2 } } } \\right ) & ( p = 2 ) \\end{cases} \\end{align*}"}
+{"id": "1242.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle - \\Delta _ p u _ p = f & \\Omega , \\\\ \\displaystyle | \\nabla u _ p | ^ { p - 2 } \\nabla u _ p \\cdot \\nu + \\lambda | u _ p | ^ { p - 2 } u _ p = g & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "3084.png", "formula": "\\begin{align*} A _ 4 & = ( \\overline { 0 , 0 , 3 , 0 , 0 , 3 } ) - ( \\overline { 0 , 0 , 3 , 0 , 0 , 3 } ) \\\\ & = ( \\overline { 0 , 0 , 0 , 0 , 0 , 0 } ) . \\end{align*}"}
+{"id": "5487.png", "formula": "\\begin{align*} F _ N ( a , 1 ; t ) = \\frac { ( a t q ) _ N } { ( t ) _ N } . \\end{align*}"}
+{"id": "2481.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = | x | ^ { a } v ^ { p } & & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta v & = | x | ^ { b } v ^ { q } f ( | \\nabla u | ) & & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1763.png", "formula": "\\begin{align*} \\mu ^ { n - \\left ( i _ { j } + \\epsilon \\right ) } \\underset { = + 1 } { \\underbrace { \\left ( - 1 \\right ) ^ { k + 1 + i _ { j } + \\epsilon + \\epsilon } } } \\det \\left ( C _ { [ n ] \\setminus \\{ i _ { 1 } , i _ { 1 } + 1 , \\dots , i _ { \\frac { m } { 2 } } , i _ { \\frac { m } { 2 } } + 1 \\} } \\right ) + o \\left ( \\mu ^ { n - \\left ( i _ { j } + \\epsilon \\right ) } \\right ) , \\end{align*}"}
+{"id": "104.png", "formula": "\\begin{align*} v = 0 \\partial A ( r _ { 1 } , r _ { 2 } ) . \\end{align*}"}
+{"id": "4677.png", "formula": "\\begin{align*} & y ^ 7 + ( 6 t ^ 4 + 6 t ^ 3 + 6 t ^ 2 + 1 2 t + 1 ) y ^ 5 + ( t ^ 8 + 2 t ^ 7 + 3 t ^ 6 + 6 t ^ 5 + t ^ 4 + 5 t + 4 ) y ^ 3 + \\\\ & ( 6 t ^ { 1 2 } + 5 t ^ { 1 1 } + 1 0 t ^ { 1 0 } + 7 t ^ 8 + 2 t ^ 7 + 3 t ^ 6 + 9 t ^ 5 + 3 t ^ 4 + 2 t ^ 3 + 6 t ^ 2 + t + 4 ) y + \\\\ & ( 1 1 t ^ { 1 4 } + 6 t ^ { 1 3 } + 1 2 t ^ { 1 2 } + 1 0 t ^ { 1 1 } + 5 t ^ { 1 0 } + 8 t ^ 9 + 6 t ^ 8 + 2 t ^ 7 + 2 t ^ 6 + 1 0 t ^ 5 + 7 t ^ 4 + 1 2 t ^ 3 + 3 t ^ 2 + 3 t + 9 ) = 0 \\end{align*}"}
+{"id": "5392.png", "formula": "\\begin{align*} ( d _ A \\vert _ D ) ^ \\dagger = d _ A ^ \\star , ( d _ A ^ \\star \\vert _ N ) ^ \\dagger = d _ A \\quad ( d ^ \\star _ A \\vert _ N ) ^ { \\dagger \\dagger } = d ^ \\star _ A \\vert _ N . \\end{align*}"}
+{"id": "8018.png", "formula": "\\begin{align*} \\begin{bmatrix} \\begin{array} { c | c } Q _ 0 & r o w ( Q _ { - j } ) _ { j \\geq 1 } \\\\ \\hline c o l ( Q _ j ) _ { j \\geq 1 } & T _ Q \\ : \\otimes \\ : I _ d \\end{array} \\end{bmatrix} - \\begin{bmatrix} \\begin{array} { c | c c } A & B ^ * & 0 \\\\ \\hline B & C & 0 \\\\ 0 & 0 & 0 \\end{array} \\end{bmatrix} \\geq 0 . \\end{align*}"}
+{"id": "4320.png", "formula": "\\begin{align*} | m _ i ( h ) | \\leq \\frac { 1 + \\rho } { \\rho } + \\frac { 1 } { \\rho ^ { i + 1 } } \\sum _ { j = i } ^ \\infty \\rho ^ j = \\frac { 2 - \\rho ^ 2 } { \\rho ( 1 - \\rho ) } . \\end{align*}"}
+{"id": "6264.png", "formula": "\\begin{align*} \\norm { z _ { j } - z _ i } \\leq \\sum _ { k = i } ^ j \\norm { z _ { k + 1 } - z _ k } \\leq \\norm { z _ i - z _ { i - 1 } } + C \\tilde { \\varphi } \\left ( H _ { p _ i } ( z _ { i } ) - F ( x ^ * ) \\right ) . \\end{align*}"}
+{"id": "1513.png", "formula": "\\begin{align*} w = q ^ 3 . \\end{align*}"}
+{"id": "816.png", "formula": "\\begin{align*} l _ 2 \\bar { \\theta } _ 1 ^ { k _ 2 - 1 } \\Big ( \\frac { K k _ 2 } { k _ 2 - 1 } \\Big ) ^ { 1 - k _ 2 } x ^ { k _ 2 } + l _ 1 = \\frac { 1 - \\theta } { \\bar { \\theta } _ 1 - \\theta } \\end{align*}"}
+{"id": "8025.png", "formula": "\\begin{align*} P = \\begin{bmatrix} F _ 0 & & & & \\\\ F _ 1 & F _ 0 \\otimes I _ d & & & \\\\ F _ 2 & F _ 1 \\otimes I _ d & F _ 0 \\otimes I _ d ^ { \\otimes 2 } & & \\\\ \\vdots & \\vdots & \\vdots & \\ddots \\\\ F _ { m - 1 } & F _ { m - 2 } \\otimes I _ d & F _ { m - 3 } \\otimes I _ d ^ { \\otimes 2 } & \\cdots & F _ 0 \\otimes I _ d ^ { \\otimes ( m - 1 ) } \\end{bmatrix} , \\ , R = F _ 0 \\otimes I _ d ^ { \\otimes m } \\end{align*}"}
+{"id": "2653.png", "formula": "\\begin{align*} \\Big ( \\frac p q \\Big ) \\Big ( \\frac q p \\Big ) = ( - 1 ) ^ { \\mu + \\nu } . \\end{align*}"}
+{"id": "1272.png", "formula": "\\begin{align*} \\tilde M = 4 A _ 1 \\hookrightarrow \\tilde D _ 4 \\end{align*}"}
+{"id": "3694.png", "formula": "\\begin{align*} N _ { K ( \\beta ) / K } \\left ( \\beta - \\frac { 1 } { c } \\right ) & = \\prod _ { ~ \\alpha \\ { \\rm o f } \\ g ( f ^ { m - 1 } ( z ) ) } \\left ( \\alpha - \\frac { 1 } { c } \\right ) = ( - 1 ) ^ t g ( \\left ( f ^ { m - 1 } \\left ( \\frac { 1 } { c } \\right ) \\right ) = ( - 1 ) ^ t g ( f ^ { ( m ) } ( 0 ) ) \\end{align*}"}
+{"id": "3499.png", "formula": "\\begin{align*} { } \\nabla \\mathrm { H } ( \\mathrm { x } ) - \\alpha \\Big ( \\nabla \\int _ \\Omega \\nabla \\mathbb { G } ^ { ( \\mathrm { k } ) } ( \\mathrm { x } , \\mathrm { y } ) \\cdot \\nabla \\mathrm { H } ( \\mathrm { y } ) d \\mathrm { y } \\Big ) = \\nabla \\mathrm { H } ^ { \\textbf { i n } } ( \\mathrm { x } ) \\end{align*}"}
+{"id": "1283.png", "formula": "\\begin{align*} ( s _ 2 \\bullet f _ 2 ) \\circ ( s _ 1 \\bullet f _ 1 ) = s _ 2 s _ 1 \\bullet ( f _ 2 \\circ f _ 1 ) , \\end{align*}"}
+{"id": "5933.png", "formula": "\\begin{align*} \\gamma ( t ) : = ( t , t ^ 2 , \\dots , t ^ k ) , | t | \\leq 1 . \\end{align*}"}
+{"id": "1133.png", "formula": "\\begin{gather*} E ( X - Y ) ^ { 2 k } \\allowbreak = \\allowbreak E ( X + \\rho Y - ( 1 + \\rho ) Y ) ^ { 2 k } \\allowbreak = \\\\ \\allowbreak \\sum _ { j = 0 } ^ { 2 k } \\binom { 2 k } { j } ( - 1 ) ^ { j } ( 1 + \\rho ) ^ { 2 k - j } E ( X + \\rho Y ) ^ { j } Y ^ { 2 k - j } \\allowbreak = \\\\ \\allowbreak \\sum _ { j = 0 } ^ { 2 k } \\binom { 2 k } { j } ( - 1 ) ^ { j } ( 1 + \\rho ) ^ { 2 k - j } E ( E ( X + \\rho Y ) ^ { j } | Y ) Y ^ { 2 k - j } \\allowbreak . \\end{gather*}"}
+{"id": "710.png", "formula": "\\begin{align*} { \\rm O } ( \\epsilon ^ 1 ) : \\rho c _ v { \\bf { I } } \\partial _ { t _ 1 } { { { \\bf { m } } } } ^ { ( 0 ) } + { \\bf { d } } _ { 1 } { { { \\bf { m } } } } ^ { ( 0 ) } = - \\frac { 1 } { \\Delta t } { \\Lambda } { { { \\bf { m } } } } ^ { ( 1 ) } + { { { \\bf { M } } } } { { { \\bf { \\bar F } } } } ^ { ( 1 ) } , \\end{align*}"}
+{"id": "3600.png", "formula": "\\begin{align*} \\sum _ i \\frac { f _ i ^ 2 } { a _ i } = \\sum _ i \\frac { f _ i ^ 2 \\tau _ i } { a _ i \\tau _ i } \\leq \\sup _ i \\tau _ i \\sum _ i \\frac { f _ i ^ 2 } { a _ i \\tau _ i } < \\infty . \\end{align*}"}
+{"id": "1947.png", "formula": "\\begin{align*} \\langle Y u , u \\rangle & = \\langle Y u _ { \\varepsilon } , u _ { \\varepsilon } \\rangle + \\langle Y u - ( Y u ) _ { \\varepsilon } , u \\rangle \\\\ & + \\langle ( Y u ) _ { \\varepsilon } - Y u _ { \\varepsilon } , u \\rangle + \\langle Y u _ { \\varepsilon } , u - u _ { \\varepsilon } \\rangle \\\\ & = : I _ 1 + I _ 2 + I _ 3 + I _ 4 . \\end{align*}"}
+{"id": "1585.png", "formula": "\\begin{align*} w _ { r _ j } t _ { r _ j } ^ 2 t _ c y _ { r _ { j } , 3 5 } ^ { - 1 } = w _ { r _ j } y _ { r _ { j } , 3 6 } ^ { - 1 } u _ { t _ c , t _ { r _ j } } . \\end{align*}"}
+{"id": "5034.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } \\frac { \\mathcal { U } ( s ) } { s | \\ln ( s ) | ^ { \\frac { 1 } { 2 } } } = 1 . \\end{align*}"}
+{"id": "2924.png", "formula": "\\begin{align*} T ( r , 1 / \\varphi ) = T ( r , \\varphi ) \\le T ( r , z b ^ * ( 1 - z A ) ^ { - 1 } a ) + \\log 2 . \\end{align*}"}
+{"id": "4311.png", "formula": "\\begin{align*} h ( i ) - \\mathbb { E } h ( \\pi ) = f ( i ) - \\sum _ { j = 0 } ^ \\infty P _ { i , j } f ( j ) , \\end{align*}"}
+{"id": "3874.png", "formula": "\\begin{align*} K _ B ( x , y ) = K _ B ( x , - y ) = K _ B ( - x , y ) = K _ B ( - x , - y ) \\end{align*}"}
+{"id": "4930.png", "formula": "\\begin{align*} \\binom { n } { 1 } + \\sum _ { i = 2 } ^ { k - 1 } \\frac { 1 + 3 { \\left ( i - 1 \\right ) } } { k - { \\left ( i - 1 \\right ) } } \\binom { n } { k - i } > 2 \\left ( \\binom { n } { 3 } + \\binom { n } { 2 } + \\binom { n } { 1 } \\right ) . \\end{align*}"}
+{"id": "3899.png", "formula": "\\begin{align*} \\omega ' ( t ) + \\lambda ( 1 + D _ t ^ { \\{ m \\} } ) \\omega ( t ) & = 0 , \\ ; t > 0 , \\\\ \\omega ( 0 ) & = 1 , \\end{align*}"}
+{"id": "8260.png", "formula": "\\begin{align*} \\begin{aligned} Z _ { s , \\bar { s } } ( t ) \\le C \\big ( 1 + Z _ { s - 1 , \\bar { s } } ( t ) \\big ) \\exp \\Big \\{ C \\int _ 0 ^ t \\Big ( \\Psi ( t , \\tau ) + \\Vert u ( \\tau ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } } \\Big ) \\dd \\tau \\Big \\} . \\end{aligned} \\end{align*}"}
+{"id": "4318.png", "formula": "\\begin{align*} m _ i ( h ) = \\frac { 1 + \\rho } { \\rho } \\left ( \\widehat { h } ( i ) + \\frac { 1 } { ( 1 + \\rho ) ^ { i + 1 } } \\sum _ { k = 0 } ^ { i - 1 } ( 1 + \\rho ) ^ k m _ k ( h ) \\right ) , \\end{align*}"}
+{"id": "7481.png", "formula": "\\begin{align*} \\psi _ m ^ { ( a ) } ( n ) & = n + ( m - 1 ) ( n - 1 ) B _ { m - 1 , n - 1 } \\ , , \\\\ & = B _ { 1 , n - 1 } + m ( m - 1 ) B _ { m , n - 2 } \\ , . \\end{align*}"}
+{"id": "7559.png", "formula": "\\begin{align*} G ( p , p _ 1 ) = \\frac { 1 } { 4 } \\log | \\lambda ( p ) | - \\frac { 1 } { 2 } \\log | z ( p ) - 1 | + c \\end{align*}"}
+{"id": "4721.png", "formula": "\\begin{align*} ( a + u ) \\cdot ( b + v ) : = a \\cdot b + ( l ( a ) v + r ( b ) u ) , \\forall a , b \\in A , u , v \\in V . \\end{align*}"}
+{"id": "2283.png", "formula": "\\begin{align*} \\dim ( W ^ m ) & \\geq \\dim \\left ( \\sum _ { i = 0 } ^ m U _ l ^ { m - i } \\left ( W _ { l + 1 } z ^ { p _ { l + 1 } } \\right ) ^ i \\right ) \\\\ & \\geq \\dim \\left ( U _ l ^ m + \\sum _ { i = 1 } ^ m ( W _ l z ^ { p _ l } ) ^ { m - i } \\left ( W _ { l + 1 } z ^ { p _ { l + 1 } } \\right ) ^ i \\right ) . \\end{align*}"}
+{"id": "4255.png", "formula": "\\begin{align*} & \\sum _ { m = 0 } ^ { N - k } \\frac { \\left ( \\frac { c q ^ { k } } { d } \\right ) _ { m } \\left ( q ^ { - ( N - k ) } \\right ) _ m ( d q ^ { N + 1 } ) ^ m } { ( q ) _ m \\left ( c q ^ { k + 1 } \\right ) _ { m } } & = \\frac { ( d q ^ { k + 1 } ) _ { \\infty } ( \\frac { c q ^ { k } } { d } ) _ { \\infty } } { ( c q ^ { k + 1 } ) _ { \\infty } ( d q ^ { N + 1 } ) _ { \\infty } } \\ ; _ { 2 } \\phi _ { 1 } \\left [ \\begin{array} { c c c } d q , & d q ^ { N + 1 } ; \\ ; & \\frac { c q ^ k } { d } \\\\ d q ^ { k + 1 } & \\end{array} \\right ] . \\end{align*}"}
+{"id": "8193.png", "formula": "\\begin{align*} \\Vert ( \\rho - 1 , u ) ^ \\ell ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha + s \\alpha } _ { 2 , 1 } } + \\Vert \\rho ^ h ( t ) - 1 \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } } _ { 2 , 1 } } + \\Vert u ^ h ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } } \\le C ( 1 + t ) ^ { - s } , \\end{align*}"}
+{"id": "5501.png", "formula": "\\begin{align*} F _ N ( a , b ; t ) & = \\frac { ( 1 - b ) ( 1 - t q ^ { N + 1 } ) } { ( 1 - t ) ( 1 - b q ^ { N + 1 } ) } + \\frac { ( 1 - q ^ { N + 1 } ) ( b - a t q ) ( 1 - b q ^ { N } ) } { ( 1 - b q ^ { N + 1 } ) ( 1 - b q ^ { N } ) ( 1 - t ) } \\sum _ { n = 0 } ^ { N } \\frac { ( q ) _ N ( a q ) _ n ( q ) _ n ( t q ) _ { N - n } ( t q ) ^ n } { ( q ) _ { N - n } ( b q ) _ n ( q ) _ n ( t q ) _ N } \\\\ & = \\frac { ( 1 - b ) ( 1 - t q ^ { N + 1 } ) } { ( 1 - t ) ( 1 - b q ^ { N + 1 } ) } + \\frac { ( 1 - q ^ { N + 1 } ) ( b - a t q ) } { ( 1 - b q ^ { N + 1 } ) ( 1 - t ) } F _ N ( a , b ; t q ) . \\end{align*}"}
+{"id": "5578.png", "formula": "\\begin{align*} ( I + J ) ^ { ( k ) } = \\sum _ { \\ell = 0 } ^ k I ^ { ( \\ell ) } \\cdot J ^ { ( k - \\ell ) } . \\end{align*}"}
+{"id": "7739.png", "formula": "\\begin{align*} S | _ { \\partial X } = \\frac { n } { n - 1 } S _ { \\hat { g } } \\end{align*}"}
+{"id": "5458.png", "formula": "\\begin{align*} \\det M = \\det \\begin{pmatrix} A & B & C \\\\ O & D & O \\\\ O & O & E \\end{pmatrix} , \\end{align*}"}
+{"id": "4927.png", "formula": "\\begin{align*} \\binom { n } { k } / \\binom { n } { ( k + r + 1 ) / 2 } & = \\frac { ( n - k + 1 ) \\cdots ( n - ( k + r + 1 ) / 2 ) } { ( k + r + 3 ) / 2 \\cdots k } \\ge \\frac { n - ( k + r + 1 ) / 2 } { ( k + r + 3 ) / 2 } \\\\ & = \\frac { 3 k + r - 1 } { k + r + 3 } \\ge \\frac { 3 k + ( k - 3 ) - 1 } { k + ( k - 3 ) + 3 } = \\frac { 2 k - 2 } { k } > \\frac { k - 1 } { k } + \\frac { k - 1 } { 2 k - 1 } . \\end{align*}"}
+{"id": "2415.png", "formula": "\\begin{align*} \\tilde { L } ( { \\rm D } ( w ) ) & = \\sum _ { N \\geq 2 } ( - 1 ) ^ { N } \\zeta ^ { \\mathfrak { m } } ( N ) \\tilde { L } ( { \\rm C u t } _ { ( \\{ 0 \\} ^ { N - 1 } ; \\emptyset ) } ( w ) ) + \\sum _ { N \\geq 2 } \\zeta ^ { \\mathfrak { m } } ( N ) \\tilde { L } ( { \\rm C u t } _ { ( \\emptyset ; \\{ 1 \\} ^ { N - 1 } ) } ( w ) ) . \\end{align*}"}
+{"id": "6777.png", "formula": "\\begin{align*} i \\partial _ t \\chi ( t ) = H ( t ) \\chi ( t ) \\chi ^ { ( k ) } ( 0 ) = \\begin{cases} \\left ( U _ N ( 0 ) \\Psi _ { N , 0 } \\right ) ^ { ( k ) } & \\ ; k \\in \\{ 1 , 2 , \\ldots , N \\} , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "6875.png", "formula": "\\begin{align*} L ( E / H , s ) = \\prod _ { \\psi } L ( E / K , \\psi , s ) \\end{align*}"}
+{"id": "5073.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ p _ t L ^ q _ z ( I _ t \\times \\mathbb { R } ^ m \\times \\mathbb { T } ^ n ) } = \\left ( \\int _ { I _ t } \\left ( \\int _ { \\mathbb { R } ^ m \\times \\mathbb { T } ^ n } | u ( t , z ) | ^ q \\ , d z \\right ) ^ { \\frac { p } { q } } \\ , d t \\right ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
+{"id": "6279.png", "formula": "\\begin{align*} & \\varphi ( t _ { j + 1 } ) = \\left ( \\norm { a _ { J _ { t _ { j + 1 } } } } _ 1 - \\lambda \\right ) t _ { j + 1 } \\\\ & \\varphi ( t _ { j + 1 } + ) = \\left ( \\norm { a _ { J _ { t _ j } } } _ 1 - \\lambda \\right ) t _ { j + 1 } + \\norm { a _ { I \\backslash J _ { t _ j } } } t _ { j + 1 } . \\end{align*}"}
+{"id": "5837.png", "formula": "\\begin{align*} \\| U \\| _ { b e r } = \\sup \\{ \\| U e _ i \\| : i = 1 , 2 \\} = 1 , \\end{align*}"}
+{"id": "7819.png", "formula": "\\begin{align*} \\frac { J ^ { w _ 1 } ( X _ { i : n } ) } { J ^ { w _ 1 } ( X _ { i : n + 1 } ) } & = \\frac { ( 2 n + 1 ) ( n - i + 1 ) } { ( n + 1 ) ( 2 n - 2 i + 1 ) } \\left ( \\frac { 2 i - 1 } { 2 i + 1 } \\right ) \\frac { E \\left ( M ( B _ { 2 i : 2 n } ) \\right ) } { E \\left ( M ( B _ { 2 i : 2 n + 2 } ) \\right ) } \\\\ & \\ge \\frac { E \\left ( M ( B _ { 2 i : 2 n } ) \\right ) } { E \\left ( M ( B _ { 2 i : 2 n + 2 } ) \\right ) } \\end{align*}"}
+{"id": "1269.png", "formula": "\\begin{align*} 6 m = 1 3 \\cdot 3 . \\end{align*}"}
+{"id": "3298.png", "formula": "\\begin{align*} \\sigma _ K = | a _ { \\hat { E } } | ^ 2 \\geq 0 \\sigma _ L = | b _ { \\hat { E } } | ^ 2 \\geq 0 \\end{align*}"}
+{"id": "3849.png", "formula": "\\begin{align*} g \\in L ^ { 1 + \\alpha } ( \\mathbb { F } ) , \\| g \\| _ { 1 + \\alpha } \\leq n ^ { \\beta } , \\mathbb { F } = \\R ~ \\mathrm { o r } ~ \\C . \\end{align*}"}
+{"id": "1465.png", "formula": "\\begin{align*} P ( \\Delta _ { \\theta , i } ^ { [ t ] } = \\tilde { q } | Q _ n ^ { [ 1 : t ] } , U _ n ^ { [ 1 : t ] } ) = P ( \\Delta = \\tilde { q } ) , \\end{align*}"}
+{"id": "4858.png", "formula": "\\begin{align*} \\frac { \\partial \\rho } { \\partial t } = \\nabla \\cdot ( \\rho \\nabla f ( x , s ) ) + \\nabla \\cdot ( D ( x , s ) \\nabla \\rho ) + K ( \\bar { \\rho } \\otimes \\mu - \\rho ) , \\end{align*}"}
+{"id": "4393.png", "formula": "\\begin{align*} d \\mu ( x ) = \\frac { 1 } { \\sqrt { 2 \\pi } } w ( x ) d x , \\end{align*}"}
+{"id": "3236.png", "formula": "\\begin{align*} w = \\frac { u + v } { 2 } - C - ( n + 1 ) \\ln 2 . \\end{align*}"}
+{"id": "6745.png", "formula": "\\begin{align*} \\| \\nabla _ { x } \\widetilde { u } \\| _ { L ^ { \\infty } } \\leq C \\| \\nabla ^ 2 _ { x } \\widetilde { u } \\| ^ { \\frac { 1 } { 2 } } \\| \\nabla ^ 3 _ { x } \\widetilde { u } \\| ^ { \\frac { 1 } { 2 } } \\leq C \\varepsilon ^ { \\frac { 1 } { 2 } - \\frac { 1 } { 2 } a } \\varepsilon ^ { - \\frac { 1 } { 2 } a } = C \\varepsilon ^ { \\frac { 1 } { 2 } - a } . \\end{align*}"}
+{"id": "2551.png", "formula": "\\begin{align*} \\alpha _ t ( x ) : = - R ^ { - 1 } B ( P _ t x + \\varphi _ t ^ \\xi ) - h ( \\mu _ t ) . \\end{align*}"}
+{"id": "1693.png", "formula": "\\begin{align*} h : = - \\Delta + \\kappa , \\end{align*}"}
+{"id": "6422.png", "formula": "\\begin{align*} V _ { \\check \\psi } \\check u ( x , \\xi ) = V _ \\psi u ( - x , - \\xi ) . \\end{align*}"}
+{"id": "4148.png", "formula": "\\begin{align*} \\dot { u } ^ M _ i ( t ) = \\frac { 1 } { M } \\sum _ { j = 1 } ^ { M } B ^ M _ { i j } \\left ( u _ j ^ M ( t ) - u _ i ^ M ( t ) \\right ) , \\ , \\ , i = 1 , 2 , . . . , M , \\end{align*}"}
+{"id": "4418.png", "formula": "\\begin{align*} \\mu ( | \\epsilon | ) = \\mu _ 0 \\abs { \\epsilon } ^ { \\alpha - 1 } , \\alpha > 0 , \\end{align*}"}
+{"id": "2913.png", "formula": "\\begin{align*} \\mathcal R + \\mathcal K = \\mathcal R , \\ \\ \\mathcal P \\mathcal R + \\mathcal K = \\mathcal P \\mathcal R \\ \\ \\mathcal R \\not \\subset \\mathcal P \\mathcal K . \\end{align*}"}
+{"id": "5936.png", "formula": "\\begin{align*} a ( p , p _ 0 ) : = ( \\frac { p - p _ 0 } { 2 k } ) ( \\frac { p _ 0 } { 2 } + \\frac { k ^ 2 + 7 k - 4 } { 2 } ) + \\frac { k } { 2 } ( \\frac { p - p _ 0 } { 2 k } ) ( \\frac { p - p _ 0 } { 2 k } + 1 ) . \\end{align*}"}
+{"id": "7414.png", "formula": "\\begin{align*} f ( q ) & = q - 6 q ^ 5 + 9 q ^ 9 + 1 0 q ^ { 1 3 } - 3 0 q ^ { 1 7 } + 1 1 q ^ { 2 5 } + 4 2 q ^ { 2 9 } - 7 0 q ^ { 3 7 } + O ( q ^ { 4 1 } ) \\in S _ 3 ^ { \\textrm { n e w } } ( \\Gamma _ 1 ( 1 6 ) ) , \\\\ g ( q ) & = q - 2 q ^ 2 - 2 q ^ 3 + 4 q ^ 4 + 4 q ^ 6 - 8 q ^ 8 - 5 q ^ 9 + 1 4 q ^ { 1 1 } - 8 q ^ { 1 2 } + O ( q ^ { 1 4 } ) \\in S _ 3 ^ { \\textrm { n e w } } ( \\Gamma _ 1 ( 8 ) ) \\end{align*}"}
+{"id": "2917.png", "formula": "\\begin{align*} \\inf \\| p ( A ) \\| ^ { 1 / { \\rm d e g } ( p ) } = { \\rm c a p } ( \\sigma ( A ) ) , \\end{align*}"}
+{"id": "903.png", "formula": "\\begin{align*} \\begin{aligned} 0 \\geq \\ , & S _ k ^ { i j } ( A \\Psi \\pm T _ \\alpha ( u - \\varphi ) ) _ { i j } \\geq A \\eta _ 0 S _ { k - 1 } - C \\Big ( S _ { k - 1 } + f ^ { 1 - 1 / ( k - 1 ) } \\Big ) \\\\ \\geq \\ , & \\frac { A } { 2 } \\eta _ 0 S _ { k - 1 } + \\frac { A } { 2 } \\eta _ 0 c _ 0 f ^ { 1 - 1 / ( k - 1 ) } - C \\Big ( S _ { k - 1 } + f ^ { 1 - 1 / ( k - 1 ) } \\Big ) > 0 \\end{aligned} \\end{align*}"}
+{"id": "7747.png", "formula": "\\begin{align*} \\hat { g } ^ E = g ^ E | _ { \\partial ( \\mathbb { R } ^ n \\times \\mathbb { S } ^ 1 ( \\lambda ) ) } = \\frac { 1 } { 4 } ( g _ { \\mathbb { S } ^ { n - 1 } } + g _ { \\mathbb { S } ^ 1 ( \\lambda ) } ) . \\end{align*}"}
+{"id": "2937.png", "formula": "\\begin{align*} \\sigma _ \\delta ( A ) \\cap \\sigma _ a ( B ) = \\emptyset \\end{align*}"}
+{"id": "4493.png", "formula": "\\begin{align*} K ( u + h ) = \\frac { 1 } { 2 } \\int _ { [ 0 , 1 ] } \\big ( u ^ 2 + 2 u h + h ^ 2 \\big ) \\end{align*}"}
+{"id": "5002.png", "formula": "\\begin{align*} S = \\left \\langle a _ 1 , a _ 2 , \\dots , a _ k \\right \\rangle = \\left \\{ n _ 1 a _ 1 + n _ 2 a _ 2 + \\dots + n _ k a _ k : n _ 1 , n _ 2 , \\dots , n _ k \\in \\N _ 0 \\right \\} . \\end{align*}"}
+{"id": "4241.png", "formula": "\\begin{align*} \\lim \\limits _ { a \\to 0 } \\left ( \\frac { z q } { a } \\right ) _ n ( a ) ^ n = \\lim \\limits _ { a \\to 0 } \\left ( 1 - \\frac { z q } { a } \\right ) \\cdots \\left ( 1 - \\frac { z q ^ { n } } { a } \\right ) ( a ) ^ n = ( - 1 ) ^ n z ^ n q ^ \\frac { n ( n + 1 ) } { 2 } . \\end{align*}"}
+{"id": "623.png", "formula": "\\begin{align*} \\widetilde { \\mu } \\ , ' ( n ) \\ r _ n ( m ) = H ' _ { m } \\ \\widetilde { r } \\ , ' _ { n } ( m + 1 ) + I ' _ m \\ \\widetilde { r } \\ , ' _ { n } ( m ) + J ' _ m \\ \\widetilde { r } \\ , ' _ { n } ( m - 1 ) \\ , , \\end{align*}"}
+{"id": "288.png", "formula": "\\begin{align*} ( \\overline { a _ { s - 1 } } , \\overline { a _ { s - 2 } } , \\ldots , \\overline { a _ { 0 } } ) = ( 0 , 0 , \\ldots , \\overline { a _ { 0 } } ) \\end{align*}"}
+{"id": "4585.png", "formula": "\\begin{align*} \\abs { \\frac { 1 } { N } \\sum _ { l = 0 } ^ { N - 1 } \\cos ( \\theta + 2 \\pi k l ) } = 0 . \\end{align*}"}
+{"id": "5544.png", "formula": "\\begin{align*} \\Tilde { c _ 2 } ( T _ 2 ) \\Tilde { \\gamma _ 2 } ( T _ 2 ) \\dfrac { \\partial T _ 2 } { \\partial t } = \\dfrac { 1 } { r ^ { \\nu } } \\dfrac { \\partial } { \\partial r } \\bigg ( \\Tilde { \\lambda _ 1 } ( T _ 2 ) r ^ { \\nu } \\dfrac { \\partial T _ 2 } { \\partial r } \\bigg ) , \\ ; \\ ; \\ ; \\beta ( t ) < r < \\infty , \\ ; \\ ; t > 0 , \\end{align*}"}
+{"id": "3912.png", "formula": "\\begin{align*} \\| [ S ( t + h ) - S ( t ) ] \\xi \\| _ { \\mathbb H ^ \\mu } & \\le h \\int _ 0 ^ 1 \\| S ' ( t + \\zeta h ) \\xi \\| _ { \\mathbb H ^ \\mu } d \\zeta \\\\ & \\le h \\| \\xi \\| _ { \\mathbb H ^ \\mu } \\int _ 0 ^ 1 \\frac { d \\zeta } { t + \\zeta h } = \\| \\xi \\| _ { \\mathbb H ^ \\mu } \\ln \\left ( 1 + \\frac h t \\right ) \\\\ & \\le \\| \\xi \\| _ { \\mathbb H ^ \\mu } \\gamma ^ { - 1 } t ^ { - \\gamma } h ^ \\gamma . \\end{align*}"}
+{"id": "8196.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot B ^ s _ { p , r } } ^ \\ell : = \\big \\| \\{ 2 ^ { j s } \\| \\dot \\Delta _ j u \\| _ { L ^ p } \\} _ { j \\leq j _ 0 } \\big \\| _ { \\ell ^ r } , \\textrm { a n d } \\| u \\| _ { \\dot B ^ s _ { p , r } } ^ h : = \\big \\| \\{ 2 ^ { j s } \\| \\dot \\Delta _ j u \\| _ { L ^ p } \\} _ { j \\geq j _ 0 } \\big \\| _ { \\ell ^ r } . \\end{align*}"}
+{"id": "7255.png", "formula": "\\begin{align*} \\over = \\sup _ { \\mu \\in M ( X ) } \\hat { F } ( \\mu ) + \\int f d \\mu , \\end{align*}"}
+{"id": "2862.png", "formula": "\\begin{align*} N = \\{ n \\in N ' \\mid ( \\mathfrak { m } _ A ) ^ k \\cdot n = 0 k \\gg 0 \\} . \\end{align*}"}
+{"id": "6605.png", "formula": "\\begin{align*} \\Gamma _ { \\theta } = \\big \\{ ( x , y ) \\in \\mathbb { R } ^ { 2 } : x \\ge 0 \\lvert y \\rvert = x \\tan \\theta \\big \\} , \\end{align*}"}
+{"id": "4395.png", "formula": "\\begin{align*} \\mathcal { J } ( f ) ( \\lambda ) : = \\hat { f } ( \\lambda ) = \\int _ 0 ^ \\infty f ( x ) \\varphi _ \\lambda ^ { ( \\alpha , \\beta ) } ( x ) d \\mu ( x ) , \\end{align*}"}
+{"id": "6675.png", "formula": "\\begin{align*} \\frac { \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( - s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { m - 1 } \\right ) } { a b \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( 1 , 0 , \\ldots , 0 ) \\right ) } \\stackrel { a \\to 0 + } { \\longrightarrow } \\infty . \\end{align*}"}
+{"id": "7484.png", "formula": "\\begin{align*} \\psi _ m ^ { ( 1 ) } ( n ) & = \\sum _ { \\nu = 1 } ^ n \\psi _ m ( n ) \\ , , \\\\ & = \\sum _ { \\nu = 1 } ^ n \\left [ B _ { 1 , \\nu - 1 } + m ( m - 1 ) B _ { m , \\nu - 2 } \\right ] \\ , , \\\\ & = \\sum _ { \\nu = 1 } ^ n B _ { 1 , \\nu - 1 } + m ( m - 1 ) \\sum _ { \\nu = 1 } ^ n B _ { m , \\nu - 2 } \\ , , \\\\ & = B _ { 2 , n - 1 } + m ( m - 1 ) B _ { m + 1 , n - 2 } \\ , , \\\\ & = B _ { 2 , n - 1 } + \\frac { m ( m - 1 ) } { m + 1 } ( n - 1 ) B _ { m , n - 1 } \\ , . \\end{align*}"}
+{"id": "562.png", "formula": "\\begin{align*} M _ n ( Q ) / n & \\le \\int _ \\R V \\ , d \\overline { Q } + \\frac 1 2 \\int _ \\R \\int _ \\R K ( x - y ) \\ , \\overline { Q } ( d x ) \\overline { Q } ( d y ) - H ( \\overline { Q } ) = M _ n ( \\overline { Q } ^ { \\otimes n } ) / n . \\end{align*}"}
+{"id": "4374.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\sum _ { i = 1 } ^ { l } \\theta _ i D _ H g ^ 0 _ i ( \\overline { x } ( T ) ) \\cdot ( x ( T ) - \\overline { x } ( T ) ) \\\\ = p ( T ) \\cdot ( x ( T ) - \\overline { x } ( T ) ) - \\sum _ { \\alpha = 1 } ^ { m } \\lambda _ \\alpha D _ H g ^ \\alpha ( \\overline { x } ( T ) ) \\cdot ( x ( T ) - \\overline { x } ( T ) ) \\\\ - \\sum _ { \\beta = 1 } ^ { q } \\mu _ \\beta D _ H h ^ \\beta ( \\overline { x } ( T ) ) \\cdot ( x ( T ) - \\overline { x } ( T ) ) \\end{array} \\right \\} \\end{align*}"}
+{"id": "7505.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial n _ + } \\left ( 2 U ^ { \\mu } + \\Re V \\right ) ( z ) = \\frac { \\partial } { \\partial n _ - } \\left ( 2 U ^ { \\mu } + \\Re V \\right ) ( z ) , z \\in D \\cap \\Sigma , \\end{align*}"}
+{"id": "3828.png", "formula": "\\begin{align*} a _ 1 = \\frac { 4 n - 1 0 - ( a _ 1 + a _ 2 ) + d } { 2 } . \\end{align*}"}
+{"id": "1512.png", "formula": "\\begin{align*} V = \\prod _ { p < w ^ 3 } \\bigl ( 1 - g ( p ) \\bigr ) . \\end{align*}"}
+{"id": "1477.png", "formula": "\\begin{align*} U _ n ( s ) = & \\sum _ { i \\in J _ w ^ { [ s ] } } \\tilde { \\Delta } _ { \\theta , i } ^ { [ s ] } \\prod _ { j \\in J _ w ^ { [ s ] } , j \\neq i } ( f _ { g ( ( s - 1 ) \\ell _ w ^ * + j ) } - \\alpha _ n ) \\\\ & \\quad + \\prod _ { j \\in J _ w ^ { [ s ] } } ( f _ { g ( ( s - 1 ) \\ell _ w ^ * + j ) } - \\alpha _ n ) z , \\end{align*}"}
+{"id": "4818.png", "formula": "\\begin{align*} \\dot X ( t ) = - \\nabla f ^ n ( X ( t ) , S ( t ) ) , \\dot S ( t ) = 0 , \\end{align*}"}
+{"id": "929.png", "formula": "\\begin{align*} u _ { n n } \\sigma _ 1 ( b ) - \\sum _ { \\alpha \\leq n - 1 } u _ { n \\alpha } ^ 2 + \\sigma _ 2 ( b ) = f . \\end{align*}"}
+{"id": "4405.png", "formula": "\\begin{align*} \\upsilon = \\tau + \\phi \\omega , 0 \\leq \\tau \\leq | \\omega | - 1 . \\end{align*}"}
+{"id": "169.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\R ^ 3 } | v | ^ k e ^ { - a | v | ^ 2 } \\d v = \\Big \\{ \\int _ { | v | \\leq 1 } + \\int _ { | v | > 1 } \\Big \\} | v | ^ k e ^ { - a | v | ^ 2 } \\d v \\ , . \\end{aligned} \\end{align*}"}
+{"id": "6479.png", "formula": "\\begin{align*} ( i _ * ) ^ { - 1 } = i ^ ! ( - ) \\cap e ( N ) ^ { - 1 } \\end{align*}"}
+{"id": "367.png", "formula": "\\begin{align*} \\{ f \\mu _ \\xi , h \\mu _ \\zeta \\} = f h \\ , \\{ \\mu _ \\xi , \\mu _ \\zeta \\} + q \\end{align*}"}
+{"id": "4954.png", "formula": "\\begin{align*} [ x \\cdot y , z _ 2 \\ldots , z _ n ] = x \\cdot [ y , z _ 2 , \\ldots , z _ n ] + [ x , z _ 2 , \\ldots , z _ n ] \\cdot y . \\end{align*}"}
+{"id": "1589.png", "formula": "\\begin{align*} T _ { s _ { 2 } } + \\rho ( s ) T _ { b _ { 2 } } = T _ { b _ { 3 } } , \\end{align*}"}
+{"id": "6642.png", "formula": "\\begin{align*} - \\Big ( 1 + \\frac { 1 } { \\varepsilon } \\Big ) ^ 2 = - 1 - \\frac { 2 } { b _ { n } \\theta } - \\frac { 1 } { b _ { n } ^ { 2 } \\theta ^ { 2 } } > \\Lambda _ n ( Q ^ { 1 + \\varepsilon } _ \\theta ) \\theta \\to 0 ^ + . \\end{align*}"}
+{"id": "714.png", "formula": "\\begin{align*} c _ s ^ 2 \\partial _ { y 1 } T = - \\frac { 1 } { \\Delta t } { \\varsigma } _ 2 { m } _ 2 ^ { ( 1 ) } , \\end{align*}"}
+{"id": "6880.png", "formula": "\\begin{align*} \\alpha _ g : = \\zeta \\mbox { a n d } \\alpha _ h : = \\alpha \\zeta ^ { - 1 } , \\end{align*}"}
+{"id": "8065.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathrm { i f } ~ \\widetilde { \\beta } _ 0 > \\gamma , & \\mathrm { f a l s e ~ a l a r m } , \\\\ \\mathrm { i f } ~ \\widetilde { \\beta } _ 0 \\leq \\gamma , & \\mathrm { n o ~ f a l s e ~ a l a r m } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3832.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 ( 2 n - 5 ) - ( a _ 2 + a _ 3 + a _ 4 ) + d } { 2 } . \\end{align*}"}
+{"id": "7486.png", "formula": "\\begin{align*} & \\psi _ m ( n ) = n + \\frac { 1 } { ( m - 2 ) ! } f _ m ( n ) \\ , , \\\\ \\intertext { w i t h $ f _ m ( n ) $ g i v e n b y } & f _ m ( n ) = \\prod _ { k = - 1 } ^ { m - 2 } ( n + k ) = \\prod _ { i = 1 } ^ m ( n - \\alpha _ i ) = \\sum _ { k = 0 } ^ m b _ { m , m - k } n ^ k \\ , , \\\\ \\intertext { w h e r e t h e $ \\alpha _ i $ a r e o f c o u r s e t h e i n t e g e r r o o t s o f t h i s p o l y n o m i a l : } & \\alpha _ i = - ( i - 2 ) , 1 \\le i \\le m \\ , . \\end{align*}"}
+{"id": "3975.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } ^ { 2 \\beta } } { \\mathrm { d } t ^ { 2 \\beta } } u ( x , t ) = \\frac { \\partial ^ { 2 } } { \\partial x ^ { 2 } } u ( x , t ) , \\ x \\in \\mathbb { R } , \\ t > 0 , \\end{align*}"}
+{"id": "2994.png", "formula": "\\begin{align*} a ' & = - a ^ 3 / 2 - a b ^ 2 / 2 , \\\\ b ' & = - a b ^ 2 \\end{align*}"}
+{"id": "2139.png", "formula": "\\begin{align*} F _ 0 ( \\vec { x } ) = \\liminf _ { T \\to \\infty } \\left \\{ \\frac { 1 } { T ^ 2 } \\int _ { ( 0 , T ) ^ 2 } f \\left ( \\vec { y } , \\nabla v ( \\vec { y } ) + \\xi \\right ) \\ , d \\vec { y } \\ ; , \\ ; v \\in W ^ { 1 , p } _ 0 ( ( 0 , T ) ^ 2 ) \\right \\} \\end{align*}"}
+{"id": "6947.png", "formula": "\\begin{align*} 1 + ( - 1 ) ^ { ( N - 1 ) d + \\chi } y ^ { d + 1 } _ { t = 0 } \\ , \\omega & = \\sum _ { k = 0 } ^ { d } \\binom { - \\chi + ( N + 1 ) d } { k } \\binom { \\chi - N d } { d - k } ( 1 - y ) ^ { \\chi - N d - d + k } \\\\ & = \\sum _ { k = 0 } ^ { d } \\binom { - \\chi + ( N + 1 ) d } { k } \\frac { ( - y ) ^ k } { ( 1 - y ) ^ { ( N + 1 ) d - \\chi } } . \\end{align*}"}
+{"id": "1080.png", "formula": "\\begin{align*} E _ { \\Delta _ { \\tau , n } } : \\ ; \\ ; y ^ { 2 } = x ( x - n \\tau ) ( x + n \\tau ^ { - 1 } ) , \\end{align*}"}
+{"id": "2439.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { \\infty } \\frac { 2 \\mu ^ { 2 r + 2 } } { ( 2 r + 2 ) ! } X _ { 1 } ^ { 2 r } = \\left ( \\frac { e ^ { \\mu X _ { 1 } / 2 } - e ^ { - \\mu X _ { 1 } / 2 } } { X _ { 1 } } \\right ) ^ { 2 } = \\mu ^ { 2 } \\Gamma _ { 1 } ( - X _ { 1 } ) ^ { - 2 } \\Gamma _ { 1 } ( X _ { 1 } ) ^ { - 2 } , \\end{align*}"}
+{"id": "3180.png", "formula": "\\begin{align*} \\| \\phi ^ n ( g ) \\| _ T & = \\inf _ { k \\in \\N } \\frac { d _ T ( f ^ n ( x ) , \\phi ^ n ( g ^ k ) f ^ n ( x ) } { k } \\\\ & \\geq C ( \\varepsilon ) \\lambda ^ n \\frac { d _ T ( x , g ^ k x ) } { k } \\\\ & = C ( \\varepsilon ) \\lambda ^ n \\| g \\| _ T \\end{align*}"}
+{"id": "8002.png", "formula": "\\begin{align*} T : = \\ : \\ , \\sum _ { 0 \\leq | v | \\leq n } q _ v L _ v \\ , + \\ , \\sum _ { 0 < | v | \\leq n } q _ v ^ * L _ v ^ * . \\end{align*}"}
+{"id": "1039.png", "formula": "\\begin{align*} x ( t ) = \\frac { 1 } { \\sqrt { 2 t + 1 } } \\end{align*}"}
+{"id": "385.png", "formula": "\\begin{align*} \\{ f h , C ^ \\infty ( M ) \\} = f \\{ h , C ^ \\infty ( M ) \\} + h \\{ f , C ^ \\infty ( M ) \\} \\end{align*}"}
+{"id": "4451.png", "formula": "\\begin{align*} \\begin{cases} u = g , & x \\in \\Gamma _ D \\\\ u \\cdot \\nu = g _ { \\nu } , & x \\in \\Gamma _ R \\end{cases} \\end{align*}"}
+{"id": "4222.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { c _ f } { n _ { \\ell _ t } } \\sum _ { X = 1 } ^ { n _ { \\ell _ t } } | O _ { q , g } ( q ^ { - X / 2 } ) | = 0 . \\end{align*}"}
+{"id": "4081.png", "formula": "\\begin{align*} J ( d _ 1 , \\ldots , d _ n ; e _ 1 , \\ldots , e _ n ) : = \\sum _ { i = 1 } ^ n e _ i \\cdot \\sum _ { 0 \\leq j \\leq i } D _ j , \\end{align*}"}
+{"id": "3305.png", "formula": "\\begin{align*} ( 1 \\otimes K _ 1 ) K _ 2 ( v \\otimes \\psi _ { n _ 2 } ) = \\lambda _ { n _ 2 } K _ 2 ( v \\otimes \\psi _ { n _ 2 } ) . \\end{align*}"}
+{"id": "118.png", "formula": "\\begin{align*} q , r , \\tilde q , \\tilde r \\geq 2 \\quad \\tfrac { 2 } { q } + \\tfrac { 1 } { r } = \\tfrac { 1 } { 2 } = \\tfrac { 2 } { \\tilde q } + \\tfrac { 1 } { \\tilde r } . \\end{align*}"}
+{"id": "3218.png", "formula": "\\begin{align*} \\mu _ 2 ( G ) > 6 - 2 \\sqrt { 5 } > \\frac { 5 } { 4 } = \\frac { 2 . 3 - 1 } { \\ell } , \\end{align*}"}
+{"id": "3748.png", "formula": "\\begin{align*} m ( x ) ( x I - A ) ^ { - 1 } = A ^ 3 + ( x - 1 0 ) A ^ 2 + ( x ^ 2 - 1 0 x - 2 4 4 ) A + ( x ^ 3 - 1 0 x ^ 2 - 2 4 4 x - 5 3 6 ) I . \\end{align*}"}
+{"id": "2431.png", "formula": "\\begin{align*} f ( \\Phi _ { p , q } ) & = \\left ( \\frac { p ' ( X _ { 1 } ) } { p ( X _ { 1 } ) } + 2 \\psi _ { 1 } ( X _ { 1 } ) \\right ) \\cdot \\Phi _ { p , q } + \\Phi _ { p , q } \\cdot \\left ( \\frac { q ' ( X _ { 1 } ) } { q ( X _ { 1 } ) } - 2 \\psi _ { 1 } ( - X _ { 1 } ) \\right ) \\\\ & = \\frac { d } { d X _ { 1 } } \\log \\left ( p ( X _ { 1 } ) \\Gamma _ { 1 } ( X _ { 1 } ) ^ { 2 } \\right ) \\cdot \\Phi _ { p , q } + \\Phi _ { p , q } \\cdot \\frac { d } { d X _ { 1 } } \\log \\left ( q ( X _ { 1 } ) \\Gamma _ { 1 } ( - X _ { 1 } ) ^ { 2 } \\right ) . \\end{align*}"}
+{"id": "8013.png", "formula": "\\begin{align*} S ( m ) = \\begin{bmatrix} A & B ^ * \\\\ B & S ( m - 1 ) \\otimes I _ d \\end{bmatrix} \\end{align*}"}
+{"id": "7905.png", "formula": "\\begin{align*} a \\ast _ R b = R ( a ) \\cdot b + a \\cdot R ( b ) , ~ a , b \\in A . \\end{align*}"}
+{"id": "2667.png", "formula": "\\begin{align*} 0 = u c \\lim _ i \\| \\varpi \\otimes \\varpi ( \\cdot ) \\xi _ i - \\xi _ i \\| \\end{align*}"}
+{"id": "5596.png", "formula": "\\begin{align*} G _ { i k } u ^ k + G _ k J ^ k _ i = 0 . \\end{align*}"}
+{"id": "6894.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial L } v ( L , s ) = \\left ( 1 - v ^ 2 ( L , s ) \\right ) \\left ( v ( L , s + 1 ) - v ( L , s - 1 ) \\right ) . \\end{align*}"}
+{"id": "4107.png", "formula": "\\begin{align*} \\int ^ { \\mathrm { B C } } ( x _ 1 , \\ldots , x _ n ) : = \\prod _ { i = 1 } ^ { n - 1 } \\int ^ { \\mathrm { B C } } ( x _ i , x _ n ) , \\end{align*}"}
+{"id": "6034.png", "formula": "\\begin{align*} S _ \\gamma ^ \\perp : X _ 1 + X _ 2 + X _ 3 + \\gamma ^ q X _ 4 + \\gamma X _ 5 = 0 \\end{align*}"}
+{"id": "4410.png", "formula": "\\begin{align*} \\upsilon = d _ 0 + d _ 1 q + \\cdots + d _ k q ^ k , d _ i \\in \\N . \\end{align*}"}
+{"id": "2762.png", "formula": "\\begin{align*} a \\cdot ( A _ { n } / P _ { k } ) - b \\cdot \\dim ( A _ n / P _ k ) = k ( n + 1 ) - 2 k ( n - k + 1 ) = k ( 2 k - n - 1 ) = 0 . \\end{align*}"}
+{"id": "3257.png", "formula": "\\begin{align*} d \\eta _ i = 2 \\alpha \\Phi _ i ^ \\mathcal { H } \\ , . \\end{align*}"}
+{"id": "2669.png", "formula": "\\begin{align*} \\int _ M \\sum _ i \\lambda _ i ^ + ( x ) \\ , d x = h _ \\mu ( f ) , \\end{align*}"}
+{"id": "5137.png", "formula": "\\begin{align*} \\Vert D \\chi _ A \\Vert ( \\Omega ) = \\sup \\left \\{ \\int _ { A } \\ , ( v ) \\ , d x : \\ , v \\in C ^ { \\infty } _ { 0 } ( \\Omega ; \\R ^ n ) , \\ , | v | \\leq 1 \\right \\} \\end{align*}"}
+{"id": "209.png", "formula": "\\begin{align*} \\phi ( x ) = \\Psi \\left ( \\frac { \\ln \\left ( \\frac { | x | } { \\sqrt { R } } \\right ) } { \\ln \\left ( \\sqrt { R } \\right ) } \\right ) , \\ , \\ , x \\in \\mathbb { R } ^ { N } . \\end{align*}"}
+{"id": "863.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\sum _ { n = 0 } ^ { \\infty } z ^ n e ^ { i t S _ n } e ^ { i { s } \\cdot { C } _ n } \\right ] = \\frac { 1 } { 1 - z \\phi _ { \\xi _ 1 , { \\eta _ 1 } } ( t , { s } ) } = f ^ { - 1 } _ { + } ( z , t , s ) f _ { - } ( z , t , \\underline { s } ) \\end{align*}"}
+{"id": "1877.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { \\infty } \\frac { 1 } { 3 ^ { l s } } Q \\left ( \\frac { x } { 3 ^ { s + 1 } } , \\frac { y } { 3 ^ { s + 1 } } \\right ) < \\infty \\end{align*}"}
+{"id": "3359.png", "formula": "\\begin{align*} y _ i = \\mathrm { R e s } _ { L K _ i / K _ i } ( \\tilde { y } _ i ) . \\end{align*}"}
+{"id": "3116.png", "formula": "\\begin{align*} A _ 1 = \\left ( a _ { n , 1 } \\right ) _ { n = 1 } ^ { \\infty } \\end{align*}"}
+{"id": "358.png", "formula": "\\begin{align*} \\chi _ 1 \\left ( x _ 1 \\right ) = \\begin{cases} \\chi _ 1 ^ - ( x _ 1 ) , & x _ 1 < 0 , \\\\ \\chi _ 1 ^ + ( x _ 1 ) , & x _ 1 > 0 , \\end{cases} \\chi _ 3 \\left ( x _ 1 \\right ) = \\begin{cases} \\chi _ 3 ^ - ( x _ 1 ) , & x _ 1 < 0 , \\\\ \\chi _ 3 ^ + ( x _ 1 ) , & x _ 1 > 0 , \\end{cases} \\end{align*}"}
+{"id": "2965.png", "formula": "\\begin{align*} { \\sum } _ { k = 0 } ^ { \\infty } \\ \\mu _ k = \\infty , { \\sum } _ { k = 0 } ^ \\infty \\ , \\mu _ k ^ { 2 p _ 1 } < \\infty , q \\geq 2 , \\ ; \\ ; q a \\geq b , \\ \\ p _ 1 > \\frac 1 2 , \\ ; \\ ; p _ 2 \\geq 1 . \\end{align*}"}
+{"id": "4982.png", "formula": "\\begin{align*} \\frac { \\partial g } { \\partial w _ { c } } = - [ ( 1 - \\lambda ) ^ { w _ { c } / r - 1 } + w _ { c } ( 1 - \\lambda ) ^ { w _ { c } / r - 1 } \\ln ( 1 - \\lambda ) \\frac { 1 } { r } ] , \\end{align*}"}
+{"id": "3429.png", "formula": "\\begin{align*} P ^ S = - ( r ^ { 1 - n } S _ r / \\rho ) Q ^ U _ { 1 , l } = - 2 U S _ r f _ { J _ { 1 , l } } ^ { ( l ) } ; \\end{align*}"}
+{"id": "875.png", "formula": "\\begin{align*} { \\textit { ( i ) } } ~ | | \\xi _ 1 | | _ { \\mathfrak { Y } } & = \\sup \\bigg \\{ | | S ( \\xi _ 1 , \\widehat { D } ) | | ~ : ~ \\widehat { D } \\in \\mathfrak { Y } \\bigg \\} \\\\ & = \\sup \\bigg \\{ | | \\sum _ { i = 1 } ^ { n } \\xi _ 1 ( d _ i ) \\overline { \\mu } ( D _ i ) | | ~ : ~ \\widehat { D } = ( D _ i , d _ i ) , ~ i = 1 , 2 , . . \\bigg \\} \\\\ & \\geq 0 \\end{align*}"}
+{"id": "6776.png", "formula": "\\begin{align*} \\int d x \\ , b _ x ^ * \\Phi _ { \\varphi _ t } ( x ) b _ x & = 2 \\int d x \\ , b _ x ^ * \\Re \\left \\{ \\int d k \\ , \\frac { 1 } { 1 + k ^ 2 } G _ x ( k ) \\overline { \\varphi _ t ( k ) } \\right \\} b _ x \\\\ & - \\int d x \\ , b _ x ^ * \\Re \\left \\{ \\int d k \\ , \\frac { i k } { \\pi ( 1 + k ^ 2 ) } G _ x ( k ) \\overline { \\varphi _ t ( k ) } \\right \\} i \\nabla _ x b _ x + \\ , . \\end{align*}"}
+{"id": "6228.png", "formula": "\\begin{align*} \\lambda _ k ( \\infty ) : = B ( 2 n + 1 ) + \\big ( \\textstyle { \\frac { \\pi m } { 2 d } } \\big ) ^ 2 , n \\in \\mathbb { N } _ 0 , \\ ; m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "3512.png", "formula": "\\begin{align*} \\Big \\Vert \\widehat { \\nabla \\mathrm { H } } \\Big \\Vert ^ 2 _ { \\mathbb { L } ^ 2 ( \\mathrm { B } ) } = \\dfrac { 1 } { | 1 - \\alpha \\lambda ^ { ( 3 ) } _ { \\mathrm { n } _ { 0 } } | ^ 2 } \\Big | \\Big \\langle \\widehat { \\nabla \\mathrm { H } } ^ { \\textbf { i n } } ; e ^ { ( 3 ) } _ { \\mathrm { n } _ { 0 } } \\Big \\rangle _ { \\mathbb { L } ^ { 2 } ( \\mathrm { B } ) } \\Big | ^ 2 + \\mathcal { O } \\Big ( 1 \\Big ) . \\end{align*}"}
+{"id": "6028.png", "formula": "\\begin{align*} \\Delta : X _ { 2 n } = X _ { 2 n + 1 } = 0 . \\end{align*}"}
+{"id": "8188.png", "formula": "\\begin{align*} \\phi ( x ) = \\phi _ \\alpha ( x ) = \\frac { c _ { \\alpha , N } } { | x | ^ { N + \\alpha } } , c _ { \\alpha , N } = \\frac { 2 ^ { \\alpha } \\Gamma ( \\frac { \\alpha + N } { 2 } ) } { \\pi ^ { N / 2 } \\Gamma ( - \\frac \\alpha 2 ) } . \\end{align*}"}
+{"id": "3000.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} x \\left ( 1 + F ^ { - 1 } _ - ( c _ 0 - \\ln x ) \\right ) , \\alpha > \\beta , \\\\ x \\left ( 1 + F ^ { - 1 } _ + ( c _ 0 - \\ln x ) \\right ) , \\alpha < \\beta . \\end{cases} \\end{align*}"}
+{"id": "4307.png", "formula": "\\begin{gather*} 2 ( 2 j + 1 ) L ^ 1 _ j ( x ) = \\ell _ { j + 1 } ( x ) - \\ell _ { j - 1 } ( x ) , \\end{gather*}"}
+{"id": "2191.png", "formula": "\\begin{align*} & \\frac { \\partial \\ ; \\ ; } { \\partial x _ 1 } \\left ( \\frac { H ^ f _ { 1 1 } H ^ f _ { 2 2 } - H ^ f _ { 1 2 } { } ^ 2 + K ( f _ 1 { } ^ 2 G - 2 f _ 1 f _ 2 F + f _ 2 { } ^ 2 E ) } { K \\Delta } \\right ) = - \\frac { r _ 2 } { f ^ 2 K ^ 2 \\Delta ^ 2 } , \\\\ & \\frac { \\partial \\ ; \\ ; } { \\partial x _ 2 } \\left ( \\frac { H ^ f _ { 1 1 } H ^ f _ { 2 2 } - H ^ f _ { 1 2 } { } ^ 2 + K ( f _ 1 { } ^ 2 G - 2 f _ 1 f _ 2 F + f _ 2 { } ^ 2 E ) } { K \\Delta } \\right ) = - \\frac { r _ 3 } { f ^ 2 K ^ 2 \\Delta ^ 2 } . \\end{align*}"}
+{"id": "5954.png", "formula": "\\begin{align*} j e _ { j } ( x _ 1 , \\ldots , x _ k ) = \\sum _ { i = 0 } ^ { j - 1 } ( - 1 ) ^ { i } e _ { j - i - 1 } ( x _ 1 , \\ldots , x _ k ) p _ { i + 1 } ( x _ 1 , \\ldots , x _ k ) . \\end{align*}"}
+{"id": "7233.png", "formula": "\\begin{align*} \\sum _ { ( j _ { 1 } , j _ { 2 } , j _ { 3 } ) \\in R ( j ) } \\bar { b } _ { j } a _ { j _ { 1 } } \\bar { a } _ { j _ { 2 } } a _ { j _ { 3 } } = ( 2 \\pi ) ^ { - 3 } \\int _ { \\mathbb { T } ^ { 2 } \\times [ 0 , 2 \\pi ] } . | v | ^ { 2 } v \\bar { g } d x d t \\end{align*}"}
+{"id": "7186.png", "formula": "\\begin{align*} \\xi ' ( x ) : = \\frac { ( 1 - q _ 1 ^ { - 1 } x ) ( 1 - q _ 2 ^ { - 1 } x ) ( 1 - q x ) } { 1 - x } . \\end{align*}"}
+{"id": "2510.png", "formula": "\\begin{align*} \\begin{gathered} ( x _ { 1 0 } , x _ { 1 1 } ) \\in \\omega ( ( x _ { 0 0 } , x _ { 0 1 } ) , T \\times T ) \\quad \\bigl ( x _ { 0 1 } , x _ { 1 1 } \\bigr ) \\in \\omega \\bigl ( ( x _ { 0 0 } , x _ { 1 0 } ) , T \\times T \\bigr ) . \\end{gathered} \\end{align*}"}
+{"id": "7914.png", "formula": "\\begin{align*} ( \\delta _ \\mathrm { m R B A } ) ^ 2 ( \\chi , \\Phi ) = ~ & \\delta _ \\mathrm { m R B A } \\big ( \\delta _ \\mathrm { H o c h } ( \\chi ) , ~ - \\widetilde { \\delta } _ \\mathrm { H o c h } ( \\Phi ) - \\Psi ^ n ( \\chi ) \\big ) \\\\ = ~ & \\big ( ( \\delta _ \\mathrm { H o c h } ) ^ 2 ( \\chi ) , ~ ( \\widetilde { \\delta } _ \\mathrm { H o c h } ) ^ 2 ( \\Phi ) + \\widetilde { \\delta } _ \\mathrm { H o c h } \\circ \\Psi ^ n ( \\chi ) - \\Psi ^ { n + 1 } \\circ \\delta _ \\mathrm { H o c h } ( \\chi ) \\big ) = 0 . \\end{align*}"}
+{"id": "7475.png", "formula": "\\begin{align*} f ^ { ( a ) } ( n ) = c _ 1 \\psi _ m ^ { ( a ) } ( n ) + c _ 2 \\psi _ { m ' } ^ { ( a ) } ( n ) \\implies f ^ { ( a ) } ( n ) = f ^ { ( a ) } ( n - 1 ) + f ^ { ( a - 1 ) } ( n ) \\ , . \\end{align*}"}
+{"id": "7640.png", "formula": "\\begin{align*} Y = \\sum _ { i = 1 } ^ { 5 } \\nu _ i P _ i + \\sum _ { i = 7 } ^ { 8 } \\nu _ i P _ i + \\mu _ 1 S _ 1 = \\sum _ { i = 0 } ^ { 1 1 } \\lambda _ i P _ i = X , \\end{align*}"}
+{"id": "315.png", "formula": "\\begin{align*} t _ { 1 } ^ { - 1 } f _ { 1 } ^ { * } \\alpha _ { 1 } = f ^ { * } u t _ { 1 } ^ { n - 1 } + f ^ { * } v t _ { 1 } ^ { m - 1 } t _ { 2 } ^ { m } \\end{align*}"}
+{"id": "2849.png", "formula": "\\begin{align*} \\mu _ { \\Delta ^ \\wedge _ w } ^ { \\mathrm { r o t } } ( x ) = \\mu _ { \\Delta ^ \\wedge _ w } ( 1 \\otimes e ^ { - \\lambda } ) . \\end{align*}"}
+{"id": "3301.png", "formula": "\\begin{gather*} ( 1 \\otimes K _ 1 ) ( v _ 1 \\otimes \\psi _ { n _ 2 } ) = \\lambda _ { n _ 2 } v _ 1 \\otimes \\psi _ { n _ 2 } , \\\\ ( L _ 1 \\otimes 1 ) ( \\phi _ { m _ 1 } \\otimes v _ 2 ) = \\mu _ { m _ 1 } \\phi _ { m _ 1 } \\otimes v _ 2 , \\end{gather*}"}
+{"id": "2796.png", "formula": "\\begin{align*} \\hat { v } _ { i } = ( - 1 ) ^ { i - 1 } v _ { i } , \\hat { \\alpha } _ i \\hat { \\beta } _ i = \\alpha _ { i + 1 } \\beta _ { i + 1 } , i = 1 , 2 , \\dots . \\end{align*}"}
+{"id": "2074.png", "formula": "\\begin{align*} S A T = \\begin{bmatrix} I _ \\ell & 0 \\\\ E & I _ { n - \\ell } \\end{bmatrix} \\end{align*}"}
+{"id": "2041.png", "formula": "\\begin{align*} \\aligned \\langle \\chi _ 0 , \\chi _ k \\rangle & = \\sum _ \\gamma \\frac { ( \\cos ( a \\gamma ) - 1 ) } { \\gamma ^ 2 } \\cdot \\frac { ( - 1 ) ^ k 2 a \\sin ( a \\gamma ) } { k \\pi + a \\gamma } \\\\ & - \\sum _ \\gamma \\frac { a \\gamma \\cos ( a \\gamma ) - \\sin ( a \\gamma ) } { a \\gamma ^ 3 } \\cdot \\frac { ( - 1 ) ^ k 2 a \\sin ( a \\gamma ) } { k \\pi + a \\gamma } . \\endaligned \\end{align*}"}
+{"id": "5116.png", "formula": "\\begin{align*} & s + \\tau \\leq \\sigma \\textmd { ( r e g u l a r i t y r e q u i r e m e n t f r o m t h e e n e r g y c o n s e r v a t i o n ) } , \\\\ & ( \\beta - \\frac { 1 } { 2 } ) + = \\tau \\textmd { ( S o b o l e v e m b e d d i n g i n 1 D ) } , \\\\ & \\beta + s = \\frac { d } { 2 } - d ( 1 - \\beta ) \\textmd { ( r a d i a l S o b o l e v e m b e d d i n g ) } , \\\\ & ( 2 + p ) \\beta + 1 = 2 + r \\textmd { ( t h e r e l a t i o n b e t w e e n $ \\beta $ a n d $ r $ ) } . \\end{align*}"}
+{"id": "155.png", "formula": "\\begin{align*} \\begin{aligned} \\langle \\L f , f \\rangle \\geq \\lambda \\| f \\| ^ 2 _ { L ^ 2 ( \\nu ) } \\end{aligned} \\end{align*}"}
+{"id": "720.png", "formula": "\\begin{align*} { \\psi _ { x , y , - 1 } } = { \\psi _ { x , y , 1 } } + \\sqrt { { { \\left ( { { \\psi _ { x + 1 , y , 0 } } - { \\psi _ { x - 1 , y , 0 } } } \\right ) } ^ 2 } + { { \\left ( { { \\psi _ { x , y + 1 , 0 } } - { \\psi _ { x , y - 1 , 0 } } } \\right ) } ^ 2 } } \\tan \\left ( { \\frac { \\pi } { 2 } - \\theta } \\right ) , \\end{align*}"}
+{"id": "4828.png", "formula": "\\begin{align*} \\nabla \\cdot ( \\rho \\nabla g ) - \\rho = - \\int _ S \\rho ( x , s ' ) \\dd \\mu ( s ' ) , \\end{align*}"}
+{"id": "1222.png", "formula": "\\begin{align*} m _ t = \\prod _ { i \\notin Q } \\bar { V } _ i \\end{align*}"}
+{"id": "5952.png", "formula": "\\begin{align*} | \\sum _ { i = 1 } ^ { k } \\xi _ { A _ i } ^ { j } - \\sum _ { i = 1 } ^ { k } \\xi _ { B _ i } ^ { j } | \\leq \\delta \\end{align*}"}
+{"id": "5260.png", "formula": "\\begin{align*} & J _ N ( p _ 1 , \\ldots , p _ m ; k _ 1 , \\ldots , k _ p ) : = \\\\ & \\min \\left ( N , \\ \\max \\left ( 0 , \\sum _ { i = 1 } ^ { p _ 1 } k _ i , \\sum _ { i = 1 } ^ { p _ 1 + p _ 2 } k _ i , \\ldots , \\sum _ { i = 1 } ^ { p _ 1 + \\ldots + p _ { m - 1 } } k _ i \\right ) + \\max \\left ( 0 , \\sum _ { i = 1 } ^ { p _ 1 } ( - k _ i ) , \\ldots , \\sum _ { i = 1 } ^ { p _ 1 + \\ldots + p _ { m - 1 } } ( - k _ i ) \\right ) \\right ) . \\end{align*}"}
+{"id": "6971.png", "formula": "\\begin{align*} \\langle T _ { d - i } ^ * V Q ^ \\frac { 1 } { 2 } h , Q ^ \\frac { 1 } { 2 } h ' \\rangle = \\langle Q ^ \\frac { 1 } { 2 } P h , Q ^ \\frac { 1 } { 2 } S _ { d - i } h ' \\rangle = \\langle Q S _ i h , h ' \\rangle = \\langle T _ i Q ^ \\frac { 1 } { 2 } h , Q ^ \\frac { 1 } { 2 } h ' \\rangle , \\end{align*}"}
+{"id": "2233.png", "formula": "\\begin{align*} H ( t ) = \\textrm { d i a g } \\left [ \\{ \\alpha _ k \\} _ { k = 1 } ^ { 1 6 } \\right ] + B \\cos ( 2 \\pi \\nu t ) + C \\cos ( 4 \\pi \\nu t ) , \\end{align*}"}
+{"id": "4634.png", "formula": "\\begin{align*} \\phi ( x , t ) & = \\int _ 0 ^ t \\lim _ { \\lambda \\to \\infty } ( \\tfrac p \\lambda \\tau ^ { p - 1 } + \\min \\{ \\phi _ - ' ( x , \\tau ) , p \\lambda \\tau ^ { p - 1 } \\} ) \\ , d \\tau \\\\ & \\le \\liminf _ { \\lambda \\to \\infty } \\int _ 0 ^ t \\tfrac p \\lambda \\tau ^ { p - 1 } + \\min \\{ \\phi _ - ' ( x , \\tau ) , p \\lambda \\tau ^ { p - 1 } \\} \\ , d \\tau = \\liminf _ { \\lambda \\to \\infty } \\phi _ \\lambda ( x , t ) . \\end{align*}"}
+{"id": "3622.png", "formula": "\\begin{align*} { \\cal A } f ( x ) = { \\sigma ^ 2 \\over 2 } f '' ( x ) + \\int _ 0 ^ \\infty \\left [ f ( x + y ) - f ( x ) \\right ] \\nu ( d y ) . \\end{align*}"}
+{"id": "7643.png", "formula": "\\begin{align*} H : = \\{ ( m , t _ 1 , t _ 2 ) \\in M _ { \\mathbb { R } } \\oplus \\mathbb { R } ^ 2 \\ | \\ t _ 1 + t _ 2 = 1 \\} \\end{align*}"}
+{"id": "6912.png", "formula": "\\begin{align*} q ( z _ i + y ) = z _ i R ( z _ i ) , z _ i \\big { | } _ { q = 0 } = \\alpha _ i , \\end{align*}"}
+{"id": "7163.png", "formula": "\\begin{align*} \\mathcal { Z } : = \\mathcal { Z } ( d ) \\subset \\mathcal { Y } ^ { \\lambda \\geq 0 } ( d ) \\end{align*}"}
+{"id": "3768.png", "formula": "\\begin{align*} g ( \\Gamma / e ) = g ( \\Gamma ) - 1 . \\end{align*}"}
+{"id": "3834.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 ( 2 n - 4 ) - ( a _ 2 + a _ 3 + a _ 4 ) + d } { 2 } , \\end{align*}"}
+{"id": "3078.png", "formula": "\\begin{align*} f _ 1 ( \\ell _ 1 ) = q _ 1 \\ell _ 1 . \\end{align*}"}
+{"id": "2164.png", "formula": "\\begin{align*} & S _ { i j k l m } + S _ { i j l m k } + S _ { i j m k l } = 0 . \\end{align*}"}
+{"id": "1646.png", "formula": "\\begin{align*} \\sum \\limits _ { \\mathsf { w } = 1 } ^ { \\mathsf { q } } \\gamma _ { \\mathsf { w } } \\ ; = \\ ; \\lim \\limits _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\ , \\mathbb { E } \\ , \\log \\ , \\left \\| \\Lambda ^ { \\mathsf { q } } ( \\mathcal { T } _ N \\cdots \\mathcal { T } _ 1 ) \\right \\| _ { \\Lambda ^ { \\mathsf { q } } \\mathbb { C } ^ { \\mathsf { L } } } \\ , , \\qquad \\mathsf { q } = 1 , \\dots , \\mathsf { L } \\ ; . \\end{align*}"}
+{"id": "5217.png", "formula": "\\begin{align*} \\hat { R } ( e _ i \\wedge e _ j ) = \\frac 1 2 \\sum _ { k , l } R _ { i j k l } e _ k \\wedge e _ l , \\end{align*}"}
+{"id": "4283.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } N _ { \\textup { S C } } ( n ) q ^ n = \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { n ( - 1 ) ^ { n - 1 } q ^ { n ( n + 1 ) / 2 } } { ( q ) _ n ( 1 + q ^ n ) } . \\end{align*}"}
+{"id": "1031.png", "formula": "\\begin{align*} \\alpha ( t _ { k _ 0 + k } ) - \\alpha ( t _ { k _ 0 } ) = \\sum _ { i = k _ 0 + 1 } ^ { k _ 0 + k } \\int _ { t _ { i - 1 } } ^ { t _ i } \\dot { \\alpha } ( t ) \\dd t \\\\ \\le \\sum _ { i = k _ 0 + 1 } ^ { k _ 0 + k } h _ i \\dot { \\alpha } ( t _ { i } ) \\le \\sum _ { i = k _ 0 + 1 } ^ { k _ 0 + k } \\frac { r } { c } \\frac { \\dot { \\alpha } ( t _ i ) } { \\dot { \\alpha } ( t _ { i - 1 } ) } . \\end{align*}"}
+{"id": "1496.png", "formula": "\\begin{align*} y _ p - y _ { p d } = H ^ { - 1 } \\min \\bigl ( \\log p , \\log \\frac { \\Delta } { d } \\bigr ) \\end{align*}"}
+{"id": "2198.png", "formula": "\\begin{align*} & K ( H _ { 1 1 } ^ f H _ { 2 2 } ^ f - H _ { 1 2 } ^ f { } ^ 2 ) \\\\ = & c K ^ 2 \\Delta - K ^ 2 ( f _ 1 { } ^ 2 G - 2 f _ 1 f _ 2 F + f _ 2 { } ^ 2 E ) \\\\ = & \\frac { b ^ 4 c } { E ^ 3 } - \\frac { b ^ 4 } { E ^ 4 } ( f _ 1 { } ^ 2 + f _ 2 { } ^ 2 E ) \\\\ = & \\frac { b ^ 4 } { E ^ 3 } \\left ( c - \\frac { 1 } { E } f _ 1 { } ^ 2 - f _ 2 { } ^ 2 \\right ) . \\end{align*}"}
+{"id": "7070.png", "formula": "\\begin{align*} I _ { S _ 1 } \\cap I _ { S _ 0 } ^ { n + 1 } = ( I _ { S _ 1 } I ^ n _ { S _ 0 } ) \\cap I ^ { n + 1 } _ { S _ 0 } = ( I _ { S _ 1 } \\cap I _ { S _ 0 } ) I ^ n _ { S _ 0 } = I _ { S _ 1 } I ^ n _ { S _ 0 } . \\end{align*}"}
+{"id": "1979.png", "formula": "\\begin{align*} ( \\lambda _ 1 ( G ) - \\lambda _ 1 ( H ) ) x _ w = 0 , \\end{align*}"}
+{"id": "1402.png", "formula": "\\begin{align*} & I _ 0 [ u _ 0 , u _ 1 ; D ] \\\\ & : = \\int _ { D } \\left [ ( | u _ 1 ( x ) | ^ 2 + | \\nabla u _ 0 ( x ) | ^ 2 + | u _ 0 ( x ) | ^ { p + 1 } ) \\langle x \\rangle ^ { \\alpha } + | u _ 0 ( x ) | ^ 2 \\langle x \\rangle ^ { - \\alpha } \\right ] \\langle x \\rangle ^ { \\lambda ( 2 - \\alpha ) } \\ , d x \\end{align*}"}
+{"id": "2854.png", "formula": "\\begin{align*} f \\circ g \\circ t ^ { - 1 } = f \\circ s ^ { - 1 } \\circ h = f \\circ k . \\end{align*}"}
+{"id": "8038.png", "formula": "\\begin{align*} E ( X ) _ i = \\{ ( u , a ) \\ , | \\ , E ( g ) ( u ) = E ( p ) ( a ) , u \\in E ( U ) _ i , a \\in E ( A ) _ i \\} . \\end{align*}"}
+{"id": "4574.png", "formula": "\\begin{align*} \\frac { u ( 1 ) } { u ( 0 ) } = \\tan \\theta , \\theta \\in [ 0 , \\pi ) . \\end{align*}"}
+{"id": "3397.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\ell r _ j s _ { g - h _ j } = 0 . \\end{align*}"}
+{"id": "3945.png", "formula": "\\begin{align*} z _ n = u \\circ \\tau _ n ^ { - 1 } - \\frac { 1 } { \\mathcal { H } ^ { d - 1 } ( \\Gamma _ n ) } \\int _ { \\Gamma _ n } u \\circ \\tau _ n ^ { - 1 } \\ , d \\mu _ { \\Gamma _ n } . \\end{align*}"}
+{"id": "911.png", "formula": "\\begin{align*} u _ { n n } \\sigma _ { k - 1 } ( b ) - \\sum _ { \\alpha = 1 } ^ { n - 1 } u _ { \\alpha n } ^ 2 \\sigma _ { k - 2 ; \\alpha } ( b ) + \\sigma _ k ( b ) = f . \\end{align*}"}
+{"id": "3004.png", "formula": "\\begin{align*} \\phi = \\phi _ { s l i p } + \\phi _ { b c } , \\end{align*}"}
+{"id": "4685.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow 1 - } \\int _ { - \\pi } ^ { \\pi } | f ( r e ^ { i \\theta } ) - f ( e ^ { i \\theta } ) | ^ { p } d m _ { \\lambda } ( \\theta ) = 0 \\end{align*}"}
+{"id": "6075.png", "formula": "\\begin{align*} \\frac { m ^ i _ k } { n ^ i _ k } & = \\frac { m ^ 1 _ k + ( 2 k - 3 ) ( i - 1 ) } { n ^ 1 _ k + ( i - 1 ) } \\\\ & < \\frac { ( 2 k - 3 ) n ^ 1 _ k + ( 2 k - 3 ) ( i - 1 ) } { n ^ 1 _ k + ( i - 1 ) } = 2 k - 3 \\\\ \\end{align*}"}
+{"id": "5308.png", "formula": "\\begin{align*} \\psi ( x ) = x - \\sum _ { \\rho \\in \\Z ( \\Gamma , t _ k ) } \\frac { x ^ { \\rho } } { \\rho } + O \\left ( \\frac { 1 - \\theta } { ( b - \\theta ) ^ { 3 } } \\left ( A + \\kappa + \\log \\frac { x + t _ k } { b - \\theta } \\right ) ^ 3 \\left ( \\frac { x } { t _ k } + x ^ b \\right ) \\right ) , \\end{align*}"}
+{"id": "6724.png", "formula": "\\begin{align*} \\Gamma ( h , g ) : = \\frac { 1 } { \\sqrt { \\mu } } Q ( \\sqrt { \\mu } h , \\sqrt { \\mu } g ) , \\mathcal { L } h : = \\Gamma ( h , \\sqrt { \\mu } ) + \\Gamma ( \\sqrt { \\mu } , h ) , \\end{align*}"}
+{"id": "4512.png", "formula": "\\begin{align*} & \\mathcal { K } = K \\circ \\mathrm { p r o j } _ 1 \\ , , \\\\ & \\mathcal { V } = V \\circ \\mathrm { p r o j } _ 1 \\ , , \\end{align*}"}
+{"id": "4566.png", "formula": "\\begin{gather*} \\| f _ n - d \\| \\geq \\max \\{ | f _ n ( x _ n ) - d ( x _ n ) | , | f _ n ( y _ n ) - d ( y _ n ) | \\} = \\\\ = \\max \\{ | 1 - d ( x _ n ) | , | d ( x _ n ) | \\} \\geq 1 / 2 . \\end{gather*}"}
+{"id": "7047.png", "formula": "\\begin{align*} \\tilde { \\mathfrak { t r } } _ 0 ^ p : = \\{ X \\in \\mathcal K ( M ) : X \\textrm { i s a t r a n s v e c t i o n a t p w i t h } X _ p \\in \\nu _ p \\} . \\ \\ \\end{align*}"}
+{"id": "4700.png", "formula": "\\begin{align*} \\| f _ s \\| _ { H _ { \\lambda } ^ p } = \\sup _ { 0 \\le r < 1 } M _ p ( f _ s ; r ) = \\sup _ { 0 \\le r < 1 } M _ p ( f ; s r ) \\le 2 ^ { 2 / p } M _ p ( f ; s ) , \\end{align*}"}
+{"id": "1858.png", "formula": "\\begin{align*} z _ { i , j } & = x _ { i , ( 1 - \\delta ( j ) ) j + \\delta ( j ) ( n + 1 - j ) } , \\mbox { \\ \\ a n d } \\\\ z _ { i , n + 1 - j } & = x _ { i , \\delta ( j ) j + ( 1 - \\delta ( j ) ) ( n + 1 - j ) } . \\end{align*}"}
+{"id": "884.png", "formula": "\\begin{align*} \\Gamma _ { k } = \\{ \\lambda \\in \\mathbb { R } ^ { n } : \\sigma _ { j } ( \\lambda ) > 0 , j = 1 , \\ldots , k \\} . \\end{align*}"}
+{"id": "5056.png", "formula": "\\begin{align*} K _ { 0 , n } ( s , f , \\chi ) = i ^ { - k } \\zeta ^ { ( N ) } ( 2 s ) \\sum _ { \\substack { q \\geq 1 \\\\ N \\mid q } } \\frac { 1 } { 2 \\pi i } \\int _ { \\Re ( u ) = \\sigma _ 0 } \\frac { \\Gamma _ \\C ( u + \\frac { k - 1 } { 2 } ) } { \\Gamma _ \\C ( - u + \\frac { k + 1 } { 2 } ) } q ^ { 2 u - 1 } \\sum _ { m = 1 } ^ \\infty \\frac { f _ m S _ { \\chi } ( m , n ; q ) } { m ^ s } ( m n ) ^ { - u } \\ , d u . \\end{align*}"}
+{"id": "7535.png", "formula": "\\begin{align*} E _ \\varphi ( F ) = \\max _ { F ' \\in \\mathcal F } E _ \\varphi ( F ' ) . \\end{align*}"}
+{"id": "2851.png", "formula": "\\begin{align*} \\mu _ { \\nabla ^ \\wedge _ w } ^ { \\mathrm { r o t } } ( x ) = \\mu _ { \\nabla ^ \\wedge _ w } ( 1 \\otimes e ^ { - \\lambda } ) . \\end{align*}"}
+{"id": "4425.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } I _ n ( v _ n ) = \\limsup _ { n \\to \\infty } I ( v _ n ) \\leq I ^ \\ast ( v ) \\liminf _ { n \\to \\infty } I _ n ( v _ n ) = \\liminf _ { n \\to \\infty } I ( v _ n ) \\geq I ^ \\ast ( v ) , \\end{align*}"}
+{"id": "2735.png", "formula": "\\begin{align*} \\alpha ( X , A ) : = { \\rm s u p } \\{ \\alpha \\in \\mathbb { R } \\ , | \\ , \\Lambda - \\alpha \\pi ^ * A \\ \\} , \\end{align*}"}
+{"id": "5736.png", "formula": "\\begin{align*} \\phi g ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } \\phi ( B _ { i j } ) x _ i x _ j . \\end{align*}"}
+{"id": "7628.png", "formula": "\\begin{align*} \\Delta _ { - K _ { \\P ^ 5 } } = \\{ m \\in M _ { \\R } \\vert \\langle m , u _ \\rho \\rangle \\geq - 1 \\rho \\in \\Sigma _ { \\P ^ 5 } ( 1 ) \\} \\subseteq M _ { \\R } , \\end{align*}"}
+{"id": "6002.png", "formula": "\\begin{align*} \\max _ { 0 \\leq l \\leq x } \\lbrace l + v _ { b ^ * } ( x - l ) - v _ { b ^ * } ( x ) \\rbrace = \\begin{cases} 0 , & x \\in [ 0 , b ^ * ] , \\\\ x - b ^ * + v _ { b ^ * } ( b ^ * ) - v _ { b ^ * } ( x ) , & x \\in ( b ^ * , \\infty ) . \\end{cases} \\end{align*}"}
+{"id": "6831.png", "formula": "\\begin{align*} E ( I _ { a } , I _ { a b } \\cup I '' _ { b } ) = \\frac 1 6 \\ell _ { a } + \\frac 1 6 ( \\ell _ { a b } + \\ell '' _ { b } ) - \\frac 1 2 \\ell _ { a b } + \\frac 1 6 \\frac { \\ell _ { a b } ^ 3 } { \\ell _ { a } ( \\ell _ { a b } + \\ell '' _ { b } ) } . \\end{align*}"}
+{"id": "6265.png", "formula": "\\begin{align*} a = ( L _ { \\nabla h } - \\sigma ) L _ { \\nabla g } ^ 2 , b = \\sigma , c = L _ { \\nabla g } . \\end{align*}"}
+{"id": "1091.png", "formula": "\\begin{align*} 2 z _ { 1 } ^ { 2 } - 1 0 z _ { 2 } ^ { 2 } = 4 p \\implies 2 z _ { 1 } ^ { 2 } \\equiv 4 p \\pmod 5 . \\end{align*}"}
+{"id": "393.png", "formula": "\\begin{align*} v _ f = h ^ 0 ( y \\partial _ y - x \\partial _ x ) + y ^ 2 h ^ y ( y ) \\partial _ y - x ^ 2 h ^ x ( x ) \\partial _ x . \\end{align*}"}
+{"id": "4761.png", "formula": "\\begin{gather*} \\partial _ 2 ^ * ( \\eth _ 1 ^ * ( a ^ * ) ) \\circ b ^ * = \\partial _ 1 ^ * ( \\eth _ 2 ^ * ( a ^ * ) ) \\circ b ^ * , \\forall a ^ * , b ^ * \\in A ^ * , \\\\ \\eth _ 2 ( \\partial _ 1 ( a ) ) \\cdot b = \\eth _ 1 ( \\partial _ 2 ( a ) ) \\cdot b , \\forall a , b \\in A . \\end{gather*}"}
+{"id": "8040.png", "formula": "\\begin{align*} \\tau _ { \\textrm { d i r } , n _ r , n _ t } = 2 \\tau _ { d , t } + \\tau _ { n _ t } + \\tau _ { n _ r } . \\end{align*}"}
+{"id": "4434.png", "formula": "\\begin{align*} Y _ { \\xi } = \\left \\{ a \\odot \\xi \\colon a \\in \\R ^ d , ~ a \\perp \\xi \\right \\} , \\end{align*}"}
+{"id": "1778.png", "formula": "\\begin{align*} A = \\left ( 1 , \\dots , i _ { 0 } - 1 , s _ { j _ { 0 } - \\left ( k - q \\right ) + 1 } , \\dots , s _ { q } \\right ) . \\end{align*}"}
+{"id": "3135.png", "formula": "\\begin{align*} \\sigma = \\prod _ { x \\in X / \\sigma } \\sigma | _ { x } . \\end{align*}"}
+{"id": "5381.png", "formula": "\\begin{align*} \\mathbb { X } _ { \\{ F , G \\} ^ { [ 1 ] } } = [ \\mathbb { X } _ F , \\mathbb { X } _ G ] . \\end{align*}"}
+{"id": "4895.png", "formula": "\\begin{align*} 3 a _ { n + 1 } = 2 a _ n + \\sum _ { k = 0 } ^ n { n + 1 \\choose k } a _ k + 3 . \\end{align*}"}
+{"id": "3480.png", "formula": "\\begin{align*} \\varphi ( \\mathrm { v } , \\mathrm { y } , t , \\tau ) : = \\displaystyle \\int _ { 0 } ^ { \\tau } \\dfrac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 ( \\mathrm { s } - \\tau ) ^ 2 } \\textbf { e x p } \\Big ( - \\dfrac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 ( \\mathrm { s } - \\tau ) } \\Big ) \\Phi ^ { \\textbf { e } } ( \\xi , \\mathrm { t } ; \\mathrm { z } , \\mathrm { s } ) d \\mathrm { s } . \\end{align*}"}
+{"id": "6039.png", "formula": "\\begin{align*} J ( \\Omega ) = j ( \\Omega , \\eta ) = \\int _ { \\Omega } \\lvert \\nabla \\eta ( x ) - A ( x ) \\rvert ^ 2 d x + \\int _ { \\Omega } \\lvert \\eta ( x ) - \\eta _ 0 ( x ) \\rvert ^ 2 d x \\end{align*}"}
+{"id": "7647.png", "formula": "\\begin{align*} I ( V ) : = \\inf \\left \\{ \\ , P ( F ) \\ , : \\ , F \\subset \\Omega , \\ , | F | = V \\ , \\right \\} . \\end{align*}"}
+{"id": "2091.png", "formula": "\\begin{align*} ( M _ 0 - Z ) ^ { - 1 } = M _ 0 ^ { - 1 } ( 1 - Z M _ 0 ^ { - 1 } ) ^ { - 1 } = M _ 0 ^ { - 1 } \\sum _ { i \\ge 0 } ( Z M _ 0 ^ { - 1 } ) ^ i . \\end{align*}"}
+{"id": "5509.png", "formula": "\\begin{align*} F _ { N } ( a , b ; t ) & = 1 - \\frac { b ( 1 - a q ) ( 1 - q ^ { N + 1 } ) ( 1 - t q ^ { N + 1 } ) } { ( 1 - t q ^ N ) ( 1 - b q ^ { N + 2 } ) ( b - a q ) } + \\frac { ( 1 - a q ) ( 1 - q ^ { N + 1 } ) ( b - a q t ) } { ( 1 - t q ^ N ) ( b - a q ) ( 1 - b q ^ { N + 2 } ) } F _ { N } ( a q , b ; t ) . \\end{align*}"}
+{"id": "3150.png", "formula": "\\begin{align*} \\tau _ b ( x ) = \\dfrac { ( 1 - | b | ^ 2 ) x + ( | x | ^ 2 + 2 x . b + 1 ) b } { | b | ^ 2 | x | ^ 2 + 2 x . b + 1 } , \\end{align*}"}
+{"id": "3994.png", "formula": "\\begin{align*} \\hat { G } _ { \\beta } ( u , t ) = E _ { \\beta , 1 } \\left ( - \\lambda \\left ( 1 - \\frac { \\ln ( 1 - ( 1 - p ) u ) } { \\ln p } \\right ) t ^ { \\beta } \\right ) . \\end{align*}"}
+{"id": "5953.png", "formula": "\\begin{align*} ( X - x _ 1 ) ( X - x _ 2 ) \\cdots ( X - x _ k ) = \\sum _ { j = 0 } ^ { k } ( - 1 ) ^ { j } e _ { j } ( x _ 1 , \\ldots , x _ k ) X ^ { k - j } \\end{align*}"}
+{"id": "4706.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ { \\pi } \\left ( h ( e ^ { i \\theta } ) - S _ { \\lambda } h ( e ^ { i \\theta } ) \\right ) \\overline { f ( e ^ { i \\theta } ) } d m _ { \\lambda } ( \\theta ) = 0 \\end{align*}"}
+{"id": "7736.png", "formula": "\\begin{align*} ( \\frac { Y ( \\partial X , [ \\hat { g } ] ) } { Y ( \\mathbb { S } ^ n , [ g _ { \\mathbb { S } } ] ) } ) ^ { \\frac { n } { 2 } } \\leq \\lim \\limits _ { t \\rightarrow + \\infty } \\frac { V o l ( \\partial B ^ { g ^ + } _ p ( t ) ) } { V o l ( \\partial B ^ \\mathbb { H } _ o ( t ) ) } = \\frac { \\mathcal { A } ( p ) } { \\omega _ n } . \\end{align*}"}
+{"id": "8136.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac 1 n \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ^ { \\otimes n } \\otimes \\overline M \\otimes \\overline A ) = \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ) . \\end{align*}"}
+{"id": "348.png", "formula": "\\begin{align*} u _ { t t } - \\Delta u + \\eta + f ( u ) = g , i n \\ H ^ { - 1 } \\ f o r \\ a . e . \\ t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "7446.png", "formula": "\\begin{align*} \\left [ \\frac { 1 } { 2 \\hat { j } + ( 2 k + 1 ) \\hat { I } } , \\ , \\tau ^ \\dagger _ \\theta \\right ] = \\tau ^ \\dagger _ \\theta \\left ( \\frac { 1 } { 2 \\hat { j } + ( 2 k + 1 ) \\hat { I } } - \\frac { 1 } { 2 \\hat { j } + ( 2 ( k - \\theta ) + 1 ) \\hat { I } } \\right ) . \\end{align*}"}
+{"id": "4666.png", "formula": "\\begin{align*} \\Psi _ { \\ell , n _ q } ( y ) = \\prod _ { j \\in \\mathcal { R } _ { \\ell , q } } \\left ( y - \\sum _ { k = 0 } ^ { n _ q - 1 } \\omega _ \\ell ^ { j q ^ k } \\right ) , \\end{align*}"}
+{"id": "7231.png", "formula": "\\begin{align*} \\| F ( a ) \\| _ { l ^ { 2 } } : = \\sup _ { \\| b \\| _ { l ^ { 2 } } = 1 } \\sum _ { ( j _ { 1 } , j _ { 2 } , j _ { 3 } ) \\in R ( j ) } \\bar { b } _ { j } a _ { j _ { 1 } } \\bar { a } _ { j _ { 2 } } a _ { j _ { 3 } } , \\end{align*}"}
+{"id": "6889.png", "formula": "\\begin{align*} \\beta ( \\nu ^ * ( \\bar { x } ) , y ) = \\bar { \\beta } ( \\bar { x } , \\nu ( y ) ) x , y \\in A . \\end{align*}"}
+{"id": "4779.png", "formula": "\\begin{gather*} u \\ast v = \\mu ( T ( u ) ) v , \\\\ u \\diamond v = \\alpha _ 1 ( u ) \\ast \\alpha _ 2 ( v ) - \\alpha _ 2 ( u ) \\ast \\alpha _ 1 ( v ) = \\mu ( T ( \\alpha _ 1 ( u ) ) ) \\alpha _ 2 ( v ) - \\mu ( T ( \\alpha _ 2 ( u ) ) ) \\alpha _ 1 ( v ) . \\end{gather*}"}
+{"id": "5553.png", "formula": "\\begin{align*} [ L _ 2 ( u _ 2 ) \\eta ^ { \\nu } u _ 2 ' ] ' + 2 \\eta ^ { \\nu + 1 } N _ 2 ( u _ 2 ) u _ 2 ' = 0 , \\ ; \\ ; \\ ; \\beta _ 0 < \\eta < \\infty ; \\end{align*}"}
+{"id": "4803.png", "formula": "\\begin{align*} \\mathcal L _ i ^ j ( \\phi _ i ( n , k ) U _ { n , k } ) & = \\phi _ { \\max ( i , j ) } ( n , k ) U _ { n , k } , \\\\ \\mathcal L _ i ^ j ( \\psi _ i ( n , k ) U _ { n , k } ) & = \\psi _ { \\max ( i , j ) } ( n , k ) U _ { n , k } . \\end{align*}"}
+{"id": "2311.png", "formula": "\\begin{align*} w ^ { ( 1 ) } _ { h + n } w ^ { ( 2 ) } = 0 \\ \\ \\mbox { f o r $ n \\in \\Q $ s u f f i c i e n t l y l a r g e } , \\end{align*}"}
+{"id": "3022.png", "formula": "\\begin{align*} \\begin{aligned} x & = F _ x ( \\varphi _ 1 , \\dot \\varphi _ 1 , \\ldots , \\varphi _ 1 ^ { ( r _ 1 - 1 ) } , \\varphi _ 2 , \\dot \\varphi _ 2 , \\ldots , \\varphi _ 2 ^ { ( r _ 2 - 1 ) } ) \\\\ u & = F _ u ( \\varphi _ 1 , \\dot \\varphi _ 1 , \\ldots , \\varphi _ 1 ^ { ( r _ 1 ) } , \\varphi _ 2 , \\dot \\varphi _ 2 , \\ldots , \\varphi _ 2 ^ { ( r _ 2 ) } ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "6011.png", "formula": "\\begin{align*} g ^ { ( q ) } ( b ; h ) & = Z ^ { ( q + r ) } ( b ) \\dfrac { \\Phi ( q + r ) } { r + q } \\Bigg [ - \\dfrac { \\rho ^ { ( q ) } _ { b } ( b ; h ) + g ^ { ( q ) } ( b ; h ) } { Z ^ { ( q ) } ( b ) } r \\int _ b ^ { \\infty } e ^ { - \\Phi ( q + r ) y } Z ^ { ( q ) } ( y ) d y - \\int _ b ^ { \\infty } e ^ { - \\Phi ( q + r ) u } h ( u ) d u \\\\ & + r \\int _ { b } ^ { \\infty } e ^ { - \\Phi ( q + r ) y } \\rho _ b ^ { ( q ) } ( y ; h ) d y \\Bigg ] + g ^ { ( q ) } ( b ; h ) + \\dfrac { r } { r + q } Z ^ { ( q + r ) } ( b ) e ^ { - \\Phi ( q + r ) b } g ^ { ( q ) } ( b ; h ) . \\end{align*}"}
+{"id": "647.png", "formula": "\\begin{align*} \\nabla _ Y \\mathcal { P } ( X ) = - \\mathcal { P } ( B ^ * ( Y , X ) ) - B ( Y , s ( X ) ) + B ^ * ( Y , t ( X ) ) . \\end{align*}"}
+{"id": "6015.png", "formula": "\\begin{align*} r \\int _ 0 ^ { b } W ^ { ( q + r ) } ( b - y ) W ^ { ( q ) } ( y ) d y = W ^ { ( q + r ) } ( b ) - W ^ { ( q ) } ( b ) . \\end{align*}"}
+{"id": "5189.png", "formula": "\\begin{align*} L _ r = ( 0 ( 0 + 1 ) + ( 1 0 ^ * 1 ( 0 + 1 ) ) ) ^ * \\enspace . \\end{align*}"}
+{"id": "908.png", "formula": "\\begin{align*} \\begin{aligned} G ^ { \\alpha \\beta } _ 0 \\sigma _ { \\alpha \\beta } \\geq \\ , & G ( \\sigma _ { \\alpha \\beta } - \\theta \\delta _ { \\alpha \\beta } ) + \\theta \\sum _ { \\alpha = 1 } ^ { n - 1 } G ^ { \\alpha \\alpha } _ 0 \\\\ \\geq \\ , & \\gamma + \\theta \\sum _ { \\alpha = 1 } ^ { n - 1 } G ^ { \\alpha \\alpha } _ 0 \\end{aligned} \\end{align*}"}
+{"id": "6368.png", "formula": "\\begin{align*} ( K _ H f ) ( t ) = \\int _ 0 ^ t K _ H ( t , s ) f ( s ) \\d s , \\end{align*}"}
+{"id": "6222.png", "formula": "\\begin{align*} H ( p ) = - \\partial _ x ^ 2 + ( p + B x ) ^ 2 - \\partial _ z ^ 2 \\end{align*}"}
+{"id": "6783.png", "formula": "\\begin{align*} i \\left [ H ^ { \\rm B } ( t ) , \\Gamma _ t \\right ] & = b ^ * ( \\psi _ t ) b ( q ( t ) \\dot { \\psi } _ t ) \\Gamma _ t + \\Gamma _ t b ( q ( t ) \\dot { \\psi } _ t ) b ( \\psi _ t ) \\end{align*}"}
+{"id": "2019.png", "formula": "\\begin{align*} \\sum _ { n \\leq e ^ t } \\frac { \\Lambda ( n ) } { \\sqrt { n } } ( t - \\log n ) = 4 e ^ { t / 2 } + O \\left ( e ^ { t / 2 } e ^ { - c \\sqrt { t } } \\ , \\right ) \\end{align*}"}
+{"id": "1638.png", "formula": "\\begin{align*} \\mathcal { R } \\ ; = \\ ; \\operatorname { d i a g } ( \\kappa _ { \\mathsf { L } } , \\ldots , \\kappa _ 1 ) \\ , , \\qquad \\qquad \\kappa _ 1 \\geq \\dots \\geq \\kappa _ { \\mathsf { L } } > 0 \\ , , \\end{align*}"}
+{"id": "5432.png", "formula": "\\begin{align*} \\log \\frac { \\Gamma \\left ( \\frac { L ^ 2 } { 2 } + 1 \\right ) ^ 2 } { \\Gamma ( L ^ 2 + 1 ) } & = \\log L - ( L ^ 2 + 1 ) \\log 2 + \\cdots , \\\\ \\log \\frac { \\Gamma \\left ( K + \\frac { L ^ 2 } { 2 } \\right ) } { \\Gamma \\left ( K - \\frac { L ^ 2 } { 2 } - 1 \\right ) } & = ( L ^ 2 + 1 ) \\log K + \\cdots , \\end{align*}"}
+{"id": "3289.png", "formula": "\\begin{align*} P _ { n ( k ) } ( m ( j ) ) = R _ k \\big ( y _ j ; \\alpha , \\beta , \\gamma , \\delta ; q ^ 2 \\big ) , k , j = 0 , \\dots , N . \\end{align*}"}
+{"id": "294.png", "formula": "\\begin{align*} \\tilde { A } _ { D ' } = ( A \\otimes _ { k } A ) \\left [ \\left ( \\frac { 1 \\otimes t _ { 1 } } { t _ { 1 } \\otimes 1 } \\right ) ^ { \\pm 1 } , \\left ( \\frac { 1 \\otimes t _ { 2 } } { t _ { 2 } \\otimes 1 } \\right ) ^ { \\pm 1 } , \\ldots , \\left ( \\frac { 1 \\otimes t _ { r ' } } { t _ { r ' } \\otimes 1 } \\right ) ^ { \\pm 1 } \\right ] . \\end{align*}"}
+{"id": "1864.png", "formula": "\\begin{align*} y _ { ( 2 k - 1 ) m + i , j } + y _ { ( 2 ( r _ 0 - k + 1 ) - 1 ) m + ( m + 1 - i ) , n + 1 - j } & = 2 M N - \\tfrac { 1 } { 2 } m n r + \\tfrac { 3 } { 2 } , \\mbox { \\ a n d } \\\\ y _ { ( 2 k ) m + i , j } + y _ { 2 ( r _ 0 - k ) m + ( m + 1 - i ) , n + 1 - j } & = \\tfrac { 1 } { 2 } m n r + \\tfrac { 3 } { 2 } . \\end{align*}"}
+{"id": "2399.png", "formula": "\\begin{align*} { \\rm e v } _ { 0 } ^ { \\mathfrak { m } } ( e _ { a _ { 1 } } \\cdots e _ { a _ { k } } ) = I ^ { \\mathfrak { m } } ( 0 , a _ { 1 } ' , \\dots a _ { k } ' , 1 ) \\end{align*}"}
+{"id": "4093.png", "formula": "\\begin{align*} \\sum _ { k | n } \\frac { 1 } { k } \\chi ^ { ( k ) } _ { n / k } = \\frac { 1 } { n } \\left [ \\left ( \\frac { \\chi _ V + \\sqrt { \\chi _ V ^ 2 - 4 } } { 2 } \\right ) ^ n + \\left ( \\frac { \\chi _ V - \\sqrt { \\chi _ V ^ 2 - 4 } } { 2 } \\right ) ^ n \\right ] \\end{align*}"}
+{"id": "2212.png", "formula": "\\begin{gather*} \\dot { x } = a x ^ 2 + v x \\end{gather*}"}
+{"id": "2751.png", "formula": "\\begin{align*} \\frac { 4 ( N + 1 ) } { 5 } = 2 d ^ * \\geq d \\geq N - 2 . \\end{align*}"}
+{"id": "2308.png", "formula": "\\begin{align*} \\log \\Lambda = - \\frac 2 d \\Bigl ( \\nu \\log \\nu + ( d - \\nu ) \\log ( d - \\nu ) \\Bigl ) + 2 + o ( 1 ) , \\end{align*}"}
+{"id": "3125.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ k ) = \\sum _ { i = 1 } ^ r F _ { Y _ i } ( \\tau _ i ^ k ) \\end{align*}"}
+{"id": "6544.png", "formula": "\\begin{align*} f ( a , b ) = ( u , a ) , f ( c , d ) = ( v , c ) , \\end{align*}"}
+{"id": "3140.png", "formula": "\\begin{align*} | X | = \\sum _ { i = 1 } ^ r q _ i \\ell _ i . \\end{align*}"}
+{"id": "5379.png", "formula": "\\begin{align*} \\{ \\hat E ^ i ( x ) , A _ j ( y ) \\} _ D = H ( x , y ) ^ i { } _ j \\end{align*}"}
+{"id": "1561.png", "formula": "\\begin{align*} \\left ( z ^ { \\prime \\prime } \\right ) ^ { - 1 } \\rho \\left ( g ^ { \\prime \\prime } \\right ) ^ { - 1 } - z \\rho \\left ( g \\right ) = \\left ( z ^ { \\prime \\prime } \\right ) ^ { - 1 } \\cdot \\left ( \\rho \\left ( g ^ { \\prime \\prime } \\right ) ^ { - 1 } - z ^ { \\prime \\prime } z \\rho \\left ( g \\right ) \\right ) . \\end{align*}"}
+{"id": "3026.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { u } _ 1 & = g _ 1 ( x , u _ 1 , u _ 2 ) \\\\ \\bar { u } _ 2 & = g _ 2 ( x , u _ 1 , u _ 2 ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "6477.png", "formula": "\\begin{align*} \\varphi : Z = X _ 0 \\dashrightarrow X _ 1 \\dashrightarrow \\cdots \\dashrightarrow X _ m = X , \\end{align*}"}
+{"id": "1700.png", "formula": "\\begin{align*} \\hat { g } ( k ) : = \\int d x \\ , g ( x ) e ^ { - 2 \\pi i k x } \\ , , k \\in \\Z \\ , . \\end{align*}"}
+{"id": "6904.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ { \\infty } q ^ d \\chi \\left ( X ^ { [ d ] } , \\wedge _ { y } L ^ { [ d ] } \\otimes ( \\wedge _ { x } M ^ { [ d ] } ) ^ { \\vee } \\right ) & = ( 1 - q ) ^ { - \\chi ( \\mathcal O _ X ) } ( 1 + q y ) ^ { \\chi ( L ) } ( 1 + q x ) ^ { \\chi ( M ^ { \\vee } ) } ( 1 - q x y ) ^ { - \\chi ( L \\otimes M ^ { \\vee } ) } . \\end{align*}"}
+{"id": "4380.png", "formula": "\\begin{align*} T ^ t f ( x ) : = \\int _ I W _ t ( x , y ) f ( y ) d y . \\end{align*}"}
+{"id": "2573.png", "formula": "\\begin{align*} \\bar \\mu _ t ^ { * , t _ 0 , \\xi } : = \\rho \\big ( - R ^ { - 1 } B \\mathbb E \\big [ \\bar U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\nu _ t ^ { * , t _ 0 , \\xi } ) | \\mathcal { F } _ t ^ { W ^ 0 } \\big ] \\big ) \\quad \\bar \\nu _ t ^ { * , t _ 0 , \\xi } : = \\mathbb E \\big [ \\bar x _ t ^ { * , t _ 0 , \\xi } | \\mathcal { F } _ t ^ { W ^ 0 } \\big ] . \\end{align*}"}
+{"id": "8090.png", "formula": "\\begin{align*} \\mathcal { H } = \\mathcal { H } _ { u u } \\oplus \\mathcal { H } _ { u \\neg u } \\oplus \\mathcal { H } _ { \\neg u u } \\oplus \\mathcal { H } _ { \\neg u \\neg u } \\end{align*}"}
+{"id": "1732.png", "formula": "\\begin{align*} F ( x , t ) & : = \\sum _ k \\int d \\eta \\ , \\frac { | c ( k , \\eta ) | } { \\left ( 1 + | \\eta + k ^ 2 | \\right ) ^ { 1 - b } } e ^ { i k x + i t \\eta } \\ , , \\\\ G ( x , t ) & : = \\sum _ k \\int d \\eta \\ , | \\tilde { v } _ \\delta ( k , \\eta ) | e ^ { 2 \\pi i k x + 2 \\pi i t \\eta } \\ , . \\end{align*}"}
+{"id": "4781.png", "formula": "\\begin{gather*} \\partial _ 1 ( e _ 1 ) = e _ 1 , \\partial _ 1 ( e _ 2 ) = e _ 2 , \\partial _ 1 ( e _ 3 ) = 2 e _ 3 ; \\\\ \\partial _ 2 ( e _ 1 ) = e _ 2 , \\partial _ 2 ( e _ 2 ) = - e _ 1 + e _ 2 , \\partial _ 2 ( e _ 3 ) = e _ 3 . \\end{gather*}"}
+{"id": "2920.png", "formula": "\\begin{align*} N _ \\infty ( r , F ) = \\sum { \\rm l o g } ^ + \\frac { r } { | b _ j | } . \\end{align*}"}
+{"id": "1850.png", "formula": "\\begin{align*} x _ { i , j } + x _ { i , j + 1 } & = a _ j , & x _ { i , n + 1 - j } + x _ { i , n - j } & = S - a _ j , \\\\ x _ { m + 1 - i , j } + x _ { m + 1 - i , j + 1 } & = a _ j , \\mbox { a n d } & x _ { m + 1 - i , n + 1 - j } + x _ { m + 1 - i , n - j } & = S - a _ j , \\mbox { \\ \\ a n d } \\end{align*}"}
+{"id": "6737.png", "formula": "\\begin{align*} ( \\frac { \\partial ^ { \\alpha } [ ( E + v \\times B ) \\cdot \\nabla _ { v } F ] } { \\sqrt { \\mu } } , \\frac { \\partial ^ { \\alpha } F } { \\sqrt { \\mu } } ) : = I _ 4 + I _ 5 + I _ 6 , \\end{align*}"}
+{"id": "3968.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\bar { \\mathcal { M } } ( t ) \\right ) = \\bar { r } _ { 1 } t , \\ \\operatorname { V a r } \\left ( \\bar { \\mathcal { M } } ( t ) \\right ) = \\bar { r } _ { 2 } t . \\end{align*}"}
+{"id": "1895.png", "formula": "\\begin{align*} q _ n ( a ) & = \\sum _ { k = 1 } ^ { r } e _ k \\cdot q _ n ( \\chi _ { i _ k , j _ k } ) \\\\ & = \\sum _ { 1 \\leq k \\leq r , j _ k = n } e _ k \\cdot \\chi _ { i _ k , j _ k } . \\end{align*}"}
+{"id": "817.png", "formula": "\\begin{align*} \\widetilde { \\theta } _ 2 = \\widetilde { \\theta } _ 2 ( N , \\mathcal { T } , \\{ x _ j \\} _ { j = 1 } ^ { N } ) : = \\theta + \\frac { 1 - \\theta } { l _ 1 + l _ 2 \\left ( \\frac { 1 } { N } \\sum \\limits _ { j = 1 } ^ N \\int _ 0 ^ { \\tau _ j } e ^ { - \\beta t } f ( x _ j ( t ) ) d t \\right ) } . \\end{align*}"}
+{"id": "2220.png", "formula": "\\begin{align*} p ( \\boldsymbol { \\mathcal { A } } ) & : = \\sum _ { k = 0 } ^ { { \\ell } } \\boldsymbol { \\mathcal { A } } ^ { k _ * } \\times \\boldsymbol { \\alpha } _ k , \\\\ p ^ { D } ( \\boldsymbol { \\mathcal { A } } ) & : = \\sum _ { k = 0 } ^ { { \\ell } } \\boldsymbol { \\alpha } _ k ^ H \\times \\boldsymbol { \\mathcal { A } } ^ { k _ * } , \\end{align*}"}
+{"id": "544.png", "formula": "\\begin{align*} g _ 1 ^ { \\prime + } ( 0 ) - g _ 2 ^ { \\prime + } ( 0 ) & = \\sum _ { i = 1 } ^ n \\int _ \\R \\big ( \\hat { f } ' _ i ( x _ i ) Q _ i ( x _ i ) - Q _ i ' ( x _ i ) \\big ) \\big ( T _ i ( x _ i ) - x _ i \\big ) d x _ i . \\end{align*}"}
+{"id": "4202.png", "formula": "\\begin{align*} \\sum _ { f ~ } \\frac { \\mu _ k ( f ) } { | f | ^ s } = \\frac { 1 - q u ^ k } { 1 - q u } \\sum _ { f ~ } \\frac { \\mu _ k ( f ) } { | f | ^ s } = \\sum _ { n = 0 } ^ \\infty a _ n u ^ n , \\end{align*}"}
+{"id": "8087.png", "formula": "\\begin{align*} 1 - \\alpha = \\tfrac { \\alpha h ^ 2 } { q ^ 2 } + \\lambda _ N \\end{align*}"}
+{"id": "5975.png", "formula": "\\begin{align*} c : = \\gamma - \\int _ { ( - 1 , 0 ) } z \\Pi ( \\mathrm { d } z ) \\end{align*}"}
+{"id": "4864.png", "formula": "\\begin{align*} \\Sigma '' : = \\Sigma ' \\cup ( W _ { ( t b ( K ' _ 1 ) - 1 , \\dots , t b ( K ' _ n ) - 1 ) } ( L ' ) ) . \\end{align*}"}
+{"id": "5968.png", "formula": "\\begin{align*} g _ { J } = 1 _ { \\sum _ { K '' \\in P _ { \\delta _ 0 } ( J ) } 1 _ { f _ { K '' } ^ { ( H , \\alpha ) } \\neq 0 } ( K '' ) \\in ( \\beta / 2 , \\beta ] } ( J ) \\sum _ { K \\in P _ { \\delta _ 0 } ( J ) } f _ { K } ^ { ( H , \\alpha ) } \\end{align*}"}
+{"id": "7294.png", "formula": "\\begin{align*} \\frac { d } { d t } \\overline { \\Upsilon } ( t ) = Q _ x ( t ) \\overline { X } ^ * ( t ) - A ^ T ( t ) \\overline { \\Upsilon } ( t ) \\end{align*}"}
+{"id": "6772.png", "formula": "\\begin{align*} a ( f ) & = \\int d ^ 3 k \\ , \\overline { f ( k ) } a _ k , a ^ * ( f ) = \\int d ^ 3 k \\ , f ( k ) a ^ * _ k , \\\\ b ( f ) & = \\int d ^ 3 x \\ , \\overline { f ( x ) } b _ x , b ^ * ( f ) = \\int d ^ 3 x \\ , f ( x ) b ^ * _ x . \\end{align*}"}
+{"id": "820.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { \\theta } _ 2 : = \\lim _ { N \\rightarrow + \\infty } \\widetilde { \\theta } _ 2 = \\theta + \\frac { 1 - \\theta } { l _ 1 + l _ 2 \\left ( \\mathbb { E } \\int _ 0 ^ { \\tau } e ^ { - \\beta t } f ( x ( t ) ) d t \\right ) } . \\end{aligned} \\end{align*}"}
+{"id": "7731.png", "formula": "\\begin{align*} \\begin{aligned} L e n g t h ( \\sigma , g ^ + ) & = \\int _ 0 ^ l | \\dot { \\sigma } ( t ) | d t = \\int _ 0 ^ l | \\nabla r | \\cdot | \\dot { \\sigma } ( t ) | d t \\\\ & \\geq \\int _ 0 ^ l | g ^ + ( \\nabla r , \\dot { \\sigma } ) | d t = \\int _ 0 ^ l | ( r \\circ \\sigma ) ' ( t ) | d t \\\\ & \\geq | \\int _ 0 ^ l ( r \\circ \\sigma ) ' ( t ) d t | = | r ( q ' ) - r ( q ) | \\end{aligned} \\end{align*}"}
+{"id": "3088.png", "formula": "\\begin{align*} a _ { k , 2 } = a _ { k , 1 } - 1 = s ( k ) - 1 = s _ 2 ( k ) \\end{align*}"}
+{"id": "2236.png", "formula": "\\begin{align*} [ \\lambda * q , p ] = [ q , \\lambda * p ] . \\end{align*}"}
+{"id": "5927.png", "formula": "\\begin{align*} \\pi _ X ^ 4 = [ \\Delta _ X ] - \\pi _ X ^ 0 - \\pi _ X ^ 2 - \\pi _ X ^ 6 - \\pi _ X ^ 8 \\ , . \\end{align*}"}
+{"id": "7150.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ d c _ i \\beta _ i \\right | : = \\sqrt { \\sum _ { i = 1 } ^ d c _ i ^ 2 } . \\end{align*}"}
+{"id": "8183.png", "formula": "\\begin{align*} \\alpha ^ 2 ( | a _ 1 | ^ 2 + | b _ 1 | ^ 2 ) = ( | a _ 1 | ^ 2 + | a _ 2 | ^ 2 ) ( | b _ 1 | ^ 2 + | b _ 2 | ^ 2 ) , \\end{align*}"}
+{"id": "7018.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } + \\Delta ^ 2 u + \\Delta \\theta = 0 , \\\\ \\theta _ t - \\Delta \\theta + \\sigma \\theta - \\Delta u _ t = 0 . \\end{cases} \\end{align*}"}
+{"id": "689.png", "formula": "\\begin{align*} g _ 1 ' : = \\frac { 1 } { 2 } ( 1 + i I ) v ' \\end{align*}"}
+{"id": "1278.png", "formula": "\\begin{align*} E = h - a _ 1 C _ 1 - \\hdots - a _ r C _ r . \\end{align*}"}
+{"id": "3618.png", "formula": "\\begin{align*} f ( X _ { T _ i } ) - f \\left ( X _ { T _ i - } + \\Delta Y _ { T _ i } \\right ) \\ge - g \\left ( X _ { T _ i } - ( X _ { T _ i - } + \\Delta Y _ { T _ i } ) \\right ) = - g ( \\xi _ j ) . \\end{align*}"}
+{"id": "5816.png", "formula": "\\begin{align*} \\omega ( T _ { \\theta } ^ { * n } ) = \\omega ( T _ { \\theta } ^ n ) = 1 \\end{align*}"}
+{"id": "2807.png", "formula": "\\begin{align*} \\left \\| \\tilde { s } _ i ^ 2 A ^ T \\tilde { u } _ i ^ A - \\tilde { c } _ i B ^ T B \\tilde { g } _ i \\right \\| _ 2 & = \\left \\| A ^ T \\tilde { u } _ i ^ A - \\tilde { c } _ i Z ^ T Z \\tilde { g } _ i \\right \\| _ 2 \\\\ & = \\left \\| A ^ T \\tilde { u } _ i ^ A - \\tilde { c } _ i R ^ T R \\tilde { g } _ i \\right \\| _ 2 = \\left \\| \\alpha _ { k + 1 } R ^ T v _ { k + 1 } e _ { k + 1 } ^ T x _ i \\right \\| _ 2 \\end{align*}"}
+{"id": "7898.png", "formula": "\\begin{align*} \\mathrm { G r } ( \\widehat { R } ) = ~ & \\{ \\widehat { R } ( a , - a ) + ( a , - a ) | ~ a \\in A \\} \\\\ = ~ & \\{ ( - R ( a ) + a , - R ( a ) - a ) | ~ a \\in A \\} \\subset A \\oplus A \\end{align*}"}
+{"id": "7809.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } { \\delta _ t \\int _ { \\Omega } { f _ { m , j } ( u ^ n ( x ) ) \\ d x } } & = \\lim _ { j \\to \\infty } { \\int _ { \\Omega } { g _ { m , j } ( x ) \\ d x } } = \\int _ { \\Omega } { g _ m ( x ) \\ d x } = \\delta _ t \\int _ { \\Omega } { f _ m ( u ^ n ( x ) ) \\ d x } . \\end{align*}"}
+{"id": "329.png", "formula": "\\begin{align*} T S ( l ) = y ( t _ l ) + \\eta ( l ) , ~ l = 1 , 2 , \\cdots , L \\end{align*}"}
+{"id": "3665.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\rho \\left ( \\partial \\widehat { \\mathcal { G } } _ { \\alpha } ( n ) \\right ) & = 2 \\alpha ( 1 - \\alpha ) ( z _ 1 + z _ 2 ) + 2 C , \\end{align*}"}
+{"id": "7838.png", "formula": "\\begin{align*} \\mathfrak { B } _ { \\mbox { \\tiny { H G S } } } = \\{ \\zeta _ 1 , . . . , \\zeta _ k \\} . \\end{align*}"}
+{"id": "2910.png", "formula": "\\begin{align*} \\mathcal Q \\mathcal A + \\mathcal K = \\mathcal Q \\mathcal A \\ \\ \\ \\ \\mathcal N + \\mathcal N \\not \\subset \\mathcal Q \\mathcal A \\end{align*}"}
+{"id": "3075.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } F _ X \\left ( \\sigma ^ { m _ r } \\right ) = \\infty . \\end{align*}"}
+{"id": "6984.png", "formula": "\\begin{align*} \\inf \\left \\{ \\int _ { X } \\left ( | \\nabla f | ^ 2 + \\alpha _ 1 f ^ 2 - \\frac { \\alpha _ 2 } { 2 } f ^ 2 \\log f ^ 2 \\right ) d m : f \\in W ^ { 1 , 2 } , \\| f \\| _ 2 = 1 \\right \\} , \\end{align*}"}
+{"id": "8223.png", "formula": "\\begin{align*} X _ 0 : = \\Vert \\sigma _ 0 \\Vert _ { \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } } + \\Vert u _ 0 \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } } . \\end{align*}"}
+{"id": "7925.png", "formula": "\\begin{align*} \\sum _ { i + j = n } \\mu _ i \\big ( \\mu _ j ( a , b ) , c \\big ) = ~ & \\sum _ { i + j = n } \\mu _ i \\big ( a , \\mu _ j ( b , c ) \\big ) , \\\\ \\sum _ { i + j + k = n } \\mu _ i \\big ( R _ j ( a ) , R _ j ( b ) \\big ) = ~ & \\sum _ { i + j + k = n } R _ i \\big ( \\mu _ j ( R _ k ( a ) , b ) ~ + ~ \\mu _ j ( a , R _ k ( b ) ) \\big ) + \\kappa ~ \\mu _ n ( a , b ) , \\end{align*}"}
+{"id": "346.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\overbrace { \\xi ' _ t \\dots \\xi ' _ t } ^ q ( X - Y ) | \\xi _ t \\| < ( 1 + \\tau ) b ^ 2 ( L ^ r + H ^ r ) L ^ r \\alpha \\end{aligned} \\end{align*}"}
+{"id": "286.png", "formula": "\\begin{align*} \\overline { d t _ { i } / t _ { i } ^ { 2 } } \\otimes \\sqrt { \\overline { a _ { 0 } t _ { i } ^ { 2 } } } = \\overline { d t _ { i } / \\prod _ { i ' \\in I } t _ { i ' } ^ { n _ { i ' } } } \\otimes \\sqrt { \\overline { a _ { 0 } \\prod _ { i ' \\in I } t _ { i ' } ^ { n _ { i ' } } \\prod _ { i ' \\in I - \\{ i \\} } t _ { i ' } ^ { n _ { i ' } } } } \\in \\Omega _ { X } ^ { 1 } ( \\log D ' ) ( R ) | _ { Z ^ { 1 / 2 } } , \\end{align*}"}
+{"id": "7607.png", "formula": "\\begin{align*} X _ \\phi = \\{ \\cup e _ i | x ^ * _ i \\geq \\phi \\} . \\end{align*}"}
+{"id": "7249.png", "formula": "\\begin{align*} 0 < \\delta _ 1 = 3 - 2 \\beta < \\frac { 3 } { 2 } . \\end{align*}"}
+{"id": "3469.png", "formula": "\\begin{align*} \\nabla _ { \\mathrm { x } } \\mathbb { I } _ { ( \\Omega ) } \\Big [ \\psi \\Big ] ( \\mathrm { x } , t ) = \\frac { 1 } { \\delta } \\Big ( \\nabla _ { \\xi } \\overline { \\mathbb { I } } _ { ( B ) } \\Big [ \\hat { \\psi } \\Big ] \\Big ) ^ \\vee . \\end{align*}"}
+{"id": "610.png", "formula": "\\begin{align*} \\sigma _ \\mathfrak { s } ( \\boldsymbol { J } ; n , p ) = \\sigma _ \\mathfrak { s } ( \\boldsymbol { J } ; q , p ) \\quad \\ | n - q | \\leq 1 \\quad \\sigma _ \\mathfrak { s } ( \\boldsymbol { J } ; n , p ) = \\sigma _ \\mathfrak { s } ( \\boldsymbol { J } ; n , m ) \\quad \\ | p - m | \\leq 1 \\ , . \\end{align*}"}
+{"id": "4978.png", "formula": "\\begin{align*} \\begin{aligned} & p _ { w _ { 1 } } = 1 - [ \\lambda w _ { r } - \\lfloor \\lambda w _ { r } \\rfloor ] \\\\ & p _ { w _ { 2 } } = 1 - p _ { w _ { 1 } } \\\\ & w _ { 1 } = \\lfloor \\lambda w _ { r } \\rfloor \\\\ & w _ { 2 } = w _ { 1 } + 1 , \\end{aligned} \\end{align*}"}
+{"id": "2664.png", "formula": "\\begin{align*} s \\cdot ( x , y , z , \\zeta ) = ( a x + b y , c x + d y , ( \\det s ) z , ( \\det s ) \\zeta ) . \\end{align*}"}
+{"id": "3405.png", "formula": "\\begin{align*} T ( \\rho , S ) = \\frac { \\partial e } { \\partial S } \\Big | _ { \\rho } = \\int ( p _ S ( \\rho , S ) / \\rho ^ 2 ) \\ , d \\rho . \\end{align*}"}
+{"id": "1906.png", "formula": "\\begin{align*} \\lim _ { q \\to { \\frac 3 2 } ^ + } \\frac 1 { r ' } = 1 . \\end{align*}"}
+{"id": "6643.png", "formula": "\\begin{align*} \\Lambda _ n ( H _ \\theta ) & \\ge \\min \\Big \\{ \\Lambda _ n ( Q ^ { 1 + \\varepsilon } _ \\theta ) , - \\Big ( 1 + \\frac { 1 } { \\varepsilon } \\Big ) ^ 2 \\Big \\} = \\Lambda _ n ( Q ^ { 1 + \\varepsilon } _ \\theta ) = - \\dfrac { 1 } { ( 2 n - 1 ) ^ 2 \\theta ^ 2 } + O \\Big ( \\dfrac { 1 } { \\theta } \\Big ) . \\end{align*}"}
+{"id": "3329.png", "formula": "\\begin{align*} w _ 2 ( j _ 1 , j _ 2 , k _ 1 , k _ 2 ) = \\frac { 1 } { | C ( n _ 1 ( k _ 1 ) , n _ 2 ( k _ 2 ) , m _ 1 ( j _ 1 ) , m _ 2 ( j _ 2 ) | ^ 2 } . \\end{align*}"}
+{"id": "217.png", "formula": "\\begin{align*} \\lambda _ { m - i } + \\left \\lfloor \\frac { k ( m - ( 2 i + 1 ) ) } { 2 } \\right \\rfloor & \\geq 1 + k i + \\left \\lfloor \\frac { k ( m - ( 2 i + 1 ) ) } { 2 } \\right \\rfloor \\\\ & = 1 + \\left \\lfloor \\frac { k ( m - 1 ) } { 2 } \\right \\rfloor , \\end{align*}"}
+{"id": "2968.png", "formula": "\\begin{align*} \\mathcal { S } = ( \\Sigma ^ { - 1 } \\mathcal { S } ) ^ { \\perp } = { } ^ { \\perp } ( \\Sigma \\mathcal { S } ) \\end{align*}"}
+{"id": "5264.png", "formula": "\\begin{align*} c _ p ( t _ 1 , \\ldots , t _ p ) : = \\frac { 1 } { N } \\ * \\kappa _ p ^ { ( N ) } ( t _ 1 \\ * N , \\ldots , t _ p \\ * N ) \\end{align*}"}
+{"id": "7425.png", "formula": "\\begin{align*} [ a _ i , a _ j ] = [ a _ i ^ \\dag , a _ j ^ \\dag ] = 0 , [ a _ i , a _ j ^ \\dag ] = \\delta _ { i j } . \\end{align*}"}
+{"id": "4470.png", "formula": "\\begin{align*} ( u v ) ' = u ' v + u v ' \\ , . \\end{align*}"}
+{"id": "1873.png", "formula": "\\begin{align*} c ( 2 x + y ) + c ( x + 2 y ) = \\frac { 9 c ( x ) c ( y ) [ c ( x ) + c ( y ) + 2 c ( x ) ^ { \\frac { 1 } { 3 } } c ( y ) ^ { \\frac { 1 } { 3 } } ( c ( x ) ^ { \\frac { 1 } { 3 } } + c ( y ) ^ { \\frac { 1 } { 3 } } ) ] } { [ 2 c ( x ) ^ { \\frac { 2 } { 3 } } + 2 c ( y ) ^ { \\frac { 2 } { 3 } } + 5 c ( x ) ^ { \\frac { 1 } { 3 } } c ( y ) ^ { \\frac { 1 } { 3 } } ] ^ { 3 } } \\end{align*}"}
+{"id": "289.png", "formula": "\\begin{align*} I _ { x } = \\{ i \\in I \\ ; | \\ ; x \\in D _ { i } \\} , \\end{align*}"}
+{"id": "5202.png", "formula": "\\begin{align*} a _ n = \\frac { 2 ^ n } { 3 } \\cdot ( 2 ^ { n - 1 } - 1 ) - \\frac { 2 ^ n } { 3 } \\cdot \\sum _ { j = 1 } ^ { n - 1 } \\binom { n - 1 } { j } \\cdot \\left ( - \\frac { 1 } { 2 } \\right ) ^ j \\enspace . \\end{align*}"}
+{"id": "678.png", "formula": "\\begin{align*} \\Im m ( a _ 0 \\overline { a _ 1 } ) = \\Im m ( a _ 2 \\overline { a _ 3 } ) ; \\end{align*}"}
+{"id": "7589.png", "formula": "\\begin{align*} \\ell ( \\gamma ) : = \\int _ \\gamma \\rho _ U ( z ) | d z | , \\operatorname { a r e a } ( B ) : = \\frac { i } { 2 } \\int _ B \\rho _ U ( z ) ^ 2 d z d \\overline { z } ; \\end{align*}"}
+{"id": "2641.png", "formula": "\\begin{align*} & \\lambda ( q ( c ( z _ 1 z _ 2 . . . z _ { m - 1 } ) ) , q ( c ( z _ 1 z _ 2 . . . z _ { m - 1 } ) ) l _ m ) = z _ m \\\\ & \\lambda ( q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } ) ) , q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } ) ) a ) = z _ { n - 1 } ' \\\\ & \\lambda ( q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } z _ { n - 1 } ' ) ) , q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } z _ { n - 1 } ' ) ) b ) = z _ n ' \\\\ & \\lambda ( q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } z _ { n - 1 } ' z _ n ' ) ) , q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } z _ { n - 1 } ' z _ n ' ) ) b ) = z _ { n + 1 } ' \\end{align*}"}
+{"id": "1709.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to \\infty } \\| \\gamma _ { \\tau , p } - \\gamma _ p \\| _ { \\mathfrak { S } ^ 1 ( \\mathfrak { h } ^ { ( p ) } ) } = 0 \\ , . \\end{align*}"}
+{"id": "2625.png", "formula": "\\begin{align*} \\lambda _ x | _ { E _ { 2 , m } } = \\lambda _ y , \\lambda _ x | _ { E _ { 2 , m + e _ i - e _ j } } = \\lambda ^ j \\lambda _ x | _ { E _ { 2 , [ m - e _ j , m + e _ i ] } } = \\phi ^ j \\end{align*}"}
+{"id": "6513.png", "formula": "\\begin{align*} C = P _ 1 ^ T P _ 0 \\end{align*}"}
+{"id": "4346.png", "formula": "\\begin{align*} \\lim _ { \\eta \\to 0 } T ^ 1 _ \\eta ( t ) = 0 \\qquad \\end{align*}"}
+{"id": "1178.png", "formula": "\\begin{align*} \\widetilde { a } _ k ( x ) h _ k '' ( x ) + \\widetilde { b } _ k ( x ) h _ k ' ( x ) + ( 2 k - 1 ) \\widetilde { c } _ k ( x ) h _ k ( x ) = 0 , \\end{align*}"}
+{"id": "7370.png", "formula": "\\begin{align*} t _ { p , s } : = ( p s ) . \\end{align*}"}
+{"id": "4185.png", "formula": "\\begin{align*} l ( C _ i \\cap \\beta ) \\mathrm { s i n } ( \\theta ) & = \\mathrm { h e i g h t } ( C _ i ) i ( \\beta , \\alpha _ i ) \\\\ \\implies i ( \\beta , \\alpha _ i ) & = \\frac { l ( C _ i \\cap \\beta ) \\mathrm { s i n } ( \\theta ) } { \\mathrm { h e i g h t } ( C _ i ) } . \\end{align*}"}
+{"id": "7931.png", "formula": "\\begin{align*} \\sum _ { i + j = n } \\varphi _ i \\big ( \\mu _ j ( a , b ) \\big ) = ~ & \\sum _ { i + j + k = n } \\mu _ i ' \\big ( \\varphi _ j ( a ) , \\varphi _ k ( b ) \\big ) , \\\\ \\sum _ { i + j = n } \\varphi _ i \\circ R _ j = ~ & \\sum _ { i + j = n } R _ i ' \\circ \\varphi _ j , \\end{align*}"}
+{"id": "2114.png", "formula": "\\begin{align*} E ( t ) : = & \\ \\varphi ( A ( t ) ) = \\mathrm { s p a n } \\{ e _ 1 + t g _ { 1 } , e _ 2 , \\ldots , e _ k \\} \\\\ E ( t ) \\cap F = & \\ \\mathrm { s p a n } \\{ e _ 1 + t g _ { 1 } , e _ 2 , \\ldots , e _ m \\} . \\end{align*}"}
+{"id": "4697.png", "formula": "\\begin{align*} \\left | r \\frac { d } { d r } \\left [ { } _ 2 \\ ! F _ { 1 } \\left ( \\lambda , \\lambda + 1 ; 2 \\lambda + 1 ; \\tilde { A } \\right ) \\right ] \\right | \\lesssim & \\frac { | 1 - z \\bar { w } | ^ 2 } { | 1 - z w | ^ 2 } \\frac { 1 - | z w | ^ 2 } { | 1 - z \\bar { w } | ^ 2 } = \\frac { 1 - | z w | ^ 2 } { | 1 - z w | ^ 2 } , z = r e ^ { i \\theta } . \\end{align*}"}
+{"id": "6578.png", "formula": "\\begin{align*} H ( C _ n ) = I _ 2 \\otimes H ( P _ { \\frac { n } { 2 } + 1 } ) \\end{align*}"}
+{"id": "5133.png", "formula": "\\begin{align*} \\mathring { A } ^ M = \\{ x \\in \\mathbb R ^ n \\ , : \\ , D ( A , x ) = 1 \\} , \\end{align*}"}
+{"id": "6782.png", "formula": "\\begin{align*} \\dot { \\Gamma } _ t & = - b ^ * ( \\psi _ t ) b ( q ( t ) \\dot { \\psi } _ t ) \\Gamma _ t - \\Gamma _ t b ^ * ( q ( t ) \\dot { \\psi } _ t ) b ( \\psi _ t ) \\end{align*}"}
+{"id": "1975.png", "formula": "\\begin{align*} \\sum _ { k \\in { E } } P _ { R _ { j , k } } & = \\sum _ { k \\in \\tilde { E } } P _ { R _ { j , k } } - \\sum _ { k \\in \\{ \\pm 1 \\} ^ d } P _ { R _ { j , k } } \\\\ & = P _ { I _ { j } ^ 1 } \\otimes P _ { I _ { j } ^ 2 } \\otimes \\cdots \\otimes P _ { I _ { j } ^ n } - P _ { I _ { j - 1 } ^ 1 } \\otimes P _ { I _ { j - 1 } ^ 2 } \\otimes \\cdots \\otimes P _ { I _ { j - 1 } ^ d } . \\end{align*}"}
+{"id": "1239.png", "formula": "\\begin{align*} \\int _ { \\mathcal { T } } \\exp ( \\varphi ( t ) ) ) \\ , d t = \\sum _ { 0 \\le m \\le \\frac { \\log \\log T } { \\log 2 } } \\int _ { \\mathcal { T } \\cap \\mathcal { P } ( m ) } \\exp ( \\varphi ( t ) ) ) \\ , d t + \\int _ { \\mathcal { T } \\cap \\left ( \\bigcap _ { m } \\mathcal { P } ( m ) ^ c \\right ) } \\exp ( \\varphi ( t ) ) ) \\ , d t . \\end{align*}"}
+{"id": "5510.png", "formula": "\\begin{align*} F ( a , b ; t ) = - \\frac { ( 1 - b ) a q } { b - a q } + \\frac { ( 1 - a q ) ( b - a t q ) } { b - a q } F ( a q , b ; t ) . \\end{align*}"}
+{"id": "4859.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial \\rho ^ { N : k } } { \\partial t } & = \\sum _ { i = 1 } ^ k \\nabla _ { x _ i } \\cdot \\Bigg ( \\rho ^ { N : k } \\nabla V _ s ( x _ i ) + \\frac 1 N \\sum _ { j = 1 } ^ k \\nabla W _ s ( x _ i , x _ j ) \\rho ^ { N : k } + \\\\ & + \\frac { N - k } N \\int _ { \\R ^ { d + 2 } } \\nabla W _ s ( x _ i , x _ { k + 1 } ) \\rho ^ { N : k + 1 } ( \\dd x _ { k + 1 } ) \\Bigg ) + K \\int _ S ( \\rho ^ { N : k } ( s ' ) - \\rho ^ { N : k } ( s ) ) \\ , \\dd \\mu ( s ' ) , \\end{aligned} \\end{align*}"}
+{"id": "5042.png", "formula": "\\begin{align*} H ^ { q } ( Y , R ^ { 1 } f ^ { \\prime } _ { \\ast } ( K ^ { m } _ { X ^ { \\prime } } ) \\otimes H ^ { l } ) = 0 \\end{align*}"}
+{"id": "5431.png", "formula": "\\begin{align*} \\widehat { H } ( s ) = ( 2 \\pi ) ^ { - s - 1 } \\frac { \\Gamma \\left ( \\frac { L ^ 2 } { 2 } + 1 \\right ) ^ 2 \\Gamma \\left ( K + \\frac { L ^ 2 } { 2 } \\right ) } { \\Gamma ( L ^ 2 + 1 ) \\Gamma \\left ( K - \\frac { L ^ 2 } { 2 } - 1 \\right ) } \\frac { \\Gamma \\left ( \\frac { K - L ^ 2 + s - 1 } { 2 } \\right ) } { \\Gamma \\left ( \\frac { K + L ^ 2 - s + 1 } { 2 } \\right ) } . \\end{align*}"}
+{"id": "2353.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in \\mathfrak { S } _ { n + 1 } } \\mathrm { s g n } ( \\sigma ) f _ { b _ { 1 } } ( x _ { a _ { \\sigma ( 1 ) } } , \\ , x _ { a _ { \\sigma ( 2 ) } } x _ { b _ { 2 } } \\cdots x _ { a _ { \\sigma ( n ) } } x _ { b _ { n } } x _ { a _ { \\sigma ( n + 1 ) } } ) = 0 . \\end{align*}"}
+{"id": "574.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } T _ { W , \\psi } ( \\mu _ n ( Q ^ n ) ) = T _ { W , \\psi } ( \\mu ) , I ( \\mu _ n ( Q ^ n ) ) \\le I ( \\mu ) , \\ \\forall n . \\end{align*}"}
+{"id": "4684.png", "formula": "\\begin{align*} D _ y u ( x , y ) = \\partial _ y u ( x , y ) + \\frac { \\lambda } { y } \\left [ u ( x , y ) - u ( x , - y ) \\right ] . \\end{align*}"}
+{"id": "5405.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\P \\big ( N ^ { 2 / 3 } ( \\lambda _ N - E _ + ) \\le r \\big ) = \\mathrm { T W } _ { \\beta } ( r ) \\ , , r \\in \\R \\ , , \\end{align*}"}
+{"id": "4341.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } u } { \\mathrm { d } t } ( t ) = \\Phi ( u ( t ) ) [ u ( t ) ] , u ( 0 ) = u ^ { i n } \\ , , \\end{align*}"}
+{"id": "6381.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { 3 } ( e _ { i } ^ { ( 1 ) } ) ^ { k _ i } ( e _ { i } ^ { ( 2 ) } ) ^ { l _ i } x ^ { n _ 1 } y ^ { n _ 2 } \\end{align*}"}
+{"id": "203.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ { N } } & | u | ^ { p } ( I ^ \\gamma _ { T - } \\varphi ) d x d t + \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ { N } } \\omega \\varphi d x d t + \\int _ { \\mathbb { R } ^ { N } } u _ 0 ( \\varphi ( 0 , x ) - k \\Delta \\varphi ( 0 , x ) ) d x \\\\ & = - \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ { N } } u \\varphi _ t d x d t + k \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ { N } } u \\Delta \\varphi _ t d x d t - \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ { N } } u \\Delta \\varphi d x d t , \\end{align*}"}
+{"id": "4696.png", "formula": "\\begin{align*} { } _ 2 \\ ! F _ { 1 } \\left ( \\lambda + 1 , \\lambda + 2 ; 2 \\lambda + 2 ; \\tilde { A } \\right ) = ( 1 - \\tilde { A } ) ^ { - 1 } { } _ 2 \\ ! F _ { 1 } \\left ( \\lambda + 1 , \\lambda ; 2 \\lambda + 2 ; \\tilde { A } \\right ) , \\end{align*}"}
+{"id": "1229.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega _ r } \\sup _ { B _ { r / 2 } ( x ) } \\abs { g - T _ x g } \\dd x & \\leq C r ^ 2 \\int _ { \\Omega } \\int _ { B _ r ( y ) \\cap \\Omega _ r } \\frac { 1 } { \\omega _ { d } r ^ { d } } \\dd x \\dd \\abs { D ^ 2 g } ( y ) \\\\ & \\leq C r ^ 2 \\abs { D ^ 2 g } ( \\Omega ) . \\end{aligned} \\end{align*}"}
+{"id": "1307.png", "formula": "\\begin{align*} f \\circ \\pi ^ 1 _ { a , b } = \\pi ^ 1 _ { a ' , b ' } \\circ ( f \\oplus g ) , g \\circ \\pi ^ 2 _ { a , b } = \\pi ^ 2 _ { a ' , b ' } \\circ ( f \\oplus g ) , \\end{align*}"}
+{"id": "5201.png", "formula": "\\begin{align*} \\sum _ { j = 1 } { n - 1 } \\binom { n - 1 } { j } 1 ^ j \\cdot 1 ^ { n - 1 - j } = 2 ^ { n - 1 } - 1 \\enspace , \\end{align*}"}
+{"id": "5939.png", "formula": "\\begin{align*} M _ a = ( \\gamma ' ( a ) \\ , \\ , \\gamma '' ( a ) \\ , \\ , \\cdots \\ , \\ , \\gamma ^ { ( k ) } ( a ) ) \\end{align*}"}
+{"id": "5292.png", "formula": "\\begin{align*} f ( x ) = \\int _ P a ( \\omega ) e ^ { 2 \\pi i \\omega x } d \\mu _ P ( \\omega ) . \\end{align*}"}
+{"id": "7008.png", "formula": "\\begin{align*} d & = \\frac { 2 ( N - 1 ) } { N + 2 } , \\\\ a & = \\frac { 2 } { N + 2 } + \\sqrt { \\frac { 4 N } { ( N + 2 ) ^ 2 } - \\varepsilon } , \\\\ b & = 2 a + d . \\end{align*}"}
+{"id": "2071.png", "formula": "\\begin{align*} \\begin{bmatrix} S _ 1 & 0 \\\\ 0 & T _ 1 ^ { - 1 } \\end{bmatrix} \\begin{bmatrix} I _ \\ell & B _ { 1 2 } \\\\ B _ { 2 1 } & B _ { 2 2 } \\end{bmatrix} \\begin{bmatrix} S _ 1 ^ { - 1 } & - B _ { 1 2 } \\\\ 0 & I _ { n - \\ell } \\end{bmatrix} = \\begin{bmatrix} I _ \\ell & 0 \\\\ D & C _ { 2 2 } \\end{bmatrix} \\end{align*}"}
+{"id": "985.png", "formula": "\\begin{align*} a _ { k - i , 0 } ^ { ( 1 ) } & = ( - 1 ) ^ i i ! \\frac { \\Gamma \\bigl ( \\frac { n } { 2 } + j \\bigr ) } { \\Gamma \\bigl ( \\frac { n - 2 k } { 2 } + i + j \\bigr ) } , & a _ { 0 , k - j } ^ { ( 1 ) } & = 0 , \\\\ a _ { 0 , k - j } ^ { ( 2 ) } & = ( - 1 ) ^ i i ! \\frac { \\Gamma \\bigl ( \\frac { n } { 2 } + i \\bigr ) \\Gamma \\bigl ( k - i \\bigr ) } { \\Gamma \\bigl ( \\frac { n - 2 k } { 2 } + i + j \\bigr ) \\Gamma \\bigl ( k - j \\bigr ) } , & a _ { k - i , 0 } ^ { ( 2 ) } & = 0 . \\end{align*}"}
+{"id": "2303.png", "formula": "\\begin{align*} \\mathfrak h ( x ) = a + \\frac b { x - c } , \\end{align*}"}
+{"id": "4157.png", "formula": "\\begin{align*} \\iota _ x y : = 2 \\ < x , y \\ > \\end{align*}"}
+{"id": "1060.png", "formula": "\\begin{align*} \\alpha _ { i j k } : = \\frac { 1 } { 2 } ( c _ { i j k } + c _ { k i j } + c _ { k j i } ) , [ e _ { i } , e _ { j } ] = c _ { i j k } e _ { k } . \\end{align*}"}
+{"id": "3875.png", "formula": "\\begin{align*} f ' ( w ) = 2 w \\left ( 1 - \\frac { 1 } { 1 + \\delta - w ^ 2 } \\right ) . \\end{align*}"}
+{"id": "1904.png", "formula": "\\begin{align*} r = \\frac { 2 q } { 2 q - 3 } , \\end{align*}"}
+{"id": "5363.png", "formula": "\\begin{align*} A : = I _ 0 , B : = I _ 1 , C : = I _ 2 , \\end{align*}"}
+{"id": "2940.png", "formula": "\\begin{align*} & ( \\lambda - A ) ^ { - 1 } C ( \\lambda - B ) ^ { - 1 } \\\\ & = \\begin{pmatrix} ( \\lambda - A _ m ) ^ { - 1 } C _ { m , n } ( \\lambda - B _ m ) ^ { - 1 } & ( \\lambda - A _ m ) ^ { - 1 } C _ { m , \\infty } ( \\lambda - B _ \\infty ) ^ { - 1 } \\\\ ( \\lambda - A _ \\infty ) ^ { - 1 } C _ { \\infty , n } ( \\lambda - B _ n ) ^ { - 1 } & ( \\lambda - A _ \\infty ) ^ { - 1 } C _ { \\infty , \\infty } ( \\lambda - B _ \\infty ) ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
+{"id": "2418.png", "formula": "\\begin{align*} { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { n } ( u * e _ { 1 } ) ) & = \\sum _ { i + j = n } { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { i } u ) * { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { j } e _ { 1 } ) = \\sum _ { i + j = n } ( - 1 ) ^ { j } { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { i } u ) * e _ { 1 } e _ { 0 } ^ { j } . \\end{align*}"}
+{"id": "7110.png", "formula": "\\begin{align*} \\chi & \\in 2 r \\textbf { W } ( d ) , \\ \\langle \\lambda , \\chi \\rangle = r \\langle \\lambda , R ( d ) ^ { \\lambda > 0 } \\rangle . \\end{align*}"}
+{"id": "6261.png", "formula": "\\begin{align*} p _ { k } & = \\left ( \\frac { ( L _ { \\nabla h } - \\sigma ) L ^ 2 _ { \\nabla g } } { 2 } \\right ) \\frac { \\lambda ^ 2 _ { k - 1 } } { \\lambda _ k } + \\frac { 1 } { 2 } \\left ( \\frac { \\sigma } { \\lambda _ k } - L _ { \\nabla g } \\right ) - p _ { k - 1 } , \\\\ M _ { 1 , k } & = p _ { k - 1 } - \\frac { \\alpha _ k + \\sigma L _ { \\nabla g } \\lambda _ { k - 1 } } { 2 \\lambda _ k } - \\frac { ( L _ { \\nabla h } - \\sigma ) L ^ 2 _ { \\nabla g } \\lambda _ { k - 1 } ^ 2 } { 2 \\lambda _ k } , \\end{align*}"}
+{"id": "5105.png", "formula": "\\begin{align*} [ \\frac { A } { A + m } , B ] = m \\frac { 1 } { A + m } [ A , B ] \\frac { 1 } { A + m } . \\end{align*}"}
+{"id": "3846.png", "formula": "\\begin{align*} \\begin{aligned} \\psi \\left ( \\l \\right ) \\vdash 2 \\l _ 1 - 1 + \\dots + 2 \\l _ { 2 + 2 i } - 1 & = 2 \\left ( \\l _ 1 + \\dots + \\l _ { 2 + 2 i } \\right ) - 2 - 2 i \\\\ & = 2 \\left ( k + 2 + i \\right ) - 2 - 2 i \\\\ & = 2 k + 2 . \\end{aligned} \\end{align*}"}
+{"id": "2953.png", "formula": "\\begin{align*} \\widehat N ^ p _ C ( F ( \\bar x ) ; \\nabla F ( \\bar x ) u ) & = \\widehat N _ { T _ C ( F ( \\bar x ) ) } ( \\nabla F ( \\bar x ) u ) \\\\ & = \\widehat N _ { \\bigcup _ { i \\in J ( F ( \\bar x ) ) } T _ { P _ i } ( F ( \\bar x ) ) } ( \\nabla F ( \\bar x ) u ) = \\bigcap \\limits _ { i \\in J ( \\bar x ; u ) } N _ { T _ { P _ i } ( F ( \\bar x ) ) } ( \\nabla F ( \\bar x ) u ) \\end{align*}"}
+{"id": "2715.png", "formula": "\\begin{align*} ( - 1 ) ^ { j + r - 1 } \\binom { j + r - 1 } { j } \\frac { 1 } { 2 } S ( n , r ) \\binom { 2 n - j } { n } \\sum _ { l = 0 } ^ { r - 1 } ( - 1 ) ^ l \\binom { 2 n - j + l } { l } \\frac { \\binom { 2 ( j + r - 1 - l ) } { j + r - 1 - l } \\binom { 2 ( n - j + l + 1 ) } { n - j + l + 1 } \\binom { n - j } { j + r - l - 1 } } { 2 \\binom { 2 n - j + l + 1 } { n } } \\end{align*}"}
+{"id": "6914.png", "formula": "\\begin{align*} P ( z ) = z R ( z ) - q ( z + y ) . \\end{align*}"}
+{"id": "2435.png", "formula": "\\begin{align*} \\Gamma _ { 1 } ( t ) ^ { - 2 } \\Gamma _ { 1 } ( - t ) ^ { - 2 } = \\sum _ { \\substack { r = 0 } } ^ { \\infty } A _ { r } t ^ { r } , \\end{align*}"}
+{"id": "6199.png", "formula": "\\begin{align*} \\Upsilon : = \\{ ( \\mu , d ) \\in \\mathbb { Z } ^ 2 | \\ 0 \\leq d \\leq m , \\ \\frac { 1 } { 2 } ( m - d ) \\leq \\mu \\leq m - d \\} . \\end{align*}"}
+{"id": "7891.png", "formula": "\\begin{align*} | \\ddot f _ { y , r } ( x _ 1 ) - \\ddot f _ { y ' , r ' } ( x _ 1 ) | = | F ( x , y ) - F ( x ' , y ' ) | \\geq | F ( x ' , y ) - F ( x , y ' ) | - | F ( x , y ) - F ( x ' , y ) | . \\end{align*}"}
+{"id": "6016.png", "formula": "\\begin{align*} \\tilde { \\delta } _ 3 & = W ^ { ( q + r ) } ( b ) \\Bigg [ - \\dfrac { \\rho ^ { ( q ) } _ { b } ( b ; h ) + \\tilde { g } ^ { ( q ) } ( b ; h ) } { W ^ { ( q ) } ( b ) } r \\int _ b ^ { \\infty } e ^ { - \\Phi ( q + r ) y } W ^ { ( q ) } ( y ) d y - \\int _ 0 ^ { b } e ^ { - \\Phi ( q + r ) u } h ( u ) d u \\\\ & \\quad + r \\int _ { b } ^ { \\infty } e ^ { - \\Phi ( q + r ) y } \\rho _ b ^ { ( q ) } ( y ; h ) d y \\Bigg ] + \\tilde { g } ^ { ( q ) } ( b ; h ) + \\int _ 0 ^ b h ( u ) W ^ { ( q + r ) } ( b - u ) d u . \\end{align*}"}
+{"id": "5593.png", "formula": "\\begin{align*} g _ y ( J y , J X ) = g _ y ( y , X ) , \\end{align*}"}
+{"id": "4972.png", "formula": "\\begin{align*} \\mathbf { P } _ { \\rm M P A } ^ { k } [ m , v ] = C _ { m , v } \\prod \\limits _ { c \\in \\mathcal { N } ( v ) } E _ { c \\rightarrow v } ^ { ( T ) } ( \\mathcal { X } _ { v } [ m ] ) , \\end{align*}"}
+{"id": "3989.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\hat { \\mathcal { M } } ( t ) } { t } & \\stackrel { d } { = } \\sum _ { j = 1 } ^ { \\infty } j \\lim _ { t \\to \\infty } \\frac { N _ { j } ( t ) } { t } \\\\ & = \\frac { \\lambda ( p - 1 ) } { p \\ln p } , \\ , \\end{align*}"}
+{"id": "2226.png", "formula": "\\begin{align*} \\boldsymbol { \\mathcal { A } } * \\boldsymbol { \\mathcal { V } } _ n = \\boldsymbol { \\mathcal { V } } _ n * \\boldsymbol { \\mathcal { T } } _ n + \\widetilde { \\boldsymbol { \\mathcal { V } } } _ n \\\\ \\boldsymbol { \\mathcal { W } } _ n * \\boldsymbol { \\mathcal { A } } = \\boldsymbol { \\mathcal { T } } _ n * \\boldsymbol { \\mathcal { W } } _ n + \\widetilde { \\boldsymbol { \\mathcal { W } } } _ n \\end{align*}"}
+{"id": "1268.png", "formula": "\\begin{align*} 1 + 4 \\cdot 5 - \\binom 4 2 = 1 5 \\ ; \\end{align*}"}
+{"id": "7807.png", "formula": "\\begin{align*} \\begin{gathered} \\cot ( \\phi ) > 0 , \\ \\ \\cot ( \\theta ) < 0 , \\ \\ \\theta \\in \\mathcal { A } \\cup \\mathcal { D } \\\\ \\ 0 < \\cot ( \\theta ) < \\cot ( \\phi ) , \\ \\ \\theta \\in \\mathcal { B } \\cup \\mathcal { E } \\\\ \\cot ( \\phi ) < \\cot ( \\theta ) , \\ \\ \\theta \\in \\mathcal { C } \\cup \\mathcal { F } \\end{gathered} \\end{align*}"}
+{"id": "6473.png", "formula": "\\begin{align*} \\sum _ { j } ( \\psi _ 2 , \\phi _ j ) _ { L ^ 2 ( \\Sigma ) } ( \\psi _ 3 , \\phi _ j ) _ { L ^ 2 ( \\Sigma ) } = \\int _ \\Sigma \\psi _ 2 ( w ) \\psi _ 3 ( w ) \\sqrt { h ( w ) } d w . \\end{align*}"}
+{"id": "6742.png", "formula": "\\begin{align*} | ( \\frac { 1 } { R \\theta } \\partial ^ { \\alpha } \\widetilde { E } , \\partial ^ { \\alpha } ( \\bar { \\rho } \\bar { u } ) ) | & = | ( \\partial ^ { \\alpha ' } \\widetilde { E } , \\partial _ { x _ i } [ \\frac { 1 } { R \\theta } \\partial ^ { \\alpha } ( \\bar { \\rho } \\bar { u } ) ] ) | \\\\ & \\leq C \\| \\partial ^ { \\alpha ' } \\widetilde { E } \\| \\| \\partial _ { x _ i } [ \\frac { 1 } { R \\theta } \\partial ^ { \\alpha } ( \\bar { \\rho } \\bar { u } ) ] \\| \\leq C \\eta _ { 0 } \\varepsilon ^ { 1 - a } . \\end{align*}"}
+{"id": "4289.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { n ( - 1 ) ^ { n - 1 } q ^ { n ( n + 1 ) / 2 } } { ( q ) _ n } + \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { n ( n + 1 ) } } { ( q ) _ n ^ { 2 } ( 1 - q ^ n ) } = \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( q ) _ n ( 1 - q ^ n ) } . \\end{align*}"}
+{"id": "921.png", "formula": "\\begin{align*} L ( x ) : = \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( y ' ) ( x _ \\beta - y _ \\beta ) . \\end{align*}"}
+{"id": "8227.png", "formula": "\\begin{align*} \\widetilde { X } ( T ) : = \\Vert \\sigma \\Vert _ { \\widetilde { L } ^ { \\infty } _ { T } ( \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } ) } + \\Vert \\widetilde { u } \\Vert _ { \\widetilde { L } ^ { \\infty } _ { T } ( \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } ) } + \\Vert \\sigma \\Vert _ { L ^ 1 _ { T } ( \\widetilde { B } ^ { \\frac { N } { 2 } + 1 , \\frac { N } { 2 } + 2 - \\alpha } ) } + \\Vert \\widetilde { u } \\Vert _ { L ^ 1 _ T ( \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } ) } . \\end{align*}"}
+{"id": "1495.png", "formula": "\\begin{align*} y _ d = \\frac { 1 } { H } \\log \\frac { \\Delta } { d } \\end{align*}"}
+{"id": "6611.png", "formula": "\\begin{align*} q _ { \\theta } ^ { \\gamma } ( v , v ) = \\int _ { U _ \\theta } | \\nabla v | ^ 2 \\dd x \\dd y - \\gamma \\int _ { \\partial U _ \\theta } | v | ^ 2 \\dd \\sigma , v \\in H ^ { 1 } ( U _ \\theta ) , \\end{align*}"}
+{"id": "2925.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\inf \\| R ^ n - K \\| ^ { 1 / n } = 0 \\end{align*}"}
+{"id": "3546.png", "formula": "\\begin{align*} \\varphi ( \\mathrm { v } , \\mathrm { y } , \\mathrm { t } , \\tau ) - \\Phi ^ { \\textbf { e } } ( \\xi , \\mathrm { t } ; \\mathrm { z } , \\tau ) = \\mathcal { O } \\Bigg ( \\sqrt { \\alpha } | \\mathrm { y } - \\mathrm { v } | \\ \\Vert \\partial _ { s } \\Phi ^ { \\textbf { e } } ( \\xi , \\mathrm { t } ; \\mathrm { z } , \\cdot ) \\Vert _ { \\mathrm { H } ^ { - \\frac { 1 } { 4 } } ( 0 , \\tau ) } \\Bigg ) . \\end{align*}"}
+{"id": "7048.png", "formula": "\\begin{align*} \\tilde { \\mathfrak { t r } } ^ p = [ \\mathfrak g , \\tilde { \\mathfrak { t r } ^ p _ 0 } ] + \\tilde { \\mathfrak { t r } } ^ p _ 0 \\end{align*}"}
+{"id": "5173.png", "formula": "\\begin{align*} T = \\left \\{ x = ( x _ 1 , x _ 2 , x _ 3 ) \\ , : \\ , x _ 1 \\in [ 0 , 1 ] \\ , , \\ , \\sqrt { x _ 2 ^ 2 + x _ 3 ^ 2 } \\le 1 \\right \\} , \\end{align*}"}
+{"id": "1878.png", "formula": "\\begin{align*} | f ( x ) - g ( x ) | \\leq \\sum _ { s = 0 } ^ { \\infty } \\frac { 1 } { 3 ^ { l s } } Q \\left ( \\frac { x } { 3 ^ { s + 1 } } , \\frac { x } { 3 ^ { s + 1 } } \\right ) \\end{align*}"}
+{"id": "6730.png", "formula": "\\begin{align*} F = M + G . \\end{align*}"}
+{"id": "4912.png", "formula": "\\begin{align*} \\Omega ^ * _ { p r o p - s m , \\frak S } ( X \\to Y ) : = \\frac { \\mathcal M ^ * _ { p r o p - s m } ( X \\to Y ) } { \\langle \\mathcal R ^ { \\frak S } _ { p r o p - s m } \\rangle ( X \\xrightarrow f Y ) } . \\end{align*}"}
+{"id": "6935.png", "formula": "\\begin{align*} { \\mathsf Z } _ { C , L , E } = { \\mathsf Z } _ { C _ 1 , L _ 1 , E _ 1 } \\cdot { \\mathsf Z } _ { C _ 2 , L _ 2 , E _ 2 } . \\end{align*}"}
+{"id": "3930.png", "formula": "\\begin{align*} \\Delta _ { \\Gamma } v _ { \\partial \\Omega } ( x ) = f ( x ) \\partial \\Omega , \\end{align*}"}
+{"id": "6861.png", "formula": "\\begin{align*} & ( \\alpha ^ { p ^ { e ' } + 1 } ( v _ 1 ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ 1 ) , \\dots , \\alpha ^ { p ^ { e ' } + 1 } ( v _ s ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ s ) , ( v _ { s + 1 } ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ { s + 1 } ) , \\dots , ( v _ n ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ n ) , 0 ) \\\\ = & ( u _ 1 g ( a _ 1 ) , \\dots , u _ s g ( a _ s ) , u _ { s + 1 } g ( a _ { s + 1 } ) , \\dots , u _ n g ( a _ n ) , 0 ) . \\end{align*}"}
+{"id": "8068.png", "formula": "\\begin{align*} r _ { i } ^ { ( k ) } = \\frac { s _ { i + k } ^ { ( k ) } } { s _ { i } ^ { ( k ) } } = \\frac { E _ { i + 2 k } - E _ { i + k } } { E _ { i + k } - E _ { i } } , \\ ; \\ ; \\ ; i , k = 1 , 2 , 3 , \\ldots . \\end{align*}"}
+{"id": "7562.png", "formula": "\\begin{align*} \\int V d \\mu _ { \\varepsilon , h } & = \\int V ( \\Phi ( \\varepsilon , p ) ) d \\mu ( p ) \\\\ & = \\int V d \\mu + \\varepsilon \\int h d V d \\mu + o ( \\varepsilon ) \\varepsilon \\to 0 . \\end{align*}"}
+{"id": "4860.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial \\rho ^ 1 } { \\partial t } & = \\nabla _ { x _ 1 } \\cdot \\Bigg ( \\rho ^ 1 \\nabla V _ s ( x _ 1 ) + \\frac 1 N \\nabla W _ s ( x _ 1 , x _ 1 ) \\rho ^ 1 + \\\\ & + \\frac { N - 1 } N \\int _ { \\R ^ { d + 2 } } \\nabla W _ s ( x _ 1 , x _ 2 ) \\rho ^ { N : 2 } ( \\dd x _ 2 ) \\Bigg ) + K \\int _ S ( \\rho ^ 1 ( s ' ) - \\rho ^ 1 ( s ) ) \\ , \\dd \\mu ( s ' ) . \\end{aligned} \\end{align*}"}
+{"id": "7787.png", "formula": "\\begin{align*} \\begin{aligned} c a p _ p ( X _ \\varepsilon ) & \\leq ( \\int _ 0 ^ \\varepsilon x ^ { \\frac { p - n - 1 } { 1 - p } } \\cdot [ V o l ( \\partial X , \\hat { g } ) + O ( x ) ) ] ^ { \\frac { 1 } { 1 - p } } d x ) ^ { 1 - p } \\\\ & = V o l ( \\partial X , \\hat { g } ) ( \\int _ 0 ^ \\varepsilon x ^ { \\frac { p - n - 1 } { 1 - p } } ( 1 + O ( x ) ) d x ) ^ { 1 - p } \\\\ & = V o l ( \\partial X , \\hat { g } ) ( \\frac { n } { p - 1 } ) ^ { p - 1 } \\varepsilon ^ { - n } + O ( \\varepsilon ^ { - n + 1 - p } ) \\end{aligned} \\end{align*}"}
+{"id": "4920.png", "formula": "\\begin{align*} \\binom { n } { k } > \\sum _ { i = 1 } ^ { k - 1 } \\left ( \\frac { j - 1 } { 2 } \\right ) ^ i \\binom { n } { k - i } > \\sum _ { i = 1 } ^ { k - r - 1 } \\frac { j - 1 } { 2 } \\binom { n } { k - i } + \\sum _ { i = k - r } ^ { k - 1 } j \\binom { n } { k - i } . \\end{align*}"}
+{"id": "515.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { X _ i } \\to \\int _ 0 ^ 1 \\mu ^ * _ u \\ , d u , . \\end{align*}"}
+{"id": "7582.png", "formula": "\\begin{align*} E _ \\varphi ( F ) \\leq E _ \\varphi [ \\mu ] = E _ { \\varphi _ m } [ \\mu ] = E _ { \\varphi _ m } ( F ) \\leq E _ \\varphi ( F ) . \\end{align*}"}
+{"id": "1444.png", "formula": "\\begin{align*} A _ { \\varepsilon } ( x ) = A _ 0 + \\frac { a _ 0 } { ( n - \\alpha ) ( 2 - \\alpha ) } \\langle x \\rangle ^ { 2 - \\alpha } - N \\ast \\left ( \\eta _ { \\varepsilon } b _ 2 \\right ) , \\end{align*}"}
+{"id": "5074.png", "formula": "\\begin{align*} ( \\mathcal { F } f ) ( \\xi ) = \\int _ { \\mathbb { R } ^ m \\times \\mathbb { T } ^ n } f ( z ) e ^ { - i z \\cdot \\xi } \\ , d z , \\end{align*}"}
+{"id": "2426.png", "formula": "\\begin{align*} \\tilde { L } ( { \\rm D } ( { \\rm r e g } _ { 0 } ( w ) ) ) & = \\sum _ { N \\geq 2 } \\zeta ^ { \\mathfrak { m } } ( N ) \\tilde { L } ( { \\rm C u t } _ { ( \\emptyset ; \\{ 1 \\} ^ { N - 1 } ) } ( { \\rm r e g } _ { 0 } ( w ) ) ) \\\\ & = \\sum _ { N \\geq 2 } \\zeta ^ { \\mathfrak { m } } ( N ) \\tilde { L } ( { \\rm r e g } _ { 0 } ( { \\rm C u t } _ { ( \\emptyset ; \\{ 1 \\} ^ { N - 1 } ) } ( w ) ) ) \\\\ & = \\sum _ { N \\geq 2 } \\zeta ^ { \\mathfrak { m } } ( N ) \\tilde { L } ( { \\rm C u t } _ { ( \\emptyset ; \\{ 1 \\} ^ { N - 1 } ) } ( w ) ) , \\end{align*}"}
+{"id": "1968.png", "formula": "\\begin{align*} \\widehat { \\mathcal { P } _ I f } ( \\xi ) + & \\widehat { \\mathcal { P } _ J f } ( \\xi ) = [ b _ I ^ 2 ( \\xi ) + b _ J ^ 2 ( \\xi ) ( \\xi ) ] \\hat { f } ( \\xi ) \\\\ & + b _ J ( \\xi ) b _ J ( 2 \\alpha ' - \\xi ) \\hat { f } ( 2 \\alpha ' - \\xi ) - b _ I ( \\xi ) b _ I ( 2 \\alpha ' - \\xi ) \\hat { f } ( 2 \\alpha ' - \\xi ) . \\end{align*}"}
+{"id": "1885.png", "formula": "\\begin{align*} \\Psi ( x ) = \\lim _ { n \\rightarrow \\infty } \\max \\{ | 2 | ^ { k } \\mu ( 0 , 2 ^ { k + 1 } x ) ; 0 \\leq k < n \\} \\end{align*}"}
+{"id": "4617.png", "formula": "\\begin{align*} \\phi ' _ + ( 0 ) = \\lim _ { t \\to 0 ^ + } \\frac { \\phi ( t ) } { t } \\le \\lim _ { t \\to 0 ^ + } L _ p t ^ { p - 1 } \\frac { \\phi ( t _ 0 ) } { t _ 0 ^ p } = 0 . \\end{align*}"}
+{"id": "3781.png", "formula": "\\begin{align*} ( 1 - 2 t c ( t ) ) ^ 2 & = 1 - 4 t \\\\ ( 1 - 2 t c ( t ) ) ^ { - 1 } & = ( 1 - 4 t ) ^ { - \\frac { 1 } { 2 } } = \\sum _ { n = 0 } ^ \\infty \\binom { - \\frac { 1 } { 2 } } { n } ( - 4 t ) ^ n \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\binom { 2 n } { n } t ^ n = \\sum _ { n = 0 } ^ \\infty ( n + 1 ) C _ n t ^ n , \\end{align*}"}
+{"id": "2823.png", "formula": "\\begin{align*} X _ s ^ * = \\left ( x - \\xi - \\frac { d } { \\gamma _ 0 } \\right ) \\frac { 1 + ( T - s ) \\rho } { 2 + T \\rho } + \\xi , s \\in [ 0 , T ) . \\end{align*}"}
+{"id": "7084.png", "formula": "\\begin{align*} p _ { H } ( Z _ r , Z _ \\bullet , Z ' _ \\bullet ) = \\begin{cases} 1 & Z ^ { \\prime } _ { \\bullet } = Z _ { \\bullet } , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "3398.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\ell r _ j s _ { g - h _ j } = 0 , \\end{align*}"}
+{"id": "3611.png", "formula": "\\begin{align*} \\tau ^ * _ n = \\inf \\left \\{ t \\ge \\tau ^ * _ { n - 1 } : \\ X _ { t - } + \\Delta Y _ t \\in \\Re \\setminus ( a , b ) \\right \\} \\end{align*}"}
+{"id": "928.png", "formula": "\\begin{align*} \\begin{aligned} F ^ { i j } \\left ( \\sqrt { b _ { n - 1 } } ( A _ 1 v - A _ 2 w ) \\pm \\nabla _ \\alpha ( u - \\varphi ) \\right ) _ { i j } \\geq \\ , & 0 \\mbox { i n } \\omega _ 1 \\\\ A \\sqrt { b _ { n - 1 } } ( A _ 1 v - A _ 2 w ) \\pm \\nabla _ \\alpha ( u - \\varphi ) \\leq \\ , & 0 \\mbox { o n } \\partial \\omega _ 1 \\end{aligned} \\end{align*}"}
+{"id": "1040.png", "formula": "\\begin{align*} \\mathcal { W } \\ , ( P ) : = \\int _ { M } \\left ( \\int _ { | \\xi | = 1 } t r \\ , \\sigma _ { - n } ( P ) ( x , \\xi ) ~ { \\mathcal V } _ { \\xi } \\right ) ~ d ^ { n } x , \\end{align*}"}
+{"id": "5791.png", "formula": "\\begin{align*} Y _ A [ k ] & = \\{ [ v ] \\in \\mathbb { P } ( V _ 6 ) | \\dim ( A \\cap F _ v ) \\geq k \\} \\end{align*}"}
+{"id": "2031.png", "formula": "\\begin{align*} \\aligned K ( t , u ) & = \\frac { 1 } { 2 } ( | t | + | u | - | t - u | ) = \\begin{cases} ~ u , & ~ t > 0 , ~ u > 0 , ~ t - u > 0 , \\\\ ~ t , & ~ t > 0 , ~ u > 0 , ~ t - u < 0 , \\\\ ~ 0 , & ~ t < 0 , ~ u > 0 , ~ t - u < 0 , \\\\ ~ - u , & ~ t < 0 , ~ u < 0 , ~ t - u < 0 , \\\\ ~ - t , & ~ t < 0 , ~ u < 0 , ~ t - u > 0 , \\\\ ~ 0 , & ~ t > 0 , ~ u < 0 , ~ t - u > 0 , \\\\ \\end{cases} \\endaligned \\end{align*}"}
+{"id": "1353.png", "formula": "\\begin{align*} & a ( r ) = \\left ( \\frac 1 { | B _ r | } \\int _ { B _ r } | \\nabla u ( x ) | ^ p \\ , d x \\right ) ^ { \\frac 1 p } \\le \\bar { C } a ( \\eta ^ { k _ 0 } ) \\le \\bar { C } \\big ( C ( \\eta ) M + 2 ^ { - k _ 0 } a ( 1 ) \\big ) \\\\ & \\qquad \\qquad \\le \\bar { C } \\big ( C ( \\eta ) M + a ( 1 ) \\big ) \\le C ( M , \\eta ) ( 1 + a ( 1 ) ) , \\end{align*}"}
+{"id": "1231.png", "formula": "\\begin{align*} | \\nu _ { p ^ { k + 1 } } | & = 1 + \\sum \\limits _ { j = 1 } ^ { p ^ { k + 1 } - 1 } \\phi ( j ) + | \\nu _ { p ^ k } | \\\\ & = 1 + \\sum \\limits _ { j = 1 } ^ { p ^ { k + 1 } - 1 } \\phi ( j ) + k + \\sum \\limits _ { i = 1 } ^ k \\sum \\limits _ { j = 1 } ^ { p ^ i - 1 } \\phi ( j ) \\\\ & = ( k + 1 ) + \\sum \\limits _ { i = 1 } ^ { k + 1 } \\sum \\limits _ { j = 1 } ^ { p ^ i - 1 } \\phi ( j ) . \\end{align*}"}
+{"id": "45.png", "formula": "\\begin{align*} \\chi _ P ( P _ a , P _ b ) & = \\left | \\sum _ { i = 1 } ^ k k P _ a ( i ) P _ b ( i ) - 1 \\right | \\\\ & = \\left | \\sum _ { i = 1 } ^ k k \\frac { 1 } { k } \\left ( 1 + \\epsilon v _ { a , i } \\right ) \\frac { 1 } { k } \\left ( 1 + \\epsilon v _ { b , i } \\right ) - 1 \\right | \\\\ & = \\left | \\sum _ { i = 1 } ^ k \\frac { 1 } { k } \\left ( 1 + \\epsilon v _ { a , i } + \\epsilon v _ { b , i } + \\epsilon ^ 2 v _ { a , i } v _ { b , i } \\right ) - 1 \\right | \\\\ & = 0 , \\end{align*}"}
+{"id": "4822.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\R ^ d \\times S } \\phi \\ , \\dd \\rho _ t - & \\int _ { \\R ^ d \\times S } \\phi \\ , \\dd \\rho _ 0 = \\\\ & = - \\int _ 0 ^ t \\int _ { \\R ^ d \\times S } ( \\nabla \\phi \\cdot \\nabla f + K \\phi ) \\ , \\dd \\rho _ s \\ , \\dd s + \\int _ 0 ^ t \\int _ { \\R ^ d \\times S } K \\phi \\ , \\dd ( \\bar \\rho _ s \\otimes \\mu ) \\ , \\dd s , \\end{aligned} \\end{align*}"}
+{"id": "4253.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\frac { q ^ k } { 1 - q ^ k } = \\sum _ { k = 1 } ^ { n } \\left [ \\begin{matrix} n \\\\ k \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { k - 1 } q ^ { k ( k + 1 ) / 2 } } { ( 1 - q ^ k ) } , \\end{align*}"}
+{"id": "3873.png", "formula": "\\begin{align*} K _ B ( x , y ) = \\frac { x I _ 0 ' ( x ) I _ 0 ( y ) - y I _ 0 ' ( y ) I _ 0 ( x ) } { x ^ 2 - y ^ 2 } , \\end{align*}"}
+{"id": "4900.png", "formula": "\\begin{align*} \\frac { 1 } { x ^ { 1 / 2 } ( 1 - x ) ^ { 1 / 2 } } \\left ( \\arcsin ( x ^ { 1 / 2 } e ^ { ( 1 - x ) t } ) - \\arcsin ( x ^ { 1 / 2 } ) \\right ) = \\sum _ { n = 0 } ^ \\infty 2 ^ n F _ n ( x , 1 / 2 ) \\frac { t ^ { n + 1 } } { ( n + 1 ) ! } . \\end{align*}"}
+{"id": "7966.png", "formula": "\\begin{align*} & t ( s _ { k + 1 } ) - t ( s _ k ) = \\int _ { s _ k } ^ { s _ { k + 1 } } t ' ( s ) d s = - 2 \\int _ { s _ k } ^ { s _ { k + 1 } } r ( s ) \\cos ( \\theta ( s ) ) d s \\\\ & = \\frac { 3 } { c } \\int _ { s _ k } ^ { s _ { k + 1 } } \\theta ' ( s ) \\cos ( \\theta ( s ) ) d s = \\frac { - 6 } { c } . \\end{align*}"}
+{"id": "2927.png", "formula": "\\begin{align*} ( w - p ( A ) ) ^ { - 1 } = \\frac { 1 } { 2 \\pi i } \\int _ \\gamma \\frac { 1 } { w - p ( \\lambda ) } ( \\lambda - A ) ^ { - 1 } d \\lambda , \\end{align*}"}
+{"id": "8030.png", "formula": "\\begin{align*} A = Q _ 0 \\mbox { a n d } B = c o l ( Q _ i ) _ { i = 1 } ^ { n } . \\end{align*}"}
+{"id": "7261.png", "formula": "\\begin{align*} \\overline { { m d i m } } _ M ( T , f + g , d ) - \\over & \\geq F ( \\mu ) + \\int f + g d \\mu - ( F ( \\mu ) + \\int f d \\mu ) \\\\ & = \\int g d \\mu . \\end{align*}"}
+{"id": "5236.png", "formula": "\\begin{align*} \\chi _ 1 | _ { I _ K } = \\prod _ { \\tau \\in J } \\omega _ \\tau ^ { a _ \\tau + 1 } \\prod _ { \\tau \\not \\in J } \\omega _ \\tau ^ { b _ \\tau } , \\chi _ 2 | _ { I _ K } = \\prod _ { \\tau \\not \\in J } \\omega _ { \\tau } ^ { a _ \\tau + 1 } \\prod _ { \\tau \\in J } \\omega _ { \\tau } ^ { b _ \\tau } \\end{align*}"}
+{"id": "3403.png", "formula": "\\begin{align*} p = p ( \\rho , S ) . \\end{align*}"}
+{"id": "1716.png", "formula": "\\begin{align*} \\psi ^ t ( x ) : = e ^ { t \\Delta } ( x ) = \\sum _ { n \\in \\Z ^ d } ( 4 \\pi t ) ^ { - 1 / 2 } e ^ { - | x - n | ^ 2 / 4 t } \\end{align*}"}
+{"id": "5481.png", "formula": "\\begin{align*} ( 1 - e ^ { i \\theta } ) F _ { N } ( 0 , e ^ { - i \\theta } ; e ^ { i \\theta } ) = ( 1 - e ^ { i \\theta } q ^ { N } ) + \\sum _ { n = 1 } ^ { N } \\begin{bmatrix} N \\\\ n \\end{bmatrix} \\frac { ( q ) _ { n } q ^ { n ^ 2 } } { ( 1 - 2 q \\cos { \\theta } + q ^ 2 ) \\cdots ( 1 - 2 q ^ n \\cos { \\theta } + q ^ { 2 n } ) } , \\end{align*}"}
+{"id": "1656.png", "formula": "\\begin{align*} \\tau _ 0 : = \\eta ( \\mathsf { A } _ 0 , \\mathsf { A } _ 1 ) \\geq 2 ^ { - 5 } \\ , \\mathsf { q } ^ { - 1 } \\ , \\pmb { \\eta } \\ , , \\quad \\qquad \\tau _ { \\mathsf { M } + 1 } : = \\eta ( \\mathsf { A } _ { \\mathsf { M } + 1 } , \\mathsf { A } _ { \\mathsf { M } + 2 } ) \\geq 2 ^ { - 5 } \\ , \\mathsf { q } ^ { - 1 } \\ , \\pmb { \\eta } \\ , , \\end{align*}"}
+{"id": "699.png", "formula": "\\begin{align*} { f _ i } \\left ( { { \\bf { x } } + { { \\bf { c } } _ i } \\Delta t , t + \\Delta t } \\right ) = { f _ i } ^ * \\left ( { { \\bf { x } } , t } \\right ) . \\end{align*}"}
+{"id": "1562.png", "formula": "\\begin{align*} v _ { 1 } v _ { 1 } ^ { - 1 } e = e , \\ldots , v _ { 4 } v _ { 4 } ^ { - 1 } e = e \\end{align*}"}
+{"id": "15.png", "formula": "\\begin{align*} w = u _ 1 + u _ 2 + \\cdots + u _ l . \\end{align*}"}
+{"id": "895.png", "formula": "\\begin{align*} 0 = H _ i = ( \\Delta u ) _ i + B x _ i \\end{align*}"}
+{"id": "2961.png", "formula": "\\begin{align*} f _ 0 ( x ) : = - \\tfrac 1 2 x ^ 2 , F ( x ) : = \\tfrac 1 2 x ^ 2 , g ( x ) : = x \\forall x \\in \\R \\end{align*}"}
+{"id": "1721.png", "formula": "\\begin{align*} \\rho ( \\Theta ( \\xi ) ) = \\frac { \\tilde { \\rho } _ { 1 } ( \\Theta ( \\xi ) ) } { \\tilde { \\rho } _ { 1 } ( \\mathrm { I } ) } \\ , , \\end{align*}"}
+{"id": "6486.png", "formula": "\\begin{align*} v ( x ) & = \\int u ( x + z ) \\ , d \\nu _ x ( z ) \\\\ & = \\int u ( x + h ( x , y ) ) \\ , d \\mu ( y ) \\\\ \\end{align*}"}
+{"id": "4522.png", "formula": "\\begin{align*} | f ( u ) | & = \\bigg | \\int _ { [ 0 , 1 ] } \\mathcal { E } _ y ' u ' \\bigg | = \\big | \\langle \\mathcal { E } _ y , u \\rangle _ { H ^ 1 } - \\langle \\mathcal { E } _ y , u \\rangle _ { L ^ 2 } \\big | \\leq \\big | \\langle \\mathcal { E } _ y , u \\rangle _ { H ^ 1 } \\big | + \\big | \\langle \\mathcal { E } _ y , u \\rangle _ { L ^ 2 } \\big | \\\\ & \\leq \\| \\mathcal { E } _ y \\| _ { H ^ 1 } \\| u \\| _ { H ^ 1 } + \\| \\mathcal { E } _ y \\| _ { L ^ 2 } \\| u \\| _ { L ^ 2 } \\leq 2 \\| \\mathcal { E } _ y \\| _ { H ^ 1 } \\| u \\| _ { H ^ 1 } \\ , . \\end{align*}"}
+{"id": "3859.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\min _ { i = - n } ^ n | \\lambda ^ z _ i | \\leq \\frac { u } { n } \\right ) \\le C _ \\alpha u ^ { \\frac { 2 \\alpha } { 1 + \\alpha } } n ^ { \\beta + 1 } , z \\in \\C , u > 0 , \\end{align*}"}
+{"id": "1984.png", "formula": "\\begin{align*} z _ w = \\begin{cases} x _ w , & w \\neq u , \\\\ x _ v , & w = u , \\end{cases} \\end{align*}"}
+{"id": "5499.png", "formula": "\\begin{align*} F _ N ( a , b ; t ) & = 1 + \\frac { ( 1 - a q ) ( 1 - q ^ { N + 1 } ) t } { ( 1 - b q ) ( 1 - t q ^ N ) } { } _ 3 \\phi _ 2 \\begin{bmatrix} q ^ { - N } , & a q ^ 2 , & q & ; q , q \\\\ q ^ { 1 - N } / t , & b q ^ 2 \\end{bmatrix} \\\\ & = 1 + \\frac { ( 1 - a q ) ( 1 - q ^ { N + 1 } ) t } { ( 1 - b q ) ( 1 - t q ^ N ) } F _ N ( a q , b q , t ) , \\end{align*}"}
+{"id": "3612.png", "formula": "\\begin{align*} \\xi ^ * _ n = \\begin{cases} \\beta - \\left ( X ( \\tau ^ * _ n - ) + \\Delta Y ( \\tau ^ * _ n ) \\right ) , & \\textrm { i f } ~ X ( \\tau ^ * _ n - ) + \\Delta Y ( \\tau ^ * _ n ) \\ge b , \\\\ \\alpha - \\left ( X ( \\tau ^ * _ n - ) + \\Delta Y ( \\tau ^ * _ n ) \\right ) , & \\textrm { i f } ~ X ( \\tau ^ * _ n - ) + \\Delta Y ( \\tau ^ * _ n ) \\le a . \\end{cases} \\end{align*}"}
+{"id": "893.png", "formula": "\\begin{align*} S _ k ^ { i j } ( \\Delta u ) _ { i j } + \\sum _ i S _ k ^ { p q , r s } u _ { p q i } u _ { r s i } = \\Delta f , \\end{align*}"}
+{"id": "7604.png", "formula": "\\begin{align*} f ( x ) = \\mathbb { E } \\left ( F ( x , v ) \\right ) . \\end{align*}"}
+{"id": "3166.png", "formula": "\\begin{align*} \\d p ( x ) ( w ) = \\lim _ { n \\to \\infty } \\frac { p ( x + t _ n w ) } { t _ n } = \\lim _ { n \\to \\infty } \\frac { \\| x + t _ n w - y _ n \\| _ { * } } { t _ n } \\in \\R , \\end{align*}"}
+{"id": "6150.png", "formula": "\\begin{align*} ( M _ { ( P ^ \\perp + z P ) U } R _ q , R _ { \\overline q } M _ { U ^ * ( P + z P ^ \\perp ) } ) & = ( M _ { ( P ^ \\perp + z P ) U } \\tau _ { \\rm B C L } \\mathfrak r _ q \\tau _ { \\rm B C L } ^ * , \\tau _ { \\rm B C L } \\mathfrak r _ { \\overline q } \\tau _ { \\rm B C L } ^ * M _ { U ^ * ( P + z P ^ \\perp ) } ) \\\\ & = \\tau _ { \\rm B C L } ( V _ 1 \\mathfrak r _ q , \\mathfrak r _ { \\overline q } V _ 2 ) \\tau _ { \\rm B C L } ^ * . \\end{align*}"}
+{"id": "2263.png", "formula": "\\begin{align*} \\sigma ' ( z _ 2 ) = \\lambda z _ 2 ^ { d _ 1 \\cdots d _ s } + , \\ \\lambda \\in k ^ * . \\end{align*}"}
+{"id": "5594.png", "formula": "\\begin{align*} \\begin{array} { l r } g _ y ( y , y ) = g _ y ( J y , J y ) , & g _ y ( y , J y ) = 0 . \\end{array} \\end{align*}"}
+{"id": "8020.png", "formula": "\\begin{align*} \\begin{bmatrix} Q _ 0 - A & r o w ( Q _ j ^ * ) _ { j \\geq m + 1 } \\\\ \\ : c o l ( Q _ j ) _ { j \\geq m + 1 } & T _ Q \\otimes I _ d ^ { m + 1 } \\end{bmatrix} \\geq 0 . \\end{align*}"}
+{"id": "3739.png", "formula": "\\begin{align*} A M & = \\begin{bmatrix} - 1 6 & 8 & 8 \\\\ 8 & - 1 6 & 8 \\\\ 8 & 8 & - 1 6 \\end{bmatrix} \\otimes J _ { 2 0 } \\end{align*}"}
+{"id": "4590.png", "formula": "\\begin{align*} \\tilde \\varphi ( n _ 0 + ( m + 1 ) N ) = \\tilde \\varphi ( n _ 0 + m N ) + \\frac { K } { ( n _ 0 + m N - b ) \\pi \\sin \\pi k } \\left ( 1 0 0 + \\varepsilon ( m ) \\right ) . \\end{align*}"}
+{"id": "6900.png", "formula": "\\begin{align*} \\frac { 2 } { q } + \\frac { d } { r } = \\frac { d } { 2 } \\end{align*}"}
+{"id": "1236.png", "formula": "\\begin{align*} \\tilde { j } = ( j _ 1 , \\ldots , j _ { \\mathcal { I } } ) , \\ , \\tilde { \\ell } = ( \\ell _ 1 , \\ldots , \\ell _ { \\mathcal { I } } ) 0 \\le j _ i , \\ell _ i \\le 1 0 0 k \\beta _ i ^ { - 3 / 4 } , \\end{align*}"}
+{"id": "8114.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { \\min } ( \\overline F _ { \\ ! n _ 1 + \\cdots + n _ \\ell } ) \\geqslant \\sum _ { i = 1 } ^ \\ell \\Big ( \\widehat { \\mu } _ { \\min } ( \\overline F _ { \\ ! n _ i } ) - \\frac 3 2 \\nu ( \\Omega _ \\infty ) \\ln ( E _ { n _ i } ) - \\int _ { \\Omega } b _ { n _ i } ( \\omega ) \\ , \\nu ( \\mathrm { d } \\omega ) \\Big ) . \\end{align*}"}
+{"id": "2408.png", "formula": "\\begin{align*} { \\rm e v } _ { 0 } ^ { \\mathfrak { m } } ( \\psi ) & = \\sum _ { n , m = 0 } ^ { \\infty } \\sum _ { \\substack { a _ { 1 } , \\dots , a _ { n } \\in \\{ 0 , z \\} \\\\ b _ { 1 } , \\dots , b _ { m } \\in \\{ 1 , 1 - z \\} } } \\big ( 1 \\otimes 1 \\otimes { \\rm e v } _ { 0 } ^ { \\mathfrak { m } } ( e _ { a _ { 1 } } \\cdots e _ { a _ { n } } ) { \\rm e v } _ { 0 } ^ { \\mathfrak { m } } ( e _ { b _ { 1 } } \\cdots e _ { b _ { m } } ) \\big ) \\cdot \\tilde { L } \\big ( { \\rm C u t } _ { ( a _ { 1 } , \\dots , a _ { n } ; b _ { 1 } , \\dots , b _ { m } ) } ( \\psi ) \\mid _ { z \\to 0 } \\big ) \\end{align*}"}
+{"id": "2217.png", "formula": "\\begin{gather*} \\dot { \\xi } = - 2 Q _ j \\xi ^ 2 + s _ j \\xi \\end{gather*}"}
+{"id": "898.png", "formula": "\\begin{align*} S _ k ^ { i j } ( T _ \\alpha u ) _ { i j } = T _ \\alpha f . \\end{align*}"}
+{"id": "1529.png", "formula": "\\begin{align*} p = \\sum _ { i = 0 } ^ { r } q _ { i } x _ { n + 1 } ^ { i } \\end{align*}"}
+{"id": "1977.png", "formula": "\\begin{align*} \\delta _ { \\square } ( U , W ) : = \\inf _ { \\phi } \\| U - W ^ { \\phi } \\| _ { \\square } , \\end{align*}"}
+{"id": "999.png", "formula": "\\begin{align*} \\alpha _ 1 = { a _ 1 } , \\qquad \\alpha _ 2 = { a _ 2 } , \\alpha _ 3 = \\frac { 2 } { 3 } ( 4 \\ , a _ 3 - a _ 1 ) . \\end{align*}"}
+{"id": "6258.png", "formula": "\\begin{align*} \\sum _ { j _ 1 = 1 } ^ m \\sum _ { j _ 2 \\cdots j _ \\ell = 0 } \\prod _ { r = 1 } ^ \\ell \\delta _ { j _ r } ( A ) = \\sum _ { j _ 1 = 1 } ^ n \\delta _ { j _ 1 } ( A ) \\sum _ { j _ 2 \\cdots j _ \\ell = 0 } \\prod _ { r = 2 } ^ \\ell \\delta _ { j _ r } ( A ) = ( 1 - \\delta ( A ) ) \\sum _ { j _ 2 \\cdots j _ \\ell = 0 } \\prod _ { r = 2 } ^ \\ell \\delta _ { j _ r } ( A ) \\end{align*}"}
+{"id": "4867.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( 2 x ) ^ { 2 n } } { n \\binom { 2 n } { n } } = \\frac { 2 x \\arcsin ( x ) } { \\sqrt { 1 - x ^ 2 } } ( | x | < 1 ) \\end{align*}"}
+{"id": "5883.png", "formula": "\\begin{align*} d \\bar { \\tilde { v } } _ { n } ^ { 1 } \\wedge d \\tilde { v } _ { n } ^ { 1 } = - d \\bar { \\tilde { v } } _ { n } ^ { 3 } \\wedge d \\tilde { v } _ { n } ^ { 3 } , \\ \\ \\ d \\bar { \\tilde { v } } _ { n } ^ { 2 } \\wedge d \\tilde { v } _ { n } ^ { 2 } = 0 , \\end{align*}"}
+{"id": "4819.png", "formula": "\\begin{align*} \\frac \\dd { \\dd t } ( X _ t , S _ t ) \\# \\rho _ t ^ n = ( X _ t , S _ t ) \\# \\left ( ( \\Delta f ^ n - K ) \\rho _ t ^ n \\right ) + K ( X _ t , S _ t ) \\# ( \\bar \\rho _ t ^ n \\otimes \\mu ) \\eqqcolon A ( t , ( X _ t , S _ t ) \\# \\rho _ t ^ n ) . \\end{align*}"}
+{"id": "524.png", "formula": "\\begin{align*} 0 \\le V _ { \\mathrm { o r i g } } - V _ { \\mathrm { d e t } } \\le \\frac { n T ^ 2 } { 2 } \\sum _ { i , j = 1 } ^ n \\int _ { \\R ^ n } | \\partial _ { i j } g | ^ 2 \\ , d \\gamma _ { y ^ * , T } , \\end{align*}"}
+{"id": "5092.png", "formula": "\\begin{align*} \\| u \\| _ { Z _ T } = \\| u \\| _ { X _ T } + \\| u \\| _ { Y _ T ^ 1 } + \\| u \\| _ { Y _ T ^ 2 } . \\end{align*}"}
+{"id": "4016.png", "formula": "\\begin{align*} \\bar { q } _ { \\beta } ( n , t ) & = \\sum _ { k = 1 } ^ { n } \\mathrm { P r } \\{ X _ { 1 } + X _ { 2 } + \\dots + X _ { k } = n \\} \\mathrm { P r } \\{ N _ { \\beta } ( t ) = k \\} \\\\ & = \\sum _ { k = 1 } ^ { n } \\sum _ { \\Theta _ { n } ^ { k } } k ! \\prod _ { j = 1 } ^ { n } \\frac { \\left ( \\rho ^ { j } \\binom { r + j - 1 } { j } \\right ) ^ { x _ { j } } } { x _ { j } ! } \\left ( \\frac { \\lambda t ^ { \\beta } ( 1 - \\rho ) ^ { r } } { 1 - ( 1 - \\rho ) ^ { r } } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } ( - \\lambda t ^ { \\beta } ) , \\end{align*}"}
+{"id": "658.png", "formula": "\\begin{align*} \\nabla _ X \\psi _ 1 = \\frac { 1 } { 2 } X _ 1 \\cdot \\psi _ 1 - \\frac { 1 } { 2 } S ( X ) \\cdot N \\cdot \\psi _ 1 \\end{align*}"}
+{"id": "4322.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\left | \\mathbb { P } ( Z _ t > j ) - \\mathbb { P } ( Z _ { t + 1 } > j ) \\right | = \\mathbb { E } Z _ { t + 1 } - \\mathbb { E } Z _ t , \\end{align*}"}
+{"id": "3136.png", "formula": "\\begin{align*} a _ k = F P \\left ( \\sigma ^ k \\right ) \\end{align*}"}
+{"id": "3419.png", "formula": "\\begin{align*} \\begin{aligned} J _ 1 \\sum _ { i \\geq 0 } ( ( { - D _ r } ) ^ { i + 1 } f ' ( J _ { k + 1 } ) ) E ^ { ( i ) } _ S ( J _ k ) & = - D _ r f ' ( J _ { k + 1 } ) \\big ( J _ { k + 1 } + D _ r E _ { \\tilde \\rho } ( J _ k ) \\big ) \\\\ & + \\sum _ { i \\geq 1 } ( ( { - D _ r } ) ^ { i + 1 } f ' ( J _ { k + 1 } ) ) \\big ( D _ r E ^ { ( i ) } _ { \\tilde \\rho } ( J _ k ) - E ^ { ( i - 1 ) } _ { \\tilde \\rho } ( J _ k ) \\big ) \\end{aligned} \\end{align*}"}
+{"id": "4237.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( \\frac { b } { a } ) _ { n } ( q ) _ { n } ( a ) _ { N - n } a ^ { n } } { ( 1 - c q ^ { n } ) ( b ) _ n ( a ) _ N } = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( \\frac { b } { c } ) _ { n - 1 } ( q ) _ n ( c q ) _ { N - n } c ^ { n - 1 } } { ( b ) _ { n - 1 } ( c q ) _ N } \\left ( \\frac { a q ^ { n - 1 } } { 1 - a q ^ { n - 1 } } - \\frac { b q ^ { n - 1 } } { 1 - b q ^ { n - 1 } } \\right ) , \\end{align*}"}
+{"id": "6986.png", "formula": "\\begin{align*} - \\Delta u = \\alpha _ 2 u \\log u + ( \\lambda - \\alpha _ 1 ) u . \\end{align*}"}
+{"id": "3165.png", "formula": "\\begin{align*} q ( x ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { l } \\max ^ { ~ ~ ~ ~ 2 } \\{ x _ i , 0 \\} + \\frac { 1 } { 2 } \\sum _ { j = l + 1 } ^ { m } x _ j ^ 2 \\end{align*}"}
+{"id": "7920.png", "formula": "\\begin{align*} \\delta _ \\mathrm { R B A } ( u ) = ~ & \\big ( \\delta _ \\mathrm { H o c h } ( u ) , - u \\big ) , \\\\ \\delta _ \\mathrm { R B A } ( f , g ) = ~ & \\big ( \\delta _ \\mathrm { H o c h } ( f ) , ~ - \\overline { \\delta } _ \\mathrm { H o c h } ( g ) - \\Phi ^ n ( f ) \\big ) , \\end{align*}"}
+{"id": "2843.png", "formula": "\\begin{align*} ( B _ s ) _ Q ^ e = ( \\delta _ s \\otimes 1 - 1 \\otimes s ( \\delta _ s ) ) \\cdot Q , ( B _ s ) _ Q ^ s = ( \\delta _ s \\otimes 1 - 1 \\otimes \\delta _ s ) \\cdot Q . \\end{align*}"}
+{"id": "7570.png", "formula": "\\begin{align*} E _ { \\varphi } ( F _ { \\varepsilon , h } ) - E _ { \\varphi } ( F ) = E _ { \\varphi } [ \\mu _ { \\varepsilon , h } ^ F ] - E _ { \\varphi } [ \\mu ] + o ( \\varepsilon ) \\end{align*}"}
+{"id": "1746.png", "formula": "\\begin{align*} w _ { n , k , m } \\left ( Y , \\mathcal { Z } \\right ) = { \\left \\lfloor \\frac { k + m - 1 } { 2 } \\right \\rfloor \\choose \\frac { m } { 2 } } . \\end{align*}"}
+{"id": "7280.png", "formula": "\\begin{align*} \\Psi _ L ( T , \\gamma ) = - \\nabla _ x \\sigma ( \\gamma ( T ) , m ^ * ( T ) ) - \\int _ \\Omega \\nabla _ m \\sigma ( \\gamma ' ( T ) , m ^ * ( T ) , \\gamma ( T ) ) \\eta ^ * ( d \\gamma ' ) , \\end{align*}"}
+{"id": "5607.png", "formula": "\\begin{align*} \\hat { \\mathbb { G } } ^ i _ { j k } = \\frac { \\partial ^ 2 \\hat { \\mathbb { G } } ^ i } { \\partial y ^ j \\partial y ^ k } = \\frac { \\partial \\hat { \\mathbb { G } } ^ i _ j } { \\partial y ^ k } . \\end{align*}"}
+{"id": "672.png", "formula": "\\begin{align*} g _ 2 = - \\frac { 1 } { \\nu } I \\widehat { g _ 1 } I + \\frac { 2 i } { \\sqrt { \\tau _ 0 } } h _ z J g _ 1 . \\end{align*}"}
+{"id": "1760.png", "formula": "\\begin{align*} { \\rm s i g n } \\left \\langle Z _ { * } , I \\backslash \\left \\{ i _ { j } + \\epsilon \\right \\} \\right \\rangle = \\left ( - 1 \\right ) ^ { k + i _ { j } + \\left ( 1 - \\epsilon \\right ) } , \\end{align*}"}
+{"id": "2928.png", "formula": "\\begin{align*} P _ \\rho = \\frac { 1 } { 2 \\pi i } \\int _ { \\gamma _ \\rho } ( \\lambda - A ) ^ { - 1 } d \\lambda . \\end{align*}"}
+{"id": "241.png", "formula": "\\begin{align*} \\nu ( x ) = \\nu [ \\phi ] ( x ) \\stackrel { d e f } { = } \\sup _ { \\lambda | : | \\lambda \\le \\lambda _ 0 } ( \\lambda x - \\phi ( \\lambda ) ) = \\phi ^ * ( x ) . \\end{align*}"}
+{"id": "467.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } L ( z ) e ^ { - \\frac { L _ j } { 2 } z } = { \\coth \\Big ( \\frac { \\sqrt 5 F _ j } { 2 } z \\Big ) } \\sinh \\Big ( \\frac { \\sqrt 5 F _ j } { 2 } z \\Big ) . \\end{align*}"}
+{"id": "1983.png", "formula": "\\begin{align*} E ( H ) = E ( G ) \\setminus \\{ u w : w \\in N _ G ( u ) \\} \\cup \\{ u w : w \\in N _ G ( v ) \\} . \\end{align*}"}
+{"id": "3669.png", "formula": "\\begin{align*} x _ i = \\begin{cases} \\frac { 1 } { t + 2 } \\pm \\epsilon & i \\in [ t ] , \\\\ \\frac { \\alpha } { t + 2 } \\pm \\epsilon & i \\in \\{ t + 1 , t + 4 \\} , \\\\ \\frac { 1 - \\alpha } { t + 2 } \\pm \\epsilon & i \\in \\{ t + 2 , t + 3 \\} . \\end{cases} \\end{align*}"}
+{"id": "8182.png", "formula": "\\begin{align*} \\alpha a _ 1 \\overline { b _ 1 } = 0 , \\end{align*}"}
+{"id": "1306.png", "formula": "\\begin{align*} ( f \\oplus g ) \\circ \\iota ^ 1 _ { a , b } = \\iota ^ 1 _ { a ' , b ' } \\circ f , ( f \\oplus g ) \\circ \\iota ^ 2 _ { a , b } = \\iota ^ 2 _ { a ' , b ' } \\circ g , \\end{align*}"}
+{"id": "3574.png", "formula": "\\begin{align*} \\hat { C } \\coloneqq \\frac { 1 } { n } \\sum _ { i = 1 } ^ n X _ i \\otimes X _ i \\quad \\quad \\hat { R } \\coloneqq \\frac { 1 } { n } \\sum _ { i = 1 } ^ n Y _ i X _ i , \\end{align*}"}
+{"id": "5905.png", "formula": "\\begin{align*} \\langle \\lambda _ { - 2 } , \\alpha _ 3 ^ \\vee \\rangle = - 2 \\ , , \\langle \\lambda _ { - 2 } , \\alpha _ 4 ^ \\vee \\rangle = 0 \\ , , \\langle \\alpha _ 3 , \\alpha _ 4 ^ \\vee \\rangle = - 1 \\ , . \\end{align*}"}
+{"id": "1094.png", "formula": "\\begin{align*} F ( a , b , c ; z ) & = \\sum _ { i = 0 } ^ \\infty \\frac { ( a ) _ i ( b ) _ i } { ( c ) _ i } \\frac { z ^ i } { i ! } \\\\ & = 1 + \\frac { a b } { c } \\frac { z } { 1 ! } + \\frac { a ( a + 1 ) b ( b + 1 ) } { c ( c + 1 ) } \\frac { z ^ 2 } { 2 ! } + \\cdots . \\end{align*}"}
+{"id": "4630.png", "formula": "\\begin{align*} \\phi ^ * \\big ( x , \\epsilon ^ { \\frac 1 { q ' } } L _ p ^ { - 1 } \\tfrac { \\phi ( x , | \\nabla \\tilde u | ) } { | \\nabla \\tilde u | } \\big ) \\le L _ q \\epsilon \\phi ^ * \\big ( x , \\tfrac { \\phi ( x , | \\nabla \\tilde u | ) } { L _ p | \\nabla \\tilde u | } \\big ) & \\le \\tfrac { L _ q \\epsilon } { L _ p } \\phi ( x , | \\nabla u | ) = \\tfrac { L _ q } { K } \\eta ( x ) \\phi ( x , | \\nabla u | ) . \\end{align*}"}
+{"id": "1461.png", "formula": "\\begin{align*} \\Phi _ { \\beta , \\varepsilon } ( t , x ; t _ 0 ) & \\ge k _ { \\beta , \\varepsilon } ( t _ 0 + t ) ^ { - \\beta } \\left ( 1 + \\frac { \\widetilde { \\gamma } _ { \\varepsilon } A _ { \\varepsilon } ( x ) } { t _ 0 + t } \\right ) ^ { - \\beta } \\\\ & \\ge c \\left ( t _ 0 + t + A _ { \\varepsilon } ( x ) \\right ) ^ { - \\beta } \\\\ & = c \\Psi ( t , x ; t _ 0 ) ^ { - \\beta } \\end{align*}"}
+{"id": "1919.png", "formula": "\\begin{align*} ( P _ 0 + \\lambda ) u _ n = _ v \\vec f _ n . \\end{align*}"}
+{"id": "7893.png", "formula": "\\begin{align*} P ( a ) \\cdot Q ( u ) = ~ & Q \\big ( P ( a ) \\cdot u + a \\cdot Q ( u ) \\big ) + \\lambda ~ \\ ! Q ( a \\cdot u ) , \\\\ Q ( u ) \\cdot P ( a ) = ~ & Q \\big ( Q ( u ) \\cdot a + u \\cdot P ( a ) \\big ) + \\lambda ~ \\ ! Q ( u \\cdot a ) , \\end{align*}"}
+{"id": "4414.png", "formula": "\\begin{align*} | \\kappa | | q | = - a \\pi _ 1 ( | \\kappa | ) = ( 1 - a ) \\pi _ 2 ( | \\kappa | ) . \\end{align*}"}
+{"id": "2135.png", "formula": "\\begin{align*} \\mathrm { c o r a n k } _ { \\Omega ( G ) } ( H ^ 1 ( G _ { \\infty } ^ { S } , E _ { p } ) ) = \\mathrm { c o r a n k } _ { \\Omega ( G ) } ( \\displaystyle \\bigoplus _ { v \\in S } { ^ { \\pm } \\widetilde { K } _ v ( E _ { p } / L _ \\infty ) } ) = 2 . \\end{align*}"}
+{"id": "7540.png", "formula": "\\begin{align*} E _ { \\varphi } ( F ) = \\max _ { F ' \\in \\mathcal F } E _ { \\varphi } ( F ' ) . \\end{align*}"}
+{"id": "4387.png", "formula": "\\begin{align*} P ( r , \\vartheta , \\varphi ) = \\frac { \\lambda ( 1 - r ^ 2 ) } { \\pi } \\int _ 0 ^ \\pi \\frac { \\sin ^ { 2 \\lambda - 1 } \\xi } { \\left ( 1 - 2 r ( \\cos \\vartheta \\cos \\varphi + \\sin \\vartheta \\sin \\varphi \\cos \\xi ) + r ^ 2 \\right ) ^ { \\lambda + 1 } } d \\xi . \\end{align*}"}
+{"id": "1857.png", "formula": "\\begin{align*} z _ { i , j } & = x _ { ( 1 - \\alpha ( i ) ) i + \\alpha ( i ) ( m + 1 - i ) , j } , \\mbox { \\ \\ a n d } \\\\ z _ { m + 1 - i , j } & = x _ { \\alpha ( i ) i + ( 1 - \\alpha ( i ) ) ( m + 1 - i ) , j } . \\end{align*}"}
+{"id": "3303.png", "formula": "\\begin{align*} \\langle v _ 1 \\otimes v _ 2 , v _ 1 ' \\otimes v _ 2 ' \\rangle _ { V _ 1 \\otimes V _ 2 } = \\langle v _ 1 , v _ 1 ' \\rangle _ { V _ 1 } \\langle v _ 2 , v _ 2 ' \\rangle _ { V _ 2 } , v _ 1 , v _ 1 ' \\in V _ 1 , v _ 2 , v _ 2 ' \\in V _ 2 . \\end{align*}"}
+{"id": "7023.png", "formula": "\\begin{align*} \\varphi ( t , x ) : = J _ 0 ( t , x ) P _ { u _ 1 } + \\left \\langle \\nabla J _ 0 ( t , x ) , M _ { u _ 1 } \\right \\rangle + H ( t , x ) P _ { u _ 1 } + J _ 1 ( t , x ) P _ { u _ 0 } - \\sigma ^ { - 1 } \\Delta J _ 0 ( t , x ) P _ { \\theta _ 0 } , \\end{align*}"}
+{"id": "4257.png", "formula": "\\begin{align*} { } _ { 2 } \\phi _ { 1 } \\left [ \\begin{matrix} a , q ^ { - N } \\\\ x \\end{matrix} \\ , ; q , q \\right ] = \\frac { \\left ( \\frac { x } { a } \\right ) _ N a ^ N } { ( x ) _ N } \\end{align*}"}
+{"id": "8186.png", "formula": "\\begin{align*} \\mathcal { D } ( u , \\rho ) ( t , x ) = - \\rho ( t , x ) \\int _ { \\R ^ N } \\phi ( x - y ) \\big ( u ( t , x ) - u ( t , y ) \\big ) \\rho ( t , y ) \\dd y . \\end{align*}"}
+{"id": "96.png", "formula": "\\begin{align*} \\| u \\| _ { C ^ { 2 } _ { r a d } } : = \\sup _ { A ( r _ { 1 } , r _ { 2 } ) } | u | + \\sup _ { A ( r _ { 1 } , r _ { 2 } ) } | \\partial _ { r } u | + \\sup _ { A ( r _ { 1 } , r _ { 2 } ) } | \\partial ^ { 2 } _ { r } u | < + \\infty , \\end{align*}"}
+{"id": "1175.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { j = 0 } ^ { k } \\dbinom { k } { j } ( 1 - q ) ^ { k - j } q ^ j \\dbinom { k } { j } \\Big ( \\frac { 1 } { 2 } \\Big ) ^ k \\bigg ) ^ { 1 / p _ k } \\leq \\Big ( \\frac { 1 - q } { 2 } \\Big ) ^ { k / p _ k } + \\Big ( \\frac { q } { 2 } \\Big ) ^ { k / p _ k } \\end{align*}"}
+{"id": "5158.png", "formula": "\\begin{align*} B \\left ( x _ { \\widetilde Q _ 1 } , \\sqrt { n } c \\ell ( \\widetilde Q _ 1 ) \\right ) \\cap B \\left ( x _ { \\widetilde Q _ 2 } , \\sqrt { n } c \\ell ( \\widetilde Q _ 2 ) \\right ) = \\emptyset \\end{align*}"}
+{"id": "7286.png", "formula": "\\begin{align*} \\frac { d } { d t } P _ 2 ( t ) = - P _ 2 ( t ) A ( t ) - A ^ T ( t ) P _ 2 ( t ) + P _ 2 ( t ) B ( t ) R ^ { - 1 } ( t ) B ( t ) P _ 2 ( t ) - Q _ x ( t ) \\end{align*}"}
+{"id": "2958.png", "formula": "\\begin{align*} \\Phi ( y ) : = \\{ z \\in \\R ^ \\ell \\ , | \\ , G ( z ) \\in D , \\ , H ( z ) = y \\} \\qquad \\forall y \\in \\R ^ m , \\end{align*}"}
+{"id": "3752.png", "formula": "\\begin{align*} ( ( x + 4 ) I + A [ \\mathcal C _ i ] ) ^ { - 1 } = - \\frac { 1 } { m _ i ( - x - 4 ) } \\sum _ { k = 1 } ^ { \\deg m _ i } D ^ k m _ i ( - x - 4 ) A [ \\mathcal C _ i ] ^ { k - 1 } \\end{align*}"}
+{"id": "7493.png", "formula": "\\begin{align*} c _ { \\mu \\mu ' } & = \\frac { 1 } { a _ { \\mu \\mu } } \\left ( \\delta _ { \\mu \\mu ' } - \\sum _ { \\kappa = 2 } ^ { \\mu - 1 } a _ { \\mu \\kappa } c _ { \\kappa \\mu ' } \\right ) \\ , , \\\\ & = ( \\mu - 2 ) ! \\left ( \\delta _ { \\mu \\mu ' } - \\sum _ { \\kappa = 2 } ^ { \\mu - 1 } a _ { \\mu \\kappa } c _ { \\kappa \\mu ' } \\right ) \\ , . \\end{align*}"}
+{"id": "5246.png", "formula": "\\begin{align*} F _ T ( \\alpha ) = T ^ { - 1 } \\ * \\sum _ { 0 < \\tilde { \\gamma } _ j , \\tilde { \\gamma _ k } \\leq T } \\exp ( i \\ * \\alpha \\ * ( \\tilde { \\gamma } _ j - \\tilde { \\gamma } _ k ) ) \\ * \\frac { 4 } { 4 + ( \\tilde { \\gamma } _ j - \\tilde { \\gamma } _ k ) ^ 2 / \\log ( T ) ^ 2 } , \\end{align*}"}
+{"id": "1917.png", "formula": "\\begin{align*} & \\int ( D _ v u ) ( - \\partial _ t U + v \\cdot D _ x U - a ^ { i j } ( t ) D _ { v _ i v _ j } U + \\lambda U ) \\ , d z \\\\ & = \\int ( D _ v ^ 2 U ) \\vec f \\ , d z - \\int u D _ x U \\ , d z = : J _ 1 + J _ 2 . \\end{align*}"}
+{"id": "4516.png", "formula": "\\begin{align*} \\{ \\Pi _ y , \\mathcal { K } \\} = - \\dd \\mathcal { K } ( \\mathbb { X } _ { \\Pi _ { \\ ! y } } \\ ! ) \\ , , \\end{align*}"}
+{"id": "4138.png", "formula": "\\begin{align*} f ( k , n , c , \\alpha ) d _ k ^ n = a _ { k , \\alpha } d _ { k - 1 } ^ n + a _ { k + 1 , \\alpha } d _ { k + 1 } ^ n , ~ \\forall k \\ge 0 , \\end{align*}"}
+{"id": "6178.png", "formula": "\\begin{align*} \\sum _ { p \\ , \\leq \\ , x } ( \\log g _ { u } ( p - 1 ) ) ^ { \\lambda } & = \\sum _ { \\substack { d \\ , \\le \\ , z \\\\ ( d , \\ , a _ { 2 } ) \\ , = \\ , 1 } } \\rho _ { \\lambda } ( d ) \\pi ( x ; \\l ( d ) , 1 ) + \\sum _ { \\substack { d \\ , \\ge \\ , z \\\\ ( d , \\ , a _ { 2 } ) \\ , = \\ , 1 } } \\rho _ { \\lambda } ( d ) \\pi ( x ; \\l ( d ) , 1 ) \\\\ & = E ' _ { 1 } + E ' _ { 2 } . \\end{align*}"}
+{"id": "4965.png", "formula": "\\begin{align*} \\begin{aligned} \\min \\limits _ { w _ { c } , r } & [ w _ { c } , r ] \\\\ s . t . & { \\rm C _ { 1 } } \\sim { \\rm C _ { 2 } } , \\\\ & { \\rm C _ { 4 } } : \\ g ( \\lambda ^ { \\star } , w _ { c } , r ) \\leq \\ln \\tau \\end{aligned} \\end{align*}"}
+{"id": "6719.png", "formula": "\\begin{align*} \\overline { M } \\equiv M _ { [ \\bar { \\rho } , \\bar { u } , \\bar { \\theta } ] } ( t , x , v ) : = \\frac { \\bar { \\rho } ( t , x ) } { \\sqrt { ( 2 \\pi R \\overline { \\theta } ( t , x ) ) ^ { 3 } } } \\exp \\big \\{ - \\frac { | v - \\overline { u } ( t , x ) | ^ 2 } { 2 R \\overline { \\theta } ( t , x ) } \\big \\} , \\end{align*}"}
+{"id": "2123.png", "formula": "\\begin{align*} \\mu _ G ^ { \\pm / \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) : = \\mu _ G ( \\mathfrak { X } ^ { \\pm / \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) ) , \\end{align*}"}
+{"id": "2541.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { \\xi , \\alpha } = & ~ [ A x _ t ^ { \\xi , \\alpha } + B \\alpha _ t + f ( \\nu _ t ) + b ( \\mu _ t ) ] d t + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ 0 ^ { \\xi , \\alpha } = & ~ \\xi , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1508.png", "formula": "\\begin{align*} c = L ( 1 , \\chi ) \\varphi ( q ) / q \\gg \\varphi ( q ) q ^ { - \\tfrac 3 2 } ( \\log q ) ^ { - 2 } , \\end{align*}"}
+{"id": "8235.png", "formula": "\\begin{align*} \\sigma \\in \\widetilde { L } ^ { \\infty } ( \\mathbb { R } ^ { + } ; \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } ) , u \\in \\widetilde { L } ^ { \\infty } ( \\mathbb { R } ^ { + } ; \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } ) \\cap L ^ { 1 } ( \\mathbb { R } ^ { + } ; { \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } } ) . \\end{align*}"}
+{"id": "6339.png", "formula": "\\begin{align*} \\int _ \\Omega \\nabla u \\cdot \\nabla \\zeta \\ , d x - \\int _ \\Omega \\left ( b u ^ + - a u ^ - \\right ) \\zeta \\ , d x = \\int _ \\Omega \\nabla z _ u \\cdot \\nabla \\zeta \\ , d x \\forall \\zeta \\in H ^ 1 _ 0 ( \\Omega ) , \\end{align*}"}
+{"id": "4343.png", "formula": "\\begin{align*} \\lim _ { \\eta \\to 0 } H _ \\eta ( u _ 1 ( t ) | u _ 2 ( t ) ) = H ( u _ 1 ( t ) | u _ 2 ( t ) ) \\ , , t \\in [ 0 , T ] \\ , . \\end{align*}"}
+{"id": "6085.png", "formula": "\\begin{align*} \\mathbb { H } _ 3 ( R ) = \\left \\{ \\left ( \\begin{array} { c c c } 1 & x & z \\\\ 0 & 1 & y \\\\ 0 & 0 & 1 \\end{array} \\right ) : x , y , z \\in R \\right \\} . \\end{align*}"}
+{"id": "5147.png", "formula": "\\begin{align*} a _ i = \\dfrac { 1 } { | Q _ i | } \\int _ { Q _ i } \\chi _ { F } ( x ) \\ , d x = \\dfrac { | F \\cap Q _ i | } { | Q _ i | } . \\end{align*}"}
+{"id": "2657.png", "formula": "\\begin{align*} R = \\frac { 1 } { 2 J ( X _ { J } ) } \\left [ 2 \\delta ( X _ { J } ) g \\wedge S + 2 \\omega ( X _ { J } ) g \\wedge g + E \\wedge g + F \\wedge S \\right ] , \\end{align*}"}
+{"id": "7752.png", "formula": "\\begin{align*} d i s t _ { g ^ + } ( y , q ) = t , \\ \\ \\ \\ d i s t _ { g ^ + } ( p _ 0 , q ) = d i s t _ { \\mathbb { S } ^ 1 ( \\lambda ) } ( \\theta , \\theta _ 0 ) = \\lambda | \\theta - \\theta _ 0 | . \\end{align*}"}
+{"id": "4687.png", "formula": "\\begin{align*} c _ n = \\int _ { - \\pi } ^ { \\pi } f ( e ^ { i \\varphi } ) \\overline { \\phi _ { n } ( e ^ { i \\varphi } ) } d m _ { \\lambda } ( \\varphi ) , n = 0 , 1 , \\cdots . \\end{align*}"}
+{"id": "63.png", "formula": "\\begin{align*} H _ { \\Psi } = H - g ( \\nabla \\Psi , \\vec \\nu ) \\end{align*}"}
+{"id": "4901.png", "formula": "\\begin{align*} P ( x , t ) = \\sum _ { n = 1 } ^ \\infty \\left ( 2 ^ { n - 1 } \\sum _ { k = 0 } ^ { n - 1 } { n \\choose k } F _ { n - k - 1 } ( x , 1 / 2 ) F _ k ( x , 1 / 2 ) \\right ) \\frac { t ^ n } { n ! } , \\end{align*}"}
+{"id": "6151.png", "formula": "\\begin{align*} { \\mathfrak r } V _ 1 V _ 2 = q \\cdot V _ 1 V _ 2 { \\mathfrak r } . \\end{align*}"}
+{"id": "3447.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\mathrm { E } | ^ 2 ( \\mathrm { y } ) d \\mathrm { y } = \\dfrac { 1 } { | 1 - \\alpha \\lambda ^ { ( 3 ) } _ { \\mathrm { n } _ { 0 } } | ^ 2 } \\Bigg [ | \\mathrm { E } ^ { \\textbf { i n } } | ^ 2 ( \\mathrm { z } ) \\Big ( \\int _ \\mathrm { \\Omega } \\mathrm { e } ^ { ( 3 ) } _ { \\mathrm { n } _ { 0 } } ( \\mathrm { x } ) d \\mathrm { x } \\Big ) ^ 2 + \\mathcal { O } \\Big ( \\delta ^ { 3 } \\Big ) \\Bigg ] \\mathrm { h } < 1 , \\end{align*}"}
+{"id": "1560.png", "formula": "\\begin{align*} \\tilde { \\rho } \\left ( v \\right ) = \\left ( \\prod _ { i = 1 } ^ { N } z _ { i } ^ { v _ { i } } \\right ) \\cdot I _ { n } . \\end{align*}"}
+{"id": "5632.png", "formula": "\\begin{align*} I _ \\gamma ( W , W ) & = \\int _ 0 ^ d \\Big [ g _ T ( D _ T ^ T W , D _ T ^ T W ) - g _ T ( R ( W , T ) T , W ) \\Big ] d t \\end{align*}"}
+{"id": "7093.png", "formula": "\\begin{gather*} w = \\frac { k _ 2 k _ 3 } { y ^ 2 } , y w = \\frac { k _ 2 k _ 3 } { y } , y ^ 2 = \\frac { k _ 2 w } { y } . \\end{gather*}"}
+{"id": "8246.png", "formula": "\\begin{align*} R _ j ( t ) : = \\Vert f _ j ( t ) \\Vert _ { L ^ 2 } + \\Vert g _ j ( t ) \\Vert _ { L ^ 2 } + \\Vert \\widetilde { g } _ j ( t ) \\Vert _ { L ^ 2 } + \\Vert \\nabla \\dot S _ { j - 1 } u ( t ) \\Vert _ { L ^ { \\infty } } X _ j ( t ) . \\end{align*}"}
+{"id": "1667.png", "formula": "\\begin{align*} \\mathtt { R } = \\left [ \\mathbf { 1 } + \\lambda \\mathtt { G } + \\lambda ^ 2 \\mathtt { H } + \\lambda ^ 3 \\mathtt { I } \\right ] ^ { - 1 } \\left ( \\left [ \\mathtt { G } + \\lambda \\mathtt { H } + \\lambda ^ 2 \\mathtt { I } \\right ] \\left [ \\mathtt { H } - \\mathtt { G } ^ 2 \\right ] + \\left [ \\mathtt { H } + \\lambda \\mathtt { I } \\right ] \\mathtt { G } - \\mathtt { I } \\right ) \\ , . \\end{align*}"}
+{"id": "5213.png", "formula": "\\begin{align*} C _ j < \\sum _ { i = 1 } ^ { s - 1 } \\left ( \\sum _ { k = j _ i } ^ { j _ { i + 1 } - 1 } ( p _ k + L + p _ { j _ { i + 1 } - 1 } ) \\right ) + \\sum _ { k = j _ s } ^ j p _ k + L + p _ j \\leq \\sum _ { i = 1 } ^ j p _ i + \\sum _ { i = 1 } ^ { s - 1 } p _ { j _ { i + 1 } - 1 } + s L + p _ j . \\end{align*}"}
+{"id": "3627.png", "formula": "\\begin{align*} f ( x ; L _ 1 , L _ 2 , L _ 3 , b ) = K _ 1 ( x - \\rho ) ^ 2 + K _ 2 ( x - \\rho ) + K _ 3 + L _ 1 e ^ { ( \\theta + c _ 1 ) x } + L _ 2 e ^ { ( \\theta + c _ 2 ) x } + L _ 3 e ^ { ( \\theta + c _ 3 ) x } . \\end{align*}"}
+{"id": "3456.png", "formula": "\\begin{align*} \\Big | u \\Big | _ { \\mathrm { H } ^ { \\mathrm { s } } \\Big ( \\mathbb { R } ; \\mathrm { L } ^ 2 ( \\partial \\Omega ) \\Big ) } ^ 2 : = \\int _ \\mathbb { R } \\int _ \\mathbb { R } \\dfrac { \\Vert u ( \\cdot , t ) - u ( \\cdot , \\tau ) \\Vert _ { \\mathrm { L } ^ 2 ( \\partial \\Omega ) } ^ 2 } { | t - \\tau | ^ { 1 + 2 \\mathrm { s } } } d t d \\tau . \\end{align*}"}
+{"id": "5169.png", "formula": "\\begin{align*} h _ { n , i } \\left ( ( \\partial C _ { n - 1 } \\setminus \\partial T _ { n , i } ) \\cup \\overline { ( \\partial T _ { n , i } \\setminus \\partial C _ { n - 1 } ) } \\right ) = \\partial C _ { n - 1 } . \\end{align*}"}
+{"id": "2642.png", "formula": "\\begin{align*} E _ { d ^ * ( x ) } = E _ { d ^ * ( y ) } \\cup E _ { d ^ * ( y ' f ' ) } \\cup E _ { [ w b ^ { - 1 } , w a ] } . \\end{align*}"}
+{"id": "5777.png", "formula": "\\begin{align*} l _ { s , t } & \\le \\left \\lfloor \\frac { ( r ^ 2 + r + 1 ) [ ( r + 1 ) s + t ] } { a _ { s , t } } \\right \\rfloor \\le \\left \\lfloor \\frac { ( r ^ 2 + r + 1 ) [ ( r + 1 ) s + t } { ( r ^ 2 + r + 1 ) s } \\right \\rfloor \\\\ & = r + 1 . \\end{align*}"}
+{"id": "7279.png", "formula": "\\begin{align*} \\partial _ t m ( t ) + \\operatorname { d i v } ( v _ E ( t , x ) m ( t ) ) = 0 . \\end{align*}"}
+{"id": "7962.png", "formula": "\\begin{align*} t ' ( s ) = - 2 r ( s ) \\cos { \\left ( \\theta ( s ) \\right ) } = 2 \\varphi _ 0 - \\frac { 2 c } { 3 } r ^ 3 ( s ) . \\end{align*}"}
+{"id": "3527.png", "formula": "\\begin{align*} \\Big \\Vert \\partial _ { \\nu } \\Tilde { \\mathrm { H } } \\Big \\Vert _ { \\mathbb { H } ^ { \\frac { 1 } { 2 } } ( \\partial \\mathrm { B } ) } & = \\mathcal { O } \\Big ( 1 \\Big ) + \\mathcal { O } \\Big ( \\delta ^ 2 \\Big ) + \\mathcal { O } \\Big ( | \\log { \\delta } | \\delta ^ { 3 - h } \\Big ) + \\mathcal { O } \\Big ( \\delta ^ { 3 - h } \\Big ) + \\mathcal { O } \\Big ( \\delta ^ { 4 - h } \\Big ) \\\\ & = \\mathcal { O } \\Big ( 1 \\Big ) . \\end{align*}"}
+{"id": "4293.png", "formula": "\\begin{align*} \\frac { ( \\frac { c } { d } ) _ { \\infty } ( d q ) _ { \\infty } } { ( q ) _ { \\infty } ( c q ) _ { \\infty } } \\sum _ { k = 1 } ^ { \\infty } \\frac { d ^ k q ^ { k ( k + 1 ) } } { ( d q ) _ k ( q ) _ k ( 1 - q ^ k ) } \\sum _ { j = 0 } ^ { \\infty } \\frac { ( d q ) _ j ( c q ^ k / d ) ^ j } { ( d q ^ { k + 1 } ) _ j ( q ) _ j } = \\frac { ( d q ) _ { \\infty } } { ( q ) _ { \\infty } } \\sum _ { k = 1 } ^ { \\infty } \\frac { ( c / d ) _ k ( d q ) ^ k } { ( q ) _ k ( 1 - q ^ k ) } \\sum _ { j = 0 } ^ { \\infty } \\frac { ( d q ) _ j c ^ j q ^ { ( j + k ) ^ 2 } } { ( d q ) _ { j + k } ( c q ) _ { j + k } ( q ) _ j } . \\end{align*}"}
+{"id": "7364.png", "formula": "\\begin{align*} ~ \\hat { \\beta } _ { k + 1 } ^ \\mathrm { H Z } : = \\max \\{ \\beta _ { k + 1 } ^ \\mathrm { H Z } , \\zeta _ { k + 1 } \\} , \\zeta _ { k + 1 } : = - \\frac { 1 } { \\norm { \\eta _ { k + 1 } } _ { x _ { k + 1 } } \\min \\{ \\zeta , \\norm { g _ { k + 1 } } _ { x _ { k + 1 } } \\} } , \\end{align*}"}
+{"id": "4462.png", "formula": "\\begin{align*} & \\sum ( - 1 ) ^ { f _ \\sigma } \\cdots E _ { i j } E _ { k l } \\cdots - \\sum ( - 1 ) ^ { f _ { \\sigma ' } } \\cdots E _ { k l } E _ { i j } \\cdots \\\\ = & \\sum ( - 1 ) ^ { f _ { \\tilde \\sigma } } \\cdots E _ { i l } \\cdots - \\sum ( - 1 ) ^ { f _ { \\tilde \\sigma ' } } \\cdots E _ { k j } \\cdots ' \\end{align*}"}
+{"id": "3661.png", "formula": "\\begin{align*} \\mathcal { F } _ { M } ( \\mathcal { G } ) = \\left \\{ F \\in \\mathcal { F } _ { \\infty } ( \\mathcal { G } ) \\colon v ( F ) \\le M \\right \\} . \\end{align*}"}
+{"id": "3090.png", "formula": "\\begin{align*} A _ 3 = \\left ( s _ 3 ( k ) \\right ) _ { k = 1 } ^ { \\infty } = ( 0 , 0 , 3 , 4 , 5 , 9 , 7 , 1 2 , 1 2 , 1 5 , 1 9 , 2 5 , \\ldots ) . \\end{align*}"}
+{"id": "6387.png", "formula": "\\begin{align*} e _ { i j } : = \\begin{cases} \\delta _ { i j } & \\mbox { i f $ j \\neq k $ } , \\\\ - 1 & \\mbox { i f $ i = k = j $ } , \\\\ [ - \\epsilon b _ { i k } ] _ + & \\mbox { i f $ i \\neq k = j $ } , \\end{cases} \\end{align*}"}
+{"id": "452.png", "formula": "\\begin{align*} B _ n ( x ) = \\sum _ { k = 0 } ^ n { \\binom n k B _ { n - k } x ^ { k } } , \\end{align*}"}
+{"id": "2466.png", "formula": "\\begin{align*} - \\Delta F _ { B } + F _ { B } = 2 J ( u , B ) \\end{align*}"}
+{"id": "1576.png", "formula": "\\begin{align*} S _ { 1 } = S _ { 0 } \\cup T \\cup U . \\end{align*}"}
+{"id": "614.png", "formula": "\\begin{align*} [ C _ { 1 2 } \\ , , \\ , C _ { 3 4 } ] = 0 \\ , , [ C _ { 1 2 } \\ , , \\ , C _ { 1 2 3 } ] = 0 \\ , , [ C _ { 2 3 } \\ , , \\ , C _ { 1 2 3 } ] = 0 \\ , , [ C _ { 2 3 } \\ , , \\ , C _ { 2 3 4 } ] = 0 \\ , , [ C _ { 3 4 } \\ , , \\ , C _ { 2 3 4 } ] = 0 \\ , . \\end{align*}"}
+{"id": "1748.png", "formula": "\\begin{align*} c _ { n , k , m } \\left ( Y , \\mathcal { Z } \\right ) = \\begin{cases} 2 w _ { n , k , m + 1 } \\left ( Y , \\mathcal { Z } \\right ) - w _ { n , k , m - 1 } \\left ( Y , \\mathcal { Z } \\right ) & { \\rm f o r \\ ; } k \\ ; { \\rm o d d } , \\\\ 2 w _ { n , k , m + 1 } \\left ( Y , \\mathcal { Z } \\right ) & { \\rm f o r \\ ; } k \\ ; { \\rm e v e n } , \\end{cases} \\end{align*}"}
+{"id": "7026.png", "formula": "\\begin{align*} \\triangle _ { \\mathrm { d i s } } = - 4 \\left [ ( \\sigma + | \\xi | ^ 2 ) ^ 2 | \\xi | ^ 2 - 2 \\sqrt { 2 } | \\xi | ^ 6 \\right ] ^ 2 - ( 1 6 \\sqrt { 2 } - 1 3 ) ( \\sigma + | \\xi | ^ 2 ) ^ 2 | \\xi | ^ 8 < 0 . \\end{align*}"}
+{"id": "3703.png", "formula": "\\begin{align*} q _ t ^ b = X _ 0 + \\int _ 0 ^ t b ( q _ s ^ b , s ) d s + W _ t , \\mathbb P a . s . \\end{align*}"}
+{"id": "233.png", "formula": "\\begin{align*} \\log \\frac { m _ t ^ { ( n ) } ( x ) } { \\pi ( x ) } & = e ^ { - \\frac { \\sigma ^ 2 } { 2 } t } \\log \\frac { m _ 0 ( x ) } { \\pi ( x ) } \\\\ & - \\int _ 0 ^ t \\frac { \\sigma ^ 2 } { 2 } e ^ { - \\frac { \\sigma ^ 2 } { 2 } ( t - s ) } \\left ( \\frac { 2 } { \\sigma ^ 2 } \\frac { \\delta F } { \\delta m } ( m _ s ^ { ( n - 1 ) } , x ) - \\operatorname { K L } ( m _ s ^ { ( n - 1 ) } | \\pi ) \\right ) d s \\ , . \\end{align*}"}
+{"id": "3866.png", "formula": "\\begin{align*} | P _ d | \\prec \\Psi ^ d , n ^ { - 1 / 4 + \\epsilon } \\leq \\Psi = ( n \\eta ) ^ { - 1 } \\leq n ^ { - \\epsilon } . \\end{align*}"}
+{"id": "7198.png", "formula": "\\begin{align*} \\sum _ { d \\geq 0 } a _ d q ^ d & = \\sum _ { 0 \\leq v _ 1 / d _ 1 < \\cdots < v _ k / d _ k < 1 } \\prod _ { i = 1 } ^ k p _ 2 \\left ( \\gcd ( d _ i , v _ i ) \\right ) q ^ { d _ i } \\\\ & = \\prod _ { \\begin{subarray} { c } 0 \\leq \\mu < 1 \\\\ \\mu = a / b , \\ \\gcd ( a , b ) = 1 \\end{subarray} } \\prod _ { k \\geq 1 } \\frac { 1 } { 1 - q ^ { b k } } . \\end{align*}"}
+{"id": "23.png", "formula": "\\begin{align*} A _ { l , u } & = \\Big \\{ i \\in [ k ] : \\frac { p _ i } { q _ i } \\in [ l , u ) \\Big \\} , \\\\ A _ { l , \\infty } & = \\Big \\{ i \\in [ k ] : \\frac { p _ i } { q _ i } \\in [ l , \\infty ] \\Big \\} . \\end{align*}"}
+{"id": "657.png", "formula": "\\begin{align*} \\nabla _ X \\psi _ 2 = - \\frac { 1 } { 2 } X _ 1 \\cdot \\psi _ 2 - \\frac { 1 } { 2 } B ( X ) \\cdot \\psi _ 2 . \\end{align*}"}
+{"id": "7870.png", "formula": "\\begin{align*} n _ 3 ( \\Delta , x ) = \\max _ { k \\in F _ B ( x ) } \\# \\{ f \\in F _ B ( x ) : \\Delta ( f , k ) \\le \\Delta \\} . \\end{align*}"}
+{"id": "1936.png", "formula": "\\begin{align*} & \\bigg ( | ( - \\Delta _ x ) ^ { 1 / 6 } u _ h - ( ( - \\Delta _ x ) ^ { 1 / 6 } u _ h ) _ { Q _ r ( z _ 0 ) } | ^ p \\bigg ) ^ { 1 / p } _ { Q _ r ( z _ 0 ) } \\leq N \\nu ^ { - 1 } ( | ( - \\Delta _ x ) ^ { 1 / 6 } u _ h | ^ p ) ^ { 1 / p } _ { Q _ { \\nu r } ( z _ 0 ) } \\\\ & \\le N \\nu ^ { - 1 } ( | ( - \\Delta _ x ) ^ { 1 / 6 } u | ^ p ) ^ { 1 / p } _ { Q _ { \\nu r } ( z _ 0 ) } + N \\nu ^ { - 1 } \\sum _ { k = 0 } ^ { \\infty } 2 ^ { - k } F _ k ( 1 ) . \\end{align*}"}
+{"id": "2916.png", "formula": "\\begin{align*} \\inf \\| p ( A ) \\| ^ { 1 / { \\rm d e g } ( p ) } = 0 , \\end{align*}"}
+{"id": "6751.png", "formula": "\\begin{align*} \\varepsilon ^ { 2 } \\sum _ { | \\alpha | = N } \\| \\frac { \\partial ^ { \\alpha } F ( 0 ) } { \\sqrt { \\mu } } \\| ^ { 2 } \\leq C ( \\eta _ { 0 } + \\varepsilon ^ { \\frac { 1 } { 2 } - a } ) \\varepsilon ^ { 2 } . \\end{align*}"}
+{"id": "273.png", "formula": "\\begin{align*} { \\rm T r } \\left ( \\frac { v + U _ t } { v - U _ t } \\cdot \\frac { w + U _ s } { w - U _ s } \\right ) = \\sum _ { k } \\frac { w + z _ k ( s ) } { w - z _ k ( s ) } \\langle u _ k ( s ) , \\frac { v + U _ t } { v - U _ t } u _ k ( s ) \\rangle . \\end{align*}"}
+{"id": "7134.png", "formula": "\\begin{align*} c ' _ { i j } = c _ { i j } + \\varepsilon , \\ c ' _ { i l } = c _ { i l } - \\varepsilon , \\ c ' _ { l j } = c _ { l j } - \\varepsilon . \\end{align*}"}
+{"id": "6853.png", "formula": "\\begin{align*} \\lambda u _ i h ( a _ i ) f ^ { p ^ { e ' } } ( a _ i ) = u _ i g ( a _ i ) , \\ s + 1 \\leq i \\leq n , \\end{align*}"}
+{"id": "661.png", "formula": "\\begin{align*} | \\Psi _ 1 | ^ 2 + | \\Psi _ 2 | ^ 2 = 1 . \\end{align*}"}
+{"id": "3275.png", "formula": "\\begin{gather*} \\rho ( m , \\alpha , \\beta , \\gamma , \\delta ; q ) = \\frac { ( \\alpha q , \\beta \\delta q , \\gamma q , \\gamma \\delta q ; q ) _ m \\big ( 1 - \\gamma \\delta q ^ { 2 m + 1 } \\big ) } { \\big ( q , \\alpha ^ { - 1 } \\gamma \\delta q , \\beta ^ { - 1 } \\gamma q , \\delta q ; q \\big ) _ m ( \\alpha \\beta q ) ^ m ( 1 - \\gamma \\delta q ) } , \\end{gather*}"}
+{"id": "1037.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\dot { x } & = \\frac { p } { t } ( z - x ) , \\\\ \\dot { z } & = - p t ^ { p - 1 } \\nabla f ( x ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "825.png", "formula": "\\begin{align*} \\begin{aligned} v ( x ) & = A x ^ { k _ 1 } + p ( x ) = \\mathbb { E } \\int _ 0 ^ { \\tau _ i ^ { * } } e ^ { - \\beta t } f ( x _ i ( t ) ) d t - \\bar { K } _ 2 \\mathbb { E } \\Big ( e ^ { - \\beta \\tau _ i ^ { * } } \\Big ) . \\end{aligned} \\end{align*}"}
+{"id": "4389.png", "formula": "\\begin{align*} u _ { x x } - u _ { t t } + q ( x ) u _ x - q ( t ) u _ t - ( r ( x ) - r ( t ) ) u = 0 . \\end{align*}"}
+{"id": "1785.png", "formula": "\\begin{align*} \\sum _ { i \\in A } \\left ( - 1 \\right ) ^ { \\# i } p _ { A \\backslash i } \\left ( W \\right ) \\langle Y , B , i \\rangle = 0 . \\end{align*}"}
+{"id": "5755.png", "formula": "\\begin{align*} h ( x _ 0 , x ) = \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } x _ i x _ j + \\sum _ { i = 1 } ^ { m } H _ { i 0 } x _ i x _ 0 + \\sum _ { i = 1 } ^ { m } H _ { 0 i } x _ 0 x _ i + A _ 0 x _ 0 x _ 0 , \\end{align*}"}
+{"id": "7362.png", "formula": "\\begin{align*} x _ { k + 1 } = \\exp _ { x _ k } ( \\alpha _ k \\eta _ k ) , \\end{align*}"}
+{"id": "6960.png", "formula": "\\begin{align*} e _ j = \\begin{cases} \\binom { N } { j } & j \\neq s + 1 \\\\ \\binom { N } { j } + ( - 1 ) ^ { s } q & j = s + 1 \\end{cases} . \\end{align*}"}
+{"id": "6009.png", "formula": "\\begin{align*} g ^ { ( q ) } ( b ; h ) = \\delta _ 1 + g ^ { ( q ) } ( b ; h ) \\delta _ 2 + \\delta _ 3 , \\end{align*}"}
+{"id": "3577.png", "formula": "\\begin{align*} \\Xi = T ( T ^ { 1 / 2 } C T ^ { 1 / 2 } + \\lambda I ) ^ { - 2 } T = T \\left [ \\sum _ i ( \\zeta _ i + \\lambda ) ^ { - 2 } \\phi _ i \\otimes \\phi _ i + \\sum _ i \\lambda ^ { - 2 } \\tilde { \\phi } _ i \\otimes \\tilde { \\phi } _ i \\right ] T , \\end{align*}"}
+{"id": "1624.png", "formula": "\\begin{align*} f ( u _ n ) - f ( v _ n ) & = \\min _ { x , y \\in M _ { n - 1 } } \\big ( f ( x ) + ( 1 - 1 / 2 ^ n ) d ( x , u _ n ) - f ( y ) + ( 1 - 1 / 2 ^ n ) d ( y , v _ n ) \\big ) \\\\ & \\ge \\min _ { x , y \\in M _ { n - 1 } } \\big ( ( 1 - 1 / 2 ^ n ) \\big ( d ( x , u _ n ) + d ( y , v _ n ) \\big ) - ( 1 - 1 / 2 ^ { n - 1 } ) d ( x , y ) \\big ) \\\\ & \\ge ( 1 - 1 / 2 ^ n ) \\min _ { x , y \\in M _ { n - 1 } } \\big ( d ( x , u _ n ) + d ( y , v _ n ) - ( 1 - 1 / 2 ^ { n + 1 } ) d ( x , y ) \\big ) \\\\ & \\ge ( 1 - 1 / 2 ^ n ) ( 1 - 1 / 2 ^ { n + 1 } ) d ( u _ n , v _ n ) \\\\ & \\ge ( 1 - 1 / 2 ^ { n - 1 } ) d ( u _ n , v _ n ) , \\end{align*}"}
+{"id": "6735.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | = N } \\{ \\| \\partial ^ { \\alpha } ( \\widetilde { \\rho } , \\widetilde { u } , \\widetilde { \\theta } , \\widetilde { E } , \\widetilde { B } ) \\| ^ { 2 } + \\| \\partial ^ { \\alpha } f \\| ^ { 2 } \\} \\leq \\varepsilon ^ { - 2 a } . \\end{align*}"}
+{"id": "6822.png", "formula": "\\begin{align*} \\partial _ { t t t } v + a \\partial _ { t t } v - b \\Delta \\partial _ t v - c \\Delta v = 0 , \\end{align*}"}
+{"id": "3532.png", "formula": "\\begin{align*} \\varepsilon _ \\mathrm { p } ( \\omega ) = \\varepsilon _ \\infty \\Bigg [ 1 + \\dfrac { \\omega _ \\mathrm { p } ^ 2 } { \\omega _ 0 ^ 2 - \\omega ^ 2 - i \\gamma \\omega } \\Bigg ] \\end{align*}"}
+{"id": "1119.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ { \\infty } h _ { n } ( x ) h _ { m } ( x ) \\exp ( - x ^ { 2 } / 2 ) d x = \\delta _ { n m } , \\end{align*}"}
+{"id": "7518.png", "formula": "\\begin{align*} G ( p , q ) - \\log \\frac { 1 } { | z _ 0 ( p ) - z _ 0 ( q ) | } = O ( 1 ) \\end{align*}"}
+{"id": "7267.png", "formula": "\\begin{align*} \\frac { d } { d t } { \\gamma } ( t ) = f ( t , \\gamma ( t ) , e _ t \\sharp \\eta , u _ K ( t , \\gamma ) ) . \\end{align*}"}
+{"id": "1022.png", "formula": "\\begin{align*} \\begin{array} { l l l } [ R / ( I , L ) ] _ 3 & \\cong [ K [ x _ 1 , x _ 2 , x _ 3 ] / ( x _ 1 ^ 3 , x _ 2 ^ 3 , x _ 3 ^ 3 , x _ 1 ^ 2 x _ 2 , x _ 1 + x _ 2 + x _ 3 ) ] _ 3 \\\\ & \\cong [ K [ x _ 1 , x _ 2 ] / ( x _ 1 ^ 3 , x _ 2 ^ 3 , x _ 1 ^ 3 + 3 x _ 1 ^ 2 x _ 2 + 3 x _ 1 x _ 2 ^ 2 + x _ 2 ^ 3 , x _ 1 ^ 2 x _ 2 ) ] _ 3 \\\\ & \\cong [ k [ x _ 1 , x _ 2 ] / ( x _ 1 ^ 3 , x _ 2 ^ 3 , x _ 1 ^ 2 x _ 2 , x _ 1 x _ 2 ^ 2 ) ] _ 3 = 0 \\end{array} \\end{align*}"}
+{"id": "5568.png", "formula": "\\begin{align*} \\sigma ( \\beta _ 0 ) : = B ( \\beta _ 0 ) \\dfrac { \\beta _ 0 ^ { 1 - \\mu - \\nu } \\Tilde { \\Phi } _ 2 ( \\beta _ 0 , + \\infty ) } { L _ { 2 m } ( - 1 + \\mu + \\nu ) } < 1 , \\end{align*}"}
+{"id": "952.png", "formula": "\\begin{align*} \\delta ^ 2 ( 1 - \\epsilon ^ 2 ) b _ { \\alpha } u _ n ( x ) \\leq u ( y ) - u ( x ) \\leq C b _ { \\alpha } ^ 2 , \\mbox { w i t h } y = ( x ' , \\delta ^ 2 b _ \\alpha ) \\in \\partial _ 2 \\omega \\end{align*}"}
+{"id": "8003.png", "formula": "\\begin{align*} M = \\begin{pmatrix} A & B ^ * \\\\ B & C \\end{pmatrix} : \\mathcal { H } _ 1 \\oplus \\mathcal { H } _ 2 \\rightarrow \\mathcal { H } _ 1 \\oplus \\mathcal { H } _ 2 \\end{align*}"}
+{"id": "7351.png", "formula": "\\begin{align*} { \\rm d i m } ( T ) & = \\sum ^ m _ { d = 0 } ( m - d - \\lceil \\frac { m - d } { 2 } \\rceil + 1 ) ( 2 d + 2 ) ^ 2 = 4 { m + 4 \\choose 4 } , \\end{align*}"}
+{"id": "7413.png", "formula": "\\begin{align*} f ( q ) = \\sum _ { k = 1 } ^ \\infty b _ k q ^ k \\in S _ 4 ^ { \\textrm { n e w } } ( \\Gamma _ 0 ( N ) ) \\end{align*}"}
+{"id": "231.png", "formula": "\\begin{align*} \\partial _ t \\log m _ t ( x ) = - \\left ( \\frac { \\delta F } { \\delta m } ( m _ t , x ) + \\frac { \\sigma ^ 2 } { 2 } \\log \\left ( \\frac { m _ t ( x ) } { \\pi ( x ) } \\right ) - \\frac { \\sigma ^ 2 } { 2 } \\operatorname { K L } ( m _ t | \\pi ) \\right ) \\ , . \\end{align*}"}
+{"id": "2130.png", "formula": "\\begin{align*} \\mathrm { c o r a n k } _ { \\Lambda ( G ) } ( H ^ 1 ( G _ { \\infty } ^ { S } , E _ { p ^ { \\infty } } ) ) = \\mathrm { c o r a n k } _ { \\Omega ( G ) } ( H ^ 1 ( G _ { \\infty } ^ { S } , E _ { p } ) ) = 2 . \\end{align*}"}
+{"id": "7427.png", "formula": "\\begin{align*} [ N , \\ , J _ z ] = 0 , [ N , \\ , J _ \\pm ] = 0 . \\end{align*}"}
+{"id": "2986.png", "formula": "\\begin{align*} f ' _ { ( x , y ) } ( t ) & = - ( k - 1 ) _ 2 ( \\deg _ W ( x ) + \\deg _ W ( y ) - 2 d ) e ^ { - t ( k - 1 ) _ 2 } \\\\ & \\ ; - ( ( k ) _ 2 - 2 ) ( W ( x , y ) - \\deg _ W ( x ) - \\deg _ W ( y ) + d ) e ^ { - t ( ( k ) _ 2 - 2 ) } \\\\ & = \\left ( ( k ) _ 2 - 2 \\right ) ( d - f _ { ( x , y ) } ( t ) ) + 2 ( k - 2 ) ( \\deg _ W ( x ) + \\deg _ W ( y ) - 2 d ) e ^ { - t ( k - 1 ) _ 2 } \\\\ & = ( 2 k - 4 ) ( g _ x ( t ) + g _ y ( t ) ) + ( k - 2 ) _ 2 d - \\left ( ( k ) _ 2 - 2 \\right ) f _ { ( x , y ) } ( t ) \\ ; . \\end{align*}"}
+{"id": "4805.png", "formula": "\\begin{align*} L ( k , m , \\lambda ) = { ( n + \\lambda ) _ m } \\times \\begin{cases} 1 & \\\\ \\frac { 1 } { ( n + \\lambda - m k ) _ { 2 m } } & \\end{cases} \\end{align*}"}
+{"id": "4140.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\left ( \\frac { 1 } { x } \\frac { d } { d x } \\right ) ^ k \\left ( \\frac { J _ { \\alpha } ( c x y ) } { ( c x y ) ^ { \\alpha } } \\right ) y ^ { \\alpha + 1 / 2 } \\psi _ { n , c } ^ { \\alpha } ( y ) d y = \\mu _ n ^ { \\alpha } ( c ) \\left ( \\frac { 1 } { x } \\frac { d } { d x } \\right ) ^ k \\left ( \\frac { \\psi _ { n , c } ^ { \\alpha } ( x ) } { ( c x ) ^ { \\alpha + 1 / 2 } } \\right ) . \\end{align*}"}
+{"id": "6038.png", "formula": "\\begin{align*} | | f | | _ { M \\dot { K } _ { p , \\vec { q } } ^ { \\alpha _ 2 , \\lambda } } & = 2 ^ { k ( \\alpha _ 2 - \\alpha _ 1 ) } \\mathop { s u p } \\limits _ { k _ 0 \\in \\mathbb { Z } } 2 ^ { - k _ { 0 } \\lambda } \\left ( \\sum _ { k = - \\infty } ^ { k _ 0 } 2 ^ { k \\alpha _ { 1 } p } | | f \\widetilde { \\chi } _ k | | _ { \\vec { q } } ^ p \\right ) ^ { \\frac { 1 } { p } } \\leq C | | f | | _ { M \\dot { K } _ { p , \\vec { q } } ^ { \\alpha _ 1 , \\lambda } } . \\end{align*}"}
+{"id": "4357.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ \\Omega \\Phi ( f _ n ( t ) ) \\ , \\mathrm { d } x = \\int _ \\Omega \\Phi ( f ( t ) ) \\ , \\mathrm { d } x \\ ; \\ ; \\ ; \\ ; \\lim _ { n \\to \\infty } \\int _ \\Omega \\Phi ( f _ n ( t _ 0 ) ) \\ , \\mathrm { d } x = \\int _ \\Omega \\Phi ( f ( t _ 0 ) ) \\ , \\mathrm { d } x \\ , . \\end{align*}"}
+{"id": "1492.png", "formula": "\\begin{align*} \\rho _ { \\ell } = \\frac { \\mu ( \\ell ) } { g ( \\ell ) } \\sum _ { d \\equiv 0 ( \\mod \\ell ) } g ( d ) y _ d . \\end{align*}"}
+{"id": "4477.png", "formula": "\\begin{align*} \\psi = \\sum _ { n = 1 } ^ { \\infty } F ( e _ n ) e _ n \\ , . \\end{align*}"}
+{"id": "56.png", "formula": "\\begin{align*} \\div X = 0 . \\end{align*}"}
+{"id": "1735.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u + ( \\Delta - \\kappa ) u = - | u | ^ 2 u \\\\ u _ 0 = \\varphi \\ , . \\end{cases} \\end{align*}"}
+{"id": "309.png", "formula": "\\begin{align*} \\mu _ { i } = \\begin{cases} e _ { i } ( e _ { i } - 1 ) & ( r _ { i - 1 } = 1 ) , \\\\ e _ { i } ^ { 2 } & ( r _ { i - 1 } = 2 ) \\end{cases} \\end{align*}"}
+{"id": "4705.png", "formula": "\\begin{align*} Q ( z , w ) = - i \\left [ 2 C ( z , w ) - P ( z , w ) - 1 - Q _ 1 ( z , w ) \\right ] \\end{align*}"}
+{"id": "6315.png", "formula": "\\begin{align*} A ^ k _ t - A _ t & : = \\int _ 0 ^ t K ( s , t ) f _ k ( s , X ^ k _ s ) \\dd s - \\int _ 0 ^ t K ( s , t ) f ( s , X _ s ) \\dd s \\\\ & \\leq \\int _ 0 ^ t | K ( s , t ) | \\big | f _ k ( s , X ^ k _ s ) - f ( s , X ^ k _ s ) \\big | \\dd s + \\int _ 0 ^ t | K ( s , t ) | \\big | f ( s , X ^ k _ s ) - f ( s , X _ s ) \\big | \\dd s \\\\ & \\leq M \\bigg ( \\sup _ { t \\in [ 0 , T ] , \\ , x \\in [ - n , n ] } | f _ k ( t , x ) - f ( t , x ) | + g _ { n } \\big ( \\| X ^ k - X \\| _ { \\infty } \\big ) \\bigg ) , \\end{align*}"}
+{"id": "2618.png", "formula": "\\begin{align*} \\alpha _ 1 : = \\{ \\{ 1 , 2 \\} , \\{ 3 , 4 \\} , \\{ 5 , 6 \\} , \\{ 7 , 8 \\} , \\ldots , \\{ 2 b - 1 , 2 b \\} \\} , \\\\ \\alpha _ 2 : = \\{ \\{ 1 , 4 \\} , \\{ 3 , 2 \\} , \\{ 5 , 6 \\} , \\{ 7 , 8 \\} , \\ldots , \\{ 2 b - 1 , 2 b \\} \\} , \\\\ \\alpha _ 3 : = \\{ \\{ 1 , 6 \\} , \\{ 3 , 4 \\} , \\{ 5 , 2 \\} , \\{ 7 , 8 \\} , \\ldots , \\{ 2 b - 1 , 2 b \\} \\} , \\\\ \\alpha _ 4 : = \\{ \\{ 1 , 8 \\} , \\{ 3 , 4 \\} , \\{ 5 , 6 \\} , \\{ 7 , 8 \\} , \\ldots , \\{ 2 b - 1 , 2 b \\} \\} . \\end{align*}"}
+{"id": "2652.png", "formula": "\\begin{align*} d ( \\lambda _ v ) = e = \\iota _ { \\star } = \\lambda _ { d ^ 0 ( v ) } \\end{align*}"}
+{"id": "2326.png", "formula": "\\begin{align*} \\zeta ( 4 ) = \\frac { \\pi ^ { 4 } } { 9 0 } , \\ : \\zeta ( 6 ) = \\frac { \\pi ^ { 6 } } { 9 4 5 } , \\ : \\zeta ( 8 ) = \\frac { \\pi ^ { 8 } } { 9 4 5 0 } , \\ : \\zeta ( 1 0 ) = \\frac { \\pi ^ { 1 0 } } { 9 3 5 5 5 } , \\ : \\zeta ( 1 2 ) = \\frac { 6 9 1 \\pi ^ { 1 2 } } { 6 3 8 5 1 2 8 7 5 } \\end{align*}"}
+{"id": "7005.png", "formula": "\\begin{align*} \\int ( \\Delta v ) ^ 2 e ^ { b v } d m = \\alpha _ 2 \\int | \\nabla v | ^ 2 e ^ { b v } d m - b \\int | \\nabla v | ^ 4 e ^ { b v } d m - ( b + 1 ) \\int ( \\Delta v ) | \\nabla v | ^ 2 e ^ { b v } d m . \\end{align*}"}
+{"id": "3453.png", "formula": "\\begin{align*} \\Big ( \\frac { 1 } { 2 } I _ { d } + \\mathcal { K } _ { \\alpha _ { \\mathrm { p } } } \\Big ) \\Big [ \\gamma ^ { \\textbf { i n t } } _ { 0 } \\mathrm { U } _ { \\mathrm { i } } \\Big ] ( \\mathrm { x } , t ) = \\mathcal { S } _ { \\alpha _ { \\mathrm { p } } } \\Big [ \\gamma ^ { \\textbf { i n t } } _ { 1 } \\mathrm { U } _ { \\mathrm { i } } \\Big ] ( \\mathrm { x } , t ) + \\gamma ^ { \\textbf { i n t } } _ { 0 } \\mathcal { V } \\Big [ f \\Big ] ( \\mathrm { x } , t ) , \\ \\ \\mathrm { x } \\in \\partial \\Omega \\end{align*}"}
+{"id": "1104.png", "formula": "\\begin{align*} \\mathbb P \\left [ \\frac 1 n \\sum _ { i = 1 } ^ n ( X _ i - \\mathbb E [ X _ i ] ) < - \\epsilon \\right ] \\le \\exp \\left ( - \\frac { n \\epsilon ^ 2 } { 2 \\sigma ^ 2 } \\right ) \\end{align*}"}
+{"id": "6359.png", "formula": "\\begin{align*} \\mathcal { S } ( \\lambda ) = \\left [ { \\begin{array} { c | c } A ( \\lambda ) & - B \\\\ \\hline C & D ( \\lambda ) \\\\ \\end{array} } \\right ] { \\mathbb C } [ \\lambda ] ^ { n + m , n + m } \\end{align*}"}
+{"id": "3830.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 n + h ^ 2 - h - 1 8 + d } { 2 } , \\end{align*}"}
+{"id": "4040.png", "formula": "\\begin{align*} \\hat { \\mathcal { V } } ( s ) = \\left [ - I ( s ) ^ { - \\delta } , I ( s ) ^ { - \\delta } \\right ] ^ { 2 k } , \\end{align*}"}
+{"id": "136.png", "formula": "\\begin{align*} s _ i ( x ) = \\left \\{ \\begin{array} { l l } s _ i x \\in X _ { k + 1 } \\subseteq ( \\Sigma X ) _ { m + 1 } & \\mbox { i f } 0 \\leq i \\leq k , \\\\ x \\in X _ { k } \\subseteq ( \\Sigma X ) _ { m } & \\mbox { i f } k + 1 \\leq i \\leq m . \\end{array} \\right . \\end{align*}"}
+{"id": "5852.png", "formula": "\\begin{align*} { } \\Big ( 1 + 2 ^ { p - 1 } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } \\Big ) \\langle f ( A ) x , x \\rangle ^ p < \\Big ( \\frac { p } { p - 1 } \\Big ) ^ { p } \\langle f ( A ) ^ p x , x \\rangle . \\end{align*}"}
+{"id": "2756.png", "formula": "\\begin{align*} 0 = \\pi _ 2 ^ * H \\cdot \\ell _ 2 = \\mu ^ * \\pi _ 2 ^ * H \\cdot \\mu ^ { - 1 } _ * ( \\ell _ 2 ) = a ( H ) - b ( H ) \\pi _ 1 ^ * H \\cdot \\mu ^ { - 1 } _ * ( \\ell _ 2 ) . \\end{align*}"}
+{"id": "3284.png", "formula": "\\begin{gather*} x _ 1 - a _ 1 x _ 2 x _ 3 = b _ 1 , ( 1 - \\delta _ { C _ 0 0 } ) x _ 2 - a _ 2 x _ 1 x _ 3 = b _ 2 , ( 1 - \\delta _ { C _ 1 0 } ) x _ 3 - a _ 3 x _ 1 x _ 2 = b _ 3 . \\end{gather*}"}
+{"id": "1377.png", "formula": "\\begin{align*} \\mathcal { U } ( t ) = U ( t ) \\begin{pmatrix} u _ 0 \\\\ u _ 1 \\end{pmatrix} + \\int _ 0 ^ t U ( t - s ) \\begin{pmatrix} 0 \\\\ - | u | ^ { p - 1 } u \\end{pmatrix} \\ , d s \\end{align*}"}
+{"id": "4115.png", "formula": "\\begin{align*} \\mathcal { L } _ c ^ { \\alpha } ( \\phi ) = \\dfrac { d } { d x } \\left [ ( 1 - x ^ 2 ) \\dfrac { d } { d x } \\phi \\right ] + \\left ( \\dfrac { \\dfrac { 1 } { 4 } - \\alpha ^ 2 } { x ^ 2 } - c ^ 2 x ^ 2 \\right ) \\phi \\end{align*}"}
+{"id": "5229.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { f - 1 } p ^ i ( s ^ { \\operatorname { m i n } } _ { i + j } - s _ { i + j } ) \\in ( p ^ f - 1 ) \\mathbb { Z } _ { \\geq 0 } \\end{align*}"}
+{"id": "5646.png", "formula": "\\begin{align*} L '' = I _ \\gamma ( V , V ) = - \\int _ 0 ^ l \\mathbf { K } ( T , V ) d t = - \\int _ 0 ^ l \\mathbf { H } ( V _ o ) d t < 0 . \\end{align*}"}
+{"id": "1197.png", "formula": "\\begin{align*} \\eta ( 1 ) = \\gamma ^ { - 1 } \\mbox { a n d } \\eta ( g ) = 1 . \\end{align*}"}
+{"id": "6478.png", "formula": "\\begin{align*} S = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , T = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} U = \\begin{pmatrix} - 1 & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "2175.png", "formula": "\\begin{align*} & h _ { i j k l } ^ 0 = - h _ { j i k l } ^ 0 = - h _ { i j l k } ^ 0 , h _ { i j k l } ^ 0 = h _ { k l i j } ^ 0 . \\end{align*}"}
+{"id": "6987.png", "formula": "\\begin{align*} - \\Delta v = | \\nabla v | ^ 2 + \\alpha _ 2 v + \\lambda - \\alpha _ 1 . \\end{align*}"}
+{"id": "4937.png", "formula": "\\begin{align*} a _ i + 2 b _ i = \\binom { n } { k - 2 - i } ~ ~ ~ \\textrm { f o r e a c h ~ } i \\in \\{ 1 , \\cdots , ( k - 5 ) / 2 - t \\} . \\end{align*}"}
+{"id": "314.png", "formula": "\\begin{align*} - F ^ { s - 1 } d f _ { 1 } ^ { * } a = \\frac { t _ { 2 } ^ { - 1 } f _ { 1 } ^ { * } \\alpha _ { 1 } d \\log t _ { 1 } + ( t _ { 2 } ^ { - 1 } f _ { 1 } ^ { * } \\alpha _ { 1 } + f _ { 1 } ^ { * } \\beta _ { 2 } ) d \\log t _ { 2 } + t _ { 1 } t _ { 2 } f _ { 1 } ^ { * } \\gamma } { t _ { 1 } ^ { n _ { 1 } } t _ { 2 } ^ { n _ { 1 } + n _ { 2 } - 1 } } \\end{align*}"}
+{"id": "4933.png", "formula": "\\begin{align*} \\binom { n } { 1 } + \\sum _ { i = 3 } ^ { k - 1 } \\frac { 3 { \\left ( i - 1 \\right ) } } { k - { \\left ( i - 1 \\right ) } } \\binom { n } { k - i } > 2 \\sum _ { i = 1 } ^ { 5 } \\binom { n } { i } . \\end{align*}"}
+{"id": "6968.png", "formula": "\\begin{align*} f ( x ) = q _ R ( x ) g ( x ) + r _ R ( x ) = g ( x ) q _ L ( x ) + r _ L ( x ) \\end{align*}"}
+{"id": "3743.png", "formula": "\\begin{align*} \\begin{bmatrix} n _ { 1 1 } & n _ { 1 2 } & n _ { 1 3 } \\\\ n _ { 2 1 } & n _ { 2 2 } & n _ { 2 3 } \\\\ n _ { 3 1 } & n _ { 3 2 } & n _ { 3 3 } \\end{bmatrix} & = \\begin{bmatrix} 2 & 1 0 & 1 0 \\\\ 1 0 & 2 & 1 0 \\\\ 1 0 & 1 0 & 2 \\end{bmatrix} . \\end{align*}"}
+{"id": "519.png", "formula": "\\begin{align*} g ( x ) = G \\bigg ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { x _ i } \\bigg ) , \\ \\ G : \\P ( \\R ) \\to \\R , \\end{align*}"}
+{"id": "7051.png", "formula": "\\begin{align*} \\label { 9 9 } \\sum _ { k = 0 } ^ r [ \\mathfrak { g } ' , \\tilde { \\mathfrak { t r } _ 0 } ^ p ] ^ { k } = \\sum _ { k = 0 } ^ r [ \\tilde { \\mathfrak { g } } , \\tilde { \\mathfrak { t r } } _ 0 ^ p ] ^ { k } . \\end{align*}"}
+{"id": "6330.png", "formula": "\\begin{align*} u = \\frac { v } { 1 - t } + \\frac { t e } { 1 - t } . \\end{align*}"}
+{"id": "2324.png", "formula": "\\begin{align*} \\zeta ( \\boldsymbol { k } ) : = \\sum _ { 0 < m _ { 1 } < \\cdots < m _ { d } } \\frac { 1 } { m _ { 1 } ^ { k _ { 1 } } \\cdots m _ { d } ^ { k _ { d } } } \\end{align*}"}
+{"id": "6963.png", "formula": "\\begin{align*} s _ { \\lambda } ( x _ 1 , \\ldots , x _ N ) = 1 . \\end{align*}"}
+{"id": "532.png", "formula": "\\begin{align*} Q _ i ( d x _ i ) & = Z _ i ^ { - 1 } e ^ { \\hat { f } _ i ( x _ i ) } \\ , d x _ i , \\hat { f } _ i : \\R \\to \\R \\cup \\{ - \\infty \\} \\\\ \\hat { f } _ i ( x _ i ) & : = \\int _ { \\R ^ { n - 1 } } f ( x _ 1 , \\ldots , x _ n ) \\ , \\prod _ { j \\neq i } Q ^ * _ j ( d x _ j ) , i \\in [ n ] . \\end{align*}"}
+{"id": "3675.png", "formula": "\\begin{align*} x _ { t + 3 } = \\frac { 1 - \\alpha } { t + 2 } \\pm 3 0 t \\delta ^ { 1 / 2 } , \\quad x _ { t + 4 } = \\frac { \\alpha } { t + 2 } \\pm 3 0 t \\delta ^ { 1 / 2 } . \\end{align*}"}
+{"id": "5422.png", "formula": "\\begin{align*} I _ 1 = O _ \\prec ( ( \\log N ) N \\Psi ^ 4 ) . \\end{align*}"}
+{"id": "7833.png", "formula": "\\begin{gather*} c ' _ 0 = ( m - 0 ) ( m - 0 - k ) c _ 0 - c _ 1 = ( 1 - 1 ) c _ 1 = 0 \\end{gather*}"}
+{"id": "7600.png", "formula": "\\begin{align*} f _ \\alpha ( \\cdot ) ( f _ H ( \\cdot ) - f _ H ( x _ \\alpha ) ) = \\sum _ { \\beta \\prec \\alpha } a _ { \\alpha } ^ { \\beta } f _ \\beta ( \\cdot ) . \\end{align*}"}
+{"id": "7706.png", "formula": "\\begin{gather*} ( d \\tilde { \\omega } ) _ { i _ 1 \\dots i _ { k + 1 } } ( x ^ \\prime , x ^ n ) = \\begin{cases} ( d \\omega ) _ { i _ 1 \\dots i _ { k + 1 } } ( x ^ \\prime , | x ^ n | ) & i _ l \\neq n , \\\\ \\operatorname { s g n } ( x ^ n ) ( d \\omega ) _ { i _ 1 \\dots i _ { k + 1 } } ( x ^ \\prime , | x ^ n | ) & i _ l = n \\end{cases} \\end{gather*}"}
+{"id": "2006.png", "formula": "\\begin{align*} Q _ \\xi ( z ) : = i \\ , \\frac { \\xi ' } { \\xi } \\left ( \\frac { 1 } { 2 } - i z \\right ) \\end{align*}"}
+{"id": "1419.png", "formula": "\\begin{align*} \\left \\| U ( t ) \\begin{pmatrix} u _ 0 \\\\ u _ 1 \\end{pmatrix} \\right \\| _ { \\mathcal { H } } & \\le e ^ { C t } \\| ( u _ 0 , u _ 1 ) \\| _ { \\mathcal { H } } \\end{align*}"}
+{"id": "4028.png", "formula": "\\begin{align*} \\begin{matrix} \\alpha _ 1 = - 2 k ( 2 k - 1 ) \\frac { b } { p - 1 } ; & \\alpha _ 2 = 4 p k ^ 2 \\frac { b ^ 2 } { ( p - 1 ) ^ 2 } ; & \\alpha _ 3 = - 2 p k ( 2 k - 1 ) \\frac { b } { p - 1 } ; \\alpha _ 4 = 4 p ( 2 p - 1 ) k ^ 2 \\frac { b ^ 2 } { ( p - 1 ) ^ 2 } . \\\\ \\end{matrix} \\end{align*}"}
+{"id": "1173.png", "formula": "\\begin{align*} & a _ k ( x ) \\Big ( \\frac { \\textup { d } } { \\textup { d } x } \\Big ) ^ 2 \\big ( ( 1 - x ) ^ { p _ k ( k - j ) / k } x ^ { p _ k j / k } \\big ) + b _ k ( x ) \\frac { \\textup { d } } { \\textup { d } x } \\big ( ( 1 - x ) ^ { p _ k ( k - j ) / k } x ^ { p _ k j / k } \\big ) \\\\ & + p _ k c _ k ( x ) ( 1 - x ) ^ { p _ k ( k - j ) / k } x ^ { p _ k j / k } \\\\ & = \\frac { p _ k ^ 2 } { k ^ 2 } \\big ( ( 1 - x ) ^ { p _ k / k } - x ^ { p _ k / k } \\big ) ( 1 - x ) ^ { p _ k ( k - j ) / k } x ^ { p _ k j / k } \\big ( j ^ 2 ( 1 - x ) ^ { p _ k / k } - ( k - j ) ^ 2 x ^ { p _ k / k } \\big ) . \\end{align*}"}
+{"id": "3346.png", "formula": "\\begin{gather*} \\Lambda _ { \\{ 1 , 4 \\} } = \\alpha \\big [ \\Lambda _ { \\{ 1 , 2 , 3 \\} } , \\Lambda _ { \\{ 2 , 3 , 4 \\} } \\big ] _ q + \\beta \\big ( \\Lambda _ { \\{ 2 , 3 \\} } \\Lambda _ { \\{ 1 , 2 , 3 , 4 \\} } + \\Lambda _ { \\{ 1 \\} } \\Lambda _ { \\{ 4 \\} } \\big ) , \\\\ \\Lambda _ { \\{ 1 , 3 , 4 \\} } = \\alpha \\big [ \\Lambda _ { \\{ 1 , 2 \\} } , \\Lambda _ { \\{ 2 , 3 , 4 \\} } \\big ] _ q + \\beta \\big ( \\Lambda _ { \\{ 2 \\} } \\Lambda _ { \\{ 1 , 2 , 3 , 4 \\} } + \\Lambda _ { \\{ 1 \\} } \\Lambda _ { \\{ 3 , 4 \\} } \\big ) . \\end{gather*}"}
+{"id": "6497.png", "formula": "\\begin{align*} \\sup _ { B _ { R _ { k } } ( x _ k ) } u & \\geq C ^ { - 1 } \\delta ^ { - \\gamma } \\left ( \\sup _ { B _ { \\delta R _ k } ( x _ k ) } u - \\inf _ { B _ { \\delta R _ k } ( x _ k ) } u \\right ) - R _ k ^ 2 \\rho \\\\ & \\geq C ^ { - 1 } \\delta ^ { - \\gamma } \\left ( u ( x _ k ) - \\inf _ { B _ { \\delta R _ k } ( x _ k ) } u - C \\delta ^ { \\gamma } \\rho \\right ) . \\\\ \\end{align*}"}
+{"id": "7252.png", "formula": "\\begin{align*} Q _ n ( X , S _ k f , d , \\epsilon ) & \\leq \\sum _ { x \\in E } ( 1 / \\epsilon ) ^ { S _ n ( S _ k f ( x ) ) } \\\\ & = \\sum _ { x \\in E } ( 1 / \\epsilon ) ^ { S _ { n k } f ( x ) } , \\end{align*}"}
+{"id": "959.png", "formula": "\\begin{align*} \\begin{aligned} L ^ \\Phi ( \\theta ) [ \\cdot ] : = & - \\big ( { \\rm d i v } _ { \\ ! z } + 2 i \\pi \\theta \\big ) \\big ( A { ( \\Phi ^ { - 1 } ( z , \\omega ) , \\omega ) } { \\big ( \\nabla _ { \\ ! \\ ! z } + 2 i \\pi \\theta \\big ) } [ \\cdot ] \\big ) \\\\ & + V { \\big ( \\Phi ^ { - 1 } \\left ( z , \\omega \\right ) , \\omega \\big ) } [ \\cdot ] . \\end{aligned} \\end{align*}"}
+{"id": "213.png", "formula": "\\begin{align*} \\mathcal { J } _ 2 = \\biggl ( \\int _ 0 ^ T \\eta ^ { - \\frac { 1 } { p - 1 } } \\left | \\eta _ t \\right | ^ { \\frac { p } { p - 1 } } d t \\biggr ) \\biggl ( \\int _ { \\mathbb { R } ^ { N } } \\phi ^ { - \\frac { 1 } { p - 1 } } \\left | \\Delta \\phi \\right | ^ { \\frac { p } { p - 1 } } d x \\biggr ) . \\end{align*}"}
+{"id": "6062.png", "formula": "\\begin{align*} \\begin{cases} \\Delta \\eta + \\lambda \\eta = 0 \\ ; \\ ; \\ ; \\ ; \\Omega \\\\ [ 0 . 2 c m ] \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\eta = 0 \\ ; \\ ; \\ ; \\ ; \\partial \\Omega \\end{cases} \\end{align*}"}
+{"id": "7649.png", "formula": "\\begin{align*} F ( \\kappa ) : = \\inf \\left \\{ \\ , P ( A ) - \\kappa | A | \\ , : \\ , A \\subset \\Omega , \\ , | A | \\ge \\pi R ^ 2 \\ , \\right \\} . \\end{align*}"}
+{"id": "6436.png", "formula": "\\begin{align*} f ( x _ 1 , y _ 1 , \\ldots , x _ k , y _ k ) = \\sum _ { n _ 1 = 0 } ^ { \\infty } c _ { n _ 1 } ( x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) p _ { n _ 1 } ^ { ( \\alpha _ 1 , \\beta _ 1 ) } ( x _ 1 , y _ 1 ) . \\end{align*}"}
+{"id": "4689.png", "formula": "\\begin{align*} A = \\frac { 4 ( { \\rm I m } z ) ( { \\rm I m } w ) } { | 1 - z w | ^ { 2 } } , \\tilde { A } = - \\frac { 4 ( { \\rm I m } z ) ( { \\rm I m } w ) } { | 1 - z \\bar { w } | ^ { 2 } } . \\end{align*}"}
+{"id": "3251.png", "formula": "\\begin{align*} v _ { j } = \\sup _ { k \\geq j } \\varphi _ { j _ { k } } . \\end{align*}"}
+{"id": "849.png", "formula": "\\begin{align*} \\mathbb { E } ( \\tau ) = M _ { \\tau } ' ( 0 ) = \\left ( - \\frac { 1 } { a ' k _ 2 \\sigma } \\right ) \\ln \\left ( \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) \\end{align*}"}
+{"id": "6682.png", "formula": "\\begin{align*} Q ( t ) = \\det \\begin{bmatrix} s _ 0 & s _ 1 & \\ldots & s _ { m - 1 } & 1 \\\\ s _ 1 & s _ 2 & \\ldots & s _ { m } & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ s _ { m } & s _ { m + 1 } & \\ldots & s _ { 2 m - 1 } & t ^ { m } \\end{bmatrix} , \\end{align*}"}
+{"id": "6688.png", "formula": "\\begin{align*} & s _ { i , k } \\in ( t _ { \\infty } ( ( s _ { i , k + 1 } , \\ldots , s _ { p - 1 } ) ) , + \\infty ) , k \\in \\mathbb { Z } \\cap [ p - N _ i - 1 , p - 2 ] , \\\\ & s _ { i , k } = t _ { \\infty } ( ( s _ { i , k + 1 } , \\ldots , s _ { k + N _ i + 1 } ) ) , k \\in \\mathbb { Z } \\cap [ - \\kappa - 1 , p - N _ i - 2 ] \\end{align*}"}
+{"id": "7483.png", "formula": "\\begin{align*} \\psi _ m ^ { ( a ) } ( n ) = B _ { a + 1 , n - 1 } + \\frac { m ( m - 1 ) } { m + a } ( n - 1 ) B _ { m + a - 1 , n - 1 } \\ , . \\end{align*}"}
+{"id": "6496.png", "formula": "\\begin{align*} a _ k = 1 + a \\frac { \\varphi ^ k - \\bar \\varphi ^ k } { \\varphi - \\bar \\varphi } . \\end{align*}"}
+{"id": "7287.png", "formula": "\\begin{align*} P _ 2 ( T ) = K _ x . \\end{align*}"}
+{"id": "1206.png", "formula": "\\begin{align*} | \\mathcal { A } _ { \\mathcal { U } } | = \\prod _ { i = 1 } ^ { t } 2 ^ { \\frac { s _ i ( s _ i - 1 ) } { 2 } } , | \\mathcal { A } _ { \\mathcal { P } } | = \\prod _ { i = 1 } ^ { t } ( s _ i ! ) ~ . \\end{align*}"}
+{"id": "1830.png", "formula": "\\begin{align*} L _ \\rhd ( \\{ y , z \\} ) L _ \\rhd ( x ) & t = \\{ y , z \\} \\rhd ( x \\rhd t ) = \\{ \\{ x , y \\} , z \\} \\rhd t + y \\rhd ( \\{ x , z \\} \\rhd t ) + x \\rhd ( y \\rhd ( z \\rhd t ) ) - z \\rhd ( x \\rhd ( y \\rhd t ) ) \\\\ & = L _ \\rhd ( \\{ \\{ x , y \\} , z \\} ) t + L _ \\rhd ( y ) L _ \\rhd ( \\{ x , z \\} ) t + L _ \\rhd ( x ) L _ \\rhd ( y ) L _ \\rhd ( z ) t - L _ \\rhd ( z ) L _ \\rhd ( x ) L _ \\rhd ( y ) t . \\end{align*}"}
+{"id": "5881.png", "formula": "\\begin{align*} \\begin{aligned} & X _ { \\tau } = x _ { n } + \\tau h \\varphi _ { 1 } ( \\tau h M ) v _ { n } + h ^ 2 \\int _ { 0 } ^ { 1 } { \\alpha } _ { \\tau \\sigma } ( h M ) F ( X _ \\sigma ) d \\sigma , \\\\ & x _ { n + 1 } = x _ { n } + h \\varphi _ 1 ( h M ) v _ { n } + h ^ 2 \\int _ { 0 } ^ { 1 } \\beta _ { \\tau } ( h M ) F ( X _ { \\tau } ) d \\tau , \\ \\ v _ { n + 1 } = \\varphi _ 0 ( h M ) v _ { n } + h \\int _ { 0 } ^ { 1 } \\gamma _ { \\tau } ( h M ) F ( X _ { \\tau } ) d \\tau , \\end{aligned} \\end{align*}"}
+{"id": "7355.png", "formula": "\\begin{align*} | \\mathcal { I } ' _ k | = \\sum ^ k _ { i = 0 } | \\mathcal { B } _ { i , i } | + \\sum ^ k _ { i = 1 } \\sum ^ i _ { l = 1 } | \\mathcal { B } _ { i , i - l } | + \\sum ^ k _ { i = 1 } \\sum ^ i _ { l = 1 } | \\mathcal { B } _ { i - l , i } | . \\end{align*}"}
+{"id": "2786.png", "formula": "\\begin{align*} \\| r _ i ^ \\mathrm { S V D } \\| _ 2 = \\sqrt { \\| A \\tilde { v } _ i - \\tilde { \\sigma } _ i \\tilde { u } _ i \\| _ 2 ^ 2 + \\| A ^ T \\tilde { u } _ i - \\tilde { \\sigma } _ i \\tilde { v } _ i \\| _ 2 ^ 2 } . \\end{align*}"}
+{"id": "3664.png", "formula": "\\begin{align*} | \\partial \\widehat { \\mathcal { G } } _ { \\alpha } ( n ) | & = \\sum _ { v _ i v _ j \\in \\partial \\mathcal { G } \\cap \\binom { W } { 2 } } | V _ i | | V _ j | + \\sum _ { v _ i \\in N _ { \\mathcal { G } } ( v _ 1 ) } | V _ i | \\left ( | V _ 1 | + | V _ 1 | ' \\right ) + \\sum _ { v _ i \\in N _ { \\mathcal { G } } ( v _ 2 ) } | V _ i | \\left ( | V _ 2 | + | V _ 2 | ' \\right ) \\\\ & + \\left ( | V _ 1 | + | V _ 1 ' | \\right ) \\left ( | V _ 2 | + | V _ 2 ' | \\right ) + | V _ 1 | | V _ 1 ' | + | V _ 2 | | V _ 2 ' | . \\end{align*}"}
+{"id": "6174.png", "formula": "\\begin{align*} \\sum _ { n = [ q _ { 1 } ^ { j _ { 1 } } , \\ldots , q _ { \\lambda } ^ { j _ { \\lambda } } ] } \\log q _ { 1 } \\cdots \\log q _ { \\lambda } & = \\sum _ { \\lambda _ 1 + \\cdots + \\lambda _ s = \\lambda } \\frac { \\lambda ! } { \\lambda _ 1 ! \\cdots \\lambda _ s ! } \\prod _ { i = 1 } ^ s ( h _ i ^ { \\lambda _ i } - ( h _ i - 1 ) ^ { \\lambda _ i } ) ( \\log q _ i ) ^ { \\lambda _ i } \\\\ & = \\rho _ { \\lambda } ( n ) . \\end{align*}"}
+{"id": "4743.png", "formula": "\\begin{align*} \\hat T ( a + u ) = T ( u ) , \\forall a \\in A , u \\in V . \\end{align*}"}
+{"id": "25.png", "formula": "\\begin{align*} I _ f ( p , q ) & = \\sum _ { i \\in A _ { 1 + \\kappa , \\infty } } q _ i f \\left ( \\frac { p _ i } { q _ i } \\right ) + \\sum _ { i \\in A _ { 1 , 1 + \\kappa } } q _ i f \\left ( \\frac { p _ i } { q _ i } \\right ) \\\\ & + \\sum _ { i \\in A _ { 1 / ( 1 + \\kappa ) , 1 } } q _ i f \\left ( \\frac { p _ i } { q _ i } \\right ) + \\sum _ { i \\in A _ { 0 , 1 / ( 1 + \\kappa ) } } q _ i f \\left ( \\frac { p _ i } { q _ i } \\right ) , \\end{align*}"}
+{"id": "8077.png", "formula": "\\begin{align*} P ^ { - 1 } A x = P ^ { - 1 } b . \\end{align*}"}
+{"id": "388.png", "formula": "\\begin{align*} \\frac { L _ \\infty ( M , \\omega ) _ { [ N ] } } { I _ { L _ \\infty } { ( N ) } } = \\frac { F _ N \\cap \\mathcal { Q } ' } { I _ \\mu + I _ N \\cap \\mathcal { Q } } ~ . \\end{align*}"}
+{"id": "7572.png", "formula": "\\begin{align*} E _ { \\varphi } ( F ) = E _ { \\varphi } [ \\mu ] \\leq E _ { \\varphi } [ \\mu ^ { \\varepsilon } _ { - \\varepsilon } ] , \\end{align*}"}
+{"id": "3454.png", "formula": "\\begin{align*} \\mathcal { S } \\mathbb { H } = ( \\frac { 1 } { 2 } I _ { d } - \\mathcal { K } ) ( \\frac { 1 } { 2 } I _ { d } + \\mathcal { K } ) . \\end{align*}"}
+{"id": "923.png", "formula": "\\begin{align*} \\begin{aligned} F ^ { i j } ( b _ { n - 1 } ( A _ 1 v - A _ 2 w ) \\pm ( u - \\varphi ) ) _ { i j } \\geq \\ , & 0 \\mbox { i n } \\omega \\\\ b _ { n - 1 } ( A _ 1 v - A _ 2 w ) \\pm ( u - \\varphi ) \\leq \\ , & 0 \\mbox { o n } \\partial \\omega \\end{aligned} \\end{align*}"}
+{"id": "7539.png", "formula": "\\begin{align*} \\mathcal F _ { \\delta } = \\overline { \\mathcal T } _ { \\delta } , \\mathcal F = \\bigcup _ { 0 < \\delta < \\delta _ 0 } \\mathcal F _ { \\delta } , \\end{align*}"}
+{"id": "2718.png", "formula": "\\begin{align*} \\binom { 2 n } { j } \\binom { 2 n - j } { u } = \\binom { 2 n } { 2 n - j - u } \\binom { j + u } { u } \\end{align*}"}
+{"id": "453.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } { n \\choose k } F _ { j k } F _ { j ( n - k ) } = \\frac { 2 ^ n L _ { j n } - 2 L _ j ^ n } { 5 } , \\end{align*}"}
+{"id": "6898.png", "formula": "\\begin{align*} \\begin{cases} ( i \\dd _ t + \\Delta ) u _ n = g _ n ( u _ n ) \\\\ u _ n ( 0 ) = u _ { n , 0 } \\in H ^ 2 ( \\R ^ d ) \\end{cases} \\end{align*}"}
+{"id": "759.png", "formula": "\\begin{align*} 0 = V _ 0 \\subset V _ 1 \\subset V _ 2 \\subset \\cdots \\subset V _ n \\subseteq V , \\end{align*}"}
+{"id": "972.png", "formula": "\\begin{align*} \\partial _ j \\xi _ \\tau ( 0 , \\omega ) = 2 \\pi i \\ ; y _ j ( \\xi ) \\ ; \\xi ( \\omega ) . \\end{align*}"}
+{"id": "3451.png", "formula": "\\begin{align*} \\Large { \\Gamma } \\Big ( 0 , \\frac { | \\xi - \\mathrm { z } | ^ 2 } { 4 \\mathrm { t } } \\Big ) = \\mathcal { E } _ 1 \\Big ( \\frac { | \\xi - \\mathrm { z } | ^ 2 } { 4 \\mathrm { t } } \\Big ) , \\end{align*}"}
+{"id": "4490.png", "formula": "\\begin{align*} \\{ f , g \\} = \\langle D _ 1 f , D _ 2 g \\rangle _ { H } - \\langle D _ 1 g , D _ 2 f \\rangle _ { H } \\ , . \\end{align*}"}
+{"id": "7954.png", "formula": "\\begin{align*} \\mbox { i f $ M $ i s u m b i l i c w i t h $ l = 3 k $ t h e n } \\ , k ( \\xi ) = \\frac { 2 n - 1 } { 2 n + 1 } H _ M ( \\xi ) \\mbox { f o r a l l } \\xi \\in M \\smallsetminus S _ M . \\end{align*}"}
+{"id": "4814.png", "formula": "\\begin{align*} & F _ n = { } _ { 0 } F _ { 2 } \\ ! \\left ( \\left . \\begin{matrix} - \\\\ \\lambda + n , 1 - \\mu - n \\end{matrix} \\right | x \\right ) : \\\\ & - F _ n ' = \\frac { ( n + \\mu ) } { ( n + \\lambda ) ( 2 n + \\lambda + \\mu ) ( 2 n + \\lambda + \\mu - 1 ) } F _ { n + 1 } + { \\frac { 2 } { ( 2 n + \\lambda + \\mu ) ( 2 n + \\lambda + \\mu - 2 ) } } F _ n \\\\ & + \\frac { ( n + \\lambda - 1 ) } { ( n + \\mu - 1 ) ( 2 n + \\lambda + \\mu - 1 ) ( 2 n + \\lambda + \\mu - 2 ) } F _ { n - 1 } ; \\\\ \\end{align*}"}
+{"id": "3034.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { a ^ 2 } { ( a + b \\bar { u } _ 2 ) ^ 3 } & = 0 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "1200.png", "formula": "\\begin{align*} G : = { \\rm G a l } ( M ' / K ' ) = { \\rm G a l } ( M ' / M _ 1 ' ) \\times { \\rm G a l } ( M ' / M _ 2 ' ) \\cong C _ 2 \\times C _ 2 \\end{align*}"}
+{"id": "518.png", "formula": "\\begin{align*} d X ^ i _ t = \\alpha _ i ( t , X _ t ) d t + d B ^ i _ t , X ^ i _ 0 = 0 , \\ \\ i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "3812.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 n + h ^ 2 - h - 1 2 } { 2 } . \\end{align*}"}
+{"id": "6305.png", "formula": "\\begin{align*} X _ t = x _ 0 ( t ) + \\int _ 0 ^ t K _ \\mu ( s , t ) \\mu ( s , X _ s ) \\dd s + \\int _ 0 ^ t K _ \\sigma ( s , t ) \\sigma ( s , X _ s ) \\dd B _ s , t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "7162.png", "formula": "\\begin{align*} \\{ x _ { i , i } - x _ { i + 1 , i + 1 } = 0 , y _ { i , i } - y _ { i + 1 , i + 1 } = 0 \\} . \\end{align*}"}
+{"id": "3869.png", "formula": "\\begin{align*} k ^ { ( r ) } _ { 0 } = \\# \\{ i : x _ i \\equiv a \\} + \\# \\{ i : x _ i \\equiv \\bar { a } \\} , k ^ { ( c ) } _ { 0 } = \\# \\{ i : y _ i \\equiv a \\} + \\# \\{ i : y _ i \\equiv \\bar a \\} , \\end{align*}"}
+{"id": "2293.png", "formula": "\\begin{align*} \\begin{cases} \\kappa _ d < \\vartheta _ d & 3 \\leq d \\leq 2 0 , \\\\ \\kappa _ d = \\vartheta _ d & d \\geq 2 1 . \\end{cases} \\end{align*}"}
+{"id": "4719.png", "formula": "\\begin{align*} \\partial ( a \\cdot b ) = \\partial ( a ) \\cdot b + a \\cdot \\partial ( b ) , \\forall a , b \\in A . \\end{align*}"}
+{"id": "6203.png", "formula": "\\begin{align*} E ^ * _ { i + k } A _ 1 E ^ * _ { i + k - 1 } A _ 1 E ^ * _ { i + k - 2 } \\cdots E ^ * _ { i + 1 } A _ 1 E ^ * _ { i } = \\big ( ( \\frac { k - 1 } { 2 } ) ! \\big ) ^ 2 \\frac { k + 1 } { 2 } M ^ { \\frac { k - 1 } { 2 } , \\frac { k - 1 } { 2 } } _ { \\frac { i + k - 1 } { 2 } , \\frac { 2 m - i } { 2 } } . \\end{align*}"}
+{"id": "5514.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\frac { ( 1 - a t q ) } { ( 1 - t ) } + \\frac { ( 1 - a q ) ( b - a t q ) t q } { ( 1 - b q ) ( 1 - t ) } F ( a q , b q ; t q ) . \\end{align*}"}
+{"id": "7083.png", "formula": "\\begin{align*} p _ { H , r _ i } ( Z _ r , Z _ { r _ i } , Z ' _ { r _ i } ) = \\begin{cases} 1 & Z ^ { \\prime } _ { r _ i } = Z _ { r _ i } , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "6758.png", "formula": "\\begin{align*} W ^ { - 1 } ( f ) = W ( - f ) , W ( f ) W ( g ) = W ( g ) W ( f ) e ^ { - 2 i \\Im \\langle f , g \\rangle } = W ( f + g ) e ^ { - i \\Im \\langle f , g \\rangle } \\end{align*}"}
+{"id": "2.png", "formula": "\\begin{align*} \\sum _ { ( d , 2 ) = 1 } M _ Z ( d ) \\left ( \\frac { d } { p } \\right ) W \\left ( \\frac { d } { X } \\right ) = \\sum _ { \\substack { \\alpha \\leq Z \\\\ ( \\alpha , 2 p ) = 1 } } \\mu ( \\alpha ) \\sum _ { ( d , 2 ) = 1 } \\left ( \\frac { d } { p } \\right ) W \\left ( \\frac { d \\alpha ^ 2 } { X } \\right ) . \\end{align*}"}
+{"id": "5469.png", "formula": "\\begin{align*} F _ { N } ( a , b ; t ) : = \\sum _ { n = 0 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( a q ) _ { n } ( t ) _ { N - n } ( q ) _ n t ^ { n } } { ( b q ) _ { n } ( t ) _ N } . \\end{align*}"}
+{"id": "2240.png", "formula": "\\begin{align*} E _ { i + 1 } = p _ i ( \\boldsymbol { \\mathcal { T } } _ n ) * E _ 1 , E _ { i + 1 } ^ D = E _ 1 ^ D * q _ i ^ D ( \\boldsymbol { \\mathcal { T } } _ n ) . \\end{align*}"}
+{"id": "2471.png", "formula": "\\begin{align*} \\Delta G ( \\omega ) = G ( \\Delta \\omega ) = \\omega - H ( \\omega ) \\end{align*}"}
+{"id": "5118.png", "formula": "\\begin{align*} \\| u \\| _ { L _ x ^ { \\frac { p d } { 2 } } H _ { y } ^ { \\frac { 1 } { 2 } + \\delta } } = o ( 1 ) . \\end{align*}"}
+{"id": "1690.png", "formula": "\\begin{align*} i \\partial _ t \\varphi ( x ) = ( - \\Delta + \\kappa ) \\varphi ( x ) - | \\varphi ( x ) | ^ 2 \\varphi ( x ) \\ , , \\end{align*}"}
+{"id": "5575.png", "formula": "\\begin{align*} \\omega _ k ( \\vec { z } \\ , ) : = 2 ^ { k n ( | J | - 1 / p _ J ) } \\chi _ { V _ k ^ J \\cap S ^ { m n - 1 } } ( \\vec { z } \\ , ) \\end{align*}"}
+{"id": "6386.png", "formula": "\\begin{align*} \\varepsilon \\Pi = B ^ { \\mathsf { T } } \\Pi = ( \\widehat { D } , 0 ) , \\end{align*}"}
+{"id": "5373.png", "formula": "\\begin{align*} V _ \\circ = V _ \\circ ^ { \\omega \\omega } . \\end{align*}"}
+{"id": "3928.png", "formula": "\\begin{align*} x \\ast _ { ( \\alpha , \\beta ) } y = \\alpha \\left ( x \\right ) \\ast \\beta \\left ( y \\right ) , \\ ; \\ ; \\ ; \\forall \\ ; \\ ; x , y \\in A . \\end{align*}"}
+{"id": "2565.png", "formula": "\\begin{align*} \\partial _ x U ( t , x , \\nu ) = P _ t , \\quad \\partial _ { x x } U ( t , x , \\nu ) = \\partial _ { x \\nu } U ( t , x , \\nu ) = 0 , \\quad \\partial _ { \\nu } U ( t , x , \\nu ) = \\partial _ { \\nu } \\Phi ( t , \\nu ) , \\quad \\partial _ { \\nu \\nu } U ( t , x , \\nu ) = \\partial _ { \\nu \\nu } \\Phi ( t , \\nu ) \\end{align*}"}
+{"id": "4539.png", "formula": "\\begin{align*} \\{ a \\in \\mathbb { R } : f ^ { - 1 } ( a , \\infty ) = \\varnothing \\} \\subset \\{ a \\in \\mathbb { R } : \\mu ( f ^ { - 1 } ( a , \\infty ) ) = 0 \\} \\ , , \\end{align*}"}
+{"id": "2121.png", "formula": "\\begin{align*} U _ I : = \\left \\{ ( x _ 1 , \\ldots , x _ n ) \\in ( K ^ \\times ) ^ n \\ \\Big | \\ \\begin{array} { l l } x _ i \\in K \\backslash R & i \\in I , \\\\ x _ i \\in R \\backslash \\{ 0 \\} & i \\notin I \\end{array} \\right \\} . \\end{align*}"}
+{"id": "356.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\theta _ q ( u _ { \\varphi ( n ) } - u ) \\| _ { L ^ \\infty ( [ 0 , T _ q ] , H ^ { - 4 } ) } = 0 . \\end{align*}"}
+{"id": "5570.png", "formula": "\\begin{align*} B ( \\beta _ 0 ) = \\dfrac { | u _ c | L ^ { 2 } _ { 2 M } ( 2 N _ { 2 M } ) ^ { \\frac { \\delta - 3 \\mu + 5 } { \\xi } } } { \\bigg [ \\exp \\bigg ( \\tfrac { 2 \\beta _ 0 ^ { \\xi } N _ { 2 M } } { L _ { 2 m } \\xi } \\bigg ) L _ { 2 m } ^ \\frac { \\delta - 3 \\mu + 5 } { \\xi } ( \\xi ) ^ { \\tfrac { 1 - \\delta - \\nu } { \\xi } } \\bigg [ \\Gamma \\bigg ( \\tfrac { 3 - \\nu - \\mu } { \\xi } \\bigg ) - \\gamma \\bigg ( \\tfrac { 3 - \\nu - \\mu } { \\xi } , \\tfrac { 2 N _ { 2 M } \\beta _ 0 ^ { \\xi } } { L _ { 2 m } \\xi } \\bigg ) \\bigg ] \\bigg ] ^ 2 } > 0 . \\end{align*}"}
+{"id": "3249.png", "formula": "\\begin{align*} \\int _ { X } \\psi ( u - v _ { j } ^ { k } ) \\theta ^ { n } _ { v _ { j } ^ { k } } \\leq \\sum _ { l = j } ^ { k } \\int _ { X } \\psi ( u - u _ { l } ) \\theta ^ { n } _ { u _ { l } } \\leq \\sum _ { l = j } ^ { k } I _ { \\psi } ( u , u _ { l } ) \\leq C ^ { - 2 j + 2 } . \\end{align*}"}
+{"id": "2514.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } T ^ { c _ 1 ( n ) } ( x _ 0 ) = x _ 1 . \\end{align*}"}
+{"id": "4530.png", "formula": "\\begin{align*} \\mathfrak { J } : = \\Big \\{ \\big \\{ \\mathrm { T r } \\ , h _ { \\gamma } , \\ ^ 2 E _ S [ f ] \\big \\} , \\ ^ 2 E _ S [ g ] \\Big \\} & + \\Big \\{ \\big \\{ ^ 2 E _ S [ f ] , ^ 2 E _ S [ g ] \\big \\} , \\mathrm { T r } \\ , h _ { \\gamma } \\Big \\} \\\\ & + \\Big \\{ \\big \\{ ^ 2 E _ S [ g ] , \\mathrm { T r } \\ , h _ { \\gamma } \\big \\} , ^ 2 E _ S [ f ] \\Big \\} \\ . \\end{align*}"}
+{"id": "8237.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\Big ( \\Vert \\sigma ( t ) \\Vert _ { \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } } + \\Vert u ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } } \\Big ) = 0 . \\end{align*}"}
+{"id": "1629.png", "formula": "\\begin{align*} \\| m _ { u _ { n + 1 } v _ { n + 1 } } - m _ { p _ i q _ i } \\| & = \\frac { d ( u _ { n + 1 } , p _ i ) + d ( q _ i , v _ { n + 1 } ) + | d ( u _ { n + 1 } , v _ { n + 1 } ) - d ( p _ i , q _ i ) | } { \\max \\big \\{ d ( u _ { n + 1 } , v _ { n + 1 } ) , d ( p _ i , q _ i ) \\big \\} } \\\\ & = \\frac { 1 + 1 + | 2 - 2 | } { \\max \\big \\{ 2 , 2 \\big \\} } \\\\ & = 1 \\end{align*}"}
+{"id": "2755.png", "formula": "\\begin{align*} a ( H ) = ( a ( H ) \\Lambda - b ( H ) \\pi ^ * _ 1 H ) \\cdot \\ell _ 1 = \\mu ^ * \\pi _ 2 ^ * H \\cdot \\ell _ 1 = \\pi _ 2 ^ * H \\cdot \\mu _ * ( \\ell _ 1 ) \\end{align*}"}
+{"id": "6863.png", "formula": "\\begin{align*} v _ i ^ { \\sqrt { q } + 1 } = v _ i '^ 2 , \\ 1 \\leq i \\leq n . \\end{align*}"}
+{"id": "2087.png", "formula": "\\begin{align*} \\| x _ 1 e _ 1 + \\ldots + x _ n e _ n \\| : = \\| ( x _ 1 , \\ldots , x _ n ) \\| = \\max _ i | x _ i | , \\end{align*}"}
+{"id": "2858.png", "formula": "\\begin{align*} \\mathsf { D } _ { \\leq 0 } & = \\{ X \\in \\mathsf { D } \\mid \\forall n \\in \\Z _ { > 0 } , \\ , H ^ n ( X ) = 0 \\} ; \\\\ \\mathsf { D } _ { \\geq 0 } & = \\{ X \\in \\mathsf { D } \\mid \\forall n \\in \\Z _ { < 0 } , \\ , H ^ n ( X ) = 0 \\} , \\end{align*}"}
+{"id": "4738.png", "formula": "\\begin{gather*} L _ A ( a ) \\eth _ k = L _ A ( \\partial _ k ( a ) ) + \\eth _ k L _ A ( a ) , \\qquad \\ ! \\partial _ k R _ A ( a ) = R _ A ( a ) \\partial _ k + R _ A ( \\partial _ k ( a ) ) , \\qquad \\ ! \\forall a \\in A . \\ ! \\ ! \\ ! \\end{gather*}"}
+{"id": "752.png", "formula": "\\begin{align*} [ a _ \\lambda b ] = \\sum _ { j \\in \\mathbb { Z _ { + } } } ( a _ { ( j ) } b ) \\frac { \\lambda ^ { j } } { j ! } . \\end{align*}"}
+{"id": "8226.png", "formula": "\\begin{align*} U _ L ( t ) : = \\int _ 0 ^ { t } \\Vert u _ L ( \\tau ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } } \\dd \\tau \\le C \\sum _ { j \\in \\mathbb { Z } } 2 ^ { j ( \\frac { N } { 2 } + 1 - \\alpha ) } ( 1 - e ^ { - \\mu t 2 ^ { j \\alpha } } ) \\Vert \\dot { \\Delta } _ j u _ 0 \\Vert _ { L ^ 2 } , \\end{align*}"}
+{"id": "3775.png", "formula": "\\begin{align*} C _ { n + 1 } = \\sum _ { i = 0 } ^ n C _ i C _ { n - i } \\end{align*}"}
+{"id": "5439.png", "formula": "\\begin{align*} & \\{ \\sum _ i r _ i \\mu _ i \\eta _ i ^ * \\ | \\ r _ i \\in R , \\mu _ i , \\eta _ i \\in E ^ * , \\mathrm { r } ( \\mu _ i ) = \\mathrm { r } ( \\eta _ i ) , w ( \\mu _ i ) = w ( \\eta _ i ) = s \\} \\ , \\cup \\\\ & \\{ \\sum _ i r _ i \\mu _ i \\eta _ i ^ * \\ | \\ r _ i \\in R , \\mu _ i , \\eta _ i \\in E ^ * , \\mathrm { r } ( \\mu _ i ) = \\mathrm { r } ( \\eta _ i ) \\in \\mathrm { S i n k } ( E ) , w ( \\mu _ i ) = w ( \\eta _ i ) < s \\} . \\end{align*}"}
+{"id": "1355.png", "formula": "\\begin{align*} & \\left | \\nabla u ( x _ 0 ) \\right | = \\lim _ { r \\rightarrow 0 } a ( r , x _ 0 ) \\le C ( M , \\eta ) ( 1 + a ( 1 ) ) = C \\Big ( 1 + \\left \\| \\nabla u \\right \\| _ { L ^ p ( B _ 1 ) } \\Big ) , \\end{align*}"}
+{"id": "7428.png", "formula": "\\begin{align*} J ^ 2 = J _ z ^ 2 + \\frac { 1 } { 2 } ( J _ + J _ - + J _ - J _ + ) , \\end{align*}"}
+{"id": "5350.png", "formula": "\\begin{align*} \\theta \\wedge \\omega = \\zeta _ \\theta \\wedge d \\theta ^ { k - n } . \\end{align*}"}
+{"id": "3402.png", "formula": "\\begin{align*} & U _ t + U U _ r + ( p _ S S _ r + p _ { \\rho } \\rho _ r ) / \\rho = 0 , \\\\ & \\rho _ t + ( U \\rho ) _ r + \\tfrac { n - 1 } { r } U \\rho = 0 , \\\\ & S _ t + U S _ r = 0 , \\end{align*}"}
+{"id": "77.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { \\Psi } u = A u + B & \\O \\\\ u = c _ { j } & ( \\partial \\O ) _ { j } \\end{cases} \\end{align*}"}
+{"id": "3541.png", "formula": "\\begin{align*} \\Phi ( \\mathrm { y } , \\tau ; \\mathrm { v } , \\mathrm { s } ) : = \\ \\begin{cases} \\dfrac { \\alpha } { 4 \\pi ( \\tau - \\mathrm { s } ) } \\textbf { e x p } \\Big ( - \\dfrac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 ( \\tau - \\mathrm { s } ) } \\Big ) , \\tau > \\mathrm { s } \\\\ 0 , \\end{cases} , \\end{align*}"}
+{"id": "975.png", "formula": "\\begin{align*} \\int _ { G ^ \\wedge } \\left ( 1 + \\gamma ( \\xi ) ^ 2 \\right ) \\ , { \\vert [ T ( f ) ] ( \\xi ) \\vert } ^ 2 d \\nu ( \\xi ) = \\int _ { G ^ \\wedge } \\vert f ( \\xi ) \\vert ^ 2 d \\nu ( \\xi ) . \\end{align*}"}
+{"id": "1132.png", "formula": "\\begin{align*} E ( X - Y ) ^ { n } \\allowbreak = \\allowbreak \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ \\frac { n ! } { ( n / 2 ) ! } \\left ( \\beta \\right ) ^ { \\left ( n / 2 \\right ) } ( 1 - \\rho ) ^ { n / 2 } & & n \\end{array} \\right . . \\end{align*}"}
+{"id": "167.png", "formula": "\\begin{align*} \\begin{aligned} \\beta ( \\arccos \\tfrac { | v _ 1 - v | } { \\sqrt { | v _ 1 - v | ^ 2 + | y | ^ 2 } } ) \\leq \\beta _ 0 | v _ 1 - v | \\big ( | v _ 1 - v | ^ 2 + | y | ^ 2 \\big ) ^ { - \\frac { 1 } { 2 } } \\ , . \\end{aligned} \\end{align*}"}
+{"id": "3993.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\mathrm { d } ^ { \\beta } } { \\mathrm { d } t ^ { \\beta } } \\hat { q } _ { \\beta } ( 0 , t ) & = - \\lambda \\hat { q } _ { \\beta } ( 0 , t ) , \\\\ \\frac { \\mathrm { d } ^ { \\beta } } { \\mathrm { d } t ^ { \\beta } } \\hat { q } _ { \\beta } ( n , t ) & = - \\lambda \\hat { q } _ { \\beta } ( n , t ) - \\frac { \\lambda } { \\ln p } \\sum _ { j = 1 } ^ { n } \\frac { ( 1 - p ) ^ { j } } { j } \\hat { q } _ { \\beta } ( n - j , t ) , \\ n \\ge 1 , \\end{aligned} \\end{align*}"}
+{"id": "1122.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } l _ { n } ( x | \\beta ) l _ { m } ( x | \\beta ) f _ { g } ( x | \\beta ) d x = \\delta _ { n , m } . \\end{align*}"}
+{"id": "7122.png", "formula": "\\begin{align*} \\chi + \\rho + \\delta = - \\sum _ { \\ell \\in T } r _ \\ell ( 3 \\mathfrak { g } _ \\ell ) + \\sum _ { i = 1 } ^ k \\psi _ i + ( w + d \\mu ) \\tau _ d . \\end{align*}"}
+{"id": "592.png", "formula": "\\begin{align*} { \\widehat { \\mathcal { Q } } _ n } ( z ) = \\frac { ( z - j ^ { ( 0 ) } + j ^ { ( 4 ) } ) ( z + j ^ { ( 0 ) } + j ^ { ( 4 ) } + 1 ) ( z - j ^ { ( 3 ) } + j _ n ^ { ( 1 2 ) } ) ( z - j ^ { ( 3 ) } - j _ n ^ { ( 1 2 ) } - 1 ) } { 2 z \\ , ( 2 z - 1 ) } \\ , . \\end{align*}"}
+{"id": "5928.png", "formula": "\\begin{align*} f _ 1 = \\tfrac { 1 } { 2 } \\cdot H ^ 4 - H ^ 2 e _ 2 \\ , , f _ 2 = - H ^ 4 + \\tfrac { 5 } { 2 } \\cdot H ^ 2 e _ 2 \\end{align*}"}
+{"id": "644.png", "formula": "\\begin{align*} \\mathcal { P } ' \\left ( \\partial _ Y \\Phi ( X ) \\right ) = - \\sum _ { i = 1 , 2 } \\sqrt { c _ i } \\ \\langle Y _ i , X \\rangle \\mathcal { P } ' ( N _ i ) + \\Phi ( \\mathcal { P } ( B ( Y , X _ T ) ) - \\mathcal { P } ( B ^ * ( Y , X _ N ) ) ) ; \\end{align*}"}
+{"id": "8057.png", "formula": "\\begin{align*} y _ { C _ u } [ \\widetilde { n } ] = & \\sum _ { k = 1 } ^ { K } ( { \\bf h } _ { u , k } { \\bf F } _ k { \\bf s } _ k e ^ { \\mathrm { j } 2 \\pi { f _ k } \\frac { \\widetilde { n } \\triangle t } { K } } \\\\ & + \\sum _ { m = 1 } ^ { M } { \\bf h } _ { u , m , k } { \\bf \\Phi } _ m { \\bf G } _ { m , k } { \\bf F } _ k { \\bf s } _ k e ^ { \\mathrm { j } 2 \\pi { f _ k } \\frac { \\widetilde { n } \\triangle t } { K } } + n _ { C _ u } [ \\widetilde { n } ] ) . \\end{align*}"}
+{"id": "4086.png", "formula": "\\begin{align*} \\sum _ { k | n } k \\cdot e _ k = 2 ^ n \\end{align*}"}
+{"id": "1932.png", "formula": "\\begin{align*} & \\widetilde z = ( r ^ 2 t + t _ 0 , r ^ 3 x + x _ 0 - r ^ 2 t v _ 0 , r v + v _ 0 ) , \\widetilde u ( z ) = u ( \\widetilde z ) , \\\\ & Y = \\partial _ t - v \\cdot D _ x , \\widetilde P _ 0 = \\partial _ t - v \\cdot D _ x - a ^ { i j } ( r ^ 2 t + t _ 0 ) D _ { v _ i v _ j } . \\end{align*}"}
+{"id": "1259.png", "formula": "\\begin{align*} M ( A / \\left | x \\right | , \\gamma , \\lambda ) = 1 . \\end{align*}"}
+{"id": "4640.png", "formula": "\\begin{align*} \\mathcal { H } _ 0 = - \\Delta _ { \\vec { x } } - \\tfrac { 3 } { 4 } \\ , \\Delta _ { \\vec { y } } . \\end{align*}"}
+{"id": "7751.png", "formula": "\\begin{align*} \\mathcal { A } ( p _ 0 ) = 2 \\omega _ { n - 1 } \\int _ 0 ^ { \\lambda \\pi } \\frac { 1 } { \\cosh ^ n y } d y . \\end{align*}"}
+{"id": "1267.png", "formula": "\\begin{align*} \\Delta = t ^ 4 ( a _ 3 '^ 4 + t ^ 5 a _ 3 '^ 3 + t ^ 6 \\hdots ) . \\end{align*}"}
+{"id": "542.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( T _ i ( x ^ \\pm _ n ) - x ^ \\pm _ n ) Q _ i ( x ^ \\pm _ n ) = 0 . \\end{align*}"}
+{"id": "2662.png", "formula": "\\begin{align*} ( W \\times T \\times L ) / D = ( V \\times U \\times T \\times L ) / D \\cong V \\times K \\end{align*}"}
+{"id": "651.png", "formula": "\\begin{align*} F _ 1 = \\frac { 1 } { \\sqrt { c _ 1 } } \\langle \\langle \\nu _ 1 \\cdot \\varphi , \\varphi \\rangle \\rangle . \\end{align*}"}
+{"id": "3040.png", "formula": "\\begin{align*} \\gamma ( x ) = x x \\in X \\setminus \\{ x _ 1 , \\ldots , x _ { \\ell } \\} \\end{align*}"}
+{"id": "2419.png", "formula": "\\begin{align*} \\sum _ { \\substack { i + j = n \\\\ j = 0 } } ( - 1 ) ^ { j } \\tilde { L } ( { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { i } u ) * e _ { 1 } e _ { 0 } ^ { j } ) & = \\tilde { L } ( { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { n } u ) * e _ { 1 } ) = \\tilde { L } ( { \\rm D } ( { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { n } u ) ) ) . \\end{align*}"}
+{"id": "3823.png", "formula": "\\begin{align*} a _ 1 + a _ 2 + \\cdots + a _ { h - 2 } = \\frac { ( h - 1 ) ( 2 n - 5 ) - ( a _ { h - 1 } + a _ h ) } { 2 } . \\end{align*}"}
+{"id": "1537.png", "formula": "\\begin{align*} F ^ { \\prime } \\left ( f _ { 2 } \\right ) \\circ F ^ { \\prime } \\left ( f _ { 1 } \\right ) \\circ \\eta _ { b _ { i } } = \\eta _ { b _ { k } } \\circ F \\left ( f _ { 2 } \\right ) \\circ F \\left ( f _ { 1 } \\right ) \\end{align*}"}
+{"id": "5366.png", "formula": "\\begin{align*} \\mathsf { I } _ \\rho = \\C \\times \\mathfrak { I } . \\end{align*}"}
+{"id": "763.png", "formula": "\\begin{align*} [ \\phi _ { \\lambda } \\psi ] _ { \\mu } a = \\phi _ { \\lambda } ( \\psi _ { \\mu - \\lambda } a ) - \\psi _ { \\mu - \\lambda } ( \\phi _ { \\lambda } a ) , \\forall \\ a \\in \\mathcal { A } , \\end{align*}"}
+{"id": "4076.png", "formula": "\\begin{align*} d H = \\Lambda \\cdot H \\end{align*}"}
+{"id": "7453.png", "formula": "\\begin{align*} \\begin{matrix} \\tau ^ \\dagger _ { - 1 } = p _ 0 ^ \\dagger \\hat { j } - 2 p _ 1 ^ \\dagger , \\tau ^ \\dagger _ { 1 } = p _ 0 ^ \\dagger ( \\hat { j } + 1 ) + 2 p _ 1 ^ \\dagger . \\end{matrix} \\end{align*}"}
+{"id": "221.png", "formula": "\\begin{align*} \\frac { \\log \\mu _ { n + 1 } ( x ) - \\log \\mu _ n ( x ) } { \\tau } = - a ( \\mu _ { n + 1 } , x ) \\ , . \\end{align*}"}
+{"id": "3557.png", "formula": "\\begin{align*} \\widetilde { I ^ n M } = \\bigcup _ { k \\ge 1 } I ^ { n + k } M : I ^ k = I ^ { n + l } : I ^ l \\ \\ l \\gg 0 . \\end{align*}"}
+{"id": "7811.png", "formula": "\\begin{align*} J ( X ) = - \\frac { 1 } { 2 } \\int _ { - \\infty } ^ { \\infty } f ^ 2 ( x ) d x = - \\frac { 1 } { 2 } E \\left ( f ( X ) \\right ) . \\end{align*}"}
+{"id": "7314.png", "formula": "\\begin{align*} a = & ( \\frac { q + 1 } 2 ) q ^ { m - 1 } + ( \\frac { q - 1 } 2 ) q ^ { m - 2 } + \\cdots + ( \\frac { q - 1 } 2 ) q ^ { m - j _ 4 + j _ 3 } + ( \\frac { q + 1 } 2 ) q ^ { m - j _ 4 + j _ 3 - 1 } \\\\ & + i _ { j _ 3 - 1 } q ^ { m - j _ 4 + j _ 3 - 2 } + \\cdots + i _ 0 q ^ { m - j _ 4 - 1 } - ( \\frac { q - 3 } 2 ) q ^ { m - j _ 4 - 2 } - i _ { m - 2 } q ^ { m - j _ 4 - 3 } - \\cdots - i _ { j _ 4 + 1 } . \\end{align*}"}
+{"id": "1647.png", "formula": "\\begin{align*} \\gamma _ { \\mathsf { q } } \\ ; = \\ ; - \\ , \\gamma _ { \\mathsf { L } - \\mathsf { q } + 1 } ^ { \\prime } \\ , , \\qquad \\mathsf { q } = 1 , \\dots , \\mathsf { L } \\ , . \\end{align*}"}
+{"id": "4916.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { r - 1 } \\frac { j } { 2 } \\binom { n } { i } + j \\binom { n } { r } + \\sum _ { i = r + 1 } ^ { k - 1 } \\frac { j - 1 } { 2 } \\binom { n } { i } < \\binom { n } { k } . \\end{align*}"}
+{"id": "6916.png", "formula": "\\begin{align*} \\frac { z _ i ( \\alpha _ i + y ) R ( z _ i ) } { \\alpha _ i ( z _ i + y ) \\prod _ { j = 1 } ^ { N } ( z _ j - \\alpha _ i ) } = ( - 1 ) ^ N \\frac { ( z _ { N + 1 } - \\alpha _ i ) } { \\alpha _ i } \\iff ( \\alpha _ i + y ) z _ i R ( z _ i ) = - ( z _ i + y ) P ( \\alpha _ i ) \\end{align*}"}
+{"id": "682.png", "formula": "\\begin{align*} \\omega : = - p _ 1 ^ * ( \\log \\sqrt { \\tau _ 0 } ) _ z d z + p _ 2 ^ * \\omega _ 0 . \\end{align*}"}
+{"id": "2614.png", "formula": "\\begin{align*} g _ n ( t , q , x , \\mu ) : = g \\big ( t , \\frac { q } { \\sqrt { n } } , x , \\mu \\big ) \\end{align*}"}
+{"id": "1888.png", "formula": "\\begin{align*} \\chi _ { i , j } ( x _ k ) = \\begin{cases} x _ k , \\quad \\quad k \\neq i , \\\\ x _ j ^ { - 1 } x _ k x _ j , k = i . \\end{cases} \\end{align*}"}
+{"id": "6280.png", "formula": "\\begin{align*} \\varphi ( t _ { j + 1 } + ) < \\varphi ( t _ { j + 1 } ) & \\Leftrightarrow \\left ( \\norm { a _ { J _ { t _ j } } } _ 1 - \\lambda \\right ) t _ { j + 1 } + \\norm { a _ { I \\backslash J _ { t _ j } } } t _ { j + 1 } < \\left ( \\norm { a _ { J _ { t _ { j + 1 } } } } _ 1 - \\lambda \\right ) t _ { j + 1 } \\\\ & \\Leftrightarrow \\norm { a _ { I \\backslash J _ { t _ j } } } < \\norm { a _ { J _ { t _ { j + 1 } } } } _ 1 - \\norm { a _ { J _ { t _ j } } } _ 1 = | a _ { j + 1 } | \\\\ & \\Leftrightarrow \\sum _ { i = j + 2 , i \\in I } | a _ i | ^ 2 < 0 , \\end{align*}"}
+{"id": "6142.png", "formula": "\\begin{align*} V ^ { * j } V _ 2 = ( V _ 2 ^ * V _ 1 ^ * ) ^ j V _ 1 = \\overline q ^ j ( V _ 1 ^ * V _ 2 ^ * ) ^ j V _ 2 = \\overline q ^ j V _ 1 ^ * ( V _ 2 ^ * V _ 1 ^ * ) ^ { * j - 1 } = \\overline q ^ j V _ 1 ^ * V ^ { * j - 1 } . \\end{align*}"}
+{"id": "6788.png", "formula": "\\begin{align*} \\sum _ { i _ 1 \\in [ d _ 1 ] , \\dots , i _ r \\in [ d _ r ] } \\lambda _ { i _ 1 , \\dots , i _ r } \\alpha _ { 1 , i _ 1 } ( x _ { I _ 1 } ) \\dots \\alpha _ { r , i _ r } ( x _ { I _ r } ) + \\sum _ { i \\in [ s ] } \\Pi _ i ( x _ { [ k ] } ) = 0 \\end{align*}"}
+{"id": "1681.png", "formula": "\\begin{align*} \\mathbb { P } _ { ( W _ 0 , v _ 0 ) = ( W , v ) } \\left ( \\aleph _ { N - n } - \\aleph _ { 0 } \\geq 2 ^ { - \\frac { 5 } { 2 } } \\lambda ^ { - 1 } \\right ) \\leq \\exp \\left [ - \\frac { ( 2 ^ { - \\frac { 5 } { 2 } } \\lambda ^ { - 1 } ) ^ 2 } { 2 ( N - n ) } \\right ] \\leq \\exp \\left [ - 2 ^ { - 6 } N ^ { - 1 } \\lambda ^ { - 2 } \\right ] \\end{align*}"}
+{"id": "2513.png", "formula": "\\begin{align*} \\phi ( x _ { 0 0 } , x _ { 0 1 } , x _ { 1 0 } , x _ { 1 1 } ) = ( x _ { 0 0 } , x _ { 1 0 } , x _ { 1 0 } , x _ { 1 1 } ) . \\end{align*}"}
+{"id": "5848.png", "formula": "\\begin{align*} { } & a _ 1 ^ p + ( \\frac { 1 } { 2 } \\displaystyle \\sum _ { k = 1 } ^ { 2 } a _ k ) ^ p + \\cdots + ( \\frac { 1 } { N } \\displaystyle \\sum _ { k = 1 } ^ { N } a _ k ) ^ p + ( \\displaystyle \\sum _ { k = 1 } ^ { N } a _ k ) ^ p \\{ \\displaystyle \\sum _ { n = { N + 1 } } ^ { \\infty } \\frac { 1 } { n ^ p } \\} < ( \\frac { p } { p - 1 } ) ^ { p } ( \\sum _ { k = 1 } ^ { N } a _ { k } ^ p ) . \\end{align*}"}
+{"id": "8050.png", "formula": "\\begin{align*} { \\bf E } _ { m , k } & = { \\bf a } _ r ( \\theta _ t , f _ k ) { \\bf b } _ m ^ T ( \\theta _ { t , i _ m } , f _ k ) , ~ { \\bf D } _ { m , k } = { \\bf a } _ r ( \\theta _ { i _ m } , f _ k ) { \\bf b } _ m ^ T ( \\theta _ { d , i _ m } , f _ k ) , \\\\ { \\bf B } _ { m , k } & = { \\bf b } _ m ( \\theta _ { t , i _ m } , f _ k ) { \\bf a } _ t ^ T ( \\theta _ { t } , f _ k ) , ~ { \\bf G } _ { m , k } = { \\bf b } _ m ( \\theta _ { d , i _ m } , f _ k ) { \\bf a } _ t ^ T ( \\theta _ { i _ m } , f _ k ) , \\end{align*}"}
+{"id": "5542.png", "formula": "\\begin{align*} T ( r , t ) = \\dfrac { \\theta ( r , t ) - \\theta _ m } { \\theta _ b - \\theta _ m } \\end{align*}"}
+{"id": "4748.png", "formula": "\\begin{align*} a \\cdot b : = a \\succ b + a \\prec b , \\forall a , b \\in A , \\end{align*}"}
+{"id": "6323.png", "formula": "\\begin{align*} \\inf _ { u \\in A } \\ , E ( u ) \\le c : = \\inf _ { h \\in \\widetilde { H } } \\ , \\sup _ { u \\in h ( Q ) } \\ , E ( u ) \\le \\sup _ { u \\in Q } \\ , E ( u ) , \\end{align*}"}
+{"id": "6000.png", "formula": "\\begin{align*} \\dfrac { \\beta Z ^ { ( \\theta ) } ( b ^ * ) - 1 + \\lambda \\tilde { C } ^ { ( \\theta , r ) } ( b ^ * ; w ' _ { + } ) } { W ^ { ( \\theta ) } ( b ^ * ) } = \\beta \\theta \\dfrac { Z ^ { ( \\theta ) } ( b ^ * ; \\Phi ( \\theta + r ) ) } { Z ^ { ( \\theta ) \\prime } ( b ^ * ; \\Phi ( \\theta + r ) ) } . \\end{align*}"}
+{"id": "112.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } h ^ { - 1 } \\| \\alpha ( 0 ) \\| _ { \\ell ^ 2 } ^ 2 = \\lim _ { h \\to 0 } \\| P _ { \\leq N } \\psi _ 0 \\| _ { L ^ 2 } ^ 2 + \\| P _ { \\leq N } \\phi _ 0 \\| _ { L ^ 2 } ^ 2 = \\| \\psi _ 0 \\| _ { L ^ 2 } ^ 2 + \\| \\phi _ 0 \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "3055.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ { \\ell } ) = \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } k C _ { \\sigma } ( k ) . \\end{align*}"}
+{"id": "7019.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } + \\Delta ^ 2 u + \\Delta \\theta = 0 , \\\\ \\theta _ t - \\Delta \\theta + \\sigma \\theta - \\Delta u _ t = 0 , \\\\ ( u , u _ t , \\theta ) ( 0 , x ) = ( u _ 0 , u _ 1 , \\theta _ 0 ) ( x ) , \\end{cases} \\end{align*}"}
+{"id": "3184.png", "formula": "\\begin{align*} h _ { g _ { u } } ( \\theta ) : = I ( g _ { u } ( \\theta ) ) - \\Theta _ { g _ { u } } ( \\theta ) I ( u ) , \\ \\ \\ \\theta \\ge 0 . \\end{align*}"}
+{"id": "2169.png", "formula": "\\begin{align*} & 4 ( R _ { i j p q } R _ { k i r p } - R _ { i j r p } R _ { k i p q } - R _ { k i p q } R _ { i j r p } + R _ { k i r p } R _ { i j p q } ) \\\\ = & 8 ( R _ { i j p q } R _ { k i r p } - R _ { i j r p } R _ { k i p q } ) . \\end{align*}"}
+{"id": "7054.png", "formula": "\\begin{align*} \\nabla _ { \\tilde { v } } \\tilde { w } = 0 . \\end{align*}"}
+{"id": "3252.png", "formula": "\\begin{align*} \\limsup _ { k \\to \\infty } \\varphi _ { j _ { k } } = \\varphi \\end{align*}"}
+{"id": "6230.png", "formula": "\\begin{align*} h _ a [ \\psi ; p ] = h _ a [ \\psi ; p _ 0 ] + q [ \\psi ] , \\end{align*}"}
+{"id": "5890.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } = v , \\ \\ \\dot { v } = v \\times \\frac { B ( x ) } { \\epsilon } + F ( x ) , \\ \\ x ( 0 ) = x _ 0 , \\ \\ v ( 0 ) = v _ 0 . \\end{aligned} \\end{align*}"}
+{"id": "7201.png", "formula": "\\begin{align*} ( \\alpha ' \\colon p ^ { \\ast } \\mathcal { F } \\rightleftarrows p ^ { \\ast } \\mathcal { G } \\colon \\beta ' ) , \\ \\alpha ' = t ^ d p ^ { \\ast } \\alpha , \\ \\beta ' = p ^ { \\ast } \\beta \\end{align*}"}
+{"id": "2865.png", "formula": "\\begin{align*} W ( t ) = \\sum _ { l = 1 } ^ { n _ 1 } \\Psi _ { k l } \\beta ^ c _ l ( t ) , k = 1 , \\dots , n . \\end{align*}"}
+{"id": "7873.png", "formula": "\\begin{align*} Y _ 1 ( f ) = \\{ x \\in E '' _ B \\cap f ^ \\delta : \\Delta ( f , k _ x ) \\le \\Delta \\} . \\end{align*}"}
+{"id": "7709.png", "formula": "\\begin{gather*} \\partial ^ { ( \\hat { \\alpha } , 0 ) } \\left ( g ^ { i _ 1 j _ 1 } \\dots g ^ { i _ { k - 1 } j _ { k - 1 } } ( \\partial ^ { m + 1 } _ n \\gamma _ { j _ 1 \\dots j _ { k - 1 } n } ) \\right ) ( p ) = 0 . \\end{gather*}"}
+{"id": "3167.png", "formula": "\\begin{align*} \\dots T _ n \\to T _ { n - 1 } \\to \\dots \\to T _ 0 = T \\end{align*}"}
+{"id": "6741.png", "formula": "\\begin{align*} ( \\frac { \\partial ^ { \\alpha _ 1 } \\widetilde { E } \\cdot \\partial ^ { \\alpha - \\alpha _ { 1 } } \\nabla _ { v } M } { \\sqrt { \\mu } } , \\frac { \\partial ^ { \\alpha } M } { \\sqrt { \\mu } } ) = ( \\partial ^ { \\alpha } \\widetilde { E } \\cdot \\nabla _ { v } M , [ \\frac { 1 } { \\mu } - \\frac { 1 } { M } ] \\partial ^ { \\alpha } M ) + ( \\partial ^ { \\alpha } \\widetilde { E } \\cdot \\nabla _ { v } M , \\frac { 1 } { M } \\partial ^ { \\alpha } M ) . \\end{align*}"}
+{"id": "256.png", "formula": "\\begin{align*} \\begin{dcases} \\dfrac { d } { d t } \\Upsilon ( y , t ) = v ( \\Upsilon ( y , t ) ) , t \\in \\mathbb { R } , \\\\ \\Upsilon ( y , 0 ) = y . \\end{dcases} \\end{align*}"}
+{"id": "7755.png", "formula": "\\begin{align*} \\mathcal { A } ( p _ 0 ) = \\int _ { \\mathbb { S } ^ { n - 1 } \\times \\mathbb { S } ^ 1 ( \\lambda ) } \\frac { 1 } { \\cosh ^ n ( \\lambda | \\theta - \\theta _ 0 | ) } d \\Theta _ { n - 1 } d \\Theta _ \\lambda . \\end{align*}"}
+{"id": "1123.png", "formula": "\\begin{align*} E X ^ { m } Y ^ { n - m } = \\sum _ { j = 0 } ^ { \\min ( m , n - m ) } \\rho ^ { j } H _ { m , j } H _ { n - m , j } . \\end{align*}"}
+{"id": "6236.png", "formula": "\\begin{align*} D ( H ( p ) ) = \\{ \\psi \\in H ^ 2 ( \\Sigma ) \\cap H ^ 1 _ 0 ( \\Sigma ) : \\ : H ( p ) \\psi \\in L ^ 2 ( \\Sigma ) \\} , \\end{align*}"}
+{"id": "7368.png", "formula": "\\begin{align*} ~ \\sum _ { k = 0 } ^ \\infty \\frac { \\norm { g _ k } _ { x _ k } ^ 4 } { \\norm { \\eta _ k } _ { x _ k } ^ 2 } < \\infty . \\end{align*}"}
+{"id": "432.png", "formula": "\\begin{align*} M _ { \\alpha } = K ( \\overline { \\alpha } ) + \\sum c _ { \\mu } ( t ) K ( \\mu ) , \\end{align*}"}
+{"id": "3871.png", "formula": "\\begin{align*} | \\mathcal { K } ^ { z _ 1 , z _ 2 } _ { p + 1 , q } | = O _ { \\prec } \\big ( n ^ { - \\frac { p + q - 3 } { 2 } } ( \\Psi ^ { p + q + 1 } + n ^ { - 1 } ) \\big ) , \\Psi = n ^ { - 1 / 8 + \\tau } , \\end{align*}"}
+{"id": "743.png", "formula": "\\begin{align*} X ( t ) & = x + \\int _ 0 ^ t A ( s , \\overline { X } ( s ) ) d s + \\int _ 0 ^ t B ( s , \\overline { X } ( s ) ) d W ( s ) \\\\ & \\quad \\ + \\int _ 0 ^ t \\int _ { \\{ | z | < 1 \\} } H ( s , \\overline { X } ( s ) , z ) \\widetilde { N } ( d s , d z ) \\\\ & \\quad \\ + \\int _ 0 ^ t \\int _ { \\{ | z | \\ge 1 \\} } J ( s , \\overline { X } ( s ) , z ) N ( d s , d z ) . \\end{align*}"}
+{"id": "755.png", "formula": "\\begin{align*} a _ \\lambda v = \\sum _ { j \\in \\mathbb { Z _ { + } } } ( a _ { ( j ) } v ) \\frac { \\lambda ^ { j } } { j ! } . \\end{align*}"}
+{"id": "3637.png", "formula": "\\begin{align*} 0 = K ' ( x ) = { \\sigma ^ 2 \\over 2 } h ''' ( x ) + \\int _ 0 ^ \\infty \\left [ h ' ( x + y ) - h ' ( x ) \\right ] \\nu ( d y ) - \\lambda h ' ( x ) + 2 ( x - \\rho ) , x \\in ( a , b ) \\end{align*}"}
+{"id": "953.png", "formula": "\\begin{align*} | F ^ { i j } ( \\nabla _ \\beta \\varphi ) _ { i j } | \\leq C \\sqrt { b _ \\alpha } \\left ( \\sum _ { i = 1 } ^ { \\alpha } F ^ { i i } + \\sum _ { i = \\alpha + 1 } ^ { n - 1 } \\frac { b _ i } { b _ \\alpha } F ^ { i i } \\right ) \\mbox { i n } \\omega _ \\epsilon . \\end{align*}"}
+{"id": "5157.png", "formula": "\\begin{align*} \\mathcal B = \\left \\{ \\widetilde Q \\in \\widetilde { \\mathcal W } \\ , : \\ , \\widetilde Q \\nsubseteq A _ 0 , \\widetilde Q \\cap B ( x , r ) \\neq \\emptyset \\right \\} \\end{align*}"}
+{"id": "2166.png", "formula": "\\begin{align*} & 2 S _ { 1 2 i j k } - 3 S _ { 1 2 i j k } + S _ { 1 2 i j k } = 0 , \\\\ & 2 S _ { 1 2 1 3 l } - 3 S _ { 1 2 1 3 l } + S _ { 1 2 1 3 l } = 0 \\end{align*}"}
+{"id": "1630.png", "formula": "\\begin{align*} W _ i ^ { ( b ) } = \\frac { ( q - 1 ) q ^ { r / 2 - b } } { N } \\left [ | { \\cal P } ( b ) | \\left ( q ^ { r / 2 } - ( - 1 ) ^ s \\right ) + ( - 1 ) ^ s u \\mu _ { ( ( \\delta - i ) \\ ! \\ ! \\ ! \\ ! \\ ! \\pmod { u } ) } ( b ) \\right ] \\ . \\end{align*}"}
+{"id": "3538.png", "formula": "\\begin{align*} \\gamma ^ { \\textbf { e x t } } _ { 1 } \\mathrm { U } _ { \\mathrm { e } } = - \\mathcal { S } _ { \\alpha _ \\mathrm { m } } ^ { - 1 } \\Big ( \\frac { 1 } { 2 } I _ { d } - \\mathcal { K } _ { \\alpha _ \\mathrm { m } } \\Big ) \\Big [ \\gamma ^ { \\textbf { e x t } } _ { 0 } \\mathrm { U } _ { \\mathrm { e } } \\Big ] . \\end{align*}"}
+{"id": "2281.png", "formula": "\\begin{align*} W = W _ 1 z ^ { p _ 1 } + W _ 2 z ^ { p _ 2 } + \\cdots + W _ l z ^ { p _ l } , \\ \\ l \\geq 2 , p _ 1 < p _ 2 < \\dots < p _ l \\end{align*}"}
+{"id": "3162.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } \\ ; \\ ; \\varphi ( x ) + \\rho \\sum _ { i = 1 } ^ { l } \\max ^ { ~ ~ ~ ~ 2 } \\{ \\ g _ i ( x ) , 0 \\} + \\rho \\sum _ { j = l + 1 } ^ { m } h ^ 2 _ j ( x ) \\mbox { s u b j e c t t o } \\ ; \\ ; x \\in \\R ^ n \\end{align*}"}
+{"id": "4973.png", "formula": "\\begin{align*} l _ { \\rm M P A } ( a _ { v } ) = \\log ( \\frac { 1 - \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\mathbf { P } _ { \\rm M P A } ^ { k } [ 0 , v ] } { \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\mathbf { P } _ { \\rm M P A } ^ { k } [ 0 , v ] } ) . \\end{align*}"}
+{"id": "1330.png", "formula": "\\begin{align*} & | q + t \\eta | \\le | q | + | \\eta | < | q | + \\frac { | q | } 2 = \\frac { 3 | q | } 2 \\\\ { \\mbox { a n d } } \\quad & | q + t \\eta | \\ge | q | - | \\eta | > | q | - \\frac { | q | } 2 = \\frac { | q | } 2 . \\end{align*}"}
+{"id": "6784.png", "formula": "\\begin{align*} \\abs { \\eqref { e q : t i m e d e r i v a t i v e n o r m e s t i m a t e 2 } } & \\leq C e ^ { C f ( t ) } \\norm { \\psi _ t } _ { H ^ 1 ( \\mathbb { R } ^ 3 ) } \\left ( N ^ { - 1 } M ^ { 5 / 8 } \\Lambda _ 1 ^ { 1 / 2 } + \\Lambda _ 1 ^ { - 1 / 2 } \\right ) = C e ^ { C f ( t ) } \\norm { \\psi _ t } _ { H ^ 1 ( \\mathbb { R } ^ 3 ) } N ^ { - 1 / 2 } M ^ { 5 / 1 6 } . \\end{align*}"}
+{"id": "6376.png", "formula": "\\begin{align*} \\tilde { X } _ \\cdot ^ h : = \\mathcal { G } _ \\mu \\left ( B _ \\cdot ^ H + ( R _ H h ) ( \\cdot ) \\right ) = \\mathcal { G } _ \\mu ( \\tilde { B } _ \\cdot ^ H ) \\end{align*}"}
+{"id": "5400.png", "formula": "\\begin{align*} \\Delta u = V ( x ) u , \\end{align*}"}
+{"id": "170.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { - a } \\int _ { | v | \\leq 1 } | v | ^ k \\d v \\leq \\int _ { | v | \\leq 1 } | v | ^ k e ^ { - a | v | ^ 2 } \\d v \\leq \\int _ { | v | \\leq 1 } | v | ^ k \\d v \\ , . \\end{aligned} \\end{align*}"}
+{"id": "5086.png", "formula": "\\begin{align*} \\frac { 2 \\sigma } { p } + \\frac { d } { q } = \\frac { d } { 2 } - \\gamma , \\frac { 2 \\sigma } { \\tilde { p } } + \\frac { d } { \\tilde { q } } = \\frac { d } { 2 } + \\gamma . \\end{align*}"}
+{"id": "3442.png", "formula": "\\begin{align*} \\nabla \\cdot \\left [ \\dfrac { 1 } { \\varepsilon } \\nabla \\mathrm { H } \\right ] + \\omega ^ { 2 } \\mu _ { m } \\mathrm { H } = 0 \\ \\ \\mathbb { R } ^ { 2 } \\end{align*}"}
+{"id": "3745.png", "formula": "\\begin{align*} - m ( f ( x ) ) I = ( A - f ( x ) I ) \\sum _ { k = 1 } ^ { \\deg m } D ^ k m ( f ( x ) ) A ^ { k - 1 } . \\end{align*}"}
+{"id": "7488.png", "formula": "\\begin{align*} b _ { m , 1 } & = - ( m - 2 ) ! \\ , , \\\\ b _ { m , 0 } & = 0 \\ , . \\end{align*}"}
+{"id": "3411.png", "formula": "\\begin{align*} & \\frac { \\delta } { \\delta U } \\Big | _ { ( U , \\rho , S ) } = \\frac { \\delta } { \\delta U } \\Big | _ { ( U , \\rho , p ) } , \\\\ & \\frac { \\delta } { \\delta \\rho } \\Big | _ { ( U , \\rho , S ) } = \\frac { \\delta } { \\delta \\rho } \\Big | _ { ( U , \\rho , p ) } + a ^ 2 \\frac { \\delta } { \\delta p } \\Big | _ { ( U , \\rho , p ) } , \\\\ & \\frac { \\delta } { \\delta S } \\Big | _ { ( U , \\rho , S ) } = ( \\partial p / \\partial S ) | _ \\rho \\frac { \\delta } { \\delta p } \\Big | _ { ( U , \\rho , p ) } . \\end{align*}"}
+{"id": "733.png", "formula": "\\begin{align*} { f _ { 1 2 } } = { f _ { 1 1 } } + \\frac { 1 } { 2 } \\left ( { { f _ 1 } - { f _ 2 } } \\right ) + \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) + \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) + \\frac { 1 } { 8 } \\left ( { 2 { F _ x } + { F _ z } } \\right ) - \\frac { 1 } { 6 } \\rho { u _ z } , \\end{align*}"}
+{"id": "539.png", "formula": "\\begin{align*} g _ 1 ^ { \\prime + } ( 0 ) & = \\sum _ { i = 1 } ^ n \\int _ { \\R ^ n } \\partial _ i f ( x ) \\big ( T _ i ( x _ i ) - x _ i \\big ) Q ( x ) d x , \\\\ g _ 2 ^ { \\prime + } ( 0 ) & = - \\sum _ { i = 1 } ^ n \\int _ { \\R } \\left ( T _ i ' ( x _ i ) - 1 \\right ) Q _ i ( x _ i ) d x _ i . \\end{align*}"}
+{"id": "7071.png", "formula": "\\begin{align*} \\omega _ { R _ Y / R _ P } = \\operatorname { H o m } _ { R _ P } ( R _ Y , R _ P ) \\end{align*}"}
+{"id": "7178.png", "formula": "\\begin{align*} \\Phi \\colon \\mathbb { T } ( d ) _ v \\stackrel { \\sim } { \\to } \\mathbb { S } ^ { } ( d ) _ { w } , \\ v = w . \\end{align*}"}
+{"id": "1379.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } | x | ^ { \\alpha } a ( x ) = a _ 0 \\end{align*}"}
+{"id": "3870.png", "formula": "\\begin{align*} \\sum _ { p + q + 1 = 3 } K _ { p + 1 , q } ^ { z } = O _ \\prec ( n ^ { - 1 / 2 } ) . \\end{align*}"}
+{"id": "1641.png", "formula": "\\begin{align*} \\mathcal { T } \\circ v \\ ; = \\ ; \\frac { \\mathcal { T } v } { \\| \\mathcal { T } v \\| } \\ ; , v \\in \\mathbb { S } _ { \\mathbb { C } } ^ { \\mathsf { L } - 1 } \\ ; , \\end{align*}"}
+{"id": "4147.png", "formula": "\\begin{align*} f ( z ) = z + o \\left ( | 1 - z | ^ 3 \\right ) z \\to 1 \\ , , \\end{align*}"}
+{"id": "5425.png", "formula": "\\begin{align*} | P _ { d } | \\prec \\frac { 1 } { N ^ { d - m _ 0 } } ( 1 - c _ 0 ) ^ { K } \\Big ( \\frac { C _ 0 } { N } \\Big ) ^ { m _ 0 - 2 } ( N \\Psi ) ^ { m _ 2 } \\sum _ { \\mathcal { J } ^ { ( \\mathrm { o f f } ) } _ d } \\prod _ { i = 1 } ^ { m _ 1 } | G _ { x _ i y _ i } | \\prec ( 1 - c _ 0 ) ^ { K } N ^ 2 \\leq N ^ { - 8 } . \\end{align*}"}
+{"id": "7381.png", "formula": "\\begin{align*} f _ 1 ( 0 + ) - f _ 1 ( 0 - ) = i \\frac { \\alpha c } { 2 } \\big ( f _ 2 ( 0 + ) + f _ 2 ( 0 - ) \\big ) f _ 2 ( 0 + ) = f _ 2 ( 0 - ) , \\end{align*}"}
+{"id": "8128.png", "formula": "\\begin{align*} \\widehat { \\mu } ^ { \\mathrm { a s y } } ( \\overline L ) = \\frac { ( \\overline L ^ { d + 1 } ) _ S } { ( d + 1 ) ( L ^ d ) } . \\end{align*}"}
+{"id": "4009.png", "formula": "\\begin{align*} \\bar { G } _ { \\beta } ( u , t ) = E _ { \\beta , 1 } \\left ( - \\lambda \\left ( 1 - \\frac { ( 1 - \\rho u ) ^ { - r } - 1 } { ( 1 - \\rho ) ^ { - r } - 1 } \\right ) t ^ { \\beta } \\right ) . \\end{align*}"}
+{"id": "5820.png", "formula": "\\begin{align*} \\Pi V \\Pi ^ * = M _ z \\oplus U , \\end{align*}"}
+{"id": "4106.png", "formula": "\\begin{align*} \\int ^ { \\mathrm { B C } } ( y _ 1 , y _ 2 ) = \\int ^ { y _ 1 } _ { y _ 2 } \\gamma _ t ^ * \\omega . \\end{align*}"}
+{"id": "983.png", "formula": "\\begin{align*} ( k - i - s - 1 ) a _ { s + 1 , t } & = \\left ( \\frac { n - 2 k } { 2 } + i + j + s + t \\right ) a _ { s , t } , \\\\ ( k - j - t - 1 ) a _ { s , t + 1 } & = \\left ( \\frac { n - 2 k } { 2 } + i + j + s + t \\right ) a _ { s , t } . \\end{align*}"}
+{"id": "4915.png", "formula": "\\begin{align*} & \\binom { n - \\ell } { j - | S | } - \\binom { n - \\ell - 1 } { j - 1 - | S | } = \\binom { n - ( \\ell + 1 ) } { j - | S | } \\end{align*}"}
+{"id": "5900.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { \\tilde { x } } _ n ( s ) = \\dot { \\tilde { v } } _ n ( s ) , \\ \\ \\dot { \\tilde { v } } _ n ( s ) = \\frac { 1 } { \\epsilon } \\tilde { B } ( \\epsilon x ( t _ n ) ) \\tilde { v } _ n ( s ) + F ( { \\tilde { x } } _ n ( s ) ) , \\ \\ { \\tilde { x } } _ n ( 0 ) = x ( t _ n ) , \\ \\ { \\tilde { v } } _ n ( 0 ) = v ( t _ n ) , \\ \\ \\ 0 < s \\leq h . \\end{aligned} \\end{align*}"}
+{"id": "4440.png", "formula": "\\begin{align*} v _ { 2 , n } ^ { \\varepsilon } = \\tilde { v } _ { 2 , n } ^ { \\varepsilon } + w _ { 2 , n } ^ { \\varepsilon } \\end{align*}"}
+{"id": "4059.png", "formula": "\\begin{align*} \\delta ( a \\rhd b ) = a \\otimes b + ( a \\otimes \\mathbf 1 + \\mathbf 1 \\otimes a ) \\rhd \\delta ( b ) , \\end{align*}"}
+{"id": "1454.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } s ^ { \\beta } \\varphi _ { \\beta , \\varepsilon } ( s ) = \\frac { \\Gamma ( \\gamma _ { \\varepsilon } ) } { \\Gamma ( \\gamma _ { \\varepsilon } - \\beta ) } . \\end{align*}"}
+{"id": "7135.png", "formula": "\\begin{align*} D _ a : = \\{ ( i , j ) \\mid \\ , \\sigma _ a + 1 \\leq j < i \\leq \\sigma _ { a + 1 } \\} . \\end{align*}"}
+{"id": "2345.png", "formula": "\\begin{align*} \\zeta ( 3 , \\{ 2 \\} ^ { n } , 1 , 2 ) = \\zeta ( \\{ 2 \\} ^ { n + 3 } ) + 2 \\zeta ( 3 , 3 , \\{ 2 \\} ^ { n } ) \\end{align*}"}
+{"id": "6701.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } \\rho ( t , x ) & \\equiv \\int _ { \\mathbb { R } ^ { 3 } } \\psi _ { 0 } ( v ) F ( t , x , v ) \\ , d v , \\\\ \\rho ( t , x ) u _ { i } ( t , x ) & \\equiv \\int _ { \\mathbb { R } ^ { 3 } } \\psi _ { i } ( v ) F ( t , x , v ) \\ , d v , \\mbox { f o r $ i = 1 , 2 , 3 $ , } \\\\ \\rho ( t , x ) [ e ( t , x ) + \\frac { 1 } { 2 } | u ( t , x ) | ^ { 2 } ] & \\equiv \\int _ { \\mathbb { R } ^ { 3 } } \\psi _ { 4 } ( v ) F ( t , x , v ) \\ , d v , \\end{array} \\right . \\end{align*}"}
+{"id": "5048.png", "formula": "\\begin{align*} g ( z ) = \\sum _ { n = 1 } ^ \\infty \\rho _ g ( n ) n ^ { \\frac { k - 1 } { 2 } } e ( n z ) \\end{align*}"}
+{"id": "5238.png", "formula": "\\begin{align*} \\varepsilon = 1 + a _ 1 \\varpi + a _ 2 \\varpi ^ 2 + a _ 3 \\varpi ^ 3 + \\dotsb , \\end{align*}"}
+{"id": "1772.png", "formula": "\\begin{align*} \\sum _ { i \\in A } \\left ( - 1 \\right ) ^ { \\# i } p _ { A \\backslash i } \\left ( W \\right ) \\langle Y , B , i \\rangle & = 0 , \\end{align*}"}
+{"id": "2929.png", "formula": "\\begin{align*} P _ { \\rho , j } = \\frac { 1 } { 2 \\pi i } \\int _ { \\gamma _ \\rho ^ j } ( \\lambda - A ) ^ { - 1 } d \\lambda \\end{align*}"}
+{"id": "4727.png", "formula": "\\begin{gather*} ( a + b ) \\star ( a ^ \\prime + b ^ \\prime ) : = ( a \\cdot _ A a ^ \\prime + r _ B ( b ^ \\prime ) a + l _ B ( b ) a ^ \\prime ) + ( b \\cdot _ B b ^ \\prime + l _ A ( a ) b ^ \\prime + r _ A ( a ^ \\prime ) b ) , \\end{gather*}"}
+{"id": "1436.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { E } ( t ; t _ 0 , x _ 0 ) & = \\int _ { B _ { t _ 0 - t } ( x _ 0 ) \\cap \\Omega } \\left ( \\partial _ t ^ 2 u - \\Delta u + u \\right ) \\partial _ t u \\ , d x \\\\ & - \\frac { 1 } { 2 } \\int _ { S _ { t _ 0 - t } \\cup S _ { b , t _ 0 - t } } ( | \\partial _ t u | ^ 2 + | \\nabla u | ^ 2 + | u | ^ 2 - 2 ( \\mathbf { n } \\cdot \\nabla u ) \\partial _ t u ) \\ , d S , \\end{align*}"}
+{"id": "5676.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { r - 1 } \\left ( 1 + \\frac { | \\partial B _ \\mathrm { i n t } ( v , i + 1 ) | } { | B _ \\mathrm { i n t } ( v , i ) | } \\right ) = | B _ \\mathrm { i n t } ( v , r ) | \\leq e ^ { C _ 1 ( p - p _ c ) r } \\end{align*}"}
+{"id": "8039.png", "formula": "\\begin{align*} { \\bf x } _ T ( t ) = [ x _ 1 ( t ) , \\cdots , x _ { N _ t } ( t ) ] ^ T = \\sum _ { k = 1 } ^ { K } { \\bf F } _ k { \\bf s } _ k e ^ { \\mathrm { j } 2 \\pi { f _ k } t } , t \\in ( 0 , \\triangle t ] , \\end{align*}"}
+{"id": "6430.png", "formula": "\\begin{align*} \\begin{bmatrix} n \\\\ k \\end{bmatrix} ( 1 - q ^ k ) = \\begin{bmatrix} n \\\\ k - 1 \\end{bmatrix} ( 1 - q ^ { n - k + 1 } ) . \\end{align*}"}
+{"id": "5496.png", "formula": "\\begin{align*} F ( a , b ; t ) = 1 + \\frac { t ( 1 - a q ) } { 1 - b q } F ( a q , b q ; t ) . \\end{align*}"}
+{"id": "6534.png", "formula": "\\begin{align*} H & = \\sum _ { \\theta _ r \\neq \\{ 1 , - 1 \\} } \\theta _ r \\left ( E _ { \\theta _ r } - E _ { - \\theta _ r } \\right ) + \\pi \\cdot E _ { - 1 } \\\\ & = \\sum _ { \\theta _ r \\neq \\{ 1 , - 1 \\} } \\theta _ r \\left ( - \\frac { 2 i } { \\sin ( \\theta ) } ( P W - W P ) \\right ) + \\pi \\cdot E _ { - 1 } . \\end{align*}"}
+{"id": "2793.png", "formula": "\\begin{align*} Q _ A = U J V ^ T , Q _ B = \\widehat { U } \\widehat { J } V ^ T , \\end{align*}"}
+{"id": "958.png", "formula": "\\begin{align*} \\sigma _ { k - 1 } ( b ) u _ { n n } ( 0 ) - \\sum _ { \\alpha \\leq n - 1 } u _ { n \\alpha } ^ 2 ( 0 ) \\sigma _ { k - 2 ; \\alpha } ( b ) + \\sigma _ k ( b ) = f ( 0 ) . \\end{align*}"}
+{"id": "5421.png", "formula": "\\begin{align*} \\sum _ { b = 1 } ^ N ( T ^ { k } ) _ { a b } = 0 , \\| T ^ { k } \\| _ { \\max } \\leq \\frac { C _ 0 } { N } , \\end{align*}"}
+{"id": "5759.png", "formula": "\\begin{align*} f ( X ) = h _ 1 ( { \\rm I d _ n } , X ) = h _ 2 ( { \\rm I d _ n } , X ) . \\end{align*}"}
+{"id": "5436.png", "formula": "\\begin{align*} \\log \\frac { \\Gamma \\left ( \\frac { L ^ 2 } { 2 } + 1 \\right ) ^ 2 } { \\Gamma ( L ^ 2 + 1 ) } = \\log L - ( L ^ 2 + 1 ) \\log 2 + \\cdots + O ( K ^ { - 1 0 0 0 } ) \\end{align*}"}
+{"id": "3047.png", "formula": "\\begin{align*} \\sigma ( x ) = \\sigma _ i ( x ) \\end{align*}"}
+{"id": "3118.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ k ) = F _ { Y _ 1 } ( \\tau _ 1 ^ k ) + F _ { X _ 1 } ( \\sigma _ 1 ^ k ) \\end{align*}"}
+{"id": "4622.png", "formula": "\\begin{align*} p \\left ( 1 - ( 2 L _ p ) ^ { - 1 / p } \\right ) = \\frac { 1 - x ^ { - h } } { h } \\leq \\left . \\frac { d x ^ s } { d s } \\right | _ { s = 0 } = \\ln x , \\end{align*}"}
+{"id": "1234.png", "formula": "\\begin{align*} & \\int _ T ^ { 2 T } \\prod _ { i = 1 } ^ r ( \\cos ( ( t + \\gamma ) \\log p _ i ) ) ^ { a _ i } \\prod _ { r + 1 \\le i \\le s } ( \\cos ( 2 ( t + \\gamma ) \\log p _ i ) ) ^ { a _ i } d t \\\\ & = ( T + \\gamma ) g ( n ) + O ( | \\gamma | ) + O ( ( p _ 1 ^ { a _ 1 } \\cdots p _ r ^ { a _ r } ) \\cdot ( p _ { r + 1 } ^ { 2 a _ { r + 1 } } \\cdots p _ { s } ^ { 2 a _ { s } } ) ) , \\end{align*}"}
+{"id": "4441.png", "formula": "\\begin{align*} \\Vert w _ { 2 , n } ^ { \\varepsilon } \\Vert _ { L _ r } = \\Vert \\Theta ( u _ { 1 , n } ^ { \\varepsilon } ) - \\Theta ( v _ 1 ) \\Vert _ { L _ r } \\longrightarrow 0 n \\to \\infty . \\end{align*}"}
+{"id": "5513.png", "formula": "\\begin{align*} F _ { N } ( a , b ; t ) = 1 + \\frac { ( 1 - a q ) ( 1 - q ^ { N + 1 } ) ( 1 - t q ^ { N + 1 } ) t } { ( 1 - t q ^ N ) ( 1 - b q ^ { N + 2 } ) ( 1 - t ) } + \\frac { ( 1 - a q ) ( 1 - q ^ { N + 1 } ) ^ 2 ( b - a t q ) t q } { ( 1 - b q ) ( 1 - t q ^ { N } ) ( 1 - b q ^ { N + 2 } ) ( 1 - t ) } F _ { N } ( a q , b q ; t q ) . \\end{align*}"}
+{"id": "4232.png", "formula": "\\begin{align*} Z ( q \\gamma _ j ^ { - 1 } ) & = \\frac { \\prod _ { m = 1 } ^ { 2 g } ( 1 - q \\gamma _ j ^ { - 1 } \\gamma _ m ) } { ( 1 - q \\gamma _ j ^ { - 1 } ) ( 1 - q ^ 2 \\gamma _ j ^ { - 1 } ) } \\\\ & = q ^ { 2 g - 3 } \\gamma _ j ^ { - 2 g + 2 } \\prod _ { m = 1 } ^ { 2 g } \\gamma _ m \\cdot \\left ( \\frac { \\prod _ { m = 1 } ^ { 2 g } ( 1 - q ^ { - 1 } \\gamma _ j \\gamma _ m ^ { - 1 } ) } { ( 1 - q ^ { - 1 } \\gamma _ j ) ( 1 - q ^ { - 2 } \\gamma _ j ) } \\right ) . \\end{align*}"}
+{"id": "7347.png", "formula": "\\begin{align*} A _ 1 & = \\big ( \\sum ^ { 2 m + 1 } _ { i = 0 } E ^ * _ i \\big ) A _ 1 \\big ( \\sum ^ { 2 m + 1 } _ { i = 0 } E ^ * _ i \\big ) = \\sum ^ { 2 m } _ { i = 0 } ( E ^ * _ { i } A _ 1 E ^ * _ { i + 1 } + E ^ * _ { i + 1 } A _ 1 E ^ * _ { i } ) , \\end{align*}"}
+{"id": "369.png", "formula": "\\begin{align*} \\lbrace \\mu _ { \\xi _ i } , h \\rbrace = \\L _ { \\xi _ i } h = 0 \\forall h \\in C ^ \\infty ( M ) ^ G ~ . \\end{align*}"}
+{"id": "5224.png", "formula": "\\begin{align*} \\Omega _ { \\tau , n } : = \\sum _ { i = 0 } ^ { f - 1 } p ^ i n _ { \\tau \\circ \\varphi ^ i } . \\end{align*}"}
+{"id": "5355.png", "formula": "\\begin{align*} P _ { i j } & : = \\frac { 1 } { n - 2 } \\left ( R _ { i j } - J g _ { i j } \\right ) , \\\\ W _ { i j k l } & : = R _ { i j k l } - P _ { i k } g _ { j l } - P _ { j l } g _ { i k } + P _ { i l } g _ { j k } + P _ { j k } g _ { i l } , \\\\ C _ { i j k } & : = \\nabla _ i P _ { j k } - \\nabla _ j P _ { i k } , \\\\ B _ { i j } & : = \\nabla ^ s C _ { s i j } + W _ { i s j t } P ^ { s t } , \\end{align*}"}
+{"id": "483.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom n k \\frac { L _ { j k - 1 } } { L _ j ^ k } B _ { n - k } = \\sqrt 5 \\ , B _ n \\Big ( \\frac { \\alpha ^ j } { L _ j } \\Big ) , \\qquad \\mbox { $ n $ o d d } , \\end{align*}"}
+{"id": "3424.png", "formula": "\\begin{align*} U ^ * _ \\epsilon + f ( S ^ * ) U ^ * _ t = 0 , \\rho ^ * _ \\epsilon + f ( S ^ * ) \\rho ^ * _ t - f ' ( S ^ * ) S ^ * _ r \\rho ^ * = 0 , S ^ * _ \\epsilon + f ( S ^ * ) S ^ * _ t = 0 . \\end{align*}"}
+{"id": "1348.png", "formula": "\\begin{align*} & \\int _ { B _ { r \\varrho } ( y _ 0 ) } \\Big ( \\left | \\nabla w ( y ) \\right | ^ p + \\chi _ { \\left \\{ w > 0 \\right \\} } ( y ) \\Big ) \\ , d y = { r ^ n } \\int _ { B _ { \\varrho } ( x _ 0 ) } \\Big ( \\left | \\nabla w ( r x ) \\right | ^ p + \\chi _ { \\left \\{ w > 0 \\right \\} } ( r x ) \\Big ) \\ , d x \\\\ & \\qquad = r ^ n \\int _ { B _ { \\varrho } ( x _ 0 ) } \\Big ( \\left | \\nabla w _ r ( x ) \\right | ^ p + \\chi _ { \\left \\{ w _ r > 0 \\right \\} } ( x ) \\Big ) \\ , d x \\end{align*}"}
+{"id": "1498.png", "formula": "\\begin{align*} K ( u ) = \\sum _ { d \\le u } h ( d ) \\bigl ( \\log \\frac { \\Delta } { d } \\bigr ) ^ 2 . \\end{align*}"}
+{"id": "5892.png", "formula": "\\begin{align*} \\begin{aligned} & \\dot { \\xi } ^ { x } _ { n } ( s ) = \\xi ^ { v } _ { n } ( s ) , \\ \\ \\dot { \\xi } ^ { v } _ { n } ( s ) = \\frac { 1 } { \\epsilon } \\tilde { B } ( x _ { n + \\frac { 1 } { 2 } } ) \\xi ^ { v } _ { n } ( s ) + F ( x ( t _ n + s ) ) - F ( \\tilde { x } _ { n } ( s ) ) + \\varsigma ^ { 0 } _ { n } ( s ) . \\end{aligned} \\end{align*}"}
+{"id": "1742.png", "formula": "\\begin{align*} \\| \\varphi \\| _ { L ^ 2 } = \\| \\hat { \\varphi } \\| _ { \\ell ^ 2 } & \\lesssim M ^ { 1 / 2 - 1 / p } \\| \\hat { \\varphi } \\| _ { \\ell ^ { p } } \\\\ & \\lesssim M ^ { 1 / 2 - 1 / p } \\| \\varphi \\| _ { L ^ { p ' } } = M ^ { 1 / 2 - 1 / p } \\ , . \\end{align*}"}
+{"id": "2719.png", "formula": "\\begin{align*} M _ { \\varphi } ( 2 n , j , 1 ; r - 1 ) = \\sum _ { u = 0 } ^ { n - j } \\binom { j + u } { u } \\bigl { ( } \\binom { 2 n } { 2 n - j - u } M _ { \\varphi } ( 2 n , j + u , 0 ; r - 1 ) \\bigr { ) } \\end{align*}"}
+{"id": "5679.png", "formula": "\\begin{align*} \\Psi ( \\{ [ \\xi A ] _ { k , A } \\} _ { k \\in { \\mathbb N } } & = \\Psi ( \\{ A _ k [ \\xi ] _ { k , A } \\} _ { k \\in { \\mathbb N } } \\\\ & = ( \\Psi \\circ \\Gamma _ A ) ( \\{ [ \\xi ] _ { k , A } \\} _ { k \\in { \\mathbb N } } \\\\ & = ( \\Gamma _ B \\circ \\Psi ) ( \\{ [ \\xi ] _ { k , A } \\} _ { k \\in { \\mathbb N } } \\\\ & = \\Gamma _ B ( \\{ [ \\eta _ k ] _ { k , B } \\} _ { k \\in { \\mathbb N } } \\\\ & = \\{ B _ k [ \\eta _ k ] _ { k , B } \\} _ { k \\in { \\mathbb N } } \\\\ & = \\{ [ \\eta _ k B ] _ { k , B } \\} _ { k \\in { \\mathbb N } } . \\end{align*}"}
+{"id": "6665.png", "formula": "\\begin{align*} \\lambda _ { - k } ' & = \\lambda _ { k } , k \\in \\mathbb { N } \\cap [ 0 , \\kappa - 1 ] \\\\ \\lambda _ { i , j } ' & = \\lambda _ { i , j } , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] , \\ j \\in \\mathbb { N } \\cap [ 1 , p ] ; \\end{align*}"}
+{"id": "7144.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\chi _ i + \\rho + d \\mu \\tau _ d & = \\sum _ { i = 1 } ^ k w _ i \\tau _ { d _ i } + \\sum _ { i = 1 } ^ k ( \\psi _ i - \\rho _ i ) + \\rho + d \\mu \\tau _ d \\\\ & = - \\frac { 3 } { 2 } \\mathfrak { g } ^ { \\lambda > 0 } + \\sum _ { i = 1 } ^ k v _ i \\tau _ { d _ i } + \\sum _ { i = 1 } ^ k ( \\psi _ i + d _ i \\mu \\tau _ { d _ i } ) . \\end{align*}"}
+{"id": "718.png", "formula": "\\begin{align*} { g _ i } \\left ( { { { \\bf { x } } _ b } , t } \\right ) = g _ i ^ { \\left ( { e q } \\right ) } \\left ( { { { \\bf { x } } _ b } , t } \\right ) + g _ i ^ { \\left ( { n e q } \\right ) } \\left ( { { { \\bf { x } } _ b } , t } \\right ) , \\end{align*}"}
+{"id": "4844.png", "formula": "\\begin{align*} \\int _ { \\R ^ { 2 d } \\times S ^ 2 } | \\nabla f ( x , s _ 1 ) - & \\nabla F ( x ) | ^ 2 \\dd \\Pi _ t ( x , y , s _ 1 , s _ 2 ) = \\int _ { \\R ^ { 2 d } \\times S ^ 2 } | \\nabla f ( x , s _ 2 ) - \\nabla F ( x ) | ^ 2 \\dd \\Pi _ t ( x , y , s _ 1 , s _ 2 ) \\\\ & \\leq \\int _ { \\R ^ { 2 d } \\times S ^ 2 } | \\nabla f ( x , s _ 2 ) - \\nabla F ( x ) | ^ 2 \\ , \\dd \\left ( \\int _ S \\rho _ t ( s _ 1 ) \\dd s _ 1 \\right ) ( x ) \\dd \\mu ( s _ 2 ) \\leq \\sigma ^ 2 . \\end{align*}"}
+{"id": "2032.png", "formula": "\\begin{align*} I _ 0 = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty + i c } ^ { \\infty + i c } \\left ( \\frac { 1 } { s - 1 } + \\frac { 1 } { s } \\right ) \\Bigl ( \\Phi _ 1 ( \\phi ; z ) \\overline { \\Phi _ 1 ( \\phi ; \\bar { z } ) } + \\Phi _ 1 ( \\phi ; - z ) \\overline { \\Phi _ 1 ( \\phi ; - \\bar { z } ) } \\Bigr ) \\ , d z , \\end{align*}"}
+{"id": "1981.png", "formula": "\\begin{align*} x _ { \\max } = x _ u < \\frac { \\sqrt { d ( u ) } } { \\lambda _ 1 } = O \\Big ( \\frac { 1 } { \\sqrt { n } } \\Big ) . \\end{align*}"}
+{"id": "1329.png", "formula": "\\begin{align*} \\int _ { B _ r ( x _ 0 ) } \\big ( \\left | \\nabla u ( x ) \\right | ^ p - \\left | \\nabla v ( x ) \\right | ^ p \\big ) \\ , d x & \\ge p \\ , \\gamma \\int _ 0 ^ 1 { s } ^ { p - 1 } \\left ( \\int _ { B _ r ( x _ 0 ) } \\left | \\nabla u ( x ) - \\nabla v ( x ) \\right | ^ p \\ , d x \\right ) \\ , d s \\\\ & = \\gamma \\int _ { B _ r ( x _ 0 ) } \\left | \\nabla u ( x ) - \\nabla v ( x ) \\right | ^ p \\ , d x , \\end{align*}"}
+{"id": "2711.png", "formula": "\\begin{align*} M _ { \\phi } ( 2 n , j , 0 ; r - 1 ) = ( - 1 ) ^ j \\frac { 1 } { 2 } S ( n , r ) \\frac { \\binom { 2 j } { j } \\binom { 2 ( n + r - j ) } { n + r - j } \\binom { 2 n - j } { n } } { 2 \\binom { 2 n + r - j } { n } } \\end{align*}"}
+{"id": "3522.png", "formula": "\\begin{align*} \\Vert \\nabla \\varphi \\Vert ^ 2 _ { \\mathbb { L } ^ 2 ( \\Omega ) } & = \\int _ { \\Omega } | \\nabla \\varphi ( \\mathrm { x } ) | ^ 2 d \\mathrm { x } \\\\ & = \\delta ^ 2 \\int _ { \\mathrm { B } } \\delta ^ { - 2 } | \\nabla _ { \\xi } \\varphi ( \\delta \\xi + z ) | ^ 2 d \\xi \\\\ & = \\Vert \\nabla \\hat { \\varphi } \\Vert ^ 2 _ { \\mathbb { L } ^ 2 ( \\mathrm { B } ) } \\end{align*}"}
+{"id": "4233.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( b / a ) _ n a ^ n } { ( 1 - q ^ n ) ( b ) _ n } = \\sum _ { n = 1 } ^ { \\infty } \\frac { a ^ n - b ^ n } { 1 - q ^ n } , \\end{align*}"}
+{"id": "4317.png", "formula": "\\begin{align*} a _ k = \\frac { \\rho } { 1 + \\rho } \\left ( \\frac { 1 } { 1 + \\rho } \\right ) ^ k , \\end{align*}"}
+{"id": "1865.png", "formula": "\\begin{align*} y _ { ( 2 k - 1 ) m + i , j } + y _ { ( 2 ( r _ 0 - k + 1 ) - 1 ) m + ( m + 1 - i ) , n + 1 - j } & = \\tfrac { 1 } { 2 } m n r + \\tfrac { 3 } { 2 } , \\mbox { \\ a n d } \\\\ y _ { ( 2 k ) m + i , j } + y _ { 2 ( r _ 0 - k ) m + ( m + 1 - i ) , n + 1 - j } & = 2 M N - \\tfrac { 1 } { 2 } m n r + \\tfrac { 3 } { 2 } . \\end{align*}"}
+{"id": "7546.png", "formula": "\\begin{align*} \\mathbb P ( \\mathcal T ) = \\frac { 1 } { Z } w ( \\mathcal T ) , Z = \\sum _ { \\mathcal T } w ( \\mathcal T ) . \\end{align*}"}
+{"id": "7804.png", "formula": "\\begin{align*} 2 \\int _ { 0 } ^ { \\theta _ { ( R , 0 ) } } \\cot ( \\theta ) \\left ( \\int _ { t _ { 1 } ( \\theta ) } ^ { s _ { 1 } ( \\theta ) } d t \\right ) d \\theta = 2 \\int _ { 0 } ^ { \\theta _ { ( R , 0 ) } } \\cot ( \\theta ) ( s _ { 1 } ( \\theta ) - t _ { 1 } ( \\theta ) ) d \\theta , \\end{align*}"}
+{"id": "3670.png", "formula": "\\begin{align*} x y = \\left ( \\frac { x + y } { 2 } \\right ) ^ 2 - \\left ( \\frac { x - y } { 2 } \\right ) ^ 2 , \\end{align*}"}
+{"id": "4007.png", "formula": "\\begin{align*} \\bar { q } ( n , t ) = \\sum _ { \\Omega ^ { n } _ { k } } \\prod _ { j = 1 } ^ { k } \\rho ^ { x _ { j } } \\binom { r + x _ { j } - 1 } { x _ { j } } \\left ( \\frac { \\lambda t ( 1 - \\rho ) ^ { r } } { 1 - ( 1 - \\rho ) ^ { r } } \\right ) ^ { k } \\frac { e ^ { - \\lambda t } } { k ! } , \\end{align*}"}
+{"id": "4594.png", "formula": "\\begin{align*} \\abs { \\sum _ { l = n _ 0 } ^ { n } \\frac { \\sin 2 \\pi \\theta ( l ) } { l - b } } \\leq \\frac { C ( E , K ) } { n _ 0 - b } , \\end{align*}"}
+{"id": "5549.png", "formula": "\\begin{align*} T _ 2 ( \\infty , t ) = T _ { \\infty } , \\end{align*}"}
+{"id": "2315.png", "formula": "\\begin{align*} \\mathbb { A } _ { n , \\Delta } = \\mu ( L \\tilde { f } ) \\frac { n \\Delta ^ 2 } { 2 } + \\sum _ { k = 1 } ^ n \\mathcal { X } _ k + \\sum _ { k = 1 } ^ n \\chi _ k . \\end{align*}"}
+{"id": "2714.png", "formula": "\\begin{align*} ( - 1 ) ^ { j + r - 1 - l } \\frac { 1 } { 2 } S ( n , r ) \\binom { 2 n - j } { n } \\frac { \\binom { 2 ( j + r - 1 - l ) } { j + r - 1 - l } \\binom { 2 ( n - j + l + 1 ) } { n - j + l + 1 } \\binom { n - j } { j + r - l - 1 } } { 2 \\binom { 2 n - j + l + 1 } { n } } \\end{align*}"}
+{"id": "2998.png", "formula": "\\begin{align*} c _ 0 : = \\ln \\beta + \\frac { 2 } { 1 - \\beta / \\alpha } , \\end{align*}"}
+{"id": "6747.png", "formula": "\\begin{align*} I ^ { 4 } _ { 9 } & = - \\sum ^ { 3 } _ { j = 1 } \\int _ { \\mathbb { R } ^ { 3 } } \\int _ { \\mathbb { R } ^ { 3 } } \\partial ^ { \\alpha } [ \\frac { 1 } { \\rho } R \\theta B _ { i j } ( \\frac { v - u } { \\sqrt { R \\theta } } ) \\frac { \\sqrt { \\mu } } { M } \\Gamma ( \\frac { G } { \\sqrt { \\mu } } , \\frac { G } { \\sqrt { \\mu } } ) ] \\ , d v \\ , \\partial ^ { \\alpha } \\partial _ { x _ j } \\widetilde { u } _ i \\ , d x . \\end{align*}"}
+{"id": "6481.png", "formula": "\\begin{align*} A = \\{ u \\geq K ^ k \\} B = \\{ u \\geq K ^ { k - 1 } \\} . \\end{align*}"}
+{"id": "4071.png", "formula": "\\begin{align*} \\mathcal { V } \\simeq \\mathcal { V } ( T ) ^ { \\nabla = 0 } \\otimes _ K \\mathcal { O } _ T . \\end{align*}"}
+{"id": "1516.png", "formula": "\\begin{align*} a _ n = \\lambda ( n ) \\end{align*}"}
+{"id": "5072.png", "formula": "\\begin{align*} \\Lambda _ f \\bigl ( s , \\tfrac { a } { q } \\bigr ) = \\omega \\chi ( \\bar { a } ) q ^ { 1 - 2 s } \\Lambda _ f \\bigl ( 1 - s , - \\tfrac { \\bar { a } } { q } \\bigr ) = \\omega ^ 2 \\chi ( - 1 ) \\Lambda _ f \\bigl ( s , \\tfrac { a } { q } \\bigr ) , \\end{align*}"}
+{"id": "1775.png", "formula": "\\begin{align*} i _ { 0 } ^ { + } = \\min \\left ( \\left [ i _ { 0 } + 1 , \\dots , n \\right ] \\cap B ^ { c } \\right ) , \\end{align*}"}
+{"id": "2009.png", "formula": "\\begin{align*} \\langle \\phi _ 1 , \\phi _ 2 \\rangle _ { G _ g , a } : = \\int _ { - a } ^ { a } \\int _ { - a } ^ { a } G _ g ( t , u ) \\phi _ 1 ( u ) \\overline { \\phi _ 2 ( t ) } \\ , d u d t \\end{align*}"}
+{"id": "735.png", "formula": "\\begin{align*} { f _ { 1 6 } } = { f _ { 1 5 } } + \\frac { 1 } { 2 } \\left ( { { f _ 3 } - { f _ 4 } } \\right ) + \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) + \\frac { 1 } { 8 } \\left ( { 2 { F _ y } + { F _ z } } \\right ) - \\frac { 1 } { 6 } \\rho { u _ z } , \\end{align*}"}
+{"id": "67.png", "formula": "\\begin{align*} i ) \\ X | _ { x _ { 0 } } \\not = 0 , i i ) \\ g ( X | _ { x _ { 0 } } , \\nabla d ( x _ { 0 } ) ) = 0 , \\end{align*}"}
+{"id": "2585.png", "formula": "\\begin{align*} k _ i ^ N ( t , \\nu ^ { N , 1 } _ { \\boldsymbol { x } } , \\ldots , \\nu ^ { N , N } _ { \\boldsymbol { x } } , \\lambda ^ { N , 1 } _ { \\boldsymbol { \\varphi } } , \\ldots , \\lambda ^ { N , N } _ { \\boldsymbol { \\varphi } } ) : = \\rho _ i ^ N \\big ( - R ^ { - 1 } B [ P _ t \\nu ^ { N , 1 } _ { \\boldsymbol { x } } + \\lambda ^ { N , 1 } _ { \\boldsymbol { \\varphi } } ] , \\ldots , - R ^ { - 1 } B [ P _ t \\nu ^ { N , N } _ { \\boldsymbol { x } } + \\lambda ^ { N , N } _ { \\boldsymbol { \\varphi } } ] \\big ) . \\end{align*}"}
+{"id": "2118.png", "formula": "\\begin{align*} N _ U ( f ) = \\sum _ { d \\in \\N } d \\ , \\# ( Y _ d \\cap H _ 1 \\cap \\ldots H _ n ) \\end{align*}"}
+{"id": "5109.png", "formula": "\\begin{align*} i \\Gamma _ { \\phi } f : = - [ \\nabla _ x \\cdot ( ( \\nabla _ x \\phi ) f ) + \\nabla _ x \\phi \\cdot \\nabla _ x f ] , \\end{align*}"}
+{"id": "4848.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial \\Pi _ t } { \\partial t } = & \\ , \\nabla _ x \\cdot ( \\Pi _ t \\nabla _ x f ( x , s _ 1 ) ) + \\nabla _ y \\cdot ( \\Pi _ t \\nabla _ y F ( y ) ) + \\\\ & K ( t ) ~ \\int _ S \\int _ S \\left ( \\Gamma ( s _ 1 , s _ 2 ) \\delta _ { x ^ * } ( y ) \\left ( \\int _ { \\R ^ d } \\Pi _ t ( x , y ' , s _ 1 ' , s _ 2 ' ) \\ , \\dd y ' \\right ) - \\Pi _ t ( x , y , s _ 1 , s _ 2 ) \\right ) \\dd s _ 1 ' \\dd s _ 2 ' , \\end{aligned} \\end{align*}"}
+{"id": "1190.png", "formula": "\\begin{align*} \\pi ( [ X ] ) \\wedge \\pi ( [ Y ] ) = [ f _ \\alpha '' X \\cap ( ( \\lambda \\setminus A _ \\alpha ) \\cup f _ \\alpha '' ( Y \\cap A _ \\alpha ) ] = [ f _ \\alpha '' X \\cap f '' _ \\alpha ( Y \\cap A _ \\alpha ) ] = [ f _ \\alpha '' ( X \\cap Y ) ) ] . \\end{align*}"}
+{"id": "6546.png", "formula": "\\begin{align*} a = a ' = y . \\end{align*}"}
+{"id": "5882.png", "formula": "\\begin{align*} \\tilde { x } ( t ) = K ^ \\textup { H } x ( t ) , \\tilde { v } ( t ) = K ^ \\textup { H } v ( t ) . \\end{align*}"}
+{"id": "5786.png", "formula": "\\begin{align*} \\alpha ( S , S ' ) = \\begin{cases} 1 & \\textnormal { i f $ S = S ' $ , } \\\\ 0 & \\textnormal { o t h e r w i s e . } \\end{cases} \\end{align*}"}
+{"id": "1862.png", "formula": "\\begin{align*} x _ { 1 , j } + x _ { 1 , j + 1 } = x _ { m , j } + x _ { m , j + 1 } . \\end{align*}"}
+{"id": "2731.png", "formula": "\\begin{align*} \\sigma ^ - _ { i j } \\bigl ( e ^ - _ { j k } \\bigr ) = e ^ - _ { i k } \\ , , \\sigma ^ - _ { i j } \\bigl ( e ^ + _ { j k } \\bigr ) = e ^ + _ { i k } \\ , , \\sigma ^ + _ { i j } \\bigl ( e ^ - _ { j k } \\bigr ) = - e ^ + _ { i k } \\ , , \\sigma ^ + _ { i j } \\bigl ( e ^ + _ { j k } \\bigr ) = - e ^ - _ { i k } \\ , , \\end{align*}"}
+{"id": "5307.png", "formula": "\\begin{align*} d ( t ) : = \\frac { ( b - \\theta ) ^ 2 } { 4 n \\left ( 1 2 \\log ( | t | + B + 5 ) + 5 1 \\log ( A + \\kappa ) + 3 1 \\log \\frac { 1 } { b - \\theta } + 1 1 3 \\right ) } . \\end{align*}"}
+{"id": "5129.png", "formula": "\\begin{align*} P ( \\widetilde A , \\R ^ n ) \\leq C P ( A , \\Omega ) \\ ; \\ ; \\ ; \\ ; \\mathcal { H } ^ { n - 1 } ( \\partial ^ M \\widetilde A \\cap \\partial \\Omega ) = 0 , \\end{align*}"}
+{"id": "1384.png", "formula": "\\begin{align*} \\widetilde { \\gamma } _ { \\varepsilon } = \\left ( \\frac { 2 - \\alpha } { n - \\alpha } + 2 \\varepsilon \\right ) ^ { - 1 } , \\gamma _ { \\varepsilon } = ( 1 - 2 \\varepsilon ) \\widetilde { \\gamma } _ { \\varepsilon } . \\end{align*}"}
+{"id": "7171.png", "formula": "\\begin{align*} \\mathfrak { g } _ { l - 1 } / \\mathfrak { g } _ l = \\bigoplus _ { j = i + l } \\mathbb { C } x _ { i , j } = \\bigoplus _ { j = i + l } \\mathbb { C } ( \\beta _ i - \\beta _ j ) . \\end{align*}"}
+{"id": "4790.png", "formula": "\\begin{align*} ( k + 1 ) \\prod _ { j = 1 } ^ q ( b _ j + k ) U _ { k + 1 } = \\prod _ { j = 1 } ^ p ( a _ j + p ) U _ k . \\end{align*}"}
+{"id": "5988.png", "formula": "\\begin{align*} U _ r ^ { 0 , b } ( t ) = X ( t ) + R ^ { { 0 , b } } _ r ( t ) , 0 \\leq t < \\widehat { T } _ b ^ { + } ( 1 ) \\end{align*}"}
+{"id": "3064.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ { \\ell } ) = q \\ell \\equiv 0 \\pmod { \\ell } . \\end{align*}"}
+{"id": "7357.png", "formula": "\\begin{align*} \\sum ^ k _ { i = 1 } \\sum ^ i _ { l = 1 } | \\mathcal { B } _ { i , i - l } | + \\sum ^ k _ { i = 1 } \\sum ^ i _ { l = 1 } | \\mathcal { B } _ { i - l , i } | = | \\{ ( i , i , t , p ) \\mid ( i , i , t , p ) \\in \\mathcal { I } ' _ { k - 1 } \\} | + | \\mathcal { I } ' _ { k - 1 } | . \\end{align*}"}
+{"id": "1350.png", "formula": "\\begin{align*} C ( \\eta ) : = 2 \\eta ^ { - n / p } . \\end{align*}"}
+{"id": "1241.png", "formula": "\\begin{align*} \\int _ { t \\in \\mathcal { T } \\cap \\mathcal { P } ( m ) } 1 \\ , d t \\leq \\int _ { t \\in \\mathcal { T } } ( 2 ^ { m / 1 0 } \\Re P _ m ( t ) ) ^ { 2 [ 2 ^ { 3 m / 4 } ] } \\ , d t . \\end{align*}"}
+{"id": "5107.png", "formula": "\\begin{align*} [ ( - \\Delta _ x ) ^ { \\sigma } , i \\Gamma _ { \\phi } ] = \\frac { \\sin ( \\pi \\sigma ) } { \\pi } \\int _ 0 ^ { \\infty } m ^ { \\sigma } \\frac { 1 } { - \\Delta _ x + m } [ - \\Delta _ x , i \\Gamma _ { \\phi } ] \\frac { 1 } { - \\Delta _ x + m } \\ , d m . \\end{align*}"}
+{"id": "311.png", "formula": "\\begin{align*} \\lambda _ { x } = \\sum _ { i = 1 } ^ { n } e _ { i } ^ { 2 } = n \\end{align*}"}
+{"id": "8164.png", "formula": "\\begin{align*} B _ j = X A _ j Y \\ \\ ( j = 0 , \\ldots , m ) . \\end{align*}"}
+{"id": "5775.png", "formula": "\\begin{align*} - 8 < a < 0 , - \\frac { 1 } { 3 } a < b < - \\frac { 3 } { 8 } a , \\frac { - 1 5 a - 8 b } { 3 2 } < c \\leq \\frac { - 3 3 a - 8 8 b + 9 3 } { 3 1 } \\end{align*}"}
+{"id": "3979.png", "formula": "\\begin{align*} \\bar { u } _ { 2 \\beta } ( x , t ) = \\begin{cases*} 2 u _ { 2 \\beta } ( x , t ) , \\ x > 0 , \\\\ 0 , \\ x < 0 , \\end{cases*} \\end{align*}"}
+{"id": "5823.png", "formula": "\\begin{align*} \\omega ( V ) = \\omega ( V ^ * ) = 1 . \\end{align*}"}
+{"id": "5354.png", "formula": "\\begin{align*} Q & = - \\frac { 1 2 } { 2 5 } ( \\mu ^ 3 - 7 \\lambda \\mu ^ 2 + 3 3 \\lambda ^ 2 \\mu - 9 \\lambda ^ 3 ) , \\\\ L _ 1 & = 1 2 ( \\mu + \\lambda ) ^ 2 ( \\mu - 3 \\lambda ) , \\\\ L _ 2 & = \\frac { 6 } { 2 5 } ( \\mu + \\lambda ) ^ 2 ( 7 \\mu - 3 3 \\lambda ) , \\\\ L _ 3 & = \\frac { 3 9 } { 2 5 } ( \\mu + \\lambda ) ^ 3 . \\end{align*}"}
+{"id": "553.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n \\big ( z _ i ^ 2 - z _ i z _ j \\big ) J _ { i j } K '' ( x _ i - x _ j ) & = \\frac 1 2 \\sum _ { i , j = 1 } ^ n ( z _ i - z _ j ) ^ 2 J _ { i j } K '' ( x _ i - x _ j ) . \\end{align*}"}
+{"id": "4184.png", "formula": "\\begin{align*} i ( w ^ i , w _ 1 ) & = \\underset { j } { \\sum } c ^ j i ( w ^ j , w _ 1 ) \\\\ i ( w ^ i , w _ 2 ) & = \\underset { j } { \\sum } c ^ j i ( w ^ j , w _ 2 ) \\\\ i ( w ^ i , w _ 3 ) & = \\underset { j } { \\sum } c ^ j i ( w ^ j , w _ 3 ) . \\\\ \\end{align*}"}
+{"id": "1395.png", "formula": "\\begin{align*} ( u _ 0 ^ { ( j ) } , u _ 1 ^ { ( j ) } ) \\in D ( \\mathcal { A } _ D ) , \\lim _ { j \\to \\infty } ( u _ 0 ^ { ( j ) } , u _ 1 ^ { ( j ) } ) = ( u _ 0 , u _ 1 ) \\ \\ \\mathcal { H } _ D \\end{align*}"}
+{"id": "4758.png", "formula": "\\begin{gather*} \\beta _ 2 ( \\partial _ 1 ( a ) \\cdot b ) - \\beta _ 1 ( \\partial _ 2 ( a ) \\cdot b ) + \\partial _ 1 ( a ) \\cdot \\partial _ 2 ( b ) - \\partial _ 2 ( a ) \\cdot \\partial _ 1 ( b ) \\\\ \\qquad { } \\overset { \\eqref { e q : q a d m 1 } } { = } \\beta _ 2 ( \\partial _ 1 ( a ) ) \\cdot b - \\beta _ 1 ( \\partial _ 2 ( a ) ) \\cdot b , \\forall a , b \\in A . \\end{gather*}"}
+{"id": "7858.png", "formula": "\\begin{align*} \\nu ( J ) \\int _ { G _ { \\theta ( J ) } ' } \\sum _ { z \\in Z _ \\delta } { \\bf 1 } _ { \\Gamma ^ { \\delta } _ z } ( \\theta ( J ) , y ) d y & = \\int _ J \\int _ { G _ { \\theta ( J ) } ' } \\sum _ { z \\in Z _ \\delta } { \\bf 1 } _ { \\Gamma ^ { \\delta } _ z } ( \\theta ( J ) , y ) d y d \\nu ( \\theta ) \\\\ & \\geq \\frac { 1 } { 2 } \\int _ J \\int _ { G _ { \\theta } ' } \\sum _ { z \\in Z _ \\delta } { \\bf 1 } _ { \\Gamma ^ { \\delta } _ z } ( \\theta , y ) d y d \\nu ( \\theta ) . \\end{align*}"}
+{"id": "274.png", "formula": "\\begin{align*} \\iota = e ^ { - ( \\log N ) ^ 2 } , \\end{align*}"}
+{"id": "1421.png", "formula": "\\begin{align*} \\mathcal { U } ( t ) & = \\begin{pmatrix} u ( t ) \\\\ v ( t ) \\end{pmatrix} = U ( t ) \\begin{pmatrix} u _ 0 \\\\ u _ 1 \\end{pmatrix} + \\int _ 0 ^ t U ( t - s ) \\begin{pmatrix} 0 \\\\ - | u ( s ) | ^ { p - 1 } u ( s ) \\end{pmatrix} \\ , d s \\end{align*}"}
+{"id": "5374.png", "formula": "\\begin{align*} V _ \\mathfrak { J } ^ \\omega = T \\C . \\end{align*}"}
+{"id": "4020.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u _ t = \\Delta u + | u | ^ { p - 1 } u , \\\\ u ( . , 0 ) = u _ 0 \\in L ^ \\infty ( \\R ^ N , \\R ) , \\end{array} \\right . \\mbox { } \\end{align*}"}
+{"id": "6242.png", "formula": "\\begin{align*} H ( p ) = - \\partial _ x ^ 2 + ( p + B x ) ^ 2 - \\partial _ z ^ 2 \\end{align*}"}
+{"id": "2890.png", "formula": "\\begin{align*} \\kappa _ 1 = a \\kappa _ 2 + b , \\end{align*}"}
+{"id": "5712.png", "formula": "\\begin{align*} \\partial _ t \\nu _ t = \\kappa ^ + [ \\nu _ t ] - \\kappa ^ - [ \\nu _ t ] , \\nu _ t \\in \\Gamma . \\end{align*}"}
+{"id": "5655.png", "formula": "\\begin{align*} \\Box f : = \\mathrm { t r } _ { g _ { \\nabla f } } H ( f ) = \\sum _ { i = 1 } ^ { 2 n } H ( f ) ( E _ i , E _ i ) \\end{align*}"}
+{"id": "534.png", "formula": "\\begin{align*} G _ 1 ( Q _ 1 , \\dots , Q _ n ) \\coloneqq \\int _ { \\R ^ n } f ( x _ 1 , \\dots , x _ n ) \\prod _ { i = 1 } ^ { n } Q _ i ( d x _ i ) , G _ 2 ( Q _ 1 , \\dots , Q _ n ) \\coloneqq H ( Q _ 1 \\times \\cdots \\times Q _ n ) . \\end{align*}"}
+{"id": "2195.png", "formula": "\\begin{align*} & f _ { 1 1 } f _ { 2 2 } - f _ { 1 2 } { } ^ 2 - \\frac { a } { E } f _ 1 f _ { 2 2 } + b f _ 2 f _ { 2 2 } + \\frac { 2 b } { E } f _ 1 f _ { 1 2 } + \\frac { b ^ 2 } { E } f _ 2 { } ^ 2 = \\frac { b ^ 2 c } { E } , \\\\ & \\frac { 1 } { E } f _ 1 { } ^ 2 + f _ 2 { } ^ 2 \\leq c \\end{align*}"}
+{"id": "2656.png", "formula": "\\begin{align*} B ( L X ) + D ( L X ) + ( n - 1 ) [ \\beta ( L X ) + \\gamma ( L X ) ] = 0 . \\end{align*}"}
+{"id": "5046.png", "formula": "\\begin{align*} { } \\Lambda _ f \\bigl ( s , \\tfrac { a } { q } \\bigr ) = \\omega \\chi ( \\bar { a } ) q ^ { 1 - 2 s } \\Lambda _ f \\bigl ( 1 - s , - \\tfrac { \\bar { a } } { q } \\bigr ) . \\end{align*}"}
+{"id": "299.png", "formula": "\\begin{align*} C _ { E } = \\tau _ { E } ^ { - 1 } ( T ^ { * } _ { X } X ( \\log E ) ) . \\end{align*}"}
+{"id": "2666.png", "formula": "\\begin{align*} A R ( G ) = \\begin{cases} A \\rtimes \\{ I \\} & d \\\\ A \\rtimes \\langle - I \\rangle & d \\end{cases} \\end{align*}"}
+{"id": "8107.png", "formula": "\\begin{align*} \\mathcal { H } = \\mathcal { H } _ { u u } \\oplus \\mathcal { H } _ { u s } \\oplus \\mathcal { H } _ { s u } \\oplus \\mathcal { H } _ { s s } , \\end{align*}"}
+{"id": "5724.png", "formula": "\\begin{align*} f ( x ) = \\phi ( \\pi _ n ( x ) ) = \\phi ( p _ 1 ( x ) , \\dots , p _ n ( x ) ) , x \\in \\R ^ \\N , \\end{align*}"}
+{"id": "375.png", "formula": "\\begin{align*} \\{ \\alpha , \\{ \\beta , \\gamma \\} \\} = \\{ \\{ \\alpha , \\beta \\} , \\gamma \\} + \\{ \\beta , \\{ \\alpha , \\gamma \\} \\} ~ , \\end{align*}"}
+{"id": "3282.png", "formula": "\\begin{gather*} \\frac { ( x _ 3 - d _ 2 ) \\big ( 1 - c _ 1 ( x _ 3 ) ^ 2 \\big ) ^ 2 - c _ 2 ( d _ 1 ) ^ 2 x _ 3 } { ( 1 - c _ 1 ( x _ 3 ) ^ 2 ) ^ 2 } = 0 . \\end{gather*}"}
+{"id": "3688.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\rho \\left ( \\partial \\mathcal { H } _ n ( \\alpha ) \\right ) = 2 \\cdot p _ { K _ { t + 4 } } ( x _ 1 , \\ldots , x _ { t + 4 } ) = \\frac { t ^ 2 + 3 t + 2 + 4 \\alpha ( 1 - \\alpha ) } { ( t + 2 ) ^ 2 } , \\end{align*}"}
+{"id": "2880.png", "formula": "\\begin{align*} \\Gamma _ { \\frac { 1 } { K } } & : = \\Big \\{ ( \\nu _ 1 , \\nu _ 2 , \\nu _ 3 ) : 2 \\nu _ 1 \\nu _ 3 = \\nu _ 2 ^ 2 , 1 - \\frac { 1 } { K } \\le \\nu _ 3 \\le 1 , \\quad \\Big | \\frac { \\nu _ 2 } { \\nu _ 3 } \\Big | \\le 1 \\Big \\} . \\end{align*}"}
+{"id": "7614.png", "formula": "\\begin{align*} | | u ^ * ( \\theta ) - u ( \\theta ) | | _ \\pi | | g _ u ( \\theta ) + v _ u ( \\theta ) | | \\geq \\mathbb { E } _ \\pi \\left ( f ( u ( \\theta ) , \\theta ) - f ( u ^ * ( \\theta ) \\theta ) \\right ) = \\varepsilon \\end{align*}"}
+{"id": "2815.png", "formula": "\\begin{align*} & \\int _ { [ t , T ] } D ^ X _ { s - } d X _ s + \\frac 1 2 \\int _ { [ t , T ] } \\Delta X _ s \\gamma _ s d X _ s = \\frac 1 2 \\left ( \\gamma _ T ^ { - 1 } ( D ^ X _ T ) ^ 2 - \\gamma _ t ^ { - 1 } d ^ 2 - \\int _ { t } ^ T ( D ^ X _ s ) ^ 2 \\nu _ s ^ 2 d \\left ( \\nu ^ { - 2 } _ s \\gamma _ s ^ { - 1 } \\right ) \\right ) \\end{align*}"}
+{"id": "6800.png", "formula": "\\begin{align*} \\alpha ( x _ { [ 5 ] } ) + \\alpha \\circ ( 3 \\ , \\ , 4 ) ( x _ { [ 5 ] } ) = \\sum _ { i \\in [ r ] } \\beta _ { i , 1 } ( x _ { I _ { i , 1 } } ) \\beta _ { i , 2 } ( x _ { I _ { i , 2 } } ) \\dots \\beta _ { i , d _ i } ( x _ { I _ { i , d _ i } } ) \\end{align*}"}
+{"id": "7331.png", "formula": "\\begin{align*} \\lim \\limits _ { m \\rightarrow \\infty } \\mu ( u ^ { m } ( \\cdot ) ) = 0 . \\end{align*}"}
+{"id": "620.png", "formula": "\\begin{align*} \\widetilde { \\lambda } ( m ) \\ r _ n ( m ) = E _ { n } \\ \\widetilde { r } _ { n + 1 } ( m ) + F _ n \\ \\widetilde { r } _ { n } ( m ) + G _ n \\ \\widetilde { r } _ { n - 1 } ( m ) \\ , , \\end{align*}"}
+{"id": "1901.png", "formula": "\\begin{align*} 1 = \\frac 1 q + \\frac 1 { q ' } + \\frac 1 3 1 = \\frac 1 r + \\frac 1 { r ' } . \\end{align*}"}
+{"id": "2373.png", "formula": "\\begin{align*} \\partial _ { \\alpha , \\beta } ( e _ { z _ { 0 } } \\cdots e _ { z _ { k + 1 } } ) & = \\sum _ { i = 1 } ^ { k } \\left ( \\Delta _ { z _ { i } , z _ { i + 1 } } ^ { \\alpha , \\beta } - \\Delta _ { z _ { i } , z _ { i - 1 } } ^ { \\alpha , \\beta } \\right ) e _ { z _ { 0 } } \\cdots \\widehat { e _ { z _ { i } } } \\cdots e _ { z _ { k + 1 } } \\end{align*}"}
+{"id": "5022.png", "formula": "\\begin{align*} \\{ \\mathcal { F } ( i ) \\} _ { i = 1 } ^ { N } : = \\{ \\mathcal { F } ( 1 ) , \\dots , \\mathcal { F } ( N - 1 ) , \\mathcal { E } ( N ) \\} , \\end{align*}"}
+{"id": "4955.png", "formula": "\\begin{align*} [ \\varphi ( x ) , \\varphi ( y ) ] = \\phi [ x , y ] . \\end{align*}"}
+{"id": "8014.png", "formula": "\\begin{align*} T _ Q = \\begin{bmatrix} Q _ 0 & Q _ { - 1 } & Q _ { - 2 } & \\cdots & \\cdots \\\\ Q _ { 1 } & Q _ 0 \\otimes I _ d & Q _ { - 1 } \\otimes I _ d & \\cdots & \\cdots \\\\ Q _ { 2 } & Q _ { 1 } \\otimes I _ d & Q _ 0 \\otimes I _ d \\otimes I _ d & \\cdots & \\cdots \\\\ \\vdots & \\vdots & \\vdots & \\ddots & & \\\\ \\end{bmatrix} , \\end{align*}"}
+{"id": "5672.png", "formula": "\\begin{align*} C ^ \\mathrm { i n t } _ { p , r } ( u , v ) = \\P _ p \\left ( u \\leftrightarrow v , d _ \\mathrm { i n t } ( u , v ) \\geq r \\right ) . \\end{align*}"}
+{"id": "2777.png", "formula": "\\begin{align*} A v _ i & = u _ i \\sigma _ i , i = 1 , \\ldots , n , \\\\ A ^ T u _ i & = v _ i \\sigma _ i , i = 1 , \\ldots , n , \\\\ A ^ T u _ i & = 0 , i = n + 1 , \\ldots , m . \\end{align*}"}
+{"id": "7485.png", "formula": "\\begin{align*} \\psi _ m ( n ) = \\sum _ { i = 0 } ^ m a _ { m i } n ^ i \\ , . \\end{align*}"}
+{"id": "3820.png", "formula": "\\begin{align*} \\l = ( 1 , \\dots , n - 7 , n - 5 , n - 4 , n - 2 , n + 1 , 2 n - 5 ) , \\end{align*}"}
+{"id": "8218.png", "formula": "\\begin{align*} Y _ j ^ 2 = \\Vert \\dot { \\Delta } _ j \\sigma \\Vert ^ 2 _ { L ^ 2 } + \\Vert \\dot { \\Delta } _ j d \\Vert ^ 2 _ { L ^ 2 } - \\delta \\frac { \\mu } { \\lambda } ( \\dot { \\Delta } _ j d | \\Lambda ^ { \\alpha - 1 } \\dot { \\Delta } _ j \\sigma ) , j \\leq j _ 0 , \\end{align*}"}
+{"id": "7468.png", "formula": "\\begin{align*} \\mathrm { I m } \\ , \\left ( N _ { \\gamma } + \\widetilde { N } _ { \\gamma } \\cdot F _ { \\gamma } \\right ) = \\real ^ { 2 m } . \\end{align*}"}
+{"id": "4707.png", "formula": "\\begin{align*} & \\int _ { - \\pi } ^ { \\pi } \\left ( P ( h ; r e ^ { i \\theta } ) - S _ { \\lambda } h ( r e ^ { i \\theta } ) \\right ) \\overline { f ( e ^ { i \\theta } ) } d m _ { \\lambda } ( \\theta ) \\\\ & = \\int _ { - \\pi } ^ { \\pi } h ( e ^ { i \\varphi } ) r e ^ { i \\varphi } \\ , \\overline { \\int _ { - \\pi } ^ { \\pi } e ^ { i \\theta } C ( e ^ { i \\theta } , r e ^ { i \\varphi } ) f ( e ^ { i \\theta } ) d m _ { \\lambda } ( \\theta ) } \\ , d m _ { \\lambda } ( \\varphi ) \\\\ & = 0 . \\end{align*}"}
+{"id": "7344.png", "formula": "\\begin{align*} { \\rm d i m } ( \\mathcal { A } ) = 4 { m + 4 \\choose 4 } . \\end{align*}"}
+{"id": "7586.png", "formula": "\\begin{align*} \\frac { h ( s ) - h ( t ) } { s - t } = \\frac { 1 } { ( z - s ) ( z - t ) } \\end{align*}"}
+{"id": "5824.png", "formula": "\\begin{align*} \\omega ( R T ) = 1 ~ ~ ~ \\mbox { a n d } ~ ~ ~ \\omega ( R ) = 1 = \\omega ( T ) . \\end{align*}"}
+{"id": "1000.png", "formula": "\\begin{align*} { \\rm D } ^ 2 \\widetilde { W } ( { \\rm D } \\phi ) . ( \\xi \\otimes \\eta , \\xi \\otimes \\eta ) = ( b _ 1 + b _ 2 ) \\| \\xi \\| ^ 2 \\| \\eta \\| ^ 2 \\sin ^ 2 \\theta + ( b _ 1 + b _ 3 ) \\| \\xi \\| ^ 2 \\| \\eta \\| ^ 2 \\cos ^ 2 \\theta , \\end{align*}"}
+{"id": "3902.png", "formula": "\\begin{align*} \\partial _ t u - ( 1 + D _ t ^ { \\{ m \\} } ) \\Delta u & = F \\ ; \\Omega , t \\in ( 0 , T ] , \\\\ u & = 0 \\ ; \\partial \\Omega , \\ ; t \\in [ 0 , T ] , \\\\ u ( 0 ) & = \\xi \\ ; \\Omega , \\end{align*}"}
+{"id": "5123.png", "formula": "\\begin{align*} \\lim _ { t _ 1 , t _ 2 \\rightarrow \\infty } \\left ( \\| | u | ^ p u \\| _ { L _ t ^ { l ' } L _ x ^ { m ' } L ^ 2 _ { y } ( [ t _ 1 , t _ 2 ] \\times \\mathbb { R } ^ d \\times \\mathbb { T } ) } + \\| | \\nabla _ x | ^ { \\sigma } ( | u | ^ p u ) \\| _ { L _ t ^ { l ' } L _ x ^ { m ' } L ^ 2 _ { y } ( [ t _ 1 , t _ 2 ] \\times \\mathbb { R } ^ d \\times \\mathbb { T } ) } \\right . \\\\ \\left . + \\| | \\partial _ { y } | ^ { \\sigma } ( | u | ^ p u ) \\| _ { L _ t ^ { l ' } L _ x ^ { m ' } L ^ 2 _ { y } ( [ t _ 1 , t _ 2 ] \\times \\mathbb { R } ^ d \\times \\mathbb { T } ) } \\right ) = 0 . \\end{align*}"}
+{"id": "1167.png", "formula": "\\begin{align*} f _ k ( x ) : = \\sum _ { j = 0 } ^ { k } \\dbinom { k } { j } ^ 2 ( 1 - x ) ^ { p _ k ( k - j ) / k } x ^ { p _ k j / k } . \\end{align*}"}
+{"id": "8026.png", "formula": "\\begin{align*} A = Q _ 0 \\mbox { a n d } B = c o l ( Q _ i ) _ { i = 1 } ^ { m } \\end{align*}"}
+{"id": "4580.png", "formula": "\\begin{align*} e ^ { 2 i \\theta } = 1 - \\frac { 2 } { 1 + i \\cot \\theta } , \\end{align*}"}
+{"id": "6699.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & \\partial _ { t } F + v \\cdot \\nabla _ { x } F - ( E + v \\times B ) \\cdot \\nabla _ { v } F = \\frac { 1 } { \\varepsilon } Q ( F , F ) , \\\\ & \\partial _ { t } E - \\nabla _ { x } \\times B = \\int _ { \\mathbb { R } ^ { 3 } } v F \\ , d v , \\\\ & \\partial _ { t } B + \\nabla _ { x } \\times E = 0 , \\\\ & \\nabla _ { x } \\cdot E = n _ b - \\int _ { \\mathbb { R } ^ { 3 } } F \\ , d v , \\nabla _ { x } \\cdot B = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "3192.png", "formula": "\\begin{align*} B ( u _ { n } ) = \\int _ { \\mathbb { R } ^ { 3 } } \\phi _ { u _ { n } } u _ { n } ^ { 2 } d x \\le C \\left \\| u _ { n } \\right \\| _ { \\frac { 1 2 } { 5 } } ^ { 4 } \\le C \\left \\| u _ { n } \\right \\| _ { 2 } ^ { 3 } \\left \\| \\nabla u _ { n } \\right \\| _ { 2 } ^ { 3 } . \\end{align*}"}
+{"id": "3120.png", "formula": "\\begin{align*} F _ { X _ 1 } ( \\sigma _ 1 ^ { \\ell _ 2 } ) = \\ell _ 2 q _ 2 . \\end{align*}"}
+{"id": "2105.png", "formula": "\\begin{align*} \\Omega _ { k , \\ell , m , n } ( F ) & : = \\{ E \\in G ( k , n ) \\mid \\dim ( E \\cap F ) \\ge m \\} , \\\\ \\Omega _ { k , \\ell , m , n } ^ o ( F ) & : = \\{ E \\in G ( k , n ) \\mid \\dim ( E \\cap F ) = m \\} . \\end{align*}"}
+{"id": "7320.png", "formula": "\\begin{align*} a _ 0 + a _ 1 \\beta ^ { i _ 1 } = - \\beta ^ { i _ 2 } . \\end{align*}"}
+{"id": "3149.png", "formula": "\\begin{align*} - \\Delta _ { \\mathbb { B } ^ { N } } u \\ , - \\ , \\lambda u \\ , = \\ , u ^ { p } , u \\in H ^ { 1 } \\left ( \\mathbb { B } ^ { N } \\right ) , \\end{align*}"}
+{"id": "1076.png", "formula": "\\begin{align*} \\sigma ( P ) ( x , \\xi ) = \\mathfrak a _ { 2 } + \\mathfrak a _ { 1 } + \\mathfrak a _ { 0 } . \\end{align*}"}
+{"id": "7482.png", "formula": "\\begin{align*} \\lim _ { n = 1 } B _ { m , n - 2 } & = \\lim _ { n = 1 } \\frac { ( n + m - 2 ) ! } { m ! \\ , ( n - 2 ) ! } \\ , , \\\\ & = \\lim _ { n = 1 } ( n - 1 ) \\frac { ( n + m - 2 ) ! } { m ! \\ , ( n - 1 ) ! } \\ , , \\\\ \\lim _ { n = 1 } B _ { m , n - 2 } & = 0 \\ , . \\end{align*}"}
+{"id": "7900.png", "formula": "\\begin{align*} R ( a ) \\cdot R ( b ) + a \\cdot b = R \\big ( R ( a ) \\cdot b + a \\cdot R ( b ) \\big ) . \\end{align*}"}
+{"id": "2549.png", "formula": "\\begin{align*} \\alpha ^ \\xi _ t : = - R ^ { - 1 } B ( P _ t x ^ \\xi _ t + \\varphi _ t ^ \\xi ) - h ( \\mu _ t ) \\end{align*}"}
+{"id": "2374.png", "formula": "\\begin{align*} \\Delta _ { x , y } ^ { \\alpha , \\beta } = \\begin{cases} 1 & \\{ \\alpha , \\beta \\} \\in \\{ \\{ x , y \\} , \\{ 1 - x , 1 - y \\} \\} \\\\ 0 & \\{ \\alpha , \\beta \\} \\notin \\{ \\{ x , y \\} , \\{ 1 - x , 1 - y \\} \\} . \\end{cases} \\end{align*}"}
+{"id": "760.png", "formula": "\\begin{align*} V _ { a , b } = \\mathbb { C } [ \\partial ] v , L _ \\lambda v = ( \\partial + a \\lambda + b ) v , { H _ i } _ \\lambda v = 0 , \\end{align*}"}
+{"id": "3396.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ m \\sum _ { j = 0 } ^ m q _ { i j } s _ { g _ 1 - i , g _ 2 - j } = 0 . \\end{align*}"}
+{"id": "2494.png", "formula": "\\begin{align*} u \\ \\mbox { i s b o u n d e d i n } \\ B _ { R } \\ \\mbox { a n d } \\ \\lim \\limits _ { | x | \\nearrow R } v ( x ) = \\infty \\end{align*}"}
+{"id": "2550.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { \\xi } = & ~ [ ( A - B ^ 2 R ^ { - 1 } P _ t ) x _ t ^ { \\xi } - B ^ 2 R ^ { - 1 } \\varphi _ t ^ \\xi - B h ( \\mu _ t ) \\\\ & + f ( \\nu _ t ) + b ( \\mu _ t ) ] d t + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ 0 ^ { \\xi } = & ~ \\xi ; \\end{aligned} \\right . \\end{align*}"}
+{"id": "7204.png", "formula": "\\begin{align*} \\mathrm { P e r f } ^ { \\mathrm { g r } } ( \\mathcal { X } \\times \\mathbb { C } ) & = \\langle \\Gamma _ G ( \\chi ) \\otimes \\mathcal { O } _ { X \\times \\mathbb { C } } ( i ) \\mid \\chi \\in M ^ + , i \\in \\mathbb { Z } \\rangle , \\\\ \\mathrm { P e r f } ^ { \\mathrm { g r } } ( \\mathcal { X } ) & = \\langle \\Gamma _ G ( \\chi ) \\otimes \\mathcal { O } _ { X } ( i ) \\mid \\chi \\in M ^ + , i \\in \\mathbb { Z } \\rangle . \\end{align*}"}
+{"id": "1101.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m U _ i < \\sum _ { i = 1 } ^ m V _ i , \\end{align*}"}
+{"id": "8187.png", "formula": "\\begin{align*} \\begin{cases} \\dot { X } _ i ( t ) = V _ i ( t ) , 1 \\leq i \\leq M , \\\\ \\dot { V } _ i ( t ) = - \\frac { 1 } { M } \\sum \\limits _ { j \\neq i } \\phi ( X _ i ( t ) - X _ j ( t ) ) ( V _ i ( t ) - V _ j ( t ) ) , \\end{cases} \\end{align*}"}
+{"id": "1713.png", "formula": "\\begin{align*} d \\tilde { \\mathbb { P } } _ { \\mathrm { G i b b s } } ^ { f } ( \\varphi ) : = \\frac { 1 } { \\tilde { z } _ { \\mathrm { G i b b s } } ^ { f } } \\ , e ^ { - H ^ { \\mathrm { W i c k } } ( \\varphi ) } \\ , f \\bigl ( : \\| \\varphi \\| _ { \\mathfrak { h } } ^ 2 : \\bigr ) \\ , d \\varphi \\ , . \\end{align*}"}
+{"id": "190.png", "formula": "\\begin{align*} Q ^ + _ { A _ 0 } ( e ^ { - i u } \\psi ) & + \\pi ^ { - } ( a \\cdot e ^ { - i u } \\psi + i d u \\cdot e ^ { - i u } \\psi ) = e ^ { - i u } Q ^ + _ { A _ 0 } \\psi - i e ^ { - i u } \\pi ^ - ( d u \\cdot \\psi ) + \\\\ & + e ^ { - i u } \\pi ^ { - } ( a \\cdot \\psi ) + i e ^ { - i u } \\pi ^ - ( d u \\cdot \\psi ) = e ^ { - i u } ( Q ^ + _ { A _ 0 } \\psi + \\pi ^ { - } ( a \\cdot \\psi ) ) . \\end{align*}"}
+{"id": "7405.png", "formula": "\\begin{align*} \\sigma ^ { - 1 } ( B _ 3 ) & \\colon \\ ; v _ 1 u = z , \\\\ \\sigma ^ { - 1 } ( B _ 6 ) & \\colon \\ ; u ( v _ 1 z - u ) = 0 ; \\end{align*}"}
+{"id": "5627.png", "formula": "\\begin{align*} J ^ i _ { k | l } & = J ^ s _ k \\hat { \\mathbb { G } } ^ i _ { s l } - J ^ i _ s \\hat { \\mathbb { G } } ^ s _ { k l } . \\end{align*}"}
+{"id": "765.png", "formula": "\\begin{align*} D + E ^ + - E ^ - = f ^ * ( f _ * D ) . \\end{align*}"}
+{"id": "4420.png", "formula": "\\begin{align*} \\sigma \\nu = 0 \\partial \\Omega . \\end{align*}"}
+{"id": "3115.png", "formula": "\\begin{align*} a _ { \\ell _ 1 , 0 } = \\ell _ 1 q _ 1 \\end{align*}"}
+{"id": "3839.png", "formula": "\\begin{align*} \\eta = & \\left ( n - 2 - \\left ( k + \\frac { - h ^ 2 + 3 h + 4 } { 2 } - i \\right ) , n - 2 - ( h - 2 + i _ { h - 4 } ) , \\ldots , n - 2 - ( 3 + i _ 1 ) \\right ) \\\\ = & \\left ( \\frac { n + d + h ^ 2 - 3 h - 9 } { 2 } + 1 , n - h - i _ { h - 4 } , \\ldots , n - 5 - i _ 1 \\right ) , \\end{align*}"}
+{"id": "4015.png", "formula": "\\begin{align*} \\bar { q } _ { \\beta } ( n , t ) = \\sum _ { \\Omega ^ { n } _ { k } } \\prod _ { j = 1 } ^ { k } \\rho ^ { x _ { j } } \\binom { r + x _ { j } - 1 } { x _ { j } } \\left ( \\frac { \\lambda t ^ { \\beta } ( 1 - \\rho ) ^ { r } } { 1 - ( 1 - \\rho ) ^ { r } } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } \\left ( - \\lambda t ^ { \\beta } \\right ) . \\end{align*}"}
+{"id": "4875.png", "formula": "\\begin{align*} F _ { n + 1 } ( x , y ) = \\left ( x ( 1 - x ) \\frac { d } { d x } + n x + y \\right ) F _ n ( x , y ) , \\end{align*}"}
+{"id": "4588.png", "formula": "\\begin{align*} \\varphi ( n _ 0 + ( m + 1 ) N ) = \\varphi ( n _ 0 + m N ) + L + \\frac { K } { ( n _ 0 + m N - b ) \\pi \\sin \\pi k } \\left ( 1 0 0 + \\varepsilon ( m ) \\right ) , \\end{align*}"}
+{"id": "1086.png", "formula": "\\begin{align*} b _ { 1 } b _ { 2 } u _ { 3 } ^ { 2 } - b _ { 2 } u _ { 3 } ^ { 2 } = p q ^ { 2 k + 1 } , \\end{align*}"}
+{"id": "6147.png", "formula": "\\begin{align*} V _ 1 g = V _ 1 V ^ n h _ n = V ^ n ( q ^ n V _ 1 h _ n ) V _ 2 g = V _ 2 V ^ n h _ n = V ^ n ( \\overline q ^ n V _ 2 h _ n ) . \\end{align*}"}
+{"id": "4842.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial \\Pi _ t } { \\partial t } = & \\ , \\nabla _ x \\cdot ( \\Pi _ t \\nabla _ x f ( x , s _ 1 ) ) + \\nabla _ y \\cdot ( \\Pi _ t \\nabla _ y F ( y ) ) + \\\\ & + K ~ \\int _ S \\int _ S ( \\Gamma ( s _ 1 , s _ 2 ) \\Pi _ t ( x , y , s _ 1 ' , s _ 2 ' ) - \\Pi _ t ( x , y , s _ 1 , s _ 2 ) ) \\dd \\mu ( s _ 1 ' ) \\dd \\mu ( s _ 2 ' ) , \\end{aligned} \\end{align*}"}
+{"id": "1291.png", "formula": "\\begin{align*} h ' \\circ \\sum _ { i = 0 } ^ { n - 1 } \\iota ^ i _ a \\circ \\pi ^ i _ a = \\sum _ { i = 0 } ^ { n - 1 } f ^ i \\circ \\pi ^ i _ a = h , \\end{align*}"}
+{"id": "1672.png", "formula": "\\begin{align*} \\mathsf { I } _ { \\mathsf { f } } : = \\left \\{ \\begin{array} { l l } \\mathsf { J } _ { \\mathsf { f } } \\ , , & \\mathsf { f } \\in \\{ 0 , \\dots , \\mathsf { F } - 1 \\} \\ , , \\\\ \\mathsf { B } \\ , , & \\mathsf { f } = \\mathsf { F } \\ , . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "2570.png", "formula": "\\begin{align*} \\alpha ^ { * , t _ 0 , x _ 0 , \\xi } _ t : = - B R ^ { - 1 } U ( t , x _ t ^ { * , t _ 0 , x _ 0 , \\xi } , \\nu _ t ^ { * , t _ 0 , \\xi } ) - h ( \\mu _ t ^ { * , t _ 0 , \\xi } ) . \\end{align*}"}
+{"id": "2221.png", "formula": "\\begin{align*} \\mathcal { K } _ n ( \\boldsymbol { \\mathcal { A } } , A ) & : = \\{ p ( \\boldsymbol { \\mathcal { A } } ) * A \\hbox { s . t . } d e g ( p ) \\leq n - 1 \\} , \\\\ \\mathcal { K } _ n ^ { D } ( B ^ D , \\boldsymbol { \\mathcal { A } } ) & : = \\{ B ^ D * p ^ { D } ( \\boldsymbol { \\mathcal { A } } ) \\hbox { s . t . } d e g ( p ^ D ) \\leq n - 1 \\} . \\end{align*}"}
+{"id": "1957.png", "formula": "\\begin{align*} F _ N ( x ) : = \\sum _ { \\ell = 1 } ^ N c _ { \\ell e _ n , R _ { 0 , k } } w _ { \\ell e _ n , R _ { 0 , k } } , G _ N ( x ) : = \\sum _ { \\ell = 1 } ^ N c _ { \\ell e _ m , R _ { 0 , k } } w _ { \\ell e _ m , R _ { 0 , k } } , \\end{align*}"}
+{"id": "1986.png", "formula": "\\begin{align*} W ( x , y ) = \\begin{cases} 0 , & ( x , y ) \\in [ 1 / 3 , 1 ] ^ 2 , \\\\ 1 , & , \\end{cases} \\end{align*}"}
+{"id": "4613.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ \\phi ( \\Omega ) } & : = \\inf \\Big \\{ \\lambda > 0 : \\varrho _ \\phi \\Big ( \\frac { u } { \\lambda } \\Big ) \\le 1 \\Big \\} . \\end{align*}"}
+{"id": "7297.png", "formula": "\\begin{align*} & X ^ * ( 0 , \\omega ) - \\overline { X } ^ * ( 0 ) = X _ 0 ( \\omega ) - \\overline { X } _ 0 , \\\\ & \\Upsilon ( T , \\omega ) - \\overline { \\Upsilon } ( T ) = - ( K _ x + K _ m ) [ X ^ * ( T , \\omega ) - \\overline { X } ^ * ( T ) ] . \\end{align*}"}
+{"id": "1081.png", "formula": "\\begin{align*} b ( x , y ) = \\begin{cases} ( x , x - 4 ^ { m } p ) & x \\neq 0 , 4 ^ { m } p , \\\\ ( - 1 , - p ) & x = 0 , \\\\ ( p , q ) & x = 4 ^ { m } p , \\\\ ( 1 , 1 ) & P = \\mathcal { O } , \\end{cases} \\end{align*}"}
+{"id": "6676.png", "formula": "\\begin{align*} \\underline { P } ( 0 ) = \\sigma \\left ( \\frac { \\underline { Q } ( t ) - \\underline { Q } ( 0 ) } { t } \\right ) = \\mathcal { D } \\left ( ( s _ { k + 1 } - a s _ { k } ) _ { k = 0 } ^ { 2 m - 1 } , ( s _ k - a s _ { k - 1 } ) _ { k = 0 } ^ { m } \\right ) , \\end{align*}"}
+{"id": "6519.png", "formula": "\\begin{align*} 2 Q - I = \\bigoplus _ { v \\in C _ 1 } G _ v = \\begin{pmatrix} G _ { v _ 1 } & & & \\\\ & G _ { v _ 2 } & & \\\\ & & \\ddots & \\\\ & & & G _ { v _ n } \\end{pmatrix} , \\end{align*}"}
+{"id": "6846.png", "formula": "\\begin{align*} \\lambda u _ i h ( a _ i ) = v _ i ^ { p ^ e + 1 } , \\ 1 \\leq i \\leq n , \\end{align*}"}
+{"id": "7472.png", "formula": "\\begin{align*} \\psi _ m ( n ) = n + ( m - 1 ) ( n - 1 ) B _ { m - 1 , n - 1 } \\ , , \\end{align*}"}
+{"id": "700.png", "formula": "\\begin{align*} { \\bf { \\hat \\Lambda } } = d i a g \\left [ { 1 , 1 , 1 , 1 , { { \\hat s } _ e } , { { \\hat s } _ \\upsilon } , { { \\hat s } _ \\upsilon } , { { \\hat s } _ \\upsilon } , { { \\hat s } _ \\upsilon } , { { \\hat s } _ \\upsilon } , { { \\hat s } _ q } , { { \\hat s } _ q } , { { \\hat s } _ q } , { { \\hat s } _ q } , { { \\hat s } _ q } , { { \\hat s } _ q } , { { \\hat s } _ \\pi } , { { \\hat s } _ \\pi } , { { \\hat s } _ \\pi } } \\right ] , \\end{align*}"}
+{"id": "6397.png", "formula": "\\begin{align*} \\pi _ { 2 1 } & = 1 , \\\\ \\pi _ { 1 3 } & = \\pi _ { 1 5 } = \\pi _ { 1 7 } = \\pi _ { 1 8 } = 0 , \\pi _ { 1 4 } = 1 , \\pi _ { 1 6 } = - 1 , \\\\ \\pi _ { 2 3 } & = \\pi _ { 2 4 } = \\pi _ { 2 6 } = \\pi _ { 2 8 } = 0 , \\pi _ { 2 5 } = 1 , \\pi _ { 2 7 } = - 1 , \\\\ X _ 1 & = \\left [ \\frac { e _ 2 e _ 3 } { e _ 4 e _ 5 e _ 7 } \\right ] , X _ 2 = \\left [ \\frac { e _ 4 e _ 6 e _ 7 ^ 2 } { e _ 1 ^ 2 e _ 8 } \\right ] . \\end{align*}"}
+{"id": "7574.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\left ( E _ { \\varphi } [ \\mu _ { \\varepsilon } ] - E _ { \\varphi } [ \\mu ^ { \\varepsilon } ] \\right ) = 0 . \\end{align*}"}
+{"id": "2487.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty \\frac { d s } { \\sqrt { F ( s ) } } < \\infty \\quad \\mbox { w h e r e } \\ ; F ( s ) = \\int _ 0 ^ s f ( t ) d t . \\end{align*}"}
+{"id": "3448.png", "formula": "\\begin{align*} \\Phi ( \\mathrm { x } , t ; \\mathrm { y } , \\tau ) : = \\ \\begin{cases} \\frac { \\alpha } { 4 \\pi ( t - \\tau ) } \\textbf { e x p } \\Bigg ( - \\frac { \\alpha | \\mathrm { x } - \\mathrm { y } | ^ 2 } { 4 ( t - \\tau ) } \\Bigg ) , \\ \\ \\ t > \\tau \\\\ 0 , \\end{cases} \\end{align*}"}
+{"id": "1655.png", "formula": "\\begin{align*} ( W _ n , v _ n ) : = \\mathcal { T } _ n \\star ( W _ { n - 1 } , v _ { n - 1 } ) \\ , , \\qquad ( W _ 0 , v _ 0 ) \\in \\mathfrak { W } \\ , , \\end{align*}"}
+{"id": "4304.png", "formula": "\\begin{align*} \\nu ( x + i y ) = \\frac { N ( x + i y ) } { { \\rm g c d } ( x , y ) } . \\end{align*}"}
+{"id": "3520.png", "formula": "\\begin{align*} \\Big \\Vert E \\Big \\Vert _ { { L } ^ 2 \\Big ( \\Omega \\Big ) } = \\mathcal { O } \\Big ( \\delta ^ { 1 - h } \\Big ) \\mathrm { h } < 1 . \\end{align*}"}
+{"id": "4521.png", "formula": "\\begin{align*} \\{ \\Pi _ y , \\mathcal { V } \\} ( \\phi , \\pi ) \\ ! & = \\ ! - \\dd _ { ( \\phi , \\pi ) } \\mathcal { V } ( \\mathbb { X } _ { \\Pi _ { \\ ! y } } \\ ! ) \\ ! = \\ ! - \\big ( \\dd _ \\phi V \\ , \\circ \\ , \\mathrm { p r o j } _ 1 \\big ) ( \\mathbb { X } _ { \\Pi _ { \\ ! y } } \\ ! ) \\ ! = \\ ! - \\dd _ \\phi V ( X _ 1 ) \\\\ \\ ! & = \\ ! - \\dd _ \\phi V ( \\mathcal { E } _ y ) \\ ! = \\ ! - \\int _ { [ 0 , 1 ] } \\ ! \\ ! \\ ! \\mathcal { E } _ y ' \\phi ' \\ , . \\end{align*}"}
+{"id": "4309.png", "formula": "\\begin{align*} { \\cal Q } = \\{ v \\in H ^ 1 ( 0 , T ; V ^ * ) \\cap L ^ 2 ( 0 , T ; V ) : \\ v ( 0 ) = 0 \\} , \\end{align*}"}
+{"id": "7426.png", "formula": "\\begin{align*} [ J _ z , \\ , J _ \\pm ] = \\pm J _ \\pm , [ J _ + , \\ , J _ - ] = 2 J _ z . \\end{align*}"}
+{"id": "128.png", "formula": "\\begin{align*} [ \\mathbf a ] = [ a _ 0 , a _ 1 , \\dots , a _ q ] \\end{align*}"}
+{"id": "1567.png", "formula": "\\begin{align*} \\left ( z _ { 3 } s ^ { \\prime } z _ { 1 } s ^ { \\prime } \\right ) ^ { - 1 } z _ { 3 } \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } \\right ) = e \\quad z _ { 3 } \\left ( z _ { 3 } s ^ { \\prime } z _ { 1 } s ^ { \\prime } \\right ) ^ { - 1 } \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } \\right ) = \\left [ z _ { 3 } , \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } \\right ) ^ { - 1 } \\right ] . \\end{align*}"}
+{"id": "6587.png", "formula": "\\begin{align*} \\dot { X } _ t = f ( X _ t ) , X _ 0 = 1 , \\end{align*}"}
+{"id": "4429.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\sup _ { v \\in X } \\sup _ { \\vert E \\vert < \\delta } \\int _ E \\vert v \\vert ^ p \\dd x = 0 . \\end{align*}"}
+{"id": "343.png", "formula": "\\begin{align*} \\begin{aligned} \\sqrt { N } | \\hat { h } - \\tilde { h } | & = \\frac { \\sqrt { N } | A _ { n , m } | } { | B _ { n , m } + \\frac { 1 } { 2 } C _ { n , m } \\zeta ( \\hat { h } - \\tilde { h } ) | } \\\\ \\end{aligned} \\end{align*}"}
+{"id": "6056.png", "formula": "\\begin{align*} \\begin{cases} \\ , \\ , \\eta _ \\epsilon \\in H ^ 1 _ 0 ( \\Omega ) \\ , \\ , \\\\ [ 0 . 3 c m ] \\int _ { \\Omega } \\nabla \\eta _ \\epsilon \\nabla v d x - \\int _ { \\Omega } k ^ 2 \\eta _ \\epsilon v d x = \\int _ { \\Omega } f _ \\epsilon v d x \\ , \\ , \\forall \\ , v \\in H ^ 1 _ 0 ( \\Omega ) . \\end{cases} \\end{align*}"}
+{"id": "3724.png", "formula": "\\begin{align*} \\partial _ t \\rho ' _ y + \\nabla ( \\rho ' _ y v ' _ y ) = 0 \\end{align*}"}
+{"id": "7465.png", "formula": "\\begin{align*} \\left ( V _ 0 + V _ 1 E _ { \\gamma } \\right ) \\cdot \\alpha + \\left ( V _ 0 E _ { \\gamma } + V _ 1 \\right ) \\cdot \\beta & = z _ C , \\\\ \\left ( - \\gamma W _ 0 + \\gamma W _ 1 E _ { \\gamma } \\right ) \\cdot \\alpha + \\left ( \\gamma W _ 0 E _ { \\gamma } - \\gamma W _ 1 \\right ) \\cdot \\beta & = z _ K \\end{align*}"}
+{"id": "6482.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ N ( \\Omega ) } = \\left ( \\int _ \\Omega | f ( x ) | ^ N \\ , d x \\right ) ^ { 1 / N } \\end{align*}"}
+{"id": "6432.png", "formula": "\\begin{align*} f ( 0 , y _ 1 , x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) = \\sum _ { n _ 1 = 0 } ^ { \\infty } c _ { n _ 1 } ( x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) y _ 1 ^ { n _ 1 } . \\end{align*}"}
+{"id": "3632.png", "formula": "\\begin{align*} f ( a ) = f ( \\alpha ) + C + c ( \\alpha - a ) \\end{align*}"}
+{"id": "2922.png", "formula": "\\begin{align*} s ( A ) = \\sum { \\rm l o g ^ + } \\sigma _ j ( A ) , \\end{align*}"}
+{"id": "2088.png", "formula": "\\begin{align*} N ( \\varphi ) : = N ( A ) = \\sigma _ 1 \\cdot \\ldots \\cdot \\sigma _ { { \\min \\{ m , n \\} } } . \\end{align*}"}
+{"id": "2296.png", "formula": "\\begin{align*} F _ 1 ( x _ 1 , x _ 2 ) = F _ 2 ( x _ 3 , x _ 4 ) = m , \\end{align*}"}
+{"id": "843.png", "formula": "\\begin{align*} J _ { i } ( \\hat { \\tau } _ i ) = \\mathbb { E } \\{ \\bar { \\theta } _ 1 \\hat { y } _ i - e ^ { - \\beta \\hat { \\tau } _ i } K \\} . \\end{align*}"}
+{"id": "5142.png", "formula": "\\begin{align*} A ' = \\bigcup _ { \\substack { Q _ i \\in \\mathcal { W } \\\\ | A \\cap Q _ i | > \\frac { 1 } { 2 } | Q _ i | } } Q _ i . \\end{align*}"}
+{"id": "2053.png", "formula": "\\begin{align*} \\begin{aligned} p ( 5 n + 4 ) & \\equiv 0 \\ ( \\mathrm { m o d } \\ 5 ) , \\\\ p ( 7 n + 5 ) & \\equiv 0 \\ ( \\mathrm { m o d } \\ 7 ) , \\\\ p ( 1 1 n + 6 ) & \\equiv 0 \\ ( \\mathrm { m o d } \\ 1 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "6.png", "formula": "\\begin{align*} \\alpha _ m : = \\min \\left \\{ \\langle Q , X \\rangle \\ , | \\ , X \\in \\R ^ { Z _ m \\times Z _ m } _ { \\geq 0 } , \\ , \\langle J , X \\rangle = 1 , \\ , X \\succeq 0 \\right \\} . \\end{align*}"}
+{"id": "4126.png", "formula": "\\begin{align*} U ( s ) = A _ n ^ { \\alpha } \\sqrt { \\sqrt { \\chi _ n ^ { \\alpha } ( c ) } s } \\ , J _ { 0 } ( \\sqrt { \\chi _ n ^ { \\alpha } ( c ) } s ) + \\sqrt { s } \\ , \\varepsilon _ { n , c } ^ { \\alpha } ( s ) \\ , ~ ~ s \\in ( 0 , S ( 0 ) ) , \\end{align*}"}
+{"id": "1169.png", "formula": "\\begin{align*} \\phi _ k \\Big ( \\frac { 1 } { k } \\Big ) & = \\frac { \\log ( 2 \\pi ) } { 2 } - \\frac { ( 2 k - p _ k ) \\log k } { k } + \\Big ( 2 k - p _ k + \\frac { p _ k } { 2 } - \\frac { 3 } { 2 } \\Big ) \\log \\Big ( 1 - \\frac { 1 } { k } \\Big ) \\\\ & > \\frac { \\log ( 2 \\pi ) } { 2 } - \\frac { \\log k } { 2 k } \\Big ( \\log _ 2 ( \\pi k ) + 1 \\Big ) - \\frac { 1 0 1 } { 2 0 0 k } \\Big ( \\log _ 2 ( \\pi k ) + 1 \\Big ) \\\\ & > \\frac { \\log ( 2 \\pi ) } { 2 } - \\frac { 6 0 1 } { 2 0 0 0 } > 0 , \\end{align*}"}
+{"id": "7374.png", "formula": "\\begin{align*} \\quad & f ( x ) : = \\sum _ { i = 1 } ^ n x _ i ^ 4 + \\sum _ { ( i , j ) \\in E } x _ i ^ 2 x _ j ^ 2 , \\\\ \\quad & x \\in \\mathbb { S } ^ { n - 1 } : = \\{ x \\in \\real ^ n : \\norm { x } = 1 \\} , \\end{align*}"}
+{"id": "2882.png", "formula": "\\begin{align*} | B _ r | \\| S _ { B _ r } f \\| _ { L ^ 2 _ { a v g } } ^ 4 & = | B _ r | ^ { - 1 } \\left ( \\int \\sum _ { \\tau } | f _ \\tau | ^ 2 W _ { B _ r } \\right ) ^ 2 \\\\ & \\lesssim \\sum _ { j = 1 } ^ { 1 0 0 0 } | B _ r | ^ { - 1 } \\left ( \\int \\sum _ { \\tau } | f _ { j , \\tau } | ^ 2 W _ { B _ r } \\right ) ^ 2 \\\\ & = \\sum _ { j = 1 } ^ { 1 0 0 0 } | B _ r | ^ { - 1 } \\left ( \\int | f _ { j } | ^ 2 W _ { B _ r } \\right ) ^ 2 . \\end{align*}"}
+{"id": "5792.png", "formula": "\\begin{align*} P _ i = \\sum _ { g \\in \\mathcal { A } _ 7 } \\chi _ { R _ i } ( g ) \\phi _ { W } ( g ) , \\end{align*}"}
+{"id": "8245.png", "formula": "\\begin{align*} \\delta : = \\begin{cases} 1 , & \\textrm { i f } \\ ; \\ ; \\bar { s } + s \\alpha \\leq \\frac { N } { 2 } + 1 - \\alpha , \\\\ \\frac { N / 2 - \\bar { s } - \\alpha + 1 } { s \\alpha } , & \\textrm { i f } \\ ; \\ ; \\bar { s } + s \\alpha > \\frac { N } { 2 } + 1 - \\alpha . \\end{cases} \\end{align*}"}
+{"id": "2759.png", "formula": "\\begin{align*} \\alpha b ( \\Gamma ) = 1 { \\rm a n d } \\alpha a ( \\Gamma ) = r . \\end{align*}"}
+{"id": "1992.png", "formula": "\\begin{align*} y _ u = \\begin{cases} 1 , & d ( u ) = 1 , \\\\ 0 , & . \\end{cases} \\end{align*}"}
+{"id": "1989.png", "formula": "\\begin{align*} \\lambda _ 1 ( G ) = \\frac { \\omega - 1 + \\sqrt { - 3 \\omega ^ 2 + ( 4 n - 2 ) \\omega + 1 } } { 2 } . \\end{align*}"}
+{"id": "703.png", "formula": "\\begin{align*} { \\bar F _ i } = - { { w } _ i } { \\bar F } , \\end{align*}"}
+{"id": "8166.png", "formula": "\\begin{align*} X : = ( t B ^ * B - A ^ * A ) _ + \\ne 0 . \\end{align*}"}
+{"id": "1908.png", "formula": "\\begin{align*} u = P _ { k _ 0 } + \\mathcal O ( t ^ \\sigma ) , \\end{align*}"}
+{"id": "4662.png", "formula": "\\begin{align*} \\deg ( F _ { { \\bf v } _ k } ) = & \\sum _ { j = 1 } ^ { n _ q } { \\bf v } _ { k , j } \\deg ( \\mathfrak { F } _ j ) = \\sum _ { j = 1 } ^ { n _ q } { \\bf v } _ { k , j } \\deg \\left ( \\phi ^ { j - 1 } \\left ( f _ 1 f _ 2 ^ 2 \\cdots f _ { \\ell - 1 } ^ { \\ell - 1 } \\right ) \\right ) \\\\ \\equiv & \\sum _ { j = 1 } ^ { n _ q } q ^ { k + 1 - j } \\sum _ { h = 1 } ^ { \\ell - 1 } h \\deg ( \\phi ^ { j - 1 } ( f _ h ) ) \\equiv \\sum _ { h = 1 } ^ { \\ell - 1 } h \\deg ( f _ h ) \\sum _ { j = 1 } ^ { n _ q } q ^ { k + 1 - j } \\bmod { \\ell } . \\end{align*}"}
+{"id": "2874.png", "formula": "\\begin{align*} \\int | f | ^ 4 \\lesssim \\sum _ { \\substack { R ^ { \\frac { 1 } { 2 } } \\le W \\le R \\\\ W \\ , } } \\sum _ { \\ell ( \\tau ) = \\frac { W } { R } } \\sum _ { U \\| U _ { \\tau , R } } | U | ^ { - 1 } \\Big ( \\int \\sum _ { \\theta \\subset \\tau } | f _ \\theta | ^ 2 W _ U \\Big ) ^ 2 . \\end{align*}"}
+{"id": "3310.png", "formula": "\\begin{align*} A = [ K _ 2 , L _ 1 \\otimes 1 ] _ q , \\end{align*}"}
+{"id": "3948.png", "formula": "\\begin{align*} \\| u _ { \\Omega _ n } - u _ { \\Omega _ n } \\circ p _ n \\| _ { L ^ 2 ( U _ h ( \\Gamma _ n ) ) } ^ 2 + \\| \\nabla u _ { \\Omega _ n } - ( \\nabla u _ { \\Omega _ n } ) \\circ p _ n \\| _ { L ^ 2 ( U _ h ( \\Gamma _ n ) ) } ^ 2 = { \\operatorname { o } } _ { h \\to 0 } ( h ) . \\end{align*}"}
+{"id": "1425.png", "formula": "\\begin{align*} d ( \\mathcal { U } , \\mathcal { W } ) = \\sup _ { t \\in [ 0 , T ] } \\| ( u ( t ) - w ( t ) , v ( t ) - z ( t ) ) \\| _ { \\mathcal { H } } \\end{align*}"}
+{"id": "1408.png", "formula": "\\begin{align*} h ( s ) = \\begin{dcases} 1 & ( p < p _ { s u b c } ( n , \\alpha ) ) , \\\\ \\log ( t _ 0 + s ) & ( p = p _ { s u b c } ( n , \\alpha ) ) , \\\\ ( t _ 0 + s ) ^ { \\frac { 1 } { 2 - \\alpha } \\left ( n - \\alpha \\frac { p + 1 } { p - 1 } \\right ) } & ( p > p _ { s u b c } ( n , \\alpha ) ) . \\end{dcases} \\end{align*}"}
+{"id": "4994.png", "formula": "\\begin{align*} h ( t ) = \\sum _ { k \\in J } f ( k ) t ^ k . \\end{align*}"}
+{"id": "5743.png", "formula": "\\begin{align*} P ^ M = \\begin{pmatrix} 0 & \\\\ & { \\rm I d } _ q \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "742.png", "formula": "\\begin{align*} \\centering \\mathbf { M } = \\left [ \\begin{array} { c c c c c c c } 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 0 & 1 & - 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & - 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & - 1 \\\\ 0 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 0 & 1 & 1 & - 1 & - 1 & 0 & 0 \\\\ 0 & 1 & 1 & 0 & 0 & - 1 & - 1 \\end{array} \\right ] . \\end{align*}"}
+{"id": "1743.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\tau ^ 2 } \\sum _ { i , j = 1 } ^ n w ( x _ i - x _ j ) \\ , . \\end{align*}"}
+{"id": "4433.png", "formula": "\\begin{align*} \\int _ { E } \\vert W u \\vert ^ p \\dd x \\leq \\int _ { E } \\vert W ( \\tau _ { a ( \\varepsilon ) } \\circ u ) \\vert ^ p \\dd x + \\int _ { E \\cap \\{ \\vert u \\vert \\geq a \\} } \\vert W ( u - \\tau _ { a ( \\varepsilon ) } \\circ u ) \\vert ^ p \\dd x < \\tfrac { \\varepsilon } { 2 } + \\tfrac { \\varepsilon } { 2 } = \\varepsilon . \\end{align*}"}
+{"id": "4929.png", "formula": "\\begin{align*} \\binom { n } { k - 1 } > \\sum _ { i = 2 } ^ { k - 1 } \\frac { j } { 2 } \\binom { n } { k - i } + \\sum _ { i = k - j - 1 } ^ { k - 1 } \\frac { j + 2 } { 2 } \\binom { n } { k - i } . \\end{align*}"}
+{"id": "6177.png", "formula": "\\begin{align*} \\sum _ { \\substack { d \\ , \\le \\ , y \\\\ ( d , \\ , a _ { 2 } ) \\ , = \\ , 1 } } \\rho _ { \\lambda } ( d ) \\Delta ( x ; \\l ( d ) , 1 ) & \\le ( \\log x ) ^ { \\lambda } \\sum _ { \\substack { d \\ , \\le \\ , y \\\\ ( d , \\ , a _ { 2 } ) \\ , = \\ , 1 } } | \\Delta ( x ; \\l ( d ) , 1 ) | \\\\ & \\le ( \\log x ) ^ { \\lambda } \\sum _ { d \\ , \\leq \\ , \\sqrt { x } / ( \\log x ) ^ { B } } \\gamma ( d ) \\ , | \\Delta ( x ; d , 1 ) | \\\\ & \\ll \\frac { x } { ( \\log x ) ^ { A - \\lambda } } . \\end{align*}"}
+{"id": "1327.png", "formula": "\\begin{align*} u _ s ( x ) - v ( x ) = s ( u ( x ) - v ( x ) ) , \\end{align*}"}
+{"id": "3609.png", "formula": "\\begin{align*} \\tau _ { \\Lambda ( \\omega ) + k } = \\infty \\quad \\hbox { a n d } \\xi _ { \\Lambda ( \\omega ) + k } = 0 \\end{align*}"}
+{"id": "3509.png", "formula": "\\begin{align*} \\int _ \\Omega \\mathrm { e } ^ { ( 2 ) } _ { \\mathrm { n } } ( \\mathrm { x ) } d \\mathrm { x } = 0 . \\end{align*}"}
+{"id": "2001.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\Psi ( t ) \\ , e ^ { i z t } \\ , d t = - \\frac { 1 } { z ^ 2 } \\frac { \\xi ' } { \\xi } \\left ( \\frac { 1 } { 2 } - i z \\right ) \\end{align*}"}
+{"id": "6197.png", "formula": "\\begin{align*} \\sum ^ k _ { i = 1 } \\sum ^ i _ { l = 1 } | \\mathcal { B } _ { i , i - l } | + \\sum ^ k _ { i = 1 } \\sum ^ i _ { l = 1 } | \\mathcal { B } _ { i - l , i } | = | \\{ ( i , i , t , p ) : ( i , i , t , p ) \\in \\mathcal { I } ' _ { k - 1 } \\} | + | \\mathcal { I } ' _ { k - 1 } | . \\end{align*}"}
+{"id": "1004.png", "formula": "\\begin{align*} \\widetilde { \\sigma } _ { j i } & = ( 2 \\ , \\mu ^ * + \\varkappa ) ( u _ { i , j } + u _ { j , i } ) + \\varkappa \\ , \\epsilon _ { j i m } \\Big ( \\underbrace { \\frac { 1 } { 2 } \\epsilon _ { m k s } u _ { s , k } } _ { r _ m } - \\vartheta _ m \\Big ) + \\lambda \\ , u _ { k , k } \\delta _ { j i } , \\\\ * m _ { j i } & = \\beta \\ , \\vartheta _ { j , i } + \\gamma \\ , \\vartheta _ { i , j } + \\alpha \\ , \\vartheta _ { k , k } \\delta _ { j i } \\ , . \\end{align*}"}
+{"id": "5834.png", "formula": "\\begin{align*} b e r ( \\sqrt { A ^ * X A } ) ) & = \\sup _ { \\lambda \\in \\Omega } | \\langle \\sqrt { A ^ * X A } \\hat { k } _ { \\lambda } , \\hat { k } _ { \\lambda } \\rangle | \\\\ & < \\sup _ { \\lambda \\in \\Omega } | \\langle { A ^ * X A } \\hat { k } _ { \\lambda } , \\hat { k } _ { \\lambda } \\rangle | ^ { 1 \\over 2 } \\\\ & = \\sqrt { b e r ( A ^ * X A } ) . \\end{align*}"}
+{"id": "7850.png", "formula": "\\begin{align*} \\rho _ \\mu ^ * \\circ \\varphi _ \\mu ^ * \\ , \\Omega _ { \\mbox { \\tiny { $ \\overline { Q } $ } } } = \\iota _ \\mu ^ * ( \\Omega _ { \\mbox { \\tiny { $ \\widetilde { Q } $ } } } - B _ { \\mbox { \\tiny { $ \\ ! \\langle \\ ! J \\mathcal { K } \\ ! \\rangle $ } } } + \\widetilde { \\bf B } ) - \\iota _ \\mu ^ * \\widetilde { \\mathcal { B } } . \\end{align*}"}
+{"id": "5546.png", "formula": "\\begin{align*} T _ 1 ( \\beta ( t ) , t ) = T _ 2 ( \\beta ( t ) , t ) = 0 \\end{align*}"}
+{"id": "4189.png", "formula": "\\begin{align*} l ( w ^ i ) = \\frac { 1 } { \\mathrm { s i n } ( \\theta ) } \\underset { l } \\sum d _ l i ( w ^ i , w _ 1 ) \\mathrm { h e i g h t } ( C _ 1 ^ l ) \\end{align*}"}
+{"id": "4728.png", "formula": "\\begin{align*} \\mathfrak { B } _ d ( a + a ^ * , b + b ^ * ) : = \\langle a , b ^ * \\rangle + \\langle a ^ * , b \\rangle , \\forall a , b \\in A , \\ a ^ * , b ^ * \\in A ^ * . \\end{align*}"}
+{"id": "8232.png", "formula": "\\begin{align*} \\sum _ { j = k } ^ \\infty \\big | \\big \\langle \\Lambda ^ \\alpha ( \\varphi _ j u ^ n h ( \\sigma ^ n ) ) - \\Lambda ^ \\alpha ( \\varphi _ j u h ( \\sigma ) ) , \\phi \\big \\rangle \\big | \\leq \\frac { \\epsilon } { 2 } . \\end{align*}"}
+{"id": "1238.png", "formula": "\\begin{align*} & \\mathcal { P } ( m ) \\\\ & : = \\left \\{ t \\in [ T , 2 T ] \\mid | \\Re P _ m ( t ) | > 2 ^ { - m / 1 0 } , | \\Re P _ n ( t ) | \\le 2 ^ { - n / 1 0 } m + 1 \\le n \\le \\frac { \\log \\log T } { \\log 2 } \\right \\} . \\end{align*}"}
+{"id": "1195.png", "formula": "\\begin{align*} d ^ { o } _ { j i k } = \\begin{cases} e ^ { ( n ) } _ { 1 , i i k } + \\nu ^ { o } _ { i k } e ^ { * } _ { 3 , i k } & \\ , \\ , \\ , j = i , \\\\ \\frac { e ^ { ( n ) } _ { 1 , j i k } } { 1 + \\nu ^ { o } _ { i k } \\gamma _ { i k } } & \\end{cases} \\end{align*}"}
+{"id": "3854.png", "formula": "\\begin{align*} H ^ { z } : = \\begin{pmatrix} 0 & X - z \\\\ X ^ * - \\overline { z } & 0 \\end{pmatrix} , G ^ { z } ( w ) : = ( H ^ { z } - w ) ^ { - 1 } , w \\in \\C \\setminus \\R , ~ z \\in \\C . \\end{align*}"}
+{"id": "5821.png", "formula": "\\begin{align*} \\omega ( M _ z \\oplus U ) = \\omega ( \\Pi V \\Pi ^ * ) = \\omega ( V ) \\leq 1 . \\end{align*}"}
+{"id": "3617.png", "formula": "\\begin{align*} U _ t = \\int _ { ( 0 , t ] \\times \\Re } e ^ { - \\lambda s } \\left \\{ f ( X _ { s - } + y ) - f ( X _ { s - } ) - f ' ( X _ { s - } ) y 1 _ { \\{ | y | < 1 \\} } \\right \\} \\left ( J ( d s , d y ) - d s \\nu ( d y ) \\right ) \\end{align*}"}
+{"id": "557.png", "formula": "\\begin{align*} H \\big ( \\mathrm { L a w } ( t _ { n + 1 } X _ { n + 1 } + ( 1 - t _ { n + 1 } ) \\widetilde { X } ) \\big ) & = H \\big ( \\widetilde { Q } \\circ ( t _ { n + 1 } T + ( 1 - t _ { n + 1 } ) \\mathrm { I d } ) ^ { - 1 } \\big ) \\\\ & \\le t _ { n + 1 } H ( Q _ { n + 1 } ) + ( 1 - t _ { n + 1 } ) H ( \\widetilde { Q } ) . \\end{align*}"}
+{"id": "2746.png", "formula": "\\begin{align*} d ^ * = ( b _ 3 ( Z ) - b _ 1 ( Z ) ) + 2 ( b _ 2 ( S ) - b _ 2 ( Z ) ) + 2 ( g ( S ) - q ( S ) ) . \\end{align*}"}
+{"id": "5314.png", "formula": "\\begin{align*} U ( \\rho _ 0 ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma - \\beta _ 0 } H ( s + \\rho _ 0 ) e ^ { L s ^ 2 + M s } d s + \\sum ^ { \\star } _ { \\rho } \\exp \\left ( L ( \\rho - \\rho _ 0 ) ^ 2 + M ( \\rho - \\rho _ 0 ) \\right ) , \\end{align*}"}
+{"id": "905.png", "formula": "\\begin{align*} \\nabla _ i u : = \\nabla _ { e _ i } u = e _ i ^ j \\partial _ j u = e _ i ^ j u _ j \\end{align*}"}
+{"id": "3286.png", "formula": "\\begin{align*} a _ s = A _ 0 , b _ t = A _ 1 , \\theta = A _ 2 , \\Omega = A _ 3 . \\end{align*}"}
+{"id": "850.png", "formula": "\\begin{align*} \\begin{aligned} M _ { \\tau } '' ( t ) & = M _ { \\tau } ( t ) \\left [ \\ln \\left ( \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) \\cdot \\frac { 1 } { k _ 2 \\sigma } \\cdot \\frac { 1 } { \\sqrt { ( a ' ) ^ { 2 } - 2 t } } \\right ] ^ { 2 } \\\\ & + M _ { \\tau } ( t ) \\left [ \\ln \\left ( \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) \\cdot \\frac { 1 } { k _ 2 \\sigma } \\cdot \\frac { 1 } { ( \\sqrt { ( a ' ) ^ { 2 } - 2 t } ) ^ { 3 } } \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "1751.png", "formula": "\\begin{align*} w _ { n , k , m } \\left ( C \\mathcal { Z } , \\mathcal { Z } \\right ) = w _ { n + 1 , k , m } \\left ( C ^ { ' } \\mathcal { Z } ^ { ' } , \\mathcal { Z } ^ { ' } \\right ) . \\end{align*}"}
+{"id": "7470.png", "formula": "\\begin{align*} S _ 3 ^ { ( 1 ) } ( n ) = \\sum _ { \\nu = 1 } ^ n \\nu ^ 3 = \\left [ \\frac { n ( n + 1 ) } { 2 } \\right ] ^ 2 \\ , . \\end{align*}"}
+{"id": "5640.png", "formula": "\\begin{align*} \\mathrm { d i a m } ( M ) \\leq d = \\frac { \\pi } { \\sqrt { \\lambda } } . \\end{align*}"}
+{"id": "3437.png", "formula": "\\begin{align*} E _ v ( a f ( b ) ) = \\sum _ { i \\geq 0 } E ^ { ( i ) } _ v ( b ) ( { - D _ z } ) ^ i ( a f ' ( b ) ) + E ^ { ( i ) } _ v ( a ) ( { - D _ z } ) ^ i f ( b ) \\end{align*}"}
+{"id": "2082.png", "formula": "\\begin{align*} U ' \\cap C _ 0 = \\bigcup _ { k \\geq 0 } U ' \\cap ( C _ k \\setminus C _ { k + 1 } ) . \\end{align*}"}
+{"id": "5987.png", "formula": "\\begin{align*} \\bar { \\pi } ^ { 0 , b } : = \\{ ( L _ r ^ { 0 , b } ( t ) , R _ r ^ { 0 , b } ( t ) ) ; t \\geq 0 \\} . \\end{align*}"}
+{"id": "7107.png", "formula": "\\begin{align*} 0 = R _ 0 \\subset R _ 1 \\subset \\cdots \\subset R _ k \\end{align*}"}
+{"id": "2107.png", "formula": "\\begin{align*} | \\Omega | = | G ( a , F ) | \\cdot \\int _ { \\Psi ^ { - 1 } ( L ) } J ( \\Psi ^ { - 1 } ) \\ , d \\Psi ^ { - 1 } ( L ) . \\end{align*}"}
+{"id": "1124.png", "formula": "\\begin{align*} H _ { j , n } = E X ^ { j } h _ { n } ( X ) = \\frac { 1 } { \\sqrt { n ! } } E X ^ { j } H _ { n } ( X ) = \\left \\{ \\begin{array} { c c c } 0 & & n > j j - n \\\\ \\frac { j ! } { 2 ^ { ( j - n ) / 2 } ( ( j - n ) / 2 ) ! \\sqrt { n ! } } & & j - n \\end{array} \\right . \\end{align*}"}
+{"id": "4139.png", "formula": "\\begin{align*} x ^ 2 ( 1 - x ^ 2 ) \\left ( \\frac { 1 } { x } \\frac { d } { d x } \\right ) ^ 2 ( U ) ( x ) = f _ 1 ( x ) \\left ( \\frac { 1 } { x } \\frac { d } { d x } \\right ) ( U ) ( x ) + f _ 2 ( x ) U ( x ) ~ ~ ~ ~ ~ x \\in ( 0 , 1 ) \\end{align*}"}
+{"id": "6057.png", "formula": "\\begin{align*} \\psi ( \\chi _ \\epsilon ( x _ 0 ) ) = \\psi ( \\chi ) + f ( \\epsilon ) D _ T J ( x _ 0 ) + \\mathcal R ( f ( \\epsilon ) ) . \\end{align*}"}
+{"id": "3562.png", "formula": "\\begin{align*} e _ 1 ( I ) \\le \\binom { e _ 0 ( I ) } { 2 } - \\binom { \\mu ( I ) - d } { 2 } - \\ell ( A / I ) + 1 , \\end{align*}"}
+{"id": "3410.png", "formula": "\\begin{align*} H = \\int _ 0 ^ \\infty \\rho ( \\tfrac { 1 } { 2 } U ^ 2 + e ) r ^ { n - 1 } \\ , d r \\end{align*}"}
+{"id": "7169.png", "formula": "\\begin{align*} \\mathcal { Z } = \\mathcal { Z } _ { d - 1 } \\hookrightarrow \\mathcal { Z } _ { d - 2 } \\hookrightarrow \\cdots \\hookrightarrow \\mathcal { Z } _ 1 \\hookrightarrow \\mathcal { Y } ( d ) ^ { \\lambda \\geq 0 } . \\end{align*}"}
+{"id": "281.png", "formula": "\\begin{align*} \\Omega ^ { 1 } _ { A } = \\Omega ^ { 1 } _ { A / A ^ { p } } = \\Omega ^ { 1 } _ { A / k } . \\end{align*}"}
+{"id": "7466.png", "formula": "\\begin{align*} \\begin{pmatrix} V _ 0 & V _ 1 \\\\ - \\gamma W _ 0 & - \\gamma W _ 1 \\end{pmatrix} \\cdot \\begin{pmatrix} \\alpha \\\\ \\beta \\end{pmatrix} + \\begin{pmatrix} V _ 1 & V _ 0 \\\\ \\gamma W _ 1 & \\gamma W _ 0 \\end{pmatrix} \\cdot \\begin{pmatrix} E _ { \\gamma } & 0 \\\\ 0 & E _ { \\gamma } \\end{pmatrix} \\cdot \\begin{pmatrix} \\alpha \\\\ \\beta \\end{pmatrix} = z . \\end{align*}"}
+{"id": "4350.png", "formula": "\\begin{align*} \\int _ \\Omega \\Phi ( f ( t ) ) \\ , \\mathrm { d } x - \\int _ \\Omega \\Phi ( f ( t _ 0 ) ) \\ , \\mathrm { d } x = \\int _ { t _ 0 } ^ t \\Big \\langle \\partial _ t f ( \\tau ) , \\Phi ' ( f ( \\tau ) ) \\Big \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } \\mathrm { d } \\tau \\ , . \\end{align*}"}
+{"id": "6343.png", "formula": "\\begin{align*} E _ { \\Lambda , ( F , G ) } f : = \\Theta _ { G } ^ * D _ { P } \\Theta _ { F } f = \\sum _ { i \\in \\Lambda } q _ i \\langle f , f _ i \\rangle g _ i , \\end{align*}"}
+{"id": "4186.png", "formula": "\\begin{align*} i ( w ^ i , ( w _ 1 ) _ l ) & = \\frac { l ( C _ 1 ^ l \\cap w ^ i ) \\mathrm { s i n } ( \\theta ) } { \\mathrm { h e i g h t } ( C _ 1 ^ l ) } \\\\ i ( w ^ j , ( w _ 1 ) _ l ) & = \\frac { l ( C _ 1 ^ l \\cap w ^ i ) \\mathrm { s i n } ( \\theta _ j ' ) } { \\mathrm { h e i g h t } ( C _ 1 ^ l ) } . \\end{align*}"}
+{"id": "6342.png", "formula": "\\begin{align*} q _ i = \\displaystyle { { \\frac { \\sum _ { j = 1 } ^ N p _ j } { \\sum _ { j = 1 } ^ N p _ j - p _ i } \\cdot \\frac { N - 1 } { n } } } ; \\ ; \\ ; \\ ; f o r \\ ; \\ ; i = 1 , 2 , . . . , N . \\ ; \\ ; \\ ; \\ ; \\end{align*}"}
+{"id": "223.png", "formula": "\\begin{align*} \\partial _ t V ^ { \\sigma } ( m _ t ) = - \\int _ { \\mathbb { R } ^ d } \\left | \\frac { \\delta F } { \\delta m } ( m _ t , x ) + \\frac { \\sigma ^ 2 } { 2 } \\log \\left ( \\frac { m _ t ( x ) } { \\pi ( x ) } \\right ) - \\frac { \\sigma ^ 2 } { 2 } \\operatorname { K L } ( m _ t | \\pi ) \\right | ^ 2 m _ t ( x ) d x \\ , . \\end{align*}"}
+{"id": "7746.png", "formula": "\\begin{align*} \\mathcal { A } ( t _ 0 , \\cdot , \\cdot ) = \\int _ { \\mathbb { S } ^ { n - 1 } \\times \\mathbb { S } ^ 1 ( \\lambda ) } e ^ { n b _ { \\gamma _ + } ( t _ 0 , w , \\theta ) } d \\Theta _ { n - 1 } d \\Theta _ \\lambda . \\end{align*}"}
+{"id": "4302.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } \\frac { q ^ { j ^ 2 } } { ( q ) _ { j } ^ { 2 } } \\sum _ { n = 1 } ^ { j } \\frac { q ^ n } { ( 1 - q ^ { n + 1 } ) ( 1 - q ^ n ) } = \\frac { q ^ 2 } { ( 1 - q ) ^ 2 ( q ) _ { \\infty } } . \\end{align*}"}
+{"id": "6301.png", "formula": "\\begin{align*} | ( ( 1 - u ) \\gamma _ i + u \\gamma ' _ i ) ( t ) - \\gamma _ i ( t ) | = u | \\gamma ' _ i ( t ) - \\gamma _ i ( t ) | < u \\epsilon \\leq \\epsilon . \\end{align*}"}
+{"id": "3559.png", "formula": "\\begin{align*} e _ 1 ( I , M ) \\leq \\binom { e _ 0 ( I , M ) - b + 1 } { 2 } + b - \\ell ( M / I M ) . \\end{align*}"}
+{"id": "2354.png", "formula": "\\begin{align*} u _ { \\sigma } = & x _ { a _ { \\sigma ( 3 ) } } x _ { b _ { 3 } } \\cdots x _ { a _ { \\sigma ( n ) } } x _ { b _ { n } } x _ { a _ { \\sigma ( n + 1 ) } } . \\end{align*}"}
+{"id": "3602.png", "formula": "\\begin{align*} J & = \\left ( \\cos ( a x ) \\frac { \\sin ( b x ) } { b } \\right ) _ 0 ^ 1 + a \\int _ 0 ^ 1 \\sin ( a x ) \\frac { \\sin ( b x ) } { b } d x \\\\ & = \\frac { 1 } { b } \\cos ( a ) \\sin ( b ) + \\frac { a } { b } \\left [ \\left ( \\sin ( a x ) \\frac { \\cos ( b x ) } { b } \\right ) _ 1 ^ 0 + \\frac { a } { b } \\int _ 0 ^ 1 \\cos ( a x ) \\cos ( b x ) d x \\right ] \\\\ & = \\frac { 1 } { b } \\cos ( a ) \\sin ( b ) - \\frac { a } { b ^ 2 } \\sin ( a ) \\cos ( b ) + \\left ( \\frac { a } { b } \\right ) ^ 2 J \\end{align*}"}
+{"id": "2389.png", "formula": "\\begin{align*} c _ { k } & \\coloneqq \\sum _ { j = 1 } ^ { k } ( b _ { j } - 1 ) , \\\\ d _ { k } & \\coloneqq \\sum _ { j = 1 } ^ { k } b _ { j } = c _ { k } + k , \\end{align*}"}
+{"id": "2561.png", "formula": "\\begin{align*} \\widetilde { \\nu } _ t ^ { * , \\xi } = \\eta _ t \\nu _ t ^ { * , \\xi } , \\widetilde { \\varphi } _ t ^ { * , \\xi } = \\eta ^ { - 1 } _ t \\varphi _ t ^ { * , \\xi } , \\quad \\widetilde { \\Lambda } _ t ^ { 0 , * , \\xi } = \\eta ^ { - 1 } _ t \\Lambda _ t ^ { 0 , * , \\xi } , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "3121.png", "formula": "\\begin{align*} F _ X ( \\sigma ^ k ) = F _ { Y _ 1 } ( \\tau _ 1 ^ k ) + F _ { Y _ 2 } ( \\tau _ 2 ^ k ) \\end{align*}"}
+{"id": "1437.png", "formula": "\\begin{align*} \\lim _ { t \\to T _ { \\max } - 0 } \\| ( u ( t ) , \\partial _ t u ( t ) ) \\| _ { \\mathcal { H } } = \\infty . \\end{align*}"}
+{"id": "4449.png", "formula": "\\begin{align*} u = g \\Gamma _ D g \\in W ^ { 1 - 1 / p } _ p ( \\Gamma _ D ; \\R ^ d ) . \\end{align*}"}
+{"id": "1271.png", "formula": "\\begin{align*} B _ i = \\sigma ( C _ i ) \\perp F \\end{align*}"}
+{"id": "6029.png", "formula": "\\begin{align*} \\ell : X _ 1 = 0 , X _ 4 = \\dots = X _ { 2 n + 1 } = 0 , \\end{align*}"}
+{"id": "3621.png", "formula": "\\begin{align*} \\lim _ { q \\to 0 } P _ x [ X ( e _ q ) \\in \\cal { A } ] = \\pi ( { \\cal A } ) \\end{align*}"}
+{"id": "3352.png", "formula": "\\begin{align*} b _ 1 \\otimes b _ 2 \\otimes \\dots \\otimes b _ s \\mbox { , w h e r e e a c h } b _ i = \\prod _ { k = 1 } ^ s ( \\prod _ { j \\in I _ { i , k } } a _ i ^ j ) , \\end{align*}"}
+{"id": "7853.png", "formula": "\\begin{align*} \\eta _ i = \\tilde f _ i ^ 1 { \\bf e } _ 1 + \\tilde f _ i ^ 2 { \\bf e } _ 2 , \\end{align*}"}
+{"id": "109.png", "formula": "\\begin{align*} \\alpha _ n ( 0 ) = h \\psi _ 0 ( h n ) + ( - 1 ) ^ n h \\phi _ 0 ( h n ) , \\end{align*}"}
+{"id": "7973.png", "formula": "\\begin{align*} \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle = \\frac { k ( \\xi ) } { k ^ 2 ( \\xi ) + \\alpha ^ 2 ( \\xi ) } \\quad \\mbox { a n d } \\left \\langle \\eta , \\xi ^ H \\right \\rangle = \\frac { \\alpha ( \\xi ) } { k ^ 2 ( \\xi ) + \\alpha ^ 2 ( \\xi ) } \\end{align*}"}
+{"id": "6408.png", "formula": "\\begin{align*} \\int _ { a } ^ { \\theta } g ( x ) \\nabla x = \\mathcal { G } ( x ) - \\mathcal { G } ( a ) . \\end{align*}"}
+{"id": "6289.png", "formula": "\\begin{align*} \\langle x , y \\rangle = \\sum _ { i = 0 } ^ { \\ell - 1 } y _ i \\langle x , \\frac { q ^ i } { q ^ \\ell - 1 } \\rangle \\end{align*}"}
+{"id": "1595.png", "formula": "\\begin{align*} V ^ { \\prime } = T _ { s _ { i } } ^ { - 1 } \\left ( 0 \\right ) . \\end{align*}"}
+{"id": "6358.png", "formula": "\\begin{align*} & P _ { 0 } ( \\lambda ) = A _ { m } , \\ , \\ , \\ , P _ { k + 1 } ( \\lambda ) = \\lambda P _ { k } ( \\lambda ) + A _ { m - k - 1 } , \\mbox { f o r } 0 \\leq k \\leq m - 1 , \\ , \\ , \\ , P _ { m } ( \\lambda ) = P ( \\lambda ) . \\end{align*}"}
+{"id": "1725.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to \\infty } a ^ \\xi _ { \\tau , m } = a ^ \\xi _ m \\end{align*}"}
+{"id": "3544.png", "formula": "\\begin{align*} \\Phi ^ { \\textbf { e } } \\Big ( \\xi , \\mathrm { t } ; \\mathrm { z } , \\tau - \\dfrac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 \\mathrm { m } ^ 2 } \\Big ) - \\Phi ^ { \\textbf { e } } \\Big ( \\xi , \\mathrm { t } ; \\mathrm { z } , \\tau \\Big ) = \\displaystyle \\int _ { \\tau } ^ { \\tau - \\frac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 \\mathrm { m } ^ 2 } } \\partial _ { s } \\Phi ^ { \\textbf { e } } \\Big ( \\xi , \\mathrm { t } ; \\mathrm { z } , \\mathrm { s } \\Big ) d \\mathrm { s } . \\end{align*}"}
+{"id": "10.png", "formula": "\\begin{align*} p _ { T _ 1 , T _ 2 } ( Z ) : = \\sum _ { \\substack { T _ 1 ' \\sim T _ 1 \\\\ T _ 2 ' \\sim T _ 2 } } \\sum _ { c , c ' \\in C _ { t } } ( c c ' ) \\prod _ { y \\in Y ( \\lambda ) } z _ { c T _ 1 ' ( y ) , c ' T _ 2 ' ( y ) } , \\end{align*}"}
+{"id": "3897.png", "formula": "\\begin{align*} \\mathbb { H } ^ { r } = \\begin{cases} H _ { 0 } ^ { r } ( \\Omega ) , & 0 \\leq r < 1 / 2 , \\\\ H _ { 0 0 } ^ { 1 / 2 } ( \\Omega ) \\varsubsetneqq H _ { 0 } ^ { 1 / 2 } ( \\Omega ) , & r = 1 / 2 , \\\\ H _ { 0 } ^ { r } ( \\Omega ) , & 1 / 2 < r \\leq 1 , \\\\ H _ { 0 } ^ { 1 } ( \\Omega ) \\cap H ^ { r } ( \\Omega ) , & 1 < r \\leq 2 , \\end{cases} \\end{align*}"}
+{"id": "4328.png", "formula": "\\begin{align*} \\sum _ { k > m } P _ { i , k } = \\left \\{ \\begin{array} { l l } 1 & m < i - 1 , \\\\ 1 - p , & i - 1 \\leq m \\leq i , \\\\ 0 , & m > i . \\end{array} \\right . \\end{align*}"}
+{"id": "7004.png", "formula": "\\begin{align*} I = \\int - \\Delta ( e ^ v ) e ^ { ( b - 1 ) v } \\Delta v d m = - \\int ( \\Delta v ) | \\nabla v | ^ 2 e ^ { b v } d m - \\int ( \\Delta v ) ^ 2 e ^ { b v } d m . \\end{align*}"}
+{"id": "3145.png", "formula": "\\begin{align*} C _ { s , t } ( x ) = \\max \\{ ( 1 - x ^ { 1 / t } ) ^ s + s \\ , x ^ { 1 / t } \\ , ( 1 - x ^ { 1 / t } ) ^ { s - 1 } , \\ , ( 1 - z ) ^ s \\} , \\end{align*}"}
+{"id": "558.png", "formula": "\\begin{align*} M _ n ( Q ) & : = \\int _ { \\R ^ n } f \\ , d Q - H ( Q ) \\\\ & = \\sum _ { i = 1 } ^ n \\int _ { \\R } V ( x ) \\ , Q _ i ( d x ) + \\frac 1 2 \\sum _ { i , j = 1 } ^ n J _ { i j } \\int _ { \\R } \\int _ { \\R } K ( x - y ) \\ , Q _ i ( d x ) Q _ j ( d y ) - \\sum _ { i = 1 } ^ n H ( Q _ i ) , \\end{align*}"}
+{"id": "7342.png", "formula": "\\begin{align*} & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ X ^ { ( i , j , t , p ) } _ { ( m + 1 , m ) } = \\{ ( y , z ) \\in X _ { ( m + 1 , m ) } \\mid \\varrho ( y , z ) = ( i , j , t , p ) \\} ; \\end{align*}"}
+{"id": "3039.png", "formula": "\\begin{align*} \\gamma ( x _ i ) = x _ { i + 1 } i \\in \\{ 1 , \\ldots , \\ell - 1 \\} , \\gamma ( x _ { \\ell } ) = x _ 1 , \\end{align*}"}
+{"id": "2803.png", "formula": "\\begin{align*} A g _ i & = c _ i u _ i ^ A , \\\\ B g _ i & = s _ i u _ i ^ B , \\\\ s _ i A ^ T u _ i ^ A & = c _ i B ^ T u _ i ^ B . \\end{align*}"}
+{"id": "6435.png", "formula": "\\begin{align*} p _ n ^ { ( \\alpha , \\beta ) } ( x , y ) = y ^ n \\mathcal { P } _ n ^ { ( \\alpha , \\beta ) } ( x / y ) , \\ , \\ , p _ n ^ { ( \\alpha , \\beta ) } ( x , 1 ) = \\mathcal { P } _ n ^ { ( \\alpha , \\beta ) } ( x ) , \\ , \\ , p _ n ^ { ( \\alpha , \\beta ) } ( 0 , y ) = y ^ n . \\end{align*}"}
+{"id": "2092.png", "formula": "\\begin{align*} f _ j = \\frac { h _ j ( x ) } { g _ j ( x ) } \\in \\mathcal { O } _ { U ' , p } , \\end{align*}"}
+{"id": "3093.png", "formula": "\\begin{align*} F _ { \\N } ( \\sigma _ i ^ k ) = \\begin{cases} | \\N | & \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "2168.png", "formula": "\\begin{align*} & R _ { i j k l } = \\alpha _ { i k } \\alpha _ { j l } - \\alpha _ { i l } \\alpha _ { j k } , \\\\ & S _ { i j k l m } = \\beta _ { i k m } \\alpha _ { j l } + \\alpha _ { i k } \\beta _ { j l m } - \\beta _ { i l m } \\alpha _ { j k } - \\alpha _ { i l } \\beta _ { j k m } \\end{align*}"}
+{"id": "8259.png", "formula": "\\begin{align*} \\begin{aligned} I _ 1 + I _ 2 \\le C \\Vert u \\Vert _ { \\widetilde { L } ^ r _ t ( \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha + \\frac { \\alpha } { r } } _ { 2 , 1 } ) } \\leq C \\Vert u \\Vert _ { \\widetilde { L } ^ \\infty _ t ( \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } ) } + C \\Vert u \\Vert _ { \\widetilde { L } ^ 1 _ t ( \\dot { B } ^ { \\frac { N } { 2 } + 1 } _ { 2 , 1 } ) } . \\end{aligned} \\end{align*}"}
+{"id": "6367.png", "formula": "\\begin{align*} K _ H ^ * ( \\mathrm { I } _ { [ 0 , t ] } e _ i ) = K _ H ( t , \\cdot ) e _ i . \\end{align*}"}
+{"id": "8247.png", "formula": "\\begin{align*} \\begin{aligned} Z _ { s , \\bar { s } } ^ \\ell ( t ) \\le C \\Vert ( \\sigma ^ \\ell _ 0 , u ^ \\ell _ 0 ) \\Vert _ { \\dot { B } ^ { \\bar { s } } _ { 2 , 1 } } & + C \\int _ 0 ^ t \\tau ^ s \\sum _ { j \\le j _ 0 } \\psi _ j ( t , \\tau ) 2 ^ { j ( \\bar { s } + s \\alpha ) } R _ j ( \\tau ) \\dd \\tau , \\end{aligned} \\end{align*}"}
+{"id": "6951.png", "formula": "\\begin{align*} \\mathsf F _ { \\chi } ( z ) = \\sum _ { d = 0 } ^ { \\infty } z ^ d \\binom { - \\chi + ( N + 1 ) d } { d } \\implies F _ \\chi ( z ) = \\sum _ { d = 0 } ^ { \\infty } z ^ d \\left ( \\left [ t ^ d \\right ] ( 1 + t ) ^ { - \\chi + ( N + 1 ) d } \\right ) . \\end{align*}"}
+{"id": "5950.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } \\tau _ { A _ i } - \\sum _ { i = 1 } ^ { k } \\tau _ { \\bar { K } _ i } + \\Box = \\sum _ { i = 1 } ^ { k } \\tau _ { B _ i } - \\sum _ { i = 1 } ^ { k } \\tau _ { \\bar { K } _ i } + \\Box \\end{align*}"}
+{"id": "920.png", "formula": "\\begin{align*} \\begin{aligned} x _ n - y _ n = \\ , & \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( y ' ) ( x _ \\beta - y _ \\beta ) + O ( | x ' - y ' | ^ 2 ) \\\\ \\geq \\ , & \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( y ' ) ( x _ \\beta - y _ \\beta ) - \\kappa | x ' - y ' | ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "4945.png", "formula": "\\begin{align*} a _ 1 & = A - \\sum _ { i = 2 } ^ { ( k - 5 ) / 2 - t } i a _ i = \\sum _ { i = 2 } ^ { ( k - 5 ) / 2 - t } \\left ( 2 t + i ( t + \\frac { k + 1 } { 2 } ) \\right ) \\frac { \\binom { n } { k - 2 - i } } { n } - \\sum _ { i = 2 } ^ { ( k - 5 ) / 2 - t } i a _ i \\\\ & \\ge \\sum _ { i = 2 } ^ { ( k - 5 ) / 2 - t } \\left [ \\left ( 2 t + i ( t + \\frac { k + 1 } { 2 } ) \\right ) \\frac { \\binom { n } { k - 2 - i } } { n } - i \\right ] \\ge \\sum _ { i = 2 } ^ { ( k - 5 ) / 2 - t } \\left ( \\frac { k + 1 } { 2 } - 1 \\right ) i \\ge 0 , \\end{align*}"}
+{"id": "235.png", "formula": "\\begin{align*} F ( m ' ) - F ( m ) = \\int _ { 0 } ^ { 1 } \\int _ { \\mathbb { R } ^ d } \\frac { \\delta F } { \\delta m } ( m + \\lambda ( m ' - m ) , a ) \\left ( m ' - m \\right ) ( d a ) \\ , d \\lambda . \\end{align*}"}
+{"id": "4437.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\sup _ { n \\in \\N } \\Vert \\tilde { v } _ n - \\bar { v } _ { n , m } \\Vert _ { L _ p } = 0 . \\end{align*}"}
+{"id": "7400.png", "formula": "\\begin{align*} f _ 1 & : = 4 x - z , \\\\ f _ 2 & : = 4 y - z , \\\\ f _ 3 & : = 4 x ^ 2 y - z ( y - t ) ^ 2 , \\\\ f _ 4 & : = 4 x y ^ 2 - z ( x - t ) ^ 2 , \\\\ f _ 5 & : = ( x - y ) ^ 2 - 2 ( x + y ) t + t ^ 2 , \\\\ f _ 6 & : = ( x - y ) ^ 2 - 2 ( x + y ) ( z + t ) + ( z + t ) ^ 2 . \\end{align*}"}
+{"id": "5807.png", "formula": "\\begin{align*} \\langle ( \\sum _ { i = 1 } ^ n A _ i ^ * X A _ i ) h , h \\rangle = \\sum _ { i = 1 } ^ n \\langle A _ i ^ * X A _ i h , h \\rangle = \\sum _ { i = 1 } ^ n \\langle X A _ i h , A _ i h \\rangle \\geq 0 . \\end{align*}"}
+{"id": "1935.png", "formula": "\\begin{align*} ( | ( - \\Delta _ x ) ^ { 1 / 6 } u _ 0 | ^ p ) ^ { 1 / p } _ { Q _ { \\nu r , R \\nu r } ( z _ 0 ) } & \\le N \\sum _ { k = 0 } ^ { \\infty } 2 ^ { - k } F _ k ( R ) , \\\\ ( | ( - \\Delta _ x ) ^ { 1 / 6 } u _ 0 | ^ p ) ^ { 1 / p } _ { Q _ { r , R r } ( z _ 0 ) } & \\le N \\nu ^ { ( 4 d + 2 ) / p } \\sum _ { k = 0 } ^ { \\infty } 2 ^ { - k } F _ k ( R ) , \\end{align*}"}
+{"id": "3413.png", "formula": "\\begin{align*} A ( r , U , p _ r / \\rho ) = \\int _ 0 ^ r \\frac { d y } { \\sqrt { U ^ 2 + \\tfrac { 2 } { n } ( 1 - ( y / r ) ^ n ) r p _ r / \\rho } } = \\int _ 0 ^ 1 \\frac { r \\ , d y } { \\sqrt { U ^ 2 + \\tfrac { 2 } { n } ( 1 - y ^ n ) r p _ r / \\rho } } . \\end{align*}"}
+{"id": "4502.png", "formula": "\\begin{align*} \\bigg | \\int _ { [ 0 , 1 ] } u ' h ' \\bigg | & = \\bigg | \\int _ { [ 0 , 1 ] } \\big ( u ' h ' + u h - u h \\big ) \\bigg | = \\big | \\langle u , h \\rangle _ { H ^ 1 } - \\langle u , h \\rangle _ { L ^ 2 } \\big | \\\\ & \\leq \\big | \\langle u , h \\rangle _ { H ^ 1 } \\big | + \\big | \\langle u , h \\rangle _ { L ^ 2 } \\big | \\leq \\| u \\| _ { H ^ 1 } \\| h \\| _ { H ^ 1 } + \\| u \\| _ { L ^ 2 } \\| h \\| _ { L ^ 2 } \\\\ & \\leq 2 \\| u \\| _ { H ^ 1 } \\| h \\| _ { H ^ 1 } \\ , , \\end{align*}"}
+{"id": "5453.png", "formula": "\\begin{align*} M = \\begin{pmatrix} J - I _ p & \\omega ^ 2 J & \\omega J \\\\ \\omega J & J - I _ q & \\omega ^ 2 J \\\\ \\omega ^ 2 J & \\omega J & J - I _ r \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "2801.png", "formula": "\\begin{align*} Q _ A V _ k Y _ k & = U _ { k + 1 } X _ { k + 1 } \\begin{bmatrix} C _ k \\\\ 0 \\end{bmatrix} , & Q _ A ^ T U _ { k + 1 } X _ { k + 1 } & = V _ k Y _ k \\begin{bmatrix} C _ k & 0 \\end{bmatrix} + \\alpha _ { k + 1 } v _ { k + 1 } e _ { k + 1 } ^ T X _ { k + 1 } , \\\\ Q _ B V _ k Y _ k & = \\widehat { U } _ { k } \\widehat { X } _ k S _ k , & Q _ B ^ T \\widehat { U } _ { k } \\widehat { X } _ k & = V _ k Y _ k S _ k + \\check { \\beta } _ { k } v _ { k + 1 } e _ { k } ^ T \\widehat { X } _ k . \\end{align*}"}
+{"id": "5522.png", "formula": "\\begin{align*} \\dim H _ l ( H , \\Pi \\otimes \\chi ^ { \\vee } ) \\le \\sum _ { j = 1 } ^ { j _ 0 } \\dim H _ l ( H , \\rho _ j \\otimes \\chi ^ { \\vee } ) . \\end{align*}"}
+{"id": "2611.png", "formula": "\\begin{align*} g _ n \\big ( t , q , x \\big ) : = \\sup _ { z \\in \\mathbb { R } ^ d } \\Big ( q \\cdot z - f _ n ( t , z , x ) \\Big ) . \\end{align*}"}
+{"id": "5783.png", "formula": "\\begin{align*} \\mathcal { D } = \\{ \\langle x \\rangle _ { \\mathbb { F } _ { q ^ m } } \\colon x \\in X \\setminus \\{ 0 \\} \\} \\end{align*}"}
+{"id": "7310.png", "formula": "\\begin{align*} M = q ^ m - q ^ { \\lceil \\frac { 2 m - 1 } { 3 } \\rceil + i } - q ^ { \\lfloor \\frac { m } { 3 } - 1 \\rfloor + i } - q ^ { i - 1 } - 1 . \\end{align*}"}
+{"id": "6697.png", "formula": "\\begin{align*} - \\frac { Q _ { i } ' ( 0 ) } { Q _ { i } '' ( 0 ) } & = \\frac { \\frac { r _ { i } ^ { 2 } \\vartheta _ { i } } { \\lambda _ { i , 2 } ^ { 2 } } - \\frac { r _ { i } } { \\lambda _ { i , 2 } ^ { 2 } } } { \\frac { r _ { i } ^ { 2 } \\vartheta _ { i } } { \\lambda _ { i , 2 } ^ { 4 } } } = \\lambda _ { i , 2 } ^ { 2 } \\frac { \\vartheta _ { i } r _ { i } - 1 } { r _ { i } ( \\vartheta _ i - 1 ) } , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] , \\end{align*}"}
+{"id": "1940.png", "formula": "\\begin{align*} g _ n = g \\phi _ n - \\vec f \\cdot D _ v \\phi _ n + u P _ 0 \\phi _ n - 2 ( a D _ v \\phi _ n ) \\cdot D _ v u . \\end{align*}"}
+{"id": "6632.png", "formula": "\\begin{align*} E _ { n } ( H _ { \\theta } ) = - \\frac { 1 } { ( 2 n - 1 ) ^ { 2 } \\theta ^ { 2 } } + O \\Big ( \\dfrac { 1 } { \\theta } \\Big ) \\theta \\rightarrow 0 ^ + . \\end{align*}"}
+{"id": "4075.png", "formula": "\\begin{align*} w ( x _ 2 ) = H ( x _ 2 ) \\circ v ( x _ 1 ) , \\end{align*}"}
+{"id": "6166.png", "formula": "\\begin{align*} & V _ 2 V _ 1 ^ * f ( z _ 1 , z _ 2 ) = M _ { z _ 2 } M _ { z _ 1 } ^ * R _ { \\overline q } f ( z _ 1 , z _ 2 ) = M _ { z _ 2 } M _ { z _ 1 } ^ * f ( \\overline q z _ 1 , \\overline q z _ 2 ) = M _ { z _ 1 } ^ * z _ 2 f ( \\overline q z _ 1 , \\overline q z _ 2 ) \\\\ & V _ 1 ^ * V _ 2 f ( z _ 1 , z _ 2 ) = M _ { z _ 1 } ^ * R _ { \\overline q } M _ { z _ 2 } f ( z _ 1 , z _ 2 ) = \\overline q M _ { z _ 1 } ^ * z _ 2 f ( \\overline q z _ 1 , \\overline q z _ 2 ) . \\end{align*}"}
+{"id": "6444.png", "formula": "\\begin{align*} \\int _ u ^ v \\frac { ( q x / u , q x / v ; q ) _ { \\infty } } { ( b x , c x ; q ) _ { \\infty } } d _ q x = \\frac { ( 1 - q ) v ( q , u / v , q v / u , b c u v ; q ) _ { \\infty } } { ( b u , b v , c u , c v ; q ) _ { \\infty } } . \\end{align*}"}
+{"id": "2783.png", "formula": "\\begin{align*} A = P _ n J _ n Q _ n ^ T , \\end{align*}"}
+{"id": "6189.png", "formula": "\\begin{align*} A _ 1 = \\big ( \\sum ^ m _ { i = 0 } E ^ * _ i \\big ) A _ 1 \\big ( \\sum ^ m _ { i = 0 } E ^ * _ i \\big ) = \\sum ^ { m - 1 } _ { i = 0 } ( E ^ * _ { i + 1 } A _ 1 E ^ * _ i + E ^ * _ i A _ 1 E ^ * _ { i + 1 } ) + E ^ * _ m A _ 1 E ^ * _ m \\end{align*}"}
+{"id": "4581.png", "formula": "\\begin{align*} \\frac { 4 } { \\pi } \\abs { \\theta _ 1 - \\theta _ 0 - \\frac { V ( n ) } { \\sin \\pi k } \\sin ^ 2 \\theta _ 0 } \\leq \\abs { e ^ { 2 i \\theta _ 1 } - e ^ { 2 i \\left ( \\theta _ 0 + \\frac { V ( n ) } { \\sin \\pi k } \\sin ^ 2 \\theta _ 0 \\right ) } } = O \\left ( \\frac { V ( n ) ^ 2 } { \\sin ^ 2 \\pi k } \\right ) . \\end{align*}"}
+{"id": "4438.png", "formula": "\\begin{align*} \\nabla ^ j w _ { n , m } = \\psi _ m \\nabla ^ j \\bar { w } _ { n , m } + \\sum _ { i = 0 } ^ { j - 1 } \\nabla ^ i \\bar { w } _ { n , m } \\otimes \\nabla ^ { j - i } \\psi _ m . \\end{align*}"}
+{"id": "4575.png", "formula": "\\begin{align*} \\frac { u ( 1 ) } { u ( 0 ) } = \\tan \\theta _ j . \\end{align*}"}
+{"id": "1566.png", "formula": "\\begin{align*} z _ { 3 } ^ { - 1 } s ^ { \\prime - 1 } z _ { 3 } s = \\left [ z _ { 3 } , s ^ { \\prime - 1 } \\right ] \\quad s ^ { \\prime - 1 } z _ { 3 } ^ { - 1 } z _ { 3 } s ^ { \\prime } = e . \\end{align*}"}
+{"id": "7030.png", "formula": "\\begin{align*} \\widehat { g } _ 1 - ( \\widehat { J } _ 0 + \\widehat { H } _ 0 + \\widehat { H } _ 1 ) \\widehat { u } _ 1 = \\sum \\limits _ { j = 0 } ^ 3 \\widehat { E } _ j \\widehat { u } _ 1 , \\end{align*}"}
+{"id": "6107.png", "formula": "\\begin{align*} \\mathcal G ( k , d ) = & \\left \\{ F \\in \\binom { [ n ] } { k } : [ d - 1 ] \\subseteq F , \\ , F \\cap [ d , k - 1 ] \\neq \\emptyset \\right \\} \\cup \\{ [ 2 , k ] \\cup \\{ i \\} : i \\in [ k + 2 , n ] \\} \\\\ & \\cup \\left \\{ F \\in \\binom { [ n ] } { k } : [ d - 1 ] \\cup [ k , k + 1 ] \\subseteq F \\right \\} \\cup \\{ [ 2 , k - 1 ] \\cup \\{ k + 1 , i \\} : i \\in [ k + 2 , n ] \\} \\\\ & \\cup \\left \\{ F \\in \\binom { [ n ] } { k } : | [ d - 1 ] \\cap F | = d - 2 , \\ , [ d , k + 1 ] \\subseteq F \\right \\} . \\end{align*}"}
+{"id": "4163.png", "formula": "\\begin{align*} \\lim _ { | k | \\to \\infty } \\chi _ k ^ { } | k | ^ { - 2 } = 0 . \\end{align*}"}
+{"id": "4813.png", "formula": "\\begin{align*} F _ n & = { } _ { 1 } F _ { 1 } \\ ! \\left ( \\left . \\begin{matrix} \\mu + n \\\\ \\lambda + n \\end{matrix} \\right | x \\right ) : & F _ n ' & = \\frac { n + \\mu } { n + \\lambda } F _ n ; \\\\ F _ n & = { } _ { 2 } F _ { 1 } \\ ! \\left ( \\left . \\begin{matrix} \\mu + n , a \\\\ \\lambda + n \\end{matrix} \\right | x \\right ) : & ( 1 - x ) F _ n ' & = \\frac { ( n + \\mu ) ( n + \\lambda - a ) } { n + \\lambda } F _ { n + 1 } + ( n + \\mu ) F _ n . \\end{align*}"}
+{"id": "2356.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { d - 1 } x _ { l _ { i + 1 } } \\cdots x _ { l _ { d } } x _ { l _ { 1 } } \\cdots x _ { l _ { i } } \\equiv \\sum _ { i = 0 } ^ { d - 1 } f _ { l _ { d } } ( S ( x _ { l _ { i + 1 } } \\cdots x _ { l _ { d - 1 } } ) , x _ { l _ { 1 } } \\cdots x _ { l _ { i } } ) = \\begin{cases} x _ { l _ { 1 } + \\cdots + l _ { d } } & d : { \\rm o d d } \\\\ 0 & d : { \\rm e v e n } , \\end{cases} \\end{align*}"}
+{"id": "7810.png", "formula": "\\begin{align*} & \\int _ { \\Omega } { f ( u _ m ^ n + 1 ) \\ d x } + \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ n \\int _ { \\Omega } f '' ( c ^ j ) ( u _ m ^ j - u _ m ^ { j - 1 } ) ^ 2 \\ d x + k \\sum _ { j = 1 } ^ n \\dfrac { 2 } { s ^ 2 } \\int _ { \\Omega } { \\norm { \\nabla [ u _ m ^ j + 1 ] ^ { s / 2 } } { } ^ 2 \\ d x } \\\\ & \\leq C k \\sum _ { j = 1 } ^ n \\int _ { \\Omega } { T ^ m ( u _ m ^ j ) ^ s \\norm { \\nabla z _ m ^ j } { } ^ 2 \\ d x } + \\int _ { \\Omega } { f ( u _ m ^ 0 + 1 ) \\ d x } . \\end{align*}"}
+{"id": "6378.png", "formula": "\\begin{align*} \\mathrm { D T } : = \\prod _ { h } \\mathrm { r o t } _ { h } , \\end{align*}"}
+{"id": "5590.png", "formula": "\\begin{align*} y ^ i R _ i - u ^ k I _ k = 2 q R , \\ \\ \\ u ^ k R _ k + y ^ i I _ i = 2 q I . \\end{align*}"}
+{"id": "6873.png", "formula": "\\begin{align*} \\left ( - \\sum _ { i } f _ { Y _ { i } } ^ { \\ast } Y _ { i } \\right ) \\cdot Y _ { i ' } = \\left ( \\sum _ { j } f _ { Z _ { j } } ^ { \\ast } Z _ { j } \\right ) \\cdot Y _ { i ' } \\ge 0 \\end{align*}"}
+{"id": "2282.png", "formula": "\\begin{align*} \\dim ( W ^ m ) \\geq \\dim \\left ( \\sum _ { i = 0 } ^ m W _ 1 ^ i W _ 2 ^ { m - i } z ^ { i p _ 1 + ( m - i ) p _ 2 } \\right ) = \\dim \\left ( \\bigoplus _ { i = 0 } ^ m W _ 1 ^ i W _ 2 ^ { m - i } z ^ { i p _ 1 + ( m - i ) p _ 2 } \\right ) \\end{align*}"}
+{"id": "769.png", "formula": "\\begin{align*} \\widehat { \\iota } ( D ) = \\frac { \\beta m + \\gamma } { m + b _ 1 + 1 } . \\end{align*}"}
+{"id": "1642.png", "formula": "\\begin{align*} e _ 1 \\ ; = \\ ; \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ 0 \\\\ 1 \\end{pmatrix} \\ ; , e _ 2 \\ ; = \\ ; \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ 1 \\\\ 0 \\end{pmatrix} \\ ; , \\ldots \\ ; , e _ { \\mathsf { L } } \\ ; = \\ ; \\begin{pmatrix} 1 \\\\ 0 \\\\ \\vdots \\\\ 0 \\\\ 0 \\end{pmatrix} \\ ; . \\end{align*}"}
+{"id": "1138.png", "formula": "\\begin{align*} x _ { n + 1 } & = ( 1 - \\beta _ n ) U y _ n + \\beta _ n y _ n , \\\\ y _ { n } & = ( 1 - \\alpha _ n ) T x _ n + \\alpha _ n u , \\end{align*}"}
+{"id": "7116.png", "formula": "\\begin{align*} _ d = ( - 1 ) ^ d p _ 3 ( d ) . \\end{align*}"}
+{"id": "125.png", "formula": "\\begin{align*} \\| S T x _ n \\| = \\| S [ ( f _ k ( x _ n ) ) _ k ] \\| \\ge M \\| ( f _ k ( x _ n ) ) _ k \\| _ \\infty \\ge M | f _ n ( x _ n ) | \\ge \\frac { M } { 2 } > 0 . \\end{align*}"}
+{"id": "5728.png", "formula": "\\begin{align*} p = \\sum _ { \\omega \\in \\mathcal { W } _ m } p _ { \\omega } \\omega , ~ p _ { \\omega } \\in \\mathbb { R } . \\end{align*}"}
+{"id": "2721.png", "formula": "\\begin{align*} \\binom { 2 n } { 2 n - j - u } \\binom { 2 n - j - u } { n } = \\binom { 2 n } { n } \\binom { n } { j + u } \\end{align*}"}
+{"id": "5878.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } { \\ddot { x } ( t ) = \\dot { x } ( t ) \\times \\frac { B ( x ( t ) ) } { \\epsilon } + F ( x ( t ) ) } , x ( 0 ) = x _ 0 , \\dot { x } ( 0 ) = \\dot { x } _ 0 , \\end{array} \\end{align*}"}
+{"id": "5586.png", "formula": "\\begin{align*} H ( v ) = H ( y _ o ) = R ( y ) + \\sqrt { - 1 } I ( y ) , \\end{align*}"}
+{"id": "4199.png", "formula": "\\begin{align*} \\lim _ { Y \\to \\infty } \\frac { 1 } { Y } \\int _ { 0 } ^ { Y } f ( e ^ { - y / 2 k } R ^ * _ k ( e ^ y ) ) \\ , d y = \\int _ { \\mathbb { R } } f ( x ) \\ , d \\nu _ k ^ * ( x ) \\end{align*}"}
+{"id": "3436.png", "formula": "\\begin{align*} E _ v = \\sum _ { j \\geq 0 } ( { - D _ z } ) ^ j \\partial _ { \\partial _ z ^ j v } , E ^ { ( i ) } _ v = \\sum _ { j \\geq 0 } { \\textstyle \\binom { i + j } { i } } ( { - D _ z } ) ^ j \\partial _ { \\partial _ z ^ { i + j } v } , i = 1 , 2 , \\ldots \\end{align*}"}
+{"id": "4070.png", "formula": "\\begin{align*} v _ { m + i } ^ * + \\sum _ { i = 1 } ^ m f _ { i j } v _ j ^ * . \\end{align*}"}
+{"id": "2598.png", "formula": "\\begin{align*} \\mathbb { E } \\Big [ \\underset { t _ 0 \\leq t \\leq T } { \\sup } | x _ t ^ i - x _ t ^ { * , i } | ^ 2 \\Big ] \\leq \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } \\mathbb { E } [ | \\xi ^ i - \\xi ^ j | ^ 2 ] + \\frac { 1 } { N ^ 2 } \\underset { i , j = 1 } { \\overset { N } { \\sum } } \\mathbb { E } [ | \\xi ^ { i } - \\xi ^ { j } | ^ 2 ] \\Big ) , \\end{align*}"}
+{"id": "3248.png", "formula": "\\begin{align*} v _ { j } ^ { l } : = \\max \\{ u _ { j } , u _ { j + 1 } , \\dots , u _ { j + l } \\} . \\end{align*}"}
+{"id": "8214.png", "formula": "\\begin{align*} h ( \\sigma ) : = \\begin{cases} \\left ( \\frac { \\gamma - 1 } { \\sqrt { \\kappa \\gamma } } \\sigma + 1 \\right ) ^ { \\frac { 1 } { \\gamma - 1 } } - 1 , \\ & \\gamma > 1 , \\\\ e ^ { \\sigma / \\sqrt { \\kappa } } - 1 , \\ & \\gamma = 1 . \\end{cases} \\end{align*}"}
+{"id": "5748.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } x _ i x _ j ^ T , ~ g ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } B _ { i j } x _ i x _ j ^ T , \\end{align*}"}
+{"id": "3174.png", "formula": "\\begin{align*} f ^ n ( \\alpha \\setminus \\beta ' _ n ) \\cap f ^ n ( \\beta ' _ n ) = \\varnothing f ^ n ( \\beta ' _ n ) \\subset [ f ^ n ( \\alpha ) ] \\end{align*}"}
+{"id": "3458.png", "formula": "\\begin{align*} \\Bigg [ \\mathrm { H } ^ { 1 , \\frac { 1 } { 2 } } \\Big ( \\partial \\Omega \\times \\mathbb { R } \\Big ) , \\mathrm { L } ^ 2 \\Big ( ( \\partial \\Omega ) _ T \\Big ) \\Bigg ] _ { \\theta = \\frac { 1 } { 2 } } = \\mathrm { H } ^ { \\frac { 1 } { 2 } , \\frac { 1 } { 4 } } \\Big ( ( \\partial \\Omega ) _ T \\Big ) . \\end{align*}"}
+{"id": "6868.png", "formula": "\\begin{align*} \\bigtriangleup k = & \\frac { ( p ^ { e ' } + n - 1 - \\deg ( h ( x ) ) ) ( \\sqrt { q } + 1 ) - ( \\sqrt { q } + n - 1 ) ( p ^ { e ' } + 1 ) } { ( p ^ { e ' } + 1 ) ( \\sqrt { q } + 1 ) } \\\\ = & \\frac { ( n - 2 ) ( \\sqrt { q } - p ^ { e ' } ) - ( \\sqrt { q } + 1 ) \\deg ( h ( x ) ) } { ( p ^ { e ' } + 1 ) ( \\sqrt { q } + 1 ) } \\\\ \\geq & 1 . \\end{align*}"}
+{"id": "5533.png", "formula": "\\begin{align*} c _ 1 ( \\theta _ 1 ) \\gamma _ 1 ( \\theta _ 1 ) \\dfrac { \\partial \\theta _ 1 } { \\partial t } = \\dfrac { 1 } { r ^ { \\nu } } \\dfrac { \\partial } { \\partial r } \\bigg ( \\lambda _ 1 ( \\theta _ 1 ) r ^ { \\nu } \\dfrac { \\partial \\theta _ 1 } { \\partial r } \\bigg ) , \\ ; \\ ; \\ ; \\alpha ( t ) < r < \\beta ( t ) , \\ ; \\ ; \\ ; 0 < \\nu < 1 , \\end{align*}"}
+{"id": "6424.png", "formula": "\\begin{align*} \\begin{matrix*} [ l ] & e _ 1 = 1 1 1 1 \\\\ & e _ 2 = 0 0 1 1 1 1 \\\\ & e _ 3 = 0 0 0 0 1 1 1 1 \\\\ & e _ 4 = 0 0 0 0 0 0 1 1 1 1 \\\\ & \\vdots \\end{matrix*} \\end{align*}"}
+{"id": "8199.png", "formula": "\\begin{align*} u v = T _ u v + T _ v u + R ( u , v ) , \\end{align*}"}
+{"id": "2483.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = v & & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta v & = | \\nabla u | ^ 2 & & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "5955.png", "formula": "\\begin{align*} ( \\xi _ { A _ 1 } - \\xi _ { B _ 1 } ) \\cdots ( \\xi _ { A _ 1 } - \\xi _ { B _ k } ) = \\sum _ { j = 0 } ^ { k } ( - 1 ) ^ { j } e _ { j } ( B ) \\xi _ { A _ { 1 } } ^ { k - j } \\end{align*}"}
+{"id": "5473.png", "formula": "\\begin{align*} { } _ 3 \\phi _ 2 \\begin{bmatrix} q ^ { - N } , & a b t / c , & b & ; q , q \\\\ b t , & b q ^ { 1 - N } / c . \\end{bmatrix} = \\frac { ( c , t ; q ) _ N } { ( c / b , b t ; q ) _ N } { } _ 3 \\phi _ 2 \\begin{bmatrix} q ^ { - N } , & a , & b & ; q , q \\\\ c , & q ^ { 1 - N } / t \\end{bmatrix} \\end{align*}"}
+{"id": "3079.png", "formula": "\\begin{align*} A ( \\ell _ i , q _ i ) = ( a _ { i , k } ) _ { k = 1 } ^ { \\infty } \\end{align*}"}
+{"id": "6618.png", "formula": "\\begin{align*} r _ j ^ \\gamma ( u , u ) = \\int _ { V _ j } | \\nabla u | ^ 2 \\dd x - \\gamma \\int _ { \\partial V _ j } | u | ^ 2 \\dd \\sigma , D ( r _ j ^ \\gamma ) = H ^ 1 ( V _ j ) , \\end{align*}"}
+{"id": "5985.png", "formula": "\\begin{align*} U _ r ^ { b } ( t ) = X ( t ) - L _ r ^ b ( t ) , t \\geq 0 , \\end{align*}"}
+{"id": "1218.png", "formula": "\\begin{align*} \\pi _ { 2 } \\circ \\pi _ { 1 } ( x ^ { ( f ) } _ i ) = x ^ { ( f ) } _ { \\pi _ { 1 } ( \\pi _ 2 ( i ) ) } = x ^ { ( T _ 1 ( f ) ) } _ { \\pi _ 2 ( i ) } = x ^ { ( g ) } _ { \\pi _ 2 ( i ) } = x _ i ^ { ( T _ 2 ( g ) ) } = x _ i ^ { ( T _ 2 \\circ T _ 1 ( f ) ) } . \\end{align*}"}
+{"id": "4159.png", "formula": "\\begin{align*} R ^ { + \\phi } ( X , Y , Z , W ) = R ^ { - \\phi } ( Z , W , X , Y ) \\end{align*}"}
+{"id": "3277.png", "formula": "\\begin{align*} & \\Delta ( Y _ K ) = K ^ 2 \\otimes Y _ K + \\big ( Y _ K - a _ s \\hat { K } ^ 2 \\big ) \\otimes 1 , \\\\ & \\Delta ( Y _ L ) = 1 \\otimes \\big ( Y _ L - b _ t \\hat { K } ^ { - 2 } \\big ) + Y _ L \\otimes K ^ { - 2 } . \\end{align*}"}
+{"id": "3690.png", "formula": "\\begin{align*} \\Gamma _ { t + 2 } ^ r = \\left \\{ \\{ u _ 1 , \\ldots , u _ { r - 3 } \\} \\cup E \\colon E \\in \\Gamma _ { t + 2 } \\right \\} . \\end{align*}"}
+{"id": "6743.png", "formula": "\\begin{align*} & ( \\partial _ { \\beta } ^ { \\alpha } [ \\frac { ( E + v \\times B ) \\cdot \\nabla _ { v } ( \\sqrt { \\mu } f ) } { \\sqrt { \\mu } } ] , \\partial _ { \\beta } ^ { \\alpha } f ) \\\\ & = ( \\partial ^ \\alpha _ \\beta [ E \\cdot \\nabla _ { v } f ] , \\partial _ { \\beta } ^ \\alpha f ) - ( \\partial ^ \\alpha _ \\beta [ \\frac { v } { 2 } \\cdot E f ] , \\partial _ { \\beta } ^ \\alpha f ) + ( \\partial ^ \\alpha _ \\beta [ v \\times B \\cdot \\nabla _ { v } f ] , \\partial _ { \\beta } ^ \\alpha f ) , \\end{align*}"}
+{"id": "6097.png", "formula": "\\begin{align*} F h ^ { m } = \\sum \\limits _ { j = 1 } ^ { r } G ' _ j \\left ( x _ j g _ j - f _ j \\textbf { z } ^ { \\beta _ j } \\right ) + { \\sum \\limits _ { i = 1 } ^ { s } H _ i ' ( { y _ i } ^ { q } - y _ i ) } . \\end{align*}"}
+{"id": "7544.png", "formula": "\\begin{align*} \\iint \\left ( h ( p ) C ( p , q ) + h ( q ) C ( q , p ) \\right ) d \\mu ( p ) d \\mu ( q ) = \\int h d V \\ , d \\mu \\end{align*}"}
+{"id": "2897.png", "formula": "\\begin{align*} \\kappa ( s ) = \\alpha \\ , \\frac { \\nu _ { n + 1 } ( s ) } { x _ { n + 1 } ( s , t ) } + \\varpi . \\end{align*}"}
+{"id": "5853.png", "formula": "\\begin{align*} \\Big ( 1 + 2 ^ { p - 1 } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } \\Big ) \\langle f ( A ) \\hat { k } _ \\lambda , \\hat { k } _ \\lambda \\rangle ^ p < \\Big ( \\frac { p } { p - 1 } \\Big ) ^ { p } \\langle f ( A ) ^ p \\hat { k } _ \\lambda , \\hat { k } _ \\lambda \\rangle . \\end{align*}"}
+{"id": "5284.png", "formula": "\\begin{align*} \\dim L _ { \\pi _ { i , j } } = 2 \\ * n + 1 - | \\pi _ { i , j } | \\end{align*}"}
+{"id": "6997.png", "formula": "\\begin{align*} u ( x _ 0 ) = \\inf _ { B ( x _ 0 , R / 2 ) } u \\geq \\frac { \\bar { \\lambda } \\tau } { 2 } > 0 , \\end{align*}"}
+{"id": "282.png", "formula": "\\begin{align*} - F ^ { s - 1 } d \\colon W _ { s } ( K ) \\rightarrow \\Omega ^ { 1 } _ { K } \\ ; ; \\ ; ( a _ { s - 1 } , \\ldots , a _ { 0 } ) \\mapsto - \\sum _ { i = 0 } ^ { s - 1 } a _ { i } ^ { p ^ { i } - 1 } d a _ { i } . \\end{align*}"}
+{"id": "7516.png", "formula": "\\begin{align*} \\theta _ 1 ( z + 1 ) = - \\theta _ 1 ( z ) , \\theta _ 1 ( z + \\tau ) = - e ^ { - \\pi i \\tau - 2 \\pi i z } \\theta _ 1 ( z ) . \\end{align*}"}
+{"id": "50.png", "formula": "\\begin{align*} \\Delta _ { \\Psi } u = \\div _ { \\Psi } ( \\nabla u ) . \\end{align*}"}
+{"id": "5865.png", "formula": "\\begin{align*} S ^ \\prime ( q ) = - \\frac { 3 } { 2 } \\frac { \\frac { 5 } { 3 } P ( q ) - P ^ \\prime ( q ) q } { q ^ 2 } < 0 . \\end{align*}"}
+{"id": "3701.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n \\mathbb E \\sum _ { i = 0 } ^ { n - 1 } ( \\Delta q ^ b _ i ) ^ 2 - n d = \\int _ 0 ^ 1 \\int _ { \\R ^ d } ( v ^ 2 - ( \\frac 1 2 \\nabla \\ln \\rho ) ^ 2 ) \\rho d x d t \\end{align*}"}
+{"id": "6550.png", "formula": "\\begin{align*} ( 2 P - I ) e _ i = e _ j . \\end{align*}"}
+{"id": "2243.png", "formula": "\\begin{align*} x d = \\sigma ( d ) x , \\ y d = \\sigma ^ { - 1 } ( d ) y , \\ y x = a , \\ x y = \\sigma ( a ) , d \\in D . \\end{align*}"}
+{"id": "5615.png", "formula": "\\begin{align*} \\delta _ { \\mu } : = \\partial _ { \\mu } - \\Gamma ^ { \\alpha } _ { ; \\mu } \\dot { \\partial } _ { \\alpha } , \\ \\ \\delta v ^ \\alpha = d v ^ \\alpha + \\Gamma ^ { \\alpha } _ { ; \\mu } d z ^ \\mu . \\end{align*}"}
+{"id": "8104.png", "formula": "\\begin{align*} T _ 2 ^ * T _ 1 = M _ z ^ * B _ r \\otimes M _ z T _ 1 T _ 2 ^ * = B _ r M _ z ^ * \\otimes M _ z . \\end{align*}"}
+{"id": "259.png", "formula": "\\begin{align*} L ( v ) = \\int _ U \\left ( \\langle W _ u ( u ^ { - 1 } ( y ) ) , v ( y ) \\rangle + W _ \\xi ( u ^ { - 1 } ( y ) ) [ \\nabla _ x u ] ^ t ( u ^ { - 1 } ( y ) ) : [ \\nabla _ y v ] ( y ) \\right ) d y . \\end{align*}"}
+{"id": "3351.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ s ( \\prod _ { j = 1 } ^ m A _ { i , j } ) \\end{align*}"}
+{"id": "5858.png", "formula": "\\begin{align*} b e r ^ p ( A ) { ( 2 + 2 ^ { p } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } ) } < ( \\frac { p } { p - 1 } ) ^ { p } \\left ( b e r ( | A | ^ { 2 p \\alpha } ) + b e r ( | A ^ * | ^ { 2 p ( 1 - \\alpha ) } ) \\right ) . \\end{align*}"}
+{"id": "3711.png", "formula": "\\begin{align*} \\mathbb { E } \\int _ 0 ^ 1 ( b ^ 2 ( q _ t ^ b , t ) + \\nabla b ( q _ t ^ b , t ) ) d t = \\int _ 0 ^ 1 \\int _ { \\mathbb { R } ^ d } ( v ^ 2 - ( \\frac 1 2 \\nabla l n \\rho ) ^ 2 ) \\rho d x d t < \\infty \\quad \\end{align*}"}
+{"id": "7513.png", "formula": "\\begin{align*} G ( p , q ) = \\log | z _ { \\infty } ( p ) | + O ( 1 ) p \\to p _ { \\infty } , \\end{align*}"}
+{"id": "1783.png", "formula": "\\begin{align*} \\left \\langle Y , B , a \\right \\rangle = 0 , \\quad { \\rm f o r } a \\in A . \\end{align*}"}
+{"id": "458.png", "formula": "\\begin{align*} F ^ 2 ( z ) = \\frac { 4 } { 5 } e ^ { L _ j z } \\sinh ^ 2 \\Big ( \\frac { \\sqrt { 5 } F _ j } { 2 } z \\Big ) , \\end{align*}"}
+{"id": "3185.png", "formula": "\\begin{align*} \\phi _ { u } ( x ) : = \\int _ { \\mathbb { R } ^ { 3 } } \\frac { 1 - e ^ { - | x - y | } } { | x - y | } u ^ { 2 } ( y ) d y . \\end{align*}"}
+{"id": "2070.png", "formula": "\\begin{align*} \\begin{bmatrix} I _ \\ell & B _ { 1 2 } \\\\ B _ { 2 1 } & B _ { 2 2 } \\end{bmatrix} : = \\begin{bmatrix} Q ^ { - 1 } & 0 \\\\ 0 & I _ { n - \\ell } \\end{bmatrix} \\begin{bmatrix} Q _ 1 & E \\\\ 0 & Q _ 2 \\end{bmatrix} \\begin{bmatrix} A _ { 1 1 } & A _ { 1 2 } \\\\ A _ { 2 1 } & A _ { 2 2 } \\end{bmatrix} \\begin{bmatrix} P _ 1 & 0 \\\\ 0 & P _ 2 \\end{bmatrix} . \\end{align*}"}
+{"id": "2819.png", "formula": "\\begin{align*} d K _ s & = \\bigg ( \\frac { \\rho ^ 2 } { \\rho + \\lambda } K _ s ^ 2 + \\frac { 2 \\lambda \\rho } { \\rho + \\lambda } K _ s - \\frac { \\lambda \\rho } { \\rho + \\lambda } \\bigg ) d s + \\sum _ { j = 1 } ^ m L _ s ^ j d W _ s ^ j , s \\in [ 0 , T ] , K _ T = \\frac 1 2 . \\end{align*}"}
+{"id": "134.png", "formula": "\\begin{align*} s _ p s _ q f = s _ p f \\qquad t _ p t _ q f = t _ p f \\end{align*}"}
+{"id": "2547.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { P } _ t + 2 A P _ t - B ^ 2 R ^ { - 1 } P ^ 2 _ t + Q = 0 , \\\\ & P _ T = G , \\end{aligned} \\right . \\end{align*}"}
+{"id": "2580.png", "formula": "\\begin{align*} \\begin{array} { l } \\mu ^ { N , 1 } _ { \\boldsymbol { \\alpha } ^ * _ t } + h _ 1 ^ N ( \\mu ^ { N , 1 } _ { \\boldsymbol { \\alpha } ^ * _ t } , \\ldots , \\mu ^ { N , N } _ { \\boldsymbol { \\alpha } ^ * _ t } ) = \\Delta ^ { N , 1 } _ { * , t } , \\\\ \\quad \\vdots \\qquad \\qquad ~ ~ \\vdots \\qquad ~ ~ ~ ~ \\vdots \\\\ \\mu ^ { N , N } _ { \\boldsymbol { \\alpha } ^ * _ t } + h _ N ^ N ( \\mu ^ { N , 1 } _ { \\boldsymbol { \\alpha } ^ * _ t } , \\ldots , \\mu ^ { N , N } _ { \\boldsymbol { \\alpha } ^ * _ t } ) = \\Delta ^ { N , N } _ { * , t } , \\end{array} \\end{align*}"}
+{"id": "1846.png", "formula": "\\begin{align*} D _ 1 & = x _ { 1 , 1 } + x _ { m , n } = \\tfrac { 1 } { 2 } m n + \\tfrac { 3 } { 2 } \\mbox { a n d } \\\\ D _ 2 & = x _ { m , 1 } + x _ { 1 , n } = \\tfrac { 1 } { 2 } m n + \\tfrac { 3 } { 2 } , \\mbox { r e s p e c t i v e l y . } \\end{align*}"}
+{"id": "4563.png", "formula": "\\begin{gather*} \\Phi _ T ( M _ { a f _ 1 + b f _ 2 } ) = \\Phi _ T ( a M _ { f _ 1 } + b M _ { f _ 2 } ) = a \\Phi _ T ( M _ { f _ 1 } ) + b \\Phi _ T ( M _ { f _ 2 } ) = \\\\ = M _ { a g _ 1 } + a R _ 1 + M _ { b g _ 2 } + b R _ 2 = M _ { a g _ 1 + b g _ 2 } + a R _ 1 + b R _ 2 \\end{gather*}"}
+{"id": "2458.png", "formula": "\\begin{align*} \\begin{aligned} & j ( u , A ) - j ( u , B ) = ( B - A ) \\abs { u } ^ 2 \\\\ & J ( u , A ) - J ( u , B ) = \\frac { 1 } { 2 } \\d ( ( A - B ) ( 1 - \\abs { u } ^ 2 ) ) . \\end{aligned} \\end{align*}"}
+{"id": "4492.png", "formula": "\\begin{align*} \\mathcal { E } ' _ x ( t ) = \\left \\{ \\begin{array} { l l } \\displaystyle \\phantom { - } \\frac { \\cosh ( 1 - x ) \\sinh t } { \\sinh 1 } \\ , , & t \\in [ 0 , x ) \\\\ \\quad & \\quad \\\\ \\displaystyle - \\frac { \\cosh x \\sinh ( 1 - t ) } { \\sinh 1 } \\ , , & t \\in ( x , 1 ] \\end{array} \\right . \\ , , \\end{align*}"}
+{"id": "2894.png", "formula": "\\begin{align*} n H = \\eta x _ { n + 1 } ^ m + \\lambda . \\end{align*}"}
+{"id": "163.png", "formula": "\\begin{align*} \\begin{aligned} f _ n = K i n e t i c ( n ) + F l u i d ( n ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "914.png", "formula": "\\begin{align*} \\varphi ( x ' , \\rho ( x ' ) ) = \\frac { 1 } { 2 } \\sum _ { \\alpha \\leq n - 1 } b _ \\alpha x _ \\alpha ^ 2 + \\frac { 1 } { 6 } \\sum _ { \\alpha , \\beta , \\gamma \\leq n - 1 } \\varphi _ { \\alpha \\beta \\gamma } ( 0 ) x _ \\alpha x _ \\beta x _ \\gamma + O ( | x ' | ^ 4 ) , \\end{align*}"}
+{"id": "71.png", "formula": "\\begin{align*} u _ + = \\max \\{ u , 0 \\} \\end{align*}"}
+{"id": "4714.png", "formula": "\\begin{align*} D _ { z } \\left ( z f ( z ) \\right ) = \\sum _ { n = 0 } ^ { \\infty } c _ { n } ( n + \\lambda + 1 ) \\phi _ { n } ( z ) , \\end{align*}"}
+{"id": "7830.png", "formula": "\\begin{align*} a _ i a ^ \\dag _ j = \\delta _ { i , j } \\left ( I - \\sum _ { k \\leq i } a ^ \\dag _ k a _ k \\right ) \\ , , \\end{align*}"}
+{"id": "632.png", "formula": "\\begin{align*} \\langle \\langle \\varphi , \\varphi ' \\rangle \\rangle = \\tau \\langle \\langle \\varphi ' , \\varphi \\rangle \\rangle \\end{align*}"}
+{"id": "7237.png", "formula": "\\begin{align*} \\vec { u } _ n ( x ) = & \\sum _ { j = 1 } ^ J g _ n ^ j e ^ { i t _ n ^ j \\Delta _ { \\mathbb { R } ^ 2 } } \\vec { \\phi } ^ j + \\vec { w } _ n ^ J ( x ) \\\\ : = & \\sum _ { j = 1 } ^ J \\frac 1 { \\lambda _ n ^ j } e ^ { i x \\xi _ n ^ j } ( e ^ { i t _ n ^ j \\Delta _ { \\mathbb { R } ^ 2 } } \\vec { \\phi } ^ j ) \\left ( \\frac { x - x _ n ^ j } { \\lambda _ n ^ j } \\right ) + \\vec { w } _ n ^ J ( x ) , \\end{align*}"}
+{"id": "6325.png", "formula": "\\begin{align*} A u = b u ^ + - a u ^ - , u \\in D \\end{align*}"}
+{"id": "4327.png", "formula": "\\begin{align*} d _ { T V } ( X , \\pi ) \\leq \\sup _ { h \\in \\mathcal { H } } \\sup _ { l \\in \\mathbb { Z } ^ + } | m _ l ( h ) | \\left | \\mathbb { E } X - \\mathbb { E } \\sum _ { j = 0 } ^ \\infty \\sum _ { k = j + 1 } ^ \\infty P _ { X , k } \\right | . \\end{align*}"}
+{"id": "4336.png", "formula": "\\begin{align*} ( f , g ) ( 0 ) = ( f ^ { i n } , g ^ { i n } ) \\ ; \\ ; \\Omega \\ , . \\end{align*}"}
+{"id": "2782.png", "formula": "\\begin{align*} \\left ( \\begin{bmatrix} 0 & A \\\\ A ^ T & 0 \\end{bmatrix} , \\begin{bmatrix} I & 0 \\\\ 0 & B ^ T B \\end{bmatrix} \\right ) , \\qquad \\left ( \\begin{bmatrix} 0 & B \\\\ B ^ T & 0 \\end{bmatrix} , \\begin{bmatrix} I & 0 \\\\ 0 & A ^ T A \\end{bmatrix} \\right ) , \\end{align*}"}
+{"id": "6255.png", "formula": "\\begin{align*} | { \\rm F i x } _ k ( A ^ { [ n ] } b ) | = \\sum _ { j _ 1 \\cdots j _ \\ell = k } | \\{ \\alpha \\in A ^ { [ n ] } \\mid b _ i ( \\alpha ) = \\alpha ( y _ i ) \\alpha ( y _ i b ) \\cdots \\alpha ( y _ i b ^ { k _ i - 1 } ) \\in { \\rm F i x } _ { j _ i } ( A ) \\} | \\end{align*}"}
+{"id": "7161.png", "formula": "\\begin{align*} ( x _ { i , i } - x _ { i + 1 , i + 1 } ) y _ { i , i + 1 } - ( y _ { i , i } - y _ { i + 1 , i + 1 } ) x _ { i , i + 1 } = 0 . \\end{align*}"}
+{"id": "6645.png", "formula": "\\begin{align*} - \\Delta f _ n = k ^ 2 f _ n - 2 \\nabla e ^ { i k x _ 1 } \\cdot \\nabla \\chi ( \\dots ) - e ^ { i k x _ 1 } \\Delta \\chi ( \\dots ) , \\end{align*}"}
+{"id": "1290.png", "formula": "\\begin{align*} \\pi _ a ^ j \\circ \\iota _ a ^ i & = ( \\iota _ 1 ^ j ) ^ { \\ast } \\circ ( \\pi _ 1 ^ i ) ^ { \\ast } = ( \\pi _ 1 ^ i \\circ \\iota _ 1 ^ j ) ^ { \\ast } , \\end{align*}"}
+{"id": "2368.png", "formula": "\\begin{align*} u = x _ { a _ { 1 } } \\cdots x _ { a _ { m } } , \\ v = x _ { b _ { 1 } } \\cdots x _ { b _ { n } } . \\end{align*}"}
+{"id": "3027.png", "formula": "\\begin{align*} \\begin{aligned} u _ 1 & = h _ 1 ( x , \\bar { u } _ 1 , \\bar { u } _ 2 ) \\\\ u _ 2 & = h _ 2 ( x , \\bar { u } _ 1 , \\bar { u } _ 2 ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "1078.png", "formula": "\\begin{align*} \\begin{aligned} \\mathfrak r _ { l k + 2 } ( 2 ) & = \\mathfrak p _ { k } \\mathfrak r _ { ( l - 1 ) k + 2 } ( 2 ) - i \\partial _ { a } \\mathfrak p _ { k } \\delta _ { a } \\mathfrak r _ { ( l - 1 ) k + 1 } ( 1 ) - \\frac { 1 } { 2 } \\partial _ { a } \\partial _ { b } \\mathfrak p _ { k } \\delta _ { a } \\delta _ { b } \\mathfrak r _ { ( l - 1 ) k } = \\cdots \\end{aligned} \\end{align*}"}
+{"id": "7193.png", "formula": "\\begin{align*} \\mathcal { S } = \\bigoplus _ { v _ 1 / d _ 1 \\leq \\cdots \\leq v _ k / d _ k } \\mathbb { F } \\cdot [ \\mathcal { E } _ { d _ 1 , v _ 1 } ] \\ast \\cdots \\ast [ \\mathcal { E } _ { d _ k , v _ k } ] . \\end{align*}"}
+{"id": "6603.png", "formula": "\\begin{align*} \\varphi _ j : = \\dfrac { \\theta _ { j + 1 } - \\theta _ j } { 2 } , \\end{align*}"}
+{"id": "6286.png", "formula": "\\begin{align*} t ^ u \\cdot L = \\sigma ^ v \\cdot L . \\end{align*}"}
+{"id": "4447.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { v \\in X } \\vert J _ n ( v ) - J ( v ) \\vert = 0 . \\end{align*}"}
+{"id": "4836.png", "formula": "\\begin{align*} \\int _ S \\int _ { B _ R } ( \\nabla g \\cdot \\nabla \\phi + \\phi ) \\ , \\dd \\rho _ R ( x , s ) = \\int _ S \\int _ { B _ R } \\phi \\ , \\dd \\bar \\rho _ R ( x ) \\dd \\mu ( s ) , \\end{align*}"}
+{"id": "7167.png", "formula": "\\begin{align*} \\mathcal { E } _ { d , v } : = p _ { \\ast } \\left ( \\mathcal { O } _ { \\mathcal { Z } } \\otimes \\mathbb { C } ( m _ 1 , \\ldots , m _ d ) \\right ) \\in D ^ b ( \\mathcal { C } ( d ) ) . \\end{align*}"}
+{"id": "8146.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac 1 n d _ \\omega ( \\xi ^ Y _ n , \\xi _ { n \\varphi ^ Y } ) = 0 . \\end{align*}"}
+{"id": "3404.png", "formula": "\\begin{align*} e ( \\rho , S ) = \\int ( p ( \\rho , S ) / \\rho ^ 2 ) \\ , d \\rho , \\end{align*}"}
+{"id": "6262.png", "formula": "\\begin{align*} \\sum _ { k = i } ^ \\infty \\norm { z _ { k + 1 } - z _ k } \\leq \\norm { z _ i - z _ { i - 1 } } + C \\tilde { \\varphi } \\left ( H _ { p _ i } ( z _ { i } ) - F ( x ^ * ) \\right ) , \\end{align*}"}
+{"id": "1596.png", "formula": "\\begin{align*} V ^ { \\prime \\prime } = \\left [ \\ker \\left ( T _ { b _ { i } } \\restriction _ { V ^ { \\prime } } - \\varphi _ { s , j , i } \\circ T _ { ( b _ { j } , s _ { i } ) } \\restriction _ { V ^ { \\prime } } \\right ) \\cap \\ker \\left ( T _ { b _ { j } } \\restriction _ { V ^ { \\prime } } - \\varphi _ { s , i , j } \\circ T _ { ( b _ { i } , s _ { i } ) } \\restriction _ { V ^ { \\prime } } \\right ) \\right ] \\end{align*}"}
+{"id": "5372.png", "formula": "\\begin{align*} V ^ \\omega \\doteq \\{ w \\in W \\ , : \\ , \\omega ( v , w ) = 0 \\ , \\forall v \\in V \\} . \\end{align*}"}
+{"id": "4146.png", "formula": "\\begin{align*} \\frac { B ' ( z ) } { B ( z ) } = \\sum \\limits _ { k = 1 } ^ n \\frac { 1 - | z _ k | ^ 2 } { \\left ( 1 - \\overline { z _ k } z \\right ) \\left ( z - z _ k \\right ) } \\ , . \\end{align*}"}
+{"id": "4187.png", "formula": "\\begin{align*} \\underset { l } \\sum l ( C _ 1 ^ l \\cap w ^ i ) & = l ( w ^ i ) \\\\ & = \\frac { 1 } { \\mathrm { s i n } ( \\theta ) } \\underset { l } \\sum i ( w ^ i , ( w _ 1 ) _ l ) \\mathrm { h e i g h t } ( C _ 1 ^ l ) \\end{align*}"}
+{"id": "5325.png", "formula": "\\begin{align*} q ^ { 3 } - q ^ { 2 } [ 2 ] _ { q } + 2 q = q ^ { 3 } - q ^ { 2 } ( q + q ^ { - 1 } ) + 2 q = q . \\end{align*}"}
+{"id": "5322.png", "formula": "\\begin{align*} \\eta ( D . T _ { i } ) = q \\eta ( D ) + \\eta ( D \\tilde { E } _ { i } ) , \\end{align*}"}
+{"id": "7635.png", "formula": "\\begin{align*} \\begin{array} { l l l } u _ { \\overline { \\nu _ 0 } } = ( 1 , 0 , 0 , 0 , 0 , 1 , 0 ) , & & u _ { \\overline { \\nu _ 4 } } = ( 0 , 0 , 0 , 0 , 1 , 0 , 1 ) , \\\\ u _ { \\overline { \\nu _ 1 } } = ( 0 , 1 , 0 , 0 , 0 , 1 , 0 ) , & & u _ { \\overline { \\nu _ 5 } } = ( - 1 , - 1 , - 1 , - 1 , - 1 , 0 , 1 ) , \\\\ u _ { \\overline { \\nu _ 2 } } = ( 0 , 0 , 1 , 0 , 0 , 1 , 0 ) , & & u _ { \\tau _ 1 } = ( 0 , 0 , 0 , 0 , 0 , 1 , 0 ) , \\\\ u _ { \\overline { \\nu _ 3 } } = ( 0 , 0 , 0 , 1 , 0 , 0 , 1 ) , & & u _ { \\tau _ 2 } = ( 0 , 0 , 0 , 0 , 0 , 0 , 1 ) . \\end{array} \\end{align*}"}
+{"id": "6414.png", "formula": "\\begin{align*} j _ { \\rho } ^ { \\gamma } ( x ) = \\rho ( x ) ^ { \\gamma - 1 } \\nabla \\rho ( x ) , \\end{align*}"}
+{"id": "7603.png", "formula": "\\begin{align*} ( \\sigma _ { - \\alpha _ i } \\sigma _ { - \\alpha _ j } ) _ { 1 \\leq i , j \\leq n } = ( J + ( \\sqrt { - 1 } ) ^ { \\dim Y } I _ \\aleph ) ( 4 I _ \\aleph - C _ \\aleph ) ^ { - 1 } L ( \\eta _ 1 , \\cdots , \\eta _ g ) . \\end{align*}"}
+{"id": "1036.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\dot { X } & = \\frac { 1 } { \\tau } ( Z - X ) , \\\\ \\dot { Z } & = - \\nabla f ( X ) \\end{aligned} \\right . \\end{align*}"}
+{"id": "1739.png", "formula": "\\begin{align*} \\sum _ { M > M _ 0 } \\sigma _ M = \\delta \\ , , \\end{align*}"}
+{"id": "7975.png", "formula": "\\begin{align*} \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle = | \\xi ^ H | ^ 2 k ( \\xi ) \\quad \\mbox { a n d } \\left \\langle \\eta , \\xi ^ H \\right \\rangle = | \\xi ^ H | ^ 2 \\alpha ( \\xi ) \\qquad \\mbox { f o r } \\xi \\in M \\smallsetminus S _ M . \\end{align*}"}
+{"id": "3963.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( Y _ { \\beta } ( t ) \\right ) & = \\frac { t ^ { \\beta } } { \\Gamma ( \\beta + 1 ) } , \\\\ \\operatorname { V a r } \\left ( Y _ { \\beta } ( t ) \\right ) & = \\left ( \\frac { 2 } { \\Gamma ( 2 \\beta + 1 ) } - \\frac { 1 } { \\Gamma ^ { 2 } ( \\beta + 1 ) } \\right ) t ^ { 2 \\beta } . \\end{align*}"}
+{"id": "2544.png", "formula": "\\begin{align*} \\mu _ t ^ { * , \\xi } = \\mathbb E [ \\alpha ^ { * , \\xi } _ t | \\mathcal { F } _ t ^ { W ^ 0 } ] \\quad \\nu _ t ^ { * , \\xi } = \\mathbb E [ x _ t ^ { \\xi , \\alpha ^ { * , \\xi } } | \\mathcal { F } _ t ^ { W ^ 0 } ] . \\end{align*}"}
+{"id": "7491.png", "formula": "\\begin{align*} \\sigma _ { a } ( n ) = \\frac { ( n + a - 1 ) ! } { ( n - 1 ) ! \\ , a ! } = B _ { a , n - 1 } \\ , . \\end{align*}"}
+{"id": "4456.png", "formula": "\\begin{align*} q = \\frac { p } { p - 1 } \\le \\frac { d p } { 2 d - 2 p } , \\end{align*}"}
+{"id": "4011.png", "formula": "\\begin{align*} \\bar { \\mathcal { M } } _ { \\beta } ( t ) \\overset { d } { = } \\bar { \\mathcal { M } } ( T _ { 2 \\beta } ( t ) ) , \\ t > 0 , \\end{align*}"}
+{"id": "6082.png", "formula": "\\begin{align*} \\tau ( x u ^ i ) = _ { k n } \\circ \\pi _ { I , n } ( x ) x \\in A - n < i < n . \\end{align*}"}
+{"id": "2785.png", "formula": "\\begin{align*} J _ k = \\begin{bmatrix} \\alpha _ 1 & \\beta _ 1 \\\\ & \\alpha _ 2 & \\beta _ 2 \\\\ & & \\ddots & \\ddots \\\\ & & & \\alpha _ { k - 1 } & \\beta _ { k - 1 } \\\\ & & & & \\alpha _ k \\\\ \\end{bmatrix} . \\end{align*}"}
+{"id": "6114.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal S _ 3 ( k , 3 ) = & \\left \\{ F \\in \\binom { [ n ] } { k } : [ 2 ] \\subseteq F , F \\cap [ 3 , k - 1 ] \\neq \\emptyset \\right \\} \\cup \\left \\{ F \\in \\binom { [ n ] } { k } : [ 2 ] \\cup [ k , k + 3 ] \\subseteq F \\right \\} \\\\ & \\cup \\{ [ 2 , k ] \\cup \\{ k + 2 \\} , \\ , [ 2 , k + 2 ] \\setminus \\{ k \\} , \\ , [ 2 , k - 1 ] \\cup \\{ k + 1 , k + 3 \\} \\} \\\\ & \\cup \\{ [ 3 , k + 1 ] \\cup \\{ 1 \\} , \\ , [ 3 , k - 1 ] \\cup \\{ 1 , k + 1 , k + 2 \\} , \\ , [ 3 , k - 1 ] \\cup \\{ 1 , k + 2 , k + 3 \\} \\} . \\end{aligned} \\end{align*}"}
+{"id": "2476.png", "formula": "\\begin{align*} \\norm { \\mu _ { k , i } } _ { W ^ { - 1 , p } ( \\Omega _ i ) } \\lesssim \\norm { \\mu _ { k , i } } _ { W ^ { - 1 , 1 } ( \\Omega _ i ) } ^ \\alpha \\abs { \\mu _ { k , i } } ( \\Omega _ i ) ^ { 1 - \\alpha } ( k = 1 , 2 , \\dots d ) . \\end{align*}"}
+{"id": "1217.png", "formula": "\\begin{align*} \\pi _ { 2 } \\circ \\pi _ { 1 } ( x ^ { ( f ) } _ i ) & = x ^ { ( f ) } _ { \\pi _ { 1 } ( \\pi _ 2 ( i ) ) } = x ^ { ( T _ 1 ( f ) ) } _ { \\pi _ 2 ( i ) } = x ^ { ( g ) } _ { \\pi _ 2 ( i ) } = \\\\ & = x _ i ^ { ( T _ 2 ( g ) ) } = x _ i ^ { ( T _ 2 \\circ T _ 1 ( f ) ) } . \\end{align*}"}
+{"id": "1680.png", "formula": "\\begin{align*} \\mathbb { P } _ { ( ( W _ 1 , v _ 1 ) , \\dots , ( W _ n , v _ n ) ) = ( x _ 1 , \\dots , x _ n ) } \\left ( \\aleph _ N - \\aleph _ n \\geq 2 ^ { - \\frac { 5 } { 2 } } \\lambda ^ { - 1 } \\right ) = \\mathbb { P } _ { ( W _ 0 , v _ 0 ) = x _ t } \\left ( \\aleph _ { N - n } - \\aleph _ 0 \\geq 2 ^ { - \\frac { 5 } { 2 } } \\lambda ^ { - 1 } \\right ) \\end{align*}"}
+{"id": "933.png", "formula": "\\begin{align*} \\begin{aligned} | u _ { \\tilde { \\nu } ( y ' ) } ( y ) | \\ , & \\leq | u _ { \\tilde { \\nu } ( 0 ) } ( 0 ) | \\\\ & + \\sum _ { \\beta = \\alpha + 1 } ^ { n - 1 } \\sup _ { \\xi \\in \\partial \\omega \\cap \\partial \\Omega } \\left ( | \\nabla _ { \\tilde { \\nu } \\tau _ \\beta } u ( \\xi ) | + \\sum _ { \\gamma = 1 } ^ { n - 1 } | \\rho _ { \\gamma \\beta } ( \\xi ) u _ \\gamma ( \\xi ) | \\right ) \\cdot | y _ \\beta | \\leq C b _ \\alpha \\end{aligned} \\end{align*}"}
+{"id": "3044.png", "formula": "\\begin{align*} \\Gamma _ { a , d } = ( x _ i ) _ { i \\in \\Z } = ( \\ldots , a + 4 d , a + 2 d , a , a + d , a + 3 d , a + 5 d , \\ldots ) . \\end{align*}"}
+{"id": "2258.png", "formula": "\\begin{align*} \\epsilon _ 0 d _ 0 + \\dots + \\epsilon _ q d _ q = \\epsilon ' _ 0 d _ 0 + \\dots + \\epsilon ' _ q d _ q \\end{align*}"}
+{"id": "8032.png", "formula": "\\begin{align*} T _ Q = T _ F ^ * T _ F \\end{align*}"}
+{"id": "533.png", "formula": "\\begin{align*} S _ i : = \\sup _ { Q _ i \\in \\P ( \\R ) } \\left ( \\int _ { \\R } \\hat { f } _ i \\ , d Q _ i - H ( Q _ i ) \\right ) . \\end{align*}"}
+{"id": "7666.png", "formula": "\\begin{align*} p _ 1 H ( p _ 1 ) V ^ { p _ 1 - 1 } = p _ 2 H ( p _ 2 ) V ^ { p _ 2 - 1 } , \\end{align*}"}
+{"id": "4515.png", "formula": "\\begin{align*} \\{ \\Phi _ x , \\mathcal { K } \\} ( \\phi , \\pi ) & = - \\dd _ { ( \\phi , \\pi ) } \\mathcal { K } ( \\mathbb { X } _ { \\Phi _ { \\ ! x } } ) = - \\big ( \\dd _ \\phi K \\circ \\mathrm { p r o j } _ 1 \\big ) ( \\mathbb { X } _ { \\Phi _ { \\ ! x } } \\ ! ) \\\\ & = - \\dd _ \\phi K ( X _ 1 ) = - \\dd _ \\phi K ( 0 ) = 0 \\ , . \\end{align*}"}
+{"id": "2081.png", "formula": "\\begin{align*} f ( y , x _ 1 , \\ldots , x _ p ) = \\Big ( y , \\ \\textstyle \\sum _ { j = 1 } ^ p t ^ { j - 1 } x _ j ^ p \\Big ) . \\end{align*}"}
+{"id": "916.png", "formula": "\\begin{align*} F ( r ) = \\sigma _ k ^ { 1 / k } ( \\lambda ( r ) ) \\mbox { w i t h } \\lambda ( r ) \\in \\Gamma _ k \\end{align*}"}
+{"id": "3467.png", "formula": "\\begin{align*} \\mathbb { I } \\Big [ \\psi \\Big ] ( \\mathrm { x } , t ) = \\int _ { \\Omega } \\Phi ( x , t ; y ) \\psi ( y ) d \\mathrm { y } . \\end{align*}"}
+{"id": "20.png", "formula": "\\begin{align*} T _ 0 & = 1 , \\\\ T _ 1 & = 1 + x , \\\\ T _ 2 & = 1 + 3 x + x ^ 2 , \\\\ T _ 3 & = 1 + 6 x + 5 x ^ 2 + x ^ 3 , \\\\ T _ 4 & = 1 + 1 0 x + 1 5 x ^ 2 + 7 x ^ 3 + x ^ 4 , \\\\ T _ 5 & = 1 + 1 5 x + 3 5 x ^ 2 + 2 8 x ^ 3 + 9 x ^ 4 + x ^ 5 . \\end{align*}"}
+{"id": "3741.png", "formula": "\\begin{align*} \\begin{bmatrix} n _ { 1 1 } & n _ { 1 2 } & n _ { 1 3 } \\\\ n _ { 2 1 } & n _ { 2 2 } & n _ { 2 3 } \\\\ n _ { 3 1 } & n _ { 3 2 } & n _ { 3 3 } \\end{bmatrix} \\begin{bmatrix} 1 & 1 & 1 \\\\ 1 & 1 & 1 \\\\ 1 & 1 & 1 \\end{bmatrix} & = \\begin{bmatrix} 2 2 & 2 2 & 2 2 \\\\ 2 2 & 2 2 & 2 2 \\\\ 2 2 & 2 2 & 2 2 \\end{bmatrix} \\end{align*}"}
+{"id": "7594.png", "formula": "\\begin{align*} \\Delta _ p \\tilde { G } ( p , q ) = \\delta _ q ( p ) - \\operatorname { a r e a } ( X ) ^ { - 1 } , \\end{align*}"}
+{"id": "6765.png", "formula": "\\begin{align*} \\mathcal G ^ { \\leq N } _ { \\perp \\psi _ t } = \\Big ( \\bigoplus _ { k = 0 } ^ N \\mathcal F _ { b , \\psi _ t } ^ { ( k ) } \\Big ) \\otimes \\mathcal F _ a = \\bigoplus _ { k = 0 } ^ N \\Big ( \\mathcal F _ { b , \\psi _ t } ^ { ( k ) } \\otimes \\mathcal F _ a \\Big ) \\ ; \\subset \\ ; \\mathcal G _ { \\perp \\psi _ t } = \\mathcal { F } _ { b , \\psi _ t } \\otimes \\mathcal { F } _ { a } \\ ; \\subset \\ ; \\mathcal { G } = \\mathcal { F } _ b \\otimes \\mathcal { F } _ a \\end{align*}"}
+{"id": "204.png", "formula": "\\begin{align*} & \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ { N } } \\omega \\varphi d x d t + \\int _ { \\mathbb { R } ^ { N } } u _ 0 ( \\varphi ( 0 , x ) - k \\Delta \\varphi ( 0 , x ) ) d x \\leq C ( p ) \\biggr ( \\mathcal { I } _ 1 + k \\mathcal { I } _ 2 + \\mathcal { I } _ 3 \\biggr ) , \\end{align*}"}
+{"id": "1500.png", "formula": "\\begin{align*} \\alpha = ( \\log w ) / \\log \\Delta . \\end{align*}"}
+{"id": "7950.png", "formula": "\\begin{align*} \\delta _ R ( \\xi ) = ( R x , R y , R ^ 2 t ) , R > 0 , \\end{align*}"}
+{"id": "4279.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } n p ( n ) q ^ n & = \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { n q ^ n } { 1 - q ^ n } . \\end{align*}"}
+{"id": "6428.png", "formula": "\\begin{align*} \\mathcal { L } ^ { ( \\alpha ) } _ n ( x ) = \\frac { ( q ^ { \\alpha + 1 } ; q ) _ n } { ( q ; q ) _ n } \\sum _ { k = 0 } ^ n \\begin{bmatrix} n \\\\ k \\end{bmatrix} \\frac { q ^ { k ^ 2 + k \\alpha } } { ( q ^ { \\alpha + 1 } ; q ) _ k } ( - x ) ^ k . \\end{align*}"}
+{"id": "6086.png", "formula": "\\begin{align*} \\phi ( x z ) = \\phi ( x ) \\phi ( z ) x \\in \\mathbb { H } _ 3 ( \\mathbb { Z } [ \\tfrac { 1 } { 2 } ] ) , z \\in Z ( \\mathbb { H } _ 3 ( \\mathbb { Z } [ \\tfrac { 1 } { 2 } ] ) ) . \\end{align*}"}
+{"id": "4618.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } \\phi _ + ' ( t ) = \\phi _ + ' ( 0 ) = 0 . \\end{align*}"}
+{"id": "4959.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { y } _ { k } & = \\sum _ { u = 1 } ^ { N } a _ { u } [ \\mathbf { x } _ { u } ^ { c } [ k ] , \\mathbf { x } _ { u } ^ { c } [ k + K \\cdot 1 ] , \\cdots , \\mathbf { x } _ { u } ^ { c } [ k + K ( L - 1 ) ] ] ^ { T } + \\mathbf { n } _ { k } \\\\ & = \\mathbf { S } \\mathbf { X } _ { k } \\mathbf { a } + \\mathbf { n } _ { k } , \\end{aligned} \\end{align*}"}
+{"id": "7762.png", "formula": "\\begin{align*} \\lambda \\sqrt { V } | _ { t = 0 } = 0 , \\ \\ \\frac { d } { d t } | _ { t = 0 } ( \\lambda \\sqrt { V } ) = 1 \\end{align*}"}
+{"id": "1875.png", "formula": "\\begin{align*} f ( 2 x + y ) + f ( 2 x - y ) = \\frac { 2 f ( x ) f ( y ) \\displaystyle { \\sum _ { \\substack { k = 0 \\\\ } } ^ { l } 2 ^ { l - k } \\binom { l } { k } f ( x ) ^ { \\frac { k } { l } } f ( y ) ^ { \\frac { l - k } { l } } } } { \\left ( 4 f ( y ) ^ { \\frac { 2 } { l } } - f ( x ) ^ { \\frac { 2 } { l } } \\right ) ^ l } \\end{align*}"}
+{"id": "6219.png", "formula": "\\begin{align*} H = - \\partial _ x ^ 2 + ( - i \\partial _ y + B x ) ^ 2 - \\partial _ z ^ 2 \\end{align*}"}
+{"id": "7766.png", "formula": "\\begin{align*} c a p _ p ( F , \\Omega ) = \\inf \\limits _ { u } \\int _ M | \\nabla u | ^ p d V _ g \\end{align*}"}
+{"id": "509.png", "formula": "\\begin{align*} y ^ * = t \\int _ { \\R ^ n } \\nabla f \\ , d \\gamma _ { y ^ * , t } , \\end{align*}"}
+{"id": "2196.png", "formula": "\\begin{align*} & \\Gamma _ { 1 1 } ^ 1 = \\frac { a } { E } , \\ ; \\ ; \\ ; \\Gamma _ { 1 1 } ^ 2 = - b , \\ ; \\ ; \\ ; \\Gamma _ { 1 2 } ^ 1 = \\frac { b } { E } , \\ ; \\ ; \\ ; \\Gamma _ { 1 2 } ^ 2 = \\Gamma _ { 2 2 } ^ 1 = \\Gamma _ { 2 2 } ^ 2 = 0 \\end{align*}"}
+{"id": "5656.png", "formula": "\\begin{align*} \\Box ^ \\perp r : = \\Box r - H ( r ) ( \\nabla r , \\nabla r ) - H ( r ) ( J ( \\nabla r ) , J ( \\nabla r ) ) . \\end{align*}"}
+{"id": "2462.png", "formula": "\\begin{align*} \\mathcal { F } : = \\left \\{ \\bar { B } _ r ( x _ 0 ) \\colon 0 < r < \\delta , \\ x _ 0 \\in M \\ \\textrm { s u c h t h a t } \\ \\abs { J _ * } ( \\partial B _ r ( x _ 0 ) ) = 0 \\right \\} \\end{align*}"}
+{"id": "6902.png", "formula": "\\begin{align*} \\begin{cases} ( i \\dd _ t + \\Delta ) v _ n = g _ n ( F _ n + v _ n ) \\\\ v _ n ( t _ 0 ) = v _ { n , 0 } \\in H ^ 2 ( \\R ^ d ) \\end{cases} \\end{align*}"}
+{"id": "4635.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla u _ \\lambda | ^ p \\ , d x & \\lesssim \\int _ \\Omega \\phi _ \\lambda ( x , | \\nabla u _ \\lambda | ) + 1 \\ , d x \\le \\int _ \\Omega \\phi _ \\lambda ( x , | \\nabla f | ) + 1 \\ , d x \\lesssim \\int _ \\Omega \\phi ( x , | \\nabla f | ) + 1 \\ , d x < \\infty , \\end{align*}"}
+{"id": "5693.png", "formula": "\\begin{gather*} \\tau ( j ) = \\begin{cases} \\sigma ( 1 ) & ( j = 1 ) \\\\ j - 1 & ( 2 \\le j \\le \\sigma ( 1 ) ) \\\\ j & ( \\sigma ( 1 ) + 1 \\le j \\le i + 1 ) . \\end{cases} \\end{gather*}"}
+{"id": "4644.png", "formula": "\\begin{align*} R ( z ) = ( \\mathcal { H } _ 0 - z ) ^ { - 1 } + G ( z ) \\Gamma ( z ) ^ { - 1 } G ( \\bar { z } ) ^ \\ast \\ , \\end{align*}"}
+{"id": "5916.png", "formula": "\\begin{align*} q ( v ) \\bigl ( w \\cdot w ^ \\prime \\bigr ) = \\epsilon ( v \\wedge w \\wedge w ' ) , \\end{align*}"}
+{"id": "2447.png", "formula": "\\begin{align*} I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( 1 , a , b ) + I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a , 1 , b ) + I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a , b , 1 ) - I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a + b + 1 ) = 0 \\end{align*}"}
+{"id": "7399.png", "formula": "\\begin{align*} \\big | E _ n ( \\alpha _ \\pm ) + 4 a ^ 2 + 2 k _ \\pm - \\mu _ n ( H ) \\big | \\leq C _ 4 \\big | a ^ { - 1 } \\ln | a | \\big | \\end{align*}"}
+{"id": "7959.png", "formula": "\\begin{align*} \\eta ( t ) = - 2 \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle \\end{align*}"}
+{"id": "2632.png", "formula": "\\begin{align*} E _ w ^ 0 & = \\{ z \\in B S ( 2 , 1 ) ^ + : z \\le w \\} , & E _ w ^ 1 & = \\{ ( z , z a ) : z , z a \\in E _ w ^ 0 \\} \\bigcup \\{ ( z , z b ) : z , z b \\in E _ w ^ 0 \\} \\\\ & & & = \\{ ( z , z l ) : z , z l \\in E _ w ^ 0 l = a , b \\} \\end{align*}"}
+{"id": "851.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mu = \\left ( - \\frac { 1 } { a ' k _ 2 \\sigma } \\right ) \\ln \\left ( \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) > 0 , \\\\ & \\varrho = \\left [ \\frac { 1 } { k _ 2 \\sigma } \\ln \\frac { K k _ 2 } { k _ 2 - 1 } \\right ] ^ { 2 } > 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2650.png", "formula": "\\begin{align*} & s ( w _ 1 w _ 2 ) = \\star = s ( w _ 2 ) \\\\ & r ( w _ 1 w _ 2 ) = \\star = r ( w _ 1 ) \\end{align*}"}
+{"id": "7113.png", "formula": "\\begin{align*} \\sum _ { d \\geq 0 } p _ 2 ( d ) q ^ d & = \\prod _ { n \\geq 1 } \\frac { 1 } { 1 - q ^ n } , \\\\ \\sum _ { d \\geq 0 } p _ 3 ( d ) q ^ d & = M ( q ) : = \\prod _ { n \\geq 1 } \\frac { 1 } { \\left ( 1 - q ^ n \\right ) ^ n } . \\end{align*}"}
+{"id": "797.png", "formula": "\\begin{align*} \\widetilde { \\theta } _ 1 = \\widetilde { \\theta } _ 1 ( N , \\mathcal { T } , \\{ x _ j \\} _ { j = 1 } ^ { N } ) : = \\theta + \\frac { 1 - \\theta } { l _ 1 + l _ 2 \\left ( \\frac { 1 } { N } \\sum \\limits _ { j = 1 } ^ N e ^ { - \\beta \\tau _ j } x _ j ( \\tau _ j ) \\right ) } . \\end{align*}"}
+{"id": "3170.png", "formula": "\\begin{align*} \\phi : \\begin{cases} u \\mapsto u \\\\ r \\mapsto s \\\\ s \\mapsto t \\\\ t \\mapsto r s t s ^ { - 1 } t ^ { - 1 } \\end{cases} \\end{align*}"}
+{"id": "1003.png", "formula": "\\begin{align*} \\widehat { \\mathbf { Q } } = \\begin{footnotesize} \\begin{pmatrix} \\widehat { \\mathbf { Q } } _ 1 & 0 \\\\ 0 & \\widehat { \\mathbf { Q } } _ 2 \\end{pmatrix} \\end{footnotesize} \\end{align*}"}
+{"id": "1815.png", "formula": "\\begin{align*} [ r ( x ^ { * } ) , r ( y ^ { * } ) ] = r ( a d ^ { * } r ( x ^ { * } ) ( y ^ { * } ) - a d ^ { * } r ( y ^ { * } ) ( x ^ { * } ) ) . \\end{align*}"}
+{"id": "405.png", "formula": "\\begin{align*} V _ { t r i v , e v e n } & = \\langle v _ 0 + v _ 2 + v _ 4 + v _ 6 \\rangle \\\\ V _ { t r i v , o d d } & = \\langle v _ 1 + v _ 3 + v _ 5 + v _ 7 \\rangle \\\\ V _ { r e f , e v e n } & = \\langle v _ 0 - v _ 2 + v _ 4 - v _ 6 \\rangle \\\\ V _ { r e f , o d d } & = \\langle v _ 1 - v _ 3 + v _ 5 - v _ 7 \\rangle \\\\ V _ { 1 , e v e n } & = \\langle v _ 0 - v _ 4 , v _ 2 - v _ 6 \\rangle \\\\ V _ { 1 , o d d } & = \\langle v _ 1 - v _ 5 , v _ 3 - v _ 7 \\rangle . \\end{align*}"}
+{"id": "4773.png", "formula": "\\begin{gather*} ( a \\diamond b - b \\diamond a ) \\ast c = a \\diamond ( b \\ast c ) - b \\ast ( a \\diamond c ) , ( a \\ast b + b \\ast a ) \\diamond c = a \\ast ( b \\diamond c ) + b \\ast ( a \\diamond c ) , \\end{gather*}"}
+{"id": "2691.png", "formula": "\\begin{align*} M _ Q ( n , j , 0 ; a ) & = \\binom { n - j } { j } \\sum _ { k = j } ^ { n - j } \\binom { n - 2 j } { k - j } \\binom { n } { k } ^ { 0 } H ( n , k , a ) \\\\ & = \\sum _ { k = j } ^ { n - j } \\binom { n - j } { j } \\binom { n - 2 j } { k - j } \\binom { a + k } { a } \\binom { a + n - k } { a } G ( n , k , a ) \\end{align*}"}
+{"id": "1438.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left [ \\frac { 1 } { 2 } \\left ( \\| \\partial _ t u ^ { ( j ) } ( t ) \\| _ { L ^ 2 } ^ 2 + \\| \\nabla u ^ { ( j ) } ( t ) \\| _ { L ^ 2 } ^ 2 \\right ) + \\frac { 1 } { p + 1 } \\| u ^ { ( j ) } ( t ) \\| _ { L ^ { p + 1 } } ^ { p + 1 } \\right ] & = - \\| \\partial _ t u ^ { ( j ) } ( t ) \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "1546.png", "formula": "\\begin{align*} & \\varphi _ { s , j , i } : \\Omega _ { b _ { j } } \\times \\Omega \\rightarrow \\Omega _ { b _ { i } } \\times \\Omega \\\\ & \\varphi _ { s , j , i } \\left ( \\omega _ { j } , \\omega \\right ) = \\left ( f _ { s , j , i } \\left ( \\omega _ { j , } X _ { s _ { i } } \\left ( \\omega \\right ) \\right ) , \\omega \\right ) . \\end{align*}"}
+{"id": "4332.png", "formula": "\\begin{align*} \\pi _ { 1 * } \\pi _ 2 ^ * ( \\psi _ j ^ * ) = \\sum a _ j ^ i \\psi _ i ^ * . \\end{align*}"}
+{"id": "1948.png", "formula": "\\begin{align*} & N _ 0 ^ { - 1 } D _ x ( - \\Delta _ x ) ^ { - s } u ( x ) = \\int D _ x u ( x - y ) \\frac { 1 } { | y | ^ { d - 2 s } } \\ , d y \\\\ & = - \\lim _ { \\varepsilon \\downarrow 0 } \\int _ { | y | > \\varepsilon } D _ y u ( x - y ) \\frac { 1 } { | y | ^ { d - 2 s } } \\ , d y = - ( d - 2 s ) \\lim _ { \\varepsilon \\downarrow 0 } \\int _ { | y | > \\varepsilon } u ( x - y ) \\frac { y } { | y | ^ { d - 2 s + 2 } } \\ , d y , \\end{align*}"}
+{"id": "5319.png", "formula": "\\begin{align*} \\beta ( v _ { a } \\otimes v _ { b } ) = \\begin{cases} q v _ { a } \\otimes v _ { b } , & ; \\\\ v _ { b } \\otimes v _ { a } , & ; \\\\ v _ { b } \\otimes v _ { a } + ( q - q ^ { - 1 } ) v _ { a } \\otimes v _ { b } , & . \\end{cases} \\end{align*}"}
+{"id": "1010.png", "formula": "\\begin{align*} \\mu ^ * + \\varkappa & = \\mu ^ { \\rm N } + \\varkappa ^ { \\rm N } , \\mu ^ * = \\mu ^ { \\rm N } - \\varkappa ^ { \\rm N } , \\lambda = \\lambda ^ { \\rm N } , \\\\ \\beta & = \\gamma ^ { \\rm N } - \\beta ^ { \\rm N } , \\ \\ \\gamma = \\gamma ^ { \\rm N } + \\beta ^ { \\rm N } , \\qquad \\alpha = \\alpha ^ { \\rm N } \\ , , \\end{align*}"}
+{"id": "5622.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\hat { \\mathbb { G } } ^ { \\alpha } _ k u ^ k = - 2 \\hat { \\mathbb { G } } ^ { \\alpha + n } & & \\hat { \\mathbb { G } } ^ { \\alpha + n } _ k u ^ k = 2 \\hat { \\mathbb { G } } ^ { \\alpha } . \\end{array} \\end{align*}"}
+{"id": "1689.png", "formula": "\\begin{align*} i \\partial _ t \\varphi ( x ) = ( - \\Delta + \\kappa ) \\varphi ( x ) + \\int d y \\ , | \\varphi ( y ) | ^ 2 \\ , w ( x - y ) \\varphi ( x ) \\ , . \\end{align*}"}
+{"id": "1660.png", "formula": "\\begin{align*} \\mathbf { A } _ { \\mathfrak { d } } ( W , v , \\mathtt { P } ) = \\operatorname { t r } \\left ( \\hat { \\Psi } ^ * \\left [ \\mathtt { X } ( W + v v ^ * , \\mathtt { P } ) - \\mathtt { X } ( W , \\mathtt { P } ) \\right ] \\hat { \\Psi } \\right ) \\ , , \\end{align*}"}
+{"id": "5384.png", "formula": "\\begin{align*} \\rho ( \\xi ) A = d _ A \\xi . \\end{align*}"}
+{"id": "4952.png", "formula": "\\begin{align*} \\{ x , y \\cdot z \\} = \\{ x , y \\} \\cdot z + y \\cdot \\{ x , z \\} - \\{ x , 1 \\} \\cdot y \\cdot z . \\end{align*}"}
+{"id": "5266.png", "formula": "\\begin{align*} c _ p ( t _ 1 , \\ldots , t _ p ) = \\sum _ { m = 1 } ^ p \\frac { ( - 1 ) ^ { m } } { m } \\ * \\sum _ { \\substack { ( p _ 1 , \\ldots , p _ m ) : \\\\ p _ 1 + \\ldots + p _ m = p , \\ p _ 1 , \\ldots p _ m \\geq 1 } } \\frac { 1 } { p _ 1 ! \\cdots p _ m ! } \\ * \\sum _ { \\sigma \\in S _ p } \\ * j ( p _ 1 , \\ldots , p _ m ; t _ { \\sigma ( 1 ) } , \\ldots , t _ { \\sigma ( p ) } ) , \\end{align*}"}
+{"id": "4553.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 4 n _ i = n , \\end{align*}"}
+{"id": "232.png", "formula": "\\begin{align*} \\log m _ t ^ { ( n ) } ( x ) & = e ^ { - \\frac { \\sigma ^ 2 } { 2 } t } \\log m _ 0 ( x ) \\\\ & - \\int _ 0 ^ t \\frac { \\sigma ^ 2 } { 2 } e ^ { - \\frac { \\sigma ^ 2 } { 2 } ( t - s ) } \\left ( \\frac { 2 } { \\sigma ^ 2 } \\frac { \\delta F } { \\delta m } ( m _ s ^ { ( n - 1 ) } , x ) - \\log \\pi ( x ) - \\operatorname { K L } ( m _ s ^ { ( n - 1 ) } | \\pi ) \\right ) d s \\ , . \\end{align*}"}
+{"id": "2154.png", "formula": "\\begin{align*} \\overline { \\alpha } ( X , Y ) : = ( \\overline { \\nabla } _ X Y ) ^ { \\perp } , \\end{align*}"}
+{"id": "931.png", "formula": "\\begin{align*} \\partial \\omega = \\partial _ 1 \\omega \\cup \\partial _ 2 \\omega \\cup \\partial _ 3 \\omega , \\end{align*}"}
+{"id": "2599.png", "formula": "\\begin{align*} \\mathbb { E } \\Big [ \\underset { t _ 0 \\leq t \\leq T } { \\sup } | \\phi ^ { N , i } ( t , \\boldsymbol { x } _ t ^ * ) - \\Psi ^ { N , i } ( t , \\boldsymbol { x } _ t ^ * ) | ^ 2 \\Big ] \\leq \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } \\mathbb { E } [ | \\xi ^ i - \\xi ^ j | ^ 2 ] + \\frac { 1 } { N ^ 2 } \\underset { i , j = 1 } { \\overset { N } { \\sum } } \\mathbb { E } [ | \\xi ^ { i } - \\xi ^ { j } | ^ 2 ] \\Big ) . \\end{align*}"}
+{"id": "7538.png", "formula": "\\begin{align*} \\mathcal T _ { \\delta } = \\left \\{ \\gamma = \\gamma _ 1 \\cup \\cdots \\cup \\gamma _ n \\mid \\begin{array} { l l } n \\in \\mathbb N , \\gamma _ j \\in \\mathcal S _ { \\delta } \\gamma \\\\ X \\setminus \\{ p _ 0 , p _ { \\infty } \\} U _ { \\delta } ( p _ 0 ) \\end{array} \\right \\} . \\end{align*}"}
+{"id": "6795.png", "formula": "\\begin{align*} \\Big ( \\alpha '' ( x _ 1 , x _ 2 , x _ { [ 3 , k ] } ) + \\alpha '' ( x _ 2 , x _ 1 , x _ { [ 3 , k ] } ) \\Big ) + \\Big ( \\alpha '' ( x _ 1 , x _ 2 , x _ { [ 3 , k ] } ) + \\alpha '' ( x _ 2 , x _ 1 , x _ { [ 3 , k ] } ) \\Big ) \\circ ( 1 \\ , \\ , 2 ) = 0 \\end{align*}"}
+{"id": "2378.png", "formula": "\\begin{align*} w _ { 1 } = q _ { 1 } + \\sum _ { s \\in S } w _ { 1 , s } e _ { s } , \\ w _ { 2 } = q _ { 2 } + \\sum _ { s \\in S } e _ { s } w _ { 2 , s } \\ \\ \\ \\ ( q _ { 1 } , q _ { 2 } \\in \\mathbb { Q } ) . \\end{align*}"}
+{"id": "4802.png", "formula": "\\begin{align*} \\mathcal C _ i ( \\phi _ i ( n , k ) U _ { n , k } ) = \\phi _ { i + 1 } ( n , k ) U _ { n , k } , \\mathcal C _ i ( \\psi _ i ( n , k ) U _ { n , k } ) = \\psi _ { i + 1 } ( n , k ) U _ { n , k } . \\end{align*}"}
+{"id": "3054.png", "formula": "\\begin{align*} f ( \\ell ) = \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } \\mu \\left ( \\frac { \\ell } { k } \\right ) g ( k ) . \\end{align*}"}
+{"id": "2861.png", "formula": "\\begin{align*} Y = \\bigsqcup _ { s \\in \\mathcal { S } } Y _ s \\end{align*}"}
+{"id": "5887.png", "formula": "\\begin{align*} \\begin{aligned} & \\alpha _ { \\tau \\sigma } ( h M ) = \\frac { \\tau - \\sigma } { 2 } \\varphi _ { 1 } ( ( \\tau - \\sigma ) h M ) , \\ \\ \\beta _ { \\tau } ( h M ) = ( 1 - \\tau ) \\varphi _ { 1 } ( ( 1 - \\tau ) h M ) , \\ \\ \\gamma _ { \\tau } ( h M ) = \\varphi _ { 0 } ( ( 1 - \\tau ) h M ) . \\end{aligned} \\end{align*}"}
+{"id": "4272.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { j } \\frac { ( d q ) ^ k ( q / d ) _ { k - 1 } ( d q ) _ { j - k } } { ( q ) _ k ( q ) _ { j - k } ( 1 - q ^ k ) } = \\frac { ( d q ) _ j } { ( 1 - 1 / d ) ( q ) _ j } \\left ( \\sum _ { k = 1 } ^ { j } \\frac { d q ^ k } { 1 - d q ^ k } - \\sum _ { k = 1 } ^ { j } \\frac { q ^ k } { 1 - q ^ k } \\right ) . \\end{align*}"}
+{"id": "8190.png", "formula": "\\begin{align*} \\Vert ( \\rho _ 0 - \\bar { \\rho } , u _ 0 ) \\Vert _ { \\dot B ^ { - s _ 0 } _ { 2 , \\infty } } ^ \\ell < \\varepsilon , s _ 0 = N \\left ( \\tfrac { 2 } { p } - \\tfrac { 1 } { 2 } \\right ) , \\end{align*}"}
+{"id": "3388.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\ell r _ j s _ { h - g _ j } = 0 \\end{align*}"}
+{"id": "7844.png", "formula": "\\begin{align*} \\widetilde { \\mathfrak { B } } ^ * _ { \\mbox { \\tiny { H G S } } } = \\{ \\mu ^ 1 , . . . , \\mu ^ k \\} , \\end{align*}"}
+{"id": "4587.png", "formula": "\\begin{align*} \\ln F ( n _ 0 + ( m + 1 ) & N ) ^ 2 = \\ln F ( n _ 0 + m N ) ^ 2 \\\\ - & \\frac { K } { 2 ( n _ 0 + m N - b ) \\sin \\pi k } ( 1 - \\cos 2 \\pi \\varphi ( n _ 0 + m N ) + \\delta ( m ) ) \\end{align*}"}
+{"id": "3865.png", "formula": "\\begin{align*} \\sum _ { p + q + 1 = 3 } | L ^ { z } _ { p + 1 , q } | = O _ { \\prec } \\big ( n ^ { - 1 / 2 } \\Psi ^ 2 + n ^ { - 1 } \\big ) . \\end{align*}"}
+{"id": "2750.png", "formula": "\\begin{align*} B = 1 , h ^ { 1 , 1 } ( S ) = b _ 2 ( S ) = 2 { \\rm a n d } A = q ( S ) = 0 . \\end{align*}"}
+{"id": "3645.png", "formula": "\\begin{align*} \\Upsilon _ \\phi ( u ) = { \\sum } _ { k , l = 1 } ^ n \\phi ( u _ { k , l } ) _ { k , l } = \\beta _ e \\phi ^ { ( n ) } ( u ) \\beta _ e ^ * ( u \\in M _ n ( X ) ) . \\end{align*}"}
+{"id": "6370.png", "formula": "\\begin{align*} D ^ L f ( \\mu ) = ( D ^ L f _ i ( \\mu ) ) \\ \\ \\mathrm { o r } \\ \\ D ^ L f ( \\mu ) = ( D ^ L f _ { i j } ( \\mu ) ) . \\end{align*}"}
+{"id": "3586.png", "formula": "\\begin{align*} \\theta _ { m i } = \\langle \\psi _ m , \\phi _ i \\rangle & = \\int \\cos ( \\omega _ m \\pi x ) \\cos ( i \\pi x ) d x \\\\ & = \\frac { i \\pi } { ( i \\pi ) ^ 2 - \\omega _ m ^ 2 \\pi ^ 2 } \\cos ( \\pi \\omega _ m ) \\sin ( i \\pi ) - \\frac { \\omega _ m \\pi } { ( i \\pi ) ^ 2 - \\omega _ m ^ 2 \\pi ^ 2 } \\sin ( \\pi \\omega _ m ) \\cos ( i \\pi ) \\\\ & = \\frac { \\pi \\omega _ m } { \\pi ^ 2 \\omega _ m ^ 2 - ( i \\pi ) ^ 2 } \\sin ( \\pi \\omega _ m ) ( - 1 ) ^ i . \\end{align*}"}
+{"id": "1572.png", "formula": "\\begin{align*} \\left ( z _ { 4 } s z _ { 1 } s \\right ) ^ { - 1 } z _ { 4 } \\left ( s z _ { 1 } s \\right ) & = e \\mbox { o r } \\\\ z _ { 4 } \\left ( z _ { 4 } s z _ { 1 } s \\right ) ^ { - 1 } \\left ( s z _ { 1 } z \\right ) & = \\left [ z _ { 4 } , \\left ( s z _ { 1 } s \\right ) ^ { - 1 } \\right ] . \\end{align*}"}
+{"id": "5790.png", "formula": "\\begin{align*} f ( L ) & = | \\{ U \\le X \\colon \\dim ( U ) = k , \\ , U \\cap A = L \\} | , \\\\ g ( L ) & = \\sum _ { A \\ge B \\ge L } f ( B ) = | \\{ U \\le X \\colon \\dim ( U ) = k , \\ , U \\cap A \\supseteq L \\} | . \\end{align*}"}
+{"id": "4201.png", "formula": "\\begin{align*} F _ { \\Phi } ( X ) : = \\sum _ { \\deg ( D ) < X } \\Phi ( D ) . \\end{align*}"}
+{"id": "4632.png", "formula": "\\begin{align*} \\ln ( 2 ) \\phi _ { B _ r } ^ + \\Big { ( } \\frac { t } { 2 } \\Big { ) } = \\int _ { t / 2 } ^ t \\tau ^ { p - 1 } \\frac { \\phi _ { B _ r } ^ + ( t / 2 ) } { \\tau ^ p } \\ , d \\tau \\le \\psi _ { B _ r } ( t ) \\le \\int _ 0 ^ t t ^ { p - 1 } L _ p \\frac { \\phi _ { B _ r } ^ + ( t ) } { t ^ p } \\ , d \\tau = L _ p \\phi _ { B _ r } ^ + ( t ) . \\end{align*}"}
+{"id": "3931.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta u _ \\Omega = h & , \\\\ u _ \\Omega = g & . \\end{array} \\right . \\end{align*}"}
+{"id": "3172.png", "formula": "\\begin{align*} \\phi ^ { - 1 } : \\begin{cases} u \\mapsto u \\\\ r \\mapsto t s r s ^ { - 1 } r ^ { - 1 } \\\\ s \\mapsto r \\\\ t \\mapsto s \\end{cases} \\end{align*}"}
+{"id": "6401.png", "formula": "\\begin{align*} e _ 1 e ' _ 1 & = q ^ 0 ( [ e ' _ 3 e _ 7 ] + q ^ 1 [ e _ 8 e _ 9 e _ { 1 0 } ] ) , \\\\ e _ 2 e ' _ 2 & = q ^ 0 ( [ e ' _ 4 e _ 8 ] + q ^ 2 ( e ' _ 3 ) ^ 2 ) , \\\\ e _ 5 e ' _ 5 & = q ^ 0 ( [ e ' _ 3 e _ { 1 3 } ] + q e ' _ 4 ) , \\\\ e _ 6 e ' _ 6 & = q ^ 0 ( [ e ' _ 4 e _ { 1 4 } ] + q ^ 2 [ e _ { 1 0 } e _ { 1 2 } e _ { 1 3 } ^ 2 ] ) . \\end{align*}"}
+{"id": "7884.png", "formula": "\\begin{align*} \\mathcal R ' = \\bigsqcup _ { \\tilde R \\in \\tilde { \\mathcal R } ' } \\mathcal R ' ( \\tilde R ) . \\end{align*}"}
+{"id": "3778.png", "formula": "\\begin{align*} c ( t ) = 1 + t c ( t ) ^ 2 . \\end{align*}"}
+{"id": "7156.png", "formula": "\\begin{align*} 0 = Q _ 0 \\subset Q _ 1 \\subset Q _ 2 \\subset \\cdots \\subset Q _ k \\end{align*}"}
+{"id": "1468.png", "formula": "\\begin{align*} S _ n = \\begin{bmatrix} \\frac { 1 } { f _ { 1 } - \\alpha _ n } \\begin{bmatrix} W _ { 1 , 1 } \\\\ \\vdots \\\\ W _ { M , 1 } \\end{bmatrix} + \\sum _ { j = 0 } ^ { \\lfloor \\frac { N } { 2 } \\rfloor - 1 } \\alpha _ n ^ j I _ { 1 , j } \\\\ \\vdots \\\\ \\frac { 1 } { f _ { y } - \\alpha _ n } \\begin{bmatrix} W _ { 1 , y } \\\\ \\vdots \\\\ W _ { M , y } \\end{bmatrix} + \\sum _ { j = 0 } ^ { \\lfloor \\frac { N } { 2 } \\rfloor - 1 } \\alpha _ n ^ j I _ { y , j } \\end{bmatrix} , \\end{align*}"}
+{"id": "6586.png", "formula": "\\begin{align*} & \\gamma : = \\mathsf { m a x } \\left \\{ \\frac { \\gamma _ j } { 2 } \\cdot \\mathsf { 1 } \\{ \\langle x , v _ j \\rangle \\neq 0 \\} ~ | ~ j = 1 , \\dots , d , \\gamma _ j \\in \\mathsf { s p e c } ( \\Gamma ) , \\Gamma v _ j = \\gamma _ j v _ j \\right \\} . \\end{align*}"}
+{"id": "3691.png", "formula": "\\begin{align*} x ^ p + y ^ q = { z ^ r } . \\end{align*}"}
+{"id": "6455.png", "formula": "\\begin{align*} \\varphi ( a , b ) \\equiv \\int _ u ^ v \\frac { ( \\gamma b x , q x / u , q x / v ; q ) _ { \\infty } } { ( b x , c x , d x ; q ) _ { \\infty } } { } _ 1 \\phi _ 2 \\left ( \\begin{gathered} \\gamma \\\\ q ^ { \\alpha + 1 } , \\gamma b x \\end{gathered} ; \\ , q , - q ^ { \\alpha + 1 } a x \\right ) d _ q x . \\end{align*}"}
+{"id": "576.png", "formula": "\\begin{align*} T _ { W , \\widetilde { K } } ( \\mu ^ t ) & = \\frac 1 2 \\int _ { [ 0 , 1 ] \\times \\R } \\int _ { [ 0 , 1 ] \\times \\R } W ( u , v ) K ( x - y ) \\mu ^ t ( d u , d x ) \\mu ^ t ( d v , d y ) \\\\ & = \\frac 1 2 \\int _ { [ 0 , 1 ] \\times \\R } \\int _ { [ 0 , 1 ] \\times \\R } W ( u , v ) K \\big ( ( 1 - t ) ( x - y ) + t ( T _ u ( x ) - T _ u ( y ) ) \\big ) \\mu ^ 0 ( d u , d x ) \\mu ^ 0 ( d v , d y ) . \\end{align*}"}
+{"id": "6956.png", "formula": "\\begin{align*} \\mathsf { c } _ { d } ( \\chi ) = \\sum _ { k = 0 } ^ { d } \\binom { - \\chi + d ( N + 1 ) } { k } \\frac { ( - y ) ^ k } { ( 1 - y ) ^ { d ( N + 1 ) } } . \\end{align*}"}
+{"id": "4976.png", "formula": "\\begin{align*} \\mathcal { C } _ { } = \\mathcal { O } ( K \\sum _ { w = 1 } ^ { w _ { r } } p _ { w } L ( M + 1 ) ^ { w } ) , \\end{align*}"}
+{"id": "6311.png", "formula": "\\begin{align*} \\mathcal { M } _ t ^ f & = f ( Z _ t ) - \\int _ 0 ^ t f ^ \\prime ( Z _ s ) \\mu ( s , X _ s ) \\dd s - \\frac { 1 } { 2 } \\int _ 0 ^ t f ^ { \\prime \\prime } ( Z _ s ) \\sigma ( s , X _ s ) ^ 2 \\dd s \\\\ & = f ( Z _ 0 ) + \\int _ 0 ^ t f ^ \\prime ( Z _ s ) \\sigma ( s , X _ s ) \\dd B _ s , \\end{align*}"}
+{"id": "5329.png", "formula": "\\begin{align*} \\bigcap _ { i = 1 } ^ b H ^ { g _ i } = 1 \\end{align*}"}
+{"id": "2788.png", "formula": "\\begin{align*} A \\tilde { V } _ k & = \\tilde { U } _ k \\tilde { \\Sigma } _ k , \\\\ A ^ T \\tilde { U } _ k & = \\tilde { V } _ k \\tilde { \\Sigma } _ k + \\beta _ k q _ { k + 1 } e _ k ^ T X _ k , \\end{align*}"}
+{"id": "6924.png", "formula": "\\begin{align*} \\bigg ( \\frac { z _ i + y } { z _ i } \\prod _ { p = 1 } ^ { r } ( 1 + z _ i x _ p ) \\bigg ) \\frac { d q } { d h _ i } = P ' ( z _ i ) \\end{align*}"}
+{"id": "6413.png", "formula": "\\begin{align*} V _ { p , q } ( \\xi ) = F ( \\xi ) ^ { \\frac { p - q } { q } } \\xi V _ { p , q } ^ { \\ast } ( \\xi ^ { \\ast } ) = F _ { \\ast } ( \\xi ^ { \\ast } ) ^ { \\frac { p - q } { q } } \\xi ^ { \\ast } . \\end{align*}"}
+{"id": "5498.png", "formula": "\\begin{align*} \\frac { ( b , q ) _ N } { ( a , q ) _ N } \\frac { ( a , q ) _ { N - n } } { ( b , q ) _ { N - n } } \\frac { a ^ n } { b ^ n } = \\frac { ( q ^ { 1 - N } / b , q ) _ n } { ( q ^ { 1 - N } / a , q ) _ n } \\end{align*}"}
+{"id": "3146.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\theta } d _ F ^ { \\theta } ( G ) \\ , d _ { F ' } ^ { \\theta } ( G ) = d _ { F , F ' } ( G ) + O ( 1 / n ) . \\end{align*}"}
+{"id": "6948.png", "formula": "\\begin{align*} \\mathsf { L } _ { d + 1 } - \\mathsf { L } _ { d } & = \\sum _ { k = 0 } ^ { d + 1 } \\binom { u } { k } \\left ( \\binom { - u + d + 1 } { d + 1 - k } - \\binom { - u + d } { d - k } \\right ) ( 1 - y ) ^ { k } = \\sum _ { k = 0 } ^ { d + 1 } \\binom { u } { k } \\binom { - u + d } { d + 1 - k } ( 1 - y ) ^ k \\\\ & = \\sum _ { k = 0 } ^ { d + 1 } \\binom { u } { d + 1 } \\binom { d + 1 } { k } ( - 1 ) ^ { d - k + 1 } ( 1 - y ) ^ { k } = \\binom { u } { d + 1 } ( - y ) ^ { d + 1 } . \\end{align*}"}
+{"id": "3350.png", "formula": "\\begin{align*} p _ x : = ( p _ { x , 1 } , \\ldots , p _ { x , r _ x } ) > 0 \\ \\ \\mbox { a n d } \\ \\ q _ x : = ( q _ { x , 1 } , \\ldots , q _ { x , s _ x } ) \\geq 0 , \\end{align*}"}
+{"id": "2267.png", "formula": "\\begin{align*} W ^ { 2 m } = ( V + k x + k y ) ^ { 2 m } \\supseteq V ^ m + V ^ m x + \\cdots + V ^ m x ^ m , m \\geq 1 , \\end{align*}"}
+{"id": "1558.png", "formula": "\\begin{align*} \\rho ( g ) - I _ n = \\rho ( h ) ^ { - 1 } ( \\rho ( h g h ^ { - 1 } ) - I _ n ) \\rho ( h ) . \\end{align*}"}
+{"id": "7329.png", "formula": "\\begin{align*} \\mu ( u ( \\cdot ) ) = \\mathbb { E } \\left [ \\int _ { 0 } ^ { T } \\Delta _ { u } \\mathcal { H } ( t ) d t \\right ] . \\end{align*}"}
+{"id": "3827.png", "formula": "\\begin{align*} \\left ( a _ { h - 1 } , a _ h \\right ) \\in \\left \\{ \\begin{aligned} & \\left ( n - 5 , n - 4 \\right ) , \\left ( n - 5 , n - 3 \\right ) , \\\\ & \\left ( n - 5 , n - 2 \\right ) , \\left ( n - 5 , n - 1 \\right ) , \\\\ & \\left ( n - 4 , n - 3 \\right ) , \\left ( n - 4 , n - 2 \\right ) , \\\\ & \\left ( n - 4 , n \\right ) , \\left ( n - 3 , n - 1 \\right ) , \\\\ & ( n - 3 , n ) , ( n - 2 , n - 1 ) , \\\\ & ( n - 2 , n ) , ( n - 1 , n ) \\end{aligned} \\right \\} . \\end{align*}"}
+{"id": "5856.png", "formula": "\\begin{align*} { } | \\langle A x , x \\rangle | ^ p { ( 1 + 2 ^ { p - 1 } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } ) } < ( \\frac { p } { p - 1 } ) ^ { p } | \\langle A x , x \\rangle | ^ p . \\end{align*}"}
+{"id": "4454.png", "formula": "\\begin{align*} \\epsilon = \\tfrac { 1 } { 2 } \\left ( \\nabla u + \\nabla u ^ T \\right ) . \\end{align*}"}
+{"id": "5317.png", "formula": "\\begin{align*} w E _ { i } = w T _ { i } - q w . \\end{align*}"}
+{"id": "6350.png", "formula": "\\begin{align*} \\mathcal { A } _ P ^ { ( 1 ) } ( U F , U G ) & = m a x _ { i = 1 } ^ N \\frac { q _ i \\bigg \\{ | \\langle U f _ i , U g _ i \\rangle | + \\| U f _ i \\| \\ ; \\| U g _ i \\| \\bigg \\} } { 2 } \\\\ & = m a x _ { i = 1 } ^ N \\frac { q _ i \\bigg \\{ | \\langle f _ i , g _ i \\rangle | + \\| f _ i \\| \\ ; \\| g _ i \\| \\bigg \\} } { 2 } \\\\ & = \\mathcal { A } _ { P } ( F , G ) . \\end{align*}"}
+{"id": "3636.png", "formula": "\\begin{align*} K ( x ) = \\int _ 0 ^ \\infty \\left [ h ( x + y ) - h ( x ) \\right ] \\nu ( d y ) - \\lambda h ( x ) + ( x - \\rho ) ^ 2 , x \\in ( - \\infty , a ) \\cup ( b , \\infty ) . \\end{align*}"}
+{"id": "5440.png", "formula": "\\begin{align*} \\langle \\ , p _ { 1 3 } , \\ , p _ { 1 4 } + \\epsilon \\ , p _ { 2 3 } , \\ , p _ { 1 2 } p _ { 3 4 } - \\epsilon p _ { 2 3 } ^ 2 \\ , \\rangle = \\langle \\ , p _ { 1 3 } , \\ , p _ { 1 4 } + \\epsilon \\ , p _ { 2 3 } \\ , \\rangle \\ , + \\ , \\hbox { i d e a l o f $ { \\rm G r } ( 2 , 4 ) $ } . \\end{align*}"}
+{"id": "2328.png", "formula": "\\begin{align*} \\zeta ( 4 ) = \\zeta ( 1 , 1 , 2 ) = \\frac { \\pi ^ { 4 } } { 9 0 } , \\ \\zeta ( 1 , 3 ) = \\frac { \\pi ^ { 4 } } { 3 6 0 } , \\ \\zeta ( 2 , 2 ) = \\frac { \\pi ^ { 4 } } { 1 2 0 } , \\end{align*}"}
+{"id": "7423.png", "formula": "\\begin{align*} \\breve { I } = N = \\sum \\limits _ { \\mu = 1 } ^ { m } a _ { \\mu } ^ { \\dagger } a _ { \\mu } = \\sum \\limits _ { \\mu = 1 } ^ { m } N _ { \\mu } . \\end{align*}"}
+{"id": "1927.png", "formula": "\\begin{align*} ( P _ 0 + \\lambda ) ( D ^ { \\alpha } _ v u ) = \\sum _ { \\widetilde \\alpha : \\ , \\widetilde \\alpha < \\alpha , | \\widetilde \\alpha | = m - 1 } c _ { \\widetilde \\alpha } D ^ { \\widetilde \\alpha } _ v D ^ { \\alpha - \\widetilde \\alpha } _ { x } u , \\end{align*}"}
+{"id": "1140.png", "formula": "\\begin{align*} \\Sigma ( k ) = \\theta \\big ( \\chi ( 2 k + 1 ) + 1 + \\lceil \\ln ( 2 L ( k + 1 ) ) \\rceil \\big ) + 1 . \\end{align*}"}
+{"id": "805.png", "formula": "\\begin{align*} \\tau _ i ^ { * } = \\inf \\{ t : \\ x _ i ( t ) \\geq x ^ * \\} , \\ \\ x ^ * = \\bar { K } _ 1 \\cdot \\frac { k _ 2 } { k _ 2 - 1 } . \\end{align*}"}
+{"id": "2792.png", "formula": "\\begin{align*} A = U _ A C G ^ { - 1 } , B = U _ B S G ^ { - 1 } , \\end{align*}"}
+{"id": "2186.png", "formula": "\\begin{align*} p ^ { \\prime } ( x _ 1 ) ^ 2 + r ^ 2 f ^ { \\prime } ( x _ 1 ) ^ 2 = 1 \\end{align*}"}
+{"id": "601.png", "formula": "\\begin{align*} \\begin{aligned} \\psi ^ { + 0 } _ { n , p } & = ( n - p + a _ { 1 2 } - a _ { 1 2 3 } + j ^ { ( 3 ) } ) \\ , \\prod _ { j = - \\infty } ^ p u _ { n , p - j } \\ , \\\\ \\rho ^ { 0 + } _ { n , p } & = ( n - p + a _ { 1 2 } - a _ { 1 2 3 } - j ^ { ( 3 ) } ) \\ , \\prod _ { k = - \\infty } ^ n u _ { n - k , p } \\ , \\end{aligned} \\end{align*}"}
+{"id": "4249.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( q ) _ n \\left ( \\frac { b } { a } \\right ) _ n ( a ) _ { N - n } a ^ n } { ( b ) _ n ( 1 - q ^ n ) ( a ) _ { N } } & = \\sum _ { n = 1 } ^ { N } \\left ( \\frac { a q ^ { n - 1 } } { 1 - a q ^ { n - 1 } } - \\frac { b q ^ { n - 1 } } { 1 - b q ^ { n - 1 } } \\right ) \\\\ & = \\sum _ { m = 1 } ^ { \\infty } ( a ^ m - b ^ m ) \\sum _ { n = 0 } ^ { N - 1 } ( q ^ n ) ^ m \\\\ & = \\sum _ { m = 1 } ^ { \\infty } \\frac { ( a ^ m - b ^ m ) ( 1 - q ^ { m N } ) } { 1 - q ^ m } . \\end{align*}"}
+{"id": "5826.png", "formula": "\\begin{align*} \\omega ( T ^ n ) = \\| T ^ n \\| = \\| T \\| ^ n = \\omega ( T ) ^ n . \\end{align*}"}
+{"id": "443.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\delta ^ 2 } ^ \\delta \\frac { 2 \\pi } { r \\log ( \\delta ^ { - 1 } ) ^ 2 } = \\frac { \\pi } { \\log ( \\delta ^ { - 1 } ) } . \\end{align*}"}
+{"id": "5020.png", "formula": "\\begin{align*} \\Omega _ J : = \\Omega \\setminus \\bigcup _ { j \\notin J } \\mathcal { E } ( j ) , \\end{align*}"}
+{"id": "4055.png", "formula": "\\begin{align*} \\frac { | \\langle f , H _ m \\rangle _ { L ^ 2 _ { \\rho _ s } } | } { \\| H _ m \\| _ { L ^ 2 _ { \\rho _ s } } } \\leq \\left \\{ \\begin{array} { r c l } C & i f & \\ell \\geq m , \\\\ 0 & i f & \\ell < m , \\end{array} \\right . \\end{align*}"}
+{"id": "5271.png", "formula": "\\begin{align*} & \\mathcal { M } ( f ) = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } } \\ * \\sum _ { m = 1 } ^ n \\ * \\frac { 1 } { m ! } \\ * \\sum _ { \\substack { ( n _ 1 , \\ldots , n _ m ) : \\\\ n _ 1 + \\ldots + n _ m = n + 1 , \\ n _ 1 , \\ldots , n _ m \\geq 1 } } \\ * \\frac { ( n + 1 ) ! } { n _ 1 ! \\cdots n _ m ! } \\times \\\\ & \\int _ { L _ { \\pi } } \\hat { f } ( t _ 1 , \\ldots , t _ n ) \\ * \\prod _ { j = 1 } ^ m c _ { n _ j } ( t _ { M _ { j - 1 } + 1 } , \\ldots , t _ { M _ j - 1 } , - t _ { M _ { j - 1 } + 1 } - \\ldots - t _ { M _ j - 1 } ) \\ * d \\lambda , \\end{align*}"}
+{"id": "5909.png", "formula": "\\begin{align*} \\wedge ^ 2 V _ 5 ^ \\vee = U _ 1 \\oplus \\cdots \\oplus U _ s \\ , , s = r + \\binom { r } { 2 } \\ , , \\end{align*}"}
+{"id": "3240.png", "formula": "\\begin{align*} I _ { \\psi } ( u , v ) = \\int _ { X } \\psi ( u - v ) ( \\theta ^ { n } _ { u } + \\theta ^ { n } _ { v } ) \\end{align*}"}
+{"id": "4056.png", "formula": "\\begin{align*} y ^ \\ell H _ n ( y , s ) = \\sum _ { j = 0 } ^ { \\left [ \\frac { \\ell + n } { 2 } \\right ] } c _ { j , \\ell , n } ( s ) H _ { n + \\ell - 2 j } ( y , s ) . \\end{align*}"}
+{"id": "1669.png", "formula": "\\begin{align*} \\mathsf { J } _ { \\mathsf { f } } : = \\min \\left \\{ \\mathsf { J } \\in \\{ \\mathsf { J } _ { \\mathsf { f } - 1 } + 1 , \\dots , \\mathsf { B } - \\mathsf { F } + \\mathsf { f } \\} : { \\kappa _ { \\mathsf { J } _ { \\mathsf { f } - 1 } } ^ 2 - \\kappa _ { \\mathsf { J } } ^ 2 } \\geq \\frac { 1 - \\phi } { \\mathsf { F } } \\ , \\big [ { \\kappa _ { \\mathsf { A } } ^ 2 - \\kappa ^ 2 _ { \\mathsf { B } } } \\big ] \\right \\} \\ , , \\mathsf { J } _ { 0 } : = \\mathsf { A } \\ , . \\end{align*}"}
+{"id": "5108.png", "formula": "\\begin{align*} [ - \\Delta _ { x } , i \\Gamma _ { \\phi } ] = 4 \\partial _ { x _ k } ( \\partial _ { x _ k x _ l } \\phi ) \\partial _ { x _ l } + \\Delta _ x ^ 2 \\phi , \\end{align*}"}
+{"id": "8095.png", "formula": "\\begin{align*} T _ 1 T _ 2 = A _ r M _ z ^ { \\alpha } \\otimes M _ z ^ { \\alpha } = r M _ z ^ { \\alpha } A _ r \\otimes M _ z ^ { \\alpha } = r ( M _ z ^ { \\alpha } A _ r \\otimes M _ z ^ { \\alpha } ) = r T _ 2 T _ 1 . \\end{align*}"}
+{"id": "5929.png", "formula": "\\begin{align*} \\pi _ X ^ 6 = [ \\Delta _ X ] - \\pi _ X ^ 0 - \\pi _ X ^ 2 - \\pi _ X ^ 4 - \\pi _ X ^ 8 - \\pi _ X ^ { 1 0 } - \\pi _ X ^ { 1 2 } \\ , . \\end{align*}"}
+{"id": "2622.png", "formula": "\\begin{align*} | \\{ j \\le 2 : j \\not = i , m _ j > 0 \\} | = \\begin{cases} 0 \\\\ 1 \\end{cases} \\end{align*}"}
+{"id": "11.png", "formula": "\\begin{align*} P _ { \\lambda } ( Z ) : = \\prod _ { k = 1 } ^ m \\left ( k ! \\ , \\det \\left ( ( z _ { i , j } ) _ { i , j = 1 } ^ { k } \\right ) \\right ) ^ { \\lambda _ { k } - \\lambda _ { k + 1 } } . \\end{align*}"}
+{"id": "4612.png", "formula": "\\begin{align*} L ^ \\phi ( \\Omega ) & : = \\big \\{ u \\in L ^ 0 ( \\Omega ) : \\lim _ { \\lambda \\to 0 ^ + } \\varrho _ \\phi ( \\lambda u ) = 0 \\big \\} \\end{align*}"}
+{"id": "4660.png", "formula": "\\begin{align*} \\mathfrak { F } _ { 1 } = f _ 1 f _ 2 ^ 2 \\cdots f _ { \\ell - 1 } ^ { \\ell - 1 } , \\end{align*}"}
+{"id": "6964.png", "formula": "\\begin{align*} H ^ { \\bullet } \\left ( \\mathsf { Q u o t } _ { d } ( E ) , \\wedge ^ { k } L ^ { [ d ] } \\right ) = \\wedge ^ { k } H ^ { \\bullet } ( E \\otimes L ) \\otimes \\mathsf { S y m } ^ { d - k } H ^ { \\bullet } ( \\mathcal O _ C ) ? \\end{align*}"}
+{"id": "489.png", "formula": "\\begin{align*} B _ n \\Big ( \\frac { 2 \\beta ^ j } { L _ j } \\Big ) - B _ n \\Big ( \\frac { F _ j \\sqrt 5 } { L _ j } \\Big ) = 0 , \\qquad \\mbox { $ n $ e v e n } . \\end{align*}"}
+{"id": "99.png", "formula": "\\begin{align*} \\varphi '' + \\Big ( ( m - 1 ) \\frac { \\sigma ' } { \\sigma } - \\Phi ' \\Big ) \\varphi ' - f ' ( u ) \\leq 0 \\textnormal { i n } \\ I = ( r _ 1 , r _ 2 ) . \\end{align*}"}
+{"id": "6489.png", "formula": "\\begin{align*} \\mathrm { T r } \\{ D ^ 2 v ( x ) \\ , A ( x ) \\} = f ( x ) , \\end{align*}"}
+{"id": "3772.png", "formula": "\\begin{align*} p ^ { \\gamma , \\nu } _ { k _ 1 , \\ldots , k _ n } = \\sum _ { r = 1 } ^ { k _ 1 - 1 } p ^ { \\gamma , \\nu } _ { r - 1 , k _ 1 - r - 1 , k _ 2 , \\ldots , k _ n } + \\gamma \\sum _ { r = 2 } ^ n k _ r p ^ { \\gamma , \\nu } _ { k _ 1 + k _ r - 2 , k _ 2 , \\ldots , \\widehat { k _ r } , \\ldots , k _ n } . \\end{align*}"}
+{"id": "5990.png", "formula": "\\begin{align*} v ( x ) = v _ { \\overline { \\pi } ^ { 0 , b ^ * } } ( x ) = v _ { b ^ * } ( x ) , x \\geq 0 . \\end{align*}"}
+{"id": "8162.png", "formula": "\\begin{align*} 0 = \\det A ( \\lambda ) = \\det \\left [ \\begin{array} { c c } \\lambda I + A _ 1 - C _ 1 C _ 1 ^ * & C _ 2 \\\\ C _ 2 ^ * & 0 \\end{array} \\right ] . \\end{align*}"}
+{"id": "4119.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { H } } ^ m _ { \\alpha } \\left ( I \\right ) = \\{ \\varphi \\in L ^ 2 ( 0 , 1 ) ; ~ ~ ~ \\left ( - \\mathcal { L } _ c ^ { \\alpha } \\right ) ^ { \\frac { m } { 2 } } ( \\varphi ) \\in L ^ 2 ( 0 , 1 ) \\} . \\end{align*}"}
+{"id": "4783.png", "formula": "\\begin{gather*} [ e _ 1 , e _ 2 ] = - [ e _ 2 , e _ 1 ] = - 3 e _ 3 , [ e _ 1 , e _ 3 ^ * ] = - [ e _ 3 ^ * , e _ 1 ] = e _ 1 ^ * + 2 e _ 2 ^ * , \\\\ [ e _ 2 , e _ 3 ^ * ] = - [ e _ 3 ^ * , e _ 2 ] = - e _ 1 ^ * + e _ 2 ^ * ; \\\\ e _ 1 \\cdot e _ 1 = e _ 2 \\cdot e _ 2 = 2 e _ 3 , e _ 1 \\cdot e _ 2 = e _ 2 \\cdot e _ 1 = e _ 3 , e _ 1 \\cdot e _ 3 ^ * = e _ 3 ^ * \\cdot e _ 1 = e _ 1 ^ * , \\\\ e _ 2 \\cdot e _ 3 ^ * = e _ 3 ^ * \\cdot e _ 2 = e _ 1 ^ * + e _ 2 ^ * . \\end{gather*}"}
+{"id": "5337.png", "formula": "\\begin{align*} S _ f = \\{ ( s , f ( s ) ) \\ , : \\ , s \\in S \\} , \\end{align*}"}
+{"id": "7767.png", "formula": "\\begin{align*} c a p _ p ( B _ r , B _ R ) = \\omega _ n [ \\int _ r ^ R ( \\sinh t ) ^ \\frac { n } { 1 - p } d t ] ^ { 1 - p } \\end{align*}"}
+{"id": "1082.png", "formula": "\\begin{align*} \\mathbb { Q } ( S , 2 ) & = \\left \\{ b \\in \\mathbb { Q } ^ * / ( \\mathbb { Q } ^ * ) ^ 2 : _ l ( b ) \\equiv 0 ~ ( \\bmod { 2 } ) ~ ~ l \\neq 2 , p , q \\right \\} \\\\ & = \\left \\{ \\pm 1 , \\ ; \\pm 2 , \\ ; \\pm p , \\ ; \\pm q , \\ ; \\pm 2 p , \\ ; \\pm 2 q , \\ ; \\pm p q , \\ ; \\pm 2 p q \\right \\} . \\end{align*}"}
+{"id": "5559.png", "formula": "\\begin{align*} u _ 2 ( \\infty ) = u _ c , \\end{align*}"}
+{"id": "5876.png", "formula": "\\begin{align*} F _ { I \\Gamma } : = ( F _ { i \\Gamma } ) _ { i \\in I } : \\R ^ n \\to \\R ^ { | I | } . \\end{align*}"}
+{"id": "6007.png", "formula": "\\begin{align*} \\tau _ b ^ - : = \\inf \\{ t > 0 : X ( t ) < b \\} , \\tau _ b ^ + : = \\inf \\{ t > 0 : X ( t ) > b \\} . \\end{align*}"}
+{"id": "7882.png", "formula": "\\begin{align*} \\# F ( \\tilde R ) \\sim \\frac { 1 } { q } \\# \\{ ( f , R ) \\in \\mathcal { G } ' \\colon R \\in \\mathcal { R } ( \\tilde R ) \\} = \\frac { 1 } { q } \\sum _ { R \\in \\mathcal { R } ( \\tilde R ) } \\# \\mathcal { G } ' ( R ) . \\end{align*}"}
+{"id": "2626.png", "formula": "\\begin{align*} \\mu | _ { E _ { 2 , m } } = \\lambda _ y = \\lambda _ x | _ { E _ { 2 , m } } . \\end{align*}"}
+{"id": "4337.png", "formula": "\\begin{align*} \\int _ \\Omega ( f ( t , x ) - f ^ { i n } ( x ) ) \\varphi ( x ) \\ \\mathrm { d } x + \\int _ 0 ^ t \\int _ \\Omega f ( s , x ) \\nabla [ a f + b g ] ( s , x ) \\cdot \\nabla \\varphi ( x ) \\ , \\mathrm { d } x \\mathrm { d } s = 0 \\end{align*}"}
+{"id": "7773.png", "formula": "\\begin{align*} \\int _ t ^ { + \\infty } [ f ' ( s ) ] ^ p A ( s ) d s = ( \\int _ t ^ { + \\infty } [ A ( s ) ] ^ { \\frac { 1 } { 1 - p } } d s ) ^ { 1 - p } \\end{align*}"}
+{"id": "5409.png", "formula": "\\begin{align*} \\mathcal { N } ( E _ 1 , E _ 2 ) : = \\# \\{ j : E _ 1 \\leq \\lambda _ j \\leq E _ 2 \\} \\ , . \\end{align*}"}
+{"id": "5166.png", "formula": "\\begin{align*} C \\subset \\partial \\bigcup _ { n \\geq 1 } T _ n \\subset \\partial \\Omega \\subset \\partial ( 0 , 1 ) ^ 3 \\cup \\bigcup _ { n \\ge 1 } \\bigcup _ { i = 1 } ^ { 8 ^ n } \\partial T _ { n , i } \\cup C . \\end{align*}"}
+{"id": "5941.png", "formula": "\\begin{align*} \\gamma ( b ) = \\gamma ( a ) + \\sum _ { j = 1 } ^ k ( j ! ) ^ { - 1 } \\gamma ^ { ( j ) } ( a ) ( b - a ) ^ j \\in \\gamma ( a ) + M _ a \\theta _ { \\delta } , \\end{align*}"}
+{"id": "5715.png", "formula": "\\begin{align*} \\Psi ^ * ( z ) : = 2 ( \\cosh ( z ) - 1 ) = e ^ z + e ^ { - z } - 2 , \\end{align*}"}
+{"id": "440.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { e ( h _ { t } ) } { t } = 2 \\nu ^ { - 1 } | \\phi | . \\end{align*}"}
+{"id": "3139.png", "formula": "\\begin{align*} L ( \\sigma ) = \\left \\{ \\ell _ 1 , \\ldots , \\ell _ r \\right \\} \\end{align*}"}
+{"id": "7127.png", "formula": "\\begin{align*} \\chi = \\sum _ { i = 1 } ^ k w _ i \\tau _ { d _ i } + \\sum _ { i = 1 } ^ k ( \\psi _ i - \\rho _ i ) . \\end{align*}"}
+{"id": "7551.png", "formula": "\\begin{align*} R _ N ( w , z ) = \\sum _ { n = 0 } ^ { N - 1 } P _ n ^ T ( w ) H _ n ^ { - 1 } P _ n ( z ) . \\end{align*}"}
+{"id": "1100.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\sum _ { i = 1 } ^ m U _ i < \\left ( \\frac { \\pi } { 4 } + \\frac { 1 9 9 \\pi } { 3 2 0 0 k } \\right ) \\| \\mathbf { x } \\| . \\end{align*}"}
+{"id": "6182.png", "formula": "\\begin{align*} \\partial ( x , y , z ) : = ( i , j , t , p ) , \\ \\ \\ i = | x \\cap y | , \\ j = | x \\cap z | , \\ t = | y \\cap z | , \\ p = | x \\cap y \\cap z | . \\end{align*}"}
+{"id": "4878.png", "formula": "\\begin{align*} E _ n ^ { ( \\mathbf { s } ) } ( x ) = \\sum _ { e \\in I _ n ^ { ( \\mathbf { s } ) } } x ^ { ( e ) } . \\end{align*}"}
+{"id": "6739.png", "formula": "\\begin{align*} I ^ 1 _ 5 = ( \\frac { \\partial ^ { \\alpha _ 1 } \\widetilde { E } \\cdot \\partial ^ { \\alpha - \\alpha _ { 1 } } \\nabla _ { v } F } { \\sqrt { \\mu } } , \\frac { \\partial ^ { \\alpha } F } { \\sqrt { \\mu } } ) , I ^ 2 _ 5 = ( \\frac { \\partial ^ { \\alpha _ 1 } \\bar { E } \\cdot \\partial ^ { \\alpha - \\alpha _ { 1 } } \\nabla _ { v } F } { \\sqrt { \\mu } } , \\frac { \\partial ^ { \\alpha } F } { \\sqrt { \\mu } } ) . \\end{align*}"}
+{"id": "2002.png", "formula": "\\begin{align*} \\aligned \\Psi ( t ) & = \\sum _ { \\gamma } \\frac { 1 - \\cos ( \\gamma t ) } { \\gamma ^ 2 } = \\sum _ { \\gamma } \\frac { 1 - e ^ { i \\gamma t } } { \\gamma ^ 2 } \\endaligned \\end{align*}"}
+{"id": "5204.png", "formula": "\\begin{align*} a _ n = \\frac { 2 ^ { 2 n - 1 } - 2 ^ n } { 3 } - \\frac { 2 - 2 ^ n } { 3 } = \\frac { 2 ^ { 2 n - 1 } - 2 } { 3 } = \\frac { 4 ^ n - 4 } { 6 } = 2 \\cdot \\frac { 4 ^ { n - 1 } - 1 } { 3 } \\enspace . \\end{align*}"}
+{"id": "7474.png", "formula": "\\begin{align*} \\psi ^ { ( a ) } _ m = \\sum _ { \\nu = 1 } ^ n \\psi _ m ^ { ( a - 1 ) } ( \\nu ) \\ , . \\end{align*}"}
+{"id": "6078.png", "formula": "\\begin{align*} \\rho ( H ) & = \\rho _ D ( V ( H ) ) + 3 ( m ( D [ V ( H ) ] ) - m ( H ) ) + 2 ( \\pi ( D [ H ] ) - \\pi ( H ) ) \\\\ & \\geq \\rho _ D ( V ( H ) ) + 3 ( m ( D [ V ( H ) ] ) - m ( H ) ) \\end{align*}"}
+{"id": "5130.png", "formula": "\\begin{align*} B ( A , r ) = \\bigcup _ { x \\in A } B ( x , r ) . \\end{align*}"}
+{"id": "2926.png", "formula": "\\begin{align*} C ( \\theta ) = \\frac { \\sqrt \\theta + 1 } { \\sqrt \\theta - 1 } + \\log \\frac { 4 e \\sqrt \\theta ( \\sqrt \\theta + 1 ) } { \\sqrt \\theta - 1 } . \\end{align*}"}
+{"id": "2651.png", "formula": "\\begin{align*} & s _ { B S } ( d ( \\lambda ) ) = \\star = d ^ 0 ( s _ { \\Lambda } ) \\\\ & r _ { B S } ( d ( \\lambda ) ) = \\star = d ^ 0 ( r _ { \\Lambda } ) \\end{align*}"}
+{"id": "4344.png", "formula": "\\begin{align*} H _ \\eta ( u _ 1 ( t ) | u _ 2 ( t ) ) - H _ \\eta ( u _ 1 ^ { i n } | u _ 2 ^ { i n } ) = T ^ 1 _ \\eta ( t ) + T ^ 2 ( t ) \\ , , \\end{align*}"}
+{"id": "3595.png", "formula": "\\begin{align*} T f = \\int _ 0 ^ 1 \\sum _ i a _ i \\phi _ i ( x ) f ( x ) \\phi _ i d x = \\sum _ i a _ i \\left [ \\int _ 0 ^ 1 \\phi _ i ( x ) f ( x ) d ( x ) \\right ] \\phi _ i = \\sum _ i a _ i \\langle f , \\phi _ i \\rangle \\phi _ i . \\end{align*}"}
+{"id": "3130.png", "formula": "\\begin{align*} \\sigma = ( \\ldots , - 4 , - 2 , 0 , 2 , 4 , 6 , \\ldots ) ( \\ldots , - 5 , - 3 , - 1 , 1 , 3 , 5 , \\ldots ) , \\end{align*}"}
+{"id": "725.png", "formula": "\\begin{align*} { f _ 6 } = { f _ 5 } - \\frac { 1 } { 3 } \\rho { u _ z } . \\end{align*}"}
+{"id": "3555.png", "formula": "\\begin{align*} P _ { I , M } ( z ) : = \\sum _ { n \\ge 0 } H _ { I , M } ( n ) z ^ n , \\end{align*}"}
+{"id": "4112.png", "formula": "\\begin{align*} i = ( f _ 1 , \\ldots , f _ n ) : V \\hookrightarrow ( \\mathbb { P } ^ 1 - \\{ 0 , 1 , \\infty \\} ) ^ n \\end{align*}"}
+{"id": "2713.png", "formula": "\\begin{align*} \\binom { 2 n - j } { r - 1 - l } \\binom { 2 n - j - r + l + 1 } { n } = \\binom { 2 n - j } { n } \\binom { n - j } { j + r - l - 1 } \\end{align*}"}
+{"id": "4909.png", "formula": "\\begin{align*} \\frak F ( f _ * [ V \\xrightarrow h X ; E ] ) ) & = \\frak F ( [ V \\xrightarrow { f \\circ h } Y ; E ] ) \\\\ & = [ Y \\xleftarrow { f \\circ h } V \\xrightarrow { g \\circ f \\circ h } Z ; E ] \\\\ & = f _ * [ X \\xleftarrow h V \\xrightarrow { g \\circ f \\circ h } Z ; E ] \\\\ & = f _ * ( \\frak F ( [ V \\xrightarrow h X ; E ] ) ) \\end{align*}"}
+{"id": "6052.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\eta - k ^ 2 \\eta = g \\ , \\ , \\ , \\ , \\Omega \\backslash K \\\\ [ 0 . 3 c m ] \\frac { \\partial \\eta } { \\partial n } = 0 \\ ; \\ , \\ , \\ , \\ , \\partial K \\\\ [ 0 . 3 c m ] \\eta = 0 \\ , \\ , \\ , \\ , \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "7846.png", "formula": "\\begin{align*} { \\bf B } : = B _ 1 + \\mathcal { B } , \\end{align*}"}
+{"id": "2189.png", "formula": "\\begin{align*} & \\Gamma ^ 1 _ { 1 1 } = \\frac { 1 } { 2 \\Delta } ( G E _ 1 - 2 F F _ 1 + F E _ 2 ) , \\Gamma ^ 2 _ { 1 1 } = - \\frac { 1 } { 2 \\Delta } ( F E _ 1 - 2 E F _ 1 + E E _ 2 ) , \\\\ & \\Gamma ^ 1 _ { 1 2 } = - \\frac { 1 } { 2 \\Delta } ( F G _ 1 - G E _ 2 ) , \\Gamma ^ 2 _ { 1 2 } = \\frac { 1 } { 2 \\Delta } ( E G _ 1 - F E _ 2 ) , \\\\ & \\Gamma ^ 1 _ { 2 2 } = - \\frac { 1 } { 2 \\Delta } ( G G _ 1 - 2 G F _ 2 + F G _ 2 ) , \\Gamma ^ 2 _ { 2 2 } = \\frac { 1 } { 2 \\Delta } ( F G _ 1 - 2 F F _ 2 + E G _ 2 ) , \\end{align*}"}
+{"id": "5641.png", "formula": "\\begin{align*} L '' = - \\int _ 0 ^ l \\mathbf { H } ( T _ o ) d t \\leq \\lambda l < 0 \\end{align*}"}
+{"id": "2991.png", "formula": "\\begin{align*} \\beta : = \\frac { \\gamma } { 8 k ^ 2 } , \\alpha : = \\frac { \\beta ^ { e ( F ) } } { 2 k ^ { k } } . \\end{align*}"}
+{"id": "4537.png", "formula": "\\begin{align*} f ^ { - 1 } ( a , \\infty ) = \\{ x \\in U : f ( x ) > a \\} \\ , . \\end{align*}"}
+{"id": "3755.png", "formula": "\\begin{align*} ( A [ \\mathcal C _ i ] - 2 I ) \\sum _ { k = 1 } ^ { \\deg m _ i } D ^ k m _ i ( 2 ) A [ \\mathcal C _ i ] ^ { k - 1 } = m _ i ( A [ \\mathcal C _ i ] ) - m _ i ( 2 ) I = 0 , \\end{align*}"}
+{"id": "2023.png", "formula": "\\begin{align*} ( \\psi _ 1 \\ast \\psi _ 2 ) ( x ) : = \\int _ { - \\infty } ^ { \\infty } \\psi _ 1 ( y ) \\psi _ 2 ( x - y ) \\ , d y , \\widetilde { \\psi } ( x ) : = \\overline { \\psi ( - x ) } . \\end{align*}"}
+{"id": "1789.png", "formula": "\\begin{align*} k _ { a } & : = j _ { a } + 1 { \\rm \\ , \\ , \\ , \\ , \\ , i f \\ , \\ , } j _ { a } \\in \\left [ i , \\bar { i } \\right ] , \\\\ k _ { a } & : = j _ { a } { \\rm \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , o t h e r w i s e } , \\end{align*}"}
+{"id": "606.png", "formula": "\\begin{align*} \\psi ^ { + 0 } _ { n , p } = \\psi ^ { - 0 } _ { n , p } = : \\psi _ { n , p } \\ , , \\rho ^ { 0 + } _ { n , p } = \\rho ^ { 0 - } _ { n , p } = : \\rho _ { n , p } \\ , , \\end{align*}"}
+{"id": "6388.png", "formula": "\\begin{align*} f _ { i j } : = \\begin{cases} \\delta _ { i j } & \\mbox { i f $ i \\neq k $ } , \\\\ - 1 & \\mbox { i f $ i = k = j $ } , \\\\ [ \\epsilon b _ { k j } ] _ + & \\mbox { i f $ i = k \\neq j $ } , \\end{cases} \\end{align*}"}
+{"id": "26.png", "formula": "\\begin{align*} 0 . 2 5 I _ f ( p , q ) & \\leq \\sum _ { i \\in A _ { 1 + \\kappa , \\infty } } p _ i f \\left ( \\frac { q _ i } { p _ i } \\right ) \\leq \\sum _ { i \\in A _ { 1 + \\kappa , \\infty } } p _ i f ( \\nu ) = p ' f ( \\nu ) , \\end{align*}"}
+{"id": "4584.png", "formula": "\\begin{align*} \\cos \\theta + \\cos ( \\theta + x ) + \\cdots + \\cos ( \\theta + ( N - 1 ) x ) = \\frac { \\sin \\left ( \\theta - \\frac { x } { 2 } + N x \\right ) - \\sin \\left ( \\theta - \\frac { x } { 2 } \\right ) } { 2 \\sin \\frac { x } { 2 } } . \\end{align*}"}
+{"id": "1287.png", "formula": "\\begin{align*} ( f + g ) ^ { \\ast } = f ^ { \\ast } + g ^ { \\ast } . \\end{align*}"}
+{"id": "1614.png", "formula": "\\begin{align*} \\left \\{ n \\in \\mathbb { N } \\mid f _ { n } = g _ { n } \\right \\} \\in \\mathcal { U } . \\end{align*}"}
+{"id": "3416.png", "formula": "\\begin{align*} C = \\int _ { V ( t ) } \\Phi \\ , r ^ { n - 1 } d r \\end{align*}"}
+{"id": "1974.png", "formula": "\\begin{align*} \\sum _ { k \\in \\tilde { E } } P _ { R _ { j , k } } & = \\sum _ { k \\in \\tilde { E } } P _ { I ^ 1 _ { j , k _ 1 } } \\otimes P _ { I ^ 2 _ { j , k _ 2 } } \\otimes \\cdots \\otimes P _ { I _ { j , k _ d } ^ n } \\\\ & = \\bigg ( \\sum _ { k _ 1 \\in E _ 2 } P _ { I ^ 1 _ { j , k _ 1 } } \\bigg ) \\otimes \\bigg ( \\sum _ { k _ 2 \\in E _ 2 } P _ { I ^ 2 _ { j , k _ 2 } } \\bigg ) \\otimes \\cdots \\otimes \\bigg ( \\sum _ { k _ d \\in E _ 2 } P _ { I ^ n _ { j , k _ d } } \\bigg ) \\\\ & = P _ { I _ { j } ^ 1 } \\otimes P _ { I _ { j } ^ 2 } \\otimes \\cdots \\otimes P _ { I _ { j } ^ d } , \\end{align*}"}
+{"id": "1534.png", "formula": "\\begin{align*} g _ { e , j , k } \\circ g _ { s , i , j } = g _ { s , j , k } \\circ g _ { e , i , j } . \\end{align*}"}
+{"id": "6607.png", "formula": "\\begin{align*} E _ { n } ( H _ { \\theta , \\alpha } ) = - \\frac { \\alpha ^ 2 } { ( 2 n - 1 ) ^ { 2 } \\theta ^ { 2 } } + O \\Big ( \\frac { 1 } { \\theta } \\Big ) \\theta \\to 0 ^ + . \\end{align*}"}
+{"id": "2109.png", "formula": "\\begin{align*} \\varphi ' ( L ) = \\mathrm { c o l u m n s p a n } ( C _ 2 C _ 1 ^ { - 1 } ) \\end{align*}"}
+{"id": "4229.png", "formula": "\\begin{align*} F _ { \\Phi } ( X ) = \\frac { q ^ { 1 - g } h } { \\zeta ( 2 ) ( 1 - q ^ { - 1 } ) ( q ^ 2 - 1 ) } q ^ { 2 X } - \\sum _ { j = 1 } ^ { 2 g } \\frac { Z ( q \\gamma _ j ^ { - 1 } ) \\gamma _ j } { Z ' ( \\gamma _ j ^ { - 1 } ) } \\frac { \\gamma _ j ^ X } { \\gamma _ j - 1 } + \\varepsilon _ { \\Phi } ( X ) , \\end{align*}"}
+{"id": "3353.png", "formula": "\\begin{align*} H ^ { i + 1 } ( K , \\mathbb { Z } ( i ) ) = 0 , \\end{align*}"}
+{"id": "4628.png", "formula": "\\begin{align*} \\nabla h & = - \\ell \\psi ( v ) ^ { - \\ell - 1 } \\eta ^ { s } \\tilde u \\psi ' ( v ) \\nabla v + s \\psi ( v ) ^ { - \\ell } \\eta ^ { s - 1 } \\tilde u \\nabla \\eta + \\psi ( v ) ^ { - \\ell } \\eta ^ s \\nabla \\tilde u . \\end{align*}"}
+{"id": "624.png", "formula": "\\begin{align*} Z g \\left ( X , Y \\right ) = g \\left ( \\nabla _ { Z } X , Y \\right ) + g \\left ( X , \\nabla _ { Z } ^ { \\ast } Y \\right ) \\end{align*}"}
+{"id": "707.png", "formula": "\\begin{align*} { \\rm O } ( \\epsilon ^ 0 ) : g _ i ^ { ( 0 ) } = g _ i ^ { ( e q ) } , \\end{align*}"}
+{"id": "1292.png", "formula": "\\begin{align*} M _ { u _ 1 \\bullet u _ 2 } = M _ { u _ 1 } \\bullet M _ { u _ 2 } . \\end{align*}"}
+{"id": "1088.png", "formula": "\\begin{align*} p ^ { 2 } \\Tilde { z } _ { 1 } ^ { 2 } - p z _ { 2 } ^ { 2 } = 4 ^ { m } p \\implies - z _ { 2 } ^ { 2 } \\equiv 4 ^ { m } \\pmod p \\end{align*}"}
+{"id": "5457.png", "formula": "\\begin{align*} \\det M = \\det \\begin{pmatrix} A & B & C \\\\ \\omega J - C & J - A & \\omega ^ 2 J - B \\\\ \\omega ^ 2 J - B & \\omega J - C & J - A \\end{pmatrix} . \\end{align*}"}
+{"id": "681.png", "formula": "\\begin{align*} H ( d G , d G ) = Q _ 0 d z ^ 2 + \\left ( \\frac { \\tau _ 0 } { 4 } + \\frac { 4 | Q _ 0 | ^ 2 } { \\tau _ 0 } \\right ) d z d \\overline { z } + \\overline { Q _ 0 } d \\overline { z } ^ 2 . \\end{align*}"}
+{"id": "4535.png", "formula": "\\begin{align*} a _ 0 = \\sqrt { \\frac { 1 } { 2 } \\coth \\frac { 1 } { 2 } } \\big ( \\tilde { f } ( 1 ) - \\tilde { f } ( 0 ) \\big ) \\ , . \\end{align*}"}
+{"id": "2264.png", "formula": "\\begin{align*} \\sigma ' ( z _ 2 ) & = \\tau _ 1 \\pi \\left ( \\lambda z _ 2 ^ { d _ 2 d _ 3 \\cdots d _ { s + 1 } } + T \\right ) \\\\ & = \\tau _ 1 \\left ( \\lambda z _ 1 ^ { d _ 2 d _ 3 \\cdots d _ { s + 1 } } \\right ) + \\tau _ 1 \\pi ( T ) \\\\ & = \\lambda ( \\alpha _ 1 z _ 1 + \\beta _ 1 ( z _ 2 ) ) ^ { d _ 2 d _ 3 \\cdots d _ { s + 1 } } + \\tau _ 1 \\pi ( T ) \\\\ & = \\lambda ' z _ 2 ^ { d _ 1 d _ 2 \\cdots d _ { s + 1 } } + T ' + \\tau _ 1 \\pi ( T ) \\end{align*}"}
+{"id": "6862.png", "formula": "\\begin{align*} \\lambda u _ i h ( a _ i ) = v _ i ^ { \\sqrt { q } + 1 } \\neq 0 , \\ 1 \\leq i \\leq n , \\end{align*}"}
+{"id": "7753.png", "formula": "\\begin{align*} \\cosh t \\cdot \\cosh ( \\lambda | \\theta - \\theta _ 0 | ) = \\cosh ( d i s t _ { g ^ + } ( p _ 0 , y ) ) . \\end{align*}"}
+{"id": "565.png", "formula": "\\begin{align*} \\big | T _ { W _ \\ell , \\phi } ( \\mu ) - T _ { W , \\phi } ( \\mu ) \\big | \\le \\sum _ { i = 1 } ^ L | c _ i | d _ \\square ( W _ \\ell , W ) . \\end{align*}"}
+{"id": "1645.png", "formula": "\\begin{align*} \\mathcal { R } \\ ; = \\ ; \\mathcal { F } ( \\Delta ^ { \\mathsf { L } } _ 1 + s ) \\mathcal { F } ^ * \\ ; , \\mathcal { P } \\ ; = \\ ; \\mathcal { F } \\Big ( \\sum _ { j = 1 } ^ { \\mathsf { L } } \\omega _ j \\ , | j \\rangle \\langle j | \\Big ) \\mathcal { F } ^ * \\ ; \\in \\ ; \\mathbb { C } ^ { \\mathsf { L } \\times \\mathsf { L } } \\ ; , \\end{align*}"}
+{"id": "2597.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\underset { j = 1 } { \\overset { N } { \\sum } } \\partial _ { x ^ j x ^ j } \\phi ^ { N , i } ( t , \\boldsymbol { x } ) \\sigma ^ 2 + \\frac { 1 } { 2 } \\underset { j , \\tau = 1 } { \\overset { N } { \\sum } } \\partial _ { x ^ j } \\partial _ { x ^ \\tau } \\phi ^ { N , i } ( t , \\boldsymbol { x } ) \\sigma _ 0 ^ 2 = \\frac { 1 } { 2 } \\partial _ { \\nu \\nu } \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } ) \\sigma _ 0 ^ 2 + O ( \\frac { 1 } { N } ) . \\end{align*}"}
+{"id": "595.png", "formula": "\\begin{align*} \\varphi ^ { 0 0 } _ { n , p } = \\frac { \\mu _ n ^ { ( 1 2 ) } + \\mu ^ { ( 1 ) } - \\mu ^ { ( 2 ) } } { 2 \\mu _ n ^ { ( 1 2 ) } } \\ , \\rho ^ { 0 0 } _ { n , p } - \\frac { 1 } { 2 \\mu _ n ^ { ( 1 2 ) } } \\big ( ( \\mu _ n ^ { ( 1 2 ) } - \\mu ^ { ( 1 ) } - \\mu ^ { ( 2 ) } - \\mu ^ { ( 0 ) } ) \\mu _ n ^ { ( 1 2 ) } + \\mu ^ { ( 0 ) } ( \\mu ^ { ( 1 ) } - \\mu ^ { ( 2 ) } ) \\big ) \\end{align*}"}
+{"id": "5484.png", "formula": "\\begin{align*} { } _ 3 \\phi _ 2 \\begin{bmatrix} q ^ { - N } , & c / b , & t & ; q , q \\\\ a t , & q ^ { 1 - N } / b \\end{bmatrix} = \\frac { ( c , t ; q ) _ N } { ( b , a t ; q ) _ N } { } _ 3 \\phi _ 2 \\begin{bmatrix} q ^ { - N } , & a , & b & ; q , q \\\\ c , & q ^ { 1 - N } / t \\end{bmatrix} . \\end{align*}"}
+{"id": "6360.png", "formula": "\\begin{align*} R ( \\lambda ) = D ( \\lambda ) + C A ( \\lambda ) ^ { - 1 } B \\in { \\mathbb F } ( \\lambda ) ^ { m , m } \\end{align*}"}
+{"id": "439.png", "formula": "\\begin{align*} \\mathcal { E } ( S ' , h \\circ \\tilde { f } ^ { - 1 } ) - \\mathcal { E } ( S , h ) = - 4 \\textrm { R e } \\int _ S \\psi \\cdot \\frac { \\mu } { 1 - | \\mu | ^ 2 } + 4 \\int _ S | \\psi | \\cdot \\frac { | \\mu | ^ 2 } { 1 - | \\mu | ^ 2 } . \\end{align*}"}
+{"id": "1331.png", "formula": "\\begin{align*} v : = \\begin{cases} \\bar { v } & \\mbox { i n } B _ { 9 / 1 0 } , \\\\ u & \\mbox { i n } B _ 1 \\setminus B _ { 9 / 1 0 } . \\end{cases} \\end{align*}"}
+{"id": "1786.png", "formula": "\\begin{align*} \\left \\langle Y , B , a \\right \\rangle = 0 , \\ , \\ , { \\rm f o r \\ , } a \\in A . \\end{align*}"}
+{"id": "5064.png", "formula": "\\begin{align*} \\frac 1 { 2 \\pi i } \\int _ { \\Re ( u ) = - \\frac 5 2 } \\widetilde { f } _ { k - 1 } ( 1 - u ) \\widetilde { g } ( u ) \\ , d u = 0 , \\quad \\widetilde { g } ( u ) = \\frac { \\sqrt { 2 } \\Gamma _ \\C ( \\frac { 1 - \\epsilon + u } { 2 } ) } { \\Gamma _ \\C ( \\frac { 2 - \\epsilon - u } { 2 } ) } \\frac { \\gamma ( - u / 2 ) } { \\gamma ( 1 + u / 2 ) } . \\end{align*}"}
+{"id": "8012.png", "formula": "\\begin{align*} T _ Q = \\begin{bmatrix} Q _ 0 & Q _ { - 1 } & Q _ { - 2 } & \\cdots & \\cdots \\\\ Q _ { 1 } & Q _ 0 \\otimes I _ d & Q _ { - 1 } \\otimes I _ d & \\cdots & \\cdots \\\\ Q _ { 2 } & Q _ { 1 } \\otimes I _ d & Q _ 0 \\otimes I _ d \\otimes I _ d & \\cdots & \\cdots \\\\ \\vdots & \\vdots & \\vdots & \\ddots & & \\\\ \\end{bmatrix} \\end{align*}"}
+{"id": "8249.png", "formula": "\\begin{align*} - \\mu \\mathcal { P } \\dot { \\Delta } _ j \\big ( \\Lambda ^ \\alpha ( u h ( \\sigma ) ) - u \\Lambda ^ \\alpha h ( \\sigma ) \\big ) + S _ { j - 1 } u \\cdot \\nabla \\mathcal { P } \\dot { \\Delta } _ j u - \\mathcal { P } \\dot { \\Delta } _ j ( u \\cdot \\nabla u ) , \\end{align*}"}
+{"id": "909.png", "formula": "\\begin{align*} \\Omega _ \\delta : = \\{ x \\in \\Omega : | x - x _ 0 | < \\delta \\} \\end{align*}"}
+{"id": "7251.png", "formula": "\\begin{align*} Q _ n ( X _ 1 \\times X _ 2 , f , d , \\epsilon ) & \\leq \\sum _ { ( x , y ) \\in E \\times F } ( 1 / \\epsilon ) ^ { S _ n f ( x , y ) } \\\\ & = \\sum _ { x \\in E } ( 1 / \\epsilon ) ^ { S _ n f _ 1 ( x ) } \\cdot \\sum _ { y \\in F } ( 1 / \\epsilon ) ^ { S _ n f _ 2 ( y ) } , \\end{align*}"}
+{"id": "5740.png", "formula": "\\begin{align*} \\sum _ { i = 1 , j = 1 } ^ { m } B _ { i j } \\otimes M ^ s _ { i j } \\succeq 0 . \\end{align*}"}
+{"id": "2784.png", "formula": "\\begin{align*} A Q _ k & = P _ k J _ k , \\\\ A ^ T P _ k & = Q _ k J _ k ^ T + \\beta _ k q _ { k + 1 } e _ k ^ T , \\end{align*}"}
+{"id": "43.png", "formula": "\\begin{align*} Z _ j \\leq \\frac { D ' } { Q _ 1 } + 1 0 \\sqrt { \\frac { D ' } { Q _ 1 ^ 2 Q _ 2 ^ 2 } } \\sqrt { \\log ( k ) } = \\frac { 1 } { Q _ 1 } \\left ( D ' + 1 0 \\frac { \\sqrt { D ' \\log k } } { Q _ 2 } \\right ) . \\end{align*}"}
+{"id": "8233.png", "formula": "\\begin{align*} \\sum _ { j = k } ^ \\infty \\big | \\big \\langle u ^ n \\Lambda ^ \\alpha ( \\varphi _ j h ( \\sigma ^ n ) ) - u \\Lambda ^ \\alpha ( \\varphi _ j h ( \\sigma ) ) , \\phi \\big \\rangle \\big | \\leq \\frac { \\epsilon } { 2 } . \\end{align*}"}
+{"id": "6070.png", "formula": "\\begin{align*} \\left ( \\omega _ { ( n _ 1 , y _ 1 ) , ( n _ 2 , y _ 2 ) , \\dots , ( n _ k , y _ k ) } \\right ) _ x & \\le \\left ( \\prod _ { j = 1 } ^ k y _ j ^ { n _ j } \\right ) \\left ( \\omega _ { \\left [ \\sum _ { j = 1 } ^ k n _ j \\right ] } \\right ) _ x \\\\ & \\le C \\left ( \\prod _ { j = 1 } ^ k y _ j ^ { n _ j } \\right ) \\frac { \\omega _ x } { x ^ { \\sum _ { j = 1 } ^ k n _ j } } , 0 \\le x < \\infty . \\end{align*}"}
+{"id": "1703.png", "formula": "\\begin{align*} \\left ( b ^ * ( g ) \\Psi \\right ) ^ { ( p ) } ( x _ 1 , \\ldots , x _ p ) & : = \\frac { 1 } { \\sqrt { p } } \\sum _ { i = 1 } ^ p g ( x _ i ) \\Psi ^ { ( p - 1 ) } ( x _ 1 , \\ldots , x _ { i - 1 } , x _ { i + 1 } , \\ldots , x _ p ) \\ , , \\\\ \\left ( b ( g ) \\Psi \\right ) ^ { ( p ) } ( x _ 1 , \\ldots , x _ p ) & : = \\sqrt { p + 1 } \\int d x \\ , \\overline { g ( x ) } \\Psi ^ { ( p + 1 ) } ( x , x _ 1 , \\ldots , x _ p ) \\ , . \\end{align*}"}
+{"id": "1160.png", "formula": "\\begin{align*} A ^ * ( Z ) = & \\pi ^ k c _ k \\sum _ { r , s } \\sum _ { D _ 1 , D _ 2 , t _ 1 , t _ 2 } \\sum _ { d _ 1 | c ( T _ 1 ) } \\sum _ { d _ 2 | c ( T _ 2 ) } ( d _ 1 d _ 2 ) ^ { k - 1 } S ( D _ 1 d _ 1 ^ { - 2 } , D _ 2 d _ 2 ^ { - 2 } ) e ( \\cdots ) \\\\ & = \\pi ^ k c _ k \\sum _ { T _ 1 , T _ 2 > 0 } \\sum _ { d _ 1 | c ( T _ 1 ) } \\sum _ { d _ 2 | c ( T _ 2 ) } ( d _ 1 d _ 2 ) ^ { k - 1 } S ( D _ 1 d _ 1 ^ { - 2 } , D _ 2 d _ 2 ^ { - 2 } ) e ( T r ( T _ 1 Z + T _ 2 Z ) ) . \\end{align*}"}
+{"id": "4423.png", "formula": "\\begin{align*} \\Gamma - \\lim _ { n \\to \\infty } I _ n = \\Gamma - \\lim _ { n \\to \\infty } I = I ^ { \\ast } . \\end{align*}"}
+{"id": "2096.png", "formula": "\\begin{align*} J ( \\varphi ) ( x ) : = N ( D _ x \\varphi ) , \\end{align*}"}
+{"id": "6231.png", "formula": "\\begin{align*} q [ \\psi ; p ] & = ( p - p _ 0 ) ^ 2 \\| \\psi \\| ^ 2 + 2 ( \\psi , ( p - p _ 0 ) ( p _ 0 + B x ) \\psi ) \\\\ [ . 3 e m ] & \\le ( p - p _ 0 ) ^ 2 ( 1 + \\delta ^ { - 1 } ) \\| \\psi \\| ^ 2 + \\delta \\| ( p _ 0 + B x ) \\psi \\| ^ 2 \\\\ [ . 3 e m ] & \\le ( p - p _ 0 ) ^ 2 ( 1 + \\delta ^ { - 1 } ) \\| \\psi \\| ^ 2 + \\delta h _ a [ \\psi ; p _ 0 ] . \\end{align*}"}
+{"id": "7595.png", "formula": "\\begin{align*} \\int \\tilde { G } ( p , q ) \\Delta f ( p ) d A ( p ) = f ( q ) - \\operatorname { a r e a } ( X ) ^ { - 1 } \\int f d A \\end{align*}"}
+{"id": "1593.png", "formula": "\\begin{align*} \\left ( f _ { n } \\left ( g _ { s , i , j } \\right ) \\right ) \\left ( w \\right ) = \\varphi _ { s , i , j } \\left ( w , 0 \\right ) \\end{align*}"}
+{"id": "4314.png", "formula": "\\begin{align*} \\mathbb { E } h ( X ) - \\mathbb { E } h ( \\pi ) & = \\mathbb { E } f ( X ) - \\mathbb { E } \\sum _ { j = 0 } ^ \\infty P _ { X , j } f ( j ) \\\\ & = \\sum _ { j = 0 } ^ \\infty f ( j ) \\left [ \\mathbb { P } ( X = j ) - \\sum _ { i = 0 } ^ \\infty \\mathbb { P } ( X = i ) \\mathbb { P } ( Z _ 1 = j | Z _ 0 = i ) \\right ] \\\\ & = \\sum _ { j = 0 } ^ \\infty \\Delta f ( j ) \\left [ \\mathbb { P } ( X > j ) - \\sum _ { k = j + 1 } ^ \\infty \\sum _ { i = 0 } ^ \\infty \\mathbb { P } ( X = i ) \\mathbb { P } ( Z _ 1 = k | Z _ 0 = i ) \\right ] , \\end{align*}"}
+{"id": "5430.png", "formula": "\\begin{align*} | K _ 3 | = O _ { \\prec } ( N ^ { - 1 / 2 } ) . \\end{align*}"}
+{"id": "2290.png", "formula": "\\begin{align*} Q ^ + _ { d , \\nu } ( X , Y ) : = \\prod _ { \\atop { 1 \\leq n \\leq d + 1 } { n \\ne \\nu } } \\left ( X ^ 2 + { \\mu } _ n Y ^ 2 \\right ) . \\end{align*}"}
+{"id": "5244.png", "formula": "\\begin{align*} S _ N ( f ) = \\sum _ { 1 \\leq i \\neq j \\leq N } f ( L _ N \\ * ( \\theta _ i - \\theta _ j ) _ c ) , \\end{align*}"}
+{"id": "8019.png", "formula": "\\begin{align*} T _ Q \\otimes I _ d - \\begin{bmatrix} C & 0 \\\\ 0 & 0 \\end{bmatrix} \\geq 0 . \\end{align*}"}
+{"id": "3916.png", "formula": "\\begin{align*} E _ 3 ( t ) & = 4 \\int _ 0 ^ h ( t + h - \\tau ) ^ { - \\delta } \\| f ( u ( \\tau ) , \\mathcal H u ( \\tau ) ) \\| ^ 2 _ { \\mathbb H ^ { \\mu - 1 - \\delta } } d \\tau \\\\ & \\le 8 { \\rho ^ * } ^ 2 [ L _ f ( \\rho ^ * ) ^ 2 + K _ f ( \\rho ^ * \\| \\ell \\| _ { L ^ 1 } ) ^ 2 \\| \\ell \\| ^ 2 _ { L ^ 1 } ] \\int _ 0 ^ h ( t + h - \\tau ) ^ { - \\delta } d \\tau . \\end{align*}"}
+{"id": "6616.png", "formula": "\\begin{align*} \\lVert f _ { n } \\rVert _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } ) } ^ { 2 } & \\ge \\int _ { n + 1 } ^ { 2 n - 1 } \\int _ { - a n + 1 } ^ { a n - 1 } e ^ { - 4 | y | } \\dd y \\dd x \\\\ & = 2 ( n - 2 ) \\int _ { 0 } ^ { a n - 1 } e ^ { - 4 y } \\dd y = 2 ( n - 2 ) \\ , \\dfrac { 1 - e ^ { 4 ( 1 - a n ) } } { 4 } \\geq C n \\end{align*}"}
+{"id": "2543.png", "formula": "\\begin{align*} \\mathcal { V } ( \\mu , \\nu ; \\alpha ^ { * , \\xi } ) = \\underset { \\alpha \\in \\mathcal { U } _ { a d } ( 0 , T ) } { \\inf } \\mathcal { J ( } \\mu , \\nu ; \\alpha ) . \\end{align*}"}
+{"id": "1313.png", "formula": "\\begin{align*} ( f \\oplus g ) ^ \\dagger = f ^ \\dagger \\oplus g ^ \\dagger , \\end{align*}"}
+{"id": "4127.png", "formula": "\\begin{align*} \\displaystyle \\sqrt { x } \\frac { M _ { \\alpha } ( x ) } { E _ { \\alpha } ( x ) } = \\begin{cases} \\sqrt { 2 x } J _ { \\alpha } ( x ) & \\mbox { i f } 0 \\leq x \\leq X _ { \\alpha } \\\\ \\sqrt { x } \\left ( J ^ 2 _ { \\alpha } ( x ) + Y ^ 2 _ { \\alpha } ( x ) \\right ) ^ { 1 / 2 } & \\mbox { i f } x \\geq X _ { \\alpha } \\end{cases} \\end{align*}"}
+{"id": "6959.png", "formula": "\\begin{align*} \\left [ q ^ { d } \\right ] \\frac { ( - 1 ) ^ { ( s - 1 ) d } } { \\prod _ { j = 1 } ^ { N } \\ , \\ , \\alpha _ j ^ { \\ell + 1 } } \\ , \\ , \\sum _ { \\vec { d } , I } \\prod _ { j \\not \\in I } ( z _ j + y ) ^ { \\ell + 1 } \\ , \\ , \\prod _ { i \\in I } \\frac { z _ i ^ { d + \\ell + N - s + 1 } } { P ' ( z _ i ) } \\prod _ { i , j \\in I , \\ , i \\ne j } ( z _ i - z _ j ) \\bigg { | } _ { \\epsilon = 0 } . \\end{align*}"}
+{"id": "4886.png", "formula": "\\begin{align*} f _ j ( x , y ) = \\frac { ( j ! ) ^ 2 } { ( x ) _ j ( y ) _ j } . \\end{align*}"}
+{"id": "1548.png", "formula": "\\begin{align*} f _ { s , i , j } \\left ( \\omega _ { i } , \\omega _ { s } \\right ) = f _ { s , i , j } \\circ X _ { \\left \\{ b _ { i } , s _ { i } \\right \\} } \\left ( \\omega ^ { \\prime } \\right ) = X _ { b _ { j } } \\left ( \\omega ^ { \\prime } \\right ) , \\end{align*}"}
+{"id": "4888.png", "formula": "\\begin{align*} & ( x - 1 ) ( x - 2 ) \\left ( f _ j ( x - 2 , y ) - f _ { j - 1 } ( x - 2 , y ) \\right ) + ( x - 1 ) ( 2 x - 5 ) f _ { j - 1 } ( x - 1 , y ) \\\\ & - ( x - 1 ) ( x - y - 1 ) f _ j ( x - 1 , y ) - ( x - 2 ) ^ 2 f _ { j - 1 } ( x , y ) \\\\ & = \\begin{cases} ( x - 1 ) ( y - 1 ) & j = 0 , \\\\ 0 & j > 0 , \\end{cases} \\end{align*}"}
+{"id": "2409.png", "formula": "\\begin{align*} p _ { { \\bf a } } = \\begin{cases} 1 & { \\bf a } = \\emptyset \\\\ - u & { \\bf a } = ( 0 ) \\\\ - u v \\cdot p _ { { \\bf a } ' } & { \\bf a } = ( 0 , z , { \\bf a } ' ) \\\\ 0 & { \\bf a } = ( 0 , 0 , { \\bf a } ' ) \\\\ ( 2 u + v ) \\cdot p _ { { \\bf a } ' } & { \\bf a } = ( z , { \\bf a } ' ) \\end{cases} \\end{align*}"}
+{"id": "7902.png", "formula": "\\begin{align*} S ( u ) \\cdot R ( a ) = S \\big ( S ( u ) \\cdot a + u \\cdot R ( a ) \\big ) + \\kappa ~ \\ ! u \\cdot a . \\end{align*}"}
+{"id": "5708.png", "formula": "\\begin{align*} \\int _ { Q } \\left [ f ( x ) - P ^ { ( s ) } _ { Q } ( f ) ( x ) \\right ] x ^ { \\gamma } \\ , d x = 0 , \\end{align*}"}
+{"id": "3144.png", "formula": "\\begin{align*} y ' = \\frac { y } { ( 1 + \\beta ) ^ s } \\frac { x + \\beta \\ , ( 1 + \\beta ) ^ { t - 1 } } { ( 1 + \\beta ) ^ t } \\ge x ' \\ge \\frac { x + \\beta ^ t } { ( 1 + \\beta ) ^ t } \\ , . \\end{align*}"}
+{"id": "3534.png", "formula": "\\begin{align*} \\Big \\Vert u \\Big \\Vert _ { \\mathrm { L } ^ \\mathrm { p } \\Big ( \\mathrm { B } \\Big ) } \\lesssim \\Big \\Vert u \\Big \\Vert _ { \\mathrm { L } ^ \\mathrm { q } \\Big ( \\mathrm { B } \\Big ) } ^ { 1 - \\mathrm { r } } \\Big \\Vert u \\Big \\Vert _ { W ^ { 1 , \\mathrm { N } } \\Big ( \\mathrm { B } \\Big ) } ^ \\mathrm { r } \\forall u \\in W ^ { 1 , \\mathrm { N } } \\Big ( \\mathrm { B } \\Big ) , \\textbf { w h e r e } \\mathrm { r } = 1 - \\frac { \\mathrm { q } } { \\mathrm { p } } \\end{align*}"}
+{"id": "3591.png", "formula": "\\begin{align*} \\| ( A + \\lambda I ) ^ { 1 / 2 } ( \\hat A + \\lambda I ) ^ { - 1 / 2 } \\| ^ 2 & = \\| ( \\hat A + \\lambda I ) ^ { - 1 / 2 } ( A + \\lambda I ) ( \\hat A + \\lambda I ) ^ { - 1 / 2 } \\| \\\\ & = \\| ( A + \\lambda I ) ^ { 1 / 2 } ( \\hat A + \\lambda I ) ^ { - 1 } ( A + \\lambda ) ^ { 1 / 2 } \\| \\\\ & = \\| ( \\hat A + \\lambda I ) ^ { - 1 / 2 } ( A + \\lambda I ) ^ { 1 / 2 } \\| ^ 2 \\end{align*}"}
+{"id": "1341.png", "formula": "\\begin{align*} \\int _ { B _ { 9 / 1 0 } } \\left | \\nabla v ( x ) - q \\right | ^ p \\ , d x \\le \\ ; & 2 ^ { p - 1 } \\left ( \\int _ { B _ { 9 / 1 0 } } \\left | \\nabla u ( x ) - q \\right | ^ p \\ , d x + \\int _ { B _ { 9 / 1 0 } } \\left | \\nabla v ( x ) - \\nabla u ( x ) \\right | ^ p \\ , d x \\right ) \\\\ \\le \\ ; & C _ 2 \\varepsilon ^ p a ^ p + C _ 1 \\varepsilon ^ { p + \\delta } + C \\sigma ( a ^ p + 1 ) , \\end{align*}"}
+{"id": "4461.png", "formula": "\\begin{align*} & \\sum ( - 1 ) ^ { f _ \\sigma } \\cdots E _ { i j } E _ { k l } \\cdots - \\sum ( - 1 ) ^ { f _ \\sigma + ( \\bar i + \\bar j ) ( \\bar k + \\bar l ) } \\cdots E _ { k l } E _ { i j } \\cdots \\\\ = & \\sum \\delta _ { j k } ( - 1 ) ^ { f _ \\sigma } \\cdots E _ { i l } \\cdots - \\sum \\delta _ { i l } ( - 1 ) ^ { f _ \\sigma + ( \\bar i + \\bar j ) ( \\bar k + \\bar l ) } \\cdots E _ { k j } \\cdots \\end{align*}"}
+{"id": "2291.png", "formula": "\\begin{align*} \\vartheta _ d = \\frac { d \\eta _ d } { d \\eta _ d + d - 2 } \\end{align*}"}
+{"id": "3833.png", "formula": "\\begin{align*} a _ 1 + a _ 2 + \\cdots + a _ { h - 3 } = \\frac { ( h - 2 ) ( 2 n - 5 ) - ( a _ { h - 2 } + a _ { h - 1 } + a _ h ) + d } { 2 } . \\end{align*}"}
+{"id": "1767.png", "formula": "\\begin{align*} { \\frac { n } { 2 } \\choose \\frac { m } { 2 } } - { \\frac { n - 2 } { 2 } \\choose \\frac { m } { 2 } - 1 } = { \\frac { n } { 2 } - 1 \\choose \\frac { m } { 2 } } , \\end{align*}"}
+{"id": "1619.png", "formula": "\\begin{align*} h ( y _ i ) & = \\max \\Big \\{ \\min _ { k \\in \\{ 1 , \\ldots , n \\} } g ( y _ k ) , g ( x _ j ) - d ( x _ j , y _ i ) \\Big \\} + a \\\\ & \\le \\max \\big \\{ g ( y _ i ) , g ( y _ i ) \\big \\} + a \\\\ & = g ( y _ i ) + a . \\end{align*}"}
+{"id": "5924.png", "formula": "\\begin{align*} B = e _ 1 ^ { - 1 } \\{ x _ 0 \\} \\ ; { } _ { e _ 2 } \\ ! \\times _ { e _ 1 } I ^ { ( 2 ) } { } _ { e _ 2 } \\ ! \\times _ { e _ 1 } I ^ { ( 2 ) } { } _ { e _ 2 } \\ ! \\times _ { e } I \\end{align*}"}
+{"id": "4053.png", "formula": "\\begin{align*} \\begin{array} { l l l } q _ - ( s ) & = & \\mathcal { K } _ { s \\tau } q _ - ( \\tau ) \\\\ & & + \\displaystyle \\int _ { \\tau } ^ { s } \\mathcal { K } _ { s s ' } \\left ( P _ - \\left [ \\mathcal { N } ( q ) + \\mathcal { D } _ s ( \\nabla q ) + \\mathcal { R } _ s ( q ) + b ' ( s ' ) \\mathcal { M } ( q ) \\right ] \\right ) d s ' . \\end{array} \\end{align*}"}
+{"id": "4108.png", "formula": "\\begin{align*} ( x _ 1 , \\ldots , x _ n ) ^ * \\mathcal { E } ^ n \\simeq \\prod _ { i = 1 } ^ { n - 1 } ( x _ i , x _ n ) ^ * \\mathcal { E } ^ 2 . \\end{align*}"}
+{"id": "5435.png", "formula": "\\begin{align*} \\left ( \\frac { K - L ^ 2 + s - 2 } { 2 } \\right ) \\left ( - 1 + \\frac { s - L ^ 2 - 1 } { K } \\right ) - \\left ( \\frac { K + L ^ 2 - s } { 2 } \\right ) \\left ( - 1 - \\frac { s - L ^ 2 - 1 } { K } \\right ) = - \\frac { s - L ^ 2 - 1 } { K } . \\end{align*}"}
+{"id": "5080.png", "formula": "\\begin{align*} P _ N f = P _ { \\leq N } f - P _ { \\leq \\frac { N } { 2 } } f . \\end{align*}"}
+{"id": "956.png", "formula": "\\begin{align*} \\begin{aligned} | F ^ { i j } ( \\nabla _ \\beta u ) _ { i j } | \\leq \\ , & | \\nabla _ \\beta f ^ { 1 / k } | + C \\left ( f ^ { 1 / k } + b _ \\alpha \\sum _ { i = 1 } ^ n F ^ { i i } \\right ) \\\\ \\leq \\ , & C m ^ { 1 / k } m ^ { - 1 / 2 ( k - 1 ) } + C b _ \\alpha \\sum _ { i = 1 } ^ n F ^ { i i } \\\\ \\leq \\ , & C b _ \\alpha ^ { - 1 / 2 } m ^ { 1 / k } + C b _ \\alpha \\sum _ { i = 1 } ^ n F ^ { i i } \\end{aligned} \\end{align*}"}
+{"id": "2752.png", "formula": "\\begin{align*} \\rm { E f f } ( \\mathbf { P } ( T ^ * _ { G / P } ) ) = \\overline { \\rm E f f } ( \\mathbf { P } ( T ^ * _ { G / P } ) ) = \\left \\langle [ \\Gamma ] , [ \\pi ^ * H ] \\right \\rangle . \\end{align*}"}
+{"id": "2716.png", "formula": "\\begin{align*} \\varphi ( 2 n , 1 , r - 1 ) = M _ { \\varphi } ( 2 n , 0 , 0 ; r - 1 ) \\end{align*}"}
+{"id": "1506.png", "formula": "\\begin{align*} \\sum _ { d \\le x } h ( d ) d = c x + O \\bigl ( q ^ { \\frac 1 4 } x ^ { \\frac 3 4 } \\bigr ) \\end{align*}"}
+{"id": "5520.png", "formula": "\\begin{align*} P \\cap \\sigma P = ( L \\cap \\sigma L ) \\ltimes U . \\end{align*}"}
+{"id": "1542.png", "formula": "\\begin{align*} F \\left ( g _ { e , k , i } \\right ) \\circ F \\left ( g _ { e , j , k } \\right ) \\circ F \\left ( g _ { e , i , j } \\right ) = \\mathrm { i d } _ { F \\left ( b _ { i } \\right ) } \\end{align*}"}
+{"id": "5877.png", "formula": "\\begin{align*} f _ { i _ 1 \\Gamma } = \\cdots = f _ { i _ r \\Gamma } = f _ { j _ 1 \\Gamma } - w _ { j _ 1 } = \\cdots = f _ { j _ s \\Gamma } - w _ { j _ s } = 0 , \\end{align*}"}
+{"id": "1953.png", "formula": "\\begin{align*} \\| f \\| _ { \\dot { B } ^ s _ { \\vec { p } q } ( \\vec { a } ) } : = \\Big ( \\sum _ { j \\in \\mathbb { Z } } ( 2 ^ { s j } \\| \\varphi _ { j } \\ast f \\| _ { \\vec { p } } ) ^ q \\Big ) ^ { 1 / q } < \\infty , \\end{align*}"}
+{"id": "7489.png", "formula": "\\begin{align*} a _ { m 0 } & = 0 \\ , , \\\\ a _ { m 1 } & = 0 \\ , , \\\\ a _ { m m } & = \\frac { 1 } { ( m - 2 ) ! } \\ , . \\end{align*}"}
+{"id": "3804.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 n - 5 + d } { 2 } , \\end{align*}"}
+{"id": "3572.png", "formula": "\\begin{align*} e _ 2 ( I , N ) = e _ 2 ( I , M ) + \\sum _ { i = 0 } ^ n \\ell \\left ( \\frac { I ^ { i + 1 } M : x } { I ^ i M } \\right ) \\ge e _ 2 ( I , M ) . \\end{align*}"}
+{"id": "2912.png", "formula": "\\begin{align*} \\mathcal A + \\mathcal F = \\mathcal A , \\ \\ \\mathcal A + \\mathcal K \\subset \\mathcal P \\mathcal K \\ \\ \\ \\ \\mathcal A + \\mathcal K \\not \\subset \\mathcal A \\mathcal A \\end{align*}"}
+{"id": "423.png", "formula": "\\begin{align*} b ^ \\dag ( y , h ) = \\bigg ( \\frac { y } { K } \\bigg ) ^ { \\frac { 1 } { \\gamma _ 2 - 1 } } . \\end{align*}"}
+{"id": "1580.png", "formula": "\\begin{align*} w _ r ^ { - 1 } y _ { r , 3 6 } u _ { t _ { c } , t _ { r } } ^ { - 1 } = e . \\end{align*}"}
+{"id": "5005.png", "formula": "\\begin{align*} z = d a = ( d - ( q + 1 ) ) a + ( q + 1 ) a , \\end{align*}"}
+{"id": "7770.png", "formula": "\\begin{align*} \\int _ X | \\nabla u | ^ p d V _ { g ^ + } = \\int _ { X \\setminus B ( t ) } [ f ' ( s ) ] ^ p | \\nabla s | ^ p d V _ { g ^ + } = \\int _ t ^ { + \\infty } [ f ' ( s ) ] ^ p A ( s ) d s \\end{align*}"}
+{"id": "3949.png", "formula": "\\begin{align*} \\nu _ { m } ( x ) = ( - \\ln | x | ) ^ m { \\rm f o r } \\ \\ 0 < | x | < \\frac 1 { e ^ 2 } { \\rm a n d } v _ { m } ( x ) = 0 { \\rm f o r } \\ | x | > 1 . \\end{align*}"}
+{"id": "1021.png", "formula": "\\begin{align*} 0 < \\Psi \\leq \\frac { 3 } { 2 } ( a _ 3 = 0 ) ( \\beta - \\gamma = 0 , \\ ) . \\end{align*}"}
+{"id": "3217.png", "formula": "\\begin{align*} D - 2 \\sqrt { D - 1 } > \\frac { 2 \\left ( \\frac { d ( d + 1 ) } { 2 } \\right ) - 1 } { 3 } = \\frac { D - 1 } { 3 } , \\end{align*}"}
+{"id": "8094.png", "formula": "\\begin{align*} T _ 1 = A _ r \\otimes M _ z ^ { \\alpha } \\ \\ \\ T _ 2 = M _ z ^ { \\alpha } \\otimes I _ { H ^ 2 ( \\mathbb { D } ) } . \\end{align*}"}
+{"id": "2216.png", "formula": "\\begin{gather*} \\begin{array} { r c l } \\dot { y } _ 1 & = & y _ 1 + 7 \\\\ \\dot { y } _ 2 & = & - 2 y _ 2 + 2 \\\\ \\dot { y } _ 3 & = & - 5 y _ 3 - 1 \\end{array} \\end{gather*}"}
+{"id": "1429.png", "formula": "\\begin{align*} \\tau _ j : = \\sup \\left \\{ t \\in [ 0 , T _ { \\mathrm { m a x } } ( u _ 0 ^ { ( j ) } , u _ 1 ^ { ( j ) } ) ) ; \\ , \\sup _ { t \\in [ 0 , T ] } \\| \\mathcal { U } ^ { ( j ) } ( t ) \\| _ { \\mathcal { H } } \\le 2 C _ 1 \\right \\} . \\end{align*}"}
+{"id": "5613.png", "formula": "\\begin{align*} \\Gamma ^ { \\alpha } _ { ; \\beta } : = G ^ { \\bar { \\tau } \\alpha } G _ { \\bar { \\tau } ; \\beta } , \\end{align*}"}
+{"id": "5817.png", "formula": "\\begin{align*} c ( T _ { \\theta } ^ { * n } ) = c ( T _ { \\theta } ^ n ) = 0 ( n \\in \\mathbb { N } ) . \\end{align*}"}
+{"id": "3312.png", "formula": "\\begin{gather*} ( L _ 1 \\otimes 1 ) \\big ( \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } \\big ) = a _ { n _ 1 } \\psi _ { n _ 1 - 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } + b _ { n _ 1 } \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } + a _ { n _ 1 + 1 } \\psi _ { n _ 1 + 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } . \\end{gather*}"}
+{"id": "3420.png", "formula": "\\begin{align*} \\begin{aligned} P ^ U & = - D _ r ( r ^ { 1 - n } E _ \\rho ( r ^ { n - 1 } \\Phi ) ) + r ^ { 1 - n } ( S _ r / \\rho ) E _ S ( r ^ { n - 1 } \\Phi ) , \\\\ P ^ \\rho & = - r ^ { 1 - n } D _ r E _ U ( r ^ { n - 1 } \\Phi ) , \\\\ P ^ S & = - r ^ { 1 - n } ( S _ r / \\rho ) E _ U ( r ^ { n - 1 } \\Phi ) . \\end{aligned} \\end{align*}"}
+{"id": "7325.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d p ^ { u } ( t ) = & - \\left [ \\left ( b _ { x } ^ { u } ( t ) \\right ) ^ { \\intercal } p ^ { u } ( t ) + \\sum \\limits _ { i = 1 } ^ { d } \\left ( \\sigma _ { x } ^ { u , i } ( t ) \\right ) ^ { \\intercal } q ^ { u , i } ( t ) + f _ { x } ^ { u } ( t ) \\right ] d t + \\sum \\limits _ { i = 1 } ^ { d } q ^ { u , i } ( t ) d W _ { i } ( t ) , t \\in \\lbrack 0 , T ] , \\\\ p ^ { u } ( T ) = & \\Phi _ { x } ( X ^ { u } ( T ) ) , \\end{array} \\right . \\end{align*}"}
+{"id": "5618.png", "formula": "\\begin{align*} \\Omega ^ { \\alpha } _ { \\beta } = \\bar \\partial \\omega ^ { \\alpha } _ { \\beta } \\end{align*}"}
+{"id": "2151.png", "formula": "\\begin{align*} \\langle d f , d f \\rangle _ { \\mathbb { R } ^ { m + k } } = g \\end{align*}"}
+{"id": "2276.png", "formula": "\\begin{align*} \\sigma ^ { m + 1 } ( z ) = \\sigma ( \\beta _ m z ^ { M ^ m } ) = \\beta _ m \\sigma ( z ) ^ { M ^ m } = \\beta _ m ( \\alpha z ^ M ) ^ { M ^ m } = \\beta _ m \\alpha ^ { M ^ m } z ^ { M ^ { m + 1 } } = \\beta _ { m + 1 } z ^ { M ^ { m + 1 } } . \\end{align*}"}
+{"id": "4019.png", "formula": "\\begin{align*} \\bar { q } _ { \\beta } ( n , t ) = \\sum _ { k = 1 } ^ { n } \\sum _ { \\Lambda _ { n } ^ { k } } k ! \\prod _ { j = 1 } ^ { n - k + 1 } \\frac { \\left ( \\rho ^ { j } \\binom { r + j - 1 } { j } \\right ) ^ { x _ { j } } } { x _ { j } ! } \\left ( \\frac { \\lambda t ^ { \\beta } ( 1 - \\rho ) ^ { r } } { 1 - ( 1 - \\rho ) ^ { r } } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } ( - \\lambda t ^ { \\beta } ) , \\end{align*}"}
+{"id": "1112.png", "formula": "\\begin{align*} ( 1 + \\rho ) ^ { k } = \\frac { 1 } { 2 ^ { k } } \\sum _ { j = 0 } ^ { 2 k } \\sum _ { m = 0 } ^ { \\left \\lfloor j / 2 \\right \\rfloor } ( 2 \\rho ) ^ { j - 2 m } \\binom { k } { j - 2 m } \\binom { k - j + 2 m } { m } . \\end{align*}"}
+{"id": "2638.png", "formula": "\\begin{align*} \\lambda ( q ( c ( z _ 1 z _ 2 . . . z _ { m - 1 } ) ) , q ( c ( z _ 1 z _ 2 . . . z _ { m - 1 } ) ) l _ m ) = z _ m \\end{align*}"}
+{"id": "3553.png", "formula": "\\begin{align*} 1 - \\alpha \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } = \\mathcal { O } ( \\delta ^ \\mathrm { h } ) \\iff \\omega ^ 2 = \\omega _ 0 ^ 2 - \\omega _ \\mathrm { p } ^ 2 \\cdot \\beta ^ { - 1 } + \\mathcal { O } ( \\delta ^ \\mathrm { h } ) . \\end{align*}"}
+{"id": "4812.png", "formula": "\\begin{align*} F _ n ' ( x ) = \\frac 2 { n } I _ 0 ( 2 \\sqrt { x } ) + \\left ( \\frac 1 { \\sqrt { x } } + \\frac { \\sqrt x } { n ( n + 1 ) } \\right ) I _ 0 ' ( 2 \\sqrt { x } ) , \\end{align*}"}
+{"id": "645.png", "formula": "\\begin{align*} \\partial _ Y \\left ( \\Phi ( \\mathcal { P } ( X ) ) \\right ) = \\Phi ( \\nabla _ Y \\mathcal { P } ( X ) ) - \\sum _ { i = 1 , 2 } \\sqrt { c _ i } \\ \\langle Y _ i , \\mathcal { P } ( X ) \\rangle N _ i \\\\ + \\Phi ( B ( Y , \\mathcal { P } ( X ) _ T ) - B ^ * ( Y , \\mathcal { P } ( X ) _ N ) ) \\end{align*}"}
+{"id": "3065.png", "formula": "\\begin{align*} F _ Y ( \\tau ^ k ) = 0 \\qquad k \\leq \\ell \\end{align*}"}
+{"id": "2985.png", "formula": "\\begin{align*} t ( K _ r , W ) = \\lim _ { n \\to \\infty } t ( K _ r , G _ n ) \\ge \\lim _ { n \\to \\infty } \\frac { \\binom { u _ n } { r } } { \\binom { n } { r } } = \\alpha ^ r = t ( K _ k , W ) ^ { r / k } \\ ; , \\end{align*}"}
+{"id": "5114.png", "formula": "\\begin{align*} \\beta + s = \\frac { d } { p } - \\frac { d } { q } \\end{align*}"}
+{"id": "6907.png", "formula": "\\begin{align*} c ( \\pi _ * ( \\mathcal { K } ^ \\vee _ i ( a _ j - a _ { i } ) ) ) & = ( 1 + h _ i ) ^ { d _ i + a _ j - a _ i + 1 } \\\\ c ( \\pi _ * ( \\mathcal { K } ^ { \\vee } _ i \\otimes \\mathcal { K } _ j ( a _ { j } - a _ { i } ) ) ) & = ( 1 + ( h _ i - h _ j ) ) ^ { d _ i - d _ j + a _ j - a _ i + 1 } . \\end{align*}"}
+{"id": "87.png", "formula": "\\begin{align*} v = 0 \\partial \\O . \\end{align*}"}
+{"id": "6297.png", "formula": "\\begin{align*} \\Lambda _ \\rho = O _ K \\cdot \\varpi _ \\rho , \\end{align*}"}
+{"id": "7490.png", "formula": "\\begin{align*} a _ { m k } = 0 , k < 2 \\lor k > m , \\ , . \\end{align*}"}
+{"id": "6476.png", "formula": "\\begin{align*} W F ^ s ( \\varphi ^ * u ) = \\varphi ^ * W F ^ s ( u ) , s \\in \\R . \\end{align*}"}
+{"id": "2645.png", "formula": "\\begin{align*} & \\lambda _ x | _ { E _ { d ^ * ( y ) } } = \\lambda _ y \\\\ & \\lambda _ x | _ { E _ { [ w b ^ { - n } , w b ^ { - n + 1 } a ] } } = \\phi _ n \\end{align*}"}
+{"id": "1457.png", "formula": "\\begin{align*} \\gamma _ { \\varepsilon } \\varphi _ { \\beta , \\varepsilon } '' ( s ) = - \\beta e ^ { - s } \\left [ - M ( \\gamma _ { \\varepsilon } - \\beta , \\gamma _ { \\varepsilon } + 1 ; s ) + M ' ( \\gamma _ { \\varepsilon } - \\beta , \\gamma _ { \\varepsilon } + 1 ; s ) \\right ] . \\end{align*}"}
+{"id": "2757.png", "formula": "\\begin{align*} \\frac { b ( \\Gamma ) } { a ( \\Gamma ) } = \\frac { \\Lambda ^ { 2 n - 1 } } { \\Lambda ^ { 2 n - 2 } \\cdot \\pi ^ * H } { \\rm a n d } H ' \\equiv \\frac { a ( \\Gamma ) } { b ( \\Gamma ) } \\Lambda ' . \\end{align*}"}
+{"id": "6511.png", "formula": "\\begin{align*} U = \\begin{pmatrix} 0 & - \\frac { 1 } { 3 } & 0 & \\frac { 2 } { 3 } & \\frac { 2 } { 3 } & 0 & 0 \\\\ [ 2 . 5 m m ] 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ [ 2 . 5 m m ] 0 & \\frac { 2 } { 3 } & 0 & \\frac { 2 } { 3 } & \\frac { 1 } { 3 } & 0 & 0 \\\\ [ 2 . 5 m m ] 0 & \\frac { 2 } { 3 } & 0 & - \\frac { 1 } { 3 } & \\frac { 2 } { 3 } & 0 & 0 \\\\ [ 2 . 5 m m ] 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ [ 2 . 5 m m ] 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ [ 2 . 5 m m ] 0 & 0 & 0 & 0 & 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "475.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { \\binom n k F _ { j k + m } B _ { n - k } ( x ) z ^ k } = \\frac { 1 } { \\sqrt 5 } \\big ( \\alpha ^ m B _ n ( x + \\alpha ^ j z ) - \\beta ^ m B _ n ( x + \\beta ^ j z ) \\big ) \\ , , \\end{align*}"}
+{"id": "7260.png", "formula": "\\begin{align*} M _ f ( T , X , d ) & = t _ f ( T , X , d ) \\\\ & = \\cap _ { n \\geq 1 } \\overline { \\{ \\mu \\in M ( X ) : F ( \\mu ) + \\int f d \\mu > \\over - \\frac { 1 } { n } \\} } . \\end{align*}"}
+{"id": "1440.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\| ( u ^ { ( j ) } ( t ) - u ( t ) , \\partial _ t u ^ { ( j ) } ( t ) - \\partial _ t u ( t ) ) \\| _ { \\mathcal { H } } = 0 \\end{align*}"}
+{"id": "3126.png", "formula": "\\begin{align*} \\sigma ( x ) = \\begin{cases} x + 1 & \\\\ x - 1 & \\end{cases} \\end{align*}"}
+{"id": "4981.png", "formula": "\\begin{align*} \\frac { \\sum \\limits _ { \\mathbf { a } \\in \\mathbb { B } ^ { N \\times 1 } \\atop | | \\mathbf { a } | | _ { 0 } = \\lambda N } | \\hat { \\mathcal { U } } _ { \\rm { a c } } ^ { \\mathbf { a } } ( \\mathbf { I } ) - \\mathcal { U } ^ { \\mathbf { a } } _ { \\rm { a c } } | } { \\binom { N } { \\lambda N } } = ( 1 - \\lambda ) N ( 1 - ( 1 - \\lambda ) ^ { \\frac { w _ { c } } { r } - 1 } ) ^ { w _ { c } } . \\end{align*}"}
+{"id": "5362.png", "formula": "\\begin{align*} \\varnothing : = I _ { \\varnothing } , \\end{align*}"}
+{"id": "2412.png", "formula": "\\begin{align*} \\Theta _ { L } & = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { { \\bf a } = ( a _ { 1 } , \\dots , a _ { n } ) \\in \\{ 0 , z \\} ^ { n } } ( 1 \\otimes 1 \\otimes { \\rm e v } _ { 0 } ^ { \\mathfrak { m } } ( e _ { a _ { 1 } } \\cdots e _ { a _ { n } } ) ) \\cdot p _ { { \\bf a } } , \\\\ \\Theta _ { R } & = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { { \\bf b } = ( b _ { 1 } , \\dots , b _ { m } ) \\in \\{ 1 , 1 - z \\} ^ { m } } ( 1 \\otimes 1 \\otimes { \\rm e v } _ { 0 } ^ { \\mathfrak { m } } ( e _ { b _ { 1 } } \\cdots e _ { b _ { m } } ) ) \\cdot q _ { { \\bf b } } . \\end{align*}"}
+{"id": "3938.png", "formula": "\\begin{align*} F _ 1 ( \\Omega _ n ) = & \\int _ { \\Gamma _ n } j _ 1 ( x , \\nabla b _ { \\Omega _ n } ( x ) , H _ { \\Gamma _ n } ( p _ n ( y ) ) ) d \\mu _ { \\Gamma _ n } ( x ) \\\\ = & \\frac { 1 } { 2 h } \\int _ { U _ h ( \\Gamma _ n ) } j _ 1 ( p _ n ( y ) , \\nabla b _ { \\Omega _ n } ( p _ n ( y ) ) , H _ { \\Gamma _ n } ( p _ n ( y ) ) ) \\ , \\det ( d _ { T _ n ^ { - 1 } ( y ) } T _ n ) \\ , d y . \\end{align*}"}
+{"id": "7384.png", "formula": "\\begin{align*} S L ( \\lambda ) \\varphi ( x ) = \\int _ \\Sigma \\frac { 1 } { 2 \\pi } K _ 0 \\bigl ( - i \\sqrt { \\lambda } | x - y | \\bigr ) \\varphi ( y ) d \\sigma ( y ) , \\varphi \\in L ^ 2 ( \\Sigma ) , x \\in \\mathbb { R } ^ 2 \\setminus \\Sigma , \\end{align*}"}
+{"id": "4620.png", "formula": "\\begin{align*} \\phi ( t ) - \\phi ( r t ) = \\int _ { r t } ^ t \\phi ' _ - ( \\tau ) \\ , d \\tau \\le ( t - r t ) \\phi ' _ - ( t ) . \\end{align*}"}
+{"id": "2527.png", "formula": "\\begin{align*} Z = \\prod _ { k \\in \\N } Z _ k , \\end{align*}"}
+{"id": "7728.png", "formula": "\\begin{align*} \\hat { g } ^ p = e ^ { - 2 s _ p } g ^ + = e ^ { - 2 ( s _ p - s _ E ) } \\hat { g } ^ E , \\end{align*}"}
+{"id": "8108.png", "formula": "\\begin{align*} \\mathcal { H } = \\bigoplus _ { A _ n \\subseteq I _ n } \\mathcal { H } _ { A _ n } \\end{align*}"}
+{"id": "5884.png", "formula": "\\begin{align*} h e ^ { { W } ^ { 1 } } \\bar { \\varphi } _ { 1 } \\big ( { W } ^ { 1 } \\big ) - \\frac { 1 } { 2 } h { W } ^ { 1 } \\bar { \\varphi } _ { 1 } \\big ( { W } ^ { 1 } \\big ) \\varphi _ { 1 } \\big ( { W } ^ { 1 } \\big ) = h e ^ { { W } ^ { 3 } } \\bar { \\varphi } _ { 1 } \\big ( { W } ^ { 3 } \\big ) - \\frac { 1 } { 2 } h { W } ^ { 3 } \\bar { \\varphi } _ { 1 } \\big ( { W } ^ { 3 } \\big ) \\varphi _ { 1 } \\big ( { W } ^ { 3 } \\big ) . \\end{align*}"}
+{"id": "3880.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } h _ 1 = f \\bar { g } _ 1 / | g | ^ 2 - u g _ 2 & \\\\ h _ 2 = f \\bar { g } _ 2 / | g | ^ 2 + u g _ 1 & \\end{array} \\right . \\end{align*}"}
+{"id": "6575.png", "formula": "\\begin{align*} H \\cong \\sum _ i \\theta _ i \\begin{pmatrix} E _ i & \\mathbf { 0 } \\\\ \\mathbf { 0 } & \\mathbf { 0 } \\end{pmatrix} + \\sum _ j \\lambda _ j \\begin{pmatrix} \\mathbf { 0 } & \\mathbf { 0 } \\\\ \\mathbf { 0 } & F _ j \\end{pmatrix} . \\end{align*}"}
+{"id": "2623.png", "formula": "\\begin{align*} \\lambda _ x | _ { E _ { 2 , m } } = \\lambda _ y , \\lambda _ x ( m + v _ i ) = f \\lambda _ x ( m + e _ i ) = s ( f ) \\end{align*}"}
+{"id": "7679.png", "formula": "\\begin{align*} g ( x ) = f \\left ( m ( x ) \\right ) \\frac { \\Phi _ n ^ { \\alpha / p } ( | m ( x ) | ) } { \\Phi _ n ^ { \\alpha / p } ( | x | ) } \\end{align*}"}
+{"id": "7032.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\to \\infty } \\int _ { | y | \\geqslant t ^ { \\alpha _ 0 } } | u _ 1 ( y ) | \\mathrm { d } y = 0 . \\end{align*}"}
+{"id": "8119.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { - \\ln \\| s _ 1 \\wedge \\cdots \\wedge s _ { r _ n } \\| _ { n \\varphi , \\det } } { n ^ 2 / 2 } = ( ( L , \\varphi ) \\cdot ( L , \\varphi ) ) \\end{align*}"}
+{"id": "6329.png", "formula": "\\begin{align*} \\Gamma ( ( X \\setminus Q ) \\times ( 0 , 1 ] ) \\cap B = \\emptyset \\end{align*}"}
+{"id": "7524.png", "formula": "\\begin{align*} C ( p , q ) = - 2 \\partial _ p G ( p , q ) d p . \\end{align*}"}
+{"id": "5935.png", "formula": "\\begin{align*} \\tau _ K : = \\{ ( \\xi _ 1 , \\dots , \\xi _ k ) \\in \\Q _ q ^ k \\colon | \\xi _ j - a ^ j | \\leq \\delta \\} \\end{align*}"}
+{"id": "1047.png", "formula": "\\begin{align*} \\begin{aligned} { \\mathfrak a } _ { 2 } = & g ^ { a b } \\xi _ { a } \\xi _ { b } , { \\mathfrak a } _ { 1 } = & \\frac { - i } { \\sqrt { ( g ) } } \\partial _ { a } \\bigl ( \\sqrt { ( g ) } g ^ { a b } \\bigr ) \\xi _ { b } , { \\mathfrak a } _ { 0 } = 0 . \\end{aligned} \\end{align*}"}
+{"id": "7956.png", "formula": "\\begin{align*} \\nu ^ H & = \\sum _ { j = 1 } ^ n \\frac { r ^ 2 x _ j - y _ j ( t - t _ 0 ) } { r R ^ 2 } X _ j + \\frac { r ^ 2 y _ j + x _ j ( t - t _ 0 ) } { r R ^ 2 } Y _ j = \\frac { r ^ 2 \\xi ^ H + ( t - t _ 0 ) J \\xi ^ H } { r R ^ 2 } \\\\ \\mbox { a n d } \\eta & = - J \\nu ^ H = \\frac { ( t - t _ 0 ) \\xi ^ H - r ^ 2 J \\xi ^ H } { r R ^ 2 } . \\end{align*}"}
+{"id": "4737.png", "formula": "\\begin{gather*} L _ A ( \\eth _ k ( a ) ) = L _ A ( a ) \\partial _ k + \\eth _ k L _ A ( a ) , \\qquad \\ ! R _ A ( \\eth _ k ( a ) ) = R _ A ( a ) \\partial _ k + \\eth _ k R _ A ( a ) , \\qquad \\ ! \\forall a \\in A . \\ ! \\ ! \\ ! \\end{gather*}"}
+{"id": "361.png", "formula": "\\begin{align*} A _ 1 & : = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix} , A _ 2 : = \\begin{pmatrix} 0 & 0 & - 1 \\\\ 0 & 0 & 0 \\\\ - 1 & 0 & 0 \\end{pmatrix} , B _ \\Gamma : = \\begin{pmatrix} 0 & 1 & 0 & 0 & - 1 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & - 1 \\end{pmatrix} , \\end{align*}"}
+{"id": "2448.png", "formula": "\\begin{align*} \\mathbb { I } _ { \\mathrm { o o e } } ^ { ( w ) } & \\coloneqq \\left \\{ ( a , b , c ) \\in \\mathbb { Z } _ { > 1 } ^ { 3 } \\mid ( a , b , c ) \\equiv ( 1 , 1 , 0 ) \\bmod { 2 } , \\ , a + b + c = w + 2 \\right \\} , \\\\ \\mathbb { I } _ { \\mathrm { e e e } } ^ { ( w ) } & \\coloneqq \\left \\{ ( a , b , c ) \\in \\mathbb { Z } _ { > 1 } ^ { 3 } \\mid ( a , b , c ) \\equiv ( 0 , 0 , 0 ) \\bmod { 2 } , \\ , a \\leq b , \\ , a < c , \\ , a + b + c = w + 2 \\right \\} , \\end{align*}"}
+{"id": "2117.png", "formula": "\\begin{align*} Y _ d : = \\{ y \\in Y \\mid \\# \\big ( U \\cap \\psi _ A ^ { - 1 } ( y ) \\big ) = d \\} , \\end{align*}"}
+{"id": "4124.png", "formula": "\\begin{align*} V '' + \\left ( ( \\lambda _ j ^ { ( \\alpha ) } ) ^ 2 + \\frac { 1 / 4 - \\alpha ^ 2 } { x ^ 2 } \\right ) V = 0 . \\end{align*}"}
+{"id": "4519.png", "formula": "\\begin{align*} \\{ \\Phi _ x , \\mathcal { K } \\} & = \\{ \\Phi _ x , \\frac { 1 } { 2 } \\int _ 0 ^ 1 \\ ! \\ ! \\Phi _ y ^ 2 \\mathrm { d } y \\} = \\int _ 0 ^ 1 \\ ! \\ ! \\Phi _ y \\{ \\Phi _ x , \\Phi _ y \\} \\ , \\mathrm { d } y = 0 \\ , , \\\\ \\{ \\Pi _ x , \\mathcal { K } \\} & = \\{ \\Pi _ x , \\frac { 1 } { 2 } \\int _ 0 ^ 1 \\ ! \\ ! \\Phi _ y ^ 2 \\mathrm { d } y \\} = \\int _ 0 ^ 1 \\ ! \\ ! \\Phi _ y \\{ \\Pi _ x , \\Phi _ y \\} \\ , \\mathrm { d } x = - \\int _ 0 ^ 1 \\ ! \\ ! \\mathcal { E } _ x ( y ) \\Phi _ y \\mathrm { d } y \\ , , \\end{align*}"}
+{"id": "622.png", "formula": "\\begin{align*} \\widetilde { \\mu } ( n ) \\ r _ n ( m ) = H _ { m } \\ \\widetilde { r } _ { n } ( m + 1 ) + I _ m \\ \\widetilde { r } _ { n } ( m ) + J _ m \\ \\widetilde { r } _ { n } ( m - 1 ) \\ , , \\end{align*}"}
+{"id": "4504.png", "formula": "\\begin{align*} \\widetilde { u } ( x ) = \\widetilde { u } ( x - y ) + \\int _ 0 ^ y u ' ( x - \\xi ) \\mathrm { d } \\xi \\ , , \\end{align*}"}
+{"id": "2245.png", "formula": "\\begin{align*} x ' \\phi ( d ) = \\phi ( \\sigma ( d ) ) x ' , y ' \\phi ( d ) = \\phi ( \\sigma ^ { - 1 } ( d ) ) y ' , y ' x ' = \\phi ( a ) , x ' y ' = \\phi ( \\sigma ( a ) ) , d \\in D . \\end{align*}"}
+{"id": "2535.png", "formula": "\\begin{align*} \\xi _ z = \\frac { 1 } { r ! } \\sum _ { j = 1 } ^ { r ! } T ^ j ( \\xi _ { i _ z , z } ) \\end{align*}"}
+{"id": "8080.png", "formula": "\\begin{align*} N \\lambda _ 1 & \\geq \\rho _ 1 \\sum _ { i = 1 } ^ N \\left ( 2 N - 2 i + 1 \\right ) = \\rho _ 1 \\left [ 2 N ^ 2 - 2 \\frac { N ( N + 1 ) } { 2 } + N \\right ] = N ^ 2 \\rho _ 1 . \\end{align*}"}
+{"id": "6196.png", "formula": "\\begin{align*} & | \\mathcal { B } _ { i , i - l } | = | \\{ ( i - 1 , i - l , t , p ) : ( i - 1 , i - l , t , p ) \\in \\mathcal { I } ' _ { k - 1 } \\} | , \\ \\ \\ \\ \\ \\\\ [ . 2 c m ] & | \\mathcal { B } _ { i - l , i } | = | \\{ ( i - l , i - 1 , t , p ) : ( i - l , i - 1 , t , p ) \\in \\mathcal { I } ' _ { k - 1 } \\} | . \\end{align*}"}
+{"id": "3105.png", "formula": "\\begin{align*} \\sigma _ 1 = \\prod _ { n = 0 } ^ { \\infty } \\alpha _ { 2 n + 1 } = ( 1 , 2 ) ( 3 , 4 ) ( 5 , 6 ) ( 7 , 8 ) \\cdots . \\end{align*}"}
+{"id": "8116.png", "formula": "\\begin{align*} ( ( L _ 0 , \\varphi _ 0 ) \\cdot ( L _ 1 , \\varphi _ 1 ) ) = ( ( L _ 0 , \\varphi _ 0 ) \\cdot ( L _ 1 , \\varphi _ 1 ) ) ' . \\end{align*}"}
+{"id": "4567.png", "formula": "\\begin{gather*} N = \\{ n \\in \\omega : l _ n ^ \\alpha \\in a _ \\alpha ^ * \\} . \\end{gather*}"}
+{"id": "275.png", "formula": "\\begin{align*} x _ { i , 1 } + x _ { i , 2 } + \\cdots + x _ { i , n } & = 1 \\textrm { \\ , f o r a l l $ i $ \\ ; \\textrm ( \\emph { r o w s u m s e q u a l o n e } ) } , \\\\ x _ { 1 , j } + x _ { 2 , j } + \\cdots + x _ { n , j } & = 1 \\textrm { \\ , f o r a l l $ j $ \\ ; ( \\emph { c o l u m n s u m s e q u a l o n e } ) } . \\end{align*}"}
+{"id": "4384.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 + } \\| T ^ t f - \\tilde { f } \\| _ p = 0 , \\end{align*}"}
+{"id": "6079.png", "formula": "\\begin{align*} \\rho _ D ( U ) & = 7 n ( D [ U ] ) - 3 m ( D [ U ] ) - 2 \\pi ( D [ U ] ) \\\\ & = \\rho ( \\Tilde { D } ) + 7 ( n ( D [ U ] ) - n ( \\tilde { D ' } ) ) - 3 ( m ( D [ U ] ) - m ( \\Tilde { D } ) ) - 2 ( \\pi ( D [ U ] ) - \\pi ( \\Tilde { D ' } ) ) \\\\ & \\leq \\rho ( \\Tilde { D ' } ) + 7 - 3 + 2 \\\\ & \\leq \\rho ( \\Tilde { D ' } ) + 6 \\end{align*}"}
+{"id": "3649.png", "formula": "\\begin{align*} Z ( f ) = \\left \\{ ( x _ 1 , \\ldots , x _ m ) \\in \\Delta _ { m - 1 } \\colon f ( x _ 1 , \\ldots , x _ m ) - \\lambda ( f ) = 0 \\right \\} . \\end{align*}"}
+{"id": "5493.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } \\frac { q ^ { 2 n } } { ( q ) _ { 2 n } } = \\frac { 1 } { ( q ) _ { 2 N } } \\sum _ { n = 0 } ^ { N } \\begin{bmatrix} N \\\\ n \\end{bmatrix} _ { q ^ 2 } ( - 1 ) ^ n ( q ^ 2 ; q ^ 2 ) _ n q ^ { n ^ 2 } . \\end{align*}"}
+{"id": "2694.png", "formula": "\\begin{align*} \\binom { a + n - k } { a } \\binom { n - k } { n - k - j } = \\binom { a + n - k } { a + j } \\binom { a + j } { j } \\end{align*}"}
+{"id": "829.png", "formula": "\\begin{align*} \\mathbb { E } \\Big ( e ^ { - \\beta \\tau _ i ^ { * } } \\Big ) = e ^ { - \\frac { \\lambda _ - } { \\sigma } \\ln \\left ( \\frac { x ^ * } { x } \\right ) } = e ^ { - k _ 1 } \\ln \\left ( \\frac { x ^ * } { x } \\right ) = \\left ( \\frac { x ^ * } { x } \\right ) ^ { - k _ 1 } . \\end{align*}"}
+{"id": "131.png", "formula": "\\begin{align*} { [ k + 1 + \\ell ] \\choose [ q ] } _ + \\to \\bigvee \\limits _ { r = 0 } ^ q \\left ( { [ k ] \\choose [ r ] } \\times { [ \\ell ] \\choose [ q - r ] } \\right ) _ + \\end{align*}"}
+{"id": "6119.png", "formula": "\\begin{align*} \\mathcal F _ 2 = \\{ F \\in \\mathcal F : | F \\cap [ d - 1 ] | = d - 2 , [ d , k - 1 ] \\subseteq F \\} . \\end{align*}"}
+{"id": "1371.png", "formula": "\\begin{align*} I _ 0 : = \\int _ { \\mathbb { R } ^ n } \\left ( | u _ 1 ( x ) | ^ 2 + | \\nabla u _ 0 ( x ) | ^ 2 + | u _ 0 ( x ) | ^ { p + 1 } + | u _ 0 ( x ) | ^ 2 \\right ) \\langle x \\rangle ^ { 2 m } \\ , d x , m > 2 \\left ( \\frac { 1 } { p - 1 } - \\frac { n } { 4 } \\right ) \\end{align*}"}
+{"id": "3708.png", "formula": "\\begin{align*} \\beta _ t ^ b : = b ( q ^ b _ t , t ) , \\end{align*}"}
+{"id": "3186.png", "formula": "\\begin{align*} I ( u ) & \\ge \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } | \\nabla u | ^ { 2 } d x - \\frac { 1 } { p } \\int _ { \\mathbb { R } ^ { 3 } } | u | ^ { p } d x \\\\ & \\ge \\frac { 1 } { 2 } \\left \\| \\nabla u \\right \\| _ { 2 } ^ { 2 } - \\frac { 1 } { p } C _ { p } \\left \\| u \\right \\| _ { 2 } ^ { \\frac { 6 - p } { 2 } } \\left \\| \\nabla u \\right \\| _ { 2 } ^ { \\frac { 3 p } { 2 } - 3 } . \\end{align*}"}
+{"id": "8066.png", "formula": "\\begin{align*} \\max _ { p } ~ \\sum _ { \\widetilde { n } = 1 } ^ { K } | { \\bf w } ^ H _ p { \\bf y } _ { R _ p } [ \\widetilde { n } ] | ^ 2 = \\widetilde { \\beta } _ 1 , \\end{align*}"}
+{"id": "7567.png", "formula": "\\begin{align*} H ( p , q ) & = h _ 1 ( z ) \\left ( \\frac { 1 } { z - w } + \\widetilde { C } ( z , w ) \\right ) + h _ 1 ( w ) \\left ( \\frac { 1 } { w - z } + \\widetilde { C } ( w , z ) \\right ) \\\\ & = \\frac { h _ 1 ( z ) - h _ 1 ( w ) } { z - w } + h _ 1 ( z ) \\widetilde { C } ( z , w ) + h _ 1 ( w ) \\widetilde { C } ( w , z ) \\end{align*}"}
+{"id": "4141.png", "formula": "\\begin{align*} U ( 0 ) = \\lim _ { x \\to 0 } A _ n ^ { \\alpha } T _ n ^ { \\alpha } ( x ) = A _ n ^ { \\alpha } \\frac { ( \\alpha + 1 ) _ n } { n ! } \\sim n ^ { \\alpha } . \\end{align*}"}
+{"id": "183.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { S } _ { \\infty m } ( q , 1 ) = \\big \\{ ( q _ { i j } ) _ { \\substack { i \\geq 1 \\\\ 0 \\leq j \\leq m } } \\big | \\ , q < q _ { i j } < 1 \\ , , q _ { i j } < q _ { i , j - 1 } ( \\forall i \\geq 1 , 0 \\leq j \\leq m ) \\ , , q _ { i + 1 , 0 } < q _ { i m } ( \\forall i \\geq 1 ) \\big \\} \\ , . \\end{aligned} \\end{align*}"}
+{"id": "1312.png", "formula": "\\begin{align*} ( \\pi ^ 1 _ { a , b } ) ^ \\dagger = \\iota ^ 1 _ { a , b } , ( \\pi ^ 2 _ { a , b } ) ^ \\dagger = \\iota ^ 2 _ { a , b } . \\end{align*}"}
+{"id": "7985.png", "formula": "\\begin{align*} f ( x , y , t ) = v ( | x | ^ 2 + | y | ^ 2 , t ) \\end{align*}"}
+{"id": "2470.png", "formula": "\\begin{align*} W ^ { - m , q } ( M , \\ , E ^ \\prime ) : = \\left ( W ^ { m , p } ( M , \\ , E ) \\right ) ^ \\prime . \\end{align*}"}
+{"id": "3222.png", "formula": "\\begin{align*} \\delta ( ( D - 1 ) G ) = ( D - 1 ) \\delta \\geq 3 ( D - 1 ) \\geq 2 D . \\end{align*}"}
+{"id": "3864.png", "formula": "\\begin{align*} \\# \\{ x _ i \\equiv a \\} = \\# \\{ y _ i \\equiv B \\} = q , \\# \\{ x _ i \\equiv B \\} = \\# \\{ y _ i \\equiv a \\} = p + 1 . \\end{align*}"}
+{"id": "3423.png", "formula": "\\begin{align*} U ^ * = U ( t , r ) , \\rho ^ * = \\rho ( t , r ) , S ^ * = S ( M ^ { - 1 } ( t , M ( t , r ) - \\epsilon ) ) \\end{align*}"}
+{"id": "2883.png", "formula": "\\begin{align*} f _ { j , \\theta } ( x ' , x _ 3 ) = \\int _ { I _ { K ^ { - 1 } } } \\int _ { \\pi _ 3 ( \\theta ) } \\widehat { f } _ { j , \\theta } ( \\nu ' , \\nu _ 3 ) e ^ { 2 \\pi i x ' \\cdot \\xi ' } e ^ { 2 \\pi i x _ 3 \\xi _ 3 } d \\nu ' d \\nu _ 3 . \\end{align*}"}
+{"id": "1480.png", "formula": "\\begin{align*} \\lambda _ 0 & = 1 - \\frac { \\tilde { D } } { \\eta } \\left ( \\lfloor \\frac { N } { 2 } \\rfloor - 1 + \\eta \\right ) , \\\\ \\lambda _ \\eta & = \\frac { \\tilde { D } } { \\eta } \\left ( \\lfloor \\frac { N } { 2 } \\rfloor - 1 + \\eta \\right ) . \\end{align*}"}
+{"id": "3599.png", "formula": "\\begin{align*} \\eta _ { i j } ^ 2 & = \\left ( \\sum _ m b _ m \\theta _ { m i } \\theta _ { m j } \\right ) ^ 2 \\leq \\sum _ m b _ m ^ 2 \\theta ^ 2 _ { m i } \\sum _ m \\theta ^ 2 _ { m j } = \\| \\phi _ j \\| ^ 2 \\sum _ m b _ m ^ 2 \\theta ^ 2 _ { m i } \\\\ & = \\sum _ m b _ m ^ 2 \\theta ^ 2 _ { m i } \\leq \\sum _ m b _ m ^ 2 \\leq \\sum _ m b _ m . \\end{align*}"}
+{"id": "1550.png", "formula": "\\begin{align*} X _ { b _ { i } } \\left ( \\omega ^ { \\prime } \\right ) = \\omega _ { i } , X _ { s _ { i } } \\left ( \\omega ^ { \\prime } \\right ) = \\omega _ { s } , \\quad X _ { s _ { j } ^ { \\prime } } \\left ( \\omega ^ { \\prime } \\right ) = \\omega _ { s ^ { \\prime } } . \\end{align*}"}
+{"id": "6438.png", "formula": "\\begin{align*} c _ { n _ 1 } ( x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) = \\frac { \\partial ^ { n _ 1 } f ( 0 , y _ 1 , x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) } { n _ 1 ! \\partial { y _ 1 } ^ { n _ 1 } } \\Big | _ { y _ 1 = 0 } . \\end{align*}"}
+{"id": "4373.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\sum _ { i \\neq j } ( \\theta ^ 1 _ i - \\theta ^ 2 _ i ) ( D _ H g _ i ^ 0 ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) + \\\\ D _ { G , 3 } f _ i ^ 0 ( T , x _ 0 ( T ) , u _ 0 ( T ) ) ) \\\\ + \\sum _ { \\alpha = 1 } ^ { m } ( \\lambda ^ 1 _ \\alpha - \\lambda ^ 2 _ \\alpha ) D _ H g ^ \\alpha ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) \\\\ + \\sum _ { \\beta = 1 } ^ { q } ( \\mu ^ 1 _ \\beta - \\mu ^ 2 _ \\beta ) D _ H h ^ \\beta ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) = 0 . \\end{array} \\right \\} \\end{align*}"}
+{"id": "3742.png", "formula": "\\begin{align*} \\begin{bmatrix} n _ { 1 1 } & n _ { 1 2 } & n _ { 1 3 } \\\\ n _ { 2 1 } & n _ { 2 2 } & n _ { 2 3 } \\\\ n _ { 3 1 } & n _ { 3 2 } & n _ { 3 3 } \\end{bmatrix} \\begin{bmatrix} 2 & - 1 & 1 \\\\ - 1 & 2 & 1 \\\\ - 1 & - 1 & 1 \\end{bmatrix} & = \\begin{bmatrix} - 1 6 & 8 & 2 2 \\\\ 8 & - 1 6 & 2 2 \\\\ 8 & 8 & 2 2 \\end{bmatrix} \\end{align*}"}
+{"id": "4897.png", "formula": "\\begin{align*} \\left ( ( 4 - e ^ t ) \\frac { d } { d t } - 2 \\right ) \\frac { 3 } { 2 } P \\left ( \\frac { 1 } { 4 } , \\frac { 2 } { 3 } t \\right ) = 3 e ^ t , \\end{align*}"}
+{"id": "7733.png", "formula": "\\begin{align*} r ( \\cdot ) = s _ { E ' } ( \\cdot ) \\ \\ o n \\ \\ X . \\end{align*}"}
+{"id": "260.png", "formula": "\\begin{align*} \\nabla _ y v ( y ) = [ \\nabla _ x \\phi ] ( u ^ { - 1 } ( y ) ) [ \\nabla _ y u ^ { - 1 } ] ( y ) = [ \\nabla _ x \\phi ] ( u ^ { - 1 } ( y ) ) [ \\nabla _ x u ] ^ { - 1 } ( u ^ { - 1 } ( y ) ) . \\end{align*}"}
+{"id": "6702.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } \\psi _ { i } ( v ) Q ( F , F ) \\ , d v = 0 , \\mbox { f o r $ i = 0 , 1 , 2 , 3 , 4 $ . } \\end{align*}"}
+{"id": "3552.png", "formula": "\\begin{align*} 1 - \\alpha \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } = \\mathcal { O } ( \\delta ^ \\mathrm { h } ) \\iff \\dfrac { \\omega _ \\mathrm { p } ^ 2 } { \\omega _ 0 ^ 2 - \\omega ^ 2 - i \\gamma \\omega } = - 1 + \\varepsilon _ \\infty ^ { - 1 } \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } - \\dfrac { \\varepsilon _ \\infty ^ { - 1 } ( \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } ) ^ 2 } { \\varepsilon _ \\mathrm { m } + \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } } + \\mathcal { O } ( \\delta ^ \\mathrm { h } ) . \\end{align*}"}
+{"id": "3432.png", "formula": "\\begin{align*} ( P ^ U , P ^ \\rho , P ^ S ) = 2 f _ { J _ { 1 , l } } ^ { ( l ) } D _ t ( U , \\rho , S ) + 2 ( 0 , \\rho D _ t f _ { J _ { 1 , l } } ^ { ( l ) } , 0 ) , \\end{align*}"}
+{"id": "6609.png", "formula": "\\begin{align*} D ( B ) = \\big \\{ f \\in H ^ { 2 } ( \\mathbb { R } \\setminus \\{ 0 \\} ) : \\ f ' ( 0 + ) = f ' ( 0 - ) , \\ f ( 0 + ) - f ( 0 - ) = - f ' ( 0 + ) \\big \\} , \\end{align*}"}
+{"id": "3805.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 n + h ^ 2 - 2 h - 8 + d } { 2 } . \\end{align*}"}
+{"id": "7663.png", "formula": "\\begin{align*} F ( \\kappa _ { E _ p } ) & = P ( E _ p ) - \\kappa _ { E _ p } | E _ p | \\\\ & > P ( E _ p ) - \\frac { \\kappa _ { E _ p } } { p } | E _ p | = P ( E _ p ) - H ( p ) | E _ p | ^ p = 0 , \\end{align*}"}
+{"id": "3002.png", "formula": "\\begin{align*} \\| f \\| _ { H ^ { r , 0 } } = \\| \\| f \\| _ { H ^ r _ x ( \\mathbb { T } ) } \\| _ { H ^ s _ y ( 0 , 1 ) } . \\end{align*}"}
+{"id": "3791.png", "formula": "\\begin{align*} \\begin{array} { l l } A _ { i , j - 1 } = \\frac { j ( 2 i + 1 ) } { i ( 2 j + 1 ) } A _ { i - 1 , j } , & i , j \\geq 1 \\\\ A _ { 0 , j } = ( 2 j + 1 ) C _ j , & j \\geq 0 \\end{array} \\end{align*}"}
+{"id": "5517.png", "formula": "\\begin{align*} r _ \\R ( x ) = \\begin{cases} \\sqrt { x } \\cdot [ W _ \\R , W _ \\R ] & ( x > 0 ) , \\\\ j \\sqrt { - x } \\cdot [ W _ \\R , W _ \\R ] & ( x < 0 ) . \\end{cases} \\end{align*}"}
+{"id": "4887.png", "formula": "\\begin{align*} \\frac { 2 ( 2 - x ) } { ( 1 - x ) ^ 2 } \\sum _ { j = 0 } ^ \\infty f _ j \\left ( 2 - \\frac { 1 } { x } , 2 - \\frac { 1 } { x } \\right ) = \\frac { 3 } { 1 - x } + \\frac { 1 - x } { ( 1 - 2 x ) ^ 2 } \\sum _ { j = 0 } ^ \\infty f _ j \\left ( 3 - \\frac { 1 } { x } , 3 - \\frac { 1 } { x } \\right ) . \\end{align*}"}
+{"id": "5930.png", "formula": "\\begin{align*} \\alpha = \\pi ^ { n ' } _ { X ' _ 0 } \\circ \\bigl ( \\sum a _ i \\cdot Z _ i \\bigr ) \\circ \\pi ^ n _ { X _ 0 } \\ , , \\alpha ^ { - 1 } = \\pi ^ n _ { X _ 0 } \\circ \\bigl ( \\sum b _ i \\cdot { } ^ { \\mathsf { t } } Z _ i \\bigr ) \\circ \\pi ^ { n ' } _ { X ' _ 0 } \\ , . \\end{align*}"}
+{"id": "7579.png", "formula": "\\begin{align*} \\iint G ( p , q ) d \\mu ( p ) d \\mu ( q ) & \\leq - \\log \\delta + \\log 4 + 2 \\log ( 2 \\delta ) + C _ 1 \\\\ & = \\log \\delta + 4 \\log 2 + C _ 1 , \\end{align*}"}
+{"id": "747.png", "formula": "\\begin{align*} \\Psi _ k = & \\int \\cdots \\int _ { s < t _ 1 < \\cdots < t _ k < t } \\sum _ { j _ 0 , \\dots , j _ k } \\int _ { H ^ k } q _ { j _ { k - 1 } , j _ { k } } ( x _ k ) \\widetilde { P ^ { ( j _ k ) } } ( t _ k , x _ k ; t , B ) \\\\ & \\times q _ { j _ { k - 2 } , j _ { k - 1 } } ( x _ { k - 1 } ) \\widetilde { P ^ { ( j _ { k - 1 } ) } } ( t _ { k - 1 } , x _ { k - 1 } ; t _ { k } , d x _ { k } ) \\cdots q _ { i , j _ { 1 } } ( x _ { 1 } ) \\widetilde { P ^ { ( j _ 1 ) } } ( t _ { 1 } , x _ { 1 } ; t _ { 2 } , d x _ { 2 } ) \\\\ & \\times \\widetilde { P ^ { ( i ) } } ( s , x ; t _ 1 , d x _ 1 ) d t _ 1 \\cdots d t _ k \\end{align*}"}
+{"id": "5339.png", "formula": "\\begin{align*} | X | = \\frac { p ^ 4 - 1 } { p - 1 } - ( p + 1 ) ^ 2 = p ( p ^ 2 - 1 ) \\end{align*}"}
+{"id": "1354.png", "formula": "\\begin{align*} & a ( r ) \\le \\left ( \\frac 1 { | B _ { \\eta ^ { k _ r + 1 } } | } \\int _ { B _ { \\eta ^ { k _ r } } } | \\nabla u ( x ) | ^ p \\ , d x \\right ) ^ { \\frac 1 p } = \\eta ^ { - n / p } a ( \\eta ^ { k _ r } ) \\\\ & \\qquad \\qquad \\le \\eta ^ { - n / p } \\big ( C ( \\eta ) M + 2 ^ { - k _ r } a ( 1 ) \\big ) \\le C ( M , \\eta ) ( 1 + a ( 1 ) ) . \\end{align*}"}
+{"id": "7496.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ m c _ { m k } g _ { k j } = \\frac { 1 } { m + 1 } { m + 1 \\choose j } B _ { m - j + 1 } \\ , , \\end{align*}"}
+{"id": "3006.png", "formula": "\\begin{align*} \\varphi ^ 0 ( y ) = y , \\varphi ^ 1 ( y ) = 1 - y . \\end{align*}"}
+{"id": "7940.png", "formula": "\\begin{align*} R _ \\Phi ( a , u ) = \\big ( R ( a ) , S ( u ) + \\Phi ( a ) \\big ) , ( a , u ) \\in A \\oplus M . \\end{align*}"}
+{"id": "3782.png", "formula": "\\begin{align*} \\widetilde { C } _ { i , j } ( 1 ) : = C _ { i , j } ( 1 ) - C _ i ( 1 ) C _ j - C _ j ( 1 ) C _ i \\end{align*}"}
+{"id": "1556.png", "formula": "\\begin{align*} \\left ( A B - B A \\right ) \\left ( B A \\right ) ^ { - 1 } = A B A ^ { - 1 } B ^ { - 1 } - I _ n . \\end{align*}"}
+{"id": "160.png", "formula": "\\begin{align*} \\begin{aligned} N _ 3 ^ { ( m / 2 ) } [ \\hat { f } ] < \\infty \\end{aligned} \\end{align*}"}
+{"id": "496.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } { n \\choose k } 2 ^ k F _ { j k } ( \\sqrt { 5 } F _ j ) ^ { n - k } \\big ( 2 ^ { 1 - ( n - k ) } - 1 \\big ) B _ { n - k } & = n F _ j 2 ^ { 1 - n } \\Big ( ( L _ j + 2 \\alpha ^ j ) ^ { n - 1 } + ( L _ j + 2 \\beta ^ j ) ^ { n - 1 } \\Big ) \\\\ & = n F _ j 2 ^ { 1 - n } \\sum _ { m = 0 } ^ { n - 1 } { n - 1 \\choose m } 2 ^ m L _ { j m } L _ j ^ { n - 1 - m } . \\end{align*}"}
+{"id": "7749.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow + \\infty } [ s _ E ( y ) - s _ { p _ 0 } ( y ) ] = \\lim \\limits _ { t \\rightarrow \\infty } [ t - d ( ( t _ 0 , w , \\theta ) , ( t , w _ 0 , \\theta _ 0 ) ) ] = b _ { \\gamma _ + } ( t _ 0 , w , \\theta ) \\end{align*}"}
+{"id": "7792.png", "formula": "\\begin{align*} k ( u ) = \\frac { 1 } { \\pi } \\frac { \\cos ( \\theta ) e ^ { - i \\theta } } { 1 - \\vert u \\vert ^ 2 } . \\end{align*}"}
+{"id": "5356.png", "formula": "\\begin{align*} P _ { [ i j ] } & = 0 , \\\\ W _ { i j k l } & = W _ { [ i j ] [ k l ] } = W _ { [ k l ] [ i j ] } , & W _ { [ i j k ] l } & = 0 , \\\\ C _ { i j k } & = C _ { [ i j ] k } , & C _ { [ i j k ] } & = 0 , \\\\ B _ { [ i j ] } & = 0 , \\end{align*}"}
+{"id": "4946.png", "formula": "\\begin{align*} b _ 1 & = \\frac { 1 } { 2 } \\left ( \\binom { n } { k - 3 } - a _ 1 \\right ) \\ge \\frac { 1 } { 2 } \\left ( \\binom { n } { k - 3 } - A \\right ) . \\end{align*}"}
+{"id": "2621.png", "formula": "\\begin{align*} \\lambda | _ { E _ { k , [ p , q ] } } ^ * ( a ) = \\lambda ( p + a ) \\end{align*}"}
+{"id": "6803.png", "formula": "\\begin{align*} \\lambda _ { A \\ , A } = \\lambda _ { B \\ , B } = \\lambda _ { B \\ , C } + \\lambda _ { A \\ , C } + \\lambda _ { A \\ , B } = 0 . \\end{align*}"}
+{"id": "2387.png", "formula": "\\begin{align*} F _ { 3 } ^ { ( 2 ) } & = \\sum _ { s = 0 } ^ { 1 } \\sum _ { k = 1 } ^ { m } \\sum _ { t = 0 } ^ { 1 } ( \\Delta _ { a _ { 2 } , t } ^ { \\alpha , \\beta } - \\Delta _ { a _ { s + 1 } , \\iota ^ { k - s } ( t ) } ^ { \\alpha , \\beta } ) Y _ { k , t } ( s + 1 ) , \\end{align*}"}
+{"id": "4041.png", "formula": "\\begin{align*} \\langle g f _ { b _ 0 } ^ { - p } , H _ { 2 k } \\rangle _ { L ^ 2 _ { \\rho _ s } } = 0 . \\end{align*}"}
+{"id": "1950.png", "formula": "\\begin{align*} \\nu : = | \\vec { a } | : = a _ 1 + \\cdots + a _ d , \\end{align*}"}
+{"id": "5810.png", "formula": "\\begin{align*} \\Pi V = M _ z \\Pi . \\end{align*}"}
+{"id": "4656.png", "formula": "\\begin{align*} \\Theta _ { f , D } = \\Theta ^ { t } _ { w _ { N _ E } ( f ) , D } \\ ; N _ E , \\end{align*}"}
+{"id": "5702.png", "formula": "\\begin{gather*} [ \\ ; , \\ ; ] ^ * ( x _ { 1 , 2 } ^ * ) ( x ) = 0 \\end{gather*}"}
+{"id": "2379.png", "formula": "\\begin{align*} \\partial _ { \\alpha , \\beta } ( w _ { 1 } e _ { a } w _ { 2 } ) = \\partial _ { \\alpha , \\beta } ( w _ { 1 } e _ { a } ) w _ { 2 } + w _ { 1 } \\partial _ { \\alpha , \\beta } ( e _ { a } w _ { 2 } ) + D _ { \\alpha , \\beta } ( w _ { 1 } , a , w _ { 2 } ) . \\end{align*}"}
+{"id": "1029.png", "formula": "\\begin{align*} \\Theta ( \\beta _ 2 ( t ) ) & = f ( y _ 2 ( t ) ) - f ^ \\star \\\\ & = f ( y _ 1 ( \\alpha ( t ) ) ) - f ^ \\star = \\Theta ( \\beta _ 1 ( \\alpha ( t ) ) ) \\end{align*}"}
+{"id": "7901.png", "formula": "\\begin{align*} R ( a ) \\cdot S ( u ) = S \\big ( R ( a ) \\cdot u + a \\cdot S ( u ) \\big ) + \\kappa ~ \\ ! a \\cdot u , \\\\ S ( u ) \\cdot R ( a ) = S \\big ( S ( u ) \\cdot a + u \\cdot R ( a ) \\big ) + \\kappa ~ \\ ! u \\cdot a . \\end{align*}"}
+{"id": "7996.png", "formula": "\\begin{align*} T _ w = T _ p ^ * T _ p \\end{align*}"}
+{"id": "3841.png", "formula": "\\begin{align*} \\left ( a _ 2 , a _ 3 , a _ 4 \\right ) = \\left \\{ ( n - 5 , n - 4 , n - 3 ) , ( n - 5 , n - 2 , n - 1 ) , ( n - 4 , n - 2 , n ) , ( n - 3 , n - 1 , n ) \\right \\} . \\end{align*}"}
+{"id": "5389.png", "formula": "\\begin{align*} \\delta _ { l } \\sigma ( g _ 1 , \\cdots , g _ { l + 1 } ) = & { \\rho } ( g _ 1 ) ( { \\sigma } ( g _ 2 , \\cdots , g _ { l + 1 } ) ) + \\\\ & + \\sum _ { i } ( - 1 ) ^ i { \\sigma } ( g _ 1 , \\cdots , g _ i g _ { i + 1 } , \\cdots , g _ { l + 1 } ) + ( - 1 ) ^ { l + 1 } { \\sigma } ( g _ 1 , \\cdots , g _ l ) . \\end{align*}"}
+{"id": "1549.png", "formula": "\\begin{align*} \\varphi _ { s , i , j } \\left ( \\omega _ { i } , \\omega \\right ) = \\left ( \\omega _ { j } , \\omega \\right ) \\quad \\quad \\varphi _ { s , j , i } \\left ( \\omega _ { j } , \\omega \\right ) = \\left ( \\omega _ { i } , \\omega \\right ) . \\end{align*}"}
+{"id": "4485.png", "formula": "\\begin{align*} \\dd _ { ( \\psi _ 1 , \\psi _ 2 ) } g ( h _ 1 , h _ 2 ) = \\langle D _ 1 g ( \\psi _ 1 , \\psi _ 2 ) , h _ 1 \\rangle _ { H } + \\langle D _ 2 g ( \\psi _ 1 , \\psi _ 2 ) , h _ 2 \\rangle _ { H } \\ , . \\end{align*}"}
+{"id": "913.png", "formula": "\\begin{align*} \\tilde { f } ( x ) \\leq \\tilde { f } ( 0 ) + \\sum _ { i = 1 } ^ n \\tilde { f } _ i ( 0 ) x _ i + C | x | ^ 2 \\leq C \\left ( \\sigma _ { k - 1 } ^ { 1 / ( k - 1 ) } ( b ) + \\sigma _ { k - 1 } ^ { 1 / 2 ( k - 1 ) } ( b ) | x ' | + x _ n + | x | ^ 2 \\right ) \\end{align*}"}
+{"id": "1308.png", "formula": "\\begin{align*} \\pi ^ 2 _ { a , b } \\circ \\iota ^ 1 _ { a , b } = 0 _ { a , b } , \\pi ^ 1 _ { a , b } \\circ \\iota ^ 2 _ { a , b } = 0 _ { b , a } , \\end{align*}"}
+{"id": "5808.png", "formula": "\\begin{align*} \\omega ( { \\sqrt { \\sum _ { i = 1 } ^ n A _ i ^ * X A _ i } } ) = \\sup _ { \\| h \\| = 1 } | \\langle { \\sqrt { \\sum _ { i = 1 } ^ n A _ i ^ * X A _ i } } h , h \\rangle | & = \\| { \\sqrt { \\sum _ { i = 1 } ^ n A _ i ^ * X A _ i } } \\| \\\\ & = \\| { \\sum _ { i = 1 } ^ n A _ i ^ * X A _ i } \\| ^ { 1 \\over 2 } \\\\ & = ( \\sup _ { \\| h \\| = 1 } | \\langle { \\sum _ { i = 1 } ^ n A _ i ^ * X A _ i } h , h \\rangle | ) ^ { 1 \\over 2 } \\\\ & = \\sqrt { \\omega ( \\sum _ { i = 1 } ^ n A _ i ^ * X A _ i ) } . \\end{align*}"}
+{"id": "5695.png", "formula": "\\begin{gather*} \\tau _ * ( \\alpha _ { ( i , 0 ) } ) = e _ { 1 , 2 } ^ { 3 } \\wedge ( \\sum _ { j = 1 } ^ { 3 } e _ { 4 , j } ^ { j } ) \\wedge \\cdots \\wedge ( \\sum _ { j = 1 } ^ { i + 1 } e _ { i + 2 , j } ^ { j } ) . \\end{gather*}"}
+{"id": "6694.png", "formula": "\\begin{align*} & Q _ i ( t ) = \\det \\begin{bmatrix} s _ { i , - 2 } & s _ { i , - 1 } & 1 \\\\ s _ { i , - 1 } & 1 & t \\\\ 1 & \\lambda _ { i , 2 } ^ { 2 } & t ^ { 2 } \\end{bmatrix} , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] , \\end{align*}"}
+{"id": "7611.png", "formula": "\\begin{align*} M ( \\theta ) = \\{ ( 0 , 1 ) , \\theta \\geq 2 \\} , \\{ ( 1 , 1 ) , \\theta \\leq 2 \\} . \\end{align*}"}
+{"id": "3099.png", "formula": "\\begin{align*} \\sigma _ 5 & = \\prod _ { n = 0 } ^ { \\infty } \\alpha _ { 5 n + 1 } \\prod _ { n = 0 } ^ { \\infty } \\beta _ { 5 n + 3 } \\\\ & = ( 1 , 2 ) ( 3 , 4 , 5 ) ( 6 , 7 ) ( 8 , 9 , 1 0 ) \\cdots \\end{align*}"}
+{"id": "3462.png", "formula": "\\begin{align*} \\Big \\Vert \\varphi \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\mathbb { R } ) } ^ 2 = \\int _ { - \\infty } ^ T | \\varphi | ^ 2 ( \\mathrm { x } , t ) d t = \\alpha \\delta ^ 2 \\int _ { - \\infty } ^ { T _ { \\delta } } | \\varphi | ^ 2 ( \\delta \\eta + \\mathrm { z } , \\alpha \\delta ^ 2 \\Tilde { \\tau } ) d \\Tilde { \\tau } = \\alpha \\delta ^ 2 \\Big \\Vert \\hat { \\varphi } \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\mathbb { R } ) } ^ 2 \\end{align*}"}
+{"id": "4782.png", "formula": "\\begin{gather*} \\partial _ 1 ^ * ( e _ 1 ^ * ) = e _ 1 ^ * , \\partial _ 1 ^ * ( e _ 2 ^ * ) = e _ 2 ^ * , \\partial _ 1 ^ * ( e _ 3 ^ * ) = 2 e _ 3 ^ * ; \\\\ \\partial _ 2 ^ * ( e _ 1 ^ * ) = - e _ 2 ^ * , \\partial _ 2 ^ * ( e _ 2 ^ * ) = e _ 1 ^ * + e _ 2 ^ * , \\partial _ 2 ^ * ( e _ 3 ^ * ) = e _ 3 ^ * . \\end{gather*}"}
+{"id": "1284.png", "formula": "\\begin{align*} \\langle s \\bullet f _ 1 , \\ldots , s \\bullet f _ n \\rangle = s \\bullet \\langle f _ 1 , \\ldots , f _ n \\rangle . \\end{align*}"}
+{"id": "5218.png", "formula": "\\begin{align*} \\mathring { R } ( e _ i \\odot e _ j ) = \\sum _ { k , l } R _ { i k l j } e _ k \\odot e _ l . \\end{align*}"}
+{"id": "6157.png", "formula": "\\begin{align*} V _ 2 ^ * V _ 1 & = ( I _ { H ^ 2 } \\otimes P U + M _ z ^ * \\otimes P ^ \\perp U ) ( I _ { H ^ 2 } \\otimes P ^ \\perp U + M _ z \\otimes P U ) \\\\ & = I _ { H ^ 2 } \\otimes P U { P ^ \\perp } U + M _ z \\otimes P U P U + M _ z ^ * \\otimes { P ^ \\perp } U { P ^ \\perp } U + I _ { H ^ 2 } \\otimes { P ^ \\perp } U P U \\end{align*}"}
+{"id": "7259.png", "formula": "\\begin{align*} F ( \\mu ) + \\int f _ 1 d \\mu + \\epsilon & \\geq \\frac { L ( f _ 1 ) } { L ( 1 ) } + \\inf _ { f \\in \\mathcal { C } + \\frac { \\epsilon } { 2 } } \\int f d \\mu + \\frac { \\epsilon } { 2 } \\\\ & = \\frac { L ( f _ 1 ) } { L ( 1 ) } + \\inf _ { g \\in \\mathcal { C } + \\frac { \\epsilon } { 2 } } \\frac { L ( g ) } { L ( 1 ) } + \\frac { \\epsilon } { 2 } \\\\ & = \\frac { 1 } { L ( 1 ) } ( L ( f _ 1 ) + \\inf _ { g \\in \\mathcal { C } + \\frac { \\epsilon } { 2 } } { L ( g ) } ) + \\frac { \\epsilon } { 2 } > 0 . \\end{align*}"}
+{"id": "5485.png", "formula": "\\begin{align*} F _ N ( a , b ; t ) = \\frac { ( a t q , q ; q ) _ N } { ( t , b q ; q ) _ N } \\sum _ { n = 0 } ^ { N } \\frac { ( b ) _ n ( t ) _ n q ^ { n } } { ( a t q ) _ n ( q ) _ n } . \\end{align*}"}
+{"id": "5588.png", "formula": "\\begin{align*} v ^ { \\alpha } H _ { \\alpha } & = \\frac 1 2 \\left ( y ^ i \\frac { \\partial } { \\partial y ^ i } - \\sqrt { - 1 } u ^ k \\frac { \\partial } { \\partial y ^ k } \\right ) \\left ( R + \\sqrt { - 1 } I \\right ) \\\\ & = \\frac 1 2 \\left ( y ^ i R ^ i + u ^ k I _ k \\right ) + \\frac { \\sqrt { - 1 } } { 2 } \\left ( y ^ i I _ i - u ^ j R _ k \\right ) . \\end{align*}"}
+{"id": "4176.png", "formula": "\\begin{align*} \\lambda \\big ( ( A - n _ 1 ) \\cap \\cdots \\cap ( A - n _ d ) \\big ) = \\mu \\big ( T ^ { - n _ 1 } Y \\cap \\cdots \\cap T ^ { - n _ d } Y \\big ) . \\end{align*}"}
+{"id": "2975.png", "formula": "\\begin{align*} [ Z _ i ] + [ X _ { i - 1 } ] & = [ Z _ { i - 1 } ] + [ X _ { i + 1 } ] , & \\ ; i = 3 , \\ldots , n - 1 \\end{align*}"}
+{"id": "4224.png", "formula": "\\begin{align*} \\alpha ^ { ( t ) } _ k ( \\theta _ { t , j } ) _ { j = 1 } ^ { g } = - \\sum _ { j = 1 } ^ g 2 \\Re \\left ( \\sum _ { \\ell = 0 } ^ { k - 1 } \\frac { Z ( \\gamma _ { j , \\ell } ^ { - 1 } ) } { k \\gamma _ { j , \\ell } ^ { 1 - k } Z ' ( \\gamma _ j ^ { - 1 } ) } \\frac { \\gamma _ { j , \\ell } } { \\gamma _ { j , \\ell } - 1 } e ^ { i ( \\theta _ { t , j } + 2 \\pi [ \\ell t ] / k ) } \\right ) \\end{align*}"}
+{"id": "5215.png", "formula": "\\begin{align*} C _ j & < p _ 1 + 3 \\sum _ { k = 2 } ^ { j _ 2 } p _ k + 2 \\sum _ { k = j _ 2 + 1 } ^ { j _ s } p _ k + \\sum _ { i = 1 } ^ { s - 1 } p _ { j _ { i + 1 } - 1 } + \\sum _ { k = j _ s + 1 } ^ { j - 1 } p _ k + 2 p _ j \\\\ & \\leq p _ 1 + 3 \\sum _ { k = 2 } ^ { j _ 2 } p _ k + 2 \\sum _ { k = j _ 2 + 1 } ^ { j _ s } p _ k + \\sum _ { i = 2 } ^ { s - 1 } p _ { j _ { i + 1 } - 1 } + \\sum _ { k = j _ s + 1 } ^ { j - 1 } p _ k + 3 p _ j \\\\ & \\leq 3 \\sum _ { k = 1 } ^ j p _ k . \\end{align*}"}
+{"id": "5250.png", "formula": "\\begin{align*} R _ { T , l } ( g ) = \\sum ^ { * } f ( \\tilde { \\gamma } _ { j _ 2 } - \\tilde { \\gamma } _ { j _ 1 } , \\ldots , \\tilde { \\gamma } _ { j _ l } - \\tilde { \\gamma } _ { j _ 1 } ) , \\end{align*}"}
+{"id": "3556.png", "formula": "\\begin{align*} e _ i ( I , M ) = \\frac { Q _ { I , M } ^ { ( i ) } ( 1 ) } { i ! } , \\end{align*}"}
+{"id": "1028.png", "formula": "\\begin{align*} \\ddot { x } + \\frac { 3 } { t } \\dot { x } + \\nabla f ( x ) = 0 \\end{align*}"}
+{"id": "6271.png", "formula": "\\begin{align*} x ^ * = \\begin{cases} 0 , & \\nabla h ( u ) = \\lambda \\omega , \\\\ - t ^ * \\norm { H _ r \\big ( p _ \\lambda ( u ) \\big ) } ^ { - 1 } H _ r \\big ( p _ \\lambda ( u ) \\big ) , & \\nabla h ( u ) \\neq \\lambda \\omega , \\end{cases} \\end{align*}"}
+{"id": "2008.png", "formula": "\\begin{align*} G _ g ( t , u ) = \\sum _ { \\gamma } \\frac { ( e ^ { i \\gamma t } - 1 ) ( e ^ { - i \\gamma u } - 1 ) } { \\gamma ^ 2 } \\end{align*}"}
+{"id": "575.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } M _ n ^ { \\psi } / n \\ge \\lim _ { n \\to \\infty } M _ n ^ { \\psi } ( Q ^ n ) / n = T _ { W , \\psi } ( \\mu ) - I ( \\mu ) . \\end{align*}"}
+{"id": "1920.png", "formula": "\\begin{align*} P _ 0 u + \\lambda u = \\div \\vec f + g , u ( - 1 , \\cdot ) = 0 . \\end{align*}"}
+{"id": "1794.png", "formula": "\\begin{align*} P _ { 1 } = \\left ( i _ { a } - 1 , i _ { a } \\right ) , \\ , P _ { 2 } = \\left ( i _ { a } + 1 , i _ { a } + 2 \\right ) , \\dots , \\ , P _ { b - a + 1 } = \\left ( i _ { b } , i _ { b } + 1 \\right ) . \\end{align*}"}
+{"id": "7667.png", "formula": "\\begin{align*} p _ 1 \\frac { P ( E _ 1 ) } { V } = p _ 2 \\frac { P ( E _ 2 ) } { V } . \\end{align*}"}
+{"id": "7153.png", "formula": "\\begin{align*} D ^ b \\left ( ( d ) \\right ) = \\Big \\langle \\boxtimes _ { i = 1 } ^ k \\mathbb { M } ( d _ i ) _ { w _ i } \\Big \\rangle , \\end{align*}"}
+{"id": "3847.png", "formula": "\\begin{align*} G ^ { z } ( w ) : = ( H ^ { z } - w ) ^ { - 1 } , H ^ { z } : = \\begin{pmatrix} 0 & X - z \\\\ X ^ * - \\overline { z } & 0 \\end{pmatrix} , w \\in \\C \\setminus \\R . \\end{align*}"}
+{"id": "4326.png", "formula": "\\begin{align*} d _ { T V } ( X , \\pi ) \\leq \\sup _ { h \\in \\mathcal { H } } \\sup _ { l \\in \\mathbb { Z } ^ + } | m _ l ( h ) | \\sum _ { i = 0 } ^ \\infty \\mathbb { P } ( X = i ) \\sum _ { j = 0 } ^ \\infty \\left | \\sum _ { k = j + 1 } ^ \\infty \\left ( Q _ { i , k } - P _ { i , k } \\right ) \\right | . \\end{align*}"}
+{"id": "6654.png", "formula": "\\begin{align*} 1 \\le \\sum _ { i = 1 } ^ { \\eta } \\lambda _ { i , 1 } ^ { 2 } \\le 2 - \\frac { 1 } { \\lambda _ { 0 } ^ { 2 } } . \\end{align*}"}
+{"id": "3877.png", "formula": "\\begin{align*} \\abs { z } ^ 2 = 1 + \\frac { \\sqrt { \\gamma } + ( x + y ^ 2 ) / \\sqrt { \\gamma } } { \\sqrt { n } } + \\frac { ( \\gamma + x ) ^ 2 } { 4 \\gamma n } \\ge 1 + \\frac { \\sqrt { \\gamma } + ( x + y ^ 2 ) / \\sqrt { \\gamma } } { \\sqrt { n } } . \\end{align*}"}
+{"id": "433.png", "formula": "\\begin{align*} e ( h ) = \\frac { 1 } { 2 } \\textrm { t r a c e } _ \\nu h ^ * \\sigma , \\end{align*}"}
+{"id": "4855.png", "formula": "\\begin{align*} \\partial _ t \\bar \\rho = \\nabla \\cdot \\nabla \\cdot ( \\bar \\rho \\Sigma ) - \\frac 1 2 \\nabla \\cdot ( \\bar \\rho \\nabla V ) . \\end{align*}"}
+{"id": "3861.png", "formula": "\\begin{align*} \\dd H ^ z _ t = - \\frac { 1 } { 2 } ( H ^ z _ t + Z ) \\dd t + \\frac { 1 } { \\sqrt { n } } \\dd \\mathcal { B } _ t , Z = \\begin{pmatrix} 0 & z I \\\\ \\overline { z } I & 0 \\end{pmatrix} , \\mathcal { B } _ t = \\begin{pmatrix} 0 & B _ t \\\\ B ^ * _ t & 0 \\end{pmatrix} \\end{align*}"}
+{"id": "7545.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial n _ + } \\left ( 2 G ^ \\mu + \\Re V \\right ) = \\ \\frac { \\partial } { \\partial n _ + } \\left ( 2 G ^ \\mu + \\Re V \\right ) D \\cap \\Sigma , \\end{align*}"}
+{"id": "4775.png", "formula": "\\begin{gather*} [ p \\otimes a , q \\otimes b ] = p q \\otimes a \\diamond b - q p \\otimes b \\diamond a , \\\\ ( p \\otimes a ) \\cdot ( q \\otimes b ) = p q \\otimes a * b + q p \\otimes b * a , \\forall p , q \\in P , a , b \\in A . \\end{gather*}"}
+{"id": "683.png", "formula": "\\begin{align*} d g _ 1 '' = ( \\log \\sqrt { \\tau _ 0 } ) _ z d z \\ g '' _ 1 + e ^ { 2 i \\theta } \\frac { 1 } { \\sqrt { \\tau _ 0 } } \\left ( Q _ 0 d z + \\frac { \\tau _ 0 } { 4 } d \\overline { z } \\right ) \\ J \\widehat { g _ 1 '' } I ; \\end{align*}"}
+{"id": "5143.png", "formula": "\\begin{align*} A _ 0 = \\bigcup _ { \\substack { \\widetilde Q \\in \\widetilde { \\mathcal W } \\\\ | c \\widetilde Q \\cap A ' | > | c \\widetilde Q \\cap ( \\Omega \\setminus A ' ) | } } \\widetilde Q . \\end{align*}"}
+{"id": "3387.png", "formula": "\\begin{align*} \\sum _ { i < \\alpha } r _ i s _ { h - g _ i } = 0 . \\end{align*}"}
+{"id": "2774.png", "formula": "\\begin{align*} A = \\sum _ { i = 1 } ^ { r } \\sigma _ i u _ i v _ i ^ T . \\end{align*}"}
+{"id": "4218.png", "formula": "\\begin{align*} \\lim _ { v \\to \\gamma ^ { - 1 } } \\frac { Z ( v ) ( - \\alpha ( v - \\alpha ^ { - 1 } ) ) ^ r } { f ( v ) ( 1 - v ) } = \\frac { ( - 1 ) ^ { r } \\alpha ^ r Z ( \\alpha ^ { - 1 } ) r ! } { f ^ { ( r ) } ( \\alpha ^ { - 1 } ) ( 1 - \\alpha ^ { - 1 } ) } . \\end{align*}"}
+{"id": "7919.png", "formula": "\\begin{align*} C ^ n _ \\mathrm { R B A } ( ( A , P ) , ( M , Q ) ) = \\begin{cases} M & n \\geq 0 , \\\\ C ^ n ( A , M ) \\oplus C ^ { n - 1 } ( A _ P , \\overline { M } ) = \\mathrm { H o m } ( A ^ { \\otimes n } , M ) \\oplus \\mathrm { H o m } ( A ^ { \\otimes n - 1 } , M ) & n \\geq 1 . \\end{cases} \\end{align*}"}
+{"id": "1594.png", "formula": "\\begin{align*} \\left ( f _ { n } \\left ( g _ { s , j , i } \\right ) \\right ) \\left ( w \\right ) = \\varphi _ { s , j , i } \\left ( w , 0 \\right ) . \\end{align*}"}
+{"id": "2014.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } 4 ( e ^ { t / 2 } + e ^ { - t / 2 } - 2 ) \\ , e ^ { i z t } \\ , d t = - \\frac { 1 } { z ^ 2 } \\ , \\left ( \\frac { 1 } { s - 1 } + \\frac { 1 } { s } \\right ) ~ \\Im ( z ) > 1 / 2 , \\end{align*}"}
+{"id": "2962.png", "formula": "\\begin{align*} \\phi ( t ) : = \\begin{cases} 0 & t = 0 , \\\\ 1 & t \\neq 0 \\end{cases} \\forall t \\in \\R , \\end{align*}"}
+{"id": "2370.png", "formula": "\\begin{align*} \\mathcal { A } _ { s , t } ^ { 0 } ( S ) = \\mathbb { Q } \\oplus \\bigoplus _ { k = 1 } ^ { \\infty } \\bigoplus _ { \\substack { a _ { 1 } , \\dots , a _ { k } \\in S \\\\ a _ { 1 } \\neq s , a _ { k } \\neq t } } \\mathbb { Q } e _ { a _ { 1 } } \\cdots e _ { a _ { k } } . \\end{align*}"}
+{"id": "723.png", "formula": "\\begin{align*} \\rho { u _ z } = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { c _ { i z } } { f _ i } + \\frac { { \\Delta t } } { 2 } { F _ z } } . \\end{align*}"}
+{"id": "2479.png", "formula": "\\begin{align*} J ( u , \\ , A ) : = \\frac { 1 } { 2 } \\d j ( u , \\ , A ) + \\frac { 1 } { 2 } F _ A \\end{align*}"}
+{"id": "5943.png", "formula": "\\begin{align*} T _ { 0 , K } = \\{ x \\in \\Q _ q ^ k \\colon | x \\cdot ( \\xi - \\gamma ( a ) ) | & \\leq 1 \\} . \\end{align*}"}
+{"id": "5377.png", "formula": "\\begin{align*} A = A _ + d \\varsigma \\end{align*}"}
+{"id": "7173.png", "formula": "\\begin{align*} \\mathcal { V } _ 0 : = \\mathcal { O } _ { \\mathfrak { g } ^ { \\oplus 2 } } \\otimes ( \\mathfrak { g } ^ { \\lambda < 0 } ) ^ { \\oplus 2 } = \\bigoplus _ { i > j } \\mathcal { O } _ { \\mathfrak { g } ^ { \\oplus 2 } } ( \\beta _ i - \\beta _ j ) ^ { \\oplus 2 } \\to \\mathfrak { g } ^ { \\oplus 2 } \\end{align*}"}
+{"id": "5367.png", "formula": "\\begin{align*} \\mathcal { E } _ A = \\mathcal { V } _ A \\oplus \\mathcal { H } _ A \\end{align*}"}
+{"id": "4944.png", "formula": "\\begin{align*} A + 2 B = \\sum _ { i = 1 } ^ { ( k - 5 ) / 2 - t } i \\binom { n } { k - 2 - i } . \\end{align*}"}
+{"id": "6839.png", "formula": "\\begin{align*} d ( P _ { 1 } , P _ { 2 } ) & : = \\min ( d ( P _ { 1 } ) , d ( P _ { 2 } ) ) , \\\\ B ( P _ { 1 } , P _ { 2 } ) & : = \\max ( B ( P _ { 1 } ) , B ( P _ { 2 } ) ) , \\\\ I ( P _ { 1 } , P _ { 2 } , u , r ) & : = \\max ( I ( P _ { 1 } , u , r ) , I ( P _ { 2 } , u , r ) ) . \\end{align*}"}
+{"id": "4284.png", "formula": "\\begin{align*} & ( q ) _ { \\infty } \\sum _ { n = 1 } ^ { \\infty } N _ { \\textup { S C } } ( n ) q ^ n + \\frac { 1 } { 2 } \\frac { ( q ) _ { \\infty } } { ( - q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( 1 - q ^ n ) ( q ) _ n } \\left ( \\frac { ( - q ) _ n } { ( q ) _ n } - 1 \\right ) \\\\ & = \\frac { 1 } { 4 } - \\frac { 1 } { 4 } \\frac { ( q ) _ { \\infty } } { ( - q ) _ { \\infty } } + \\frac { 1 } { 2 } \\frac { ( q ) _ { \\infty } } { ( - q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - q ) _ n } { ( q ) _ n } \\frac { q ^ n } { 1 - q ^ n } . \\end{align*}"}
+{"id": "1248.png", "formula": "\\begin{align*} \\begin{aligned} & \\left [ \\int _ \\Omega | \\nabla u _ p | ^ { ( p - 1 ) s } d x + \\int _ { \\partial \\Omega } \\lambda | u _ p | ^ { ( p - 1 ) s } d \\mathcal H ^ { N - 1 } \\right ] ^ { \\frac 1 s } \\\\ & \\le \\left [ \\int _ \\Omega | \\nabla u _ p | ^ { p } d x + \\int _ { \\partial \\Omega } \\lambda | u _ p | ^ { p } d \\mathcal H ^ { N - 1 } \\right ] ^ { \\frac { p - 1 } p } \\Lambda ^ { \\frac 1 s - \\frac { p - 1 } p } \\le M ( f , g , \\lambda ) \\Lambda ^ { \\frac 1 s } \\end{aligned} \\end{align*}"}
+{"id": "3685.png", "formula": "\\begin{align*} & \\left \\{ \\{ 1 , t + 1 , t + 4 \\} , \\{ 1 , t + 2 , t + 4 \\} , \\{ 1 , t + 1 , t + 2 \\} \\} \\right \\} \\\\ = & \\left \\{ \\{ \\phi ( u _ 1 ' ) , \\phi ( u _ { t + 1 } ' ) , \\phi ( u _ { t + 4 } ' ) \\} , \\{ \\phi ( u _ 1 ' ) , \\phi ( u _ { t + 2 } ) , \\phi ( u _ { t + 4 } ' ) \\} , \\{ \\phi ( v ) , \\phi ( u _ { t + 1 } ) , \\phi ( u _ { t + 2 } ) \\} \\} \\right \\} \\subset \\Gamma _ { t + 2 } . \\end{align*}"}
+{"id": "6127.png", "formula": "\\begin{align*} & \\ | \\mathcal H ( 2 d + 1 , d , 2 ) | - | \\mathcal G ( 2 d + 1 , d ) | \\\\ = & \\ \\sum _ { i = d + 2 } ^ { 2 d } \\sum _ { j = d + 2 } ^ { i } \\binom { n - j } { d } - \\binom { n - 2 d - 2 } { d } - 2 n + 3 d + 5 \\\\ \\geq & \\ \\sum _ { i = d + 2 } ^ { d + 3 } \\sum _ { j = d + 2 } ^ { i } \\binom { n - j } { d } - \\binom { n - 2 d - 2 } { d } - 2 n + 3 d + 5 \\\\ > & \\ 2 \\binom { n - d - 2 } { d } - 2 n + 3 d + 5 > 4 ( n - d - 2 ) - 2 n + 3 d + 5 = 2 n - d - 3 > 0 , \\end{align*}"}
+{"id": "2907.png", "formula": "\\begin{align*} \\varphi ( z ) = \\sum _ { j = 1 } ^ d \\delta _ j ( z ) f _ j ( p ( z ) ) \\end{align*}"}
+{"id": "3590.png", "formula": "\\begin{align*} \\lambda = n ^ { - \\tfrac { b } { 1 + b } } , \\end{align*}"}
+{"id": "277.png", "formula": "\\begin{align*} ( d _ { n - 1 } , d _ { n - 2 } , \\ldots , d _ 0 ) & = ( r _ 1 , r _ 1 + r _ 2 , \\ldots , r _ 1 + r _ 2 + \\cdots + r _ n ) , \\\\ ( d _ { - n + 1 } , d _ { - n + 2 } , \\ldots , d _ 0 ) & = ( c _ 1 , c _ 1 + c _ 2 , \\ldots , c _ 1 + c _ 2 + \\cdots + c _ n ) . \\end{align*}"}
+{"id": "3214.png", "formula": "\\begin{align*} \\tilde { x } ^ T L ( G ) \\tilde { x } = x ^ T L ( H ) x = \\mu _ 2 ( H ) . \\end{align*}"}
+{"id": "7069.png", "formula": "\\begin{align*} I _ { S _ 1 } \\cap I _ { S _ 0 } = I _ { S _ 1 } I _ { S _ 0 } . \\end{align*}"}
+{"id": "5444.png", "formula": "\\begin{align*} P \\cdot [ \\ , \\wedge _ \\ell ( { \\rm I d } _ n + z T ) \\ , ] _ i \\ , \\ , = \\ , \\ , 0 { \\rm f o r } \\ , \\ , \\ , i = 1 , 2 , \\ldots , \\ell . \\end{align*}"}
+{"id": "3648.png", "formula": "\\begin{align*} \\lambda ( f ) = \\max \\left \\{ f ( x _ 1 , \\ldots , x _ m ) \\colon ( x _ 1 , \\ldots , x _ m ) \\in \\Delta _ { m - 1 } \\right \\} , \\end{align*}"}
+{"id": "4778.png", "formula": "\\begin{align*} u \\ast v : = \\mu ( T ( u ) ) v , \\forall u , v \\in V . \\end{align*}"}
+{"id": "5076.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": "2461.png", "formula": "\\begin{align*} \\abs { J _ * } \\ ! \\left ( V \\setminus \\bigcup _ { j \\in \\N } U _ j \\right ) = 0 \\end{align*}"}
+{"id": "6198.png", "formula": "\\begin{align*} | \\mathcal { I } ' _ k | & = \\sum ^ k _ { i = 0 } | \\mathcal { B } _ { i , i } | + | \\{ ( i , i , t , p ) : ( i , i , t , p ) \\in \\mathcal { I } ' _ { k - 1 } \\} | + | \\mathcal { I } ' _ { k - 1 } | \\\\ [ 0 . 1 c m ] & = \\frac { ( k + 1 ) ( k + 2 ) ( k + 3 ) } { 6 } + { k + 3 \\choose 4 } \\ \\ \\ \\ \\ \\\\ & = { k + 4 \\choose 4 } , \\end{align*}"}
+{"id": "2825.png", "formula": "\\begin{align*} d \\gamma _ s ^ { - 1 } = \\gamma _ s ^ { - 1 } \\left ( - ( \\mu _ s - \\sigma _ s ^ 2 ) d s - \\sigma _ s d W ^ 1 _ s \\right ) , s \\in [ 0 , T ] . \\end{align*}"}
+{"id": "8034.png", "formula": "\\begin{align*} X _ m = \\{ ( u , a ) \\ , | \\ , g _ m ( u ) = p _ m ( a ) , u \\in U _ m , a \\in A _ m \\} , \\end{align*}"}
+{"id": "5135.png", "formula": "\\begin{align*} \\partial ^ M A = \\{ x \\in \\mathbb R ^ n \\ , : \\ , \\overline { D } ( A , x ) > 0 \\overline { D } ( \\mathbb R ^ n \\setminus A , x ) > 0 \\} . \\end{align*}"}
+{"id": "4832.png", "formula": "\\begin{align*} \\begin{gathered} \\int _ S \\int _ { B _ R } \\rho ( x , s ) \\ , \\dd x \\dd \\mu ( s ) = \\int _ S \\int _ { B _ R } \\phi \\rho ( x , s ) \\ , \\dd x \\dd \\mu ( s ) = ( \\phi , \\rho ) _ { L ^ 2 ( S , L ^ 2 ( B _ R ) ; \\mu ) } = \\\\ = ( L ^ * \\phi , \\rho ) _ { L ^ 2 ( S , L ^ 2 ( B _ R ) ; \\mu ) } = ( \\phi , L \\rho ) _ { L ^ 2 ( S , L ^ 2 ( B _ R ) ; \\mu ) } . \\end{gathered} \\end{align*}"}
+{"id": "7633.png", "formula": "\\begin{align*} \\mathcal { X } _ { L T } = [ U _ { \\Sigma _ { L T } } / G ] = [ \\C ^ 6 \\setminus \\{ 0 \\} / \\left ( \\C ^ \\ast \\times G _ { 8 1 } \\right ) ] , \\end{align*}"}
+{"id": "5465.png", "formula": "\\begin{align*} F ( a , b , t , N ) : = \\sum _ { n = 0 } ^ { N } \\frac { ( a q ) _ n } { ( b q ) _ n } t ^ n . \\end{align*}"}
+{"id": "4129.png", "formula": "\\begin{align*} \\delta _ n ^ 1 = \\sqrt { \\frac { 2 ( \\alpha ^ 2 - 1 / 4 ) } { \\chi ^ { \\alpha } _ n ( c ) } } < x _ n ^ * < \\delta _ n ^ 2 = \\frac { \\pi + \\frac { \\pi } { 2 } \\alpha - \\frac { 3 } { 4 } } { \\sqrt { \\chi _ n ^ \\alpha ( c ) + 1 / 4 - \\alpha ^ 2 } } . \\end{align*}"}
+{"id": "6786.png", "formula": "\\begin{align*} \\norm { \\int _ { \\abs { k } \\leq \\Lambda } d k \\ , G _ x ( k ) a _ k ^ * \\psi _ t ( x ) \\Psi } ^ 2 & = \\norm { \\int _ { \\abs { k } \\leq \\Lambda } d k \\ , G _ x ( k ) a _ { - k } \\psi _ t ( x ) \\Psi } ^ 2 + 4 \\pi \\Lambda \\abs { \\psi _ t ( x ) } ^ 2 \\norm { \\Psi } ^ 2 \\end{align*}"}
+{"id": "5196.png", "formula": "\\begin{align*} \\frac { 2 } { 3 } \\cdot \\frac { 1 } { X + 1 } - \\frac { 1 } { 3 } \\cdot \\frac { 1 } { 2 X + 1 } = \\sum _ { k = 0 } ^ \\infty \\left ( \\frac { 1 } { 3 } \\cdot ( 2 ( - 1 ) ^ k + 2 ^ k ) \\right ) X ^ k = \\sum _ { k = 0 } ^ \\infty \\ell _ k X ^ k \\enspace , \\end{align*}"}
+{"id": "572.png", "formula": "\\begin{align*} T _ { W , \\psi } ( \\mu _ \\infty ) - I ( \\mu _ \\infty ) & \\ge \\limsup _ { n \\to \\infty } \\big ( T _ { W _ { n J } , \\psi } ( \\mu _ n ( Q ^ n ) ) - I ( \\mu _ n ( Q ^ n ) ) \\big ) = \\limsup _ { n \\to \\infty } M _ n ^ \\psi ( Q ^ n ) / n . \\end{align*}"}
+{"id": "4674.png", "formula": "\\begin{align*} a _ { P ^ n } & = \\begin{cases} a _ P a _ { P ^ { n - 1 } } - p a _ { P ^ { n - 2 } } , & P \\nmid N _ E , \\\\ a _ P a _ { P ^ { n - 1 } } , & P \\mid N _ E , \\end{cases} \\end{align*}"}
+{"id": "2703.png", "formula": "\\begin{align*} \\binom { n - j } { a + j - l } \\binom { n - a - 2 j + l } { a - l } = \\binom { n - j } { a - l } \\binom { n - ( j + a - l ) } { j + a - l } \\end{align*}"}
+{"id": "1552.png", "formula": "\\begin{align*} \\varphi _ { s , i , j } \\left ( \\omega _ { i } , \\omega \\right ) = \\left ( f _ { s , i , j } \\left ( \\omega _ { i } , \\omega _ { s } \\right ) , \\omega \\right ) = \\left ( X _ { b _ { j } } \\left ( \\omega ^ { \\prime } \\right ) , \\omega \\right ) \\end{align*}"}
+{"id": "4874.png", "formula": "\\begin{align*} F _ n ( x , y ) = \\sum _ { \\pi \\in \\mathfrak { S } _ n } x ^ { \\mathrm { e x c } ( \\pi ) } y ^ { \\mathrm { c y c } ( \\pi ) } , ( n > 0 ) . \\end{align*}"}
+{"id": "7775.png", "formula": "\\begin{align*} \\mathcal { C } h ( X , g ^ + ) = \\inf \\limits _ { \\Omega } \\frac { V o l ( \\partial \\Omega ) } { V o l ( \\Omega ) } \\end{align*}"}
+{"id": "6700.png", "formula": "\\begin{align*} Q ( F _ 1 , F _ 2 ) ( v ) = \\int _ { \\mathbb { R } ^ { 3 } } \\int _ { \\mathbb { S } ^ { 2 } } | ( v - v _ { * } ) \\cdot \\omega | \\{ F _ 1 ( v ' ) F _ 2 ( v ' _ { * } ) - F _ 1 ( v ) F _ 2 ( v _ { * } ) \\} \\ , d \\omega \\ , d v _ { * } , \\end{align*}"}
+{"id": "5237.png", "formula": "\\begin{align*} \\gamma = \\mathrm { r e s } _ v \\left ( \\left ( L ^ \\mathrm { A H } ( \\varepsilon ( v ) ) \\frac { d L ^ { \\mathrm { A H } } ( \\eta ( v ) ) } { d v } - L ^ { \\mathrm { A H } } ( \\varepsilon ( v ) ) d _ { \\mathrm { l o g } } ( B ( v ) ) + L ^ { \\mathrm { A H } } ( \\eta ( v ) ) d _ { \\mathrm { l o g } } ( A ( v ) ) \\right ) \\left ( \\frac { 1 } { z ( v ) ^ p - 1 } \\right ) \\right ) , \\end{align*}"}
+{"id": "1232.png", "formula": "\\begin{align*} \\nu _ { 1 5 , Z ( \\mathbb { Z } _ { 1 5 } ) } = \\big \\{ & \\{ 3 , 3 \\} , \\{ 3 , 5 \\} , \\{ 3 , 6 \\} , \\{ 3 , 9 \\} , \\{ 3 , 1 0 \\} , \\{ 3 , 1 2 \\} , \\{ 5 , 5 \\} , \\{ 5 , 6 \\} , \\\\ & \\{ 5 , 9 \\} , \\{ 5 , 1 0 \\} , \\{ 5 , 1 2 \\} , \\{ 6 , 9 \\} , \\{ 9 , 1 0 \\} , \\{ 9 , 1 2 \\} \\big \\} . \\end{align*}"}
+{"id": "52.png", "formula": "\\begin{align*} \\S = \\{ v \\in C ^ { 2 } ( \\overline \\O ) : v | _ { ( \\partial \\O ) _ { j } } = \\phi _ { j } \\} . \\end{align*}"}
+{"id": "7624.png", "formula": "\\begin{align*} g \\cdot ( p , t ) = ( g \\cdot p , \\chi ( g ) t ) , \\ \\ g \\in G , \\ ( p , t ) \\in \\C ^ r \\times \\C . \\end{align*}"}
+{"id": "4861.png", "formula": "\\begin{align*} \\partial _ t \\rho = \\nabla _ x \\cdot \\left ( \\rho \\left ( \\nabla V _ s ( x ) + \\int _ { \\R ^ { d + 2 } } \\nabla W _ s ( x , y ) \\ , \\bar \\rho ( d y ) \\right ) \\right ) + K \\int _ S ( \\rho ( s ' ) - \\rho ( s ) ) \\ , \\dd \\mu ( s ' ) . \\end{align*}"}
+{"id": "1380.png", "formula": "\\begin{align*} & I _ 0 [ u _ 0 , u _ 1 ] \\\\ & : = \\int _ { \\Omega } \\left [ ( | u _ 1 ( x ) | ^ 2 + | \\nabla u _ 0 ( x ) | ^ 2 + | u _ 0 ( x ) | ^ { p + 1 } ) \\langle x \\rangle ^ { \\alpha } + | u _ 0 ( x ) | ^ 2 \\langle x \\rangle ^ { - \\alpha } \\right ] \\langle x \\rangle ^ { \\lambda ( 2 - \\alpha ) } \\ , d x \\\\ & < \\infty \\end{align*}"}
+{"id": "8015.png", "formula": "\\begin{align*} T _ Q = \\begin{bmatrix} Q _ 0 & r o w ( Q _ { - j } ) _ { j \\geq 1 } \\\\ c o l ( Q _ j ) _ { j \\geq 1 } & T _ Q \\ : \\otimes \\ : I _ d \\end{bmatrix} . \\end{align*}"}
+{"id": "2405.png", "formula": "\\begin{align*} { \\rm C u t } _ { ( a _ { 1 } , \\dots , a _ { n } ; b _ { 1 } , \\dots , b _ { m } ) } ( e _ { s _ { 1 } } \\cdots e _ { s _ { t } } ) = \\begin{cases} w ' & e _ { s _ { 1 } } \\cdots e _ { s _ { t } } \\ e _ { a _ { 1 } } \\cdots e _ { a _ { n } } w ' e _ { b _ { 1 } } \\cdots e _ { b _ { m } } \\\\ { \\rm o t h e r w i s e } & 0 . \\end{cases} \\end{align*}"}
+{"id": "4047.png", "formula": "\\begin{align*} e _ b ( y ) = ( p - 1 ) ^ { - 1 } \\left ( \\sum _ { \\ell = 0 } ^ { L } \\left ( - \\frac { b y ^ { 2 k } } { p - 1 } \\right ) ^ \\ell + \\left ( - \\frac { b y ^ { 2 k } } { p - 1 } \\right ) ^ { L + 1 } e _ b ( y ) \\right ) , \\forall L \\in \\mathbb { N } ^ * . \\end{align*}"}
+{"id": "7208.png", "formula": "\\begin{align*} _ d : = H ^ \\cdot ( \\mathcal { B } P S _ d ) \\hookrightarrow H ^ \\cdot \\left ( ^ d ( \\mathbb { C } ^ 2 ) , { } ^ p \\tau ^ { \\leq 0 } R \\pi _ * \\omega _ { \\mathcal { C } ( d ) } \\right ) \\hookrightarrow H ^ { \\mathrm { B M } } _ \\cdot ( \\mathcal { C } ( d ) ) . \\end{align*}"}
+{"id": "2699.png", "formula": "\\begin{align*} \\binom { n - j } { k - j } \\binom { n - k } { a + j - l } = \\binom { n - j } { a + j - l } \\binom { n - a - 2 j + l } { k - j } \\end{align*}"}
+{"id": "3784.png", "formula": "\\begin{align*} C _ { i _ 1 , \\ldots , i _ k , i ' _ 1 , \\ldots , i ' _ { k ' } } ( 1 ) = & C _ { i _ 1 , \\ldots , i _ k } ( 1 ) C _ { i ' _ 1 , \\ldots , i ' _ { k ' } } + C _ { i ' _ 1 , \\ldots , i ' _ { k ' } } ( 1 ) C _ { i _ 1 , \\ldots , i _ k } \\\\ & + \\sum _ { r = 1 } ^ k \\sum _ { s = 1 } ^ { k ' } \\widetilde { C } _ { i _ r , i ' _ s } ( 1 ) C _ { i _ 1 , \\ldots , \\widehat { i _ r } , \\ldots , i _ k } C _ { i ' _ 1 , \\ldots , \\widehat { i ' _ s } , \\ldots , i ' _ { k ' } } \\end{align*}"}
+{"id": "1189.png", "formula": "\\begin{align*} \\pi ^ * ( \\eta ^ { - 1 } C ) = ^ * \\nu ^ { - 1 } C . \\end{align*}"}
+{"id": "2018.png", "formula": "\\begin{align*} \\frac { \\xi ' } { \\xi } \\left ( \\frac { 1 } { 2 } - i z \\right ) = i b + i \\sum _ \\gamma \\left ( \\frac { 1 } { z - \\gamma } + \\frac { 1 } { \\gamma } \\right ) , \\end{align*}"}
+{"id": "7815.png", "formula": "\\begin{align*} \\Lambda _ X ^ { w _ 1 } ( u ) & = w _ 1 ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) \\\\ & \\ge w _ 2 ( G ^ { - 1 } ( u ) ) g ( G ^ { - 1 } ( u ) ) \\\\ = & \\Lambda _ Y ^ { w _ 2 } ( u ) . \\end{align*}"}
+{"id": "2609.png", "formula": "\\begin{align*} 0 & = e _ y ( y , v ) = f ^ 2 ( y , v ) = f ( y , y ) f ( y , v ) + \\sum _ { y < u < v } f ( y , u ) f ( u , v ) \\\\ & = \\pm f ( y , v ) + \\sum _ { y < u < v } f ( y , u ) f ( u , v ) , \\end{align*}"}
+{"id": "3272.png", "formula": "\\begin{align*} R _ n \\big ( y _ 0 ; \\alpha , \\beta , \\gamma , \\delta ; q ^ 2 \\big ) = 1 . \\end{align*}"}
+{"id": "5348.png", "formula": "\\begin{align*} \\begin{aligned} \\theta \\wedge \\omega \\wedge d \\theta ^ { n + 1 - k } & = \\Gamma \\omega \\wedge d \\theta ^ { n + 2 - k } , & & , \\\\ \\theta \\wedge \\omega & = \\Gamma \\omega \\wedge d \\theta , & & . \\end{aligned} \\end{align*}"}
+{"id": "6807.png", "formula": "\\begin{align*} \\sigma ( x _ { [ 5 ] } ) = \\sum _ { i \\in [ q ] , j \\in [ t ] } & \\lambda _ { i j } \\Big ( \\sum _ { \\{ a , b , c \\} \\in \\binom { [ 4 ] } { 3 } } \\tau _ { i } ( x _ a , x _ b , x _ c ) \\tilde { \\gamma } _ i ( x _ { [ 4 ] \\setminus \\{ a , b , c \\} } , x _ 5 ) \\Big ) \\\\ & + \\sum _ { i \\in [ s _ 3 ] } \\Big ( \\sum _ { \\{ a , b \\} \\in \\binom { [ 4 ] } { 2 } } \\tilde { \\beta } '' _ { i } ( x _ a , x _ b , x _ 5 ) \\gamma '' _ i ( x _ { [ 4 ] \\setminus \\{ a , b \\} } ) \\Big ) . \\end{align*}"}
+{"id": "5286.png", "formula": "\\begin{align*} \\begin{cases} & \\sum _ { j = 1 } ^ { n + 1 } t _ { i \\ * ( n + 1 ) + j } = 0 , \\ \\ \\forall \\ 0 \\leq i \\leq m - 1 , \\\\ & \\sum _ { j \\in B } t _ j = 0 , \\ \\ \\forall B \\in \\pi . \\end{cases} \\end{align*}"}
+{"id": "2945.png", "formula": "\\begin{align*} h ( x ) = \\begin{cases} 1 & x = 0 , \\\\ 0 & x \\neq 0 , \\end{cases} \\Psi ( x ) = \\begin{cases} \\R & x = 0 , \\\\ \\varnothing & x \\neq 0 \\end{cases} \\forall x \\in \\R . \\end{align*}"}
+{"id": "7822.png", "formula": "\\begin{align*} \\Lambda _ Z ^ { w _ 1 } ( u ) = w _ 1 ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) = \\frac { ( - 1 ) ^ m ( 1 - u ) ( l n ( 1 - u ) ) ^ m } { \\lambda ^ { m - 1 } } , 0 < u < 1 . \\end{align*}"}
+{"id": "821.png", "formula": "\\begin{align*} J _ i ( \\tau _ i ) = \\mathbb { E } \\Bigg \\{ \\bar { \\theta } _ 2 \\int _ 0 ^ { \\tau _ i } e ^ { - \\beta t } f ( x _ i ( t ) ) d t - e ^ { - \\beta \\tau _ i } K \\Bigg \\} = \\bar { \\theta } _ 2 \\mathbb { E } \\Bigg \\{ \\int _ 0 ^ { \\tau _ i } e ^ { - \\beta t } f ( x _ i ( t ) ) d t - e ^ { - \\beta \\tau _ i } \\bar { K } _ 2 \\Bigg \\} . \\end{align*}"}
+{"id": "8239.png", "formula": "\\begin{align*} Z _ { s , \\bar { s } } ^ \\ell ( t ) : = t ^ s \\Vert ( \\sigma , u ) ^ \\ell ( t ) \\Vert _ { \\dot { B } ^ { \\bar { s } + s \\alpha } _ { 2 , 1 } } = t ^ s \\sum _ { j \\leq j _ 0 } 2 ^ { j ( \\bar { s } + s \\alpha ) } \\| ( \\dot \\Delta _ j \\sigma , \\dot \\Delta _ j u ) ( t ) \\| _ { L ^ 2 } = t ^ s \\sum _ { j \\leq j _ 0 } 2 ^ { j ( \\bar { s } + s \\alpha ) } X _ j ( t ) , \\end{align*}"}
+{"id": "6532.png", "formula": "\\begin{align*} H = - i \\sum _ r \\log ( e ^ { i \\theta _ r } ) E _ r = \\sum _ r \\theta _ r E _ { \\theta _ r } , \\end{align*}"}
+{"id": "788.png", "formula": "\\begin{align*} \\phi ^ { L } ( z ^ { L , N _ { L } } ) ( \\cdot ) : = \\lambda h ( \\cdot ) + ( 1 - \\lambda ) z ^ { L , N _ { L } } ( \\cdot ) \\end{align*}"}
+{"id": "701.png", "formula": "\\begin{align*} \\rho { c _ v } \\frac { { \\partial T } } { { \\partial t } } + \\rho { c _ v } { \\bf { u } } \\cdot \\nabla T = \\nabla \\cdot \\left ( { \\lambda \\nabla T } \\right ) - T { \\left ( { \\frac { { \\partial { p _ { E O S } } } } { { \\partial T } } } \\right ) _ \\rho } \\nabla \\cdot { \\bf { u } } . \\end{align*}"}
+{"id": "4857.png", "formula": "\\begin{align*} \\frac { \\partial \\rho } { \\partial t } = \\nabla \\cdot ( \\rho \\nabla f ( x , s ) ) + D ( s ) \\Delta \\rho + K ( \\bar { \\rho } \\otimes \\mu - \\rho ) , \\end{align*}"}
+{"id": "1691.png", "formula": "\\begin{align*} \\varphi ( \\cdot ) \\equiv \\varphi ( \\cdot , t ) : = S _ t \\varphi _ 0 ( \\cdot ) \\in \\mathfrak { h } \\ , . \\end{align*}"}
+{"id": "7593.png", "formula": "\\begin{align*} \\tilde { G } ( p , q ) = \\tilde { G } ( q , p ) p \\neq q , \\end{align*}"}
+{"id": "2918.png", "formula": "\\begin{align*} { \\rm c a p } ( \\sigma ( A + K ) ) = { \\rm c a p } ( \\sigma ( A ) ) . \\end{align*}"}
+{"id": "5910.png", "formula": "\\begin{align*} A = \\left ( \\begin{smallmatrix} \\lambda & * & & & \\\\ & \\lambda & * & & \\\\ & & \\lambda & * & \\\\ & & & \\lambda & * \\\\ & & & & \\lambda \\end{smallmatrix} \\right ) \\ , , \\end{align*}"}
+{"id": "7945.png", "formula": "\\begin{align*} \\chi ^ { s ' } ( a , b ) = ~ & s ' ( a ) \\cdot _ { E ' } s ' ( b ) - s ' ( a \\cdot b ) \\\\ = ~ & ( \\varphi s ) ( a ) \\cdot _ { E ' } ( \\varphi s ) ( b ) - ( \\varphi s ) ( a \\cdot b ) \\\\ = ~ & \\varphi \\big ( s ( a ) \\cdot _ E s ( b ) - s ( a \\cdot b ) \\big ) = \\varphi ( \\chi ^ s ( a , b ) ) = \\chi ^ s ( a , b ) \\end{align*}"}
+{"id": "3194.png", "formula": "\\begin{align*} & B ( \\alpha _ { n } ( u _ { n } - u ) ) - B ( u _ { n } - u ) = ( \\alpha _ { n } ^ { 4 } - 1 ) B ( u _ { n } - u ) = o ( 1 ) \\\\ & C ( \\alpha _ { n } ( u _ { n } - u ) ) - C ( u _ { n } - u ) = ( \\alpha _ { n } ^ { p } - 1 ) C ( u _ { n } - u ) = o ( 1 ) . \\end{align*}"}
+{"id": "6859.png", "formula": "\\begin{align*} \\beta \\lambda u _ i h ( a _ i ) f ^ { p ^ { e ' } } ( a _ i ) = u _ i g ( a _ i ) = \\lambda u _ i h ( a _ i ) f ^ { p ^ { e ' } } ( a _ i ) , \\ 1 \\leq i \\leq s . \\end{align*}"}
+{"id": "6117.png", "formula": "\\begin{align*} \\begin{aligned} | \\mathcal B ^ { ( d ) } | \\geq \\left \\lceil k - d - \\frac { 1 } { 2 } - C \\frac { \\binom { n - d - 1 } { k - d - 1 } } { \\binom { n - d } { k - d } } \\right \\rceil = \\left \\lceil k - d - \\frac { 1 } { 2 } - C \\frac { ( k - d ) } { n - d } \\right \\rceil \\geq k - d , \\end{aligned} \\end{align*}"}
+{"id": "719.png", "formula": "\\begin{align*} g _ i ^ { \\left ( { n e q } \\right ) } \\left ( { { { \\bf { x } } _ b } , t } \\right ) = { g _ i } \\left ( { { { \\bf { x } } _ f } , t } \\right ) - g _ i ^ { \\left ( { e q } \\right ) } \\left ( { { { \\bf { x } } _ f } , t } \\right ) . \\end{align*}"}
+{"id": "5539.png", "formula": "\\begin{align*} - \\lambda _ 1 ( \\theta _ m ) \\dfrac { \\partial \\theta _ 1 ( \\beta ( t ) , t ) } { \\partial r } = - \\lambda _ 2 ( \\theta _ m ) \\dfrac { \\partial \\theta _ 2 ( \\beta ( t ) , t ) } { \\partial r } + l _ m \\gamma _ m \\dfrac { d \\beta } { d t } , \\end{align*}"}
+{"id": "2798.png", "formula": "\\begin{align*} \\begin{bmatrix} I _ m & 0 \\end{bmatrix} Q V _ k & = U _ { k + 1 } J _ k , & Q Q ^ T \\begin{bmatrix} U _ { k + 1 } \\\\ 0 \\end{bmatrix} & = Q V _ k J _ k ^ T + \\alpha _ { k + 1 } Q v _ { k + 1 } e _ { k + 1 } ^ T , \\\\ \\begin{bmatrix} 0 & I _ p \\end{bmatrix} Q V _ k & = \\widehat { U } _ { k } \\check { J } _ k , & Q Q ^ T \\begin{bmatrix} 0 \\\\ \\widehat { U } _ { k } \\end{bmatrix} & = Q V _ k \\check { J } _ k ^ T + \\check { \\beta } _ { k } Q v _ { k + 1 } e _ { k } ^ T , \\end{align*}"}
+{"id": "5474.png", "formula": "\\begin{align*} F _ N ( a , b ; t ) = \\frac { ( 1 - t q ^ N ) ( 1 - b ) } { ( 1 - b q ^ N ) ( 1 - t ) } \\sum _ { n = 0 } ^ { N } \\frac { ( q ^ { - N } ) _ n ( a t q / b ) _ n ( q ) _ n q ^ n } { ( t q ) _ n ( q ^ { 1 - N } / b ) _ n ( q ) _ n } . \\end{align*}"}
+{"id": "1728.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u + ( \\Delta - \\kappa ) u = \\left ( w * | u | ^ 2 \\right ) u \\\\ u _ 0 = \\varphi . \\end{cases} \\end{align*}"}
+{"id": "4551.png", "formula": "\\begin{align*} T I ( G ) + \\overline { T I } ( G ) = ( n - 1 ) \\sum _ { u \\in V ( G ) } f ( d _ u ) = ( n - 1 ) H _ f ( G ) \\ , . \\end{align*}"}
+{"id": "1146.png", "formula": "\\begin{align*} \\lambda _ { k , S ; ( n , g ) } ( l , r ) : = \\frac { \\Gamma _ n ( k - \\frac { n + g + 1 } { 2 } ) ( 4 l - S ^ { - 1 } [ r ] ) ^ { - k + ( n + g + 1 ) / 2 } } { ( \\pi ) ^ { n k - n ( n + g + 1 ) / 2 } ( \\det 2 S ) ^ { n / 2 } } . \\end{align*}"}
+{"id": "7581.png", "formula": "\\begin{align*} E _ { \\varphi _ m } [ \\mu ] & = \\left ( \\iint _ { F _ m \\times F _ m } + \\iint _ { ( X \\times X ) \\setminus ( F _ m \\times F _ m ) } \\right ) [ G ( p , q ) + \\frac 1 2 ( \\varphi _ m ( p ) + \\varphi _ m ( q ) ) ] d \\mu ( p ) d \\mu ( q ) \\\\ & \\geq \\mu ( F _ m ) ^ 2 E _ { \\varphi _ m } [ \\tilde \\mu ] + ( M + 1 ) ( 1 - \\mu ( F _ m ) ^ 2 ) . \\end{align*}"}
+{"id": "2581.png", "formula": "\\begin{align*} \\{ \\lambda _ { t } ^ { N , j } \\} _ { 1 \\leq j \\leq N } \\subset \\Big \\{ \\lambda \\in \\mathbb C \\ , : \\ , \\left | \\lambda - 1 \\right | \\leq \\frac { 1 } { N - 1 } \\sum _ { j \\not = i } h ' ( \\mu ^ { N , j } _ { \\boldsymbol { \\alpha } ^ * _ t } ) \\Big \\} . \\end{align*}"}
+{"id": "594.png", "formula": "\\begin{align*} \\frac { \\psi ^ { + 0 } _ { n , p - 1 } } { \\psi ^ { + 0 } _ { n , p } } \\ \\frac { \\rho ^ { 0 + } _ { n , p } } { \\rho ^ { 0 + } _ { n - 1 , p } } = \\frac { ( n - p + a _ { 1 2 } - a _ { 1 2 3 } + j ^ { ( 3 ) } + 1 ) ( n - p + a _ { 1 2 } - a _ { 1 2 3 } - j ^ { ( 3 ) } ) } { ( n - p + a _ { 1 2 } - a _ { 1 2 3 } + j ^ { ( 3 ) } ) ( n - p + a _ { 1 2 } - a _ { 1 2 3 } - j ^ { ( 3 ) } - 1 ) } \\ . \\end{align*}"}
+{"id": "3434.png", "formula": "\\begin{align*} ( P ^ U , P ^ \\rho , P ^ S ) = - A _ U f _ { J _ { 2 , l } } ^ { ( l ) } D _ r ( U , \\rho , S ) - \\big ( A _ r f _ { J _ { 2 , l } } ^ { ( l ) } , ( D _ r ( A _ U f _ { J _ { 2 , l } } ^ { ( l ) } ) + \\tfrac { n - 1 } { r } A _ U ) \\rho , 0 \\big ) \\end{align*}"}
+{"id": "3560.png", "formula": "\\begin{align*} e _ 1 ( I , M ) \\leq \\binom { e _ 0 ( I , M ) - b } { 2 } + b + 1 - \\ell ( M / I M ) . \\end{align*}"}
+{"id": "7317.png", "formula": "\\begin{align*} r _ j = \\begin{cases} \\frac { q ^ m - 1 } 2 + q ^ { j - 1 } - q ^ { j + 1 } + 1 & { \\rm i f } ~ 1 \\leq j \\leq m - 2 , \\\\ \\frac { q ^ m - 1 } 2 + q ^ { m - 2 } + 2 & { \\rm i f } ~ j = m - 1 . \\end{cases} \\end{align*}"}
+{"id": "5442.png", "formula": "\\begin{align*} P \\cdot \\wedge _ \\ell ( { \\rm I d } _ n + z T ) \\ , \\ , = \\ , \\ , P . \\end{align*}"}
+{"id": "7730.png", "formula": "\\begin{align*} d i s t _ { g ^ + } ( q , E ' ) = d i s t _ { g ^ + } ( q , q ' ) \\end{align*}"}
+{"id": "8046.png", "formula": "\\begin{align*} { \\bf S } _ { \\textrm { d i r } } = [ { \\mathbf s } _ { \\textrm { d i r } } [ 1 ] , \\cdots , { \\mathbf s } _ { \\textrm { d i r } } [ K ] ] \\in \\mathbb { C } ^ { N _ r \\times K } . \\end{align*}"}
+{"id": "8010.png", "formula": "\\begin{align*} S _ { M \\otimes I _ d } ( I ) \\otimes I _ 2 = S _ { M \\otimes I _ d } ( J ) . \\end{align*}"}
+{"id": "7899.png", "formula": "\\begin{align*} & \\big ( - R ( a ) + a , - R ( a ) - a \\big ) \\bullet \\big ( - R ( b ) + b , - R ( b ) - b \\big ) \\\\ & = \\big ( R ( a ) \\cdot R ( b ) - R ( a ) \\cdot b - a \\cdot R ( b ) + a \\cdot b , ~ R ( a ) \\cdot R ( b ) + R ( a ) \\cdot b + a \\cdot R ( b ) + a \\cdot b \\big ) \\end{align*}"}
+{"id": "2241.png", "formula": "\\begin{align*} E _ 1 ^ D * q _ { k + 1 } ^ D ( \\boldsymbol { \\mathcal { T } } _ n ) & = E _ { k + 1 } ^ D * \\boldsymbol { \\mathcal { T } } _ n - \\boldsymbol { \\alpha } _ { k } \\times E _ { k + 1 } ^ D - \\boldsymbol { \\beta } _ { k } \\times E _ { k } ^ D , \\\\ p _ { k + 1 } ( \\boldsymbol { \\mathcal { T } } _ n ) \\times \\boldsymbol { \\beta } _ { k + 1 } & = \\boldsymbol { \\mathcal { T } } _ n * E _ { k + 1 } - E _ { k + 1 } \\times \\boldsymbol { \\alpha } _ { k } - E _ k . \\end{align*}"}
+{"id": "2875.png", "formula": "\\begin{align*} W _ { U _ { \\tau _ k , R } } ( x ) = W \\circ R _ { \\tau _ k } ( x ) \\end{align*}"}
+{"id": "1032.png", "formula": "\\begin{align*} \\{ E ( t _ { i _ j } ) \\} _ { j = 1 } ^ { \\infty } = \\{ E ( t _ i ) \\ , | \\ , E ( t _ i ) > 1 + \\varepsilon ) \\} _ { i = k _ 0 + 1 } ^ { \\infty } , \\end{align*}"}
+{"id": "7194.png", "formula": "\\begin{align*} \\bigoplus _ { d \\geq 0 } G _ T ( \\mathcal { C } ( d ) ) = \\bigoplus _ { v _ 1 / d _ 1 < \\cdots < v _ k / d _ k } K ( \\mathbb { T } _ T ( d _ 1 ) _ { v _ 1 } \\boxtimes \\cdots \\boxtimes \\mathbb { T } _ T ( d _ k ) _ { v _ k } ) . \\end{align*}"}
+{"id": "7439.png", "formula": "\\begin{align*} \\begin{matrix} p ^ \\dagger _ 0 = 2 a _ 0 ^ { \\dagger } , \\\\ \\displaystyle p ^ \\dagger _ k = \\frac { 1 } { \\prod _ { i = 1 } ^ { k } \\sqrt { ( s + i ) ( s - i + 1 ) } } \\left ( a _ { - k } ^ { \\dagger } J _ + ^ k + a _ k ^ \\dagger J _ - ^ k \\right ) , \\\\ \\displaystyle m ^ \\dagger _ k = \\frac { 1 } { \\prod _ { i = 1 } ^ { k } \\sqrt { ( s + i ) ( s - i + 1 ) } } \\left ( a _ { - k } ^ { \\dagger } J _ + ^ k - a _ k ^ \\dagger J _ - ^ k \\right ) . \\end{matrix} \\end{align*}"}
+{"id": "5779.png", "formula": "\\begin{align*} a _ { s , t } & \\le l _ { s , t } [ r ^ 2 + r + 1 + ( r + 1 ) s + t - l _ { s , t } - 1 ] + 1 \\\\ & \\le r [ r ^ 2 + r + 1 + ( r + 1 ) s + t - r - 1 ] + 1 \\\\ & = r [ r ^ 2 + ( r + 1 ) s + t ] + 1 \\end{align*}"}
+{"id": "5248.png", "formula": "\\begin{align*} R _ { T , l } ( g ) = \\sum ^ { * } g ( \\tilde { \\gamma } _ { j _ 1 } , \\ldots , \\tilde { \\gamma } _ { j _ l } ) , \\end{align*}"}
+{"id": "7285.png", "formula": "\\begin{align*} P _ 1 ( T ) = K _ x + K _ m , \\end{align*}"}
+{"id": "7700.png", "formula": "\\begin{align*} F _ 0 M _ f = { \\rm a d j } ( Z ) \\cdot \\big [ \\tfrac { 1 } { f } \\big ] \\end{align*}"}
+{"id": "133.png", "formula": "\\begin{align*} { [ k + 1 + \\ell ] \\choose [ q ] } = \\sum _ { i = - 1 } ^ { q } { [ k ] \\choose [ i ] } \\cdot { [ \\ell ] \\choose [ q - 1 - i ] } = { [ k ] \\choose [ q ] } + { [ \\ell ] \\choose [ q ] } + \\sum _ { i = 0 } ^ { q - 1 } { [ k ] \\choose [ i ] } \\cdot { [ \\ell ] \\choose [ q - 1 - i ] } , \\end{align*}"}
+{"id": "1943.png", "formula": "\\begin{align*} \\int \\zeta ^ q ( t ) \\ , d t = 1 , | \\zeta ' | \\le N _ 0 ( R _ 0 R _ 1 ) ^ { - 2 - 2 / q } . \\end{align*}"}
+{"id": "6469.png", "formula": "\\begin{align*} & H ^ { 2 \\theta } ( \\mathbb { R } ^ 2 \\times \\Sigma ^ 2 ) & = \\{ u \\in \\mathcal S ' ( \\mathbb { R } ^ 2 \\times \\Sigma ^ 2 ) ; \\sum _ { j , k } \\int _ { \\mathbb { R } ^ 2 } ( | \\xi _ 0 | ^ 2 + | \\eta _ 0 | ^ 2 + \\lambda _ j ^ 2 + \\lambda _ k ^ 2 ) ^ { 2 \\theta } | \\mathcal F u _ { j k } ( \\xi _ 0 , \\eta _ 0 ) | ^ 2 d \\xi _ 0 d \\eta _ 0 < \\infty \\} , \\end{align*}"}
+{"id": "6540.png", "formula": "\\begin{align*} U ^ 2 = \\exp \\left ( \\gamma ( U - U ^ T ) \\right ) \\end{align*}"}
+{"id": "2433.png", "formula": "\\begin{align*} A _ { r } = \\begin{cases} \\frac { 2 \\mu ^ { r } } { ( r + 2 ) ! } & r , \\\\ 0 & r , \\end{cases} \\end{align*}"}
+{"id": "827.png", "formula": "\\begin{align*} \\tau _ i ^ { * } = \\inf \\{ t : \\ W _ i ( t ) \\geq a ' t + b ' \\} \\end{align*}"}
+{"id": "6021.png", "formula": "\\begin{align*} L ^ { \\pi ^ y } _ { r } ( t ) : = L ^ { \\pi ^ { ( \\varepsilon ) } } _ { r } ( t ) , R ^ { \\pi ^ y } _ { r } ( t ) : = \\left ( R ^ { \\pi ^ { ( \\varepsilon ) } } _ { r } ( t ) - y + x \\right ) \\vee 0 , t \\geq 0 . \\end{align*}"}
+{"id": "512.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { X _ i } \\to Q , . \\end{align*}"}
+{"id": "2284.png", "formula": "\\begin{align*} \\dim ( W ^ m ) \\geq c ' \\dim ( W ) \\left ( \\frac m 5 \\right ) ^ { n + 1 } = \\frac { c ' } { 5 ^ { n + 1 } } \\dim ( W ) m ^ { n + 1 } = c _ { n + 1 } \\dim ( W ) m ^ { n + 1 } , m \\geq 2 . \\end{align*}"}
+{"id": "2840.png", "formula": "\\begin{align*} r \\cdot m \\cdot r ' = r m w ( r ' ) \\end{align*}"}
+{"id": "7907.png", "formula": "\\begin{align*} a ~ { \\cdot } _ S ~ u : = R ( a ) \\cdot u + a \\cdot S ( u ) u ~ { \\cdot } _ S ~ a : = S ( u ) \\cdot a + u \\cdot R ( a ) , \\end{align*}"}
+{"id": "1768.png", "formula": "\\begin{align*} \\begin{cases} 2 w _ { n , k , 2 } \\left ( Y , \\mathcal { Z } \\right ) = s + 1 & { \\rm f o r \\ ; } k \\ ; { \\rm o d d } , \\\\ 2 w _ { n , k , 2 } \\left ( Y , \\mathcal { Z } \\right ) = s & { \\rm f o r \\ ; } k \\ ; { \\rm e v e n } . \\end{cases} \\end{align*}"}
+{"id": "4252.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( q ) _ { n - 1 } a ^ n } { ( 1 - q ^ n ) ( a ) _ n } & = \\sum _ { n = 1 } ^ { \\infty } \\frac { a q ^ { n - 1 } } { ( 1 - a q ^ { n - 1 } ) ^ 2 } \\\\ & = \\sum _ { m = 1 } ^ { \\infty } m \\left ( \\frac { a } { q } \\right ) ^ m \\sum _ { n = 1 } ^ { \\infty } q ^ { m n } \\\\ & = \\sum _ { m = 1 } ^ { \\infty } \\frac { m a ^ m } { 1 - q ^ m } . \\end{align*}"}
+{"id": "4483.png", "formula": "\\begin{align*} \\imath _ { \\mathbb { X } _ f } \\Omega = \\dd f \\ , , \\end{align*}"}
+{"id": "1736.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\| f \\| _ { L ^ p } \\leq | \\langle f , \\psi \\rangle | - | \\langle f , \\psi - \\varphi \\rangle | \\leq | \\langle f , \\psi \\rangle - \\langle f , \\psi - \\varphi \\rangle | = | \\langle f , \\varphi \\rangle | , \\end{align*}"}
+{"id": "2449.png", "formula": "\\begin{align*} p _ { n } \\coloneqq \\frac { 4 1 } { 2 1 6 } n ^ { 4 } - \\frac { 9 1 } { 1 0 8 } n ^ { 3 } + \\frac { 1 7 } { 1 2 } n ^ { 2 } + \\begin{cases} - \\frac { 1 1 } { 3 } n + 9 & n \\equiv 0 \\bmod { 6 } \\\\ - \\frac { 1 0 7 } { 2 7 } n + \\frac { 1 6 1 } { 2 7 } & n \\equiv 2 \\bmod { 6 } \\\\ - \\frac { 1 3 9 } { 2 7 } n + \\frac { 1 6 9 } { 2 7 } & n \\equiv 4 \\bmod { 6 } \\end{cases} \\end{align*}"}
+{"id": "6965.png", "formula": "\\begin{align*} \\vec { r } = \\vec { c } + \\vec { e } \\end{align*}"}
+{"id": "2600.png", "formula": "\\begin{align*} | \\phi ^ { N , i } ( t _ 0 , \\boldsymbol { \\xi } ) - \\Psi ^ { N , i } ( t _ 0 , \\boldsymbol { \\xi } ) | \\leq \\frac { C } { N } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 + \\frac { 1 } { N ^ 2 } \\underset { i , j = 1 } { \\overset { N } { \\sum } } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "4158.png", "formula": "\\begin{align*} \\nabla ^ b _ \\pm = \\nabla ^ b _ 0 \\pm \\psi \\end{align*}"}
+{"id": "7835.png", "formula": "\\begin{gather*} \\phi _ d ^ \\ast \\ , \\Delta _ { k - m } = A + T B + T ^ 2 C \\end{gather*}"}
+{"id": "2978.png", "formula": "\\begin{align*} [ Z _ i ] & = - [ X _ { i - 1 } ] + [ X _ { i + 1 } ] + [ Z _ { i - 1 } ] & \\ ; i = 3 , \\ldots , n - 1 \\\\ [ Y _ i ] & = - [ X _ { i + 1 } ] + [ X _ i ] & \\ ; i = 2 , \\ldots , n - 1 \\\\ [ Z _ 2 ] & = [ Y _ 1 ] + [ X _ 3 ] \\\\ [ Y _ n ] & = [ Z _ { n - 1 } ] - [ X _ { n - 1 } ] \\end{align*}"}
+{"id": "468.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\lfloor { n } / { 2 } \\rfloor } \\binom { n } { 2 k } ( 5 F _ j ^ 2 ) ^ k \\left ( \\frac { F _ { j ( n - 2 k + 1 ) } } { n - 2 k + 1 } { B _ { 2 k } } - \\frac { F _ j L _ j ^ { n - 2 k } } { 2 ^ n } \\right ) = 0 \\ , , \\end{align*}"}
+{"id": "3328.png", "formula": "\\begin{gather*} \\sum _ { j _ 1 = 0 } ^ { N _ 1 } \\sum _ { j _ 2 = 0 } ^ { N _ 2 } w _ 2 ( j _ 1 , j _ 2 , k _ 1 , k _ 2 ) P _ { n _ 1 ( k _ 1 ) , n _ 2 ( k _ 2 ) } ( m _ 1 ( j _ 1 ) , m _ 2 ( j _ 2 ) ) \\overline { P _ { n _ 1 ( k _ 1 ' ) , n _ 2 ( k _ 2 ' ) } ( m _ 1 ( j _ 1 ) , m _ 2 ( j _ 2 ) ) } \\\\ \\qquad { } = \\delta _ { k _ 1 , k _ 1 ' } \\delta _ { k _ 2 , k _ 2 ' } , \\end{gather*}"}
+{"id": "5097.png", "formula": "\\begin{align*} u ( N ^ { - 1 } x , N ^ { - 1 } y , N ^ { - 2 } \\mathfrak { m } ( N ) ^ { - 1 } t ) = N ^ { d - n } \\int _ { B _ 1 ^ { d - n } } \\sum _ { m \\in N ^ { - 1 } \\mathbb { Z } ^ n \\cap B _ 1 ^ { n } } \\hat { g } ( N \\xi , N m ) e ^ { i ( x \\cdot \\xi + y \\cdot m + t ( \\frac { \\mathfrak { m } ( N \\xi ) } { N ^ 2 \\mathfrak { m } ( N ) } + \\frac { \\mathfrak { m } ( N m ) } { N ^ 2 \\mathfrak { m } ( N ) } ) ) } \\ , d \\xi . \\end{align*}"}
+{"id": "810.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\lambda _ + = a ' + \\sqrt { ( a ' ) ^ 2 + 2 \\beta } , \\\\ & \\lambda _ - = a ' - \\sqrt { ( a ' ) ^ 2 + 2 \\beta } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "262.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\lambda ^ j _ i > \\lambda ^ j _ { i + 1 } \\Longleftrightarrow i \\in A ^ j \\\\ \\sigma _ { I ^ 1 } \\cup \\cdots \\cup \\sigma _ { I ^ k } = [ p t ] \\Longrightarrow \\frac 1 r \\sum _ { j = 1 } ^ k | \\lambda ^ j _ { I ^ j } | \\leq \\frac 1 n \\sum _ { j = 1 } ^ k | \\lambda ^ j | \\end{array} \\right . \\end{align*}"}
+{"id": "7386.png", "formula": "\\begin{align*} g = ( - \\Delta - \\lambda ) ^ { - 1 } f + \\alpha \\Psi _ { \\lambda } ( I - \\alpha \\lambda S ( \\lambda ) ) ^ { - 1 } \\Psi _ { \\overline { \\lambda } } ^ { \\ast } f , \\end{align*}"}
+{"id": "2760.png", "formula": "\\begin{align*} \\frac { b } { a } = \\alpha _ { G / P } = { \\rm i n d e x } ( G / P ) \\cdot \\alpha ( G / P , - K _ { G / P } ) \\leq \\frac { { \\rm i n d e x } ( G / P ) } { \\dim ( G / P ) } . \\end{align*}"}
+{"id": "2960.png", "formula": "\\begin{align*} H ( z _ 1 , z _ 2 , z _ 3 ) : = ( z _ 1 , z _ 3 ) , G ( z _ 1 , z _ 2 , z _ 3 ) : = ( z _ 1 , z _ 2 , z _ 2 , z _ 3 ) \\qquad \\forall ( z _ 1 , z _ 2 , z _ 3 ) \\in \\R ^ \\ell , \\end{align*}"}
+{"id": "4659.png", "formula": "\\begin{align*} \\mathcal { L } ( \\chi , u ) = \\prod _ { 1 \\leq j \\leq 2 g / ( \\ell - 1 ) } ( 1 - \\gamma _ j u ) , \\end{align*}"}
+{"id": "6562.png", "formula": "\\begin{align*} U ^ 2 = \\exp ( i H ) , \\end{align*}"}
+{"id": "5267.png", "formula": "\\begin{align*} & ( i ) \\ \\ c _ p ( t _ 1 , \\ldots , t _ p ) , \\ p > 1 , \\ \\ \\ \\sum _ { i = 1 } ^ p t _ i = 0 . \\\\ & ( i i ) \\ \\ c _ p ( t _ 1 , \\ldots , t _ p ) = 0 \\ \\ \\ p > 1 \\ \\ \\ \\ \\sum _ { i = 1 } ^ p t _ i \\neq 0 . \\\\ & ( i i i ) \\ \\ c _ 1 ( 0 ) = 1 \\ \\ \\ \\ c _ 1 ( t ) = 0 \\ \\ \\ \\ t \\neq 0 . \\end{align*}"}
+{"id": "5111.png", "formula": "\\begin{align*} \\nabla ( \\abs { u } ^ { p + 2 } ) = \\frac { p + 2 } { p } \\nabla ( \\abs { u } ^ p ) \\abs { u } ^ 2 , \\end{align*}"}
+{"id": "3825.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 n + h ^ 2 - 1 7 } { 2 } . \\end{align*}"}
+{"id": "8176.png", "formula": "\\begin{align*} \\langle ( I + | \\lambda | ^ 2 T ) ^ { - 1 } ( I + \\lambda U T ^ { 1 / 2 } ) a , ( I + \\lambda U T ^ { 1 / 2 } ) a \\rangle = \\| a \\| ^ 2 . \\end{align*}"}
+{"id": "4240.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( q ) _ n \\left ( \\frac { c } { d } \\right ) _ n ( - z d ) ^ n q ^ \\frac { n ( n + 1 ) } { 2 } } { ( z q ) _ n ( c q ) _ n } = \\frac { z } { c } ( c - d ) \\displaystyle \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( q ) _ n \\left ( \\frac { z d q } { c } \\right ) _ { n - 1 } ( c q ) _ { N - n } ( c q ) ^ n } { ( z q ) _ n ( c q ) _ N } . \\end{align*}"}
+{"id": "7299.png", "formula": "\\begin{align*} \\inf _ { v \\in V } \\sigma ( v ) = 0 . \\end{align*}"}
+{"id": "4713.png", "formula": "\\begin{align*} \\Phi _ { j , r , \\theta } ( s , \\varphi ) = \\frac { \\left ( 1 - r s + | \\sin \\theta | + | \\sin \\varphi | \\right ) ^ { - 2 \\lambda } } { \\left ( 1 - r s + \\left | \\sin ( \\theta - \\varphi ) / 2 \\right | \\right ) ^ { j + 2 } } . \\end{align*}"}
+{"id": "4243.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { z ^ n c ^ n q ^ { n ^ 2 } } { ( z q ) _ n ( c q ) _ n } = z \\sum _ { n = 1 } ^ { \\infty } \\frac { ( c q ) ^ n } { ( z q ) _ n } . \\end{align*}"}
+{"id": "3774.png", "formula": "\\begin{align*} C _ { i _ 1 , \\ldots , i _ k } = \\prod _ { r = 1 } ^ k \\left [ \\frac { 1 } { i _ r + 1 } \\binom { 2 i _ r } { i _ r } \\right ] = \\prod _ { r = 1 } ^ k C _ { i _ r } . \\end{align*}"}
+{"id": "3821.png", "formula": "\\begin{align*} a _ 1 = \\frac { 3 n - 1 2 - d } { 2 } . \\end{align*}"}
+{"id": "871.png", "formula": "\\begin{align*} \\Gamma ( \\alpha ) = \\lim _ { n \\rightarrow \\infty } \\frac { ( n - 1 ) ! \\ , n ^ { \\alpha } } { \\alpha ( \\alpha + 1 ) ( \\alpha + 2 ) \\cdots ( \\alpha + n - 1 ) } , \\ , \\alpha \\in { \\mathbb R } \\setminus \\{ 0 , - 1 , - 2 , \\ldots \\} , \\end{align*}"}
+{"id": "1649.png", "formula": "\\begin{align*} e ^ { \\lambda \\mathtt { P } } \\cdot Q = Q + \\lambda \\mathtt { X } ( Q , \\mathtt { P } ) + \\lambda ^ 2 \\mathtt { Y } ( Q , \\mathtt { P } ) + \\lambda ^ 3 \\ , \\mathtt { Z } ^ { ( \\lambda ) } ( Q , \\mathtt { P } ) \\ , . \\end{align*}"}
+{"id": "4562.png", "formula": "\\begin{gather*} M _ { g _ 1 - g _ 2 } = M _ { g _ 1 } - M _ { g _ 2 } = R _ 1 - \\Phi _ T ( M _ { f _ 1 } ) - R _ 2 + \\Phi _ T ( M _ { f _ 2 } ) = \\\\ = R _ 1 - R _ 2 - \\Phi _ T ( M _ { f _ 1 } - M _ { f _ 2 } ) \\end{gather*}"}
+{"id": "38.png", "formula": "\\begin{align*} \\P \\left \\{ Y \\geq x \\right \\} + \\sum _ { j = 1 } ^ { D - 2 } 2 ^ { j - 1 } \\P \\left \\{ 2 ^ { - j } Y \\geq x \\right \\} & \\leq 1 + \\sum _ { j = 1 } ^ { D - 2 } 2 ^ { j - 1 } = 2 ^ { D - 2 } . \\end{align*}"}
+{"id": "7449.png", "formula": "\\begin{align*} p _ 0 ^ { \\dagger } = 2 a _ 0 ^ { \\dagger } , \\sqrt { 2 } p _ 1 ^ { \\dagger } = a _ 1 ^ { \\dagger } J _ - + a _ { - 1 } ^ { \\dagger } J _ + , \\end{align*}"}
+{"id": "6240.png", "formula": "\\begin{align*} D ( H ) = \\{ \\psi \\in H ^ 2 ( \\Omega ) \\cap H ^ 1 _ 0 ( \\Omega ) : \\ : H \\psi \\in L ^ 2 ( \\Omega ) \\} , \\end{align*}"}
+{"id": "4800.png", "formula": "\\begin{align*} \\mathcal F _ { c , i } ( \\phi _ { i - 1 } ( n , k ) U _ { n , k } ) & = ( k + c ) \\phi _ i ( n , k ) U _ { n , k } , \\\\ \\mathcal F _ { c + 1 , i } ( \\psi _ { i - 1 } ( n , k ) U _ { n , k } ) & = ( k + c ) \\psi _ i ( n , k ) U _ { n , k } . \\end{align*}"}
+{"id": "8086.png", "formula": "\\begin{align*} \\alpha = \\frac { \\left ( \\lambda _ 1 - 1 \\right ) h ^ 2 } { \\lambda _ 1 \\left ( h ^ 2 + q ^ 2 \\right ) } . \\end{align*}"}
+{"id": "7212.png", "formula": "\\begin{align*} \\dim U = \\dim Z ( d ) ^ { \\mathrm { c l } } = \\frac { d ( d + 1 ) } { 2 } + 1 , \\ \\dim \\left ( Z ( d ) ^ { \\mathrm { c l } } \\setminus U \\right ) < \\dim Z ( d ) ^ { \\mathrm { c l } } . \\end{align*}"}
+{"id": "3937.png", "formula": "\\begin{align*} \\mathcal { H } ^ { d - 1 } ( \\Gamma _ n ) & = \\frac { 1 } { 2 h } \\int _ { U _ { h - t } ( \\Gamma _ \\infty ) } \\det ( d T _ n ) \\ , d y + \\frac { 1 } { 2 h } \\int _ { U _ h ( \\Gamma _ n ) \\setminus U _ { h - t } ( \\Gamma _ \\infty ) } \\det ( d T _ n ) \\ , d y \\end{align*}"}
+{"id": "1113.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } ( - 1 ) ^ { j } \\binom { n } { j } \\frac { \\left ( \\beta \\right ) ^ { \\left ( j + m \\right ) } } { \\left ( \\beta \\right ) ^ { \\left ( j \\right ) } } = ( - 1 ) ^ { n } n ! \\binom { m } { n } \\frac { \\left ( \\beta \\right ) ^ { \\left ( m \\right ) } } { \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } } . \\end{align*}"}
+{"id": "5398.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } d X ^ { g _ n } ( t ) - L \\Psi ( X ^ { g _ n } ( t ) ) d t = \\int _ Z f ( s , X ^ { g _ n } ( s ) , z ) ( g _ n ( s , z ) - 1 ) \\nu ( d z ) d t , \\\\ X ^ { g _ n } ( 0 ) = x \\in L ^ 2 ( \\mu ) , \\end{array} \\right . \\end{align*}"}
+{"id": "2344.png", "formula": "\\begin{align*} \\zeta ( \\{ \\{ 2 \\} ^ { m } , 1 , \\{ 2 \\} ^ { m } , 3 \\} ^ { n } , \\{ 2 \\} ^ { m } ) = q _ { n , m } \\cdot \\pi ^ { { \\rm w t } } , \\end{align*}"}
+{"id": "4639.png", "formula": "\\begin{align*} L ^ 2 _ { \\mathrm { s y m } } ( \\mathbb { R } ^ 6 ) : = \\left \\{ \\psi \\in L ^ 2 ( \\mathbb { R } ^ 6 ) \\ , \\big | \\ ; \\psi ( \\vec { x } , \\vec { y } ) = \\psi ( - \\vec { x } , \\vec { y } ) = \\psi \\ ! \\left ( \\tfrac { 1 } { 2 } \\ , \\vec { x } + \\vec { y } , \\tfrac { 3 } { 4 } \\ , \\vec { x } - \\tfrac { 1 } { 2 } \\ , \\vec { y } \\right ) \\right \\} \\ ! . \\end{align*}"}
+{"id": "5630.png", "formula": "\\begin{align*} D ^ T _ { \\frac { \\partial } { \\partial x ^ j } } \\frac { \\partial } { \\partial x ^ i } = \\hat { \\mathbb { G } } ^ k _ { i j } ( T ) \\frac { \\partial } { \\partial x ^ k } . \\end{align*}"}
+{"id": "4786.png", "formula": "\\begin{align*} x ( A _ m ( n ) p _ { n + m } ( x ) + \\dots + A _ 0 ( n ) p _ n ( x ) ) = B _ \\ell ( n ) p _ { n + \\ell } ( x ) + \\dots + B _ 0 ( n ) p _ n ( x ) , \\end{align*}"}
+{"id": "3907.png", "formula": "\\begin{align*} \\| S * g ( t ) \\| ^ 2 _ { \\mathbb H ^ \\mu } = \\sum _ { n = 1 } ^ \\infty \\lambda ^ \\mu _ n \\left ( \\int _ 0 ^ t \\omega ( t - \\tau , \\lambda _ n ) g _ n ( \\tau ) d \\tau \\right ) ^ 2 , \\ ; g _ n ( \\tau ) = ( g ( \\tau ) , e _ n ) . \\end{align*}"}
+{"id": "1538.png", "formula": "\\begin{align*} F ^ { \\prime } \\left ( f _ { 3 } \\right ) = F ^ { \\prime } \\left ( f _ { 2 } \\circ f _ { 1 } \\right ) \\end{align*}"}
+{"id": "4572.png", "formula": "\\begin{align*} H u ( n ) = ( \\Delta + V ) u ( n ) = u ( n + 1 ) + u ( n - 1 ) + V ( n ) u ( n ) , \\end{align*}"}
+{"id": "4589.png", "formula": "\\begin{align*} \\ln F ( n _ 0 + ( m + 1 ) & N ) ^ 2 = \\ln F ( n _ 0 + m N ) ^ 2 \\\\ - & \\frac { K } { 2 ( n _ 0 + m N - b ) \\sin \\pi k } ( 1 - \\cos 2 \\pi \\tilde \\varphi ( n _ 0 + m N ) + \\delta ( m ) ) \\end{align*}"}
+{"id": "2056.png", "formula": "\\begin{align*} f \\left ( \\frac { a z + b } { c z + d } \\right ) = \\chi ( d ) ( c z + d ) ^ k f ( z ) \\end{align*}"}
+{"id": "1105.png", "formula": "\\begin{align*} \\lambda ^ * = - 3 \\sqrt 2 \\frac { \\sqrt { 1 + a ^ 2 } } { 2 + \\frac { 1 9 7 } { 1 9 9 } a ^ 2 } \\eta . \\end{align*}"}
+{"id": "6981.png", "formula": "\\begin{align*} - \\int _ { X } \\langle \\nabla f , \\nabla g \\rangle d m = \\int _ { X } g d \\mu , g \\in \\mathrm { L i p } _ { b s } . \\end{align*}"}
+{"id": "2463.png", "formula": "\\begin{align*} \\abs { J _ * } \\ ! ( V ) = \\sum _ { j \\in \\N } \\abs { J _ * } \\ ! ( U _ j ) \\end{align*}"}
+{"id": "5062.png", "formula": "\\begin{align*} f _ n ( y ) = \\frac { \\mathbf { 1 } _ { ( 0 , 1 ) } ( y ) } { \\sqrt { 1 - y ^ 2 } } \\begin{cases} \\cos ( n \\arcsin { y } ) & 2 \\mid n , \\\\ \\sin ( n \\arcsin { y } ) & 2 \\nmid n . \\end{cases} \\end{align*}"}
+{"id": "3281.png", "formula": "\\begin{gather*} x _ 2 = \\frac { d _ 1 } { 1 - c _ 1 ( x _ 3 ) ^ 2 } . \\end{gather*}"}
+{"id": "7063.png", "formula": "\\begin{align*} f _ a = \\sum _ { i } \\tilde { a } _ j X _ j \\mod ( X _ { k } , \\ldots , X _ d ) ^ 2 . \\end{align*}"}
+{"id": "4034.png", "formula": "\\begin{align*} M = \\frac { 2 k p } { p - 1 } . \\end{align*}"}
+{"id": "3766.png", "formula": "\\begin{align*} b ( \\Gamma / e ) = b ( \\Gamma ) - 1 . \\end{align*}"}
+{"id": "4528.png", "formula": "\\begin{align*} \\{ A _ \\gamma ( t ) , ^ 2 E _ S [ f ] \\} = \\frac { \\tau _ k } { 2 } \\ , \\dot { \\gamma } ^ a ( t ) \\int _ { \\mathbf { x } \\in S } \\mathrm { d } S ^ { b c } ( \\mathbf { x } ) f ^ k ( \\mathbf { x } ) \\mathcal { E } ^ { ( 3 ) } _ { \\gamma ( t ) } ( \\mathbf { x } ) v _ { a b c } ( \\mathbf { x } ) \\ , . \\end{align*}"}
+{"id": "5704.png", "formula": "\\begin{gather*} D : \\Lambda \\to \\Lambda , s _ { \\lambda } \\mapsto s _ { \\langle \\lambda \\rangle } , \\end{gather*}"}
+{"id": "5188.png", "formula": "\\begin{align*} I _ { n , k } = 2 ^ { n - k } \\enspace . \\end{align*}"}
+{"id": "374.png", "formula": "\\begin{align*} \\tilde { l } _ k ( ( v _ 1 , \\alpha _ 1 ) , \\dots , ( v _ k , \\alpha _ k ) ) = \\varsigma ( k ) \\iota ( v _ 1 \\wedge \\dots \\wedge v _ k ) ~ \\omega \\end{align*}"}
+{"id": "1869.png", "formula": "\\begin{align*} M _ i & = m _ 1 m _ 2 \\cdots m _ i \\mbox { a n d } \\\\ N _ i & = n _ 1 n _ 2 \\cdots n _ i . \\end{align*}"}
+{"id": "2219.png", "formula": "\\begin{align*} \\frac { d } { d t } \\boldsymbol { \\mathsf { U } } ( t ) = \\boldsymbol { \\mathsf { A } } ( t ) \\boldsymbol { \\mathsf { U } } ( t ) , \\boldsymbol { \\mathsf { U } } ( { a } ) = I _ N , t \\in I = [ { a } , b ] , \\end{align*}"}
+{"id": "1804.png", "formula": "\\begin{align*} J ( x , y , [ x , z ] ) = [ J ( x , y , z ) , x ] , \\end{align*}"}
+{"id": "4780.png", "formula": "\\begin{align*} u \\ast ' v & \\overset { \\hphantom { ( 0 . 0 0 ) } } { = } \\mu ( T ( u ) ) v , \\\\ u \\diamond ' v & \\overset { \\hphantom { ( 0 . 0 0 ) } } { = } \\rho _ \\mu ( T ( u ) ) v = \\mu ( \\partial _ 1 ( T ( u ) ) ) \\alpha _ 2 ( v ) - \\mu ( \\partial _ 2 ( T ( u ) ) ) \\alpha _ 1 ( v ) \\\\ & \\overset { \\eqref { e q : o o p 2 } } { = } \\mu ( T ( \\alpha _ 1 ( u ) ) ) \\alpha _ 2 ( v ) - \\mu ( T ( \\alpha _ 2 ( u ) ) ) \\alpha _ 1 ( v ) . \\end{align*}"}
+{"id": "8062.png", "formula": "\\begin{align*} g ( { \\bf x } _ r ) = { \\bf x } _ r ^ T { \\bf H } _ r { \\bf x } _ r , \\end{align*}"}
+{"id": "1117.png", "formula": "\\begin{align*} f _ { N } ( x , y | \\rho ) = \\frac { 1 } { 2 \\pi ( 1 - \\rho ^ { 2 } ) } \\exp \\left ( - \\frac { x ^ { 2 } - 2 \\rho x y + y ^ { 2 } } { 2 ( 1 - \\rho ^ { 2 } ) } \\right ) , \\end{align*}"}
+{"id": "8023.png", "formula": "\\begin{align*} \\begin{bmatrix} \\Tilde { A } & r o w ( Q _ j ^ * ) _ { j = 1 } ^ m - X \\\\ c o l ( Q _ j ) _ { j = 1 } ^ m - X & S ( m - 1 ) \\otimes I _ d \\end{bmatrix} \\leq S ( m ) = \\begin{bmatrix} A & B ^ * \\\\ B & C \\end{bmatrix} . \\end{align*}"}
+{"id": "5836.png", "formula": "\\begin{align*} b e r ( U ) = \\sup \\{ | \\langle U e _ 1 , e _ 1 \\rangle | , | \\langle U e _ 2 , e _ 2 \\rangle | \\} = 0 , \\end{align*}"}
+{"id": "5051.png", "formula": "\\begin{align*} { } F _ k ( s , x ) = \\frac 1 { 2 \\pi i } \\int _ { \\Re ( u ) = \\sigma _ 1 } \\frac { \\Gamma _ \\C ( u + \\frac { k - 1 } { 2 } ) \\gamma ( 1 - s - u ) } { \\Gamma _ \\C ( - u + \\frac { k + 1 } { 2 } ) \\gamma ( s + u ) } x ^ u \\ , d u . \\end{align*}"}
+{"id": "2066.png", "formula": "\\begin{align*} N ( A ) : = \\sigma _ 1 \\cdot \\ldots \\cdot \\sigma _ { \\min \\{ m , n \\} } = \\max \\{ | \\det A _ { I , J } | : | I | = | J | = \\min \\{ n , m \\} \\} . \\end{align*}"}
+{"id": "6283.png", "formula": "\\begin{align*} \\tilde { f } _ * \\big ( \\Psi ^ { - 1 } \\mathbf { m } \\big ) = U \\cdot ( \\Psi ' ) ^ { - 1 } \\mathbf { m ' } . \\end{align*}"}
+{"id": "1413.png", "formula": "\\begin{align*} \\mathcal { A } = \\begin{pmatrix} 0 & 1 \\\\ \\Delta & - a ( x ) \\end{pmatrix} \\end{align*}"}
+{"id": "1814.png", "formula": "\\begin{align*} [ r _ { 1 2 } , r _ { 1 3 } ] & = \\sum _ { i , j } [ x _ i , x _ j ] \\otimes y _ i \\otimes y _ j , \\\\ [ r _ { 1 3 } , r _ { 2 3 } ] & = \\sum _ { i , j } x _ i \\otimes x _ j \\otimes [ y _ i , y _ j ] , \\\\ [ r _ { 1 2 } , r _ { 2 3 } ] & = \\sum _ { i , j } x _ i \\otimes [ y _ i , x _ j ] \\otimes y _ j . \\end{align*}"}
+{"id": "4245.png", "formula": "\\begin{align*} ( a q ) _ N \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { n a ^ n q ^ { n ^ 2 } } { ( a q ) _ n } = \\displaystyle \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( q ) _ n ( - 1 ) ^ { n - 1 } a ^ n q ^ \\frac { n ( n + 1 ) } { 2 } } { 1 - q ^ n } . \\end{align*}"}
+{"id": "2841.png", "formula": "\\begin{align*} B _ { s } : = R \\otimes _ { R ^ s } R . \\end{align*}"}
+{"id": "7300.png", "formula": "\\begin{align*} \\inf _ { v } \\sigma ( v ) = 0 . \\end{align*}"}
+{"id": "390.png", "formula": "\\begin{align*} \\sum _ i g _ i \\underline { \\xi } _ i = \\sum _ i g _ i v _ { \\mu _ { \\xi _ i } } = v _ { \\sum _ i g _ i \\underline { \\xi } _ i } - \\sum _ i \\mu _ i v _ { g _ i } \\ , . \\end{align*}"}
+{"id": "4120.png", "formula": "\\begin{align*} \\begin{cases} \\norm { \\left ( - \\mathcal { L } _ c ^ { \\alpha } \\right ) ^ m ( \\varphi ) } ^ 2 _ { L ^ 2 ( 0 , 1 ) } = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\left ( \\chi _ n ^ { \\alpha } ( c ) \\right ) ^ { 2 m } | a _ n ( \\varphi ) | ^ 2 \\\\ \\norm { \\left ( - \\mathcal { L } _ c ^ { \\alpha } \\right ) ^ { m + \\frac { 1 } { 2 } } ( \\varphi ) } ^ 2 _ { L ^ 2 ( 0 , 1 ) } = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\left ( \\chi _ n ^ { \\alpha } ( c ) \\right ) ^ { 2 m + 1 } | a _ n ( \\varphi ) | ^ 2 \\end{cases} . \\end{align*}"}
+{"id": "2397.png", "formula": "\\begin{align*} s = \\overrightarrow { u } _ { a _ { 1 } } , \\ t = \\begin{cases} \\overrightarrow { v } _ { a _ { n } } & \\bullet = { \\rm e v } \\\\ \\overrightarrow { v } _ { \\iota ( a _ { n } ) } & \\bullet = { \\rm o d } . \\end{cases} \\end{align*}"}
+{"id": "5128.png", "formula": "\\begin{align*} B V _ l ( \\Omega ) = \\{ u \\in L ^ { 1 } _ { l o c } ( \\Omega ) : \\ , \\| D u \\| ( \\Omega ) < \\infty \\} . \\end{align*}"}
+{"id": "8160.png", "formula": "\\begin{align*} \\widehat { \\deg } ( \\overline L ) & \\leqslant \\widehat { \\mu } _ { \\max } ( \\overline F ^ \\vee ) + \\sum _ { i = 1 } ^ d \\widehat { \\mu } _ { \\max } ( \\overline E _ d ) = \\widehat { \\mu } ( \\overline F ^ \\vee ) + \\sum _ { i = 1 } ^ d \\widehat { \\mu } _ { \\max } ( \\overline E _ d ) \\\\ & = - \\widehat { \\mu } ( \\overline F ) + \\sum _ { i = 1 } ^ d \\widehat { \\mu } _ { \\max } ( \\overline E _ d ) \\end{align*}"}
+{"id": "7367.png", "formula": "\\begin{align*} \\gamma : = \\frac { \\mu ( 1 - c _ 2 ) } { 2 ( 1 - c _ 1 ) } . \\end{align*}"}
+{"id": "7711.png", "formula": "\\begin{align*} R _ { i j k l } [ g ^ + ] = | d \\rho | ^ 2 _ { g } ( g ^ + _ { i k } g ^ + _ { j l } - g ^ + _ { i l } g ^ + _ { j k } ) + O ( \\rho ^ { - 3 } ) \\end{align*}"}
+{"id": "5144.png", "formula": "\\begin{align*} \\widetilde A = A \\cup A _ 0 . \\end{align*}"}
+{"id": "1812.png", "formula": "\\begin{align*} \\rho ( [ [ x , y ] , z ] ) = \\rho ( x ) \\rho ( y ) \\rho ( z ) - \\rho ( z ) \\rho ( x ) \\rho ( y ) + \\rho ( y ) \\rho ( [ z , x ] ) - \\rho ( [ y , z ] ) \\rho ( x ) . \\end{align*}"}
+{"id": "6141.png", "formula": "\\begin{align*} V ^ { * j } V _ 1 = ( V _ 2 ^ * V _ 1 ^ * ) ^ j V _ 1 = V _ 2 ^ * ( V _ 1 ^ * V _ 2 ^ * ) ^ { j - 1 } = q ^ { j - 1 } V _ 2 ^ * V ^ { * j - 1 } \\end{align*}"}
+{"id": "6686.png", "formula": "\\begin{align*} Q ( t ) = \\det \\begin{bmatrix} s _ 0 & s _ 1 & \\ldots & s _ { m - 1 } & 1 \\\\ s _ 1 & s _ 2 & \\ldots & s _ { m } & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ s _ { m } & s _ { m + 1 } & \\ldots & s _ { 2 m - 1 } & t ^ { m } \\end{bmatrix} , \\end{align*}"}
+{"id": "1907.png", "formula": "\\begin{align*} a _ { k + 1 } = \\min \\left \\{ \\sigma , ( k + 1 ) \\bigg ( \\frac 1 2 - \\frac 3 { 2 p } \\bigg ) + a _ 0 \\right \\} . \\end{align*}"}
+{"id": "525.png", "formula": "\\begin{align*} \\sup _ { Q \\in \\P ( \\R ^ n ) } \\left ( \\int _ { \\R ^ n } f \\ , d Q - H ( Q ) \\right ) = \\log Z \\in ( - \\infty , \\infty ) , \\end{align*}"}
+{"id": "4806.png", "formula": "\\begin{align*} \\mathcal G _ { c , i _ 0 + 1 } ( \\phi _ { i _ 0 } ( n , k ) U _ { n , k } ) & = B _ { i _ 0 + 1 } ( n , c ) ( k + c ) \\phi _ i ( n , k ) U _ { n , k } = 0 , \\\\ \\mathcal G _ { c + 1 , i _ 0 + 1 } ( \\psi _ { i _ 0 } ( n , k ) U _ { n , k } ) & = B _ { i _ 0 + 1 } ( n , c + 1 ) ( k + c ) \\psi _ i ( n , k ) U _ { n , k } = 0 . \\end{align*}"}
+{"id": "2193.png", "formula": "\\begin{align*} & f _ { 1 1 } = H ^ f _ { 1 1 } + \\Gamma ^ 1 _ { 1 1 } f _ 1 + \\Gamma ^ 2 _ { 1 1 } f _ 2 , \\\\ & f _ { 1 2 } = H ^ f _ { 1 2 } + \\Gamma ^ 1 _ { 1 2 } f _ 1 + \\Gamma ^ 2 _ { 1 2 } f _ 2 \\end{align*}"}
+{"id": "4491.png", "formula": "\\begin{align*} \\{ f , g \\} = \\int _ { \\mathbb { R } ^ n } \\left ( \\frac { \\delta f } { \\delta \\phi ( x ) } \\frac { \\delta g } { \\delta \\pi ( x ) } - \\frac { \\delta g } { \\delta \\phi ( x ) } \\frac { \\delta f } { \\delta \\pi ( x ) } \\right ) \\mathrm { d } ^ n x \\ , , \\end{align*}"}
+{"id": "3224.png", "formula": "\\begin{align*} \\mu _ 2 ( ( D - 1 ) G ) > ( D - 1 ) \\frac { 1 } { \\delta + 1 } \\left ( 2 + \\frac { 1 } { D - 1 } \\right ) = \\frac { 2 D - 1 } { \\delta + 1 } = \\frac { 2 D - 1 } { \\ell } . \\end{align*}"}
+{"id": "649.png", "formula": "\\begin{align*} A \\circ \\widetilde \\Phi \\circ \\widetilde { \\mathcal { P } } \\circ \\widetilde \\Phi ^ { - 1 } \\circ A ^ { - 1 } = \\mathcal { P } ' \\end{align*}"}
+{"id": "8031.png", "formula": "\\begin{align*} Q _ 0 & = \\sum _ { j = 0 } ^ n F _ j ^ * F _ j ; \\\\ Q _ { - j } & = \\sum _ { k = j } ^ { n } F _ k ^ * ( F _ { k - j } \\otimes I _ d ^ { \\otimes j } ) \\mbox { f o r } 1 \\leq j \\leq n . \\end{align*}"}
+{"id": "2517.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } T ^ { c ( n ) } ( x _ { 0 0 } , x _ { 0 1 } ) = ( x _ { 1 0 } , x _ { 1 1 } ) , \\end{align*}"}
+{"id": "6733.png", "formula": "\\begin{align*} \\overline { G } = \\varepsilon \\frac { \\sqrt { R } } { \\sqrt { \\theta } } \\sum ^ { 3 } _ { j = 1 } \\frac { \\partial \\bar { \\theta } } { \\partial x _ { j } } A _ { j } ( \\frac { v - u } { \\sqrt { R \\theta } } ) + \\varepsilon \\sum ^ { 3 } _ { j = 1 } \\sum ^ { 3 } _ { i = 1 } \\frac { \\partial \\bar { u } _ { j } } { \\partial x _ { i } } B _ { i j } ( \\frac { v - u } { \\sqrt { R \\theta } } ) . \\end{align*}"}
+{"id": "2654.png", "formula": "\\begin{align*} \\lfloor x \\rfloor + \\Big \\lfloor x + \\frac 1 n \\Big \\rfloor + \\ldots + \\Big \\lfloor x + \\frac { n - 1 } n \\Big \\rfloor = \\lfloor n x \\rfloor . \\end{align*}"}
+{"id": "1533.png", "formula": "\\begin{align*} g _ { s ^ { \\prime \\prime } , k , i } \\circ g _ { s ^ { \\prime } , j , k } \\circ g _ { s , i , j } = \\mathrm { i d } _ { b _ { i } } . \\end{align*}"}
+{"id": "7258.png", "formula": "\\begin{align*} F ( \\mu ) + \\epsilon & = \\inf _ { f \\in \\mathcal { A } } \\int f d \\mu + \\frac { \\epsilon } { 2 } + \\frac { \\epsilon } { 2 } \\\\ & = \\inf _ { f \\in \\mathcal { A } + \\frac { \\epsilon } { 2 } } \\int f d \\mu + \\frac { \\epsilon } { 2 } \\geq \\inf _ { f \\in \\mathcal { C } + \\frac { \\epsilon } { 2 } } \\int f d \\mu + \\frac { \\epsilon } { 2 } . \\end{align*}"}
+{"id": "4349.png", "formula": "\\begin{align*} \\Big [ t \\mapsto I ( t ) : = \\int _ \\Omega \\Phi ( f ( t ) ) \\ , \\mathrm { d } x \\Big ] \\in { \\rm C } ( [ 0 , T ] , \\R ) \\end{align*}"}
+{"id": "1089.png", "formula": "\\begin{align*} 2 u _ { 1 } ^ { 2 } - 2 p u _ { 2 } ^ { 2 } = 4 ^ { m } p ^ { 2 k + 1 } \\implies u _ { 1 } \\equiv 0 \\pmod p , \\end{align*}"}
+{"id": "4726.png", "formula": "\\begin{gather*} ( 1 + \\theta _ k ) r ( \\partial _ k ( a ) ) v = 0 , ( 1 + \\theta _ k ) l ( \\partial _ k ( a ) ) v = 0 , \\forall a \\in A , \\ v \\in V , \\ 1 \\leq k \\leq m . \\end{gather*}"}
+{"id": "4570.png", "formula": "\\begin{gather*} \\int f _ M d \\mu _ n = 0 . \\end{gather*}"}
+{"id": "5914.png", "formula": "\\begin{align*} L = E _ 8 ( - 1 ) ^ { \\oplus 2 } \\oplus U ^ { \\oplus 2 } \\oplus I ( - 2 ) ^ { \\oplus 2 } \\ , , \\end{align*}"}
+{"id": "6779.png", "formula": "\\begin{align*} A ( t ) = H ( t ) + \\widetilde { C } \\left ( \\mathcal { N } + 1 \\right ) \\end{align*}"}
+{"id": "3653.png", "formula": "\\begin{align*} p _ { \\mathcal { G } } ( X _ 1 , \\ldots , X _ m ) = \\sum _ { E \\in \\mathcal { G } } \\prod _ { i \\in E } X _ i . \\end{align*}"}
+{"id": "396.png", "formula": "\\begin{align*} v _ f = \\big ( y f ^ { x y | x } ( x ) + x y ( f ^ { x y | x } ) ' ( x ) + y f ^ { x y | y } ( y ) \\big ) \\partial _ y - \\big ( x f ^ { x y | y } ( y ) + x y ( f ^ { x y | y } ) ' ( y ) + x f ^ { x y | x } ( x ) \\big ) \\partial _ x . \\end{align*}"}
+{"id": "5245.png", "formula": "\\begin{align*} ( x - y ) _ c = \\begin{cases} x - y & - \\pi \\leq x - y < \\pi , \\\\ x - y - 2 \\ * \\pi & \\pi \\leq x - y < 2 \\ * \\pi , \\\\ x - y + 2 \\ * \\pi & - 2 \\ * \\pi < x - y < - \\pi . \\end{cases} \\end{align*}"}
+{"id": "3193.png", "formula": "\\begin{align*} \\left \\| u _ { n } - u \\right \\| _ { 2 } ^ { 2 } + \\left \\| u \\right \\| _ { 2 } ^ { 2 } = \\left \\| u _ { n } \\right \\| _ { 2 } ^ { 2 } + o ( 1 ) , \\end{align*}"}
+{"id": "2319.png", "formula": "\\begin{align*} \\d X _ t = - \\boldsymbol A X _ t \\d t + \\sigma \\d W _ t , t \\geq 0 , \\end{align*}"}
+{"id": "1110.png", "formula": "\\begin{align*} \\min _ { \\xi , \\eta } p ( \\xi , \\eta ) & = \\min \\left \\{ p \\left ( \\frac { 4 7 } { 5 0 } , 1 \\right ) , p \\left ( \\frac { 4 7 } { 5 0 } , \\frac { 9 9 } { 1 0 0 } \\right ) \\right \\} \\\\ & \\ge 0 . 0 0 4 6 . \\end{align*}"}
+{"id": "6811.png", "formula": "\\begin{align*} \\tilde { \\sigma } ( d , d , a _ 3 , \\dots , a _ k ) = \\sigma ' ( d , a _ 3 , \\dots , a _ k ) . \\end{align*}"}
+{"id": "5182.png", "formula": "\\begin{align*} \\left ( \\int _ { 2 c Q } | \\nabla E f ( x ) | ^ s \\ , \\d x \\right ) ^ \\frac { p } { s } \\ge \\left ( C ( 2 c ) ^ { n - s } \\ell ( Q ) ^ { n - s } \\right ) ^ \\frac { p } { s } = C ' \\ell ( Q ) ^ { n - p } \\ell ( Q ) ^ { ( \\frac { p } { s } - 1 ) n } \\end{align*}"}
+{"id": "4648.png", "formula": "\\begin{align*} \\hat { f } ^ { \\ , \\sharp } _ \\beta ( x ) = \\frac { 1 } { \\beta } \\ , \\hat { h } \\ ! \\left ( \\tfrac { x } { \\beta } \\right ) \\ ! \\end{align*}"}
+{"id": "3734.png", "formula": "\\begin{align*} \\mathbf n C ( \\operatorname { D e c k } ( p ) ) = C ( p ^ \\prime ) . \\end{align*}"}
+{"id": "3563.png", "formula": "\\begin{align*} e _ 1 ( I ) \\le \\binom { e _ 0 ( I ) } { 2 } - \\binom { \\mu ( I ) - 1 } { 2 } - \\ell ( A / I ) + 1 . \\end{align*}"}
+{"id": "5256.png", "formula": "\\begin{align*} T _ { N , s } = \\sum _ { m = 1 } ^ N e ^ { i \\ * s \\ * \\theta _ m } = T r ( U ^ s ) \\end{align*}"}
+{"id": "1493.png", "formula": "\\begin{align*} W = \\sum _ d h ( d ) y _ d ^ 2 - \\sum _ { p < z } g ( p ) \\sum _ { ( d , p ) = 1 } h ( d ) ( y _ d - y _ { p d } ) ^ 2 ; \\end{align*}"}
+{"id": "4853.png", "formula": "\\begin{align*} \\frac { \\partial \\bar \\rho _ 1 } { \\partial t } = \\nabla \\cdot \\left ( \\int _ S \\rho _ 1 \\nabla f \\ , \\dd \\mu \\right ) . \\end{align*}"}
+{"id": "6554.png", "formula": "\\begin{align*} U ^ { n - 1 } = I , \\end{align*}"}
+{"id": "5591.png", "formula": "\\begin{align*} g = g _ { i j } ( y ) d x ^ i \\otimes d x ^ j . \\end{align*}"}
+{"id": "1315.png", "formula": "\\begin{align*} f \\otimes ( g _ 1 + g _ 2 ) = ( f \\otimes g _ 1 ) + ( f \\otimes g _ 2 ) , ( f _ 1 + f _ 2 ) \\otimes g = ( f _ 1 \\otimes g ) + ( f _ 2 \\otimes g ) , \\end{align*}"}
+{"id": "566.png", "formula": "\\begin{align*} M _ n ^ \\psi ( Q ) & : = \\sum _ { i = 1 } ^ n \\int _ { \\R } V ( x ) \\ , Q _ i ( d x ) + \\sum _ { i , j = 1 } ^ n J _ { i j } \\int _ { \\R } \\int _ { \\R } \\psi ( x , y ) \\ , Q _ i ( d x ) Q _ j ( d y ) - \\sum _ { i = 1 } ^ n H ( Q _ i ) \\\\ & = \\sum _ { i , j = 1 } ^ n J _ { i j } \\int _ { \\R } \\int _ { \\R } \\psi ( x , y ) \\ , Q _ i ( d x ) Q _ j ( d y ) - \\sum _ { i = 1 } ^ n H ( Q _ i \\ , | \\ , \\rho ) . \\end{align*}"}
+{"id": "7617.png", "formula": "\\begin{align*} S : = \\C [ x _ \\rho \\ \\vert \\ \\rho \\in \\Sigma ( 1 ) ] . \\end{align*}"}
+{"id": "7583.png", "formula": "\\begin{align*} \\left | \\int \\varphi _ m d \\mu _ 2 - \\int \\varphi _ m d \\mu _ 1 \\right | = \\left | \\int ( \\widetilde { \\varphi } _ m - \\varphi _ m ) d \\mu _ 1 \\right | \\leq \\frac { \\varepsilon } { 3 } \\end{align*}"}
+{"id": "2546.png", "formula": "\\begin{align*} \\alpha _ t ^ { \\xi } = - R ^ { - 1 } B y _ t ^ { \\xi } - h ( \\mu _ t ) . \\end{align*}"}
+{"id": "1125.png", "formula": "\\begin{align*} E ( X - Y ) ^ { n } = \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ \\frac { n ! } { ( n / 2 ) ! } ( 1 - \\rho ) ^ { n / 2 } & & n \\end{array} \\right . . \\end{align*}"}
+{"id": "4381.png", "formula": "\\begin{align*} D _ x = \\Delta _ x - | x | ^ 2 , \\end{align*}"}
+{"id": "4983.png", "formula": "\\begin{align*} \\frac { \\partial \\ln ( R _ { \\rm F A } ) } { \\partial \\lambda } = \\frac { - 1 } { \\lambda ( 1 - \\lambda ) } + \\frac { w _ { c } ( \\frac { w _ { c } } { r } - 1 ) ( 1 - \\lambda ) ^ { \\frac { w _ { c } } { r } - 2 } } { 1 - ( 1 - \\lambda ) ^ { \\frac { w _ { c } } { r } - 1 } } , \\end{align*}"}
+{"id": "3753.png", "formula": "\\begin{align*} ( x + 6 ) ( ( x + 4 ) I + A [ \\mathcal C _ i ] ) ^ { - 1 } = \\frac { 1 } { q _ i ( - x - 4 ) } \\sum _ { k = 1 } ^ { \\deg m _ i } D ^ k m _ i ( - x - 4 ) A [ \\mathcal C _ i ] ^ { k - 1 } \\end{align*}"}
+{"id": "1372.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t ^ 2 u - \\Delta u + a ( x ) \\partial _ t u = f ( u ) , & t > 0 , x \\in \\Omega , \\\\ u ( t , x ) = 0 , & t > 0 , x \\in \\partial \\Omega , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\ \\partial _ t u ( 0 , x ) = u _ 1 ( x ) , & x \\in \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "2634.png", "formula": "\\begin{align*} \\lambda | ^ * _ { [ w _ 1 , w _ 2 ] } ( a ) = \\lambda ( w _ 1 a ) \\end{align*}"}
+{"id": "1604.png", "formula": "\\begin{align*} | A | = \\lambda H ( X _ A ) \\le \\lambda \\sum _ { e \\in A } H ( X _ e ) = \\lambda \\sum _ { e \\in A } \\frac { 1 } { \\lambda } = | A | . \\end{align*}"}
+{"id": "1343.png", "formula": "\\begin{align*} | \\bar { q } | = | \\nabla v ( 0 ) - q | \\le C \\varepsilon a . \\end{align*}"}
+{"id": "9.png", "formula": "\\begin{align*} \\phi ( \\pi \\cdot e _ { \\sigma } ) = \\phi ( e _ { \\pi \\sigma \\pi ^ { - 1 } } ) = \\phi ( e _ { ( \\pi \\sigma _ 1 \\ , \\pi \\sigma _ 2 \\ , \\ldots \\ , \\pi \\sigma _ m ) } ) = ( \\overline { \\pi r } , 1 ) = \\pi \\cdot \\phi ( e _ { \\sigma } ) , \\end{align*}"}
+{"id": "4198.png", "formula": "\\begin{align*} \\left ( A ^ { \\tau } _ { j , \\varepsilon } \\mathcal { G } \\right ) ( X ) = e ^ { - 2 \\tau } \\underset { Z \\in \\tilde { \\mathcal { N } } \\left ( \\mathcal { Q } _ { j , \\varepsilon } , X , \\tau \\right ) } { \\sum } \\mathcal { G } ( Z ) \\end{align*}"}
+{"id": "583.png", "formula": "\\begin{align*} 2 x _ { 1 2 3 4 } : = w _ { 1 2 , 3 , 4 } - w _ { 1 , 3 , 4 } - w _ { 2 , 3 , 4 } = w _ { 1 , 2 3 , 4 } - w _ { 1 , 2 , 4 } - w _ { 1 , 3 , 4 } = w _ { 1 , 2 , 3 4 } - w _ { 1 , 2 , 3 } - w _ { 1 , 2 , 4 } \\ , . \\end{align*}"}
+{"id": "1382.png", "formula": "\\begin{align*} ( f , g ) _ { H ^ k } & = \\sum _ { | \\alpha | \\le k } ( \\partial ^ { \\alpha } f , \\partial ^ { \\alpha } g ) _ { L ^ 2 } , \\| f \\| _ { H ^ k } = \\sqrt { ( f , f ) _ { H ^ k } } , \\end{align*}"}
+{"id": "5163.png", "formula": "\\begin{align*} C _ { } = C ( p ) \\Vert E \\Vert ^ { \\frac { 2 p } { 2 - p } } . \\end{align*}"}
+{"id": "4520.png", "formula": "\\begin{align*} \\{ \\Phi _ x , \\mathcal { V } \\} ( \\phi , \\pi ) & = - \\dd _ { ( \\phi , \\pi ) } \\mathcal { V } ( \\mathbb { X } _ { \\Phi _ { \\ ! x } } \\ ! ) = - \\big ( \\dd _ \\phi V \\circ \\mathrm { p r o j } _ 1 \\big ) ( \\mathbb { X } _ { \\Phi _ { \\ ! x } } \\ ! ) \\\\ & = - \\dd _ \\phi V ( X _ 1 ) = - \\dd _ \\phi V ( 0 ) = 0 \\ , , \\end{align*}"}
+{"id": "1184.png", "formula": "\\begin{align*} \\mathcal { R } _ \\lambda ( s _ i ) v _ T = \\pm \\frac { 1 } { a _ { i + 1 } - a _ { i } } v _ { T } + \\sqrt { 1 - \\frac { 1 } { ( a _ { i + 1 } - a _ { i } ) ^ 2 } } v _ { T ' } , \\end{align*}"}
+{"id": "5976.png", "formula": "\\begin{align*} \\Phi ( q ) : = \\sup \\{ \\lambda \\geq 0 : \\psi _ X ( \\lambda ) = q \\} . \\end{align*}"}
+{"id": "2474.png", "formula": "\\begin{align*} * ( \\omega \\otimes u ) : = ( * \\omega ) \\otimes u \\end{align*}"}
+{"id": "1098.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\sum _ { i = 1 } ^ m V _ i > \\left ( \\frac { \\pi } { 4 } + \\frac { 1 9 9 \\pi } { 3 2 0 0 k } \\right ) \\| \\mathbf { x } \\| . \\end{align*}"}
+{"id": "6020.png", "formula": "\\begin{align*} C ^ { ( \\theta , r ) } ( b ; w ) + \\rho ^ { ( \\theta ) } _ { b } ( b ; w ) & = \\dfrac { Z ^ { ( \\theta ) } ( b ) } { \\theta } \\bigg ( \\dfrac { \\Xi ^ { ( \\theta , r ) } ( b , w ' _ { + } ) } { Z ^ { ( \\theta ) } ( b ; \\Phi ( \\theta + r ) ) } + w ( 0 ) \\bigg ) , \\end{align*}"}
+{"id": "7046.png", "formula": "\\begin{align*} \\mathfrak { t r } _ 0 ^ p : = \\{ X \\in \\mathcal K ^ G ( M ) : X \\textrm { i s a t r a n s v e c t i o n a t p w i t h } X _ p \\in \\nu _ p \\} \\end{align*}"}
+{"id": "3426.png", "formula": "\\begin{align*} S ^ * = S ( \\sigma , r ) , \\sigma = t - \\epsilon f ( S ^ * ) . \\end{align*}"}
+{"id": "2392.png", "formula": "\\begin{align*} \\partial _ { \\alpha , \\alpha } ( w ) = - \\Delta _ { a _ { 1 } , a _ { 1 } } ^ { \\alpha , \\alpha } w _ { 1 } + \\Delta _ { a _ { n } , a _ { n } } ^ { \\alpha , \\alpha } w _ { 2 } \\end{align*}"}
+{"id": "1922.png", "formula": "\\begin{align*} & \\| \\zeta _ 0 ( - \\Delta _ x ) ^ { 1 / 6 } u _ { \\varepsilon } - ( - \\Delta _ x ) ^ { 1 / 6 } ( u _ { \\varepsilon } \\zeta _ 0 ) \\| _ { L _ p ( Q _ { 1 , R } ) } \\\\ & \\le N ( d ) R ^ { - 1 } \\sum _ { k = 0 } ^ { \\infty } 2 ^ { - k - 3 d k / p } \\| u _ { \\varepsilon } \\| _ { L _ { p } ( Q _ { 1 , 2 ^ k R } ) } . \\end{align*}"}
+{"id": "1673.png", "formula": "\\begin{align*} \\mathsf { A } = \\mathsf { J } _ 0 < \\mathsf { J } _ 1 < \\dots < \\mathsf { J } _ { \\mathsf { F } - 1 } < \\mathsf { J } _ { \\mathsf { F } } \\leq \\mathsf { B } \\end{align*}"}
+{"id": "1328.png", "formula": "\\begin{align*} & \\int _ { B _ r ( x _ 0 ) } \\Big ( \\left | \\nabla u ( x ) \\right | ^ p - \\left | \\nabla v ( x ) \\right | ^ p \\Big ) \\ , d x \\\\ = \\ ; & p \\int _ 0 ^ 1 \\frac { 1 } { s } \\bigg ( \\int _ { B _ r ( x _ 0 ) } \\left ( \\left | \\nabla u _ s ( x ) \\right | ^ { p - 2 } \\nabla u _ s ( x ) - \\left | \\nabla v ( x ) \\right | ^ { p - 2 } \\nabla v ( x ) \\right ) \\cdot \\nabla ( u _ s - v ) ( x ) \\ , d x \\bigg ) \\ , d s . \\end{align*}"}
+{"id": "6352.png", "formula": "\\begin{align*} \\psi ( g \\cdot g ) = \\sum _ { k \\in G } f ( g ^ { - 1 } k ) \\psi ( k ) \\end{align*}"}
+{"id": "537.png", "formula": "\\begin{align*} \\hat { f } _ i ( x _ i ) = \\int _ { \\R ^ { n - 1 } } f ( x _ 1 , \\ldots , x _ n ) \\ , \\prod _ { j \\neq i } Q _ j ( d x _ j ) \\ge - c _ 1 e ^ { c _ 2 x _ i ^ 2 } \\prod _ { j \\neq i } \\int _ { \\R } e ^ { c _ 2 x _ j ^ 2 } Q _ j ( x _ j ) \\ , d x _ j \\end{align*}"}
+{"id": "4788.png", "formula": "\\begin{align*} \\pi ( x ) p _ n ' ( x ) = E _ { - s } ( n ) p _ { n - s } ( x ) + \\dots + E _ { t } ( n ) p _ { n + t } ( x ) , \\end{align*}"}
+{"id": "5773.png", "formula": "\\begin{align*} - \\frac { 1 } { 3 } a < b < - \\frac { 3 } { 8 } a . \\end{align*}"}
+{"id": "7630.png", "formula": "\\begin{align*} \\{ \\rho _ i \\ , \\ , \\vert \\ , \\ , i \\in I , I \\subseteq \\{ 0 , \\dots , 5 \\} , | I | = 5 \\} . \\end{align*}"}
+{"id": "2222.png", "formula": "\\begin{align*} \\boldsymbol { \\mathcal { W } } _ { n } * \\boldsymbol { \\mathcal { V } } _ { n } = \\boldsymbol { \\mathcal { I } } _ * \\in \\mathbb { R } ^ { n \\times n \\times M \\times M } , \\end{align*}"}
+{"id": "107.png", "formula": "\\begin{align*} \\Delta _ { \\Psi } d = - \\frac { d } { d r } \\log ( \\theta _ { \\Psi } ) = H _ { \\Psi } . \\end{align*}"}
+{"id": "7664.png", "formula": "\\begin{align*} R _ { p } [ E ] = \\frac { P ( E ) } { | E | ^ { p } } . \\end{align*}"}
+{"id": "7055.png", "formula": "\\begin{align*} \\mathcal K ( M ) = \\mathcal K ( \\mathbb R ^ r ) \\oplus \\mathcal K ( M ' ) \\end{align*}"}
+{"id": "6628.png", "formula": "\\begin{align*} U : L ^ { 2 } ( \\Omega _ { \\frac { \\pi } { 4 } } ) \\rightarrow L ^ { 2 } ( \\Omega _ { \\theta } ) , U v ( x , y ) = \\sqrt { \\tan \\theta } v ( x , y \\tan \\theta ) , \\end{align*}"}
+{"id": "6858.png", "formula": "\\begin{align*} & ( \\beta \\lambda u _ 1 h ( a _ 1 ) f ^ { p ^ { e ' } } ( a _ 1 ) , \\dots , \\beta \\lambda u _ s h ( a _ s ) f ^ { p ^ { e ' } } ( a _ s ) , \\lambda u _ { s + 1 } h ( a _ { s + 1 } ) f ^ { p ^ { e ' } } ( a _ { s + 1 } ) , \\dots , \\lambda u _ n h ( a _ n ) f ^ { p ^ { e ' } } ( a _ n ) , \\\\ & f _ { k - 1 } ^ { p ^ { e ' } } ) = ( u _ 1 g ( a _ 1 ) , \\dots , u _ s g ( a _ s ) , u _ { s + 1 } g ( a _ { s + 1 } ) , \\dots , u _ n g ( a _ n ) , - g _ { n - k } ) . \\end{align*}"}
+{"id": "8123.png", "formula": "\\begin{align*} ( ( L , \\varphi ) ^ { d + 1 } ) & \\geqslant ( ( L , \\varphi _ 0 ) ^ { d + 1 } ) - \\frac 1 2 \\nu ( \\Omega _ \\infty ) ( d + 1 ) \\ln \\binom { r + \\delta - 1 } { \\delta } \\\\ & \\qquad - \\frac 1 2 \\nu ( \\Omega _ \\infty ) \\delta ( d + 1 ) \\ln ( r ) - \\nu ( \\Omega _ \\infty ) \\ln ( \\delta ! ) \\\\ & \\geqslant ( ( L , \\varphi _ 0 ) ^ { d + 1 } ) - \\nu ( \\Omega _ \\infty ) \\delta ( d + 1 ) \\ln ( r ) - \\nu ( \\Omega _ \\infty ) \\ln ( \\delta ! ) , \\end{align*}"}
+{"id": "4718.png", "formula": "\\begin{align*} D _ { z } \\left ( z \\widetilde { K } _ { \\lambda } ( z , w ) \\right ) = K _ { \\lambda , 1 } ( z , w ) , \\end{align*}"}
+{"id": "2696.png", "formula": "\\begin{align*} \\binom { a + n - k } { a + j } \\binom { a + k } { a } = \\sum _ { l = 0 } ^ { a } \\binom { n - k } { a + j - l } \\binom { k - j } { a - l } \\binom { n - j + l } { l } \\end{align*}"}
+{"id": "968.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { H ^ s _ \\gamma ( G ) } : = \\left ( \\int _ { G ^ \\wedge } \\left ( 1 + \\gamma ( \\xi ) ^ 2 \\right ) ^ s \\vert \\widehat { f } ( \\xi ) \\vert ^ 2 d \\nu ( \\xi ) \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "5494.png", "formula": "\\begin{align*} \\frac { ( t ) _ N ( b q ) _ N } { ( q ) _ N } F _ N ( 0 , b ; t ) = \\sum _ { n = 0 } ^ { N } \\frac { ( b ) _ n ( t ) _ n q ^ { n } } { ( q ) _ n } , \\end{align*}"}
+{"id": "5423.png", "formula": "\\begin{align*} K : = \\lceil - \\frac { 1 0 } { \\log ( 1 - c _ 0 ) } \\log N \\rceil , \\end{align*}"}
+{"id": "3417.png", "formula": "\\begin{align*} E _ U ( r ^ { n - 1 } \\Phi ) = 0 , D _ r ( r ^ { 1 - n } E _ \\rho ( r ^ { n - 1 } \\Phi ) ) = r ^ { 1 - n } ( S _ r / \\rho ) E _ S ( r ^ { n - 1 } \\Phi ) \\end{align*}"}
+{"id": "3296.png", "formula": "\\begin{align*} [ X , Y ] = X Y - Y X . \\end{align*}"}
+{"id": "4883.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n ( - 1 ) ^ k B _ { n - k } ^ { ( - k ) } = 0 \\end{align*}"}
+{"id": "2712.png", "formula": "\\begin{align*} ( - 1 ) ^ { j + r - 1 - l } \\frac { 1 } { 2 } S ( n , r ) \\frac { \\binom { 2 ( j + r - 1 - l ) } { j + r - 1 - l } \\binom { 2 ( n - j + l + 1 ) } { n - j + l + 1 } \\binom { 2 n - j - r + l + 1 } { n } } { 2 \\binom { 2 n - j + l + 1 } { n } } \\end{align*}"}
+{"id": "3505.png", "formula": "\\begin{align*} ( 1 - \\alpha ) \\Big \\langle \\nabla \\mathrm { H } ; \\mathrm { e } ^ { ( 2 ) } _ { \\mathrm { n } } \\Big \\rangle _ { \\mathbb { L } ^ { 2 } ( \\Omega ) } \\ = \\ \\Big \\langle \\nabla \\mathrm { H } ^ { \\textbf { i n } } ; \\mathrm { e } ^ { ( 2 ) } _ { \\mathrm { n } } \\Big \\rangle _ { \\mathbb { L } ^ { 2 } \\Big ( \\Omega \\Big ) } + \\Big \\langle \\mathbb { T } \\Big [ \\nabla \\mathrm { H } \\Big ] ; \\ \\mathrm { e } ^ { ( 2 ) } _ \\mathrm { n } \\Big \\rangle _ { \\mathbb { L } ^ { 2 } ( \\Omega ) } . \\end{align*}"}
+{"id": "3824.png", "formula": "\\begin{align*} a _ 1 = \\frac { 4 n - 1 0 - ( a _ 2 + a _ 3 ) } { 2 } , \\end{align*}"}
+{"id": "5036.png", "formula": "\\begin{align*} \\min \\{ | x | , 1 \\} K ( x ) \\in L ^ 1 ( \\R ^ n ) , \\quad K ( x ) = K ( - x ) \\ , \\ \\forall x \\in \\R ^ n . \\end{align*}"}
+{"id": "2346.png", "formula": "\\begin{align*} \\Bbbk \\star \\emptyset = \\emptyset \\star \\Bbbk & = \\Bbbk \\\\ ( \\Bbbk , 2 ) \\star \\Bbbk ' = \\Bbbk \\star ( \\Bbbk ' , 2 ) & = ( \\Bbbk \\star \\Bbbk ' , 2 ) \\\\ ( \\Bbbk , 3 ) \\star ( \\Bbbk ' , 3 ) & = ( \\Bbbk \\star ( \\Bbbk ' , 3 ) , 3 ) + ( ( \\Bbbk , 3 ) \\star \\Bbbk ' , 3 ) + ( \\Bbbk \\star \\Bbbk ' , 2 , 2 , 2 ) . \\end{align*}"}
+{"id": "6748.png", "formula": "\\begin{align*} E _ \\alpha ( t ) = & \\sum _ { | \\alpha | \\leq N - 1 } ( \\sum ^ { 3 } _ { i , j = 1 } \\int _ { \\mathbb { R } ^ { 3 } } \\int _ { \\mathbb { R } ^ { 3 } } \\big \\{ \\partial ^ { \\alpha } [ \\frac { 1 } { \\rho } \\partial _ { x _ j } ( R \\theta B _ { i j } ( \\frac { v - u } { \\sqrt { R \\theta } } ) \\frac { \\varepsilon \\sqrt { \\mu } } { M } f ) ] \\partial ^ { \\alpha } \\widetilde { u } _ i \\big \\} \\ , d v \\ , d x - \\widetilde { E } ( t ) ) . \\end{align*}"}
+{"id": "3255.png", "formula": "\\begin{align*} d \\eta _ i = 2 \\alpha \\Phi _ i + 2 ( \\alpha - \\delta ) \\eta _ j \\wedge \\eta _ k \\end{align*}"}
+{"id": "4014.png", "formula": "\\begin{align*} \\bar { q } _ { \\beta } ( n , t ) = \\begin{cases*} E _ { \\beta , 1 } \\left ( - \\lambda t ^ { \\beta } \\right ) , \\ n = 0 , \\\\ \\displaystyle \\sum _ { \\Omega _ { n } } \\prod _ { j = 1 } ^ { n } \\frac { \\left ( \\rho ^ { j } \\binom { r + j - 1 } { j } \\right ) ^ { x _ { j } } } { x _ { j } ! } z _ { n } ! \\left ( \\frac { \\lambda t ^ { \\beta } ( 1 - \\rho ) ^ { r } } { 1 - ( 1 - \\rho ) ^ { r } } \\right ) ^ { z _ { n } } E _ { \\beta , z _ { n } \\beta + 1 } ^ { z _ { n } + 1 } \\left ( - \\lambda t ^ { \\beta } \\right ) , \\ n \\ge 1 . \\end{cases*} \\end{align*}"}
+{"id": "2106.png", "formula": "\\begin{align*} a = m , \\ a _ 1 = k - m , \\ b = p = n + m - k - \\ell , \\ b _ 1 = n - k - p . \\end{align*}"}
+{"id": "6569.png", "formula": "\\begin{align*} \\hat { E } = \\bigoplus _ i \\hat { E } _ i , \\hat { F } = \\bigoplus _ j \\hat { F } _ j . \\end{align*}"}
+{"id": "2853.png", "formula": "\\begin{align*} \\Xi _ { s , ! } ^ \\wedge : = ( \\overline { \\imath } _ s ) _ * ( \\overline { \\imath } _ s ) ^ * \\Xi _ ! ^ \\wedge , \\Xi _ { s , * } ^ \\wedge : = ( \\overline { \\imath } _ s ) _ * ( \\overline { \\imath } _ s ) ^ ! \\Xi _ * ^ \\wedge . \\end{align*}"}
+{"id": "1899.png", "formula": "\\begin{align*} u ( x , t ) = P _ 0 + \\sum _ { k = 0 } ^ { k _ 0 - 1 } \\mathcal O ( t ^ { a _ k } ) + \\mathcal O ( t ^ { \\sigma } ) = \\sum _ { k = - 1 } ^ { k _ 0 } \\mathcal O ( t ^ { a _ k } ) , \\end{align*}"}
+{"id": "1714.png", "formula": "\\begin{align*} \\| \\mathcal { A } \\| _ { \\mathfrak { S } ^ q ( \\mathcal { H } ) } : = \\begin{cases} \\left ( \\mathrm { T r } \\ , | \\mathcal { A } | ^ q \\right ) ^ { 1 / q } & q < \\infty \\\\ \\sup \\mathrm { s p e c } \\ , | \\mathcal { A } | & q = \\infty \\ , , \\end{cases} \\end{align*}"}
+{"id": "7793.png", "formula": "\\begin{align*} k ( u ) = \\frac { i } { 2 \\pi } \\frac { 1 } { I m ( u ) } . \\end{align*}"}
+{"id": "6816.png", "formula": "\\begin{align*} g ( s ) = \\frac 1 \\varepsilon \\ , e ^ { - \\frac { s } \\varepsilon } , \\quad \\varepsilon > 0 . \\end{align*}"}
+{"id": "1275.png", "formula": "\\begin{align*} w _ 1 + w _ 2 + w _ 3 \\in 2 \\tilde M . \\end{align*}"}
+{"id": "3315.png", "formula": "\\begin{gather*} P _ { n _ 1 , n _ 2 } ( m _ 1 , m _ 2 ) = C ( n _ 1 , n _ 2 , m _ 1 , m _ 2 ) \\big \\langle \\phi _ { m _ 1 } \\otimes \\phi _ { m _ 2 } ^ { m _ 1 } , \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } \\big \\rangle _ { V _ 1 \\otimes V _ 2 } , \\end{gather*}"}
+{"id": "2094.png", "formula": "\\begin{align*} \\Omega _ X = | \\sigma _ 1 \\wedge \\ldots \\wedge \\sigma _ n | , \\end{align*}"}
+{"id": "2161.png", "formula": "\\begin{align*} & R _ { i j } R _ { k l } = \\overline { R } _ { i j } \\overline { R } _ { k l } = R _ { i j k l } , \\\\ & S _ { i j } S _ { k l } S _ m = S _ { i j k l m } , \\end{align*}"}
+{"id": "3984.png", "formula": "\\begin{align*} \\Theta _ { n } ^ { k } = \\left \\{ ( x _ { 1 } , x _ { 2 } , \\dots , x _ { n } ) : \\sum _ { j = 1 } ^ { n } x _ { j } = k , \\ \\sum _ { j = 1 } ^ { n } j x _ { j } = n , \\ x _ { j } \\in \\mathbb { N } \\cup \\{ 0 \\} \\right \\} . \\end{align*}"}
+{"id": "7849.png", "formula": "\\begin{align*} \\varphi _ \\mu : = \\varphi _ 0 \\circ \\overline { \\textup { s h i f t } } _ \\mu : \\widetilde { J } ^ { - 1 } ( \\mu ) / F \\to T ^ * \\overline { Q } , \\end{align*}"}
+{"id": "8200.png", "formula": "\\begin{align*} { T } _ { u } v = \\sum _ j { \\dot { S } _ { j - 1 } u } { \\dot { \\Delta } _ j v } , \\textrm { a n d } { R } ( u , v ) = \\sum _ j \\dot { \\Delta } _ j u \\widetilde { \\dot { \\Delta } } _ j v , \\widetilde { \\dot { \\Delta } } _ j : = \\dot { \\Delta } _ { j - 1 } + \\dot { \\Delta } _ j + \\dot { \\Delta } _ { j + 1 } . \\end{align*}"}
+{"id": "4833.png", "formula": "\\begin{align*} \\begin{gathered} r ( L ) \\int _ S \\int _ { B _ R } \\rho ( x , s ) \\ , \\dd x \\dd \\mu ( s ) = \\int _ S \\int _ { B _ R } \\phi ( r ( L ) \\rho ) ( x , s ) \\ , \\dd x \\dd \\mu ( s ) = ( \\phi , r ( L ) \\rho ) _ { L ^ 2 ( S , L ^ 2 ( B _ R ) ; \\mu ) } \\\\ = ( \\phi , L \\rho ) _ { L ^ 2 ( S , L ^ 2 ( B _ R ) ; \\mu ) } , \\end{gathered} \\end{align*}"}
+{"id": "5220.png", "formula": "\\begin{align*} D _ i = \\begin{pmatrix} - ( \\l _ i + \\mu _ 1 ) + \\frac { 1 } { 1 2 } S & & \\\\ & - ( \\l _ i + \\mu _ 2 ) + \\frac { 1 } { 1 2 } S & \\\\ & & - ( \\l _ i + \\mu _ 3 ) + \\frac { 1 } { 1 2 } S \\end{pmatrix} , \\end{align*}"}
+{"id": "3767.png", "formula": "\\begin{align*} b ( \\Gamma / e ) = b ( \\Gamma ) - 2 . \\end{align*}"}
+{"id": "7718.png", "formula": "\\begin{align*} s _ p ( \\cdot ) = d i s t _ { g ^ + } ( p , \\cdot ) \\end{align*}"}
+{"id": "963.png", "formula": "\\begin{align*} f ( x + k , \\omega ) = f ( x , \\tau ( k ) \\omega ) . \\end{align*}"}
+{"id": "4894.png", "formula": "\\begin{align*} a _ n = \\left ( \\frac { 2 } { 3 } \\right ) ^ n p _ n \\left ( \\frac { 1 } { 4 } \\right ) . \\end{align*}"}
+{"id": "6468.png", "formula": "\\begin{align*} \\omega ^ { ( 2 ) } _ { G } ( t , x ; s , y ) = \\sum _ j \\lambda _ j ^ { - 1 } e ^ { i \\lambda _ j ( t - s ) } \\phi _ j ( x ) \\phi _ j ( y ) . \\end{align*}"}
+{"id": "298.png", "formula": "\\begin{align*} C _ { D ' } = \\bigcup _ { I '' \\subset I ' } T ^ { * } _ { D _ { I '' } } X \\subset T ^ { * } X \\end{align*}"}
+{"id": "833.png", "formula": "\\begin{align*} \\begin{aligned} & - \\frac { p ' ( x ^ { * } ) } { k _ 1 ( x ^ { * } ) ^ { k _ 1 - 1 } } x ^ { k _ 1 } + p ( x ) - \\left ( - \\frac { p ' ( x ^ { * } ) } { k _ 1 ( x ^ { * } ) ^ { k _ 1 - 1 } } ( x ^ { * } ) ^ { k _ 1 } + p ( x ^ { * } ) \\right ) \\left ( \\frac { x ^ { * } } { x } \\right ) ^ { - k _ 1 } \\\\ & = \\frac { 1 } { l _ 2 } \\left [ \\frac { 1 - \\theta } { - \\left ( \\frac { K } { - \\frac { p ' ( x ^ { * } ) } { k _ 1 ( x ^ { * } ) ^ { k _ 1 - 1 } } ( x ^ { * } ) ^ { k _ 1 } + p ( x ^ { * } ) } + \\theta \\right ) } - l _ 1 \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "4296.png", "formula": "\\begin{align*} \\frac { ( q ) _ { N } } { ( z q ) _ { N } ( z ^ { - 1 } q ) _ { N } } = \\frac { 1 } { ( q ) _ N } + ( 1 - z ) \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ n ( q ) _ n q ^ { n ( n + 1 ) / 2 } } { ( q ) _ { n + N } } \\left ( \\frac { 1 } { 1 - z q ^ n } - \\frac { 1 } { z - q ^ n } \\right ) , \\end{align*}"}
+{"id": "6918.png", "formula": "\\begin{align*} \\left [ q ^ d \\right ] \\frac { ( - 1 ) ^ { ( N - 1 ) d } } { \\mathsf V _ { N + 1 } } \\ , \\begin{vmatrix} z _ 1 ^ { d + N } & z _ 1 ^ { d + N - 1 } & \\cdots & z _ 1 ^ { d + 1 } \\\\ z _ 2 ^ { d + N } & z _ 2 ^ { d + N - 1 } & \\cdots & z _ 2 ^ { d + 1 } \\\\ \\vdots & \\vdots & \\cdots & \\vdots \\\\ z _ N ^ { d + N } & z _ N ^ { d + N - 1 } & \\cdots & z _ N ^ { d + 1 } \\end{vmatrix} \\prod _ { i = 1 } ^ { N } \\bigg ( \\frac { \\alpha _ i - z _ { N + 1 } } { \\alpha _ i } \\bigg ) ^ { b _ i } \\bigg { | } _ { \\epsilon = 0 } . \\end{align*}"}
+{"id": "1471.png", "formula": "\\begin{align*} A _ n ( s ) = & ( S _ n ^ { [ s ] } ) ^ T Q _ n ( s ) , s \\in \\{ 1 , \\dotsc , \\gamma _ r \\} \\\\ = & \\sum _ { i = 1 } ^ { \\ell _ r ^ * } \\left ( \\frac { 1 } { f _ { g ( ( s - 1 ) \\ell _ r ^ * + i ) } - \\alpha _ n } W _ { \\theta , i } ^ { [ s ] } \\right ) 1 _ { \\{ i \\in J _ r ^ { [ s ] } \\} } \\\\ & \\quad + P _ { \\alpha _ n } ( \\lfloor \\frac { N } { 2 } \\rfloor ) . \\end{align*}"}
+{"id": "5749.png", "formula": "\\begin{align*} g ( \\tilde X ^ M ) = \\sum _ { i = 1 , j = 1 } ^ { m } B _ { i j } \\otimes \\begin{pmatrix} M ^ s _ { i j } & 0 \\\\ 0 & 0 \\end{pmatrix} \\succeq 0 . \\end{align*}"}
+{"id": "7340.png", "formula": "\\begin{align*} & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ X ^ { ( i , j , t , p ) } _ { ( m , m ) } = \\{ ( y , z ) \\in X _ { ( m , m ) } \\mid \\varrho ( y , z ) = ( i , j , t , p ) \\} . \\ \\end{align*}"}
+{"id": "1808.png", "formula": "\\begin{align*} r _ { 1 2 } = \\sum _ i x _ i \\otimes y _ i \\otimes 1 , r _ { 1 3 } = \\sum _ { i } x _ i \\otimes 1 \\otimes y _ i , r _ { 2 3 } = \\sum _ i 1 \\otimes x _ i \\otimes y _ i , \\end{align*}"}
+{"id": "591.png", "formula": "\\begin{align*} j _ p ^ { ( 1 2 3 ) } = p + a _ { 1 2 3 } , \\end{align*}"}
+{"id": "5208.png", "formula": "\\begin{align*} H _ s \\leq \\sum _ { i = j _ s } ^ { j _ { s + 1 } - 1 } ( a _ i + b _ i ) + L . \\end{align*}"}
+{"id": "6346.png", "formula": "\\begin{align*} \\mathcal { A } _ { P } ^ { ( 1 ) } ( U F ) & = \\mathcal { A } _ { P } ^ { ( 1 ) } ( U F , U G ) = \\textit { m a x } _ { i = 1 } ^ N \\left \\{ q _ i \\dfrac { { | \\langle U f _ i , U g _ i \\rangle | + \\| U f _ i \\| \\ ; \\| U g _ i \\| } } { 2 } \\right \\} \\leq \\textit { m a x } _ { i = 1 } ^ N \\left \\{ q _ i \\dfrac { { | \\langle U f _ i , U g '' _ i \\rangle | + \\| U f _ i \\| \\ ; \\| U g '' _ i \\| } } { 2 } \\right \\} \\end{align*}"}
+{"id": "7265.png", "formula": "\\begin{align*} f : B _ 0 & \\longrightarrow B _ { m - 1 } \\\\ b & \\longmapsto \\begin{cases} b + m - 1 & b < y m \\\\ b - 1 & b > y m \\end{cases} \\end{align*}"}
+{"id": "4315.png", "formula": "\\begin{align*} d _ { T V } ( X , \\pi ) \\leq \\sup _ { h \\in \\mathcal { H } } \\sup _ { l \\in \\mathbb { Z } ^ + } | m _ l ( h ) | \\sum _ { j = 0 } ^ \\infty \\left | \\mathbb { P } ( X > j ) - \\sum _ { k = j + 1 } ^ \\infty \\sum _ { i = 0 } ^ \\infty \\mathbb { P } ( X = i ) P _ { i , k } \\right | . \\end{align*}"}
+{"id": "5071.png", "formula": "\\begin{align*} { } f \\bigg | \\begin{pmatrix} a H & \\frac { a \\bar { a } H ^ 2 - 1 } { q H } \\\\ q H & \\bar { a } H \\\\ \\end{pmatrix} = i ^ { - \\ell } \\omega \\chi ( \\bar { a } ) f . \\end{align*}"}
+{"id": "3922.png", "formula": "\\begin{align*} \\| | u | ^ { p - 1 } v \\| _ { L ^ q } & \\le \\| | u | ^ { p - 1 } \\| _ { L ^ { \\hat q } } \\| v \\| _ { L ^ { \\hat p } } \\\\ & = \\| u \\| ^ { p - 1 } _ { L ^ { ( p - 1 ) \\hat q } } \\| v \\| _ { L ^ { \\hat p } } . \\end{align*}"}
+{"id": "2727.png", "formula": "\\begin{align*} A u t ( p t ^ { * k } ) \\cong \\left ( \\prod _ { i = 1 } ^ k A u t ( p t ) \\right ) \\rtimes S y m ( k ) \\cong S y m ( k ) . \\end{align*}"}
+{"id": "6695.png", "formula": "\\begin{align*} & Q _ { i } ' ( 0 ) = \\det \\begin{bmatrix} s _ { i , - 2 } & s _ { i , - 1 } & 0 \\\\ s _ { i , - 1 } & 1 & 1 \\\\ 1 & \\lambda _ { i , 2 } ^ { 2 } & 0 \\end{bmatrix} = - \\lambda _ { i , 2 } ^ { 2 } s _ { i , - 2 } + s _ { i , - 1 } , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] , \\end{align*}"}
+{"id": "7939.png", "formula": "\\begin{align*} ( a , u ) \\cdot _ \\ltimes ( b , v ) = ( a \\cdot b , ~ a \\cdot v + u \\cdot a + \\chi ( a , b ) ) , ( a , u ) , ( b , v ) \\in A \\oplus M . \\end{align*}"}
+{"id": "2972.png", "formula": "\\begin{align*} K _ 0 ^ { \\operatorname { a d d } } ( \\mathcal { S } ) = G _ 0 ( \\mathcal { S } ) / ( [ X \\oplus Y ] - [ X ] - [ Y ] \\ ; \\vline \\ ; X , \\ ; Y \\in \\mathcal { S } ) \\end{align*}"}
+{"id": "3698.png", "formula": "\\begin{align*} X _ t = X _ 0 + \\int _ 0 ^ t b [ \\rho , \\nabla S ] ( X _ s , s ) d s + W _ t , \\mathbb P a . s . \\end{align*}"}
+{"id": "5535.png", "formula": "\\begin{align*} \\theta _ 1 ( \\alpha ( t ) , t ) = \\theta _ b , \\end{align*}"}
+{"id": "1490.png", "formula": "\\begin{align*} W = \\sum _ { d _ 1 } \\sum _ { d _ 2 } \\rho _ { d _ 1 } \\rho _ { d _ 2 } \\bigl ( g ( [ d _ 1 , d _ 2 ] ) - \\sum _ { p < z } g ( [ p , d _ 1 , d _ 2 ] ) \\bigr ) \\end{align*}"}
+{"id": "3723.png", "formula": "\\begin{align*} \\begin{cases} \\rho ' _ y : = \\rho + y g \\\\ v ' _ y : = \\nabla S + X _ y ^ g \\end{cases} \\end{align*}"}
+{"id": "2909.png", "formula": "\\begin{align*} M = \\begin{pmatrix} A & C \\\\ & B \\end{pmatrix} \\end{align*}"}
+{"id": "5344.png", "formula": "\\begin{align*} m _ 1 m _ 1 & = 0 , \\\\ m _ 1 m _ 2 & = m _ 2 ( m _ 1 \\otimes 1 + 1 \\otimes m _ 1 ) , \\\\ m _ 1 m _ 3 & = m _ 2 ( 1 \\otimes m _ 2 - m _ 2 \\otimes 1 ) - m _ 3 ( m _ 1 \\otimes 1 ^ { \\otimes 2 } + 1 \\otimes m _ 1 \\otimes 1 + 1 ^ { \\otimes 2 } \\otimes m _ 1 ) . \\end{align*}"}
+{"id": "4603.png", "formula": "\\begin{align*} & \\ln R ( n + N ) ^ 2 \\\\ = & \\ln R ( n ) ^ 2 - \\sum _ { l = 0 } ^ { N - 1 } \\frac { V ( n + l ) } { \\sin \\pi k } \\sin 2 \\pi \\theta ( n + l ) + \\frac { O ( 1 ) } { ( n - b ) ^ 2 } \\\\ = & \\ln R ( n ) ^ 2 - K _ 1 \\sum _ { l = 0 } ^ { N - 1 } \\frac { \\sin 2 \\pi \\theta ( n + l ) + \\sin 2 \\pi \\tilde { \\theta } ( n + l ) + 1 0 0 } { ( n - b ) \\sin \\pi k } \\sin 2 \\pi \\theta ( n + l ) \\\\ & + \\frac { O ( 1 ) } { ( n - b ) ^ 2 } . \\end{align*}"}
+{"id": "3476.png", "formula": "\\begin{align*} \\textbf { e r r } ^ { ( 2 ) } : = \\displaystyle \\frac { 1 } { \\alpha _ \\mathrm { m } } \\int _ { 0 } ^ t \\int _ { \\partial \\Omega } \\gamma ^ { \\textbf { i n t } } _ { 1 , \\mathrm { y } } \\Phi ^ { \\textbf { e } } ( \\xi , t ; \\mathrm { y } , \\tau ) \\gamma ^ { \\textbf { i n t } } _ { 0 } \\mathrm { U } _ { \\mathrm { i } } ( \\mathrm { y } , \\tau ) d \\sigma _ { \\mathrm { y } } d \\tau . \\end{align*}"}
+{"id": "3479.png", "formula": "\\begin{align*} \\Big ( \\mathcal { K } _ { \\alpha _ \\mathrm { p } } - \\mathcal { K } _ \\Big ) \\Big [ \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\cdot ) \\Big ] ( \\mathrm { y } , \\tau ) = \\int _ { \\partial \\Omega } \\frac { ( \\mathrm { y } - \\mathrm { v } ) \\cdot \\nu _ { \\mathrm { v } } } { 2 \\pi | \\mathrm { y } - \\mathrm { v } | ^ 2 } \\Big [ \\varphi ( \\mathrm { v } , \\mathrm { y } , \\mathrm { t } , \\tau ) - \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) \\Big ] d \\sigma _ \\mathrm { v } \\end{align*}"}
+{"id": "474.png", "formula": "\\begin{align*} \\sum _ { k = c _ 1 } ^ { c _ 2 } { L _ { i w _ k + m } v _ k z ^ { w _ k } } = \\alpha ^ m h ( \\alpha ^ i z ) - \\beta ^ m h ( \\beta ^ i z ) \\ , . \\end{align*}"}
+{"id": "7348.png", "formula": "\\begin{align*} \\Upsilon : = \\{ ( \\mu , d ) \\in \\mathbb { Z } ^ 2 | \\ 0 \\leq d \\leq m , \\ \\frac { 1 } { 2 } ( m - d ) \\leq \\mu \\leq m - d \\} . \\end{align*}"}
+{"id": "4710.png", "formula": "\\begin{align*} K _ { \\lambda , 1 } ( z , w ) & = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( n + \\lambda + 2 ) ( n + \\lambda + 1 ) } { \\lambda + 1 } \\phi _ { n } ( z ) \\overline { \\phi _ { n } ( w ) } , \\\\ K _ { \\lambda , 2 } ( z , w ) & = \\frac { 1 } { 2 \\lambda + 2 } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\Gamma ( n + \\lambda + 4 ) } { \\Gamma ( n + \\lambda + 1 ) } \\phi _ { n } ( z ) \\overline { \\phi _ { n } ( w ) } . \\end{align*}"}
+{"id": "8204.png", "formula": "\\begin{align*} 0 < s < \\frac { N } { p } , s = \\frac { N } { p } r = 1 . \\end{align*}"}
+{"id": "2584.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { * , i } = & ~ \\Big [ A x _ t ^ { * , i } - B ^ 2 R ^ { - 1 } y _ t ^ { * , i } - B h \\big ( \\rho _ i ^ N ( \\Delta ^ { N , 1 } _ { * , t } , \\ldots , \\Delta ^ { N , N } _ { * , t } ) \\big ) \\Big ] d t + \\sigma d W _ t ^ i + \\sigma _ 0 d W ^ 0 _ t , \\\\ x ^ { * , i } _ 0 = & ~ \\xi ^ i . \\end{aligned} \\right . \\end{align*}"}
+{"id": "814.png", "formula": "\\begin{align*} \\bar { \\theta } _ 1 = x ^ { - 1 } l _ 2 ^ { - \\frac { 1 } { k _ 2 } } \\Big ( \\frac { K k _ 2 } { k _ 2 - 1 } \\Big ) ^ { \\frac { k _ 2 - 1 } { k _ 2 } } . \\end{align*}"}
+{"id": "4940.png", "formula": "\\begin{align*} & ~ ~ ~ \\left ( \\frac { 1 } { j - 1 } + \\left ( \\frac { 1 } { j - 1 } \\right ) ^ 2 + \\cdots \\right ) \\cdot \\frac { j } { 2 } \\cdot \\binom { n } { k } + \\frac { j } { 2 } \\cdot \\frac { 1 } { ( j - 1 ) ^ 2 } \\cdot \\binom { n } { k } \\\\ & < \\left ( \\frac { j } { 2 ( j - 2 ) } + \\frac { j } { 2 ( j - 1 ) ^ 2 } \\right ) \\binom { n } { k } < \\binom { n } { k } \\end{align*}"}
+{"id": "3182.png", "formula": "\\begin{align*} I ( u ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } | \\nabla u | ^ { 2 } d x + \\frac { 1 } { 4 } \\int _ { \\mathbb { R } ^ { 3 } } \\phi _ { u } u ^ { 2 } d x - \\frac { 1 } { p } \\int _ { \\mathbb { R } ^ { 3 } } | u | ^ { p } d x \\end{align*}"}
+{"id": "2336.png", "formula": "\\begin{align*} a _ { i + 1 } = \\begin{cases} a _ { i } & i \\in \\{ l _ { 1 } , l _ { 1 } + l _ { 2 } , \\ldots , l _ { 1 } + \\cdots + l _ { d - 1 } \\} \\\\ 1 - a _ { i } & \\mathrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "5668.png", "formula": "\\begin{align*} T ^ i V ^ j \\hat { \\mathbb { G } } ^ { k } _ { i j } ( T ) = V ^ j \\hat { \\mathbb { G } } ^ { k } _ { j } ( T ) = 2 \\hat { \\mathbb { G } } ^ { i } ( T ) J ^ k _ i . \\end{align*}"}
+{"id": "4039.png", "formula": "\\begin{align*} P _ { 2 k } ( q ( s _ 0 ) ) = 0 , \\| g \\| _ { L ^ \\infty _ M } \\le I ^ { - \\delta } ( s _ 0 ) , \\end{align*}"}
+{"id": "3619.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n e ^ { - \\lambda T _ i } \\left [ f \\left ( X _ { T _ i } \\right ) - f \\left ( X _ { T _ i - } + \\Delta Y _ { T _ i } \\right ) \\right ] \\ge - \\sum _ { j \\le n , \\tau _ j \\le t \\wedge S _ m } e ^ { - \\lambda \\tau _ j } g ( \\xi _ j ) \\end{align*}"}
+{"id": "1543.png", "formula": "\\begin{align*} F \\left ( g _ { s , j , k } \\right ) \\circ F \\left ( g _ { e , i , j } \\right ) = F ^ { \\prime } \\left ( s \\right ) ^ { \\left ( j , k , i \\right ) } = F ^ { \\prime } \\left ( s \\right ) ^ { \\left ( i , j , k \\right ) } = F \\left ( g _ { e , j , k } \\right ) \\circ F \\left ( g _ { s , i , j } \\right ) . \\end{align*}"}
+{"id": "7744.png", "formula": "\\begin{align*} \\mathcal { A } ( o ) = 2 ^ n \\cdot \\lim \\limits _ { t \\rightarrow + \\infty } e ^ { - n t } \\omega _ n \\sinh ^ n t = \\omega _ n . \\end{align*}"}
+{"id": "5899.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } = v , \\ \\ \\dot { v } = v \\times \\frac { B ( \\epsilon x ) } { \\epsilon } + F ( x ) , \\ \\ x ( 0 ) = x _ 0 , \\ \\ v ( 0 ) = v _ 0 , \\end{aligned} \\end{align*}"}
+{"id": "1882.png", "formula": "\\begin{align*} | \\Lambda ( x , y ) | & = \\lim _ { m \\rightarrow \\infty } \\left | \\frac { 1 } { 3 ^ l } \\right | ^ { m } \\left | \\Lambda \\left ( \\frac { x } { 3 ^ { m } } , \\ \\frac { y } { 3 ^ { m } } \\right ) \\right | \\\\ & \\leq \\lim _ { m \\rightarrow \\infty } \\left | \\frac { 1 } { 3 ^ l } \\right | ^ { m } G \\left ( \\frac { x } { 3 ^ { m } } , \\ \\frac { y } { 3 ^ { m } } \\right ) \\\\ & = 0 . \\end{align*}"}
+{"id": "7319.png", "formula": "\\begin{align*} \\kappa = \\begin{cases} 1 & { \\rm i f } ~ q ^ m \\equiv 1 \\pmod 4 , \\\\ 2 & { \\rm i f } ~ q ^ m \\equiv 3 \\pmod 4 . \\end{cases} \\end{align*}"}
+{"id": "3033.png", "formula": "\\begin{align*} \\begin{aligned} u _ 1 & = h ( x , \\bar { u } _ 1 , \\bar { u } _ 2 ) \\\\ u _ 2 & = \\bar { u } _ 2 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "1934.png", "formula": "\\begin{align*} & ( | ( - \\Delta _ x ) ^ { 1 / 6 } \\widetilde u _ 0 | ^ p ) ^ { 1 / p } _ { Q _ { 1 , R } } \\\\ & \\le N ( d , p , \\delta ) ( 2 \\nu r ) \\sum _ { k = 0 } ^ { \\infty } 2 ^ { - k } \\big ( \\sum _ { i = 1 } ^ d ( | \\widetilde f _ i | ^ p ) ^ { 1 / p } _ { Q _ { 1 , 2 ^ { k + 1 } R } } + \\lambda ^ { - 1 / 2 } ( | \\widetilde g | ^ p ) ^ { 1 / p } _ { Q _ { 1 , 2 ^ { k + 1 } R } } \\big ) . \\end{align*}"}
+{"id": "3635.png", "formula": "\\begin{align*} K ( x ) = { \\sigma ^ 2 \\over 2 } h '' ( x ) + \\int _ 0 ^ \\infty \\left [ h ( x + y ) - h ( x ) \\right ] \\nu ( d y ) - \\lambda h ( x ) + ( x - \\rho ) ^ 2 = 0 , x \\in ( a , b ) \\end{align*}"}
+{"id": "8029.png", "formula": "\\begin{align*} T _ Q : = \\ : Q _ 0 \\otimes I _ { \\ell ^ 2 ( \\mathcal { F } ^ + _ d ) } \\ , + \\ , \\sum _ { 0 < | v | \\leq n } Q _ v \\otimes L _ v \\ , + \\ , \\sum _ { 0 < | v | \\leq n } Q _ v ^ * \\otimes L _ v ^ * \\ , \\end{align*}"}
+{"id": "6668.png", "formula": "\\begin{align*} \\mathcal { E } ( \\mathbf { t } , \\mathbf { f } ) & = \\det \\begin{bmatrix} f _ 0 & t _ 0 & t _ 1 & \\ldots & t _ { \\frac { N + 1 } { 2 } - 1 } \\\\ f _ 1 & t _ 1 & t _ 2 & \\ldots & t _ { \\frac { N } { 2 } } \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ f _ { \\frac { N + 1 } { 2 } } & t _ { \\frac { N + 1 } { 2 } } & t _ { \\frac { N + 1 } { 2 } + 1 } & \\ldots & t _ { N } \\end{bmatrix} \\end{align*}"}
+{"id": "6806.png", "formula": "\\begin{align*} \\alpha ( d , d , x _ 3 , x _ 4 , y _ 5 ) = & \\sum _ { i \\in [ q ] , j \\in [ t ] } \\lambda _ { i j } \\tau _ i ( d , d , x _ 3 ) \\tilde { \\gamma } _ j ( x _ 4 , y _ 5 ) + \\sum _ { i \\in [ q ] , j \\in [ t ] } \\lambda ' _ { i j } \\tau _ i ( d , d , x _ 4 ) \\tilde { \\gamma } _ j ( x _ 3 , y _ 5 ) + \\sum _ { i \\in [ s _ 3 ] } \\beta '' _ { i } ( d , y _ 5 ) \\gamma '' _ i ( x _ 3 , x _ 4 ) \\end{align*}"}
+{"id": "1158.png", "formula": "\\begin{align*} S = \\frac { ( 4 \\pi ) ^ { 2 k - 3 } } { \\Gamma ( 2 k - 3 ) } \\Big ( 1 + \\frac { p } { p + 1 } C ( p ) + O ( k ^ { - \\alpha } p ^ { - \\alpha / 2 } D ^ { - \\alpha } + p ^ { - 9 / 1 6 + \\epsilon } k ^ { - 5 / 2 4 } D ^ { 7 / 8 + \\epsilon } ) \\Big ) . \\end{align*}"}
+{"id": "6433.png", "formula": "\\begin{align*} c _ { n _ 1 } ( x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) = \\frac { \\partial ^ { n _ 1 } f ( 0 , y _ 1 , x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) } { n _ 1 ! \\partial { y _ 1 } ^ { n _ 1 } } \\Big | _ { y _ 1 = 0 } . \\end{align*}"}
+{"id": "7598.png", "formula": "\\begin{align*} X _ \\alpha \\cap Y = \\overline { B ^ - \\cdot x _ \\alpha } \\cap Y \\subset \\overline { B ^ - \\cdot x _ \\beta } \\cap Y = X _ \\beta \\cap Y , \\end{align*}"}
+{"id": "7211.png", "formula": "\\begin{align*} \\mathcal { Z } = \\mathcal { Z } ( d ) & = Z / B = Z ( d ) / B , \\\\ \\mathcal { C } ( d ) & = D ( d ) / G L ( d ) . \\end{align*}"}
+{"id": "431.png", "formula": "\\begin{align*} Z ( \\mathbb { Z } \\mathfrak { S } _ { n } ) = \\langle e _ { 1 } ( L _ { 1 } , \\dots , L _ { n } ) , \\dots , e _ { n } ( L _ { 1 } , \\dots , L _ { n } ) \\rangle . \\end{align*}"}
+{"id": "1386.png", "formula": "\\begin{align*} s \\varphi '' ( s ) + ( \\gamma _ { \\varepsilon } + s ) \\varphi ' ( s ) + \\beta \\varphi ( s ) = 0 . \\end{align*}"}
+{"id": "5428.png", "formula": "\\begin{align*} n : = n _ 1 + n _ 2 , d : = \\# \\{ 1 \\leq i \\leq n _ 1 : x _ i \\neq y _ i \\} + n _ 2 , \\end{align*}"}
+{"id": "2949.png", "formula": "\\begin{align*} \\Phi ( x ) : = Q \\cap G ^ { - 1 } ( x ) \\qquad \\forall x \\in \\R ^ n . \\end{align*}"}
+{"id": "1066.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { g } ^ { \\Delta ^ { ( s ) } } ( V , W ) & : = \\mathcal { W } ( \\nabla ^ { ( s ) } _ { V } \\nabla ^ { ( s ) } _ { W } ( \\Delta ^ { ( s ) } ) ^ { - n - 2 } ) = 2 ^ { m } \\mathcal { g } ^ { \\Delta } ( V , W ) , \\\\ \\mathcal { G } ^ { \\Delta ^ { ( s ) } } ( V , W ) & : = \\mathcal { W } ( \\nabla ^ { ( s ) } _ { V } \\nabla ^ { ( s ) } _ { W } ( \\Delta ^ { ( s ) } ) ^ { - n } ) = 2 ^ { m } \\mathcal { G } ^ { \\Delta } ( V , W ) . \\end{aligned} \\end{align*}"}
+{"id": "3958.png", "formula": "\\begin{align*} q ( n , t ) = \\sum _ { \\Omega _ { n } } \\prod _ { j = 1 } ^ { n } \\frac { ( \\alpha t \\theta ^ { j } / j ! ) ^ { x _ { j } } } { x _ { j } ! } e ^ { - \\alpha t \\left ( e ^ { \\theta } - 1 \\right ) } , \\end{align*}"}
+{"id": "3492.png", "formula": "\\begin{align*} \\textbf { e r r } ^ { ( 4 ) } = \\mathcal { O } \\Bigg ( \\varepsilon ^ { \\frac { 2 3 } { 4 } } \\sqrt [ 4 ] { \\mathcal { K } ^ { ( \\mathrm { T _ 0 } ) } _ { \\mathrm { r } } \\mathcal { S } ^ { ( \\mathrm { T _ 0 } ) } _ { \\mathrm { r } } } \\frac { 1 } { | \\xi - z | ^ { \\frac { 7 } { 2 } - 2 \\mathrm { r } } } \\Big \\Vert | \\mathrm { E } | ^ { 2 } \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\Omega ) } \\Bigg ) . \\end{align*}"}
+{"id": "6936.png", "formula": "\\begin{align*} \\mathsf Z _ { C , L , E } = \\mathsf A ^ { \\chi ( C , \\mathcal O _ C ) } \\cdot { \\mathsf B } ^ { \\deg L } \\cdot \\mathsf C ^ { \\deg E } , \\end{align*}"}
+{"id": "3762.png", "formula": "\\begin{align*} \\{ a , b c \\} = \\{ a , b \\} c + ( - 1 ) ^ { ( a + 1 ) b } b \\{ a , c \\} . \\end{align*}"}
+{"id": "4988.png", "formula": "\\begin{align*} \\Phi ( p , q ) : = \\frac { \\Lambda ( p , q ) } { p ^ 0 q ^ 0 } \\mathcal { S } ( p , q ) \\end{align*}"}
+{"id": "5460.png", "formula": "\\begin{align*} ( n , k , d _ K ( n , k ) ) = & ( 1 2 , 8 , 4 ) , ( 1 3 , 9 , 4 ) , ( 2 5 , 2 0 , 4 ) , ( 2 6 , 5 , 1 6 ) , ( 2 6 , 1 8 , 6 ) , \\\\ & ( 2 6 , 2 1 , 4 ) , ( 2 7 , 5 , 1 7 ) , ( 2 7 , 2 2 , 4 ) , ( 2 8 , 6 , 1 7 ) , ( 2 8 , 2 3 , 4 ) , \\\\ & ( 2 9 , 2 4 , 4 ) , ( 3 0 , 2 4 , 4 ) , ( 3 0 , 2 5 , 4 ) . \\end{align*}"}
+{"id": "2382.png", "formula": "\\begin{align*} Y _ { k , t } ( s ) = e _ { a _ { 1 } } W ( t ; c _ { k } + k + s - 2 ) g _ { k , s } . \\end{align*}"}
+{"id": "2062.png", "formula": "\\begin{align*} \\| ( x _ 1 , \\ldots , x _ n ) \\| : = \\max _ i | x _ i | . \\end{align*}"}
+{"id": "5652.png", "formula": "\\begin{align*} D ^ { \\nabla f } _ { \\frac { \\partial } { \\partial x ^ j } } \\frac { \\partial } { \\partial x ^ i } = \\hat { \\mathbb { G } } ^ k _ { i j } ( \\nabla f ) \\frac { \\partial } { \\partial x ^ k } . \\end{align*}"}
+{"id": "4450.png", "formula": "\\begin{align*} \\int _ \\Omega \\epsilon \\cdot \\varphi \\dd x = \\int _ { \\partial \\Omega } g ( \\varphi \\cdot \\nu ) \\dd \\mathcal { H } ^ { d - 1 } . \\end{align*}"}
+{"id": "7619.png", "formula": "\\begin{align*} \\mathcal { X } _ \\Sigma : = \\left [ U _ \\Sigma / G _ \\Sigma \\right ] \\end{align*}"}
+{"id": "8179.png", "formula": "\\begin{align*} \\| C a \\| ^ 2 + \\| C b \\| ^ 2 = \\| a \\| ^ 2 \\| b \\| ^ 2 . \\end{align*}"}
+{"id": "1796.png", "formula": "\\begin{align*} c _ { n + 1 , k , m } \\left ( Y , \\mathcal { Z } ^ { ' } \\right ) = c _ { n , k , m } \\left ( Y , \\mathcal { Z } \\right ) . \\end{align*}"}
+{"id": "7520.png", "formula": "\\begin{align*} E _ \\varphi \\left [ \\mu \\right ] = \\iint G ( p , q ) d \\mu ( p ) d \\mu ( q ) + \\int \\varphi d \\mu . \\end{align*}"}
+{"id": "6276.png", "formula": "\\begin{align*} t ^ * = \\frac { 1 } { \\alpha \\left ( 1 + \\norm { x ^ * } ^ 2 \\right ) ^ { - 1 / 2 } + \\beta } . \\end{align*}"}
+{"id": "4799.png", "formula": "\\begin{align*} \\phi _ i ( n , k ) = \\prod _ { j = 0 } ^ { i - 1 } \\alpha ( n + j , k ) , \\psi _ i ( n , k ) = \\psi _ 0 ( n , k ) \\phi _ i ( n , k - 1 ) , \\end{align*}"}
+{"id": "1216.png", "formula": "\\begin{align*} h ( V ) = g ( A _ 2 \\cdot V + b _ 2 ) = f ( A _ 1 ( A _ 2 \\cdot V + b _ 2 ) + b _ 1 ) = f ( A _ 1 A _ 2 \\cdot V + A _ 1 b _ 2 + b _ 1 ) . \\end{align*}"}
+{"id": "6731.png", "formula": "\\begin{align*} F = M + \\overline { G } + \\sqrt { \\mu } f . \\end{align*}"}
+{"id": "1099.png", "formula": "\\begin{align*} \\P \\left ( \\frac { 1 } { m } \\sum _ { i = 1 } ^ m U _ i \\geq \\left ( \\frac { \\pi } { 4 } + \\frac { 1 9 9 \\pi } { 3 2 0 0 k } \\right ) \\| \\mathbf { x } \\| \\right ) \\leq 2 \\exp \\left ( - \\frac { c _ 8 m } { k } \\right ) . \\end{align*}"}
+{"id": "3141.png", "formula": "\\begin{align*} a _ { \\ell _ 1 } = q _ 1 \\ell _ 1 . \\end{align*}"}
+{"id": "4902.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty b _ n ( x ) x ^ n = \\sum _ { j = 0 } ^ \\infty \\frac { j ! ( x + 1 ) _ j x ^ { 2 j } } { ( 1 - x ) ^ 2 ( 1 - 2 x ) ^ 2 \\cdots ( 1 - ( j + 1 ) x ) ^ 2 } . \\end{align*}"}
+{"id": "2771.png", "formula": "\\begin{gather*} - \\frac { i } { 4 } e ^ { \\phi ' } F ^ { ' \\alpha \\hat { \\beta } } = - \\frac { i } { 4 } e ^ { \\phi } F ^ { \\alpha \\hat { \\beta } } - \\epsilon ^ { \\alpha } { \\hat { \\epsilon } } ^ { \\hat { \\beta } } C ^ { - 1 } \\\\ \\phi ^ { ' } = \\phi + \\frac { 1 } { 2 } \\log C \\end{gather*}"}
+{"id": "5958.png", "formula": "\\begin{align*} | j e _ { J + 1 } ( B ) - j e _ { J + 1 } ( A ) | & = | \\sum _ { i = 0 } ^ { J } ( - 1 ) ^ { i } ( e _ { J - i } ( B ) p _ { i + 1 } ( B ) - e _ { J - i } ( A ) p _ { i + 1 } ( A ) ) | \\\\ & \\leq \\max _ { 0 \\leq i \\leq J } | e _ { J - i } ( B ) p _ { i + 1 } ( B ) - e _ { J - i } ( A ) p _ { i + 1 } ( A ) | . \\end{align*}"}
+{"id": "5300.png", "formula": "\\begin{align*} f _ \\theta ( x ) = \\int _ { P \\cap \\theta } a ( \\omega ) e ^ { 2 \\pi i \\omega x } d \\mu _ P ( \\omega ) . \\end{align*}"}
+{"id": "2146.png", "formula": "\\begin{align*} X _ p ^ \\perp = \\left \\{ u \\in W ^ { 1 , p } _ \\sharp ( Y ^ m ) | { \\bf R } { \\bf R } ^ T \\ ; \\nabla _ y u = \\vec { 0 } \\right \\} \\end{align*}"}
+{"id": "3967.png", "formula": "\\begin{align*} \\bar { G } ( u , t ) = \\exp \\left ( - \\lambda t \\left ( 1 - \\frac { ( 1 - \\rho u ) ^ { - r } - 1 } { ( 1 - \\rho ) ^ { - r } - 1 } \\right ) \\right ) . \\end{align*}"}
+{"id": "4330.png", "formula": "\\begin{align*} \\mathbb { E } \\sum _ { j = 0 } ^ \\infty \\sum _ { k = j + 1 } ^ \\infty \\left ( P _ { X , k } - Q _ { X , k } \\right ) & = \\mathbb { P } ( X = n ) \\sum _ { j = 0 } ^ n \\left ( P _ { n , n + 1 } - \\sum _ { k = j + 1 } ^ n \\nu _ k P _ { n , n + 1 } \\right ) \\\\ & = \\mathbb { P } ( X = n ) \\left ( n + 1 - \\mathbb { E } \\nu \\right ) P _ { n , n + 1 } , \\end{align*}"}
+{"id": "6705.png", "formula": "\\begin{align*} P _ { 0 } h \\equiv \\sum _ { i = 0 } ^ { 4 } \\langle h , \\frac { \\chi _ { i } } { M } \\rangle \\chi _ { i } , \\mbox { a n d } P _ { 1 } h \\equiv h - P _ { 0 } h , \\end{align*}"}
+{"id": "7867.png", "formula": "\\begin{align*} \\lambda R = f ^ { \\lambda \\delta } ( \\lambda I \\cap J ) . \\end{align*}"}
+{"id": "4583.png", "formula": "\\begin{align*} \\abs { \\frac { 1 } { N } \\sum _ { l = 0 } ^ { N - 1 } \\sin ( \\theta \\pm \\nu \\pi k l ) } \\leq \\varepsilon . \\end{align*}"}
+{"id": "3516.png", "formula": "\\begin{align*} \\widehat { \\nabla \\mathrm { H } } - \\alpha \\ \\nabla \\int _ \\mathrm { B } \\nabla \\mathbb { G } ^ { ( 0 ) } ( \\mathrm { \\xi } , \\mathrm { \\eta } ) \\cdot \\widehat { \\nabla \\mathrm { H } } ( \\mathrm { \\eta } ) d \\mathrm { \\eta } = \\widehat { \\nabla \\mathrm { H } } ^ { \\textbf { i n } } + \\alpha \\delta ^ 2 \\mathbb { T } _ \\delta \\Big [ \\widehat { \\nabla \\mathrm { H } } \\Big ] . \\end{align*}"}
+{"id": "4615.png", "formula": "\\begin{align*} \\phi ^ * ( x , t ) : = \\sup _ { s \\ge 0 } ( s t - \\phi ( x , s ) ) . \\end{align*}"}
+{"id": "5571.png", "formula": "\\begin{align*} \\dfrac { 2 l _ b \\gamma _ b ( \\alpha _ 0 ) ^ { \\nu + 1 } } { \\theta _ b - \\theta _ m } + \\dfrac { u _ c } { \\Phi _ 2 [ \\beta _ 0 , + \\infty , u _ 2 ] } = \\dfrac { 2 l _ m \\gamma _ m ( \\beta _ 0 ) ^ { \\nu + 1 } } { \\theta _ b - \\theta _ m } \\end{align*}"}
+{"id": "4246.png", "formula": "\\begin{align*} \\lim \\limits _ { b \\to a } \\left ( \\frac { b } { a } \\right ) _ { n } \\sum _ { k = 0 } ^ { n - 1 } \\left ( \\frac { q ^ k } { 1 - \\frac { b q ^ k } { a } } \\right ) & = \\lim \\limits _ { b \\to a } \\left [ \\left ( \\frac { b } { a } \\right ) _ n \\frac { 1 } { 1 - \\frac { b } { a } } + \\left ( \\frac { b } { a } \\right ) _ n \\left ( \\frac { q } { 1 - \\frac { b q } { a } } + \\cdots + \\frac { q ^ { n - 1 } } { 1 - \\frac { b q ^ { n - 1 } } { a } } \\right ) \\right ] \\\\ & = ( q ) _ { n - 1 } . \\end{align*}"}
+{"id": "2156.png", "formula": "\\begin{align*} & - g ( R ( X , Y ) Z , W ) = \\alpha ( X , Z ) \\alpha ( Y , W ) - \\alpha ( X , W ) \\alpha ( Y , Z ) , \\\\ & ( \\nabla _ Z \\alpha ) ( X , Y ) = ( \\nabla _ Y \\alpha ) ( X , Z ) . \\end{align*}"}
+{"id": "3306.png", "formula": "\\begin{align*} K _ 2 ( v \\otimes \\psi _ { n _ 2 } ) = w \\otimes \\psi _ { n _ 2 } . \\end{align*}"}
+{"id": "6102.png", "formula": "\\begin{align*} \\mathcal A ( k , d ) = \\left \\{ A \\in \\binom { [ n ] } { k } : | A \\cap [ d + 1 ] | \\geq d \\right \\} \\end{align*}"}
+{"id": "4807.png", "formula": "\\begin{align*} \\mathcal G _ { b _ { i _ 0 + 1 } - 1 , i _ 0 + 1 } \\mathcal F _ { b _ { i _ 0 } - 1 , i _ 0 } \\cdots \\mathcal F _ { b _ { 1 } - 1 , 1 } ( U _ { n , k } ) & = 0 , \\\\ \\mathcal G _ { a _ { i _ 0 + 1 } , i _ 0 + 1 } \\mathcal F _ { a _ { i _ 0 } , i _ 0 } \\cdots \\mathcal F _ { a _ { 1 } , 1 } ( U _ { n , k - 1 } ) & = 0 , \\end{align*}"}
+{"id": "634.png", "formula": "\\begin{align*} \\partial _ X \\langle \\langle \\varphi , \\varphi ' \\rangle \\rangle = \\langle \\langle \\nabla _ X \\varphi , \\varphi ' \\rangle \\rangle + \\langle \\langle \\varphi , \\nabla _ X \\varphi ' \\rangle \\rangle \\end{align*}"}
+{"id": "1322.png", "formula": "\\begin{align*} \\frac { \\sum _ { \\substack { u \\sim v \\\\ u \\in E } } \\mathbf { \\mathrm { x } } _ u } { \\lambda \\mathbf { \\mathrm { x } } _ v } \\geq \\frac { \\lambda \\mathbf { \\mathrm { x } } _ v - ( k - 1 ) - ( 2 k + 1 ) \\eta } { \\lambda \\mathbf { \\mathrm { x } } _ v } \\ge 1 - \\dfrac { ( k - 1 ) + ( 2 k + 2 ) \\eta } { k - \\frac { 1 } { 1 6 k ^ 2 } } \\ge \\frac { 4 } { 5 k } , \\end{align*}"}
+{"id": "7442.png", "formula": "\\begin{align*} \\sigma _ { k - 1 } ^ \\theta = \\frac { \\theta ^ 2 + \\theta + k ^ 2 + k } { ( s + k ) ( s - k + 1 ) } \\sigma _ { k } ^ \\theta + \\hat { j } \\frac { 2 \\theta \\sigma _ { k } ^ \\theta } { ( s + k ) ( s - k + 1 ) } - \\sigma _ { k } ^ \\theta - \\frac { ( \\hat { j } + k + 1 ) ( \\hat { j } - k ) } { ( s + k ) ( s - k + 1 ) } \\sigma _ { k + 1 } ^ \\theta . \\end{align*}"}
+{"id": "3881.png", "formula": "\\begin{align*} \\sum _ { \\mu , \\nu = 1 } ^ n \\frac { \\partial ^ 2 \\log | g | ^ 2 } { \\partial z _ \\mu \\partial \\bar { z } _ \\nu } \\xi _ \\mu \\bar { \\xi } _ \\nu \\ge | g | ^ { - 4 } \\sum _ { j < k } \\left | \\sum _ { \\mu = 1 } ^ n \\left ( g _ j \\frac { \\partial g _ k } { \\partial z _ \\mu } - g _ k \\frac { \\partial g _ j } { \\partial z _ \\mu } \\right ) \\xi _ \\mu \\right | ^ 2 \\end{align*}"}
+{"id": "6347.png", "formula": "\\begin{align*} q _ i \\| f _ i \\| \\ ; \\| \\tfrac { 1 } { A } f _ i + \\epsilon _ { 1 } ^ { i } h _ i \\| = q _ i \\sqrt { \\frac { 1 } { A ^ 2 } \\| f _ i \\| ^ 4 + ( \\epsilon _ { 1 } ^ { i } ) ^ 2 \\| f _ i \\| ^ 2 \\| h _ i \\| ^ 2 + \\frac { 2 } { A } \\epsilon _ { 1 } ^ { i } \\| f _ i \\| ^ 2 R e ( \\langle f _ i , h _ i \\rangle ) } \\ ; \\ ; < \\ ; q _ i \\frac { 1 } { A } \\| f _ i \\| ^ 2 = m a x _ { i = 1 } ^ N \\frac { q _ i } { A } \\| f _ i \\| ^ 2 . \\end{align*}"}
+{"id": "7696.png", "formula": "\\begin{align*} \\int _ { \\mathbb { D } } | \\hat f ( z ) | ^ { \\alpha } ( 1 - | z | ^ 2 ) ^ { \\alpha - 2 } { d x d y } & = \\int _ { - 1 } ^ { 1 } d x \\int _ { - \\sqrt { 1 - x ^ 2 } } ^ { \\sqrt { 1 - x ^ 2 } } ( 1 - x ^ 2 - y ^ 2 ) ^ { \\alpha - 2 } f ^ { \\alpha } ( x ) d y \\\\ & = \\int _ { - 1 } ^ 1 \\frac { \\sqrt { \\pi } \\left ( 1 - x ^ 2 \\right ) ^ { \\alpha - \\frac { 3 } { 2 } } \\Gamma [ \\alpha - 1 ] } { \\Gamma \\left [ \\alpha - \\frac { 1 } { 2 } \\right ] } f ^ { \\alpha } ( x ) d x . \\end{align*}"}
+{"id": "6168.png", "formula": "\\begin{align*} V _ i V _ j f ( \\underline { z } ) = R _ { q ^ { d - i } } M _ { z _ i } R _ { q ^ { d - j } } M _ { z _ j } f ( \\underline { z } ) = q ^ { d - j } R _ { q ^ { d - i } } z _ i z _ j f ( q ^ { d - j } \\underline { z } ) = q ^ { 3 d - 2 i - j } z _ i z _ j f ( q ^ { 2 d - i - j } \\underline { z } ) \\end{align*}"}
+{"id": "1840.png", "formula": "\\begin{align*} f g ( p ^ k ) & = f ( p ^ k ) g ( p ^ k ) = \\pi ( A ^ k u ) \\pi ( B ^ k v ) \\\\ & = \\pi \\left ( ( A ^ k \\otimes B ^ k ) ( u \\otimes v ) \\right ) \\\\ & = \\pi ( ( A \\otimes B ) ^ k ( u \\otimes v ) ) \\end{align*}"}
+{"id": "4281.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\textup { s p t } ( n ) q ^ n = \\sum _ { n = 1 } ^ { \\infty } n p ( n ) q ^ n - \\frac { 1 } { 2 } \\sum _ { n = 1 } ^ { \\infty } N _ 2 ( n ) q ^ n , \\end{align*}"}
+{"id": "7290.png", "formula": "\\begin{align*} X ^ * ( 0 , \\omega ) = X _ 0 ( \\omega ) . \\end{align*}"}
+{"id": "5199.png", "formula": "\\begin{align*} a _ n & = \\sum _ { k = 2 } ^ n 2 ^ { n - k } \\cdot \\binom { n - 1 } { k - 1 } \\cdot \\frac { 2 ^ k + 2 \\cdot ( - 1 ) ^ k } { 3 } = \\\\ & = \\frac { 2 ^ n } { 3 } \\sum _ { k = 2 } ^ { n } \\binom { n - 1 } { k - 1 } + \\frac { 2 ^ { n + 1 } } { 3 } \\sum _ { k = 2 } ^ { n } \\binom { n - 1 } { k - 1 } \\cdot \\left ( - \\frac { 1 } { 2 } \\right ) ^ k \\enspace . \\end{align*}"}
+{"id": "4026.png", "formula": "\\begin{align*} w ( y , s ) = f _ { b ( s ) } ( y ) \\left ( 1 + e _ { b ( s ) } ( y ) q ( y , s ) \\right ) \\end{align*}"}
+{"id": "5947.png", "formula": "\\begin{align*} x _ { k + 1 } ^ { j } + \\cdots + x _ { s } ^ { j } - y _ { k + 1 } ^ { j } - \\cdots - y _ { s } ^ { j } & = y _ { 1 } ^ { j } + \\cdots + y _ { k } ^ { j } - x _ { 1 } ^ { j } - \\cdots - x _ { k } ^ { j } , 1 \\leq j \\leq k \\end{align*}"}
+{"id": "2132.png", "formula": "\\begin{align*} \\mathrm { c o r a n k } _ { \\Lambda ( G ) } ( \\displaystyle \\bigoplus _ { v \\in S _ p } J _ v ^ { \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) ) = \\mathrm { c o r a n k } _ { \\Lambda ( G ) } ( H ^ 1 ( G _ { \\infty } ^ { S } , E _ { p ^ { \\infty } } ) = 2 . \\end{align*}"}
+{"id": "3934.png", "formula": "\\begin{align*} \\int _ { - t } ^ \\top \\int _ { \\Gamma _ n } f ( x ) \\ , d \\mu _ { \\Gamma _ n } ( x ) d t = \\int _ { U _ t ( \\Gamma _ n ) } f \\circ p _ n ( y ) \\det ( d _ { T _ n ^ { - 1 } ( y ) } T _ n ) \\ , d y . \\end{align*}"}
+{"id": "4559.png", "formula": "\\begin{align*} 2 \\xi _ 2 = \\frac { 2 ( 3 ^ { \\alpha + 1 } - 2 ^ { 2 \\alpha + 1 } - 1 ) } { 3 } < \\xi _ 1 = - \\frac { ( 2 ^ \\alpha - 2 ) ( 2 ^ \\alpha - 1 ) } { 3 } < \\frac { \\xi _ 2 } { 2 } = \\frac { 3 ^ { \\alpha + 1 } - 2 ^ { 2 \\alpha + 1 } - 1 } { 6 } \\end{align*}"}
+{"id": "7693.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 1 + } \\| f \\| _ { \\alpha , \\alpha p } = \\| f \\| _ { p } . \\end{align*}"}
+{"id": "608.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathfrak { s } \\ , ( z _ 1 , z _ 2 , z _ 3 , z _ 4 , z _ 0 ) = ( z _ 0 , z _ 4 , z _ 3 , z _ 2 , z _ 1 ) \\ , , \\\\ & \\mathfrak { t } \\ , ( z _ 1 , z _ 2 , z _ 3 , z _ 4 , z _ 0 ) = ( z _ 2 , - z _ 3 - 1 , - z _ 1 - 1 , z _ 4 , z _ 0 ) \\ , , \\\\ & \\mathfrak { i } \\ , ( z _ 1 , z _ 2 , z _ 3 , z _ 4 , z _ 0 ) = ( z _ 1 , z _ 2 , z _ 3 , z _ 0 , z _ 4 ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "6571.png", "formula": "\\begin{align*} P A P ^ T = B . \\end{align*}"}
+{"id": "7293.png", "formula": "\\begin{align*} \\frac { d } { d t } \\overline { X } ^ * ( t ) = A ( t ) \\overline { X } ^ * ( t ) + B ( t ) R ^ { - 1 } ( t ) B ( t ) \\overline { \\Upsilon } ( t ) , \\end{align*}"}
+{"id": "7086.png", "formula": "\\begin{gather*} h _ { { R _ Z ^ H } } ( Z , Z _ \\bullet , Z _ \\bullet ' ) = Z _ 5 ^ 5 + Z _ 5 Z _ { 1 3 } Z _ 7 ' + Z _ { 1 4 } Z _ { 1 1 } ' + Z _ 5 Z _ 7 Z _ { 1 3 } ' + Z _ { 1 1 } Z _ { 1 4 } ' , h _ H ( Z ) = 5 Z _ 5 ^ 5 . \\end{gather*}"}
+{"id": "2425.png", "formula": "\\begin{align*} \\tilde { L } ( { \\rm D } ( w ) ) = \\sum _ { N \\geq 2 } \\zeta ^ { \\mathfrak { m } } ( N ) \\tilde { L } ( { \\rm C u t } _ { ( \\emptyset ; \\{ 1 \\} ^ { N - 1 } ) } ( w ) ) \\end{align*}"}
+{"id": "1198.png", "formula": "\\begin{align*} \\nu = [ h , \\eta ] = h \\eta h ^ { - 1 } \\eta ^ { - 1 } = \\rho \\eta ^ { g } \\rho ^ { - 1 } \\eta ^ { - 1 } \\in \\Gamma ^ G . \\end{align*}"}
+{"id": "8149.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } ( \\overline L ) = \\inf _ { x \\in X ^ { ( 0 ) } } \\frac { h _ { \\overline L } ( x ) } { [ K ( x ) : K ] } . \\end{align*}"}
+{"id": "4695.png", "formula": "\\begin{align*} r \\frac { d } { d r } \\left [ { } _ 2 \\ ! F _ { 1 } \\left ( \\lambda , \\lambda + 1 ; 2 \\lambda + 1 ; \\tilde { A } \\right ) \\right ] = \\frac { \\lambda ( \\lambda + 1 ) } { 2 \\lambda + 1 } { } _ 2 \\ ! F _ { 1 } \\left ( \\lambda + 1 , \\lambda + 2 ; 2 \\lambda + 2 ; \\tilde { A } \\right ) r \\frac { d } { d r } \\tilde { A } , \\end{align*}"}
+{"id": "1170.png", "formula": "\\begin{align*} \\frac { 3 ^ { 9 ( 2 k - p _ k ) / 5 + 1 } } { 1 0 ^ { 2 k - p _ k + 1 } } & > \\frac { 3 ^ { ( 9 / 1 0 ) \\log _ 2 ( \\pi k ) + 1 } } { 1 0 ^ { ( 1 / 2 ) \\log _ 2 ( \\pi k ) + 4 1 / 4 0 } } = \\frac { 3 } { 1 0 ^ { 4 1 / 4 0 } } ( \\pi k ) ^ { ( 9 / 1 0 ) \\log _ 2 3 - ( 1 / 2 ) \\log _ 2 1 0 } \\\\ & > \\frac { 1 } { 4 } ( \\pi k ) ^ { - 1 / 4 } > ( 2 \\pi k ) ^ { - 1 / 2 } , \\end{align*}"}
+{"id": "2631.png", "formula": "\\begin{align*} B S ( m , n ) = \\langle a , b \\mid a b ^ m = b ^ n a \\rangle \\end{align*}"}
+{"id": "6071.png", "formula": "\\begin{align*} \\max _ { j = 1 , \\dots , M } \\ell _ j \\le N + 1 , \\max _ { s \\in \\{ m _ 1 , \\dots , m _ { \\ell _ j } \\} , \\ , j = 1 \\dots , M } y ^ { s } _ j \\le L , \\quad \\sum _ { j = 1 } ^ N \\sum _ { s \\in \\{ m _ 1 , \\dots , m _ { \\ell _ j } \\} } n ^ { s } _ j = N . \\end{align*}"}
+{"id": "7642.png", "formula": "\\begin{align*} \\begin{aligned} w & = u _ 1 ( c _ 0 x _ 0 ^ 3 x _ 6 ^ 3 + c _ 1 x _ 1 ^ 3 x _ 7 ^ 3 + c _ 2 x _ 2 ^ 3 x _ 8 ^ 3 - 3 \\lambda _ 1 x _ 3 x _ 4 x _ 5 x _ 6 x _ 7 x _ 8 ) \\\\ & { } + u _ 2 ( c _ 3 x _ 3 ^ 3 x _ 9 ^ 3 + c _ 4 x _ 4 ^ 3 x _ { 1 0 } ^ 3 + c _ 5 x _ 5 ^ 3 x _ { 1 1 } ^ 3 - 3 \\lambda _ 2 x _ 0 x _ 1 x _ 2 x _ 9 x _ { 1 0 } x _ { 1 1 } ) , \\end{aligned} \\end{align*}"}
+{"id": "7981.png", "formula": "\\begin{align*} t ' ( s ) = - \\frac { 2 c ( 2 n - 1 ) } { ( 2 n + 1 ) } r ^ { 3 } ( s ) \\leq 0 . \\end{align*}"}
+{"id": "5683.png", "formula": "\\begin{align*} X = \\{ c _ 0 F _ 0 + c _ 1 F _ 1 + . . . + c _ n F _ n = 0 \\} \\end{align*}"}
+{"id": "7512.png", "formula": "\\begin{align*} G ( p , q ) = - \\log | z ( p ) | + O ( 1 ) p \\to q , \\end{align*}"}
+{"id": "7613.png", "formula": "\\begin{align*} \\frac { | | x _ { k - 1 } ( \\theta ) - x ^ { \\dagger } _ { k - 1 , \\varepsilon } ( \\theta ) | | _ \\pi ^ 2 } { 2 \\eta _ k t } \\leq \\frac { \\varepsilon _ { k - 1 } ^ 2 } { 2 \\varepsilon _ k \\alpha ^ 2 } = \\varepsilon _ k \\end{align*}"}
+{"id": "745.png", "formula": "\\begin{align*} V _ n ' ( | X ( t ) - Y ( t ) | _ H ^ 2 ) & = ( 1 - r ) | X ( t ) - Y ( t ) | _ H ^ { - 2 r } , \\\\ V _ n '' ( | X ( t ) - Y ( t ) | _ H ^ 2 ) & = - r ( 1 - r ) | X ( t ) - Y ( t ) | _ H ^ { - 2 r - 2 } . \\end{align*}"}
+{"id": "363.png", "formula": "\\begin{align*} \\{ h , f _ 1 \\ , f _ 2 \\} = \\{ h , f _ 1 \\} f _ 2 + f _ 1 \\{ h , f _ 2 \\} \\end{align*}"}
+{"id": "3341.png", "formula": "\\begin{align*} K _ 2 \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } = \\lambda _ { n _ 1 } \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } , \\end{align*}"}
+{"id": "2809.png", "formula": "\\begin{align*} d D ^ X _ s = - D ^ X _ s d R _ s + \\gamma _ s d X _ s , s \\in [ t , T ] , D ^ X _ { t - } = d . \\end{align*}"}
+{"id": "7064.png", "formula": "\\begin{align*} f _ a ( X _ 0 , \\cdots , X _ i , ( X _ { i + 1 } + \\tilde { z } _ { i + 1 } ) X _ i , \\cdots , ( X _ d + \\tilde { z } _ d ) X _ i ) = : X _ i g _ a ( { X } _ { 0 } , \\dots , { X } _ d ) . \\end{align*}"}
+{"id": "5980.png", "formula": "\\begin{align*} W _ { b } ^ { ( q , r ) } ( x ) = W ^ { ( q ) } ( x ) , Z _ { b } ^ { ( q , r ) } ( x ) = Z ^ { ( q ) } ( x ) , \\overline { Z } _ { b } ^ { ( q , r ) } ( x ) = \\overline { Z } ^ { ( q ) } ( x ) , \\quad \\ x \\in [ 0 , b ] . \\end{align*}"}
+{"id": "6869.png", "formula": "\\begin{align*} \\bigcup _ { j = 1 } ^ { w } H _ j = \\left \\{ a _ 1 , a _ 2 , \\dots , a _ n \\right \\} . \\end{align*}"}
+{"id": "1157.png", "formula": "\\begin{align*} \\Delta _ 1 ( m , n ) = \\delta ( m , n ) + O ( ( m n ) ^ { 3 / 8 + \\epsilon } k ^ { - 1 3 / 1 2 } ) . \\end{align*}"}
+{"id": "6425.png", "formula": "\\begin{align*} ( a ; q ) _ 0 = 1 , \\ , \\ , ( a ; q ) _ n = \\prod _ { k = 0 } ^ { n - 1 } ( 1 - a q ^ k ) , \\ , \\ , ( a ; q ) _ { \\infty } = \\prod _ { k = 0 } ^ { \\infty } ( 1 - a q ^ k ) . \\end{align*}"}
+{"id": "5495.png", "formula": "\\begin{align*} \\frac { ( t ) _ N ( t ^ { - 1 } q ) _ N } { ( q ) _ N } F _ N ( 0 , t ^ { - 1 } ; t ) = \\sum _ { n = 0 } ^ { N } \\frac { ( t ^ { - 1 } ) _ n ( t ) _ n q ^ { n } } { ( q ) _ n } . \\end{align*}"}
+{"id": "6493.png", "formula": "\\begin{align*} A : = A _ { k } : = \\{ u > K ^ k \\} \\cap Q _ 1 B : = A _ { k - 1 } : = \\{ u > K ^ { k - 1 } \\} \\cap Q _ 1 . \\end{align*}"}
+{"id": "6241.png", "formula": "\\begin{align*} H = \\int ^ \\oplus _ \\mathbb { R } H ( p ) \\ , \\mathrm { d } p , \\end{align*}"}
+{"id": "5874.png", "formula": "\\begin{align*} \\xi = \\xi _ 1 \\cap \\cdots \\cap \\xi _ k \\end{align*}"}
+{"id": "2502.png", "formula": "\\begin{align*} \\delta = \\lim _ { N \\to \\infty } \\dfrac { | A \\cap \\Phi _ N | } { | \\Phi _ N | } \\end{align*}"}
+{"id": "1925.png", "formula": "\\begin{align*} \\| I _ 2 \\| _ { L _ p ( Q _ r ) } \\le N ( d , p ) \\sum _ { k = 0 } ^ { \\infty } 2 ^ { - 2 k } ( | h | ^ p ) ^ { 1 / p } _ { Q _ { 1 , 2 ^ k } } . \\end{align*}"}
+{"id": "3498.png", "formula": "\\begin{align*} \\mathrm { H } ( \\mathrm { x } ) - \\alpha \\int _ \\Omega \\nabla \\mathbb { G } ^ { ( \\mathrm { k } ) } ( \\mathrm { x } , \\mathrm { y } ) \\cdot \\nabla \\mathrm { H } ( \\mathrm { y } ) d \\mathrm { y } = \\mathrm { H } ^ { \\textbf { i n } } ( \\mathrm { x } ) , \\end{align*}"}
+{"id": "7789.png", "formula": "\\begin{align*} C _ { g } ( z , w ) = - \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } \\frac { g ( u ) } { \\overline { u - w } ( u - z ) } d a ( u ) , g \\in \\mathcal { G } _ 1 , z , w \\in \\mathbb { C } . \\end{align*}"}
+{"id": "6089.png", "formula": "\\begin{align*} \\sum _ { i = - k } ^ { k } d ( \\gamma ^ i ( x ) , \\gamma ^ i ( y ) ) \\leq d ( x , y ) \\sum _ { i = - k } ^ k L ^ { | i | } \\leq C L ^ { - ( 2 k + 1 / 2 ) } \\sum _ { i = - k } ^ k L ^ { | i | } \\leq \\frac { 2 C \\sqrt { L } } { L - 1 } . \\end{align*}"}
+{"id": "4133.png", "formula": "\\begin{align*} \\partial _ c \\chi _ { n } ^ { \\alpha } ( c ) = 2 c \\int _ 0 ^ 1 x ^ 2 \\left ( \\psi _ { n , c } ^ { ( \\alpha ) } ( x ) \\right ) ^ 2 d x . \\end{align*}"}
+{"id": "6055.png", "formula": "\\begin{align*} f _ \\epsilon ( x ) = \\begin{cases} f ( x ) \\ ; \\ ; s i \\ ; \\ ; x \\in \\Omega \\backslash \\overline { \\omega _ \\epsilon } \\\\ [ 0 . 2 c m ] \\gamma f ( x ) \\ ; \\ ; s i \\ ; \\ ; x \\in \\omega _ \\epsilon ; \\end{cases} \\end{align*}"}
+{"id": "7896.png", "formula": "\\begin{align*} R ( a ) \\cdot R ( b ) = R \\big ( R ( a ) \\cdot b + a \\cdot R ( b ) \\big ) + \\kappa ~ \\ ! a \\cdot b , ~ a , b \\in A . \\end{align*}"}
+{"id": "2838.png", "formula": "\\begin{align*} [ M _ \\lambda : \\mathsf { L } ( \\lambda ) ] = 1 [ M _ \\lambda : \\mathsf { L } ( \\mu ) ] \\neq 0 \\Rightarrow \\mu \\preceq \\lambda , \\end{align*}"}
+{"id": "493.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { n \\choose k } 2 ^ k F _ { j k } ( \\sqrt { 5 } F _ j ) ^ { n - k } B _ { n - k } ( \\alpha ) n F _ j 2 ^ { 1 - n } \\Big ( ( \\sqrt { 5 } F _ j + L _ { j + 3 } ) ^ { n - 1 } + ( - \\sqrt { 5 } F _ j + L _ { j + 3 } ) ^ { n - 1 } \\Big ) , \\end{align*}"}
+{"id": "839.png", "formula": "\\begin{align*} \\widetilde { \\theta } _ 1 = \\widetilde { \\theta } _ 1 ( y _ i ^ { * } , y _ { - i } ^ { * } ) = \\theta + \\frac { ( 1 - \\theta ) } { l _ 1 + \\frac { l _ 2 y _ i ^ { * } } { N } + l _ 2 y _ { - i } ^ { * } } \\end{align*}"}
+{"id": "5162.png", "formula": "\\begin{align*} \\Vert E \\Vert ^ p 4 \\pi r ^ { 2 - p } \\geq ( 2 \\pi ) ^ { 1 - p } \\int _ { 2 r } ^ { C r } t ^ { 1 - p } = \\frac { ( 2 \\pi ) ^ { 1 - p } } { 2 - p } \\left ( ( C r ) ^ { 2 - p } - ( 2 r ) ^ { 2 - p } \\right ) . \\end{align*}"}
+{"id": "1949.png", "formula": "\\begin{align*} | t ^ { \\vec { a } } x | _ { \\vec { a } } = t | x | _ { \\vec { a } } , \\ ; \\ ; \\ ; \\ ; x \\in \\mathbb { R } ^ d , \\ ; t > 0 . \\end{align*}"}
+{"id": "1704.png", "formula": "\\begin{align*} [ b ( g _ 1 ) , b ^ * ( g _ 2 ) ] = \\langle g _ 1 , g _ 2 \\rangle _ { \\mathfrak { h } } , [ b ( g _ 1 ) , b ( g _ 2 ) ] = [ b ^ * ( g _ 1 ) , b ^ * ( g _ 2 ) ] = 0 \\ , , \\end{align*}"}
+{"id": "3730.png", "formula": "\\begin{align*} \\begin{cases} \\Delta \\phi + \\lambda m ( x ) \\phi = 0 & \\ ; \\ ; \\Omega , \\\\ \\ ; \\phi = 0 & \\ ; \\ ; \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "165.png", "formula": "\\begin{align*} \\begin{aligned} | \\mathfrak { a } ( | v | ) | + | \\mathfrak { b } ( | v | ) | \\leq C ( 1 + | v | ) \\ \\forall v \\in \\R ^ 3 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "2061.png", "formula": "\\begin{align*} | X | : = \\int _ X \\Omega _ X . \\end{align*}"}
+{"id": "1544.png", "formula": "\\begin{align*} P \\left ( \\bigcap _ { e \\in E } X _ { e } ^ { - 1 } \\left ( A _ { e } \\right ) \\right ) = \\prod _ { e \\in E } P \\left ( X _ { e } ^ { - 1 } \\left ( A _ { e } \\right ) \\right ) . \\end{align*}"}
+{"id": "4269.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { n ( - d ) ^ { n - 1 } ( q / d ) _ { n - 1 } q ^ { n ( n + 1 ) / 2 } } { ( q ) _ { n } ^ 2 } & = \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { n q ^ n ( d q ) _ { n - 1 } } { ( q ) _ n } \\\\ & \\quad - \\frac { ( d q ) _ { \\infty } } { ( q ) _ { \\infty } } \\sum _ { j = 1 } ^ { \\infty } \\frac { q ^ { j ^ 2 } } { ( q ) _ { j } ^ { 2 } } \\sum _ { n = 1 } ^ { j } \\frac { q ^ n } { ( 1 - d q ^ n ) ( 1 - q ^ n ) } . \\end{align*}"}
+{"id": "4175.png", "formula": "\\begin{align*} n _ 1 t = n _ 2 t = \\cdots = n _ k t t \\in N ^ { - 1 } \\Z . \\end{align*}"}
+{"id": "1999.png", "formula": "\\begin{align*} \\bigg | \\sum _ { u \\in N ( v ) } z _ u \\bigg | = | ( q _ n - d ( v ) ) z _ v | > \\Omega ( \\sqrt { n } ) . \\end{align*}"}
+{"id": "873.png", "formula": "\\begin{align*} \\delta _ 1 ( \\alpha , \\beta ) = 1 + \\varepsilon ( \\alpha , \\beta ) , \\mbox { w h e r e } \\varepsilon ( \\alpha , \\beta ) = \\frac { ( r - \\alpha - \\beta ) ( \\alpha - \\beta ) } { ( \\beta + 1 ) ( r - \\beta + 1 ) } . \\end{align*}"}
+{"id": "6316.png", "formula": "\\begin{align*} X ^ n _ t = x _ 0 ( t ) + \\int _ 0 ^ t K _ \\mu ( s , t ) \\mu _ n ( s , X ^ n _ s ) \\dd s + \\int _ 0 ^ t K _ \\sigma ( s , t ) \\sigma _ n ( s , X ^ n _ s ) \\dd B _ s , t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "200.png", "formula": "\\begin{align*} 2 ^ { 2 r + s + m } = 2 ^ { 2 r + 2 k + s } \\ , \\ , ( a _ f ( j ) ) \\end{align*}"}
+{"id": "1765.png", "formula": "\\begin{align*} { \\rm s i g n } \\left \\langle Z _ { * } , I \\backslash \\left \\{ n \\right \\} \\right \\rangle = + 1 \\quad { \\rm a n d } \\quad { \\rm s i g n } \\left \\langle Z _ { * } , I \\backslash \\left \\{ 1 \\right \\} \\right \\rangle = \\left ( - 1 \\right ) ^ { k + 1 } . \\end{align*}"}
+{"id": "2152.png", "formula": "\\begin{align*} | \\widetilde { R } | : = \\begin{vmatrix} R _ { 1 2 1 2 } & R _ { 1 2 1 3 } & R _ { 1 2 2 3 } \\\\ R _ { 1 2 1 3 } & R _ { 1 3 1 3 } & R _ { 1 3 2 3 } \\\\ R _ { 1 2 2 3 } & R _ { 1 3 2 3 } & R _ { 2 3 2 3 } \\end{vmatrix} \\geq 0 \\end{align*}"}
+{"id": "6426.png", "formula": "\\begin{align*} \\eta _ { x _ i } ^ r \\{ f ( x _ 1 , \\cdots , x _ n ) \\} = f ( x _ 1 , \\cdots , x _ { i - 1 } , q ^ r x _ i , x _ { i + 1 } , \\cdots , x _ n ) . \\end{align*}"}
+{"id": "4741.png", "formula": "\\begin{gather*} \\partial _ k P _ r ( a ) = \\partial _ k r ^ \\sharp ( \\varphi ( a ) ) , \\\\ P _ r \\partial _ k ( a ) = r ^ \\sharp ( \\varphi ( \\partial _ k ( a ) ) ) = \\sum _ i \\mathfrak { B } ( \\partial _ k ( a ) , a _ i ) b _ i = \\sum _ i \\mathfrak { B } ( a , \\hat { \\partial _ k } ( a _ i ) ) b _ i \\\\ \\hphantom { P _ r \\partial _ k ( a ) } { } = \\sum _ i \\langle \\varphi ( a ) , \\hat { \\partial _ k } ( a _ i ) \\rangle b _ i = r ^ \\sharp \\hat { \\partial _ k } ^ * ( \\varphi ( a ) ) . \\end{gather*}"}
+{"id": "7523.png", "formula": "\\begin{align*} E _ { \\varphi } ( F ) = \\inf _ { \\mu \\in \\mathcal E ^ 1 ( F ) } E _ { \\varphi } \\left [ \\mu \\right ] . \\end{align*}"}
+{"id": "7253.png", "formula": "\\begin{align*} \\hat { F } ( \\mu ) = \\lim _ { k \\to \\infty } \\limsup _ { n \\to \\infty } { F } ( \\mu _ k ^ { ( n ) } ) , \\hat { F } ( \\nu ) = \\lim _ { k \\to \\infty } \\limsup _ { n \\to \\infty } { F } ( \\nu _ k ^ { ( n ) } ) . \\end{align*}"}
+{"id": "205.png", "formula": "\\begin{align*} ( I ^ \\gamma _ { T - } \\psi ) ( t ) = \\frac { \\Gamma ( m + 1 ) } { \\Gamma ( \\gamma + m + 1 ) } T ^ { \\gamma } \\biggl ( 1 - \\frac { t } { T } \\biggr ) ^ { m + \\gamma } , \\ , \\ , \\ , t \\in [ 0 , T ) , \\end{align*}"}
+{"id": "6111.png", "formula": "\\begin{align*} \\begin{aligned} | \\mathcal S ( k , 3 ) | = \\binom { n - 2 } { k - 2 } - \\binom { n - k + 1 } { k - 2 } + \\binom { n - k - 2 } { k - 5 } + 6 \\sim ( k - 3 ) \\binom { n } { k - 3 } \\end{aligned} \\end{align*}"}
+{"id": "4600.png", "formula": "\\begin{align*} V ( n ) = K _ 1 ( E , A ) \\frac { \\sin 2 \\pi \\theta ( n ) + \\sin 2 \\pi \\tilde { \\theta } ( n ) + 1 0 0 } { n - b } . \\end{align*}"}
+{"id": "1255.png", "formula": "\\begin{align*} - \\frac { 1 } { r ^ { N - 1 } } \\left ( r ^ { N - 1 } | u _ p ' ( r ) | ^ { p - 2 } u _ p ' ( r ) \\right ) ^ { ' } = \\frac { A } { r } , \\end{align*}"}
+{"id": "415.png", "formula": "\\begin{align*} \\sum _ { \\substack { U \\pmod { D } \\\\ U \\equiv S \\pmod { Q } } } \\psi ( U ) = \\prod _ { i = 1 } ^ { m } \\sum _ { \\substack { U _ i \\pmod { P _ i ^ { \\alpha _ i } } \\\\ U _ i \\equiv S \\pmod { P _ i ^ { \\beta _ i } } } } \\psi _ { P _ i ^ { \\alpha _ i } } ( U _ i ) . \\end{align*}"}
+{"id": "581.png", "formula": "\\begin{align*} & [ C _ { I J } , C _ { J K } ] = [ C _ { J K } , C _ { I K } ] = [ C _ { I K } , C _ { I J } ] \\ , , \\\\ & \\big [ C _ { J K } , [ C _ { I J } , C _ { J K } ] \\big ] = 2 C _ { I K } C _ { J K } - 2 C _ { J K } C _ { I J } + 2 ( C _ J - C _ K ) ( C _ { I J K } - C _ I ) \\ , , \\end{align*}"}
+{"id": "2498.png", "formula": "\\begin{align*} B _ 1 + \\cdots + B _ k = \\{ b _ 1 + \\cdots + b _ k : b _ 1 \\in B _ 1 , \\dots , b _ k \\in B _ k \\} \\end{align*}"}
+{"id": "5648.png", "formula": "\\begin{align*} \\mathbf { R i c } ^ \\perp ( y ) = \\mathbf { R i c } ( y ) - \\mathbf { K } ( y , J y ) = \\mathbf { R i c } ( y ) - \\mathbf { H } ( y _ o ) . \\end{align*}"}
+{"id": "4890.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty f _ j \\left ( 2 - \\frac { 1 } { x } , 2 - \\frac { 1 } { x } \\right ) = \\frac { 1 - x } { 2 - x } + \\frac { ( 1 - x ) ^ 2 } { ( 1 - 2 x ) ( 2 - x ) } \\sum _ { j = 0 } ^ \\infty f _ j \\left ( 3 - \\frac { 1 } { x } , 2 - \\frac { 1 } { x } \\right ) . \\end{align*}"}
+{"id": "2369.png", "formula": "\\begin{align*} ( t , v ' ) = \\begin{cases} ( 1 , x _ { b _ { 1 } - 1 } x _ { b _ { 2 } } \\cdots x _ { b _ { n } } ) & b _ { 1 } > 1 \\\\ ( 0 , x _ { b _ { 2 } } \\cdots x _ { b _ { n } } ) & b _ { 1 } = 1 . \\end{cases} \\end{align*}"}
+{"id": "4339.png", "formula": "\\begin{align*} H ( u _ 1 | u _ 2 ) : = \\int _ \\Omega \\left \\{ \\bigg [ f _ 1 \\ln \\Big ( \\frac { f _ 1 } { f _ 2 } \\Big ) - ( f _ 1 - f _ 2 ) \\bigg ] + \\cfrac { b } { c } \\bigg [ g _ 1 \\ln \\Big ( \\frac { g _ 1 } { g _ 2 } \\Big ) - ( g _ 1 - g _ 2 ) \\bigg ] \\right \\} \\ , \\mathrm { d } x \\ , , \\end{align*}"}
+{"id": "5012.png", "formula": "\\begin{align*} \\| u _ { i n } \\| ^ 2 - \\Big \\| u _ { i n } - \\sum _ { s = 1 } ^ { m } \\phi _ i ( g _ s ) \\ , ( w _ { i } \\circ g _ s ^ { - 1 } ) ( \\ , \\cdot \\ , - g _ s \\xi _ { r n } ) \\Big \\| ^ 2 + m \\| w _ { i } \\| ^ 2 + o _ n ( 1 ) \\quad i \\in I _ { Q _ r } , \\end{align*}"}
+{"id": "5032.png", "formula": "\\begin{align*} \\partial ^ * ( E \\cap F ) & \\subseteq ( F ^ { ( 1 ) } \\cap \\partial ^ * E ) \\cup ( E ^ { ( 1 ) } \\cap \\partial ^ * F ) , \\\\ \\partial ^ * ( E \\cup F ) & \\subseteq ( F ^ { ( 0 ) } \\cap \\partial ^ * E ) \\cup ( E ^ { ( 0 ) } \\cap \\partial ^ * F ) . \\end{align*}"}
+{"id": "1553.png", "formula": "\\begin{align*} \\varphi _ { s ^ { \\prime } , i , j } \\left ( \\omega _ { i } , \\omega \\right ) = f _ { s ^ { \\prime } , i , j } \\left ( \\omega _ { i } , X _ { s _ { i } ^ { \\prime } } \\left ( \\omega \\right ) \\right ) = f _ { s ^ { \\prime } , i , j } \\left ( \\omega _ { i } , X _ { s _ { i } ^ { \\prime } } \\left ( \\omega ^ { \\prime } \\right ) \\right ) = \\varphi _ { s ^ { \\prime } , i , j } \\left ( \\omega _ { i } , \\omega ^ { \\prime } \\right ) . \\end{align*}"}
+{"id": "140.png", "formula": "\\begin{align*} ( d _ i y ) ( [ \\mathbf { a } ] \\otimes [ \\mathbf { b } ] ) = y ( d ^ i [ \\mathbf { a } ] \\otimes [ \\mathbf { b } ] ) = 0 , \\end{align*}"}
+{"id": "7806.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { 1 } { 2 \\pi } \\int _ { \\mathcal { U } } ( - \\cot ( \\theta ) + i ) ( g _ { \\phi } ( u ( t , \\theta ) ) - h ( t , \\theta ) ) d t d \\theta + \\\\ \\frac { 1 } { 2 \\pi } \\int _ { \\mathcal { L } } ( \\cot ( \\theta ) - i ) ( g _ { \\phi } ( u ( t , \\theta ) ) - h ( t , \\theta ) ) d t d \\theta = 0 , \\end{gathered} \\end{align*}"}
+{"id": "1816.png", "formula": "\\begin{align*} B ( [ x , y ] , z ) = B ( x , [ y , z ] ) . \\end{align*}"}
+{"id": "7012.png", "formula": "\\begin{align*} L _ { w } ( f _ { + } ) \\geq \\chi _ { \\{ f > 0 \\} } L _ { w } f = \\chi _ { \\{ f > 0 \\} } \\left ( L ^ { a c } _ w f \\cdot m _ w + L ^ { s } _ w f \\right ) \\geq \\chi _ { \\{ f > 0 \\} } L ^ { a c } _ w f \\cdot m _ w . \\end{align*}"}
+{"id": "775.png", "formula": "\\begin{align*} \\left | \\Psi ( A ) ( u ) \\right | = \\left | \\sum _ { j = 1 } ^ { n } A ( x _ { j } , y _ { j } ) \\right | \\leq \\sum _ { j = 1 } ^ { n } \\left | A ( x _ { j } , y _ { j } ) \\right | \\leq \\left \\| A \\right \\| _ { X , Y ; \\ell _ { 1 } } \\cdot \\left \\| ( x _ { j } ) _ { j = 1 } ^ { n } \\right \\| _ { X ( E ) } \\cdot \\left \\| ( y _ { j } ) _ { j = 1 } ^ { n } \\right \\| _ { Y ( F ) } . \\end{align*}"}
+{"id": "2870.png", "formula": "\\begin{align*} g _ { M ' } ^ 3 ( \\tau , r ) & \\leqslant 4 ( g _ 0 ( r ^ { - 1 } ) ) ^ { - 2 m ' _ 0 } ( M ' \\vee C _ { m ' _ 0 } ) + 2 r \\\\ & + 2 ( C _ { m ' _ 0 } ( g ( r ^ { - 1 } ) ) + 1 ) \\big ( \\ln \\ln ( r ^ { - 1 } \\vee 3 ) \\big ) ^ { - 1 } + r \\exp \\big ( C \\tau \\ln \\ln ( r ^ { - 1 } \\vee 3 ) \\big ) . \\end{align*}"}
+{"id": "2593.png", "formula": "\\begin{align*} \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\Psi } ( t , \\boldsymbol { x } ) ) } ^ { N , i } : = k _ i ^ N ( t , \\nu ^ { N , 1 } _ { \\boldsymbol { x } } , \\ldots , \\nu ^ { N , N } _ { \\boldsymbol { x } } , \\lambda ^ { N , 1 } _ { \\boldsymbol { \\Psi } ( t , \\boldsymbol { x } ) } , \\ldots , \\lambda ^ { N , N } _ { \\boldsymbol { \\Psi } ( t , \\boldsymbol { x } ) } ) , \\end{align*}"}
+{"id": "1959.png", "formula": "\\begin{align*} s = t + \\frac { \\nu } { p _ { } } - \\frac { \\nu } { \\tau _ { } } . \\end{align*}"}
+{"id": "1871.png", "formula": "\\begin{align*} f ( 2 x + y ) + f ( 2 x - y ) = \\frac { 2 f ( x ) f ( y ) [ 4 f ( y ) + f ( x ) ] } { ( 4 f ( y ) - f ( x ) ) ^ { 2 } } \\end{align*}"}
+{"id": "1192.png", "formula": "\\begin{align*} ( b ^ { [ j ] } \\wedge a ) ( \\alpha ) ( \\xi ) = \\begin{cases} b ( \\alpha ) ( \\xi ) & \\alpha < \\beta , \\\\ x _ j ( \\beta ) ( \\xi ) & \\alpha = \\beta \\xi \\le \\zeta , \\\\ b ( \\beta ) ( \\xi ) & \\alpha = \\beta \\xi > \\zeta , \\\\ a ( \\alpha ) ( \\xi ) & \\end{cases} \\end{align*}"}
+{"id": "4578.png", "formula": "\\begin{align*} Y ( n ) = R ( n ) \\begin{pmatrix} \\sin ( \\pi \\theta ( n ) - \\pi k ) \\\\ \\cos ( \\pi \\theta ( n ) - \\pi k ) \\end{pmatrix} . \\end{align*}"}
+{"id": "2491.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\triangle u & = | x | ^ a v ^ p & & \\quad \\mbox { i n } \\ B _ { R } , \\\\ \\triangle v & = | x | ^ { b } v ^ q | \\nabla u | ^ { s } & & \\quad \\mbox { i n } \\ B _ { R } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2120.png", "formula": "\\begin{align*} \\varphi \\colon ( K ^ \\times ) ^ n \\to ( K ^ \\times ) ^ { t - 1 } , \\ , \\varphi ( x ) = ( x ^ { a _ 2 } , \\ldots , x ^ { a _ t } ) . \\end{align*}"}
+{"id": "2492.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { v ' ( r ) ^ { 2 } } { 2 } & \\leq \\int _ { 0 } ^ { r } t ^ { b } v ^ { q } ( t ) v ' ( t ) f ( w ( t ) ) d t \\\\ & \\leq r ^ { b } f ( w ( r ) ) \\int _ { 0 } ^ { r } \\left ( \\frac { v ^ { q + 1 } ( t ) } { q + 1 } \\right ) ' d t \\\\ & \\leq C f ( w ( r ) ) v ^ { q + 1 } ( r ) \\quad \\mbox { f o r a l l } \\ 0 < r < R , \\end{aligned} \\end{align*}"}
+{"id": "5738.png", "formula": "\\begin{align*} M ^ s = \\begin{pmatrix} M ^ s _ { 1 1 } & \\cdots & M ^ s _ { 1 m } \\\\ \\vdots & \\ddots & \\vdots \\\\ M ^ s _ { m 1 } & \\cdots & M ^ s _ { m m } \\\\ \\end{pmatrix} , \\end{align*}"}
+{"id": "127.png", "formula": "\\begin{align*} M _ k & > \\| g _ { v - f } \\| _ { L ^ p ( V ' _ k ) } , \\\\ V _ k & = \\{ x \\in W : f ( x ) < v ( x ) + k \\} , \\\\ W ' _ k & = \\{ x \\in V ' _ k : f ( x ) < v ( x ) + k + M _ k \\} . \\end{align*}"}
+{"id": "4928.png", "formula": "\\begin{align*} \\binom { n } { k - 1 } = \\sum _ { i = 2 } ^ { k - 1 } { \\left ( \\frac { j } { 2 } \\right ) } ^ { i - 1 } \\binom { n } { k - i } + { \\left ( \\frac { j } { 2 } \\right ) } ^ { k - 2 } \\binom { n } { 1 } + \\sum _ { i = 2 } ^ { k - 1 } { \\left ( \\frac { j } { 2 } \\right ) } ^ { i - 2 } \\frac { 1 + { \\left ( j + 1 \\right ) } { \\left ( i - 1 \\right ) } } { k - { \\left ( i - 1 \\right ) } } \\binom { n } { k - i } . \\end{align*}"}
+{"id": "784.png", "formula": "\\begin{align*} u ( c , C ) = \\frac { c ^ { 1 - \\alpha } } { 1 - \\alpha } C ^ { \\gamma \\alpha } \\end{align*}"}
+{"id": "7412.png", "formula": "\\begin{align*} f ( q ) = \\sum _ { k = 1 } ^ \\infty b _ k q ^ k \\in S _ 3 ^ { \\textrm { n e w } } ( \\Gamma ( N ) ) \\end{align*}"}
+{"id": "199.png", "formula": "\\begin{align*} 2 ^ { 2 r + s + m - 1 } ( 1 - d ) = 2 ^ { 2 r + 2 k + s - 1 } \\ , a _ f ( 1 - d ) . \\end{align*}"}
+{"id": "2836.png", "formula": "\\begin{align*} \\omega ( \\psi _ { w } ) = a ( \\gamma ( w ) ) ^ { - 1 } d I ( \\psi _ { w } ) [ w ] \\end{align*}"}
+{"id": "3561.png", "formula": "\\begin{align*} e _ 1 ( I , M ) = \\binom { e _ 0 ( I , M ) - b } { 2 } , \\end{align*}"}
+{"id": "5497.png", "formula": "\\begin{align*} F _ N ( a , b ; t ) = 1 + \\frac { t ( 1 - a q ) ( 1 - q ^ { N + 1 } ) } { ( 1 - b q ) ( 1 - t q ^ N ) } F _ N ( a q , b q ; t ) . \\end{align*}"}
+{"id": "295.png", "formula": "\\begin{align*} R ^ { ( D ' \\subset D ) } = \\sum _ { i ' \\in I ' } n _ { i ' } D _ { i ' } ^ { ( D ' \\subset D ) } + \\sum _ { i ' \\in I - I ' } ( n _ { i ' } - 1 ) D _ { i ' } ^ { ( D ' \\subset D ) } . \\end{align*}"}
+{"id": "3302.png", "formula": "\\begin{align*} \\big \\langle \\phi _ { m _ 1 } , \\phi _ { m _ 1 ' } \\big \\rangle _ { V _ 1 } = \\delta _ { m _ 1 m _ 1 ' } \\big \\langle \\psi _ { n _ 2 } , \\psi _ { n _ 2 ' } \\big \\rangle _ { V _ 2 } = \\delta _ { n _ 2 n _ 2 ' } . \\end{align*}"}
+{"id": "3337.png", "formula": "\\begin{gather*} \\psi _ { n + 1 } = \\big ( \\beta ^ + L + \\gamma ^ + [ K , L ] _ q + \\delta ^ + B + \\varepsilon ^ + D _ 1 \\big ) \\psi _ n , \\\\ \\psi _ { n - 1 } = \\big ( \\beta ^ - L + \\gamma ^ - [ K , L ] _ q + \\delta ^ - B + \\varepsilon ^ - D _ 1 \\big ) \\psi _ n . \\end{gather*}"}
+{"id": "5907.png", "formula": "\\begin{align*} q _ 1 = x _ { 2 3 } x _ { 4 5 } - x _ { 2 4 } x _ { 3 5 } + x _ { 2 5 } x _ { 3 4 } \\ , , \\ldots , q _ 5 = x _ { 1 2 } x _ { 3 4 } - x _ { 1 3 } x _ { 2 4 } + x _ { 1 4 } x _ { 2 3 } \\ , . \\end{align*}"}
+{"id": "7111.png", "formula": "\\begin{align*} \\chi = \\sum _ { \\beta \\in \\mathcal { W } } c _ { \\beta } \\beta + c \\tau _ d . \\end{align*}"}
+{"id": "7521.png", "formula": "\\begin{align*} E [ \\nu ] = \\iint G ( p , q ) d \\nu ( p ) d \\nu ( q ) \\geq 0 , \\end{align*}"}
+{"id": "5598.png", "formula": "\\begin{align*} g _ { i j } J ^ i _ p J ^ j _ q = g _ { p q } . \\end{align*}"}
+{"id": "3010.png", "formula": "\\begin{align*} \\phi _ { b c } = \\phi _ { b c , S } + \\phi _ { b c , T } + \\phi _ { b c , R } , \\end{align*}"}
+{"id": "934.png", "formula": "\\begin{align*} \\begin{aligned} u ( x ) \\geq u ( x _ 0 ) - C b _ \\alpha | x - x _ 0 | = \\ , & u ( x _ 0 ) - C b _ \\alpha ( x _ n - \\rho ( 0 , \\tilde { x } ) ) \\\\ \\geq \\ , & u ( x _ 0 ) - C \\epsilon _ 0 \\delta ^ 2 b _ \\alpha ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "8106.png", "formula": "\\begin{align*} \\mathcal { H } = \\mathcal { H } _ { u u } \\oplus \\mathcal { H } _ { u \\neg u } \\oplus \\mathcal { H } _ { \\neg u u } \\oplus \\mathcal { H } _ { \\neg u \\neg u } , \\end{align*}"}
+{"id": "7837.png", "formula": "\\begin{align*} T Q = D \\oplus W \\mbox { a n d } W \\subset V . \\end{align*}"}
+{"id": "4852.png", "formula": "\\begin{align*} \\int _ S \\rho _ 1 \\nabla f \\ , \\dd \\mu = \\bar \\rho _ 1 \\nabla F + \\int _ S \\nabla \\cdot ( \\rho _ 0 ( \\nabla f - \\nabla F ) ) \\nabla f \\ , \\dd \\mu . \\end{align*}"}
+{"id": "351.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\lim _ { m \\rightarrow \\infty } \\int _ { 0 } ^ { T } \\int _ { t } ^ { T } ( f ( u ^ n ) - f ( u ^ m ) , u ^ n _ t - u ^ m _ t ) d \\tau d t = 0 . \\end{align*}"}
+{"id": "5661.png", "formula": "\\begin{align*} r _ { 1 1 } : = T _ o ^ \\alpha T _ o ^ \\beta \\frac { \\partial ^ 2 r } { \\partial z ^ \\alpha \\partial z ^ \\beta } - 2 \\mathbb { G } ^ \\alpha ( T _ o ) \\frac { \\partial r } { \\partial z ^ \\alpha } \\end{align*}"}
+{"id": "570.png", "formula": "\\begin{align*} C \\le M _ n ^ { \\psi } ( Q ^ n ) / n = T _ { W _ { n J } , \\psi } ( \\mu _ n ( Q ^ n ) ) - I ( \\mu _ n ( Q ^ n ) ) \\le - I ( \\mu _ n ( Q ^ n ) ) . \\end{align*}"}
+{"id": "1356.png", "formula": "\\begin{align*} \\{ u = 0 \\} \\cap B _ { s / 1 0 0 } \\ne \\varnothing \\end{align*}"}
+{"id": "3035.png", "formula": "\\begin{align*} \\begin{aligned} u _ 1 & = b ( x ) \\bar { u } _ 2 \\\\ u _ 2 & = \\bar { u } _ 2 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "8132.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac 1 n \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ^ { \\otimes n } \\otimes ( A , \\varphi _ 1 ) ) = \\lim _ { n \\rightarrow + \\infty } \\frac 1 n \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ^ { \\otimes n } \\otimes ( A , \\varphi _ 2 ) ) . \\end{align*}"}
+{"id": "2828.png", "formula": "\\begin{align*} & \\int _ { [ t , T ] } D _ { s - } d X _ s + \\frac 1 2 \\int _ { [ t , T ] } \\Delta X _ s \\gamma _ s d X _ s \\\\ & = \\frac 1 2 \\left ( \\gamma _ T ^ { - 1 } D _ T ^ 2 - \\gamma _ t ^ { - 1 } d ^ 2 - \\int _ t ^ T D _ s ^ 2 \\left ( d \\gamma _ s ^ { - 1 } + \\gamma _ s ^ { - 1 } d [ R ] _ s - 2 \\gamma _ s ^ { - 1 } d R _ s - 2 d [ \\gamma ^ { - 1 } , R ] _ s \\right ) \\right ) . \\end{align*}"}
+{"id": "3364.png", "formula": "\\begin{gather*} \\forall \\beta , \\ ; \\ ; \\mathrm { R e s } _ { L _ { \\alpha \\beta } / K _ { \\alpha } } ( x _ { \\alpha } ) = y _ { \\alpha \\beta } , \\\\ \\forall \\gamma , \\ ; \\ ; \\mathrm { R e s } _ { M _ { \\alpha \\gamma } / K _ { \\alpha } } ( x _ { \\alpha } ) = z _ { \\alpha \\gamma } . \\end{gather*}"}
+{"id": "3299.png", "formula": "\\begin{align*} \\psi _ { n _ 2 } : = \\psi _ { n _ 2 ( k ) } \\lambda _ { n _ 2 } : = \\lambda _ { n _ 2 ( k ) } \\end{align*}"}
+{"id": "3503.png", "formula": "\\begin{align*} \\Big \\langle \\nabla \\mathrm { H } ^ { \\textbf { i n } } ; \\mathrm { e } ^ { ( 1 ) } _ { \\mathrm { n } } \\Big \\rangle _ { \\mathbb { L } ^ { 2 } \\Big ( \\Omega \\Big ) } = 0 . \\end{align*}"}
+{"id": "7431.png", "formula": "\\begin{align*} [ H ^ n , \\ , p ^ \\dagger ] = p ^ \\dagger ( ( H + P ) ^ n - H ^ n ) H ^ n p ^ \\dagger = p ^ \\dagger ( H + P ) ^ n . \\end{align*}"}
+{"id": "8145.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } ( \\overline L ) = \\inf _ { Y \\in \\Theta _ X } \\widehat { \\mu } _ { \\max } ^ { \\mathrm { a s y } } ( \\overline L | _ Y ) = \\inf _ { Y \\in \\Theta _ X } \\frac { h _ { \\overline L } ( Y ) } { ( \\dim ( Y ) + 1 ) \\deg _ L ( Y ) } . \\end{align*}"}
+{"id": "6967.png", "formula": "\\begin{align*} \\vec { r } = \\vec { c } + \\vec { e } \\end{align*}"}
+{"id": "428.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } | \\Lambda _ { \\leq n } ( m ) | t ^ { n } = \\frac { m ! t ^ { m } } { ( 1 - t ) ^ { m } } \\prod _ { n = 1 } ^ { \\infty } \\frac { 1 } { 1 - t ^ { n } } . \\end{align*}"}
+{"id": "2733.png", "formula": "\\begin{align*} w = \\prod _ { I _ 1 } \\sigma _ i \\in W ( B C _ { d _ 1 + d _ 2 } ) \\ , , \\end{align*}"}
+{"id": "5511.png", "formula": "\\begin{align*} F _ { N } ( a , b ; t ) & = 1 + \\frac { ( 1 - a q ) ( 1 - q ^ { N + 1 } ) ( 1 - t q ^ { N + 1 } ) } { ( 1 - t q ^ N ) ( 1 - b q ^ { N + 2 } ) ( 1 - t ) } \\large \\left ( t - \\frac { ( b - a q t ) } { ( b - a q ) } + \\frac { ( b - a q t ) ( 1 - b ) } { ( b - a q ) ( 1 - b q ^ { N + 1 } ) } \\large \\right ) \\\\ & \\quad + \\frac { ( 1 - a q ) ( 1 - q ^ { N + 1 } ) ^ 2 ( b - a q t ) ( b - a t q ^ 2 ) } { ( 1 - t q ^ N ) ( b - a q ) ( 1 - b q ^ { N + 2 } ) ( 1 - b q ^ { N + 1 } ) ( 1 - t ) } F _ { N } ( a q , b ; t q ) . \\end{align*}"}
+{"id": "1557.png", "formula": "\\begin{align*} A = \\left [ \\begin{matrix} 0 & & & 1 \\\\ 1 & 0 \\\\ & \\ddots & \\ddots \\\\ & & 1 & 0 \\end{matrix} \\right ] \\end{align*}"}
+{"id": "5678.png", "formula": "\\begin{align*} \\Phi ^ * ( K ) \\geq \\frac { 1 } { 2 } \\sup \\left \\{ \\alpha \\in [ 0 , p ] : \\alpha ^ { - \\alpha } ( 1 - \\alpha ) ^ { - ( 1 - \\alpha ) } \\left [ \\frac { p } { 1 - p } \\right ] ^ { \\alpha } < e ^ { \\zeta ( p ) } \\right \\} \\end{align*}"}
+{"id": "6364.png", "formula": "\\begin{align*} \\mathbb { L } _ { \\sigma } ( \\lambda ) : = \\lambda \\mathbb { M } _ { d } - \\mathbb { M } _ { \\sigma ^ { - 1 } ( 1 ) } \\cdots \\mathbb { M } _ { \\sigma ^ { - 1 } ( m ) } = : \\lambda \\mathbb { M } _ { d } - \\mathbb { M } _ { \\sigma } . \\end{align*}"}
+{"id": "4735.png", "formula": "\\begin{align*} ( \\alpha _ 1 , \\alpha _ 2 ) ( \\beta _ 1 , \\beta _ 2 ) = ( \\beta _ 1 \\alpha _ 1 , \\alpha _ 2 \\beta _ 2 ) , \\forall \\alpha _ 1 , \\alpha _ 2 , \\beta _ 1 , \\beta _ 2 \\in { \\rm G L } ( V ) . \\end{align*}"}
+{"id": "4010.png", "formula": "\\begin{align*} \\bar { \\mathcal { M } } _ { \\beta } ( t ) \\stackrel { d } { = } \\bar { \\mathcal { M } } ( Y _ { \\beta } ( t ) ) , \\ t \\ge 0 . \\end{align*}"}
+{"id": "8174.png", "formula": "\\begin{align*} b ^ * C T ^ n C ^ * b = a ^ * T ^ { n + 1 } a \\ \\ ( n = 0 , 1 , 2 , \\ldots ) . \\end{align*}"}
+{"id": "4865.png", "formula": "\\begin{align*} \\zeta _ { C B } ( s ) = \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { n ^ s { 2 n \\choose n } } ( s \\in \\mathbb { C } ) . \\end{align*}"}
+{"id": "4995.png", "formula": "\\begin{align*} D _ { G } ( J , N ) \\leq [ G : H ] ^ N \\cdot \\# \\bigg \\{ \\vec { v } = ( v _ 1 , v _ 2 , \\ldots , v _ N ) \\in \\Z _ n ^ N : \\Re \\bigg ( \\prod _ { j = 1 } ^ N h \\big ( e _ n ( v _ j ) \\big ) \\bigg ) \\geq 1 \\bigg \\} . \\end{align*}"}
+{"id": "4303.png", "formula": "\\begin{align*} \\Delta = a d - b c \\in \\Z . \\end{align*}"}
+{"id": "8001.png", "formula": "\\begin{align*} R _ i ^ * L _ w R _ j & = R _ i ^ * R _ j L _ w \\ : \\\\ & = \\delta _ { i j } L _ w \\\\ \\end{align*}"}
+{"id": "6259.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { d \\mid n } \\phi ( d ) z ^ { n / d } = \\left ( 1 - \\frac { 1 } { p _ 1 } \\right ) z + \\frac { 1 } { p _ 1 } z ^ { p _ 1 } , \\end{align*}"}
+{"id": "600.png", "formula": "\\begin{align*} Z ^ { + + } _ { n , p } = \\frac { ( n - p + a _ { 1 2 } - a _ { 1 2 3 } + j ^ { ( 3 ) } + 1 ) ( n - p + a _ { 1 2 } - a _ { 1 2 3 } - j ^ { ( 3 ) } ) } { ( n - p + a _ { 1 2 } - a _ { 1 2 3 } + j ^ { ( 3 ) } ) ( n - p + a _ { 1 2 } - a _ { 1 2 3 } - j ^ { ( 3 ) } - 1 ) } \\ , . \\end{align*}"}
+{"id": "4847.png", "formula": "\\begin{align*} \\frac { \\partial \\nu } { \\partial t } = \\frac { 1 } K \\nabla \\cdot ( \\nu \\nabla f ( x , s ) ) + \\int _ S ( \\nu ( x , s ' ) - \\nu ( x , s ) ) \\dd \\mu ( s ' ) , \\end{align*}"}
+{"id": "2522.png", "formula": "\\begin{align*} F ^ 1 x & = ( x _ { 1 0 0 } , x _ { 1 0 1 } , x _ { 1 1 0 } , x _ { 1 1 1 } ) \\\\ F ^ 2 x & = ( x _ { 0 1 0 } , x _ { 0 1 1 } , x _ { 1 1 0 } , x _ { 1 1 1 } ) \\\\ F ^ 3 x & = ( x _ { 0 0 1 } , x _ { 0 1 1 } , x _ { 1 0 1 } , x _ { 1 1 1 } ) \\end{align*}"}
+{"id": "7039.png", "formula": "\\begin{align*} \\mathfrak u ^ p : = \\{ Z \\in \\mathcal K ^ G ( M ) : \\nabla _ { \\nu _ p } Z \\subset \\nu _ p \\} . \\end{align*}"}
+{"id": "5165.png", "formula": "\\begin{align*} \\Omega = ( 0 , 1 ) ^ 3 \\setminus \\overline { \\bigcup _ { n \\geq 1 } T _ n } . \\end{align*}"}
+{"id": "1166.png", "formula": "\\begin{align*} \\mathbb { P } ( S _ k = 0 ) ^ { 1 / p _ k } \\leq \\mathbb { P } ( S _ k = - k ) ^ { 1 / p _ k } + \\mathbb { P } ( S _ k = k ) ^ { 1 / p _ k } \\end{align*}"}
+{"id": "5101.png", "formula": "\\begin{align*} \\Gamma _ { \\phi } : = i ( \\nabla _ x \\cdot \\nabla _ x \\phi + \\nabla _ x \\phi \\cdot \\nabla _ x ) , \\end{align*}"}
+{"id": "1730.png", "formula": "\\begin{align*} \\psi ( t ) = \\begin{cases} 1 & \\mathrm { i f } \\ , | t | \\leq 2 \\\\ 0 & \\mathrm { i f } \\ , | t | > 4 . \\end{cases} \\end{align*}"}
+{"id": "5721.png", "formula": "\\begin{align*} \\Pi _ n ( \\dd \\nu ) \\bar \\kappa _ n ( \\nu , \\dd \\eta ) = \\Pi _ n ( \\dd \\nu ) \\bar \\kappa _ n ( \\nu , \\dd \\eta ) . \\end{align*}"}
+{"id": "535.png", "formula": "\\begin{align*} M _ i ( t ) = Q ^ * _ i \\circ ( ( 1 - t ) \\mathrm { I d } + t T _ i ) ^ { - 1 } , \\end{align*}"}
+{"id": "1443.png", "formula": "\\begin{align*} N ( x ) = \\begin{dcases} \\frac { | x | } { 2 } & ( n = 1 ) , \\\\ \\frac { 1 } { 2 \\pi } \\log \\frac { 1 } { | x | } & ( n = 2 ) , \\\\ \\frac { \\Gamma ( n / 2 + 1 ) } { n ( n - 2 ) \\pi ^ { n / 2 } } | x | ^ { 2 - n } & ( n \\ge 3 ) . \\end{dcases} \\end{align*}"}
+{"id": "7671.png", "formula": "\\begin{align*} G ( w ) & : = \\int _ { - r } ^ { r } \\sqrt { 1 + ( w ' ) ^ 2 } \\ , \\mathrm { d } x - \\kappa \\int _ { - r } ^ r w \\ , \\mathrm { d } x . \\end{align*}"}
+{"id": "6958.png", "formula": "\\begin{align*} \\textrm { N } _ { \\vec d , I } = \\bigoplus _ { i \\in I , \\ , \\ , j \\in [ N ] , \\ , \\ , i \\ne j } \\pi _ * \\left ( \\mathcal { K } _ i ^ { \\vee } \\right ) - \\bigoplus _ { i , j \\in I , \\ , \\ , i \\ne j } \\pi _ * \\left ( \\mathcal { K } ^ { \\vee } _ i \\otimes \\mathcal { K } _ j \\right ) . \\end{align*}"}
+{"id": "7303.png", "formula": "\\begin{align*} \\Psi ( x , v ) = \\int _ { 0 } ^ { \\infty } \\ , \\exp \\left ( - s \\sigma ( v ) \\right ) \\varphi _ { 0 } ( x - s v , v ) \\d s , \\forall ( x , v ) \\in \\O \\times V \\end{align*}"}
+{"id": "5302.png", "formula": "\\begin{align*} \\left | \\R ( x ) \\right | \\leq A x ^ { \\theta } ( \\kappa , A > 0 , ~ 0 < \\theta < 1 ~ \\textrm { c o n s t a n t s } , ~ x \\geq 1 ~ \\textrm { a r b i t r a r y } ) . \\end{align*}"}
+{"id": "2084.png", "formula": "\\begin{align*} D \\widetilde \\varphi = \\left ( \\begin{array} { c | c c c } 1 & 0 & \\cdots & 0 \\\\ \\hline a _ 2 & & & \\\\ \\vdots & & D \\widetilde { \\varphi } _ t & \\\\ a _ n & & & \\end{array} \\right ) . \\end{align*}"}
+{"id": "7172.png", "formula": "\\begin{align*} \\mathcal { V } _ { l } : = \\bigoplus _ { i = 1 } ^ { d - l } \\mathcal { O } _ { Z _ { l - 1 } } ( \\beta _ i - \\beta _ { i + l } ) \\to Z _ { l - 1 } . \\end{align*}"}
+{"id": "2885.png", "formula": "\\begin{align*} U _ { \\tau , R } : = \\{ x \\in \\R ^ 3 : | x \\cdot { \\bf { c } } ( l \\sigma ^ { - 1 } R ^ { - 1 / 2 } ) | \\le \\sigma ^ { - 2 } \\quad & \\quad | x \\cdot { \\bf { n } } ( l \\sigma ^ { - 1 } R ^ { - 1 / 2 } ) | \\le R \\\\ & \\qquad | x \\cdot { \\bf { t } } ( l \\sigma ^ { - 1 } R ^ { - 1 / 2 } ) | \\le \\sigma ^ { - 1 } R ^ { 1 / 2 } \\} . \\end{align*}"}
+{"id": "2208.png", "formula": "\\begin{align*} \\mathfrak { h } _ 3 & : \\ ; [ e _ 1 , e _ 2 ] = e _ 3 , \\\\ \\mathfrak { r } _ { 3 , 1 } & : \\ ; [ e _ 1 , e _ 2 ] = e _ 2 , \\ ; \\ ; [ e _ 1 , e _ 3 ] = e _ 3 . \\end{align*}"}
+{"id": "4383.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } - D _ x \\right ) u ( x , t ) = 0 \\end{align*}"}
+{"id": "2563.png", "formula": "\\begin{align*} \\widetilde h ( t , x , y ) = h ( \\widetilde k ( t , x , y ) ) \\quad \\widetilde b ( t , x , y ) = b ( \\widetilde k ( t , x , y ) ) . \\end{align*}"}
+{"id": "6521.png", "formula": "\\begin{align*} R \\cdot ( a , b ) = ( b , a ) \\end{align*}"}
+{"id": "6838.png", "formula": "\\begin{align*} \\abs { P ( x ) } _ { v } \\le \\sharp H \\max _ { h = ( h _ { 1 } , \\dotsc , h _ { n } ) \\in H } \\Big ( \\abs { a _ { h } } _ { v } \\prod _ { i = 1 } ^ n \\abs { x _ { i } } _ { v } ^ { h _ { i } } \\Big ) , \\end{align*}"}
+{"id": "2298.png", "formula": "\\begin{align*} Q _ { d , \\nu } ^ + \\circ \\gamma = Q _ { d , \\nu } ^ + . \\end{align*}"}
+{"id": "7240.png", "formula": "\\begin{align*} \\int _ { J ^ { \\alpha } } \\big ( N ( t ) ^ 3 + \\epsilon _ 3 \\| \\vec { u } ( t ) \\| ^ 4 _ { L _ x ^ 4 h ^ { 1 } ( \\mathbb { R } ^ 2 \\times \\mathbb { Z } ) } \\big ) d t = 2 \\epsilon _ 3 . \\end{align*}"}
+{"id": "6577.png", "formula": "\\begin{align*} U ( C _ n ) = I _ 2 \\otimes U ( P _ { \\frac { n } { 2 } + 1 } ) \\end{align*}"}
+{"id": "2582.png", "formula": "\\begin{align*} \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } ^ * _ t } = \\rho _ i ^ N ( \\Delta _ { * , t } ^ { N , 1 } , \\ldots , \\Delta _ { * , t } ^ { N , N } ) , \\end{align*}"}
+{"id": "6336.png", "formula": "\\begin{align*} A u = b u ^ + - a u ^ - + f ( u ) , u \\in D , \\end{align*}"}
+{"id": "3254.png", "formula": "\\begin{gather*} \\varphi _ i \\ , \\xi _ j = \\xi _ k , \\eta _ i \\circ \\varphi _ j = \\eta _ k , \\varphi _ i \\circ \\varphi _ j = \\varphi _ k + \\xi _ i \\otimes \\eta _ j \\end{gather*}"}
+{"id": "1838.png", "formula": "\\begin{align*} R _ p ( \\varphi , s ) & = \\sum _ { k = 0 } ^ \\infty \\frac { \\varphi ( p ^ k ) } { p ^ { k s } } = 1 + \\sum _ { k = 1 } ^ \\infty \\frac { p ^ k - p ^ { k - 1 } } { p ^ { k s } } \\\\ & = 1 + \\frac { p - 1 } { p ^ s } \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { p ^ { k ( s - 1 ) } } = 1 + \\frac { p - 1 } { p ^ s - p } \\\\ & = \\frac { p ^ s - 1 } { p ^ s - p } . \\end{align*}"}
+{"id": "5986.png", "formula": "\\begin{align*} L _ r ^ b ( t ) : = \\sum _ { T _ b ^ + ( i ) \\leq t } \\left ( U _ r ^ { b } ( T _ b ^ + ( i ) - ) - b \\right ) , t \\geq 0 , \\end{align*}"}
+{"id": "2254.png", "formula": "\\begin{align*} u ^ { - 1 } x ^ m d = u ^ { - 1 } \\sigma ^ m ( d ) x ^ m = u ^ { - 1 } u d u ^ { - 1 } x ^ m = d u ^ { - 1 } x ^ m \\end{align*}"}
+{"id": "5309.png", "formula": "\\begin{align*} \\nu : = N ( b - r ; \\tau - r , \\tau + r ) \\ll r \\log \\tau , \\end{align*}"}
+{"id": "4270.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\alpha ) _ n ( \\beta ) _ n } { ( \\gamma ) _ n ( q ) _ n } z ^ n = \\frac { ( \\alpha z ) _ { \\infty } } { ( z ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\alpha ) _ n ( \\gamma / \\beta ) _ n ( - \\beta z ) ^ n q ^ { n ( n - 1 ) / 2 } } { ( \\gamma ) _ n ( \\alpha z ) _ n ( q ) _ n } \\end{align*}"}
+{"id": "6470.png", "formula": "\\begin{align*} & \\| \\psi \\omega ^ { ( 2 ) } _ { G } \\| ^ 2 _ { H ^ { - \\frac { 1 } { 2 } - \\epsilon } ( M \\times M ) } \\\\ & = \\sum _ { j = k } \\int _ { \\mathbb { R } ^ 2 } ( | \\xi _ 0 | ^ 2 + | \\eta _ 0 | ^ 2 + \\lambda _ j ^ 2 + \\lambda _ k ^ 2 ) ^ { - \\frac { 1 } { 2 } - \\epsilon } \\Big | { \\mathcal F _ { ( t , s ) \\to ( \\xi _ 0 , \\eta _ 0 ) } } \\Big ( \\frac { \\psi ( t , s ) } { \\lambda _ j } e ^ { i { \\lambda _ j ( t - s ) } } \\Big ) ( \\xi _ 0 , \\eta _ 0 ) \\Big | ^ 2 d \\xi _ 0 d \\eta _ 0 . \\end{align*}"}
+{"id": "5770.png", "formula": "\\begin{align*} - \\frac { 5 8 ( 5 a + 6 b ) } { 7 1 } < c \\leq - \\frac { 5 8 ( 5 a + 6 b ) - 3 3 5 } { 7 1 } . \\end{align*}"}
+{"id": "5385.png", "formula": "\\begin{align*} \\rho ( \\xi ) ( A ) = d _ A \\lambda ; \\rho ( \\xi ) ( B ) = d _ A \\tau + [ \\lambda , B ] \\end{align*}"}
+{"id": "4759.png", "formula": "\\begin{gather*} [ a + b , a ^ \\prime + b ^ \\prime ] : = ( [ a , a ^ \\prime ] _ A + \\rho _ B ( b ) a ^ \\prime - \\rho _ B ( b ^ \\prime ) a ) + ( [ b , b ^ \\prime ] _ B + \\rho _ A ( a ) b ^ \\prime - \\rho _ A ( a ^ \\prime ) b ) , \\\\ ( a + b ) \\cdot ( a ^ \\prime + b ^ \\prime ) : = ( a \\cdot _ A a ^ \\prime + \\mu _ B ( b ) a ^ \\prime + \\mu _ B ( b ^ \\prime ) a ) + ( b \\cdot _ B b ^ \\prime + \\mu _ A ( a ) b ^ \\prime + \\mu _ A ( a ^ \\prime ) b ) , \\end{gather*}"}
+{"id": "3.png", "formula": "\\begin{align*} \\sum _ { ( d , 2 ) = 1 } \\left ( \\frac { d } { p } \\right ) W \\left ( \\frac { d \\alpha ^ 2 } { X } \\right ) = \\sum _ { d } \\left ( \\frac { d } { p } \\right ) W \\left ( \\frac { d \\alpha ^ 2 } { X } \\right ) - \\left ( \\frac { 2 } { p } \\right ) \\sum _ { d } \\left ( \\frac { d } { p } \\right ) W \\left ( \\frac { 2 d \\alpha ^ 2 } { X } \\right ) . \\end{align*}"}
+{"id": "3443.png", "formula": "\\begin{align*} \\mathrm { E } ( \\mathrm { x } ) = \\begin{pmatrix} \\partial _ { \\mathrm { x _ 2 } } \\mathrm { H } ( \\mathrm { x } ) \\\\ - \\partial _ { \\mathrm { x _ 1 } } \\mathrm { H } ( \\mathrm { x } ) \\end{pmatrix} . \\end{align*}"}
+{"id": "6557.png", "formula": "\\begin{align*} E _ { \\lambda _ k } = \\frac { 1 } { n - 1 } f f ^ * . \\end{align*}"}
+{"id": "6721.png", "formula": "\\begin{align*} \\sqrt { \\mu } f ( t , x , v ) = G ( t , x , v ) - \\overline { G } ( t , x , v ) , \\end{align*}"}
+{"id": "6612.png", "formula": "\\begin{align*} Q ^ \\gamma _ \\theta \\simeq \\gamma ^ 2 Q _ \\theta , Q _ \\theta : = Q ^ 1 _ \\theta . \\end{align*}"}
+{"id": "5722.png", "formula": "\\begin{align*} \\partial _ t G _ t [ \\nu ] = V [ G _ t [ \\nu ] ] . \\end{align*}"}
+{"id": "2990.png", "formula": "\\begin{align*} S & : = \\{ ( x , y ) \\in \\Omega ^ 2 : W ( x , y ) > 0 \\} , J : = \\{ x \\in \\Omega : I ( x ) > 0 \\} , \\\\ P & : = \\{ ( x , y , z ) \\in \\Omega ^ 3 : x \\in J , ( y , z ) \\in P _ x \\} , \\\\ Q & : = \\{ ( x , y , z ) \\in \\Omega ^ 3 : x \\in J , ( y , z ) \\in S _ x ^ 2 \\setminus S \\} . \\end{align*}"}
+{"id": "4394.png", "formula": "\\begin{align*} \\mathcal { F } ( T ^ t f ( x ) ) ( z ) = e ^ { - b ^ 2 t z ^ 2 } \\mathcal { F } ( f ) ( z ) . \\end{align*}"}
+{"id": "3246.png", "formula": "\\begin{align*} v _ { k } = \\max \\{ u _ { 1 } , \\dots , u _ { k } \\} . \\end{align*}"}
+{"id": "6696.png", "formula": "\\begin{align*} & Q _ { i } '' ( 0 ) = \\det \\begin{bmatrix} s _ { i , - 2 } & s _ { i , - 1 } & 0 \\\\ s _ { i , - 1 } & 1 & 0 \\\\ 1 & \\lambda _ { i , 2 } ^ { 2 } & 1 \\end{bmatrix} = s _ { i , - 2 } - s _ { i , - 1 } ^ { 2 } , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] . \\end{align*}"}
+{"id": "761.png", "formula": "\\begin{align*} \\phi _ { \\lambda } ( \\partial a ) = ( \\partial + \\lambda ) \\phi _ { \\lambda } ( a ) , \\forall \\ a \\in \\mathcal { A } . \\end{align*}"}
+{"id": "4642.png", "formula": "\\begin{align*} \\gamma _ c = \\frac { 4 } { 3 } - \\frac { \\sqrt { 3 } } { \\pi } \\approx 0 . 7 8 2 0 0 4 . \\end{align*}"}
+{"id": "7183.png", "formula": "\\begin{align*} \\bigoplus _ { d \\geq 0 } K _ T ( B G L ( d ) ) = \\bigoplus _ { d \\geq 0 } \\mathbb { K } \\left [ z _ 1 ^ { \\pm 1 } , \\ldots , z _ d ^ { \\pm 1 } \\right ] ^ { \\mathfrak { S } _ d } \\stackrel { \\cong } { \\to } \\bigoplus _ { d \\geq 0 } K _ T ( \\mathcal { Y } ( d ) ) . \\end{align*}"}
+{"id": "6556.png", "formula": "\\begin{align*} f _ k = \\begin{pmatrix} 1 & \\lambda _ k ^ { - 1 } & \\lambda _ k & \\lambda _ k ^ { - 2 } & \\lambda _ k ^ { 2 } & \\cdots & \\lambda _ k ^ { - ( n - 2 ) / 2 } & \\lambda _ k ^ { ( n - 2 ) / 2 } \\end{pmatrix} ^ T , \\end{align*}"}
+{"id": "7829.png", "formula": "\\begin{align*} \\begin{array} { l l } a ^ \\dagger _ i a ^ \\dagger _ j = a _ j a _ i = 0 & \\ , \\ , i \\geq j \\ , , \\\\ \\end{array} \\end{align*}"}
+{"id": "3414.png", "formula": "\\begin{align*} \\begin{aligned} U ( t , r ) & \\to U ( t , r ) + \\epsilon P ^ U | _ { ( U ( t , r ) , \\rho ( t , r ) , S ( t , r ) ) } + O ( \\epsilon ^ 2 ) , \\\\ \\rho ( t , r ) & \\to \\rho ( t , r ) + \\epsilon P ^ \\rho | _ { ( U ( t , r ) , \\rho ( t , r ) , S ( t , r ) ) } + O ( \\epsilon ^ 2 ) , \\\\ S ( t , r ) & \\to S ( t , r ) + \\epsilon P ^ S | _ { ( U ( t , r ) , \\rho ( t , r ) , S ( t , r ) ) } + O ( \\epsilon ^ 2 ) , \\end{aligned} \\end{align*}"}
+{"id": "1479.png", "formula": "\\begin{align*} \\min & \\sum _ { i = \\eta _ 1 , \\eta _ 2 } \\lambda _ i \\frac { N } { \\lfloor \\frac { N } { 2 } \\rfloor - 1 + i } \\\\ & \\sum _ { i = \\eta _ 1 , \\eta _ 2 } \\lambda _ i \\frac { i } { \\lfloor \\frac { N } { 2 } \\rfloor - 1 + i } \\leq \\tilde { D } \\\\ & \\lambda _ { \\eta _ 1 } + \\lambda _ { \\eta _ 2 } = 1 \\\\ & \\lambda _ { \\eta _ 1 } , \\lambda _ { \\eta _ 2 } \\geq 0 . \\end{align*}"}
+{"id": "4607.png", "formula": "\\begin{align*} \\ln R _ j ( n ) ^ 2 & \\leq \\ln R _ j ( n _ 0 ) ^ 2 - \\sum _ { l = n _ 0 } ^ { n - 1 } \\frac { V ( l ) } { \\sin \\pi k _ j } \\sin 2 \\pi \\theta _ j ( l ) + \\sum _ { l = n _ 0 } ^ { n - 1 } \\frac { O ( 1 ) } { ( l - b ) ^ 2 } \\\\ & = \\ln R _ j ( n _ 0 ) ^ 2 + \\frac { O ( 1 ) } { n _ 0 - b } - \\frac { K _ 1 } { \\sin \\pi k _ j } \\sum _ { l = n _ 0 } ^ { n - 1 } \\frac { \\sin 2 \\pi \\theta ( l ) + \\sin 2 \\pi \\tilde { \\theta } ( l ) + 1 0 0 } { l - b } \\sin 2 \\pi \\theta _ j ( l ) . \\\\ & = \\ln R _ j ( n _ 0 ) ^ 2 + \\frac { O ( 1 ) } { n _ 0 - b } , \\end{align*}"}
+{"id": "1449.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\frac { M ( b , c ; s ) } { s ^ { b - c } e ^ s } = \\frac { \\Gamma ( c ) } { \\Gamma ( b ) } , \\end{align*}"}
+{"id": "4460.png", "formula": "\\begin{align*} & x = \\frac { C _ 1 ^ 2 - C _ 1 + C _ 2 } { 2 C _ 1 } , y = \\frac { C _ 1 ^ 2 + C _ 1 - C _ 2 } { 2 C _ 1 } , \\\\ & \\sum _ { k = 0 } ^ { \\infty } C _ k z ^ k = \\frac { 1 - \\frac { ( 1 - ( x + 1 ) z ) ( y z + 1 ) } { ( 1 - x z ) ( 1 - ( 1 - y ) z ) } } { z } = \\frac { C _ 1 z } { \\left ( 1 - \\frac { \\left ( - C _ 1 ^ 2 + C _ 1 + C _ 2 \\right ) z } { 2 C _ 1 } \\right ) \\left ( 1 - \\frac { \\left ( C _ 1 ^ 2 - C _ 1 + C _ 2 \\right ) z } { 2 C _ 1 } \\right ) } \\end{align*}"}
+{"id": "1991.png", "formula": "\\begin{align*} x _ 0 = \\frac { 2 n - 3 - \\sqrt { n ^ 2 - n + 1 } } { 3 } , ~ x _ 1 = \\frac { 2 n - 3 + \\sqrt { n ^ 2 - n + 1 } } { 3 } . \\end{align*}"}
+{"id": "3197.png", "formula": "\\begin{align*} I ( u _ { \\theta } ) & = A ( u _ { \\theta } ) + B ( u _ { \\theta } ) + C ( u _ { \\theta } ) \\\\ & \\le \\theta ^ { 2 - 2 \\beta } A ( u ) + \\theta ^ { 4 - \\beta } H ( u ) + \\theta ^ { ( 1 - \\frac { 3 } { 2 } \\beta ) p + 3 \\beta } C ( u ) \\\\ & = \\theta ^ { 2 } \\left ( I ( u ) + ( \\theta ^ { - 2 \\beta } - 1 ) A ( u ) + ( \\theta ^ { ( 1 - \\frac { 3 } { 2 } \\beta ) p + 3 \\beta - 2 } - 1 ) C ( u ) + \\theta ^ { 2 - \\beta } H ( u ) - B ( u ) \\right ) \\\\ & = \\theta ^ { 2 } ( I ( u ) + f ( \\theta , u ) ) , \\end{align*}"}
+{"id": "4114.png", "formula": "\\begin{align*} \\mathcal { H } B ^ \\alpha _ c = \\left \\{ f \\in L ^ 2 ( 0 , \\infty ) , \\ , \\ , \\mbox { S u p p o r t } \\mathcal { H } ^ { \\alpha } f \\subseteq [ 0 , c ] \\right \\} \\end{align*}"}
+{"id": "7658.png", "formula": "\\begin{align*} F ( \\kappa _ 1 ) & = P ( E _ { \\kappa _ 1 } ) - \\kappa _ 1 | E _ { \\kappa _ 1 } | \\le P ( E _ { \\kappa _ 2 } ) - \\kappa _ 1 | E _ { \\kappa _ 2 } | \\\\ & < P ( E _ { \\kappa _ 2 } ) - \\kappa _ 2 | E _ { \\kappa _ 2 } | = F ( \\kappa _ 2 ) . \\end{align*}"}
+{"id": "1606.png", "formula": "\\begin{align*} P _ { \\{ e , e ' \\} } ( \\omega _ { e } , \\omega _ { e ' } ) = P _ e ( \\omega _ { e } ) P _ { e ' } ( \\omega _ { e ' } ) > 0 . \\end{align*}"}
+{"id": "5801.png", "formula": "\\begin{align*} \\| | A | \\| ^ 2 = \\sup _ { \\| x \\| = 1 } \\| | A | x \\| ^ 2 = \\sup _ { \\| x \\| = 1 } | \\langle A ^ * A x , x \\rangle | = \\| A ^ * A \\| = \\| A \\| ^ 2 . \\end{align*}"}
+{"id": "4211.png", "formula": "\\begin{align*} \\sum _ { X = 1 } ^ \\infty X ^ { r - 1 } u ^ { X } = \\frac { 1 } { ( 1 - u ) ^ r } \\sum _ { k = 0 } ^ { r - 1 } A ( r - 1 , k ) u ^ { k + 1 } , \\end{align*}"}
+{"id": "2337.png", "formula": "\\begin{align*} I _ { \\mathrm { b l } } ^ { \\mathfrak { m } } ( 3 , 4 , 5 ) = I ^ { \\mathfrak { m } } ( \\overbrace { 0 , 1 , 0 } ^ { 3 } , \\overbrace { 0 , 1 , 0 , 1 } ^ { 4 } , \\overbrace { 1 , 0 , 1 , 0 , 1 } ^ { 5 } ) . \\end{align*}"}
+{"id": "3052.png", "formula": "\\begin{align*} \\mu ( k ) = \\begin{cases} 1 & \\\\ ( - 1 ) ^ r & \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "3633.png", "formula": "\\begin{align*} f ( b ) = f ( \\beta ) + D + d ( b - \\beta ) . \\end{align*}"}
+{"id": "7974.png", "formula": "\\begin{align*} & \\frac { 1 } { k ^ 2 ( \\xi ) + \\alpha ^ 2 ( \\xi ) } = \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle ^ 2 + \\left \\langle \\eta , \\xi ^ H \\right \\rangle ^ 2 \\\\ & = \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle ^ 2 + \\left \\langle \\eta , \\xi ^ H \\right \\rangle ^ 2 + \\sum _ { j = 1 } ^ { n - 1 } \\left \\langle V _ j , \\xi ^ H \\right \\rangle ^ 2 + \\left \\langle W _ j , \\xi ^ H \\right \\rangle ^ 2 = | \\xi ^ H | ^ 2 , \\end{align*}"}
+{"id": "4824.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ( t , x ( t ) , s ) = & e ^ { \\int _ 0 ^ t ( \\Delta f ( x ( u ) , s ) - K ) \\ , \\dd u } \\rho _ 0 ( x , s ) + \\\\ & + K \\int _ 0 ^ t e ^ { \\int _ r ^ t ( \\Delta f ( x ( u ) , s ) - K ) \\ , \\dd u } \\int _ S \\rho ( r , x ( r ) , s ' ) \\ , \\dd \\mu ( s ' ) \\dd r . \\end{aligned} \\end{align*}"}
+{"id": "4958.png", "formula": "\\begin{align*} \\mathbf { y } = \\sum _ { u = 1 } ^ { N } \\mathbf { x } _ { u } ^ { c } \\cdot a _ { u } + \\mathbf { n } . \\end{align*}"}
+{"id": "7541.png", "formula": "\\begin{align*} X : w ^ 2 = z ( z - x _ 1 ) ( z - x _ 2 ) \\end{align*}"}
+{"id": "3643.png", "formula": "\\begin{align*} \\varphi ^ * ( x ) : = \\varphi ( x ^ * ) ^ * ( x \\in X ) . \\end{align*}"}
+{"id": "3195.png", "formula": "\\begin{align*} \\left \\| u _ { n } - u _ { m } \\right \\| _ { p } \\le \\left \\| u _ { n } - u _ { m } \\right \\| _ { 2 } ^ { \\frac { 6 - p } { 2 p } } \\left \\| u _ { n } - u _ { m } \\right \\| _ { 2 } ^ { \\frac { 3 } { 2 } - 3 p } = o ( 1 ) . \\end{align*}"}
+{"id": "1422.png", "formula": "\\begin{align*} \\mathbf { \\Phi } ( \\mathcal { U } ) ( t ) & = U ( t ) \\begin{pmatrix} u _ 0 \\\\ u _ 1 \\end{pmatrix} + \\int _ 0 ^ t U ( t - s ) \\begin{pmatrix} 0 \\\\ - | u ( s ) | ^ { p - 1 } u ( s ) \\end{pmatrix} \\ , d s . \\end{align*}"}
+{"id": "618.png", "formula": "\\begin{align*} \\lambda ( m ) \\ , r _ n ( m ) = A _ { n } \\ r _ { n + 1 } ( m ) - ( A _ n + C _ n ) \\ , r _ n ( m ) + C _ n \\ r _ { n - 1 } ( m ) \\ , , \\end{align*}"}
+{"id": "7687.png", "formula": "\\begin{align*} u _ b ( x ) = \\exp \\left \\{ \\frac { b ( 2 - n ) r ^ 2 } { ( n - 1 ) n } F \\left [ \\begin{array} { c } 1 , 1 , 2 - \\frac { n } { 2 } \\\\ 2 , 1 + \\frac { n } { 2 } \\end{array} ; r ^ 2 \\right ] \\right \\} \\left ( 1 - r ^ 2 \\right ) ^ { \\frac { b } { ( n - 1 ) } } . \\end{align*}"}
+{"id": "3470.png", "formula": "\\begin{align*} \\Big \\Vert \\nabla _ { \\mathrm { x } } \\mathbb { I } _ { ( \\Omega ) } \\Big [ \\psi \\Big ] \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\Omega \\times \\mathbb { R _ + } ) } ^ 2 = \\alpha \\delta ^ 2 \\Big \\Vert \\nabla _ { \\xi } \\mathbb { I } _ { ( B ) } \\Big [ \\hat { \\psi } \\Big ] \\Big \\Vert _ { \\mathrm { L } ^ 2 ( B \\times \\mathbb { R _ + } ) } ^ 2 , \\end{align*}"}
+{"id": "1041.png", "formula": "\\begin{align*} \\mathcal { W } \\ , ( \\Delta ^ { - m } ) = v _ { n - 1 } \\ , v o l ( M ) , \\end{align*}"}
+{"id": "899.png", "formula": "\\begin{align*} \\Psi = \\rho ( x ' ) - x _ n - \\theta | x ' | ^ 2 + K x _ n ^ 2 . \\end{align*}"}
+{"id": "49.png", "formula": "\\begin{align*} \\div _ { \\Psi } X = e ^ { \\Psi } \\div ( e ^ { - \\Psi } X ) \\end{align*}"}
+{"id": "6926.png", "formula": "\\begin{align*} z _ i R ( z _ i ) & = q ( z _ i + y ) \\prod _ { p = 1 } ^ { r } ( 1 + z _ i x _ p ) , \\\\ \\prod _ { j = 1 } ^ { N } ( z _ j - \\alpha _ i ) & = ( - 1 ) ^ { N + 1 } \\frac { P ( \\alpha _ i ) } { z _ { N + 1 } - \\alpha _ i } = ( - 1 ) ^ N q \\frac { ( \\alpha _ i + y ) \\prod _ { p = 1 } ^ { r } ( 1 + \\alpha _ i x _ p ) } { z _ { N + 1 } - \\alpha _ i } . \\end{align*}"}
+{"id": "413.png", "formula": "\\begin{align*} S _ H ( D , M , Q , S ) : = \\{ K \\pmod H : ( K , H ) = 1 , K \\equiv Q \\pmod D , K \\equiv S \\pmod M \\} \\end{align*}"}
+{"id": "4208.png", "formula": "\\begin{align*} H ( u ) = \\sum _ { X = 1 } ^ \\infty A ( X ) u ^ { X - 1 } \\end{align*}"}
+{"id": "5187.png", "formula": "\\begin{align*} r _ i ( x ) = q _ { i + 1 } ( x ) r _ { i + 1 } ( x ) + r _ { i + 2 } ( x ) \\enspace , \\end{align*}"}
+{"id": "7993.png", "formula": "\\begin{align*} w ( e ^ { i t } ) = \\sum _ { j = - m } ^ m c _ j e ^ { i j t } \\end{align*}"}
+{"id": "2288.png", "formula": "\\begin{align*} \\kappa _ d = \\begin{cases} \\displaystyle { \\frac { 1 2 } { 1 9 } } & d = 3 , \\\\ \\displaystyle { \\frac 3 { ( d - 2 ) \\sqrt d + 3 } } & 4 \\leq d \\leq 8 , \\\\ \\displaystyle { \\frac 1 { d - 1 } } & d \\geq 9 . \\end{cases} \\end{align*}"}
+{"id": "6619.png", "formula": "\\begin{align*} T _ \\Gamma \\ge \\bigoplus _ { j = 1 } ^ M R ^ 2 _ j J \\end{align*}"}
+{"id": "5468.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( a q ) _ n ( a t q / b ) _ n } { ( b q ) _ n ( t ) _ { n + 1 } } ( 1 - a t q ^ { 2 n + 1 } ) ( b t ) ^ n q ^ { n ^ 2 } . \\end{align*}"}
+{"id": "4854.png", "formula": "\\begin{align*} \\partial _ t \\bar \\rho & = \\partial _ t \\left ( \\bar \\rho _ 0 + \\frac 1 K \\bar \\rho _ 1 \\right ) + O \\left ( \\frac 1 { K ^ 2 } \\right ) = \\\\ & = \\nabla \\cdot \\left ( \\left ( \\bar \\rho _ 0 + \\frac 1 K \\bar \\rho _ 1 \\right ) \\nabla F \\right ) + \\frac 1 K \\nabla \\cdot ( \\nabla \\cdot ( \\bar \\rho _ 0 \\Sigma ) ) - \\frac 1 { 2 K } \\nabla \\cdot ( \\bar \\rho _ 0 \\nabla V ) + O \\left ( \\frac 1 { K ^ 2 } \\right ) \\end{align*}"}
+{"id": "3062.png", "formula": "\\begin{align*} g ( \\ell ) = \\frac { 1 } { \\ell } \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } \\mu \\left ( \\frac { \\ell } { k } \\right ) f ( k ) \\end{align*}"}
+{"id": "5974.png", "formula": "\\begin{align*} X ( t ) = c t - S ( t ) , \\ , \\ , t \\geq 0 , \\end{align*}"}
+{"id": "1951.png", "formula": "\\begin{align*} \\| f \\| _ { \\vec { p } } : = \\left ( \\int _ \\mathbb { R } \\cdots \\left ( \\int _ \\mathbb { R } \\left ( \\int _ \\mathbb { R } | f ( x _ 1 , \\dots , x _ d ) | ^ { p _ 1 } d x _ 1 \\right ) ^ { \\frac { p _ 2 } { p _ 1 } } d x _ 2 \\right ) ^ { \\frac { p _ 3 } { p _ 2 } } \\cdots d x _ d \\right ) ^ { \\frac { 1 } { p _ d } } < \\infty . \\end{align*}"}
+{"id": "4097.png", "formula": "\\begin{align*} D _ j = \\sum _ { ( m _ k ) \\in S _ j } \\prod _ { k = 1 } ^ n \\binom { d _ k + m _ k - 1 } { m _ k } \\end{align*}"}
+{"id": "3030.png", "formula": "\\begin{align*} \\begin{aligned} u _ 1 & = \\bar { u } _ 2 \\\\ u _ 2 & = h ( x , \\bar { u } _ 1 , \\bar { u } _ 2 ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "555.png", "formula": "\\begin{align*} H \\bigg ( \\mathrm { L a w } \\bigg ( \\sum _ { i = 1 } ^ n t _ i X _ i \\bigg ) \\bigg ) \\le \\sum _ { i = 1 } ^ n t _ i H ( Q _ i ) . \\end{align*}"}
+{"id": "8097.png", "formula": "\\begin{align*} T _ 2 T _ 1 ( z _ 1 ^ { m _ 1 } z _ 2 ^ { m _ 2 } \\eta ) = M _ { z _ 2 } ^ { \\alpha _ 2 } D [ U ] ( \\alpha _ 1 z _ 1 ^ { m _ 1 + 1 } z _ 2 ^ { m _ 2 } \\eta ) = \\alpha _ 1 \\alpha _ 2 z _ 1 ^ { m _ 1 + 1 } z _ 2 ^ { m _ 2 + 1 } U ^ { m _ 1 + 1 } \\eta , \\end{align*}"}
+{"id": "1750.png", "formula": "\\begin{align*} Y = C \\mathcal { Z } \\quad { \\rm a n d } Y ^ { ' } = C ^ { ' } \\mathcal { Z } ^ { ' } . \\end{align*}"}
+{"id": "1905.png", "formula": "\\begin{align*} \\frac 1 { r ' } = 1 - \\frac 1 r = \\frac 3 { 2 q } ; \\frac 1 3 + \\frac 1 q + \\frac 1 { q ' } = 1 , \\end{align*}"}
+{"id": "922.png", "formula": "\\begin{align*} w ( x ) : = x _ n - y _ n - L ( x ) + \\kappa | x ' - y ' | ^ 2 \\geq 0 \\mbox { o n } \\partial \\omega \\cap \\partial \\Omega . \\end{align*}"}
+{"id": "6257.png", "formula": "\\begin{align*} \\delta ( A ) \\sum _ { j _ 2 , \\dots , j _ \\ell \\geq 0 } \\prod _ { r = 2 } ^ \\ell \\delta _ { j _ r } ( A ) = \\delta ( A ) \\prod _ { r = 2 } ^ \\ell \\sum _ { j _ r = 0 } ^ m \\delta _ { j _ r } ( A ) = \\delta ( A ) \\prod _ { r = 2 } ^ \\ell 1 = \\delta ( A ) \\end{align*}"}
+{"id": "7114.png", "formula": "\\begin{align*} W = z [ x , y ] \\in \\mathbb { C } [ Q ] / [ \\mathbb { C } [ Q ] , \\mathbb { C } [ Q ] ] , \\end{align*}"}
+{"id": "2608.png", "formula": "\\begin{align*} f ( u , y ) = \\mp \\sum _ { u < v < y } f ( u , v ) f ( v , y ) . \\end{align*}"}
+{"id": "1482.png", "formula": "\\begin{align*} A _ d = g ( d ) X + r _ d { \\rm i f } \\ , \\ , d \\mid P ( z ) \\end{align*}"}
+{"id": "6553.png", "formula": "\\begin{align*} U ^ 7 = I . \\end{align*}"}
+{"id": "7702.png", "formula": "\\begin{align*} \\tfrac { x y ^ 2 z ^ 2 } { f ^ 2 } = - ( \\partial _ x x + \\partial _ y y + \\partial _ z z + 1 ) \\cdot \\tfrac { 1 } { f } . \\end{align*}"}
+{"id": "68.png", "formula": "\\begin{align*} X ( u ) ( x _ { 0 } ) = g ( X | _ { x _ { 0 } } , \\nabla u ( x _ { 0 } ) ) = 0 . \\end{align*}"}
+{"id": "7385.png", "formula": "\\begin{align*} S ( \\lambda ) \\varphi ( x ) = \\int _ { \\Sigma } \\frac { 1 } { 2 \\pi } K _ 0 \\bigl ( - i \\sqrt { \\lambda } | x - y | \\bigr ) \\varphi ( y ) d \\sigma ( y ) , \\varphi \\in L ^ 2 ( \\Sigma ) , ~ x \\in \\Sigma . \\end{align*}"}
+{"id": "639.png", "formula": "\\begin{align*} d F ( X ) = \\langle \\langle X \\cdot \\varphi , \\varphi \\rangle \\rangle , \\end{align*}"}
+{"id": "1109.png", "formula": "\\begin{align*} \\xi ^ * ( \\eta ) \\ge \\xi ^ * \\left . \\left ( \\frac { 9 9 } { 1 0 0 } \\right ) \\right | _ { a ^ 2 = \\frac { 9 } { 1 0 } } \\geq 0 . 9 4 4 4 3 7 > \\frac { 4 7 } { 5 0 } . \\end{align*}"}
+{"id": "7580.png", "formula": "\\begin{align*} \\iint G ( p , q ) d \\mu ( p ) d \\mu ( q ) \\leq - \\log \\left ( \\frac { \\eta } { 4 C } \\right ) + C _ j = - \\log \\eta + \\log 4 + \\log C + C _ j \\end{align*}"}
+{"id": "5660.png", "formula": "\\begin{align*} c t _ { \\lambda } ( t ) = \\left \\{ \\begin{array} { l c } \\sqrt { \\lambda } \\cot ( \\sqrt { \\lambda } t ) , & \\lambda > 0 , \\\\ 1 / t , & \\lambda = 0 , \\\\ \\sqrt { - \\lambda } \\coth ( \\sqrt { - \\lambda } t ) & \\lambda < 0 , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "7705.png", "formula": "\\begin{gather*} \\omega _ { i _ 1 \\dots i _ k } ( x ^ \\prime , 0 ) = 0 x ^ \\prime i _ l = n 1 \\leq l \\leq k . \\end{gather*}"}
+{"id": "8109.png", "formula": "\\begin{align*} \\textstyle \\mathcal W _ 2 ^ 2 ( \\mu , \\nu ) : = \\inf _ { \\pi \\in \\Pi ( \\mu , \\nu ) } \\int | x - y | ^ 2 \\ , \\pi ( d x , d y ) , \\end{align*}"}
+{"id": "4062.png", "formula": "\\begin{align*} t \\rightarrow s = \\sum _ { s ' \\hbox { \\tiny { v e r t e x o f } } s } t \\rightarrow _ { s ' } s , \\end{align*}"}
+{"id": "7646.png", "formula": "\\begin{align*} H ( p ) : = \\inf \\left \\{ \\ , \\frac { P ( F ) } { | F | ^ p } \\ , : \\ , F \\subset \\Omega \\ , , \\ , | F | > 0 \\ , \\right \\} , \\end{align*}"}
+{"id": "6014.png", "formula": "\\begin{align*} r \\int _ 0 ^ { b } e ^ { - \\Phi ( q + r ) y } W ^ { ( q ) } ( y ) d y = 1 - r \\int _ b ^ { \\infty } e ^ { - \\Phi ( q + r ) y } W ^ { ( q ) } ( y ) d y . \\end{align*}"}
+{"id": "3947.png", "formula": "\\begin{align*} \\int _ D \\langle \\nabla E _ { \\Omega _ n } ( u _ { \\Omega _ n } ) , \\nabla \\varphi \\rangle - f \\varphi = 0 . \\end{align*}"}
+{"id": "2033.png", "formula": "\\begin{align*} I _ 1 = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty + i c } ^ { \\infty + i c } \\frac { \\zeta ' } { \\zeta } ( s ) \\Bigl ( \\Phi _ 1 ( \\phi ; z ) \\overline { \\Phi _ 1 ( \\phi ; \\bar { z } ) } + \\Phi _ 1 ( \\phi ; - z ) \\overline { \\Phi _ 1 ( \\phi ; - \\bar { z } ) } \\Bigr ) \\ , d z , \\end{align*}"}
+{"id": "7678.png", "formula": "\\begin{align*} \\Delta _ h e ^ u & = ( 1 - | x | ^ 2 ) ^ 2 | \\nabla u | ^ 2 e ^ u \\\\ & + e ^ u ( ( 1 - | x | ^ 2 ) ^ 2 \\Delta u ( x ) + 2 ( n - 2 ) ( 1 - | x | ^ 2 ) \\sum _ { i = 1 } ^ { n } x _ { i } \\frac { \\partial u } { \\partial x _ { i } } ( x ) ) \\\\ & = ( 1 - | x | ^ 2 ) ^ 2 | \\nabla u | ^ 2 e ^ u + e ^ u \\Delta _ h u \\ge 0 . \\end{align*}"}
+{"id": "5906.png", "formula": "\\begin{align*} & \\lambda _ 1 = [ - 1 , 0 , 0 , 0 , 1 ] \\ , , \\lambda _ 2 = [ - 1 , 0 , 0 , 1 , 0 ] \\ , , \\lambda _ 3 = [ 0 , - 1 , 0 , 0 , 1 ] \\ , , \\\\ & \\lambda _ 4 = [ 0 , - 1 , 0 , 1 , 0 ] \\ , , \\lambda _ 5 = [ 0 , 0 , - 1 , 0 , 1 ] \\ , , \\lambda _ 6 = [ 0 , 0 , - 1 , 1 , 0 ] \\ , . \\end{align*}"}
+{"id": "3031.png", "formula": "\\begin{align*} \\begin{aligned} u _ 1 & = \\bar { u } _ 2 \\\\ u _ 2 & = b ( x ) \\bar { u } _ 2 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "122.png", "formula": "\\begin{align*} \\| f \\| = \\sup \\{ | f ( x ) | : x \\in B _ X \\} = \\sup \\{ | y ( f ) | : y \\in B _ { X '' } \\} \\ \\ \\ \\ ( \\forall f \\in X ' ) \\end{align*}"}
+{"id": "6993.png", "formula": "\\begin{align*} \\int \\langle \\nabla u , \\nabla \\phi \\rangle d m = \\int \\left ( \\frac { \\alpha _ 2 } { 2 } \\phi u \\log u ^ 2 + ( \\alpha _ 2 - \\alpha _ 1 - \\beta ) \\phi u \\right ) d m , \\phi \\in W ^ { 1 , 2 } . \\end{align*}"}
+{"id": "4693.png", "formula": "\\begin{align*} \\left | r \\frac { d } { d r } \\left [ { } _ 2 \\ ! F _ { 1 } \\left ( \\lambda , \\lambda ; 2 \\lambda + 1 ; A \\right ) \\right ] \\right | \\lesssim \\frac { 1 } { | 1 - z w | } \\ln \\left ( \\frac { | 1 - z w | ^ 2 } { | 1 - z \\bar { w } | ^ 2 } + 2 \\right ) \\lesssim \\frac { 1 } { | 1 - z \\overline { w } | } , z = r e ^ { i \\theta } , \\end{align*}"}
+{"id": "4808.png", "formula": "\\begin{align*} \\mathcal D _ { i _ 0 } ( \\phi _ { i _ 0 } ( n , k ) U _ { n , k } ) & = C _ { i _ 0 } \\phi _ { i _ 0 + 1 } ( n , k ) U _ { n , k } = 0 , \\\\ \\mathcal D _ { i _ 0 } ( \\psi _ { i _ 0 } ( n , k ) U _ { n , k } ) & = C _ { i _ 0 } \\psi _ { i _ 0 + 1 } ( n , k ) U _ { n , k } = 0 . \\end{align*}"}
+{"id": "6229.png", "formula": "\\begin{align*} ( \\psi _ \\varepsilon , H ( p ) \\psi _ \\varepsilon ) = \\lambda _ { n , m } \\big ( 1 + 2 \\varepsilon \\mathrm { R e } \\ , ( \\varphi , \\psi _ { n , m } ) \\big ) + \\varepsilon ^ 2 \\| H ( p ) ^ { 1 / 2 } \\psi _ { n , m } \\| ^ 2 . \\end{align*}"}
+{"id": "5506.png", "formula": "\\begin{align*} F _ { N } ( a , b ; t ) & = \\frac { 1 - t q ^ N } { 1 - t } + \\frac { ( 1 - t q ^ N ) ( b - a ) ( 1 - q ^ N ) t q } { ( 1 - t ) ( 1 - b q ) ( 1 - t q ) } \\sum _ { n = 0 } ^ { N - 1 } \\frac { ( q ^ { 1 - N } ) _ { n } q ^ { N n + n } ( b q / a ) _ { n } ( a t ) ^ { n } } { ( b q ^ 2 ) _ { n } ( t q ^ 2 ) _ { n } } \\\\ & = \\frac { 1 - t q ^ { N + 1 } } { 1 - t } + \\frac { ( 1 - t q ^ { N + 1 } ) ( b - a ) ( 1 - q ^ { N + 1 } ) t q } { ( 1 - t ) ( 1 - b q ) ( 1 - t q ) } \\sum _ { n = 0 } ^ { N } \\frac { ( q ^ { - N } ) _ { n } q ^ { ( N + 1 ) n + n } ( b q / a ) _ { n } ( a t ) ^ { n } } { ( b q ^ 2 ) _ { n } ( t q ^ 2 ) _ { n } } . \\end{align*}"}
+{"id": "6895.png", "formula": "\\begin{align*} \\varphi _ i ( x ) : = \\frac { \\varphi ( x - i ) } { \\sum _ { k \\in \\mathbb { Z } ^ d } \\varphi ( x - k ) } \\end{align*}"}
+{"id": "4169.png", "formula": "\\begin{align*} u _ 1 u _ 2 = ( 2 b _ 1 + 1 ) ( 2 b _ 2 + 1 ) & = ( 2 d + 1 + 2 \\sqrt { d ^ 2 + d } ) ( 2 d + 1 - 2 \\sqrt { d ^ 2 + d } ) \\\\ & = ( 2 d + 1 ) ^ 2 - 4 ( d ^ 2 + d ) = 1 . \\end{align*}"}
+{"id": "1410.png", "formula": "\\begin{align*} \\int _ 0 ^ t ( t _ 0 + s ) ^ { \\lambda - \\frac { p + 1 } { p - 1 } } h ( s ) \\ , d s & \\le C \\begin{dcases} 1 & \\left ( \\lambda < \\frac { 2 } { p - 1 } \\right ) , \\\\ ( \\log ( t _ 0 + t ) ) ^ 2 & \\left ( \\lambda = \\frac { 2 } { p - 1 } \\right ) , \\\\ ( t _ 0 + t ) ^ { \\lambda - \\frac { 2 } { p - 1 } } \\log ( t _ 0 + t ) & \\left ( \\lambda > \\frac { 2 } { p - 1 } \\right ) ; \\end{dcases} \\end{align*}"}
+{"id": "3487.png", "formula": "\\begin{align*} \\mathrm { U } _ { \\mathrm { e } } ( \\xi , t ) & = \\frac { \\gamma _ { \\mathrm { p } } } { \\gamma _ { \\mathrm { m } } } \\frac { 1 } { \\alpha _ \\mathrm { m } } \\Bigg [ - \\int _ { 0 } ^ t \\int _ { \\partial \\Omega } \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) \\gamma ^ { \\textbf { i n t } } _ { 1 } \\mathrm { U } _ { \\mathrm { i } } ( \\mathrm { y } , \\tau ) d \\sigma _ \\mathrm { y } d \\tau + \\textbf { e r r } ^ { ( 1 ) } + \\textbf { e r r } ^ { ( 2 ) } \\Bigg ] . \\end{align*}"}
+{"id": "603.png", "formula": "\\begin{align*} N = j ^ { ( 1 ) } + j ^ { ( 2 ) } - j ^ { ( 3 ) } + j ^ { ( 4 ) } + j ^ { ( 0 ) } \\ , . \\end{align*}"}
+{"id": "4702.png", "formula": "\\begin{align*} \\| d \\nu \\| _ { { \\frak B _ { \\lambda } } } = \\sup c _ { \\lambda } \\left | \\int _ { - \\pi } ^ { \\pi } h ( \\theta ) | \\sin \\theta | ^ { 2 \\lambda } d \\nu ( \\theta ) \\right | , \\end{align*}"}
+{"id": "5434.png", "formula": "\\begin{align*} \\log \\left ( \\frac { K \\pm ( s - L ^ 2 - 1 ) } { 2 e } \\right ) = \\log \\left ( \\frac { K } { 2 } \\right ) - 1 + \\sum _ { j = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { j + 1 } } { j } \\left ( \\pm \\frac { s - L ^ 2 - 1 } { K } \\right ) ^ { j } . \\end{align*}"}
+{"id": "1005.png", "formula": "\\begin{align*} { \\sigma } _ { j i } & = \\mu ^ * ( u _ { i , j } + u _ { j , i } ) + \\varkappa \\ , ( u _ { i , j } - \\epsilon _ { j i s } \\vartheta _ s ) + \\lambda \\ , u _ { k , k } \\delta _ { j i } , \\\\ * m _ { j i } & = \\beta \\ , \\vartheta _ { j , i } + \\gamma \\ , \\vartheta _ { i , j } + \\alpha \\ , \\vartheta _ { k , k } \\delta _ { j i } \\ , \\end{align*}"}
+{"id": "3319.png", "formula": "\\begin{align*} p _ { n _ 1 ( 0 ) } ^ { V _ 1 } ( m _ 1 ; n _ 2 ) = 1 p _ { n _ 1 ( - 1 ) } ^ { V _ 1 } ( m _ 1 ; n _ 2 ) = 0 , \\end{align*}"}
+{"id": "7035.png", "formula": "\\begin{align*} \\nu _ q ^ { ( 1 ) } = \\nu _ q + \\hat { \\nu } _ q = \\nu _ q + \\mathrm { s p a n } \\ , \\{ \\nabla _ v Z : Z \\in \\mathcal K ^ G ( M ) , v \\in \\nu _ q \\} \\end{align*}"}
+{"id": "7436.png", "formula": "\\begin{align*} \\sum _ { \\mu = 1 } ^ { n } T _ \\mu ( \\sum _ { \\eta = 1 } ^ { n } \\alpha _ { \\mu \\eta } \\sigma _ \\eta - \\sigma _ \\mu P ) = 0 , \\end{align*}"}
+{"id": "2674.png", "formula": "\\begin{align*} \\varphi ( 2 n , m , 0 ) = \\sum _ { k = 0 } ^ { 2 n } ( - 1 ) ^ k \\binom { 2 n } { k } ^ m C _ k C _ { 2 n - k } \\end{align*}"}
+{"id": "3605.png", "formula": "\\begin{align*} \\sum _ m m ^ { - ( 1 + \\delta ) } \\left ( \\frac { m + b / a } { m + ( b - j ) / a } \\right ) ^ 2 & \\leq \\sum _ m m ^ { - ( 1 + \\delta ) } \\left ( \\frac { m + b / a } { m } \\right ) ^ 2 \\\\ & = \\sum _ m m ^ { - ( 1 + \\delta ) } \\left ( 1 + \\frac { b } { a m } \\right ) ^ 2 \\\\ & \\lesssim \\sum _ m m ^ { - ( 1 + \\delta ) } \\left [ 1 + \\frac { 1 } { m ^ 2 } \\right ] \\coloneqq C _ 2 < \\infty . \\end{align*}"}
+{"id": "828.png", "formula": "\\begin{align*} \\mathbb { E } \\Big ( e ^ { - \\beta \\tau _ i ^ { * } } \\Big ) = e ^ { - \\lambda b ' } \\end{align*}"}
+{"id": "4550.png", "formula": "\\begin{align*} H _ f ( G ) = \\sum _ { v \\in V ( G ) } f ( d _ v ) . \\end{align*}"}
+{"id": "269.png", "formula": "\\begin{align*} K _ { t , x } ( z , y ) = \\sum _ { i , j } \\Psi _ j ( y ) ( A ^ { - 1 } ) _ { j , i } T ( x _ i , z ) = \\sum _ i T ( x _ i , z ) \\frac { \\det ( i A \\Psi ( y ) ) } { \\det ( A ) } . \\end{align*}"}
+{"id": "6448.png", "formula": "\\begin{align*} L _ n ^ { ( \\alpha ) } ( x ) \\equiv \\sum _ { k = 0 } ^ n \\begin{bmatrix} n \\\\ k \\end{bmatrix} \\frac { q ^ { k ^ 2 + k \\alpha } } { ( q ^ { \\alpha + 1 } ; q ) _ k } ( - x ) ^ k = L _ n ^ { ( \\alpha ) } ( x , 1 ) . \\end{align*}"}
+{"id": "6792.png", "formula": "\\begin{align*} \\tilde { \\sigma } ( x _ { [ k ] } ) = \\sum _ { i \\in [ q ] } ( \\mu _ { i , \\ell } + \\mu _ { i , \\ell ' } ) \\pi _ i ( x _ { [ k ] } ) + \\sum _ { i \\in [ t ] } ( \\nu _ { i , \\ell } + \\nu _ { i , \\ell ' } ) \\rho _ i ( x _ { [ k ] } ) . \\end{align*}"}
+{"id": "6006.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } v _ n ^ + = \\lim _ { n \\to \\infty } v _ n ^ - = \\Gamma V = V . \\end{align*}"}
+{"id": "987.png", "formula": "\\begin{align*} D _ { 2 k ; w _ 1 , w _ 2 } ^ { e ^ { 2 \\Upsilon } g } \\bigl ( u \\otimes v \\bigr ) = e ^ { ( w _ 1 + w _ 2 - 2 k ) \\Upsilon } D _ { 2 k ; w _ 1 , w _ 2 } ^ { g } \\bigl ( e ^ { - w _ 1 \\Upsilon } u \\otimes e ^ { - w _ 2 \\Upsilon } v \\bigr ) \\end{align*}"}
+{"id": "730.png", "formula": "\\begin{align*} \\delta x = - \\frac { 1 } { 2 } \\left ( { { f _ 1 } - { f _ 2 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) - \\frac { 1 } { 4 } { F _ x } , \\end{align*}"}
+{"id": "2722.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } S ( n , r ) \\binom { n } { j + u } ( - 1 ) ^ { j + u + r - 1 } \\binom { j + u + r - 1 } { j + u } \\sum _ { l = 0 } ^ { r - 1 } ( - 1 ) ^ l \\binom { 2 n - j - u + l } { l } \\cdot \\\\ \\cdot \\binom { 2 ( j + u + r - 1 - l ) } { j + u + r - 1 - l } \\binom { n - j - u } { j + u + r - l - 1 } \\bigl { ( } \\frac { 1 } { 2 } S ( n , n - j - u + l + 1 ) \\bigr { ) } \\end{align*}"}
+{"id": "1017.png", "formula": "\\begin{align*} { \\mathbf { A } } = \\mathbf { A n t i } ( \\vartheta _ 1 , \\vartheta _ 2 , \\vartheta _ 3 ) : = \\begin{footnotesize} \\begin{pmatrix} 0 & - \\vartheta _ 3 & \\vartheta _ 2 \\\\ \\vartheta _ 3 & 0 & - \\vartheta _ 1 \\\\ - \\vartheta _ 2 & \\vartheta _ 1 & 0 \\end{pmatrix} \\end{footnotesize} \\in \\mathfrak { s o } ( 3 ) , \\end{align*}"}
+{"id": "5057.png", "formula": "\\begin{align*} \\sum _ { \\substack { q \\geq 1 \\\\ N \\mid q } } \\frac { 1 } { 2 \\pi i } \\int _ { \\Re ( u ) = \\sigma _ 0 } \\frac { \\Gamma _ \\C ( u + \\frac { k - 1 } { 2 } ) } { \\Gamma _ \\C ( - u + \\frac { k + 1 } { 2 } ) } q ^ { 2 u - 1 } n ^ { - u } \\sum _ { \\substack { a \\bmod { q } \\\\ a \\bar { a } \\equiv 1 \\bmod { q } } } \\chi ( a ) e ( n \\bar { a } / q ) L _ f \\bigl ( s + u , \\tfrac { a } { q } \\bigr ) \\ , d u . \\end{align*}"}
+{"id": "6573.png", "formula": "\\begin{align*} H \\cong A \\oplus B = \\begin{pmatrix} A & \\mathbf { 0 } \\\\ \\mathbf { 0 } & B \\end{pmatrix} , \\end{align*}"}
+{"id": "2098.png", "formula": "\\begin{align*} [ i ] _ q : = \\sum _ { j = 0 } ^ { i - 1 } q ^ j , [ k ] _ q ! : = \\prod _ { i = 1 } ^ k [ i ] _ q , \\begin{bmatrix} n \\\\ k \\end{bmatrix} _ q : = \\frac { [ n ] _ q ! } { [ k ] _ q ! \\ [ n - k ] _ q ! } . \\end{align*}"}
+{"id": "6484.png", "formula": "\\begin{align*} \\nu _ x ( E ) = \\nu _ x ( - E ) . \\end{align*}"}
+{"id": "6101.png", "formula": "\\begin{align*} \\left \\langle ( \\omega ^ * , x ^ * ) , ( \\omega , x ) \\right \\rangle = \\langle \\omega ^ * , x \\rangle + \\langle \\omega , x ^ * \\rangle , \\end{align*}"}
+{"id": "2896.png", "formula": "\\begin{align*} H ( s , t ) = \\frac { \\kappa ( s ) } { n } . \\end{align*}"}
+{"id": "180.png", "formula": "\\begin{align*} \\begin{aligned} | \\zeta _ \\shortparallel | ^ 2 \\geq \\tfrac { 1 } { 2 } | \\zeta _ \\shortparallel - \\tfrac { 1 } { 2 } V _ \\shortparallel | ^ 2 - \\tfrac { 1 } { 4 } | V _ \\shortparallel | ^ 2 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "6761.png", "formula": "\\begin{align*} \\Phi _ { \\varphi } ( x ) & = \\int d k \\ , \\abs { k } ^ { - 1 } \\left ( e ^ { 2 \\pi i k x } \\varphi ( k ) + e ^ { - 2 \\pi i k x } \\overline { \\varphi ( k ) } \\right ) \\ , \\varphi \\in L ^ 2 ( \\mathbb { R } ^ 3 ) , \\\\ \\mu ( t ) & = \\frac { 1 } { 2 } \\sqrt { \\alpha } \\int d x \\ , \\Phi _ { \\varphi _ t } ( x ) \\abs { \\psi ( t , x ) } ^ 2 \\\\ h ( t ) & = - \\Delta + \\sqrt { \\alpha } \\Phi _ { \\varphi _ t } - \\mu ( t ) \\end{align*}"}
+{"id": "3628.png", "formula": "\\begin{align*} f ' ( a ) = - c \\end{align*}"}
+{"id": "4715.png", "formula": "\\begin{align*} ( 1 - | z | ^ 2 ) D _ { z } \\left ( z f ( z ) \\right ) = 2 ( \\widetilde { P } _ { \\lambda , 2 } f ) ( z ) - 2 ( 1 - | z | ^ 2 ) f ( z ) . \\end{align*}"}
+{"id": "3363.png", "formula": "\\begin{align*} K _ 1 \\otimes _ K K _ \\alpha & = \\prod _ \\beta L _ { \\alpha \\beta } \\\\ K _ 2 \\otimes _ K K _ \\alpha & = \\prod _ \\gamma M _ { \\alpha \\gamma } \\\\ L _ { \\alpha \\beta } \\otimes _ { K _ \\alpha } M _ { \\alpha \\gamma } & = \\prod _ { \\delta } N _ { \\alpha \\beta \\gamma \\delta } \\end{align*}"}
+{"id": "7851.png", "formula": "\\begin{align*} \\widetilde \\Omega = \\Omega _ { \\mbox { \\tiny { $ \\widetilde { Q } $ } } } - B _ { \\mbox { \\tiny { $ \\ ! \\langle \\ ! J \\mathcal { K } \\ ! \\rangle $ } } } = - d ( p _ \\theta d \\theta + p _ \\psi d \\psi + p _ \\phi d \\phi ) + r \\tfrac { F ( \\phi ) } { \\sin ^ 2 \\phi } ( p _ \\theta - p _ \\psi ) d \\theta \\wedge d \\phi , \\end{align*}"}
+{"id": "5326.png", "formula": "\\begin{align*} D ^ + _ n & = \\{ X _ n > 0 \\} , \\ , \\ , \\ , D ^ - _ n = \\{ X _ n < 0 \\} , \\ , \\ , \\ , D ^ 0 _ n = \\{ X _ n = 0 \\} , \\ , \\ , n \\geq 0 , \\\\ D _ n & = D ^ + _ n ( D ^ + _ { n - 1 } ) ^ c \\cup D ^ - _ n ( D ^ - _ { n - 1 } ) ^ c \\cup D ^ 0 _ n ( D ^ 0 _ { n - 1 } ) ^ c , \\ , \\ , n \\geq 1 , \\end{align*}"}
+{"id": "2665.png", "formula": "\\begin{align*} \\dot { v } ( s ) = \\begin{cases} \\frac { 1 } { | G / G _ 0 | } \\sum _ { t G _ 0 \\in G / G _ 0 } u ( t ^ { - 1 } s t ) & s \\in G _ 0 \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "4103.png", "formula": "\\begin{align*} d \\widetilde { G } = \\Lambda G + \\Theta G . \\end{align*}"}
+{"id": "3673.png", "formula": "\\begin{align*} | x _ { t + 1 } - x _ { t + 4 } | & = | y _ { 1 2 } - y _ { 2 4 } | \\le 2 0 t \\delta ^ { 1 / 2 } , \\quad | x _ { t + 2 } - x _ { t + 3 } | & = | y _ { 1 2 } - y _ { 1 3 } | \\le 2 0 t \\delta ^ { 1 / 2 } . \\end{align*}"}
+{"id": "1079.png", "formula": "\\begin{align*} E : y ^ { 2 } = x ( x - 4 ^ { m } p ) ( x + p ) \\end{align*}"}
+{"id": "6817.png", "formula": "\\begin{align*} \\partial _ t \\int _ 0 ^ \\infty g ( s ) \\nabla u ( t - s ) d s & = \\frac { 1 } { \\varepsilon } \\partial _ t \\ , \\bigg ( e ^ { - \\frac { t } \\varepsilon } \\int _ { - \\infty } ^ t e ^ { \\frac { s } \\varepsilon } \\nabla u ( s ) d s \\bigg ) \\\\ & = \\frac { 1 } { \\varepsilon } \\nabla u ( t ) - \\frac { 1 } { \\varepsilon } \\int _ 0 ^ \\infty g ( s ) \\nabla u ( t - s ) d s , \\end{align*}"}
+{"id": "7229.png", "formula": "\\begin{align*} i v _ { t } + \\Delta v = 0 , v ( 0 , x ) = v _ { 0 } = \\sum _ { j } a _ { j } e ^ { i j x } . \\end{align*}"}
+{"id": "4893.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { ( 2 n ) ^ { - 1 } ( 2 x e ^ t ) ^ { 2 n } } { { 2 n \\choose n } } = \\frac { x e ^ t \\arcsin ( x e ^ t ) } { ( 1 - x ^ 2 e ^ { 2 t } ) ^ { 1 / 2 } } . \\end{align*}"}
+{"id": "1131.png", "formula": "\\begin{gather*} g ( t ) \\allowbreak = \\allowbreak \\frac { 1 } { ( 1 - t ^ { 2 } ) ^ { \\beta } } \\left ( 1 + \\frac { \\rho t ^ { 2 } } { 1 - t ^ { 2 } } \\right ) ^ { - \\beta } \\allowbreak = \\\\ = \\allowbreak \\frac { 1 } { ( 1 - t ^ { 2 } ( 1 - \\rho ) ) ^ { \\beta } } = \\sum _ { n \\geq 0 } t ^ { 2 n } ( 1 - \\rho ) ^ { n } \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } / n ! . \\end{gather*}"}
+{"id": "144.png", "formula": "\\begin{align*} ( d _ { i } y ) ( [ \\mathbf { a } ] \\otimes [ \\mathbf { b } ] ) = 0 , \\end{align*}"}
+{"id": "1458.png", "formula": "\\begin{align*} \\gamma _ { \\varepsilon } ( \\gamma _ { \\varepsilon } + 1 ) \\varphi _ { \\beta , \\varepsilon } '' ( s ) = \\beta ( \\beta + 1 ) e ^ { - s } M ( \\gamma _ { \\varepsilon } - \\beta , \\gamma _ { \\varepsilon } + 2 ; s ) . \\end{align*}"}
+{"id": "2453.png", "formula": "\\begin{align*} & 3 \\sum _ { \\substack { ( a , b , c ) \\equiv ( 1 , 1 , 0 ) \\bmod { 2 } \\\\ a + b + c = w + 2 , a , b > 1 } } r { } _ { a , b , c } I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a , b , c ) - \\sum _ { \\substack { ( a , b , c ) \\equiv ( 0 , 0 , 0 ) \\bmod { 2 } \\\\ a + b + c = w + 2 } } r _ { a , b , c } ' I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( a , b , c ) \\\\ & \\overset { ? } { = } \\frac { - 2 2 0 w ^ { 6 } + 2 7 5 1 w ^ { 5 } - 1 0 3 7 5 w ^ { 4 } + 1 6 6 2 0 w ^ { 3 } - 3 5 6 2 0 w ^ { 2 } - 7 5 3 6 w + 2 8 5 1 2 0 } { 1 4 4 0 } I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( w + 2 ) \\end{align*}"}
+{"id": "7420.png", "formula": "\\begin{align*} a ^ \\dagger \\left | n \\right > = \\sqrt { n + 1 } \\left | n + 1 \\right > , a \\left | n + 1 \\right > = \\sqrt { n + 1 } \\left | n \\right > , a \\left | 0 \\right > = 0 , \\end{align*}"}
+{"id": "6535.png", "formula": "\\begin{align*} S = \\sum _ { \\theta _ r \\neq \\{ 1 , - 1 \\} } \\theta _ r \\left ( - \\frac { 2 } { \\sin ( \\theta _ r ) } ( P W - W P ) \\right ) \\end{align*}"}
+{"id": "1643.png", "formula": "\\begin{align*} \\mathcal { T } \\cdot Q \\ ; = \\ ; \\mathcal { T } \\Phi \\left ( \\Phi ^ * \\mathcal { T } ^ * \\mathcal { T } \\Phi \\right ) ^ { - 1 } \\Phi ^ * \\mathcal { T } ^ * \\ ; . \\end{align*}"}
+{"id": "4809.png", "formula": "\\begin{align*} \\mathcal D _ { i _ 0 } \\mathcal C _ { i _ 0 - 1 } \\cdots \\mathcal C _ { p } ( \\psi _ { p } ( n , k ) U _ { n , k } ) = 0 , \\end{align*}"}
+{"id": "3842.png", "formula": "\\begin{align*} \\eta = \\left ( \\frac { n + \\left ( h ^ 2 - 2 h - 1 5 \\right ) + d } { 2 } + i , n - h - 1 - i _ 1 , \\dots , n - 6 - i _ { h - 4 } \\right ) , \\end{align*}"}
+{"id": "7143.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\chi _ i - \\sigma _ J + \\rho + d \\mu \\tau _ d \\in \\textbf { V } ( d ) . \\end{align*}"}
+{"id": "4816.png", "formula": "\\begin{align*} \\frac { \\partial \\rho } { \\partial t } = \\nabla \\cdot ( \\rho \\nabla f ( x , s ) ) + K ( \\bar { \\rho } \\otimes \\mu - \\rho ) . \\end{align*}"}
+{"id": "1181.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 / d & \\sqrt { 1 - 1 / d ^ 2 } \\\\ \\sqrt { 1 - 1 / d ^ 2 } & - 1 / d \\end{pmatrix} , \\end{align*}"}
+{"id": "7045.png", "formula": "\\begin{align*} \\mathrm { d } _ p \\ell _ t ( u ) & = \\mathrm { d } _ p \\ell _ t ( Z . p ) = \\tfrac { \\mathrm { d } \\ , } { \\mathrm { d } s } _ 0 \\ell _ t ( \\mathrm { E x p } ( s Z ) p ) \\\\ & = \\tfrac { \\mathrm { d } \\ , } { \\mathrm { d } s } _ 0 L _ t ( \\mathrm { E x p } ( s Z ) ) p = h _ t ( Z ) . p = \\mathrm { e } ^ { - t B } u . \\end{align*}"}
+{"id": "7843.png", "formula": "\\begin{align*} \\widetilde J _ \\eta ( \\alpha _ x ) = \\langle \\widetilde J , \\eta \\rangle ( \\alpha _ x ) : = \\langle \\widetilde J ( \\alpha _ x ) , \\eta ( x ) \\rangle , \\end{align*}"}
+{"id": "5018.png", "formula": "\\begin{align*} P ( B _ r ) = 2 \\pi r , \\end{align*}"}
+{"id": "7575.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } E [ \\mu _ { \\varepsilon } - \\mu ^ { \\varepsilon } ] = 0 . \\end{align*}"}
+{"id": "662.png", "formula": "\\begin{align*} ( T _ 1 - f _ 1 ) \\cdot ( T _ 2 + f _ 2 ) \\cdot \\Psi _ 1 = i \\Psi _ 1 \\end{align*}"}
+{"id": "4964.png", "formula": "\\begin{align*} \\begin{aligned} \\ln ( R _ { \\rm F A } ( \\lambda , w _ { c } , r ) ) & \\leq \\ln ( \\frac { 1 - \\lambda } { \\lambda } ) - w _ { c } ( 1 - \\lambda ) ^ { \\frac { w _ { c } } { r } - 1 } , \\end{aligned} \\end{align*}"}
+{"id": "201.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\large \\displaystyle u _ t - k \\Delta u _ t - \\Delta u = | u | ^ { p } + \\omega ( x ) , \\ , \\ ( t , x ) \\in ( 0 , \\infty ) \\times \\mathbb { R } ^ N , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\ , x \\in \\mathbb { R } ^ N , \\end{array} \\right . \\end{align*}"}
+{"id": "2624.png", "formula": "\\begin{align*} E _ { 2 , m + e _ i } = E _ { 2 , m } \\cup ( E _ { 2 , m + e _ i - e _ j } ) \\cup ( E _ { 2 , [ m - e _ j , m + e _ i ] } ) \\end{align*}"}
+{"id": "316.png", "formula": "\\begin{align*} t _ { 2 } ^ { - 1 } f _ { 1 } ^ { * } \\alpha _ { 1 } & = f ^ { * } u t _ { 1 } ^ { n } t _ { 2 } ^ { n - 1 } + f ^ { * } v t _ { 2 } ^ { m - 1 } \\end{align*}"}
+{"id": "5782.png", "formula": "\\begin{align*} | \\mathcal { A } | = \\frac { | X \\setminus \\{ 0 \\} | } { q ^ m - 1 } = \\frac { q ^ N - 1 } { q ^ m - 1 } . \\end{align*}"}
+{"id": "961.png", "formula": "\\begin{align*} \\mathcal { L } _ \\Phi : = { \\big \\{ F ( z , \\omega ) = f ( \\Phi ^ { - 1 } ( z , \\omega ) , \\omega ) ; f \\in L ^ 2 _ { \\rm l o c } ( \\mathbb { R } ^ n ; L ^ 2 ( \\Omega ) ) \\ ; \\ ; \\big \\} } \\end{align*}"}
+{"id": "4194.png", "formula": "\\begin{align*} i ( w ^ i , w _ 1 ) = \\underset { j } \\sum c ^ j i ( w ^ j , w _ 1 ) , \\end{align*}"}
+{"id": "7337.png", "formula": "\\begin{align*} & X _ { ( m , m ) } = { S \\choose m } \\times { S \\choose m } , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ X _ { ( m , m + 1 ) } = { S \\choose m } \\times { S \\choose m + 1 } , \\\\ & X _ { ( m + 1 , m ) } = { S \\choose m + 1 } \\times { S \\choose m } , \\ \\ \\ \\ \\ \\ \\ \\ X _ { ( m + 1 , m + 1 ) } = { S \\choose m + 1 } \\times { S \\choose m + 1 } . \\end{align*}"}
+{"id": "5650.png", "formula": "\\begin{align*} \\nabla f = \\mathcal { L } ^ { - 1 } ( d f ) \\end{align*}"}
+{"id": "7429.png", "formula": "\\begin{align*} [ H , \\ , p ^ \\dagger ] = p ^ \\dagger P H p ^ \\dagger = p ^ \\dagger ( P + H ) . \\end{align*}"}
+{"id": "7301.png", "formula": "\\begin{align*} \\mathcal { V } ( t ) = \\sum _ { n = 0 } ^ { \\infty } U _ { n } ( t ) \\end{align*}"}
+{"id": "4063.png", "formula": "\\begin{align*} \\delta ( P \\searrow Q ) & = \\frac { 1 } { | m i n ( Q ) | } \\left ( \\sum \\limits _ { \\underset { } { v \\in X _ 2 , \\ , v \\in \\hbox { \\tiny { m i n } } ( Q ) } } P \\otimes Q + \\sum \\limits _ { \\underset { J \\circledcirc Q } { } } ( P \\otimes \\mathbf 1 ) \\searrow ( J \\otimes Q \\backslash J ) + \\sum \\limits _ { \\underset { J \\circledcirc Q } { } } ( \\mathbf 1 \\otimes P ) \\searrow ( J \\otimes Q \\backslash J ) \\right ) . \\end{align*}"}
+{"id": "3716.png", "formula": "\\begin{align*} \\tilde D A ^ Q ( \\rho , v ) ( \\delta \\rho , \\delta v ) = 0 , \\forall ( \\delta \\rho , \\delta v ) \\end{align*}"}
+{"id": "4234.png", "formula": "\\begin{align*} ( a ) _ 0 & : = ( a ; q ) _ 0 = 1 , \\\\ ( a ) _ n & : = ( a ; q ) _ n = ( 1 - a ) ( 1 - a q ) \\cdots ( 1 - a q ^ { n - 1 } ) , n \\geq 1 , \\\\ ( a ) _ { \\infty } & : = ( a ; q ) _ { \\i } = \\lim _ { n \\to \\i } ( a ; q ) _ n , | q | < 1 . \\end{align*}"}
+{"id": "6728.png", "formula": "\\begin{align*} T _ { m a x } = \\frac { 1 } { 4 C _ 1 } \\frac { 1 } { \\eta _ { 0 } \\varepsilon ^ a + \\varepsilon ^ { \\frac { 1 } { 2 } - a } } , \\mbox { f o r } a \\in [ 0 , \\frac { 1 } { 2 } ) , \\end{align*}"}
+{"id": "2983.png", "formula": "\\begin{align*} W ^ f ( x , y ) : = W \\left ( f ( x ) , f ( y ) \\right ) \\ ; . \\end{align*}"}
+{"id": "5545.png", "formula": "\\begin{align*} T _ 1 ( \\alpha ( t ) , t ) = 1 , \\end{align*}"}
+{"id": "6679.png", "formula": "\\begin{align*} \\mathcal { D } \\left ( ( s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { 2 m - 1 } , ( 1 , 0 , \\ldots , 0 ) \\right ) & \\stackrel { a \\to 0 + } { \\longrightarrow } \\mathcal { D } \\left ( ( s _ { k + 1 } ) _ { k = 0 } ^ { 2 m - 1 } , ( 1 , 0 , \\ldots , 0 ) \\right ) \\\\ & = ( - 1 ) ^ { m } \\mathcal { E } \\left ( ( s _ { k + 1 } ) _ { k = 0 } ^ { 2 m - 1 } , ( 1 , 0 , \\ldots , 0 ) \\right ) . \\end{align*}"}
+{"id": "5784.png", "formula": "\\begin{align*} \\lambda _ i u _ 1 + \\dots + \\lambda _ i u _ { \\ell } = \\lambda _ i w = u _ 1 ^ i + \\dots + u _ { \\ell } ^ i \\end{align*}"}
+{"id": "5131.png", "formula": "\\begin{align*} \\overline { D } ( A , \\Omega , x ) = \\limsup _ { r \\searrow 0 } \\frac { | A \\cap B ( x , r ) | } { | B ( x , r ) \\cap \\Omega | } , \\end{align*}"}
+{"id": "6143.png", "formula": "\\begin{align*} \\tau _ { \\rm B C L } h = \\begin{bmatrix} D _ { V _ 1 ^ * } \\\\ D _ { V _ 2 ^ * } V _ 1 ^ * \\end{bmatrix} ( I - z V ^ * ) ^ { - 1 } h \\oplus \\lim _ { n \\to \\infty } ( V _ 1 V _ 2 ) ^ n ( V _ 2 ^ * V _ 1 ^ * ) ^ n h ; \\end{align*}"}
+{"id": "445.png", "formula": "\\begin{align*} | Q ( \\eta _ \\delta W ) - Q ( W ) | & \\leq \\int _ { \\delta ^ 2 \\leq | z | \\leq \\delta } | \\eta _ \\delta ( d W ) ^ N + ( W d \\eta _ \\delta ) ^ N | ^ 2 + \\int _ { | z | \\leq \\delta } | ( d W ) ^ N | ^ 2 + \\int _ S ( 1 - \\eta _ \\delta ^ 2 ) | W | ^ 2 | k | ^ 2 \\\\ & = \\int _ { \\delta ^ 2 \\leq | z | \\leq \\delta } | \\eta _ \\delta ( d W ) ^ N + ( W d \\eta _ \\delta ) ^ N | ^ 2 + O ( \\delta ^ 2 ) , \\end{align*}"}
+{"id": "32.png", "formula": "\\begin{align*} \\nu _ j = \\begin{cases} \\gamma _ j - 1 , & j \\geq j ^ * \\\\ 1 - \\gamma _ j , & \\end{cases} \\end{align*}"}
+{"id": "3491.png", "formula": "\\begin{align*} \\sqrt [ 4 ] { \\int _ { 0 } ^ { \\mathrm { T _ 0 } } \\int _ { 0 } ^ { \\mathrm { T _ 0 } } | \\partial _ { t } \\Phi ^ { \\textbf { e } } ( \\xi , t ; z , \\tau ) | ^ 2 d \\tau d t } = \\mathcal { O } \\Bigg ( \\sqrt [ 4 ] { \\mathcal { S } ^ { ( \\mathrm { T _ 0 } ) } _ { \\mathrm { r } } } \\frac { 1 } { | \\xi - z | ^ { 2 - 2 \\mathrm { r } } } \\Bigg ) , \\end{align*}"}
+{"id": "1601.png", "formula": "\\begin{align*} \\left ( f _ { n } \\left ( g _ { s , i , j } \\right ) \\right ) \\left ( T _ { b _ { i } } \\left ( v \\right ) \\right ) = \\varphi _ { s , i , j } \\left ( T _ { b _ { i } } \\left ( v \\right ) , 0 \\right ) = \\varphi _ { s , i , j } \\left ( T _ { ( b _ { i } , s _ { i } ) } \\left ( v \\right ) \\right ) = T _ { b _ { j } } \\left ( v \\right ) . \\end{align*}"}
+{"id": "3926.png", "formula": "\\begin{align*} L _ f ^ * & = | ( g , \\kappa ) ^ { - 1 } | \\| g \\| [ ( 1 + m _ 0 ) \\| \\kappa \\| _ { \\mathbb H ^ 1 } + L _ 1 \\| \\kappa \\| ] + L _ 1 , \\\\ K _ f ^ * & = | ( g , \\kappa ) ^ { - 1 } | \\| g \\| \\| \\kappa \\| _ { \\mathbb H ^ 1 } . \\end{align*}"}
+{"id": "466.png", "formula": "\\begin{align*} \\frac { \\sqrt 5 } { 2 } F ( z ) e ^ { - \\frac { L _ j } { 2 } z } = \\tanh \\Big ( \\frac { \\sqrt 5 F _ j } { 2 } z \\Big ) \\cosh \\Big ( \\frac { \\sqrt 5 F _ j } { 2 } z \\Big ) . \\end{align*}"}
+{"id": "5737.png", "formula": "\\begin{align*} K = \\{ \\mathbf { J } ( \\phi ) ~ | ~ \\phi : \\mathbb { R } ^ { q \\times q } \\rightarrow \\mathbb { R } ^ { q \\times q } ~ { \\rm i s ~ a ~ c o m p l e t e l y ~ p o s i t i v e ~ l i n e a r ~ m a p } , | | \\mathbf { J } ( \\phi ) | | = 1 \\} , \\end{align*}"}
+{"id": "3431.png", "formula": "\\begin{align*} P ^ S = 2 S _ t f _ { J _ { 1 , l } } ^ { ( l ) } , P ^ \\rho = 2 D _ t ( \\rho f _ { J _ { 1 , l } } ^ { ( l ) } ) , P ^ U = 2 U _ t f _ { J _ { 1 , l } } ^ { ( l ) } , \\end{align*}"}
+{"id": "7578.png", "formula": "\\begin{align*} \\iint G ( p , q ) d \\mu ( p ) d \\mu ( q ) & \\leq \\iint \\log \\frac { 1 } { | z _ { \\infty } ( p ) ^ { - 1 } - z _ { \\infty } ( q ) ^ { - 1 } | } d \\mu ( p ) d \\mu ( q ) + C _ 1 \\\\ & = \\iint \\log \\frac { 1 } { | z ^ { - 1 } - w ^ { - 1 } | } d \\omega ( z ) d \\omega ( w ) + C _ 1 \\\\ & = \\iint \\log \\frac { | z w | } { | z - w | } d \\omega ( z ) d \\omega ( w ) + C _ 1 \\\\ & \\leq - \\log \\frac { \\delta } { 4 } + 2 \\int \\log | z | d \\omega ( z ) + C _ 1 . \\end{align*}"}
+{"id": "2834.png", "formula": "\\begin{align*} ( u _ s ^ n - H _ s ^ n ) d \\gamma _ s ^ { - \\frac 1 2 } & = ( u _ s ^ n - H _ s ^ n ) \\gamma _ s ^ { - \\frac 1 2 } \\left ( - \\frac 1 2 \\mu _ s + \\frac 3 8 \\sigma _ s ^ 2 \\right ) d s - ( u _ s ^ n - H _ s ^ n ) \\gamma _ s ^ { - \\frac 1 2 } \\frac 1 2 \\sigma _ s d W _ s ^ 1 . \\end{align*}"}
+{"id": "4597.png", "formula": "\\begin{align*} \\theta ( l + 1 ) - \\theta ( l ) - k = \\frac { O ( K ) } { ( l - b ) \\sin \\pi k } . \\end{align*}"}
+{"id": "3982.png", "formula": "\\begin{align*} q _ { \\beta } ( n , t ) = \\sum _ { \\Omega ^ { n } _ { k } } \\prod _ { j = 1 } ^ { k } \\frac { \\theta ^ { x _ { j } } } { x _ { j } ! } \\left ( \\alpha t ^ { \\beta } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } \\left ( - \\alpha ( e ^ { \\theta } - 1 ) t ^ { \\beta } \\right ) . \\end{align*}"}
+{"id": "6794.png", "formula": "\\begin{align*} \\alpha '' ( x _ 1 , x _ 2 , x _ 3 , \\dots , x _ k ) + \\alpha '' ( x _ 2 , x _ 1 , x _ 3 , \\dots , x _ k ) = \\sum _ { i \\in [ s ] } \\gamma _ i ( x _ 1 , x _ { I _ i } ) \\gamma ' _ i ( x _ 2 , x _ { I ' _ i } ) \\delta _ { i , 1 } ( x _ { J _ { i , 1 } } ) \\cdots \\delta _ { i , d _ i } ( x _ { J _ { i , d _ i } } ) \\end{align*}"}
+{"id": "6463.png", "formula": "\\begin{align*} p ( x , D _ x ) u = ( 2 \\pi ) ^ { - n / 2 } \\int _ { \\mathbb { R } ^ n } e ^ { i x \\cdot \\xi } p ( x , \\xi ) ( { \\cal { F } } { u } ) ( \\xi ) d ^ n \\xi , u \\in \\mathcal S ( \\mathbb R ^ n ) . \\end{align*}"}
+{"id": "4605.png", "formula": "\\begin{align*} \\sin 2 \\pi \\tilde { \\theta } ( n + l ) & = \\sin 2 \\pi \\left ( \\tilde \\theta ( n ) + \\tilde { k } l + \\frac { O ( 1 ) } { 1 + n - b } \\right ) \\\\ & = \\sin ( 2 \\pi \\tilde { \\theta } ( n ) - 2 \\pi k l ) + \\frac { O ( 1 ) } { 1 + n - b } . \\end{align*}"}
+{"id": "8079.png", "formula": "\\begin{align*} \\rho _ N + \\sum _ { j = 1 } ^ { N - 1 } \\rho _ j = \\sum _ { i = 1 } ^ N \\rho _ i . \\end{align*}"}
+{"id": "7187.png", "formula": "\\begin{align*} ( - 1 ) ^ { d _ 1 d _ 2 } q ^ { - d _ 1 d _ 2 } \\prod _ { \\begin{subarray} { c } 1 \\leq i \\leq d _ 1 \\\\ d _ 1 < j \\leq d \\end{subarray} } z _ i ^ { - 1 } z _ j \\cdot \\prod _ { \\begin{subarray} { c } 1 \\leq i \\leq d _ 1 \\\\ d _ 1 < j \\leq d \\end{subarray} } \\xi ' ( z _ i z _ j ^ { - 1 } ) = \\prod _ { \\begin{subarray} { c } 1 \\leq i \\leq d _ 1 \\\\ d _ 1 < j \\leq d \\end{subarray} } \\xi ( z _ i z _ j ^ { - 1 } ) . \\end{align*}"}
+{"id": "2489.png", "formula": "\\begin{align*} \\int _ { 1 } ^ \\infty \\frac { d s } { \\Big ( \\displaystyle \\int _ 0 ^ s F ( t ) d t \\Big ) ^ { p / ( 2 p - q + 1 ) } } = \\infty . \\end{align*}"}
+{"id": "1586.png", "formula": "\\begin{align*} H ( X _ A \\mid X _ C ) + H ( X _ B \\mid X _ C ) - H ( X _ { A \\cup B } \\mid X _ C ) = 0 , \\end{align*}"}
+{"id": "1314.png", "formula": "\\begin{align*} ( f + g ) ^ \\dagger = f ^ \\dagger + g ^ \\dagger , 0 _ { a , b } ^ \\dagger = 0 _ { b , a } \\end{align*}"}
+{"id": "2197.png", "formula": "\\begin{align*} H _ { 1 1 } ^ f = f _ { 1 1 } - \\frac { a } { E } f _ 1 + b f _ 2 , \\ ; \\ ; \\ ; H _ { 1 2 } ^ f = f _ { 1 2 } - \\frac { b } { E } f _ 1 , \\ ; \\ ; \\ ; H _ { 2 2 } ^ f = f _ { 2 2 } . \\end{align*}"}
+{"id": "7729.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { A } ( p ) & = 2 ^ n \\int _ { \\partial X } d V _ { \\hat { g } ^ p } = 2 ^ n \\int _ { \\partial X } e ^ { - n ( s _ p - s _ E ) } d V _ { \\hat { g } ^ E } \\\\ & \\leq 2 ^ n \\int _ { \\partial X } d V _ { \\hat { g } ^ E } = \\mathcal { A } ( E ) . \\end{aligned} \\end{align*}"}
+{"id": "8041.png", "formula": "\\begin{align*} & f _ { D , m , 4 } = \\frac { f _ c } { c } \\left ( \\frac { \\langle { \\bf v } , { \\bf p } _ d - { \\bf p } _ { i _ m } \\rangle } { \\| { \\bf p } _ d - { \\bf p } _ { i _ m } \\| _ 2 } + \\frac { \\langle { \\bf v } , { \\bf p } _ d - { \\bf p } _ { i _ m } \\rangle } { \\| { \\bf p } _ d - { \\bf p } _ { i _ m } \\| _ 2 } \\right ) , \\ ; \\forall m . \\end{align*}"}
+{"id": "2616.png", "formula": "\\begin{align*} \\partial _ t \\mu ^ \\phi = - \\partial _ x \\Big \\{ \\Big [ b ( t , \\cdot ) + \\sigma \\phi ( t , \\cdot ) \\Big ] \\mu ^ \\phi \\Big \\} . \\end{align*}"}
+{"id": "5854.png", "formula": "\\begin{align*} \\Big ( 1 + 2 ^ { p - 1 } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } \\Big ) \\langle f ( A ) \\hat { k } _ \\lambda , \\hat { k } _ \\lambda \\rangle ^ p < \\Big ( \\frac { p } { p - 1 } \\Big ) ^ { p } \\sup _ { \\lambda \\in \\Omega } | \\langle f ( A ) ^ p \\hat { k } _ \\lambda , \\hat { k } _ \\lambda \\rangle | . \\end{align*}"}
+{"id": "2386.png", "formula": "\\begin{align*} F _ { 3 } ^ { ( 1 ) } = \\sum _ { s = 0 } ^ { 1 } \\sum _ { s ' = 0 } ^ { 1 } \\sum _ { 1 \\leq j < k \\leq m } e _ { a _ { 1 } } D _ { \\alpha , \\beta } ( U ( b _ { 1 } + \\cdots + b _ { j } + s - 1 ) , \\iota ^ { c _ { j } } ( a _ { s + 1 } ) , U ( b _ { j + 1 } + \\cdots + b _ { k } + s ' - 1 ) ) g _ { k , s + s ' } , \\end{align*}"}
+{"id": "6384.png", "formula": "\\begin{align*} \\sum _ { k \\in I } b _ { k i } \\pi _ { k j } = \\delta _ { i j } \\widehat { d } _ j \\end{align*}"}
+{"id": "7250.png", "formula": "\\begin{align*} { { m d i m } } _ M ( T _ 1 \\times T _ 2 , X _ 1 \\times X _ 2 , f , d ) = { { m d i m } } _ M ( T _ 1 , X _ 1 , f _ 1 , d _ 1 ) + { { m d i m } } _ M ( T _ 2 , X _ 2 , f _ 2 , d _ 2 ) , \\end{align*}"}
+{"id": "1319.png", "formula": "\\begin{align*} \\lambda \\mathrm { x } _ v = \\sum _ { u \\sim v } \\mathrm { x } _ u \\leq d ( v ) . \\end{align*}"}
+{"id": "3943.png", "formula": "\\begin{align*} w _ n = v _ n \\circ \\tau _ n - \\frac { 1 } { \\mathcal { H } ^ { d - 1 } ( \\Gamma _ \\infty ) } \\int _ { \\Gamma _ \\infty } v _ n \\circ \\tau _ n \\ , d \\mu _ { \\Gamma _ \\infty } . \\end{align*}"}
+{"id": "6220.png", "formula": "\\begin{align*} D ( H ) = \\{ \\psi \\in H ^ 2 ( \\Omega ) : \\ , H \\psi \\in L ^ 2 ( \\Omega ) , \\ , \\psi ( \\vec x ) = 0 \\ ; \\ ; \\ ; \\ ; x \\in \\partial \\Omega \\setminus W , \\ ; \\partial _ z \\psi ( \\vec x ) = 0 \\ ; \\ ; \\ ; \\ ; x \\in W \\} \\end{align*}"}
+{"id": "2424.png", "formula": "\\begin{align*} - \\tilde { L } ( { \\rm D } ( w ) ) = - \\sum _ { N \\geq 2 } \\zeta ^ { \\mathfrak { m } } ( N ) \\tilde { L } ( { \\rm C u t } _ { ( \\emptyset ; \\{ 1 \\} ^ { N - 1 } ) } ( w ) ) + \\tilde { L } ( { \\rm D } ( { \\rm r e g } _ { 0 } ( \\tau ( w ) ) ) ) , \\end{align*}"}
+{"id": "5886.png", "formula": "\\begin{align*} & \\beta _ { \\tau } ( W ) = ( 1 - \\tau ) \\varphi _ { 1 } \\big ( ( 1 - \\tau ) W \\big ) , \\gamma _ { \\tau } ( W ) = \\varphi _ { 0 } \\big ( ( 1 - \\tau ) W \\big ) , \\\\ & \\bar { \\alpha } _ { \\tau \\sigma } ( W ) - \\alpha _ { \\sigma \\tau } ( W ) = ( \\tau - \\sigma ) \\varphi _ { 1 } \\big ( - ( \\tau - \\sigma ) W \\big ) , \\end{align*}"}
+{"id": "3923.png", "formula": "\\begin{align*} L _ f ( \\rho ) = 2 C \\rho ^ { p - 1 } , \\ ; K _ f ( \\rho ) = C _ \\theta \\| \\chi \\| _ \\infty . \\end{align*}"}
+{"id": "2050.png", "formula": "\\begin{align*} \\lambda _ { n + 1 } = ( n + 1 ) a _ { n + 1 } - \\sum _ { j = 1 } ^ { n } a _ { n - j + 1 } \\lambda _ j \\end{align*}"}
+{"id": "3304.png", "formula": "\\begin{align*} K _ 1 \\psi _ { n _ 2 } = \\lambda _ { n _ 2 } \\psi _ { n _ 2 } . \\end{align*}"}
+{"id": "421.png", "formula": "\\begin{align*} \\lim \\limits _ { y \\rightarrow 0 } v _ y ( y , h ) = - \\infty , ~ ~ \\mathrm { a n d } ~ ~ \\lim \\limits _ { h \\rightarrow \\infty } \\frac { h } { v _ y ( y , h ) } = - C _ \\infty , \\end{align*}"}
+{"id": "3702.png", "formula": "\\begin{align*} A ^ Q ( \\rho , v ) : = \\int _ { \\R ^ d } \\int _ 0 ^ 1 ( v ^ 2 - ( \\frac 1 2 \\nabla l n \\rho ) ^ 2 ) \\rho d t d x \\end{align*}"}
+{"id": "1577.png", "formula": "\\begin{align*} R _ { 1 } = R _ { 0 } \\cup R _ { T } . \\end{align*}"}
+{"id": "7102.png", "formula": "\\begin{align*} N _ n = \\sum _ { k \\geq 1 , k | n } \\frac { 1 } { k ^ 2 } \\Omega _ { n / k } . \\end{align*}"}
+{"id": "368.png", "formula": "\\begin{align*} \\left ( g \\cdot \\mu _ \\xi \\right ) ( x ) = \\mu _ \\xi ( g ^ { - 1 } \\cdot x ) = \\mu _ { g ^ { - 1 } \\cdot \\xi } ( x ) \\end{align*}"}
+{"id": "7597.png", "formula": "\\begin{align*} G ( p , q ) = 2 \\pi [ \\tilde { G } ( p , q ) - \\tilde { G } ( p , p _ \\infty ) - \\tilde { G } ( q , p _ \\infty ) ] \\end{align*}"}
+{"id": "3720.png", "formula": "\\begin{align*} = \\int _ 0 ^ 1 \\int _ { \\R ^ d } ( 2 v - \\nabla \\lambda ) \\rho \\delta v d x d t + [ \\lambda \\rho \\delta v ] _ { - \\infty } ^ { + \\infty } \\end{align*}"}
+{"id": "1644.png", "formula": "\\begin{align*} Q _ n \\ ; : = \\ ; \\mathcal { T } _ n \\cdot Q _ { n - 1 } \\ , , Q _ 0 \\in \\mathbb { G } _ { \\mathsf { L } , \\mathsf { q } } \\ , , n \\in \\mathbb { N } \\ , . \\end{align*}"}
+{"id": "486.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom n k \\frac { 2 ^ k F _ { j k } } { L _ j ^ { k } } B _ { n - k } = \\frac { n } { \\sqrt 5 } \\Big ( \\frac { \\sqrt 5 F _ j } { L _ j } \\Big ) ^ { n - 1 } , \\qquad \\mbox { $ n $ e v e n } , \\end{align*}"}
+{"id": "2770.png", "formula": "\\begin{gather*} \\alpha _ { \\lambda } ( s _ 1 ) = e ^ { i t } s _ 1 \\\\ \\alpha _ { \\lambda } ( s _ 2 ) = e ^ { i \\lambda t } s _ 2 \\end{gather*}"}
+{"id": "360.png", "formula": "\\begin{align*} v _ 1 = \\frac { 1 } { \\epsilon _ 1 \\omega } \\left ( f _ 1 - k v _ 3 \\right ) . \\end{align*}"}
+{"id": "2864.png", "formula": "\\begin{align*} ( d / d t ) v _ k + i \\lambda _ k v _ k = 0 , \\ ; \\ ; 1 \\leqslant k \\leqslant n . \\end{align*}"}
+{"id": "2127.png", "formula": "\\begin{align*} \\mathrm { c o r a n k } _ { \\Lambda ( G ) } H ^ { 1 } ( G _ { \\infty } ^ { S } , E _ { p ^ { \\infty } } ) = \\mathrm { c o r a n k } _ { \\Omega ( G ) } H ^ { 1 } ( G _ { \\infty } ^ { S } , E _ { p } ) . \\end{align*}"}
+{"id": "237.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t u - \\Delta u = f , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\end{array} \\right . \\end{align*}"}
+{"id": "2464.png", "formula": "\\begin{align*} V : = \\bigcup _ { K \\in \\mathcal { T } _ { n - 2 } } V _ K \\end{align*}"}
+{"id": "1717.png", "formula": "\\begin{align*} \\int \\mathbb { W } ^ { \\tau } _ { x , \\tilde { x } } ( d \\omega ) f ( \\omega ( t _ 1 ) , \\ldots , \\omega ( t _ n ) ) & \\\\ = \\int d x _ 1 \\ , \\ldots \\ , d x _ n & \\ , \\psi ^ { t _ 1 } ( x _ 1 - \\tilde { x } ) \\psi ^ { t _ 2 - t _ 1 } ( x _ 2 - x _ 1 ) \\ldots \\\\ & \\times \\psi ^ { t _ { n } - t _ { n - 1 } } ( x _ n - x _ { n - 1 } ) \\psi ^ { \\tau - t _ { n } } ( x - x _ n ) f ( x _ 1 , \\ldots , x _ n ) \\ , . \\end{align*}"}
+{"id": "1224.png", "formula": "\\begin{align*} S = S ( 1 , \\cdot , \\cdot ) = ( F ( 1 , \\cdot , \\cdot ) , G ( 1 , \\cdot , \\cdot ) ) , \\tilde S \\coloneqq ( \\log ( F ) , \\log ( G ) ) , \\end{align*}"}
+{"id": "3935.png", "formula": "\\begin{align*} \\sup _ { x \\in \\Gamma _ \\infty } \\sup _ { \\substack { v \\in T _ x \\Gamma _ n \\\\ \\| v \\| = 1 } } \\| d _ x p _ n ( v ) - v \\| \\xrightarrow [ n \\to \\infty ] { } 0 . \\end{align*}"}
+{"id": "4215.png", "formula": "\\begin{align*} \\sum _ { X = 1 } ^ \\infty ( Q _ k ( X ) - M T _ k ( X ) + c q ^ { X / 2 k } X ^ { r - 1 } ) ( 1 - \\cos ( \\phi - ( X - 1 ) \\theta ) ) u ^ { X - 1 } \\end{align*}"}
+{"id": "942.png", "formula": "\\begin{align*} \\begin{aligned} u ( \\bar { x } ) \\leq \\ , & u ( x ^ * ) + C \\epsilon _ 0 \\delta ^ 2 b _ { \\alpha } ^ 2 + C b _ { \\alpha } \\bar { x } _ n + C \\delta ^ 3 b _ \\alpha ^ 2 \\\\ \\leq \\ , & u ( x ^ * ) + C \\left ( \\epsilon _ 0 + \\delta \\right ) \\delta ^ 2 b _ \\alpha ^ 2 \\leq \\left ( 1 + \\frac { c _ 0 } { 3 } \\right ) u ( x ^ * ) \\end{aligned} \\end{align*}"}
+{"id": "5146.png", "formula": "\\begin{align*} u = S _ { \\mathcal W } \\chi _ { A ' } \\in L ^ { 1 , p } ( \\Omega ) . \\end{align*}"}
+{"id": "7304.png", "formula": "\\begin{align*} d \\geq \\begin{cases} \\delta + 1 & { \\rm i f ~ } \\delta \\equiv \\frac { q + 1 } 2 \\pmod { q } , \\\\ \\delta & { \\rm o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "5261.png", "formula": "\\begin{align*} & J _ N ( p _ 1 , \\ldots , p _ m ; k _ 1 , \\ldots , k _ p ) = \\\\ & \\max \\left ( 0 , \\sum _ { i = 1 } ^ { p _ 1 } k _ i , \\sum _ { i = 1 } ^ { p _ 1 + p _ 2 } k _ i , \\ldots , \\sum _ { i = 1 } ^ { p _ 1 + \\ldots + p _ { m - 1 } } k _ i \\right ) + \\max \\left ( 0 , \\sum _ { i = 1 } ^ { p _ 1 } ( - k _ i ) , \\ldots , \\sum _ { i = 1 } ^ { p _ 1 + \\ldots + p _ { m - 1 } } ( - k _ i ) \\right ) . \\end{align*}"}
+{"id": "3042.png", "formula": "\\begin{align*} \\gamma _ { i , \\ell } = ( i , i + 1 , i + 2 , \\ldots , i + \\ell - 1 ) \\end{align*}"}
+{"id": "5083.png", "formula": "\\begin{align*} \\| u \\| _ { S ^ { s } _ { p , q } } : = \\| | \\nabla | ^ { - d ( 1 - \\sigma ) ( \\frac { 1 } { 2 } - \\frac { 1 } { q } ) } u \\| _ { L ^ p _ { t \\in I } W _ { x } ^ { s , q } } \\end{align*}"}
+{"id": "3613.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } E _ x \\left [ e ^ { - \\lambda t } | X _ { t } | \\right ] = 0 \\end{align*}"}
+{"id": "5299.png", "formula": "\\begin{align*} f ( x ) = \\int _ P a ( \\omega ) e ^ { 2 \\pi i \\omega x } d \\mu _ P ( \\omega ) . \\end{align*}"}
+{"id": "5931.png", "formula": "\\begin{align*} x _ 1 ^ j + x _ { 2 } ^ { j } + \\dots + x _ s ^ j & = y _ 1 ^ j + y _ { 2 } ^ { j } + \\dots + y _ s ^ j , 1 \\leq j \\leq k \\end{align*}"}
+{"id": "5288.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ 2 } = \\| \\hat f \\| _ { L ^ 2 } . \\end{align*}"}
+{"id": "806.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\tau _ i ^ { * } = \\inf \\{ t : \\ W _ i ( t ) \\geq a ' t + b ' \\} , \\\\ & a ' = \\frac { \\sigma } { 2 } - \\frac { \\alpha } { \\sigma } , b ' = \\frac { 1 } { \\sigma } \\ln \\left ( \\frac { x ^ * } { x } \\right ) = \\frac { 1 } { \\sigma } \\ln \\left ( \\frac { \\bar { K } _ 1 } { x } \\cdot \\frac { k _ 2 } { k _ 2 - 1 } \\right ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "4549.png", "formula": "\\begin{align*} 0 = \\mathrm { d i v } _ w \\tilde { E } ^ i + \\epsilon ^ i _ { \\phantom { i } j k } \\imath _ { \\tilde { E } ^ k } A ^ j = \\big ( \\mathrm { d i v } _ v \\hat { E } ^ i + \\epsilon ^ i _ { \\phantom { i } j k } \\imath _ { \\hat { E } ^ k } A ^ j \\big ) \\left ( \\frac { v } { w } \\right ) \\ , . \\end{align*}"}
+{"id": "1142.png", "formula": "\\begin{align*} \\lim \\nolimits _ { k \\to \\infty } C ( l ' , r ' ; P _ { k , S } ^ { l , r } ) = \\delta _ S ( l , r ; \\ , l ' , r ' ) , \\end{align*}"}
+{"id": "991.png", "formula": "\\begin{align*} { \\rm e } ^ * _ { j i } = u _ { i , j } + \\epsilon _ { i j m } \\ , \\vartheta _ m , \\qquad \\mathfrak { K } _ { j i } = \\vartheta _ { j , i } . \\end{align*}"}
+{"id": "1006.png", "formula": "\\begin{align*} \\varepsilon _ { i j } & = \\frac { 1 } { 2 } ( u _ { i , j } + u _ { j , i } ) , \\gamma _ { [ i j ] } = u _ { [ j , i ] } - P _ { [ i j ] } , \\kappa _ { i [ j k ] } = P _ { [ j k ] , i } , \\end{align*}"}
+{"id": "2174.png", "formula": "\\begin{align*} \\overline { \\beta } _ { i k m } \\alpha _ { j l } ^ 0 + \\alpha _ { i k } ^ 0 \\overline { \\beta } _ { j l m } - \\overline { \\beta } _ { i l m } \\alpha _ { j k } ^ 0 - \\alpha _ { i l } ^ 0 \\overline { \\beta } _ { j k m } = \\overline { S } _ { i j k l m } . \\end{align*}"}
+{"id": "522.png", "formula": "\\begin{align*} \\partial _ i g ( x ) = \\frac { 1 } { n } D _ m G \\bigg ( \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\delta _ { x _ k } , x _ i \\bigg ) , \\partial _ { i j } g ( x ) = \\frac { 1 } { n ^ 2 } D _ m ^ 2 G \\bigg ( \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\delta _ { x _ k } , x _ i , x _ j \\bigg ) , \\ \\ i \\neq j . \\end{align*}"}
+{"id": "1733.png", "formula": "\\begin{align*} I & \\lesssim \\int \\int d x \\ , d t \\ , F \\overline { G } G \\overline { G } = \\left | \\int \\int d x \\ , d t \\ , F \\overline { G } G \\overline { G } \\right | \\\\ & \\leq \\| F \\| _ { L ^ 4 _ { t , x } } \\| G \\| _ { L ^ 4 _ { t , x } } ^ 3 \\ , . \\end{align*}"}
+{"id": "1446.png", "formula": "\\begin{align*} | N \\ast ( \\eta _ { \\varepsilon } b _ 2 ) ( x ) | & \\le C \\begin{cases} 1 + \\log \\langle x \\rangle & ( n = 2 ) \\\\ \\langle x \\rangle ^ { 2 - n } & ( n = 1 , n \\ge 3 ) \\end{cases} , | \\nabla N \\ast ( \\eta _ { \\varepsilon } b _ 2 ) ( x ) | \\le C \\langle x \\rangle ^ { 1 - n } \\end{align*}"}
+{"id": "4724.png", "formula": "\\begin{gather*} r ( a ) \\beta _ k ( v ) = r ( \\partial _ k ( a ) ) v + \\beta _ k ( r ( a ) v ) , \\\\ l ( a ) \\beta _ k ( v ) = l ( \\partial _ k ( a ) ) v + \\beta _ k ( l ( a ) v ) , \\forall a \\in A , \\ v \\in V . \\end{gather*}"}
+{"id": "4638.png", "formula": "\\begin{align*} \\begin{cases} \\vec { r } _ k = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ 3 \\epsilon _ { i j k } ( \\vec { x } _ i - \\vec { x } _ j ) , \\\\ \\vec { \\rho } _ k = \\frac { 3 } { 2 } \\ , \\vec { x } _ k - \\frac { 1 } { 2 } \\sum _ { \\ell = 1 } ^ 3 \\vec { x } _ \\ell , \\end{cases} k \\in \\{ 1 , 2 , 3 \\} \\end{align*}"}
+{"id": "1409.png", "formula": "\\begin{align*} \\int _ 0 ^ t ( t _ 0 + s ) ^ { \\lambda - \\frac { p + 1 } { p - 1 } } h ( s ) \\ , d s & \\le C \\begin{dcases} 1 & \\left ( \\lambda < \\frac { 2 } { p - 1 } \\right ) , \\\\ \\log ( t _ 0 + t ) & \\left ( \\lambda = \\frac { 2 } { p - 1 } \\right ) , \\\\ ( t _ 0 + t ) ^ { \\lambda - \\frac { 2 } { p - 1 } } & \\left ( \\lambda > \\frac { 2 } { p - 1 } \\right ) ; \\end{dcases} \\end{align*}"}
+{"id": "2955.png", "formula": "\\begin{align*} T _ C ( F ( \\bar x ) ) = \\prod _ { i = 1 } ^ \\ell T _ { \\mathcal Q _ { m _ i } } ( F _ i ( \\bar x ) ) . \\end{align*}"}
+{"id": "7590.png", "formula": "\\begin{align*} d ( p , q ) : = \\inf \\left \\{ \\ell ( \\gamma ) \\mid \\right \\} . \\end{align*}"}
+{"id": "1958.png", "formula": "\\begin{align*} t = \\alpha - \\sum _ { j = 1 } ^ d \\frac { a _ j } { \\tau _ j } + \\frac { \\nu } { \\tau _ { } } , \\end{align*}"}
+{"id": "5719.png", "formula": "\\begin{align*} L _ n ( x _ 1 , \\dots , x _ N ) : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ N \\delta _ { x _ i } . \\end{align*}"}
+{"id": "7682.png", "formula": "\\begin{align*} - \\mu ' ( t ) = \\int _ { u = t } | \\nabla u | ^ { - 1 } \\frac { 2 ^ n d \\mathcal { H } ^ { n - 1 } ( x ) } { ( 1 - | x | ^ 2 ) ^ n } \\end{align*}"}
+{"id": "3758.png", "formula": "\\begin{align*} \\det P & = \\frac { ( x - 1 2 ) ( x - 2 ) ( x + 2 ) ^ 5 ( x + 4 ) ^ { 1 0 } ( x + 6 ) ^ 5 } { ( x - 2 2 ) ( x + 8 ) } ; \\\\ P ^ { - 1 } & = \\frac { I _ { 2 0 } } { x + 4 } - \\frac { 1 0 J _ { 2 0 } } { ( x - 1 2 ) ( x - 2 ) ( x + 6 ) } + \\frac { X ^ 2 - ( x + 4 ) X } { ( x + 2 ) ( x + 4 ) ( x + 6 ) } . \\end{align*}"}
+{"id": "2010.png", "formula": "\\begin{align*} \\mathfrak { C } _ 0 ( a ) = \\left \\{ \\phi \\in C _ c ^ \\infty ( \\R ) \\ , \\left | ~ { \\rm s u p p } \\ , \\phi \\subset [ - a , a ] , ~ \\int _ { - a } ^ { a } \\phi ( t ) \\ , d t = 0 \\right . \\right \\} , \\end{align*}"}
+{"id": "8194.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\big ( \\Vert \\rho ( t ) - 1 \\Vert _ { \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } } + \\Vert u ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } } \\big ) = 0 . \\end{align*}"}
+{"id": "7803.png", "formula": "\\begin{align*} - \\int _ { \\frac { \\pi } { 2 } } ^ { \\pi } \\cot ( \\theta ) \\left ( \\int _ { t _ { 1 } ( \\theta ) } ^ { t _ { 2 } ( \\theta ) } d t \\right ) d \\theta = - \\int _ { \\frac { \\pi } { 2 } } ^ { \\pi } \\cot ( \\theta ) 2 ( \\pi - \\theta ) d \\theta \\end{align*}"}
+{"id": "2617.png", "formula": "\\begin{align*} \\widehat \\Phi ^ { \\phi ^ \\varepsilon } ( t ) \\overset { d } { = } \\Phi ^ { q _ \\varepsilon } ( t ) \\end{align*}"}
+{"id": "5335.png", "formula": "\\begin{align*} F ^ * ( G ) = T \\times C _ { p _ 1 } \\times \\cdots \\times C _ { p _ k } , \\end{align*}"}
+{"id": "412.png", "formula": "\\begin{align*} & \\left | \\{ K \\pmod H : ( K , H ) = 1 K \\equiv 1 \\pmod D \\} \\right | \\\\ & \\quad = \\left | \\{ K \\pmod { H } : ( K , H ) = 1 K \\equiv Q \\pmod D \\} \\right | . \\end{align*}"}
+{"id": "7120.png", "formula": "\\begin{align*} A \\mapsto A ' , \\ ( d _ i , w _ i ) _ { i = 1 } ^ k \\mapsto ( d _ i , v _ i ) _ { i = 1 } ^ k . \\end{align*}"}
+{"id": "5035.png", "formula": "\\begin{align*} f ( \\varepsilon ) = \\frac { \\mathcal { U } ( \\varepsilon ) } { \\varepsilon } . \\end{align*}"}
+{"id": "7549.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\oint _ { \\gamma } P ( w ) \\frac { A ^ L ( w ) } { w ^ { M + N } } R _ N ( w , z ) d w = P ( z ) \\end{align*}"}
+{"id": "2867.png", "formula": "\\begin{align*} A _ { k j } = \\begin{cases} \\sum _ { l = 1 } ^ { n _ 1 } \\Psi _ { k l } \\bar \\Psi _ { j l } , & \\lambda _ k = \\lambda _ j , \\\\ 0 , & . \\end{cases} \\end{align*}"}
+{"id": "4216.png", "formula": "\\begin{align*} \\phi = \\pi - \\arg \\left ( \\frac { ( - 1 ) ^ r \\alpha ^ r Z ( \\alpha ^ { - 1 } ) } { f ^ { ( r ) } ( \\alpha ^ { - 1 } ) ( 1 - \\alpha ^ { - 1 } ) } \\right ) . \\end{align*}"}
+{"id": "411.png", "formula": "\\begin{align*} \\ker ( w ) & = \\left \\{ K \\pmod H \\in ( \\mathbb { A } / H ) ^ \\times : w ( K ) = 1 \\right \\} \\\\ & = \\left \\{ K \\pmod H : ( K , H ) = 1 K \\equiv 1 \\pmod D \\right \\} . \\end{align*}"}
+{"id": "2348.png", "formula": "\\begin{align*} \\alpha _ { i } ( f , g ) = c \\delta _ { 1 , i } + \\sum _ { j \\in f ^ { - 1 } ( i ) } a _ { j } + \\sum _ { j \\in f ^ { - 1 } ( i ) } b _ { j } . \\end{align*}"}
+{"id": "2366.png", "formula": "\\begin{align*} B ( x _ { b _ { 1 } } \\cdots x _ { b _ { m } } ) = e _ { a _ { 1 } } \\cdots e _ { a _ { b _ { 1 } + \\cdots + b _ { m } } } \\end{align*}"}
+{"id": "240.png", "formula": "\\begin{align*} S [ \\xi ] ( u ) \\stackrel { d e f } { = } { \\bf P } ( \\ \\min _ { i \\in D } \\xi _ i > u \\ ) , \\ u \\ge 1 . \\end{align*}"}
+{"id": "585.png", "formula": "\\begin{align*} \\mu ^ { ( i ) } = j ^ { ( i ) } ( j ^ { ( i ) } + 1 ) \\ , , \\ i = 0 , 1 , \\dots , 4 \\ , , \\mu _ n ^ { ( 1 2 ) } = j _ n ^ { ( 1 2 ) } ( j _ n ^ { ( 1 2 ) } + 1 ) \\quad \\mu _ p ^ { ( 1 2 3 ) } = j _ p ^ { ( 1 2 3 ) } ( j _ p ^ { ( 1 2 3 ) } + 1 ) . \\end{align*}"}
+{"id": "926.png", "formula": "\\begin{align*} \\begin{aligned} | F ^ { i j } ( \\nabla _ \\alpha u ) _ { i j } | \\leq \\ , & | \\nabla _ \\alpha f ^ { 1 / k } | + C \\sum _ { i , j = 1 } ^ n F ^ { i j } u _ { i j } + C | D u | \\sum _ { i = 1 } ^ n F ^ { i i } \\\\ \\leq \\ , & C f ^ { \\frac { 1 } { k } - \\frac { 1 } { 2 ( k - 1 ) } } + C b _ { n - 1 } \\sum _ { i = 1 } ^ n F ^ { i i } \\\\ \\leq \\ , & C b _ { n - 1 } ^ { 1 / 2 - 1 / k } + C b _ { n - 1 } \\sum _ { i = 1 } ^ n F ^ { i i } \\end{aligned} \\end{align*}"}
+{"id": "7082.png", "formula": "\\begin{align*} ( f ) _ \\infty = g \\left ( \\frac { 1 } { t } \\right ) ( 1 + d _ { > 0 } ( t ) ) \\in R _ { X , \\infty } , \\end{align*}"}
+{"id": "3826.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 n + h ^ 2 - 1 8 } { 2 } . \\end{align*}"}
+{"id": "6877.png", "formula": "\\begin{align*} L ( E , \\chi _ 1 , 1 ) \\ne 0 , L ' ( E , \\chi _ 2 , 1 ) \\ne 0 , \\mbox { a n d } \\chi _ 2 ( p ) = \\alpha . \\end{align*}"}
+{"id": "5226.png", "formula": "\\begin{align*} g \\cdot \\sum _ { i \\geq 0 } f _ i v ^ i = \\sum _ { i \\geq 0 } g ( f _ i ) \\omega ( g ) ^ i v ^ i \\end{align*}"}
+{"id": "2639.png", "formula": "\\begin{align*} z _ { n - 1 } ' & = \\lambda ( d ( z ) a ^ { - 1 } b ^ { - 1 } , d ( z ) a ^ { - 1 } b ^ { - 1 } a ) \\\\ z _ n ' & = \\lambda ( d ( z ) a ^ { - 1 } b ^ { - 1 } a , d ( z ) a ^ { - 1 } b ^ { - 1 } a b ) \\\\ z _ { n + 1 } ' & = \\lambda ( d ( z ) a ^ { - 1 } b ^ { - 1 } a b , d ( z ) a ^ { - 1 } b ^ { - 1 } a b ^ 2 ) \\\\ z ' & = z _ 1 . . . z _ { n - 2 } z _ { n - 1 } ' z _ n ' z _ { n + 1 } ' \\end{align*}"}
+{"id": "6727.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } & \\partial _ { t } \\widetilde { E } - \\nabla _ { x } \\times \\widetilde { B } = \\rho u - \\bar { \\rho } \\bar { u } , \\\\ & \\partial _ { t } \\widetilde { B } + \\nabla _ { x } \\times \\widetilde { E } = 0 , \\\\ & \\nabla _ { x } \\cdot \\widetilde { E } = - \\widetilde { \\rho } , \\nabla _ { x } \\cdot \\widetilde { B } = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "4104.png", "formula": "\\begin{align*} \\int ^ { \\mathrm { B C } } ( y _ 1 , y _ 2 ) = \\beta ( y _ 1 , y _ 2 ; t ) . \\end{align*}"}
+{"id": "1635.png", "formula": "\\begin{align*} \\phi _ { r } ( x , t ) = \\phi _ { ( S ) } \\left ( x + \\frac { \\phi _ { r - 1 } ^ { ( 1 ) } ( x , t ) - x } { 2 } , t \\right ) . \\end{align*}"}
+{"id": "5507.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } \\frac { ( q ^ { - N } ) _ { n } q ^ { ( N + 1 ) n + n } ( b q / a ) _ { n } ( a t ) ^ { n } } { ( b q ^ 2 ) _ { n } ( t q ^ 2 ) _ { n } } = \\frac { ( 1 - t q ) } { ( 1 - t q ^ { N + 1 } ) } F _ N ( a , b q ; t q ) . \\end{align*}"}
+{"id": "6462.png", "formula": "\\begin{align*} P \\phi = \\partial _ { t t } \\phi - \\Delta _ h \\phi + m ^ 2 \\phi \\end{align*}"}
+{"id": "3391.png", "formula": "\\begin{align*} 0 & = \\sum _ { g \\in \\Z ^ n } r ^ * ( g ) s _ { h - g } = \\sum _ { g _ 1 , \\ldots , g _ n \\in \\Z } r ^ * ( g _ 1 , \\ldots , g _ n ) s _ { ( h _ 1 - g _ 1 , \\ldots , h _ n - g _ n ) } \\\\ & = \\sum _ { g _ 1 , \\ldots , g _ n \\in \\Z } r ^ * ( h _ 1 - g _ 1 , \\ldots , h _ n - g _ n ) s _ { ( g _ 1 , \\ldots , g _ n ) } \\\\ & = \\sum _ { i \\in \\N } r ^ * ( h _ 1 - i ^ 2 , h _ 2 + i , h _ 3 , \\ldots , h _ n ) s _ { ( i ^ 2 , - i , 0 , \\ldots , 0 ) } \\\\ & = \\sum _ { i = 0 } ^ \\infty r ^ * ( h _ 1 - i ^ 2 , h _ 2 + i , h _ 3 , \\ldots , h _ n ) . \\end{align*}"}
+{"id": "4469.png", "formula": "\\begin{align*} \\tilde { u } ( b ) - \\tilde { u } ( a ) = \\int _ b ^ a u ' ( t ) \\mathrm { d } t \\ , , \\forall b , a \\in \\bar { I } \\ , . \\end{align*}"}
+{"id": "2285.png", "formula": "\\begin{align*} G _ 1 ( X _ 1 , X _ 2 ) : = F _ 1 ( D X _ 1 , D X _ 2 ) \\end{align*}"}
+{"id": "2395.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 } f ( \\alpha ) = 0 . \\end{align*}"}
+{"id": "3603.png", "formula": "\\begin{align*} \\eta _ { i j } = \\sum _ m b _ m \\theta _ { m i } \\theta _ { m j } = \\frac { 1 } { \\pi ^ 2 } \\sum _ m b _ m \\frac { \\omega _ m } { \\omega _ m ^ 2 - i ^ 2 } \\frac { \\omega _ m } { \\omega _ m ^ 2 - j ^ 2 } \\sin ^ 2 ( \\pi \\omega _ m ) ( - 1 ) ^ { i + j } \\end{align*}"}
+{"id": "6272.png", "formula": "\\begin{align*} \\min _ { 0 \\leq t \\leq R } \\bigg \\{ \\varphi ( t ) = \\alpha \\sqrt { 1 + t ^ 2 } + \\frac { \\beta } { 2 } t ^ 2 - \\norm { H _ r ( p _ \\lambda ( u ) } t \\bigg \\} , \\end{align*}"}
+{"id": "6213.png", "formula": "\\begin{align*} \\textsf { W } _ \\nu = \\mathcal { W } _ { ( \\lfloor \\frac { \\nu + 1 } { 2 } \\rfloor , d ) } + \\mathcal { W } _ { ( \\lfloor \\frac { \\nu + 1 } { 2 } \\rfloor + 1 , d ) } + \\cdots + \\mathcal { W } _ { ( \\nu , d ) } \\ \\ ( ) \\end{align*}"}
+{"id": "1114.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { n } \\left ( - 1 \\right ) ^ { m } \\sum _ { k = 0 } ^ { m } \\binom { m } { k } \\binom { n - m } { k } \\rho ^ { k } = \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ ( 1 - \\rho ) ^ { n / 2 } & & n \\end{array} \\right . . \\end{align*}"}
+{"id": "5938.png", "formula": "\\begin{align*} T _ { \\delta } : = \\delta ^ { - 1 } \\O \\times \\delta ^ { - 2 } \\O \\times \\dots \\times \\delta ^ { - k } \\O . \\end{align*}"}
+{"id": "7763.png", "formula": "\\begin{align*} \\lambda = \\frac { 2 r _ m } { 3 r _ m ^ 2 + 1 } . \\end{align*}"}
+{"id": "3990.png", "formula": "\\begin{align*} \\hat { q } ( n , t ) = \\sum _ { \\Omega _ { n } } \\prod _ { j = 1 } ^ { n } \\frac { \\left ( ( 1 - p ) ^ { j } / j \\right ) ^ { x _ { j } } } { x _ { j } ! } \\left ( \\frac { - \\lambda t } { \\ln p } \\right ) ^ { z _ { n } } e ^ { - \\lambda t } , \\end{align*}"}
+{"id": "1795.png", "formula": "\\begin{align*} \\mathcal { L } \\left ( t \\right ) = \\left \\{ \\mathfrak { C } _ { 1 } \\left ( t \\right ) , \\dots , \\mathfrak { C } _ { n _ { 0 } } \\left ( t \\right ) \\right \\} , \\end{align*}"}
+{"id": "5912.png", "formula": "\\begin{align*} M _ Q \\left ( \\begin{smallmatrix} a _ 1 \\\\ a _ 2 \\\\ a _ 3 \\\\ a _ 4 \\\\ a _ 5 \\end{smallmatrix} \\right ) = 0 \\ , . \\end{align*}"}
+{"id": "5277.png", "formula": "\\begin{align*} \\sigma ^ 2 ( f ) = \\frac { 1 } { ( 2 \\pi ) ^ { n } } \\ * \\sum _ { \\pi } ^ * \\prod _ { B \\in \\pi } \\ * \\int _ { L _ { \\pi } } \\ f ( t ^ { ( 1 ) } ) \\ * f ( t ^ { ( 2 ) } ) \\ * \\prod _ { B \\in \\pi } c _ { | B | } ( t _ i : i \\in B ) \\ \\ * d \\lambda , \\end{align*}"}
+{"id": "1453.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\frac { M ( b , c ; s ) } { s ^ { b - c } e ^ s } & = \\lim _ { s \\to \\infty } \\frac { \\frac { d ^ m } { d s ^ m } M ( b , c ; s ) } { \\frac { d ^ m } { d s ^ m } ( s ^ { b - c } e ^ s ) } = \\frac { ( b ) _ m } { ( c ) _ m } \\lim _ { s \\to \\infty } \\frac { M ( b + m , c + m ; s ) } { s ^ { b - c } e ^ s + o ( s ^ { b - c } e ^ s ) } \\\\ & = \\frac { ( b ) _ m \\Gamma ( c + m ) } { ( c ) _ m \\Gamma ( b + m ) } = \\frac { \\Gamma ( c ) } { \\Gamma ( b ) } . \\end{align*}"}
+{"id": "2260.png", "formula": "\\begin{align*} d _ l & = ( \\epsilon ' _ 0 - \\epsilon _ 0 ) d _ 0 + \\dots + ( \\epsilon ' _ { l - 1 } - \\epsilon _ { l - 1 } ) d _ { l - 1 } \\\\ & \\leq d _ 0 + \\dots + d _ { l - 1 } \\leq \\sum _ { i = 0 } ^ { l - 1 } \\left ( \\frac 1 r \\right ) ^ { l - i } d _ l = \\frac { 1 - \\left ( \\frac 1 r \\right ) ^ { l } } { r - 1 } d _ l < d _ l , \\end{align*}"}
+{"id": "1403.png", "formula": "\\begin{align*} | \\nabla ( \\chi _ j u _ 0 ) | ^ 2 = \\chi _ j ^ 2 | \\nabla u _ 0 | ^ 2 + 2 ( \\nabla \\chi _ j \\cdot \\nabla u _ 0 ) \\chi _ j u _ 0 + | \\nabla \\chi _ j | ^ 2 | u _ 0 | ^ 2 , \\end{align*}"}
+{"id": "7159.png", "formula": "\\begin{align*} \\mathcal { Y } ( d ) ^ { \\lambda \\geq 0 } : = \\left ( \\mathfrak { g } ^ { \\lambda \\geq 0 } \\right ) ^ { \\oplus 2 } \\Big / G L ( V ) ^ { \\lambda \\geq 0 } \\end{align*}"}
+{"id": "6655.png", "formula": "\\begin{align*} Q ( t ) = \\det \\begin{bmatrix} s _ 0 & s _ 1 & \\ldots & s _ { m - 1 } & 1 \\\\ s _ 1 & s _ 2 & \\ldots & s _ { m } & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ s _ { m } & s _ { m + 1 } & \\ldots & s _ { 2 m - 1 } & t ^ { m } \\end{bmatrix} , \\end{align*}"}
+{"id": "2969.png", "formula": "\\begin{align*} X ^ { \\perp } = \\{ y \\in \\mathcal { T } \\ ; \\vline \\ ; \\mathrm { H o m } ( x , y ) = 0 \\ ; \\ ; x \\in X \\} \\end{align*}"}
+{"id": "8071.png", "formula": "\\begin{align*} p | _ 1 ( z = \\sum _ { j = 2 } ^ N D _ j + b ) = \\rho _ 1 g D _ 1 . \\end{align*}"}
+{"id": "7701.png", "formula": "\\begin{align*} f \\left ( 1 - \\sum _ { i = 1 } ^ n p _ i ( x ) x _ i - \\sum _ { i = 1 } ^ n q _ { i , i } ( x ) \\right ) = - \\sum _ { i , j } ^ n q _ { i , j } ( x ) x _ i \\frac { \\partial f } { \\partial x _ j } . \\end{align*}"}
+{"id": "3523.png", "formula": "\\begin{align*} \\Vert \\nabla \\varphi \\Vert ^ 4 _ { \\mathbb { L } ^ 4 ( \\Omega ) } & = \\int _ { \\Omega } | \\nabla \\varphi ( \\mathrm { x } ) | ^ 4 d \\mathrm { x } \\\\ & = \\delta ^ 2 \\int _ { \\mathrm { B } } \\delta ^ { - 4 } | \\nabla _ { \\xi } \\varphi ( \\delta \\xi + z ) | ^ 4 d \\xi \\\\ & = \\delta ^ { - 2 } \\Vert \\nabla \\hat { \\varphi } \\Vert ^ 4 _ { \\mathbb { L } ^ 4 ( \\mathrm { B } ) } , \\end{align*}"}
+{"id": "8005.png", "formula": "\\begin{align*} M = \\begin{pmatrix} A & B ^ * \\\\ B & C \\end{pmatrix} : \\mathcal { H } _ 1 \\oplus \\mathcal { H } _ 2 \\rightarrow \\mathcal { H } _ 1 \\oplus \\mathcal { H } _ 2 \\end{align*}"}
+{"id": "5707.png", "formula": "\\begin{align*} [ b , \\mathcal { M } ] ( f ) ( x ) : = b ( x ) \\mathcal { M } ( f ) ( x ) - \\mathcal { M } ( b f ) ( x ) . \\end{align*}"}
+{"id": "7879.png", "formula": "\\begin{align*} \\mathcal G ( f ) = \\{ R \\in \\mathcal R : ( f , R ) \\in \\mathcal G \\} . \\end{align*}"}
+{"id": "6277.png", "formula": "\\begin{align*} \\frac { v _ i - p _ i } { \\lambda } = u _ i \\in \\partial | \\cdot | ( x _ i ^ * ) = \\partial | \\cdot | ( - t ^ * v _ i ) = - \\partial | \\cdot | ( v _ i ) \\Rightarrow 0 \\in \\partial | \\cdot | ( v _ i ) + \\frac { v _ i - p _ i } { \\lambda } , \\end{align*}"}
+{"id": "7738.png", "formula": "\\begin{align*} S = - 2 n \\frac { \\Delta _ g x } { x } + t r _ g ( R i c [ g ^ + ] + n g ^ + ) = - 2 n \\frac { \\Delta _ g x } { x } + o ( 1 ) . \\end{align*}"}
+{"id": "1168.png", "formula": "\\begin{align*} \\phi _ k ' ( t ) & = - ( 2 k - p _ k ) \\log \\frac { 1 - t } { t } + \\frac { 1 - 2 t } { 2 ( 1 - t ) t } , \\\\ \\phi _ k '' ( t ) & = \\frac { 2 k - p _ k } { ( 1 - t ) t } - \\frac { 2 t ^ 2 - 2 t + 1 } { 2 ( 1 - t ) ^ 2 t ^ 2 } \\end{align*}"}
+{"id": "5413.png", "formula": "\\begin{align*} H ^ { ( 2 ) } ( t ) : = \\mathrm { e } ^ { - \\frac { t } { 2 } } W ^ { S } + \\sqrt { 1 - \\mathrm { e } ^ { - t } } H \\ , , t \\in \\R ^ + . \\end{align*}"}
+{"id": "5861.png", "formula": "\\begin{align*} f ^ p ( t ) \\Big ( 1 + 2 ^ { p - 1 } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } \\Big ) & < \\Big ( \\frac { p } { p - 1 } \\Big ) ^ { p } f ^ p ( t ) . \\end{align*}"}
+{"id": "6091.png", "formula": "\\begin{align*} u _ g u _ h u _ g ^ { - 1 } = \\sigma ( g , h ) \\sigma ( g h , g ^ { - 1 } ) \\sigma ( g , g ^ { - 1 } ) ^ { - 1 } u _ h = \\chi ( g _ 1 h _ 2 - h _ 1 g _ 2 ) u _ h . \\end{align*}"}
+{"id": "1698.png", "formula": "\\begin{align*} \\mu : = \\bigotimes _ { k \\in \\N } \\mu _ k . \\end{align*}"}
+{"id": "2776.png", "formula": "\\begin{align*} Z : = \\begin{bmatrix} A \\\\ B \\end{bmatrix} = Q R = \\begin{bmatrix} Q _ A \\\\ Q _ B \\end{bmatrix} R , \\end{align*}"}
+{"id": "1722.png", "formula": "\\begin{align*} A ^ \\xi ( \\zeta ) & : = \\tilde { \\rho } _ { \\zeta } ( \\Theta ( \\xi ) ) \\ , . \\end{align*}"}
+{"id": "3709.png", "formula": "\\begin{align*} b [ \\rho , v ] : = v + \\frac 1 2 \\nabla \\log \\rho , \\end{align*}"}
+{"id": "1008.png", "formula": "\\begin{align*} \\| \\boldsymbol { \\kappa } \\| ^ 2 = \\| { \\rm D } \\mathbf { A } \\| ^ 2 = 2 \\ , \\| { \\rm D } { \\rm a x l } ( \\mathbf { A } ) \\| ^ 2 = 2 \\ , \\| \\boldsymbol { \\mathfrak { K } } \\| ^ 2 & = 2 \\ , \\| { \\rm D } \\vartheta \\| ^ 2 \\end{align*}"}
+{"id": "2343.png", "formula": "\\begin{align*} \\sum _ { m _ { 0 } + \\cdots + m _ { 2 n } = m } Z ( m _ { 0 } , \\dots , m _ { 2 n } ) = \\frac { 1 } { 2 n + 1 } \\binom { m + 2 n } { m } \\cdot \\frac { \\pi ^ { \\mathrm { w t } } } { ( \\mathrm { w t } + 1 ) ! } \\end{align*}"}
+{"id": "1774.png", "formula": "\\begin{align*} i _ { 0 } ^ { - } = \\max \\left ( \\left [ 1 , \\dots , i _ { 0 } - 1 \\right ] \\cap B ^ { c } \\right ) , \\end{align*}"}
+{"id": "3362.png", "formula": "\\begin{align*} m t _ { i j } = m \\mathrm { R e s } _ { \\sigma _ { i j } : K _ i \\hookrightarrow K _ { i j } } ( \\tilde { y } _ i ) \\end{align*}"}
+{"id": "2390.png", "formula": "\\begin{align*} X _ { k , s } ' & = - \\sum _ { l = 0 } ^ { k - 1 } ( - 1 ) ^ { k - l } X _ { l , s } U ( d _ { k } - d _ { l } - 1 ) \\\\ & \\ \\ + \\begin{cases} X _ { k , s - 1 } + \\sum _ { l = 0 } ^ { k - 1 } ( - 1 ) ^ { k - l } X _ { l , s - 1 } U ( d _ { k } - d _ { l } ) & s > 1 \\\\ \\delta _ { k , 0 } & s = 1 , \\end{cases} \\end{align*}"}
+{"id": "2055.png", "formula": "\\begin{align*} k : = \\frac 1 2 \\sum _ { \\delta | N } r _ \\delta \\in \\mathbb { Z } , \\end{align*}"}
+{"id": "2029.png", "formula": "\\begin{align*} K _ N ( a ) = \\left \\{ \\psi \\in K ( a ) \\left | ~ \\int _ { - a } ^ { a } \\psi ( x ) \\exp \\left ( \\frac { \\pi i n x } { a } \\right ) \\ , d x = 0 ~ \\right . \\right \\} . \\end{align*}"}
+{"id": "4722.png", "formula": "\\begin{gather*} \\alpha _ k ( l ( a ) v ) = l ( \\partial _ k ( a ) ) v + l ( a ) \\alpha _ k ( v ) , \\\\ \\alpha _ k ( r ( a ) v ) = r ( \\partial _ k ( a ) ) v + r ( a ) \\alpha _ k ( v ) , \\forall a \\in A , v \\in V . \\end{gather*}"}
+{"id": "5397.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } d X ^ g ( t ) - L \\Psi ( X ^ g ( t ) ) d t = \\int _ Z f ( s , X ^ g ( s ) , z ) ( g ( s , z ) - 1 ) \\nu ( d z ) d t , \\\\ X ^ g ( 0 ) = x \\in L ^ 2 ( \\mu ) , \\end{array} \\right . \\end{align*}"}
+{"id": "6830.png", "formula": "\\begin{align*} 2 E ( I _ { a } , I _ { b } ) & = - \\frac 2 3 \\log ( s ) - \\frac 1 3 \\log ( r ) + \\frac 1 3 \\log ( u ) + \\frac 2 3 \\log ( t ) \\\\ & = \\frac 1 3 \\ , \\ell _ { a } + \\frac 1 3 \\ , \\ell _ { b } + d _ { a b } . \\end{align*}"}
+{"id": "684.png", "formula": "\\begin{align*} d \\sigma - ( \\log \\sqrt { \\tau _ 0 } ) _ z d z \\sigma = e ^ { 2 i \\theta } \\frac { 1 } { \\sqrt { \\tau _ 0 } } \\left ( Q _ 0 d z + \\frac { \\tau _ 0 } { 4 } d \\overline { z } \\right ) \\ J \\widehat { \\sigma } I \\end{align*}"}
+{"id": "253.png", "formula": "\\begin{align*} \\langle \\nabla y _ \\ell , \\partial _ k \\Q ^ t \\partial _ k \\Q \\ , x \\rangle = \\left \\langle \\nabla y _ \\ell , \\sum _ { i = 1 } ^ d ( \\partial _ k f _ i ) ^ 2 w ^ i \\right \\rangle , \\end{align*}"}
+{"id": "3390.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\ell r _ j s _ { h - g _ j } = 0 . \\end{align*}"}
+{"id": "7244.png", "formula": "\\begin{align*} \\| \\vec { u } \\| ^ 2 _ { \\tilde { X } _ { k _ { \\ast } } ( [ 0 , T ] ) } : = \\sup _ { 0 \\leq j \\leq k _ { \\ast } } \\sup _ { G _ k ^ { j } \\subset [ 0 , T ] } \\| \\vec { u } \\| ^ 2 _ { X ( G _ k ^ { j } ) } . \\end{align*}"}
+{"id": "3811.png", "formula": "\\begin{align*} a _ 1 = \\frac { 4 n - 8 - ( a _ 2 + a _ 3 ) } { 2 } , \\end{align*}"}
+{"id": "3952.png", "formula": "\\begin{align*} \\mathcal { L } \\left ( \\frac { \\mathrm { d } ^ { \\beta } } { \\mathrm { d } t ^ { \\beta } } f ( t ) ; s \\right ) = s ^ { \\beta } \\tilde { f } ( s ) - s ^ { \\beta - 1 } f ( 0 ) , \\ s > 0 . \\end{align*}"}
+{"id": "7532.png", "formula": "\\begin{align*} F _ { \\varepsilon , h } = \\Phi ( \\varepsilon , F ) \\end{align*}"}
+{"id": "7393.png", "formula": "\\begin{align*} i \\partial _ { \\overline { z } } \\left ( \\Psi _ { \\lambda } \\varphi \\right ) _ { \\pm } = 2 \\partial _ { \\overline { z } } \\partial _ { z } \\left ( S L ( \\lambda ) \\varphi \\right ) _ { \\pm } = \\frac { 1 } { 2 } \\Delta \\left ( S L ( \\lambda ) \\varphi \\right ) _ { \\pm } = - \\frac { \\lambda } { 2 } \\left ( S L ( \\lambda ) \\varphi \\right ) _ { \\pm } . \\end{align*}"}
+{"id": "5885.png", "formula": "\\begin{align*} \\sum \\limits _ { J = 1 } ^ { 3 } \\Big ( h e ^ { { W } ^ { J } } \\bar { \\varphi } _ { 1 } \\big ( { W } ^ { J } \\big ) - \\frac { 1 } { 2 } h { W } ^ { J } \\bar { \\varphi } _ { 1 } \\big ( { W } ^ { J } \\big ) \\varphi _ { 1 } \\big ( { W } ^ { J } \\big ) \\Big ) d \\bar { \\tilde { v } } _ { n } ^ { J } \\wedge d \\tilde { v } _ { n } ^ { J } = 0 . \\end{align*}"}
+{"id": "4606.png", "formula": "\\begin{align*} \\ln R ( n + N ) ^ 2 = \\ln R ( n ) ^ 2 - \\frac { K _ 1 N } { 2 ( n - b ) \\sin \\pi k } ( 1 - \\cos 2 \\pi \\varphi ( n ) + \\delta ( n ) ) , \\end{align*}"}
+{"id": "8172.png", "formula": "\\begin{align*} ( a ^ * + \\overline { \\lambda } b ^ * C ) \\left ( I - | \\lambda | ^ 2 T + | \\lambda | ^ 4 T ^ 2 - | \\lambda | ^ 6 T ^ 3 + \\ldots \\right ) ( a + \\lambda C ^ * b ) = \\| a \\| ^ 2 . \\end{align*}"}
+{"id": "97.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ \\Psi u = f ( u ) & A ( r _ { 1 } , r _ { 2 } ) \\\\ u \\equiv c _ { 1 } & \\{ r _ { 1 } \\} \\times N \\\\ u \\equiv c _ { 2 } & \\{ r _ { 2 } \\} \\times N . \\end{cases} \\end{align*}"}
+{"id": "887.png", "formula": "\\begin{align*} \\sum _ i S ^ { i i } _ k = ( n - k + 1 ) S _ { k - 1 } \\geq c _ 0 S _ k ^ { 1 - 1 / ( k - 1 ) } S _ 1 ^ { 1 / ( k - 1 ) } \\end{align*}"}
+{"id": "6629.png", "formula": "\\begin{align*} \\| v \\| ^ 2 _ s : = s _ \\theta ( v , v ) + \\big ( 1 - \\Lambda _ 1 ( S _ \\theta ) \\big ) \\| v \\| ^ 2 _ { L ^ 2 ( \\Omega _ { \\frac { \\pi } { 4 } } ) } \\end{align*}"}
+{"id": "19.png", "formula": "\\begin{align*} \\lambda _ n = ( \\lambda { m } ) ^ n . \\end{align*}"}
+{"id": "8216.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t \\Lambda ^ { \\alpha - 1 } \\dot { \\Delta } _ j \\sigma + \\Lambda ^ { \\alpha - 1 } ( \\dot S _ { j - 1 } v \\cdot \\nabla \\dot { \\Delta } _ j \\sigma ) + \\lambda \\Lambda ^ \\alpha { \\dot { \\Delta } _ j d } = \\Lambda ^ { \\alpha - 1 } f _ j . \\end{aligned} \\end{align*}"}
+{"id": "7452.png", "formula": "\\begin{align*} \\begin{matrix} [ J ^ 2 , \\ , \\tau ^ \\dagger _ { - 1 } ] = - \\tau ^ \\dagger _ { - 1 } 2 \\hat { j } , [ J ^ 2 , \\ , \\tau ^ \\dagger _ { 1 } ] = \\tau ^ \\dagger _ { 1 } 2 ( \\hat { j } + 1 ) \\\\ \\end{matrix} \\end{align*}"}
+{"id": "7330.png", "formula": "\\begin{align*} u _ { \\tau \\varepsilon } ( t ) = \\left \\{ \\begin{array} [ c ] { l l } v ( t ) , & t \\in E _ { \\tau \\varepsilon } \\\\ u ( t ) , & t \\in \\lbrack 0 , T ] \\backslash E _ { \\tau \\varepsilon } , \\end{array} \\right . \\end{align*}"}
+{"id": "465.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\binom { n } { k } 2 ^ k \\left ( ( - 1 ) ^ k \\frac { L _ { j k } } { L ^ k _ j } \\Bigl ( \\frac { L _ j } { \\sqrt 5 F _ j } \\Bigr ) ^ n - \\frac { 1 + ( - 1 ) ^ n } { n - k + 1 } B _ k \\right ) = 1 + ( - 1 ) ^ n \\ , , \\end{align*}"}
+{"id": "6896.png", "formula": "\\begin{align*} F _ 0 ( r ) = ( 2 \\pi ) ^ { \\frac { d } { 2 } } i ^ { - k } r ^ { - \\frac { d - 2 } { 2 } } \\int _ 0 ^ \\infty f _ 0 ( s ) J _ { \\nu ( k ) } ( r s ) s ^ { \\frac { d } { 2 } } d s \\end{align*}"}
+{"id": "4797.png", "formula": "\\begin{align*} \\psi _ 0 ( n , k ) = ( \\epsilon _ 1 \\epsilon _ 2 ) ^ m m ^ { m ( \\epsilon \\epsilon _ 1 - \\epsilon _ 2 ) } \\frac { ( n + \\lambda + \\epsilon _ 2 k m - \\delta _ { \\epsilon _ 2 , 1 } m ) _ { m } ^ { \\epsilon _ 2 } } { ( n + \\mu + \\epsilon _ 1 k m - \\delta _ { \\epsilon _ 1 , 1 } m ) _ { m } ^ { \\epsilon \\epsilon _ 1 } } , \\end{align*}"}
+{"id": "5240.png", "formula": "\\begin{align*} H _ t ( a ) = \\left \\{ \\begin{array} { l r } t & \\\\ 0 & \\end{array} \\right . \\end{align*}"}
+{"id": "3680.png", "formula": "\\begin{align*} V _ i ' = \\begin{cases} V _ i \\cap N ( v ) \\cap \\left ( \\bigcap _ { u \\in U _ 1 \\setminus V _ i } N ( u ) \\right ) & i \\in [ t + 4 ] \\setminus I _ { s m a l l } , \\\\ V _ i \\cap \\left ( \\bigcap _ { u \\in U _ 1 \\setminus V _ i } N ( u ) \\right ) & i \\in I _ { s m a l l } . \\end{cases} \\end{align*}"}
+{"id": "1754.png", "formula": "\\begin{align*} I ^ { \\pm } = I _ { k _ { 0 } } ^ { \\pm } \\sqcup \\bigsqcup _ { i \\neq k _ { 0 } } I _ { i } . \\end{align*}"}
+{"id": "5946.png", "formula": "\\begin{align*} \\widehat { 1 _ { T _ { 0 , K } } } ( \\xi ) & = \\int _ { M _ a ^ { - T } T _ { \\delta } } \\chi ( - x \\cdot \\xi ) d x \\\\ & = \\det ( M _ a ) ^ { - 1 } \\int _ { T _ { \\delta } } \\chi ( - M _ a ^ { - T } y \\cdot \\xi ) d y = \\det ( M _ a ) ^ { - 1 } \\delta ^ { - k ( k + 1 ) / 2 } 1 _ { \\theta _ { \\delta } } ( M _ a ^ { - 1 } \\xi ) \\end{align*}"}
+{"id": "6734.png", "formula": "\\begin{align*} \\frac { \\partial \\overline { G } } { \\partial v _ { k } } = \\varepsilon \\frac { \\sqrt { R } } { \\sqrt { \\theta } } \\sum ^ { 3 } _ { j = 1 } \\frac { \\partial \\bar { \\theta } } { \\partial x _ { j } } \\partial _ { v _ { k } } A _ { j } ( \\frac { v - u } { \\sqrt { R \\theta } } ) \\frac { 1 } { \\sqrt { R \\theta } } + \\varepsilon \\sum ^ { 3 } _ { i , j = 1 } \\frac { \\partial \\bar { u } _ { j } } { \\partial x _ { i } } \\partial _ { v _ { k } } B _ { i j } ( \\frac { v - u } { \\sqrt { R \\theta } } ) \\frac { 1 } { \\sqrt { R \\theta } } , \\end{align*}"}
+{"id": "1978.png", "formula": "\\begin{align*} p ( H ) = \\lambda _ 1 ( H ) + \\lambda _ 1 ( \\overline { H } ) \\geq \\lambda _ 1 ( G ) + \\lambda _ 1 ( \\overline { G } ) + 2 ( - 1 ) ^ { a _ { u v } } \\cdot ( x _ u x _ v - \\overline { x } _ u \\overline { x } _ v ) = p ( G ) . \\end{align*}"}
+{"id": "2320.png", "formula": "\\begin{align*} \\langle b _ { \\theta _ 0 } ( x ) , x \\slash \\lVert x \\rVert \\rangle & = \\sum _ { i = 1 } ^ n \\lVert x \\rVert ( \\alpha _ i + \\lVert x \\rVert ) ^ { - ( 1 + q _ i ) } \\langle x \\slash \\lVert x \\rVert , A _ i ( \\theta _ 0 ) x \\slash \\lVert x \\rVert \\rangle \\\\ & \\leq \\sum _ { i = 1 } ^ { k _ 0 } \\lVert x \\rVert ( \\alpha _ i + \\lVert x \\rVert ) ^ { - ( 1 + q _ i ) } \\lambda _ { \\max } ( A _ i ( \\theta _ 0 ) ) . \\end{align*}"}
+{"id": "4388.png", "formula": "\\begin{align*} \\varphi _ n ^ { ( \\alpha ) } ( x ) = \\sqrt { \\frac { \\Gamma ( n + 1 ) } { \\Gamma ( n + \\alpha + 1 ) } } e ^ { - \\frac { x ^ 2 } { 2 } } L _ n ^ { ( \\alpha ) } ( x ^ 2 ) \\sqrt { 2 x } \\end{align*}"}
+{"id": "4406.png", "formula": "\\begin{align*} \\upsilon = \\nu _ 0 + \\cdots + \\nu _ k q ^ k , \\nu _ i \\in \\mathfrak { N } _ h . \\end{align*}"}
+{"id": "7205.png", "formula": "\\begin{align*} K _ i ^ { \\mathrm { t o p } } ( \\mathrm { P e r f } ^ { \\mathrm { g r } } ( \\mathcal { X } \\times \\mathbb { C } ) ) = K _ i ^ { \\mathrm { t o p } } ( \\mathrm { P e r f } ^ { \\mathrm { g r } } ( \\mathcal { X } ) ) = \\begin{cases} \\bigoplus _ { j \\in \\mathbb { Z } } K ( B G ) _ { \\mathbb { Q } } \\cdot e _ j , & i = 0 , \\\\ 0 , & i = 1 . \\end{cases} \\end{align*}"}
+{"id": "1747.png", "formula": "\\begin{align*} c _ { n , k , m } \\left ( Y , \\mathcal { Z } \\right ) = \\begin{cases} \\frac { 2 k + m - 1 } { m + 1 } { \\frac { k + m - 2 } { 2 } \\choose \\frac { m - 1 } { 2 } } & { \\rm f o r \\ ; } k \\ ; { \\rm o d d } , \\\\ 2 { \\frac { k + m - 1 } { 2 } \\choose \\frac { m + 1 } { 2 } } & { \\rm f o r \\ ; } k \\ ; { \\rm e v e n . } \\end{cases} \\end{align*}"}
+{"id": "998.png", "formula": "\\begin{align*} a _ 1 = \\alpha _ 1 , a _ 2 = \\alpha _ 2 , { a _ 3 } = \\frac { 2 \\ , \\alpha _ 1 + 3 \\ , \\alpha _ 3 } { 8 } \\end{align*}"}
+{"id": "7972.png", "formula": "\\begin{align*} & \\eta ( \\left \\langle \\eta , \\xi ^ H \\right \\rangle ) = 1 + \\left \\langle \\nabla _ \\eta \\eta , \\xi ^ H \\right \\rangle = 1 + \\left \\langle \\nabla _ \\eta J \\eta , J \\xi ^ H \\right \\rangle = 1 + \\left \\langle \\nabla _ \\eta \\nu ^ H , J \\xi ^ H \\right \\rangle \\\\ & = 1 + l ( \\xi ) \\left \\langle \\eta , J \\xi ^ H \\right \\rangle = 1 - l ( \\xi ) \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle . \\end{align*}"}
+{"id": "4195.png", "formula": "\\begin{align*} w ^ i = \\sum _ k a _ { k j } w ^ j \\end{align*}"}
+{"id": "648.png", "formula": "\\begin{align*} ( \\partial - \\nabla ) _ X Y = - \\sum _ { i = 1 , 2 } \\sqrt { c _ i } \\ \\langle X _ i , Y \\rangle \\nu _ i + B ( X , Y _ T ) - B ^ * ( X , Y _ N ) \\end{align*}"}
+{"id": "7390.png", "formula": "\\begin{align*} \\gamma _ D ^ + ( S L ( \\lambda ) \\varphi ) _ { + } = \\gamma _ D ^ - ( S L ( \\lambda ) \\varphi ) _ { - } \\partial _ { \\nu } ( S L ( \\lambda ) \\varphi ) _ { + } - \\partial _ { \\nu } ( S L ( \\lambda ) \\varphi ) _ { - } = \\varphi \\end{align*}"}
+{"id": "7401.png", "formula": "\\begin{align*} f : = f _ 1 f _ 2 f _ 3 f _ 4 . \\end{align*}"}
+{"id": "7389.png", "formula": "\\begin{align*} ( I - \\alpha c ^ 2 M _ 3 \\mathcal { C } _ { \\lambda + c ^ 2 / 2 } ) ^ { - 1 } M _ 3 = M _ 3 ( I - \\alpha c ^ 2 M _ 3 \\mathcal { C } _ { \\lambda + c ^ 2 / 2 } M _ 3 ) ^ { - 1 } . \\end{align*}"}
+{"id": "2837.png", "formula": "\\begin{align*} e ( s ) = I _ { s } ( \\psi _ { n } ) + o ( 1 ) = \\sum _ { i = 1 } ^ { p } I _ { s } ( \\phi _ { i } ) + o ( 1 ) \\end{align*}"}
+{"id": "7394.png", "formula": "\\begin{align*} E _ n ( \\alpha ) = - \\frac { \\alpha ^ 2 } { 4 } + \\mu _ n ( H ) + \\mathcal { O } ( \\alpha ^ { - 1 } \\ln | \\alpha | ) , \\alpha \\rightarrow - \\infty . \\end{align*}"}
+{"id": "6989.png", "formula": "\\begin{align*} F ( f ) : = F _ { \\alpha _ 1 , \\alpha _ 2 } ( f ) = \\int _ { X } \\left ( | \\nabla f | ^ 2 + \\alpha _ 1 f ^ 2 - \\frac { \\alpha _ 2 } { 2 } f ^ 2 \\log f ^ 2 \\right ) d m , f \\in \\mathcal { F } . \\end{align*}"}
+{"id": "5138.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty \\psi _ i = \\chi _ \\Omega . \\end{align*}"}
+{"id": "7099.png", "formula": "\\begin{align*} _ d : = _ { \\mathbb { C } ^ 3 , d } = ( - 1 ) ^ d p _ 3 ( d ) . \\end{align*}"}
+{"id": "4546.png", "formula": "\\begin{align*} \\tilde { E } ^ i = \\hat { E } ^ i \\left ( \\frac { v } { w } \\right ) \\ , . \\end{align*}"}
+{"id": "5117.png", "formula": "\\begin{align*} \\| f ( t ) \\| _ { L ^ p _ { t > t _ 0 } } : = ( \\int _ { t _ 0 } ^ { \\infty } | f ( t ) | ^ { p } \\ , d t ) ^ { \\frac { 1 } { p } } \\end{align*}"}
+{"id": "313.png", "formula": "\\begin{align*} - F ^ { s - 1 } d f _ { 1 } ^ { * } a = \\frac { ( t _ { 1 } ^ { - 1 } f _ { 1 } ^ { * } \\alpha _ { 1 } + t _ { 2 } f _ { 1 } ^ { * } \\beta _ { 2 } ) d \\log t _ { 1 } + t _ { 2 } f ^ { * } \\beta _ { 2 } d \\log t _ { 2 } + t _ { 1 } t _ { 2 } f _ { 1 } ^ { * } \\gamma } { t _ { 1 } ^ { n _ { 1 } + n _ { 2 } - 1 } t _ { 2 } ^ { n _ { 2 } } } \\end{align*}"}
+{"id": "949.png", "formula": "\\begin{align*} y _ n = \\rho ( x ) - \\rho ( x _ 0 ) \\geq \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( x ' _ 0 ) y _ \\beta + \\kappa | y ' | ^ 2 , \\end{align*}"}
+{"id": "948.png", "formula": "\\begin{align*} u ( x ) \\geq l ( y ' ) + \\frac { 1 } { 4 } \\sum _ { \\beta = 1 } ^ { n - 1 } b _ \\beta y _ \\beta ^ 2 - C b _ \\alpha | \\hat { y } | ^ 2 \\end{align*}"}
+{"id": "1771.png", "formula": "\\begin{align*} \\left ( - 1 \\right ) ^ { k + m } \\left ( \\sum _ { \\alpha = 1 } ^ { k + 1 } \\left ( - 1 \\right ) ^ { \\alpha } p _ { A , l _ { \\alpha } } \\left ( C \\right ) p _ { l _ { 1 } , \\dots , \\hat { l _ { \\alpha } } , \\dots , l _ { k + 1 } } \\left ( C \\right ) \\right ) . \\end{align*}"}
+{"id": "635.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } B ( X ) : = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ p e _ j \\cdot B ( X , e _ j ) \\ \\in C l ( T M \\oplus E \\oplus \\mathcal { E } _ 2 ) \\end{align*}"}
+{"id": "3836.png", "formula": "\\begin{align*} \\eta = \\left ( \\frac { n + \\left ( h ^ 2 - 3 h - 9 \\right ) + d } { 2 } + i , n - h - i _ 1 , \\dots , n - 5 - i _ { h - 4 } \\right ) , \\end{align*}"}
+{"id": "4682.png", "formula": "\\begin{align*} P _ { 0 } ( z , w ) & = \\frac { 1 } { | 1 - z w | ^ { 2 \\lambda } } { } _ 2 \\ ! F _ { 1 } \\Big ( { \\lambda , \\lambda \\atop 2 \\lambda + 1 } ; \\frac { 4 ( { \\rm I m } z ) ( { \\rm I m } w ) } { | 1 - z w | ^ { 2 } } \\Big ) \\\\ & = \\frac { 1 } { | 1 - z \\bar { w } | ^ { 2 \\lambda } } { } _ 2 \\ ! F _ { 1 } \\Big ( { \\lambda , \\lambda + 1 \\atop 2 \\lambda + 1 } ; - \\frac { 4 ( { \\rm I m } z ) ( { \\rm I m } w ) } { | 1 - z \\bar { w } | ^ { 2 } } \\Big ) , \\end{align*}"}
+{"id": "2068.png", "formula": "\\begin{align*} S A T = \\begin{bmatrix} I _ \\ell & 0 \\\\ D & I _ { n - \\ell } \\end{bmatrix} , \\end{align*}"}
+{"id": "4398.png", "formula": "\\begin{align*} T _ d ^ \\eta g ( \\xi ) = \\int _ 0 ^ \\infty a ( \\xi , \\eta , \\zeta ) g ( \\zeta ) d \\nu ( \\zeta ) , \\end{align*}"}
+{"id": "4367.png", "formula": "\\begin{align*} D _ H G ^ 0 ( ( \\sigma _ 1 ^ 0 , . . . , \\sigma _ l ^ 0 , x _ 0 ) ( T ) ) = w _ i ^ 1 + D _ H g _ i ^ 0 ( x _ 0 ( T ) ) \\circ w ^ 2 . \\end{align*}"}
+{"id": "1155.png", "formula": "\\begin{align*} \\epsilon _ { f \\otimes \\chi _ D } = \\begin{cases} - \\lambda _ f ( p ) p ^ { 1 / 2 } & ( p , D ) = 1 D \\equiv \\square \\mod 4 p ; \\\\ 1 & p | D . \\end{cases} \\end{align*}"}
+{"id": "6491.png", "formula": "\\begin{align*} A \\subset Q _ 1 = \\bigcup _ { Q \\in \\mathcal Q _ B } Q \\cup \\bigcup _ { Q \\in \\mathcal G _ L } Q . \\end{align*}"}
+{"id": "1867.png", "formula": "\\begin{align*} y _ { ( 2 k - 1 ) m + i , n ^ { + } _ 0 } & = M N - x _ { i , n ^ { + } _ 0 } + ( k - \\tfrac { 1 } { 2 } ) ( m n ) + \\tfrac { 3 } { 2 } \\mbox { \\ i f } i + n ^ { + } _ 0 \\mbox { i s o d d a n d } \\\\ y _ { ( 2 k ) m + i , n ^ { + } _ 0 } & = x _ { i , n ^ { + } _ 0 } + k ( m n ) \\mbox { \\ i f } i + n ^ { + } _ 0 \\mbox { i s e v e n . } \\end{align*}"}
+{"id": "1365.png", "formula": "\\begin{align*} \\partial _ t v - \\Delta v = 0 , t > 0 , x \\in \\mathbb { R } ^ n . \\end{align*}"}
+{"id": "7791.png", "formula": "\\begin{align*} k ( u ) = - \\frac { 1 } { \\pi } \\frac { 1 } { \\overline { u + 1 } ( u - 1 ) } , \\ : u \\neq \\pm 1 \\end{align*}"}
+{"id": "481.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\binom n k \\frac { F _ { j k - 1 } } { L _ j ^ k } B _ { n - k } = B _ n \\Big ( \\frac { \\alpha ^ j } { L _ j } \\Big ) , \\qquad \\mbox { $ n $ e v e n } , \\end{align*}"}
+{"id": "6480.png", "formula": "\\begin{align*} X ^ { r h T } = \\bigsqcup _ { \\rho } X ^ { r h T } _ \\rho \\end{align*}"}
+{"id": "4536.png", "formula": "\\begin{align*} \\mathcal { E } _ x ( t ) = 1 + \\frac { 1 } { \\sinh 1 } \\sinh \\left ( x - \\frac { 1 } { 2 } \\right ) \\sinh \\left ( t - \\frac { 1 } { 2 } \\right ) + 2 \\sum _ { k = 1 } ^ \\infty \\frac { \\cos \\big ( 2 \\pi k ( x - t ) \\big ) } { 1 + 4 \\pi ^ 2 k ^ 2 } \\ , , \\end{align*}"}
+{"id": "3226.png", "formula": "\\begin{align*} \\lambda _ 2 ( G ) \\geq \\lambda _ 2 ( H ) \\geq \\lambda _ 1 ( H ) = \\min \\{ \\lambda _ 1 ( H _ 1 ) , \\lambda _ 1 ( G _ 2 ) \\} . \\end{align*}"}
+{"id": "7118.png", "formula": "\\begin{align*} D ^ b ( \\mathcal { X } ( d ) ) = \\bigoplus _ { w \\in \\mathbb { Z } } D ^ b ( \\mathcal { X } ( d ) ) _ w . \\end{align*}"}
+{"id": "6528.png", "formula": "\\begin{align*} H = \\sum _ { \\theta _ r } ( \\theta _ r + 2 k _ r \\pi ) E _ { \\theta _ r } , \\end{align*}"}
+{"id": "1219.png", "formula": "\\begin{align*} \\pi ( m _ t ) = m _ { t _ 1 } + \\ldots + m _ { t _ s } , \\end{align*}"}
+{"id": "3988.png", "formula": "\\begin{align*} \\hat { \\mathcal { M } } ( t ) \\stackrel { d } { = } \\sum _ { j = 1 } ^ { \\infty } j N _ { j } ( t ) , \\end{align*}"}
+{"id": "8084.png", "formula": "\\begin{align*} \\chi _ 0 = \\frac { \\lambda _ N q ^ 2 + h ^ 2 } { q ^ 2 + h ^ 2 } , \\ \\ \\ \\chi _ 1 = \\frac { \\lambda _ 1 q ^ 2 + h ^ 2 } { q ^ 2 + h ^ 2 } . \\end{align*}"}
+{"id": "449.png", "formula": "\\begin{align*} B _ n ( 1 - x ) = ( - 1 ) ^ n B _ n ( x ) , \\end{align*}"}
+{"id": "3067.png", "formula": "\\begin{align*} F _ Y ( \\tau ^ k ) = F _ X ( \\sigma ^ k ) k \\not \\equiv 0 \\pmod { \\ell } . \\end{align*}"}
+{"id": "940.png", "formula": "\\begin{align*} \\begin{aligned} u ( \\bar { x } ) & = \\frac { 1 } { 2 } \\sum _ { \\beta = 1 } ^ { n - 1 } b _ \\beta \\bar { x } _ \\beta ^ 2 + + \\frac { 1 } { 6 } \\sum _ { \\xi , \\beta , \\gamma } \\varphi _ { \\xi \\beta \\gamma } ( 0 ) \\bar { x } _ \\xi \\bar { x } _ \\beta \\bar { x } _ \\gamma + O ( | \\bar { x } ' | ^ 4 ) \\\\ & \\leq u ( x _ 0 ) + \\frac { 1 } { 2 } \\sum _ { \\beta = 1 } ^ { \\alpha } b _ \\beta \\bar { x } _ \\beta ^ 2 + C \\delta ^ 3 b _ \\alpha ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "1777.png", "formula": "\\begin{align*} a _ { j } = { \\rm m a x } \\left ( j , s _ { j - \\left ( k - q \\right ) + 1 } \\right ) , \\quad { \\rm f o r } \\quad 1 \\leq j \\leq k - 1 , \\end{align*}"}
+{"id": "5078.png", "formula": "\\begin{align*} \\eta _ 1 ( \\xi ) = \\begin{cases} 1 , \\ | \\xi | \\le 1 , \\\\ 0 , \\ | \\xi | \\ge 2 , \\end{cases} \\end{align*}"}
+{"id": "4495.png", "formula": "\\begin{align*} \\left | \\int _ { [ 0 , 1 ] } u h \\right | & = \\left | \\int _ { [ 0 , 1 ] } u h + u ' h ' - u ' h ' \\right | = \\big | \\langle u , h \\rangle _ { H ^ 1 } - \\langle u ' , h ' \\rangle _ { L ^ 2 } \\big | \\\\ & \\leq \\big | \\langle u , h \\rangle _ { H ^ 1 } \\big | + \\big | \\langle u ' , h ' \\rangle _ { L ^ 2 } \\big | \\leq \\| u \\| _ { H ^ 1 } \\| h \\| _ { H ^ 1 } + \\| u ' \\| _ { L ^ 2 } \\| h ' \\| _ { L ^ 2 } \\\\ & \\leq 2 \\| u \\| _ { H ^ 1 } \\| h \\| _ { H ^ 1 } \\ , , \\end{align*}"}
+{"id": "5136.png", "formula": "\\begin{align*} \\mathcal H ^ s ( A ) = \\lim _ { \\delta \\searrow 0 } \\mathcal H _ \\delta ^ s ( A ) , \\end{align*}"}
+{"id": "5681.png", "formula": "\\begin{align*} c _ { i - 1 } u _ { i - 1 } ^ { d - 1 } u _ i = c _ i u _ i ^ { d - 1 } u _ { i + 1 } . \\end{align*}"}
+{"id": "3695.png", "formula": "\\begin{align*} i \\partial _ t \\psi + \\frac 1 2 \\nabla ^ 2 \\psi = 0 , \\psi ( x , 0 ) = \\psi _ o \\end{align*}"}
+{"id": "245.png", "formula": "\\begin{align*} g ^ * ( \\vec { x } ) = g ^ * ( x ) \\stackrel { d e f } { = } \\sup _ { \\lambda \\in R ^ d } [ ( \\lambda , x ) - g ( \\lambda ) ] . \\end{align*}"}
+{"id": "1276.png", "formula": "\\begin{align*} E = H - ( C + 2 C ' + C '' ) . \\end{align*}"}
+{"id": "1426.png", "formula": "\\begin{align*} T _ { \\mathrm { m a x } } & = \\sup \\left \\{ T \\in ( 0 , \\infty ] ; \\ , { } ^ { \\exists } \\mathcal { U } = \\begin{pmatrix} u \\\\ v \\end{pmatrix} \\in C ( [ 0 , T ) ; \\ , \\mathcal { H } ) \\ \\mbox { s a t i s f i e s \\eqref { a p p : 1 : i n t e q } } \\right \\} . \\end{align*}"}
+{"id": "2256.png", "formula": "\\begin{align*} u ^ { - 1 } x ^ m u y ^ m = x ^ m y ^ m = \\sigma ^ m ( b ) = u b u ^ { - 1 } = b , \\end{align*}"}
+{"id": "6001.png", "formula": "\\begin{align*} C ^ 1 _ 0 = \\dfrac { r ( \\beta - 1 ) + \\theta \\beta } { \\theta \\Phi ( \\theta + r ) } , \\end{align*}"}
+{"id": "4213.png", "formula": "\\begin{align*} \\sum _ { X = 1 } ^ \\infty \\mathrm { M T } _ k ( X ) u ^ { X - 1 } = \\frac { d q } { 1 - q u } \\end{align*}"}
+{"id": "5416.png", "formula": "\\begin{align*} \\Im G _ { a a } ( t , z ) = \\sum _ { j = 1 } ^ N \\frac { \\eta | \\langle \\mathbf { e } _ a , \\mathbf { u } _ j ( t ) \\rangle | ^ 2 } { | \\lambda _ j ( t ) - z | ^ 2 } \\prec \\frac { \\eta } { N } \\sum _ { j = 1 } ^ N \\frac { 1 } { | \\lambda _ j ( t ) - z | ^ 2 } = \\Im m _ N ( t , z ) , \\end{align*}"}
+{"id": "4905.png", "formula": "\\begin{align*} \\frac { 2 - x } { ( 1 - x ) ^ 2 } \\sum _ { j = 0 } ^ \\infty f _ j \\left ( 2 - \\frac { 1 } { x } , 2 - \\frac { 1 } { x } \\right ) = \\frac { 1 } { 2 } \\frac { 3 } { 1 - x } + \\frac { 1 - x } { 2 ( 1 - 2 x ) ^ 2 } \\sum _ { j = 0 } ^ \\infty f _ j \\left ( 3 - \\frac { 1 } { x } , 3 - \\frac { 1 } { x } \\right ) . \\end{align*}"}
+{"id": "2586.png", "formula": "\\begin{align*} \\left . \\begin{aligned} & \\boldsymbol { y } _ t ^ { * } : = ( y _ t ^ { * , i } ) _ { 1 \\leq i \\leq N } : = ( P _ t x _ t ^ { * , i } + \\varphi _ t ^ { * , i } ) _ { 1 \\leq i \\leq N } , \\\\ & \\boldsymbol { z } _ t ^ { * } : = ( z _ t ^ { * , i , j } ) _ { 1 \\leq i , j \\leq N } : = ( P _ t \\sigma + \\Lambda _ t ^ { * , i , j } ) _ { 1 \\leq i , j \\leq N } , \\boldsymbol { z } _ t ^ { 0 , * } : = ( z _ t ^ { 0 , * , i } ) _ { 1 \\leq i \\leq N } : = ( P _ t \\sigma _ 0 + \\Lambda _ t ^ { 0 , * , i } ) _ { 1 \\leq i \\leq N } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7910.png", "formula": "\\begin{align*} ( \\delta _ \\mathrm { H o c h } f ) ( a _ 1 , \\ldots , a _ { n + 1 } ) = ~ & ( - 1 ) ^ { n + 1 } ~ a _ 1 \\cdot f ( a _ 2 , \\ldots , a _ { n + 1 } ) + f ( a _ 1 , \\ldots , a _ n ) \\cdot a _ { n + 1 } \\\\ ~ & + \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i + n + 1 } ~ f ( a _ 1 , \\ldots , a _ i \\cdot a _ { i + 1 } , \\ldots , a _ { n + 1 } ) , \\end{align*}"}
+{"id": "8207.png", "formula": "\\begin{align*} \\begin{aligned} \\Big \\| \\Lambda ^ \\alpha ( u v ) - \\sum _ { | k | < \\alpha _ 1 } \\frac { 1 } { k ! } \\partial ^ k u \\ , \\Lambda ^ { \\alpha , k } v - \\sum _ { | m | \\le \\alpha _ 2 } \\frac { 1 } { m ! } \\partial ^ { m } v \\ , \\Lambda ^ { \\alpha , m } u \\Big \\| _ { L ^ p } \\lesssim _ { \\alpha , \\alpha _ 1 , \\alpha _ 2 , p , N } \\| \\Lambda ^ { \\alpha _ 1 } u \\| _ { L ^ p } \\| \\Lambda ^ { \\alpha _ 2 } v \\| _ { \\operatorname { B M O } } , \\end{aligned} \\end{align*}"}
+{"id": "3883.png", "formula": "\\begin{align*} ( 1 + \\delta ) \\| T ^ \\ast \\omega \\| _ { \\varphi _ 1 } ^ 2 + \\| S \\omega \\| _ { \\varphi _ 3 } ^ 2 \\ge \\int _ { \\Omega _ \\varepsilon } \\sum _ { j , k = 1 } ^ n \\left ( \\frac { \\partial ^ 2 \\varphi } { \\partial z _ j \\partial \\bar { z } _ k } + \\frac 1 2 \\frac { \\partial ^ 2 \\kappa \\circ s } { \\partial z _ j \\partial \\bar { z } _ k } \\right ) \\omega _ j \\bar { \\omega } _ k e ^ { - \\varphi ' } . \\end{align*}"}
+{"id": "1107.png", "formula": "\\begin{align*} \\min _ { \\xi , \\eta } p ( \\xi , \\eta ) = \\min \\left \\{ \\min _ { \\xi } p \\left ( \\xi , \\frac 1 2 \\right ) , \\min _ { \\xi } p \\left ( \\xi , \\frac { 9 9 } { 1 0 0 } \\right ) \\right \\} . \\end{align*}"}
+{"id": "7196.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { F } } K _ T ( \\mathcal { D T } ( d ) ) \\otimes _ { \\mathbb { K } } \\mathbb { F } = p _ 3 ( d ) . \\end{align*}"}
+{"id": "7779.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow + \\infty } \\frac { c a p _ p ( B ^ { g ^ + } _ q ( t ) ) } { e ^ { n t } } = \\frac { 1 } { 2 ^ n } ( \\frac { n } { p - 1 } ) ^ { p - 1 } \\mathcal { A } ( q ) \\end{align*}"}
+{"id": "973.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ G \\partial _ j \\zeta ( 0 , \\omega ) \\ ; \\overline { f ( 0 , \\omega ) } d \\mu ( \\omega ) & = \\int _ { G ^ \\wedge } 2 \\pi i \\ ; y _ j ( \\xi ) \\widehat { \\zeta ( 0 , \\xi ) } \\ ; \\overline { \\widehat { f ( 0 , \\xi ) } } d \\nu ( \\xi ) \\\\ [ 5 p t ] & = - \\int _ { G ^ \\wedge } \\widehat { \\zeta ( 0 , \\xi ) } \\ ; \\overline { 2 \\pi i \\ ; y _ j ( \\xi ) \\widehat { f ( 0 , \\xi ) } } d \\nu ( \\xi ) . \\end{aligned} \\end{align*}"}
+{"id": "6595.png", "formula": "\\begin{align*} \\mathbb { E } [ | X _ t ( x ) | ^ 2 ] = | \\exp ( t \\tilde A ) \\exp ( \\frac { 1 } { 2 } ( t \\alpha - t ^ 2 \\beta + ( t ^ 3 - p _ \\Gamma t ) \\Gamma ) ) x | ^ 2 . \\end{align*}"}
+{"id": "7681.png", "formula": "\\begin{align*} P ^ n - c ^ { n / 2 } V ^ { n - 1 } = P ^ n - n ^ n \\omega _ n V ^ { n - 1 } \\ge ( n - 1 ) ^ n V ^ n . \\end{align*}"}
+{"id": "3733.png", "formula": "\\begin{align*} \\sum _ { \\mathfrak f ( x ) \\in \\operatorname { D e c k } ( p ) } n _ { \\mathfrak f } \\cdot \\mathfrak f ( x ) = p ^ \\prime ( x ) . \\end{align*}"}
+{"id": "1055.png", "formula": "\\begin{align*} \\mathcal { W } ( f V W \\Delta ^ { - n - 1 } ) = \\mathcal { g } ^ { \\Delta } ( f V , W ) = \\mathcal { g } ^ { \\Delta } ( V , f W ) , \\end{align*}"}
+{"id": "5364.png", "formula": "\\begin{align*} A B : = I _ { 0 1 } , A C : = I _ { 0 2 } , B C : = I _ { 1 2 } , \\end{align*}"}
+{"id": "5254.png", "formula": "\\begin{align*} \\hat { f } ( \\xi ) = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } } \\ * \\int _ { \\mathbb { R } ^ n } f ( x ) e ^ { - i \\ * \\xi \\cdot x } \\ * d x . \\end{align*}"}
+{"id": "7651.png", "formula": "\\begin{align*} F ( \\kappa ) : = \\inf \\left \\{ \\ , P ( A ) - \\kappa | A | \\ , : \\ , A \\subset \\Omega , \\ , | A | \\ge \\pi R ^ 2 \\ , \\right \\} . \\end{align*}"}
+{"id": "3843.png", "formula": "\\begin{align*} \\phi ( \\eta ) = \\left ( 4 + i _ { h - 4 } , \\dots , h - 1 + i _ 1 , \\frac { n - h ^ 2 + 2 h + 1 1 - d } { 2 } - i \\right ) . \\end{align*}"}
+{"id": "2871.png", "formula": "\\begin{align*} d x = b ( x ) d \\tau + \\sigma ( x ) d \\beta ( \\tau ) , \\ ; \\ ; \\ ; x \\in \\mathbb { R } ^ l , \\ ; \\tau \\geqslant 0 , \\end{align*}"}
+{"id": "2363.png", "formula": "\\begin{align*} s _ { k } = \\sum _ { f ( j ) \\leq k } ( b _ { j } - 1 ) \\end{align*}"}
+{"id": "85.png", "formula": "\\begin{align*} g ( \\nabla \\Psi , \\nabla X ( u ) ) = D _ { X } g ( \\nabla \\Psi , \\nabla u ) . \\end{align*}"}
+{"id": "750.png", "formula": "\\begin{align*} V i r = \\mathbb { C } [ \\partial ] L , [ L _ \\lambda L ] = ( \\partial + 2 \\lambda ) L . \\end{align*}"}
+{"id": "605.png", "formula": "\\begin{align*} \\begin{aligned} & j ^ { ( 1 ) } \\geq j ^ { ( 2 ) } \\geq j ^ { ( 4 ) } \\ , , j ^ { ( 0 ) } \\geq j ^ { ( 4 ) } \\ , , j ^ { ( i ) } \\geq 0 \\ , , i = 0 , 1 , . . . , 4 \\ , , \\\\ & j ^ { ( 1 ) } + j ^ { ( 2 ) } + j ^ { ( 0 ) } + j ^ { ( 4 ) } \\geq j ^ { ( 3 ) } \\geq j ^ { ( 1 ) } + j ^ { ( 2 ) } + j ^ { ( 0 ) } - j ^ { ( 4 ) } \\ , . \\end{aligned} \\end{align*}"}
+{"id": "5889.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } \\ddot { x } = \\dot { x } \\times \\frac { B ( x ) } { \\epsilon } + F ( x ) , x ( 0 ) = x _ 0 , \\dot { x } ( 0 ) = \\dot { x } _ 0 . \\end{array} \\end{align*}"}
+{"id": "6689.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ { \\eta } \\lambda _ { i , 1 } ^ { 2 } s _ { i , - k - 1 } = \\frac { 1 } { \\prod _ { j = 0 } ^ { k - 1 } \\lambda _ { - j } ^ { 2 } } , k \\in \\mathbb { N } \\cap [ 0 , \\kappa - 1 ] , \\\\ & \\sum _ { i = 1 } ^ { \\eta } \\lambda _ { i , 1 } ^ { 2 } s _ { i , - \\kappa - 1 } \\le \\frac { 1 } { \\prod _ { j = 0 } ^ { \\kappa - 1 } \\lambda _ { - j } ^ { 2 } } , \\\\ & \\sup _ { i \\in \\mathbb { N } \\cap [ 1 , \\eta ] } - \\frac { Q _ { i } ^ { ( K _ i - 1 ) } ( 0 ) } { Q _ { i } ^ { ( K _ i ) } ( 0 ) } < \\infty , \\end{align*}"}
+{"id": "2030.png", "formula": "\\begin{align*} \\Phi _ 1 ( \\phi , z ) : = \\frac { \\widehat { \\phi } ( z ) - \\widehat { \\phi } ( 0 ) } { z } = \\int _ { - a } ^ { a } \\phi ( t ) \\frac { e ^ { i z t } - 1 } { z } \\ , d t . \\end{align*}"}
+{"id": "2739.png", "formula": "\\begin{align*} \\mu _ H ^ { { \\rm m a x } } ( { \\rm S y m ^ m } E ) = \\mu _ H ( { \\rm S y m ^ m } E ) = \\frac { m c _ 1 ( E ) \\cdot H ^ { n - 1 } } { r } , \\end{align*}"}
+{"id": "1886.png", "formula": "\\begin{align*} \\lim _ { j \\rightarrow \\infty } \\lim _ { n \\rightarrow \\infty } \\max \\{ | 2 | ^ { k } \\mu ( 0 , \\ 2 ^ { k + 1 } x ) ; j \\leq k < n + j \\} = 0 \\end{align*}"}
+{"id": "84.png", "formula": "\\begin{align*} [ \\Delta , X ] = 0 . \\end{align*}"}
+{"id": "530.png", "formula": "\\begin{align*} f ( x ) \\le a - b | x | ^ 2 , x \\in \\R ^ n , \\ a : = f ( 0 ) + \\kappa ^ { - 1 } | \\nabla f ( 0 ) | ^ 2 , \\ b : = \\kappa / 4 . \\end{align*}"}
+{"id": "5617.png", "formula": "\\begin{align*} \\begin{array} { l l } \\Gamma ^ { \\alpha } _ { \\beta ; \\mu } : = G ^ { \\bar { \\tau } \\alpha } \\delta _ { \\mu } \\left ( G _ { \\beta \\bar { \\tau } } \\right ) , & C ^ { \\alpha } _ { \\beta \\gamma } : = G ^ { \\bar { \\tau } \\alpha } G _ { \\beta \\bar { \\tau } \\gamma } . \\end{array} \\end{align*}"}
+{"id": "1946.png", "formula": "\\begin{align*} \\begin{aligned} & | \\eta _ j | \\le 1 , | D _ v \\eta _ j | \\le N 2 ^ { - j } R ^ { - 1 } , | D ^ 2 _ v \\eta _ j | \\le N 2 ^ { - 2 j } R ^ { - 2 } , \\\\ & | D _ x \\eta _ j | \\le N 2 ^ { - 3 j } R ^ { - 3 } , | \\partial _ t \\eta _ j | \\le N 2 ^ { - 2 j } R ^ { - 2 } . \\end{aligned} \\end{align*}"}
+{"id": "1305.png", "formula": "\\begin{align*} 0 _ { a ' , b } \\circ f = 0 _ { a , b } , f \\circ 0 _ { b , a } = 0 _ { b , a ' } , \\end{align*}"}
+{"id": "7123.png", "formula": "\\begin{align*} \\chi + \\rho + \\delta = - r ( 3 \\mathfrak { g } ^ { \\lambda > 0 } ) + \\psi , \\end{align*}"}
+{"id": "3068.png", "formula": "\\begin{align*} f _ { j } ( d \\ell _ j ) - f _ { j } ( \\ell _ j ) = f _ { j } ( d \\ell _ j ) - q _ j \\ell _ j < 0 \\end{align*}"}
+{"id": "55.png", "formula": "\\begin{align*} \\phi _ { t } ^ { \\ast } g = g . \\end{align*}"}
+{"id": "5525.png", "formula": "\\begin{align*} \\dim H _ l ( H , ( \\rho _ j / \\rho _ { j , k } ) \\otimes \\chi ^ { \\vee } ) \\le \\sum _ { a = 0 } ^ { k - 1 } d _ { l , j , a } . \\end{align*}"}
+{"id": "2149.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\mathrm { d i v } \\ ; \\int _ { Y ^ m } \\sigma \\left ( \\vec { x } , \\vec { y } , \\nabla u ( \\vec { x } ) + { \\bf R } ^ T \\nabla _ { \\vec { y } } u _ 1 ( \\vec { x } , \\vec { y } ) \\right ) \\ ; \\mathrm { d } \\vec { y } & = f ( \\vec { x } ) \\ ; , \\vec { x } \\in \\Omega \\\\ \\left . u \\right | _ { \\partial \\Omega } & = 0 \\end{aligned} \\right . \\end{align*}"}
+{"id": "6725.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { \\mu } } L _ { M } ( \\sqrt { \\mu } f ) = \\mathcal { L } f + \\Gamma ( \\frac { M - \\mu } { \\sqrt { \\mu } } , f ) + \\Gamma ( f , \\frac { M - \\mu } { \\sqrt { \\mu } } ) . \\end{align*}"}
+{"id": "3444.png", "formula": "\\begin{align*} \\Delta \\mathrm { E } + \\omega ^ 2 \\mu _ \\mathrm { m } \\varepsilon \\mathrm { E } = 0 \\mathbb { R } ^ 2 , \\end{align*}"}
+{"id": "3607.png", "formula": "\\begin{align*} \\eta _ { i j } = ( i j ) ^ { - \\min \\left ( 1 , \\tfrac { 1 + \\delta } { 2 } \\right ) } . \\end{align*}"}
+{"id": "2496.png", "formula": "\\begin{align*} \\xi _ { t } = h ( \\xi ) \\end{align*}"}
+{"id": "2090.png", "formula": "\\begin{align*} M Q = P \\begin{bmatrix} I _ k \\\\ 0 \\\\ D \\end{bmatrix} , \\end{align*}"}
+{"id": "6966.png", "formula": "\\begin{align*} \\vec { r } = \\vec { c } + \\vec { e } \\end{align*}"}
+{"id": "4459.png", "formula": "\\begin{align*} \\varphi ( C _ 1 ) & = x _ 1 + x _ 2 , \\\\ \\varphi ( C _ 2 ) & = ( x _ 1 + x _ 2 ) ( x _ 1 - x _ 2 + 1 ) , \\\\ \\varphi ( C _ 3 ) & = ( x _ 1 + x _ 2 ) ( x _ 1 ^ 2 - x _ 1 x _ 2 + x _ 2 ^ 2 + x _ 1 - 2 x _ 2 + 1 ) \\\\ & = \\varphi ( C _ 1 ) ( \\frac { 3 \\varphi ( C _ 2 ) ^ 2 } { 4 \\varphi ( C _ 1 ) ^ 2 } + \\frac { \\varphi ( C _ 1 ) ^ 2 } { 4 } - \\frac { \\varphi ( C _ 1 ) } { 2 } + \\frac { 1 } { 4 } ) , \\\\ \\dots \\end{align*}"}
+{"id": "5127.png", "formula": "\\begin{align*} \\Vert D u \\Vert ( \\Omega ) = \\sup \\left \\{ \\int _ { \\Omega } u \\ , ( v ) \\ , d x \\ , : \\ , v \\in C ^ { \\infty } _ { 0 } ( \\Omega ; \\R ^ n ) , \\ , | v | \\leq 1 \\right \\} \\end{align*}"}
+{"id": "3510.png", "formula": "\\begin{align*} \\Bigg | \\Big \\langle \\widehat { \\nabla \\mathrm { H } } ^ { \\textbf { i n } } ; \\mathrm { e } ^ { ( 2 ) } _ { \\mathrm { n } _ { 0 } } \\Big \\rangle _ { \\mathbb { L } ^ { 2 } ( \\mathrm { B } ) } \\Bigg | ^ 2 = \\mathcal { O } \\Big ( \\delta ^ 2 \\Big ) . \\end{align*}"}
+{"id": "2103.png", "formula": "\\begin{align*} D _ x A ^ { 1 1 } D _ x ^ { - 1 } + D _ x B ^ { 1 1 } = C ^ { 1 1 } . \\end{align*}"}
+{"id": "3046.png", "formula": "\\begin{align*} F _ { \\N } ( \\sigma _ 1 ^ k ) = F _ { \\N } ( \\sigma _ 2 ^ k ) = F _ { \\N } ( \\sigma _ 3 ^ k ) = F _ { \\N } ( \\sigma _ 4 ^ k ) = \\begin{cases} 0 & \\\\ \\infty & \\end{cases} \\end{align*}"}
+{"id": "330.png", "formula": "\\begin{align*} \\tilde { g } _ { i } ( t + 1 ) = \\mathcal { F } _ { h ( t + 1 ) } ^ { - 1 } \\circ \\mathcal { F } _ { h ( t ) } \\left ( g _ { i } ( t ) \\right ) , \\end{align*}"}
+{"id": "6348.png", "formula": "\\begin{align*} q _ j \\| f _ j \\| \\ ; \\| \\tfrac { 1 } { A } f _ j + \\epsilon _ { 2 } ^ { j } h _ j \\| = q _ j \\sqrt { \\frac { 1 } { A ^ 2 } \\| f _ j \\| ^ 4 + ( \\epsilon _ { 2 } ^ { j } ) ^ 2 \\| f _ j \\| ^ 2 \\| h _ j \\| ^ 2 + \\frac { 2 } { A } \\epsilon _ { 2 } ^ { j } \\| f _ i \\| ^ 2 R e ( \\langle f _ j , h _ j \\rangle ) } \\ ; \\ ; < \\ ; q _ j \\frac { 1 } { A } \\| f _ i \\| ^ 2 < m a x _ { i = 1 } ^ N \\frac { q _ i } { A } \\| f _ i \\| ^ 2 . \\end{align*}"}
+{"id": "6211.png", "formula": "\\begin{align*} M ^ { 0 , 0 } _ { \\frac { 2 m - i - k } { 2 } , \\frac { i + k - 2 } { 2 } } M ^ { \\frac { k - 2 } { 2 } , \\frac { k - 2 } { 2 } } _ { \\frac { i + k - 2 } { 2 } , \\frac { 2 m - i } { 2 } } = \\frac { k } { 2 } M ^ { \\frac { 2 m - k } { 2 } , \\frac { 2 m - i - k } { 2 } } _ { \\frac { 2 m - i - k } { 2 } , \\frac { 2 m - i } { 2 } } \\end{align*}"}
+{"id": "1030.png", "formula": "\\begin{align*} h _ k \\dot { \\alpha } ( t _ { k - 1 } ) \\le \\frac { r } { c } ( k = k _ 0 + 1 , k _ 0 + 2 , \\ldots ) . \\end{align*}"}
+{"id": "454.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } { n \\choose k } L _ { j k } L _ { j ( n - k ) } = 2 ^ n L _ { j n } + 2 L _ j ^ n , \\end{align*}"}
+{"id": "307.png", "formula": "\\begin{align*} f : X ^ { \\prime } = X _ { s } \\xrightarrow { f _ { s } } X _ { s - 1 } \\xrightarrow { f _ { s - 1 } } \\cdots \\xrightarrow { f _ { 1 } } X _ { 0 } = X , \\end{align*}"}
+{"id": "2054.png", "formula": "\\begin{align*} \\left ( \\sum _ { n = 0 } ^ \\infty a ( n ) q ^ n \\right ) \\ | \\ U ( j ) = \\sum _ { n = 0 } ^ \\infty a ( j n ) q ^ n . \\end{align*}"}
+{"id": "6195.png", "formula": "\\begin{align*} | \\mathcal { I } ' _ k | = \\sum ^ k _ { i = 0 } | \\mathcal { B } _ { i , i } | + \\sum ^ k _ { i = 1 } \\sum ^ i _ { l = 1 } | \\mathcal { B } _ { i , i - l } | + \\sum ^ k _ { i = 1 } \\sum ^ i _ { l = 1 } | \\mathcal { B } _ { i - l , i } | . \\end{align*}"}
+{"id": "5932.png", "formula": "\\begin{align*} \\widehat { f } ( \\xi ) : = \\int _ { \\Q _ q ^ k } f ( x ) \\chi ( - x \\cdot \\xi ) d x \\end{align*}"}
+{"id": "5017.png", "formula": "\\begin{align*} w ( x ) : = \\begin{cases} \\| x \\| ^ { - \\frac 3 2 } & \\| x \\| \\leq 1 \\\\ e ^ { - \\| x \\| } & \\| x \\| > 1 \\end{cases} , \\end{align*}"}
+{"id": "621.png", "formula": "\\begin{align*} \\widetilde { \\lambda } \\ , ' ( m ) \\ r _ n ( m ) = E ' _ { n } \\ \\widetilde { r } \\ , ' _ { n + 1 } ( m ) + F ' _ n \\ \\widetilde { r } \\ , ' _ { n } ( m ) + G \\ , ' _ n \\ \\widetilde { r } \\ , ' _ { n - 1 } ( m ) \\ , , \\end{align*}"}
+{"id": "7157.png", "formula": "\\begin{align*} \\ast = p _ { \\ast } q ^ { \\ast } \\colon D ^ b ( \\mathcal { C } ( d _ 1 ) ) \\boxtimes D ^ b ( \\mathcal { C } ( d _ 2 ) ) \\to D ^ b ( \\mathcal { C } ( d ) ) \\end{align*}"}
+{"id": "6392.png", "formula": "\\begin{align*} ( q ^ { r / 2 } M ^ { ( \\omega ) } ( \\alpha ) ) ^ \\dagger : = q ^ { - r / 2 } M ^ { ( \\omega ) } ( \\alpha ) \\end{align*}"}
+{"id": "6835.png", "formula": "\\begin{align*} 0 & = \\sum _ { v \\in M _ { K } } \\sum _ { i = 1 } ^ n N _ { v } \\log ( \\abs { u _ { i } } _ { v } ) \\\\ & = \\sum _ { v \\in M _ { K } } \\sum _ { i = 1 } ^ n N _ { v } ( m _ { i , v } + M _ { i , v } ) \\\\ & \\le \\sum _ { v \\in M _ { K } } N _ { v } ( m _ { v } + n M _ { v } ) , \\end{align*}"}
+{"id": "6284.png", "formula": "\\begin{align*} \\Psi ^ { - 1 } \\cdot \\tilde { f } ( \\mathbf { m } ) = \\tilde { f } _ * \\big ( \\Psi ^ { - 1 } \\mathbf { m } \\big ) . \\end{align*}"}
+{"id": "7713.png", "formula": "\\begin{align*} \\mathcal { A } ( p ) = 2 ^ n \\cdot \\lim \\limits _ { t \\rightarrow + \\infty } e ^ { - n t } \\cdot V o l ( \\partial B _ p ^ { g ^ + } ( t ) , g ^ + ) \\end{align*}"}
+{"id": "5311.png", "formula": "\\begin{align*} Z _ a ( x ) : = \\sum _ { \\zeta ( \\rho ) = 0 , \\rho = \\beta + i \\gamma \\atop a < \\Re \\rho = \\beta , | \\gamma | < x } \\frac { x ^ \\beta } { | \\rho | } , Z ( x ) : = \\sum _ { \\zeta ( \\rho ) = 0 , \\rho = \\beta + i \\gamma \\atop \\theta < \\Re \\rho = \\beta , | \\gamma | < x } \\frac { x ^ \\beta } { | \\rho | } . \\end{align*}"}
+{"id": "6099.png", "formula": "\\begin{align*} \\textbf { t } ^ { \\alpha _ 1 } F _ { \\alpha _ 1 } + \\cdots + \\textbf { t } ^ { \\alpha _ l } F _ { \\alpha _ l } = \\sum \\limits _ { j = 1 } ^ { r } G _ j \\left ( x _ j \\textbf { t } ^ { \\beta _ j } g _ j - f _ j \\textbf { z } ^ { \\beta _ j } \\textbf { t } ^ { \\beta _ j } \\right ) + { \\sum \\limits _ { i = 1 } ^ { s } H _ i ( { y _ i } ^ { q } - y _ i ) } + H ( h w - 1 ) , \\end{align*}"}
+{"id": "6318.png", "formula": "\\begin{align*} \\hat { X } ^ k _ t = x _ 0 ( t ) + \\int _ 0 ^ t K _ \\mu ( s , t ) \\mu _ { n _ k } ( s , \\hat { X } ^ k _ s ) \\dd s + \\int _ 0 ^ t K _ \\sigma ( s , t ) \\dd \\hat { M } ^ k _ s , t \\in [ 0 , T ] , \\tilde { \\P } \\end{align*}"}
+{"id": "378.png", "formula": "\\begin{align*} [ u , w ] = \\big [ u , { \\textstyle \\sum _ i } f _ i \\underline \\xi _ i \\big ] = \\sum _ { i = 1 } ^ k \\ , [ u , f _ i \\underline \\xi _ i ] = \\sum _ { i = 1 } ^ k \\ ; f _ i [ u , { \\underline \\xi } _ i ] + ( u f ) \\underline \\xi _ i ~ . \\end{align*}"}
+{"id": "5315.png", "formula": "\\begin{align*} [ k ] _ { t } ! = [ k ] _ { t } [ k - 1 ] _ { t } \\dotsb [ 2 ] _ { t } [ 1 ] _ { t } . \\end{align*}"}
+{"id": "4826.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { ( \\R ^ d ) ^ 2 } \\int _ { S ^ 2 } ( | x _ 1 - x _ 2 | ^ 2 + | s _ 1 - s _ 2 | ^ 2 ) \\ , \\dd \\Pi _ t ( x _ 1 , s _ 1 , x _ 2 , s _ 2 ) \\\\ & \\leq e ^ { 3 L t } \\int _ { ( \\R ^ d ) ^ 2 } \\int _ { S ^ 2 } ( | x _ 1 - x _ 2 | ^ 2 + | s _ 1 - s _ 2 | ^ 2 ) \\ , \\dd \\Pi _ 0 ( x _ 1 , s _ 1 , x _ 2 , s _ 2 ) + K \\frac { e ^ { 3 L t } - 1 } { 3 L } W _ 2 ^ 2 ( \\mu _ 1 , \\mu _ 2 ) , \\end{aligned} \\end{align*}"}
+{"id": "1069.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathcal { g } _ { D } ( v , w ) = 2 ^ { m } v _ { n - 1 } \\int _ { M } g ( v , w ) ~ v o l _ { g } , \\\\ & \\mathcal { G } _ { D } ( v , w ) = 2 ^ { m } \\frac { v _ { n - 1 } } { 6 } \\int _ { M } G ( v , w ) ~ v o l _ { g } , \\end{aligned} \\end{align*}"}
+{"id": "917.png", "formula": "\\begin{align*} F ^ { i j } = \\frac { \\partial F ( D ^ 2 u ) } { \\partial u _ { i j } } . \\end{align*}"}
+{"id": "6815.png", "formula": "\\begin{align*} \\partial _ t u - \\omega \\kappa \\Delta u - ( 1 - \\omega ) \\kappa \\ , \\int _ 0 ^ \\infty g ( s ) \\Delta u ( t - s ) d s = 0 , \\end{align*}"}
+{"id": "3594.png", "formula": "\\begin{align*} \\| \\Xi \\| & = \\left \\Vert T ( T ^ { 1 / 2 } C T ^ { 1 / 2 } + \\lambda I ) ^ { - 2 } T \\right \\Vert \\\\ & = \\sup _ { i } \\frac { \\mu _ i ^ 2 } { ( \\mu _ i \\xi _ i + \\lambda ) ^ 2 } \\lesssim \\sup _ i \\frac { i ^ { - 2 t } } { ( i ^ { - ( t + c ) } + \\lambda ) ^ 2 } = \\left [ \\sup _ i \\frac { i ^ { - t } } { i ^ { - ( t + c ) } + \\lambda ) } \\right ] ^ 2 \\\\ & \\leq \\left [ \\lambda ^ { \\tfrac { t - ( t + c ) } { t + c } } \\right ] ^ 2 = \\lambda ^ { - \\tfrac { - 2 c } { t + c } } , \\end{align*}"}
+{"id": "1203.png", "formula": "\\begin{align*} | \\mathcal { U } ( s _ i ) | = 2 ^ { \\frac { s _ i ( s _ i - 1 ) } { 2 } } , | \\mathcal { P } ( s _ i ) | = s _ i ! ~ . \\end{align*}"}
+{"id": "5614.png", "formula": "\\begin{align*} \\mathbb { G } ^ { \\alpha } = \\frac 1 2 \\Gamma ^ { \\alpha } _ { ; \\beta } v ^ { \\beta } . \\end{align*}"}
+{"id": "3399.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\ell r _ j s _ { g - h _ j } = 0 . \\end{align*}"}
+{"id": "189.png", "formula": "\\begin{align*} Q ^ + _ { A _ 0 } \\psi + \\pi ^ - ( a \\cdot \\psi ) = 0 , \\ , \\ , \\ , d ^ { + } a + F ^ + _ { A _ 0 } = \\rho ^ { - 1 } ( \\mu ( \\psi ) ) . \\end{align*}"}
+{"id": "7548.png", "formula": "\\begin{align*} A ( z ) = A _ 0 ( z ) A _ 1 ( z ) \\cdots A _ { p - 1 } ( z ) . \\end{align*}"}
+{"id": "5846.png", "formula": "\\begin{align*} b e r ( S ) = 0 . \\end{align*}"}
+{"id": "6821.png", "formula": "\\begin{align*} \\varepsilon \\partial _ { t t t } u _ 1 + \\partial _ { t t } u _ 1 - \\varepsilon \\omega \\kappa \\Delta \\partial ^ 2 _ t u _ 1 - ( \\kappa + \\varepsilon \\kappa _ 1 ) \\Delta \\partial _ t u _ 1 - \\kappa _ 1 \\Delta u _ 1 = 0 . \\end{align*}"}
+{"id": "2396.png", "formula": "\\begin{align*} \\frac { d } { d \\alpha } f ( \\alpha ) = 0 . \\end{align*}"}
+{"id": "426.png", "formula": "\\begin{align*} K ( \\lambda ) K ( \\mu ) = \\sum _ { \\nu \\in \\mathcal { P } ^ { } } f _ { \\lambda , \\mu } ^ { \\nu } ( t ) K ( \\nu ) . \\end{align*}"}
+{"id": "6626.png", "formula": "\\begin{align*} \\Omega _ { \\theta } = \\{ ( x , y ) \\in \\mathbb { R } ^ { 2 } : x < y \\tan \\theta \\} . \\end{align*}"}
+{"id": "3322.png", "formula": "\\begin{align*} \\mathbf { N } = \\{ ( l _ 1 , l _ 2 ) \\colon l _ 1 = 0 , \\dots , N _ 1 l _ 2 = 0 , \\dots , N _ 2 \\} . \\end{align*}"}
+{"id": "3013.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = f ( x , u ) \\end{aligned} \\end{align*}"}
+{"id": "2872.png", "formula": "\\begin{align*} \\begin{cases} | \\mathcal { K } ( c ' , x ) | \\leqslant ( c ' + 1 ) C , & | x | < 1 , \\\\ \\mathcal { K } ( c ' , x ) \\leqslant ( - \\alpha + \\frac { \\beta } { | x | } + \\frac { C } { | x | } + c ' C ) , & | x | \\geqslant 1 , \\end{cases} \\end{align*}"}
+{"id": "5560.png", "formula": "\\begin{align*} L _ i ( u _ i ) \\eta ^ { \\nu } u _ i ' ( \\eta ) = v _ i ( \\eta ) , \\ ; \\ ; i = 1 , 2 \\end{align*}"}
+{"id": "1600.png", "formula": "\\begin{align*} V ^ { \\prime } = T _ { s _ { i } } ^ { - 1 } \\left ( 0 \\right ) \\cap T _ { s _ { j } ^ { \\prime } } ^ { - 1 } \\left ( 0 \\right ) \\cap T _ { s _ { k } ^ { \\prime \\prime } } ^ { - 1 } \\left ( 0 \\right ) = T _ { ( s _ { i } , s _ { j } ^ { \\prime } , s _ { k } ^ { \\prime \\prime } ) } ^ { - 1 } \\left ( 0 \\right ) . \\end{align*}"}
+{"id": "1056.png", "formula": "\\begin{align*} \\mathcal { W } ( f V W \\Delta ^ { - n } ) = \\mathcal { G } ^ { \\Delta } ( f V , W ) = \\mathcal { G } ^ { \\Delta } ( V , f W ) \\end{align*}"}
+{"id": "280.png", "formula": "\\begin{align*} H ^ { 1 } ( K , \\mathbf { Q } / \\mathbf { Z } ) = \\varinjlim _ { m } H ^ { 1 } ( K , \\mathbf { Z } / m \\mathbf { Z } ) . \\end{align*}"}
+{"id": "3063.png", "formula": "\\begin{align*} f ( \\ell ) = \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } k g ( k ) = \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } k C _ { \\sigma } ( k ) = F _ X ( \\sigma ^ { \\ell } ) . \\end{align*}"}
+{"id": "6538.png", "formula": "\\begin{align*} U : = \\left ( 2 \\widehat { M } \\widehat { M } ^ T - I \\right ) \\left ( 2 \\widehat { N } \\widehat { N } ^ T - I \\right ) , \\end{align*}"}
+{"id": "3478.png", "formula": "\\begin{align*} \\mathrm { U } _ { \\mathrm { e } } ( \\xi , \\mathrm { t } ) = - \\frac { \\gamma _ { \\mathrm { p } } } { \\gamma _ { \\mathrm { m } } } \\frac { 1 } { \\alpha _ \\mathrm { m } } \\int _ { 0 } ^ t \\int _ { \\partial \\Omega } \\Phi ( \\xi , t ; \\mathrm { z } , \\tau ) \\gamma ^ { \\textbf { i n t } } _ { 1 } \\mathrm { U } _ { \\mathrm { i } } ( \\mathrm { y } , \\tau ) d \\sigma _ \\mathrm { y } d \\tau + \\textbf { e r r } ^ { ( 1 ) } + \\textbf { e r r } ^ { ( 2 ) } . \\end{align*}"}
+{"id": "1945.png", "formula": "\\begin{align*} & \\mathsf { f } _ n = \\vec f \\phi _ n - ( a D _ v \\phi _ n ) u , \\\\ & \\mathsf { g } _ n = g \\phi _ n - \\vec f \\cdot D _ v \\phi _ n + u ( \\partial _ t \\phi _ n - v \\cdot D _ x \\phi _ n ) - ( a D _ v \\phi _ { n } ) \\cdot D _ v u , \\end{align*}"}
+{"id": "7392.png", "formula": "\\begin{align*} \\widehat { \\Psi } _ \\lambda : = - 2 i \\gamma _ D \\partial _ { \\overline { z } } ( - \\Delta - \\overline { \\lambda } ) ^ { - 1 } . \\end{align*}"}
+{"id": "1183.png", "formula": "\\begin{align*} ( \\mathcal { R } _ \\lambda ( \\sigma s _ i ) v _ T , v _ T ) = \\pm ( \\mathcal { R } _ \\lambda ( \\sigma ) v _ T , v _ T ) . \\end{align*}"}
+{"id": "7158.png", "formula": "\\begin{align*} \\ast = p _ { \\ast } q ^ { \\ast } \\colon D ^ b _ T ( \\mathcal { C } ( d _ 1 ) ) \\boxtimes D ^ b _ T ( \\mathcal { C } ( d _ 2 ) ) \\to D ^ b _ T ( \\mathcal { C } ( d ) ) . \\end{align*}"}
+{"id": "6200.png", "formula": "\\begin{align*} V = \\sum _ { ( \\mu , d ) \\in \\Upsilon } \\mathcal { W } _ { ( \\mu , d ) } \\ \\ \\ \\ \\ ( ) \\end{align*}"}
+{"id": "1072.png", "formula": "\\begin{align*} \\Delta _ { h } = \\sum _ { a = 1 } ^ { 4 } \\chi ^ { - 1 } \\cdot \\delta _ { a } \\cdot \\chi \\cdot \\delta _ { a } \\cdot \\chi ^ { - 1 } , \\end{align*}"}
+{"id": "1204.png", "formula": "\\begin{align*} | \\mathcal { A } _ { \\mathcal { U } } | = \\prod _ { i = 1 } ^ { t } 2 ^ { \\frac { s _ i ( s _ i - 1 ) } { 2 } } , | \\mathcal { A } _ { \\mathcal { P } } | = \\prod _ { i = 1 } ^ { t } s _ i ! ~ . \\end{align*}"}
+{"id": "5794.png", "formula": "\\begin{align*} { } \\displaystyle \\sum _ { n = 1 } ^ { \\infty } w _ n ^ { H } \\left ( \\sum _ { k = 1 } ^ { n } a _ k \\right ) ^ 2 \\leq \\sum _ { n = 1 } ^ { \\infty } { a _ n } ^ 2 , \\end{align*}"}
+{"id": "6740.png", "formula": "\\begin{align*} \\partial ^ { \\alpha - \\alpha _ { 1 } } \\nabla _ { v } F = \\partial ^ { \\alpha - \\alpha _ { 1 } } \\nabla _ { v } M + \\partial ^ { \\alpha - \\alpha _ { 1 } } \\nabla _ { v } \\overline { G } + \\sqrt { \\mu } \\partial ^ { \\alpha - \\alpha _ { 1 } } \\nabla _ { v } f - \\frac { v } { 2 } \\sqrt { \\mu } \\partial ^ { \\alpha - \\alpha _ { 1 } } f . \\end{align*}"}
+{"id": "3247.png", "formula": "\\begin{align*} \\int _ { X } \\psi ( w - \\phi ) \\theta ^ { n } _ { \\phi } = \\lim _ { k \\to \\infty } \\int _ { X } \\psi ( w _ { k } - \\phi ) \\theta ^ { n } _ { \\phi } \\leq \\int _ { X } \\psi ( v _ { k } - \\phi ) \\theta ^ { n } _ { \\phi } \\leq \\tilde { C } . \\end{align*}"}
+{"id": "4579.png", "formula": "\\begin{align*} \\theta ( n + 1 ) = \\theta ( n ) + k + \\sin ^ 2 \\pi \\theta ( n ) \\frac { V ( n ) } { \\pi \\sin \\pi k } + O \\left ( \\frac { V ( n ) ^ 2 } { \\sin ^ 2 \\pi k } \\right ) . \\end{align*}"}
+{"id": "6770.png", "formula": "\\begin{align*} H ^ { \\rm { B } } ( t ) & = \\int d x \\ , b ^ { * } _ x h ( t ) b _ x + \\mathcal { N } _ a + \\left ( \\int d x \\int d k \\ , K ( t , k , x ) \\big ( a _ k ^ { * } + a _ { - k } \\big ) b ^ * _ x + \\right ) . \\end{align*}"}
+{"id": "1616.png", "formula": "\\begin{align*} d \\left ( F \\left ( \\varphi _ { 1 } \\right ) , F \\left ( \\varphi _ { 2 } \\right ) \\right ) = \\lim _ { n \\rightarrow \\mathcal { U } } d \\left ( f _ { n } \\left ( \\varphi _ { 1 } \\right ) , f _ { n } \\left ( \\varphi _ { 2 } \\right ) \\right ) = 0 \\end{align*}"}
+{"id": "2048.png", "formula": "\\begin{align*} \\sum _ \\rho \\rho ^ { - m } = \\sum _ { k = 1 } ^ { m } ( - 1 ) ^ { k + 1 } \\binom { m } { k } \\lambda _ k \\end{align*}"}
+{"id": "2943.png", "formula": "\\begin{align*} T _ \\Omega ( \\bar z ) : = \\{ w \\in \\R ^ n \\ , | \\ , \\exists t _ k \\downarrow 0 , \\ , \\exists w _ k \\to w , \\ , \\forall k \\in \\N \\colon \\ , \\bar z + t _ k w _ k \\in \\Omega \\} \\end{align*}"}
+{"id": "7953.png", "formula": "\\begin{align*} \\left \\langle [ Z _ 1 , Z _ 2 ] , \\nu ^ H \\right \\rangle = \\frac { - 4 \\left \\langle \\nu , T \\right \\rangle } { | \\mathcal { P } _ H ( \\nu ) | } \\left \\langle J ( Z _ 1 ) , Z _ 2 \\right \\rangle \\quad \\mbox { f o r e v e r y } Z _ 1 , Z _ 2 \\in \\mathcal { H } \\cap T M . \\end{align*}"}
+{"id": "8007.png", "formula": "\\begin{align*} S _ { M \\otimes I _ d } ( \\{ 0 , d \\} ) & = S _ { M \\otimes I _ d } ( S _ { M \\otimes I _ d } ( N ) ; \\{ 0 , d \\} ) \\\\ S _ { M \\otimes I _ d } ( \\{ 1 , d + 1 \\} ) & = S _ { M \\otimes I _ d } ( S _ { M \\otimes I _ d } ( N ) ; \\{ 1 , d + 1 \\} ) \\\\ & \\vdots \\\\ S _ { M \\otimes I _ d } ( \\{ d - 1 , d + ( d - 1 ) \\} ) & = S _ { M \\otimes I _ d } ( S _ { M \\otimes I _ d } ( N ) ; \\{ d - 1 , d + ( d - 1 ) \\} ) \\\\ \\end{align*}"}
+{"id": "6666.png", "formula": "\\begin{align*} ( t _ { \\infty } ( \\mathbf { s } ) , + \\infty ) = \\bigcup _ { 0 < a < b } ( t _ { a , b } ( \\mathbf { s } ) , T _ { a , b } ( \\mathbf { s } ) ) \\end{align*}"}
+{"id": "6065.png", "formula": "\\begin{align*} a _ \\epsilon ( \\eta _ \\epsilon , v ) = \\int _ { \\Omega _ \\epsilon } \\nabla \\eta _ \\epsilon \\nabla v d x - \\int _ { \\Omega _ \\epsilon } k ^ 2 \\eta _ \\epsilon v d x \\end{align*}"}
+{"id": "6165.png", "formula": "\\begin{align*} V _ 2 V _ 1 ^ * ( 1 ) = M _ z M _ z ^ * R _ { \\overline q } ( 1 ) = 0 V _ 1 ^ * V _ 2 ( 1 ) = M _ z ^ * R _ { \\overline q } M _ z ( 1 ) = \\overline q . \\end{align*}"}
+{"id": "3504.png", "formula": "\\begin{align*} \\Big \\langle \\nabla \\mathrm { H } ; \\mathrm { e } ^ { ( 1 ) } _ { \\mathrm { n } } \\Big \\rangle _ { \\mathbb { L } ^ { 2 } ( \\Omega ) } = \\Big \\langle \\mathbb { T } \\Big [ \\nabla \\mathrm { H } \\Big ] ; \\ \\mathrm { e } ^ { ( 1 ) } _ \\mathrm { n } \\Big \\rangle _ { \\mathbb { L } ^ 2 ( \\Omega ) } . \\end{align*}"}
+{"id": "1434.png", "formula": "\\begin{align*} \\Lambda ( t _ 0 , x _ 0 ) & = \\{ ( t , x ) \\in ( 0 , T ) \\times \\Omega ; \\ , 0 < t < t _ 0 , | x - x _ 0 | < t _ 0 - t \\} \\\\ & = \\bigcup _ { t \\in ( 0 , t _ 0 ) } \\left ( \\{ t \\} \\times ( B _ { t _ 0 - t } ( x _ 0 ) \\cap \\Omega ) \\right ) ) . \\end{align*}"}
+{"id": "1889.png", "formula": "\\begin{align*} \\widetilde { \\chi } _ { i , j } ( x _ k ) = \\begin{cases} 0 , \\quad \\quad k \\neq i , \\\\ [ x _ i , x _ j ] , k = i . \\end{cases} \\end{align*}"}
+{"id": "6135.png", "formula": "\\begin{align*} \\left ( \\begin{bmatrix} R _ q \\otimes P ^ \\perp U + M _ z R _ q \\otimes P U & 0 \\\\ 0 & W _ 1 \\end{bmatrix} , \\begin{bmatrix} R _ { \\overline q } \\otimes U ^ * P + R _ { \\overline q } M _ z \\otimes U ^ * P ^ \\perp & 0 \\\\ 0 & W _ 2 \\end{bmatrix} \\right ) \\end{align*}"}
+{"id": "1499.png", "formula": "\\begin{align*} H ^ 2 W \\ge K ( \\Delta ) \\bigl ( 1 - \\sum _ { w < p < z } g ( p ) \\bigr ) - \\sum _ { p \\le w } g ( p ) ( \\log p ) ^ 2 \\sum _ { \\substack { d < \\Delta \\\\ ( d , p ) = 1 } } h ( d ) . \\end{align*}"}
+{"id": "1831.png", "formula": "\\begin{align*} L _ \\rhd ( \\{ x , z \\} ) [ y , t ] & = \\{ x , z \\} \\rhd [ y , t ] = x \\rhd [ z \\rhd y , t ] - [ z \\rhd ( x \\rhd y ) , t ] - [ x \\rhd ( z \\rhd t ) , y ] + z \\rhd [ x \\rhd t , y ] \\\\ & = L _ \\rhd ( x ) [ L _ \\rhd ( z ) y , t ] - [ L _ \\rhd ( z ) L _ \\rhd ( x ) y , t ] - [ L _ \\rhd ( x ) L _ \\rhd ( z ) t , y ] + L _ \\rhd ( z ) [ L _ \\rhd ( x ) t , y ] . \\end{align*}"}
+{"id": "5989.png", "formula": "\\begin{align*} U _ r ^ { 0 , b } ( t ) = X ( t ) - L _ r ^ { 0 , b } ( t ) + R _ r ^ { 0 , b } ( t ) , t \\geq 0 , \\end{align*}"}
+{"id": "889.png", "formula": "\\begin{align*} \\begin{aligned} 0 \\geq & S ^ { i j } _ k ( u _ 1 + B | x | ^ 2 ) _ { i j } = f _ 1 + 2 B \\sum _ i S ^ { i i } _ k \\\\ \\geq & - A f ^ { 1 - 1 / ( k - 1 ) } + 2 c _ 0 B S _ 1 ^ { 1 / ( k - 1 ) } S _ k ^ { 1 - 1 / ( k - 1 ) } \\\\ = & - A f ^ { 1 - 1 / ( k - 1 ) } + 2 c _ 0 B ( \\Delta u ) ^ { 1 / ( k - 1 ) } f ^ { 1 - 1 / ( k - 1 ) } . \\end{aligned} \\end{align*}"}
+{"id": "3086.png", "formula": "\\begin{align*} s _ n ( k ) = \\sum _ { \\substack { d | k \\\\ d \\geq n } } d . \\end{align*}"}
+{"id": "5659.png", "formula": "\\begin{align*} \\Box ^ \\perp r | _ q & = \\sum _ { i = 1 } ^ { 2 n - 2 } H ( r ) ( E _ i , E _ i ) | _ q = \\sum _ { i = 1 } ^ { 2 n - 2 } I _ \\gamma ( J _ i , J _ i ) \\leq \\sum _ { i = 1 } ^ { 2 n - 2 } I _ \\gamma ( W _ i , W _ i ) \\\\ & = \\frac { 1 } { ( s _ \\lambda ( r ) ) ^ 2 } \\int _ 0 ^ r \\Big ( ( 2 n - 2 ) ( s ' _ \\lambda ( t ) ) ^ 2 - \\mathbf { R i c } ^ \\perp ( T ) ( s _ \\lambda ( t ) ) ^ 2 \\Big ) d t . \\end{align*}"}
+{"id": "5647.png", "formula": "\\begin{align*} d ( \\tilde \\gamma ( 0 , s ) , \\tilde \\gamma ( l , s ) ) = d ( \\tilde \\gamma ( 0 , s ) , f ( \\tilde \\gamma ( 0 , s ) ) ) \\geq l . \\end{align*}"}
+{"id": "317.png", "formula": "\\begin{align*} F ( t ) - t = \\left ( \\frac { t _ { 2 } } { t _ { 1 } ^ { n } } , \\frac { t _ { 3 } } { t _ { 1 } ^ { p n } } \\right ) , \\end{align*}"}
+{"id": "416.png", "formula": "\\begin{align*} \\varphi _ k ( H , M ) = \\varphi ( H ) ^ k \\prod _ { P \\mid M } \\left ( 1 - \\frac { 1 } { | P | - 1 } + \\frac { 1 } { ( | P | - 1 ) ^ 2 } - \\cdots + \\frac { ( - 1 ) ^ { k - 1 } } { ( | P | - 1 ) ^ { k - 1 } } \\right ) \\end{align*}"}
+{"id": "1494.png", "formula": "\\begin{align*} \\sum _ d h ( d ) y _ d = 1 . \\end{align*}"}
+{"id": "2530.png", "formula": "\\begin{align*} T ( x , y , w , z ) = ( x + \\alpha , y + x , w + \\alpha , z + w ) \\end{align*}"}
+{"id": "3292.png", "formula": "\\begin{gather*} w ( j , k ) : = w ( j , k , \\alpha _ 0 , \\alpha _ 1 , \\alpha _ 2 , \\alpha _ 3 ; q ) = \\frac { \\rho \\big ( j , \\alpha , \\beta , \\gamma , \\delta ; q ^ 2 \\big ) } { h _ n \\big ( \\alpha , \\beta , \\gamma , \\delta : q ^ 2 \\big ) } . \\end{gather*}"}
+{"id": "5041.png", "formula": "\\begin{align*} H ^ { q } ( Y , f ^ { \\prime } _ { \\ast } R ^ { 1 } \\varepsilon _ { \\ast } ( K ^ { m } _ { X } ) \\otimes H ^ { l } ) = 0 \\end{align*}"}
+{"id": "6973.png", "formula": "\\begin{align*} \\operatorname { R a n } \\Phi = \\{ X \\in \\mathcal B ( \\mathcal H ) : P ^ * X P = X \\} = \\mathcal T ( P ) . \\end{align*}"}
+{"id": "5839.png", "formula": "\\begin{align*} b e r ( S ) = \\sup \\{ | \\langle S e _ i , e _ i \\rangle | : i \\in \\mathbb { Z } _ { + } \\} = \\sup \\{ | \\langle e _ { i + 1 } , e _ i \\rangle | : i \\in \\mathbb { Z } _ { + } \\} = 0 , \\end{align*}"}
+{"id": "1612.png", "formula": "\\begin{align*} = \\dim \\left ( \\sum _ { e \\in S } W _ { e } \\right ) , \\end{align*}"}
+{"id": "2804.png", "formula": "\\begin{align*} s _ i ^ 2 A ^ T u _ i ^ A = c _ i B ^ T B g _ i , \\\\ c _ i ^ 2 B ^ T u _ i ^ B = s _ i A ^ T A g _ i , \\end{align*}"}
+{"id": "5666.png", "formula": "\\begin{align*} 4 \\frac { \\partial ^ 2 r } { \\partial z ^ \\alpha \\partial z ^ \\beta } & = \\frac { \\partial ^ 2 r } { \\partial x ^ \\alpha \\partial x ^ \\beta } - \\frac { \\partial ^ 2 r } { \\partial x ^ { \\alpha + n } \\partial x ^ { \\beta + n } } - \\sqrt { - 1 } \\left ( \\frac { \\partial ^ 2 r } { \\partial x ^ { \\alpha + n } \\partial x ^ \\beta } + \\frac { \\partial ^ 2 r } { \\partial x ^ \\alpha \\partial x ^ { \\beta + n } } \\right ) . \\end{align*}"}
+{"id": "1990.png", "formula": "\\begin{align*} \\lambda _ 1 ( G ) + \\lambda _ 1 ( \\overline { G } ) = \\frac { \\sqrt { - 3 \\omega ^ 2 + ( 4 n - 2 ) \\omega + 1 } - \\omega } { 2 } + n - \\frac { 3 } { 2 } . \\end{align*}"}
+{"id": "328.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } \\dot { s } _ i = \\varepsilon z _ i - \\kappa s _ i - \\gamma ( f _ i - 1 ) \\\\ \\dot { f } _ i = s _ i \\\\ \\tau \\dot { v } _ i = f _ i - v _ i ^ { \\frac { 1 } { \\alpha } } \\\\ \\tau \\dot { q } _ i = \\frac { f _ i ( 1 - ( 1 - \\rho ) ^ { \\frac { 1 } { f _ i } } ) } { \\rho } - \\frac { v _ i ^ { \\frac { 1 } { \\alpha } } q _ i } { v _ i } \\\\ y _ i = V _ 0 [ k _ 1 ( 1 - q _ i ) + k _ 2 ( 1 - \\frac { q _ i } { v _ i } ) + k _ 3 ( 1 - v _ i ) ] \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "6930.png", "formula": "\\begin{align*} e _ j = \\binom { N } { j } + ( - 1 ) ^ { j - 1 } q \\left [ z ^ { N + 1 - j } \\right ] ( y + z ) ( 1 + z x _ 1 ) \\cdots ( 1 + z x _ r ) . \\end{align*}"}
+{"id": "296.png", "formula": "\\begin{align*} A & ^ { ( D ' \\subset D , R ) } \\\\ & = A ^ { ( D ' \\subset D ) } \\left [ \\frac { I _ { \\delta } } { \\prod _ { i \\in I } t _ { i } ^ { n _ { i } } \\otimes 1 } , \\frac { ( 1 \\otimes t _ { 1 } ) - ( t _ { 1 } \\otimes 1 ) } { ( t _ { 1 } \\otimes 1 ) ( \\prod _ { i \\in I } t _ { i } ^ { n _ { i } } \\otimes 1 ) } , \\ldots , \\frac { ( 1 \\otimes t _ { r ' } ) - ( t _ { r ' } \\otimes 1 ) } { ( t _ { r ' } \\otimes 1 ) ( \\prod _ { i \\in I } t _ { i } ^ { n _ { i } } \\otimes 1 ) } \\right ] . \\end{align*}"}
+{"id": "456.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 0 \\\\ n - k \\equiv 0 \\ , ( \\mathrm { m o d } \\ , 2 ) } } ^ n \\binom { n } { k } \\big ( 2 ^ k L _ { j k } - 2 L _ j ^ k \\big ) ( \\sqrt { 5 } F _ j ) ^ { n - k } \\frac { B _ { n - k + 2 } } { n - k + 2 } = \\frac { 2 ^ { n + 2 } L _ { j ( n + 2 ) } - 2 L _ j ^ { n + 2 } } { 5 ( n + 1 ) ( n + 2 ) F _ j ^ 2 } - L _ j ^ n , \\end{align*}"}
+{"id": "5863.png", "formula": "\\begin{align*} \\vartheta D s = D e + p D \\left ( \\frac { 1 } { \\varrho } \\right ) , \\end{align*}"}
+{"id": "8051.png", "formula": "\\begin{align*} { \\widetilde { \\bf s } } _ { \\textrm { i n d } , m } [ f _ k ] = & ( \\widetilde { \\alpha } _ { m , 2 } { \\bf E } _ { m , k } { \\bf \\Phi } _ m { \\bf G } _ { m , k } + \\widetilde { \\alpha } _ { m , 3 } { \\bf D } _ { m , k } { \\bf \\Phi } _ m { \\bf B } _ { m , k } \\\\ & + \\widetilde { \\alpha } _ { m , 4 } { \\bf D } _ { m , k } { \\bf \\Phi } _ m { \\bf W } _ { m , k } { \\bf \\Phi } _ m { \\bf G } _ { m , k } ) { \\bf F } _ k { \\bf s } _ k , \\end{align*}"}
+{"id": "3593.png", "formula": "\\begin{align*} \\sup _ i \\left [ \\frac { \\lambda _ i ( A ) } { \\lambda + \\lambda _ i ( A ) } \\right ] \\geq \\frac { \\sup _ i \\lambda _ i ( A ) } { \\lambda + \\| A \\| } = \\frac { \\| A \\| } { \\lambda + \\| A \\| } . \\end{align*}"}
+{"id": "7067.png", "formula": "\\begin{align*} b = x _ { j _ 0 } a _ 1 + z a _ 2 \\in I _ { S _ 1 } \\cap ( I _ { S _ 2 } I _ { S _ 0 } ^ { m - 1 } ) \\end{align*}"}
+{"id": "5211.png", "formula": "\\begin{align*} \\sum _ { k = j _ { s - 1 } + 1 } ^ { j _ s } p _ k > L , \\enspace s \\geq 2 , \\end{align*}"}
+{"id": "7660.png", "formula": "\\begin{align*} H ( p ) = \\frac { P ( E _ p ) } { | E _ p | ^ { p } } \\le \\frac { P ( F ) } { | F | ^ { p } } . \\end{align*}"}
+{"id": "862.png", "formula": "\\begin{gather*} ( \\mathcal { E } _ k ) _ { k , n + 1 } = 1 \\ ; \\ ; \\\\ ( \\mathcal { E } _ k ) _ { i , j } = 0 , \\ ; \\ ; i \\neq k \\ ; ; \\ ; j \\neq n + 1 \\end{gather*}"}
+{"id": "5015.png", "formula": "\\begin{align*} \\mathrm { e } ^ { 2 \\pi \\mathrm { i } j / m } ( z _ 1 , z _ 2 , y ) & : = ( \\mathrm { e } ^ { 2 \\pi \\mathrm { i } j / m } z _ 1 , \\mathrm { e } ^ { 2 \\pi \\mathrm { i } j / m } z _ 2 , y ) , \\qquad \\qquad \\tau ( z _ { 1 } , z _ { 2 } , y ) : = ( z _ { 2 } , z _ { 1 } , y ) , \\\\ \\alpha ( z _ 1 , z _ 2 , y ) & : = ( z _ 1 , z _ 2 , \\alpha y ) \\quad \\alpha \\in O ( N - 4 ) . \\end{align*}"}
+{"id": "3845.png", "formula": "\\begin{align*} \\eta = & \\left ( n - 2 - \\left ( k + \\frac { - h ^ 2 + 2 h + 1 1 } { 2 } - i \\right ) , n - 2 - ( h - 1 + i _ { h - 4 } ) , \\ldots , n - 2 - ( 4 + i _ 1 ) \\right ) \\\\ = & \\left ( \\frac { 2 n - 2 k + h ^ 2 - 2 h - 1 5 } { 2 } + i , n - h - 1 - i _ { h - 4 } , \\dots , n - 6 - i _ 1 \\right ) . \\end{align*}"}
+{"id": "6326.png", "formula": "\\begin{align*} n _ { l - 1 } ( a , b ) = \\inf _ { w \\in M _ { l - 1 } \\cap S } \\ , I ( \\theta ( w , a , b ) + w , a , b ) \\end{align*}"}
+{"id": "5030.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} - \\Delta _ p u & = 1 , & & \\\\ u & = 0 , & & \\end{aligned} \\end{cases} \\end{align*}"}
+{"id": "7626.png", "formula": "\\begin{align*} w : = \\sum _ { j = 1 } ^ t u _ j f _ j . \\end{align*}"}
+{"id": "51.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { \\Psi } u = f ( u ) & \\O \\\\ u = \\phi _ { j } & ( \\partial \\O ) _ { j } \\end{cases} \\end{align*}"}
+{"id": "4366.png", "formula": "\\begin{align*} \\begin{array} { l l } G _ { i _ 0 } ^ 0 ( X ( T ) ) & = \\int _ { 0 } ^ { T } f _ { i _ 0 } ^ 0 ( s , x ( s ) , u ( s ) ) d s + g _ { i _ 0 } ^ 0 ( x ( T ) ) \\\\ \\null & > G _ { i _ 0 } ^ 0 ( X _ 0 ( T ) ) = \\int _ { 0 } ^ { T } f _ { i _ 0 } ^ 0 ( s , x _ 0 ( s ) , u _ 0 ( s ) ) d s + g _ { i _ 0 } ^ 0 ( x _ 0 ( T ) ) . \\end{array} \\end{align*}"}
+{"id": "3071.png", "formula": "\\begin{align*} C _ { \\sigma } ( k ) = 0 \\qquad \\end{align*}"}
+{"id": "6310.png", "formula": "\\begin{align*} X _ t = x _ 0 ( t ) + \\int _ 0 ^ t K _ \\mu ( s , t ) \\dd A _ s + \\int _ 0 ^ t K _ \\sigma ( s , t ) \\dd M _ s , t \\in [ 0 , T ] , \\quad \\P . \\end{align*}"}
+{"id": "7152.png", "formula": "\\begin{align*} \\eta _ i : = \\langle \\lambda _ i , ( \\mathcal { X } ^ f ( d ) ^ { \\vee } ) ^ { \\lambda _ i > 0 } \\rangle = - \\langle \\lambda _ i , \\mathcal { X } ^ f ( d ) ^ { \\lambda _ i < 0 } \\rangle = k + 3 k ( d - k ) . \\end{align*}"}
+{"id": "5799.png", "formula": "\\begin{align*} \\pi ( f ) = \\int _ { \\sigma ( A ) } f d \\mathcal { P } . \\end{align*}"}
+{"id": "5151.png", "formula": "\\begin{align*} D ( A ' , \\Omega , x ) = 1 D ( A , \\Omega , x ) = 1 , \\end{align*}"}
+{"id": "2203.png", "formula": "\\begin{align*} & R _ { 1 2 1 2 } R _ { 1 3 1 3 } - R _ { 1 2 1 3 } { } ^ 2 + ( b - c ) ^ 2 ( a + d ) ^ 2 \\\\ = & \\{ a d - ( b + c ) ^ 2 \\} \\{ a d + ( 3 b - c ) ( b - 3 c ) \\} . \\end{align*}"}
+{"id": "1821.png", "formula": "\\begin{align*} [ x + a , y + b ] _ { \\rho } = & [ x , y ] + \\rho ( x ) b - \\rho ( y ) a + [ a , b ] _ V , \\forall x , y \\in A , a , b \\in V . \\end{align*}"}
+{"id": "1668.png", "formula": "\\begin{align*} \\left [ \\Phi ^ * ( e ^ { \\lambda \\mathtt { P } } ) ^ * e ^ { \\lambda \\mathtt { P } } \\Phi \\right ] ^ { - 1 } & = \\mathbf { 1 } - \\Phi ^ * \\big [ \\lambda \\mathtt { U } ( \\mathtt { P } ) + \\lambda ^ 2 \\mathtt { V } ( Q , \\mathtt { P } ) \\big ] \\Phi + \\lambda ^ 3 \\ , \\mathtt { R } ^ { ( \\lambda ) } ( \\Phi , \\mathtt { P } ) \\ , , \\end{align*}"}
+{"id": "3253.png", "formula": "\\begin{gather*} \\varphi \\ , \\xi = 0 , \\varphi ^ 2 = - \\mathrm { i d } + \\xi \\otimes \\eta , g ( \\varphi X , \\varphi Y ) = g ( X , Y ) - \\eta ( X ) \\eta ( Y ) \\ , , X , Y \\in T M \\ , . \\end{gather*}"}
+{"id": "6420.png", "formula": "\\begin{align*} \\mu _ s ( x , \\xi ) = 1 + | x | + | \\xi | ^ { \\frac 1 s } . \\end{align*}"}
+{"id": "3511.png", "formula": "\\begin{align*} \\Big \\langle \\widehat { \\nabla \\mathrm { H } } ^ { \\textbf { i n } } ; \\mathrm { e } ^ { ( 3 ) } _ { \\mathrm { n } _ { 0 } } \\Big \\rangle _ { \\mathbb { L } ^ 2 ( \\mathrm { B } ) } = \\nabla \\mathrm { H } ^ { \\textbf { i n } } ( \\mathrm { z } ) \\int _ \\mathrm { B } \\Tilde { \\mathrm { e } } ^ { ( 3 ) } _ { \\mathrm { n } _ { 0 } } ( \\mathrm { x } ) d \\mathrm { x } + \\mathcal { O } ( \\delta ) . \\end{align*}"}
+{"id": "4335.png", "formula": "\\begin{align*} \\nabla f \\cdot \\mathbf { n } = \\nabla g \\cdot \\mathbf { n } = 0 \\ ; \\ ; ( 0 , \\infty ) \\times \\partial \\Omega \\ , , \\end{align*}"}
+{"id": "4771.png", "formula": "\\begin{gather*} \\theta _ 2 \\partial _ 1 ( a ) b = \\theta _ 1 \\partial _ 2 ( a ) b , \\qquad \\ ! \\theta _ 2 \\mu ( \\partial _ 1 ( a ) ) v = \\theta _ 1 \\mu ( \\partial _ 2 ( a ) ) v , \\qquad \\ ! \\theta _ 2 \\alpha _ 1 ( \\mu ( a ) v ) = \\theta _ 1 \\alpha _ 2 ( \\mu ( a ) v ) , \\ ! \\ ! \\ ! \\ ! \\end{gather*}"}
+{"id": "7396.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to - \\infty } \\lambda \\mu _ n ( S ( \\lambda ) ) = - \\infty \\end{align*}"}
+{"id": "197.png", "formula": "\\begin{align*} \\lambda _ V = ( 2 - h ) ^ { 2 r + 2 k } \\ , 2 ^ { s - 1 } ( 1 - d ) = 2 ^ { s - 1 } \\ , 2 ^ { 2 r + 2 k } ( 1 - d ) = 2 ^ { 2 r + 2 k + s - 1 } ( 1 - d ) . \\end{align*}"}
+{"id": "4506.png", "formula": "\\begin{align*} \\| \\varphi _ n - u \\| _ \\infty = \\| \\varphi _ n - \\widetilde { u } \\| _ \\infty = \\sup _ { \\mathbf { x } \\in \\mathbb { R } ^ 3 } | \\varphi _ n ( \\mathbf { x } ) - \\widetilde { u } ( \\mathbf { x } ) | \\end{align*}"}
+{"id": "2475.png", "formula": "\\begin{align*} \\star \\omega = ( * \\omega ) ^ \\# , \\star S = * ( S ^ \\flat ) \\end{align*}"}
+{"id": "5991.png", "formula": "\\begin{align*} v ^ { L R } _ { b } ( x ) & = - C ^ 1 _ b \\left ( Z _ b ^ { ( \\theta , r ) } ( x ) - r Z ^ { ( \\theta ) } ( b ) \\overline { W } ^ { ( r + \\theta ) } ( x - b ) \\right ) - r \\overline { \\overline { W } } ^ { ( r + \\theta ) } ( x - b ) \\\\ & + \\beta \\left ( \\overline { Z } _ b ^ { ( \\theta , r ) } ( x ) + \\dfrac { \\psi _ X ' ( 0 + ) } { \\theta } - r \\overline { Z } ^ { ( \\theta ) } ( b ) \\overline { W } ^ { ( \\theta + r ) } ( x - b ) \\right ) , \\end{align*}"}
+{"id": "5569.png", "formula": "\\begin{align*} \\Tilde { \\Phi } _ 2 ( \\beta _ 0 , + \\infty ) = \\tfrac { 2 } { L _ { 2 m } ^ { 2 } } \\bigg [ \\tilde { N _ 2 } \\tfrac { \\beta _ 0 ^ { 3 - 2 \\mu - \\nu } } { ( 3 - 2 \\mu - \\nu ) ( 1 - \\mu - \\nu ) } + \\tfrac { \\tilde { L _ 2 } N _ { 2 M } } { L _ { 2 m } } \\tfrac { \\beta _ 0 ^ { \\delta - 3 \\mu + 3 - \\nu } } { ( \\delta - 3 \\mu + 3 - \\nu ) ( 1 - \\mu - \\nu ) } + \\tfrac { \\tilde { L _ 2 } } { L _ { 2 m } } \\tfrac { \\beta ^ { 1 - \\mu - \\nu } } { { \\mu + \\nu - 1 } } \\bigg ] , \\end{align*}"}
+{"id": "6040.png", "formula": "\\begin{align*} J ( K ) = \\int _ { \\Omega \\backslash K } \\lvert \\nabla u _ K ( x ) - A ( x ) \\rvert ^ 2 d x + \\int _ { \\Omega \\backslash K } \\lvert u _ K ( x ) - u _ 0 ( x ) \\rvert ^ 2 d x \\end{align*}"}
+{"id": "176.png", "formula": "\\begin{align*} \\begin{aligned} K f = K ^ { 1 - \\chi } f + K ^ \\chi f \\ , , \\ K ^ \\Lambda f = K _ 1 ^ \\Lambda f - K _ 2 ^ \\Lambda f \\textrm { f o r } \\Lambda = 1 - \\chi \\textrm { a n d } \\chi \\ , , \\end{aligned} \\end{align*}"}
+{"id": "2111.png", "formula": "\\begin{align*} S ' _ A = \\begin{bmatrix} 0 & 0 & I _ m & 0 & 0 \\\\ 0 & 0 & 0 & I _ { k - m } & 0 \\\\ A _ { 1 1 } & A _ { 1 2 } - D & 0 & D & 0 \\\\ A _ { 2 1 } & A _ { 2 2 } & 0 & 0 & I _ { \\ell - k } \\\\ A _ { 3 1 } & A _ { 3 2 } & 0 & 0 & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "3610.png", "formula": "\\begin{align*} I ( x , T , \\Xi ) = E _ x \\left [ \\int _ 0 ^ \\infty e ^ { - \\lambda t } \\phi ( X _ t ) d t + \\sum _ { n = 1 } ^ \\infty e ^ { - \\lambda \\tau _ n } g ( \\xi _ n ) \\right ] \\end{align*}"}
+{"id": "6879.png", "formula": "\\begin{align*} V _ { g h } = V _ g \\otimes V _ h \\simeq \\mathrm { I n d } ^ { \\Q } _ K ( \\psi ) \\oplus \\mathrm { I n d } ^ { \\Q } _ K ( \\xi ) \\mbox { a s } G _ { \\Q } \\mbox { - m o d u l e s } \\end{align*}"}
+{"id": "7670.png", "formula": "\\begin{align*} \\mathcal { C } & : = \\left \\{ w \\in H ^ 1 ( - r , r ) \\ | \\ w - f _ E \\in H ^ 1 _ 0 ( - r , r ) , \\ w \\leq f _ { \\Omega } \\ \\right \\} \\end{align*}"}
+{"id": "7498.png", "formula": "\\begin{align*} E _ \\varphi \\left [ \\mu \\right ] = \\iint \\log \\frac { 1 } { | s - t | } d \\mu ( s ) d \\mu ( t ) + \\int \\varphi d \\mu . \\end{align*}"}
+{"id": "1870.png", "formula": "\\begin{align*} f ( x + y ) = \\frac { f ( x ) f ( y ) } { f ( x ) + f ( y ) } , \\end{align*}"}
+{"id": "5845.png", "formula": "\\begin{align*} 0 \\leq c ( D ) & = \\inf \\{ | \\langle D x , x \\rangle | : x \\in \\ell ^ 2 ( \\mathbb { Z } _ { + } ) , \\| x \\| = 1 \\} \\\\ & \\leq \\inf \\{ | \\langle D e _ i , e _ i \\rangle | : i \\in \\mathbb { Z } _ { + } \\} \\\\ & = \\inf \\{ | \\langle \\lambda _ i e _ { i } , e _ i \\rangle | : i \\in \\mathbb { Z } _ { + } \\} \\\\ & = \\inf _ { i } { | \\lambda _ i | } = 0 . \\end{align*}"}
+{"id": "8234.png", "formula": "\\begin{align*} \\Vert \\delta h \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } - 1 } _ { 2 , 1 } } = \\| h ( \\sigma _ 1 ) - h ( \\sigma _ 2 ) \\| _ { \\dot { B } ^ { \\frac { N } { 2 } - 1 } _ { 2 , 1 } } \\lesssim \\big ( 1 + \\| ( \\sigma _ 1 , \\sigma _ 2 ) \\| _ { L ^ \\infty _ t ( \\dot { B } ^ { \\frac { N } { 2 } } _ { 2 , 1 } ) } \\big ) ^ { [ \\frac { N } { 2 } + 1 ] } \\Vert \\delta \\sigma \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } } _ { 2 , 1 } } \\le C \\Vert \\delta \\sigma \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } } _ { 2 , 1 } } . \\end{align*}"}
+{"id": "3455.png", "formula": "\\begin{align*} \\Big | u \\Big | _ { \\mathrm { L } ^ 2 \\Big ( \\mathbb { R } ; \\mathrm { H } ^ { \\mathrm { r } } ( \\partial \\Omega ) \\Big ) } ^ 2 : = \\int _ { \\partial \\Omega } \\int _ { \\partial \\Omega } \\dfrac { \\Vert u ( \\mathrm { x } , \\cdot ) - u ( \\mathrm { y } , \\cdot ) \\Vert _ { \\mathrm { L } ^ 2 ( \\mathbb { R } ) } ^ 2 } { | \\mathrm { x } - \\mathrm { y } | ^ { n - 1 + 2 \\mathrm { r } } } d \\sigma _ \\mathrm { x } d \\sigma _ \\mathrm { y } , \\end{align*}"}
+{"id": "5547.png", "formula": "\\begin{align*} \\Tilde { \\lambda _ 1 } ( 1 ) \\dfrac { \\partial T _ 1 } { \\partial r } \\bigg | _ { r = \\alpha ( t ) } = - l _ b \\gamma _ b \\alpha ' ( t ) / ( \\theta _ b - \\theta _ m ) , \\end{align*}"}
+{"id": "3725.png", "formula": "\\begin{align*} \\lim _ { y \\to 0 } \\frac 1 y ( A ^ Q ( \\rho ' _ y , v ' _ y ) - A ^ Q ( \\rho , \\nabla S ) ) = \\tilde D A ^ Q ( \\rho , \\nabla S ) ( \\delta \\rho , \\delta v ) = 0 \\end{align*}"}
+{"id": "584.png", "formula": "\\begin{align*} w _ { 1 , 2 , 3 } = 0 \\ , w _ { 1 , 2 , 4 } = 0 \\ , w _ { 1 , 3 , 4 } = 0 \\ , w _ { 2 , 3 , 4 } = 0 \\ , x _ { 1 2 3 4 } = 0 \\ , . \\end{align*}"}
+{"id": "4102.png", "formula": "\\begin{align*} \\widetilde { G } = \\exp ( \\sum N _ i \\cdot \\log ( t _ i ) ) \\cdot G , \\end{align*}"}
+{"id": "932.png", "formula": "\\begin{align*} u ( x ) \\geq \\left ( \\frac { 1 } { 2 } - C \\delta \\right ) \\sum _ { \\beta = \\alpha + 1 } ^ { n - 1 } b _ \\beta x _ \\beta ^ 2 - C b _ \\alpha | \\hat { x } | ^ 2 \\geq \\frac { 1 } { 4 } \\sum _ { \\beta = \\alpha + 1 } ^ { n - 1 } b _ \\beta x _ \\beta ^ 2 - C b _ \\alpha | \\hat { x } | ^ 2 \\end{align*}"}
+{"id": "1374.png", "formula": "\\begin{align*} ( 1 + t ) E [ u ] ( t ) + \\int _ { \\mathbb { R } ^ n } a ( x ) | u ( t , x ) | ^ 2 \\ , d x & \\le C \\begin{dcases} ( 1 + t ) ^ { - \\frac { 4 } { 2 - \\alpha } \\left ( \\frac { 1 } { p - 1 } - \\frac { n - \\alpha } { 4 } \\right ) } & ( p > p _ { s u b c } ( n , \\alpha ) ) , \\\\ ( 1 + t ) ^ { - \\frac { 2 } { p - 1 } } \\log ( 2 + t ) & ( p = p _ { s u b c } ( n , \\alpha ) ) , \\\\ ( 1 + t ) ^ { - \\frac { 2 } { p - 1 } } & ( p < p _ { s u b c } ( n , \\alpha ) ) , \\end{dcases} \\end{align*}"}
+{"id": "1824.png", "formula": "\\begin{align*} & [ T ( a ) , T ( b ) ] = T ( \\rho ( T ( a ) ) b - \\rho ( T ( b ) ) a + \\lambda [ a , b ] _ V , \\end{align*}"}
+{"id": "740.png", "formula": "\\begin{align*} \\rho \\left ( { \\bf { x } } \\right ) = \\frac { { { \\rho _ l } + { \\rho _ v } } } { 2 } - \\frac { { { \\rho _ l } - { \\rho _ v } } } { 2 } \\tanh \\frac { { 2 \\left ( { \\left | { { \\bf { x } } - { { \\bf { x } } _ 0 } } \\right | - { R _ 0 } } \\right ) } } { W } , \\end{align*}"}
+{"id": "5562.png", "formula": "\\begin{align*} u _ 2 ( \\eta ) = u _ { c } \\dfrac { \\Phi _ 2 [ \\beta _ 0 , \\eta , L _ 2 ( u _ 2 ) , N _ 2 ( u _ 2 ) ] } { \\Phi _ 2 [ \\beta _ 0 , \\infty , L _ 2 ( u _ c ) , N _ 2 ( u _ c ) ] } , \\ ; \\ ; \\ ; \\beta _ 0 \\leq \\eta < \\infty ; \\end{align*}"}
+{"id": "858.png", "formula": "\\begin{align*} \\begin{aligned} & \\varphi ' ( K ) = K ^ { - \\frac { 2 a ' } { k _ 2 \\sigma } - 1 } \\left [ 1 - \\frac { 2 a ' } { k _ 2 \\sigma } \\ln \\left ( \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) \\right ] \\end{aligned} \\end{align*}"}
+{"id": "2361.png", "formula": "\\begin{align*} \\zeta ^ { \\mathfrak { m } } ( \\Bbbk ) = L _ { B } ( \\theta ( \\Bbbk ) ) \\ \\ \\ ( \\Bbbk \\in W _ { 2 , 3 } ) . \\end{align*}"}
+{"id": "6459.png", "formula": "\\begin{align*} \\omega ^ { ( 2 ) } _ { \\mu } ( f _ 1 , f _ 2 ) = \\lambda \\left ( \\binom { \\rho _ 0 G f _ 1 } { \\rho _ 1 G f _ 1 } , \\binom { \\rho _ 0 G f _ 2 } { \\rho _ 1 G f _ 2 } \\right ) \\end{align*}"}
+{"id": "6851.png", "formula": "\\begin{align*} & ( \\alpha ^ { p ^ { e ' } + 1 } ( v _ 1 ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ 1 ) , \\dots , \\alpha ^ { p ^ { e ' } + 1 } ( v _ s ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ s ) , ( v _ { s + 1 } ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ { s + 1 } ) , \\dots , ( v _ n ' ) ^ { p ^ { e ' } + 1 } f ^ { p ^ { e ' } } ( a _ n ) ) \\\\ = & ( u _ 1 g ( a _ 1 ) , \\dots , u _ s g ( a _ s ) , u _ { s + 1 } g ( a _ { s + 1 } ) , \\dots , u _ n g ( a _ n ) ) . \\end{align*}"}
+{"id": "3425.png", "formula": "\\begin{align*} U ^ * = U ( \\sigma , r ) , \\rho ^ * = \\rho ( \\sigma , r ) \\big ( 1 + \\epsilon S _ t ( \\sigma , r ) f ' ( S ^ * ) \\big ) ^ { S _ r ( \\sigma , r ) / S _ t ( \\sigma , r ) } , \\end{align*}"}
+{"id": "5684.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial x _ 0 } ( P ) = \\ldots = \\frac { \\partial { F } } { \\partial x _ n } ( P ) = 0 . \\end{align*}"}
+{"id": "1757.png", "formula": "\\begin{align*} Z _ { * } = Z _ { n } + \\mu Z _ { n - 1 } + \\cdots + \\mu ^ { m - 1 } Z _ { n - m + 1 } \\end{align*}"}
+{"id": "6516.png", "formula": "\\begin{align*} U = ( 2 P - I ) ( 2 Q - I ) . \\end{align*}"}
+{"id": "5625.png", "formula": "\\begin{align*} \\hat { \\mathbb { G } } ^ i _ { s k } u ^ k + \\hat { \\mathbb { G } } ^ i _ k J ^ k _ s = 2 J ^ i _ k \\hat { \\mathbb { G } } ^ k _ s . \\end{align*}"}
+{"id": "6606.png", "formula": "\\begin{align*} H _ { \\theta , \\alpha } : = T _ { \\Gamma _ \\theta , \\alpha } , H _ { \\theta } : = H _ { \\theta , - 1 } . \\end{align*}"}
+{"id": "1694.png", "formula": "\\begin{align*} h : = \\sum _ { k \\in \\N } \\lambda _ k u _ k u ^ * _ k , \\end{align*}"}
+{"id": "3564.png", "formula": "\\begin{align*} e _ 1 ( I ) \\le \\binom { e _ 0 ( I ) } { 2 } - \\binom { g - 1 } { 2 } - \\ell ( A / \\tilde { I } ) + 1 , \\end{align*}"}
+{"id": "4083.png", "formula": "\\begin{align*} N > \\frac { J ( d _ 1 , \\ldots , d _ n ; e _ 1 , \\ldots , e _ n ) } { \\sum _ { i = 1 } ^ n ( e _ i - d _ i ) } , \\end{align*}"}
+{"id": "7210.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { Q } } K _ 0 ^ { \\mathrm { t o p } } ( \\mathbb { S } ( d ) _ w ) = \\dim _ { \\mathbb { Q } } K _ 0 ^ { \\mathrm { t o p } } ( \\mathbb { T } ( d ) _ w ) \\geq p _ 2 ( ( d , w ) ) . \\end{align*}"}
+{"id": "5343.png", "formula": "\\begin{align*} \\sum _ { r + s + t = n } ( - 1 ) ^ { r + s t } m _ { r + t + 1 } ( 1 ^ { \\otimes r } \\otimes m _ s \\otimes 1 ^ { \\otimes t } ) = 0 \\end{align*}"}
+{"id": "4435.png", "formula": "\\begin{align*} \\Gamma - \\lim _ { n \\to \\infty } I _ n ^ X = \\Gamma - \\lim _ { n \\to \\infty } I ^ X . \\end{align*}"}
+{"id": "6791.png", "formula": "\\begin{align*} \\phi ( x _ { [ k ] } ) = \\sum _ { i \\in [ s ] } \\tilde { \\beta } _ i ( x _ I ) \\tilde { \\gamma } _ i ( x _ { [ k ] \\setminus I } ) + & \\sum _ { i \\in [ r _ 2 ] } \\beta ^ { ( 2 ) } _ { i , 1 } ( x _ { I ^ { ( 2 ) } _ { i , 1 } } ) \\dots \\beta ^ { ( 2 ) } _ { i , d ^ { ( 2 ) } _ i } ( x _ { I ^ { ( 2 ) } _ { i , d ^ { ( 2 ) } _ i } } ) \\\\ + & \\sum _ { i \\in [ r _ 3 ] } \\beta ^ { ( 3 ) } _ { i , 1 } ( x _ { I ^ { ( 3 ) } _ { i , 1 } } ) \\dots \\beta ^ { ( 3 ) } _ { i , d ^ { ( 3 ) } _ i } ( x _ { I ^ { ( 3 ) } _ { i , d ^ { ( 3 ) } _ i } } ) \\end{align*}"}
+{"id": "5003.png", "formula": "\\begin{align*} \\langle A \\rangle = \\left \\{ \\sum \\limits _ { i = 1 } ^ k n _ i a _ i : k \\in \\N _ 0 , n _ i \\in \\N _ 0 , a _ i \\in A \\right \\} , \\end{align*}"}
+{"id": "2420.png", "formula": "\\begin{align*} \\tilde { L } ( { \\rm D } ( e _ { 0 } ^ { n } u ) ) = \\sum _ { N \\geq 2 } ( - 1 ) ^ { N } \\zeta ^ { \\mathfrak { m } } ( N ) \\tilde { L } ( { \\rm C u t } _ { ( \\{ 0 \\} ^ { N - 1 } ; \\emptyset ) } ( e _ { 0 } ^ { n } u ) ) + \\tilde { L } ( { \\rm D } ( { \\rm r e g } _ { 0 } ( e _ { 0 } ^ { n } u ) ) ) \\end{align*}"}
+{"id": "3900.png", "formula": "\\begin{align*} \\omega ( t ) + \\lambda \\int _ 0 ^ t ( 1 + m ( t - \\tau ) ) \\omega ( \\tau ) d \\tau = 1 , \\end{align*}"}
+{"id": "5727.png", "formula": "\\begin{align*} f ( X ) = \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } \\otimes X _ i X _ j \\quad g ( X ) = \\sum _ { i = 1 , j = 1 } ^ { m } B _ { i j } \\otimes X _ i X _ j . \\end{align*}"}
+{"id": "2377.png", "formula": "\\begin{align*} D _ { \\alpha , \\beta } ( w _ { 1 } , a , w _ { 2 } ) = \\sum _ { s \\in S } \\Delta _ { a , s } ^ { \\alpha , \\beta } ( w _ { 1 } e _ { s } w _ { 2 , s } - w _ { 1 , s } e _ { s } w _ { 2 } ) \\end{align*}"}
+{"id": "4668.png", "formula": "\\begin{align*} y ^ \\ell + \\sum _ { m = 0 } ^ { \\ell - 1 } a _ { m } y ^ { m } : = \\Psi _ { \\ell , n _ q } ( y ) ^ { n _ q } ( y - n _ q ) . \\end{align*}"}
+{"id": "1960.png", "formula": "\\begin{align*} \\beta = s + \\sum _ { j = 1 } ^ d \\frac { a _ j } { p _ j } - \\frac { \\nu } { p _ { } } . \\end{align*}"}
+{"id": "5429.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ { m } } & \\sum _ { v _ 1 , \\ldots , v _ m = 1 } ^ N c _ { v _ 1 , \\ldots , v _ m } ( t , z ) \\prod ^ { n _ 1 } _ { i = 1 } G _ { x _ { i } y _ { i } } ( t , z ) \\prod _ { l = 1 } ^ { n _ 2 } ( G _ { w _ { l } w _ { l } } - m _ { s c } ) ( t , z ) \\\\ = & : \\frac { 1 } { N ^ { m } } \\sum _ { \\mathcal { I } _ m } c _ { \\mathcal { I } _ m } \\prod ^ { n _ 1 } _ { i = 1 } G _ { x _ i y _ i } \\prod _ { l = 1 } ^ { n _ 2 } ( G _ { w _ { l } w _ { l } } - m _ { s c } ) , t \\in \\R ^ + , z \\in \\C ^ + , \\end{align*}"}
+{"id": "2923.png", "formula": "\\begin{align*} T _ 1 ( r , F ) = T _ 1 ( r , F ^ { - 1 } ) \\end{align*}"}
+{"id": "31.png", "formula": "\\begin{align*} p _ i = \\begin{cases} 0 . 5 \\tilde { q } _ i ( 1 + \\tilde { \\delta } _ i ) , & i \\in [ m ] \\\\ 0 . 5 \\tilde { q } _ { i - m } ( 1 - \\tilde { \\delta } _ { i - m } ) , & i \\in [ k ] \\setminus [ m ] . \\end{cases} \\end{align*}"}
+{"id": "2037.png", "formula": "\\begin{align*} \\tau _ j ^ \\pm = \\frac { ( - 1 ) ^ { j - 1 } } { \\sqrt { j ( j + 1 ) } } \\left ( \\sum _ { k = 1 } ^ { j } ( - 1 ) ^ { k - 1 } \\chi _ { \\pm ( N + k ) } - j ( - 1 ) ^ { ( j + 1 ) - 1 } \\chi _ { \\pm ( N + j + 1 ) } \\right ) \\end{align*}"}
+{"id": "80.png", "formula": "\\begin{gather*} g ( X , \\nabla \\Psi ) \\equiv c o n s t \\\\ \\Updownarrow \\\\ \\div _ \\Psi ( X ) = \\div ( X ) - g ( X , \\nabla \\Psi ) \\equiv c o n s t . \\end{gather*}"}
+{"id": "6543.png", "formula": "\\begin{align*} A \\big ( \\overrightarrow { H } \\big ) _ { ( a , b ) , ( c , d ) } = \\begin{cases} 1 , \\quad c \\in f _ { a b } a \\not \\in f _ { c d } , \\\\ [ 2 . 5 m m ] - 1 , \\quad a \\in f _ { c d } c \\not \\in f _ { a b } , \\\\ [ 2 . 5 m m ] 0 , \\quad \\end{cases} \\end{align*}"}
+{"id": "4478.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } | F ( e _ n ) | ^ 2 < + \\infty \\end{align*}"}
+{"id": "4501.png", "formula": "\\begin{align*} \\lim _ { h \\rightarrow 0 } \\frac { 1 } { \\| h \\| } _ { \\ ! H ^ 1 } \\ ! \\bigg | V ( u + h ) - V ( u ) - \\int _ { [ 0 , 1 ] } u ' h ' \\bigg | = \\lim _ { h \\rightarrow 0 } \\frac { 1 } { 2 \\| h \\| } _ { \\ ! H ^ 1 } \\ ! \\bigg | \\int _ { [ 0 , 1 ] } ( h ' ) ^ 2 \\bigg | = 0 \\ , . \\end{align*}"}
+{"id": "2606.png", "formula": "\\begin{align*} | \\phi ^ { N , i } ( t , \\boldsymbol { x } _ t ) - \\Phi ( t , \\bar { \\nu } _ t ^ i ) | = | \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } _ t } ) - \\Phi ( t , \\bar { \\nu } _ t ^ i ) | \\leq C | \\nu ^ { N , i } _ { \\boldsymbol { x } _ t } - \\bar { \\nu } _ t ^ i | . \\end{align*}"}
+{"id": "6377.png", "formula": "\\begin{align*} & I _ { 0 + } ^ \\alpha f ( x ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ 0 ^ x \\frac { f ( y ) } { ( x - y ) ^ { 1 - \\alpha } } \\d y . \\end{align*}"}
+{"id": "1880.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\left | \\frac { 1 } { 3 ^ l } \\right | ^ { m } G \\left ( \\frac { x } { 3 ^ { m + 1 } } , \\ \\frac { y } { 3 ^ { m + 1 } } \\right ) = 0 \\end{align*}"}
+{"id": "6555.png", "formula": "\\begin{align*} \\lambda _ k = \\left ( e ^ { \\frac { 2 \\pi i } { n - 1 } } \\right ) ^ k \\end{align*}"}
+{"id": "900.png", "formula": "\\begin{align*} \\begin{aligned} \\Psi \\leq & - \\frac { \\theta } { 2 } | x ' | ^ 2 , \\mbox { o n } \\partial _ 1 \\omega _ \\delta \\\\ \\Psi \\leq & - \\frac { \\delta ^ 2 } { 2 } , \\mbox { o n } \\partial _ 2 \\omega _ \\delta \\\\ \\Psi \\leq & - \\frac { \\theta \\delta ^ 2 } { 2 } , \\mbox { o n } \\partial _ 3 \\omega _ \\delta . \\end{aligned} \\end{align*}"}
+{"id": "22.png", "formula": "\\begin{align*} w _ \\Gamma ( x ) = j p ( x ) / q ( x ) \\in [ \\gamma _ { j } , \\gamma _ { j + 1 } ) . \\end{align*}"}
+{"id": "7519.png", "formula": "\\begin{align*} E \\left [ \\mu \\right ] = \\iint G ( p , q ) d \\mu ( p ) d \\mu ( q ) . \\end{align*}"}
+{"id": "3959.png", "formula": "\\begin{align*} \\hat { \\mathcal { M } } ( t ) = Y ( t ) . \\end{align*}"}
+{"id": "1062.png", "formula": "\\begin{align*} \\Delta ^ { ( s ) } : = \\nabla ^ { ( s ) * } \\nabla ^ { ( s ) } = - \\nabla ^ { ( s ) } _ { e _ { i } } \\nabla ^ { ( s ) } _ { e _ { i } } + \\nabla ^ { ( s ) } _ { \\nabla _ { e _ { i } } e _ { i } } = - \\nabla ^ { ( s ) } _ { e _ { i } } \\nabla ^ { ( s ) } _ { e _ { i } } + \\alpha _ { i i j } \\nabla ^ { ( s ) } _ { e _ { j } } . \\end{align*}"}
+{"id": "4663.png", "formula": "\\begin{align*} D _ 1 ( t ) = f _ 1 ^ * ( u ( t ) , v ( t ) ) , \\dots , D _ { \\ell - 1 } ( t ) = f _ { \\ell - 1 } ^ * ( u ( t ) , v ( t ) ) . \\end{align*}"}
+{"id": "278.png", "formula": "\\begin{align*} & E _ 1 : = \\overline { W _ 1 } , E _ 2 : = \\overline { W _ 2 } \\setminus \\left ( \\overline { W _ 1 } \\cup U _ 1 \\right ) , E _ 3 : = \\overline { W _ 3 } \\setminus \\left ( \\overline { W _ 1 } \\cup U _ 1 \\cup \\overline { W _ 2 } \\cup U _ 2 \\right ) , \\dots , \\\\ & E _ m : = \\overline { W _ m } \\setminus \\left ( \\overline { W _ 1 } \\cup U _ 1 \\cup \\overline { W _ 2 } \\cup U _ 2 \\cup \\dots \\cup \\overline { W _ { m - 1 } } \\cup U _ { m - 1 } \\right ) . \\end{align*}"}
+{"id": "7864.png", "formula": "\\begin{align*} E _ \\delta = \\{ \\theta \\in J / 4 : | h ( \\theta ) | \\le \\delta \\} \\end{align*}"}
+{"id": "8173.png", "formula": "\\begin{align*} a ^ * T ^ n C ^ * b = 0 \\ \\ ( n = 0 , 1 , 2 , \\ldots ) \\ \\ \\mbox { a n d } \\end{align*}"}
+{"id": "5754.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } x _ i x _ j + \\sum _ { i = 1 } ^ { m } A _ i x _ i + A _ 0 , \\end{align*}"}
+{"id": "3112.png", "formula": "\\begin{align*} A _ { n + 1 } = \\left ( a _ { k , n + 1 } \\right ) _ { k = 1 } ^ { \\infty } \\end{align*}"}
+{"id": "4509.png", "formula": "\\begin{align*} D _ 1 \\Phi _ x ( \\phi , \\pi ) = \\mathcal { E } _ x \\ , . \\end{align*}"}
+{"id": "105.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ \\Psi ( - v ) \\geq f ' ( u ) ( - v ) & A ( r _ { 1 } , r _ { 2 } ) \\\\ ( - v ) = 0 & \\partial A ( r _ { 1 } , r _ { 2 } ) . \\end{cases} \\end{align*}"}
+{"id": "1599.png", "formula": "\\begin{align*} f _ { n } \\left ( g _ { s , j , i } \\right ) \\circ f _ { n } \\left ( g _ { s , i , j } \\right ) \\restriction _ { T _ { b _ { i } } \\left ( V ^ { \\prime \\prime } \\right ) } = \\mathrm { i d } _ { T _ { b _ { i } } \\left ( V ^ { \\prime \\prime } \\right ) } , \\end{align*}"}
+{"id": "18.png", "formula": "\\begin{align*} c _ { n - 1 , k - 1 } = \\sum _ { l = k - 1 } ^ { n - 1 } a _ { n - 1 , l } b _ { l , k - 1 } , ~ ~ ~ c _ { n - 1 , k } = \\sum _ { l = k } ^ { n - 1 } a _ { n - 1 , l } b _ { l , k } . \\end{align*}"}
+{"id": "1620.png", "formula": "\\begin{align*} \\big | g ( u _ i ) - g ( x ) \\big | & = \\big | ( 1 - \\varepsilon ) f ( v _ i ) + ( 1 - \\varepsilon ) d ( u _ i , v _ i ) - ( 1 - \\varepsilon ) f ( x ) \\big | \\\\ & \\le ( 1 - \\varepsilon ) \\big ( d ( u _ i , v _ i ) + | f ( x ) - f ( v _ i ) | \\big ) \\\\ & \\le ( 1 - \\varepsilon ) \\big ( d ( u _ i , v _ i ) + d ( x , v _ i ) \\big ) \\\\ & \\le d ( x , u _ i ) . \\end{align*}"}
+{"id": "5934.png", "formula": "\\begin{align*} \\theta _ I : = \\Big \\{ \\gamma ( a ) + \\sum _ { j = 1 } ^ k t _ j \\gamma ^ { ( j ) } ( a ) \\in \\Q _ q ^ k \\colon | t _ j | \\leq | I | ^ j \\Big \\} \\end{align*}"}
+{"id": "30.png", "formula": "\\begin{align*} \\delta _ i = \\begin{cases} \\tilde { \\delta } _ i , & i \\in [ m ] \\\\ - \\tilde { \\delta } _ { i - m } , & i \\in [ k ] \\setminus [ m ] . \\end{cases} \\end{align*}"}
+{"id": "242.png", "formula": "\\begin{align*} \\forall i \\in D \\ \\Rightarrow \\ \\beta _ i : = \\ | | \\xi _ i | | B \\phi \\in ( 0 , \\infty ) . \\end{align*}"}
+{"id": "4869.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( 2 n ) ^ k ( 2 x ) ^ { 2 n } } { \\binom { 2 n } { n } } = \\frac { x } { ( 1 - x ^ 2 ) ^ { k + \\frac { 3 } { 2 } } } \\left ( x \\sqrt { 1 - x ^ 2 } p _ k ( x ^ 2 ) + \\arcsin ( x ) q _ k ( x ^ 2 ) \\right ) . \\end{align*}"}
+{"id": "2577.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } _ t ^ * : = ( \\alpha _ t ^ { * , i } ) _ { 1 \\leq i \\leq N } = \\left ( - R ^ { - 1 } B y _ t ^ { * , i } - h ( \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } ^ * _ t } ) \\right ) _ { 1 \\leq i \\leq N } \\end{align*}"}
+{"id": "6308.png", "formula": "\\begin{align*} \\mathcal { A } ^ f \\colon [ 0 , T ] \\times \\R \\times \\R \\to \\R \\quad \\mathcal { A } ^ f ( t , x , z ) : = \\mu ( t , x ) f ^ \\prime ( z ) + \\frac { 1 } { 2 } \\sigma ( t , x ) ^ 2 f ^ { \\prime \\prime } ( z ) . \\end{align*}"}
+{"id": "7108.png", "formula": "\\begin{align*} \\textbf { W } ( d ) : = \\frac { 3 } { 2 } [ 0 , \\beta _ i - \\beta _ j ] + \\mathbb { R } \\tau _ d \\subset M ( d ) _ { \\mathbb { R } } , \\end{align*}"}
+{"id": "5095.png", "formula": "\\begin{align*} g ( x , y ) = \\sum _ { l \\in \\mathbb { Z } ^ n , | l | \\leq N } \\int _ { [ - N , N ] ^ { d - n } } \\hat { g _ l } ( \\xi ) e ^ { 2 \\pi i ( x \\cdot \\xi + y \\cdot l ) } \\ , d \\xi . \\end{align*}"}
+{"id": "6677.png", "formula": "\\begin{align*} - \\frac { \\underline { P } ( 0 ) } { \\underline { Q } ( 0 ) } & = \\frac { \\mathcal { D } \\left ( ( s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { 2 m - 1 } , ( - a s _ { k - 1 } ) _ { k = 0 } ^ { m } \\right ) } { a \\mathcal { D } \\left ( ( s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { 2 m - 1 } , ( 1 , 0 , \\ldots , 0 ) \\right ) } \\\\ & + \\frac { \\mathcal { D } \\left ( ( s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { 2 m - 1 } , ( s _ k ) _ { k = 0 } ^ { m } \\right ) } { a \\mathcal { D } \\left ( ( s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { 2 m - 1 } , ( 1 , 0 , \\ldots , 0 ) \\right ) } \\end{align*}"}
+{"id": "1431.png", "formula": "\\begin{align*} \\int _ 0 ^ t U ( t - s ) \\begin{pmatrix} 0 \\\\ - | u | ^ { p - 1 } u ( s ) \\end{pmatrix} \\ , d s = \\int _ 0 ^ t U ( s ) \\begin{pmatrix} 0 \\\\ - | u | ^ { p - 1 } u ( t - s ) \\end{pmatrix} \\ , d s \\end{align*}"}
+{"id": "815.png", "formula": "\\begin{align*} l _ 2 \\bar { \\theta } _ 1 ^ { k _ 2 - 1 } \\Big ( \\frac { K k _ 2 } { k _ 2 - 1 } \\Big ) ^ { 1 - k _ 2 } x ^ { k _ 2 } + l _ 1 = \\frac { 1 - \\theta } { \\bar { \\theta } _ 1 - \\theta } \\end{align*}"}
+{"id": "2583.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } _ t ^ * = ( \\alpha _ t ^ { * , i } ) _ { 1 \\leq i \\leq N } = \\left ( - R ^ { - 1 } B y _ t ^ { * , i } - h \\big ( \\rho _ i ^ N ( \\Delta ^ { N , 1 } _ { * , t } , \\ldots , \\Delta ^ { N , N } _ { * , t } ) \\big ) \\right ) _ { 1 \\leq i \\leq N } . \\end{align*}"}
+{"id": "6827.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ n \\alpha _ n ^ i \\partial _ t ^ { i + n + 1 } u _ { n } + \\partial _ t ^ { n + 1 } u _ { n } - \\sum _ { i = 1 } ^ { 2 n } \\beta _ n ^ i \\Delta \\partial _ t ^ i u _ n \\\\ & \\quad - \\omega _ { n + 1 } \\kappa _ n \\Delta u _ { n } - ( 1 - \\omega _ { n + 1 } ) \\kappa _ n \\int _ 0 ^ \\infty g _ { n + 1 } ( s ) \\Delta u _ { n } ( t - s ) d s = 0 . \\end{align*}"}
+{"id": "6726.png", "formula": "\\begin{align*} - \\langle \\mathcal { L } g , g \\rangle \\geq c _ 1 | g | ^ { 2 } _ { \\nu } = c _ 1 \\int _ { \\mathbb { R } ^ { 3 } } \\nu ( v ) | g | ^ { 2 } \\ , d v , \\end{align*}"}
+{"id": "832.png", "formula": "\\begin{align*} A x ^ { k _ 1 } + p ( x ) - ( A ( x ^ { * } ) ^ { k _ 1 } + p ( x ^ { * } ) ) \\left ( \\frac { x ^ { * } } { x } \\right ) ^ { - k _ 1 } = \\frac { 1 } { l _ 2 } \\left [ \\frac { 1 - \\theta } { - \\left ( \\frac { K } { A ( x ^ { * } ) ^ { k _ 1 } + p ( x ^ { * } ) } + \\theta \\right ) } - l _ 1 \\right ] . \\end{align*}"}
+{"id": "4095.png", "formula": "\\begin{align*} \\chi _ n = - n \\sum _ { k < n , k | n } \\frac { k } { n } \\chi _ { k } ^ { ( n / k ) } . \\end{align*}"}
+{"id": "4879.png", "formula": "\\begin{align*} E _ n ^ { ( \\mathbf { s } ) } ( x ) = k ^ n F _ n \\left ( x , \\frac { 1 } { k } \\right ) . \\end{align*}"}
+{"id": "4923.png", "formula": "\\begin{align*} & ~ ~ \\left ( \\frac { k - 1 } { 2 k } + \\left ( \\frac { k - 1 } { 2 k } \\right ) ^ 2 + \\cdots \\right ) \\binom { n } { ( k + r ) / 2 } + \\frac { 1 } { 2 } \\binom { n } { ( k + r ) / 2 } + \\frac { k - 1 } { 2 k } \\binom { n } { ( k + r ) / 2 } \\\\ & < \\frac { k - 1 } { k + 1 } \\binom { n } { ( k + r ) / 2 } + \\frac { 2 k - 1 } { 2 k } \\binom { n } { ( k + r ) / 2 } = \\frac { 4 k ^ 2 - k - 1 } { 2 k ^ 2 + 2 k } \\binom { n } { ( k + r ) / 2 } \\le \\binom { n } { k } . \\end{align*}"}
+{"id": "3113.png", "formula": "\\begin{align*} | X | = \\sum _ { k = 1 } ^ n \\ell _ k q _ { \\ell _ k } \\end{align*}"}
+{"id": "5604.png", "formula": "\\begin{align*} \\hat { \\mathbb { G } } ^ k _ i = \\frac { \\partial \\hat { \\mathbb { G } } ^ k } { \\partial y ^ i } . \\end{align*}"}
+{"id": "5875.png", "formula": "\\begin{align*} ( F _ 1 , ~ F _ 2 ) = \\big ( \\max ( a - 1 , ~ 2 a + b + 2 , ~ 3 a + 2 b - 5 ) , ~ \\max ( a + b , ~ 2 a + 2 b , ~ a + 2 b ) \\big ) . \\end{align*}"}
+{"id": "7124.png", "formula": "\\begin{align*} \\chi = \\sum _ { i = 1 } ^ k \\chi _ i , \\ \\chi _ i \\in M ( d _ i ) , \\ w _ i : = \\langle 1 _ { d _ i } , \\chi _ i \\rangle . \\end{align*}"}
+{"id": "4763.png", "formula": "\\begin{align*} [ a , b ] & \\overset { \\hphantom { ( 2 . 7 ) } } { = } \\partial _ 1 ( a ) \\cdot \\partial _ 2 ( b ) - \\partial _ 2 ( a ) \\cdot \\partial _ 1 ( b ) \\\\ & \\overset { \\eqref { e q : q a d m 1 } } { = } - \\eth _ 2 ( \\partial _ 1 ( a ) \\cdot b ) + \\eth _ 1 ( \\partial _ 2 ( a ) \\cdot b ) + \\eth _ 2 ( \\partial _ 1 ( a ) ) \\cdot b - \\eth _ 1 ( \\partial _ 2 ( a ) ) \\cdot b . \\end{align*}"}
+{"id": "1106.png", "formula": "\\begin{align*} \\phi ( \\lambda ^ * , \\eta ) & = \\frac { 5 9 7 } { 5 0 } \\eta ^ 2 \\frac { - 1 - \\frac { 4 0 0 } { 1 9 9 } a ^ 2 } { 2 + \\frac { 1 9 7 } { 1 9 9 } a ^ 2 } + \\frac { 1 9 9 } { 5 0 } \\frac { 1 + a ^ 2 } { 2 } - \\frac { 2 0 1 } { 1 0 0 } \\\\ \\ & \\ge \\frac { 5 9 7 } { 3 5 0 } \\frac { - 1 - \\frac { 4 0 0 } { 1 9 9 } a ^ 2 } { 2 + \\frac { 1 9 7 } { 1 9 9 } a ^ 2 } + \\frac { 1 9 9 } { 5 0 } \\frac { 1 + a ^ 2 } { 2 } - \\frac { 2 0 1 } { 1 0 0 } . \\end{align*}"}
+{"id": "1294.png", "formula": "\\begin{align*} \\pi _ a ^ i \\circ \\pi ^ 1 _ { a , b } \\circ \\iota ^ 1 _ { a , b } \\circ \\iota ^ { j _ 1 } _ a = \\begin{cases} \\mathbf { 1 } _ { a ^ i } , & i = j _ 1 , \\\\ 0 _ { a ^ { j _ 1 } , a ^ i } , & , \\end{cases} \\quad \\pi _ a ^ i \\circ \\pi ^ 1 _ { a , b } \\circ \\iota ^ 2 _ { a , b } \\circ \\iota ^ { j _ 2 } _ b = 0 _ { b ^ { j _ 2 } , a ^ i } . \\end{align*}"}
+{"id": "664.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = - \\frac { 1 } { 2 } ( \\sqrt { | c _ 1 | } \\ X _ 1 \\cdot \\nu _ 1 + \\sqrt { | c _ 2 | } \\ X _ 2 \\cdot \\nu _ 2 ) \\cdot \\varphi - \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi \\end{align*}"}
+{"id": "3999.png", "formula": "\\begin{align*} \\hat { q } _ { \\beta } ( n , t ) = \\sum _ { \\Omega ^ { n } _ { k } } \\prod _ { j = 1 } ^ { k } \\frac { ( 1 - p ) ^ { x _ { j } } } { x _ { j } } \\left ( \\frac { - \\lambda t ^ { \\beta } } { \\ln p } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } \\left ( - \\lambda t ^ { \\beta } \\right ) , \\end{align*}"}
+{"id": "4238.png", "formula": "\\begin{align*} \\left ( \\frac { q ^ { - N } } { x } \\right ) _ { n } = \\frac { ( - 1 ) ^ n ( x q ^ { N - n + 1 } ) _ n } { x ^ n q ^ { N n } } q ^ { \\frac { n ( n - 1 ) } { 2 } } \\end{align*}"}
+{"id": "6133.png", "formula": "\\begin{align*} R _ q f ( \\underline { z } ) : = f ( q \\underline { z } ) f \\in H ^ 2 ( \\mathbb D ^ d ) , \\end{align*}"}
+{"id": "3760.png", "formula": "\\begin{align*} 3 8 4 \\alpha ^ 2 + 1 3 2 \\beta ^ 2 + 6 2 4 \\alpha \\beta - 1 2 0 0 0 \\alpha - 6 2 4 0 \\beta + 6 8 4 0 0 & = 0 \\\\ 3 2 \\alpha ^ 2 - 1 6 \\beta ^ 2 - 5 6 \\alpha \\beta + 5 6 0 \\alpha + 6 8 0 \\beta - 6 0 0 0 & = 0 . \\end{align*}"}
+{"id": "5788.png", "formula": "\\begin{align*} c _ i = \\sum _ { j = 1 } ^ m \\Gamma ( c ) _ { i j } \\ , \\beta _ j \\mbox { f o r a l l $ i \\in \\{ 1 , \\ldots , n \\} $ } . \\end{align*}"}
+{"id": "2225.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } _ { k } = W _ { k } ^ D * \\boldsymbol { \\mathcal { A } } * V _ { k } , \\ ; \\ ; \\boldsymbol { \\beta } _ { k + 1 } = W _ { k + 1 } ^ D * \\boldsymbol { \\mathcal { A } } * V _ { k } . \\end{align*}"}
+{"id": "5972.png", "formula": "\\begin{align*} \\int _ { B _ { \\delta ^ { - 1 } } } | g _ { I _ 1 } \\dots g _ { I _ k } | ^ 2 \\lesssim _ { \\kappa } ( \\delta ^ { k - 1 } ) ^ k \\prod _ { j = 1 } ^ k \\int _ { B _ { \\delta ^ { - 1 } } } | g _ { I _ j } | ^ 2 = | B _ { \\delta ^ { - 1 } } | ^ { - ( k - 1 ) } \\prod _ { j = 1 } ^ k \\int _ { B _ { \\delta ^ { - 1 } } } \\Big ( \\sum _ { K _ j \\in P _ { \\delta } ( I _ j ) } | g _ { K _ j } | ^ 2 \\Big ) , \\end{align*}"}
+{"id": "3285.png", "formula": "\\begin{gather*} x _ 1 - a _ 1 x _ 2 x _ 3 = 0 , \\\\ - a _ 2 x _ 1 x _ 3 = 0 , \\\\ - a _ 3 x _ 1 x _ 2 = b _ 3 . \\end{gather*}"}
+{"id": "674.png", "formula": "\\begin{align*} a = - \\frac { 1 } { \\mu } \\left ( \\partial _ y \\mu \\ d x - \\partial _ x \\mu \\ d y \\right ) = ( \\frac { 1 } { \\mu } d \\mu - \\frac { 2 } { \\mu } \\mu _ z d z ) I \\end{align*}"}
+{"id": "2737.png", "formula": "\\begin{align*} \\sigma _ { \\Gamma } ( D ) = \\lim _ { \\epsilon \\to 0 ^ + } \\inf \\ , \\{ _ { \\Gamma } D ' \\ , | \\ , D ' \\geq 0 \\ { \\rm a n d } \\ D ' \\sim _ { \\R } D + \\epsilon A \\} , \\end{align*}"}
+{"id": "3271.png", "formula": "\\begin{align*} P _ n ( 0 ) = 1 . \\end{align*}"}
+{"id": "3506.png", "formula": "\\begin{align*} \\Big \\Vert \\nabla \\mathrm { H } \\Big \\Vert _ { \\mathbb { L } ^ { 2 } ( \\Omega ) } = \\mathcal { O } \\Big ( \\delta ^ { 1 - \\mathrm { h } } \\Big ) 0 < \\mathrm { h } < 1 . \\end{align*}"}
+{"id": "3110.png", "formula": "\\begin{align*} Y _ i \\subseteq X _ 1 \\setminus \\bigcup _ { j = 1 } ^ { i - 1 } Y _ j \\end{align*}"}
+{"id": "5761.png", "formula": "\\begin{align*} \\begin{cases} d y _ 1 \\in \\Lambda ( x _ 1 , \\cdots , x _ { n - 1 } ) \\\\ d y _ j = x _ n \\cdot \\underset { i } { \\sum } \\alpha ^ j _ i x _ i , \\alpha ^ i _ j \\in \\mathbb { Q } j \\geq 2 . \\end{cases} \\end{align*}"}
+{"id": "736.png", "formula": "\\begin{align*} { f _ { 1 7 } } = { f _ { 1 8 } } - \\frac { 1 } { 2 } \\left ( { { f _ 3 } - { f _ 4 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) + \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) - \\frac { 1 } { 8 } \\left ( { 2 { F _ y } - { F _ z } } \\right ) - \\frac { 1 } { 6 } \\rho { u _ z } . \\end{align*}"}
+{"id": "513.png", "formula": "\\begin{align*} d _ \\square ( W _ 1 , W _ 2 ) : = \\sup _ { S , T \\subset [ 0 , 1 ] } \\bigg | \\int _ { S \\times T } \\big ( W _ 1 ( u , v ) - W _ 2 ( u , v ) \\big ) d u d v \\bigg | , \\end{align*}"}
+{"id": "966.png", "formula": "\\begin{align*} \\widehat { f } ( \\xi ) : = \\int _ G f ( x ) \\ , \\overline { \\xi ( x ) } \\ , d \\mu ( x ) \\end{align*}"}
+{"id": "2239.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } _ { k } = [ q _ { k } , \\lambda * p _ { k } ] , \\boldsymbol { \\beta } _ { k + 1 } = [ q _ { k + 1 } , \\lambda * p _ { k } ] . \\end{align*}"}
+{"id": "5616.png", "formula": "\\begin{align*} \\omega ^ { \\alpha } _ { \\beta } = \\Gamma ^ { \\alpha } _ { \\beta ; \\mu } d z ^ { \\mu } + C ^ { \\alpha } _ { \\beta \\gamma } \\delta v ^ { \\gamma } , \\end{align*}"}
+{"id": "4949.png", "formula": "\\begin{align*} a _ i + 2 b _ i = \\binom { n } { k - 2 - i } \\mbox { ~ ~ f o r e a c h ~ } i \\in \\{ 1 , \\cdots , ( k - 5 ) / 2 - t \\} . \\end{align*}"}
+{"id": "6790.png", "formula": "\\begin{align*} \\phi ( x _ { [ k ] } ) = \\sum _ { i \\in [ r _ 1 ] } \\beta ^ { ( 1 ) } _ { i , 1 } ( x _ I ) \\gamma _ i ( x _ { [ k ] \\setminus I } ) + & \\sum _ { i \\in [ r _ 2 ] } \\beta ^ { ( 2 ) } _ { i , 1 } ( x _ { I ^ { ( 2 ) } _ { i , 1 } } ) \\dots \\beta ^ { ( 2 ) } _ { i , d ^ { ( 2 ) } _ i } ( x _ { I ^ { ( 2 ) } _ { i , d ^ { ( 2 ) } _ i } } ) \\\\ + & \\sum _ { i \\in [ r _ 3 ] } \\beta ^ { ( 3 ) } _ { i , 1 } ( x _ { I ^ { ( 3 ) } _ { i , 1 } } ) \\dots \\beta ^ { ( 3 ) } _ { i , d ^ { ( 3 ) } _ i } ( x _ { I ^ { ( 3 ) } _ { i , d ^ { ( 3 ) } _ i } } ) . \\end{align*}"}
+{"id": "7784.png", "formula": "\\begin{align*} g = d x ^ 2 + g _ x = d x ^ 2 + \\hat { g } + O ( x ) \\end{align*}"}
+{"id": "3291.png", "formula": "\\begin{gather*} \\delta _ { n ( k ) , n ( k ' ) } = \\sum _ { j = 0 } ^ N w ( j , k ) P _ { n ( k ) } ( m ( j ) ) \\overline { P _ { n ( k ' ) } ( m ( j ) ) } , \\end{gather*}"}
+{"id": "4871.png", "formula": "\\begin{align*} \\left ( \\frac { 2 } { 3 } \\right ) ^ n p _ n \\left ( \\frac { 1 } { 4 } \\right ) = \\sum _ { k = 0 } ^ n B _ { n - k } ^ { ( - k ) } . \\end{align*}"}
+{"id": "147.png", "formula": "\\begin{align*} \\widetilde { x } ( [ 0 , \\ldots , k + 1 ] \\otimes [ 1 , \\ldots , \\ell ] ) = x ( [ 0 , \\ldots , k ] \\otimes [ 0 , 1 , \\ldots , \\ell ] ) = 0 . \\end{align*}"}
+{"id": "612.png", "formula": "\\begin{align*} g _ I = \\big ( g _ { \\{ i \\} } \\cup g _ { I \\setminus \\{ i \\} } \\big ) \\setminus \\big ( g _ { \\{ i \\} } \\cap g _ { I \\setminus \\{ i \\} } \\big ) \\ , , \\forall i \\in I . \\end{align*}"}
+{"id": "5233.png", "formula": "\\begin{align*} \\overline { r } ^ \\vee | _ { G _ { K _ \\infty } } \\cong \\begin{pmatrix} \\psi _ 1 \\prod _ \\tau \\omega _ \\tau ^ { - t _ \\tau } & c ' \\\\ 0 & \\psi _ 2 \\prod _ \\tau \\omega _ \\tau ^ { - s _ \\tau } \\end{pmatrix} \\end{align*}"}
+{"id": "2295.png", "formula": "\\begin{align*} \\begin{cases} \\vartheta _ d > 2 / ( d + 1 ) & d = 4 , \\ , 5 , \\\\ \\vartheta _ d < 2 / ( d + 1 ) & d \\geq 6 . \\end{cases} \\end{align*}"}
+{"id": "7921.png", "formula": "\\begin{align*} \\Phi ^ n ( f ) ( a _ 1 , \\ldots , a _ n ) = f \\big ( P ( a _ 1 ) , \\ldots , P ( a _ n ) \\big ) - \\sum _ { k = 0 } ^ { n - 1 } \\lambda ^ { n - k - 1 } \\sum _ { 1 \\leq i _ 1 < \\cdots < i _ k \\leq n } ~ Q \\circ f \\big ( a _ 1 , \\ldots , P ( a _ { i _ 1 } ) , \\ldots , P ( a _ { i _ k } ) , \\ldots , a _ n \\big ) . \\end{align*}"}
+{"id": "291.png", "formula": "\\begin{align*} \\alpha _ { i ' } ' = \\begin{cases} f ^ { * } \\alpha _ { i ' } & ( i ' \\neq i ) , \\\\ \\sum _ { i '' = 1 } ^ { r '' } f ^ { * } \\alpha _ { i '' } & ( i ' = i ) \\end{cases} \\end{align*}"}
+{"id": "780.png", "formula": "\\begin{align*} \\left \\| Q _ { L } \\circ T _ { \\varphi } \\circ I _ { M } \\right \\| _ { X ; Y ^ { \\rm d u a l } } & = \\| \\Psi ^ { - 1 } ( T ) \\| = \\left \\| \\varphi | _ { M \\otimes _ { \\alpha _ { X ; Y } } L ^ \\perp } \\circ ( I d _ M \\otimes Q _ L ^ * ) \\right \\| \\\\ & \\leq \\left \\| \\varphi \\right \\| \\cdot \\left \\| I _ { M } \\otimes Q _ { L } ^ * \\right \\| \\leq \\left \\| \\varphi \\right \\| . \\end{align*}"}
+{"id": "2655.png", "formula": "\\begin{align*} A ( X _ 1 ) K ( X _ 2 , X _ 5 , X _ 3 , X _ 4 ) + A ( X _ 2 ) K ( X _ 1 , X _ 5 , X _ 3 , X _ 4 ) = 0 \\end{align*}"}
+{"id": "3495.png", "formula": "\\begin{align*} \\textbf { e r r } ^ { ( 2 ) } = \\mathcal { O } \\Bigg ( \\delta ^ { 7 } \\sqrt { \\mathcal { K } ^ { ( \\mathrm { T _ 0 } ) } _ { \\mathrm { r } } } \\frac { 1 } { | \\xi - \\cdot | ^ { 2 - 2 \\mathrm { r } } } \\Big \\Vert | \\mathrm { E } | ^ { 2 } \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\Omega ) } \\Bigg ) . \\end{align*}"}
+{"id": "7480.png", "formula": "\\begin{align*} B _ { a , b } = \\sum _ { \\beta = 1 } ^ b B _ { a - 1 , \\beta } \\ , . \\end{align*}"}
+{"id": "1182.png", "formula": "\\begin{align*} | ( \\mathcal { R } _ \\lambda ( \\sigma ) v _ T , v _ T ) | = \\prod _ { j } \\frac { 1 } { | a _ { i _ j + 1 } - a _ { i _ j } | } , \\end{align*}"}
+{"id": "4498.png", "formula": "\\begin{align*} \\delta V = \\int _ 0 ^ 1 \\frac { \\delta V } { \\delta u } \\delta u \\ , , \\end{align*}"}
+{"id": "3681.png", "formula": "\\begin{align*} \\left \\{ \\phi ( \\{ u _ { t + 1 } , u _ { t + 4 } \\} ) , \\phi ( \\{ u _ { t + 2 } , u _ { t + 3 } \\} ) \\right \\} = \\left \\{ \\{ t + 1 , t + 4 \\} , \\{ t + 2 , t + 3 \\} \\right \\} . \\end{align*}"}
+{"id": "7827.png", "formula": "\\begin{align*} J ^ { w _ 1 } ( \\textbf { X } _ { R S S } ^ { ( n ) } ) & = - \\frac { Q _ n } { 2 } \\prod _ { i = 1 } ^ { n } E \\left ( \\Lambda _ X ^ { w _ 1 } ( B _ { 2 i - 1 : 2 n - 1 } ) \\right ) \\\\ & \\le - \\frac { Q _ n } { 2 } \\prod _ { i = 1 } ^ { n } E \\left ( \\Lambda _ Y ^ { w _ 2 } ( B _ { 2 i - 1 : 2 n - 1 } ) \\right ) \\\\ & = J ^ { w _ 2 } ( \\textbf { Y } _ { R S S } ^ { ( n ) } ) , \\end{align*}"}
+{"id": "1923.png", "formula": "\\begin{align*} \\begin{aligned} \\| D _ x u \\| _ { L _ p ( Q _ r ) } & \\leq N \\sum _ { k = 0 } ^ { \\infty } 2 ^ { - 2 k } ( | ( - \\Delta _ x ) ^ { 1 / 6 } u | ^ p ) _ { Q _ { 1 , 2 ^ k } } ^ { 1 / p } , \\end{aligned} \\end{align*}"}
+{"id": "4629.png", "formula": "\\begin{align*} \\nabla \\tilde u \\cdot \\nabla h & = - \\psi ( v ) ^ { - \\ell - 1 } \\eta ^ { s } [ \\ell \\psi ' ( v ) v - \\psi ( v ) ] | \\nabla \\tilde u | ^ 2 + s \\psi ( v ) ^ { - \\ell } \\eta ^ { s - 1 } \\tilde u \\ , \\nabla \\tilde u \\cdot \\nabla \\eta . \\end{align*}"}
+{"id": "6121.png", "formula": "\\begin{align*} F _ { 2 , 1 } = [ 3 , k - 1 ] \\cup \\{ 1 \\} \\cup e _ 3 , & F _ { 2 , 2 } = [ 3 , k - 1 ] \\cup \\{ 1 \\} \\cup e _ 4 , \\\\ F _ { 2 , 3 } = [ 3 , k - 1 ] \\cup \\{ 1 \\} \\cup e _ 5 \\ & F _ { 2 , 4 } = [ 3 , k - 1 ] \\cup \\{ 1 \\} \\cup e _ 6 . \\end{align*}"}
+{"id": "4113.png", "formula": "\\begin{align*} \\{ ( x , y , \\frac { 1 - x } { 1 - x y } , 1 - x y , \\frac { 1 - y } { 1 - x y } ) : ( x , y ) \\in ( \\mathbb { P } ^ 1 - \\{ 0 , 1 , \\infty \\} ) ^ 2 - \\{ x y = 1 \\} \\} , \\end{align*}"}
+{"id": "7675.png", "formula": "\\begin{align*} ( \\Delta _ h u ) ( m ( x ) ) = \\Delta _ h ( u \\circ m ) ( x ) , \\end{align*}"}
+{"id": "6520.png", "formula": "\\begin{align*} 2 P - I = \\bigoplus _ { v \\in C _ 0 } \\frac { 1 } { 2 } J _ 2 - I = \\begin{pmatrix} \\frac { 1 } { 2 } J _ 2 - I & & & \\\\ [ 3 m m ] & \\frac { 1 } { 2 } J _ 2 - I & & \\\\ [ 3 m m ] & & \\ddots & \\\\ [ 3 m m ] & & & \\frac { 1 } { 2 } J _ 2 - I \\end{pmatrix} , \\end{align*}"}
+{"id": "4152.png", "formula": "\\begin{align*} \\frac { d } { d t } \\frac { \\int _ 0 ^ 1 u ^ 2 ( x , t ) d x } { 2 } = \\int _ 0 ^ 1 \\Big ( \\int _ { [ 0 , 1 ] } W ( x , y ) ( u ( y , t ) - u ( x , t ) ) d y \\Big ) u ( x , t ) d x \\leq 2 C { \\int _ 0 ^ 1 u ^ 2 ( x , t ) d x } \\end{align*}"}
+{"id": "5102.png", "formula": "\\begin{align*} \\Gamma _ { \\phi } f : = i [ \\nabla _ x \\cdot ( ( \\nabla _ x \\phi ) f ) + \\nabla _ x \\phi \\cdot \\nabla _ x f ] . \\end{align*}"}
+{"id": "2104.png", "formula": "\\begin{align*} \\varpi ^ { x _ i - x _ j } a ^ { 1 1 } _ { i j } + \\varpi ^ { x _ i } b ^ { 1 1 } _ { i j } = c ^ { 1 1 } _ { i j } , 1 \\le i , j \\le k . \\end{align*}"}
+{"id": "4817.png", "formula": "\\begin{align*} \\frac { \\partial \\nu } { \\partial t } = \\nabla \\cdot ( \\nu \\nabla f ( x , s ) ) + K \\int _ S ( \\nu ( x , s ' ) - \\nu ( x , s ) ) \\dd \\mu ( s ' ) , \\end{align*}"}
+{"id": "1115.png", "formula": "\\begin{gather*} \\sum _ { m = 0 } ^ { n } \\left ( - 1 \\right ) ^ { m } ( 2 m - 1 ) ! ! ( 2 n - 2 m - 1 ) ! ! \\sum _ { k = 0 } ^ { m } \\frac { 1 } { k ! ( m - k ) ! ( n - m - k ) ! } \\frac { x ^ { k } } { ( 2 k - 1 ) ! ! } \\\\ = \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ \\frac { ( n - 1 ) ! ! } { ( n / 2 ) ! } ( 2 - x ) ^ { n / 2 } & & n \\end{array} \\right . . \\end{gather*}"}
+{"id": "1568.png", "formula": "\\begin{align*} \\left ( z _ { 3 } s ^ { \\prime } z _ { 1 } \\right ) ^ { - 1 } s ^ { \\prime - 1 } \\left ( z _ { 3 } s ^ { \\prime } z _ { 1 } s ^ { \\prime } \\right ) & = \\left [ \\left ( z _ { 3 } s ^ { \\prime } z _ { 1 } \\right ) ^ { - 1 } , s ^ { \\prime - 1 } \\right ] \\mbox { a n d } \\\\ s ^ { \\prime - 1 } \\left ( z _ { 3 } s ^ { \\prime } z _ { 1 } \\right ) ^ { - 1 } \\left ( z _ { 3 } s ^ { \\prime } z _ { 1 } s ^ { \\prime } \\right ) & = e . \\end{align*}"}
+{"id": "8152.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { \\min } ( \\overline E _ m ) = \\widehat { \\mu } _ { \\max } ( f _ * ( \\overline L ^ { \\otimes m } ) ) . \\end{align*}"}
+{"id": "7197.png", "formula": "\\begin{align*} K \\left ( \\mathbb { T } _ T ( d _ 1 ) _ { v _ 1 } \\boxtimes \\cdots \\boxtimes \\mathbb { T } _ T ( d _ k ) _ { v _ k } \\right ) \\otimes _ { \\mathbb { K } } \\mathbb { F } = \\bigotimes _ { i = 1 } ^ k K _ T ( \\mathbb { T } ( d _ i ) _ { v _ i } ) \\otimes _ { \\mathbb { K } } \\mathbb { F } , \\end{align*}"}
+{"id": "5787.png", "formula": "\\begin{align*} \\rho ( \\mathcal { C } ) = m - \\min \\{ h : \\exists \\ , U \\le X \\mid \\dim ( U ) = m , \\ , U \\cap S _ \\infty = \\{ 0 \\} , \\ , U \\mbox { ~ i s ~ } ( \\mathcal { A } _ \\mathcal { C } , h ) \\mbox { - s c a t t e r e d } \\} . \\end{align*}"}
+{"id": "110.png", "formula": "\\begin{align*} \\alpha _ n ( 0 ) = h [ P _ { \\leq N } \\psi _ 0 ] ( h n ) + ( - 1 ) ^ n h [ P _ { \\leq N } \\phi _ 0 ] ( h n ) . \\end{align*}"}
+{"id": "7447.png", "formula": "\\begin{align*} \\left [ \\frac { 1 } { 2 \\hat { j } + ( 2 k + 1 ) \\hat { I } } , \\ , \\tau ^ \\dagger _ \\theta \\right ] = \\left ( \\frac { 1 } { 2 \\hat { j } + ( 2 k + 1 ) \\hat { I } } - \\frac { 1 } { 2 \\hat { j } + ( 2 ( k - \\theta ) + 1 ) \\hat { I } } \\right ) \\tau ^ \\dagger _ \\theta . \\end{align*}"}
+{"id": "7202.png", "formula": "\\begin{align*} ( \\gamma ' \\colon \\mathcal { P } _ 0 \\rightleftarrows \\mathcal { Q } _ 0 \\colon \\delta ' ) , \\ \\gamma ' = t ^ { - d } \\gamma _ 0 , \\ \\delta ' = \\delta _ 0 \\end{align*}"}
+{"id": "4918.png", "formula": "\\begin{align*} \\binom { n - 1 } { k } & > \\frac { j - 3 } { 2 } \\binom { n - 1 } { k - 1 } + ( j - 1 ) \\sum _ { i = r + 1 } ^ { k - 2 } \\binom { n - 1 } { i } + \\frac { 5 j - 1 } { 2 } \\binom { n - 1 } { r } + 2 j \\binom { n - 1 } { r - 1 } \\end{align*}"}
+{"id": "2044.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } \\ , \\mu _ n = \\sum _ { j = 1 } ^ { n + 1 } \\left [ \\sum _ { k = 1 } ^ { n - j + 2 } k \\ , 2 ^ { k - 1 } \\binom { n - k + 2 } { j } \\right ] ( - 1 ) ^ { j + 1 } \\lambda _ j \\end{align*}"}
+{"id": "2704.png", "formula": "\\begin{align*} \\binom { a + j } { j } \\sum _ { l = 0 } ^ { a } \\binom { n - j + l } { l } \\binom { n - j } { a - l } \\bigl { ( } \\binom { n - ( j + a - l ) } { j + a - l } \\sum _ { k = a + j - l } ^ { n - ( a + j - l ) } \\binom { n - 2 ( a + j - l ) } { k - ( j + a - l ) } G ( n , k , a ) \\bigr { ) } \\end{align*}"}
+{"id": "7338.png", "formula": "\\begin{align*} \\varrho ( y , z ) : = ( i , j , t , p ) , \\ \\ \\ i = | x _ 0 \\cap y | , \\ j = | x _ 0 \\cap z | , \\ t = | y \\cap z | , \\ p = | x _ 0 \\cap y \\cap z | . \\end{align*}"}
+{"id": "7500.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { E _ \\varphi \\left [ \\mu _ { \\varepsilon , h } \\right ] - E _ \\varphi \\left [ \\mu \\right ] } { \\varepsilon } = 0 \\end{align*}"}
+{"id": "141.png", "formula": "\\begin{align*} ( d _ i y ) ( [ \\mathbf { a ' } , \\mathbf { a } ] \\otimes [ 0 , \\mathbf { b } ] ) = y ( d ^ i [ \\mathbf { a ' } , \\mathbf { a } ] \\otimes [ 0 , \\mathbf { b } ] ) = 0 , \\end{align*}"}
+{"id": "4703.png", "formula": "\\begin{align*} \\| d \\nu \\| _ { { \\frak B _ { \\lambda } } } & = \\sup \\lim _ { r \\rightarrow 1 - } c _ { \\lambda } \\left | \\int _ { - \\pi } ^ { \\pi } P ( h , r e ^ { i \\theta } ) | \\sin \\theta | ^ { 2 \\lambda } d \\nu ( \\theta ) \\right | \\\\ & = \\sup \\lim _ { r \\rightarrow 1 - } \\left | \\int _ { - \\pi } ^ { \\pi } h ( \\varphi ) P ( d \\nu , r e ^ { i \\varphi } ) d m _ { \\lambda } ( \\varphi ) \\right | , \\end{align*}"}
+{"id": "1215.png", "formula": "\\begin{align*} h ( V ) & = g ( A _ 2 \\cdot V + b _ 2 ) = f ( A _ 1 ( A _ 2 \\cdot V + b _ 2 ) + b _ 1 ) = \\\\ & = f ( A _ 1 A _ 2 \\cdot V + A _ 1 b _ 2 + b _ 1 ) . \\end{align*}"}
+{"id": "161.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t F _ \\kappa + v \\cdot \\nabla _ x F _ \\kappa = \\tfrac { 1 } { \\kappa } \\mathcal { B } ( F _ \\kappa , F _ \\kappa ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "6625.png", "formula": "\\begin{align*} \\widetilde { h } _ \\theta \\big ( ( v , u ) , ( v , u ) \\big ) : = h _ { \\theta } ^ { N } ( v , v ) + h _ { \\theta } ^ { D } ( u , u ) , D ( \\widetilde { h } _ \\theta ) : = D ( h _ { \\theta } ^ { N } ) \\times D ( h _ { \\theta } ^ { D } ) , \\end{align*}"}
+{"id": "227.png", "formula": "\\begin{align*} \\partial _ t V ^ { \\sigma } ( m _ t ) & = \\int _ { \\mathbb { R } ^ d } a ( m _ t , x ) \\partial _ t m _ t ( x ) d x = \\int _ { \\mathbb { R } ^ d } a ( m _ t , x ) \\div \\left ( \\nabla a ( m _ t , x ) m _ t ( x ) \\right ) d x \\\\ & = - \\int _ { \\mathbb { R } ^ d } \\left | \\nabla a ( m _ t , x ) \\right | ^ 2 m _ t ( d x ) \\ , . \\end{align*}"}
+{"id": "5862.png", "formula": "\\begin{align*} f ^ p ( \\widetilde { A } ( \\lambda ) ) \\Big ( 1 + 2 ^ { p - 1 } \\displaystyle \\sum _ { n = 3 } ^ { \\infty } \\frac { 1 } { n ^ p } \\Big ) & < \\Big ( \\frac { p } { p - 1 } \\Big ) ^ { p } \\widetilde { f ^ p ( A ) } ( \\lambda ) . \\end{align*}"}
+{"id": "2963.png", "formula": "\\begin{align*} \\min \\{ f _ 0 ( x ) \\ , | \\ , F _ i ( x ) = 0 \\ , \\forall i \\in I _ 0 ( \\bar x ) \\} \\end{align*}"}
+{"id": "882.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\sigma _ k \\big ( \\lambda ( D ^ { 2 } u ) \\big ) & = f \\ ; \\ ; \\mbox { i n } ~ \\Omega , \\\\ u & = \\varphi \\ ; \\ ; \\mbox { o n } ~ \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "3850.png", "formula": "\\begin{align*} \\gamma _ n : = \\frac { \\log n - 5 \\log \\log n - \\log ( 2 \\pi ^ 4 ) } { 2 } . \\end{align*}"}
+{"id": "2266.png", "formula": "\\begin{align*} \\deg ( \\sigma '^ { m + 1 } ( z _ 2 ) ) & = \\deg ( ( \\tau _ 1 \\pi \\tau _ 2 \\cdots \\pi \\tau _ s \\pi ) ^ { m + 1 } ( z _ 2 ) ) \\\\ & = ( d _ 1 \\cdots d _ s ) ^ { m + 1 } \\\\ & \\geq ( d _ 1 \\cdots d _ s ) ^ { m } = 2 \\deg ( \\sigma '^ { m } ( z _ 2 ) ) . \\end{align*}"}
+{"id": "8056.png", "formula": "\\begin{align*} { y } _ { C _ u } ( t ) = & \\sum _ { k = 1 } ^ { K } ( { \\bf h } _ { u , k } ^ T { \\bf F } _ k { \\bf s } _ k e ^ { \\mathrm { j } 2 \\pi { f _ k } t } \\ ! + \\ ! \\sum _ { m = 1 } ^ { M } { \\bf h } _ { u , m , k } ^ T { \\bf \\Phi } _ m { \\bf G } _ { m , k } { \\bf F } _ k { \\bf s } _ k e ^ { \\mathrm { j } 2 \\pi { f _ k } t } \\\\ & + n _ { C _ u } ( t ) ) , \\end{align*}"}
+{"id": "1541.png", "formula": "\\begin{align*} F ^ { \\prime } \\left ( g _ { s , 1 , 2 } \\right ) & = F ^ { \\prime } \\left ( g _ { s , 2 , 3 } \\right ) = F ^ { \\prime } \\left ( g _ { s , 3 , 1 } \\right ) = F \\left ( s \\right ) \\mbox { a n d } \\\\ F ^ { \\prime } \\left ( g _ { s , 2 , 1 } \\right ) & = F ^ { \\prime } \\left ( g _ { s , 3 , 2 } \\right ) = F ^ { \\prime } \\left ( g _ { s , 1 , 3 } \\right ) = F \\left ( s \\right ) ^ { - 1 } \\end{align*}"}
+{"id": "264.png", "formula": "\\begin{align*} I _ q = \\{ ( i _ 1 - 1 ) q + 1 , \\ldots , i _ 1 q , ( i _ 2 - 1 ) q + 1 , \\ldots , i _ 2 q , \\ldots , ( i _ r - 1 ) q + 1 , \\ldots , i _ r q \\} \\end{align*}"}
+{"id": "5855.png", "formula": "\\begin{align*} b e r ^ 2 ( f ( A ) ) < \\displaystyle \\frac { 2 } { w _ 1 + 4 \\sum _ { n = 1 } ^ { \\infty } w _ { n + 1 } } b e r ( f ( A ) ^ 2 ) , \\end{align*}"}
+{"id": "7862.png", "formula": "\\begin{align*} \\frac { | k ( \\theta ) | } { d ( f , g ) } \\le \\frac { | k ( \\theta _ 0 ) | } { d ( f , g ) } + \\left | \\frac { k ( \\theta ) } { d ( f , g ) } - \\frac { k ( \\theta _ 0 ) } { d ( f , g ) } \\right | < \\frac \\kappa 2 + | l ( \\theta ) - l ( \\theta _ 0 ) | < \\frac \\kappa 2 + \\frac \\kappa 2 = \\kappa . \\end{align*}"}
+{"id": "7930.png", "formula": "\\begin{align*} \\varphi _ t \\big ( \\mu _ t ( a , b ) \\big ) = \\mu _ t ' \\big ( \\varphi _ t ( a ) , \\varphi _ t ( b ) \\big ) ~ ~ ~ ~ ~ ~ ~ ~ \\varphi _ t \\circ R _ t = R _ t ' \\circ \\varphi _ t , ~ a , b \\in A . \\end{align*}"}
+{"id": "4031.png", "formula": "\\begin{align*} e _ b ( y ) = \\left ( p - 1 + b y ^ { 2 k } \\right ) ^ { - 1 } . \\end{align*}"}
+{"id": "4899.png", "formula": "\\begin{align*} p _ n ( x ) = 2 ^ n \\sum _ { k = 0 } ^ n { n + 1 \\choose k } F _ { n - k } ( x , 1 / 2 ) F _ k ( x , 1 / 2 ) . \\end{align*}"}
+{"id": "3159.png", "formula": "\\begin{align*} f ( x , y ) \\ , \\ , = \\ , \\ , Q [ f _ { 1 } , f _ { 2 } ] ( x , y ) \\end{align*}"}
+{"id": "615.png", "formula": "\\begin{align*} \\psi _ { n , p } & = \\Big [ { \\frac { n ( 2 j ^ { ( 1 ) } + 1 - n ) ( 2 j ^ { ( 2 ) } + 1 - n ) ( 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } + 2 - n ) ( N - n - p + 2 j ^ { ( 3 ) } + 2 ) ( N - n - p + 1 ) } { ( 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } + 2 - 2 n ) ^ 2 ( 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } + 1 - 2 n ) ( 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } + 3 - 2 n ) } } \\\\ & \\qquad \\times { ( p - n - N + 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } + 1 ) ( n - p - N + 2 j ^ { ( 0 ) } + 2 j ^ { ( 4 ) } ) } \\Big ] ^ { \\frac 1 2 } \\ , , \\end{align*}"}
+{"id": "1988.png", "formula": "\\begin{align*} S _ n & : = \\{ u \\in V ( G _ n ) : | f _ n ( u ) - \\alpha _ 1 | < \\varepsilon _ 0 , ~ \\ , | \\overline { f } _ n ( u ) - \\alpha _ 2 | < \\varepsilon _ 0 \\} , \\\\ L _ n & : = \\{ u \\in V ( G _ n ) : | f _ n ( u ) - \\beta _ 1 | < \\varepsilon _ 0 , ~ \\ , | \\overline { f } _ n ( u ) - \\beta _ 2 | < \\varepsilon _ 0 \\} , \\\\ T _ n & : = V ( G _ n ) \\setminus ( S _ n \\cup L _ n ) . \\end{align*}"}
+{"id": "1194.png", "formula": "\\begin{align*} y _ { i k } = { \\bf { w } } ^ { H } _ { i } { \\bf { h } } _ { i k } s _ { i } + \\sum _ { j \\neq i } { \\bf { w } } ^ { H } _ { j } { \\bf { h } } _ { i k } s _ { j } + n _ { i k } \\end{align*}"}
+{"id": "2902.png", "formula": "\\begin{align*} \\kappa = \\frac { \\alpha } { f \\sqrt { 1 + { f ' } ^ 2 } } + \\varpi \\end{align*}"}
+{"id": "551.png", "formula": "\\begin{align*} \\partial _ { i i } f ( x ) & = V '' ( x _ i ) + \\sum _ { j \\neq i } J _ { i j } K '' ( x _ i - x _ j ) , \\partial _ { i j } f ( x ) = - J _ { i j } K '' ( x _ i - x _ j ) . \\end{align*}"}
+{"id": "3531.png", "formula": "\\begin{align*} \\Delta \\mathrm { E } + \\omega ^ 2 \\mu _ \\mathrm { m } \\varepsilon \\mathrm { E } = 0 \\end{align*}"}
+{"id": "5818.png", "formula": "\\begin{align*} c ( T _ { \\theta } ^ n ) = 0 ( n \\in \\mathbb { N } ) . \\end{align*}"}
+{"id": "5045.png", "formula": "\\begin{align*} \\Lambda _ f ( s , \\alpha ) = \\gamma ( s ) \\sum _ { n = 1 } ^ \\infty \\frac { f _ n e ( n \\alpha ) } { n ^ s } . \\end{align*}"}
+{"id": "598.png", "formula": "\\begin{align*} 1 - 3 Z ^ { + + } _ { n - 1 , p } + 3 Z ^ { + + } _ { n - 1 , p } Z ^ { + + } _ { n , p } - Z ^ { + + } _ { n - 1 , p } Z ^ { + + } _ { n , p } Z ^ { + + } _ { n + 1 , p } = 0 \\ , , \\quad Z ^ { + + } _ { n , p } = \\frac { \\psi ^ { + 0 } _ { n , p - 1 } \\ \\rho ^ { 0 + } _ { n , p } } { \\psi ^ { + 0 } _ { n , p } \\ \\rho ^ { 0 + } _ { n - 1 , p } } \\ . \\end{align*}"}
+{"id": "690.png", "formula": "\\begin{align*} \\underline { u } = \\frac { 1 } { 2 } \\sum _ { 1 \\leq j < k \\leq N } \\langle u ( e _ j ) , e _ k \\rangle \\ e _ j \\cdot e _ k . \\end{align*}"}
+{"id": "17.png", "formula": "\\begin{align*} a _ { n , k } = \\lambda ^ m a _ { n - 1 , k - 1 } + ( \\frac { \\lambda ^ m - 1 } { \\lambda - 1 } ) ^ n a _ { n - 1 , k } . \\end{align*}"}
+{"id": "1987.png", "formula": "\\begin{align*} \\mathcal { S } _ n & : = \\{ x \\in [ 0 , 1 ] : | f _ n ( x ) - \\alpha _ 1 | < \\varepsilon _ 0 , ~ \\ , | \\overline { f } _ n ( x ) - \\alpha _ 2 | < \\varepsilon _ 0 \\} , \\\\ \\mathcal { L } _ n & : = \\{ x \\in [ 0 , 1 ] : | f _ n ( x ) - \\beta _ 1 | < \\varepsilon _ 0 , ~ \\ , | \\overline { f } _ n ( x ) - \\beta _ 2 | < \\varepsilon _ 0 \\} , \\\\ \\mathcal { T } _ n & : = [ 0 , 1 ] \\setminus ( \\mathcal { S } _ n \\cup \\mathcal { L } _ n ) . \\end{align*}"}
+{"id": "8112.png", "formula": "\\begin{gather*} \\widehat { \\deg } ( \\overline V ) = - \\int _ { \\mathbb R } t \\ , \\mathrm { d } ( \\dim _ K ( \\mathcal F ^ t ( \\overline V ) ) ) , \\\\ \\widehat { \\deg } _ + ( \\overline V ) = - \\int _ 0 ^ { + \\infty } t \\ , \\mathrm { d } ( \\dim _ K ( \\mathcal F ^ t ( \\overline V ) ) ) = \\int _ { 0 } ^ { + \\infty } \\dim _ K ( \\mathcal F ^ t ( \\overline V ) ) \\ , \\mathrm { d } t . \\end{gather*}"}
+{"id": "6608.png", "formula": "\\begin{align*} b ( f , f ) = \\int _ { \\mathbb { R } } \\lvert f ' \\rvert ^ { 2 } \\dd x - \\lvert f ( 0 + ) - f ( 0 - ) \\rvert ^ { 2 } , D ( b ) = H ^ { 1 } ( \\mathbb { R } \\setminus \\{ 0 \\} ) , \\end{align*}"}
+{"id": "857.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { D } _ { K } ( p | | q ) & = \\int \\ln \\frac { p ( t ) } { q ( t ) } \\cdot p ( t ) d t \\\\ & = R ( K ) + \\int \\ln p ( t ) p ( t ) d t = R ( K ) + \\end{aligned} \\end{align*}"}
+{"id": "1288.png", "formula": "\\begin{align*} \\iota ^ i _ a = \\iota ^ { \\lfloor i / n _ 2 \\rfloor } _ 1 \\otimes \\iota ^ { i \\ , n _ 2 } _ 2 , \\quad \\pi ^ i _ a = \\pi ^ { \\lfloor i / n _ 2 \\rfloor } _ 1 \\otimes \\pi ^ { i \\ , n _ 2 } _ 2 . \\end{align*}"}
+{"id": "1685.png", "formula": "\\begin{align*} d \\mathbb { P } _ { \\mathrm { G i b b s } } ( \\varphi ) : = \\frac { 1 } { z _ { \\mathrm { G i b b s } } } \\ , e ^ { - H ( \\varphi ) } \\ , d \\varphi \\ , . \\end{align*}"}
+{"id": "4452.png", "formula": "\\begin{align*} \\nabla u _ 1 + \\nabla u _ 1 ^ T = \\nabla u _ 2 + \\nabla u _ 2 ^ T , \\end{align*}"}
+{"id": "120.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\mathcal R \\bigl [ \\eqref { a h d u h } \\bigr ] = \\eqref { d u h a m e l 1 } \\end{align*}"}
+{"id": "886.png", "formula": "\\begin{align*} S ^ { i j } _ k = \\frac { \\partial S _ k [ D ^ 2 u ] } { \\partial r _ { i j } } . \\end{align*}"}
+{"id": "7464.png", "formula": "\\begin{align*} V _ 0 u ( 0 ) + V _ 1 u ( \\ell ) = z _ C , W _ 0 u ' ( 0 ) - W _ 1 u ' ( \\ell ) = z _ K . \\end{align*}"}
+{"id": "3074.png", "formula": "\\begin{align*} F _ X \\left ( \\sigma ^ { m _ r } \\right ) \\geq \\sum _ { j = 1 } ^ r | X _ { i _ j } | = \\sum _ { j = 1 } ^ r \\ell _ { i _ j } \\end{align*}"}
+{"id": "2145.png", "formula": "\\begin{align*} W ^ { 1 , p } _ \\sharp ( Y ^ m ) = X _ p \\oplus X _ p ^ \\perp \\end{align*}"}
+{"id": "6780.png", "formula": "\\begin{align*} U _ N ( t ) b ^ * ( f ) b ( g ) U _ N ( t ) ^ * & = b ^ * ( f ) b ( g ) \\ ; f , g \\in L ^ 2 _ { \\perp \\psi _ t } ( \\mathbb { R } ^ 3 ) \\end{align*}"}
+{"id": "2868.png", "formula": "\\begin{align*} \\| \\mu _ 1 - \\mu _ 2 \\| _ { L , M } ^ * : = \\sup _ { f \\in C _ b ( M ) , \\ , | f | _ L \\leqslant 1 } \\Big | \\langle f , \\mu _ 1 \\rangle - \\langle f , \\mu _ 2 \\rangle \\Big | \\le 2 , \\end{align*}"}
+{"id": "4592.png", "formula": "\\begin{align*} \\frac { 2 } { 3 } \\leq & \\sum _ { m = m _ { 2 l } } ^ { m _ { 2 l + 1 } } \\frac { K } { ( n _ 0 + m N - b ) \\pi \\sin \\pi k } ( 1 0 0 + \\varepsilon ( m ) ) \\\\ \\leq & 1 0 1 \\frac { K } { N \\pi \\sin \\pi k } \\sum _ { m = m _ { 2 l } } ^ { m _ { 2 l + 1 } } \\frac { 1 } { \\frac { n _ 0 - b } { N } + m } \\\\ \\leq & 1 0 2 C _ k \\ln \\frac { m _ { 2 l + 1 } N + n _ 0 - b } { m _ { 2 l } N + n _ 0 - b } , \\end{align*}"}
+{"id": "7641.png", "formula": "\\begin{align*} \\sum \\nu _ i + \\mu _ 1 = \\sum _ { i = 0 } ^ { 1 1 } \\lambda _ i = 1 . \\end{align*}"}
+{"id": "1847.png", "formula": "\\begin{align*} x _ { i - 1 , j - 1 } & + x _ { i - 1 , j } + x _ { i , j - 1 } + x _ { i , j } = S \\\\ & \\mbox { a n d } \\\\ x _ { m + 2 - i , n + 2 - j } & + x _ { m + 2 - i , n + 1 - j } + x _ { m + 1 - i , n + 2 - j } + x _ { m + 1 - i , n + 1 - j } = S , \\end{align*}"}
+{"id": "2421.png", "formula": "\\begin{align*} \\tilde { L } ( { \\rm D } ( w ) ) = \\sum _ { N \\geq 2 } ( - 1 ) ^ { N } \\zeta ^ { \\mathfrak { m } } ( N ) \\tilde { L } ( { \\rm C u t } _ { ( \\{ 0 \\} ^ { N - 1 } ; \\emptyset ) } ( w ) ) + \\tilde { L } ( { \\rm D } ( { \\rm r e g } _ { 0 } ( w ) ) ) \\end{align*}"}
+{"id": "2780.png", "formula": "\\begin{align*} H ( A ) = \\begin{bmatrix} 0 & A \\\\ A ^ T & 0 \\end{bmatrix} , \\end{align*}"}
+{"id": "3097.png", "formula": "\\begin{align*} \\sigma _ 4 ^ k = \\begin{cases} \\delta _ { 3 , 0 } ^ k & \\\\ \\delta _ { 3 , 0 } ^ k \\prod _ { n \\in \\Z } \\alpha _ { 3 n + 1 } & \\end{cases} \\end{align*}"}
+{"id": "430.png", "formula": "\\begin{align*} \\mathsf { S u p } ^ { m } ( g ) \\cap \\mathsf { S u p } ^ { m } ( h ) = \\emptyset \\end{align*}"}
+{"id": "8089.png", "formula": "\\begin{align*} \\chi _ 1 = \\frac { \\lambda _ 1 q ^ 2 + h ^ 2 } { q ^ 2 + h ^ 2 } = \\frac { \\lambda _ 1 \\left ( \\frac { q } { h } \\right ) ^ 2 + 1 } { \\left ( \\frac { q } { h } \\right ) ^ 2 + 1 } . \\end{align*}"}
+{"id": "7771.png", "formula": "\\begin{align*} \\begin{aligned} 1 & = - \\int _ t ^ { + \\infty } f ' ( s ) d s = \\int _ t ^ { + \\infty } - f ' ( s ) [ A ( s ) ] ^ { \\frac { 1 } { p } } [ A ( s ) ] ^ { - \\frac { 1 } { p } } d s \\\\ & \\leq ( \\int _ t ^ { + \\infty } [ f ' ( s ) ] ^ p A ( s ) d s ) ^ { \\frac { 1 } { p } } ( \\int _ t ^ { + \\infty } [ A ( s ) ] ^ { \\frac { 1 } { 1 - p } } d s ) ^ { 1 - \\frac { 1 } { p } } \\end{aligned} \\end{align*}"}
+{"id": "2773.png", "formula": "\\begin{align*} A = U \\Sigma V ^ T , \\end{align*}"}
+{"id": "7875.png", "formula": "\\begin{align*} t ' = C _ R t \\Delta / \\delta . \\end{align*}"}
+{"id": "8159.png", "formula": "\\begin{align*} \\widehat { \\deg } ( \\overline Q ) \\geqslant \\sum _ { i = 1 } ^ d \\widehat { \\mu } _ { \\min } ( \\overline E _ i ) . \\end{align*}"}
+{"id": "2820.png", "formula": "\\begin{align*} K _ s = \\frac { 1 } { 2 + ( T - s ) \\rho } , s \\in [ 0 , T ] . \\end{align*}"}
+{"id": "1432.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { U } _ { N L } ( t ) & = \\mathcal { A } \\mathcal { U } _ { N L } ( t ) + \\begin{pmatrix} 0 \\\\ - | u | ^ { p - 1 } u ( t ) \\end{pmatrix} . \\end{align*}"}
+{"id": "24.png", "formula": "\\begin{align*} A _ { l , u } & = \\left \\{ i \\in [ k ] : \\frac { p _ i } { q _ i } \\in [ l , u ) \\right \\} \\ , \\ , \\ , \\ , \\\\ A _ { l , \\infty } & = \\left \\{ i \\in [ k ] : \\frac { p _ i } { q _ i } \\in [ l , \\infty ] \\right \\} . \\end{align*}"}
+{"id": "5809.png", "formula": "\\begin{align*} \\Pi ( V _ s \\oplus V _ u ) = ( M _ z \\oplus V _ u ) \\Pi , \\end{align*}"}
+{"id": "138.png", "formula": "\\begin{align*} x ( [ \\mathbf { a } ] \\otimes [ \\mathbf { b } ] ) = 0 . \\end{align*}"}
+{"id": "3956.png", "formula": "\\begin{align*} \\Omega ^ { n } _ { k } = \\left \\{ ( x _ { 1 } , x _ { 2 } , \\dots , x _ { k } ) : \\sum _ { i = 1 } ^ { k } x _ { i } = n , \\ k \\le n , \\ x _ { i } \\ge 1 \\right \\} . \\end{align*}"}
+{"id": "6887.png", "formula": "\\begin{align*} X = X _ n \\to X _ { n - 1 } \\to \\cdots \\to X _ 1 \\to \\textrm { \\rm S p e c } ( F ) . \\end{align*}"}
+{"id": "8221.png", "formula": "\\begin{align*} \\begin{aligned} \\Vert \\dot \\Delta _ j \\mathbb { P } u ( t ) \\Vert _ { L ^ 2 } + \\mu 2 ^ { j \\alpha } \\int _ 0 ^ t \\Vert \\dot \\Delta _ j \\mathbb { P } u \\Vert _ { L ^ 2 } \\dd \\tau \\le 2 \\Vert \\dot \\Delta _ j \\mathbb { P } u _ 0 \\Vert _ { L ^ 2 } + C \\int _ 0 ^ t \\Big ( \\Vert \\widetilde { g } _ j \\Vert _ { L ^ 2 } + \\Vert \\nabla v \\Vert _ { L ^ { \\infty } } \\Vert \\dot \\Delta _ j \\mathbb { P } u \\Vert _ { L ^ 2 } \\Big ) \\dd \\tau . \\end{aligned} \\end{align*}"}
+{"id": "5096.png", "formula": "\\begin{align*} u ( x , y , t ) = e ^ { i t \\mathfrak { m } ( \\nabla / i ) } g . \\end{align*}"}
+{"id": "7262.png", "formula": "\\begin{align*} \\int g + \\epsilon d \\mu & = - \\int - ( g + \\epsilon ) d \\mu \\\\ & \\geq - \\overline { { m d i m } } _ M ( T , f - ( g + \\epsilon ) , d ) + \\over \\\\ & \\geq - \\overline { { m d i m } } _ M ( T , f - \\inf ( g + \\epsilon ) , d ) + \\over \\\\ & = \\inf ( g + \\epsilon ) > 0 . \\end{align*}"}
+{"id": "1710.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to \\infty } \\mathcal { Z } _ { \\tau } = z \\ , . \\end{align*}"}
+{"id": "3914.png", "formula": "\\begin{align*} \\| \\Phi ( u ) ( t + h ) & - \\Phi ( u ) ( t ) \\| ^ 2 _ { \\mathbb H ^ \\mu } \\le 2 \\| [ S ( t + h ) - S ( t ) ] \\xi \\| ^ 2 _ { \\mathbb H ^ \\mu } \\\\ & + 4 \\int _ 0 ^ t \\tau ^ { - \\delta } \\| D _ h f ( u ) ( t - \\tau ) \\| ^ 2 _ { \\mathbb H ^ { \\mu - 1 - \\delta } } d \\tau \\\\ & + 4 \\int _ t ^ { t + h } \\tau ^ { - \\delta } \\| f ( u ( t + h - \\tau ) , \\mathcal H u ( t + h - \\tau ) ) \\| ^ 2 _ { \\mathbb H ^ { \\mu - 1 - \\delta } } d \\tau \\\\ & = E _ 1 ( t ) + E _ 2 ( t ) + E _ 3 ( t ) . \\end{align*}"}
+{"id": "5629.png", "formula": "\\begin{align*} \\left ( \\nabla _ { y ^ { \\mathcal { H } } } J \\right ) y = \\nabla _ { y ^ { \\mathcal { H } } } \\left ( J y \\right ) = 0 . \\end{align*}"}
+{"id": "3232.png", "formula": "\\begin{align*} \\theta ^ { n } _ { \\varphi _ { j } } = e ^ { \\varphi _ { j } - u _ { j } } \\theta ^ { n } _ { u _ { j } } + e ^ { \\varphi _ { j } - v _ { j } } \\theta ^ { n } _ { v _ { j } } . \\end{align*}"}
+{"id": "5831.png", "formula": "\\begin{align*} T T _ 2 = T _ 2 T . \\end{align*}"}
+{"id": "3038.png", "formula": "\\begin{align*} \\begin{aligned} y _ 1 & = \\bar { u } _ 1 \\\\ y _ 2 & = ( \\bar { u } _ 1 ^ 2 x _ 1 - 2 \\bar { u } _ 1 x _ 2 + 2 x _ 3 ) \\bar { u } _ { 1 , 1 } - 2 \\bar { u } _ 1 x _ 1 + 2 x _ 2 \\ , , \\end{aligned} \\end{align*}"}
+{"id": "6985.png", "formula": "\\begin{align*} \\lambda = \\int _ { X } \\left ( | \\nabla u | ^ 2 + \\alpha _ 1 u ^ 2 - \\frac { \\alpha _ 2 } { 2 } u ^ 2 \\log u ^ 2 \\right ) d m . \\end{align*}"}
+{"id": "5872.png", "formula": "\\begin{align*} f _ { I \\Gamma } : = ( f _ { i \\Gamma } ) _ { i \\in I } . \\end{align*}"}
+{"id": "4036.png", "formula": "\\begin{align*} \\| g \\| _ { L ^ \\infty _ M } = \\| ( 1 + | y | ^ M ) ^ { - 1 } g \\| _ { L ^ \\infty } , \\end{align*}"}
+{"id": "173.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi _ 2 ( \\gamma ) : = \\{ b _ 0 | \\ , - 2 ( 2 - b _ 0 - \\gamma ) > - 3 \\ , , 2 \\gamma > - 3 \\} . \\end{aligned} \\end{align*}"}
+{"id": "7036.png", "formula": "\\begin{align*} \\nu ^ { ( 2 ) } = \\nu ^ { ( 1 ) } + \\hat { \\nu ^ { ( 1 ) } } = \\nu + \\hat { \\nu } + \\hat { \\hat { \\nu } } . \\end{align*}"}
+{"id": "4503.png", "formula": "\\begin{align*} 0 \\leq \\frac { 1 } { \\| h \\| } _ { \\ ! H ^ 1 } \\ ! \\bigg | \\int _ { [ 0 , 1 ] } ( h ' ) ^ 2 \\bigg | \\leq \\frac { 1 } { \\| h \\| } _ { \\ ! H ^ 1 } \\ ! \\bigg | \\int _ { [ 0 , 1 ] } \\big ( h ^ 2 + ( h ' ) ^ 2 \\big ) \\bigg | = \\| h \\| _ { H ^ 1 } \\ , , \\end{align*}"}
+{"id": "5378.png", "formula": "\\begin{align*} ( A , \\ , [ u ] , \\ , E _ ) \\triangleleft g = ( g ^ { - 1 } A g + g ^ { - 1 } d g , \\ , [ u g ] , \\ , g ^ { - 1 } E _ g ) \\end{align*}"}
+{"id": "7683.png", "formula": "\\begin{align*} | \\partial A _ t | _ h ^ 2 & = \\left ( \\int _ { \\partial A _ t } \\frac { 2 ^ { n - 1 } d S } { ( 1 - | x | ^ 2 ) ^ { n - 1 } } \\right ) ^ { 2 } \\\\ & \\le \\int _ { \\partial A _ t } | \\nabla u | ^ { - 1 } \\frac { 2 ^ { n } d S } { ( 1 - | x | ^ 2 ) ^ n } \\int _ { \\partial A _ t } \\frac { | \\nabla u | 2 ^ { n - 2 } d S } { ( 1 - | x | ^ 2 ) ^ { n - 2 } } . \\end{align*}"}
+{"id": "5664.png", "formula": "\\begin{align*} 2 \\mathbb { G } ^ \\alpha ( T _ o ) \\frac { \\partial r } { \\partial z ^ \\alpha } & = ( \\hat { \\mathbb { G } } ^ { \\alpha } ( T ) + \\sqrt { - 1 } \\hat { \\mathbb { G } } ^ { \\alpha + n } ( T ) ) ( \\frac { \\partial r } { \\partial x ^ \\alpha } - \\sqrt { - 1 } \\frac { \\partial r } { \\partial x ^ { \\alpha + n } } ) \\\\ & = \\hat { \\mathbb { G } } ^ { k } ( T ) \\frac { \\partial r } { \\partial x ^ k } - \\sqrt { - 1 } \\hat { \\mathbb { G } } ^ { i } ( T ) J ^ k _ i \\frac { \\partial r } { \\partial x ^ k } . \\end{align*}"}
+{"id": "3381.png", "formula": "\\begin{align*} W _ { \\mathcal { M } } : = \\int [ \\Gamma _ \\gamma ] d \\mathcal { M } ( \\gamma ) , \\end{align*}"}
+{"id": "7275.png", "formula": "\\begin{align*} \\operatorname { p } ^ 1 \\sharp \\nu ^ * ( t ) = m ^ * ( t ) \\qquad \\forall t \\in [ 0 , T ] ; \\end{align*}"}
+{"id": "166.png", "formula": "\\begin{align*} \\begin{aligned} v ' = \\xi + v \\ , , \\ v _ 1 = v + \\xi + y \\ , . \\end{aligned} \\end{align*}"}
+{"id": "8238.png", "formula": "\\begin{align*} X _ j ( t ) : = Y _ j ( t ) + \\Vert \\dot \\Delta _ j \\mathbb { P } u ( t ) \\Vert _ { L ^ 2 } \\approx \\begin{cases} \\Vert \\dot { \\Delta } _ j \\sigma ( t ) \\Vert _ { L ^ 2 } + \\Vert \\dot { \\Delta } _ j u ( t ) \\Vert _ { L ^ 2 } , & \\textrm { f o r } \\ ; \\ ; j \\leq j _ 0 , \\\\ \\Vert \\Lambda ^ { \\alpha - 1 } \\dot { \\Delta } _ j \\sigma ( t ) \\Vert _ { L ^ 2 } + \\Vert \\dot { \\Delta } _ j u ( t ) \\Vert _ { L ^ 2 } , & \\textrm { f o r } \\ ; \\ ; j > j _ 0 . \\end{cases} \\end{align*}"}
+{"id": "6146.png", "formula": "\\begin{align*} h \\mapsto \\begin{bmatrix} D _ { V ^ * } ( I - z V ^ * ) ^ { - 1 } h \\\\ \\lim _ { n } V ^ n V ^ { * n } h \\end{bmatrix} . \\end{align*}"}
+{"id": "4188.png", "formula": "\\begin{align*} l ( w ^ j ) = \\frac { 1 } { \\mathrm { s i n } ( \\theta _ j ' ) } \\underset { l } \\sum i ( w ^ j , ( w _ 1 ) _ l ) \\mathrm { h e i g h t } ( C _ 1 ^ l ) \\end{align*}"}
+{"id": "2791.png", "formula": "\\begin{align*} Q _ A = U _ A C W ^ T , Q _ B = U _ B S W ^ T , \\end{align*}"}
+{"id": "3533.png", "formula": "\\begin{align*} \\mathrm { E } ( \\mathrm { x } ) - \\omega ^ 2 \\mu _ \\mathrm { m } \\Big ( \\varepsilon _ \\mathrm { p } ( \\omega , \\gamma ) - \\varepsilon _ \\mathrm { m } \\Big ) \\int _ \\Omega \\mathbb { G } ^ { ( \\mathrm { k } ) } ( \\mathrm { x } , \\mathrm { y } ) \\mathrm { E } ( \\mathrm { y } ) d \\mathrm { y } = \\mathrm { E } ^ { \\textbf { i n } } ( \\mathrm { x } ) \\end{align*}"}
+{"id": "7020.png", "formula": "\\begin{align*} \\begin{cases} w _ { t t } + \\Delta ^ 2 w = 0 , \\\\ ( w , w _ t ) ( 0 , x ) = ( w _ 0 , w _ 1 ) ( x ) , \\end{cases} \\end{align*}"}
+{"id": "4054.png", "formula": "\\begin{align*} \\left | \\langle f , H _ m \\rangle _ { L ^ 2 _ { \\rho _ s } } \\right | \\leq \\left \\{ \\begin{array} { r c l } C I ^ { - 2 m } & i f & m \\leq \\ell \\\\ 0 & i f & m > \\ell \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "8083.png", "formula": "\\begin{align*} q = C _ I k F r . \\end{align*}"}
+{"id": "61.png", "formula": "\\begin{align*} f ( x _ 1 , . . . , x _ n ) = \\sqrt { x _ 1 ^ 2 + . . . + x _ { n - 1 } ^ 2 } = | x ' | \\end{align*}"}
+{"id": "1829.png", "formula": "\\begin{align*} x \\circ y = x \\rhd y + \\frac { 1 } { 2 } [ x , y ] . \\end{align*}"}
+{"id": "5459.png", "formula": "\\begin{align*} \\det M = \\det \\begin{pmatrix} J - A & \\omega ^ 2 J - B & \\omega J - C \\\\ C & A & B \\\\ B & C & A \\end{pmatrix} . \\end{align*}"}
+{"id": "3539.png", "formula": "\\begin{align*} \\gamma ^ { \\textbf { e x t } } _ { 1 } \\mathrm { U } _ { \\mathrm { e } } = - \\mathbb { H } _ { \\alpha _ \\mathrm { m } } \\Big [ \\gamma ^ { \\textbf { e x t } } _ { 0 } \\mathrm { U } _ { \\mathrm { e } } \\Big ] ( \\mathrm { x } , t ) + \\Big ( \\frac { 1 } { 2 } I _ { d } - \\mathcal { K } _ { \\alpha _ \\mathrm { m } } ^ { * } \\Big ) \\Big [ \\gamma ^ { \\textbf { e x t } } _ { 1 } \\mathrm { U } _ { \\mathrm { e } } \\Big ] ( \\mathrm { x } , t ) . \\end{align*}"}
+{"id": "1185.png", "formula": "\\begin{align*} ( \\mathcal { R } _ \\lambda ( \\sigma s _ i ) v _ T , v _ T ) = \\pm \\frac { 1 } { a _ { i + 1 } - a _ { i } } ( \\mathcal { R } _ \\lambda ( \\sigma ) v _ T , v _ T ) . \\end{align*}"}
+{"id": "7373.png", "formula": "\\begin{align*} \\quad & f ( x ) : = x ^ \\top A x , \\\\ \\quad & x \\in \\mathbb { S } ^ { n - 1 } : = \\{ x \\in \\real ^ n : \\norm { x } = 1 \\} , \\end{align*}"}
+{"id": "767.png", "formula": "\\begin{align*} E ^ + - E ^ { - } = f ^ * ( f _ * D ) - D = f ^ * ( f _ * N ) - N + f ^ * ( f _ * P ) - P = E _ N ^ + - E _ N ^ { - } + E _ P . \\end{align*}"}
+{"id": "1103.png", "formula": "\\begin{align*} \\mathbf { Y } = \\mathbf { F X F } ^ \\top , \\end{align*}"}
+{"id": "3210.png", "formula": "\\begin{align*} h _ { g _ { u } } ( \\theta ) & = I ( g _ { u } ( \\theta ) ) - \\theta ^ { 2 } I ( u ) \\\\ & = ( \\theta ^ { 2 - 2 \\beta } - \\theta ^ { 2 } ) A ( u ) + ( \\theta ^ { ( 1 - \\frac { 3 } { 2 } \\beta ) p + 3 \\beta } - \\theta ^ { 2 } ) C ( u ) - \\theta ^ { 2 } B ( u ) \\\\ & \\ \\ \\ + \\frac { 1 } { 4 } \\theta ^ { 4 - \\beta } \\int _ { \\mathbb { R } ^ { 3 } } \\int _ { \\mathbb { R } ^ { 3 } } \\frac { 1 - e ^ { - \\theta ^ { \\beta } | x - y | } } { | x - y | } u ( x ) ^ { 2 } u ( y ) ^ { 2 } d x d y , \\end{align*}"}
+{"id": "3530.png", "formula": "\\begin{align*} \\Big \\Vert | \\mathrm { E } | ^ 2 \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\Omega ) } = \\mathcal { O } \\Big ( \\delta ^ { - 1 } \\Big ) . \\end{align*}"}
+{"id": "2229.png", "formula": "\\begin{align*} \\boldsymbol { \\mathcal { A } } _ { k , \\ell , : , : } : = \\boldsymbol { \\nu } _ { k , \\ell } , k , \\ell = 1 , \\dots , N , \\end{align*}"}
+{"id": "8169.png", "formula": "\\begin{align*} F ( \\lambda ) = E ( \\lambda ) ^ { 1 / 2 } [ I - D ( \\lambda ) ] E ( \\lambda ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "3761.png", "formula": "\\begin{align*} \\langle d x , y \\rangle + ( - 1 ) ^ x \\langle x , d y \\rangle = 0 . \\end{align*}"}
+{"id": "4262.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\frac { q ^ k } { 1 - q ^ k } - \\sum _ { k = 1 } ^ { n - 1 } \\frac { x q ^ k } { 1 - x q ^ k } = \\frac { x } { 1 - x } - \\frac { 1 } { ( x ) _ n } \\sum _ { k = 1 } ^ { n } \\left [ \\begin{matrix} n \\\\ k \\end{matrix} \\right ] \\frac { \\left ( \\frac { q } { x } \\right ) _ k ( x ) _ { n - k } x ^ k } { 1 - q ^ k } . \\end{align*}"}
+{"id": "3374.png", "formula": "\\begin{align*} m _ { d + 1 } = \\alpha _ 1 q + \\dots + \\alpha _ d q + \\beta \\ell \\ , . \\end{align*}"}
+{"id": "6845.png", "formula": "\\begin{align*} \\lambda u _ i h ( a _ i ) = v _ i ^ { p ^ e + 1 } , \\ 1 \\leq i \\leq n , \\end{align*}"}
+{"id": "366.png", "formula": "\\begin{align*} I _ \\mu = \\Big \\langle \\mu _ \\xi \\Big \\rangle _ { \\xi \\in \\mathfrak { g } } ^ { } \\\\ = \\left \\{ \\left . \\sum _ { i = 1 } ^ n f _ i ~ \\mu _ { \\xi _ i } \\right | ~ n \\geq 0 , ~ f _ i \\in C ^ \\infty ( M ) , ~ \\xi _ i \\in \\mathfrak { g } , ~ 1 \\leq i \\leq n \\right \\} ~ . \\end{align*}"}
+{"id": "7639.png", "formula": "\\begin{align*} Y = \\sum _ { i = 1 } ^ { 5 } \\nu _ i P _ i + \\sum _ { i = 7 } ^ { 8 } \\nu _ i P _ i + \\mu _ 1 S _ 1 . \\end{align*}"}
+{"id": "7509.png", "formula": "\\begin{align*} F _ { \\varepsilon , h } = \\{ x + \\varepsilon h ( x ) \\mid x \\in F \\} . \\end{align*}"}
+{"id": "8208.png", "formula": "\\begin{align*} \\begin{aligned} \\Big \\| \\Lambda ^ { \\alpha } ( u v ) - \\sum _ { | k | \\le \\alpha _ 1 } \\frac { 1 } { k ! } \\partial ^ k u \\ , \\Lambda ^ { \\alpha , k } v - \\sum _ { | m | \\le \\alpha _ 2 } \\frac { 1 } { m ! } \\partial ^ { m } v \\ , \\Lambda ^ { \\alpha , m } u \\Big \\| _ { L ^ p } \\lesssim _ { \\alpha , \\alpha _ 1 , \\alpha _ 2 , p , p _ 1 , p _ 2 , N } \\| \\Lambda ^ { \\alpha _ 1 } u \\| _ { L ^ { p _ 1 } } \\| \\Lambda ^ { \\alpha _ 2 } v \\| _ { L ^ { p _ 2 } } , \\end{aligned} \\end{align*}"}
+{"id": "7128.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k v _ i \\tau _ { d _ i } = - \\sum _ { \\ell \\in T } 3 \\left ( r _ \\ell - \\frac { 1 } { 2 } \\right ) \\mathfrak { g } _ \\ell + w \\tau _ d \\end{align*}"}
+{"id": "2493.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ( r ) & = u ( 0 ) + \\int _ { 0 } ^ { r } t ^ { 1 - N } \\left ( \\int _ { 0 } ^ { t } s ^ { a } v ^ { p } ( s ) d s \\right ) d t , & & r > 0 , \\\\ v ( r ) & = v ( 0 ) + \\int _ { 0 } ^ { r } t ^ { 1 - N } \\left ( \\int _ { 0 } ^ { t } s ^ { b } v ^ { q } ( s ) f ( | u ' ( s ) | ) d s \\right ) d t , & & r > 0 , \\\\ u ( 0 ) & > 0 , v ( 0 ) > 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "4149.png", "formula": "\\begin{align*} \\| W \\| _ \\Box = \\sup _ { S , T \\subseteq [ 0 , 1 ] } \\left | \\int _ { S \\times T } W ( x , y ) d x d y \\right | , \\end{align*}"}
+{"id": "1428.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\| \\mathcal { U } ^ { ( j ) } ( t ) - \\mathcal { U } ( t ) \\| _ { \\mathcal { H } } = 0 . \\end{align*}"}
+{"id": "5316.png", "formula": "\\begin{align*} u \\leq _ { j _ { 1 } } u _ { 1 } \\leq _ { j _ { 2 } } \\dotsb \\leq _ { j _ { r } } u _ { r } = v . \\end{align*}"}
+{"id": "4057.png", "formula": "\\begin{align*} \\begin{array} { l l l } | \\tilde \\theta ( z ) | & \\leq & C e ^ { ( \\tau - \\sigma ) } e ^ { - ( [ M ] + 1 ) \\frac { \\tau - \\sigma } { 2 } } I ^ { - M } ( 1 + | z | ^ M ) | q _ - | _ { \\sigma } \\mbox { f o r } 0 \\leq m \\leq [ M ] + 1 . \\end{array} \\end{align*}"}
+{"id": "501.png", "formula": "\\begin{align*} \\exp ^ { H ( \\xi ) } = R ( H _ 1 ^ { \\mu _ 1 } ( \\xi ) H _ 2 ^ { \\mu _ 2 } ( \\xi ) ) \\prod _ { j = 3 } ^ K H _ j ^ { \\mu _ j } ( \\xi ) , \\ \\xi \\in V , \\end{align*}"}
+{"id": "342.png", "formula": "\\begin{align*} | \\hat { h } - \\tilde { h } | = \\lim _ { n , m \\rightarrow \\infty } \\frac { | A _ { n , m } | } { | B _ { n , m } + \\frac { 1 } { 2 } C _ { n , m } \\zeta ( \\hat { h } - \\tilde { h } ) | } = 0 \\quad . \\end{align*}"}
+{"id": "7017.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } m \\left ( \\left | f _ k ( x ) - f ( x ) \\right | > \\varepsilon \\right ) = 0 . \\end{align*}"}
+{"id": "7727.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { A } ( E ) & = 2 ^ n \\cdot \\lim \\limits _ { t \\rightarrow + \\infty } e ^ { - n t } V o l ( \\partial B _ E ^ { g ^ + } ( t ) , g ^ + ) \\\\ & = 2 ^ n \\cdot \\lim \\limits _ { t \\rightarrow + \\infty } V o l ( \\partial B _ E ^ { g ^ + } ( t ) , g ^ E | _ { \\partial B _ E ( t ) } ) \\\\ & = 2 ^ n \\cdot V o l ( \\partial X , \\hat { g } ^ E ) \\end{aligned} \\end{align*}"}
+{"id": "902.png", "formula": "\\begin{align*} \\begin{aligned} 0 = \\ , & ( A \\Psi \\pm T _ \\alpha ( u - \\varphi ) ) _ n \\\\ = \\ , & - A + 2 A K x _ n \\\\ & \\pm \\left [ ( u - \\varphi ) _ { \\alpha n } + \\sum _ { \\beta < n } B _ { \\alpha \\beta } ( x _ { \\beta } ( u - \\varphi ) _ { n n } - x _ n ( u - \\varphi ) _ { \\beta n } ) - \\sum _ { \\beta < n } B _ { \\alpha \\beta } ( u - \\varphi ) _ { \\beta } \\right ] \\\\ < \\ , & 0 , \\end{aligned} \\end{align*}"}
+{"id": "4347.png", "formula": "\\begin{align*} H ( u _ 1 ( t ) | u _ 2 ( t ) ) - H ( u _ 1 ^ { i n } | u _ 2 ^ { i n } ) = T ^ 2 ( t ) \\qquad \\end{align*}"}
+{"id": "520.png", "formula": "\\begin{align*} V _ { \\mathrm { o r i g } } = \\sup _ { Q \\in \\P ( \\R ^ n ) } \\left ( \\int _ { \\R ^ n } g \\ , d Q - \\frac { 1 } { n } H ( Q \\ , | \\ , \\gamma _ T ) \\right ) = \\frac { 1 } { n } \\log \\int _ { \\R ^ n } e ^ { n g } \\ , d \\gamma _ T , \\end{align*}"}
+{"id": "1916.png", "formula": "\\begin{align*} P _ 0 u _ { ( \\varepsilon ) } + \\lambda u _ { ( \\varepsilon ) } = g _ { \\varepsilon } , \\end{align*}"}
+{"id": "964.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac 1 { t ^ n | A | } \\int _ { A _ t } f ( x ) \\ , d x = M ( f ) , \\end{align*}"}
+{"id": "34.png", "formula": "\\begin{align*} q ' _ j & = \\sum _ { i \\in A _ j } q _ i = \\sum _ { i \\in [ m ] : \\tilde { \\delta } _ i \\in [ \\nu _ j , \\nu _ { j + 1 } ) } q _ i = \\sum _ { i \\in [ m ] : \\tilde { \\delta } _ i \\in [ \\nu _ j , \\nu _ { j + 1 } ) } 0 . 5 \\tilde { q } _ i = 0 . 5 \\P \\{ \\tilde { X } \\in [ \\nu _ j , \\nu _ { j + 1 } ) \\} . \\end{align*}"}
+{"id": "3587.png", "formula": "\\begin{align*} \\eta _ { i j } = & \\sum _ m b _ m \\theta _ { m i } \\theta _ { m j } = \\frac { 1 } { \\pi ^ 2 } \\sum _ m b _ m \\frac { \\omega _ m } { \\omega _ m ^ 2 - i ^ 2 } \\frac { \\omega _ m } { \\omega _ m ^ 2 - j ^ 2 } \\sin ^ 2 ( \\pi \\omega _ m ) ( - 1 ) ^ { i + j } \\\\ \\stackrel { ( * ) } { \\lesssim } & ( i j ) ^ { - \\emph { m i n } \\left ( 1 , \\frac { \\delta + 1 } { 2 } \\right ) } , \\end{align*}"}
+{"id": "4352.png", "formula": "\\begin{align*} I ' ( t ) = \\frac { \\mathrm { d } } { \\mathrm { d } t } \\int _ \\Omega \\Phi ( f ( t ) ) \\ , \\mathrm { d } x = \\int _ \\Omega \\Phi ' ( f ( t ) ) \\partial _ t f ( t ) \\ , \\mathrm { d } x , t \\in \\mathcal { I } \\ , . \\end{align*}"}
+{"id": "1963.png", "formula": "\\begin{align*} \\alpha = s + \\sum _ { j = 1 } ^ d \\frac { a _ j } { \\tau _ j } - \\frac { \\nu } { \\tau _ { } } . \\end{align*}"}
+{"id": "6100.png", "formula": "\\begin{align*} Z _ 2 = \\{ ( \\omega , x ) \\in \\ell _ \\infty \\times \\ell _ 2 : \\exists f \\in \\mathcal C : f ( 1 / 2 ) = x \\quad \\mathrm { a n d } f ' ( 1 / 2 ) = \\omega \\} \\end{align*}"}
+{"id": "7906.png", "formula": "\\begin{align*} R ( a ) \\ast _ R R ( b ) = ~ & R ^ 2 ( a ) \\cdot R ( b ) + R ( a ) \\cdot R ^ 2 ( b ) \\\\ = ~ & R ( R ( a ) \\ast _ R b ) + \\kappa ~ \\ ! R ( a ) \\cdot b + R ( a \\ast _ R R ( b ) ) + \\kappa ~ \\ ! a \\cdot R ( b ) \\\\ = ~ & R \\big ( R ( a ) \\ast _ R b + a \\ast _ R R ( b ) \\big ) + \\kappa ~ \\ ! a \\ast _ R b . \\end{align*}"}
+{"id": "3654.png", "formula": "\\begin{align*} Z ( \\mathcal { G } ) = Z ( p _ { \\mathcal { G } } ) = \\left \\{ ( x _ 1 , \\ldots , x _ n ) \\in \\Delta _ { m - 1 } \\colon p _ { \\mathcal { G } } ( x _ 1 , \\ldots , x _ m ) = \\lambda ( \\mathcal { G } ) \\right \\} . \\end{align*}"}
+{"id": "1511.png", "formula": "\\begin{align*} \\nu = 1 - \\delta ( w , \\sqrt x ) - \\tfrac 1 2 \\ge \\tfrac 1 4 . \\end{align*}"}
+{"id": "6176.png", "formula": "\\begin{align*} \\sum _ { p \\ , \\leq \\ , x } ( \\log g _ { u } ( p - 1 ) ) ^ { \\lambda } & = \\sum _ { \\substack { d \\ , \\le \\ , y \\\\ ( d , \\ , a _ { 2 } ) \\ , = \\ , 1 } } \\rho _ { \\lambda } ( d ) \\pi ( x ; \\l ( d ) , 1 ) + \\sum _ { \\substack { d \\ , \\ge \\ , y \\\\ ( d , \\ , a _ { 2 } ) \\ , = \\ , 1 } } \\rho _ { \\lambda } ( d ) \\pi ( x ; \\l ( d ) , 1 ) \\\\ & = E _ { 1 } + E _ { 2 } . \\end{align*}"}
+{"id": "4172.png", "formula": "\\begin{align*} ( \\partial _ N P ) ( n ) & = P ( n + N ) P ( n ) ^ { - 1 } , \\\\ ( \\partial ' _ N P ) ( n ) & = P ( n + N ) ^ { - 1 } P ( n ) . \\end{align*}"}
+{"id": "1769.png", "formula": "\\begin{align*} 2 w _ { n , k , 2 } \\left ( Y , \\mathcal { Z } \\right ) = \\left | \\sum _ { x \\in l \\cap P \\left ( Y , \\mathcal { Z } \\right ) } \\epsilon \\left ( x \\right ) \\right | , \\end{align*}"}
+{"id": "4654.png", "formula": "\\begin{align*} L _ f ( \\chi , 1 ) = \\tau _ \\chi ^ { - 1 } \\chi ( \\Theta _ { f , D } ) , \\end{align*}"}
+{"id": "59.png", "formula": "\\begin{align*} | \\nabla r | = 1 \\end{align*}"}
+{"id": "4348.png", "formula": "\\begin{align*} \\frac { d } { c } = \\frac { b } { a } + \\frac { a d - b c } { a c } \\ , , \\end{align*}"}
+{"id": "6126.png", "formula": "\\begin{align*} & \\ | \\mathcal H ( k , d , 2 ) | - | \\mathcal H ( k , d , l ) | \\\\ = & \\ \\sum _ { i = d + 2 } ^ { 2 d } \\sum _ { j = d + 2 } ^ { i } \\binom { n - j } { k - d - 1 } - \\sum _ { i = d + 2 } ^ l \\binom { n - d - i + 1 } { k - d } - ( d - 1 ) \\binom { n - d - l + 1 } { k - d - l + 2 } \\\\ < & \\ \\frac { d ( d - 1 ) } { 2 } \\binom { n - d - 2 } { k - d - 1 } - \\binom { n - d - l + 1 } { k - d } . \\end{align*}"}
+{"id": "5334.png", "formula": "\\begin{align*} G = F ( G ) { : } L , \\end{align*}"}
+{"id": "1866.png", "formula": "\\begin{align*} y _ { i , j } + y _ { m r + 1 - i , n + 1 - j } & = \\tfrac { 1 } { 2 } m n r + \\tfrac { 3 } { 2 } , \\mbox { \\ i f } i + j \\mbox { i s e v e n , a n d } \\\\ y _ { i , j } + y _ { m r + 1 - i , n + 1 - j } & = 2 M N - \\tfrac { 1 } { 2 } m n r + \\tfrac { 3 } { 2 } , \\mbox { \\ i f } i + j \\mbox { i s o d d . } \\end{align*}"}
+{"id": "3750.png", "formula": "\\begin{align*} P & = ( x + 4 ) I + \\frac { 1 0 ( x + 6 ) J } { ( x - 2 2 ) ( x + 8 ) } + A [ \\mathcal T ] \\\\ Q & = \\mathbf { u } + \\frac { 9 x + 8 2 } { ( x - 2 2 ) ( x + 8 ) } \\mathbf { 1 } \\\\ R & = ( x + 4 ) + \\frac { 1 0 ( x + 6 ) } { ( x - 2 2 ) ( x + 8 ) } = \\frac { x ^ 3 - 1 0 x ^ 2 - 2 2 2 x - 6 4 4 } { ( x - 2 2 ) ( x + 8 ) } \\end{align*}"}
+{"id": "7878.png", "formula": "\\begin{align*} E _ 2 = \\bigcup _ { x \\in E _ 2 } Y ' _ 2 ( R _ x ) . \\end{align*}"}
+{"id": "3433.png", "formula": "\\begin{align*} P ^ S & = - ( r ^ { 1 - n } S _ r / \\rho ) Q ^ U _ { 2 , l } = - S _ r A _ U f _ { J _ { 2 , l } } ^ { ( l ) } , \\\\ P ^ \\rho & = - r ^ { 1 - n } D _ r Q ^ U _ { 1 , l } = - D _ r ( \\rho A _ U f _ { J _ { 2 , l } } ^ { ( l ) } ) - \\tfrac { n - 1 } { r } \\rho A _ U f _ { J _ { 2 , l } } ^ { ( l ) } , \\\\ P ^ U & = ( r ^ { 1 - n } S _ r / \\rho ) Q ^ S _ { 1 , l } - D _ r ( r ^ { 1 - n } Q ^ \\rho _ { 1 , l } ) = - ( A _ r + U _ r A _ U ) f _ { J _ { 2 , l } } ^ { ( l ) } \\end{align*}"}
+{"id": "7006.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int \\Delta ( g ) | \\nabla f | ^ 2 d m = \\frac { a ^ 2 d } { 2 } \\int e ^ { ( 2 a + d ) v } ( \\Delta v ) | \\nabla v | ^ 2 d m + \\frac { a ^ 2 d ^ 2 } { 2 } \\int e ^ { ( 2 a + d ) v } | \\nabla v | ^ 4 d m , \\end{align*}"}
+{"id": "6771.png", "formula": "\\begin{align*} \\Psi _ N = W \\big ( \\sqrt { N } \\varphi \\big ) \\sum _ { k = 0 } ^ N \\psi ^ { \\otimes ( N - k ) } \\otimes _ s \\chi ^ { ( k ) } \\in \\mathcal { H } ^ { ( N ) } . \\end{align*}"}
+{"id": "1351.png", "formula": "\\begin{align*} k _ 0 + 1 \\not \\in { \\mathcal { K } } . \\end{align*}"}
+{"id": "7915.png", "formula": "\\begin{align*} \\begin{cases} a \\cdot \\chi ( b , c ) - \\chi ( a \\cdot b , c ) + \\chi ( a , b \\cdot c ) - \\chi ( a , b ) \\cdot c = 0 , \\\\ \\\\ R ( a ) \\cdot \\Phi ( b ) - S ( a \\cdot \\Phi ( b ) ) - \\Phi \\big ( R ( a ) \\cdot b + a \\cdot R ( b ) \\big ) + \\Phi ( a ) \\cdot R ( b ) - S ( \\Phi ( a ) \\cdot b ) \\\\ + \\chi ( R ( a ) , R ( b ) ) - \\kappa ~ \\ ! \\chi ( a , b ) - R \\big ( \\chi ( R ( a ) , b ) + \\chi ( a , R ( b ) ) \\big ) = 0 . \\end{cases} \\end{align*}"}
+{"id": "8197.png", "formula": "\\begin{align*} \\| u \\| _ { \\widetilde { L } ^ q _ T ( \\dot { B } ^ s _ { p , r } ) } : = \\big { \\Vert } \\big \\{ 2 ^ { j s } \\Vert { \\dot { \\Delta } _ j u } \\Vert _ { L ^ q _ T ( L ^ p ) } \\big \\} _ { j \\in \\mathbb { Z } } \\big { \\Vert } _ { \\ell ^ r ( \\mathbb { Z } ) } , \\end{align*}"}
+{"id": "3955.png", "formula": "\\begin{align*} \\mathrm { P r } \\{ \\xi ( t ) = n \\} = \\sum _ { \\Omega ^ { n } _ { k } } \\prod _ { j = 1 } ^ { k } \\lambda _ { x _ j } \\frac { t ^ { k } } { k ! } e ^ { - \\sum _ { j = 1 } ^ { \\infty } t \\lambda _ j } , \\end{align*}"}
+{"id": "344.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 Z } | \\sqrt { Z ' } - \\sqrt { Z } | ^ 2 & = \\frac { 1 } { 2 Z } \\frac { | Z - Z ' | ^ 2 } { | \\sqrt { Z } + \\sqrt { Z ' } | ^ 2 } \\\\ & \\leqslant K _ 1 | Z - Z ' | ^ 2 \\\\ & = K _ 1 | \\int \\exp ( - \\varrho ( x ) ) \\rho _ 0 ( x ) d x - \\int \\exp ( - \\varrho ' ( x ) ) \\rho _ 0 ( x ) d x | ^ 2 \\\\ & \\leqslant K _ 1 \\int | \\exp ( - \\varrho ( x ) ) - \\exp ( - \\varrho ' ( x ) ) | ^ 2 \\rho _ 0 ( x ) d x \\end{aligned} \\end{align*}"}
+{"id": "6125.png", "formula": "\\begin{align*} & \\ | \\mathcal H ( k , d , 2 ) | - | \\mathcal H ( k , d , l ) | \\\\ \\geq & \\ ( d - 1 ) \\binom { n - d - 1 } { k - d } - \\sum _ { i = 3 } ^ { d + 1 } \\binom { n - d - i + 1 } { k - d } - ( d - 1 ) \\binom { n - d - l + 1 } { k - d - l + 2 } \\\\ = & \\ \\sum _ { i = 3 } ^ { d + 1 } \\left ( \\binom { n - d - 1 } { k - d } - \\binom { n - d - i + 1 } { k - d } \\right ) - ( d - 1 ) \\binom { n - d - l + 1 } { k - d - l + 2 } . \\end{align*}"}
+{"id": "4221.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { 1 } { k } \\sum _ { n = 1 } ^ k f ( E _ { M , k } ( n ) ) & = \\lim _ { k \\to \\infty } \\frac { 1 } { k } \\sum _ { n = 1 } ^ k [ f \\circ \\alpha _ k ] ( \\varphi ( n ) ) = \\int _ { \\R } [ f \\circ \\alpha _ k ] ( x ) \\ , d \\mu _ G ( x ) = \\int _ { \\R } f ( x ) \\ , d \\nu _ k ( x ) , \\end{align*}"}
+{"id": "2128.png", "formula": "\\begin{align*} \\mathrm { c o r a n k } _ { \\Lambda ( G ) } H ^ { 1 } ( L _ { \\infty , w } , E _ { p ^ { \\infty } } ) = \\mathrm { c o r a n k } _ { \\Omega ( G ) } H ^ { 1 } ( L _ { \\infty , w } , E _ { p } ) . \\end{align*}"}
+{"id": "5081.png", "formula": "\\begin{align*} P _ { \\leq N } : = \\sum _ { M \\leq N } P _ M , P _ { > N } : = \\sum _ { M > N } P _ M . \\end{align*}"}
+{"id": "604.png", "formula": "\\begin{align*} ( \\mathcal { F } _ N ) = \\begin{pmatrix} N + 2 \\\\ 2 \\end{pmatrix} \\ , . \\end{align*}"}
+{"id": "7823.png", "formula": "\\begin{align*} \\Lambda _ V ^ { w _ 1 } ( u ) & = w _ 1 ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) \\\\ & = \\alpha ( 1 - u ) ^ { \\frac { \\alpha - m + 1 } { \\alpha } } . \\end{align*}"}
+{"id": "4548.png", "formula": "\\begin{align*} \\mathrm { d i v } _ w \\tilde { E } ^ i = \\left ( \\frac { \\mathrm { d } \\imath _ { \\tilde { E } ^ i } w } { w } \\right ) = \\left ( \\frac { \\mathrm { d } \\imath _ { \\hat { E } ^ i } v } { w } \\right ) = \\left ( \\frac { \\mathrm { d } \\imath _ { \\hat { E } ^ i } v } { v } \\right ) \\left ( \\frac { v } { w } \\right ) = \\mathrm { d i v } _ v \\hat { E } ^ i \\left ( \\frac { v } { w } \\right ) \\ , . \\end{align*}"}
+{"id": "3191.png", "formula": "\\begin{align*} C ( u _ { n } - u ) + C ( u ) = C ( u _ { n } ) + o ( 1 ) . \\end{align*}"}
+{"id": "5589.png", "formula": "\\begin{align*} y ^ i R ^ i + u ^ k I _ k = 2 p R , \\ \\ \\ y ^ i I _ i - u ^ j R _ k = 2 p I . \\end{align*}"}
+{"id": "3777.png", "formula": "\\begin{align*} A _ { j _ 1 , j _ 2 } = ( 2 j _ 2 + 1 ) \\sum _ { r = 0 } ^ { j _ 1 } ( r + 1 ) C _ r C _ { j _ 1 + j _ 2 - r } . \\end{align*}"}
+{"id": "2280.png", "formula": "\\begin{align*} \\sigma ^ m ( z _ 1 ) = c z _ 1 ^ { M ^ m [ 1 , 1 ] } z _ 2 ^ { M ^ m [ 2 , 1 ] } \\dots z _ n ^ { M ^ m [ n , 1 ] } \\not \\in V ^ l . \\end{align*}"}
+{"id": "6798.png", "formula": "\\begin{align*} \\alpha ( x _ { [ 4 ] } ) + \\alpha \\circ ( 3 \\ , \\ , 4 ) ( x _ { [ 4 ] } ) = \\sum _ { i \\in [ r ] \\setminus \\mathcal { I } } \\beta _ { i , 1 } ( x _ { I _ { i , 1 } } ) \\beta _ { i , 2 } ( x _ { I _ { i , 2 } } ) \\end{align*}"}
+{"id": "7059.png", "formula": "\\begin{align*} \\ ; _ { 2 } \\phi _ { 1 } ( q ^ { a } , q ^ { b } ; q ^ { c } ; q , x ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( q ^ { a } ; q ) _ { k } ( q ^ { b } ; q ) _ { k } } { ( q ^ { c } ; q ) _ { k } ( q ; q ) _ { k } } x ^ { k } , \\end{align*}"}
+{"id": "5115.png", "formula": "\\begin{align*} p = 1 , p = \\infty , q = 1 , q = \\infty , \\frac { 1 } { p } - \\frac { 1 } { q } = s \\end{align*}"}
+{"id": "3083.png", "formula": "\\begin{align*} A _ 3 & = ( \\overline { 0 , 4 , 3 , 4 , 0 , 7 } ) - ( \\overline { 0 , 4 , 0 , 4 , 0 , 4 } ) \\\\ & = ( \\overline { 0 , 0 , 3 , 0 , 0 , 3 } ) . \\end{align*}"}
+{"id": "4001.png", "formula": "\\begin{align*} \\mathrm { P r } \\{ X _ { 1 } + X _ { 2 } + \\dots + X _ { k } = n \\} & = \\underset { m _ j \\in \\mathbb { N } } { \\underset { m _ { 1 } + m _ { 2 } + \\dots + m _ { k } = n } { \\sum } } \\prod _ { j = 1 } ^ { k } \\mathrm { P r } \\{ X _ { j } = m _ j \\} \\\\ & = \\underset { m _ j \\in \\mathbb { N } } { \\underset { m _ { 1 } + m _ { 2 } + \\dots + m _ { k } = n } { \\sum } } \\left ( \\frac { - 1 } { \\ln p } \\right ) ^ { k } \\prod _ { j = 1 } ^ { k } \\frac { ( 1 - p ) ^ { m _ { j } } } { m _ { j } } , \\end{align*}"}
+{"id": "5628.png", "formula": "\\begin{align*} J ^ i _ { k | l } y ^ k y ^ l & = J ^ s _ k \\hat { \\mathbb { G } } ^ i _ { s l } y ^ l y ^ k - J ^ i _ s \\hat { \\mathbb { G } } ^ s _ { k l } y ^ l y ^ k \\\\ & = J ^ s _ k \\hat { \\mathbb { G } } ^ i _ s y ^ k - J ^ i _ s \\hat { \\mathbb { G } } ^ s _ k y ^ k = \\hat { \\mathbb { G } } ^ i _ s u ^ s - 2 J ^ i _ s \\hat { \\mathbb { G } } ^ s . \\end{align*}"}
+{"id": "731.png", "formula": "\\begin{align*} \\delta y = - \\frac { 1 } { 2 } \\left ( { { f _ 3 } - { f _ 4 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) + \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) - \\frac { 1 } { 4 } { F _ y } , \\end{align*}"}
+{"id": "7397.png", "formula": "\\begin{align*} \\alpha _ { \\pm } = - 2 | a | \\left ( \\sqrt { 1 + \\frac { k _ { \\pm } } { a ^ 2 } } + 1 \\right ) = - 4 | a | - \\frac { k _ { \\pm } } { | a | } + f _ { \\pm } ( a ) \\end{align*}"}
+{"id": "3395.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\ell r _ j s _ { g - h _ j } = 0 \\end{align*}"}
+{"id": "8124.png", "formula": "\\begin{align*} \\operatorname { \\widehat { v o l } } _ { \\chi } ( L , \\varphi ) = \\lim _ { n \\rightarrow + \\infty } \\frac { \\widehat { \\deg } ( H ^ 0 ( X , L ^ { \\otimes n } ) , \\xi _ n ' ) } { n ^ { d + 1 } / ( d + 1 ) ! } . \\end{align*}"}
+{"id": "2155.png", "formula": "\\begin{align*} ( \\nabla _ Z ^ { \\perp } \\overline { \\alpha } ) ( X , Y ) : = \\nabla ^ { \\perp } _ Z \\overline { \\alpha } ( X , Y ) - \\overline { \\alpha } ( \\nabla _ Z X , Y ) - \\overline { \\alpha } ( X , \\nabla _ Z Y ) \\end{align*}"}
+{"id": "5488.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow { 1 } } ( 1 - t ) F _ N ( a , b ; t ) & = \\frac { ( 1 - b ) ( 1 - t q ^ N ) } { ( 1 - b q ^ N ) } F _ N ( a / b , 1 ; b ) \\\\ & = \\frac { ( 1 - b ) ( 1 - t q ^ N ) } { ( 1 - b q ^ N ) } \\cdot \\frac { ( a q ) _ N } { ( b ) _ N } \\\\ & = \\frac { ( 1 - q ^ N ) ( a q ) _ N } { ( b q ) _ N } , \\end{align*}"}
+{"id": "4091.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n ( e _ i - d _ i ) = 2 - r + \\sum _ { i \\leq n / 2 } e _ { 2 i } . \\end{align*}"}
+{"id": "4776.png", "formula": "\\begin{align*} a \\diamond b : = \\partial _ 1 ( a ) \\ast \\partial _ 2 ( b ) - \\partial _ 2 ( a ) \\ast \\partial _ 1 ( b ) , \\forall a , b \\in A . \\end{align*}"}
+{"id": "2887.png", "formula": "\\begin{align*} p \\Delta \\left ( H - \\frac { \\mu } { 2 } \\right ) ^ { p - 1 } + 2 p \\left ( H - \\frac { \\mu } { 2 } \\right ) ^ { p - 1 } \\left ( 2 H ^ 2 - K \\right ) - 4 H \\left ( \\left ( H - \\frac { \\mu } { 2 } \\right ) ^ { p } + \\varsigma \\right ) = 0 , \\end{align*}"}
+{"id": "5798.png", "formula": "\\begin{align*} B e r ( A ) & = \\{ \\widetilde { A } ( z ) : z \\in \\Omega \\} , \\\\ b e r ( A ) & = \\displaystyle \\sup _ { z \\in \\Omega } | \\widetilde { A } ( z ) | . \\end{align*}"}
+{"id": "548.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\mathcal { W } _ 2 ^ 2 ( P _ { S _ i } , Q _ { S _ i } ) \\le \\mathcal { W } _ 2 ^ 2 ( P _ { S _ 1 \\cup \\cdots \\cup S _ m } , Q _ { S _ 1 \\cup \\cdots \\cup S _ m } ) \\le \\mathcal { W } _ 2 ^ 2 ( P , Q ) . \\end{align*}"}
+{"id": "619.png", "formula": "\\begin{align*} n ( n + \\alpha + \\beta + 1 ) \\ , r _ n ( m ) = B _ m \\ r _ { n } ( m + 1 ) - ( B _ m + D _ m ) \\ , r _ n ( m ) + D _ m \\ r _ n ( m - 1 ) \\ , , \\end{align*}"}
+{"id": "6153.png", "formula": "\\begin{align*} V _ 2 V _ 1 ^ * & = ( R _ { \\overline q } \\otimes U ^ * P + R _ { \\overline q } M _ z \\otimes U ^ * P ^ \\perp ) ( R _ { \\overline q } \\otimes U ^ * P ^ \\perp + R _ { \\overline q } M _ z ^ * \\otimes U ^ * P ) \\\\ & = R _ { \\overline q } ^ 2 \\otimes U ^ * P U ^ * P ^ \\perp + R _ { \\overline q } ^ 2 M _ z ^ * \\otimes U ^ * P U ^ * P + R _ { \\overline q } M _ z R _ { \\overline q } \\otimes U ^ * P ^ \\perp U ^ * P ^ \\perp \\\\ & \\quad + R _ { \\overline q } M _ z R _ { \\overline q } M _ z ^ * \\otimes U ^ * P ^ \\perp U ^ * P \\end{align*}"}
+{"id": "7068.png", "formula": "\\begin{align*} I _ { S _ 1 } \\cap I _ { S _ 0 } ^ n = I _ { S _ 1 } I ^ n _ { S _ 0 } . \\end{align*}"}
+{"id": "7199.png", "formula": "\\begin{align*} \\{ ( k , a , b ) \\in \\mathbb { Z } _ { \\geq 0 } ^ 3 : d = b k , \\gcd ( a , b ) = 1 , 0 \\leq a < b \\} \\stackrel { \\cong } { \\to } \\{ 0 , 1 , \\ldots , d - 1 \\} \\end{align*}"}
+{"id": "6580.png", "formula": "\\begin{align*} P U _ 1 P ^ { - 1 } = \\sum _ r e ^ { i \\theta _ r } P E _ r P ^ { - 1 } = \\sum _ r e ^ { i \\theta _ r } E _ r ' = U _ 2 . \\end{align*}"}
+{"id": "1788.png", "formula": "\\begin{align*} \\bar { i } : = \\min \\left \\{ I ^ { c } \\cap \\left \\{ j _ { a } + 2 , \\dots , n \\right \\} \\right \\} , \\end{align*}"}
+{"id": "2506.png", "formula": "\\begin{align*} \\{ n \\in \\Z : T ^ n a \\in E \\} = A . \\end{align*}"}
+{"id": "3545.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { \\frac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 \\tau } } \\textbf { e x p } ( - \\mathrm { m } ^ 2 ) d ( \\mathrm { m } ^ 2 ) = 1 - \\textbf { e x p } \\Big ( \\frac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 \\tau } \\Big ) = 1 - \\Big [ 1 - \\frac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 \\tau } + \\frac { 1 } { 2 ! } \\Big ( \\frac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 \\tau } \\Big ) ^ 2 - . . . \\Big ] . \\end{align*}"}
+{"id": "6764.png", "formula": "\\begin{align*} \\mathcal { F } _ { a } = W ^ * \\big ( \\sqrt { N } \\varphi _ t \\big ) \\mathcal { F } . \\end{align*}"}
+{"id": "6594.png", "formula": "\\begin{align*} 1 2 [ A , \\Gamma ] & = [ A , C ^ 2 ] + [ A , C C ^ * ] + [ A , C ^ * C ] + [ A , ( C ^ * ) ^ 2 ] = O . \\end{align*}"}
+{"id": "5160.png", "formula": "\\begin{align*} D ( \\widetilde A , x ) = D ( A ' , \\Omega , x ) \\in \\{ 0 , 1 \\} \\end{align*}"}
+{"id": "5551.png", "formula": "\\begin{align*} \\alpha ( t ) = 2 \\alpha _ 0 \\sqrt { t } , \\ ; \\ ; \\ ; \\beta ( t ) = 2 \\beta _ 0 \\sqrt { t } , \\end{align*}"}
+{"id": "2934.png", "formula": "\\begin{align*} A X - X B = C \\end{align*}"}
+{"id": "5813.png", "formula": "\\begin{align*} \\Pi V \\Pi ^ * = M _ z . \\end{align*}"}
+{"id": "3713.png", "formula": "\\begin{align*} A ^ Q ( \\rho , v ) = I ( b [ \\rho , v ] ) \\end{align*}"}
+{"id": "2919.png", "formula": "\\begin{align*} m _ \\infty ( r , F ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } { \\rm l o g } ^ + \\| F ( r e ^ { i \\theta } ) \\| d \\theta , \\end{align*}"}
+{"id": "4862.png", "formula": "\\begin{align*} N = \\begin{cases} 0 , & [ \\Sigma ] ^ 2 \\geq 0 , \\\\ - [ \\Sigma ] ^ 2 , & [ \\Sigma ] ^ 2 < 0 \\end{cases} . \\end{align*}"}
+{"id": "4239.png", "formula": "\\begin{align*} \\frac { \\left ( q ^ { - N } \\right ) _ { n } } { \\left ( \\frac { q ^ { - N } } { x } \\right ) _ { n } } = \\frac { \\left ( q ^ { N - n + 1 } \\right ) _ { n } } { \\left ( x q ^ { N - n + 1 } \\right ) _ { n } } x ^ { n } = \\frac { \\left ( q \\right ) _ { N } \\left ( x q \\right ) _ { N - n } } { \\left ( q \\right ) _ { N - n } \\left ( x q \\right ) _ { N } } x ^ { n } , \\end{align*}"}
+{"id": "1942.png", "formula": "\\begin{align*} & I : = ( | a - \\bar a | ^ { p _ 1 } \\ , | D _ v u | ^ { p _ 1 } ) ^ { 1 / p _ 1 } _ { Q _ { 2 \\nu r , 2 ^ { k + 1 } ( 2 \\nu r ) } } \\\\ & \\le ( | a - \\bar a | ^ { p _ 1 \\alpha _ 1 } ) ^ { 1 / ( p _ 1 \\alpha _ 1 ) } _ { Q _ { 2 \\nu r , 2 ^ { k + 1 } ( 2 \\nu r ) } } \\ , ( | D _ v u | ^ { p _ 1 \\alpha } ) ^ { 1 / ( p _ 1 \\alpha ) } _ { Q _ { 2 \\nu r , 2 ^ { k + 1 } ( 2 \\nu r ) } } = : I _ 1 ^ { 1 / ( p _ 1 \\alpha _ 1 ) } I _ 2 ^ { 1 / ( p _ 1 \\alpha ) } . \\end{align*}"}
+{"id": "7645.png", "formula": "\\begin{align*} u _ 2 ^ 3 x _ 0 ^ 3 x _ 1 ^ 3 x _ 2 ^ 3 x _ 9 ^ 3 x _ { 1 0 } ^ 3 x _ { 1 1 } ^ 3 u _ 2 ^ 3 x _ 0 ^ 3 x _ 1 ^ 3 x _ 2 ^ 3 x _ 9 ^ 3 x _ { 1 0 } ^ 3 x _ { 1 1 } ^ 3 = ( u _ 2 x _ 0 x _ 1 x _ 2 x _ 9 x _ { 1 0 } x _ { 1 1 } ) ^ 6 \\in \\langle \\partial w \\rangle \\end{align*}"}
+{"id": "3815.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 n + h ^ 2 - h - 1 3 + d } { 2 } , \\end{align*}"}
+{"id": "4740.png", "formula": "\\begin{gather*} P _ r ( a ) \\cdot P _ r ( b ) = P _ r ( a \\cdot P _ r ( b ) + P _ r ( a ) \\cdot b ) , \\forall a , b \\in A , \\\\ \\partial _ k P _ r = P _ r \\partial _ k , \\forall k = 1 , \\dots , m . \\end{gather*}"}
+{"id": "4821.png", "formula": "\\begin{align*} \\frac { \\partial \\rho ^ n } { \\partial t } = \\nabla \\cdot ( \\rho ^ n \\nabla f ^ n ( x , s ) ) + K ( \\bar { \\rho ^ n } \\otimes \\mu - \\rho ^ n ) , \\end{align*}"}
+{"id": "2629.png", "formula": "\\begin{align*} \\Lambda = \\cup _ { m \\in \\mathbb { N } ^ 2 } \\Lambda ^ m . \\end{align*}"}
+{"id": "2305.png", "formula": "\\begin{align*} \\vert L _ { d , p } ( x , y ) \\vert = x \\cdot \\vert x - y \\vert \\cdot \\vert x - 2 y \\vert \\cdots \\vert x - ( d - 2 ) y \\vert \\cdot \\vert x - p y \\vert . \\end{align*}"}
+{"id": "5006.png", "formula": "\\begin{align*} l = ( z - 2 g - a + r ) / a \\ge ( 2 g - 2 g - a + r ) / a = ( r - a ) / a > - 1 . \\end{align*}"}
+{"id": "2136.png", "formula": "\\begin{align*} \\mathrm { r a n k } _ { \\Lambda ( G ) } ( \\mathfrak { X } ^ { \\pm / \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) ) = \\mathrm { r a n k } _ { \\Omega ( G ) } ( \\mathfrak { X } ^ { \\pm / \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) / p \\mathfrak { X } ^ { \\pm / \\pm } ( E _ { p ^ { \\infty } } / L _ \\infty ) ) = 0 \\end{align*}"}
+{"id": "6813.png", "formula": "\\begin{align*} \\partial _ t u - \\kappa \\Delta u = 0 . \\end{align*}"}
+{"id": "7495.png", "formula": "\\begin{align*} & S ^ { ( 1 ) } _ 2 ( n ) = \\sum _ { \\nu = 1 } ^ n \\nu ^ 2 = \\psi ^ { ( 1 ) } _ 2 ( n ) = \\left [ 1 + \\frac { 2 } { 3 } ( n - 1 ) \\right ] B _ { 2 , n - 1 } = \\frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 ) \\ , . \\\\ & S ^ { ( 1 ) } _ 3 ( n ) = \\sum _ { \\nu = 1 } ^ n \\nu ^ 3 = \\psi ^ { ( 1 ) } _ 3 ( n ) = B _ { 2 , n - 1 } + \\frac { 3 } { 2 } ( n - 1 ) B _ { 3 , n - 1 } = \\left [ \\frac { n ( n + 1 ) } { 2 } \\right ] ^ 2 \\ , . \\end{align*}"}
+{"id": "5901.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ { n } \\tilde { x } _ { i t } \\tilde { v } _ { i t } = \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ { n } \\tilde { x } _ { i t } v _ { i t } - \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ { n } \\sum _ { s = 1 } ^ { T } \\tilde { x } _ { i t } { v } _ { i s } . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "279.png", "formula": "\\begin{align*} \\left | \\{ 0 \\leq k < n \\mid g ^ \\prime ( T ^ { k N } x ) \\neq * \\} \\right | & \\leq \\sum _ { k = 0 } ^ { n - 1 } 1 _ { Y \\setminus ( E _ 1 \\cup \\dots \\cup E _ m ) } ( S ^ { k N } y ) \\\\ & \\leq \\sum _ { k = 0 } ^ { n N - 1 } 1 _ { Y \\setminus ( E _ 1 \\cup \\dots \\cup E _ m ) } ( S ^ { k } y ) \\\\ & < \\delta n N \\end{align*}"}
+{"id": "1378.png", "formula": "\\begin{align*} 1 < p < \\infty \\ ( n = 1 , 2 ) , 1 < p \\le \\frac { n } { n - 2 } \\ ( n \\ge 3 ) , \\end{align*}"}
+{"id": "4934.png", "formula": "\\begin{align*} \\left ( \\frac { k - 1 } { 2 } + t \\right ) x + \\left ( \\frac { k + 1 } { 2 } + t \\right ) y = \\binom { n } { ( k + 1 ) / 2 + t } + \\cdots + \\binom { n } { k } , \\end{align*}"}
+{"id": "7309.png", "formula": "\\begin{align*} M = \\left [ ( q - 1 ) q ^ { m - 1 } - q ^ { \\lceil \\frac { 2 m - 1 } { 3 } \\rceil } - q ^ { \\lfloor \\frac { m } { 3 } - 1 \\rfloor } - 1 \\right ] q ^ i \\bmod { ( q ^ m - 1 ) } , \\end{align*}"}
+{"id": "4259.png", "formula": "\\begin{align*} T _ 1 & = \\frac { d - c } { c } \\sum _ { n = 1 } ^ N \\frac { \\left ( \\frac { d q } { c } \\right ) _ { n - 1 } ( c q ) _ { N - n } ( c q ) ^ n } { ( q ) _ n ( c q ) _ N ( q ) _ { N - n } } \\\\ T _ 2 & = \\frac { d - c } { c } \\sum _ { n = 1 } ^ N \\frac { \\left ( \\frac { d q } { c } \\right ) _ { n - 1 } ( c q ) _ { N - n } ( c q ) ^ n } { ( q ) _ n ( c q ) _ N ( q ) _ { N - n } } \\left ( - \\sum _ { k = 1 } ^ { n - 1 } \\frac { q ^ k d / c } { 1 - q ^ k d / c } + \\sum _ { k = 1 } ^ n \\frac { q ^ k } { 1 - q ^ k } \\right ) . \\end{align*}"}
+{"id": "3428.png", "formula": "\\begin{align*} & Q ^ U _ { 1 , l } \\equiv E _ U ( r ^ { n - 1 } \\rho f ( J _ { 1 , l } ) ) = 2 r ^ { n - 1 } \\rho U f _ { J _ { 1 , l } } ^ { ( l ) } , \\\\ & Q ^ \\rho _ { 1 , l } \\equiv E _ \\rho ( r ^ { n - 1 } \\rho f ( J _ { 1 , l } ) ) = r ^ { n - 1 } \\big ( U ^ 2 f _ { J _ { 1 , l } } ^ { ( l ) } + f - \\sum _ { 0 \\leq i \\leq l } J _ { 1 , l - i } f _ { J _ { 1 , l } } ^ { ( i ) } \\big ) , \\\\ & Q ^ S _ { 1 , l } \\equiv E _ S ( r ^ { n - 1 } \\rho f ( J _ { 1 , l } ) ) = - \\tfrac { 2 } { n } p ' D _ r ( r ^ n f _ { J _ { 1 , l } } ^ { ( l ) } ) . \\end{align*}"}
+{"id": "2769.png", "formula": "\\begin{gather*} \\lim _ { n \\to \\infty } \\max _ { | t | \\leq 1 } | | \\alpha _ t ( u _ n ) - e ^ { i p t } u _ n | | = 0 . \\end{gather*}"}
+{"id": "6749.png", "formula": "\\begin{align*} | E _ \\alpha ( t ) | \\leq C \\eta \\sum _ { | \\alpha | \\leq N - 1 } \\| \\partial ^ { \\alpha } ( \\widetilde { \\rho } , \\widetilde { u } , \\widetilde { \\theta } ) \\| ^ 2 + C _ { \\eta } \\varepsilon ^ 2 \\sum _ { | \\alpha | = N } ( \\| \\partial ^ { \\alpha } ( \\widetilde { u } , \\widetilde { \\theta } ) \\| ^ 2 + \\| \\partial ^ { \\alpha } f \\| ^ 2 ) + C _ { \\eta } \\varepsilon \\varepsilon ^ { 2 - 2 a } . \\end{align*}"}
+{"id": "1653.png", "formula": "\\begin{align*} f _ { n _ 2 } & \\leq ( 1 - g ) ^ { n _ 2 - n _ 1 } f _ { n _ 1 } + h \\sum _ { n = n _ 1 } ^ { n _ 2 - 1 } ( 1 - g ) ^ { n _ 2 - 1 - n } \\\\ & \\leq ( 1 - g ) ^ { n _ 2 - n _ 1 } f _ { n _ 1 } + h \\sum _ { k \\in \\mathbb { N } _ 0 } ( 1 - g ) ^ k \\\\ & \\leq f _ { n _ 1 } \\exp \\left [ - g ( n _ 2 - n _ 1 ) \\right ] + g ^ { - 1 } h \\ , . \\end{align*}"}
+{"id": "7948.png", "formula": "\\begin{align*} B ( x ) \\diamond B ( y ) = B \\big ( x \\diamond B ( x ) \\diamond y \\diamond B ( x ) ^ { - 1 } \\big ) , x , y \\in \\mathfrak { g } . \\end{align*}"}
+{"id": "4353.png", "formula": "\\begin{align*} J _ 1 ( t ) & : = \\int _ \\Omega \\int _ 0 ^ 1 \\Big | \\Phi ' ( ( 1 - s ) f ( t _ 0 ) + s f ( t ) ) \\Big [ \\frac { f ( t ) - f ( t _ 0 ) } { t - t _ 0 } - \\partial _ t f ( t _ 0 ) \\Big ] \\Big | \\ , \\mathrm { d } s \\mathrm { d } x \\ , , \\\\ [ 1 e x ] J _ 2 ( t ) & : = \\int _ \\Omega \\int _ 0 ^ 1 \\Big | [ \\Phi ' ( ( 1 - s ) f ( t _ 0 ) + s f ( t ) ) - \\Phi ' ( f ( t _ 0 ) ) ] \\partial _ t f ( t _ 0 ) \\Big | \\ , \\mathrm { d } s \\mathrm { d } x \\ , . \\end{align*}"}
+{"id": "726.png", "formula": "\\begin{align*} { f _ { 1 2 } } = { f _ { 1 1 } } - f _ { 1 1 } ^ { \\left ( { e q } \\right ) } + f _ { 1 2 } ^ { \\left ( { e q } \\right ) } - \\delta x - \\delta z , \\end{align*}"}
+{"id": "3094.png", "formula": "\\begin{align*} F _ { \\N } ( \\sigma _ 2 ^ 2 ) = F _ { \\N } ( \\sigma _ 3 ^ 2 ) = | \\N | \\end{align*}"}
+{"id": "5631.png", "formula": "\\begin{align*} L '' = I _ \\gamma ( U _ \\perp , U _ \\perp ) + g _ T ( D ^ T _ U U , T ) | _ { t = 0 } ^ d . \\end{align*}"}
+{"id": "6019.png", "formula": "\\begin{align*} \\rho ^ { ( \\theta ) \\prime } _ { b } ( x ; w ) = w ( 0 ) W ^ { ( \\theta ) } ( x ) - w ( b ) W ^ { ( \\theta ) } ( x - b ) + \\rho ^ { ( \\theta ) } _ { b } ( x ; w ' _ { + } ) , x \\geq 0 . \\end{align*}"}
+{"id": "4368.png", "formula": "\\begin{align*} D _ H G ^ \\alpha ( ( \\sigma _ 1 ^ 0 , . . . , \\sigma _ l ^ 0 , x _ 0 ) ( T ) ) = D _ H g ^ \\alpha ( x _ 0 ( T ) ) \\circ w ^ 2 . \\end{align*}"}
+{"id": "1270.png", "formula": "\\begin{align*} N _ v ^ { ( i ) } = \\max \\left \\{ \\begin{matrix} r ; \\ ; \\exists \\ , \\\\ \\end{matrix} \\right \\} . \\end{align*}"}
+{"id": "6582.png", "formula": "\\begin{align*} H _ 1 = \\sum _ { \\theta _ r } \\theta _ r E _ r , H _ 2 = \\sum _ { \\theta _ r } \\theta _ r E _ r ' , \\end{align*}"}
+{"id": "7644.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ 2 ( 3 u _ 1 x _ i ^ 3 x _ { 6 + i } ^ 3 - 3 \\lambda u _ 2 x _ 0 x _ 1 x _ 2 x _ 9 x _ { 1 0 } x _ { 1 1 } ) \\in \\langle \\partial w \\rangle . \\end{align*}"}
+{"id": "404.png", "formula": "\\begin{align*} [ v , w ] = [ q ^ 2 \\partial _ q , v ] = q ^ 2 \\frac { \\partial w ^ 1 } { \\partial q } \\partial _ { \\sigma _ 1 } + \\frac { \\partial w ^ 2 } { \\partial q } \\partial _ { \\sigma _ 2 } + \\Big ( \\frac { \\partial w ^ 1 } { \\partial q } - 2 q w ^ q \\Big ) \\partial _ { q } ~ . \\end{align*}"}
+{"id": "3008.png", "formula": "\\begin{align*} \\widetilde { \\phi } _ { b c , R } ^ 0 = \\phi _ { b c , R } ^ 0 + g ^ 0 , g ^ 0 = y ( \\phi ^ 0 _ { b c , S } + \\phi ^ 0 _ { b c , T } ) . \\end{align*}"}
+{"id": "6637.png", "formula": "\\begin{align*} h _ \\theta ( u , u ) & \\ge \\int _ { \\Omega _ + } | \\nabla u _ + | ^ 2 \\dd x + \\int _ { \\Omega _ - } | \\nabla u _ - | ^ 2 \\dd x \\\\ & - ( 1 + \\varepsilon ) \\int _ { \\Gamma _ \\theta } | u _ + | ^ 2 \\dd \\sigma - \\big ( 1 + \\frac { 1 } { \\varepsilon } \\big ) \\int _ { \\Gamma _ \\theta } | u _ - | ^ 2 \\dd \\sigma \\\\ & = q ^ { 1 + \\varepsilon } _ \\theta ( u _ + , u _ + ) - r ( u _ - , u _ - ) , \\end{align*}"}
+{"id": "3936.png", "formula": "\\begin{align*} \\left | x - y - r _ 0 \\operatorname { s i g n } t \\nabla b _ { \\Omega _ \\infty } ( y ) \\right | & = \\left | t \\nabla b _ { \\Omega _ n } ( y ) - r _ 0 \\operatorname { s i g n } t \\nabla b _ { \\Omega _ \\infty } ( y ) \\right | \\\\ & \\leq \\frac { t } { 2 } + \\left | t - r _ 0 \\operatorname { s i g n } t \\right | < r _ 0 . \\end{align*}"}
+{"id": "7146.png", "formula": "\\begin{align*} \\chi = \\sum _ { i = 1 } ^ k \\chi _ i \\chi _ a \\in M ( d _ a ) 1 \\leq a \\leq k , \\end{align*}"}
+{"id": "7049.png", "formula": "\\begin{align*} \\tilde { \\mathfrak { t r } } ^ p . p = \\nu ^ { ( 1 ) } _ p . \\end{align*}"}
+{"id": "7105.png", "formula": "\\begin{align*} \\tau _ d : = \\frac { 1 } { d } \\sum _ { j = 1 } ^ d \\beta _ j \\in M _ { \\mathbb { R } } . \\end{align*}"}
+{"id": "505.png", "formula": "\\begin{align*} P ( d x ) = Z ^ { - 1 } e ^ { f ( x ) } \\mu ^ { \\otimes n } ( d x ) , \\end{align*}"}
+{"id": "6429.png", "formula": "\\begin{align*} L _ n ^ { ( \\alpha ) } ( x , y ) = \\frac { ( q ; q ) _ n } { ( q ^ { \\alpha + 1 } ; q ) _ n } y ^ n \\mathcal { L } ^ { ( \\alpha ) } _ n ( x / y ) , \\ , \\ , L _ n ^ { ( \\alpha ) } ( x , 1 ) = \\frac { ( q ; q ) _ n } { ( q ^ { \\alpha + 1 } ; q ) _ n } \\mathcal { L } ^ { ( \\alpha ) } _ n ( x ) , \\ , \\ , L _ n ^ { ( \\alpha ) } ( 0 , y ) = y ^ n . \\end{align*}"}
+{"id": "1570.png", "formula": "\\begin{align*} \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } z _ { 2 } \\right ) ^ { - 1 } z _ { 2 } \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } \\right ) = \\left [ z _ { 2 } ^ { - 1 } , s ^ { \\prime } z _ { 1 } s ^ { \\prime } \\right ] \\mbox { a n d } \\\\ z _ { 2 } \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } z _ { 2 } \\right ) ^ { - 1 } \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } \\right ) = e . \\end{align*}"}
+{"id": "224.png", "formula": "\\begin{align*} \\kappa = \\left ( \\frac { 2 R } { \\lambda r } \\right ) ^ { - 1 } \\ , . \\end{align*}"}
+{"id": "7336.png", "formula": "\\begin{align*} ( B \\phi - \\phi B ) W = 0 \\ \\ \\ . \\end{align*}"}
+{"id": "252.png", "formula": "\\begin{align*} \\partial _ \\ell \\Q ^ t \\partial _ \\ell \\Q \\ , x = \\sum _ { i = 1 } ^ d ( \\partial _ \\ell f _ i ) ^ 2 w ^ i , \\Q ^ t \\partial ^ 2 _ { \\ell k } \\Q \\ , x = \\sum _ { i = 1 } ^ d \\left [ \\partial ^ 2 _ { \\ell k } f _ { i } [ w ^ i ] ^ \\perp - \\partial _ \\ell f _ { i } \\partial _ k f _ { i } w ^ i \\right ] , \\end{align*}"}
+{"id": "6395.png", "formula": "\\begin{align*} \\alpha _ { i j } ^ k = \\begin{cases} d & \\mbox { i f $ d _ i = d _ j \\neq d _ k $ } , \\\\ 1 & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "1398.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } ( u _ 0 ^ { ( j ) } , u _ 1 ^ { ( j ) } ) = ( u _ 0 , u _ 1 ) H _ 0 ^ 1 ( \\Omega ) \\times L ^ 2 ( \\Omega ) . \\end{align*}"}
+{"id": "8102.png", "formula": "\\begin{align*} T _ 1 = B _ r \\otimes M _ z \\ \\ \\ T _ 2 = M _ z \\otimes I _ { H ^ 2 ( \\mathbb { D } ) } . \\end{align*}"}
+{"id": "2434.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in \\mathbb { Z } / d \\mathbb { Z } } L _ { B } ( x _ { l _ { \\sigma + 1 } } \\cdots x _ { l _ { \\sigma + d } } ) \\equiv \\begin{cases} L _ { B } ( x _ { l _ { 1 } + \\cdots + l _ { d } } ) & d : { \\rm o d d } \\\\ 0 & d : { \\rm e v e n } , \\end{cases} \\end{align*}"}
+{"id": "764.png", "formula": "\\begin{align*} \\epsilon ( \\pi ^ * D , y ) = \\epsilon ( D , \\pi ( y ) ) . \\end{align*}"}
+{"id": "2007.png", "formula": "\\begin{align*} g ( t ) : = - \\Psi ( t ) \\end{align*}"}
+{"id": "6590.png", "formula": "\\begin{align*} \\mathbb { E } [ | X _ t ( x ) | ^ 2 ] & = x ^ * \\exp ( t A ^ * ) \\exp ( t \\alpha - t ^ 2 \\beta + t ^ 3 \\Gamma ) \\exp ( t A ) x \\\\ & = | \\exp ( t A ) \\exp ( t \\alpha - t ^ 2 \\beta + t ^ 3 \\Gamma ) x | ^ 2 \\textrm { w i t h } \\end{align*}"}
+{"id": "3736.png", "formula": "\\begin{align*} \\mathfrak { f } _ 1 ( x ) & = ( x + 5 ) ^ { 4 1 } ( x - 1 1 ) ^ { 1 3 } ( x - 1 3 ) ^ 3 ( x ^ 2 - 2 3 x + 1 0 6 ) , \\\\ \\mathfrak { f } _ 2 ( x ) & = ( x + 5 ) ^ { 4 1 } ( x - 1 1 ) ^ { 1 3 } ( x - 1 3 ) ^ 2 ( x ^ 3 - 3 6 x ^ 2 + 4 0 5 x - 1 3 8 2 ) , \\\\ \\mathfrak { f } _ 3 ( x ) & = ( x + 5 ) ^ { 4 1 } ( x - 1 1 ) ^ { 1 3 } ( x - 1 3 ) ^ 2 ( x - 1 7 ) ( x ^ 2 - 1 9 x + 8 2 ) . \\end{align*}"}
+{"id": "1956.png", "formula": "\\begin{align*} f * H _ R = \\sum _ { i \\in F _ R } \\phi _ i * \\psi _ i * f * H _ R , \\end{align*}"}
+{"id": "7845.png", "formula": "\\begin{align*} T Q = H \\ ! \\mbox { \\tiny { o r } } \\oplus V = H \\ ! \\mbox { \\tiny { o r } } \\oplus S \\oplus W , \\end{align*}"}
+{"id": "4911.png", "formula": "\\begin{align*} \\Omega ^ * _ { \\frak S } ( X \\to Y ) : = \\frac { \\mathcal M ^ * ( X \\to Y ) } { \\langle \\mathcal R ^ { \\frak S } \\rangle ( X \\xrightarrow f Y ) } . \\end{align*}"}
+{"id": "6864.png", "formula": "\\begin{align*} ( v _ i '' ) ^ { p ^ { e ' } + 1 } = \\lambda u _ i h ( a _ i ) , \\ 1 \\leq i \\leq n . \\end{align*}"}
+{"id": "7903.png", "formula": "\\begin{align*} ( a \\odot f ) ( u ) = f ( u \\cdot a ) ~ ~ ~ ~ ~ ~ ( f \\odot a ) ( u ) = f ( a \\cdot u ) , a \\in A , f \\in \\mathrm { E n d } ( M ) , u \\in M . \\end{align*}"}
+{"id": "5185.png", "formula": "\\begin{align*} \\left ( \\int _ { 5 c \\widetilde Q } | \\nabla E u ( x ) | ^ s \\ , \\d x \\right ) ^ \\frac { p } { s } \\ge \\left ( C ( 5 c ) ^ { n - s } \\ell ( \\widetilde Q ) ^ { n - s } \\right ) ^ \\frac { p } { s } = C ' \\ell ( \\widetilde Q ) ^ { n - p } \\ell ( \\widetilde Q ) ^ { ( \\frac { p } { s } - 1 ) n } \\end{align*}"}
+{"id": "7137.png", "formula": "\\begin{align*} \\frac { 3 } { 2 } \\mathfrak { g } ^ { \\lambda < 0 } + \\sum _ { i = 1 } ^ k v _ i \\tau _ { d _ i } + d \\mu \\tau _ d \\in \\textbf { V } ( d ) . \\end{align*}"}
+{"id": "2237.png", "formula": "\\begin{align*} [ q _ i , p _ j ] = \\delta _ { i j } I _ M , \\end{align*}"}
+{"id": "4005.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\bar { \\mathcal { M } } ( t ) } { t } & \\stackrel { d } { = } \\sum _ { j = 1 } ^ { \\infty } j \\lim _ { t \\to \\infty } \\frac { N _ { j } ( t ) } { t } \\\\ & = \\frac { \\lambda r \\rho } { ( 1 - \\rho ) ( 1 - ( 1 - \\rho ) ^ { r } ) } , \\ , \\end{align*}"}
+{"id": "6494.png", "formula": "\\begin{align*} \\tilde u ( y ) = \\frac { 1 } { K ^ { k - 1 } } \\ , u ( x _ 0 + 2 ^ { - \\ell } y ) \\end{align*}"}
+{"id": "5328.png", "formula": "\\begin{align*} \\lefteqn { \\sup _ { T _ j \\leq i < T _ { j + 1 } } X _ i I \\{ T _ j = s , T _ { j + 1 } = t \\} I \\{ D ^ + _ { T _ j } \\} } \\\\ & \\leq \\lambda ^ { m a x } ( U _ { T _ j } + M ^ { a b s } _ { { T _ j } , T _ { j + 1 } ^ - } ) I \\{ T _ j = s , T _ { j + 1 } = t \\} I \\{ D ^ + _ { T _ j } \\} . \\end{align*}"}
+{"id": "7333.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d X _ { 1 } ( t ) = & b _ { x } ( t ) X _ { 1 } ( t ) d t + \\sum \\limits _ { i = 1 } ^ { d } \\left [ \\sigma _ { x } ^ { i } ( t ) X _ { 1 } ( t ) + \\hat { \\sigma } ^ { i } ( t ) \\mathrm { 1 } _ { E _ { \\tau \\varepsilon } } ( t ) \\right ] d W _ { i } ( t ) , t \\in \\lbrack 0 , T ] , \\\\ X _ { 1 } ( 0 ) = & 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "8033.png", "formula": "\\begin{align*} \\sum _ { i + j = l } ( - 1 ) ^ i d _ i d _ j = 0 . \\end{align*}"}
+{"id": "6828.png", "formula": "\\begin{align*} \\alpha _ 1 ^ 1 \\partial _ { t t t } u _ 1 + \\partial _ { t t } u _ 1 - \\beta _ 1 ^ 1 \\Delta \\partial _ t u _ 1 - \\omega _ { 2 } \\kappa _ 1 \\Delta u _ { 1 } - ( 1 - \\omega _ { 2 } ) \\kappa _ 1 \\int _ 0 ^ \\infty g _ { 2 } ( s ) \\Delta u _ { 1 } ( t - s ) d s = 0 . \\end{align*}"}
+{"id": "4088.png", "formula": "\\begin{align*} \\sum _ { k | n } k \\cdot e _ k = \\frac { 1 } { n } ( ( g + \\sqrt { g ^ 2 - 1 } ) ^ n + ( g - \\sqrt { g ^ 2 - 1 } ) ^ n ) . \\end{align*}"}
+{"id": "6190.png", "formula": "\\begin{align*} \\partial _ H ( x , y ) = 2 m - 2 i , \\ \\partial _ H ( x , z ) = 2 m - 2 j , \\ \\partial _ H ( y , z ) = 2 m - 2 t , \\end{align*}"}
+{"id": "6579.png", "formula": "\\begin{align*} U _ 1 = \\sum _ { \\theta _ r } e ^ { i \\theta _ r } E _ r , U _ 2 = \\sum _ { \\theta _ r } e ^ { i \\theta _ r } E _ r ' \\end{align*}"}
+{"id": "4768.png", "formula": "\\begin{gather*} [ T ( u ) , T ( v ) ] = T ( \\rho ( T ( u ) ) v - \\rho ( T ( v ) ) u ) , \\\\ T ( u ) \\cdot T ( v ) = T ( \\mu ( T ( u ) ) v + \\mu ( T ( v ) ) u ) , \\forall u , v \\in V . \\end{gather*}"}
+{"id": "5964.png", "formula": "\\begin{align*} f _ { K } ^ { ( H ) } = \\sum _ { \\alpha \\in 2 ^ { \\N } \\cap [ 1 , \\delta _ { 0 } ^ { - k ( k - 1 ) / 2 } ] } f _ { K } ^ { ( H , \\alpha ) } \\end{align*}"}
+{"id": "3571.png", "formula": "\\begin{align*} P _ { I , N } ( z ) = \\frac { H _ { I , N } ( 0 ) + \\sum _ { i = 0 } ^ { p - 1 } ( H _ { I , N } ( i ) - H _ { I , N } ( i - 1 ) ) z ^ i + ( e _ 0 - H _ { I , N } ( p - 1 ) ) z ^ { p } } { 1 - z } . \\end{align*}"}
+{"id": "6391.png", "formula": "\\begin{align*} A ' = \\begin{cases} \\displaystyle { A _ k ^ { - 1 } \\left ( \\prod _ { j \\in I } A _ j ^ { [ b _ { j k } ] _ + } + \\prod _ { j \\in I } A _ j ^ { [ - b _ { j k } ] _ + } \\right ) } & \\mbox { i f $ i = k $ } , \\\\ A _ i & \\mbox { i f $ i \\neq k $ } , \\end{cases} \\end{align*}"}
+{"id": "2069.png", "formula": "\\begin{align*} \\begin{bmatrix} Q _ 1 & E \\\\ 0 & Q _ 2 \\end{bmatrix} \\begin{bmatrix} A _ { 1 1 } & A _ { 1 2 } \\\\ A _ { 2 1 } & A _ { 2 2 } \\end{bmatrix} \\begin{bmatrix} P _ 1 & 0 \\\\ 0 & P _ 2 \\end{bmatrix} = \\begin{bmatrix} ( Q _ 1 A _ { 1 1 } + E Q _ 2 A _ { 2 1 } ) P _ 1 & * \\\\ Q _ 2 A _ { 2 1 } P _ 2 & * \\end{bmatrix} = \\begin{bmatrix} H _ 1 + E H _ 2 & * \\\\ H _ 2 & * \\end{bmatrix} . \\end{align*}"}
+{"id": "3515.png", "formula": "\\begin{align*} \\nabla \\mathrm { H } ( \\mathrm { x } ) - \\alpha \\ \\nabla \\int _ \\Omega \\nabla \\mathbb { G } ^ { ( 0 ) } ( \\mathrm { x } , \\mathrm { y } ) \\cdot \\nabla \\mathrm { H } ( \\mathrm { y } ) d \\mathrm { y } = \\nabla \\mathrm { H } ^ { \\textbf { i n } } ( \\mathrm { x } ) + \\alpha \\mathbb { T } \\Big [ \\nabla \\mathrm { H } \\Big ] . \\end{align*}"}
+{"id": "5752.png", "formula": "\\begin{align*} \\tilde f ( x _ 0 , x ) = \\sum _ { i = 1 , j = 1 } ^ { m } a _ { i j } x _ i x _ j + \\sum _ { i = 1 } ^ { m } a _ i x _ i x _ 0 + a _ 0 x _ 0 ^ 2 . \\end{align*}"}
+{"id": "4747.png", "formula": "\\begin{gather*} ( a \\prec b ) \\prec c = a \\prec ( b \\prec c + b \\succ c ) , ( a \\succ b ) \\prec c = a \\succ ( b \\prec c ) , \\\\ ( a \\prec b + a \\succ b ) \\succ c = a \\succ ( b \\succ c ) \\end{gather*}"}
+{"id": "7078.png", "formula": "\\begin{align*} P ( x , y _ s ) : = { } & \\left | \\begin{matrix} A _ { 2 , 1 } - y _ s & A _ { 2 , 2 } & \\cdots & A _ { 2 , r - 1 } \\\\ A _ { 3 , 1 } & A _ { 3 , 2 } - y _ s & \\cdots & A _ { 3 , r - 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ { r - 1 , 1 } & A _ { r - 1 , 2 } & \\cdots & A _ { r - 1 , r - 1 } - y _ s \\\\ \\end{matrix} \\right | \\\\ = { } & y _ s ^ { r - 2 } + B _ 1 y _ s ^ { r - 3 } + \\cdots + B _ { r - 3 } y _ s + B _ { r - 4 } , \\end{align*}"}
+{"id": "6977.png", "formula": "\\begin{align*} \\tilde X T _ j Q ^ \\frac { 1 } { 2 } h = \\tilde X Q ^ \\frac { 1 } { 2 } S _ j h = Q ^ \\frac { 1 } { 2 } X S _ j h = Q ^ \\frac { 1 } { 2 } S _ j X h = T _ j Q ^ \\frac { 1 } { 2 } X h = T _ j \\tilde X Q ^ \\frac { 1 } { 2 } h \\end{align*}"}
+{"id": "5729.png", "formula": "\\begin{align*} & \\inf \\{ \\| M _ 1 - M _ 2 \\| ~ | ~ M _ 1 \\in C , \\ M _ 2 \\in D \\} \\\\ = & \\inf \\{ \\| M _ 1 - M _ 2 \\| ~ | ~ M _ 1 \\in C , \\ M _ 2 \\in \\{ A - \\lambda B ~ | ~ \\lambda \\leq \\lambda _ 0 \\} \\} = d > 0 . \\end{align*}"}
+{"id": "7764.png", "formula": "\\begin{align*} - 1 - \\frac { m } { r ^ 3 } \\leq K [ g ^ m ] \\leq - 1 + \\frac { 2 m } { r ^ 3 } , \\ \\ \\ R i c [ g ^ m ] = - 3 g ^ m . \\end{align*}"}
+{"id": "1165.png", "formula": "\\begin{align*} p _ k = \\log _ 2 \\binom { 2 k } { k } . \\end{align*}"}
+{"id": "5195.png", "formula": "\\begin{align*} \\frac { X - 1 } { ( X + 1 ) ( 2 X - 1 ) } = \\frac { 2 } { 3 } \\cdot \\frac { 1 } { X + 1 } - \\frac { 1 } { 3 } \\cdot \\frac { 1 } { 2 X + 1 } \\enspace . \\end{align*}"}
+{"id": "3933.png", "formula": "\\begin{align*} \\int _ { \\Gamma _ n } f ( x ) \\ , d \\mu _ { \\Gamma _ n } ( x ) = \\frac { 1 } { 2 t } \\int _ { U _ t ( \\Gamma _ n ) } f \\circ p _ n ( y ) \\ , \\det ( d _ { T _ n ^ { - 1 } ( y ) } T _ n ) \\ , d y . \\end{align*}"}
+{"id": "8141.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac 1 n \\operatorname { \\widehat { \\mu } _ { \\max } ^ { \\mathrm { a s y } } } ( \\overline L ^ { \\otimes n } \\otimes \\overline A ) = \\operatorname { \\widehat { \\mu } _ { \\max } ^ { \\mathrm { a s y } } } ( \\overline L ) . \\end{align*}"}
+{"id": "5864.png", "formula": "\\begin{align*} & P ( 0 ) = 0 , \\ ; P ^ \\prime ( q ) > 0 , q > 0 , \\\\ & 0 < \\frac { \\frac { 5 } { 3 } P ( q ) - P ^ \\prime ( q ) q } { q } < c q < 0 , \\lim \\limits _ { q \\rightarrow \\infty } \\frac { P ( q ) } { q ^ { \\frac { 5 } { 3 } } } = \\overline { p } > 0 . \\end{align*}"}
+{"id": "4340.png", "formula": "\\begin{align*} M ( X ) = ( m _ { j k } ( X ) ) _ { 1 \\le j , k \\le 2 } : = \\begin{pmatrix} a X _ 1 & b X _ 1 \\\\ c X _ 2 & d X _ 2 \\end{pmatrix} \\ , , X = ( X _ 1 , X _ 2 ) \\in \\mathbb { R } ^ 2 \\ , . \\end{align*}"}
+{"id": "3657.png", "formula": "\\begin{align*} & p ( x _ 1 , \\ldots , x _ { i - 1 } , \\frac { x _ i + x _ j } { 2 } , x _ { i + 1 } , \\ldots , x _ { j - 1 } , \\frac { x _ i + x _ j } { 2 } , x _ { j + 1 } , \\ldots , x _ m ) \\\\ & = p _ 1 + p _ 2 \\left ( \\frac { x _ i + x _ j } { 2 } + \\frac { x _ i + x _ j } { 2 } \\right ) + p _ 3 \\left ( \\frac { x _ i + x _ j } { 2 } \\right ) ^ 2 \\\\ & \\ge p _ 1 + p _ 2 ( x _ i + x _ j ) + p _ 3 x _ i x _ j = \\lambda ( p ) , \\end{align*}"}
+{"id": "778.png", "formula": "\\begin{align*} \\left | \\Phi ( T ) ( u ) \\right | \\leq \\sum _ { j = 1 } ^ { n } \\left | \\psi _ { j } ( T ( x _ { j } ) ) \\right | \\leq \\left \\| T \\right \\| _ { X ; Y ^ { \\rm d u a l } } \\cdot \\left \\| ( x _ { j } ) _ { j = 1 } ^ { n } \\right \\| _ { X ( E ) } \\cdot \\left \\| ( \\psi _ { j } ) _ { j = 1 } ^ { n } \\right \\| _ { Y ( F ^ * ) } . \\end{align*}"}
+{"id": "6667.png", "formula": "\\begin{align*} \\mathcal { D } ( \\mathbf { t } , \\mathbf { f } ) & = \\det \\begin{bmatrix} t _ 0 & t _ 1 & \\ldots & t _ { \\frac { N + 1 } { 2 } - 1 } & f _ 1 \\\\ t _ 1 & t _ 2 & \\ldots & t _ { \\frac { N + 1 } { 2 } } & f _ 2 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ t _ { \\frac { N + 1 } { 2 } } & t _ { \\frac { N } { 2 } + 1 } & \\ldots & t _ { N } & f _ { \\frac { N + 1 } { 2 } } \\end{bmatrix} \\end{align*}"}
+{"id": "5815.png", "formula": "\\begin{align*} c ( V ^ { * } ) = c ( V ) = c ( M _ z ) = 0 . \\end{align*}"}
+{"id": "1696.png", "formula": "\\begin{align*} u _ k : = e ^ { 2 \\pi i k x } \\end{align*}"}
+{"id": "1684.png", "formula": "\\begin{align*} - \\log \\ , \\det \\left ( \\Phi _ n ^ * \\mathcal { T } _ { n + 1 } ^ * \\mathcal { T } _ { n + 1 } \\Phi _ n \\right ) = \\log \\ , \\det \\left ( \\mathcal { T } _ { n + 1 } ^ { - 1 } ( \\mathcal { T } _ { n + 1 } ^ * ) ^ { - 1 } \\right ) - \\log \\ , \\det \\left ( ( \\Phi _ n ^ { \\perp } ) ^ * \\mathcal { T } _ { n + 1 } ^ { - 1 } ( \\mathcal { T } _ { n + 1 } ^ * ) ^ { - 1 } \\Phi _ n ^ { \\perp } \\right ) \\end{align*}"}
+{"id": "3471.png", "formula": "\\begin{align*} \\Big \\Vert \\partial _ { t } ^ { \\frac { 1 } { 2 } } \\mathbb { I } _ { ( \\Omega ) } \\Big [ \\psi \\Big ] \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\Omega \\times \\mathbb { R _ + } ) } ^ 2 = \\alpha \\delta ^ 2 \\Big \\Vert \\partial _ { \\Tilde { t } } ^ { \\frac { 1 } { 2 } } \\mathbb { I } _ { ( B ) } \\Big [ \\hat { \\psi } \\Big ] \\Big \\Vert _ { \\mathrm { L } ^ 2 ( B \\times \\mathbb { R _ + } ) } ^ 2 . \\end{align*}"}
+{"id": "5059.png", "formula": "\\begin{align*} L _ f \\bigl ( 1 - s - u , - \\tfrac { \\bar { a } } { q } \\bigr ) \\ll 1 \\quad \\frac { \\Gamma _ \\C ( u + \\frac { k - 1 } { 2 } ) } { \\Gamma _ \\C ( - u + \\frac { k + 1 } { 2 } ) } \\frac { \\gamma ( 1 - s - u ) } { \\gamma ( s + u ) } \\ll | u | ^ { - 2 \\Re ( s ) } \\quad \\Re ( u ) = \\sigma _ 1 . \\end{align*}"}
+{"id": "1803.png", "formula": "\\begin{align*} \\mathcal { R } ( x ) \\mathcal { R } ( y ) = \\mathcal { R } ( \\mathcal { R } ( x ) y + x \\mathcal { R } ( y ) + \\lambda x y ) , \\end{align*}"}
+{"id": "6184.png", "formula": "\\begin{align*} X _ { ( i , j , t , p ) } = \\{ ( x , y , z ) \\in X \\times X \\times X \\mid \\partial ( x , y , z ) = ( i , j , t , p ) \\} \\ \\end{align*}"}
+{"id": "1188.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\deg _ p ( N _ k ' ) = \\lim _ { k \\to \\infty } \\deg _ p ( N _ k '' ) ; \\end{align*}"}
+{"id": "5829.png", "formula": "\\begin{align*} \\Pi T \\Pi ^ * = M _ z . \\end{align*}"}
+{"id": "993.png", "formula": "\\begin{align*} \\mathcal { A } = & \\int _ { 0 } ^ { T } \\int _ \\Omega \\bigg ( \\frac { 1 } { 2 } \\ , \\rho \\ , \\| \\dot { u } \\| ^ 2 + \\ , \\rho \\ , j \\ , \\| \\dot { \\vartheta } \\| ^ 2 - W ) \\ , d v \\ , d t , \\end{align*}"}
+{"id": "1894.png", "formula": "\\begin{align*} \\widetilde { J } _ { n , m } \\left ( h \\cdot J ^ { m + 1 } I A _ n \\right ) ( x _ i ) = \\phi _ { n , m + 1 } ^ { - 1 } \\left ( w _ { i , m + 1 \\cdot \\Gamma ^ { m + 2 } F ( X _ n ) } \\right ) \\end{align*}"}
+{"id": "4655.png", "formula": "\\begin{align*} L _ f ( \\chi , s ) = L _ f ( \\chi \\chi _ { s - 1 } , 1 ) = \\tau _ { \\chi \\chi _ { s - 1 } } ^ { - 1 } \\ ; ( \\chi \\chi _ { s - 1 } ) ( \\Theta _ { f , D } ) . \\end{align*}"}
+{"id": "6558.png", "formula": "\\begin{align*} U = \\sum _ { k = 0 } ^ { n - 2 } \\lambda _ k E _ { \\lambda _ k } , \\end{align*}"}
+{"id": "4840.png", "formula": "\\begin{align*} \\rho = \\sum _ { n = 0 } ^ { + \\infty } \\frac 1 { K ^ n } \\rho _ n . \\end{align*}"}
+{"id": "5318.png", "formula": "\\begin{align*} w \\preceq _ { i _ { 1 } } w _ { 1 } \\preceq _ { i _ { 2 } } w _ { 2 } \\preceq _ { i _ { 3 } } \\dotsb \\preceq _ { i _ { t } } w _ { t } = w ' . \\end{align*}"}
+{"id": "1245.png", "formula": "\\begin{align*} \\| v \\| _ \\lambda = \\int _ \\Omega | \\nabla v | \\ , d x + \\int _ { \\partial \\Omega } \\lambda ( x ) | v | \\ , d \\mathcal H ^ { N - 1 } \\end{align*}"}
+{"id": "4355.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\| f _ n ( t ) - f ( t ) \\| _ 2 = 0 \\ , . \\end{align*}"}
+{"id": "4568.png", "formula": "\\begin{gather*} B = \\{ n \\in N : l _ n ^ \\alpha \\in b _ \\alpha ^ * \\} . \\end{gather*}"}
+{"id": "5103.png", "formula": "\\begin{align*} [ ( - \\partial ^ 2 _ y ) ^ \\sigma , i \\Gamma _ { \\phi } ] = 0 , \\end{align*}"}
+{"id": "5438.png", "formula": "\\begin{align*} \\widetilde { h } ^ + ( t ) = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ \\infty \\Omega \\left ( y \\frac { K ^ 2 } { t ^ 3 } \\right ) A ( y , t , K ) e ^ { i B ( y , t , K ) } \\ , d y + O ( K ^ { - 1 0 0 } ) , \\end{align*}"}
+{"id": "6450.png", "formula": "\\begin{align*} \\Delta _ { x , \\alpha } = \\eta _ x - \\frac { x q ^ { \\alpha + 1 } } { 1 - q ^ { \\alpha + 1 } } \\lambda _ { \\alpha } \\eta _ x . \\end{align*}"}
+{"id": "3133.png", "formula": "\\begin{align*} F P ( \\sigma ^ k ) = F P ( \\sigma ^ { - k } ) = k \\end{align*}"}
+{"id": "7062.png", "formula": "\\begin{align*} y = \\sum _ { t \\in T _ 2 } x _ t a _ t . \\end{align*}"}
+{"id": "3057.png", "formula": "\\begin{align*} \\ell C _ { \\sigma } ( \\ell ) = \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } \\mu \\left ( \\frac { \\ell } { k } \\right ) F _ X ( \\sigma ^ k ) . \\end{align*}"}
+{"id": "5490.png", "formula": "\\begin{align*} F _ N ( b / t , 0 ; t ) = \\frac { ( b q ) _ N ( q ) _ N } { ( t ) _ N } \\sum _ { n = 0 } ^ { N } \\frac { ( t ) _ n q ^ n } { ( b q ) _ n ( q ) _ n } , \\end{align*}"}
+{"id": "194.png", "formula": "\\begin{align*} & d _ { ( \\psi , a ) } \\mathcal { F } ( \\phi , b ) = \\lim _ { t \\to 0 } \\frac { \\mathcal { F } ( \\gamma _ t ) - \\mathcal { F } ( \\psi , a ) } { t } \\\\ & = \\left ( Q ^ + \\phi + \\lim _ { t \\to 0 } \\frac { \\pi ^ { - } ( a _ t \\cdot \\psi _ t - a \\cdot \\psi ) } { t } , \\ , \\ , d ^ * b , \\ , \\ , d ^ + b - \\lim _ { t \\to 0 } \\frac { \\rho ^ { - 1 } ( \\mu ( \\psi _ t ) - \\mu ( \\psi ) ) } { t } \\right ) . \\end{align*}"}
+{"id": "2497.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & Y ( t _ { j } ) \\leq Y _ 2 , \\\\ & Z ( t _ j ) \\leq Z _ 2 , \\\\ & W ( t _ j ) \\leq W _ 2 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3789.png", "formula": "\\begin{align*} \\widetilde { C } _ { i , j } ( 1 ) = 2 \\left ( j C _ { i + j - 1 } + \\sum _ { r = 0 } ^ { i - 1 } \\widetilde { C } _ { r , j } ( 1 ) C _ { i - 1 - r } \\right ) . \\end{align*}"}
+{"id": "3400.png", "formula": "\\begin{align*} ( x _ 1 y ) x _ 2 + ( x _ 2 y ) x _ 1 = x _ 1 ( y x _ 2 ) + x _ 2 ( y x _ 1 ) . \\end{align*}"}
+{"id": "230.png", "formula": "\\begin{align*} \\partial _ t m _ t ( x ) = - \\left ( \\frac { \\delta F } { \\delta m } ( m _ t , x ) + \\frac { \\sigma ^ 2 } { 2 } \\log \\left ( \\frac { m _ t ( x ) } { \\pi ( x ) } \\right ) - \\frac { \\sigma ^ 2 } { 2 } \\operatorname { K L } ( m _ t | \\pi ) \\right ) m _ t ( x ) \\ , , \\end{align*}"}
+{"id": "6485.png", "formula": "\\begin{align*} \\nu _ x = h ( x , \\cdot ) \\# \\mu \\end{align*}"}
+{"id": "1293.png", "formula": "\\begin{align*} ( M _ u ) _ { i , j } & = \\pi _ { b \\otimes a } ^ i \\circ \\sigma _ { a , b } \\circ \\iota _ { a \\otimes b } ^ j \\\\ & = ( \\pi _ { b } ^ { i _ 1 } \\otimes \\pi _ { a } ^ { i _ 2 } ) \\circ \\sigma _ { a , b } \\circ ( \\iota _ { a } ^ { j _ 1 } \\otimes \\iota _ { b } ^ { j _ 2 } ) \\\\ & = \\begin{cases} \\sigma _ { a ^ { j _ 1 } , b ^ { j _ 2 } } , & j _ 1 = i _ 2 , \\ ; j _ 2 = i _ 1 , \\\\ 0 _ { a ^ { j _ 1 } \\otimes b ^ { j _ 2 } , b ^ { i _ 1 } \\otimes a ^ { i _ 2 } } , & . \\end{cases} \\end{align*}"}
+{"id": "4987.png", "formula": "\\begin{align*} p ^ { \\mu } q _ { \\mu } : = - p ^ 0 q ^ 0 + \\sum _ { i = 1 } ^ 3 p _ i q _ i . \\end{align*}"}
+{"id": "6064.png", "formula": "\\begin{align*} \\begin{cases} \\ , \\ , \\eta \\in H ^ 1 _ 0 ( \\Omega ) \\ , \\ , \\ , \\\\ [ 0 . 3 c m ] \\int _ { \\Omega } \\nabla \\eta \\nabla v d x - \\int _ { \\Omega } \\lambda \\eta v d x = 0 \\ , \\ , \\forall \\ , v \\in H ^ 1 _ 0 ( \\Omega ) \\end{cases} \\end{align*}"}
+{"id": "5170.png", "formula": "\\begin{align*} h = \\lim _ { n \\to \\infty } h _ 1 \\circ \\cdots \\circ h _ n . \\end{align*}"}
+{"id": "4338.png", "formula": "\\begin{align*} \\int _ \\Omega ( g ( t , x ) - g ^ { i n } ( x ) ) \\varphi ( x ) \\ \\mathrm { d } x + \\int _ 0 ^ t \\int _ \\Omega g ( s , x ) \\nabla [ c f + d g ] ( s , x ) \\cdot \\nabla \\varphi ( x ) \\ , \\mathrm { d } x \\mathrm { d } s = 0 \\ , . \\end{align*}"}
+{"id": "3258.png", "formula": "\\begin{align*} 0 = \\eta _ 1 ( \\overline { [ X , Y ] } _ p ) = - d \\eta _ 1 ( \\overline { X } _ p , \\varphi _ 1 \\overline { X } _ p ) = 2 \\alpha \\left \\| ( \\overline { X } _ p ) _ \\mathcal { H } \\right \\| ^ 2 \\ , , \\end{align*}"}
+{"id": "2589.png", "formula": "\\begin{align*} | k _ i ^ N ( t , \\nu ^ { N , 1 } _ { \\boldsymbol { x } } , \\ldots , \\nu ^ { N , N } _ { \\boldsymbol { x } } , \\lambda ^ { N , 1 } _ { \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) } , \\ldots , \\lambda ^ { N , N } _ { \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) } ) - k \\big ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } , \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } ) \\big ) | = O \\Big ( \\frac { 1 } { N } \\Big ( 1 + \\frac { 1 } { N } \\underset { j \\neq i } \\sum | x ^ i - x ^ j | \\Big ) \\Big ) ; \\end{align*}"}
+{"id": "3771.png", "formula": "\\begin{align*} p ^ { \\gamma , \\nu } _ { k _ 1 , \\ldots , k _ n } & = \\sum _ { g \\geq 0 , b \\geq 1 } \\eta ^ { g , b } _ { k _ 1 , \\ldots , k _ n } \\gamma ^ g \\nu ^ b , \\\\ p ^ \\nu _ { k _ 1 , \\ldots , k _ n } & = \\sum _ { b \\geq 1 } \\eta ^ { * , b } _ { k _ 1 , \\ldots , k _ n } \\nu ^ b . \\end{align*}"}
+{"id": "2936.png", "formula": "\\begin{align*} \\begin{pmatrix} A & C \\\\ & B \\end{pmatrix} = \\begin{pmatrix} I & \\\\ & B \\end{pmatrix} \\begin{pmatrix} I & C \\\\ & I \\end{pmatrix} \\begin{pmatrix} A \\\\ & I \\end{pmatrix} . \\end{align*}"}
+{"id": "1002.png", "formula": "\\begin{align*} \\frac { d ^ 2 \\ , } { d \\ , \\tau ^ 2 } W ( \\mathbf { e } + \\tau \\ , \\xi \\otimes \\eta , \\boldsymbol { \\mathfrak { K } } + \\tau \\ , \\zeta \\otimes \\eta ) \\bigg | _ { \\tau = 0 } > 0 \\forall \\ \\eta , \\xi , \\zeta \\in \\mathbb { R } ^ 3 , \\ \\ \\lVert \\eta \\rVert = \\lVert \\xi \\rVert = \\lVert \\zeta \\rVert = 1 . \\end{align*}"}
+{"id": "7534.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { E _ { \\varphi } ( F _ { \\varepsilon , h } ) - E _ { \\varphi } ( F ) } { \\varepsilon } = \\lim _ { \\varepsilon \\to 0 } \\frac { E _ { \\varphi } [ \\mu ^ F _ { \\varepsilon , h } ] - E _ { \\varphi } [ \\mu ^ F ] } { \\varepsilon } \\end{align*}"}
+{"id": "5898.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { \\tilde { x } } _ n ( s ) = \\dot { \\tilde { v } } _ n ( s ) , \\ \\dot { \\tilde { v } } _ n ( s ) = \\frac { 1 } { \\epsilon } \\tilde { B } ( x ( t _ n ) ) \\tilde { v } _ n ( s ) + F ( { \\tilde { x } } _ n ( s ) ) , \\ { \\tilde { x } } _ n ( 0 ) = x ( t _ n ) , \\ { \\tilde { v } } _ n ( 0 ) = v ( t _ n ) , \\ 0 < s \\leq h . \\end{aligned} \\end{align*}"}
+{"id": "320.png", "formula": "\\begin{align*} F ( t ) - t = \\left ( t _ { 1 } ' , \\frac { 1 } { t _ { 1 } ' } , \\frac { t _ { 3 } ' } { t _ { 1 } '^ { p } t _ { 2 } '^ { p ^ { 2 } + p } } \\right ) \\end{align*}"}
+{"id": "6409.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\theta _ { 0 } } ^ { - w } \\mathcal { H } ( \\theta ) = \\mathcal { J } ^ { w } _ { \\theta _ { 0 } } \\mathcal { H } ( \\theta ) = \\frac { 1 } { \\Gamma ( w ) } \\int _ { \\theta _ { 0 } } ^ { \\theta } ( \\theta - x ) ^ { w - 1 } \\mathcal { H } ( x ) \\nabla x . \\end{align*}"}
+{"id": "4092.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n ( e _ i - d _ i ) \\geq 2 - r + \\sum _ { i \\leq n / 2 } 4 ^ i / 2 i - 2 ^ { i + 1 / 2 } / i ^ { 1 / 2 } \\end{align*}"}
+{"id": "4430.png", "formula": "\\begin{align*} \\tau _ { a } ( z ) = \\begin{cases} z , & \\vert z \\vert < a \\\\ 0 , & \\vert z \\vert \\geq a . \\end{cases} \\end{align*}"}
+{"id": "1364.png", "formula": "\\begin{align*} E _ { L } ( t ) : = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ n } ( | \\partial _ t u ( t , x ) | ^ 2 + | \\nabla u ( t , x ) | ^ 2 ) \\ , d x . \\end{align*}"}
+{"id": "4598.png", "formula": "\\begin{align*} \\abs { ( e ^ { 2 \\pi k i } - 1 ) \\sum _ { l = n _ 0 } ^ { n } \\frac { e ^ { 2 \\pi i \\theta ( l ) } } { l - b } } \\leq & \\frac { 2 } { n _ 0 - b } + O \\left ( \\frac { K } { \\sin \\pi k } \\right ) \\sum _ { l = n _ 0 } ^ { \\infty } \\frac { 1 } { ( l - b ) ^ 2 } + \\sum _ { l = n _ 0 } ^ { \\infty } \\frac { 1 } { ( l - b ) ^ 2 } \\\\ \\leq & \\frac { C ( E , K ) } { n _ 0 - b } . \\end{align*}"}
+{"id": "5844.png", "formula": "\\begin{align*} \\omega ( D ) = \\| D \\| . \\end{align*}"}
+{"id": "2743.png", "formula": "\\begin{align*} g ( X , L ) : = \\frac { ( K _ X + ( n - 1 ) L ) \\cdot L ^ { n - 1 } } { 2 } + 1 . \\end{align*}"}
+{"id": "2765.png", "formula": "\\begin{align*} a \\cdot ( C _ n / P _ k ) - b \\cdot \\dim ( C _ n / P _ k ) & = ( 2 n - k ) ( 2 n - k + 1 ) - k ( 4 n - 3 k + 1 ) \\\\ & \\geq ( 2 n - 2 k + 1 ) ( 2 n - 2 k ) \\geq 0 \\end{align*}"}
+{"id": "6932.png", "formula": "\\begin{align*} T _ m = ( - 1 ) ^ { ( N - 1 ) m } \\left [ t ^ { m } \\right ] \\frac { 1 } { ( 1 - t ) ( 1 - x _ 1 y t ) \\cdots ( 1 - x _ r y t ) } . \\end{align*}"}
+{"id": "8101.png", "formula": "\\begin{align*} T _ 2 T _ 1 ^ * ( z _ 1 ^ { m _ 1 } z _ 2 ^ { m _ 2 } \\eta ) & = \\begin{cases} M _ { z _ 2 } ^ { \\alpha _ 2 } D [ U ] ( \\bar { \\alpha _ 1 } z _ 1 ^ { m _ 1 - 1 } z _ 2 ^ { m _ 2 } \\eta ) & \\\\ 0 & \\end{cases} \\\\ & = \\begin{cases} \\bar { \\alpha _ 1 } \\alpha _ 2 z _ 1 ^ { m _ 1 - 1 } z _ 2 ^ { m _ 2 + 1 } U ^ { m _ 1 - 1 } \\eta & \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "6565.png", "formula": "\\begin{align*} \\frac { 2 } { n - 1 } \\sum _ { r = 1 } ^ { \\frac { n } { 2 } - 1 } \\frac { 2 \\pi r } { ( n - 1 ) } \\sin \\left ( \\frac { 2 \\pi r } { n - 1 } \\alpha \\right ) \\end{align*}"}
+{"id": "3245.png", "formula": "\\begin{align*} \\int _ { \\{ u > \\phi - L \\} } \\psi ( u _ { j } - u ) \\theta ^ { n } _ { u _ { j } } = \\int _ { \\{ u > \\phi - L \\} } \\psi ( u _ { j } ^ { L } - u ^ { L } ) \\theta ^ { n } _ { u _ { j } ^ { L } } . \\end{align*}"}
+{"id": "2534.png", "formula": "\\begin{align*} \\tilde { \\xi } ( g \\gamma ) = \\tilde { \\xi } ( g ) , \\qquad g \\in G ^ \\circ , \\gamma \\in \\Gamma ^ \\circ , \\end{align*}"}
+{"id": "4427.png", "formula": "\\begin{align*} u _ i = \\sum _ { \\xi \\in \\Z ^ d } \\hat { u } _ i ( \\xi ) e ^ { - 2 \\pi i \\xi \\cdot x } , i = 1 , 2 , \\end{align*}"}
+{"id": "7448.png", "formula": "\\begin{align*} \\begin{matrix} J _ z = \\sum \\limits _ { \\mu = - 1 } ^ { 1 } \\mu a _ { \\mu } ^ { \\dagger } a _ { \\mu } , J _ + = ( J _ - ) ^ { \\dagger } = \\sum \\limits _ { \\mu = - 1 } ^ { \\mu = 0 } \\sqrt { ( \\mu + 2 ) ( 1 - \\mu ) } a _ { \\mu + 1 } ^ { \\dagger } a _ { \\mu } . \\end{matrix} \\end{align*}"}
+{"id": "4227.png", "formula": "\\begin{align*} J _ 0 ( z ) = \\sum _ { m = 0 } ^ \\infty \\frac { ( - 1 ) ^ m ( z / 2 ) ^ { 2 m } } { ( m ! ) ^ 2 } \\end{align*}"}
+{"id": "4173.png", "formula": "\\begin{align*} \\partial ' _ { a b } P ( n ) = \\partial ' _ a P \\big ( a ( b - 1 ) + n \\big ) \\ \\partial ' _ a P \\big ( a ( b - 2 ) + n \\big ) \\ \\cdots \\ \\partial ' _ a P \\big ( a + n \\big ) \\ \\partial ' _ a P \\big ( n \\big ) . \\end{align*}"}
+{"id": "5200.png", "formula": "\\begin{align*} a _ n = \\frac { 2 ^ n } { 3 } \\sum _ { j = 1 } ^ { n - 1 } \\binom { n - 1 } { j } + \\frac { 2 ^ { n + 1 } } { 3 } \\sum _ { j = 1 } ^ { n - 1 } \\binom { n - 1 } { j } \\cdot \\left ( - \\frac { 1 } { 2 } \\right ) ^ { j + 1 } \\enspace . \\end{align*}"}
+{"id": "3177.png", "formula": "\\begin{align*} \\rho _ i : = e _ i \\cdot h _ i e _ i \\cdot h _ i ^ 2 e _ i \\dots \\end{align*}"}
+{"id": "5180.png", "formula": "\\begin{align*} \\mathcal E _ 1 ( O , \\gamma ) = \\inf _ \\Gamma \\mathcal { H } ^ { n - 1 } ( \\Gamma ) , \\end{align*}"}
+{"id": "2866.png", "formula": "\\begin{align*} \\langle f \\rangle ( a ) = \\lim _ { { T ' } \\to \\infty } \\frac { 1 } { { T ' } } \\int _ 0 ^ { T ' } f ( \\Phi _ { - t \\Lambda } a ) d t , a \\in \\C ^ n . \\end{align*}"}
+{"id": "1367.png", "formula": "\\begin{align*} a ( x ) \\partial _ t v - \\Delta v = 0 \\end{align*}"}
+{"id": "3887.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 ^ + } \\int _ { \\Omega \\setminus B _ r ( 0 ) } f \\ , d \\gamma _ { \\mu _ 1 , \\mu _ 2 } = + \\infty , \\end{align*}"}
+{"id": "7908.png", "formula": "\\begin{align*} ( a , u ) \\ltimes _ { R \\oplus S } ( b , v ) = ~ & ( R ( a ) , S ( u ) ) \\ltimes _ { R \\oplus S } ( b , v ) ~ + ~ ( a , u ) \\ltimes _ { R \\oplus S } ( R ( b ) , S ( v ) ) \\\\ = ~ & \\big ( R ( a ) \\cdot b + a \\cdot R ( b ) , ~ R ( a ) \\cdot v + S ( u ) \\cdot b + a \\cdot S ( v ) + u \\cdot R ( b ) \\big ) \\\\ = ~ & \\big ( a \\ast _ R b , ~ a ~ { \\cdot } _ S ~ v + u ~ { \\cdot } _ S ~ b \\big ) . \\end{align*}"}
+{"id": "4204.png", "formula": "\\begin{align*} \\sum _ { \\deg ( D ) = N } \\mu _ k ( D ) + \\sum _ { j = 1 } ^ { 2 g } \\sum _ { \\ell = 0 } ^ { k - 1 } \\frac { Z ( \\gamma _ { j , \\ell } ^ { - 1 } ) } { k \\gamma _ { j , \\ell } ^ { 1 - k } Z ' ( \\gamma _ j ^ { - 1 } ) } \\gamma _ { j , \\ell } ^ { N + 1 } - \\frac { q ^ { N } } { \\zeta ( k ) } \\frac { q ^ { - g } h } { ( 1 - q ^ { - 1 } ) } . \\end{align*}"}
+{"id": "2411.png", "formula": "\\begin{align*} { \\rm C u t } _ { ( { \\bf a } ; { \\bf b } ) } ( \\psi ) \\mid _ { z \\to 0 } = p _ { { \\bf a } } \\cdot ( \\psi \\mid _ { z \\to 0 } ) \\cdot q _ { { \\bf b } } . \\end{align*}"}
+{"id": "7826.png", "formula": "\\begin{align*} \\Lambda _ X ^ { w _ 1 } ( u ) & = w _ 1 ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) \\\\ & \\ge w _ 2 ( G ^ { - 1 } ( u ) ) g ( G ^ { - 1 } ( u ) ) \\\\ = & \\Lambda _ Y ^ { w _ 2 } ( u ) . \\end{align*}"}
+{"id": "5918.png", "formula": "\\begin{align*} \\bigl ( \\wedge ^ 3 U \\bigr ) ^ \\perp = u \\wedge \\bigl ( \\wedge ^ 2 V _ 6 ^ \\vee \\bigr ) = F _ u \\ , , F _ u : = \\bigl \\{ \\xi \\in \\wedge ^ 3 V _ 6 ^ \\vee \\bigm | \\xi \\wedge u = 0 \\bigr \\} \\ , . \\end{align*}"}
+{"id": "2574.png", "formula": "\\begin{align*} \\begin{aligned} & \\bar y _ t ^ { * , t _ 0 , \\xi } : = \\bar U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) , \\bar z _ t ^ { * , t _ 0 , \\xi } : = \\sigma \\partial _ x \\bar U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) , \\\\ & \\bar z _ t ^ { 0 , * , t _ 0 , \\xi } : = \\sigma _ 0 [ \\partial _ x \\bar U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) + \\partial _ \\nu U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) ] . \\end{aligned} \\end{align*}"}
+{"id": "6888.png", "formula": "\\begin{align*} '' E ^ { p 0 } _ 2 = H ^ p _ h ( H ^ 0 _ v ( A ^ { S _ * U . } ) ) \\longrightarrow H ^ p ( X , A ) , \\end{align*}"}
+{"id": "602.png", "formula": "\\begin{align*} a _ { 1 2 } = - j ^ { ( 1 ) } - j ^ { ( 2 ) } - 1 \\ , , a _ { 1 2 3 } = - j ^ { ( 4 ) } - j ^ { ( 0 ) } - 1 \\ , , \\end{align*}"}
+{"id": "7943.png", "formula": "\\begin{align*} \\chi ^ s ( a , b ) = s ( a ) \\cdot _ E s ( b ) - s ( a \\cdot b ) ~ ~ ~ ~ ~ ~ \\Phi ^ s ( a ) = U s ( a ) - s R ( a ) , a , b \\in A . \\end{align*}"}
+{"id": "8061.png", "formula": "\\begin{align*} & { \\bf \\Xi } _ { p , t } = \\mathrm { b l k d i a g } ( { \\bf \\Xi } _ { p , t , 1 } , \\cdots , { \\bf \\Xi } _ { p , t , k } ) , \\ ; { \\bf \\Xi } _ { p , c } = \\mathrm { b l k d i a g } ( { \\bf \\Xi } _ { p , c , 1 } , \\cdots , { \\bf \\Xi } _ { p , c , k } ) , \\\\ & { \\bf R } _ { u } = \\mathrm { b l k d i a g } ( { \\bf R } _ { 1 , u } , \\cdots , { \\bf R } _ { u , k } ) , \\ ; \\widetilde { \\bf R } _ { u } = \\mathrm { b l k d i a g } ( \\widetilde { \\bf R } _ { 1 , u } , \\cdots , \\widetilde { \\bf R } _ { u , k } ) . \\end{align*}"}
+{"id": "5705.png", "formula": "\\begin{gather*} H ^ A _ i ( \\pi _ 1 ( U \\Sigma _ g ) , \\Q ) = \\Q [ 0 ] \\oplus H [ 1 ] . \\end{gather*}"}
+{"id": "5491.png", "formula": "\\begin{align*} F _ N ( b / t , 0 ; t ) = \\frac { ( 1 - t q ^ N ) } { ( 1 - t ) } \\sum _ { n = 0 } ^ { N } \\begin{bmatrix} N \\\\ n \\end{bmatrix} \\frac { ( q ) _ n ( - b ) ^ n q ^ { n ( n + 1 ) / 2 } } { ( t q ) _ n } . \\end{align*}"}
+{"id": "3205.png", "formula": "\\begin{align*} I _ { \\rho _ { n } ^ { 2 } } & \\le I ( \\frac { \\rho _ { n } } { \\rho } v _ { n } ) \\\\ & = I ( v _ { n } ) + o ( 1 ) \\\\ & = I _ { \\rho ^ { 2 } } + o ( 1 ) . \\end{align*}"}
+{"id": "8185.png", "formula": "\\begin{align*} P ( \\rho ) = \\kappa \\rho ^ { \\gamma } , \\kappa > 0 , \\ , \\ , \\gamma \\ge 1 . \\end{align*}"}
+{"id": "8058.png", "formula": "\\begin{align*} \\mathop { { \\mathrm { m a x i m i z e } } } \\limits _ { { \\bf w } _ p , { \\bf \\Phi } _ m , { \\bf F } _ k } & { \\quad } \\min _ { p } \\{ \\mathrm { S I N R } _ { R _ p } \\} \\\\ \\textrm { s u b j e c t t o } & { \\quad } \\| { \\bf F } _ k \\| _ F ^ 2 \\leq \\mathcal { P } _ k , \\forall k , \\\\ & { \\quad } \\mathrm { S I N R } _ { C _ u } \\geq \\xi , \\forall u , \\\\ & { \\quad } | { \\bf \\Phi } _ m ( i , i ) | = 1 , \\forall i , \\forall m , \\end{align*}"}
+{"id": "2072.png", "formula": "\\begin{align*} S \\begin{bmatrix} A _ { 1 1 } \\\\ A _ { 2 1 } \\end{bmatrix} T _ { 1 1 } = \\begin{bmatrix} I _ { \\ell } \\\\ D \\end{bmatrix} . \\end{align*}"}
+{"id": "6708.png", "formula": "\\begin{align*} \\partial _ { t } ( M + G ) + v \\cdot \\nabla _ { x } ( M + G ) - ( E + v \\times B ) \\cdot \\nabla _ { v } ( M + G ) = \\frac { 1 } { \\varepsilon } L _ { M } G + \\frac { 1 } { \\varepsilon } Q ( G , G ) , \\end{align*}"}
+{"id": "2257.png", "formula": "\\begin{align*} z ^ { \\epsilon _ 0 } x z ^ { \\epsilon _ 1 } x \\cdots z ^ { \\epsilon _ { p - 1 } } x z ^ { \\epsilon _ q } & = z ^ { \\epsilon _ 0 } \\sigma ( z ^ { \\epsilon _ 1 } ) \\sigma ^ 2 ( z ^ { \\epsilon _ 2 } ) \\cdots \\sigma ^ q ( z ^ { \\epsilon _ q } ) x ^ q \\\\ & = z ^ { \\epsilon _ 0 } ( \\sigma ( z ) ) ^ { \\epsilon _ 1 } ( \\sigma ^ 2 ( z ) ) ^ { \\epsilon _ 2 } \\cdots ( \\sigma ^ q ( z ) ) ^ { \\epsilon _ q } x ^ q \\end{align*}"}
+{"id": "5603.png", "formula": "\\begin{align*} \\hat { \\mathbb { G } } ^ i = \\frac 1 4 g ^ { i l } ( G _ { l ; k } y ^ k - G _ { ; l } ) \\end{align*}"}
+{"id": "1863.png", "formula": "\\begin{align*} \\{ y _ { i , j } : ( i , j ) \\in \\mathrm { G r i d } ( m r , n ) \\mbox { a n d } i + j \\mbox { i s e v e n } \\} & = \\{ 1 , 2 , \\ldots , \\tfrac { 1 } { 2 } m n r + \\tfrac { 1 } { 2 } \\} , \\mbox { a n d } \\\\ \\{ y _ { i , j } : ( i , j ) \\in \\mathrm { G r i d } ( m r , n ) \\mbox { a n d } i + j \\mbox { i s o d d } \\} & = \\{ M N - \\tfrac { 1 } { 2 } m n r + \\tfrac { 3 } { 2 } , \\\\ & M N - \\tfrac { 1 } { 2 } m n r + \\tfrac { 5 } { 2 } , \\allowbreak \\ldots , M N \\} . \\end{align*}"}
+{"id": "434.png", "formula": "\\begin{align*} \\mathcal { E } ( S , h ) = \\int _ S e ( h ) d A , \\end{align*}"}
+{"id": "3477.png", "formula": "\\begin{align*} \\textbf { e r r } ^ { ( 2 ) } : = \\mathcal { O } \\Bigg ( \\Big \\Vert \\gamma ^ { \\textbf { i n t } } _ { 0 } \\mathrm { U } _ { \\mathrm { i } } \\Big \\Vert _ { \\mathrm { H } ^ { \\frac { 1 } { 2 } , \\frac { 1 } { 4 } } \\Big ( \\partial \\Omega \\times \\mathbb { R } \\Big ) } \\Big \\Vert \\gamma ^ { \\textbf { i n t } } _ { 1 , \\mathrm { y } } \\Phi ^ { \\textbf { e } } ( \\xi , t ; \\mathrm { y } , \\cdot ) \\Big \\Vert _ { \\mathrm { L } ^ 2 \\Big ( \\partial \\Omega \\times ( 0 , \\mathrm { t } ) \\Big ) } \\Bigg ) . \\end{align*}"}
+{"id": "599.png", "formula": "\\begin{align*} 1 - 3 Z ^ { - + } _ { n - 1 , p } + 3 Z ^ { - + } _ { n - 1 , p } Z ^ { - + } _ { n , p } - Z ^ { - + } _ { n - 1 , p } Z ^ { - + } _ { n , p } Z ^ { - + } _ { n + 1 , p } = 0 \\ , , \\quad Z ^ { - + } _ { n , p } = \\frac { \\psi ^ { - 0 } _ { n , p } \\rho ^ { 0 + } _ { n , p } } { \\psi ^ { - 0 } _ { n , p - 1 } \\rho ^ { 0 + } _ { n - 1 , p } } \\ . \\end{align*}"}
+{"id": "7983.png", "formula": "\\begin{align*} \\eta \\left ( \\varphi _ v \\right ) ( \\xi ) = \\frac { \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle } { | \\xi ^ H | ^ 3 } \\left ( - 2 k ( \\xi ) | \\xi ^ H | ^ 2 - \\left \\langle \\nu ^ H , \\xi ^ H \\right \\rangle + 3 k ( \\xi ) | \\xi ^ H | ^ 2 \\right ) = 0 . \\end{align*}"}
+{"id": "5228.png", "formula": "\\begin{align*} \\Omega _ { \\tau , \\sigma , J , x } - \\Omega _ { \\tau , \\sigma , J ' , x ' } - ( p ^ f - 1 ) ( x _ \\tau - x _ \\tau ' ) = \\Lambda _ { \\tau } + p \\Lambda _ { \\tau \\circ \\varphi } + ( p ^ f - 1 ) ( s _ \\tau - x _ \\tau + x ' _ \\tau - s _ \\tau ' ) . \\end{align*}"}
+{"id": "7024.png", "formula": "\\begin{align*} \\begin{cases} \\widehat { u } _ { t t t } + ( \\sigma + | \\xi | ^ 2 ) \\widehat { u } _ { t t } + 2 | \\xi | ^ 4 \\widehat { u } _ t + ( \\sigma + | \\xi | ^ 2 ) | \\xi | ^ 4 \\widehat { u } = 0 , \\\\ ( \\widehat { u } , \\widehat { u } _ t , \\widehat { u } _ { t t } ) ( 0 , \\xi ) = ( \\widehat { u } _ 0 , \\widehat { u } _ 1 , \\widehat { u } _ 2 ) ( \\xi ) , \\end{cases} \\end{align*}"}
+{"id": "970.png", "formula": "\\begin{align*} \\tau ( x ) \\omega : = \\varphi ( x ) \\ , \\omega , \\end{align*}"}
+{"id": "7272.png", "formula": "\\begin{align*} \\frac { d } { d t } X ( t , \\omega ) = f ( t , X ( t , \\omega ) , m ( t ) , u _ L ( t , \\omega ) ) \\end{align*}"}
+{"id": "2816.png", "formula": "\\begin{align*} D ^ X _ r & = \\gamma _ r X _ r + \\nu _ r ^ { - 1 } \\left ( d - \\gamma _ t x - \\int _ t ^ r X _ s d ( \\nu _ s \\gamma _ s ) \\right ) , r \\in [ t , T ] . \\end{align*}"}
+{"id": "4271.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } \\frac { ( d q ) _ { j } ( q ^ { k } / d ) ^ { j } } { ( d q ^ { k + 1 } ) _ j ( q ) _ j } = \\frac { ( q ^ { k + 1 } ) _ { \\infty } } { ( q ^ { k } / d ) _ { \\infty } } \\sum _ { j = 0 } ^ { \\infty } \\frac { ( d q ) _ { j } q ^ { j ^ 2 + 2 j k } } { ( d q ^ { k + 1 } ) _ j ( q ^ { k + 1 } ) _ j ( q ) _ j } . \\end{align*}"}
+{"id": "3797.png", "formula": "\\begin{align*} h & \\leq \\lfloor \\l _ t / 2 \\rfloor - \\l _ t + n + j - 1 \\\\ & = n + j - 1 - \\lceil \\l _ t / 2 \\rceil . \\end{align*}"}
+{"id": "3892.png", "formula": "\\begin{align*} { \\bf c } ( \\tau ) = - \\tau ( N - 2 - \\mu _ 1 + \\tau ) + \\mu _ 2 > 0 , \\end{align*}"}
+{"id": "3597.png", "formula": "\\begin{align*} T ^ { 1 / 2 } ( T ^ { 1 / 2 } C T ^ { 1 / 2 } ) = \\sum _ { i j } a _ j \\sqrt { a _ i } \\eta _ { i j } \\phi _ j \\otimes \\phi _ i . \\end{align*}"}
+{"id": "4667.png", "formula": "\\begin{align*} \\Psi _ { \\ell , 2 } ( y ) = \\sum _ { n = 0 } ^ \\frac { \\ell - 1 } { 2 } ( - 1 ) ^ { \\left \\lfloor \\frac { \\ell - 1 - 2 n } { 4 } \\right \\rfloor } \\binom { \\left \\lfloor \\frac { \\ell - 1 + 2 n } { 4 } \\right \\rfloor } { n } y ^ n . \\end{align*}"}
+{"id": "7969.png", "formula": "\\begin{align*} \\eta ( k ) = ( l - 2 k ) \\alpha , \\eta ( \\alpha ) = k ^ 2 - \\alpha ^ 2 - k l . \\end{align*}"}
+{"id": "1318.png", "formula": "\\begin{align*} \\lambda ^ 2 & \\leq \\max _ { v \\in V } \\left \\{ \\sum _ { u \\sim v } A ^ 2 _ { v , u } \\right \\} = \\max _ { v \\in V } \\left \\{ \\sum _ { u \\sim v } d ( u ) \\right \\} \\\\ & = \\max _ { v \\in V } \\left \\{ d ( v ) + 2 e ( N _ 1 ( v ) ) + e ( N _ 1 ( v ) , N _ 2 ( v ) ) \\right \\} \\\\ & \\leq 2 e ( G _ { n , k } ' ) \\leq ( 4 k + 2 ) n \\leq 5 k n , \\end{align*}"}
+{"id": "6874.png", "formula": "\\begin{align*} \\Gamma \\left ( \\mathfrak { X } , \\ , \\frac { 1 + I _ { Y } I _ { Z } ^ { f _ { Z } } } { 1 + I _ { Y } ^ { f _ { Y } } I _ { Z } ^ { f _ { Z } } } \\right ) = 0 . \\end{align*}"}
+{"id": "3542.png", "formula": "\\begin{align*} \\mathcal { K } [ f ] ( \\mathrm { y } , \\tau ) : = \\frac { 1 } { \\alpha } \\displaystyle \\int _ { 0 } ^ { \\tau } \\int _ { \\partial \\Omega } \\partial _ { \\nu _ \\mathrm { y } } \\Phi ( \\mathrm { y } , \\tau ; \\mathrm { v } , \\mathrm { s } ) f ( \\mathrm { v } , \\mathrm { s } ) d \\sigma _ { \\mathrm { v } } d \\mathrm { s } . \\end{align*}"}
+{"id": "5454.png", "formula": "\\begin{align*} \\det M = \\det \\begin{pmatrix} A & B & C \\\\ C & A & B \\\\ B & C & A \\end{pmatrix} . \\end{align*}"}
+{"id": "1469.png", "formula": "\\begin{align*} Q _ n ( s ) = \\ ! \\begin{bmatrix} e _ M ( \\theta ) 1 _ { \\{ 1 \\in J _ r ^ { [ s ] } \\} } \\ ! + \\ ! ( f _ { g ( ( s - 1 ) \\ell _ r ^ * + 1 ) } - \\alpha _ n ) \\tilde { Z } _ { s , 1 } \\\\ \\vdots \\\\ e _ M ( \\theta ) 1 _ { \\{ \\ell _ r ^ * \\in J _ r ^ { [ s ] } \\} } \\ ! + \\ ! ( f _ { g ( s \\ell _ r ^ * ) } - \\alpha _ n ) \\tilde { Z } _ { s , \\ell _ r ^ * } \\end{bmatrix} , \\end{align*}"}
+{"id": "1149.png", "formula": "\\begin{align*} p _ { k , S } ( n , r ) = \\frac { 2 ^ { g / 2 } ( 2 \\pi D ) ^ \\ell } { \\Gamma ( \\ell ) ( \\det 2 S ) ^ { \\ell - 1 / 2 } } \\big ( 2 + O \\big ( \\frac { D ^ { g / 2 + \\epsilon } } { \\ell ^ { g / 2 + 1 / 3 } ( \\det 2 S ) ^ { g / 2 + 1 / 2 + \\epsilon } } \\big ) \\big ) . \\end{align*}"}
+{"id": "3087.png", "formula": "\\begin{align*} A _ 1 = \\left ( s ( k ) \\right ) _ { k = 1 } ^ { \\infty } = ( 1 , 3 , 4 , 7 , 6 , 1 2 , 8 , 1 5 , 1 3 , 1 8 , 2 0 , 2 8 , \\ldots ) . \\end{align*}"}
+{"id": "1902.png", "formula": "\\begin{align*} r = \\frac { 2 q } { 2 q - 3 } , \\end{align*}"}
+{"id": "3548.png", "formula": "\\begin{align*} \\partial _ { \\mathrm { i j } } \\Big ( \\mathbb { G } ^ { ( \\mathrm { k } ) } - \\mathbb { G } ^ { ( 0 ) } \\Big ) ( \\mathrm { x } , \\mathrm { y } ) = 2 a \\delta _ { \\mathrm { i j } } \\log | x - y | + 2 \\mathrm { a } \\dfrac { ( \\mathrm { x } - \\mathrm { y } ) _ { \\mathrm { i } } ( \\mathrm { x } - \\mathrm { y } ) _ { \\mathrm { j } } } { | \\mathrm { x } - \\mathrm { y } | ^ { 2 } } + 2 \\mathrm { b } \\delta _ { \\mathrm { i j } } + \\mathcal { O } ( | \\mathrm { x } - \\mathrm { y } | ) , \\mathrm { x } \\sim \\mathrm { y } , \\end{align*}"}
+{"id": "1050.png", "formula": "\\begin{align*} g ^ { a b } = \\delta _ { a b } + \\frac { 1 } { 3 } R _ { a c b d } x ^ { c } x ^ { d } + o ( { \\mathbf { x ^ { 2 } } } ) , \\end{align*}"}
+{"id": "1083.png", "formula": "\\begin{align*} & b _ 1 z _ 1 ^ 2 - b _ 2 z _ 2 ^ 2 = 4 ^ { m } p , \\\\ & b _ 1 z _ 1 ^ 2 - b _ 1 b _ 2 z _ 3 ^ 2 = - p , \\end{align*}"}
+{"id": "3037.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } _ 1 & = \\tfrac { \\bar { u } _ 2 } { \\bar { u } _ 1 } \\\\ \\dot { x } _ 2 & = \\bar { u } _ 2 \\\\ \\dot { x } _ 3 & = x _ 1 + \\tfrac { 1 } { 2 } \\bar { u } _ 1 \\bar { u } _ 2 \\ , , \\end{aligned} \\begin{aligned} \\dot { \\bar { u } } _ 1 & = \\bar { u } _ { 1 , 1 } \\\\ \\dot { \\bar { u } } _ { 1 , 1 } & = \\bar { u } _ { 1 , 2 } \\\\ \\dot { \\bar { u } } _ { 1 , 2 } & = \\bar { u } _ { 1 , 3 } \\\\ \\dot { \\bar { u } } _ { 1 , 3 } & = \\bar { u } _ { 1 , 4 } \\end{aligned} \\end{align*}"}
+{"id": "3813.png", "formula": "\\begin{align*} a _ 1 = \\frac { 4 n - 8 - ( a _ 1 + a _ 2 ) + d } { 2 } . \\end{align*}"}
+{"id": "5567.png", "formula": "\\begin{align*} \\epsilon ( \\alpha _ 0 , \\beta _ 0 ) = \\dfrac { 2 L _ { 1 M } ^ { \\frac { 5 - \\nu } { 2 } } L _ { 1 m } ^ { \\nu - 2 } \\Tilde { \\Phi } _ 1 ( \\alpha _ 0 , \\beta _ 0 ) } { N _ { 1 m } ^ { \\frac { 1 - \\nu } { 2 } } \\bigg [ \\gamma \\bigg ( \\tfrac { 1 - \\nu } { 2 } , \\beta _ 0 ^ 2 \\tfrac { N _ { 1 M } } { L _ { 1 m } } \\bigg ) - \\gamma \\bigg ( \\tfrac { 1 - \\nu } { 2 } , \\alpha _ 0 ^ 2 \\tfrac { N _ { 1 M } } { L _ { 1 m } } \\bigg ) \\bigg ] } < 1 , \\end{align*}"}
+{"id": "7840.png", "formula": "\\begin{align*} \\widetilde J _ i : = { \\bf i } _ { ( \\eta _ i ) _ { T ^ * \\widetilde { Q } } } \\Theta _ { \\mbox { \\tiny { $ \\widetilde Q $ } } } , \\end{align*}"}
+{"id": "3997.png", "formula": "\\begin{align*} \\hat { \\mathcal { M } } _ { \\beta } ( t ) \\stackrel { d } { = } \\sum _ { i = 1 } ^ { N _ { \\beta } ( t ) } X _ { i } , \\ t \\ge 0 , \\end{align*}"}
+{"id": "4117.png", "formula": "\\begin{align*} \\mathcal { H } ^ r _ { \\alpha } \\left ( 0 , \\infty \\right ) = \\{ u \\in L ^ 2 ( 0 , \\infty ) ; ~ ~ ~ \\int _ 0 ^ { \\infty } ( 1 + y ^ 2 ) ^ r | \\mathcal { H } ^ { \\alpha } ( u ) ( y ) | ^ 2 d y < \\infty \\} \\end{align*}"}
+{"id": "4602.png", "formula": "\\begin{align*} V ( n ) = K _ 1 ( E , A ) \\frac { \\sin 2 \\pi \\theta ( n ) + 1 0 0 } { n - b } . \\end{align*}"}
+{"id": "4791.png", "formula": "\\begin{align*} \\left ( \\vartheta ( \\vartheta + b _ { 1 } - 1 ) \\cdots ( \\vartheta + b _ { q } - 1 ) - z ( \\vartheta + a _ { 1 } ) \\cdots ( \\vartheta + a _ { p } ) \\right ) w = 0 , \\end{align*}"}
+{"id": "7407.png", "formula": "\\begin{align*} B ^ { ( 1 ) } = - 2 K ^ { ( 1 ) } , \\end{align*}"}
+{"id": "1048.png", "formula": "\\begin{align*} g _ { a b } = \\delta _ { a b } - \\frac { 1 } { 3 } R _ { a c b d } x ^ { c } x ^ { d } + o ( { \\mathbf { x ^ { 2 } } } ) , \\end{align*}"}
+{"id": "5148.png", "formula": "\\begin{align*} \\mathcal V _ 0 = \\left \\{ \\widetilde Q \\in \\widetilde { \\mathcal W } \\ , : \\ , \\widetilde Q \\subset A _ 0 , \\partial ^ M A _ 0 \\cap \\widetilde Q \\neq \\emptyset \\right \\} . \\end{align*}"}
+{"id": "7554.png", "formula": "\\begin{align*} Z = Y \\begin{pmatrix} E & 0 \\\\ 0 & E \\end{pmatrix} \\end{align*}"}
+{"id": "3623.png", "formula": "\\begin{align*} e ^ { \\theta x } \\int _ b ^ \\infty \\left [ f ( b ) + d ( z - b ) \\right ] \\theta e ^ { - \\theta z } d z = 0 , \\end{align*}"}
+{"id": "6909.png", "formula": "\\begin{align*} e _ { \\mathbb { C } ^ * } ( \\pi _ * ( \\mathcal { K } ^ \\vee _ i ( a _ j - a _ { i } ) ) ) & = ( h _ i + w _ i \\epsilon - w _ j \\epsilon ) ^ { d _ i + a _ j - a _ i + 1 } \\\\ e _ { \\mathbb { C } ^ * } ( \\pi _ * ( \\mathcal { K } ^ { \\vee } _ i \\otimes \\mathcal { K } _ j ( a _ { j } - a _ { i } ) ) ) & = ( h _ i + w _ i \\epsilon - h _ j - w _ j \\epsilon ) ^ { d _ i - d _ j + a _ j - a _ i + 1 } . \\end{align*}"}
+{"id": "8228.png", "formula": "\\begin{align*} \\widetilde { X } ( t ) \\lesssim \\Vert \\sigma _ 0 \\Vert _ { \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } } + \\int _ 0 ^ t \\Vert \\nabla u ( \\tau ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } } _ { 2 , 1 } } \\widetilde { X } ( \\tau ) \\dd \\tau + \\Vert \\widetilde { F } \\Vert _ { L ^ 1 _ t ( \\widetilde { B } ^ { \\frac { N } { 2 } + 1 - \\alpha , \\frac { N } { 2 } } ) } + \\Vert \\widetilde { G } \\Vert _ { L ^ 1 _ t ( \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } ) } . \\end{align*}"}
+{"id": "5125.png", "formula": "\\begin{align*} ( i \\partial _ t + ( - \\Delta _ { x } ) ^ { \\sigma } + ( - \\partial _ { y } ^ { 2 } ) ^ { \\sigma } ) u = \\mu | u | ^ { \\frac { 4 \\sigma } { d + 1 - 2 \\sigma } } u , u ( 0 , x , y ) = u _ 0 ( x , y ) \\in H ^ { \\sigma } ( \\mathbb { R } ^ d \\times \\mathbb { T } ) . \\end{align*}"}
+{"id": "462.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 0 \\\\ n - k \\equiv 0 \\ , ( \\mathrm { m o d } \\ , 2 ) } } ^ n \\binom { n } { k } \\big ( 2 ^ k L _ { k } - 2 \\big ) ( \\sqrt { 5 } ) ^ { n - k } \\frac { B _ { n - k + 2 } } { n - k + 2 } = \\frac { 2 ^ { n + 2 } L _ { n + 2 } - 2 } { { 5 } ( n + 1 ) ( n + 2 ) } - 1 \\ , , \\end{align*}"}
+{"id": "1381.png", "formula": "\\begin{align*} ( f , g ) _ { L ^ 2 } & : = \\int _ { \\Omega } f ( x ) g ( x ) \\ , d x . \\end{align*}"}
+{"id": "3209.png", "formula": "\\begin{align*} 0 = & 2 A ( u ) + B ( u ) - \\frac { 6 - 3 p } { 2 } C ( u ) - \\frac { 1 } { 4 } \\int _ { \\mathbb { R } ^ { 3 } } \\int _ { \\mathbb { R } ^ { 3 } } e ^ { - | x - y | } u ^ { 2 } ( x ) u ^ { 2 } ( y ) d x d y . \\end{align*}"}
+{"id": "409.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq a , b _ 1 , b _ 2 , \\ldots , b _ s \\leq n \\\\ ( a , n ) = 1 } } ( a - 1 , b _ 1 , \\ldots , b _ s , n ) \\chi ( a ) \\lambda _ 1 ( b _ 1 ) \\cdots \\lambda _ s ( b _ s ) = \\varphi ( n ) \\sigma _ s \\left ( ( \\frac { n } { d } , w _ 1 , \\ldots , w _ s ) \\right ) , \\end{align*}"}
+{"id": "1335.png", "formula": "\\begin{align*} \\left | B _ { 9 / 1 0 } \\cap \\left \\{ u = 0 \\right \\} \\right | \\le C _ 1 \\varepsilon ^ { p + \\delta } , \\end{align*}"}
+{"id": "7886.png", "formula": "\\begin{align*} \\Phi ( x , 0 ) = x _ 2 + O ( x ^ 3 ) , \\end{align*}"}
+{"id": "5088.png", "formula": "\\begin{align*} u ( x , y ) = \\sum _ { j } u _ j ( x ) \\phi _ j ( y ) , \\end{align*}"}
+{"id": "885.png", "formula": "\\begin{align*} S _ k [ r ] = \\sigma _ k ( \\lambda ( r ) ) \\end{align*}"}
+{"id": "1779.png", "formula": "\\begin{align*} \\sum _ { i = a _ { j _ { 0 } } + 1 } ^ { a _ { j _ { 0 } + 1 } - 1 } p _ { A , i } \\left ( C \\right ) \\left \\langle Y , B , i \\right \\rangle + \\sum _ { i = a _ { j _ { 0 } + 1 } + 1 } ^ { a _ { j _ { 0 } + 2 } - 1 } p _ { A , i } \\left ( C \\right ) \\left \\langle Y , B , i \\right \\rangle + \\cdots + \\sum _ { i = a _ { k - 1 } + 1 } ^ { n } p _ { A , i } \\left ( C \\right ) \\left \\langle Y , B , i \\right \\rangle = 0 , \\end{align*}"}
+{"id": "2227.png", "formula": "\\begin{align*} ( \\boldsymbol { \\mathcal { A } } * \\boldsymbol { \\mathcal { V } } _ n ) _ { i _ 1 , i _ 2 , : , : } = \\sum _ { k = 1 } ^ { N } \\boldsymbol { \\mathcal { A } } _ { i _ 1 , k , : , : } ( \\boldsymbol { \\mathcal { V } } _ n ) _ { k , i _ 2 , : , : } = \\sum _ { k = 1 } ^ { N } \\boldsymbol { \\mathcal { A } } _ { i _ 1 , k , : , : } ( { V } _ { i _ 2 } ) _ { k , : , : } = ( \\boldsymbol { \\mathcal { A } } * { V } _ { i _ 2 } ) _ { i _ 1 , : , : } \\end{align*}"}
+{"id": "6183.png", "formula": "\\begin{align*} \\mathcal { I } _ m = \\big \\{ ( i , j , t , p ) \\mid \\ & 0 \\leq i , j \\leq m , \\ \\mathrm { m a x } \\{ i + j - m , m - 1 - i - j \\} \\leq t \\leq m - | i - j | , \\\\ & \\mathrm { m a x } \\{ 0 , i + j - m , i + t - m , j + t - m \\} \\leq p \\leq \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mathrm { m i n } \\{ i , j , t , i + j + t + 1 - m \\} \\big \\} \\end{align*}"}
+{"id": "2248.png", "formula": "\\begin{align*} W _ i = k \\{ 1 , z ^ { \\pm 1 } , z ^ j x , z ^ { - j } x ^ { - 1 } , z ^ j s : 0 \\leq j \\leq i \\} . \\end{align*}"}
+{"id": "6306.png", "formula": "\\begin{align*} X _ t = x _ 0 ( t ) + \\int _ 0 ^ t K _ \\mu ( s , t ) \\mu ( s , X _ s ) \\dd s + \\int _ 0 ^ t K _ \\sigma ( s , t ) \\sigma ( s , X _ s ) \\dd B _ s , t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "1474.png", "formula": "\\begin{align*} \\tilde { U } _ n & = U _ n \\times \\tilde { Q } _ n \\\\ & = \\begin{bmatrix} \\frac { \\Delta _ { \\theta , 1 } } { f _ { 1 } - \\alpha _ n } e _ M ( \\theta ) 1 _ { \\{ 1 \\in J _ w \\} } + P _ { \\alpha _ n } ( \\lfloor \\frac { N } { 2 } \\rfloor - 1 ) \\\\ \\vdots \\\\ \\frac { \\Delta _ { \\theta , y } } { f _ { y } - \\alpha _ n } e _ M ( \\theta ) 1 _ { \\{ y \\in J _ w \\} } + P _ { \\alpha _ n } ( \\lfloor \\frac { N } { 2 } \\rfloor - 1 ) \\end{bmatrix} , \\end{align*}"}
+{"id": "1610.png", "formula": "\\begin{align*} W _ { e } ^ { \\prime } = \\mathrm { s p a n } \\left \\{ b _ { 1 } ^ { e } , \\ldots , b _ { c _ { n } } ^ { e } \\right \\} . \\end{align*}"}
+{"id": "214.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\eta ^ { - \\frac { 1 } { p - 1 } } \\left | \\eta _ t \\right | ^ { \\frac { p } { p - 1 } } d t = C T ^ { - \\frac { N } { 2 } + 1 } . \\end{align*}"}
+{"id": "7783.png", "formula": "\\begin{align*} \\begin{aligned} c a p _ p ( X _ \\varepsilon ) & \\leq \\int _ X | \\nabla ^ { g ^ + } u | _ { g ^ + } ^ p d V _ { g ^ + } = ( \\int _ 0 ^ \\varepsilon T ^ { \\frac { 1 } { 1 - p } } ( x ) d x ) ^ { 1 - p } \\\\ & = ( \\int _ 0 ^ \\varepsilon x ^ { \\frac { p - n - 1 } { 1 - p } } \\cdot [ V o l ( \\Sigma _ x ) ] ^ { \\frac { 1 } { 1 - p } } d x ) ^ { 1 - p } \\end{aligned} \\end{align*}"}
+{"id": "8053.png", "formula": "\\begin{align*} \\widetilde { \\bf y } _ { R _ p } [ f _ k ] = \\widetilde { \\bf s } _ { { \\mathrm { d i r } } _ { p } } [ f _ k ] + \\widetilde { \\bf s } _ { \\mathrm { i n d } _ p } [ f _ k ] + \\widetilde { \\bf c } _ { \\mathrm { d i r } } [ f _ k ] + \\widetilde { \\bf c } _ { \\mathrm { i n d } } [ f _ k ] + \\widetilde { \\bf n } _ R [ f _ k ] , \\end{align*}"}
+{"id": "6338.png", "formula": "\\begin{align*} \\int _ \\Omega \\nabla u \\cdot \\nabla \\zeta \\ , d x - \\int _ \\Omega \\left ( b u ^ + - a u ^ - \\right ) \\zeta \\ , d x = \\int _ \\Omega \\nabla z _ u \\cdot \\nabla \\zeta \\ , d x \\forall \\zeta \\in H ^ 1 _ 0 ( \\Omega ) , \\end{align*}"}
+{"id": "2767.png", "formula": "\\begin{align*} G = \\sqrt { \\hat { A } ( X ) } \\omega . \\end{align*}"}
+{"id": "3153.png", "formula": "\\begin{align*} \\begin{gathered} - \\Delta _ { \\mathbb { B } ^ { N } } w - \\lambda w = | w | ^ { p - 1 } w , \\ ; w > 0 \\ ; \\mathbb { B } ^ { N } , w \\in H ^ { 1 } \\left ( \\mathbb { B } ^ { N } \\right ) . \\end{gathered} \\end{align*}"}
+{"id": "7106.png", "formula": "\\begin{align*} \\left [ X ^ { \\lambda \\geq 0 } \\right ] - \\left [ \\mathfrak { g } ^ { \\lambda \\geq 0 } \\right ] \\in K _ 0 ( T ( d ) ) = M \\end{align*}"}
+{"id": "7976.png", "formula": "\\begin{align*} S _ M = M \\cap L _ v . \\end{align*}"}
+{"id": "3382.png", "formula": "\\begin{align*} I _ 2 ( \\varepsilon ) = \\langle \\log | J ( \\lambda , z ) | \\widetilde { W } _ { \\mathcal { M } } , d d ^ c \\phi \\rangle = \\int _ { \\mathrm { S u p p } \\mathcal { M } \\setminus S _ \\varepsilon } d \\mathcal { M } ( \\gamma ) \\left ( \\int _ B \\log | J ( \\lambda , \\gamma ( \\lambda ) ) | d d ^ c ( \\phi \\circ \\gamma ) \\right ) = 0 , \\end{align*}"}
+{"id": "2455.png", "formula": "\\begin{align*} - \\Delta F _ * + F _ * = 2 \\pi J _ * \\end{align*}"}
+{"id": "168.png", "formula": "\\begin{align*} \\begin{aligned} \\textrm { t h e i n t e g r a l } \\int _ { \\R ^ 3 } | v | ^ k e ^ { - a | v | ^ 2 } \\d v \\textrm { c o n v e r g e s , i f a n d o n l y i f } k > - 3 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "3406.png", "formula": "\\begin{align*} U _ t + U U _ r + p _ r / \\rho = 0 , \\end{align*}"}
+{"id": "326.png", "formula": "\\begin{align*} \\begin{array} { l l } I _ { u , i } & = g _ { u , i } ( V _ { u , i } - V _ i ) J _ { u , i } \\\\ \\dot { J } _ { u , i } & = - \\frac { J _ { u , i } } { \\tau _ { u , i } } + \\sum _ { k , j } w _ { i j } ^ u \\delta ( t - t _ k ^ j ) \\end{array} \\end{align*}"}
+{"id": "5387.png", "formula": "\\begin{align*} \\boldsymbol { s } = ( j ^ \\infty ) ^ * ( P _ \\beta ( x ^ i , u ^ \\alpha , u _ I ^ \\alpha ) d _ V u ^ \\beta \\wedge d x ^ 1 \\wedge \\cdots \\wedge d x ^ { } . \\end{align*}"}
+{"id": "3728.png", "formula": "\\begin{align*} \\underset { T : \\ ; T \\sharp \\rho _ o = \\rho _ 1 } { i n f } \\int _ { \\mathbb { R } ^ d } \\mid T x - x \\mid ^ 2 \\rho _ 0 ( x ) d x \\end{align*}"}
+{"id": "3440.png", "formula": "\\begin{align*} \\begin{cases} \\nabla \\times \\textbf { E } = - \\mu \\frac { \\partial } { \\partial t } \\textbf { H } \\\\ [ 1 0 p t ] \\nabla \\times \\textbf { H } = \\varepsilon \\frac { \\partial } { \\partial t } \\textbf { E } , \\end{cases} \\end{align*}"}
+{"id": "2604.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\underset { i = 1 } { \\overset { N } { \\sum } } \\mathbb { E } \\Big [ \\sup _ { 0 \\leq s \\leq t } | x _ s ^ i - x _ s ^ { * , i } | ^ 2 \\Big ] \\leq \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N ^ 2 } \\sum _ { i , j = 1 } ^ N \\mathbb E \\big [ | \\xi ^ i - \\xi ^ j | ^ 2 \\big ] \\Big ) . \\end{align*}"}
+{"id": "3370.png", "formula": "\\begin{align*} \\ln ( | E - & \\lambda v ( x + i \\rho / 2 + j _ n \\omega ) - v _ 1 ( j _ n ) | + 1 ) \\\\ & \\leq \\ln ( 2 | E - \\lambda v ( x + i y _ 0 + j _ n \\omega ) - \\lambda \\eta ( x + j _ n \\omega ) - v _ 1 ( j _ n ) | + 1 ) . \\end{align*}"}
+{"id": "1848.png", "formula": "\\begin{align*} ( x _ { i - 1 , j - 1 } + x _ { m + 2 - i , n + 2 - j } ) & + ( x _ { i - 1 , j } + x _ { m + 2 - i , n + 1 - j } ) \\\\ + ( x _ { i , j - 1 } + x _ { m + 1 - i , n + 2 - j } ) & + ( x _ { i , j } + x _ { m + 1 - i , n + 1 - j } ) = 2 S . \\end{align*}"}
+{"id": "3606.png", "formula": "\\begin{align*} \\sum _ m \\frac { \\omega ^ 2 _ m b _ m } { ( \\omega _ m + j ) ^ 2 ( \\omega _ m - j ) ^ 2 } \\lesssim \\begin{cases} j ^ { - 2 } , ~ ~ j < b , \\\\ j ^ { - \\min ( 1 + \\delta , 2 ) } , ~ ~ j > b \\end{cases} \\lesssim j ^ { - \\min ( 1 + \\delta , 2 ) } . \\end{align*}"}
+{"id": "7441.png", "formula": "\\begin{align*} \\sigma _ { s - 1 } ^ \\theta = \\hat { j } \\frac { \\theta } { s } + \\frac { \\theta ^ 2 + \\theta + s ^ 2 - s } { 2 s } . \\end{align*}"}
+{"id": "3908.png", "formula": "\\begin{align*} \\| S * g ( t ) \\| ^ 2 _ { \\mathbb H ^ \\mu } & \\le \\sum _ { n = 1 } ^ \\infty \\int _ 0 ^ t \\omega ( t - \\tau , \\lambda _ 1 ) \\lambda _ n ^ { \\mu - 1 } | g _ n ( \\tau ) | ^ 2 d \\tau \\\\ & = \\int _ 0 ^ t \\omega ( t - \\tau , \\lambda _ 1 ) \\| g ( \\tau ) \\| ^ 2 _ { \\mathbb H ^ { \\mu - 1 } } d \\tau . \\end{align*}"}
+{"id": "1026.png", "formula": "\\begin{align*} \\sigma _ r ( C _ t ) = \\overline { \\cup _ { p _ 1 , \\ldots , p _ r \\in C _ t } \\langle p _ 1 , \\ldots , p _ r \\rangle } . \\end{align*}"}
+{"id": "7558.png", "formula": "\\begin{align*} \\int _ { \\gamma _ X } f _ j \\frac { \\lambda ^ L } { z ^ { M + N } } \\omega = 0 , \\end{align*}"}
+{"id": "6122.png", "formula": "\\begin{align*} \\mathcal F _ { 2 } ^ { ( s ) } = \\{ F \\in \\mathcal F : ( [ d + 1 ] \\setminus \\{ s \\} ) \\subseteq F , s \\notin F , | F \\cap \\varepsilon _ 1 | \\geq 2 , | F \\cap \\varepsilon _ 2 | \\geq 2 \\} . \\end{align*}"}
+{"id": "4694.png", "formula": "\\begin{align*} \\left | r \\frac { d } { d r } \\left [ C ( z , w ) \\right ] \\right | \\lesssim & \\frac { 1 } { | 1 - z \\overline { w } | ^ 2 } \\frac { 1 } { | 1 - z w | ^ { 2 \\lambda } } \\asymp \\frac { | 1 - z \\overline { w } | ^ { - 2 } } { ( | 1 - z \\overline { w } | + | 1 - z w | ) ^ { 2 \\lambda } } , z = r e ^ { i \\theta } . \\end{align*}"}
+{"id": "3942.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 ( h - t ) } \\int _ { U _ { h - t } ( \\Gamma _ \\infty ) } j _ 1 ( p _ n ( y ) , \\nabla b _ { \\Omega _ n } ( p _ n ( y ) ) , H _ { \\Gamma _ n } ( p _ n ( y ) ) ) \\ , \\det ( d T _ \\infty ) \\ , d y \\\\ & = \\frac { 1 } { 2 ( h - t ) } \\int _ { U _ { h - t } ( \\Gamma _ \\infty ) } j _ 1 ( p _ n ( y ) , \\nabla b _ { \\Omega _ n } ( p _ n ( y ) ) , \\operatorname { T r } \\nabla ^ 2 b _ { \\Omega _ n } ( y ) ) \\ , \\det ( d T _ \\infty ) \\ , d y + \\operatorname { O } ( h ) . \\end{align*}"}
+{"id": "2822.png", "formula": "\\begin{align*} \\Gamma _ s = \\exp \\left ( - \\rho \\int _ 0 ^ s \\frac { \\lambda } { \\lambda + \\rho } + \\theta _ r d r \\right ) = \\exp \\left ( - \\frac { \\rho } { \\lambda + \\rho } \\left ( \\lambda s + \\rho \\int _ 0 ^ s K _ r d r \\right ) \\right ) , s \\in [ 0 , T ] . \\end{align*}"}
+{"id": "5191.png", "formula": "\\begin{align*} G ( X ) = \\frac { 1 } { 1 - \\left ( X ( X + X ) + ( X \\cdot \\frac { 1 } { 1 - X } \\cdot X ( X + X ) \\right ) } = \\frac { 1 - X } { 1 - X - 2 X ^ 2 } \\enspace . \\end{align*}"}
+{"id": "7932.png", "formula": "\\begin{align*} \\mu _ 1 ( a , b ) + \\varphi _ 1 ( a \\cdot b ) = ~ & \\mu _ 1 ' ( a , b ) + \\varphi _ 1 ( a ) \\cdot b + a \\cdot \\varphi _ 1 ( b ) , \\\\ R _ 1 + \\varphi _ 1 \\circ R = ~ & R _ 1 ' + R \\circ \\varphi _ 1 , . \\end{align*}"}
+{"id": "1412.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} u \\\\ v \\end{pmatrix} , \\begin{pmatrix} w \\\\ z \\end{pmatrix} \\right ) _ { \\mathcal { H } } = ( u , w ) _ { H ^ 1 } + ( v , z ) _ { L ^ 2 } . \\end{align*}"}
+{"id": "6781.png", "formula": "\\begin{align*} C ^ { - 1 } A - C B & \\leq H ^ { \\rm B } ( t ) \\leq C A , \\\\ \\pm i \\left [ H ^ { \\rm B } ( t ) , B \\right ] & \\leq C A , \\\\ \\rm \\frac { d } { d t } H ^ { \\rm B } ( t ) & \\leq C \\left ( \\norm { \\psi _ t } _ { H ^ 3 ( \\mathbb { R } ^ 3 ) } ^ 2 + \\norm { \\varphi _ t } _ { L _ 2 ^ 2 ( \\mathbb { R } ^ 3 ) } ^ 2 \\right ) A . \\end{align*}"}
+{"id": "2914.png", "formula": "\\begin{align*} { \\rm c l } \\ { \\mathcal N } \\subset { \\rm c l } \\ ( { \\mathcal N + \\mathcal K } ) \\subset { \\rm c l } \\mathcal A = \\mathcal B i \\mathcal Q \\mathcal T . \\end{align*}"}
+{"id": "4942.png", "formula": "\\begin{align*} b _ i = \\left \\lfloor \\binom { n } { k - 2 - i } / 2 \\right \\rfloor ~ \\mbox { a n d } ~ a _ i = \\binom { n } { k - 2 - i } - 2 b _ i \\in \\{ 0 , 1 \\} . \\end{align*}"}
+{"id": "7346.png", "formula": "\\begin{align*} A _ 1 = \\sum ^ { 2 m } _ { \\stackrel { i = 0 } { i \\ e v e n } } ( \\mathcal { R } ^ { m , \\frac { 2 m - i } { 2 } } _ { \\frac { 2 m - i } { 2 } , \\frac { 2 m - i } { 2 } } + \\mathcal { L } ^ { m , \\frac { 2 m - i } { 2 } } _ { \\frac { 2 m - i } { 2 } , \\frac { 2 m - i } { 2 } } ) + \\sum ^ { 2 m } _ { \\stackrel { i = 0 } { i \\ o d d } } ( \\mathcal { R } ^ { m , \\frac { 2 m - 1 - i } { 2 } } _ { \\frac { 2 m - 1 - i } { 2 } , \\frac { 2 m + 1 - i } { 2 } } + \\mathcal { L } ^ { m , \\frac { 2 m - 1 - i } { 2 } } _ { \\frac { 2 m + 1 - i } { 2 } , \\frac { 2 m - 1 - i } { 2 } } ) \\end{align*}"}
+{"id": "2893.png", "formula": "\\begin{align*} H = \\frac { \\alpha } { n } \\frac { \\nu _ { n + 1 } } { x _ { n + 1 } } + \\frac { \\varpi } { n } , \\end{align*}"}
+{"id": "6035.png", "formula": "\\begin{align*} & \\pi : X _ 2 = X _ 4 = X _ 6 = 0 \\mbox { a n d } \\pi ^ \\tau : X _ 1 = X _ 5 = X _ 6 = 0 , \\end{align*}"}
+{"id": "7391.png", "formula": "\\begin{align*} S L ( \\lambda ) \\varphi = ( - \\Delta - \\lambda ) ^ { - 1 } \\gamma _ D ' \\varphi S ( \\lambda ) \\varphi = \\gamma _ D ( - \\Delta - \\lambda ) ^ { - 1 } \\gamma _ D ' \\varphi \\end{align*}"}
+{"id": "7602.png", "formula": "\\begin{align*} ( \\sigma _ { - \\alpha _ i } \\eta _ j ) _ { 1 \\leq i \\leq n , 1 \\leq k \\leq g } = ( j ^ * ( ( ( 4 I _ \\aleph - C _ \\aleph ) ^ { - 1 } \\sigma _ { - \\alpha _ i , X } ) \\eta _ j ^ X ) ) _ { 1 \\leq i \\leq n , 1 \\leq k \\leq g } , \\end{align*}"}
+{"id": "6499.png", "formula": "\\begin{align*} \\sup _ { B _ { R _ { k } } ( x _ k ) } u & > C ^ { - 1 } \\delta ^ { - \\gamma } \\left ( u ( x _ k ) - u ( x _ k ) / 2 \\right ) \\\\ & = C ^ { - 1 } \\delta ^ { - \\gamma } u ( x _ k ) / 2 \\\\ & > C ^ { - 1 } \\delta ^ { - \\gamma } M _ { k - 1 } / 2 \\left ( \\frac { 1 } { M _ { k _ 0 } } \\sup _ { B _ 3 } u + \\inf _ { B _ 1 } u + \\rho \\right ) \\\\ & = M _ k \\left ( \\frac { 1 } { M _ { k _ 0 } } \\sup _ { B _ 3 } u + \\inf _ { B _ 1 } u + \\rho \\right ) , \\end{align*}"}
+{"id": "4072.png", "formula": "\\begin{align*} \\mathcal { V } \\mapsto \\mathcal { V } ( ] \\overline { x } [ ) ^ { \\nabla = 0 } \\end{align*}"}
+{"id": "5406.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N S _ { i j } = 1 , 1 \\leq j \\leq N . \\end{align*}"}
+{"id": "6224.png", "formula": "\\begin{align*} \\lambda _ { n , m } = B ( 2 n + 1 ) + \\big ( \\textstyle { \\frac { \\pi m } { d } } \\big ) ^ 2 , n \\in \\mathbb { N } _ 0 , \\ ; m \\in \\mathbb { N } \\end{align*}"}
+{"id": "5966.png", "formula": "\\begin{align*} f _ { K } ^ { ( H , \\alpha ) } = \\sum _ { \\beta \\in 2 ^ { \\N } \\cap [ 1 , \\nu / \\delta _ 0 ] } f _ { K } ^ { ( H , \\alpha , \\beta ) } \\end{align*}"}
+{"id": "787.png", "formula": "\\begin{align*} \\mathbb { E } U _ i \\left ( ( 1 - \\lambda ) X ^ { i } _ { T } + \\lambda \\left ( X ^ { i } _ { T } - \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } X ^ { j } _ { T } \\right ) \\right ) \\end{align*}"}
+{"id": "957.png", "formula": "\\begin{align*} v = \\frac { 1 } { \\sqrt { b _ \\alpha } } \\sum _ { \\beta = \\alpha + 1 } ^ { n - 1 } b _ \\beta y _ \\beta ^ 2 + \\sqrt { b _ \\alpha } \\bar { y } _ n . \\end{align*}"}
+{"id": "5027.png", "formula": "\\begin{align*} F ^ t = \\{ u < t \\} = X \\setminus \\{ u \\ge t \\} \\subset X \\setminus \\{ u \\ge 0 \\} \\subset \\Omega . \\end{align*}"}
+{"id": "4922.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { ( k + r ) / 2 - 1 } \\binom { n } { i } \\right ) + \\frac { 1 } { 2 } \\binom { n } { ( k + r ) / 2 } + \\binom { n } { r } < \\binom { n } { k } \\end{align*}"}
+{"id": "2101.png", "formula": "\\begin{align*} g \\cdot \\begin{bmatrix} I _ k & 0 & 0 & 0 \\\\ 0 & I _ { \\ell - k } & 0 & 0 \\\\ D _ x & 0 & I _ k & 0 \\\\ 0 & 0 & 0 & I _ { n - k - \\ell } \\end{bmatrix} , \\end{align*}"}
+{"id": "4670.png", "formula": "\\begin{align*} C _ F : y ^ \\ell + \\sum _ { r = 1 } ^ { \\frac { \\ell - 1 } { 2 } } a _ { 2 r - 1 } ( \\mathfrak { F } _ 1 \\mathfrak { F } _ 2 ) ^ { \\frac { \\ell + 1 } { 2 } - r } y ^ { 2 r - 1 } - \\mathfrak { F } _ 1 \\mathfrak { F } _ 2 ( \\mathfrak { F } _ 1 ^ { \\ell - 2 } + \\mathfrak { F } _ 2 ^ { \\ell - 2 } ) = 0 . \\end{align*}"}
+{"id": "2330.png", "formula": "\\begin{align*} \\zeta ( \\{ 1 , 3 \\} ^ { n } ) = \\frac { 2 \\pi ^ { 4 n } } { ( 4 n + 2 ) ! } \\end{align*}"}
+{"id": "4645.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\ , + \\infty } Q [ u _ n ] = - \\infty , \\end{align*}"}
+{"id": "4282.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { n ( - q ) _ { n - 1 } q ^ { n ( n + 1 ) / 2 } } { ( q ) _ n ^ { 2 } } = \\sum _ { n = 1 } ^ { \\infty } \\frac { n q ^ n ( - q ) _ { n - 1 } } { ( q ) _ n } - ( - q ) _ { \\infty } \\sum _ { j = 1 } ^ { \\infty } \\frac { q ^ { j ^ 2 } } { ( q ) _ { j } ^ { 2 } } \\sum _ { n = 1 } ^ { j } \\frac { q ^ n } { ( 1 - q ^ { 2 n } ) } . \\end{align*}"}
+{"id": "2416.png", "formula": "\\begin{align*} \\tilde { L } ( { \\rm D } ( w ) ) = 0 \\end{align*}"}
+{"id": "7422.png", "formula": "\\begin{align*} X = ( x _ { i j } ) \\mapsto \\breve { X } = \\sum _ { i , j = 1 } ^ { m } x _ { i , j } a _ i ^ \\dagger a _ j , [ \\breve { X } , \\ , \\breve { Y } ] = \\breve { [ X , \\ , Y ] } . \\end{align*}"}
+{"id": "7352.png", "formula": "\\begin{align*} \\partial _ H ( x _ 0 , y ) = 2 m - 2 i , \\ \\partial _ H ( x _ 0 , z ) = 2 m - 2 j , \\ \\partial _ H ( y , z ) = 2 m - 2 t , \\end{align*}"}
+{"id": "4013.png", "formula": "\\begin{align*} \\bar { q } _ { \\beta } ( n , t ) = \\begin{cases*} E _ { \\beta , 1 } \\left ( - \\lambda t ^ { \\beta } \\right ) , \\ n = 0 , \\\\ \\displaystyle \\rho ^ { n } \\sum _ { j = 1 } ^ { n } \\sum _ { m = 1 } ^ { j } ( - 1 ) ^ { m } \\binom { j } { m } \\left ( \\frac { - \\lambda t ^ { \\beta } } { ( 1 - \\rho ) ^ { - r } - 1 } \\right ) ^ { j } \\binom { r m + n - 1 } { n } E _ { \\beta , j \\beta + 1 } ^ { j + 1 } \\left ( - \\lambda t ^ { \\beta } \\right ) , \\ n \\ge 1 . \\end{cases*} \\end{align*}"}
+{"id": "6651.png", "formula": "\\begin{align*} \\lambda _ { - k } & = 1 , k \\in \\mathbb { N } \\\\ \\lambda _ { i , j } & = 1 , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] , \\ j \\in \\mathbb { N } _ 2 \\\\ \\sum _ { i = 1 } ^ { \\eta } \\lambda _ { i , 1 } ^ { 2 } & = 1 . \\end{align*}"}
+{"id": "5249.png", "formula": "\\begin{align*} \\rho _ l ( x _ 1 , \\ldots , x _ l ) = d e t ( K ( x _ i , x _ j ) ) _ { 1 \\leq i , j \\leq l } , \\ \\ K ( x , y ) = \\frac { s i n ( \\pi \\ * ( x - y ) ) } { \\pi ( x - y ) } , \\end{align*}"}
+{"id": "6456.png", "formula": "\\begin{align*} \\int _ u ^ v \\frac { ( q x / u , q x / v , \\gamma b x ; q ) _ { \\infty } } { ( d x , b x , c x ; q ) _ { \\infty } } d _ q x & = \\frac { ( 1 - q ) v ( q , u / v , q v / u ; q ) _ { \\infty } ( d c u v , b c u v , \\gamma b / c ; q ) _ { \\infty } } { ( d u , d v , b u , b v , c u , c v ; q ) _ { \\infty } } \\\\ & \\times { } _ 3 \\phi _ 2 \\left ( \\begin{gathered} c u , c v , c d u v / \\gamma \\\\ d c u v , b c u v \\end{gathered} ; \\ , q , \\frac { \\gamma b } { c } \\right ) . \\end{align*}"}
+{"id": "883.png", "formula": "\\begin{align*} \\sigma _ { k } ( \\lambda ) = \\sum _ { i _ { 1 } < \\ldots < i _ { k } } \\lambda _ { i _ { 1 } } \\ldots \\lambda _ { i _ { k } } , \\ \\ k = 1 , \\ldots , n , \\end{align*}"}
+{"id": "2898.png", "formula": "\\begin{align*} \\kappa ( s ) = \\alpha \\ , \\frac { \\nu _ { n + 1 } ( s ) } { x _ { n + 1 } ( s ) } + \\varpi . \\end{align*}"}
+{"id": "1163.png", "formula": "\\begin{align*} W ( F ) ( \\tau , \\tau ' ) : = F _ { z = 0 } . \\end{align*}"}
+{"id": "1007.png", "formula": "\\begin{align*} 0 & = ( \\lambda ^ { \\rm M } + \\mu ^ { \\rm M } - \\mu _ { \\rm c } ^ { \\rm M } ) u _ { j , j i } + ( \\mu ^ { \\rm M } + \\mu _ { \\rm c } ^ { \\rm M } ) u _ { i , j j } - 2 \\mu _ { \\rm c } ^ { \\rm M } P _ { [ j i ] , j } , \\\\ 0 & = ( \\beta _ 1 ^ { \\rm M } + \\beta _ 3 ^ { \\rm M } ) ( P _ { [ k i ] , k j } + P _ { [ j k ] , k i } ) + 2 \\beta _ 2 ^ { \\rm M } P _ { [ i j ] , k k } - 2 \\mu _ { \\rm c } ^ { \\rm M } P _ { [ i j ] } + \\mu _ { \\rm c } ^ { \\rm M } ( u _ { j , i } - u _ { i , j } ) . \\end{align*}"}
+{"id": "7721.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { A } ( p ) & = 2 ^ n \\cdot \\lim \\limits _ { t \\rightarrow + \\infty } e ^ { - n t } V o l ( \\partial B _ p ^ { g ^ + } ( t ) , g ^ + ) \\\\ & = 2 ^ n \\cdot \\lim \\limits _ { t \\rightarrow + \\infty } V o l ( \\partial B ( t ) , g ^ p | _ { \\partial B ( t ) } ) \\\\ & = 2 ^ n \\cdot V o l ( \\partial X , \\hat { g } ^ p ) \\end{aligned} \\end{align*}"}
+{"id": "3330.png", "formula": "\\begin{align*} \\alpha = \\alpha _ 0 + \\alpha _ 1 + \\alpha _ 2 \\hat { \\alpha } = \\alpha _ 0 - \\alpha _ 1 + \\alpha _ 2 . \\end{align*}"}
+{"id": "4228.png", "formula": "\\begin{align*} 1 - \\gamma _ m ^ { - 1 } \\gamma _ { j , \\ell } & = 1 + O _ g ( q ^ { - 1 / 2 + 1 / 2 k } ) , \\\\ \\frac { 1 } { 1 - \\gamma _ { j , \\ell } ^ { - 1 } } & = 1 + O _ g ( q ^ { - 1 / 2 k } ) , \\\\ \\frac { 1 } { 1 - q ^ { - 1 } \\gamma _ { j , \\ell } } & = 1 + O _ g ( q ^ { - 1 + 1 / 2 k } ) . \\end{align*}"}
+{"id": "4300.png", "formula": "\\begin{align*} R _ 1 ( q , N ) = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( - 1 ) ^ { n + 1 } ( q ) _ n q ^ { n ( 3 n + 1 ) / 2 } } { ( q ) _ { n + N } ( 1 - q ^ n ) } , \\end{align*}"}
+{"id": "7345.png", "formula": "\\begin{align*} \\mathcal { A } = \\mathcal { A } _ 1 + \\mathcal { A } _ 2 + \\mathcal { A } _ 3 \\ \\ . \\end{align*}"}
+{"id": "5582.png", "formula": "\\begin{align*} v ^ { \\alpha } = y ^ { \\alpha } + \\sqrt { - 1 } y ^ { \\alpha + n } \\end{align*}"}
+{"id": "5714.png", "formula": "\\begin{align*} \\partial _ t \\nu _ t = \\kappa ^ + [ \\nu _ t ] - \\kappa ^ - [ \\nu _ t ] , \\end{align*}"}
+{"id": "7659.png", "formula": "\\begin{align*} P ( E _ p ) = \\inf \\{ \\ , P ( F ) \\ , : \\ , F \\subset \\Omega \\ , , | F | = | E _ p | \\ , \\} . \\end{align*}"}
+{"id": "5016.png", "formula": "\\begin{align*} \\begin{cases} \\begin{aligned} - \\Delta _ p u & = 1 , & & \\\\ u & = 0 , & & \\end{aligned} \\end{cases} \\end{align*}"}
+{"id": "538.png", "formula": "\\begin{align*} g _ 1 ( t ) & : = \\int _ { \\R ^ n } f \\big ( ( 1 - t ) x _ 1 + t T _ 1 ( x _ 1 ) , \\ldots , ( 1 - t ) x _ n + t T _ n ( x _ 1 ) \\big ) \\prod _ { i = 1 } ^ n Q _ i ( d x _ i ) , \\\\ g _ 2 ( t ) & : = H \\big ( M _ 1 ( t ) \\times \\cdots \\times M _ n ( t ) \\big ) = \\sum _ { i = 1 } ^ n H ( M _ i ( t ) ) . \\end{align*}"}
+{"id": "5357.png", "formula": "\\begin{align*} J & = P _ s { } ^ s , & W _ { i s j } { } ^ s & = 0 , & C _ { s i } { } ^ s & = 0 . \\end{align*}"}
+{"id": "2380.png", "formula": "\\begin{align*} W ( t ; b ) = \\begin{cases} e _ { \\iota ^ { 0 } ( t ) } e _ { \\iota ^ { 1 } ( t ) } \\cdots e _ { \\iota ^ { b - 1 } ( t ) } & b \\geq 0 \\\\ 0 & b < 0 . \\end{cases} \\end{align*}"}
+{"id": "2364.png", "formula": "\\begin{align*} V _ { k } = \\begin{cases} 1 & f ^ { - 1 } ( \\{ k \\} ) = \\emptyset \\\\ U ( i _ { k + 1 } - i _ { k } - 1 + \\sum _ { f ( j ) = k } b _ { j } ) & f ^ { - 1 } ( \\{ k \\} ) \\neq \\emptyset . \\end{cases} \\end{align*}"}
+{"id": "8130.png", "formula": "\\begin{align*} ( ( L _ { 0 } , \\varphi _ 0 ^ { ( n ) } ) \\cdots & ( L _ d , \\varphi _ d ^ { ( n ) } ) ) _ S \\geqslant \\sum _ { i = 0 } ^ d \\delta _ i \\frac { \\widehat { \\mu } _ { \\min } ( E _ { i , n } , \\xi _ { i , n } ) } { n } \\\\ & - \\frac 3 2 \\nu ( \\Omega _ \\infty ) \\sum _ { i = 0 } ^ d \\frac { \\delta _ i } { n } \\ln ( r _ { i , n } + 1 ) - \\nu ( \\Omega _ \\infty ) \\sum _ { i = 0 } ^ d \\frac { \\delta _ i } { n } \\ln ( n ^ d \\delta _ i ) . \\end{align*}"}
+{"id": "2244.png", "formula": "\\begin{align*} ( a + A _ { i - 1 } ) ( b + F _ { j - 1 } ) = a b + F _ { i + j - 1 } \\end{align*}"}
+{"id": "7881.png", "formula": "\\begin{align*} F ( \\tilde R ) = \\{ f \\in F _ B \\colon \\mathcal { G } ' ( f ) \\cap \\mathcal { R } ( \\tilde R ) \\neq \\varnothing \\} . \\end{align*}"}
+{"id": "5895.png", "formula": "\\begin{align*} \\xi ^ { v } _ { n } ( h ) = \\mathcal { O } ( h ^ 3 / \\epsilon ^ 2 ) . \\end{align*}"}
+{"id": "904.png", "formula": "\\begin{align*} m : = \\inf _ { \\partial \\Omega } \\frac { G ( ( D ^ 2 u ) ' ) } { \\widetilde { f } } , \\end{align*}"}
+{"id": "4407.png", "formula": "\\begin{align*} \\upsilon = m + n q , m , n \\in \\Z . \\end{align*}"}
+{"id": "4636.png", "formula": "\\begin{align*} \\int _ \\Omega \\phi ( x , | \\nabla u _ \\infty | ) \\ , d x = \\lim _ { \\lambda \\to \\infty } \\int _ \\Omega \\min \\{ \\phi ( x , | \\nabla u _ \\infty | ) , \\lambda | \\nabla u _ \\infty | ^ p \\} \\ , d x \\le \\limsup _ { \\lambda \\to \\infty } \\int _ \\Omega \\phi _ \\lambda ( x , | \\nabla u _ \\infty | ) \\ , d x , \\end{align*}"}
+{"id": "2511.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } T ^ { c _ 1 ( n ) } x _ { 0 0 } = x _ { 1 0 } , \\textup { a n d } \\lim _ { m \\to \\infty } T ^ { c _ 2 ( m ) } x _ { 1 0 } = x _ { 1 1 } . \\end{align*}"}
+{"id": "4417.png", "formula": "\\begin{align*} \\epsilon = \\frac 1 2 \\left ( \\nabla u + \\nabla u ^ T \\right ) , \\end{align*}"}
+{"id": "6132.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { i = 3 } ^ { k - 1 } \\binom { n - i } { k - 3 } - ( k - \\frac { 7 } { 2 } ) \\binom { n - 3 } { k - 3 } & = \\sum _ { i = 3 } ^ { k - 1 } \\left ( \\binom { n - i } { k - 3 } - \\binom { n - 3 } { k - 3 } \\right ) + \\frac { 1 } { 2 } \\binom { n - 3 } { k - 3 } \\\\ & = \\frac { 1 } { 2 } \\binom { n - 3 } { k - 3 } - \\sum _ { i = 4 } ^ { k - 1 } \\sum _ { j = 4 } ^ { i } \\binom { n - j } { k - 4 } \\\\ & > \\frac { 1 } { 2 } \\binom { n - 3 } { k - 3 } - \\frac { ( k - 3 ) ( k - 4 ) } { 2 } \\binom { n - 4 } { k - 4 } > 0 . \\end{aligned} \\end{align*}"}
+{"id": "3407.png", "formula": "\\begin{align*} p _ t + U p _ r + a ^ 2 \\rho ( U _ r + \\tfrac { n - 1 } { r } U ) = 0 , \\end{align*}"}
+{"id": "3547.png", "formula": "\\begin{align*} \\mathbb { T } \\Big [ \\mathrm { H } \\Big ] ( \\mathrm { x } ) = \\int _ { \\Omega } \\nabla \\nabla \\cdot \\Big ( \\mathbb { G } ^ { ( \\mathrm { k } ) } - \\mathbb { G } ^ { ( 0 ) } \\Big ) ( \\mathrm { x } , \\mathrm { y } ) \\mathrm { H } ( \\mathrm { y } ) d \\mathrm { y } . \\end{align*}"}
+{"id": "2554.png", "formula": "\\begin{align*} \\alpha ^ { * , \\xi } _ t : = - R ^ { - 1 } B ( P _ t x ^ { * , \\xi } _ t + \\varphi _ t ^ { * , \\xi } ) - h ( \\mu _ t ^ { * , \\xi } ) . \\end{align*}"}
+{"id": "5094.png", "formula": "\\begin{align*} \\frac { 1 } { m ' } = \\frac { 1 } { m } + \\frac { p } { r } , \\frac { 1 } { l ' } > \\frac { 1 } { l } + \\frac { p } { q } . \\end{align*}"}
+{"id": "4419.png", "formula": "\\begin{align*} u = 0 \\partial \\Omega \\end{align*}"}
+{"id": "5489.png", "formula": "\\begin{align*} \\frac { 1 } { ( b q ) _ { N } } = ( 1 - q ^ { N } ) \\sum _ { n = 0 } ^ { N } \\begin{bmatrix} N \\\\ n \\end{bmatrix} \\frac { ( q ) _ n ( b ) ^ n q ^ { n ^ 2 } } { ( b q ) _ n ( q ) _ n } . \\end{align*}"}
+{"id": "5556.png", "formula": "\\begin{align*} u _ 2 ( \\beta _ 0 ) = 0 , \\end{align*}"}
+{"id": "8043.png", "formula": "\\begin{align*} { \\bf s } _ { \\textrm { d i r } } ( t ) & = [ s _ { \\textrm { d i r } _ 1 } ( t ) , \\cdots , s _ { \\textrm { d i r } _ { N _ r } } ( t ) ] ^ T = \\sum _ { k = 1 } ^ { K } \\widetilde { \\alpha } _ { 1 , k } { \\bf a } _ r ( \\theta _ t , f _ k ) { \\bf a } _ t ^ T ( \\theta _ t , f _ k ) { \\bf F } _ k { \\bf s } _ k e ^ { \\mathrm { j } 2 \\pi { f _ k } t } , \\end{align*}"}
+{"id": "6766.png", "formula": "\\begin{align*} i \\partial _ t \\chi _ { \\le N } ( t ) = H ^ { \\leq N } ( t ) \\chi _ { \\le N } ( t ) \\end{align*}"}
+{"id": "3061.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } \\mu \\left ( \\frac { \\ell } { k } \\right ) f ( k ) \\equiv 0 \\pmod { \\ell } . \\end{align*}"}
+{"id": "675.png", "formula": "\\begin{align*} [ S ] = \\mu \\left ( \\frac { 1 } { 2 } d \\overline { z } + ( \\alpha - I \\beta ) d z \\right ) J \\end{align*}"}
+{"id": "3012.png", "formula": "\\begin{align*} C _ n : = ( p - 1 ) \\sum _ { i = 0 } ^ { n - 1 } p ^ { ( d - 1 ) ( n - 1 - i ) } B _ i , \\end{align*}"}
+{"id": "3895.png", "formula": "\\begin{align*} \\partial _ t u - ( 1 + m _ 0 \\partial _ t ^ \\alpha ) \\Delta u = f . \\end{align*}"}
+{"id": "5597.png", "formula": "\\begin{align*} 0 = g _ { i k } u ^ k + g _ { j k } y ^ j J ^ k _ i = g _ { i k } J ^ k _ j y ^ j + g _ { j k } y ^ j J ^ k _ i \\end{align*}"}
+{"id": "1631.png", "formula": "\\begin{align*} A ( T ) = 1 + \\frac { q ^ r - 1 } { u } \\sum _ { i = 0 } ^ { u - 1 } T ^ { W _ i ^ { ( b ) } } \\ ; . \\end{align*}"}
+{"id": "7326.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } H ( t , x , p , q , u ) = & p ^ { \\intercal } b ( t , x , u ) + \\left \\langle q , \\sigma ( t , x , u ) \\right \\rangle + f ( t , x , u ) , \\\\ & ( t , x , p , q , u ) \\in \\lbrack 0 , T ] \\times \\mathbb { R } ^ { n } \\times \\mathbb { R } ^ { n } \\times \\mathbb { R } ^ { n \\times d } \\times U , \\end{array} \\end{align*}"}
+{"id": "2648.png", "formula": "\\begin{align*} x _ { \\lambda } x _ { \\mu } x _ { \\nu } = ( x _ { \\lambda } x _ { \\mu } ) x _ { \\nu } ( \\lambda \\mu ) \\nu \\end{align*}"}
+{"id": "6506.png", "formula": "\\begin{align*} \\binom { n - 1 } { r - 1 } \\le \\binom { n - r - 1 } { r - 1 } + \\binom { n - r - s } { r - 1 } . \\end{align*}"}
+{"id": "4096.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n e _ i \\cdot \\sum _ { 0 \\leq j \\leq i } D _ j , \\end{align*}"}
+{"id": "3111.png", "formula": "\\begin{align*} a _ { \\ell _ n , n } = \\ell _ n q _ n \\end{align*}"}
+{"id": "1405.png", "formula": "\\begin{align*} \\frac { | \\nabla \\Theta | ^ 2 } { a ( x ) \\Theta } & = ( 2 - \\alpha ) ^ 2 \\frac { \\langle x \\rangle ^ { - 2 \\alpha } | x | ^ 2 } { a ( x ) ( t _ 0 + t + \\langle x \\rangle ^ { 2 - \\alpha } ) } \\le \\frac { ( 2 - \\alpha ) ^ 2 } { a _ 0 } \\end{align*}"}
+{"id": "2401.png", "formula": "\\begin{align*} { \\rm e v } _ { 0 } ^ { \\mathfrak { m } } ( e _ { a _ { 1 } } \\cdots e _ { a _ { k } } ) = I ^ { \\mathfrak { m } } ( \\infty ' , a _ { 1 } ' , \\dots , a _ { k } ' , 1 ) \\end{align*}"}
+{"id": "4823.png", "formula": "\\begin{align*} \\frac { \\partial \\phi } { \\partial t } = - \\nabla \\phi \\cdot \\nabla f + K \\int _ S \\phi ( x , s ' ) \\ , \\dd \\mu ( s ' ) - K \\phi + \\psi \\end{align*}"}
+{"id": "355.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\| u ( t ) \\| _ { L ^ 2 } = 0 . \\end{align*}"}
+{"id": "7831.png", "formula": "\\begin{align*} & a _ i a _ j = 0 \\ , \\ , { \\rm i f } \\ , \\ , i \\leq j \\\\ & a _ i a _ j a _ j ^ \\dag = \\alpha _ { i , j } a _ i \\\\ & a ^ \\dag _ i a _ i = a _ { i - 1 } a ^ \\dag _ { i - 1 } - a _ i a ^ \\dag _ i \\\\ & a _ i a ^ \\dag _ i a _ j = a _ j - \\alpha _ { i , j } \\sum _ { j < k \\leq i } a ^ \\dag _ k a _ k a _ j \\ , \\ , \\end{align*}"}
+{"id": "7522.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } E [ \\mu _ n - \\mu ] = 0 , \\end{align*}"}
+{"id": "5009.png", "formula": "\\begin{align*} \\Gamma _ r : = G _ { \\xi _ { r n } } = K _ { h _ { r } } r = 1 , \\ldots , t . \\end{align*}"}
+{"id": "5847.png", "formula": "\\begin{align*} \\Big ( 1 + N ^ { p - 1 } \\displaystyle \\sum _ { n = N + 1 } ^ { \\infty } \\frac { 1 } { n ^ p } \\Big ) \\langle f ( A ) x , x \\rangle ^ p < \\Big ( \\frac { p } { p - 1 } \\Big ) ^ { p } \\langle f ( A ) ^ p x , x \\rangle . \\end{align*}"}
+{"id": "1845.png", "formula": "\\begin{align*} S ( i , j ) & = \\tfrac { 1 } { 2 } m n + \\tfrac { 3 } { 2 } \\mbox { i f } i + j \\mbox { i s e v e n , a n d } \\\\ S ( i , j ) & = \\tfrac { 3 } { 2 } m n + \\tfrac { 3 } { 2 } \\mbox { i f } i + j \\mbox { i s o d d . } \\end{align*}"}
+{"id": "7257.png", "formula": "\\begin{align*} & \\overline { { m d i m } } _ M ( T , - ( c g + 1 + \\overline { { m d i m } } _ M ( T , X , d ) ) , d ) \\\\ = & \\overline { { m d i m } } _ M ( T , - c g , d ) - 1 - \\overline { { m d i m } } _ M ( T , X , d ) \\\\ \\leq & \\overline { { m d i m } } _ M ( T , 0 , d ) - 1 - \\overline { { m d i m } } _ M ( T , X , d ) = - 1 < 0 , \\end{align*}"}
+{"id": "5623.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } J ^ i _ { \\alpha } = \\delta ^ i _ { \\alpha + n } , \\\\ J ^ i _ { \\alpha + n } = - \\delta ^ i _ { \\alpha } \\end{array} \\right . \\end{align*}"}
+{"id": "4192.png", "formula": "\\begin{align*} l ( w ^ j ) \\mathrm { s i n } ( \\theta _ j ' ) = \\mathcal { C } i ( w ^ j , w _ 1 ) . \\end{align*}"}
+{"id": "1225.png", "formula": "\\begin{align*} L \\coloneqq S ' ( \\phi , \\psi ) = ( L _ 1 , L _ 2 ) , \\tilde L \\coloneqq \\tilde S ' ( \\phi , \\psi ) = ( \\tilde L _ 1 , \\tilde L _ 2 ) , \\end{align*}"}
+{"id": "4155.png", "formula": "\\begin{align*} \\chi ( E ) & = h ^ 0 ( X , E ) = r + s \\\\ \\chi ( E \\otimes I _ Z ) & = \\mathrm { h } ^ 0 ( X , E \\otimes I _ Z ) = r + s - r k . \\end{align*}"}
+{"id": "7417.png", "formula": "\\begin{align*} \\hat { H } = \\hbar \\omega \\left ( \\hat { N } + \\frac { 1 } { 2 } \\right ) . \\end{align*}"}
+{"id": "8100.png", "formula": "\\begin{align*} T _ 1 ^ * T _ 2 ( z _ 1 ^ { m _ 1 } z _ 2 ^ { m _ 2 } \\eta ) & = [ M _ { z _ 1 } ^ { \\alpha _ 1 } ] ^ * ( \\alpha _ 2 z _ 1 ^ { m _ 1 } z _ 2 ^ { m _ 2 + 1 } U ^ { m _ 1 } \\eta ) \\\\ & = \\begin{cases} \\bar { \\alpha _ 1 } \\alpha _ 2 z _ 1 ^ { m _ 1 - 1 } z _ 2 ^ { m _ 2 + 1 } U ^ { m _ 1 } \\eta & \\\\ 0 & , \\end{cases} \\end{align*}"}
+{"id": "7136.png", "formula": "\\begin{align*} \\chi + \\rho + \\delta = \\frac { 3 } { 2 } \\mathfrak { g } ^ { \\lambda < 0 } + \\sum _ { i = 1 } ^ k v _ i \\tau _ { d _ i } + d \\mu \\tau _ d + \\sum _ { i = 1 } ^ k \\psi _ i \\in \\textbf { V } ( d ) . \\end{align*}"}
+{"id": "6833.png", "formula": "\\begin{align*} E _ { 0 } = \\frac 1 6 \\frac { m _ { a } ^ 2 } { \\ell _ { a } } + \\frac 1 6 \\frac { m _ { b } ^ 2 } { \\ell _ { b } } - \\frac 1 3 \\ell _ { a b } \\frac { m _ { a } } { \\ell _ { a } } \\frac { m _ { b } } { \\ell _ { b } } . \\end{align*}"}
+{"id": "6945.png", "formula": "\\begin{align*} \\left [ q ^ d \\right ] \\sum _ { j = 0 } ^ { N } & ( - 1 ) ^ j \\binom { N } { j } \\left [ t ^ { \\ell + j } \\right ] \\frac { 1 } { ( 1 - y t ) ( 1 - t ) ^ N - q t ^ N } + ( - 1 ) ^ { N + 1 } \\left [ q ^ { d - 1 } \\right ] \\left [ t ^ \\ell \\right ] \\frac { 1 } { ( 1 - y t ) ( 1 - t ) ^ N - q t ^ N } \\\\ & = ( - 1 ) ^ N \\left [ q ^ d \\right ] \\left [ t ^ { \\ell + N } \\right ] \\frac { ( 1 - t ) ^ N } { ( 1 - y t ) ( 1 - t ) ^ N - q t ^ N } + ( - 1 ) ^ { N + 1 } \\left [ q ^ { d - 1 } \\right ] \\left [ t ^ \\ell \\right ] \\frac { 1 } { ( 1 - y t ) ( 1 - t ) ^ N - q t ^ N } . \\end{align*}"}
+{"id": "1677.png", "formula": "\\begin{align*} | E ^ { ( \\lambda ) } | & \\leq \\left [ 1 + 2 ^ 5 \\lambda \\left ( 1 + 2 ^ 3 ( 1 + 2 \\ , \\lambda + 2 \\ , \\lambda ^ 2 ) \\right ) ( 2 + 2 \\ , \\lambda ) \\right ] \\left [ 2 ^ 6 + 2 ^ 4 \\right ] \\\\ & \\qquad \\ ; + 2 ^ 4 \\lambda \\left [ 1 + 2 ^ 3 ( 1 + 2 \\ , \\lambda + 2 \\ , \\lambda ^ 2 ) \\right ] \\left [ 2 ^ 2 + 2 ^ 5 \\right ] \\\\ & \\leq 2 ^ { 1 0 } - 1 \\ , . \\end{align*}"}
+{"id": "5275.png", "formula": "\\begin{align*} \\begin{cases} & \\sum _ { i = 1 } ^ { n + 1 } k _ i = 0 , \\\\ & \\sum _ { i = 1 } ^ { n + 1 + i } k _ i = 0 , \\\\ & \\sum _ { i \\in B } k _ i = 0 , \\ \\ \\forall B \\in \\pi . \\end{cases} \\end{align*}"}
+{"id": "327.png", "formula": "\\begin{align*} V _ i ( t ) = V _ { r e s t } , t \\in [ t _ k ^ i , t _ k ^ i + T _ { r e f } ] . \\end{align*}"}
+{"id": "1256.png", "formula": "\\begin{align*} - u _ { p } ' ( r ) = \\left ( \\frac { A } { N - 1 } \\right ) ^ { \\frac { 1 } { p - 1 } } . \\end{align*}"}
+{"id": "3449.png", "formula": "\\begin{align*} \\gamma _ p = \\overline { \\gamma } _ p \\ ; \\delta ^ { - 2 } ~ ~ \\mbox { a n d } ~ ~ \\rho _ { \\mathrm { p } } \\mathrm { c } _ { \\mathrm { p } } \\sim 1 , ~ ~ \\mbox { s u c h t h a t } ~ ~ \\gamma _ m < \\sqrt { \\overline { \\gamma } _ p \\ ; \\rho _ { \\mathrm { p } } \\mathrm { c } _ { \\mathrm { p } } } , ~ ~ \\delta \\ll 1 . \\end{align*}"}
+{"id": "3250.png", "formula": "\\begin{align*} \\int _ { X } \\psi ( u - v ) \\theta ^ { n } _ { v } \\leq \\liminf _ { j \\to \\infty } \\int _ { X } \\psi ( u - v _ { j } ) \\theta ^ { n } _ { v _ { j } } \\leq \\liminf _ { j \\to \\infty } C ^ { - 2 j + 2 } = 0 . \\end{align*}"}
+{"id": "1111.png", "formula": "\\begin{align*} \\frac { ( 2 k ) ! } { k ! } ( 1 + \\rho ) ^ { k } = \\sum _ { j = 0 } ^ { k } \\binom { 2 k } { 2 j } ( 1 + \\rho ) ^ { 2 k - 2 j } ( 1 - \\rho ^ { 2 } ) ^ { j } ( 2 j - 1 ) ! ! ( 2 k - 2 j - 1 ) ! ! . \\end{align*}"}
+{"id": "5039.png", "formula": "\\begin{align*} \\dot \\gamma ( t ) = \\sum _ { i = 1 } ^ k u _ i ( t ) X _ i ( \\gamma ( t ) ) . \\end{align*}"}
+{"id": "2519.png", "formula": "\\begin{align*} x = \\bigl ( \\pi _ 1 ( \\tilde { x } _ { 0 0 } ) , \\pi _ 1 ( \\tilde { x } _ { 0 1 } ) , \\pi _ 1 ( \\tilde { x } _ { 1 0 } ) , \\pi _ 1 ( \\tilde { x } _ { 1 1 } ) \\bigr ) \\end{align*}"}
+{"id": "4144.png", "formula": "\\begin{align*} \\lim \\limits _ { | z | \\to 1 } \\left ( 1 - | z | ^ 2 \\right ) \\frac { | B ' ( z ) | } { 1 - | B ( z ) | ^ 2 } = 1 \\ , . \\end{align*}"}
+{"id": "4442.png", "formula": "\\begin{align*} \\Vert \\tilde { v } _ { 2 , n } ^ { \\varepsilon } - v _ { 2 , n } ^ { \\varepsilon } \\Vert _ { L _ r } = \\Vert w _ { 2 , n } ^ { \\varepsilon } \\Vert _ { L _ r } \\longrightarrow 0 n \\to \\infty , \\end{align*}"}
+{"id": "5145.png", "formula": "\\begin{align*} ( S _ { \\mathcal W } v ) ( x ) = \\sum _ { i = 1 } ^ \\infty \\psi _ i ( x ) \\frac { 1 } { | Q _ i | } \\int _ { Q _ i } v ( y ) \\ , d ( y ) . \\end{align*}"}
+{"id": "888.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ n u ^ 2 _ { 1 l } = 4 B ^ 2 \\sum _ l x ^ 2 _ l = 4 B ^ 2 | x | ^ 2 \\geq 4 B ^ 2 \\delta _ 0 . \\end{align*}"}
+{"id": "7912.png", "formula": "\\begin{align*} \\delta _ \\mathrm { m R B A } ( u ) = ~ & ( \\delta _ \\mathrm { H o c h } ( u ) , - u ) , ~ u \\in M , \\\\ \\delta _ \\mathrm { m R B A } ( \\chi , \\Phi ) = ~ & \\big ( \\delta _ \\mathrm { H o c h } ( \\chi ) , ~ - \\widetilde { \\delta } _ \\mathrm { H o c h } ( \\Phi ) - \\Psi ^ n ( \\chi ) \\big ) , ~ ( \\chi , \\Phi ) \\in C ^ n _ \\mathrm { m R B A } ( ( A , R ) , ( M , S ) ) . \\end{align*}"}
+{"id": "5411.png", "formula": "\\begin{align*} F ( x ) = 1 , \\mbox { i f } | x | \\leq 1 / 9 ; F ( x ) = 0 , \\mbox { i f } | x | \\geq 2 / 9 , \\end{align*}"}
+{"id": "881.png", "formula": "\\begin{align*} ( d _ 1 : = d _ 1 '' , \\ldots , d _ n '' : = d _ 3 ) . \\end{align*}"}
+{"id": "5403.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n _ 1 } \\int _ { B _ { \\rho } ( y _ i ) } u ^ 2 ( x , 0 ) d x \\le e ^ { M _ 1 ( 1 - \\beta ) } \\sum _ { i = 1 } ^ { n _ 1 } \\left ( \\int _ { B _ { \\rho / 2 } ( y _ i ) } u ^ 2 ( x , 0 ) d x \\right ) ^ { \\beta } \\left ( N \\int _ { Q _ { 4 } } u ^ 2 ( x , t ) \\right ) ^ { 1 - \\beta } . \\end{align*}"}
+{"id": "5637.png", "formula": "\\begin{align*} D ^ T _ T W = s _ \\lambda ' ( t ) V = \\sqrt { \\lambda } \\cos ( \\sqrt { \\lambda } t ) V . \\end{align*}"}
+{"id": "7185.png", "formula": "\\begin{align*} f \\ast g : = \\frac { 1 } { a ! b ! } \\mathrm { S y m } \\left ( f g \\cdot \\prod _ { \\substack { 1 \\leq i \\le a , \\\\ a < j \\leq a + b } } \\xi ( z _ i z _ j ^ { - 1 } ) \\right ) , \\end{align*}"}
+{"id": "7665.png", "formula": "\\begin{align*} \\kappa _ { E ^ i _ p } V _ i ^ { 1 - p } = p H ( p ) \\ , . \\end{align*}"}
+{"id": "4547.png", "formula": "\\begin{align*} \\imath _ { \\tilde { E } ^ i } w = \\imath _ { \\hat { E } ^ i } v \\ , . \\end{align*}"}
+{"id": "693.png", "formula": "\\begin{align*} f _ i ^ { \\left ( { e q } \\right ) } \\left ( { { \\bf { x } } , t } \\right ) = { { \\hat w } _ i } \\rho \\left [ { 1 + \\frac { { { { \\bf { u } } ^ { e q } } \\cdot { { \\bf { c } } _ i } } } { { \\hat c _ s ^ 2 } } + \\frac { { { { \\left ( { { { \\bf { u } } ^ { e q } } \\cdot { { \\bf { c } } _ i } } \\right ) } ^ 2 } } } { { 2 \\hat c _ s ^ 4 } } - \\frac { { \\left | { { { \\bf { u } } ^ { e q } } } \\right | } } { { 2 \\hat c _ s ^ 2 } } } \\right ] , \\end{align*}"}
+{"id": "631.png", "formula": "\\begin{align*} \\log \\frac { 1 } { x } - \\gamma + \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n - 1 } } { n \\cdot n ! } x ^ { n } \\ \\ ( \\gamma : \\rm { E u l e r ' s \\ c o n s t a n t } ) ; \\end{align*}"}
+{"id": "7175.png", "formula": "\\begin{align*} \\chi = \\sum _ { i = 1 } ^ d m _ i \\beta _ i = \\sum _ { i = 1 } ^ { d - 1 } \\left ( \\frac { v i } { d } + 1 - \\left \\lceil \\frac { v i } { d } \\right \\rceil \\right ) ( \\beta _ { i + 1 } - \\beta _ i ) + \\frac { v } { d } \\sum _ { i = 1 } ^ d \\beta _ i . \\end{align*}"}
+{"id": "8044.png", "formula": "\\begin{align*} { \\bf a } _ t ( \\theta , f _ k ) \\ ! = \\ ! [ 1 , \\ ! e ^ { - \\mathrm { j } v _ t ( \\theta , f _ k ) } , \\ ! \\cdots \\ ! , \\ ! e ^ { - \\mathrm { j } ( N _ t - 1 ) v _ t ( \\theta , f _ k ) } ] ^ T , \\\\ { \\bf a } _ r ( \\theta , f _ k ) \\ ! = \\ ! [ 1 , \\ ! e ^ { - \\mathrm { j } v _ r ( \\theta , f _ k ) } , \\ ! \\cdots \\ ! , \\ ! e ^ { - \\mathrm { j } ( N _ r - 1 ) v _ r ( \\theta , f _ k ) } ] ^ T , \\end{align*}"}
+{"id": "7419.png", "formula": "\\begin{align*} \\begin{matrix} \\hat { N } \\left | n \\right > = n \\left | n \\right > , \\hat { N } ( a ^ \\dagger \\left | n \\right > ) = a ^ \\dagger ( \\hat { N } + \\hat { I } ) \\left | n \\right > = ( n + 1 ) ( a ^ \\dagger \\left | n \\right > ) , \\\\ \\hat { N } ( a \\left | n \\right > ) = a ( \\hat { N } - \\hat { I } ) \\left | n \\right > = ( n - 1 ) ( a \\left | n \\right > ) , a \\left | 0 \\right > = 0 = \\hat { N } \\left | 0 \\right > , \\end{matrix} \\end{align*}"}
+{"id": "2971.png", "formula": "\\begin{align*} K _ 0 ^ { \\operatorname { a d d } } ( \\mathcal { T } ) = G _ 0 ( \\mathcal { T } ) / ( [ Y ] - [ X ] - [ Z ] \\ ; \\vline \\ ; \\ ; X \\rightarrow Y \\rightarrow Z \\rightarrow \\Sigma X \\ ; \\mathcal { T } ) \\end{align*}"}
+{"id": "6400.png", "formula": "\\begin{align*} e _ 3 e ' _ 3 & = q ^ 0 ( [ e _ 1 e _ 5 ] + q [ e _ 2 e _ { 1 1 } ] ) , \\\\ e _ 4 e ' _ 4 & = q ^ 0 ( [ e _ 2 e _ 6 ] + q ^ 2 [ e _ 5 ^ 2 e _ { 1 0 } ] ) \\end{align*}"}
+{"id": "1686.png", "formula": "\\begin{align*} d \\mathbb { P } _ { \\mathrm { G i b b s } } ^ { f } ( \\varphi ) : = \\frac { 1 } { z _ { \\mathrm { G i b b s } } ^ { f } } \\ , e ^ { - H ( \\varphi ) } \\ , f ( \\| \\varphi \\| _ { \\mathfrak { h } } ^ 2 ) \\ , d \\varphi \\ , , \\end{align*}"}
+{"id": "659.png", "formula": "\\begin{align*} \\nabla _ X \\Psi _ 1 = - \\frac { 1 } { 2 } \\left ( X - \\langle X , T _ 1 \\rangle ( T _ 1 - f _ 1 ) - \\langle X , T _ 2 \\rangle ( T _ 2 - f _ 2 ) \\right ) \\cdot \\Psi _ 2 - \\frac { 1 } { 2 } S ( X ) \\cdot \\Psi _ 1 \\end{align*}"}
+{"id": "287.png", "formula": "\\begin{align*} \\overline { d ( u t _ { i } ) / ( u t _ { i } ) ^ { 2 } } \\otimes \\sqrt { \\overline { a _ { 0 } ( u t _ { i } ) ^ { 2 } } } & = \\overline { d t _ { i } / t _ { i } ^ { 2 } } \\otimes \\sqrt { \\overline { a _ { 0 } t _ { i } ^ { 2 } } } + \\overline { t _ { i } d u / u t _ { i } ^ { 2 } } \\otimes \\sqrt { \\overline { a _ { 0 } t _ { i } ^ { 2 } } } \\\\ & = \\overline { d t _ { i } / t _ { i } ^ { 2 } } \\otimes \\sqrt { \\overline { a _ { 0 } t _ { i } ^ { 2 } } } \\end{align*}"}
+{"id": "6641.png", "formula": "\\begin{align*} \\Lambda _ n ( Q ^ { 1 + \\varepsilon } _ \\theta ) = - \\dfrac { 1 } { ( 2 n - 1 ) ^ 2 \\theta ^ 2 } + O \\Big ( \\dfrac { 1 } { \\theta } \\Big ) \\theta \\to 0 ^ + . \\end{align*}"}
+{"id": "894.png", "formula": "\\begin{align*} S _ k ^ { i j } H _ { i j } = - \\sum _ i S _ k ^ { p q , r s } u _ { p q i } u _ { r s i } + B ( n - k + 1 ) S _ { k - 1 } + \\Delta f . \\end{align*}"}
+{"id": "3394.png", "formula": "\\begin{align*} 0 & = \\sum _ { i = 0 } ^ { m + 1 } u ( ( m + j + 2 ) ^ 2 + j - ( i + j + 1 ) ^ 2 , i ) \\\\ & = u ( ( m + j + 2 ) ^ 2 + j - ( ( m + 1 ) + j + 1 ) ^ 2 , m + 1 ) \\\\ & = u ( j , m + 1 ) , \\end{align*}"}
+{"id": "4487.png", "formula": "\\begin{align*} \\Omega ( \\mathbb { X } _ f , \\mathbb { Y } ) = \\dd f ( \\mathbb { Y } ) \\ , , \\end{align*}"}
+{"id": "389.png", "formula": "\\begin{align*} [ v _ f , \\underline { \\xi } ] = \\sum _ i g _ i ~ \\underline { \\xi } _ i \\end{align*}"}
+{"id": "3672.png", "formula": "\\begin{align*} \\delta & \\ge \\frac { t } { 6 ( t + 2 ) } \\sum _ { i = 1 } ^ { t } \\left ( x _ i - \\frac { 1 } { t + 2 } \\right ) ^ 2 + \\sum _ { i = t + 1 } ^ { t + 2 } \\left ( y _ i - \\frac { 1 } { t + 2 } \\right ) ^ 2 \\\\ & + ( \\sigma _ 1 - x _ { t } ) \\left ( \\frac { y _ { 1 2 } - y _ { 3 4 } } { 2 } \\right ) ^ 2 + x _ { t } \\left ( \\frac { y _ { 1 3 } - y _ { 2 4 } } { 2 } \\right ) ^ 2 . \\end{align*}"}
+{"id": "4052.png", "formula": "\\begin{align*} \\mathcal { R } _ s ( q ) = I ( s ) ^ { - 2 } y ^ { 2 k - 2 } \\left ( \\alpha _ 1 + \\alpha _ 2 y ^ { 2 k } e _ b + ( \\alpha _ 3 + \\alpha _ 4 y ^ { 2 k } e _ b ) q \\right ) , \\end{align*}"}
+{"id": "8148.png", "formula": "\\begin{align*} \\widehat { \\deg } ( f _ * ( \\overline L ^ { \\otimes n } ) ) \\geqslant \\sum _ { j = 1 } ^ n \\sum _ { i = 1 } ^ r \\widehat { \\deg } ( \\overline { I _ { i , j } / I _ { i - 1 , j } } ) + \\widehat { \\deg } ( s ) \\sum _ { k = 0 } ^ { n - 1 } h ^ 0 ( L ^ { \\otimes k } ) . \\end{align*}"}
+{"id": "3409.png", "formula": "\\begin{align*} a ^ 2 = - F _ \\rho / F _ p = \\frac { \\partial p } { \\partial \\rho } \\Big | _ { S = F ( \\rho , p ) } . \\end{align*}"}
+{"id": "6991.png", "formula": "\\begin{align*} G ( t , \\beta ) = F ( u + t \\phi ) + \\beta \\left ( \\| u + t \\phi \\| _ 2 ^ 2 - 1 \\right ) . \\end{align*}"}
+{"id": "376.png", "formula": "\\begin{align*} \\{ \\alpha , \\beta \\} = \\L _ { v _ \\alpha } \\beta \\end{align*}"}
+{"id": "117.png", "formula": "\\begin{align*} q , r , \\tilde q , \\tilde r \\geq 2 \\quad \\tfrac { 2 } { q } + \\tfrac { 1 } { r } = \\tfrac { 1 } { 2 } = \\tfrac { 2 } { \\tilde q } + \\tfrac { 1 } { \\tilde r } . \\end{align*}"}
+{"id": "6098.png", "formula": "\\begin{align*} F = \\sum \\limits _ { j = 1 } ^ { r } G _ j \\left ( x _ j g _ j - f _ j \\textbf { z } ^ { \\beta _ j } \\right ) + { \\sum \\limits _ { i = 1 } ^ { s } H _ i ( { y _ i } ^ { q } - y _ i ) } + H ( g _ 1 \\cdots g _ r w - 1 ) , \\end{align*}"}
+{"id": "4066.png", "formula": "\\begin{align*} \\sigma _ 1 ' & = \\sigma _ { \\mid I } = \\sigma _ 1 , \\\\ \\sigma _ 2 ' & = \\sigma _ { \\mid P \\setminus I } = \\sigma _ 2 . \\end{align*}"}
+{"id": "906.png", "formula": "\\begin{align*} \\nabla _ { i j } u : = \\nabla _ { e _ i } \\nabla _ { e _ j } u = e _ i ^ k e _ j ^ l \\partial _ { k } \\partial _ l u = e _ i ^ k e _ j ^ l u _ { k l } . \\end{align*}"}
+{"id": "6509.png", "formula": "\\begin{align*} U ( t ) : = \\exp ( i t H ) , \\quad ( t \\in \\mathbb { R } ) . \\end{align*}"}
+{"id": "8054.png", "formula": "\\begin{align*} & { \\bf h } _ { u , k } = \\sum _ { l = 1 } ^ { L } \\sum _ { k = 1 } ^ { K } { \\alpha _ { l } } e ^ { - \\mathrm { j } \\frac { 2 \\pi k d } { K } } { \\bf a } _ t ( \\phi _ { l } , f _ k ) r ( k T _ s - \\tau _ l ) , \\ ; \\in \\mathbb { C } ^ { { N _ t } \\times 1 } , \\end{align*}"}
+{"id": "4849.png", "formula": "\\begin{align*} \\partial _ t \\eta = \\nabla _ y \\cdot ( \\eta \\nabla _ y F ( y ) ) + K ( \\delta _ { x ^ * } \\otimes \\mu - \\eta ) . \\end{align*}"}
+{"id": "7274.png", "formula": "\\begin{align*} \\frac { d } { d t } \\gamma ( t ) = f ( t , \\gamma ( t ) , m ( t ) , u _ E ( t , \\gamma ( t ) ) ) \\end{align*}"}
+{"id": "6162.png", "formula": "\\begin{align*} T ^ n = \\overline q ^ { x _ n } T _ 1 ^ n T _ 2 ^ n = q ^ { y _ n } T _ 2 ^ n T _ 1 ^ n n \\geq 1 , \\end{align*}"}
+{"id": "783.png", "formula": "\\begin{align*} J _ i ( \\tau _ i ) = \\mathbb { E } \\left \\{ e ^ { - \\beta \\tau _ i } x _ i ( \\tau _ i ) - e ^ { - \\beta \\tau _ i } K \\right \\} . \\end{align*}"}
+{"id": "4170.png", "formula": "\\begin{align*} \\begin{gathered} R _ T ( x , U ) - n = R _ T ( T ^ n x , U ) = R _ T ( x , T ^ { - n } U ) , \\frac { R _ T ( x , U ) } { n } = R _ { T ^ n } ( x , U ) , \\\\ R _ { T ^ { n _ 1 } \\times \\cdots \\times T ^ { n _ d } } \\big ( ( x _ 1 , \\ldots , x _ d ) , U _ 1 \\times \\cdots \\times U _ d \\big ) = \\bigcap _ { i = 1 } ^ d R _ { T ^ { n _ i } } ( x _ i , U _ i ) . \\end{gathered} \\end{align*}"}
+{"id": "191.png", "formula": "\\begin{align*} d ^ * d u + d d ^ * u = \\Delta u = d ^ * ( - i a ) \\end{align*}"}
+{"id": "5626.png", "formula": "\\begin{align*} J ^ i _ { k | l } & : = \\frac { \\delta J ^ i _ k } { \\delta x ^ l } + J ^ s _ k \\hat { \\mathbb { G } } ^ i _ { s l } - J ^ i _ s \\hat { \\mathbb { G } } ^ s _ { k l } . \\end{align*}"}
+{"id": "3659.png", "formula": "\\begin{align*} \\hat { p } ( \\vec { y } ) = p ( ( x _ 1 + x _ 2 ) / 2 , ( x _ 1 + x _ 2 ) / 2 , x _ 3 , \\ldots , x _ m ) . \\end{align*}"}
+{"id": "4723.png", "formula": "\\begin{gather*} \\varphi ( l _ 1 ( a ) v ) = l _ 2 ( a ) \\varphi ( v ) , \\varphi ( r _ 1 ( a ) v ) = r _ 2 ( a ) \\varphi ( v ) , \\\\ \\varphi ( \\alpha _ { 1 k } ( v ) ) = \\alpha _ { 2 k } ( \\varphi ( v ) ) , \\forall a \\in A , \\ v \\in V . \\end{gather*}"}
+{"id": "4676.png", "formula": "\\begin{align*} & y ^ 5 + ( 5 4 t ^ 4 + 1 8 t ^ 3 + 3 4 t ^ 2 + 1 8 t + 3 9 ) y ^ 3 + ( 5 t ^ 8 + 2 3 t ^ 7 + 4 4 t ^ 6 + 2 0 t ^ 5 + 3 5 t ^ 4 + 3 0 t ^ 3 + 1 7 t ^ 2 + 3 3 t + 2 1 ) y \\\\ & + ( 5 7 t ^ { 1 0 } + 1 8 t ^ 9 + 2 4 t ^ 8 + 5 8 t ^ 7 + 1 4 t ^ 6 + 9 t ^ 5 + 4 1 t ^ 4 + 1 7 t ^ 3 + 3 8 t ^ 2 + 4 8 t + 4 4 ) = 0 \\end{align*}"}
+{"id": "394.png", "formula": "\\begin{align*} f ( x , y ) = f ^ 0 + x f ^ x ( x ) + y f ^ y ( y ) + x y f ^ { x y | x } ( x ) + x y f ^ { x y | y } ( y ) . \\end{align*}"}
+{"id": "3060.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 1 \\\\ k | \\ell } } ^ { \\ell } \\mu \\left ( \\frac { \\ell } { k } \\right ) f ( k ) \\geq 0 \\end{align*}"}
+{"id": "2140.png", "formula": "\\begin{align*} L ( f ) = \\lim _ { T \\to + \\infty } \\frac { 1 } { { ( 2 T ) } ^ n } \\int _ { { ] - T ; T [ } ^ n } f ( \\vec { x } ) \\ , \\mathrm { d } \\vec { x } = \\int _ { Y ^ m } g ( \\vec { y } ) \\ , \\mathrm { d } \\vec { y } = [ g ] \\end{align*}"}
+{"id": "8074.png", "formula": "\\begin{align*} \\mathcal { A } _ { i j } = \\rho _ { \\min \\{ i , j \\} } . \\end{align*}"}
+{"id": "6552.png", "formula": "\\begin{align*} U = \\begin{pmatrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "7316.png", "formula": "\\begin{align*} \\left ( \\frac { q ^ m - 1 } { 2 } - q ^ { m - 1 } + 2 \\mu \\right ) q ^ j = ( q ^ m + 1 ) f + L . \\end{align*}"}
+{"id": "2144.png", "formula": "\\begin{align*} - \\int _ { Y ^ m } \\left ( { \\bf R } { \\bf R } ^ T \\nabla _ y \\right ) \\cdot \\vec { \\phi } ( \\vec { y } ) \\ ; { \\theta } ( \\vec { y } ) \\ ; \\mathrm { d } \\vec { y } = \\int _ { Y ^ m } \\vec { \\phi } ( \\vec { y } ) \\cdot \\left ( { \\bf R } { \\bf R } ^ T \\nabla _ y \\right ) \\ ; { \\theta } ( \\vec { y } ) \\ ; \\mathrm { d } \\vec { y } \\end{align*}"}
+{"id": "2478.png", "formula": "\\begin{align*} - \\Delta v + v = f \\textrm { i n t h e s e n s e o f } \\mathcal { D } ^ \\prime ( M ) \\end{align*}"}
+{"id": "2717.png", "formula": "\\begin{align*} M _ { \\varphi } ( 2 n , j , 1 ; r - 1 ) = \\sum _ { u = 0 } ^ { n - j } \\binom { 2 n } { j } \\binom { 2 n - j } { u } M _ { \\varphi } ( 2 n , j + u , 0 ; r - 1 ) \\end{align*}"}
+{"id": "300.png", "formula": "\\begin{align*} [ \\tau _ { E } ^ { - 1 } ( T ^ { * } _ { X } X ( \\log E ) ) ] = \\sum _ { \\Theta ' \\subset \\Theta } [ T ^ { * } _ { E _ { \\Theta ' } } X ] \\end{align*}"}
+{"id": "6636.png", "formula": "\\begin{align*} 2 a b = 2 \\cdot \\sqrt { \\varepsilon } a \\cdot \\frac { b } { \\sqrt { \\varepsilon } } \\leq \\varepsilon a ^ { 2 } + \\dfrac { b ^ { 2 } } { \\varepsilon } . \\end{align*}"}
+{"id": "3001.png", "formula": "\\begin{align*} \\left . U \\right | _ { t = 0 } = \\left ( u _ { 0 } \\left ( x , \\frac { y } { \\varepsilon } \\right ) , \\varepsilon v _ { 0 } \\left ( x , \\frac { y } { \\varepsilon } \\right ) \\right ) = U _ { 0 } ^ { \\varepsilon } \\mathcal { S } ^ { \\varepsilon } . \\end{align*}"}
+{"id": "28.png", "formula": "\\begin{align*} I _ f ( p ' , q ' ) \\geq \\sum _ { j = 0 } ^ { D - 1 } q '' _ j f \\left ( \\frac { p '' _ j } { q '' _ j } \\right ) . \\end{align*}"}
+{"id": "836.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & A = - \\frac { p ' ( x ^ { * } ) } { k _ 1 ( x ^ { * } ) ^ { k _ 1 - 1 } } , \\\\ & \\bar { \\theta } _ 2 = - \\frac { K } { A ( x ^ { * } ) ^ { k _ 1 } + p ( x ^ { * } ) } = \\frac { K k _ 1 } { k _ 1 p ( x ^ { * } ) - p ' ( x ^ { * } ) x ^ { * } } . \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "7174.png", "formula": "\\begin{align*} \\sigma = \\sum _ { i > j } a _ { i , j } ( \\beta _ i - \\beta _ j ) , \\ - 1 \\leq a _ { i , j } \\leq 2 , 0 \\leq a _ { i + 1 , i } \\leq 2 . \\end{align*}"}
+{"id": "4932.png", "formula": "\\begin{align*} \\binom { n } { k - 2 } > \\sum _ { i = 3 } ^ { k - 1 } \\frac { j } { 2 } \\binom { n } { k - i } + \\sum _ { i = k - 2 j - 1 } ^ { k - 1 } \\frac { j + 2 } { 2 } \\binom { n } { k - i } . \\end{align*}"}
+{"id": "6090.png", "formula": "\\begin{align*} r : = \\sup _ { \\mu \\in \\mathcal C } \\int f d \\mu < \\int f d \\nu . \\end{align*}"}
+{"id": "6644.png", "formula": "\\begin{align*} f _ n : x = ( x _ 1 , x _ 2 ) \\mapsto e ^ { i k x _ 1 } \\chi \\big ( | x - a _ n | - ( n - 1 ) \\big ) \\in C _ c ^ { \\infty } ( \\mathbb { R } ^ 2 \\setminus \\Gamma ) . \\end{align*}"}
+{"id": "267.png", "formula": "\\begin{align*} h _ N ( t , x ) : = \\xi _ N ( f _ x , t ) , \\end{align*}"}
+{"id": "7661.png", "formula": "\\begin{align*} \\kappa _ { E _ p } = p H ( p ) | E _ p | ^ { p - 1 } = p \\frac { P ( E _ p ) } { | E _ p | ^ p } | E _ p | ^ { p - 1 } = p \\frac { P ( E _ p ) } { | E _ p | } \\ge p H ( 1 ) \\ , . \\end{align*}"}
+{"id": "4873.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } A _ n ( x ) \\frac { t ^ n } { n ! } = \\frac { 1 - x } { e ^ { t ( x - 1 ) } - x } . \\end{align*}"}
+{"id": "5577.png", "formula": "\\begin{align*} \\Omega _ k : = \\omega _ k - \\alpha _ k \\chi _ { ( V _ { 2 } ^ J ) ^ c \\cap { S ^ { m n - 1 } } } , \\end{align*}"}
+{"id": "1130.png", "formula": "\\begin{gather*} m _ { n } ( t ) = \\allowbreak \\sqrt { \\frac { n ! } { \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } } } \\frac { 1 } { ( 1 - t ) ^ { \\beta } } \\frac { 1 } { \\Gamma ( \\beta ) } \\times \\\\ \\int _ { 0 } ^ { \\infty } \\left ( \\sum _ { j = 0 } ^ { n } \\left ( \\frac { - t y } { 1 - t } \\right ) ^ { j } L _ { n - j } ( x | ( \\beta + j ) ) / j ! \\right ) y ^ { \\beta - 1 } \\exp ( - y ) d y . \\end{gather*}"}
+{"id": "2607.png", "formula": "\\begin{align*} 0 & = e _ y ( u , y ) = f ^ 2 ( u , y ) = f ( u , y ) f ( y , y ) + \\sum _ { u < v < y } f ( u , v ) f ( v , y ) \\\\ & = \\pm f ( u , y ) + \\sum _ { u < v < y } f ( u , v ) f ( v , y ) , \\end{align*}"}
+{"id": "5025.png", "formula": "\\begin{align*} F _ k : = \\bigcup _ { j = 1 } ^ k E _ j , G _ k : = \\bigcap _ { j = 1 } ^ k E _ j , \\end{align*}"}
+{"id": "7618.png", "formula": "\\begin{align*} B ( \\Sigma ) = \\langle x ^ { \\hat { \\sigma } } \\ \\vert \\ \\sigma \\in \\Sigma \\rangle \\subseteq S . \\end{align*}"}
+{"id": "4294.png", "formula": "\\begin{align*} C _ 1 ( q ) & : = \\sum _ { n = 1 } ^ { \\infty } \\overline { M } _ 1 ( n ) q ^ n , \\\\ R _ 1 ( q ) & : = \\sum _ { n = 1 } ^ { \\infty } \\overline { N } _ 1 ( n ) q ^ n , \\end{align*}"}
+{"id": "6876.png", "formula": "\\begin{align*} N = p M \\mbox { w i t h } p \\nmid M , \\chi _ K ( p ) = - 1 , \\qquad \\chi _ K ( M ) = 1 . \\end{align*}"}
+{"id": "7621.png", "formula": "\\begin{align*} \\nabla _ j : = \\{ n \\in N _ { \\R } \\ \\vert \\ \\langle m , n \\rangle \\geq - \\delta _ { i j } m \\in \\Delta _ i , 1 \\le i \\le r \\} . \\end{align*}"}
+{"id": "2306.png", "formula": "\\begin{align*} \\lambda _ { d , p } ( t ) = ( t - \\nu ) \\cdot ( t - p ) \\cdot \\lambda _ \\nu ^ - ( t ) \\cdot \\lambda _ { d , \\nu } ^ + ( t ) , \\end{align*}"}
+{"id": "3259.png", "formula": "\\begin{gather*} [ K _ 0 , K _ 1 ] _ q = K _ 2 , \\\\ [ K _ 1 , K _ 2 ] _ q = B K _ 1 + C _ 0 K _ 0 + D _ 0 , \\\\ [ K _ 2 , K _ 0 ] _ q = B K _ 0 + C _ 1 K _ 1 + D _ 1 , \\end{gather*}"}
+{"id": "406.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { \\substack { a = 1 \\\\ ( a , n ) = 1 } } ^ n ( a - 1 , n ) = \\varphi ( n ) \\sigma _ 0 ( n ) , \\end{align*}"}
+{"id": "3970.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\mathcal { M } ( t ) } { t } & \\stackrel { d } { = } \\sum _ { j = 1 } ^ { \\infty } j \\lim _ { t \\to \\infty } \\frac { N _ { j } ( t ) } { t } \\\\ & = \\alpha \\theta e ^ { \\theta } , \\ . \\end{align*}"}
+{"id": "2180.png", "formula": "\\begin{align*} & \\nabla _ { X _ i } X _ j = \\nabla ^ B _ { X _ i } X _ j , \\\\ & \\nabla _ { X _ i } Y _ p = \\nabla _ { Y _ p } X _ i = f ^ { - 1 } f _ i Y _ p , \\\\ & \\nabla _ { Y _ p } Y _ q = \\nabla ^ F _ { Y _ p } Y _ q - f \\ , h _ { p q } \\ , { \\rm g r a d } \\ : f , \\end{align*}"}
+{"id": "6032.png", "formula": "\\begin{align*} & \\tau : X _ 1 ' = X _ 2 ^ q , X _ 2 ' = X _ 1 ^ q , X _ 3 ' = X _ 3 , X _ 4 ' = X _ 5 ^ q , X _ 5 ' = X _ 4 ^ q , X _ 6 ' = X _ 6 . \\end{align*}"}
+{"id": "6051.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\eta ' = ( k ^ 2 ) ' \\eta + k ^ 2 \\eta ' \\ , \\ , \\ , \\ , \\Omega \\\\ [ 0 . 5 c m ] \\frac { \\partial \\eta ' } { \\partial n } = \\left ( - \\frac { \\partial ^ 2 \\eta } { \\partial n ^ 2 } \\right ) V \\cdot n + \\nabla \\eta \\cdot \\nabla _ \\Gamma ( V \\cdot n ) \\ , \\ , \\ , \\ , \\partial \\Omega \\\\ [ 0 . 5 c m ] \\int _ { \\partial \\Omega } \\eta ^ 2 ( V \\cdot n ) d \\sigma + 2 \\int _ \\Omega \\eta \\eta ' d x = 0 . \\end{cases} \\end{align*}"}
+{"id": "7632.png", "formula": "\\begin{align*} t _ 2 ^ 3 t _ 5 ^ 3 = t _ 0 t _ 1 t _ 2 t _ 3 t _ 4 t _ 5 = 1 . \\end{align*}"}
+{"id": "2485.png", "formula": "\\begin{align*} \\int _ { 1 } ^ \\infty \\frac { d s } { \\Big ( \\displaystyle \\int _ 0 ^ s F ( t ) d t \\Big ) ^ { p / ( 2 p - q + 1 ) } } < \\infty \\quad \\mbox { w h e r e } \\ ; F ( t ) = \\int _ 0 ^ t f ( k ) d k . \\end{align*}"}
+{"id": "638.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = \\frac { 1 } { 2 } ( \\sqrt { c _ 1 } \\ X _ 1 \\cdot \\nu _ 1 + \\sqrt { c _ 2 } \\ X _ 2 \\cdot \\nu _ 2 ) \\cdot \\varphi - \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi \\end{align*}"}
+{"id": "3788.png", "formula": "\\begin{align*} \\widetilde { C } _ { i , j } ( 1 ) & = \\left \\{ \\begin{array} { c l } \\frac { i j } { ( i + j ) } \\binom { 2 i } { i } \\binom { 2 j } { j } , & i + j > 0 \\\\ 0 , & i + j = 0 \\end{array} \\right \\} , \\\\ A _ { i , j } & = \\frac { ( 2 i + 1 ) ! ( 2 j + 1 ) ! } { ( i + j + 1 ) ( i ! j ! ) ^ 2 } = \\frac { ( 2 i + 1 ) ( 2 j + 1 ) } { ( i + j + 1 ) } \\binom { 2 i } { i } \\binom { 2 j } { j } . \\end{align*}"}
+{"id": "6768.png", "formula": "\\begin{align*} q ( t , x , y ) = \\delta ( x - y ) - \\psi _ t ( x ) \\overline { \\psi _ t ( y ) } \\end{align*}"}
+{"id": "4345.png", "formula": "\\begin{align*} T ^ 1 _ \\eta ( t ) & : = \\eta ^ 2 \\int _ 0 ^ t \\int _ \\Omega \\nabla ( a f _ 1 + b g _ 1 ) \\cdot \\frac { \\nabla f _ 1 } { ( f _ 1 + \\eta ) ^ 2 } \\ , \\mathrm { d } x \\mathrm { d } s \\\\ [ 1 e x ] & + \\eta ^ 2 \\cfrac { b } { c } \\int _ 0 ^ t \\int _ \\Omega \\nabla ( c f _ 1 + d g _ 1 ) \\cdot \\frac { \\nabla g _ 1 } { ( g _ 1 + \\eta ) ^ 2 } \\ , \\mathrm { d } x \\mathrm { d } s \\end{align*}"}
+{"id": "284.png", "formula": "\\begin{align*} - F ^ { s - 1 } d a = - \\sum _ { i = 0 } ^ { s - 1 } a _ { i } ^ { p ^ { i } - 1 } d a _ { i } \\end{align*}"}
+{"id": "795.png", "formula": "\\begin{align*} \\max _ { \\tau _ i \\in \\mathcal { S } ^ { i } } { J } _ i ( \\tau _ i ) = \\max _ { \\tau _ i \\in \\mathcal { S } ^ { i } } \\mathbb { E } \\left \\{ \\int _ 0 ^ { \\tau _ i } e ^ { - \\beta t } f ( x _ i ( t ) ) d t - e ^ { - \\beta \\tau _ i } K \\right \\} . \\end{align*}"}
+{"id": "6915.png", "formula": "\\begin{align*} \\frac { d q } { d h _ i } = \\frac { d q } { d z _ i } \\cdot \\frac { d z _ i } { d h _ i } = z _ i \\frac { d q } { d z _ i } = \\frac { z _ i } { z _ i + y } P ' ( z _ i ) . \\end{align*}"}
+{"id": "4906.png", "formula": "\\begin{align*} c _ n ( E ) = [ V ( s ) \\xrightarrow { i _ { V ( s ) } } X ; { \\bf 0 } ] \\end{align*}"}
+{"id": "5483.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { N } \\begin{bmatrix} N \\\\ n \\end{bmatrix} _ { q ^ m } \\frac { ( q ^ m , q ^ m ) _ { n } ( q ^ r ) _ { N - n } q ^ { r n } } { ( q ^ r ) _ { N } ( 1 - q ^ r ) ( 1 - q ^ { m + r } ) \\cdots ( 1 - q ^ { n m + r } ) } \\\\ & = ( 1 - q ^ { N + r } ) \\sum _ { n = 0 } ^ { N } \\begin{bmatrix} N \\\\ n \\end{bmatrix} _ { q ^ m } \\frac { ( q ^ m , q ^ m ) _ { n } q ^ { m n ^ 2 + 2 r n } } { [ ( 1 - q ^ r ) ( 1 - q ^ { m + r } ) \\cdots ( 1 - q ^ { n m + r } ) ] ^ 2 } \\end{align*}"}
+{"id": "8016.png", "formula": "\\begin{align*} S ( m ) = \\begin{bmatrix} A & B ^ * \\\\ B & C \\end{bmatrix} : \\mathcal { H } _ 0 \\oplus \\mathcal { H } _ 1 \\oplus \\mathcal { H } _ 2 \\oplus \\cdots \\oplus \\mathcal { H } _ m \\rightarrow \\mathcal { H } _ 0 \\oplus \\mathcal { H } _ 1 \\oplus \\mathcal { H } _ 2 \\oplus \\cdots \\ , \\oplus \\ , \\mathcal { H } _ m . \\end{align*}"}
+{"id": "5795.png", "formula": "\\begin{align*} { } \\displaystyle \\sum _ { n = 1 } ^ { \\infty } w _ n \\Big ( \\sum _ { k = 1 } ^ { n } a _ k \\Big ) ^ 2 \\leq \\sum _ { n = 1 } ^ { \\infty } { a _ n } ^ 2 , \\end{align*}"}
+{"id": "702.png", "formula": "\\begin{align*} g _ i ^ { ( e q ) } = { { w } _ i } T , \\end{align*}"}
+{"id": "2391.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { m } X _ { k , s } \\iota _ { \\mathcal { A } } ^ { c _ { k } } ( Y _ { k , s } ' ) = G + F _ { 2 } \\end{align*}"}
+{"id": "6602.png", "formula": "\\begin{align*} V _ { j } : = \\big \\{ ( r , \\theta ) : \\theta \\in ( \\theta _ { j } , \\theta _ { j + 1 } ) , \\ r > 0 \\big \\} , \\theta _ { M + 1 } : = 2 \\pi + \\theta _ { 1 } . \\end{align*}"}
+{"id": "6706.png", "formula": "\\begin{align*} \\langle h ( v ) , \\psi _ { i } ( v ) \\rangle = 0 , \\mbox { f o r $ i = 0 , 1 , 2 , 3 , 4 $ } . \\end{align*}"}
+{"id": "5970.png", "formula": "\\begin{align*} b ' ( p ) & = ( 1 - \\frac { 1 } { k } ) ^ { - \\frac { p } { 2 k } } [ 1 + \\frac { p - k ( k + 1 ) } { 2 k } \\log ( 1 - \\frac { 1 } { k } ) ^ { - 1 } ] \\\\ & = ( 1 - \\frac { 1 } { k } ) ^ { - \\frac { p } { 2 k } } [ 1 + \\frac { p - k ( k + 1 ) } { 2 k } ( \\log k - \\log ( k - 1 ) ) ] \\\\ & \\geq ( 1 - \\frac { 1 } { k } ) ^ { - \\frac { p } { 2 k } } [ 1 + \\frac { 2 - k ( k + 1 ) } { 2 k } ( \\log k - \\log ( k - 1 ) ) ] \\\\ & \\geq ( 1 - \\frac { 1 } { k } ) ^ { - \\frac { p } { 2 k } } [ 1 + \\frac { 2 - k ( k + 1 ) } { 2 k } \\frac { 1 } { k - 1 } ] , \\end{align*}"}
+{"id": "1726.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to \\infty } \\alpha _ { \\tau , m } ^ \\xi ( \\zeta ) = \\alpha _ m ^ \\xi ( \\zeta ) \\end{align*}"}
+{"id": "1186.png", "formula": "\\begin{align*} \\lim \\limits _ { i \\to \\infty } \\left | \\ , ^ { k _ i } \\ ! \\lambda / \\left ( \\ , ^ { k _ i } \\ ! \\lambda _ 1 \\right ) \\right | = \\infty . \\end{align*}"}
+{"id": "265.png", "formula": "\\begin{align*} B _ t = \\sum _ k X _ k \\tilde { B } _ t ^ k \\end{align*}"}
+{"id": "2891.png", "formula": "\\begin{align*} p \\frac { d ^ 2 } { d s ^ 2 } \\left ( \\left ( \\kappa - \\mu \\right ) ^ { p - 1 } \\right ) + p \\kappa ^ 2 \\left ( \\kappa - \\mu \\right ) ^ { p - 1 } - \\kappa \\left ( \\left ( \\kappa - \\mu \\right ) ^ p + \\sigma \\right ) = 0 . \\end{align*}"}
+{"id": "7176.png", "formula": "\\begin{align*} \\mathcal { K } : = ( \\mathcal { O } _ { \\mathcal { C } ( d ) } \\otimes _ { \\mathcal { O } _ { \\mathcal { Y } ( d ) } } \\mathcal { O } _ { \\mathcal { X } ( d ) } , d _ { \\mathcal { K } } ) , \\ d _ { \\mathcal { K } } = d _ { s } \\otimes 1 + \\kappa , \\end{align*}"}
+{"id": "3575.png", "formula": "\\begin{align*} \\lambda = n ^ { - ~ \\tfrac { t + c } { 1 + c + 2 t ( 1 - \\alpha ) } } . \\end{align*}"}
+{"id": "1665.png", "formula": "\\begin{align*} \\Phi ^ * ( e ^ { \\lambda \\mathtt { P } } ) ^ * e ^ { \\lambda \\mathtt { P } } \\Phi & = \\mathbf { 1 } + \\Phi ^ * \\big [ \\lambda \\mathtt { U } ( \\mathtt { P } ) + \\lambda ^ 2 \\mathtt { V } ( \\mathbf { 0 } , \\mathtt { P } ) \\big ] \\Phi + \\lambda ^ 3 \\mathtt { S } ^ { ( \\lambda ) } ( \\Phi , \\mathtt { P } ) \\ , , \\end{align*}"}
+{"id": "5026.png", "formula": "\\begin{align*} F : = \\bigcup _ { j } E _ j , G : = \\bigcap _ { j } E _ j . \\end{align*}"}
+{"id": "1618.png", "formula": "\\begin{align*} f ( x ) - f ( y ) & = f ( x ) - f ( u _ 0 ) + f ( v _ 0 ) - f ( y ) + f ( u _ 0 ) - f ( v _ 0 ) \\\\ & > - d ( x , u _ 0 ) - d ( v _ 0 , y ) + ( 1 - \\alpha / 2 + \\delta ) d ( u _ 0 , v _ 0 ) \\\\ & > d ( x , y ) - \\alpha / 2 d ( u _ 0 , v _ 0 ) \\\\ & \\ge ( 1 - \\alpha ) d ( x , y ) . \\end{align*}"}
+{"id": "3588.png", "formula": "\\begin{align*} \\psi _ { 2 ^ { j } + \\ell - 1 } ( x ) = \\begin{cases} + 2 ^ { \\tfrac { j } { 2 } } , & ~ x \\in \\left [ \\frac { \\ell - 1 } { 2 ^ j } , \\frac { \\ell - 1 / 2 } { 2 ^ j } \\right ] \\\\ - 2 ^ { \\tfrac { j } { 2 } } , & x \\in \\left [ \\frac { \\ell - 1 / 2 } { 2 ^ j } , \\frac { \\ell } { 2 ^ j } \\right ] \\\\ 0 , & \\end{cases} . \\end{align*}"}
+{"id": "4957.png", "formula": "\\begin{align*} [ x , y , z ] = \\mathfrak D ( x ) \\cdot [ y , z ] - \\mathfrak D ( y ) \\cdot [ x , z ] + \\mathfrak D ( z ) \\cdot [ x , y ] . \\end{align*}"}
+{"id": "5178.png", "formula": "\\begin{align*} \\int _ Q | \\nabla u ( x ) | ^ p \\ , \\d x = \\int _ { ( x _ 1 , \\dots , x _ { n - 1 } ) } \\int _ { y _ 1 } ^ { y _ 2 } | \\nabla u ( x _ 1 , \\dots , x _ { n - 1 } , t ) | ^ p \\ , \\d t \\ge \\ell ( Q ) ^ { n - 1 } c ^ p \\ell ( Q ) ^ { 1 - p } = c ^ p \\ell ( Q ) ^ { n - p } . \\end{align*}"}
+{"id": "5184.png", "formula": "\\begin{align*} \\partial ^ M \\widetilde A = ( \\partial ^ M A \\cap \\Omega ) \\cup ( \\partial ^ M A _ 0 \\setminus \\overline \\Omega ) \\cup H \\end{align*}"}
+{"id": "1773.png", "formula": "\\begin{align*} \\left \\langle Y , B , i _ { 0 } \\right \\rangle = 0 { \\rm \\ , \\ , \\ , \\ , a n d \\ , \\ , \\ , \\ , } { \\rm s i g n } \\left ( \\left \\langle Y , B , i _ { 0 } ^ { - } \\right \\rangle \\right ) = { \\rm s i g n } \\left ( \\left \\langle Y , B , i _ { 0 } ^ { + } \\right \\rangle \\right ) \\neq 0 , \\end{align*}"}
+{"id": "4265.png", "formula": "\\begin{align*} T _ 2 ^ { * } = \\frac { 1 } { ( c q ) _ N } \\sum _ { k = 1 } ^ N \\frac { \\left ( \\frac { c q } { d } \\right ) _ k ( d q ) ^ k ( d q ) _ { N - k } } { ( q ) _ k ( 1 - q ^ k ) ( q ) _ { N - k } } , \\end{align*}"}
+{"id": "5806.png", "formula": "\\begin{align*} \\omega ( { \\sqrt { \\sum _ { i = 1 } ^ n A _ i ^ * X A _ i } } ) = { \\sqrt { \\omega ( \\sum _ { i = 1 } ^ n A _ i ^ * X A _ i ) } } . \\end{align*}"}
+{"id": "7424.png", "formula": "\\begin{align*} J _ z = \\sum \\limits _ { \\mu = - s } ^ { s } \\mu a _ { \\mu } ^ { \\dagger } a _ { \\mu } , J _ + = \\sum \\limits _ { \\mu = - s } ^ { \\mu = s - 1 } \\sqrt { ( s + \\mu + 1 ) ( s - \\mu ) } a _ { \\mu + 1 } ^ { \\dagger } a _ { \\mu } = ( J _ - ) ^ { \\dagger } . \\end{align*}"}
+{"id": "4111.png", "formula": "\\begin{align*} H ^ 0 ( X , \\Omega _ { X | K } ( - x _ 1 - \\ldots - x _ { n - 1 } ) ) = H ^ 0 ( X , \\Omega _ { X | K } ( - x _ 1 - \\ldots - x _ n ) ) , \\end{align*}"}
+{"id": "7587.png", "formula": "\\begin{align*} \\iint \\frac { h ( s ) - h ( t ) } { s - t } d \\mu ( s ) d \\mu ( t ) = \\left [ \\int \\frac { d \\mu ( s ) } { z - s } \\right ] ^ 2 \\end{align*}"}
+{"id": "1063.png", "formula": "\\begin{align*} e _ { i } = \\frac { \\partial } { \\partial x ^ { i } } - \\frac { 1 } { 6 } R _ { i j k \\ell } \\ , x ^ { j } x ^ { k } \\frac { \\partial } { \\partial x ^ { \\ell } } + o ( \\mathbf { x ^ { 2 } ) } \\end{align*}"}
+{"id": "8111.png", "formula": "\\begin{align*} \\forall \\ , i \\in \\{ 1 , \\ldots , d + 1 \\} , e _ i ( \\overline L ) = \\sup _ { \\begin{subarray} { c } \\\\ \\operatorname { c o d i m } ( Y ) \\geqslant i \\end{subarray} } \\ ; \\inf _ { x \\in ( X \\setminus Y ) ^ { ( 0 ) } } h _ { \\overline L } ( x ) . \\end{align*}"}
+{"id": "5911.png", "formula": "\\begin{align*} A \\circ x _ { i j } x _ { l m } = ( a _ i + a _ j + a _ l + a _ m ) \\cdot x _ { i j } x _ { l m } \\ , . \\end{align*}"}
+{"id": "642.png", "formula": "\\begin{align*} \\mathcal { D } + \\mathcal { E } = 0 . \\end{align*}"}
+{"id": "1843.png", "formula": "\\begin{align*} R _ p ( J _ n ^ 2 , 4 n ) & = \\sum _ { k = 0 } ^ \\infty \\frac { J _ n ^ 2 ( p ^ k ) } { p ^ { 4 n k } } = 1 + \\sum _ { k = 1 } ^ \\infty \\frac { ( p ^ { n k } - p ^ { n ( k - 1 ) } ) ^ 2 } { p ^ { 4 n k } } \\\\ & = 1 + ( 1 - 2 p ^ { - n } + p ^ { - 2 n } ) \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { p ^ { 2 n k } } \\\\ & = 1 + \\frac { p ^ { 2 n } - 2 p ^ { n } + 1 } { p ^ { 4 n } - p ^ { 2 n } } \\\\ & = \\frac { p ^ { 4 n } - 2 p ^ n + 1 } { p ^ { 4 n } - p ^ { 2 n } } \\end{align*}"}
+{"id": "3446.png", "formula": "\\begin{align*} \\omega ^ 2 = \\omega _ 0 ^ 2 + \\omega _ \\mathrm { p } ^ 2 \\dfrac { \\varepsilon _ \\mathrm { m } + \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } } { \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } ( 1 - \\varepsilon _ \\infty ^ { - 1 } \\varepsilon _ \\mathrm { m } ) + \\varepsilon _ \\mathrm { m } } + \\mathcal { O } ( \\delta ^ \\mathrm { h } ) \\mbox { a n d } \\gamma \\omega \\sim \\delta ^ h . \\end{align*}"}
+{"id": "6804.png", "formula": "\\begin{align*} 0 = \\lambda _ { C \\ , D } + \\lambda _ { A \\ , D } + \\lambda _ { A \\ , C } = \\lambda _ { C \\ , D } + \\lambda _ { B \\ , D } + \\lambda _ { B \\ , C } = & \\lambda _ { A \\ , B } + \\lambda _ { B \\ , C } + \\lambda _ { A \\ , C } = \\lambda _ { A \\ , B } + \\lambda _ { B \\ , D } + \\lambda _ { A \\ , D } \\\\ = & \\lambda _ { A \\ , A } + \\lambda _ { A \\ , C } + \\lambda _ { A \\ , C } = \\lambda _ { C \\ , C } + \\lambda _ { A \\ , C } + \\lambda _ { A \\ , C } . \\end{align*}"}
+{"id": "5756.png", "formula": "\\begin{align*} ~ H _ { i 0 } + H _ { 0 i } = A _ { i } , i = 1 , \\cdots , m . \\end{align*}"}
+{"id": "859.png", "formula": "\\begin{align*} K \\in \\arg \\max F ( K , \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\delta _ { \\tau _ i } ( 0 , x ] ) \\end{align*}"}
+{"id": "5710.png", "formula": "\\begin{align*} \\partial _ t u _ t ( x ) = \\int _ { \\R ^ d } m ( y , x ) \\ , u _ t ( y ) \\ , \\dd y - u _ t ( x ) \\int _ { \\R ^ d } c ( x , y ) \\ , u _ t ( y ) \\ , \\dd y . \\end{align*}"}
+{"id": "3801.png", "formula": "\\begin{align*} a _ 1 + a _ 2 + \\dots + a _ { h - 2 } = \\frac { ( h - 1 ) ( 2 n - 2 ) + d - ( a _ { h - 1 } + a _ h ) } { 2 } . \\end{align*}"}
+{"id": "3376.png", "formula": "\\begin{align*} q \\alpha _ 1 + \\dots + q \\alpha _ d = p _ { d + 1 } - \\beta \\ , . \\end{align*}"}
+{"id": "234.png", "formula": "\\begin{align*} \\log \\frac { \\pi ( x ) } { m _ t ^ { ( n ) } ( x ) } & = - e ^ { - \\frac { \\sigma ^ 2 } { 2 } t } \\log \\frac { m _ 0 ( x ) } { \\pi ( x ) } \\\\ & - \\int _ 0 ^ t \\frac { \\sigma ^ 2 } { 2 } e ^ { - \\frac { \\sigma ^ 2 } { 2 } ( t - s ) } \\left ( - \\frac { 2 } { \\sigma ^ 2 } \\frac { \\delta F } { \\delta m } ( m _ s ^ { ( n - 1 ) } , x ) + \\operatorname { K L } ( m _ s ^ { ( n - 1 ) } | \\pi ) \\right ) d s \\ , . \\end{align*}"}
+{"id": "6204.png", "formula": "\\begin{align*} E ^ * _ { i + k } A _ 1 E ^ * _ { i + k - 1 } A _ 1 E ^ * _ { i + k - 2 } \\cdots E ^ * _ { i + 1 } A _ 1 E ^ * _ { i } = \\big ( ( \\frac { k } { 2 } ) ! \\big ) ^ 2 M ^ { \\frac { 2 m - k } { 2 } , \\frac { 2 m - i - k } { 2 } } _ { \\frac { 2 m - i - k } { 2 } , \\frac { 2 m - i } { 2 } } . \\end{align*}"}
+{"id": "1448.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\frac { M ( b , c ; s ) } { s ^ { b - c } e ^ s } = \\frac { \\Gamma ( c ) } { \\Gamma ( b ) } . \\end{align*}"}
+{"id": "2932.png", "formula": "\\begin{align*} \\widehat { \\sigma ( M _ C ) } = \\widehat { \\sigma ( M _ 0 ) } . \\end{align*}"}
+{"id": "1128.png", "formula": "\\begin{align*} x ^ { j } \\allowbreak = \\allowbreak j ! \\sum _ { k = 0 } ^ { j } ( - 1 ) ^ { k } \\frac { ( \\beta ) ^ { ( j ) } } { ( j - k ) ! ( \\beta ) ^ { ( k ) } } L _ { k } ^ { ( \\beta ) } ( x ) , \\end{align*}"}
+{"id": "2877.png", "formula": "\\begin{align*} \\# \\{ \\tau _ { s _ k } \\in { \\bf { S } } _ { s _ k } : f _ { \\tau _ { s _ k } } \\not \\equiv 0 \\} \\prod _ { i = 1 } ^ { k - 1 } \\max _ { \\tau _ { s _ { i + 1 } } \\in { \\bf { S } } _ { s _ { i + 1 } } } \\# \\{ \\tau _ { s _ i } \\in { \\bf { S } } _ { s _ i } : \\tau _ { s _ i } \\subset \\tau _ { s _ { i + 1 } } , f _ { \\tau _ { s _ i } } \\not \\equiv 0 \\} . \\end{align*}"}
+{"id": "251.png", "formula": "\\begin{align*} \\Q ^ t \\partial _ \\ell \\Q \\nabla y _ k = y _ k ^ { - 1 } \\partial _ \\ell f _ k [ w ^ k ] ^ \\perp , \\Q ^ t \\partial _ \\ell \\Q \\ , x = \\sum _ { i = 1 } ^ d \\partial _ \\ell f _ i [ w ^ i ] ^ \\perp , \\end{align*}"}
+{"id": "2884.png", "formula": "\\begin{align*} \\tilde { \\nu _ 1 } ( 0 ) = 0 \\qquad \\frac { d } { d \\tilde { \\nu _ 2 } } \\tilde { \\nu _ 1 } ( 0 ) = 0 . \\end{align*}"}
+{"id": "6003.png", "formula": "\\begin{align*} \\hat { f } ' ( + \\infty , i ) = \\sum _ { j \\neq i } \\dfrac { \\lambda _ { i j } } { { \\lambda } _ i } f ' ( + \\infty , j ) \\leq 1 . \\end{align*}"}
+{"id": "340.png", "formula": "\\begin{align*} p ( X ( t ) | Y _ { t - 1 } ) = \\int p ( X ( t ) | X ( t - 1 ) ) p ( X ( t - 1 ) | Y _ { t - 1 } ) d X ( t - 1 ) . \\end{align*}"}
+{"id": "2738.png", "formula": "\\begin{align*} \\mu _ H ^ { { \\rm m a x } } ( { \\rm S y m ^ m } E ) \\geq \\mu _ H ( \\det ( E ) ^ { \\otimes ( a m ) } ) = a m c _ 1 ( E ) \\cdot H ^ { n - 1 } . \\end{align*}"}
+{"id": "6512.png", "formula": "\\begin{align*} C _ { i , j } = 1 \\end{align*}"}
+{"id": "5239.png", "formula": "\\begin{align*} \\gamma = \\mathrm { r e s } _ v \\left ( \\left ( L ^ \\mathrm { A H } ( \\varepsilon ( v ) ) \\frac { d L ^ { \\mathrm { A H } } ( \\eta ( v ) ) } { d v } - L ^ { \\mathrm { A H } } ( \\varepsilon ( v ) ) d _ { \\mathrm { l o g } } ( B ( v ) ) + L ^ { \\mathrm { A H } } ( \\eta ( v ) ) d _ { \\mathrm { l o g } } ( A ( v ) ) \\right ) \\left ( \\frac { 1 } { z ( v ) ^ p - 1 } \\right ) \\right ) . \\end{align*}"}
+{"id": "8151.png", "formula": "\\begin{align*} ( \\pi ^ * ( \\overline M ) ^ { d + 1 } ) _ S = ( \\deg \\pi ) ( \\overline M ^ { d + 1 } ) _ S \\quad ( \\pi ^ * ( M ) ^ d ) = ( \\deg \\pi ) ( M ^ d ) . \\end{align*}"}
+{"id": "7016.png", "formula": "\\begin{align*} - \\int _ X \\langle \\nabla g , \\nabla g \\rangle d m _ w = L _ w g ( g ) \\geq 0 , \\end{align*}"}
+{"id": "7876.png", "formula": "\\begin{align*} R _ x = f ^ \\delta _ { k _ x } ( I ) , \\end{align*}"}
+{"id": "2842.png", "formula": "\\begin{align*} \\delta _ s \\delta _ s = \\delta _ s ( \\delta _ s + s ( \\delta _ s ) ) - \\delta _ s s ( \\delta _ s ) \\end{align*}"}
+{"id": "2613.png", "formula": "\\begin{align*} & f _ n \\big ( t , ( \\partial _ x v _ n \\Sigma ) ( t , x ) , x \\big ) = ( \\partial _ x v _ n \\Sigma \\cdot q ) ( t , x ) - g _ n \\big ( t , q ( t , x ) , x \\big ) . \\end{align*}"}
+{"id": "6431.png", "formula": "\\begin{align*} f ( x _ 1 , y _ 1 , \\ldots , x _ k , y _ k ) = \\sum _ { n _ 1 = 0 } ^ { \\infty } c _ { n _ 1 } ( x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) L _ { n _ 1 } ^ { ( \\alpha _ 1 ) } ( x _ 1 , y _ 1 ) . \\end{align*}"}
+{"id": "3764.png", "formula": "\\begin{align*} g ( \\Gamma ) : = \\frac { | V ( \\Gamma ) | + | E ( \\Gamma ) | - b ( \\Gamma ) } { 2 } \\end{align*}"}
+{"id": "4936.png", "formula": "\\begin{align*} a _ 0 + \\sum _ { i = 1 } ^ { ( k - 5 ) / 2 - t } ( a _ i + b _ i ) = y = \\sum _ { i = 0 } ^ { ( k - 5 ) / 2 - t } \\frac { k - 2 + 2 t + i ( ( k - 1 ) / 2 + t ) } { n } \\binom { n } { k - 2 - i } . \\end{align*}"}
+{"id": "4884.png", "formula": "\\begin{align*} 3 b _ { n + 1 } = 2 b _ n + \\sum _ { k = 0 } ^ n { n + 1 \\choose k } b _ k + 3 . \\end{align*}"}
+{"id": "3551.png", "formula": "\\begin{align*} 1 - \\alpha \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } = \\mathcal { O } ( \\delta ^ \\mathrm { h } ) \\iff \\varepsilon _ \\mathrm { p } = \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } - \\dfrac { \\Big ( \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } \\Big ) ^ 2 } { \\varepsilon _ \\mathrm { m } + \\lambda _ \\mathrm { n _ 0 } ^ { ( 3 ) } } + \\mathcal { O } ( \\delta ^ \\mathrm { h } ) \\end{align*}"}
+{"id": "5089.png", "formula": "\\begin{align*} \\| u \\| _ { X _ T } = \\| u \\| _ { L _ t ^ q L _ x ^ r H _ y ^ { \\frac { 1 } { 2 } + \\delta } ( [ - T , T ] \\times \\mathbb { R } ^ d \\times \\mathbb { T } ) } , \\end{align*}"}
+{"id": "3687.png", "formula": "\\begin{align*} x _ i = \\begin{cases} \\frac { 1 } { t + 2 } \\pm 3 0 t \\epsilon ^ { 1 / 2 } & i \\in [ t ] , \\\\ \\frac { \\alpha } { t + 2 } \\pm 3 0 t \\epsilon ^ { 1 / 2 } & i \\in \\{ t + 1 , t + 4 \\} , \\\\ \\frac { 1 - \\alpha } { t + 2 } \\pm 3 0 t \\epsilon ^ { 1 / 2 } & i \\in \\{ t + 2 , t + 3 \\} . \\end{cases} \\end{align*}"}
+{"id": "3678.png", "formula": "\\begin{align*} \\mathcal { H } [ U ] = \\mathcal { G } [ U ] \\cong K _ { t + 2 } ^ { 3 } \\quad L ( u _ i ) [ U ] = L _ { \\mathcal { G } } ( u _ i ) [ U ] = L _ { \\mathcal { G } } ( u _ i ' ) [ U ] = L ( u _ i ' ) [ U ] . \\end{align*}"}
+{"id": "1944.png", "formula": "\\begin{align*} \\mathcal { P } u _ s ( z ) + \\lambda u _ s ( z ) & = D _ { v _ i } [ f ^ i ( z ) \\zeta ( v _ d - s ) - a ^ { i d } ( z ) \\zeta ' ( v _ d - s ) u ( z ) ] \\\\ & - f _ d ( z ) \\zeta ' ( v _ d - s ) + g \\zeta ( v _ d - s ) - a ^ { d j } ( z ) \\zeta ' ( v _ d - s ) D _ { v _ { j } } u ( z ) . \\end{align*}"}
+{"id": "4839.png", "formula": "\\begin{align*} W _ 2 ( \\pi _ S \\# \\rho _ { \\infty , 1 } , \\pi _ S \\# \\rho _ { \\infty , 2 } ) ^ 2 \\leq \\int _ { \\R ^ { 2 d } \\times S ^ 2 } | s _ 1 - s _ 2 | ^ 2 & \\dd \\Pi _ t ( x , y , s _ 1 , s _ 2 ) = \\\\ & \\left ( \\int _ { \\R ^ { 2 d } \\times S ^ 2 } | s _ 1 - s _ 2 | ^ 2 \\dd \\Pi _ 0 ( x , y , s _ 1 , s _ 2 ) \\right ) e ^ { - K t } , \\end{align*}"}
+{"id": "5830.png", "formula": "\\begin{align*} T T _ 1 = ( T _ 1 T _ 2 ) T _ 1 = T _ 1 ( T _ 1 T _ 2 ) = T _ 1 T . \\end{align*}"}
+{"id": "5608.png", "formula": "\\begin{align*} \\Omega ^ i _ j = \\frac 1 2 R ^ i _ { j k l } d x ^ k \\wedge d x ^ l + B ^ i _ { j k l } \\delta y ^ l \\wedge d x ^ k \\end{align*}"}
+{"id": "2250.png", "formula": "\\begin{align*} b = b _ m = \\sigma ^ { - ( m - 1 ) } ( a ) \\sigma ^ { - ( m - 2 ) } ( a ) \\cdots \\sigma ^ { - 1 } ( a ) a . \\end{align*}"}
+{"id": "7933.png", "formula": "\\begin{align*} R _ 1 - R _ 1 ' = - \\Psi ^ 1 ( \\varphi _ 1 ) . \\end{align*}"}
+{"id": "2025.png", "formula": "\\begin{align*} \\langle D \\psi _ 1 , D \\psi _ 2 \\rangle _ { G _ g , a } = W ( \\psi _ 1 \\ast \\widetilde { \\psi _ 2 } ) \\end{align*}"}
+{"id": "6383.png", "formula": "\\begin{align*} T ( Y _ { i } , X _ { 1 } , X _ { 4 } ) = T ( Y _ { 1 } , X _ { 3 } , X _ { k } ) = T ( Y _ { 1 } , X _ { 4 } , X _ { k } ) = T ( Y _ { 2 } , X _ { 4 } , X _ { k } ) = 0 \\end{align*}"}
+{"id": "7961.png", "formula": "\\begin{align*} & t ( s ) = \\gamma _ 3 ( s ) , r ( s ) = \\left ( \\gamma ^ 2 _ 1 ( s ) + \\gamma ^ 2 _ 2 ( s ) \\right ) ^ { \\frac { 1 } { 2 } } = | \\xi ^ H | , \\quad \\mbox { a n d } \\\\ & \\begin{cases} \\cos { \\left ( \\theta ( s ) \\right ) } = \\left \\langle \\nu ^ H , \\frac { \\xi ^ H } { | \\xi ^ H | } \\right \\rangle , \\\\ \\sin { \\left ( \\theta ( s ) \\right ) } = \\left \\langle \\eta , \\frac { \\xi ^ H } { | \\xi ^ H | } \\right \\rangle . \\end{cases} \\end{align*}"}
+{"id": "5913.png", "formula": "\\begin{align*} t _ 1 \\ , x _ { 2 4 } x _ { 3 4 } + t _ 2 \\ , x _ { 1 2 } x _ { 1 5 } + t _ 3 \\ , x _ { 1 3 } x _ { 3 5 } + t _ 4 \\ , x _ { 2 5 } ^ 2 + t _ 5 \\ , x _ { 2 5 } x _ { 4 5 } + t _ 6 \\ , x _ { 1 4 } x _ { 1 5 } + t _ 7 \\ , x _ { 4 5 } ^ 2 + t _ 8 \\ , x _ { 2 3 } x _ { 2 4 } = 0 \\ , , \\end{align*}"}
+{"id": "5515.png", "formula": "\\begin{align*} \\Pi \\mapsto H _ 0 ( H , \\Pi ) = \\Pi _ H : = \\Pi / \\sum _ { h \\in H } ( h - 1 ) \\Pi . \\end{align*}"}
+{"id": "1486.png", "formula": "\\begin{align*} J \\le \\prod _ { p | P ( z ) } ( 1 + h ( p ) ) = V ( z ) ^ { - 1 } \\end{align*}"}
+{"id": "444.png", "formula": "\\begin{align*} \\int _ \\Sigma | ( d ( \\eta _ \\delta W ) ) ^ N | ^ 2 & = \\int _ \\Sigma | \\eta _ \\delta ( d W ) ^ N + ( W d \\eta _ \\delta ) ^ N | ^ 2 \\\\ & = \\int _ { \\delta ^ 2 \\leq | z | \\leq \\delta } | \\eta _ \\delta ( d W ) ^ N + ( W d \\eta _ \\delta ) ^ N | ^ 2 + \\int _ { | z | \\geq \\delta } | ( d W ) ^ N | ^ 2 , \\end{align*}"}
+{"id": "727.png", "formula": "\\begin{align*} { f _ { 1 3 } } = { f _ { 1 4 } } - f _ { 1 4 } ^ { \\left ( { e q } \\right ) } + f _ { 1 3 } ^ { \\left ( { e q } \\right ) } + \\delta x - \\delta z , \\end{align*}"}
+{"id": "6457.png", "formula": "\\begin{align*} ( \\square _ g + m ^ 2 ) \\phi : = g ^ { \\mu \\nu } \\nabla _ { \\mu } \\nabla _ { \\nu } \\phi + m ^ { 2 } \\phi = 0 \\end{align*}"}
+{"id": "137.png", "formula": "\\begin{align*} x ( [ \\mathbf { a ' } , \\mathbf { a } ] \\otimes [ 0 , \\mathbf { b } ] ) = 0 \\end{align*}"}
+{"id": "6245.png", "formula": "\\begin{align*} \\mathcal { Z } ( G ) = \\mathcal { Z } ( G ; x _ 1 , \\dots , x _ n ) = \\frac { 1 } { | G | } \\sum _ { g \\in G } x _ 1 ^ { c _ 1 ( g ) } \\cdots x _ n ^ { c _ n ( g ) } . \\end{align*}"}
+{"id": "6621.png", "formula": "\\begin{align*} \\Omega _ + : = \\{ ( x , y ) : | \\arg ( x + i y ) | < \\theta \\} , \\Omega _ - : = \\{ ( x , y ) : | \\arg ( x + i y ) | > \\theta \\} , \\end{align*}"}
+{"id": "6307.png", "formula": "\\begin{align*} \\mathcal { M } ^ f _ t : = f ( Z _ t ) - \\int _ 0 ^ t \\mathcal { A } ^ f ( s , X _ s , Z _ s ) \\dd s , t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "3349.png", "formula": "\\begin{align*} \\partial ( A ^ { ( p ) } ) = \\sum _ { B ^ { ( p + 1 ) } } \\left ( \\sum _ { b ^ { ( p ) } \\subset B } \\left ( \\iota _ { b , B } \\sum _ { \\gamma \\in \\overline { \\Gamma } ( b , A ) } \\mu ( \\gamma ) \\right ) \\right ) \\cdot B . \\end{align*}"}
+{"id": "3852.png", "formula": "\\begin{align*} \\Omega _ 1 : = \\left [ 1 + \\sqrt { \\frac { \\gamma _ n } { 4 n } } - \\frac { C _ n } { \\sqrt { 4 n \\gamma _ n } } , ~ 1 + \\sqrt { \\frac { \\gamma _ n } { 4 n } } + \\frac { C _ n } { \\sqrt { 4 n \\gamma _ n } } \\right ] \\times \\left [ - \\frac { n ^ { \\tau / 2 } } { n ^ { 1 / 4 } } , ~ \\frac { n ^ { \\tau / 2 } } { n ^ { 1 / 4 } } \\right ] , \\end{align*}"}
+{"id": "8127.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } ( \\overline L { } ^ { \\otimes m } ) = m \\operatorname { \\widehat { \\mu } _ { \\min } ^ { \\mathrm { a s y } } } ( \\overline L ) . \\end{align*}"}
+{"id": "5997.png", "formula": "\\begin{align*} \\mathbb { E } _ b \\left [ e ^ { - q \\tau _ { 0 } ^ - ( r ) } \\right ] & = Z ^ { ( q ) } ( b ) - q \\dfrac { Z ^ { ( q ) } ( b , \\Phi ( q + r ) ) } { Z ^ { ( q ) \\prime } ( b , \\Phi ( q + r ) ) } W ^ { ( q ) } ( b ) . \\end{align*}"}
+{"id": "6690.png", "formula": "\\begin{align*} \\underline { Q } _ { i } ( t ) & = \\det \\begin{bmatrix} 1 & 1 \\\\ \\lambda _ { i , 2 } ^ { 2 } & t \\end{bmatrix} , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] , \\\\ \\underline { P } _ { i } ( 0 ) & = \\sigma \\left ( \\frac { \\underline { Q } _ { i } ( t ) - \\underline { Q } _ { i } ( 0 ) } { t } \\right ) , i \\in \\mathbb { N } \\cap [ 1 , \\eta ] . \\end{align*}"}
+{"id": "6623.png", "formula": "\\begin{align*} h _ { \\theta } ( u , u ) = h _ { \\theta } ^ { N } ( \\Pi _ + u , \\Pi _ + u ) + h _ { \\theta } ^ { D } ( \\Pi _ - u , \\Pi _ - u ) \\end{align*}"}
+{"id": "93.png", "formula": "\\begin{align*} \\Psi ( r , \\xi ) = \\Phi ( r ) + \\Gamma ( \\xi ) \\end{align*}"}
+{"id": "5113.png", "formula": "\\begin{align*} \\overline { \\partial _ { x _ k } u _ m } x _ k x _ l \\partial _ { x _ l } u _ m & \\leq \\frac { 1 } { 2 } \\abs { \\partial _ { x _ k } u _ m x _ k } ^ 2 + \\frac { 1 } { 2 } \\abs { \\partial _ { x _ l } u _ m x _ l } ^ 2 \\\\ & \\leq \\frac { 1 } { 2 } \\abs { \\nabla _ { \\vec { e } } u _ m } ^ 2 \\abs { x } ^ 2 + \\frac { 1 } { 2 } \\abs { \\nabla _ { \\vec { e } } u _ m } ^ 2 \\abs { x } ^ 2 \\\\ & = \\abs { \\nabla _ x u _ m } ^ 2 \\abs { x } ^ 2 . \\end{align*}"}
+{"id": "6746.png", "formula": "\\begin{align*} I _ { 7 } \\leq & - \\varepsilon \\sum ^ { 3 } _ { j = 1 } ( \\frac { 1 } { \\rho } \\mu ( \\theta ) ( \\partial ^ { \\alpha } \\partial _ { x _ { j } } \\widetilde { u } _ { i } + \\partial ^ { \\alpha } \\partial _ { x _ { i } } \\widetilde { u } _ { j } - \\frac { 2 } { 3 } \\delta _ { i j } \\nabla _ { x } \\cdot \\partial ^ { \\alpha } \\widetilde { u } ) , \\partial ^ { \\alpha } \\partial _ { x _ { j } } \\widetilde { u } _ { i } ) \\\\ & + C [ \\eta _ 0 ( 1 + t ) ^ { - \\vartheta } + \\varepsilon ^ { \\frac { 1 } { 2 } - a } ] \\varepsilon ^ { 2 - 2 a } . \\end{align*}"}
+{"id": "2630.png", "formula": "\\begin{align*} s ( \\mu \\nu ) = ( \\mu \\nu ) ( d ( \\mu \\nu ) ) = ( \\mu \\nu ) | ^ * _ { E _ { 2 , [ d ( \\mu ) , d ( \\mu ) + d ( \\nu ) ] } } ( d ( \\nu ) ) = \\nu ( d ( \\nu ) ) = s ( \\nu ) \\end{align*}"}
+{"id": "5711.png", "formula": "\\begin{align*} \\begin{aligned} A ^ i _ t & \\to A ^ i _ t + A ^ { N _ t + 1 } _ t & \\mbox { w i t h r a t e } \\quad & m \\left ( X _ t ^ i , X _ t ^ { N _ 1 + 1 } \\right ) \\gamma \\left ( X _ t ^ { N _ 1 + 1 } \\right ) , \\\\ A ^ i _ t + A ^ j _ t & \\to A ^ j _ t & \\mbox { w i t h r a t e } & n ^ { - 1 } c \\left ( X ^ i _ t , X ^ j _ t \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "2299.png", "formula": "\\begin{align*} & | Q | = \\\\ & ( x ^ 2 - \\mu _ 1 y ^ 2 ) ( x ^ 2 - \\mu _ 2 y ^ 2 ) \\cdots ( x ^ 2 - \\mu _ n y ^ 2 ) ( \\mu _ { n + 1 } y ^ 2 - x ^ 2 ) \\cdots ( \\mu _ { d + 1 } y ^ 2 - x ^ 2 ) | x ^ 2 - \\mu _ { \\nu } y ^ 2 | ^ { - 1 } \\end{align*}"}
+{"id": "2003.png", "formula": "\\begin{align*} G _ g ( t , u ) : = g ( t - u ) - g ( t ) - g ( - u ) + g ( 0 ) . \\end{align*}"}
+{"id": "3107.png", "formula": "\\begin{align*} \\sigma ( x ) = x + 2 \\end{align*}"}
+{"id": "7009.png", "formula": "\\begin{align*} L f ( g ) : = - \\int \\langle \\nabla f , \\nabla g \\rangle d m , g \\in W ^ { 1 , 2 } \\cap L ^ { \\infty } . \\end{align*}"}
+{"id": "7676.png", "formula": "\\begin{align*} \\Phi _ n ( r ) = \\exp \\left \\{ \\frac { ( n - 1 ) ( 2 - n ) r ^ 2 } { n } F \\left [ \\begin{array} { c } 1 , 1 , 2 - \\frac { n } { 2 } \\\\ 2 , 1 + \\frac { n } { 2 } \\end{array} ; r ^ 2 \\right ] \\right \\} \\left ( 1 - r ^ 2 \\right ) ^ { n - 1 } . \\end{align*}"}
+{"id": "3592.png", "formula": "\\begin{align*} \\| ( A + \\lambda I ) ^ { 1 / 2 } ( \\hat A + \\lambda I ) ^ { - 1 / 2 } \\| ^ 2 & = \\| ( A + \\lambda I ) ^ { 1 / 2 } ( \\hat A + \\lambda I ) ^ { - 1 } ( A + \\lambda ) ^ { 1 / 2 } \\| \\\\ & = \\left \\| \\left [ I - ( A + \\lambda I ) ^ { - 1 / 2 } ( A - \\hat { A } ) ( A + \\lambda I ) ^ { - 1 / 2 } \\right ] ^ { - 1 } \\right \\| \\\\ & \\leq \\frac { 1 } { 1 - \\left \\| ( A + \\lambda I ) ^ { - 1 / 2 } ( A - \\hat { A } ) ( A + \\lambda I ) ^ { - 1 / 2 } \\right \\| } . \\end{align*}"}
+{"id": "3644.png", "formula": "\\begin{align*} \\Theta _ F ( x ) _ { k , l } : = F ( e ^ { k , l } \\otimes x ) ( x \\in X ; k , l = 1 , \\dots , n ) . \\end{align*}"}
+{"id": "7307.png", "formula": "\\begin{align*} \\overline { M ' _ i } = ( \\mathop { q - 2 } \\limits _ { m - 1 } , \\underbrace { q - 1 , \\ldots , q - 1 , } _ { m - 2 - j _ 1 } \\mathop { q - 2 } \\limits _ { j _ 1 } , \\ldots , \\mathop { q - 2 } _ { j _ { i - 2 } } , \\underbrace { q - 1 , \\ldots , q - 1 } _ { j _ { i - 2 } - j _ { i - 1 } - 1 } , \\mathop { q - 2 } _ { j _ { i - 1 } } , \\underbrace { q - 1 , \\ldots , q - 1 } _ { j _ { i - 1 } } ) . \\end{align*}"}
+{"id": "2273.png", "formula": "\\begin{align*} \\sigma ( z ) : = ( \\sigma ( z _ 1 ) , \\sigma ( z _ 2 ) , \\dots , \\sigma ( z _ n ) ) = ( \\alpha _ 1 z ^ { M [ - , 1 ] } , \\alpha _ 2 z ^ { M [ - , 2 ] } , \\dots , \\alpha _ n z ^ { M [ - , n ] } ) = \\alpha z ^ M . \\end{align*}"}
+{"id": "3383.png", "formula": "\\begin{align*} T _ { \\Omega } ( \\gamma , \\eta ) : = \\{ ( \\lambda , z ) \\in \\Omega \\times \\C ^ k \\colon | z - \\gamma ( \\lambda ) | < \\eta \\} . \\end{align*}"}
+{"id": "7860.png", "formula": "\\begin{align*} L = \\sup _ { z \\in Z _ \\delta } \\sup _ { \\theta \\in I } \\left | \\frac { d } { d \\theta } \\rho _ \\theta ( z ) \\right | \\lesssim 1 . \\end{align*}"}
+{"id": "4745.png", "formula": "\\begin{gather*} ( ( \\eth _ k + \\beta _ k ) ( a + u ) ) \\cdot ( b + v ) = \\eth _ k ( a ) \\cdot b + r ( b ) \\beta _ k ( u ) + l ( \\eth _ k ( a ) ) v , \\\\ ( a + u ) \\cdot ( ( \\partial _ k + \\alpha _ k ) ( b + v ) ) = a \\cdot \\partial _ k ( b ) + l ( a ) \\alpha _ k ( v ) + r ( \\partial _ k ( b ) ) u , \\\\ ( \\eth _ k + \\beta _ k ) ( ( a + u ) \\cdot ( b + v ) ) = \\eth _ k ( a \\cdot b ) + \\beta _ k ( l ( a ) v ) + \\beta _ k ( r ( b ) u ) . \\end{gather*}"}
+{"id": "236.png", "formula": "\\begin{align*} \\dot { B } _ { p , r } ^ { s } = \\Big \\{ f \\in \\mathcal { S } ' _ h ( \\mathbb { R } ^ { d } ) : \\ ; \\| f \\| _ { \\dot { B } _ { p , r } ^ { s } ( \\mathbb { R } ^ { d } ) } < \\infty \\Big \\} , \\end{align*}"}
+{"id": "4943.png", "formula": "\\begin{align*} B = \\left ( \\frac { k - 3 } { 2 } + t \\right ) A - t y . \\end{align*}"}
+{"id": "2869.png", "formula": "\\begin{align*} \\| \\mathcal { D } ( a ( \\tau ; \\mu ) ) - \\mu ^ 0 \\| _ L ^ * \\leqslant 2 \\frac { M ' } { M _ 1 ( \\tau ) ^ { 2 m _ 0 ' } } + \\tilde g _ { M _ 1 ( \\tau ) } ( \\tau ) = : { \\hat g } _ { M ' } ( \\tau ) . \\end{align*}"}
+{"id": "708.png", "formula": "\\begin{align*} { \\rm O } ( \\epsilon ^ 1 ) : \\rho c _ v \\partial _ { t _ 1 } g _ i ^ { ( 0 ) } + d _ { 1 i } g _ i ^ { ( 0 ) } = - \\frac { 1 } { \\Delta t } \\left ( { { { \\bf { M } } } } ^ { - 1 } { \\Lambda } { { { \\bf { M } } } } \\right ) _ { i j } g _ j ^ { ( 1 ) } + { { w } _ i \\bar F ^ { ( 1 ) } } , \\end{align*}"}
+{"id": "4399.png", "formula": "\\begin{align*} \\mathcal { J } ( T ^ t f ) ( \\lambda ) = \\varphi _ \\lambda ^ { ( \\alpha , \\beta ) } ( t ) \\mathcal { J } ( f ) ( \\lambda ) \\end{align*}"}
+{"id": "5741.png", "formula": "\\begin{align*} f ( X ) = \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } \\otimes M ^ s _ { i j } \\quad g ( X ) = \\sum _ { i = 1 , j = 1 } ^ { m } B _ { i j } \\otimes M ^ s _ { i j } . \\end{align*}"}
+{"id": "6919.png", "formula": "\\begin{align*} \\left [ q ^ d \\right ] \\sum _ { k \\geq 0 } ( - 1 ) ^ { ( N - 1 ) d + k } \\binom { \\chi } { k } s _ { \\lambda _ k } ( z _ 1 , \\ldots , z _ { N + 1 } ) . \\end{align*}"}
+{"id": "4061.png", "formula": "\\begin{align*} L _ { [ x , y ] } = [ L _ x , L _ y ] , \\end{align*}"}
+{"id": "1994.png", "formula": "\\begin{align*} 1 = \\sum _ { u \\in T } x _ u ^ 2 + \\sum _ { u \\in V ( G ) \\setminus T } x _ u ^ 2 < \\frac { | T | } { 4 n } + ( n - | T | ) \\cdot \\frac { n } { ( n - 3 ) ^ 2 } , \\end{align*}"}
+{"id": "5058.png", "formula": "\\begin{align*} \\sum _ { \\substack { a \\bmod { q } \\\\ a \\bar { a } \\equiv 1 \\bmod { q } } } \\chi ( a ) e ( n \\bar { a } / q ) L _ f \\bigl ( s + u , \\tfrac { a } { q } \\bigr ) = \\omega q ^ { 1 - 2 s - 2 u } \\frac { \\gamma ( 1 - s - u ) } { \\gamma ( s + u ) } \\sum _ { \\substack { a \\bmod { q } \\\\ a \\bar { a } \\equiv 1 \\bmod { q } } } e ( n \\bar { a } / q ) L _ f \\bigl ( 1 - s - u , - \\tfrac { \\bar { a } } { q } \\bigr ) , \\end{align*}"}
+{"id": "301.png", "formula": "\\begin{align*} C _ { D ' \\subset D } = C _ { D ' } \\cup \\bigcup _ { i \\in I - I ' } T ^ { * } _ { D _ { i } } X \\subset T ^ { * } X \\end{align*}"}
+{"id": "2944.png", "formula": "\\begin{align*} h ( x ) : = \\inf \\limits _ { y \\in \\R ^ m } \\varphi ( x , y ) \\forall x \\in \\R ^ n . \\end{align*}"}
+{"id": "6076.png", "formula": "\\begin{align*} 2 m ( D ) \\ge ( 2 k - 1 ) n ( D - S ) + ( 2 k - 2 ) | S | = ( 2 k - 1 ) n ( D ) - | S | . \\end{align*}"}
+{"id": "6209.png", "formula": "\\begin{align*} M ^ { 0 , 0 } _ { \\frac { i + k - 1 } { 2 } , \\frac { 2 m - i - k + 1 } { 2 } } M ^ { \\frac { 2 m - k + 1 } { 2 } , \\frac { 2 m - i - k + 1 } { 2 } } _ { \\frac { 2 m - i - k + 1 } { 2 } , \\frac { 2 m - i } { 2 } } = \\frac { k + 1 } { 2 } M ^ { \\frac { k - 1 } { 2 } , \\frac { k - 1 } { 2 } } _ { \\frac { i + k - 1 } { 2 } , \\frac { 2 m - i } { 2 } } \\end{align*}"}
+{"id": "7936.png", "formula": "\\begin{align*} \\delta _ \\mathrm { m R B A } ( \\varphi _ 1 + \\delta _ \\mathrm { H o c h } ( a ) , 0 ) = ~ & \\big ( \\delta _ \\mathrm { H o c h } ( \\varphi _ 1 ) , ~ - \\Psi ^ 1 ( \\varphi _ 1 ) - \\Psi ^ 1 \\circ \\delta _ \\mathrm { H o c h } ( a ) \\big ) \\\\ = ~ & \\big ( \\delta _ \\mathrm { H o c h } ( \\varphi _ 1 ) , ~ - \\Psi ^ 1 ( \\varphi _ 1 ) - \\widetilde { \\delta } _ \\mathrm { H o c h } ( a ) \\big ) = \\delta _ \\mathrm { m R B A } ( \\varphi _ 1 , a ) . \\end{align*}"}
+{"id": "6923.png", "formula": "\\begin{align*} \\left [ q ^ d \\right ] ( - 1 ) ^ { ( N - 1 ) d } f ( z _ { N + 1 } ) \\ , \\prod _ { i = 1 } ^ { N } \\ , \\frac { z _ i ^ { d + 1 } } { P ' ( z _ i ) } \\prod _ { 1 \\leq i \\neq j \\leq N } ( z _ i - z _ j ) \\bigg { | } _ { \\epsilon = 0 } \\end{align*}"}
+{"id": "3244.png", "formula": "\\begin{align*} \\int _ { X } \\psi ( u _ { j } - u ) \\theta ^ { n } _ { u _ { j } } = \\int _ { X \\setminus O } \\psi ( u _ { j } - u ) \\theta ^ { n } _ { u _ { j } } + \\int _ { O } \\psi ( u _ { j } - u ) \\theta ^ { n } _ { u _ { j } } . \\end{align*}"}
+{"id": "1136.png", "formula": "\\begin{gather*} = \\sum _ { s = 0 } ^ { n } \\binom { n } { s } y ^ { n - s } ( 1 - \\rho ) ^ { s } \\sum _ { t = s } ^ { n } ( - 1 ) ^ { n - t } \\frac { ( n - s ) ! } { ( t - s ) ! t ! } \\rho ^ { t - s } ( \\beta + t - s ) ^ { ( s ) } \\\\ = \\sum _ { s = 0 } ^ { n } \\binom { n } { s } y ^ { n - s } ( 1 - \\rho ) ^ { s } \\sum _ { m = 0 } ^ { n - s } ( - 1 ) ^ { n - m - s } \\binom { n - s } { m } \\rho ^ { m } ( \\beta + m ) ^ { ( s ) } . \\end{gather*}"}
+{"id": "8244.png", "formula": "\\begin{align*} \\psi _ j ( t , \\tau ) : = e ^ { - \\bar { \\mu } 2 ^ { j \\alpha } ( t - \\tau ) } t ^ s \\tau ^ { - s } \\end{align*}"}
+{"id": "6309.png", "formula": "\\begin{align*} \\mathcal { M } ^ f _ t : = f ( Z _ t ) - \\int _ 0 ^ t \\mathcal { A } ^ f ( s , X _ s , Z _ s ) \\dd s , t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "4565.png", "formula": "\\begin{gather*} \\| f \\| \\leq \\| M _ f - \\Phi _ T ^ { - 1 } ( R ) \\| = \\| \\Phi _ T ^ { - 1 } \\circ \\Phi _ T ( M _ f - \\Phi _ T ^ { - 1 } ( R ) ) \\| = \\| \\Phi _ T ^ { - 1 } ( M _ g + R - R ) \\| = \\\\ = \\| \\Phi _ T ^ { - 1 } ( M _ g ) \\| \\leq \\| \\Phi _ T ^ { - 1 } \\| \\| M _ g \\| = \\| \\Phi _ T ^ { - 1 } \\| \\| g \\| . \\end{gather*}"}
+{"id": "5124.png", "formula": "\\begin{align*} ( i \\partial _ t + ( - \\Delta _ { x } ) ^ { \\sigma } + ( - \\partial _ { y } ^ { 2 } ) ^ { \\sigma } ) u = \\mu | u | ^ { \\frac { 4 \\sigma } { d } } u , u ( 0 , x , y ) = u _ 0 ( x , y ) \\in H ^ { \\sigma } ( \\mathbb { R } ^ d \\times \\mathbb { T } ) , \\end{align*}"}
+{"id": "3279.png", "formula": "\\begin{gather*} x _ 2 - a _ 2 ( b _ 1 + a _ 1 x _ 2 x _ 3 ) x _ 3 = b _ 2 , x _ 3 - a _ 3 ( b _ 1 + a _ 1 x _ 2 x _ 3 ) x _ 2 = b _ 3 , \\end{gather*}"}
+{"id": "6145.png", "formula": "\\begin{align*} ( R _ { q } \\otimes P ^ \\perp U + M _ z R _ { q } \\otimes P U ) ( z ^ n \\otimes \\xi ) = q ^ n z ^ n \\otimes P ^ \\perp U \\xi + q ^ n z ^ { n + 1 } \\otimes P U \\xi \\end{align*}"}
+{"id": "6574.png", "formula": "\\begin{align*} A = \\sum _ i \\theta _ i E _ i , B = \\sum _ j \\lambda _ j F _ j , \\end{align*}"}
+{"id": "4795.png", "formula": "\\begin{align*} \\frac { U _ { n + 1 , k } } { U _ { n , k } } = a ( n , k ) , \\frac { U _ { n , k - 1 } } { U _ { n , k } } = \\psi _ 0 ( n , k ) \\frac { ( b _ 1 + k - 1 ) \\dotsm ( b _ q + k - 1 ) k } { ( a _ 1 + k - 1 ) \\dotsm ( a _ p + k - 1 ) } , \\end{align*}"}
+{"id": "7968.png", "formula": "\\begin{align*} V _ j ( k ) = W _ j ( k ) = 0 = V _ j ( l ) = W _ j ( l ) \\quad \\mbox { f o r } j \\in \\{ 1 , \\ldots , n - 1 \\} \\end{align*}"}
+{"id": "1392.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t ^ 2 u - \\Delta u + a ( x ) \\partial _ t u + | u | ^ { p - 1 } u = 0 , & t \\in ( 0 , T ] , x \\in D , \\\\ u ( t , x ) = 0 , & t \\in ( 0 , T ] , x \\in \\partial D , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\ \\partial _ t u ( 0 , x ) = u _ 1 ( x ) , & x \\in D \\end{array} \\right . \\end{align*}"}
+{"id": "4377.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\sum _ { i = 1 } ^ { l } \\theta _ i D _ H g ^ 0 _ i ( \\overline { x } ( T ) ) \\cdot ( x ( T ) - \\overline { x } ( T ) ) \\\\ = p ( T ) \\cdot ( x ( T ) - \\overline { x } ( T ) ) - \\sum _ { \\alpha = 1 } ^ { m } \\lambda _ \\alpha D _ H g ^ \\alpha ( \\overline { x } ( T ) ) \\cdot ( x ( T ) - \\overline { x } ( T ) ) \\\\ - \\sum _ { \\beta = 1 } ^ { q } \\mu _ \\beta D _ H h ^ \\beta ( \\overline { x } ( T ) ) \\cdot ( x ( T ) - \\overline { x } ( T ) ) \\end{array} \\right \\} \\end{align*}"}
+{"id": "5449.png", "formula": "\\begin{align*} W _ { \\mu _ { \\rm m a x } } \\ , \\ , = \\ , \\ , \\bigcup _ { \\mu \\in \\mathcal { P } } W _ { \\mu } \\ , \\ , = \\ , \\ , \\mathrm { G r } ( \\ell , n ) ^ T . \\end{align*}"}
+{"id": "3689.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\rho \\left ( \\mathcal { H } _ n ( \\alpha ) \\right ) = 6 \\cdot p _ { \\Gamma _ { t + 2 } } ( x _ 1 , \\ldots , x _ { t + 4 } ) = \\frac { t ( t + 1 ) } { ( t + 2 ) ^ 2 } . \\end{align*}"}
+{"id": "4614.png", "formula": "\\begin{align*} \\| u \\| _ { W ^ { 1 , \\phi } ( \\Omega ) } : = \\| u \\| _ { L ^ \\phi ( \\Omega ) } + \\| \\nabla u \\| _ { L ^ \\phi ( \\Omega ) } . \\end{align*}"}
+{"id": "7487.png", "formula": "\\begin{align*} a _ { m , m - k } = \\frac { b _ { m , m - k } } { ( m - 2 ) ! } + \\delta _ { 1 k } , \\ , , m \\ge 2 \\ , , \\end{align*}"}
+{"id": "2124.png", "formula": "\\begin{align*} \\mu _ G ^ { \\pm / \\pm } ( ( E _ { 1 } ) _ { p ^ { \\infty } } / L _ \\infty ) = 0 \\iff \\mu _ G ^ { \\pm / \\pm } ( ( E _ { 2 } ) _ { p ^ { \\infty } } / L _ \\infty ) = 0 \\end{align*}"}
+{"id": "2957.png", "formula": "\\begin{align*} C : = H ( Q ) , Q : = \\{ z \\in \\R ^ \\ell \\ , | \\ , G ( z ) \\in D \\} \\end{align*}"}
+{"id": "8099.png", "formula": "\\begin{align*} T _ 1 T _ 2 = \\widetilde { U } T _ 2 T _ 1 \\end{align*}"}
+{"id": "6857.png", "formula": "\\begin{align*} ( v _ i ' ) ^ { p ^ { e ' } + 1 } = \\lambda u _ i h ( a _ i ) , 1 \\leq i \\leq n . \\end{align*}"}
+{"id": "6975.png", "formula": "\\begin{align*} R _ i : = \\pi ( Q S _ i ) \\mbox { f o r } 1 \\leq i \\leq d - 1 \\quad U = \\pi ( Q P ) . \\end{align*}"}
+{"id": "8178.png", "formula": "\\begin{align*} \\langle a _ j , b _ j \\rangle = 0 \\ \\ ( j = 1 , \\ldots , m ) . \\end{align*}"}
+{"id": "4845.png", "formula": "\\begin{align*} \\begin{aligned} \\Pi _ t ' = \\Gamma ( & s _ 1 , s _ 2 ) \\int _ { S ^ 2 } \\Pi _ t ' ( x , y , s _ 1 ' , s _ 2 ' ) \\ , \\dd s _ 1 ' \\dd s _ 2 ' = \\Gamma ( s _ 1 , s _ 2 ) \\Sigma _ t ( x , y ) = \\\\ & = \\int _ { S ^ 2 } \\mu ( s _ 1 ) \\delta _ { s _ 1 - s _ 2 } \\Pi _ t ^ K ( x , y , s _ 1 ' , s _ 2 ' ) \\ , \\dd s _ 1 ' \\dd s _ 2 ' = \\int _ { S ^ 2 } \\Gamma ( s _ 1 , s _ 2 ) \\Pi _ t ^ K ( x , y , s _ 1 ' , s _ 2 ' ) \\ , \\dd s _ 1 ' \\dd s _ 2 ' . \\end{aligned} \\end{align*}"}
+{"id": "1805.png", "formula": "\\begin{align*} [ y , z ] \\rhd ( x \\rhd t ) + [ [ x , y ] , z ] \\rhd t + y \\rhd ( [ x , z ] \\rhd t - x \\rhd ( y \\rhd ( z \\rhd t ) ) + z \\rhd ( x \\rhd ( y \\rhd t ) ) = 0 , \\end{align*}"}
+{"id": "7380.png", "formula": "\\begin{align*} f _ { + } - f _ { - } = - \\frac { \\alpha } { 2 } \\big ( \\partial _ { \\nu } f _ { + } + \\partial _ { \\nu } f _ { - } + i \\partial _ { t } f _ { + } + i \\partial _ { t } f _ { - } \\big ) \\partial _ { \\overline { z } } f _ { + } = \\partial _ { \\overline { z } } f _ { - } \\Sigma , \\end{align*}"}
+{"id": "5532.png", "formula": "\\begin{align*} 2 \\lambda = \\eta , \\begin{cases} k + 1 \\in | l | - 2 \\Z _ { \\ge 0 } & \\\\ k - 1 \\in | l | + 2 \\Z _ { \\ge 0 } & \\end{cases} \\end{align*}"}
+{"id": "5462.png", "formula": "\\begin{align*} F ( a , b ; t ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( a q ) _ n } { ( b q ) _ n } t ^ n , \\end{align*}"}
+{"id": "14.png", "formula": "\\begin{align*} \\beta _ m & = \\min \\left \\{ \\sum _ { \\omega \\in \\Omega _ m ' } | \\omega | x _ { \\omega } q _ { \\omega } \\ , \\ , : \\ , \\ , x _ { \\omega } \\geq 0 \\ , \\ , \\forall \\omega \\in \\Omega _ m ' , \\ , \\sum _ { \\omega \\in \\Omega _ m ' } | \\omega | x _ { \\omega } = 1 , \\ , \\sum _ { \\omega \\in \\Omega _ m ' } x _ { \\omega } A _ { \\omega } \\succeq 0 \\right \\} \\end{align*}"}
+{"id": "4191.png", "formula": "\\begin{align*} l ( w ^ i ) \\mathrm { s i n } ( \\theta ) = \\mathcal { C } i ( w ^ i , w _ 1 ) \\end{align*}"}
+{"id": "6716.png", "formula": "\\begin{align*} \\hat { A } _ { j } ( v ) = \\frac { | v | ^ { 2 } - 5 } { 2 } v _ { j } \\mbox { a n d } \\hat { B } _ { i j } ( v ) = v _ { i } v _ { j } - \\frac { 1 } { 3 } \\delta _ { i j } | v | ^ { 2 } , \\mbox { f o r } i , j = 1 , 2 , 3 . \\end{align*}"}
+{"id": "637.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = \\frac { 1 } { 2 } ( \\sqrt { c _ 1 } \\ X _ 1 \\cdot \\nu _ 1 + \\sqrt { c _ 2 } \\ X _ 2 \\cdot \\nu _ 2 ) \\cdot \\varphi - \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi . \\end{align*}"}
+{"id": "3014.png", "formula": "\\begin{align*} \\begin{aligned} y & = \\varphi ( x , u , \\dot { u } , \\ldots , u ^ { ( q ) } ) \\end{aligned} \\end{align*}"}
+{"id": "1497.png", "formula": "\\begin{align*} H ^ 2 W = K ( \\Delta ) - \\sum _ { p < z } g ( p ) \\sum _ { \\substack { d < \\Delta \\\\ ( d , p ) = 1 } } h ( d ) \\bigl \\{ \\min \\bigl ( \\log p , \\log \\frac { \\Delta } { d } \\bigr ) \\bigr \\} ^ 2 \\end{align*}"}
+{"id": "5730.png", "formula": "\\begin{align*} \\mathbf { J } ( \\phi ) = \\sum _ { a , b = 1 } ^ { s } \\phi ( E _ { a b } ) \\otimes E _ { a b } = \\begin{pmatrix} J _ { 1 1 } & \\cdots & J _ { 1 t } \\\\ \\vdots & \\ddots & \\vdots \\\\ J _ { t 1 } & \\cdots & J _ { t t } \\end{pmatrix} , \\end{align*}"}
+{"id": "2313.png", "formula": "\\begin{align*} L u = f , \\end{align*}"}
+{"id": "7225.png", "formula": "\\begin{align*} \\aligned & ( i \\partial _ t + \\Delta _ { x } ) u _ j = \\sum _ { ( j _ 1 , j _ 2 , j _ 3 ) \\in R ( j ) } u _ { j _ { 1 } } \\bar { u } _ { j _ { 2 } } u _ { j _ { 3 } } , \\\\ & R ( j ) = \\{ ( j _ 1 , j _ 2 , j _ 3 ) \\in ( \\mathbb { Z } ^ 2 ) ^ 3 : j _ 1 - j _ 2 + j _ 3 = j a n d | j _ 1 | ^ 2 - | j _ 2 | ^ 2 + | j _ 3 | ^ 2 = | j | ^ 2 \\} \\endaligned \\end{align*}"}
+{"id": "1967.png", "formula": "\\begin{align*} \\widehat { \\mathcal { P } _ I f } ( \\xi ) : = b _ I ( \\xi ) \\bigl [ b _ I ( \\xi ) \\hat { f } ( \\xi ) + b _ I ( 2 \\alpha _ I - \\xi ) \\hat { f } ( 2 \\alpha _ I - \\xi ) - b _ I ( 2 \\alpha ' _ I - \\xi ) \\hat { f } ( 2 \\alpha ' _ I - \\xi ) \\bigr ] . \\end{align*}"}
+{"id": "4631.png", "formula": "\\begin{align*} \\psi _ { B _ r } ( t ) : = \\int _ 0 ^ t \\tau ^ { p - 1 } \\sup _ { s \\in ( 0 , \\tau ] } \\frac { \\phi _ { B _ r } ^ + ( s ) } { s ^ p } \\ , d \\tau \\quad t \\ge 0 . \\end{align*}"}
+{"id": "5716.png", "formula": "\\begin{align*} \\Psi ( s ) = s \\log \\left ( \\frac { s + \\sqrt { s ^ 2 + 4 } } { 2 } \\right ) - \\sqrt { s ^ 2 + 4 } + 2 \\end{align*}"}
+{"id": "4455.png", "formula": "\\begin{align*} \\epsilon = \\tfrac { 1 } { 2 } \\left ( \\nabla u + \\nabla u ^ T \\right ) , \\end{align*}"}
+{"id": "2310.png", "formula": "\\begin{align*} ( 1 - H ^ * ) ^ n \\ 1 & = \\sum _ { j = 0 } ^ n ( - 1 ) ^ j \\binom { n } { j } \\ ( H ^ * ) ^ j \\ 1 \\\\ \\\\ & = \\sum _ { j = 0 } ^ n ( - 1 ) ^ j \\binom { n } { j } \\big ( ( - 1 ) ^ j \\frac { ( \\log x ) ^ j } { j ! } \\big ) \\\\ \\\\ & = \\sum _ { j = 0 } ^ n \\binom { n } { j } \\frac { ( \\log x ) ^ j } { j ! } . \\end{align*}"}
+{"id": "429.png", "formula": "\\begin{align*} [ t ^ { n } ] ( t ^ { m } F ( t ) G ( t ) ) = | \\Lambda _ { n } ( m ) | . \\end{align*}"}
+{"id": "6246.png", "formula": "\\begin{align*} \\frac { | \\{ g \\in G \\mid \\textup { $ \\varphi ( g ) $ i s a d e r a n g e m e n t } \\} | } { | G | } = \\frac { | \\{ h \\in H \\mid \\textup { $ h $ i s a d e r a n g e m e n t } \\} | } { | H | } \\end{align*}"}
+{"id": "3483.png", "formula": "\\begin{align*} \\varphi ( \\mathrm { v } , \\mathrm { y } , \\mathrm { t } , \\tau ) - \\Phi ^ { \\textbf { e } } ( \\xi , \\mathrm { t } ; \\mathrm { z } , \\tau ) = \\mathcal { O } \\Bigg ( \\sqrt { \\alpha } | \\mathrm { y } - \\mathrm { v } | \\ \\Vert \\partial _ \\mathrm { s } \\Phi ^ { \\textbf { e } } ( \\xi , \\mathrm { t } ; \\mathrm { z } , \\cdot ) \\Vert _ { \\mathrm { H } ^ { - \\frac { 1 } { 4 } } ( 0 , t ) } \\Bigg ) , \\end{align*}"}
+{"id": "82.png", "formula": "\\begin{align*} \\Delta _ { \\Psi } X ( u ) = X ( \\Delta _ { \\Psi } u ) . \\end{align*}"}
+{"id": "3985.png", "formula": "\\begin{align*} q _ { \\beta } ( n , t ) = \\sum _ { k = 1 } ^ { n } \\underset { m _ j \\in \\mathbb { N } } { \\underset { m _ { 1 } + m _ { 2 } + \\dots + m _ { k } = n } { \\sum } } \\prod _ { j = 1 } ^ { k } \\frac { \\theta ^ { m _ { j } } } { m _ { j } ! } \\left ( \\alpha t ^ { \\beta } \\right ) ^ { k } E _ { \\beta , k \\beta + 1 } ^ { k + 1 } \\left ( - \\alpha ( e ^ { \\theta } - 1 ) t ^ { \\beta } \\right ) . \\end{align*}"}
+{"id": "219.png", "formula": "\\begin{align*} m ^ { \\sigma , * } ( x ) = \\frac { 1 } { Z } \\exp { \\left ( - \\frac { 2 } { \\sigma ^ 2 } \\left ( \\frac { \\delta F } { \\delta m } ( m ^ { \\sigma , * } , x ) + U ( x ) \\right ) \\right ) } \\ , , \\end{align*}"}
+{"id": "7457.png", "formula": "\\begin{align*} L _ + = \\tau ^ \\dagger _ { - 1 } \\frac { 1 } { 2 \\sqrt { 2 } \\sqrt { \\hat { j } + 1 } } \\frac { \\sqrt { 2 ( \\hat { j } + 1 ) - 1 } } { \\sqrt { 2 ( \\hat { j } + 1 ) + 1 } } . \\end{align*}"}
+{"id": "7034.png", "formula": "\\begin{align*} \\mathcal { D } ^ { ( 1 ) } = \\mathcal { D } + \\hat { \\mathcal { D } } \\end{align*}"}
+{"id": "7088.png", "formula": "\\begin{align*} p _ X ^ { ( j ) } ( x , y , y ' ) : = \\frac { \\big ( \\delta _ { y , y ' } f _ X ^ { ( j ) } \\big ) ( x , y , y ' ) } { \\big ( f _ { X , y } ^ { ( j ) } \\big ) ( x , y ) } . \\end{align*}"}
+{"id": "3024.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi _ 1 ^ { ( r _ 1 + 1 ) } & = \\bar { u } _ 1 ^ { ( r _ 1 - \\rho _ 1 + 1 ) } \\\\ & \\vdotswithin { = } \\\\ \\varphi _ 1 ^ { ( n + \\rho _ 1 ) } & = \\bar { u } _ 1 ^ { ( n ) } \\end{aligned} \\end{align*}"}
+{"id": "7669.png", "formula": "\\begin{align*} E \\cap ( [ - r , r ] \\times \\R ) & = \\{ ( x , y ) \\in \\R ^ 2 \\ | \\ - L \\leq y \\leq f _ E ( x ) \\} \\\\ \\Omega \\cap ( [ - r , r ] \\times \\R ) & = \\{ ( x , y ) \\in \\R ^ 2 \\ | \\ - L \\leq y \\leq f _ \\Omega ( x ) \\} \\end{align*}"}
+{"id": "786.png", "formula": "\\begin{align*} \\frac { e ^ { - \\beta \\tau _ i } x _ i ( \\tau _ i ) } { l _ 1 + l _ 2 \\left ( \\frac { 1 } { N } \\sum \\limits _ { j = 1 } ^ N e ^ { - \\beta \\tau _ j } x _ j ( \\tau _ j ) \\right ) } \\end{align*}"}
+{"id": "6707.png", "formula": "\\begin{align*} F = M + G , P _ { 0 } F = M , P _ { 1 } F = G . \\end{align*}"}
+{"id": "6010.png", "formula": "\\begin{align*} r \\int _ 0 ^ { b } W ^ { ( q + r ) } ( b - y ) Z ^ { ( q ) } ( y ) d y = Z ^ { ( q + r ) } ( b ) - Z ^ { ( q ) } ( b ) . \\end{align*}"}
+{"id": "7277.png", "formula": "\\begin{align*} \\nabla _ x H ( t , x , \\psi , \\operatorname { p } ^ 1 \\sharp \\nu , u ) & = \\psi \\nabla _ x f ( t , x , \\operatorname { p } ^ 1 \\sharp \\nu , u ) - \\nabla _ x f _ 0 ( t , x , \\operatorname { p } ^ 1 \\sharp \\nu , u ) , \\\\ \\nabla _ \\psi H ( t , x , \\psi , \\operatorname { p } ^ 1 \\sharp \\nu , u ) & = f ( t , x , \\operatorname { p } ^ 1 \\sharp \\nu , u ) , \\end{align*}"}
+{"id": "57.png", "formula": "\\begin{align*} \\mathcal { V } _ p = T _ p \\Sigma _ t \\ \\ \\ \\ \\forall p \\in \\Sigma _ t \\end{align*}"}
+{"id": "4680.png", "formula": "\\begin{align*} D _ { z } \\phi _ { n } ^ { \\lambda } ( z ) = \\sqrt { n ( n + 2 \\lambda ) } \\phi _ { n - 1 } ^ { \\lambda } ( z ) \\hbox { a n d } D _ { z } ( z \\phi _ { n - 1 } ^ { \\lambda } ( z ) ) = ( n + \\lambda ) \\phi _ { n - 1 } ^ { \\lambda } ( z ) . \\end{align*}"}
+{"id": "698.png", "formula": "\\begin{align*} \\psi = \\sqrt { \\frac { { 2 \\Delta { t ^ 2 } \\left ( { { p _ { E O S } } - \\rho \\hat c _ s ^ 2 } \\right ) } } { { G \\Delta { x ^ 2 } } } } . \\end{align*}"}
+{"id": "3976.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\beta } ( t ) \\overset { d } { = } \\mathcal { M } ( T _ { 2 \\beta } ( t ) ) , \\ t > 0 , \\end{align*}"}
+{"id": "4925.png", "formula": "\\begin{align*} \\binom { n } { r } + \\sum _ { i = 1 } ^ { ( k + r - 1 ) / 2 } \\binom { n } { i } < \\binom { n } { k } \\end{align*}"}
+{"id": "7756.png", "formula": "\\begin{align*} 0 < \\mathcal { A } ( 0 _ n , \\theta _ 0 ) < 2 \\omega _ { n - 1 } \\int _ 0 ^ { + \\infty } \\frac { 1 } { \\cosh ^ n y } d y = \\omega _ n . \\end{align*}"}
+{"id": "4135.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left [ x ^ { 2 \\alpha + 1 } ( 1 - x ^ 2 ) \\frac { d } { d x } [ K _ n ^ { \\alpha } ] \\right ] + \\gamma _ n ^ { \\alpha } ( 0 ) x ^ { 2 \\alpha + 1 } K _ n ^ { \\alpha } = 0 , \\end{align*}"}
+{"id": "48.png", "formula": "\\begin{align*} \\Delta _ { \\Psi } u = e ^ { \\Psi } \\div ( e ^ { - \\Psi } \\nabla u ) = \\Delta u - g ( \\nabla \\Psi , \\nabla u ) . \\end{align*}"}
+{"id": "503.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n f _ i ( s ) \\bar { f _ i } ( t ) = \\mu \\log K ( G ( s ) , \\bar { G } ( t ) ) ~ ~ ( s , t ) \\in U \\times \\hbox { c o n j } ( { U } ) , \\end{align*}"}
+{"id": "7780.png", "formula": "\\begin{align*} c a p _ p ( B ( t ) ) = O ( e ^ { n t } ) \\Leftrightarrow c a p _ p ( X _ { \\varepsilon ( t ) } ) = O ( e ^ { n \\varepsilon ( t ) } ) , \\ t \\rightarrow + \\infty . \\end{align*}"}
+{"id": "5896.png", "formula": "\\begin{align*} e ^ { x } _ { n + 1 } = \\tilde { e } ^ { x } _ { n } + \\xi ^ { x } _ { n } ( h ) , \\ \\ \\ e ^ { v } _ { n + 1 } = \\tilde { e } ^ { v } _ { n } + \\xi ^ { v } _ { n } ( h ) , \\end{align*}"}
+{"id": "7949.png", "formula": "\\begin{align*} \\xi \\circ \\xi ' = ( x , y , t ) \\circ ( x ' , y ' , t ' ) = \\left ( x + x ' , y + y ' , t + t ' + 2 \\sum _ { k = 1 } ^ n ( x _ k y ' _ k - y _ k x ' _ k ) \\right ) , \\end{align*}"}
+{"id": "3327.png", "formula": "\\begin{gather*} \\sum _ { j _ 1 = 0 } ^ { N _ 1 } \\sum _ { j _ 2 = 0 } ^ { N _ 2 } \\big \\langle \\phi _ { m _ 1 ( j _ 1 ) } \\otimes \\phi _ { m _ 2 ( j _ 2 ) } ^ { m _ 1 ( j _ 1 ) } , \\psi _ { n _ 1 ( k _ 1 ) } ^ { n _ 2 ( k _ 2 ) } \\otimes \\psi _ { n _ 2 ( k _ 2 ) } \\big \\rangle _ { V _ 1 \\otimes V _ 2 } \\\\ \\qquad { } \\times \\overline { \\big \\langle \\phi _ { m _ 1 ( j _ 1 ) } \\otimes \\phi _ { m _ 2 ( j _ 2 ) } ^ { m _ 1 ( j _ 1 ) } , \\psi _ { n _ 1 ( k _ 1 ' ) } ^ { n _ 2 ( k _ 2 ' ) } \\otimes \\psi _ { n _ 2 ( k _ 2 ' ) } \\big \\rangle } _ { V _ 1 \\otimes V _ 2 } . \\end{gather*}"}
+{"id": "3795.png", "formula": "\\begin{align*} \\begin{cases} \\l _ t \\leq 2 n - 5 & n - d , \\\\ \\l _ t \\leq 2 n - 4 & n - d . \\end{cases} \\end{align*}"}
+{"id": "3196.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } \\phi _ { u _ { n } } u _ { n } ( u _ { n } - u _ { m } ) d x & \\le C \\left \\| \\phi _ { u _ { n } } \\right \\| _ { 6 } \\left \\| u _ { n } \\right \\| _ { 2 } \\left \\| u _ { n } - u _ { m } \\right \\| _ { 3 } \\\\ & \\le C \\left \\| u _ { n } \\right \\| ^ { 2 } \\left \\| u _ { n } \\right \\| _ { 2 } \\left \\| u _ { n } - u _ { m } \\right \\| _ { 3 } \\\\ & = o ( 1 ) . \\end{align*}"}
+{"id": "451.png", "formula": "\\begin{align*} { B _ n ( m x ) } = { m ^ { n - 1 } } \\sum _ { k = 0 } ^ { m - 1 } { B _ n \\Big ( { x + \\frac { k } { m } } \\Bigr ) } , m \\geq 1 . \\end{align*}"}
+{"id": "6908.png", "formula": "\\begin{align*} c ( \\pi _ * ( \\mathcal { K } ^ \\vee _ i ( a _ j - a _ { i } ) ) ) & = ( 1 + ( h _ i + w _ i \\epsilon - w _ j \\epsilon ) ) ^ { d _ i + a _ j - a _ i + 1 } \\\\ c ( \\pi _ * ( \\mathcal { K } ^ { \\vee } _ i \\otimes \\mathcal { K } _ j ( a _ { j } - a _ { i } ) ) ) & = ( 1 + ( h _ i + w _ i \\epsilon - h _ j - w _ j \\epsilon ) ) ^ { d _ i - d _ j + a _ j - a _ i + 1 } . \\end{align*}"}
+{"id": "3332.png", "formula": "\\begin{align*} 2 x _ 1 = 2 j _ 1 + 2 j _ 2 - \\hat { \\alpha } - N _ 1 - N _ 2 - 1 , 2 x _ 2 = 2 j _ 1 - \\hat { \\alpha } - N _ 1 + N _ 2 - 1 , \\end{align*}"}
+{"id": "7715.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow + \\infty } \\frac { c a p _ p ( B ^ \\mathbb { H } _ o ( t ) ) } { e ^ { n t } } = \\frac { 1 } { 2 ^ n } ( \\frac { n } { p - 1 } ) ^ { p - 1 } \\omega _ n . \\end{align*}"}
+{"id": "5401.png", "formula": "\\begin{align*} \\begin{cases} \\int _ { \\R ^ n } \\mathcal G ( x , t ; y , 0 ) d y = 1 \\\\ \\int _ { \\R ^ n } \\mathcal G ( x , t ; y , 0 ) \\phi ( y ) d y \\to \\phi ( x ) \\ \\end{cases} \\end{align*}"}
+{"id": "4591.png", "formula": "\\begin{align*} & \\tilde \\varphi ( n _ 0 + ( m _ { 2 l + 1 } + 1 ) N ) \\\\ = & \\tilde \\varphi ( n _ 0 + m _ { 2 l } N ) + \\sum _ { m = m _ { 2 l } } ^ { m _ { 2 l + 1 } } \\frac { K } { ( n _ 0 + m N - b ) \\pi \\sin \\pi k } ( 1 0 0 + \\varepsilon ( m ) ) . \\end{align*}"}
+{"id": "4739.png", "formula": "\\begin{gather*} r ^ \\sharp ( a ^ * ) \\cdot r ^ \\sharp ( b ^ * ) = r ^ \\sharp \\big ( R ^ * _ A ( r ^ \\sharp ( a ^ * ) ) b ^ * + L ^ * _ A ( r ^ \\sharp ( b ^ * ) ) a ^ * \\big ) , \\forall a ^ * , b ^ * \\in A ^ * , \\\\ \\partial _ k r ^ \\sharp = r ^ \\sharp \\eth _ k ^ * , \\forall k = 1 , \\dots , m . \\end{gather*}"}
+{"id": "7378.png", "formula": "\\begin{align*} f _ { + } - f _ { - } = - \\frac { \\alpha } { 2 } \\big ( \\partial _ { \\nu } f _ { + } + \\partial _ { \\nu } f _ { - } \\big ) \\partial _ { \\nu } f _ { + } = \\partial _ { \\nu } f _ { - } \\Sigma ; \\end{align*}"}
+{"id": "6249.png", "formula": "\\begin{align*} \\delta ( G , [ n ] ) = \\sum _ { j = 0 } ^ { t - 1 } \\frac { ( - 1 ) ^ j } { j ! } + ( n - t ) ! \\sum _ { j = t } ^ { n } \\frac { ( - 1 ) ^ j } { j ! ( n - j ) ! } . \\end{align*}"}
+{"id": "410.png", "formula": "\\begin{align*} \\sum _ { \\substack { a _ 1 , a _ 2 , \\ldots , a _ k = 1 \\\\ ( a _ 1 \\cdots a _ k , n ) = 1 \\\\ ( a _ 1 + \\cdots + a _ k , n ) = 1 } } ^ n ( a _ 1 + \\cdots + a _ k - 1 , n ) = \\varphi _ k ( n ) \\sigma _ 0 ( n ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\end{align*}"}
+{"id": "5269.png", "formula": "\\begin{align*} \\begin{cases} & \\sum _ { i = 1 } ^ { n + 1 } k _ i = 0 , \\\\ & \\sum _ { i \\in B } k _ i = 0 , \\ \\ \\forall B \\in \\pi , \\end{cases} \\end{align*}"}
+{"id": "4166.png", "formula": "\\begin{align*} \\begin{aligned} | m + j | ^ { - 4 } & \\le ( | m | - | j | ) ^ { - 4 } & & = | m | ^ { - 4 } ( 1 - | j | / | m | ) ^ { - 4 } \\\\ & \\le | m | ^ { - 4 } ( 1 + { 3 0 } | j | / | m | ) & & \\le | m | ^ { - 4 } ( 1 + { 3 0 \\cdot 2 ^ r } | m | ^ { r - 1 } ) . \\end{aligned} \\end{align*}"}
+{"id": "6312.png", "formula": "\\begin{align*} & \\int _ 0 ^ t | K _ { \\mu } ( s , t ' ) - K _ { \\mu } ( s , t ) | ^ { \\frac { p } { p - 1 } } \\dd s + \\int _ t ^ { t ' } | K _ { \\mu } ( s , t ' ) | ^ { \\frac { p } { p - 1 } } \\dd s \\leq C _ p | t ' - t | ^ { \\frac { \\gamma p } { p - 1 } } , \\\\ & \\int _ 0 ^ t | K _ { \\sigma } ( s , t ' ) - K _ { \\sigma } ( s , t ) | ^ { \\frac { 2 p } { p - 2 } } \\dd s + \\int _ t ^ { t ' } | K _ { \\sigma } ( s , t ' ) | ^ { \\frac { 2 p } { p - 2 } } \\dd s \\leq C _ p | t ' - t | ^ { \\frac { 2 \\gamma p } { p - 2 } } . \\end{align*}"}
+{"id": "6849.png", "formula": "\\begin{align*} v _ i ^ { p ^ e + 1 } = v _ i ^ { \\mu ( p ^ { e ' } + 1 ) + \\nu ( p ^ h - 1 ) } = ( v _ i ^ { \\mu } ) ^ { p ^ { e ' } + 1 } = ( v _ i ' ) ^ { p ^ { e ' } + 1 } . \\end{align*}"}
+{"id": "5982.png", "formula": "\\begin{align*} U _ r ^ { \\pi } ( t ) : = X ( t ) - L _ r ^ { \\pi } ( t ) + R _ r ^ { \\pi } ( t ) , t \\geq 0 . \\end{align*}"}
+{"id": "3565.png", "formula": "\\begin{align*} \\frac { \\tilde { l } ( \\tilde { l } + 1 ) } { 2 } + \\ell ( A / \\tilde { I } ) \\ge \\tilde { l } + \\ell ( A / \\tilde { I } ) = \\ell ( A / I ) . \\end{align*}"}
+{"id": "5979.png", "formula": "\\begin{align*} Z ^ { ( q ) \\prime } ( x , \\Phi ( q + r ) ) & : = \\dfrac \\partial { \\partial x } Z ^ { ( q ) } ( x , \\Phi ( q + r ) ) = \\Phi ( q + r ) Z ^ { ( q ) } ( x , \\Phi ( q + r ) ) - r W ^ { ( q ) } ( x ) , x > 0 . \\end{align*}"}
+{"id": "6061.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } \\Delta \\eta _ \\epsilon + \\lambda _ \\epsilon \\eta _ \\epsilon = 0 \\ , \\ , \\ , \\ , \\Omega _ \\epsilon \\\\ \\eta _ \\epsilon = 0 \\ , \\ , \\ , \\ , \\partial \\Omega \\\\ \\eta _ \\epsilon = 0 \\ , \\ , \\ , \\ , \\partial \\omega _ \\epsilon . \\end{array} \\right . \\end{align*}"}
+{"id": "7563.png", "formula": "\\begin{align*} \\left . \\frac { d } { d \\varepsilon } \\right | _ { \\varepsilon = 0 } G ( \\Phi ( \\varepsilon , p ) , \\Phi ( \\varepsilon , q ) ) = - \\Re ( h ( p ) C ( p , q ) + h ( q ) C ( q , p ) ) \\end{align*}"}
+{"id": "2556.png", "formula": "\\begin{align*} | \\rho ^ \\prime | = \\left | \\frac { 1 } { 1 + h ^ \\prime } \\right | \\leq \\frac { 1 } { \\varepsilon _ 0 } . \\end{align*}"}
+{"id": "5433.png", "formula": "\\begin{align*} A ( y , t , K ) & = \\frac { L } { ( t y ) ^ \\frac { 1 } { 2 } } \\left ( \\frac { a _ 1 K ^ 2 y } { t ^ 3 } \\right ) ^ { \\frac { \\sigma } { 2 } } \\left ( \\frac { a _ 2 K } { t } \\right ) ^ { 2 i t } b _ 1 ( y , t , K ) , \\\\ B ( y , t , K ) & = \\frac { a _ 3 t ^ 3 } { K ^ 2 } + \\frac { y } { 2 } \\left ( \\log \\left ( \\frac { a _ 4 K ^ 2 y } { t ^ 3 } \\right ) + b _ 2 ( y , t , K ) \\right ) , \\end{align*}"}
+{"id": "6412.png", "formula": "\\begin{align*} p ( \\theta ) = \\phi ( p ) + \\frac { 1 } { \\Gamma ( w ) } \\int _ { 0 } ^ { \\theta } \\big ( \\theta - \\alpha ( x ) ) ^ { w - 1 } \\mathcal { L } ( x , p ( x ) , ^ { C } D ^ { w } p ( x ) \\big ) \\nabla x . \\end{align*}"}
+{"id": "3461.png", "formula": "\\begin{align*} \\hat { \\varphi } ( \\eta , \\Tilde { \\tau } ) = \\varphi ^ { \\Lambda } ( \\eta , \\Tilde { \\tau } ) : = \\varphi ( \\delta \\eta + \\mathrm { z } , \\alpha \\delta ^ 2 \\Tilde { \\tau } ) , \\check { \\psi } ( \\mathrm { x } , t ) = \\psi ^ \\vee ( \\mathrm { x } , t ) : = \\psi \\Bigg ( \\frac { \\mathrm { x } - \\mathrm { z } } { \\delta } , \\frac { t } { \\alpha \\delta ^ 2 } \\Bigg ) \\end{align*}"}
+{"id": "208.png", "formula": "\\begin{align*} \\eta ( t ) = \\nu \\left ( \\frac { t } { T } \\right ) , \\ , \\ , \\ , t > 0 , \\end{align*}"}
+{"id": "2331.png", "formula": "\\begin{align*} \\zeta ( \\{ \\{ 2 \\} ^ { m } , 1 , \\{ 2 \\} ^ { m } , 3 \\} ^ { n } , \\{ 2 \\} ^ { m } ) \\stackrel { ? } { = } \\frac { 1 } { 2 n + 1 } \\frac { \\pi ^ { \\mathrm { w t } } } { ( \\mathrm { w t } + 1 ) ! } \\quad \\left ( = \\frac { 1 } { 2 n + 1 } \\zeta ( \\{ 2 \\} ^ { { \\rm w t } / 2 } ) \\right ) \\end{align*}"}
+{"id": "1423.png", "formula": "\\begin{align*} \\left \\| U ( t ) \\begin{pmatrix} u _ 0 \\\\ u _ 1 \\end{pmatrix} \\right \\| _ { \\mathcal { H } } & \\le C _ 0 \\| ( u _ 0 , u _ 1 ) \\| _ { \\mathcal { H } } \\end{align*}"}
+{"id": "5268.png", "formula": "\\begin{align*} L _ { \\pi } : = \\{ t = ( t _ 1 , \\ldots , t _ { n + 1 } ) \\in \\R ^ { n + 1 } : \\sum _ { i = 1 } ^ { n + 1 } t _ i = 0 ; \\ \\ \\sum _ { i \\in B } t _ i = 0 \\ \\forall B \\in \\pi \\} . \\end{align*}"}
+{"id": "6225.png", "formula": "\\begin{align*} \\phi _ { n , m } ( x , z ) = \\sqrt { \\textstyle { \\frac { 2 } { d } } } \\ , h _ n \\big ( x + \\textstyle { \\frac { p } { B } } \\big ) \\ , \\sin \\textstyle { \\frac { \\pi m z } { d } } , \\end{align*}"}
+{"id": "3288.png", "formula": "\\begin{gather*} p _ 0 + p _ 1 = 2 \\alpha _ 0 , p _ 2 + p _ 3 = \\alpha _ 1 , p _ 0 - p _ 1 = 2 \\alpha _ 3 , p _ 2 - p _ 3 = 2 \\alpha _ 2 . \\end{gather*}"}
+{"id": "824.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & A ( x ^ * ) ^ { k _ 1 } + p ( x ^ * ) = - \\bar { K } _ 2 , \\\\ & k _ 1 A ( x ^ * ) ^ { k _ 1 - 1 } + p ' ( x ^ * ) = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "4941.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k - 2 } \\frac { j } { 2 } \\binom { n } { i } < \\binom { n } { k } . \\end{align*}"}
+{"id": "6173.png", "formula": "\\begin{align*} \\langle h , f _ d + V _ d f _ { d - 1 } + \\cdots + V _ d V _ { d - 1 } \\cdots V _ { i + 1 } V _ { i - 1 } \\cdots V _ 3 f _ 2 + V _ d V _ { d - 1 } \\cdots V _ { i + 1 } V _ { i - 1 } \\cdots V _ 2 f _ 1 \\rangle = 0 , \\end{align*}"}
+{"id": "6920.png", "formula": "\\begin{align*} e _ j = \\begin{cases} \\binom { N } { j } & j \\ne N , N + 1 \\\\ 1 + ( - 1 ) ^ { N - 1 } q & j = N \\\\ ( - 1 ) ^ N q y & j = N + 1 . \\\\ \\end{cases} \\end{align*}"}
+{"id": "5077.png", "formula": "\\begin{align*} \\left ( \\mathcal { F } e ^ { i t \\phi ( \\nabla / i ) } f \\right ) ( \\xi ) = e ^ { - i t \\phi ( \\xi ) } ( \\mathcal { F } f ) ( \\xi ) . \\end{align*}"}
+{"id": "1679.png", "formula": "\\begin{align*} ( W _ N , v _ N ) \\in \\mathfrak { A } _ { \\mathsf { m } + 1 } ^ { \\sigma , \\tau } \\ , \\wedge \\ , \\mathfrak { O } _ { 0 , N } = \\bigsqcup \\limits _ { n = 1 } ^ { N - 1 } \\Big [ \\ , ( W _ N , v _ N ) \\in \\mathfrak { A } _ { \\mathsf { m } + 1 } ^ { \\sigma , \\tau } \\ , \\wedge \\ , \\mathfrak { O } _ { 0 , N } \\ , \\wedge \\ , n = n _ { \\textnormal { m i n } } \\ , \\Big ] \\ , . \\end{align*}"}
+{"id": "4400.png", "formula": "\\begin{align*} T _ d ^ \\eta \\hat { f } ( \\lambda ) = \\mathcal { J } ( f \\varphi _ \\eta ^ { ( \\alpha , \\beta ) } ) ( \\lambda ) . \\end{align*}"}
+{"id": "3693.png", "formula": "\\begin{align*} g _ { 1 } ( z ) = z + \\frac { 1 } { m } { \\rm a n d } g _ { 2 } ( z ) = z ^ 2 - \\frac { z } { m } + \\frac { 1 } { m ^ 2 } . \\end{align*}"}
+{"id": "2122.png", "formula": "\\begin{align*} N = [ R ^ \\times : ( R ^ \\times ) ^ s ] = \\| s \\| _ K \\cdot \\# \\{ x \\in K \\mid x ^ s = 1 \\} , \\end{align*}"}
+{"id": "5745.png", "formula": "\\begin{align*} \\begin{aligned} E ^ T ( { \\rm I d } _ q \\otimes P ^ M ) f ( X ^ M ) ( { \\rm I d } _ q \\otimes P ^ M ) E & = E ^ T \\left ( \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } \\otimes P ^ M { X ^ M _ i } { X ^ M _ j } P ^ M \\right ) E \\\\ & = E ^ T \\left ( \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } \\otimes M ^ s _ { i j } \\right ) E < 0 . \\end{aligned} \\end{align*}"}
+{"id": "6449.png", "formula": "\\begin{align*} y ^ n L _ n ^ { ( \\alpha ) } ( x / y ) = L _ n ^ { ( \\alpha ) } ( x , y ) . \\end{align*}"}
+{"id": "436.png", "formula": "\\begin{align*} \\phi ( h ) = \\sum _ { i = 1 } ^ n \\phi ( h _ i ) . \\end{align*}"}
+{"id": "3756.png", "formula": "\\begin{align*} \\tau _ i ( - 6 ) = \\frac { 1 } { q _ i ( 2 ) } \\cdot \\frac { q _ i ( 2 ) } { | \\mathcal C _ i | } J _ { | \\mathcal C _ i | } = \\frac { J _ { | \\mathcal C _ i | } } { | \\mathcal C _ i | } \\end{align*}"}
+{"id": "4321.png", "formula": "\\begin{align*} \\sum _ { k = j + 1 } ^ \\infty \\sum _ { i = 0 } ^ \\infty \\mathbb { P } ( Z _ t = i ) P _ { i , k } = \\sum _ { k = j + 1 } ^ \\infty \\mathbb { P } ( Z _ { t + 1 } = k ) = \\mathbb { P } ( Z _ { t + 1 } > j ) . \\end{align*}"}
+{"id": "4306.png", "formula": "\\begin{gather*} \\phi ^ 0 _ { 2 m + j } = L ^ { m + 1 } _ { j + m - 1 } , j = 2 , \\ldots , n - 2 m . \\end{gather*}"}
+{"id": "1334.png", "formula": "\\begin{align*} \\int _ { B _ { 9 / 1 0 } } \\left | \\nabla u ( x ) - \\nabla v ( x ) \\right | ^ p \\ , d x \\le C \\left | \\left \\{ u = 0 \\right \\} \\cap B _ { 9 / 1 0 } \\right | + C \\sigma ( a ^ p + 1 ) , \\end{align*}"}
+{"id": "2201.png", "formula": "\\begin{align*} & R _ { 1 2 1 2 } = - a ^ 2 - ( b + c ) ( 3 b - c ) , \\\\ & R _ { 1 2 1 3 } = - 2 ( a c + b d ) , \\\\ & R _ { 1 2 2 3 } = R _ { 1 3 2 3 } = 0 , \\\\ & R _ { 1 3 1 3 } = ( b + c ) ( b - 3 c ) - d ^ 2 , \\\\ & R _ { 2 3 2 3 } = ( b + c ) ^ 2 - a d . \\end{align*}"}
+{"id": "756.png", "formula": "\\begin{align*} a _ { ( n ) } \\cdot v = 0 , n \\geq N , \\end{align*}"}
+{"id": "3256.png", "formula": "\\begin{align*} [ \\xi _ i , \\xi _ j ] = 2 \\delta \\xi _ k . \\end{align*}"}
+{"id": "5843.png", "formula": "\\begin{align*} \\| D \\| _ { b e r } = \\sup \\{ \\| D e _ i \\| : i \\in \\mathbb { Z } _ { + } \\} = \\sup _ { i } { | \\lambda _ i | } = \\| D \\| . \\end{align*}"}
+{"id": "2067.png", "formula": "\\begin{align*} N ( A ) = \\| A \\| . \\end{align*}"}
+{"id": "4396.png", "formula": "\\begin{align*} v ( \\lambda ) = \\left | \\frac { 2 ^ { \\varrho - i \\lambda } \\Gamma ( i \\lambda ) \\Gamma ( \\alpha + 1 ) } { \\Gamma \\left ( \\frac { \\varrho + i \\lambda } { 2 } \\right ) \\Gamma \\left ( \\frac { \\varrho + i \\lambda } { 2 } - \\beta \\right ) } \\right | ^ { - 2 } . \\end{align*}"}
+{"id": "4439.png", "formula": "\\begin{align*} \\begin{cases} \\mathrm { s p t } ( \\tilde { u } _ { 1 , n } ^ { \\varepsilon } - u _ 1 ) \\subset \\Omega _ n & \\\\ \\mathrm { s p t } ( \\tilde { v } _ { 2 , n } ^ { \\varepsilon } - v _ 2 ) \\subset \\Omega _ n \\end{cases} \\end{align*}"}
+{"id": "1134.png", "formula": "\\begin{gather*} H _ { j , n } = \\frac { \\sqrt { n ! } } { \\Gamma ( \\beta ) \\sqrt { \\Gamma ( n + \\beta ) } } \\sum _ { k = 0 } ^ { n } \\frac { ( - 1 ) ^ { k } } { k ! } \\frac { ( \\beta ) ^ { ( n ) } } { ( n - k ) ! ( \\beta ) ^ { ( k ) } } \\int _ { 0 } ^ { \\infty } x ^ { k + j } x ^ { \\beta - 1 } \\exp ( - x ) d x \\\\ = \\frac { \\sqrt { n ! } } { \\sqrt { \\Gamma ( n + \\beta ) } } \\sum _ { k = 0 } ^ { n } \\frac { ( - 1 ) ^ { k } } { k ! } \\frac { ( \\beta ) ^ { ( n ) } ( \\beta ) ^ { ( k + j ) } } { ( n - k ) ! ( \\beta ) ^ { ( k ) } } . \\end{gather*}"}
+{"id": "6048.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta p - k ^ 2 p = 2 ( \\eta - \\eta _ 0 ) - 2 \\Delta \\eta + 2 \\nabla \\cdot A \\ , \\ , \\ , \\ , \\Omega \\\\ [ 0 . 3 c m ] \\frac { \\partial p } { \\partial n } = 0 \\ , \\ , \\ , \\ , \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "1711.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } d x \\ , U ( x ) = \\int _ { \\Lambda } d x \\ , U ( x ) = - 1 \\ , . \\end{align*}"}
+{"id": "5214.png", "formula": "\\begin{align*} C _ j < p _ 1 + 2 \\sum _ { k = 2 } ^ { j _ s } p _ k + \\sum _ { i = 1 } ^ { s - 1 } p _ { j _ { i + 1 } - 1 } + \\sum _ { k = j _ s + 1 } ^ { j - 1 } p _ k + 2 p _ j + L . \\end{align*}"}
+{"id": "5793.png", "formula": "\\begin{align*} { } \\displaystyle \\sum _ { n = 1 } ^ { \\infty } \\Big ( \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } a _ k \\Big ) ^ p \\leq \\Big ( \\frac { p } { p - 1 } \\Big ) ^ p \\sum _ { n = 1 } ^ { \\infty } { a _ n } ^ p , \\end{align*}"}
+{"id": "437.png", "formula": "\\begin{align*} e ( f ) = \\frac { 1 } { 2 } \\textrm { t r a c e } _ \\nu g _ { i j } ( f ) . \\end{align*}"}
+{"id": "6890.png", "formula": "\\begin{align*} \\mathbb P _ { \\sf { P l a n } } ( \\lbrace \\lambda \\rbrace ) : = { \\rm e } ^ { - L ^ 2 } L ^ { 2 | \\lambda | } \\left ( \\frac { \\dim \\lambda } { | \\lambda | ! } \\right ) ^ 2 , \\lambda \\in \\mathbb Y . \\end{align*}"}
+{"id": "1761.png", "formula": "\\begin{align*} \\left \\langle Z _ { * } , i _ { 1 } , i _ { 1 } + 1 , \\dots , \\widehat { i _ { j } + \\epsilon } , \\dots , i _ { \\frac { m } { 2 } } , i _ { \\frac { m } { 2 } } + 1 \\right \\rangle = \\det \\left ( \\begin{matrix} C \\\\ V _ { * } \\\\ I _ { i _ { 1 } , i _ { 1 } + 1 , \\ldots , \\widehat { i _ { j } + \\epsilon } , \\ldots , i _ { \\frac { m } { 2 } } , i _ { \\frac { m } { 2 } } + 1 } \\end{matrix} \\right ) \\det ( \\mathcal { Z } ) , \\end{align*}"}
+{"id": "7065.png", "formula": "\\begin{align*} Y _ { M _ 1 , h ' } \\cap Y _ { M _ 2 , h ' } \\subset \\tilde { p } _ { h ' } ^ { - 1 } ( Y _ { M _ 1 , h ' - 1 } ) \\cap \\tilde { p } _ { h ' } ^ { - 1 } ( Y _ { M _ 2 , h ' - 1 } ) = \\emptyset . \\end{align*}"}
+{"id": "7918.png", "formula": "\\begin{align*} a \\ast _ P b : = P ( a ) \\cdot b ~ + ~ a \\cdot P ( b ) + \\lambda ~ \\ ! a \\cdot b , a , b \\in A . \\end{align*}"}
+{"id": "5505.png", "formula": "\\begin{align*} ( q ^ { 1 - N } ) _ { n } = \\frac { ( - 1 ) ^ n q ^ { n ( n - 1 ) / 2 } ( q ) _ { N - 1 } } { ( q ) _ { N - n - 1 } q ^ { ( N - 1 ) n } } . \\end{align*}"}
+{"id": "4153.png", "formula": "\\begin{align*} \\Delta = \\sqrt { a _ { 1 2 } ^ 2 + a _ { 1 3 } ^ 2 + a _ { 2 3 } ^ 2 - a _ { 1 2 } a _ { 1 3 } - a _ { 1 2 } a _ { 2 3 } - a _ { 1 3 } a _ { 2 3 } } = \\sqrt { \\frac { ( a _ { 1 2 } - a _ { 1 3 } ) ^ 2 + ( a _ { 1 2 } - a _ { 2 3 } ) ^ 2 + ( a _ { 1 3 } - a _ { 2 3 } ) ^ 2 } { 2 } } \\end{align*}"}
+{"id": "4891.png", "formula": "\\begin{align*} P ( x , t ) & : = \\sum _ { n = 0 } ^ \\infty p _ { n - 1 } ( x ) \\frac { t ^ n } { n ! } = \\frac { e ^ { ( 1 - x ) t } \\left ( \\arcsin ( x ^ { 1 / 2 } e ^ { ( 1 - x ) t } ) - \\arcsin ( x ^ { 1 / 2 } ) \\right ) } { x ^ { 1 / 2 } ( 1 - x e ^ { 2 ( 1 - x ) t } ) ^ { 1 / 2 } } . \\end{align*}"}
+{"id": "4531.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\mathrm { d } t \\int _ t ^ 1 \\mathrm { d } s \\ , F ( t , s ) = \\int _ 0 ^ 1 \\mathrm { d } s \\int _ 0 ^ s \\mathrm { d } t \\ , F ( t , s ) \\ , \\\\ \\int _ 0 ^ 1 \\mathrm { d } t \\int _ 0 ^ t \\mathrm { d } s \\ , F ( t , s ) = \\int _ 0 ^ 1 \\mathrm { d } s \\int _ s ^ 1 \\mathrm { d } t \\ , F ( t , s ) \\ . \\end{align*}"}
+{"id": "7148.png", "formula": "\\begin{align*} \\mathbb { D } ( d ; \\delta ) _ w = \\left \\langle \\boxtimes _ { i = 1 } ^ k \\mathbb { M } ( d _ i ) _ { w _ i } \\right \\rangle \\end{align*}"}
+{"id": "3220.png", "formula": "\\begin{align*} \\mu _ 2 ( G ) > \\frac { 1 } { \\delta + 1 } \\left ( 2 + \\frac { 1 } { \\binom { d + 1 } { 2 } - 1 } \\right ) , \\end{align*}"}
+{"id": "6317.png", "formula": "\\begin{align*} A _ t ^ n : = \\int _ 0 ^ t \\mu _ n ( s , X ^ n _ s ) \\dd s M _ t ^ n : = \\int _ 0 ^ t \\sigma _ n ( s , X ^ n _ s ) \\dd B _ s , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "7073.png", "formula": "\\begin{align*} \\tau _ { R _ Y / R _ P } = \\mu ( \\Delta _ \\tau ) \\circ \\tau . \\end{align*}"}
+{"id": "1433.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\| u ^ { ( j ) } ( t ) - u ( t ) \\| _ { H ^ 1 } = 0 , \\\\ \\lim _ { j \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\| \\partial _ t u ^ { ( j ) } ( t ) - v ( t ) \\| _ { L ^ 2 } = 0 , \\end{align*}"}
+{"id": "3412.png", "formula": "\\begin{align*} \\quad & \\frac { d } { d t } \\int _ { V ( t ) } \\rho \\ , r ^ { n - 1 } d r = 0 , \\\\ \\quad & \\frac { d } { d t } \\int _ { V ( t ) } \\rho f ( S ) \\ , r ^ { n - 1 } d r = 0 , \\\\ \\quad & \\frac { d } { d t } \\int _ { V ( t ) } \\rho ( \\tfrac { 1 } { 2 } U ^ 2 + e ) \\ , r ^ { n - 1 } d r = - ( r ^ { n - 1 } p U ) \\Big | _ { \\partial V ( t ) } , \\end{align*}"}
+{"id": "6801.png", "formula": "\\begin{align*} \\phi + \\phi \\circ ( 1 \\ , \\ , 3 ) + \\phi \\circ ( 1 \\ , \\ , 4 ) = 0 \\end{align*}"}
+{"id": "6561.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { ( n - 2 ) / 2 } \\frac { 2 k \\pi } { n - 1 } \\sin \\left ( \\frac { 2 k \\pi } { n - 1 } \\cdot b \\right ) = \\frac { \\pi \\csc \\left ( \\frac { b \\pi } { 2 a + 1 } \\right ) \\left ( - 2 ( a + 1 ) + \\sin \\left ( \\frac { 2 b \\pi ( a + 1 ) } { 2 a + 1 } \\right ) \\csc \\left ( \\frac { b \\cdot \\pi } { 2 a + 1 } \\right ) \\right ) } { 4 a + 2 } . \\end{align*}"}
+{"id": "457.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 0 \\\\ n - k \\equiv 0 \\ , ( \\mathrm { m o d } \\ , 2 ) } } ^ n \\binom { n } { k } \\big ( 2 ^ k L _ { j k } + 2 L _ j ^ k \\big ) ( \\sqrt { 5 } F _ j ) ^ { n - k } \\frac { 2 ^ { n - k + 2 } - 1 } { n - k + 2 } B _ { n - k + 2 } = L _ j ^ n , \\end{align*}"}
+{"id": "2702.png", "formula": "\\begin{align*} \\binom { a + j } { j } \\sum _ { l = 0 } ^ { a } \\binom { n - j + l } { l } \\binom { n - j } { a + j - l } \\binom { n - a - 2 j + l } { a - l } \\sum _ { k = a + j - l } ^ { n - ( a + j - l ) } \\binom { n - 2 a - 2 j + 2 l } { k - j - a + l } G ( n , k , a ) \\end{align*}"}
+{"id": "239.png", "formula": "\\begin{align*} Q ( u ) = Q [ D ] ( u ) \\stackrel { d e f } { = } { \\bf P } \\left [ \\max _ { i = 1 , 2 , \\ldots , d } \\xi _ i > u \\right ] \\end{align*}"}
+{"id": "2761.png", "formula": "\\begin{align*} a \\cdot { \\rm i n d e x } ( G / P ) = b \\cdot \\dim ( G / P ) . \\end{align*}"}
+{"id": "2987.png", "formula": "\\begin{align*} \\Gamma _ k ^ \\tau = 1 - e ^ { - ( k ) _ 2 \\tau } \\cdot ( 1 - W ) \\ ; . \\end{align*}"}
+{"id": "4301.png", "formula": "\\begin{align*} C _ 1 ( q , N ) - R _ 1 ( q , N ) = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{array} { c } N \\\\ n \\end{array} \\right ] \\frac { ( - 1 ) ^ { n + 1 } ( q ) _ n q ^ { n ( n + 1 ) / 2 } ( 1 - q ^ { n ^ 2 } ) } { ( q ) _ { n + N } ( 1 - q ^ n ) } . \\end{align*}"}
+{"id": "2178.png", "formula": "\\begin{align*} & \\overline { r } _ 1 = \\{ R _ { 1 2 1 3 } ( R _ { 2 3 2 3 } - c ) - R _ { 1 2 2 3 } R _ { 1 3 2 3 } \\} S _ { 1 2 1 2 1 } - \\{ R _ { 1 2 1 3 } R _ { 1 3 2 3 } - R _ { 1 2 2 3 } ( R _ { 1 3 1 3 } - c ) \\} S _ { 1 2 1 2 2 } \\\\ & - \\{ ( R _ { 1 2 1 2 } - c ) ( R _ { 2 3 2 3 } - c ) - R _ { 1 2 2 3 } { } ^ 2 \\} S _ { 1 2 1 3 1 } + \\{ ( R _ { 1 2 1 2 } - c ) R _ { 1 3 2 3 } - R _ { 1 2 1 3 } R _ { 1 2 2 3 } \\} S _ { 1 2 1 3 2 } \\\\ & + \\{ ( R _ { 1 2 1 2 } - c ) R _ { 1 3 2 3 } - R _ { 1 2 1 3 } R _ { 1 2 2 3 } \\} S _ { 1 2 2 3 1 } - \\{ ( R _ { 1 2 1 2 } - c ) ( R _ { 1 3 1 3 } - c ) - R _ { 1 2 1 3 } { } ^ 2 \\} S _ { 1 2 2 3 2 } . \\end{align*}"}
+{"id": "3321.png", "formula": "\\begin{align*} \\frac { g ( m _ 1 , m _ 2 ) } { \\big \\langle \\phi _ { m _ 2 } ^ { m _ 1 } , \\psi _ { n _ 2 ( 0 ) } \\big \\rangle _ { V _ 2 } } = \\frac { 1 } { \\big \\langle \\phi _ { m _ 2 ( 0 ) } ^ { m _ 1 } , \\psi _ { n _ 2 ( 0 ) } \\big \\rangle _ { V _ 2 } } . \\end{align*}"}
+{"id": "372.png", "formula": "\\begin{align*} \\iota ( v _ { \\alpha _ 1 } \\wedge \\dots \\wedge v _ { \\alpha _ k } ) ~ \\omega = \\iota _ { v _ { \\alpha _ k } } \\ldots \\iota _ { v _ { \\alpha _ 1 } } \\omega = \\omega ( v _ { \\alpha _ 1 } , \\ldots , v _ { \\alpha _ k } , \\cdot , \\cdots , \\cdot ) . \\end{align*}"}
+{"id": "195.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { \\pi ^ { - } ( a _ t \\cdot \\psi _ t - a \\cdot \\psi ) } { t } & = \\pi ^ { - } \\left ( \\lim _ { t \\to 0 } \\frac { a _ t \\cdot \\psi _ t - a _ t \\cdot \\psi } { t } + \\lim _ { t \\to 0 } \\frac { a _ t - a } { t } \\cdot \\psi \\right ) \\\\ & = \\pi ^ { - } ( b \\cdot \\phi + a \\cdot \\psi ) . \\end{align*}"}
+{"id": "7230.png", "formula": "\\begin{align*} \\| a \\| _ { l ^ { 2 } } = \\| v _ { 0 } \\| _ { l ^ { 2 } } , \\| a \\| _ { h ^ { s } } = \\| v _ { 0 } \\| _ { H ^ { s } } . \\end{align*}"}
+{"id": "491.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { n \\choose k } 2 ^ k F _ { j k } ( \\pm \\sqrt { 5 } F _ j ) ^ { n - k } & B _ { n - k } ( x ) \\\\ = n & F _ j \\Big ( ( \\pm \\sqrt { 5 } F _ j x + L _ { j } ) ^ { n - 1 } + ( \\pm \\sqrt { 5 } F _ j ( x - 1 ) + L _ { j } ) ^ { n - 1 } \\Big ) . \\end{align*}"}
+{"id": "1634.png", "formula": "\\begin{align*} \\phi _ { ( I S ) } ( x , t ) = \\phi _ { ( S ) } \\left ( x + \\frac { \\phi _ { ( I S ) } ^ { ( 1 ) } ( x , t ) - x } { 2 } , t \\right ) . \\end{align*}"}
+{"id": "7467.png", "formula": "\\begin{align*} N _ { \\gamma } \\coloneqq \\begin{pmatrix} V _ 0 & V _ 1 \\\\ - \\gamma W _ 0 & - \\gamma W _ 1 \\end{pmatrix} , \\widetilde { N } _ { \\gamma } \\coloneqq \\begin{pmatrix} V _ 1 & V _ 0 \\\\ \\gamma W _ 1 & \\gamma W _ 0 \\end{pmatrix} , F _ { \\gamma } \\coloneqq \\begin{pmatrix} E _ { \\gamma } & 0 \\\\ 0 & E _ { \\gamma } \\end{pmatrix} , \\end{align*}"}
+{"id": "7227.png", "formula": "\\begin{align*} \\begin{aligned} & F ( a ) : = \\{ f _ { j } \\} _ { j \\in \\mathbb { Z } ^ { 2 } } , \\\\ & f _ { j } : = \\sum _ { ( j _ 1 , j _ 2 , j _ 3 ) \\in R ( j ) } a _ { j _ { 1 } } \\bar { a } _ { j _ { 2 } } a _ { j _ { 3 } } . \\end{aligned} \\end{align*}"}
+{"id": "5279.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ s ( - 1 ) ^ k \\frac { s ! } { k ! \\ * ( s - k ) ! } = 0 . \\end{align*}"}
+{"id": "7627.png", "formula": "\\begin{align*} [ V _ p / S ] \\rightarrow [ V _ p ^ x / S ] = X _ p . \\end{align*}"}
+{"id": "5713.png", "formula": "\\begin{align*} \\mathrm { g r a d } _ { \\Gamma } \\ , F ( \\nu , x ) = ( \\nabla g ) \\left ( \\langle 1 , \\nu \\rangle , \\langle f _ 1 , \\nu \\rangle , \\dots , \\langle f _ m , \\nu \\rangle \\right ) \\cdot ( 1 , f _ 1 ( x ) , \\dots , f _ m ( x ) ) ^ \\top . \\end{align*}"}
+{"id": "4021.png", "formula": "\\begin{align*} \\lim _ { t \\to T _ { m a x } } \\left \\| u ( \\cdot , t ) \\right \\| _ { L ^ \\infty } = + \\infty . \\end{align*}"}
+{"id": "7599.png", "formula": "\\begin{align*} \\textrm { $ f ( z _ \\alpha ) - f ( z _ \\beta ) = 0 \\ ( { \\rm m o d } \\ \\chi _ C ) $ . } \\end{align*}"}
+{"id": "6567.png", "formula": "\\begin{align*} \\mathcal { H } = \\ell ^ 2 ( \\Omega ) = \\{ \\psi : \\Omega \\rightarrow \\mathbb { C } \\mid \\sum _ { \\omega \\in \\Omega } \\abs { \\psi ( \\omega ) } ^ 2 < \\infty \\} = \\bigoplus _ i \\mathcal { C } _ i = \\bigoplus _ j \\mathcal { D } _ j . \\end{align*}"}
+{"id": "3119.png", "formula": "\\begin{align*} F _ { X _ 1 } ( \\sigma _ 1 ^ k ) = \\ell _ 1 q _ 1 - \\ell _ 1 q _ 1 = 0 \\end{align*}"}
+{"id": "2775.png", "formula": "\\begin{align*} A = U _ A C G ^ { - 1 } , B = U _ B S G ^ { - 1 } , \\end{align*}"}
+{"id": "4846.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\int _ { \\R ^ d \\times S } | x - x ^ * | ^ 2 \\ , \\dd \\rho _ t ( x , s ) & = - 2 \\int _ { \\R ^ d \\times S } ( x - x ^ * ) \\cdot ( \\nabla f ( x , s ) - \\nabla f ( x ^ * , s ) ) \\ , \\dd \\rho _ t ( x , s ) \\leq \\\\ & \\leq - 2 m \\int _ { \\R ^ d \\times S } | x - x ^ * | ^ 2 \\ , \\dd \\rho _ t ( x , s ) \\end{align*}"}
+{"id": "896.png", "formula": "\\begin{align*} x _ n = \\rho ( x ' ) = \\frac { 1 } { 2 } \\sum _ { \\alpha , \\beta < n } B _ { \\alpha \\beta } x _ \\alpha x _ \\beta + O ( | x ' | ^ 3 ) , \\end{align*}"}
+{"id": "5561.png", "formula": "\\begin{align*} u _ 1 ( \\eta ) = 1 - \\dfrac { \\Phi _ 1 [ \\alpha _ 0 , \\eta , L _ 1 ( u _ 1 ) , N _ 1 ( u _ 1 ) ] } { \\Phi _ 1 [ \\alpha _ 0 , \\beta _ 0 , L _ 1 ( 0 ) , N _ 1 ( 0 ) ] } , \\ ; \\ ; \\ ; \\alpha _ 0 \\leq \\eta \\leq \\beta _ 0 ; \\end{align*}"}
+{"id": "907.png", "formula": "\\begin{align*} G ^ { \\alpha \\beta } _ 0 = \\frac { \\partial G } { \\partial r _ { \\alpha \\beta } } ( \\nabla _ { \\alpha \\beta } u ( x _ 0 ) ) , 1 \\leq \\alpha , \\beta \\leq n - 1 . \\end{align*}"}
+{"id": "6292.png", "formula": "\\begin{align*} y _ { i '' } '' = y _ { i ' } ' . \\end{align*}"}
+{"id": "2171.png", "formula": "\\begin{align*} \\overline { \\beta } _ { i k m } \\alpha _ { j l } ^ 0 + \\alpha _ { i k } ^ 0 \\overline { \\beta } _ { j l m } - \\overline { \\beta } _ { i l m } \\alpha _ { j k } ^ 0 - \\alpha _ { i l } ^ 0 \\overline { \\beta } _ { j k m } = \\overline { S } _ { i j k l m } , \\end{align*}"}
+{"id": "8125.png", "formula": "\\begin{align*} & \\quad \\ ; ( ( n L , n \\varphi ) ^ { d + 1 } ) + \\frac { 1 } { 2 } \\nu ( \\Omega _ \\infty ) ( d + 1 ) n ^ d ( L ^ d ) \\ln ( 2 r _ n ) \\\\ & \\geqslant ( ( n L , n \\widetilde \\varphi _ n ) ^ { d + 1 } ) - \\nu ( \\Omega _ \\infty ) n ^ d ( L ^ d ) \\Big ( ( d + 1 ) \\ln ( r _ n ) + \\ln ( n ^ d ( L ^ d ) ) \\Big ) , \\end{align*}"}
+{"id": "6498.png", "formula": "\\begin{align*} \\inf _ { B _ { \\delta R _ k } ( x _ k ) } u + C \\delta ^ { \\gamma } \\rho & \\leq C ( \\delta R _ k ) ^ { - \\lambda } \\inf _ { B _ 1 } u + C \\rho + C \\delta ^ { \\gamma } \\rho \\\\ & < 2 C ( \\delta R _ k ) ^ { - 2 \\lambda } \\inf _ { B _ 1 } u + \\frac { M _ { k - 1 } } { 2 } \\rho \\\\ & = \\frac { M _ { k - 1 } } { 2 } \\left ( \\inf _ { B _ 1 } u + \\rho \\right ) \\\\ & < u ( x _ k ) / 2 , \\\\ \\end{align*}"}
+{"id": "6210.png", "formula": "\\begin{align*} E ^ * _ { i + k - 1 } A _ 1 E ^ * _ { i + k - 2 } A _ 1 E ^ * _ { i + k - 3 } \\cdots E ^ * _ { i + 1 } A _ 1 E ^ * _ { i } = \\big ( ( \\frac { k - 2 } { 2 } ) ! \\big ) ^ 2 \\frac { k } { 2 } M ^ { \\frac { k - 2 } { 2 } , \\frac { k - 2 } { 2 } } _ { \\frac { i + k - 2 } { 2 } , \\frac { 2 m - i } { 2 } } . \\end{align*}"}
+{"id": "8143.png", "formula": "\\begin{gather*} \\delta _ { i } ' = ( M L _ 1 \\cdots L _ { i - 1 } L _ { i + 1 } \\cdots L _ d ) \\\\ \\delta _ { i , n } = ( L _ { 0 , n } L _ 1 \\cdots L _ { i - 1 } L _ { i + 1 } \\cdots L _ d ) = n \\delta _ i + \\delta _ { i , n } ' . \\end{gather*}"}
+{"id": "492.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { k = 0 } ^ n { n \\choose k } 2 ^ k F _ { j k } ( \\sqrt { 5 } F _ j ) ^ { n - k } B _ { n - k } ( x ) \\frac { z ^ n } { n ! } & = F _ j z e ^ { \\big ( \\frac { \\sqrt { 5 } F _ j } { 2 } ( 2 x - 1 ) + L _ j \\big ) z } \\Big ( e ^ { \\frac { \\sqrt { 5 } F _ j } { 2 } z } + e ^ { - \\frac { \\sqrt { 5 } F _ j } { 2 } z } \\Big ) \\\\ & = F _ j z \\Big ( e ^ { ( \\sqrt { 5 } F _ j x + L _ j ) z } + e ^ { ( \\sqrt { 5 } F _ j ( x - 1 ) + L _ j ) z } \\Big ) . \\end{align*}"}
+{"id": "7638.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { 5 } \\nu _ i + \\mu _ 1 + \\mu _ 2 = \\sum _ { i = 0 } ^ { 1 1 } \\lambda _ i = 1 . \\end{align*}"}
+{"id": "2806.png", "formula": "\\begin{align*} A \\tilde { g } _ i & = \\tilde { c } _ i \\tilde { u } _ i ^ A , & A ^ T \\tilde { u } _ i ^ A & = R ^ T \\left ( \\tilde { c } _ i R \\tilde { g } _ i + \\alpha _ { k + 1 } v _ { k + 1 } e _ { k + 1 } ^ T x _ i \\right ) , \\\\ B \\tilde { g } _ i & = \\tilde { s } _ i \\tilde { u } _ i ^ B , & B ^ T \\tilde { u } _ i ^ B & = R ^ T \\left ( \\tilde { s } _ i R \\tilde { g } _ i + \\check { \\beta } _ { k } v _ { k + 1 } e _ { k } ^ T \\hat { x } _ i \\right ) . \\end{align*}"}
+{"id": "4766.png", "formula": "\\begin{align*} [ a , b ] = \\eth _ 2 ( a \\cdot \\partial _ 1 ( b ) ) - \\eth _ 1 ( a \\cdot \\partial _ 2 ( b ) ) , \\forall a , b \\in A . \\end{align*}"}
+{"id": "6671.png", "formula": "\\begin{align*} \\frac { \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( a s ^ { ( 1 ) } _ { k - 1 } ) _ { k = 0 } ^ { m - 1 } \\right ) } { a b \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( 1 , 0 , \\ldots , 0 ) \\right ) } \\stackrel { a \\to 0 + } { \\longrightarrow } \\frac { \\mathcal { D } \\left ( ( s ^ { ( 1 ) } _ { k + 1 } ) _ { k = 0 } ^ { 2 m - 3 } , ( s ^ { ( 1 ) } _ { k - 1 } ) _ { k = 0 } ^ { m - 1 } \\right ) } { b \\mathcal { D } \\left ( ( - s ^ { ( 1 ) } _ { k + 1 } ) _ { k = 0 } ^ { 2 m - 3 } , ( 1 , 0 , \\ldots , 0 ) \\right ) } \\end{align*}"}
+{"id": "8219.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } Y _ j + \\bar { \\mu } 2 ^ { j \\alpha } Y _ j \\le C \\Big ( \\Vert f _ j \\Vert _ { L ^ 2 } + \\Vert g _ j \\Vert _ { L ^ 2 } + \\Vert \\nabla v \\Vert _ { L ^ { \\infty } } Y _ j \\Big ) , \\end{aligned} \\end{align*}"}
+{"id": "6559.png", "formula": "\\begin{align*} H = \\sum _ { k = 0 } ^ { ( n - 2 ) / 2 } \\frac { 2 k \\pi } { n - 1 } \\left ( E _ { \\lambda _ k } - \\overline { E _ { \\lambda _ k } } \\right ) . \\end{align*}"}
+{"id": "826.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } \\int _ 0 ^ { \\tau _ i ^ { * } } e ^ { - \\beta t } f ( x _ i ( t ) ) d t & = A x ^ { k _ 1 } + p ( x ) + \\bar { K } _ 2 \\mathbb { E } \\Big ( e ^ { - \\beta \\tau _ i ^ { * } } \\Big ) . \\end{aligned} \\end{align*}"}
+{"id": "1738.png", "formula": "\\begin{align*} \\| G \\| _ { L ^ 1 } & = \\int _ { y > 0 } d y \\ , y \\ , \\mu \\left ( | G | > y \\right ) \\ , \\\\ & \\leq \\int _ { y > 1 } d y \\ , y \\ , \\mu \\left ( | G | > y \\right ) + 1 , \\end{align*}"}
+{"id": "1333.png", "formula": "\\begin{align*} J _ p ( v , B _ 1 ) = \\ ; & \\int _ { B _ 1 } \\Big ( | \\nabla v ( x ) | ^ p + \\chi _ { \\{ v > 0 \\} } ( x ) \\Big ) \\ , d x \\\\ \\le \\ ; & \\int _ { B _ { 9 / 1 0 } } | \\nabla v ( x ) | ^ p \\ , d x + \\int _ { B _ 1 \\setminus B _ { 9 / 1 0 } } | \\nabla v ( x ) | ^ p \\ , d x + | B _ 1 | \\\\ \\le \\ ; & \\int _ { B _ 1 } \\left | \\nabla u \\right | ^ p \\ , d x + \\left | B _ 1 \\right | \\le a ^ p + \\left | B _ 1 \\right | . \\end{align*}"}
+{"id": "1985.png", "formula": "\\begin{align*} \\overline { z } _ w = \\begin{cases} \\overline { x } _ w , & w \\neq u , \\\\ \\overline { x } _ v , & w = u . \\end{cases} \\end{align*}"}
+{"id": "4006.png", "formula": "\\begin{align*} \\bar { q } ( n , t ) = \\sum _ { \\Omega _ { n } } \\prod _ { j = 1 } ^ { n } \\frac { \\left ( \\rho ^ { j } \\binom { r + j - 1 } { j } \\right ) ^ { x _ { j } } } { x _ { j } ! } \\left ( \\frac { \\lambda t ( 1 - \\rho ) ^ { r } } { 1 - ( 1 - \\rho ) ^ { r } } \\right ) ^ { z _ { n } } e ^ { - \\lambda t } , \\end{align*}"}
+{"id": "7565.png", "formula": "\\begin{align*} E _ \\varphi [ \\mu _ { \\varepsilon , h } ] - E _ \\varphi [ \\mu ] - \\varepsilon \\Re D _ { V , h } ( \\mu ) = o ( \\varepsilon ) \\varepsilon \\to 0 , \\end{align*}"}
+{"id": "2904.png", "formula": "\\begin{align*} L [ \\nu _ { n + 1 } ] = - m \\eta x _ { n + 1 } ^ { m - 1 } . \\end{align*}"}
+{"id": "7800.png", "formula": "\\begin{align*} k ( u ( t , \\theta ) ) = \\frac { - 1 } { 2 \\pi } \\frac { \\cot \\theta - i } { \\csc ^ 3 ( \\theta ) ( \\sin t + \\cos \\theta ) } . \\end{align*}"}
+{"id": "5312.png", "formula": "\\begin{align*} S ( x ) \\le \\begin{cases} x ^ \\theta \\log ^ 6 ( x + 1 ) & \\textrm { i f } \\theta = \\theta _ 0 ; \\\\ Z _ \\alpha ( x ) + O \\left ( \\frac { 1 - \\theta } { ( \\alpha - \\theta ) ^ 3 } \\log ^ 3 \\frac { x + 1 } { \\alpha - \\theta } ~ x ^ \\alpha \\right ) \\ll \\frac { 1 - \\theta } { ( \\alpha - \\theta ) ^ 3 } \\log ^ 3 \\frac { x + 1 } { \\alpha - \\theta } ~ x ^ { \\theta _ 0 } & \\textrm { i f } \\theta < \\alpha < \\theta _ 0 . \\end{cases} \\end{align*}"}
+{"id": "2110.png", "formula": "\\begin{align*} S _ A = \\begin{bmatrix} I _ m & 0 & I _ m & 0 & 0 \\\\ 0 & I _ { k - m } & 0 & I _ { k - m } & 0 \\\\ A _ { 1 1 } & A _ { 1 2 } & 0 & D & 0 \\\\ A _ { 2 1 } & A _ { 2 2 } & 0 & 0 & I _ { \\ell - k } \\\\ A _ { 3 1 } & A _ { 3 2 } & 0 & 0 & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "1836.png", "formula": "\\begin{align*} \\{ x , y \\} = x \\circ y - y \\circ x , x \\rhd y = x \\succ y - y \\prec x , \\end{align*}"}
+{"id": "1147.png", "formula": "\\begin{align*} \\lim \\nolimits _ { k \\to \\infty } C ( l ' , r ' ; P _ { k , S } ^ { l , r } ) = \\delta _ S ( l , r ; \\ , l ' , r ' ) , \\end{align*}"}
+{"id": "7091.png", "formula": "\\begin{align*} \\Omega ( P , Q ) = \\frac { { \\rm d } t _ P { \\rm d } t _ Q } { ( t _ P - t _ Q ) ^ 2 } ( 1 + d _ { > 0 } ( t _ P , t _ Q ) ) . \\end{align*}"}
+{"id": "6829.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ n \\alpha _ n ^ i \\partial _ t ^ { i + n + 2 } u _ { n + 1 } + \\partial _ t ^ { n + 2 } u _ { n + 1 } - \\sum _ { i = 1 } ^ { 2 n } \\beta _ n ^ i \\Delta \\partial _ t ^ { i + 1 } u _ { n + 1 } - \\omega _ { n + 1 } \\kappa _ n \\Delta \\partial _ t u _ { n + 1 } \\\\ & \\quad - ( 1 - \\omega _ { n + 1 } ) \\kappa _ n \\int _ 0 ^ \\infty g _ { n + 1 } ( s ) \\Delta \\partial _ t u _ { n + 1 } ( t - s ) d s - \\kappa _ { n + 1 } \\Delta u _ { n + 1 } = 0 . \\end{align*}"}
+{"id": "5414.png", "formula": "\\begin{align*} G _ { i j } = G _ { j i } , \\qquad 1 \\leq i < j \\leq N . \\end{align*}"}
+{"id": "1591.png", "formula": "\\begin{align*} & T _ { s _ { 3 } '' } + \\rho ( s '' ) T _ { s _ { 2 } ' } + \\rho ( s '' ) \\rho ( s ' ) T _ { s _ { 1 } } \\\\ = & \\left [ T _ { b _ { 1 } } - \\rho ( s '' ) T _ { b _ { 3 } } \\right ] + \\rho ( s '' ) \\left [ T _ { b _ { 3 } } - \\rho ( s ' ) T _ { b _ { 2 } } \\right ] + \\rho ( s '' ) \\rho ( s ' ) \\left [ T _ { b _ { 2 } } - \\rho ( s ) T _ { b _ { 1 } } \\right ] \\\\ = & T _ { b _ { 1 } } - \\rho ( s '' ) \\rho ( s ' ) \\rho ( s ) T _ { b _ { 1 } } = \\left [ \\mathrm { i d } _ { W } - \\rho ( s '' ) \\rho ( s ' ) \\rho ( s ) \\right ] T _ { b _ { 1 } } . \\end{align*}"}
+{"id": "7072.png", "formula": "\\begin{align*} ( a b \\otimes 1 ) - ( 1 \\otimes a b ) = ( b \\otimes 1 ) ( a \\otimes 1 - 1 \\otimes a ) + ( 1 \\otimes a ) ( b \\otimes 1 - 1 \\otimes b ) , \\end{align*}"}
+{"id": "1298.png", "formula": "\\begin{align*} \\mathrm { T r } ( f ) = \\varepsilon _ a \\circ ( f \\otimes a ^ \\ast ) \\circ \\sigma _ { a ^ * , a } \\circ \\eta _ a . \\end{align*}"}
+{"id": "308.png", "formula": "\\begin{align*} I _ { x _ { i } } = \\{ i ' \\in I _ { i } \\ ; | \\ ; x _ { i } \\in D _ { i ' } ^ { ( i ) } \\} . \\end{align*}"}
+{"id": "1539.png", "formula": "\\begin{align*} F ^ { \\prime } \\left ( g _ { s , 1 , 2 } \\right ) = & F \\left ( g _ { e , 2 , 1 } \\circ g _ { s , 1 , 2 } \\right ) \\overset { ( 1 ) } { = } F \\left ( g _ { e , 3 , 1 } \\circ g _ { e , 2 , 3 } \\circ g _ { s , 1 , 2 } \\right ) \\\\ \\overset { ( 2 ) } { = } & F \\left ( g _ { e , 1 , 3 } \\circ g _ { s , 2 , 3 } \\circ g _ { e , 1 , 2 } \\right ) = F ^ { \\prime } \\left ( g _ { s , 2 , 3 } \\right ) . \\end{align*}"}
+{"id": "6233.png", "formula": "\\begin{align*} D ( H ) = \\{ \\psi \\in H ^ 2 ( \\Omega ) \\cap H ^ 1 _ 0 ( \\Omega ) : \\ : H \\psi \\in L ^ 2 ( \\Omega ) \\} , ; \\end{align*}"}
+{"id": "1230.png", "formula": "\\begin{align*} E ( x , y ) = f ^ * ( y ) - f ^ * ( \\nabla f ( x ) ) - x \\cdot ( y - \\nabla f ( x ) ) \\geq \\frac { 1 } { 2 M } \\abs { y - \\nabla f ( x ) } ^ 2 \\end{align*}"}
+{"id": "3814.png", "formula": "\\begin{align*} a _ 1 + a _ 2 + \\cdots + a _ { h - 2 } = \\frac { ( h - 1 ) ( 2 n - 4 ) - ( a _ { h - 1 } + a _ h ) + d } { 2 } . \\end{align*}"}
+{"id": "7689.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 1 + } \\| f \\| _ { \\alpha , p } = \\| f \\| _ { p } . \\end{align*}"}
+{"id": "5061.png", "formula": "\\begin{align*} \\frac 1 { 2 \\pi i } \\int _ { \\Re ( u ) = - \\frac 5 2 } \\frac { \\Gamma _ \\C ( \\frac { k - 1 + u } { 2 } ) \\gamma ( - u / 2 ) } { 2 \\Gamma _ \\C ( \\frac { k + 1 - u } { 2 } ) \\gamma ( 1 + u / 2 ) } \\ , d u = 0 \\quad k \\ge 4 k \\equiv \\epsilon \\ ; ( 2 ) . \\end{align*}"}
+{"id": "1286.png", "formula": "\\begin{align*} ( a ^ \\ast \\otimes f ) \\circ \\eta _ a = ( f ^ \\ast \\otimes b ) \\circ \\eta _ b , \\varepsilon _ a \\circ ( a \\otimes f ^ { \\ast } ) = \\varepsilon _ b \\circ ( f \\otimes b ^ \\ast ) . \\end{align*}"}
+{"id": "6152.png", "formula": "\\begin{align*} & R _ { \\overline q } M _ z R _ { \\overline q } ( z ^ n ) = \\overline q ^ { 2 n + 1 } z ^ n , R _ { \\overline q } M _ z R _ { \\overline q } M _ z ^ * ( z ^ n ) = \\overline q ^ { 2 n - 1 } z ^ n \\\\ & R _ { \\overline q } M _ z ^ * R _ { \\overline q } ( z ^ n ) = \\overline q ^ { 2 n - 1 } z ^ { n - 1 } , R _ { \\overline q } M _ z ^ * R _ { \\overline q } M _ z ( z ^ n ) = \\overline q ^ { 2 n + 1 } z ^ n . \\end{align*}"}
+{"id": "4109.png", "formula": "\\begin{align*} d - N _ 0 \\cdot \\frac { d u } { u } - \\sum _ { i = 1 } ^ n N _ i \\cdot \\frac { d t _ i } { d _ i } . \\end{align*}"}
+{"id": "5156.png", "formula": "\\begin{align*} D ( A ' , \\Omega , x ) = 0 D ( \\widetilde A , x ) = 0 . \\end{align*}"}
+{"id": "7874.png", "formula": "\\begin{align*} \\# \\{ f \\in F _ B : x \\in Y _ 1 ( f ) \\} = \\sum _ { f \\in F _ B } { \\bf 1 } _ { Y _ 1 ( f ) } ( x ) \\sim \\mu _ 2 , \\forall x \\in E _ 2 . \\end{align*}"}
+{"id": "7312.png", "formula": "\\begin{align*} a = i q ^ j \\bmod { ( q ^ m + 1 ) } . \\end{align*}"}
+{"id": "6407.png", "formula": "\\begin{align*} \\mathrm { d e g } _ \\infty = ( 2 \\varpi _ 1 + \\varpi _ 2 , 2 \\varpi _ 1 + \\varpi _ 2 , 2 \\varpi _ 1 + \\varpi _ 2 , 2 \\varpi _ 1 + \\varpi _ 2 ) \\in \\mathsf { P } _ + ^ { \\oplus 4 } . \\end{align*}"}
+{"id": "8067.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathrm { i f } ~ \\widetilde { \\beta } _ 1 > \\gamma , & \\mathrm { d e t e c t i o n } , \\\\ \\mathrm { i f } ~ \\widetilde { \\beta } _ 1 \\leq \\gamma , & \\mathrm { n o ~ d e t e c t i o n } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3211.png", "formula": "\\begin{align*} h ' _ { g _ { u } } ( 1 ) = & - 2 \\beta A ( u ) + \\left ( ( 1 - \\frac { 3 } { 2 } \\beta ) p + 3 \\beta - 2 \\right ) C ( u ) + ( 2 - \\beta ) B ( u ) \\\\ & + \\frac { \\beta } { 4 } \\int _ { \\mathbb { R } ^ { 3 } } \\int _ { \\mathbb { R } ^ { 3 } } e ^ { - | x - y | } u ^ { 2 } ( x ) u ^ { 2 } ( y ) d x d y . \\end{align*}"}
+{"id": "2799.png", "formula": "\\begin{align*} x _ { j + 1 } = \\underset { x \\in \\mathbb { R } ^ n } { \\arg \\min } \\| Z x - \\tilde { u } _ { j + 1 } \\| . \\end{align*}"}
+{"id": "2277.png", "formula": "\\begin{align*} \\sigma ^ m ( z _ i ) \\in V : = k \\{ z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\dots z _ n ^ { i _ n } : - N \\leq i _ j \\leq N , 1 \\leq j \\leq n \\} , m \\geq 1 , \\end{align*}"}
+{"id": "6331.png", "formula": "\\begin{align*} \\inf _ { u \\in A } \\ , E ( u ) \\le c : = \\inf _ { h \\in \\widetilde { H } } \\ , \\sup _ { u \\in h ( Q ) } \\ , E ( u ) \\le \\sup _ { u \\in Q } \\ , E ( u ) , \\end{align*}"}
+{"id": "1648.png", "formula": "\\begin{align*} & \\mathtt { X } ( Q , \\mathtt { P } ) \\ ; = \\ ; Q ^ { \\perp } \\mathtt { P } Q + Q \\mathtt { P } ^ * Q ^ { \\perp } \\ , , \\\\ & \\mathtt { Y } ( Q , \\mathtt { P } ) \\ ; = \\ ; Q ^ { \\perp } \\mathtt { P } Q \\mathtt { P } ^ * Q ^ { \\perp } - Q \\mathtt { P } ^ * Q ^ { \\perp } \\mathtt { P } Q + \\frac { 1 } { 2 } \\left [ Q ^ { \\perp } \\mathtt { P } ( Q ^ { \\perp } - Q ) \\mathtt { P } Q + Q \\mathtt { P } ^ * ( Q ^ { \\perp } - Q ) \\mathtt { P } ^ * Q ^ { \\perp } \\right ] \\ , . \\end{align*}"}
+{"id": "7977.png", "formula": "\\begin{align*} \\varphi _ h ( \\xi ) & = \\frac { 2 n - 1 } { 2 n + 1 } H _ M ( \\xi ) | \\xi ^ H | ^ 2 - k ( \\xi ) | \\xi ^ H | ^ 2 = | \\xi ^ H | ^ 2 \\left ( \\frac { ( 2 n - 2 ) k ( \\xi ) + l ( \\xi ) } { 2 n + 1 } - k ( \\xi ) \\right ) \\\\ & = \\frac { l ( \\xi ) - 3 k ( \\xi ) } { 2 n + 1 } | \\xi ^ H | ^ 2 , \\end{align*}"}
+{"id": "3787.png", "formula": "\\begin{align*} C _ { i _ 1 , \\ldots , i _ k } ( 1 ) = & \\sum _ { r = 0 } ^ { i _ 1 - 1 } C _ { r , i _ 1 - 1 - r , i _ 2 , \\ldots , i _ k } ( 1 ) + C _ { i _ 2 , \\ldots , i _ k } \\left ( C _ { i _ 1 } ( 1 ) - \\sum _ { r = 0 } ^ { i _ 1 - 1 } C _ { r , i _ 1 - 1 - r } ( 1 ) \\right ) \\\\ & + 2 \\sum _ { r = 2 } ^ k i _ r C _ { i _ 1 + i _ r - 1 , i _ 2 , \\ldots , \\widehat { i _ r } , \\ldots , i _ k } \\end{align*}"}
+{"id": "7742.png", "formula": "\\begin{align*} \\lim \\limits _ { \\delta \\rightarrow 0 } \\delta ( 1 + \\frac { 2 } { a ( \\delta ) } ) ^ { \\frac { 1 } { 2 } } = \\lim \\limits _ { \\delta \\rightarrow 0 } ( \\frac { 2 \\delta ^ 2 } { a ( \\delta ) } ) ^ { \\frac { 1 } { 2 } } = ( \\frac { 4 n ( n - 1 ) } { \\min S _ { \\hat { g } } } ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
+{"id": "4254.png", "formula": "\\begin{align*} _ { 2 } \\phi _ { 1 } \\left [ \\begin{array} { c c c } \\alpha , & \\beta \\ ; ; & z \\\\ \\gamma & \\end{array} \\right ] = \\frac { ( \\beta ) _ { \\infty } ( \\alpha z ) _ { \\infty } } { ( \\gamma ) _ { \\infty } ( z ) _ { \\infty } } \\ ; _ { 2 } \\phi _ { 1 } \\left [ \\begin{array} { c c c } \\frac { \\gamma } { \\beta } , & z \\ ; ; & \\beta \\\\ \\alpha z & \\end{array} \\right ] . \\end{align*}"}
+{"id": "4554.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 4 i \\ , n _ i = 2 m , \\end{align*}"}
+{"id": "5550.png", "formula": "\\begin{align*} T ( r , t ) = u ( \\eta ) , \\ ; \\ ; \\ ; \\eta = \\dfrac { r } { 2 \\sqrt { t } } , \\end{align*}"}
+{"id": "2075.png", "formula": "\\begin{align*} S \\begin{bmatrix} A _ { 1 1 } \\\\ A _ { 2 1 } \\end{bmatrix} T _ { 1 1 } = \\begin{bmatrix} I _ \\ell \\\\ E \\end{bmatrix} . \\end{align*}"}
+{"id": "1321.png", "formula": "\\begin{align*} 2 e ( S _ 1 ) \\alpha + 2 e ( L ) + e ( L _ 1 , S _ 1 ) \\alpha + e ( N _ 1 , S _ 2 ) \\alpha \\le ( 4 k + 2 ) n \\alpha + 2 \\binom { | L | } { 2 } + ( 2 k + 1 ) n \\alpha + ( 2 k + 1 ) n \\alpha \\le 1 1 k n \\alpha , \\end{align*}"}
+{"id": "6722.png", "formula": "\\begin{align*} { ( \\widetilde { \\rho } , \\widetilde { u } , \\widetilde { \\theta } , \\widetilde { E } , \\widetilde { B } ) ( t , x ) = ( \\rho - \\bar { \\rho } , u - \\bar { u } , \\theta - \\bar { \\theta } , E - \\overline { E } , B - \\bar { B } ) ( t , x ) , } \\end{align*}"}
+{"id": "4457.png", "formula": "\\begin{align*} \\epsilon = \\tfrac { 1 } { 2 } \\left ( \\nabla u + \\nabla u ^ T \\right ) , \\end{align*}"}
+{"id": "4533.png", "formula": "\\begin{align*} f ( x ) = & \\frac { a _ 0 } { \\sqrt { \\sinh 1 } } \\sinh \\left ( x - \\frac { 1 } { 2 } \\right ) + b _ 0 \\\\ & + \\sum _ { k = 1 } ^ \\infty \\sqrt { \\frac { 2 } { 1 + 4 \\pi ^ 2 k ^ 2 } } \\big ( a _ k \\sin ( 2 \\pi k x ) + b _ k \\cos ( 2 \\pi k x ) \\big ) \\ , , \\end{align*}"}
+{"id": "6950.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ { \\infty } z ^ d \\cdot \\left ( \\left [ t ^ d \\right ] \\Phi ( t ) ^ { d } \\cdot \\Psi ( t ) \\right ) = \\frac { \\Psi ( t ) } { \\Phi ( t ) } \\cdot \\frac { d t } { d z } \\ , . \\end{align*}"}
+{"id": "4351.png", "formula": "\\begin{align*} \\Big [ t \\mapsto I ( t ) : = \\int _ \\Omega \\Phi ( f ( t ) ) \\ , \\mathrm { d } x \\Big ] \\in { \\rm C } ^ 1 ( \\mathcal { I } , \\R ) \\end{align*}"}
+{"id": "1637.png", "formula": "\\begin{align*} \\mathcal { T } _ n \\ ; = \\ ; e ^ { \\lambda \\mathcal { P } _ n } \\mathcal { R } \\ ; \\in \\ ; \\textnormal { G L } ( \\mathsf { L } , \\mathbb { C } ) \\end{align*}"}
+{"id": "8103.png", "formula": "\\begin{align*} T _ 1 T _ 2 = B _ r M _ z \\otimes M _ z = r M _ z B _ r \\otimes M _ z = r ( M _ z B _ r \\otimes M _ z ) = r T _ 2 T _ 1 . \\end{align*}"}
+{"id": "2230.png", "formula": "\\begin{align*} { \\mathbf { s } _ n } : = \\frac { 1 } { h } \\left ( \\boldsymbol { \\theta } \\times \\left ( R _ \\ast ( \\boldsymbol { \\mathcal { T } } _ n ) \\right ) _ { 1 , 1 , : , : } \\right ) \\mathbf { e } _ 1 \\approx \\mathbf { \\hat { s } } , \\end{align*}"}
+{"id": "1336.png", "formula": "\\begin{align*} C \\varepsilon a \\ge \\left ( \\int _ { B _ { 9 / 1 0 } \\cap \\left \\{ u = 0 \\right \\} } \\left | \\ell ( x ) \\right | ^ { p ^ * } \\ , d x \\right ) ^ { 1 / p ^ * } \\ge c _ 1 a \\left | B _ { 9 / 1 0 } \\cap \\left \\{ u = 0 \\right \\} \\right | ^ { 1 / p ^ * } , \\end{align*}"}
+{"id": "4012.png", "formula": "\\begin{align*} \\bar { \\mathcal { M } } _ { \\beta } ( t ) \\stackrel { d } { = } \\sum _ { i = 1 } ^ { N _ { \\beta } ( t ) } X _ { i } , \\ t \\ge 0 , \\end{align*}"}
+{"id": "4177.png", "formula": "\\begin{align*} R \\big ( \\pi ^ * x ^ * , \\pi ^ * ( \\sigma ^ * ) ^ { - 1 } B ( y , \\delta ) \\big ) & = R \\big ( \\tau ^ * \\pi ^ * x ^ * , \\tau ^ * \\pi ^ * ( \\sigma ^ * ) ^ { - 1 } B ( y , \\delta ) \\big ) \\\\ & = R \\big ( \\pi x , \\pi B ( y , \\delta ) \\big ) \\\\ & \\supseteq R \\big ( \\pi x , B ( \\pi y , \\delta ' ) \\big ) . \\end{align*}"}
+{"id": "6952.png", "formula": "\\begin{align*} \\mathsf F _ { \\chi } ( z ) = \\frac { ( 1 + t ) ^ { - \\chi + 1 } } { 1 - N t } z = \\frac { t } { ( 1 + t ) ^ { N + 1 } } . \\end{align*}"}
+{"id": "5216.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { j _ 2 } C _ k & = \\left ( C _ 1 + C _ { j _ 2 } \\right ) + \\sum _ { k = 2 } ^ { j _ 2 - 1 } C _ j \\\\ & \\leq \\left ( 2 p _ 1 + L + \\sum _ { k = 1 } ^ { j _ 2 } p _ k + p _ { j _ 2 - 1 } + p _ { j _ 2 } + 2 L \\right ) + \\sum _ { k = 2 } ^ { j _ 2 - 1 } C _ j ^ { O P T _ f } \\\\ & \\leq 1 . 5 \\left ( C _ 1 ^ { O P T _ f } + C _ { j _ 2 } ^ { O P T _ f } \\right ) + \\sum _ { k = 2 } ^ { j _ 2 - 1 } C _ j ^ { O P T _ f } \\leq 1 . 5 \\sum _ { k = 1 } ^ { j _ 2 } C _ j ^ { O P T _ f } \\end{align*}"}
+{"id": "6298.png", "formula": "\\begin{align*} \\frac { 1 } { t } = \\big ( \\theta + ( t - \\theta ) \\big ) ^ { - 1 } = \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n \\cdot ( \\frac { 1 } { \\theta } ) ^ { n + 1 } \\cdot ( t - \\theta ) ^ n . \\end{align*}"}
+{"id": "3631.png", "formula": "\\begin{align*} f ' ( \\beta ) = d \\end{align*}"}
+{"id": "6419.png", "formula": "\\begin{align*} ( 2 + \\lambda ) F \\left ( V _ { p , q } ( \\xi ) - V _ { p , q } ( \\eta ) \\right ) = ( 2 + \\lambda ) F ( \\xi ) ^ { \\frac { p - q } { q } } F ( \\eta ) ^ { \\frac { p - q } { q } } F \\left ( F ( \\eta ) ^ { \\frac { q - p } { q } } \\xi - F ( \\xi ) ^ { \\frac { q - p } { q } } \\eta \\right ) . \\end{align*}"}
+{"id": "5529.png", "formula": "\\begin{align*} \\tau ^ 1 = \\widehat { \\bigotimes _ { 1 \\le i \\le n _ 1 } } \\tau ^ 1 _ i , \\quad \\tau ^ 2 = \\widehat { \\bigotimes _ { 1 \\le i \\le n _ 2 } } \\tau ^ 2 _ { i , 1 } \\hat { \\otimes } \\tau ^ 2 _ { i , 2 } , \\quad \\tau ^ 3 = \\widehat { \\bigotimes _ { 1 \\le i \\le n _ 3 } } \\tau ^ 3 _ i \\end{align*}"}
+{"id": "7768.png", "formula": "\\begin{align*} c a p _ p ( B _ t ) = \\frac { 1 } { 2 ^ n } ( \\frac { n } { p - 1 } ) ^ { p - 1 } \\omega _ n e ^ { n t } + o ( e ^ { n t } ) , \\ \\ t \\rightarrow + \\infty . \\end{align*}"}
+{"id": "1764.png", "formula": "\\begin{align*} { \\rm s i g n } \\left \\langle Z _ { * } , I \\backslash \\left \\{ i _ { j } + \\epsilon \\right \\} \\right \\rangle = \\left ( - 1 \\right ) ^ { i _ { j } + \\left ( 1 - \\epsilon \\right ) } , \\end{align*}"}
+{"id": "6911.png", "formula": "\\begin{align*} \\sum _ { ( d _ 1 , \\ldots , d _ N ) } q _ 1 ^ { d _ 1 } \\cdots q _ N ^ { d _ N } \\left [ h _ 1 ^ { d _ 1 } \\ldots h _ N ^ { d _ N } \\right ] \\left ( \\Phi _ 1 ( h _ 1 ) ^ { d _ 1 + 1 } \\cdots \\Phi _ N ^ { d _ N + 1 } ( h _ N ) \\cdot \\Psi ( h _ 1 , \\ldots , h _ N ) \\right ) = \\frac { \\Psi } { J } \\end{align*}"}
+{"id": "3740.png", "formula": "\\begin{align*} \\begin{bmatrix} n _ { 1 1 } & n _ { 1 2 } & n _ { 1 3 } \\\\ n _ { 2 1 } & n _ { 2 2 } & n _ { 2 3 } \\\\ n _ { 3 1 } & n _ { 3 2 } & n _ { 3 3 } \\end{bmatrix} \\begin{bmatrix} 2 & - 1 & - 1 \\\\ - 1 & 2 & - 1 \\\\ - 1 & - 1 & 2 \\end{bmatrix} & = \\begin{bmatrix} - 1 6 & 8 & 8 \\\\ 8 & - 1 6 & 8 \\\\ 8 & 8 & - 1 6 \\end{bmatrix} \\end{align*}"}
+{"id": "4998.png", "formula": "\\begin{align*} d = Q - 1 + \\phi ( r ^ s ) > \\phi ( r ^ s ) > \\frac { r ^ s } { 2 } > \\frac { 1 } { 2 } ( n / 2 ) ^ { s / ( s + 1 ) } > \\frac { 1 } { 8 } n ^ { 1 - \\epsilon / 3 } . \\end{align*}"}
+{"id": "947.png", "formula": "\\begin{align*} \\left | \\frac { \\partial ^ 2 \\varphi } { \\partial x _ \\beta \\partial x _ \\mu } ( x ' , \\rho ( x ' ) ) \\right | \\leq C \\delta \\sqrt { \\tilde { b } _ \\beta \\tilde { b } _ \\mu } , \\ \\beta , \\mu = 1 , \\ldots , n - 1 ; \\beta \\neq \\mu \\end{align*}"}
+{"id": "3147.png", "formula": "\\begin{align*} c ^ { ( w _ s , w _ t ) } _ { s , t } = \\lim _ { n \\to \\infty } \\min \\left \\{ w _ s \\ , \\frac { k _ s ( \\overline { G } ) } { { n \\choose s } } + w _ t \\ , \\frac { k _ t ( G ) } { { n \\choose t } } : | G | = n \\right \\} \\end{align*}"}
+{"id": "5633.png", "formula": "\\begin{align*} I _ \\gamma ( W , W ) & = \\int _ 0 ^ d \\Big [ g _ T ( D _ T ^ T W , D _ T ^ T W ) - g _ T ( W , W ) \\cdot \\mathbf { K } ( T , W ) \\Big ] d t \\end{align*}"}
+{"id": "7129.png", "formula": "\\begin{align*} A = ( d _ i , w _ i ) _ { i = 1 } ^ k , \\ A ' = ( d _ i , v _ i ) _ { i = 1 } ^ k . \\end{align*}"}
+{"id": "5420.png", "formula": "\\begin{align*} T ^ { k } = S ^ { k } - \\Pi , T S ^ { k } = T ^ { k + 1 } . \\end{align*}"}
+{"id": "2824.png", "formula": "\\begin{align*} X _ s ^ * & = \\left ( x - E _ 0 [ \\xi ] - \\frac { d } { \\gamma _ 0 } \\right ) \\frac { 1 + ( T - s ) \\rho } { 2 + T \\rho } + E _ s [ \\xi ] - \\int _ 0 ^ s \\frac { 1 + ( T - s ) \\rho } { 2 + ( T - r ) \\rho } d E _ r [ \\xi ] \\\\ & = \\left ( x - E _ 0 [ \\xi ] - \\frac { d } { \\gamma _ 0 } \\right ) \\frac { 1 + ( T - s ) \\rho } { 2 + T \\rho } + E _ 0 [ \\xi ] + \\int _ 0 ^ s \\left ( 1 - \\frac { 1 + ( T - s ) \\rho } { 2 + ( T - r ) \\rho } \\right ) d E _ r [ \\xi ] . \\end{align*}"}
+{"id": "6570.png", "formula": "\\begin{align*} \\hat { U } = \\hat { F } \\hat { E } . \\end{align*}"}
+{"id": "3658.png", "formula": "\\begin{align*} \\vec { y } = \\alpha \\cdot ( \\frac { x _ 1 + x _ 2 } { 2 } , 0 , 0 , \\frac { x _ 1 + x _ 2 } { 2 } , x _ 3 , \\ldots , x _ m ) + ( 1 - \\alpha ) \\cdot ( 0 , \\frac { x _ 1 + x _ 2 } { 2 } , \\frac { x _ 1 + x _ 2 } { 2 } , 0 , x _ 3 , \\ldots , x _ m ) . \\end{align*}"}
+{"id": "811.png", "formula": "\\begin{align*} \\mathbb { E } \\Big ( e ^ { - \\beta \\tau ^ { * } } \\Big ) = e ^ { - \\lambda _ { + } b ' } . \\end{align*}"}
+{"id": "1201.png", "formula": "\\begin{align*} \\pi _ { ( A , b ) } = \\pi _ { ( L , b ) } \\circ \\pi _ { ( U , 0 ) } \\circ \\pi _ { ( P , 0 ) } , \\end{align*}"}
+{"id": "8212.png", "formula": "\\begin{align*} \\begin{aligned} \\Vert \\Lambda ^ \\alpha ( u v ) - u \\Lambda ^ \\alpha v \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - r } _ { 2 , 1 } } \\lesssim _ { \\alpha , r , r _ 1 , N } \\Vert u \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 + \\alpha - r _ 1 } _ { 2 , 1 } } \\Vert v \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + r _ 1 - r } _ { 2 , 1 } } . \\end{aligned} \\end{align*}"}
+{"id": "1068.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { g } _ { D } ( v , w ) & : = \\mathcal { W } \\bigl ( \\hat v \\hat w D ^ { - n } \\bigr ) , \\\\ \\mathcal { G } _ { D } ( v , w ) & : = \\mathcal { W } \\bigl ( \\hat v ( D \\hat w + \\hat w D ) D ^ { - n + 1 } \\bigr ) \\\\ & \\ , \\ , = \\mathcal { W } \\bigl ( ( D \\hat v + \\hat v D ) \\hat w D ^ { - n + 1 } \\bigr ) , \\end{aligned} \\end{align*}"}
+{"id": "1366.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t ^ 2 u - \\Delta u + a ( x ) \\partial _ t u = 0 , & t > 0 , x \\in \\Omega , \\\\ u ( t , x ) = 0 , & t > 0 , x \\in \\partial \\Omega , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\ \\partial _ t u ( 0 , x ) = u _ 1 ( x ) , & x \\in \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "7248.png", "formula": "\\begin{align*} 0 < \\delta _ 2 = 2 - 2 \\beta < \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "5554.png", "formula": "\\begin{align*} u _ 1 ( \\alpha _ 0 ) = 1 , \\end{align*}"}
+{"id": "1317.png", "formula": "\\begin{align*} \\eta & < \\left \\{ \\frac { 1 } { 1 0 k } , \\frac { 1 } { ( 2 k + 2 ) ( 1 6 k ^ 2 ) } \\cdot \\left ( 1 6 k ^ 2 - 1 - \\frac { 4 ( 1 6 k ^ 3 - 1 ) } { 5 k } \\right ) \\right \\} \\\\ \\epsilon & < \\min \\left \\{ \\eta , \\frac { \\eta } { 2 } , \\frac { 1 } { 8 k ^ 3 } , \\frac { \\eta } { 3 2 k ^ 3 + 2 } \\right \\} \\\\ \\alpha & < \\min \\left \\{ \\eta , \\frac { \\epsilon ^ 2 } { 2 2 k } \\right \\} . \\end{align*}"}
+{"id": "7584.png", "formula": "\\begin{align*} E _ \\varphi ( F _ 2 ) = E _ { \\varphi _ m } ( F _ 2 ) & \\leq \\iint G ( p , q ) d \\mu _ 2 ( p ) d \\mu _ 2 ( q ) + \\int \\varphi _ m d \\mu _ 2 \\\\ & \\leq \\iint G ( p , q ) d \\mu _ 1 ( p ) d \\mu _ 1 ( q ) + \\int \\varphi _ m d \\mu _ 1 + \\varepsilon \\\\ & = E _ { \\varphi _ m } ( F _ 1 ) + \\varepsilon = E _ \\varphi ( F _ 1 ) + \\varepsilon . \\end{align*}"}
+{"id": "1012.png", "formula": "\\begin{align*} \\mu ^ * & = \\mu ^ { \\rm N } - \\varkappa ^ { \\rm N } , \\qquad \\ , \\varkappa = 2 \\varkappa ^ { \\rm N } , \\qquad \\lambda = \\lambda ^ { \\rm N } , \\\\ \\beta & = \\gamma ^ { \\rm N } - \\beta ^ { \\rm N } , \\ \\gamma = \\gamma ^ { \\rm N } + \\beta ^ { \\rm N } , \\ , \\qquad \\alpha = \\alpha ^ { \\rm N } \\ , . \\end{align*}"}
+{"id": "3169.png", "formula": "\\begin{align*} \\pi _ 1 ( \\Gamma , \\tau ) = \\left \\langle \\bigcup _ { v \\in V ( \\Gamma ) } G _ v \\cup ( t _ e ) _ { e \\in E ( \\Gamma ) } | \\bigcup _ { v \\in V ( \\Gamma ) } R _ v , \\bigcup _ { e \\in E ( \\Gamma ) } R _ e , R _ { \\tau } \\right \\rangle \\end{align*}"}
+{"id": "1692.png", "formula": "\\begin{align*} \\mathcal { Q } _ { \\mathrm { G i b b s } } ^ { f } ( X ^ 1 , \\ldots , X ^ m ; t _ 1 , \\ldots , t _ m ) : = \\int d \\mathbb { P } _ { \\mathrm { G i b b s } } ^ { f } ( \\varphi ) \\ , X ^ 1 ( S _ { t _ 1 } \\varphi ) \\ , \\cdots \\ , X ^ m ( S _ { t _ m } \\varphi ) \\ , , \\end{align*}"}
+{"id": "5174.png", "formula": "\\begin{align*} U _ { } = \\left \\{ x = ( x _ 1 , x _ 2 , x _ 3 ) \\ , : \\ , x _ 1 \\in ( 0 , 1 ) \\ , , \\ , \\sqrt { x _ 2 ^ 2 + x _ 3 ^ 2 } < 2 \\right \\} , \\end{align*}"}
+{"id": "3960.png", "formula": "\\begin{align*} \\bar { \\mathcal { M } } ( t ) = Y ( t ) . \\end{align*}"}
+{"id": "3769.png", "formula": "\\begin{align*} g ( \\Gamma / e ) = g ( \\Gamma ) . \\end{align*}"}
+{"id": "4068.png", "formula": "\\begin{align*} X ^ N _ S : = \\underbrace { X \\times _ S X \\times _ S \\ldots \\times _ S X } _ . \\end{align*}"}
+{"id": "5833.png", "formula": "\\begin{align*} \\| T \\| _ { b e r } = \\sup _ { \\lambda \\in \\Omega } \\| T \\hat { k } _ { \\lambda } \\| . \\end{align*}"}
+{"id": "7160.png", "formula": "\\begin{align*} \\sum _ { i \\leq a \\leq j } x _ { i , a } y _ { a , j } = \\sum _ { i \\leq a \\leq j } y _ { i , a } x _ { a , j } , \\end{align*}"}
+{"id": "1654.png", "formula": "\\begin{align*} \\mathcal { T } \\cdot ( W + v v ^ * ) = \\mathcal { T } \\cdot W + \\left [ ( ( \\mathcal { T } \\cdot W ) ^ { \\perp } \\mathcal { T } ) \\circ v \\right ] \\left [ ( ( \\mathcal { T } \\cdot W ) ^ { \\perp } \\mathcal { T } ) \\circ v \\right ] ^ * \\ , . \\end{align*}"}
+{"id": "6363.png", "formula": "\\begin{align*} T _ { \\sigma } ( \\lambda ) : = \\lambda N _ { d _ D } - N _ { \\sigma ^ { - 1 } ( 1 ) } \\cdots N _ { \\sigma ^ { - 1 } ( d _ D ) } = \\lambda N _ { d _ D } - N _ { \\sigma } . \\end{align*}"}
+{"id": "3992.png", "formula": "\\begin{align*} E ( \\hat { \\mathcal { M } } ( t ) ) = \\hat { r } _ { 1 } t , \\ \\operatorname { V a r } ( \\hat { \\mathcal { M } } ( t ) ) = \\hat { r } _ { 2 } t , \\ \\operatorname { C o v } ( \\hat { \\mathcal { M } } ( s ) , \\hat { \\mathcal { M } } ( t ) ) = \\hat { r } _ { 2 } \\min \\{ s , t \\} . \\end{align*}"}
+{"id": "6617.png", "formula": "\\begin{align*} - \\Delta f _ { n } ^ { + } = ( k ^ { 2 } - 4 ) f ^ { + } _ { n } & + ( 4 \\widetilde \\chi _ { n } ' ( y ) - \\widetilde \\chi _ { n } '' ( y ) ) \\chi _ { n } ( x ) e ^ { i k x - 2 y } \\\\ & - \\big ( 2 i k \\chi ' _ { n } ( x ) + \\chi '' _ { n } ( x ) \\big ) \\widetilde { \\chi } _ { n } ( y ) e ^ { i k x - 2 y } . \\end{align*}"}
+{"id": "6563.png", "formula": "\\begin{align*} \\abs { U ( t ) _ { a , b } } = 1 . \\end{align*}"}
+{"id": "3710.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } n \\mathbb { E } \\sum _ { i = 0 } ^ { n - 1 } ( \\triangle q _ i ^ b ) ^ 2 - n d = \\mathbb { E } \\int _ 0 ^ 1 ( b ^ 2 ( q ^ b _ t , t ) + \\nabla b ( q _ t ^ b , t ) ) d t \\quad \\end{align*}"}
+{"id": "7888.png", "formula": "\\begin{align*} \\mathcal { F } = \\{ f _ { y , r } \\colon ( y , r ) \\in Y \\times R \\} \\subset C ^ 2 ( I ) \\end{align*}"}
+{"id": "4236.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( b / a ) _ n a ^ n } { ( 1 - c q ^ n ) ( b ) _ n } = \\frac { ( b / c ) _ { \\infty } } { ( b ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( c ) _ n ( b / c ) ^ n } { ( q ) _ n } \\sum _ { m = 1 } ^ { \\infty } \\frac { a ^ m - b ^ m } { 1 - c q ^ { m + n } } . \\end{align*}"}
+{"id": "979.png", "formula": "\\begin{align*} \\tau ( x ) \\omega : = \\omega + \\Lambda x - \\left \\lfloor \\omega + \\Lambda x \\right \\rfloor \\end{align*}"}
+{"id": "5183.png", "formula": "\\begin{align*} \\| \\nabla u \\| ^ { p } _ { L ^ p ( \\Omega ) } \\lesssim \\sum _ { Q _ i } \\sum _ { Q _ j \\cap Q _ i \\neq \\emptyset } \\ell ( Q _ i ) ^ { 1 - p } P ( A , Q _ i \\cup Q _ j ) \\lesssim \\sum _ { Q _ i } \\ell ( Q _ i ) ^ { 1 - p } P ( A , Q _ i ) . \\end{align*}"}
+{"id": "955.png", "formula": "\\begin{align*} m ^ { - 1 / 2 ( k - 1 ) } \\leq ( b _ \\alpha ^ { k - 1 } ) ^ { - 1 / 2 ( k - 1 ) } = b _ \\alpha ^ { - 1 / 2 } . \\end{align*}"}
+{"id": "1211.png", "formula": "\\begin{align*} P = \\begin{bmatrix} P _ 1 & & & & 0 \\\\ & 1 & & & \\\\ & & 1 & & \\\\ & & & 1 & \\\\ 0 & & & & P _ 5 \\\\ \\end{bmatrix} \\end{align*}"}
+{"id": "5099.png", "formula": "\\begin{align*} \\tilde { E } f = \\int _ { B _ 1 ^ { d - n } } \\int _ { B _ 1 ^ { n } } f ( \\xi _ 1 , \\xi _ 2 ) e ^ { i ( x \\cdot \\xi _ 1 + y \\cdot \\xi _ 2 + t ( \\frac { \\mathfrak { m } ( N \\xi _ 1 ) } { N ^ 2 \\mathfrak { m } ( N ) } + \\frac { \\mathfrak { m } ( N \\xi _ 2 ) } { N ^ 2 \\mathfrak { m } ( N ) } ) ) } \\ , d \\xi _ 1 d \\xi _ 2 . \\end{align*}"}
+{"id": "4276.png", "formula": "\\begin{align*} F _ d ( q ) = \\frac { 1 } { ( q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { n q ^ n ( d q ) _ { n - 1 } } { ( q ) _ n } . \\end{align*}"}
+{"id": "6179.png", "formula": "\\begin{align*} \\sum _ { \\substack { ( n , \\ , a _ { 2 } ) \\ , = \\ , 1 \\\\ [ 1 p t ] \\omega ( n ) \\ , \\le \\ , \\lambda \\\\ P ( n ) \\ , \\ge \\ , y _ u } } \\frac { \\rho _ { \\lambda } ( n ) } { \\varphi \\big ( \\l ( n ) \\big ) } \\le 2 C _ u ( \\lambda + 1 ) \\Gamma ( \\lambda + 1 , \\log w _ u ) \\le 2 C _ u ( \\lambda + 1 ) ! . \\end{align*}"}
+{"id": "5639.png", "formula": "\\begin{align*} I _ \\gamma ( W , W ) & = \\int _ 0 ^ d [ g _ T ( D _ T ^ T W , D _ T ^ T W ) - g _ T ( W , W ) \\mathbf { K } ( T , W ) ] d t \\\\ & = \\int _ 0 ^ d [ g _ T ( D _ T ^ T W , D _ T ^ T W ) - g _ T ( W , W ) \\mathbf { H } ( T _ o ) ] d t \\\\ & \\leq \\int _ 0 ^ d [ ( s _ \\lambda ' ( t ) ) ^ 2 - \\lambda ( s _ \\lambda ( t ) ) ^ 2 ] d t = \\frac { 1 } { 2 } \\sqrt { \\lambda } \\sin ( 2 \\sqrt { \\lambda } d ) = 0 . \\end{align*}"}
+{"id": "5281.png", "formula": "\\begin{align*} \\begin{cases} & \\sum _ { j = 1 } ^ { n + 1 } k _ { i \\ * ( n + 1 ) + j } = 0 , \\ \\forall \\ 0 \\leq i \\leq m - 1 , \\\\ & \\sum _ { i \\in B } k _ i = 0 , \\ \\ \\forall B \\in \\pi . \\end{cases} \\end{align*}"}
+{"id": "729.png", "formula": "\\begin{align*} { f _ { 1 7 } } = { f _ { 1 8 } } - f _ { 1 8 } ^ { \\left ( { e q } \\right ) } + f _ { 1 7 } ^ { \\left ( { e q } \\right ) } + \\delta y - \\delta z \\end{align*}"}
+{"id": "2805.png", "formula": "\\begin{align*} \\left \\| r ^ \\mathrm { G S V D } \\right \\| _ 2 = \\sqrt { \\left \\| \\tilde { s } _ i ^ 2 A ^ T \\tilde { u } _ i ^ A - \\tilde { c } _ i B ^ T B \\tilde { g } _ i \\right \\| _ 2 ^ 2 + \\left \\| \\tilde { c } _ i ^ 2 B ^ T \\tilde { u } _ i ^ B - \\tilde { s } _ i A ^ T A \\tilde { g } _ i \\right \\| _ 2 ^ 2 } . \\end{align*}"}
+{"id": "4704.png", "formula": "\\begin{align*} M _ p ( f _ n - f ; r ) ^ p = \\lim _ { m \\rightarrow \\infty } \\int _ { - \\pi } ^ { \\pi } | f _ n ( r e ^ { i \\theta } ) - f _ m ( r e ^ { i \\theta } ) | ^ p \\ , d m _ { \\lambda } ( \\theta ) \\le \\liminf _ { m \\rightarrow \\infty } \\| f _ n - f _ m \\| ^ p _ { H _ { \\lambda } ^ { p } } , \\end{align*}"}
+{"id": "210.png", "formula": "\\begin{align*} \\Psi ( s ) = \\left \\{ \\begin{array} { l } 1 , \\ , \\ , \\ , \\ , - \\infty < s \\leq 0 , \\\\ 0 , \\ , \\ , \\ , \\ , s \\geq 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "7058.png", "formula": "\\begin{align*} [ \\chi ] _ { q } = \\frac { 1 - q ^ { \\chi } } { 1 - q } , \\chi \\in \\Bbb { N } _ { 0 } . \\end{align*}"}
+{"id": "7842.png", "formula": "\\begin{align*} \\langle \\widetilde J ( \\alpha _ x ) , \\nu \\rangle = { \\bf i } _ { \\nu _ { T ^ * \\ ! \\widetilde Q } } \\Theta _ { \\mbox { \\tiny { $ \\widetilde Q $ } } } { \\mbox { \\tiny { $ ( \\alpha _ x ) $ } } } , \\end{align*}"}
+{"id": "5585.png", "formula": "\\begin{align*} u = u ^ i \\frac { \\partial } { \\partial x ^ i } = J ^ i _ k y ^ k \\frac { \\partial } { \\partial x ^ i } . \\end{align*}"}
+{"id": "6865.png", "formula": "\\begin{align*} & ( \\beta \\lambda u _ 1 h ( a _ 1 ) f ^ { p ^ { e ' } } ( a _ 1 ) , \\dots , \\beta \\lambda u _ s h ( a _ s ) f ^ { p ^ { e ' } } ( a _ s ) , \\lambda u _ { s + 1 } h ( a _ { s + 1 } ) f ^ { p ^ { e ' } } ( a _ { s + 1 } ) , \\dots , \\lambda u _ n h ( a _ n ) f ^ { p ^ { e ' } } ( a _ n ) ) \\\\ = & ( u _ 1 g ( a _ 1 ) , \\dots , u _ s g ( a _ s ) , u _ { s + 1 } g ( a _ { s + 1 } ) , \\dots , u _ n g ( a _ n ) ) . \\end{align*}"}
+{"id": "7629.png", "formula": "\\begin{align*} Q _ { 1 , \\lambda } = x _ 0 ^ 3 + x _ 1 ^ 3 + x _ 2 ^ 3 - 3 \\lambda x _ 3 x _ 4 x _ 5 , Q _ { 2 , \\lambda } = x _ 3 ^ 3 + x _ 4 ^ 3 + x _ 5 ^ 3 - 3 \\lambda x _ 0 x _ 1 x _ 2 . \\end{align*}"}
+{"id": "1311.png", "formula": "\\begin{align*} \\varepsilon ^ \\dagger = \\sigma _ { a ^ \\ast , a } \\circ \\eta , \\end{align*}"}
+{"id": "5235.png", "formula": "\\begin{align*} \\Omega _ { \\tau , t - s } = \\sum _ { i = 0 } ^ { f - 1 } p ^ i \\left ( a _ { \\tau \\circ \\varphi ^ i } + e - 2 s _ { \\tau \\circ \\varphi ^ i } \\right ) = \\sum _ { i = 0 } ^ { f - 1 } p ^ i ( a _ { \\tau \\circ \\varphi ^ i } + 1 ) ( - 1 ) ^ { \\tau \\circ \\varphi ^ i \\in J } + \\sum _ { i = 0 } ^ { f - 1 } p ^ i ( e - 1 - 2 x _ { \\tau \\circ \\varphi ^ i } ) , \\end{align*}"}
+{"id": "8035.png", "formula": "\\begin{align*} i _ m ( a ) & = \\begin{cases} ( a , 0 , u _ m ( a ) ) , & 0 \\leq m \\leq r , \\\\ a , & m > r \\end{cases} \\quad \\\\ p _ m ( a , b ' , b ) & = \\begin{cases} b , & 0 \\leq m \\leq r , \\\\ u _ m ( a ) , & m > r . \\end{cases} \\end{align*}"}
+{"id": "6046.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta p - k ^ 2 p = 2 ( \\eta - \\eta _ 0 ) - 2 \\Delta \\eta + 2 \\nabla \\cdot A \\ , \\ , \\ , \\ , \\Omega \\\\ [ 0 . 3 c m ] p = 0 \\ , \\ , \\ , \\ , \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "8027.png", "formula": "\\begin{align*} S ( m ) = \\begin{bmatrix} A & B ^ * \\\\ B & S ( m - 1 ) \\otimes I _ d \\ : \\end{bmatrix} . \\end{align*}"}
+{"id": "4938.png", "formula": "\\begin{align*} t y + \\sum _ { i = 1 } ^ { ( k - 5 ) / 2 - t } i b _ i - \\left ( \\frac { k - 3 } { 2 } + t \\right ) \\cdot \\sum _ { i = 1 } ^ { ( k - 5 ) / 2 - t } i a _ i = 0 . \\end{align*}"}
+{"id": "184.png", "formula": "\\begin{align*} Q ^ + _ A \\psi = 0 , \\ , \\ , \\ , \\ , F ^ + _ A = \\rho ^ { - 1 } ( \\mu ( \\psi ) ) , \\end{align*}"}
+{"id": "1352.png", "formula": "\\begin{align*} a _ 0 : = \\frac { M } 2 { \\mbox { a n d } } a _ 1 : = \\eta ^ { - n / p } \\big ( C ( \\eta ) M + a ( 1 ) \\big ) . \\end{align*}"}
+{"id": "6421.png", "formula": "\\begin{align*} \\lambda ( \\mu x , \\mu ^ s \\xi ) = \\mu \\lambda ( x , \\xi ) \\end{align*}"}
+{"id": "401.png", "formula": "\\begin{align*} \\pi \\circ \\Psi _ t = \\psi _ t \\circ \\pi . \\end{align*}"}
+{"id": "1478.png", "formula": "\\begin{align*} C _ R ^ { [ 2 ] } & = \\frac { N } { \\ell _ r ^ * } , C _ W ^ { [ 2 ] } = \\frac { \\gamma _ w \\times N } { \\gamma _ w \\times \\ell _ w ^ * } = \\frac { N } { \\ell _ w ^ * } . \\end{align*}"}
+{"id": "7782.png", "formula": "\\begin{align*} \\begin{aligned} 1 & = \\int _ 0 ^ \\varepsilon f ' ( x ) d x = \\int _ 0 ^ \\varepsilon f ' ( x ) T ^ { \\frac { 1 } { p } } ( x ) T ^ { - \\frac { 1 } { p } } ( x ) d x \\\\ & \\leq ( \\int _ 0 ^ \\varepsilon [ f ' ( x ) ] ^ p T ( x ) d x ) ^ { \\frac { 1 } { p } } ( \\int _ 0 ^ \\varepsilon T ^ { \\frac { 1 } { 1 - p } } ( x ) d x ) ^ { 1 - \\frac { 1 } { p } } \\end{aligned} \\end{align*}"}
+{"id": "2689.png", "formula": "\\begin{align*} \\binom { a } { b } \\binom { b } { c } = \\binom { a } { c } \\binom { a - c } { b - c } \\end{align*}"}
+{"id": "106.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ \\Psi u = f ( u ) & A ( r _ { 0 } , + \\infty ) \\\\ u \\equiv c _ { 0 } & \\{ r _ { 0 } \\} \\times N . \\end{cases} \\end{align*}"}
+{"id": "5390.png", "formula": "\\begin{align*} \\L _ { \\overline { \\rho } ( \\xi ) } \\boldsymbol { E L } = d \\langle \\overline { \\boldsymbol { \\mu } } , \\xi \\rangle , \\overline { \\boldsymbol { \\mu } } \\approx 0 \\end{align*}"}
+{"id": "4401.png", "formula": "\\begin{align*} \\Delta _ t ^ r f ( x ) = ( I - T ^ t ) ^ r f = \\sum _ { k = 0 } ^ r ( - 1 ) ^ { r - k } \\binom { r } { k } ( T ^ t ) ^ k f ( x ) , \\end{align*}"}
+{"id": "2206.png", "formula": "\\begin{align*} & R _ { 1 2 1 2 } = R _ { 1 3 1 3 } = R _ { 2 3 2 3 } = - a ^ 2 , \\\\ & R _ { 1 2 1 3 } = R _ { 1 2 2 3 } = R _ { 1 3 2 3 } = 0 , \\end{align*}"}
+{"id": "6809.png", "formula": "\\begin{align*} \\sigma ( x _ { [ 5 ] } ) = \\sum _ { i \\in [ q ] , j \\in [ t ] } & \\lambda _ { i j } \\Big ( \\sum _ { \\{ a , b , c \\} \\in \\binom { [ 5 ] } { 3 } } \\tau _ { i } ( x _ a , x _ b , x _ c ) \\tilde { \\gamma } _ i ( x _ { [ 5 ] \\setminus \\{ a , b , c \\} } ) \\Big ) . \\end{align*}"}
+{"id": "5735.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } x _ i x _ j \\quad g ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } B _ { i j } x _ i x _ j , \\end{align*}"}
+{"id": "2000.png", "formula": "\\begin{align*} \\aligned \\Psi ( t ) & : = 4 ( e ^ { t / 2 } + e ^ { - t / 2 } - 2 ) - \\sum _ { n \\leq e ^ t } \\frac { \\Lambda ( n ) } { \\sqrt { n } } ( t - \\log n ) \\\\ & + \\frac { t } { 2 } \\left [ \\frac { \\Gamma ' } { \\Gamma } \\left ( \\frac { 1 } { 4 } \\right ) - \\log \\pi \\right ] + \\frac { 1 } { 4 } \\left ( C - e ^ { - t / 2 } \\Phi ( e ^ { - 2 t } , 2 , 1 / 4 ) \\right ) , \\endaligned \\end{align*}"}
+{"id": "6488.png", "formula": "\\begin{align*} M = \\sup _ { x \\in \\Omega } u ( x ) - v ( x ) \\end{align*}"}
+{"id": "1152.png", "formula": "\\begin{align*} \\sum _ { \\lambda , \\mu } \\exp \\Big ( \\frac { - \\pi ( u ^ 2 + v ^ 2 ) } { v } S [ \\lambda ] - \\frac { \\pi } { v } S [ \\mu ] - \\frac { 2 \\pi } { v } u \\lambda ^ t S \\mu \\Big ) = \\sum _ { \\lambda , \\mu } \\exp \\Big ( - \\pi v S [ \\lambda ] - \\frac { \\pi } { v } S [ u \\lambda + \\mu ] \\Big ) . \\end{align*}"}
+{"id": "3342.png", "formula": "\\begin{gather*} L _ 2 ( L _ 1 \\otimes 1 ) \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } = ( L _ 1 \\otimes 1 ) L _ 2 \\psi _ { n _ 1 } ^ { n _ 2 } \\otimes \\psi _ { n _ 2 } . \\end{gather*}"}
+{"id": "1258.png", "formula": "\\begin{align*} u _ p ( R ) = \\left \\{ \\frac { 1 } { \\lambda } \\left [ \\frac { A } { N - 1 } + \\gamma \\right ] \\right \\} ^ { \\frac { 1 } { p - 1 } } \\end{align*}"}
+{"id": "7911.png", "formula": "\\begin{align*} C ^ n _ \\mathrm { m R B A } ( ( A , R ) , ( M , S ) ) = \\begin{cases} M & n = 0 , \\\\ C ^ n ( A , M ) \\oplus C ^ { n - 1 } ( A _ R , \\widetilde { M } ) = \\mathrm { H o m } ( A ^ { \\otimes n } , M ) \\oplus \\mathrm { H o m } ( A ^ { \\otimes n - 1 } , { M } ) & n \\geq 1 . \\end{cases} \\end{align*}"}
+{"id": "5500.png", "formula": "\\begin{align*} F _ N ( a , b ; t ) = \\frac { ( 1 - b ) ( 1 - t q ^ { N + 1 } ) } { ( 1 - t ) ( 1 - b q ^ { N + 1 } ) } + \\frac { ( 1 - q ^ { N + 1 } ) ( b - a t q ) } { ( 1 - b q ^ { N + 1 } ) ( 1 - t ) } F _ N ( a , b ; t q ) . \\end{align*}"}
+{"id": "1393.png", "formula": "\\begin{align*} \\mathcal { A } _ D = \\begin{pmatrix} 0 & 1 \\\\ \\Delta & - a ( x ) \\end{pmatrix} \\end{align*}"}
+{"id": "1540.png", "formula": "\\begin{align*} F ^ { \\prime } \\left ( g _ { s , 3 , 1 } \\right ) = & F \\left ( g _ { s , 3 , 1 } \\circ g _ { e , 1 , 3 } \\right ) = F \\left ( g _ { s , 3 , 1 } \\circ g _ { e , 2 , 3 } \\circ g _ { e , 1 , 2 } \\right ) \\\\ = & F \\left ( g _ { e , 3 , 1 } \\circ g _ { s , 2 , 3 } \\circ g _ { e , 1 , 2 } \\right ) = F ^ { \\prime } \\left ( g _ { s , 2 , 3 } \\right ) . \\end{align*}"}
+{"id": "2562.png", "formula": "\\begin{align*} \\widetilde k ( t , x , y ) : = \\rho ( - R ^ { - 1 } B P _ t \\eta _ t ^ { - 1 } x - R ^ { - 1 } B \\eta _ t y ) \\quad \\end{align*}"}
+{"id": "865.png", "formula": "\\begin{align*} P ( z , t ) = \\sum _ { n = 0 } ^ { \\infty } z ^ { n } p _ { n } ( t ) , & Q ( z , t ) = \\sum _ { n = 0 } ^ { \\infty } z ^ { n } q _ { n } ( t ) \\ , , \\\\ P ^ { * } ( z , t ) = \\sum _ { n = 0 } ^ { \\infty } z ^ { n } p _ { n } ^ { * } ( t ) , \\qquad & Q ^ { * } ( z , t ) = \\sum _ { n = 0 } ^ { \\infty } z ^ { n } q _ { n } ^ { * } ( t ) \\ , , \\end{align*}"}
+{"id": "802.png", "formula": "\\begin{align*} \\beta v ( x ) - \\alpha x v ' ( x ) - \\frac { 1 } { 2 } \\sigma ^ 2 x ^ 2 v '' ( x ) = 0 , \\end{align*}"}
+{"id": "5270.png", "formula": "\\begin{align*} \\dim L _ { \\pi } = n + 1 - | \\pi | . \\end{align*}"}
+{"id": "4523.png", "formula": "\\begin{align*} \\mathcal { V } = \\frac { 1 } { 2 } \\int _ 0 ^ 1 ( \\Phi _ x ' ) ^ 2 \\mathrm { d } x \\ , , \\end{align*}"}
+{"id": "4785.png", "formula": "\\begin{align*} \\pi ( x ) p _ n ' ( x ) = \\alpha _ n p _ { n + 1 } + \\beta _ n p _ n + \\gamma _ n p _ { n - 1 } , \\end{align*}"}
+{"id": "2270.png", "formula": "\\begin{align*} \\alpha ^ b & : = \\alpha _ 1 ^ { b _ 1 } \\alpha _ 2 ^ { b _ 2 } \\dots \\alpha _ n ^ { b _ n } \\in L _ n , \\\\ \\alpha ^ M & : = ( \\alpha ^ { M [ - , 1 ] } , \\alpha ^ { M [ - , 2 ] } , \\dots , \\alpha ^ { M [ - , n ] } ) \\in L _ n ^ n , \\\\ \\alpha \\beta & : = ( \\alpha _ 1 \\beta _ 1 , \\alpha _ 2 \\beta _ 2 , \\dots , \\alpha _ n \\beta _ n ) \\in L _ n ^ n . \\end{align*}"}
+{"id": "4298.png", "formula": "\\begin{align*} C _ 1 ( q , N ) & = \\lim _ { z \\to 1 } z \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n + 1 } ( q ) _ n q ^ { n ( n + 1 ) / 2 } } { ( q ) _ { n + N } } \\frac { 1 - q ^ n } { ( 1 - z q ^ n ) ^ 2 } \\\\ & = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n + 1 } ( q ) _ n q ^ { n ( n + 1 ) / 2 } } { ( q ) _ { n + N } ( 1 - q ^ n ) } . \\end{align*}"}
+{"id": "3393.png", "formula": "\\begin{align*} 0 & = \\sum _ { i = 0 } ^ \\infty u ( ( m + j + 1 ) ^ 2 + j - i ^ 2 , - j - 1 + i ) \\\\ & = \\sum _ { i = 0 } ^ \\infty u ( ( m + j + 1 ) ^ 2 + j - ( i + j + 1 ) ^ 2 , i ) \\\\ & = \\sum _ { i = 0 } ^ m u ( ( m + j + 1 ) ^ 2 + j - ( i + j + 1 ) ^ 2 , i ) . \\end{align*}"}
+{"id": "7760.png", "formula": "\\begin{align*} V ^ { - 1 } d r ^ 2 = d t ^ 2 , \\ \\ i . e . \\ \\ t = \\int _ { r _ m } ^ r [ V ( \\tau ) ] ^ { - 1 / 2 } d \\tau . \\end{align*}"}
+{"id": "3183.png", "formula": "\\begin{align*} & A ( u ) : = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } | \\nabla u | ^ { 2 } d x \\ \\ \\ \\ \\ \\ \\ \\ B ( u ) : = \\frac { 1 } { 4 } \\int _ { \\mathbb { R } ^ { 3 } } \\phi _ { u } | u | ^ { 2 } d x \\\\ & C ( u ) : = - \\frac { 1 } { p } \\int _ { \\mathbb { R } ^ { 3 } } | u | ^ { p } d x \\ \\ \\ \\ \\ \\ \\ \\ T ( u ) : = B ( u ) + C ( u ) \\end{align*}"}
+{"id": "5526.png", "formula": "\\begin{align*} \\dim H _ l ( H , \\rho _ j \\otimes \\chi ^ { \\vee } ) = \\dim \\varprojlim _ k H _ l ( H , ( \\rho _ j / \\rho _ { j , k } ) \\otimes \\chi ^ { \\vee } ) \\le \\sum _ { k = 0 } ^ { \\infty } d _ { l , j , k } \\end{align*}"}
+{"id": "3682.png", "formula": "\\begin{align*} | L ( v ) | / n ^ 2 & \\le \\binom { t } { 2 } \\left ( \\frac { 1 } { t + 2 } + \\epsilon \\right ) ^ 2 + 2 \\times 1 0 2 t ^ 2 \\epsilon ^ { 1 / 4 } + 2 \\epsilon ^ { 1 / 4 } \\\\ & = \\frac { t ( t + 1 ) } { 2 ( t + 2 ) ^ 2 } - \\frac { t } { ( t + 2 ) ^ 2 } + \\frac { t ( t - 1 ) } { t + 2 } \\epsilon + \\frac { t ( t - 1 ) } { 2 } \\epsilon ^ 2 + 2 0 4 t ^ 2 \\epsilon ^ { 1 / 4 } + 2 \\epsilon ^ { 1 / 4 } \\\\ & < 3 ( \\lambda - \\delta ) , \\end{align*}"}
+{"id": "4165.png", "formula": "\\begin{align*} | m + j | ^ { - 4 } & \\ge ( | m | + | j | ) ^ { - 4 } & & = | m | ^ { - 4 } ( 1 + | j | / | m | ) ^ { - 4 } \\\\ & \\ge | m | ^ { - 4 } ( 1 - 4 | j | / | m | ) & & \\ge | m | ^ { - 4 } ( 1 - 2 ^ { 2 + r } | m | ^ { r - 1 } ) . \\end{align*}"}
+{"id": "5838.png", "formula": "\\begin{align*} \\omega ( U ) = \\| U \\| = 1 , ~ ~ c ( U ) = 0 . \\end{align*}"}
+{"id": "3142.png", "formula": "\\begin{align*} c _ { s , t } = \\lim _ { n \\to \\infty } \\min \\left \\{ \\frac { k _ s ( \\overline { G } ) } { { n \\choose s } } + \\frac { k _ t ( G ) } { { n \\choose t } } : | G | = n \\right \\} . \\end{align*}"}
+{"id": "3313.png", "formula": "\\begin{align*} \\big \\{ \\phi _ { m _ 2 ( j ) } ^ { m _ 1 } \\big \\} _ { j = 0 } ^ { N _ 2 } , \\end{align*}"}
+{"id": "4731.png", "formula": "\\begin{gather*} \\partial _ k ^ * ( a ^ * ) \\circ b ^ * = a ^ * \\circ \\eth _ k ^ * ( b ^ * ) + \\partial _ k ^ * ( a ^ * \\circ b ^ * ) , \\\\ a ^ * \\circ \\partial _ k ^ * ( b ^ * ) = \\eth _ k ^ * ( a ^ * ) \\circ b ^ * + \\partial _ k ^ * ( a ^ * \\circ b ^ * ) , \\forall k = 1 , \\dots , m , \\end{gather*}"}
+{"id": "1414.png", "formula": "\\begin{align*} \\left ( \\mathcal { A } \\begin{pmatrix} u \\\\ v \\end{pmatrix} , \\begin{pmatrix} u \\\\ v \\end{pmatrix} \\right ) _ { \\mathcal { H } } \\le \\left \\| ( u , v ) \\right \\| _ { \\mathcal { H } } ^ 2 \\end{align*}"}
+{"id": "7555.png", "formula": "\\begin{align*} Z _ + = Z _ - \\begin{pmatrix} \\sigma & 0 \\\\ 0 & \\sigma \\end{pmatrix} , \\end{align*}"}
+{"id": "1937.png", "formula": "\\begin{align*} \\lambda ^ { 1 / 2 } ( | u _ 0 | ^ p ) ^ { 1 / p } _ { Q _ { \\nu r } ( z _ 0 ) } + ( | D _ v u _ 0 | ^ p ) ^ { 1 / p } _ { Q _ { \\nu r } ( z _ 0 ) } & \\le N \\sum _ { k = 0 } ^ { \\infty } 2 ^ { - k ^ 2 / 8 } F _ k ( 1 ) , \\\\ \\lambda ^ { 1 / 2 } ( | u _ 0 | ^ p ) ^ { 1 / p } _ { Q _ { r } ( z _ 0 ) } + ( | D _ v u _ 0 | ^ p ) ^ { 1 / p } _ { Q _ { r } ( z _ 0 ) } & \\le N \\nu ^ { ( 4 d + 2 ) / p } \\sum _ { k = 0 } ^ { \\infty } 2 ^ { - k ^ 2 / 8 } F _ k ( 1 ) . \\end{align*}"}
+{"id": "2163.png", "formula": "\\begin{align*} & R _ { i j } = - R _ { j i } , \\overline { R } _ { i j } = - \\overline { R } _ { j i } , \\\\ & S _ { i j } = - S _ { j i } , S _ { i j } S _ k + S _ { j k } S _ i + S _ { k i } S _ j = 0 \\end{align*}"}
+{"id": "2729.png", "formula": "\\begin{align*} \\sigma ( \\left ( z _ 1 , z _ 2 , \\dots , z _ k \\right ) ) & = \\left ( z _ 1 + 1 , z _ 2 , \\dots , z _ k \\right ) \\\\ s _ i ( \\left ( z _ 1 , z _ 2 , \\dots , z _ k \\right ) ) & = \\left ( z _ { \\tau _ i ( 1 ) } , z _ { \\tau _ i ( 2 ) } , \\dots , z _ { \\tau _ i ( k ) } \\right ) \\end{align*}"}
+{"id": "1033.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } ( t _ { k _ 0 + k } - t _ { k _ 0 } ) = \\lim _ { k \\to \\infty } \\sum _ { i = k _ 0 + 1 } ^ { k _ 0 + k } h _ i \\le \\lim _ { k \\to \\infty } \\frac { r } { c } \\sum _ { i = k _ 0 + 1 } ^ { k _ 0 + k } \\frac { 1 } { \\dot { \\alpha } ( t _ { i - 1 } ) } . \\end{align*}"}
+{"id": "8150.png", "formula": "\\begin{align*} \\widehat { \\mu } ^ { \\mathrm { a s y } } ( \\overline M ) = \\frac { ( \\overline M ^ { d + 1 } ) _ S } { ( M ^ d ) ( \\dim Y + 1 ) } \\quad \\widehat { \\mu } ^ { \\mathrm { a s y } } ( \\pi ^ * ( \\overline M ) ) = \\frac { ( \\pi ^ * ( \\overline M ) ^ { d + 1 } ) _ S } { ( \\pi ^ * ( M ) ^ d ) ( \\dim X + 1 ) } , \\end{align*}"}
+{"id": "6094.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\omega } | \\chi ( g ) - _ n ( \\phi _ n ( g ) ) | = 0 . \\end{align*}"}
+{"id": "3977.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\beta } ( t ) \\stackrel { d } { = } \\sum _ { i = 1 } ^ { N ( Y _ { \\beta } ( t ) ) } X _ { i } \\stackrel { d } { = } \\sum _ { i = 1 } ^ { N _ { \\beta } ( t ) } X _ { i } , \\ t \\ge 0 , \\end{align*}"}
+{"id": "7332.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } & \\mathbb { E } \\left [ \\int _ { 0 } ^ { T } \\mathcal { H } ( t , X ^ { m } ( t ) , p ^ { m } ( t ) , q ^ { m } ( t ) , P ^ { m } ( t ) , v ^ { m } ( t ) , u ^ { m } ( t ) ) d t \\right ] \\\\ = & \\inf \\limits _ { u ( \\cdot ) \\in \\mathcal { U } [ 0 , T ] } \\mathbb { E } \\left [ \\int _ { 0 } ^ { T } \\mathcal { H } ( t , X ^ { m } ( t ) , p ^ { m } ( t ) , q ^ { m } ( t ) , P ^ { m } ( t ) , u ( t ) , u ^ { m } ( t ) ) d t \\right ] . \\end{array} \\end{align*}"}
+{"id": "1439.png", "formula": "\\begin{align*} u ( t ) = u _ 0 + \\int _ 0 ^ t \\partial _ t u ( s ) \\ , d s , \\end{align*}"}
+{"id": "5620.png", "formula": "\\begin{align*} \\mathbf { H } ( v ) = \\frac { 2 } { G ^ 2 } R _ { \\alpha \\bar { \\beta } ; \\mu \\bar { \\nu } } v ^ { \\alpha } v ^ { \\bar { \\beta } } v ^ { \\mu } v ^ { \\bar { \\nu } } \\end{align*}"}
+{"id": "785.png", "formula": "\\begin{align*} u ( c , C ) = \\frac { ( c / C ) ^ { 1 - \\alpha } } { 1 - \\alpha } \\end{align*}"}
+{"id": "6638.png", "formula": "\\begin{align*} r ( v , v ) = \\int _ { \\Omega _ - } | \\nabla v | ^ 2 \\dd x - \\big ( 1 + \\frac { 1 } { \\varepsilon } \\big ) \\int _ { \\Gamma _ \\theta } | v | ^ 2 \\dd \\sigma , D ( r ) = H ^ 1 ( \\Omega _ - ) . \\end{align*}"}
+{"id": "4733.png", "formula": "\\begin{gather*} ( \\phi + \\psi ^ * ) ( a + a ^ * ) \\star ( \\phi + \\psi ^ * ) ( b + b ^ * ) = ( \\phi + \\psi ^ * ) ( ( a + a ^ * ) \\star ( b + b ^ * ) ) , \\end{gather*}"}
+{"id": "5880.png", "formula": "\\begin{align*} \\begin{aligned} x ( t _ n + h ) & = x ( t _ n ) + h \\varphi _ 1 ( h M ) v ( t _ n ) + h ^ 2 \\int _ { 0 } ^ 1 ( 1 - \\tau ) \\varphi _ 1 ( ( 1 - \\tau ) h M ) F ( x ( t _ n + h \\tau ) ) d \\tau , \\\\ v ( t _ n + h ) & = \\varphi _ { 0 } ( h M ) v ( t _ n ) + h \\int _ { 0 } ^ 1 \\varphi _ { 0 } ( ( 1 - \\tau ) h M ) F ( x ( t _ n + h \\tau ) ) d \\tau , \\end{aligned} \\end{align*}"}
+{"id": "1759.png", "formula": "\\begin{align*} w _ { k , m } = \\sum _ { I } { \\rm s i g n } \\left \\langle I \\right \\rangle \\times w _ { I } ^ { e l e } , \\end{align*}"}
+{"id": "2318.png", "formula": "\\begin{align*} \\mathcal L _ \\infty ( b ) = \\int _ { \\R ^ d } \\left ( b ^ \\top ( x ) a _ 0 ^ { - 1 } ( x ) \\left ( b ( x ) - 2 b _ 0 ( x ) \\right ) \\right ) \\rho _ 0 ( x ) \\d x . \\end{align*}"}
+{"id": "6140.png", "formula": "\\begin{align*} \\| D _ { V ^ * } h \\| ^ 2 = \\| D _ { V _ 1 ^ * } V _ 2 ^ * h \\| ^ 2 + \\| D _ { V _ 2 ^ * } h \\| ^ 2 = \\| D _ { V _ 1 ^ * } h \\| ^ 2 + \\| D _ { V _ 2 ^ * } V _ 1 ^ * h \\| ^ 2 . \\end{align*}"}
+{"id": "1067.png", "formula": "\\begin{align*} D = i \\gamma ^ { j } \\nabla ^ { ( s ) } _ { e _ { j } } , \\end{align*}"}
+{"id": "7854.png", "formula": "\\begin{align*} E = \\bigcup _ { a \\in A } \\{ a \\} \\times B _ a , \\end{align*}"}
+{"id": "5503.png", "formula": "\\begin{align*} F _ N ( a , b ; t ) = \\frac { ( 1 - t q ^ { N + 1 } ) } { ( 1 - t ) } + \\frac { ( 1 - q ^ { N + 1 } ) ( b - a ) t q } { ( 1 - b q ) ( 1 - t ) } F _ N ( a , b q ; t q ) . \\end{align*}"}
+{"id": "3290.png", "formula": "\\begin{gather*} \\alpha = - q ^ { \\alpha _ 0 + \\alpha _ 1 + \\alpha _ 2 + \\alpha _ 3 } , \\beta = q ^ { \\alpha _ 0 - \\alpha _ 1 - \\alpha _ 2 + \\alpha _ 3 } , \\gamma = q ^ { 2 \\alpha _ 3 } , \\delta = - q ^ { 2 \\alpha _ 1 } . \\end{gather*}"}
+{"id": "2215.png", "formula": "\\begin{gather*} \\begin{array} { c c l l } \\dot { x } _ 1 & = & x _ 1 ^ 2 + 7 x _ 2 ^ 2 - 4 x _ 1 x _ 2 & + 5 x _ 1 \\\\ & & & \\\\ \\dot { x } _ 2 & = & - 2 x _ 2 ^ 2 + x _ 1 x _ 2 + 2 x _ 1 x _ 3 - 7 x _ 2 x _ 3 & + 2 x _ 2 \\\\ & & & \\\\ \\dot { x } _ 3 & = & - x _ 2 ^ 2 - 7 x _ 3 ^ 2 - 4 x _ 2 x _ 3 & - x _ 3 \\end{array} \\end{gather*}"}
+{"id": "7220.png", "formula": "\\begin{align*} j = j _ 1 , j _ 2 = j _ 3 \\textmd { o r } ( j - j _ 1 ) \\cdot ( j _ 1 - j _ 2 ) = 0 . \\end{align*}"}
+{"id": "54.png", "formula": "\\begin{align*} ( L _ { X } g ) ( Y , Z ) = g ( \\nabla _ { Y } X , Z ) + g ( \\nabla _ { Z } X , Y ) = 0 . \\end{align*}"}
+{"id": "5443.png", "formula": "\\begin{align*} \\wedge _ \\ell ( { \\rm I d } _ n + z T ) \\ , \\ , \\ , = \\ , \\ , \\ , \\wedge _ \\ell \\ , { \\rm I d } _ n \\ , \\ , + \\ , \\ , \\sum _ { i = 1 } ^ \\ell \\ , [ \\ , \\wedge _ \\ell ( { \\rm I d } _ n + z T ) \\ , ] _ i \\ , z ^ i , \\end{align*}"}
+{"id": "7772.png", "formula": "\\begin{align*} f ( s ) = - ( \\int _ t ^ { + \\infty } [ A ( \\tau ) ] ^ { \\frac { 1 } { 1 - p } } d \\tau ) ^ { - 1 } \\cdot \\int _ t ^ s ( [ A ( \\tau ) ] ^ { \\frac { 1 } { 1 - p } } d \\tau + 1 \\end{align*}"}
+{"id": "499.png", "formula": "\\begin{align*} \\Omega = \\left \\{ ( \\xi , z ) \\in \\mathbb { C } ^ { d _ { 0 } } \\times D : | | \\xi | | ^ { 2 } < \\varphi ( z ) \\right \\} \\end{align*}"}
+{"id": "8063.png", "formula": "\\begin{align*} { \\bf x } _ r = \\left [ \\begin{matrix} \\Re \\{ { \\bf x } \\} \\\\ \\Im \\{ { \\bf x } \\} \\end{matrix} \\right ] , { \\bf H } _ r = \\left [ \\begin{matrix} \\Re \\{ { \\bf H } \\} & - \\Im \\{ { \\bf H } \\} \\\\ \\Im \\{ { \\bf H } \\} & \\Re \\{ { \\bf H } \\} \\end{matrix} \\right ] . \\end{align*}"}
+{"id": "1303.png", "formula": "\\begin{align*} f _ 1 + ( f _ 2 + f _ 3 ) = ( f _ 1 + f _ 2 ) + f _ 3 , f _ 1 + f _ 2 = f _ 2 + f _ 1 , f + 0 _ { a , a ' } = f , \\end{align*}"}
+{"id": "1609.png", "formula": "\\begin{align*} \\left ( \\pi _ { e _ { 1 } } \\circ \\psi \\right ) \\circ T _ { C \\setminus \\left \\{ e _ { 1 } \\right \\} } = \\pi _ { e _ { 1 } } \\circ T _ { C } = T _ { e _ { 1 } } , \\end{align*}"}
+{"id": "3929.png", "formula": "\\begin{align*} F _ 1 ( \\Omega ) = & \\int _ { \\partial \\Omega } j _ 1 ( x , \\nu ( x ) , H _ { \\partial \\Omega } ( x ) ) \\ , d \\mu _ { \\partial \\Omega } ( x ) , \\end{align*}"}
+{"id": "3521.png", "formula": "\\begin{align*} \\Vert \\nabla \\varphi \\Vert _ { \\mathbb { L } ^ 2 ( \\Omega ) } = \\Vert \\nabla \\hat { \\varphi } \\Vert _ { \\mathbb { L } ^ 2 ( \\mathrm { B } ) } \\Vert \\nabla \\varphi \\Vert _ { \\mathbb { L } ^ 4 ( \\Omega ) } = \\delta ^ { - \\frac { 1 } { 2 } } \\Vert \\nabla \\hat { \\varphi } \\Vert _ { \\mathbb { L } ^ 4 ( \\mathrm { B } ) } . \\end{align*}"}
+{"id": "6417.png", "formula": "\\begin{align*} F _ { \\ast } \\left ( j _ { F } ^ { p } ( x ) - j _ { F } ^ { p } ( y ) \\right ) = \\left ( \\sum _ { i } \\rho _ { \\ast } \\left ( j _ { \\rho } ^ { p } ( x _ { i } ) - j _ { \\rho } ^ { p } ( y _ { i } ) \\right ) ^ { p ^ { \\prime } } \\right ) ^ { \\frac { 1 } { p ^ { \\prime } } } . \\end{align*}"}
+{"id": "6988.png", "formula": "\\begin{align*} \\lambda _ { \\varepsilon } = \\inf \\left \\{ \\frac { 2 } { \\alpha _ { \\varepsilon } } \\int _ { X } | \\nabla f | ^ 2 d m + \\int _ { X } \\left ( \\varepsilon f ^ 2 - f ^ 2 \\log f ^ 2 \\right ) d m : f \\in W ^ { 1 , 2 } , \\| f \\| _ 2 = 1 \\right \\} , \\end{align*}"}
+{"id": "5371.png", "formula": "\\begin{align*} \\C = J _ { c } ^ { - 1 } ( 0 ) , \\end{align*}"}
+{"id": "915.png", "formula": "\\begin{align*} v : = - d + \\frac { 1 } { 8 b _ { n - 1 } } d ^ 2 . \\end{align*}"}
+{"id": "3974.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\beta } ( t ) \\stackrel { d } { = } \\mathcal { M } ( Y _ { \\beta } ( t ) ) , \\ t \\ge 0 . \\end{align*}"}
+{"id": "1455.png", "formula": "\\begin{align*} \\varphi _ { \\beta , \\varepsilon } ' ( s ) = e ^ { - s } \\left [ - M ( \\gamma _ { \\varepsilon } - \\beta , \\gamma _ { \\varepsilon } ; s ) + M ' ( \\gamma _ { \\varepsilon } - \\beta , \\gamma _ { \\varepsilon } ; s ) \\right ] \\end{align*}"}
+{"id": "7979.png", "formula": "\\begin{align*} t ' ( s ) = - 2 r ( s ) \\cos { \\left ( \\theta ( s ) \\right ) } = 2 \\phi _ 0 r ^ { 2 - 2 n } ( s ) - \\frac { 2 c ( 2 n - 1 ) } { ( 2 n + 1 ) } r ^ { 3 } ( s ) . \\end{align*}"}
+{"id": "6983.png", "formula": "\\begin{align*} \\frac { d } { d t } P _ t f = \\Delta P _ t f , t \\in ( 0 , \\infty ) . \\end{align*}"}
+{"id": "3751.png", "formula": "\\begin{align*} P ^ { - 1 } = ( ( x + 4 ) I + A [ \\mathcal T ] ) ^ { - 1 } - \\frac { 1 0 J } { ( x - 1 2 ) ( x - 2 ) ( x + 6 ) } . \\end{align*}"}
+{"id": "6526.png", "formula": "\\begin{align*} H = - i \\sum _ r \\log ( e ^ { i \\theta _ r } ) E _ { \\theta _ r } = \\sum _ r \\theta _ r E _ { \\theta _ r } . \\end{align*}"}
+{"id": "3578.png", "formula": "\\begin{align*} \\sum _ i \\frac { \\left \\Vert \\sum _ j \\mu _ j \\langle \\phi _ i , \\psi _ j \\rangle \\psi _ j \\right \\Vert ^ 2 } { ( \\zeta _ i + \\lambda ) ^ { 2 } } & = \\sum _ i \\frac { \\mu ^ 2 _ i } { ( \\zeta _ i + \\lambda ) ^ 2 } \\sum _ j \\frac { \\mu ^ 2 _ j } { \\mu ^ 2 _ i } \\langle \\phi _ i , \\psi _ j \\rangle ^ 2 \\\\ & \\le \\sum _ i \\frac { \\mu ^ 2 _ i } { ( \\zeta _ i + \\lambda ) ^ 2 } \\sup _ i \\frac { 1 } { \\mu ^ 2 _ i } \\sum _ j \\mu ^ 2 _ j \\langle \\phi _ i , \\psi _ j \\rangle ^ 2 . \\end{align*}"}
+{"id": "3343.png", "formula": "\\begin{gather*} f _ A ( a , b , c ) = 0 a b = q ^ 2 b a \\ \\ c b = q ^ 2 b c , \\\\ f _ A ( a , b , c ) + f _ A ( c , b , a ) = 0 a b = q ^ 2 b a , a c = q ^ { - 4 } c a \\ \\ b c = c b . \\end{gather*}"}
+{"id": "5785.png", "formula": "\\begin{align*} \\frac { h m n } { h + 1 } + \\frac { m } { h + 1 } - m + h - t m h & = \\frac { 1 } { h + 1 } \\left ( h m n + m - m ( h + 1 ) + h ( h + 1 ) - ( h + 1 ) t m h \\right ) \\\\ & = \\frac { 1 } { h + 1 } \\left ( r h m + m - m ( h + 1 ) + h ( h + 1 ) \\right ) \\\\ & = \\frac { h } { h + 1 } \\left ( m ( r - 1 ) + h + 1 \\right ) > 0 \\end{align*}"}
+{"id": "4787.png", "formula": "\\begin{align*} \\left ( C _ r ( n ) p _ { n + r } ( x ) + \\dots + C _ 0 ( n ) p _ n ( x ) \\right ) ' = D _ s ( n ) p _ { n + s } ( x ) + \\dots + D _ 0 ( n ) p _ n ( x ) , \\end{align*}"}
+{"id": "387.png", "formula": "\\begin{align*} L _ \\infty ( M , \\omega ) _ { [ N ] } = & ~ F _ N \\cap \\mathcal { Q } ' \\\\ I _ { L _ \\infty } { ( N ) } = & ~ I _ \\mu + ( I _ N \\cap \\mathcal { Q } ) , \\end{align*}"}
+{"id": "4029.png", "formula": "\\begin{align*} f _ b ( y ) = \\left ( p - 1 + b y ^ { 2 k } \\right ) ^ { - \\frac { 1 } { p - 1 } } , \\ ; \\ ; k > 1 , \\end{align*}"}
+{"id": "715.png", "formula": "\\begin{align*} c _ s ^ 2 \\partial _ { z 1 } T = - \\frac { 1 } { \\Delta t } { \\varsigma } _ 3 { m } _ 3 ^ { ( 1 ) } , \\end{align*}"}
+{"id": "7504.png", "formula": "\\begin{align*} 2 U ^ { \\mu } ( z ) + \\Re V ( z ) = c _ j , z \\in \\Sigma _ j , \\end{align*}"}
+{"id": "1074.png", "formula": "\\begin{align*} G _ { a b } = 4 \\left ( k ^ { - 2 } k _ { a } k _ { b } + k ^ { - 1 } k _ { a b } \\right ) + 8 \\delta _ { a b } k ^ { - 2 } k _ { c } k _ { c } - 4 \\delta _ { a b } k ^ { - 1 } k _ { c c } . \\end{align*}"}
+{"id": "6899.png", "formula": "\\begin{align*} g ( u _ 1 + u _ 2 ) - g ( u _ 1 ) = \\int _ 0 ^ 1 [ g _ z ( u _ 1 + \\theta u _ 2 ) u _ 2 + g _ { \\bar { z } } ( u _ 1 + \\theta u _ 2 ) \\bar { u } _ 2 ] d \\theta \\end{align*}"}
+{"id": "5805.png", "formula": "\\begin{align*} \\sup _ { \\| x \\| = 1 } | \\langle A x , x \\rangle | ^ { p } \\leq \\sup _ { \\| x \\| = 1 } \\langle A ^ * A x , x \\rangle ^ { p \\over 2 } = \\| A ^ * A \\| ^ { p \\over 2 } = \\| A \\| ^ p = \\| | A | \\| ^ p = \\| | A | ^ p \\| , \\end{align*}"}
+{"id": "1445.png", "formula": "\\begin{align*} \\Delta A _ { \\varepsilon } ( x ) & = b _ 1 ( x ) + \\eta _ { \\varepsilon } ( x ) b _ 2 ( x ) = a ( x ) - ( 1 - \\eta _ { \\varepsilon } ) b _ 2 ( x ) , \\end{align*}"}
+{"id": "341.png", "formula": "\\begin{align*} \\lim _ { m , n \\rightarrow \\infty } ( \\hat { h } - \\tilde { h } ) \\biggl ( B _ { n , m } + \\frac { 1 } { 2 } \\zeta ( \\hat { h } - \\tilde { h } ) C _ { n , m } \\biggr ) = 0 \\quad . \\end{align*}"}
+{"id": "1527.png", "formula": "\\begin{align*} & d _ \\mathrm { h a m m } ( \\sigma , \\sigma ' ) = d _ \\mathrm { h a m m } ( \\sigma \\circ \\tau , \\sigma ' \\circ \\tau ) , \\\\ & d _ \\mathrm { h a m m } ( \\sigma , \\sigma ' ) = d _ \\mathrm { h a m m } ( \\tau \\circ \\sigma , \\tau \\circ \\sigma ' ) . \\end{align*}"}
+{"id": "7707.png", "formula": "\\begin{gather*} ( d \\tilde { \\star } \\tilde { \\omega } ) _ { i _ 1 \\dots i _ { n - k + 1 } } ( x ^ \\prime , x ^ n ) = \\begin{cases} ( d \\star \\omega ) _ { i _ 1 \\dots i _ { n - k + 1 } } ( x ^ \\prime , | x ^ n | ) & i _ l = n , \\\\ \\operatorname { s g n } ( x ^ n ) ( d \\star \\omega ) _ { i _ 1 \\dots i _ { n - k + 1 } } ( x ^ \\prime , | x ^ n | ) & i _ l \\neq n \\end{cases} , \\end{gather*}"}
+{"id": "5888.png", "formula": "\\begin{align*} \\begin{aligned} & \\alpha _ { \\tau \\sigma } ( h M ) = \\Big ( \\frac { 1 } { 6 } + \\frac { \\tau - \\sigma } { 2 } \\Big ) \\varphi _ { 1 } ( ( \\tau - \\sigma ) h M ) , \\\\ & \\beta _ { \\tau } ( h M ) = ( 1 - \\tau ) \\varphi _ { 1 } ( ( 1 - \\tau ) h M ) , \\quad \\gamma _ { \\tau } ( h M ) = \\varphi _ { 0 } ( ( 1 - \\tau ) h M ) , \\end{aligned} \\end{align*}"}
+{"id": "3642.png", "formula": "\\begin{align*} \\beta _ e ( \\alpha \\otimes \\gamma ) \\beta _ e ^ * = { \\sum } _ { i , j = 1 } ^ n \\alpha _ { i , j } \\gamma _ { i , j } ( \\alpha , \\gamma \\in M _ n ) . \\end{align*}"}
+{"id": "5921.png", "formula": "\\begin{align*} f _ 1 = \\tfrac { 1 } { 2 } \\cdot H ^ 4 - H ^ 2 e _ 2 \\ , , f _ 2 = - H ^ 4 + \\tfrac { 5 } { 2 } \\cdot H ^ 2 e _ 2 \\end{align*}"}
+{"id": "1719.png", "formula": "\\begin{align*} \\rho _ \\tau ( \\Theta _ \\tau ( \\xi ) ) = \\frac { \\tilde { \\rho } _ { \\tau , 1 } ( \\Theta _ \\tau ( \\xi ) ) } { \\tilde { \\rho } _ { \\tau , 1 } ( \\mathrm { I } ) } , \\end{align*}"}
+{"id": "331.png", "formula": "\\begin{align*} \\hat { h } _ { n , m } = \\arg \\min L ( h , Y _ 1 , \\cdots , Y _ n , \\Theta _ m ) , \\end{align*}"}
+{"id": "4754.png", "formula": "\\begin{align*} \\rho _ \\mu ( a ) : = \\mu ( \\partial _ 1 ( a ) ) \\alpha _ 2 - \\mu ( \\partial _ 2 ( a ) ) \\alpha _ 1 , \\forall a \\in A . \\end{align*}"}
+{"id": "6750.png", "formula": "\\begin{align*} - \\frac { 1 } { \\varepsilon } ( \\mathcal { L } \\partial ^ { \\alpha } f , \\frac { \\partial ^ { \\alpha } F } { \\sqrt { \\mu } } ) = - \\frac { 1 } { \\varepsilon } ( \\mathcal { L } \\partial ^ { \\alpha } f , \\frac { \\partial ^ { \\alpha } M } { \\sqrt { \\mu } } ) - \\frac { 1 } { \\varepsilon } ( \\mathcal { L } \\partial ^ { \\alpha } f , \\partial ^ { \\alpha } f ) - \\frac { 1 } { \\varepsilon } ( \\mathcal { L } \\partial ^ { \\alpha } f , \\frac { \\partial ^ { \\alpha } \\overline { G } } { \\sqrt { \\mu } } ) . \\end{align*}"}
+{"id": "3201.png", "formula": "\\begin{align*} I _ { \\rho ^ { 2 } } & = I _ { \\frac { \\rho ^ { 2 } } { \\mu ^ { 2 } } \\mu ^ { 2 } } < \\frac { \\rho ^ { 2 } } { \\mu ^ { 2 } } I _ { \\mu ^ { 2 } } \\\\ & = \\frac { \\rho ^ { 2 } - \\mu ^ { 2 } + \\mu ^ { 2 } } { \\mu ^ { 2 } } I _ { \\mu ^ { 2 } } \\\\ & = \\frac { \\rho ^ { 2 } - \\mu ^ { 2 } } { \\mu ^ { 2 } } I _ { \\frac { \\mu ^ { 2 } } { \\rho ^ { 2 } - \\mu ^ { 2 } } \\rho ^ { 2 } - \\mu ^ { 2 } } + I _ { \\mu ^ { 2 } } \\le I _ { \\mu ^ { 2 } } + I _ { \\rho ^ { 2 } - \\mu ^ { 2 } } . \\end{align*}"}
+{"id": "5893.png", "formula": "\\begin{align*} \\begin{aligned} \\varsigma ^ { 0 } _ { n } ( s ) = \\frac { 1 } { \\epsilon } \\tilde { B } _ { n } ^ { ( 1 ) } ( s - h / 2 ) v ( t _ n ) + \\frac { 1 } { \\epsilon } \\tilde { B } _ { n } ^ { ( 1 ) } s ( s - h / 2 ) \\dot { v } ( t _ n ) + \\mathcal { O } ( h ^ 2 / \\epsilon ^ 2 ) . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "4448.png", "formula": "\\begin{align*} v _ { n _ k } : = \\begin{cases} 0 , & x \\notin \\Sigma _ { n _ k } \\\\ z _ { n _ k } , & x \\in \\Sigma _ { n _ k } , \\end{cases} \\end{align*}"}
+{"id": "5259.png", "formula": "\\begin{align*} \\kappa _ p ^ { ( N ) } ( k _ 1 , \\ldots , k _ p ) = \\sum _ { m = 1 } ^ p \\frac { ( - 1 ) ^ { m } } { m } \\ * \\sum _ { \\substack { ( p _ 1 , \\ldots , p _ m ) : \\\\ p _ 1 + \\ldots + p _ m = p , \\ p _ 1 , \\ldots p _ m \\geq 1 } } \\frac { 1 } { p _ 1 ! \\cdots p _ m ! } \\ * \\sum _ { \\sigma \\in S _ p } \\ * J _ N ( p _ 1 , \\ldots , p _ m ; k _ { \\sigma ( 1 ) } , \\ldots , k _ { \\sigma ( p ) } ) , \\end{align*}"}
+{"id": "1860.png", "formula": "\\begin{align*} y _ { ( 2 k - 1 ) m + 2 i - 1 , j } + y _ { ( 2 k - 1 ) m + 2 i - 1 , j + 1 } & = 2 M N + 3 - ( x _ { 2 i - 1 , j } + x _ { 2 i - 1 , j + 1 } ) \\mbox { a n d } \\\\ y _ { ( 2 k - 1 ) m + 2 i , j } + y _ { ( 2 k - 1 ) m + 2 i , j + 1 } & = 2 M N + 3 - ( x _ { 2 i , j } + x _ { 2 i , j + 1 } ) . \\end{align*}"}
+{"id": "6600.png", "formula": "\\begin{align*} [ u ] ( s ) : = \\lim _ { t \\to 0 ^ + } u \\big ( s + t N ( s ) \\big ) - u \\big ( s - t N ( s ) \\big ) , \\end{align*}"}
+{"id": "2322.png", "formula": "\\begin{align*} \\mathcal C ( s , c _ 0 ) & \\coloneqq \\left \\{ \\zeta \\in \\R ^ { N } : \\| \\zeta \\| _ 1 \\le ( 1 + c _ 0 ) \\left \\| \\zeta _ { | \\mathcal I _ s ( \\zeta ) } \\right \\| _ 1 \\right \\} , \\\\ \\mathcal S _ 1 ( s ) & \\coloneqq \\left \\{ \\theta \\in \\R ^ N : \\| \\theta \\| _ 0 = s \\right \\} \\quad \\mathcal S _ 2 ( s , \\theta ) \\coloneqq \\left \\{ \\eta \\in \\R ^ N : \\theta - \\eta \\in \\mathcal C ( s , c _ 0 ) \\right \\} . \\end{align*}"}
+{"id": "580.png", "formula": "\\begin{align*} & [ C _ I , C _ J ] = 0 I \\cap J = \\emptyset \\ \\ \\ \\ I \\subseteq J \\ , , \\\\ & C _ I = \\frac { 1 } { 2 } \\sum _ { i , j \\in I \\atop i \\neq j } C _ { i j } - ( | I | - 2 ) \\sum _ { i \\in I } C _ i \\ , , \\end{align*}"}
+{"id": "6872.png", "formula": "\\begin{align*} & \\left ( - \\sum _ { i } m ( j - 1 ) _ { i } Y _ { i } \\right ) \\cdot Y _ { i ( j ) } \\\\ & = \\left ( - \\sum _ { i } m _ { i } Y _ { i } \\right ) \\cdot Y _ { i ( j ) } - Y _ { i ( 1 ) } \\cdot Y _ { i ( j ) } - \\dots - Y _ { i ( j - 1 ) } \\cdot Y _ { i ( j ) } . \\end{align*}"}
+{"id": "3517.png", "formula": "\\begin{align*} \\sum _ { \\mathrm { n } \\in \\mathbb { N } } \\Big | \\Big \\langle \\widehat { \\nabla \\mathrm { H } } ; \\mathrm { e } ^ { ( 1 ) } _ \\mathrm { n } \\Big \\rangle _ { \\mathbb { L } ^ 2 ( \\mathrm { B } ) } \\Big | ^ 2 = | \\alpha | ^ 2 \\delta ^ { 4 } \\sum _ { \\mathrm { n } \\in \\mathbb { N } } \\Big | \\Big \\langle \\mathbb { T } _ \\delta \\Big [ \\widehat { \\nabla \\mathrm { H } } \\Big ] ; \\mathrm { e } ^ { ( 3 ) } _ \\mathrm { n } \\Big \\rangle _ { \\mathbb { L } ^ 2 ( \\mathrm { B } ) } \\Big | ^ 2 \\end{align*}"}
+{"id": "7328.png", "formula": "\\begin{align*} \\Delta _ { u } \\mathcal { H } ( t ) = \\mathcal { H } ( t , X ^ { u } ( t ) , p ^ { u } ( t ) , q ^ { u } ( t ) , P ^ { u } ( t ) , v ( t ) , u ( t ) ) - \\mathcal { H } ( t , X ^ { u } ( t ) , p ^ { u } ( t ) , q ^ { u } ( t ) , P ^ { u } ( t ) , u ( t ) , u ( t ) ) , \\end{align*}"}
+{"id": "4651.png", "formula": "\\begin{align*} \\chi _ F = \\chi _ { F _ 1 } \\chi ^ 2 _ { F _ 2 } \\cdots \\chi ^ { \\ell - 1 } _ { F _ { \\ell - 1 } } \\end{align*}"}
+{"id": "5190.png", "formula": "\\begin{align*} \\mathcal { F } _ L = \\sum _ { k = 0 } ^ { \\infty } \\ell _ k X ^ k \\enspace , \\end{align*}"}
+{"id": "7955.png", "formula": "\\begin{align*} | \\mathcal { P } _ H ( \\nu ) | = \\frac { 2 r R ^ 2 } { \\sqrt { 4 r ^ 2 R ^ 4 + ( t - t _ 0 ) ^ 2 } } \\quad \\mbox { a n d } \\left \\langle \\nu , T \\right \\rangle = \\frac { t - t _ 0 } { \\sqrt { 4 r ^ 2 R ^ 4 + ( t - t _ 0 ) ^ 2 } } , \\end{align*}"}
+{"id": "6649.png", "formula": "\\begin{align*} 1 + \\prod _ { j = 0 } ^ { k - 1 } \\lambda _ { - j } ^ { 2 } \\sum _ { i = 1 } ^ { \\eta } \\lambda _ { i , 1 } ^ { 2 } s _ { i , - k - 1 } = \\lambda _ { - k + 1 } ^ { 2 } , k \\in \\mathbb { N } \\cap [ 1 , \\kappa - 1 ] , \\end{align*}"}
+{"id": "571.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } T _ { W _ { n J } , \\psi } ( \\mu _ n ( Q ^ n ) ) & \\le \\limsup _ { n \\to \\infty } T _ { W _ { n J } , \\psi _ m } ( \\mu _ n ( Q ^ n ) ) \\le \\limsup _ { n \\to \\infty } T _ { W , \\psi _ m } ( \\mu _ n ( Q ^ n ) ) = T _ { W , \\psi _ m } ( \\mu _ \\infty ) , \\end{align*}"}
+{"id": "6954.png", "formula": "\\begin{align*} \\mathsf B ^ N = 1 + \\sum _ { d = 1 } ^ { \\infty } ( - 1 ) ^ { d + 1 } \\frac { N } { d } \\cdot \\binom { ( N + 1 ) ( d - 1 ) } { d - 1 } \\cdot \\left ( \\frac { q y } { ( 1 - y ) ^ { N + 1 } } \\right ) ^ { d } . \\end{align*}"}
+{"id": "7720.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow + \\infty } V o l ( \\partial B ( t ) , g ^ p | _ { \\partial B ( t ) } ) = V o l ( \\partial X , \\hat { g } ^ p ) \\end{align*}"}
+{"id": "6080.png", "formula": "\\begin{align*} \\Sigma ( D ) = \\sum _ { v \\in V ( D ) } w ( v ) = 7 n ( D ) - 3 m ( D ) - \\sum _ { v \\in V ( D ) } \\epsilon ( v ) \\end{align*}"}
+{"id": "912.png", "formula": "\\begin{align*} \\{ u _ { \\alpha \\beta } ( 0 ) \\} _ { 1 \\leq \\alpha , \\beta \\leq n - 1 } = \\mathrm { d i a g } \\{ b _ 1 , \\ldots , b _ { n - 1 } \\} . \\end{align*}"}
+{"id": "6128.png", "formula": "\\begin{align*} \\begin{aligned} \\binom { n } { j } - \\binom { n - i } { j } = \\sum _ { h = 1 } ^ { i } \\binom { n - h } { j - 1 } , \\end{aligned} \\end{align*}"}
+{"id": "6927.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { N } \\frac { 1 + \\alpha _ i x _ p } { 1 + z _ i x _ p } = \\frac { R \\left ( - 1 / x _ p \\right ) } { P \\left ( - 1 / x _ p \\right ) } \\left ( - \\frac { 1 } { x _ p } - z _ { N + 1 } \\right ) = ( 1 + z _ { N + 1 } x _ p ) . \\end{align*}"}
+{"id": "6218.png", "formula": "\\begin{align*} E ^ * _ \\nu \\mathcal { W } _ { ( \\mu - 1 , d ) } = E ^ * _ \\nu E _ { \\mu - 1 } P ( E _ { \\mu - 2 } ) \\mathcal { L } _ \\nu . \\end{align*}"}
+{"id": "2946.png", "formula": "\\begin{align*} \\mathbb V _ { \\delta , \\rho } ( w ) : = \\left \\{ w ' \\in \\mathbb { B } _ \\delta ( 0 ) \\ , \\middle | \\ , \\big \\| \\| w \\| w ' - \\| w ' \\| w \\big \\| \\leq \\rho \\| w ' \\| \\| w \\| \\right \\} . \\end{align*}"}
+{"id": "3138.png", "formula": "\\begin{align*} a _ { k } = F P \\left ( \\sigma ^ { k } \\right ) = | X | \\end{align*}"}
+{"id": "5171.png", "formula": "\\begin{align*} f _ { n , i } ^ j ( U _ { n , i } ^ j ) = U _ { } f _ { n , i } ^ j ( T _ { n , i } ^ j ) = T , \\end{align*}"}
+{"id": "6399.png", "formula": "\\begin{align*} e _ 1 e ' _ 1 & = q ^ { - 1 / 2 } ( [ e _ 2 e _ 3 ] + q [ e _ 4 e _ 5 e _ 7 ] ) , \\\\ e _ 2 e ' _ 2 & = q ^ { - 1 } ( [ e _ 4 e _ 6 e _ 7 ^ 2 ] + q ^ 2 [ e _ 1 ^ 2 e _ 8 ] ) \\end{align*}"}
+{"id": "7052.png", "formula": "\\begin{align*} \\tilde { \\mathfrak g } = \\mathfrak g ' \\oplus \\mathfrak h ' , \\end{align*}"}
+{"id": "2619.png", "formula": "\\begin{align*} & E ^ 0 _ { k , m } = \\{ n \\in \\mathbb { N } ^ k : 0 \\le n \\le m \\} \\\\ & E ^ 1 _ { k , m } = \\{ n + v _ i : n , n + e _ i \\in E ^ 0 _ { k , m } \\} \\\\ & r ( n + v _ i ) = n , s ( n + v _ i ) = n + e _ i , c ( n + v _ i ) = c _ i \\end{align*}"}
+{"id": "7306.png", "formula": "\\begin{align*} \\overline { M _ { i + 1 } } = ( \\mathop { q - 2 } \\limits _ { m - 1 } , \\underbrace { q - 1 , \\ldots , q - 1 , } _ { m - 2 - j _ 1 } \\mathop { q - 2 } \\limits _ { j _ 1 } , \\ldots , \\mathop { q - 2 } _ { j _ { i - 1 } } , \\underbrace { q - 1 , \\ldots , q - 1 } _ { j _ { i - 1 } - j _ i - 1 } , \\mathop { q - 2 } _ { j _ i } , \\underbrace { q - 1 , \\ldots , q - 1 } _ { j _ i } ) , \\end{align*}"}
+{"id": "4947.png", "formula": "\\begin{align*} a _ 0 & = y - \\sum _ { i = 1 } ^ { ( k - 5 ) / 2 - t } a _ i - \\sum _ { i = 1 } ^ { ( k - 5 ) / 2 - t } b _ i \\\\ & \\ge y - A - \\frac { 1 } { 2 } \\sum _ { i = 2 } ^ { ( k - 5 ) / 2 - t } \\binom { n } { k - 2 - i } - \\frac { 1 } { 2 } \\left ( \\binom { n } { k - 3 } - \\left ( A - \\sum _ { i = 2 } ^ { ( k - 5 ) / 2 - t } i \\right ) \\right ) \\\\ & = y - \\frac { 1 } { 2 } A - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { ( k - 5 ) / 2 - t } \\binom { n } { k - 2 - i } - \\frac { \\left ( ( k - 1 ) / 2 - t \\right ) \\left ( ( k - 7 ) / 2 - t \\right ) } { 4 } . \\end{align*}"}
+{"id": "6581.png", "formula": "\\begin{align*} P E _ r P ^ { - 1 } = E _ r ' . \\end{align*}"}
+{"id": "1077.png", "formula": "\\begin{align*} \\sigma ( P ^ { - 1 } ) ( x , \\xi ) = \\mathfrak b _ { 2 } + \\mathfrak b _ { 3 } + \\mathfrak b _ { 4 } + . . . , \\end{align*}"}
+{"id": "4755.png", "formula": "\\begin{align*} \\rho _ \\mu ( a ) & \\overset { \\hphantom { ( 2 . 3 ) } } { = } \\mu ( \\partial _ 1 ( a ) ) \\alpha _ 2 - \\mu ( \\partial _ 2 ( a ) ) \\alpha _ 1 \\\\ & \\overset { \\eqref { e q : r e p d 1 } } { = } ( \\alpha _ 1 \\mu ( a ) - \\mu ( a ) \\alpha _ 1 ) \\alpha _ 2 - ( \\alpha _ 2 \\mu ( a ) - \\mu ( a ) \\alpha _ 2 ) \\alpha _ 1 = \\alpha _ 1 \\mu ( a ) \\alpha _ 2 - \\alpha _ 2 \\mu ( a ) \\alpha _ 1 . \\end{align*}"}
+{"id": "7592.png", "formula": "\\begin{align*} \\Delta f = - \\star d \\star d f \\end{align*}"}
+{"id": "4753.png", "formula": "\\begin{align*} [ a , b ] : = \\partial _ 1 ( a ) \\cdot \\partial _ 2 ( b ) - \\partial _ 2 ( a ) \\cdot \\partial _ 1 ( b ) , \\forall a , b \\in A . \\end{align*}"}
+{"id": "6356.png", "formula": "\\begin{align*} R ( \\lambda ) = D ( \\lambda ) + C A ( \\lambda ) ^ { - 1 } B \\in { \\mathbb C } ( \\lambda ) ^ { m , m } . \\end{align*}"}
+{"id": "7053.png", "formula": "\\begin{align*} \\left [ \\mathfrak h ' , \\sum _ { k = 0 } ^ r [ \\tilde { \\mathfrak g } , \\tilde { \\mathfrak { t r } ^ p _ 0 } ] ^ k \\right ] \\subset \\sum _ { k = 0 } ^ r [ \\tilde { \\mathfrak g } , \\tilde { \\mathfrak { t r } ^ p _ 0 } ] ^ k \\end{align*}"}
+{"id": "7219.png", "formula": "\\begin{align*} j = j _ 1 , j _ 2 = j _ 3 \\textmd { o r } j = j _ 3 , j _ 2 = j _ 1 . \\end{align*}"}
+{"id": "2017.png", "formula": "\\begin{align*} \\frac { \\Gamma ' } { \\Gamma } ( w ) = - \\gamma _ 0 - \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { w + n } - \\frac { 1 } { n + 1 } \\right ) , \\end{align*}"}
+{"id": "4708.png", "formula": "\\begin{align*} \\sup _ { x \\in S , \\| x \\| \\le 1 } | L ( x ) | = \\min _ { \\widetilde { L } \\in S ^ { \\perp } } \\| L + \\widetilde { L } \\| , \\end{align*}"}
+{"id": "2403.png", "formula": "\\begin{align*} { \\rm e v } _ { 1 / 2 } ^ { \\mathfrak { m } } ( e _ { a _ { 1 } } \\cdots e _ { a _ { k } } ) = I ^ { \\mathfrak { m } } ( 0 , a _ { 1 } ' , \\dots , a _ { k } ' , 1 ) \\end{align*}"}
+{"id": "2548.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\varphi _ t ^ \\xi = & - [ ( A - B ^ 2 R ^ { - 1 } P _ t ) \\varphi _ t ^ \\xi + P _ t f ( \\nu _ t ) + P _ t b ( \\mu _ t ) \\\\ & + Q l ( \\nu _ t ) - P _ t B h ( \\mu _ t ) ] d t + \\Lambda _ t ^ { 0 , \\xi } d W ^ 0 _ t , \\\\ \\varphi _ T ^ \\xi = & ~ G g ( \\nu _ T ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2323.png", "formula": "\\begin{align*} \\langle b ( x ) , \\nabla V ( x ) \\rangle & = \\gamma \\lVert x \\rVert ^ { \\gamma - 1 } \\langle b ( x ) , x \\slash \\lVert x \\rVert \\rangle \\leq - r \\gamma \\lVert x \\rVert ^ { \\gamma - 1 - q } = - \\phi \\circ V ( x ) , \\end{align*}"}
+{"id": "5732.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } a _ { i j } x _ i x _ j \\quad g ( x ) = \\sum _ { i = 1 , j = 1 } ^ { m } b _ { i j } x _ i x _ j , \\end{align*}"}
+{"id": "3957.png", "formula": "\\begin{align*} \\mathcal { M } ( t ) \\stackrel { d } { = } Y ( t ) . \\end{align*}"}
+{"id": "3021.png", "formula": "\\begin{align*} \\begin{aligned} x _ i & = F _ { x _ i } ( \\varphi , \\dot { \\varphi } , \\ldots , \\varphi ^ { ( r - 1 ) } ) \\ , , & i & = 1 , \\ldots , n \\\\ u _ j & = F _ { u _ j } ( \\varphi , \\dot { \\varphi } , \\ldots , \\varphi ^ { ( r ) } ) \\ , , & j & = 1 , 2 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "847.png", "formula": "\\begin{align*} M _ { \\tau } ( t ) : = \\mathbb { E } \\Big ( e ^ { t \\tau } \\Big ) = e ^ { - \\lambda b ' } = e ^ { - \\frac { \\lambda } { \\sigma } \\ln \\left ( \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) ^ { \\frac { 1 } { k _ 2 } } } = \\left ( \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) ^ { - \\frac { \\lambda _ + ( t ) } { k _ 2 \\sigma } } \\end{align*}"}
+{"id": "2524.png", "formula": "\\begin{align*} F ^ 1 ( \\phi _ i x ) = F ^ i x F _ 1 ( \\phi _ i x ) = F _ i x \\end{align*}"}
+{"id": "3430.png", "formula": "\\begin{align*} P ^ \\rho = - r ^ { 1 - n } D _ r Q ^ U _ { 1 , l } = - 2 ( ( \\rho U ) _ r + \\tfrac { n - 1 } { r } \\rho U ) f _ { J _ { 1 , l } } ^ { ( l ) } - 2 \\rho U D _ r f _ { J _ { 1 , l } } ^ { ( l ) } ; \\end{align*}"}
+{"id": "8085.png", "formula": "\\begin{align*} 1 + \\tfrac { \\alpha \\lambda _ 1 q ^ 2 } { h ^ 2 } = \\left ( 1 - \\alpha \\right ) \\lambda _ 1 . \\end{align*}"}
+{"id": "3202.png", "formula": "\\begin{align*} & A ( g _ { u } ( \\theta ) ) = \\theta ^ { 2 - 2 \\beta } A ( u ) , \\\\ & C ( g _ { u } ( \\theta ) ) = \\theta ^ { ( 1 - \\frac { 3 } { 2 } \\beta ) p + 3 \\beta } C ( u ) , \\\\ & B ( g _ { u } ( \\theta ) ) = \\theta ^ { 4 - \\beta } \\int _ { \\mathbb { R } ^ { 3 } } \\int _ { \\mathbb { R } ^ { 3 } } \\frac { 1 - e ^ { - \\theta ^ { \\beta } } | x - y | } { | x - y | } u ( x ) ^ { 2 } u ( y ) ^ { 2 } d x d y . \\end{align*}"}
+{"id": "3132.png", "formula": "\\begin{align*} \\sigma = ( 1 ) ( 2 , 3 ) ( 4 , 5 , 6 ) ( 7 , 8 , 9 , 1 0 ) \\cdots \\left ( \\frac { n ^ 2 - n } { 2 } + 1 , \\frac { n ^ 2 - n } { 2 } + 1 , \\frac { n ^ 2 - n } { 2 } + 1 , \\ldots , \\frac { n ^ 2 - n } { 2 } + n , \\right ) \\cdots \\end{align*}"}
+{"id": "2200.png", "formula": "\\begin{align*} & \\nabla _ { e _ 1 } e _ 1 = 0 , \\nabla _ { e _ 1 } e _ 2 = ( b - c ) e _ 3 , \\nabla _ { e _ 1 } e _ 3 = - ( b - c ) e _ 2 , \\\\ & \\nabla _ { e _ 2 } e _ 1 = - a e _ 2 - ( b + c ) e _ 3 , \\nabla _ { e _ 2 } e _ 2 = a e _ 1 , \\nabla _ { e _ 2 } e _ 3 = ( b + c ) e _ 1 , \\\\ & \\nabla _ { e _ 3 } e _ 1 = - ( b + c ) e _ 2 - d e _ 3 , \\nabla _ { e _ 3 } e _ 2 = ( b + c ) e _ 1 , \\nabla _ { e _ 3 } e _ 3 = d e _ 1 , \\end{align*}"}
+{"id": "113.png", "formula": "\\begin{align*} \\int f ( x ) \\ , \\overline { g ( x ) } \\ , d x = h \\sum _ n f ( n h ) \\ , \\overline { g ( n h ) } . \\end{align*}"}
+{"id": "5750.png", "formula": "\\begin{align*} E '^ T f ( \\tilde X ^ M ) { E ' } = E '^ T \\left ( \\sum _ { i = 1 , j = 1 } ^ { m } A _ { i j } \\otimes \\begin{pmatrix} M ^ s _ { i j } & 0 \\\\ 0 & 0 \\end{pmatrix} \\right ) { E ' } < 0 . \\end{align*}"}
+{"id": "1938.png", "formula": "\\begin{align*} & A _ 1 = \\sum _ { j = 0 } ^ { \\infty } 2 ^ { - 2 j } ( | ( - \\Delta _ x ) ^ { 1 / 6 } u _ 0 | ^ p ) ^ { 1 / p } _ { Q _ { \\nu r , 2 ^ j \\nu r } ( z _ 0 ) } , \\\\ & A _ 2 = \\lambda ^ { 1 / 2 } ( | u _ 0 | ^ p ) ^ { 1 / p } _ { Q _ { \\nu r } ( z _ 0 ) } + ( | D _ v u _ 0 | ^ p ) ^ { 1 / p } _ { Q _ { \\nu r } ( z _ 0 ) } . \\end{align*}"}
+{"id": "6941.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { N } ( - y ) ^ m \\begin{vmatrix} e _ { N + 1 - m } & e _ { N + 2 - m } & e _ { N + 3 - m } & \\cdots & e _ { N + \\ell - m } \\\\ e _ 0 & e _ 1 & e _ 2 & \\cdots & e _ { \\ell - 1 } \\\\ 0 & e _ 0 & e _ 1 & \\cdots & e _ { \\ell } \\\\ \\vdots & \\vdots & \\vdots & \\cdots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & e _ 1 \\end{vmatrix} . \\end{align*}"}
+{"id": "4851.png", "formula": "\\begin{align*} \\frac { \\partial \\rho _ 0 } { \\partial t } = \\nabla \\cdot ( \\rho _ 0 \\nabla f ) + \\bar \\rho _ 1 \\otimes \\mu - \\rho _ 1 . \\end{align*}"}
+{"id": "7206.png", "formula": "\\begin{align*} K ^ { \\mathrm { t o p } } _ 0 ( \\mathbb { S } ^ { \\mathrm { g r } } _ A ) \\cong \\bigotimes _ { i = 1 } ^ k K ^ { \\mathrm { t o p } } _ 0 \\left ( \\mathbb { S } ^ { \\mathrm { g r } } ( d _ i ) _ { w _ i } \\right ) , \\ K ^ { \\mathrm { t o p } } _ 1 ( \\mathbb { S } ^ { \\mathrm { g r } } _ A ) = 0 . \\end{align*}"}
+{"id": "8255.png", "formula": "\\begin{align*} Z _ { s , \\bar { s } } ( t ) = Z _ { s , \\bar { s } } ^ \\ell ( t ) + Z _ s ^ h ( t ) \\le C , \\end{align*}"}
+{"id": "3803.png", "formula": "\\begin{align*} \\left ( n - 3 \\right ) + \\left ( n - 2 \\right ) = 2 n - 5 = d + 2 n - 3 , \\\\ \\left ( n - 3 \\right ) + \\left ( n - 1 \\right ) = 2 n - 4 = d + 2 n - 3 , \\\\ \\left ( n - 2 \\right ) + n = 2 n - 2 = d + 2 n - 3 , \\\\ \\left ( n - 1 \\right ) + n = 2 n - 1 = d + 2 n - 3 . \\end{align*}"}
+{"id": "4205.png", "formula": "\\begin{align*} Q _ k ( X ) & = \\frac { q ^ { 1 - g } h } { \\zeta ( k ) ( q - 1 ) ^ 2 } q ^ X - \\sum _ { j = 1 } ^ { 2 g } \\sum _ { \\ell = 0 } ^ { k - 1 } \\frac { Z ( \\gamma _ { j , \\ell } ^ { - 1 } ) \\gamma _ { j , \\ell } } { k \\gamma _ { j , \\ell } ^ { 1 - k } Z ' ( \\gamma _ j ^ { - 1 } ) } \\frac { \\gamma _ { j , \\ell } ^ X } { \\gamma _ { j , \\ell } - 1 } + \\varepsilon _ { k } ( X ) , \\end{align*}"}
+{"id": "2892.png", "formula": "\\begin{align*} p \\Delta \\left ( H - \\frac { \\mu } { n } \\right ) ^ { p - 1 } + p \\lvert A \\rvert ^ 2 \\left ( H - \\frac { \\mu } { n } \\right ) ^ { p - 1 } - n ^ 2 H \\left ( \\left ( H - \\frac { \\mu } { n } \\right ) ^ p + \\varsigma \\right ) = 0 , \\end{align*}"}
+{"id": "4953.png", "formula": "\\begin{align*} z \\cdot [ x _ 1 , \\ldots , x _ n ] = \\frac { 1 } { n } \\sum \\limits _ { i = 1 } ^ n [ x _ 1 , \\ldots , z \\cdot x _ i , \\ldots , x _ n ] . \\end{align*}"}
+{"id": "4986.png", "formula": "\\begin{align*} E [ d _ { c } ^ { l } ] = E [ W _ { u } ^ { l } ] + E [ a _ { u } ] . \\end{align*}"}
+{"id": "3940.png", "formula": "\\begin{align*} \\left \\| \\frac { 2 ( h - t ) } { 2 h } \\frac { \\det ( d T _ n ) } { \\det ( d T _ \\infty ) } - 1 \\right \\| _ { L ^ \\infty ( U _ { h } ( \\Gamma _ \\infty ) ) } = { \\operatorname { o } } _ { h \\to 0 } ( 1 ) + \\operatorname { O } \\left ( \\frac { t } { h } \\right ) . \\end{align*}"}
+{"id": "3489.png", "formula": "\\begin{align*} \\textbf { e r r } ^ { ( 3 ) } : = \\mathcal { O } \\Bigg ( \\varepsilon ^ 4 \\sqrt { \\mathcal { K } ^ { ( \\mathrm { T _ 0 } ) } _ { \\mathrm { r } } } \\frac { 1 } { | \\xi - z | ^ { 2 - 2 \\mathrm { r } } } \\Big \\Vert | \\mathrm { E } | ^ { 2 } \\Big \\Vert _ { \\mathrm { L } ^ 2 ( \\Omega ) } \\Bigg ) . \\end{align*}"}
+{"id": "5523.png", "formula": "\\begin{align*} d _ { l , j , k } : = \\dim H _ l ( H , ( \\rho _ { j , k } / \\rho _ { j , k + 1 } ) \\otimes \\chi ^ { \\vee } ) . \\end{align*}"}
+{"id": "6669.png", "formula": "\\begin{align*} \\overline { P } ( 0 ) & = \\sigma \\left ( \\frac { \\overline { Q } ( t ) - \\overline { Q } ( 0 ) } { t } \\right ) = \\mathcal { D } \\left ( ( s _ { k } ' ) _ { k = 0 } ^ { 2 m - 3 } , ( ( s _ { k - 1 } ' ) ) _ { k = 0 } ^ { m - 1 } \\right ) , \\end{align*}"}
+{"id": "2558.png", "formula": "\\begin{align*} \\mu _ t ^ { * , \\xi } = k ( t , \\nu _ t ^ { * , \\xi } , \\varphi _ t ^ { * , \\xi } ) . \\end{align*}"}
+{"id": "4290.png", "formula": "\\begin{align*} \\frac { - 1 } { ( - q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { n q ^ { n ( n + 1 ) / 2 } } { ( q ) _ n } + \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ n q ^ { n ( n + 1 ) } } { ( q ^ 2 ; q ^ 2 ) _ n ( 1 - q ^ n ) } = \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - q ) ^ n } { ( q ) _ n ( 1 - q ^ n ) } . \\end{align*}"}
+{"id": "7805.png", "formula": "\\begin{align*} \\begin{gathered} s _ { 1 } ( \\theta ) - t _ { 1 } ( \\theta ) = \\arcsin ( \\frac { M \\sin ^ 2 ( \\theta ) - 1 } { \\cos ( \\theta ) } ) - \\arcsin ( - \\cos ( \\theta ) ) \\leq \\\\ \\frac { 1 } { ( 1 - \\cos ^ 2 ( \\theta ) ) ^ { \\frac { 1 } { 2 } } } \\frac { ( M - 1 ) \\sin ^ { 2 } ( \\theta ) } { \\cos ( \\theta ) } = \\\\ \\frac { ( M - 1 ) \\sin ( \\theta ) } { \\cos ( \\theta ) } , \\end{gathered} \\end{align*}"}
+{"id": "7984.png", "formula": "\\begin{align*} 1 & = \\left \\langle \\nu ^ H , \\frac { \\xi ^ H } { | \\xi ^ H | } \\right \\rangle ^ 2 + \\left \\langle \\eta , \\frac { \\xi ^ H } { | \\xi ^ H | } \\right \\rangle ^ 2 \\\\ & = \\left ( \\frac { ( 2 n - 1 ) c } { 2 n + 1 } | \\xi ^ H | ^ 2 \\right ) ^ 2 + \\left ( \\frac { ( 2 n - 1 ) c } { 2 n + 1 } \\left ( t - t _ 1 - \\frac { 2 n + 1 } { ( 2 n - 1 ) c } \\right ) \\right ) ^ 2 \\end{align*}"}
+{"id": "3314.png", "formula": "\\begin{gather*} L _ 2 \\big ( \\phi _ { m _ 1 } \\otimes \\phi _ { m _ 2 } ^ { m _ 1 } \\big ) = \\mu _ { m _ 2 } \\phi _ { m _ 1 } \\otimes \\phi _ { m _ 2 } ^ { m _ 1 } . \\end{gather*}"}
+{"id": "1144.png", "formula": "\\begin{align*} ( \\phi | _ { k , S } \\gamma ) ( \\tau , z ) : = J _ { k , S } ( \\gamma , ( \\tau , z ) ) \\phi ( \\gamma ( \\tau , z ) ) , \\end{align*}"}
+{"id": "8180.png", "formula": "\\begin{align*} \\| C ^ 2 a \\| ^ 2 + \\| C ^ 2 b \\| ^ 2 = \\| a \\| ^ 2 \\| C b \\| ^ 2 + \\| b \\| ^ 2 \\| C a \\| ^ 2 . \\end{align*}"}
+{"id": "7935.png", "formula": "\\begin{align*} ( \\mu _ 1 , R _ 1 ) = \\delta _ \\mathrm { m R B A } ( \\varphi _ 1 , a ) , ( \\varphi _ 1 , a ) \\in C ^ 1 _ \\mathrm { m R B A } ( ( A , R ) , ( A , R ) ) . \\end{align*}"}
+{"id": "3223.png", "formula": "\\begin{align*} \\ell = \\max \\left \\{ \\frac { \\delta ( ( D - 1 ) G ) + 1 } { m } , 2 \\right \\} = \\max \\left \\{ \\delta + \\frac { 1 } { D - 1 } , 2 \\right \\} = \\delta + 1 . \\end{align*}"}
+{"id": "7478.png", "formula": "\\begin{align*} \\sum _ { \\boldsymbol { \\nu } _ a = 1 } ^ { n } \\nu ^ m & = \\sum _ { \\boldsymbol { \\nu } = 1 } ^ n \\sum _ k c _ { m k } \\psi _ m ( \\nu ) \\ , , \\\\ & = \\sum _ k c _ { m k } \\sum _ { \\boldsymbol { \\nu } = 1 } ^ n \\psi _ m ( \\nu ) \\ , , \\\\ & = \\sum _ k c _ { m k } \\psi _ m ^ { ( a ) } ( n ) \\ , . \\end{align*}"}
+{"id": "616.png", "formula": "\\begin{align*} \\rho _ { n , p } & = \\Big [ { \\frac { p ( 2 j ^ { ( 4 ) } + 1 - p ) ( 2 j ^ { ( 0 ) } + 1 - p ) ( 2 j ^ { ( 4 ) } + 2 j ^ { ( 0 ) } + 2 - p ) ( N - n - p + 2 j ^ { ( 3 ) } + 2 ) ( N - n - p + 1 ) } { ( 2 j ^ { ( 0 ) } + 2 j ^ { ( 4 ) } + 2 - 2 p ) ^ 2 ( 2 j ^ { ( 0 ) } + 2 j ^ { ( 4 ) } + 1 - 2 p ) ( 2 j ^ { ( 0 ) } + 2 j ^ { ( 4 ) } + 3 - 2 p ) } } \\\\ & \\qquad \\times { ( p - n - N + 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } ) ( n - p - N + 2 j ^ { ( 0 ) } + 2 j ^ { ( 4 ) } + 1 ) } \\Big ] ^ { \\frac 1 2 } \\ , , \\end{align*}"}
+{"id": "7126.png", "formula": "\\begin{align*} \\chi = - \\sum _ { \\ell \\in T } r _ l ( 3 \\mathfrak { g } _ l ) + \\frac { 1 } { 2 } \\mathfrak { g } ^ { \\lambda > 0 } + w \\tau _ d + \\sum _ { i = 1 } ^ k ( \\psi _ i - \\rho _ i ) . \\end{align*}"}
+{"id": "2826.png", "formula": "\\begin{align*} d \\gamma _ s ^ { \\frac 1 2 } & = \\gamma _ s ^ { \\frac 1 2 } \\left ( \\frac 1 2 \\mu _ s - \\frac { 1 } { 8 } \\sigma _ s ^ 2 \\right ) d s + \\frac 1 2 \\gamma _ s ^ { \\frac 1 2 } \\sigma _ s d W ^ 1 _ s , s \\in [ 0 , T ] , \\end{align*}"}
+{"id": "1127.png", "formula": "\\begin{align*} H _ { j , n } = \\frac { \\sqrt { n ! } } { \\Gamma ( \\beta ) \\sqrt { \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } } } \\int _ { 0 } ^ { \\infty } x ^ { j } L _ { n } ^ { ( \\beta ) } ( x ) x ^ { \\beta - 1 } \\exp ( - x ) d x . \\end{align*}"}
+{"id": "1120.png", "formula": "\\begin{align*} f _ { G } ( x , y | \\rho ) = f _ { g } ( x | \\beta ) f _ { g } ( y | \\beta ) \\sum _ { j \\geq 0 } \\rho ^ { n } l _ { n } ( x | \\beta ) l _ { n } ( y | \\beta ) , \\end{align*}"}
+{"id": "5725.png", "formula": "\\begin{align*} \\nabla f ( x ) : = \\left ( \\frac { \\partial \\phi } { \\partial z _ 1 } ( \\pi _ n ( x ) ) , \\dots , \\frac { \\partial \\phi } { \\partial z _ n } ( \\pi _ n ( x ) ) , 0 , 0 , \\dots \\right ) . \\end{align*}"}
+{"id": "6291.png", "formula": "\\begin{align*} \\sum _ { i ' = 0 } ^ { \\ell ' - 1 } y _ { i ' } ' \\cdot \\frac { q ^ { i ' } } { q ^ { \\ell ' } - 1 } = y = \\sum _ { i = 0 } ^ { \\ell - 1 } y _ i \\cdot \\frac { q ^ i } { q ^ \\ell - 1 } = \\sum _ { i = 0 } ^ { \\ell - 1 } y _ i \\cdot \\sum _ { j = 0 } ^ { \\ell ' - 1 } \\frac { q ^ { i + \\ell j } } { q ^ { \\ell \\ell ' } - 1 } = \\sum _ { i '' = 0 } ^ { \\ell \\ell ' - 1 } y _ { i '' } '' \\cdot \\frac { q ^ { i '' } } { q ^ { \\ell \\ell ' } - 1 } , \\end{align*}"}
+{"id": "3584.png", "formula": "\\begin{align*} \\mathcal { \\tilde { H } } = \\left \\{ f ( x ) = \\sum _ i f _ i \\phi _ i ( x ) , x \\in [ 0 , 1 ] : \\sum _ i \\frac { f _ i ^ 2 } { a _ i \\tau _ i } < \\infty \\right \\} , \\end{align*}"}
+{"id": "3773.png", "formula": "\\begin{align*} p ^ \\nu _ { k _ 1 , \\ldots , k _ n } = \\sum _ { r = 1 } ^ { k _ 1 - 1 } p ^ \\nu _ { r - 1 , k _ 1 - r - 1 , k _ 2 , \\ldots , k _ n } + \\sum _ { r = 2 } ^ n k _ r p ^ \\nu _ { k _ 1 + k _ r - 2 , k _ 2 , \\ldots , \\widehat { k _ r } , \\ldots , k _ n } . \\end{align*}"}
+{"id": "6137.png", "formula": "\\begin{align*} D _ { V ^ * } h \\mapsto \\begin{bmatrix} D _ { V _ 1 ^ * } \\\\ D _ { V _ 2 ^ * } V _ 1 ^ * \\end{bmatrix} h ; \\quad \\mbox { a n d } \\end{align*}"}
+{"id": "2789.png", "formula": "\\begin{align*} A ^ T \\tilde { U } _ k = \\begin{bmatrix} \\tilde { V } _ k & q _ { k + 1 } \\end{bmatrix} \\tilde { J } _ k ^ T , \\tilde { J } _ k = \\begin{bmatrix} \\tilde { \\sigma } _ 1 & & & & \\rho _ 1 \\\\ & \\tilde { \\sigma } _ 2 & & & \\rho _ 2 \\\\ & & \\ddots & & \\vdots \\\\ & & & \\tilde { \\sigma } _ k & \\rho _ k \\\\ \\end{bmatrix} , \\end{align*}"}
+{"id": "1734.png", "formula": "\\begin{align*} \\| F \\| _ { L ^ 4 _ { t , x } } \\lesssim \\| F \\| _ { X ^ { 0 , 3 / 8 } } \\leq \\| F \\| _ { X ^ { 0 , b - 1 } } = \\| c \\| _ { \\ell ^ 2 _ k L ^ 2 _ \\eta } = 1 \\ , . \\end{align*}"}
+{"id": "5427.png", "formula": "\\begin{align*} H ^ { ( 2 ) } ( t ) = \\mathrm { e } ^ { - \\frac { t } { 2 } } W ^ { S } + \\sqrt { 1 - \\mathrm { e } ^ { - t } } H ^ { S } , \\end{align*}"}
+{"id": "321.png", "formula": "\\begin{align*} \\theta = [ \\theta _ { 1 } , \\cdots , \\theta _ { n } ] ^ { \\top } \\end{align*}"}
+{"id": "967.png", "formula": "\\begin{align*} \\gamma ( x ) : = p ( x , e ) \\equiv p ( x , 1 ) . \\end{align*}"}
+{"id": "3157.png", "formula": "\\begin{align*} \\left ( m ^ { 1 } { \\bf v } ^ { \\prime } _ { 1 } + m ^ { 2 } { \\bf v } ^ { \\prime } _ { 2 } + m ^ { 3 } { \\bf u } ^ { \\prime } _ { 1 } + m ^ { 4 } { \\bf u } ^ { \\prime } _ { 2 } \\right ) \\cdot { \\bf l } \\ , \\ , \\ , = \\ , \\ , \\ , 0 \\end{align*}"}
+{"id": "3294.png", "formula": "\\begin{align*} \\Delta ( C _ 0 ) = C _ 0 ( 1 \\otimes 1 ) \\ \\ \\Delta ( C _ 1 ) = C _ 1 ( 1 \\otimes 1 ) . \\end{align*}"}
+{"id": "1826.png", "formula": "\\begin{align*} \\psi ( [ a , b ] _ T ) = ~ & \\psi \\big ( \\rho ( T ( a ) ) b - \\rho ( T ( b ) ) a + \\lambda [ a , b ] _ V \\big ) \\\\ = ~ & \\rho ( \\phi ( T ( a ) ) ) ( \\psi ( b ) ) - \\rho ( \\phi ( T ( b ) ) ) ( \\psi ( a ) ) + \\lambda [ \\psi ( a ) , \\psi ( b ) ] _ V \\\\ = ~ & \\rho ( T ' ( \\psi ( a ) ) ) ( \\psi ( b ) ) - \\rho ( T ' ( \\psi ( b ) ) ) ( \\psi ( a ) ) + \\lambda [ \\psi ( a ) , \\psi ( b ) ] _ V = [ \\psi ( a ) , \\psi ( b ) ] _ { T ' } . \\end{align*}"}
+{"id": "506.png", "formula": "\\begin{align*} \\log Z = \\log \\int _ { \\R ^ n } e ^ f \\ , d \\mu ^ { \\otimes n } = \\sup _ { Q \\in \\P ( \\R ^ n ) } \\left ( \\int _ { \\R ^ n } f \\ , d Q - H ( Q \\ , | \\ , \\mu ^ { \\otimes n } ) \\right ) , \\end{align*}"}
+{"id": "1387.png", "formula": "\\begin{align*} \\beta \\varphi _ { \\beta , \\varepsilon } ( s ) + s \\varphi _ { \\beta , \\varepsilon } ' ( s ) = \\beta \\varphi _ { \\beta + 1 , \\varepsilon } ( s ) . \\end{align*}"}
+{"id": "2881.png", "formula": "\\begin{align*} \\Big \\{ ( \\nu _ 1 , \\nu _ 2 , \\nu _ 3 ) : \\nu _ 1 = \\frac { 1 } { 2 } \\nu _ 2 ^ 2 , 1 - \\frac { 1 } { K } \\le \\nu _ 3 \\le 1 , \\Big | \\frac { \\nu _ 2 } { \\nu _ 3 } \\Big | \\le 1 \\Big \\} \\end{align*}"}
+{"id": "5612.png", "formula": "\\begin{align*} \\mathbf { R i c } ( y ) = \\frac { g ^ { i k } R _ { i k } } { G ( y ) } = \\sum _ { j = 1 } ^ { 2 n - 1 } \\mathbf { K } ( y , e _ i ) \\end{align*}"}
+{"id": "6684.png", "formula": "\\begin{align*} Q ( t ) = \\det \\begin{bmatrix} s _ { - 1 } & s _ { 0 } & \\ldots & s _ { K - 2 } & 1 \\\\ s _ { 0 } & s _ { 1 } & \\ldots & s _ { K - 1 } & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ s _ { K - 1 } & s _ { K } & \\ldots & s _ { 2 K - 2 } & t ^ { K } \\end{bmatrix} , \\end{align*}"}
+{"id": "5181.png", "formula": "\\begin{align*} \\mathcal W = \\{ Q \\ , : \\ , Q \\cap \\Gamma \\ne \\emptyset , Q \\R ^ n \\setminus \\overline { \\Omega } \\} . \\end{align*}"}
+{"id": "792.png", "formula": "\\begin{align*} \\mathcal { J } _ i ( \\tau _ i , \\tau _ { - i } ) = \\mathbb { E } \\left \\{ e ^ { - \\beta \\tau _ i } \\Bigg [ \\frac { x _ i ( \\tau _ i ) } { l _ 1 + l _ 2 \\left ( \\frac { 1 } { N } \\sum \\limits _ { j = 1 } ^ N e ^ { - \\beta \\tau _ j } x _ j ( \\tau _ j ) \\right ) } - K \\Bigg ] \\right \\} . \\end{align*}"}
+{"id": "5584.png", "formula": "\\begin{align*} \\left ( g _ { i j } \\right ) = \\frac 1 2 \\left ( \\frac { \\partial ^ 2 G ^ o } { \\partial y ^ i \\partial y ^ j } \\right ) . \\end{align*}"}
+{"id": "3326.png", "formula": "\\begin{gather*} \\sum _ { j _ 1 = 0 } ^ { N _ 1 } \\sum _ { j _ 2 = 0 } ^ { N _ 2 } \\big \\langle \\psi _ { n _ 1 ( k _ 1 ' ) } ^ { n _ 2 ( k _ 2 ' ) } , \\phi _ { m _ 1 ( j _ 1 ) } \\big \\rangle _ { V _ 1 } \\big \\langle \\phi _ { m _ 1 ( j _ 1 ) } , \\psi _ { n _ 1 ( k _ 1 ) } ^ { n _ 2 ( k _ 2 ) } \\big \\rangle _ { V _ 1 } \\big \\langle \\psi _ { n _ 2 ( k _ 2 ' ) } , \\phi _ { m _ 2 ( j _ 2 ) } ^ { m _ 1 ( j _ 1 ) } \\big \\rangle _ { V _ 2 } \\big \\langle \\phi _ { m _ 2 ( j _ 2 ) } ^ { m _ 1 ( j _ 1 ) } , \\psi _ { n _ 2 ( k _ 2 ) } \\big \\rangle _ { V _ 2 } . \\end{gather*}"}
+{"id": "6585.png", "formula": "\\begin{align*} [ A , B ] = O \\mbox { a n d } [ A , B ^ * ] = O . \\end{align*}"}
+{"id": "1280.png", "formula": "\\begin{align*} \\ulcorner f \\urcorner = ( a ^ \\ast \\otimes f ) \\circ \\eta _ a , \\llcorner f \\lrcorner = \\varepsilon _ b \\circ ( f \\otimes b ^ \\ast ) . \\end{align*}"}
+{"id": "7234.png", "formula": "\\begin{align*} \\Im ( \\sum _ { ( j , j _ 1 , j _ 2 , j _ 3 ) \\in \\mathcal { R } } u _ j \\bar { u } _ { j _ 1 } u _ { j _ 2 } \\bar { u } _ { j _ 3 } ) = 0 . \\end{align*}"}
+{"id": "2046.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 - 2 X ) ^ 2 ( 1 - X ) ^ 2 } \\sum _ { n = 0 } ^ { \\infty } \\lambda _ { n + 1 } \\left ( \\frac { X } { X - 1 } \\right ) ^ n = \\sum _ { n = 0 } ^ { \\infty } \\mu _ n \\frac { X ^ n } { n ! } . \\end{align*}"}
+{"id": "3543.png", "formula": "\\begin{align*} \\varphi ( \\mathrm { v } , \\mathrm { y } , \\mathrm { t } , \\tau ) : = \\displaystyle \\int _ { 0 } ^ { \\tau } \\dfrac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 ( \\mathrm { s } - \\tau ) ^ 2 } \\textbf { e x p } \\Big ( - \\dfrac { \\alpha | \\mathrm { y } - \\mathrm { v } | ^ 2 } { 4 ( \\mathrm { s } - \\tau ) } \\Big ) \\Phi ^ { \\textbf { e } } ( \\xi , \\mathrm { t } ; \\mathrm { z } , \\mathrm { s } ) d \\mathrm { s } . \\end{align*}"}
+{"id": "2484.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta _ { p } u & = v ^ { m } | \\nabla u | ^ { \\alpha } & & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta _ { p } v & = v ^ { \\beta } | \\nabla u | ^ q & & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "2873.png", "formula": "\\begin{align*} N _ j ( t ) - N _ j ( 0 ) = \\sum _ { k = 1 } ^ d \\int _ 0 ^ t \\Psi _ { j k } ( s ) d \\beta ^ c _ k ( s ) \\forall \\ , 0 \\le t \\le T , \\ ; j = 1 , \\dots , d . \\end{align*}"}
+{"id": "3558.png", "formula": "\\begin{align*} [ H ^ 0 _ { G _ + } ( G _ I ) M ) ) ] _ n = \\frac { \\widetilde { I ^ { n + 1 } M } \\cap I ^ n M } { I ^ { n + 1 } M } . \\end{align*}"}
+{"id": "6634.png", "formula": "\\begin{align*} \\begin{aligned} h _ { \\theta } ( J v , J v ) & = \\int _ { \\Omega _ + } | \\nabla v | ^ 2 \\dd x - \\int _ { \\Gamma _ \\theta } | v | ^ 2 \\dd \\sigma \\\\ & \\equiv \\int _ { \\Omega _ + } | \\nabla v | ^ 2 \\dd x - \\int _ { \\partial \\Omega _ + } | v | ^ 2 \\dd \\sigma \\equiv q _ \\theta ( v , v ) ; \\end{aligned} \\end{align*}"}
+{"id": "3840.png", "formula": "\\begin{align*} a _ 1 = \\frac { 2 \\left ( 2 n - 5 \\right ) + d - \\left ( a _ 2 + a _ 3 + a _ 4 \\right ) } { 2 } , \\end{align*}"}
+{"id": "3884.png", "formula": "\\begin{align*} \\Gamma _ { \\mu _ 1 , \\mu _ 2 } ( x ) = | x | ^ { \\tau _ { + } ( \\mu _ 1 , \\mu _ 2 ) } , \\end{align*}"}
+{"id": "7817.png", "formula": "\\begin{align*} J ^ { w _ 1 } ( X _ { i : n } ) & = - \\frac { n c _ { i , n } } { 2 } \\int _ { 0 } ^ { 1 } \\Lambda _ { X } ^ { w _ 1 } ( u ) \\phi _ { 2 i - 1 : 2 n - 1 } ( u ) d u \\\\ & = - \\frac { c _ { i , n } ( 2 i - 1 ) } { 4 } \\int _ { 0 } ^ { 1 } w _ 1 ( F ^ { - 1 } ( u ) ) r \\left ( F ^ { - 1 } ( u ) \\right ) \\phi _ { 2 i : 2 n } ( u ) d u \\\\ & = - \\frac { c _ { i , n } ( 2 i - 1 ) } { 4 } E \\left ( M ( B _ { 2 i : 2 n } ) \\right ) , \\end{align*}"}
+{"id": "4235.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( b / a ) _ n a ^ n } { ( 1 - c q ^ n ) ( b ) _ n } = \\sum _ { m = 0 } ^ { \\infty } \\frac { ( b / c ) _ m c ^ m } { ( b ) _ m } \\left ( \\frac { a q ^ m } { 1 - a q ^ m } - \\frac { b q ^ m } { 1 - b q ^ m } \\right ) . \\end{align*}"}
+{"id": "4043.png", "formula": "\\begin{align*} \\partial _ s q _ n = P _ { n } ( \\partial _ s q ) + \\tilde L , \\end{align*}"}
+{"id": "2830.png", "formula": "\\begin{align*} K _ s = \\int _ t ^ s K _ r \\nu _ r ^ { - 1 } \\gamma _ r ^ { - 1 } d ( \\nu _ r \\gamma _ r ) , s \\in [ t , T ] . \\end{align*}"}
+{"id": "626.png", "formula": "\\begin{align*} C \\left ( X , Y , Z \\right ) \\equiv \\left ( \\nabla _ { Z } g \\right ) \\left ( X , Y \\right ) = Z g \\left ( X , Y \\right ) - g \\left ( \\nabla _ { Z } X , Y \\right ) - g \\left ( X , \\nabla _ { Z } Y \\right ) \\end{align*}"}
+{"id": "7989.png", "formula": "\\begin{align*} \\left ( c _ 0 d \\right ) ^ { t - 1 } \\geq \\binom { n } { t - a } \\cdot r \\cdot \\left ( \\frac { 2 \\rho r } { a } \\right ) ^ { 3 t } . \\end{align*}"}
+{"id": "2532.png", "formula": "\\begin{align*} S _ \\beta ( x , y , z ) = ( x + \\alpha , y + x , z + x + \\beta ) \\end{align*}"}
+{"id": "980.png", "formula": "\\begin{align*} ( 2 w _ 1 + n - 2 s - 2 ) a _ { s + 1 , t } & = - ( 2 w _ 1 + 2 w _ 2 + n - 2 k - 2 s - 2 t ) a _ { s , t } , \\\\ ( 2 w _ 2 + n - 2 t - 2 ) a _ { s , t + 1 } & = - ( 2 w _ 1 + 2 w _ 2 + n - 2 k - 2 s - 2 t ) a _ { s , t } \\end{align*}"}
+{"id": "7076.png", "formula": "\\begin{align*} f _ X ( x , y ) = 0 , \\end{align*}"}
+{"id": "2350.png", "formula": "\\begin{align*} f _ { c } ( x _ { a } u , x _ { b } v ) = & x _ { c } f _ { a } ( u , x _ { b } v ) + x _ { c } f _ { b } ( x _ { a } u , v ) - f _ { c + a + b } ( u , v ) \\quad ( u , v \\in \\mathfrak { X } ) , \\end{align*}"}
+{"id": "1659.png", "formula": "\\begin{align*} { \\tau } _ { \\mathsf { m } } : = \\eta ( \\mathsf { A } _ { \\mathsf { m } } , \\mathsf { A } _ { \\mathsf { m } + 1 } ) \\geq 2 ^ 4 \\lambda \\ , , \\qquad \\qquad \\mathsf { m } = 1 , \\dots , \\mathsf { M } \\ , . \\end{align*}"}
+{"id": "4540.png", "formula": "\\begin{align*} \\sup _ { x \\in U } \\chi = \\mathrm { i n f } \\{ a \\in \\mathbb { R } : f ^ { - 1 } ( a , \\infty ) = \\varnothing \\} = 1 \\ , , \\end{align*}"}
+{"id": "101.png", "formula": "\\begin{align*} \\varphi ( t ) = 1 + A ( t ) - B ( t ) \\end{align*}"}
+{"id": "3848.png", "formula": "\\begin{align*} X = \\left ( X ^ { ( n ) } ( u ) : n \\in \\N , u \\in U ^ { ( n ) } \\right ) \\quad Y = \\left ( Y ^ { ( n ) } ( u ) : n \\in \\N , u \\in U ^ { ( n ) } \\right ) \\end{align*}"}
+{"id": "7543.png", "formula": "\\begin{align*} C ^ { ( 2 , - 1 ) } ( u , v ; a ) \\frac { d v } { d u ^ 2 } = \\frac { 1 } { u - v } + O ( 1 ) \\end{align*}"}
+{"id": "4664.png", "formula": "\\begin{align*} S _ f & : = \\{ x \\in \\mathbb { F } _ q [ t ] ^ m : f ( x ) B \\} , \\\\ \\mu _ { S _ f } & : = \\lim \\limits _ { N \\rightarrow \\infty } \\frac { | \\{ b \\in S _ f : | b | < N \\} | } { N ^ m } . \\end{align*}"}
+{"id": "6247.png", "formula": "\\begin{align*} ( \\alpha , b ) ( \\beta , c ) = ( \\alpha \\beta ^ { b ^ { - 1 } } , b c ) \\qquad \\textup { w h e r e $ \\alpha , \\beta \\in A ^ Y $ , $ b , c \\in B $ , } \\end{align*}"}
+{"id": "3639.png", "formula": "\\begin{align*} K '' ( x ) = \\theta e ^ { \\theta ( x - a ) } \\int _ 0 ^ \\infty [ h ' ( a + z ) + c ] \\nu ( d z ) + 2 , x < a . \\end{align*}"}
+{"id": "2995.png", "formula": "\\begin{align*} f ' ( x ) & = \\frac { b ' \\left ( a ^ { - 1 } ( x ) \\right ) } { a ' \\left ( a ^ { - 1 } ( x ) \\right ) } = - \\frac { a \\left ( a ^ { - 1 } ( x ) \\right ) b ( a ^ { - 1 } ( x ) ) ^ 2 } { a \\left ( a ^ { - 1 } ( x ) \\right ) ^ 3 / 2 + a \\left ( a ^ { - 1 } ( x ) \\right ) b \\left ( a ^ { - 1 } ( x ) \\right ) ^ 2 / 2 } \\\\ & = \\frac { 2 x f ( x ) ^ 2 } { x ^ 3 + x f ( x ) ^ 2 } = \\frac { 2 } { ( x / f ( x ) ) ^ 2 + 1 } \\ ; . \\end{align*}"}
+{"id": "3357.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { L _ { 1 } K _ \\mathcal { I } / L _ { 1 } } ( x _ 1 ) = \\mathrm { R e s } _ { L K _ \\mathcal { I } / L } ( x _ 1 ) = z , \\end{align*}"}
+{"id": "3089.png", "formula": "\\begin{align*} A _ 2 = \\left ( s _ 2 ( k ) \\right ) _ { k = 1 } ^ { \\infty } = ( 0 , 2 , 3 , 6 , 5 , 1 1 , 7 , 1 4 , 1 2 , 1 7 , 1 9 , 2 7 , \\ldots ) . \\end{align*}"}
+{"id": "4525.png", "formula": "\\begin{align*} ^ 2 E _ S [ f ] = \\int _ { \\mathbf { x } \\in S } \\mathrm { d } S ^ { b c } ( \\mathbf { x } ) f ^ i ( \\mathbf { x } ) \\hat { \\mathfrak { E } } _ { \\mathbf { x } i } ^ a v _ { a b c } ( \\mathbf { x } ) \\ , , \\end{align*}"}
+{"id": "347.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } - \\Delta u = G ( t ) , ( x , t ) \\in \\Omega \\times \\mathbb { R } ^ + , \\\\ u | _ { \\partial \\Omega } = 0 , t \\in \\mathbb { R } ^ + , \\\\ \\xi _ u ( 0 ) = \\xi _ 0 , x \\in \\Omega . \\end{cases} \\end{align*}"}
+{"id": "3501.png", "formula": "\\begin{align*} \\partial _ { \\mathrm { i j } } \\Big ( \\mathbb { G } ^ { ( \\mathrm { k } ) } - \\mathbb { G } ^ { ( 0 ) } \\Big ) ( \\mathrm { x } , \\mathrm { y } ) = 2 a \\delta _ { \\mathrm { i j } } \\log | x - y | + 2 \\mathrm { a } \\dfrac { ( \\mathrm { x } - \\mathrm { y } ) _ { \\mathrm { i } } ( \\mathrm { x } - \\mathrm { y } ) _ { \\mathrm { j } } } { | \\mathrm { x } - \\mathrm { y } | ^ { 2 } } + 2 \\mathrm { b } \\delta _ { \\mathrm { i j } } + \\mathcal { O } ( | \\mathrm { x } - \\mathrm { y } | ) , \\mathrm { x } \\sim \\mathrm { y } . \\end{align*}"}
+{"id": "4421.png", "formula": "\\begin{align*} \\left [ ( a _ 1 + a _ 2 ) ^ p + ( b _ 1 + b _ 2 ) ^ q \\right ] ^ { 1 / p } & \\leq \\left [ ( a _ 1 + a _ 2 ) ^ p + \\bigl ( b _ 1 ^ { q / p } + b _ 2 ^ { q / p } \\bigr ) ^ p \\right ] ^ { 1 / p } \\\\ & \\leq \\left [ a _ 1 ^ p + \\bigl ( b _ 1 ^ { q / p } \\bigr ) ^ p \\right ] ^ { 1 / p } + \\left [ a _ 2 ^ p + \\bigl ( b _ 2 ^ { q / p } \\bigr ) ^ p \\right ] ^ { 1 / p } \\\\ & = \\bigl ( a _ 1 ^ p + b _ 1 ^ q \\bigr ) ^ { 1 / p } + \\bigl ( a _ 2 ^ p + b _ 2 ^ q \\bigr ) ^ { 1 / p } . \\end{align*}"}
+{"id": "2172.png", "formula": "\\begin{align*} & \\gamma ^ 1 _ { i k } \\alpha _ { j l } ^ 0 + \\alpha _ { i k } ^ 0 \\gamma ^ 1 _ { j l } - \\gamma ^ 1 _ { i l } \\alpha _ { j k } ^ 0 - \\alpha _ { i l } ^ 0 \\gamma ^ 1 _ { j k } \\\\ = & T _ { i j k l } + \\frac { 1 } { | A _ 0 | } ( \\alpha _ { i k } ^ 0 \\alpha _ { j l } ^ 0 - \\alpha _ { i l } ^ 0 \\alpha _ { j k } ^ 0 ) ( \\overline { \\overline { \\alpha } } _ 1 ^ { \\ , 0 } T _ { 2 3 } - \\overline { \\overline { \\alpha } } _ 2 ^ { \\ , 0 } T _ { 1 3 } + \\overline { \\overline { \\alpha } } _ 3 ^ { \\ , 0 } T _ { 1 2 } ) ^ 2 . \\end{align*}"}
+{"id": "3231.png", "formula": "\\begin{align*} \\int _ { \\{ v < u \\} } \\left ( e ^ { \\lambda u } - e ^ { \\lambda v } \\right ) e ^ { - w } \\mu = 0 . \\end{align*}"}
+{"id": "1093.png", "formula": "\\begin{align*} \\mathbf x ^ { t } \\gets \\underset { \\mathbf z : \\operatorname { s u p p } ( \\mathbf z ) = \\hat { S } ^ { t } } { \\arg \\min } f ( \\mathbf z ) , \\end{align*}"}
+{"id": "8153.png", "formula": "\\begin{align*} ( d + 1 ) \\widehat { \\mu } ^ { \\mathrm { a s y } } ( \\overline L ) \\geqslant \\sum _ { i = 1 } ^ { d + 1 } e _ i ( \\overline L ) . \\end{align*}"}
+{"id": "7033.png", "formula": "\\begin{align*} ( [ X , X ' ] ) ^ p = ( B ' ( v ) - B ( v ' ) , R _ { v , v ' } - [ B , B ' ] ) . \\end{align*}"}
+{"id": "5448.png", "formula": "\\begin{align*} A \\ , \\ , = \\ , \\ , \\begin{pmatrix} A _ 0 & A _ 1 & A _ 2 & \\cdots & A _ { r - 2 } & A _ { r - 1 } \\\\ 0 & A _ 0 & A _ 1 & \\cdots & A _ { r - 3 } & A _ { r - 2 } \\\\ 0 & 0 & A _ 0 & \\cdots & A _ { r - 4 } & A _ { r - 3 } \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & A _ 0 & A _ 1 \\\\ 0 & 0 & 0 & \\cdots & 0 & A _ 0 \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "7003.png", "formula": "\\begin{align*} P _ t u ( x ) \\geq u ( x ) \\exp \\left ( - \\alpha _ 2 t \\log u ( x ) - ( \\lambda - \\alpha _ 1 ) t \\right ) = e ^ { - ( \\lambda - \\alpha _ 1 ) t } u ( x ) ^ { 1 - \\alpha _ 2 t } . \\end{align*}"}
+{"id": "7266.png", "formula": "\\begin{align*} \\frac { 1 } { h _ n } \\| Z _ \\nu ^ { h _ n } ( t ) - & X ^ * ( t ) - h _ n Y _ \\nu ( t ) ) \\| _ { L ^ p } \\leq ( a _ n ^ { ( 1 ) } + a _ n ^ { ( 2 ) } + a _ n ^ { ( 3 ) } ) \\\\ & + ( C _ x + C _ m ) \\int _ { s + h _ n } ^ t \\frac { 1 } { h _ n } \\| Z _ \\nu ^ { h _ n } ( \\tau ) - X ^ * ( \\tau ) - h _ n Y _ \\nu ( \\tau ) ) \\| _ { L ^ p } d \\tau . \\end{align*}"}
+{"id": "7497.png", "formula": "\\begin{align*} B _ { m - j + 1 } = \\frac { ( m - j + 1 ) ! \\ , j ! } { m ! } \\sum _ { k = 2 } ^ m c _ { m k } g _ { k j } \\ , . \\end{align*}"}
+{"id": "2294.png", "formula": "\\begin{align*} \\begin{cases} \\vartheta _ { 2 d } > 1 / ( d + 1 ) & d = 2 , \\ , 3 , \\\\ \\vartheta _ { 2 d } < 1 / ( d + 1 ) & d \\geq 4 , \\end{cases} \\end{align*}"}
+{"id": "1151.png", "formula": "\\begin{align*} \\mathbb B _ { k , S , g } ( \\tau , z ) = v ^ { k } e ^ { - 4 \\pi S [ y ^ t ] / v } B _ { k , S , g } ( \\tau , z ; \\tau , z ) . \\end{align*}"}
+{"id": "3872.png", "formula": "\\begin{align*} f ( w ) : = w ^ 2 + \\log ( | z | ^ 2 - w ^ 2 ) . \\end{align*}"}
+{"id": "6018.png", "formula": "\\begin{align*} \\tilde { C } ^ { ( q , r ) } ( b ; h ) = \\tilde { g } ^ { ( q ) } ( b ; h ) = \\dfrac { \\Xi ^ { ( q , r ) } ( b ; h ) - \\dfrac { \\rho ^ { ( q ) } _ { b } ( b ; h ) } { W ^ { ( q ) } ( b ) } Z ^ { ( q ) } ( b ; \\Phi ( q + r ) ) } { \\displaystyle \\dfrac { Z ^ { ( q ) } ( b ; \\Phi ( q + r ) ) } { W ^ { ( q ) } ( b ) } - \\dfrac { r } { \\Phi ( q + r ) } } , \\end{align*}"}
+{"id": "2915.png", "formula": "\\begin{align*} p _ j ( z ) = z ^ j + a _ 1 z ^ { j - 1 } + { \\cdots } + a _ j \\end{align*}"}
+{"id": "391.png", "formula": "\\begin{align*} L _ \\infty ( M , \\omega ) _ { [ N ] } = F _ N \\cap \\mathcal { Q ' } . \\end{align*}"}
+{"id": "2982.png", "formula": "\\begin{align*} t ^ { \\mathbf { A } } ( F , W ) = \\left ( \\prod _ { i j \\in E } d ( A _ i , A _ j ) \\pm \\alpha | R | ^ { | R | } \\right ) \\prod _ { i \\in R } \\pi ( A _ i ) \\ ; . \\end{align*}"}
+{"id": "855.png", "formula": "\\begin{align*} \\begin{aligned} K \\in \\arg \\min \\mathcal { D } _ { K } ( p | | q ) \\end{aligned} \\end{align*}"}
+{"id": "7451.png", "formula": "\\begin{align*} \\begin{matrix} [ J ^ 2 , \\ , p _ 0 ^ \\dagger ] = 2 p _ 0 ^ \\dagger + 4 p _ 1 ^ \\dagger , [ J ^ 2 , \\ , p _ 1 ^ \\dagger ] = p _ 0 ^ \\dagger \\hat { j } ( \\hat { j } + 1 ) . \\end{matrix} \\end{align*}"}
+{"id": "2578.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d y ^ { * , i } _ t = & - [ A y ^ { * , i } _ t + Q x ^ { * , i } _ t + Q l ( \\nu ^ { N , i } _ { \\boldsymbol { x } ^ * _ t } ) ] d t + \\sum _ { j = 1 } ^ N z _ t ^ { * , i , j } d W ^ j _ t + z ^ { 0 , * , i } d W ^ 0 _ t , \\\\ y ^ { * , i } _ T = & G ( x _ T ^ { * , i } + g ( \\nu ^ { N , i } _ { \\boldsymbol { x } ^ * _ T } ) ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3318.png", "formula": "\\begin{align*} p _ { n _ 1 } ^ { V _ 1 } ( m _ 1 ; n _ 2 ) : = \\frac { \\big \\langle \\phi _ { m _ 1 } , \\psi _ { n _ 1 } ^ { n _ 2 } \\big \\rangle _ { V _ 1 } } { \\big \\langle \\phi _ { m _ 1 } , \\psi _ { n _ 1 ( 0 ) } ^ { n _ 2 } \\big \\rangle _ { V _ 1 } } , \\end{align*}"}
+{"id": "7871.png", "formula": "\\begin{align*} n _ 3 ( 6 t , x ) = \\# F _ B ( x ) . \\end{align*}"}
+{"id": "4203.png", "formula": "\\begin{align*} \\frac { Z ( u ) } { Z ( u ^ k ) } & = \\frac { ( 1 - u ^ k ) ( 1 - q u ^ k ) } { ( 1 - u ) ( 1 - q u ) } \\frac { \\prod _ { j = 1 } ^ { 2 g } ( 1 - \\gamma _ j u ) } { \\prod _ { j = 1 } ^ { 2 g } ( 1 - \\gamma _ j u ^ k ) } . \\end{align*}"}
+{"id": "6639.png", "formula": "\\begin{align*} \\begin{aligned} \\Lambda _ n ( H _ \\theta ) & \\ge \\Lambda _ n ( Q ^ { 1 + \\varepsilon } _ \\theta \\oplus R ) = \\Lambda _ n ( Q ^ { 1 + \\varepsilon } _ \\theta \\oplus Q ^ { 1 + \\frac { 1 } { \\varepsilon } } _ { \\pi - \\theta } ) \\\\ & \\ge \\min \\Big \\{ \\Lambda _ n ( Q ^ { 1 + \\varepsilon } _ \\theta ) , - \\Big ( 1 + \\frac { 1 } { \\varepsilon } \\Big ) ^ 2 \\Big \\} . \\end{aligned} \\end{align*}"}
+{"id": "1485.png", "formula": "\\begin{align*} h ( p ) = \\frac { g ( p ) } { 1 - g ( p ) } . \\end{align*}"}
+{"id": "1424.png", "formula": "\\begin{align*} M _ { T , K } : = \\left \\{ \\mathcal { U } = \\begin{pmatrix} u \\\\ v \\end{pmatrix} \\in C ( [ 0 , T ] ; \\mathcal { H } ) ; \\ , \\sup _ { t \\in [ 0 , T ] } \\| ( u ( t ) , v ( t ) ) \\| _ { \\mathcal { H } } \\le K \\right \\} . \\end{align*}"}
+{"id": "3043.png", "formula": "\\begin{align*} x _ i = \\begin{cases} a + ( 2 i - 1 ) d & \\\\ a - 2 i d & \\end{cases} \\end{align*}"}
+{"id": "6337.png", "formula": "\\begin{align*} \\sup _ { u \\in B } \\ , E ( u ) = 0 . \\end{align*}"}
+{"id": "3998.png", "formula": "\\begin{align*} \\hat { q } _ { \\beta } ( n , t ) = \\begin{cases*} E _ { \\beta , 1 } \\left ( - \\lambda t ^ { \\beta } \\right ) , \\ n = 0 , \\\\ \\displaystyle \\sum _ { \\Omega _ { n } } \\prod _ { j = 1 } ^ { n } \\frac { \\left ( ( 1 - p ) ^ { j } / j \\right ) ^ { x _ { j } } } { x _ { j } ! } z _ { n } ! \\left ( \\frac { - \\lambda t ^ { \\beta } } { \\ln p } \\right ) ^ { z _ { n } } E _ { \\beta , z _ { n } \\beta + 1 } ^ { z _ { n } + 1 } \\left ( - \\lambda t ^ { \\beta } \\right ) , \\ n \\ge 1 , \\end{cases*} \\end{align*}"}
+{"id": "6757.png", "formula": "\\begin{align*} W ( f ) = \\exp \\left ( \\int d k \\ , \\big ( f ( k ) a ^ * _ k - \\overline { f ( k ) } a _ k \\big ) \\right ) \\end{align*}"}
+{"id": "3757.png", "formula": "\\begin{align*} P & = ( x + 4 ) I _ { 2 0 } + \\frac { 1 0 ( x + 6 ) } { ( x - 2 2 ) ( x + 8 ) } J _ { 2 0 } + X ; \\\\ Q & = B + \\frac { 9 x + 8 2 } { ( x - 2 2 ) ( x + 8 ) } J _ { 2 0 , s } ; \\\\ R & = ( x + 4 ) I _ { s } + \\frac { 1 0 ( x + 6 ) } { ( x - 2 2 ) ( x + 8 ) } J _ { s } . \\end{align*}"}
+{"id": "1598.png", "formula": "\\begin{align*} \\left ( f _ { n } \\left ( g _ { s , j , i } \\right ) \\right ) \\left ( w \\right ) = T _ { b _ { i } } \\left ( v \\right ) . \\end{align*}"}
+{"id": "6221.png", "formula": "\\begin{align*} H = \\int ^ \\oplus _ \\mathbb { R } H ( p ) \\ , \\mathrm { d } p , \\end{align*}"}
+{"id": "3204.png", "formula": "\\begin{align*} \\frac { 1 } { n } & > I ( w _ { n } ) \\\\ & \\ge \\frac { 1 } { 2 } \\left \\| \\nabla w _ { n } \\right \\| _ { 2 } ^ { 2 } - \\frac { 1 } { p } \\left \\| w _ { n } \\right \\| _ { p } ^ { p } \\\\ & \\ge \\frac { 1 } { 2 } \\left \\| \\nabla w _ { n } \\right \\| _ { 2 } ^ { 2 } - C \\rho _ { n } ^ { \\frac { 6 - p } { 2 } } \\left \\| \\nabla w _ { n } \\right \\| _ { 2 } ^ { \\frac { 3 ( p - 2 ) } { 2 } } . \\end{align*}"}
+{"id": "2063.png", "formula": "\\begin{align*} \\| A \\| = \\max _ { i , j } | a _ { i j } | . \\end{align*}"}
+{"id": "7824.png", "formula": "\\begin{align*} \\Lambda _ Y ^ { w _ 1 } ( u ) = \\frac { w _ 1 ( \\phi ( F ^ { - 1 } ( u ) ) ) } { \\phi ^ \\prime ( F ^ { - 1 } ( u ) ) } f ( F ^ { - 1 } ( u ) ) \\leq w _ 1 ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) = \\Lambda _ X ^ { w _ 1 } ( u ) . \\end{align*}"}
+{"id": "1793.png", "formula": "\\begin{align*} { \\rm s i g n } \\left ( \\left \\langle Y , B , i \\right \\rangle \\right ) = { \\rm s i g n } \\left ( \\left \\langle Y , B , \\bar { i } \\right \\rangle \\right ) \\neq 0 . \\end{align*}"}
+{"id": "5306.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { c - i \\infty } ^ { c + i \\infty } e ^ { a s ^ 2 + b s } d s = \\frac { 1 } { 2 \\sqrt { \\pi a } } \\exp \\left ( - \\frac { b ^ 2 } { 4 a } \\right ) . \\end{align*}"}
+{"id": "7790.png", "formula": "\\begin{align*} I _ { g } = C _ { g } ( 1 , - 1 ) = - \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } \\frac { g ( u ) } { \\overline { u + 1 } ( u - 1 ) } d a ( u ) , \\ g \\in \\mathcal { G } _ 1 . \\end{align*}"}
+{"id": "3276.png", "formula": "\\begin{gather*} \\hat { K } \\hat { K } ^ { - 1 } = 1 = \\hat { K } ^ { - 1 } \\hat { K } , \\hat { K } \\hat { E } = q \\hat { E } \\hat { K } , \\hat { K } \\hat { F } = q ^ { - 1 } \\hat { F } \\hat { K } , \\hat { E } \\hat { F } - \\hat { F } \\hat { E } = \\frac { \\hat { K } ^ 2 - \\hat { K } ^ { - 2 } } { q - q ^ { - 1 } } . \\end{gather*}"}
+{"id": "1650.png", "formula": "\\begin{align*} \\mathsf { d } ( Q ) \\ ; = \\ ; \\operatorname { t r } \\left [ \\hat { \\alpha } ^ * Q \\hat { \\alpha } \\right ] \\ ; . \\end{align*}"}
+{"id": "1802.png", "formula": "\\begin{align*} i _ { 1 } = i _ { 2 } = \\cdots = i _ { r } \\mod 2 . \\end{align*}"}
+{"id": "2725.png", "formula": "\\begin{align*} \\varphi ( 2 n , t + 1 , r - 1 ) = M _ { \\varphi } ( 2 n , 0 , t ; r - 1 ) \\end{align*}"}
+{"id": "5304.png", "formula": "\\begin{align*} \\psi ( x ) = x + O ( x \\exp ( - c \\sqrt { \\log x } ) ) \\end{align*}"}
+{"id": "7305.png", "formula": "\\begin{align*} \\overline { M _ i } = ( \\mathop { q - 2 } \\limits _ { m - 1 } , \\underbrace { q - 1 , \\ldots , q - 1 , } _ { m - 2 - l _ 1 } \\mathop { q - 2 } \\limits _ { l _ 1 } , \\ldots , \\mathop { q - 2 } _ { l _ { i - 2 } } , \\underbrace { q - 1 , \\ldots , q - 1 } _ { l _ { i - 2 } - l _ { i - 1 } - 1 } , \\mathop { q - 2 } _ { l _ { i - 1 } } , \\underbrace { q - 1 , \\ldots , q - 1 } _ { l _ { i - 1 } } ) , \\end{align*}"}
+{"id": "2234.png", "formula": "\\begin{align*} H ( t ) = & \\textrm { d i a g } \\left [ \\{ \\alpha _ k \\} _ { k = 1 } ^ { 1 6 } \\right ] + B ( 0 . 5 + \\cos ( 4 t ) + \\sin ( 1 0 t ) - 0 . 4 \\sin ( 1 6 t ) ) \\\\ & + C ( \\sin ( 4 t ) + \\cos ( 8 t ) + 2 \\sin ( 1 2 t ) ) , \\end{align*}"}
+{"id": "2287.png", "formula": "\\begin{align*} \\Phi _ p ( 1 , 1 ) = p \\end{align*}"}
+{"id": "3128.png", "formula": "\\begin{align*} \\left ( F P ( \\sigma ) , F P ( \\sigma ^ 2 ) \\right ) = ( 0 , \\infty ) . \\end{align*}"}
+{"id": "6855.png", "formula": "\\begin{align*} f ( x ) = r ( x ) \\prod _ { i = 1 } ^ { s } ( x - a _ i ) , \\end{align*}"}
+{"id": "2341.png", "formula": "\\begin{align*} s _ { k } ( x _ { i } w ) & = x _ { i + k } w \\quad ( w \\in \\mathfrak { X } ) , \\\\ s _ { k } ( 1 ) & = 0 . \\end{align*}"}
+{"id": "7226.png", "formula": "\\begin{align*} \\begin{aligned} & \\| a \\| _ { l ^ { 2 } } ^ { 2 } : = \\sum _ { j \\in \\mathbb { Z } ^ { 2 } } | a _ { j } | ^ { 2 } , \\\\ & \\| a \\| _ { h ^ { s } } ^ { 2 } : = \\sum _ { j \\in \\mathbb { Z } ^ { 2 } } \\langle j \\rangle ^ { 2 s } | a _ { j } | ^ { 2 } , \\end{aligned} \\end{align*}"}
+{"id": "2500.png", "formula": "\\begin{align*} \\Phi _ N = \\{ L _ N , L _ N + 1 , \\dots , M _ N - 1 \\} \\end{align*}"}
+{"id": "4128.png", "formula": "\\begin{align*} \\sup _ { x \\in ( 0 , 1 ) } \\left | U \\left ( S ( x ) \\right ) \\right | = \\left | U \\left ( S ( x _ n ^ * ) \\right ) \\right | \\leq \\left [ \\left | U ( S ( \\delta _ n ^ 1 ) ) \\right | ^ 2 + \\frac { 1 } { \\chi _ n ^ { \\alpha } ( c ) + \\theta _ \\alpha ( S ( \\delta _ n ^ 1 ) ) } \\left | U ' ( S ( \\delta _ n ^ 1 ) ) \\right | ^ 2 \\right ] ^ { 1 / 2 } , \\end{align*}"}
+{"id": "901.png", "formula": "\\begin{align*} W : = \\max _ { \\overline { \\omega } _ \\delta } ( A \\Psi \\pm T _ \\alpha ( u - \\varphi ) ) \\end{align*}"}
+{"id": "1349.png", "formula": "\\begin{align*} \\widetilde { \\kappa } : = \\kappa s ^ { \\beta } , \\end{align*}"}
+{"id": "398.png", "formula": "\\begin{align*} \\theta _ \\eta ( v _ 1 , \\ldots , v _ n ) = \\eta ( \\pi _ * v _ 1 , \\ldots , \\pi _ * v _ n ) \\end{align*}"}
+{"id": "4555.png", "formula": "\\begin{align*} H _ f ( G ) & = \\frac { 1 } { 3 } \\Big ( 4 f ( 1 ) - f ( 4 ) \\Big ) n + \\frac { 2 } { 3 } \\Big ( f ( 4 ) - f ( 1 ) \\Big ) m \\\\ [ 4 m m ] & \\ + \\left ( f ( 2 ) - \\frac { 2 } { 3 } f ( 1 ) - \\frac { 1 } { 3 } f ( 4 ) \\right ) n _ 2 + \\left ( f ( 3 ) - \\frac { 1 } { 3 } f ( 1 ) - \\frac { 2 } { 3 } f ( 4 ) \\right ) n _ 3 \\ , . \\end{align*}"}
+{"id": "7096.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { F } } K _ { T } ( \\mathcal { D T } ( d ) ) \\otimes _ { \\mathbb { K } } \\mathbb { F } = p _ 3 ( d ) . \\end{align*}"}
+{"id": "3127.png", "formula": "\\begin{align*} \\sigma = ( 1 , 2 ) ( 3 , 4 ) ( 5 , 6 ) ( 7 , 8 ) \\cdots . \\end{align*}"}
+{"id": "6648.png", "formula": "\\begin{align*} 1 + \\sum _ { i = 1 } ^ { \\eta } \\lambda _ { i , 1 } ^ { 2 } s _ { i , - 1 } = \\sum _ { i = 1 } ^ { \\eta } \\lambda _ { i , 1 } ^ { 2 } , \\end{align*}"}
+{"id": "1608.png", "formula": "\\begin{align*} H ( X _ e ) + H ( X _ { C \\setminus \\{ e , e ' \\} } ) = H ( X _ { C \\setminus \\{ e ' \\} } ) = H ( X _ { C \\setminus \\{ e \\} } ) = H ( X _ { e ' } ) + H ( X _ { C \\setminus \\{ e , e ' \\} } ) . \\end{align*}"}
+{"id": "47.png", "formula": "\\begin{align*} f ' ( z ) = \\frac { 1 } { 2 \\sqrt { z } } - \\frac { 1 } { 2 \\sqrt { 1 - z } } = \\frac { 1 } { 2 } \\frac { \\sqrt { 1 - z } - \\sqrt { z } } { \\sqrt { z ( 1 - z ) } } \\geq 0 , \\end{align*}"}
+{"id": "872.png", "formula": "\\begin{align*} \\frac { \\phi ( \\alpha ) } { \\phi ( \\beta ) } = \\frac { \\Gamma ( 1 + \\beta ) \\ , \\Gamma ( 1 + r - \\beta ) } { \\Gamma ( 1 + \\alpha ) \\ , \\Gamma ( 1 + r - \\alpha ) } = \\lim _ { n \\rightarrow \\infty } \\prod _ { i = 1 } ^ n \\ , \\delta _ i ( \\alpha , \\beta ) , \\ , \\mbox { w h e r e } \\end{align*}"}
+{"id": "3950.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } ^ { \\beta } } { \\mathrm { d } t ^ { \\beta } } p _ { \\beta } ( n , t ) = - ( \\lambda _ { 1 } + \\lambda _ { 2 } + \\dots + \\lambda _ { k } ) p _ { \\beta } ( n , t ) + \\sum _ { j = 1 } ^ { \\min \\{ n , k \\} } \\lambda _ { j } p _ { \\beta } ( n - j , t ) , \\ n \\ge 0 , \\end{align*}"}
+{"id": "3384.png", "formula": "\\begin{align*} s _ n = \\sum _ { j = 1 } ^ m r _ j s _ { n - j } \\end{align*}"}
+{"id": "1339.png", "formula": "\\begin{align*} | B _ { 9 / 1 0 } \\cap \\left \\{ u = 0 \\right \\} | = 0 , \\end{align*}"}
+{"id": "7292.png", "formula": "\\begin{align*} \\Upsilon ( T , \\omega ) = - ( K _ x + K _ m ) X ^ * ( T , \\omega ) + K _ m \\overline { X } ^ * ( T ) . \\end{align*}"}
+{"id": "6917.png", "formula": "\\begin{align*} \\left [ q ^ { d } \\right ] ( - 1 ) ^ { ( N - 1 ) d } \\prod _ { i = 1 } ^ { N } \\bigg ( \\frac { \\alpha _ i - z _ { N + 1 } } { \\alpha _ i } \\bigg ) ^ { b _ i } \\frac { z _ i ^ { d + 1 } } { P ' ( z _ i ) } \\cdot \\prod _ { 1 \\le i \\ne j \\le N } ( z _ i - z _ j ) \\bigg { | } _ { \\epsilon = 0 } . \\end{align*}"}
+{"id": "3160.png", "formula": "\\begin{align*} F ( { \\bf z } ) \\ , \\ , = \\ , \\ , f \\left ( x ( { \\bf z } ) , y ( { \\bf z } ) , a _ { 1 } ( { \\bf z } ) , a _ { 2 } ( { \\bf z } ) \\right ) \\end{align*}"}
+{"id": "5804.png", "formula": "\\begin{align*} | \\langle A x , x \\rangle | ^ { p } \\leq \\| A x \\| ^ { p } = \\langle A x , A x \\rangle ^ { p \\over 2 } = \\langle A ^ * A x , x \\rangle ^ { p \\over 2 } . \\end{align*}"}
+{"id": "3666.png", "formula": "\\begin{align*} C & = \\sum _ { v _ i v _ j \\in \\partial \\mathcal { G } \\cap \\binom { W } { 2 } } z _ i z _ j + \\sum _ { v _ i \\in N _ { \\mathcal { G } } ( v _ 1 ) } z _ i z _ 1 + \\sum _ { v _ i \\in N _ { \\mathcal { G } } ( v _ 2 ) } z _ i z _ 2 + z _ 1 z _ 2 \\end{align*}"}
+{"id": "4610.png", "formula": "\\begin{align*} \\frac { \\phi ( x , s ) } { s ^ r } = s ^ { p - r } \\frac { \\phi ( x , s ) } { s ^ { p } } \\le s ^ { p - r } L _ p \\frac { \\phi ( x , t ) } { t ^ { p } } = L _ p \\Big { ( } \\frac { s } { t } \\Big { ) } ^ { p - r } \\frac { \\phi ( x , t ) } { t ^ r } \\le L _ p \\frac { \\phi ( x , t ) } { t ^ r } . \\end{align*}"}
+{"id": "2501.png", "formula": "\\begin{align*} A = \\{ n \\in \\N : T ^ n a \\in E \\} . \\end{align*}"}
+{"id": "8211.png", "formula": "\\begin{align*} \\Lambda ^ { \\alpha } ( u v ) - u \\Lambda ^ { \\alpha } v = \\big ( \\Lambda ^ { \\alpha } T _ { u } v - T _ { u } ( \\Lambda ^ { \\alpha } v ) \\big ) + ( \\Lambda ^ { \\alpha } T _ { v } u - T _ { \\Lambda ^ { \\alpha } v } u ) + \\big ( \\Lambda ^ { \\alpha } R ( u , v ) - R ( u , \\Lambda ^ { \\alpha } v ) \\big ) = : \\sum _ { j = 1 } ^ 3 \\Pi _ j . \\end{align*}"}
+{"id": "696.png", "formula": "\\begin{align*} \\rho { \\bf { u } } = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { { \\bf { c } } _ i } { f _ i } } + \\frac { 1 } { 2 } { \\bf { F } } \\Delta t . \\end{align*}"}
+{"id": "6698.png", "formula": "\\begin{align*} S _ 1 & = \\sup _ { i \\in I _ 1 } - \\frac { Q _ { i } ^ { ( K _ i - 1 ) } ( 0 ) } { Q _ { i } ^ { ( K _ i ) } ( 0 ) } \\\\ S _ 2 & = \\sup _ { i \\in I _ 2 } \\inf \\left \\{ - \\frac { ( Q _ { i , s _ { i , - \\kappa - 2 } } ) ^ { ( K _ i - 1 ) } ( 0 ) } { Q _ { i } ^ { ( K _ i ) } ( 0 ) } \\ ! : s _ { i , - \\kappa - 2 } \\in ( t _ { \\infty } ( \\mathbf { s } _ { i } ) , \\infty ) \\right \\} . \\end{align*}"}
+{"id": "4361.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\sum _ { \\alpha = 1 } ^ { m } \\lambda _ \\alpha D _ H g ^ \\alpha ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) \\\\ + \\sum _ { \\beta = 1 } ^ { q } \\mu _ \\beta D _ H h ^ \\beta ( x _ 0 ( T ) ) \\circ D _ { G , 3 } f ( T , x _ 0 ( T ) , u _ 0 ( T ) ) = 0 . \\end{array} \\right \\} \\end{align*}"}
+{"id": "3920.png", "formula": "\\begin{align*} F _ 2 ( t ) & \\le 6 \\int _ 0 ^ t ( { L _ f ^ * } ^ 2 + { K _ f ^ * } ^ 2 \\| \\ell \\| ^ 2 _ { L ^ 1 } + \\epsilon ) \\| u ( t + h - \\tau ) - u ( t - \\tau ) \\| ^ 2 _ { \\mathbb H ^ \\mu } d \\tau \\\\ & = 6 \\int _ 0 ^ t ( { L _ f ^ * } ^ 2 + { K _ f ^ * } ^ 2 \\| \\ell \\| ^ 2 _ { L ^ 1 } + \\epsilon ) \\| u ( \\tau + h ) - u ( \\tau ) \\| ^ 2 _ { \\mathbb H ^ \\mu } d \\tau , \\end{align*}"}
+{"id": "1135.png", "formula": "\\begin{gather*} \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ \\frac { n ! } { ( n / 2 ) ! } \\left ( \\beta \\right ) ^ { \\left ( n / 2 \\right ) } ( 1 - \\rho ) ^ { n / 2 } & & n \\end{array} \\right . = E ( - X + Y ) ^ { n } \\allowbreak \\\\ = \\sum _ { m = 0 } ^ { n } ( - 1 ) ^ { m } \\binom { n } { m } E X ^ { m } Y ^ { n - m } . \\allowbreak \\end{gather*}"}
+{"id": "2162.png", "formula": "\\begin{align*} & ( R _ { 1 2 } S _ { 2 3 } - R _ { 1 3 } S _ { 1 2 } ) ^ 2 S _ 3 \\\\ = & R _ { 1 2 } { } ^ 2 S _ { 2 3 } { } ^ 2 S _ 3 - 2 R _ { 1 2 } R _ { 1 3 } S _ { 2 3 } S _ { 1 2 } S _ 3 + R _ { 1 3 } { } ^ 2 S _ { 1 2 } { } ^ 2 S _ 3 \\\\ = & R _ { 1 2 1 2 } S _ { 2 3 2 3 3 } - 2 R _ { 1 2 1 3 } S _ { 1 2 2 3 3 } + R _ { 1 3 1 3 } S _ { 1 2 1 2 3 } . \\end{align*}"}
+{"id": "7863.png", "formula": "\\begin{align*} \\Delta ( f , g ) = \\min _ { \\theta \\in J / 2 } | f ( \\theta ) - g ( \\theta ) | + | f ' ( \\theta ) - g ' ( \\theta ) | , \\end{align*}"}
+{"id": "1780.png", "formula": "\\begin{align*} \\left \\langle Y , B , i \\right \\rangle = 0 , { \\rm f o r } i \\in \\left [ n \\right ] \\backslash A . \\end{align*}"}
+{"id": "7273.png", "formula": "\\begin{align*} u _ L ( t , \\omega ) = u _ E ( t , X ( t , \\omega ) ) . \\end{align*}"}
+{"id": "6024.png", "formula": "\\begin{align*} | V ( x , i ) - V _ { \\tilde { \\pi } } ( x , i ) | & \\leq | V ( x , i ) - V ( x _ { j ^ * } , i ) | \\\\ & + | V ( x _ { j ^ * } , i ) - V _ { \\pi ^ { i , j ^ * } } ( x _ { j ^ * } , i ) | + | V _ { \\pi ^ { i , j ^ * } } ( x _ { j ^ * } , i ) - V _ { \\tilde { \\pi } } ( x , i ) | \\\\ & \\leq ( 2 + \\beta ) \\varepsilon \\leq ( 2 + \\beta ) \\varepsilon . \\end{align*}"}
+{"id": "8253.png", "formula": "\\begin{align*} \\widetilde { R } _ j ( t ) : = 2 ^ { j ( \\alpha - 1 ) } \\Vert f _ j ( t ) \\Vert _ { L ^ 2 } + \\Vert g _ j ( t ) \\Vert _ { L ^ 2 } + \\Vert \\widetilde { g } _ j ( t ) \\Vert _ { L ^ 2 } + \\Vert \\nabla \\dot S _ { j - 1 } u ( t ) \\Vert _ { L ^ \\infty } X _ j ( t ) . \\end{align*}"}
+{"id": "3576.png", "formula": "\\begin{align*} \\lambda = n ^ { - ~ \\tfrac { b } { 1 + b + 2 b \\nu } } , \\end{align*}"}
+{"id": "4424.png", "formula": "\\begin{align*} \\limsup _ { m \\to \\infty } \\sup _ { n \\in \\N } \\vert I _ m ( v _ n ) - I ( v _ n ) \\vert = 0 . \\end{align*}"}
+{"id": "2065.png", "formula": "\\begin{align*} \\mu ( X ) = \\lim _ { k \\to \\infty } \\pi _ k ( X ) q ^ { - k n } \\end{align*}"}
+{"id": "6237.png", "formula": "\\begin{align*} \\lambda _ { n , m _ 1 , m _ 2 } = B ( 2 n + 1 ) + \\big ( \\textstyle { \\frac { \\pi m _ 1 } { d _ 1 } } \\big ) ^ 2 + \\big ( \\textstyle { \\frac { \\pi m _ 2 } { d _ 2 } } \\big ) ^ 2 , n \\in \\mathbb { N } _ 0 , \\ ; m _ 1 , m _ 2 \\in \\mathbb { N } \\end{align*}"}
+{"id": "7239.png", "formula": "\\begin{align*} \\sum \\limits _ { | j | > K ( \\eta ) } \\langle j \\rangle ^ { 2 } \\| u _ { j } \\| _ { L ^ 2 } ^ { 2 } & < \\eta , \\\\ \\sum \\limits _ { | j | = 0 } ^ { K ( \\eta ) } \\langle j \\rangle ^ { 2 } \\int _ { | x - x ( t ) | \\geq \\frac { R ( \\eta ) } { N ( t ) } } | u _ { j } ( x ) | ^ { 2 } \\mathrm { d } x & < \\eta , \\\\ \\sum \\limits _ { | j | = 0 } ^ { K ( \\eta ) } \\langle j \\rangle ^ { 2 } \\int _ { | \\xi - \\xi ( t ) | \\geq R ( \\eta ) N ( t ) } | \\hat { u } _ { j } ( \\xi ) | ^ { 2 } \\mathrm { d } \\xi & < \\eta . \\end{align*}"}
+{"id": "3422.png", "formula": "\\begin{align*} U ^ * _ \\epsilon = 0 , \\rho ^ * _ \\epsilon = 0 , S ^ * _ \\epsilon + r ^ { 1 - n } S ^ * _ r / \\rho ^ * = 0 . \\end{align*}"}
+{"id": "2954.png", "formula": "\\begin{align*} C : = \\prod _ { i = 1 } ^ \\ell \\mathcal Q _ { m _ i } \\end{align*}"}
+{"id": "2450.png", "formula": "\\begin{align*} p _ { a , b , c } = ( a - b ) ( a + b - 6 c + 3 ) \\end{align*}"}
+{"id": "8236.png", "formula": "\\begin{align*} \\Vert ( \\sigma , u ) ^ \\ell ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha + s \\alpha } _ { 2 , 1 } } + \\Vert \\sigma ^ h ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } } _ { 2 , 1 } } + \\Vert u ^ h ( t ) \\Vert _ { \\dot { B } ^ { \\frac { N } { 2 } + 1 - \\alpha } _ { 2 , 1 } } \\le C ( 1 + t ) ^ { - s } . \\end{align*}"}
+{"id": "550.png", "formula": "\\begin{align*} A _ 2 ^ 2 \\le \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mathcal { W } _ 1 ^ 2 ( P _ i , Q ^ * _ i ) \\le \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mathcal { W } _ 2 ^ 2 ( P _ i , Q ^ * _ i ) \\le \\frac { 2 R _ f } { \\kappa n } . \\end{align*}"}
+{"id": "6355.png", "formula": "\\begin{align*} { S } ( \\lambda ) = \\left [ { \\begin{array} { c | c } A ( \\lambda ) & - B \\\\ \\hline C & D ( \\lambda ) \\\\ \\end{array} } \\right ] { \\mathbb C } [ \\lambda ] ^ { n + m , n + m } \\end{align*}"}
+{"id": "5723.png", "formula": "\\begin{align*} \\dot z _ t = \\mathbf { W } _ t ( z _ t ) \\end{align*}"}
+{"id": "791.png", "formula": "\\begin{align*} \\mathcal { J } _ i ( \\tau _ i , \\tau _ { - i } ) = \\mathbb { E } \\left \\{ \\theta e ^ { - \\beta \\tau _ i } x _ i ( \\tau _ i ) + ( 1 - \\theta ) \\frac { e ^ { - \\beta \\tau _ i } x _ i ( \\tau _ i ) } { l _ 1 + l _ 2 \\left ( \\frac { 1 } { N } \\sum \\limits _ { j = 1 } ^ N e ^ { - \\beta \\tau _ j } x _ j ( \\tau _ j ) \\right ) } - e ^ { - \\beta \\tau _ i } K \\right \\} . \\end{align*}"}
+{"id": "182.png", "formula": "\\begin{align*} \\begin{aligned} \\langle \\partial _ { t , x } ^ \\beta f , \\phi \\rangle = \\partial _ { t , x } ^ \\beta \\langle f , \\phi \\rangle - \\langle f , \\partial _ { t , x } ^ \\beta \\phi \\rangle = - \\langle f , \\partial _ { t , x } ^ \\beta \\phi \\rangle \\ , , \\end{aligned} \\end{align*}"}
+{"id": "7283.png", "formula": "\\begin{align*} u ^ * ( t , \\omega ) = - R ^ { - 1 } ( t ) B ^ T ( t ) \\big [ P _ 1 ( t ) ( X ^ * ( t , \\omega ) - \\overline { X } ^ * ( t ) ) + P _ 2 ( t ) \\overline { X } ^ * ( t ) \\big ] , \\end{align*}"}
+{"id": "7450.png", "formula": "\\begin{align*} \\begin{matrix} [ p _ 0 , \\ , p _ 0 ^ \\dagger ] = 4 , [ p _ 1 , \\ , p _ 1 ^ \\dagger ] = 2 \\hat { j } ( \\hat { j } + 1 ) - J _ z ( 2 J _ z + 1 ) + ( N - N _ 0 ) ( J _ z - 2 ) , \\\\ [ p _ 1 , \\ , p _ 0 ^ \\dagger ] = 2 ( N - N _ 0 ) , [ p _ 0 , \\ , p _ 1 ^ \\dagger ] = 2 ( N - N _ 0 ) , \\end{matrix} \\end{align*}"}
+{"id": "2931.png", "formula": "\\begin{align*} M _ C = \\begin{pmatrix} A & C \\\\ & B \\end{pmatrix} \\end{align*}"}
+{"id": "7856.png", "formula": "\\begin{align*} 1 = \\mathbb { P } ( Z ) = \\mathbb { P } \\big ( Z \\cap \\rho _ \\theta ^ { - 1 } ( \\rho _ \\theta ( Z ) \\big ) = \\mathbb { P } \\Big ( Z \\cap \\bigcup _ { k \\geq k _ 0 } \\rho _ \\theta ^ { - 1 } ( G _ { \\theta , k } ) \\Big ) . \\end{align*}"}
+{"id": "2011.png", "formula": "\\begin{align*} \\mu _ n : = \\int _ { 0 } ^ { \\infty } 4 ^ { - 1 } e ^ { - t / 2 } \\ , \\Psi ( t ) \\cdot t ^ n \\ , d t \\end{align*}"}
+{"id": "6266.png", "formula": "\\begin{align*} p _ { k } = \\frac { a } { 2 } \\left ( \\frac { \\lambda _ { k - 1 } ^ 2 } { \\lambda _ k } \\right ) + \\frac { 1 } { 2 } \\left ( \\frac { b } { \\lambda _ k } - c \\right ) - p _ { k - 1 } , ~ M _ { 1 , k } = p _ { k - 1 } - \\frac { \\alpha _ k + b c \\lambda _ { k - 1 } } { 2 \\lambda _ k } - \\frac { a \\lambda _ { k - 1 } ^ 2 } { 2 \\lambda _ k } . \\end{align*}"}
+{"id": "4161.png", "formula": "\\begin{align*} \\begin{cases} \\dfrac { d } { d t } g _ t & = - P _ t ^ s \\\\ \\dfrac { d } { d t } b _ t & = P _ t ^ a \\end{cases} \\end{align*}"}
+{"id": "2325.png", "formula": "\\begin{align*} \\zeta ( 2 ) = \\frac { \\pi ^ { 2 } } { 6 } \\end{align*}"}
+{"id": "4476.png", "formula": "\\begin{align*} F ( v ) = \\sum _ { n = 1 } ^ { \\infty } \\langle v , e _ n \\rangle _ { \\mathcal { H } } F ( e _ n ) = \\langle \\psi , v \\rangle _ { \\mathcal { H } } \\end{align*}"}
+{"id": "6678.png", "formula": "\\begin{align*} \\mathcal { D } \\left ( ( s ^ { ( 1 ) } _ { k } ) _ { k = 0 } ^ { 2 m - 1 } , ( s _ k ) _ { k = 0 } ^ { m } \\right ) & \\stackrel { a \\to 0 + } { \\longrightarrow } \\mathcal { D } \\left ( ( s _ { k + 1 } ) _ { k = 0 } ^ { 2 m - 1 } , ( s _ k ) _ { k = 0 } ^ { m } \\right ) \\\\ & = ( - 1 ) ^ { m } \\mathcal { D } \\left ( ( s _ { k } ) _ { k = 0 } ^ { 2 m - 1 } , ( s _ k ) _ { k = m } ^ { 2 m } \\right ) , \\end{align*}"}
+{"id": "7375.png", "formula": "\\begin{align*} \\Big ( 1 - \\frac { d ^ \\prime } { 2 } + \\frac { k _ 0 } { q _ 0 } + \\frac { k _ 1 } { q _ 1 } + \\dots + \\frac { k _ l } { q _ l } \\Big ) = \\frac { \\chi ( S ) } { p } . \\end{align*}"}
+{"id": "1569.png", "formula": "\\begin{align*} \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } z _ { 2 } ^ { i } \\right ) ^ { - 1 } z _ { 2 } \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } z _ { 2 } ^ { i - i } \\right ) & = \\left [ \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } z _ { 2 } ^ { i } \\right ) ^ { - 1 } , z _ { 2 } \\right ] \\mbox { a n d } \\\\ z _ { 2 } \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } z _ { 2 } ^ { i } \\right ) ^ { - 1 } \\left ( s ^ { \\prime } z _ { 1 } s ^ { \\prime } z _ { 2 } ^ { i - i } \\right ) & = e . \\end{align*}"}
+{"id": "6934.png", "formula": "\\begin{align*} T = \\bigg ( 1 - \\sum _ { j = 0 } ^ { r } ( - 1 ) ^ j e _ { N + 1 } ^ j e _ { N - j } t ^ { j + 1 } \\bigg ) ^ { - 1 } = \\frac { 1 } { ( 1 - ( - 1 ) ^ { N - 1 } t ) \\prod _ { p = 1 } ^ { r } ( 1 - ( - 1 ) ^ { N - 1 } x _ p y t ) } . \\end{align*}"}
+{"id": "4569.png", "formula": "\\begin{gather*} \\sup \\{ f _ m : m \\in M \\backslash D \\} = \\sup \\{ f _ m : m \\in M \\} - \\sum _ { m \\in D } f _ m . \\end{gather*}"}
+{"id": "3020.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } _ i & = f _ i ( x , u ) \\ , , & i = 1 , \\ldots , n \\end{aligned} \\end{align*}"}
+{"id": "1881.png", "formula": "\\begin{align*} g ( x ) = \\lim _ { m \\rightarrow \\infty } \\frac { 1 } { 3 ^ { l m } } f \\left ( \\frac { x } { 3 ^ { m } } \\right ) \\end{align*}"}
+{"id": "852.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb { E } ( \\tau - t _ 0 ) ^ { 2 } = \\mathbb { E } \\tau ^ { 2 } - 2 t _ 0 \\mathbb { E } ( \\tau ) + t _ 0 ^ { 2 } \\\\ & = \\frac { 1 } { ( k _ 2 \\sigma a ' ) ^ { 2 } } \\left ( \\ln \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) ^ { 2 } + \\left ( \\frac { 2 t _ 0 } { k _ 2 \\sigma a ' } - \\frac { 1 } { k _ 2 \\sigma ( a ' ) ^ { 3 } } \\right ) \\left ( \\ln \\frac { K k _ 2 } { k _ 2 - 1 } \\right ) + t _ 0 ^ { 2 } \\end{aligned} \\end{align*}"}
+{"id": "4924.png", "formula": "\\begin{align*} \\binom { n } { k } / \\binom { n } { ( k + r ) / 2 } & = \\frac { ( n - k + 1 ) \\cdots ( n - ( k + r ) / 2 ) } { ( \\frac { k + r } { 2 } + 1 ) \\cdots k } \\ge \\frac { n - ( k + r ) / 2 } { ( k + r ) / 2 + 1 } \\\\ & = \\frac { 3 k + r } { k + r + 2 } \\ge \\frac { 3 k + ( k - 2 ) } { k + ( k - 2 ) + 2 } = \\frac { 2 k - 1 } { k } . \\end{align*}"}
+{"id": "5543.png", "formula": "\\begin{align*} \\Tilde { c _ 1 } ( T _ 1 ) \\Tilde { \\gamma _ 1 } ( T _ 1 ) \\dfrac { \\partial T _ 1 } { \\partial t } = \\dfrac { 1 } { r ^ { \\nu } } \\dfrac { \\partial } { \\partial r } \\bigg ( \\Tilde { \\lambda _ 1 } ( T _ 1 ) r ^ { \\nu } \\dfrac { \\partial T _ 1 } { \\partial r } \\bigg ) , \\ ; \\ ; \\ ; \\alpha ( t ) < r < \\beta ( t ) , \\ ; \\ ; t > 0 , \\end{align*}"}
+{"id": "7060.png", "formula": "\\begin{align*} \\mathbb { D } _ { x , q } f ( x ) = \\frac { f ( x ) - f ( q x ) } { ( 1 - q ) x } , x \\neq 0 . \\end{align*}"}
+{"id": "3910.png", "formula": "\\begin{align*} \\| S * g ( t ) \\| ^ 2 _ { \\mathbb H ^ \\mu } & \\le \\sum _ { n = 1 } ^ \\infty \\lambda _ n ^ { \\mu - 1 - \\delta } \\int _ 0 ^ t ( t - \\tau ) ^ { - \\delta } | g _ n ( \\tau ) | ^ 2 d \\tau \\\\ & = \\int _ 0 ^ t ( t - \\tau ) ^ { - \\delta } \\| g ( \\tau ) \\| ^ 2 _ { \\mathbb H ^ { \\mu - 1 - \\delta } } d \\tau . \\end{align*}"}
+{"id": "2942.png", "formula": "\\begin{align*} N _ \\Omega ( \\bar z ; w ) : = \\{ z ^ * \\in \\R ^ n \\ , | \\ , \\exists t _ k \\downarrow 0 , \\ , \\exists w _ k \\to w , \\ , \\exists z _ k ^ * \\to z ^ * , \\ , \\forall k \\in \\N \\colon \\ , z _ k ^ * \\in \\widehat N _ \\Omega ( \\bar z + t _ k w _ k ) \\} \\end{align*}"}
+{"id": "70.png", "formula": "\\begin{align*} \\nabla u ( x _ { 0 } ) \\in \\mathrm { s p a n } { \\nabla d ( x _ { 0 } ) } = ( T _ { x } \\Sigma _ { d ( x _ { 0 } ) } ) ^ { \\perp } . \\end{align*}"}
+{"id": "3045.png", "formula": "\\begin{align*} \\Gamma _ { 3 , 3 } = ( x _ i ) _ { i \\in \\Z } = ( \\ldots , 1 5 , 9 , 3 , 6 , 1 2 , 1 8 , \\ldots ) . \\end{align*}"}
+{"id": "7825.png", "formula": "\\begin{align*} \\frac { J ^ { w _ 1 } ( \\textbf { X } _ { R S S } ^ { ( n ) } ) } { J ^ { w _ 1 } ( \\textbf { X } _ { S R S } ^ { ( n ) } ) } \\leq \\frac { n ^ { 2 n } } { ( n - 1 ) ^ { 2 ( n - 1 ) ( n - 2 ) } } \\prod _ { i = 2 } ^ { n - 1 } \\left ( \\binom { n - 1 } { i - 1 } ^ 2 ( i - 1 ) ^ { 2 i - 2 } ( n - i ) ^ { 2 n - 2 i } \\right ) . \\end{align*}"}
+{"id": "420.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\rightarrow + \\infty } u _ x ( x , h ) = 0 , ~ ~ \\mathrm { a n d } ~ ~ \\lim \\limits _ { x \\rightarrow + \\infty , ~ ( x , h ) \\in \\mathcal { D } _ 2 } \\frac { h } { x } = C _ \\infty , \\end{align*}"}
+{"id": "7571.png", "formula": "\\begin{align*} E _ { \\varphi } ( F _ { \\varepsilon , h } ) = E _ { \\varphi } [ \\mu ^ { \\varepsilon } ] \\leq E _ { \\varphi } [ \\mu _ \\varepsilon ] , \\end{align*}"}
+{"id": "5395.png", "formula": "\\begin{align*} X ^ g ( t ) = x + L \\int _ 0 ^ t \\Psi ( X ^ g ( s ) ) d s + \\int _ 0 ^ t \\int _ Z f ( s , X ^ g ( s ) , z ) \\big ( g ( s , z ) - 1 \\big ) \\nu ( d z ) d s , ~ ~ \\forall ~ t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "2444.png", "formula": "\\begin{align*} B _ { n } \\mathcal { Z } ^ { \\mathfrak { m } } = \\left \\langle I _ { { \\rm b l } } ^ { \\mathfrak { m } } ( l _ { 0 } , \\dots , l _ { m } ) \\mid m \\leq n \\right \\rangle _ { \\mathbb { Q } } . \\end{align*}"}
+{"id": "986.png", "formula": "\\begin{align*} ( k - s - 1 ) a _ { s + 1 , t } & = ( s + t ) a _ { s , t } , \\\\ ( k - t - 1 ) a _ { s , t + 1 } & = ( s + t ) a _ { s , t } . \\end{align*}"}
+{"id": "6660.png", "formula": "\\begin{align*} \\underline { Q } ( t ) = ( t - a ) \\det \\begin{bmatrix} \\tilde { s } _ 0 & \\tilde { s } _ 1 & \\ldots & \\tilde { s } _ { m - 1 } & 1 \\\\ \\tilde { s } _ 1 & \\tilde { s } _ 2 & \\ldots & \\tilde { s } _ { m } & t \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ \\tilde { s } _ { m } & \\tilde { s } _ { m + 1 } & \\ldots & \\tilde { s } _ { 2 m - 1 } & t ^ { m } \\end{bmatrix} , \\end{align*}"}
+{"id": "6327.png", "formula": "\\begin{align*} m _ l ( a , b ) = \\sup _ { v \\in N _ l \\cap S } \\ , I ( v + \\tau ( v , a , b ) , a , b ) . \\end{align*}"}
+{"id": "7182.png", "formula": "\\begin{align*} \\chi _ 1 + \\chi _ 2 - \\sigma + \\rho \\in \\textbf { W } ( d ) _ v = \\textbf { W } ( d ) _ w . \\end{align*}"}
+{"id": "4364.png", "formula": "\\begin{align*} \\begin{array} { l l } \\underline { d } ( \\sigma _ 1 , . . . , \\sigma _ l , x ) ( t ) & = ( \\underline { d } \\sigma _ 1 ( t ) , . . . , \\underline { d } \\sigma _ l ( t ) , \\underline { d } x ( t ) ) \\\\ \\null & = ( f _ 1 ^ 0 ( t , x ( t ) , u ( t ) ) , . . . , f ^ 0 _ l ( t , x ( t ) , u ( t ) ) , f ( t , x ( t ) , u ( t ) ) ) \\\\ \\null & = F ( t , ( \\sigma _ 1 , . . . , \\sigma _ l , x ) ( t ) , u ( t ) ) \\end{array} \\end{align*}"}
+{"id": "4616.png", "formula": "\\begin{align*} \\phi ^ * \\Big ( \\frac { \\phi ( t ) } { L t } \\Big ) = \\sup _ { s \\ge 0 } \\Big ( s \\frac { \\phi ( t ) } { L t } - \\phi ( s ) \\Big ) \\le \\max \\bigg \\{ \\frac { \\phi ( t ) } { L } , \\sup _ { s > t } \\Big ( s \\frac { \\phi ( t ) } { L t } - \\phi ( s ) \\Big ) \\bigg \\} . \\end{align*}"}
+{"id": "4116.png", "formula": "\\begin{align*} \\mathcal { H } ^ r _ { \\alpha } \\left ( I \\right ) = \\{ u \\in L ^ 2 ( I ) ; ~ ~ ~ x ^ { ( \\alpha + k + 1 / 2 ) } \\left ( \\frac { 1 } { x } \\frac { d } { d x } \\right ) ^ { ( k ) } \\left ( x ^ { - ( \\alpha + 1 / 2 ) } u \\right ) \\in L ^ 2 ( I ) ~ \\ , , \\forall k \\leq r \\} . \\end{align*}"}
+{"id": "4881.png", "formula": "\\begin{align*} \\left ( ( 2 x t - 1 ) \\frac { d } { d t } + 2 x ( 1 - x ) \\frac { d } { d x } + 1 \\right ) Q ( x , t ) = 0 \\end{align*}"}
+{"id": "3243.png", "formula": "\\begin{align*} I _ { \\psi } ( u , v ) = I _ { \\psi } ( \\max ( u , v ) , u ) + I _ { \\psi } ( \\max ( u , v ) , v ) . \\end{align*}"}
+{"id": "5662.png", "formula": "\\begin{align*} H ( r ) ( V , V ) = - 4 r _ { 1 1 } \\end{align*}"}
+{"id": "2249.png", "formula": "\\begin{align*} \\dim ( W ^ m ) = \\sum _ { i = 0 } ^ m \\dim ( W _ i ^ { m - i } ) \\leq \\sum _ { i = 0 } ^ m ( 2 i ( m - i ) + 1 ) ( 2 ( m - i ) + 1 ) ( m - i + 1 ) \\sim m ^ 5 , \\end{align*}"}
+{"id": "7421.png", "formula": "\\begin{align*} \\left | n \\right > = \\frac { 1 } { \\sqrt { n ! } } ( a ^ \\dagger ) ^ n \\left | 0 \\right > . \\end{align*}"}
+{"id": "2682.png", "formula": "\\begin{align*} M _ Q ( n , j , 0 ; a ) = \\binom { a + j } { a } \\sum _ { l = 0 } ^ { a } \\binom { n - j + l } { l } \\binom { n - j } { a - l } M _ R ( n , j + a - l , 0 ; a ) \\end{align*}"}
+{"id": "1102.png", "formula": "\\begin{align*} m = | M _ 1 | + | M _ 2 | + | M _ 3 | + | M _ 4 | \\geq C k \\log ^ 3 n . \\end{align*}"}
+{"id": "4079.png", "formula": "\\begin{align*} \\dim H ^ 1 _ f ( G _ { \\Q } , U _ n ( x ) ) = \\begin{cases} \\dim H ^ 1 _ f ( G _ { \\Q _ p } , U _ n ) , n \\in 1 + 2 \\cdot \\mathbb { Z } _ { > 0 } \\\\ 0 , n \\in 2 \\cdot \\mathbb { Z } _ { > 0 } \\end{cases} . \\end{align*}"}
+{"id": "1344.png", "formula": "\\begin{align*} [ u ] ^ p _ { \\mathcal { L } ^ { p , \\lambda } ( \\Omega ) } : = \\sup _ { \\stackrel { x _ 0 \\in \\Omega } { \\rho > 0 } } \\rho ^ { - \\lambda } \\int _ { \\Omega _ { x _ 0 , \\rho } } \\left | u ( x ) - u _ { x _ 0 , \\rho } \\right | ^ p \\ , d x < + \\infty , \\end{align*}"}
+{"id": "1143.png", "formula": "\\begin{align*} [ ( \\lambda _ 1 , \\mu _ 1 ) , \\kappa _ 1 ] \\cdot [ ( \\lambda _ 2 , \\mu _ 2 ) , \\kappa _ 2 ] : = [ ( \\lambda _ 1 + \\lambda _ 2 , \\mu _ 1 + \\mu _ 2 ) , \\kappa _ 1 + \\kappa _ 2 + \\lambda _ 1 \\mu _ 2 ^ t - \\mu _ 1 \\lambda _ 2 ^ t ] . \\end{align*}"}
+{"id": "5132.png", "formula": "\\begin{align*} \\underline { D } ( A , \\Omega , x ) = \\liminf _ { r \\searrow 0 } \\frac { | A \\cap B ( x , r ) | } { | B ( x , r ) \\cap \\Omega | } . \\end{align*}"}
+{"id": "8189.png", "formula": "\\begin{align*} \\mathcal { D } ( u , \\rho ) = - \\rho \\big ( \\Lambda ^ \\alpha ( \\rho u ) - u \\Lambda ^ \\alpha \\rho \\big ) = - \\rho \\big ( [ \\Lambda ^ \\alpha , u ] \\rho \\big ) . \\end{align*}"}
+{"id": "3524.png", "formula": "\\begin{align*} \\mathrm { H } ( \\mathrm { x } ) - \\alpha \\int _ \\Omega \\nabla \\mathbb { G } ^ { ( 0 ) } ( \\mathrm { x } , \\mathrm { y } ) \\cdot \\nabla \\mathrm { H } ( \\mathrm { y } ) d \\mathrm { y } = \\mathrm { H } ^ { \\textbf { i n } } ( \\mathrm { x } ) + \\alpha \\int _ \\Omega \\nabla \\Big ( \\mathbb { G } ^ { ( \\mathrm { k } ) } - \\mathbb { G } ^ { ( 0 ) } \\Big ) ( \\mathrm { x } , \\mathrm { y } ) \\cdot \\nabla \\mathrm { H } ( \\mathrm { y } ) d \\mathrm { y } . \\end{align*}"}
+{"id": "1522.png", "formula": "\\begin{align*} r ( S ) = \\frac { 1 } { c } \\dim \\left ( \\sum _ { e \\in S } W _ { e } \\right ) . \\end{align*}"}
+{"id": "789.png", "formula": "\\begin{align*} \\max _ { \\tau _ i , \\tau _ { - i } \\in \\mathcal { S } } \\mathcal { J } _ i ( \\tau _ i , \\tau _ { - i } ) = \\max _ { \\tau _ i , \\tau _ { - i } \\in \\mathcal { S } } \\mathbb { E } \\left \\{ { C } _ { \\theta } ( \\tau _ i , \\tau _ { - i } ) - e ^ { - \\beta \\tau _ i } K \\right \\} \\end{align*}"}
+{"id": "6093.png", "formula": "\\begin{align*} J _ c = \\left \\{ \\sum _ { g \\in F } f _ g u _ g : f _ g \\in I _ c \\right \\} \\end{align*}"}
+{"id": "2218.png", "formula": "\\begin{gather*} \\begin{array} { c c l l } \\dot { x } _ 1 & = & - x _ 1 ^ 2 - x _ 2 ^ 2 - x _ 3 ^ 2 - 2 \\eta x _ 1 x _ 2 & + x _ 1 \\\\ & & & \\\\ \\dot { x } _ 2 & = & - 2 x _ 1 x _ 2 - \\eta x _ 1 ^ 2 - \\eta x _ 2 ^ 2 + \\eta x _ 3 ^ 2 & - \\beta x _ 3 + x _ 2 \\\\ & & & \\\\ \\dot { x } _ 3 & = & - 2 x _ 1 x _ 3 - 2 \\eta x _ 2 x _ 3 & + \\beta x _ 2 + x _ 3 \\end{array} \\end{gather*}"}
+{"id": "7189.png", "formula": "\\begin{align*} K _ T ( \\mathbb { T } ( d ) _ v ) \\otimes _ { \\mathbb { K } } \\mathbb { F } = \\bigoplus _ { \\begin{subarray} { c } ( d , v ) = ( d _ 1 , v _ 1 ) + \\cdots + ( d _ k , v _ k ) , \\\\ v / d = v _ i / d _ i \\end{subarray} } \\mathbb { F } \\cdot [ \\mathcal { E } _ { d _ 1 , v _ 1 } ] \\ast \\cdots \\ast [ \\mathcal { E } _ { d _ k , v _ k } ] . \\end{align*}"}
+{"id": "4977.png", "formula": "\\begin{align*} \\sum _ { w = 1 } ^ { w _ { r } } p _ { w } ( M + 1 ) ^ { w } \\approx p _ { w _ { 1 } } ( M + 1 ) ^ { w _ { 1 } } + p _ { w _ { 2 } } ( M + 1 ) ^ { w _ { 2 } } , \\end{align*}"}
+{"id": "1955.png", "formula": "\\begin{align*} R = \\delta _ R ( \\mathcal { R } _ 0 ) + \\xi _ R , \\end{align*}"}
+{"id": "2247.png", "formula": "\\begin{align*} W ^ m = \\sum _ { i = 0 } ^ m W _ i ^ { m - i } d ^ i , \\ \\ W _ i : = V + \\delta ( V ) + \\cdots + \\delta ^ i ( V ) \\end{align*}"}