diff --git "a/data_tmp/process_14/tokenized_finally.jsonl" "b/data_tmp/process_14/tokenized_finally.jsonl"
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-{"id": "680.png", "formula": "\\begin{align*} \\mu _ N ( d x ) : = \\frac { 1 } { Z } \\exp \\left ( - H ( x ) \\right ) d x . \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} \\gamma _ J = a _ { 2 J } b _ { 0 J } , \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} A _ \\epsilon & = A _ \\epsilon + ( n + 1 ) H _ 0 V _ \\epsilon \\\\ & = - ( n + 1 ) a _ n \\int _ { z _ 1 } ^ { z _ 2 } \\Big ( H ( z , \\epsilon ) - H _ 0 \\Big ) h ^ n h _ \\epsilon d z + a _ n \\left [ \\frac { h ^ n h _ z h _ \\epsilon } { \\sqrt { 1 + h _ z ^ 2 } } \\right ] _ { z _ 1 } ^ { z _ 2 } . \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\max _ { x \\in \\Lambda _ j } \\vert A _ j ( x ) \\vert = \\infty \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} \\mathcal { T } = \\frac { \\tau _ 0 } { 1 + 2 C _ r ( 1 + \\| e ^ { \\tau _ 0 A } \\overline { v } _ 0 \\| _ { H ^ r } ^ 2 + \\| e ^ { \\tau _ 0 A } \\widetilde { v } _ 0 \\| _ { H ^ r } ^ 2 ) } > 0 , \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} K _ { ( i _ 1 \\dots i _ d , i _ { d + 1 } ) } = 0 \\ , . \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} \\frac { \\ell _ t } { L } = 1 + \\Pr \\left ( Q _ t = A B \\right ) . \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} { _ { 0 } \\mathcal { F } _ 0 } ^ { ( d ) } \\left ( \\begin{matrix} \\\\ \\end{matrix} ; \\mathbf { 1 } + \\mathbf { z } , \\mathbf { u } \\right ) & = e ^ { E _ { 0 } ( \\mathbf { z } ) } { _ { 0 } \\mathcal { F } _ 0 } ^ { ( d ) } \\left ( \\begin{matrix} \\\\ \\end{matrix} ; \\mathbf { z } , \\mathbf { u } \\right ) = { _ { 0 } \\mathcal { F } _ 0 } ^ { ( d ) } \\left ( \\begin{matrix} \\\\ \\end{matrix} ; \\mathbf { z } , \\mathbf { u } \\right ) e ^ { | \\mathbf { u } | } . \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} \\big \\lbrace \\omega \\in \\Omega \\ : s \\mapsto X ( \\omega , s ) \\ s . t . \\ V ( \\omega , 0 ) = - c , \\ N ( \\omega , t ) = n , \\ X ( \\omega , t ) = 2 \\beta - x \\big \\rbrace , \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} c & = a + b \\\\ d & = a b , \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} M _ 1 \\leq C ( \\| \\nabla u \\| _ { L ^ \\infty } + \\| | D | ^ s \\omega \\| _ { L ^ { 2 } } + 1 ) \\| | D | ^ s X \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} W ( \\phi ^ a _ x , g ; x ) & = \\phi _ x ( x - 0 ) g ' ( x ) - \\phi _ x ' ( x - 0 ) g ( x ) , \\\\ W ( \\phi ^ b _ x , g ; x ) & = \\phi _ x ( x + 0 ) g ' ( x ) - \\phi _ x ' ( x + 0 ) g ( x ) . \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} \\Psi ( ( A ^ { 1 , 0 } ) ^ g ) = e ^ { \\frac { i } { \\hbar } S ^ { 1 , 0 } _ { g W Z W } ( A ^ { 1 , 0 } , g ) } \\Psi ( A ^ { 1 , 0 } ) \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} p _ { t } ^ n ( x , y ) = \\int _ { K _ n } p _ { s } ^ n ( x , z ) p _ { t - s } ^ n ( z , y ) \\ , d m ( z ) . \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} & B _ { 3 } ( y ) = \\int _ { 0 } ^ { \\infty } \\frac { d \\rho } { ( 1 + 2 ( \\rho ^ { 2 } + y ^ { 2 } ) + ( \\rho ^ { 2 } - y ^ { 2 } ) ^ { 2 } ) ^ { 1 / 2 } } \\leq C \\int _ { 0 } ^ { 2 y } \\frac { d \\rho } { y } + \\int _ { 2 y } ^ { \\infty } \\frac { d \\rho } { ( 1 + \\rho ^ { 2 } ) } \\leq C \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} G ( z , t ) = e ^ { - t ( c _ { 1 } \\{ ( \\lambda _ { 1 } + \\mu ( 1 - z ) ) ^ { \\alpha _ { 1 } } - { \\lambda _ { 1 } } ^ { \\alpha _ { 1 } } \\} + c _ { 2 } \\{ ( \\lambda _ { 2 } + \\mu ( 1 - z ) ) ^ { \\alpha _ { 2 } } - { \\lambda _ { 2 } } ^ { \\alpha _ { 2 } } \\} ) } , \\ ; | z | \\leq 1 , \\ ; \\ ; \\mu \\leq \\frac { \\lambda _ { i } } { 2 } , \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} \\tfrac 1 2 ( x _ 1 - 1 ) ^ 2 + \\tfrac 1 2 ( x _ 2 - 1 ) ^ 2 & \\ , \\to \\ , \\min \\\\ x _ 1 \\ , \\leq \\ , 0 \\ , \\lor \\ , x _ 2 & \\ , \\leq \\ , 0 . \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} \\psi ^ * \\varphi ^ * ( A _ { \\overline k } ) = ( F ^ e _ { X \\times _ k \\overline k } ) ^ * ( A _ { \\overline k } ) = A _ { \\overline k } ^ q . \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} & \\frac { c _ { b } } { 4 \\pi } \\int _ { 0 } ^ { \\pi } d \\theta \\int _ { 0 } ^ { \\infty } d \\xi \\left ( \\frac { \\sin ( \\xi t _ { + } ) } { t _ { + } ^ { 3 } } + \\frac { \\sin ( \\xi t _ { - } ) } { t _ { - } ^ { 3 } } \\right ) \\partial _ { \\xi } ^ { 3 } \\left ( \\frac { \\xi ^ { 2 } \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\right ) \\\\ & = \\frac { 2 b } { t ^ { 3 } \\log ^ { b } ( t ) } + E _ { \\partial _ { 1 2 } v _ { 2 } , 1 } ( t , r ) \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} \\| f \\mid W ^ 1 _ p ( \\Omega ) \\| = \\| f \\mid L _ p ( \\Omega ) \\| + \\| \\nabla f \\mid L _ p ( \\Omega ) \\| . \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} M _ D ( x , z _ 0 ) & = \\int _ { D \\setminus U } M _ D ( y , z _ 0 ) \\omega _ U ^ x ( d y ) . \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\varepsilon ^ { n } _ k = \\begin{cases} 1 k = n , \\\\ 0 k \\neq n , \\end{cases} k \\in \\mathbb Z ^ c . \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} \\Delta _ A ( f , a ^ 2 ) = \\| ~ ( x ^ 2 - a ^ 2 ) f \\| _ { L ^ 2 ( \\mathbb { R } ) } = \\left ( \\int _ { \\mathbb { R } } ( x ^ 2 - a ^ 2 ) ^ 2 | f ( x ) | ^ 2 d x \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} w _ \\pm ( z ) = \\left ( ( \\beta + 1 ) c _ \\beta \\int _ { \\pm 1 } ^ z ( \\hat z ^ 2 - 1 ) ^ { \\beta } \\hat z ^ { - 2 - 2 \\beta } \\ , d \\hat z \\right ) ^ { \\frac 1 { \\beta + 1 } } , \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A \\left ( \\left ( x \\right ) \\right ) u = F \\left ( u , \\bar { u } \\right ) , x \\in R ^ { n } , t \\in \\left [ 0 , T \\right ] \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} F ( t ) = \\dfrac { d _ R ^ 2 } { 1 4 4 \\pi D _ \\theta t } \\exp \\bigg ( \\dfrac { - ( G _ z - W _ z t ) ^ 2 } { 4 D _ \\theta t } \\bigg ) \\cdot \\bigg ( 4 \\sum _ { k = 0 } ^ { 3 } \\lambda ( k , t ) + 2 \\sum _ { k = 1 } ^ { 3 } \\xi ( k , t ) + 2 \\omega ( t _ s ) \\bigg ) , \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} F ( X \\otimes Z _ 5 ) & = ( X _ 1 \\oplus X _ 4 ) \\oplus ( X _ 1 \\oplus X _ 3 \\oplus X _ 4 \\oplus X _ 5 ) , \\\\ F ( X \\otimes Z _ 5 ) & = ( X _ 1 \\oplus X _ 3 \\oplus X _ 4 ) \\oplus ( X _ 1 \\oplus X _ 4 \\oplus X _ 5 ) . \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} \\frac { u _ 1 } { \\sin \\phi _ 1 } = \\frac { u _ 2 } { \\sin \\phi _ 2 } + \\frac { u _ 3 } { \\sin \\phi _ 3 } \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} \\mu = \\frac { \\| \\eta f \\| _ { H ^ { \\beta - s } ( \\mathbb { R } ^ { n } \\times \\{ 0 \\} ) } } { \\| x _ { n + 1 } ^ { \\frac { 2 s - 1 } { 2 } } \\tilde { w } \\| _ { L ^ { 2 } ( C _ { \\overline { s } , 2 } ^ { + } ) } } . \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} L = \\lim _ { t \\to \\infty } \\sup _ { \\zeta \\in [ t , \\infty ] } v ( \\zeta ) \\le a L \\mu ^ { - 1 } + \\sup _ { t \\in \\mathbb R ^ + } \\omega ( \\cdot , \\mu ) * b ( t ) + ( v _ 0 + a M + a \\mu ^ { - 1 } ) \\varepsilon , \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} R ( z ) = z ^ 3 \\circ \\frac { z ^ 2 - 4 } { z - 1 } \\circ \\frac { z ^ 2 + 2 } { z + 1 } = \\frac { z ( z - 8 ) ^ 3 } { ( z + 1 ) ^ 3 } \\circ z ^ 3 . \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} F ( u ) = F _ 2 ( u ) : = ( - a u _ 1 + c u _ 2 , G ( u _ 1 ) - b u _ 2 ) , \\ ; \\ ; \\mathbf { u ^ * } = ( K _ 1 , K _ 2 ) \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} f ^ { \\varepsilon , a , n } ( 0 , x , v ) = f _ 0 ( x , v ) , \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} Q ( A _ p , A _ q ) \\ = \\ & 1 \\ p < q \\in [ 0 , n ] , \\\\ Q ( A _ { n + p } , A _ { n + q } ) \\ = \\ & 1 \\ p < q \\in [ n ] . \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} { f _ { Y \\left | X \\right . } } \\left ( { y \\left | x \\right . } \\right ) = \\frac { 1 } { { \\sqrt { 2 \\pi \\left ( { 1 + x { \\varsigma ^ 2 } } \\right ) { \\sigma ^ 2 } } } } { e ^ { - \\frac { { { { \\left ( { y - x } \\right ) } ^ 2 } } } { { 2 \\left ( { 1 + x { \\varsigma ^ 2 } } \\right ) { \\sigma ^ 2 } } } } } , \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} y _ 0 = j _ p ( x ) , y _ { 2 n + 1 } = \\Pi _ { M ^ \\perp } ^ { p ^ * } y _ { 2 n } , y _ { 2 n } = \\Pi _ { N ^ \\perp } ^ { p ^ * } y _ { 2 n - 1 } , \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} \\mathsf { \\hat { t } } _ { \\mathsf { j } } = \\mathsf { \\hat { q } } _ { \\mathsf { j } } \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} Z ^ { ( k ) } = \\underbrace { Y \\times _ X \\ldots \\times _ X Y } _ { k } , \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} \\sum \\delta _ j ^ { \\beta } = \\sum \\left ( \\delta _ j ^ { \\beta } \\ , j ^ a \\right ) \\ , j ^ { - a } \\leq \\left ( \\sum \\delta _ j ^ 2 \\ , j ^ { \\frac { 2 a } { \\beta } } \\right ) ^ { \\frac { \\beta } { 2 } } \\left ( \\sum j ^ { - \\frac { 2 a } { 2 - \\beta } } \\right ) ^ { \\frac { 2 - \\beta } { 2 } } \\ , . \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} S _ { } [ g , A ] = \\frac { 1 } { 2 } \\int _ { \\Sigma } \\d { } ^ 2 z ( g ^ { - 1 } D g ) ^ \\mu E _ { \\mu \\nu } ( g ^ { - 1 } \\bar { D } g ) ^ \\nu , \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } b _ { \\ell } ( n ) q ^ n = \\prod _ { n = 1 } ^ { \\infty } \\frac { ( 1 - q ^ { \\ell n } ) } { ( 1 - q ^ n ) } . \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} e ^ { - K } = i \\langle \\Omega , \\overline { \\Omega } \\rangle . \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} | | \\gamma ^ V ( t ) | | ^ 2 \\leq - \\lambda _ 1 | | \\gamma ^ V ( t ) | | ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ N ) } \\leq - \\lambda _ 1 | | \\gamma ( t ) | | ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ N ) } = - \\lambda _ 1 ( t L ) ^ N | | \\tilde { u } | | ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ N ) } . \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} M _ { N } = \\left ( \\begin{array} { r r r r r r r r r r r r r r r r r r r } 1 & 0 & 1 & 0 & - 1 & 0 & - 1 & 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 \\\\ - 1 & 0 & - 1 & 0 & 1 & 0 & 1 & 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 \\end{array} \\right ) , \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} R = 4 \\langle S u , ( I I ) u \\rangle + 4 \\langle A u , ( I ) u \\rangle + 4 \\langle A u , ( I I I ) u \\rangle + 4 \\langle ( I ) u , ( I I ) u \\rangle + 4 \\langle ( I I ) u , ( I I I ) u \\rangle . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} t _ { + / - } = t \\pm r \\cos ( \\theta ) \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} \\left ( w _ { 1 } ^ { \\top } G \\left ( E _ { 1 } \\right ) w _ { 1 } - d _ { 1 1 } + E _ { 1 } \\right ) \\left ( w _ { 2 } ^ { \\top } G \\left ( E _ { 1 } \\right ) w _ { 2 } - d _ { 2 2 } + E _ { 1 } \\right ) & = 1 + o \\left ( 1 \\right ) \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} S = | G | \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} q ^ * ( t ) = & q ^ * ( T ) \\exp \\Bigl [ - \\int _ t ^ T \\frac { \\lambda ^ * ( s ) + \\lambda _ 1 ^ * ( s ) + c ^ * ( s ) } { s } d s \\Bigr ] \\\\ & - \\int _ t ^ T \\exp \\Bigl [ - \\int _ t ^ { \\tau } \\frac { \\lambda ^ * ( s ) + \\lambda _ 1 ^ * ( s ) + c ^ * ( s ) } { s } d s \\Bigr ] \\frac { \\gamma ^ * ( \\tau ) u ^ * ( \\tau ) } { \\tau } d \\tau . \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} L ( \\overline { v _ { M } } ) ( 0 ) = \\int _ { 0 } ^ { \\infty } \\overline { y } ( \\xi ) \\lim _ { r \\rightarrow 0 } L \\left ( \\frac { \\phi ( r , \\xi ) } { \\sqrt { r } } \\right ) \\chi _ { \\leq 1 } \\left ( \\frac { \\xi } { M } \\right ) \\rho ( \\xi ) d \\xi = 0 \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} \\P & \\left [ | X _ { t } ^ { i } - v _ { i } t | \\geq \\frac { \\epsilon } { d } u , t \\in [ 0 , u ] \\right ] \\\\ & = \\P \\left [ | Y _ { t } - \\bar { v } _ { i } t | \\geq \\frac { \\epsilon } { d } u , t \\in [ 0 , ( p ( e _ { i } ) + p ( - e _ { i } ) ) u ] \\right ] . \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} W = \\bigcup _ { y \\in [ \\eta ( x ) ] ^ c } V _ { x y } . \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} \\Gamma _ 0 & = \\sum _ { k = 0 } ^ \\infty ( A ' ) ^ k A ^ k = A ' \\Gamma _ 0 A + I _ 2 \\\\ \\Omega _ 0 & = \\sum _ { k = 0 } ^ \\infty ( B ' ) ^ { - k } ( B ' ) ^ { - 1 } T ' T B ^ { - 1 } B ^ { - k } = ( B ' ) ^ { - 1 } \\Omega _ 0 B ^ { - 1 } + ( B ' ) ^ { - 1 } T ' T B ^ { - 1 } \\\\ \\Gamma _ k & = \\Gamma _ 0 A ^ k = ( \\Gamma _ 0 A \\Gamma _ 0 ^ { - 1 } ) ^ k \\Gamma _ 0 \\\\ \\Omega _ k & = ( B ' ) ^ { - k } \\Omega _ 0 \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} \\sum _ { u _ k : \\ , \\ , d ( u _ 0 , u _ k ) = k } S ( u _ 0 , u _ { \\ell } , u _ k ) \\le ( 2 t ) ^ { 2 ( k - \\ell ) } \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} ( G _ { 1 3 } ) & = ( G _ 1 ) + ( G _ 3 ) \\\\ & \\leq ( G ) . \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\Big | \\frac { d ^ j } { d z ^ j } \\log \\Big ( \\frac { \\prod _ { w \\in W _ n } ( z - w ) } { \\prod _ { u \\in U _ n } ( z - u ) } \\Big ) \\Big | _ { z = \\l _ n } \\Big | < \\infty , \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} \\rho ( ( x , s ) , ( y , t ) ) = \\inf \\{ ( u - s ) + ( u - t ) : ( \\exists z \\in X ) B _ { \\leq s } ( x ) \\cup B _ { \\leq t } ( y ) \\subseteq B _ { \\leq u } ( z ) \\} . \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { g ( t ) } t = - c _ 0 . \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} d ( x , x ' ) | A ' - B ' | = v _ \\ell ^ { - 1 } v _ { \\ell - k } | W ( z ) - W ( 0 ) | \\leq | z | \\sup _ { | \\zeta | \\leq 1 } | W ' ( \\zeta ) | . \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} U _ { x } V _ { y , x } = V _ { x , y } U _ { x } , V _ { U _ { x } y , y } = V _ { x , U _ { y } x } , U _ { U _ { x } y } = U _ { x } U _ { y } U _ { x } \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} \\frac { Q ( A , B ) } { 2 | A | \\cdot | B | } \\ = \\ P ( a > b ) - \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} C _ 1 = [ \\emptyset , 1 , \\emptyset , \\emptyset , \\emptyset , 2 , \\emptyset , \\emptyset , 1 0 , \\emptyset ] , C _ 2 = [ 8 , \\emptyset , \\emptyset , 1 4 , 1 6 , \\emptyset , \\emptyset , \\emptyset , \\emptyset , \\emptyset ] , C _ 3 = [ \\emptyset , \\emptyset , 1 9 , \\emptyset , \\emptyset , \\emptyset , 2 5 , \\emptyset , \\emptyset , 2 7 ] . \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} \\Pi _ s ( F _ i ) = - \\Pi ( F _ { i s } ) = - \\Pi ( \\nabla _ { F _ i } F _ s ) \\ , , \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} | F _ 3 | \\cdot b _ { i + 1 } = h _ i \\le D \\cdot | A _ 1 \\cap \\cdots \\cap A _ i \\setminus ( A _ { i + 1 } \\cup \\cdots \\cup A _ { k - 1 } ) | . \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} \\forall \\hat { x } \\in \\hat { E } \\quad \\hat { x } = \\sup \\{ x \\in E : x \\leqslant \\hat { x } \\} = \\inf \\{ x \\in E : \\hat { x } \\leqslant x \\} \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} | g r a p h _ { \\Sigma } ( v _ j ) | \\llcorner U ' = V _ j \\llcorner U ' \\ \\ \\ j \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} \\partial _ { t } f + \\mathbf { u } \\cdot \\nabla _ { z , \\upsilon } f = - \\gamma ^ { 2 } \\partial _ { z } \\rho + \\Delta _ { t } f , \\Delta _ { t } \\phi = f , \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} K ( x , y , t ) & : = x y ( 1 - t S ( x , y ) ) . \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} \\| \\nabla u ( t ^ 0 ) \\| _ p = \\lim \\limits _ { t \\rightarrow t ^ { 0 } } \\| \\nabla u \\| _ p \\geq r _ * > 0 , \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} - \\partial _ { t t } u + \\partial _ { r r } u + \\frac { 1 } { r } \\partial _ { r } u = \\frac { \\sin ( 2 u ) } { 2 r ^ { 2 } } \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} g ( s ) & = \\ , ( 0 , 1 , 2 , 0 , 1 , 3 , { \\bf 2 } , 5 , 5 , { \\bf 2 } , 7 , 3 , { \\bf 2 } , 3 ) \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , { \\bf 2 } , 3 , { \\bf 2 } , 5 , 5 , { \\bf 2 } , 8 , 3 , { \\bf 2 } , 3 ) \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , { \\bf 2 } , 3 , 6 , 5 , 5 , { \\bf 2 } , 8 , 3 , { \\bf 2 } , 3 ) \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , { \\bf 2 } , 3 , 6 , 5 , 5 , 8 , { 6 } , 3 , { \\bf 2 } , 3 ) = f _ { 5 , 3 } ( s ) = f _ 5 ^ * ( s ) . \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} \\begin{cases} u _ t = ( D _ x ^ \\alpha u ) _ x + f \\quad & \\\\ u = g \\quad & \\end{cases} \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} \\prod _ { t = 1 } ^ r \\left ( \\frac { 1 } { x _ { i _ t } - x _ { j _ t } } \\right ) = \\frac { 1 } { \\prod _ { b + 1 \\leqslant i \\leqslant n } \\prod _ { 1 \\leqslant j \\leqslant b } ( x _ i - x _ j ) } , \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } v _ { 4 } ^ { 0 } ( t , r ) | \\leq \\begin{cases} \\frac { C } { t ^ { 3 } \\log ^ { 2 N + b - 2 } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { t ^ { 2 } \\log ^ { 3 N + b } ( t ) } , r \\geq \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} \\mathcal { L } _ L ^ { \\lambda } f : = \\sum \\limits _ { \\ell _ 1 + \\ell _ 2 + \\ell _ 3 \\leq L } \\mathcal { S } _ { \\lambda \\mu _ { \\ell } } \\left ( \\alpha _ { \\ell } \\right ) p _ { \\ell } \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} \\tilde { I } _ 1 = \\frac { 2 x y _ x - ( y + x ^ 2 ) } { y _ x } \\ , , \\tilde { I } _ 2 = \\frac { 9 ( y + x ^ 2 ) y _ x ^ 2 - 4 x ( x ^ 2 + 9 y ) y _ x + 1 2 y ( x ^ 2 + y ) } { y _ x } \\ , , y _ x \\neq 0 \\ , . \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} A ^ p \\sigma _ h B ^ p & \\ge A ^ { 1 / 2 } h ( C ^ p ) h ( C ) ^ { 1 - p } A ^ { 1 / 2 } \\\\ & \\geq \\lambda _ { \\min } \\left ( \\frac { h ( C ^ p ) } { h ( C ) ^ p } \\right ) A ^ { 1 / 2 } h ( C ) A ^ { 1 / 2 } \\\\ & = \\lambda _ { \\min } \\left ( \\frac { h ( C ^ p ) } { h ( C ) ^ p } \\right ) ( A \\sigma _ h B ) . \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} \\frac 1 2 \\Delta \\eta ^ { 2 } & = \\eta \\eta ' \\Delta \\rho + \\eta \\eta '' + ( \\eta ' ) ^ { 2 } \\\\ & \\geq ( n - 1 ) i _ R ( z ) \\eta \\eta ' + \\eta \\eta '' + ( \\eta ' ) ^ { 2 } \\\\ & \\geq - \\frac { 4 } { R } \\left ( ( n - 1 ) i _ R ( z ) + \\frac { 2 } { R } \\right ) \\eta \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} 1 - \\varepsilon \\leq | \\tau ( x \\prod ^ n _ { t = 1 } P _ { S _ t } ( a ^ t _ k ) ) | . \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} n _ { t + 1 } \\in \\bigcap _ { \\ell = 0 } ^ d \\left [ B _ 1 ^ \\ell \\cap \\bigcap _ { s \\in \\{ 1 , . . . , 2 ^ \\ell \\} \\setminus \\{ 2 ^ \\ell \\} } \\bigcap _ { 1 \\leq j _ 1 < \\cdots < j _ s \\leq t } B _ { s + 1 } ^ \\ell ( n _ { j _ 1 } , . . . , n _ { j _ s } ) \\right ] \\in p . \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} M & = \\left ( \\begin{array} { c c c c c } 0 & 0 & \\cdots & 0 & 0 \\\\ 0 & m _ { 2 , 2 } & \\cdots & m _ { 2 , c } & m _ { 2 , c + 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\cdots & m _ { c , c } & m _ { c , c + 1 } \\\\ 0 & 0 & \\cdots & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} \\mathcal { M } ( G : H ) : = \\{ \\nu \\in M ( G ) : \\nu _ h = \\nu \\ \\forall h \\in H \\} . \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} L _ { \\Sigma } u _ 0 = 0 \\ \\ B _ { 2 r _ 2 } \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} G = G ( d , s , t ) + \\frac { d } { 2 } ( A '' - A ) + ( R - \\rho - 1 ) . \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} \\rho _ { s c } ( x ) = \\frac { 1 } { 2 \\pi } \\sqrt { \\left ( 4 - x ^ 2 \\right ) _ + } , \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} d ^ * ( \\eta ( x ) , \\eta ( z ) ) = d ( x , z ) \\leq d ( x , y ) + d ( y , z ) = d ^ * ( \\eta ( x ) , \\eta ( y ) ) + d ^ * ( \\eta ( y ) , \\eta ( z ) ) . \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} p _ \\mu ( x ) = N _ { \\alpha } [ 1 + b _ \\alpha ( x - \\mu ) ^ 2 ] ^ { \\frac { 1 } { \\alpha - 1 } } _ + , \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} \\frac { d ^ a } { d X ^ a } \\sum _ { k = 0 } ^ { n - 1 } T _ { \\o _ k } ( 1 , R ( X ) ) X ^ k \\Big | _ { X = \\l } = \\frac { d ^ a } { d X ^ a } \\sum _ { k = 0 } ^ { n - 1 } T _ { \\o ' _ k } ( 1 , R ( X ) ) X ^ k \\Big | _ { X = \\l } \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} D _ \\alpha u _ t & = D _ \\alpha [ \\rho ( Q \\ast u - u ) ] - D _ \\alpha ( \\mathbf { u } \\cdot \\nabla u ) \\\\ & = D _ \\alpha [ \\rho ( Q \\ast u - u ) ] - [ D _ \\alpha ( \\mathbf { u } \\cdot \\nabla u ) - \\mathbf { u } \\cdot \\nabla D _ \\alpha u ] - \\mathbf { u } \\cdot \\nabla D _ \\alpha u . \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} v _ R ( r , \\omega ) : = u ( r R , \\omega ) / c _ R \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} \\mathfrak g = ( h _ \\delta ^ \\perp \\cap \\mathfrak t ) \\oplus \\bigoplus \\limits _ { \\alpha \\in Y ( \\delta ) } V ( \\alpha ) . \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} \\partial _ t \\Delta _ q \\omega + \\Delta _ q ( u \\cdot \\nabla \\omega ) = \\partial _ 1 \\Delta _ q \\theta . \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N w _ j g ( \\mathbf { x } _ j ) = \\int _ { \\mathbb { S } ^ 2 } g { \\rm { d } } \\omega \\quad \\forall g \\in \\mathbb { P } _ { 2 L } \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} w ^ { ( 1 ) } ( \\theta ) = - 2 b _ \\alpha S ( \\theta ) ^ { - 1 } \\boldsymbol { \\Sigma } ^ { - 1 } \\boldsymbol { \\mu } , f ^ { ( 1 ) } ( { \\bf { x } } ) = { \\bf { x } } , w ^ { ( 2 ) } ( \\theta ) = b _ \\alpha S ( \\theta ) ^ { - 1 } \\rm { v e c } ( \\boldsymbol { \\Sigma } ^ { - 1 } ) , f ^ { ( 2 ) } ( { \\bf { x } } ) = \\rm { v e c } ( { \\bf { x } } { \\bf { x } } ^ \\top ) . \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} \\frac { 3 } { ( 3 - \\kappa ) ( 1 + \\mu ) ^ 2 } = \\frac { 3 } { 3 + \\kappa } > \\frac { 3 - \\kappa } { 3 + \\kappa } = \\frac { 1 } { ( 1 + \\mu ) ^ 2 } \\ , . \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} ( \\bar { A } ) ^ { - 1 } = P A ^ { - 1 } N P ^ * . \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} \\Delta f = \\mathrm { t r } ( \\mathrm { H e s s } _ f ) - \\langle \\nabla \\phi , \\nabla f \\rangle \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} \\partial _ t u - \\mathcal { L } u = f ( u , \\overline u ) ( t , x ) \\in \\mathbb { R } \\times \\Omega \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} I ^ { \\varphi ^ t } ( x ) : = \\{ l \\in \\mathcal Q \\ , | \\ , \\varphi ^ t ( G _ l ( x ) , H _ l ( x ) ) = 0 \\} \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} \\phi ( - q ^ { { \\frac { j - i } { 2 } } } x _ { j - 1 } x _ { j - 2 } \\cdots x _ { i } ) = \\displaystyle \\sum _ { m \\geq 0 } \\frac { q ^ { \\frac { ( 2 - ( j - i ) ) m ^ { 2 } } { 2 } } x _ { i } ^ { m } x _ { i + 1 } ^ { m } \\cdots x _ { j - 1 } ^ { m } } { ( q ) _ { m } } . \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } \\partial _ { j } a _ { j k } \\partial _ { k } \\bigg ] \\tilde { w } & = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ \\tilde { w } & = w \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} \\sum _ { s = m } ^ { n } K _ { s } ^ { ( 3 ) } = \\frac { 1 } { 3 } \\left ( K _ { n + 2 } ^ { ( 3 ) } + 2 K _ { n } ^ { ( 3 ) } + K _ { m } ^ { ( 3 ) } - K _ { m + 2 } ^ { ( 3 ) } \\right ) , \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} \\| w \\| _ U ^ 2 = \\frac { 1 } { 2 \\i } W _ a ( \\bar w , w ) . \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} f ( z ) : = e ^ { - z ^ 2 / 2 } \\frac { ( 1 - e ^ { - z ^ 2 } ) } { z ^ 2 } . \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} I _ { r } ( \\theta ) = \\frac { r \\sin \\theta } { \\theta } \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t ^ { 2 } } { | r e ^ { i \\theta } - t | ^ { 2 } } \\ , d \\sigma ( t ) , \\quad \\theta \\in ( 0 , \\pi ] , \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} \\omega ^ \\chi _ \\varphi : = ( d _ 1 \\cdots d _ k f ^ \\varphi ) | _ \\Delta \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} \\mathsf { R m i n } ( s ) = \\{ s _ i : s _ i < s _ j \\mbox { f o r a l l } j > i \\} . \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} a ( n - 1 ) = 2 ^ { n - 2 } + \\sum _ { i = 1 } ^ { n - 2 } a ( i ) . \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} r _ { 2 } * ( K _ { 2 } * x ) + r _ { 2 } * x & = r _ { 2 } * H _ { 2 } \\\\ ( r _ { 2 } * K _ { 2 } ) * x + r _ { 2 } * x & = r _ { 2 } * H _ { 2 } \\\\ H _ { 2 } - x = K _ { 2 } * x & = r _ { 2 } * H _ { 2 } \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} C _ 0 : = \\sup _ { t > 0 } \\big [ | \\bar h ( t ) - c _ 0 t | + | \\underline h ( t ) - c _ 0 t | \\big ] < \\infty . \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} \\omega ' = f , \\int \\omega ( \\theta ) d \\theta = 0 . \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} a _ { 0 } = - \\frac { 1 } { 2 } + i \\int _ { \\mathbb { T } } \\frac { \\Im ( \\alpha t ) } { \\left | 1 - \\alpha t \\right | ^ { 2 } } \\ , d \\mu _ { 1 } ( t ) \\quad a _ { 1 } = - \\overline { \\alpha } \\int _ { \\mathbb { T } } \\frac { 1 } { \\left | 1 - \\alpha t \\right | ^ { 2 } } \\ , d \\mu _ { 1 } ( t ) . \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} c ' ( s _ 0 ) \\hat { h } _ z ( z , s _ 0 ) + \\hat { h } _ s ( z , s _ 0 ) = c ' ( s _ 0 ) h ( z , s _ 0 ) + h _ s ( z , s _ 0 ) , \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} \\overline { X } \\otimes X = \\oplus _ i ( \\overline { X } _ i \\otimes X _ i ) , X \\otimes \\overline { X } = \\oplus _ i ( X _ i \\otimes \\overline { X } _ i ) . \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} ( L \\bar { L } + \\lambda ) f = 0 . \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} J _ { \\tau , 0 } = \\left ( \\begin{array} { c c c c } 0 & 0 & 0 & 0 \\\\ 0 & - K & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} t { \\phi } _ { p , \\alpha } ' ( t ) = p t ^ { p } ( 1 + \\log ( t ) ) ^ { \\alpha } + \\alpha t ^ { p } ( 1 + \\log ( t ) ) ^ { \\alpha - 1 } . \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} & m _ { [ i _ { 2 } - 1 , i _ { 2 } - 1 ] } m _ { [ i _ { 2 } , i _ { 2 } ] } + \\sum _ { s = i _ { 2 } } ^ { j _ { 2 } - 1 } m _ { [ s , s ] } m _ { [ s + 1 , s + 1 ] } + m _ { [ j _ { 2 } , j _ { 2 } ] } m _ { [ j _ { 2 } + 1 , j _ { 2 } + 1 ] } \\\\ & = ( 2 ( j _ { 2 } - i _ { 2 } ) + 2 ) m _ { [ i _ { 2 } , j _ { 2 } ] } m _ { [ i _ { 1 } , n ] } + \\cdots , \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} 0 & = \\nabla f ( x _ k ) + \\sum \\limits _ { i \\in I ^ g ( x _ k ) } \\lambda _ i ^ k \\nabla g _ i ( x _ k ) + \\sum \\limits _ { j \\in \\mathcal P } \\rho _ j ^ k \\nabla h _ j ( x _ k ) \\\\ & + \\sum \\limits _ { l \\in I ^ { \\varphi ^ { t _ k } _ \\textup { F B } } ( x _ k ) } \\xi ^ k _ l \\left ( \\alpha ^ k _ l \\nabla G _ l ( x _ k ) + \\beta ^ k _ l \\nabla H _ l ( x _ k ) \\right ) \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} | C | \\leq \\frac { \\eta N _ t - N _ t + t } { \\eta N _ t - N _ t + Q _ t } = \\frac { \\eta - 1 + t / N _ t } { \\eta - 1 + Q _ t / N _ t } . \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} \\frac { | \\chi ( a ) | } { \\| a \\| } = \\frac { \\Bigm | \\sum \\limits _ { n \\in \\mathbb Z ^ c } a _ n u ^ n \\Bigm | } { \\| a \\| } \\le \\frac { \\sum \\limits _ { n \\in \\mathbb Z ^ c } | a _ n | \\cdot | u ^ n | } { \\| a \\| } \\le \\frac { \\sum \\limits _ { n \\in \\mathbb Z ^ c } | a _ n | \\gamma g ( n ) } { \\sum \\limits _ { n \\in \\mathbb Z ^ c } | a _ n | g ( n ) } = \\gamma . \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} \\frac { 1 } { | Q | } \\int _ Q \\left ( \\frac { \\sum _ { j \\in \\mathbb { N } } h _ j ( x ) \\chi _ { Q _ j } ( x ) } { \\lambda } \\right ) ^ { p ( x ) } \\ , \\d x & = \\sum _ { j \\in \\mathbb { N } } \\frac { | Q _ j | } { | Q | } \\frac { 1 } { | Q _ j | } \\int _ { Q _ j } \\left ( \\frac { h _ j ( x ) } { \\lambda } \\right ) ^ { p ( x ) } \\ , \\d x \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} H ( \\mu _ { \\lambda } ^ { ( n _ { j + 1 } ) } ; ( n _ { j + 1 } + 1 ) ^ { - C ( n _ { j + 1 } + 1 ) } ) = n _ { j + 1 } H ( p ) . \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\| P _ * ( r , k ) - \\Pi ( k ) \\cdot \\chi _ { E _ s ^ c } \\| _ { 2 , \\sigma } = 0 \\ , . \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} S _ { t } + \\left [ S , K \\right ] = - 2 i \\tilde { \\gamma } \\left ( \\nabla \\varphi . \\partial _ { t } \\nabla + \\Delta \\varphi \\partial _ { t } \\right ) + \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} A _ h ^ { w / 2 } = \\frac { i } { 2 \\pi } \\int _ \\gamma \\tilde \\lambda ^ { w / 2 } ( A _ h - \\tilde \\lambda ) ^ { - 1 } \\ , \\dd \\tilde \\lambda \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( \\xi ) = \\lim _ { r \\uparrow 1 } \\eta _ { \\mu } ( r \\xi ) , \\xi \\in \\mathbb { T } , \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} L _ n = \\zeta ( 3 ) + \\frac { c _ 1 } { n } + \\frac { c _ 2 + o ( 1 ) } { n ^ { 4 / 3 } } \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} m _ 1 & = \\frac { b _ 0 } { a _ 0 } , \\\\ \\xi _ 1 & = \\frac { 1 } { a _ 0 } \\left ( \\frac { a _ 2 b _ 0 } { a _ 0 } - ( a _ 1 + b _ 2 ) + \\frac { a _ 0 b _ 1 } { b _ 0 } \\right ) . \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} G _ { 1 ^ m } ^ { ( k ) } = \\sum _ { a \\geq m } ( - 1 ) ^ { a - m } \\binom { a - 1 } { m - 1 } m _ { 1 ^ a } = \\sum _ { i \\geq 0 } ( - 1 ) ^ i \\binom { m + i - 1 } { m - 1 } e _ { m + i } = G _ { 1 ^ m } \\ , , \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} B _ { n , \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } + \\omega _ { j } \\mathbf { 1 } \\mid { \\boldsymbol { \\omega } } ) - B _ { n , \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } \\mid { \\boldsymbol { \\omega } } ) = & \\sum _ { i = 1 } ^ { r } B _ { n - 1 , \\mathbf { m } _ { i } } ^ { ( d ) } ( z \\mid \\widehat { { \\boldsymbol { \\omega } } } ( j ) ) \\left ( m _ { i } + \\frac { d } { 2 } ( r - i ) \\right ) h _ { - , i } ^ { ( d ) } ( \\mathbf { m } ) . \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} ( T \\eta ) _ n = \\eta _ n + \\min \\left \\{ W _ { n - 1 } , J - \\eta _ n \\right \\} - \\min \\left \\{ \\eta _ n , K - W _ { n - 1 } \\right \\} . \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} b a ( \\mathbf { 1 } - e ) & = ( \\mathbf { 1 } - a b ) ( \\mathbf { 1 } - e ) = \\mathbf { 1 } - e - a b + a b e = \\mathbf { 1 } - a b = b a . \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} [ \\rho _ a , \\rho _ b ] = C ^ c { } _ { a b } ( X ) \\rho _ c , C ^ c { } _ { a b } ( X ) \\in C ^ \\infty ( M ) , \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} \\begin{aligned} & 0 \\leq \\mathbf { x } ( i ) \\leq \\tau \\forall i \\in \\{ 1 , 2 , . . . , N \\} \\\\ & | | \\mathbf { x } | | _ 2 = 1 . \\end{aligned} \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} | v _ { 4 , c } ( t , r ) | \\leq C | \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) | \\begin{cases} \\frac { 1 } { t ^ { 2 } r ^ { 3 } \\log ^ { 3 b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { \\log ( r ) } { \\log ^ { 2 b } ( t ) r ^ { 4 } | t - r | } + \\frac { \\log ^ { 2 b \\alpha } ( t ) } { t ^ { 2 } r ^ { 3 } \\log ^ { 3 b + 1 } ( t ) } , r \\geq \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} \\partial _ t u - ( 1 + \\gamma \\partial _ t ^ \\alpha ) \\Delta u & = F \\ ; \\Omega , t > 0 , \\\\ u & = 0 \\ ; \\partial \\Omega , t \\ge 0 , \\\\ u ( \\cdot , 0 ) & = \\xi \\ ; \\Omega , \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} \\eta ^ \\rightarrow ( U ) = \\mu ^ { - 1 } ( U \\cap \\mu ( X / { \\equiv } ) ) . \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} c = B ^ { 2 } - C ^ { 2 } + h ^ { 2 } + I = A ^ 2 + B ^ 2 - C ^ 2 + I . \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} \\partial ( i n t ( \\Upsilon ) ) \\setminus \\left ( \\Lambda _ { 1 } \\cup \\Lambda _ { 2 } \\right ) = \\left \\{ \\Gamma ( t ) : t \\in \\mathbb { R } \\right \\} , \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} u = \\nabla ^ { \\perp } \\Delta ^ { - 1 } \\omega . \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} \\phi ( - \\frac { q } { 2 } x _ { n - 1 } ) \\cdots \\phi ( - \\frac { q } { 2 } x _ 1 ) = \\sum _ { k _ 1 , \\cdots , k _ { n - 1 } \\geq 0 } \\frac { q ^ { k _ 1 ^ 2 / 2 + \\cdots + k _ { n - 1 } ^ 2 / 2 - k _ 1 k _ 2 - \\cdots - k _ { n - 2 } k _ { n - 1 } } x _ 1 ^ { k _ 1 } \\cdots x _ { n - 1 } ^ { k _ { n - 1 } } } { ( q ) _ { k _ 1 } \\cdots ( q ) _ { k _ { n - 1 } } } . \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} \\lim _ { \\Lambda _ { 1 } \\uparrow \\uparrow V } \\psi _ { \\Lambda _ { 1 } } ( b _ { \\Lambda } ) = \\sum _ { i , j \\in I } \\prod _ { x \\in \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * b _ { x } \\right ) \\prod _ { x \\in \\Lambda _ { 1 } \\setminus \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * \\right ) \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} \\begin{cases} \\phi _ i ' ( x - \\underline h ( t ) ) \\leq - \\epsilon _ 1 & \\mbox { f o r } x \\in [ \\underline h ( t ) - K _ 0 , \\underline h ( t ) ] , \\ t \\geq 0 , \\\\ \\phi _ i ' ( - x - \\underline h ( t ) ) \\leq - \\epsilon _ 1 & \\mbox { f o r } x \\in [ - \\underline h ( t ) , - \\underline h ( t ) + K _ 0 ] , \\ t \\geq 0 . \\end{cases} \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} X / { \\equiv } = \\eta ^ \\rightarrow ( X ) = \\eta ^ \\rightarrow ( B \\cup B ^ c ) = \\eta ^ \\rightarrow ( B ) \\cup \\eta ^ \\rightarrow ( B ^ c ) . \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} u _ 2 ( x ) v _ 2 ( y ) - u _ 2 ( y ) v _ 2 ( x ) = u _ 1 ( x ) v _ 1 ( y ) - u _ 1 ( y ) v _ 1 ( x ) , x , y \\in ] a , b [ . \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} C _ { \\tau _ 0 } e ^ { \\widetilde { K } ( \\mathcal { T } ) } = | \\Omega _ 0 | , \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} \\phi _ t ^ * \\left ( \\sum u _ { i } \\sigma _ i \\right ) = \\sum e ^ { \\lambda _ i t } u _ { i } \\sigma _ i = \\sum u _ { i } ' \\sigma _ i \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} \\frac { 1 } { z } G _ { \\mu } \\left ( \\frac { 1 } { z } \\right ) = \\frac { 1 } { 1 - \\eta _ { \\mu } ( z ) } , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} \\sum _ { k \\in J _ i } z ^ i _ k = 1 . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} \\ddot { w } + \\eta ( t ) \\dot { w } = - \\partial \\Phi ( w ) , \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} \\min \\limits _ { p \\in \\mathbb { P } _ L } ~ ~ \\left \\{ \\frac 1 2 \\sum _ { j = 1 } ^ N w _ j \\left ( p ( \\mathbf { x } _ j ) - f ( \\mathbf { x } _ j ) \\right ) ^ 2 \\right \\} \\quad \\quad \\sum _ { \\ell = 1 } ^ { d } | \\alpha _ { \\ell } | \\leq \\eta . \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} \\lambda _ n = n - \\langle u | 1 \\rangle - \\sum _ { k = n + 1 } ^ \\infty \\gamma _ k \\ . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} \\ell + h ^ { \\vee } = \\frac { k + h ^ { \\vee } + 1 } { k + h ^ { \\vee } } . \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} M _ { i i } = \\frac { ( v _ M ) _ i + \\sum _ { j \\neq i } ( N _ M ) _ { i j } ( u _ M ) _ j } { ( u _ M ) _ i } . \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} \\gamma . \\mathbf { z } & = \\left ( [ v ] , \\tfrac { k _ { 1 } + k _ { 2 } \\theta + w - v } { b } , k _ { 2 } \\right ) , \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} - 4 \\int _ { t } ^ { \\infty } d s \\frac { \\lambda _ { 0 , 0 } '' ( s ) } { 1 + s - t } = - \\frac { 4 b } { t ^ { 2 } \\log ^ { b } ( t ) } + E _ { \\lambda _ { 0 , 0 } } \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { \\bf 2 } , { 3 } , { \\bf 2 } , { 3 } , 8 , 8 , { \\bf 4 } ) & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { 2 } , { 3 } , { 3 } , { \\bf 4 } , 8 , 8 , { \\bf 4 } ) \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { 2 } , { 3 } , { 3 } , { \\bf 4 } , 4 , { \\bf 4 } ) \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { 2 } , { 5 } , { 3 } , { \\bf 5 } , 5 , { \\bf 4 } ) = f _ { 5 , 4 } ( s ) = f _ 5 ^ * ( s ) . \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} \\mathfrak { F } _ { C S } = \\left ( \\mathcal { F } _ { C S } , L ^ { \\bullet } _ { C S } , \\alpha _ { C S } ^ { \\bullet } , Q _ { C S } \\right ) \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} T _ { f , k } : H ^ 0 ( X , L ^ { \\otimes k } \\otimes G ) \\to H ^ 0 ( X , L ^ { \\otimes k } \\otimes G ) , T _ { f , k } ( h ) : = P _ k \\circ M _ f , \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} A _ i & = ( Z _ { 1 i } , { ( Z _ { 1 i } ^ 2 - 1 ) } / { 2 } ) ^ \\tau , \\\\ B _ i & = ( ( Z _ { 2 i } ^ 2 - 1 ) / 2 , ( Z _ { 1 i } ^ 2 - 1 ) ( Z _ { 2 i } ^ 2 - 1 ) / 2 , Z _ { 1 i } ( Z _ { 2 i } ^ 2 - 1 ) / 2 , - ( Z _ { 2 i } ^ 4 - 6 Z _ { 2 i } ^ 2 + 3 ) / 1 2 ) ^ \\tau . \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} \\Sigma ' = \\prod _ { c \\in G / H } \\Sigma ' _ c , \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A \\left ( x \\right ) u + V \\left ( x , t \\right ) u = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , T \\right ] , \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} c _ \\beta = \\frac { 2 } { \\Gamma \\left ( - \\beta - 1 / 2 \\right ) \\Gamma ( \\beta + 1 ) } \\left ( \\frac { \\pi } { \\sin \\pi ( - 2 \\beta - 1 ) } \\right ) ^ { 1 / 2 } > 0 \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} \\int b ( S v \\cdot g - v \\overline { S ^ * \\overline { g } } ) = \\int ( b \\cdot S v \\cdot g - S ( b v ) \\cdot \\overline { \\overline { g } } ) = \\int g \\cdot [ b , S ] v , \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} { \\rm s g n } \\left ( f ( 0 , y ; \\mu ) \\right ) = { \\rm s g n } ( y ) , ~ y , \\mu \\in \\mathbb { R } , \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} F ( x , y ) & = \\begin{pmatrix} x + c y + f _ 1 ( x , y ) \\\\ y + f _ 2 ( x , y ) \\end{pmatrix} , \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} \\eta ( t ) = \\sup _ { s \\in ( 0 , t ] } \\frac { s ^ \\nu } { | B _ s ( o ) | } , \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} 0 = \\int _ a ^ c G _ { \\phi , \\psi } ( c , y ) ( L _ { \\phi , \\psi } f ) ( y ) \\d y . \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} \\sum _ { j \\in \\alpha } n _ j + \\sum _ { j \\in \\beta } n _ j = \\sum _ { j \\in \\gamma } n _ j \\alpha \\cup \\beta = \\gamma \\alpha \\cap \\beta = \\emptyset . \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} S _ { Y M } ( A ) = - \\frac 1 2 \\int _ { M } t r ( F _ { \\mu \\nu } ( x ) F ^ { \\mu \\nu } ( x ) ) V o l ( d x ) , \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} f & = ( x y + b ^ { 1 / 2 } x ) ^ 2 + ( x ^ 2 + a ^ { 1 / 2 } x ) ^ 2 x \\\\ & = ( x ^ 2 + a ^ { 1 / 2 } x ) ^ 4 + ( x ) ^ 4 x + ( y + b ^ { 1 / 2 } + b ^ { 1 / 4 } ) ^ 4 x ^ 2 + ( a ^ { 1 / 4 } ) ^ 4 x ^ 3 \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} - \\Delta u = g ( u ) , \\mathbb { R } ^ N N \\geq 3 , \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} ( I _ { t \\times 4 } + N _ 2 ( | \\xi | ) ) ^ { - 1 } = I _ { 4 \\times 4 } - N _ 2 ( | \\xi | ) ( I _ { 4 \\times 4 } + N _ 2 ( | \\xi | ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} \\beta = \\frac { 1 } { | \\Omega | } ( \\epsilon ^ { - 1 } f ^ n ( \\overline u _ h ^ { n } + \\alpha ) , 1 ) _ { \\mathcal T _ h } . \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} \\alpha . f \\it { m } _ \\lambda \\alpha . g = \\alpha . \\big ( f \\it { m } _ \\lambda g \\big ) , \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; f . \\alpha \\it { m } _ \\lambda g . \\alpha = \\big ( f \\it { m } _ \\lambda g \\big ) . \\alpha \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} { \\mathbf V } _ { j } ^ \\ell = ( { { \\mathbf V } _ j ^ \\ell } ^ + , { { \\mathbf V } _ j ^ \\ell } ^ - ) = \\Big ( \\big ( { \\varphi _ { j 1 } ^ { \\ell } } ^ { + } , { \\varphi _ { j 2 } ^ { \\ell } } ^ { + } \\big ) , \\ , \\big ( { \\varphi _ { j 1 } ^ { \\ell } } ^ { - } , { \\varphi _ { j 2 } ^ { \\ell } } ^ { - } \\big ) \\Big ) . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} \\mathcal { L } _ X d = d \\mathcal { L } _ X , \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} ( g ^ { i j } ) _ s = - g ^ { i k } ( g _ { k m } ) _ s g ^ { m j } = 2 g ^ { i k } A ^ V _ { k m } g ^ { m j } \\ , . \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} f _ { n , \\alpha } ( \\pi _ { 1 } , \\ldots , \\pi _ { k } ) : = \\left \\{ \\begin{array} { r l } ( \\pi _ { 1 } , \\ldots , \\pi _ { k - 1 } , \\bar { \\pi } _ { k } ) ~ ~ ~ & ( \\pi _ { 1 } , \\ldots , \\pi _ { k } ) \\in \\mathcal { A } _ { n , \\alpha } , \\\\ ( \\pi _ { 1 } , \\ldots , \\pi _ { k - 1 } , \\pi _ { k } ) ~ ~ ~ & , \\end{array} \\right . \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} ( \\partial _ { t } ^ + e _ h ^ { u ^ n } , e _ h ^ { u ^ n } ) _ { \\mathcal T _ h } = \\frac { \\| e _ h ^ { u ^ n } \\| ^ 2 _ { \\mathcal { T } _ h } - \\| e _ h ^ { u ^ { n - 1 } } \\| ^ 2 _ { \\mathcal { T } _ h } + ( \\Delta t ) ^ 2 \\| \\partial _ { t } ^ + e _ h ^ { u ^ n } \\| ^ 2 _ { \\mathcal { T } _ h } } { 2 \\Delta t } . \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} \\Big ( \\frac { \\eta } { N ^ 3 } \\Big ) ^ c \\Big ( \\frac { \\eta } { ( N - 1 ) ^ 3 } \\Big ) ^ c \\dots \\Big ( \\frac { \\eta } { 1 ^ 3 } \\Big ) ^ c = \\Big [ \\frac { \\eta ^ N } { ( N ! ) ^ 3 } \\Big ] ^ c \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} \\max _ { t \\in [ 0 , 1 ] } J ( \\gamma ( t ) ) = J ( \\tilde { u } ) . \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} \\partial _ { w _ { 0 , a , b } } = \\left ( \\frac { 1 } { x _ { k _ 1 } - x _ { k _ 1 + 1 } } - \\frac { s _ { k _ 1 } } { x _ { k _ 1 } - x _ { k _ 1 + 1 } } \\right ) \\ldots \\left ( \\frac { 1 } { x _ { k _ { a b } } - x _ { k _ { a b } + 1 } } - \\frac { s _ { k _ { a b } } } { x _ { k _ { a b } } - x _ { k _ { a b } + 1 } } \\right ) . \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} \\lambda _ { n K + k } ( u ) = \\lambda _ { n K } ( u ) + k \\ . \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} \\mathcal A : = \\{ \\textbf { I P O P T } , \\textbf { S c h o l t e s S C } , \\textbf { S c h o l t e s C C } , \\textbf { s m o o t h e d F B } , \\textbf { o f f s e t K S } \\} \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} H _ { 2 n } & = 2 ^ { 4 n ^ 2 - 1 } \\frac { ( 2 n + 2 ) ! } { ( 4 n + 1 ) ! } \\prod _ { k = 1 } ^ { 2 n - 1 } \\frac { k ! ^ 2 } { ( 2 k + 1 ) ! ^ 2 } , \\\\ H _ { 2 n + 1 } & = 2 ^ { 4 n ( n + 1 ) } \\frac { ( 2 n + 1 ) ( 2 n + 2 ) ! } { ( 4 n + 3 ) ! } \\prod _ { k = 1 } ^ { 2 n } \\frac { k ! ^ 2 } { ( 2 k + 1 ) ! ^ 2 } . \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} e ^ 3 = [ 2 0 , 1 1 , 1 , 2 , 4 , 3 , 1 , 5 , 1 , 2 , 1 6 , \\dots ] , \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} \\| \\bullet \\| = \\| \\bullet \\| _ { L ^ { 2 } ( \\mathbb { R } _ { + } ^ { n + 1 } ) } , & \\| \\bullet \\| _ { 0 } = \\| \\bullet \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { n } \\times \\{ 0 \\} ) } \\\\ \\langle \\bullet , \\bullet \\rangle = \\langle \\bullet , \\bullet \\rangle _ { L ^ { 2 } ( \\mathbb { R } _ { + } ^ { n + 1 } ) } , & \\langle \\bullet , \\bullet \\rangle _ { 0 } = \\langle \\bullet , \\bullet \\rangle _ { L ^ { 2 } ( \\mathbb { R } ^ { n } \\times \\{ 0 \\} ) } \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} & \\mathfrak { G } _ { s } : = \\{ g _ { s } ( s _ { 1 } , s _ { 2 } , s _ { 3 } ) ( t _ { 1 } , t _ { 2 } , t _ { 3 } ) \\\\ & \\{ ( t _ { 1 } , t _ { 2 } , t _ { 3 } ) ^ { \\pm 1 } , ( t _ { 1 } , t _ { 1 } / t _ { 3 } , t _ { 1 } / t _ { 2 } ) ^ { \\pm 1 } , ( t _ { 2 } / t _ { 3 } , t _ { 2 } , t _ { 2 } / t _ { 1 } ) ^ { \\pm 1 } , ( t _ { 3 } / t _ { 2 } , t _ { 3 } / t _ { 1 } , t _ { 3 } ) ^ { \\pm 1 } \\} \\} \\simeq C _ { 2 } ^ { 3 } , \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} \\| u \\| _ { \\mathcal { H } } = \\inf \\left \\{ \\tau > 0 : \\rho _ { \\mathcal { H } } \\left ( \\frac { u } { \\tau } \\right ) \\leq 1 \\right \\} , \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} \\psi ( x , k ) = \\frac { \\overline { P _ * } ( 2 x , k ) e ^ { i k x } - P _ * ( 2 x , k ) e ^ { - i k x } } { 2 i } \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} \\frac { d } { d \\alpha } \\left ( - \\log ( \\det ( \\Delta _ { C _ { \\alpha } } ) ) \\right ) = \\frac { 1 } { 3 \\pi } + \\frac { \\pi } { 3 \\alpha ^ 2 } - \\frac { \\gamma _ e } { 1 2 \\pi } \\left ( \\frac { 4 \\pi ^ 2 } { \\alpha ^ 2 } - 1 \\right ) - \\frac { 1 } { 2 \\pi } \\sum _ { k = 1 } ^ { \\lceil \\frac { \\pi } { \\alpha } - 1 \\rceil } \\frac { \\log \\left | \\sin \\left ( k \\alpha / 2 \\right ) \\right | } { \\sin ^ 2 \\left ( k \\alpha / 2 \\right ) } . \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} d _ 2 = O ( 3 ^ { N _ T } / \\sqrt { T ^ d } ) , d _ 3 = O _ { T \\to + \\infty } ( N _ T ^ { - 1 / 1 2 } + T ^ { - 1 / 4 } + ( \\log T ) ^ { - 1 / 2 } ) . \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} u ^ { \\ell + 1 } = e ^ { \\tau \\mathcal { L } } \\left [ \\left ( 1 + \\tau u ^ \\ell \\left ( \\Psi ( \\tau ) \\overline { u ^ \\ell } \\right ) \\right ) u ^ \\ell \\right ] + \\mathcal { R } ( \\tau , u ^ \\ell ) \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} E [ W _ q ( f ) ( A ) ^ 2 ] = 2 ^ n . \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} { \\cal J } \\left [ f _ X ( x ) \\right ] = { \\int _ 0 ^ A { f _ X ( x ) } \\ln ( { f _ X ( x ) } ) { \\rm { d } } x + \\frac { 1 } { 2 } \\int _ 0 ^ A { \\ln \\left ( { 1 + { \\varsigma ^ 2 } x } \\right ) f _ X ( x ) } { \\rm { d } } x } . \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} \\begin{aligned} \\mu ( \\xi ) & = \\frac { ( n - 1 ) } { n } \\Big \\{ n \\cos ^ 2 ( \\xi ) + 2 \\cos ( \\xi ) [ - ( 1 - \\xi ^ 2 ) \\cos ( \\xi ) + n \\xi \\sin ( \\xi ) ] + \\\\ & - n ( 1 - \\xi ^ 2 ) \\sin ^ 2 ( \\xi ) \\Big \\} \\end{aligned} \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} g \\circ \\bigg ( 1 - \\frac { 1 } { z _ { i + 1 } } \\bigg ) \\bigg ( 1 + \\frac { z _ { i + 1 } } { z _ i } s _ i \\bigg ) = 0 . \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} \\lambda _ { \\min } ^ { 1 - p } ( P _ { f } ( A , B ) ) \\bigg \\| { f ( C ^ p ) \\over f ( C ) ^ p } \\bigg \\| _ \\infty K ( \\xi , 2 p - 1 ) = 1 \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { u } = i \\left [ \\Delta \\tilde { u } + A \\left ( x \\right ) \\tilde { u } + \\tilde { V } \\left ( x , t \\right ) \\tilde { u } + \\tilde { F } \\left ( x , t \\right ) \\right ] , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} H _ m > 1 + \\sum ^ { m - 1 } _ { i = 1 } H _ i . \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} [ V ^ b ( D ) ] - [ e _ 1 ] \\in K _ 0 ( \\mathcal { I } _ G ( M ) ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; e _ 1 : = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\int _ { B \\setminus B _ \\epsilon ( 0 ) } | f ( x , ( u _ j ) _ + ) u _ j - f ( x , u _ + ) u | d x = 0 \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} & \\mathbb { P } _ { n _ { l } } \\left [ - M \\le \\log \\left ( Y _ { n _ { l } } \\right ) \\le M \\right ] \\ge 1 - \\frac { \\delta } { 1 0 0 } \\\\ & \\mathbb { P } _ { n _ { l } } \\left [ - M \\le \\left \\{ \\sum _ { i = 1 } ^ { k } \\frac { 2 \\mu _ { i } W _ { n _ { l } , i } - \\mu _ { i } ^ 2 } { 2 \\sigma _ { i } ^ 2 } \\right \\} \\le M \\right ] \\ge 1 - \\frac { \\delta } { 1 0 0 } \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} \\mu ( \\eta ^ \\rightarrow ( U ) ) & = \\{ \\mu ( \\eta ( x ) ) \\mid x \\in U \\} \\\\ & = U \\cap \\{ \\mu ( \\eta ( x ) ) \\mid x \\in X \\} \\\\ & = U \\cap \\mu ( X / { \\equiv } ) , \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} \\frac { \\partial _ G \\Delta } { | G _ \\Delta | } - \\frac { \\partial _ G \\Sigma } { n } & \\le \\sum _ { m \\ge 0 } \\frac { 6 ( c + 1 ) \\log ( ( 4 / 3 ) ^ m t ) + 9 } { ( 4 / 3 ) ^ m t } \\\\ & = \\frac { 6 ( c + 1 ) \\log ( 4 / 3 ) } { t } \\sum _ { m \\ge 0 } m ( 3 / 4 ) ^ m + \\frac { 6 ( c + 1 ) \\log t + 9 } { t } \\sum _ { m \\ge 0 } ( 3 / 4 ) ^ m . \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} \\frac { | r | ^ { n - 1 } } { ( n - 1 ) ! } | c _ { n - 1 , n } | - \\sum _ { i = 2 } ^ n \\frac { | r | ^ { n - i } } { ( n - i ) ! } | c _ { n - i , n } | > \\sum _ { i = 1 } ^ j | c _ { j - i , j } | \\frac { | r | ^ { j - i } } { ( j - i ) ! } \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} f \\left ( - \\frac { 3 \\lambda } { 2 E r } \\right ) = - \\frac { 2 7 \\lambda ^ 2 } { 1 6 E ^ 2 } . \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} v _ { 3 , 1 , a } ^ { \\lambda } ( t , r ) = \\frac { - 1 } { r } \\int _ { t } ^ { t + 6 r } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { ( s - t ) } \\lambda '' ( s ) \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( - 1 - \\rho ^ { 2 } + r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} \\mathcal { C } _ { J , K } ^ { r e v } : = \\left \\{ \\eta \\in \\mathcal { C } ^ { c a n } _ { J , K } : \\ : R \\eta \\in \\mathcal { C } ^ { c a n } _ { J , K } \\right \\} \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} \\mu ^ { \\sigma } _ N ( d x ) : = \\frac { 1 } { Z } \\exp \\left ( \\sigma \\sum _ { i = 1 } ^ N x _ i - H ( x ) \\right ) d x . \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} \\lambda _ j ( \\mu ) = \\min _ { \\substack { V \\subset H ^ 2 ( \\Omega ) \\\\ { \\rm d i m } V = j } } \\max _ { \\substack { v \\in V \\\\ \\frac { \\partial v } { \\partial \\nu } \\ne 0 } } \\frac { \\mathcal { Q } _ { \\mu , D } ( v , v ) } { \\int _ { \\partial \\Omega } \\left ( \\frac { \\partial v } { \\partial \\nu } \\right ) ^ 2 d \\sigma } , \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} \\mu _ { ( l ) } ( \\lambda ) = \\frac { \\left ( 3 l ^ 4 + 2 ( N - 2 ) l ^ 3 - ( N + 1 ) l ^ 2 - ( N - 2 ) l - \\xi _ { ( l ) } \\lambda \\right ) } { \\left ( \\eta _ { ( l ) } - \\lambda \\right ) } . \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} K _ { 1 } ( w , \\lambda ( t ) ) = \\int _ { 0 } ^ { \\infty } \\frac { r d r } { \\lambda ( t ) ^ { 2 } ( 1 + \\frac { r ^ { 2 } } { \\lambda ( t ) ^ { 2 } } ) ^ { 3 } } \\int _ { 0 } ^ { w } \\frac { \\rho d \\rho } { w } \\left ( 1 + \\frac { r ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( r ^ { 2 } - 1 - \\rho ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } \\right ) \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} \\bar { \\alpha } _ { 1 6 } ( i ) = ( \\widetilde { S q } ^ 0 ) ^ i ( \\lambda _ 7 ^ 2 \\lambda _ 0 b ^ { [ 2 ] } + ( \\lambda _ 3 ^ 2 \\lambda _ 9 + \\lambda _ 7 \\lambda _ 5 \\lambda _ 3 ) b ^ { [ 1 ] } ) , i \\geq 0 . \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} D = d H - \\sum _ { i = 1 } ^ s m _ i E _ i . \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} \\mathbb { K } _ { n } ^ { ( j ) } ( x ) = p ^ { n - 1 } \\left ( - N \\right ) _ { n - 1 } h _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) \\ , _ { 3 } F _ { 2 } \\left ( \\begin{array} { c | c } - n , - x , f _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) & \\\\ & p ^ { - 1 } \\\\ - N , f _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) - 1 & \\end{array} \\right ) , \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} \\phi ( p , q ) = \\phi ^ { \\bar { 0 } } ( p , q ) - \\phi ^ { \\bar { 1 } } ( p , q ) , \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} L f = g . \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} \\| ~ ( x - a ) ^ 2 f \\| _ { L ^ 2 ( \\mathbb { R } ) } = \\left ( \\int _ { \\mathbb { R } } ( x - a ) ^ 4 | f ( x ) | ^ 2 d x \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} Q _ { i j } ( u , v ) = \\left \\{ \\begin{array} { l l } 0 & \\mbox { i f } i = j , \\\\ ( u - v ) ^ { h _ { i j } } ( v - u ) ^ { h _ { j i } } & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} | \\partial _ { t r } v _ { 4 } ( t , r ) | \\leq \\begin{cases} \\frac { C } { t ^ { 3 } \\log ^ { 3 b + 2 N - 2 } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { t ^ { 3 5 / 1 2 } \\log ^ { 2 b - 1 } ( t ) } , r \\geq \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} | u | _ { C } : = \\sup _ { x \\in O } | u ( x ) | _ F < \\infty . \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} c & = a + b \\\\ d & = a b \\\\ e & = ( a + \\alpha ) ( b + \\beta ) , \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} \\Phi ( z ) = z \\exp ( u ( z ) ) , \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} \\int _ B | u _ j ( x ) - u ( x ) | ^ { \\ 2 m s + | x | ^ \\alpha } d x & = \\int _ B \\Big | \\frac { u _ j ( x ) - u ( x ) } { b _ j } \\Big | ^ { \\ 2 m s + | x | ^ \\alpha } b _ j ^ { \\ 2 m s + x | ^ \\alpha } d x \\\\ & \\leq b _ j ^ { \\ 2 m s } \\int _ B \\Big | \\frac { u _ j ( x ) - u ( x ) } { b _ j } \\Big | ^ { \\ 2 m s + | x | ^ \\alpha } d x \\\\ & \\leq b _ j ^ { \\ 2 m s } \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} \\begin{cases} A _ m ( x ) = B _ { m - 1 } ( x ) + x ^ 2 A _ { m - 1 } ( x ) , & ( m \\ge 3 ) , \\\\ B _ m ( x ) = x A _ { m } ( x ) + B _ { m - 1 } ( x ) , & ( m \\ge 2 ) . \\end{cases} \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} G _ { \\mu _ { 1 } } \\left ( F _ { \\rho _ { 1 } } ( x ) \\right ) = a _ { 0 } + a _ { 1 } \\alpha + \\frac { a _ { 1 } } { \\sqrt [ 3 ] { \\left | c _ { 3 } \\right | } } \\exp \\left ( \\frac { 2 \\pi i - \\theta i } { 3 } \\right ) ( x - c _ { 0 } ) ^ { 1 / 3 } + O \\left ( \\left | x - c _ { 0 } \\right | ^ { 2 / 3 } \\right ) \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} w _ i ( x ) > 0 \\mbox { f o r } x < 0 , \\ ; i = 1 , . . . , m _ 0 . \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} L ( x , \\lambda ) = f ( x ) + \\lambda ^ T g ( x ) . \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} \\rho _ \\alpha ( V ) = \\rho _ \\alpha ( M ) \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} \\sum _ { m , n \\in \\mathbb { Z } } \\mu ( T _ m ( \\phi _ 1 ) \\cap T _ n ( \\phi _ 2 ) ) x ^ m y ^ n = \\frac { 3 } { 6 4 } x y ^ { - 1 } + \\frac { 1 5 } { 6 4 } x ^ { - 1 } y + \\frac { 3 5 } { 1 9 2 } x ^ { - 2 } + \\frac { 7 } { 1 9 2 } y ^ { - 2 } + \\frac { 7 } { 3 2 } x ^ { - 1 } y ^ { - 1 } + \\frac { 9 } { 3 2 } . \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} - \\partial _ { t t } v _ { 4 , 1 , 2 } + \\partial _ { r r } v _ { 4 , 1 , 2 } + \\frac { 1 } { r } \\partial _ { r } v _ { 4 , 1 , 2 } - \\frac { v _ { 4 , 1 , 2 } } { r ^ { 2 } } = v _ { 4 , c } ^ { \\lambda _ { 1 } } - v _ { 4 , c } ^ { \\lambda _ { 2 } } \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} \\int _ { - M } ^ 0 | x | ^ { \\sigma } \\psi ( x ) d x = \\int _ { \\R } \\int _ 0 ^ M x ^ { \\sigma } J ( y ) \\psi ( - x ) d x d y . \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} \\left < \\hat { H } , H \\right > & = \\left < \\sum _ { A \\in \\left [ b \\right ] ^ d } \\hat { w } _ A h _ { d , b , A } , \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { B \\in \\left [ b \\right ] ^ d } h _ { d , b , B } \\ 1 \\left ( X _ i \\in \\Lambda _ { d , b , B } \\right ) \\right > \\\\ & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { A \\in \\left [ b \\right ] ^ d } \\hat { w } _ A \\ 1 \\left ( X _ i \\in \\Lambda _ B \\right ) b ^ d = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\hat { H } ( X _ i ) . \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} h _ { i j } : = X _ { ( i ; j ) } - \\frac { 2 } { n + 1 } \\mathrm { d i v } X \\ , g _ { i j } \\ , . \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } ^ k ( ( b , j ) ) = ( b + k \\delta _ { ( a , i ) } , j + k ( r \\delta _ { ( a , i ) } + 1 ) ) \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} \\langle \\Omega , \\overline { \\Omega } \\rangle & = X ^ 0 \\bar { F } _ 0 - \\bar { X } ^ 0 F _ 0 + X ^ 1 \\bar { F } _ 1 - \\bar { X } ^ 1 F _ 1 . \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} ( & U _ h \\mapsto U _ j = U ^ { ( 1 ) } \\mapsto \\ldots \\mapsto U ^ { ( K ) } , \\\\ & ( ( a _ h , a _ j ) , ( a _ h ^ { ( 1 ) } , a _ j ^ { ( 1 ) } ) , \\ldots , ( a _ h ^ { ( K - 1 ) } , a _ j ^ { ( K - 1 ) } ) ) , \\\\ & ( b , b ^ { \\prime } = b ^ { ( 1 ) } , \\ldots , b ^ { ( K ) } ) ) \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} & \\sum _ { \\substack { r = 0 \\\\ } } ^ { 2 n - 1 } x _ { r / 2 } y ^ * _ { r / 2 + t \\bmod n } + \\sum _ { \\substack { r = 0 \\\\ } } ^ { 2 n - 1 } y _ { ( r - 1 ) / 2 } x ^ * _ { ( r + 1 ) / 2 + t \\bmod n } \\\\ & = R _ { X , Y } ( t ) + R _ { Y , X } ( t + 1 ) . \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} - \\partial _ { t t } v _ { 5 } + \\partial _ { r r } v _ { 5 } + \\frac { 1 } { r } \\partial _ { r } v _ { 5 } - \\frac { v _ { 5 } } { r ^ { 2 } } = N _ { 2 } ( f _ { v _ { 5 } } ) ( t , r ) \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} \\Gamma _ { 0 } & : = \\{ w \\in \\C : \\ \\Re w = 0 , \\ \\Im w \\leq 0 \\} , \\\\ \\quad \\Gamma _ { n } & : = \\{ w \\in \\C : \\ \\Re w = t _ { n } \\gamma , \\ \\Im w \\leq y _ { n } - 1 \\} . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} \\chi ^ * ( \\Phi ( D ) ) & = \\chi _ \\infty ( \\Phi ( D ) ) + d _ \\infty ( \\Phi ( D ) ) = d _ \\infty ( \\Phi ( D ) ) \\le ( \\ell + \\zeta ) M d _ \\infty ( D ) \\\\ & \\le ( \\ell + \\zeta ) M [ d _ \\infty ( D ) + \\chi _ \\infty ( D ) ] = ( \\ell + \\zeta ) M \\chi ^ * ( D ) . \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} \\eta _ { \\nu ^ { \\boxtimes k } } = \\eta _ { \\nu } \\circ \\eta _ { \\mu } . \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} \\begin{aligned} n _ { 2 r , 7 } ( 0 , 1 ) = & \\frac { 7 - 7 ^ { 2 r + 1 } } { 2 4 } , \\ ; \\ ; \\ ; n _ { 2 r , 5 } ( 4 , - 3 ) = \\frac { 1 7 \\cdot 7 ^ { 2 r } - 1 7 } { 2 4 } , \\\\ n _ { 2 r , 7 } ( 8 , - 7 ) = & \\frac { 4 1 \\cdot 7 ^ { 2 r } - 4 1 } { 2 4 } . \\end{aligned} \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} W _ { \\alpha } \\doteqdot \\left ( R , _ { \\alpha } - i n A _ { \\alpha \\beta } , ^ { \\beta } \\right ) = 0 \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} \\sigma \\bigl ( F ( t , \\sigma ( Y ) ) \\bigr ) = ( - 1 ) ^ { \\mathrm { s i g n } ( \\sigma ) } \\ , F ( t , Y ) \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} \\bigcup _ { 1 \\leq j \\leq k - 2 } \\bigcup _ { 1 \\leq i \\leq l - 1 } ( D ( V _ i , \\widetilde { V } _ { l - 1 + j } ) \\cup D ( V _ { l - 1 + j } , \\widetilde { V } _ i ) ) = ( ( k + 1 ) ( k - 2 ) ) \\boxtimes ( \\Z _ { k - 1 } \\times ( \\Z _ { t k + 1 } \\backslash \\{ 0 \\} ) ) . \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} b _ 0 ( v , v , A _ 0 w ) + b _ 0 ( v , w , A _ 0 v ) + b _ 0 ( w , v , A _ 0 v ) = 0 \\ ; , \\forall \\ v , w \\in D ( A _ 0 ) \\ ; , \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} a & = 2 , & b & = 0 . 0 5 , & c & = 0 . 2 5 , & d & = 0 . 3 , \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ k ( - 1 ) ^ { k - i } c ( k , i ) q ^ i = ( q ) _ k . \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} \\theta _ i ( b ) = \" \\mbox { N o } \\varphi _ { \\alpha _ j } \\ ; ( j < i ) \\mbox { e n c o d e s } \\varphi _ { \\alpha _ i } ( \\bar { x } , b ) \" \\mbox { a n d } \\varphi ' _ { \\alpha _ i } ( \\bar { x } , y ) = \\varphi _ { \\alpha _ i } ( \\bar { x } , y ) \\wedge \\theta _ i ( y ) . \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} \\Delta ^ { k } ( \\mu ) = \\{ ( i , j ) \\in \\Delta ^ + _ { \\ell } \\mid k - \\mu _ i + i < j \\} \\ , . \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} Z ( t ) & = B ( S _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( t ) ) , \\\\ W ( t ) & = B ( E _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( t ) ) , \\ ; \\ ; t > 0 . \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\bar M _ t ^ { \\bar g } : = & \\bar g ( Y _ t ) - \\bar g ( Y _ 0 ) - \\sum _ { i = 1 } ^ m \\frac { 1 } { 2 } \\int _ 0 ^ t \\Big ( \\bar \\nabla ^ 2 \\bar g ( Y _ s ) \\big ( Z _ s ^ i , Z _ s ^ i \\big ) + \\left \\langle \\bar \\nabla \\bar g ( Y _ s ) , \\bar A ( Y _ s ) ( Z _ s , Z _ s ) \\right \\rangle \\Big ) d s \\\\ & + \\int _ 0 ^ t \\langle \\bar \\nabla \\bar g ( Y _ s ) , \\bar f ( Y _ s , Z _ s ) \\rangle d s \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} p _ \\theta ( { \\bf { x } } ) = \\big [ h + F ( \\theta ) + w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\big ] ^ { \\frac { 1 } { \\alpha - 1 } } . \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} \\kappa _ g = \\frac { e ^ { q _ 2 - q _ 1 } - 2 } { \\left ( e ^ { q _ 2 - q _ 1 } + 1 \\right ) ^ 2 } \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} & \\int _ { \\mathbb R } \\tilde \\phi _ 1 ( x _ 1 , x _ 2 , x _ 3 ) d x _ 1 = 0 \\quad { \\rm f o r \\ e a c h \\ f i x e d \\ } ( x _ 2 , x _ 3 ) \\ { \\rm i n \\ } \\mathbb R ^ 2 ; \\\\ & \\int _ { \\mathbb R ^ 2 } \\tilde \\phi _ 1 ( x _ 1 , x _ 2 , x _ 3 ) d x _ 2 d x _ 3 = 0 \\quad { \\rm f o r \\ e a c h \\ f i x e d \\ } x _ 1 \\ { \\rm i n \\ } \\mathbb R \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} K _ Z + ( p - 1 ) \\sum _ { i = 1 } ^ { \\ell ( X / k ) } C _ i \\sim f ^ * K _ X \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N } \\frac { 1 } { \\log K _ { j } \\log \\log K _ { j } } \\le C \\left ( 1 + \\frac { 1 } { n _ { 1 } } \\sum _ { j = 1 } ^ { N } \\log K _ { j } \\right ) . \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} \\bar { \\alpha } ( f ) : = \\bar { \\alpha ( \\bar f ) } . \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} \\xi = l ( 2 l ^ 2 + ( N - 1 ) l - N + 2 ) , \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} A _ { \\varepsilon } ( H , x ) = \\min \\left \\{ \\bigcup _ { k = 0 } ^ x A _ { \\varepsilon } ( H _ 1 , k ) \\oplus A _ { \\varepsilon } ( H _ 2 , x - k ) \\right \\} x = 0 , 1 , \\ldots , B , \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} \\partial _ { j } \\partial _ { k } = & e ^ { - 2 t } \\bigg ( \\omega _ { j } \\omega _ { k } \\partial _ { t } ^ { 2 } + \\omega _ { j } \\Omega _ { k } \\partial _ { t } + \\omega _ { k } \\Omega _ { j } \\partial _ { t } + ( \\delta _ { j k } - 2 \\omega _ { j } \\omega _ { k } ) \\partial _ { t } \\\\ & \\qquad + \\frac { 1 } { 2 } \\Omega _ { j } \\Omega _ { k } + \\frac { 1 } { 2 } \\Omega _ { k } \\Omega _ { j } - \\frac { 1 } { 2 } \\omega _ { j } \\Omega _ { k } - \\frac { 1 } { 2 } \\omega _ { k } \\Omega _ { j } \\bigg ) . \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} \\frac { d f _ r } { d x } ( x ) = \\frac { 1 } { \\log ( 1 + r ) ( 1 + x ) } - \\frac { 1 } { r + x } \\geq \\frac { 1 } { r ( 1 + x ) } - \\frac { 1 } { r + x } = \\frac { ( 1 - r ) x } { r ( 1 + x ) ( r + x ) } \\geq 0 \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} 0 \\geq \\bar { \\nabla } ^ 2 _ { X , X } \\phi ( p ) & = \\bar { \\nabla } ^ 2 _ { X , X } ( 1 + \\varphi ^ 2 - \\cot ^ 2 \\theta f ^ 2 ) ( p ) \\\\ & = 2 ( X \\cdot \\bar { \\nabla } \\varphi ) ^ 2 - 2 \\cot ^ 2 \\theta ( X \\cdot \\bar { \\nabla } f ) ^ 2 . \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} \\tilde { g } ( \\omega ) : = e ^ { i \\phi ( y ; \\omega ) } g ( \\omega ) \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} t _ 0 ( p ) : = \\inf \\big \\{ t \\ge 0 \\colon p + i t \\in \\Omega \\big \\} . \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} A ^ 0 _ n & = 0 . \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} \\tau = \\underset { E _ 0 \\subset \\Omega , \\mathbb P ( E _ 0 ) = 0 } { s u p } ( \\underset { \\Omega - E _ 0 } { i n f } \\sigma ) \\geq T + \\varepsilon _ 0 > T . \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c c c } D - \\lambda & & W ^ { \\top } \\\\ \\\\ W & & B - \\lambda \\\\ \\\\ \\end{array} \\right ] \\left [ \\begin{array} { c } q \\\\ - G \\left ( \\lambda \\right ) W q \\end{array} \\right ] = \\vec { 0 } . \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} \\chi _ h ( \\Gamma , V ) = \\sum _ { ( T ) } \\chi _ { o r b } ( C ( T ) ) T r ( T ^ { - 1 } , V ) \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} \\hat { q } _ { \\mathsf { k , m } } = \\frac { \\exp \\left [ - \\frac { \\alpha - 1 } { 2 } \\beta \\lambda _ { \\mathsf { m } } \\right ] } { \\nu _ { \\mathsf { k } } + b _ { \\mathsf { m } } } \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} \\alpha = \\frac { \\lambda _ L } { \\omega _ L } + \\frac { \\lambda _ R } { \\omega _ R } . \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\theta } H _ \\pm ( z , t , \\theta ) = \\mp ( 2 \\pi i ) \\cdot \\log \\Lambda \\left ( \\pm \\frac { z } { 2 \\pi i t } , \\frac { 1 } { 2 } \\mp \\theta \\b 1 \\right ) . \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} Y _ t = \\xi - \\sum _ { i = 1 } ^ m \\int _ t ^ T Z _ s ^ i d B _ s ^ i - \\sum _ { i = 1 } ^ m \\frac { 1 } { 2 } \\int _ t ^ T \\bar A ( Y _ s ) ( Z _ s ^ i , Z _ s ^ i ) d s + \\int _ t ^ T \\bar f ( Y _ s , Z _ s ) d s . \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} z ' ( e ^ { i \\theta } ) = \\frac { d } { d \\theta } R ( e ^ { i \\theta } ) e ^ { i \\theta } = [ R ' ( e ^ { i \\theta } ) + i R ( e ^ { i \\theta } ) ] e ^ { i \\theta } , \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } \\left ( \\frac { \\cos ( 2 Q _ { 1 } ( R ) ) - 1 } { R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\right ) v _ { 3 } ( t , R \\lambda ( t ) ) \\phi _ { 0 } ( R ) R d R \\\\ & = \\frac { 1 6 } { \\lambda ( t ) } \\int _ { t } ^ { \\infty } K _ { 3 } ( s - t , \\lambda ( t ) ) \\lambda '' ( s ) d s + \\int _ { 0 } ^ { \\infty } \\left ( \\frac { \\cos ( 2 Q _ { 1 } ( R ) ) - 1 } { R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\right ) E _ { 5 } ( t , R \\lambda ( t ) ) \\phi _ { 0 } ( R ) R d R \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} P _ k ( \\varepsilon ) = \\int _ { 0 } ^ 1 P ( t \\varepsilon ) p _ { k + 1 } ^ { ( m ) } ( t ) d t , \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} T [ \\sigma ( \\xi + \\xi _ 0 , \\eta + \\eta _ { 0 } ) ] ( f , g ) ( x ) = E _ { - \\xi _ 0 - \\eta _ 0 } ( x ) T [ \\sigma ] ( E _ { \\xi _ 0 } f , E _ { \\eta _ 0 } g ) ( x ) . \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} \\| w \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { n } \\times \\{ 0 \\} ) } & = \\bigg [ \\int _ { \\mathbb { R } ^ { n } \\times \\{ 0 \\} } ( \\langle \\xi \\rangle ^ { 2 - 2 s } | \\hat { w } | ^ { 2 } ) ^ { s } ( \\langle \\xi \\rangle ^ { - 2 s } | \\hat { w } | ^ { 2 } ) ^ { 1 - s } \\ , d \\xi \\bigg ] ^ { \\frac { 1 } { 2 } } \\\\ & \\le ( \\mu ^ { 1 - s } \\| w \\| _ { H ^ { 1 - s } ( \\mathbb { R } ^ { n } \\times \\{ 0 \\} ) } ) ^ { s } ( \\mu ^ { - s } \\| w \\| _ { H ^ { - s } ( \\mathbb { R } ^ { n } \\times \\{ 0 \\} ) } ) ^ { 1 - s } \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} \\vec { \\mathbb { I } } ^ 2 & = \\langle \\langle \\mathbb { I } ^ 2 _ { \\gamma } \\colon \\gamma \\leq \\delta \\rangle , \\langle \\dot { \\mathbb { J } } ^ 2 _ { \\gamma } \\colon \\gamma < \\delta \\rangle \\rangle \\\\ \\dot { \\vec { \\mathbb { I } } } ^ 1 & = \\langle \\langle \\dot { \\mathbb { I } } ^ 1 _ { \\gamma } \\colon \\gamma \\leq \\delta \\rangle , \\langle \\dot { \\mathbb { J } } ^ 1 _ { \\gamma } \\colon \\gamma < \\delta \\rangle \\rangle \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} \\tilde { x } & = \\frac { x } { \\mu ^ 2 } , & \\tilde { y } & = \\frac { y - \\zeta ( \\mu ) } { \\mu } , \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} \\Psi = \\prod _ { i = 1 } ^ { N } r _ i ^ { - ( d _ i - 1 ) / 2 } R ( r _ 1 , \\cdots , r _ N ) \\prod _ { i = 1 } ^ { N } Y _ i ( \\Omega _ i ) , \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} \\frac { d } { d \\alpha } \\zeta _ { S _ \\alpha } ' ( 0 ) = \\frac { 5 } { 2 4 \\pi } + \\frac { 1 } { 1 2 } ( \\gamma _ e - \\log 2 ) \\left ( - \\frac { \\pi } { \\alpha ^ 2 } + \\frac 1 \\pi \\right ) + \\int _ 1 ^ \\infty \\frac { \\frac { \\pi } { \\alpha ^ 2 } e ^ { \\frac \\pi \\alpha t } } { ( e ^ { \\pi t / \\alpha } - 1 ) ^ 2 } \\frac { d t } { e ^ t - 1 } \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} \\theta ( a ^ \\epsilon b ^ { [ t ] } ) = \\left \\{ \\begin{array} { l l } a b ^ { [ p ( t + 1 ) - 1 ] } , & \\epsilon = 1 , \\\\ 0 , & \\epsilon = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} \\mathbb { G } _ { N } ' : = \\frac { 1 } { \\sqrt { N } } \\sum _ { i = 1 } ^ { N } \\left ( \\frac { S _ { i , N } } { \\pi _ { i , N } } - 1 \\right ) \\delta _ { Y _ { i } } . \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} \\hat { h } _ { s z } ( z _ 2 , s ) = 0 . \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} r _ i = \\sqrt { \\sum _ { x \\in \\mathcal { B } _ i } x ^ 2 } , ~ ~ ~ i = 1 , 2 , \\cdots , N , \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} \\frac { 6 } { r ^ 2 } = \\lambda _ 2 \\leq \\frac { \\int _ { S ^ 2 ( r ) } | \\nabla u | ^ 2 \\ , d S } { \\int _ { S ^ 2 ( r ) } u ^ 2 \\ , d S } , \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} \\zeta _ 1 = { a _ 2 } { b _ 3 } - { a _ 3 } { b _ 2 } , \\zeta _ 2 = { a _ 3 } { b _ 1 } - { a _ 1 } { b _ 3 } , \\zeta _ 3 = { a _ 1 } { b _ 2 } - { a _ 2 } { b _ 1 } \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} n _ { 2 r - 1 , \\ell } ( c , d ) & = - c \\left ( \\dfrac { \\ell ^ { 2 r } - 1 } { 2 4 } \\right ) - \\ell \\cdot d \\left ( \\dfrac { \\ell ^ { 2 r - 2 } - 1 } { 2 4 } \\right ) . \\\\ n _ { 2 r , \\ell } ( c , d ) & = - c \\left ( \\dfrac { \\ell ^ { 2 r } - 1 } { 2 4 } \\right ) - \\ell \\cdot d \\left ( \\dfrac { \\ell ^ { 2 r } - 1 } { 2 4 } \\right ) . \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\alpha ^ - ( p ) : = \\lim _ { t \\to + \\infty } \\alpha ^ - _ { \\Omega , p } ( t ) > 0 \\mathrm { a n d } \\alpha ^ + ( p ) : = \\lim _ { t \\to + \\infty } \\alpha ^ + _ { \\Omega , p } ( t ) > 0 , \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} A = \\sum _ { k + | \\alpha | \\leq j \\leq m } \\hat x ^ { - j } a _ { j k \\alpha } ( h , h \\hat x , y ) ( \\hat x D _ { \\hat x } ) ^ k D _ y ^ \\alpha . \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} g ( s ) = \\left \\{ \\begin{array} { l l } - V _ { 0 , \\infty } s + f _ 0 ( s ) , \\ s \\geq 0 , \\\\ - g ( - s ) , \\ s < 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} X _ 1 : = A , X _ 2 : = C _ 2 , X _ 3 : = M _ 0 , X _ 4 : = \\bar M _ 0 . \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} \\mathcal { L } _ L f ( \\mathbf { x } _ j ) = f ( \\mathbf { x } _ j ) , 1 \\leq j \\leq N , \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} | k ' ( x ( r ) ) | & = \\left | \\frac { ( 1 / 2 \\pi \\beta ) f ' ( r ) } { ( x ' ( r ) / x ( r ) ) x ( r ) } \\right | \\\\ & \\le \\frac { 1 } { \\pi f ( r ) ^ { 2 } x ( r ) } \\frac { r | f ' ( r ) | } { \\sqrt { 1 + r ^ { 2 } f ' ( r ) ^ { 2 } } } \\\\ & \\le \\frac { 1 } { \\pi f ( r ) ^ { 2 } x ( r ) } . \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} \\lim _ { k } { | \\phi _ k ( x y ) - \\phi _ { k } ( y x ) | } = 0 , \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} \\varepsilon ^ a _ 3 = \\varepsilon ^ b _ 1 \\varepsilon ^ c _ 2 C ^ a { } _ { b c } . \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } F _ n ^ { ( 1 ) } ( z ) = \\frac { P _ 0 ^ { ( 1 ) } ( z ) } { ( z - \\l _ 0 ) ^ { k _ 1 } } , \\lim _ { n \\to \\infty } F _ n ^ { ( 2 ) } ( z ) = \\frac { P _ 0 ^ { ( 2 ) } ( z ) } { ( z - \\l _ 0 ) ^ { k _ 2 } } \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} S _ { B F } ^ { 1 , 0 } [ ( A , B ) ^ { ( g , \\tau ) } ] - S _ { B F } ^ { 1 , 0 } [ ( A , B ) ] = S _ { \\tau F } ^ { 1 , 0 } [ A , B , g , \\tau ] . \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} \\varphi _ { T _ { \\rm f o c u s } ( r ) } ( 0 , 1 ) & = 0 , \\\\ \\psi _ { T _ { \\rm f o c u s } ( r ) } ( 0 , 1 ) & = \\frac { 1 } { r } \\ , P _ { \\rm f o c u s } ( r ) . \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} & ( - \\Delta ) ^ { s } _ { p } u + a ( x ) \\vert u \\vert ^ { p - 2 } u + ( - \\Delta ) ^ { s } _ { q } u + b ( x ) \\vert u \\vert ^ { q - 2 } u + \\mu ( x ) \\vert u \\vert ^ { r - 2 } u \\\\ & \\quad = \\lambda h ( x ) \\vert u \\vert ^ { m - 2 } u , x \\in { \\mathbb { R } } ^ { N } , \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} \\langle \\left . \\frac { ( \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 ) } { r ^ { 2 } } v _ { 1 } \\right \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle & = \\frac { - 1 } { \\lambda ( t ) ^ { 2 } } \\int _ { 0 } ^ { \\infty } v _ { 1 } ( t , R \\lambda ( t ) ) \\frac { 8 } { ( 1 + R ^ { 2 } ) ^ { 2 } } \\frac { 2 R } { 1 + R ^ { 2 } } R d R \\\\ & = - \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { 0 } ^ { \\infty } v _ { 1 } ( t , R \\lambda ( t ) ) \\frac { R ^ { 2 } } { ( 1 + R ^ { 2 } ) ^ { 3 } } d R \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} A _ { h , \\tilde h , \\tilde \\omega } = \\tilde h ^ m A _ h - \\tilde \\omega , \\tilde h = | \\tilde \\lambda | ^ { - 1 / m } , \\ \\tilde \\omega = \\frac { \\tilde \\lambda } { | \\tilde \\lambda | } , \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} t p _ { \\nu _ { \\beta } } \\left ( t \\right ) = \\frac { \\left ( t r \\right ) ^ { \\frac { 1 } { \\beta - 1 } } } { \\pi } \\frac { r \\sin \\left ( \\frac { \\beta } { \\beta - 1 } f ( r ) \\right ) } { \\left | 1 - \\eta _ { \\nu } \\left ( r e ^ { i f ( r ) } \\right ) \\right | ^ { 2 } } , \\quad r > 0 , \\ ; \\eta _ { \\nu } \\left ( r e ^ { i f ( r ) } \\right ) \\neq 1 . \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} Q ( z , \\overline { z } ) = \\left ( q _ { 1 } ( z , \\overline { z } ) , q _ { 2 } ( z , \\overline { z } ) , \\dots , q _ { N } ( z , \\overline { z } ) \\right ) , \\mbox { f o r : } q _ { l } ( z , \\overline { z } ) = z _ { l } \\overline { z } _ { \\tau \\left ( l \\right ) } + \\lambda _ { l } \\left ( z _ { l } z _ { \\sigma \\left ( l \\right ) } + \\overline { z } _ { l } \\overline { z } _ { \\sigma \\left ( l \\right ) } \\right ) , \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( p _ 1 - p _ 2 \\right ) = \\frac { 1 } { 2 } \\left ( H _ 0 ^ 2 - y ^ 2 \\right ) e ^ { - z } . \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} \\ell _ R ( S / J ( D ) ) = \\sum _ { i = 1 } ^ t \\ell _ R ( S _ { m _ i } / J ( D _ i ) ) = \\sum _ { i = 1 } ^ t [ S / m _ i : R / m _ R ] \\ell _ { S _ { m _ i } } ( S _ { m _ i } / J ( D _ i ) ) . \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} S : = \\left \\{ f \\in \\mathcal { B } ^ { \\alpha } ( \\mathbb { R } ^ 2 ) : ~ \\| f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = 1 \\right \\} . \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} Q _ { R } ( x ) : = \\max \\left \\{ K _ R ( x ) , \\frac { I _ R ( x ) } { R } , \\frac { 1 } { R ^ 2 } \\right \\} . \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} D ^ m T = - D T D ^ m r ( T ) \\left ( D T \\right ) ^ { \\otimes m } - D T \\mathcal { P } _ m ( R , T ) & & m \\ge 2 . \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} \\overline { V } _ 0 = \\overline { v } _ 0 , \\ ; \\ ; \\ ; ( U _ \\pm ) _ 0 = ( u _ \\pm ) _ 0 = \\frac { 1 } { 2 } ( \\widetilde { v } _ 0 \\pm i \\widetilde { v } _ 0 ^ \\perp ) . \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } \\right ) \\langle x , x \\rangle ^ \\frac { 1 } { 2 } + \\langle x , x \\rangle ^ \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } = \\sqrt { n } \\langle x , x \\rangle ^ \\frac { 1 } { 2 } ( 2 - c _ x ) \\langle x , x \\rangle ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} z ^ { 2 } - a \\ , x ^ { 2 } - b \\ , y ^ { 2 } = c , a , b , c \\in \\mathbb { R } , a \\ , b \\neq 0 , c > 0 , \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} \\int _ { M } \\left ( R - \\frac { 1 } { 2 } T o r - \\frac { 1 } { 2 } T o r ^ { \\prime } \\right ) \\left ( \\gamma , \\gamma \\right ) d \\mu + \\int _ { M } \\left \\vert \\gamma _ { 1 , 1 } \\right \\vert ^ { 2 } d \\mu - \\int _ { M } T o r \\left ( d _ { b } u , \\gamma \\right ) d \\mu = 0 . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\max \\{ C _ { \\tilde { m } , 1 } ( \\varepsilon ) , C _ { \\tilde { m } , 2 } ( \\varepsilon ) , C _ { \\tilde { m } , 3 } ( \\varepsilon ) \\} = 0 . \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} F _ { \\mu _ { 1 } } ( F _ { \\rho _ { 1 } } ( z ) ) = F _ { \\mu _ { 2 } } ( F _ { \\rho _ { 2 } } ( z ) ) = F _ { \\rho _ { 1 } } ( z ) + F _ { \\rho _ { 2 } } ( z ) - z , z \\in \\mathbb { H } . \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} \\varphi ' ( U _ { \\ell } ) { R _ { \\ell } } \\partial _ j U _ { \\ell , k } = \\partial _ j ( { R _ { \\ell } } \\varphi ' ( U _ { \\ell } ) U _ { \\ell , k } ) - 2 \\sqrt { R _ { \\ell } } U _ { \\ell , k } \\varphi ' ( U _ { \\ell } ) \\partial _ j \\sqrt { R _ { \\ell } } - R _ { \\ell } U _ { \\ell , k } \\varphi '' ( U _ { \\ell } ) \\partial _ { j } U _ { \\ell } , \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} f _ L ( 0 , 0 ; \\mu ) & = 0 , & g _ L ( 0 , 0 ; \\mu ) & = 0 , \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} \\alpha y ( t ) = F ( t ) + \\int _ { t } ^ { \\infty } \\frac { y ( s ) } { \\log ( \\lambda _ { 0 } ( s ) ) } \\left ( \\frac { 1 } { 1 + s - t } + \\frac { 1 } { ( \\lambda _ { 0 } ( t ) ^ { 1 - \\alpha } + s - t ) ( 1 + s - t ) ^ { 3 } } \\right ) d s , t \\geq T _ { 0 } \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} p ' & = \\frac { p } { \\varepsilon ( p ) } , \\\\ A ' & = \\frac { \\gamma ( p ) } { \\varepsilon ( p ) } = \\frac { \\gamma ( p ) \\ , p ' } { p } . \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} x ( 0 , \\mu ) = x _ 0 ( \\mu ) \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} ( \\phi _ 2 ^ { \\mathbb I } , \\phi _ 0 ^ { \\mathbb I } ) = c _ 1 ( \\Phi _ 2 , \\ , \\Phi _ 0 ) , ( - \\phi _ 2 ^ { \\mathbb R } , \\phi _ 0 ^ { \\mathbb R } ) = c _ 2 ( \\Phi _ 2 , \\ , \\Phi _ 0 ) , \\phi _ 1 = i ( c _ 3 \\ , U ^ + , c _ 4 \\ , U ^ - ) , \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} S ^ { ( 3 ) } _ { 4 , 4 } = - i w ^ { ( 3 ) } _ 4 = - 1 2 i \\lambda _ 3 a _ 1 . \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} ( G ) & \\leq ( G _ 1 ) + ( G _ 2 ) = ( G _ C ) \\\\ & = 1 + ( G _ 2 ) \\leq 1 + ( G ) . \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\Theta ( \\varepsilon ) : = \\Big \\{ ( y ^ I , z ^ I ) \\in \\Re ^ p \\times \\Re ^ p : \\Psi _ { \\varepsilon } ( y ^ I , z ^ I ) = 0 \\Big \\} . \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} \\forall \\delta > 0 \\ \\Rightarrow \\sum _ { k = 1 } ^ { \\infty } T _ k \\left ( \\ k ^ { \\delta k } \\ \\right ) < \\infty , \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} c _ { i , 0 } = ( - x ) ^ { i - 1 } c , c _ { i , 1 } = ( - x ) ^ { i - 2 } ( 2 ( i - 1 ) c - x ) . \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} \\psi _ { n , n } ^ { \\ast } ( Q ) = \\frac { L _ { n , n } ^ { \\ast } ( q ) } { q } \\geq - \\frac { L _ { n , 2 } ( q ) } { q } - O ( q ^ { - 1 } ) = ( \\gamma - \\epsilon _ { 3 } ) - O ( q ^ { - 1 } ) . \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} \\langle \\sigma _ k , D u _ k \\rangle = F ^ * _ k ( \\sigma _ k ) + F _ k ( D u _ k ) \\ge C ( L ) | \\sigma _ k | ^ { q ' } + L _ 0 | D u _ k | ^ p - \\mu _ k \\ge - \\mu _ k , \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} \\sup \\limits _ { t \\in \\left [ 0 , 1 \\right ] } \\left \\Vert e ^ { \\frac { \\gamma \\left \\vert x \\right \\vert ^ { p } } { 2 } } \\upsilon \\left ( . , t \\right ) \\right \\Vert _ { X } \\leq C _ { 0 } k ^ { C _ { p } } a _ { k } ^ { 2 } = C _ { 0 } k ^ { C _ { p } } e ^ { a \\left ( k , p \\right ) } \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} \\ell = \\limsup _ { r \\to 0 } \\frac { G ( r ) } r , \\ ; M = \\sup _ { t \\in [ 0 , T ] } \\omega ( \\cdot , \\lambda _ 1 ) * p ( t ) . \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} - \\partial _ { t t } v _ { 1 } + \\partial _ { r r } v _ { 1 } + \\frac { 1 } { r } \\partial _ { r } v _ { 1 } - \\frac { v _ { 1 } } { r ^ { 2 } } = - 2 \\lambda '' ( t ) \\frac { r } { 1 + r ^ { 2 } } \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} \\| Q _ L u \\| _ { L _ t ^ p L _ { x , y } ^ q } \\lesssim L ^ { \\frac { 2 } { 3 p } + \\frac { 1 } { q } } \\| Q _ L u \\| _ { L _ { x , y , t } ^ 2 } , \\textnormal { i f } \\ \\ \\frac { 2 } { p } + \\frac { 2 } { q } = 1 , \\ p \\geq 4 . \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} w _ 0 = w _ 0 ( z ) = \\left ( - ( 2 \\beta + 1 ) c _ \\beta \\int _ 0 ^ z ( \\hat z ^ 2 - 1 ) ^ { \\beta } \\hat z ^ { - 2 - 2 \\beta } \\ , d \\hat z \\right ) ^ { - \\frac 1 { 2 \\beta + 1 } } \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} P _ * ( r _ 2 , k ) - P _ * ( r _ 1 , k ) = - \\int _ { r _ 1 } ^ { r _ 2 } A ( r ) P ( r , k ) d r \\ , . \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} h = h _ { \\b + 1 , \\gamma } , \\ P = e _ { \\gamma } e _ { \\gamma } ^ { \\top } , \\ R = \\frac { 1 } { A } = G _ { \\b , \\gamma - 1 } \\left ( E + \\i \\eta \\right ) , \\ , S = \\frac { 1 } { A + B } = G _ { \\b , \\gamma } \\left ( E + \\i \\eta \\right ) . \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} r _ { A } = \\inf \\lbrace r > 0 : \\exists s \\in B _ { \\ell ^ d _ 2 } \\ , \\ , A \\subset B _ r ( s ) \\rbrace . \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} x _ 3 ^ 2 = F _ 1 ( x _ 0 , x _ 1 , x _ 2 ) F _ 2 ( x _ 0 , x _ 1 , x _ 2 ) G _ 1 ( x _ 0 , x _ 1 , x _ 2 ) G _ 2 ( x _ 0 , x _ 1 , x _ 2 ) + F ( x _ 0 , x _ 1 , x _ 2 ) ^ 2 , \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\left ( \\int _ { n } ^ { n + 1 } | f | ^ 2 d r \\right ) \\left ( \\int _ n ^ { n + 1 } \\left ( \\int _ { \\R } \\frac { | P ( r , k ) | ^ 2 } { \\kappa ( k ) } d \\sigma ( k ) \\right ) d r \\right ) \\stackrel { \\eqref { s d _ d 1 } } { \\le } K \\| f \\| _ 2 ^ 2 \\ , , \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} \\lambda \\eta ( x ) = \\eta ( \\lambda x ) & \\lambda \\in K , x \\in V , \\\\ \\eta ( x ) + \\eta ( y ) = \\eta ( x + y ) & x , y \\in V , \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} 2 ^ { 1 / v } \\lambda ^ { 1 - v } \\Gamma ( 1 / v ) b _ { n } ^ { v - 1 } \\exp \\left ( \\frac { b _ { n } ^ { v } } { 2 \\lambda ^ { v } } \\right ) = n . \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} 0 & \\leq \\Big ( \\ln A _ { n , m } - \\frac { n - 2 m } 2 \\ln ( \\varepsilon + \\frac { | x | ^ 2 } { \\varepsilon } ) \\Big ) | x | ^ \\alpha \\\\ & \\leq \\Big ( \\ln A _ { n , m } - \\frac { n - 2 m } 2 \\ln \\varepsilon \\Big ) a _ \\varepsilon ^ \\alpha \\\\ & = O ( \\varepsilon ^ { \\alpha / 2 } ( - \\ln \\varepsilon ) ) \\\\ & = o ( 1 ) _ { \\varepsilon \\searrow 0 } . \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\nabla v _ n ( x ) \\cdot \\nabla \\varphi ( x ) \\ ; d x + \\int _ { \\mathbb { R } ^ N } V _ 0 ( x ) v _ n ( x ) \\varphi ( x ) \\ ; d x = \\int _ { \\mathbb { R } ^ N } \\dfrac { f _ 0 ( u _ n ( x ) ) } { u _ n ( x ) } v _ n ( x ) \\varphi ( x ) \\ ; d x + o _ n ( 1 ) . \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} \\psi _ t = ( \\psi _ { t - 1 } \\times i d ) \\circ \\rho _ t . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} L ( s , \\chi ) = \\sum _ { n = 1 } ^ \\infty \\frac { \\chi ( n ) } { n ^ s } = \\prod _ { p \\ } \\Big ( 1 - \\frac { \\chi ( p ) } { p ^ s } \\Big ) ^ { - 1 } . \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} F ( \\theta ) = c _ 0 + c _ 1 w _ 1 ( \\theta ) + \\cdots + c _ k w _ k ( \\theta ) . \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} x ^ { \\# } = \\begin{pmatrix} b c - | x | ^ 2 & \\bar { y } \\bar { x } - c z & z x - b \\bar { y } \\\\ x y - c \\bar { z } & a c - | y | ^ 2 & \\bar { z } \\bar { y } - a x \\\\ \\bar { x } \\bar { z } & y z - a x & a b - n ( z ) \\end{pmatrix} . \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} 0 \\geqslant \\Re \\langle P _ n x , H _ n P _ n x \\rangle = \\Re \\langle P _ n x , P _ n H x \\rangle = \\Re \\langle P _ n x , H x \\rangle \\to \\Re \\langle x , H x \\rangle \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} | v _ { 1 } ^ { \\lambda } + v _ { 2 } + v _ { 3 } ^ { \\lambda } | \\leq \\begin{cases} \\frac { C r } { t ^ { 2 } \\log ^ { b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C \\log ( r ) } { | t - r | } , t > r \\geq \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} P ^ b _ { Q } : = \\left ( \\begin{array} { c c } { } ^ b S _ { + } ^ 2 & { } ^ b S _ { + } ( I + { } ^ b S _ { + } ) Q ^ b \\\\ { } ^ b S _ { - } D ^ + & I - { } ^ b S _ { - } ^ 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { M } ( - 1 ) ^ k [ 4 k - 1 ] \\frac { ( q ^ { - 1 } ; q ^ 2 ) _ k ^ 4 ( q ^ { - 3 } ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k ^ 4 ( q ^ { 4 } ; q ^ 2 ) _ k } q ^ { k ^ 2 + 7 k } \\equiv \\frac { \\Omega _ q ( n ) } { [ 2 ] ^ 2 [ 4 ] [ 6 ] } \\sum _ { k = 0 } ^ { ( n - 3 ) / 2 } \\frac { ( q ^ 3 ; q ^ 2 ) _ k ^ 3 } { ( q ^ 2 , q ^ 6 , q ^ 8 ; q ^ 2 ) _ k } q ^ { 2 k } . \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} h _ 0 ( Q ) \\cap ( \\partial B _ \\rho \\cap X ) = \\{ \\gamma ^ X ( t ) : t \\in [ 0 , 1 ] \\} \\cap \\partial B _ \\rho = \\{ \\gamma ^ X ( t _ 0 ) \\} , \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} \\gamma = a _ { 2 L } a _ { 3 R } - a _ { 3 L } a _ { 2 R } \\ , . \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} P _ { } ^ e = { \\Pr \\big ( { x _ T } = { 0 } \\big ) } \\Pr \\bigg ( { x _ D } = { 1 } ~ \\bigg | ~ { x _ T } = { 0 } \\bigg ) + { \\Pr \\big ( { x _ T } = { 1 } \\big ) } \\Pr \\bigg ( { x _ D } = { 0 } ~ \\bigg | ~ { x _ T } = { 1 } \\bigg ) . \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} & F ( t ; x , y , 1 , u , 1 , v ) \\\\ & = \\frac { x v t ^ 2 ( 1 - y r ) } { ( 1 - y t u + t u v ( y r - 1 ) ) ( 1 - y t u v ) ( t u x + y ^ { - 1 } - t u ) } \\\\ & \\quad + \\frac { t x ( 1 - y t u ) ( 1 - t u v ( y - y r + 1 ) ) } { ( t u x + y ^ { - 1 } - t u ) ( 1 - y t u + t u v ( y r - 1 ) ) ( 1 - y t u v ) } F ( t ; x , y - y r + 1 , 1 , u , 1 , v ) . \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} & | - 4 b \\int _ { t } ^ { t + \\lambda _ { 0 , 0 } ( t ) ^ { 1 - \\alpha } } \\frac { d x } { x ^ { 2 } \\log ^ { b + 1 } ( x ) ( \\lambda _ { 0 , 0 } ( t ) ^ { 1 - \\alpha } + x - t ) ( 1 + x - t ) ^ { 3 } } | \\\\ & \\leq C \\int _ { t } ^ { t + \\lambda _ { 0 , 0 } ( t ) ^ { 1 - \\alpha } } \\frac { d x } { x ^ { 2 } \\log ^ { b + 1 } ( x ) } \\frac { 1 } { \\lambda _ { 0 , 0 } ( t ) ^ { 1 - \\alpha } } \\\\ & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} ( - P ) _ { B } ^ { \\alpha } \\phi = \\frac { \\sin \\alpha \\pi } { \\pi } \\int _ { 0 } ^ { \\infty } \\lambda ^ { \\alpha - 1 } \\bigg [ ( \\lambda - P ) ^ { - 1 } - \\frac { \\lambda } { \\lambda ^ { 2 } + 1 } \\bigg ] ( - P ) \\phi \\ , d \\lambda + \\sin \\frac { \\alpha \\pi } { 2 } ( - P ) \\phi . \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} f B ( e , x ) g = f B ( e ^ 2 , x ) g = f e B ( e , x ) g + f B ( e , x ) e g = 0 . \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} \\ln ( P _ n ) \\ \\leq \\ [ ( N n + 1 ) & ( \\ln ( N n + 1 ) - 1 ) + 1 ] - n [ N ( \\ln ( N ) - 1 ) + 1 ] \\\\ = \\ ( N n + 1 ) & ( \\ln ( n + ( 1 / N ) ) + \\ln ( N ) - n . \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} p _ t ( x ) = ( 1 - t ) p ^ * ( x ) + t p ( x ) . \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} ( 1 + Q ( s ) ) ^ { - 1 } = 1 - Q ( s ) + Q ( s ) ( 1 + Q ( s ) ) ^ { - 1 } Q ( s ) . \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} [ n ] _ 2 ! : = \\prod _ { i = 1 } ^ n ( 2 ^ i - 1 ) . \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} Y _ i ( \\Omega _ i ) = \\prod _ { k = 1 } ^ { d _ i - 1 } h ^ i _ k ( \\phi ^ i _ k ) , ~ ~ ~ i = 1 , 2 , \\cdots , N . \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} \\int _ { t } ^ { \\infty } d x | K ( x - t , z _ { 1 } ) - K ( x - t , z _ { 2 } ) | & \\leq | z _ { 1 } - z _ { 2 } | \\int _ { 0 } ^ { 1 } d q \\int _ { 0 } ^ { \\infty } d w | \\partial _ { 2 } K ( w , z _ { 2 } + ( z _ { 1 } - z _ { 2 } ) q ) | \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} \\mathcal H _ 0 = \\{ \\tilde \\psi \\in \\mathcal H _ * \\ , : \\ , \\langle \\tilde \\psi , Z _ 0 \\rangle _ * = 0 \\} . \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} P _ n ( A , B ) & = A P _ { n - 1 } ( B ^ { - 1 } , A ^ { - 1 } ) A = A B ^ { - 1 } P _ { n - 2 } ( A , B ) B ^ { - 1 } A , n \\ge 2 , \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} c _ 1 = - \\frac { \\int _ { z _ 1 } ^ { z _ 2 } h _ 0 ^ n \\varphi _ 2 d z } { \\int _ { z _ 1 } ^ { z _ 2 } h _ 0 ^ n \\varphi _ 1 d z } c _ 2 , \\ ; \\ ; \\ ; c _ 2 \\neq 0 , \\ ; \\ ; \\ ; c _ i = 0 \\ ; \\ ; \\ ; ( i \\geq 3 ) . \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} \\min _ p \\ , & B _ { \\alpha } ( p , q ) \\\\ \\mbox { s u b j e c t t o } & \\sum \\limits _ x p ( x ) f _ i ( x ) = a _ i , i \\in \\{ 1 , \\ldots , k \\} , \\\\ & \\sum \\limits _ x p ( x ) = 1 , \\\\ & p ( x ) \\ge 0 \\quad \\forall x \\in \\mathbb { S } . \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} \\left ( j _ { n } ^ { ( 3 ) } \\right ) ^ { 2 } - 9 \\left ( J _ { n } ^ { ( 3 ) } \\right ) ^ { 2 } = 2 ^ { n + 2 } j _ { n - 3 } ^ { ( 3 ) } , \\ n \\geq 3 . \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} M _ i = \\textup { b l k d i a g } ( m _ { i , 1 } , m _ { i , 2 } , \\dots , m _ { i , p } ) \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} f & = ( x y + A ^ 2 x ^ 2 + B ^ 4 x ) ^ 2 + ( x ^ 2 + B ^ 2 x ) ^ 2 x \\\\ & = ( y + ( A ^ 2 + A ) x + B ^ 4 ) ^ 4 + x ^ 4 x + ( x + B ^ 2 ) ^ 4 x ^ 2 + B ^ 4 x ^ 3 \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} t _ 1 ^ { \\kappa - p / 2 + 1 } \\int _ { t _ 1 } ^ { 1 / 2 } \\frac { \\displaystyle | Z _ N ( t ) - \\sigma B _ N ( t ) | ^ p } { t ^ { \\kappa } } d t = O _ P \\left ( ( N t _ 1 ) ^ { ( - 1 / 2 + \\zeta ) p } \\right ) = o _ P ( 1 ) . \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} \\mathcal { A } _ { N , K } = \\left \\{ \\max _ { 0 \\leq n \\leq N } \\max \\left \\{ \\max _ { 0 \\leq m \\leq n } ( S _ m - S _ n ) , \\ : \\max _ { n \\leq m \\leq N } ( S _ m - S _ N - S _ n ) \\right \\} \\leq K \\right \\} . \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} 2 b _ \\alpha N _ { \\theta , \\alpha } ^ { \\alpha - 1 } [ c _ 0 \\boldsymbol { \\Sigma } ^ { - 1 } \\boldsymbol { \\mu } - \\boldsymbol { \\Sigma } ^ { - 1 } \\tilde { c } ] = \\bf { 0 } , \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} \\lambda _ 1 = \\frac { 1 - \\sqrt { 1 7 } } { 4 } , \\lambda _ 2 = \\frac { 1 + \\sqrt { 1 7 } } { 4 } \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} \\Big ( \\sum _ { \\vert \\alpha \\vert = m } \\Vert x _ { \\alpha } \\Vert ^ { q } \\Big ) ^ { 1 / q } \\leq C ^ { m } \\bigg ( \\int _ { \\mathbb { T } ^ { n } } \\Vert P ( z ) \\Vert ^ { q } d z \\bigg ) ^ { 1 / q } \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} U S f = \\sum _ { j \\in J } U S g _ j = \\sum _ { j \\in J } S U g _ j = S U f . \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} \\beta = \\left ( \\frac { d \\xi _ L } { d \\mu } - \\frac { d \\xi _ R } { d \\mu } \\middle ) \\right | _ { \\mu = 0 } , \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} P _ { Q } : = \\left ( \\begin{array} { c c } S _ { + } ^ 2 & S _ { + } ( I + S _ { + } ) Q \\\\ S _ { - } D ^ + & I - S _ { - } ^ 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} t = \\operatorname { H } ( s ) = \\int _ { 0 } ^ { s } \\frac { d x } { [ G _ 0 : G _ x ] } , \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} \\frac { \\partial \\lambda _ { \\ker } } { \\partial t } = r ( t ) . \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} \\beta e ^ { - \\beta } + \\frac { 1 } { 2 } ( 2 \\beta ^ 2 - \\beta ) e ^ { - 2 \\beta } > \\sum _ { k = 3 } ^ \\infty \\frac { k ^ { k - 2 } } { k ! } \\Bigg [ k - 1 - k \\beta \\Bigg ] \\beta ^ { k - 1 } e ^ { - k \\beta } . \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} \\sum _ { i = m + 3 } ^ L G _ i + 2 \\leq \\sum _ { i = m + 4 } ^ L H _ i . \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} \\left [ - \\hat { L } ^ 2 _ i + f _ i ( \\Omega _ i ) \\right ] Y _ i ( \\Omega _ i ) = \\lambda _ i Y _ i ( \\Omega _ i ) \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} \\left \\{ 3 ^ a ( u _ 1 ^ b u _ 2 ^ c ) ^ 2 \\beta ^ d : a , c \\in \\mathbb { Z } _ { \\geq 0 } , b \\in \\mathbb { Z } , d = 0 , 1 , 2 , \\textnormal { a n d } c x _ 1 ^ { - 1 } + y _ { 1 , d } x _ 1 ^ { - 1 } \\geq b \\geq c x _ 2 - y _ { 2 , d } \\right \\} . \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} E _ 0 ( q ) & = E _ 1 ( q ) = E _ 2 ( q ) = 1 , E _ 3 ( q ) = 2 , \\\\ E _ 4 ( q ) & = q + 4 , \\\\ E _ 5 ( q ) & = 2 q ^ 2 + 5 q + 9 , \\\\ E _ 6 ( q ) & = q ^ 4 + 5 q ^ 3 + 1 4 q ^ 2 + 2 0 q + 2 1 . \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha = & \\ , \\ , ( e ^ { - ( q _ 1 - q _ 2 ) } - e ^ { - ( q _ 1 + q _ 2 - 2 q _ 3 ) } ) d q _ 1 \\\\ & - ( 1 - e ^ { - 2 ( q _ 1 - q _ 3 ) } ) d q _ 2 \\\\ & + ( e ^ { - ( q _ 2 - q _ 3 ) } - e ^ { - ( 2 q _ 1 - q _ 2 - q _ 3 ) } ) d q _ 3 , \\end{aligned} \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} = \\binom { 2 k + 1 } { k } \\frac { c _ 1 ^ k \\ , c _ 2 ^ { k + 1 } } { ( c _ 1 + c _ 2 ) ^ { 2 k + 1 } } + \\Bigl ( 1 - \\frac { c _ 1 } { c _ 2 } \\Bigr ) \\sum _ { j = 0 } ^ { k - 1 } \\binom { 2 k + 1 } { j } \\frac { c _ 1 ^ { j } \\ , c _ 2 ^ { 2 k + 1 - j } } { ( c _ 1 + c _ 2 ) ^ { 2 k + 1 } } = \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\max \\{ D _ p ( M , x _ n ) , D _ p ( N , x _ n ) \\} = 0 \\Rightarrow \\lim _ { n \\to \\infty } D _ p ( M \\cap N , x _ n ) = 0 , \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} \\dot { z } = \\beta \\{ P _ { \\Omega } [ z - \\alpha G ( z ) ] - z \\} , \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} \\pi R \\ = \\ \\bar \\pi ( R ) \\ = \\ \\{ ( \\pi ( i ) , \\pi ( j ) ) : ( i , j ) \\in R \\} . \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } K _ n ^ { - 1 } \\int _ { \\R ^ d _ + \\cap B ( z _ 0 , 1 / n ) } ( z _ d \\vee y _ d ) ^ { - ( \\beta _ 1 - \\beta _ 2 ) } \\left ( \\log \\left ( 1 + \\frac { z _ d \\vee y _ d } { z _ d \\wedge y _ d } \\right ) \\right ) ^ { \\beta _ 3 } \\left ( \\log \\left ( \\frac { 1 } { z _ d \\vee y _ d } \\right ) \\right ) ^ { \\beta _ 4 } d y \\\\ & = z _ d ^ { - ( \\beta _ 1 - \\beta _ 2 ) } \\ , . \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} | x | < h ( n ) : = c _ 4 n ^ { - g } \\frac { ( 2 n ) ! } { | \\alpha | ^ n \\ , n ! } \\quad | x e ^ \\alpha - y | \\ge c _ 4 n ^ { - g + 1 } \\frac { | \\alpha | ^ n } { 4 ^ n \\ , n ! } \\ , . \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} l ( a ) & = l ( a b a ) \\precsim l ( a b ) \\precsim r ( b ) \\sim l ( b ) . \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} i v _ n & = i f _ n \\cdot \\left ( x _ 1 + \\ldots + x _ n \\right ) + i g _ n m ^ 2 , n \\geq 2 . \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} \\lim _ { v \\uparrow \\infty } \\int _ { 0 } ^ { T } \\int _ Y | y - \\eta ( t ) | _ Y ^ 2 \\bar { f } ( y ) \\bar { \\mu } ( d y ) d t = 0 . \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} \\int _ { G } c ( g ^ { - 1 } x ) d \\mu ( g ) = 1 , \\quad \\mbox { f o r a l l } ~ x \\in M , \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} \\kappa ( a , b , c , d ) = \\begin{vmatrix} b _ 1 & \\boxed { d _ 1 } \\\\ b _ 2 & d _ 2 \\end{vmatrix} ^ { - 1 } \\begin{vmatrix} b _ 1 & \\boxed { c _ 1 } \\\\ b _ 2 & c _ 2 \\end{vmatrix} \\begin{vmatrix} a _ 1 & \\boxed { c _ 1 } \\\\ a _ 2 & c _ 2 \\end{vmatrix} ^ { - 1 } \\begin{vmatrix} a _ 1 & \\boxed { d _ 1 } \\\\ a _ 2 & d _ 2 \\end{vmatrix} , \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} 2 N \\leq \\sum _ { i = m + 4 } ^ { L } H _ i + 2 . \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} F ( t ; x , y , 0 , u , z , v ) & = \\frac { y ^ 2 x z ^ 2 v t ^ 2 } { ( 1 - y t u ) ( y - y z r + z ) } + \\frac { z ( y t u x - y t u + 1 ) } { y - y z r + z } F ( t ; x , y , 1 , u , z , v ) \\\\ & \\quad + \\frac { y ^ 2 u ^ 2 v t ^ 2 z ^ 2 ( 1 - v ) ( y t u x - y t u + 1 ) } { ( 1 - y t u ) ( y - y z r + z ) } F ( t ; x , y , 1 , u , 1 , v ) . \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} \\nu _ a ( L ) = \\nu _ a ( L ^ U ) , \\nu _ b ( L ) = \\nu _ b ( L ^ U ) . \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} m _ { 1 , 2 } = m _ { 4 , 5 } & = m _ { 7 , 9 } , & m _ { 1 , 5 } = m _ { 1 , 6 } & = 0 , \\\\ m _ { 1 , 3 } = m _ { 4 , 6 } & = - \\frac { m _ { 8 , 9 } } { m _ { 7 , 9 } } , & 1 - m _ { 7 , 9 } ^ { 2 } + \\frac { m _ { 8 , 9 } ^ { 2 } } { m _ { 7 , 9 } } & = 0 . \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} q ( h ) = B . g ( { h } ) B \\equiv \\frac { 1 } { 2 } ( g ( { h } - 2 ) ( \\mu - { h } ^ 2 + 2 { h } ) + g ( { h } + 2 ) ( \\mu - { h } ^ 2 - 2 { h } ) ) , \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} - i w ^ { ( s ) } _ n & = - i \\lambda _ s B _ { n , s } ( 1 ! a _ 0 , 2 ! a _ 1 , 3 ! a _ 2 , 4 ! a _ 3 , \\ldots ) , w _ n ^ { ( s ) } = 0 \\ \\forall n < s . \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} & \\frac { - c _ { b } r } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\sin ^ { 2 } ( \\theta ) d \\theta \\left ( ( b - 1 ) \\int _ { 0 } ^ { \\frac { t _ { + } } { 2 } } \\frac { d u \\sin ( u ) } { u \\log ^ { b } ( \\frac { t _ { + } } { u } ) t _ { + } ^ { 2 } } \\right ) = - \\frac { c _ { b } r } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\sin ^ { 2 } ( \\theta ) d \\theta ( b - 1 ) \\left ( \\frac { \\pi } { 2 \\log ^ { b } ( t _ { + } ) t _ { + } ^ { 2 } } + ( t , r , \\theta ) \\right ) \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} \\langle g , f _ 0 \\rangle _ { \\alpha } = ( \\gamma + 2 ) \\langle \\widehat { A } g , f _ 0 \\rangle _ { \\alpha } \\Leftrightarrow \\langle g , f _ 0 \\rangle _ { \\alpha } = ( \\gamma + 2 ) \\langle g , \\widehat { A } f _ 0 \\rangle _ { \\alpha } , \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} K ( \\Psi ^ { i } ; \\Psi ^ { i } ; \\gamma ) = K ( \\Psi ^ i _ 0 ; \\Psi ^ i _ 0 ; \\gamma ) = K ( \\Psi ^ i _ 1 ; \\Psi ^ i _ 1 ; \\gamma ) = \\cdots = K ( \\Psi ^ i _ { i - 1 } ; \\Psi ^ i _ { i - 1 } ; \\gamma ) = K ( \\Psi ^ { i + 1 } ; \\Psi ^ { i + 1 } ; \\gamma ) \\ , , \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} \\alpha ^ k _ l + \\beta ^ k _ l = 1 + 1 - \\frac { G _ l ( x _ k ) + H _ l ( x _ k ) } { \\sqrt { G _ l ^ 2 ( x _ k ) + H _ l ^ 2 ( x _ k ) + 2 t _ k } } = 1 - \\frac { \\varphi ^ t _ \\textup { F B } ( G _ l ( x _ k ) , H _ l ( x _ k ) ) } { \\sqrt { G _ l ^ 2 ( x _ k ) + H _ l ^ 2 ( x _ k ) + 2 t _ k } } \\geq 1 \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} y ( t , \\mu ) = \\sum _ { i = 1 } ^ { \\infty } w _ i ( t ) \\Phi _ i ( \\mu ) \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} G = G _ { 0 } + R _ { 3 } + 4 R _ { 2 } = ( R _ { 1 1 } : 2 R _ { 3 } ) + R _ { 3 } + 4 R _ { 2 } = ( \\zeta _ { 6 } + \\zeta _ { 6 } ^ { 5 } ) ( \\zeta _ { 1 1 } + \\zeta _ { 1 1 } ^ { 1 0 } ) + \\sum _ { i \\neq 1 , 1 0 } \\zeta _ { 1 1 } ^ { i } + R _ { 3 } + 4 R _ { 2 } . \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} e _ R ( \\mathcal I ( 1 ) ^ { [ d _ 1 ] } , \\ldots , \\mathcal I ( r ) ^ { [ d _ r ] } ; M ) = e _ R ( \\mathcal I ( 1 ) ^ { [ d _ 1 ] } , \\ldots , \\mathcal I ( s ) ^ { [ d _ s ] } ; M ) > 0 \\mbox { i f } d _ { s + 1 } + \\cdots + d _ r = 0 \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N - 1 } B _ { \\mathbf { m } } ^ { ( d ) } \\left ( \\mathbf { z } + \\frac { i } { N } \\mathbf { 1 } \\right ) & = N ^ { 1 - | \\mathbf { m } | } B _ { \\mathbf { m } } ^ { ( d ) } ( N \\mathbf { z } ) \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} f _ i & = P _ D \\lambda _ i , & f _ i ^ { p R , q R } = P _ { D _ { p R } } [ \\mathbf { 1 } _ { D _ { p R , q R } } P _ D ^ * \\lambda _ i ] , \\enskip & \\widetilde { f } _ i ^ { p R , q R } = P _ { D _ { p R } } [ \\mathbf { 1 } _ { D _ { q R } ^ R } P _ D ^ * \\lambda _ i ] , \\\\ f _ i ^ * & = P _ D ^ * \\lambda _ i , & f _ i ^ { p R , q R * } = P _ { D _ { p R } } ^ * [ \\mathbf { 1 } _ { D _ { p R , q R } } P _ D ^ * \\lambda _ i ] , \\enskip & \\widetilde { f } _ i ^ { p R , q R * } = P _ { D _ { p R } } ^ * [ \\mathbf { 1 } _ { D _ { q R } ^ R } P _ D ^ * \\lambda _ i ] . \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} d _ 1 \\le & \\sqrt { \\sum _ { n = N _ T + 1 } ^ { + \\infty } \\frac { \\phi ^ 2 ( u ) H _ { n - 1 } ^ 2 ( u ) } { n ! T ^ d } \\int _ { [ - T , T ] ^ d } \\rho ^ n ( t ) \\prod _ { j = 1 } ^ d ( T - | t _ j | ) \\ , d t } \\cr \\le & K \\sqrt { \\phi ( u ) } \\sqrt { \\int _ { \\mathbb R ^ d } | \\rho ( t ) | \\ , d t } \\sqrt { \\sum _ { n = N _ T + 1 } ^ { + \\infty } \\frac { 1 } { n ( n - 1 ) ^ { 1 / 6 } } } \\cr = & O ( N _ T ^ { - 1 / 1 2 } ) , \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty | \\widehat { \\phi ^ { ( 2 ) } } ( t \\xi _ 2 , t \\xi _ 3 ) | ^ 2 { d t \\over t } = 1 \\mbox { f o r a l l } ( \\xi _ 2 , \\xi _ 3 ) \\in \\Bbb { R } ^ 2 \\backslash \\{ 0 \\} . \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} D _ 4 = K = 1 , C _ 1 = C _ 3 = 0 . \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} x _ 0 : = x , x _ { 2 n + 1 } : = x _ { 2 n } - P _ M x _ { 2 n } \\quad x _ { 2 n } : = x _ { 2 n - 1 } - P _ N x _ { 2 n - 1 } \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} 0 < \\kappa : = K ( 0 ) L ^ { - \\frac q 2 } ( 0 ) \\le K ( t ) L ^ { - \\frac q 2 } ( t ) \\le \\frac 1 q L ' ( t ) L ^ { - \\frac q 2 } ( t ) = \\frac { 2 } { ( 2 - q ) q } \\Big ( L ^ { \\frac { 2 - q } { 2 } } ( t ) \\Big ) ' . \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} t = \\beta F _ { \\mu } ( t ) + ( 1 - \\beta ) F _ { \\nu } ( F _ { \\mu } ( t ) ) , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} Z _ m ' = \\{ ( x _ i , w ( x _ i ) ) \\mid i \\in [ 1 , k ] \\} , Z _ m '' = \\{ ( x _ i , w ( x _ { i + 1 } ) ) \\mid i \\in [ 1 , k - 1 ] \\} , Z _ m ''' = \\{ ( x _ k , w ( x _ { 1 } ) ) \\} . \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} T ^ { ( 1 ) } & = \\frac { 1 } { \\omega } \\ , \\Psi ^ { ( 1 ) } _ { T ^ { ( 0 ) } } , \\\\ P ^ { ( 1 ) } & = \\Phi ^ { ( 1 ) } _ { T ^ { ( 0 ) } } + \\frac { \\lambda } { \\omega } \\ , \\Psi ^ { ( 1 ) } _ { T ^ { ( 0 ) } } . \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} \\psi _ { k } = \\left \\{ \\begin{array} { l l } 0 & \\\\ \\frac { 1 } { \\frac { k } { 2 } + 1 } \\binom { k } { \\frac { k } { 2 } } & . \\end{array} \\right . \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} - 4 \\int _ { t } ^ { \\infty } \\frac { \\lambda _ { 0 , 1 } '' ( s ) d s } { 1 + s - t } = \\frac { 4 b ^ { 2 } \\log ( \\log ( t ) ) } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } + E _ { v _ { 3 , i p } , 0 1 } \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} \\tau ( \\nabla a ) = d \\tau ( a ) \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\sup _ { Q _ { y } } N ( \\epsilon \\lVert F \\rVert _ { L _ { 2 } ( Q _ { y } ) } , \\mathcal { F } , L _ { 2 } ( Q _ { y } ) ) = \\sup _ { Q _ { y , x } } N ( \\epsilon \\lVert F \\circ \\phi _ { y } \\rVert _ { L _ { 2 } ( Q _ { y , x } ) } , \\mathcal { F } \\circ \\phi _ { y } , L _ { 2 } ( Q _ { y , x } ) ) , \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} e _ R ( \\mathcal I ( j ) ; R ) = e _ R ( \\{ I ( m D ( j ) ) \\} ; R ) = \\sum _ { i = 1 } ^ t [ S / m _ i : R / m _ R ] e _ { S _ { m _ i } } ( \\{ J ( m D ( j ) _ i ) \\} ; S _ { m _ i } ) \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} \\mathbb { L } _ { B R S T } ^ { \\mathrm { I } } & = \\frac 1 2 \\left ( A d A + d c A + c d c \\right ) + \\frac 1 6 A [ A , A ] - \\frac { 1 } { 1 2 } c [ c , c ] \\\\ \\mathbb { L } _ { B R S T } ^ { \\mathrm { I I } } & = \\frac 1 2 \\left ( A d A + c d A \\right ) + \\frac 1 6 A [ A , A ] - \\frac 1 4 A [ c , c ] - \\frac { 1 } { 1 2 } c [ c , c ] \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} h _ { i } : = ( h _ { i , m } ) _ { m } \\qquad ; i , m \\in I \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} \\delta ( p , q ) = \\begin{cases} 0 & p = q \\\\ \\infty & p \\neq q \\end{cases} , \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} \\mu ^ { I } : = \\mu _ { m } ^ { n ( I ) } | _ { I } . \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} & \\phi ( - q ^ { 1 / 2 } x _ 3 ) \\phi ( - q ^ { 1 / 2 } x _ 1 ) \\phi ( - q ^ { 1 / 2 } x _ 2 ) \\\\ & = \\phi ( - q ^ { 1 / 2 } x _ 3 ) \\phi ( - q ^ { 1 / 2 } x _ 2 ) \\phi ( - q x _ 1 x _ 2 ) \\phi ( - q ^ { 1 / 2 } x _ 1 ) \\\\ & = \\phi ( - q ^ { 1 / 2 } x _ 2 ) \\phi ( - q x _ 3 x _ 2 ) \\phi ( - q ^ { 1 / 2 } x _ 3 ) \\phi ( - q x _ 1 x _ 2 ) \\phi ( - q ^ { 1 / 2 } x _ 1 ) \\\\ & = \\phi ( - q ^ { 1 / 2 } x _ 2 ) \\phi ( - q x _ 3 x _ 2 ) \\phi ( - q x _ 1 x _ 2 ) \\phi ( - q ^ { 3 / 2 } x _ 3 x _ 1 x _ 2 ) \\phi ( - q ^ { 1 / 2 } x _ 3 ) \\phi ( - q ^ { 1 / 2 } x _ 1 ) \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} h _ 0 & = q _ 0 ^ 2 \\alpha ^ 2 - 2 q _ 0 q _ 1 \\alpha - 2 p ^ 1 q _ 0 - q _ 1 ^ 2 \\\\ h _ 1 & = - 2 q _ 0 q _ 1 \\alpha ^ 2 + ( 6 q _ 1 ^ 2 + 2 q _ 0 p ^ 1 ) \\alpha + 2 p ^ 1 q _ 1 \\\\ h _ 2 & = - p ^ 1 q _ 0 \\alpha ^ 2 + 3 ( p ^ 1 q _ 1 + p ^ 0 q _ 0 ) \\alpha + 2 ( p ^ 1 ) ^ 2 - 3 p ^ 0 q _ 1 . \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { ( x - 2 k t ) / ( 2 \\sqrt t ) } e ^ { i u ^ 2 } d u = \\left ( \\int _ { 0 } ^ { ( x - 2 k t ) / ( 2 \\sqrt t ) } e ^ { i u ^ 2 } d u \\right ) \\Bigl ( \\mu _ - ( ( x - 2 k t ) / ( 2 \\sqrt t ) ) + \\mu _ 0 ( \\cdot ) + \\mu _ + ( \\cdot ) \\Bigr ) \\ , . \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} p _ { k + 1 } = ( x + a _ k ) p _ k - p _ { k - 1 } \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } x _ n = j _ { p ^ * } ( \\Pi ^ { p * } _ { M \\perp \\cap N ^ \\perp } j _ p ( x ) ) = ( I - P _ { ( M ^ \\perp \\cap N ^ \\perp ) ^ \\perp } ) x \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} b \\tau ^ r _ c & = \\tau ^ r _ { \\delta c } + I ^ * \\sigma _ { c } , \\\\ B \\tau ^ r _ c & = 0 . \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} \\widehat \\Gamma ( x ) = P [ \\widehat \\Gamma ] ( x ) , \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\begin{bmatrix} K _ 1 ^ - \\\\ W _ 1 \\end{bmatrix} x _ 1 ( T ) = \\lim _ { T \\rightarrow \\infty } \\begin{bmatrix} 0 & 0 & D _ 3 \\end{bmatrix} x _ 1 ( T ) = D _ 3 x _ { 1 , 3 } ( T ) = 0 . \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} E _ { \\rm r e g } & = E - \\alpha A \\\\ A _ { \\rm r e g } & = A + \\beta E \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} | A + _ G B | , | A \\cdot _ G B | = \\frac { 1 } { \\sqrt 2 } | G | ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} g ( \\lambda ) = \\int _ 0 ^ { \\infty } ( 1 - e ^ { - t \\lambda } ) \\frac { e ^ { - t } } { t } d t . \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} \\hat { x } = \\{ A \\in \\mathfrak { m } : x \\in A \\} . \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} \\widetilde { b } = \\widetilde { \\tau } ( \\widetilde { g } ) - \\widetilde { g } . \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} \\mathcal { A } A \\mathcal { B } = \\left ( \\begin{array} { c c c c c c } \\tilde { a _ 1 } & & & & & \\\\ & \\ddots & & & & \\\\ & & \\tilde { a _ s } & & & \\\\ & & & 0 & & \\\\ & & & & \\ddots & \\\\ & & & & & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} \\frac { ( d - a ) ^ 2 } { 4 c ^ 2 } + \\frac { b } { c } = \\frac { ( d + a ) ^ 2 - 4 a d } { 4 c ^ 2 } + \\frac { 4 b c } { 4 c ^ 2 } = \\frac { ( d + a ) ^ 2 - 4 ( a d - b c ) } { 4 c ^ 2 } = \\frac { D } { 4 c ^ 2 } . \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} e _ R ( \\mathcal I ( j ) ^ { [ 2 ] } ; R ) = \\sum _ { i = 1 } ^ t - [ S / m _ i : R / m _ R ] ( \\Delta ( j ) _ i ^ 2 ) \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} X \\in L ^ \\infty ( [ 0 , t ] ; C ^ s ) , ~ \\omega \\in L ^ \\infty ( [ 0 , t ] ; & H ^ s ) , ~ \\partial _ X \\omega \\in L ^ \\infty ( [ 0 , t ] ; L ^ p ) . \\\\ \\theta \\in L ^ \\infty ( [ 0 , t ] ; H ^ { 1 + s } ) & , ~ \\partial _ 1 \\theta \\in L ^ 2 ( [ 0 , t ] ; H ^ { 1 + s } ) . \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} \\Delta _ s : = \\Phi _ s ^ { T _ s } , \\Gamma _ s : = \\Phi _ s ^ { G L _ s } , \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} \\partial _ { r } v _ { 1 } ( t , r ) = \\int _ { t } ^ { \\infty } \\frac { \\lambda '' ( s ) } { 1 + s - t } d s + E _ { \\partial _ { r } v _ { 1 } } ( t , r ) \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} ( \\partial _ { t } ^ + \\phi _ h ^ n , \\phi _ h ^ n ) _ { \\mathcal { T } _ h } = \\frac { 1 } { 2 \\Delta t } \\left ( \\| \\phi _ h ^ n \\| ^ 2 _ { \\mathcal { T } _ h } - \\| \\phi _ h ^ { n - 1 } \\| ^ 2 _ { \\mathcal { T } _ h } + \\| \\phi _ h ^ n - \\phi _ h ^ { n - 1 } \\| ^ 2 _ { \\mathcal { T } _ h } \\right ) , \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} \\| e _ h ^ { u ^ { m } } \\| ^ 2 _ { \\mathcal { T } _ h } + 2 \\Delta t \\sum _ { n = 1 } ^ { m } \\| e _ h ^ { \\phi ^ { n } } \\| ^ 2 _ { \\mathcal { T } _ h } + ( \\Delta t ) ^ 2 \\sum _ { n = 1 } ^ { m } \\| \\partial _ { t } ^ + e _ h ^ { u ^ { n } } \\| ^ 2 _ { \\mathcal { T } _ h } & \\le C ( \\epsilon ) \\big ( ( \\Delta t ) ^ 2 + h ^ { 2 ( k + 2 ) } \\big ) . \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} = \\sum _ { j = 0 } ^ k \\frac { k ! ^ 2 } { j ! ( 2 k + 1 - j ) ! } \\Bigl [ ( c t - \\beta ) ^ { j } ( c t + \\beta ) ^ { 2 k + 1 - j } - ( c t - \\beta ) ^ { 2 k + 1 - j } ( c t + \\beta ) ^ { j } \\Bigr ] \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} K _ 2 ( \\theta ) : = 2 \\max \\left \\{ ( 4 C _ 2 / K _ 1 ) ^ { 1 / \\omega _ * } \\theta ^ { \\beta / \\omega _ * } , ( c _ 0 + 1 ) t _ 0 ( \\theta ) + C _ 1 \\right \\} . \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} L _ { 2 } ( h , u , T ) : & = | T \\cap E _ { h } ( u ) | \\\\ L _ { 1 } ( h , u , T ) : & = \\frac { | \\partial ( T \\cap E _ { h } ( u ) ) | _ { 1 } } { 2 } , \\\\ L _ { 0 } ( h , u , T ) : & = \\sharp \\mbox { c o n n e c t e d c o m p o n e n t s i n } T \\cap E _ { h } ( u ) - \\sharp \\mbox { h o l e s i n } T \\cap E _ { h } ( u ) . \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} ( N - 2 ) H _ { n - k - 1 } & = ( N - 2 ) H _ { n - k - 2 } + ( N - 2 ) H _ { n - k - 3 } + N ( N - 2 ) H _ { n - 2 k - 4 } , \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} F ( t ; x , y , 1 , u , z , v ) & = \\frac { x z v t ^ 2 ( 1 - y r ) ( y t u v ( 1 - z ) + z ) } { ( 1 - y t u ) ( 1 - t u v ( y - y r + 1 ) ) ( t u x + y ^ { - 1 } - t u ) } \\\\ & \\quad + \\frac { t x ( y - y z r + z ) } { ( y - y r + 1 ) ( t u x + y ^ { - 1 } - t u ) } F ( t ; x , y - y r + 1 , 1 , u , z , v ) \\\\ & \\quad - \\frac { y u ^ 2 v t ^ 2 z ( 1 - v ) ( y r - 1 ) ( y t u v ( 1 - z ) + z ) } { ( 1 - y t u ) ( 1 - t u v ( y - y r + 1 ) ) } F ( t ; x , y , 1 , u , 1 , v ) . \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} D = m E . \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p ( \\cdot ) } \\left ( Q , \\frac { \\d x } { | Q | } \\right ) } : = \\inf \\left \\{ \\lambda > 0 : \\frac { 1 } { | Q | } \\int _ { Q } \\left ( \\frac { | f ( x ) | } { \\lambda } \\right ) ^ { p ( x ) } \\ , \\d x \\leq 1 \\right \\} . \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( 1 - \\sigma ) D ^ 2 u : D ^ 2 \\varphi + \\sigma \\Delta u \\Delta \\varphi d x - \\lambda \\int _ { \\partial \\Omega } \\frac { \\partial u } { \\partial \\nu } \\frac { \\partial \\varphi } { \\partial \\nu } d \\sigma = \\mu ( \\lambda ) \\int _ { \\partial \\Omega } u \\varphi d \\sigma \\ , , \\ \\ \\ \\forall \\varphi \\in H ^ 2 ( \\Omega ) , \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} | \\partial _ { r } N _ { 2 } ( f _ { v _ { 5 } } ) ( t , r ) | & \\leq \\frac { C } { r ^ { 3 } t ^ { 3 } \\log ^ { 4 N + 7 b - 2 } ( t ) } + \\frac { C \\lambda ( t ) \\log ( r ) } { r ^ { 3 } t ^ { 3 / 2 } | t - r | \\log ^ { 3 b - 1 + \\frac { 5 N } { 2 } } ( t ) } + \\frac { C \\log ^ { 2 } ( r ) } { r ^ { 2 } t ^ { 3 / 2 } ( t - r ) ^ { 2 } \\log ^ { 3 b - 1 + \\frac { 5 N } { 2 } } ( t ) } \\\\ & + \\frac { C \\log ^ { 3 } ( r ) } { r ^ { 2 } ( t - r ) ^ { 4 } } , t > r \\geq \\frac { t } { 2 } \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} \\ ! { \\kappa \\Sigma ^ 1 _ 1 B o r L o c } | = \\nexists x \\in X , \\ , a _ i \\in A _ i , \\ , b _ j \\in B _ j \\ , ( \\bigwedge _ i ( a _ i \\le x ) \\wedge \\bigwedge _ j ( x \\le b _ j ) ) . \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} \\P \\left [ | | X _ { t } - \\vec { v } t | | \\geq \\epsilon u , t \\in [ 0 , u ] \\right ] \\leq \\sum _ { i = 1 } ^ { d } \\P \\left [ | X _ { t } ^ { i } - v _ { i } t | \\geq \\frac { \\epsilon } { d } u , t \\in [ 0 , u ] \\right ] , \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = n \\} \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} s _ \\gamma = \\det ( h _ { \\gamma _ i + j - i } ) _ { 1 \\le i , j \\le \\ell } \\ , . \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} \\mathbb { P } _ \\eta \\left ( \\mathcal { T } _ t ( y ) \\geq \\frac { 1 - p } { 4 } t , \\mathcal { T } _ t ( 0 ) \\leq \\frac { 1 - p } { 4 } \\delta ^ d t \\right ) \\qquad \\\\ \\leq \\sum _ { i = 1 } ^ d \\mathbb { P } _ \\eta \\left ( \\mathcal { T } _ t ( y ^ { ( i - 1 ) } ) \\geq \\delta ^ { i - 1 } \\frac { 1 - p } { 4 } t , \\mathcal { T } _ t ( y ^ { ( i ) } ) \\leq \\frac { 1 - p } { 4 } \\delta ^ i t \\right ) . \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} \\left | \\left < T x , y \\right > \\right | = \\left | \\left < U | T | x , y \\right > \\right | = \\left | \\left < | T | ^ { \\frac { 1 } { 2 } } x , | T | ^ { \\frac { 1 } { 2 } } U ^ * y \\right > \\right | . \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} N = n _ 1 + \\cdots + n _ t M = m _ 1 + \\cdots + m _ t . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} \\langle D , H _ { 1 2 3 } \\rangle = 8 - ( 3 + 3 + 3 ) = - 1 < 0 . \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} Y _ p = \\sum _ { p \\leq i < k \\leq D } L ^ 2 _ { i k } - \\bigg ( \\sum _ { i = p } ^ { D } x _ i ^ 2 \\bigg ) \\bigg ( \\sum _ { k = p } ^ { D - 1 } \\frac { \\alpha _ i } { x _ k ^ 2 } \\bigg ) , ~ ~ ~ p = 1 , 2 , \\cdots , D - 1 , \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} U _ R ^ { ( \\alpha ) } = \\left \\{ f \\in \\overline { B } _ R ^ { ( \\alpha ) } : ~ \\| f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = 1 \\right \\} \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} \\frac { 1 } { \\| f \\| _ 2 \\sqrt { \\mu ( X ) } } \\int _ { X } | f ( x ) | \\ , d \\mu ( x ) = 1 - \\frac { c _ f } { 2 } \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} \\bar { \\rho } ( L _ { - 1 } ) = z ^ 2 , \\bar { \\rho } ( L _ 0 ) = \\frac 1 { 2 } z \\partial + \\frac 1 4 , \\bar { \\rho } ( L _ { 1 } ) = \\frac 1 { 4 } \\partial ^ 2 \\ , . \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} \\begin{cases} \\dd J _ i ( x ) \\approx | x | ^ { - \\gamma } & \\mbox { i f $ i \\in \\{ 1 , . . . , m _ 0 \\} $ a n d $ \\mu _ i \\not = 0 $ } , \\\\ [ 2 m m ] \\dd J _ i ( x ) \\leq C | x | ^ { - \\omega } & \\mbox { i f $ i \\in \\{ 1 , . . . , m _ 0 \\} $ a n d $ \\mu _ i = 0 $ } . \\end{cases} \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} | \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( t , \\xi ) | & \\leq \\frac { C \\log ^ { 3 } ( t ) } { t ^ { 2 } } , \\frac { t } { 2 } \\leq \\frac { 1 } { \\xi } \\leq t - \\sqrt { t } \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} 4 9 j _ { n } ^ { ( 3 ) } = 4 3 K _ { n } ^ { ( 3 ) } + 8 K _ { n - 1 } ^ { ( 3 ) } + 3 6 K _ { n - 2 } ^ { ( 3 ) } , \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} q ^ { \\prime } = q \\frac { ( n + 1 ) k } { n ( k + 1 ) } , \\qquad \\psi ^ { \\prime } = \\Phi _ { k , n } ( \\psi ) \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} \\{ s \\in S : \\rho _ q ( s ) \\in \\widehat { A } \\} = \\rho _ q ^ { - 1 } ( A ) \\in p . \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} \\bar { \\mathcal { R } } _ { \\epsilon } = \\inf \\left \\{ u \\in \\left [ \\frac { 2 } { \\epsilon } , \\infty \\right ) \\cap \\N : | | X _ { t } - t \\vec { v } | | \\leq \\epsilon t , t \\geq u \\right \\} \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} D _ { G \\slash K } : = \\sum _ { i = 1 } ^ { \\dim ( \\mathfrak { p } ) } X _ i \\otimes c ( X _ i ) , \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( V _ 2 ^ { ( 1 ) } - V _ 2 ^ { ( 2 ) } ) v ^ { ( 1 ) } v ^ { ( 2 ) } v ^ { ( 3 ) } d x = 0 \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} R _ { 1 2 1 2 } = \\frac { \\left ( 3 { { e } ^ { 3 q _ 3 + 2 q _ 2 } } + 2 { { e } ^ { 2 q _ 3 + 3 q _ 2 } } - 2 { { e } ^ { q _ 3 + 4 q _ 2 } } - 2 { { e } ^ { 5 q _ 2 } } \\right ) { { e } ^ { 2 q _ 1 } } + \\left ( - { { e } ^ { 2 q _ 3 + 2 q _ 2 } } + { { e } ^ { q _ 3 + 3 q _ 2 } } + { { e } ^ { 4 q _ 2 } } \\right ) { { e } ^ { 3 q _ 1 } } } { ( e ^ { q _ 1 } + e ^ { q _ 2 } ) \\Delta _ 1 } , \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} | | | \\P | | | ^ 2 _ k : = \\sum _ { g \\in \\Gamma } | | | P ( g ) | | | ^ 2 ( 1 + | g | ) ^ { 2 k } \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} P _ L ( \\tilde { r } ; 0 ) & = - P _ { \\rm a f f i n e } \\left ( - \\tilde { r } ; \\lambda _ L , \\omega _ L , \\frac { \\beta } { a _ 2 } \\right ) , \\\\ T _ L ( \\tilde { r } ; 0 ) & = T _ { \\rm a f f i n e } \\left ( - \\tilde { r } ; \\lambda _ L , \\omega _ L , \\frac { \\beta } { a _ 2 } \\right ) , \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} \\begin{cases} D ^ { \\alpha _ 2 } _ 0 y ( t ) = f ( t , y ( t ) , y ' ( t ) ) , ~ ~ t \\in [ 0 , b ] , \\\\ y ( 0 ) = y _ 0 , \\ y ( b ) = y _ b , \\end{cases} \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} \\limsup _ { m \\to \\infty } \\mathbb { P } _ { n _ { l _ m } } \\left ( \\left | \\log ( Y _ { n _ { l _ m } } ) - \\sum _ { k = 1 } ^ { K } \\frac { 2 \\mu _ { k } W _ { n _ { l _ m } , k } - \\mu _ { k } ^ { 2 } } { 2 \\sigma _ { k } ^ { 2 } } \\right | \\ge \\epsilon \\right ) \\le \\delta . \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} G _ m - \\sum _ { i = L + 1 } ^ { m - 1 } G _ i \\geq H _ m - \\sum _ { i = L + 1 } ^ { m - 1 } H _ i \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} g ( f ) = \\kappa \\bigg ( \\prod _ { 1 \\leq i < j \\leq \\ell } \\left ( 1 - \\frac { z _ i } { z _ j } \\bigg ) \\cdot f \\right ) \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} a _ 0 ( n _ i a _ i a _ j ' - n _ j a _ j a _ i ' ) = ( i n _ j - j n _ i ) a _ i a _ j , \\\\ ( \\ a _ 0 ( i a _ i a _ j ' - j a _ j a _ i ' ) = ( i n _ j - j n _ i ) a _ i a _ j a _ 0 ' ) . \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} \\psi _ { \\mu } ( z ) = \\int _ { \\mathbb { R } _ { + } } \\frac { t z } { 1 - t z } \\ , d \\mu ( t ) , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} \\mathcal { \\tilde { K } } _ k = \\bigcap _ { \\forall R _ j \\in \\gamma _ k } \\mathcal { K } _ j . \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} H _ { L + 2 } = H _ { L + 1 } + H _ { m + 2 } + \\dots + H _ { 3 } + 2 N & \\leq 1 + \\sum _ { i = 1 } ^ { L + 1 } H _ { i } \\\\ \\iff H _ { m + 2 } + \\dots + H _ { 3 } + 2 N & \\leq 1 + \\sum _ { i = 1 } ^ { L } H _ { i } . \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} | J _ { 0 } + e ^ { - \\lambda \\tau } J _ { \\tau } - \\lambda I | = 0 . \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} \\{ \\partial ( n _ { j _ 1 } , . . . , n _ { j _ { 2 ^ \\ell - 1 } } , n ) \\ , | \\ , 1 \\leq j _ { 1 } < j _ 2 < j _ 3 < \\cdots < j _ { 2 ^ \\ell - 1 } \\leq M \\} \\cap E = \\emptyset . \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 2 } v _ { 2 } ( t , r ) & = \\frac { - c _ { b } } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\sin ^ { 2 } ( \\theta ) d \\theta \\int _ { 0 } ^ { \\infty } d \\xi r \\left ( \\frac { \\sin ( \\xi t _ { + } ) } { t _ { + } ^ { 4 } } + \\frac { \\sin ( \\xi t _ { - } ) } { t _ { - } ^ { 4 } } \\right ) \\partial _ { \\xi } ^ { 4 } \\left ( \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) \\cdot \\xi ^ { 3 } } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\right ) \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} z = 0 \\mbox { o n } \\Gamma . \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} \\dim \\mu _ { \\l _ 0 , \\tau } = \\min \\Big \\{ 1 , \\frac { H ( p ) } { \\log \\l _ 0 ^ { - 1 } } \\Big \\} \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} E _ { 7 ( 7 ) } ~ = ~ \\mathcal { N } _ { 6 } ^ { 7 ( 7 ) - } \\oplus E _ { 6 ( 6 ) } \\oplus s o ( 1 , 1 ) \\oplus \\mathcal { N } _ { 6 } ^ { 7 ( 7 ) + } \\newline \\mathbf { , } \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big \\| | x | ^ { - \\frac s 2 } u \\Big \\| _ 2 ^ 2 = - 2 I ( u ) . \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} G _ { m + 1 } - 2 G _ m + H _ m - \\sum _ { i = L + 1 } ^ { m - 1 } H _ i \\geq H _ { m + 1 } - 2 H _ m + H _ m - \\sum _ { i = L + 1 } ^ { m - 1 } H _ i = H _ { m + 1 } - \\sum _ { i = L + 1 } ^ { m } H _ i , \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} \\phi ( s ) = c _ 1 \\left ( ( s + \\lambda _ 1 ) ^ { \\alpha _ 1 } - \\lambda _ 1 ^ { \\alpha _ 1 } \\right ) + c _ 2 \\left ( ( s + \\lambda _ 2 ) ^ { \\alpha _ 2 } - \\lambda _ 2 ^ { \\alpha _ 2 } \\right ) . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} B ^ k ( A - A ^ 2 S ' ) ^ k & = B ^ k ( A ^ k - A ^ { k + 1 } S ' ) = B ^ k A ^ k - B ^ k A ^ { k - 1 } A ^ 2 S ' = B ^ k A ^ k - B ^ k A ^ { k - 1 } A ^ k { T ' } ^ { k - 1 } \\\\ & = B ^ k A ^ k - B ^ k A ^ { 2 k - 1 } { T ' } ^ { k - 1 } = B ^ k A ^ k - ( B ^ k A ^ { k } ) ^ k { T ' } ^ { k - 1 } = B ^ k A ^ k - ( B ^ k A ^ { k } ) ^ 2 S ' . \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} { \\rm c h } [ W _ { B _ 2 } ( \\Lambda _ 0 ) ] ( \\tau ) = { \\rm c h } [ W _ { A _ 2 } ( \\Lambda _ 0 ) ] ( \\tau ) \\cdot { \\rm c h } [ W _ { A _ 1 } ( \\Lambda _ 0 ) ] ( \\tau ) , \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} | \\partial _ { t } \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) \\left ( v _ { 1 } + v _ { 2 } + v _ { 3 } \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | & \\leq \\frac { C } { t ^ { 3 } \\log ^ { 2 b + 2 N + 1 } ( t ) } + \\frac { C } { t ^ { 2 } \\log ^ { 1 + b \\alpha + b + 2 N } ( t ) } \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} s ( v ) : = 2 ^ n e ^ { - \\| v \\| ^ 2 } v \\in H _ p ^ * \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} & { } _ 3 \\phi _ 2 \\ ! \\left [ \\begin{matrix} ( 1 - r ) ^ j , 1 - a , b \\\\ e , ( 1 - r ) ^ { j + 1 } ( 1 - a ) b / c e \\end{matrix} ; 1 - r , 1 - r \\right ] \\\\ & = \\frac { ( ( 1 - r ) / e , ( 1 - r ) ( 1 - a ) b / c e ; 1 - r ) _ j } { ( ( 1 - r ) ( 1 - a ) / e , ( 1 - r ) b / c e ; 1 - r ) _ j } \\ ; { } _ 3 \\phi _ 2 \\ ! \\left [ \\begin{matrix} ( 1 - r ) ^ j , 1 - a , c \\\\ c e / b , ( 1 - r ) ^ { j + 1 } ( 1 - a ) / e \\end{matrix} ; 1 - r , 1 - r \\right ] . \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} K ( \\Psi ; M ; \\mu ) = \\prod _ { j \\in M } ( 1 - L _ j ) \\prod _ { ( i , j ) \\in \\Delta ^ + \\setminus \\Psi } ( 1 - R _ { i j } ) k _ \\mu \\\\ = \\prod _ { j \\in M \\setminus y } ( 1 - L _ j ) \\prod _ { ( i , j ) \\in \\Delta ^ + \\setminus \\Psi } ( 1 - R _ { i j } ) ( k _ \\mu - k _ { \\mu - \\epsilon _ { y } } ) \\ , , \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} \\rho ( L _ i ) \\ , = \\ , h ( z ) ^ i ( \\rho ( L _ 0 ) + c _ i ( z ) ) i = - 1 , 1 \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} \\langle \\gamma ^ I , \\gamma ^ J \\rangle = \\langle \\gamma _ I , \\gamma _ J \\rangle = 0 , \\ \\langle \\gamma ^ I , \\gamma _ J \\rangle = \\delta ^ I _ J . \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\left ( \\frac { ( r ^ { 2 } - 1 - ( s - t ) ^ { 2 } ) } { \\sqrt { \\beta } ( 1 + r ^ { 2 } + ( s - t ) ^ { 2 } + \\sqrt { \\beta } ) } \\right ) = \\frac { - 1 } { 2 ( s - t ) ^ { 2 } } \\left ( 1 + O \\left ( \\frac { 1 + r ^ { 2 } } { ( s - t ) ^ { 2 } } \\right ) \\right ) \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( F \\left ( ( S ^ { \\varepsilon , L , K } _ n ) _ { n = 0 } ^ { \\lfloor L / \\varepsilon ^ 2 \\rfloor } \\right ) \\right ) = \\mathbf { E } \\left ( F \\left ( ( S ^ { \\varepsilon } _ n ) _ { n = 0 } ^ { \\lfloor L / \\varepsilon ^ 2 \\rfloor } \\right ) \\ : \\vline \\ : \\mathcal { A } _ { \\varepsilon , L , K } , \\ : S ^ \\varepsilon _ { \\lfloor L / \\varepsilon ^ 2 \\rfloor } > 0 \\right ) , \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} P _ L ( \\tilde { r } ; 0 ) & = - P _ { \\rm a f f i n e } \\left ( - \\tilde { r } ; \\lambda , \\omega , \\frac { \\beta } { a _ 2 } \\right ) , \\\\ T _ L ( \\tilde { r } ; 0 ) & = T _ { \\rm a f f i n e } \\left ( - \\tilde { r } ; \\lambda , \\omega , \\frac { \\beta } { a _ 2 } \\right ) , \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} \\lambda ^ 2 = \\frac { m } { n } \\frac { 1 - n ^ 2 } { 1 + m ^ 2 } \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} S _ t : = \\int _ 0 ^ t ( 1 - 2 \\eta _ s ) d s ; \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} D ^ m k _ u & = A _ u ^ { - 1 } \\left ( D k _ u ( R ) D ^ m r + D ^ m k _ u ( R ) \\left ( D R \\right ) ^ { \\otimes m } + \\mathcal { P } _ m ( k _ u , R ) \\right ) \\\\ & - A _ u ^ { - 1 } \\left ( D g _ u ( K { } { } ) D ^ m K { } { } + D ^ m g _ u ( K { } { } ) \\left ( D K { } { } \\right ) ^ { \\otimes m } + \\mathcal { P } _ m ( g _ u , K { } { } ) \\right ) \\\\ & = f _ { m , 2 } + A _ u ^ { - 1 } \\left ( D k _ u ( R ) D ^ m r + D ^ m k _ u ( R ) ( D R ) ^ { \\otimes m } - D g _ u ( K { } { } ) D ^ m K { } { } \\right ) , \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} \\alpha ^ 1 _ { d _ 1 - 1 } = \\bigg [ 2 \\sum _ { a = 1 } ^ { d _ 1 - 1 } J _ a + \\frac { d _ 1 - 2 } { 2 } + A + J _ 1 \\bigg ] ^ 2 - \\frac { ( d _ 1 - 2 ) ^ 2 } { 4 } , ~ ~ ~ J _ a = 0 , 1 , 2 , \\cdots , \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} \\langle f , 1 \\rangle & = \\langle \\theta _ { i _ 1 } \\cdots \\theta _ { i _ r } \\overline { \\theta } _ { j _ 1 } \\cdots \\overline { \\theta } _ { j _ s } f ^ \\prime , 1 \\rangle \\\\ & : = w ( i _ 1 , \\dots , i _ r ) \\delta _ { r , s } \\delta _ { i _ 1 , j _ r } \\cdots \\delta _ { i _ r , j _ 1 } \\langle f ^ \\prime , 1 \\rangle \\\\ & = w ( i ) \\delta _ { r , s } \\delta _ { i , j ^ T } \\langle f ^ \\prime , 1 \\rangle \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} \\begin{cases} D ^ { \\alpha _ 1 } _ 0 y _ s ( t ) = w _ s ( t ) , & y _ s ( 0 ) = - b _ 1 / a _ 1 , \\\\ D ^ { 1 - \\alpha _ 1 } _ 0 w _ s ( t ) = z _ s ( t ) , & w _ s ( 0 ) = 0 , \\\\ D ^ { \\alpha _ 2 - 1 } _ 0 z _ s ( t ) = f _ s ( t , y ( t ) , w ( t ) ) , & z _ s ( 0 ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} \\varepsilon ( p ) & : = \\frac { \\beta p } { 2 ( p - 1 ) } , \\\\ \\gamma ( p ) & : = \\frac { 2 \\chi } { p } - \\frac { K \\beta p } { 2 ( p - 1 ) } . \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} \\vec { a } = D \\vec { d } . \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} \\| \\Delta _ q \\theta \\| _ { L ^ 2 } ^ 2 \\leq b _ q 2 ^ { - 2 q s } \\| \\theta \\| _ { H ^ s } ^ 2 . \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} \\psi ( \\beta ) \\tau \\psi ( \\beta ^ { - 1 } ) = \\psi ( \\beta ) ^ j \\tau . \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} L _ i [ U ] = \\left \\langle M L _ i [ U ] \\right \\rangle . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} \\frac { d } { d \\sigma } A _ N ( \\sigma ) = m . \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} \\mathcal { C } _ { J , K } ^ { \\exists } & : = \\left \\{ \\eta \\in \\mathcal { C } _ { J , K } : \\ : \\mbox { t h e r e e x i s t s a B B S ( $ J $ , $ K $ ) c a r r i e r f o r $ \\eta $ } \\right \\} ; \\\\ \\mathcal { C } _ { J , K } ^ { \\exists ! } & : = \\left \\{ \\eta \\in \\mathcal { C } _ { J , K } : \\ : \\mbox { t h e r e e x i s t s a u n i q u e B B S ( $ J $ , $ K $ ) c a r r i e r f o r $ \\eta $ } \\right \\} . \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} L u _ n = \\sum _ { 2 \\leq j + \\alpha \\leq n } \\ ! \\ ! a _ { j , \\alpha } ( t , x ) \\sum _ { | \\vec { k } | + | \\vec { m } | = n } u _ { k _ 1 } \\cdots u _ { k _ j } \\frac { \\partial u _ { m _ 1 } } { \\partial x } \\cdots \\frac { \\partial u _ { m _ { \\alpha } } } { \\partial x } \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} x : = \\left ( \\frac { \\delta } { m ( y ) } \\right ) ^ \\frac { 1 } { p } y . \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} \\left ( A + P - \\varkappa _ { \\mathsf { k } } \\right ) ^ { \\mathsf { n } ^ { \\ast } } \\mathsf { q } _ { \\mathsf { k } } = 0 . \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} \\partial _ t h = Q ( f , h ) + Q ( h , \\mu ) . \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} \\psi _ { k , j } ^ { \\ast } ( Q ^ { \\prime } ) & \\leq \\Phi _ { k , n } ( \\psi _ { n , 1 } ^ { \\ast } ( Q ) ) = ( \\psi _ { n , 1 } ^ { \\ast } ( Q ) - 1 ) \\frac { k + 1 } { k } \\frac { n } { n + 1 } + 1 , 1 \\leq j \\leq k - n , \\\\ \\psi _ { k , k - n + j } ^ { \\ast } ( Q ^ { \\prime } ) & \\leq \\Phi _ { n , k } ( \\psi _ { n , j } ^ { \\ast } ( Q ) ) = ( \\psi _ { n , j } ^ { \\ast } ( Q ) - 1 ) \\frac { k + 1 } { k } \\frac { n } { n + 1 } + 1 , 1 \\leq j \\leq n . \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} [ \\omega ' ] _ { A _ { p ' } } = [ \\omega ] ^ { 1 / ( p - 1 ) } _ { A _ p } . \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} \\sup _ { 0 < t \\leq 1 - \\frac { 1 } { n } } \\left \\| \\max _ { 1 \\leq i , j \\leq d } \\left | c ( \\nu ) ^ { i j } _ { t + \\frac { 1 } { n } } - c ( \\nu ) ^ { i j } _ t \\right | \\right \\| _ 2 = O ( n ^ { - \\gamma } ) \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} \\Pi _ s \\Pi ( W ) = ( \\Pi _ s \\Pi ( W ) ) ^ T = - F _ j \\ , g ^ { i j } \\ , \\langle \\Pi ( \\nabla _ { F _ i } F _ s ) , W \\rangle \\ , . \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} \\psi : \\bigoplus _ { i = 1 , 2 , 3 } H ^ 0 ( D - E _ i ) \\to H ^ 0 ( D ) \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} & V a r \\left [ \\left ( \\frac { 1 } { n } \\right ) ^ { k } \\sum _ { w \\in \\mathcal { W } _ { 1 , 2 k } } X _ { w } \\right ] \\\\ & = 4 k ^ { 2 } \\psi _ { 2 k } ^ { 2 } \\left ( 1 + O \\left ( \\frac { k ^ 2 } { n } \\right ) \\right ) + \\mathbf { E } \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } ( ( b , j ) ) = ( b - a r + i , j + i r - a r ^ 2 + 1 ) \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{gather*} \\varepsilon ^ { - ( m ) } _ k = \\begin{cases} 1 k = ( 0 , \\dots , 0 , \\underset { m } { - 1 } , 0 , \\dots , 0 ) , \\\\ 0 k \\neq ( 0 , \\dots , 0 , \\underset { m } { - 1 } , 0 , \\dots , 0 ) \\end{cases} \\end{gather*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\mathcal { R } ^ { \\Sigma } _ { A , B , P } ( \\rho ) \\{ \\mathrm { P a i r } _ { A , B } \\} & = \\rho \\{ \\mathrm { P a i r } _ { A , B } ^ { - 1 } \\} \\{ \\mathrm { P a i r } _ { A , B } \\} \\\\ & = \\rho \\{ \\mathrm { P a i r } _ { A , B } ^ { - 1 } \\bullet \\mathrm { P a i r } _ { A , B } \\} \\\\ & = \\rho \\{ \\mathrm { i d } _ { \\Sigma ( \\Sigma ( \\Gamma , A ) , B ) } \\} \\\\ & = \\rho . \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} A _ { r , \\ell } ( c , d ) : = \\sum _ { i = 0 } ^ { r - 1 } \\theta _ { \\ell } ( \\lambda _ { i } , \\mu _ i ) , \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} y ( t ) = \\frac { F ( t ) } { \\alpha } - \\int _ { t } ^ { \\infty } \\frac { F ( s ) } { \\alpha } r ( - t , - s ) d s \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} r & = A _ c k _ c + g _ c \\circ K { } { } - k _ c \\circ ( A _ c + r ) , \\\\ k _ u & = A _ u ^ { - 1 } k _ u \\circ ( A _ c + r ) - A _ u ^ { - 1 } g _ u \\circ K { } { } , \\\\ k _ s & = A _ s k _ s \\circ ( A _ c + r ) ^ { - 1 } + g _ s \\circ K { } { } \\circ ( A _ c + r ) ^ { - 1 } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} \\widehat { q } _ 1 = x _ 1 , \\widehat { q } _ 2 = x _ 2 , \\widehat { p } _ 1 = - i \\frac { \\partial } { \\partial x _ 1 } , \\widehat { p } _ 2 = - i \\frac { \\partial } { \\partial x _ 2 } , \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} M _ 1 = \\frac { 1 } { \\lambda ^ * } \\boldsymbol { \\pi } { \\mathbf 1 } = \\frac { 1 } { \\lambda ^ * } , M _ 2 = 2 \\frac { 1 } { \\lambda ^ * } \\boldsymbol { \\pi } ( - C ) ^ { - 1 } { \\mathbf 1 } . \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} \\sigma \\bigl ( F ( t , \\sigma ( Y ) ) \\bigr ) = F ( t , Y ) \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} \\dot { x } = \\beta \\{ P _ { X } [ x - \\alpha F ( x ) ] - x \\} \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} \\psi _ { k , k - n + 1 } ^ { \\ast } ( Q ^ { \\prime } ) = \\frac { L _ { k , k - n + 1 } ^ { \\ast } ( q ^ { \\prime } ) } { q ^ { \\prime } } \\leq \\min _ { 1 \\leq u \\leq k - n + 1 } \\frac { L _ { k , R _ { u } } ^ { \\ast } ( q ^ { \\prime } ) } { q ^ { \\prime } } \\leq \\Phi _ { k , n } ( \\psi _ { n , 1 } ^ { \\ast } ( Q ) ) \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} u _ n - u _ n ^ \\ast & = ( e _ n - l ( e _ n ) ) - ( e _ n - l ( e _ n ) ) ^ \\ast = e _ n - l ( e _ n ) - e _ n ^ \\ast + l ( e _ n ) = e _ n - e _ n ^ \\ast \\to e - e ^ \\ast = 0 , \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} E ( \\mu ) \\dot { x } ( t , \\mu ) & = f ( x ( t , \\mu ) , \\mu ) \\\\ [ 1 e x ] y ( t , \\mu ) & = g ( x ( t , \\mu ) , \\mu ) \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} q _ 0 & = - \\lambda \\left ( \\lambda ^ 2 + 7 \\omega ^ 2 \\right ) , \\\\ q _ 1 & = \\omega \\left ( \\lambda ^ 2 + 3 \\omega ^ 2 \\right ) , \\\\ q _ 2 & = - 2 \\lambda \\omega ^ 2 , \\\\ q _ 3 & = 6 \\omega ^ 3 . \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} \\mathcal { F } f ( \\omega ) : = \\int _ { \\mathbb { R } ^ d } f ( t ) e ^ { - 2 \\pi i t \\cdot \\omega } d t \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} L ( H , x ) = \\min \\left \\{ \\bigcup _ { k = 0 } ^ x L ( H _ 1 , k ) \\oplus L ( H _ 2 , x - k ) \\right \\} x = 0 , 1 , \\ldots , B , \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial \\psi } { \\partial X ^ 1 } , \\cdots , \\frac { \\partial \\psi } { \\partial X ^ n } \\right ) - \\frac { \\mathbf { i } } { 2 \\pi r } ( ( q '^ t \\mathcal { B } ) _ 1 - ( q ^ t \\mathcal { B } ) _ 1 , \\cdots , ( q '^ t \\mathcal { B } ) _ n - ( q ^ t \\mathcal { B } ) _ n ) \\psi = 0 , \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} \\Phi ( z ) = \\gamma z \\exp H _ { \\sigma } ( z ) , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { l } \\exp ( i \\xi ^ 2 ) \\widehat h ( - k + \\xi / \\sqrt t ) d \\xi = \\int _ { 0 } ^ { l } \\exp ( i \\xi ^ 2 ) \\widehat h ( - k + \\xi / \\sqrt t ) \\mu _ 0 d \\xi + \\int _ { 0 } ^ { l } \\exp ( i \\xi ^ 2 ) \\widehat h ( - k + \\xi / \\sqrt t ) ( 1 - \\mu _ 0 ) d \\xi \\ , . \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} \\delta ( \\Delta { f } ) & = \\frac { d } { d t } ( g ^ { i j } f _ { i j } ) - \\frac { d } { d t } ( g ^ { i j } \\Gamma _ { i j } ^ k f _ k ) \\\\ & = 2 u \\langle h , f \\rangle + \\Delta \\dot f + 2 u \\langle \\nabla H , \\nabla f \\rangle + 2 \\ , h ( \\nabla { u } , \\nabla { f } ) - 2 H \\langle \\nabla u , \\nabla f \\rangle , \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} L ( H , x ) = \\begin{cases} \\{ ( u ^ 1 ( a ) , u ^ 2 ( a ) ) \\} , & x = 0 \\\\ \\{ ( 0 , 0 ) \\} , & \\end{cases} \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} \\begin{array} { l } \\Delta _ x ( f , a ) = \\| ( \\widehat { x } - a \\widehat { I } ) f \\| _ { L ^ 2 ( \\mathbb { R } ) } = \\left ( \\int _ { \\mathbb { R } } ( x - a ) ^ 2 | f ( x ) | ^ 2 d x \\right ) ^ { \\frac { 1 } { 2 } } \\\\ \\\\ \\Delta _ { \\xi } ( f , b ) = \\| ( \\widehat { \\xi } - b \\widehat { I } ) f \\| _ { L ^ 2 ( \\mathbb { R } ) } = \\left ( \\int _ { \\mathbb { R } } ( \\xi - b ) ^ 2 | \\widetilde { f } ( \\xi ) | ^ 2 d \\xi . \\right ) ^ { \\frac { 1 } { 2 } } \\end{array} \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} \\inf _ { | \\xi | = 1 } \\int _ { | y | \\leq N _ \\nu } | y \\cdot \\xi | ^ 2 \\nu ( d y ) > 0 . \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} \\hat { J } ^ { \\sigma } _ { p , v } ( s ; Q ) = r ^ { 2 ( 1 - \\sigma ) } \\hat { J } ^ { \\sigma } _ { p , v ^ { ( r ) } } ( s / r ; r ^ { 2 ( \\sigma - 1 ) } Q ) \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} K _ 2 \\geq c _ 0 + \\frac { C ^ 2 } { ( c _ 0 / 2 ) ^ \\alpha } \\sum _ { i = 1 } ^ m \\mu _ i . \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} \\mathcal { M } _ { J , K } ^ { r e v } & : = \\left \\{ \\mu _ { J , K } \\in \\mathcal { M } _ { J , K } : \\ : \\mu _ { J , K } ^ { \\otimes \\mathbb { Z } } \\in \\mathcal { P } _ { J , K } ^ { r e v } \\right \\} , \\\\ \\mathcal { M } _ { J , K } ^ { i n v } & : = \\left \\{ \\mu _ { J , K } \\in \\mathcal { M } _ { J , K } : \\ : \\mu _ { J , K } ^ { \\otimes \\mathbb { Z } } \\in \\mathcal { P } _ { J , K } ^ { i n v } \\right \\} ; \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} X _ { \\pm } ( x ) : = a \\otimes \\left [ - 2 \\pi \\ , \\mathsf { M } ^ { \\mathbb { R } } \\mp \\frac { 1 } { b } \\frac { \\partial \\ ; } { \\partial r } \\pm \\frac { \\partial \\ ; } { \\partial \\Theta } \\right ] ( \\Phi ) + \\left [ 2 \\pi \\ , \\mathsf { M } ^ { \\mathbb { Z } } \\pm \\frac { \\partial \\ ; } { \\partial \\Theta } \\right ] ( a ) \\otimes \\Phi \\ \\in \\ C _ { c } ^ { \\infty } ( \\mathcal { A } _ { \\theta } ) \\odot C _ { c } ^ { \\infty } ( \\mathcal { Z } _ { b } ) , \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} a \\# b \\sim \\sum _ { k = 0 } ^ \\infty \\left ( \\frac { i } { 2 } \\right ) ^ k ( a \\# b ) _ k ( a \\# b ) _ k = \\sum _ { | \\alpha | + | \\beta | = k } \\frac { 1 } { \\alpha ! \\beta ! } ( - 1 ) ^ { | \\beta | } ( \\partial ^ \\alpha _ x \\partial ^ \\beta _ \\xi a ) ( \\partial ^ \\beta _ x \\partial ^ \\alpha _ \\xi b ) \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} \\operatorname { I n d _ c } ( D ) : = [ P _ { Q } ] - [ e _ 1 ] \\in K _ 0 ( \\mathcal { A } _ G ^ c ( M , E ) ) \\equiv K _ 0 ( \\mathcal { S } _ G ^ c ( M , E ) ) \\ ; \\ ; \\ ; \\ ; e _ 1 : = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} c ( A , B , C ; K , L , M ) = \\frac { A K } { K B } \\cdot \\frac { B L } { L C } \\cdot \\frac { C M } { M A } = 1 . \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} \\widetilde { q } = 0 , \\ \\ \\widetilde { q } _ { \\Gamma } = 0 , \\ \\ \\nabla \\widetilde { p } = 0 . \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} W ( h , k ) = \\phi ( h ) \\psi ( k ) - \\psi ( h ) \\phi ( k ) . \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} \\kappa ( x , y , z , t ) = \\mu \\cdot \\kappa ( x ' , y ' , z ' , t ' ) \\cdot \\mu ^ { - 1 } \\ . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} G _ m \\leq 1 + \\sum ^ { m - 1 } _ { i = 1 } G _ i = 1 + \\sum ^ { L } _ { i = 1 } G _ i + \\sum _ { i = L + 1 } ^ { m - 1 } G _ i = 1 + \\sum ^ { L } _ { i = 1 } H _ i + \\sum _ { i = L + 1 } ^ { m - 1 } G _ i . \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} h _ t ( x , \\xi ) = \\frac { 1 } { ( \\cosh t ) ^ n } e ^ { - ( \\| x \\| ^ 2 + \\| \\xi \\| ^ 2 ) \\tanh t } \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} ( x ^ { \\# } , y ) = \\frac { ( x , x , y ) } { 2 } . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} I _ { \\ell } = D _ \\ell ( ( n _ k ) _ { k \\in \\N } ) . \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} P ^ r _ \\mathcal { M } ( z ) = \\arg \\min _ { v \\in T _ z \\mathcal { M } } \\norm { z - v } ^ 2 _ r . \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ s _ p u + c ( x ) u \\geq 0 , & x \\in \\Omega , \\\\ u > 0 , & x \\in \\Omega , \\\\ u = 0 , & x \\in \\mathbb { R } ^ n \\setminus \\Omega , \\end{cases} \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} a _ j = \\sqrt { \\mu _ j ( \\lambda ) + b } \\cdot \\hat a _ j \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} P _ { n } ( x ) _ { i } & = \\sqrt { \\tfrac { n } { n + 2 } } ( x _ { n + 1 } + \\theta ) x _ { i } , \\\\ \\\\ 0 & = n x _ { n + 1 } ^ { 2 } - 2 \\theta x _ { n + 1 } - | x | ^ { 2 } = ( n + 1 ) x _ { n + 1 } ^ { 2 } - 2 \\theta x _ { n + 1 } - 1 . \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} P ( \\varepsilon z ) = \\sum _ { k = 0 } ^ { m / 2 } \\sum _ { A \\in \\mathcal { A } _ k } \\sum _ { l = k } ^ { m / 2 } \\sum _ { \\gamma \\in \\Gamma _ { A , l } } \\sum _ { \\beta \\in B _ { A , l } } x _ { 2 \\beta + 2 \\gamma + 1 _ A } \\varepsilon _ A z ^ { 2 \\beta + 2 \\gamma + 1 _ A } . \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} \\partial U _ t \\cap M _ { \\pm } = \\bar { \\Sigma } _ t ^ { \\pm , e x t } \\cup \\bigcup _ { p \\in S i n g ( \\Sigma ) } \\bar { \\Sigma } _ { t , p } ^ { \\pm , i n t } \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} & \\nabla _ { \\ ! x } \\cdot \\Big ( \\big \\{ \\beta _ \\varepsilon ( v ) + [ v \\ ! - \\ ! \\beta _ \\varepsilon ( v ) ] \\eta _ \\varepsilon ( x ) \\big \\} \\psi \\Big ) = 0 , \\\\ [ 5 p t ] & \\nabla _ { \\ ! v } \\cdot \\big ( \\mathbf { B } \\psi \\big ) = \\mathbf { B } \\cdot \\nabla _ { \\ ! v } \\psi + ( \\nabla _ { \\ ! v } \\cdot \\mathbf { B } ) \\psi = 0 , \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} F _ J ( x , y ; \\mu ) = \\begin{bmatrix} f _ J ( x , y ; \\mu ) \\\\ g _ J ( x , y ; \\mu ) \\end{bmatrix} , \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} \\| u \\| ^ * _ { k ; \\Omega } : = \\| u \\| _ { C ^ k _ * ( \\Omega ) } : = \\sum _ { j = 0 } ^ k \\sup _ { \\Omega } \\rho ^ { j - 1 } \\cdot | \\nabla ^ j u | \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} \\vec h _ a = c W _ a ( g , h ) \\vec f _ a + c W _ a ( h , f ) \\vec g _ a \\quad c = - 1 / W _ a ( f , g ) . \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\| u ( \\cdot , t ) \\| & \\le \\sup _ { t \\in \\mathbb R ^ + } \\int _ 0 ^ t \\omega ( t - s , \\lambda _ 1 ) \\| p \\| _ \\infty d s \\\\ & = \\| p \\| _ \\infty \\lambda _ 1 ^ { - 1 } \\sup _ { t \\in \\mathbb R ^ + } ( 1 - \\omega ( t , \\lambda _ 1 ) ) = \\| p \\| _ \\infty \\lambda _ 1 ^ { - 1 } . \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} \\| \\partial _ X u \\| _ { C ^ s } \\leq C ( \\| \\nabla u \\| _ { L ^ \\infty } \\| X \\| _ { C ^ s } + \\| \\partial _ X \\omega \\| _ { C ^ { s - 1 } } ) . \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} T _ g \\rho _ H ( \\tau ) = \\rho _ H ( \\tau ) T _ { g \\circ \\alpha ( \\tau ) } . \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} & ~ ~ ~ ~ ~ ~ ~ G ~ ~ \\underbrace { \\Rightarrow } _ { ~ \\ref { t h e o r e m 2 } } ~ ~ \\tilde { G } ~ \\underbrace { \\Rightarrow } _ { } ~ G _ R \\\\ & { ( \\tilde { G } ) } ~ \\leq ~ ( G _ R ) . \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} L \\varphi ( x ) = \\mathrm { P . V . } \\int _ { \\R ^ d } ( \\varphi ( y ) - \\varphi ( x ) ) j ( | x - y | ) d y , \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} & \\Bigl | \\int _ { \\R ^ 3 \\times \\R ^ 3 } { h ( \\tau _ 1 + \\tau _ 2 , \\xi _ 1 + \\xi _ 2 , \\eta _ 1 + \\eta _ 2 ) f ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) g ( \\tau _ 2 , \\xi _ 2 , \\eta _ 2 ) } d \\sigma _ 1 d \\sigma _ 2 \\Bigr | \\\\ & \\lesssim A ^ { \\frac { 1 } { 2 } } N _ 1 ^ { - 2 } ( L _ 0 L _ 1 L _ 2 ) ^ { \\frac { 1 } { 2 } } \\| f \\| _ { L ^ 2 } \\| g \\| _ { L ^ 2 } \\| h \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} ( G ) = ( G ^ \\prime ) . \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} P _ { x } ( \\tau _ { B ( x , r ) } ^ N \\le t ) = P _ { x } ( \\tau _ { B ( x , r ) } \\le t ) . \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} L = \\phi \\left [ c + c \\frac { L [ \\phi ^ { - 1 } ] ' ( L ) } { \\phi ^ { - 1 } ( L ) } \\right ] , \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} \\hat { H } = - \\sum _ { i = 1 } ^ { D } \\frac { \\partial ^ 2 } { \\partial x _ i ^ 2 } + \\omega ^ 2 r ^ 2 + \\frac { 1 } { x _ 1 ^ 2 + x _ 2 ^ 2 } F ^ 1 _ 1 ( \\phi ^ 1 _ 1 ) \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} R _ k ^ { } = \\log _ 2 ( 1 + \\gamma _ k ) ~ ~ , \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} C ^ { 2 } . { h } ^ { n } & \\equiv a _ { n } h ^ { n } + a _ { n - 2 } h ^ { n - 2 } + \\cdots + a _ { 0 } \\\\ \\\\ a _ { n } & = - 4 n ^ { 2 } \\\\ k & = 2 , 4 , \\ldots , k - 3 \\ ( k - 2 ) \\\\ a _ { n - k } & = - \\frac { 1 } { 2 } \\bigg ( - \\mu { n \\choose k } 4 ^ { k } + 2 { n \\choose k + 1 } 4 ^ { k + 1 } + { n \\choose k + 2 } 4 ^ { k + 2 } \\bigg ) . \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} E _ 1 & = \\{ ( x _ 1 , x _ 2 , x _ 3 ) \\in R : b ( x _ 1 , x _ 2 , x _ 3 ) \\geq m _ { \\tilde R } ( b ) \\} , \\\\ E _ 2 & = \\{ ( x _ 1 , x _ 2 , x _ 3 ) \\in R : b ( x _ 1 , x _ 2 , x _ 3 ) < m _ { \\tilde R } ( b ) \\} . \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} \\nabla _ A \\Bar { U } ( x , t ) = R \\nabla _ A U ( x , - t ) + i \\bigl ( A ( x , t ) - R A ( x - t ) \\bigr ) U ( x , - t ) , \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} 0 \\in \\beta _ { \\ell } - \\alpha _ { \\ell } + \\lambda \\mu _ { \\ell } \\partial ( | \\beta _ { \\ell } | ) \\quad \\forall \\ell = 1 , \\ldots , { d } , \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} P _ { } ^ e = \\sum _ { q \\in Y } P _ q ^ e \\prod _ { k \\in Y , k \\neq q } P _ k ^ c + \\sum _ { q \\in Y } P _ q ^ c \\prod _ { k \\in Y , k \\neq q } P _ k ^ e , \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} w = w - H w + H w = x - y + H w = x \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} \\langle H , H \\rangle & = n - 1 , \\\\ \\langle H , E _ i \\rangle & = 0 , \\\\ \\langle E _ i , E _ j \\rangle & = - \\delta _ { i , j } . \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} 1 = \\langle f _ { n + 1 } | S f _ n \\rangle ^ 2 \\frac { \\gamma _ { n + 1 } ^ 2 } { | \\langle f _ { n + 1 } | 1 \\rangle | ^ 2 } \\sum _ { p = 0 } ^ \\infty \\frac { | \\langle f _ p | 1 \\rangle | ^ 2 } { ( \\lambda _ p - \\lambda _ n - 1 ) ^ 2 } \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} I _ 4 ( Q ) = - ( p ^ 0 q _ 0 ) ^ 2 \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} P _ { M } x : = \\{ y \\in M \\colon \\| x - y \\| = d ( x , M ) \\} \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} \\alpha ( Q ( j ) , Q ( i ) ) & = \\alpha ( 1 , Q ( i ) ) - \\alpha ( 1 , Q ( j ) ) \\\\ & = | [ 1 , Q ( i ) ) \\cap Q ( J ) | - | [ 1 , Q ( j ) ) \\cap Q ( J ) | \\\\ & = Q ( i ) - 1 - | [ 1 , Q ( i ) ) \\cap Q ( I \\sqcup I + 1 ) | - | [ 1 , Q ( j ) ) \\cap Q ( J ) | \\\\ & = Q ( i ) - 1 - | [ 1 , i ) \\cap ( I \\sqcup I + 1 ) | - | [ 1 , j ) \\cap J | \\\\ & = Q ( i ) - i + | [ 1 , i ) \\cap J | - | [ 1 , j ) \\cap J | \\ , \\cdot \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } D ^ 2 u - u g = 0 , & \\\\ \\frac { \\partial u } { \\partial \\nu } - \\coth \\theta u = 0 , & . \\end{array} \\right . \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} \\left ( { \\rm O p } ( a ) ( f ) \\right ) ( g ) : = \\int _ G \\int _ \\mathfrak { g ^ * } \\chi ( g h ^ { - 1 } ) e ^ { i \\left < \\xi , \\exp ^ { - 1 } ( g h ^ { - 1 } ) \\right > } a ( \\xi ) d \\xi d h , \\quad \\mbox { f o r } ~ f \\in C ^ \\infty _ c ( G ) , \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} f _ 0 = ( \\gamma + 2 ) \\widehat { A } f _ 0 \\Leftrightarrow \\left ( \\widehat { A } ^ { - 1 } - 2 \\widehat { I } \\right ) f _ 0 = \\gamma f _ 0 . \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\mathcal { E } \\emph { ( } \\theta \\emph { ) } = \\displaystyle \\int _ { M } P _ { 0 } \\lambda \\cdot \\lambda d \\mu _ { 0 } + \\displaystyle \\int _ { M } Q _ { 0 } \\lambda d \\mu _ { 0 } \\leq \\beta ^ { 2 } , \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} \\dot { V } ( z ) & = \\nabla V ( z ) \\dot { z } \\\\ & = - [ ( \\nabla { L } ( x , \\lambda ^ * ) - \\nabla { L } ( x ^ * , \\lambda ) ) + z - z ^ * ] ^ T ( z - \\tilde { z } ) \\\\ & = - [ G ( z ) + z - z ^ * ] ^ T ( z - \\tilde { z } ) \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} | | L ^ { * } g | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } = \\int _ { 0 } ^ { \\infty } \\left ( ( g ' ( r ) ) ^ { 2 } + \\frac { 4 g ( r ) ^ { 2 } } { r ^ { 2 } ( 1 + r ^ { 2 } ) } \\right ) r d r \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} \\rho ( ( x _ 1 , t _ 1 ) , ( x _ 2 , t _ 2 ) ) : = | \\tanh ( t _ 1 ) - \\tanh ( t _ 2 ) | \\vee \\Bigl | \\frac { \\tanh ( x _ 1 ) } { 1 + | t _ 1 | } - \\frac { \\tanh ( x _ 2 ) } { 1 + | t _ 2 | } \\Bigr | . \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} S _ 1 & = \\sum _ { i = 1 } ^ k ( - 1 ) ^ { k - i } \\sum _ { \\substack { 2 \\leq j \\leq k \\\\ j \\equiv k \\ ( { \\rm m o d } \\ p ) } } ( j - 1 ) \\frac { ( k - 2 ) ! } { ( k - j ) ! } p ( k - j , i - 1 ) q ^ { i - 1 } \\\\ & = ( - 1 ) ^ { k - 1 } ( k - 2 ) ! \\sum _ { \\substack { 2 \\leq j \\leq k \\\\ j \\equiv k \\ ( { \\rm m o d } \\ p ) } } ( j - 1 ) ( - 1 ) ^ { \\frac { k - j } { p } } \\binom { q / p } { ( k - j ) / p } \\\\ & = ( - 1 ) ^ { k - 1 } ( k - 2 ) ! \\sum _ { 0 \\leq \\ell \\leq \\lfloor \\frac { k - 2 } { p } \\rfloor } ( k - 1 - p \\ell ) ( - 1 ) ^ { \\ell } \\binom { q / p } { \\ell } \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} \\widetilde f ( 0 ^ + ) = f ' ( \\infty ) , \\qquad { \\widetilde f } \\ , ' ( \\infty ) = f ( 0 ^ + ) . \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} f ( t ) = f _ 0 + \\int _ 0 ^ t [ Q ^ { + } ( s , f ( s ) ) - Q ^ { - } ( s , f ( t ) ) ] d s , \\forall t \\geq 0 \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} ( M + N ) ( p ) = - 2 a ^ 2 - 2 a ^ 3 m - \\frac { a ^ 3 } { \\epsilon } + \\frac { a ^ 5 } { 2 \\epsilon ^ 3 } + \\frac { m a ^ 6 } { 2 \\epsilon ^ 3 } + \\frac { a ^ 6 } { 4 \\epsilon ^ 4 } , \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { H _ n } { r _ 1 ^ n } = 0 \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\frac { \\ell _ R ( R / I ( m D ) ) } { m ^ d } = \\mu ( { \\rm V o l } ( \\Delta ( 0 ) ) - { \\rm V o l } ( \\Delta ( D ) ) ) . \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} \\hat { H } _ { c o u l } = - \\sum _ { i = 1 } ^ { D } \\frac { \\partial ^ 2 } { \\partial x _ i ^ 2 } - \\frac { \\eta } { r } + \\frac { 1 } { x _ 1 ^ 2 + x _ 2 ^ 2 } F ^ 1 _ 1 ( \\phi ^ 1 _ 1 ) , \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} T _ 2 ^ { - 1 } B _ 0 ^ { ( 1 ) } ( | \\xi | ) T _ 2 = B _ 0 ^ { ( 1 ) } ( | \\xi | ) + T _ 2 ^ { - 1 } \\left [ B _ 0 ^ { ( 1 ) } ( | \\xi | ) , N _ 2 ( | \\xi | ) \\right ] . \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} | F _ 2 | \\ge \\frac { \\frac m 2 - t } { \\binom { t } { k - 1 } } . \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { k - 1 } ( - 1 ) ^ { j - i } \\binom { m - k + j - s - 1 } { j - s } \\binom { k - m + s - i - 1 } { s - i } \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} R _ n = \\max _ { k \\ge n } \\frac { p p _ k } { | p - p _ k | } . \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} & \\underline h ( t ) : = ( K _ 1 t + \\theta ) ^ { 1 / ( { \\gamma } - 1 ) } , \\ \\ t \\geq 0 , \\\\ & \\underline U ( t , x ) : = K _ 2 \\frac { \\underline h ( t ) - | x | } { \\underline h ( t ) } { \\Theta } , \\ \\ \\ t \\geq 0 , \\ x \\in [ - \\underline h ( t ) , \\underline h ( t ) ] , \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} e _ s ^ \\perp k _ m ^ { ( r ) } = \\sum _ { i = 0 } ^ m \\binom { r + i - 1 } { i } ( h _ { m - i } e _ s ^ \\perp + h _ { m - i - 1 } e _ { s - 1 } ^ \\perp ) = k _ m ^ { ( r ) } e _ s ^ \\perp + k _ { m - 1 } ^ { ( r ) } e _ { s - 1 } ^ \\perp \\ , . \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} \\psi [ \\Pi ^ \\R f ] ( \\cdot , \\xi ) = \\Pi \\big ( \\psi [ f ] ( \\cdot , \\xi ) \\big ) \\ . \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} \\mathcal { O } = \\mathbb { Z } \\times \\lbrace a _ 1 , \\dots , a _ m \\rbrace \\ , , \\ , \\mathcal { O } ' = \\mathbb { Z } _ { > 0 } \\lbrace a _ 1 , \\dots , a _ m \\rbrace \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} G ^ { \\left ( N + 1 \\right ) } _ { 0 ; 1 , 0 , \\dots , 0 , \\dots , 0 } ( z ) = G ^ { \\left ( N + 1 \\right ) } _ { 0 ; 0 , 0 , \\dots , 1 , \\dots , 0 } ( z ) = \\dots = G ^ { \\left ( N + 1 \\right ) } _ { 0 ; 0 , 0 , \\dots , 0 , \\dots , 1 } ( z ) = 0 , \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} \\int f ( W _ { n _ { k _ l } , 1 } , \\ldots , W _ { n _ { k _ l } , m } ) Y _ { n _ { k _ l } } \\mathrm { d } \\mathbb { P } _ { n _ { k _ l } } = \\int f ( W _ { n _ { k _ l } , 1 } , \\ldots , W _ { n _ { k _ l } , m } ) \\mathrm { d } \\mathbb { Q } _ { n _ { k _ l } } \\to \\int f ( Z _ 1 ' , \\ldots , Z _ { m } ' ) \\mathrm { d } Q . \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} \\| f \\| _ { \\alpha } ^ 2 : = 2 \\| f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } + \\| \\widehat { u } f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 + \\| \\widehat { v } f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\widetilde { T } } ( | x | ^ { - s } u _ t , \\omega ) d t + \\int _ 0 ^ { \\widetilde { T } } ( \\xi , \\nabla \\omega ) d t = \\int _ 0 ^ { \\widetilde { T } } ( | u | ^ { q - 2 } u \\ln | u | , \\omega ) d t . \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} & \\Gamma _ { n } : = \\{ w \\in \\C : \\ \\Re w = t _ { n } \\sin \\alpha _ n , \\ \\Im w \\leq y _ { n } - 1 \\} , \\\\ & \\Lambda _ { n } : = \\{ w \\in \\C : \\ \\Re w = - t _ { n } \\sin \\alpha _ n , \\ \\Im w \\leq y _ { n } \\} , \\\\ & \\Gamma : = \\{ w \\in \\C : \\ \\Re w = 0 , \\ \\Im w \\leq 0 \\} . \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} B ( | \\xi | ; \\rho , \\theta ) = \\frac { 1 } { 2 } \\left ( | \\xi | ^ { 2 \\rho } + | \\xi | ^ { 2 \\theta } \\right ) B _ 0 + i | \\xi | B _ 1 . \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} \\begin{cases} \\big ( - \\alpha _ 1 { \\eta ^ + } ' \\big ) ' \\ , + \\ , a _ 1 \\eta ^ + \\ , + \\ , b _ 1 \\zeta ^ + \\ , = \\ , 0 , \\\\ [ 2 m m ] \\big ( - \\alpha _ 2 { \\eta ^ - } ' \\big ) ' \\ , + \\ , a _ 1 \\eta ^ - \\ , + \\ , b _ 2 \\zeta ^ - \\ , = \\ , 0 , \\\\ [ 2 m m ] \\big ( - \\beta _ 1 { \\zeta ^ + } ' \\big ) ' \\ , + \\ , d _ 1 \\zeta ^ + \\ , + \\ , b _ 1 \\eta ^ + \\ , + \\ , b _ 3 \\zeta ^ - \\ , = \\ , 0 , \\\\ [ 2 m m ] \\big ( - \\beta _ 2 { \\zeta ^ - } ' \\big ) ' \\ , + \\ , d _ 2 \\zeta ^ - \\ , + \\ , b _ 2 \\eta ^ - \\ , + \\ , b _ 3 \\zeta ^ + \\ , = \\ , 0 . \\end{cases} \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} ( x _ 1 + x _ 2 ) \\otimes y = x _ 1 \\otimes y + x _ 2 \\otimes y , \\\\ x \\otimes ( y _ 1 + y _ 2 ) = x \\otimes y _ 1 + x \\otimes y _ 2 , \\\\ \\alpha ( x \\otimes y ) = ( \\alpha x ) \\otimes y = x \\otimes ( \\alpha y ) . \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} \\omega ' ( t ) + \\mu ( 1 + \\gamma \\partial _ t ^ \\alpha ) \\omega ( t ) & = 0 , \\ ; t > 0 , \\\\ \\omega ( 0 ) & = 1 , \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} & \\Bigl | \\int _ { \\R ^ 3 \\times \\R ^ 3 } { h ( \\tau _ 1 + \\tau _ 2 , \\xi _ 1 + \\xi _ 2 , \\eta _ 1 + \\eta _ 2 ) f ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) g ( \\tau _ 2 , \\xi _ 2 , \\eta _ 2 ) } d \\sigma _ 1 d \\sigma _ 2 \\Bigr | \\\\ & \\lesssim A ^ { \\frac { 1 } { 2 } } N _ 1 ^ { - 2 } ( L _ 0 L _ 1 L _ 2 ) ^ { \\frac { 1 } { 2 } } \\| f \\| _ { L ^ 2 } \\| g \\| _ { L ^ 2 } \\| h \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} \\left \\| \\Lambda f ( t ) \\right \\| + \\int _ 0 ^ t \\Delta ( s , f ( s ) ) d s = \\left \\| \\Lambda f _ 0 \\right \\| , \\forall t \\geq 0 , \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} Q _ { K + u } ( x , y , z ) = & \\dfrac { \\mathfrak { c } ( m - 4 ) } { 2 } \\left ( a ^ { \\prime } \\left ( x - \\ell _ { 0 } d - \\dfrac { d } { \\mathfrak { c } } \\right ) ^ { 2 } + b ^ { \\prime } \\left ( y - \\ell _ { 0 } d - \\dfrac { d } { \\mathfrak { c } } \\right ) ^ { 2 } + c ^ { \\prime } \\left ( z - \\ell _ { 0 } d - \\dfrac { d } { \\mathfrak { c } } \\right ) ^ { 2 } \\right ) . \\textsc { } \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} \\| | \\nabla _ x | ^ { \\frac { 1 } { 2 p } } | \\nabla _ y | ^ { \\frac { 1 } { 2 p } } Q _ L u \\| _ { L _ t ^ p L _ { x , y } ^ q } & \\lesssim L ^ { \\frac { 1 } { 2 } } \\| Q _ L u \\| _ { L ^ 2 _ { x , y , t } } , \\textnormal { i f } \\ \\ \\frac { 2 } { p } + \\frac { 2 } { q } = 1 , \\ p > 2 , \\\\ \\| Q _ L u \\| _ { L _ t ^ p L _ { x , y } ^ q } & \\lesssim L ^ { \\frac { 1 } { 2 } } \\| Q _ L u \\| _ { L ^ 2 _ { x , y , t } } , \\textnormal { i f } \\ \\ \\frac { 3 } { p } + \\frac { 2 } { q } = 1 , \\ p > 3 . \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} r _ i = \\begin{cases} 0 , & i = 1 , \\ldots , z ( t ) - 1 , \\\\ n _ i , & i = z ( t ) + 1 , \\ldots , t , \\end{cases} \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} \\begin{array} { l } ( x - 1 ) ^ { n - 2 r _ 1 + k _ 4 } p _ 4 ( x ) ( u ^ 2 ( x - 1 ) ^ { r _ 1 } + u ^ 3 ( x - 1 ) ^ { k _ 4 } p _ 4 ( x ) ) - ( x - 1 ) ^ { n - r _ 1 } g _ 1 ( x ) , \\\\ = u ^ 3 ( x - 1 ) ^ { n - 2 r _ 1 + 2 k _ 4 } p _ 4 ^ 2 ( x ) - u ^ 3 ( x - 1 ) ^ { n - r _ 1 + k _ 5 } p _ 5 ( x ) , \\\\ = u ^ 3 ( x - 1 ) ^ { \\tau _ 1 } \\tilde h _ 1 ( x ) , \\end{array} \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} I \\leq C d _ q 2 ^ { - \\sigma q } ( \\sqrt { q + 2 } V ( t ) + 1 ) \\| \\rho \\| _ { B ^ { \\sigma } _ { 2 , r } } \\| \\Delta _ q \\rho \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} \\alpha _ q ( \\eta ) \\leq 1 - \\eta \\big ( 1 - \\frac { 1 } { \\hat { n } q ^ { \\hat { m } } } \\big ) ^ { - 1 } \\eta \\leq 1 - \\frac { 1 } { \\hat { n } q ^ { \\hat { m } } } \\ \\ \\ \\ \\alpha _ q ( \\eta ) = 0 \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ { \\R } \\rho ( X _ { s - } , y ) \\nu ( d y , d s ) = \\sum _ { k = 1 } ^ { N _ t } \\rho ( X _ { \\tau _ { k } - } ) = \\int \\limits _ 0 ^ t \\rho ( X _ { s - } ) d N _ s . \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\frac { ( 1 + \\epsilon ) F ( v ) - F ( ( 1 + \\epsilon ) v ) } { \\epsilon } = & \\lim _ { \\epsilon \\to 0 } \\frac { \\epsilon F ( v ) - [ F ( v + \\epsilon v ) - F ( v ) ] } { \\epsilon } \\\\ = & F ( v ) - v [ \\nabla F ( v ) ] ^ T \\succeq 2 c _ 1 \\mathbf { 1 } \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} d F _ { \\mu _ t } ( v _ t ) _ y : = \\int _ { f ^ { - 1 } ( { y } ) } d f _ x ( v _ { t , x } ) \\ d \\mu _ t ^ y ( x ) \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} \\Delta ^ { I } \\boldsymbol { x } = - 2 H \\boldsymbol { n } . \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } F _ { f _ { n _ k } } ( x , y ) = \\frac { f ( x ) - f ( y ) } { | x - y | ^ { ( d + \\alpha ) / 2 } } \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} C ^ k _ { { \\rm d i f f } , \\lambda } ( G ) : = \\{ c \\in C ^ \\infty ( G ^ { \\times ( k + 1 ) } , \\mathbb { C } ) , ~ c ( g g _ 0 , \\ldots , g g _ k ) = c ( g _ 0 , \\ldots , g _ k ) , ~ c ( g _ 0 , \\ldots , g _ k ) = ( - 1 ) ^ k c ( g _ k , g _ 0 , \\ldots , g _ { k - 1 } ) \\} , \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} \\tilde { \\boldsymbol { z } } = \\frac { 1 } { L P } \\boldsymbol { Y } \\tilde { \\boldsymbol { x } } ^ { * } = \\boldsymbol { A } \\boldsymbol { s } + \\boldsymbol { h } _ d + \\boldsymbol { w } , \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} u _ { 1 } ( t , x ) & = - \\frac { \\lambda '' ( s ) } { 2 \\pi } \\int _ { B ( 0 , t - s ) } \\frac { \\log ( 1 + | y + x | ^ { 2 } ) } { ( ( t - s ) ^ { 2 } - | y | ^ { 2 } ) ^ { 1 / 2 } } d y \\\\ & = - \\frac { \\lambda '' ( s ) } { 2 \\pi } \\int _ { 0 } ^ { t - s } \\rho d \\rho \\int _ { 0 } ^ { 2 \\pi } d \\theta \\frac { \\log ( 1 + | x | ^ { 2 } + 2 | x | \\rho \\cos ( \\theta ) + \\rho ^ { 2 } ) } { ( ( t - s ) ^ { 2 } - \\rho ^ { 2 } ) ^ { 1 / 2 } } \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\int _ 0 ^ { \\widetilde { T } } \\int _ \\Omega | u ^ m | ^ q \\ln | u ^ m | d x d t = \\int _ 0 ^ { \\widetilde { T } } \\int _ \\Omega | u | ^ q \\ln | u | d x d t . \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} \\begin{cases} - \\hat { v } _ { \\infty } \\cdot L _ { \\Sigma } \\hat { v } _ \\infty \\leq 0 \\ & \\Sigma ; \\\\ - L _ { \\Sigma } \\hat { v } _ \\infty = 0 \\ & \\big ( \\{ \\hat { u } _ \\infty ^ - < \\hat { v } _ \\infty < \\hat { u } _ \\infty ^ + \\} \\cap \\Omega _ 0 \\big ) \\cup \\Sigma \\setminus \\Omega _ 0 . \\end{cases} \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} - i w ^ { ( s ) } _ n & = - i \\frac { \\lambda _ s } { s ! } n ! \\underbrace { \\sum _ { j = 0 } ^ { n - s } \\sum _ { k = 0 } ^ { n - s - j } \\cdots \\sum _ { l = 0 } ^ { n - s - j - k - \\ldots } } _ { s - 1 } \\underbrace { a _ j a _ k \\cdots a _ l a _ { n - s - j - k - \\ldots - l } } _ { s a } . \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} \\begin{cases} D ^ { \\alpha _ 1 } _ 0 y ( t ) = w ( t ) , & y ( 0 ) = y _ 0 = \\frac { \\gamma _ 1 - b _ 1 s } { a _ 1 } \\\\ D ^ { 1 - \\alpha _ 1 } _ 0 w ( t ) = z ( t ) , & w ( 0 ) = 0 \\\\ D ^ { \\alpha _ 2 - 1 } _ 0 z ( t ) = f ( t , y ( t ) , w ( t ) ) , & z ( 0 ) = s . \\end{cases} \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} X C X - X D - A X + B = 0 , \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} W + \\sum _ { k = 1 } ^ { K - 1 } \\frac { 2 \\mu _ { k + 1 } Z _ { k } - \\mu _ { k + 1 } ^ 2 } { 4 ( k + 1 ) } \\stackrel { d } { \\to } M \\{ n _ { l _ { q } } \\} . \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} C _ { \\mu } \\frac { 1 } { T } \\sum _ { t = 1 } ^ T \\Delta _ t \\leq \\frac { \\rho \\tau \\Theta _ 1 + ( 1 - \\rho ) \\Theta _ 1 + 2 \\sigma ^ 2 \\rho ^ 2 T } { \\gamma T } \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} \\lim _ { x \\uparrow c _ { 0 } ^ { - 1 } } \\frac { q _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( x ) } { \\left | x - c _ { 0 } ^ { - 1 } \\right | ^ { 1 / 2 } } = \\frac { a _ { 1 } } { \\pi \\sqrt { \\left | c _ { 2 } \\right | } } \\sin \\frac { \\theta } { 2 } , \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( \\sum _ { i _ 1 , \\ldots , i _ n = 1 } ^ N p _ { i _ n } \\cdots p _ { i _ 1 } \\left \\| A _ { i _ n } \\cdots A _ { i _ 1 } \\right \\| ^ p \\right ) ^ { \\frac { 1 } { n } } \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} \\partial _ t u - D \\Delta u = \\gamma u ( 1 - | u | ^ 2 ) , u _ { / t = 0 } = u _ { 0 } \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} A _ { n + 1 , n + 1 } = \\begin{pmatrix} n & n + 1 \\\\ n & n + 1 + \\alpha \\end{pmatrix} \\begin{pmatrix} n - \\alpha & n \\\\ n & n \\end{pmatrix} A _ { n , n } = n C _ { n + 1 } A _ { n , n } , \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} | \\Phi ( \\xi _ 1 ' , \\eta _ 1 ' , \\xi _ 2 , \\eta _ 2 ) | & = | \\xi _ 1 ' \\xi _ 2 ( \\xi _ 1 ' + \\xi _ 2 ) + \\eta _ 1 ' \\eta _ 2 ( \\eta _ 1 ' + \\eta _ 2 ) | \\leq 2 ^ 4 A ^ { - 1 } N _ 1 ^ 3 , \\\\ | F ( \\xi _ 1 ' , \\eta _ 1 ' , \\xi _ 2 , \\eta _ 2 ) | & = | \\xi _ 1 ' \\eta _ 2 + \\xi _ 2 \\eta _ 1 ' + 2 ( \\xi _ 1 ' \\eta _ 1 ' + \\xi _ 2 \\eta _ 2 ) | \\leq 2 ^ 4 A ^ { - 1 } N _ 1 ^ 2 , \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} c h _ i ( E _ { ( r , A , r ' , \\mathcal { U } , p , q ) } ) = c h _ i ( E _ { ( s , B , s ' , \\mathcal { V } , u , v ) } ) . \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} \\sqrt { \\omega } \\lambda ( t ) T ( y ) ( t , \\omega ) = - \\lambda ( t ) \\int _ { t } ^ { \\infty } \\sin ( ( t - x ) \\sqrt { \\omega } ) \\left ( F _ { 2 } ( y ) ( x , \\omega ) - \\mathcal { F } ( \\sqrt { \\cdot } F ( x , \\cdot \\lambda ( x ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) - \\mathcal { F } ( \\sqrt { \\cdot } F _ { 3 } ( x , \\cdot \\lambda ( x ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) \\right ) d x \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} a _ { n + 1 } \\leq 2 a _ n = a _ n + a _ n \\leq a _ n + a _ { n - 1 } + \\cdots + a _ 1 + 1 , \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} I & = \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } \\big | \\bar { f _ 0 } \\big ( X ( t _ 0 ) , V ( t _ 0 ) \\big ) \\big | d x d v \\\\ & \\leq e ^ { C T _ 1 } \\ ! \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } \\big | f _ 0 ( x , v ) \\big | d x d v , \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} L _ a ^ \\# = L _ b , & L _ b ^ \\# = L _ a , \\\\ L _ { \\min } \\subset L _ a \\subset L _ { \\max } , & L _ { \\min } \\subset L _ b \\subset L _ { \\max } . \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} Z _ 0 ' & = \\{ ( 2 , 1 ) , ( 1 , 8 ) , ( - 3 , 1 5 ) \\} , & & Z _ 0 '' = \\{ ( 2 , 8 ) , ( 1 , 1 5 ) \\} , & & Z _ 0 ''' = \\{ ( - 3 , 1 ) \\} , \\\\ Z _ 1 ' & = \\{ ( 6 , 2 ) , ( 4 , 1 4 ) , ( 0 , 1 7 ) \\} , & & Z _ 1 '' = \\{ ( 6 , 1 4 ) , ( 4 , 1 7 ) \\} , & & Z _ 1 ''' = \\{ ( 0 , 2 ) \\} , \\\\ Z _ 2 ' & = \\{ ( 9 , 1 0 ) , ( 8 , 1 3 ) , ( 5 , 1 6 ) , ( 3 , 1 9 ) \\} , & & Z _ 2 '' = \\{ ( 9 , 1 3 ) , ( 8 , 1 6 ) , ( 5 , 1 9 ) \\} , & & Z _ 2 ''' = \\{ ( 3 , 1 0 ) \\} . \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} a _ { 2 J } = \\frac { \\partial f _ J } { \\partial y } ( 0 , 0 ; 0 ) . \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} \\frac { K ( t , s ) } { K ( u , s ) } & = \\frac { \\frac { 1 } { 1 - s + t } + \\frac { 1 } { ( \\lambda _ { 0 } ( - t ) ^ { 1 - \\alpha } - s + t ) ( 1 - s + t ) ^ { 3 } } } { \\frac { 1 } { 1 - s + u } + \\frac { 1 } { ( \\lambda _ { 0 } ( - u ) ^ { 1 - \\alpha } - s + u ) ( 1 - s + u ) ^ { 3 } } } \\\\ & \\leq \\frac { 1 - s + u } { 1 - s + t } + \\frac { ( \\lambda _ { 0 } ( - u ) ^ { 1 - \\alpha } - s + u ) ( 1 - s + u ) ^ { 3 } } { ( \\lambda _ { 0 } ( - t ) ^ { 1 - \\alpha } - s + t ) ( 1 - s + t ) ^ { 3 } } \\\\ & \\leq 2 , s \\leq u \\leq t \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} C = \\{ s e ^ { i \\theta } : \\theta \\in [ f ( r ) + \\varepsilon , \\pi ] , s \\in J \\} \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} I _ a = \\sum _ { i , j = 1 } ^ { 2 } \\int _ { \\Omega } u _ i \\partial _ { i } \\partial _ j ^ 2 \\theta \\partial _ j ^ 2 \\theta \\dd x + \\sum _ { i , j \\neq k = 1 } ^ { 2 } \\int _ { \\Omega } u _ i \\partial _ { i } \\partial _ j ^ 2 \\theta \\partial _ k ^ 2 \\theta \\dd x = I _ { a _ 1 } + I _ { a _ 2 } \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} B \\sigma _ h A = B ^ { 1 / 2 } h ( W ) B ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} \\delta _ i ( \\beta _ 0 , \\beta _ 1 , \\eta ) = \\sum _ { s + 2 t + 2 k = 1 } ^ { 4 } \\frac { \\beta _ 0 ^ s \\beta _ 1 ^ t ( \\eta ^ 2 - 1 ) ^ k } { s ! t ! k ! } \\delta ^ { ( s , t , k ) } _ i + \\epsilon _ { i n } ^ { ( 2 ) } \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} ( q ; q ) _ \\infty \\sum _ { n = 0 } ^ \\infty a _ k ( n ) q ^ n = ( q ^ k ; q ^ k ) _ \\infty \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { k n } + 2 q ^ { 2 k n } + \\dots + ( k - 1 ) q ^ { ( k - 1 ) k n } } { 1 + q ^ { k n } + q ^ { 2 k n } + \\dots + q ^ { ( k - 1 ) k n } } , \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} u _ { / t = 0 } = u _ { 0 } \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} r _ \\alpha = \\sum _ { \\beta \\succcurlyeq \\alpha } ( - 1 ) ^ { \\ell ( \\alpha ) - \\ell ( \\beta ) } h _ { \\widetilde \\beta } . \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} | \\mu ^ { k } _ { i , \\pm } | = \\left | 1 - \\frac { \\Delta t _ k } { 2 } \\left ( \\Delta t _ k b ^ { k } _ i \\lambda ^ { k } _ i + \\eta \\right ) \\pm i \\frac { \\Delta t _ k } { 2 } \\sqrt { 4 b ^ { k } _ i \\lambda ^ { k } _ i - \\left ( \\Delta t _ k b ^ { k } _ i \\lambda ^ { k } _ i + \\eta \\right ) ^ 2 } \\right | = \\sqrt { 1 - \\eta \\Delta t _ k } . \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} J _ 2 = - 2 \\int _ { 2 a t } ^ { 2 b t } A ( 2 x ) P ( 2 x , k ) \\left ( \\int _ { 2 a t } ^ x \\frac { e ^ { i ( u ^ 2 / ( 4 t ) - k u ) } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) d u \\right ) d x \\ , . \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} e ^ { F ( x , y ) } = \\frac { u ' ( x ) v ' ( y ) } { ( u ( x ) - v ( y ) ) ^ 2 } , \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} [ x _ 1 , x _ 2 , x _ 3 ] _ \\tau = \\tau ( x _ 1 ) [ x _ 2 , x _ 3 ] - ( - 1 ) ^ { \\bar x _ 1 \\bar x _ 2 } \\tau ( x _ 2 ) [ x _ 1 , x _ 3 ] + ( - 1 ) ^ { \\bar x _ 3 ( \\bar x _ 1 + \\bar x _ 2 ) } \\tau ( x _ 3 ) [ x _ 1 , x _ 2 ] . \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} H _ { j + 1 } = H _ j + \\dots + H _ { j - k + 1 } + 2 H _ { j - k - 1 } + H _ { j - k - 1 } + H _ { j - k - 2 } + \\dots + H _ 1 + 1 \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} \\| A ^ { r - \\frac { 1 } { 2 } } & e ^ { \\tau A } \\partial _ t \\overline { v } \\| \\leq C _ r \\Big ( \\| e ^ { \\tau A } \\overline { v } \\| _ { H ^ { r } } \\| e ^ { \\tau A } \\overline { v } \\| _ { H ^ { r + \\frac { 1 } { 2 } } } + \\| e ^ { \\tau A } \\widetilde { v } \\| _ { H ^ { r } } \\| e ^ { \\tau A } \\widetilde { v } \\| _ { H ^ { r + \\frac { 1 } { 2 } } } \\Big ) , \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} X _ { P _ n } = X _ { ( P _ n , ( 1 ^ n ) ) } = \\sum _ { S \\subseteq E ( G ) } ( - 1 ) ^ { | S | } p _ { \\lambda ( ( V ( P _ n ) , S ) , ( 1 ^ n ) ) } = \\sum _ { \\beta \\vDash n } ( - 1 ) ^ { n - \\ell ( \\beta ) } p _ { \\widetilde \\beta } . \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} \\inf _ { s \\in [ 0 , 1 ] } U _ s ( f , g ) = f ! g \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\sup _ { s \\in [ 0 , 1 ] } U _ s ( f , g ) = L ( f , g ) , \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} & \\lambda _ { 1 } ( t ) E _ { 0 , 1 } ( \\lambda _ { 1 } ( t ) , \\lambda _ { 1 } ' ( t ) , \\lambda _ { 1 } '' ( t ) ) - \\lambda _ { 2 } ( t ) E _ { 0 , 1 } ( \\lambda _ { 2 } ( t ) , \\lambda _ { 2 } ' ( t ) , \\lambda _ { 2 } '' ( t ) ) \\\\ & \\leq \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\log ^ { b + 1 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} K _ { p , q } ( \\varphi ; \\Omega ) = \\| K _ p \\mid L _ { \\kappa } ( \\Omega ) \\| < \\infty , \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} \\underline U _ t = & - \\epsilon ' ( t ) [ \\Phi ( x - \\underline h ( t ) ) + \\Phi ( - x - \\underline h ( t ) ) - \\mathbf { u } ^ * ] \\\\ & - ( 1 - \\epsilon ) h ' ( t ) [ \\Phi ' ( x - \\underline h ( t ) ) + \\Phi ' ( - x - \\underline h ( t ) ) ] \\\\ = & \\alpha ( t + \\theta ) ^ { - \\alpha - 1 } [ \\Phi ( x - \\underline h ( t ) ) + \\Phi ( - x - \\underline h ( t ) ) - \\mathbf { u } ^ * ] \\\\ & - ( 1 - \\epsilon ) [ c _ 0 + \\delta ' ( t ) ] [ \\Phi ' ( x - \\underline h ( t ) ) + \\Phi ' ( - x - \\underline h ( t ) ) ] , \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} \\overline { K } ( x _ 0 , x _ 1 , y _ 0 , y _ 1 , t ) = { x _ 1 ^ 2 y _ 1 ^ 2 K ( \\frac { x _ 0 } { x _ 1 } , \\frac { y _ 0 } { y _ 1 } , t ) } = x _ 0 x _ 1 y _ 0 y _ 1 - t \\sum _ { i , j = 0 } ^ 2 d _ { i - 1 , j - 1 } x _ 0 ^ { i } x _ 1 ^ { 2 - i } y _ 0 ^ j y _ 1 ^ { 2 - j } . \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} 1 + \\sum ^ { L } _ { i = 1 } H _ i \\geq G _ m - \\sum _ { i = L + 1 } ^ { m - 1 } G _ i . \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} \\varlimsup _ { \\rho \\to + 0 } \\biggl [ \\ , \\lim _ { \\sigma \\to + 0 } \\Bigl ( \\frac { 1 } { \\rho ^ 2 } \\sup _ { ( 0 , \\sigma ) \\times D _ { \\rho } } | u ( t , x ) | \\Bigr ) \\biggr ] = 0 . \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} \\int _ 0 ^ t f \\left ( v , e ^ { \\xi \\mathcal { A } } \\overline v \\right ) d \\xi = t f ( v , \\overline v ) . \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} \\alpha _ { i j } ^ { ( p ) } & \\coloneqq A _ { i j } \\bullet \\begin{pmatrix} I _ d \\\\ ( U ^ { ( p ) } ) ^ \\top \\end{pmatrix} \\begin{pmatrix} I _ d \\\\ ( V ^ { ( p ) } ) ^ \\top \\end{pmatrix} ^ \\top - d _ { i j } ^ 2 , \\\\ \\alpha _ { i k } ^ { ( p ) } & \\coloneqq A _ { i k } \\bullet \\begin{pmatrix} I _ d \\\\ ( U ^ { ( p ) } ) ^ \\top \\end{pmatrix} \\begin{pmatrix} I _ d \\\\ ( V ^ { ( p ) } ) ^ \\top \\end{pmatrix} ^ \\top - d _ { i k } ^ 2 . \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} P _ { X } = \\arg \\min _ { v \\in X } \\norm { x - v } . \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} f _ n & = B _ { n - 2 , 1 } ( 2 ! a _ 1 , 3 ! a _ 2 , \\ldots ) + B _ { n - 2 , 2 } ( 2 ! a _ 1 , 3 ! a _ 2 , \\ldots ) , \\\\ c _ { n - 2 } & = B _ { n , 2 } \\left ( 1 , 2 ! a _ 1 , \\ldots \\right ) , \\\\ g _ n & : = n f _ n - c _ { n - 2 } = \\frac { n ( n - 2 ) ! } { 2 } \\sum _ { k = 0 } ^ { n - 2 } a _ { n - k - 2 } a _ k ( n - k - 2 ) k . \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} u ( r , \\omega ) = \\sum _ { k \\geq 1 } ( u _ k ( r ) + v _ k ( r ) ) w _ k ( \\omega ) \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} \\underline { x } ^ { \\prime } = ( x _ { 0 } , x _ { 1 } , \\ldots , x _ { n } , 0 , 0 , \\ldots , 0 ) \\in \\mathbb { Z } ^ { k + 1 } , \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} F ( s ) = \\int _ 0 ^ { + \\infty } \\ln | 1 - \\tau ^ 2 | \\tau ^ { - ( 1 + s ) } d \\tau . \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} S = \\sqrt { I _ 4 } . \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} X \\times Y = \\{ ( x , y ) : x \\in X , y \\in Y \\} , \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} 2 c _ { b } \\lambda ( t ) \\int _ { 0 } ^ { \\infty } d \\xi \\frac { \\sin ( t \\xi ) } { t ^ { 2 } } \\left ( \\partial _ { 2 2 } \\psi _ { v _ { 2 } } \\lambda ' ( t ) ^ { 2 } \\right ) = 2 c _ { b } \\lambda ( t ) \\int _ { 0 } ^ { \\infty } d \\xi \\frac { \\cos ( t \\xi ) } { t ^ { 3 } } \\partial _ { 1 2 2 } \\psi _ { v _ { 2 } } ( \\xi , \\lambda ( t ) ) \\lambda ' ( t ) ^ { 2 } \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} C ^ { ( 1 ) } _ { 0 } = \\frac { \\sum _ { w \\in W } ( - 1 ) ^ { \\ell ( w ) } e ^ { w ( \\rho ) } / ( 1 - e ^ { w ( \\omega _ 1 ) } ) } { \\Delta } , \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} \\mathbf { A } f ( t ) = t ^ { - \\alpha } \\left ( b t f ' ( t ) + \\int _ 0 ^ { \\infty } ( f ( t e ^ { y } ) - f ( t ) ) m ( d e ^ { - y } ) \\right ) \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} { \\rm c h } [ W ( \\Lambda _ 0 ) ] \\ \\ _ { = } ^ { ? } \\sum _ { n _ 1 , n _ 2 , n _ 3 , n _ 4 , n _ 5 \\geq 0 } \\frac { q ^ { n _ 1 ^ 2 + n _ 2 ^ 2 + ( n _ 3 + n _ 5 ) ^ 2 + n _ 4 ^ 2 + n _ 3 ^ 2 + ( 2 n _ 3 + n _ 5 ) ( n _ 1 ) + n _ 4 ( n _ 1 + n _ 2 ) + n _ 3 n _ 4 } } { ( q ) _ { n _ 1 } ( q ) _ { n _ 2 } ( q ) _ { n _ 3 } ( q ) _ { n _ 4 } ( q ) _ { n _ 5 } } \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} & | | \\left ( \\partial _ { r } + \\frac { 1 } { r } \\right ) v _ { 4 , c } ( t , r ) | | _ { L ^ { 2 } ( r d r ) } \\leq | | \\partial _ { r } v _ { 4 , c } ( t , r ) | | _ { L ^ { 2 } ( r d r ) } + | | \\frac { v _ { 4 , c } ( t , r ) } { r } | | _ { L ^ { 2 } ( r d r ) } \\\\ & \\leq \\frac { C } { t ^ { 2 } \\log ^ { 3 b + 3 N } ( t ) } \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} h = h _ d + \\alpha u + \\beta v \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\leq \\frac { \\binom { n } { k } } { m \\binom { n - m } { k - 1 } } \\leq 4 . \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} L _ { k l } = x _ k p _ l - x _ l p _ k , ~ ~ ~ p _ a = \\frac { \\partial } { \\partial x _ a } . \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} & P _ { d } \\left \\{ \\sup _ { 1 \\leq i \\leq k } \\sup _ { f , g \\in \\mathcal { F } _ { i } } | \\mathbb { H } _ { N } ' f - \\mathbb { H } _ { N } ' g | > \\epsilon \\right \\} < \\eta + \\widetilde { D } _ { N } \\\\ & N = 1 , 2 , \\dots \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} \\Psi ( z ) = z G ( z ) , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} g ( \\gamma ) = \\begin{cases} \\mu ( \\gamma ) & \\gamma \\in ( \\mu ) , \\\\ f ( \\gamma ) & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} \\lim _ { s \\to - \\infty } \\limsup _ { \\varepsilon \\rightarrow 0 } \\mathbf { P } \\left ( M ^ { \\varepsilon } _ { s / b _ { \\varepsilon } } > S ^ { \\varepsilon } _ { t / b _ \\varepsilon } \\right ) = 0 , \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} \\Phi _ i ( \\mu ) = \\Psi _ { i _ 1 } ^ { ( 1 ) } ( \\mu _ 1 ) \\cdot \\Psi _ { i _ 2 } ^ { ( 2 ) } ( \\mu _ 2 ) \\cdot \\ldots \\cdot \\Psi _ { i _ { q - 1 } } ^ { ( q - 1 ) } ( \\mu _ { q - 1 } ) \\cdot \\Psi _ { i _ q } ^ { ( q ) } ( \\mu _ { q } ) , \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} u \\left ( x , t \\right ) = U \\left ( x , 1 - t \\right ) u \\left ( \\frac { x } { 1 - t } , \\frac { t } { 1 - t } \\right ) \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} \\left [ \\Re \\frac { 1 + \\eta _ { \\mu _ { 1 } } ( z ) } { 1 - \\eta _ { \\mu _ { 1 } } ( z ) } \\right ] \\int _ { \\mathbb { T } } \\frac { | 1 - \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } } { | t - \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } } \\ , d \\sigma ( t ) = \\beta \\Re \\frac { 1 + \\eta _ { \\nu _ { 1 } } ( z ) } { 1 - \\eta _ { \\nu _ { 1 } } ( z ) } . \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } \\| \\omega ( t ) \\| _ { H ^ s } ^ 2 \\leq C ( 1 + \\| \\nabla u \\| _ { L ^ \\infty } ) \\times ( \\| \\omega \\| _ { H ^ s } ^ 2 + \\| \\theta \\| _ { H ^ { 1 + s } } ^ 2 ) . \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} N ( a ) N ( b ) = N ( a N ( b ) + N ( a ) b - N ( a b ) ) , ~ a , b \\in A . \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} \\xi _ s & = \\xi _ s ( h _ 0 , \\tilde { h } _ 0 ) = K + \\tilde { h } _ 0 + s ( h _ 0 - \\tilde { h } _ 0 ) , \\\\ \\varphi & = \\varphi ( h _ 0 , \\tilde { h } _ 0 ) = \\int _ 0 ^ 1 p ' \\circ \\xi _ s ^ x \\ , d s + ( K ^ y + { h } _ 0 ^ y ) \\ , \\int _ 0 ^ 1 q ' \\circ \\xi _ s ^ x \\ , d s + \\int _ 0 ^ 1 ( D _ 1 u + D _ 1 g ) \\circ \\xi _ s \\ , d s , \\\\ \\psi & = \\psi ( h _ 0 , \\tilde { h } _ 0 ) = q \\circ ( K ^ x + \\tilde { h } _ 0 ^ x ) + \\int _ 0 ^ 1 ( D _ 2 u + D _ 2 g ) \\circ \\xi _ s \\ , d s , \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} 2 M _ n - S _ n - ( 2 M _ { n - 1 } - S _ { n - 1 } ) & = 2 Y _ n + 2 S _ n - J - S _ n - ( 2 Y _ { n - 1 } + 2 S _ { n - 1 } - J ) + S _ { n - 1 } \\\\ & = 2 Y _ n - 2 Y _ { n - 1 } + S _ n - S _ { n - 1 } , \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} t \\ge \\frac { ( \\frac m 2 - t ) b _ { i + 1 } } { \\binom { t } { k - 1 } ( b _ i + b _ { i + 1 } ) \\binom { b _ { k - 1 } + b _ k } { b _ k } } , \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} \\mathbf { u } ( t , z , \\upsilon ) = ( 0 , g ) ^ T + v ^ \\prime \\nabla _ { z , \\upsilon } ^ { \\perp } P _ { \\neq } \\phi = ( 0 , g ) ^ T + h \\nabla _ { z , \\upsilon } ^ { \\perp } P _ { \\neq } \\phi + \\nabla _ { z , \\upsilon } ^ { \\perp } P _ { \\neq } \\phi , \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} B _ { \\mathbf { m } } ^ { ( d ) } \\left ( \\mathbf { 0 } \\right ) = B _ { | \\mathbf { m } | } . \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} H _ s ( x ) : = \\sum _ { i = 1 } ^ N \\left ( \\frac { 1 } { 2 } x _ i ^ 2 + \\sum _ { j : 1 \\leq | j - i | \\leq R } M _ { i j } x _ i x _ j + s \\psi _ b ( x _ i ) \\right ) \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} P ( x ; 2 , 1 ) & = \\frac { - 2 + 2 e ^ { 2 x i } } { ( 2 + 2 i ) - ( 2 - 2 i ) e ^ { 2 x i } } = \\frac { \\tan ( 2 x ) + \\sec ( 2 x ) - 1 } { 2 } . \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} \\Lambda ( A ) = \\min _ { i , j \\in [ n ] } \\sum _ { \\ell = 1 } ^ n \\min \\{ a _ { i \\ell } , a _ { j \\ell } \\} . \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} \\Phi _ n ( S ) = \\left ( \\left [ \\left \\langle S b _ { d + k , k } , b _ { d + j , j } \\right \\rangle \\right ] _ { j , k = \\max \\{ 0 , - d \\} } ^ { n - 1 } \\right ) _ { d = - n + 1 } ^ { \\infty } . \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} \\big ( \\sigma ^ + - f ' ( 0 ) \\big ) | | e | | _ { L ^ 2 ( \\mathbb { R } ^ N ) } ^ 2 \\leq | | e | | ^ 2 = 1 < a _ 0 | | e | | ^ 2 _ { L ^ 2 ( \\mathbb { R } ^ N ) } . \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} & | | \\partial _ { r } v _ { 3 } ( t ) | | _ { L ^ { 2 } ( r d r ) } \\leq C \\int _ { t } ^ { \\infty } | | F _ { 0 , 1 } ( s ) | | _ { L ^ { 2 } ( r d r ) } d s \\leq C \\int _ { t } ^ { \\infty } \\frac { \\sqrt { \\log ( \\log ( s ) ) } } { s ^ { 2 } \\log ^ { b + 1 } ( s ) } d s \\\\ & \\leq \\frac { C \\sqrt { \\log ( \\log ( t ) ) } } { t \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} I ( h _ 0 ( v ) ) = - \\dfrac { 1 } { 2 } | | v | | ^ 2 - \\int _ { \\mathbb { R } ^ N } F _ 0 ( | v | ) \\ ; d x \\leq 0 . \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} 0 & = \\nabla f ( \\bar x ) + \\sum \\limits _ { i \\in I ^ g ( \\bar x ) } \\bar \\lambda _ i \\nabla g _ i ( \\bar x ) + \\sum \\limits _ { j \\in \\mathcal P } \\bar \\rho _ j \\nabla h _ j ( \\bar x ) \\\\ & + \\sum \\limits _ { l \\in I ^ { \\varphi _ \\textup { K S } } ( \\bar x ) } \\bar \\xi _ l \\bigl ( H _ l ( \\bar x ) \\nabla G _ l ( \\bar x ) + G _ l ( \\bar x ) \\nabla H _ l ( \\bar x ) \\bigr ) \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} 0 = \\iota _ R \\omega = \\sum _ { i = 0 } ^ n x _ i F _ i = \\sum _ { j = 1 } ^ r \\left ( \\sum _ { i = 0 } ^ n x _ i f _ { i j } \\right ) G _ j \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} \\begin{aligned} - C \\leq ( 1 - x ) ^ { - s } I _ 4 ' - ( 1 - x ) ^ { - s } \\int _ { 1 } ^ { \\infty } \\frac { \\left [ ( 1 + k ) ^ s - 1 \\right ] ^ { p - 1 } } { k ^ { 1 + s p } } \\mathrm { d } k \\leq C . \\end{aligned} \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} ( s _ { 1 } , s _ { 2 } , t _ { 1 } , t _ { 2 } ) = ( \\widehat { s } _ { 1 } , \\widehat { s } _ { 2 } , \\widehat { t } _ { 1 } , \\widehat { t } _ { 2 } ) . \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} B _ { R , \\l , K } ^ { ( n ) } = \\Big ( A _ { R } ^ { ( n ) } ( \\l ) , \\frac { d } { d X } A _ { R } ^ { ( n ) } ( \\l ) , \\ldots , \\frac { d ^ { K - 1 } } { d X ^ { K - 1 } } A _ { R } ^ { ( n ) } ( \\l ) \\Big ) . \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} 0 \\longrightarrow N \\otimes N ^ { \\vee } \\longrightarrow N ^ { \\vee \\vee } \\otimes N ^ { \\vee } = \\mathcal { O } _ { X \\times S } \\longrightarrow \\mathcal { O } _ { Z } \\longrightarrow 0 \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} f _ i Q _ i L _ s ^ { \\frac { p - 1 } { 2 } ( 2 n + 1 ) } = f _ i R _ { s ; i _ 1 , \\dots , i _ k } q _ { s ; 0 } ^ { n + [ \\frac { k } { 2 } ] } . \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} H ^ 0 ( \\mathcal { I } _ Z ( d ) ) ^ { \\oplus \\binom { n + 1 } { 2 } } \\rightarrow H ^ 0 ( \\Omega ^ 1 _ { \\mathbb { P } ^ n } ( d + 2 ) \\otimes \\mathcal { I } _ Z ) \\stackrel { \\phi } { \\longrightarrow } H ^ 1 ( \\Omega ^ 2 _ { \\mathbb { P } ^ n } ( d + 2 ) \\otimes \\mathcal { I } _ Z ) \\rightarrow H ^ 1 ( \\mathcal { I } _ Z ( d ) ) ^ { \\oplus \\binom { n + 1 } { 2 } } \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} | \\partial _ { t } E _ { 5 } ( t , r ) | & \\leq \\frac { C r } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } + \\frac { C r \\sup _ { x \\geq t } \\left ( \\frac { | e ''' ( x ) | x } { \\lambda ( x ) ^ { 3 - 2 \\alpha } } \\right ) } { t \\log ^ { ( 3 - 2 \\alpha ) b } ( t ) } \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} R ^ { - 1 } ( S ) = ( R ^ { - 1 } ( S ) \\setminus i ( S ) ) \\sqcup i ( S ) . \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} f _ { k + 7 } + 8 N - 5 \\leq f _ { k + 7 } + 4 N - 5 + \\sum _ { i = 1 } ^ { k + 3 } ( f _ { i + 1 } - 1 ) + f _ 1 , \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} H _ n = a _ 1 r _ 1 ^ n + \\sum _ { i = 2 } ^ k q _ i ( n ) r _ i ^ n , \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} w ( \\epsilon ) w ( \\overline { \\epsilon } ) & = 1 \\ \\\\ \\pi ( \\overline { \\epsilon } ) & = \\overline { \\pi ( \\epsilon ) } . \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ t m _ i u _ i & = \\sum _ { i = 1 } ^ { j - 1 } m _ i n _ i - \\sum _ { i = 1 } ^ { j - 1 } m _ i \\alpha _ i + \\sum _ { i = j } ^ t m _ i u _ i \\leq \\tilde { K } - m _ j \\delta - m _ j \\sum _ { i = 1 } ^ { j - 1 } \\alpha _ i + m _ j \\sum _ { i = j } ^ t u _ i \\\\ & = \\tilde { K } - m _ j \\left ( \\sum _ { i = j } ^ t n _ i - \\sum _ { i = 1 } ^ t \\alpha _ i + \\sum _ { i = 1 } ^ { j - 1 } \\alpha _ i - \\sum _ { i = j } ^ t u _ i \\right ) = \\tilde { K } . \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} \\Psi _ { \\Theta , v } = \\{ \\psi _ { \\Theta , v ; j } ( x ) = \\Theta x + v _ j : j = 1 , \\ldots , m \\} \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} \\gamma _ 0 ( H ^ 2 ( \\Omega ) ) = \\gamma _ 0 ( H ^ 2 _ { \\lambda , N } ( \\Omega ) ) = \\mathcal S ^ { \\frac { 3 } { 2 } } ( \\partial \\Omega ) \\ ( = \\mathcal S ^ { \\frac { 3 } { 2 } } _ { \\lambda } ( \\partial \\Omega ) ) \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} { \\mathcal M } _ { \\frak z } f ( x _ 1 , x _ 2 , x _ 3 ) = \\sup _ { R \\ni ( x _ 1 , x _ 2 , x _ 3 ) } \\frac { 1 } { | R | } \\int _ { R } | f ( u _ 1 , u _ 2 , u _ 3 ) | d u _ 1 d u _ 2 d u _ 3 , \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} u _ j ( r , \\omega ) : = m a x \\{ r , \\frac { 1 } { j } \\} ^ { - ( n - 2 ) / 2 } \\cdot w _ 1 ( \\omega ) \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} \\beta _ 1 = \\xi _ 2 - \\xi _ 3 , \\beta _ 2 = \\xi _ 3 - \\xi _ 1 , \\quad \\beta _ 3 = \\xi _ 1 - \\xi _ 2 , \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} | \\partial _ { t } \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) v _ { 4 } \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | & \\leq \\frac { C } { t ^ { 3 } \\log ^ { 2 b + 2 N } ( t ) } + \\frac { C } { t ^ { 2 } \\log ^ { b + 2 N - 1 } ( t ) } \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} { } ^ b \\mathcal { A } ^ \\epsilon _ G ( M ) : = ( \\mathcal { A } ( G ) \\hat { \\otimes } \\ , { } ^ b \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } + ( \\mathcal { A } ( G ) \\hat { \\otimes } \\ , \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} \\mathbb { P } _ \\eta ( \\mathcal { G } ^ c ) \\leq \\mathbb { E } _ \\eta \\left ( \\prod _ { k = d \\lfloor \\alpha t \\rfloor } ^ { \\lfloor \\beta t \\rfloor } ( \\mathbb { 1 } _ { \\mathcal { U } _ k } \\mathbb { 1 } _ { \\mathcal { G } _ { k } ^ c } ) \\right ) . \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} \\partial _ { r } v _ { 1 } ( t , r ) + \\partial _ { r } v _ { 2 } ( t , r ) = \\int _ { t } ^ { \\infty } \\frac { \\lambda '' ( s ) d s } { 1 + s - t } - \\frac { b } { t ^ { 2 } \\log ^ { b } ( t ) } + E _ { \\partial _ { r } v _ { 1 } } ( t , r ) + E _ { \\partial _ { r } v _ { 2 } } ( t , r ) \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} ( f ^ { - 1 } g ) '' = 2 f ^ { - 1 } f ' f ^ { - 1 } f ' f ^ { - 1 } g - 2 f ^ { - 1 } f ' f ^ { - 1 } g ' - f ^ { - 1 } f '' f ^ { - 1 } g + f ^ { - 1 } g '' . \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} K = \\begin{bmatrix} 0 & 0 & 0 & 0 \\\\ 0 & K _ { 1 , 2 2 } & K _ { 1 , 2 3 } & K _ { 1 2 , 2 } \\\\ 0 & K _ { 1 , 2 3 } ^ { \\top } & K _ { 1 , 3 3 } & K _ { 1 2 , 3 } \\\\ 0 & K _ { 1 2 , 2 } ^ { \\top } & K _ { 1 2 , 3 } ^ { \\top } & K _ 2 \\end{bmatrix} \\ , . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} \\psi _ { \\Lambda _ 1 } ( a _ \\Lambda ) = \\sum _ { i \\in I } \\prod _ { x \\in \\Lambda } \\hbox { T r } ( h _ { x , i } h _ { x , i } ^ { * } a _ x ) \\beta _ { i i } ^ { | \\Lambda _ 1 \\setminus \\Lambda | } + \\sum _ { i \\ne j } \\prod _ { x \\in \\Lambda } \\hbox { T r } ( h _ { x , i } h _ { x , j } ^ { * } a _ x ) c ^ { | \\Lambda _ 1 \\setminus \\Lambda | } , \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} \\int _ { t } ^ { \\infty } \\frac { R H S _ { 3 } ( z ) } { 4 \\alpha } r _ { 2 } ( - t , - z ) d z = \\frac { R H S _ { 3 } ( t ) } { 4 \\alpha } - e ''' ( t ) , t \\geq T _ { 0 } \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} { } _ \\alpha \\phi _ \\beta \\ ! \\left [ \\begin{matrix} a _ 1 , \\ldots , a _ \\alpha \\\\ b _ 1 , \\ldots , b _ \\beta \\end{matrix} ; q , z \\right ] : = \\sum _ { k = 0 } ^ \\infty \\frac { ( a _ 1 , \\ldots , a _ \\alpha ; q ) _ k } { ( q , b _ 1 , \\ldots , b _ \\beta ; q ) _ k } \\left ( ( - 1 ) ^ k q ^ { \\binom k 2 } \\right ) ^ { 1 + \\beta - \\alpha } z ^ k . \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} \\vartheta : = ( 4 e ^ { L h } B L + 9 B L ) h ^ 2 + ( 3 e ^ { L h } + 4 ) \\varepsilon h + e ^ { L h } \\delta \\ , , \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} & | \\frac { 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } - \\sqrt { ( 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } ) ^ { 2 } + 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } } { - 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } | \\leq C \\frac { R ^ { 2 } \\lambda ( t ) ^ { 2 } } { R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\leq C , \\\\ & R \\lambda ( t ) > \\frac { w } { 2 } , w \\geq 1 \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} \\phi _ { - } : = \\inf _ { x \\in C _ { s , 1 / 8 } ^ { + } } \\phi ( x ) \\quad \\quad \\phi _ { + } : = \\sup _ { x \\in C _ { s , 1 / 4 } ^ { + } \\setminus C _ { s , 3 / 1 6 } ^ { + } } \\phi ( x ) . \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} Y _ { i t } & = X _ { 1 i t } + \\lambda _ { i 1 } f _ { t 1 } + \\lambda _ { i 2 } f _ { t 2 } + E _ { i t } , \\\\ X _ { 1 i t } & = \\frac { 1 } { 2 } \\lambda _ { i 1 } f _ { t 1 } + \\lambda _ { i 2 } f _ { t 2 } + E _ { 1 i t } , \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} a _ k ( n ) & = a _ k ( n - 1 ) + a _ k ( n - 3 ) - a _ k ( n - 6 ) - a _ k ( n - 1 0 ) \\\\ & \\quad \\ , + a _ k ( n - 1 5 ) + a _ k ( n - 2 1 ) - a _ k ( n - 2 8 ) - a _ k ( n - 3 6 ) + \\cdots . \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} f ( x ) = \\alpha u ( x ) + \\beta v ( x ) + \\int _ x ^ b \\big ( u ( x ) v ( y ) - v ( x ) u ( y ) \\big ) g ( y ) \\d y , \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} \\forall l \\in I ^ { \\varphi _ \\textup { K S } } ( \\bar x ) \\colon \\xi _ l : = \\begin{cases} \\mu _ l / H _ l ( \\bar x ) & l \\in I ^ { 0 + } ( \\bar x ) , \\\\ \\nu _ l / G _ l ( \\bar x ) & l \\in I ^ { + 0 } ( \\bar x ) , \\\\ 0 & l \\in I ^ { 0 0 } ( \\bar x ) , \\end{cases} \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} I I = \\sum _ { i , j , k = 1 } ^ 2 \\int _ { \\Omega } \\partial _ j ^ 2 u _ i \\partial _ i \\theta \\partial _ k ^ 2 \\theta \\dd x + I _ b \\ ; . \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} \\frac { d } { d t } \\| \\theta ( t ) \\| _ { H ^ { 1 + s } } ^ 2 + \\| \\partial _ 1 \\theta \\| _ { H ^ { 1 + s } } ^ 2 \\leq C ( 1 + \\| \\nabla u \\| _ { L ^ \\infty } + \\| \\partial _ 1 \\theta \\| _ { L ^ \\infty } ) \\times ( \\| \\omega \\| _ { H ^ s } ^ 2 + \\| \\theta \\| _ { H ^ { 1 + s } } ^ 2 ) . \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ { t t } v _ { 2 } + \\partial _ { r r } v _ { 2 } + \\frac { 1 } { r } \\partial _ { r } v _ { 2 } - \\frac { v _ { 2 } } { r ^ { 2 } } = 0 \\\\ v _ { 2 } ( 0 ) = 0 \\\\ \\partial _ { t } v _ { 2 } ( 0 ) = v _ { 2 , 0 } \\end{cases} \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} R ( \\lambda ) + Q ( \\lambda ) e ^ { - \\lambda \\tau } = 0 , \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} Z ( b ) = Z _ M ( b ) = \\pi ^ { - 1 } ( b U ( M ) ) L ( b ) = L _ M ( b ) = \\{ | z | \\mid z \\in Z ( b ) \\} . \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} \\| u \\| _ { 1 , \\mathcal { H } , 0 } = \\| \\nabla u \\| _ { \\mathcal { H } } , \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} P _ { m } \\left ( z ; \\frac { d } { 2 } \\right ) & = z ^ { m } , P _ { m } ^ { \\mathrm { i p } } \\left ( z ; \\frac { d } { 2 } \\right ) = \\begin{cases} z ( z - 1 ) \\cdots ( z - m + 1 ) & ( m \\not = 0 ) \\\\ 1 & ( m = 0 ) \\end{cases} . \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} g ( ( T \\times T ^ 2 ) ^ n ( a , b , a , b ) ) & = ( n \\alpha , n ^ 2 \\alpha + 2 n a , 2 n \\alpha , 4 n ^ 2 \\alpha + 4 n a ) \\\\ & = ( S _ { 2 a } \\times S _ { 2 a } ^ 2 ) ^ n ( 0 , 0 , 0 , 0 ) \\\\ & = ( S _ { 2 a } \\times S _ { 2 a } ^ 2 ) ^ n ( g ( a , b , a , b ) ) , \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} \\det \\left ( W ^ { \\top } G \\left ( \\lambda \\right ) W - D + \\lambda I _ { 2 } \\right ) = 0 \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} { \\rm I d } ( [ \\lambda 1 + a , A ^ \\sharp ] ) = { \\rm I d } ( [ a , A ] ) \\subseteq A . \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} f _ { 1 } = - \\frac { 1 } { 2 } \\log \\left ( 1 - 2 \\beta J \\right ) + \\beta ( J ' - J ) + \\frac { 1 } { 4 } \\log \\left ( 1 - 4 \\beta ^ 2 \\right ) . \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ i = m . \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} \\alpha _ q ( \\eta ) : = \\limsup _ { t \\rightarrow \\infty } \\frac { \\log _ q \\big ( A _ q ( \\eta N _ t ) \\big ) } { D _ t } . \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} & V ^ { ( k _ 2 ) } = A ^ { ( 1 ) } p ^ { ( 1 ) } m ( b ) s ^ { ( k _ 2 ) } B ^ { ( k _ 2 ) } = W ^ { ( 1 ) } , \\\\ & c ^ { ( k _ 2 ) } = p ^ { ( 1 ) } m ( b ) s ^ { ( k _ 2 ) } = d ^ { ( 1 ) } . \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} ( \\rho ( \\tau ) K _ z ) ( w ) = K _ z ( \\alpha ( \\tau ^ { - 1 } ) ( w ) ) = K _ { \\alpha ( \\tau ) ( z ) } ( \\alpha ( \\tau ) ( \\alpha ( \\tau ^ { - 1 } ) ( w ) ) ) = K _ { \\alpha ( \\tau ) ( z ) } ( w ) . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} \\boldsymbol { B } ( \\phi , \\phi ) = 0 , \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( F \\left ( ( S ^ { L , K } _ t ) _ { t \\in [ 0 , L ] } \\right ) \\right ) = \\mathbf { E } \\left ( F \\left ( ( S _ t ) _ { t \\in [ 0 , L ] } \\right ) \\ : \\vline \\ : \\mathcal { A } _ { L , K } , \\ : S _ L > 0 \\right ) , \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} J _ 1 = \\int _ K \\left ( \\int _ { K } \\frac { ( f ( y ) - f ( x ) ) \\tilde { \\psi } ( y ) } { | x - y | ^ { d + \\alpha } } d y \\right ) ^ 2 d x . \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} & | \\frac { - c _ { b } r } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\sin ^ { 2 } ( \\theta ) d \\theta ( b - 1 ) ( t , r , \\theta ) | \\leq C r \\int _ { 0 } ^ { \\pi } \\frac { d \\theta } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\\\ & \\leq \\frac { C r } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} & K ( x _ 0 , x _ 1 , y _ 0 , y _ 1 , t _ 0 , t _ 1 ) = x _ { { 0 } } x _ { { 1 } } y _ { { 0 } } y _ { { 1 } } - \\\\ & t \\left ( d _ { { - 1 , - 1 } } { x _ { { 1 } } } ^ { 2 } { y _ { { 1 } } } ^ { 2 } + d _ { { - 1 , 0 } } { x _ { { 1 } } } ^ { 2 } y _ { { 0 } } y _ { { 1 } } + d _ { { 0 , - 1 } } x _ { { 0 } } x _ { { 1 } } { y _ { { 1 } } } ^ { 2 } + d _ { { 0 , 0 } } x _ { { 0 } } x _ { { 1 } } y _ { { 0 } } y _ { { 1 } } + d _ { { 0 , 1 } } x _ { { 0 } } x _ { { 1 } } { y _ { { 0 } } } ^ { 2 } + d _ { { 1 , 1 } } { x _ { { 0 } } } ^ { 2 } { y _ { { 0 } } } ^ { 2 } \\right ) . \\\\ \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} \\phi ( K ) = \\begin{bmatrix} Q _ { 1 , 1 1 } & Q _ { 1 , 1 2 } & Q _ { 1 , 1 3 } & Q _ { 1 2 , 1 } \\\\ Q _ { 1 , 1 2 } ^ { \\top } & * & * & * \\\\ Q _ { 1 , 1 3 } ^ { \\top } & * & * & * \\\\ Q _ { 1 2 , 1 } ^ { \\top } & * & * & * \\end{bmatrix} \\ , . \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} \\kappa ( G _ { n , q } ) = \\sum _ { k = 1 } ^ { k _ 0 } A _ k + \\sum _ { k = 3 } ^ { k _ 0 } B _ k + R , \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} \\begin{cases} D ^ { \\alpha _ 1 } _ 0 y ( t ) = w ( t ) , & ~ y ( 0 ) = y _ 0 , \\ , y ( b ) = y _ b , \\\\ D ^ { 1 - \\alpha _ 1 } _ 0 w ( t ) = z ( t ) , & ~ w ( 0 ) = 0 , \\\\ D ^ { \\alpha _ 2 - 1 } _ 0 z ( t ) = f ( t , y ( t ) , w ( t ) ) , & ~ z ( 0 ) = y ^ { ( 1 ) } ( 0 ) . \\end{cases} \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} M \\le \\prod _ { k = 1 } ^ s \\sup \\{ | z - \\alpha _ k | _ p \\ , ; \\ , z \\in B \\} ^ { n _ k } = \\prod _ { k = 1 } ^ s \\max \\{ | \\alpha _ i - \\alpha _ k | _ v , \\ , \\delta \\} ^ { n _ k } . \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} & \\left ( \\cdots \\left ( \\varphi _ { k _ 1 l _ 1 } ^ { r _ 2 ' \\cdots r _ s ' } \\right ) \\cdots \\right ) _ { k _ s l _ s } ( X _ 1 , \\cdots , X _ j + 2 \\pi , \\cdots , X _ n , Y _ 1 , \\cdots , Y _ n ) \\\\ & = \\left ( \\cdots \\left ( \\varphi _ { k _ 1 l _ 1 } ^ { r _ 2 ' \\cdots r _ s ' } \\right ) \\cdots \\right ) _ { k _ s l _ s } ( X _ 1 , \\cdots , X _ n , Y _ 1 , \\cdots , Y _ n ) . \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } & \\partial _ t \\rho + u \\cdot \\nabla \\rho = f , \\\\ & \\rho ( 0 , x ) = \\rho _ 0 ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} p ( z ) & = ( z - b ) ( a - b ) ^ { - 1 } f ( a ) + ( z - a ) ( b - a ) ^ { - 1 } f ( b ) \\cr & = \\left ( b ( b - a ) ^ { - 1 } f ( a ) a ( a - b ) ^ { - 1 } f ( b ) \\right ) + z \\left ( ( a - b ) ^ { - 1 } f ( a ) - a ( b - a ) ^ { - 1 } f ( b ) \\right ) . \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} g : = g ( a _ 3 , a _ 4 , b _ 3 , b _ 4 , c _ 1 , c _ 3 , d _ 2 , d _ 4 ) \\mapsto \\left ( \\left ( \\begin{array} { c c } 2 a _ 3 + a _ 4 & b _ 3 \\\\ 2 c _ 1 + c _ 3 & 2 d _ 2 + d _ 4 \\end{array} \\right ) , \\left ( \\begin{array} { c c } a _ 4 & b _ 3 - 2 b _ 4 \\\\ c _ 3 & d _ 4 \\end{array} \\right ) \\right ) \\ , , \\\\ \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} \\begin{aligned} & \\rho _ t ^ { k + 1 } + \\nabla \\cdot \\left ( \\rho ^ { k + 1 } Q \\ast \\mathbf { u } ^ k \\right ) = 0 , \\\\ & \\mathbf { u } _ t ^ { k + 1 } + ( \\mathbf { u } ^ k \\cdot \\nabla ) \\mathbf { u } ^ { k + 1 } = \\rho ^ { k } \\left ( Q \\ast \\mathbf { u } ^ { k + 1 } - \\mathbf { u } ^ { k + 1 } \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\infty } \\sup _ { 0 \\leq t \\leq T } \\int \\| \\bar { x } - \\bar { \\eta } ^ { \\nu } ( t , \\cdot ) \\| _ { H ^ { - 1 } } f ( t , x ) \\mu _ { N , m } ( d x ) = 0 . \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} ( \\omega _ { 1 } \\rtimes u ) ( V ^ { l _ { 1 } } U ^ { k _ { 1 } } ) \\left ( z ^ { 0 } \\otimes \\varepsilon _ { 0 } \\right ) = \\lambda ^ { l _ { 1 } k _ { 1 } } \\ , z ^ { l _ { 1 } } \\otimes \\varepsilon _ { k _ { 1 } } , \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} \\bigg [ - \\sum _ { i = 1 } ^ { N } \\frac { \\partial ^ 2 } { \\partial r _ i ^ 2 } + \\omega ^ 2 r ^ 2 + \\sum _ { i = 1 } ^ { N } \\frac { 1 } { r _ i ^ 2 } ( \\lambda _ i + \\frac { 1 } { 4 } ( d _ i - 1 ) ( d _ i - 3 ) ) \\bigg ] R ( r _ 1 , \\cdots , r _ N ) = E R ( r _ 1 , \\cdots , r _ N ) . \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} & \\frac { 1 } { C _ { 1 } \\xi \\log ^ { 2 } ( \\xi ) } \\leq \\rho ( \\xi ) \\leq \\frac { C _ { 1 } } { \\xi \\log ^ { 2 } ( \\xi ) } , 0 < \\xi < \\frac { 1 } { 2 e ^ { 2 } } \\\\ & \\frac { \\xi } { C _ { 1 } } \\leq \\rho ( \\xi ) \\leq C _ { 1 } \\xi , \\xi > \\frac { 1 } { 2 e ^ { 2 } } \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} p _ { n + 1 } ( \\underline x _ { n + 1 } ) = ( 1 + x _ 1 ) ( 1 + x _ { n + 1 } ) \\cdot \\sum _ { I \\subseteq [ 2 , n ] } \\left ( \\prod _ { i \\in I } x _ i \\right ) s _ n ^ I ( \\underline x _ n ) . \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} K = \\begin{bmatrix} 0 & 0 & 0 & K _ { 1 2 , 1 } \\\\ 0 & K _ { 1 , 2 2 } ^ - & K _ { 1 , 2 3 } ^ - & K _ { 1 2 , 2 } \\\\ 0 & K _ { 1 , 2 3 } ^ { - \\top } & K _ { 1 , 3 3 } ^ + & K _ { 1 2 , 3 } \\\\ K _ { 1 2 , 1 } ^ { \\top } & K _ { 1 2 , 2 } ^ { \\top } & K _ { 1 2 , 3 } ^ { \\top } & K _ 2 \\end{bmatrix} \\ , . \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} 0 & = \\sum \\limits _ { i \\in I ^ g ( x ) } \\lambda _ i \\nabla g _ i ( x ) + \\sum \\limits _ { j \\in \\mathcal P } \\rho _ j \\nabla h _ j ( x ) + \\sum \\limits _ { l \\in I ^ { \\varphi ^ t _ \\textup { F B } } ( x ) } \\xi _ l \\left ( \\alpha _ l \\nabla G _ l ( x ) + \\beta _ l \\nabla H _ l ( x ) \\right ) \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} \\Theta _ q ( a , b , n ) & = \\frac { ( b - q ^ n ) ( a b - 1 - a ^ 2 + a q ^ n ) } { ( a - b ) ( 1 - a b ) } \\frac { ( b q ) ^ { ( n + 1 ) / 2 } ( q ^ { - 2 } / b ; q ^ 2 ) _ { ( n + 1 ) / 2 } } { ( b q ^ 2 ; q ^ 2 ) _ { ( n + 1 ) / 2 } } \\\\ [ 5 p t ] & + \\frac { ( 1 - a q ^ n ) ( a - q ^ n ) } { ( a - b ) ( 1 - a b ) } \\frac { ( b ; q ^ 2 ) _ 2 ( q ^ { - 1 } ; q ^ 2 ) _ { ( n + 1 ) / 2 } ^ 2 } { ( q ^ { - 1 } ; q ^ 2 ) _ 2 ( q ^ 2 / a , a q ^ { 2 } ; q ^ 2 ) _ { ( n + 1 ) / 2 } } . \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} \\begin{aligned} x _ { \\frac { n } { 4 } } & = \\frac { \\sin \\left ( \\beta - \\frac { \\pi } { n } \\right ) } { \\sin \\frac { 2 \\pi } { n } } = - \\frac { 1 } { 2 } , \\\\ y _ { \\frac { n } { 4 } } & = \\frac { \\sin \\left ( \\frac { \\pi } { n } - \\beta \\right ) } { \\sin \\frac { 2 \\pi } { n } } = \\frac { 1 } { 2 } . \\end{aligned} \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} 2 a ( n ) - 2 \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { n } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - m ^ 2 ) - 2 \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { \\frac { n } { 2 } } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - 2 m ^ 2 ) = 0 , \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 \\} = \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} 0 = C _ s \\int _ 0 ^ { \\frac { 1 } { 2 \\sqrt { N } } } \\frac { \\left ( 1 + \\tau ^ 2 + \\frac { 2 } { \\sqrt { N } } \\tau \\right ) ^ { - \\tilde \\gamma / 2 } + \\left ( 1 + \\tau ^ 2 - \\frac { 2 } { \\sqrt { N } } \\tau \\right ) ^ { - \\tilde \\gamma / 2 } - 2 } { \\tau ^ { 1 + 2 s } } \\ , d \\tau + C _ s O ( 1 ) , \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} \\log | \\Phi ( r \\zeta ) | = [ 1 - T ( r \\zeta ) ] \\log r , \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { n } K _ { - s } ^ { ( 3 ) } = \\frac { 1 } { 3 } \\left ( K _ { n + 2 } ^ { ( 3 ) } + 2 K _ { n } ^ { ( 3 ) } \\right ) - 2 ^ { n + 1 } - 2 ^ { - n } + 3 . \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ { \\varepsilon , k } } | H | = \\int _ { M } | H | + \\int _ { M ' } | H | + \\sum _ { i = 1 } ^ N \\int _ { S _ { \\varepsilon , k , i } } | H | = 2 \\int _ { M } | H | + \\sum _ { i = 1 } ^ N \\int _ { S _ { \\varepsilon , k , i } } | H | . \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} \\left ( A - \\nu _ { \\mathsf { k } } \\right ) \\mathsf { q = \\hat { p } } _ { \\mathsf { k } } . \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} - 1 & = \\Big ( u + \\tfrac { d - a } { 2 c } v \\Big ) ^ 2 - \\tfrac { D } { 4 c ^ 2 } v ^ 2 \\\\ & = \\Big ( u + \\tfrac { d - a } { 2 c } v \\Big ) ^ 2 - \\Big ( \\tfrac { \\sqrt { D } } { 2 c } v \\Big ) ^ 2 \\\\ & = \\Big ( u + \\tfrac { d - a + \\sqrt { D } } { 2 c } v \\Big ) \\Big ( u + \\tfrac { d - a - \\sqrt { D } } { 2 c } v \\Big ) \\\\ & = \\tilde u \\tilde v . \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\lambda _ { 4 } ( \\xi ) \\geq \\frac { 6 \\sqrt { 5 } - 5 } { 3 1 } = 0 . 2 7 1 5 \\ldots > \\frac { 1 } { 4 } . \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} h \\circ \\varphi _ t = \\Phi _ t \\circ h \\quad \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} b _ 1 ( u , \\theta , \\phi ) : = \\langle B _ 1 ( u , \\theta ) , \\phi \\rangle _ { V _ 1 ' } : = ( ( u \\cdot \\nabla ) \\theta , \\phi ) , u \\in \\mathcal { V } , \\ \\ \\theta , \\phi \\in \\mathcal { F } _ 2 . \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} & f _ k ( q ) = ( q ; q ) _ \\infty ( q ; q ^ 2 ) _ \\infty \\sum _ { n = 0 } ^ \\infty a _ k ( n ) q ^ n \\\\ & \\qquad \\ = ( q ; q ^ 2 ) _ \\infty ( q ^ { k } ; q ^ { k } ) _ \\infty \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { k n } + 2 q ^ { 2 k n } + \\dots + ( k - 1 ) q ^ { ( k - 1 ) k n } } { 1 + q ^ { k n } + q ^ { 2 k n } + \\dots + q ^ { ( k - 1 ) k n } } . \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} D ^ * \\coloneqq \\begin{cases} \\overline { D } , & , \\\\ \\overline { D } \\cup \\{ \\infty \\} , & , \\end{cases} \\enskip \\partial ^ * D \\coloneqq \\begin{cases} \\partial { D } , & , \\\\ \\partial { D } \\cup \\{ \\infty \\} , & , \\end{cases} \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} x _ i x _ { i + 1 } = q x _ { i + 1 } x _ i , \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} \\| \\nabla X \\| _ { L ^ \\infty _ t ( { L ^ 2 } ) } ^ 2 \\leq C . \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} \\mathfrak { g } = \\{ \\xi _ { 1 } , \\xi _ { 2 } , \\xi _ { 3 } , \\xi _ { 4 } , \\{ \\xi _ { 1 } , \\xi _ { 2 } \\} , \\{ \\xi _ { 1 } , \\xi _ { 3 } \\} , \\{ \\xi _ { 2 } , \\xi _ { 4 } \\} , \\{ \\xi _ { 3 } , \\xi _ { 4 } \\} \\} . \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} { P _ * } ( 4 b t , k ) \\int _ { 2 a t } ^ { 2 b t } \\frac { e ^ { i u ^ 2 / ( 4 t ) } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) e ^ { - i k u } d u = \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} \\Omega _ \\mu = \\{ x \\in \\R ^ 3 | \\ x _ 0 ^ 2 + \\frac { x _ 1 ^ 2 + x _ 2 ^ 2 } { e ^ { 2 \\nu } } < 1 \\} , \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} m _ \\lambda * f _ { k _ 1 } * \\ldots * f _ { k _ r } = e _ { \\lambda _ { 1 } } * e _ { \\lambda _ 2 } * \\ldots * e _ { \\lambda _ { n - r } } * f _ { k _ 1 } * \\ldots * f _ { k _ r } + \\mbox { l o w e r t e r m s } . \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } & \\partial _ t u + u \\cdot \\nabla u - \\nu _ 1 \\partial _ 1 ^ 2 u - \\nu _ 2 \\partial _ 2 ^ 2 u = - \\nabla p + \\theta e _ 2 , \\ \\\\ & \\partial _ t \\theta + u \\cdot \\nabla \\theta - \\kappa _ 1 \\partial _ 1 ^ 2 \\theta - \\kappa _ 2 \\partial _ 2 ^ 2 \\theta = 0 , \\\\ & \\nabla \\cdot u = 0 , \\\\ & u ( 0 , x ) = u _ 0 ( x ) , \\theta ( 0 , x ) = \\theta _ 0 ( x ) . \\end{array} \\right . \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} 1 + \\sum _ { n \\geq 1 } A _ n ( t ) \\frac { x ^ n } { n ! } = \\frac { 1 - t } { 1 - t e ^ { ( 1 - t ) x } } . \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} X _ n = P _ n [ X ] = P _ n [ \\overline { D ( H ) } ] \\subseteq \\overline { P _ n [ D ( H ) ] } \\subseteq X _ n , \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\ell _ j ( x ) : = p ( x ) p ( x _ j ) ^ { - 1 } , p ( x ) = p _ { \\{ x _ 0 , \\ldots , x _ n \\} \\setminus \\{ x _ j \\} } ( x ) , \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} \\int _ { X \\times Y } c ( x , y ) d \\gamma = \\min _ { \\tilde { \\gamma } \\in \\Pi ( \\mu , \\nu ) } \\int _ { X \\times Y } c ( x , y ) d \\tilde \\gamma . \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} F ( x ; - 1 , \\beta ) = \\frac 1 { 1 + \\beta } \\left ( \\left ( \\log \\frac 2 { x } \\right ) ^ { 1 + \\beta } - \\left ( \\log 2 \\right ) ^ { 1 + \\beta } \\right ) . \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} | \\O | = \\binom { n } { 3 } = 3 ^ { a - 1 } ( n - 1 ) ( n - 2 ) . \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} w _ n & = \\widehat { w } _ n - q _ { n - 1 } w _ { n - 1 } - \\cdots - q _ 1 w _ 1 \\\\ & = a _ { n , 1 } u _ 1 + \\cdots + a _ { n , n } u _ n , \\\\ \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} \\rho ( L _ 2 ) = \\rho ' ( L _ 2 ) + \\alpha h ( z ) ^ 2 \\ , . \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} 9 9 9 6 & = 5 8 ^ 2 + 1 4 ^ 2 + 6 ^ 2 + 8 0 ^ 2 \\ \\ 5 8 + 3 \\times 1 4 = 1 0 ^ 2 , \\\\ 9 9 9 9 9 9 9 9 & = 1 3 9 ^ 2 + 1 9 ^ 2 + 6 8 6 6 ^ 2 + 7 2 6 9 ^ 2 \\ \\ 1 3 9 + 3 \\times 1 9 = 1 4 ^ 2 , \\\\ 3 9 9 9 9 9 9 9 9 9 & = 2 3 4 7 ^ 2 + 1 8 ^ 2 + 1 2 6 7 1 ^ 2 + 1 5 2 9 5 ^ 2 \\ \\ 2 3 4 7 + 3 \\times 1 8 = 4 9 ^ 2 . \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} D _ i = \\{ ( x , x ^ { \\varphi _ { i , 1 } } , \\ldots , x ^ { \\varphi _ { i , m - 1 } } ) \\ , : \\ , x \\in T \\} \\cong T \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} g ( x , t ) = g _ { m , n } ( x , t ) = ( c _ 1 t - x ) ^ m ( c _ 2 t + x ) ^ n \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} m _ { k + 1 } = a _ { 1 1 } a ^ { ( 2 ) } _ { 2 2 } \\cdots a ^ { ( 2 k - 2 ) } _ { k k } a ^ { ( 2 k ) } _ { k + 1 k + 1 } . \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} W _ n = W _ { n - 1 } - \\min \\left \\{ W _ { n - 1 } , J - \\eta _ n \\right \\} + \\min \\left \\{ \\eta _ n , K - W _ { n - 1 } \\right \\} , \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} H _ { n + 1 } = H _ n + N H _ { n - k - 1 } , \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} x & = \\theta - ( \\theta ^ * + b \\dot { \\theta } ) , \\\\ y & = \\dot { \\theta } . \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} | g | _ B = \\sup \\{ | g ( z ) | _ p \\ , ; \\ , z \\in B \\} . \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} \\partial _ t ( \\partial _ k h ) = Q ( f , \\partial _ k h ) + Q ( \\partial _ k f , h ) + Q ( \\partial _ k h , \\mu ) + Q ( h , \\partial _ k \\mu ) , \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} \\varepsilon _ 1 ^ 2 = 0 = \\varepsilon _ 2 ^ 2 \\qquad \\textrm { a n d } \\qquad \\alpha \\varepsilon _ 1 = \\varepsilon _ 2 \\alpha . \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} \\mathrm { d i m } ~ \\mathrm { H o m } _ { G _ { n } } ( \\pi , \\pi ' ) & \\geq \\mathrm { d i m } ~ \\mathrm { H o m } _ { G _ { n } } ( \\mathrm { i n d } _ { R _ { n + 1 - i } } ^ { G _ { n } } \\pi ^ { [ i ] } \\boxtimes \\psi _ { i - 1 } , \\pi ' ) \\\\ & = \\mathrm { d i m } ~ \\mathrm { H o m } _ { G _ { n + 1 - i } } ( \\pi ^ { [ i ] } , { } ^ { ( i - 1 ) } \\pi ' ) \\neq 0 . \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} \\sum _ { k = 3 } ^ \\infty \\frac { k ^ { k - 2 } } { k ! } \\Bigg [ k - 1 - k \\beta \\Bigg ] \\big ( \\beta e ^ { - \\beta } \\big ) ^ { k - 2 } & < \\frac { 2 e ^ 2 } { 3 ^ { 5 / 2 } \\sqrt { 2 \\pi } } \\sum _ { k = 3 } ^ \\infty \\big ( \\beta e ^ { 1 - \\beta } \\big ) ^ { k - 2 } \\\\ & = \\frac { 2 e ^ 2 } { 3 ^ { 5 / 2 } \\sqrt { 2 \\pi } } \\frac { \\beta e ^ { 1 - \\beta } } { 1 - \\beta e ^ { 1 - \\beta } } . \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} = p \\mathbb { P } _ \\eta ( \\tau _ x \\leq t ) + \\eta ( x ) \\mathbb { P } _ \\eta ( \\tau _ x > t ) = p - p \\mathbb { P } _ \\eta ( \\tau _ x > t ) + \\eta ( x ) \\mathbb { P } _ \\eta ( \\tau _ x > t ) \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} I ( t ) & = \\left ( \\int _ 0 ^ T + \\int _ T ^ t \\right ) \\omega ( t - s , \\lambda _ 1 ) p ( s ) G ( \\| u [ \\xi ] ( \\cdot , s - \\rho ( s ) ) \\| ) d \\tau \\\\ & \\le G ( 2 \\eta ) \\int _ 0 ^ T \\omega ( t - s , \\lambda _ 1 ) p ( s ) d s + \\varepsilon \\int _ T ^ t \\omega ( t - s , \\lambda _ 1 ) p ( s ) d s \\\\ & \\le G ( 2 \\eta ) \\omega ( t - T , \\lambda _ 1 ) \\int _ 0 ^ T p ( s ) d s + \\varepsilon M \\\\ & \\le [ G ( 2 \\eta ) + M ] \\varepsilon , \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} \\Psi ( z ) = 1 + \\left | c _ { 3 } \\right | e ^ { i \\theta } ( z - 1 ) ^ { 3 } + \\sum _ { n = 4 } ^ { \\infty } c _ { n } ( z - 1 ) ^ { n } \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} [ e _ a , e _ b ] _ Q : = C ^ c { } _ { a b } ( X ) e _ c . \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} k ( x ) ^ { 3 } = \\left | \\int _ { x _ { 0 } } ^ { x } 3 k ( s ) ^ { 2 } k ' ( s ) \\ , d s \\right | . \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} \\dim H ^ 0 ( X _ { n , s } , \\mathcal { O } ( D ) ) = \\sum _ { I , \\sigma _ t } ( - 1 ) ^ { | I | } { { n + k _ { I , \\sigma _ t } - r _ { I , \\sigma _ t } - 1 } \\choose n } , \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} \\mathcal { M } ( t ) : = \\frac { 1 } { 2 } \\norm { \\langle \\nu ( t ) , \\dot { \\Gamma } ( t ) \\rangle } ^ 2 + \\mathcal { V } ( \\Gamma ( t ) ) , \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} ( K T x ) _ k = \\sum \\limits _ { r \\in \\mathbb Z ^ c } \\Bigl ( \\sum \\limits _ { m \\in \\mathbb Z ^ c } a _ { k m } b _ { k - m , r - m } \\Bigr ) x _ { k - r } , k \\in \\mathbb Z ^ c . \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} \\varprojlim _ { i , j \\in \\N } \\Sigma _ { i j } = \\varprojlim _ { j \\in \\N } \\Sigma _ { j } \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} Y = Y _ { M , m } = \\left \\{ ( y _ 1 , \\cdots , y _ M ) ; \\ \\frac { 1 } { M } \\sum _ { l = 1 } ^ M y _ l = m \\right \\} . \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} 0 \\longrightarrow R \\xrightarrow { \\begin{pmatrix} x \\\\ y \\\\ z ^ 2 \\end{pmatrix} } R ^ 3 \\xrightarrow { \\begin{pmatrix} 0 & z ^ 2 & - y \\\\ - z ^ 2 & 0 & x \\\\ y & - x & 0 \\end{pmatrix} } R ^ 3 \\xrightarrow { \\begin{pmatrix} x & y & z ^ 2 \\end{pmatrix} } R \\longrightarrow R / I \\longrightarrow 0 \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} \\left . \\frac { d } { d \\gamma } \\left ( - \\log \\det ( \\Delta _ { C _ { \\gamma } } ) \\right ) \\right | _ { \\gamma = \\alpha } = \\left . \\frac { d } { d \\gamma } \\left ( - 2 \\log \\det ( \\Delta _ { S _ { \\gamma } } ) \\right ) \\right | _ { \\gamma = \\frac \\alpha 2 } \\frac 1 2 = \\left . \\frac { d } { d \\gamma } \\left ( - \\log \\det ( \\Delta _ { S _ { \\gamma } } ) \\right ) \\right | _ { \\gamma = \\frac \\alpha 2 } . \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} \\mathcal { B } ^ n = H ^ { 1 } ( I _ n ) , \\mathcal { A } ^ n ( f , g ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { d } \\int _ { I _ n } \\partial _ i f \\partial _ i g \\ , d m , f , g \\in \\mathcal { B } ^ n . \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y _ l } \\bar { H } ( y ) = \\frac { 1 } { M } y _ l + \\frac { 1 } { N } \\sum _ { j \\in B ( l ) } s _ j + \\frac { 1 } { N } \\mathbb { E } _ { \\mu _ { N , m } ( d x | y ) } \\left [ \\sum _ { i = 1 } ^ N \\sum _ { j \\in B ( l ) } M _ { i j } X _ i + \\sum _ { i \\in B ( l ) } \\delta \\psi ' ( X _ i ) \\right ] . \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} S _ 1 & = ( - 1 ) ^ { k - 1 + \\lfloor \\frac { k - 2 } { p } \\rfloor } ( k - 2 ) ! \\left ( ( k - 1 ) \\binom { q / p - 1 } { \\lfloor ( k - 2 ) / p \\rfloor } - q \\binom { q / p - 2 } { \\lfloor ( k - 2 ) / p \\rfloor - 1 } \\right ) \\\\ & = ( - 1 ) ^ { k - 1 + \\lfloor \\frac { k - 2 } { p } \\rfloor } ( k - 2 ) ! \\left ( \\frac { q \\{ k - 2 \\} _ p - p ( k - 2 ) } { q - p } + 1 \\right ) \\binom { q / p - 1 } { \\lfloor ( k - 2 ) / p \\rfloor } . \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} \\dot { x } & = y - y ^ 2 , \\\\ \\dot { y } & = - x , \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} \\psi _ { k , k - n + 2 } ^ { \\ast } ( Q ^ { \\prime } ) = \\frac { L _ { k , k - n + 1 } ^ { \\ast } ( q ^ { \\prime } ) } { q ^ { \\prime } } \\leq \\frac { L _ { k , R _ { k - n + 2 } } ^ { \\ast } ( q ^ { \\prime } ) } { q ^ { \\prime } } \\leq \\Phi _ { k , n } ( \\psi _ { n , 2 } ^ { \\ast } ( Q ) ) . \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} F _ { b } ( \\xi ) = \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) \\partial _ { \\xi } ^ { 2 } \\left ( \\xi ^ { 2 } \\frac { K _ { 1 } ( \\xi \\lambda ( t ) ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\right ) + \\psi _ { v _ { 2 } } ( \\xi , \\lambda ( t ) ) \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} u & = \\sum _ { i = n - r } ^ { n - 1 } ( - 1 ) ^ i \\binom { 2 n - 1 - r } { i } e _ { r + 1 - n + j , j + 1 } \\\\ v & = \\sum _ { i = 0 } ^ { r - 1 } ( - 1 ) ^ i \\binom { r - 1 } { i } e _ { n - r - 1 - i , i + 1 } , \\\\ \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} \\int _ { \\mathcal { L } _ { p } \\left ( 0 , \\ , A \\exp \\left ( B \\ , \\omega ( a ) \\right ) \\right ) } & G ( x , y ) \\ , | f ( y ) | \\ , d y = \\int _ { \\mathcal { L } _ { p } \\left ( 0 , \\ , ( 2 A ) ^ { - 1 } \\exp ( - B \\omega ( m _ { 0 } ) ) \\right ) } G ( x , y ) \\ , | f ( y ) | \\ , d y \\\\ & \\quad + \\int _ { \\mathcal { L } _ { p } \\left ( ( 2 A ) ^ { - 1 } \\exp ( - B \\omega ( m _ { 0 } ) ) , \\ , A \\exp \\left ( B \\ , \\omega ( a ) \\right ) \\right ) } G ( x , y ) \\ , | f ( y ) | \\ , d y \\ , . \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} ( \\sum _ { s = i _ { 1 } } ^ { j _ { 2 } } m _ { s , s + 1 } ) m _ { i _ { 1 } , n + 1 } = - ( j _ { 2 } - i _ { 1 } + 1 ) m _ { i _ { 2 } , j _ { 2 } + 1 } m _ { i _ { 1 } , n + 1 } + \\cdots , \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} K _ { - n } ^ { ( 3 ) } = K _ { n } ^ { ( 3 ) } + 2 ^ { - n } - 2 ^ { n } , \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} \\frac { 1 } { 2 t _ n } \\left ( u _ n \\Phi ( x ) u _ n ^ { - 1 } - \\Phi ( x ) \\right ) & = - \\Phi ( x _ n ) + \\frac { 1 } { 2 t _ n } \\sum \\limits _ { k = 1 } ^ { n - 1 } \\left ( u _ n \\Phi ( t _ k x _ k ) u _ n ^ { - 1 } - \\Phi ( t _ k x _ k ) \\right ) . \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} \\mathcal { C } _ { n } : = \\{ ( s _ { 1 } , \\cdots , s _ { n } ) \\in \\mu _ { \\infty } ^ { n } : s _ { i } \\neq s _ { j } 1 \\leq i < j \\leq n \\} . \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} s ( f , \\Upsilon , \\mathcal { P } ^ 1 _ n ) & = \\sum _ { k } f ( x _ k ^ { n , 1 } ) \\ , | I ^ 1 _ k | _ \\Upsilon = \\sum _ { k } f ( x _ k ^ { n , 1 } ) \\ , \\int _ { I _ k ^ 1 } \\lambda \\ , \\psi \\\\ & = \\sum _ { k } f ( x _ k ^ { n , 1 } ) \\ , \\gamma \\int _ { I _ k } \\psi = \\gamma \\ , S ( f , \\Psi , \\mathcal P _ n ) \\ , , \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} P _ 1 = t ^ { \\frac { p } { p - 1 } } ( 1 - \\xi ^ { \\sigma _ 1 } ) \\exp \\left \\{ \\frac { \\tau \\log \\xi } { p - 1 } \\right \\} = t ^ { \\frac { p } { p - 1 } } ( 1 - \\xi ) \\xi ^ { \\frac { \\tau } { p - 1 } } . \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} \\rho ( L _ 1 ) \\ , = \\ , \\rho ( L _ { - 1 } ) ^ { - 1 } \\left ( \\rho ( L _ 0 ) ^ 2 - \\rho ( L _ 0 ) - \\mu ( \\mu + 1 ) \\right ) . \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} | \\{ ( a + b ) ( b + d ) = 1 ~ : ~ a \\in A , \\ , d \\in D , \\ , b \\in \\o \\cdot [ N ] \\} | \\ll \\sqrt { | A | | D | } N ^ { 2 / 3 } \\cdot | D | ^ { 1 / 6 l } \\ , . \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} & I = \\frac { 1 6 } { \\lambda ( t ) ^ { 3 } } \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) \\partial _ { 2 } K ( s - t , \\lambda ( t ) ) \\lambda ' ( t ) \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} L _ u ( S f _ k ) = ( \\lambda _ k + 1 ) S f _ k - \\langle S f _ k \\ , | \\ , u \\rangle 1 \\ . \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} \\mathbb { P } _ { n _ { l _ q } } \\left ( \\left | \\log \\left ( \\frac { d \\tilde { \\mathbb { Q } } _ { n _ { l _ { q } } } } { d \\mathbb { P } _ { n _ { l _ { q } } } } \\right ) - \\sum _ { k = 1 } ^ { m _ { n _ { l _ q } } } \\frac { 2 \\mu _ { k } ( C _ { n _ { l _ q } , k } - ( n - 1 ) \\mathbb { I } _ { k = 2 } ) - \\mu _ { k } ^ { 2 } } { 4 k } \\right | \\ge \\epsilon \\right ) \\le \\delta . \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} L _ { 1 } ( u ) ( t , r ) = \\frac { \\sin ( 2 u ( t , r ) ) } { 2 r ^ { 2 } } \\left ( \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) \\left ( \\cos ( 2 v _ { c o r r } ) - 1 \\right ) - \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\sin ( 2 v _ { c o r r } ) \\right ) \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} x ^ * ( c ) = \\left ( \\displaystyle \\frac { 3 c } { 1 + 4 c } , \\displaystyle \\frac { 3 c } { 1 + 4 c } \\right ) ^ T . \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} \\rho ^ * \\Psi _ { \\mathrm { G F } } \\ , = \\ , - 2 ( 1 - 6 c + 6 c ^ 2 ) v ( h ( z ) ) \\Psi \\ , = \\ , ( 1 - 3 \\rho ( { \\mathcal { C } } ) ) v ( h ( z ) ) \\Psi \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} Z _ l = \\sum _ { 1 \\leq i < k \\leq l } L ^ 2 _ { i k } - \\bigg ( \\sum _ { i = 1 } ^ { l } x _ i ^ 2 \\bigg ) \\bigg ( \\sum _ { k = 1 } ^ { l } \\frac { \\alpha _ i } { x _ i ^ 2 } \\bigg ) , ~ ~ ~ l = 2 , \\cdots , D - 1 , \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} B ^ k A ^ k - ( B ^ k A ^ k ) ^ 2 S & = B ^ k A ^ k - ( B ^ k A ^ k ) ^ 2 B ^ k T ^ 2 A ^ K = B ^ k A ^ k - B ^ k ( A ^ k B ^ k A ^ k ) B ^ k T ^ 2 A ^ k \\\\ & = B ^ k A ^ k - B ^ k A ^ { k + 2 } T ^ 2 = B ^ k ( I - A ^ 2 T ^ 2 ) A ^ k = B ^ k ( I - A T ) A ^ k \\\\ & = B ^ k Q A ^ k = B ^ k Q ^ k A ^ k = B ^ k ( Q A ) ^ k . \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} \\left [ \\int _ { \\mathbb { R } } \\frac { 1 + t ^ { 2 } } { | 1 - t G _ { \\mu _ { 1 } } ( F _ { \\rho _ { 1 } } ( x _ { 0 } ) ) | ^ { 2 } } \\ , d \\sigma ( t ) \\right ] \\left [ \\int _ { \\mathbb { R } } \\frac { d \\mu _ { 1 } ( t ) } { ( F _ { \\rho _ { 1 } } ( x _ { 0 } ) - t ) ^ { 2 } } \\right ] = 1 . \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} \\Gamma ( f ) = \\lambda _ \\Gamma ( f ) = \\int _ X f ( x ) d \\lambda _ \\Gamma ( x ) , \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} \\mathfrak { p } _ { N } ( \\mathbf { s } _ { N } ) : = \\mathfrak { p } _ { N } ( \\mathbf { s } _ { N } ; \\mathbf { X } _ { N } ) \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} \\lim _ { q _ 1 - q _ 2 \\rightarrow 0 } \\kappa _ g ( q _ 1 , q _ 2 ) = - \\frac { 1 } { 4 } \\ ; . \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} g _ { \\rm s l i d e } = \\left . \\frac { f _ L g _ R - f _ R g _ L } { f _ L - f _ R } \\right | _ { x = 0 } \\ , , \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} E _ { 0 } ( \\mathbf { z } ) B _ { \\mathbf { m } } ^ { ( d ) } \\left ( \\mathbf { z } \\right ) = \\sum _ { i = 1 } ^ { r } B _ { \\mathbf { m } _ { i } } ^ { ( d ) } \\left ( \\mathbf { z } \\right ) \\left ( m _ { i } + \\frac { d } { 2 } ( r - i ) \\right ) h _ { - , i } ^ { ( d ) } ( \\mathbf { m } ) . \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} x = \\sum _ { n = 1 } ^ \\infty \\frac { w _ n } { \\beta ^ n } \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} Q _ { \\infty } = e ^ { - 4 \\lambda _ { \\infty } } ( Q _ { 0 } ) _ { \\ker } . \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} a _ { 0 L } & = f _ L ( 0 , 0 ) , & a _ { 0 R } & = f _ R ( 0 , 0 ) . \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} 0 & = \\nabla f ( \\bar x ) + \\sum \\limits _ { i \\in I ^ g ( \\bar x ) } \\lambda _ i \\nabla g _ i ( \\bar x ) + \\sum \\limits _ { j \\in \\mathcal P } \\rho _ j \\nabla h _ j ( \\bar x ) \\\\ & + \\sum \\limits _ { l \\in I ^ { 0 + } ( \\bar x ) } \\mu _ l \\nabla G _ l ( \\bar x ) + \\sum \\limits _ { l \\in I ^ { + 0 } ( \\bar x ) } \\nu _ l \\nabla H _ l ( \\bar x ) \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} Y = \\beta X + Z . \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} \\| \\Lambda f _ 0 \\| = \\| \\Lambda f ( t _ * ) \\| + \\int _ 0 ^ { t _ * } \\Delta ( s , f ( s ) ) d s < \\| \\Lambda F ( t _ * ) \\| + \\int _ 0 ^ { t _ * } \\Delta ( s , F ( s ) ) d s , \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} A - B & = P _ \\natural A ( I - P _ \\natural ) + ( I - P _ \\natural ) A P _ \\natural \\\\ & = [ P _ \\natural , A ] ( I - P _ \\natural ) + ( I - P _ \\natural ) [ A , P _ \\natural ] \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} v _ { 5 } ( t , r ) = \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { \\infty } d \\xi J _ { 1 } ( r \\xi ) \\sin ( ( t - s ) \\xi ) \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( s , \\xi ) \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } ( f \\mu _ { N , m } ) = \\nabla \\cdot \\left ( A \\nabla f \\mu _ { N , m } \\right ) . \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} u ^ 1 _ t - u ^ 2 _ t & = - \\int _ 0 ^ t \\nabla \\cdot e ^ { ( t - s ) \\Delta } \\left \\{ u ^ 1 _ s F ( K \\ast u ^ 1 _ s ) - u ^ 2 _ s F ( K \\ast u ^ 2 _ s ) \\right \\} ~ d s \\\\ & = - \\int _ 0 ^ t \\nabla \\cdot e ^ { ( t - s ) \\Delta } \\left \\{ ( u ^ 1 _ s - u ^ 2 _ s ) F ( K \\ast u ^ 1 _ s ) + u ^ 2 _ s ( F ( K \\ast u ^ 1 _ s ) - F ( K \\ast u ^ 2 _ s ) ) \\right \\} ~ d s . \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} \\begin{array} { l } ( \\epsilon _ 4 , \\epsilon _ 5 , \\epsilon _ 6 ) \\\\ = \\begin{cases} ( \\min \\{ n - r + k _ 2 , n - r + k _ 1 - r _ 1 + k _ 4 \\} + \\hat e _ 2 , & \\mbox { i f ~ } n - r + k _ 1 > r _ 1 , \\\\ ~ ~ ~ ~ ~ ~ ~ n - r + k _ 3 , n - r + k _ 1 - r _ 1 + k _ 5 ) & \\\\ ( \\min \\{ k _ 4 , r _ 1 - k _ 1 + k _ 2 \\} + \\hat e _ 2 , r _ 1 - k _ 1 + k _ 3 , k _ 5 ) & \\mbox { i f ~ } n - r + k _ 1 \\le r _ 1 , \\end{cases} \\end{array} \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} \\left ( A \\mathsf { p } \\right ) _ { \\mathsf { m } } : = \\sum _ { \\mathsf { n = 1 } } ^ { + \\infty } a _ { \\mathsf { m , n } } p _ { \\mathsf { n } } \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} \\mathfrak { \\widehat D } _ { k , m } = ( m - k ) a ^ { k - 2 } X ^ d + \\frac { ( k - 3 ) ( k - 2 m ) } { 2 } a ^ { k - 4 } X ^ { 2 d } - \\frac { ( k - 4 ) ( k - 5 ) ( k - 3 m ) } { 3 ! } a ^ { k - 6 } X ^ { 3 d } + \\cdots . \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} \\begin{cases} d ( x , z ) \\leq d ( x , y ) + d ( y , z ) , \\\\ d ( y , z ) \\leq d ( y , x ) + d ( x , z ) , \\end{cases} \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} D = E ^ 0 \\circ E ^ 1 [ p ] \\circ \\cdots \\circ E ^ { j - 1 } \\left [ p ^ { j - 1 } \\right ] \\circ T ^ j \\left [ p ^ j \\right ] \\circ F . \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\dot { u } ( t ) = \\operatorname { d i v } \\left ( \\frac { \\nabla u ( t ) } { \\abs { \\nabla u ( t ) } } \\right ) , \\ ; & \\Omega \\times ( 0 , \\infty ) \\\\ u ( 0 ) = u _ 0 , \\ ; & \\Omega \\times 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} u _ 1 \\to v _ 1 \\ \\ R , u _ 1 = v _ 1 \\ \\ u _ 2 \\to v _ 2 \\ \\ S . \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} v _ { 1 } ( t , r ) = \\int _ { t } ^ { \\infty } d s \\frac { \\lambda '' ( s ) } { r } \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\left ( 1 + \\frac { r ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( 1 + r ^ { 2 } + \\rho ^ { 2 } ) ^ { 2 } - 4 r ^ { 2 } \\rho ^ { 2 } } } \\right ) \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} x ( t , \\mu ) = \\sum _ { i = 1 } ^ { \\infty } v _ i ( t ) \\Phi _ i ( \\mu ) \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} d ( x _ 1 , y _ 1 ) \\leq d ( x _ 1 , x _ 2 ) + d ( x _ 2 , y _ 1 ) = d ( x _ 2 , y _ 1 ) \\leq d ( x _ 2 , y _ 2 ) + d ( y _ 2 , y _ 1 ) = d ( x _ 2 , y _ 2 ) , \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} M _ { q } ( t ) \\sim \\left \\{ \\begin{array} { l l l } \\frac { \\Gamma ( 1 + q ) } { \\Gamma ( 1 + q \\alpha ) } t ^ { q \\alpha } , \\ ; & \\mbox { a s } \\ ; t \\rightarrow 0 , \\\\ \\frac { \\lambda ^ { q ( 1 - \\alpha ) } } { \\alpha ^ { q } } t ^ { q } , \\ ; \\lambda > 0 , & \\mbox { a s } \\ ; t \\rightarrow \\infty , \\\\ \\frac { \\Gamma ( 1 + q ) } { \\Gamma ( 1 + q \\alpha ) } t ^ { q \\alpha } , \\ ; \\lambda = 0 , & \\mbox { a s } \\ ; t \\rightarrow \\infty . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 k } { \\partial t ^ 2 } - c _ 1 c _ 2 \\frac { \\partial ^ 2 k } { \\partial x ^ 2 } + ( c _ 1 - c _ 2 ) \\frac { \\partial ^ 2 k } { \\partial x \\partial t } = \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} [ D - T _ u , T _ { w _ \\lambda } T _ { \\overline w _ \\lambda } ] - T _ { D | w _ \\lambda | ^ 2 } = 0 \\ . \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} \\operatorname { I m } ( \\rho ^ \\gamma ) \\ , = \\ , \\langle \\partial , z \\partial - c , z ^ 2 \\partial - 2 c z \\rangle \\ , . \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} \\mathcal Q _ { \\mu , D } ( T _ { \\mu , D } ( u _ { \\mu } ) , \\varphi ) = \\int _ { \\partial \\Omega } \\frac { \\partial u _ { \\mu } } { \\partial \\nu } \\frac { \\partial \\varphi } { \\partial \\nu } d \\sigma \\ , , \\ \\ \\ \\forall \\varphi \\in H ^ 2 ( \\Omega ) , \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} X ^ { ( - i ) } f ^ { ( \\ell ) } = ( X f ) ^ { ( \\ell - i ) } \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} ( \\psi _ s ^ M ) ^ \\# = \\delta _ 1 ( \\Sigma ^ { - 1 } P _ { s - 1 } \\otimes M ) \\circ \\cdots \\circ \\delta _ 1 ( \\Sigma ^ { - s } M ) . \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} p _ { L } ( 1 + \\varepsilon ) > \\varepsilon ^ 4 \\binom { L - 1 } { 3 } - \\left \\lceil L ( L + 1 ) / 4 \\right \\rceil - 1 > 0 . \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} D ( H _ { n , \\mathcal { H } } ) = \\{ P \\mathcal { H } ^ { - 1 } P _ n z \\ : ; \\ : z \\in D ( H ) \\} . \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} H _ { L } = c _ 1 H _ { L - 1 } + \\cdots + \\left ( c _ { L - 1 } + 1 \\right ) H _ 1 = c _ 1 G _ { L - 1 } + \\cdots + \\left ( c _ { L - 1 } + 1 \\right ) G _ 1 = ( G _ { L } - 1 ) + G _ 1 = G _ { L } . \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} [ ( x , a ) , ( y , b ) , ( z , c ) ] _ 1 = ( [ x , y , z ] , \\rho ( x , y ) c - ( - 1 ) ^ { \\bar y \\bar z } \\rho ( x , z ) b + ( - 1 ) ^ { \\bar x ( \\bar y + \\bar z ) } \\rho ( y , z ) a ) , \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} g \\left ( x , t \\right ) = \\theta _ { r } \\left ( x \\right ) \\theta \\left ( \\psi \\left ( x , t \\right ) \\right ) \\tilde { u } \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} | r ^ { 2 } \\partial _ { r } ^ { 2 } v _ { 5 } ( t , r ) | & \\leq \\frac { C r } { t ^ { 4 } \\log ^ { 3 b } ( t ) } + \\frac { C r ^ { 2 } } { t ^ { 4 } \\log ^ { 3 N + b - 2 } ( t ) } + \\frac { C r } { t ^ { 7 / 2 } \\log ^ { \\frac { 5 N } { 2 } + 3 b - 3 } ( t ) } + \\frac { C r } { t ^ { 7 / 2 } \\log ^ { \\frac { 5 N } { 2 } + 3 b - 3 } ( t ) } \\\\ & \\leq \\frac { C r } { t ^ { 3 } \\log ^ { 3 N + b - 2 } ( t ) } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} b & = \\tau ( g ) - g . \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} f \\left ( x _ 1 , \\ldots , x _ d \\right ) = \\sum _ { i = 1 } ^ k w _ i f _ { i , 1 } \\left ( x _ 1 \\right ) f _ { i , 2 } \\left ( x _ 2 \\right ) \\cdots f _ { i , d } \\left ( x _ d \\right ) . \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} \\phi ( - q ^ { \\frac { j - i } { 2 } } x _ { j - 1 } x _ { j - 1 } \\cdots x _ { i } ) = \\displaystyle \\sum _ { m _ { i , j } \\geq 0 } ( - 1 ) ^ { m _ { i , j } } \\frac { q ^ { \\frac { m _ { i , j } ( m _ { i , j } - 1 ) } { 2 } } ( - q ^ { \\frac { j - i } { 2 } } x _ { j - 1 } x _ { j - 2 } \\cdots x _ { i } ) ^ { m _ { i , j } } } { ( q ) _ { m _ { i , j } } } . \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} n \\geq \\sum \\limits _ { i = 1 } ^ { t } ( 2 a _ i + 3 ) = 2 | I | + 3 t , \\geq 2 | I | + 6 , \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} x _ a ^ { ( 0 ) } = 0 & \\ \\ \\mbox { f o r } w _ a > 0 , \\\\ x _ a ^ { ( 1 ) } = 0 & \\ \\ \\mbox { f o r } w _ a > 1 , \\\\ \\ldots & \\\\ x _ a ^ { ( r - 1 ) } = 0 & \\ \\ \\mbox { f o r } w _ a > r - 1 . \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} \\phi ( u ) = b u + u \\int _ 0 ^ { 1 } \\ r ^ { u - 1 } m ( r ) d r = b u + u \\int _ 0 ^ { \\infty } \\ e ^ { - u y } m ( e ^ { - y } ) d y \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} \\overline { H } \\ , \\overline { H } ^ { - 1 } \\subseteq \\overline { H } \\ , \\overline { H ^ { - 1 } } \\subseteq \\overline { H H ^ { - 1 } } = \\overline { H } . \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} \\gamma _ + \\psi = ( 1 \\ ! - \\ ! a ) \\mathcal { R ^ * } \\ ! \\left [ \\gamma _ - \\psi \\right ] \\rightarrow \\mathcal { R ^ * } \\ ! \\left [ \\gamma _ - \\psi \\right ] . \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} \\delta ( \\Delta f ) & = \\Delta \\dot { f } + 2 u \\langle h , \\ , f \\rangle + 2 u \\langle \\nabla H , \\nabla f \\rangle \\\\ & + 2 \\ , h ( \\nabla { u } , \\nabla { f } ) - 2 H \\langle \\nabla u , \\nabla f \\rangle , \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { l l l l } \\beta J & \\beta J & \\ldots & \\beta J \\\\ 2 \\beta ^ 2 \\sigma _ { 1 } & 2 \\beta ^ 2 \\sigma _ { 2 } & \\ldots & 2 \\beta ^ 2 \\sigma _ { n } \\end{array} \\right ) . \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} ( c _ { k _ 4 } - c _ { k _ 3 } ) - ( c _ { k _ 2 } - c _ { k _ 1 } ) = \\sum _ { j \\in \\alpha ( k _ 1 , k _ 2 , k _ 3 , k _ 4 ) } n _ j . \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} \\mathfrak { \\widehat G } _ k = ( 3 - k ) a ^ { k - 2 } X ^ d + \\frac { ( k - 3 ) ( k - 6 ) } { 2 } a ^ { k - 4 } X ^ { 2 d } - \\frac { ( k - 4 ) ( k - 5 ) ( k - 9 ) } { 3 ! } a ^ { k - 6 } X ^ { 3 d } + \\cdots . \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\kappa _ \\sigma = \\frac { \\langle Q \\zeta , \\zeta \\rangle } { \\langle E \\zeta , \\zeta \\rangle } , \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} \\norm { f } _ p = \\left ( \\int _ S { \\abs f } ^ p \\ , \\mathrm { d } \\mu \\right ) ^ \\frac { 1 } { p } . \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} \\mathfrak { F } = ( \\mathcal { F } , L ^ \\bullet , \\theta ^ \\bullet , Q ) \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} | 1 + \\frac { R ^ { 2 } \\lambda ( t ) ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( R ^ { 2 } \\lambda ( t ) ^ { 2 } - 1 - \\rho ^ { 2 } ) ^ { 2 } + 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } } | & = | 1 + \\frac { ( - 1 - \\frac { 1 } { \\rho ^ { 2 } } + \\frac { R ^ { 2 } \\lambda ( t ) ^ { 2 } } { \\rho ^ { 2 } } ) } { \\sqrt { 1 - 2 ( \\frac { R ^ { 2 } \\lambda ( t ) ^ { 2 } } { \\rho ^ { 2 } } - \\frac { 1 } { \\rho ^ { 2 } } ) + \\frac { ( R ^ { 2 } \\lambda ( t ) ^ { 2 } + 1 ) ^ { 2 } } { \\rho ^ { 4 } } } } | \\\\ & \\leq C \\frac { ( R \\lambda ( t ) + 1 ) ^ { 2 } } { \\rho ^ { 2 } } \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} & \\partial _ t u - \\alpha \\Delta u = \\gamma u ( 1 - | u | ^ 2 ) , \\ , \\alpha , \\ , \\gamma \\in \\mathbb { C } , \\mbox { R e } \\alpha \\geq 0 , \\\\ & \\mathcal { L } = \\alpha \\Delta , \\ , \\ , \\mathcal { A } = - 2 i \\mbox { I m } \\alpha \\Delta , \\ , f ( u , \\overline { u } ) = \\gamma u ( 1 - u \\overline { u } ) \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} \\psi ( t ) = \\| \\xi \\| _ \\infty + ( 1 + \\| \\xi \\| _ \\infty ) \\int _ 0 ^ t p ( s ) d s + \\int _ 0 ^ t p ( s ) \\psi ( s ) d s , \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} \\int _ { y \\in X } \\varphi ( x ) \\psi ( x ) d \\mu ( x ) & = \\int _ { x \\in X } \\varphi ( x ) \\left ( \\int _ { y \\in X } \\Phi ( x , y ) d \\pi _ 2 ( y ) \\right ) d \\pi _ 1 ( x ) \\\\ & = \\int _ { x \\in X } \\varphi ( x ) \\frac { d \\pi _ 1 } { d \\mu } ( x ) \\left ( \\int _ { y \\in X } \\Phi ( x , y ) d \\pi _ 2 ( y ) \\right ) d \\mu ( x ) \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { j = 0 } ^ k b _ j ( - d ) ^ j : b _ j \\in \\{ a _ 1 , \\ldots , a _ d \\} \\right \\} . \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} u _ \\varepsilon ( x ) \\geq u _ \\varepsilon ( R \\varepsilon ) = c ^ { - \\frac 1 { n - 1 } } \\frac n { \\alpha _ n } \\big [ - \\log \\varepsilon - \\log ( - \\log \\varepsilon ) \\big ] \\geq 1 \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} \\Phi _ { 1 , n + 1 } ( x ) = P _ { 1 } [ \\Phi _ { 1 , n } ] ( x ) \\preceq P _ { 1 } [ \\Phi _ { 2 , n } ] ( x ) \\preceq P _ { 2 } [ \\Phi _ { 2 , n } ] ( x ) = \\Phi _ { 2 , n + 1 } ( x ) \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} \\beta _ { E , A , p } ( R ) \\sim _ { A , p } \\beta _ { E , 3 , 2 } ( R ) = : \\beta _ { E } ( R ) . \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c l } a _ { 1 1 } X _ 1 ^ { d _ 1 } & + a _ { 1 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { 1 t } X _ t ^ { d _ t } = b _ 1 \\\\ a _ { 2 1 } X _ 1 ^ { d _ 1 } & + a _ { 2 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { 2 t } X _ t ^ { d _ t } = b _ 2 \\\\ \\ ; \\vdots & & \\quad \\vdots \\\\ a _ { n 1 } X _ 1 ^ { d _ 1 } & + a _ { n 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { n t } X _ t ^ { d _ t } = b _ n , \\end{array} \\right . \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} & \\frac { 1 } { r ^ p } \\int _ { S ^ 2 ( r ) } \\bigg ( \\frac { p ( p - 1 ) r ^ 2 } { 4 r ^ 4 } ( - 2 u ) ^ 2 + \\frac { 2 ( p ^ 2 - p - 1 ) } { r ^ 2 } u ( - 2 u ) + \\frac { ( p - 1 ) ( p - 2 ) } { r ^ 2 } u ^ 2 \\bigg ) \\ , \\ , d S \\\\ & = \\frac { 1 } { r ^ { p + 2 } } \\int _ { S ^ 2 ( r ) } \\bigg ( p ( p - 1 ) - 2 ( p ^ 2 - p - 1 ) + ( p - 1 ) ( p - 2 ) \\bigg ) u ^ 2 \\ , d S \\\\ & = \\frac { 1 } { r ^ { p + 2 } } \\int _ { S ^ 2 ( r ) } 2 u ^ 2 ( 2 - p ) \\ , d S < 0 , p > 2 , \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} \\Delta _ L f ( x ) = \\int _ 0 ^ 1 K _ L ( x ; t ) d t . \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} \\ell ^ { ( \\alpha ) } ( \\mu ) : = \\sum \\limits _ { i = 1 } ^ n [ 1 + b _ \\alpha ( X _ i - \\mu ) ^ 2 ] ~ { \\bf 1 } ( \\mu \\in I _ i ) . \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} p _ 0 + a p _ 1 + a ^ 2 p _ 2 & = f ( a ) , \\cr p _ 1 + ( b - a ) ^ { - 1 } ( b ^ 2 - a ^ 2 ) p _ 2 & = ( b - a ) ^ { - 1 } ( f ( b ) - f ( a ) ) , \\cr \\{ ( c - a ) ^ { - 1 } ( c ^ 2 - a ^ 2 ) - ( b - a ) ^ { - 1 } ( b ^ 2 - a ^ 2 ) \\} p _ 2 & = ( c - a ) ^ { - 1 } ( f ( c ) - f ( a ) ) - ( b - a ) ^ { - 1 } ( f ( b ) - f ( a ) ) , \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} \\sigma & = \\sin ( \\theta ) \\ , \\cos ( \\varphi ) \\ , \\sigma _ 1 + \\sin ( \\theta ) \\ , \\sin ( \\varphi ) \\ , \\sigma _ 2 + \\cos ( \\theta ) \\ , \\sigma _ 3 \\\\ \\bar { \\sigma } & = \\cos ( \\theta ) \\cos ( \\varphi ) \\ , \\sigma _ 1 + \\cos ( \\theta ) \\sin ( \\varphi ) \\ , \\sigma _ 2 - \\sin ( \\theta ) \\ , \\sigma _ 3 \\\\ \\hat { \\sigma } & = - \\sin ( \\varphi ) \\ , \\sigma _ 1 + \\cos ( \\varphi ) \\ , \\sigma _ 2 \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{align*} ( & X ^ { ( 1 ) } \\ldots \\mapsto X ^ { ( K ) } , \\\\ & ( ( x _ h ^ { ( 1 ) } , x _ j ^ { ( 1 ) } ) , \\ldots , ( x _ h ^ { ( K - 1 ) } , x _ j ^ { ( K - 1 ) } ) ) , \\\\ & ( e ^ { ( 1 ) } , \\ldots , e ^ { ( K ) } ) ) , \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} f \\left ( M + m - \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } { { a } _ { i } } } \\right ) \\le f \\left ( M \\right ) + f \\left ( m \\right ) - \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( { { a } _ { i } } \\right ) } \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { \\ell _ R ( R / I ( 1 ) _ { t n _ 1 } \\cdots I ( r ) _ { t n _ r } ) } { t ^ d } = \\lim _ { t \\rightarrow \\infty } \\frac { \\ell _ R ( R / I ( 1 ) _ { t n _ 1 } \\cdots I ( s ) _ { t n _ s } ) } { t ^ d } \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} \\langle \\widehat { A } f , f \\rangle _ { \\alpha } = \\| f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 > 0 , \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} h _ i ( \\beta ) : = \\inf _ { x \\leq 0 } w _ i ^ \\beta ( x ) > 0 \\mbox { f o r } \\beta < \\beta _ i ^ * , \\ ; h _ i ( \\beta _ i ^ * ) = 0 . \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} \\{ - \\frac { n } { 2 } \\} \\cup \\{ - \\frac { 1 } { 2 } \\pm \\frac { 1 } { 2 } \\sqrt { 1 + n ( n - 2 ) + 2 ( n + 2 k ) ( n + 2 k - 2 ) } , k = 1 , 2 , \\ldots \\} . \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} W ^ 2 + ( t + s ) U ^ 4 = h G \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} | - \\lambda ( t ) \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) E _ { 5 } \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} T _ i ( x ) ^ { - 1 } = \\left ( D R _ i ^ { - 1 } ( x ) \\right ) ^ { - 1 } = D R _ i ( R _ i ^ { - 1 } ( x ) ) = A _ c + D r _ i ( R _ i ^ { - 1 } ( x ) ) . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} - \\frac { 1 } { \\omega } \\sum _ { j = 1 } ^ { \\infty } \\int _ { 0 } ^ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } r ^ { 2 j } \\omega ^ { j } \\lambda ( t ) ^ { 2 j } \\phi _ { j } ( r ^ { 2 } ) F _ { 4 } ( t , r \\lambda ( t ) ) d r \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} A _ { r , R } ^ { + } & : = \\{ x \\in \\mathbb { R } _ { + } ^ { n + 1 } : r \\le | x | \\le R \\} \\\\ A _ { r , R } ' & : = \\{ ( x ' , 0 ) \\in \\mathbb { R } ^ { n } \\times \\{ 0 \\} : r \\le | ( x ' , 0 ) | \\le R \\} . \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} v = ( v _ { 1 } , \\dots , v _ { d } ) = \\sum _ { y \\sim 0 } p ( y ) y \\in \\R ^ d . \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} \\begin{aligned} \\theta _ { 1 3 } ( \\lambda , \\mu ) & = \\theta _ { 1 3 } ( \\lambda - 1 3 , \\mu ) , \\\\ \\theta _ { 1 3 } ( \\lambda , \\mu + 1 ) & = \\theta _ { 1 3 } ( \\lambda + 2 , \\mu ) . \\end{aligned} \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} S ^ { p ' } _ { - } ( \\Omega ) = \\{ \\sigma \\in L ^ { p ' } ( \\Omega ) : \\ , \\ , { \\rm d i v } \\sigma \\le \\ , 0 \\mathcal { D } ' ( \\Omega ) \\} , \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} y _ x = \\frac { x ^ 2 + y } { 2 x - \\tilde { c } _ 1 } \\ , . \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} P & = | A | ^ 4 - 2 | A ^ 2 | ^ 2 + \\sum _ { i , j , k , \\ell } \\left \\{ 2 \\ , \\langle A _ { j \\ell } , A _ { i k } \\rangle \\ , \\langle A _ { \\ell k } , A _ { i j } \\rangle \\ , - \\langle A _ { i j } , A _ { k \\ell } \\rangle ^ 2 \\right \\} \\\\ & = \\left ( \\sum _ { i } \\lambda _ i ^ 2 \\right ) ^ 2 - 2 \\sum _ i \\lambda _ i ^ 4 + 2 \\sum _ { i } \\lambda _ i ^ 4 - \\sum _ { i , k } \\lambda _ i ^ 2 \\lambda _ k ^ 2 = 0 \\ , . \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} A ^ 1 _ { l + 2 } \\big | _ { x _ 1 = x _ p \\neq 0 } & = - i x _ p b _ { l + 1 } . \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} | | x - y | | \\leq | | x - v t | | + | | y - v t | | \\leq \\sqrt { t \\left ( 1 - \\sum _ { i = 1 } ^ { d } | v _ { i } | \\right ) } , \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} \\sum _ { \\mathsf { m = 1 } } ^ { + \\infty } \\left ( \\sum _ { \\mathsf { n = 1 } } ^ { + \\infty } r _ { \\mathsf { n , m } } \\right ) ^ { 2 } = Z _ { \\left ( \\alpha - 1 \\right ) \\beta } Z _ { \\frac { \\alpha + 1 } { 2 } \\beta } ^ { 2 } < + \\infty \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} \\begin{cases} f _ 1 '' + a f _ 1 ' + b f _ 1 & \\ ! \\ ! \\ ! \\ ! = 0 \\\\ f _ 2 '' + a f _ 2 ' + b f _ 2 & \\ ! \\ ! \\ ! \\ ! = 0 . \\end{cases} \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} \\mathcal { O } ( D , \\varepsilon ) : = \\{ \\Omega \\subset D \\ | \\ \\} . \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} g ( 0 , 0 ; 0 ) = 0 . \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} & \\int _ { \\mathbb R } \\tilde \\phi _ 2 ( x _ 1 , x _ 2 , x _ 3 ) d x _ 2 = 0 \\quad { \\rm f o r \\ e a c h \\ f i x e d \\ } ( x _ 1 , x _ 3 ) \\ { \\rm i n \\ } \\mathbb R ^ 2 ; \\\\ & \\int _ { \\mathbb R ^ 2 } \\tilde \\phi _ 2 ( x _ 1 , x _ 2 , x _ 3 ) d x _ 1 d x _ 3 = 0 \\quad { \\rm f o r \\ e a c h \\ f i x e d \\ } x _ 2 \\ { \\rm i n \\ } \\mathbb R . \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} t \\ , \\frac { \\partial w } { \\partial t } = t ^ { - d } a ( t , x ) & + ( \\lambda ( t , x ) - d ) w + b ( t , x ) \\frac { \\partial w } { \\partial x } \\\\ & \\qquad + \\sum _ { j + \\alpha \\geq 2 } a _ { j , \\alpha } ( t , x ) t ^ { d ( j + \\alpha - 1 ) } w ^ j \\Bigl ( \\frac { \\partial w } { \\partial x } \\Bigr ) ^ { \\alpha } . \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{align*} J _ 0 ( u ) [ \\varphi ] = ( \\gamma _ 0 ( u ) , \\gamma _ 0 ( \\varphi ) ) _ { \\partial \\Omega } \\ , , \\ \\ \\ \\forall u , \\varphi \\in H ^ 2 ( \\Omega ) . \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} H _ { m + 2 } + H _ { m + 3 } + ( 2 L - 3 ( m + 1 ) ) + \\sum _ { i = 2 ( L - m ) + 1 } ^ L G _ i \\leq \\sum _ { i = 2 ( L - m ) } ^ L H _ i . \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} \\lambda _ { 2 } ( \\xi ) = 1 , \\widehat { \\lambda } _ { 2 } ( \\xi ) = \\frac { \\sqrt { 5 } - 1 } { 2 } = 0 . 6 1 8 0 \\ldots , \\lambda _ { 3 } ( \\xi ) = \\frac { 1 } { \\sqrt { 5 } } = 0 . 4 4 7 2 \\ldots , \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} { } ^ b \\Psi ^ { - \\infty } _ { G , c } ( M ) = \\{ T _ { \\widetilde { k } } , \\ ; \\widetilde { k } \\in ( C ^ \\infty _ c ( G ) \\hat { \\otimes } \\ , { } ^ b \\Psi ^ { - \\infty } ( S ) ) ^ { K \\times K } \\} \\ , . \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} T _ H ( \\lambda _ q ^ { * ^ G } ) ( \\psi ) & = \\int _ G \\psi _ q ( x ) d \\lambda _ q ^ { * ^ G } ( x ) \\\\ & = \\int _ G \\psi _ q ( x ^ { - 1 } ) d \\overline { \\lambda _ q } ( x ) \\\\ & = \\left ( \\int _ G \\overline { \\psi _ q ( x ^ { - 1 } ) } d \\lambda _ q ( x ) \\right ) ^ { - } \\\\ & = \\left ( \\int _ { G / H } T _ H ( \\psi _ q ^ { * ^ G } ) ( x H ) d \\lambda ( x H ) \\right ) ^ { - } = \\lambda ^ { \\ast ^ { G / H } } ( \\psi ) . \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} \\Psi ( \\xi ) = \\phi ( | \\xi | ^ 2 ) = \\int _ { \\R ^ d } \\left ( 1 - \\cos ( \\xi \\cdot x ) \\right ) J ( d x ) , \\xi \\in \\R ^ d , \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} & \\Sigma _ { 1 } + \\Sigma _ { 2 } = - \\tfrac { m } { n } | x | ^ { 2 } , & & c _ { 1 } \\Sigma _ { 1 } + c _ { 2 } \\Sigma _ { 2 } = \\tfrac { m } { n } \\left ( \\tfrac { m } { n } - 1 \\right ) | x | ^ { 2 } . \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} | | \\frac { \\lambda ( t ) | v _ { 5 } ( t , r ) \\partial _ { r } v _ { 5 } ( t , r ) | } { r ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) } \\vert _ { r = R \\lambda ( t ) } | | _ { L ^ { 2 } ( R d R ) } & \\leq C \\lambda ( t ) | | \\frac { v _ { 5 } ( t , r ) } { r ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) } | | _ { L ^ { \\infty } } \\cdot | | ( \\partial _ { 2 } v _ { 5 } ) ( t , \\cdot \\lambda ( t ) ) | | _ { L ^ { 2 } ( R d R ) } \\\\ & \\leq \\frac { C } { t ^ { 2 1 / 4 } \\log ^ { b - 6 + \\frac { 5 N } { 2 } } ( t ) } \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} \\dim ( \\hat { \\sigma } ( X ) ) ^ 2 = 3 ^ a ( u _ 1 ^ { - ( b + c + d ) + 2 } u _ 2 ^ { b + 2 } ) ^ 2 \\beta ^ d \\qquad \\qquad \\dim ( \\hat { \\sigma ^ 2 } ( X ) ) ^ 2 = 3 ^ a ( u _ 1 ^ { c - 2 } u _ 2 ^ { - ( b + c + d ) + 4 } ) ^ 2 \\beta ^ d . \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} \\mathcal { R } ^ \\epsilon _ G ( M ) : = ( \\mathcal { A } ( G ) \\hat { \\otimes } \\ , \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} & w ' = \\left ( { w ' } _ { 1 } , \\dots , { w ' } _ { N } , \\frac { { w ' } _ { N + 1 } } { G ^ { \\left ( N + 1 \\right ) } _ { 0 ; 1 , 0 , \\dots , 0 , \\dots , 0 } ( z ) } , { w ' } _ { N + 2 } , \\dots , { w ' } _ { N ' } \\right ) , \\\\ & { z ' } = \\left ( { z ' } _ { 1 } , \\dots , { z ' } _ { N } , \\frac { { z ' } _ { N + 1 } } { \\gamma _ { N + 1 , 1 } } , { z ' } _ { N + 2 } , \\dots , { z ' } _ { N ' } \\right ) , \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} | \\partial _ { t } v _ { 3 } ( t , r ) | \\leq \\begin{cases} \\frac { C r \\log ( \\log ( t ) ) } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { r t \\log ^ { b + 1 } ( t ) } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} \\lim _ { \\xi \\rightarrow \\overline { c _ { 0 } } ^ { - } } \\frac { p _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( \\xi ) } { \\left | \\xi - \\overline { c _ { 0 } } \\right | ^ { 1 / 2 } } = \\frac { 2 \\left | a _ { 1 } \\right | } { \\sqrt { \\left | c _ { 2 } \\right | } } \\cos \\left ( \\frac { \\theta } { 2 } - \\frac { 3 \\pi } { 4 } \\right ) , \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} \\partial _ r [ w ( \\theta ) ] ^ \\top \\Big [ \\frac { \\mathbb { E } _ \\theta [ f ( \\textbf { X } ) + c ] } { h + c _ 0 + w ( \\theta ) ^ \\top \\mathbb { E } _ \\theta [ f ( \\textbf { X } ) + c ] } - \\frac { \\bar { f } + c } { h + c _ 0 + w ( \\theta ) ^ \\top [ \\bar { f } + c ] } \\Big ] = 0 , \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} d _ { 1 , 0 } d _ { - 1 , 0 } - d _ { - 1 , 1 } d _ { 1 , - 1 } = 0 . \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} f ( B ) & = f ^ \\lambda ( B ) . \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} g _ i ( x ^ * ) & \\leq 0 , ~ \\forall ^ { m } _ { i = 1 } \\\\ \\lambda ^ * _ i & \\geq 0 , ~ \\forall ^ { m } _ { i = 1 } \\\\ \\lambda ^ * _ i g _ i ( x ^ * ) & = 0 , ~ \\forall ^ { m } _ { i = 1 } \\\\ \\nabla f ( x ^ * ) + \\lambda ^ { * T } \\nabla g ( x ^ * ) & = 0 . \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} Z ( F ) : = \\{ ( x , y , z ) \\in \\mathbb R ^ 3 : F ( x , y , z ) = 0 \\} . \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} b = - ( f _ 1 '' ( f _ 1 ' ) ^ { - 1 } - f _ 2 '' ( f _ 2 ' ) ^ { - 1 } ) ( f _ 1 ( f _ 1 ' ) ^ { - 1 } - f _ 2 ( f _ 2 ' ) ^ { - 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} c _ 1 H _ { L - 1 } + c _ 2 H _ { L - 2 } + \\cdots + c _ { L - 1 } H _ { 1 } + 1 = H _ L \\leq 1 + H _ 1 + H _ 2 + \\cdots + H _ { L - 1 } . \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} P ( \\tilde { r } ; \\mu ) = P _ L \\big ( \\tilde { \\phi } ( \\tilde { r } ) ; \\mu \\big ) , \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} \\int _ { \\alpha t } ^ { \\beta t } \\frac { e ^ { i x ^ 2 / ( 4 t ) } } { \\sqrt t } \\widehat h ( x / ( 2 t ) ) e ^ { i x k } d x = \\frac { e ^ { - i t k ^ 2 } } { \\sqrt t } \\int _ { \\alpha t } ^ { \\beta t } \\exp \\left ( i \\left ( \\frac { x } { 2 \\sqrt t } + k \\sqrt t \\right ) ^ 2 \\right ) \\widehat h ( x / ( 2 t ) ) d x \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} \\hat { H } = - \\sum _ { i = 1 } ^ { D } \\frac { \\partial ^ 2 } { \\partial x _ i ^ 2 } + \\omega ^ 2 r ^ 2 + \\frac { \\beta _ 1 } { x _ 1 ^ 2 + \\cdots + x ^ 2 _ { n _ 1 } } + \\cdots + \\frac { \\beta _ N } { x ^ 2 _ { n _ { N - 1 } + 1 } + \\cdots + x _ D ^ 2 } . \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} \\| X ( t ) \\| _ { C ^ s } \\leq C ( t ) e ^ { C \\int _ 0 ^ t \\| \\nabla u ( \\tau ) \\| _ { L ^ \\infty } ~ d \\tau } . \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} | e ''' ( t ) | & \\leq C \\sup _ { x \\geq t } | R H S _ { 3 } ( x ) | \\\\ & \\leq \\frac { C } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } + \\frac { C \\sup _ { x \\geq t } \\left ( x ^ { 3 / 2 } | e ''' ( x ) | \\right ) } { t ^ { 3 / 2 } \\sqrt { \\log ( \\log ( t ) ) } } \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} P ( z ) = \\displaystyle \\sum _ { l = 1 } ^ { N } A _ { l } ( z , \\overline { z } ) q _ { l } ( z , \\overline { z } ) + P _ { 0 } ( z , \\overline { z } ) , \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} \\dot s ( t ) = A ( t ) ( e ^ { - s } ( t ) - 1 ) \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} & m _ { [ i _ { 1 } - 1 , i _ { 1 } - 1 ] } m _ { [ i _ { 1 } , i _ { 1 } ] } + \\sum _ { s = i _ { 1 } } ^ { j _ { 2 } - 1 } m _ { [ s , s ] } m _ { [ s + 1 , s + 1 ] } + m _ { [ j _ { 2 } , j _ { 2 } ] } m _ { [ j _ { 2 } + 1 , j _ { 2 } + 1 ] } \\\\ & = ( 2 ( j _ { 2 } - i _ { 1 } ) + 2 ) m _ { [ i _ { 2 } , j _ { 2 } ] } m _ { [ i _ { 1 } , n ] } + \\cdots , \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } P \\bigg ] \\tilde { u } = - x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j = 1 } ^ { n } \\partial _ { j } f _ { j } \\quad B _ { 2 r } ^ { + } . \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} \\Theta _ n ( f ) = \\begin{cases} f _ { [ 1 ] } & n = 1 \\\\ f _ { [ 1 ] } + \\cdots + f _ { [ n ] } + f _ { [ n + 1 ] } & n \\geq 2 \\end{cases} \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} & | - \\frac { 1 } { r } \\int _ { t } ^ { t + 6 r } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { ( s - t ) } \\lambda '' ( s ) \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( - 1 - \\rho ^ { 2 } + r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) | \\\\ & \\leq \\frac { 1 } { r } \\int _ { t } ^ { t + 6 r } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { ( s - t ) } | \\lambda '' ( s ) | \\cdot 2 \\\\ & \\leq C r \\sup _ { x \\geq t } | \\lambda '' ( x ) | \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} \\min _ { \\tau + 1 \\leq t \\leq T } \\Pi _ t & \\leq \\frac { 4 ( 1 + \\gamma ^ 2 \\tau ) } { T - \\tau } ( L + \\L \\rho + \\mu ( 1 - \\rho ) ) ^ 2 \\sum _ { t = 1 } ^ { T - \\tau } \\Delta _ t + \\frac { 2 } { T - \\tau } \\sum _ { t = 1 } ^ T \\Phi _ t \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} d \\hat { X } ( t ) = \\hat { b } ( t ) d t + \\hat { \\sigma } ( t ) d \\tilde { B } ( t ) + \\sigma ( X ( t ) ) \\rho ( t ) d t - d \\hat { \\eta } ( t ) \\ , . \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} ( g \\otimes f ) ( p ^ B _ { Y _ 1 \\otimes X _ 1 } ) = ( p ^ B _ { Y _ 2 \\otimes X _ 2 } ) ( g \\otimes f ) \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} B _ { k , 0 } ( n ) & = \\{ \\lambda = ( \\lambda _ 1 ^ { m _ 1 } , \\lambda _ 2 ^ { m _ 2 } , \\ldots , \\lambda _ \\ell ^ { m _ \\ell } , a ^ b ) \\in \\mathcal { P } ( n ) \\mid k \\nmid \\lambda _ i \\ , \\forall i , \\ , k \\mid a , \\ , k \\mid b \\} , \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} T _ { q ^ p } ( a ) = T _ { q + \\epsilon - 1 } ( a ) , \\ ; \\ ; U _ { q ^ p - \\epsilon ( q - 1 ) - 2 } ( a ) = 0 \\ ; \\ ; \\ ; m o d \\ ; \\ ; \\Phi _ p ( q ) . \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} l ( a , b ) = R ( a ) b + R ( R ( a ) b ) - R ( a b ) , r ( b , a ) = b R ( a ) + R ( b R ( a ) ) - R ( b a ) , ~ a \\in ( A , * ) , b \\in A . \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} G _ 1 ( x _ 0 , \\dots , x _ 3 ) G _ 2 ( x _ 0 , \\dots , x _ 3 ) + F _ 1 ( x _ 0 , \\dots , x _ 3 ) F _ 2 ( x _ 0 , \\dots , x _ 3 ) F _ 3 ( x _ 0 , \\dots , x _ 3 ) F _ 4 ( x _ 0 , \\dots , x _ 3 ) = 0 , \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} [ \\ ! [ \\sigma , D U ] \\ ! ] _ { u _ 0 } ( \\overline { \\Omega } ) = \\int _ { \\Omega } ( U - u _ 0 ) \\ , d ( - { \\rm d i v } \\sigma ) + \\int _ { \\Omega } \\langle \\sigma , D u _ 0 \\rangle \\ , d x . \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} g ^ { x y } = ( g _ { x y } ) ^ { - 1 } \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} u ( x , t ) = \\cos ( ( t + 1 ) ( x + 2 ) ) . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\left | \\int _ 0 ^ a f ( r ) P ( r , k ) d r \\right | ^ 2 { d \\sigma } = \\int _ 0 ^ a | f ( r ) | ^ 2 d r \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} q = \\left \\{ \\begin{array} { l l } \\pm q _ { s , 0 } ^ { j _ 1 - n } \\cdots q _ { s , s - 1 } ^ { j _ s } S t _ s ( m ) & \\delta = 0 , \\\\ \\pm q _ { s , 0 } ^ { j _ 1 - n - 1 } \\cdots q _ { s , s - 1 } ^ { j _ s } L _ s ^ { \\frac { p - 1 } { 2 } } S t _ s ( m ) & \\delta = 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{gather*} c _ { 1 , 0 } = c _ 1 + c _ 2 , c _ { 2 , 0 } = c _ 1 c _ 2 - c _ 1 x _ 1 - c _ 2 x _ 2 , \\\\ c _ { 2 , 1 } = 3 c _ { 1 , 0 } - ( x _ 1 + x _ 2 ) , c _ { 3 , 1 } = x _ 1 ^ 2 + x _ 2 ^ 2 - ( x _ 1 + x _ 2 ) c _ { 1 , 0 } + 4 c _ { 2 , 0 } . \\end{gather*}"}
-{"id": "3158.png", "formula": "\\begin{align*} f ( p ) = \\inf _ { a \\in A } \\varphi ( a ) \\cdot \\frac { d ( a , p ) } { d ( p , A ) } , \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} \\frac { x _ { k + \\frac { n } { 2 } + 1 } - x _ { k } } { y _ { k + \\frac { n } { 2 } + 1 } - y _ { k } } = \\tan \\frac { 2 k \\pi } { n } = \\frac { x _ { k } } { y _ { k } - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} f ( m , H ( \\vec { \\bf b } ) ) \\ge \\left \\{ \\begin{array} { l c } m ^ { \\frac 1 { k ( b _ k + 1 ) } } \\left ( \\frac { b _ { k - 1 } } { 4 ( b _ { k - 2 } + 2 b _ { k - 1 } ) } \\right ) ^ { \\frac 1 k } , & k \\ge 4 , \\\\ m ^ { \\frac 1 { b _ 3 + 2 } } \\left ( \\frac { b _ 2 } { 4 ( b _ 1 + 2 b _ 2 ) } \\right ) ^ { \\frac { b _ 3 + 1 } { b _ 3 + 2 } } , & k = 3 . \\end{array} \\right . \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} \\psi _ { n , 2 } ^ { \\ast } ( Q ) \\leq \\max _ { i = 1 , 2 } \\frac { L _ { n , P _ { i } } ^ { \\ast } ( q ) } { q } \\leq \\frac { n + 1 } { n } \\frac { 1 } { 1 + \\widehat { w } _ { n } ( \\xi ) } - \\frac { 1 } { n } + \\epsilon _ { 1 } = \\frac { n - \\widehat { w } _ { n } ( \\xi ) } { n ( 1 + \\widehat { w } _ { n } ( \\xi ) ) } + \\epsilon _ { 1 } . \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} \\mathbf { x } = \\begin{pmatrix} x & X \\\\ Y & y \\end{pmatrix} , x , y \\in \\mathbb { R } , ~ X , Y \\in \\mathfrak { J } . \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } \\psi _ { k , j } ^ { \\ast } ( Q ^ { \\prime } ) \\leq \\frac { k ( 1 - n ) \\lambda _ { n } + ( k n + n - k ^ { 2 } - k ) \\widehat { \\lambda } _ { n } + k ^ 2 - k n + k - 1 } { k ( n - 1 ) ( \\lambda _ { n } + 1 ) } + \\epsilon _ { 5 } . \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} d = \\begin{cases} D _ { k k } & D _ { k k } \\neq 0 , \\\\ \\infty & D _ { k k } = 0 . \\end{cases} \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} c _ { \\mathsf { p , } \\alpha , \\beta } \\exp \\left [ - \\frac { \\alpha + 1 } { 2 } \\beta \\lambda _ { \\mathsf { m } } \\right ] = \\left ( \\nu _ { \\mathsf { k } } + b _ { \\mathsf { m } } \\right ) p _ { \\mathsf { m } } \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} u ( z ) = \\frac { 1 + \\eta _ { \\mu _ { 1 } } ( z ) } { 1 - \\eta _ { \\mu _ { 1 } } ( z ) } & = 1 + 2 \\psi _ { \\mu _ { 1 } } ( z ) \\\\ & = \\int _ { \\mathbb { T } } \\left [ 1 + \\frac { 2 \\xi z } { 1 - \\xi z } \\right ] \\ , d \\mu _ { 1 } ( \\xi ) \\\\ & = \\int _ { \\mathbb { T } } \\left [ \\frac { 1 + \\xi z } { 1 - \\xi z } \\right ] \\ , d \\mu _ { 1 } ( \\xi ) \\\\ & = \\int _ { \\mathbb { T } } \\left [ \\frac { \\xi + z } { \\xi - z } \\right ] \\ , d \\mu _ { 1 } ( 1 / \\xi ) \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} f ( x ) & = \\int \\limits _ { D \\setminus D _ n } P _ { D _ n } ( x , y ) f ( y ) d y \\\\ & = \\int \\limits _ { D _ n } M _ { D _ n } ( x , v ) \\left ( G _ { D _ n } ( x _ 0 , v ) \\int \\limits _ { D \\setminus D _ n } j ( | v - y | ) f ( y ) d y \\right ) d v \\\\ & = \\int \\limits _ { D _ n } M _ { D _ n } ( x , v ) \\eta _ { D _ n } f ( d v ) , \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} \\sum _ { t = t _ 0 } ^ { \\infty } \\alpha ( t ) = \\infty ~ ~ \\mbox { a n d } ~ ~ \\sum _ { t = t _ 0 } ^ { \\infty } \\alpha ^ 2 ( t ) < \\infty . \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} \\Pi _ s ( W ) = - \\Pi \\ , \\left ( \\nabla _ { W ^ T } \\ , F _ s \\right ) - F _ j \\ , g ^ { i j } \\ , \\langle \\Pi ( \\nabla _ { F _ i } F _ s ) , W \\rangle \\ , . \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} A = n I _ { m } , D = ( 1 0 ^ 4 n + m ) I _ n - 1 0 ^ 4 \\boldsymbol { 1 } _ { n \\times n } , B _ l = \\boldsymbol { 1 } _ { m } , B _ r = \\boldsymbol { 1 } _ { n } , C _ l = \\boldsymbol { 1 } _ { n } , C _ r = \\boldsymbol { 1 } _ { m } , \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} q = \\left \\lceil \\frac { C n _ { j } ( \\log n _ { j } ) ^ { 2 } } { \\log \\lambda ^ { - 1 } } \\right \\rceil , \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} & | | \\sqrt { \\omega } \\lambda ( x ) \\left ( K \\left ( y ( x , \\frac { \\cdot } { \\lambda ( x ) ^ { 2 } } ) \\right ) ( \\omega \\lambda ( x ) ^ { 2 } ) \\right ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } \\\\ & \\leq C \\left ( | | y ( x ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } + | | \\sqrt { \\omega } \\lambda ( x ) y ( x ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } \\right ) \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\int \\| \\bar { x } - \\zeta _ 0 \\| _ { H ^ { - 1 } } ^ 2 f _ 0 ( x ) \\mu _ { N , m } ( d x ) = 0 . \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} s ^ { h ' } _ { h + 1 } + \\sum _ { h ' < h '' < h + 1 } t ^ { h '' } _ { j , h } s ^ { h ' } _ { h '' } + t ^ { h ' } _ { h } = \\begin{cases} 0 , & h \\neq j - i - 1 h \\neq h ' \\\\ c _ { i + 1 } & h \\neq j - i - 1 h = h ' \\\\ P ( i , h ' , j ) , & \\mbox { i f } h = j - i - 1 h \\neq h ' \\\\ P ( i , h ' , j ) + c _ { i + 1 } , & \\mbox { i f } h = j - i - 1 h = h ' \\\\ \\end{cases} \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} \\phi ( - q ^ { \\frac { j - i } { 2 } } x _ { j - 1 } x _ { j - 2 } \\cdots x _ { i } ) & = \\displaystyle \\sum _ { m \\geq 0 } ( - 1 ) ^ { m } \\frac { q ^ { \\frac { m ( m - 1 ) } { 2 } } ( - q ^ { \\frac { j - i } { 2 } } x _ { j - 1 } \\cdots x _ { i } ) ^ { m } } { ( q ) _ { m } } \\\\ & = \\displaystyle \\sum _ { m \\geq 0 } \\frac { q ^ { \\frac { m ^ { 2 } } { 2 } } q ^ { \\frac { m ( j - i ) } { 2 } - \\frac { m } { 2 } } ( x _ { j - 1 } \\cdots x _ { i } ) ^ { m } } { ( q ) _ { m } } , \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} \\mathfrak { a } ^ { \\dagger } = \\Theta _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) \\nabla + \\Lambda _ { n } ^ { \\left ( j \\right ) } \\left ( x ; 1 , 2 \\right ) I , \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} ( E \\rho ) ^ + = ( \\rho ^ T E ^ T E \\rho ) ^ { - 1 } \\rho ^ T E ^ T \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} C ^ * _ 2 ( g , u ) = \\Psi \\left ( \\frac { u } { \\sigma _ g } \\right ) , C ^ * _ 1 ( g , u ) = - \\frac { \\sqrt { 2 \\pi \\lambda } } { 4 \\ , \\sigma _ g } \\Psi ' \\left ( \\frac { u } { \\sigma _ g } \\right ) , C ^ * _ 0 ( g , u ) = \\frac { \\lambda } { 2 \\pi \\ , \\sigma _ g ^ 2 } \\Psi ^ { '' } \\left ( \\frac { u } { \\sigma _ g } \\right ) . \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} \\phi ( a , [ b , c ] ) = ( - 1 ) ^ { x ( y + z ) } \\phi ( [ b , c ] , a ) . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} \\langle 1 | f _ 0 \\rangle = ( 1 - | \\langle 1 | f _ N \\rangle | ^ 2 ) ^ { 1 / 2 } = \\frac { \\sqrt N } { \\sqrt { N + \\gamma _ N } } \\ , . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} A + _ G A = \\{ 2 \\sqrt i : i \\in [ n ] \\} \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\big < J ' ( \\phi _ n ) - J ' ( \\phi ) , \\phi _ n - \\phi \\big > = 0 . \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} \\langle u , v \\rangle = \\langle u ^ + , v ^ + \\rangle _ { \\mathcal R } \\ , + \\ , \\langle u ^ - , v ^ - \\rangle _ { \\mathcal R } \\quad \\langle u ^ \\pm , v ^ \\pm \\rangle _ { \\mathcal R } = { \\mathbf { R e } } \\int _ { \\mathbb { R } ^ 2 } u ^ { \\pm } \\overline { v ^ \\pm } , \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} \\bar { \\pi } _ { k } ( t ) : = \\left \\{ \\begin{array} { l l } \\pi _ { k } ( t ) & \\varsigma _ { \\pi _ { k } } \\leq t \\leq t _ { n , k } , \\\\ \\pi _ { k } ( t _ { n , k } ) + \\dfrac { t - t _ { n , k } } { s _ { n , k } - t _ { n , k } } \\big [ \\pi _ { i _ { 0 } } ( s _ { n , k } ) - \\pi _ { k } ( t _ { n , k } ) \\big ] & t _ { n , k } < t < s _ { n , k } , \\\\ \\pi _ { i _ { 0 } } ( t ) & t \\geq s _ { n , k } . \\end{array} \\right . \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} u ( t , \\cdot ) _ { / \\partial \\Omega } = 0 . \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} \\sigma _ 1 = 1 , \\sigma _ { k + 1 } = \\frac { \\log \\left ( \\xi ^ { \\sigma _ k } - \\xi ^ \\tau \\right ) } { \\log \\xi } , \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} \\Sigma _ { L , \\ , N } ^ \\alpha & = \\Sigma _ { 0 , \\ , N } ^ { \\alpha _ 0 } \\times \\prod _ { i = 1 } ^ { L } D \\Sigma _ { i - 1 , \\ , N } ^ { \\alpha _ { i } } , L \\in \\{ 1 , \\ , \\dots , r \\} , \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} 4 n ( 8 - \\mu ) + 6 { l \\choose 2 } 4 ^ { 2 } + { l \\choose 3 } 4 ^ { 3 } + 4 b _ { l - 2 } ( l ) = 4 ( l + 2 ) b _ { l - 2 } , \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} \\begin{cases} u ( t _ i ) = \\displaystyle \\sum _ { j = 0 } ^ { k } \\frac { ( \\Delta t _ i ) ^ j } { j ! } \\frac { d ^ j u } { d t ^ j } ( t _ 0 ) + \\mathcal { O } ( \\Delta t _ i ) ^ { k + 1 } \\\\ u ( t _ { i + 1 } ) = \\displaystyle \\sum _ { j = 0 } ^ { k } \\frac { ( \\Delta t _ { i + 1 } ) ^ j } { j ! } \\frac { d ^ j u } { d t ^ j } ( t _ 0 ) + \\mathcal { O } ( \\Delta t _ { i + 1 } ) ^ { k + 1 } , \\end{cases} \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} \\tilde { x } & = \\frac { x } { \\mu } , & \\tilde { y } & = \\frac { y } { \\mu } , \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} a _ { n } = \\lim _ { \\varepsilon \\downarrow 0 } \\frac { \\psi _ { \\mu _ { 1 } } ^ { ( n ) } ( \\alpha + i \\varepsilon ) } { n ! } = \\lim _ { \\varepsilon \\downarrow 0 } \\int _ { \\mathbb { R } _ { + } } \\frac { t ^ { n } } { ( 1 - t ( \\alpha + i \\varepsilon ) ) ^ { n + 1 } } \\ , d \\mu _ { 1 } ( t ) = \\int _ { \\mathbb { R } _ { + } } \\frac { t ^ { n } } { ( 1 - \\alpha t ) ^ { n + 1 } } \\ , d \\mu _ { 1 } ( t ) . \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} ( R \\eta ) _ n = \\eta _ { 1 - n } , \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { r } } \\phi ( r , \\xi ) = \\frac { 2 \\left ( a ( \\xi ) \\psi ^ { + } ( r , \\xi ) \\right ) } { \\sqrt { r } } , r ^ { 2 } \\xi > 4 \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} & Q A = d _ A c ; \\ \\ Q B = [ c , B ] ; \\ \\ Q c = \\frac 1 2 [ c , c ] ; \\\\ & Q A ^ \\dag = d _ A \\star F _ A + [ c , A ^ \\dag ] ; \\ \\ Q B ^ \\dag = F _ A + \\star B + [ c , B ^ \\dag ] ; \\ \\ Q c ^ \\dag = d _ A A ^ \\dag + [ c , c ^ \\dag ] ; \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} L _ C = \\partial _ r ^ 2 + \\frac { n - 1 } { r } \\partial _ r + \\frac { 1 } { r ^ 2 } ( \\Delta _ S + | A _ S | ^ 2 ) \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} \\frac { 1 } { k + h ^ { \\vee } } + \\frac { 1 } { \\ell + h ^ { \\vee } } = 1 . \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} & \\lambda _ 1 ( t , x ) = R _ 1 ( t , x , u ( t , x ) , q ( t . x ) ) \\in X _ 0 ( ( 0 , T _ 1 ] \\times D _ { R _ 1 } ) ) , \\\\ & b _ 1 ( t , x ) = R _ 2 ( t , x , u ( t , x ) , q ( t . x ) ) \\in X _ 0 ( ( 0 , T _ 1 ] \\times D _ { R _ 1 } ) ) \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} & \\delta _ r ^ { - 1 } \\widetilde { T } _ r - \\delta _ r ^ { - 1 } \\widetilde { T } _ s = \\\\ & = \\mu [ U ] \\left ( x ^ { ( 1 ) } \\otimes \\ldots \\otimes t ^ { ( i _ 0 ) } _ r \\otimes \\ldots \\otimes x ^ { ( m ) } \\right ) - \\mu [ U ] \\left ( x ^ { ( 1 ) } \\otimes \\ldots \\otimes t ^ { ( i _ 0 ) } _ s \\otimes \\ldots \\otimes x ^ { ( m ) } \\right ) = \\\\ & = \\mu [ U ] \\left ( x ^ { ( 1 ) } \\otimes \\ldots \\otimes \\left ( t ^ { ( i _ 0 ) } _ r - t ^ { ( i _ 0 ) } _ s \\right ) \\otimes \\ldots \\otimes x ^ { ( m ) } \\right ) . \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} [ \\partial P _ 1 ] _ \\alpha f = ( P _ 1 f ) ' . \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} ( A - \\lambda ) ^ { - 1 } = | \\lambda | ^ { - 1 } R ( h , \\omega ) , h = | \\lambda | ^ { - 1 / m } , \\ \\omega = \\lambda / | \\lambda | , \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} \\Delta D ^ { ( 2 ) } _ { 0 } = \\sum _ { w ' \\in W ^ { \\{ 1 , 3 , 4 \\} } } ( - 1 ) ^ { \\ell ( w ' ) } w ' \\left ( \\sum _ { w \\in W _ { \\{ 1 , 3 , 4 \\} } } ( - 1 ) ^ { \\ell ( w ) } e ^ { w ( \\rho ) } D ^ { ( 2 ) } _ { w ; 0 } \\right ) \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} E _ { 0 , 1 } ( \\lambda ( t ) , \\lambda ' ( t ) , \\lambda '' ( t ) ) & = 2 \\frac { \\lambda ' ( t ) ^ { 2 } } { \\lambda ( t ) ^ { 2 } } + \\frac { 2 \\lambda '' ( t ) } { \\lambda ( t ) } - \\frac { 4 \\alpha \\lambda '' ( t ) \\log ( \\lambda ( t ) ) } { \\lambda ( t ) } \\left ( \\frac { 1 } { - 1 + \\lambda ( t ) ^ { 2 \\alpha } } + 1 \\right ) \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} ( \\mathcal { C } ) : \\exists \\ , a , A > 0 , \\forall \\ , \\ell \\geq 0 , \\nu ( \\forall \\ , x \\in \\{ - \\lfloor \\ell \\rfloor , \\dots , 0 \\} ^ d , \\eta ( x ) = 1 ) \\leq A e ^ { - a \\ell } . \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } \\left ( \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) - 1 } { r ^ { 2 } } \\right ) v _ { 5 } \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle \\right ) | \\leq \\frac { C } { t ^ { 4 } \\log ^ { 3 N - b - 2 } ( t ) } \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} H _ { n + 1 } = H _ n + N H _ { n - k - 1 } \\leq H _ n + ( N - 2 ) H _ { n - k - 1 } + H _ { n - k - 1 } + H _ { n - k - 2 } + \\dots + H _ 1 + 1 . \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} \\hat X & = \\sum _ { n = 0 } ^ \\infty \\sum _ { k = 0 } ^ n \\alpha _ { n - k } \\alpha _ k \\left ( - \\partial _ \\mu \\partial ^ \\mu - m ^ 2 \\right ) \\left ( - \\partial _ \\mu \\partial ^ \\mu \\right ) ^ { n } = \\sum _ { k = 0 } ^ \\infty \\beta _ k \\left ( - \\partial _ \\mu \\partial ^ \\mu \\right ) ^ k \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} \\eta _ { \\rho _ { 1 } } ( \\mathbb { H } ) = \\Omega _ { \\rho _ { 1 } } \\cap \\mathbb { H } \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} y \\in \\bigcap _ { \\substack { F ^ c \\in \\tau \\\\ x \\in F } } F \\cap \\bigcap _ { \\substack { U \\in \\tau \\\\ x \\in U } } U = \\overline { \\{ x \\} } \\cap \\bigcap _ { \\substack { U \\in \\tau \\\\ x \\in U } } U , \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} ( R _ \\tau f ) ( z ) = f ( \\tau ^ { - 1 } z ) . \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} ( q ; q ) _ \\infty = \\sum _ { n = - \\infty } ^ { \\infty } ( - 1 ) ^ n q ^ { n ( 3 n - 1 ) / 2 } , \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} \\alpha _ { \\Omega , 0 } ^ - ( t ) = \\begin{cases} \\pi / 2 , & ~ t \\in ( 0 , 1 ] , \\\\ \\arcsin ( 1 / t ) , & ~ t \\in ( 1 , \\sqrt { 2 } ] , \\\\ \\pi / 2 , & ~ t \\in ( \\sqrt { 2 } , + \\infty ) , \\end{cases} \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} m _ \\lambda = e _ { \\lambda _ { 1 } } * e _ { \\lambda _ 2 } * \\ldots * e _ { \\lambda _ n } + \\mbox { l o w e r t e r m s } . \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} - \\Im G _ { \\mu _ { 2 } } ( x ) = \\pi p _ { \\mu _ { 2 } } ( x ) = \\sqrt { \\frac { \\pi } { 2 } } e ^ { - x ^ { 2 } / 2 } , x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} H _ { L + 1 } - G _ { L + 1 } = 2 c _ 1 \\geq 2 = 2 ( 1 ) = 2 \\left ( H _ { L } - G _ { L } \\right ) , \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} t \\frac { \\partial w } { \\partial t } = \\lambda _ b ( t , x ) w + b ( t , x ) \\frac { \\partial w } { \\partial x } \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} \\alpha = \\frac { \\partial } { \\partial y } \\left ( f _ L g _ R - f _ R g _ L \\middle ) \\right | _ { x = y = 0 } \\ , . \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} [ x , y , a z ] = ( - 1 ) ^ { \\bar a ( \\bar x + \\bar y ) } a [ x , y , z ] + \\rho ( x , y ) a z , ~ ~ \\forall x , y , z \\in \\mathcal H ( L ) , \\ a \\in \\mathcal H ( A ) . \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} \\partial _ { t } v _ { 4 } ( t , r ) = \\int _ { t } ^ { \\infty } d x \\int _ { 0 } ^ { \\infty } d \\xi J _ { 1 } ( r \\xi ) \\cos ( ( t - x ) \\xi ) \\xi \\widehat { v _ { 4 , c } } ( x , \\xi ) \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} \\pm \\left ( \\zeta _ { 5 } + \\zeta _ { 5 } ^ { 2 } + \\zeta _ { 5 } ^ { 3 } + \\zeta _ { 5 } ^ { 4 } + \\sum _ { i = 1 } ^ { 1 0 } ( \\zeta _ { 3 } + \\zeta _ { 3 } ^ { 2 } ) \\zeta _ { 1 1 } ^ { i } \\right ) \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\| u _ n - u \\| _ { L ^ 2 ( \\R ^ d ) } & \\le \\lim _ { R \\to \\infty } \\Big ( \\sup _ { n } \\| \\chi ( | x | > R ) u _ n \\| _ { L ^ 2 ( \\R ^ d ) } + \\| \\chi ( | x | > R ) u \\| _ { L ^ 2 ( \\R ^ d ) } \\Big ) = 0 . \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} f ( I + p H + M ( p ) ) = \\int _ { - \\alpha } ^ \\alpha f ( 1 + p t ) \\ , d E _ p ( t ) . \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} I ^ { [ k ] } _ { V S } & = \\sum _ j L ^ { [ k ] } _ j J \\left ( \\sqrt { j } J ^ { - 1 } ( I ^ { [ k ] } _ { C V } ) \\right ) . \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} { \\rm c h } [ W ( \\Lambda _ 0 ) ] \\ \\ _ { = } ^ { ? } \\sum _ { n _ 1 , n _ 2 , n _ 3 , n _ 4 \\geq 0 } \\frac { q ^ { n _ 1 ^ 2 + n _ 2 ^ 2 + n _ 3 ^ 2 + n _ 4 ^ 2 + n _ 1 n _ 2 + n _ 1 n _ 4 } } { ( q ) _ { n _ 1 } ( q ) _ { n _ 2 } ( q ) _ { n _ 3 } ( q ) _ { n _ 4 } } \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} q = \\ln \\ ( \\frac { e ^ { r + \\alpha } } { ( e ^ \\alpha + 1 ) ( x - y ) } + 1 \\ ) - \\frac { e ^ { r + \\alpha } } { ( e ^ \\alpha + 1 ) ( x - y ) } - 1 . \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} N _ { \\Z / n \\Z } ( a _ 1 , a _ 2 , \\ldots , a _ k ; b ) & = \\sum _ { i = 2 } ^ k ( - 1 ) ^ { k - i } c ( k , i ) n ^ { i - 1 } + ( - 1 ) ^ { k - 1 } c ( k , 1 ) \\gcd ( \\sum _ { i = 1 } ^ k a _ i , n ) \\\\ & = \\frac { 1 } { n } ( n ) _ k + ( - 1 ) ^ { k - 1 } ( k - 1 ) ! \\Big ( \\gcd ( \\sum _ { i = 1 } ^ k a _ k , n ) - 1 \\Big ) . \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} \\langle M _ { \\widetilde { a } } b _ { p , q } , b _ { j , k } \\rangle = \\langle \\widetilde { a } b _ { p , q } , b _ { j , k } \\rangle = \\delta _ { p - q , j - k } \\beta _ { a , p - q , q , k } . \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } R d R \\left ( \\frac { \\cos ( 2 Q _ { 1 } ( R ) ) - 1 } { R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\right ) \\phi _ { 0 } ( R ) v _ { 2 } ( t , R \\lambda ( t ) ) = \\frac { 4 b } { \\lambda ( t ) t ^ { 2 } \\log ^ { b } ( t ) } + E _ { v _ { 2 } , i p } ( t , \\lambda ( t ) ) \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} H ^ i ( Y , ( \\Psi _ e ) _ * ( \\omega _ { Y _ e } \\otimes H _ e '^ { p ^ e d } \\otimes A _ e '^ { p ^ e } ) \\otimes H '^ { - i } ) = 0 \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} & { \\mathfrak { n } } _ 1 ( \\lambda _ 1 ) = \\frac { 1 } { \\sqrt { 1 + 9 \\xi _ 1 ^ 4 + 9 \\eta _ 1 ^ 4 } } \\left ( - 1 , \\ 3 \\xi _ 1 ^ 2 , \\ 3 \\eta _ 1 ^ 2 \\right ) , \\\\ & { \\mathfrak { n } } _ 2 ( \\lambda _ 2 ) = \\frac { 1 } { \\sqrt { 1 + 9 \\xi _ 2 ^ 4 + 9 \\eta _ 2 ^ 4 } } \\left ( - 1 , \\ 3 \\xi _ 2 ^ 2 , \\ 3 \\eta _ 2 ^ 2 \\right ) , \\\\ & { \\mathfrak { n } } _ 3 ( \\lambda _ 3 ) = \\frac { 1 } { \\sqrt { 1 + 9 \\xi ^ 4 + 9 \\eta ^ 4 } } \\left ( - 1 , \\ 3 \\xi ^ 2 , \\ 3 \\eta ^ 2 \\right ) . \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} \\forall l \\in I ^ { 0 0 } ( \\bar x ) \\colon \\mu _ l = 0 \\ , \\land \\ , \\nu _ l = 0 . \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{gather*} A = a _ n \\int _ { z _ 1 } ^ { z _ 2 } \\sqrt { 1 + h _ z ( z , \\epsilon ) ^ 2 } \\ ; h ( z , \\epsilon ) ^ n d z , \\\\ V = v _ { n + 1 } \\int _ { z _ 1 } ^ { z _ 2 } h ( z , \\epsilon ) ^ { n + 1 } d z . \\end{gather*}"}
-{"id": "3879.png", "formula": "\\begin{align*} \\Sigma \\cap B ^ M _ { r _ 0 } \\to ( 0 , + \\infty ) \\times S \\ \\ \\ x \\mapsto ( r , \\omega ) = { \\big ( } d i s t _ { \\Sigma } ( p , x ) , \\Pi ( x ) { \\big ) } \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} \\sqcup _ { i \\in I } \\Lambda _ i = ( \\lambda _ { i , j } ) _ { ( i , j ) \\in \\cup _ { i \\in I } \\{ i \\} \\times J _ i } . \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} \\rho = \\frac { w _ - } { w _ + } , \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} ( \\partial _ { t } ^ + e _ h ^ { u ^ n } , \\Pi _ W e _ h ^ { u ^ n } ) _ { \\mathcal { T } _ h } + ( e _ h ^ { u ^ n } , e _ h ^ { \\phi ^ n } ) _ { \\mathcal { T } _ h } = ( \\partial _ { t } ^ + u _ { I h } ^ n - \\partial _ t u ^ n , \\Pi _ W e _ h ^ { u ^ n } ) _ { \\mathcal { T } _ h } . \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} \\mathcal Q _ { \\sigma } ( v _ 0 , \\varphi ) = \\lambda _ 0 ( 0 ) ( \\gamma _ 1 ( v _ 0 ) , \\gamma _ 1 ( \\varphi ) ) \\ , , \\ \\ \\ \\forall \\varphi \\in H ^ 2 ( B ) , \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} \\sum _ { j \\in \\alpha ( k _ 3 , k _ 4 ) } n _ j = \\sum _ { j \\in \\alpha ( k _ 1 , k _ 2 , k _ 3 , k _ 4 ) } n _ j + \\sum _ { j \\in \\alpha ( k _ 1 , k _ 2 ) } n _ j . \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} U = \\dfrac { \\sqrt { R } U } { \\sqrt { R } } \\mathbf { 1 } _ { \\sqrt { R } > 0 } . \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} f ( B ) & = \\min _ { \\alpha < \\infty } \\ - f ^ \\alpha ( B ) = \\ - f ^ \\alpha ( B ) . \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} | \\partial _ { t } \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) v _ { 3 } \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | & \\leq \\frac { C \\log ( \\log ( t ) ) } { t ^ { 3 } \\log ^ { 2 } ( t ) } + \\frac { C } { t ^ { 2 } \\log ^ { - b + 1 + b \\alpha } ( t ) } \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} G = \\xi _ { 1 } ( \\zeta _ { 6 } + \\zeta _ { 6 } ^ { 5 } + \\zeta _ { 1 7 } + \\cdots + \\zeta _ { 1 7 } ^ { 1 6 } ) + \\xi _ { 2 } ( \\zeta _ { 6 } + \\zeta _ { 6 } ^ { 5 } + \\zeta _ { 5 } + \\cdots + \\zeta _ { 5 } ^ { 4 } ) \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} \\psi _ { n , n + 1 } ^ { \\ast } ( Q ) = \\frac { L _ { n , n + 1 } ^ { \\ast } ( q ) } { q } \\geq - \\frac { L _ { n , 1 } ( q ) } { q } - O ( q ^ { - 1 } ) \\geq ( \\alpha - \\epsilon ) - O ( q ^ { - 1 } ) \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} \\Big ( \\sum _ { \\vert \\alpha \\vert = m } \\Vert x _ { \\alpha } \\Vert ^ { q } \\Big ) ^ { \\frac { 1 } { q } } \\leq C ^ { m } \\Big ( \\int _ { \\mathbb { T } ^ { n } } \\Big \\Vert \\sum _ { \\vert \\alpha \\vert = m } x _ { \\alpha } z ^ { \\alpha } \\Big \\Vert ^ { 2 } d z \\Big ) ^ { \\frac { 1 } { 2 } } \\ , . \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} z _ 1 = z _ 2 = z _ 3 , \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} N ^ { \\prime } ( \\eta _ { \\mu _ { 1 } } ( \\alpha ) ) = - \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t ^ { 2 } } { ( \\eta _ { \\mu _ { 1 } } ( \\alpha ) - t ) ^ { 2 } } \\ , d \\sigma ( t ) . \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{gather*} \\Psi _ i ^ + = \\lbrace \\mu \\in \\Psi _ i \\mid a _ \\mu > 0 \\rbrace , \\Psi _ i ^ - = \\lbrace \\mu \\in \\Psi _ i \\mid a _ \\mu < 0 \\rbrace , \\\\ \\Psi _ j ^ + = \\lbrace \\nu \\in \\Psi _ j \\mid b _ \\nu > 0 \\rbrace , \\Psi _ j ^ - = \\lbrace \\nu \\in \\Psi _ j \\mid b _ \\nu < 0 \\rbrace \\end{gather*}"}
-{"id": "5325.png", "formula": "\\begin{align*} & | | \\lambda ( t ) \\sqrt { \\omega } \\mathcal { F } ( \\sqrt { \\cdot } ( N ( u _ { 1 } ) - N ( u _ { 2 } ) ) ( t , \\cdot \\lambda ( t ) ) ) ( \\omega \\lambda ( t ) ^ { 2 } ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } \\\\ & \\leq C \\frac { | | y _ { 1 } - y _ { 2 } | | _ { Z } } { t ^ { 4 } \\log ^ { \\epsilon - b } ( t ) } \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} X = 3 ^ { 1 / 2 } 2 ^ { - 1 / 3 } \\begin{pmatrix} - \\tfrac { 1 } { \\sqrt { 3 } } x + y & z _ { 3 } & \\bar { z } _ { 1 } \\\\ \\bar { z } _ { 3 } & - \\tfrac { 1 } { \\sqrt { 3 } } x - y & z _ { 2 } \\\\ z _ { 1 } & \\bar { z } _ { 2 } & \\tfrac { 2 } { \\sqrt { 3 } } x \\end{pmatrix} . \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\xi _ 6 ^ { 5 n } \\sum _ { a = 0 } ^ n \\xi _ 6 ^ { - 4 a } \\sum _ { b = 0 } ^ a \\xi _ 6 ^ { - b } ( b + 1 ) \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} a _ i = \\mu _ i h ( z ) ^ { i + n _ i } ( - h ' ( z ) ) ^ { - n _ i } \\ , . \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } d s \\frac { \\lambda '' ( s ) } { ( s - t ) } = \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } d s \\frac { \\lambda '' ( s ) } { 1 + s - t } + E _ { 2 , \\partial _ { r } v _ { 1 } } ( t , r ) \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} J _ { \\tau , 1 } = \\left ( \\begin{array} { c c c c } 0 & 0 & 0 & 0 \\\\ 0 & - K & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} & \\int _ { C } \\int _ { C } | g ( t ) | | K ( t , s ) | | f ( s ) | d s d t \\leq \\int _ { C } \\frac { | g ( t ) | } { 2 \\alpha } \\left ( \\int _ { a } ^ { t } \\frac { d s } { 1 - s + t } + \\frac { 1 } { \\lambda _ { 0 } ( - t ) ^ { 1 - \\alpha } } \\int _ { - \\infty } ^ { t } \\frac { d s } { ( 1 - s + t ) ^ { 3 } } \\right ) d t \\\\ & \\leq \\frac { \\log ( 1 + d - a ) + \\frac { 1 } { 2 \\lambda _ { 0 } ( - a ) ^ { 1 - \\alpha } } } { 2 \\alpha } \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} \\tau _ { L U } ( a ) : = \\begin{cases} L & a < L , \\\\ a & L \\leq a \\leq U \\\\ U & a > U . \\end{cases} \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} v ( z ) = \\log \\gamma + \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t \\eta _ { \\mu _ { 1 } } ( z ) } { \\eta _ { \\mu _ { 1 } } ( z ) - t } \\ , d \\sigma ( t ) , z \\in \\mathbb { C } \\setminus \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} A ' ( \\sigma _ { \\infty } ) - A ' ( \\sigma _ N ) = A _ N ' ( \\sigma _ N ) - A ' ( \\sigma _ N ) \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} ( f b ) _ { i , j } = \\sum _ { k \\in \\Z _ d } f _ { i , k } b _ { k , j } \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 p } { \\partial t ^ 2 } - c _ 1 c _ 2 \\frac { \\partial ^ 2 p } { \\partial x ^ 2 } + ( c _ 1 - c _ 2 ) \\frac { \\partial ^ 2 p } { \\partial x \\partial t } = - \\frac { 2 \\alpha } { t } \\Bigl [ \\frac { \\partial p } { \\partial t } + \\frac { ( c _ 1 - c _ 2 ) } { 2 } \\frac { \\partial p } { \\partial x } \\Bigr ] \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} \\left [ S _ { \\alpha } , K _ { \\alpha } \\right ] = \\frac { 2 } { r ^ { 2 } } \\Delta - \\frac { 4 \\alpha ^ { 2 } } { r ^ { 4 } } \\left \\vert \\frac { x } { r } + \\varphi e _ { 1 } \\right \\vert ^ { 2 } - \\frac { 1 } { 2 } + \\frac { 2 i \\varphi ^ { \\prime } } { r } \\partial _ { x _ { 1 } } + \\left [ A , K _ { \\alpha } \\right ] , \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} f ! _ \\lambda g = \\big ( ( 1 - \\lambda ) f ^ * + \\lambda g ^ * \\big ) ^ * = \\big ( f . ( 1 - \\lambda ) \\Box g . \\lambda \\big ) ^ { * * } . \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} P _ 1 ( 1 , g , \\omega ) = \\delta Q ( 1 , g , \\omega ) \\ , . \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} a _ 1 \\ = \\ c _ 1 , \\ a _ 2 \\ = \\ c _ 2 , \\ \\cdots , \\ a _ { s - 1 } \\ = \\ c _ { s - 1 } \\ { \\rm { a n d } } \\ a _ s < c _ s , \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} x _ 0 : = x , x _ { 2 n + 1 } : = P _ M x _ { 2 n } \\quad x _ { 2 n } : = P _ N x _ { 2 n - 1 } \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} r _ { \\mathsf { m , n } } = c _ { \\mathsf { m , n } } \\exp \\left [ - \\frac { \\beta } { 2 } \\left ( \\lambda _ { \\mathsf { m } } - \\lambda _ { \\mathsf { n } } \\right ) \\right ] , \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\sum _ { k = 0 } ^ n B _ { n , k } \\left ( x _ 1 , x _ 2 , \\ldots \\right ) u ^ k \\frac { t ^ n } { n ! } & = \\exp \\left ( u \\sum _ { j = 1 } ^ \\infty x _ j \\frac { t ^ j } { j ! } \\right ) . \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} D a ( \\phi ) = - a ( D \\phi ) = - a ( \\lambda \\phi ) = - \\lambda a ( \\phi ) . \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} \\int _ { \\R ^ { d } } \\Theta ( x ) \\Phi ( x - \\alpha ) \\ , d x = \\begin{cases} 1 & \\ ; \\ ; \\alpha = 0 \\in \\Z ^ { d } , \\\\ 0 & \\ ; \\ ; \\alpha \\in \\Z ^ { d } \\setminus \\{ 0 \\} . \\end{cases} \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} a \\gamma ' ( t _ 0 ) = \\frac { a f ( \\beta ) } { f ' ( \\gamma ( t _ 0 ) ) } = \\frac { a f ( z _ 0 ) } { t _ 0 f ' ( z _ 0 ) } = \\left ( \\sum _ { \\ell = 1 } ^ s \\frac { t _ 0 n _ \\ell } { a ( z _ 0 - \\alpha _ \\ell ) } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} \\| x \\| ^ 2 = | \\varphi ( x , x + \\lambda y ) | \\leq \\| \\varphi \\| \\| x \\| \\| x + \\lambda y \\| , \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} f ( 0 , 0 ; \\mu ) = g ( 0 , 0 ; \\mu ) , \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} \\limsup _ { a \\to \\infty } \\left ( \\int _ 0 ^ { a } \\left ( { \\rm R i c } _ M ( s ) \\right ) ^ { \\frac { 1 } { 2 } } d s - \\frac { 1 } { 2 } \\log a \\right ) = \\infty , \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} & | \\phi ( x ) | ^ { 2 } + h ( x , v ) ^ { 2 } = | x | ^ { 2 } , & & h ( u , \\phi ( x ) ) + c h ( x , v ) = 0 , & & | u | ^ { 2 } + c ^ { 2 } = 1 , \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} \\frac { p } { 2 } \\nabla \\cdot ( | \\nabla u | ^ { p - 2 } \\nabla u ) + u ( 1 - | u | ^ 2 ) = 0 . \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} F _ { \\nu , x } ( t ) = \\mathbb { P } _ \\nu ( \\tau _ x > t ) = \\int _ { \\{ 0 , 1 \\} ^ { \\mathbb { Z } ^ d } } \\mathbb { P } _ \\eta ( \\tau _ x > t ) \\mathrm { d } \\nu ( \\eta ) \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{align*} \\ln ( { { f _ X } ( x ) } ) + 1 + \\frac { 1 } { 2 } \\ln ( 1 + { \\varsigma ^ 2 } x ) = c + b x , \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} \\nabla _ { ( \\phi ^ { - 1 } ) ^ * \\mathcal { L } _ { ( r , A , p ' , q ' ) } } \\psi = \\psi \\nabla _ { ( \\phi ^ { - 1 } ) ^ * \\mathcal { L } _ { ( r , A , p , q ) } } . \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} E _ k : = H ^ 0 ( X , L ^ { \\otimes k } \\otimes G ) , \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} \\gamma = \\left ( \\frac { \\partial f _ L } { \\partial y } \\frac { \\partial f _ R } { \\partial \\mu } - \\frac { \\partial f _ L } { \\partial \\mu } \\frac { \\partial f _ R } { \\partial y } \\right ) \\bigg | _ { x = y = \\mu = 0 } . \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} B _ 1 = \\int _ { \\R ^ 2 } \\partial _ k X \\cdot \\nabla u \\cdot \\partial _ k X ~ d x \\leq \\| \\nabla u \\| _ { L ^ \\infty } \\| \\nabla X \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} B _ i = V \\circ U , B _ j = U \\circ V , \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} \\mu _ n = \\frac { \\nu + P \\nu + \\ldots + P ^ { n - 1 } \\nu } { n } \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( q ^ { - 1 } ; q ^ 2 ) _ k ^ 4 } { ( q ^ 2 ; q ^ 2 ) _ k ^ 4 } q ^ { 4 k } \\equiv - ( 1 + 3 q + q ^ 2 ) [ n ] ^ 4 \\pmod { [ n ] ^ 4 \\Phi _ n ( q ) } , \\\\ [ 5 p t ] & \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a ; q ^ 2 ) _ k ( q ^ { - 1 } ; q ^ 2 ) _ k ^ 2 } { ( q ^ { 2 } / a , a q ^ 2 ; q ^ 2 ) _ k ( q ^ 2 ; q ^ 2 ) _ k ^ 2 } q ^ { 4 k } \\equiv 0 \\pmod { [ n ] ^ 2 ( 1 - a q ^ n ) ( a - q ^ n ) } , \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} \\mu _ n ( p ) = p ( n ) + \\nu _ n ( p ( 0 ) , \\dots , p ( n - 1 ) ) . \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} \\left ( \\frac { \\eta _ { \\mu } ( z ) } { z } \\right ) ^ { k } = \\left ( \\frac { \\eta _ { \\nu ^ { \\boxtimes k } } ( z ) } { z } \\right ) ^ { k - 1 } , z \\in \\mathbb { D } ; \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} p D _ p ( y , x ) = \\| y \\| ^ p - \\| x \\| ^ p - p \\langle j _ p ( x ) , y - x \\rangle \\geq \\frac p \\rho K C ' 2 ^ { - \\rho } R ^ { p - \\rho } \\| y - x \\| ^ \\rho , \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} - 2 \\int _ { 0 } ^ { \\frac { t } { 2 } } \\frac { \\sin ( u ) ( b - 1 ) d u } { t ^ { 3 } u \\log ^ { b } ( \\frac { t } { u } ) } = \\frac { - 2 ( b - 1 ) } { t ^ { 3 } } \\frac { \\pi } { 2 } \\left ( \\frac { 1 } { \\log ^ { b } ( t ) } + O \\left ( \\frac { 1 } { \\log ^ { b + 1 } ( t ) } \\right ) \\right ) \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} S _ { t o t } ^ + [ \\mathcal { A } ] = S _ { C S } [ \\mathcal { A } ] + F ^ + + S ^ + _ { g W Z W } [ g , A ] \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} \\left [ \\widehat { q } _ 1 , \\widehat { p } _ 1 \\right ] = \\left [ \\widehat { q } _ 2 , \\widehat { p } _ 2 \\right ] = i , \\left [ \\widehat { q } _ 1 , \\widehat { q } _ 2 \\right ] = i \\theta , \\left [ \\widehat { p } _ 1 , \\widehat { p } _ 2 \\right ] = i \\eta , \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} \\frac { 8 - 3 \\mu } { 4 } = \\frac { - 3 \\mu + 4 0 } { 2 } \\mbox { h e n c e } \\mu = 2 4 . \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} ( \\# S ^ { g ^ n _ Z } _ Z ) ^ * ( \\{ ( z , S ) \\in Z \\times \\# S ^ { Y ^ n } \\mid S ( \\vec { y } ) \\} ) & : = \\{ ( z , R ) \\in Z \\times \\# S ^ { X ^ n } \\mid \\bigvee _ { \\vec { x } \\in X ^ n } ( ( g ^ n _ Z ( z , \\vec { x } ) = ( z , \\vec { y } ) ) \\wedge R ( \\vec { x } ) ) \\} . \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\Big ( { h _ { 1 1 } ^ 1 } ^ + { U ^ + } ' + { h _ { 1 1 } ^ 1 } ^ - { U ^ - } ' + { h _ { 1 2 } ^ 1 } ^ + \\frac { U ^ + } { r } + { h _ { 1 2 } ^ 1 } ^ - \\frac { U ^ - } { r } \\Big ) r \\ , { \\mathrm d } r = 0 . \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} & J [ u , p ] ( x ) = \\frac { \\alpha ( u ( 0 ) - u ( x ) ) + ( \\alpha + 1 ) p x } { x ^ { \\alpha + 1 } \\Gamma ( 1 - \\alpha ) } , \\\\ & K _ { ( a , b ) } [ u , p ] ( x ) = \\frac { \\alpha ( \\alpha + 1 ) } { \\Gamma ( 1 - \\alpha ) } \\int _ a ^ b [ u ( x - z ) - u ( x ) + p z ] \\frac { d z } { z ^ { \\alpha + 2 } } , \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} H ( p ) \\ge & \\dim \\mu _ R = \\lim _ N \\frac { H ( A _ { R } ; N ) } { N } \\\\ = & \\lim _ N \\frac { 1 } { N } \\sum _ { l = 1 } ^ { N } H ( A _ { R } ; l | l - 1 ) \\ge H ( A _ { R } ^ { ( n ) } ; k | k - 1 ) , \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} \\begin{cases} D ^ { 1 - \\epsilon } _ 0 \\tilde { y } ( t ) = \\tilde { z } ( t ) , & \\tilde { y } ( 0 ) = y _ 0 , \\\\ D ^ { \\alpha _ 2 - 1 } _ 0 \\tilde { z } ( t ) = f ( t , \\tilde { y } ( t ) , \\tilde { z } ( t ) ) . & \\tilde { z } ( 0 ) = s . \\end{cases} \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} \\int _ { a } ^ { b } f ' ( x ) \\d x & = f ( b ) - f ( a ) . \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} \\begin{array} { l } \\mathcal V = \\Bigl \\{ v ( \\cdot , \\cdot ) \\in C ( [ 0 , t _ f ] \\times \\overline { D } ) \\Bigl | v ( t , x ) > S \\ast ( \\frac { \\kappa ^ 2 } { 4 } | v | ^ r - \\frac { c _ 1 ^ 2 + c _ 2 ^ 2 } { 2 } v ) ( t , x ) + I ( t , x ) \\Bigr \\} , \\end{array} \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} \\mathbf { T } ( s , t ) = \\frac 1 { \\big ( 1 + s ^ 2 \\big ) ^ { \\frac 1 t } } , s \\geq t > 0 . \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla u | ^ { 2 p - 2 } \\langle \\nabla u , \\nabla \\Delta _ { p , f } u \\rangle \\ , d \\mu = - ( p - 1 ) \\lambda _ { 1 ; p , f } \\int _ { \\Omega } | u | ^ { p - 2 } | \\nabla u | ^ { 2 p } \\ , d \\mu , \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} \\begin{aligned} & r i c _ g - \\frac { s c a l _ g } { n } g = \\frac { n - 2 } { h } \\Bigg [ ( \\nabla ^ 2 h ) _ { g _ 0 } - \\frac { ( \\Delta h ) _ { g _ 0 } } { n } g _ 0 + \\\\ & + d h \\otimes d ( \\log \\rho _ \\kappa ) + d ( \\log \\rho _ \\kappa ) \\otimes d h - \\frac { 2 } { n } g _ 0 \\left ( ( \\nabla h ) _ { g _ 0 } , ( \\nabla \\log \\rho _ \\kappa ) _ { g _ 0 } \\right ) g _ 0 \\Bigg ] , \\end{aligned} \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} g _ { { \\bf i } _ m } ( [ a , b ] ) \\cap g _ { { \\bf i } _ n } ( [ a , b ] ) = \\emptyset \\quad \\ ; \\ ; m , n \\in \\{ 1 , \\ldots , \\ell \\} , m \\neq n . \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} F ^ n ( w ) = F ( w ) + ( D ^ n - D ) \\circ w , \\ \\ \\ w \\in \\R _ + ^ m . \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} \\int _ { m = A n ^ { 1 / 3 } } ^ { n ^ { 1 / 3 } \\log ^ { ( 2 ) } n + 1 } \\frac { 2 \\beta C m ^ { 1 / 2 } } { n ^ { 1 / 2 } } e ^ { - \\frac { \\gamma } { 2 } m ^ { 3 / 2 } n ^ { - 1 / 2 } + \\lambda m n ^ { - 1 / 3 } } d m \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} Q : = \\frac { I - \\exp ( - \\frac { 1 } { 2 } D ^ - D ^ + ) } { D ^ - D ^ + } D ^ + \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} a ( n ) = \\sum _ { i = 1 } ^ { n - 1 } a ( i ) + ( 2 ^ { k + 1 } - 1 ) ( 2 ^ { n - 1 } + 1 ) \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} \\zeta _ { 6 } + \\zeta _ { 6 } ^ { 5 } + \\zeta _ { 5 } + \\zeta _ { 5 } ^ { 2 } + \\zeta _ { 5 } ^ { 3 } + \\zeta _ { 5 } ^ { 4 } = 0 , \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} z \\in \\mathbb { C } ^ { n - 1 } \\rightsquigarrow \\sum _ { \\substack { \\alpha \\in F \\\\ \\alpha _ { n } = k } } x _ { \\alpha } z _ { 1 } ^ { \\alpha _ { 1 } } \\cdots z _ { n - 1 } ^ { \\alpha _ { n - 1 } } , \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} t _ { 1 , \\mathrm { m u l t } } = { - \\omega _ 1 ^ \\mathrm { s p } - \\omega _ 2 ^ \\mathrm { s p } } t _ { 2 , \\mathrm { m u l t } } = { \\omega _ 1 ^ \\mathrm { s p } - \\omega _ 2 ^ \\mathrm { s p } } . \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} N ( u ) ( t , R \\lambda ( t ) ) & = \\left ( \\frac { \\sin ( 2 v ( t , R ) ) - 2 v ( t , R ) } { 2 R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\right ) \\cos ( 2 Q _ { 1 } ( R ) ) \\\\ & + \\left ( \\frac { \\cos ( 2 v ( t , R ) ) - 1 } { 2 R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\right ) \\sin ( 2 Q _ { 1 } ( R ) + 2 v _ { c o r r } ( t , R \\lambda ( t ) ) ) \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} v ' ( t ) + \\mu ( 1 + \\gamma \\partial _ t ^ \\alpha ) v ( t ) & = g ( t ) , t > 0 , \\\\ v ( 0 ) & = v _ 0 , \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} w ^ * ( t ) = & \\psi ( x _ 0 ) \\exp \\Bigl [ - \\int _ { t } ^ { T } \\frac { \\lambda ^ * ( s ) } { s } d s \\Bigr ] \\\\ & - \\int _ t ^ T \\exp \\Bigl [ - \\int _ t ^ { \\tau } \\frac { \\lambda ^ * ( s ) } { s } d s \\Bigr ] \\frac { g ^ * ( \\tau ) } { \\tau } d \\tau . \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} \\bar { \\partial } ( \\varphi ^ { r _ 1 ' \\cdots r _ s ' } ) = O . \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } r { U ^ { \\pm } } ^ 2 { \\mathfrak h } _ 1 ^ { \\pm } = 0 , \\int _ 0 ^ { \\infty } r { U ^ { \\pm } } ^ 2 | { \\mathfrak h } _ 1 ^ \\pm | ^ 2 ( 1 + r ^ { 2 + \\sigma } ) \\ , { \\mathrm d } r \\leq C \\int _ { { \\mathbb R } ^ 2 } | h | ^ 2 ( 1 + r ^ { 2 + \\sigma } ) . \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} \\eta _ j = \\min _ { \\substack { V \\subset \\mathcal { H } ^ 2 _ { 0 , D } ( \\Omega ) \\\\ { \\rm d i m } V = j } } \\max _ { \\substack { v \\in V \\\\ u \\ne 0 } } \\frac { \\mathcal Q _ { \\sigma } ( v , v ) } { \\int _ { \\partial \\Omega } \\left ( \\frac { \\partial v } { \\partial \\nu } \\right ) ^ 2 d \\sigma } , \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} d - 2 = \\sum _ { i = 1 } ^ { j - 1 } n _ i + \\delta ' \\ \\ \\dim ( C ' ) = \\dim ( C ) = \\sum _ { i = j } ^ t n _ i m _ i - m _ j \\delta = \\sum _ { i = j } ^ t \\tilde { n } _ i m _ i - \\delta ' m _ j , \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} p ( x , 0 ) = \\delta ( x ) \\ , \\ \\ \\ \\ p _ t ( x , 0 ) = 0 \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} [ X , Y ] = f ' ( x ) \\partial _ y . \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} = - e ^ { - \\lambda t } \\sum _ { k = 0 } ^ \\infty I _ { 2 k + 2 } \\Bigl ( \\frac { 2 \\lambda } { c _ 1 + c _ 2 } \\sqrt { ( c _ 1 t - \\beta ) ( c _ 2 t + \\beta ) } \\Bigr ) \\Bigl ( \\frac { c _ 2 } { c _ 1 } \\Bigr ) ^ { 2 k + 1 } \\Biggl ( \\sqrt { \\frac { c _ 1 t - \\beta } { c _ 2 t + \\beta } } \\Biggr ) ^ { 2 k + 2 } \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} \\langle \\overline { D } _ x \\rangle = \\bigg \\langle \\partial _ x + y ^ i _ x \\partial _ { y ^ i } - \\bigg ( \\Gamma _ { 1 1 } ^ k + ( 2 \\Gamma _ { 1 m } ^ k - \\delta ^ k _ m \\Gamma _ { 1 1 } ^ 1 ) y ^ m _ x + ( \\Gamma _ { j m } ^ k - 2 \\delta ^ k _ m \\Gamma _ { 1 j } ^ 1 ) y ^ j _ x y ^ m _ x - \\Gamma _ { i j } ^ 1 \\ , y ^ i _ x y ^ j _ x y ^ k _ x \\bigg ) \\partial _ { y ^ k _ x } \\bigg \\rangle \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} & x _ { i _ { 1 } , i _ { 2 } } y _ { i _ { 3 } , i _ { 4 } } = x _ { i _ { \\sigma _ { 1 } } , i _ { \\sigma _ { 2 } } } y _ { i _ { \\sigma _ { 3 } } , i _ { { 4 } } } , \\\\ & x _ { i _ { 1 } , i _ { 2 } } z _ { i _ { 3 } } = x _ { i _ { \\sigma _ { 1 } } , i _ { \\sigma _ { 2 } } } z _ { i _ { \\sigma _ { 3 } } } \\\\ & y _ { i _ { 1 } , i _ { 3 } } z _ { i _ { 2 } } = y _ { i _ { 2 } , i _ { 3 } } z _ { i _ { 1 } } \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} f ^ { ( 1 ) } ( \\xi _ { 0 } ) = \\overline { \\xi _ { 0 } } f ( \\xi _ { 0 } ) \\left [ 1 - \\xi _ { 0 } H ^ { ( 1 ) } ( \\xi _ { 0 } ) \\right ] ; \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} F ( x , y , z ) = 0 , \\ ( y , z ) \\in \\Omega , \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} \\Lambda _ { 1 } = \\left \\{ ( \\lambda , \\hat { \\mu } _ { 1 } ) \\in \\mathbb { R } ^ { 2 } : \\lambda \\leq \\hat { \\lambda } _ { 1 } \\right \\} ~ \\mbox { a n d } ~ \\Lambda _ { 2 } = \\left \\{ ( \\hat { \\lambda } _ { 1 } , \\mu ) \\in \\mathbb { R } ^ { 2 } : \\mu \\leq \\hat { \\mu } _ { 1 } \\right \\} . \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} & | \\frac { - 1 } { 2 \\omega } \\int _ { 0 } ^ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } r \\phi _ { 0 } ( r ) F _ { 4 } ( t , r \\lambda ( t ) ) d r | = | \\frac { 1 } { 2 \\omega } \\int _ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } ^ { \\infty } \\phi _ { 0 } ( r ) F _ { 4 } ( t , r \\lambda ( t ) ) r d r | = 0 \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} B ^ { \\circ \\alpha } & : = \\max \\{ A \\in \\infty \\Sigma ^ 0 _ \\alpha ( X ) \\mid A \\le B \\} , \\\\ \\ - B ^ \\alpha & : = \\min \\{ C \\in \\infty \\Pi ^ 0 _ \\alpha ( X ) \\mid B \\le C \\} . \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} F ( X \\otimes X ' ) = F ( X ) \\otimes F ( X ' ) = X _ j \\otimes X _ j = X _ 0 \\oplus X _ k \\oplus X _ 3 . \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} D ^ m [ f _ 1 \\circ f _ 2 \\circ f _ 3 ] ( x ) & = D f _ 1 ( z ) \\left ( D f _ 2 ( y ) \\right ) \\left ( D ^ m f _ 3 ( x ) \\right ) + D f _ 1 ( z ) \\left ( D ^ m f _ 2 ( y ) \\right ) \\left ( D f _ 3 ( x ) \\right ) ^ { \\otimes m } \\\\ & + D ^ m f _ 1 ( z ) \\left ( D f _ 2 ( y ) \\right ) ^ { \\otimes m } \\left ( D f _ 3 ( x ) \\right ) ^ { \\otimes m } + \\mathcal { P } _ m ( f _ 1 , f _ 2 ) ( y ) \\left ( D f _ 3 ( x ) \\right ) ^ { \\otimes m } \\\\ & + \\mathcal { P } _ m ( f _ 1 \\circ f _ 2 , f _ 3 ) ( x ) , \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} \\Theta _ { m + n - 1 } ( f \\circ _ i g ) = \\Theta _ m ( f ) \\circ _ i \\Theta _ n ( g ) \\Theta _ 1 ( \\mathrm { i d } ) = \\mathrm { i d } _ A , \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { \\partial \\phi } { \\partial \\nu } & = u \\frac { \\partial u } { \\partial \\nu } + D ^ 2 u ( \\nu , D u ) \\\\ & = \\varphi ( f + D ^ 2 u ( \\nu , \\nu ) ) + D ^ 2 u ( \\nu , \\bar { \\nabla } f ) \\\\ & = \\varphi ( f + D ^ 2 u ( \\nu , \\nu ) ) + \\nabla _ { \\bar { \\nabla } f } \\nabla u \\cdot \\nu \\\\ & = \\varphi ( f + D ^ 2 u ( \\nu , \\nu ) ) + \\bar { \\nabla } f \\cdot \\bar { \\nabla } \\varphi - h ( \\bar { \\nabla } f , \\bar { \\nabla } f ) . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} \\rho ( L _ { - 1 } ) = - \\partial & \\widehat \\rho ( L _ { - 1 } ) = q ^ { - 1 } \\\\ \\rho ( L _ { 0 } ) = - z \\partial - c & \\widehat \\rho ( L _ 0 ) = - q \\partial _ q - c + 1 \\\\ \\rho ( L _ { 1 } ) = - z ^ 2 \\partial - 2 c z & \\widehat \\rho ( L _ 1 ) = q ^ 3 \\partial _ q ^ 2 + 2 c q ^ 2 \\partial _ q \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} c ( A ; \\gamma \\alpha ) = d \\big ( ( \\chi + \\lambda ) d A \\big ) & = d \\big ( \\chi d A \\big ) + d \\big ( \\lambda d A \\big ) \\\\ & = c ( A ; \\gamma ) + c ( A ; \\alpha ) = c ( A ; \\alpha \\gamma ) , \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} \\vec { \\mathbb { I } } = \\langle \\langle \\mathbb { I } _ { \\gamma } \\colon \\gamma \\leq \\delta \\rangle , \\langle \\dot { \\mathbb { J } } _ { \\gamma } \\colon \\gamma < \\delta \\rangle \\rangle \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} \\nu _ { N , 1 } ^ { \\sigma } = \\mu _ N ^ { \\sigma } \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} & \\log \\left ( Z _ { n } ( \\beta ) \\right ) + \\frac { 1 } { 2 } \\log \\left ( 1 - 2 \\beta J \\right ) - ( n - 1 ) \\beta ^ 2 + \\beta ( J - J ' ) - \\beta C _ { n , 1 } - \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\sum _ { k = 2 } ^ { m _ { n } } \\frac { 2 ( 2 \\beta ) ^ { k } \\left ( C _ { n , k } - ( n - 1 ) \\mathbb { I } _ { k = 2 } \\right ) - ( 2 \\beta ) ^ { 2 k } } { 4 k } \\stackrel { p } { \\to } 0 . \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} \\mathcal { T } S : = \\theta ^ \\tau ( T S ) . \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{align*} ( \\Im c _ m ) ^ 2 - ( \\Im c _ { m + 1 } ) ^ 2 \\ , & = ( y _ { m + 1 } - y _ m ) \\big ( 2 t _ { n + 1 } - y _ { m } - y _ { m + 1 } + 2 \\kappa \\big ) \\\\ & > \\ , m ( m + 1 ) ! \\cdot \\big ( 2 t _ { n + 1 } - t _ { m + 1 } - t _ { m + 2 } \\big ) \\\\ & \\ge \\ , m ( m + 1 ) ! \\cdot \\big ( ( n + 1 ) ! - n ! \\big ) \\ , = \\ , m n ( m + 1 ) ! \\ , n ! , \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} Q : = \\frac { I - \\exp ( - \\frac { 1 } { 2 } D ^ - D ^ + ) } { D ^ - D ^ + } D ^ + \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} E _ k & \\ge \\int _ { \\R ^ d } | \\nabla u _ a | ^ 2 - a \\int _ { \\R ^ d } | u _ a | ^ { 2 + 4 / d } \\\\ & = ( a ^ * - a ) ^ { - 2 / p } \\left [ \\int _ { \\R ^ d } | \\nabla v _ a | ^ 2 - a \\int _ { \\R ^ d } | v _ a | ^ { 2 + 4 / d } \\right ] . \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { \\| a _ n \\| _ X } { n ^ \\sigma } \\leq C \\| D \\| _ { \\mathcal { H } _ { p } ( X ) } , \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} j _ { n } ^ { ( 3 ) } - 3 J _ { n } ^ { ( 3 ) } = 2 j _ { n - 3 } ^ { ( 3 ) } , \\ n \\geq 3 , \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} P = \\left ( \\begin{array} { c c } 1 - p _ 0 & p _ 0 \\\\ 1 - p _ 1 & p _ 1 \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} m _ { s c } ( z ) = \\int _ \\R \\frac { \\rho _ { s c } ( x ) } { x - z } \\d x . \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} \\begin{aligned} & r _ N = r \\cos \\theta _ { N - 1 } \\\\ & r _ { N - 1 } = r \\sin \\theta _ { N - 1 } \\cos \\theta _ { N - 2 } \\\\ & \\cdots \\cdots \\\\ & r _ 2 = r \\sin \\theta _ { N - 1 } \\sin \\theta _ { N - 2 } \\cdots \\sin \\theta _ 2 \\cos \\theta _ 1 \\\\ & r _ 1 = r \\sin \\theta _ { N - 1 } \\sin \\theta _ { N - 2 } \\cdots \\sin \\theta _ 2 \\sin \\theta _ 1 . \\end{aligned} \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} \\partial _ { t } \\upsilon = S \\upsilon + K \\upsilon + \\left ( a + i b \\right ) \\left ( V f + e ^ { \\gamma \\varphi } F \\right ) R ^ { n } \\times \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} B ( x , v ) = \\tau ( v ) \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} E _ { u , v } \\colon v ( x ^ 3 + y ^ 3 + z ^ 3 ) = 3 u x y z . \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } \\left ( - \\lambda ( t ) \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) - 1 } { r ^ { 2 } } \\right ) E _ { 5 } \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle \\right ) | \\leq \\frac { C } { t ^ { 4 } \\log ^ { b + 1 } ( t ) } + \\frac { C \\sup _ { x \\geq t } \\left ( \\frac { | e '''' ( x ) | x } { \\lambda ( x ) ^ { 3 - 2 \\alpha } } \\right ) } { t \\log ^ { ( 3 - 2 \\alpha ) b } ( t ) } \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} \\begin{aligned} C _ { k } & : 2 ( 3 ^ { k - 2 } ) \\times 2 ( 3 ^ { k - 2 } ) \\mbox { s q u a r e m a t r i x ( i . e . , $ 2 \\times 2 , 6 \\times 6 , 1 8 \\times 1 8 , \\dots $ ) } \\\\ A _ { k } & : 2 ( 3 ^ { k - 2 } ) \\times 2 ( 3 ^ { k - 1 } ) \\mbox { r e c t a n g u l a r w i d e m a t r i x } \\\\ B _ { k } & : 2 ( 3 ^ { k - 1 } ) \\times 2 ( 3 ^ { k - 2 } ) \\mbox { r e c t a n g u l a r t a l l m a t r i x . } \\end{aligned} \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} c = ( h + 1 ) ^ { 2 } + 4 y x = h ^ { 2 } + 1 + 2 x y + 2 y x . \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} \\| f _ 1 \\| _ 2 ^ 2 + \\| f _ 2 \\| _ 2 ^ 2 = \\| { \\cal F } \\| _ { 2 , 2 \\sigma } ^ 2 \\ , . \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} \\lambda _ { 1 } = \\lambda ' _ { 1 } = \\tilde { \\lambda } _ { 1 } = \\tilde { \\lambda ' } _ { 1 } , \\lambda _ { 2 } = \\lambda ' _ { 2 } = \\tilde { \\lambda } _ { 2 } = \\tilde { \\lambda ' } _ { 2 } , \\dots , \\lambda _ { \\tilde { N } } = \\lambda ' _ { \\tilde { N } } = \\tilde { \\lambda } _ { \\tilde { N } } = \\tilde { \\lambda ' } _ { \\tilde { N } } . \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{bmatrix} \\tau & 1 \\\\ - \\delta & 0 \\end{bmatrix} \\begin{bmatrix} x \\\\ y \\end{bmatrix} - { \\rm s g n } ( x ) \\begin{bmatrix} b _ 1 \\\\ b _ 2 \\end{bmatrix} . \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} s ^ { \\bullet } & \\ , = ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { \\bf 2 } , { 3 } , { \\bf 2 } , { 3 } , 8 , 8 , 4 ) \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { 2 } , { 3 } , { 3 } , { \\bf 4 } , 8 , 8 , { \\bf 4 } ) , \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} \\langle \\Pi _ k ^ \\partial u _ h , \\mu _ h \\rangle _ { E } = \\langle u _ h , \\mu _ h \\rangle _ { E } , \\forall \\mu _ h \\in \\mathcal { P } ^ k ( E ) \\ \\textup { a n d } \\ E \\in \\partial K . \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = F ( x , y ; \\mu ) , { \\rm ~ u n t i l ~ } x = 0 , \\ , y > 0 , \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} \\Im F _ { \\rho _ { 1 } } ( z ) = \\Im ( F _ { \\rho _ { 1 } ' } ( F _ { \\rho _ { 1 } '' } ( z ) ) \\ge \\Im ( F _ { \\rho _ { 1 } '' } ( z ) ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} V = \\Big [ \\frac { \\kappa } { 6 } ( 1 + 4 g ) + \\frac { h ^ 2 } { 6 \\kappa } + \\frac { 3 } { 8 \\kappa } g ^ { i j } h _ i h _ j \\Big ] ( p ^ 0 ) ^ 2 + \\frac { 6 } { \\kappa } ( q _ 0 ) ^ 2 + \\frac { 2 } { \\kappa } h p ^ 0 q _ 0 . \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} \\mathbf { P } \\left ( ( \\eta _ n ) _ { n = 1 } ^ { M } = ( x _ n ) _ { n = 1 } ^ { M } \\ : \\vline \\ : \\hat { W } _ N = \\hat { W } _ 0 , \\ : \\max _ { 1 \\leq n \\leq N } \\hat { W } _ n \\leq K \\right ) . \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} C = \\frac { 1 } { 2 \\ln ( 2 ) } \\displaystyle \\int _ 0 ^ \\infty s e ^ { - s } M ^ { ( c ) } _ { \\gamma _ 1 } ( s ) M ^ { ( c ) } _ { \\gamma _ 2 } ( s ) d s , \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} \\O ^ { ( 0 ) } ( x ) - \\O ^ { ( 0 ) } ( x ' ) = \\int _ \\gamma d \\O ^ { ( 0 ) } \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} G ( u ) = ( g _ i ( u ) ) : = ( 1 + \\epsilon ) F ( u ) - F ( ( 1 + \\epsilon ) u ) , \\ \\ u , v \\in \\R ^ m . \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} \\tilde { x } _ i = \\arg \\min _ { c \\in \\mathcal { C } } | c - \\gamma \\hat { x _ i } | ^ 2 , i = 1 , \\ldots , N . \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} A ^ { p - { 1 \\over 2 } } h ( \\widetilde C ) ^ { 2 p - 1 } A ^ { p - { 1 \\over 2 } } \\succ _ { \\log } ( A ^ { 1 / 2 } h \\bigl ( \\widetilde C \\bigr ) A ^ { 1 / 2 } \\bigr ) ^ { 2 p - 1 } = P _ f ( A , B ) ^ { 2 p - 1 } . \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} V _ g f ( x , \\omega ) = \\langle f , \\pi ( x , \\omega ) g \\rangle _ { L ^ 2 ( \\mathbb { R } ^ d ) } = \\int _ { \\mathbb { R } ^ d } f ( t ) \\overline { g ( t - x ) } e ^ { - 2 \\pi i t \\cdot \\omega } d t . \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} & | | \\xi \\overline { y } ( \\xi ) | | _ { L ^ { 2 } ( \\rho ( \\xi ) d \\xi ) } ^ { 2 } = | | L ^ { * } L \\overline { v } | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} x _ 0 : = x , x _ { 2 n + 1 } : = ( I - P _ M ) x _ { 2 n } \\quad x _ { 2 n } : = ( I - P _ N ) x _ { 2 n - 1 } \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} B = U _ r \\circ U _ { r - 1 } \\circ . . . \\circ U _ 1 , \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} N ^ N \\prod _ { j = 1 } ^ p f ( \\beta _ j ) ^ { m _ j } = \\prod _ { i = 1 } ^ s \\Big ( n _ i ^ { n _ i } \\prod _ { k \\neq i } ( \\alpha _ i - \\alpha _ k ) ^ { n _ k } \\Big ) . \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} \\sum _ { w \\in S _ \\ell } ( - 1 ) ^ { \\ell ( w ) } \\mathbf { z } ^ { \\gamma + \\rho - w \\rho } = \\prod _ { i < j } \\Big ( 1 - \\frac { z _ i } { z _ j } \\Big ) \\mathbf { z } ^ \\gamma \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} \\mathrm { N } ^ A _ n : = \\{ i \\in \\mathrm { N } ^ A : \\exists m < n \\ , ( c _ i = c ^ A _ m ) \\} , \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int \\limits _ { D \\cap { B ( x , 2 \\delta ) } ^ c } G _ D ( x _ n , y ) j ( | y - z _ n | ) d y = \\int \\limits _ { D \\cap { B ( x , 2 \\delta ) } ^ c } G _ D ( x , y ) j ( | y - z | ) d y . \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} A _ 3 ( 9 ) & = \\{ ( ( 9 ) , 9 , 0 ) , \\ , ( ( 6 , 3 ) , 6 , 0 ) , \\ , ( ( 6 , 3 ) , 3 , 0 ) , \\ , ( ( 6 , 2 , 1 ) , 6 , 0 ) , \\\\ & \\quad \\ ( ( 5 , 3 , 1 ) , 3 , 0 ) , \\ , ( ( 4 , 3 , 2 ) , 3 , 0 ) , \\ , ( ( 4 , 3 , 1 ^ 2 ) , 3 , 0 ) , \\\\ & \\quad \\ ( ( 3 ^ 2 , 2 , 1 ) , 3 , 0 ) , \\ , ( ( 3 ^ 2 , 2 , 1 ) , 3 , 1 ) , \\ , ( ( 3 , 2 ^ 2 , 1 ^ 2 ) , 3 , 0 ) \\} \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} \\partial _ { 0 , ( a , b ) } P _ 1 = S _ { a , b } ^ \\star [ \\partial _ 0 P _ 1 \\frac 1 \\alpha ] _ \\alpha S _ { a , b } , \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} & \\widetilde { a } _ { \\mu , \\nu } = \\iint _ { \\R ^ n \\times \\R ^ n } \\Theta ( \\xi - \\mu , \\eta - \\nu ) \\sigma _ { A , \\Phi } ( \\xi , \\eta ) \\ , d \\xi d \\eta \\\\ & = \\sum _ { \\mu ^ { \\prime } , \\nu ^ { \\prime } } a _ { \\mu ^ { \\prime } , \\nu ^ { \\prime } } \\iint _ { \\R ^ n \\times \\R ^ n } \\Theta ( \\xi - \\mu , \\eta - \\nu ) \\Phi ( \\xi - \\mu ^ { \\prime } , \\eta - \\nu ^ { \\prime } ) \\ , d \\xi d \\eta . \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} 2 ^ { X } & = \\{ A \\subseteq X : A \\} , \\\\ C ( X ) & = \\{ A \\subseteq X : A \\} . \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} \\sigma _ { \\sigma _ y ( x ) } = \\sigma _ { \\sigma _ z ( x ) } . \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} v _ k ( r ; c _ k ^ + , c _ k ^ - ) = \\int _ S u ( r , \\omega ) w _ k ( \\omega ) \\ d \\omega \\leq ( \\int _ S u ( r , \\omega ) ^ 2 \\ d \\omega ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} x _ 0 = x , x _ { 2 n + 1 } = \\Pi _ { M } ^ p x _ { 2 n } , x _ { 2 n } = \\Pi _ { N } ^ p x _ { 2 n - 1 } , \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} w _ { i j } ( t ) \\triangleq \\begin{cases} a _ { i j } ( t ) b _ { i j } ( t ) , & i \\not = j \\cr 1 - \\sum _ { j \\not = i } a _ { i j } ( t ) b _ { i j ( t ) } , & i = j \\end{cases} . \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} P \\{ M ( t ) = \\mathcal { T } ( t ) = c t \\ | \\ V ( 0 ) = c \\} = e ^ { - \\lambda t } . \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} v _ k ( r ) = v _ k ( r ; c _ k ^ + , c _ k ^ - ) : = \\begin{cases} c _ k ^ + r ^ { \\gamma ^ + _ k } + c _ k ^ - r ^ { \\gamma ^ - _ k } = r ^ { - \\frac { n - 2 } { 2 } } ( c _ k ^ + r ^ { b _ k } + c _ k ^ - r ^ { - b _ k } ) , & b _ k \\neq 0 ; \\\\ r ^ { - \\frac { n - 2 } { 2 } } ( c _ k ^ + + c _ k ^ - \\log r ) , & b _ k = 0 . \\end{cases} \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} ( h ^ 2 \\Delta _ g - 1 ) u = f , \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} \\left ( \\rho ( L _ 0 ) + \\frac { 1 } { 2 } \\right ) ^ 2 - \\left ( \\mu + \\frac { 1 } { 2 } \\right ) ^ 2 = h ( z ) ( \\rho ( L _ 0 ) + c ) h ( z ) ^ { - 1 } ( \\rho ( L _ 0 ) - c ) \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ { n + 1 } ( 1 + x _ i ) \\right ) \\sum _ { I \\subseteq [ 2 , n ] } k _ { n - | I | } ( \\underline x _ n ^ I ) \\sum _ { J \\subseteq I } ( - 1 ) ^ { | J | } \\left ( \\prod _ { j \\in J } \\frac { x _ j } { 1 + x _ j } \\right ) \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} \\Omega ( z ) = \\frac { 1 - \\mu } { \\lambda } + \\sum _ { k = 2 } ^ { \\infty } c _ k z ^ k , \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} ( l + 1 ) \\mu = 4 n ( l + 1 ) ^ { 2 } . \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} \\varphi _ { T ( y ; \\mu ) } \\left ( \\tfrac { 1 } { y } R ( y ; \\mu ) , \\Theta ( y ; \\mu ) \\right ) & = \\frac { 1 } { y } \\ , P ( y ; \\mu ) , \\\\ \\psi _ { T ( y ; \\mu ) } \\left ( \\tfrac { 1 } { y } R ( y ; \\mu ) , \\Theta ( y ; \\mu ) \\right ) & = - \\frac { 3 \\pi } { 2 } , \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} { \\phi } _ { p , \\alpha } ( t ) = t ^ { p } ( 1 + \\log ^ { + } ( t ) ) ^ { \\alpha } , \\alpha > 0 , p \\geq 1 , \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\| u ( \\cdot , t ) \\| _ { L _ 1 ( K ) } = 0 . \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} \\small \\begin{aligned} P ( A | A ) = \\frac { \\beta + \\alpha ( 1 - \\alpha - \\beta ) ^ t } { \\alpha + \\beta } , ~ & P ( B | A ) = \\frac { \\alpha - \\alpha ( 1 - \\alpha - \\beta ) ^ t } { \\alpha + \\beta } , \\\\ P ( A | B ) = \\frac { \\beta - \\beta ( 1 - \\alpha - \\beta ) ^ t } { \\alpha + \\beta } , ~ & P ( B | B ) = \\frac { \\alpha + \\beta ( 1 - \\alpha - \\beta ) ^ t } { \\alpha + \\beta } . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} \\Upsilon = \\left \\{ ( \\lambda , \\mu ) \\in \\mathbb { R } ^ { 2 } : \\eqref { p q } ~ \\mbox { a d m i t s a t l e a s t o n e p o s i t i v e s o l u t i o n } \\right \\} , \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} \\xi _ { N , M } = \\xi _ { N , M } ( c , c ' ) & = \\left ( 1 + M ^ { - 1 } + A ^ { - 1 } M c - A ^ { - 1 } c ' \\right ) N , \\\\ \\eta _ { N , M } = \\eta _ { N , M } ( c , c ' ) & = \\Bigl ( - ( \\sqrt { 2 } + 1 ) ^ { \\frac { 2 } { 3 } } ( \\sqrt { 3 } - \\sqrt { 2 } ) + ( \\sqrt { 2 } + 1 ) ^ { \\frac { 2 } { 3 } } M ^ { - 1 } \\\\ & + ( \\sqrt { 2 } + 1 ) ^ { \\frac { 2 } { 3 } } A ^ { - 1 } M c - ( \\sqrt { 2 } - 1 ) ^ { \\frac { 2 } { 3 } } A ^ { - 1 } c ' \\Bigr ) N . \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} \\lim _ { k } { \\| z _ k - ( u z ) z _ k ( u z ) ^ * \\| _ 2 } = 0 , \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} a _ { i j } ^ { ( 2 k ) } = \\begin{cases} ( - 1 ) ^ { k + i } \\dfrac { m ^ { 1 \\dots k } _ { 1 \\dots i - 1 i + 1 \\dots k j } } { m _ { k } } \\medskip & \\mbox { i f } 1 \\leq i \\leq k , k + 1 \\leq j \\leq n \\\\ \\dfrac { m ^ { 1 \\dots k i } _ { 1 \\dots k j } } { m _ { k } } & \\mbox { i f } k + 1 \\leq i \\leq m , k + 1 \\leq j \\leq n \\end{cases} . \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} B ( e _ { x y } , e _ { u v } ) & = B ( e _ { x y } , e _ { y v } ) = B ( e _ x e _ { x y } e _ y , e _ y e _ { y v } e _ v ) = e _ x B ( e _ x , e _ { x y } ) e _ y e _ { y v } e _ v \\\\ & = B ( e _ x , e _ { x y } ) ( x , y ) e _ { x v } \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} d _ T ( f ) = ( - 1 ) ^ n ~ \\delta _ { \\mathrm { H o c h } } ( f ) . \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} ( i _ { Z ^ { ( 2 ) } } ) ^ ! _ { i _ { Y ^ 4 } } \\circ ( p _ { 1 2 } \\times p _ { 2 3 } ) ^ ! = ( p _ T ) ^ ! _ { p _ { 1 2 } \\times p _ { 2 3 } } \\circ ( i _ { Z \\times Z } ) ^ ! _ { i _ { Y ^ 4 } } . \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} \\omega _ { 2 } \\rtimes v ( a ) ( z ^ { 0 } \\otimes \\varepsilon _ { 0 } ) = \\sum _ { n , m } a _ { n , m } z ^ { n } \\otimes \\varepsilon _ { - m } , \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} \\mathcal { C } _ 0 ( G : H ) : = \\{ f \\in \\mathcal { C } _ 0 ( G ) : R _ h f = f \\ \\forall h \\in H \\} , \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} \\left ( A - \\nu _ { \\mathsf { k } } \\right ) \\mathsf { q = \\gamma \\hat { p } } _ { \\mathsf { k } } \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} \\Delta f = \\mathrm { t r } ( \\mathrm { H e s s } _ f ) . \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} \\begin{array} { l } \\bullet ~ ( x - 1 ) ^ { \\epsilon _ 1 } h _ 1 ( x ) = 0 , \\\\ \\bullet ~ ( \\epsilon _ 2 , \\epsilon _ 3 ) = ( n - r + k _ 3 = 5 , n - 2 r + k _ 1 + k _ 2 = 1 ) , \\\\ \\end{array} \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} \\big \\lbrace \\omega \\in \\Omega \\ : s \\mapsto X ( \\omega , s ) \\ s . t . \\ V ( \\omega , 0 ) = + c , \\ N ( \\omega , t ) = n , \\ \\max _ { 0 \\le s \\le t } X ( \\omega , s ) > \\beta , \\ X ( \\omega , t ) = x \\big \\rbrace , \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} H _ n ( f ) = v _ 0 ^ n v _ 1 ^ { n - 1 } v _ 2 ^ { n - 2 } \\cdots v _ { n - 2 } ^ 2 v _ { n - 1 } . \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\frac { q ^ { n ( n + a ) } } { ( q ; q ) _ n } = \\frac { 1 } { ( q ^ { 1 + a } ; q ^ 5 ) _ \\infty ( q ^ { 4 - a } ; q ^ 5 ) _ \\infty } \\ , \\cdot \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} { \\rm d i m } \\ , i ( S ) = { \\rm d i m } \\ , S \\le 2 . \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = \\langle \\widehat { A } u , v \\rangle _ { \\alpha } , \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} \\pi ( \\xi _ 1 ) \\pi ( \\xi _ 2 ) = c ( \\xi _ 1 , \\xi _ 2 ) \\overline { c ( \\xi _ 2 , \\xi _ 1 ) } \\pi ( \\xi _ 2 ) \\pi ( \\xi _ 1 ) = c _ s ( \\xi _ 1 , \\xi _ 2 ) \\pi ( \\xi _ 2 ) \\pi ( \\xi _ 1 ) , \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} P ( \\tilde { r } ; \\mu ) = P _ L \\left ( P _ R ( \\tilde { r } ; \\mu ) ; \\mu \\right ) , \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{gather*} \\forall x , y , z \\ , ( J ( x , y , z ) - > ( x \\le z ) \\wedge ( y \\le z ) \\wedge ( x \\wedge y = \\bot ) ) , \\\\ \\forall x , y , z , w \\ , ( J ( x , y , z ) \\wedge ( x \\le w ) \\wedge ( y \\le w ) - > ( z \\le w ) ) , \\\\ \\forall x , y \\ , ( ( x \\wedge y = \\bot ) - > \\exists ! _ \\le ^ { x , y } z \\ , J ( x , y , z ) ) \\end{gather*}"}
-{"id": "2519.png", "formula": "\\begin{align*} A \\zeta \\langle B \\zeta , \\zeta \\rangle - B \\zeta \\langle A \\zeta , \\zeta \\rangle = 0 . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} \\int _ { J } x ( s ) K ( t , s ) d s + x ( t ) = H ( t ) , t \\in J \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } ^ k ( ( b , j ) ) = ( b + k \\delta _ { ( a , i ) } , j + k ( r \\delta _ { ( a , i ) } + 1 ) ) , \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} S ^ { ( 3 ) } _ { 4 , 3 } & = 6 \\cdot S ^ { ( 3 ) } _ { 4 , 3 , } = 1 2 i \\lambda _ 3 a _ 1 . \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} ( [ \\delta _ { \\varepsilon _ 1 } , \\delta _ { \\varepsilon _ 2 } ] - \\delta _ { [ \\varepsilon _ 1 , \\varepsilon _ 2 ] _ Q } ) A = 0 \\Longrightarrow R _ { { } ^ Q \\nabla ^ \\pm } = 0 . \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} e ^ { u } \\frac { u ^ { m } } { m ! } & = \\sum _ { n \\geq 0 } \\binom { n } { m } \\frac { u ^ { n } } { n ! } . \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} e ^ { - C _ \\varphi / h } \\| \\langle x \\rangle ^ { - s } \\mathbf { 1 } _ { \\le M } v \\| ^ 2 _ { L ^ 2 } + \\| \\langle x \\rangle ^ { - s } \\mathbf { 1 } _ { \\ge M } v \\| ^ 2 _ { L ^ 2 } & \\le \\frac { C } { h ^ 2 } \\| \\langle x \\rangle ^ { s } ( P ( h ) - E \\pm i \\varepsilon ) v \\| ^ 2 _ { L ^ 2 } + \\frac { C \\varepsilon } { h } \\| v \\| ^ 2 _ { L ^ 2 } , \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } & \\partial _ t u + u \\cdot \\nabla u = - \\nabla p + \\theta e _ 2 , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ x \\in { \\mathbb { R } ^ 2 } , ~ t > 0 \\\\ & \\partial _ t \\theta + u \\cdot \\nabla \\theta - \\kappa \\partial _ 1 ^ 2 \\theta = 0 , \\\\ & \\nabla \\cdot u = 0 , \\\\ & u ( 0 , x ) = u _ 0 ( x ) , \\theta ( 0 , x ) = \\theta _ 0 ( x ) , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} = \\sum _ { j = 0 } ^ { k } \\Biggl [ \\binom { 2 k } { j } - \\binom { 2 k } { j - 1 } \\Biggr ] \\frac { c _ 1 ^ { j } \\ , c _ 2 ^ { 2 k - j } } { ( c _ 1 + c _ 2 ) ^ { 2 k } } \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} \\Delta t \\sum _ { n = 1 } ^ N \\| e _ h ^ { u ^ n } \\| _ { \\mathcal { T } _ h } ^ 2 \\leq C \\big ( ( \\Delta t ) ^ 2 + h ^ { 2 ( k + 2 ) } \\big ) . \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} p ^ { 2 } \\mid a ^ { \\prime } b ^ { \\prime } c ^ { \\prime } \\mid d ( K ) \\mid d ( L ) = \\mathfrak { c } ^ { 3 } ( m - 2 ) ^ { 3 } a b c / 6 4 . \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} 0 = h ^ * K _ X \\cdot C = ( K _ W + D _ W + c C ) \\cdot C = D _ W \\cdot C = ( a C + b F _ W ) \\cdot C = - 2 + b . \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} \\alpha ^ - _ { \\Omega , p } ( t ) : = \\min \\big \\{ \\pi , \\ , \\sup \\{ \\alpha > 0 : p + i t e ^ { i \\theta } \\in \\Omega \\mathrm { \\ f o r \\ a l l \\ } \\theta \\in [ 0 , \\alpha ] \\} \\big \\} \\in ( 0 , \\pi ] . \\hphantom { { } ^ - } \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} \\sup _ { u \\in Q } I ( u ) = \\max _ { u \\in Q } I ( u ) < + \\infty . \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} E _ { \\alpha } ^ { ( n ) } = \\{ \\eta \\in ( 0 , 1 ) : \\dim \\mu _ { \\eta } < \\alpha P ( \\eta ) = 0 0 \\ne P \\in \\mathcal { P } _ { L _ 0 } ^ { ( n ) } \\} . \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) \\le \\beta \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k \\} = \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} \\frac { d u } { d t } ( t _ 0 ) = \\displaystyle \\sum _ { i = 0 } ^ { k } \\alpha _ { k , i } \\left ( u ( t _ i ) + u ( t _ { i + 1 } ) + u ( t _ { i + 2 } ) + u ( t _ { i + 3 } ) \\right ) + \\epsilon \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} - ( \\nabla + i { \\boldsymbol k } ) \\cdot ( \\nabla + i { \\boldsymbol k } ) u ( x ) = \\left ( \\frac { \\omega } { c } \\right ) ^ 2 \\epsilon ( x , \\omega ) u ( x ) ~ ~ { \\rm i n } ~ D _ 0 \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} G ^ 2 ( t ) & = g _ i ^ 2 + o ( 1 ) , \\\\ G _ t G & = \\frac 1 2 g _ i \\bar { g } _ { \\bar { t } } ( R _ i ) - \\frac 1 2 g _ i \\bar { g } _ { \\bar { t } } ( R _ i ) \\frac { t } { \\varepsilon _ i } + o ( 1 ) , \\\\ G _ t ^ 2 + G _ { t t } G & = - \\frac 1 2 g _ i \\bar { g } _ { \\bar { t } } ( R _ i ) \\cdot \\frac { 1 } { \\varepsilon _ i } + O ( 1 ) , \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} \\frac { c ^ 2 + 1 } { 2 } = \\boldsymbol { \\pi } C { \\mathbf 1 } \\boldsymbol { \\pi } C ^ { - 1 } { \\mathbf 1 } = \\big ( \\sum _ { i = 1 } ^ p \\pi _ i c _ i \\big ) \\big ( \\sum _ { i = 1 } ^ p \\pi _ i \\frac { 1 } { c _ i } \\big ) . \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} \\theta _ { 2 , i } \\delta ( \\varepsilon ) + C _ i ( \\varepsilon ) = \\theta _ { 2 , i } \\delta ( \\varepsilon ) + ( 1 - \\theta _ { 2 , i } ) \\frac { C _ i ( \\varepsilon ) } { 1 - \\theta _ { 2 , 1 } } \\le \\theta _ { 2 , i } \\delta ( \\varepsilon ) + ( 1 - \\theta _ { 2 , i } ) \\delta ( \\varepsilon ) = \\delta ( \\varepsilon ) . \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} \\psi ( x , y ) = \\varphi ( x , y ) , \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} { \\equiv } = \\bigcap _ { U \\in \\mathcal { U } } U . \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} U ( \\omega , \\lambda ; \\omega _ 0 ) : = \\{ x \\in X | \\kappa _ { \\omega } < 0 \\ , \\ , a n d \\ , \\ , ( - \\kappa _ { \\omega } \\omega ) ^ n \\le \\lambda \\omega _ 0 ^ n \\ , \\ , a t \\ , \\ , x \\} \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} d z = \\frac { d w } { ( - H ) \\sqrt { - U ( w ) } } . \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} \\sum _ { k \\in \\Z } ( - 1 ) ^ { k } \\int _ { \\R } \\theta ( x ) \\widetilde { \\phi } ( x - k ) \\ , d x = 0 \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 u } { \\partial t ^ 2 } - c ^ 2 \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } = \\frac { 2 m } { t } \\frac { \\partial u } { \\partial t } \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} C B . g ( { h } ) = & \\frac { 1 } { 2 } ( x - y ) \\big ( ( \\mu - h ^ { 2 } + 2 h ) g ( { h } - 2 ) - ( - \\mu + h ^ { 2 } + 2 h ) g ( { h } + 2 ) \\big ) \\\\ & \\equiv \\frac { 1 } { 2 } \\big ( ( \\mu - h ^ { 2 } + 6 h - 8 ) g ( h - 4 ) - ( - \\mu + h ^ { 2 } - 2 h ) + g ( h ) x \\\\ & - ( - \\mu - h ^ { 2 } - 2 h ) g ( h ) y - ( - \\mu + h ^ { 2 } + 6 h + 8 ) g ( h + 4 ) y \\big ) . \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} P ( \\ell ) : = \\ ; & \\mbox { a l l t h e p r i m e f a c t o r s o f $ \\ell $ } , \\\\ P _ { D } ( \\ell ) : = \\ ; & \\mbox { a l l t h e p r i m e f a c t o r s o f $ \\ell $ i n $ \\mathbb { P } ( D ) $ } , \\\\ P _ { m } ( \\ell _ { 1 } , \\ell _ { 2 } ) : = \\ ; & P _ { - 4 \\ell _ { 2 } } ( \\ell _ { 1 } ) \\backslash P ( m - 2 ) , \\\\ G _ { m } ( \\ell _ { 1 } , \\ell _ { 2 } ) : = \\ ; & P ( \\gcd ( \\ell _ { 1 } , \\ell _ { 2 } ) ) \\backslash P ( 2 ( m - 2 ) ) . \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} \\begin{aligned} r i c _ g & = \\kappa ( n - 1 ) g _ \\kappa + \\\\ & + h ^ { - 2 } \\Big \\{ ( n - 2 ) h ( \\nabla ^ 2 h ) _ { g _ \\kappa } + \\Big [ h ( \\Delta h ) _ { g _ \\kappa } - ( n - 1 ) \\| ( \\nabla h ) _ { g _ \\kappa } \\| ^ 2 \\Big ] g _ \\kappa \\Big \\} , \\end{aligned} \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} \\mathcal { B } ( \\mathbb { R } ^ 2 ) = M _ m ^ 2 ( \\mathbb { R } ^ 2 ) . \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} \\hat { A } ( z ) = h _ 2 ( z ) + t ^ 2 J ^ { h _ 1 } ( z ) e _ 2 ( z ) . \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} & G ( x , y ) \\asymp \\\\ & \\frac { 1 } { | x - y | ^ { d - \\alpha } } \\left ( \\frac { x _ d \\wedge y _ d } { | x - y | } \\wedge 1 \\right ) ^ p \\left ( \\frac { x _ d \\vee y _ d } { | x - y | } \\wedge 1 \\right ) ^ { p - { a _ p } _ + } \\log \\left ( 2 + { \\bf 1 } _ { a _ p \\le 0 } \\frac { | x - y | } { ( x _ d \\vee y _ d ) \\wedge | x - y | } \\right ) ^ { \\beta _ 4 + { \\bf 1 } _ { a _ p = 0 } } . \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{gather*} A ( B C ) = ( A B ) C , \\\\ \\alpha ( A B ) = ( \\alpha A ) B = A ( \\alpha B ) , \\\\ ( A + B ) C = A C + B C , \\ ; A ( B + C ) = A B + A C . \\end{gather*}"}
-{"id": "3496.png", "formula": "\\begin{align*} Z _ { \\beta } : = \\sum _ { \\mathsf { m = 1 } } ^ { \\mathsf { + \\infty } } \\exp \\left [ - \\beta \\lambda _ { \\mathsf { m } } \\right ] < + \\infty \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\partial _ { t } u _ { m } = i \\left [ \\Delta u _ { m } + \\sum \\limits _ { j = 1 } ^ { N } a _ { m j } u _ { j } + \\sum \\limits _ { j = 1 } ^ { N } b _ { m j } u _ { j } \\right ] , x \\in R ^ { n } , t \\in \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} b _ { 1 } = \\int _ { \\mathbb { R } } \\frac { 1 + t ^ { 2 } } { ( 1 - t a _ { 0 } ) ^ { 2 } } \\ , d \\sigma ( t ) . \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} ( \\eta _ 1 ^ \\sigma , \\cdots , \\eta _ N ^ \\sigma ) ^ T = M ^ { - 1 } ( \\sigma , \\cdots , \\sigma ) ^ T . \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} \\partial _ { t } v _ { 4 } ( t , r ) & = - \\int _ { t } ^ { \\infty } d s \\frac { v _ { 4 , s } ( t , r ) } { ( s - t ) } - \\frac { 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { B _ { s - t } ( 0 ) } \\frac { d A ( y ) } { \\sqrt { ( s - t ) ^ { 2 } - | y | ^ { 2 } } } _ { v _ { 4 , 1 } } \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} \\dot { q _ 1 } - e ^ { - ( q _ 1 - q _ 2 ) } \\dot { q _ 2 } = 0 , \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} u * v = u \\cdot T ( v ) + T ( u ) \\cdot v + H ( T u , T v ) , ~ ~ u , v \\in M . \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} \\mathrm { S } _ { \\mathrm { S c h u r } } & = 2 k \\nabla ^ 2 f ( x ) - 2 A ^ T A - q _ 1 I \\\\ & - ( A \\nabla ^ 2 f ( x ) ) ^ T ( A A ^ T ) ^ { - 1 } A \\nabla ^ 2 f ( x ) . \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} G _ { \\mu } ( x ) = \\lim _ { y \\downarrow 0 } G _ { \\mu } ( x + i y ) , x \\in \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} ( \\partial _ { 2 } \\phi ( r , \\xi ) ) _ { 1 } = \\left ( \\frac { a ( \\xi ) i r } { \\xi ^ { 3 / 4 } } e ^ { i r \\sqrt { \\xi } } \\sigma ( r \\sqrt { \\xi } , r ) \\right ) \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} [ \\omega ] _ { \\sim _ n } = \\{ \\omega ' \\in \\Omega : \\omega \\sim _ n \\omega ' \\} . \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} T _ 1 ( u ) - T _ 1 ' ( u ) = d _ T ( a ) ( u ) , ~ u \\in M . \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} \\ss _ 1 = ( \\mu , ~ \\sigma _ + ^ 2 - 1 , \\beta _ 0 ^ 2 + ( \\eta ^ 2 - 1 ) ) ^ \\tau ; ~ ~ \\ss _ 2 = ( 0 , 0 , \\beta _ 0 ^ 4 ) ^ \\tau . \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} ( ( - P ) ^ { s } + q ) u = 0 \\quad \\mathbb { R } ^ { n } , P = \\sum _ { j , k = 1 } ^ { n } \\partial _ { j } a _ { j k } ( x ) \\partial _ { k } \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} \\lambda = \\frac { m } { n } \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } & \\partial _ t X + u \\cdot \\nabla X = \\partial _ X u , \\\\ & X ( 0 , x ) = X _ 0 ( x ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} \\boldsymbol { y } _ l = ( \\beta \\boldsymbol { G } \\boldsymbol { \\Theta } \\boldsymbol { S } \\boldsymbol { h } _ r + \\boldsymbol { h } _ d ) x _ l + \\boldsymbol { w } _ l , \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} J _ n ( x , y ) = \\frac { \\psi _ { \\gamma _ n } ( x ) \\psi _ { \\gamma _ n } ( y ) } { | x - y | ^ { ( d + \\alpha ) } } , J ( x , y ) = \\frac { \\psi _ { 0 } ( x ) \\psi _ { 0 } ( y ) } { | x - y | ^ { ( d + \\alpha ) } } . \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} \\psi ( x , y ) & = \\varphi ^ { - 1 } ( x + \\varphi ( y ) ) < \\varphi ^ { - 1 } ( x + \\varphi ( c ) ) < \\varphi ^ { - 1 } ( \\varphi ( c ) ) = c \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} \\begin{aligned} & F ^ 1 _ 1 ( \\phi ^ 1 _ 1 ) = \\frac { A ( A - 3 ) + B ^ 2 } { \\cos ^ 2 3 \\phi ^ 1 _ 1 } - \\frac { B ( 2 A - 3 ) \\sin 3 \\phi ^ 1 _ 1 } { \\cos ^ 2 3 \\phi ^ 1 _ 1 } + 9 \\bigg [ \\frac { 2 ( 2 A - 3 ) } { 2 A - 3 - 2 B \\sin 3 \\phi ^ 1 _ 1 } - \\frac { 2 [ ( 2 A - 3 ) ^ 2 - 4 B ^ 2 ] } { ( 2 A - 3 - 2 B \\sin 3 \\phi ^ 1 _ 1 ) ^ 2 } \\bigg ] \\\\ & G ^ 1 _ 2 ( \\phi ^ 1 _ 2 ) = G ^ 1 _ 3 ( \\phi ^ 1 _ 3 ) = \\cdots = G ^ 1 _ { d _ i - 1 } ( \\phi ^ 1 _ { d _ i - 1 } ) = 0 \\end{aligned} \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} \\gamma _ k ( 0 ) = \\gamma _ k ( L _ k ) = ( 0 , 0 ) , \\partial _ s \\gamma _ k ( 0 ) = - \\partial _ s \\gamma _ k ( L _ k ) = ( 1 , 0 ) , \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} D _ \\ell ( ( n _ k ) _ { k \\in \\N } ) = \\{ \\partial ( n _ { j _ 1 } , . . . , n _ { j _ { 2 ^ \\ell } } ) \\ , | \\ , j _ 1 < \\cdots < j _ { 2 ^ \\ell } \\} . \\index { $ D _ \\ell ( ( n _ k ) _ { k \\in \\N } ) $ } \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} v ( t ) & \\le \\omega ( t , \\mu ) v _ 0 + \\int _ 0 ^ t \\omega ( t - s , \\mu ) [ a \\sup _ { \\zeta \\in [ s - \\rho ( s ) , s ] } v ( \\zeta ) + b ( s ) ] d s , \\ ; t > 0 , \\\\ v ( s ) & = \\psi ( s ) , s \\in [ - \\tau , 0 ] , \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} \\big ( \\phi _ t = \\mathrm { i d } _ A + t ( \\mathrm { a d } ^ l _ a - \\mathrm { a d } ^ r _ a ) + \\sum _ { i \\geq 2 } t ^ i \\phi _ i , ~ \\psi _ t = \\mathrm { i d } _ A + t ( l _ a - r _ a + H ( a , T - ) - H ( T - , a ) ) + \\sum _ { i \\geq 2 } t ^ i \\psi _ i \\big ) \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} B ^ 1 = \\bar J \\oplus \\bar \\pi ^ * ( \\bar I ^ 1 ) . \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} Q ( A , B ) \\ = \\ \\sum _ { i \\in [ k ] } \\ Q ( A _ i , B _ i ) . \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} \\rho _ { \\phi } ( N ) = \\rho _ { \\phi \\circ \\pi _ * } ( N ' ) = \\rho _ { \\phi \\circ \\pi _ * } ( M ' ) = \\rho _ { \\phi } ( M ) . \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} \\tilde { \\chi } _ A ( n _ 1 , n _ 2 ) & = - \\frac { 1 } { 7 2 0 } \\frac { 1 } { 6 } ( n _ 1 + 2 ) ( n _ 2 + 1 ) ( ( n _ 1 + 2 ) ^ 2 - ( n _ 2 + 1 ) ^ 2 ) \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} \\mathcal { C } _ { J , K } ^ { i n v } : = \\left \\{ \\eta \\in \\mathcal { C } _ { J , K } : \\ : T ^ t \\eta \\in \\mathcal { C } ^ { r e v } _ { J , K } , \\ : \\forall t \\in \\mathbb { Z } \\right \\} , \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} \\mathcal { D } _ { J , K } ^ { - 1 } : = \\theta ^ { - 1 } \\circ \\mathcal { D } _ { K , J } \\circ \\theta ^ { - 1 } , \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} & t ^ { ( i _ r ) } _ r \\otimes x ^ { ( i _ s ) } - x ^ { ( i _ r ) } \\otimes t ^ { ( i _ s ) } _ s = \\sum _ { j = 2 } ^ { n _ r } \\beta _ j b _ j ^ { ( i _ r ) } \\otimes t ^ { ( i _ s ) } _ s - \\sum _ { k = 2 } ^ { n _ s } \\eta _ k t ^ { ( i _ r ) } _ r \\otimes c ^ { ( i _ s ) } _ k . \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} \\sum _ { I } q _ I h ^ I = 0 . \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} R ( r ) = U _ { 1 , r } ( s , \\gamma ) ( \\partial _ s U _ { 1 , 0 } ( s , \\gamma _ r ) - \\Delta _ L U _ { 1 , 0 } ( s , \\gamma _ r ) ) . \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} \\phi ( x y ) = x y H = ( x H ) ( y H ) = \\phi ( x ) \\phi ( y ) . \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} R ^ { m + r } ( x ) T ( x ) = 1 + x ^ { \\operatorname { d e g } R + \\operatorname { d e g } S + 1 } P ( x ) = 1 + x ^ { n + k - l } P ( x ) . \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} B _ { n , \\mathbf { m } } ^ { ( d ) } ( \\vert { \\boldsymbol { \\omega } } \\vert \\mathbf { 1 } - \\mathbf { z } \\mid { \\boldsymbol { \\omega } } ) = & ( - 1 ) ^ { | \\mathbf { m } | } B _ { n , \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } \\mid { \\boldsymbol { \\omega } } ) . \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d \\hat { x } _ { t } & = ( F _ { t } \\hat { x } _ { t } + f _ { t } + \\theta _ { t } ^ { \\ast } ) d t + P _ { t } G _ { t } R _ { t } ^ { - 1 } d \\hat { I } _ { t } , \\\\ \\hat { x } _ { t } ( 0 ) & = x _ { 0 } \\end{array} \\right . \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} o _ n ( 1 ) = I ' ( u _ n ) \\big ( u _ n ^ + - u _ n ^ - \\big ) + \\int _ { \\mathbb { R } ^ N } f _ 0 ( u _ n ( x ) ) \\big ( u _ n ^ + ( x ) - u _ n ^ - ( x ) \\big ) \\ ; d x = | | u _ n | | ^ 2 , \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} ( \\sigma ^ 0 , \\sigma ^ 1 , \\ldots , \\sigma ^ 4 ) : = \\ ( \\frac { \\omega ^ 0 } { H _ 1 } , \\frac { \\omega ^ 1 } { H _ 4 } , H _ 4 \\omega ^ 2 , \\frac { \\omega ^ 3 } { H _ 1 H _ 4 } , H _ 4 ( \\omega ^ 4 + \\gamma ) \\ ) . \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} t _ 3 = \\begin{cases} \\min \\{ n - r _ 1 + k _ 4 , \\tau _ 2 \\} & \\mbox { i f $ \\tilde h _ 2 ( x ) $ i s a u n i t } , \\\\ n - r _ 1 + k _ 4 & \\mbox { i f $ \\tilde h _ 2 ( x ) = 0 $ } , \\end{cases} \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} L _ { r , \\ell } : = \\sum _ { \\nu _ r = \\mu _ r } C _ { \\mu _ 0 , \\mu _ 1 } ^ { \\lambda _ 0 } C _ { \\mu _ 1 , \\mu _ 2 } ^ { \\lambda _ 1 } \\cdots C _ { \\mu _ { r - 1 } , \\nu _ r } ^ { \\lambda _ { r - 1 } } * g _ { \\ell } ^ { \\nu _ r } . \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} w ( t , x ) = & \\psi ( \\phi ( T ; t , x ) ) \\exp \\Bigl [ - \\int _ { t } ^ { T } \\frac { \\lambda ( s , \\phi ( s ; t , x ) ) } { s } d s \\Bigr ] \\\\ & - \\int _ t ^ T \\exp \\Bigl [ - \\int _ t ^ { \\tau } \\frac { \\lambda ( s , \\phi ( s ; t , x ) ) } { s } d s \\Bigr ] \\frac { g ( \\tau , \\phi ( \\tau ; t , x ) ) } { \\tau } d \\tau . \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} \\xi : = ( \\xi _ 1 , \\cdots , \\xi _ n ) ^ t , \\ \\theta : = ( \\theta _ 1 , \\cdots , \\theta _ n ) ^ t , \\ \\xi ' : = ( \\xi ' _ 1 , \\cdots , \\xi ' _ n ) ^ t , \\ \\theta ' : = ( \\theta ' _ 1 , \\cdots , \\theta ' _ n ) ^ t \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} \\mu ( d x ) : = \\frac { 1 } { Z } \\exp \\left ( - H ( x ) \\right ) d x . \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { 0 , p } = \\left \\Vert u \\right \\Vert _ { L ^ p ( \\mathbb { R } ^ d ) } \\beta \\leq \\gamma , \\left \\Vert u \\right \\Vert _ { \\beta , p } \\leq \\left \\Vert u \\right \\Vert _ { \\gamma , p } . \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\phi _ { T _ { \\rm f o c u s } ( r ) } ( 0 , r ) = \\left ( 0 , P _ { \\rm f o c u s } ( r ) \\right ) . \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} ( \\partial _ { t } ^ + \\phi _ h ^ n , \\phi _ h ^ n ) _ { \\mathcal { T } _ h } + \\epsilon \\| \\partial _ { t } ^ + u _ h ^ n \\| ^ 2 _ { \\mathcal { T } _ h } = \\epsilon ^ { - 1 } ( \\partial _ { t } ^ + f ^ n ( u _ h ^ n ) , \\phi _ h ^ n ) _ { \\mathcal { T } _ h } . \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} V _ { i j } ^ { \\perp } & = \\nabla _ { F _ j } ^ { \\perp } \\nabla _ { F _ i } V = \\nabla _ { F _ j } ^ { \\perp } \\nabla _ { F _ i } ^ { \\perp } V - \\nabla _ { F _ j } ^ { \\perp } \\left ( A ^ V _ { i k } F _ k \\right ) = \\nabla _ { F _ j } ^ { \\perp } \\nabla _ { F _ i } ^ { \\perp } V - A ^ V _ { i k } \\ , A _ { j k } \\ , . \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} 4 c _ { b } \\lambda ( t ) \\int _ { 0 } ^ { \\infty } d \\xi \\frac { \\xi \\cos ( t \\xi ) } { t ^ { 2 } } \\partial _ { 2 } \\psi _ { v _ { 2 } } \\lambda ' ( t ) & = - 4 c _ { b } \\lambda ( t ) \\int _ { 0 } ^ { \\infty } d \\xi \\frac { \\cos ( t \\xi ) } { t ^ { 4 } } \\partial _ { \\xi } ^ { 2 } \\left ( \\xi \\partial _ { 2 } \\psi _ { v _ { 2 } } \\right ) \\lambda ' ( t ) \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\delta } ( f ) : = \\sup _ { x \\neq y \\in \\R ^ d } \\frac { | f ( x ) - f ( y ) | } { | x - y | ^ \\delta } . \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} b _ i + \\sum _ { r = i + 1 } ^ { k - 1 } ( - 1 ) ^ { r - i } \\binom { \\left \\lceil \\frac { \\alpha m } k \\right \\rceil - k + r - i - 1 } { r - i } b _ r \\ge 0 . \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\max \\{ C _ { 1 , 1 } ( \\varepsilon ) , C _ { 1 , 2 } ( \\varepsilon ) , C _ { 1 , 3 } ( \\varepsilon ) \\} = 0 . \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} a ( n ) = 2 ^ { n - 1 } + 2 a ( n - 1 ) - 2 ^ { n - 2 } = 2 a ( n - 1 ) + 2 ^ { n - 2 } . \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} \\norm { D u } = \\int _ \\Omega \\abs { \\nabla u ( x ) } d x , \\ ; u \\in W _ 1 ^ 1 ( \\Omega ) . \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} \\gamma ^ { \\ast } = \\frac { \\widehat { w } _ { n } ( n - w _ { n } ) + ( n + 1 ) ( w _ { n } - \\widehat { w } _ { n } ) } { n \\widehat { w } _ { n } ( 1 + w _ { n } ) } . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} H _ 1 : = \\begin{bmatrix} A _ 1 & - B _ 1 B _ 1 ^ { \\top } \\\\ - Q _ 1 & - A _ 1 ^ { \\top } \\end{bmatrix} . \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k b _ j D ^ j : b _ j \\in \\{ a _ 1 , \\ldots , a _ d \\} \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} u ( \\cdot , t ) = S ( t ) \\xi ( \\cdot , 0 ) + \\int _ 0 ^ t S ( t - s ) f ( s , u _ \\rho ( \\cdot , s ) ) d s , \\ ; t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} \\mathcal { N } _ 2 ( \\xi , \\mathcal { A } , v ) = e ^ { - \\xi \\mathcal { A } } G \\left ( e ^ { \\xi \\mathcal { A } } \\overline v \\right ) \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} \\mathcal { F } ( M ) : = \\int _ M \\mathcal { E } ( H , K ) \\ , d S , \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { \\beta } : = \\int _ { \\R ^ d } \\left ( 1 + | \\xi | ^ { 2 } \\right ) ^ { \\beta } \\mathcal { F } u ( \\xi ) ~ \\overline { \\mathcal { F } v } ( \\xi ) ~ d \\xi \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} a ( t , x ) & = \\langle f ( t , x , v ) , \\chi _ 0 \\rangle \\\\ b ^ j ( t , x ) & = \\langle f ( t , x , v ) , \\chi _ j \\rangle , \\ j = 1 , 2 , 3 , \\\\ c ( t , x ) & = \\langle f ( t , x , v ) , \\chi _ 4 \\rangle , \\\\ d ( t , x , v ) & = ( I - P ) f ( t , x , v ) , \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} d J ( \\chi ; A ) = \\alpha ( \\chi ; A ) _ { \\ , | \\S } , d Q _ \\Sigma ( \\chi ; A ) = \\int _ \\Sigma \\alpha ( \\chi ; A ) _ { \\ , | \\S } . \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} A _ { k } = \\bigcup _ { m = ( x , t ) \\in M _ { k } } \\bar { E } _ { k } ( m ) \\cap D _ { k } ( m ) , \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} \\mathcal { R } _ { \\epsilon } = \\sup \\Big \\{ | | X _ { t } - t \\vec { v } | | : t \\in [ 0 , \\bar { \\mathcal { R } } _ { \\epsilon } ] \\Big \\} , \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{cases} F _ L ( x , y ; \\mu ) , & x < 0 , \\\\ F _ R ( x , y ; \\mu ) , & x > 0 , \\end{cases} \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} \\Phi ( z ) = \\gamma + z + N _ { \\sigma } ( z ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} \\vec { \\mathbb { I } } ^ 2 & = \\langle \\langle \\mathbb { I } ^ 2 _ { \\gamma } \\colon \\gamma \\leq \\delta \\rangle , \\langle \\dot { \\mathbb { J } } ^ 2 _ { \\gamma } \\colon \\gamma < \\delta \\rangle \\rangle \\\\ \\dot { \\vec { \\mathbb { I } } } ^ 1 & = \\langle \\langle \\dot { \\mathbb { I } } ^ 1 _ { \\gamma } \\colon \\gamma \\leq \\delta \\rangle , \\langle \\dot { \\mathbb { J } } ^ 1 _ { \\gamma } \\colon \\gamma < \\delta \\rangle \\rangle \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\hat { U } _ { j , h } ( t ) = \\frac { \\hat { f } _ { j , h } } { ( \\delta + t ( \\lambda _ { j , h } - \\delta ) ) ^ { \\alpha } } \\mbox { w h i c h i m p l i e s } \\ \\ \\hat { U } _ { j , h } ( 1 ) = \\lambda _ { j , h } ^ { - \\alpha } \\hat { f } _ { j , h } . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in \\mathfrak { S } _ n } W _ 1 ( \\sigma ) = \\begin{cases} E _ n ; & \\\\ 0 . & \\end{cases} \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} V _ g f ( x , \\omega ) = \\langle f , \\pi ( x , \\omega ) \\overline { g } \\rangle . \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} t \\binom { t } { k - 1 } \\ge \\frac { ( \\frac m 2 - t ) b _ { i + 1 } } { ( b _ i + b _ { i + 1 } ) \\binom { b _ { k - 1 } + b _ k } { b _ k } } . \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} p _ { k + 1 } = t _ { k } p _ k - p _ { k - 1 } \\\\ q _ { k + 1 } = t _ { k } q _ k - q _ { k - 1 } . \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\left ( \\int _ { \\epsilon } ^ { t / 2 } d y \\frac { e ^ { - y } } { y \\log ^ { a } ( \\frac { t } { i y } ) } \\right ) = \\int _ { 0 } ^ { t / 2 } d y \\frac { e ^ { - y } \\sin ( a \\tan ^ { - 1 } ( \\frac { \\pi } { 2 ( \\log ( t ) - \\log ( y ) ) } ) ) } { y ( ( \\log ( t ) - \\log ( y ) ) ^ { 2 } + \\frac { \\pi ^ { 2 } } { 4 } ) ^ { a / 2 } } \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} r _ n [ D , T ] = \\{ r _ n ( S ) : S \\in [ D , T ] \\} . \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} X _ { J , K } ( t ) \\left \\{ \\begin{array} { l } \\leq \\inf \\left \\{ n \\geq 1 : \\ : \\sum _ { m = 1 } ^ { n } ( T ^ t \\eta ) _ m \\geq t \\left ( m _ { K , J } + \\varepsilon \\right ) \\right \\} , \\\\ \\geq \\inf \\left \\{ n \\geq 1 : \\ : \\sum _ { m = 1 } ^ { n } ( T ^ t \\eta ) _ m \\geq t \\left ( m _ { K , J } - \\varepsilon \\right ) \\right \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} { \\cal J } \\left [ f _ X ( x ) \\right ] = { \\int _ 0 ^ \\infty { f _ X ( x ) } \\ln \\left ( { f _ X ( x ) } \\right ) { \\rm { d } } x + \\frac { 1 } { 2 } \\int _ 0 ^ \\infty { \\ln \\left ( { 1 + { \\varsigma ^ 2 } x } \\right ) f _ X ( x ) } { \\rm { d } } x } . \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} \\Pi ^ q _ { M _ 1 } \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix} x \\\\ y \\\\ 0 \\end{pmatrix} , \\Pi ^ q _ { M _ 2 } \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix} x \\\\ 0 \\\\ z \\end{pmatrix} \\qquad \\Pi ^ q _ { M _ 1 \\cap M _ 2 } \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix} x \\\\ 0 \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ q _ G ( \\Sigma _ T ) } & \\leq C \\Big [ \\big ( \\| \\frac { | u _ 1 - u _ 2 | } { G } \\| _ { C ^ 0 } + \\| \\frac { \\rho | \\nabla ( u _ 1 - u _ 2 ) | } { G } \\| _ { C ^ 0 } \\big ) \\cdot ( 1 + \\| u _ 2 \\| _ { W _ G ^ { 2 , q } } \\cdot \\frac { T } { \\theta } ) + \\| u _ 1 - u _ 2 \\| _ { W _ G ^ { 2 , q } } \\Big ] \\\\ & \\leq C _ 1 ' \\| u _ 1 - u _ 2 \\| _ { X _ T } \\cdot T ^ { - 1 } \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} s _ { \\mathsf { m , n } } : = - \\frac { r _ { \\mathsf { m , n } } } { b _ { \\mathsf { m } } } \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} \\begin{bmatrix} x ^ * \\\\ y ^ * \\end{bmatrix} = \\frac { \\kappa } { \\omega } \\begin{bmatrix} a _ 2 \\\\ - a _ 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } ( \\alpha _ \\varepsilon , x _ \\varepsilon , u _ \\varepsilon ( x _ \\varepsilon ) ) = ( \\alpha , \\hat { x } , u ( \\hat { x } ) ) . \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} D ( H _ 1 ) & = \\bigl \\{ x \\in X \\ : ; \\ : ( ( \\mathcal { H } _ e x _ e ) ' ) _ { e \\in E } \\in X \\mathcal { H } _ e x _ e | _ { \\partial ( a ( e ) , b ( e ) ) } = 0 e \\in E \\bigr \\} \\\\ D ( H _ 2 ) & = \\bigl \\{ x \\in X \\ : ; \\ : ( P _ 0 \\mathcal { H } _ e x _ e ) _ { e \\in E } \\in X \\bigr \\} . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} V = \\left ( \\begin{array} { c c } e ^ { - D ^ - D ^ + } & e ^ { - \\frac { 1 } { 2 } D ^ - D ^ + } \\left ( \\frac { I - e ^ { - D ^ - D ^ + } } { D ^ - D ^ + } \\right ) D ^ - \\\\ e ^ { - \\frac { 1 } { 2 } D ^ + D ^ - } D ^ + & I - e ^ { - D ^ + D ^ - } \\end{array} \\right ) \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} [ V ^ b ( D ) ] - [ e _ 1 ] \\in K _ 0 ( \\mathcal { A } ^ { \\infty , \\delta } _ G ( M ) ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; e _ 1 : = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} F ( x ) = { v _ 0 x ^ { k _ 0 } \\over { 1 + u _ 1 ( x ) x - \\displaystyle { \\strut v _ 1 x ^ { k _ 0 + k _ 1 + \\delta } \\over { 1 + u _ 2 ( x ) x - \\displaystyle { \\strut v _ 2 x ^ { k _ 1 + k _ 2 + \\delta } \\over { 1 + u _ 3 ( x ) x - \\displaystyle { \\ddots } } } } } } } \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} w _ n ^ + ( r , \\theta ) = U _ n ^ + ( r ) e ^ { i n _ + \\theta } , w _ n ^ - ( r , \\theta ) = U _ n ^ - ( r ) e ^ { i n _ - \\theta } , \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} J - \\frac { \\partial J } { \\partial h _ z } h _ z = : C = { \\rm c o n s t . } , \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} { } ^ Q \\widetilde { \\Omega } ^ + _ a = ( \\widetilde { K } _ + ) ^ { - 1 } \\tilde { \\rho } _ a ( \\widetilde { K } _ + ) = ( \\tilde { \\rho } ^ T E \\tilde { \\rho } ) ^ + \\tilde { \\rho } _ a ( \\tilde { \\rho } ^ T E \\tilde { \\rho } ) , \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} | | v _ { 6 } ( t , \\cdot \\lambda ( t ) ) | | _ { L ^ { 2 } ( R d R ) } = \\lambda ( t ) | | y _ { 0 } ( t ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} { \\rm T r } ^ \\chi _ s ( P ) = \\frac { 1 } { ( 2 \\pi ) ^ { \\dim M } } \\int _ { T ^ * M } \\chi ( x ) { \\rm t r } _ s ( \\sigma _ { t ^ { - 1 } } ( P ) ) ( x , \\xi ) d x d \\xi , \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} \\begin{aligned} k _ L & = 1 , & k _ R & = 3 , \\\\ b _ L & = b , & b _ R & = b , \\end{aligned} \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} 1 + \\cos \\phi = \\frac { 2 n ( 1 - m ^ 2 ) } { ( n + m ) ( 1 - m n ) } \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} \\lim _ { k } { | \\phi _ k ( y x ) - \\phi _ k ( x y ) | } = 0 , \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} g _ p & = ( 1 - \\rho ^ { p + 1 } ) ( 1 - \\rho ^ n ) ^ { n - 2 - p } \\left ( \\frac { \\rho } { 1 - \\rho ^ { n + 1 } } \\right ) ^ { n - 1 - p } \\\\ & = \\frac { \\left ( \\sum _ { m = 0 } ^ p \\rho ^ { m + 1 } \\right ) \\left ( \\sum _ { \\ell = 0 } ^ { n - 1 } \\rho ^ { \\ell + 1 } \\right ) ^ { n - 2 - p } } { \\left ( \\sum _ { k = 0 } ^ { n } \\rho ^ k \\right ) ^ { n - 1 - p } } , \\ ; \\ ; \\ ; ( 0 \\leq p \\leq n - 1 ) . \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { L } \\varepsilon ^ { n } \\binom { L - 1 } { n - 1 } > \\varepsilon ^ 4 \\binom { L - 1 } { 3 } , \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} v ' ( 0 ) = 0 , v ' ( j ^ + ) = \\frac 1 2 v ' ( j ^ - ) , j \\in \\{ 1 , 2 , 3 , \\ldots \\} . \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { 1 , \\infty } } = \\sup _ { t > 0 } t f ^ { * * } ( t ) = \\sup _ { t > 0 } \\int _ { 0 } ^ { t } f ^ { * } ( s ) d s = \\int _ { 0 } ^ { \\infty } f ^ { * } ( s ) d s = \\| f \\| _ { L ^ { 1 } } . \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} m & = \\frac { 1 - \\sigma ^ 2 } { 1 + \\varepsilon \\sigma + \\sigma ^ 2 } \\\\ n & = \\frac { \\sigma ( 2 + \\varepsilon \\sigma ) } { 1 + \\varepsilon \\sigma + \\sigma ^ 2 } \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} \\exp ( - A ) \\exp ( B ) \\exp ( A ) = \\exp \\big ( B - [ A , B ] + \\cdots \\big ) , \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} \\partial _ { t r } v _ { 4 } ( t , r ) & = \\frac { ( t \\partial _ { t } ^ { 2 } v _ { 4 } + r \\partial _ { t r } v _ { 4 } ) } { r } - \\frac { t } { r } \\partial _ { t } ^ { 2 } v _ { 4 } \\\\ & = \\frac { V ( \\partial _ { t } v _ { 4 } ) } { r } - \\frac { t } { r } \\partial _ { t } ^ { 2 } v _ { 4 } \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} h = \\sum _ { A \\in \\left [ B \\right ] ^ { \\times d } } w _ A \\prod _ { i = 1 } ^ d h _ { d , B , A _ i } . \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} \\partial _ t \\bar { f } + w ' \\cdot \\nabla _ { \\ ! y ' } \\bar { f } = \\nabla _ { w ' } \\cdot \\big ( \\mathbb { A } \\nabla _ { \\ ! w ' } \\bar { f } \\big ) + \\mathbb { B } \\cdot \\nabla _ { \\ ! w ' } \\bar { f } , \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} \\tau : \\Pi \\longrightarrow \\hat { \\Pi } : = \\Pi _ q \\big ( ( n _ 1 - u _ 1 ) \\times m _ 1 \\mid \\cdots \\mid ( n _ t - u _ t ) \\times m _ t \\big ) . \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} S _ { t } + \\left [ S , K \\right ] = \\gamma \\partial _ { t } ^ { 2 } \\varphi + 2 \\gamma ^ { 2 } a \\nabla \\varphi . \\nabla \\varphi _ { t } - 2 i b \\gamma \\left ( 2 \\nabla \\varphi _ { t } . \\nabla + \\Delta \\varphi _ { t } \\right ) - \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} \\sum _ { m = - \\infty } ^ \\infty ( - 1 ) ^ m a _ k ( n - m ^ 2 ) = a ( n ) - 2 \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { n } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - m ^ 2 ) . \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} f ( t , t _ 1 , t _ 2 , t _ 3 ) = \\frac { ( t _ 1 - t ) ( t _ 3 - t _ 2 ) } { ( t _ 2 - t _ 1 ) ( t - t _ 3 ) } \\left ( 1 + ( t _ 2 - t ) ( t _ 3 - t _ 1 ) \\left ( \\frac 1 6 ( z ' ( 0 ) ) ^ { - 1 } z ''' ( 0 ) - \\frac 1 4 ( ( z ( 0 ) ' ) ^ { - 1 } z ( 0 ) '' ) ^ 2 \\right ) \\right ) \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} A & : = \\sum _ { j = 1 } ^ J Q _ j , \\\\ \\Delta ^ Q _ j & : = \\frac { Q _ j } { A } , j = 1 , \\dots , J - 1 , \\\\ \\Delta ^ { E } _ j & : = \\frac { { E } _ j } { L - A } , j = 1 , \\dots , J - 1 . \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 ) = \\left ( \\frac { 4 } { \\pi ^ 2 a b } \\right ) ^ { \\frac { 1 } { 4 } } \\exp \\left [ - \\frac { 1 } { a } ( x _ 1 - x _ 1 ^ { ( 0 ) } ) ^ 2 - \\frac { 1 } { b } ( x _ 2 - x _ 2 ^ { ( 0 ) } ) ^ 2 \\right ] \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} w _ { N } ( X _ { i } ) : = \\min \\left \\{ w ( X _ { i } ) ; \\frac { 1 } { c _ { N } } \\sum _ { j = 1 } ^ { N } w ( X _ { j } ) \\right \\} , i = 1 , 2 , \\dots , N , \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} J ^ { \\perp } = \\{ a \\in \\mathfrak { g } | \\phi ( a , J ) = \\{ 0 \\} \\} \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} R ^ { ( \\gamma ) , 0 } _ 1 \\varphi = \\int _ 0 ^ \\infty \\mathrm { e } ^ { - t } { } _ h P _ t ^ { ( \\gamma ) } \\varphi d t = \\frac { u _ { \\lambda _ \\gamma + 1 } \\ast \\psi _ \\gamma } { \\psi _ \\gamma } = \\frac { u _ { \\lambda _ \\gamma } - u _ { \\lambda _ \\gamma + 1 } } { u _ { \\lambda _ \\gamma } } = 1 - w _ { 1 , \\lambda _ \\gamma } . \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} & \\frak R ( x _ 1 , x _ 2 , x _ 3 , v _ 1 , v _ 2 , v _ 3 ) = \\sum \\limits _ { j , k } \\frak R _ { j , k } ( x _ 1 , x _ 2 , x _ 3 , v _ 1 , v _ 2 , v _ 3 ) . \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} e = [ 2 , ( 1 , 2 n , 1 ) _ { n = 1 } ^ \\infty ] = [ 2 , 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , \\dots ] . \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} T ^ 4 _ { 1 3 } = T ^ 4 _ { 2 3 } = T ^ 4 _ { 3 4 } = \\bar T ^ 4 _ 3 = T ^ 4 _ 3 = R ^ 4 _ 3 = 0 ; \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} Z _ { t } = X _ { n + 1 } \\leadsto \\ldots \\leadsto X _ 1 = U . \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} & c _ 2 \\le \\P _ x ( T _ { W _ y } < \\tau _ U ) = G ^ U \\mu ( x ) = \\int _ U G ^ U ( x , z ) \\mu ( d z ) \\le \\int _ U G ( x , z ) \\mu ( d z ) \\\\ & \\le c _ 3 G ( x , y ) \\mu ( U ) = c _ 3 G ( x , y ) \\mathrm { C a p } ^ { Y ^ U } ( W _ y ) \\ , . \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} ( - q ; q ) _ \\infty = \\dfrac { 1 } { ( q ; q ^ 2 ) _ \\infty } , \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} 1 - \\displaystyle \\frac { [ y ^ I ] _ i } { \\sqrt { [ y ^ I ] _ i ^ 2 + [ z ^ I ] _ i ^ 2 + 2 \\varepsilon ^ 2 } } = \\displaystyle \\frac { [ z ^ I ] _ i } { [ y ^ I ] _ i + [ z ^ I ] _ i } , \\ 1 - \\displaystyle \\frac { [ z ^ I ] _ i } { \\sqrt { [ y ^ I ] _ i ^ 2 + [ z ^ I ] _ i ^ 2 + 2 \\varepsilon ^ 2 } } = \\displaystyle \\frac { [ y ^ I ] _ i } { [ y ^ I ] _ i + [ z ^ I ] _ i } . \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} \\mathsf { \\hat { s } } _ { \\mathsf { k } } = \\left \\{ \\begin{array} { c } \\left \\Vert \\tilde { Q } \\mathsf { \\hat { p } } _ { \\mathsf { 1 } } \\right \\Vert _ { 2 } ^ { - 2 } \\tilde { Q } \\mathsf { \\hat { p } } _ { \\mathsf { 1 } } \\mathsf { k = 1 , } \\\\ \\\\ \\mathsf { \\hat { q } } _ { \\mathsf { k } } \\mathsf { k } \\in \\left \\{ 2 , 3 , . . . \\right \\} , \\end{array} \\right . \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} \\frac { ( 8 - \\mu ) { l \\choose 2 } + 2 4 { l \\choose 3 } + 1 6 { l \\choose 4 } } { 2 ( l ) } = \\frac { ( 8 - \\mu ) l + 2 4 { l \\choose 2 } + 1 6 { l \\choose 3 } } { 2 } . , \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} P \\left ( \\left \\| \\mathbb { I } _ n \\right \\| _ { \\ell _ \\infty } > L \\sqrt { \\frac { \\log ( d + r ) } { n } } \\right ) & \\leq \\sum _ { i , j = 1 } ^ { d + r } P \\left ( \\sqrt { n } \\left | \\mathbb { I } _ n ^ { i j } \\right | > L \\sqrt { \\log ( d + r ) } \\right ) \\leq 2 ( d + r ) ^ { 2 - C L ^ 2 } . \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} w _ { 2 } ( \\xi ) \\geq \\widehat { w } _ { 2 } ( \\xi ) ^ 2 - \\widehat { w } _ { 2 } ( \\xi ) , \\widehat { w } _ { 2 } ( \\xi ) = \\frac { 1 } { 1 - \\widehat { \\lambda } _ { 2 } ( \\xi ) } \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} S ^ { ( 3 ) } _ 4 & = S ^ { ( 3 ) } _ { 4 , 4 } + S ^ { ( 3 ) } _ { 4 , 3 } = 0 . \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} | | | \\Phi | | | ^ 2 _ m : = \\int _ G | | \\Phi ( g ) | | ^ 2 _ { 1 } ( 1 + L ( g ) ) ^ { 2 m } d g \\ , . \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} \\Lambda _ { \\underline { \\xi } } ^ { \\ast } = \\{ ( x _ { 0 } + \\xi _ { 1 } x _ { 1 } + \\cdots + \\xi _ { N } x _ { N } , x _ { 1 } , \\ldots , x _ { N } ) \\in \\mathbb { R } ^ { N + 1 } : x _ { j } \\in \\mathbb { Z } \\} . \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} A ( z ) = \\begin{pmatrix} 1 & 1 \\\\ 0 & e ^ { 4 \\pi i z } \\end{pmatrix} , z \\in \\Delta . \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k + 1 \\} = \\Bigl ( \\frac { c _ 2 } { c _ 1 + c _ 2 } \\Bigr ) ^ { k + 1 } \\sum _ { j = 0 } ^ k A _ j ^ { ( k ) } \\Bigl ( \\frac { c _ 1 } { c _ 1 + c _ 2 } \\Bigr ) ^ { j } \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} X _ { w } : = A _ { i _ { 0 } , i _ { 1 } } \\ldots A _ { i _ { 2 k } , i _ { 0 } } . \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} F ( t , x , v ) | _ { \\gamma _ - } = F ( t , x , v - 2 n _ x ( n _ x \\cdot v ) ) = F ( t , x , R _ x v ) , \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} \\phi ^ { - 1 } ( \\phi ( U ) ) & = \\{ x \\in G \\mid x H = u H u \\in U \\} \\\\ & = \\{ x \\in G \\mid x \\in u H u \\in U \\} \\\\ & = U \\ ! H , \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} F ( t ) = L ( t ) - C _ 1 - C _ 2 J ( u ( t ) ) , \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} H _ { L + k } = H _ { L + k - 1 } + H _ { k + m } + \\dots + H _ { k + 1 } + N H _ { k } \\leq \\sum _ { i = 1 } ^ { L + k - 1 } H _ { i } + 1 . \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} 1 - \\theta _ { n , p } ( x ) = \\frac { ( 1 - p ) x ^ 2 } { 2 \\log \\frac { n } { 2 } } - \\frac { ( 1 - p ) [ 3 ( 1 - p ) x - 4 ( 1 - 2 p ) ] x ^ { 3 } } { 2 4 ( \\log \\frac { n } { 2 } ) ^ { 2 } } + o ( \\frac { 1 } { ( \\log \\frac { n } { 2 } ) ^ { 2 } } ) . \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} \\sup _ { | z - c | \\leq R } \\sum _ { k = n } ^ \\infty | a _ k z ^ k | & \\leq \\sum _ { k = n } ^ \\infty C _ 2 ( 2 R ) ^ k \\exp \\left ( - \\gamma _ 2 k ^ { \\frac { d } { d - 1 } } \\right ) \\\\ & \\leq \\sum _ { k = n } ^ \\infty C _ 3 \\exp \\left ( - \\gamma _ 3 k ^ { \\frac { d } { d - 1 } } \\right ) \\leq C _ 4 \\exp \\left ( - \\gamma _ 4 n ^ { \\frac { d } { d - 1 } } \\right ) \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} Z ^ i _ n : = Y _ { 1 ^ n } \\times _ { Y _ { \\sigma _ i } } Y _ { 1 ^ n } , Z ^ { \\sigma _ i } _ n : = \\overline { Z ^ i _ n \\setminus Y _ { 1 ^ n } } \\subset Z _ n . \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } a _ 1 & b _ 1 \\\\ c _ 1 & d _ 1 \\end{array} \\right ) \\ast \\left ( \\begin{array} { c c } a _ 2 & b _ 2 \\\\ c _ 2 & d _ 2 \\end{array} \\right ) = \\left ( \\begin{array} { c c c c } a _ 1 & 0 & b _ 1 & 0 \\\\ 0 & a _ 2 & 0 & b _ 2 \\\\ c _ 1 & 0 & d _ 1 & 0 \\\\ 0 & c _ 2 & 0 & d _ 2 \\end{array} \\right ) , \\\\ \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} \\frac { \\sin \\left ( \\beta - \\frac { \\pi } { n } \\right ) } { \\sin \\frac { 2 \\pi } { n } } = - \\frac { 1 } { 2 } \\Rightarrow \\beta = \\beta _ 0 ( n ) = \\frac { \\pi } { n } - \\arcsin \\left ( \\frac { 1 } { 2 } \\sin \\frac { 2 \\pi } { n } \\right ) = \\frac { \\pi ^ 3 } { 2 n ^ 3 } + \\frac { \\pi ^ 5 } { 8 n ^ 5 } + O \\left ( \\frac { 1 } { n ^ 7 } \\right ) . \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} n _ i ' = \\begin{cases} n _ i , & i \\neq j , \\\\ n _ j - \\delta , & i = j , \\end{cases} \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} \\Lambda ( \\mu ) = \\ln ( \\gamma ( \\mu ) ) + \\frac { \\lambda ( \\mu ) } { \\omega ( \\mu ) } \\left ( \\phi ( \\mu ) + \\frac { 3 \\pi } { 2 } \\right ) . \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} F \\circ K = K \\circ R . \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} { \\mathcal Q } _ A \\nabla _ { \\lambda } { \\mathcal Q } _ B = { \\mathcal Q } _ { A \\nabla _ { \\lambda } B } , \\ ; { \\mathcal Q } _ A ! _ { \\lambda } { \\mathcal Q } _ B = { \\mathcal Q } _ { A ! _ { \\lambda } B } , \\ ; { \\mathcal Q } _ A \\sharp _ { \\lambda } { \\mathcal Q } _ B = { \\mathcal Q } _ { A \\sharp _ { \\lambda } B } , \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} b _ { 1 } = \\frac { 1 6 - \\mu } { 2 } . \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} e _ { i j } ( x ) = \\int _ 0 ^ 1 \\partial _ i f _ j ( k \\Phi ( x ) + t W ^ { \\beta ^ * } ( x ) ) d t . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} q ^ 2 + 1 = ( q + \\sqrt { 2 q } + 1 ) ( q - \\sqrt { 2 q } + 1 ) , \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} | M ^ N _ t ( x ) - M _ s ^ N ( x ) | & \\leq \\left | \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ s ^ t \\nabla e ^ { ( t - u ) \\Delta } V ^ N ( X _ u ^ { i , N } - x ) \\cdot d W ^ i _ u \\right | \\\\ & + \\left | \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ 0 ^ s \\nabla e ^ { ( s - u ) \\Delta } \\left [ e ^ { ( t - s ) \\Delta } V ^ N ( X _ u ^ { i , N } - x ) - V ^ N ( X _ u ^ { i , N } - x ) \\right ] \\cdot d W ^ i _ u \\right | \\\\ & = : | I ^ N _ { s , t } ( x ) | + | I I ^ N _ { s , t } ( x ) | . \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} F ^ { - k } \\{ x \\} = \\{ y \\} , \\ , \\ , F ^ { - k } \\{ x ' \\} = \\{ y ' \\} \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} \\partial _ { 2 } K ( x , \\lambda ( t ) ) = \\int _ { 0 } ^ { \\infty } d R \\frac { R } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\int _ { 0 } ^ { x } \\rho d \\rho \\left ( \\frac { 1 } { \\sqrt { x ^ { 2 } - \\rho ^ { 2 } } } - \\frac { 1 } { x } \\right ) \\left ( \\frac { 4 \\lambda ( t ) R ^ { 2 } ( 1 + \\rho ^ { 2 } + \\lambda ( t ) ^ { 2 } R ^ { 2 } ) } { ( 4 \\lambda ( t ) ^ { 2 } R ^ { 2 } + ( 1 + \\rho ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } ) ^ { 2 } ) ^ { 3 / 2 } } \\right ) \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} K ^ { \\ast } ( Q ) = \\{ \\underline { y } \\in \\mathbb { R } ^ { N + 1 } : \\vert \\underline { y } \\cdot \\underline { z } \\vert \\leq 1 , \\underline { z } \\in K ( Q ) \\} \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} \\displaystyle b ^ { k d } D _ { k , m } ( a , X ^ d ) = D _ { k , m } ( a b ^ d , ( X b ^ 2 ) ^ { d } ) . \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} \\gamma _ { l ' i } \\overline { \\gamma _ { \\tau ' \\left ( l ' \\right ) j } } = \\left \\{ \\begin{matrix} G ^ { \\left ( l ' \\right ) } _ { 0 ; 0 , \\dots , 1 , \\dots , 0 } ( z ) , & \\mbox { f o r a l l $ i , j = 1 , \\dots , N $ w i t h $ j = \\tau \\left ( i \\right ) $ , } \\\\ \\quad \\quad \\quad \\quad \\quad 0 , & \\quad \\mbox { f o r a l l $ i , j = 1 , \\dots , N $ w i t h $ j \\neq \\tau \\left ( i \\right ) $ , } \\end{matrix} \\right . \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} \\mathcal K ( x _ 1 - y _ 1 , x _ 2 - y _ 2 , x _ 3 - y _ 3 ) & \\geq \\frac { 1 } { | 7 \\ell ( I ) | ^ { 2 } | 7 \\ell ( J ) | ^ { 2 } + ( 4 9 \\ell ( S ) ) ^ 2 } \\\\ & = { 1 \\over 2 \\cdot 4 9 ^ 2 \\ell ( I ) ^ 2 \\ell ( J ) ^ 2 } \\\\ & = { 1 \\over 2 \\cdot 4 9 ^ 2 | R | } , \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} \\lim _ { x \\uparrow c _ { 0 } } \\frac { p _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ( x ) } { \\left | x - c _ { 0 } \\right | ^ { 1 / 2 } } = \\frac { a _ { 1 } } { \\pi \\sqrt { \\left | c _ { 2 } \\right | } } \\cos \\frac { \\theta } { 2 } , \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} & \\bar \\nabla { \\rm d i s t } _ N ( P _ N ( p ) ) = \\gamma ' ( 0 ) \\in T _ { P _ N ( p ) } ^ { \\bot } N , \\\\ & \\bar \\nabla { \\rm d i s t } _ N ( p ) = \\gamma ' ( { \\rm d i s t } _ N ( p ) ) = \\gamma ' ( 0 ) , \\\\ & | \\bar \\nabla { \\rm d i s t } _ N ( p ) | = 1 . \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} \\tilde { n } _ i = \\begin{cases} n _ i , & i \\neq s , \\\\ n _ s - 1 , & i = s . \\end{cases} \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( V _ m ^ { ( 1 ) } - V _ m ^ { ( 2 ) } ) v ^ { ( 1 ) } \\cdots v ^ { ( m + 1 ) } d x = 0 \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} p _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ( x ) = \\frac { - \\Im a _ { 2 k } } { 2 \\pi c _ { 2 } ^ { k + 1 / 2 } } \\left | x - c _ { 0 } \\right | ^ { k - 1 / 2 } \\left [ 1 + o ( 1 ) \\right ] \\qquad ( x \\rightarrow c _ { 0 } ^ { + } ) . \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} ( A ^ \\dag ) ^ g = & g ^ { - 1 } A ^ \\dag g \\\\ ( c ) ^ g = & g ^ { - 1 } c g \\\\ ( c ^ \\dag ) ^ g = & g ^ { - 1 } c ^ \\dag g . \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} \\lbrace \\omega \\in \\Omega \\ : \\ V ( \\omega , 0 ) = - c , \\ N ( \\omega , t ) = n , \\ \\mathcal { T } ( \\omega , t ) > \\beta \\rbrace , \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} Q = ( \\tau ' \\times \\Pi _ { 2 i + k - n } ) \\boxtimes ( \\tau '' \\times \\Pi _ { n - i - k } ) , \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} \\boldsymbol { B } ( \\psi _ j ^ \\ell , \\psi _ j ^ \\ell ) = { \\mathbb M } ( \\varphi _ j ^ \\ell , \\varphi _ j ^ \\ell ) = \\pi \\mathcal { B } _ j ^ \\ell ( { \\mathbf V } _ { j } ^ \\ell , { \\mathbf V } _ { j } ^ \\ell ) . \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} R _ { 1 3 1 3 } = \\frac { { { e } ^ { q _ 3 + 3 q _ 2 + 2 q _ 1 } } } { \\Delta _ 1 } , R _ { 1 2 1 3 } = - \\frac { { { e } ^ { 2 q _ 3 + 2 q _ 2 + 2 q _ 1 } } } { \\Delta _ 1 } , R _ { 1 2 2 3 } = \\frac { { { e } ^ { 2 q _ 3 + 3 q _ 2 + q _ 1 } } } { \\Delta _ 1 } , R _ { 1 3 2 3 } = - \\frac { { { e } ^ { q _ 3 + 4 q _ 2 + q _ 1 } } } { \\Delta _ 1 } , \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} ( \\mathcal { K } ) ^ c & = ( \\R ^ 2 \\times \\R ^ 2 ) \\setminus \\mathcal { K } , \\ ( \\tilde { \\mathcal { K } } ) ^ c = ( \\R ^ 3 \\times \\R ^ 3 ) \\setminus \\tilde { \\mathcal { K } } \\\\ ( \\mathcal { K } ' ) ^ c & = ( \\R ^ 2 \\times \\R ^ 2 ) \\setminus \\mathcal { K } ' , ( \\tilde { \\mathcal { K } } ' ) ^ c = ( \\R ^ 3 \\times \\R ^ 3 ) \\setminus \\tilde { \\mathcal { K } } ' . \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} | \\partial _ { t } v _ { 1 } ( t , r ) | \\leq \\begin{cases} \\frac { C r } { t ^ { 3 } \\log ^ { b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { r t \\log ^ { b + 1 } ( t ) } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} x \\vert _ { D _ 0 } = y \\vert _ { D _ 0 } \\mbox { i m p l i e s } x = y \\mbox { f o r a l l } x \\in \\Sigma . \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} p _ { 0 } = 0 , & p _ 1 = 1 , p _ { k + 1 } = ( x + a _ k ) p _ k - p _ { k - 1 } , k \\geq 1 \\\\ U _ { 0 } = 0 , & U _ 1 = 1 , U _ { k + 1 } = x U _ k - U _ { k - 1 } , k \\geq 1 \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} \\Phi _ 0 ( z ) : = \\big [ g r a p h _ { \\Sigma } \\big ( \\bar { t } \\cdot ( z _ 1 \\phi _ 1 + z _ 2 \\phi + . . . + z _ I \\phi _ I ) \\big ) \\big ] \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} K _ { H , z } ( w ) = \\sum _ { j = 0 } ^ \\infty \\overline { e _ j ( z ) } e _ j ( w ) . \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { y \\in S ( x ) } w ( x , y ) f ( y ) \\ \\ \\ \\ x \\in \\bigcup ^ { N - 1 } _ { n = 0 } T _ n . \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} \\frac { \\gamma _ { 1 ^ n } } { \\gamma _ { 1 ^ n } ^ { s _ i } } = \\frac { \\gamma _ { 1 ^ n } / \\gamma _ { \\sigma _ i } } { ( \\gamma _ { 1 ^ n } / \\gamma _ { \\sigma _ i } ) ^ { s _ i } } = - \\frac { x _ i - x _ { i + 1 } + \\Delta _ { i , i + 1 } } { x _ { i + 1 } - x _ i + \\Delta _ { i , i + 1 } } . \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} L _ { \\Sigma } u _ 0 = 0 \\ \\ B _ { r _ 2 } ; \\ \\ \\ \\ u _ 0 ( r , \\cdot ) \\begin{cases} = 0 \\ & r > r _ 2 \\\\ \\geq 0 \\ & 0 < r < r _ 2 \\end{cases} \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} \\psi ^ \\bot = \\sum _ { j \\geq 2 } \\psi _ j ^ 1 + \\sum _ { j \\geq 2 } \\psi _ j ^ 2 , \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} \\left | \\tau + \\frac { P _ { 1 , 1 } ( \\lambda ) } { P _ { 1 , 2 } ( \\lambda ) } \\right | = \\left | \\frac { Q _ { 1 } ( \\lambda , 1 , \\tau ) } { P _ { 1 , 2 } ( \\lambda ) } \\right | \\le r ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\left ( \\mbox { e s s ~ s u p } _ { | z | \\ge R } | f ( z ) | \\right ) = 0 , \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} \\Big | \\bar \\nabla \\big ( { \\rm d i s t } ^ 2 _ N \\big ) ( p ) \\Big | ^ 2 = 4 { \\rm d i s t } ^ 2 _ N ( p ) , \\ \\ \\ \\forall \\ p \\in B ( N , 3 \\delta _ 0 ) . \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} ( x _ 3 + F ) ( x _ 3 - F ) = F _ 1 F _ 2 G _ 1 G _ 2 , \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} \\beta _ { E } ( R ) : = \\beta _ { E , 3 , 2 } . \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\mathbb { E } ( S _ { \\alpha , \\lambda } ( t ) ) = \\alpha \\lambda ^ { \\alpha - 1 } t , \\ ; \\ ; \\mathbb { E } ( S _ { \\alpha , \\lambda } ( t ) ) ^ 2 = \\alpha ( 1 - \\alpha ) \\lambda ^ { \\alpha - 2 } t + ( \\alpha \\lambda ^ { \\alpha - 1 } t ) ^ 2 , \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{gather*} \\Xi _ n ^ w \\circ j ^ * ( b ' ) = h \\circ \\Xi _ n ^ { \\preccurlyeq w } ( b ' ) = \\xi ^ { \\mu \\lambda } _ { 1 , \\beta _ w ^ { - 1 } } h \\circ \\Xi _ n \\circ i _ * ( b ' ) = \\xi ^ { \\mu \\lambda } _ { ( w , \\beta _ w ^ { - 1 } \\beta _ w P ) } = \\xi ^ { \\mu \\lambda } _ { ( w , P ) } . \\end{gather*}"}
-{"id": "7323.png", "formula": "\\begin{align*} \\cos \\phi = \\frac { u _ 1 ^ 2 + u _ 2 ^ 2 - u _ 3 ^ 2 } { 2 u _ 1 u _ 2 } = \\frac { 2 ( u _ 1 ^ 2 + u _ 2 ^ 2 ) - 2 u _ 3 ^ 2 } { 4 u _ 1 u _ 2 } \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} \\mathbf { z } _ { k + 1 } = A _ k \\mathbf { z } _ k , \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} I ^ { \\tilde G } ( \\bar x ) & : = \\{ l \\in \\mathcal Q \\ , | \\ , \\tilde G _ l ( \\bar x ) = 0 \\ , \\land \\ , \\tilde H _ l ( \\bar x ) \\neq 0 \\} , \\\\ I ^ { \\tilde H } ( \\bar x ) & : = \\{ l \\in \\mathcal Q \\ , | \\ , \\tilde G _ l ( \\bar x ) \\neq 0 \\ , \\land \\ , \\tilde H _ l ( \\bar x ) = 0 \\} , \\\\ I ^ { \\tilde G \\tilde H } ( \\bar x ) & : = \\{ l \\in \\mathcal Q \\ , | \\ , \\tilde G _ l ( \\bar x ) = 0 \\ , \\land \\ , \\tilde H _ l ( \\bar x ) = 0 \\} . \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} A ^ \\circ : = \\{ y \\in \\R ^ n \\ , | \\ , \\forall x \\in A \\colon \\ , x \\cdot y \\leq 0 \\} \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} \\mathbb { Q } _ { n } ' : = \\frac { 1 } { 2 ^ { n } } \\sum _ { \\sigma \\in \\{ - 1 , 1 \\} ^ { n } } \\mathbb { Q } _ { n , \\sigma } ' . \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} \\begin{aligned} X & { } = \\frac { B + W + y Z } { \\sqrt { 1 + y ^ 2 + \\lVert W \\rVert ^ 2 } } , & \\xi _ 1 & { } = \\frac { - y B + Z } { \\sqrt { 1 + y ^ 2 } } , & \\xi _ 2 & { } = \\frac { \\lVert W \\rVert ^ 2 B - ( 1 + y ^ 2 ) W + y \\lVert W \\rVert ^ 2 Z } { \\lVert W \\rVert \\sqrt { ( 1 + y ^ 2 ) ( 1 + y ^ 2 + \\lVert W \\rVert ^ 2 ) } } . \\end{aligned} \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} \\alpha _ { k , 0 } Y ^ n & + ( \\alpha _ { k , 0 } + \\alpha _ { k , 1 } ) Y ^ { n + 1 } + ( \\alpha _ { k , 0 } + \\alpha _ { k , 1 } + \\alpha _ { k , 2 } ) Y ^ { n + 2 } \\\\ & + \\displaystyle \\sum _ { j = 3 } ^ { k } \\left ( \\alpha _ { k , j - 3 } + \\alpha _ { k , j - 2 } + \\alpha _ { k , j - 1 } + \\alpha _ { k , j } \\right ) Y ^ { n + j } + ( \\alpha _ { k , k - 2 } + \\alpha _ { k , k - 1 } + \\alpha _ { k , k } ) Y ^ { n + k + 1 } \\\\ & + ( \\alpha _ { k , k - 1 } + \\alpha _ { k , k } ) Y ^ { n + k + 2 } + \\alpha _ { k , k } Y ^ { n + k + 3 } = f ( t _ n , Y ^ n ) \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} | \\{ ( \\xi _ 1 , \\eta _ 1 ) \\ | \\ ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) \\in E ( \\tau _ 2 , \\xi _ 2 , \\eta _ 2 ) \\} | = & \\int _ { \\theta _ 1 } \\int _ { r _ 1 } { \\chi } _ { E ( \\tau _ 2 , \\xi _ 2 , \\eta _ 2 ) } ( | \\xi _ 1 | , \\theta _ 1 ) r _ 1 d r _ 1 d \\theta _ 1 \\\\ \\lesssim & A ^ { - 1 } M N _ 1 ^ { - 1 } \\max ( L _ 1 , L _ 2 ) . \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} g \\circ f : = \\bigoplus _ { \\substack { v \\in J \\\\ w \\in K } } g _ { v w } \\circ f _ { v w } . \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} b _ { 1 } = \\begin{cases} - ( 1 - \\eta _ { \\mu _ { 1 } } ( \\alpha ) ) ^ { 2 } \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t ^ { 2 } } { ( \\eta _ { \\mu _ { 1 } } ( \\alpha ) - t ) ^ { 2 } } \\ , d \\sigma ( t ) & a _ { 0 } \\neq - 1 , \\\\ - \\int _ { [ 0 , + \\infty ] } ( 1 + t ^ { 2 } ) \\ , d \\sigma ( t ) & a _ { 0 } = - 1 . \\end{cases} \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} y = P _ n x - P _ n H x = P _ n ( 1 - H ) x \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} \\dot { y } ^ 2 = \\dot { x } ^ 2 ( y _ x ) ^ 2 = k \\ , \\frac { 2 x - \\tilde { c } _ 1 } { ( x ^ 2 + y ) ^ 2 } \\ , \\frac { ( x ^ 2 + y ) ^ 2 } { ( 2 x - \\tilde { c } _ 1 ) ^ 2 } = \\frac { k } { ( 2 x - \\tilde { c } _ 1 ) } \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} \\omega ( m + 1 ) - \\omega ( m ) \\sim \\begin{cases} C \\ , m ^ { \\gamma _ 1 / 2 } & \\quad \\hbox { i f } \\gamma _ 1 \\geq - 2 \\\\ C m ^ { - 1 } & \\quad \\hbox { i f } \\gamma _ 1 < - 2 \\ , . \\end{cases} \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} \\int _ { H I } e ^ { s x } \\overline { G } ( s , t ) d s = - \\int _ { r } ^ { - \\lambda _ 2 + R } e ^ { - x \\lambda _ 2 } e ^ { - w x } e ^ { t ( c _ 1 \\lambda ^ { \\alpha _ 1 } + c _ 2 \\lambda _ 2 ^ { \\alpha _ 2 } ) } e ^ { - t [ c _ 1 ( \\lambda _ 1 - \\lambda _ 2 + w e ^ { - i \\pi } ) ^ { \\alpha _ 1 } + c _ 2 ( w ^ { \\alpha _ 2 } e ^ { - i \\pi \\alpha _ 2 } ) ] } d w . \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & \\ , \\rightarrow \\ , \\min & & & \\\\ g _ i ( x ) & \\ , \\leq \\ , 0 & \\qquad & i \\in \\mathcal M & \\\\ h _ j ( x ) & \\ , = \\ , 0 & \\qquad & j \\in \\mathcal P & \\\\ \\varphi ( G _ l ( x ) , H _ l ( x ) ) & \\ , \\leq \\ , 0 & \\qquad & l \\in \\mathcal Q & \\end{aligned} \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} J _ { 0 } = 0 , \\ J _ { 1 } = 1 , \\ J _ { n + 2 } = J _ { n + 1 } + 2 J _ { n } , \\ n \\geq 0 . \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} C : = \\{ ( a , b ) \\in \\R ^ 2 \\ , | \\ , a , b \\geq 0 \\ , \\land \\ , a b = 0 \\} . \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} \\epsilon _ 1 : = \\min _ { 1 \\leq i \\leq m } \\inf _ { x \\in [ - K _ 0 , 0 ] } | \\phi _ i ' ( x ) | > 0 . \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} \\alpha = \\frac { A } { 3 } - \\frac { B } { 3 } - \\frac { 1 } { 2 } , ~ ~ ~ \\beta = \\frac { A } { 3 } + \\frac { B } { 3 } - \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} W _ { y , y ' } ( z ) = U _ { y } ( z ) - U _ { y ' } ( z ) . \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} N _ \\lambda & = \\bigcup _ { 0 \\leq i < j \\leq r - 1 } [ \\tilde { \\lambda } _ i + 1 , \\tilde { \\lambda } _ { i + 1 } ] \\times [ \\tilde { \\lambda } _ j + 1 , \\tilde { \\lambda } _ { j + 1 } ] , \\\\ I _ \\lambda & = N _ \\lambda \\cup \\left ( \\bigcup _ i [ \\tilde { \\lambda } _ i + 1 , \\tilde { \\lambda } _ { i + 1 } ] ^ 2 \\right ) \\setminus \\left \\{ ( i , i ) : 1 \\leq i \\leq n \\right \\} . \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} \\gamma _ { p ^ k } ( a ) & = - \\delta ( \\gamma _ { p ^ { k - 1 } } ( a ) ) + \\frac { d ^ { m ^ { k - 1 } } - ( d - p ) ^ { m ^ { k - 1 } } } { p } \\cdot \\frac { \\phi ( \\gamma _ { p ^ { k - 1 } } ( a ) ) } { d ^ { m ^ { k - 1 } } } \\\\ & = \\frac { 1 } { p } \\cdot \\left ( \\gamma _ { p ^ { k - 1 } } ( a ) ^ p - ( d - p ) ^ { m ^ { k - 1 } } \\cdot \\frac { \\phi ( \\gamma _ { p ^ { k - 1 } } ( a ) ) } { d ^ { m ^ { k - 1 } } } \\right ) , \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} T ^ 2 _ { 1 2 } = T ^ 2 _ { 1 3 } = T ^ 2 _ { 1 4 } = T ^ 2 _ 1 = \\bar T ^ 2 _ 1 = R ^ 2 _ 1 = 0 ; \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} \\| [ b _ 0 , T ] ( f ) \\| _ { L ^ 2 ( \\mathbb R ^ 3 ) } & = \\| b _ 0 T ( f ) - T ( b _ 0 f ) \\| _ { L ^ 2 ( \\mathbb R ^ 3 ) } \\\\ & \\leq \\| b _ 0 T _ 1 ( f ) - T _ 1 ( b _ 0 f ) \\| _ { L ^ 2 ( \\mathbb R ^ 3 ) } + \\| b _ 0 T _ 2 ( f ) - T _ 2 ( b _ 0 f ) \\| _ { L ^ 2 ( \\mathbb R ^ 3 ) } , \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| \\Delta _ q X ( t ) \\| _ { L ^ 2 } ^ 2 & = - \\int _ { \\R ^ 2 } \\Delta _ q ( u \\cdot \\nabla X ) \\cdot \\Delta _ q X ~ d \\tau + \\int _ { \\R ^ 2 } \\Delta _ q \\partial _ X u \\cdot \\Delta _ q X ~ d \\tau . \\\\ \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\mathcal { T } _ { 3 , M } = \\big \\lbrace & ( 0 : 0 : 0 : t ^ 5 : t ^ 5 : 2 : t ^ 2 : 1 ) + ( 0 : 0 : 0 : t ^ 3 : t ^ 3 : 2 : t ^ 6 : 1 ) , \\\\ & ( 0 : 0 : 0 : t ^ 3 : t ^ 7 : t ^ 7 : t ^ 5 : 1 ) + ( 0 : 0 : 0 : t ^ 5 : t : t : t 3 : 1 ) , \\\\ & ( 0 : 0 : 0 : t ^ 5 : t ^ 6 : t ^ 5 : 1 : 0 ) + ( 0 : 0 : 0 : t ^ 3 : t ^ 2 : t ^ 3 : 1 : 0 ) \\big \\rbrace , \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} B ( e x f , g y h ) & = B ( e x f , g ) g y h + g B ( e x f , g y h ) = - e B ( g , x ) f g y h + g B ( e x f , g y h ) \\\\ & = g B ( e x f , g y h ) = g e x f B ( f , g y h ) + g B ( e x f , g y h ) f \\\\ & = g e x f g B ( f , y ) h + g B ( e x f , g y h ) f = g B ( e x f , g y h ) f . \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} \\binom { n + k - 1 } { n - 1 } \\leq 2 ^ { n + k } , \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} X _ { k i t } = \\sum _ { j = 1 } ^ { r _ N } \\lambda _ { i j } f _ { t j } \\delta _ { k j } + E _ { k i t } , \\ k \\in \\{ 1 , \\dots , K \\} . \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} C B . f ( { h } ) = & \\frac { 1 } { 2 } ( x - y ) ( f ( { h } - 2 ) + f ( { h } + 2 ) ) { B } \\\\ & \\equiv \\frac { 1 } { 2 } ( f ( h - 4 ) x B + f ( h ) x B - f ( h ) y B - f ( h + 4 ) y B ) . \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} \\lambda _ i = l _ i ( l _ i + d _ i - 2 ) + \\beta _ i , ~ ~ ~ l _ i = 0 , 1 , 2 , \\cdots , i = 1 , 2 , \\cdots , N . \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} 2 \\langle S u , A u \\rangle = \\langle [ S , A ] u , u \\rangle + 2 \\tau \\langle S u , ( \\partial _ { n + 1 } \\phi ) u \\rangle _ { 0 } - \\langle A u , \\partial _ { n + 1 } u \\rangle _ { 0 } + \\langle \\partial _ { n + 1 } ( A u ) , u \\rangle _ { 0 } . \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} V ( D ) = \\left ( \\begin{array} { c c } e ^ { - D ^ - D ^ + } & e ^ { - \\frac { 1 } { 2 } D ^ - D ^ + } \\left ( \\frac { I - e ^ { - D ^ - D ^ + } } { D ^ - D ^ + } \\right ) D ^ - \\\\ e ^ { - \\frac { 1 } { 2 } D ^ + D ^ - } D ^ + & I - e ^ { - D ^ + D ^ - } \\end{array} \\right ) \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} \\underbrace { \\mu _ { n } \\boxtimes \\cdots \\boxtimes \\mu _ { n } } _ { n } = \\mu , n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} p _ n ( x ) ^ { \\alpha - 1 } = q ( x ) ^ { \\alpha - 1 } + F _ n + \\theta _ { n } ^ { T } f ( x ) \\quad \\forall x \\in \\mathbb { S } \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} \\psi ( x ) = 1 \\ \\mbox { i n } [ - ( l _ 2 - l _ 1 ) , l _ 2 - l _ 1 ] , \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} [ A _ 1 \\otimes A _ 2 , C \\otimes C ] = [ A _ 1 \\otimes C , A _ 2 \\otimes C ] = [ A _ 2 \\oplus A _ 3 \\oplus B \\oplus C , A _ 1 \\oplus A _ 3 \\oplus B \\oplus C ] = 3 . \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\lambda _ { 0 } ( t ) < \\lambda _ { i } ( t ) < 2 \\lambda _ { 0 } ( t ) < \\frac { 1 } { 2 } , i = 1 , 2 \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} x B & = \\frac { 1 } { 2 } ( B ^ { 2 } + C B ) \\\\ & = \\frac { 1 } { 2 } ( B ^ { 2 } + B C + 2 h ) \\\\ & \\equiv \\frac { 1 } { 2 } ( \\mu - { h } ^ { 2 } + 2 { h } ) , \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} L _ { 2 , \\ell } ( \\tau ) : & = U _ { \\ell } \\left ( \\phi _ { \\ell } ^ d ( \\tau ) L _ 1 ( \\tau ) \\right ) , \\\\ L _ { 2 , \\ell } ( \\tau ) & = \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ { \\ell \\cdot n } ) ^ d ( 1 - q ^ n ) ^ c \\sum _ { m \\geq \\left \\lceil \\frac { \\delta _ { \\ell } \\cdot d + \\lceil \\frac { \\delta _ { \\ell } \\cdot c } { \\ell } \\rceil } { \\ell } \\right \\rceil } ^ { \\infty } p _ { [ 1 ^ c \\ell ^ d ] } ( \\ell ^ 2 m - \\delta _ { \\ell } \\cdot \\ell \\cdot d - \\delta _ { \\ell } \\cdot c ) q ^ m . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} u ( x , t ) & = \\sum _ { n = 1 } ^ \\infty u _ n ( t ) \\varphi _ n ( x ) , \\\\ F ( x , t ) & = \\sum _ { n = 1 } ^ \\infty F _ n ( t ) \\varphi _ n ( x ) , \\ ; \\xi ( x ) = \\sum _ { n = 1 } ^ \\infty \\xi _ n \\varphi _ n ( x ) . \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} a ^ { ( 2 k ) } _ { k + 1 k + 1 } = \\frac { m _ { k + 1 } } { m _ { k } } . \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} \\Tilde { \\boldsymbol { H } } _ r = \\sqrt { \\frac { \\kappa } { 1 + \\kappa } } \\boldsymbol { a } _ { M } ( \\vartheta ^ { } ) \\boldsymbol { a } _ { N _ t } ^ H ( \\vartheta ^ { } ) + \\sqrt { \\frac { 1 } { 1 + \\kappa } } \\boldsymbol { H } ^ { } , \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} & - \\frac { 1 } { 4 n } ( h - 2 n ) \\big ( r ( h - 2 ) - r ( h + 2 ) \\big ) \\\\ & = - \\frac { 1 } { 4 n } ( ( h - 2 n ) r ( h - 2 ) - ( h - 2 n ) r ( h + 2 ) ) \\\\ & \\overset { b y ( \\ref { r h - 2 & r h + 2 } ) } { { = \\joinrel = \\joinrel = } } - \\frac { 1 } { 4 n } ( ( h - 2 n ) r ( h - 2 ) - ( h + 2 n ) r ( h - 2 ) ) \\\\ & = r ( h - 2 ) . \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} A \\cdot M = A M A ^ T . \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} W ^ { \\beta ^ * } ( x ) \\succeq \\mathbf { 0 } \\mbox { f o r } x \\leq 0 , \\ ; \\ \\ w _ { i _ 0 } ^ { \\beta ^ * } ( x _ { i _ 0 } ^ 0 ) = 0 . \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} f = g _ 1 + g { \\rm o n } N ' : = r ^ { - 1 } ( N ) ; \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} \\hat { \\sigma } ( n ) = \\prod _ { \\epsilon _ j \\neq 0 } \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} \\nu = F ^ { ( \\alpha ) } \\left [ f _ { \\nu } \\right ] \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} a _ k ( n ) & = a _ k ( n - 1 ) + a _ k ( n - 2 ) - a _ k ( n - 4 ) - a _ k ( n - 8 ) \\\\ & \\quad \\ , + a _ k ( n - 9 ) + a _ k ( n - 1 8 ) - a _ k ( n - 1 6 ) - a _ k ( n - 3 2 ) + \\cdots , \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} C ( A , X ) & = \\{ B \\in C ( X ) : A \\subset B \\} , \\\\ C ( p , X ) & = \\{ A \\in C ( X ) : p \\in A \\} , \\\\ K ( X ) & = \\{ C ( x , X ) : x \\in X \\} . \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} = \\left ( \\prod _ { i = 1 } ^ { n + 1 } ( 1 + x _ i ) \\right ) \\sum _ { I \\subseteq [ 2 , n ] } k _ { n - | I | } ( \\underline x _ n ^ I ) \\left ( \\prod _ { i \\in I } \\frac { 1 } { 1 + x _ i } \\right ) \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} c _ { \\mathsf { m , n } } = \\exp \\left [ - \\frac { \\alpha \\beta } { 2 } \\left ( \\lambda _ { \\mathsf { m } } + \\lambda _ { \\mathsf { n } } \\right ) \\right ] \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) \\le \\beta \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k \\} = \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} & L _ { 0 1 2 } ^ { \\max } : = \\max ( L _ 0 , L _ 1 , L _ 2 ) , N _ { 0 1 2 } ^ { \\max } : = \\max ( N _ 0 , N _ 1 , N _ 2 ) , \\\\ & w _ { N _ 0 , L _ 0 } : = Q _ { L _ 0 } P _ { N _ 0 } w , \\ u _ { N _ 1 , L _ 1 } : = Q _ { L _ 1 } P _ { N _ 1 } u , \\ v _ { N _ 2 , L _ 2 } : = Q _ { L _ 2 } P _ { N _ 2 } v . \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} ( g ^ { i j } ) _ s \\ , \\Pi \\left ( F _ { i j } \\right ) = 2 g ^ { i k } A ^ V _ { k m } g ^ { m j } \\ , A _ { i j } \\ , . \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} K _ { \\alpha ( \\tau ) ( z ) } ( \\alpha ( \\tau ) ( w ) ) = K _ z ( w ) \\qquad ( \\tau \\in G , \\ z , w \\in \\Omega ) . \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 h } { \\partial t ^ 2 } - c _ 1 c _ 2 \\frac { \\partial ^ 2 h } { \\partial x ^ 2 } + ( c _ 1 - c _ 2 ) \\frac { \\partial ^ 2 h } { \\partial x \\partial t } = - \\frac { m + n + 2 } { t } \\frac { \\partial h } { \\partial t } - \\frac { 1 } { t } \\Bigl [ ( c _ 1 - c _ 2 ) ( m + n + 1 ) - ( c _ 1 m - c _ 2 n ) \\Bigr ] \\frac { \\partial h } { \\partial x } \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} f _ a ^ { - 1 } ( U \\times V ) = \\begin{cases} \\emptyset & a \\not \\in V , \\\\ U & a \\in V . \\end{cases} \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} - 4 \\int _ { t } ^ { \\infty } \\frac { \\lambda _ { 0 , 0 } '' ( x ) d x } { ( \\log ^ { ( \\alpha - 1 ) b } ( t ) + x - t ) ( 1 + x - t ) ^ { 3 } } = E _ { v _ { 3 } , i p , f } + \\frac { 4 b ^ { 2 } ( \\alpha - 1 ) \\log ( \\log ( t ) ) } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} M _ D ( x , z _ 0 ) & = \\int _ { D \\setminus U } M _ D ( y , z _ 0 ) \\omega _ U ^ x ( d y ) + \\frac { P _ U ( x , z _ 0 ) } { P _ D ( x _ 0 , z _ 0 ) } \\\\ & > \\int _ { D \\setminus U } M _ D ( y , z _ 0 ) \\omega _ U ^ x ( d y ) , x \\in U . \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} P _ n ( F ) A = \\bar { P } _ n ( F ) L A \\xrightarrow { \\sim } \\bar { P } _ n ( F ) A \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} F _ { \\mu _ { 1 } } ^ { \\langle - 1 \\rangle } ( z ) + \\gamma + N _ { \\sigma } ( z ) = F _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ^ { \\langle - 1 \\rangle } ( z ) \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} A C ( g _ t ^ \\tau ( x ) ) = A C ( g _ s ^ \\tau ( x ) ) A C ( g _ t ^ \\tau ( x ) ) \\cap A C ( g _ s ^ \\tau ( x ) ) = \\emptyset . \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} \\alpha ^ 1 _ 1 = ( A + 3 J _ 1 ) ^ 2 , ~ ~ ~ J _ 1 = 0 , 1 , 2 \\cdots . \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} & ( f \\circ _ i ( g \\circ _ j h ) ) ( [ r ] ; a _ 1 , \\ldots , a _ { m + n + p - 2 } ) \\\\ & = f ( [ r ] ; a _ 1 , \\ldots , a _ { i - 1 } , ( g \\circ _ j h ) ( [ 1 ] + \\cdots + [ n + p ] ; a _ i , \\ldots , a _ { i + n + p - 2 } ) , \\ldots , a _ { m + n + p - 2 } ) . \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} \\forall l \\in I ^ { G H } ( \\bar x , \\bar y , \\bar z ) \\colon \\mu _ l = 0 \\ , \\land \\ , \\nu _ l = 0 . \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} \\mu _ { m } ^ { n ( I ) + 1 } ( J ) = \\mu _ { m } ^ { n ( J ) } ( J ) = \\ell ( J ) ^ { d } . \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} u '' + u | u | ^ { p - 2 } = \\lambda u \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} R ^ 4 _ 4 = - R ^ 2 _ 2 . \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} C \\ge ( - K _ { F _ { \\alpha , i , \\beta } } ) ^ { r _ { \\alpha , i } } = ( - m _ \\alpha K _ { X _ \\alpha } ) ^ { r _ { \\alpha , i } } \\cdot F _ { \\alpha , i , \\beta } = ( H _ \\alpha + D _ \\alpha ) ^ { r _ { \\alpha , i } } \\cdot F _ { \\alpha , i , \\beta } \\ge H _ \\alpha ^ { r _ { \\alpha , i } } \\cdot F _ { \\alpha , i , \\beta } . \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} f ( a , b , z ) = f ( b , a , q / z ) & = f ( b , a p , q / z ) - a f ( b , a p , q ^ 2 / z ) \\\\ & = f ( a p , b , z ) - a f ( a p , b , z / q ) , \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} x _ 1 ( 1 - x _ 1 ) \\leq t l = 1 , \\ldots , n \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} \\| A \\| _ { \\ell _ w } : = \\left \\{ \\begin{array} { l l } \\{ \\sum _ { i = 1 } ^ l \\sum _ { j = 1 } ^ { k } | A ^ { i j } | ^ w \\} ^ { 1 / w } & w < \\infty , \\\\ \\max _ { 1 \\leq i \\leq l } \\max _ { 1 \\leq j \\leq k } | A ^ { i j } | & w = \\infty . \\end{array} \\right . \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } a _ { j k } \\partial _ { j } \\partial _ { k } \\bigg ] \\tilde { u } & = f \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { u } & = V \\tilde { u } \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} n _ { i } + n _ { - 1 } - 1 \\ , = \\ , n _ { i - 1 } \\forall i \\geq 0 \\ , . \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} A ^ G = \\prod _ { c \\in G / H } A ^ c , B ^ G = \\prod _ { c \\in G / H } B ^ c \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} f ( 0 , 0 ; 0 ; 0 ) = g ( 0 , 0 ; 0 ; 0 ) = 0 , \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} \\rho ( L _ { 0 } ) \\ , & = \\ , - \\frac { h ( z ) } { h ' ( z ) } \\partial + b ( z ) \\ , , \\\\ \\rho ( L _ i ) \\ , & = \\ , h ( z ) ^ i \\left ( \\rho ( L _ 0 ) + i \\lambda \\right ) P ( \\rho ( L _ 0 ) - \\lambda - i , i ) \\ , , \\ \\ i \\neq 0 , \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} \\nabla _ { 1 } \\boldsymbol { x } = \\boldsymbol { e _ { 1 } } , \\ \\ \\ \\nabla _ { 2 } \\boldsymbol { x } = \\boldsymbol { e _ { 2 } } , \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} C . ( r ( h ) + \\frac { i } { 2 n } ( C . r ( h ) ) ) = & C . r ( h ) + \\frac { i } { 2 n } ( C ^ { 2 } r ( h ) ) \\\\ \\equiv & C . r ( h ) + \\frac { i } { 2 n } ( - 4 n ^ { 2 } r ( h ) ) \\\\ \\equiv & ( C . r ( h ) - i ( 2 n r ( h ) ) ) \\\\ \\equiv & 2 n i \\big ( r ( h ) + \\frac { i } { 2 n } C . r ( h ) \\big ) \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} Q ( d x ^ { \\Lambda _ 2 } | y ) = - \\log \\int _ { \\substack { \\frac { 1 } { K - R } \\sum _ { i \\in \\Lambda _ 1 ^ { ( l ) } } x _ i = \\tilde { y } _ l \\\\ l = 1 , \\cdots , M } } \\exp \\left ( - H ( x ) \\right ) \\mathcal { L } ( d x ^ { \\Lambda _ 1 } ) , \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} \\Theta ^ { } _ 1 ( \\Lambda ) ( 0 ) = 0 . \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} s _ 0 = 4 , \\ ; \\ ; \\ ; s _ { n + 1 } = s _ n ^ 2 - 2 , \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} M ^ { m + n } & = \\left ( Q \\setminus \\coprod \\limits _ { i = 1 } ^ { + \\infty } ( B _ { \\frac 4 5 r _ i } ( o _ i ) \\times _ { g _ i } S ^ { n - 1 } ) \\right ) \\cup _ { \\textrm { I d } } \\coprod \\limits _ { i = 1 } ^ { + \\infty } P _ i \\\\ & = \\left ( ( \\mathbb { R } ^ { m + 1 } \\setminus \\coprod \\limits _ { i = 1 } ^ { + \\infty } D _ i ^ { m + 1 } ) \\times S ^ { n - 1 } \\right ) \\cup _ { \\textrm { I d } } \\coprod \\limits _ { i = 1 } ^ { + \\infty } ( S ^ m \\times D ^ n ) _ i , \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} ( M , g ) = ( M ^ n _ { \\kappa } , h ^ { - 2 } g _ { \\kappa } ) , \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} \\sin \\phi = \\frac { 2 n ( 1 - m ^ 2 ) } { ( n + m ) ( 1 - m n ) } \\lambda \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} \\mathfrak M ( H ^ \\infty ( S ' ) ) = S _ E \\sqcup \\hat r ^ { - 1 } ( \\partial S ) . \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} u * v = u \\cdot T ( v ) + T ( u ) \\cdot v + H ( T u , T v ) , ~ u , v \\in M . \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} M _ n = \\min \\left \\{ \\max \\left \\{ M _ { n - 1 } , \\tilde { S } _ n \\right \\} , \\tilde { S } _ n + K - J \\right \\} , \\qquad \\forall n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} Q _ { \\infty } = 0 . \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} f _ { Q , \\mu } & \\leq \\log \\left ( 2 \\| M _ \\mu \\| _ { L ^ 1 \\to L ^ { 1 , \\infty } } ^ { 1 / 2 } \\right ) \\leq \\log \\left ( \\frac { ( w ( x ) \\chi _ Q ( x ) ) ^ { 1 / 2 } } { ( w _ { Q , \\mu } ) ^ { 1 / 2 } } \\right ) \\\\ & \\leq \\log \\left ( \\frac { M ( w \\chi _ Q ) ( x ) ^ { 1 / 2 } } { ( w _ { Q , \\mu } ) ^ { 1 / 2 } } \\right ) = \\log v ( x ) = b ( x ) \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} \\int _ 0 ^ s \\frac { d \\norm { v ( t ) } ^ 2 } { 2 d t } + \\eta \\norm { v ( t ) } ^ 2 d t = \\int _ 0 ^ s ( \\operatorname { d i v } ( p ) - \\operatorname { d i v } ( \\bar { p } ) , \\phi ^ s _ v ( t ) ) d t . \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} B = ( C \\times \\{ 0 \\} ) \\cup ( D \\times \\{ 1 \\} ) , \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} D _ { k , m } ( a , X ) : = \\sum _ { i = 0 } ^ { \\lfloor \\frac k 2 \\rfloor } \\frac { k - m i } { k - i } \\dbinom { k - i } { i } ( - X ) ^ { i } a ^ { k - 2 i } , \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} \\overline { i n t ( \\Upsilon ) } = \\displaystyle \\bigcup _ { t \\in \\mathbb { R } } \\left \\{ ( \\lambda , \\mu ) \\in L ( t ) : ( \\lambda , \\mu ) \\leq \\Gamma ( t ) \\right \\} . \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} & - r \\int _ { t + 6 r } ^ { \\infty } d s \\lambda '' ( s ) ( s - t ) \\left ( \\frac { 1 } { ( \\lambda ( t ) ^ { 2 - 2 \\alpha } + ( s - t ) ^ { 2 } ) } - \\frac { 1 } { ( \\lambda ( s ) ^ { 2 - 2 \\alpha } + ( s - t ) ^ { 2 } ) } \\right ) \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} P _ { d } \\left \\{ \\mathbb { G } _ { N } ' f - \\mathbb { G } _ { N } ' g > x \\right \\} & = P _ { d } \\left \\{ e ^ { \\lambda \\sum _ { i = 1 } ^ { N } Z _ { i , N } ( f , g ) } > e ^ { \\lambda x } \\right \\} \\\\ & \\leq e ^ { - \\lambda x } E _ { d } e ^ { \\lambda \\sum _ { i = 1 } ^ { N } Z _ { i , N } ( f , g ) } \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} D _ p ( \\Pi _ { M \\cap N } ^ p y , y ) < \\frac { \\varepsilon m ( y ) } { \\delta } = \\frac { \\varepsilon } { \\delta } \\max \\{ D _ p ( \\Pi _ M ^ p y , y ) , D _ p ( \\Pi _ N ^ p y , y ) \\} . \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} I = i _ 0 + i _ 1 + \\cdots + i _ { 2 l + 1 } \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { \\ell } v _ i ' = \\sum \\limits _ { i = 3 } ^ { \\ell } v _ i ' \\leq 4 ( \\ell - 2 ) < n , \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} _ { \\mathbb C } \\begin{pmatrix} \\Lambda _ 1 & \\hdots & \\Lambda _ { m + 1 } \\cr 1 & \\hdots & 1 \\end{pmatrix} = m + 1 . \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} f ' ( t ) + f ( t ) = C \\log ( 1 - B e ^ { - t } ) f ( 0 ) = 0 , \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\widetilde { T } } ( | x | ^ { - s } u ^ m _ t , \\omega ) d t + \\int _ 0 ^ { \\widetilde { T } } ( | \\nabla u ^ m | ^ { p - 2 } \\nabla u ^ m , \\nabla \\omega ) d t = \\int _ 0 ^ { \\widetilde { T } } ( | u ^ m | ^ { q - 2 } u ^ m \\ln | u ^ m | , \\omega ) d t , \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} \\Re \\langle x , H \\mathcal { H } x \\rangle _ { \\mathcal { H } } = \\Re \\langle x , \\mathcal { H } H \\mathcal { H } x \\rangle = \\Re \\langle \\mathcal { H } x , H \\mathcal { H } x \\rangle \\leqslant 0 \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} { \\Pr } \\bigg ( { x _ R } = 1 \\bigg ) = \\Pr \\bigg ( x _ T = 1 \\bigg ) { \\Pr } \\bigg ( { { x } _ R } = 1 ~ \\bigg | ~ { x _ T } = 1 \\bigg ) + \\Pr \\bigg ( x _ T = 0 \\bigg ) { \\Pr } \\bigg ( { { x } _ R } = 1 ~ \\bigg | ~ { x _ T } = 0 \\bigg ) . \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} \\textnormal { d i a m } ( S _ i ) \\leq 2 ^ { - 8 0 } A ^ { - 1 } \\textnormal { f o r } \\ i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} e ^ z = \\sum _ { k = 0 } ^ \\infty \\frac { z ^ k } { k ! } \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} \\int _ { B } \\exp \\big ( \\alpha _ n & | u _ \\varepsilon | ^ { \\frac n { n - 1 } + | x | ^ \\alpha } \\big ) d x \\\\ & \\geq | B | + | B | \\exp \\big ( \\sum _ { i = 1 } ^ { n - 1 } \\frac 1 i \\big ) + O ( R ^ { - \\frac n { n - 1 } } ) \\\\ & + \\frac { c ^ { - \\frac n { n - 1 } - a ^ \\alpha } \\alpha _ n ^ { n - 1 } } { ( n - 1 ) ! } \\Big ( \\int _ { B _ a } \\Big ( - \\frac n { \\alpha _ n } \\log | x | \\Big ) ^ { n + ( n - 1 ) | x | ^ \\alpha } d x + o ( 1 ) \\Big ) . \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} D _ 1 : = D - H _ { 1 2 3 } = 3 H - 2 E _ 1 - 2 E _ 2 - 2 E _ 3 - E _ 4 - \\ldots - E _ { 8 - i } \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} \\Delta _ { q _ 1 } ( f , < q _ 1 > _ f ) \\Delta _ { p _ 1 } ( f , < p _ 1 > _ f ) = \\frac { 1 } { 2 } \\left ( \\frac { 1 } { 2 } \\mu ^ 2 E ^ 2 a + \\frac { \\mu ^ 2 \\theta ^ 2 } { \\lambda ^ 2 a b } + \\frac { \\eta ^ 2 E ^ 2 a ^ 2 b } { 3 2 \\mu ^ 2 } + \\frac { \\theta ^ 2 \\eta ^ 2 } { 1 6 \\mu ^ 2 \\lambda ^ 2 } \\right ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} \\inf _ { x \\in M } | B _ 1 ( x ) | = \\upsilon > 0 ; \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} T ( \\tilde { r } ; \\mu ) = T _ R ( \\tilde { r } ; \\mu ) + T _ L \\left ( P _ R ( \\tilde { r } ; \\mu ) ; \\mu \\right ) . \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} \\lim _ { \\omega _ { n + 1 } \\rightarrow 0 } \\omega _ { n + 1 } ^ { 1 - 2 s } \\Omega _ { n + 1 } \\overline { u } = \\tilde { V } \\overline { u } , \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} \\mathcal E ( t ) + \\int _ 0 ^ t \\mathcal D ( s ) \\ , \\dd s & \\leq C ( \\mathcal E ( 0 ) ) , \\\\ \\mathcal E _ { \\mathrm { B D } } ( t ) + \\int _ 0 ^ t \\mathcal D _ { \\mathrm { B D } } ( s ) \\ , \\dd s & \\leq C ' ( \\mathcal E ( 0 ) , \\mathcal E _ { \\mathrm { B D } } ( 0 ) ) , \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} A ^ { j _ 1 } _ { n _ 1 } \\Big | _ { x _ e = 0 } \\cdot \\frac { i } { x _ e } \\cdot A ^ { j _ 2 } _ { n _ 2 } \\Big | _ { x _ e = 0 } . \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} \\begin{array} { l @ { { } { } } l } H _ { L + 1 } & = c _ 1 H _ { L } + \\cdots + \\left ( c _ { L - 1 } + 1 \\right ) H _ 2 + \\left ( c _ { L } - 1 \\right ) H _ 1 \\\\ & = c _ 1 \\left ( G _ { L } + 1 \\right ) + \\cdots + \\left ( c _ { L - 1 } + 1 \\right ) G _ 2 + \\left ( c _ { L } - 1 \\right ) G _ 1 \\\\ & = c _ 1 + G _ 2 - G _ 1 + \\sum _ { i = 1 } ^ { L } c _ { i } G _ { L + 1 - i } \\\\ & = c _ 1 + c _ 1 + G _ { L + 1 } \\\\ & = 2 c _ 1 + G _ { L + 1 } . \\\\ \\end{array} \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} \\frac { | \\partial _ { R } ( v _ { c o r r } ( x , R \\lambda ( x ) ) ) | } { ( 1 + R ^ { 2 } ) \\lambda ( x ) ^ { 2 } } \\leq \\begin{cases} \\frac { C \\log ( \\log ( x ) ) } { x ^ { 2 } \\log ( x ) ( 1 + R ^ { 2 } ) } , R \\lambda ( x ) \\leq \\log ^ { N } ( x ) \\\\ \\frac { 1 } { x ^ { 2 } ( 1 + R ^ { 2 } ) } , \\log ^ { N } ( x ) \\leq R \\lambda ( x ) \\leq \\frac { x } { 2 } \\\\ \\frac { \\log ^ { b } ( x ) } { \\sqrt { x } } \\cdot \\frac { 1 } { 1 + R ^ { 2 } } , R \\lambda ( x ) > \\frac { x } { 2 } \\end{cases} \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} A _ { \\alpha , \\beta } = A _ \\beta ^ { - 1 } A _ { - \\alpha } , D _ { \\alpha , \\beta } = D _ \\alpha ^ { - 1 } D _ { - \\beta } , \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} \\ddot { \\Gamma } ( t ) + \\left ( \\eta \\operatorname { I d } + \\nu ( t ) \\otimes \\dot { \\nu } ( t ) \\right ) \\dot { \\Gamma } ( t ) = - \\operatorname { d i v } \\left ( \\nu ( t ) \\right ) \\nu ( t ) . \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} \\eta _ { \\nu _ { n } ^ { \\boxtimes k _ { n } } } ( z ) = \\eta _ { \\mu _ { n } } ( z ) \\left ( \\frac { \\eta _ { \\mu _ { n } } ( z ) } { z } \\right ) ^ { 1 / ( k _ { n } - 1 ) } , z \\in \\mathbb { \\overline { D } } , \\ ; n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} \\varepsilon > D _ p ( \\Pi _ { M \\cap N } ^ p x , x ) = \\frac { \\delta } { m ( y ) } D _ p ( \\Pi _ { M \\cap N } ^ p y , y ) , \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} f g = \\sum _ { x \\le y } \\left ( \\sum _ { x \\le z \\le y } f ( x , z ) g ( z , y ) \\right ) e _ { x y } , \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} \\partial Q = ( \\partial B _ R \\cap V ) \\oplus \\{ r e : 0 \\leq r \\leq R \\} \\cup ( \\bar { B } _ R \\cap V ) \\cup ( \\bar { B } _ R \\cap V ) \\oplus \\{ R e \\} . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ N w _ j g ( \\mathbf { x } _ j ) \\approx \\int _ { \\Omega } g \\omega \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} \\\\ \\sum _ { k = 1 } ^ n C _ f ^ n \\int \\limits _ { [ t , T ] } ^ { \\sim } \\int \\limits _ { [ s _ 1 , T ] } ^ { \\sim } \\cdots \\int \\limits _ { [ s _ { k - 1 } , T ] } ^ { \\sim } v _ { n - k } ( s _ k ) d s _ k \\cdots d s _ 1 & \\leq \\left ( \\dfrac { C _ 1 } { \\sqrt { M } } \\right ) ^ { n } e ^ { C _ f \\sqrt { M } ( T - t ) } \\sum _ { k = 1 } ^ n { \\dfrac { ( C _ f \\sqrt { M } ( T - t ) ) ^ k } { k ! C _ 1 ^ k } } \\\\ & \\leq \\left ( \\dfrac { C _ 1 } { \\sqrt { M } } \\right ) ^ { n } e ^ { C _ f \\sqrt { M } ( T - t ) \\left ( 1 + \\frac { 1 } { C _ 1 } \\right ) } \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} \\begin{aligned} & R _ N | _ { t = 0 } = R _ 0 , \\\\ & \\left [ \\int _ { \\mathbb T ^ d _ { \\ell } } R _ N U _ N \\cdot \\phi \\right ] | _ { t = 0 } = \\int _ { \\mathbb T ^ d _ { \\ell } } R _ 0 U _ 0 \\cdot \\phi , \\forall \\phi \\in ( X _ N ) ^ d . \\end{aligned} \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} \\int _ B | \\nabla \\Delta ^ k u | ^ 2 d x & \\Big ( \\frac { n - 2 } 2 \\Big ) ^ 2 \\Big ( \\prod _ { i = 0 } ^ { k - 2 } c _ { n , 2 + 4 i } \\Big ) \\int _ B \\frac { ( \\Delta u ) ^ 2 } { | x | ^ { 4 k - 2 } } d x . \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} { \\rm c o d i m } ( \\eta ) = B ' ( { \\bf m } ) - \\frac 1 2 { \\mathbf { k } ^ { \\top } A \\mathbf { k } } . \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} \\langle \\rho ( - p ) \\rho ( p ) \\rangle = \\langle \\phi ( - p ) \\phi ( p ) \\rangle + \\frac { i } { x _ p } \\sum _ { l = 1 } ^ \\infty \\left ( A ^ 1 _ { l + 2 } \\right ) ^ 2 \\cdot \\frac { 1 } { ( l + 1 ) ! } M ^ { ( l ) } ( p ) \\frac { i } { x _ p } . \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} \\partial _ t \\omega + u \\cdot \\nabla \\omega = \\partial _ 1 \\theta . \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} d Q ( [ \\chi , [ \\eta , \\zeta ] ] ; A ) + d Q ( [ \\eta , [ \\zeta , \\chi ] ] ; A ) + d Q ( [ \\zeta , [ \\chi , \\eta ] ] ; A ) = d Q \\big ( [ \\chi , [ \\eta , \\zeta ] ] + [ \\eta , [ \\zeta , \\chi ] ] + [ \\zeta , [ \\chi , \\eta ] ] ; A \\big ) = 0 , \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} c _ n \\alpha \\in I _ C = ( 1 / 2 , 1 / 2 + \\ell ) \\mbox { a n d } d _ n \\alpha \\in I _ D = ( 0 , \\ell ) \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} ( 0 , + \\infty ) = \\bigcup _ { t \\in G } - t + A . \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} W _ a ( f , g ) W _ a ( h , k ) + W _ a ( g , h ) W _ a ( f , k ) + W _ a ( h , f ) W _ a ( g , k ) = 0 , \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} \\Lambda ^ i _ { L , \\ , R } ( f _ 1 ^ i , \\ , \\dots , \\ , f _ { L - 1 } ^ i ) = \\sum _ { j = 1 } ^ { L - 1 } \\ , P _ { L , \\ , j } \\ , f _ j ^ i \\circ R , i = x , \\ , y , \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} ( T x ) _ k = \\sum \\limits _ { m \\in \\mathbb Z ^ c } b _ { k m } x _ { k - m } , k \\in \\mathbb Z ^ c , \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} F _ 0 ( s ) : = \\displaystyle \\int _ { 0 } ^ { s } f _ 0 ( t ) \\ , d t \\hat { F } _ 0 ( s ) : = f _ 0 ( s ) s - 2 F _ 0 ( s ) = \\hat { F } ( s ) s \\in \\mathbb { R } , \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} \\| \\widehat { A } f \\| _ { L ^ 2 ( \\mathbb { R } ) } = \\| \\widehat { B } f \\| _ { L ^ 2 ( \\mathbb { R } ) } \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} y _ l = \\frac { 1 } { K _ l } \\sum _ { i \\in B ( l ) } x _ i l \\in \\{ 1 , \\cdots , M \\} . \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} \\theta ( h ) = g h g ^ { - 1 } h \\in \\Sigma _ 1 . \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} S ( t , g , h ) : = \\int _ 0 ^ t V ^ { t - s } B ( s , g , h ) d s , t \\geq 0 . \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} \\tau ( \\mathfrak { X } ) ( g ) = ( \\mathfrak { X } ( g ) ) ( 1 _ G ) = ( g ^ { - 1 } ( \\mathfrak { X } ( 1 _ G ) ) ) ( 1 _ G ) = ( \\mathfrak { X } ( 1 _ G ) ) ( g ) . \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} \\Delta _ 1 = ( e ^ { q _ 2 } - e ^ { q _ 1 } ) ( e ^ { q _ 3 } - e ^ { q _ 2 } ) ( e ^ { q _ 2 } + e ^ { q _ 1 } ) ^ 2 ( e ^ { q _ 3 } + e ^ { q _ 2 } ) ^ 2 . \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} S : = F ( \\beta \\N \\setminus \\N ) \\subset \\mathfrak M ( H ^ \\infty ( V ) ) \\setminus V . \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} v _ { 2 } ( t , r ) = - \\frac { c _ { b } r } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\sin ^ { 2 } ( \\theta ) d \\theta \\int _ { 0 } ^ { \\infty } d \\xi \\partial _ { \\xi } ^ { 2 } \\left ( \\frac { \\xi \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\right ) \\left ( \\frac { \\sin ( \\xi t _ { + } ) } { t _ { + } ^ { 2 } } + \\frac { \\sin ( \\xi t _ { - } ) } { t _ { - } ^ { 2 } } \\right ) \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} v _ { 2 } ( t , r ) = \\frac { - c _ { b } r } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\sin ^ { 2 } ( \\theta ) d \\theta \\int _ { 0 } ^ { \\infty } d \\xi \\left ( \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) \\left ( \\frac { b - 1 } { \\xi \\log ^ { b } ( \\frac { 1 } { \\xi } ) } + \\frac { b ( b - 1 ) } { \\xi \\log ^ { b + 1 } ( \\frac { 1 } { \\xi } ) } \\right ) + \\psi ( \\xi ) \\right ) \\left ( \\frac { \\sin ( \\xi t _ { + } ) } { t _ { + } ^ { 2 } } + \\frac { \\sin ( \\xi t _ { - } ) } { t _ { - } ^ { 2 } } \\right ) \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} \\ell = \\limsup _ { r \\to 0 } \\frac { G ( r ) } { r } , \\ ; M = \\sup _ { t \\ge 0 } \\int _ 0 ^ t \\omega ( t - \\tau , \\lambda _ 1 ) p ( \\tau ) d \\tau . \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} T _ { k + 1 } = T _ k \\cup \\{ d _ k \\} \\cup \\{ \\psi _ { k + 1 } ( s ) : s \\in E _ k \\} \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} I ( u ) = \\dfrac { 1 } { 2 } \\left [ r ^ 2 - R ^ 2 \\right ] - \\int _ { \\mathbb { R } ^ N } F _ 0 ( u ) \\ ; d x \\leq \\dfrac { 1 } { 2 } \\left [ r ^ 2 - R ^ 2 \\right ] \\leq 0 . \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} \\begin{aligned} & x _ { n _ i } = r _ i \\cos \\phi ^ { i } _ { d _ i - 1 } \\\\ & x _ { n _ i - 1 } = r _ i \\sin \\phi ^ { i } _ { d _ i - 1 } \\cos \\phi ^ { i } _ { d _ i - 2 } \\\\ & \\cdots \\cdots \\\\ & x _ { n _ { i - 1 } + 1 } = r _ i \\sin \\phi ^ { i } _ { d _ i - 1 } \\cdots \\sin \\phi ^ { i } _ 2 \\sin \\phi ^ { i } _ 1 . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} \\frac { \\mathbb { E } _ \\theta [ f ( \\textbf { X } ) ] } { \\mathbb { E } _ \\theta [ h ( \\textbf { X } ) + w ( \\theta ) ^ \\top f ( \\textbf { X } ) ] } - \\frac { \\bar { f } } { \\overline { h } + w ( \\theta ) ^ \\top \\bar { f } } = 0 . \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } R ( t ) & \\leq \\lim _ { t \\to \\infty } \\left ( \\frac { t } { \\sigma ( t ) } \\right ) ^ \\beta R ( t _ { \\sigma ( t ) } ) + c ( \\sigma ( t ) ) t _ { \\sigma ( t ) + 1 } ^ { \\beta } = 0 , \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} - 2 \\beta ^ { 3 / 2 } h ' ( \\beta ) = \\sum _ { k = 1 } ^ \\infty \\frac { k ^ { k - 2 } } { k ! } \\Bigg [ \\int _ 0 ^ \\beta x ^ { k - 3 / 2 } e ^ { - k x } \\dd x - 2 \\beta ^ { k - 1 / 2 } e ^ { - k \\beta } \\Bigg ] . \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} \\varphi ( g _ 0 , g _ 1 , g _ 2 ) : = { \\rm A r e a } ( \\Delta ( g _ 0 \\cdot i , g _ 1 \\cdot i , g _ 2 \\cdot i ) ) , \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta u - n u = 0 , & ; \\\\ u = f , & , \\end{array} \\right . \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} \\epsilon ^ { - 1 } \\left ( f ^ n ( u _ { h , 1 } ^ n ) - f ^ n ( u _ { h , 2 } ^ n ) , U _ h ^ n \\right ) _ { \\mathcal T _ h } & = \\epsilon ^ { - 1 } \\left ( ( u _ { h , 1 } ^ n ) ^ 3 - ( u _ { h , 2 } ^ n ) ^ 3 , U _ h ^ n \\right ) _ { \\mathcal T _ h } \\\\ & = \\epsilon ^ { - 1 } \\left ( ( u _ { h , 1 } ^ n ) ^ 2 + u _ { h , 1 } ^ n u _ { h , 2 } ^ n + ( u _ { h , 2 } ^ n ) ^ 2 , ( U _ h ^ n ) ^ 2 \\right ) _ { \\mathcal T _ h } \\\\ & \\geq 0 . \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} & Q _ 1 ( x _ 1 , x _ 2 ) = M _ L ^ { ( 1 ) } ( x _ 1 , x _ 2 ) \\cdot P _ 1 ( x _ 1 , x _ 2 ) \\cdot M _ R ^ { ( 1 ) } ( x _ 1 , x _ 2 ) , \\\\ & Q _ 2 ( x _ 1 , x _ 2 ) = M _ L ^ { ( 2 ) } ( x _ 1 , x _ 2 ) \\cdot P _ 2 ( x _ 1 , x _ 2 ) \\cdot M _ R ^ { ( 2 ) } ( x _ 1 , x _ 2 ) . \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} \\hat { H } _ { c o u l } = - \\sum _ { i = 1 } ^ { D } \\frac { \\partial ^ 2 } { \\partial x _ i ^ 2 } - \\frac { \\eta } { r } + \\frac { \\alpha _ 1 } { x _ 1 ^ 2 + \\cdots + x ^ 2 _ { n _ 1 } } + \\cdots + \\frac { \\alpha _ { N - 1 } } { x ^ 2 _ { n _ { N - 2 } + 1 } + \\cdots + x ^ 2 _ { n _ { N - 1 } } } , \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} \\begin{aligned} \\omega _ 6 & = w ( t _ 1 w + t _ 2 y ) d x + w ( t _ 0 w + t _ 3 x ) d y - ( t _ 0 y w + t _ 1 x w + ( t _ 2 + t _ 3 ) x y ) d w , \\\\ \\omega _ 7 & = w ( t _ 1 w + t _ 2 z ) d y + w ( t _ 0 w - t _ 2 y + t _ 3 z ) d z - ( t _ 0 z w + t _ 1 y w + t _ 3 z ^ 2 ) d w . \\end{aligned} \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} \\| \\overline { V } ( t ) \\| _ { H ^ r } \\leq W ( t ) \\leq W ( 0 ) ^ { e ^ { C _ r t } } = ( \\| \\overline { V } _ 0 \\| _ { H ^ r } + e ) ^ { e ^ { C _ r t } } \\leq ( M + e ) ^ { e ^ { C _ r t } } = : \\theta _ { M , r } ( t ) . \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} \\gamma ( \\mathcal { V } ) : = K ^ - P _ { \\mathcal { V } } + K ^ + ( I _ n - P _ { \\mathcal { V } } ) , \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} T _ H ( f ) ( x H ) = \\int _ H f ( x h ) d h . \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} & | \\partial _ { \\xi } \\left ( \\psi _ { v _ { 2 } } ( \\xi , \\lambda _ { 1 } ( t ) ) - \\psi _ { v _ { 2 } } ( \\xi , \\lambda _ { 2 } ( t ) ) \\right ) | \\leq \\sup _ { x \\in [ \\frac { \\lambda _ { 0 } ( t ) } { 2 } , \\frac { 1 } { 2 } ] } | \\partial _ { 1 2 } \\psi _ { v _ { 2 } } ( \\xi , x ) | | \\lambda _ { 1 } ( t ) - \\lambda _ { 2 } ( t ) | \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\pi _ { i , N } ( 1 - \\pi _ { i , N } ) & = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\pi _ { i , N } - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\pi _ { i , N } ^ { 2 } \\\\ & \\overset { P ( a s ) } { \\rightarrow } \\frac { \\theta E w _ { \\theta } ( X _ { 1 } ) } { E w ( X _ { 1 } ) } - \\frac { \\theta ^ { 2 } E w _ { \\theta } ^ { 2 } ( X _ { 1 } ) } { [ E w ( X _ { 1 } ) ] ^ { 2 } } . \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} P = x _ { 4 } ^ { 3 } - x _ { 4 } ( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } ) + 2 \\sqrt { 5 } x _ { 1 } x _ { 2 } x _ { 3 } . \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} \\int _ { z _ 1 } ^ { z _ 2 } u h ^ n d z = \\frac { V ' ( s ) } { ( n + 1 ) a _ n H ' ( s ) } , \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ z ( - 1 ) ^ t \\binom { x } { t } \\binom { y - t + 1 } { z - t } = \\sum _ { t = 0 } ^ z ( - 1 ) ^ t \\binom { x } { t } \\binom { y - t } { z - t } + \\sum _ { t = 0 } ^ { z - 1 } ( - 1 ) ^ t \\binom { x } { t } \\binom { y - t } { z - t - 1 } . \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} & | 2 c _ { b } ( \\lambda _ { 0 } ( t ) + e _ { 2 } ( t ) ) \\int _ { 0 } ^ { \\infty } d \\xi \\frac { \\cos ( t \\xi ) } { t ^ { 3 } } \\left ( \\partial _ { \\xi } \\psi _ { v _ { 2 } } ( \\xi , \\lambda _ { 1 } ( t ) ) - \\partial _ { \\xi } \\psi _ { v _ { 2 } } ( \\xi , \\lambda _ { 2 } ( t ) ) \\right ) | \\\\ & \\leq \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 3 } \\sqrt { \\log ( \\log ( t ) ) } } \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} & { 1 \\over | R | } \\int _ R | b ( x _ 1 , x _ 2 , x _ 3 ) - b _ R | d x _ 1 d x _ 2 d x _ 3 \\\\ & = { 1 \\over a ^ 4 } \\int _ { a } ^ { a + a ^ 2 } \\int _ { a } ^ { 2 a } \\int _ { a } ^ { 2 a } \\bigg | x _ 1 - { 3 a \\over 2 } \\bigg | \\ , d x _ 1 d x _ 2 d x _ 3 \\\\ & = { a \\over 4 } . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} B _ { G , \\ ; L + m } - B _ { G , \\ ; L + m - 1 } = 2 G _ { L + m - 1 } - G _ { L + m } . \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} \\forall l \\in I ^ { \\tilde G \\tilde H } ( \\bar x ) \\colon \\tilde \\mu _ l \\tilde \\nu _ l = 0 , \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} q ( v ) = \\begin{cases} - v , & v \\le \\frac { a } { 2 } , \\\\ v - a , & \\frac { a } { 2 } \\le v \\le \\frac { a + 1 } { 2 } , \\\\ 1 - v , & v \\ge \\frac { a + 1 } { 2 } . \\end{cases} \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} H _ 1 ( \\tilde { \\xi } ) = \\tilde { \\xi } ^ 3 + \\frac { 1 } { 2 } ( 1 - 3 \\sqrt { 2 } + 2 \\sqrt { 3 } & + \\sqrt { 6 } ) \\tilde { \\xi } ^ 2 + \\frac { \\sqrt { 3 } } { 4 } ( 8 + 4 \\sqrt { 2 } - 5 \\sqrt { 3 } - 4 \\sqrt { 6 } ) \\tilde { \\xi } \\\\ & + \\frac { 3 \\sqrt { 3 } } { 8 } ( \\sqrt { 2 } + 1 ) ( \\sqrt { 3 } - \\sqrt { 2 } ) ( 3 + 2 \\sqrt { 2 } - 2 \\sqrt { 3 } ) , \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} \\Omega _ h & : = \\{ x \\in \\Omega \\ , : \\ , \\mathrm { d i s t } ( x , \\partial \\Omega ) > h \\} , \\\\ K _ { \\epsilon , h } & : = \\{ k \\in \\Z ^ n \\ , : \\ , \\epsilon ( Y + k ) \\subset \\Omega _ h \\} , \\\\ \\Omega _ { \\epsilon , h } & : = \\mathrm { i n t } \\bigcup _ { k \\in K _ { \\epsilon , h } } \\epsilon \\big ( \\overline { Y } + k \\big ) , \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} \\psi _ t ( \\hat { x } , \\hat { t } ) \\le \\psi _ x ( \\hat { x } , \\hat { t } ) + f ( \\hat { x } , \\hat { t } ) \\quad \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} \\sqrt { n } \\sum _ { k = 1 } ^ { \\log n } \\frac { k ^ { k - 2 } } { k ! } \\int _ { 1 0 0 0 \\log n } ^ \\infty x ^ { k - 1 } e ^ { - k x } \\frac { \\dd x } { \\sqrt { x } } & \\leq \\sqrt { n } \\sum _ { k = 1 } ^ { \\log n } \\frac { k ^ { k - 2 } } { k ! } \\int _ { 1 0 0 0 \\log n } ^ \\infty x ^ { k - 1 } e ^ { - k x } \\dd x \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} \\begin{aligned} f _ L ( 0 , \\xi _ L ( \\mu ) ; \\mu ) & = 0 , & f _ R ( 0 , \\xi _ R ( \\mu ) ; \\mu ) & = 0 , \\\\ g _ L ( 0 , \\xi _ L ( \\mu ) ; \\mu ) & = 0 , & g _ R ( 0 , \\xi _ R ( \\mu ) ; \\mu ) & = 0 , \\end{aligned} \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{align*} H _ { 2 \\alpha } ( r , \\phi , r ' , \\phi ' , t ) = \\frac 1 2 \\left ( \\widetilde H _ D + \\widetilde H _ N \\right ) . \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} \\begin{cases} ( u _ \\varepsilon ) _ t = ( D _ x ^ \\alpha u _ \\varepsilon ) _ x + f _ \\varepsilon \\quad & \\\\ u _ \\varepsilon = g _ \\varepsilon \\quad & \\end{cases} \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} \\partial _ p F ^ { g ' } ( x , z , p ) & = : ( a ^ { i j } ( x , z , p ) p _ i + b ^ j ( x , z , p ) \\big ) \\partial _ j \\\\ \\partial _ z F ^ { g ' } ( x , z , p ) & = : - \\big ( | A _ { \\Sigma } | ^ 2 + R i c _ M ( \\nu , \\nu ) \\big ) z + \\frac { 1 } { 2 } t r _ { \\Sigma } \\nabla ^ M _ { \\nu } \\beta + c ( x , z , p ) \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} g = \\dd x ^ 2 + x ^ 2 k ( x , y , \\dd y ) , \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} \\mathbf { P } \\left ( ( \\eta ^ N _ n ) _ { n = 1 } ^ { N } = ( x _ n ) _ { n = 1 } ^ { N } \\right ) = \\frac { 1 } { Z } \\exp \\left ( - \\sum _ { k = 0 } ^ \\infty \\beta _ k f _ k \\left ( ( x _ n ) _ { n = 1 } ^ { N } \\right ) \\right ) \\mathbf { 1 } _ { \\{ f _ 0 \\left ( ( x _ n ) _ { n = 1 } ^ { N } \\right ) < N / 2 \\} } , \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} \\hat { F } \\circ \\hat { G } ( S ) = & \\bigcap _ { 0 \\in U \\in \\hat { G } ( S ) } { \\rm c l } _ { T ^ { \\max } ( \\hat { X } ) } [ I ( U ) ] \\\\ = & \\bigcap _ { 0 \\in V \\in \\hat { T } } { \\rm c l } _ { T ^ { \\max } ( \\hat { X } ) } [ I ( X ) \\cap V ] . \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} \\alpha = a _ { 0 L } b _ { 2 R } + a _ { 2 L } b _ { 0 R } - a _ { 0 R } b _ { 2 L } - a _ { 2 R } b _ { 0 L } \\ , . \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 p } { \\partial t ^ 2 } - c _ 1 c _ 2 \\frac { \\partial ^ 2 p } { \\partial x ^ 2 } + ( c _ 1 - c _ 2 ) \\frac { \\partial ^ 2 p } { \\partial x \\partial t } = - 2 \\lambda \\frac { \\partial p } { \\partial t } + \\lambda ( c _ 2 - c _ 1 ) \\frac { \\partial p } { \\partial x } \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} \\kappa _ L ( r ) : = \\left ( l \\left ( r ^ { - 1 } \\right ) \\right ) ^ { - 1 } , \\kappa _ U ( r ) : = l ( r ) . \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} x = y \\vert _ { M ^ { i + j } } = p _ { i j k } ( y \\vert _ { M ^ { k + j } } ) \\subset p _ { i j k } ( A _ { k j } ) . \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} \\hat { E } \\dot { \\hat { v } } ( t ) & = \\hat { f } ( \\hat { v } ( t ) ) \\\\ [ 1 e x ] \\hat { w } ( t ) & = \\hat { g } ( \\hat { v } ( t ) ) \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} & \\partial _ { t } \\left ( \\int _ { 0 } ^ { \\frac { 1 } { 2 } } d \\xi \\left ( \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) - 1 \\right ) \\frac { \\sin ( t \\xi ) } { t ^ { 2 } } \\left ( \\frac { ( b - 1 ) } { \\xi \\log ^ { b } ( \\frac { 1 } { \\xi } ) } + \\frac { b ( b - 1 ) } { \\xi \\log ^ { b + 1 } ( \\frac { 1 } { \\xi } ) } \\right ) \\right ) \\\\ & = \\frac { - \\sin ( \\frac { t } { 2 } ) } { t ^ { 3 } } \\left ( \\frac { b - 1 } { \\log ^ { b } ( 2 ) } + \\frac { b ( b - 1 ) } { \\log ^ { b + 1 } ( 2 ) } \\right ) + \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} \\frac { \\beta _ { \\zeta } ( j t + k ) } { \\beta _ { \\zeta } ( j t ) } = \\lim _ { q \\to \\zeta } \\frac { \\beta _ { q } ( j t + k ) } { \\beta _ { q } ( j t ) } = \\beta _ { \\zeta } ( k ) , \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} & F ( \\Phi ( - x - \\underline h ( t ) ) ) = F ( \\Phi ( - x - \\underline h ( t ) ) ) - F ( \\mathbf { u } ^ * ) \\preceq L \\frac { C } { h ( t ) ^ \\alpha } \\mathbf { 1 } \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} T _ H ^ * ( \\beta ) = \\beta _ q , \\ \\forall \\beta \\in \\mathcal { C } _ 0 ( G / H ) ^ * , \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} d \\alpha \\wedge \\alpha = 0 . \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} ( A _ i \\phi , \\psi ) _ { H _ i } = ( ( \\phi , \\psi ) ) , \\phi , \\psi \\in D ( A _ i ) . \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} A _ j ^ \\dag ( \\varphi _ i ) & = P ( \\varphi _ i \\theta _ j ) = \\varphi _ i \\theta _ j = w ( i ) ^ { - 1 / 2 } \\ , \\theta _ i \\ , \\theta _ j = w ( i ) ^ { - 1 / 2 } \\ , \\theta _ { ( i , j ) } \\\\ & = \\left ( \\dfrac { w ( i , j ) } { w ( i ) } \\right ) ^ { 1 / 2 } \\ ! \\ ! \\varphi _ { ( i , j ) } . \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} j _ p ( x ) = D \\left ( \\frac { 1 } { p } \\| \\cdot \\| ^ { p } \\right ) ( x ) \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} N \\prod _ { j = 1 } ^ p ( \\beta _ j - \\alpha _ k ) ^ { m _ j } = n _ k \\prod _ { i \\neq k } ( \\alpha _ i - \\alpha _ k ) ( 1 \\le k \\le s ) . \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} \\bold { m } _ { i } \\left ( \\sum _ { l = j } ^ { n } m _ { j , l } C _ { l } \\right ) = \\left ( Y ^ { ( 1 ) } _ { i , j } , \\ldots , Y ^ { ( n ) } _ { i , j } \\right ) M _ { \\underline { a } , \\underline { b } } \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} B _ { G , L + m + 1 } - B _ { G , L + m } = 2 G _ { L + m } - G _ { L + m + 1 } . \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} | \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) | & \\leq \\begin{cases} 2 , \\rho \\leq 4 r \\\\ | \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + 1 | + | - 1 + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) | , \\rho > 4 r \\end{cases} \\\\ & \\leq \\begin{cases} 2 , \\rho \\leq 4 r \\\\ \\frac { r ^ { 2 } } { \\rho ^ { 2 } } , \\rho > 4 r \\end{cases} \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} \\begin{cases} h ( t ) \\leq C t ^ { 1 / ( { \\gamma } - 1 ) } & { \\rm i f } \\ { \\gamma } \\in ( 1 , 2 ) , \\\\ h ( t ) \\leq C t \\ln t & { \\rm i f } \\ { \\gamma } = 2 . \\end{cases} \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} G ( s , r , \\rho ) & = \\int _ { 0 } ^ { 2 \\pi } d \\theta \\frac { v _ { 4 , c } ( s , \\sqrt { r ^ { 2 } + 2 r \\rho \\cos ( \\theta ) + \\rho ^ { 2 } } ) } { \\sqrt { r ^ { 2 } + 2 r \\rho \\cos ( \\theta ) + \\rho ^ { 2 } } } \\left ( r + \\rho \\cos ( \\theta ) \\right ) \\\\ & s \\geq t , r \\geq 0 , s - t \\geq \\rho \\geq 0 \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} F ^ { \\leq } \\circ K - K \\circ R = 0 [ 0 , \\rho ) . \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} | \\dot { \\gamma } | ( t ) = \\lim _ { h \\rightarrow 0 } \\frac { d ( \\gamma ( t + h ) , \\gamma ( t ) ) } { | h | } . \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} \\begin{cases} r _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( x , 0 ) = \\delta ( x ) \\\\ \\lim _ { | x | \\rightarrow \\infty } r _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( x , t ) = 0 , \\\\ \\lim _ { | x | \\rightarrow \\infty } \\frac { \\partial } { \\partial x } r _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( x , t ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} D = \\{ d _ n \\} _ { n \\in \\N } \\mbox { w i t h } d _ n = s _ n ^ 2 \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} \\nabla ^ T _ { E _ j } x ^ T = E _ j + A ^ { x ^ { \\perp } } ( E _ j , E _ k ) \\ , E _ k \\ , . \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} C _ G ( a ) & = \\left \\{ y \\in G \\mid \\pi _ i \\left ( y ^ { - 1 } a y \\right ) = a _ i \\ \\forall i = 1 , \\dots , d \\right \\} \\\\ & = \\left \\{ y \\in G \\mid h _ i ( y ) = 0 \\ \\forall i = 1 , \\dots , d \\right \\} , \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} \\begin{cases} D ^ { 1 - \\epsilon } _ 0 \\tilde { y } ( t ) = \\tilde { z } ( t ) , & \\tilde { y } ( 0 ) = \\frac { \\gamma _ 1 - b _ 1 s } { a _ 1 } , \\\\ D ^ { \\alpha _ 2 - 1 } _ 0 \\tilde { z } ( t ) = f ( t , \\tilde { y } ( t ) , \\tilde { z } ( t ) ) . & \\tilde { z } ( 0 ) = s . \\end{cases} \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} \\alpha ^ 1 _ { d _ 1 - 1 } = \\bigg [ 2 \\sum _ { a = 1 } ^ { d _ 1 - 1 } J _ a + \\frac { d _ 1 - 2 } { 2 } + A + J _ 1 \\bigg ] ^ 2 - \\frac { ( d _ 1 - 2 ) ^ 2 } { 4 } , ~ ~ ~ J _ a = 0 , 1 , 2 , \\cdots , \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} X = \\partial _ 1 + \\cdots + \\partial _ n \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} T ^ 4 _ 0 = - T ^ 2 _ 0 . \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} \\mathcal { N } ( \\mathcal { X } ) : = \\mathcal { \\Delta } ( \\mathbf { x } ) - \\Phi ^ { 2 } . \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} \\{ ( x - x _ 0 ) \\times \\omega \\} \\cdot n _ x = 0 , \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} & \\max _ { \\bar { \\boldsymbol { \\theta } } } \\bar { \\boldsymbol { \\theta } } ^ { \\rm H } ( \\boldsymbol { R } + \\boldsymbol { V } ) \\bar { \\boldsymbol { \\theta } } \\\\ & ~ ~ \\textrm { s . t . } \\quad ~ | \\theta _ n | = 1 , \\forall n = 1 , \\ldots , N , \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t / 2 } \\frac { \\sin ( u ) } { u \\log ^ { a } ( t / u ) } d u = \\lim _ { \\epsilon \\rightarrow 0 } \\left ( \\int _ { \\epsilon } ^ { t / 2 } d y \\frac { e ^ { - y } } { y \\log ^ { a } ( \\frac { t } { i y } ) } \\right ) - \\left ( \\int _ { 0 } ^ { \\frac { \\pi } { 2 } } i d \\theta \\frac { e ^ { \\frac { i t } { 2 } \\cos ( \\theta ) } e ^ { - \\frac { t } { 2 } \\sin ( \\theta ) } } { \\log ^ { a } ( 2 e ^ { - i \\theta } ) } \\right ) \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} D _ { \\mathcal { H } , \\pm } \\big ( a . _ { \\mathcal { H } _ { - b } } \\Psi \\bigr ) = a . _ { \\mathcal { H } _ { - b } } \\bigl [ D _ { \\mathcal { H } , \\pm } \\big ( \\Psi \\bigr ) \\bigr ] + b \\cdot \\left ( \\mathsf { M } ^ { \\mathbb { Z } } \\mp \\frac { 1 } { 2 \\pi } \\frac { \\partial \\ ; } { \\partial \\Theta } \\right ) ( a ) . _ { \\mathcal { H } _ { - b } } \\Psi . \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} P \\big [ \\sigma _ { \\{ R \\} } ^ { z } < \\sigma ^ { z } _ { \\{ 0 \\} } \\big ] = g _ { \\{ 0 \\} } ( z , R ) / g _ { \\{ 0 \\} } ( R , R ) \\leq C ' a ( z ) / R , \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} \\nu _ k ( f ) : = \\left ( \\int _ G | f ( g ) | ^ 2 ( 1 + L ( g ) ) ^ { 2 k } d g \\right ) ^ { \\frac { 1 } { 2 } } \\ , . \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} w _ { 0 , a , b } ( i ) = \\begin{cases} i + b & \\mbox { i f } 1 \\leqslant i \\leqslant a , \\\\ i - a & \\mbox { i f } a < i \\leqslant n . \\end{cases} \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} N \\leq \\sum _ { i = m + 3 } ^ { L - 1 } H _ i \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } ( x _ \\varepsilon , t _ \\varepsilon , y _ \\varepsilon , s _ \\varepsilon , u ^ * ( x _ \\varepsilon , t _ \\varepsilon ) , v _ * ( y _ \\varepsilon , s _ \\varepsilon ) ) = ( \\hat { x } , \\hat { t } , \\hat { x } , \\hat { t } , u ^ * ( \\hat { x } , \\hat { t } ) , v _ * ( \\hat { x } , \\hat { t } ) ) , \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 f } { \\partial t _ 3 \\partial t _ 2 } ( 0 , t _ 1 , 0 , t _ 1 ) = - ( z ( 0 ) - z ( t _ 1 ) ) ^ { - 1 } z ' ( t _ 1 ) ( z ( 0 ) - z ( t _ 1 ) ) ^ { - 1 } z ' ( 0 ) , \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} \\log ^ + | x | = \\left \\{ \\begin{array} { l l } \\log x & { \\rm i f } \\ ; \\ ; x \\geq 1 \\\\ 0 & { \\rm i f } \\ ; \\ ; x < 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } v _ { 5 } ( t , r ) | & \\leq \\frac { C } { t ^ { 4 } \\log ^ { 3 N + b - 2 } ( t ) } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} K _ X ^ 2 = p ^ { \\epsilon ( X / k ) } ( K _ Z + D ) ^ 2 . \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} \\mathcal { X } _ { n } = \\ , & \\{ f : S \\rightarrow \\C \\ | \\ f \\in \\textrm { H o l } ( S ) , \\ ; f ( ( 0 , \\rho ) ) \\subset \\R , \\ ; \\| f \\| _ n : = \\sup _ { z \\in S } \\frac { | f ( z ) | } { | z | ^ n } < \\infty \\} , \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } \\| \\Delta _ q \\omega ( t ) \\| _ { L ^ 2 } ^ 2 + \\| \\partial _ 1 \\Delta _ q \\omega \\| _ { L ^ 2 } ^ 2 = \\int _ { \\R ^ 2 } \\partial _ 1 \\Delta _ q \\theta \\Delta _ q \\omega ~ d x - \\int _ { \\R ^ 2 } \\Delta _ q ( u \\cdot \\nabla \\omega ) \\Delta _ q \\omega ~ d x . \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{gather*} \\lVert v \\rVert _ \\pi = \\inf \\Bigl \\{ \\sum \\limits _ { k = 1 } ^ n \\| e _ k \\| \\cdot \\| x _ k \\| : \\ , v = \\sum \\limits _ { k = 1 } ^ n e _ k \\otimes x _ k \\Bigr \\} , \\end{gather*}"}
-{"id": "2622.png", "formula": "\\begin{align*} P ( z , \\overline { z } ) = \\frac { \\varphi _ { p } ( z , \\overline { z } ) - \\overline { \\varphi _ { p } ( z , \\overline { z } ) } } { 2 \\sqrt { - 1 } } , \\quad \\mbox { f o r g i v e n $ p \\in \\mathbb { N } ^ { \\star } $ . } \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} ( 1 + z ^ { - 1 } \\sum \\limits _ { n , l = 0 } ^ { \\infty } z ^ { - n } q \\bar { x } ^ { n } x ^ { l } p ) ^ { - 1 } = ( 1 - z ^ { - 1 } \\sum \\limits _ { n , l = 0 } ^ { \\infty } z ^ { - n } q x ^ { l } \\bar { x } ^ { n } p ) ^ { - 1 } . \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} C = \\{ c _ n \\} _ { n \\in \\N } \\mbox { w i t h } c _ n = 2 n s _ n + n ^ 2 \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} \\mathcal R _ A ( v , \\epsilon ) = \\{ n \\in \\N \\ , | \\ , \\mu ( A \\cap T ^ { - v ( n ) } A ) > \\mu ^ 2 ( A ) - \\epsilon \\} \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta u & = u ^ { 2 ^ \\star + | x | ^ \\alpha - 1 } & & B , \\\\ u & > 0 & & B , \\\\ u & = 0 & & \\partial B , \\end{aligned} \\right . \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} \\langle j _ p ( \\Pi _ M ^ p x _ n ) - j _ p ( x _ n ) , z \\rangle = 0 \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} \\begin{array} { l l l } H _ { m _ s } ( n _ 1 , \\ldots , n _ r ) & = & \\frac { 1 } { m _ s ^ d } \\left ( \\lim _ { m \\rightarrow \\infty } \\frac { \\ell _ R ( R / I ( m n _ 1 F _ { m _ s , 1 } ) \\cdots I ( m n _ r F _ { m _ s , r } ) ) } { m ^ d } \\right ) \\\\ & = & \\frac { - 1 } { m _ s ^ d d ! } ( ( - n _ 1 F _ { m _ s , 1 } - \\cdots - n _ r F _ { m _ s , r } ) ^ d ) \\end{array} \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} \\gamma \\cdot Z = - \\bar { Z } , \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\to + \\infty } \\int _ { \\mathbb { R } ^ N } \\left ( | \\nabla v _ n ( x ) | ^ 2 + V _ { 0 , \\infty } v ^ 2 _ n ( x ) \\right ) d x = \\lim _ { n \\to + \\infty } \\int _ { \\mathbb { R } ^ N } \\left ( | \\nabla v _ n ( x ) | ^ 2 + V _ 0 ( x ) v ^ 2 _ n ( x ) \\right ) d x . \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} \\| \\chi _ { \\{ x \\in Q : | f ( x ) - f _ Q | \\geq t \\} } \\| _ { L ^ { p ( \\cdot ) } \\left ( Q , \\frac { \\d x } { | Q | } \\right ) } \\leq 1 / 2 ^ r = e ^ { - C ( n , p ^ + ) t / \\| f \\| _ { \\mathrm { B M O } } } , \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} \\beta & = a _ 3 b _ 2 - a _ 2 b _ 3 \\ , , \\\\ \\gamma & = a _ 4 b _ 2 - a _ 2 b _ 4 \\ , . \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} \\int _ B | \\nabla ^ m u _ j | ^ 2 d x & = 2 c + \\int _ B \\frac { 2 } { \\ 2 m s + r ^ \\alpha } | u _ j | ^ { \\ 2 m s + | x | ^ \\alpha } d x + o ( 1 ) _ { j \\nearrow + \\infty } \\\\ & \\leq 2 c + \\frac { 2 } { \\ 2 m s } \\int _ B ( u _ j ) _ + ^ { \\ 2 m s + | x | ^ \\alpha } d x + o ( 1 ) _ { j \\nearrow + \\infty } . \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} B _ Y \\equiv B _ { Y , \\eta } : = \\sup \\left \\{ N \\in \\mathbb { Z } : \\ : \\min \\{ J , K \\} \\le \\eta _ { n } + Y _ { n - 1 } \\le \\max \\{ J , K \\} , \\ : \\forall n \\le N \\right \\} , \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} \\begin{array} { c } V ( x ) = - ( \\eta x - a ) ^ 2 - F ^ 2 \\mu ^ 4 x ^ 4 - 2 F \\mu ^ 2 ( \\eta x - a ) x ^ 2 + \\\\ \\\\ + 4 8 \\exp \\left ( - 2 \\sqrt { 3 } x - 2 \\sqrt { 3 } \\mu ^ 2 E x ^ 2 + \\frac { \\sqrt { 3 } \\theta a } { \\mu \\lambda } \\right ) , \\end{array} \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} b = & \\frac { c _ { Q , n } } { u _ Q ^ n } + \\ldots + \\frac { c _ { Q , 2 } } { u _ Q ^ 2 } + \\frac { c _ { Q , 1 } } { u _ Q } + \\tilde { f } \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} \\zeta _ { L , p } ^ { * } ( s ) = \\sum _ { I \\subseteq T ^ { * } } c _ { p , \\ , I } ^ { * } P _ { I } ^ { * } ( p , \\ , p ^ { - s } ) , \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} N ( z ) = u _ { \\sigma } \\left ( \\frac { z } { 1 - z } \\right ) = \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t z } { z - t } \\ , d \\sigma ( t ) , z \\in \\mathbb { C } \\setminus \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\hat { V } _ { \\rm c y l } & = \\frac { v _ { n + 1 } r ^ { n + 1 } L } { \\frac 1 2 v _ { n + 2 } L ^ { n + 2 } } = \\frac { 2 v _ { n + 1 } } { v _ { n + 2 } } \\left ( \\frac { r } { L } \\right ) ^ { n + 1 } , \\ ; \\ ; \\ ; ( 0 \\leq r < + \\infty ) , \\\\ \\hat { A } _ { \\rm c y l } & = \\frac { a _ n r ^ n L } { \\frac 1 2 a _ { n + 1 } R ^ { n + 1 } } = \\frac { 2 a _ n } { a _ { n + 1 } } \\left ( \\frac { r } { L } \\right ) ^ n \\left ( \\frac { L } { R } \\right ) ^ { n + 1 } , \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} \\hat { H } = - \\sum _ { i = 1 } ^ { D } \\frac { \\partial ^ 2 } { \\partial x _ i ^ 2 } + \\omega ^ 2 r ^ 2 + \\sum _ { i = 1 } ^ { N } \\frac { 1 } { r _ i ^ 2 } f _ i ( \\Omega _ i ) . \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} \\psi _ { \\nu _ { 1 } } ( z ) = \\frac { 1 } { 2 \\beta } \\int _ { [ 0 , + \\infty ] } \\frac { t \\eta _ { \\mu _ { 1 } } ( z ) } { 1 - t \\eta _ { \\mu _ { 1 } } ( z ) } \\ , \\left ( \\frac { 1 } { t } + t \\right ) \\ , d \\sigma ( 1 / t ) , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} & r \\int _ { 0 } ^ { 1 } d \\beta \\int _ { t } ^ { \\infty } d s \\int _ { B _ { s - t } ( 0 ) \\cap ( B _ { \\frac { s } { 2 } } ( - \\beta x ) ) ^ { c } } \\frac { d A ( y ) } { ( s - t ) } | \\partial _ { 2 } v _ { 4 , c } ( s , | \\beta x + y | ) | \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} r _ { \\mathsf { m , n } } p _ { \\beta , \\mathsf { n , G i b b s } } = r _ { \\mathsf { n , m } } p _ { \\beta , \\mathsf { m , G i b b s } } \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\Big ( \\frac { 1 } { \\lambda _ n + \\lambda + 1 } - \\frac { 1 } { \\lambda _ n + \\lambda } \\Big ) = - \\ \\frac { \\displaystyle { \\sum _ { n = 0 } ^ \\infty \\frac { | \\langle 1 | f _ n \\rangle | ^ 2 } { ( \\lambda _ n + \\lambda ) ^ 2 } } } { \\displaystyle { \\sum _ { n = 0 } ^ \\infty \\frac { | \\langle 1 | f _ n \\rangle | ^ 2 } { \\lambda _ n + \\lambda } } } = \\frac { d } { d \\lambda } \\log \\mathcal H _ \\lambda ( u ) \\ . \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} | | - \\frac { 1 } { \\omega } \\sum _ { j = 1 } ^ { \\infty } \\int _ { 0 } ^ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } r ^ { 2 j } \\omega ^ { j } \\lambda ( t ) ^ { 2 j } \\phi _ { j } ( r ^ { 2 } ) F _ { 4 } ( t , r \\lambda ( t ) ) d r | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } \\leq \\frac { C ( \\log ( \\log ( t ) ) ) ^ { 2 } } { t ^ { 2 } \\log ^ { b + 1 - 2 b \\alpha } ( t ) } \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} H ( i ) = i L + [ 0 , L ) ^ { d } . \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{bmatrix} \\lambda x + y + \\eta S ( x ) \\\\ - x + \\lambda y - \\mu + \\nu S ( x ) \\end{bmatrix} . \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} \\left | x - y \\right | \\le \\tilde { \\epsilon } & \\Rightarrow \\left | \\log ( x ) - \\log ( y ) \\right | \\le \\epsilon \\\\ \\Leftrightarrow \\left | x - y \\right | > \\tilde { \\epsilon } & \\Leftarrow \\left | \\log ( x ) - \\log ( y ) \\right | > \\epsilon . \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} { } ^ b { \\rm T r } _ { \\chi } ( A ) = { } ^ b { \\rm T r } _ { S } \\left ( \\Phi _ A ( e ) \\right ) . \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} \\mathcal { O } _ A ' ( n ) = \\{ f \\in \\mathcal { O } _ A ( n ) | ~ f _ { [ n + 1 ] } = 0 \\} . \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} \\mathcal { B } ( M ) : = \\frac { 1 } { 4 } \\int _ M ( \\kappa _ 1 - \\kappa _ 2 ) ^ 2 \\ , d S = \\int _ M H ^ 2 - K + k _ 0 \\ , \\ , d S , \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} t \\ , \\frac { d u } { d t } = f ( t , u ) \\mbox { u n d e r $ f ( 0 , 0 ) = 0 $ } \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} j ( E ' ) & = \\frac { 2 7 u ^ 3 ( 9 u ^ 3 - 8 v ^ 3 ) ^ 3 } { v ^ 9 ( u - v ) ( u ^ 2 + u v + v ^ 2 ) } , \\\\ \\smallskip j ( E ) & = \\frac { 2 7 u ^ 3 ( u + 2 v ) ^ 3 ( u ^ 2 - 2 u v + 4 v ^ 2 ) ^ 3 } { v ^ 3 ( u - v ) ^ 3 ( u ^ 2 + u v + v ^ 2 ) ^ 3 } , \\\\ \\smallskip j ( E ^ { \\prime \\prime } ) & = \\frac { 3 ( u + 2 v ) ^ 3 ( u ^ 3 + 7 8 u ^ 2 v + 8 4 u v ^ 2 + 8 0 v ^ 3 ) ^ 3 } { v ( u - v ) ^ 9 ( u ^ 2 + u v + v ^ 2 ) } . \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} \\left \\| \\frac { P f } { \\| P f \\| } - f \\right \\| ^ 2 & = 1 + 1 - \\left \\langle \\frac { P f } { \\| P f \\| } , f \\right \\rangle - \\left \\langle f , \\frac { P f } { \\| P f \\| } \\right \\rangle = 2 - \\left \\langle \\frac { P ^ 2 f } { \\| P f \\| } , f \\right \\rangle - \\left \\langle f , \\frac { P ^ 2 f } { \\| P f \\| } \\right \\rangle \\\\ & = 2 - 2 \\| P f \\| . \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} 2 L ( t ) = \\Big \\| | x | ^ { - \\frac s 2 } u \\Big \\| _ 2 ^ 2 \\le \\widetilde { C } \\| \\nabla u \\| _ p ^ p + \\widetilde { C } . \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} \\lambda ( t ) = \\lambda _ { 0 } ( t ) + e _ { 0 } ( t ) , | | e _ { 0 } | | _ { X } \\leq 1 \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} H _ { L } = c _ 1 H _ { L - 1 } + \\cdots + \\left ( c _ { L - 1 } + 1 \\right ) H _ { 1 } + 1 = c _ 1 G _ { L - 1 } + \\cdots + \\left ( c _ { L - 1 } + 1 \\right ) G _ 1 + 1 = G _ { L } + G _ 1 = G _ { L } + 1 . \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} \\Big ( \\sum _ { \\vert \\alpha \\vert = m } \\Vert x _ { \\alpha } \\Vert ^ { q } \\Big ) ^ { \\frac { 1 } { q } } \\leq c ^ { m } \\| P \\| _ p . \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} E _ n = \\{ \\omega \\in \\Omega - \\Omega _ { \\delta } : \\sigma _ n ( \\omega ) < t _ 0 \\} , \\mbox { f o r a l l } n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} \\begin{cases} \\dot { A } ( t ) = d _ { A ( t ) } \\gamma _ t \\\\ \\dot { B } ( t ) = - [ \\gamma _ t , B ( t ) ] + d _ { A ( t ) } \\beta _ t . \\end{cases} \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} ( N - 2 ) H _ { n - k } = ( N - 2 ) H _ { n - k - 1 } + N ( N - 2 ) H _ { n - 2 k - 2 } . \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\int _ { F G } e ^ { s x } \\overline { G } ( s , t ) d s = - \\int _ { r } ^ { - r + \\lambda _ 2 - \\lambda _ 1 } e ^ { - x \\lambda _ 1 } e ^ { - w x } e ^ { t ( c _ 1 \\lambda _ 1 ^ { \\alpha _ 1 } + c _ 2 \\lambda _ 2 ^ { \\alpha _ 2 } ) } e ^ { - t [ c _ 1 ( w ^ { \\alpha _ 1 } e ^ { - i \\pi \\alpha _ 1 } ) + c _ 2 ( \\lambda _ 2 - \\lambda _ 1 + w e ^ { - i \\pi } ) ^ { \\alpha _ 2 } ] } d w . \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} \\begin{cases} d X _ { t } ^ { i , N } = F _ { A } \\bigg ( \\frac { 1 } { N } \\displaystyle \\sum _ { k = 1 } ^ { N } ( K \\ast V ^ { N } ) ( X _ { t } ^ { i , N } - X _ { t } ^ { k , N } ) \\bigg ) \\ ; d t + \\sqrt { 2 } \\ ; d W _ { t } ^ { i } , t \\leq T , ~ 1 \\leq i \\leq N , \\\\ X _ { 0 } ^ { i , N } , ~ 1 \\leq i \\leq N , \\{ W ^ i , ~ 1 \\leq i \\leq N \\} , \\end{cases} \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} \\begin{aligned} \\sqrt { \\eta _ 2 } \\ , R _ \\eta \\in L ^ \\infty ( 0 , T ; H ^ { 2 s + 1 } ( \\mathbb T ^ d _ { \\ell } ) ) , \\sqrt { \\eta _ 2 } \\ , \\Delta ^ { s + 1 } R _ \\eta \\in L ^ 2 ( 0 , T ; L ^ 2 ( \\mathbb T ^ d _ { \\ell } ) ) , \\\\ \\eta _ 1 ^ { \\frac { 1 } { \\alpha } } \\ , R _ \\eta ^ { - 1 } \\in L ^ \\infty ( 0 , T ; L ^ { \\alpha } ( \\mathbb T ^ d _ { \\ell } ) ) , \\sqrt { \\eta _ 1 } \\ , \\nabla R _ \\eta ^ { - \\frac { \\alpha } { 2 } } \\in L ^ 2 ( 0 , T ; L ^ 2 ( \\mathbb T ^ d _ { \\ell } ) ) . \\end{aligned} \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} \\langle \\langle S , \\varphi \\Phi \\rangle \\rangle & = \\langle \\langle \\Delta _ y \\pi , \\varphi \\Phi \\rangle \\rangle \\\\ & = ( - 1 ) ^ 2 \\langle \\langle \\pi , \\Delta _ y ( \\varphi \\Phi ) \\rangle \\rangle \\\\ & = \\langle \\langle \\pi , \\varphi \\Delta _ y \\Phi \\rangle \\rangle \\\\ & = \\iint _ { \\mathbb { R } ^ { 2 n } } \\varphi ( x ) ( \\Delta _ y \\Phi ) ( x , y ) d \\pi ( x , y ) \\\\ & = \\int _ { \\mathbb { R } ^ { n } } \\varphi ( x ) ( \\Delta _ y \\Phi ) ( x , x ) d x . \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} d _ 0 ( ( \\{ S _ 1 , \\ldots , S _ k \\} , p ) , ( \\{ T _ 1 , \\ldots , T _ k \\} , q ) ) = \\sum _ { i = 1 } ^ { k } | p _ i - q _ i | + \\| S _ i - T _ i \\| . \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} \\tilde { f } ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) & = f ( N _ 1 ^ 3 \\tau _ 1 , N _ 1 \\xi _ 1 , N _ 1 \\eta _ 1 ) , \\\\ \\tilde { g } ( \\tau _ 2 , \\xi _ 2 , \\eta _ 2 ) & = g ( N _ 1 ^ 3 \\tau _ 2 , N _ 1 \\xi _ 2 , N _ 1 \\eta _ 2 ) , \\\\ \\tilde { h } ( \\tau , \\xi , \\eta ) & = h ( N _ 1 ^ 3 \\tau , N _ 1 \\xi , N _ 1 \\eta ) . \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} \\| \\widehat { u } f _ { \\nu } \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 + \\| \\widehat { v } f _ { \\nu } \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 = \\nu \\| f _ { \\nu } \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} H = - \\frac { 1 } { L } \\int _ { w _ - } ^ { w _ + } \\frac { 1 } { \\sqrt { - U ( w ) } } d w , \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} A _ 0 \\ = \\ & \\{ 2 n + 1 , 3 n + 2 , 4 n + 3 \\} , \\\\ A _ j \\ = \\ & \\{ 2 n + 1 - j , 3 n + 2 - j , 4 n + 3 + 2 j \\} , \\\\ A _ n \\ = \\ & \\{ n + 1 , 2 n + 2 , 6 n + 3 \\} , \\\\ A _ { n + 1 } \\ = \\ & \\{ n , 4 n + 2 , 4 n + 4 \\} , \\\\ A _ { n + j } \\ = \\ & \\{ n + 1 - j , 4 n + 3 - j , 4 n + 2 + 2 j \\} , \\\\ A _ { 2 n } \\ = \\ & \\{ 1 , 3 n + 3 , 6 n + 2 \\} , \\\\ \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} \\int _ \\R \\left | \\int _ 0 ^ a f ( r ) P ( r , k ) d r \\right | ^ 2 d \\sigma ( k ) = \\int _ { 0 } ^ a | f ( r ) | ^ 2 d r \\ , . \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} \\epsilon _ 2 : = \\frac { \\epsilon _ 1 \\delta } { 2 m L } , \\ \\mbox { w i t h $ L = L ( \\mathbf { u ^ * } ) $ i s g i v e n b y \\eqref { 1 . 3 a } . } \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} \\frac { d } { d t } \\| \\nabla X ( t ) \\| _ { L ^ 2 } ^ 2 \\leq C ( \\| \\nabla u \\| _ { L ^ \\infty } + \\| \\nabla \\omega \\| _ { L ^ 2 } + \\| \\partial _ 1 \\nabla \\omega \\| _ { L ^ 2 } ) \\times ( \\| X \\| _ { L ^ 2 } ^ 2 + \\| \\nabla X \\| _ { L ^ 2 } ^ 2 ) . \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} B . \\big ( r ( h ) + & \\frac { i } { 2 n } ( C . r ( h ) ) \\big ) = B . r ( h ) + \\frac { i } { 2 n } ( B C . r ( h ) ) \\\\ \\equiv & \\frac { ( - 1 ) } { 2 n } h C . r ( h ) + i h r ( h ) \\\\ \\equiv & i h \\big ( \\frac { ( - 1 ) } { 2 n i } C . r ( h ) + r ( h ) \\big ) \\\\ = & i h \\big ( r ( h ) + \\frac { i } { 2 n } C . r ( h ) \\big ) \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} T ^ * ( \\beta ) ( a ) : = \\beta ( T ( a ) ) , \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} r _ 1 \\star _ P r _ 2 : = r _ 1 P ( r _ 2 ) + P ( r _ 1 ) r _ 2 + \\lambda r _ 1 r _ 2 \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} \\widehat { f } ( \\xi ) = \\int _ { 0 } ^ { \\infty } f ( r ) J _ { 1 } ( r \\xi ) r d r \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} \\nabla G _ r ( z ) = \\begin{bmatrix} k \\nabla ^ 2 f ( x ) - A ^ T A & k A ^ T \\\\ A \\nabla ^ 2 f ( x ) - k A & A A ^ T \\end{bmatrix} \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} \\Phi _ t ( x ) ( y ) : = p ( x , y , t ) . \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} \\mathcal { R } _ { 1 , 0 } ( t , u ) = \\int _ 0 ^ t e ^ { ( t - \\xi ) \\mathcal { L } } \\Big ( f \\left ( u ( \\xi ) , \\overline u ( \\xi ) \\right ) - f \\left ( e ^ { \\xi \\mathcal { L } } u _ 0 , e ^ { \\xi \\overline { \\mathcal { L } } } \\overline u _ 0 \\right ) \\Big ) d \\xi . \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} \\tan ( x ) + \\sec ( x ) = \\sum E _ n \\frac { x ^ n } { n ! } . \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} w ^ { ( 1 ) } ( \\theta ) = [ w _ 1 ( \\theta ) , \\ldots , w _ d ( \\theta ) ] ^ \\top , f ^ { ( 1 ) } ( { \\bf { x } } ) = [ f _ 1 ( { \\bf { x } } ) , \\ldots , f _ d ( { \\bf { x } } ) ] ^ \\top , \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} T ( y _ { 1 } ) - T ( y _ { 2 } ) = - \\int _ { t } ^ { \\infty } \\frac { \\sin ( ( t - x ) \\sqrt { \\omega } ) } { \\sqrt { \\omega } } \\left ( F _ { 2 } ( y _ { 1 } ) - F _ { 2 } ( y _ { 2 } ) - \\left ( \\mathcal { F } ( \\sqrt { \\cdot } ( F _ { 3 } ( u ( y _ { 1 } ) ) - F _ { 3 } ( u ( y _ { 2 } ) ) ) ( x , \\cdot \\lambda ( x ) ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) \\right ) \\right ) d x \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} \\tilde F = \\begin{pmatrix} F & 0 \\\\ 0 & F \\end{pmatrix} \\colon H \\oplus H \\to H ^ \\prime \\oplus H ^ \\prime \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} \\gamma = a _ { 2 L } b _ { 2 R } - a _ { 2 R } b _ { 2 L } \\ , . \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} B ( | \\xi | ; \\rho , \\theta ) = \\frac { 1 } { 2 } \\left ( | \\xi | ^ { 2 \\rho } + | \\xi | ^ { 2 \\theta } \\right ) B _ 0 + i | \\xi | B _ 1 \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} { \\mathbb D } ( \\Phi _ 2 , \\Phi _ 0 ) = 0 . \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} p _ { \\nu _ { \\beta } } ( t ) = \\frac { \\beta } { \\beta - 1 } \\frac { \\left | F _ { \\mu } ( t ) \\right | ^ { 2 } } { \\left | F _ { \\nu } \\left ( F _ { \\mu } ( t ) \\right ) \\right | ^ { 2 } } p _ { \\mu } ( t ) , t \\in I \\cup \\left \\{ t _ { 0 } \\right \\} . \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} \\left ( x ^ { - b } G ( x ) \\right ) ' = - b x ^ { - b - 1 } G ( x ) + x ^ { - b } G ' ( x ) . \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} \\delta ( t ) : = K _ 1 + \\frac { c _ 0 } { 1 - \\beta } [ ( t + \\theta ) ^ { 1 - \\beta } - \\theta ^ { 1 - \\beta } ] . \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} \\sum \\limits _ { \\mu \\in \\Psi _ i } a _ \\mu \\widehat \\mu _ * + \\rho _ i = \\sigma = \\sum \\limits _ { \\nu \\in \\Psi _ j } b _ \\nu \\widehat \\nu _ * + \\rho _ j \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} I ( p , q ) : = \\int p ( \\textbf { { x } } ) \\ln \\frac { p ( \\textbf { { x } } ) } { q ( \\textbf { { x } } ) } d \\textbf { { x } } . \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} \\mu _ { n + 1 } : = \\Big ( 1 - \\frac { \\gamma _ { n + 1 } } { \\lambda _ { n + 1 } - \\lambda _ 0 } \\Big ) \\prod _ { 1 \\leq p \\ne n + 1 } \\frac { \\displaystyle { \\Big ( 1 - \\frac { \\gamma _ p } { \\lambda _ p - \\lambda _ { n + 1 } } \\Big ) } } { \\displaystyle { \\Big ( 1 - \\frac { \\gamma _ p } { \\lambda _ p - \\lambda _ { n } - 1 } \\Big ) } } \\ . \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} h ( r ) = \\Phi ( r e ^ { i f ( r ) } ) , r > 0 , \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} \\lambda _ { 0 , 1 } ( t ) & = \\int _ { t } ^ { \\infty } \\int _ { t _ { 1 } } ^ { \\infty } \\frac { - b ^ { 2 } \\log ( \\log ( t _ { 2 } ) ) } { t _ { 2 } ^ { 2 } \\log ^ { b + 2 } ( t _ { 2 } ) } d t _ { 2 } d t _ { 1 } = \\frac { - b ^ { 2 } \\log ( \\log ( t ) ) } { ( b + 1 ) \\log ^ { b + 1 } ( t ) } + O \\left ( \\frac { 1 } { \\log ^ { b + 1 } ( t ) } \\right ) \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} g = g ^ { - 2 } \\oplus g ^ { - 1 } \\oplus g ^ { 0 } \\oplus g ^ { + 1 } \\oplus g ^ { + 2 } \\ , . \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} y ^ \\prime = ( n - 2 ) ( x ^ 2 + x ^ \\prime ) - 2 x y - y ^ 2 , \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} H _ { p } ( \\mathbb { T } ^ { \\infty } , X ) = \\Big \\{ f \\in L _ { p } ( \\mathbb { T } ^ { \\infty } , X ) \\ , : \\ , \\ , \\hat { f } ( \\alpha ) = 0 \\ , , \\ , \\ , \\ , \\ , \\forall \\alpha \\in \\mathbb { Z } ^ { ( \\mathbb { N } ) } \\setminus \\mathbb { N } _ { 0 } ^ { ( \\mathbb { N } ) } \\Big \\} \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} \\tilde { f } _ K ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) & = \\tilde { f } ( \\tau _ 1 , \\xi _ 1 , K ^ { \\frac { 1 } { 2 } } \\eta _ 1 ) , \\\\ \\tilde { g } _ K ( \\tau _ 2 , \\xi _ 2 , \\eta _ 2 ) & = \\tilde { g } ( \\tau _ 2 , \\xi _ 2 , K ^ { \\frac { 1 } { 2 } } \\eta _ 2 ) , \\\\ \\tilde { h } _ K ( \\tau , \\xi , \\eta ) & = \\tilde { h } ( \\tau , \\xi , K ^ { \\frac { 1 } { 2 } } \\eta ) . \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} B _ \\alpha ( p , q ) = B _ \\alpha ( p , p ^ * ) + B _ \\alpha ( p ^ * , q ) \\quad \\forall p \\in \\mathbb { L } . \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} x ^ { \\# } = ( B ( x _ 0 ) , \\alpha x _ 0 ^ * ) ( x , y ) & = \\alpha \\beta + B ( x _ 0 * , y _ 0 ) , \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} \\nabla ^ 2 ( a ) = [ \\theta , a ] \\nabla ( \\theta ) = 0 \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } L _ { u ^ { m - 1 } } ( u ^ m ) = 0 , \\\\ u ^ m ( 0 ) = u _ 0 , \\dot { u } ^ m ( 0 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} J _ { i } ^ n ( x ) : = \\begin{cases} J _ { i } ( x ) & \\mbox { f o r } | x | \\leq n , \\\\ \\frac { n } { | x | } J _ { i } ( x ) & \\mbox { f o r } | x | \\geq n . \\end{cases} \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} x _ { 1 + 2 + \\ldots + ( s - 1 ) } & = : x _ p \\neq 0 . \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} \\left ( \\sum _ { n = 1 } ^ \\infty r ^ { \\Omega ( n ) } \\| a _ n \\| ^ q \\right ) ^ { \\frac { 1 } { q } } \\leq C \\| D \\| _ { \\mathcal { H } _ p ( X ) } . \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} \\begin{bmatrix} K _ 1 ^ - \\\\ W _ 1 \\end{bmatrix} = \\begin{bmatrix} 0 & D _ 2 & D _ 3 \\end{bmatrix} \\ , . \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} \\overline D _ { | \\ker D } = - \\frac 1 2 \\tau L ( e _ 0 ) _ { | \\ker D } , \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} \\partial ( u _ 1 ^ { \\epsilon _ 1 } v _ 1 ^ { i _ 1 } & \\cdots u _ s ^ { \\epsilon _ s } v _ s ^ { i _ s } ) = \\\\ & \\left \\{ \\begin{array} { l l } ( - 1 ) ^ { \\epsilon _ 1 + \\cdots + \\epsilon _ { s - 1 } } u _ 1 ^ { \\epsilon _ 1 } v _ 1 ^ { i _ 1 } \\cdots u _ { s - 1 } ^ { \\epsilon _ { s - 1 } } v _ { s - 1 } ^ { i _ { s - 1 } } , & \\epsilon _ s = - i _ s = 1 ; \\\\ 0 , & , \\end{array} \\right . \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} I _ 2 & = - \\partial _ x \\left ( \\left \\langle A ( v ) \\left ( \\partial _ x v , \\partial _ x v \\right ) , \\partial _ x v \\right \\rangle \\right ) + \\left \\langle A ( v ) \\left ( \\partial _ x v , \\partial _ x v \\right ) , \\partial _ { x x } ^ 2 v \\right \\rangle \\\\ & = \\left \\langle A ( v ) \\left ( \\partial _ x v , \\partial _ x v \\right ) , \\partial _ { x x } ^ 2 v \\right \\rangle . \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} \\nabla \\Delta \\log R = \\frac { \\nabla \\Delta R } { R } - \\frac { \\Delta R \\nabla R } { R ^ 2 } - 2 \\frac { \\nabla ^ 2 R \\nabla R } { R ^ 2 } + 2 \\frac { | \\nabla R | ^ 2 \\nabla R } { R ^ 3 } , \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} \\bar { A } = W ^ \\top A V , \\bar { B } = W ^ \\top B , \\bar { C } = C V , \\bar { E } = W ^ \\top E V , \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} \\left | \\sum _ { l = 0 } ^ k \\left ( - h R _ { \\gamma \\gamma } \\right ) ^ l \\right | \\le C . \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} \\mathcal { D } = \\left ( \\begin{array} { c c } - b & \\partial _ x - a \\\\ - \\partial _ x - a & b \\end{array} \\right ) \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} ( 0 , s _ { 1 } , \\cdots , s _ { m } , s _ { m + 1 } ) = ( 0 , s _ { 1 } , \\cdots , s _ { m } , \\widehat { s } _ { 3 } ) \\sim ( 0 , t _ { 1 } , \\cdots , t _ { m } , \\widehat { t } _ { 3 } ) = ( 0 , t _ { 1 } , \\cdots , t _ { m } , t _ { m + 1 } ) . \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} P _ { d } \\{ \\mathbb { H } _ { N } ' \\in K ^ { \\delta } \\} + \\widetilde { A } _ { N } \\geq 1 - \\eta \\quad N = 1 , 2 , \\dots , \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} \\hat { x } _ { t } = \\bar { x } _ { t } + \\int _ { 0 } ^ { t } P _ { t } R _ { s } ^ { t } \\theta _ { s } ^ { \\ast } d s . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} \\rho ( [ q _ 1 , q _ 2 ] _ Q ) = [ \\rho ( q _ 1 ) , \\rho ( q _ 2 ) ] _ { T M } \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} H _ { L - m + k + 1 } = H _ { L - m + k } + H _ { k + 1 } + \\dots + H _ { 1 } + 1 . \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} & \\int _ { \\R ^ d _ + } G ( x , y ) f ( y ) \\ , d y = G f ( x ) = r ^ { \\alpha } G f ^ { ( r ) } ( x / r ) = r ^ { \\alpha } \\int _ { \\R ^ d _ + } G ( x / r , y ) f ^ { ( r ) } ( y ) \\ , d y \\\\ & = r ^ { \\alpha - d } \\int _ { \\R ^ d _ + } G ( x / r , z / r ) f ^ { ( r ) } ( z / r ) \\ , d z = r ^ { \\alpha - d } \\int _ { \\R ^ d _ + } G ( x / r , y / r ) f ( y ) \\ , d y \\ , . \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} L ( f , g ) = : \\int _ 0 ^ 1 f \\sharp _ t g d t \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} & \\lambda _ c ( t , x ) = \\lambda ( t , x ) + c _ 1 ( t , x ) , \\\\ & b _ c ( t , x ) = b ( t , x ) + c _ 2 ( t , x ) . \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} ( P v ) | _ \\Omega = v \\mbox { i n } \\Omega , \\| P v \\| _ { W ^ { k , r } ( \\tilde { \\Omega } ) } \\le C _ { l , r , \\Omega } \\| v \\| _ { W ^ { k , r } ( \\Omega ) } , \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} \\| \\exp ( \\alpha \\log A + ( 1 - \\alpha ) \\log B ) \\| _ \\infty & = \\lim _ { p / 2 \\ge q \\searrow 0 } \\| ( P _ { t ^ n h } ( A ^ q , B ^ q ) ) \\| _ \\infty ^ { 1 / q } \\\\ & \\le \\| P _ { t ^ n h } ( A ^ p , B ^ p ) \\| _ \\infty ^ { 1 / p } < 1 . \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} - \\lambda _ { 1 2 } b v + \\lambda _ { 1 3 } \\sqrt { 1 + b v ^ { 2 } } = 0 . \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} - \\partial _ { t t } u + \\partial _ { r r } u + \\frac { 1 } { r } \\partial _ { r } u = 0 \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} Y ^ n _ t : = \\lambda _ 0 R ^ n _ { \\lambda _ 0 } h ( X ^ n _ t ) , Z ^ n _ t : = \\lambda _ 0 \\left ( \\lambda _ 0 R ^ n _ { \\lambda _ 0 } h - h \\right ) ( X _ t ^ n ) . \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} \\frac { d } { d z } \\left ( \\frac { \\partial J } { \\partial h _ z } \\right ) - \\frac { \\partial J } { \\partial h } = 0 , \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} \\partial _ t v + P _ S ( v \\cdot \\nabla v + w \\partial _ z v ) + \\Omega \\widetilde { v } ^ \\perp = 0 . \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} \\widetilde { g } ( x ) = \\lim _ { r \\to 0 } \\frac { 1 } { \\lambda ^ N ( B ( x , r ) ) } \\int _ { B ( x , r ) } g ( y ) d y \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} \\leq \\frac { \\mathbb { E } _ \\eta \\left ( \\left . \\int _ { t / 2 } ^ { t } \\mathbb { 1 } _ { \\{ \\eta _ s ( x ) = 1 \\} } \\mathrm { d } s \\right | \\mathcal { F } _ { k } \\right ) } { \\frac { t } { 2 } - \\frac { 1 - p } { 4 } t } = \\frac { \\int _ { t / 2 } ^ { t } \\mathbb { P } _ \\eta ( \\eta _ s ( x ) = 1 | \\mathcal { F } _ { k } ) \\mathrm { d } s } { \\left ( 1 - \\frac { 1 - p } { 2 } \\right ) \\frac { t } { 2 } } . \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} \\mathfrak { X } ( g ) = \\mathfrak { X } ( s _ 1 \\dots s _ n ) = s _ 1 ^ { - 1 } \\mathfrak { X } ( s _ 2 \\dots s _ n ) = \\dots = s _ n ^ { - 1 } \\dots s _ 1 ^ { - 1 } ( \\mathfrak { X } ( 1 _ G ) ) = g ^ { - 1 } \\mathfrak { X } ( 1 _ G ) . \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} ( a b - b a ) ( a c - c a ) = 0 , ~ b , c \\in A . \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} v ^ { \\lambda } _ { 3 , 2 } = v ^ { \\lambda } _ { 3 } - v ^ { \\lambda } _ { 3 , 1 } \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} 0 = \\bar { \\nabla } \\phi ( p ) = 2 \\varphi ( p ) \\bar { \\nabla } \\varphi ( p ) - 2 \\cot ^ 2 \\theta f ( p ) \\bar { \\nabla } f ( p ) . \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} \\left | \\int _ { 2 ^ N } ^ { 2 ^ N + n } f ( r ) P ( r , k ) d r \\right | ^ 2 = \\left | \\sum _ { j = 1 } ^ N \\int _ { 2 ^ N + n _ { j - 1 } } ^ { 2 ^ N + n _ j } f ( r ) P ( r , k ) d r \\right | ^ 2 \\le \\\\ N \\sum _ { j = 1 } ^ N \\left | \\int _ { 2 ^ N + n _ { j - 1 } } ^ { 2 ^ N + n _ j } f ( r ) P ( r , k ) d r \\right | ^ 2 \\leq N \\sum _ { j = 1 } ^ N \\sum _ { p = 0 } ^ { 2 ^ j - 1 } \\left | \\int _ { 2 ^ N + p 2 ^ { N - j } } ^ { 2 ^ N + ( p + 1 ) 2 ^ { N - j } } f ( r ) P ( r , k ) d r \\right | ^ 2 \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} I _ { p , \\alpha } = \\frac { \\int _ { \\Omega } | u | ^ { \\alpha - p } | \\nabla u | ^ { 2 p } \\ , d \\mu } { \\int _ { \\Omega } | u | ^ { \\alpha } | \\nabla u | ^ { p } \\ , d \\mu } . \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} 2 + 2 \\delta \\xi _ t + \\xi _ t ' = \\theta , \\ t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} y ( t , \\xi ) = \\mathcal { F } ( \\sqrt { \\cdot } u ( t , \\cdot \\lambda ( t ) ) ) ( \\xi \\lambda ( t ) ^ { 2 } ) \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} \\frac { d \\lambda } { d \\tau } = \\frac { \\lambda [ - R ( \\lambda ) \\bar { R ' } ( \\lambda ) e ^ { \\lambda \\tau } e ^ { \\bar { \\lambda } \\tau } + Q ( \\lambda ) \\bar { Q ' } ( \\lambda ) - \\tau | Q ( \\lambda ) | ^ { 2 } ] } { | R ' ( \\lambda ) e ^ { \\lambda \\tau } + Q ' ( \\lambda ) - \\tau Q ( \\lambda ) | ^ { 2 } } . \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} & \\bigl ( \\mathcal { T } _ { k _ 1 } ^ { A , d } \\times \\mathcal { T } _ { k } ^ { A , d } \\bigr ) \\cap ( \\mathfrak { D } _ { j _ 1 } ^ A \\times \\mathfrak { D } _ { j } ^ A ) \\not = \\emptyset , \\\\ | F ( \\xi _ 1 , \\eta _ 1 , \\xi , \\eta ) | & \\geq A ^ { - 1 } d ^ { - 1 } N _ 1 ^ 2 \\ \\ \\textnormal { f o r a n y } \\ ( \\xi _ 1 , \\eta _ 1 ) \\times ( \\xi , \\eta ) \\in \\mathcal { T } _ { k _ 1 } ^ { A , d } \\times \\mathcal { T } _ { k } ^ { A , d } . \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} g ( T ) = \\sum _ { i = 1 } ^ s ( \\frac { \\phi ( 2 ^ { \\epsilon } p _ i ^ { a _ i } ) } 2 - 1 ) + \\sum _ { \\iota = 1 } ^ \\rho ( \\frac { \\phi ( 2 ^ { \\epsilon _ \\iota } ) } 2 - 1 ) + c ( T ) \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} k _ a ^ { ( b - r ) } = k _ a ^ { ( b - r + 1 ) } - k _ { a - 1 } ^ { ( b - r + 1 ) } = \\cdots = \\sum _ { i = 0 } ^ r ( - 1 ) ^ i \\binom { r } { i } k _ { a - i } ^ { ( b ) } \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} \\Psi ^ { ( 0 ) } _ t = \\phi - \\omega t . \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} \\langle j _ p ( \\Pi _ M ^ p x ) - j _ p ( x ) , z \\rangle = 0 , \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} C _ { \\beta , l } : = \\{ \\psi \\in h _ \\alpha ^ \\beta ( \\tau ( \\kappa _ \\alpha ) ) ( d _ \\beta ^ \\tau ) : ( H _ \\beta ^ l ) ^ \\tau ( \\psi ) = ( H _ \\beta ^ l ) ^ r ( \\psi \\circ \\tau ) \\} \\in h _ \\alpha ^ \\beta ( \\tau ( \\kappa _ \\alpha ) ) ( d _ \\beta ^ \\tau ) , \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { j } \\partial _ { r } ^ { k } v _ { 2 } ( t , r ) | \\leq \\frac { C } { | t - r | ^ { 1 + j + k } } \\log ( \\log ( r ) ) , t \\neq r \\geq \\frac { t } { 2 } , b = 1 \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} w _ i ( t , x ) : = v _ i ( t , x ) + \\epsilon e ^ { A t } \\ \\mbox { f o r } \\ ( t , x ) \\in [ 0 , T ] \\times \\R , \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} \\partial _ { z } K ( w , z ) = \\int _ { 0 } ^ { \\infty } d R \\int _ { 0 } ^ { w } d \\rho \\left ( \\frac { 4 p R ^ 3 z \\left ( p ^ 2 + R ^ 2 z ^ 2 + 1 \\right ) } { \\left ( R ^ 2 + 1 \\right ) ^ 3 \\left ( \\left ( p ^ 2 - R ^ 2 z ^ 2 + 1 \\right ) ^ 2 + 4 R ^ 2 z ^ 2 \\right ) ^ { 3 / 2 } } \\right ) \\left ( \\frac { 1 } { \\sqrt { w ^ { 2 } - \\rho ^ { 2 } } } - \\frac { 1 } { w } \\right ) \\geq 0 \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} ( h ^ 2 \\Delta _ g - 1 ) u = f ; \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} \\begin{cases} \\underline h ( t ) : = ( c _ 0 - \\delta ) t + K , \\ \\ \\ t \\geq 0 , \\\\ \\underline U ( t , x ) : = ( 1 - \\epsilon ) \\big [ \\Phi ( x - \\underline h ( t ) ) + \\Phi ( - x - \\underline h ( t ) ) - \\mathbf { u } ^ * \\big ] , \\ \\ \\ t \\geq 0 , \\ x \\in [ - \\underline h ( t ) , \\underline h ( t ) ] , \\end{cases} \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = F ( x , y ) = \\begin{bmatrix} f ( x , y ) \\\\ g ( x , y ) \\end{bmatrix} . \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} [ T ^ { p ^ v } \\textbf { a } ] _ i & \\equiv \\sum _ { j = 0 } ^ { p ^ v } \\binom { p ^ v } { j } \\textbf { a } _ { i + j } \\equiv \\sum _ { j = 0 } ^ { p } \\binom { p ^ v } { j p ^ { v - 1 } } \\textbf { a } _ { i + j p ^ { v - 1 } } \\equiv \\sum _ { j = 0 } ^ { p } \\binom { p } { j } \\textbf { a } _ { i + j p ^ { v - 1 } } \\pmod { p ^ 2 } , \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } \\| \\Delta _ q \\rho \\| _ { L ^ 2 } ^ 2 = - \\int _ { \\R ^ 2 } \\Delta _ q ( v \\cdot \\nabla \\rho ) \\Delta _ q \\rho ~ d x + \\int _ { \\R ^ 2 } \\Delta _ q f \\Delta _ q \\rho ~ d x \\triangleq I + I I . \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} \\phi ( - q ^ { 1 / 2 } y ) \\phi ( - q ^ { 1 / 2 } x ) = \\phi ( - q ^ { 1 / 2 } x ) \\phi ( - q y x ) \\phi ( - q ^ { 1 / 2 } y ) \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} \\Im \\frac { 1 } { 1 - \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) } = \\Im ( 1 + \\psi _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) ) = \\int _ { \\mathbb { R } _ { + } } \\frac { t r \\sin ( f ( r ) ) } { | 1 - t r e ^ { i f ( r ) } | ^ { 2 } } \\ , d \\mu _ { 1 } ( t ) . \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} G ( z ) = \\frac { 1 } { H - z } \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} C _ { k , 1 } ( n ) & = \\{ \\lambda = ( \\lambda _ 1 ^ { m _ 1 } , \\lambda _ 2 ^ { m _ 2 } , \\ldots , \\lambda _ \\ell ^ { m _ \\ell } , a ^ b ) \\in \\mathcal { P } ( n ) \\mid m _ i < k \\ , \\forall i , \\ , b \\geq k , \\ , k \\nmid a \\} . \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} \\frac { G ( K _ 1 ) } { K _ 1 } = \\frac { a b } { c } , \\ ; \\ ; K _ 2 = \\frac { G ( K _ 1 ) } { b } . \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} ( 1 + w _ { N } ( \\underline { \\xi } ) ) \\Big ( \\frac { 1 } { N } + \\underline { \\psi } _ { N , 1 } ^ { \\ast } \\Big ) = ( 1 + \\widehat { w } _ { N } ( \\underline { \\xi } ) ) \\Big ( \\frac { 1 } { N } + \\overline { \\psi } _ { N , 1 } ^ { \\ast } \\Big ) = \\frac { N + 1 } { N } . \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} & ( 1 + a ) \\big ( Q _ { \\Sigma } ( v , v ) + C '' \\| v \\| ^ 2 _ { L ^ 2 ( \\Sigma ) } \\big ) \\\\ \\geq \\ & a \\int _ { \\Sigma } | \\nabla v | ^ 2 + \\Big ( Q _ { \\Sigma } ( v , v ) - \\vartheta \\int _ { \\Sigma } v ^ 2 / \\rho ^ 2 + C '' \\| v \\| ^ 2 _ { L ^ 2 ( \\Sigma ) } \\Big ) \\geq a \\int _ { \\Sigma } | \\nabla v | ^ 2 \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} a ^ j _ 3 > a ^ { n + j } _ 3 > a ^ { n + p } _ 3 & \\ 1 \\leq p \\leq j , \\\\ a ^ { n + j } _ 3 > a ^ { j - 1 } _ 3 > a ^ q _ 3 & \\ 0 \\leq q \\leq j - 1 . \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} \\alpha _ 1 : = E _ 1 - E _ 2 , \\alpha _ 2 & : = E _ 2 - E _ 3 , \\quad \\ldots \\alpha _ { s - 1 } : = E _ { s - 1 } - E _ { s } , \\\\ \\alpha _ { s } & : = H - \\sum _ { i = 1 } ^ { n + 1 } E _ i \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} | \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( t , \\xi ) | & \\leq \\frac { C \\xi } { t } \\left ( \\log ^ { 3 } ( t ) + | \\log ( \\xi ) | ^ { 3 } \\right ) , \\frac { 1 } { \\xi } \\geq t + \\sqrt { t } \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} Q _ { 0 1 } ( u , v ) = Q _ { 1 0 } ( u , v ) = ( u - v ) ^ 2 , \\frac { Q _ { 0 1 } ( y _ 3 , y _ 2 ) - Q _ { 0 1 } ( y _ 1 , y _ 2 ) } { y _ 3 - y _ 1 } = y _ 1 - 2 y _ 2 + y _ 3 . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } \\frac { \\partial v _ { k , \\mu } } { \\partial \\nu } \\frac { \\partial v _ { l , \\mu } } { \\partial \\nu } d \\sigma = 0 \\ \\ \\ { \\rm i f \\ } k \\ne l \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} \\begin{cases} ( 1 ) & t \\odot ( x \\oplus y ) = ( t \\odot x ) \\oplus ( t \\odot y ) \\\\ ( 2 ) & ( t _ 1 \\oplus t _ 2 ) \\odot x = ( t _ 1 \\odot x ) \\oplus ( t _ 2 \\odot x ) \\\\ ( 3 ) & t _ 2 \\odot ( t _ 1 \\odot x ) = ( t _ 1 \\odot t _ 2 ) \\odot x \\\\ ( 4 ) & { \\sf 1 } \\odot x = x \\end{cases} \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} J _ \\mathcal { C } ( s ; t ) = J _ \\mathcal { C } ( t ; t ) \\exp \\ ! \\left ( \\int _ t ^ s { \\rm t r } \\left [ \\mathbf { M } ( \\tau ) \\right ] d \\tau \\right ) , \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} k = 2 n , \\ ; n \\neq 3 , 1 - \\frac { n + 2 } { n ^ 2 - n } < \\lambda _ { n } ( \\xi ) \\leq 1 . \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} m n ^ 3 ( 1 - m ^ 2 ) & = 1 - n ^ 2 \\\\ 1 + m ^ 3 n ^ 3 & = n ^ 2 ( 1 + m n ) \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} \\mathcal { E } ( \\lambda _ { \\infty } ) - \\mathcal { E } ( \\lambda _ { 0 } ) = - \\displaystyle \\int _ { 0 } ^ { \\infty } \\displaystyle \\int _ { M } e ^ { 4 \\lambda } [ Q - e ^ { - 4 \\lambda } ( Q _ { 0 } ) _ { \\ker } ] ^ { 2 } d \\mu d t . \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} | \\langle j _ p ( x ^ * ) - j _ p ( x ) , z - x ^ * \\rangle | = & \\left | \\left \\langle j _ p ( x ^ * ) - j _ p ( x _ N ) - \\sum _ { k = 1 } ^ N j _ p ( x _ k ) - j _ p ( x _ { k - 1 } ) , z - x ^ * \\right \\rangle \\right | \\\\ = & \\langle j _ p ( x ^ * ) - j _ p ( x _ N ) , z - x ^ * \\rangle \\xrightarrow [ ] { N \\to \\infty } 0 , \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} \\Phi ( z ) = \\gamma _ \\beta + z + N _ { \\sigma _ \\beta } ( z ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} A ( p ) ( u , v ) : & = \\bar \\nabla _ { \\bar X } \\bar Y ( p ) - \\nabla _ X Y ( p ) \\\\ & = \\bar \\nabla _ { \\bar X } \\bar Y ( p ) - ( \\nabla _ { \\bar X } { \\bar Y } ) ^ T ( p ) , \\ \\ \\forall \\ u , v \\in T _ p N , \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} \\phi _ t ^ * ( u _ 1 \\sigma _ 1 + u _ 2 \\sigma _ 2 ) = e ^ { \\lambda t } \\ \\left ( u _ 1 \\cos ( t ) \\sigma _ 1 - u _ 1 \\sin ( t ) \\sigma _ 2 + u _ 2 \\sin ( t ) \\sigma _ 1 + u _ 2 \\cos ( t ) \\sigma _ 2 \\right ) . \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} I _ { n + 1 } ( x ) = I _ { n - 1 } ( x ) - \\frac { 2 n } { x } I _ n ( x ) . \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\partial _ s U _ { 1 , 0 } ( s , \\gamma ) = \\Delta _ L U _ { 1 , 0 } ( s , \\gamma ) . \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} 0 = \\lim _ { t \\rightarrow \\infty } \\displaystyle \\int _ { M } e ^ { 4 \\lambda } [ Q - e ^ { - 4 \\lambda } ( Q _ { 0 } ) _ { \\ker } ] ^ { 2 } d \\mu = \\displaystyle \\int _ { M } e ^ { 4 \\lambda _ { \\infty } } [ Q _ { \\infty } - e ^ { - 4 \\lambda _ { \\infty } } ( Q _ { 0 } ) _ { \\ker } ] ^ { 2 } d \\mu _ { \\infty } \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} m ^ 2 + m n + n ^ 2 = 1 \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} \\varphi \\ast _ { G / H } \\psi ( x H ) = \\int _ { G / H } \\varphi ( y H ) J _ 1 \\psi ( y ^ { - 1 } x H ) d \\mu ( y H ) . \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} \\pi _ \\lambda = \\pi ( \\cdot , \\lambda ) \\bar \\pi _ \\lambda = \\bar \\pi ( \\cdot , \\lambda ) , \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} P _ n ( 1 - H ) x = P _ n ( 1 - H ) ( P _ n x + w _ k ) . \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} K ( \\Psi ; M ; \\gamma ) : = K ( \\hat { \\Psi } ; \\hat { M } ; \\gamma ) \\ , . \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } \\left ( \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) - 1 } { r ^ { 2 } } \\right ) v _ { 3 } \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle \\right ) | \\leq \\frac { C } { t ^ { 3 } \\log ^ { 1 + 2 \\alpha b - 3 b } ( t ) } \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} \\Lambda & : = [ N ] = \\{ 1 , 2 , \\cdots , N \\} , \\\\ \\Lambda _ { 1 } & : = \\bigcup _ { l \\in \\mathbb { Z } } \\Lambda \\cap \\left ( \\left [ 1 , K - R \\right ] + ( l - 1 ) K \\right ) = \\bigcup _ { l = 1 } ^ { M } \\Lambda _ 1 ^ { ( l ) } , \\\\ \\Lambda _ { 2 } & : = \\bigcup _ { l \\in \\mathbb { Z } } \\Lambda \\cap \\left ( \\left [ K - R + 1 , K \\right ] + ( l - 1 ) K \\right ) = \\bigcup _ { l = 1 } ^ { M } \\Lambda _ 2 ^ { ( l ) } , \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} ( \\sigma _ J \\times \\sigma _ K ) \\circ F _ { J , K } = F _ { J , K } \\circ ( \\sigma _ J \\times \\sigma _ K ) , \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} \\lambda _ { ( l ) } ( \\mu ) = \\frac { \\left ( 3 l ^ 4 + 2 ( N - 2 ) l ^ 3 - ( N + 1 ) l ^ 2 - ( N - 2 ) l - \\eta _ { ( l ) } \\mu \\right ) } { \\left ( \\xi _ { ( l ) } - \\mu \\right ) } . \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} \\psi _ { r _ 1 , r _ 2 } ( x _ 1 - y _ 1 , x _ 2 - y _ 2 , x _ 3 - y _ 3 ) = r _ 1 ^ { - 2 } r _ 2 ^ { - 2 } \\psi ^ { ( 1 ) } \\Big ( { x _ 1 - y _ 1 \\over r _ 1 } \\Big ) \\psi ^ { ( 2 ) } \\Big ( { x _ 2 - y _ 2 \\over r _ 2 } , { x _ 3 - y _ 3 \\over r _ 1 r _ 2 } \\Big ) , \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} A ' ( \\sigma ) = \\lim _ { N \\to \\infty } A _ N ' ( \\sigma ) \\sigma \\in \\mathbb { R } . \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} \\hat { h } _ { s z } ( 0 , s ) = 0 . \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} \\| f _ { \\vec { S } _ { \\ell } } \\| _ { L ^ 2 ( B ^ { n - 1 } ) } ^ 2 \\lesssim _ { \\varepsilon } r _ { \\ell } ^ { \\ell / 2 } \\Big ( \\prod _ { i = 1 } ^ { \\ell } r _ i ^ { - 1 / 2 } D _ i ^ { \\delta } \\Big ) R ^ { O ( \\varepsilon _ { \\circ } ) } \\| f _ { \\vec { S } _ { \\ell } } ^ { \\# } \\| _ { 2 } ^ 2 \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} f _ 5 ^ * ( s ^ \\bullet ) = f _ { 5 , 4 } ( s ^ { \\bullet } ) = ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { 2 } , { \\bf 3 } , 7 , { \\bf 3 } , { 7 } , { 7 } , { 4 } ) . \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} R _ { T , \\gamma } = \\min \\left \\{ \\frac { G d _ 0 } { \\sqrt { \\gamma T } } , \\frac { 2 \\beta d _ 0 ^ 2 } { \\gamma T } , \\frac { G ^ 2 } { { \\gamma } \\alpha T } , \\beta d _ 0 ^ 2 \\left ( 1 - \\gamma \\frac { \\alpha } { \\beta } \\right ) ^ T \\right \\} . \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} \\Lambda _ { \\underline { \\xi } } = \\{ ( x , \\xi _ { 1 } x - y _ { 1 } , \\ldots , \\xi _ { N } x - y _ { N } ) : \\ ; x , y _ { j } \\in \\mathbb { Z } \\} . \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y _ l } \\bar { H } ( y ) = \\frac { 1 } { M } y _ l + \\frac { 1 } { N } \\sum _ { j \\in B ( l ) } s _ j + \\frac { 1 } { N } \\mathbb { E } _ { \\mu _ { N , m } ( d x | y ) } \\left [ \\sum _ { i = 1 } ^ N \\sum _ { j \\in B ( l ) } M _ { i j } X _ i + \\sum _ { i \\in B ( l ) } \\delta \\psi ' ( X _ i ) \\right ] . \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} T = \\sum _ { j = 1 } ^ { \\widetilde { k } - 1 } \\delta ^ { \\prime } _ j T _ j ^ { \\prime } + \\sum _ { j = \\widetilde { k } + 1 } ^ k \\delta _ j T _ j , \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} X _ { ( P _ { \\ell ( \\alpha ) } , \\alpha ) } X _ { ( P _ { \\ell ( \\beta ) } , \\beta ) } = X _ { ( P _ { \\ell ( \\alpha ) } \\mid P _ { \\ell ( \\beta ) } , \\alpha \\cdot \\beta ) } = X _ { ( P _ { \\ell ( \\alpha \\cdot \\beta ) } , \\alpha \\cdot \\beta ) } + X _ { ( P _ { \\ell ( \\alpha \\odot \\beta ) } , \\alpha \\odot \\beta ) } . \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} B _ { k , k } \\cdot D \\cdot W ' = I _ k . \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} S ^ { ( s ) } _ { n , k } & = - i w ^ { ( s ) } _ k \\sum _ { P ( k ) } S ^ { ( s ) } _ { n , k , } = - i w ^ { ( s ) } _ k \\sum _ { P ( k ) } \\prod _ { i = 1 } ^ k b _ { k _ i } = B _ { n , k } \\left ( b _ 1 , b _ 2 , \\ldots \\right ) . \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} u _ \\lambda = \\left ( 1 - \\frac { 5 0 } { 5 1 } \\lambda \\right ) \\begin{pmatrix} 1 \\\\ 0 \\\\ \\frac { 1 } { 2 } \\end{pmatrix} + \\frac { 5 0 } { 5 1 } \\lambda \\begin{pmatrix} 1 \\\\ 1 \\\\ \\frac { 1 0 1 } { 1 0 0 } \\end{pmatrix} \\in M _ 2 \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} \\alpha = \\chi _ { { \\rm f o l d } , L } ( 0 ) - \\chi _ { { \\rm f o l d } , R } ( 0 ) . \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} D _ 1 : = 2 H - E _ 1 - \\ldots - E _ 9 , D _ 2 : = 2 H - E _ 1 - \\ldots - E _ { 1 4 } \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} & x _ { \\epsilon _ 1 + \\epsilon _ 2 } ^ 2 = 0 , x _ { \\epsilon _ 1 - \\epsilon _ 2 } ^ 2 = 0 , x _ { \\epsilon _ 1 } ^ 2 - x _ { \\epsilon _ 1 + \\epsilon _ 2 } x _ { \\epsilon _ 1 - \\epsilon _ 2 } = 0 , x _ { \\epsilon _ 2 } x _ { \\epsilon _ 1 + \\epsilon _ 2 } = 0 , \\\\ & x _ { \\epsilon _ 1 } x _ { \\epsilon _ 1 - \\epsilon _ 2 } = 0 , x _ { \\epsilon _ 1 } x _ { \\epsilon _ 1 + \\epsilon _ 2 } = 0 , x _ { \\epsilon _ 2 } ^ 3 = 0 , x _ { \\epsilon _ 2 } ^ 2 x _ { \\epsilon _ 1 } = 0 . \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} \\hat { p } _ { \\mathsf { k , m } } = \\frac { \\exp \\left [ - \\frac { \\alpha + 1 } { 2 } \\beta \\lambda _ { \\mathsf { m } } \\right ] } { \\nu _ { \\mathsf { k } } + b _ { \\mathsf { m } } } . \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} \\gamma ( t ) = t ^ 8 + 1 8 a _ 4 t ^ 4 + 1 0 8 a _ 6 t ^ 2 - 2 7 a _ 4 ^ 2 . \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} r ( \\mu _ { J , K } ) : = \\min _ { a : \\ : \\mu _ { J , K } ( a ) > 0 } \\min \\left \\{ a , \\sigma _ J ( a ) \\right \\} , \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} g ( q ) = E ( q ) ^ { - 1 } , \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} r ( \\alpha ) : = \\Big ( \\frac { \\alpha } { B _ \\alpha ^ { q + \\alpha } } \\Big ) ^ { \\frac { 1 } { q + \\alpha - p } } , \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} \\alpha = \\frac { \\lambda _ L } { \\omega _ L } + \\frac { \\lambda _ R } { \\omega _ R } . \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta | A _ { \\Sigma } | ^ 2 = | \\nabla A _ { \\Sigma } | ^ 2 - | A _ { \\Sigma } | ^ 4 \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} C . u = & C \\big ( f ( h ) ( r ( h ) + \\frac { i } { 2 n } ( C . r ( h ) ) ) \\big ) \\\\ \\equiv & \\big ( f ^ { \\prime \\prime } ( h ) B + f ^ { \\prime } ( h ) C \\big ) \\big ( r ( h ) + \\frac { i } { 2 n } ( C . r ( h ) ) \\big ) \\\\ \\equiv & i h f ^ { \\prime \\prime } ( h ) ( r ( h ) + \\frac { i } { 2 n } C . r ( h ) ) - 2 n i f ^ { \\prime } ( h ) ( r ( h ) + \\frac { i } { 2 n } C . r ( h ) ) \\\\ \\equiv & \\big ( i h f ^ { \\prime \\prime } ( h ) - 2 n i f ^ { \\prime } ( h ) \\big ) \\big ( r ( h ) + \\frac { i } { 2 n } C . r ( h ) \\big ) , \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} \\abs { V _ \\Gamma } - \\abs { E _ \\Gamma } + \\abs { \\Gamma } & = 1 . \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} \\Omega _ \\mathbf { F } [ \\varphi ] ( p ) = \\sup _ { a \\in A } \\varphi ( a ) \\cdot \\mathbf { F } ^ * ( a , p ) , \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} J \\partial _ x \\phi = ( \\lambda A + B ) \\phi , \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} a _ 0 ( n _ i a _ i a _ j ' - n _ j a _ j a _ i ' ) = ( i n _ j - j n _ i ) a _ i a _ j , \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} \\Psi ( z ) = \\gamma + z + N _ { \\sigma } ( F _ { \\mu _ { 1 } } ( z ) ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} | | \\Phi _ A | | _ { k , \\ell } = | | \\sum _ i f _ i \\otimes a _ i | | _ { k , \\ell } \\leq \\sum _ i | | f _ i ( 1 + L ) ^ \\ell | | _ { L ^ 2 ( G ) } | | a _ i | | _ k . \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} \\hat { y } ( t ) - \\tilde { y } ( t ) = J ^ { 1 - \\epsilon } _ 0 ( w ( t ) - \\tilde { z } ( t ) ) . \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} R = \\frac { 1 } { \\overline { \\lim } _ { n \\to \\infty } \\sqrt [ n ] { \\sigma _ n } } . \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} & \\rho ( L _ { - 1 } ^ { \\gamma } ) \\ , = \\ , h ( z ) ^ { - 1 } ( L - c ) \\ , \\overset { \\gamma } \\cdots \\ , h ( z ) ^ { - 1 } ( L - c ) \\ , = \\ , h ( z ) ^ { - \\gamma } P ( L - c , \\gamma ) , \\\\ & \\rho ( L _ { 1 } ^ { \\alpha } ) \\ , = \\ , h ( z ) ( L + c ) \\ , \\overset { \\alpha } \\cdots \\ , h ( z ) ( L + c ) \\ , = \\ , h ( z ) ^ { \\alpha } P ( L + c - \\alpha + 1 , \\alpha ) , \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} f \\left ( x _ 1 , \\ldots , x _ d \\right ) = f _ 1 \\left ( x _ 1 \\right ) f _ 2 \\left ( x _ 2 \\right ) \\cdots f _ d \\left ( x _ d \\right ) . \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} w _ { \\lambda , \\lambda _ \\gamma } ( x ) : = \\left \\lbrace \\begin{aligned} & \\frac { u _ { \\lambda + \\lambda _ \\gamma } ( x ) } { \\psi _ \\gamma ( x ) } , x \\neq 0 ; \\\\ & 1 , \\qquad \\qquad \\ ; \\ ; \\ , x = 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} E _ { } ( u , v ) = \\pi \\int _ { 0 } ^ { \\infty } \\left ( v ^ { 2 } + \\frac { \\sin ^ { 2 } ( u ) } { r ^ { 2 } } + \\left ( \\partial _ { r } u \\right ) ^ { 2 } \\right ) r d r \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} & \\int _ { \\frac { 1 } { 2 } } ^ { \\infty } d w | K _ { 3 } ( w , \\lambda ( t ) ) + \\frac { 1 } { 4 ( \\lambda ( t ) ^ { 1 - \\alpha } + w ) ( 1 + w ) ^ { 3 } } | \\\\ & \\leq \\int _ { \\frac { 1 } { 2 } } ^ { \\infty } d w | K _ { 3 } ( w , \\lambda ( t ) ) | + \\int _ { \\frac { 1 } { 2 } } ^ { \\infty } \\frac { d w } { ( \\lambda ( t ) ^ { 1 - \\alpha } + w ) ( 1 + w ) ^ { 3 } } \\\\ & \\leq C \\int _ { \\frac { 1 } { 2 } } ^ { \\infty } d w \\frac { w | \\lambda ( t ) ^ { 2 - 2 \\alpha } - 1 | } { w ^ { 4 } } + C \\\\ & \\leq C \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} & 2 ( \\cos ( 2 \\overline { v } _ { 1 } ( t , R ) ) - 1 ) \\partial _ { R } \\overline { v } _ { 1 } ( t , R ) - 2 ( \\cos ( 2 \\overline { v } _ { 2 } ( t , R ) ) - 1 ) \\partial _ { R } \\overline { v } _ { 2 } ( t , R ) \\\\ & = 2 \\left ( \\cos ( 2 \\overline { v } _ { 1 } ( t , R ) ) - 1 \\right ) \\left ( \\partial _ { R } ( \\overline { v } _ { 1 } - \\overline { v } _ { 2 } ) \\right ) + 2 \\partial _ { R } \\overline { v } _ { 2 } \\left ( \\cos ( 2 \\overline { v } _ { 1 } ) - \\cos ( 2 \\overline { v } _ { 2 } ) \\right ) \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } d \\xi \\frac { \\sin ( t \\xi ) } { t ^ { 2 } } \\left ( \\psi _ { v _ { 2 } } ( \\xi , \\lambda _ { 1 } ( t ) ) - \\psi _ { v _ { 2 } } ( \\xi , \\lambda _ { 2 } ( t ) ) \\right ) = \\int _ { 0 } ^ { \\infty } d \\xi \\frac { \\cos ( t \\xi ) } { t ^ { 3 } } \\partial _ { \\xi } \\left ( \\psi _ { v _ { 2 } } ( \\xi , \\lambda _ { 1 } ( t ) ) - \\psi _ { v _ { 2 } } ( \\xi , \\lambda _ { 2 } ( t ) ) \\right ) \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} \\delta t \\sum _ { n = 1 } ^ m | | \\phi _ h ^ n | | _ { \\mathcal { T } _ h } ^ 2 \\leq \\frac { C } { \\epsilon } . \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} \\mathcal { L } ^ { - 1 } \\left [ \\frac { 1 } { ( s + \\lambda ) ^ { 1 - \\alpha } } \\right ] = t ^ { - \\alpha } M ^ { 1 - \\alpha } _ { 1 , 1 - \\alpha } ( - \\lambda t ) . \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} & | - 2 \\int _ { t } ^ { t + 2 ( r + 1 ) } d s \\lambda '' ( s ) ( s - t ) \\left ( \\frac { ( r ^ { 2 } - 1 - ( s - t ) ^ { 2 } ) } { \\sqrt { \\beta } ( 1 + r ^ { 2 } + ( s - t ) ^ { 2 } + \\sqrt { \\beta } ) } \\right ) | \\\\ & \\leq C \\sup _ { x \\geq t } \\left ( | \\lambda '' ( x ) | \\right ) \\int _ { t } ^ { t + 2 ( r + 1 ) } d s \\frac { | s - t | } { 1 + r ^ { 2 } } \\\\ & \\leq C \\sup _ { x \\geq t } \\left ( | \\lambda '' ( x ) | \\right ) \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} f ^ { ( k ) } _ 2 ( z ) = \\frac { a _ 0 - e ^ { - z ^ 2 } ( a _ 0 + a _ 2 z ^ 2 + \\ldots + a _ { 2 k } z ^ { 2 k } ) } { z ^ { k + 2 } } , \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} v ( \\beta ) = e ^ \\beta + \\beta - \\frac { 1 } { 2 } - \\frac { 3 e ^ 3 } { 5 \\sqrt { 2 \\pi } } \\beta e ^ { - \\beta } \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} \\sum _ { j \\in \\beta } n _ j + \\sum _ { j \\in \\lambda _ 1 \\cup \\rho _ 2 } n _ j = \\sum _ { j \\in \\lambda _ 2 \\cup \\rho _ 1 } n _ j . \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} | \\partial _ { t r } v _ { 1 } ( t , r ) | \\leq \\begin{cases} \\frac { C } { t ^ { 3 } \\log ^ { b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { r ^ { 2 } t \\log ^ { b + 1 } ( t ) } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} J _ { n } ( x ) = \\frac { 1 } { \\Gamma ( n + \\frac { 1 } { 2 } ) \\sqrt { \\pi } } \\left ( \\frac { x } { 2 } \\right ) ^ { n } \\int _ { 0 } ^ { \\pi } \\cos ( x \\cos ( \\theta ) ) \\sin ^ { 2 n } ( \\theta ) d \\theta \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} R = 4 \\langle S \\overline { v } , ( I I ) \\overline { v } \\rangle - 4 \\langle A \\overline { v } , ( I ) \\overline { v } \\rangle - 4 \\langle A \\overline { v } , ( I I I ) \\overline { v } \\rangle + 4 \\langle ( I ) \\overline { v } , ( I I ) \\overline { v } \\rangle + 4 \\langle ( I I ) \\overline { v } , ( I I I ) \\overline { v } \\rangle . \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\int _ X ( 2 \\pi c _ 1 ( K _ X ) ) ^ n = \\int _ X ( - R i c ( \\hat \\omega _ i ) ) ^ n > 0 , \\ , \\ , a s \\ , \\ , i \\to \\infty , \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} \\| ( 1 + | x | ^ 2 ) \\ , | J | _ 2 \\ , | h | _ 2 \\| _ { L ^ 1 } & \\leq \\| ( 1 + | x | ^ 2 ) \\ , | J | _ 2 \\| _ { L ^ 2 } \\ , \\| | h | _ 2 \\| _ { L ^ 2 } \\leq C \\ , \\| J \\| _ { L ^ 2 } \\ , \\| h \\| _ { W ^ { 2 , 2 } } \\\\ & \\leq C \\ , \\| U \\| _ { L ^ 2 } \\ , \\left ( \\| U \\| _ { L ^ 2 } ^ 2 + \\| \\phi _ { U } \\| _ { L ^ 2 } \\right ) = C \\ , \\| U \\| _ { L ^ 2 } ^ 3 + C \\ , \\| U \\| _ { L ^ 2 } \\ , \\| \\phi _ { U } \\| _ { L ^ 2 } \\ , . \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\downarrow \\gamma _ { c r } } \\frac { \\log \\lambda _ 0 ( \\gamma ) } { \\log \\gamma } = \\frac { \\alpha } { d - \\alpha } , \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} \\zeta _ \\beta ( 0 ) = - \\frac { 1 3 } { 1 2 } + \\frac 1 { 6 ( \\beta + 1 ) } - \\frac 1 { 1 2 ( 2 \\beta + 1 ) } . \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} P _ { m } ( x ) = 2 S _ { m } \\left ( \\frac { x } { 2 } \\right ) . \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} M & = \\left ( \\begin{array} { c c c | c c c | c } 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & m _ { 5 , 5 } & m _ { 5 , 6 } & m _ { 5 , 7 } \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} \\sigma & = \\frac { n } { 1 + m } \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} | | | \\Phi | | | ^ 2 _ m : = \\int _ G | | \\Phi ( g ) | | ^ 2 _ { b } ( 1 + L ( g ) ) ^ { 2 m } d g \\ , . \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} & | | \\frac { \\sin ( 2 ( v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } ) ) } { r ^ { 2 } } \\left ( - \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } + 2 v _ { 5 } ) \\partial _ { r } ( Q _ { \\frac { 1 } { \\lambda ( t ) } } + v _ { 5 } ) + \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\partial _ { r } Q _ { \\frac { 1 } { \\lambda ( t ) } } \\right ) \\vert _ { r = R \\lambda ( t ) } | | _ { L ^ { 2 } ( R d R ) } \\\\ & \\leq \\frac { C \\log ^ { 4 } ( t ) } { t ^ { 2 1 / 4 } } \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} \\psi ( x , x ) = \\| x \\| ^ 2 ( x \\in X ) . \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} A \\left ( \\int _ X f ( x ) \\ , d x \\right ) = \\int _ X A ( f ( x ) ) \\ , d x . \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} Q ( A , B ) \\ = \\ 2 \\cdot | \\{ ( a , b ) \\in A \\times B : a > b \\} | - | A | \\cdot | B | . \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} [ w ] _ { A _ { p , \\mathcal { B } } } : = \\sup _ { B \\in \\mathcal { B } } \\left ( \\frac { 1 } { \\mu ( B ) } \\int _ B w \\ , d \\mu \\right ) \\left ( \\frac { 1 } { \\mu ( B ) } \\int _ B w ^ { 1 - p ' } \\ , d \\mu \\right ) ^ { p - 1 } , 1 < p < \\infty . \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} I _ { 2 , 2 } = \\frac { \\int _ { \\Omega } | \\nabla u | ^ { 4 } \\ , d \\mu } { \\int _ { \\Omega } | u | ^ { 2 } | \\nabla u | ^ { 2 } \\ , d \\mu } . \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} X ' = X _ n \\to \\cdots \\to X _ 0 = X \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} C \\Phi _ 5 ( q ) = \\sum _ { n = 0 } ^ { \\infty } \\bigg ( { p _ { [ 1 ^ 0 5 ^ 1 ] } ( n ) + 2 5 p _ { [ 1 ^ 6 5 ^ { - 5 } ] } ( n - 1 ) } \\bigg ) q ^ n . \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} x \\times y = ( x + y ) ^ \\# - x ^ \\# - y ^ \\# \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} D \\cdot U ( r _ 2 / 2 ) \\sup _ S w _ 1 = m ^ { ( s ) } ( r _ 2 / 2 ) \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} \\begin{aligned} y ( t _ { n + 1 } ) & = y ( 0 ) + \\frac { 1 } { \\Gamma ( 1 - \\epsilon ) } \\int _ { t _ 0 } ^ { t _ { n + 1 } } ( t _ { n + 1 } - \\tau ) ^ { - \\epsilon } z ( \\tau ) d \\tau , \\\\ z ( t _ { n + 1 } ) & = z ( 0 ) + J _ 0 ^ { \\alpha _ 2 - 1 } f ( t _ { n + 1 } ) + \\frac { 1 } { \\Gamma ( \\alpha _ 2 ) } \\int _ { t _ 0 } ^ { t _ { n + 1 } } ( t _ { n + 1 } - \\tau ) ^ { \\alpha _ 2 - 1 } \\Bigl { ( } c ( \\tau ) y ( \\tau ) + b ( \\tau ) z ( \\tau ) \\Bigr { ) } d \\tau . \\end{aligned} \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} \\sin \\phi _ 3 & = \\frac { 4 A } { 2 u _ 1 u _ 2 } \\\\ \\cos \\phi _ 3 & = \\frac { u _ 1 ^ 2 + u _ 2 ^ 2 - u _ 3 ^ 2 } { 2 u _ 1 u _ 2 } \\\\ 1 + \\cos \\phi _ 3 & = \\frac { ( u _ 1 + u _ 2 ) ^ 2 - u _ 3 ^ 2 } { 2 u _ 1 u _ 2 } \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\chi ( X , \\mathcal { O } _ X ( D ) ) = \\dim H ^ 0 ( X , \\mathcal { O } _ X ( D ) ) \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} ( x _ { i } , \\bar { x } _ { i } , p _ { i } , q _ { i } ) \\mapsto z ^ { - w _ { 0 } \\lambda } ( 1 + z ^ { - 1 } \\sum \\limits _ { n , l = 0 } ^ { \\infty } z ^ { - n } q \\bar { x } ^ { n } x ^ { l } p ) , \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { \\ell } v _ i ' = \\sum \\limits _ { i = 3 } ^ { \\ell } v _ i ' \\leq \\sum \\limits _ { i = 3 } ^ { \\ell } ( 5 - v _ i ) = 5 ( \\ell - 2 ) - n + v _ 1 + v _ 2 < n , \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} \\kappa ( g x \\lambda _ 1 , g y \\lambda _ 2 , g z \\lambda _ 3 , g t \\lambda _ 4 ) = \\lambda _ 3 ^ { - 1 } \\kappa ( x , y , z , t ) \\lambda _ 3 \\ . \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} T _ \\gamma ( x , y ) = ( x , \\gamma \\ , y ) . \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} \\left | \\frac { \\Psi ' ( z ( t ) ) } { \\Psi ( z ( t ) ) } \\right | & = \\left | \\frac { 1 } { R ( t ) t } + \\beta \\int _ { \\mathbb { T } } \\frac { 2 \\xi } { ( \\xi - R ( t ) t ) ^ { 2 } } d \\mu _ { 1 } ( \\overline { \\xi } ) \\right | \\\\ & = \\frac { 1 } { R ( t ) } \\left | 1 + \\beta \\int _ { \\mathbb { T } } \\frac { 2 \\xi R ( t ) t } { ( \\xi - R ( t ) t ) ^ { 2 } } d \\mu _ { 1 } ( \\overline { \\xi } ) \\right | \\\\ & \\ge \\frac { 1 } { R ( t ) } \\left [ 1 - \\frac { 2 R ( t ) \\log R ( t ) } { R ( t ) ^ { 2 } - 1 } \\right ] . \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} | J \\psi ( x H ) | & = \\left | \\int _ H \\psi ( h x H ) d h \\right | \\\\ & \\le \\int _ H | \\psi ( h x H ) | d h \\\\ & \\le \\int _ H \\| \\psi \\| _ { \\sup } d h \\\\ & = \\| \\psi \\| _ { \\sup } \\left ( \\int _ H d h \\right ) = \\| \\psi \\| _ { \\sup } , \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} \\beta C _ { n _ { l _ { q } } , 1 } + \\sum _ { k = 2 } ^ { m _ { n _ { l _ { q } } } } \\frac { 2 \\mu _ { k } \\left ( C _ { n _ { l _ { q } } , k } - ( n _ { l _ { q } } - 1 ) \\mathbb { I } _ { k = 2 } \\right ) - \\mu _ { k } ^ 2 } { 4 k } \\left | \\mathbb { P } _ { n _ { l _ q } } \\right . \\stackrel { d } { \\to } M \\{ n _ { l _ { q } } \\} . \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} \\varepsilon : = ( 2 A ) ^ { - 1 } \\exp ( - B \\omega ( m ) ) , \\quad \\quad \\delta : = \\exp ( B \\omega ( m ) - B \\omega ( m + 1 ) ) . \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} \\omega _ H ( U , V ) : = \\omega ( U ^ H , V ^ H ) , \\ , \\ , U , V \\in T B . \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} | | \\gamma ^ V ( t ) | | ^ 2 = - ( t L ) ^ { N - 2 } \\int _ { \\mathbb { R } ^ N } \\left [ | \\nabla \\tilde { u } ^ V ( x ) | ^ 2 + ( t L ) ^ 2 V _ 0 ( t L x + y ) ( \\tilde { u } ^ V ( x ) ) ^ 2 \\right ] d x . \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} ( f g ) '' = f '' g + 2 f ' g ' + f g '' , \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\to + \\infty } \\int _ { \\mathbb { R } ^ N } \\hat { F } _ 0 ( u _ n ( x ) ) \\ ; d x = + \\infty . \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} A ( p ) ( u , u ) = \\sum _ { i , j = 1 } ^ L \\frac { \\partial ^ 2 P _ N } { \\partial p _ i \\partial p _ j } ( p ) u _ i u _ j , \\ p \\in N , \\ u = ( u _ 1 , \\cdots , u _ L ) \\in T _ p N . \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } g _ n ( x ) \\ge c _ 2 x _ d ^ { \\beta _ 2 + \\alpha } \\log ( r _ 0 / x _ d ) = c _ 2 x _ d ^ { p } \\log ( r _ 0 / x _ d ) \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} \\cos \\phi = \\frac { ( n - m ) ( 1 + m n ) } { ( n + m ) ( 1 - m n ) } \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} & - \\tau \\langle u , ( \\partial _ { n + 1 } ^ { 4 } \\phi ) u \\rangle + ( 2 s + 1 ) ( 2 s - 1 ) \\tau \\langle u , x _ { n + 1 } ^ { - 3 } ( \\partial _ { n + 1 } \\phi ) u \\rangle \\\\ = & 2 \\tau ( 1 - 2 s ) ( 1 + 2 s ) ^ { 2 } \\| x _ { n + 1 } ^ { - 1 - s } u \\| ^ { 2 } - 8 \\tau \\mathring { c } _ { s } \\| x _ { n + 1 } ^ { - 1 } u \\| ^ { 2 } . \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( 1 - \\sigma ) D ^ 2 \\tilde v _ j : D ^ 2 \\varphi + \\sigma \\Delta \\tilde v _ j \\Delta \\varphi d x = \\eta \\int _ { \\partial \\Omega } \\frac { \\partial \\tilde v _ j } { \\partial \\nu } \\frac { \\partial \\varphi } { \\partial \\nu } d \\sigma \\ , , \\ \\ \\ \\forall \\varphi \\in \\mathcal { H } ^ 2 _ { 0 , D } ( \\Omega ) , \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} z + F _ { \\mu } ^ { \\langle - 1 \\rangle } ( z ) = F _ { \\mu _ { 1 } } ^ { \\langle - 1 \\rangle } ( z ) + F _ { \\mu _ { 2 } } ^ { \\langle - 1 \\rangle } ( z ) \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} \\begin{aligned} u ( t , x ) = & \\int _ D K ( t , x - y ) ( J + T + 1 ) ( 1 + u _ 0 ( y ) ) d y + \\int _ D S ( t , x - y ) ( J + T + 1 ) v _ 0 ( y ) d y \\\\ & + c _ 1 ( S \\ast u ) ( t , x ) + \\int _ 0 ^ t \\int _ D S ( t - s , x - y ) ( c _ 2 u + f ( u ) ) \\dot { W } ( d s , d y ) , \\end{aligned} \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} \\begin{aligned} & \\lambda _ 1 = \\frac { e ^ { q _ 3 - q _ 2 } - 2 e ^ { 2 q _ 3 - 2 q _ 2 } } { ( 1 + e ^ { q _ 3 - q _ 2 } ) ^ 2 } , \\\\ & \\lambda _ 2 = \\frac { e ^ { q _ 3 - q _ 1 } } { ( 1 + e ^ { q _ 2 - q _ 1 } ) ( 1 + e ^ { q _ 3 - q _ 2 } ) } , \\\\ & \\lambda _ 3 = \\frac { e ^ { q _ 2 - q _ 1 } - 2 e ^ { 2 q _ 2 - 2 q _ 1 } } { ( 1 + e ^ { q _ 2 - q _ 1 } ) ^ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} | | \\overline { v } ( r ) | | _ { L ^ { 2 } ( r d r ) } = | | \\mathcal { F } ^ { - 1 } ( \\overline { y } ) ( r ) | | _ { L ^ { 2 } ( d r ) } = | | \\overline { y } | | _ { L ^ { 2 } ( \\rho ( \\xi ) d \\xi ) } \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} M ^ { m + n } & = \\left ( Q \\setminus \\coprod \\limits _ { i = 1 } ^ { + \\infty } ( B _ { \\frac 4 5 r _ i } ( o _ i ) \\times _ { g _ i } S ^ { n - 1 } ) \\right ) \\cup _ { \\textrm { I d } } \\coprod \\limits _ { i = 1 } ^ { + \\infty } P _ i . \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} D _ { i j } = \\begin{pmatrix} a _ { i , i } & a _ { i , j } \\\\ a _ { i , j } & a _ { j , j } \\end{pmatrix} \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} \\bar { \\gamma } _ { X } = P _ { X } \\left ( \\frac { \\lambda _ W } { 4 \\pi d _ 0 } \\right ) ^ { 2 } \\left ( \\frac { d _ 0 } { d _ { X D } } \\right ) ^ { \\eta } . \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\frac { \\partial ^ 2 } { \\partial a ^ 2 } \\zeta ' _ B ( 0 ; a , 1 , 1 ) - 2 \\left ( \\frac 1 { 1 2 } - \\zeta ' _ R ( - 1 ) \\right ) \\frac 1 { a ^ 3 } - \\sum _ { k = 2 } ^ { N - 1 } \\left ( 1 - \\frac 1 k \\right ) B _ { 2 k } \\zeta _ R ( 2 k - 1 ) a ^ { 2 k - 3 } \\right | \\\\ \\leq \\left ( 1 - \\frac 1 N \\right ) | B _ { 2 N } \\zeta _ R ( 2 N - 1 ) | a ^ { 2 N - 3 } ; \\end{aligned} \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} ( \\nabla ^ \\pm q _ 1 = 0 \\ \\& \\ \\nabla ^ \\pm q _ 2 = 0 \\Rightarrow \\nabla ^ \\pm [ q _ 1 , q _ 2 ] _ Q = 0 ) \\Longleftrightarrow \\nabla ^ \\pm _ \\mu ( T _ { \\nabla ^ \\pm } ) ^ a { } _ { b c } = 0 , \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} \\Psi _ a = E \\rho ( E \\rho ) ^ + \\Psi _ a , \\det ( \\rho ^ T E \\rho ) \\neq 0 , \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} ( T \\eta ) _ n : = \\mathbf { 1 } _ { \\{ ( T S ) _ n - ( T S ) _ { n - 1 } = - 1 \\} } , \\forall n \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} \\partial _ { 2 } \\phi ( r , \\xi ) = \\partial _ { \\xi } \\left ( \\widetilde { \\phi _ { 0 } } ( r ) + \\frac { 1 } { \\sqrt { r } } \\sum _ { j = 1 } ^ { \\infty } ( r ^ { 2 j } \\xi ^ { j } \\phi _ { j } ( r ^ { 2 } ) ) \\right ) \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} \\frac { d } { d t } A ^ { g _ t } = d _ { A ^ { g _ t } } \\gamma _ t . \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} E _ { 0 } ( \\mathbf { z } ) { _ { 0 } \\mathcal { F } _ 0 } ^ { ( d ) } \\left ( \\begin{matrix} \\\\ \\end{matrix} ; \\mathbf { z } , \\mathbf { u } \\right ) & = { _ { 0 } \\mathcal { F } _ 0 } ^ { ( d ) } \\left ( \\begin{matrix} \\\\ \\end{matrix} ; \\mathbf { z } , \\mathbf { u } \\right ) | \\mathbf { u } | . \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} x _ 1 ^ { ( 0 ) } = - \\frac { \\lambda } { 2 E } , \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n ' - 1 } T _ { \\omega _ j } ( Y _ 1 , Y _ 2 ) X ^ j - \\sum _ { j = 0 } ^ { n ' - 1 } T _ { \\omega _ j ' } ( Y _ 1 , Y _ 2 ) X ^ j , \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} t \\ , \\frac { \\partial u } { \\partial t } = a ( t , x ) + \\lambda ( t , x ) u + b ( t , x ) \\frac { \\partial u } { \\partial x } + R \\Bigl ( t , x , u , \\frac { \\partial u } { \\partial x } \\Bigr ) \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} 0 \\longrightarrow R \\xrightarrow { \\begin{pmatrix} - y \\\\ - x \\\\ z \\end{pmatrix} } R ^ 3 \\xrightarrow { \\begin{pmatrix} z & 0 & y \\\\ - x & y & 0 \\\\ 0 & - x z & - x ^ 2 \\end{pmatrix} } R ^ 3 \\xrightarrow { \\begin{pmatrix} x ^ 2 & x z & y \\end{pmatrix} } R \\longrightarrow R / I \\longrightarrow 0 \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} H _ m - \\sum _ { i = L + 1 } ^ { m - 1 } H _ i & > G _ m - \\sum _ { i = L + 1 } ^ { m - 1 } G _ i . \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} \\omega _ { + } = \\sqrt { 2 K b - b ^ { 2 } } . \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} \\langle \\gamma _ 1 , \\gamma _ 2 \\rangle = \\sum _ { \\alpha , \\beta } \\langle \\alpha , \\beta \\rangle \\ , w _ 1 ( \\alpha ) w _ 2 ( \\beta ) , \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} 0 \\leq \\xi ( x ) \\leq 1 , \\ \\ \\ \\xi ( x ) = 1 \\ { \\rm f o r } \\ | x | < \\tilde { \\epsilon } , \\ \\xi ( x ) = 0 \\ { \\rm f o r } \\ | x | > 2 \\tilde { \\epsilon } . \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} \\delta _ i ( \\beta _ 0 , \\beta _ 1 , \\eta ) = \\ss _ 2 ^ \\tau B _ i + \\epsilon _ { i n } ^ { ( 3 ) } \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} T _ { q ^ p - r ^ p } ( a ) = T _ { q - r } ( a ) , \\ ; \\ ; U _ { q ^ p - r ^ p - \\epsilon ( q - r ) - 1 } ( a ) = 0 \\ ; \\ ; \\ ; m o d \\ ; \\ ; \\Phi _ p ( q , r ) . \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} f ^ { \\prime \\kappa } ( g ^ { \\prime * } ( B ) ) = g ^ * ( f ^ \\kappa ( B ) ) . \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} t \\ , \\frac { \\partial u } { \\partial t } = \\lambda ( t , x ) u + b ( t , x ) \\frac { \\partial u } { \\partial x } + R \\Bigl ( t , x , u , \\frac { \\partial u } { \\partial x } \\Bigr ) \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} \\Xi _ n ( [ Z _ n ^ i ] ) = \\xi ^ { 1 ^ n , \\sigma _ i } _ { ( 1 , \\gamma _ { \\sigma _ i } ) } * \\xi ^ { \\sigma _ i , 1 ^ n } _ { ( 1 , \\gamma _ { \\sigma _ i } ) } = \\xi _ { ( 1 , \\gamma _ { \\sigma _ i } ) } + \\xi _ { ( s _ i , \\gamma _ { \\sigma _ i } ) } . \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} \\Psi ( z ) = \\gamma + z + N _ { \\sigma } ( F _ { \\mu _ { 1 } } ( z ) ) , z \\in \\mathbb { H } . \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} f _ R ( 0 , 0 ; \\mu ) = 0 , \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d ^ 4 - \\left ( \\left ( | \\xi | ^ { 2 \\rho } + | \\xi | ^ { 2 \\theta } \\right ) ^ 2 + \\left ( a ^ 2 + b ^ 2 \\right ) | \\xi | ^ 2 \\right ) d ^ 2 + a ^ 2 b ^ 2 | \\xi | ^ 4 = 0 , \\\\ & i d \\left ( | \\xi | ^ { 2 \\rho } + | \\xi | ^ { 2 \\theta } \\right ) \\left ( 2 d ^ 2 - \\left ( a ^ 2 + b ^ 2 \\right ) | \\xi | ^ 2 \\right ) = 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} \\Omega _ \\Sigma = \\int _ \\Sigma \\langle c , \\bar \\dd A ^ { 1 , 0 } - i \\hbar \\ ; \\dd _ { A ^ { 1 , 0 } } \\frac { \\delta } { \\delta A ^ { 1 , 0 } } \\rangle - i \\hbar \\frac 1 2 \\langle [ c , c ] , \\frac { \\delta } { \\delta c } \\rangle \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\frac { \\delta ^ + _ { \\Omega , p } ( t _ n ) } { \\delta ^ - _ { \\Omega , p } ( t _ n ) } = 0 , \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} F _ { \\nu _ { \\beta } } = F _ { \\nu } \\circ F _ { \\mu } \\quad \\mathbb { H } . \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} C \\Phi _ 7 ( q ) = \\sum _ { n = 0 } ^ { \\infty } \\bigg ( p _ { [ 1 ^ 0 7 ^ 1 ] } ( n ) + 7 ^ 2 p _ { [ 1 ^ 4 7 ^ { - 3 } ] } ( n - 1 ) + 7 ^ 3 p _ { [ 1 ^ 8 7 ^ { - 7 } ] } ( n - 2 ) \\bigg ) q ^ n . \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} \\sum _ { g \\in \\Gamma } \\| \\R ( g ) \\| ( 1 + | g | ) ^ { 2 k } & \\leq \\sum _ { g \\in \\Gamma } \\| \\R ( g ) \\| _ { { \\rm H S } } ( 1 + | g | ) ^ { 2 k } \\\\ & = \\sum _ { g \\in \\Gamma } \\int _ { M \\times M } | K _ { \\R ( g ) } | ^ 2 d { \\rm v o l } _ { M \\times M } ( 1 + | g | ) ^ { 2 k } < \\infty \\\\ \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} A _ N ' ( \\sigma _ N ) = A ' ( \\sigma _ { \\infty } ) = m . \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} \\varphi _ { M , \\gamma , n } ( \\bar { x } _ n ) : = \\bigwedge _ { \\beta < \\gamma } \\varphi _ { M , \\beta , n } ( \\bar { x } _ n ) . \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} m _ j + \\ldots + m _ s & = ( n + 1 ) d - ( m _ { 1 } + \\ldots + m _ { j - 1 } ) - 2 - c \\\\ & = ( n + 2 - j ) d + ( d - m _ 1 ) + \\dots + ( d - m _ { j - 1 } ) - 2 - c \\\\ & \\geq ( n + 2 - j ) d - c + ( j - 3 ) . \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} \\beta = \\left ( \\frac { \\partial ^ 2 f } { \\partial \\mu \\partial x } + \\frac { \\partial ^ 2 g } { \\partial \\mu \\partial y } \\right ) \\bigg | _ { x = y = \\mu = 0 } . \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\sqrt { R } \\ge 0 ( 0 , \\infty ) \\times \\R ^ d , \\sqrt { R } U = 0 \\{ \\sqrt { R } = 0 \\} . \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} \\sup _ { h _ 0 \\in \\Gamma _ 0 } \\sup _ { u \\in Q _ 0 } I ( h _ 0 ( u ) ) = \\sup _ { u \\in \\partial Q } I ( h _ 0 ( u ) ) = \\omega < \\alpha \\leq c . \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\widehat { W } = \\begin{bmatrix} ( L - m ) I & - I \\\\ 0 & I \\end{bmatrix} \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} \\mu ^ { I } ( I ) = \\ell ( I ) ^ { d } . \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} d _ 0 : = \\begin{cases} 3 2 t ^ 4 { } \\\\ 8 s t ^ 4 { } \\\\ \\max ( 1 2 ( s + 1 ) ^ 2 , s ^ 3 ) { } . \\end{cases} \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} [ \\check { e } _ i , \\check { e } _ j ] _ { X ^ { * * } Q } : = \\check { C } ^ k { } _ { i j } \\check { e } _ k , \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} = \\sum _ { j = 0 } ^ { k } \\Biggl [ \\binom { 2 k + 1 } { j } - \\binom { 2 k + 1 } { j - 1 } \\Biggr ] \\frac { c _ 1 ^ { j } \\ , c _ 2 ^ { 2 k + 1 - j } } { ( c _ 1 + c _ 2 ) ^ { 2 k + 1 } } \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} V = \\left ( \\begin{array} { c c } e ^ { - D ^ - D ^ + } & e ^ { - \\frac { 1 } { 2 } D ^ - D ^ + } \\left ( \\frac { I - e ^ { - D ^ - D ^ + } } { D ^ - D ^ + } \\right ) D ^ - \\\\ e ^ { - \\frac { 1 } { 2 } D ^ + D ^ - } D ^ + & I - e ^ { - D ^ + D ^ - } \\end{array} \\right ) \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} & ( { h } ^ { 2 } - 2 { h } - \\mu ) r ( { h } - 4 ) - 4 { h } f ( { h } ) - ( { h } ^ { 2 } + 2 { h } - \\mu ) r ( { h } + 4 ) ) \\\\ & = ( - 8 n h - 4 h ) r ( h ) . \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} \\mathcal { L } _ { x } ( s ) & : = \\{ y \\in M \\ , : \\ , G ( x , y ) = s \\} , \\\\ \\mathcal { L } _ { x } ( a , b ) & : = \\{ y \\in M \\ , : \\ , a < G ( x , y ) < b \\} . \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} - b = a f _ 1 ' f _ 1 ^ { - 1 } + f _ 1 '' f _ 1 ^ { - 1 } = a f _ 2 ' f _ 2 ^ { - 1 } + f _ 2 '' f _ 2 ^ { - 1 } . \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} f ( x ) = x ^ { L } - \\sum _ { i = 1 } ^ { L } c _ { i } x ^ { L - i } , \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} \\int _ { 2 a t } ^ x \\frac { e ^ { i ( u ^ 2 / ( 4 t ) - k u ) } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) d u = \\int _ { 2 a t } ^ 0 \\frac { e ^ { i ( u ^ 2 / ( 4 t ) - k u ) } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) d u + \\int _ { 0 } ^ x \\frac { e ^ { i ( u ^ 2 / ( 4 t ) - k u ) } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) d u \\ , . \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} \\lVert A \\rVert _ { \\mathfrak S _ 1 } = \\inf \\sum _ { i = 1 } ^ \\infty \\lVert a _ i \\rVert \\cdot \\lVert y _ i \\rVert , \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} \\partial _ { r } v _ { 1 } ( t , r ) = \\int _ { t } ^ { \\infty } \\frac { \\lambda '' ( s ) } { 1 + s - t } d s + E _ { \\partial _ { r } v _ { 1 } } ( t , r ) \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} | \\partial _ { r } ^ { 2 } v _ { 2 } ( t , r ) | & \\leq \\frac { C r } { t ^ { 4 } \\log ^ { b } ( t ) } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} \\mathsf { \\hat { r } } _ { \\mathsf { k , m } } = - \\frac { \\nu _ { \\mathsf { k } } \\exp \\left [ - \\beta \\lambda _ { \\mathsf { m } } \\right ] } { Z _ { \\frac { \\alpha + 1 } { 2 } \\beta } \\left ( \\nu _ { \\mathsf { k } } + b _ { \\mathsf { m } } \\right ) } \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} { \\left . { \\frac { { { \\rm { d } } \\rho ( \\varepsilon ) } } { { { \\rm { d } } \\varepsilon } } } \\right | _ { \\varepsilon = 0 } } = \\int _ 0 ^ A { \\eta ( x ) \\left \\{ \\ln ( { f _ X ( x ) } ) + 1 + \\frac { 1 } { 2 } \\ln ( 1 + { \\varsigma ^ 2 } x ) \\right \\} { \\rm { d } } x } = 0 . \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} \\left ( \\mathsf { \\hat { p } } _ { \\mathsf { j } } , \\mathsf { \\hat { q } } _ { \\mathsf { 1 } } \\right ) _ { 2 } = \\delta _ { \\mathsf { j , 1 } } \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} \\mathcal { C } _ { J , K } ^ { r } : = \\left \\{ \\eta \\in \\mathcal { C } _ { J , K } : \\ : \\liminf _ { n \\to - \\infty } \\min \\left \\{ \\eta _ n , J - \\eta _ n \\right \\} = r \\right \\} . \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} \\mathcal { B } _ { b } ( E ) & = \\{ f : E \\to \\R \\mid f \\| f \\| _ { E , \\infty } < \\infty \\} , \\\\ C ( E ) & = \\{ f : E \\to \\R \\mid f E \\} , \\\\ C _ { b } ( E ) & = C ( E ) \\cap \\mathcal { B } _ { b } ( E ) , \\\\ C _ { c } ( E ) & = \\{ f : E \\to \\R \\mid f E \\} . \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} & \\mathfrak { G } _ { t } : = \\{ g _ { t } ( t _ { 1 } , t _ { 2 } , t _ { 3 } ) ( s _ { 1 } , s _ { 2 } , s _ { 3 } ) \\\\ & \\{ ( s _ { 1 } , s _ { 2 } , s _ { 3 } ) ^ { \\pm 1 } , ( s _ { 1 } , s _ { 1 } / s _ { 3 } , s _ { 1 } / s _ { 2 } ) ^ { \\pm 1 } , ( s _ { 2 } / s _ { 3 } , s _ { 2 } , s _ { 2 } / s _ { 1 } ) ^ { \\pm 1 } , ( s _ { 3 } / s _ { 2 } , s _ { 3 } / s _ { 1 } , s _ { 3 } ) ^ { \\pm 1 } \\} \\} \\simeq C _ { 2 } ^ { 3 } \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} \\nabla \\vert X \\vert ^ 2 = 2 \\ , \\vert X \\vert \\ , \\nabla \\vert X \\vert . \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} \\Psi ( T _ { n , \\widetilde { a } } ) _ d = \\bigl [ \\beta _ { a , d , j , k } \\bigr ] _ { j , k = \\max \\{ 0 , - d \\} } ^ { n - 1 } . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} \\theta _ { n } ^ { \\prime } ( \\tau ) = \\beta \\varphi _ { n } ^ { 2 } \\left ( \\tau \\right ) \\leq \\beta , \\theta _ { n } ^ { \\left ( j \\right ) } ( \\tau ) = \\beta C _ { j } n ^ { 1 - j } , j = 1 , 2 , . . . . \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} \\log \\left ( \\frac { d \\mathbb { Q } _ { n } } { d \\mathbb { P } _ { n } } \\right ) - \\sum _ { k = 2 } ^ { m _ { n } } \\frac { 2 \\mu _ { k } \\left ( C _ { n , k } - ( n - 1 ) \\mathbb { I } _ { k = 2 } \\right ) - \\mu _ { k } ^ 2 } { 4 k } \\left | \\mathbb { P } _ { n } \\right . \\stackrel { p } { \\to } 0 . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} u _ 1 & = ( 1 - m n ) u \\\\ u _ 2 & = ( m + n ) u \\\\ u _ 3 & = [ 1 + m n - ( n - m ) ] u \\\\ u _ 4 & = [ 1 + m n + ( n - m ) ] u \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} 0 \\leq K _ { 1 } ( w , \\lambda ( t ) ) & = \\int _ { 0 } ^ { \\infty } \\frac { R d R } { 2 w ( 1 + R ^ { 2 } ) ^ { 3 } } \\left ( \\frac { 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } w ^ { 2 } } { 1 + R ^ { 2 } \\lambda ( t ) ^ { 2 } + w ^ { 2 } + \\sqrt { ( 1 + ( R \\lambda ( t ) + w ) ^ { 2 } ) ( 1 + ( R \\lambda ( t ) - w ) ^ { 2 } ) } } \\right ) \\\\ & \\leq C \\int _ { 0 } ^ { \\infty } \\frac { R ^ { 3 } d R } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\frac { \\lambda ( t ) ^ { 2 } w } { 1 + w ^ { 2 } } \\leq C \\lambda ( t ) ^ { 2 } \\frac { w } { 1 + w ^ { 2 } } \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} \\Omega _ n : = \\left \\{ x \\in \\mathbb { R } ^ N : \\dfrac { f _ 0 ( u _ n ( x ) ) } { u _ n ( x ) } \\leq V _ { 0 , \\infty } \\right \\} , \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} h ( t ) = \\frac { \\eta _ { \\mu } ( t ) } { | \\eta _ { \\mu } ( t ) | } , \\ h _ { n } ( t ) = \\frac { \\eta _ { \\mu _ { n } } ( t ) } { | \\eta _ { \\mu _ { n } } ( t ) | } , t \\in \\mathbb { T } , n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} \\Delta ^ { I } n _ { 3 } = \\Delta ^ { I } \\Big ( - \\frac { \\sqrt { \\omega } } { \\sqrt { \\Phi } } \\Big ) = \\lambda _ { 3 1 } \\Big ( - \\frac { a u } { \\sqrt { \\Phi } } \\Big ) + \\lambda _ { 3 2 } \\Big ( - \\frac { b v } { \\sqrt { \\Phi } } \\Big ) + \\lambda _ { 3 3 } \\Big ( \\frac { \\sqrt { \\omega } } { \\sqrt { \\Phi } } \\Big ) . \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} \\partial _ t f + v \\cdot \\nabla _ { \\ ! x } f = \\bar { A } _ g f \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} \\begin{array} { l @ { { } \\iff { } } l } g ( r ) > 0 & 0 = r f ( r ) < g ( r ) \\\\ & r \\left ( r ^ { L } - \\sum _ { i = 1 } ^ { L } c _ { i } r ^ { L - i } - m \\right ) < r ^ { L + 1 } - \\sum _ { i = 1 } ^ { L } c _ { i } r ^ { L + 1 - i } - m \\\\ & m < r m \\\\ & r > 1 . \\end{array} \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} \\Delta _ j f ( x ^ j _ k ) & = \\frac { 1 } { \\mu ( Q ^ j _ k ) } \\sum _ { \\{ i : d ( x ^ j _ i , x ^ j _ k ) < C \\delta ^ j \\} } \\left ( f ( x ^ j _ i ) - f ( x ^ j _ k ) \\right ) \\left ( \\mu ( Q ^ j _ k ) + \\mu ( Q ^ j _ i ) \\right ) H ^ j _ { k i } \\\\ & = \\sum _ { \\{ i : d ( x ^ j _ i , x ^ j _ k ) < C \\delta ^ j \\} } \\left ( f ( x ^ j _ i ) - f ( x ^ j _ k ) \\right ) \\left ( 1 + \\frac { \\mu ( Q ^ j _ i ) } { \\mu ( Q ^ j _ k ) } \\right ) H ^ j _ { k i } . \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} E ' [ W _ q ( f ) ( A ) ^ 2 ] = ( 2 ^ { 2 n } - I _ f ^ 2 ) / ( 2 ^ n - 1 ) , \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} \\sum _ { a = 1 } ^ { k } \\sum _ { i = a } ^ { a + m - 1 } H _ { i } \\geq \\sum _ { i = 1 } ^ { k + m - 1 } H _ { i } + \\sum _ { i = 2 } ^ { k + m - 2 } H _ { i } . \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial \\zeta } { \\partial t } = \\frac { \\partial ^ 2 } { \\partial \\theta ^ 2 } \\varphi ' ( \\zeta ) , \\\\ \\zeta ( 0 , \\cdot ) = \\zeta _ 0 , \\end{cases} \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} | \\frac { \\phi ( r , \\xi ) } { r ^ { 3 / 2 } \\langle \\log ( r ) \\rangle } | \\leq C \\frac { | a ( \\xi ) | \\sqrt { \\xi } } { \\langle \\log ( \\xi ) \\rangle } + C \\begin{cases} \\frac { 1 } { \\langle \\log ( \\xi ) \\rangle } , \\xi \\geq 1 \\\\ 1 , \\xi \\leq 1 \\end{cases} \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} \\int _ { B _ \\delta } \\tau _ i \\tau _ j | \\tau | ^ { - ( k + 2 s ) } d \\tau = \\delta _ { i j } \\int _ { B _ \\delta } \\tau _ 1 ^ 2 | \\tau | ^ { - ( k + 2 s ) } d \\tau = \\delta _ { i j } k ^ { - 1 } \\int _ { B _ \\delta } | \\tau | ^ { 2 - k - 2 s } d \\tau = \\delta _ { i j } k ^ { - 1 } | \\mathbb S ^ { k - 1 } | \\frac { \\delta ^ { 2 - 2 s } } { 2 - 2 s } . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} Q ^ { ( 1 ) } _ m = \\sum _ { k = 0 } ^ { m } \\chi \\left ( L ( k \\omega _ 1 ) \\right ) . \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{align*} a _ { f } ( t ) = | \\{ x \\in \\R ^ { n } : | f ( x ) | > t \\} | . \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} \\Theta ^ { } : \\begin{pmatrix} r \\\\ k _ u \\\\ k _ s \\end{pmatrix} \\mapsto \\begin{pmatrix*} [ l ] A _ c k _ c + g _ c \\circ K { } { } - k _ c \\circ ( A _ c + r ) \\\\ A _ u ^ { - 1 } k _ u \\circ ( A _ c + r ) - A _ u ^ { - 1 } g _ u \\circ K { } { } \\\\ A _ s k _ s \\circ ( A _ c + r ) ^ { - 1 } + g _ s \\circ K { } { } \\circ ( A _ c + r ) ^ { - 1 } \\end{pmatrix*} . \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} \\langle f , g \\rangle & = \\langle \\theta _ { i _ 1 } \\cdots \\theta _ { i _ r } \\overline { \\theta } _ { j _ 1 } \\cdots \\overline { \\theta } _ { j _ s } f ^ \\prime , \\ , \\theta _ { k _ 1 } \\cdots \\theta _ { k _ t } \\overline { \\theta } _ { l _ 1 } \\cdots \\overline { \\theta } _ { l _ u } g ^ \\prime \\rangle \\\\ & : = w ( i , l ^ T ) \\delta _ { r + u , s + t } \\delta _ { ( i , l ^ T ) , ( k , j ^ T ) } \\ , \\langle f ^ \\prime , g ^ \\prime \\rangle , \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d x _ { t } & = ( F _ { t } x _ { t } + f _ { t } ) d t + d w _ { t } , \\\\ x ( 0 ) & = x _ { 0 } , \\\\ d m _ { t } & = ( G _ { t } { x } _ { t } + g _ { t } ) d t + d v _ { t } , \\\\ m ( 0 ) & = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} & i \\rightsquigarrow i + 1 \\rightsquigarrow i + 2 \\rightsquigarrow i + 3 i \\rightsquigarrow i + 3 i = 1 , 5 , \\\\ & 4 \\rightrightarrows 8 , 2 \\rightrightarrows 6 , 5 \\rightrightarrows 1 , 7 \\rightrightarrows 3 . \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } a _ 1 & b _ 1 \\\\ c _ 1 & d _ 1 \\end{array} \\right ) \\dotplus \\left ( \\begin{array} { c c } a _ 2 & b _ 2 \\\\ c _ 2 & d _ 2 \\end{array} \\right ) = \\left ( \\begin{array} { c c c c } a _ 1 & b _ 1 & 0 & 0 \\\\ c _ 1 & d _ 1 & 0 & 0 \\\\ 0 & 0 & a _ 2 & b _ 2 \\\\ 0 & 0 & c _ 2 & d _ 2 \\end{array} \\right ) , \\\\ \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} c _ { \\mathsf { q , } \\alpha , \\beta } : = \\sum _ { \\mathsf { n = 1 } } ^ { \\mathsf { + \\infty } } \\exp \\left [ - \\frac { \\alpha - 1 } { 2 } \\beta \\lambda _ { \\mathsf { n } } \\right ] q _ { \\mathsf { n } } \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{align*} F = \\sum _ { i , j , k } d _ { i j k } \\frac { X ^ i X ^ j X ^ k } { X ^ 0 } ; \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} \\underset { t \\rightarrow t _ 1 - } { l i m } \\underset { x \\in \\overline { D } } { s u p } | u ( t , x ) | = + \\infty \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} \\left \\langle \\frac { \\nabla \\dot { u } ( \\Gamma ( t ) , t ) } { \\abs { \\nabla u ( \\Gamma ( t ) , t ) } } , \\dot { \\Gamma } ( t ) \\right \\rangle \\langle \\nabla u ( \\Gamma ( t ) , t ) , \\nu ( t ) \\rangle = \\langle \\nabla \\dot { u } ( \\Gamma ( t ) , t ) , \\dot { \\Gamma } ( t ) \\rangle \\ ; . \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\infty } \\sup _ { 0 \\leq t \\leq T } \\int \\| \\bar { x } - \\bar { \\eta } ^ { \\nu } ( t , \\cdot ) \\| _ { H ^ { - 1 } } f ( t , x ) \\mu _ { N , m } ( d x ) = 0 . \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} \\vec { \\mathbb { I } } ^ 2 & = \\langle \\langle \\mathbb { I } ^ 2 _ { \\gamma } \\colon \\gamma \\leq \\delta \\rangle , \\langle \\dot { \\mathbb { J } } ^ { 2 } _ { \\gamma } \\colon \\gamma < \\delta \\rangle \\rangle \\\\ \\dot { \\vec { \\mathbb { I } } } ^ 1 & = \\langle \\langle \\dot { \\mathbb { I } } ^ 1 _ { \\gamma } \\colon \\gamma \\leq \\delta \\rangle , \\langle \\dot { \\mathbb { J } } ^ { 1 } _ { \\gamma } \\colon \\gamma < \\delta \\rangle \\rangle \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} J _ 1 ( v ) [ \\varphi ] = ( \\gamma _ 1 ( v ) , \\gamma _ 1 ( \\varphi ) ) _ { \\partial \\Omega } \\ , , \\ \\ \\ \\forall v , \\varphi \\in H ^ 2 ( \\Omega ) . \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} \\mathcal { V } ( M ) = \\int _ { M \\times [ 0 , t ] } \\ * r ^ * ( d V ) \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} \\lim _ { n } \\big ( U ( f , \\Psi , \\mathcal { P } _ n ) - L ( f , \\Psi , \\mathcal { P } _ n ) \\big ) = 0 \\ , , \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} \\gamma ( k , p , q ) = ( k + \\delta _ { p q } , p ) \\ , \\ , , \\mu ( k , p , q ) = ( k , q ) \\ , \\cdot \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} \\begin{cases} ( z - h \\bar { x } ) ^ { - 1 } p ( \\tilde { w } ) = t x ( u ) - y u \\\\ ( y - t x ) ^ { - 1 } p ( w ) = h \\bar { x } ( u ) - z u \\\\ w - \\tilde { w } = t h q ( u ) \\end{cases} \\Rightarrow \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { \\Phi } ^ { ( 0 ) } _ t \\\\ \\dot { \\Psi } ^ { ( 0 ) } _ t \\end{bmatrix} & = A \\begin{bmatrix} \\Phi ^ { ( 0 ) } _ t \\\\ \\Psi ^ { ( 0 ) } _ t \\end{bmatrix} , & \\begin{bmatrix} \\Phi ^ { ( 0 ) } _ 0 \\\\ \\Psi ^ { ( 0 ) } _ 0 \\end{bmatrix} & = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} s _ n - c _ 1 s _ { n - 1 } + c _ 2 s _ { n - 2 } + \\dots + ( - 1 ) ^ n n c _ n & = 0 . \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} g ( s ) & = \\left ( c _ { 1 } \\alpha _ { 1 } \\lambda _ { 1 } ^ { \\alpha _ 1 - 1 } + c _ { 2 } \\alpha _ { 2 } \\lambda _ { 2 } ^ { \\alpha _ 2 - 1 } \\right ) s ^ { - p } + \\left ( c _ { 1 } ( \\alpha _ { 1 } - 1 ) \\lambda _ { 1 } ^ { \\alpha _ 1 - 2 } + c _ { 2 } ( \\alpha _ { 2 } - 1 ) \\lambda ^ { \\alpha _ 2 - 2 } \\right ) s ^ { p - 1 } + . . . . \\\\ & = \\sum _ { k = 0 } ^ { \\infty } { b _ { k } s ^ { k + \\gamma + 1 } } , \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} \\varrho ( x , y ) : = \\inf \\{ \\ell ( \\gamma ) \\colon \\gamma \\colon [ 0 , 1 ] \\to A \\gamma ( 0 ) = x , \\ , \\gamma ( 1 ) = y \\} , \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} w _ i \\rightsquigarrow w _ { i + 1 } w _ { i + 1 } \\rightsquigarrow w _ i i = 0 , 1 , \\ldots , l - 1 , \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\xi _ 5 ^ { n } \\sum _ { a = 0 } ^ n \\xi _ 5 ^ { a } \\sum _ { b = 0 } ^ a \\xi _ 5 ^ b \\sum _ { c = 0 } ^ b \\xi _ 5 ^ c . \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} \\| x \\| \\coloneqq \\| \\langle x , x \\rangle \\| ^ \\frac { 1 } { 2 } = \\left \\| \\sum _ { i = 1 } ^ { n } a _ i a _ i ^ * \\right \\| ^ \\frac { 1 } { 2 } , \\forall x = ( a _ 1 , \\dots , a _ n ) \\in \\mathcal { A } ^ n \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} | S ' \\cap \\left \\{ v _ { i , j } , u _ { l , q } , \\ , : \\ , i , l \\in \\{ 1 , \\dots , n \\} , j \\in \\{ 1 , \\dots , k + 2 \\} , q \\in \\{ 1 , \\dots , k + 3 \\} \\right \\} | = n ( 2 k + 5 ) . \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} ( x - y ) z _ { x y } = \\pm 2 \\sqrt { - z _ x z _ y } . \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} & | Q _ { \\Sigma } ( \\sum _ i a _ i \\phi _ i , \\sum _ i a _ i \\phi _ i ) - Q _ { \\Sigma } ( \\sum _ i a _ i \\psi _ i , \\sum _ i a _ i \\psi _ i ) | \\\\ = \\ & | \\sum _ { 1 \\leq i , j \\leq I } a _ i a _ j ( Q _ { \\Sigma } ( \\phi _ i , \\phi _ j ) - Q _ { \\Sigma } ( \\psi _ i , \\psi _ j ) ) | \\leq C ( \\Sigma , I ) \\varepsilon \\cdot \\sum _ i a _ i ^ 2 \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} \\hat H = p _ 1 p _ 2 \\left ( e ^ { - | q _ 1 - q _ 2 | } - 1 \\right ) \\ ; \\ ; \\mbox { f o r } \\ ; \\ ; n = 2 \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} M = \\max \\Big \\{ \\| e ^ { \\tau _ { 1 0 } A } ( \\overline { v } _ 0 ) _ 1 \\| _ { H ^ r } ^ 2 + \\| e ^ { \\tau _ { 1 0 } A } ( \\widetilde { v } _ 0 ) _ 1 \\| _ { H ^ r } ^ 2 , \\| e ^ { \\tau _ { 2 0 } A } ( \\overline { v } _ 0 ) _ 2 \\| _ { H ^ r } ^ 2 + \\| e ^ { \\tau _ { 2 0 } A } ( \\widetilde { v } _ 0 ) _ 2 \\| _ { H ^ r } ^ 2 \\Big \\} . \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} \\phi _ { s , t } ( x _ 1 , x _ 2 , x _ 3 ) : = { 1 \\over s ^ 2 t ^ 2 } \\phi ^ { ( 1 ) } ( { x _ 1 \\over s } ) \\phi ^ { ( 2 ) } ( { x _ 2 \\over t } , { x _ 3 \\over s t } ) . \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} a _ k ( n ) = \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { n } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - m ^ 2 ) + \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { \\frac { n } { 2 } } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - 2 m ^ 2 ) . \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} \\partial _ { r } ^ { 2 } v _ { 2 } ( t , r ) = I _ { r r } + I I _ { r r } \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} | \\alpha ( n ) - \\alpha ( m ) | & = \\int _ { m \\leq d ( x , x _ { 0 } ) \\leq n } | f ( x ) | d { \\rm v o l } ( x ) \\\\ & \\leq C \\int _ { m \\leq d ( x , x _ { 0 } ) \\leq n } e ^ { - q d ( x , x _ { 0 } ) } d { \\rm v o l } ( x ) \\quad \\mbox { f o r a l l } ~ q \\in \\mathbb { N } , \\\\ & \\leq C ' e ^ { - q m + n p } , \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\lambda + b - K + K e ^ { - \\lambda \\tau } = 0 . \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\int _ { \\Delta } | J _ 1 | ^ 2 d \\sigma = 0 , \\lim _ { t \\to \\infty } \\int _ { \\Delta } | J _ 2 | ^ 2 d \\sigma = 0 \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} K ^ { \\alpha _ \\varepsilon } _ { ( 0 , x _ \\varepsilon - a ) } [ \\varphi , p _ \\varepsilon ] ( x _ \\varepsilon ) = \\int _ 0 ^ { x _ \\varepsilon - a } \\frac 1 { z ^ { \\alpha _ \\varepsilon } } \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\varphi '' ( x _ \\varepsilon - s t z ) t \\ , d s d t . \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} G _ { m + 1 } - \\sum _ { i = L + 1 } ^ { m } G _ i = G _ { m + 1 } - \\sum _ { i = L + 1 } ^ { m - 1 } G _ i - G _ m & \\geq G _ { m + 1 } - 2 G _ m + H _ m - \\sum _ { i = L + 1 } ^ { m - 1 } H _ i . \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} \\frac { 2 c _ { b } } { \\lambda ( t ) } \\int _ { 0 } ^ { 1 / 2 } d \\xi \\frac { \\sin ( t \\xi ) } { t ^ { 2 } } \\left ( \\frac { b - 1 } { \\xi \\log ^ { b } ( \\frac { 1 } { \\xi } ) } + \\frac { b ( b - 1 ) } { \\xi \\log ^ { b + 1 } ( \\frac { 1 } { \\xi } ) } \\right ) = \\frac { 2 c _ { b } } { \\lambda ( t ) } \\int _ { 0 } ^ { t / 2 } d u \\frac { \\sin ( u ) } { t ^ { 2 } } \\left ( \\frac { b - 1 } { u \\log ^ { b } ( \\frac { t } { u } ) } + \\frac { b ( b - 1 ) } { u \\log ^ { b + 1 } ( \\frac { t } { u } ) } \\right ) \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} \\psi _ \\Lambda ( . ) : = \\langle \\Psi _ { \\Lambda } , ( . ) \\Psi _ { \\Lambda } \\rangle \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{gather*} e _ 1 : = v w \\frac { \\partial } { \\partial u } + u \\frac { \\partial } { \\partial v } , e _ 2 : = u \\frac { \\partial } { \\partial u } + v \\frac { \\partial } { \\partial v } , \\\\ e _ 3 : = v ^ 2 \\frac { \\partial } { \\partial u } + 2 u \\frac { \\partial } { \\partial w } , e _ 4 : = v \\frac { \\partial } { \\partial v } - 2 w \\frac { \\partial } { \\partial w } . \\end{gather*}"}
-{"id": "3008.png", "formula": "\\begin{align*} & P _ { L + 1 , \\ , 1 } ( t ) = P ' _ { L , \\ , 1 } ( t ) , \\\\ & P _ { L + 1 , \\ , j } ( t ) = P ' _ { L , \\ , j } + P _ { L , \\ , j - 1 } D R , 2 \\leq j \\leq L - 1 , \\\\ & P _ { L + 1 , \\ , L } ( t ) = P _ { L , \\ , L - 1 } D R - L ( D R ) ^ { L - 1 } D ^ 2 R , \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} x _ 0 = \\mu \\cos u \\cosh v , \\ x _ 1 = \\mu \\sin u \\sinh v \\cos \\phi , \\ x _ 2 = \\mu \\sin u \\sinh v \\sin \\phi , \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} D = D ' + D '' . \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} g _ { 5 , 3 } ( ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { 2 } , { \\bf 3 } , { \\bf 3 } ) ) = ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { 2 } , { \\bf 3 } , 7 , { \\bf 3 } ) . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} \\big ( \\varphi , \\psi \\big ) ( x , t ) = \\big ( \\varphi ( x , t ) , \\psi ( t ) \\big ) : = \\big ( \\dfrac { \\tanh ( x ) } { 1 + \\vert t \\vert } , \\tanh ( t ) \\big ) , \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } d _ 0 ( ( \\Gamma _ m , p ) , ( S _ 0 , p ) ) = 0 \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} \\rho ( x ) & = \\sum _ { n = 1 } ^ \\infty \\frac { b _ n } { n ! } \\phi ^ n ( x ) . \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} \\int _ { { \\Bbb R } ^ 2 } \\phi ( x _ 1 , x _ 2 , x _ 3 ) d x _ 1 d x _ 2 = \\int _ { { \\Bbb R } ^ 2 } \\phi ( x _ 1 , x _ 2 , x _ 3 ) d x _ 2 d x _ 3 = \\int _ { { \\Bbb R } ^ 2 } \\phi ( x _ 1 , x _ 2 , x _ 3 ) d x _ 3 d x _ 1 = 0 . \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} D = \\left ( \\begin{array} { c c } \\lambda _ 1 & 0 \\\\ 0 & \\lambda _ 2 \\end{array} \\right ) \\mbox { a n d } C = \\left ( \\begin{array} { c c } - \\sigma _ 1 - \\lambda _ 1 & \\sigma _ 1 \\\\ \\sigma _ 2 & - \\sigma _ 2 - \\lambda _ 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} E _ { 7 ( - 5 ) } ~ = \\mathcal { N } _ { 2 } ^ { 7 ( - 5 ) - } \\oplus ~ s o ( 7 , 3 ) \\oplus s u ( 2 ) \\oplus s o ( 1 , 1 ) \\oplus \\mathcal { N } _ { 2 } ^ { 7 ( - 5 ) + } \\newline \\mathbf { , } \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} \\left ( \\frac { a _ 1 ' } { r _ 1 ' } ( l _ 1 - k _ 1 ) + I _ { k _ 1 l _ 1 } ^ 1 , \\cdots , \\frac { a _ s ' } { r _ s ' } ( l _ s - k _ s ) + I _ { k _ s l _ s } ^ s , I ^ { s + 1 } , \\cdots , I ^ n \\right ) = 0 . \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} & \\langle e _ { a , 1 } ( t , R \\lambda ( t ) ) , \\phi _ { 0 } ( R ) \\rangle _ { L ^ { 2 } ( R d R ) } \\\\ & = - \\frac { 4 } { \\lambda ( t ) } \\int _ { t } ^ { \\infty } \\frac { \\lambda '' ( s ) } { 1 + s - t } d s - 2 \\int _ { 0 } ^ { \\infty } \\sin ( t \\xi ) \\widehat { v _ { 2 , 0 } } ( \\xi ) \\xi ^ { 2 } K _ { 1 } ( \\xi \\lambda ( t ) ) d \\xi + \\frac { 4 \\lambda '' ( t ) \\log ( \\lambda ( t ) ) } { \\lambda ( t ) } + f _ { 1 } ( \\lambda ( t ) , \\lambda ' ( t ) , \\lambda '' ( t ) ) \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} K { } { } = \\iota + \\begin{pmatrix} k _ c \\\\ k _ u \\\\ k _ s \\end{pmatrix} \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} - \\gamma : = \\frac { \\lambda _ { n } - ( n + 1 ) \\widehat { \\lambda } _ { n } + 1 } { n ( 1 + \\lambda _ { n } ) } . \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} \\mathcal { S } ^ { l i n } : = \\left \\{ S \\in \\mathcal { S } ^ { 0 } \\ : : \\ : \\lim _ { | n | \\rightarrow \\infty } \\frac { S _ n } { n } = c \\mbox { f o r s o m e } c > 0 \\right \\} , \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} & x _ { 1 } x _ { 2 } x _ { 3 } + x _ { 1 } x _ { 4 } x _ { 5 } + x _ { 1 } x _ { 6 } x _ { 7 } + x _ { 2 } x _ { 4 } x _ { 6 } + x _ { 2 } x _ { 5 } x _ { 7 } + x _ { 3 } x _ { 5 } x _ { 6 } + x _ { 3 } x _ { 4 } x _ { 7 } . \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} \\tilde { F } \\circ ( \\tilde { K } + \\Delta ) - ( \\tilde { K } + \\Delta ) \\circ R = 0 . \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} H _ { L + k - 1 } = H _ { L + k - 2 } + H _ { k + m - 1 } + \\dots + H _ { k } + N H _ { k - 1 } . \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} \\frac { 1 } { A + B } = \\frac { 1 } { A } - \\frac { 1 } { A } B \\frac { 1 } { A } + \\frac { 1 } { A } B \\frac { 1 } { A } B \\frac { 1 } { A } - \\left ( \\frac { 1 } { A } B \\right ) ^ { 3 } \\frac { 1 } { A } \\cdots \\pm \\left ( \\frac { 1 } { A } B \\right ) ^ { k } \\frac { 1 } { A + B } . \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} | v _ { 1 } ^ { \\lambda } + v _ { 2 } + v _ { 3 } ^ { \\lambda } | \\leq \\begin{cases} \\frac { C r } { t ^ { 2 } \\log ^ { b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { \\sqrt { r } } , t - \\sqrt { t } \\leq r \\leq t + \\sqrt { t } \\\\ \\frac { C \\log ( r ) } { | t - r | } , \\frac { t } { 2 } \\leq r \\leq t - \\sqrt { t } , r \\geq t + \\sqrt { t } \\end{cases} \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} p = E ^ { - 1 } \\dot { q } = g ( \\dot { q } , . ) , \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} \\partial _ t \\Delta _ q X + \\Delta _ q ( u \\cdot \\nabla X ) = \\Delta _ q \\partial _ X u . \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} P ( i ) = i - | \\{ j \\in J / j < i \\ , , \\phi ( j , i ) < 0 \\} | \\ , \\cdot \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} \\sum _ { x = 0 } ^ { \\infty } \\dfrac { ( N _ x + 1 ) ^ { 1 - 2 \\rho } } { ( x + 1 ) ^ { 2 \\alpha } } < + \\infty , { \\mbox { a . s . o n $ E $ } . } \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t / 2 } \\frac { d y } { y } \\frac { e ^ { - y } } { \\log ^ { a + 1 } ( t / y ) } = \\frac { 1 } { a \\log ^ { a } ( t ) } + O \\left ( \\frac { 1 } { \\log ^ { a + 1 } ( t ) } \\right ) \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ \\infty \\left ( { a \\atop b + \\alpha n } \\right ) v ^ { b + \\alpha n } = \\frac { 1 } { \\alpha } ( 1 + v ) ^ a , | v | = 1 , ~ | \\arg v | < \\pi , ~ 0 < \\alpha \\le 1 , \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} \\left < p _ i , p _ j \\right > = \\delta _ { i j } , \\quad 1 \\leq i , j \\leq d , \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} E ( \\mu ) \\dot { x } ( t , \\mu ) & = A ( \\mu ) x ( t , \\mu ) + B ( \\mu ) u ( t ) \\\\ [ 1 e x ] y ( t , \\mu ) & = C ( \\mu ) x ( t , \\mu ) . \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} \\chi ( X , \\mathcal { O } _ { X } ( D ) ) = \\binom { 7 } { 4 } - 7 \\binom { 5 } { 4 } = 0 \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} \\dot { x } ^ 2 = \\frac { k } { ( x ^ 2 + y ) \\ , y _ x } = \\frac { 2 x - \\tilde { c } _ 1 } { ( x ^ 2 + y ) ^ 2 } \\ , k \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} V _ { \\ell } = R ^ { 1 / 4 } _ { \\ell } U _ { \\ell } \\dfrac { \\phi ( R _ { \\ell } ) } { R ^ { 1 / 4 } _ { \\ell } } \\in L ^ 4 _ { \\rm l o c } ( ( 0 , \\infty ) \\times \\mathbb T ^ d _ { \\ell } ) . \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} \\left \\vert \\nabla e ^ { ( t - u ) \\Delta } V ^ { N } ( x ) \\right \\vert ^ { 2 } & = \\frac { C } { ( t - u ) ^ 2 } \\left | \\int _ { \\R ^ d } ( x - y ) \\ , g _ { 2 ( t - u ) } ( x - y ) \\ , V ^ N ( y ) \\ , d y \\right | ^ 2 \\\\ & \\leq \\frac { C } { ( t - u ) ^ 2 } \\int _ { \\R ^ d } \\left | x - y \\right | ^ 2 \\ , V ^ N ( y ) ^ 2 \\ , g _ { 2 ( t - u ) } ( x - y ) \\ , d y . \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} f _ k ( { x } ) = f _ 1 \\left ( H ^ { k - 1 } ( { x } ) \\right ) = f _ 1 \\left ( \\left ( \\mathbf { 1 } _ { \\{ H ^ { k - 1 } _ W ( \\Delta ( { w } ) ) _ n = 1 \\} } \\right ) _ { n = 1 } ^ l \\right ) , \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} p ( d ) : = \\left \\{ \\begin{array} { c l } \\frac { 2 d } { d - 2 } & \\mbox { i f } d > 2 \\\\ \\infty & \\mbox { i f } d \\leq 2 \\end{array} \\right . . \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} \\ss _ 2 = ( \\beta _ 1 ^ 2 , ~ \\beta _ 0 \\beta _ 1 , ~ \\beta _ 0 ^ 4 ) ^ \\tau . \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} \\omega ^ \\frac { Q - \\epsilon } { 2 } = \\delta \\ ; \\ ; m o d \\ ; \\ ; Q , \\ ; \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\int _ { - \\infty } ^ { - 1 } \\big [ u _ i ^ * - \\phi _ i ( x ) \\big ] | x | ^ { \\alpha - 1 } d x = \\infty . \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} | v _ { 3 } ( t , r ) | & \\leq \\frac { 1 } { r } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } | \\lambda '' ( s ) | \\cdot 2 \\\\ & \\leq \\frac { C } { r } \\int _ { t } ^ { \\infty } d s | \\lambda '' ( s ) | ( s - t ) \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} f ( u _ { n + 1 } ) - f ( u _ { n } ) = f ' ( \\xi _ n ) ( u _ { n + 1 } - u _ { n } ) = \\xi _ n ^ { - \\alpha } ( u _ { n } - u _ { n + 1 } ) , \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} \\Delta ^ { I } n _ { 3 } = \\Delta ^ { I } \\Big ( - \\frac { 1 } { \\sqrt { g } } \\Big ) = \\lambda _ { 3 1 } \\Big ( - \\frac { a u } { \\sqrt { g } } \\Big ) + \\lambda _ { 3 2 } \\Big ( - \\frac { b v } { \\sqrt { g } } \\Big ) + \\lambda _ { 3 3 } \\Big ( \\frac { 1 } { \\sqrt { g } } \\Big ) . \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} \\overline { \\lim } _ { n \\to \\infty } \\sqrt [ n ] { | \\xi _ n | } = \\overline { \\lim } _ { n \\to \\infty } \\sqrt [ n ] { \\sigma _ n } . \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} Q ( A , B ) \\ = \\ \\sum _ { ( i , j ) \\in [ k ] \\times [ \\ell ] } \\ Q ( A _ i , B _ j ) . \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} \\mathcal { L } _ \\circ ^ \\beta = \\{ p \\in \\mathcal { L } ^ \\beta : \\exists A \\in p \\ , \\ , \\ , \\ , \\lambda ( A ) = 0 \\} , \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} \\lim _ { t \\to + 0 } \\Bigl ( \\sup _ { x \\in D _ R } | t ^ { - d } u ( t , x ) | \\Bigr ) = 0 \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} B ( e , f ) = B ( e ^ 2 , f ) = e B ( e , f ) + B ( e , f ) e = e f B ( e , f ) + f B ( e , f ) e = f B ( e , f ) e . \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} \\frac { \\partial \\sqrt { I _ 4 ( p , q ) } } { \\partial q _ I } = \\frac { 1 } { 2 \\sqrt { I _ 4 ( p , q ) } } \\frac { \\partial I _ 4 } { \\partial q _ I } . \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} \\Sigma _ E = \\pi ( \\Sigma _ { i j } ) = \\pi \\left ( \\bigcap _ { k \\geq i } p _ { i j k } ( A _ { k j } ) \\right ) = \\bigcap _ { n \\in \\N } \\pi ( p _ { i j k } ( A _ { k j } ) ) . \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial ^ 2 u } { \\partial \\nu ^ 2 } = \\lambda \\frac { \\partial u } { \\partial \\nu } , & { \\rm o n \\ } \\partial B , \\\\ - { \\rm d i v } _ { \\partial \\Omega } ( D ^ 2 u \\cdot \\nu ) _ { \\partial \\Omega } - \\frac { \\partial \\Delta u } { \\partial \\nu } = \\mu u , & { \\rm o n \\ } \\partial B , \\end{cases} \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 1 ^ - } \\sup _ { x \\in [ 0 , l ] } | J ^ { 1 - \\alpha } f '' ( x ) - J ^ 0 f '' ( x ) | = 0 . \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} \\big \\{ x \\ ! \\in \\ ! \\bar { \\Omega } : { \\rm d i s t } ( x , \\partial \\Omega ) < \\delta _ 0 \\big \\} \\subset \\bigcup _ { i = 1 } ^ n B _ { \\delta _ i } ( x _ i ) . \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} \\frac { 1 } { \\log ( \\log ( t ) ) } \\int _ { t } ^ { \\infty } \\frac { d s } { s ^ { 2 } \\log ^ { b + 1 } ( s ) ( 1 + s - t ) ^ { 3 } } | \\frac { 1 } { \\lambda _ { 1 } ( t ) ^ { 1 - \\alpha } + s - t } - \\frac { 1 } { \\lambda _ { 2 } ( t ) ^ { 1 - \\alpha } + s - t } | \\leq \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { ( \\log ( \\log ( t ) ) ) ^ { 3 / 2 } t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} v _ { 4 } = v _ { 4 } ^ { 0 } ( t , r ) + v _ { 4 } ^ { 1 } ( t , r ) \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} \\nu _ k ( a ) = \\left ( \\sum _ { g \\in \\Gamma } ( 1 + | g | ) ^ { 2 k } | a ( g ) | ^ 2 \\right ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} h _ { C , C _ 1 , D , D _ 1 , E , E _ 1 , E _ 2 } ( x ) = & H _ { C , C _ 1 , D , D _ 1 , E , E _ 1 , E _ 2 } ( x ) \\\\ & \\indent + H _ { C , C _ 1 , D , D \\cup \\{ 0 \\} , E , E _ 1 , E _ 2 } ( x ) h _ { C , C , D , D , E , E , E } H _ { C , C \\cup \\{ 0 \\} , D , D _ 1 , E , E _ 1 , E _ 2 } ( x ) . \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} x _ p & : = p ^ 2 - m ^ 2 . \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} A \\cdot _ G A = \\{ i - j : i , j \\in [ n ] \\} \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} \\mathcal { L } _ { x } \\left ( g _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( x , t ) \\right ) = e ^ { - t \\left ( c _ 1 \\left ( ( s + \\lambda _ 1 ) ^ { \\alpha _ 1 } - \\lambda _ 1 ^ { \\alpha _ 1 } \\right ) + c _ 2 \\left ( ( s + \\lambda _ 2 ) ^ { \\alpha _ 2 } - \\lambda _ 2 ^ { \\alpha _ 2 } \\right ) \\right ) } = { \\overline { G } } ( s , t ) ( s a y ) . \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} - \\partial _ { t t } v _ { 4 } + \\partial _ { r r } v _ { 4 } + \\frac { 1 } { r } \\partial _ { r } v _ { 4 } - \\frac { v _ { 4 } } { r ^ { 2 } } = v _ { 4 , c } \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} v ' + 2 \\delta r - \\sigma v ^ \\frac { 1 } { 2 } = ( 4 \\delta ^ \\frac { 1 } { 2 } - \\sigma ) \\delta ^ \\frac { 1 } { 2 } r = 0 . \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} T ( \\zeta ) = \\lim _ { r \\uparrow 1 } T ( r \\zeta ) = 2 \\int _ { \\mathbb { T } } \\frac { d \\sigma ( t ) } { | t - \\zeta | ^ { 2 } } . \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{align*} H ( u ) = | | ( 1 - P ) h _ 0 ( u ) | | e + P ( h _ 0 ( u ) ) = : H _ 0 ( u ) \\not = \\rho e , \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} r _ { a ^ { - 1 } } ( r _ a ( x ) ) = x a a ^ { - 1 } = x = x a ^ { - 1 } a = r _ a ( r _ a ^ { - 1 } ( x ) ) . \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} \\det { } _ { T B \\otimes E _ k ^ * } \\big ( ( \\Theta ^ { E _ k } ) ^ T / 2 \\pi \\big ) ^ { 1 / N _ k } = \\det { } \\big ( \\Theta ^ { E _ k } ( \\partial z _ i , \\overline { \\partial z } _ j ) ^ T / 2 \\pi \\big ) ^ { 1 / N _ k } \\cdot m ! \\cdot \\nu _ B , \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} \\frac { d } { d x _ i } \\tilde { \\Psi } _ l ^ c = \\frac { d } { d x _ i } \\left ( H ( \\bar { x } ^ { B ( l ) } ) + \\tilde { \\Psi } _ l ^ c \\right ) i \\in \\Lambda _ 2 ^ { ( l ) } . \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ { \\lfloor { N / 2 } \\rfloor } ( a _ k + a _ { N - k } ) U _ k ^ 2 \\right | \\leq \\frac { C ' } { N } \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} h ( x ) + \\frac { 1 } { 2 } h ' ( x ) - h '' ( x ) = 0 , \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} F ( x , y ; \\mu ) = \\begin{bmatrix} f ( x , y ; \\mu ) \\\\ g ( x , y ; \\mu ) \\end{bmatrix} , \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} V _ i ( \\bar { x } _ 0 , \\ldots , \\bar { x } _ \\lambda ) = ( \\bar { x } _ 0 - \\bar { x } _ i , \\ldots , \\bar { x } _ { i - 1 } - \\bar { x } _ i , - \\bar { x } _ i , \\bar { x } _ { i + 1 } - \\bar { x } _ i , \\ldots , x _ \\lambda - \\bar { x } _ i ) \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} S _ 1 = \\frac { X ^ { 1 / 2 } } { 3 d } + O ( 1 ) . \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} \\partial _ { x _ i } Z = 0 , \\ i = 1 , \\cdots , n - 1 . \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} F ( X , Y , Z ) = X ( Y - X ) - Z . \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} \\omega = \\sum _ { j = 1 } ^ r G _ j d L _ j . \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} \\tilde A _ { h , \\tilde h } = e ^ { - m \\hat { \\tilde h } s _ { \\tilde h } } D _ { s _ { \\tilde h } } ^ j \\hat { \\tilde h } ^ { m - j } P _ { m - j } ( h , h e ^ { \\hat { \\tilde h } s _ { \\tilde h } } \\hat x ' , y , \\hat { \\tilde h } { } ^ { - 1 } D _ { Y _ { \\tilde h } } ) \\bigl ( \\delta ( s _ { \\tilde h } ) \\delta ( Y _ { \\tilde h } ) \\bigr ) \\cdot \\hat { \\tilde h } { } ^ { - n } . \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} r + 2 s + N _ 0 ( n - 2 ) = N _ 2 ( n ) , \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} \\eta _ { \\rho _ { 1 } } ( \\mathbb { H } ) = \\{ r e ^ { i \\theta } : r > 0 , f ( r ) < \\theta < \\pi \\} \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} \\frac { d \\mathrm { R e } ( \\lambda ) } { d \\tau } \\Big \\vert _ { \\tau = \\tau _ { j } } = \\frac { \\omega _ { + } ^ { 2 } } { ( \\cos \\omega _ { + } \\tau _ { j } - K \\tau _ { j } ) ^ { 2 } + ( \\sin \\omega _ { + } \\tau _ { j } ) ^ { 2 } } > 0 ; \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} I _ \\mathcal { F } ( M ) ( u , u ) = \\delta ^ 2 \\mathcal { F } ( M ) , \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} \\underline U ( t , \\pm \\underline h ( t ) ) = ( 1 - \\epsilon ( t ) ) [ \\Phi ( - 2 \\underline h ( t ) ) - \\mathbf { u } ^ * ] \\prec \\mathbf { 0 } \\mbox { f o r } \\ t \\geq 0 . \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c _ 2 \\} = \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} G ( t ; x , q , u , z , 1 ) & = G ( t ; x , q , u , z ) = G ( t ; u , z , x , q ) \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} P ^ { ( n , k ) } & = \\left \\lbrace \\left \\lbrace P _ j \\right \\rbrace _ j : P _ 1 \\sqcup \\ldots \\sqcup P _ k = \\left \\lbrace 1 , \\ldots , n \\right \\rbrace , \\abs { P _ j } \\geq 1 \\right \\rbrace \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} R i c ( T , T ) & = - m \\frac { \\bar { u } _ { \\bar { t } \\bar { t } } } { \\bar { u } } - ( n - 1 ) \\frac { \\bar { g } _ { \\bar { t } \\bar { t } } } { \\bar { g } } \\\\ & \\ge ( n - 1 ) l _ i ^ 2 - m ( \\frac 1 N + \\frac 1 { N ^ 2 } ) l _ i ^ 2 \\\\ & \\ge \\frac m 2 l _ i ^ 2 - m ( \\frac 1 5 + \\frac 1 { 2 5 } ) l _ i ^ 2 \\\\ & > 0 , \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} \\sum _ { x = 0 } ^ { \\infty } { \\dfrac { ( N _ x + 1 ) ^ { - \\rho / p + ( 1 - 2 \\rho ) / q } } { x + 1 } } \\leq \\left ( \\sum _ { x = 0 } ^ { \\infty } \\dfrac { ( N _ x + 1 ) ^ { - \\rho } } { ( x + 1 ) ^ { \\alpha } } \\right ) ^ { 1 / p } \\left ( \\sum _ { x = 0 } ^ { \\infty } \\dfrac { ( N _ x + 1 ) ^ { 1 - 2 \\rho } } { ( x + 1 ) ^ { 2 \\alpha } } \\right ) ^ { 1 / q } < + \\infty . \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} H ^ { \\circ 2 n + 1 } ( z , g ^ { \\circ n ^ 2 } ( w ) ) = H ^ { \\circ 2 n + 1 } ( z , w _ { n ^ 2 } ) = ( h _ { n ^ 2 + 2 n } \\circ \\ldots \\circ h _ { n ^ 2 + 1 } \\circ h _ { n ^ 2 } ( z ) , w _ { n ^ 2 + 2 n + 1 } ) \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} \\mathbb { G } _ { N } f : = \\frac { 1 } { \\sqrt { N } } \\sum _ { i = 1 } ^ { N } ( f ( Y _ { i } ) - P _ { y } f ) , f \\in \\mathcal { F } . \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} \\limsup \\limits _ { n \\rightarrow \\infty } d ( \\mu _ n , \\mathbf { 0 } ) & \\leq \\limsup \\limits _ { n \\rightarrow \\infty } d _ { r , \\emptyset , 0 } ( \\mu _ n , \\mathbf { 0 } ) = \\limsup \\limits _ { n \\rightarrow \\infty } I _ r ( \\mu _ n ) + I _ r ( \\mathbf { 0 } ) + 2 ^ { - r } = 2 ^ { - r } . \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} ( a ) u _ 1 ' = u _ 1 = 0 \\quad \\frac { u _ 2 ' } { u _ 2 } > 0 \\qquad \\qquad ( b ) \\frac { u _ 1 ' } { u _ 1 } > 0 \\quad \\frac { 1 } { u _ 1 ' } e ^ \\frac { u _ 2 ' } { u _ 1 ' } = \\frac { 1 } { u _ 1 } e ^ \\frac { u _ 2 } { u _ 1 } = : u \\ , . \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} U _ n ^ \\pm \\geq 0 r \\geq 0 , U _ n ^ \\pm \\rightarrow t ^ { \\pm } \\mbox { a s } r \\rightarrow \\infty , \\\\ [ 2 m m ] U _ n ^ \\pm ( 0 ) = 0 \\mbox { i f } n _ { \\pm } \\neq 0 , { U _ n ^ \\pm } ' ( 0 ) = 0 \\mbox { i f } n _ { \\pm } = 0 . \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} | f ' ( t ) | & = \\left | \\frac { f ' ( t ) } { f ( t ) } \\right | = \\left | \\frac { \\Psi ' ( z ( t ) ) } { \\Psi ( z ( t ) ) } \\right | | z ' ( t ) | . \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} f _ { 5 , 1 } ( ( 0 , 1 , 2 , 0 , 1 , 2 , 2 ) ) = ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , 2 ) ; \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} \\mathfrak { F } _ { Y M } = \\left ( \\mathcal { F } _ { Y M } , L ^ \\bullet _ { Y M } , \\theta _ { Y M } ^ \\bullet , Q \\right ) \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} p _ \\theta ( { \\bf { x } } ) = S ( \\theta ) ^ { \\frac { 1 } { \\alpha - 1 } } N _ { \\theta , \\alpha } \\big [ 1 + b _ \\alpha S ( \\theta ) ^ { - 1 } \\lbrace \\rm { v e c } ^ \\top ( \\boldsymbol { \\Sigma } ^ { - 1 } ) \\rm { v e c } ( { \\bf { x } } { \\bf { x } } ^ \\top ) - 2 ( \\boldsymbol { \\Sigma } ^ { - 1 } \\boldsymbol { \\mu } ) ^ \\top { \\bf { x } } \\rbrace \\big ] ^ { \\frac { 1 } { \\alpha - 1 } } . \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} K _ { 3 } ( w , \\lambda ( t ) ) & = \\left ( \\frac { w } { 1 + w ^ { 2 } } - \\frac { w } { \\lambda ( t ) ^ { 2 - 2 \\alpha } + w ^ { 2 } } \\right ) \\frac { w ^ { 4 } } { 4 ( w ^ { 2 } + 3 6 \\lambda ( t ) ^ { 2 } ) ^ { 2 } } \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} P ( z ) = \\sum _ { | \\alpha | \\le m } x _ \\alpha z ^ \\alpha . \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} p _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ( x ) = \\frac { - a _ { 1 } } { \\pi \\sqrt { c _ { 2 } } } \\left | x - c _ { 0 } \\right | ^ { 1 / 2 } \\left [ 1 + o ( 1 ) \\right ] \\qquad ( x \\rightarrow c _ { 0 } ^ { - } ) , \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} \\forall ( a , b ) \\in \\R ^ 2 \\colon \\varphi ^ t _ \\textup { K S } ( a , b ) : = \\varphi _ \\textup { K S } ( a , b ) - t = \\begin{cases} a b - t & a + b \\geq 0 \\\\ - \\tfrac 1 2 ( a ^ 2 + b ^ 2 + 2 t ) & a + b < 0 . \\end{cases} \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\star : = \\max \\Big \\{ j \\in \\{ 1 , \\ldots , \\ell \\} : \\ , i _ { j } \\leq n \\Big \\} . \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} \\xi _ i ( 1 ) = \\xi _ i ( 0 ) + \\xi _ i ' ( \\zeta _ i ) \\mbox { f o r s o m e } \\zeta _ i \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} Q ( x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } , x _ { 3 } , y _ { 3 } ) = x _ { 1 } x _ { 2 } x _ { 3 } - x _ { 1 } y _ { 2 } y _ { 3 } - y _ { 1 } x _ { 2 } y _ { 3 } - y _ { 1 } y _ { 2 } x _ { 3 } . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} & \\frak R _ { j , k } ( x _ 1 , x _ 2 , x _ 3 , v _ 1 , v _ 2 , v _ 3 ) \\\\ & : = \\sum _ { R = I \\times J \\times S \\in \\mathcal R ^ N _ { \\frak z } ( j , k ) } \\int _ { R } \\psi _ { j , k } ( x _ 1 - y _ 1 , x _ 2 - y _ 2 , x _ 3 - y _ 3 ) \\\\ & \\times ( \\psi _ { j , k } ( y _ 1 - v _ 1 , y _ 2 - v _ 2 , y _ 3 - v _ 3 ) - \\psi _ { j , k } ( x _ I - v _ 1 , x _ J - v _ 2 , x _ S - v _ 3 ) ) d y _ 1 d y _ 2 d y _ 3 , \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} & \\int _ { \\mathbb R ^ 2 } \\sum _ { k \\in \\Bbb Z } 2 ^ { 2 k } \\big \\vert \\big ( f ( \\cdot , \\cdot , \\cdot ) \\ast _ 1 \\psi ^ { ( 1 ) } _ { j } ( y _ 1 ) \\big ) \\ast _ { 2 , 3 } \\psi ^ { ( 2 ) } _ { j , k } ( y _ 2 , y _ 3 ) \\big \\vert ^ 2 w _ { y _ 1 , 2 ^ { - j } } ( R _ { 2 ^ { - k } , 2 ^ { - j - k } } ( y _ 2 , y _ 3 ) ) { d y _ 2 \\ , d y _ 3 } \\\\ & = \\int _ { \\mathbb R ^ 2 } \\big \\vert S _ { j } ( F _ { y _ 1 , 2 ^ { - j } } ) ( y _ 2 , y _ 3 ) \\big \\vert ^ 2 w _ { y _ 1 , 2 ^ { - j } } ( y _ 2 , y _ 3 ) { d y _ 2 \\ , d y _ 3 } , \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} a = - 2 f _ 1 ' f _ 1 ^ { - 1 } - f _ 1 \\varphi '' ( \\varphi ' ) ^ { - 1 } f _ 1 ^ { - 1 } . \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\prod _ { s = i _ { 1 } } ^ { j _ { 2 } } q ^ { \\frac { m ^ { 2 } _ { s , s + 1 } } { 2 } } = q ^ { ( j _ { 2 } - i _ { 1 } + 1 ) m _ { i _ { 2 } , j _ { 2 } + 1 } m _ { i _ { 1 } , n + 1 } } \\cdots . \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} { \\mathbb T } _ { T E } ^ h ( \\omega ) = S ^ h + i A ^ { 1 , h } - i A ^ { 2 , h } + | { \\boldsymbol k } | ^ 2 M ^ h - \\left ( \\frac { \\omega } { c } \\right ) ^ 2 M ^ h _ \\epsilon \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} c ( \\phi _ i ^ { ( k ) } ) '' ( 0 ^ - ) = - \\sum _ { j = 1 } ^ m \\partial _ j f _ i ( \\mathbf { 0 } ) ( \\phi _ j ^ { ( k ) } ) ' ( 0 ^ - ) > 0 . \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} P _ n ^ + \\lbrace \\mathcal { T } ( t ) > \\beta \\rbrace + P _ n ^ - \\lbrace \\mathcal { T } ( t ) > \\beta \\rbrace & = 2 P \\lbrace \\mathcal { T } ( t ) > \\beta \\ | \\ N ( t ) = n \\rbrace = P _ n ^ + \\lbrace M ( t ) > \\beta \\rbrace \\\\ & = P _ n ^ + \\lbrace M ( t ) > \\beta , \\ \\mathcal { T } ( t ) > \\beta \\rbrace + P _ n ^ + \\lbrace M ( t ) > \\beta , \\mathcal { T } ( t ) \\le \\beta \\rbrace , \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} b _ { 0 } = \\frac { 8 - \\mu } { 4 } . \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} \\begin{aligned} \\theta _ { 5 } ( \\lambda , \\mu ) & = \\theta _ { 5 } ( \\lambda + 5 , \\mu ) , \\\\ \\theta _ { 5 } ( \\lambda , \\mu + 1 ) & = \\theta _ { 5 } ( \\lambda + 6 , \\mu ) . \\end{aligned} \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} \\Phi \\circ [ v _ 1 , v _ 2 ] _ { V } = [ \\Phi \\circ v _ 1 , \\Phi \\circ v _ 2 ] _ { Q } , \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} A _ { N } ( \\sigma ) : = \\frac { 1 } { N } a _ N = \\frac { 1 } { N } \\log \\int _ { \\mathbb { R } ^ N } \\exp \\left ( \\sigma \\sum _ { i = 1 } ^ { N } x _ i - H _ N ( x ) \\right ) d x \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} \\int _ { G } f ( x ) d \\lambda _ q ( x ) = \\int _ { G / H } T _ H ( f ) ( x H ) d \\lambda ( x H ) , \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu ( A ) } \\int h d m = \\frac { 1 } { \\mu ( A ) } \\int 1 d \\mu = 1 \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} y _ { m + 1 } - y _ m \\ , > y _ { m + 1 } - t _ { m + 1 } \\ , > ( m + 1 ) \\ , ( m + 1 ) ! \\ , - \\ , ( m + 1 ) ! \\ , = \\ , m ( m + 1 ) ! \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} | E _ { 4 } | & \\leq C r \\sup _ { x \\geq t } | \\lambda '' ( x ) | \\int _ { 6 r } ^ { \\infty } d w \\frac { r ^ { 2 } w } { w ^ { 2 } } \\left ( \\frac { 1 } { 1 + w ^ { 2 } } + \\frac { 1 } { \\lambda ( t + w ) ^ { 2 - 2 \\alpha } + w ^ { 2 } } \\right ) \\\\ & \\leq C r \\sup _ { x \\geq t } | \\lambda '' ( x ) | \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} I ^ { [ k ] } _ { V C } & = \\sum _ i \\lambda ^ { [ k ] } _ j I ^ { j , [ k ] } _ { V C } . \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} \\mathrm { d i s t } ( x , M ) = \\| P _ M x - x \\| \\leq \\| \\Pi _ M ^ p x - x \\| \\leq C _ 1 D _ p ( \\Pi _ M ^ p x , x ) ^ \\frac { 1 } { \\rho } . \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} d P _ { n + 1 } ( x , x _ { n + 1 } ) & = - \\sum _ { i = 1 } ^ { n } x _ { n + 1 } x _ { i } d x _ { i } + \\sqrt { \\tfrac { n + 2 } { n } } d P _ { n } ( x ) + \\left ( \\tfrac { n } { 2 } x _ { n + 1 } ^ { 2 } - \\tfrac { 1 } { 2 } | x | ^ { 2 } \\right ) d x _ { n + 1 } \\\\ & = - \\sum _ { i = 1 } ^ { n } x _ { n + 1 } x _ { i } d x _ { i } + \\sqrt { \\tfrac { n + 2 } { n } } d P _ { n } ( x ) + \\left ( \\tfrac { n + 1 } { 2 } x _ { n + 1 } ^ { 2 } - \\tfrac { 1 } { 2 } \\right ) d x _ { n + 1 } . \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} \\mathcal { N } ( t , s , \\xi , v , \\overline v ) = e ^ { ( t - s ) \\mathcal { L } } f \\left ( e ^ { s \\mathcal { L } } v , e ^ { s \\mathcal { L } } e ^ { \\xi \\mathcal { A } } \\overline v \\right ) . \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} F ' ( s ) & = - \\int _ 0 ^ { + \\infty } \\frac { \\ln | 1 - \\tau ^ 2 | \\ln \\tau } { \\tau ^ { 1 + s } } d \\tau \\\\ & = \\frac 2 s \\int _ 0 ^ { + \\infty } \\frac { \\tau ^ { 1 - s } \\ln \\tau } { 1 - \\tau ^ 2 } d \\tau - \\frac 1 s F ( s ) \\\\ & \\leq - \\frac 2 s I - \\frac 1 s F ( s ) , \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} A ^ * _ i \\ & = \\ \\{ N n - a + 1 : a \\in \\ A _ i \\} , \\\\ A ^ { ( M ) } _ i \\ & = \\ \\{ N n ( q - 1 ) + a : a \\in A _ i , q \\in [ M ] \\} , \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} \\langle h _ { j } , h _ { i } \\rangle = T _ { i , j } \\qquad ; \\qquad \\forall i , j \\in I . \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} u _ \\lambda ( x ) = \\int _ 0 ^ \\infty \\mathrm { e } ^ { - \\lambda t } t ^ { - \\frac { d } { \\alpha } } p _ 1 \\left ( \\frac { x } { t ^ { 1 / \\alpha } } \\right ) d t , x \\neq 0 . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } \\psi _ { k , j } ^ { \\ast } ( Q ^ { \\prime } ) \\leq ( k - n ) \\frac { ( k + 1 ) n } { k ( n + 1 ) } \\psi _ { n , 1 } ^ { \\ast } ( Q ) + \\frac { ( k + 1 ) n } { k ( n + 1 ) } \\sum _ { j = 1 } ^ { n } \\psi _ { n , j } ^ { \\ast } ( Q ) + \\frac { k - n } { n + 1 } . \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} \\hat { E } _ { \\Lambda } \\left ( \\bigotimes _ { x \\in \\Lambda } b _ { x } \\right ) = \\diamond _ { x \\in \\Lambda } \\hat { E } _ { x } ( b _ { x } ) \\qquad ; b _ { x } \\in \\mathcal { B } _ { x } \\ , \\ x \\in \\Lambda \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} u ^ N _ t ( x ) & = u ^ N _ 0 ( x ) - \\int _ 0 ^ t \\langle \\mu _ s ^ N , \\nabla V ^ N ( x - \\cdot ) \\cdot F \\big ( K \\ast u ^ N _ s ( \\cdot ) \\big ) \\rangle \\ d s \\\\ & - \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ 0 ^ t \\nabla V ^ N ( x - X _ s ^ { i , N } ) \\cdot d W ^ i _ s + \\int _ 0 ^ t \\Delta u ^ N _ s ( x ) \\ d s . \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} [ T _ v , T _ { \\overline w } ] = \\langle \\ , \\cdot \\ , | w \\rangle v - H _ v H _ w \\ . \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} \\| f \\| = c _ \\beta \\left ( \\int _ { \\Bbb C } | z ^ 2 - 1 | ^ { 2 \\beta } | z | ^ { - 4 - 4 \\beta } | f ( z , \\bar z ) | ^ 2 \\frac { d z \\wedge d \\bar z } { - 2 i } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} \\rho _ t + \\nabla \\cdot ( \\rho \\mathbf { v } ) = 0 . \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} q _ { i j } \\circ p _ { i , j + 1 } = p _ { i j } \\circ q _ { i + 1 , j } i , j \\in \\N . \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} \\overline { X } = \\{ ( x _ 1 , x _ 2 , \\ldots , x _ k ) \\in X : x _ i \\neq x _ j , \\ \\forall \\ , i \\neq j \\} . \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} \\frac { d _ n } { c _ n } = \\frac { s _ n ^ 2 } { 2 n s _ n + n ^ 2 } > \\frac { 1 } { \\ell } + 1 \\mbox { a n d } \\frac { c _ { n + 1 } } { d _ n } = \\frac { 2 ( n + 1 ) s _ { n + 1 } + ( n + 1 ) ^ 2 } { s _ n ^ 2 } > \\frac { 1 } { \\ell } + 1 \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} { \\phi } _ { p , \\alpha } ( t ) = t ^ { p } ( 1 + \\log ^ { + } ( t ) ) ^ { \\alpha } = t ^ { p } ( 1 + \\log ( t ) ) ^ { \\alpha } , \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} & \\frac { \\partial w } { \\partial x _ 1 } = e ^ { i \\theta } \\Big [ U ' \\cos \\theta - i \\frac { U } { r } \\sin \\theta \\ , \\Big ] , \\\\ [ 1 m m ] & \\frac { \\partial w } { \\partial x _ 2 } = e ^ { i \\theta } \\Big [ U ' \\sin \\theta + i \\frac { U } { r } \\cos \\theta \\ , \\Big ] . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} | \\Phi ( r \\zeta ) | = r \\exp \\Re H _ { \\sigma } ( r \\zeta ) = r \\exp \\left [ \\int _ { \\mathbb { T } } \\frac { 1 - r ^ { 2 } } { | t - r \\zeta | ^ { 2 } } \\ , d \\sigma ( t ) \\right ] , r \\in ( 0 , 1 ) , \\zeta \\in \\mathbb { T } , \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} & ( M _ 0 \\cdot M _ 1 M _ 0 M _ 1 ) ^ r = M _ 0 ( M _ 1 M _ 0 ) ^ { 2 r - 1 } M _ 1 , ( M _ 0 \\cdot M _ 1 M _ 0 M _ 1 ) ^ r M _ 0 = M _ 0 ( M _ 1 M _ 0 ) ^ { 2 r } , \\\\ & ( M _ 1 M _ 0 M _ 1 \\cdot M _ 0 ) ^ r = ( M _ 1 M _ 0 ) ^ { 2 r } , ( M _ 1 M _ 0 M _ 1 \\cdot M _ 0 ) ^ r M _ 1 M _ 0 M _ 1 = ( M _ 1 M _ 0 ) ^ { 2 r + 1 } M _ 1 , \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} | ( x + y ) ^ { \\sigma } - x ^ { \\sigma } | \\leq \\sum _ { k = 1 } ^ n c _ k | y | ^ k x ^ { \\sigma - k } + c _ { n + 1 } | y | ^ { \\sigma } \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} \\left ( \\int _ { ( x - 2 k t ) / ( 2 \\sqrt t ) } ^ \\infty e ^ { i u ^ 2 } d u \\right ) \\mu _ { + } ( ( x - 2 k t ) / ( 2 \\sqrt t ) ) = \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } H _ s ( n _ 1 , \\ldots , n _ s ) = F ( n _ 1 , \\ldots , n _ r ) . \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} ( A + g ) \\circ K { } { } - K { } { } \\circ ( A _ c + r ) = 0 . \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} \\lambda _ k & = \\zeta ( \\mathfrak n _ 1 ) \\times \\langle \\mathfrak m _ 1 \\rangle \\times \\ldots \\times \\zeta ( \\mathfrak n _ { k } ) \\times \\langle \\mathfrak m _ { k } \\rangle \\times \\zeta ( \\mathfrak n _ { k + 1 } + \\ldots + \\mathfrak n _ { r + 1 } + \\mathfrak m _ { k + 1 } + \\ldots + \\mathfrak m _ { r } ) \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} \\forall t \\in I \\colon u _ \\textup { d } ( t ) : = - 2 0 \\ , \\sin ( \\pi t / 3 ) v _ \\textup { d } ( t ) : = 1 0 \\ , \\cos ( \\pi t / 2 ) . \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} \\| R \\| \\geq | R ( \\hat { x } _ L , \\hat { x } _ R ) | = \\bigl | \\hat { T } ( \\hat { x } _ L , \\hat { x } _ R ) \\bigr | + \\eta \\bigl | \\hat { T } ( \\hat { x } _ L , \\hat { x } _ R ) \\bigr | ^ 2 > \\left ( 1 - \\frac { \\eta ^ 3 } { 2 ^ 4 } \\right ) + \\eta \\left ( 1 - \\frac { \\eta ^ 3 } { 2 ^ 4 } \\right ) ^ 2 . \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} \\Omega = X ^ I \\gamma _ I + F _ I \\gamma ^ I \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} P _ { 2 } = ( T - 3 ) ( T + 3 ) ^ { 3 } Q _ { 2 } , \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} B _ { 1 } = [ a , a + n ] ^ { d } \\times [ - n , 0 ] B _ { 2 } = [ 1 , n ] ^ { d } \\times [ T , T + n ] . \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} b _ \\ell : = \\left | \\bigcap _ { i \\in I } A _ i \\setminus \\bigcup _ { i \\in [ k ] \\setminus I } A _ i \\right | . \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} I ' = \\langle & E _ { 1 , 2 } ^ { 2 } , E _ { 1 , 3 } ^ { 2 } , E _ { 1 , 4 } ^ { 2 } , E _ { 2 , 3 } ^ { 2 } , E _ { 2 , 4 } ^ { 2 } , E _ { 3 , 4 } ^ { 2 } , E _ { 1 , 2 } E _ { 1 , 3 } , E _ { 1 , 2 } E _ { 1 , 4 } , E _ { 1 , 3 } E _ { 1 , 4 } , E _ { 2 , 3 } E _ { 2 , 4 } , \\\\ & E _ { 1 , 4 } E _ { 2 , 3 } , E _ { 1 , 3 } E _ { 2 , 3 } , E _ { 1 , 4 } E _ { 2 , 4 } , E _ { 1 , 4 } E _ { 3 , 4 } , E _ { 2 , 4 } E _ { 3 , 4 } \\rangle . \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} s _ 0 = s _ { 0 , 0 } + s _ { 0 , 1 } t _ x + s _ { 0 , 2 } t _ x ^ 2 + O ( t _ x ^ 3 ) , s _ 1 = s _ { 1 , 0 } + s _ { 1 , 1 } t _ x + s _ { 1 , 2 } t _ x ^ 2 + O ( t _ x ^ 3 ) ; \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} \\bar { \\rho } ( L _ { - 1 } ) = z , \\bar { \\rho } ( L _ 0 ) = z \\partial + b , \\bar { \\rho } ( L _ { 1 } ) = z \\partial ^ 2 + 2 b \\ , \\partial \\ , . \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} \\R ^ n _ k = [ 0 , \\infty ) _ x ^ k \\times \\R _ y ^ { n - k } \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} \\left ( \\frac { C _ { n , k _ { 1 } } - ( n - 1 ) \\mathbb { I } _ { k _ { 1 } = 2 } - \\mu _ { k _ 1 } } { \\sqrt { 2 k _ { 1 } } } , \\ldots , \\frac { C _ { n , k _ { l } } - \\mu _ { k _ { l } } } { \\sqrt { 2 k _ { l } } } \\right ) \\stackrel { d } { \\to } N _ { l } ( 0 , I _ { l } ) . \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} \\int _ B ( \\Delta ^ k u ) ^ 2 d x \\geq \\Big ( \\prod _ { i = 0 } ^ { k - 2 } c _ { n , 4 i } \\Big ) \\int _ B \\frac { ( \\Delta u ) ^ 2 } { | x | ^ { 4 ( k - 1 ) } } d x \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} \\partial _ { t r } v _ { 2 } ( t , r ) = \\frac { c _ { b } } { 2 } \\int _ { 0 } ^ { \\infty } \\cos ( t \\xi ) \\xi ^ { 2 } ( J _ { 0 } ( r \\xi ) - J _ { 2 } ( r \\xi ) ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) d \\xi } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} & { h _ 0 } ^ { \\pm } = e ^ { i \\theta } \\Big [ \\ , { h _ { 0 1 } } ^ { \\pm } \\ , + \\ , i { h _ { 0 2 } } ^ { \\pm } \\ , \\Big ] , \\\\ [ 1 m m ] & { h _ j ^ 1 } ^ { \\pm } = e ^ { i \\theta } \\Big [ \\ , { h _ { j 1 } ^ 1 } ^ { \\pm } \\sin { j \\theta } \\ , + \\ , i { h _ { j 2 } ^ 1 } ^ { \\pm } \\cos { j \\theta } \\ , \\Big ] , \\\\ [ 1 m m ] & { h _ j ^ 2 } ^ { \\pm } = e ^ { i \\theta } \\Big [ \\ , { h _ { j 1 } ^ 2 } ^ { \\pm } \\cos { j \\theta } \\ , + \\ , i { h _ { j 2 } ^ 2 } ^ { \\pm } \\sin { j \\theta } \\ , \\Big ] . \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\phi _ { j } ^ { ( 2 ) } ( x _ n ) } { \\phi _ { j } ^ { ( 1 ) } ( x _ n ) } = p > 1 . \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} e ( z ) : = ( r ( z ) , z ) , z \\in S _ { o } ' . \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} w ( t , \\phi ( t ; T , x _ 0 ) ) = & \\psi ( x _ 0 ) \\exp \\Bigl [ - \\int _ { t } ^ { T } \\frac { \\lambda ( s , \\phi ( s ; T , x _ 0 ) ) } { s } d s \\Bigr ] \\\\ & - \\int _ t ^ T \\exp \\Bigl [ - \\int _ t ^ { \\tau } \\frac { \\lambda ( s , \\phi ( s ; T , x _ 0 ) ) } { s } d s \\Bigr ] \\frac { g ( \\tau , \\phi ( \\tau ; T , x _ 0 ) ) } { \\tau } d \\tau . \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} \\sigma _ k ^ { h 1 , 2 } & = h ^ n \\nabla _ y \\Phi ( h \\vec { k } , h \\vec { k } ) \\cdot \\sum _ { 0 < \\abs { \\vec { j } - \\vec { k } } < \\tfrac { 1 } { h } } \\frac { h \\vec { j } - h \\vec { k } } { \\abs { h \\vec { j } - h \\vec { k } } ^ { n + \\alpha } } \\\\ & = h ^ { 1 - \\alpha } \\nabla _ y \\Phi ( h \\vec { k } , h \\vec { k } ) \\cdot \\sum _ { 0 < | \\vec { i } | < \\tfrac { 1 } { h } } \\frac { \\vec { i } } { | \\vec { i } | ^ { n + \\alpha } } . \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} - d i v _ C ( \\nabla _ C v + b _ 0 ) + | A _ C | ^ 2 v + b _ 1 = 0 \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} P _ { n + 1 } ( x , x _ { n + 1 } ) & \\leq \\tfrac { 1 } { 6 } \\left ( ( n + 3 ) x _ { n + 1 } ^ { 3 } - 3 x _ { n + 1 } + ( n - 1 ) \\sqrt { \\tfrac { n + 2 } { n } } | x | ^ { 3 } \\right ) \\\\ & = \\tfrac { 1 } { 6 } \\left ( ( n + 3 ) x _ { n + 1 } ^ { 3 } - 3 x _ { n + 1 } + ( n - 1 ) \\sqrt { \\tfrac { n + 2 } { n } } ( 1 - x _ { n + 1 } ^ { 2 } ) ^ { \\tfrac { 3 } { 2 } } \\right ) \\leq \\tfrac { n } { 6 } , \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} v = \\frac { u _ 3 } { p ( 1 + q ^ 2 } \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} \\| \\lambda \\| _ { M ( G / H ) } = \\| \\lambda ^ { \\ast ^ { G / H } \\ast ^ { G / H } } \\| _ { M ( G / H ) } \\le \\| \\lambda ^ { \\ast ^ { G / H } } \\| _ { M ( G / H ) } , \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} P _ { L , S } \\coloneqq \\biggl \\{ \\left ( H _ n \\right ) [ c _ 1 , \\ldots , c _ { L } ] \\ \\bigg | \\ \\sum _ { i = 1 } ^ { L } c _ { i } = S + 1 \\biggr \\} . \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { M } ( - 1 ) ^ k [ 4 k - 1 ] \\frac { ( q ^ { - 1 } ; q ^ 2 ) _ k ^ 5 } { ( q ^ 2 ; q ^ 2 ) _ k ^ 5 } q ^ { k ^ 2 + 5 k } \\equiv \\frac { \\Omega _ q ( n ) } { [ 2 ] ^ 3 [ 4 ] } \\sum _ { k = 0 } ^ { ( n - 3 ) / 2 } \\frac { ( q ^ 3 ; q ^ 2 ) _ k ^ 3 } { ( q ^ 2 ; q ^ 2 ) _ k ( q ^ 6 ; q ^ 2 ) _ k ^ 2 } q ^ { 2 k } . \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} & F ( t ; x , y , 1 , u , z , v ) \\\\ & = \\sum _ { k = 0 } ^ { \\infty } z ( \\delta _ k t u v ( 1 - z ) + z ) H ( t ; x , \\delta _ k , 1 , u , 1 , v ) \\prod _ { i = 0 } ^ { k - 1 } \\frac { t x ( \\delta _ i - \\delta _ i z r + z ) } { ( t u x + \\delta _ i ^ { - 1 } - t u ) ( \\delta _ i - \\delta _ i r + 1 ) } . \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} c _ s : ( G \\times \\widehat { G } ) \\times ( G \\times \\widehat { G } ) & \\to \\mathbb { T } \\\\ ( \\xi _ 1 , \\xi _ 2 ) & \\mapsto \\overline { \\omega _ 2 ( x _ 1 ) } \\omega _ 1 ( x _ 2 ) . \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} \\mathcal { K } _ { \\psi } ( \\Omega ) : = \\left \\{ z \\in u _ 0 + { W ^ { 1 , p } _ 0 ( \\Omega ) } : z \\ge \\psi \\ , \\ , \\textnormal { a . e . i n $ \\Omega $ } \\right \\} \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} f \\diamond g ( x ) = : \\sup _ { x ^ * \\in \\partial f ( x ) } \\Big \\{ \\Re e \\langle x ^ * , x \\rangle - g ^ * ( x ^ * ) \\Big \\} , \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} \\bigg [ \\omega _ { n + 1 } ^ { 1 - 2 s } \\partial _ { t } ^ { 2 } + \\sum _ { j = 1 } ^ { n + 1 } \\Omega _ { j } \\omega _ { n + 1 } ^ { 1 - 2 s } \\Omega _ { j } - \\omega _ { n + 1 } ^ { 1 - 2 s } \\frac { ( n - 2 s ) ^ { 2 } } { 4 } \\bigg ] \\overline { u } + R \\overline { u } = \\tilde { f } , \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} S _ F [ A , \\widetilde { X } ] = \\int _ \\Sigma \\langle \\widetilde { X } , F \\rangle : = \\int _ \\Sigma \\widetilde { X } _ a F ^ a , \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} \\overline { \\nabla _ t } ^ 2 _ { e _ i , e _ j } u + a ( t ) u h ( t ) + u \\bar { g } ( t ) = 0 . \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} ( \\nu \\ast _ G \\nu ' ) ^ { * ^ G } = { \\nu ' } ^ { * ^ G } \\ast _ G \\nu ^ { * ^ G } , \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} U _ { 1 2 , n } : = \\{ n U _ { 1 2 } \\} . \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} q ( i ) = \\eta _ i ( x _ i ) = \\eta _ i ( z ( i ) ) = f ( \\eta ( z ) ) ( i ) \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ t u ( t , x ) = \\Delta u ( t , x ) - \\nabla \\cdot \\big ( u ( t , x ) ~ K \\ast _ { x } u ( t , x ) \\big ) , t > 0 , ~ x \\in \\R ^ d , \\\\ & u ( 0 , x ) = u _ 0 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} u ^ N _ { t } ( x ) - u _ { t } ( x ) & = e ^ { t \\Delta } ( u ^ N _ { 0 } - u _ { 0 } ) ( x ) + \\int _ 0 ^ t \\nabla \\cdot e ^ { ( t - s ) \\Delta } \\left ( u _ { s } F ( K \\ast u _ s ) - u ^ { N } _ { s } F ( K \\ast u ^ { N } _ s ) \\right ) ( x ) ~ d s \\\\ & + E _ { t } ( x ) - M ^ N _ { t } ( x ) , \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} \\int _ 0 ^ t f \\left ( v , e ^ { \\xi \\mathcal { A } } \\overline v \\right ) d \\xi = \\int _ 0 ^ t \\mathcal { B } \\left ( F ( v ) \\cdot G \\left ( e ^ { \\xi \\mathcal { A } } \\overline v \\right ) \\right ) d \\xi . \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} \\sum _ { i , j \\in I } \\beta _ { V , i , j } = 1 . \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} v = \\sqrt { \\tfrac { n } { k ( n + 1 - k ) } } \\sum _ { i \\in I } v _ { i } , \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} \\lim _ { \\xi \\rightarrow \\overline { c _ { 0 } } ^ { - } } \\frac { p _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( \\xi ) } { \\left | \\xi - \\overline { c _ { 0 } } \\right | ^ { 1 / 3 } } = \\frac { - 2 \\left | a _ { 1 } \\right | } { \\sqrt [ 3 ] { \\left | c _ { 3 } \\right | } } \\cos \\left ( \\frac { \\theta } { 3 } - \\frac { 7 \\pi } { 6 } \\right ) , \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} P ( x ) & = \\det \\begin{pmatrix} x _ { 1 } & x _ { 2 } & x _ { 3 } \\\\ x _ { 4 } & x _ { 5 } & x _ { 6 } \\\\ x _ { 7 } & x _ { 8 } & x _ { 9 } \\end{pmatrix} \\\\ & = x _ { 1 } x _ { 5 } x _ { 9 } + x _ { 2 } x _ { 6 } x _ { 7 } + x _ { 3 } x _ { 4 } x _ { 8 } - x _ { 1 } x _ { 6 } x _ { 8 } - x _ { 2 } x _ { 4 } x _ { 9 } - x _ { 3 } x _ { 5 } x _ { 7 } , \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{cases} F _ L ( x , y ) , & x < 0 , \\\\ F _ R ( x , y ) , & x > 0 , \\end{cases} \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} | \\tilde { F } ( \\xi _ 2 ' , \\eta _ 2 ' ) | = & \\Bigl | \\frac { 3 } { 2 } \\ , \\xi _ 1 ' \\ , \\eta _ 1 ' + 2 \\ , \\xi _ 2 ' \\ , \\eta _ 2 ' \\Bigr | \\\\ \\geq & 2 | \\xi _ 2 ' \\ , \\eta _ 2 ' | - \\frac { 3 } { 2 } | \\xi _ 1 ' \\ , \\eta _ 1 ' | \\\\ \\geq & 2 ^ { - 5 4 } N _ 1 ^ 2 , \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} \\partial _ s U _ { 1 , 0 } ( s , \\gamma ) = - \\int _ 0 ^ 1 U _ { 1 , t } ( s , \\gamma ) \\partial _ s A _ \\mu ( s , \\gamma ( t ) ) \\dot { \\gamma } ^ \\mu ( t ) U _ { t , 0 } ( s , \\gamma ) d t . \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} \\bigg ( \\frac { 1 } { 2 } \\Delta + \\big ( 2 H ^ 2 - K + 2 k _ 0 \\big ) \\bigg ) \\mathcal { E } _ H + \\big ( \\nabla \\cdot \\tilde { \\nabla } + 2 H K \\big ) \\mathcal { E } _ K - 2 H \\mathcal { E } = 0 , \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} w ^ { ( 1 ) } ( \\eta ) = - 2 c _ \\beta S ( \\eta ) ^ { - 1 } \\cdot \\boldsymbol { \\Sigma } ^ { - 1 } \\boldsymbol { \\mu } , f ^ { ( 1 ) } ( { \\bf { x } } ) = { \\bf { x } } , w ^ { ( 2 ) } ( \\eta ) = c _ \\beta S ( \\eta ) ^ { - 1 } \\cdot \\rm { v e c } ( \\boldsymbol { \\Sigma } ^ { - 1 } ) , f ^ { ( 2 ) } ( { \\bf { x } } ) = \\rm { v e c } ( { \\bf { x } } { \\bf { x } } ^ \\top ) . \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} \\liminf _ { x \\to x _ { 0 } , x \\in I } \\frac { p _ { \\nu _ { 1 } \\boxplus \\nu _ { 2 } } ( x ) } { p _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ( x ) } & = \\liminf _ { x \\to x _ { 0 } , x \\in I } \\frac { 1 } { \\beta } \\int _ { \\mathbb { R } } \\frac { 1 + t ^ { 2 } } { | 1 - t G _ { \\mu _ { 1 } } ( F _ { \\rho _ { 1 } } ( x ) ) | ^ { 2 } } \\ , d \\sigma ( t ) \\\\ & \\ge \\frac { 1 } { \\beta } \\int _ { \\mathbb { R } } \\frac { 1 + t ^ { 2 } } { | 1 - t G _ { \\mu _ { 1 } } ( F _ { \\rho _ { 1 } } ( x _ { 0 } ) ) | ^ { 2 } } \\ , d \\sigma ( t ) > 0 , \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} I _ 2 \\le c _ 3 x _ d ^ p \\begin{cases} y _ d ^ { 2 \\alpha - p + \\beta _ 1 + \\beta _ 2 } \\left ( \\log ( 1 / { y _ d } ) \\right ) ^ { \\beta _ 4 } & 2 \\alpha - 2 p + \\beta _ 1 + \\beta _ 2 < 0 ; \\\\ y _ d ^ p \\left ( \\log ( 1 / y _ d ) \\right ) ^ { \\beta _ 4 + 1 } & 2 \\alpha - 2 p + \\beta _ 1 + \\beta _ 2 = 0 . \\end{cases} \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} | S ( X ) | = | S _ 1 ( X ) | + | S _ 2 ( X ) | - | S _ 1 ( X ) \\cap S _ 2 ( X ) | . \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} S ^ * ( L _ u + \\lambda I d ) S = L _ u + ( \\lambda + 1 ) I d \\ , \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} \\alpha = \\left ( \\sigma _ { { \\rm f o l d } , L } - \\sigma _ { { \\rm f o l d } , R } \\middle ) \\right | _ { \\mu = 0 } \\ , , \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} \\zeta _ t = \\inf \\{ s > 0 ; \\ : \\chi _ s > t \\} . \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} \\left | \\frac { x ' ( r ) } { x ( r ) } \\right | & = \\left | \\frac { ( d / d r ) \\Psi ( r e ^ { i f ( r ) } ) } { \\Psi ( r e ^ { i f ( r ) } ) } \\right | \\\\ & \\ge \\frac { f ( r ) ^ { 2 } } { 2 \\beta } \\frac { \\sqrt { 1 + r ^ { 2 } f ' ( r ) ^ { 2 } } } { r } , \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} \\left ( j _ { n - 3 } ^ { ( 3 ) } \\right ) ^ { 2 } + 3 J _ { n } ^ { ( 3 ) } j _ { n } ^ { ( 3 ) } = 4 ^ { n } , \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} D _ p ( z , \\Pi _ N x _ n ) = D _ p ( z , x _ n ) - D _ p ( \\Pi _ N ^ p x _ n , x _ n ) \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} a _ 1 & = 1 , \\\\ a _ { 3 k } & = - ( 4 k - 1 ) ^ 2 ( 2 k - 1 ) x ^ 3 , \\\\ a _ { 3 k + 1 } & = - 4 k ^ 2 x ^ 2 , \\\\ a _ { 3 k + 2 } & = - ( 4 k + 1 ) ^ 2 ( 2 k + 1 ) x ^ 3 ; \\\\ b _ 0 & = 0 , \\\\ b _ { 3 k } & = - ( 6 k - 1 ) x + 1 , \\\\ b _ { 3 k + 1 } & = - ( 6 k + 1 ) x + 1 , \\\\ b _ { 3 k + 2 } & = - 4 ( 2 k + 1 ) ^ 2 x ^ 2 - 2 ( 2 k + 1 ) x + 1 . \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} B _ \\alpha ( p , q _ n ) = B _ \\alpha ( p , p ^ * ) + B _ \\alpha ( p ^ * , q _ n ) \\forall n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} B = \\left \\{ \\begin{array} { c } \\eta _ { t } ( x ) \\geq L ^ { d + 4 } , \\\\ t \\in [ 0 , L ] x \\in [ - L ^ { d + 6 } , L ^ { d + 6 } ] ^ { d } \\end{array} \\right \\} , \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) \\le \\beta \\ | \\ V ( 0 ) = - c _ 2 \\} = \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { ( p ^ r + 1 ) / 2 } ( - 1 ) ^ k ( 4 k - 1 ) \\frac { ( - \\frac { 1 } { 2 } ) _ k ^ 5 } { k ! ^ 5 } \\equiv \\frac { p ^ r ( p ^ { 2 r } - p ^ { 4 r } - 1 ) } { 1 6 ( p ^ { 2 r } - 1 ) } \\sum _ { k = 0 } ^ { ( p ^ r - 3 ) / 2 } \\frac { ( \\frac { 3 } { 2 } ) _ k ^ 3 } { k ! ( k + 2 ) ! ^ 2 } \\pmod { p ^ { r + 3 } } , \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\Big \\| | x | ^ { - \\frac s 2 } u _ \\tau ^ m \\Big \\| _ 2 ^ 2 d \\tau + J ( u ^ m ( x , t ) ) = J ( u ^ m ( x , 0 ) ) < d \\quad I ( u ^ m ( x , 0 ) ) > 0 . \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 f } { \\partial t _ 3 \\partial t _ 2 } ( 0 , t _ 1 , 0 , t _ 1 ) = t _ 1 ^ { - 2 } ( 1 + 6 t _ 1 ^ 2 S ( Z ) + . . . ) + . . . \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} \\tau ^ { n + 1 } ( c ) ( 0 ) & = \\tau ( \\tau ^ { n } ( c ) ) ( 0 ) = \\tau ( d ) ( 0 ) = d ( 1 ) - d ( 0 ) ^ 2 \\\\ & = c ( n + 1 ) + \\nu _ k ( c ( 1 ) , \\dots , c ( n ) ) - \\mu _ n ( c ( 0 ) , \\dots , c ( k ) ) ^ 2 \\\\ & = c ( n + 1 ) + \\nu _ { n + 1 } ( c ( 0 ) , \\dots , c ( n ) ) , \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} q _ 1 \\sim q _ 2 \\ \\Leftrightarrow \\ x _ a ^ { ( w _ a ) } ( q _ 1 ) = x _ a ^ { ( w _ a ) } ( q _ 2 ) \\ \\ \\mbox { f o r } \\ a = 1 , \\ldots , n . \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( q ^ { - 1 } ; q ^ 2 ) _ k ^ 5 ( q ^ { - 3 } ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k ^ 5 ( q ^ { 4 } ; q ^ 2 ) _ k } q ^ { 9 k } \\equiv \\frac { \\Omega _ q ( n ) [ 5 ] [ 7 ] } { [ 2 ] ^ 3 [ 4 ] ^ 2 [ 6 ] } \\sum _ { k = 0 } ^ { ( n - 3 ) / 2 } \\frac { ( q ^ 3 ; q ^ 2 ) _ k ^ 3 ( q ^ 9 ; q ^ 2 ) _ k } { ( q ^ 2 , q ^ 6 , q ^ 6 , q ^ 8 ; q ^ 2 ) _ k } q ^ { 2 k } . \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} D u _ 1 = v _ 1 + C u _ 2 , A u _ 2 = v _ 2 + B u _ 1 , \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} b & = \\tau ( \\tilde { g } ) - \\tilde { g } . \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} \\Lambda ( \\mu ) = \\sigma _ { { \\rm f o l d } , L } ( \\mu ) - \\sigma _ { { \\rm f o l d } , R } ( \\mu ) . \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} \\Delta ( t , x ) \\preceq & \\ ( 1 - \\epsilon ) \\big ( \\delta \\big [ \\Phi ' ( x - \\underline h ( t ) ) + \\Phi ' ( - x - \\underline h ( t ) ) \\big ] + 2 m L \\epsilon _ 2 \\mathbf { 1 } \\big ) \\\\ \\preceq & \\ ( 1 - \\epsilon ) \\big ( - \\delta \\epsilon _ 1 + 2 m L \\epsilon _ 2 \\big ) \\mathbf { 1 } = \\mathbf { 0 } . \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} \\mathbb { 1 } _ { \\mathcal { U } _ k } \\mathbb { E } _ \\eta \\left ( \\mathbb { 1 } _ { \\mathcal { G } _ { k } ^ c } | \\mathcal { F } _ { k } \\right ) \\leq \\mathbb { 1 } _ { \\mathcal { U } _ k } \\prod _ { x \\in H _ k \\cap E } e ^ { - c _ 3 ' } = \\mathbb { 1 } _ { \\mathcal { U } _ k } e ^ { - c _ 3 ' | H _ k \\cap E | } . \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} J ( u ) = \\dfrac { 1 } { 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla u ( x ) | ^ 2 \\ ; d x - \\int _ { \\mathbb { R } ^ N } G ( u ( x ) ) \\ ; d x , \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} C : = A + _ G B , \\ , \\ , \\ , \\ , D : = A \\cdot _ G B , \\ , \\ , \\ , \\ , E : = ( A + \\alpha ) \\cdot _ G ( B + \\beta ) . \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} C \\Phi _ 7 ( q ) = \\prod _ { n = 1 } ^ { \\infty } \\frac { 1 } { 1 - q ^ { 7 n } } + 4 9 \\cdot q \\prod _ { n = 1 } ^ { \\infty } \\frac { ( 1 - q ^ { 7 n } ) ^ 3 } { ( 1 - q ^ n ) ^ 4 } + 3 4 3 \\cdot q ^ 2 \\prod _ { n = 1 } ^ { \\infty } \\frac { ( 1 - q ^ { 7 n } ) ^ 7 } { ( 1 - q ^ n ) ^ 8 } . \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} M = ( M _ { n p } ) _ { n , p \\ge 0 } , M _ { n p } \\equiv M _ { n p } ( u ) = \\langle S ^ * f _ p | f _ n \\rangle = \\langle f _ p | S f _ n \\rangle \\ . \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} \\rho ( x ) : = m i n \\ \\{ d i s t _ { \\Sigma } ( x , p ) : p \\in S i n g ( \\Sigma ) \\} \\cup \\{ 2 \\tau \\} \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ t f \\left ( v , e ^ { \\xi \\mathcal { A } } \\overline v \\right ) d \\xi = t \\mathcal { B } ( F ( v ) \\cdot \\varphi _ 1 ( t \\mathcal { A } ) G ( \\overline v ) ) + \\mathcal { R } _ { 1 , 2 } ( t ) \\end{aligned} \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} X _ { ( P _ { \\ell ( \\alpha ) } , \\alpha ) } & = \\sum _ { S \\subseteq E ' } ( - 1 ) ^ { | S | } X _ { P _ { \\alpha ^ c } + S } \\\\ & = \\sum _ { \\textup { s e t } ( \\alpha ) \\subseteq J \\subseteq [ | \\alpha | - 1 ] } ( - 1 ) ^ { | J - \\textup { s e t } ( \\alpha ) | } X _ { N _ { | \\alpha | } + \\{ v _ i v _ { i + 1 } \\mid i \\in J \\} } . \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} \\big \\langle \\delta \\mathbf { N } , \\mathbf { r } _ { i j } \\big \\rangle = - \\big \\langle u ^ l \\mathbf { r } _ l , ( h _ { i j } \\mathbf { N } + \\Gamma _ { i j } ^ k \\mathbf { r } _ k - g _ { i j } k _ 0 \\mathbf { r } ) \\big \\rangle = g _ { i j } k _ 0 u . \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} \\xi _ n ^ { f , \\mu } ( \\omega ) = \\int _ { [ 0 , 1 ] } f ( g _ { i _ 1 , \\ldots , i _ n } ( x ) ) \\mu ( d x ) \\ ; \\ ; \\omega = ( i _ 1 , i _ 2 , \\ldots ) . \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} G = G _ { 0 } + 2 G _ { 1 } + \\cdots + 2 G _ { k } = G _ { 0 } + \\xi _ { 1 } G _ { 1 } + \\overline { \\xi _ { 1 } G _ { 1 } } + \\cdots + \\xi _ { k } G _ { k } + \\overline { \\xi _ { k } G _ { k } } , \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\frac { \\rho d \\rho } { ( 1 + 2 ( \\rho ^ { 2 } + r ^ { 2 } ) \\lambda ( t ) ^ { 2 \\alpha - 2 } + ( \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } \\lambda ( t ) ^ { 4 \\alpha - 4 } ) } & \\leq \\int _ { 0 } ^ { \\lambda ( t ) ^ { 1 - \\alpha } } \\rho d \\rho + \\int _ { \\lambda ( t ) ^ { 1 - \\alpha } } ^ { \\infty } \\frac { d \\rho } { \\rho ^ { 3 } \\lambda ( t ) ^ { 4 \\alpha - 4 } } \\\\ & \\leq C \\lambda ( t ) ^ { 2 - 2 \\alpha } \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} \\widetilde { C } ^ c { } _ { a b } = ( K ^ { - 1 } ) ^ c { } _ { d } ( K ^ e { } _ { a } K ^ f { } _ { b } & C ^ d { } _ { e f } + K ^ e { } _ { a } \\rho ^ \\mu _ e \\partial _ \\mu K ^ d { } _ { b } - K ^ e { } _ { b } \\rho ^ \\mu _ e \\partial _ \\mu K ^ d { } _ { a } ) . \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} F _ { \\nu } ( z ) = \\frac { \\beta } { \\beta - 1 } z + \\frac { 1 } { 1 - \\beta } \\Phi ( z ) , \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} h ( t ) = \\begin{bmatrix} a _ 3 \\Phi ^ { ( 0 ) ^ { \\scriptstyle 2 } } _ t + a _ 4 \\Phi ^ { ( 0 ) } _ t \\Psi ^ { ( 0 ) } _ t + a _ 5 \\Psi ^ { ( 0 ) ^ { \\scriptstyle 2 } } _ t \\\\ b _ 3 \\Phi ^ { ( 0 ) ^ { \\scriptstyle 2 } } _ t + b _ 4 \\Phi ^ { ( 0 ) } _ t \\Psi ^ { ( 0 ) } _ t + b _ 5 \\Psi ^ { ( 0 ) ^ { \\scriptstyle 2 } } _ t \\end{bmatrix} . \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} \\begin{cases} A _ { 1 } A _ { 2 } A _ { 3 } B _ { 1 } B _ { 2 } B _ { 3 } = A _ { 1 } ' A _ { 2 } ' A _ { 3 } ' B _ { 1 } ' B _ { 2 } ' B _ { 3 } ' = - 1 , \\\\ A _ { 2 } B _ { 3 } A _ { 2 } ' B _ { 3 } ' = A _ { 3 } B _ { 1 } A _ { 3 } ' B _ { 1 } ' = 1 , \\\\ A _ { 3 } B _ { 2 } \\overline { A _ { 3 } ' B _ { 2 } ' } = A _ { 1 } B _ { 3 } \\overline { A _ { 1 } ' B _ { 3 } ' } = 1 , \\end{cases} \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} \\tilde \\xi _ { j + 1 } : = \\min _ { \\substack { u \\in U _ j ^ { \\perp } \\cap \\mathcal H ^ 2 _ { 0 , N } ( \\Omega ) \\\\ u \\ne 0 } } \\frac { \\mathcal Q _ { \\sigma } ( u , u ) } { \\int _ { \\partial \\Omega } u ^ 2 d \\sigma } > \\mu . \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} \\forall k \\in \\N \\colon ( \\tilde \\lambda ^ k _ { I ^ g ( \\bar x ) } , \\tilde \\rho ^ k , \\tilde \\xi ^ k _ { \\mathcal I ( \\bar x ) } ) : = \\frac { ( \\lambda ^ k _ { I ^ g ( \\bar x ) } , \\rho ^ k , \\xi _ { \\mathcal I ( \\bar x ) } ^ k ) } { \\Vert ( \\lambda ^ k _ { I ^ g ( \\bar x ) } , \\rho ^ k , \\xi _ { \\mathcal I ( \\bar x ) } ^ k ) \\Vert _ 2 } . \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\Delta _ { q _ 1 } ( f , < q _ 1 > _ f ) \\Delta _ { p _ 1 } ( f , < p _ 1 > _ f ) \\to \\frac { \\theta \\eta } { 8 \\mu \\lambda } = \\frac { \\xi } { 4 ( 1 + \\sqrt { 1 - \\xi } ) } \\leq \\frac { \\xi } { 4 } < \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} \\frac { d ^ a } { d X ^ a } \\sum _ { j = 0 } ^ { n ' - 1 } T _ { \\o _ j } ( 1 , R ( X ) ) X ^ j \\Big | _ { X = \\eta } = \\frac { d ^ a } { d X ^ a } \\sum _ { j = 0 } ^ { n ' - 1 } T _ { \\o ' _ j } ( 1 , R ( X ) ) X ^ j \\Big | _ { X = \\eta } \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} & J [ \\xi _ 1 ^ { y , s , \\varepsilon } , p ] ( \\hat { x } , \\hat { t } ) + K _ { ( 0 , \\hat { x } ) } [ \\xi _ 1 ^ { y , s , \\varepsilon } , p ] ( \\hat { x } , \\hat { t } ) \\\\ & = - ( M _ 1 + M _ 2 + \\| f \\| _ \\infty ) ( J [ \\rho , \\rho ' ( \\hat { x } ) ] ( \\hat { x } , \\hat { t } ) + K _ { ( 0 , \\hat { x } ) } [ \\rho , \\rho ' ( \\hat { x } ) ] ( \\hat { x } , \\hat { t } ) ) \\\\ & = - ( M _ 1 + M _ 2 + \\| f \\| _ \\infty ) ( D _ x ^ \\alpha \\rho ) _ x ( \\hat { x } ) \\\\ & = M _ 1 + M _ 2 + \\| f \\| _ \\infty . \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} \\alpha = \\frac { A } { 3 } - \\frac { B } { 3 } - \\frac { 1 } { 2 } , ~ ~ ~ \\beta = \\frac { A } { 3 } + \\frac { B } { 3 } - \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} Y _ t = \\rho _ t \\left ( Q _ t , W _ { 1 , t } , \\ldots , W _ { N , t } \\right ) . \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle J _ { \\infty } ( B _ { i . j } ) \\twoheadrightarrow g r ^ { i , j } ( J _ { \\infty } ( B _ { i . j - 1 } ) ) & \\mbox { i f } j - 1 \\geq 1 \\\\ J _ { \\infty } ( B _ { i . j } ) \\twoheadrightarrow g r ^ { i . j } ( J _ { \\infty } ( B _ { i - 1 . i - 1 } ) ) & \\mbox { i f } j = 1 \\\\ \\end{cases} , \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} e ( \\lambda _ I ) = e ( I ) = i _ 1 - i _ 2 - \\cdots - i _ s . \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} \\mu _ { m } ^ { m - 1 } | _ { I } = \\ell ( I ) ^ { d } \\frac { \\mu _ { m } ^ { m } | _ { I } } { \\mu _ { m } ^ { m } ( I ) } < \\frac { 1 } { 2 } \\mu _ { m } ^ { m } | _ { I } \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} A = 2 \\lambda n ( 1 - m ^ 2 ) u ^ 2 \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} C : = A + _ G B , \\ , \\ , \\ , \\ , D : = A + _ G \\lambda B , \\ , \\ , \\ , \\ , E : = A \\cdot _ G B . \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} \\frac { p _ n } { q _ n } = [ a _ 0 , a _ 1 , \\dots , a _ n ] \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} \\partial _ t f + v \\cdot \\nabla _ { \\ ! x } f = \\bar { A } _ g f + \\bar { K } _ g f \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} ( \\mathcal { A } ( G ) \\hat { \\otimes } \\ , { } ^ b \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } \\ , , ( \\mathcal { A } ( G ) \\hat { \\otimes } \\ , \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} I _ { 1 } = [ \\frac { 1 } { \\xi } , \\infty ) , I _ { 2 } = [ t + \\sqrt { t } , \\frac { 1 } { \\xi } ] , I _ { 3 } = [ t - \\sqrt { t } , t + \\sqrt { t } ] , I _ { 4 } = [ \\frac { t } { 2 } , t - \\sqrt { t } ] , I _ { 5 } = [ 0 , \\frac { t } { 2 } ] \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} \\tilde A _ { h , \\tilde h } = \\tilde h ^ m h ^ m x ^ { - m } ( x D _ x ) ^ j P _ { m - j } ( h , x , y , D _ y ) . \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} s ( r ) \\int _ { | y | \\leq r } ( 1 \\wedge | r ^ { - 1 } y | ^ 2 ) \\nu ( d y ) & = w ( r ) \\int _ { | y | \\leq r } | r ^ { - 1 } y | ^ 2 \\nu ( d y ) \\\\ & = \\int _ { | y | \\leq 1 } | y | ^ 2 w ( r ) \\nu ( r \\ , d y ) \\leq N , \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} \\frac { \\partial \\zeta } { \\partial t } = \\frac { \\partial ^ 2 } { \\partial \\theta ^ 2 } \\varphi ' ( \\zeta ) \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} g ^ { - 1 } ( f ^ { - 1 } ( A \\cap K ) , f ( A \\cap L ) ) & = g ^ { - 1 } ( f ^ { - 1 } ( \\{ q \\} ) , f ( A \\cap L ) ) \\\\ & = Y \\cup f ( A \\cap L ) = \\varphi ( A \\cap Y ) \\cup f ( A \\cap L ) . \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} r ' = s ' . \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} \\frac { 8 - \\mu } { 4 } = 2 0 - \\mu , \\mbox { h e n c e } \\mu = 2 4 . \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} | \\nabla ^ n ( f ^ { - 1 } ) | ^ 2 \\lesssim \\sum _ { j = 1 } ^ { n } \\sum _ { i _ 1 + \\cdots + i _ j = n } \\frac { | \\nabla ^ { i _ 1 } f | ^ 2 \\cdots | \\nabla ^ { i _ j } f | ^ 2 } { f ^ { 2 ( j + 1 ) } } , \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} f ( m , H ( \\vec { \\bf b } ) ) \\ge \\max _ { 1 \\le i \\le k - 2 } \\left ( { \\frac { m b _ { i + 1 } } { 2 ( b _ i + b _ { i + 1 } ) \\binom { b _ { k - 1 } + b _ k } { b _ k } } } \\right ) ^ { \\frac 1 k } . \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( h ( r ) ) = r e ^ { i f ( r ) } , r > 0 . \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} X : \\ x _ 0 x _ 1 x _ 2 x _ 3 - ( x _ 0 + x _ 1 + x _ 3 ) ( x _ 0 ^ { 3 } + x _ 0 ^ { 2 } x _ 1 + x _ 1 ^ { 3 } + x _ 1 x _ 2 ^ { 2 } + x _ 2 ^ { 3 } + x _ 0 x _ 1 x _ 3 + x _ 2 x _ 3 ^ { 2 } ) = 0 . \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} \\psi _ { v _ { 2 } } ( \\xi , \\lambda ( t ) ) = 2 \\chi _ { \\leq \\frac { 1 } { 4 } } ' ( \\xi ) \\partial _ { \\xi } \\left ( \\frac { \\xi ^ { 2 } K _ { 1 } ( \\xi \\lambda ( t ) ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\right ) + \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } '' ( \\xi ) \\xi ^ { 2 } K _ { 1 } ( \\xi \\lambda ( t ) ) } { \\log ^ { b + 1 } ( \\frac { 1 } { \\xi } ) } \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} b > - ( 1 - \\varepsilon ) \\mu , \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} P _ { n } ( x ) & = \\tfrac { \\sqrt { n ( n + 1 ) } } { 6 } \\sum _ { j = 2 } ^ { n } \\tfrac { 1 } { \\sqrt { j ( j + 1 ) } } \\sum _ { i = 1 } ^ { j - 1 } ( x _ { j } ^ { 3 } - 3 x _ { j } x _ { i } ^ { 2 } ) , \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} I _ { j } : = \\{ i \\in \\lbrace 1 , \\ldots , n \\rbrace : x _ i \\in A _ j \\} \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} \\langle j _ p ( x - P _ M x ) , P _ M x \\rangle = \\langle \\Pi ^ { p ^ * } _ { M ^ \\perp } j _ p ( x ) , P _ M x \\rangle = 0 \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} u _ t - \\operatorname { d i v } \\left ( D \\nabla u \\right ) = q ( x , t ) , \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} f ^ \\kappa ( B ) & : = f ^ { \\ @ B _ \\kappa } ( B ) = \\min \\{ C \\in \\ @ B _ \\kappa ( Y ) \\mid B \\le f ^ * ( C ) \\} ; \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} r _ { \\mathsf { m , n } } = \\exp \\left [ - \\frac { 1 } { 2 } \\left ( \\left ( \\alpha + 1 \\right ) \\lambda _ { \\mathsf { m } } + \\left ( \\alpha - 1 \\right ) \\lambda _ { \\mathsf { n } } \\right ) \\beta \\right ] \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} S ^ 2 ( M ) = \\sum _ { x = 0 } ^ { \\infty } Z _ x & \\le \\left ( 1 + 2 \\sum _ { k = 1 } ^ { \\infty } \\dfrac { 1 } { ( \\delta _ { 2 k } ) ^ 2 } \\right ) \\cdot \\left ( \\sum _ { x = 0 } ^ { \\infty } \\dfrac { 1 } { f ( 0 , x ) ^ 2 } \\right ) = : \\lambda < + \\infty \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} \\int _ { a _ 1 } ^ { b _ 1 } ( L f ( x ) g ( x ) - f ( x ) L g ( x ) ) \\d x & = \\int _ { a _ 1 } ^ { b _ 1 } ( f ( x ) g ' ( x ) - f ' ( x ) g ( x ) ) ' \\d x \\\\ & = W ( f , g ; a _ 1 ) - W ( f , g ; b _ 1 ) . \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} \\wedge ^ N ( V \\otimes W ) = \\bigoplus _ { a _ 1 + \\dots + a _ n = N } \\wedge ^ { a _ 1 } V \\otimes \\dots \\otimes \\wedge ^ { a _ n } V . \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} C ' : = f _ 2 ( C _ Y ) \\subseteq \\Pi _ 2 ^ { ( t ) } . \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} \\nabla ^ 2 F ( \\tilde { y } _ k ) = ( \\tilde { L } - m ) \\Big ( \\nabla ^ 2 \\phi \\big ( ( \\nabla \\phi ) ^ { - 1 } ( \\tilde { y } _ k ) \\big ) \\Big ) ^ { - 1 } - I . \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} \\int _ { B C } e ^ { s x } \\overline { G } ( s , t ) d s = \\int _ { r } ^ { - \\lambda _ 2 + R } e ^ { - x \\lambda _ 2 } e ^ { - w x } e ^ { t ( c _ 1 \\lambda ^ { \\alpha _ 1 } + c _ 2 \\lambda _ 2 ^ { \\alpha _ 2 } ) } e ^ { - t [ c _ 1 ( \\lambda _ 1 - \\lambda _ 2 + w e ^ { i \\pi } ) ^ { \\alpha _ 1 } + c _ 2 ( w ^ { \\alpha _ 2 } e ^ { i \\pi \\alpha _ 2 } ) ] } d w . \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} \\tt = ( \\mu , \\beta _ 0 ^ 2 / 2 + ( \\sigma _ + ^ 2 - 1 ) , \\beta _ 0 ^ 2 , 0 , 0 ) ^ \\tau . \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} J ( u ( x , t ) ) \\le J ( u ( x , t _ 0 ) ) = J ( u _ 0 ) - \\int _ 0 ^ { t _ 0 } \\Big \\| | x | ^ { - \\frac s 2 } u _ t \\Big \\| _ 2 ^ 2 d t < d 0 \\leq t < t _ 0 , \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} \\varphi ^ R _ { T _ R ( r ; \\mu ) } \\left ( \\mu , r \\right ) & = - \\mu \\\\ \\psi ^ R _ { T _ R ( r ; \\mu ) } \\left ( \\mu , r \\right ) & = P _ R ( r ; \\mu ) , \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} \\inf _ { | \\xi | = 1 } \\int _ { | y | \\leq N _ \\nu } | y \\cdot \\xi | ^ 2 \\nu ( d y ) > 0 , \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} A ^ \\top M E + E ^ \\top M A + F = 0 \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} D ^ { \\alpha } _ x I _ Z ( u _ 0 ) ( t , x ) = I _ Z ( D ^ { \\alpha } _ x u _ 0 ) ( t , x ) , D ^ { \\alpha } _ x F _ Z ( f ) ( t , x ) = F _ Z ( D ^ { \\alpha } _ x f ) ( t , x ) . \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} \\delta ( h _ { i j } ) = \\delta \\langle \\mathbf { N } , \\mathbf { r } _ { i j } \\rangle = \\langle \\delta \\mathbf { N } , \\mathbf { r } _ { i j } \\rangle + \\langle \\mathbf { N } , \\delta \\mathbf { r } _ { i j } \\rangle . \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} B _ 3 = - \\int _ { \\R ^ 2 } \\partial _ k u \\cdot \\nabla X \\cdot \\partial _ k X ~ d x \\leq \\| \\nabla u \\| _ { L ^ \\infty } \\| \\nabla X \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} \\begin{bmatrix} - 1 . 7 2 8 7 & - 1 . 1 7 6 1 & - 1 . 0 5 9 7 & - 0 . 3 2 3 6 & - 0 . 2 3 4 0 \\\\ 0 . 4 7 0 6 & 0 . 4 7 1 2 & 0 . 5 4 3 5 & 0 . 6 3 0 9 & 0 . 7 5 3 3 \\\\ 0 . 8 0 2 0 & 0 . 9 2 3 7 & 1 . 1 3 9 4 & 1 . 4 3 7 3 & 1 . 5 3 5 1 \\\\ 1 . 6 9 4 1 & 8 . 6 5 0 1 & 1 0 . 7 2 5 4 & 1 2 . 7 6 9 4 & 1 3 . 0 3 4 9 . \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} p _ { n + 1 } ( \\underline x _ { n + 1 } ) = \\sum _ { I \\subseteq [ 2 , n + 1 ] } k _ { n + 1 - | I | } ( \\underline x _ { n + 1 } ^ I ) . \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} \\| f \\mid L ^ 1 _ p ( \\Omega ) \\| = \\| \\nabla f \\mid L _ p ( \\Omega ) \\| . \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} A _ h = h ^ 2 \\Delta _ g + 1 . \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} \\Delta _ { p , f } \\phi : = e ^ { f } { \\rm d i v } ( e ^ { - f } | \\nabla \\phi | ^ { p - 2 } \\nabla \\phi ) , \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} I ' = \\langle & E _ { 1 , 2 } ^ { 2 } , E _ { 1 , 3 } ^ { 2 } , E _ { 1 , 4 } ^ { 2 } , E _ { 2 , 3 } ^ { 2 } , E _ { 2 , 4 } ^ { 2 } , E _ { 3 , 4 } ^ { 2 } , E _ { 1 , 2 } E _ { 1 , 3 } , E _ { 1 , 2 } E _ { 1 , 4 } , E _ { 1 , 3 } E _ { 1 , 4 } , E _ { 2 , 3 } E _ { 2 , 4 } , \\\\ & E _ { 1 , 3 } E _ { 2 , 4 } , E _ { 1 , 3 } E _ { 2 , 3 } , E _ { 1 , 4 } E _ { 2 , 4 } , E _ { 1 , 4 } E _ { 3 , 4 } , E _ { 2 , 4 } E _ { 3 , 4 } \\rangle , \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{gather*} J _ 1 ( L \\bar { u } ) = J ( L \\bar { u } , 0 ) < 0 \\ , \\ , J _ 2 ( L \\bar { v } ) = J ( 0 , L \\bar { v } ) < 0 . \\end{gather*}"}
-{"id": "4432.png", "formula": "\\begin{align*} 1 - ( n + 1 - k ) \\frac { d - 1 } { ( n + 1 - k ) d - c + ( k - 2 ) } = \\frac { n - 1 - c } { d ( n + 1 - k ) - c + ( k - 2 ) } \\geq 0 . \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} J ( u ) = \\frac 1 q I ( u ) + \\frac { q - p } { p q } \\| \\nabla u \\| _ p ^ p + \\frac { 1 } { q ^ 2 } \\| u \\| _ q ^ q . \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} \\lambda ( t ) = \\frac { 1 } { \\log ^ { b } ( t ) } + e ( t ) , | e ( t ) | \\leq \\frac { C } { \\log ^ { b } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } , | e ^ { ( j ) } ( t ) | \\leq \\begin{cases} \\frac { C } { t ^ { j } \\log ^ { b + 1 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } , j = 1 , 2 \\\\ \\frac { C } { t ^ { j } \\log ^ { b + 1 } ( t ) } , j = 3 , 4 \\end{cases} \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} \\lim _ { \\mu \\rightarrow - \\infty } \\lambda _ j ( \\mu ) = \\eta _ j , \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} J ^ { x } = \\{ q \\in \\mathfrak { g } _ { \\bar { 0 } } | \\phi ^ { x } ( q , \\mathfrak { g } _ { \\bar { 0 } } ) = \\{ 0 \\} \\} , \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} & { \\alpha } ^ 2 - \\frac { 3 } { 4 } ( \\sqrt { 2 } - 1 ) ^ 2 N ^ 2 + \\frac { 1 } { 2 } \\sqrt { C ( M _ 0 ) \\left ( \\frac { 9 } { 4 } - \\frac { \\alpha ^ 2 } { N ^ 2 } \\right ) } M _ 0 ^ { - \\frac { 1 } { 2 } } N ^ 2 = 0 , \\\\ & { \\beta } ^ 2 - \\frac { 3 } { 4 } ( \\sqrt { 2 } - 1 ) ^ 2 N ^ 2 - \\frac { 1 } { 2 } \\sqrt { C ( M _ 0 ) \\left ( \\frac { 9 } { 4 } - \\frac { \\beta ^ 2 } { N ^ 2 } \\right ) } M _ 0 ^ { - \\frac { 1 } { 2 } } N ^ 2 = 0 . \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} \\| f \\| _ { H ^ { - 1 } } ^ 2 = \\int _ { \\mathbb { T } ^ 1 } \\omega ^ 2 ( \\theta ) d \\theta , \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} | 3 \\Phi ( \\xi _ 1 , \\eta _ 1 , - \\xi _ 2 , - \\eta _ 2 ) & + \\tau _ 2 - \\xi _ 2 ^ 3 - \\eta _ 2 ^ 3 | \\lesssim \\max ( L _ 0 , L _ 1 ) , \\\\ | \\partial _ { \\eta _ 1 } \\Phi ( \\xi _ 1 , \\eta _ 1 , - \\xi _ 2 , - \\eta _ 2 ) | & = | ( \\eta _ 1 + ( \\eta _ 2 - \\eta _ 1 ) ) \\ , ( \\eta _ 1 - ( \\eta _ 2 - \\eta _ 1 ) ) | \\\\ & \\geq ( | \\eta _ 1 | - | \\eta _ 2 - \\eta _ 1 | ) ^ 2 \\gtrsim A ^ { - 2 } N _ 1 ^ 2 . \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} \\bigcup _ \\rho \\ , Z _ \\rho = A _ { i j } , \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} q _ { \\mathsf { k } , \\mathsf { m } } = c _ { \\mathsf { q , } \\alpha , \\beta } \\hat { p } _ { \\mathsf { k , m } } - \\frac { \\hat { p } _ { \\mathsf { k , m } } } { \\nu _ { \\mathsf { k } } + b _ { \\mathsf { m } } } \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} & | \\partial _ { r } v _ { 5 } ( t , r ) | \\leq \\frac { C } { \\sqrt { r } } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { 1 } d \\xi \\sqrt { \\xi } | \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( s , \\xi ) | + \\frac { C } { \\sqrt { r } } \\int _ { t } ^ { \\infty } d s | | \\xi \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( s ) | | _ { L ^ { 2 } ( \\xi d \\xi ) } \\\\ & \\leq \\frac { C \\log ^ { 3 } ( t ) } { \\sqrt { r } t ^ { 3 / 2 } } \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} \\mathcal { A } B \\mathcal { B } = \\mathcal { A } ( k A ) \\mathcal { B } = \\left ( \\begin{array} { c c c c c c c } k \\tilde { a _ 1 } & & & & & \\\\ & \\ddots & & & & \\\\ & & k \\tilde { a _ t } & & & \\\\ & & & 0 & & \\\\ & & & & \\ddots & \\\\ & & & & & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} q ( x ) = \\begin{cases} 0 , & x < R _ c \\ , , \\\\ \\frac { b x } { 1 + b h x } , & x > R _ c \\ , , \\end{cases} \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { m - 1 } K _ j ~ \\longrightarrow ~ F _ { m - 1 } ^ { \\varphi } \\stackrel { u } { \\longrightarrow } B \\left ( \\prod _ { j = 1 } ^ { m - 1 } Z _ j \\right ) . \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} \\phi ^ I & = \\frac { \\partial S ( p , q ) } { \\partial q _ I } , \\\\ \\psi _ I & = - \\frac { \\partial S ( p , q ) } { \\partial p ^ I } . \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} \\lim _ { a \\uparrow a ^ * } & ( a ^ * - a ) ^ { - p _ { j _ 0 } / p } K ( x _ { j _ 0 } + ( a ^ * - a ) ^ { 1 / p } x ) | v _ a ( { x + ( a ^ * - a ) ^ { - 1 / p } x _ { j _ 0 } } ) | ^ { 2 + 4 / d } \\\\ & = \\lambda _ { j _ 0 } | x | ^ { p _ { j _ 0 } } \\Big ( b ^ { d / 2 } Q _ 0 ( b ( x + x _ 0 ) ) \\Big ) ^ { 2 + 4 / d } . \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} S _ T ( u ) : = \\int _ { [ 0 , T ] ^ d } 1 _ { ( X ( t ) \\ge u ) } \\ , d t \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} P ( r ; \\mu ) = \\sqrt { r ^ 2 + 4 \\kappa \\mu + \\frac { 4 \\alpha } { 3 } \\ , r ^ 3 + E } , \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} a = \\left ( \\frac { y } { B ^ 2 } + B ^ 2 \\right ) ^ 2 + \\frac { y } { B ^ 2 } + B ^ 2 ; \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} ( q ^ { 2 / 3 } - p ^ { 2 / 3 } | \\tau | ^ 2 ) ^ 2 - p ^ { 2 / 3 } q ^ { 2 / 3 } ( \\tau - \\Bar { \\tau } ) ^ 2 & = q ^ { 4 / 3 } + p ^ { 4 / 3 } | \\tau | ^ 4 - p ^ { 2 / 3 } q ^ { 2 / 3 } ( \\tau ^ 2 + \\Bar { \\tau } ^ 2 ) \\\\ & = ( p ^ { 2 / 3 } \\Bar { \\tau } ^ 2 - q ^ { 2 / 3 } ) \\cdot ( p ^ { 2 / 3 } \\tau ^ 2 - q ^ { 2 / 3 } ) \\\\ & = ( p ^ { 1 / 3 } \\Bar { \\tau } + q ^ { 1 / 3 } ) \\cdot ( p ^ { 1 / 3 } \\Bar { \\tau } - q ^ { 1 / 3 } ) \\cdot ( p ^ { 1 / 3 } \\tau - q ^ { 1 / 3 } ) \\cdot ( p ^ { 1 / 3 } \\tau + q ^ { 1 / 3 } ) \\\\ & = | q ^ { 1 / 3 } + p ^ { 1 / 3 } \\tau | ^ 2 \\cdot | q ^ { 1 / 3 } - p ^ { 1 / 3 } \\tau | ^ 2 , \\\\ \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} ( L ^ p , \\ell ^ q ) ^ \\ast = ( L ^ { p ^ { \\prime } } , \\ell ^ { q ^ { \\prime } } ) , 1 / p + 1 / p ^ { \\prime } = 1 , 1 / q + 1 / q ^ { \\prime } = 1 , \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} \\left \\Vert e ^ { k \\left \\vert x \\right \\vert ^ { p } } u \\left ( x , 0 \\right ) \\right \\Vert _ { X } = a _ { k } \\leq a _ { 2 } e ^ { 2 a _ { 1 } k ^ { \\frac { q } { q - p } } } = a _ { 2 } e ^ { 2 a _ { 1 } k ^ { \\frac { 1 } { 2 - p } } } . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} P _ f ( A , B + L _ n ) & = ( B + L _ n ) ^ { 1 / 2 } f \\bigl ( ( B + L _ n ) ^ { - 1 / 2 } A ( B + L _ n ) ^ { - 1 / 2 } \\bigr ) ( B + L _ n ) ^ { 1 / 2 } , \\\\ P _ { f _ \\delta } ( A , B + L _ n ) & = ( B + L _ n ) ^ { 1 / 2 } f \\bigl ( ( B + L _ n ) ^ { - 1 / 2 } A ( B + L _ n ) ^ { - 1 / 2 } + \\delta I \\bigr ) ( B + L _ n ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} g ( x ) = \\int _ a ^ x \\phi _ x ^ a ( y ) ( L _ \\bullet g ) ( y ) \\d y + \\int _ x ^ b \\phi _ x ^ b ( y ) ( L _ \\bullet g ) ( y ) \\d y . \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} w _ i ^ \\beta ( - 1 ) = \\psi _ i ( \\beta - 1 ) - k \\phi _ i ( - 1 ) \\to - k \\phi _ i ( - 1 ) < 0 \\mbox { a s } \\beta \\to \\infty . \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} C _ { 3 , 0 } ( 9 ) = & \\{ ( 3 ^ 3 ) \\} \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} k ( x , a ) = ( k x , k a ) , ( x , a ) + ( y , b ) = ( x + y , a + b ) , a ( y , b ) = ( a y , a b ) , \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} G _ 0 ( F ( T ) , T _ 0 ) = & \\{ \\ , U + F ( T ) \\mid U \\in T _ 0 \\ , \\} \\\\ = & \\{ \\ , ( X ' \\cap V ) + F ( T ) \\mid V \\in T \\ , \\} \\\\ = & \\ , T . \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} \\sup _ { \\partial B _ r } \\varphi _ 1 / G \\leq C ( \\Sigma ) \\inf _ { \\partial B _ r } \\varphi _ 1 / G = C ( \\Sigma ) \\inf _ { A _ { r , r _ 2 } } \\varphi _ 1 / G \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} D _ { H } = \\hbox { d i a g } ( h ^ { 0 } _ { 1 } , \\dots , h ^ { 0 } _ { d } ) \\quad ; h ^ { 0 } _ { j } \\in \\mathbb { R } \\ , \\ j \\in \\{ 1 , \\dots , d \\} . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} \\lim _ n s ( f , \\Psi , \\mathcal { P } ^ 1 _ n ) = u ( f , \\Psi ) , \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} & ( L _ { 1 } ( u _ { 1 } ) - L _ { 1 } ( u _ { 2 } ) ) ( t , r ) \\\\ & = \\left ( \\frac { \\sin ( 2 u _ { 1 } ( t , r ) ) - \\sin ( 2 u _ { 2 } ( t , r ) ) } { 2 r ^ { 2 } } \\right ) \\left ( \\cos ( 2 Q _ { 1 } ( \\frac { r } { \\lambda ( t ) } ) ) \\left ( \\cos ( 2 v _ { c o r r } ( t , r ) ) - 1 \\right ) - \\sin ( 2 Q _ { 1 } ( \\frac { r } { \\lambda ( t ) } ) ) \\sin ( 2 v _ { c o r r } ( t , r ) ) \\right ) \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} t p _ { \\mu } ( t ) = \\frac { 1 } { \\pi } \\frac { h ^ { \\langle - 1 \\rangle } ( 1 / t ) \\sin f ( h ^ { \\langle - 1 \\rangle } ( 1 / t ) ) } { | 1 - h ^ { \\langle - 1 \\rangle } ( 1 / t ) e ^ { i f ( h ^ { \\langle - 1 \\rangle } ( 1 / t ) ) } | ^ { 2 } } , t \\notin D _ { \\mu } . \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} U _ { \\pi ( x ) } = U _ x . \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} B _ { H , \\ ; L + m } - B _ { H , \\ ; L + m - 1 } = 2 H _ { L + m - 1 } - H _ { L + m } . \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} \\varphi ( m ) = \\sup _ { \\sigma \\in \\mathbb { R } } \\left ( \\sigma m - A ( \\sigma ) \\right ) = \\sigma _ { \\infty } m - A ( \\sigma _ { \\infty } ) . \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { N \\to \\infty } P \\left \\{ \\int ^ { 1 } _ { 1 - \\delta } \\frac { | Z _ N ( t ) - \\sigma B _ N ( t ) | ^ p } { w ( t ) } d t > \\varepsilon \\right \\} = 0 . \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} \\Delta _ { i , 1 } f ( x ) & = \\frac { 1 } { g ( x ) } \\frac { \\partial } { \\partial x _ i } \\left [ g ( x ) \\left ( f ( F ( x ) ) - f ( x ) \\right ) \\right ] - \\frac { \\partial ( f ( y ) - f ( x ) ) } { \\partial x _ i } ( x , F ( x ) ) \\\\ & = \\nabla f ( F ( x ) ) \\cdot \\frac { \\partial F } { \\partial x _ i } ( x ) - \\frac { \\partial f } { \\partial x _ i } ( x ) + \\frac { \\partial \\log g ( x ) } { \\partial x _ i } ( f ( F ( x ) ) - f ( x ) ) + \\frac { \\partial f } { \\partial x _ i } ( x ) , \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} z \\eta _ { \\mu _ { 1 } } ( \\eta _ { \\rho _ { 1 } } ( z ) ) = z \\eta _ { \\mu _ { 2 } } ( \\eta _ { \\rho _ { 2 } } ( z ) ) = \\eta _ { \\rho _ { 1 } } ( z ) \\eta _ { \\rho _ { 2 } } ( z ) , z \\in \\mathbb { D } . \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} \\displaystyle \\mathcal { P } _ n ^ + = \\lbrace \\omega \\in \\Omega \\ : s \\mapsto X ( \\omega , s ) \\ s . t . \\ V _ 0 ( \\omega ) = + c , \\ N ( \\omega , t ) = n , \\ \\max _ { 0 \\le s \\le t } X ( \\omega , s ) > \\beta , \\ X ( \\omega , t ) = x \\rbrace \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\log \\{ 0 . 5 \\phi ( Z _ { 2 i } ; \\beta _ 0 + \\beta _ 1 Z _ { 1 i } , \\eta ) + 0 . 5 \\phi ( - Z _ { 2 i } ; \\beta _ 0 + \\beta _ 1 Z _ { 1 i } , \\eta ) \\} \\leq - n \\log M _ 0 = - 4 n . \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} \\| e ^ { \\tau ( t ) A } \\overline { v } ( t ) \\| _ { H ^ r } ^ 2 + \\| e ^ { \\tau ( t ) A } \\widetilde { v } ( t ) \\| _ { H ^ r } ^ 2 & + 2 \\int _ 0 ^ t \\| A ^ { r + \\frac { 1 } { 2 } } e ^ { \\tau ( s ) A } \\overline { v } ( s ) \\| ^ 2 + \\| A ^ { r + \\frac { 1 } { 2 } } e ^ { \\tau ( s ) A } \\widetilde { v } ( s ) \\| ^ 2 d s \\\\ & \\leq \\| e ^ { \\tau _ 0 A } \\overline { v } _ 0 \\| _ { H ^ r } ^ 2 + \\| e ^ { \\tau _ 0 A } \\widetilde { v } _ 0 \\| _ { H ^ r } ^ 2 . \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} f _ L ( 0 , 0 ; \\mu ) & = 0 , & f _ R ( 0 , 0 ; \\mu ) & = 0 , \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} D _ { k , m } ( a , X ) = m E _ { k } ( a , X ) - ( m - 1 ) D _ { k } ( a , X ) . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} \\ker \\left ( A + P \\right ) = \\left \\{ 0 \\right \\} \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta u + n u = 0 , & ; \\\\ u = f , & , \\end{array} \\right . \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} R i c ( \\omega _ i ( t ) ) = - \\omega ( t ) + t \\hat \\omega _ i . \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} \\rho _ \\alpha ( V ) = \\bar \\rho _ { \\phi } ( M ) . \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} Q ( x , \\lambda ) = \\lambda - g ( x ) . \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} \\mathcal { L } _ k ^ \\beta = \\{ p \\in \\mathcal { L } ^ \\beta : \\forall A \\in p \\ , \\ , \\ , \\ , \\lambda ( A ) > k \\} \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} e = l ( a b ) & \\sim l ( b a ) \\sim r ( b a ) \\stackrel { \\eqref { b a e } } { \\precsim } \\mathbf { 1 } - e . \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} \\lVert ( \\alpha , A ) \\rVert = | \\alpha | + \\lVert A \\rVert . \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} F ^ * ( \\zeta ) : = \\sup _ { \\xi \\in \\mathbb { R } ^ n } \\left ( \\langle \\zeta , \\xi \\rangle - F ( \\xi ) \\right ) { \\forall \\ , } \\zeta \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} 0 & = \\det \\left ( J - \\mu - e _ { k } e _ { k } ^ { \\top } J \\right ) \\\\ & = \\det \\left ( J - \\mu \\right ) \\det \\left ( I _ { n } - e _ { k } e _ { k } ^ { \\top } J G _ { J } \\left ( \\mu \\right ) \\right ) \\\\ & = \\det \\left ( J - \\mu \\right ) \\left ( 1 - e _ { k } ^ { \\top } J G _ { J } \\left ( \\mu \\right ) e _ { k } \\right ) . \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} d \\theta _ { 1 } ^ { 1 } = d \\sigma \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} \\partial _ x Y _ \\mu ^ * ( x ) J Y _ \\mu ( x ) = ( \\mu - \\bar \\mu ) Y _ \\mu ^ * ( x ) A ( x ) Y _ \\mu ( x ) , Y _ \\mu ^ * ( a ) J Y _ \\mu ( a ) = J . \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} G : = \\left \\{ \\left ( \\frac { u \\sqrt v } { \\sqrt w } , \\frac { z \\sqrt w } { \\sqrt v } \\right ) : v , w , u , z v , w \\leq n ^ { 1 / 5 } , u , z \\leq n ^ { 3 / 5 } \\right \\} \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} \\Lambda _ { 1 ; f } \\lambda _ { 1 ; f } & \\leq \\frac { \\int _ { \\Omega } ( \\Delta _ f \\varphi ) ^ 2 \\ , d \\mu } { \\int _ { \\Omega } | \\nabla \\varphi | ^ 2 \\ , d \\mu } \\frac { \\int _ { \\Omega } | \\nabla \\varphi | ^ 2 \\ , d \\mu } { \\int _ { \\Omega } \\varphi ^ 2 \\ , d \\mu } \\\\ & = \\frac { \\int _ { \\Omega } ( \\Delta _ f \\varphi ) ^ 2 \\ , d \\mu } { \\int _ { \\Omega } \\varphi ^ 2 \\ , d \\mu } \\\\ & \\leq \\frac { 1 6 } { 3 } \\lambda _ { 1 ; f } ^ 2 , \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} \\eta _ { \\nu _ { n } ^ { \\boxtimes k _ { n } } } = \\eta _ { \\nu _ { n } } \\circ \\eta _ { \\mu _ { n } } , n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} y \\in \\bigcap _ { j = 1 } ^ { \\ell } \\tilde { Q } _ { I _ j } . \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} ( B _ 1 , B _ 2 ) = ( B _ 3 , B _ 4 ) = ( 1 , 0 ) . \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} \\Theta ^ { [ 2 ] } : \\begin{pmatrix} \\rho \\\\ \\kappa _ u \\\\ \\kappa _ s \\end{pmatrix} \\mapsto \\begin{pmatrix*} [ l ] A _ c D k _ c + D g _ c ( K { } { } ) \\kappa - D k _ c \\left ( R \\right ) P _ \\rho \\\\ - A _ u ^ { - 1 } D g _ u ( K { } { } ) \\kappa + A _ u ^ { - 1 } \\kappa _ u ( R ) P _ \\rho \\\\ A _ s \\kappa _ s ( T ) Q _ \\rho + D g _ s ( K { } { } \\circ T ) \\kappa ( T ) Q _ \\rho \\end{pmatrix*} \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} 9 9 9 9 7 & = ( - 9 8 ) ^ 2 + 3 4 ^ 2 + 1 1 9 ^ 2 + 2 7 4 ^ 2 \\ \\ - 9 8 + 3 \\times 3 4 = 4 , \\\\ 9 9 9 9 9 & = ( - 2 9 ) ^ 2 + 1 0 ^ 2 + 3 3 ^ 2 + 3 1 3 ^ 2 \\ \\ - 2 9 + 3 \\times 1 0 = 1 . \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} | \\Theta _ \\mathbf { F } [ \\varphi ] ( p ) | & = | \\Omega _ \\mathbf { F } [ \\varphi ^ + ] ( p ) - \\Omega _ \\mathbf { F } [ \\varphi ^ - ] ( p ) | \\\\ & \\leq \\sup _ { a \\in A } | \\varphi ^ + ( a ) - \\varphi ^ - ( a ) | \\cdot \\mathbf { F } ^ * ( a , p ) \\\\ & = \\sup _ { a \\in A } | \\varphi ( a ) | \\cdot \\mathbf { F } ^ * ( a , p ) \\leq \\sup _ { a \\in A } | \\varphi ( a ) | = \\| \\varphi \\| . \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} D _ { 3 } ( \\Pi ^ { 3 } _ { M _ 2 } v _ \\lambda , v _ \\lambda ) \\leq D _ { 3 } ( u _ \\lambda , v _ \\lambda ) = \\frac { 1 5 2 \\lambda ^ { 3 } } { 3 9 7 9 5 3 } . \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} T _ 0 = \\sup \\{ s \\in ( 0 , T ] : w _ i ( t , x ) > 0 \\ { \\rm f o r \\ a l l } \\ ( t , x ) \\in [ 0 , s ] \\times \\R , \\ ; 1 \\leq i \\leq m \\} . \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} F _ { 5 } ( t , r ) & = N _ { 2 } ( v _ { 5 } ) + \\frac { \\sin ( 2 ( v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } ) ) } { 2 r ^ { 2 } } \\left ( \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } + 2 v _ { 5 } ) - \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\right ) \\\\ & + \\left ( \\frac { \\cos ( 2 ( v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } ) ) - 1 } { 2 r ^ { 2 } } \\right ) \\left ( \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } + 2 v _ { 5 } ) - \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\right ) \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} P _ \\varepsilon = \\left ( \\begin{array} { c c } 1 - \\varepsilon \\lambda _ 0 & \\varepsilon \\lambda _ 0 \\\\ \\varepsilon \\lambda _ 1 & 1 - \\varepsilon \\lambda _ 1 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} C _ T ( n ) : = \\inf _ M \\frac { 1 } { d ( M ) } \\int _ M | H | = \\pi , \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} u ^ N _ t ( x ) = e ^ { t \\Delta } u ^ N _ 0 ( x ) - \\int _ 0 ^ t \\nabla \\cdot e ^ { ( t - s ) \\Delta } \\langle \\mu _ s ^ N , V ^ N ( x - \\cdot ) F \\big ( K \\ast u ^ N _ s ( \\cdot ) \\big ) \\rangle \\ d s \\\\ - \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\nabla V ^ N ( x - X _ s ^ { i , N } ) \\cdot d W ^ i _ s , \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} C ( S _ 1 , S _ k ) = \\sum _ { j = 1 } ^ { k - 1 } C ( S _ j , S _ { j + 1 } ) . \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} J ^ { \\sigma } _ u ( r , s ) : = \\int _ { A _ { r , s } } u ^ 2 ( t , \\omega ) t ^ { - n - 2 \\sigma } \\ d x \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} \\phi ( \\xi - \\mu ) \\widehat { f } ( \\xi ) = F ( \\mu ) \\theta ( \\xi - \\mu ) , \\phi ( \\eta - \\nu ) \\widehat { g } ( \\xi ) = G ( \\nu ) \\theta ( \\eta - \\nu ) \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} | \\{ ( a + b ) ( b + d ) = 1 ~ : ~ a \\in A , \\ , b \\in B , \\ , d \\in D \\} | \\ll \\sqrt { | A | | D | } N \\cdot N ^ { - C _ 2 \\d / \\tau } \\ , . \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} a _ { l , J , l ' } = \\overline { b _ { l , J , l ' } } , \\quad \\mbox { f o r a l l $ J \\in \\mathbb { N } ^ { N } $ h a v i n g l e n g t h $ p - 1 \\geq 2 $ a n d $ l , l ' = 1 , \\dots , N $ . } \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} | N _ { 2 } ( f _ { v _ { 5 } } ) ( t , r ) | \\leq \\begin{cases} \\frac { C r } { ( \\lambda ( t ) ^ { 2 } + r ^ { 2 } ) t ^ { 4 } \\log ^ { 3 b } ( t ) } + \\frac { C r } { t ^ { 6 } \\log ^ { 3 b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C \\log ^ { 3 } ( r ) } { r ^ { 2 } | t - r | ^ { 3 } } + \\frac { C } { r ^ { 7 / 2 } t ^ { 5 / 2 } \\log ^ { 3 N + 7 b - 2 } ( t ) } , \\frac { t } { 2 } \\leq r \\leq t - \\sqrt { t } r \\geq t + \\sqrt { t } \\\\ \\frac { C } { r ^ { 7 / 2 } } , t - \\sqrt { t } \\leq r \\leq t + \\sqrt { t } \\end{cases} \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} v _ { 3 } ( t , r ) = - \\frac { 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\int _ { 0 } ^ { 2 \\pi } d \\theta \\frac { F _ { 0 , 1 } ( s , \\sqrt { r ^ { 2 } + \\rho ^ { 2 } + 2 r \\rho \\cos ( \\theta ) } ) ( r + \\rho \\cos ( \\theta ) ) } { \\sqrt { r ^ { 2 } + \\rho ^ { 2 } + 2 r \\rho \\cos ( \\theta ) } } \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} \\partial _ { t } u = i \\left [ \\Delta u + L u + q \\left ( x , t \\right ) u \\right ] , x \\in R ^ { n } , y \\in G , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} Y B Y - Y A - D Y + C = 0 . \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} \\langle \\widehat { p } _ 1 ^ 2 f , f \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = \\frac { \\mu ^ 2 } { a } + \\left ( \\frac { \\eta } { 2 \\mu } \\right ) ^ 2 \\left ( ( x _ 2 ^ { ( 0 ) } ) ^ 2 + \\frac { b } { 4 } \\right ) \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} \\big ( f ^ * + g ^ * \\big ) ^ * = f \\Box g \\ ; \\Longleftrightarrow \\ ; f \\Box g \\in \\Gamma _ 0 ( E ) . \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} K ( \\xi , p ) : = \\frac { \\xi ^ p - \\xi } { ( p - 1 ) ( \\xi - 1 ) } \\left ( \\frac { p - 1 } { p } \\cdot \\frac { \\xi ^ p - 1 } { \\xi ^ p - \\xi } \\right ) ^ p \\mbox { f o r $ \\xi > 1 $ a n d $ p \\in \\mathbb { R } $ } , \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} \\beta _ { \\Lambda ^ c ; i , j } : = \\lim _ { \\Lambda _ { 1 } \\uparrow \\uparrow V } \\prod _ { x \\in \\Lambda _ { 1 } \\setminus \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * \\right ) = \\lim _ { \\Lambda _ { 1 } \\uparrow \\uparrow V } e ^ { \\sum _ { x \\in \\Lambda _ { 1 } \\setminus \\Lambda } H _ { x , i , j } } ; \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} A [ g , \\bar { g } , \\hat { g } ] = \\begin{psmallmatrix} H _ x & I _ x & J _ x \\\\ H _ y & I _ y & J _ y \\\\ H _ { p _ x } & I _ { p _ x } & J _ { p _ x } \\\\ H _ { p _ y } & I _ { p _ y } & J _ { p _ y } \\end{psmallmatrix} \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} p ( x , t ) = \\frac { \\Gamma ( 2 \\alpha ) } { \\Gamma ( \\alpha ) ^ 2 } \\frac { ( c _ 1 t - x ) ^ { \\alpha - 1 } ( c _ 2 t + x ) ^ { \\alpha - 1 } } { \\bigl [ ( c _ 1 + c _ 2 ) t \\bigr ] ^ { 2 \\alpha - 1 } } \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} \\| D \\| _ { \\mathcal { H } _ { p ' } ( X ) } \\leq C \\Big ( \\sum _ { n = 1 } ^ \\infty \\| a _ n \\| _ X ^ p \\Big ) ^ { \\frac { 1 } { p } } ; \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} f ( t , t _ 1 , t _ 2 , t _ 3 ) = 1 - ( t _ 2 - t ) ( t _ 3 - t _ 1 ) ( z ( t ) - z ( t _ 1 ) ) ^ { - 1 } z ' ( t _ 1 ) ( z ( t ) - z ( t _ 1 ) ) ^ { - 1 } z ' ( t ) + o ( ( t _ 2 - t ) ( t _ 1 - t _ 3 ) ) . \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\left ( \\langle x , x \\rangle ^ \\frac { - 1 } { 2 } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } - \\frac { 1 } { \\sqrt { n } } \\right ) \\left ( \\langle x , x \\rangle ^ \\frac { - 1 } { 2 } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } - \\frac { 1 } { \\sqrt { n } } \\right ) ^ * = c _ x . \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} t _ 3 = \\begin{cases} \\min \\{ r _ 1 , \\tau _ 1 \\} & \\mbox { i f $ \\tilde h _ 1 ( x ) $ i s a u n i t } , \\\\ r _ 1 & \\mbox { i f $ \\tilde h _ 1 ( x ) = 0 $ } , \\end{cases} \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} \\overline { \\nabla } _ X \\psi = \\nabla _ X \\psi - \\frac 1 2 e _ 0 \\cdot W ( X ) \\cdot \\psi \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} \\| x ^ { j _ s } _ k - P _ { R _ { j _ s } } ( x ^ { j _ s } _ k ) \\| _ 2 \\leq \\frac { 2 \\varepsilon } { n ( \\prod _ { i \\in I } 2 ^ { | F _ i | | K _ i | } ) ( \\prod ^ { s - 1 } _ { v = 1 } | R _ { j _ v } | ) ( \\prod ^ r _ { v = s + 1 } | 2 ^ { | F _ { j _ v } | | K _ { j _ v } | } ) } . \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} p _ \\theta ( { \\bf { x } } ) = Z ( \\theta ) [ h ( { \\bf { x } } ) + \\widetilde { w } ( \\theta ) ^ \\top \\widetilde { f } ( { \\bf { x } } ) ] ^ { \\frac { 1 } { \\alpha - 1 } } \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} \\sum _ { \\substack { 2 < p _ 1 , p _ 2 \\le x \\\\ p _ 1 + p _ 2 = n } } 1 \\geq \\delta \\frac { x } { \\log ^ { 2 } x } \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} \\frac { 1 } { K } \\sum _ { i \\in B ( l ) } x _ i = y _ l , l = 1 , \\cdots , M . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} K = \\{ 0 _ q \\} \\times \\Re ^ p _ + \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} P _ { n } ( x ) = | x | ^ { 3 } P _ { n } ( \\tfrac { x } { | x | } ) \\leq | x | ^ { 3 } \\sup _ { | z | = 1 } P _ { n } ( z ) \\leq \\tfrac { n - 1 } { 6 } | x | ^ { 3 } , \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} 0 \\neq \\gamma _ { l ' i } \\overline { \\gamma _ { \\tau ' \\left ( l ' \\right ) \\tau ( i ) } } = G ^ { \\left ( 1 \\right ) } _ { 0 ; 0 , \\dots , 1 , \\dots , 0 } ( z ) \\Longleftrightarrow i = i _ { l } , \\quad \\mbox { f o r a l l $ l = 1 , \\dots , N $ . } \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} \\underline { \\psi } _ { k , 1 } = - \\overline { \\psi } _ { k , k + 1 } ^ { \\ast } \\leq \\frac { h } { k + 1 - h } \\underline { \\psi } _ { k , h } ^ { \\ast } , \\ ; k \\leq 2 n - 2 . \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} P : = \\bigwedge _ { x \\in X ; y \\ne y ' \\in Y } \\neg ( [ x | - > y ] \\wedge [ x | - > y ' ] ) \\wedge \\bigwedge _ { x \\ne x ' \\in X ; y \\in Y } \\neg ( [ x | - > y ] \\wedge [ x ' | - > y ] ) \\in \\kappa \\Pi ^ 0 _ 1 ( \\# S ^ { X \\times Y } ) \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} 0 \\geqslant \\Re \\langle x - y , H _ 0 x - H \\mathcal { H } y \\rangle _ \\mathcal { H } = \\Re \\langle \\mathcal { H } x - \\mathcal { H } y , H _ 0 x - H \\mathcal { H } y \\rangle . \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} \\left [ \\displaystyle \\sum _ { i = 1 } ^ p [ y ^ I ] _ i \\nabla ^ 2 c _ i ( x ) - A A ^ T - { \\cal J } c ( x ) ^ T { \\cal J } _ { z ^ I } \\Psi _ { \\varepsilon } ( y ^ I , z ^ I ) { \\cal J } c ( x ) \\right ] \\xi _ 1 = 0 . \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} e _ j ^ \\perp & = e _ j - \\pi _ { u _ 1 } ( e _ j ) - \\pi _ { u _ 2 } ( e _ j ) . \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} \\left ( F ( z , w ) ; G ( z , w ) \\right ) = \\left ( F _ { 1 } ( z , w ) , F _ { 2 } ( z , w ) , \\dots , F _ { N } ( z , w ) , \\dots , F _ { N ' } ( z , w ) ; G _ { 1 } ( z , w ) , G _ { 2 } ( z , w ) , \\dots , G _ { N } ( z , w ) , \\dots , G _ { N ' } ( z , w ) \\right ) . \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} \\tilde { h } _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( x , s ) = \\frac { \\phi ( s ) } { s } e ^ { - x \\phi ( s ) } , \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k - 1 \\} = P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k \\} = \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} & T _ 1 = 1 - v w - v = 1 - v _ i v _ m v _ f - v _ i v _ m v _ f w , \\\\ & T _ 2 = 1 - v w - v = 1 - v _ i ^ { \\prime } v _ m ^ { \\prime } v _ f ^ { \\prime } - v _ i ^ { \\prime } v _ m ^ { \\prime } v _ f ^ { \\prime } w . \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} [ \\boldsymbol { m } _ { 1 } , e _ { 1 } ] & = - ( m _ { 1 , 2 } e _ { 3 } + m _ { 1 , 3 } e _ { 4 } + \\cdots + m _ { 1 , c } e _ { c + 1 } ) , \\\\ [ \\boldsymbol { m } _ { i } , e _ { 1 } ] & = - ( m _ { i , i } e _ { i + 1 } + m _ { i , i + 1 } e _ { i + 2 } + \\cdots + m _ { i , c } e _ { c + 1 } ) , \\\\ [ \\boldsymbol { m } _ { 1 } , e _ { j } ] & = m _ { 1 , 1 } e _ { j + 1 } , \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} y ^ k _ { x x } = - \\Gamma _ { 1 1 } ^ k - ( 2 \\Gamma _ { 1 i } ^ k - \\delta ^ k _ i \\Gamma _ { 1 1 } ^ 1 ) y ^ i _ x - ( \\Gamma _ { i j } ^ k - 2 \\delta ^ k _ i \\Gamma _ { 1 j } ^ 1 ) y ^ i _ x y ^ j _ x + \\Gamma _ { i j } ^ 1 \\ , y ^ i _ x y ^ j _ x y ^ k _ x \\ , , k = 2 , \\dots , N \\ , , \\ , \\ , \\ , i , j \\geq 2 \\ , , \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} K ( \\Psi ; M ; \\gamma ) + K ( \\Psi ; M ; s _ i \\gamma - \\epsilon _ { i } + \\epsilon _ { i + 1 } ) = 0 \\ , . \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} K _ { \\alpha ( \\tau ) ( z ) } ( \\alpha ( \\tau ) ( w ) ) & = ( \\rho ( \\tau ^ { - 1 } ) K _ { \\alpha ( \\tau ) ( z ) } ) ( w ) = \\langle \\rho ( \\tau ^ { - 1 } ) K _ { \\alpha ( \\tau ) ( z ) } , K _ w \\rangle _ { L ^ 2 ( \\Omega ) } \\\\ & = \\langle K _ { \\alpha ( \\tau ) ( z ) } , \\rho ( \\tau ) K _ w \\rangle _ { L ^ 2 ( \\Omega ) } = \\overline { ( \\rho ( \\tau ) K _ w ) ( \\alpha ( \\tau ) ( z ) ) } = \\overline { K _ w ( z ) } = K _ z ( w ) . \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} \\psi _ { \\Lambda } ( b _ { \\Lambda } ) = \\sum _ { i , j \\in I } \\prod _ { x \\in \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * b _ { x } \\right ) . \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} \\big ( \\phi _ t = \\mathrm { i d } _ A + t ( \\mathrm { a d } ^ l _ a - \\mathrm { a d } ^ r _ a ) , ~ \\psi _ t = \\mathrm { i d } _ M + t ( l _ a - r _ a + H ( a , T - ) - H ( T - , a ) ) \\big ) \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} G ^ { B ^ + _ 1 } ( x , y ) \\ge \\frac { c _ 1 x ^ p _ d } { | x - y | ^ { d + \\alpha + \\beta _ 1 + \\beta _ 2 } } \\begin{cases} y _ d ^ { 2 \\alpha - p + \\beta _ 1 + \\beta _ 2 } \\left ( \\log ( | x - y | / y _ d ) \\right ) ^ { \\beta _ 4 } & 2 \\alpha - 2 p + \\beta _ 1 + \\beta _ 2 < 0 ; \\\\ y _ d ^ p \\left ( \\log ( | x - y | / y _ d ) \\right ) ^ { \\beta _ 4 + 1 } & 2 \\alpha - 2 p + \\beta _ 1 + \\beta _ 2 = 0 . \\end{cases} \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ { N } \\tilde g ( | x | ^ 2 + \\tau ^ 2 + 2 \\tau | x | | \\langle \\hat x , \\xi _ i \\rangle | ) + \\tilde g ( | x | ^ 2 + \\tau ^ 2 - 2 \\tau | x | | \\langle \\hat x , \\xi _ i \\rangle | ) \\\\ \\geq & N \\Big { ( } \\tilde g ( | x | ^ 2 + \\tau ^ 2 + 2 \\tau | x | \\frac { 1 } { \\sqrt { N } } ) + \\tilde g ( | x | ^ 2 + \\tau ^ 2 - 2 \\tau | x | \\frac { 1 } { \\sqrt { N } } ) \\Big { ) } . \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} A ^ { \\ast } \\mathsf { q } = \\nu _ { \\mathsf { k } } \\mathsf { q } \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} \\left | \\frac { \\Psi ' ( z ) } { \\Psi ( z ) } \\right | = \\left | \\frac { 1 } { z } + \\beta u ' ( z ) \\right | = \\left | \\frac { 1 } { z } + \\beta \\int _ { \\mathbb { T } } \\frac { 2 \\xi } { ( \\xi - z ) ^ { 2 } } d \\mu _ { 1 } ( \\overline { \\xi } ) \\right | , \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} \\hat { \\psi } _ { \\Lambda } ( b _ { \\Lambda } ) : = \\sum _ { i , j \\in I } \\prod _ { x \\in \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * b _ { x } \\right ) \\beta _ { \\Lambda ; i , j } \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} v ( x ) = \\sum _ { j = 1 } ^ \\ell a _ j x ^ { 2 j - 1 } . \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} T _ m ( \\phi ) : = \\{ s \\in F ^ * / F ^ { * 2 } : c ( \\phi _ s ) = \\ell ^ m \\} \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} \\norm f _ { H } ^ { 2 } : = \\int _ { \\R } f ^ { 2 } ( x ) \\eta ( x ) \\ , d x + \\int _ { \\R } ( f ' ( x ) ) ^ { 2 } \\psi ( x ) \\ , d x \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} f ( s , \\xi ) = - d _ 1 | s | ^ { q _ 1 - 2 } s + d _ 2 | \\xi | ^ { p - 1 } \\quad s \\in \\R \\xi \\in \\R ^ N , \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} - \\Delta \\Phi _ c = & \\partial _ t c ( t ) \\ \\ \\Omega , \\\\ \\frac { \\partial \\Phi _ c } { \\partial n } = & 0 \\ \\ \\partial \\Omega . \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} I _ { 1 2 } \\le c _ 9 \\begin{cases} y _ d ^ { 2 \\alpha - p + \\beta _ 1 + \\beta _ 2 } ( \\log ( 1 / y _ d ) ) ^ { \\beta _ 4 } & 2 \\alpha - 2 p + \\beta _ 1 + \\beta _ 2 < 0 ; \\\\ y _ d ^ p \\left ( \\log ( 1 / y _ d ) \\right ) ^ { \\beta _ 4 + 1 } & 2 \\alpha - 2 p + \\beta _ 1 + \\beta _ 2 = 0 . \\end{cases} \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\left | \\frac { | a _ i | } { \\| x \\| _ 2 } - \\frac { 1 } { \\sqrt { n } } \\right | ^ 2 = c _ x . \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} \\phi _ { p , q } ( \\lambda ) = & d e t ( I _ 4 \\lambda - D _ { B _ 1 ^ c } ) \\\\ = & { \\lambda } ^ { 4 } + \\left ( - q + 4 - p \\right ) { \\lambda } ^ { 3 } \\\\ & + \\left ( - 8 \\ , p q + 2 \\ , p + 2 \\ , q + 4 \\right ) { \\lambda } ^ { 2 } \\\\ & + \\left ( - 1 4 \\ , p q + 6 \\ , p + 6 \\ , q \\right ) \\lambda - 5 \\ , p q + 2 \\ , p + 2 \\ , q . \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} \\langle L _ { \\max } f _ 1 | f _ 2 \\rangle - \\langle f _ 1 | L _ { \\max } f _ 2 \\rangle = W ( f _ 1 , f _ 2 ; b ) - W ( f _ 1 , f _ 2 ; a ) . \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\langle N P ^ * y , ( I d + ( M _ { i j } ) ) N P ^ * y \\rangle & = \\frac { N } { 2 } \\langle y , P N P ^ * y + P ( M _ { i j } ) N P ^ * y \\rangle _ Y \\\\ & = \\frac { N } { 2 } \\langle y , ( I d + P ( M _ { i j } ) N P ^ * y \\rangle _ Y . \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} \\left ( \\int _ 0 ^ 1 e ^ { - \\rho \\eta } d \\eta \\right ) ^ { 1 / 2 } \\left ( \\int _ 0 ^ 1 \\alpha ^ 2 ( x - \\eta ) d \\eta \\right ) ^ { 1 / 2 } + \\sum _ { j = 1 } ^ { \\infty } e ^ { - \\rho j / 2 } \\left ( \\int _ j ^ { j + 1 } \\alpha ^ 2 ( x - \\eta ) d \\eta \\right ) ^ { 1 / 2 } \\lesssim \\frac { \\| A \\| _ { \\rm S t } } { \\rho ^ { 1 / 2 } } \\ , . \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} 0 < s \\leq q \\cdot \\frac { \\lambda _ { n } - \\widehat { \\lambda } _ { n } } { 1 + \\lambda _ { n } } + \\epsilon _ { 2 } q . \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} R _ k = R _ { k - 1 } \\setminus ( A _ k \\cup B _ k ) \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\P _ x ( T _ { W _ y } < \\tau _ U ) \\ge c _ 1 \\frac { | W _ y | } { | U | } = c _ 2 > 0 \\ , . \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} \\mathcal { B } _ { \\Lambda } : = \\mathcal { B } ( \\mathcal { H } _ { \\Lambda } ) \\equiv \\bigotimes _ { x \\in \\Lambda } \\mathcal { B } ( \\mathcal { H } _ { x } ) = : \\bigotimes _ { x \\in \\Lambda } \\mathcal { B } _ { x } . \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} r _ + & : = \\inf \\{ r : R > r , s p t V \\cap R \\cdot \\Sigma = \\emptyset \\} \\\\ r _ - & : = \\sup \\{ r : R < r , s p t V \\cap R \\cdot \\Sigma = \\emptyset \\} \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} A ^ \\dagger _ n b _ { p , n - 1 } = b _ { p , n } . \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} C ( { \\bf m } ) + E ( { \\bf m } ) - ( k _ 1 ^ 2 / 2 + \\cdots + k _ n ^ 2 / 2 - k _ 1 k _ 2 - . . - k _ { n - 1 } k _ n ) = B ( { \\bf m } ) - \\frac 1 2 { \\mathbf { k } ^ { \\top } A \\mathbf { k } } . \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} \\hat { d } _ { \\Pi } ( { \\hat { \\pi } _ 1 } , { \\hat { \\pi } _ 2 } ) : = | \\tanh ( { \\varsigma _ { { \\pi _ 1 } } } ) - \\tanh ( { \\varsigma _ { { \\pi _ 2 } } } ) | \\vee \\mathop { \\sup } \\limits _ { t \\ge { \\varsigma _ { { \\pi _ 1 } } } \\wedge { \\varsigma _ { { \\pi _ 2 } } } } \\left | { \\frac { { \\tanh ( \\hat { \\pi } _ { 1 } ( t ) ) } } { { 1 + | t | } } - \\frac { { \\tanh ( \\hat { \\pi } _ { 2 } ( t ) ) } } { { 1 + | t | } } } \\right | , \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} I _ 1 & = \\frac { 1 } { 2 } \\partial _ { x x } ^ 2 \\left ( | \\partial _ x v | ^ 2 \\right ) - | \\partial _ { x x } ^ 2 v | ^ 2 . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} [ \\delta _ { \\varepsilon _ 1 } , \\delta _ { \\varepsilon _ 2 } ] X = \\delta _ { [ \\varepsilon _ 1 , \\varepsilon _ 2 ] _ Q } X \\Longleftrightarrow [ \\rho ( \\varepsilon _ 1 ) , \\rho ( \\varepsilon _ 2 ) ] X = \\rho ( [ \\varepsilon _ 1 , \\varepsilon _ 2 ] _ Q ) X . \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} \\Big ( \\int _ { \\mathbb { T } } \\Big \\| \\sum _ { k = 1 } ^ N x _ k z ^ { k } \\Big \\| ^ { p ' } d z \\Big ) ^ { \\frac { 1 } { p ' } } \\leq C \\Big ( \\sum _ { k = 1 } ^ N \\big \\| x _ k \\big \\| ^ p \\Big ) ^ { \\frac { 1 } { p } } \\ , . \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} \\begin{aligned} T & = ( \\mu + \\nu ) \\bar { g } ( \\cdot , X ) \\otimes \\bar { g } ( \\cdot , X ) + \\nu \\bar { g } \\\\ & = \\sec ^ 2 ( \\xi ( x ) ) \\left [ \\nu ( \\xi ( x ) ) x ^ * g _ 0 + \\mu ( \\xi ( x ) ) \\left ( e ^ { - \\xi ( x ) \\sqrt { n - 1 } } \\right ) ^ 2 d t ^ 2 \\right ] , \\end{aligned} \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} & \\lim _ { t \\to 0 } \\int _ { \\Omega } \\sqrt { R } ( t , y ) \\psi ( y ) \\ , \\dd y = \\int _ { \\Omega } \\sqrt { R _ 0 } ( y ) \\psi ( y ) \\ , \\dd y , \\\\ & \\lim _ { t \\to 0 } \\int _ { \\Omega } \\sqrt { R } ( t , y ) ( \\sqrt { R } U ) ( t , y ) \\psi ( y ) \\ , \\dd y = \\int _ { \\Omega } \\Lambda _ 0 ( y ) \\psi ( y ) \\ , \\dd y . \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} A x : = P _ 1 ( \\mathcal { H } x ) ' + P _ 0 \\mathcal { H } x , \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} | | \\sqrt { \\omega } \\lambda ( t ) \\mathcal { F } ( \\sqrt { \\cdot } L _ { 1 } ( u ) ( t , \\cdot \\lambda ( t ) ) ) ( \\omega \\lambda ( t ) ^ { 2 } ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } ^ { 2 } & = \\frac { 1 } { \\lambda ( t ) ^ { 2 } } | | L ( L _ { 1 } ( u ) ( t , \\cdot \\lambda ( t ) ) ) | | _ { L ^ { 2 } ( R d R ) } ^ { 2 } \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} R _ v = \\max _ { 1 \\le k , \\ell \\le s } | \\alpha _ k - \\alpha _ \\ell | _ v \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} | A + _ G B | = n . \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} \\alpha _ { 0 } + \\int _ { [ 0 , + \\infty ) } \\frac { 1 } { t } \\ , d \\rho ( t ) = \\lim _ { z \\uparrow 0 } F ( z ) \\le 0 . \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} 6 K _ { n } ^ { ( 3 ) } = 5 j _ { n } ^ { ( 3 ) } + 3 j _ { n - 1 } ^ { ( 3 ) } - 5 j _ { n - 2 } ^ { ( 3 ) } , \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} \\frac { a _ 1 ' } { r _ 1 ' } ( l _ 1 - 1 ) + I _ { 1 l _ 1 } ^ 1 = 0 . \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} \\mathrm { i n d e x } ( S ^ 7 \\times \\R , \\hat { h } ) = & - \\mathrm { i n d e x } ( D ( V _ k ) \\cup ( - D ( V _ k ) ) , \\hat { g } _ k \\cup \\hat { g } _ k ) \\\\ & + \\mathrm { i n d e x } ( ( S ^ 7 \\times ( - \\infty , - 1 ] ) \\cup ( - D ( V _ k ) ) , \\hat { h } \\cup \\hat { g } _ k ) \\\\ & + \\mathrm { i n d e x } ( D ( V _ k ) \\cup ( S ^ 7 \\times [ - 1 , \\infty ) ) , \\hat { g } _ k \\cup \\hat { h } ) . \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\log n } \\frac { k ^ { k - 2 } } { k ! } \\int _ 0 ^ { \\infty } x ^ { k - 3 / 2 } e ^ { - k x } \\dd x = \\sum _ { k = 1 } ^ { \\log n } \\frac { k ^ { k - 2 } } { k ! } \\frac { \\sqrt { k } } { k ^ k } \\Gamma \\left ( k - \\frac { 1 } { 2 } \\right ) = \\sum _ { k = 1 } ^ { \\log n } \\frac { 1 } { k ^ { 3 / 2 } } \\frac { \\Gamma \\left ( k - \\frac { 1 } { 2 } \\right ) } { k ! } . \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\left < v , z \\right > = \\int _ { \\Omega } v z \\omega , \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} \\partial _ { \\xi } ^ { 3 } \\left ( \\frac { \\xi ^ { 2 } } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) \\right ) = \\left ( \\frac { ( b - 1 ) b ( b + 1 ) } { \\xi \\log ^ { b + 2 } ( \\frac { 1 } { \\xi } ) } + \\frac { 3 b ( b - 1 ) } { \\xi \\log ^ { b + 1 } ( \\frac { 1 } { \\xi } ) } + \\frac { 2 ( b - 1 ) } { \\xi \\log ^ { b } ( \\frac { 1 } { \\xi } ) } \\right ) \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) + \\psi ( \\xi ) \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} \\Delta ^ { I } \\boldsymbol { n } = g r a d ^ { I } 2 H + ( 4 H ^ { 2 } - 2 K ) \\boldsymbol { n } . \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} \\langle U _ T ^ n f , f \\rangle = \\int _ \\mathbb T e ^ { 2 \\pi i n x } \\text d \\gamma ( x ) . \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\prod _ { w \\in W _ n } ( \\l _ n - w ) } { \\prod _ { u \\in U _ n } ( \\l _ n - u ) } = 1 . \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} S \\leq C _ 1 + C _ 2 : = C < \\infty \\mbox { f o r a l l } M > 0 . \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} \\Pr [ Z \\in A ] = \\frac 1 2 \\Pr [ X \\in A ] + \\frac 1 2 \\Pr [ Y \\in A ] \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} T _ \\nabla ( q _ 1 , q _ 2 ) = \\nabla _ { q _ 1 } q _ 2 - \\nabla _ { q _ 2 } q _ 1 - [ q _ 1 , q _ 2 ] _ Q . \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} \\| g \\star \\rho _ \\delta - F _ { f _ \\delta } \\| _ { H } = \\| g \\star \\rho _ \\delta - F _ f \\star \\rho _ \\delta \\| _ { H } = \\| ( g - F _ f ) \\star \\rho _ \\delta \\| _ { H } \\lesssim \\| g - F _ f \\| _ { H } . \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\xi _ { 1 2 } ^ n \\sum _ { a = 0 } ^ n \\xi _ { 3 } ^ { a } \\sum _ { b = 0 } ^ a \\xi _ { 6 } ^ { b } \\sum _ { c = 0 } ^ b \\xi _ { 3 } ^ { c } . \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sum _ { i = 1 } ^ N \\int _ { S _ { \\varepsilon _ k , k , i } } | H | = \\frac { 1 } { 2 } \\ell ( \\partial M ) \\lim _ { k \\to \\infty } \\int _ { \\gamma _ k } | \\kappa | d s = \\frac { \\pi } { 2 } \\ell ( \\partial M ) . \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} J _ { \\infty } ( R ) & = \\displaystyle \\lim _ { \\underset { m } { \\rightarrow } } R _ { m } \\\\ & = \\frac { \\mathbb { C } [ x _ { j , ( - 1 - i ) } \\ ; | \\ ; 0 \\leq i , \\ ; 1 \\leq j \\leq n ] } { ( T ^ { j } f _ { i } | i = 1 , \\ldots n , \\ ; j \\in \\mathbb { N } ) } . \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} F _ \\gamma ( x ) = \\displaystyle \\int _ { 0 } ^ { \\infty } F _ { \\gamma _ 1 } \\left ( x \\left ( \\frac { { \\cal C } } { y } + 1 \\right ) \\right ) f _ { \\gamma _ 2 } ( y ) \\mathrm { d } y , \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} K ( x , y , t ) Q ( x , y , t ) & = x y - F ^ 1 ( x , t ) - F ^ 2 ( y , t ) + t d _ { - 1 , - 1 } Q ( 0 , 0 , t ) \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} \\frac { \\Phi ( \\eta _ { \\mu _ { 1 } } ( z ) ) } { \\eta _ { \\mu _ { 1 } } ( z ) } z = \\eta _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ^ { \\langle - 1 \\rangle } ( \\eta _ { \\mu _ { 1 } } ( z ) ) = \\eta _ { \\rho _ { 1 } } ^ { \\langle - 1 \\rangle } ( z ) \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} \\phi ( x ) & = \\sum _ { j = 0 } ^ \\infty a _ j \\rho ^ { j + 1 } ( x ) , a _ 0 = 1 \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} H ( \\mu _ { \\lambda } ; \\l ^ n ) = H ( \\mu _ { \\lambda } ^ { ( n ) } ; \\l ^ n ) + O ( 1 ) . \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} H _ 4 = - \\epsilon . \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} l ( e ) e = e , \\ , \\ , e l ( e ) = l ( e ) . \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} \\pi _ x = \\left ( \\frac { 1 - 2 p } { 1 - p } \\right ) \\left ( \\frac { p } { 1 - p } \\right ) ^ x , \\forall x \\in \\mathbb { Z } _ + . \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} \\phi ( \\mu ) = \\Theta ( 0 ; \\mu ) . \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} & ( a _ 1 , b _ 1 ) . ( b _ 2 , a _ 2 ) = ( a _ 3 , b _ 3 ) . ( b _ 4 , a _ 4 ) \\ , , & ( a _ 1 , b _ 1 ) . ( a _ 2 , - b _ 2 ) = ( a _ 3 , b _ 3 ) . ( a _ 4 , - b _ 4 ) , \\\\ & \\left | \\left | ( a _ 1 , b _ 1 ) \\right | \\right | ^ 2 - \\left | \\left | ( a _ 3 , b _ 3 ) \\right | \\right | ^ 2 = 1 \\ , , & \\left | \\left | ( a _ 4 , b _ 4 ) \\right | \\right | ^ 2 - \\left | \\left | ( a _ 2 , b _ 2 ) \\right | \\right | ^ 2 = 1 \\ , , \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} q c o n f \\left ( \\mathfrak { J } _ { 3 } ^ { \\mathbb { O } _ { s } } \\right ) \\supset ^ { n s } g _ { D = 1 1 } \\oplus \\left . s l ( D - 2 , \\mathbb { R } ) \\right \\vert _ { D = 1 1 } . \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} 0 \\leq & \\beta \\leq \\beta _ c \\Rightarrow \\Lambda ( \\beta ) = 0 , \\\\ & \\beta > \\beta _ c \\Rightarrow \\Lambda ( \\beta ) > 0 . \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} h _ { 1 i } ( 1 ) - h _ { 2 i } ( 1 ) = \\sum _ { j = 1 } ^ 2 \\frac { 1 } { j ! } \\left ( h _ { 1 i } ^ { ( j ) } ( 0 ) - h _ { 2 i } ^ { ( j ) } ( 0 ) \\right ) + \\frac { 1 } { 3 ! } \\left ( h _ { 1 i } ^ { ( 3 ) } ( \\tau _ 1 ) - h _ { 2 i } ^ { ( 3 ) } ( \\tau _ 2 ) \\right ) , \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} R ^ 2 _ 2 = K + \\frac { 1 } { 2 } ( R ^ 3 _ 3 - R ^ 1 _ 1 ) + \\frac { 1 } { 2 A } ( \\epsilon S _ 1 - S _ 3 ) \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} ( \\Phi \\circ H _ { s ( z ) } ) ( \\rho ) = \\Phi \\circ H _ { s ( z ) } \\circ \\rho \\circ H _ { s ( z ) ^ { - 1 } } \\circ \\Phi ^ { - 1 } = ( H _ { s ( \\phi ( z ) ) } \\circ \\Phi ) ( \\rho ) \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} \\norm { \\eta ( x ) } ^ * = \\norm { x } & x \\in V . \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} = e ^ { - \\lambda t } \\Biggl [ I _ { 0 } \\Bigl ( \\frac { 2 \\lambda t \\sqrt { \\alpha } } { 1 + \\alpha } \\Bigr ) + \\frac { 1 } { \\sqrt { \\alpha } } I _ { 1 } \\Bigl ( \\frac { 2 \\lambda t \\sqrt { \\alpha } } { 1 + \\alpha } \\Bigr ) + ( 1 - \\alpha ) \\sum _ { r = 2 } ^ \\infty I _ { r } \\Bigl ( \\frac { 2 \\lambda t \\sqrt { \\alpha } } { 1 + \\alpha } \\Bigr ) \\Bigl ( \\frac { 1 } { \\sqrt { \\alpha } } \\Bigr ) ^ { r } \\ \\Biggr ] \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { L } c _ { i } H _ { L + m - i } - \\sum _ { i = 1 } ^ { L } c _ { i } G _ { L + m - i } \\geq 2 \\left ( \\sum _ { i = 1 } ^ { L } c _ { i } H _ { L + m - 1 - i } - \\sum _ { i = 1 } ^ { L } c _ { i } G _ { L + m - 1 - i } \\right ) . \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} \\psi = \\psi _ 0 + \\sum _ { j = 1 } ^ \\infty { \\psi _ j ^ 1 } + \\sum _ { j = 1 } ^ \\infty { \\psi _ j ^ 2 } , \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d X ( t ) & = b ( X ( t ) ) d t + \\sigma ( X ( t ) ) d \\tilde { B } ( t ) + \\sigma ( X ( t ) ) \\rho ( t ) d t - d \\eta _ { X } ( t ) , \\\\ X ( 0 ) & = x . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} = \\frac { \\beta } { ( 1 - u ) \\sqrt { c ^ 2 t ^ 2 - u ( c ^ 2 t ^ 2 - \\beta ^ 2 ) } } \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} E _ n ( - 1 ) = \\sum _ { k = 0 } ^ { n - 1 } \\binom { n - k - 1 } { k } k ! \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} \\partial _ { t t } z - \\Delta z + m ^ 2 z = g ( z ) , z ( 0 ) = u _ { 0 } , \\partial _ { t } z ( 0 ) = u _ { 1 } , \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} Q ^ { \\pm } ( t , 0 ) = 0 a n d \\Delta ( t , 0 ) = 0 a . e . o n { \\mathbb R } _ { + } . \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} ( & ( U ^ { ( 1 ) } = V ^ { ( k _ 1 ) } \\mapsto \\ldots \\mapsto V ^ { ( k _ 2 ) } ) , \\\\ & ( ( a _ h ^ { ( k _ 1 ) } , a _ j ^ { ( k _ 1 ) } ) , \\ldots , ( a _ h ^ { ( k _ 2 - 1 ) } , a _ j ^ { ( k _ 2 - 1 ) } ) ) , \\\\ & ( b ^ { ( 1 ) } = c ^ { ( 1 ) } , \\ldots , c ^ { ( k _ 2 ) } ) ) \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} S & = \\{ ( a _ i , d _ i ) \\in \\Z ^ 2 : 1 \\leq i \\leq r \\} , \\\\ S ' & = \\{ ( a _ i - ( d _ i - 1 ) k , d _ i ) \\in \\Z ^ 2 : 1 \\leq i \\leq r \\} , \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} \\| \\gamma \\| ^ 2 = \\| \\overline { \\gamma } \\| ^ 2 = \\inf _ { ( \\omega , \\overline { \\omega } ) } \\| \\omega \\| \\| \\overline { \\omega } \\| \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} f _ { l o g } ( r ) & = ( 1 + r ) \\ln ( 1 + r ) + ( 1 - r ) \\ln ( 1 - r ) - k r ^ 2 , r \\in ( - 1 , 1 ) , \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} ( \\epsilon ^ { - 1 } f ^ n ( u ^ n _ h ) , \\Delta _ h u ^ n _ h ) _ { \\mathcal { T } _ h } - \\epsilon \\| \\Delta _ h u _ h ^ n \\| ^ 2 _ { \\mathcal { T } _ h } - ( \\phi _ h ^ n , \\Delta _ h u ^ n _ h ) _ { \\mathcal { T } _ h } = 0 . \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} | \\partial _ { t } \\langle \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) F _ { 0 , 2 } \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | \\leq \\frac { C } { t ^ { 3 } \\log ^ { 2 b + 1 - 2 b \\alpha + 2 N } ( t ) } + \\frac { C | e ''' ( t ) | } { \\log ^ { b - 2 b \\alpha + 2 N } ( t ) } \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} \\frac { s ^ { \\ast } } { q } = \\frac { L _ { n , 2 } ^ { \\ast } ( q + s ^ { \\ast } ) } { q } - \\frac { L _ { n , 1 } ^ { \\ast } ( q ) } { q } \\leq ( \\beta ^ { \\ast } + \\epsilon _ { 1 } ) \\frac { q + s ^ { \\ast } } { q } - \\alpha ^ { \\ast } + \\epsilon = \\beta ^ { \\ast } - \\alpha ^ { \\ast } + \\beta ^ { \\ast } \\frac { s ^ { \\ast } } { q } + \\epsilon _ { 2 } \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} \\widetilde { p } _ n ( { \\bf { x } } ) = \\frac { 1 } { n h _ n ^ d } \\sum \\limits _ { i = 1 } ^ n \\xi \\Big ( \\frac { { \\bf { x } } - \\textbf { X } _ i } { h _ n } \\Big ) , \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} U \\tilde { R } ( - e _ { n + 1 } ) U ^ { - 1 } & = \\frac 1 2 \\tilde { R } ( e _ { n + 1 } ) \\tilde { R } ( e _ n + e _ { n + 1 } ) \\tilde { R } ( - e _ { n + 1 } ) \\tilde { R } ( e _ n + e _ { n + 1 } ) \\tilde { R } ( e _ { n + 1 } ) \\\\ & = \\frac 1 2 ( \\tilde { R } ( e _ { n + 1 } ) + \\tilde { R } ( e _ n ) + \\tilde { R } ( e _ n ) - \\tilde { R } ( e _ { n + 1 } ) ) = \\tilde { R } ( e _ n ) . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{gather*} \\left ( \\frac { d w } { d z } \\right ) ^ 2 + ( - H ) ^ 2 U ( w ) = 0 , \\\\ U ( w ) : = 1 - \\left ( \\frac { w ^ n } { K + w ^ { n + 1 } } \\right ) ^ 2 , \\ ; \\ ; \\ ; K : = ( - H ) ^ n C . \\end{gather*}"}
-{"id": "9466.png", "formula": "\\begin{align*} T _ { p _ i } = \\prod _ { j = 1 } ^ { v _ i } C _ { p _ i ^ { b _ { i j } } } \\ { \\rm a n d } \\ T _ 2 = \\prod _ { \\iota = 1 } ^ \\rho C _ { 2 ^ { \\epsilon _ \\iota } } \\times C _ { 2 ^ { \\epsilon } } ^ \\sigma \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} F \\circ ( K + \\Delta ) - ( K + \\Delta ) \\circ R = 0 , \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} f ( t ) = U ^ t f _ 0 + \\int _ 0 ^ t U ^ { t - s } [ Q ^ { + } ( s , f ( s ) ) - Q ^ { - } ( s , f ( s ) ) ] d s , t \\geq 0 , \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} \\partial _ { t } u = \\Delta u + A \\left ( x \\right ) u + V \\left ( x , t \\right ) u , \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} v = v ^ { T } + v ^ { \\bot } , \\ \\ v ^ { T } \\in T _ p N , \\ v ^ { \\bot } \\in T _ p ^ { \\bot } N , \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} i _ t = s _ { k _ 1 } s _ { k _ 2 } \\ldots s _ { k _ { t - 1 } } ( k _ t + 1 ) , j _ t = s _ { k _ 1 } s _ { k _ 2 } \\ldots s _ { k _ { t - 1 } } ( k _ t ) . \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} & \\lim _ { T \\to 0 } \\sup _ { ( 0 , T ] \\times D _ R } | b _ 1 ( t , x ) | = o ( R ) \\mbox { ( a s $ R \\longrightarrow + 0 $ ) } , \\\\ & \\lim _ { T \\to 0 } \\sup _ { ( 0 , T ] \\times D _ R } | ( \\partial b _ 1 / \\partial x ) ( t , x ) | = o ( 1 ) \\mbox { ( a s $ R \\longrightarrow + 0 $ ) } . \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} v _ 0 & = 1 , \\\\ v _ { 3 k } & = 4 k ^ 2 , \\\\ v _ { 3 k + 1 } & = ( 4 k + 1 ) ^ 2 ( 2 k + 1 ) , \\\\ v _ { 3 k + 2 } & = ( 4 k + 3 ) ^ 2 ( 2 k + 1 ) . \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} ( A ^ \\dagger _ n f ) ( z ) = \\frac { 1 } { \\sqrt { n } } ( A ^ \\dagger f ) ( z ) = \\frac { 1 } { \\sqrt { n } } \\left ( \\overline { z } - \\frac { \\partial } { \\partial z } \\right ) f ( z ) . \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} \\tilde { u } \\left ( x , t \\right ) = \\left ( \\sqrt { \\alpha \\beta } \\rho ^ { - 1 } \\left ( t \\right ) \\right ) ^ { \\frac { n } { 2 } } u \\left ( \\sqrt { \\alpha \\beta } x \\rho ^ { - 1 } \\left ( t \\right ) , \\beta t \\rho ^ { - 1 } \\left ( t \\right ) \\right ) e ^ { \\varphi } . \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} \\phi ( \\mathfrak { g } , [ J , J ] ) ) = \\phi ( [ \\mathfrak { g } , J ] , J ) \\subset \\phi ( J ^ { \\perp } , J ) = \\{ 0 \\} . \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { B \\setminus B _ a } \\exp \\big ( \\alpha _ n | u _ j | ^ { \\frac n { n - 1 } + | x | ^ \\alpha } \\big ) d x & = | B \\setminus B _ a | + o ( 1 ) _ { j \\searrow 0 } \\\\ & = \\int _ { B \\setminus B _ a } \\exp \\big ( \\alpha _ n | u _ j | ^ { \\frac n { n - 1 } } \\big ) d x + o ( 1 ) _ { j \\searrow 0 } \\end{aligned} \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} \\Sigma ^ + _ T : = \\Sigma _ T ^ { + , e x t } \\cup ( \\bigcup _ { p \\in S i n g ( \\Sigma ) } \\Sigma _ { T , p } ^ { + , i n t } \\ ) \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} & ( x ) = 2 \\bigg ( 0 . 1 6 8 { e ^ { - 1 . 7 5 2 { x ^ 2 } } } + 0 . 1 4 4 { e ^ { - 1 . 0 5 { x ^ 2 } } } + 0 . 0 0 2 { e ^ { - 1 . 2 0 6 { x ^ 2 } } } \\bigg ) . \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} \\{ x \\mapsto ( \\l _ 0 x + ( T _ j ( 1 , \\tau _ 0 ) , 0 , \\ldots , 0 , T _ j ( 0 , 1 ) ) : j = 1 , \\ldots , m \\} . \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} R _ 0 ( \\l _ 0 ) = \\lim _ { n \\to \\infty } \\frac { F _ n ^ { ( 1 ) } ( \\l _ n ) } { F _ n ^ { ( 2 ) } ( \\l _ n ) } = \\lim _ { n \\to \\infty } R _ n ( \\l _ n ) \\frac { \\prod _ { w \\in W _ n } ( \\l _ n - w ) } { \\prod _ { u \\in U _ n } ( \\l _ n - u ) } = \\tau _ 0 , \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} 1 = [ X \\otimes X ' , X \\otimes X ' ] = [ X \\otimes X ^ \\ast , X ' \\otimes ( X ' ) ^ \\ast ] . \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} E _ { } ( u , \\partial _ { t } u ) & = \\pi \\left ( | | \\partial _ { t } u | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } + | | \\frac { \\sin ( u ) } { r } | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } + | | \\partial _ { r } u | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } \\right ) \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} \\phi = \\{ \\phi _ { a _ 0 , a _ 1 } \\} : \\check C _ 1 ( \\mathcal R ^ 2 ; \\omega ) \\to \\mathcal R ^ * , \\ \\phi _ { a _ 0 , a _ 1 } ( x ) = \\kappa ( a _ 1 , a _ 0 , x , \\omega ) , \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} & \\lim _ { h \\to 0 ^ + } \\int _ { \\mathbb { R } ^ n } \\varphi ( x ) \\sum _ { \\vec { k } \\in \\mathbb { Z } ^ n } \\mathcal { X } _ { Q ( h \\vec { k } ) } \\left ( \\sum _ { \\abs { \\vec { j } - \\vec { k } } > \\tfrac { 1 } { h } } h ^ n \\frac { \\Phi ( h \\vec { k } , h \\vec { j } ) } { \\abs { h \\vec { k } - h \\vec { j } } ^ { n + \\alpha } } \\right ) d x \\\\ & = \\int _ { \\mathbb { R } ^ n } \\varphi ( x ) \\int _ { \\{ y : \\abs { x - y } \\geq 1 \\} } \\frac { \\Phi ( x , y ) } { \\abs { x - y } ^ { n + \\alpha } } d y d x . \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( q ^ { - 1 } ; q ^ 2 ) _ k ^ 4 ( c q ^ { - 1 } , d q ^ { - 1 } ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k ^ 4 ( q ^ 2 / c , q ^ 2 / d ; q ^ 2 ) _ k } \\bigg ( \\frac { q ^ 7 } { c d } \\bigg ) ^ k \\\\ [ 5 p t ] \\ : & \\ : \\equiv \\frac { \\Omega _ q ( n ) ( 1 - q ) ^ 2 ( q ^ 3 / c d ; q ^ 2 ) _ 2 } { ( 1 + q ) ^ 2 ( q ^ 2 / c , q ^ 2 / d ; q ^ 2 ) _ 2 } \\sum _ { k = 0 } ^ { ( n - 3 ) / 2 } \\frac { ( q ^ 3 ; q ^ 2 ) _ k ^ 3 ( q ^ 7 / c d ; q ^ 2 ) _ k } { ( q ^ 2 , q ^ 6 , q ^ 6 / c , q ^ 6 / d ; q ^ 2 ) _ k } q ^ { 2 k } , \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { H _ n } { r _ 1 ^ n } = C \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} X / _ G Y = \\left \\{ - \\frac { 1 + \\lambda 2 ^ { j - i } } { 1 + 2 ^ { j - i } } : i , j \\in \\mathbb N , i , j \\leq n \\right \\} . \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & \\ , \\to \\ , \\min & & & \\\\ g _ i ( x ) & \\ , \\leq \\ , 0 & \\qquad & i \\in \\mathcal M & \\\\ h _ j ( x ) & \\ , = \\ , 0 & & j \\in \\mathcal P & \\\\ 0 \\ , \\leq \\ , \\bar G _ l ( x ) \\ , \\perp \\ , \\bar H _ l ( x ) & \\ , \\geq \\ , 0 & & l \\in \\mathcal Q . & \\end{aligned} \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} x ^ 2 + ( y + a x ) ^ 2 = z ^ 2 + ( a x ) ^ 2 \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} I _ 4 ( Q ) = - 4 p ^ 0 I _ 3 ( q ) . \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} m = m ( \\alpha ) = \\frac { \\sin \\alpha } { 1 + \\cos \\alpha } = \\tan \\bigg ( \\frac { \\alpha } { 2 } \\bigg ) \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} \\phi = \\frac { \\partial w } { \\partial { x _ 1 } } = \\Big ( \\frac { \\partial w ^ + } { \\partial { x _ 1 } } , \\frac { \\partial w ^ - } { \\partial { x _ 1 } } \\Big ) \\quad \\phi = \\frac { \\partial w } { \\partial { x _ 2 } } = \\Big ( \\frac { \\partial w ^ + } { \\partial { x _ 2 } } , \\frac { \\partial w ^ - } { \\partial { x _ 2 } } \\Big ) , \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} p _ \\theta ( { \\bf { x } } ) = Z ( \\theta ) [ 1 + \\widetilde { w } ( \\theta ) ^ \\top \\widetilde { f } ( { \\bf { x } } ) ] ^ { \\frac { 1 } { \\alpha - 1 } } \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} \\hat { \\pi } _ { n } ( t ) : = \\dfrac { 1 } { n } \\Big [ \\sum _ { i = 1 } ^ { \\lfloor n ^ { 2 } t \\rfloor } Z _ { i } + ( n ^ { 2 } t - \\lfloor n ^ { 2 } t \\rfloor ) Z _ { \\lfloor n ^ { 2 } t \\rfloor + 1 } \\Big ] , \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = \\nabla f ( x _ k ) + \\sum \\limits _ { i \\in I ^ g ( \\bar x ) } \\lambda _ i ^ k \\nabla g _ i ( x _ k ) + \\sum \\limits _ { j \\in \\mathcal P } \\rho _ j ^ k \\nabla h _ j ( x _ k ) \\\\ & + \\sum \\limits _ { l \\in \\mathcal I ( \\bar x ) } \\xi ^ k _ l \\left ( \\alpha ^ k _ l \\nabla G _ l ( x _ k ) + \\beta ^ k _ l \\nabla H _ l ( x _ k ) \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} { \\mathcal L } ( \\psi ^ \\bot ) = h ^ \\bot . \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} \\rho ( f , g ) = \\sum ^ \\infty _ { n = 0 } \\frac { 1 } { 2 ^ n } \\cdot \\frac { d ( f ( x _ n ) , g ( x _ n ) ) } { 1 + d ( f ( x _ n ) , g ( x _ n ) ) } , \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} \\xi _ j = l ( 2 l ^ 2 + ( N - 1 ) l - N + 2 ) \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} _ { v _ { 4 , 1 } } & = \\frac { - \\partial _ { 2 } v _ { 4 , c } ( s , | x + y | ) } { | x + y | ^ { 2 } ( s - t ) } \\left ( ( x + y ) \\cdot y \\right ) \\left ( \\hat { x } \\cdot ( x + y ) \\right ) \\\\ & + \\frac { v _ { 4 , c } ( s , | x + y | ) } { ( s - t ) | x + y | } \\left ( - y \\cdot \\hat { x } + \\frac { \\left ( \\hat { x } \\cdot ( x + y ) \\right ) \\left ( y \\cdot ( y + x ) \\right ) } { | x + y | ^ { 2 } } \\right ) \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} ( h + ( 2 n - 2 ) ) r ( h - 4 ) = ( h - ( 2 n + 2 ) ) r ( h ) . \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} G ( s , r , \\rho ) & = \\int _ { 0 } ^ { 2 \\pi } d \\theta \\frac { v _ { 4 , c } ( s , \\sqrt { r ^ { 2 } + 2 r \\rho \\cos ( \\theta ) + \\rho ^ { 2 } } ) } { \\sqrt { r ^ { 2 } + 2 r \\rho \\cos ( \\theta ) + \\rho ^ { 2 } } } \\left ( r + \\rho \\cos ( \\theta ) \\right ) \\\\ & s \\geq t , r \\geq 0 , s - t \\geq \\rho \\geq 0 \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} T ( \\tilde { r } ; \\mu ) = T _ L \\big ( \\tilde { \\phi } ( \\tilde { r } ) ; \\mu \\big ) , \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} v _ { 5 , 2 } ( t , r ) = v _ { 5 } ( t , r ) - v _ { 5 , 1 } ( t , r ) \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} \\beta ^ n & = a _ { n - 1 } \\beta ^ { n - 1 } - a _ { n - 2 } \\beta ^ { n - 2 } + \\dots \\\\ & \\le m ( \\beta ^ { n - 1 } + \\beta ^ { n - 3 } + \\dots + 1 ) \\\\ & = m \\frac { \\beta ^ { n + 1 } - 1 } { \\beta ^ 2 - 1 } < \\beta ^ n \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} \\mathrm { I n d e x } \\ , T _ { \\alpha \\otimes I _ F } & = \\mathrm { I n d e x } \\ , T _ \\alpha \\ , \\cdot \\ , \\mathrm { d i m \\ , K e r } \\ , B ^ + + \\mathrm { I n d e x } \\ , T ^ - _ \\alpha \\ , \\cdot \\ , \\mathrm { d i m \\ , K e r } \\ , B ^ - + \\mathrm { I n d e x } \\ , Q \\\\ & = \\mathrm { I n d e x } \\ , T _ \\alpha \\ , \\cdot \\ , \\mathrm { I n d e x } \\ , B + \\mathrm { I n d e x } \\ , Q \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} ( \\theta + d d ^ c \\varphi ) ^ n = e ^ \\varphi \\mu \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} ( f \\ast _ G g ) ^ { * ^ G } = g ^ { * ^ G } \\ast _ G f ^ { * ^ G } , \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} \\psi _ Y ( w ) : = \\begin{cases} \\displaystyle - \\frac { 1 } { L } & ( Y = H ) \\\\ \\displaystyle \\frac { v _ { n + 1 } w ^ { n + 1 } } { ( - H ) ^ { n + 2 } } & ( Y = V ) \\\\ \\displaystyle \\frac { a _ n w ^ { n } \\sqrt { 1 - U ( w ) } } { ( - H ) ^ { n + 1 } } & ( Y = A ) \\\\ \\end{cases} . \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} \\sum _ { 0 < s \\leq t } G ' ( X _ { s - } ) \\Delta X ( s ) & = \\sum _ { 0 < s \\leq t } G ' ( X _ { s - } ) \\rho ( X _ { s - } , \\Delta L _ s ) \\mathbf { 1 } _ D ( s ) \\\\ & = \\int \\limits _ 0 ^ t \\int \\limits _ { \\mathbb { R } } G ' ( X _ { s - } ) \\rho ( X _ { s - } , y ) \\nu ( d y , d s ) \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} \\Phi _ A ( g ) : = \\sum _ i \\lambda _ i f _ i ( g ) a _ i \\in \\Psi ^ { - \\infty } ( S ) , g \\in G , \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { \\ell } v _ i ' = \\sum \\limits _ { i = 2 } ^ { \\ell } v _ i ' \\leq \\sum \\limits _ { i = 2 } ^ { \\ell } ( 5 - v _ i ) = 5 ( \\ell - 1 ) - n + v _ 1 < n , \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} \\| \\nabla \\omega \\| _ { L ^ \\infty _ t ( L ^ 2 ) } ^ 2 + \\| \\nabla \\theta \\| _ { L ^ \\infty _ t ( L ^ 2 ) } ^ 2 + \\| \\partial _ 1 \\nabla \\omega \\| _ { L ^ 2 _ t ( L ^ 2 ) } ^ 2 \\leq C . \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} H _ 1 : = \\rho F - H | _ { A _ i } \\in H ^ \\infty ( A _ i , H ^ \\infty ( S _ o ' ) ) = H ^ \\infty ( A _ i \\times S _ o ' ) . \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} \\mathbb { E } _ { x _ 0 } \\tau _ D = \\int _ D G _ D ( x _ 0 , y ) d y = \\infty , . \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} r i c _ g - \\frac { s c a l _ g } { n } g = \\frac { 1 } { f } \\Big ( ( \\nabla ^ 2 f ) _ g - \\frac { ( \\Delta f ) _ g } { n } g \\Big ) \\quad \\mbox { o n } M , \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} ( 1 + \\lambda _ { N } ( \\underline { \\xi } ) ) ( 1 + \\underline { \\psi } _ { N , 1 } ) = ( 1 + \\widehat { \\lambda } _ { k } ( \\underline { \\xi } ) ) ( 1 + \\overline { \\psi } _ { N , 1 } ) = \\frac { N + 1 } { N } , \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} \\begin{aligned} \\nu ( \\xi ) & = \\frac { ( n - 1 ) \\cos ^ 2 ( \\xi ) } { n } \\Big \\{ [ \\tan ( \\xi ) + \\sqrt { n - 1 } ] ^ 2 + \\sec ^ 2 ( \\xi ) + \\\\ & + ( n - 2 ) \\tan ( \\xi ) [ \\tan ( \\xi ) - \\sqrt { n - 1 } ] \\Big \\} - \\frac { n - 2 } { n } \\mu ( \\xi ) , \\end{aligned} \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} { } ^ b \\Psi ^ { - \\infty } _ { G , c } ( M ) = \\{ T _ { \\widetilde { k } } , \\ ; \\widetilde { k } \\in ( C ^ \\infty _ c ( G ) \\hat { \\otimes } \\ , { } ^ b \\Psi ^ { - \\infty } ( S ) ) ^ { K \\times K } \\} \\ , . \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} J _ { 0 , 0 } = \\left ( \\begin{array} { c c c c } \\frac { 1 } { b } - a & 0 & 1 & 1 \\\\ 0 & - b + K & 0 & 0 \\\\ - 1 & 0 & - c & 0 \\\\ - \\frac { d } { b } & 0 & 0 & - k \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} A _ 1 ( t , x ) : = & \\alpha ( t + \\theta ) ^ { - \\alpha - 1 } [ \\Phi ( x - \\underline h ( t ) ) + \\Phi ( - x - \\underline h ( t ) ) - \\mathbf { u } ^ * ] , \\\\ A _ 2 ( t , x ) : = & - ( 1 - \\epsilon ) \\delta ' ( t ) [ \\Phi ' ( x - \\underline h ( t ) ) + \\Phi ' ( - x - \\underline h ( t ) ) ] \\\\ & + ( 1 - \\epsilon ) [ F ( \\Phi ( x - \\underline h ( t ) ) ) + F ( \\Phi ( - x - \\underline h ( t ) ) ) ] - F ( \\underline U ( t , x ) ) . \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} & | \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) K ( s - t , \\lambda ( t ) ) | \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\\\ & | \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { t } ^ { \\infty } d x \\lambda '' ( x ) \\left ( K _ { 1 } ( x - t , \\lambda ( t ) ) - \\frac { \\lambda ( t ) ^ { 2 } } { 4 ( 1 + x - t ) } \\right ) | \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} \\sup _ k \\left \\| \\Delta \\left ( \\sum _ { i \\ge 0 } ^ k e ^ { - \\lambda _ i t } \\phi _ i ^ 2 \\right ) \\right \\| _ { H ^ { 1 , 2 } } = \\sup _ k \\left \\| 2 \\sum _ { i \\ge 0 } ^ k e ^ { - \\lambda _ i t } \\left ( - \\lambda _ i \\phi _ i ^ 2 + | \\nabla \\phi _ i | ^ 2 \\right ) \\right \\| _ { H ^ { 1 , 2 } } < \\infty . \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} \\Vert w \\Vert _ 2 ^ 2 = \\Vert u \\Vert _ 2 ^ 2 , \\Vert w ' \\Vert _ 2 ^ 2 \\leq \\Vert u ' \\Vert _ 2 ^ 2 , \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} & \\tilde { K } ( t ) = \\begin{pmatrix} t ^ 2 + \\cdots \\\\ \\gamma ^ { - 1 } \\ , K _ { k + 1 } ^ y \\ , t ^ { k + 1 } + \\cdots \\end{pmatrix} , \\\\ & \\tilde { K } ( t ) = \\begin{pmatrix} t + \\cdots \\\\ \\gamma ^ { - 1 } \\ , K _ l ^ y \\ , t ^ l + \\cdots \\end{pmatrix} , \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} | \\frac { - 3 2 \\lambda ' ( t ) } { \\lambda ( t ) ^ { 3 } } \\int _ { t } ^ { \\infty } d x \\lambda '' ( x ) K ( x - t , \\lambda ( t ) ) | & \\leq \\frac { C \\log ^ { 3 b } ( t ) } { t ^ { 3 } \\log ^ { 2 b + 2 } ( t ) } \\int _ { t } ^ { \\infty } | K ( x - t , \\lambda ( t ) ) | d x \\\\ & \\leq \\frac { C } { t ^ { 3 } \\log ^ { b + 2 } ( t ) } \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} a _ j = \\sqrt { \\frac { \\mu _ j ( \\lambda ) } { \\mathcal Q _ { \\lambda , N } ( u _ { j , \\lambda } , u _ { j , \\lambda } ) } } \\hat a _ j \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} ( H _ 0 ^ 2 - h ^ 2 ) e ^ { - z } = 4 H _ 1 - H _ 0 ^ 2 - h ^ 2 , \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} & Q = [ - 1 / 2 , 1 / 2 ] ^ { n } , \\\\ & a Q = [ - a / 2 , a / 2 ] ^ n , a \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ 3 } v d \\boldsymbol { x } = \\hat { v } _ 0 = 0 . \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} T ^ { - 1 } \\eta : = R T R \\eta \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} r + K * r = K , r + r * K = K \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} S _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( t ) = S _ { \\alpha _ 1 , \\lambda _ 1 } ( c _ 1 t ) + S _ { \\alpha _ 2 , \\lambda _ 2 } ( c _ 2 t ) , \\ ; c _ 1 , \\ ; c _ 2 \\geq 0 . \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} A + _ G A : = \\{ a + b : ( a , b ) \\in G \\} . \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} | | & \\leq 4 \\int _ { t } ^ { 2 t } d s \\frac { | | \\lambda _ { 0 , 0 } ''' | | _ { L ^ { \\infty } ( t , 2 t ) } ( s - t ) } { 1 + s - t } + \\frac { 4 } { 1 + t } \\int _ { 2 t } ^ { \\infty } d s | \\lambda _ { 0 , 0 } '' ( s ) | \\\\ & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} \\sum _ { k \\in \\Bbb { Z } } | \\widehat { \\psi ^ { ( 2 ) } } ( 2 ^ { k } \\xi _ 2 , 2 ^ k \\xi _ 3 ) | ^ 2 = 1 \\mbox { f o r a l l } ( \\xi _ 2 , \\xi _ 3 ) \\in \\Bbb { R } ^ 2 \\backslash \\{ 0 \\} . \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} R = \\begin{bmatrix} \\big ( \\frac { \\partial g } { \\partial x } \\big ) ^ T \\frac { \\partial g } { \\partial x } & \\big ( - \\frac { \\partial g } { \\partial x } \\big ) ^ T \\\\ - \\frac { \\partial g } { \\partial x } & k I \\end{bmatrix} . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} { \\rm s u p p } \\varphi \\subset \\bar { B } \\ \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } \\varphi d x d v = 1 . \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} E \\dot { x } ( t ) & = A x ( t ) + B u ( t ) \\\\ [ 1 e x ] y ( t ) & = C x ( t ) \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} \\int _ { B } ( u _ j ) _ + ^ { \\ 2 m s + | x | ^ \\alpha } d x = l ^ 2 + o ( 1 ) _ { j \\nearrow + \\infty } \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} \\mathbb { L } : = \\Big \\{ p \\in \\mathcal { P } : \\sum \\limits _ { x \\in ( \\mathbb { L } ) } p ( x ) f _ i ( x ) = a _ i , i \\in \\{ 1 , \\dots , k \\} \\Big \\} . \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} v _ { 2 } ( t , r ) = c _ { b } \\int _ { 0 } ^ { \\infty } d \\xi \\sin ( t \\xi ) J _ { 1 } ( r \\xi ) \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) \\cdot \\begin{cases} \\frac { 1 } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } , b \\neq 1 \\\\ \\log ( \\log ( \\frac { 1 } { \\xi } ) ) , b = 1 \\end{cases} \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} E g ( \\widehat { \\mathbb { H } } ' \\mathbf { f } ) = E g ( \\mathbb { H } ' \\mathbf { f } ) \\quad g \\in C _ { b } ( \\mathbb { R } ^ { r } ) , \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} & \\delta ^ 2 _ 0 ( x _ i ) = x _ i + \\epsilon _ i & & \\delta ^ 2 _ 1 ( x _ i ) = x _ i & & \\delta ^ 2 _ 2 ( x _ i ) = x _ i \\\\ & \\delta ^ 2 _ 0 ( x _ i + \\epsilon _ i ) = x _ i + \\tau _ i & & \\delta ^ 2 _ 1 ( x _ i + \\epsilon _ i ) = x _ i + \\tau _ i & & \\delta ^ 2 _ 2 ( x _ i + \\epsilon _ i ) = x _ i + \\epsilon _ i . \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} h ^ 2 \\Delta _ { g _ 0 } + 1 = ( h D _ x ) ^ 2 - i ( n - 1 ) h x ^ { - 1 } h D _ x + h ^ 2 x ^ { - 2 } \\Delta _ k + 1 , D = \\frac { 1 } { i } \\partial ; \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} \\nu ^ { \\ast ^ G } ( f ) : = \\int _ G f ( x ^ { - 1 } ) d \\overline { \\nu } ( x ) , \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} t \\ , \\frac { \\partial u } { \\partial t } = F \\Bigl ( t , x , u , \\frac { \\partial u } { \\partial x } \\Bigr ) , \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} b ^ { [ t ] } \\beta ^ { \\epsilon } P ^ k = \\binom { t - ( p - 1 ) k - \\epsilon } { k } a ^ { \\epsilon } b ^ { [ t - ( p - 1 ) k - \\epsilon ] } , \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} \\lambda = m n \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} \\Phi : = \\frac { \\varphi '' } { \\varphi ' } = ( \\log | \\varphi ' | ) ' , \\mathcal { W } : = \\frac { w } { w ' } = \\frac { 1 } { ( \\log | w | ) ' } , \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t / 2 } \\frac { \\sin ( u ) } { u \\log ^ { a } ( t / u ) } d u = \\frac { \\pi } { 2 \\log ^ { a } ( t ) } + O \\left ( \\frac { 1 } { \\log ^ { a + 1 } ( t ) } \\right ) \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u = F \\left ( u , \\bar { u } \\right ) , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} \\sum _ { 0 < s \\le t } \\left ( G ( X _ s ) - G ( X _ { s - } ) \\right ) = \\int _ 0 ^ t \\int _ \\R \\left ( G ( X _ { s - } + \\rho ( X _ { s - } , y ) ) - G ( X _ { s - } ) \\right ) \\nu ( d y , d s ) . \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} [ s _ { 1 } , s _ { 2 } , s _ { 3 } , t _ { 1 } , t _ { 2 } , t _ { 3 } ] = \\sigma \\cdot g \\cdot [ s _ { 1 } ' , s _ { 2 } ' , s _ { 3 } ' , t _ { 1 } ' , t _ { 2 } ' , t _ { 3 } ' ] \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} h _ C ( m ) & = \\max \\{ | x _ 2 ^ { i m } - x ^ { ( i + 1 ) m } _ 2 | : i = 0 , 1 , \\cdots , \\lfloor n _ C / m \\rfloor - 1 \\} \\\\ & = \\frac { m \\pi } { 2 n _ C } \\sin ( \\xi ) \\ ; , \\ \\ \\xi \\in \\left ( \\frac { 2 m i - 1 } { 2 n _ C } \\pi , \\frac { 2 m ( i + 1 ) - 1 } { 2 n _ C } \\pi \\right ) \\\\ & \\approx \\frac { m \\pi } { 2 n _ C } \\\\ \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} ( \\lambda \\ast _ { G / H } \\lambda ' ) ^ { \\ast ^ { G / H } } = T _ H \\left ( { \\lambda ' } _ q ^ { \\ast ^ { G } } \\ast _ { G } \\lambda _ q ^ { \\ast ^ { G } } \\right ) = \\lambda '^ { \\ast ^ { G / H } } \\ast _ { G / H } \\lambda ^ { \\ast ^ { G / H } } . \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\prod _ { y \\in \\Lambda ^ { ' } } \\mathrm { T r } ( h _ { y , i } h ^ \\ast _ { y , j } b _ y ) = : \\alpha ^ { ( | \\Lambda ^ { ' } | ) } ( J ^ { - 1 } _ { \\Lambda ^ { ' } } ( b _ { \\Lambda ^ { ' } } ) ) \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} G = ( R _ { p } : H _ { 1 } , \\cdots , H _ { j } ) + G _ { 1 } + \\cdots + G _ { k } , \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} f ( z ) = z \\exp \\left ( - H _ { \\sigma } ( z ) \\right ) , z \\in \\mathbb { D } . \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} \\sum _ { k } f ( x _ k ^ { n , 1 } ) \\ , \\psi ( x _ k ^ { n , 1 } ) \\ , \\phi ( \\alpha + & | I _ k ^ n | ) \\\\ = \\sum _ { k } f ( x _ k ^ { n , 1 } ) & \\psi ( x _ k ^ { n , 1 } ) \\ , \\Big ( \\frac { \\phi ( \\alpha + | I _ k ^ n | ) } { | I _ k ^ n | } - \\phi ' _ + ( \\alpha ) \\Big ) \\ , | I _ k ^ n | \\\\ & + \\phi ' _ + ( \\alpha ) \\ , \\sum _ { k } f ( x _ k ^ { n , 1 } ) \\psi ( x _ k ^ { n , 1 } ) \\ , | I _ k ^ n | = C _ n + D _ n , \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} c = \\liminf _ { r \\uparrow 1 } \\frac { 1 - \\left | \\eta _ { \\mu _ { 1 } } ( r \\alpha ) \\right | } { 1 - r } \\in ( 0 , + \\infty ) . \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} \\Delta ^ { I } \\boldsymbol { x } = - \\nabla _ { 1 } \\nabla _ { 1 } \\boldsymbol { x } - \\nabla _ { 2 } \\nabla _ { 2 } \\boldsymbol { x } - q _ { 2 } \\nabla _ { 1 } \\boldsymbol { x } + q _ { 1 } \\nabla _ { 2 } \\boldsymbol { x } . \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} D \\ge \\frac { | F _ 3 | \\cdot b _ { i + 1 } } { b _ i + b _ { i + 1 } } \\ge \\frac { ( \\frac m 2 - t ) b _ { i + 1 } } { \\binom { t } { k - 1 } \\binom { b _ { k - 1 } + b _ k } { b _ k } ( b _ i + b _ { i + 1 } ) } . \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} F ( \\Phi ^ { c _ 1 } ( x ) ) - F ( \\Phi ^ { c _ 2 } ( x ) ) = W ( x ) E ( x ) \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} H _ { n , \\mathcal { H } } ^ { \\star } \\tilde { x } = H ^ { \\star } \\mathcal { H } \\tilde { x } \\ ; \\ ; \\tilde { x } \\in D ( H _ { n , \\mathcal { H } } ^ { \\star } ) \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} \\begin{aligned} & ( \\nabla ^ 2 f ) _ g - \\frac { ( \\Delta f ) _ g } { n } g = ( \\nabla ^ 2 f ) _ { g _ \\kappa } - \\frac { ( \\Delta f ) _ { g _ \\kappa } } { n } g _ \\kappa + \\\\ & + d f \\otimes d ( \\log h ) + d ( \\log h ) \\otimes d f - \\frac { 2 } { n } g _ \\kappa ( ( \\nabla f ) _ { g _ \\kappa } , ( \\nabla \\log h ) _ { g _ \\kappa } ) g _ \\kappa . \\end{aligned} \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} Q _ { x } ^ { N + 1 } ( T _ { B ( x , r ) } ^ { N + 1 , N } \\le t ) = Q _ { x } ^ { N + 1 } ( T _ { B ( x , r ) } ^ { N + 1 } \\le t ) . \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( \\xi ' , \\eta ' ) & = \\xi _ { N , M } { \\xi ' } ^ 2 + \\eta _ { N , M } { \\eta ' } ^ 2 - \\frac { { \\xi _ { N , M } } ^ 3 + { \\eta _ { N , M } } ^ 3 } { 4 } , \\\\ \\tilde { F } ( \\xi ' , \\eta ' ) & = 2 \\xi ' \\eta ' + \\frac { 3 } { 2 } \\ , \\xi _ { N , M } \\ , \\eta _ { N , M } . \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} x & = s _ 2 - \\lambda _ 2 \\frac { a + \\lambda _ 2 } { 1 + a \\lambda _ 2 } s _ 2 \\\\ x & = \\frac { s _ 2 } { 1 + a \\lambda _ 2 } ( 1 - \\lambda _ 2 ^ 2 ) \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } b _ { i _ 1 , \\ldots , i _ r } ( s ) = b _ { i _ 1 , \\ldots , i _ r } . \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} ( N - s ) ! \\prod _ { j = 1 } ^ p \\big | f ( \\beta _ j ) \\big | ^ { m _ j } \\le \\prod _ { i = 1 } ^ r \\Big ( ( n _ i - 1 ) ! \\prod _ { k \\neq i } | \\alpha _ i - \\alpha _ k | ^ { n _ k - 1 } \\Big ) . \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} & R \\Bigl ( t , x , u _ 0 + U , \\frac { \\partial u _ 0 } { \\partial x } + \\frac { \\partial U } { \\partial x } \\Bigr ) - R \\Bigl ( t , x , u _ 0 , \\frac { \\partial u _ 0 } { \\partial x } \\Bigr ) \\\\ & = c _ 1 ( t , x ) U + c _ 2 ( t , x ) \\frac { \\partial U } { \\partial x } + \\sum _ { j + \\alpha \\geq 2 } c _ { j , \\alpha } ( t , x ) U ^ j \\Bigl ( \\frac { \\partial U } { \\partial x } \\Bigr ) ^ { \\alpha } \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} T ( y ; \\mu ) = 2 \\mu + T _ R ( y ; \\mu ) + T _ L \\left ( P _ R ( y ; \\mu ) ; \\mu \\right ) . \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\frac { x } { 1 + x } = x - \\frac { x ^ 2 } { 1 + x } = x - \\phi _ 1 ( x ) , \\forall x > 0 , \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} P = P _ 4 \\circ P _ 3 \\circ P _ 2 \\circ P _ 1 \\ , , \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} v _ { 4 , 1 , 2 } ( t , r ) = \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { \\infty } d \\xi J _ { 1 } ( r \\xi ) \\sin ( ( t - s ) \\xi ) \\left ( \\widehat { v _ { 4 , c } ^ { \\lambda _ { 1 } } - v _ { 4 , c } ^ { \\lambda _ { 2 } } } ( s , \\xi ) \\right ) \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} \\prod _ { j = r + 1 } ^ { r + s } ( 1 - L _ j ) ^ r k _ { ( 0 ^ r , \\gamma ) } = k _ \\gamma \\ , . \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} \\lim _ { \\beta \\to + \\infty } F ( \\beta ) = + \\infty . \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} G _ \\rightarrow g ( x ) = v ( x ) \\int _ a ^ x u ( y ) g ( y ) \\d y - u ( x ) \\int _ a ^ x v ( y ) g ( y ) \\d y . \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} P ( ( 1 + t ) ^ { - 1 } , t ) = \\sum _ r \\gamma _ { r } \\frac { 1 } { ( 1 + t ) ^ { n _ r } } + \\sum _ s \\delta _ { s } t ^ { n _ s } . \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} v _ { 3 } ( t , r ) = \\frac { - 1 } { r } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\lambda '' ( s ) \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} \\begin{aligned} A ( r ) - ( 1 + \\eta ) B ( r ) & \\geq w ' \\Big [ E + ( \\varphi ' ) ^ 2 ( 1 + 2 \\mathcal { W } \\Phi - ( 1 + \\eta ) \\mathcal { W } \\Phi ^ 2 \\min ( \\mathcal { W } , \\tfrac { h } { 4 \\varphi ' } ) ) - V - \\mathcal { W } V ' \\Big ] . \\end{aligned} \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} \\begin{pmatrix} A _ c & 0 & 0 \\\\ 0 & A _ u & 0 \\\\ 0 & 0 & A _ s \\end{pmatrix} \\begin{pmatrix} \\operatorname { I d } + k _ c \\\\ k _ u \\\\ k _ s \\end{pmatrix} + \\begin{pmatrix} g _ c \\circ K { } { } \\\\ g _ u \\circ K { } { } \\\\ g _ s \\circ K { } { } \\end{pmatrix} - \\begin{pmatrix} A _ c + r + k _ c \\circ ( A _ c + r ) \\\\ k _ u \\circ ( A _ c + r ) \\\\ k _ s \\circ ( A _ c + r ) \\end{pmatrix} & = 0 . \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} F _ { \\mu } ( t ) = x + i f ( x ) \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} \\left \\{ P _ { k } ( z , \\overline { z } ) \\right \\} _ { k = 1 , \\dots , \\left [ \\frac { p } { 2 } \\right ] } , \\left \\{ R _ { k } ( z , \\overline { z } ) \\right \\} _ { k = 1 , \\dots , \\left [ \\frac { p } { 2 } \\right ] } , \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} p _ \\theta ( { \\bf { x } } ) & = Z ( \\theta ) \\exp \\big [ h ( { \\bf { x } } ) + w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\big ] . \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} \\alpha ^ { + } _ { \\Omega , 0 } ( t _ { n + 1 } ) = \\pi / 4 \\quad \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} K _ { 3 } ( w , \\lambda ( t ) ) & = \\left ( \\frac { w } { 1 + w ^ { 2 } } - \\frac { w } { \\lambda ( t ) ^ { 2 - 2 \\alpha } + w ^ { 2 } } \\right ) \\frac { w ^ { 4 } } { 4 ( w ^ { 2 } + 3 6 \\lambda ( t ) ^ { 2 } ) ^ { 2 } } \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} | | f | | _ { X } = _ { t \\geq T _ { 0 } } \\left ( | f ( t ) | b \\log ^ { b } ( t ) \\sqrt { \\log ( \\log ( t ) ) } + | f ' ( t ) | t \\log ^ { b + 1 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } + | f '' ( t ) | t ^ { 2 } \\log ^ { b + 1 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } \\right ) \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} S ^ * T _ u S = T _ u \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} E _ { n , 2 } ( e _ i ) = \\delta _ { i , 2 } e _ n , \\ \\ 1 \\leq i \\leq n , \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} \\begin{aligned} & \\forall l \\in I ^ { 0 + } ( \\bar x ) \\cap I ^ { \\varphi ^ t _ \\textup { F B } } ( x ) \\colon \\quad & & \\alpha _ l \\neq 0 \\quad & & \\beta _ l \\approx 0 & \\\\ & \\forall l \\in I ^ { + 0 } ( \\bar x ) \\cap I ^ { \\varphi ^ t _ \\textup { F B } } ( x ) \\colon \\quad & & \\alpha _ l \\approx 0 \\quad & & \\beta _ l \\neq 0 \\end{aligned} \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} \\| ( B + L _ n ) ^ { - 1 / 2 } A ( B + L _ n ) ^ { - 1 / 2 } \\| _ \\infty & = \\| ( B + L _ n ) ^ { - 1 / 2 } B ^ { 1 / 2 } W B ^ { 1 / 2 } ( B + L _ n ) ^ { - 1 / 2 } \\| _ \\infty \\\\ & \\le c \\| ( B + L _ n ) ^ { - 1 / 2 } B ( B + L _ n ) ^ { - 1 / 2 } \\| _ \\infty \\\\ & = c \\| B ^ { 1 / 2 } ( B + L _ n ) ^ { - 1 } B ^ { 1 / 2 } \\| _ \\infty \\le c , \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} H : = U ^ * D _ { H } U . \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} \\partial _ s \\mathcal { N } ( t , s , \\xi , v , \\overline v ) = e ^ { ( t - s ) \\mathcal { L } } \\mathcal { C } [ f , \\mathcal { L } ] \\left ( e ^ { s \\mathcal { L } } v , e ^ { s \\mathcal { L } } e ^ { \\xi \\mathcal { A } } \\overline v \\right ) . \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } \\frac { \\lambda '' ( s ) r } { ( s - t ) } d s & = \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } \\frac { \\lambda '' ( s ) r } { 1 + s - t } d s + \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } \\lambda '' ( s ) r \\left ( \\frac { 1 } { s - t } - \\frac { 1 } { 1 + s - t } \\right ) d s \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} D _ \\alpha u _ 1 = \\alpha v _ 1 + u _ 1 + \\alpha C u _ 2 \\ge 0 , A _ \\beta u _ 2 = \\beta v _ 2 + u _ 2 + \\beta B u _ 1 \\ge 0 . \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} h ( z ) ( \\rho ( L _ 0 ) + c ) h ( z ) ^ { - 1 } ( \\rho ( L _ 0 ) - c ) & = ( \\rho ( L _ 0 ) + c + 1 ) ( \\rho ( L _ 0 ) - c ) \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} \\begin{cases} | \\vec x | \\leq t ^ { 1 / m } \\\\ | \\Theta \\vec x - \\vec y | \\leq t ^ { - \\gamma / n } \\end{cases} , \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} \\tfrac { 1 } { 6 } r ^ { 3 } + \\tfrac { 1 } { 2 } r | v | ^ { 2 } + P ( v ) & = \\hat { P } ( v , r ) = \\hat { Q } ( \\phi ( v ) + r u , | v | ^ { 2 } ) \\\\ & = \\hat { Q } ( r u , | v | ^ { 2 } ) = \\tfrac { 1 } { 6 } | v | ^ { 6 } + \\tfrac { 1 } { 2 } r ^ { 2 } | v | ^ { 2 } | u | ^ { 2 } + r ^ { 3 } Q ( u ) , \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} \\alpha = - \\alpha _ { 0 } - \\int _ { ( 0 , + \\infty ) } \\frac { 1 } { t } \\ , d \\rho ( t ) , \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} \\min \\limits _ { p \\in \\mathbb { P } _ L } ~ ~ \\left \\{ \\frac 1 2 \\sum _ { j = 1 } ^ N w _ j \\left ( p ( \\mathbf { x } _ j ) - f ( \\mathbf { x } _ j ) \\right ) ^ 2 \\right \\} \\quad p ( \\mathbf { x } ) = \\sum _ { \\ell = 1 } ^ { d } \\alpha _ { \\ell } p _ { \\ell } ( \\mathbf { x } ) \\in \\mathbb { P } _ L . \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} \\lim _ { n } U ( f , \\Psi , \\mathcal Q _ n ) = \\lim _ { n } L ( f , \\Psi , { \\mathcal Q } _ n ) = \\int _ I f \\ , d \\Psi . \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} X = \\left ( \\begin{array} { c c c } \\alpha _ { 1 } & x _ { 3 } & \\overline { x } _ { 2 } \\\\ \\overline { x } _ { 3 } & \\alpha _ { 2 } & x _ { 1 } \\\\ x _ { 2 } & \\overline { x } _ { 1 } & \\alpha _ { 3 } \\end{array} \\right ) , ~ \\alpha _ { 1 } , \\alpha _ { 2 } , \\alpha _ { 3 } \\in \\mathbb { R } , ~ x _ { 1 } , x _ { 2 } , x _ { 3 } \\in \\mathbb { A } \\mathbb { A } _ { s } , \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} \\delta ^ 2 \\mathcal { V } ( M ) = \\int _ M - 2 H u ^ 2 \\ , d S , \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} N _ { \\sigma } ( z ) = \\int _ { \\mathbb { R } } \\frac { 1 + t z } { z - t } \\ , d \\sigma ( t ) , z \\in \\mathbb { H } . \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} K _ { 1 , 1 } \\le C _ \\alpha \\int _ \\delta ^ { \\bar { x } \\wedge \\bar { y } } C z ^ 2 \\frac { d z } { z ^ { \\alpha + 2 } } = \\frac { C _ \\alpha C } { 1 - \\alpha } ( ( \\bar { x } \\wedge \\bar { y } ) ^ { 1 - \\alpha } - \\delta ^ { 1 - \\alpha } ) , \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} \\sup _ { D _ { \\bar { p } R } } \\frac { f _ 1 ^ { \\bar { p } R / 2 , \\bar { q } R } } { f _ 2 ^ { \\bar { p } R / 2 , \\bar { q } R } } - m _ { \\bar { q } R } = \\sup _ { D _ { \\bar { p } R } } \\frac { g } { f _ 2 ^ { \\bar { p } R / 2 , \\bar { q } R } } & \\le c ^ 4 \\inf _ { D _ { \\bar { p } R } } \\frac { g } { f _ 2 ^ { \\bar { p } R / 2 , \\bar { q } R } } \\\\ & = c ^ 4 \\left ( \\inf _ { D _ { \\bar { p } R } } \\frac { f _ 1 ^ { \\bar { p } R / 2 , \\bar { q } R } } { f _ 2 ^ { \\bar { p } R / 2 , \\bar { q } R } } - m _ { \\bar { q } R } \\right ) , \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} | C | \\leq \\bigg \\lfloor \\frac { | \\Pi | } { V _ r ( \\Pi ) } \\bigg \\rfloor , \\ r = \\lfloor ( d - 1 ) / 2 \\rfloor . \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} p _ t ( x ) = t ^ { - d / \\alpha } p _ 1 ( x / t ^ { 1 / \\alpha } ) , t > 0 , x \\neq 0 , \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} C ^ + = \\phi _ { 0 } ^ + + \\phi _ { 2 } ^ + , C ^ - = \\phi _ { 0 } ^ - + \\phi _ { 2 } ^ - , D ^ + = \\phi _ { 0 } ^ + - \\phi _ { 2 } ^ + , D ^ - = \\phi _ { 0 } ^ - - \\phi _ { 2 } ^ - . \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} | \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( t , \\xi ) | & \\leq \\frac { C \\log ^ { 3 } ( t ) } { t ^ { 2 } } , t - \\sqrt { t } \\leq \\frac { 1 } { \\xi } \\leq t + \\sqrt { t } \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} | u ( r ) | & = \\Big | - \\int _ r ^ 1 u ' ( s ) d s \\Big | \\\\ & \\leq \\Big ( \\int _ r ^ 1 | u ' ( s ) | ^ 2 s ^ { \\beta - 1 } d s \\Big ) ^ { 1 / 2 } \\Big ( \\int _ r ^ 1 s ^ { 1 - \\beta } d s \\Big ) ^ { 1 / 2 } \\\\ & \\leq \\Big ( a \\frac { r ^ { 2 - \\beta } - 1 } { \\beta - 2 } \\Big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{gather*} J _ 1 ( \\bar { u } ) = c _ 1 , \\ , J _ 2 ( \\bar { v } ) = c _ 2 \\ , \\ , J _ 1 ' ( \\bar { u } ) = 0 = J _ 2 ' ( \\bar { v } ) . \\end{gather*}"}
-{"id": "8167.png", "formula": "\\begin{align*} \\frac { \\partial _ r [ F ( \\theta ) ] + \\partial _ r [ w ( \\theta ) ] ^ \\top \\mathbb { E } _ \\theta [ f ( \\textbf { X } ) ] } { \\mathbb { E } _ \\theta [ h + F ( \\theta ) + w ( \\theta ) ^ \\top f ( \\textbf { X } ) ] } = \\frac { \\partial _ r [ F ( \\theta ) ] + \\partial _ r [ w ( \\theta ) ] ^ \\top \\bar { f } } { h + F ( \\theta ) + w ( \\theta ) ^ \\top \\bar { f } } . \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} H _ { n + 1 } = 1 + \\sum _ 1 ^ { n } H _ { i } & = 1 + \\sum _ 1 ^ { n - 1 } H _ { i } + H _ { n } = 2 H _ { n } . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} f ( z ) = f ( \\beta ) ( 1 + h ( z ) ^ \\ell ) \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} \\left ( \\frac { y } { x } \\right ) ^ 2 + \\frac { y } { x } = x + a + \\frac { b } { x ^ 2 } = x ^ 2 + x + a + x ^ 2 + \\frac { B ^ 8 } { x ^ 2 } , \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} - \\beta : = \\frac { 1 - n \\widehat { \\lambda } _ { n } } { n ( 1 + \\widehat { \\lambda } _ { n } ) } . \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\frac { a ' _ k } { a _ k } - k \\frac { a ' _ 1 } { a _ 1 } - ( n _ k - k n _ 1 ) \\frac { a ' _ 0 } { a _ 0 } \\ , = \\ , 0 \\ , . \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} & \\P \\big \\{ Z _ { 1 } ^ { x , y } Z _ { 1 } ^ { y , z } = 0 \\big \\} \\\\ & ~ \\geq \\P \\big \\{ h _ { 1 } ( ( x , 0 ) ) = h _ { 1 } ( ( y , 0 ) ) = ( x + \\lceil \\tfrac { y - x } { 2 } \\rceil , \\lceil \\tfrac { y - x } { 2 } \\rceil ) , h _ { 1 } ( ( z , 0 ) ) = ( z , \\lceil \\tfrac { y - x } { 2 } \\rceil ) \\big \\} \\\\ & ~ \\geq ( 1 - p ) ^ { 3 ( \\lceil \\tfrac { y - x } { 2 } \\rceil + 1 ) ^ { 2 } - 5 } p ^ { 2 } \\geq ( 1 - p ) ^ { 3 ( m _ { 0 } + 1 ) ^ { 2 } - 5 } p ^ { 2 } . \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} \\langle \\widehat { p } _ 1 f , f \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = \\int _ { \\mathbb { R } ^ 2 } \\left [ \\left ( - i \\mu \\frac { \\partial } { \\partial x _ 1 } + \\frac { \\eta } { 2 \\mu } x _ 2 \\right ) f ( x _ 1 , x _ 2 ) \\right ] \\overline { f ( x _ 1 , x _ 2 ) } d x _ 1 d x _ 2 = \\frac { \\eta } { 2 \\mu } x _ 2 ^ { ( 0 ) } \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} H _ { n + 1 } \\ = \\ c _ 1 H _ n + \\cdots + c _ L H _ { n + 1 - L } , \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} v _ { s } ( t , r ) = \\frac { \\lambda '' ( s ) } { r } \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\left ( 1 + \\frac { r ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( 1 + r ^ { 2 } + \\rho ^ { 2 } ) ^ { 2 } - 4 r ^ { 2 } \\rho ^ { 2 } } } \\right ) \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} f ( \\eta _ S ( x ) ) = f ( \\eta _ S ( y ) ) & \\iff \\eta ( x ) = \\eta ( y ) \\\\ & \\iff x \\in U y \\in U & U X \\\\ & \\iff x \\in U \\cap S y \\in U \\cap S & U X \\\\ & \\iff \\eta _ S ( x ) = \\eta _ S ( y ) . \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c l } a _ { 1 1 } h _ 1 ( X _ 1 ) & + a _ { 1 2 } h _ 2 ( X _ 2 ) + \\cdots + & a _ { 1 t } h _ t ( X _ t ) = g _ 1 ( X _ 1 , \\ldots , X _ k ) \\\\ a _ { 2 1 } h _ 1 ( X _ 1 ) & + a _ { 2 2 } h _ 2 ( X _ 2 ) + \\cdots + & a _ { 2 t } h _ t ( X _ t ) = g _ 2 ( X _ 1 , \\ldots , X _ k ) \\\\ \\ ; \\vdots & & \\quad \\vdots \\\\ a _ { n 1 } h _ 1 ( X _ 1 ) & + a _ { n 2 } h _ 2 ( X _ 2 ) + \\cdots + & a _ { n t } h _ t ( X _ t ) = g _ n ( X _ 1 , \\ldots , X _ k ) \\end{array} \\right . \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} { \\mathbb M } ( \\varphi , \\varphi ) = { \\mathbb M } ( \\varphi _ 0 , \\varphi _ 0 ) + \\sum _ { j = 1 } ^ { \\infty } { \\mathbb M } ( \\varphi _ j ^ 1 , \\varphi _ j ^ 1 ) + \\sum _ { j = 1 } ^ { \\infty } { \\mathbb M } ( \\varphi _ j ^ 2 , \\varphi _ j ^ 2 ) . \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ \\infty \\binom { a } { \\alpha n } ^ l = \\int _ { - \\infty } ^ \\infty \\binom { a } { \\alpha x } ^ l d x , 0 < \\alpha \\le \\frac { 2 } { l } . \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} \\Xi _ n ( P ) & = \\left ( p _ 1 ^ * a _ \\lambda ^ * ( \\gamma _ \\lambda ) \\cdot p _ 2 ^ * a _ \\lambda ^ * ( \\gamma _ \\lambda ) \\right ) \\cdot \\Delta _ * \\left ( a _ \\lambda ^ * ( \\gamma _ \\lambda ) ^ { - 1 } \\cdot i ^ * _ { Y _ \\lambda } ( P ) \\right ) \\\\ & = \\Delta _ * \\left ( a _ \\lambda ^ * ( \\gamma _ \\lambda ) \\cdot a _ \\lambda ^ * ( P ) \\right ) = \\xi ^ { \\lambda \\lambda } _ { ( 1 , \\gamma _ \\lambda P ) } , \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} \\Phi ( \\xi _ 1 , \\eta _ 1 , \\xi _ 2 , \\eta _ 2 ) & = \\xi _ 1 \\xi _ 2 ( \\xi _ 1 + \\xi _ 2 ) + \\eta _ 1 \\eta _ 2 ( \\eta _ 1 + \\eta _ 2 ) , \\\\ F ( \\xi _ 1 , \\eta _ 1 , \\xi _ 2 , \\eta _ 2 ) & = \\xi _ 1 \\eta _ 2 + \\xi _ 2 \\eta _ 1 + 2 ( \\xi _ 1 \\eta _ 1 + \\xi _ 2 \\eta _ 2 ) . \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} \\begin{aligned} T \\triangleq & \\int ^ { + \\infty } _ \\alpha [ \\lambda _ 1 \\alpha ^ 2 + \\beta ^ 2 - \\lambda _ 1 s ^ 2 + \\frac { \\kappa ^ 2 } { 2 + 2 r } ( s ^ { ( r + 1 ) } - \\alpha ^ { ( r + 1 ) } ) ] ^ { \\frac { - 1 } { 2 } } d s . \\end{aligned} \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } \\partial _ { j } a _ { j k } \\partial _ { k } \\bigg ] \\overline { v } & = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ \\overline { v } & = 0 \\quad C _ { \\overline { s } , 1 } ' . \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} H _ { } ( x ) & = \\sum _ { l = 1 } ^ { M } \\left ( \\sum _ { i \\in B ( l ) } \\left ( \\psi ( x _ i ) + s _ i x _ i + \\frac { 1 } { 2 } \\sum _ { \\substack { j \\in B ( l ) , \\\\ 1 \\leq | j - i | \\leq R } } M _ { i j } x _ i x _ j \\right ) \\right ) \\\\ & = \\sum _ { l = 1 } ^ { M } H _ K ( x ^ { B ( l ) } ) . \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} \\big ( \\alpha . f \\big ) ^ * = f ^ * . \\alpha \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\big ( f . \\alpha \\big ) ^ * = \\alpha . f ^ * . \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} D h _ m ( \\rho ) & = D ^ 2 T \\left ( \\operatorname { I d } , ( \\rho \\circ T ) ( D T ) ^ { \\otimes m } \\right ) + D T \\left ( D \\rho ( T ) \\right ) \\left ( D T , ( D T ) ^ { \\otimes m } \\right ) \\\\ & + D T ( \\rho \\circ T ) \\sum _ { i = 0 } ^ { m - 1 } \\left ( ( D T ) ^ { \\otimes i } , D ^ 2 T , ( D T ) ^ { \\otimes m - 1 - i } \\right ) . \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} V _ 1 ( z ) & = ( L ( x ^ * , \\lambda ^ * ) - L ( x ^ * , \\lambda ) ) + ( L ( x , \\lambda ^ * ) - L ( x ^ * , \\lambda ^ * ) ) \\\\ & ~ ~ ~ + \\frac { 1 } { 2 } \\norm { z - z ^ * } ^ 2 _ r . \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} v ( t ) & \\le d ( t ) + c \\sup _ { \\zeta \\in [ - \\tau , t ] } v ( \\zeta ) , \\ ; t > 0 , \\\\ v ( s ) & = \\psi ( s ) , \\ ; s \\in [ - \\tau , 0 ] , \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} B ' ( { \\bf m } ) = \\sum _ { \\substack { 1 \\leq i _ 1 < j _ 1 \\leq n \\\\ 1 \\leq i _ 2 < j _ 2 \\leq n \\\\ 1 \\leq i _ 1 \\leq i _ 2 \\leq n \\\\ 1 \\leq j _ 1 \\leq j _ 2 \\leq n \\\\ j _ { 1 } > i _ 2 } } m _ { i _ { 1 } , j _ { 1 } } m _ { i _ { 2 } , j _ { 2 } } , \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} \\frac { 1 } { N } | p _ n - U _ n | \\leq | \\sin ( \\theta ) | | p _ n - U _ n | \\leq | \\delta _ n | \\leq \\sum _ { j = 1 } ^ { n - 1 } | a _ j p _ j | . \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} f ( t ) = \\frac { 1 } { \\Psi ( z ( t ) ) } , \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} \\tilde \\omega = \\sum _ { 1 \\le n < p \\le N } a _ { n p } ( \\gamma _ 1 , \\dots , \\gamma _ { N } ) \\ , d \\gamma _ n \\wedge d \\gamma _ p \\ . \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} ( g _ { i j } ) _ s = \\langle F _ { s i } , F _ j \\rangle + \\langle F _ i , F _ { s j } \\rangle = - 2 \\ , \\langle V , F _ { i j } \\rangle = - 2 A ^ V _ { i j } \\ , . \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} G _ { \\mu } ( z ) = \\int _ { \\mathbb { R } _ { + } } \\frac { d \\mu ( t ) } { z - t } , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{gather*} g _ i I _ { \\lambda } \\cap s _ i I _ { \\mu , i - 1 } = ( g _ i I _ { \\lambda } \\cap I _ { \\mu , i } ) \\sqcup M ^ i _ { t + 1 , t } , \\\\ g _ i N _ { \\lambda } \\cup N _ { \\mu , i } = ( g _ i N _ { \\lambda } \\cup s _ i N _ { \\mu , i - 1 } ) \\sqcup M ^ i _ { t , t + 1 } . \\end{gather*}"}
-{"id": "7710.png", "formula": "\\begin{align*} F ( \\zeta , c ; x , y ) = y ^ 6 - 9 x y ^ 4 + 2 7 x ^ 2 y ^ 2 - 2 7 x ^ 3 + 4 c ^ 2 - ( 3 6 x y + 4 y ^ 3 ) \\ , c + ( 1 8 x y ^ 2 - 5 y ^ 4 - 9 x ^ 2 - 8 c y ) \\ , \\zeta + 4 y ^ 2 \\ , \\zeta ^ 2 . \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} \\alpha = \\frac { \\lambda _ L } { \\omega _ L } + \\frac { \\lambda _ R } { \\omega _ R } , \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\mathop { \\rm e s s \\ , i n f } _ { \\Omega \\times ( t , \\infty ) } f = \\infty . \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} 2 \\int _ { t } ^ { t + \\frac { r } { 6 } } \\frac { \\lambda ''' ( s ) } { r } ( s - t ) d s & = \\frac { 2 } { r } \\left ( \\frac { r } { 6 } \\lambda '' ( t + \\frac { r } { 6 } ) - \\lambda ' ( t + \\frac { r } { 6 } ) + \\lambda ' ( t ) \\right ) \\\\ & = \\frac { 2 \\lambda ' ( t ) } { r } + E _ { \\partial _ { t } v _ { 1 } , 1 } ( t , r ) \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} f ( t & ) = f _ 0 + \\int _ 0 ^ t [ Q ^ { + } ( s , f ( s ) ) - Q ^ { - } ( s , f ( s ) ) ] d s \\\\ + & \\int _ 0 ^ t \\left [ a \\left ( \\left \\| \\Lambda f ( s ) \\right \\| + \\int _ 0 ^ s \\Delta ( \\tau , f ( \\tau ) ) d \\tau \\right ) - a ( \\left \\| \\Lambda f _ 0 \\right \\| ) \\right ] \\Lambda f ( s ) d s \\forall t \\geq 0 , \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} 1 6 n ^ { 3 } + 2 4 n ^ { 2 } + 8 n - 6 n \\mu = 4 n ^ 3 - 4 n - 3 n \\mu + 3 \\mu \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} b _ 2 & = \\frac { i } { x _ { 1 + 2 } } \\cdot 2 i a _ 1 \\left ( x _ 1 + x _ 2 + x _ { 1 + 2 } \\right ) \\Big | _ { x _ 1 = 0 = x _ 2 } = - 2 a _ 1 \\\\ b _ 3 & = - 6 a _ 2 + 1 2 a _ 1 ^ 2 . \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} I _ { i b \\beta } \\subset K _ { i b \\beta c } \\cap R = K _ { i a \\beta } \\cap R \\subset m _ R ^ { i a \\overline c } \\subset m _ R ^ i \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} - \\frac { 1 } { \\omega } \\int _ { 0 } ^ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } \\widetilde { \\phi _ { 0 } } ( r ) \\sqrt { r } F _ { 4 } ( t , r \\lambda ( t ) ) d r = \\frac { - 1 } { 2 \\omega } \\int _ { 0 } ^ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } r \\phi _ { 0 } ( r ) F _ { 4 } ( t , r \\lambda ( t ) ) d r \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} P _ n x _ k - H _ n P _ n x _ k = P _ n x _ k - H P _ n x _ k = P _ n x _ k - P _ n H x _ k = P _ n \\left ( x _ k - H x _ k \\right ) \\to P _ n y _ n = y _ n \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} m ( x ) : = \\max \\{ D _ p ( \\Pi _ M ^ p x , x ) , D _ p ( \\Pi _ N ^ p x , x ) \\} . \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} g ( X , Y , Z ) = g ( X + Z , \\ Y , \\ - Z ) , \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( z ) ^ { k } = z \\eta _ { \\nu ^ { \\boxtimes k } } ( z ) ^ { k - 1 } , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} M & = \\left ( \\begin{array} { c c | c | c c } m _ { 1 , 1 } & m _ { 1 , 2 } & m _ { 1 , 3 } & m _ { 1 , 4 } & m _ { 1 , 5 } \\\\ 0 & m _ { 2 , 2 } & m _ { 2 , 3 } & m _ { 2 , 4 } & m _ { 2 , 5 } \\\\ \\hline 0 & 0 & m _ { 3 , 3 } & m _ { 3 , 4 } & m _ { 3 , 5 } \\\\ \\hline 0 & 0 & 0 & m _ { 4 , 4 } & m _ { 4 , 5 } \\\\ 0 & 0 & 0 & 0 & m _ { 5 , 5 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} c _ 1 ( h , \\varepsilon ) = \\begin{cases} 1 + h / \\varepsilon & ( \\varepsilon > 0 ) , \\\\ 1 & ( \\varepsilon = 0 ) . \\end{cases} \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} v _ \\varepsilon ( x ) ^ { | x | ^ \\alpha } & = A _ { n , m } ^ { | x | ^ \\alpha } \\Big ( \\frac { \\varepsilon } { \\varepsilon ^ 2 + | x | ^ 2 } \\Big ) ^ { \\frac { n - 2 m } { 2 } | x | ^ \\alpha } \\\\ & = \\exp \\Big [ \\Big ( \\ln A _ { n , m } - \\frac { n - 2 m } 2 \\ln \\big ( \\varepsilon + \\frac { | x | ^ 2 } { \\varepsilon } \\big ) \\Big ) | x | ^ \\alpha \\Big ] . \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { ( n + 1 ) / 2 } [ 4 k - 1 ] \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a , q ^ { - 1 } / b , c q ^ { - 1 } , d q ^ { - 1 } , q ^ { - 1 } ; q ^ 2 ) _ k } { ( q ^ 2 / a , a q ^ 2 , b q ^ 2 , q ^ 2 / c , q ^ 2 / d , q ^ 2 ; q ^ 2 ) _ k } \\bigg ( \\frac { b q ^ 7 } { c d } \\bigg ) ^ k \\equiv 0 \\pmod { \\Phi _ n ( q ) } . \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} \\lim _ { v \\to \\infty } \\sup _ { 0 \\leq t \\leq T } \\frac { 1 } { N } \\int ( x - N P ^ { t } \\eta ) \\cdot A ^ { - 1 } ( x - N P ^ { t } \\eta ) f ( t , x ) \\mu ( d x ) = 0 . \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} \\langle y , ( 1 - H ) u \\rangle = 0 \\mbox { f o r a l l } u \\in D ( H ) . \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} \\mathcal { F } _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) \\nabla ^ { 2 } \\mathbb { K } _ { n } ^ { ( j ) } \\left ( x \\right ) + \\mathcal { G } _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) \\nabla \\mathbb { K } _ { n } ^ { ( j ) } \\left ( x \\right ) + \\mathcal { H } _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) \\mathbb { K } _ { n } ^ { ( j ) } \\left ( x \\right ) = 0 , \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} I : q c o n f \\left ( J _ { 3 } ^ { \\mathbb { H } _ { s } } \\right ) \\supset \\left . s l \\left ( q + 4 , \\mathbb { R } \\right ) \\right \\vert _ { q = 4 } ; \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} L u _ 1 = a ( t , x ) , \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} M = \\begin{bmatrix} 1 - \\alpha & \\alpha \\\\ \\beta & 1 - \\beta \\end{bmatrix} , \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} \\gamma = \\sum _ { I = ( \\epsilon _ 1 , i _ 1 , \\dots , \\epsilon _ s , i _ s ) \\in \\mathcal { I } } \\omega _ I ( - 1 ) ^ { s [ \\frac { | m | } { 2 } ] } u _ 1 ^ { \\epsilon _ 1 } v _ 1 ^ { ( p - 1 ) i _ 1 - \\epsilon _ 1 } \\cdots u _ s ^ { \\epsilon _ s } v _ s ^ { ( p - 1 ) i _ s - \\epsilon _ s } S _ s ( m ) \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} \\widetilde { \\varrho } = \\frac { 1 } { 2 ( d + 1 ) } , \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} \\Upsilon _ { n } = \\begin{pmatrix} Q _ { Z } \\otimes Q _ { Z } \\\\ Q _ { Z } \\otimes Q _ { Z } \\end{pmatrix} . \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} B _ 1 \\ & = \\ \\{ 2 , 4 , 6 \\} , \\\\ B _ 2 \\ & = \\ \\{ 2 , 3 , 8 \\} , \\\\ B _ 3 \\ & = \\ \\{ 1 , 5 , 7 \\} , \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} \\begin{aligned} & \\partial ^ 2 _ t u = \\partial ^ 2 _ x u + c _ 1 u + ( c _ 2 u + f ( u ) ) \\dot { W } ( t , x ) , t > 0 , x \\in \\overline { D } , \\\\ & u ( 0 , x ) = ( J + T + 1 ) ( 1 + u _ 0 ( x ) ) , \\partial _ t u ( 0 , x ) = ( J + T + 1 ) v _ 0 ( x ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} \\Sigma _ { 0 , \\ , N } ^ { \\alpha _ 0 } & = \\{ f \\in \\Sigma _ { 0 , \\ , N } \\ | \\ \\| f \\| _ { \\Sigma _ { 0 , \\ , N } } \\leq \\alpha _ 0 \\} , \\\\ D \\Sigma _ { i - 1 , \\ , N } ^ { \\alpha _ i } & = \\{ f \\in D \\Sigma _ { i - 1 , \\ , N } \\ | \\ \\| f \\| _ { D \\Sigma _ { i - 1 , \\ , N } } \\leq \\alpha _ i \\} , i \\in \\{ 1 , \\ , \\dots , r \\} , \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } & \\frac { d } { d t } \\psi ( x , t ) = u ( \\psi ( t , x ) , t ) , \\\\ & \\psi ( 0 , x ) = x . \\end{array} \\right . \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} a ^ 2 + b ^ 2 = c ^ 2 \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} & \\dim H ^ 0 ( X , \\mathcal { O } _ X ( D ) ) = 1 \\\\ & \\dim H ^ 1 ( X , \\mathcal { O } _ X ( D ) ) = 0 . \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} \\mathbb { P } _ x ( X _ { \\tau _ D } \\in A ) = \\omega _ D ^ x ( A ) , A \\in \\mathcal { B } ( \\R ^ d ) . \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} ( x _ { i } , \\bar { x } _ { i } , p _ { i } , q _ { i } ) \\mapsto z ^ { - w _ { 0 } \\lambda } ( 1 + z ^ { - 1 } \\sum \\limits _ { n , l = 0 } ^ { \\infty } z ^ { - n } q \\bar { x } ^ { n } x ^ { l } p ) \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} \\mathcal B _ N ( \\Omega ) : = \\left \\{ u \\in \\mathcal H ^ 2 _ { 0 , N } ( \\Omega ) : \\int _ { \\Omega } \\Delta u \\Delta \\varphi d x = 0 \\ , , \\forall \\varphi \\in H ^ 2 _ 0 ( \\Omega ) \\right \\} . \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} p D _ p ( y , x ) = \\| y \\| ^ p - \\| x \\| ^ p - p \\langle j _ p ( x ) , y - x \\rangle \\leq \\frac p \\sigma K C R ^ { p - \\sigma } \\| y - x \\| ^ \\sigma , \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} \\begin{aligned} & K ( t , x ) \\triangleq { \\textstyle \\sum \\limits _ { n \\in Z } } \\frac { 1 } { 2 } ( \\delta ( x + t + n J ) + \\delta ( x - t + n J ) ) , \\\\ & S ( t , x ) \\triangleq { \\textstyle \\sum \\limits _ { n \\in Z } } \\frac { 1 } { 2 } I _ { [ - t , t ] } ( x + n J ) , \\end{aligned} \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} \\Omega _ { \\rho _ { 1 } } = \\{ r t : t \\in \\mathbb { T } , 0 \\le r < R ( t ) \\} \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} b _ { p , q } = \\sqrt { \\frac { \\min \\{ p , q \\} ! } { \\max \\{ p , q \\} ! } } \\sum _ { s = 0 } ^ { \\min \\{ p , q \\} } \\binom { \\max \\{ p , q \\} } { s } \\ , \\frac { ( - 1 ) ^ s } { ( \\min \\{ p , q \\} - s ) ! } m _ { p - s , q - s } . \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} \\mathrm { E L } _ { A K S Z } \\coloneqq \\mathrm { C r i t } ( S ^ { A K S Z } ) = \\mathrm { d g M a p } ( T [ 1 ] I , \\mathcal { F } ^ { ( 1 ) } ) . \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} \\frac { d } { d t } \\| \\Delta _ q \\rho ( t ) \\| _ { L ^ 2 } \\leq \\| \\Delta _ q f \\| _ { L ^ 2 } + C d _ q 2 ^ { - \\sigma q } ( \\sqrt { q + 2 } V ( t ) + 1 ) \\| \\rho \\| _ { B ^ { \\sigma } _ { 2 , r } } . \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} & = F ( x , y ; \\mu ) , { \\rm ~ f o r ~ } x < 0 , \\\\ y & \\mapsto \\phi ( y ; \\mu ) , { \\rm ~ w h e n ~ } x = 0 , \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} \\tau _ { L U } g ( x ) : = \\tau _ { L U } ( g ( x ) ) , L < U , \\ x \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} R _ { \\ell } U _ { \\ell } & = \\sqrt { R } _ { \\ell } ( \\sqrt { R _ { \\ell } } U _ { \\ell } ) \\in L ^ { 2 } _ { \\rm l o c } ( ( 0 , \\infty ) \\times \\mathbb T ^ d _ { \\ell } ) \\ , , \\\\ R _ { \\ell } U _ { \\ell } \\otimes U _ { \\ell } & = \\sqrt { R } _ { \\ell } \\ , R _ { \\ell } ^ { 1 / 4 } U _ { \\ell } \\otimes R _ { \\ell } ^ { 1 / 4 } U _ { \\ell } \\in L ^ { 1 } _ { \\rm l o c } ( ( 0 , \\infty ) ; L ^ 2 ( \\mathbb T ^ d _ { \\ell } ) ) . \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} \\big ( \\varphi ( y ) \\big ) _ k = f ( y _ k ) , \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} \\sum _ { k \\in \\Z , \\ ; k \\ , } \\widetilde { \\phi } ( x - k ) = \\sum _ { k \\in \\Z , \\ ; k \\ , } \\widetilde { \\phi } ( x - k ) \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} \\int _ B \\frac { ( u _ j ) _ + ^ { \\ 2 m s + | x | ^ \\alpha } } { \\ 2 m s + | x | ^ \\alpha } d x = \\frac { l ^ 2 } 2 - c + o ( 1 ) _ { j \\nearrow + \\infty } . \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} & | | \\mathcal { F } ( \\sqrt { \\cdot } ( N ( u _ { 1 } ) - N ( u _ { 2 } ) ) ( t , \\cdot \\lambda ( t ) ) ) ( \\omega \\lambda ( t ) ^ { 2 } ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } \\leq \\frac { C } { \\lambda ( t ) } \\frac { | | y _ { 1 } - y _ { 2 } | | _ { Z } } { t ^ { 4 } \\log ^ { \\epsilon } ( t ) } \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} & | \\int _ { t } ^ { t + 2 ( r + 1 ) } d s \\frac { \\lambda '' ( s ) } { r } \\int _ { 0 } ^ { s - t } d \\rho \\frac { \\rho } { ( s - t ) } \\left ( 1 + \\frac { r ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( r ^ { 2 } - 1 - \\rho ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } \\right ) | \\\\ & \\leq C r \\sup _ { x \\geq t } \\left ( | \\lambda '' ( x ) | \\right ) \\int _ { t } ^ { t + 2 ( r + 1 ) } \\frac { ( s - t ) } { 1 + r ^ { 2 } } d s \\\\ & \\leq C r \\sup _ { x \\geq t } \\left ( | \\lambda '' ( x ) | \\right ) \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} D ^ m [ f _ 1 \\circ f _ 2 ] ( x ) & = D f _ 1 ( f _ 2 ( x ) ) D ^ m f _ 2 ( x ) + D ^ m f _ 1 ( f _ 2 ( x ) ) \\left ( D f _ 2 ( x ) \\right ) ^ { \\otimes m } + \\mathcal { P } _ m ( f _ 1 , f _ 2 ) ( x ) \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} \\max _ { \\xi \\in ( 0 , 1 ) } ( 1 - \\xi ) \\xi ^ { \\frac { 1 } { p - 1 } } = \\frac { p - 1 } { p } p ^ { - \\frac { 1 } { p - 1 } } \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} \\partial _ { r } v _ { 4 } ( t , r ) = \\int _ { t } ^ { \\infty } d x \\int _ { 0 } ^ { \\infty } d \\xi \\xi J _ { 1 } ' ( r \\xi ) \\sin ( ( t - x ) \\xi ) \\widehat { v _ { 4 , c } } ( x , \\xi ) \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} H = g ( \\dot \\gamma , \\dot \\gamma ) \\ , , I _ i = I _ i ( \\dot \\gamma , \\dot \\gamma ) \\ , . \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} | y - r x | ^ 2 & = | r - r _ 0 | ^ 2 | x | ^ 2 + | y _ \\perp | ^ 2 \\gtrsim ( 1 - r ) ^ 2 | x | ^ 2 + | y _ \\perp | ^ 2 . \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} ( f | g ) & : = \\int _ a ^ b \\overline { f ( x ) } g ( x ) \\d x , \\\\ \\langle f | g \\rangle & : = \\int _ a ^ b f ( x ) g ( x ) \\d x = ( \\bar f | g ) . \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} \\Pi u ( z ) = \\langle ( I d - z M _ N ) ^ { - 1 } X _ N \\ , | \\ , Y _ N \\rangle _ { \\C ^ { N + 1 } } \\ , \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} \\tau ( t ) = \\Big ( \\tau _ 0 - \\int _ 0 ^ t \\frac { e ^ { \\widetilde { K } _ 0 ( s ) } } { \\sqrt { | \\Omega _ 0 | - e ^ { \\widetilde { K } _ 0 ( s ) } } } d s - \\int _ 0 ^ t \\frac { e ^ { \\widetilde { K } _ 0 ( s ) } } { | \\Omega _ 0 | } d s \\Big ) e ^ { - \\int _ 0 ^ t \\widetilde { K } _ 0 ( s ) d s } > 0 . \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} \\frac { \\mu ( a ) } { \\mu ( 0 ) } = \\frac { \\nu ( a ) } { \\nu ( 0 ) } \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} | A _ { r _ 1 } \\cap \\cdots \\cap A _ { r _ i } | = \\sum _ { G ' \\supseteq G } | U _ { G ' } | = \\sum _ { G ' \\supseteq G } \\left | \\bigcap _ { X \\in G ' } X \\setminus \\bigcup _ { X \\not \\in G ' } X \\right | . \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} & \\tilde { \\mathbf { u } } _ t + ( \\mathbf { v } \\cdot \\nabla ) \\tilde { \\mathbf { u } } = - ( \\tilde { \\mathbf { v } } \\cdot \\nabla ) \\mathbf { v } + \\tilde { w } ( Q \\ast \\mathbf { v } - \\mathbf { v } ) + w ( Q \\ast \\tilde { \\mathbf { u } } - \\tilde { \\mathbf { u } } ) . \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} f ( x , y , z , t ) = \\dfrac { 1 } { \\sqrt { ( 4 \\pi D _ \\theta t ) ^ 3 } } \\exp \\bigg ( - \\dfrac { ( x + G _ x - W _ x t ) ^ 2 + ( y + G _ y - W _ y t ) ^ 2 + ( z + G _ z - W _ z t ) ^ 2 } { 4 D _ \\theta t } \\bigg ) , \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} q _ 0 q _ i = p ^ 0 p ^ i = p ^ i q _ j = 0 \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} ( x - c _ { 0 } ) ^ { 1 / 3 } = \\begin{cases} e ^ { i \\pi / 3 } \\left | x - c _ { 0 } \\right | ^ { 1 / 3 } & x < c _ { 0 } , \\\\ \\left | x - c _ { 0 } \\right | ^ { 1 / 3 } & x > c _ { 0 } . \\end{cases} \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} \\begin{aligned} \\log \\det ( \\Delta \\ ! \\ ! \\restriction _ { | w _ \\pm | \\leq \\epsilon } ) = - \\frac 1 3 \\log \\epsilon & + \\frac { 2 \\beta + 1 } { 3 ( \\beta + 1 ) } \\log 2 + \\frac 1 { 3 ( \\beta + 1 ) } \\log c _ \\beta \\\\ & - 2 \\zeta ' _ R ( - 1 ) - \\frac 5 { 1 2 } - \\frac 1 2 \\log ( 2 \\pi ) + o ( 1 ) \\quad \\epsilon \\to 0 ^ + . \\end{aligned} \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { \\varepsilon \\searrow 0 } \\Theta ( \\varepsilon ) = \\Theta ( 0 ) . \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} { } _ h \\mathbf { P } _ { \\gamma } ^ x ( \\zeta _ h < \\infty , { } _ h W ^ { \\alpha , ( \\gamma ) } _ { \\zeta _ h - } = 0 ) = { } _ h \\mathbf { P } _ { \\gamma } ^ x ( \\zeta _ h < \\infty ) = \\mathbf { P } ^ x _ \\gamma ( \\sigma _ 0 < \\infty ) = 1 , \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} \\mathcal { E } ( \\tau , t _ { n } ) = u ( t _ { n + 1 } ) - \\Phi ^ \\tau _ { , 1 } ( u ( t _ { n } ) ) . \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} c _ { n } = b _ { 1 } a _ { n } + \\sum _ { j = 2 } ^ { n - 1 } b _ { j } \\left ( \\sum _ { k _ { 1 } + k _ { 2 } + \\cdots + k _ { j } = n } a _ { k _ { 1 } } a _ { k _ { 2 } } \\cdots a _ { k _ { j } } \\right ) + b _ { n } a _ { 1 } ^ { n } , n \\geq 4 , \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = c _ 1 t \\ | \\ V ( 0 ) = c _ 1 \\} = e ^ { - \\lambda t } \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} a _ { n + 1 } \\leq 1 + \\sum _ { i = 1 } ^ { n } a _ i . \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} h = h _ 0 + \\sum _ { j = 1 } ^ \\infty { h _ j ^ 1 } + \\sum _ { j = 1 } ^ \\infty { h _ j ^ 2 } , \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\xi _ { 4 } ^ { n } \\sum _ { a = 0 } ^ n \\xi _ { 1 2 } ^ { a } \\sum _ { b = 0 } ^ a \\xi _ { 3 } ^ { b } \\sum _ { c = 0 } ^ b \\xi _ { 1 2 } ^ { c } . \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } p _ { i , N } ( 1 - p _ { i , N } ) \\overset { P ( a s ) } { \\rightarrow } C \\quad C > 0 . \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} S h = A ^ { - 1 } S f + \\langle A S h | 1 \\rangle A ^ { - 1 } 1 \\ . \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} m ( x , \\omega ) = \\sqrt { | \\psi ( x ) | ^ 2 + | \\phi ( \\omega ) | ^ 2 } . \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} f = \\left ( a \\chi _ 0 + \\sum _ { j = 1 } ^ 3 b ^ j \\chi _ j + c \\chi _ 4 \\right ) + d . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} \\beta = \\frac { d \\lambda _ L } { d \\mu } ( 0 ) . \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\tilde { \\phi } ( a d _ { k } ( d ( a ) ) , a d _ { k } ( b ) ) = \\tilde { \\phi } ( d , a d _ { k } [ a , b ] ) , \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} r ' : = r _ 1 ' \\cdots r _ s ' \\in \\mathbb { N } . \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} \\frac { 1 } { | Q | } \\int _ { Q } \\left ( \\frac { | \\chi _ Q ( x ) | } { \\lambda _ 0 } \\right ) ^ { p ( x ) } \\ , \\d x = \\frac { 1 } { | Q | } \\int _ { Q } \\left ( | \\chi _ Q ( x ) | \\right ) ^ { p ( x ) } \\ , \\d x = 1 , \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\xi _ j ( \\mu ) = \\frac { 1 } { f _ j } \\left ( \\frac { g _ j } { f _ j } \\frac { \\partial f _ j } { \\partial y } - \\left ( \\frac { \\partial f _ j } { \\partial x } + \\frac { \\partial g _ j } { \\partial y } \\right ) + \\frac { f _ j } { g _ j } \\frac { \\partial g _ j } { \\partial x } \\middle ) \\right | _ { x = y = 0 } , \\\\ \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} F _ \\Phi ( v _ 1 , v _ 2 ) = \\bar { \\nabla } _ { \\rho ( v _ 1 ) } \\Phi ( v _ 2 ) - \\bar { \\nabla } _ { \\rho ( v _ 2 ) } \\Phi ( v _ 1 ) - \\Phi ( [ v _ 1 , v _ 2 ] _ V ) - T _ { \\nabla } ( \\Phi ( v _ 1 ) , \\Phi ( v _ 2 ) ) . \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} L _ { \\min } = L _ \\mathrm { c } ^ { \\# \\# } = L _ { \\max } ^ \\# . \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 ) = \\sum \\limits _ { j , k } \\sum _ { R = I \\times J \\times S \\in \\mathcal R ^ N _ { \\frak z } ( j , k ) } | R | \\widetilde \\psi _ { j , k } ( x _ 1 , x _ 2 , x _ 3 , x _ I , x _ J , x _ S ) \\ \\psi _ { j , k } * f ( x _ I , x _ J , x _ S ) , \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} \\max \\Big \\{ \\frak { h } _ \\mu ( \\beta , m ) : \\mu \\Big \\} = \\max _ { ( x _ 1 , \\cdots , x _ { m - 2 } ) \\in D _ { m , a } } f _ a ( x _ 1 , \\cdots , x _ { m - 2 } ) \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 \\} = \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{align*} h : = e ^ { H / 2 } W ^ * \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} \\left | p _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ' ( f ( t ) ) \\right | & = \\frac { | ( \\log R ) ' ( t ) | } { 2 \\pi \\beta | f ' ( t ) | } \\\\ & \\le \\frac { 1 } { 2 \\pi \\beta } \\frac { 1 } { 1 - \\frac { 2 R ( t ) \\log R ( t ) } { R ( t ) ^ { 2 } - 1 } } . \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} \\min _ { v \\in { \\mathcal { K } _ { \\psi } ( \\Omega ) } } \\int _ { \\Omega } G ( D v ) \\ , d x = \\max _ { \\sigma \\in S ^ { p ' } _ { - } ( \\Omega ) } \\left ( [ \\ ! [ \\sigma , D \\psi ] \\ ! ] _ { u _ 0 } ( \\overline { \\Omega } ) - \\int _ { \\Omega } G ^ * ( \\sigma ) \\ , d x \\right ) \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} \\frac { d \\mathbb { Q } _ { n , \\sigma } ' } { d \\mathbb { P } _ { n } } = \\exp \\left \\{ \\sum _ { i < j } \\left ( \\frac { 2 \\beta } { \\sqrt { n } } \\sigma _ { i } \\sigma _ { j } A _ { i , j } - \\frac { 2 \\beta ^ 2 } { n } \\right ) \\right \\} . \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} \\forall ( a , b ) \\in \\mathbb R ^ 2 \\colon \\varphi ( a , b ) = 0 \\ , \\Longleftrightarrow \\ , a , b \\geq 0 \\ , \\land \\ , a b = 0 \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} \\tilde { \\mbox { A u x } } \\left [ J \\right ] = \\begin{pmatrix} \\mbox { A u x } \\left [ J \\right ] & \\mbox { O } _ { N ^ { p - 1 } } & \\dots & \\mbox { O } _ { N ^ { p - 1 } } \\\\ \\mbox { O } _ { N ^ { p - 1 } } & \\mbox { A u x } \\left [ J \\right ] & \\dots & \\mbox { O } _ { N ^ { p - 1 } } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\mbox { O } _ { N ^ { p - 1 } } & \\mbox { O } _ { N ^ { p - 1 } } & \\dots & \\mbox { A u x } \\left [ J \\right ] \\end{pmatrix} . \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} \\begin{cases} \\bigl | { \\xi _ 2 } + \\frac { 1 } { 2 } N - \\frac { \\sqrt { 3 } } { 2 } ( \\sqrt { 2 } - 1 ) N \\bigr | \\leq 2 ^ { 1 2 } { A ' } ^ { - \\frac { 1 } { 2 } } N , \\\\ | \\eta _ 2 - ( \\sqrt { 2 } + 1 ) ^ { \\frac { 2 } { 3 } } \\xi _ 2 + ( \\sqrt { 3 } - 1 ) ( \\sqrt { 2 } - 1 ) ^ { \\frac { 1 } { 3 } } N | \\leq 2 ^ { 2 5 } { A ' } ^ { - 1 } N , \\end{cases} \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} h ( r ) = \\Phi ( r e ^ { i f ( r ) } ) = \\lim _ { n \\to \\infty } \\Phi _ { n } ( r e ^ { i f _ { n } ( r ) } ) = \\lim _ { n \\to \\infty } h _ { n } ( r ) , \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} \\overline { \\mathcal { O } _ { \\mathcal { G } } ( x ) } = H P ( X ) . \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} W ( f _ 1 , f _ 2 ; a ) & : = \\lim \\limits _ { d \\searrow a } W ( f _ 1 , f _ 2 ; d ) , \\\\ W ( f _ 1 , f _ 2 ; b ) & : = \\lim \\limits _ { d \\nearrow b } W ( f _ 1 , f _ 2 ; d ) , \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} g ' _ r ( t ) + 2 \\pi \\ , r \\omega ( r e ^ { 2 \\pi i t } ) [ i e ^ { 2 \\pi i t } ] g _ r ( t ) & = 0 & & , \\\\ g _ r ( 0 ) & = \\operatorname { i d } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} D ( t _ { s + 1 } ) & \\leq D ( t _ { s } ) \\prod _ { \\ell = t _ { s } + 1 } ^ { t _ { s + 1 } } a ( \\ell ) + \\sum _ { \\tau = t _ s } ^ { t _ { s + 1 } - 1 } b ( \\tau ) \\prod _ { \\ell = \\tau + 2 } ^ { t _ { s + 1 } } a ( \\ell ) \\cr & \\leq D ( t _ { s } ) \\lambda + c ( s ) , \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} \\alpha = \\left ( \\chi _ { { \\rm f o c u s } , L } - \\frac { \\sigma _ { { \\rm f o l d } , R } } { 3 } \\middle ) \\right | _ { \\mu = 0 } , \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} \\sigma [ \\theta , \\varphi ] = \\sin ( \\theta ) \\ , \\cos ( \\varphi ) \\ , \\sigma _ 1 + \\sin ( \\theta ) \\ , \\sin ( \\varphi ) \\ , \\sigma _ 2 + \\cos ( \\theta ) \\ , \\sigma _ 3 \\ , , \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} V ( D ) = \\left ( \\begin{array} { c c } e ^ { - D ^ - D ^ + } & e ^ { - \\frac { 1 } { 2 } D ^ - D ^ + } \\left ( \\frac { I - e ^ { - D ^ - D ^ + } } { D ^ - D ^ + } \\right ) D ^ - \\\\ e ^ { - \\frac { 1 } { 2 } D ^ + D ^ - } D ^ + & I - e ^ { - D ^ + D ^ - } \\end{array} \\right ) \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} \\beta _ { \\Lambda , \\Lambda _ { 0 } ; i , j } : = \\prod _ { x \\in \\Lambda \\setminus \\Lambda _ { 0 } } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * \\right ) = e ^ { \\sum _ { x \\in \\Lambda \\setminus \\Lambda _ { 0 } } H _ { x , i , j } } . \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} G _ \\bullet ( x , x - 0 ) - G _ \\bullet ( x , x + 0 ) & = 0 , \\\\ \\partial _ 2 G _ \\bullet ( x , x - 0 ) - \\partial _ 2 G _ \\bullet ( x , x + 0 ) & = 1 ; \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} \\Pi _ M ^ p x = \\underset { m \\in M } { \\mathrm { a r g \\ m i n } } \\ D _ p ( m , x ) \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} Y _ n = F ^ { ( 2 ) } _ { J , K } ( \\eta _ n , Y _ { n - 1 } ) , \\forall n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} \\Psi ( z ) & = \\gamma z \\exp \\left [ \\beta \\frac { \\eta _ { \\mu _ { 1 } } ( z ) + 1 } { \\eta _ { \\mu _ { 1 } } ( z ) - 1 } \\right ] \\\\ & = \\gamma z \\exp \\beta \\left [ 1 - \\frac { 2 } { 1 - \\eta _ { \\mu _ { 1 } } ( z ) } \\right ] \\\\ & = e ^ { - \\beta } \\gamma z \\exp \\left [ - 2 \\beta \\psi _ { \\mu _ { 1 } } ( z ) \\right ] . \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} \\frac { \\partial w ^ \\pm } { \\partial { x _ 1 } } = \\frac { 1 } { 2 } \\Big ( { U ^ \\pm } ' - \\frac { U ^ \\pm } { r } \\Big ) e ^ { 2 i \\theta } \\ , + \\ , \\frac { 1 } { 2 } \\Big ( { U ^ \\pm } ' + \\frac { U ^ \\pm } { r } \\Big ) , \\\\ [ 1 m m ] \\frac { \\partial w ^ \\pm } { \\partial { x _ 2 } } = \\frac { - i } { 2 } \\Big ( { U ^ \\pm } ' - \\frac { U ^ \\pm } { r } \\Big ) e ^ { 2 i \\theta } \\ , + \\ , \\frac { i } { 2 } \\Big ( { U ^ \\pm } ' + \\frac { U ^ \\pm } { r } \\Big ) . \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} \\begin{array} { c } \\left \\{ \\left ( i \\mu \\frac { \\partial } { \\partial x _ 1 } + \\frac { \\eta x _ 2 } { 2 \\mu } \\right ) ^ 2 - \\left ( i \\mu \\frac { \\partial } { \\partial x _ 2 } - \\frac { \\eta x _ 1 } { 2 \\mu } \\right ) ^ 2 \\right . \\\\ \\\\ \\left . - 4 8 \\exp \\left [ - 2 \\sqrt { 3 } \\left ( \\lambda x _ 2 + \\frac { i \\theta } { 2 \\lambda } \\frac { \\partial } { \\partial x _ 1 } \\right ) \\right ] \\right \\} \\psi ( x _ 1 , x _ 2 ) = 0 \\end{array} \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} \\left ( \\frac { ( r ^ { 2 } - 1 - ( s - t ) ^ { 2 } ) } { \\sqrt { \\beta } ( 1 + r ^ { 2 } + ( s - t ) ^ { 2 } + \\sqrt { \\beta } ) } \\right ) & = \\frac { - \\left ( 1 + \\frac { 1 - r ^ { 2 } } { ( s - t ) ^ { 2 } } \\right ) } { ( s - t ) ^ { 2 } } \\left ( \\frac { 1 } { \\sqrt { 1 + q } } \\cdot \\frac { 1 } { 1 + y + \\sqrt { 1 + q } } \\right ) \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} \\mathfrak { a } ^ { \\dagger } \\left [ \\mathfrak { a } \\left ( \\mathbb { K } _ { n } ^ { ( j ) } ( x ) \\right ) \\right ] = \\mathfrak { a } ^ { \\dagger } \\left [ \\Lambda _ { n } ^ { \\left ( j \\right ) } \\left ( x ; 1 , 1 \\right ) \\mathbb { K } _ { n - 1 } ^ { ( j ) } ( x ) \\right ] . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} \\Xi _ n ( \\tau _ i ) = \\widetilde { \\xi } _ { ( 1 , \\gamma _ { \\sigma _ i } / \\gamma _ { 1 ^ n } - 1 ) } + \\widetilde { \\xi } _ { ( s _ i , \\gamma _ { \\sigma _ i } / \\gamma _ { 1 ^ n } ) } = \\widetilde { \\xi } _ { 1 , \\frac { \\Delta _ { i , i + 1 } } { x _ { i + 1 } - x _ i } } + \\widetilde { \\xi } _ { s _ i , 1 + \\frac { \\Delta _ { i , i + 1 } } { x _ { i + 1 } - x _ i } } . \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} ( a Z , x ) \\cdot ( b Z , y ) = ( Z , [ a , b ] ) , \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} F ^ b ( x ) = \\psi _ { a b } ^ { - 1 } ( x ) F ^ a ( x ) \\psi _ { a b } ( x ) . \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} b & = \\tau ( g ) - g . \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} e _ { S _ { m _ i } } ( \\{ J ( m D ( 1 ) _ i ) \\} ; S _ { m _ i } ) = e _ { S _ { m _ i } } ( \\{ J ( m D ( 2 ) _ i ) \\} ; S _ { m _ i } ) \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) \\le \\beta \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k \\} = \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} N _ 0 ( n ) = [ n ] _ 2 ! , N _ 1 ( n ) = 2 ^ { n - 1 } [ n - 1 ] _ 2 ! , N _ 2 ( n ) = \\begin{cases} 1 & ( n \\le 2 ) \\\\ 2 ^ { n - 3 } ( 2 ^ n - 1 ) [ n - 2 ] _ 2 ! & ( n \\ge 3 ) . \\end{cases} \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} \\mathbf { E } \\le 2 ^ { 4 k + 2 } \\left ( C _ { 3 } k \\right ) ^ { C _ { 4 } k } \\sum _ { t = 1 } ^ { 2 k - 1 } \\left ( C _ { 1 } ( 4 k + 2 ) \\right ) ^ { C _ { 2 } } \\left ( \\frac { \\left ( 4 k + 2 \\right ) ^ { 6 } } { n } \\right ) ^ { 2 k - t } \\to 0 \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} J \\overline \\Psi ( t ) = \\left [ \\begin{array} { c c c } I _ s & 0 & 0 \\\\ \\phi _ 1 & \\phi _ 2 & \\phi _ 3 \\end{array} \\right ] , ~ ~ ~ J \\Phi ( t ) = \\left [ \\begin{array} { c c c } I _ s & 0 & 0 \\\\ 0 & I _ { r - s } & 0 \\\\ \\phi _ 1 & \\phi _ 2 & \\phi _ 3 \\end{array} \\right ] \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} \\frac { d x } { d t } = - \\frac { b ( t , x ) } { t } , x \\bigr | _ { t = t _ 0 } = x _ 0 . \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} f ( 0 , 0 ) = g ( 0 , 0 ) = 0 , \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} s _ 1 = \\frac { a + \\lambda _ 2 } { 1 + a \\lambda _ 2 } s _ 2 \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} \\left | \\langle \\Pi _ 0 ( \\partial _ { z _ j } ^ T ) , \\nabla _ { E _ i } E _ i \\rangle \\right | & = \\left | \\langle v ^ T , \\nabla _ { E _ i } E _ i \\rangle \\right | = \\left | \\langle \\Pi ( v ) , \\nabla _ { E _ i } E _ i \\rangle \\right | \\leq \\delta \\ , | v | \\ , | A | \\leq 2 \\ , \\delta \\ , | A | \\ , . \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} r _ \\alpha = \\omega ( r _ { \\alpha ^ c } ) = \\sum _ { \\beta \\succcurlyeq \\alpha ^ c } ( - 1 ) ^ { \\ell ( \\alpha ^ c ) - \\ell ( \\beta ) } e _ { \\widetilde \\beta } . \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} | J _ { 0 } + e ^ { - \\lambda \\tau } J _ { \\tau } - \\lambda I | = 0 \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( 1 / t ) = r e ^ { i f ( r ) } , \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} L _ { s , i } : = [ 0 , \\dots , \\hat { i } , \\dots , s ] ; L _ { s } : = L _ { s , s } ; q _ { s , i } : = L _ { s , i } / L _ { s } , \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} \\Big \\{ [ \\xi _ a ] ^ * _ i [ \\xi _ b ] ^ * _ i = 0 \\Big \\} \\mbox { \\ o r \\ } \\Big \\{ [ \\xi _ a ] ^ * _ i \\leq 0 \\mbox { a n d } [ \\xi _ b ] ^ * _ i \\leq 0 \\Big \\} \\forall i \\in \\beta , \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} e ^ { ( 1 + z ) u } & = e ^ { \\partial _ { z } } e ^ { z u } = e ^ { z u } e ^ { u } . \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} P g : = \\sum _ { \\varphi _ { i } \\in \\Phi } \\langle \\varphi _ { i } , g \\rangle \\ , \\varphi _ { i } . \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} n _ 1 \\in \\bigcap _ { \\ell = 0 } ^ d B _ 1 ^ \\ell \\in p . \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} f \\left ( \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } { { a } _ { i } } } \\right ) \\le \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( { { a } _ { i } } \\right ) } \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} B ( e , f ) = B ( e , f ^ 2 ) = f B ( e , f ) + B ( e , f ) f = f B ( e , f ) . \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} P ^ { * } _ { X } = \\frac { A _ { X } ^ { b } } { A _ { \\cal F } ^ { b } + A _ { \\cal R } ^ { b } } P _ { t o t } , X \\in \\{ \\cal F , \\cal R \\} , \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} \\mathcal { A } ' _ k & = \\{ A \\subseteq [ n ] : \\ | A | = 2 k + 1 \\} , \\\\ \\Gamma ' _ { A , l } & = \\big \\{ \\gamma \\in \\N _ 0 ^ n : \\sum _ { i \\in A } \\gamma _ i = l - k \\ \\gamma _ i = 0 i \\in A ^ c \\big \\} , \\\\ B ' _ { A , l } & = \\big \\{ \\beta \\in \\N _ 0 ^ n : \\sum _ { i \\in A } \\beta _ i = ( m - 1 ) / 2 - l \\ \\beta _ i = 0 i \\in A \\big \\} . \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} s ^ { ( \\alpha ) } ( { \\bf { x } } ; \\theta ) : = \\nabla \\ln p ^ { ( \\alpha ) } _ \\theta ( { \\bf { x } } ) = \\nabla \\ln \\frac { p _ \\theta ( { \\bf { x } } ) ^ \\alpha } { \\| p _ \\theta \\| ^ \\alpha } = \\nabla \\big [ \\ln p _ \\theta ( { \\bf { x } } ) ^ \\alpha - \\ln \\| p _ \\theta \\| ^ \\alpha \\big ] = \\alpha \\big [ s ( { \\bf { x } } ; \\theta ) - \\nabla \\ln \\| p _ \\theta \\| \\big ] . \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} \\begin{aligned} R _ c & = 1 6 , & r & = 1 , & h & = 1 , \\\\ k & = 0 . 4 5 , & \\delta & = 0 . 3 6 , & K & = 5 0 , \\end{aligned} \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} b ' _ k & = b _ k \\forall \\ ; k < s - 1 \\\\ b ' _ { s - 1 } & = b _ { s - 1 } + \\frac { \\lambda _ s } { x _ { 1 + \\ldots + ( s - 1 ) } } . \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} \\hat { h } _ { s z } ( 0 , s _ 0 ) = - c ' ( s _ 0 ) h _ { z z } ( 0 , s _ 0 ) + c ' ( s _ 0 ) h _ z ( 0 , s _ 0 ) + h _ { s z } ( 0 , s _ 0 ) . \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} \\langle A \\tilde x , P _ 1 ^ { - 1 } \\mathcal { H } x \\rangle = \\langle P _ 1 ^ { - 1 } \\mathcal { H } \\tilde x , A ^ \\star x \\rangle , \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} & { } _ 3 \\phi _ 2 \\ ! \\left [ \\begin{matrix} ( 1 - r ) ^ j , b , c \\\\ d , e \\end{matrix} ; 1 - r , 1 - r \\right ] \\\\ & = \\frac { ( ( 1 - r ) / e ; 1 - r ) _ j } { ( ( 1 - r ) b c / d e ; 1 - r ) _ j } \\ ; { } _ 3 \\phi _ 2 \\ ! \\left [ \\begin{matrix} ( 1 - r ) ^ j , d / b , d / c \\\\ d , d e / b c \\end{matrix} ; 1 - r , 1 - r \\right ] . \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} \\widehat { q } ( z ) = \\mathcal { Z } \\big ( q \\big ) ( z ) = \\sum \\limits _ { k = 0 } ^ { \\infty } q _ k z ^ { - k } , \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} P ^ { v } ( x ) = \\tfrac { 1 } { 6 } \\left ( h ( x , v ) ^ { 3 } - \\tfrac { 3 } { n + 2 } | x | ^ { 2 } | v | ^ { 2 } h ( x , v ) \\right ) . \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} d Y ( t ) = \\bar { b } ( Y ( t ) ) \\ , d t + \\bar { \\sigma } ( Y ( t ) ) \\ , d B ( t ) \\ , , \\ , \\ , Y ( 0 ) = F ( x ) \\ , , \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} \\| S h \\| = \\| S \\| \\mbox { a n d } \\langle S h , T h \\rangle = 0 . \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} T ( \\tilde { r } ; \\mu ) = T _ R ( \\tilde { r } ; \\mu ) + T _ L \\left ( P _ R ( \\tilde { r } ; \\mu ) ; \\mu \\right ) . \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} a _ 0 ( i a _ i a _ j ' - j a _ j a _ i ' ) = ( i n _ j - j n _ i ) a _ i a _ j a _ 0 ' . \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} \\left | \\int _ 0 ^ a e ^ { i u ^ 2 } G _ 1 ( \\epsilon u ) d u \\right | \\le C _ { ( g ) } ( \\epsilon + \\epsilon | a | ) = C _ { ( g ) } ( \\epsilon + \\nu ) \\ , . \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} d ( \\lambda _ { n - 1 } ) & = \\sum _ { i + j = n } \\binom { i + j } { i } \\lambda _ { i - 1 } \\lambda _ { j - 1 } , \\\\ d ( \\mu _ { n - 1 } ) & = \\sum _ { i + j = n } \\binom { i + j } { i } ( \\lambda _ { i - 1 } \\mu _ { j - 1 } - \\mu _ { i - 1 } \\lambda _ { j - 1 } ) , \\\\ d ( \\sigma \\tau ) & = ( - 1 ) ^ { \\deg \\sigma } \\sigma d ( \\tau ) + d ( \\sigma ) \\tau . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\zeta _ t } ( z ) = \\frac { t ^ { z \\alpha } } { \\phi _ { \\alpha } ' ( 0 ^ + ) } \\frac { \\Gamma ( z ) } { W _ { \\phi _ { \\alpha } } ( z ) } . \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} e ^ { \\frac { r _ i ' } { r } \\mathbf { i } ( q ' - q ) ^ t \\mathcal { B } _ i } = 1 \\ ( i = 1 , \\cdots , s ) , \\\\ \\ e ^ { \\frac { \\mathbf { i } } { r } ( q ' - q ) ^ t \\mathcal { B } _ i } = 1 \\ ( i = s + 1 , \\cdots , n ) . \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} H ( \\nu ; r _ 1 | r _ 2 ) = H ( \\nu ; r _ 1 ) - H ( \\nu ; r _ 2 ) , \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} \\{ B _ { a , b } ^ { ( j ) } \\mid a = 1 , \\ldots , n _ j - \\delta , \\ , b = 1 , \\ldots , m _ j \\} \\cup \\{ B _ { a , b } ^ { ( i ) } \\mid i = j + 1 , \\ldots , t , \\ , a = 1 , \\ldots , n _ i , \\ , b = 1 , \\ldots , m _ i \\} \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} H _ m : = ( H _ { 1 m } \\ , H _ { 2 m } ) { \\rm a n d } G _ m : = \\left ( \\ ! \\ ! \\ ! \\begin{array} { c } G _ { 1 m } \\\\ G _ { 2 m } \\end{array} \\ ! \\ ! \\ ! \\right ) . \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} \\max _ { 1 \\le n \\le N } \\| e _ h ^ { u ^ n } \\| ^ 2 _ { \\mathcal { T } _ h } + \\Delta t \\sum _ { n = 1 } ^ { N } \\| e _ h ^ { \\phi ^ n } \\| ^ 2 _ { \\mathcal { T } _ h } + \\le C ( ( \\Delta t ) ^ 2 + h ^ { 2 ( k + 2 ) } ) . \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} D _ i ^ { \\prime } = \\bigoplus _ { \\substack { i + 1 \\leq j \\leq n - 1 \\\\ 1 \\leq k \\leq j - i } } D _ j ^ { ( k ) } , D _ i ^ { + } = \\bigoplus _ { \\substack { i + 2 \\leq j \\leq n - 1 \\\\ 2 \\leq k \\leq j - i } } D _ j ^ { ( k ) } , D _ i ^ { - } = \\bigoplus _ { \\substack { i + 2 \\leq j \\leq n - 1 \\\\ 1 \\leq k \\leq j - i - 1 } } D _ j ^ { ( k ) } . \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} S ^ * L _ u S = L _ u + I d \\ , . \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{gather*} A \\mathbf 1 = \\mathbf 1 A = A \\end{gather*}"}
-{"id": "5048.png", "formula": "\\begin{align*} \\lambda ( t ) = \\lambda _ { 0 } ( t ) + e _ { 0 } ( t ) , | | e _ { 0 } | | _ { X } \\leq 1 \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} b = \\frac { 1 } { 1 - \\lambda } ( c - d ) , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , a = \\frac { 1 } { 1 - \\lambda } ( d - \\lambda c ) . \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu ( B ) } \\int _ { B } e ^ { \\varphi ( x ) - \\avg { \\varphi } _ { B } } \\ , d \\mu \\geq e ^ { \\frac { 1 } { \\mu ( B ) } \\int _ { B } \\varphi ( x ) - \\avg { \\varphi } _ { B } \\ , d \\mu } = 1 \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} F ( z ) = \\int _ { \\R } h ( x ) e ^ { z x } \\eta ( x ) \\ , d x + \\int _ { \\R } h ' ( x ) z e ^ { z x } \\psi ( x ) \\ , d x . \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} \\Psi ( z ) = z G ( z ) = F ( \\eta _ { \\mu _ { 1 } } ( z ) ) = \\gamma z \\exp \\left [ \\beta u ( z ) \\right ] , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} ( h ^ { 2 } + & 2 h - \\mu ) ( h + 4 ) ^ { n } = h ^ { n + 2 } - 2 ( 2 n + 1 ) h ^ { n + 1 } \\\\ & + \\sum _ { k = 0 } ^ { n - 2 } \\bigg ( - \\mu { n \\choose k } ( 4 ) ^ { k } + 2 { n \\choose k + 1 } ( 4 ) ^ { k + 1 } \\\\ & + { n \\choose k + 2 } ( 4 ) ^ { k + 2 } \\bigg ) h ^ { n - k } \\\\ & + ( - \\mu n ( 4 ) ^ { n - 1 } + 2 ( 4 ) ^ { n } ) h - \\mu ( 4 ) ^ { n } . \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} F ( \\overline { z } ) = \\overline { F ( z ) } , z \\in \\mathcal { S } _ { 1 } . \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} X _ { 1 + 2 } & = \\beta _ 0 + \\beta _ 1 \\left ( k _ 1 + k _ 2 \\right ) _ \\mu \\left ( k _ 1 + k _ 2 \\right ) ^ \\mu + \\beta _ 2 \\left ( \\left ( k _ 1 + k _ 2 \\right ) _ \\mu \\left ( k _ 1 + k _ 2 \\right ) ^ \\mu \\right ) ^ 2 + \\ldots . \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\frac { p _ { \\nu _ { 1 } \\boxtimes \\nu _ { 2 } } ( \\xi ) } { p _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( \\xi ) } = \\frac { | 1 - \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } } { \\beta } \\int _ { \\mathbb { T } } \\frac { d \\sigma ( t ) } { | t - \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } } , \\quad \\xi \\in \\Gamma . \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} & \\int _ { - \\infty } ^ \\infty \\frac { \\left ( b q ^ x , a q ^ { - x } ; p \\right ) _ \\infty } { \\left ( - q ^ x , - q ^ { 1 - x } ; q \\right ) _ \\infty } \\ , e ^ { i x y } \\ , d x \\\\ & = \\frac { 2 \\pi i / \\log q } { \\sinh \\frac { \\pi y } { \\log q } } \\frac { ( - q , - q , e ^ { i y } , q e ^ { - i y } ; q ) _ \\infty } { ( q , q , - e ^ { i y } , - q e ^ { - i y } ; q ) _ \\infty } \\sum _ { n = - \\infty } ^ \\infty \\frac { ( b q ^ n , a q ^ { - n } ; p ) _ \\infty } { ( - q ^ n , - q ^ { 1 - n } ; q ) _ \\infty } \\ , e ^ { i n y } . \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} x = \\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix} , h = \\begin{bmatrix} 1 & 0 \\\\ 0 & - 1 \\end{bmatrix} , y = \\begin{bmatrix} 0 & 0 \\\\ 1 & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} V _ j \\llcorner M \\setminus B ^ M _ { s _ j } ( S i n g ( \\Sigma ) ) = | g r a p h _ { \\Sigma } ( v _ j ) | \\llcorner M \\setminus B ^ M _ { s _ j } ( S i n g ( \\Sigma ) ) \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} \\begin{cases} \\Phi ' ( x - \\underline h ( t ) ) < - \\epsilon _ 1 \\mathbf { 1 } & \\mbox { f o r } x \\in [ \\underline h ( t ) - K _ 0 , \\underline h ( t ) ] , \\\\ \\Phi ' ( - x - \\underline h ( t ) ) < - \\epsilon _ 1 \\mathbf { 1 } & \\mbox { f o r } x \\in [ - \\underline h ( t ) , - \\underline h ( t ) + K _ 0 ] . \\end{cases} \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} \\Lambda ( \\mu ) = \\frac { m _ 2 ( \\mu ) m _ 4 ( \\mu ) } { m _ 1 ( \\mu ) m _ 3 ( \\mu ) } , \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} M \\int _ M ^ \\infty J ( y ) d y \\leq \\min \\left \\{ \\int _ 0 ^ \\infty J ( y ) , M _ 2 \\right \\} : = M _ 3 < \\infty \\mbox { f o r a l l } M > 0 . \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} I ( z ) = \\gamma + \\int _ { \\mathbb { R } } \\frac { z + t } { 1 - t z } \\ , d \\sigma ( t ) , z \\notin \\mathbb { R } , \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} | \\Lambda _ { \\lambda , \\lambda ' } ( \\psi ) | & = \\left | \\int _ { G / H } \\int _ { G / H } \\left ( \\int _ H \\psi ( x h y H ) d h \\right ) d \\lambda ( x H ) d \\lambda ' ( y H ) \\right | \\\\ & \\le \\int _ { G / H } \\int _ { G / H } \\left | \\int _ H \\psi ( x h y H ) d h \\right | d | \\lambda | ( x H ) d | \\lambda ' | ( y H ) \\\\ & \\le \\int _ { G / H } \\int _ { G / H } \\int _ H \\left | \\psi ( x h y H ) \\right | d h d | \\lambda | ( x H ) d | \\lambda ' | ( y H ) \\le \\| \\psi \\| _ { \\sup } \\| \\lambda \\| \\| \\lambda ' \\| . \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} \\ * r ( \\ * x , t ) : = \\ * r _ t ( \\ * x ) = \\ * r _ 0 ( \\ * x ) + t \\ , u ( \\ * x ) \\ * N , \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{gather*} g ( k ) = g ( l n _ \\varepsilon + m ) \\le \\underbrace { g ( n _ \\varepsilon ) g ( n _ \\varepsilon ) \\dots g ( n _ \\varepsilon ) } _ { l } \\cdot g ( m ) = [ g ( n _ \\varepsilon ) ] ^ l g ( m ) . \\end{gather*}"}
-{"id": "5121.png", "formula": "\\begin{align*} & | \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) \\partial _ { 2 } K ( s - t , \\lambda ( t ) ) \\lambda ' ( t ) | \\leq \\frac { C } { t ^ { 3 } \\log ^ { 2 b + 2 } ( t ) } \\int _ { t } ^ { \\infty } d s | \\partial _ { 2 } K ( s - t , \\lambda ( t ) ) | \\\\ & \\leq \\frac { C } { t ^ { 3 } \\log ^ { 3 b + 2 } ( t ) } \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} \\bar { \\eta } ^ v \\rightharpoonup \\eta _ * * L _ t ^ { \\infty } ( L _ { \\theta } ^ 2 ) = ( L _ t ^ 1 ( L _ { \\theta } ^ 2 ) ) ^ * . \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} \\mbox { A u x } = \\begin{pmatrix} 1 & \\dots & 1 & \\dots & 1 \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\ 1 & \\dots & 1 & \\dots & 1 \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\ 1 & \\dots & 1 & \\dots & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} ( \\mu _ { \\varphi } ) _ q = \\sigma _ { \\varphi _ q } . \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} u _ 1 & = \\sin ( \\phi _ 2 + \\phi _ 3 ) w \\\\ u _ 2 & = \\sin \\phi _ 2 \\cdot w \\\\ u _ 3 & = \\sin \\phi _ 3 \\cdot w \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} V ^ { l _ { 1 } } U ^ { k _ { 1 } } . \\left ( \\Phi \\otimes ( z ^ { 0 } \\otimes \\varepsilon _ { 0 } ) \\right ) = \\Phi \\otimes ( \\omega _ { 1 } \\rtimes u ) ( V ^ { l _ { 1 } } U ^ { k _ { 1 } } ) \\left ( z ^ { 0 } \\otimes \\varepsilon _ { 0 } \\right ) . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} | v _ { 5 } ( t , r ) | \\leq \\begin{cases} \\frac { C r } { t ^ { 7 / 2 } \\log ^ { 3 b - 3 + \\frac { 5 N } { 2 } } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C \\log ^ { 4 } ( t ) } { \\sqrt { r } t ^ { 3 / 2 } } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} X _ i ' : = \\bigwedge _ { j \\succ i } f _ { j i } ^ \\kappa ( X _ j ) . \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} T ^ 4 _ { 1 3 } = T ^ 4 _ { 2 3 } = T ^ 4 _ { 3 4 } = \\bar T ^ 4 _ 3 = T ^ 4 _ 3 = R ^ 4 _ 3 = 0 . \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\sum _ { a = 0 } ^ n \\sum _ { b = 0 } ^ a \\sum _ { c = 0 } ^ b \\xi _ 3 ^ { n + b } = \\xi _ 3 ^ { n } \\sum _ { a = 0 } ^ n \\sum _ { b = 0 } ^ a \\xi _ 3 ^ { b } ( b + 1 ) . \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} \\lambda _ n ( G ^ c ) & = x ^ T D ( G ^ c ) x \\\\ & = x ^ T ( J _ n - I _ n ) x + x ^ T A ( G ) x \\\\ & \\ge x ^ T ( J _ n - I _ n ) x + x ^ T A ( T _ 1 ( n - 4 , 1 ) ) x \\\\ & = x ^ T D ( T _ 1 ^ c ( n - 4 , 1 ) ) x . \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} z - 1 = \\begin{cases} \\left | z - 1 \\right | e ^ { i \\pi / 2 } e ^ { i \\tau / 2 } & z = e ^ { i \\tau } , \\ ; 0 < \\tau < \\pi / 4 , \\\\ \\left | z - 1 \\right | e ^ { i 3 \\pi / 2 } e ^ { i \\tau / 2 } & z = e ^ { i \\tau } , \\ ; 7 \\pi / 4 < \\tau < 2 \\pi . \\end{cases} \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} M : = \\left [ I - Y _ k Z _ k - \\gamma ^ 2 T _ k ( I - Z _ k Y _ k ) ^ { - 1 } S _ k \\right ] ^ { - 1 } \\equiv K ( I - Y _ k Z _ k ) ^ { - 1 } \\ge 0 . \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } \\| \\partial _ k \\omega ( t ) \\| _ { L ^ 2 } ^ 2 + \\| \\partial _ 1 \\partial _ k \\omega \\| _ { L ^ 2 } ^ 2 & = \\int _ { \\R ^ 2 } \\partial _ 1 \\partial _ k \\theta \\partial _ k \\omega ~ d x - \\int _ { \\R ^ 2 } \\partial _ k u \\cdot \\nabla \\omega \\partial _ k \\omega ~ d x \\\\ & \\triangleq N _ 1 + N _ 2 . \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} \\left | \\frac { \\partial } { \\partial y _ l } \\bar { H } _ { } ( \\mathbf { 0 } ) \\right | & = \\left | \\frac { 1 } { N } \\mathbb { E } _ { \\mu _ { } ( d x | 0 ) } \\left [ \\sum _ { i , j \\in B ( l ) } M _ { i j } X _ i + \\sum _ { i \\in B ( l ) } \\delta \\psi ' ( X _ i ) \\right ] \\right | \\\\ & \\overset { L e m m a ~ \\ref { l _ c e _ m o m e n t _ e s t i m a t e } } { \\lesssim } \\frac { 1 } { N } \\left ( 2 K R ^ 2 + K \\right ) \\sim \\frac { 1 } { M } . \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} \\overline D \\psi - D \\psi & = \\sum _ { i = 1 } ^ n e _ i \\cdot ( \\overline \\nabla _ { e _ i } - \\nabla _ { e _ i } ) \\psi \\\\ & = - \\frac 1 2 \\sum _ { i = 1 } ^ n e _ i \\cdot e _ 0 \\cdot W ( e _ i ) \\cdot \\psi \\\\ & = \\phantom { - } \\frac 1 2 \\sum _ { i , j = 1 } ^ n g ( W ( e _ i ) , e _ j ) e _ i \\cdot e _ j \\cdot e _ 0 \\cdot \\psi \\\\ & = - \\frac 1 2 \\sum _ { i = 1 } ^ n g ( W ( e _ i ) , e _ i ) e _ 0 \\cdot \\psi . \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} \\widetilde { d } ( x , y ) = \\sum ^ \\infty _ { n = 0 } \\frac { 1 } { 2 ^ n } \\frac { d ( x _ n , y _ n ) } { 1 + d ( x _ n , y _ n ) } \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } \\frac { ( \\lambda ' ( t ) ) ^ { 2 } } { r \\lambda ( t ) } \\left ( \\mathcal { F } ^ { - 1 } ( y _ { 0 } ( t , \\frac { \\cdot } { \\lambda ( t ) ^ { 2 } } ) ) ( \\frac { r } { \\lambda ( t ) } ) \\right ) ^ { 2 } r d r = ( \\lambda ' ( t ) ) ^ { 2 } \\lambda ( t ) ^ { 2 } \\int _ { 0 } ^ { \\infty } | y _ { 0 } ( t , \\omega ) | ^ { 2 } \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega \\\\ & \\leq C \\frac { \\lambda ' ( t ) ^ { 2 } \\lambda ( t ) ^ { 2 } } { t ^ { 4 } \\log ^ { \\epsilon } ( t ) } \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} R _ \\nabla ( q _ 1 , q _ 2 ) ( v ) = \\nabla _ { q _ 1 } \\nabla _ { q _ 2 } v - \\nabla _ { q _ 2 } \\nabla _ { q _ 1 } v - \\nabla _ { [ q _ 1 , q _ 2 ] _ Q } v , \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} C _ 4 = 0 . \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} \\Psi ( \\lambda ) = d e t ( I _ 5 - D _ { L '^ c } ) = & { \\lambda } ^ { 5 } - \\left ( 2 \\ , n - 1 0 \\right ) { \\lambda } ^ { 4 } \\\\ & - \\left ( - 2 8 + 1 0 \\ , n \\right ) { \\lambda } ^ { 3 } - 1 0 \\ , n { \\lambda } ^ { 2 } - \\left ( - 4 \\ , n + 4 8 \\right ) \\lambda . \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} \\mathcal { T } _ k & = \\begin{cases} \\mathcal { T } _ { k - 1 } , & \\mbox { i f } \\mathcal { T } _ { k - 1 } \\in \\mathbb { T } \\\\ \\mathcal { T } _ { k - 1 } \\cup \\{ u _ k \\} , & \\mbox { i f } \\mathcal { T } _ { k - 1 } \\not \\in \\mathbb { T } \\end{cases} , \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} \\| u \\| _ { q + \\alpha } ^ { q + \\alpha } = \\| u \\| _ { q + \\alpha } ^ { p } \\| u \\| _ { q + \\alpha } ^ { q + \\alpha - p } \\le B _ \\alpha ^ { q + \\alpha } \\| \\nabla u \\| _ p ^ p \\Big ( \\frac { p q } { q - p } J ( u _ 0 ) \\Big ) ^ { \\frac { q + \\alpha - p } { p } } . \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} \\Psi ( \\tau ) = \\varphi _ 1 \\big ( \\tau \\mathcal { A } \\big ) . \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} h ( \\tilde { r } ) = \\lambda _ R T _ R ( \\tilde { r } ; 0 ) + \\lambda _ L T _ L \\left ( P _ R ( \\tilde { r } ; 0 ) ; 0 \\right ) . \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} \\int _ { M } \\left ( R - \\frac { 1 } { 2 } T o r - \\frac { 1 } { 2 } T o r ^ { \\prime } \\right ) \\left ( \\gamma , \\gamma \\right ) d \\mu + \\int _ { M } \\left \\vert \\gamma _ { 1 , 1 } \\right \\vert ^ { 2 } d \\mu + \\frac { 3 } { 4 } \\int _ { M } Q u d \\mu + \\frac { 1 } { 2 } \\int _ { M } \\left ( P u \\right ) u d \\mu = 0 . \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t t } u _ { 4 , s } - \\Delta u _ { 4 , s } = 0 \\\\ u _ { 4 , s } ( s , x ) = 0 \\\\ \\partial _ { t } u _ { 4 , s } ( s , x ) = - \\int _ { | x | } ^ { \\infty } v _ { 4 , c } ( s , q ) d q \\end{cases} \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} \\langle \\tilde { L } u , \\varphi \\rangle = \\langle L u , \\varphi \\rangle , \\varphi \\in C _ c ^ \\infty ( D ) . \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} a _ { n } = ( - 1 ) ^ { n } \\int _ { \\mathbb { R } } \\frac { d \\mu _ { 1 } ( t ) } { ( \\alpha - t ) ^ { n + 1 } } , n = 0 , 1 , 2 , \\cdots , 2 k - 1 , \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} \\int \\limits _ 0 ^ t G ' ( X _ { s - } ) d X _ s & = \\int \\limits _ 0 ^ t G ' ( X _ { s - } ) \\mu ( X _ { s - } ) d s + \\int _ 0 ^ t G ' ( X _ { s - } ) \\sigma ( X _ { s - } ) d W _ s \\\\ & \\quad + \\int \\limits _ 0 ^ t \\int \\limits _ { \\mathbb { R } } G ' ( X _ { s - } ) \\rho ( X _ { s - } , y ) \\nu ( d y , d s ) . \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} H ( \\Psi ; \\gamma ) & = \\prod _ { ( i , j ) \\in \\Delta ^ + \\setminus \\Psi } ( 1 - { R } _ { i j } ) h _ \\gamma \\ , , \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} 1 & = - b _ 2 b _ 1 + b _ 2 ^ 2 + a _ 2 ^ 2 + b _ 1 a _ 2 + b _ 1 ^ 2 + a _ 1 ^ 2 + a _ 1 a _ 2 + a _ 1 b _ 2 \\\\ & = \\left ( \\frac { a _ 1 + a _ 2 } { \\sqrt { 2 } } \\right ) ^ 2 + \\left ( \\frac { a _ 2 + b _ 1 } { \\sqrt { 2 } } \\right ) ^ 2 + \\left ( \\frac { b _ 1 - b _ 2 } { \\sqrt { 2 } } \\right ) ^ 2 + \\left ( \\frac { b _ 2 + a _ 1 } { \\sqrt { 2 } } \\right ) ^ 2 . \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} K _ { 2 3 } ( t , x , y ) | _ { t = \\log \\left ( \\frac { 1 + s } { 1 - s } \\right ) } & \\lesssim ( 1 - s ) ^ { n + 1 } \\frac { | x + y | + | x - y | } { { s } ^ { ( n - 1 ) / 2 } } \\\\ & \\times \\exp \\Big ( - \\frac { 1 } { 8 } ( s | x + y | ^ 2 + \\frac { 1 } { s } | x - y | ^ 2 ) \\Big ) \\ , e ^ { - ( | y | ^ 2 - | x | ^ 2 ) / 4 } \\ , e ^ { | y | ^ 2 - | x | ^ 2 } \\\\ & \\lesssim \\frac { ( 1 - s ) ^ { n } } { { s } ^ { n / 2 } } \\exp \\Big ( - \\frac { | ( 1 + s ) y - ( 1 - s ) x | ^ 2 } { 1 6 s } \\Big ) \\ , e ^ { | y | ^ 2 - | x | ^ 2 } , \\quad ( x , y ) \\in N ^ c . \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} \\mathcal C & = ( h + 1 ) ^ 2 + 4 f e = 4 \\left ( \\left ( L _ 0 - \\frac { 1 } { 2 } \\right ) ^ 2 - L _ { - 1 } L _ 1 \\right ) = \\\\ & = ( h - 1 ) ^ 2 + 4 e f = 4 \\left ( \\left ( L _ 0 + \\frac { 1 } { 2 } \\right ) ^ 2 - L _ { 1 } L _ { - 1 } \\right ) . \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} C ^ \\infty _ M = C ^ \\infty _ { M , ( 0 ) } \\supset C ^ \\infty _ { M , ( 1 ) } \\supset \\cdots \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} z ^ { 4 } + p z ^ { 3 } + q z ^ { 2 } + u z + v = 0 . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} Q _ a ( x , y ) : = \\begin{cases} ( x - y ) V ( y ) , & x > y > a , \\\\ 0 & \\end{cases} \\\\ T _ a ( x , y ) : = \\begin{cases} ( x - y ) , & x > y > a , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} \\Theta _ { 2 } & \\leq C 2 ^ { - 2 ( 1 + s ) q } b _ q \\| \\partial _ 1 \\theta \\| _ { L ^ \\infty } \\| \\omega \\| _ { H ^ s } \\| \\theta \\| _ { H ^ { 1 + s } } \\\\ & \\quad + C 2 ^ { - 2 ( 1 + s ) q } b _ q ( \\| \\omega \\| _ { H ^ s } + \\| \\theta \\| _ { H ^ { 1 + s } } ) \\| \\partial _ 1 \\theta \\| _ { H ^ { 1 + s } } . \\\\ \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{cases} \\begin{bmatrix} \\lambda _ L x + y \\\\ - x + \\lambda _ L y \\end{bmatrix} , & x < 0 , \\\\ \\begin{bmatrix} \\lambda _ R x + y + \\mu \\\\ - x + \\lambda _ R ( y + \\mu ) \\end{bmatrix} , & x > 0 . \\end{cases} \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} & U _ k ' \\cdot \\varphi ^ { r _ 1 ' \\cdots , r _ s ' } ( X _ 1 , \\cdots , X _ n , Y _ 1 , \\cdots , Y _ n ) \\\\ & = \\varphi ^ { r _ 1 ' \\cdots r _ s ' } ( X _ 1 , \\cdots , X _ n , Y _ 1 , \\cdots , Y _ k + 2 \\pi , \\cdots , Y _ n ) \\cdot U _ k ' . \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} \\eta _ { \\mu _ { n } } = \\eta _ { \\nu _ { n } } ^ { \\langle - 1 \\rangle } \\circ \\eta _ { \\nu _ { n } ^ { \\boxtimes k _ { n } } } \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} a _ n = ( n + 1 ) v _ { n + 1 } = \\frac { 2 \\pi ^ { \\frac { n + 1 } { 2 } } } { \\Gamma ( \\frac { n + 1 } { 2 } ) } . \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} \\mathcal Q ( g ) ( t ) = \\int _ 0 ^ t S ( t - s ) g ( s ) d s . \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} \\Delta ( w , \\eta ) = \\frac { e ^ { - \\zeta ' ( - 1 ) } \\cdot G ( w + \\eta + 1 ) \\cdot e ^ { - \\frac { w ^ 2 } { 4 } + \\frac { \\eta ^ 2 } { 2 } - \\frac { \\eta } { 2 } + \\frac { 1 } { 1 2 } } } { \\Gamma ( w + \\eta ) ^ { w + \\eta } \\cdot w ^ { - \\frac { ( w + \\eta ) ^ 2 } { 2 } + \\frac { w + \\eta } { 2 } - \\frac { 1 } { 1 2 } } } , \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} H ^ 0 _ { \\mathrm { H o c h } } ( M , A ) = \\{ a \\in A | ~ a T ( u ) - T ( u ) a + T H ( T u , a ) - T H ( a , T u ) = T ( a \\cdot u - u \\cdot a ) , ~ \\forall u \\in M \\} . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} \\sum _ { \\mathsf { m = 1 } } ^ { + \\infty } \\sum _ { \\mathsf { n = 1 } } ^ { + \\infty } r _ { \\mathsf { m , n } } ^ { 2 } = Z _ { \\left ( \\alpha - 1 \\right ) \\beta } Z _ { \\left ( \\alpha + 1 \\right ) \\beta } < + \\infty \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} A ^ b ( x ) = \\psi _ { a b } ^ { - 1 } ( x ) A ^ a ( x ) \\psi _ { a b } ( x ) + \\psi _ { a b } ^ { - 1 } ( x ) d \\psi _ { a b } ( x ) . \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} \\forall x , y \\in E _ + \\quad \\norm x \\vee y \\norm = \\max \\{ \\norm x \\norm , \\norm y \\norm \\} \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} M _ D ( x , z _ 0 ) = \\begin{cases} \\frac { P _ D ( x , z _ 0 ) } { P _ D ( x _ 0 , z _ 0 ) } , & , \\\\ \\frac { \\mathbb { E } _ { x } \\tau _ D } { \\mathbb { E } _ { x _ 0 } \\tau _ D } , & \\end{cases} \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} \\begin{aligned} \\bar G _ l ( x ) & \\ , \\geq \\ , 0 & \\qquad & l \\in \\mathcal Q & \\\\ \\bar H _ l ( x ) & \\ , \\geq \\ , 0 & & l \\in \\mathcal Q & \\\\ \\bar G ( x ) \\cdot \\bar H ( x ) & \\ , = \\ , 0 . & & & \\end{aligned} \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} D _ { 3 } ( \\Pi ^ { 3 } _ { M _ 1 } v _ \\lambda , v _ \\lambda ) \\leq D _ { 3 } ( w _ \\lambda , v _ \\lambda ) = \\frac { 1 4 8 \\lambda ^ { 3 } } { 3 5 2 9 4 7 } \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} \\lim _ { v \\uparrow \\infty } \\sup _ { 0 \\leq t \\leq T } \\frac { 1 } { N _ v } \\int ( x - N _ v P _ v ^ { t } \\eta ) \\cdot A _ v ^ { - 1 } ( x - N _ v P _ v ^ { t } \\eta ) f ( t , x ) \\mu ( d x ) = 0 , \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} \\partial _ t \\Delta _ q \\theta + \\Delta _ q ( u \\cdot \\nabla \\theta ) - \\partial _ 1 ^ 2 \\Delta _ q \\theta = 0 . \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} X _ { \\alpha \\circ ( G , w ) } & = U _ { ( G , w ) } ( r _ { ( \\alpha _ 1 ) } ) U _ { ( G , w ) } ( r _ { ( \\alpha _ 2 , \\dots , \\alpha _ { \\ell } ) } ) - U _ { ( G , w ) } ( r _ { ( \\alpha _ 1 + \\alpha _ 2 , \\dots , \\alpha _ { \\ell } ) } ) \\\\ & = U _ { ( G , w ) } ( r _ { ( \\alpha _ 1 ) } r _ { ( \\alpha _ 2 , \\dots , \\alpha _ { \\ell } ) } - r _ { ( \\alpha _ 1 + \\alpha _ 2 , \\dots , \\alpha _ { \\ell } ) } ) \\\\ & = U _ { ( G , w ) } ( r _ \\alpha ) \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} W _ { \\b , \\gamma + 1 } = W _ { \\b , \\gamma } + h _ { \\b + 1 , \\gamma + 1 } e _ { \\gamma + 1 } e _ { \\gamma + 1 } ^ { \\top } \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} t _ 1 ^ { \\kappa - p / 2 - 1 } & \\left | \\int _ { t _ 1 } ^ { s _ 1 } \\frac { | W ( t ) | ^ p } { ( t ( 1 - t ) ) ^ \\kappa } d t - \\int _ { t _ 1 } ^ { s _ 1 } \\frac { | W ( t ) | ^ p } { t ^ \\kappa } d t \\right | \\\\ & \\leq \\left | 1 - \\frac { 1 } { ( 1 - s _ 1 ) ^ \\kappa } \\right | \\int _ { t _ 1 } ^ { s _ 1 } t _ 1 ^ { \\kappa - p / 2 - 1 } \\frac { | W ( t ) | ^ p } { t ^ \\kappa } d t \\\\ & = o _ P ( 1 ) . \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\mu ^ N = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\delta _ { X ^ { i , N } } ~ , \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} u _ t - \\Delta u = 0 , u ( t = 0 ) = \\varphi ( x ) , x \\in \\R ^ d , \\ , t \\geq 0 . \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} \\arg \\eta _ { \\rho _ { 1 } } ( z ) = \\arg ( \\eta _ { \\rho _ { 1 } ' } ( \\eta _ { \\rho _ { 1 } '' } ( z ) ) \\ge \\arg \\eta _ { \\rho _ { 1 } '' } ( z ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} \\rho _ { L \\oplus M } ( a ( x _ 1 + m _ 1 ) , x _ 2 + m _ 2 ) = \\rho _ { L \\oplus M } ( a x _ 1 + a m _ 1 ) , x _ 2 + m _ 2 ) = \\rho ( a x _ 1 , x _ 2 ) = ( - 1 ) ^ { \\bar a \\bar x _ 1 } \\rho ( x _ 1 , a x _ 2 ) \\\\ = ( - 1 ) ^ { \\overline a ~ \\overline { x _ 1 + m _ 1 } } \\rho _ { L \\oplus M } ( x _ 1 + m _ 1 , a x _ 2 + a m _ 2 ) = ( - 1 ) ^ { \\overline a ~ \\overline { x _ 1 + m _ 1 } } \\rho _ { L \\oplus M } ( x _ 1 + m _ 1 , a ( x _ 2 + m _ 2 ) ) . \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} \\mathbf { F } ( \\mathbf { X } , \\mathbf { Y } ) = \\mathbf { X } + \\mathbf { Y } + \\mathbf { G } ( \\mathbf { X } , \\mathbf { Y } ) , \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} \\left \\| x \\right \\| \\geqslant 0 , ~ \\left \\| x \\right \\| = 0 \\Leftrightarrow x = 0 ; \\\\ \\left \\| \\alpha x \\right \\| \\leqslant \\left \\| x \\right \\| , ~ \\forall ~ \\alpha \\in \\mathbb { C } , | \\alpha | \\le 1 ; \\\\ \\lim _ { \\alpha \\rightarrow 0 } \\left \\| \\alpha x \\right \\| = 0 ; \\\\ \\left \\| x + y \\right \\| \\le \\left \\| x \\right \\| + \\left \\| y \\right \\| . \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} { } ^ b \\Psi ^ { - \\infty , \\epsilon } _ { G , c } ( M ) = \\{ T _ { \\widetilde { k } } , \\ ; \\widetilde { k } \\in ( C ^ \\infty _ c ( G ) \\hat { \\otimes } \\ , { } ^ b \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } \\} \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} & \\langle \\varphi _ i , \\varphi _ k \\overline { \\theta } _ j \\rangle = ( w ( i ) w ( k ) ) ^ { - 1 / 2 } \\langle \\theta _ i , \\theta _ k \\overline { \\theta } _ j \\rangle \\\\ & = ( w ( i ) w ( k ) ) ^ { - 1 / 2 } \\ , w ( k ) \\ , \\delta _ { r + 1 , t } \\ , \\delta ( ( i , j ^ T ) , k ) \\ , \\langle 1 , 1 \\rangle \\\\ & = \\left ( \\dfrac { w ( k ) } { w ( i ) } \\right ) ^ { 1 / 2 } \\ ! \\ ! \\ ! \\delta _ { r + 1 , t } \\ , \\delta ( ( i , j ) , k ) , \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\overline { v _ { M } } ( r ) = 0 \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} \\phi ( Q ( j ) , Q ( i ) ) & = l _ j - 2 l _ { i + 1 } - \\Delta ( j , i + 1 ) - \\Delta ( i + 1 - | [ 1 , i ) \\cap J | + | [ 1 , j ) \\cap J | , i + 1 ) \\\\ & = l _ j - 2 l _ i - \\Delta ( j , i ) - \\Delta ( i + 1 - | [ 1 , i ) \\cap J | + | [ 1 , j ) \\cap J | , i ) \\ , \\cdot \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} u ( t , r ) = Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) + \\sum _ { k = 1 } ^ { 6 } v _ { k } ( t , r ) \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} | | \\gamma ^ X ( 1 ) | | ^ 2 = L ^ { N - 2 } \\int _ { \\mathbb { R } ^ N } \\left [ | \\nabla \\tilde { u } ^ X ( x ) | ^ 2 + ( L ) ^ 2 V ( L x + y ) ( \\tilde { u } ^ X ( x ) ) ^ 2 \\right ] d x > \\rho ^ 2 , \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} \\tilde { x } & = \\frac { x } { \\mu } , & \\tilde { y } & = \\frac { y - \\zeta _ L ( \\mu ) } { \\mu } , \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n - 1 } t _ i = t , \\prod _ { i = 1 } ^ { n - 1 } \\binom { m - 1 + t _ i } { m - 1 } \\not \\equiv 0 \\pmod p . \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} \\sin ( \\theta ) p _ n ( x ) = | \\phi _ { n } | \\sin ( n \\theta - \\arg ( \\phi _ n ) ) \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} v ( t , x ) = \\psi ( \\phi ( 0 ; t , x ) ) \\exp \\Bigl [ \\int _ 0 ^ t \\frac { \\lambda ( s , \\phi ( s ; t , x ) ) } { s } d s \\Bigr ] . \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} | v _ { 1 } ( t , r ) | & = | \\int _ { t } ^ { \\infty } d s \\frac { \\lambda '' ( s ) } { r } \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\left ( 1 + \\frac { r ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( r ^ { 2 } - 1 - \\rho ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } \\right ) | \\\\ & \\leq \\frac { C } { r } \\int _ { t } ^ { \\infty } | \\lambda '' ( s ) | ( s - t ) d s \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} p _ { t } ^ { U } ( x , d y ) = p _ { t } ^ { U } ( x , y ) \\ , d m ( y ) t > 0 x \\in U . \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} h ( R , \\omega ) = c R ^ { \\gamma _ 1 ^ - } w _ 1 ( \\omega ) + O ( R ^ { \\gamma _ 1 ^ - - \\epsilon } ) \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} K ( x , y , z ) = \\sum _ { j , k \\in \\mathbb Z } 2 ^ { - 2 j - 2 k } \\psi _ { j , k } \\Big ( { x \\over 2 ^ j } , { y \\over 2 ^ k } , { z \\over 2 ^ { j + k } } \\Big ) \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} & F ( t ; x , y , 1 , u , 1 , v ) \\\\ & = \\sum _ { m = 0 } ^ { \\infty } \\frac { r v ( 1 - y r ) ( 1 - r ) ^ m } { ( x - x u + u ( 1 - y r ) ( 1 - r ) ^ m ) ( 1 - y t u v ) } \\\\ & \\quad \\times \\prod _ { i = 0 } ^ { m } \\frac { x ( 1 - ( 1 - y r ) ( 1 - r ) ^ i ) ( x - x u + u ( 1 - y r ) ( 1 - r ) ^ i ) } { ( x - u ( x - 1 ) ( 1 - y r ) ( 1 - r ) ^ i ) ( x - x u + u ( 1 - r v ) ( 1 - y r ) ( 1 - r ) ^ i ) } , \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{gather*} \\begin{cases} \\lim _ { t \\to 0 } \\mathbf { F } ( t , t ) = 1 , & \\\\ \\lim _ { t \\to 0 } \\mathbf { F } ( s , t ) = 0 , & \\end{cases} \\\\ \\mathbf { F } ( \\cdot , t ) \\ \\\\ \\ \\mathbf { F } ( s , t ) \\ \\end{gather*}"}
-{"id": "1031.png", "formula": "\\begin{align*} P ( \\lambda ) = \\alpha _ { k , 0 } \\lambda ^ { k + 1 } + \\displaystyle \\sum _ { j = 1 } ^ { k } \\left ( \\alpha _ { k , j - 1 } + \\alpha _ { k , j } \\right ) \\lambda ^ { k + 1 - j } + \\alpha _ { k , k } \\lambda ^ 0 \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} \\partial _ { t } v _ { 4 } ( t , r ) = \\frac { - 1 } { 2 \\pi } \\int _ { 0 } ^ { \\infty } d w \\int _ { 0 } ^ { w } \\frac { \\rho d \\rho } { \\sqrt { w ^ { 2 } - \\rho ^ { 2 } } } \\int _ { 0 } ^ { 2 \\pi } d \\theta \\frac { \\partial _ { 1 } v _ { 4 , c } ( t + w , \\sqrt { r ^ { 2 } + \\rho ^ { 2 } + 2 r \\rho \\cos ( \\theta ) } ) } { \\sqrt { r ^ { 2 } + \\rho ^ { 2 } + 2 r \\rho \\cos ( \\theta ) } } ( r + \\rho \\cos ( \\theta ) ) \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} \\widetilde { f } ( \\xi ) : = ( 2 \\pi ) ^ { - d / 2 } \\int _ { \\mathbb { R } ^ d } f ( x ) e ^ { - i x \\cdot \\xi } d x . \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} \\phi _ k \\in \\mathcal S : = \\Big \\{ \\psi \\in H _ { l o c } ^ 1 ( \\mathbb R _ + , \\mathbb C ^ 2 ) \\cap L _ r ^ 2 ( \\mathbb R _ + , \\mathbb C ^ 2 ) \\ , : \\ , \\int _ 0 ^ \\infty \\Big [ \\ , | \\psi ' | ^ 2 + \\frac { 1 } { r ^ 2 } | \\psi | ^ 2 \\ , \\Big ] r \\ , { \\mathrm d } r < + \\infty \\Big \\} . \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} & \\langle H _ { n - 1 - i } , H _ { n - 1 - i } \\rangle = i \\\\ \\frac { 1 } { n - 1 } & \\langle H _ { n - 1 - i } , - K _ { X _ { n , s } } \\rangle = i + 2 \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} K _ { n , z } ( w ) = e ^ { \\overline { z } w } L _ { n - 1 } ^ { ( 1 ) } ( | w - z | ^ 2 ) . \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} \\left | \\bigcup \\limits _ { 1 \\leq i \\ne j \\leq m } D ( B _ i , \\widetilde { B } _ j ) \\right | = \\sum \\limits _ { 1 \\leq i \\leq m } \\sum \\limits _ { 1 \\leq j \\leq m \\atop j \\ne i } \\sum _ { b \\in B _ i } | D ( \\{ b \\} , \\widetilde { B } _ j ) | = \\sum \\limits _ { 1 \\leq i \\leq m } \\sum \\limits _ { 1 \\leq j \\leq m \\atop j \\ne i } \\sum _ { b \\in B _ i } \\widetilde { k } = \\widetilde { k } a ( m - 1 ) . \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a , q ^ { - 1 } / b , c q ^ { - 1 } , d q ^ { - 1 } , q ^ { - 1 } ; q ^ 2 ) _ k } { ( q ^ { 2 } / a , a q ^ 2 , b q ^ 2 , q ^ 2 / c , q ^ { 2 } / d , q ^ 2 ; q ^ 2 ) _ k } \\bigg ( \\frac { b q ^ 7 } { c d } \\bigg ) ^ k \\\\ [ 5 p t ] \\ : & \\ : \\equiv [ n ] \\Theta _ q ( a , b , n ) \\sum _ { k = 0 } ^ { ( n + 1 ) / 2 } \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a , q ^ { - 1 } / b , q ^ 3 / c d ; q ^ 2 ) _ k } { ( q ^ 2 , q ^ { - 2 } / b , q ^ 2 / c , q ^ 2 / d ; q ^ 2 ) _ k } q ^ { 2 k } , \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} \\begin{array} { l l } V _ o = C N _ { C _ { 1 2 } ( 1 , 3 ) } ( V _ 1 ) \\cup C N _ { C _ { 1 2 } ( 1 , 3 ) } ( V _ 3 ) \\mbox { o r } \\\\ V _ o = C N _ { C _ { 1 2 } ( 1 , 3 ) } ( V _ 1 ) \\cup C N _ { C _ { 1 2 } ( 1 , 3 ) } ( V _ 4 ) , \\end{array} \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} A _ 0 ^ { ( 0 ) } = 1 , \\ \\ \\ A _ k ^ { ( k ) } = A _ { k - 1 } ^ { ( k ) } , \\ \\ \\ A _ j ^ { ( k ) } = \\sum _ { i = 0 } ^ j A _ i ^ { ( k - 1 ) } , \\ \\ \\ \\ k > j \\ge 0 \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} F _ { \\lambda , t } : = \\chi ( d _ { \\lambda } ^ { t } ) , \\textup { w h e r e } d _ { \\lambda } ^ { t } = \\begin{bmatrix} 0 & \\lambda ( \\mathsf { M } - t ) - \\frac { \\partial } { \\partial r } \\\\ \\lambda ( \\mathsf { M } - t ) + \\frac { \\partial } { \\partial r } & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} & ( T u , u + B ( T u ) ) \\cdot _ H ( T v , v + B ( T v ) ) \\\\ & = ( T ( u ) T ( v ) , ( T u ) \\cdot v + ( T u ) \\cdot B ( T v ) + u \\cdot ( T v ) + B ( T u ) \\cdot T v + H ( T u , T v ) ) \\\\ & = ( T ( u ) T ( v ) , ( T u ) \\cdot v + u \\cdot ( T v ) + B ( T u , T v ) + H ( T u , T v ) ) \\in \\tau _ B ( \\mathrm { G r } ( T ) ) . \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} \\hat { A } _ { \\rm u n d } = \\frac { A ( s ) } { \\frac 1 2 a _ { n + 1 } R ^ { n + 1 } \\vert _ { R = L ( \\hat { V } _ { \\rm u n d } ) ^ { \\frac { 1 } { n + 2 } } } } . \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} \\Pr { \\prod _ { i = 1 } ^ d \\norm [ 1 ] { x _ i } _ 2 > ( \\sqrt { n } + s ) ^ d } \\le \\Pr { \\frac { 1 } { d } \\sum _ { i = 1 } ^ d \\big ( \\norm [ 1 ] { x _ i } _ 2 - \\sqrt { n } \\big ) > s } . \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} \\Omega : = \\C ~ \\big \\backslash \\bigcup _ { n = 0 } ^ { \\infty } \\Gamma _ { n } . \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} K = - \\log ( - i \\langle v , \\bar { v } \\rangle ) \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} & T ( e ) ' ( t ) = - \\int _ { t } ^ { \\infty } T ( e ) '' ( x ) d x \\\\ & | T ( e ) ' ( t ) | \\leq \\frac { 1 } { 1 0 0 } \\int _ { t } ^ { \\infty } \\frac { d x } { \\sqrt { \\log ( \\log ( x ) ) } x ^ { 2 } \\log ^ { b + 1 } ( x ) } \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} f ^ * ( \\mathcal L _ 1 ) \\cdot \\ldots \\cdot f ^ * ( \\mathcal L _ d ) = \\mathcal L _ 1 \\cdot \\ldots \\cdot \\mathcal L _ d \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} A _ k ' ( n ) & = \\{ ( \\lambda = ( \\lambda _ 1 ^ { m _ 1 } , \\lambda _ 2 ^ { m _ 2 } , \\ldots , \\lambda _ \\ell ^ { m _ \\ell } ) , \\lambda _ { i _ 0 } , m ) \\mid \\\\ & \\lambda \\in \\mathcal { P } ( n ) , \\ , 1 \\leq i _ 0 \\leq \\ell , \\ , m _ i < k \\ , \\forall i \\not = i _ 0 , \\ , m _ { i _ 0 } \\ge k , \\ , k \\mid \\lambda _ { i _ 0 } , \\ , 0 \\le m < k \\} . \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} k _ m ^ { ( r ) } = \\sum _ { i = 0 } ^ m \\binom { r + i - 1 } { i } h _ { m - i } \\ , , \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} \\partial _ { t } Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) = \\frac { - 2 r \\lambda ' ( t ) } { r ^ { 2 } + \\lambda ( t ) ^ { 2 } } \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} P ( z ) = \\sum _ { | \\alpha | = m } x _ \\alpha z ^ \\alpha , \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} \\langle E _ k , \\nabla _ { E _ j } ^ T x ^ T \\rangle & = \\langle E _ k , \\nabla _ { E _ j } x ^ T \\rangle = \\langle E _ k , \\nabla _ { E _ j } ( x - x ^ { \\perp } ) \\rangle = \\delta _ { j k } - \\langle E _ k , \\nabla _ { E _ j } x ^ { \\perp } \\rangle \\\\ & = \\delta _ { j k } + \\langle \\nabla _ { E _ j } E _ k , x ^ { \\perp } \\rangle = \\delta _ { j k } + A ^ { x ^ { \\perp } } ( E _ j , E _ k ) \\ , . \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} H _ { n } ^ { C W } ( \\sigma ) : = \\frac { J } { n } \\sum _ { i , j = 1 } ^ { n } \\sigma _ { i } \\sigma _ { j } = \\frac { J } { n } \\left ( \\sum _ { i = 1 } ^ { n } \\sigma _ { i } \\right ) ^ 2 . \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} \\ell _ { a + c - 1 } = \\ell _ { J - a + c - 1 } \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} \\psi _ { s } ' ( x _ { n + 1 } ) = ( 1 - 2 s ) x _ { n + 1 } ^ { - 2 s } - 1 < 0 0 \\le x _ { n + 1 } \\le 1 / 2 . \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} L ( \\psi ) = \\lim _ { N \\to \\infty } \\frac { 1 } { | J _ N | } \\sum _ { n \\in J _ N } 1 _ A ( n ) \\psi ( s _ n ) \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} G _ 1 - G _ 2 = K , \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d X ( t ) & = b ( X ( t ) ) d t + \\sigma ( X ( t ) ) d B ( t ) - d \\eta ( t ) , \\\\ X ( 0 ) & = x \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} f ( a , b , z ) = f ( a , b p , z ) - b f ( a , b p , q z ) , \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} \\lim _ { y \\uparrow \\infty } \\frac { F ( i y ) } { i y } = 1 . \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} S _ { \\varepsilon , k , i } ( \\sigma , s ) : = c _ i ( \\sigma ) + \\varepsilon x _ k ( s ) e _ 2 ( \\sigma ) + \\varepsilon y _ k ( s ) e _ 3 ( \\sigma ) . \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} A _ { \\varepsilon } ( H , x ) = \\begin{cases} \\{ ( ( 1 + \\varepsilon ) ^ i , ( 1 + \\varepsilon ) ^ j , \\gamma ' ) \\} , & x = 0 \\\\ \\{ ( 0 , 0 , \\hat { \\gamma } ) \\} , & \\end{cases} \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} f ' ( z ) = ( N - s ) ( z - \\alpha _ 1 ) ^ { n _ 1 - 2 } \\cdots ( z - \\alpha _ r ) ^ { n _ r - 2 } ( z - \\beta _ 1 ) ^ { m _ 1 } \\cdots ( z - \\beta _ p ) ^ { m _ p } \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} E _ { 1 , 1 } \\cdot \\Lambda _ { a , b , c } & = c \\Lambda _ { a , b , c } , \\\\ E _ { 2 , 2 } \\cdot \\Lambda _ { a , b , c } & = ( b - c ) \\Lambda _ { a , b , c } , \\\\ E _ { 3 , 3 } \\cdot \\Lambda _ { a , b , c } & = ( a - b ) \\Lambda _ { a , b , c } , \\\\ E _ { 4 , 4 } \\cdot \\Lambda _ { a , b , c } & = ( n - a ) \\Lambda _ { a , b , c } , \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} T _ s ( H ) & = n H - ( n - 1 ) \\sum _ { i = 1 } ^ { n + 1 } E _ i \\\\ T _ s ( E _ j ) & = H - \\sum _ { i \\neq j , i = 1 } ^ { n + 1 } E _ { i } j \\leq n + 1 \\\\ T _ s ( E _ j ) & = E _ j \\qquad j > n + 1 \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} C : = \\{ ( \\varphi _ 1 ( \\alpha _ 1 ) , . . . , \\varphi _ t ( \\alpha _ t ) ) \\mid ( \\alpha _ 1 , . . . , \\alpha _ t ) \\in H \\} . \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = ( - 1 ) ^ { n } \\sum _ { a = 0 } ^ n \\sum _ { b = 0 } ^ a ( - 1 ) ^ { b } ( b + 1 ) = ( - 1 ) \\sum _ { a = 0 } ^ \\frac { n + 1 } { 2 } \\left ( ( a + 1 ) - \\frac { 2 a + 2 } { 2 } \\right ) = 0 \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} \\widetilde { \\tau } ( t ) = \\tau _ 0 - 2 t C _ r ( 1 + M ) , \\ ; \\ ; \\ ; \\widetilde { \\mathcal { T } } = \\frac { \\tau _ 0 } { 1 + 2 C _ r ( 1 + M ) } . \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} [ \\partial \\lambda ] _ \\alpha f = ( \\lambda f ) ' . \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\hat { \\psi } ( a _ { \\Lambda } J _ t ( b _ { \\Lambda ^ { ' } } ) ) = \\sum _ { i , j } \\left ( \\prod _ { x \\in \\Lambda } \\mathrm { T r } ( h _ { x , i } h ^ \\ast _ { x , j } a _ x ) \\right ) \\beta _ { \\Lambda ^ c , i , j } \\alpha ^ { ( | \\Lambda ^ { ' } | ) } ( J ^ { - 1 } _ { \\Lambda ^ { ' } } ( b _ { \\Lambda ^ { ' } } ) ) \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\phi \\partial _ { t } a d x = \\int _ { \\Omega } ( b \\cdot \\nabla _ { x } ) \\phi d x . \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} P ( z , \\overline { z } ) = \\frac { \\varphi _ { p } ( z , \\overline { z } ) + \\overline { \\varphi _ { p } ( z , \\overline { z } ) } } { 2 } , \\quad \\mbox { f o r g i v e n $ p \\in \\mathbb { N } ^ { \\star } $ . } \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\rho _ i \\rho _ j = \\rho _ j \\rho _ i ( i , j = 1 , \\dots , n ) , ( \\rho _ 0 \\rho _ k ) ^ 2 = ( \\rho _ k \\rho _ 0 ) ^ 2 ( k = 1 , \\dots , n ) . \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sqrt [ n ] { \\sigma _ n } = 0 , \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} x _ { n _ 1 , . . . , n _ k } = \\frac 1 { x _ { n _ 1 , . . . , n _ { k - 1 } } + n _ k } . \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} \\left ( \\frac { C _ { n , k _ { 1 } } - ( n - 1 ) \\mathbb { I } _ { k _ { 1 } = 2 } - \\mu _ { k _ { 1 } } } { \\sqrt { 2 k _ { 1 } } } , \\ldots , \\frac { C _ { n , k _ { l } } - \\mu _ { k _ { l } } } { \\sqrt { 2 k _ { l } } } \\right ) \\stackrel { d } { \\to } N _ { l } ( 0 , I _ { l } ) . \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} f _ { n , 2 } ( \\cdot \\ , u ) : = \\big ( f _ n ( \\cdot \\ , u ) - f _ n ^ - ( \\cdot \\ , u ) \\big ) / 2 i \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} \\int _ { X } \\left | \\frac { | f ( x ) | } { \\| f \\| _ 2 } - \\frac { 1 } { \\sqrt { \\mu ( X ) } } \\right | ^ 2 \\ , d \\mu ( x ) = c _ f . \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} \\mathbb { E } ^ { S C } _ { \\sigma _ { 0 } , \\tilde { \\sigma } _ { 0 } } \\big [ \\ , \\Vert S _ { t + 1 } - \\tilde { S } _ { t + 1 } \\Vert _ { 1 } - \\Vert S _ { t } - \\tilde { S } _ { t } \\Vert _ { 1 } \\ , \\vert \\ , \\mathcal { F } _ { t } \\ , \\big ] \\leq - \\frac { 1 } { n } \\ , \\big ( \\tilde { S } _ { t } ^ { 2 } - S _ { t } ^ { 2 } \\big ) - \\frac { 1 } { n } \\ , \\big ( \\tilde { S } _ { t } ^ { 3 } - S _ { t } ^ { 3 } \\big ) = - \\frac { 1 } { 2 n } \\ , \\Vert S _ { t } - \\tilde { S } _ { t } \\Vert _ { 1 } . \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} I ( u ) = \\dfrac { 1 } { 2 } \\Big ( | | u ^ + | | ^ 2 - | | u ^ - | | ^ 2 \\Big ) - \\int _ { \\mathbb { R } ^ N } F _ 0 ( u ( x ) ) \\ , d x , \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} T \\cong \\prod _ { i = 1 } ^ { s _ 0 } T _ { p _ i } \\times T _ 2 . \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} U \\left ( \\phi _ { \\ell } ( \\tau ) ^ { \\lambda } J _ { \\ell , \\mu } \\right ) = \\sum _ { \\nu } C _ { \\mu , \\nu } ^ { \\lambda } J _ { \\ell , \\nu } . \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} \\eta ^ + = \\Phi _ { 0 } ^ + + \\Phi _ { 2 } ^ + = { \\frac { U ^ + } { r } } , \\qquad \\eta ^ - = \\Phi _ { 0 } ^ - + \\Phi _ { 2 } ^ - = { \\frac { U ^ - } { r } } , \\\\ [ 1 m m ] \\zeta ^ + = \\Phi _ { 0 } ^ + - \\Phi _ { 2 } ^ + = { U ^ + } ' , \\qquad \\zeta ^ - = \\Phi _ { 0 } ^ - - \\Phi _ { 2 } ^ - = { U ^ - } ' . \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} S \\left ( \\sum _ { j \\in J } g _ j \\right ) = \\sum _ { j \\in J } A _ j g _ j \\qquad ( g _ j \\in W _ j ) . \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} \\frac { d } { d t } L _ u = [ B _ u , L _ u ] \\ , . \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} \\lim _ { N - M \\to \\infty } \\frac { 1 } { N - M } \\sum _ { n = M } ^ { N - 1 } \\psi ( P ( n ) ) \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} \\frac { x } { p } = \\frac { 1 - \\frac { m ^ { 1 / 2 } } { c n ^ { 1 / 2 } } } { 1 - \\frac { c m ^ { 1 / 2 } } { 2 n ^ { 1 / 2 } } + O ( m n ^ { - 1 } ) } = 1 - \\frac { m ^ { 1 / 2 } } { c n ^ { 1 / 2 } } + \\frac { c m ^ { 1 / 2 } } { 2 n ^ { 1 / 2 } } + O ( m n ^ { - 1 } ) \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} \\sigma _ { \\alpha ^ k } ( X , T ) = \\sigma _ { \\alpha ^ k } ( \\Sigma , J ) = - \\sum _ { j = 1 } ^ { n - 1 } \\sigma _ { \\omega ^ { d j } } ( Y , K ) + \\sum _ { j = 0 } ^ { n - 1 } \\sigma _ { \\omega ^ { d j + k } } ( Y , K ) . \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} u _ { 4 , s } ( t , x ) = \\frac { 1 } { 2 \\pi } \\int _ { B _ { s - t } ( x ) } \\frac { \\left ( \\int _ { | y | } ^ { \\infty } v _ { 4 , c } ( s , q ) d q \\right ) } { \\sqrt { ( s - t ) ^ { 2 } - | y - x | ^ { 2 } } } d y = \\frac { 1 } { 2 \\pi } \\int _ { B _ { s - t } ( 0 ) } \\frac { \\left ( \\int _ { | z + x | } ^ { \\infty } v _ { 4 , c } ( s , q ) d q \\right ) } { \\sqrt { ( s - t ) ^ { 2 } - | z | ^ { 2 } } } d z \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} \\left ( F _ { _ { 1 ; 0 , \\dots , 0 } } ^ { \\left ( 1 \\right ) } ( z ) , \\dots , F _ { _ { 1 ; 0 , \\dots , 0 } } ^ { \\left ( N \\right ) } ( z ) , F _ { _ { 1 ; 0 , \\dots , 0 } } ^ { \\left ( N + 1 \\right ) } ( z ) , \\dots , F _ { _ { 1 ; 0 , \\dots , 0 } } ^ { \\left ( N ' \\right ) } ( z ) \\right ) = \\left ( z _ { 1 } , \\dots , z _ { N } , 0 , \\dots , 0 \\right ) , \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} { \\rm R i c } _ V : = { \\rm R i c } - \\frac { 1 } { 2 } L _ V g . \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} \\alpha \\rho ^ { s - 2 } \\cdot C ^ { s - 1 } \\left ( \\alpha \\rho ^ { s - 2 } \\right ) + 1 & = C \\left ( \\alpha \\rho ^ { s - 2 } \\right ) . \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } | P _ * ( r , k ) - 1 | ^ 2 d \\sigma = \\int _ 0 ^ r | A ( \\rho ) | ^ 2 d \\rho \\ , . \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} 0 = \\lambda _ { 0 , 1 } \\leqslant \\lambda _ { 0 , 2 } \\leqslant \\cdots \\leqslant \\lambda _ { 0 , m } \\leqslant \\cdots , \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} \\bar { \\rho } ( L _ { - 1 } ) = { a + z } , \\ , \\bar { \\rho } ( L _ 0 ) = ( a + z ) \\partial , \\ , \\bar { \\rho } ( L _ { 1 } ) = ( a + z ) \\partial ^ 2 - c ( c + 1 ) ( a + z ) ^ { - 1 } \\ , . \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} \\langle \\widehat { A } f , g \\rangle _ { \\alpha } = \\langle f , g \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = \\overline { \\langle g , f \\rangle } _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = \\overline { \\langle \\widehat { A } g , f \\rangle } _ { \\alpha } = \\langle f , \\widehat { A } g \\rangle _ { \\alpha } . \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} \\lceil \\eta N _ t \\rceil - 3 = \\sum _ { j = 1 } ^ { \\ell ( t ) } n _ j + \\delta ( t ) = N _ { \\ell ( t ) } + \\delta ( t ) . \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} \\Phi _ { k , n } ( t ) : = ( t - 1 ) \\frac { ( k + 1 ) n } { k ( n + 1 ) } + 1 = \\frac { n ( k + 1 ) } { ( n + 1 ) k } t + \\frac { k - n } { k ( n + 1 ) } . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} x & = X - Y , \\\\ y & = - X - Y + 2 . \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} e ^ { - \\tau \\phi _ { - } } = \\frac { \\| x _ { n + 1 } ^ { \\frac { 1 - 2 s } { 2 } } \\tilde { w } \\| _ { L ^ { 2 } ( C _ { s , 1 / 2 } ^ { + } ) } ^ { \\frac { \\phi _ { - } } { \\phi _ { + } - \\phi _ { - } } ( 1 - \\alpha ) } } { \\lim _ { x _ { n + 1 } \\rightarrow 0 } \\| x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { w } \\| _ { L ^ { 2 } ( C _ { s , 1 / 2 } ' ) } ^ { \\frac { \\phi _ { - } } { \\phi _ { + } - \\phi _ { - } } ( 1 - \\alpha ) } } . \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum \\limits _ { j = 1 } ^ n p _ \\theta ( \\textbf { X } _ j ) ^ { \\alpha - 2 } \\partial _ r [ p _ \\theta ( \\textbf { X } _ j ) ] = \\int p _ \\theta ( { \\bf { x } } ) ^ { \\alpha - 1 } \\partial _ r [ p _ \\theta ( { \\bf { x } } ) ] d { \\bf { x } } . \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} \\frac { d \\tilde { x } } { d \\tau } & = a _ 4 \\sqrt { \\tilde { x } } + a _ 2 \\tilde { y } , \\\\ \\frac { d \\tilde { y } } { d \\tau } & = 0 . \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} A ( z ) = \\begin{pmatrix} z & 0 \\\\ 0 & z \\end{pmatrix} , \\ z \\in \\C ^ \\ast \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} \\bar { P } _ { { \\rm { i n 2 o n } } } ^ { \\rm { e } } = \\int \\limits _ 0 ^ \\infty { P _ { { \\rm { i n 2 o n } } } ^ { \\rm { e } } ( \\gamma ) } { f _ \\gamma } ( \\gamma ) { \\rm { d } } \\gamma . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} x _ 0 : = x , x _ { 2 n + 1 } : = \\Pi ^ p _ { M } x _ { 2 n } \\quad x _ { 2 n } : = \\Pi ^ p _ { N } x _ { 2 n - 1 } \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n + 1 } \\omega _ { j } \\Omega _ { j } & = 0 \\quad \\quad \\sum _ { j = 1 } ^ { n + 1 } \\Omega _ { j } \\omega _ { j } = n \\quad \\mathcal { S } _ { + } ^ { n } , \\\\ \\sum _ { j = 1 } ^ { n } \\omega _ { j } \\Omega _ { j } & = 0 \\quad \\quad \\sum _ { j = 1 } ^ { n } \\Omega _ { j } \\omega _ { j } = n \\quad \\partial \\mathcal { S } _ { + } ^ { n } . \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} B ( e _ { x y } , e _ { u v } ) & = B ( e _ x e _ { x y } e _ y , e _ u e _ { u v } e _ v ) = e _ x e _ u B ( e _ { x y } , e _ { u v } ) e _ y e _ v \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta , \\ \\bigcup _ { k = 1 } ^ \\infty \\{ N ( t ) = 2 k \\} \\ | \\ V ( 0 ) = c _ 1 \\} = \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} \\sup _ { 0 < s < \\frac { 1 } { 8 ( 1 + | x | ) ^ 2 } } K _ { 2 3 } ( t , x , y ) | _ { t = \\log \\left ( \\frac { 1 + s } { 1 - s } \\right ) } & \\lesssim \\sup _ { 0 < s < \\frac { 1 } { 8 ( 1 + | x | ) ^ 2 } } \\frac { | x | } { s ^ { ( n - 1 ) / 2 } } \\exp \\Big ( - \\frac { c } { s ( 1 + | x | ) ^ 2 } \\Big ) e ^ { | y | ^ 2 - | x | ^ 2 } \\\\ & \\lesssim | x | ( 1 + | x | ) ^ { n - 1 } e ^ { | y | ^ 2 - | x | ^ 2 } \\lesssim ( 1 + | x | ) ^ n e ^ { | y | ^ 2 - | x | ^ 2 } , \\ , ( x , y ) \\in N ^ c . \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} d \\theta ^ { 1 } & = \\theta ^ { 1 } \\wedge \\theta _ { 1 } { } ^ { 1 } + \\theta \\wedge \\tau ^ { 1 } \\\\ \\tau ^ { 1 } & \\equiv 0 \\mod \\theta ^ { \\bar { 1 } } \\\\ 0 & = \\theta _ { 1 } { } ^ { 1 } + \\theta _ { \\bar { 1 } } { } ^ { \\bar { 1 } } , \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} 2 H = \\alpha t ^ 2 + ( \\alpha ^ 2 - 4 \\alpha ) t + \\beta ^ 2 + 2 \\alpha + ( 2 \\alpha - 1 ) \\beta . \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} T _ j \\llcorner U = [ g r a p h _ { \\Sigma } ( u _ j ) ] \\llcorner U \\ \\ \\ j > > 1 \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} J _ 2 ( \\chi ^ + _ 2 , \\chi ^ - _ 2 ) = & \\ , \\frac { 1 } { 2 } \\int _ 0 ^ \\infty \\Big [ \\ , \\big | { \\chi ^ + _ 2 } ' ( r ) \\big | ^ 2 { U ^ + } ^ 2 + \\big | { \\chi ^ - _ 2 } ' ( r ) \\big | ^ 2 { U ^ - } ^ 2 + 2 A _ + { U ^ + } ^ 4 { \\chi ^ + _ 2 } ^ 2 + 2 A _ - { U ^ - } ^ 4 { \\chi ^ - _ 2 } ^ 2 \\ , \\Big ] r \\ , { \\mathrm d } r \\\\ [ 1 m m ] & + \\int _ 0 ^ \\infty \\Big [ \\ , 4 B { U ^ + } ^ 2 { U ^ - } ^ 2 \\chi ^ + _ 2 \\chi ^ - _ 2 + { \\mathfrak h } _ 2 ^ + \\chi ^ + _ 2 { U ^ + } ^ 2 + { \\mathfrak h } _ 2 ^ - \\chi ^ - _ 2 { U ^ - } ^ 2 \\ , \\Big ] r \\ , { \\mathrm d } r . \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} \\Psi _ { \\Lambda } : = \\sum _ { i \\in I } \\bigotimes _ { x \\in \\Lambda } h _ { x , i } \\in \\bigotimes _ { x \\in \\Lambda } \\mathcal { H } _ { x } = : \\mathcal { H } _ { \\Lambda } \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} \\frac { \\displaystyle \\sum _ { i = 0 } ^ { k } \\alpha _ { k , i } \\left ( ( \\Delta t _ i ) ^ j + ( \\Delta t _ { i + 1 } ) ^ j \\right ) } { j ! } = \\begin{cases} 1 , j = 1 , \\\\ 0 , j \\neq 1 . \\end{cases} \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} c _ { 3 } = \\lambda b _ { 1 } \\left ( \\left [ \\int _ { \\mathbb { R } } \\frac { 1 } { ( \\alpha - t ) ^ { 2 } } \\frac { d \\mu _ { 1 } ( t ) } { \\lambda } \\right ] ^ { 2 } - \\int _ { \\mathbb { R } } \\frac { 1 } { ( \\alpha - t ) ^ { 4 } } \\frac { d \\mu _ { 1 } ( t ) } { \\lambda } \\right ) < 0 . \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} | U _ { G ' } | = \\left \\{ \\begin{array} { c c } d _ { | G ' | } , & i \\le | G ' | < k , \\\\ 0 , & k \\le | G ' | < m , \\\\ d _ k , & | G ' | = m . \\end{array} \\right . \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} \\Phi \\left [ \\gamma ( L _ n ) \\right ] = \\gamma \\left ( K ^ { - 1 } L _ n \\right ) \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} m ^ { ( s ) } ( r ) \\leq D \\cdot U ( r ) \\sup _ S w _ 1 = m ^ { ( s ) } ( r _ 2 / 2 ) \\cdot \\frac { U ( r ) } { U ( r _ 2 / 2 ) } \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} & - 4 \\int _ { t } ^ { \\infty } \\frac { \\lambda _ { 0 , 0 } '' ( s ) } { 1 + s - t } d s + \\frac { 4 b } { t ^ { 2 } \\log ^ { b } ( t ) } = E _ { \\lambda _ { 0 , 0 } } \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} \\sigma _ i = ( 1 , \\ldots , \\underbrace { 2 } _ { } , \\ldots , 1 ) . \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } v _ { 1 } ( t , r ) | \\leq \\begin{cases} \\frac { C r } { t ^ { 4 } \\log ^ { b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { r t ^ { 2 } \\log ^ { b + 1 } ( t ) } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} A d m ^ { \\ell } _ { \\Z } = A d m ^ { k + 1 } _ { \\Z } = P _ + ^ { p + q - h ^ { \\vee } } , A d m ^ { \\check { \\ell } } _ { \\Z } = A d m ^ { k } _ { \\Z } = P _ + ^ { p - h ^ { \\vee } } , \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} \\partial _ t f + v \\cdot \\nabla _ x f + L f = \\Gamma ( g , f ) . \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} & \\sigma _ { \\sigma _ { ( a , i ) } ( ( a , i ) ) } ( ( b , j ) ) = \\sigma _ { ( a + \\delta _ { ( a , i ) } , i + r \\delta _ { ( a , i ) } + 1 ) } ( ( b , j ) ) = \\\\ & ( b - r ( a + \\delta _ { ( a , i ) } ) + i + r \\delta _ { ( a , i ) } + 1 , j + r ( i + r \\delta _ { ( a , i ) } + 1 ) - r ^ 2 ( a + \\delta _ { ( a , i ) } ) + 1 ) = \\\\ & ( b - r a + i + 1 , j + r i + r - r ^ 2 a + 1 ) = ( b + \\delta _ { ( a , i ) } + 1 , j + r ( \\delta _ { ( a , i ) } + 1 ) + 1 ) . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} v _ { 3 , 2 } ( t , r ) = \\frac { - 1 } { r } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\rho d \\rho & \\left ( \\frac { 1 } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } - \\frac { 1 } { ( s - t ) } \\right ) \\lambda '' ( s ) \\\\ & \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( - 1 - \\rho ^ { 2 } + r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} n - a _ { N + 1 } \\geq 1 + \\sum _ { i = 1 } ^ { N } { a _ i - a _ { N + 1 } } \\geq 0 . \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} \\sum _ { i = j } ^ { \\infty } \\delta _ i ^ 2 \\leq 3 \\left ( F ( \\Sigma _ { i - 1 } ) - \\lim _ { t \\to \\infty } F ( \\Sigma _ t ) \\right ) \\leq C \\ , j ^ { - \\rho } \\ , . \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} & \\frak R ( x _ 1 , x _ 2 , x _ 3 , v _ 1 , v _ 2 , v _ 3 ) \\\\ & = \\sum \\limits _ { j , k } \\sum _ { R = I \\times J \\times S \\in \\mathcal R ^ N _ { \\frak z } ( j , k ) } \\int _ { R } \\psi _ { j , k } ( x _ 1 - y _ 1 , x _ 2 - y _ 2 , x _ 3 - y _ 3 ) \\\\ & \\times ( \\psi _ { j , k } ( y _ 1 - v _ 1 , y _ 2 - v _ 2 , y _ 3 - v _ 3 ) - \\psi _ { j , k } ( x _ I - v _ 1 , x _ J - v _ 2 , x _ S - v _ 3 ) ) d y _ 1 d y _ 2 d y _ 3 . \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} \\psi ( z ) = \\log 2 - \\log ( 1 + | z | ^ 2 ) , \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ N X _ i ( t ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N X _ i ( 0 ) = m . \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} ( x + y ) z _ { x y } - ( { W _ 0 } ( e ^ { p } ) + 1 ) ( { W _ { - 1 } } ( - e ^ q ) + 1 ) = 0 , \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} ( n - 2 ) \\frac { h ^ { \\prime \\prime } } { h } - 2 \\frac { h ^ \\prime } { h } \\frac { f ^ \\prime } { f } - \\frac { f ^ { \\prime \\prime } } { f } = 0 , \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} \\mathcal { V } : = \\{ u \\in W _ 0 ^ { 1 , p } ( \\Omega ) | I ( u ) < 0 , ~ J ( u ) < d \\} . \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} 2 ( l _ i - l _ { i ' } ) & = l ' _ i - l ' _ { i ' } \\\\ & \\geq \\beta ( i , i ' ) + \\Delta ( i , i ' ) \\\\ & = i ' - i - 1 + \\Delta ( i , i ' ) \\\\ & \\geq 2 \\Delta ( i , i ' ) - 1 \\ , \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { L ^ 2 _ { \\sigma } ( E ) } : = \\int _ E | f ( x ) | ^ 2 \\cdot | x | ^ { - 2 \\sigma - n } \\ d x \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} \\int _ B | \\nabla ^ m u _ \\varepsilon | ^ 2 d x = t _ \\varepsilon ^ { \\ 2 m s - 2 } \\int _ B t _ \\varepsilon ^ { | x | ^ \\alpha } | u _ \\varepsilon | ^ { \\ 2 m s + | x | ^ \\alpha } d x . \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} \\tilde { x } & = \\frac { x } { \\mu } , & \\tilde { y } & = \\frac { y - \\zeta ( \\mu ) } { \\mu } , \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { h ( t ) } t = c _ 0 . \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} K = a c , \\ \\ \\ \\ H = \\frac { a + c } { 2 } . \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} q _ 1 = \\frac { ( n - 1 ) d ^ 2 - c - e } { ( n + 1 ) d - c - 2 } \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} & ( I _ { 3 p } \\otimes e _ 1 ) ^ \\top M _ \\Delta ( M _ + , M _ - , M _ 0 ) ( I _ { 3 p } \\otimes e _ 1 ) \\\\ = ~ & ( I _ { 3 p } \\otimes e _ 1 ) ^ \\top ( M _ \\Delta ( m _ + , m _ - , m _ 0 ) \\otimes I _ p ) ( I _ { 3 p } \\otimes e _ 1 ) \\\\ = ~ & ( I _ { 3 p } \\otimes e _ 1 ) ^ \\top ( M _ \\Delta ( m _ + , m _ - , m _ 0 ) \\otimes e _ 1 ) \\\\ = ~ & M _ \\Delta ( m _ + , m _ - , m _ 0 ) . \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} \\lambda _ n ( G ^ c ) & = x ^ T D ( G ^ c ) x \\\\ & = x ^ T ( J _ n - I _ n ) x + x ^ T A ( G ) x \\\\ & \\ge x ^ T ( J _ n - I _ n ) x + x ^ T A ( B _ 2 ( p , q ) ) x \\\\ & = x ^ T D ( B _ 2 ^ c ( p , q ) ) x . \\end{align*}"}
-{"id": "3266.png", "formula": "\\begin{align*} & \\delta _ r ^ { - 1 } \\widetilde { T } _ r = \\mu [ U ] \\left ( x ^ { ( 1 ) } \\otimes \\ldots \\otimes t ^ { ( i _ r ) } _ r \\otimes \\ldots \\otimes x ^ { ( m ) } \\right ) , \\\\ & \\delta _ s ^ { - 1 } \\widetilde { T } _ s = \\mu [ U ] \\left ( x ^ { ( 1 ) } \\otimes \\ldots \\otimes t ^ { ( i _ s ) } _ s \\otimes \\ldots \\otimes x ^ { ( m ) } \\right ) . \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} C _ { \\tilde \\epsilon } : = \\dd \\min _ { 1 \\leq i \\leq m } \\min _ { x \\in [ - 2 \\tilde { \\epsilon } , 0 ] } | \\phi _ i ' ( x ) | . \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} M \\int _ M ^ \\infty J ( y ) d y \\leq M ^ { 1 - \\alpha } \\int _ M ^ \\infty y ^ \\alpha J ( y ) d y \\leq \\int _ 1 ^ \\infty y ^ \\alpha J ( y ) d y : = M _ 2 \\mbox { f o r } M \\geq 1 , \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} p ( t , x ) = \\frac { 1 } { ( 2 \\pi ) ^ { d } } \\int _ { \\R ^ d } \\cos ( x \\cdot \\xi ) e ^ { - t \\Psi ( \\xi ) } \\ , d \\xi , t > 0 , \\ , x \\in \\R ^ d . \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} \\langle \\delta \\mathbf { N } , \\mathbf { r } _ j \\rangle = \\langle c ^ i \\mathbf { r } _ i , \\mathbf { r } _ j \\rangle = c ^ i = - \\langle \\mathbf { N } , \\delta \\mathbf { r } _ j \\rangle = - \\langle \\mathbf { N } , u _ i \\mathbf { N } + u \\mathbf { N } _ i \\rangle = - u _ i , \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} [ h , L _ 1 ] = L _ 1 , [ h , f ] = - f , [ L _ 1 , f ] = 2 h . \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} \\Lambda : = \\sum _ { i = 1 } ^ { k } p _ i \\log \\lambda _ i > 0 \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} x \\in D ( B ) = \\{ x \\in X \\ : ; \\ : ( x ' _ e ) _ { e \\in E } \\in X , \\ , x _ { ( \\frac 1 { k + 1 } , \\frac 1 { k } ) } \\bigl ( \\textstyle { \\frac 1 { k + 1 } } \\bigr ) = x _ { ( { \\frac 1 { k + 2 } } , \\frac 1 { k + 1 } ) } \\bigl ( { \\frac 1 { k + 2 } } \\bigr ) , \\ , x _ { ( \\frac 1 2 , 1 ) } ( 1 ) = 0 \\} . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} M _ { \\Gamma } ( t ) = \\frac { 1 } { 4 \\pi i } \\log \\sideset { } { ^ { \\star } } \\prod _ { \\psi } \\left ( \\frac { L ( 1 + 2 i t , \\psi ) L ( 1 + 2 i t , \\overline { \\psi } ) } { L ( 1 - 2 i t , \\psi ) L ( 1 - 2 i t , \\overline { \\psi } ) } \\right ) + O ( t \\log t ) . \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} { \\rm d i m } \\ , \\left ( \\bigsqcup _ { 1 \\le k \\le n _ 0 } R ^ { - 1 } ( \\partial \\Delta _ k ) \\right ) = 2 { \\rm a n d } H ^ 2 \\left ( \\bigsqcup _ { 1 \\le k \\le n _ 0 } R ^ { - 1 } ( \\partial \\Delta _ k ) , \\Z \\right ) = 0 . \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} \\hat { Z } _ l = \\sum _ { 1 \\leq i < j } ^ { n _ l } L _ { i j } ^ 2 - \\bigg ( \\sum _ { i = 1 } ^ { l } r _ i ^ 2 \\bigg ) \\bigg ( \\sum _ { i = 1 } ^ { l } \\frac { 1 } { r _ i ^ 2 } f _ i ( \\Omega _ i ) \\bigg ) , ~ ~ ~ l = 2 , \\cdots , N . \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} \\P ( X _ { n + 1 } = X _ n + 1 \\ , | \\ , X _ 0 , \\ldots , X _ n ) & = 1 - \\P ( X _ { n + 1 } = X _ n - 1 \\ , | \\ , X _ 0 , \\ldots , X _ n ) \\\\ & = \\frac { w _ n ( X _ n ) } { w _ n ( X _ n - 1 ) + w _ n ( X _ n ) } . \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} T _ { \\Phi ( \\varepsilon ) } ( x , y , z ^ I ) = \\left \\{ d \\in \\Re ^ n \\times \\Re ^ { q + p } \\times \\Re ^ p : { \\cal J } F _ \\varepsilon ( x , y , z ^ I ) d = 0 \\right \\} \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} v _ n x v _ n ^ { - 1 } - x & = \\sum \\limits _ { k = n + 1 } ^ \\infty \\left ( v _ n y _ k v _ n ^ { - 1 } - y _ k \\right ) + \\sum \\limits _ { k = 1 } ^ n \\left ( v _ n y _ k v _ n ^ { - 1 } - y _ k \\right ) \\\\ \\stackrel { \\eqref { i n v v } } { = } & \\sum \\limits _ { k = 1 } ^ { n } \\left ( v _ n y _ k v _ n ^ { - 1 } - y _ k \\right ) \\stackrel { \\eqref { v v v v } } { = } - 2 y _ n + \\sum \\limits _ { k = 1 } ^ { n - 1 } \\left ( v _ n y _ k v _ n ^ { - 1 } - y _ k \\right ) \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} [ \\rho ( L _ { - 1 } ) , T ] \\ , & = \\ , 0 \\\\ [ \\rho ( L _ { 0 } ) , T ] \\ , & = \\ , - 2 T \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} u ( \\beta ) = \\left ( e ^ \\beta + \\beta - \\frac { 1 } { 2 } \\right ) ( 1 - \\beta e ^ { 1 - \\beta } ) - \\frac { 2 e ^ 3 } { 3 ^ { 5 / 2 } \\sqrt { 2 \\pi } } \\beta e ^ { - \\beta } \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} \\hat p \\circ \\tilde s \\circ e ^ * | _ { U \\cup \\widetilde U _ 1 } = \\hat r _ u | _ { U \\cup \\widetilde U _ 1 } ; \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} & | \\int _ { t } ^ { \\infty } d s \\frac { \\lambda '' ( s ) } { r } \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\frac { 4 r \\left ( p ^ 2 + r ^ 2 + 1 \\right ) } { \\left ( \\left ( p ^ 2 - r ^ 2 + 1 \\right ) ^ 2 + 4 r ^ 2 \\right ) ^ { 3 / 2 } } | \\\\ & \\leq C _ { x \\geq t } \\left ( | \\lambda '' ( x ) | ( 1 + ( x - t ) ^ { 2 } ) \\right ) \\frac { 1 } { r ^ { 2 } } \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} J \\psi ( x H ) : = \\int _ H \\psi ( h x H ) d h , \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} h _ n ( x _ 1 , \\dots , x _ n ) = \\prod _ { r = 1 } ^ n \\left ( \\prod _ { s = 1 } ^ r ( 1 + x _ s ) - 1 \\right ) , \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} \\int _ { D \\setminus U } M _ D ( y , \\infty ) \\omega _ U ^ x ( d y ) & = \\frac { 1 } { \\mathbb { E } _ { x _ 0 } \\tau _ D } \\mathbb { E } _ x \\big [ \\mathbb { E } _ { X _ { \\tau _ U } } \\tau _ D \\big ] = \\frac { 1 } { \\mathbb { E } _ { x _ 0 } \\tau _ D } \\mathbb { E } _ x \\left [ \\int _ { \\tau _ U } ^ { \\tau _ D } \\mathbf 1 d t \\right ] \\\\ & < \\frac { \\mathbb { E } _ { x } \\tau _ D } { \\mathbb { E } _ { x _ 0 } \\tau _ D } = M _ D ( x , \\infty ) , \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} \\epsilon _ 1 : = \\inf _ { 1 \\leq i \\leq m } \\inf _ { x \\in [ - K _ 0 , 0 ] } { | \\phi _ i ' ( x ) | } > 0 . \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} f _ { k + 1 } ( x ) D ^ { ( k ) } \\left ( G ( x ) \\right ) + \\dots + f _ 2 ( x ) D ^ { ( 1 ) } \\left ( G ( x ) \\right ) + f _ 1 ( x ) G ( x ) + f _ 0 ( x ) = 0 \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} a _ { n + 1 } \\leq 1 + \\sum _ { i = 1 } ^ { n } a _ i . \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} H _ { k + m } \\leq \\sum _ { i = 1 } ^ { k + m - 1 } H _ { i } + 1 . \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} \\tau ( c ) ( n ) = c ( n + 1 ) - c ( n ) ^ 2 c \\in \\R ^ { \\Z } n \\in \\Z . \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} \\mathbb { A } & = A ( t ) + a ( t ) d t \\\\ \\mathbb { c } & = c ( t ) + \\gamma ( t ) d t \\\\ \\mathbb { t } & = \\tau ( t ) + j ( t ) d t \\\\ \\mathbb { B } & = B ( t ) + b ( t ) d t \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} \\alpha _ 1 = c ; \\alpha _ { j } = c - ( \\alpha _ 1 ^ 2 + \\alpha _ 2 ^ 2 + \\cdots + \\alpha _ { j - 1 } ^ 2 ) ; j = 2 , \\cdots , p - 1 . \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} r = \\log _ p [ k : k ^ p ] = \\log _ p [ K ( B ) : K ( B ) ^ p ] = \\dim B . \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} \\begin{aligned} \\mu ( \\xi ) & = \\frac { n - 1 } { 2 } \\Big \\{ - n \\cosh ^ 2 ( \\xi ) + 2 \\cosh ( \\xi ) [ ( 1 + \\xi ^ 2 ) \\cosh ( \\xi ) + n \\xi \\sinh ( \\xi ) ] + \\\\ & - n ( 1 + \\xi ^ 2 ) \\sinh ^ 2 ( \\xi ) \\Big \\} , \\end{aligned} \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} ( h + 2 n ) r ( h - 2 ) = ( h - 2 n ) r ( h + 2 ) \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} \\Gamma _ { \\mathfrak { m } } \\left ( H _ { 0 } ( F ) \\otimes _ { R } B \\right ) = 0 . \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} \\kappa _ \\omega ( x ) : = \\rho ( H ^ \\omega _ x ) \\cdot H ^ \\omega _ x , \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} h = e _ 1 + f _ 1 = e _ 2 + f _ 4 = e _ 3 + f _ 2 = e _ 4 + f _ 3 . \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} F ^ { ( \\alpha ) } \\left [ f \\right ] = \\| f \\| _ { \\alpha } ^ 2 - 2 \\| f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 = \\| f \\| _ { \\alpha } ^ 2 - 2 , \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} ( 2 k + \\delta _ { p q } ) _ { p q } = ( k + \\delta _ { p q } ) _ p + k _ q \\ , \\cdot \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} H _ 2 ( p , q ) = \\frac { 1 } { 3 } ( p _ 1 ^ 2 + p _ 2 ^ 2 + p _ 3 ^ 2 + 3 e ^ { ( q _ 1 - q _ 2 ) } ( p _ 1 ^ 2 p _ 2 + p _ 1 p _ 2 ^ 2 ) + 3 e ^ { ( q _ 2 - q _ 3 ) } ( p _ 2 ^ 2 p _ 3 + p _ 2 p _ 3 ^ 2 ) \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} \\mathcal { B } \\left ( F ( v ) \\cdot G \\left ( e ^ { \\xi \\mathcal { A } } \\overline v \\right ) \\right ) = \\mathcal { B } \\left ( F ( v ) \\cdot e ^ { \\xi \\mathcal { A } } \\mathcal { N } _ 2 ( \\xi , \\mathcal { A } , v ) \\right ) . \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} \\mathcal { D } ( L _ a ) & : = \\{ f \\in \\mathcal { D } ( L _ { \\max } ) \\ \\mid \\ W ( f , g ; a ) = 0 g \\in \\mathcal { D } ( L _ { \\max } ) \\} , \\\\ \\mathcal { D } ( L _ b ) & : = \\{ f \\in \\mathcal { D } ( L _ { \\max } ) \\ \\mid \\ W ( f , g ; b ) = 0 g \\in \\mathcal { D } ( L _ { \\max } ) \\} . \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} f = f _ 0 ^ 4 + f _ 1 ^ 4 g + f _ 2 ^ 4 g ^ 2 + f _ 3 ^ 4 g ^ 3 = ( f _ 0 ^ 2 + f _ 2 ^ 2 g ) ^ 2 + ( f _ 1 ^ 2 + f _ 3 ^ 2 g ) ^ 2 g ( f _ i \\in k ^ { 1 / 4 } \\cdot k ( X ) ) . \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} \\begin{bmatrix} D _ { - \\beta } & \\alpha C \\\\ \\beta B & A _ { - \\alpha } \\end{bmatrix} \\begin{bmatrix} u _ 1 \\\\ u _ 2 \\end{bmatrix} = \\begin{bmatrix} D _ \\alpha & - \\beta C \\\\ - \\alpha B & A _ \\beta \\end{bmatrix} \\begin{bmatrix} u _ 1 \\\\ u _ 2 \\end{bmatrix} - \\gamma \\begin{bmatrix} v _ 1 \\\\ v _ 2 \\end{bmatrix} , \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} \\int _ { X } \\left | \\frac { | f ( x ) | } { \\| f \\| _ 2 } - \\frac { 1 } { \\sqrt { \\mu ( X ) } } \\right | ^ 2 \\ , d \\mu ( x ) & = \\frac { 1 } { \\| f \\| _ 2 ^ 2 } \\int _ { X } | f ( x ) | ^ 2 \\ , d \\mu ( x ) + \\frac { 1 } { \\mu ( X ) } \\int _ { X } \\ , d \\mu ( x ) \\\\ & \\quad - 2 \\frac { 1 } { \\| f \\| _ 2 \\sqrt { \\mu ( X ) } } \\int _ { X } | f ( x ) | \\ , d \\mu ( x ) \\\\ & = 2 \\left ( 1 - \\frac { 1 } { \\| f \\| _ 2 \\sqrt { \\mu ( X ) } } \\int _ { X } | f ( x ) | \\ , d \\mu ( x ) \\right ) \\\\ & = c _ f \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} Q ( B _ 2 , B _ 3 ) \\ = \\ Q ( B _ 3 , & B _ 4 ) \\ = \\ Q ( B _ 4 , B _ 2 ) \\ = \\\\ Q ( B _ 3 , B _ 1 ) \\ = \\ Q ( & B _ 1 , B _ 4 ) \\ = \\ 1 , \\\\ Q ( B _ 1 ^ - , B _ 2 ^ - ) \\ & = \\ 0 , \\\\ Q ( B _ 0 , B _ 1 ^ - ) \\ = \\ & Q ( B _ 0 , B _ 2 ^ - ) \\ = \\ 2 . \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} \\alpha ( n ) : = \\int _ { B _ { n } ( x _ { 0 } ) } | f ( x ) | d { \\rm v o l } ( x ) \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} \\int _ { G } c ( g ^ { - 1 } x ) d \\mu ( g ) = 1 , \\quad \\mbox { f o r a l l } ~ x \\in M , \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} \\begin{aligned} A ( t ) & = p _ 2 ( t ) + e ^ { q _ 3 ( t ) - q _ 2 ( t ) } p _ 3 ( t ) = \\\\ & \\frac { 1 } { 2 } \\left ( H _ 0 \\left ( 1 + e ^ { q _ 3 - q _ 2 } \\right ) + \\sqrt { ( 1 - e ^ { q _ 3 ( t ) - q _ 2 ( t ) } ) ( 4 H _ 1 - ( 1 + e ^ { q _ 3 ( t ) - q _ 2 ( t ) } ) H _ 0 ^ 2 ) } \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} H _ { T _ u w } = H _ { \\Pi u } T _ { \\overline w } + T _ u H _ w - \\langle w | \\ , \\cdot \\rangle \\Pi u \\ . \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} \\frac { d } { d t } L ( t ) & = - I ( u ( t ) ) \\le \\int _ \\Omega | u | ^ q \\ln | u | d x < \\frac { 1 } { \\alpha } \\| u \\| _ { q + \\alpha } ^ { q + \\alpha } \\\\ & < \\frac 1 \\alpha \\Big [ C _ * ^ { \\theta ( q + \\alpha ) } \\alpha ^ { - \\frac { \\theta ( q + \\alpha ) } { p } } [ ( \\Omega ) ] ^ { \\frac { s ( 1 - \\theta ) ( q + \\alpha ) } { 2 } } \\Big ] ^ { \\frac { p } { p - \\theta ( q + \\alpha ) } } 2 ^ \\kappa L ^ \\kappa ( t ) . \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} { \\mu _ k ^ { } } ^ { } = { e _ k ^ { } } ^ { - 1 } , \\\\ { \\mu _ l ^ { } } ^ { } = { e _ l ^ { } } ^ { - 1 } . \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} u ( t ) = e ^ { t \\mathcal { L } } u _ 0 + \\int _ 0 ^ t e ^ { ( t - \\xi ) \\mathcal { L } } f \\left ( u ( \\xi ) , \\overline u ( \\xi ) \\right ) d \\xi . \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} \\partial _ r [ w ( \\theta ) ] ^ \\top \\big ( \\mathbb { E } _ \\theta [ f ( \\textbf { X } ) ] - \\bar { f } \\big ) = 0 ~ ~ r \\in \\{ 1 , \\dots , k \\} . \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} \\left | \\ddot x _ k e _ 2 ^ \\perp + \\ddot y _ k e _ 3 ^ \\perp \\right | ^ 2 & = \\left | \\left ( \\ddot x _ k ( 1 - \\dot x _ k ^ 2 ) - \\ddot y _ k \\dot x _ k \\dot y _ k \\right ) e _ 2 + \\left ( \\ddot y _ k ( 1 - \\dot y _ k ^ 2 ) - \\ddot x _ k \\dot x _ k \\dot y _ k \\right ) e _ 3 \\right | ^ 2 + o ( 1 ) \\\\ & = | \\ddot x _ k \\dot y _ k - \\ddot y _ k \\dot x _ k | ^ 2 + o ( 1 ) = | \\kappa | ^ 2 + o ( 1 ) , \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} \\begin{aligned} \\gamma ( 1 - \\rho _ n ) - ( 1 - \\rho _ n \\alpha ^ { - 1 } ) \\leq 0 , \\\\ \\gamma ( 1 - \\sigma _ m \\alpha ) - ( 1 - \\sigma _ m ) \\leq 0 , \\end{aligned} \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N } \\psi _ { N , j } ^ { \\ast } ( Q ) = \\psi _ { N , 1 } ( Q ) + O ( q ^ { - 1 } ) , \\sum _ { j = 1 } ^ { N } \\psi _ { N , j } ( Q ) = \\psi _ { N , 1 } ^ { \\ast } ( Q ) + O ( q ^ { - 1 } ) . \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} \\tau ( x , y ) = \\tau _ y ( x ) = \\sigma ^ { - 1 } _ { \\sigma _ x ( y ) } ( x ) , \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} _ { v _ { 4 , 2 } ^ { 1 } } & = \\frac { - \\partial _ { 1 2 } v _ { 4 , c } ^ { 1 } ( s , | x + y | ) } { | x + y | ^ { 2 } ( s - t ) } ( ( x + y ) \\cdot y ) ( \\widehat { x } \\cdot ( x + y ) ) \\\\ & + \\frac { \\partial _ { 1 } v _ { 4 , c } ^ { 1 } ( s , | x + y | ) } { ( s - t ) | x + y | } \\left ( - y \\cdot \\widehat { x } + \\frac { ( \\widehat { x } \\cdot ( x + y ) ) ( y \\cdot ( y + x ) ) } { | x + y | ^ { 2 } } \\right ) \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} \\triangle _ { m , a } ( x _ { 1 } , \\cdots , x _ { n } ) : = \\sum _ { i = 1 } ^ { n } a _ { i } p _ { m } ( x _ { i } ) . \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} \\max _ { i = 1 , 2 } H ( P _ { i } ) \\leq M , \\max _ { i = 1 , 2 } | P _ { i } ( \\xi ) | < M ^ { - \\widehat { w } _ { n } ( \\xi ) + \\epsilon } . \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} \\| f _ n - f _ m \\| _ { \\alpha } = \\left ( 2 \\| f _ n - f _ m \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 + \\| \\widehat { q _ 1 } ( f _ n - f _ m ) \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 + \\| \\widehat { q _ 2 } ( f _ n - f _ m ) \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 \\right ) ^ { 1 / 2 } < \\epsilon \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} e _ R ( \\mathcal I ( 1 ) _ t ^ { [ d _ 1 ] } , \\ldots , \\mathcal I ( r ) _ t ^ { [ d _ r ] } ; R ) = \\frac { 1 } { s _ t ^ d } e _ R ( ( I ( 1 ) _ { t , s _ t } ) ^ { [ d _ 1 ] } , \\ldots , ( I ( r ) _ { t , s _ t } ) ^ { [ d _ r ] } ; R ) \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} \\sup _ { u \\in Q } I ( h _ 0 ( u ) ) = \\max _ { u \\in Q } I ( h _ 0 ( u ) ) < + \\infty . \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\max \\{ C _ { 2 , 1 } ( \\varepsilon ) , C _ { 2 , 2 } ( \\varepsilon ) , C _ { 2 , 3 } ( \\varepsilon ) \\} = 0 . \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} f _ L ( 0 , 0 ; \\mu ) & = 0 , & g _ L ( 0 , 0 ; \\mu ) & = 0 , \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} \\Gamma : V = \\bigoplus _ { g \\in G } V _ g . \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c l } a _ { 1 1 } X _ 1 ^ { d _ 1 } & + a _ { 1 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { 1 t } X _ t ^ { d _ t } = g _ 1 ( X _ 1 , \\ldots , X _ k ) \\\\ a _ { 2 1 } X _ 1 ^ { d _ 1 } & + a _ { 2 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { 2 t } X _ t ^ { d _ t } = g _ 2 ( X _ 1 , \\ldots , X _ k ) \\\\ \\ ; \\vdots & & \\quad \\vdots \\\\ a _ { n 1 } X _ 1 ^ { d _ 1 } & + a _ { n 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { n t } X _ t ^ { d _ t } = g _ n ( X _ 1 , \\ldots , X _ k ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} g _ 2 ( x ) = u ^ 2 ( x - 1 ) ^ { r _ 2 } + u ^ 3 ( x - 1 ) ^ { r _ 6 } p _ 6 ( x ) . \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} S _ { r , \\ell } = \\sum _ { k = 0 } ^ { \\ell - 1 } \\phi _ { \\ell } ^ r ( \\zeta _ { \\ell } ^ k q ^ { \\frac { 1 } { \\ell } } ) . \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} \\Psi ^ { - 1 } ( \\{ P \\in \\mathrm { S p e c } ( k [ X _ { 1 } , . . . , X _ { n } ] ) : f _ { 1 } , . . . . , f _ { m } \\in P \\} ) = \\left \\{ p \\in S _ { n } ^ { K } ( k ) : \\bigwedge _ { i = 1 } ^ { m } f _ { i } ( \\overline { v } ) = 0 \\in p \\right \\} \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} a ( z ) = | z | ^ { - 1 } \\ , \\left ( I d - \\frac { z \\otimes z } { | z | ^ 2 } \\right ) . \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a , q ^ { - 1 } / b , c q ^ { - 1 } , d q ^ { - 1 } , q ^ { - 1 } ; q ^ 2 ) _ k } { ( q ^ 2 / a , a q ^ 2 , b q ^ 2 , q ^ 2 / c , q ^ 2 / d , q ^ 2 ; q ^ 2 ) _ k } \\bigg ( \\frac { b q ^ 7 } { c d } \\bigg ) ^ k \\equiv 0 \\pmod { [ n ] } . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( 1 / x ) \\left [ x \\eta _ { \\mu } ( 1 / x ) \\right ] ^ { 1 / ( k - 1 ) } = \\eta _ { \\nu ^ { \\boxtimes k } } ( 1 / x ) , x \\in ( 0 , + \\infty ) , \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} \\left ( - \\partial _ { t t } + \\partial _ { r r } + \\frac { 1 } { r } \\partial _ { r } - \\frac { 1 } { r ^ { 2 } } \\right ) \\left ( V ( \\partial _ { t } v _ { 4 } ) \\right ) = V \\left ( \\partial _ { t } v _ { 4 , c } \\right ) + 2 \\partial _ { t } v _ { 4 , c } \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ t \\widetilde { u } ( t , x ) = \\Delta \\widetilde { u } ( t , x ) - \\nabla \\cdot \\left ( \\widetilde { u } ( t , x ) F _ { A } ( K \\ast \\widetilde { u } ( t , x ) ) \\right ) , t > 0 , ~ x \\in \\R ^ d \\\\ & \\widetilde { u } ( 0 , x ) = u _ 0 ( x ) . \\end{cases} \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} & \\Phi ^ { - 1 } ( ( \\phi ^ { - 1 } ) ^ { - 1 } ( L _ { ( r , A , p ' ) } ) ) = ( \\phi ^ { - 1 } ) ^ { - 1 } ( L _ { ( r , A , p ) } ) , \\\\ & \\Phi ^ * ( \\phi ^ { - 1 } ) ^ * \\mathcal { L } _ { ( r , A , p ' , q ' ) } \\cong ( \\phi ^ { - 1 } ) ^ * \\mathcal { L } _ { ( r , A , p , q ) } . \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} G _ \\leftrightarrow ( x , y ) : = u ( x ) v ( y ) - u ( y ) v ( x ) \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} \\xi _ L ( 0 ) = \\xi _ R ( 0 ) = 0 . \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} S ^ { ( s ) } _ n & = - i \\lambda _ s \\sum _ { k = s } ^ n k ! [ \\rho ^ k ] \\phi ^ s ( \\rho ) \\cdot B _ { n , k } \\left ( b _ 1 , b _ 2 , \\ldots \\right ) \\\\ & = - i \\lambda _ s [ \\phi ^ n ] \\left ( \\big ( \\phi ( \\rho ( \\phi ) ) \\big ) ^ s \\right ) = - i \\lambda _ s [ \\phi ^ n ] \\phi ^ s = - i \\lambda _ s \\delta _ { n s } . \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} \\widehat { v _ { 2 , 0 } } ( \\xi ) = - \\frac { 1 } { \\xi \\pi } \\int _ { 0 } ^ { \\infty } F ( t ) \\sin ( t \\xi ) d t \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} \\| f \\| _ { M _ m ^ { r , s } ( \\mathbb { R } ^ d ) } = \\| m V _ g f \\| _ { L _ { x , \\omega } ^ { r , s } ( \\mathbb { R } ^ { 2 d } ) } = \\left ( \\int _ { \\mathbb { R } ^ d } \\left ( \\int _ { \\mathbb { R } ^ d } | V _ g f ( x , \\omega ) m ( x , \\omega ) | ^ r d x \\right ) ^ { \\frac { s } { r } } d \\omega \\right ) ^ { \\frac { 1 } { s } } < \\infty . \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} \\norm { X } _ 2 = \\norm [ 1 ] { x _ 1 \\otimes \\cdots \\otimes x _ d } _ 2 = \\prod _ { i = 1 } ^ d \\norm [ 1 ] { x _ i } _ 2 . \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} \\left | T _ { \\eqref { e _ n + m _ t e n s o r _ d i f f } } \\right | \\leq C ( | \\sigma | + 1 ) \\sum _ { k = 1 } ^ { N _ 1 + N _ 2 } \\exp \\left ( - C ( k , I _ { N _ 1 , N _ 2 } ) \\right ) = C \\left ( | \\sigma | + 1 \\right ) . \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} | ( \\theta , w ) | = \\sum _ { \\alpha } \\bigl ( 1 + \\log _ 2 ( w ( \\alpha ) + 1 ) \\bigr ) , \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} b ^ i _ 2 > b ^ j _ 3 \\ \\ b ^ i _ 1 > b ^ j _ 3 & \\Longrightarrow Q ( A _ i , A _ j ) \\geq 2 \\cdot 6 - 9 = 3 , \\\\ b ^ i _ 1 \\ > \\ b ^ j _ 2 & \\Longrightarrow Q ( A _ i , A _ j ) \\geq 2 \\cdot 6 - 9 = 3 . \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} | V ( G ) \\cap \\Delta | & = | V ( G ) \\cap \\Delta _ { 0 , 1 } | + | G _ { \\Delta _ { 1 , k } } | + | V ( G ) \\cap \\Delta _ { k , k + 1 } | \\\\ & \\le 2 c ( \\partial _ G \\Delta _ { 0 , 1 } + \\partial _ G \\Delta _ { k , k + 1 } ) + 8 c + 4 + f ( \\lceil 8 c + 4 \\rceil ) \\\\ & \\le 2 c \\partial _ G \\Delta + 8 c ^ 2 + 1 6 c + 4 + f ( \\lceil 8 c + 4 \\rceil ) \\le g ( \\partial _ G \\Delta - 2 ) - 2 , \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 f } { \\partial t _ 3 \\partial t _ 2 } ( 0 , t _ 1 , 0 , t _ 1 ) = t _ 1 ^ { - 2 } \\left ( 1 + \\frac { t _ 1 ^ 2 ( z ' ( 0 ) ) ^ { - 1 } z ''' ( 0 ) } { 6 } - \\frac { t _ 1 ^ 2 ( ( z ' ( 0 ) ) ^ { - 1 } z '' ( 0 ) ) ^ 2 } { 4 } + . . . \\right ) \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\left \\| ( { P _ * } ( 4 b t , k ) - \\Pi ( k ) \\cdot \\chi _ { E _ s ^ c } ) \\int _ { 2 a t } ^ { 2 b t } \\frac { e ^ { i u ^ 2 / ( 4 t ) } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) e ^ { - i k u } d u \\right \\| _ { 2 , \\sigma } = 0 \\ , . \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} \\Omega _ { k } \\omega _ { j } = \\delta _ { j k } - \\omega _ { k } \\omega _ { j } , \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} \\widetilde { \\varphi } _ 1 ^ P ( \\lambda ^ 1 _ { p - 1 } a b ^ { [ k p + p - 2 ] } ) = \\left [ \\beta Q ^ { p } a b ^ { [ k p + p - 2 ] } \\right ] . \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} \\mathcal { A T } _ k = \\{ r _ k ( T ) : T \\in \\mathcal { T } \\} . \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} \\delta _ { \\varepsilon } X = & \\check { \\rho } ( \\varepsilon ) ( X ) , \\\\ \\delta _ { \\varepsilon } A ^ i = & d \\varepsilon ^ i + \\check { C } ^ i { } _ { j k } A ^ j \\varepsilon ^ k + ( \\check { \\Omega } ^ + ) ^ i { } _ { \\alpha j } \\varepsilon ^ j D _ + \\sigma ^ \\alpha + ( \\check { \\Omega } ^ - ) ^ i { } _ { \\alpha j } \\varepsilon ^ j D _ - \\sigma ^ \\alpha , \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 \\overline { s } } \\partial _ { n + 1 } v \\bigg | _ { C _ { \\overline { s } , 1 / 2 } ' } & = \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 \\overline { s } } ( \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { w } ) \\bigg | _ { C _ { \\overline { s } , 1 / 2 } ' } \\\\ & = - \\lim _ { x _ { n + 1 } \\rightarrow 0 } \\sum _ { j , k = 1 } ^ { n } \\partial _ { j } a _ { j k } \\partial _ { k } \\tilde { w } \\bigg | _ { C _ { \\overline { s } , 1 / 2 } ' } = 0 . \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} N _ t = \\sum _ { i = 1 } ^ t n _ i , D _ t = \\dim ( \\Pi ^ { ( t ) } ) = \\sum _ { i = 1 } ^ t n _ i m _ i . \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} \\eta ^ \\leftarrow ( B ' ) = \\eta ^ { - 1 } ( B ' ) = \\{ x \\in X \\mid \\eta ( x ) \\in B ' \\} \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} a = b = c = d = 1 , \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} C = \\bigcup _ { \\zeta < \\xi } { C _ { \\zeta } } \\cup \\bigg \\{ \\sup { \\bigcup _ { \\zeta < \\xi } { C _ { \\zeta } } } \\bigg \\} \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} i a _ { k - 2 } ( k - 1 ) ! X _ e \\frac { i } { X _ e } i a _ { j - 2 } ( j - 1 ) ! X _ e & = - i a _ { j - 2 } a _ { k - 2 } ( j - 1 ) ! ( k - 1 ) ! X _ e . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} & { \\rm c h } [ W ( \\Lambda _ 0 ) ] ( q ) \\leq H S _ { q } ( J _ { \\infty } ( R _ { W ( \\Lambda _ 0 ) } ) ) = H S _ { q } ( g r ^ { G } ( J _ { \\infty } ( R _ { W ( \\Lambda _ 0 ) } ) ) ) \\\\ & \\leq H S _ { q } ( g r ^ { G } ( J _ { \\infty } ( A ) ) , \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} \\begin{cases} [ g ^ n ( t ) , h ^ n ( t ) ] \\subset [ g ( t + T ) , h ( t + T ) ] , & t \\geq 0 , \\\\ U ^ n ( t , x ) \\preceq U ( t + T , x ) , & t \\geq 0 , \\ ; x \\in [ g ^ n ( t ) , h ^ n ( t ) ] . \\end{cases} \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} u _ { t } = e ^ { t \\Delta } u _ 0 - \\int _ 0 ^ t \\nabla \\cdot e ^ { ( t - s ) \\Delta } ( u _ { s } F _ { A } ( K \\ast u _ { s } ) ) \\ d s , 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} 1 - 2 \\beta r e ^ { i f ( r ) } \\psi ' _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) & = \\frac { 2 \\beta } { f ( r ) } \\Im \\frac { 1 } { 1 - \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) } - 2 \\beta r e ^ { i f ( r ) } \\psi ' _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) \\\\ & = 2 \\beta \\int _ { \\mathbb { R } _ { + } } \\left [ \\frac { 1 } { f ( r ) } \\frac { t r \\sin ( f ( r ) ) } { | 1 - t r e ^ { i f ( r ) } | ^ { 2 } } - \\frac { t r e ^ { i f ( r ) } } { ( 1 - t r e ^ { i f ( r ) } ) ^ { 2 } } \\right ] \\ , d \\mu _ { 1 } ( t ) . \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} B = M + \\sqrt { \\frac { p \\left ( n + 2 \\right ) } { \\left ( 1 - p \\right ) } } l l ^ { \\top } , \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} ( T \\eta ) _ n + W _ n = \\eta _ n + W _ { n - 1 } . \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} \\overline { \\widehat { w } } ( \\xi ) = \\limsup _ { n \\to \\infty } \\frac { \\widehat { w } _ { n } ( \\xi ) } { n } , \\qquad \\overline { \\lambda } ( \\xi ) = \\limsup _ { n \\to \\infty } n \\lambda _ { n } ( \\xi ) . \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} I ^ { [ 1 ] } _ { S V } & = \\frac { 1 } { 2 } J \\left ( \\sqrt { 2 F ^ { [ 1 ] } _ { 0 0 } \\left ( m \\right ) } \\right ) + \\frac { 1 } { 2 } J \\left ( \\sqrt { 2 F ^ { [ 1 ] } _ { 0 1 } \\left ( m \\right ) } \\right ) , \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} \\mu _ { N , m } ( d x | y ) = \\mu _ { N , m } ( d x ^ { \\Lambda _ 1 } | x ^ { \\Lambda _ 2 } , y ) \\bar { \\mu } _ { N , m } ( d x ^ { \\Lambda _ 2 } | y ) . \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} \\partial ( v S _ s ( m ) ) = ( - 1 ) ^ { \\deg v + 1 } \\sum _ { \\epsilon , \\ell } ( - 1 ) ^ \\ell v _ { \\epsilon , \\ell } S _ { s - 1 } ( \\beta ^ { 1 - \\epsilon } P ^ \\ell m ) . \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} G ( p , y ) & = G ( p , y _ 0 ) + \\int _ { a } ^ { r ( y ) } \\langle \\nabla G ( p , \\gamma ( s ) ) , \\dot { \\gamma } ( s ) \\rangle \\ , d s \\\\ & \\leq G ( p , y _ 0 ) + C \\int _ { a } ^ { r ( y ) } \\sqrt { Q _ { \\frac { r ( \\gamma ( s ) ) } { 4 } } \\big ( r ( \\gamma ( s ) ) \\big ) } G ( p , \\gamma ( s ) ) \\ , d s . \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} \\| f \\| _ 1 = \\left ( 1 - \\frac { c _ f } { 2 } \\right ) \\sqrt { \\mu ( X ) } \\| f \\| _ 2 . \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} f _ i ( u ) = \\sum _ { j = 1 } ^ m a _ j ( u _ j - u _ j ^ * ) \\mbox { f o r s o m e c o n s t a n t s $ a _ j $ , $ j = 1 , . . . , m $ } . \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} e = \\left ( \\frac { 1 } { 1 - \\lambda } \\right ) ^ 2 ( c - d ) ( d - \\lambda c ) . \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} & { \\rm C o e f f } _ { m _ { 1 } , \\cdots m _ { n - 1 } } ( \\sum _ { ( \\lambda _ { 1 } \\cdots \\lambda _ { n - 1 } ) \\in \\mathbb { N } ^ { n - 1 } } M _ { 1 } x _ 1 ^ { \\lambda _ 1 } \\cdots x _ { n - 1 } ^ { \\lambda _ { n - 1 } } ) \\\\ & = { \\rm C o e f f } _ { m _ { 1 } , \\cdots m _ { n - 1 } } ( \\sum _ { ( k _ { 1 } \\cdots k _ { n - 1 } ) \\in \\mathbb { N } ^ { n - 1 } } N _ { 1 } x _ 1 ^ { k _ 1 } \\cdots x _ { n - 1 } ^ { k _ { n - 1 } } ) \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} 2 H _ { k + 1 } = 2 ^ { k + 1 } > 2 ^ { k + 1 } - 1 = H _ { k + 2 } . \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} \\Psi _ e : Y _ e \\to Y , Y : = X \\times _ k \\kappa , Y _ e : = X _ e \\times _ k \\kappa . \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} ( X , \\N [ X ] \\to M ) \\otimes ( Y , \\N [ Y ] \\to N ) = ( X \\wedge Y , \\N [ X \\wedge Y ] \\to M \\otimes N ) \\to ( Z , \\N [ Z ] \\to L ) \\ , , \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} b _ { p , q } ( z ) = \\begin{cases} \\sqrt { \\frac { q ! } { p ! } } ( - 1 ) ^ { q } \\ , z ^ { p - q } L _ q ^ { ( p - q ) } ( | z | ^ 2 ) , & \\ p \\ge q ; \\\\ [ 1 e x ] \\sqrt { \\frac { p ! } { q ! } } ( - 1 ) ^ { p } \\ , \\overline { z } ^ { q - p } L _ p ^ { ( q - p ) } ( | z | ^ 2 ) , & \\ p \\le q . \\end{cases} \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} \\langle \\sigma _ k , D u _ k \\rangle = F _ k ^ * ( \\sigma _ k ) + F _ k ( D u _ k ) , \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} \\partial \\Omega _ { \\rho _ { 1 } } \\cap \\mathbb { T } = \\{ t \\in \\mathbb { T } : R ( t ) = 1 \\} \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} P ( x ) & = x _ { 1 } x _ { 2 } x _ { 3 } + x _ { 1 } x _ { 4 } x _ { 5 } + x _ { 2 } x _ { 4 } x _ { 6 } + x _ { 3 } x _ { 5 } x _ { 6 } \\\\ & = \\tfrac { 1 } { 2 } ( x _ { 1 } + x _ { 6 } ) ( x _ { 2 } + x _ { 5 } ) ( x _ { 3 } + x _ { 4 } ) + \\tfrac { 1 } { 2 } ( x _ { 1 } - x _ { 6 } ) ( x _ { 2 } - x _ { 5 } ) ( x _ { 3 } - x _ { 4 } ) . \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} , \\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix} \\right ) \\ \\ \\left ( \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} \\right ) , \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} | 3 \\Phi ( \\xi _ 1 , \\eta _ 1 , \\xi , \\eta ) - \\tau + \\xi ^ 3 + \\eta ^ 3 | & = | 3 \\xi _ 1 \\xi ( \\xi _ 1 + \\xi ) + 3 \\eta _ 1 \\eta ( \\eta _ 1 + \\eta ) - \\tau + \\xi ^ 3 + \\eta ^ 3 | \\\\ & = | ( \\tau _ 1 - \\xi _ 1 ^ 3 - \\eta _ 1 ^ 3 ) - ( \\tau _ 1 + \\tau - ( \\xi _ 1 + \\xi ) ^ 3 - ( \\eta _ 1 + \\eta ) ^ 3 ) | \\\\ & \\lesssim \\max ( L _ 1 , L _ 2 ) . \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} U ( t , x ) \\preceq U ^ n ( t , x ) = \\Psi ^ { c _ n } ( x - c _ n t ) \\ \\mbox { f o r } \\ t \\geq 0 , \\ x \\in \\R , \\ n > N . \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} \\inf \\{ R : s \\cdot P _ + \\cap s p t ( V ) = \\emptyset , \\ \\forall s > R \\} \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} \\underline { x } ^ { \\prime } _ { i } = ( 0 , \\ldots , 0 , x _ { 0 } , x _ { 1 } , \\ldots , x _ { n } , 0 , 0 , \\ldots , 0 ) \\in \\mathbb { Z } ^ { k + 1 } , \\qquad 1 \\leq i \\leq k - n + 1 , \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\left ( \\log | f _ X / f _ Y | \\right ) } { \\partial X \\partial Y } = \\frac { 2 } { ( Y - 2 X ) ^ 2 } . \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} B _ { \\alpha } ( p , q ) : = \\frac { \\alpha } { 1 - \\alpha } \\int p ( \\textbf { { x } } ) q ( \\textbf { { x } } ) ^ { \\alpha - 1 } d \\textbf { { x } } - \\frac { 1 } { 1 - \\alpha } \\int p ( \\textbf { { x } } ) ^ { \\alpha } d \\textbf { { x } } + \\int q ( \\textbf { { x } } ) ^ { \\alpha } d \\textbf { { x } } . \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} \\frac { t { \\phi } _ { p , \\alpha } ' ( t ) } { { \\phi } _ { p , \\alpha } ( t ) } = p + \\frac { \\alpha } { 1 + \\log ( t ) } , t > 1 , \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} & | - 4 b \\int _ { t + \\log ^ { ( \\alpha - 1 ) b } ( t ) + \\frac { 1 } { 2 } } ^ { \\infty } \\frac { 1 } { x ^ { 2 } \\log ^ { b + 1 } ( x ) } \\frac { 1 } { ( \\log ^ { ( \\alpha - 1 ) b } ( t ) + x - t ) } \\frac { d x } { ( 1 + x - t ) ^ { 3 } } | \\\\ & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\int _ { t + \\log ^ { ( \\alpha - 1 ) b } ( t ) + \\frac { 1 } { 2 } } ^ { \\infty } \\frac { d x } { ( x - t ) ^ { 4 } } \\\\ & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { \\ell } v _ i ' = \\sum \\limits _ { i = 3 } ^ { \\ell } v _ i ' \\leq \\sum \\limits _ { i = 3 } ^ { \\ell } ( 5 - v _ i ) = 5 ( \\ell - 2 ) - n + v _ 1 + v _ 2 < n , \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} u ( x , y , t ) = \\cos \\left ( x y ( 5 - 2 \\vert t - \\tfrac { 5 } { 2 } \\vert ) \\right ) + y ^ { 1 0 - 4 | t - 5 / 2 | } \\sin \\left ( 4 x ( 5 - 2 \\vert t - \\tfrac { 5 } { 2 } \\vert ) \\right ) , \\ , \\ , ( x , y ) \\in [ - 1 , 1 ] ^ 2 \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\prod _ { \\kappa = s } ^ { t - 1 } \\frac { \\beta ( \\kappa ) } { \\beta ( \\kappa + 1 ) } \\theta \\leq \\lim _ { t \\to \\infty } \\hat { \\theta } ^ { t - \\hat { \\tau } } \\prod _ { \\kappa = s } ^ { \\hat { \\tau } - 1 } \\frac { \\beta ( \\kappa ) } { \\beta ( \\kappa + 1 ) } \\theta = 0 . \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} A _ 2 = - a _ n \\int _ { z _ 1 } ^ { z _ 2 } h _ 1 \\mathcal { L } h _ 1 d z + a _ n \\left [ h _ 1 \\sigma ( z ) h _ { 1 z } \\right ] _ { z _ 1 } ^ { z _ 2 } . \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} U = Z _ 1 \\overset { \\phi _ 1 } { \\longmapsto } \\ldots \\overset { \\phi _ { t } } { \\longmapsto } Z _ { t + 1 } = W \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 ) = \\sum \\limits _ { j , k } \\psi _ { j , k } * \\psi _ { j , k } * f ( x _ 1 , x _ 2 , x _ 3 ) \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c l } a _ { 1 1 } X _ 1 ^ { d _ 1 } & + a _ { 1 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { 1 t } X _ t ^ { d _ t } = g _ 1 ( X _ 1 , \\ldots , X _ k ) \\\\ a _ { 2 1 } X _ 1 ^ { d _ 1 } & + a _ { 2 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { 2 t } X _ t ^ { d _ t } = g _ 2 ( X _ 1 , \\ldots , X _ k ) \\\\ \\ ; \\vdots & & \\quad \\vdots \\\\ a _ { n 1 } X _ 1 ^ { d _ 1 } & + a _ { n 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { n t } X _ t ^ { d _ t } = g _ n ( X _ 1 , \\ldots , X _ k ) . \\end{array} \\right . \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} Y _ n = K - Y _ { n - 1 } , \\qquad \\forall n \\le N , \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} k ( y ) = \\frac { ( y _ d \\wedge 1 ) ^ { \\beta _ 1 } ( y _ d \\vee 1 ) ^ { \\beta _ 2 } } { | y | ^ { d + \\alpha + \\beta _ 1 + \\beta _ 2 } } ( 1 + | \\log ( y _ d ) | ) ^ { \\beta _ 3 } \\left ( \\log \\left ( 1 + \\frac { | y | } { y _ d \\vee 1 } \\right ) \\right ) ^ { \\beta _ 4 } . \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} \\int _ { A _ r } | u _ a | ^ 2 & \\le \\left ( C ( a ^ * - a ) ^ { 1 - 2 / p } \\right ) ^ { d / ( d + 2 ) } \\left ( C r ^ { d ( 1 - p / 2 ) } \\right ) ^ { 2 / ( d + 2 ) } \\\\ & = C \\left [ \\frac { r } { ( a ^ * - a ) ^ { 1 / p } } \\right ] ^ { \\frac { d } { d + 2 } ( 2 - p ) } . \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} n = ( m ^ 2 - 2 y ) ^ 2 + y ^ 2 + z ^ 2 + w ^ 2 . \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\frac { d y } { y } \\frac { e ^ { - y } } { \\log ^ { a + 1 } ( t / y ) } = \\int _ { 0 } ^ { 1 } \\frac { d y } { y \\log ^ { a + 1 } ( t / y ) } + \\int _ { 0 } ^ { 1 } d y \\frac { \\left ( e ^ { - y } - 1 \\right ) } { y \\log ^ { a + 1 } ( \\frac { t } { y } ) } \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} ( A _ { i j } ) _ s = - F _ k \\ , \\langle \\nabla _ { F _ k } ^ { \\perp } V , A _ { i j } \\rangle + \\nabla _ { F _ j } ^ { \\perp } \\nabla _ { F _ i } ^ { \\perp } V - A ^ V _ { i k } \\ , A _ { j k } \\ , . \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} \\frac { d \\mathbb { Q } _ { n , \\sigma } } { d \\mathbb { P } _ { n } } : = \\exp \\left \\{ \\sum _ { i < j } \\left ( \\frac { 2 \\beta } { \\sqrt { n } } \\sigma _ { i } \\sigma _ { j } A _ { i , j } - \\frac { 2 \\beta ^ 2 } { n } \\right ) + \\frac { \\beta J } { n } \\left ( \\sum _ { i = 1 } ^ { n } \\sigma _ { i } \\right ) ^ 2 \\right \\} . \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} \\mathcal { A } = \\{ C \\in \\mathfrak { m } : \\ t h e r e \\ i s \\ s o m e \\ B \\in p \\ w i t h \\ A \\cap B \\subseteq C \\} . \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} & ( \\mathrm { i d } + B \\circ T ) ( u ) * _ B ( \\mathrm { i d } + B \\circ T ) ( v ) \\\\ & = ( T u ) ( \\mathrm { i d } + B \\circ T ) ( v ) + ( \\mathrm { i d } + B \\circ T ) ( u ) ( T v ) + H ( T u , T v ) \\\\ & = T ( u ) \\cdot v + u \\cdot T ( v ) + ( T u ) \\cdot B ( T v ) + B ( T u ) \\cdot ( T v ) + H ( T u , T v ) \\\\ & = u * v + B ( ( T u ) ( T v ) ) \\\\ & = u * v + B T ( u * v ) = ( \\mathrm { i d } + B \\circ T ) ( u * v ) . \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} \\dot { q _ i } = \\frac { \\partial H } { \\partial p _ i } , \\dot { p _ i } = - \\frac { \\partial H } { \\partial q _ i } , i = 1 , \\ldots , n , \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} \\langle A ^ { \\star } y , x \\rangle = \\langle y , A x \\rangle = \\langle y , P _ 1 \\partial x \\rangle = \\langle P _ 1 y , \\partial x \\rangle . \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} u _ n \\Phi ( x ) u _ n ^ { - 1 } - \\Phi ( x ) = & \\Phi \\left ( v _ n x v _ n ^ { - 1 } - x \\right ) = - 2 \\Phi ( y _ n ) + \\sum \\limits _ { k = 1 } ^ { n - 1 } \\left ( u _ n \\Phi ( y _ k ) u _ n ^ { - 1 } - \\Phi ( y _ k ) \\right ) . \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} 0 = \\langle S f , h _ k \\rangle _ H = \\langle h _ k , h _ k \\rangle _ H + \\sum _ { q \\in J \\setminus \\{ k \\} } \\langle h _ q , h _ k \\rangle _ H = \\| h _ k \\| _ H ^ 2 . \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} \\left \\{ a : \\ : \\mu _ { J , K } ( a ) > 0 \\right \\} = m \\mathbb { Z } _ + \\cap \\{ 0 , 1 , \\dots , J \\} , \\quad \\left \\{ a : \\ : \\mu _ { K , J } ( a ) > 0 \\right \\} = m \\mathbb { Z } _ + \\cap \\{ 0 , 1 , \\dots , K \\} , \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} v _ { s , t } ( r , \\omega ) = \\begin{cases} v ( r , \\omega ) , \\ & t \\leq r \\leq 1 \\\\ ( \\frac { \\log t \\cdot \\log s } { \\log t - \\log s } - \\frac { \\log t } { \\log t - \\log s } \\log r ) r ^ { - \\frac { n - 2 } { 2 } } w _ 1 ( \\omega ) , \\ & s \\leq r \\leq t \\\\ 0 , \\ & 0 < r < s \\end{cases} \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\dot { X } & = a X - b X Y , \\\\ \\dot { Y } & = - c Y + d X Y , \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} | \\Delta \\eta _ j | = | \\Delta _ M \\eta _ j - \\nabla _ M ^ 2 \\eta _ j ( \\nu , \\nu ) | \\leq n | \\nabla ^ 2 _ M \\eta _ j | \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} \\check { \\rho } ( \\varepsilon ) = \\check { \\rho } ( \\sigma , \\epsilon ) : = \\sigma . \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} \\beta [ f ] = \\overline { \\lim } _ { n \\to \\infty } \\left \\{ \\ n ^ { 1 / \\rho [ f ] } \\ \\sqrt [ n ] { \\sigma _ n } \\ \\right \\} . \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } \\frac { \\lambda '' ( s ) r } { 1 + s - t } d s = \\int _ { t } ^ { \\infty } \\frac { \\lambda '' ( s ) r } { 1 + s - t } d s - \\int _ { t } ^ { t + 2 ( r + 1 ) } \\frac { \\lambda '' ( s ) r } { 1 + s - t } d s \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} | | f | | _ { X } = _ { t \\geq T _ { 0 } } \\left ( | f ( t ) | b \\log ^ { b } ( t ) \\sqrt { \\log ( \\log ( t ) ) } + | f ' ( t ) | t \\log ^ { b + 1 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } + | f '' ( t ) | t ^ { 2 } \\log ^ { b + 1 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } \\right ) \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} \\begin{aligned} ( \\bigvee C , b ) & < | \\{ ( c , b ) \\mid c \\in C \\} & & , \\\\ ( a , \\bigvee C ) & < | \\{ ( a , c ) \\mid c \\in C \\} & & , \\end{aligned} \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} B _ \\alpha ( p , q ) = \\frac { \\alpha } { 1 - \\alpha } \\sum \\limits _ { x \\in \\mathbb { S } } p ( x ) q ( x ) ^ { \\alpha - 1 } - \\frac { 1 } { 1 - \\alpha } \\sum \\limits _ { x \\in \\mathbb { S } } p ( x ) ^ { \\alpha } + \\sum \\limits _ { x \\in \\mathbb { S } } q ( x ) ^ { \\alpha } . \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} \\langle f | g \\rangle : = ( J f | g ) . \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} { \\rm c l } ( P ^ \\mu ( M ) ) \\cap A = { \\rm c l } ( A \\cap P ^ \\mu ( M ) ) \\cap A . \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\bar { E } \\dot { \\bar { x } } ( t ) & = \\bar { A } \\bar { x } ( t ) + \\bar { B } u ( t ) \\\\ [ 1 e x ] \\bar { y } ( t ) & = \\bar { C } \\bar { x } ( t ) \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} E _ { 0 } ( \\mathbf { z } ) B _ { n , \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } \\mid { \\boldsymbol { \\omega } } ) = & \\sum _ { i = 1 } ^ { r } B _ { n , \\mathbf { m } _ { i } } ^ { ( d ) } ( z \\mid { \\boldsymbol { \\omega } } ) \\left ( m _ { i } + \\frac { d } { 2 } ( r - i ) \\right ) h _ { - , i } ^ { ( d ) } ( \\mathbf { m } ) . \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } ( ( b , j ) ) = ( b + \\delta _ { ( a , i ) } , j + r \\delta _ { ( a , i ) } + 1 ) . \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} ( a , c ) = ( 0 , 1 ) ( a , c ) = ( 1 , 0 ) . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty H _ n ( x ) y ^ n & = \\frac { 1 + x y ( 1 + y ) + 2 x y ^ 3 ( 1 + y ) - x ^ 2 y ^ 3 ( 1 + y ) ^ 2 ) } { 1 - x y ^ 2 ( 1 + y ) } \\\\ & = - 2 y + x y + x y ^ 2 + \\frac { 1 + 2 y } { 1 - x y ^ 2 ( 1 + y ) } . \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} c . 1 + c _ 0 F ( \\theta ) + \\sum \\limits _ { i = 1 } ^ { d } c _ i w _ i ( \\theta ) + \\sum \\limits _ { i = 1 } ^ d \\sum \\limits _ { j = i } ^ d c _ { i j } w _ { i j } ( \\theta ) = 0 \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} \\hat { J } _ { n _ { N - 1 } + 1 } = \\hat { T } _ N = \\hat { L } ^ 2 _ N . \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} \\liminf _ { n \\to - \\infty } \\min \\{ Y _ n , K - Y _ n \\} = r , \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} \\psi _ { j , k } ( x _ 1 , x _ 2 , x _ 3 ) : = 2 ^ { - 2 ( j + k ) } \\psi ^ { ( 1 ) } ( 2 ^ { - j } x _ 1 ) \\psi ^ { ( 2 ) } ( 2 ^ { - k } x _ 2 , 2 ^ { - ( j + k ) } x _ 3 ) . \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} h ( z ) = \\frac 1 z , b ( z ) = \\frac { 1 - d } 2 \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } v _ { 4 } ( t , r ) | \\leq \\begin{cases} \\frac { C } { t ^ { 3 } \\log ^ { b + 2 N - 2 } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { t ^ { 2 } \\log ^ { 3 N + b } ( t ) } , r \\geq \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} ( a - d ) u + b v + c w = 0 , \\ \\ u ^ 2 + v w = - 1 . \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} \\tilde { x } & = \\frac { x } { \\mu } , & \\tilde { y } & = \\frac { y - \\zeta ( \\mu ) } { \\mu } , \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\dot { \\mathbf { z } } ( t ) = F ( \\mathbf { z } ( t ) ) & \\textmd { i n } \\ \\Omega \\times ( 0 , \\infty ) , \\\\ \\mathbf { z } ( 0 ) = [ u _ 0 , 0 ] ^ \\top & \\textmd { i n } \\ \\Omega \\times 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} k + h ^ { \\vee } = \\frac { p } { q } , p , q \\in \\Z _ { \\geq 1 } , \\ ( p , q ) = 1 , \\ p \\geq h ^ { \\vee } . \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} y _ { n + 1 } - 1 \\ , = \\ , t _ { n + 1 } \\sqrt { ( n + 2 ) ^ 2 - \\gamma ^ 2 } \\ , - \\ , 1 \\ , > \\ , t _ { n + 1 } \\quad . \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} X = \\{ x \\in \\mathbb { R } ^ n | g _ i ( x ) \\leq 0 , \\forall ^ m _ { i = 1 } \\} , \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} \\phi ( \\mathfrak { g } ^ { r } , \\mathfrak { g } ^ { s } ) = \\{ 0 \\} , \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} J ( \\psi ) = \\frac { 1 } { 2 } \\boldsymbol { B } ( \\psi , \\psi ) - \\langle \\psi , h \\rangle , \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} h _ { d , b , A } \\triangleq \\prod _ { i = 1 } ^ d h _ { 1 , b , A _ i } , \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} \\begin{aligned} f ^ { \\prime } ( h ) & = \\frac { 1 } { 2 } ( f ( h - 2 ) + f ( h + 2 ) ) \\\\ f ^ { \\prime \\prime } ( h ) & = \\frac { 1 } { 2 } ( f ( h - 2 ) - f ( h + 2 ) ) , \\end{aligned} \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} \\delta \\mathcal { V } = \\int _ M u \\ , d S = 0 . \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} ( A _ 1 , A _ 2 , \\dots , A _ m ) \\ \\longmapsto \\ \\phi \\big ( \\exp \\big ( H + \\sum _ { j = 1 } ^ m p _ j \\log A _ j \\big ) \\big ) \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} f ^ \\Gamma ( B ) : = \\min \\{ C \\in \\Gamma ( Y ) \\mid B \\le f ^ * ( C ) \\} , . \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) \\le \\beta \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k + 1 \\} = \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} 1 + \\sum _ { n \\geq 1 } A _ n ( i ) \\frac { x ^ n } { n ! } = \\frac { 1 - i } { 1 - i e ^ { ( 1 - i ) x } } = \\frac { \\tan ( ( 1 + i ) x ) + \\sec ( ( 1 + i ) x ) - i } { 1 - i } . \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} \\dim \\mu _ { \\l _ 0 , \\tau _ 0 } = \\min \\{ 1 , H ( p ) / \\log \\l ^ { - 1 } _ 0 \\} . \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} b _ { n - 2 } = \\frac { - n \\mu + 8 { n \\choose 2 } + 1 6 { n \\choose 3 } } { 2 } . \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} G ( \\tilde { r } ) = \\frac { \\tilde { r } \\left ( \\tilde { \\gamma } - P _ R \\right ) } { P _ R \\left ( \\tilde { \\gamma } - \\tilde { r } \\right ) } . \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} L _ { B V } ^ { a , \\mathrm { I } } \\Big \\vert _ { A ^ \\dag = c ^ \\dag = 0 } & = \\mathbb { L } _ { B R S T } ^ { \\mathrm { I } } \\\\ L _ { B V } ^ { b , \\mathrm { I } } \\Big \\vert _ { A ^ \\dag = c ^ \\dag = 0 } & = \\mathbb { L } _ { B R S T } ^ { \\mathrm { I } } + c F _ A + ( d - \\mathcal { L } _ Q ) ( c A ) \\approx \\mathbb { L } _ { B R S T } ^ { \\mathrm { I I } } + ( d - \\mathcal { L } _ Q ) ( c A ) \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} \\mho _ \\mathbf { G } [ \\varphi ] ( p ) = \\inf _ { a \\in A } \\varphi ( a ) \\cdot \\mathbf { G } ^ * ( a , p ) , p \\in X \\setminus A . \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} E ( u , v ) = \\pi \\int _ { 0 } ^ { \\infty } \\left ( v ^ { 2 } + ( \\partial _ { r } u ) ^ { 2 } + \\frac { u ^ { 2 } } { r ^ { 2 } } \\right ) r d r \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { k } P _ { 2 k } [ 2 r ] r \\psi _ { 2 r } = 0 . \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} \\sup _ { | | v | | \\leq 1 } \\left | \\Big ( I _ \\infty ' ( u _ n ) - { I } ' ( { u } _ n ) \\Big ) v \\right | = \\sup _ { | | v | | \\leq 1 } \\left | \\int _ { \\mathbb { R } ^ N } \\Big ( V _ { 0 , \\infty } - V _ 0 ( x ) \\Big ) { u } _ n ( x ) v ( x ) \\ ; d x \\right | \\to 0 . \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} \\mathcal { F } ^ { i } & = H _ { - i } \\left ( \\check { C } ( \\underline { a } ) \\otimes _ { R } - \\right ) \\\\ & = H _ { - i } \\left ( \\left ( \\varinjlim \\Sigma ^ { - n } K ^ { R } ( \\underline { a } ^ { k } ) \\right ) \\otimes _ { R } - \\right ) \\\\ & \\cong \\varinjlim H _ { n - i } \\left ( K ^ { R } ( \\underline { a } ^ { k } ) \\otimes _ { R } - \\right ) \\\\ & \\cong \\varinjlim H _ { n - i } \\left ( \\underline { a } ^ { k } ; - \\right ) \\\\ \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d \\hat { x } _ { t } & = ( F _ { t } \\hat { x } _ { t } + f _ { t } + \\widehat { \\theta _ { t } ^ { \\ast } } ) d t + P _ { t } G _ { t } R _ { t } ^ { - 1 } d \\hat { I } _ { t } , \\\\ \\hat { x } _ { t } ( 0 ) & = x _ { 0 } \\end{array} \\right . \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} \\mu ( d x ) = \\frac { 1 } { Z } \\exp \\left ( - H ( x ) \\right ) d x \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} g ( x , y ) = c ( x , y ) h ( x , y ) \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} H \\left ( \\left \\lfloor r ^ { - 1 } \\sum _ { k = 0 } ^ { n - 1 } a _ { \\xi _ k } \\lambda ^ { k } + s \\right \\rfloor \\right ) \\le H ( \\mu _ { \\lambda } ^ { ( n ) } ; r ) < n H ( p ) . \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} \\int \\limits _ E f \\circ \\varphi ( x ) | J ( x , \\varphi ) | ~ d x = \\int \\limits _ { \\mathbb R ^ n \\setminus \\varphi ( S ) } f ( y ) N _ f ( E , y ) ~ d y \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} v ( x ) = x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { w } ( x ) \\quad f = \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { w } = c ^ { - 1 } _ { s } ( - P ) ^ { s } u , \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} \\Phi ( z ) = \\frac { z ^ { k } } { \\eta _ { \\nu } ( z ) ^ { k - 1 } } , z \\in \\mathbb { C } \\backslash \\mathbb { R _ { + } } , \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} \\begin{cases} D ^ { \\alpha _ 1 } _ 0 y ( t ) = w ( t ) , & y ( 0 ) = y _ 0 , \\\\ D ^ { 1 - \\alpha _ 1 } _ 0 w ( t ) = z ( t ) , & w ( 0 ) = 0 , \\\\ D ^ { \\alpha _ 2 - 1 } _ 0 z ( t ) = f ( t , y ( t ) , w ( t ) ) , & z ( 0 ) = s , \\end{cases} \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} \\eta _ U | u | ( D ) & = \\int _ D G _ U ( x _ 0 , z ) \\left ( \\int _ { D \\setminus U } j ( | z - y | ) | u ( y ) | d y \\right ) d z \\\\ & = \\int _ { D \\setminus U } P _ U ( x _ 0 , y ) | u ( y ) | d y . \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} | ( N ( u _ { 1 } ) - N ( u _ { 2 } ) ) ( t , R \\lambda ( t ) ) | & \\leq C \\frac { | \\overline { v } _ { 1 } - \\overline { v } _ { 2 } | \\left ( \\overline { v } _ { 1 } ^ { 2 } + \\overline { v } _ { 2 } ^ { 2 } \\right ) } { R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\\\ & + C \\frac { | \\overline { v } _ { 1 } - \\overline { v } _ { 2 } | \\left ( | \\overline { v } _ { 1 } | + | \\overline { v } _ { 2 } | \\right ) } { R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\left ( | Q _ { 1 } ( R ) | + | v _ { c o r r } ( t , R \\lambda ( t ) ) | \\right ) \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} \\Theta ( L _ i ) = ( - 1 ) ^ { i + 1 } L _ { - i } \\ , , \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} P \\lbrace \\max _ { 0 \\le k \\le n } S _ k > \\beta , S _ n = x \\ | \\ S _ 1 = 1 \\rbrace & = P \\lbrace \\max _ { 0 \\le k \\le n - 1 } S _ k > \\beta - 1 , S _ { n - 1 } = x - 1 \\rbrace \\\\ & = P \\lbrace S _ { n - 1 } = 2 \\beta - x + 1 \\rbrace = P \\lbrace S _ n = 2 \\beta - x \\ | \\ S _ 1 = - 1 \\rbrace , \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} \\Lambda _ { d , b , A } \\triangleq \\prod _ { i = 1 } ^ d \\left [ \\frac { A _ i - 1 } { b } , \\frac { A _ i } { b } \\right ) . \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} L _ { \\Sigma } ^ { s } : = L _ { \\Sigma } - s | A _ { \\Sigma } | ^ 2 = \\Delta _ { \\Sigma } + ( 1 - s ) | A _ { \\Sigma } | + R i c ( \\nu , \\nu ) \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} & \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } \\big | F ( t _ * , x , v ) \\big | \\cdot \\chi _ { \\left \\{ ( x , v ) : x _ { \\ ! \\perp } ^ { 2 } + \\left | \\beta _ \\varepsilon ( v ) \\right | ^ 2 \\geq \\delta \\right \\} } d x d v \\\\ = & \\ ; \\ ; \\lim _ { k \\rightarrow \\infty } \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } F ( t _ * , x , v ) \\varphi _ k ( x , v ) d x d v \\leq \\ ; \\ ; \\| f _ 0 \\| _ { L ^ 1 ( \\Omega \\times \\mathbb { R } ^ 3 ) } . \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} \\frac { \\partial \\omega _ t } { \\partial t } = - R i c ( \\omega _ t ) - \\omega _ t , \\omega _ { | _ { t = 0 } } = \\omega _ 0 . \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} T ( x ) = 2 \\ln | x + a | \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} \\bigg | \\int \\left ( \\bar { f } ( x , y ) - \\tilde { f } ( x , y ) \\right ) g ( x , y ) d x d y \\bigg | = 0 , \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} \\partial _ { t } f = S f + K f + \\left ( a + i b \\right ) \\left ( V f + e ^ { \\gamma \\varphi } F \\right ) R ^ { n } \\times \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} \\begin{aligned} p _ { [ 1 ^ 0 1 1 ^ 1 ] } \\bigg ( 1 1 ^ { 2 r } m + \\frac { 1 1 - 1 1 ^ { 2 r + 1 } } { 2 4 } \\bigg ) & \\equiv 0 \\pmod { 1 1 ^ { 2 r } } , \\\\ p \\bigg ( 1 1 ^ { 2 r + 1 } m + \\frac { 1 - 1 1 ^ { 2 r + 2 } } { 2 4 } \\bigg ) & \\equiv 0 \\pmod { 1 1 ^ { 2 r + 1 } } . \\end{aligned} \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} k = m _ { 1 } + \\dots + m _ { n + 1 } - ( n - 1 ) d \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} d _ i : = \\sum _ { j = i } ^ { k - 1 } ( - 1 ) ^ { j - i } \\binom { m - k + j - i - 1 } { j - i } b _ j , \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} | \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { t } ^ { \\infty } d x \\lambda ''' ( x ) K ( x - t , \\lambda ( t ) ) | & \\leq C \\sup _ { x \\geq t } | \\lambda ''' ( x ) | \\\\ & \\leq \\frac { C } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } + C \\sup _ { x \\geq t } | e ''' ( x ) | \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} V _ { 0 } + V _ { 1 } x = ( s _ { 1 } , s _ { 2 } , t _ { 1 } , t _ { 2 } ) = ( \\widehat { s } _ { 1 } , \\widehat { s } _ { 2 } , \\widehat { t } _ { 1 } , \\widehat { t } _ { 2 } ) = \\widehat { V } _ { 0 } + \\widehat { V } _ { 1 } \\widehat { x } , \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} ( c - a ) ^ { - 1 } ( c ^ 2 - a ^ 2 ) = ( b - a ) ^ { - 1 } ( b ^ 2 - a ^ 2 ) . \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} X _ t = x _ 0 + \\int _ 0 ^ t a ( s , X _ s ) \\ , d s + \\int _ 0 ^ t b ( s , X _ s ) \\ , d W _ s \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} H _ { \\bar { l } } ( \\bar { x } ^ { B ( l ) } ) & = H ( x ) - H _ l ( x ^ { B ( l ) } | \\bar { x } ^ { B ( l ) } ) \\\\ & = \\sum _ { i \\notin B ( l ) } \\psi ( x _ i ) + \\frac { 1 } { 2 } \\sum _ { i , j \\notin B ( l ) } M _ { i j } x _ i x _ j . \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} A _ { k , m } = ( a ^ { ( m ) } _ { i j } ) _ { 1 \\le i , j \\le k } , \\ B _ { k , m } = ( b ^ { ( m ) } _ { i j } ) _ { 1 \\le i , j \\le k } , W _ { k - 1 , m } = ( w ^ { ( m ) } _ { i j } ) _ { 1 \\le i , j \\le k - 1 } , \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} \\langle \\left [ ( \\widehat x ) ^ { 2 k } , ( \\widehat \\xi ) ^ { 2 k } \\right ] f , f \\rangle _ { L ^ 2 ( \\mathbb { R } ) } = ( - i ) ^ { 2 k } \\sqrt { \\frac { 2 a } { \\pi } } \\int _ { \\mathbb { R } } e ^ { - a x ^ 2 } \\left ( x ^ { 2 k } \\frac { d ^ { 2 k } } { d x ^ { 2 k } } - \\frac { d ^ { 2 k } } { d x ^ { 2 k } } x ^ { 2 k } \\right ) e ^ { - a x ^ 2 } d x \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} \\mathfrak { F } _ { 1 Y M } = \\left ( \\mathcal { F } _ { 1 Y M } , L ^ \\bullet _ { 1 Y M } , \\theta _ { 1 Y M } ^ \\bullet , Q \\right ) \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} T Y = ( T Y / \\pi ^ * T B ) \\oplus T ^ H Y , \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} H ^ \\infty _ L ( G ) : = \\left \\{ f \\in L ^ 2 ( G ) , ~ g \\mapsto ( 1 + L ( g ) ) ^ k f ( g ) \\in L ^ 2 ( G ) ~ \\forall k \\right \\} \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} \\dot { z } = ( 1 - e ^ { - z } ) ( p _ 1 - p _ 2 ) . \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} F _ { \\rm a p p l y } ( y ) = \\nu _ 1 y + \\nu _ 2 y ^ 2 , \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} g = \\left ( \\begin{array} { c c c c } 2 a _ 3 + a _ 4 & 0 & 2 b _ 3 & b _ 3 \\\\ a _ 3 & a _ 4 & b _ 3 & b _ 4 \\\\ c _ 1 & c _ 3 & 2 d _ 2 + d _ 4 & d _ 2 \\\\ c _ 3 & - 2 c _ 3 & 0 & d _ 4 \\end{array} \\right ) . \\\\ \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} \\tilde { u } \\left ( x , t \\right ) = \\left ( \\sqrt { \\alpha \\beta } \\sigma \\left ( t \\right ) \\right ) ^ { \\frac { n } { 2 } } u \\left ( \\sqrt { \\alpha \\beta } x \\sigma \\left ( t \\right ) , \\beta t \\sigma \\left ( t \\right ) \\right ) e ^ { \\eta } . \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} & \\phi ( - q ^ { 3 / 2 } x _ 3 x _ 1 x _ 2 ) \\\\ & = \\sum \\frac { q ^ { \\frac { n _ { 1 , 4 } ^ { 2 } } { 2 } + n _ { 1 , 4 } } ( x _ 3 x _ 1 x _ 2 ) ^ { n _ { 1 , 4 } } } { ( q ) _ { n _ { 1 , 4 } } } \\\\ & = \\sum \\frac { q ^ { - \\frac { n _ { 1 , 4 } ^ { 2 } } { 2 } } ( x _ 2 ) ^ { n _ { 1 , 4 } } ( x _ 3 ) ^ { n _ { 1 , 4 } } ( x _ 1 ) ^ { n _ { 1 , 4 } } } { ( q ) _ { n _ { 1 , 4 } } } . \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} G = \\pm ( \\zeta _ { 6 } + \\zeta _ { 6 } ^ { 5 } + \\zeta _ { 1 9 } + \\cdots + \\zeta _ { 1 9 } ^ { 2 2 } ) + \\xi + ( - \\xi ) + \\overline { \\xi } + ( - \\overline { \\xi } ) , \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} \\Omega _ { \\mu } \\cap \\mathbb { H } = \\{ r e ^ { i \\theta } : r > 0 , \\theta \\in ( f ( r ) , \\pi ) \\} . \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} \\mathbb { E } ( e ^ { - s S _ { \\alpha } ( t ) } ) = e ^ { - t s ^ { \\alpha } } , \\ ; s > 0 , \\ ; \\alpha \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} \\widetilde { \\Gamma } ^ a _ { b c } - \\Gamma ^ a _ { b c } = \\delta ^ a _ b \\alpha _ c + \\delta ^ a _ c \\alpha _ b \\ , . \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} r ( P ) ( \\mathcal { N } ) : = i _ \\theta ^ * P ( \\mathcal { N } \\times \\mathbb { T } ^ 1 ) \\ , . \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} \\begin{cases} \\underline h ^ n ( t ) : = ( c ^ n - \\delta ) t + K , \\ \\ \\ t \\geq 0 , \\\\ \\underline U ^ n ( t , x ) : = ( 1 - \\epsilon ) [ \\Phi ^ n ( x - \\underline h ^ n ( t ) ) + \\Phi ^ n ( - x - \\underline h ^ n ( t ) ) - \\mathbf { u _ n ^ * } ] , \\ \\ \\ t \\geq 0 , \\ x \\in [ - \\underline h ^ n ( t ) , \\underline h ^ n ( t ) ] , \\end{cases} \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} \\Phi ( P _ f ( A , B ) ) & = \\Phi ( B ) ^ { 1 / 2 } \\Psi ( f ( B ^ { - 1 / 2 } A B ^ { - 1 / 2 } ) ) \\Phi ( B ) ^ { 1 / 2 } \\\\ & \\ge \\Phi ( B ) ^ { 1 / 2 } f ( \\Psi ( B ^ { - 1 / 2 } A B ^ { - 1 / 2 } ) ) \\Phi ( B ) ^ { 1 / 2 } = P _ f ( \\Phi ( A ) , \\Phi ( B ) ) , \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} e _ R ( \\mathcal I ( j ) ^ { [ 1 ] } , \\mathcal I ( k ) ^ { [ 1 ] } ; R ) = \\sum _ { i = 1 } ^ t - [ S / m _ i : R / m _ R ] ( \\Delta ( j ) _ i \\cdot \\Delta ( k ) _ i ) . \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} 1 4 7 J _ { n } ^ { ( 3 ) } = 1 3 K _ { n } ^ { ( 3 ) } + 4 8 K _ { n - 1 } ^ { ( 3 ) } + 2 0 K _ { n - 2 } ^ { ( 3 ) } , \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} ( q ^ 2 + 1 - 2 ) \\cdot 1 = | \\{ ( R , c ) : R \\mbox { i s a p o i n t } , c \\mbox { i s a c i r c l e t h r o u g h } P , Q , R \\} | = a _ 2 \\cdot ( q + 1 - 2 ) , \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} { \\cal L } \\psi _ i ( z ) & = - \\lambda _ i \\psi _ i ( z ) , z _ 1 \\le z \\le z _ 2 , \\\\ \\psi _ { i } ( z _ 1 ) & = \\psi _ { i } ( z _ 2 ) = 0 , \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} \\mathcal { E } = \\mathbb { Z } \\times \\lbrace a _ 1 , \\dots , a _ m \\rbrace ^ 2 \\ , , \\ , \\mathcal { E } ' = \\mathbb { Z } _ { > 0 } \\lbrace a _ 1 , \\dots , a _ m \\rbrace ^ 2 \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} \\psi ( \\check { X } ) = \\lambda e ^ { \\frac { \\mathbf { i } } { 2 \\pi r } ( q ' - q ) ^ t \\mathcal { B } \\check { X } } , \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} \\delta ( \\tau ( x _ i ) ) & = \\delta ( \\tau ' ( x _ i ) + \\epsilon _ i ) = \\delta ( \\tau ' ( x _ i ) ) + \\delta ( \\epsilon _ i ) + \\sum _ { i = 1 } ^ { p - 1 } p ^ { - 1 } \\binom { p } { i } \\tau ' ( x _ i ) ^ { i } \\epsilon _ i ^ { p - i } , \\\\ \\tau ( \\delta ( x _ i ) ) & = \\sum _ { I } \\tau ' ( \\delta ( x _ i ) ^ { ( I ) } ) \\cdot \\epsilon _ 1 ^ { I _ 1 } \\cdots \\epsilon _ n ^ { I _ n } , \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} & | \\partial _ { t } \\left ( \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { t } ^ { \\infty } d x \\lambda '' ( x ) K ( x - t , \\lambda ( t ) ) \\right ) | \\leq \\frac { C } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } + C \\sup _ { x \\geq t } | e ''' ( x ) | \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} C ^ \\infty ( E ) _ { ( i ) } = \\{ f \\in C ^ \\infty ( E ) | \\ \\kappa _ t ^ * f = O ( t ^ i ) \\} . \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\eta ^ { ( \\beta ) } } [ \\textbf { X } ] = \\mathbb { E } _ { \\widetilde { p } _ n ^ { ( \\beta ) } } [ { \\bf { X } } ] ~ ~ ~ \\mathbb { E } _ { \\eta ^ { ( \\beta ) } } [ ( \\textbf { X } \\textbf { X } ^ T ) ] = \\mathbb { E } _ { \\widetilde { p } _ n ^ { ( \\beta ) } } [ ( \\textbf { X } \\textbf { X } ^ T ) ] . \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} \\sqrt { r } \\tilde { \\phi } _ { \\sqrt { \\xi } } ( r ) = f ( \\sqrt { \\xi } ) \\phi ( r , \\xi ) \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} y & = 2 \\lambda _ 2 s _ 2 \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} D = \\sum _ { x \\in X ^ \\circ } \\tfrac { e _ x - 1 } { 2 } [ x ] . \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y _ l } \\langle P ^ * y , s \\rangle = \\frac { \\partial } { \\partial y _ l } \\left ( \\frac { 1 } { N } \\sum _ { k = 1 } ^ { M } \\sum _ { j \\in B ( k ) } s _ j y _ k \\right ) = \\frac { 1 } { N } \\sum _ { j \\in B ( l ) } s _ j . \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} P _ t ^ \\mathcal { A } ( f ) : = \\frac { t } { 2 \\sqrt { \\pi } } \\int _ 0 ^ \\infty \\frac { e ^ { - t ^ 2 / 4 u } } { u ^ { 3 / 2 } } T _ u ^ \\mathcal { A } ( f ) \\ , d u , t > 0 . \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} [ P ^ b _ { Q } ] - [ e _ 1 ] \\in K _ 0 ( \\mathcal { A } ^ \\infty _ G ( M ) ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; e _ 1 : = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} t ^ { h ' } _ { h } + \\sum _ { h ' - 1 < h '' < h } s ^ { h '' } _ { h } t ^ { h ' } _ { h '' } + s ^ { h ' - 1 } _ { h } = 0 \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\sum _ { a = 0 } ^ n \\sum _ { b = 0 } ^ a \\sum _ { c = 0 } ^ b \\xi _ 4 ^ { n + 2 b } = \\xi _ 4 ^ { n } \\sum _ { a = 0 } ^ n \\sum _ { b = 0 } ^ a ( - 1 ) ^ { b } ( b + 1 ) . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} Q ( B _ 1 ^ + , B _ 2 ^ + ) \\ & = \\ 0 , \\\\ Q ( B _ 1 ^ + , B _ 0 ) \\ = \\ Q ( B _ 2 ^ + , & B _ 0 ) \\ = \\ 2 . \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 P } { \\partial \\tilde { r } ^ 2 } \\left ( \\tilde { r } ^ * ; 0 \\right ) = \\frac { \\tilde { \\gamma } } { \\tilde { r } ^ * \\left ( \\tilde { \\gamma } - \\tilde { r } ^ * \\right ) } - \\frac { \\tilde { \\gamma } \\ , \\frac { \\partial P _ R } { \\partial \\tilde { r } } } { P _ R \\left ( \\tilde { \\gamma } - P _ R \\right ) } + 2 \\frac { d h } { d \\tilde { r } } . \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} p _ \\theta ( { \\bf { x } } ) = S ( \\theta ) ^ { \\frac { 1 } { \\alpha - 1 } } [ h + [ w ( \\theta ) / S ( \\theta ) ] ^ \\top f ( { \\bf { x } } ) ] ^ { \\frac { 1 } { \\alpha - 1 } } , \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} x = H - ( r * H ) \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} \\begin{array} { c c l } \\int _ { M } \\gamma _ { 1 , 0 } u _ { \\overline { 1 } } d \\mu & = & - \\int _ { M } \\gamma _ { 1 } u _ { \\overline { 1 } 0 } d \\mu \\\\ & = & - \\int _ { M } \\gamma _ { 1 } \\left ( u _ { 0 \\overline { 1 } } - A _ { \\overline { 1 } \\overline { 1 } } u _ { 1 } \\right ) d \\mu \\\\ & = & \\int _ { M } A _ { \\overline { 1 } \\overline { 1 } } u _ { 1 } \\gamma _ { 1 } d \\mu . \\end{array} \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} & \\delta \\int _ M \\mathcal { E } ( H , K ) \\ , d S \\\\ & = \\int _ M \\bigg ( \\frac { 1 } { 2 } \\mathcal { E } _ H + 2 H \\mathcal { E } _ K \\bigg ) \\Delta u + \\bigg ( ( 2 H ^ 2 - K + 2 k _ 0 ) \\mathcal { E } _ H + 2 H K \\mathcal { E } _ K - 2 H \\mathcal { E } \\bigg ) u \\\\ & \\phantom { \\int _ M h } - \\mathcal { E } _ K \\langle h , \\ , u \\rangle \\ , \\ , d S , \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} \\langle h ^ \\pm , i w ^ \\pm \\rangle = \\Big \\langle h ^ \\pm , \\frac { \\partial w ^ \\pm } { \\partial { x _ 1 } } \\Big \\rangle = \\Big \\langle h ^ \\pm , \\frac { \\partial w ^ \\pm } { \\partial { x _ 2 } } \\Big \\rangle = 0 , \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} Q ^ { ( k + 1 , j ) } : = \\left \\lbrace \\left \\lbrace P _ { i _ 1 } , \\ldots , P _ { i _ { j } } \\right \\rbrace \\subseteq \\left \\lbrace P _ 1 , \\ldots , P _ k , ( 1 + \\ldots + n ) \\right \\rbrace \\right \\rbrace . \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} \\left . \\begin{array} { @ { } l l @ { } } & w _ i \\rightrightarrows w _ { i + 1 } i = 0 , 1 , \\ldots , l - 1 \\\\ & w _ { j + 1 } \\rightrightarrows w _ j j = 0 , 1 , \\ldots , l - 1 . \\end{array} \\right \\} \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} \\Sigma _ E = ( \\tau ' ( \\Sigma ' ) ) _ E = \\varphi ( \\Sigma ' _ { M E } ) = \\varphi \\left ( \\bigcap _ { n \\in \\N } C _ n \\right ) = \\bigcap _ { n \\in \\N } \\varphi ( C _ n ) \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} \\dot { x } & = y , \\\\ \\dot { y } & = \\begin{cases} - k _ L x - b _ L y + F _ { \\rm a p p l y } ( y ) , & x < 0 , \\\\ - k _ L x - k _ R ( x + \\hat { x } ) - ( b _ L + b _ R ) y + F _ { \\rm a p p l y } ( y ) , & x > 0 . \\end{cases} \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} a _ 1 ' & = - a _ 1 a _ 2 \\frac { b _ { 3 } } { b _ { 1 } } , \\cr a _ 2 ' & = - a _ { 3 } a _ { 4 } { b _ { 1 } b _ { 5 } } , \\cr a _ j ' & = - a _ { 2 j - 1 } a _ { 2 j } { b _ { 2 j - 3 } } { b _ { 2 j + 1 } } ; \\cr b _ 0 ' & = \\frac { b _ 0 b _ 1 + a _ 1 } { b _ 1 } , \\cr b _ 1 ' & = b _ 1 b _ 2 b _ 3 + a _ 3 b _ 1 + a _ 2 b _ 3 , \\\\ b _ j ' & = b _ { 2 j - 1 } b _ { 2 j } b _ { 2 j + 1 } + a _ { 2 j + 1 } b _ { 2 j - 1 } + a _ { 2 j } b _ { 2 j + 1 } . \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} { } ^ b { \\rm T r } _ { \\chi } ( A ) = { } ^ b { \\rm T r } _ { S } \\left ( \\Phi _ A ( e ) \\right ) . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} m m ^ * \\geq \\frac { 1 } { \\| \\iota \\| ^ 2 } m ( ( \\iota \\iota ^ * ) \\otimes 1 _ A ) m ^ * = \\frac { 1 } { \\| \\iota \\| ^ 2 } 1 _ A . \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} | p _ n - U _ n | \\leq ( N + 2 C ) \\frac { C } { N } = C + \\frac { 2 C ^ 2 } { N } \\leq C + C = 2 C . \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} V _ { g \\cdot \\gamma } ( g \\cdot z ) = V _ { \\gamma } ( z ) . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} & | - 1 6 \\int _ { t } ^ { \\infty } \\lambda ''' ( s ) \\left ( K _ { 3 } ( s - t , \\lambda ( t ) ) - K _ { 3 , 0 } ( s - t , \\lambda ( t ) ) \\right ) d s | \\\\ & \\leq C \\sup _ { x \\geq t } | \\lambda ''' ( x ) | \\int _ { t } ^ { \\infty } | K _ { 3 } ( s - t , \\lambda ( t ) ) - K _ { 3 , 0 } ( s - t , \\lambda ( t ) ) | d s \\\\ & \\leq \\frac { C } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } + C \\sup _ { x \\geq t } | e ''' ( x ) | \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} M ^ * : = \\sup _ { x \\geq X } \\phi ( x ) = \\max _ { x \\in [ X , X + r ] } \\phi ( x ) < \\infty . \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a , q ^ { - 1 } / b , c q ^ { - 1 } , d q ^ { - 1 } , q ^ { - 1 } ; q ^ 2 ) _ k } { ( q ^ { 2 } / a , a q ^ 2 , b q ^ 2 , q ^ 2 / c , q ^ { 2 } / d , q ^ 2 ; q ^ 2 ) _ k } \\bigg ( \\frac { b q ^ 7 } { c d } \\bigg ) ^ k \\\\ [ 5 p t ] \\ : & \\ : \\equiv [ n ] ( b q ) ^ { ( n + 1 ) / 2 } \\frac { ( q ^ { - 2 } / b ; q ^ 2 ) _ { ( n + 1 ) / 2 } } { ( b q ^ 2 ; q ^ 2 ) _ { ( n + 1 ) / 2 } } \\sum _ { k = 0 } ^ { ( n + 1 ) / 2 } \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a , q ^ { - 1 } / b , q ^ 3 / c d ; q ^ 2 ) _ k } { ( q ^ 2 , q ^ { - 2 } / b , q ^ 2 / c , q ^ 2 / d ; q ^ 2 ) _ k } q ^ { 2 k } . \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} K _ { ( n + 1 ) , z } = \\sum _ { p = 0 } ^ \\infty \\overline { b _ { p , n } ( z ) } b _ { p , n } = \\sum _ { p = 0 } ^ \\infty \\alpha _ p b _ { p , n } = \\sum _ { p = 0 } ^ \\infty \\alpha _ p A _ { n - 1 } ^ \\dagger b _ { p , n - 1 } . \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} X _ t ^ D \\coloneqq \\begin{cases} X _ t , & t < \\tau _ D , \\\\ \\partial , & t \\ge \\tau _ D , \\end{cases} \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} \\lambda _ { 1 , r } = \\inf _ { u \\in W ^ { 1 , r } ( \\Omega ) } \\left \\{ \\int _ \\Omega | \\nabla u | ^ r \\ , d x : \\int _ \\Omega | u | ^ r \\ , d x = 1 \\right \\} , \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} I I : c o n f \\left ( J _ { 3 } ^ { \\mathbb { O } _ { s } } \\right ) \\supset \\left . s l \\left ( q , \\mathbb { R } \\right ) \\right \\vert _ { q = 8 } . \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} | - \\left ( \\int _ { 0 } ^ { \\frac { \\pi } { 2 } } i d \\theta \\frac { e ^ { \\frac { i t } { 2 } \\cos ( \\theta ) } e ^ { - \\frac { t } { 2 } \\sin ( \\theta ) } } { \\log ^ { a } ( 2 e ^ { - i \\theta } ) } \\right ) | \\leq C \\int _ { 0 } ^ { \\frac { \\pi } { 2 } } d \\theta \\frac { e ^ { - \\frac { t } { 2 } \\sin ( \\theta ) } } { \\log ^ { a } ( 2 ) } \\leq C \\int _ { 0 } ^ { \\frac { \\pi } { 2 } } d \\theta e ^ { - \\frac { t } { 2 } \\frac { 2 } { \\pi } \\theta } = O \\left ( \\frac { 1 } { t } \\right ) \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} f = ( \\delta _ 0 + \\pi _ r + \\pi ^ { \\ast 2 } _ r + \\dots , \\pi ^ { \\ast ( k - 1 ) } _ r ) \\ast \\Pi _ r \\ast ( f , \\lambda ) + \\pi ^ { \\ast k } _ r \\ast f . \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} \\frac { | B _ t | } { | B _ s | } \\ge \\frac { 1 } { \\tilde C ^ 2 } \\left ( \\frac { t } { s } \\right ) ^ b = C _ 1 \\left ( \\frac { t } { s } \\right ) ^ b . \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} \\int _ { 0 } ^ 1 t ^ { i - 1 } p _ j ^ { ( m ) } ( t ) \\ , d t = \\delta _ { i j } , \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} \\partial _ { r } v _ { 4 } ( t , r ) + \\frac { v _ { 4 } ( t , r ) } { r } = \\int _ { t } ^ { \\infty } d x \\int _ { 0 } ^ { \\infty } \\xi d \\xi J _ { 0 } ( r \\xi ) \\sin ( ( t - x ) \\xi ) \\widehat { v _ { 4 , c } } ( x , \\xi ) \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } \\right ) ^ 2 \\leq n \\sum _ { i = 1 } ^ { n } a _ i a _ i ^ * , \\forall n \\in \\mathbb { N } , \\forall a _ 1 , \\dots , a _ n \\in \\mathcal { A } . \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} H _ { L - m + 2 } = H _ { L - m + 1 } + H _ 2 + H _ 1 + 1 = ( L - m + 2 ) + 2 + 1 + 1 = L - m + \\sum _ { a = 1 } ^ { 2 } \\frac { a ( a + 1 ) } { 2 } + 2 . \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} i _ { 0 } : = \\min \\big \\{ i \\in \\{ 1 , \\ldots , k - 1 \\} : \\vert \\pi _ { i } ( t _ { n , k } ) - \\pi _ { k } ( t _ { n , k } ) \\vert \\leq n ^ { - \\alpha } \\big \\} , \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} \\hat { Z } _ l = \\sum _ { 1 \\leq i < j } ^ { n _ l } L _ { i j } ^ 2 - \\bigg ( \\sum _ { a = 1 } ^ { l } r _ a ^ 2 \\bigg ) \\bigg ( \\sum _ { a = 1 } ^ { l } \\frac { 1 } { r _ a ^ 2 } g _ a ( \\Omega _ a ) \\bigg ) , ~ ~ ~ l = 2 , \\cdots , N - 1 . \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} \\rho _ t ( t , x ) = - K _ 4 \\xi ' \\bar h ' ( t ) \\epsilon ( t ) \\hat V ^ * + K _ 4 \\xi \\epsilon ' ( t ) \\hat V ^ * \\preceq \\xi _ * K _ 4 \\epsilon ( t ) \\hat V ^ * \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} & p = \\sum _ { j = 1 } ^ { n _ 1 } \\alpha _ j a _ j + \\alpha a = \\sum _ { j = 1 } ^ { n _ 1 } \\alpha _ j a _ j + \\alpha c _ 1 c , \\\\ & q = \\sum _ { j = 1 } ^ { n _ 2 } \\beta _ j b _ j + \\beta b = \\sum _ { j = 1 } ^ { n _ 2 } \\beta _ j b _ j + \\beta c c _ 2 . \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} - 4 \\int _ { t } ^ { \\infty } \\frac { e '''' ( s ) d s } { \\log ( \\lambda _ { 0 , 0 } ( s ) ) ( 1 + s - t ) } + 4 \\alpha e '''' ( t ) - 4 \\int _ { t } ^ { \\infty } \\frac { e '''' ( s ) d s } { \\log ( \\lambda _ { 0 , 0 } ( s ) ) ( \\lambda _ { 0 , 0 } ( t ) ^ { 1 - \\alpha } + s - t ) ( 1 + s - t ) ^ { 3 } } = R H S _ { 4 } ( t ) \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} \\pi _ { \\mathbf { 0 } } ^ { ( n ) } ( \\sigma , \\gamma ) ( t ) = \\dfrac { 1 } { n \\sigma } \\Big [ \\sum _ { i = 1 } ^ { N ( n ^ { 2 } \\gamma t ) } X _ { i } + \\dfrac { n ^ { 2 } \\gamma t - R _ { N ( n ^ { 2 } \\gamma t ) } } { R _ { N ( n ^ { 2 } \\gamma t ) + 1 } - R _ { N ( n ^ { 2 } \\gamma t ) } } X _ { N ( n ^ { 2 } \\gamma t ) + 1 } \\Big ] . \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} & P _ { d } \\left \\{ \\sup _ { f , g \\in \\mathcal { F } : \\rho ( f , g ) < \\delta } | \\mathbb { H } _ { N } ' f - \\mathbb { H } _ { N } ' g | > \\epsilon \\right \\} < \\eta + \\widetilde { C } _ { N } \\\\ & N = 1 , 2 , \\dots \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} \\left \\vert \\sum _ { j = 1 } ^ { N + 1 } \\psi _ { N , j } ( Q ) \\right \\vert \\leq \\frac { C _ { N } } { q } , \\left \\vert \\sum _ { j = 1 } ^ { N + 1 } L _ { N , j } ( q ) \\right \\vert \\leq C _ { N } , \\qquad q > 0 , \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} a ( n ) = 2 ^ { n - 1 } + \\sum _ { i = 1 } ^ { n - 1 } a ( i ) \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} P ( t , x ) = ( p _ i ( t , x ) ) : = \\mathbf { u } ^ * - \\Phi ( x - \\underline h ( t ) ) , \\ \\ Q ( t , x ) = ( q _ i ( t , x ) ) : = \\mathbf { u } ^ * - \\Phi ( - x - \\underline h ( t ) ) . \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} \\Theta ^ { } _ 1 ( \\Lambda ) ( 0 ) & = A _ c k _ c ( 0 ) + g _ c ( K { } { } ( 0 ) ) - k _ c ( ( A _ c + r ) ( 0 ) ) . \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\dot { u } ( t ) = \\abs { \\nabla u ( t ) } \\operatorname { d i v } \\left ( \\frac { \\nabla u ( t ) } { \\abs { \\nabla u ( t ) } } \\right ) , & \\R ^ 2 \\times ( 0 , \\infty ) ; \\\\ u ( 0 ) = u _ 0 , & \\R ^ 2 \\times 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} \\frac { \\Phi ( z ) } { z } \\eta _ { \\mu _ { 1 } } ^ { - 1 } ( z ) = \\eta _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ^ { \\langle - 1 \\rangle } ( z ) , \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} \\mathbf { R } ( s , t ) = \\frac t s , s \\geq t > 0 . \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} \\gamma ^ - _ j ( t ) = h ^ { - 1 } \\left ( \\rho ^ { 2 j + 1 } ( 1 - t ) ^ { 1 / \\ell } \\right ) ( 1 - \\delta < t \\le 1 ) . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} \\displaystyle \\int _ { M } ( P _ { 0 } ^ { k } \\lambda ^ { \\perp } ) ( P _ { 0 } ^ { k + 1 } \\lambda ^ { \\perp } ) \\ , d \\mu _ { 0 } = \\displaystyle \\int _ { M } ( P _ { 0 } ^ { k } \\lambda ^ { \\perp } ) ( P _ { 0 } P _ { 0 } ^ { k } \\lambda ^ { \\perp } ) d \\mu _ { 0 } \\geq \\Upsilon \\displaystyle \\int _ { M } ( P _ { 0 } ^ { k } \\lambda ^ { \\perp } ) ^ { 2 } d \\mu _ { 0 } . \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } \\int _ { \\Omega } \\dfrac { F _ 0 ( R e ( x ) + v ( x ) ) } { ( R e ( x ) + v ( x ) ) ^ 2 } ( R e ( x ) + v ( x ) ) ^ 2 \\ ; d x = \\dfrac { a _ 0 } { 2 } \\int _ { \\Omega } ( R ^ 2 e ^ 2 ( x ) + v ^ 2 ( x ) ) \\ ; d x , \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} ( t \\hat \\omega - R i c ( \\omega _ 0 ) + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ( t ) ) ^ n = e ^ { \\varphi ( t ) } \\omega _ 0 ^ n , \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } N \\ , \\epsilon _ N = 0 . \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} \\Big ( \\sum _ { n = 1 } ^ { \\infty } \\vert a _ { n } \\vert ^ { q } \\Big ) ^ { \\frac { 1 } { q } } \\leq C \\Big \\Vert \\sum a _ { n } n ^ { - s } \\Big \\Vert _ { \\mathcal { H } _ { q ' } } \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} v _ { 5 } ( t , r ) & = \\frac { - r } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\int _ { 0 } ^ { 1 } \\partial _ { 2 } G _ { 5 } ( s , r \\beta , \\rho ) d \\beta \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} f ( m , H ( b _ 1 , \\ldots , b _ k ) ) \\ge \\left \\{ \\begin{array} { l c } m ^ { \\frac 1 { k ( b _ k + 1 ) } } \\left ( \\frac { b _ { k - 1 } } { 4 ( b _ { k - 2 } + 2 b _ { k - 1 } ) } \\right ) ^ { \\frac 1 k } , & k \\ge 4 , \\\\ m ^ { \\frac 1 { b _ 3 + 2 } } \\left ( \\frac { b _ 2 } { 4 ( b _ 1 + 2 b _ 2 ) } \\right ) ^ { \\frac { b _ 3 + 1 } { b _ 3 + 2 } } , & k = 3 . \\end{array} \\right . \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} c _ 2 ( E ) = \\sum _ i ( \\Omega ^ { 4 } ) _ { i i } = \\sum _ { i , j } d e t ( \\partial ^ { 2 } h _ { i j } ) d z ^ { 1 } \\wedge d \\bar { z } ^ { 1 } \\wedge d z ^ { 2 } \\wedge d \\bar { z } ^ { 2 } = \\sum _ { i , j } d e t ( D _ { i j } ) d e t ( \\bar D _ { i j } ) d z ^ { 1 } \\wedge d \\bar { z } ^ { 1 } \\wedge d z ^ { 2 } \\wedge d \\bar { z } ^ { 2 } \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} \\nabla { G } _ r + \\nabla { G } ^ T _ r & = R ^ { - 1 } \\nabla G + \\nabla G ^ T R ^ { - 1 } , \\\\ & \\geq \\nu \\mathrm { I } , \\forall z \\in \\mathcal { M } , \\forall t \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ) _ m ( q ) _ n } = \\sum _ { n _ 2 \\geq 0 } \\frac { q ^ { ( n - n _ 2 ) ( m - n _ 2 ) } } { ( q ) _ { m - n _ 2 } ( q ) _ { n - n _ 2 } ( q ) _ { n _ 2 } } . \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} \\rho [ g ] = \\overline { \\lim } _ { n \\to \\infty } \\left \\{ \\ \\frac { n \\ \\ln n } { | \\ln | c _ n | \\ | } \\ \\right \\} , \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ { \\Sigma _ k } | H | & = 2 \\int _ M | H | + \\frac { \\pi } { 2 } \\ell ( \\partial M ) , \\\\ \\lim _ { k \\to \\infty } d ( \\Sigma _ k ) & = d ( M ) . \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} E _ { 5 } ^ { \\lambda } ( t , r ) = v _ { 3 , 1 , a } ^ { \\lambda } ( t , r ) + v _ { 3 , 1 , b , i , 1 } ^ { \\lambda } ( t , r ) + v _ { 3 , 1 , b , i i } ^ { \\lambda } ( t , r ) + v _ { 3 , 2 } ^ { \\lambda } ( t , r ) \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} E = E _ 0 \\cup \\cdots \\cup E _ k \\cup \\{ \\{ \\alpha _ 0 , \\alpha _ 1 \\} , \\dots , \\{ \\alpha _ 0 , \\alpha _ k \\} \\} . \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} | \\sum _ { i = 1 } ^ n x _ i c _ { n - i , n } \\frac { r ^ { n - i } } { ( n - i ) ! } | & \\geq | x _ 1 | \\frac { | r | ^ { n - 1 } } { ( n - 1 ) ! } | c _ { n - 1 , n } | - \\sum _ { i = 2 } ^ n | x _ i | \\frac { | r | ^ { n - i } } { ( n - i ) ! } | c _ { n - i , n } | \\\\ & \\geq | x _ 1 | ( \\frac { | r | ^ { n - 1 } } { ( n - 1 ) ! } | c _ { n - 1 , n } | - \\sum _ { i = 2 } ^ n \\frac { | r | ^ { n - i } } { ( n - i ) ! } | c _ { n - i , n } | ) . \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} \\sum _ { P \\in Q ^ { ( 5 , 3 ) } } X _ { P } & = X _ { 1 + 2 + 3 } + X _ { 1 + 2 + 4 } + X _ { 1 + 2 + 5 } + X _ { 1 + 3 + 4 } + X _ { 1 + 3 + 5 } \\\\ & + X _ { 1 + 4 + 5 } + X _ { 2 + 3 + 4 } + X _ { 2 + 3 + 5 } + X _ { 2 + 4 + 5 } + X _ { 3 + 4 + 5 } . \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} - \\Delta \\psi = \\left ( \\frac { \\omega } { c } \\right ) ^ 2 \\epsilon ( x , \\omega ) \\psi . \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} \\Phi _ 1 ( y ) = ( h a ) y ( h a ) ^ { - 1 } \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} { \\small \\begin{alignedat} { 2 } g = \\left ( \\begin{array} { c c c | c c c c } r & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & r & 0 & 0 & 0 & 0 & 0 \\\\ c & - c & r & 0 & 0 & 0 & 0 \\\\ \\hline 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ - c & 0 & 0 & a & r & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & c & 0 & 0 & 0 & b & r \\end{array} \\right ) , \\end{alignedat} a , b , c \\in \\R ; ~ r \\ne 0 . } \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} E _ L ( f ) \\leq C ( k , \\zeta , s ) \\| f \\| _ { k , \\zeta } L ^ { - k - \\zeta } = \\mathcal { O } ( L ^ { - k - \\zeta } ) , \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ 1 { \\rm e } ^ { i x \\xi } \\ , \\psi [ f ] ( x , \\xi ) \\ , d \\xi \\ , \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} f \\in \\mathcal { S } ^ { m } _ W \\Longrightarrow f ( D ) = f ( D ) ^ \\# + f ( D ) ^ b , \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} ( n _ { k } - n _ { 1 } ) - ( n _ { k } - n _ 2 ) = n _ 2 - n _ 1 \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} \\mathcal O _ { X , p } = \\mathcal O _ { Y _ 0 , p } \\overset { \\sigma _ 1 } { \\rightarrow } \\mathcal O _ { Y _ 1 , p } = \\mathcal O _ { Y _ 0 , p } / ( a _ 1 ) \\overset { \\sigma _ 2 } { \\rightarrow } \\cdots \\overset { \\sigma _ { d - 1 } } { \\rightarrow } \\mathcal O _ { Y _ { d - 1 } , p } = \\mathcal O _ { Y _ { d - 2 } , p } / ( a _ { d - 1 } ) . \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} - \\mu { n \\choose 2 } 4 ^ { 2 } + 2 { n \\choose 3 } 4 ^ { 3 } + { n \\choose 4 } 4 ^ { 4 } + 8 b _ { n - 2 } ( n - 2 ) ^ { 2 } = 8 b _ { n - 2 } n ^ { 2 } \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} \\mathcal H _ \\lambda ( u ) : = \\langle ( L _ u + \\lambda I d ) ^ { - 1 } 1 \\ , | \\ , 1 \\rangle \\ . \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} \\mathcal { L } _ { t } \\left ( c + \\frac { \\partial } { \\partial t } \\right ) ^ { \\nu } f ( x , t ) = ( c + s ) ^ { \\nu } \\mathcal { L } _ { t } { f ( x , t ) } - ( c + s ) ^ { \\nu - 1 } f ( x , 0 ) , \\ ; s > 0 . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} & \\mathbb { E } \\left [ \\boldsymbol { s } \\boldsymbol { s } ^ { \\rm H } \\right ] = \\rho ^ 2 \\textbf { 1 } \\cdot \\textbf { 1 } ^ { \\rm T } + \\rho ( 1 - \\rho ) \\mathbf { I } \\mathbb { E } \\left [ \\boldsymbol { s } ^ { \\rm H } \\right ] = \\rho \\textbf { 1 } , \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} t ^ * m = m ( 1 _ A \\otimes t ^ * ) . \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\sum _ { a = 0 } ^ n \\sum _ { b = 0 } ^ a \\sum _ { c = 0 } ^ b ( - 1 ) ^ { n - b } = ( - 1 ) ^ { n } \\sum _ { a = 0 } ^ n \\sum _ { b = 0 } ^ a ( - 1 ) ^ { b } ( b + 1 ) . \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} \\hat { y } ^ { ( m ) } ( t , \\mu ) = \\displaystyle { \\sum _ { i = 1 } ^ m } \\hat { w } _ i ( t ) \\Phi _ i ( \\mu ) , \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} \\begin{aligned} - C \\leq ( 1 - x ) ^ { - s } I _ 3 ' - ( 1 - x ) ^ { - s } \\int _ { 0 } ^ { 1 } \\frac { \\left [ 1 - \\left ( 1 - k \\right ) ^ s \\right ] ^ { p - 1 } - \\left [ \\left ( 1 + k \\right ) ^ s - 1 \\right ] ^ { p - 1 } } { k ^ { 1 + s p } } \\mathrm { d } k \\leq C , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} J _ 2 = \\int _ { 2 a t } ^ { 2 b t } 2 A ( 2 x ) P ( 2 x , k ) \\left ( \\int _ { 2 a t } ^ x \\frac { e ^ { i u ^ 2 / ( 4 t ) } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) e ^ { - i k u } d u \\right ) d x \\ , . \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} m = \\log ^ { - 1 - \\rho } ( r + e ) , m = ( r + 1 ) ^ { - \\rho } , \\qquad \\rho > 0 . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} | c _ n | ^ 2 ~ & = ~ ( t _ { n + 1 } - y _ n + \\kappa ) ^ 2 + ( t _ n \\sin \\alpha _ n ) ^ 2 \\\\ [ . 3 e x ] & = ~ \\Big ( t _ { n + 1 } - t _ { n + 1 } \\sqrt { 1 - \\big ( \\tfrac { t _ { n } } { t _ { n + 1 } } \\sin \\alpha _ n \\big ) ^ 2 } + \\kappa \\Big ) ^ 2 + ( t _ n \\sin \\alpha _ n ) ^ 2 \\\\ [ . 3 e x ] & < ~ \\big ( ( n - 1 ) ! \\big ) ^ 2 \\Big ( \\tfrac 1 { n ( n + 1 ) } + \\tfrac 1 { ( n - 1 ) ! } \\Big ) ^ 2 + \\big ( ( n - 1 ) ! \\big ) ^ 2 \\\\ [ . 3 e x ] & < ~ 3 \\big ( ( n - 1 ) ! \\big ) ^ 2 \\ , < ~ ( t _ { n + 1 } \\sin \\alpha _ { n + 1 } ) ^ 2 \\ , < \\ , t ^ 2 _ { n + 1 } , \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} U _ { \\ell } ( h ( z ) ) = \\frac { 1 } { \\ell } \\sum _ { k = 0 } ^ { \\ell - 1 } h \\bigg ( \\zeta _ { \\ell } ^ k q ^ { \\frac { 1 } { \\ell } } \\bigg ) . \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} \\psi \\mapsto \\Lambda _ { \\lambda , \\lambda ' } ( \\psi ) : = \\int _ { G / H } \\int _ { G / H } \\left ( \\int _ H \\psi ( x h y H ) d h \\right ) d \\lambda ( x H ) d \\lambda ' ( y H ) , \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} u v = T _ { u } v + T _ { v } u + R ( u , v ) \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} c _ T ( g ) = \\sum _ { i , j = 1 } ^ N \\log _ 2 ( | \\langle T , g ( T ) \\rangle _ { i j } + \\delta _ { i j } | + 1 ) , \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A \\left ( x \\right ) u + V \\left ( x , t \\right ) u = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} P _ n ( 1 - H ) ( P _ n x + w _ k ) & = P _ n ( P _ n x + w _ k ) - P _ n H ( P _ n x + w _ k ) \\\\ & = P _ n x - P _ n H x \\\\ & = P _ n ( 1 - H ) x . \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} \\mathcal { F } ^ { A K S Z } = \\mathrm { M a p } ( T [ 1 ] I , \\mathcal { F } ^ { ( 1 ) } ) \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} T ( u ) T ( v ) = T ( u \\cdot T ( v ) + T ( u ) \\cdot v + H ( T u , T v ) ) , ~ u , v \\in M . \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} | | F _ { 6 } ( t , r ) | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } & \\leq C \\int _ { \\frac { t } { 4 } } ^ { \\frac { t } { 2 } } \\frac { \\lambda ( t ) ^ { 4 } r ( v _ { 4 } ( t , r ) ^ { 2 } + v _ { 5 } ( t , r ) ^ { 2 } ) } { r ^ { 8 } } d r + C \\int _ { \\frac { t } { 2 } } ^ { \\infty } \\frac { \\lambda ( t ) ^ { 4 } ( v _ { 4 } ^ { 2 } + v _ { 5 } ^ { 2 } ) r d r } { r ^ { 8 } } \\\\ & \\leq \\frac { C } { t ^ { 8 } \\log ^ { 1 0 b + 4 N - 2 } ( t ) } \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} \\beta _ { m a x } = c _ 1 t \\frac { ( c _ 1 ^ 2 - c _ 2 ^ 2 - c _ 1 c _ 2 ) } { 2 c _ 1 - c _ 2 } = c _ 1 t \\Bigl ( 1 + \\frac { c _ 1 ( c _ 1 + c _ 2 ) } { c _ 2 ^ 2 - 2 c _ 1 ^ 2 } \\Bigr ) \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} f = \\sum _ { j = 0 } ^ \\infty \\langle f , b _ { j , n - 1 } \\rangle b _ { j , n - 1 } . \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} h ( x ) \\eta ( x ) - ( h ' ( x ) \\psi ( x ) ) ' = 0 \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\int _ { D E } e ^ { s x } \\overline { G } ( s , t ) d s = \\int _ { r } ^ { - r + \\lambda _ 2 - \\lambda _ 1 } e ^ { - x \\lambda _ 1 } e ^ { - w x } e ^ { t ( c _ 1 \\lambda _ 1 ^ { \\alpha _ 1 } + c _ 2 \\lambda _ 2 ^ { \\alpha _ 2 } ) } e ^ { - t [ c _ 1 ( w ^ { \\alpha _ 1 } e ^ { i \\pi \\alpha _ 1 } ) + c _ 2 ( \\lambda _ 2 - \\lambda _ 1 + w e ^ { i \\pi } ) ^ { \\alpha _ 2 } ] } d w . \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} P ^ 2 _ { H \\Z P } ( 1 + a _ 1 \\tilde { t } + a _ 2 \\tilde { t } ^ 2 + \\cdots ) & = P ^ 2 ( 1 ) + P ^ 2 ( a _ 1 \\tilde { t } ) + P ^ 2 ( a _ 2 \\tilde { t } ^ 2 ) + \\cdots + ( ) \\\\ & = 1 + a _ 1 ^ 2 \\tilde { t } ( \\tilde { t } + \\tilde { z } ) + a _ 2 ^ 2 \\tilde { t } ^ 2 ( \\tilde { t } + \\tilde { z } ) ^ 2 + \\cdots + ( ) , \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} \\begin{array} { l } \\partial ^ 2 _ t u = \\partial ^ 2 _ x u + \\frac { \\kappa ^ 2 } { 4 } | u | ^ r - \\frac { c _ 1 ^ 2 + c _ 2 ^ 2 } { 2 } u , t > 0 , x \\in D , \\\\ u ( t , x ) = 0 , \\mbox { f o r } ~ x \\in \\partial D , ~ t \\geq 0 , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\partial _ t u ( 0 , x ) = v _ 0 ( x ) , \\end{array} \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\begin{bmatrix} x _ 1 ( \\xi , t ) \\\\ x _ 2 ( \\xi , t ) \\end{bmatrix} = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} \\frac { \\partial } { \\partial \\xi } \\Bigl ( \\begin{bmatrix} \\rho ( \\xi ) ^ { - 1 } & 0 \\\\ 0 & T ( \\xi ) \\end{bmatrix} \\begin{bmatrix} x _ 1 ( \\xi , t ) \\\\ x _ 2 ( \\xi , t ) \\end{bmatrix} \\Bigr ) \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} \\begin{array} { l } \\mathop { \\min } \\limits _ { { t _ { } } } \\ , \\ , \\ , { \\rm { P } } _ { } ^ e \\\\ \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , t _ { s } = t _ { } + t _ { } . \\end{array} \\ \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} { \\mathbf E } _ p ( u ) = \\int _ { \\mathbb { R } ^ 2 } | \\nabla u | ^ p + \\frac { 1 } { 2 } ( 1 - | u | ^ 2 ) ^ 2 , \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} \\psi _ { \\varepsilon } ( a , b ) = a + b - \\sqrt { a ^ 2 + b ^ 2 + 2 \\varepsilon ^ 2 } . \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} { e ^ { c - 1 } } = \\frac { { { \\varsigma ^ 2 } } } { { 2 \\left ( { \\sqrt { 1 + { \\varsigma ^ 2 } A } - 1 } \\right ) } } . \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} \\omega = \\sum _ { i = 0 } ^ 3 \\omega _ i \\equiv \\frac 1 2 \\left ( A d A + d x _ 1 A + x _ 1 d x _ 2 \\right ) + \\frac 1 6 A [ A , A ] - \\frac { 1 } { 1 2 } x _ 1 [ x _ 2 , x _ 3 ] , \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} G _ t G & = \\frac { 3 ( e ^ 2 - g _ i ^ 2 ) + 6 e f \\varepsilon _ i } { 8 \\varepsilon _ i ^ 3 } t ^ 2 - \\frac { e f } { 2 \\varepsilon _ i } t - \\frac { 3 ( e ^ 2 - g _ i ^ 2 ) + 2 e f \\varepsilon _ i } { 8 \\varepsilon _ i } , \\\\ G _ t ^ 2 + G _ { t t } G & = \\frac { 3 ( e ^ 2 - g _ i ^ 2 ) + 6 e f \\varepsilon _ i } { 4 \\varepsilon _ i ^ 3 } t - \\frac { e f } { 2 \\varepsilon _ i } . \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} \\Sigma ( p , X ) = \\begin{cases} { o r d } ( v , X ) + { o r d } ( w , X ) , & v \\neq w , \\\\ { o r d } ( v , X ) , & v = w . \\end{cases} \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} \\b A = \\begin{pmatrix} A & \\tfrac { 1 } { \\ell } e \\\\ \\tfrac { - \\epsilon } { \\ell } e ^ t & 0 \\end{pmatrix} , \\b F = d \\b A + \\b A ^ 2 = \\begin{pmatrix} F & \\tfrac { 1 } { \\ell } T \\\\ \\tfrac { - \\epsilon } { \\ell } T ^ t & 0 \\end{pmatrix} = \\begin{pmatrix} R - \\tfrac { \\epsilon } { \\ell ^ 2 } e e ^ t & \\tfrac { 1 } { \\ell } D ^ A e \\\\ \\tfrac { - \\epsilon } { \\ell } ( D ^ A e ) ^ t & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} \\begin{aligned} G _ { m } ( a ^ { \\prime } , b ^ { \\prime } , c ^ { \\prime } ) \\backslash B _ { m } ( a ^ { \\prime } , b ^ { \\prime } , c ^ { \\prime } ) \\subseteq G _ { m } ( a , b , c ) \\backslash B _ { m } ( a , b , c ) . \\end{aligned} \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} \\varrho _ p ( A _ 1 , \\ldots , A _ N ) : = \\lim _ { n \\to \\infty } \\left ( \\sum _ { i _ 1 , \\ldots , i _ n = 1 } ^ N \\left \\| A _ { i _ n } \\cdots A _ { i _ 1 } \\right \\| ^ p \\right ) ^ { \\frac { 1 } { n } } \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} \\frac { u _ { m - 2 } ' } { u _ m ' } = \\frac { u _ { m - 2 } } { u _ m } + t \\ , \\frac { u _ { m - 1 } } { u _ m } + \\frac 1 2 \\ , t ^ 2 = \\frac { u _ { m - 2 } } { u _ m } + t \\ , \\left ( \\frac { u _ { m - 1 } } { u _ m } + \\frac { t } { 2 } \\right ) \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} s ( z ) : = ( r _ u ( z ) , n ) , z \\in a ^ n ( U ' ) , n \\in \\Z , \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} 0 = \\frac { \\tilde \\gamma } { ( 4 N ) ^ { 1 - s } } \\Big { ( } - 1 + \\frac { \\tilde \\gamma + 2 } { N } \\Big { ) } \\frac { C _ s } { 2 - 2 s } + C _ s O ( \\frac { 1 } { 3 - 2 s } ) + C _ s O ( 1 ) , \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} a _ { n _ 0 + 1 } > a _ { n _ 0 + 1 } - 1 > \\sum _ { i = 1 } ^ { n _ 0 } a _ i , \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} Z _ t = G ( x _ 0 ) + \\int _ 0 ^ t \\tilde \\mu ( Z _ { s - } ) d s + \\int _ 0 ^ t \\tilde \\sigma ( Z _ { s - } ) d W _ s + \\int _ 0 ^ t \\int _ \\R \\tilde \\rho ( Z _ { s - } , y ) \\nu ( d y , d s ) , \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} \\mathcal { M } ( G / H ) : = \\{ \\lambda \\in M ( G / H ) : \\lambda _ h = \\lambda \\ \\forall h \\in H \\} , \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} \\mathbf { P } \\left ( ( \\eta ^ { N , M a r } _ n ) _ { n = 1 } ^ { M } = ( x _ n ) _ { n = 1 } ^ { M } \\right ) = \\frac { \\mathbf { E } \\left ( \\nu ( \\eta ^ { M a r } _ 0 ) ^ { - 1 } \\mathbf { 1 } _ { \\left \\{ ( \\eta ^ { M a r } _ n ) _ { n = 1 } ^ { M } = ( x _ n ) _ { n = 1 } ^ { M } , \\ : \\eta ^ { M a r } _ N = \\eta ^ { M a r } _ 0 , \\ : S _ N > 0 \\right \\} } \\right ) } { \\mathbf { E } \\left ( \\nu ( \\eta ^ { M a r } _ 0 ) ^ { - 1 } \\mathbf { 1 } _ { \\left \\{ \\ : \\eta ^ { M a r } _ N = \\eta ^ { M a r } _ 0 , \\ : S _ N > 0 \\right \\} } \\right ) } . \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} f _ { i _ 0 i _ { j + 1 } } \\circ s \\circ \\iota _ { i _ j } = f _ { i _ j i _ { j + 1 } } : A _ { i _ j } - > A _ { i _ { j + 1 } } . \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} ( g _ 1 \\otimes g _ 2 \\otimes x ) * ( h _ 1 \\otimes h _ 2 \\otimes y ) = \\delta _ { g _ 2 , h _ 1 } ( g _ 1 \\otimes h _ 2 \\otimes x y \\gamma ^ { g _ 2 } ) , \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} \\delta \\langle h , \\ , f \\rangle & = \\langle h , \\ , \\dot { f } \\rangle + \\langle \\ , u , \\ , f \\rangle + 3 u \\langle h ^ 2 , \\ , f \\rangle \\\\ & + u k _ 0 \\Delta f + 2 \\ , h ^ 2 ( \\nabla u , \\nabla f ) + \\frac { 1 } { 2 } u \\langle \\nabla | h | ^ 2 , \\nabla f \\rangle - | h | ^ 2 \\langle \\nabla u , \\nabla f \\rangle , \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} f ( z ) = f ( z , \\omega ) = f [ \\{ \\xi _ k \\} ] ( z ) \\stackrel { d e f } { = } \\sum _ { k = 0 } ^ { \\infty } \\xi _ k \\ z ^ k . \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} A _ j ( v ) = v ^ j \\frac { | v | ^ 2 - 5 } { \\sqrt { 1 0 } } \\sqrt { \\mu } , \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} a \\wedge \\neg a = \\bot , & & a \\vee \\neg a = \\top \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} - \\partial _ { t t } u + \\partial _ { r r } u + \\frac { 1 } { r } \\partial _ { r } u - \\frac { u } { r ^ { 2 } } = 0 \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} I = \\sum _ { i , j , k = 1 } ^ { 2 } \\int _ { \\Omega } u _ i \\partial _ { i } \\partial _ j ^ 2 \\theta \\partial _ k ^ 2 \\theta \\dd x + \\sum _ { i , j , k = 1 } ^ { 2 } \\int _ { \\Omega } \\partial _ j u _ i \\partial _ { i , j } \\theta \\partial _ k ^ 2 \\theta \\dd x = I _ a + I _ b \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} | E _ { i n t , 2 } | & \\leq \\frac { 1 } { 2 b ^ { 2 } ( \\log ( \\log ( t ) ) ) ^ { 3 / 2 } \\log ^ { b } ( t ) } + \\frac { 1 } { \\sqrt { \\log ( \\log ( t ) ) } \\log ^ { b + 1 } ( t ) } + \\frac { 1 } { 2 ( b + 1 ) ( \\log ( \\log ( t ) ) ) ^ { 3 / 2 } \\log ^ { b + 1 } ( t ) } \\\\ & \\leq \\frac { 1 } { 1 0 0 } \\frac { 1 } { b \\log ^ { b } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } , t \\geq T _ { 0 } \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} f ( v , w ) = \\mathcal { B } \\left ( F ( v ) \\cdot G ( w ) \\right ) , F , \\ , G : \\ , \\mathbb { C } \\rightarrow \\mathbb { C } ^ J , \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} Q ( g , h ) = \\nabla \\cdot { \\Big ( } [ a * g ] \\ ; \\nabla h - [ a * \\nabla g ] \\ ; h { \\Big ) } , \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} - 4 \\int _ { t } ^ { \\infty } \\frac { \\lambda _ { 0 , 0 } '' ( x ) d x } { ( \\lambda _ { 0 , 0 } ( t ) ^ { 1 - \\alpha } + x - t ) ( 1 + x - t ) ^ { 3 } } & = - 4 \\int _ { t } ^ { \\infty } \\frac { b } { x ^ { 2 } \\log ^ { b + 1 } ( x ) } \\frac { d x } { ( \\lambda _ { 0 , 0 } ( t ) ^ { 1 - \\alpha } + x - t ) ( 1 + x - t ) ^ { 3 } } + E _ { v _ { 3 , i p } } \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} \\mu _ { J , K } \\times \\mu _ { K , J } \\circ F _ { J , K } ^ { - 1 } = \\mu _ { J , K } \\times \\mu _ { K , J } , \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} \\bar { F } ( x , y ) = \\begin{pmatrix} x + c \\ , y \\\\ y + p ( x ) + y q ( x ) + u ( x , y ) + g ( x , y ) \\end{pmatrix} , \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} h _ i ( x , G ( x ) ) \\equiv 0 \\mod x ^ { N _ i } , \\ ( 1 \\leq i \\leq k ) N _ 1 + \\dots + N _ k = N . \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} c _ { n } = 2 p v ^ { - 1 } \\lambda ^ { v } b _ { n } ^ { p - v } , d _ { n } = b _ { n } ^ { p } , \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\iota _ \\mu : U _ \\mu : = \\lambda ^ { - 1 } ( \\mu ) \\hookrightarrow U \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} S _ 1 + S _ 2 = ( - 1 ) ^ { k - 1 + \\lfloor \\frac { k - 1 } { p } \\rfloor } ( k - 2 ) ! ( k - 1 - q 1 _ A ) \\binom { q / p - 1 } { \\lfloor ( k - 1 ) / p \\rfloor } , \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} & I V = \\frac { - 4 8 \\lambda ' ( t ) } { \\lambda ( t ) ^ { 4 } } \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) \\left ( K ( s - t , \\lambda ( t ) ) + K _ { 1 } ( s - t , \\lambda ( t ) ) \\right ) \\\\ & | I V | \\leq \\frac { C } { t ^ { 3 } \\log ( t ) } \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} \\# J _ { n s } ( 1 3 ) ( \\mathbb { F } _ { 5 } ) & = 3 \\cdot 7 \\cdot 1 1 \\cdot 1 3 ^ 2 \\cdot 2 9 , \\\\ \\# J _ { n s } ( 1 3 ) ( \\mathbb { F } _ { 1 9 } ) & = 2 ^ 2 \\cdot 7 \\cdot 8 3 \\cdot 9 7 \\cdot 1 1 3 \\cdot 8 8 3 , \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} A ( t , x ) : = & ( 1 + \\epsilon ) F ( \\Phi ( x - \\bar h ( t ) ) ) - F ( ( 1 + \\epsilon ) \\Phi ( x - \\bar h ( t ) ) ) \\\\ & - ( 1 + \\epsilon ) \\delta ' ( t ) \\Phi ' ( x - \\bar h ( t ) ) - \\beta ( t + \\theta ) ^ { - \\beta - 1 } \\Phi ( x - \\underline h ( t ) ) . \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} h \\big ( \\varphi _ t ( z ) \\big ) = h ( z ) + i t , \\quad \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} \\mathcal T _ X ( \\bar x ) = \\bigcup \\limits _ { I \\subset I ^ { 0 0 } ( \\bar x ) } \\mathcal L _ { X ( \\bar x , I ) } ( \\bar x ) , \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} T ( u ) \\stackrel { d e f } { = } \\sup _ { k } { \\bf P } ( | \\eta _ k | / \\sigma _ k > u ) , \\ \\ u \\ge 1 \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} A + _ G \\lambda A : = \\{ a + \\lambda b : ( a , b ) \\in G \\} , \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} V ( \\omega , \\lambda ; \\omega _ 0 ) : = \\{ x \\in X | \\kappa _ { \\omega } < 0 \\ , \\ , a n d \\ , \\ , ( - \\kappa _ { \\omega } \\omega ) ^ n > \\lambda \\omega _ 0 ^ n \\ , \\ , a t \\ , \\ , x \\} , \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} W ( \\bar w , w ) _ a = | m | ^ 2 W _ a ( \\bar u , u ) + \\bar { m } W _ a ( \\bar u , v ) + m W _ a ( \\bar v , u ) + W _ a ( \\bar v , v ) = 2 \\i \\Im m . \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} ( x + y ) z _ { x y } - \\mathcal { W } ( z _ x , z _ y ) = 0 , \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} F ( x , y ) = - 2 \\ln | x - y | ( x \\ne y ) . \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} \\partial _ { t } \\phi ( R _ { \\ell } ) + U _ { \\ell } \\cdot \\nabla \\phi ( R _ { \\ell } ) = - \\dfrac { 1 } { \\tau ^ 2 } \\phi ' ( R _ { \\ell } ) \\sqrt { R _ { \\ell } } { \\rm T r a c e } \\mathbf { T } _ { N , \\ell } . \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} \\begin{array} { c l } \\displaystyle \\min _ { x , y , z ^ I } & f ( x ) \\\\ { \\rm s . t . } & H ( x , y , z ^ I ) = 0 . \\end{array} \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} G _ r ( z ) = \\begin{bmatrix} \\nabla ^ r _ x { L } ( x , \\lambda ) \\\\ - \\nabla ^ r _ \\lambda { L } ( x , \\lambda ) \\end{bmatrix} , \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} \\mathcal T _ X ( \\bar x ) = \\bigcup \\limits _ { I \\subset I ^ { 0 0 } ( \\bar x ) } \\mathcal T _ { X ( \\bar x , I ) } ( \\bar x ) \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} h ( r , \\omega ) = r ^ { \\gamma } ( c + c ' \\log r ) w _ j ( \\omega ) + O _ 2 ( r ^ { \\gamma - \\epsilon } ) \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} V = \\bigoplus _ { g \\in G } V _ g . \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} \\eta _ { \\varphi } ( D _ { \\partial } ) : = 2 c _ p \\ , \\left [ \\sum _ { i = 0 } ^ { 2 p } \\int _ 0 ^ { \\infty } \\sigma _ \\varphi ( p _ t , \\dots , [ \\dot { p } _ t , p _ t ] , \\dots , p _ t ) d t \\right ] \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} p _ n ( k ) = 2 + \\frac { 6 } { 2 + \\nu ^ { 3 / 2 } } n ^ { - 1 } + O ( n ^ { - 2 } ) . \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} \\tilde { D } _ { \\pm } \\bigl ( T _ { x } ( f \\otimes c ) \\bigr ) = & T _ { X _ { \\pm } ( x ) } ( f \\otimes c ) + T _ { x } D _ { 2 , \\pm } \\left ( f \\otimes c \\right ) \\bigr ) . \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} \\vline \\quad \\begin{aligned} & & & 2 \\sum _ { j \\in E _ { \\mathrm { s s } } [ i ] } \\alpha _ { i j } + \\sum _ { k \\in E _ { \\mathrm { s a } } [ i ] } \\alpha _ { i k } \\\\ & & & \\frac { 1 } { 2 } \\sum _ { j \\in E _ { \\mathrm { s s } } [ i ] } \\alpha _ { i j } ^ 2 + \\frac { 1 } { 2 } \\sum _ { k \\in E _ { \\mathrm { s a } } [ i ] } \\alpha _ { i k } ^ 2 = f ( U ^ { ( 0 ) } , V ^ { ( 0 ) } ) . \\end{aligned} \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} \\omega = ( t _ 1 x z + t _ 3 y ^ 2 + t _ 4 x z + t _ 5 x y ) d x + ( t _ 0 y z + t _ 2 x z ) d y - ( t _ 0 y ^ 2 + ( t _ 1 + t _ 2 ) x y + t _ 4 x ^ 2 ) d z . \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} \\prec _ t = \\sum _ { i \\geq 0 } t ^ i \\prec _ i , \\succ _ t = \\sum _ { i \\geq 0 } t ^ i \\succ _ i \\curlyvee _ t = \\sum _ { i \\geq 0 } t ^ i \\curlyvee _ i ( \\prec _ 0 = \\prec , ~ \\succ _ 0 = \\succ \\curlyvee _ 0 = \\curlyvee ) \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} K ( \\Psi ; M ; \\gamma ) : = \\prod _ { j \\in M } ( 1 - L _ j ) \\prod _ { ( i , j ) \\in \\Psi } ( 1 - R _ { i j } ) ^ { - 1 } g _ \\gamma \\ , , \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} \\sum _ { s = i _ { 1 } } ^ { j _ { 2 } } m ^ { 2 } _ { s , s + 1 } = 2 ( j _ { 2 } - i _ { 1 } + 1 ) m _ { i _ { 2 } , j _ { 2 } + 1 } m _ { i _ { 1 } , n + 1 } + \\cdots , \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} \\lim _ n S ( f , \\Psi , \\mathcal { P } _ n ) = \\int _ I f d \\Psi . \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} \\nabla _ { 1 } \\boldsymbol { e _ { 2 } } = - q _ { 1 } \\boldsymbol { e _ { 1 } } + b \\boldsymbol { n } , \\ \\ \\ \\ \\nabla _ { 2 } \\boldsymbol { e _ { 2 } } = - q _ { 2 } \\boldsymbol { e _ { 1 } } + c \\boldsymbol { n } , \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} F _ { i _ 1 } \\cap \\ldots \\cap F _ { i _ \\ell } = \\left \\{ \\begin{array} { l c } \\varnothing , & \\ell > L , \\\\ \\neq \\varnothing , & \\ell \\le L . \\end{array} \\right . \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} | \\partial _ { t } \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) f \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | & \\leq C \\int _ { 0 } ^ { \\infty } \\frac { r | \\lambda ' ( t ) | } { ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) ^ { 3 } } | f ( t , r ) | r d r + C \\int _ { 0 } ^ { \\infty } \\frac { r \\lambda ( t ) } { ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) ^ { 3 } } | \\partial _ { t } f ( t , r ) | r d r \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} { } ^ h ( g , A ) = ( h g , h A h ^ { - 1 } + \\d h h ^ { - 1 } ) . \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} D = D _ 1 + D _ 2 . \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} \\mathfrak h ^ \\perp = \\mathfrak p _ u \\oplus ( \\mathfrak d ^ \\perp \\cap \\mathfrak c ) \\oplus ( \\mathfrak h ^ \\perp \\cap \\mathfrak p ^ + _ u ) . \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} u ( 0 , t ) = u ( l , t ) = 0 \\quad \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} B _ \\lambda ^ Q ( d F ) = B _ \\lambda ( v _ F ) = [ z + \\lambda { K _ 1 } , v _ F ] . \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} \\Omega _ { \\rho _ { 1 } } = \\{ r t : t \\in \\mathbb { T } , 0 \\le r < R ( t ) \\} \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} \\mu _ { m } ^ { m - 1 } | _ { I } = \\mu _ { m } ^ { m } | _ { I } . \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} P ( f , n ) \\ , : = \\ , \\begin{cases} f ( f + 1 ) \\cdot \\ldots \\cdot ( f + n - 1 ) & n > 0 \\\\ 1 & n = 0 \\\\ \\frac 1 { ( f + n ) ( f + n + 1 ) \\dots ( f - 1 ) } & n \\leq - 1 \\end{cases} \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} \\partial _ t f + v \\cdot \\nabla _ { x } f + L f = \\Gamma ( f , f ) , \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} \\Pr \\bigg ( { x _ D } = 0 ~ \\bigg | ~ { x _ T } = 1 \\bigg ) & = \\Pr \\bigg ( { x _ R } = 0 ~ \\bigg | ~ { x _ T } = 1 \\bigg ) \\times \\Pr \\bigg ( { x _ D } = 0 ~ \\bigg | ~ { x _ R } = 0 , { x _ T } = 1 \\bigg ) \\\\ & + \\Pr \\bigg ( { x _ R } = 1 ~ \\bigg | ~ { x _ T } = 1 \\bigg ) \\times \\Pr \\bigg ( { x _ D } = 0 ~ \\bigg | ~ { x _ R } = 1 , { x _ T } = 1 \\bigg ) . \\ \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} & \\mathcal { T } _ k ^ A : = \\{ ( \\xi , \\eta ) \\in \\R ^ 2 \\ | \\ ( \\xi , \\eta ) \\in A ^ { - 1 } N _ 1 \\bigl ( [ k _ { ( 1 ) } , k _ { ( 1 ) } + 1 ) \\times [ k _ { ( 2 ) } , k _ { ( 2 ) } + 1 ) \\bigr ) \\} , \\\\ & \\tilde { \\mathcal { T } } _ k ^ A : = \\R \\times \\mathcal { T } _ k ^ A . \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} \\partial _ { t } ^ { j } \\partial _ { r } ^ { k } v _ { 2 } ( t , r ) = \\frac { \\partial _ { t } ^ { j } \\partial _ { r } ^ { k } \\left ( \\frac { - b r } { t ^ { 2 } } \\right ) } { \\log ^ { b } ( t ) } + E _ { \\partial _ { t } ^ { j } \\partial _ { r } ^ { k } v _ { 2 } } ( t , r ) \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} \\widetilde { F } ( x _ { 1 } , y _ { 1 } , z _ { 1 } ) & = ( a + b + c ) ( 2 m ^ { \\prime } - 1 ) - 2 ( m ^ { \\prime } - 1 ) ( a + b + c ) - n \\\\ & = a + b + c - n \\equiv 0 \\pmod { 8 } , \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} Q ( A ^ * _ i , A ^ * _ j ) \\ & = \\ - Q ( A _ i , A _ j ) , \\\\ Q ( A ^ { ( M ) } _ i , A ^ { ( M ) } _ j ) \\ & = \\ M \\cdot Q ( A _ i , A _ j ) . \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} { \\phi } _ { p , \\alpha } ( s t ) = s ^ { p } t ^ { p } ( 1 + \\log ^ + ( s t ) ) ^ { \\alpha } = s ^ { p } t ^ { p } ( 1 + \\log ( s t ) ) ^ { \\alpha } \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} | | P | | _ b ^ 2 : = \\| \\chi P \\| ^ 2 _ 1 + \\| \\phi [ \\mathcal { V } , P ] \\| ^ 2 _ 1 + \\| [ \\mathcal { V } , P ] \\| ^ 2 _ 1 + \\| [ \\phi , P ] \\| ^ 2 + \\| P \\| ^ 2 \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\xi _ n ^ { f , \\mu } ( \\omega ) = \\int _ { [ 0 , 1 ] } f ( x ) \\omega ( \\mu ) ( d x ) \\ ; \\ ; f \\in C ( [ 0 , 1 ] ) . \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} ( X ^ { m } ) ^ n - 1 & = \\prod _ { d _ 1 | m } \\prod _ { d _ 2 | n } \\Phi _ { d _ 1 \\cdot d _ 2 } ( X ) \\\\ & = \\left [ \\prod _ { d _ 1 | m } \\prod _ { d _ 2 | n , d _ 2 < n } \\Phi _ { d _ 1 \\cdot d _ 2 } ( X ) \\right ] \\cdot \\left [ \\prod _ { d _ 1 | m } \\Phi _ { d _ 1 \\cdot n } ( X ) \\right ] \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} \\sum _ { A _ 1 , \\ldots , A _ { k - 2 } \\in F _ 1 } \\deg _ G ( A _ 1 , \\ldots , A _ { k - 2 } ) \\ge \\frac { \\binom k 2 } { k } \\cdot \\frac { ( k - 1 ) ^ { k - 1 } } { k ^ { k - 1 } } \\cdot \\frac { ( m / 2 ) ^ k } { t ^ { k - 1 } } = \\frac { ( k - 1 ) ^ { k } } { 2 k ^ { k - 1 } } \\cdot \\frac { ( m / 2 ) ^ k } { t ^ { k - 1 } } . \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} \\left \\{ ( \\xi _ 2 , \\eta _ 2 ) \\in \\R ^ 2 \\setminus \\bigcup _ { k _ 2 \\in \\check { Z } _ A } \\mathcal { T } _ { k _ 2 } ^ A \\ , \\left | \\ , \\begin{aligned} & | \\Phi ( \\xi _ 1 ' , \\eta _ 1 ' , \\xi _ 2 , \\eta _ 2 ) | \\leq 2 ^ 4 A ^ { - 1 } N _ 1 ^ 3 , \\\\ & | F ( \\xi _ 1 ' , \\eta _ 1 ' , \\xi _ 2 , \\eta _ 2 ) | \\leq 2 ^ 4 A ^ { - 1 } N _ 1 ^ 2 . \\end{aligned} \\right . \\right \\} \\subset \\bigcup _ { \\ell = 1 } ^ 4 \\mathcal { T } _ { k _ { 2 , { ( \\ell ) } } } ^ { 2 ^ { - 2 0 0 } A } . \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty | y _ i | ( t ) \\cdot | a _ i | \\cdot | x | , \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} ( D ^ \\alpha _ x u ) _ x ( x ) = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } & \\left ( \\frac { \\alpha ( u ( 0 ) - u ( x ) ) + ( \\alpha + 1 ) u ' ( x ) x } { x ^ { \\alpha + 1 } } \\right . \\\\ & \\quad \\left . + \\alpha ( \\alpha + 1 ) \\int _ 0 ^ { x } \\frac { u ( y ) - u ( x ) - u ' ( x ) ( y - x ) } { ( x - y ) ^ { \\alpha + 2 } } d y \\right ) . \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} h ( x , G ( x ) ) = 0 \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} V _ { 2 ( r - s ) + 1 } ^ { 1 } \\cap V _ { 2 s - 1 } ^ { 2 } = \\{ 0 \\} , \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\widetilde { p } _ n ^ { ( \\beta ) } } [ { X _ i X _ j } ] = \\mathbb { E } _ { \\widetilde { p } _ n } [ X _ i { X } _ j ] & = \\frac { 1 } { n ^ { 1 / 2 } 2 ^ d } \\sum _ { l = 1 } ^ n \\frac { ( 2 n ^ { - 1 / 2 d } ) ^ { d - 2 } [ 4 n ^ { - 1 / 2 d } X _ { i l } ] [ 4 n ^ { - 1 / 2 d } X _ { j l } ] } { 4 } \\\\ & = \\frac { 1 } { n } \\sum _ { l = 1 } ^ n X _ { i l } X _ { j l } ~ ~ i , j \\in \\{ 1 , \\dots , d \\} ~ ~ i < j . \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} \\| \\mathcal { C } ( a ) \\| _ { \\mathsf { S _ 1 } } = \\frac 1 2 \\sqrt { \\| a \\| _ 2 ^ 2 + 2 \\Lambda ( a ) } + \\frac 1 2 \\sqrt { \\| a \\| _ 2 ^ 2 - 2 \\Lambda ( a ) } . \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} ( U = U ^ { ( 1 ) } \\mapsto \\ldots \\mapsto U ^ { ( K ) } , ( ( a _ h ^ { ( 1 ) } , a _ j ^ { ( 1 ) } ) , \\ldots , ( a _ h ^ { ( K - 1 ) } , a _ j ^ { ( K - 1 ) } ) ) , ( b = b ^ { ( 1 ) } , \\ldots , b ^ { ( K ) } ) ) \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} b _ B : = \\frac { 1 } { \\mu ( B ) } \\int _ B b ( x ) \\ , d \\mu ; \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} \\min \\{ J , K \\} \\le T ^ t \\eta _ n + T ^ t W _ { n - 1 } = T ^ { t + 1 } \\eta _ n + T ^ t W _ n \\le \\max \\{ J , K \\} , \\forall n \\le N . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} \\bigcup _ { \\{ j : 1 \\leq j \\leq m , \\ , j \\ne i \\} } D ( B _ i , B _ j ) = \\lambda _ i \\boxtimes ( G \\backslash \\{ 0 \\} ) , \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} ( - P ) _ { B } ^ { s } u = c _ { s } ' \\lim _ { \\epsilon \\rightarrow 0 _ { + } } \\int _ { \\epsilon } ^ { \\infty } ( 1 - e ^ { t P } ) u \\ , \\frac { d t } { t ^ { 1 + s } } \\quad c _ { s } ' , \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} s _ { n + L } = c _ { L - 1 } s _ { n + L - 1 } + \\dots + c _ 0 s _ n , 0 \\leq n \\leq N - L - 1 . \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} Q _ d ( x , y ) : = \\begin{cases} ( y - x ) V ( y ) , & x < y < d , \\\\ ( x - y ) V ( y ) , & x > y > d , \\\\ 0 & \\end{cases} \\\\ T _ d ( x , y ) : = \\begin{cases} ( y - x ) , & x < y < d , \\\\ ( x - y ) , & x > y > d , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{cases} \\begin{bmatrix} y \\\\ 1 \\end{bmatrix} , & x < 0 , \\\\ \\begin{bmatrix} - 2 ( y - 1 ) \\\\ - 1 \\end{bmatrix} , & x > 0 , \\end{cases} \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} Y _ t ^ x = h ( B _ T + x ) - \\sum _ { i = 1 } ^ m \\int _ t ^ T Z _ s ^ { x , i } d B _ s ^ i - \\sum _ { i = 1 } ^ m \\frac { 1 } { 2 } \\int _ t ^ T \\bar A ( Y _ s ^ x ) ( Z _ s ^ { x , i } , Z _ s ^ { x , i } ) d s + \\int _ t ^ T \\bar f ( Y _ s ^ x , Z _ s ^ x ) d s . \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} N _ 1 \\leq \\frac 1 2 \\| \\partial _ 1 \\nabla \\omega \\| _ { L ^ 2 } ^ 2 + \\frac 1 2 \\| \\nabla \\theta \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} P _ L ( \\tilde { r } ; 0 ) & = - P _ { \\rm a f f i n e } \\left ( - \\tilde { r } ; \\lambda , \\omega , - b _ 3 \\right ) , \\\\ T _ L ( \\tilde { r } ; 0 ) & = T _ { \\rm a f f i n e } \\left ( - \\tilde { r } ; \\lambda , \\omega , - b _ 3 \\right ) , \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} L _ { \\rho ^ i ( e ) } = L ^ i _ { \\rho ( e ) } , \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} \\partial _ { r } v _ { 5 } ( t , r ) = \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { \\infty } d \\xi \\xi J _ { 1 } ' ( r \\xi ) \\sin ( ( t - s ) \\xi ) \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( s , \\xi ) \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} \\left ( A + P - \\varkappa _ { \\mathsf { k } } \\right ) ^ { \\mathsf { n } ^ { \\ast } } \\mathsf { q } _ { \\mathsf { k } , \\gamma } = \\gamma \\left ( 1 - \\varkappa _ { \\mathsf { k } } \\right ) ^ { \\mathsf { n } ^ { \\ast } } \\mathsf { \\hat { p } } _ { \\beta , \\mathsf { G i b b s } } \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} G ( d , s , t ) = \\frac { d ^ 2 } { 2 s } + \\frac { d } { 2 s } \\left ( 2 H - 2 - s \\right ) + \\rho + 1 . \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} ( J _ p ) ^ \\perp \\cap \\ker _ p ( \\pi _ * ) = { \\bf 0 } . \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } a _ m t ^ m = \\frac { 1 } { ( 1 - t ) \\left ( 1 - t x _ 1 \\right ) \\left ( 1 - t ^ 2 x _ 2 \\right ) \\left ( 1 - t x _ 3 \\right ) \\left ( 1 - t ^ 2 x _ 3 \\right ) \\left ( 1 - t x _ 4 \\right ) { } ^ 2 } . \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} p ^ { ( \\alpha ) } _ \\theta ( { \\bf { x } } ) = \\Big ( { Z ( \\theta ) ^ { \\alpha } } \\big / ~ { \\| p _ \\theta \\| ^ \\alpha } \\Big ) ~ \\Big [ h ( { \\bf { x } } ) + w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\Big ] ^ { \\frac { \\alpha } { 1 - \\alpha } } = Z ' ( \\theta ) \\Big [ h ( { \\bf { x } } ) + w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\Big ] ^ { \\frac { 1 } { \\frac { 1 } { \\alpha } - 1 } } , \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} p _ { \\theta } ( { \\bf { x } } ) = N _ { \\theta , \\alpha } \\big [ 1 + b _ \\alpha ( { \\bf { x } } - \\boldsymbol { \\mu } ) ^ \\top \\boldsymbol { \\Sigma } ^ { - 1 } ( { \\bf { x } } - \\boldsymbol { \\mu } ) \\big ] _ + ^ { \\frac { 1 } { \\alpha - 1 } } , \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} \\tau ^ r _ \\varphi ( A _ 0 , \\ldots , A _ k ) : = \\int _ { G ^ { \\times k } } { } ^ b { \\rm T r } _ S \\left ( \\Phi _ { A _ 0 } ( ( g _ 1 \\cdots g _ k ) ^ { - 1 } ) \\circ \\Phi _ { A _ 1 } ( g _ 1 ) \\circ \\ldots \\circ \\Phi _ { A _ k } ( g _ k ) \\right ) \\varphi ( e , g _ 1 , g _ 1 g _ 2 , \\ldots , g _ 1 \\cdots g _ k ) d g _ 1 \\cdots d g _ k \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} \\chi ( X _ { 3 , 9 } , \\mathcal { O } _ { X _ { 3 , 9 } } ( D _ 1 ) ) = \\binom { 5 } { 2 } - 9 = 1 . \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} \\langle j _ p ( \\Pi _ M ^ p x _ m ) - j _ p ( x _ m ) , z \\rangle = 0 \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} \\begin{aligned} & \\bigg [ - \\frac { \\partial ^ 2 } { \\partial { \\phi ^ i _ j } ^ 2 } - ( j - 1 ) \\cot \\phi ^ i _ j \\frac { \\partial } { \\partial \\phi ^ i _ j } + G ^ i _ j ( \\phi ^ i _ j ) + \\frac { \\alpha ^ i _ { j - 1 } } { \\sin ^ 2 \\phi ^ i _ j } \\bigg ] h ^ i _ j ( \\phi ^ i _ j ) = \\alpha ^ i _ { j } h ^ i _ j ( \\phi ^ i _ j ) , \\\\ & i = 1 , 2 , \\cdots , N ~ ~ ~ j = 1 , 2 , \\cdots , d _ i - 1 \\\\ & \\lambda _ i = \\alpha ^ i _ { d _ i - 1 } , ~ ~ ~ \\alpha ^ i _ 0 = 0 . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} \\Phi ( \\zeta ) = \\frac { \\langle A \\zeta , \\zeta \\rangle } { \\langle B \\zeta , \\zeta \\rangle } = \\frac { \\langle B B ^ { - 1 } A \\zeta , \\zeta \\rangle } { \\langle B \\zeta , \\zeta \\rangle } = \\lambda \\frac { \\langle B \\zeta , \\zeta \\rangle } { \\langle B \\zeta , \\zeta \\rangle } = \\lambda . \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} \\chi ( \\nu ) = I n t ( [ N ] , [ N ] ) \\geq - 2 , \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 b \\log ( \\log ( T _ { 0 , 2 } ) ) } + \\frac { b } { \\log ( T _ { 0 , 2 } ) } + \\frac { b } { 2 ( b + 1 ) \\log ( T _ { 0 , 2 } ) \\log ( \\log ( T _ { 0 , 2 } ) ) } \\leq \\frac { 1 } { 1 0 0 } \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} s ( \\mu ) = \\frac { a _ { 2 R } b _ { 0 L } } { a _ { 2 R } b _ { 0 L } - a _ { 2 L } b _ { 0 R } } . \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} \\mathcal T _ X ( \\bar x ) ^ \\circ = \\bigcap \\limits _ { I \\subset I ^ { 0 0 } ( \\bar x ) } \\mathcal L _ { X ( \\bar x , I ) } ( \\bar x ) ^ \\circ \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} \\begin{cases} G _ i \\leq H _ i , & \\\\ G _ i \\leq H _ { i - 1 } - 1 , & . \\end{cases} \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} \\begin{aligned} a & \\mapsto 0 \\\\ b & \\mapsto \\frac 1 2 \\tilde f _ 1 \\left ( \\theta ' - \\frac 1 2 \\theta \\right ) { \\tilde f _ 1 } ^ { - 1 } \\end{aligned} \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} H ^ 2 _ { \\mu , D } ( \\Omega ) = \\left \\{ v \\in H ^ 2 ( \\Omega ) : \\mathcal { Q } _ { \\mu , D } ( v , \\varphi ) = 0 \\ , , \\ \\forall \\varphi \\in \\mathcal { H } ^ 2 _ { 0 , N } ( \\Omega ) \\right \\} . \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} \\partial _ { r } v _ { 2 } ( t , r ) = I _ { r } + I I _ { r } \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} ( x ^ { - 1 } y ) ^ 2 + ( x ^ { - 1 } y ) + ( B ^ 4 x ^ { - 1 } ) ^ 2 + ( B ^ 4 x ^ { - 1 } ) = x + B ^ 4 x ^ { - 1 } + A ^ 4 , \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n ^ { 1 / 6 } } Y ( m , k ) p ^ k \\leq ( m - 1 ) ! + C ( m - 1 ) ! ( m ^ 3 p ) ^ { 1 / 2 } \\sinh ( ( m ^ 3 p ) ^ { 1 / 2 } ) , \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} \\left \\{ G = \\sum _ { j = 1 } ^ { \\infty } \\hat \\alpha _ j \\hat u _ { j , 0 } : \\left ( \\frac { \\hat \\alpha _ j } { \\sqrt { \\mu _ j ( 0 ) } } \\right ) _ { j \\geq N + 2 } \\in l ^ 2 \\right \\} , \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} u _ t = ( a ( u ) \\nabla u ) + g ( x , u ) \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} C = \\left \\{ \\begin{array} { c } L ^ { d + 6 } , 0 = x _ { 0 } \\sim x _ { 1 } \\sim \\cdots \\sim x _ { L ^ { d + 6 } } \\\\ X _ 0 , X _ { 1 } , \\dots , X _ { L ^ { d + 6 } - 1 } \\\\ X _ { i } x _ { i } x _ { i + 1 } X _ { i - 1 } x _ { i } L \\end{array} \\right \\} . \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\dot { \\Gamma } ( t ) = - \\operatorname { d i v } \\left ( \\nu ( t ) \\right ) \\nu ( t ) \\ ; , \\\\ \\Gamma ( 0 ) = \\Gamma _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} \\left | r \\cos \\theta \\int _ { [ 0 , + \\infty ] } \\frac { 1 - t ^ { 2 } } { | r e ^ { i \\theta } - t | ^ { 2 } } \\ , d \\sigma ( t ) \\right | & \\le r \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t ^ { 2 } } { | r e ^ { i \\theta } - t | ^ { 2 } } \\ , d \\sigma ( t ) \\\\ & = \\frac { \\theta } { \\sin \\theta } I _ { r } ( \\theta ) \\le \\frac { \\pi } { 2 } I _ { r } ( f ( r ) ) < 2 \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} \\abs { \\textup { I } _ 1 } & = \\abs { ( - \\Delta \\tilde u - \\tilde { f } , \\chi ) _ { \\Omega _ h \\backslash \\Omega } } \\\\ & \\le \\norm { \\Delta \\tilde u + \\tilde { f } } _ { \\Omega _ h \\setminus \\Omega } \\norm { \\chi } _ { \\Omega _ h \\setminus \\Omega } \\le C h ( \\norm { u } _ { 2 , \\Omega } + \\norm { f } _ { \\Omega } ) \\norm { \\chi } _ { h } . \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} G _ 1 : = \\rho F - G | _ S \\in H ^ \\infty ( S , H ^ \\infty ( S _ o ' ) ) = H ^ \\infty ( S \\times S _ o ' ) . \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} v ( r e ^ { i \\theta } ) = \\theta \\left [ 1 - \\frac { r \\sin \\theta } { \\theta } \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t ^ { 2 } } { | r e ^ { i \\theta } - t | ^ { 2 } } \\ , d \\sigma ( t ) \\right ] , \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} \\partial _ c \\min \\left \\{ [ y ^ I ] ^ * , [ z ^ I ] ^ * \\right \\} = \\left [ { \\rm D i a g } ( v _ a ) { \\rm D i a g } ( v _ b ) \\right ] , \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} \\mbox { A u x } _ { p } = \\mbox { A u x } \\left [ p - 2 , \\dots , 0 \\right ] \\mbox { A u x } \\left [ p - 3 , 1 , \\dots , 0 \\right ] \\dots \\mbox { A u x } \\left [ 0 , \\dots , 1 , p - 3 \\right ] \\mbox { A u x } \\left [ 0 , \\dots , p - 2 \\right ] , \\quad \\mbox { f o r a l l $ p \\geq 3 $ . } \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} \\begin{aligned} D ^ { ( 2 ) } _ { w ; 0 } = \\frac { 1 } { \\left ( 1 - e ^ { w \\left ( \\lambda _ 1 \\right ) } \\right ) \\left ( 1 - e ^ { w \\left ( \\lambda _ 2 \\right ) } \\right ) \\left ( 1 - e ^ { w \\left ( \\lambda _ 3 \\right ) } \\right ) { } ^ 2 \\left ( 1 - e ^ { w \\left ( \\lambda _ 4 \\right ) } \\right ) { } ^ 2 } \\end{aligned} . \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} \\varphi ( t ) / r + | x ( t ) | & \\leq \\varphi ( t ) / r + | x _ 0 | + B ( \\varphi ( t _ 0 ) - \\varphi ( t ) ) \\\\ & = \\varphi ( t _ 0 ) / r + | x _ 0 | - ( 1 / r - B ) ( \\varphi ( t _ 0 ) - \\varphi ( t ) ) \\\\ & \\leq \\varphi ( t _ 0 ) / r + | x _ 0 | < R . \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} - \\Delta \\phi _ b = & b ( t ) , \\ \\ \\Omega , \\\\ \\phi _ b \\cdot n = & 0 , \\ \\ \\partial \\Omega \\\\ \\partial _ n \\phi _ b = & ( \\partial _ n \\phi _ b \\cdot n ) n \\ \\ \\partial \\Omega . \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} y ( t , \\xi ) = \\mathcal { F } ( \\sqrt { \\cdot } u ( t , \\cdot \\lambda ( t ) ) ) ( \\xi \\lambda ( t ) ^ { 2 } ) \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} B ^ { - 1 } A \\zeta = \\lambda \\zeta \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\sum _ { a = 0 } ^ n \\sum _ { b = 0 } ^ a \\sum _ { c = 0 } ^ b \\xi _ 3 ^ { 2 n - a - b } = \\xi _ 3 ^ { 2 n } \\sum _ { a = 0 } ^ n \\xi _ 3 ^ { - a } \\sum _ { b = 0 } ^ a \\xi _ 3 ^ { - b } ( b + 1 ) . \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} Q _ { 2 3 2 3 } = \\frac { 3 e ^ { 2 q _ 3 + q _ 2 - 3 q _ 1 } + 2 e ^ { 2 q _ 3 - 2 q _ 1 } - 2 e ^ { 2 q _ 3 - q _ 2 - q _ 1 } - 2 e ^ { 2 q _ 3 - 2 q _ 2 } - e ^ { q _ 3 + q _ 2 - 2 q _ 1 } + e ^ { q _ 3 - q _ 1 } + e ^ { q _ 3 - q _ 2 } } { ( 1 + e ^ { q _ 3 - q _ 2 } ) \\Delta _ 3 } , \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} | q ^ * ( t ) | & \\leq r _ 2 \\Bigl ( \\frac { t } { T } \\Bigr ) ^ a + \\int _ t ^ T \\Bigl ( \\frac { t } { \\tau } \\Bigr ) ^ a \\frac { L r _ 1 ( \\tau / T ) ^ a } { \\tau } d \\tau \\\\ & = r _ 2 \\Bigl ( \\frac { t } { T } \\Bigr ) ^ a + L r _ 1 \\Bigl ( \\frac { t } { T } \\Bigr ) ^ a \\log \\Bigl ( \\frac { T } { t } \\Bigr ) \\mbox { o n $ ( t _ { \\xi } , T ] $ } . \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} x & = u - c v - d , \\\\ y & = u - a v - b . \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} E ( v _ { 5 } ( t ) , \\partial _ { t } v _ { 5 } ( t ) ) & \\leq C \\left ( \\int _ { t } ^ { \\infty } | | N _ { 2 } ( f _ { v _ { 5 } } ) ( s ) | | _ { L ^ { 2 } ( r d r ) } d s \\right ) ^ { 2 } \\leq \\frac { C \\log ^ { 6 } ( t ) } { t ^ { 7 / 2 } } \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} \\mathbf { u } ( t , z , \\upsilon ) = \\Big ( \\begin{matrix} 0 \\\\ g \\end{matrix} \\Big ) + v ' \\nabla _ { z , \\upsilon } ^ { \\perp } P _ { \\neq } \\phi . \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} R ( \\alpha , \\beta ) = \\begin{cases} 1 & \\ ; \\ ; ( \\alpha , \\beta ) = ( 0 , 0 ) \\\\ 0 & \\ ; \\ ; ( \\alpha , \\beta ) \\neq ( 0 , 0 ) . \\end{cases} \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } a _ { 2 , 1 } = 0 , \\ \\ a _ { 2 , 2 } \\alpha _ i = 0 , \\ 4 \\leq i \\leq n - 1 , \\\\ b _ n = a _ { 2 , 2 } \\alpha _ n . \\end{array} \\right . \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} V _ { s } ( x ) : = \\begin{cases} | x | ^ { - s } & s \\in ( 0 , d ) \\\\ - \\log | x | & s = 0 \\end{cases} , x \\in \\R ^ d . \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} \\Sigma _ { i j } \\coloneqq \\Sigma _ { M ^ { i + j } } = \\{ x \\vert _ { M ^ { i + j } } : x \\in \\Sigma \\} \\subset A ^ { M ^ { i + j } } . \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} S _ { j , n } u = \\sum _ { k = 0 } ^ { n - 1 } u _ { j + k } \\circ F ^ k \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} V \\cap \\Gamma = \\gamma ^ + _ j ( [ 1 , 1 + \\delta ) ) \\cup \\gamma ^ + _ k ( [ 1 , 1 + \\delta ) ) = h ( P ) P = [ 0 , \\epsilon ) \\rho ^ { 2 j } \\cup [ 0 , \\epsilon ) \\rho ^ { 2 k } . \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} | | \\left ( \\partial _ { r } + \\frac { 1 } { r } \\right ) v _ { 4 } | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } & = \\int _ { 0 } ^ { \\infty } \\left ( ( \\partial _ { r } v _ { 4 } ) ^ { 2 } + \\frac { \\partial _ { r } ( v _ { 4 } ^ { 2 } ) } { r } + \\frac { v _ { 4 } ^ { 2 } } { r ^ { 2 } } \\right ) r d r \\\\ & = | | \\partial _ { r } v _ { 4 } | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } + | | \\frac { v _ { 4 } } { r } | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} \\mathcal { R } _ { 2 } ( \\tau , t _ { n } ) = u ( t _ { n + 1 } ) - \\Phi ^ \\tau _ { , 2 } ( u ( t _ { n } ) ) \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} \\hat H = H _ 2 - H _ 0 H _ 1 + \\frac { 1 } { 6 } H _ 0 ^ 3 \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} d _ { { N } _ { b } , + } : = - \\tfrac { 1 } { b } \\tfrac { \\partial \\ ; } { \\partial r } + \\tfrac { \\partial \\ ; } { \\partial \\Theta } - 2 \\pi \\mathsf { M } , \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} \\langle \\tilde { L } ( P _ D ^ * \\lambda ) , \\varphi \\rangle = \\langle P _ D ^ * \\lambda , L \\varphi \\rangle & = \\int _ D P _ D \\lambda ( x ) L \\varphi ( x ) d x + \\int _ { D ^ c } L \\varphi ( x ) \\lambda ( d x ) \\\\ & \\eqqcolon I _ 1 + I _ 2 . \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} \\frac { p ^ 0 } { q _ 0 } = \\frac { P ^ 0 } { Q _ 0 } . \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} | I _ 1 | \\leq & \\int _ { - \\infty } ^ 0 \\int _ { - y } ^ { M - y } | y | ^ \\sigma J ( y ) \\psi ( - x ) d x d y + \\int _ 0 ^ M \\int _ { 0 } ^ { M - y } | y | ^ \\sigma J ( y ) \\psi ( - x ) d x d y \\\\ \\leq & \\ 2 \\int _ 0 ^ \\infty \\psi ( - x ) d x \\int _ 0 ^ \\infty y ^ \\sigma J ( y ) d y : = \\tilde C _ 1 < \\infty . \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < c _ 1 t \\ | \\ V ( 0 ) = c _ 1 \\} = e ^ { - \\lambda t } ( e ^ { \\lambda t } - 1 ) = 1 - e ^ { - \\lambda t } = \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} u _ { t } = e ^ { t \\Delta } u _ 0 - \\int _ 0 ^ t \\nabla \\cdot ( e ^ { ( t - s ) \\Delta } ( u _ { s } \\ , K \\ast u _ { s } ) ) \\ , d s , 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c , \\ N ( t ) = 2 k + 1 \\} = \\frac { \\beta } { c t } \\sum _ { j = 0 } ^ { k } \\binom { 2 j } { j } \\Bigl ( \\frac { \\sqrt { c ^ 2 t ^ 2 - \\beta ^ 2 } } { 2 c t } \\Bigr ) ^ { 2 j } \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} R \\nabla \\ ( \\frac { \\Delta \\sqrt { R } } { \\sqrt { R } } \\ ) = { \\rm d i v } ( \\sqrt { R } \\nabla ^ 2 \\sqrt { R } - \\nabla \\sqrt { R } \\otimes \\nabla \\sqrt { R } ) , \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} 0 = D [ Q _ { \\tilde { \\rho } } ( P _ { \\tilde { \\rho } } \\circ T ) ] = D Q _ { \\tilde { \\rho } } ( P _ { \\tilde { \\rho } } \\circ T ) + Q _ { \\tilde { \\rho } } D P _ { \\tilde { \\rho } } ( T ) D T . \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} H _ 0 & = 1 , \\\\ H _ { 4 k } & = \\frac { ( - 1 ) ^ { k } k ^ 2 ( 2 k - 1 ) ! ^ 2 } { 2 ^ { 8 k ^ 2 - 4 k - 2 } } \\prod _ { j = 1 } ^ { 2 k - 1 } ( 2 j + 1 ) ! ^ 4 \\\\ H _ { 4 k + 1 } & = \\frac { ( - 1 ) ^ { k } ( 2 k ) ! ^ 2 } { 2 ^ { 8 k ^ 2 } ( 4 k + 1 ) ! ^ 2 } \\prod _ { j = 1 } ^ { 2 k } ( 2 j + 1 ) ! ^ 4 \\\\ H _ { 4 k + 2 } & = \\frac { ( - 1 ) ^ { k } ( 2 k + 1 ) ! ^ 2 } { 2 ^ { 8 k ^ 2 + 4 k } } \\prod _ { j = 1 } ^ { 2 k } ( 2 j + 1 ) ! ^ 4 \\\\ H _ { 4 k + 3 } & = 0 \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} v _ { 4 , s } ( t , r ) = \\frac { - 1 } { 2 \\pi } \\int _ { B _ { s - t } ( 0 ) } \\frac { d A ( y ) } { \\sqrt { ( s - t ) ^ { 2 } - | y | ^ { 2 } } } \\frac { v _ { 4 , c } ( s , | x + y | ) \\left ( ( x + y ) \\cdot \\widehat { x } \\right ) } { | x + y | } \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} & - \\mu { n \\choose 2 } 4 ^ { 2 } & + 2 { n \\choose 3 } 4 ^ { 3 } + { n \\choose 4 } 4 ^ { 4 } + 8 b _ { n - 2 } n ^ { 2 } - 3 2 b _ { n - 2 } n + 3 2 b _ { n - 2 } \\\\ & = 8 b _ { n - 2 } n ^ { 2 } , \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} \\mu ^ { \\sigma } _ N ( d x ) : = \\frac { 1 } { Z } \\exp \\left ( \\sigma \\sum _ { i = 1 } ^ N x _ i - H ( x ) \\right ) d x . \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} { \\rm R i c } _ f : = { \\rm R i c } + { \\rm H e s s } ( f ) . \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} D ^ m \\Theta ^ { } ( \\mathcal { M } ) - \\Theta ^ { [ m + 1 ] } ( D ^ m \\mathcal { M } ) & = \\int _ 0 ^ 1 D \\mathcal { G } ( \\mathcal { N } ( s ) , D ^ m \\mathcal { M } ) s \\begin{pmatrix} \\Lambda - \\mathcal { M } \\\\ \\vdots \\\\ D ^ { m - 1 } \\left ( \\Lambda - \\mathcal { M } \\right ) \\\\ 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } \\rho _ i ( \\mathcal { B } _ { i } ^ { \\epsilon _ i } ) = 1 \\mathbf { P } , \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t g _ B & = - 2 R i c _ B + 2 p \\frac { \\nabla ^ 2 \\phi } { \\phi } \\\\ \\partial _ t \\phi & = \\Delta \\phi - ( p - 1 ) \\frac { 1 - | \\nabla \\phi | ^ 2 } { \\phi } \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} | - \\lambda ( t ) \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) v _ { 5 } \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | & \\leq \\frac { C } { t ^ { 7 / 2 } \\log ^ { 3 b - 3 + \\frac { 5 N } { 2 } } ( t ) } \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} \\\\ v _ n ( t ) \\leq \\left ( \\dfrac { a } { \\sqrt { M } } \\right ) ^ { n - 1 } \\left ( \\dfrac { C _ { \\varphi } } { \\sqrt { M } } \\sum _ { k = 0 } ^ { n - 2 } \\dfrac { \\sqrt { M } ^ k C _ f ^ k ( T - t ) ^ k } { a ^ k k ! } \\binom { n - 2 } { k } + \\dfrac { C _ { \\varphi } + C _ 0 T } { \\sqrt { M } } \\sum _ { k = 1 } ^ { n - 1 } \\dfrac { \\sqrt { M } ^ k C _ f ^ k ( T - t ) ^ k } { a ^ k k ! } \\binom { n - 2 } { k - 1 } \\right ) \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} R ( \\alpha , \\beta ) = \\iint _ { \\R ^ n \\times \\R ^ n } \\Theta ( \\xi , \\eta ) \\Phi ( \\xi - \\alpha , \\eta - \\beta ) \\ , d \\xi d \\eta , \\alpha , \\beta \\in \\Z ^ n , \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial a ^ 2 } \\zeta ' _ B ( 0 ; a , 1 , 1 ) - 2 \\left ( \\frac 1 { 1 2 } - \\zeta ' _ R ( - 1 ) \\right ) \\frac 1 { a ^ 3 } = \\int _ 0 ^ \\infty \\frac { \\partial ^ 2 } { \\partial a ^ 2 } F ( a t ) \\frac 1 { t ( e ^ { t } - 1 ) } \\ , d t . \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} \\begin{bmatrix} u \\\\ v \\end{bmatrix} \\mapsto \\mathrm { i } \\begin{bmatrix} 0 & - 1 \\\\ 1 & 0 \\end{bmatrix} \\begin{bmatrix} u ' \\\\ v ' \\end{bmatrix} + \\begin{bmatrix} 0 & 0 \\\\ 0 & \\mathrm { i } \\end{bmatrix} \\begin{bmatrix} u \\\\ v \\end{bmatrix} \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} K _ { 1 } ( w , \\lambda ( t ) ) & = \\int _ { 0 } ^ { \\infty } d R \\frac { R } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\frac { R ^ { 2 } \\lambda ( t ) ^ { 2 } } { w } \\\\ & - \\int _ { 0 } ^ { \\infty } d R \\frac { R } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\frac { 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } { 2 w \\left ( \\sqrt { \\left ( - R ^ { 2 } \\lambda ( t ) ^ { 2 } + w ^ 2 + 1 \\right ) ^ 2 + 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } - R ^ { 2 } \\lambda ( t ) ^ { 2 } + w ^ 2 + 1 \\right ) } \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} \\mathfrak { S } _ c ( \\mathfrak R ) = \\mathfrak { S } _ c ( \\mathfrak { G } _ { n i l } ^ { ( 2 ) } ) \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} ~ \\Big \\lbrace I _ L , I _ R , \\tfrac { 1 } { j - k + 1 } \\sum _ { i = k } ^ { j } X _ { i } \\Big \\rbrace , \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} \\delta _ { \\Omega , p } ^ - ( t ) : = \\min \\big \\{ t , ~ \\inf \\{ | z - ( p + i t ) | \\colon \\Re z \\leq \\Re p , ~ z \\in \\C \\setminus \\Omega \\} \\big \\} \\in [ 0 , t ] . \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} & r \\int _ { 0 } ^ { 1 } d \\beta \\int _ { t } ^ { \\infty } d s \\int _ { B _ { s - t } ( 0 ) \\cap B _ { \\frac { s } { 2 } } ( - \\beta x ) } \\frac { d A ( y ) } { ( s - t ) } \\frac { | v _ { 4 , c } ( s , | \\beta x + y | ) | } { | \\beta x + y | } \\leq \\frac { C r } { t ^ { 2 } \\log ^ { 3 b + 2 N - 1 } ( t ) } \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { N \\to \\infty } P \\left \\{ \\int _ { 0 } ^ \\delta \\frac { | Z _ N ( t ) - \\sigma B _ N ( t ) | | \\sigma B _ N ( t ) | ^ { p - 1 } } { w ( t ) } d t > \\varepsilon \\right \\} = 0 \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} \\begin{array} { l } g _ 0 ( x ) = ( x - 1 ) ^ { t _ 0 } + u ( x - 1 ) ^ { k _ 1 } p _ 1 ( x ) + u ^ 2 ( x - 1 ) ^ { k _ 2 } p _ 2 ( x ) + u ^ 3 ( x - 1 ) ^ { k _ 3 } p _ 3 ( x ) , \\\\ g _ 1 ( x ) = u ( x - 1 ) ^ { t _ 1 } + u ^ 2 ( x - 1 ) ^ { k _ 4 } p _ 4 ( x ) + u ^ 3 ( x - 1 ) ^ { k _ 5 } p _ 5 ( x ) , \\\\ g _ 2 ( x ) = u ^ 2 ( x - 1 ) ^ { t _ 2 } + u ^ 3 ( x - 1 ) ^ { k _ 6 } p _ 6 ( x ) , \\\\ g _ 3 ( x ) = u ^ 3 ( x - 1 ) ^ { t _ 3 } , \\end{array} \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} | \\beta _ 0 \\beta _ 1 ( \\eta ^ 2 - 1 ) | \\leq | \\beta _ 0 | \\{ \\beta _ 1 ^ 2 + ( \\eta ^ 2 - 1 ) ^ 2 \\} = o _ p ( | \\ss _ 2 | ) , \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} 1 - G _ 1 \\big ( ( \\alpha ^ * _ n x + \\beta ^ * _ n ) ^ { \\frac { 1 } { p } } \\big ) = n ^ { - 1 } e ^ { - x } \\left \\{ 1 - \\frac { ( 1 - p ) x ^ 2 } { 2 \\log \\frac { n } { 2 } } + \\frac { ( 1 - p ) [ 3 ( 1 - p ) x - 4 ( 1 - 2 p ) ] x ^ { 3 } } { 2 4 ( \\log \\frac { n } { 2 } ) ^ { 2 } } + o ( \\frac { 1 } { ( \\log \\frac { n } { 2 } ) ^ { 2 } } ) \\right \\} . \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} & \\mathbf { B } ( i , j ) = \\mathbf { A } ^ { \\mathsf { T } } ( i , j ) \\circ \\mathbf { A } ( i , j ) = 1 , \\ a n d \\\\ & \\mathbf { B } ( j , i ) = \\mathbf { A } ^ { \\mathsf { T } } ( j , i ) \\circ \\mathbf { A } ( j , i ) = 1 . \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} { \\Pr } \\bigg ( g _ { T , R } ^ \\theta ~ \\bigg | ~ { x _ T } = 0 \\bigg ) & \\sim \\mathcal { N } \\bigg ( { { \\mu } _ { { { 0 } _ { T , R } } } } , { \\sigma } _ { { 0 } _ { T , R } } ^ 2 \\bigg ) , \\\\ { \\Pr } \\bigg ( g _ { T , R } ^ \\theta ~ \\bigg | ~ { x _ T } = 1 \\bigg ) & \\sim \\mathcal { N } \\bigg ( { \\mu _ { { 1 } _ { T , R } } } , \\sigma _ { { 1 } _ { T , R } } ^ 2 \\bigg ) , \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} 1 - G _ v \\big ( ( \\alpha ^ * _ n x + \\beta ^ * _ n ) ^ { \\frac { 1 } { p } } \\big ) = & n ^ { - 1 } e ^ { - x } \\left \\{ 1 - \\frac { ( 1 - v ^ { - 1 } ) ^ 3 ( \\log \\log n ) ^ { 2 } } { 2 \\log n } \\right . \\\\ & \\left . + \\frac { ( 1 - v ^ { - 1 } ) ^ { 2 } ( 1 - \\log 2 \\Gamma ( { 1 } / { v } ) + x ) \\log \\log n } { \\log n } + o \\big ( \\frac { \\log \\log n } { \\log n } \\big ) \\right \\} . \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\infty } T _ { \\eqref { e _ 3 7 _ 2 } } = \\int _ 0 ^ T \\int _ { \\mathbb { T } ^ 1 } \\varphi ( \\eta _ * + \\xi ) \\beta ( t ) d \\theta d t . \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} Q _ { \\Sigma } ( v , v ) + | \\mu ' | \\sum _ { i = 1 } ^ I ( \\int _ { \\Sigma } v \\cdot \\psi _ i \\ ) ^ 2 \\geq \\lambda '' \\| v \\| _ { L ^ 2 ( \\Sigma ) } ^ 2 \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} C B . ( { h } ^ { l } B ) & \\equiv ( a _ { l + 1 } h ^ { l + 1 } + a _ { l - 1 } h ^ { l - 1 } + \\cdots + a _ { 1 } h ) B \\\\ \\\\ a _ { l + 1 } & = 2 ( l + 2 ) \\\\ k & = 1 , 3 , \\ldots , k - 2 \\ ( k - 1 ) \\\\ a _ { l - k } & = \\frac { 1 } { 2 } \\bigg ( ( 8 - \\mu ) { l \\choose k } 4 ^ { k } + 6 { l \\choose k + 1 } 4 ^ { k + 1 } + { l \\choose k + 2 } 4 ^ { k + 2 } \\bigg ) . \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} P _ R ( \\tilde { r } ; 0 ) & = P _ { \\rm a f f i n e } \\left ( \\tilde { r } ; \\lambda _ R , \\omega _ R , - \\tfrac { \\beta _ R } { a _ { 2 R } } \\right ) , \\\\ T _ R ( \\tilde { r } ; 0 ) & = T _ { \\rm a f f i n e } \\left ( \\tilde { r } ; \\lambda _ R , \\omega _ R , - \\tfrac { \\beta _ R } { a _ { 2 R } } \\right ) , \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} \\delta ( d S ) = d ( \\delta \\mathcal { A } ) = d \\int _ M - 2 H u \\ , d S = - 2 H u \\ , d S . \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ { p , f } u = - \\lambda _ { p , f } | u | ^ { p - 2 } u , \\ \\ { \\rm i n } \\ \\Omega , \\\\ u | _ { \\partial \\Omega } = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} w _ i ( \\theta ) = - 2 b _ \\alpha N _ { \\theta , \\alpha } ^ { \\alpha - 1 } \\sum \\limits _ { j = 1 } ^ d \\sigma ^ { i j } \\mu _ j , f _ i ( { \\bf { x } } ) = x _ i , \\quad ~ i \\in \\{ 1 , \\ldots , d \\} , \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} F ^ { ( \\alpha ) } [ f _ 0 ] = _ { f \\in S } F ^ { ( \\alpha ) } \\left [ f \\right ] . \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} ( D ^ { 2 } F _ { p } ) ( \\omega ) = p | \\omega | ^ { p - 2 } \\left ( I _ { \\R ^ { 2 l } } + ( p - 2 ) \\frac { \\omega } { | \\omega | } \\otimes \\frac { \\omega } { | \\omega | } \\right ) , \\omega \\in ( \\R ^ { 2 } ) ^ { l } \\setminus \\{ 0 \\} \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} \\psi _ { \\mu } ( z ) = \\int _ { \\mathbb { T } } \\frac { t z } { 1 - t z } \\ , d \\mu ( t ) , z \\in \\mathbb { D } . \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} & \\log \\left ( Z _ { n } ( \\beta ) \\right ) + \\frac { 1 } { 2 } \\log \\left ( 1 - 2 \\beta J \\right ) - ( n - 1 ) \\beta ^ 2 + \\beta ( J - J ' ) - \\beta C _ { n , 1 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - \\sum _ { k = 2 } ^ { m _ { n } } \\frac { 2 \\mu _ { k } \\left ( C _ { n , k } - ( n - 1 ) \\mathbb { I } _ { k = 2 } \\right ) - \\mu _ { k } ^ 2 } { 4 k } \\left | \\mathbb { P } _ { n } \\right . \\stackrel { p } { \\to } 0 . \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} \\| ( - P ) ^ { s } u \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } ^ { 2 } = ( ( - P ) ^ { 2 s } u , u ) _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } \\le \\| ( - P ) ^ { 2 s } u \\| _ { H ^ { - 2 s } ( \\mathbb { R } ^ { n } ) } \\| u \\| _ { H ^ { 2 s } ( \\mathbb { R } ^ { n } ) } \\le C \\| u \\| _ { H ^ { 2 s } ( \\mathbb { R } ^ { n } ) } ^ { 2 } . \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} \\mathbb { E } _ \\eta ( \\eta _ t ( x ) ) = \\mathbb { E } _ \\eta ( \\eta _ t ( x ) | \\tau _ x \\leq t ) \\mathbb { P } _ \\eta ( \\tau _ x \\leq t ) + \\mathbb { E } _ \\eta ( \\eta _ t ( x ) | \\tau _ x > t ) \\mathbb { P } _ \\eta ( \\tau _ x > t ) \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} & { \\mathbf 1 } _ { | y | \\le 1 / 2 } \\le \\chi \\le { \\mathbf 1 } _ { | y | < 1 } , \\\\ & \\operatorname { s u p p } ( \\zeta ) \\subset B ( 0 , 1 ) , \\int _ { \\mathbb R ^ d } \\zeta ( y ) { \\rm d } y = 1 , \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} B _ { n , k } \\left ( x _ 1 , x _ 2 , x _ 3 , \\ldots \\right ) & = \\sum _ P x _ { \\abs { P _ 1 } } \\cdots x _ { \\abs { P _ k } } \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} x & { } = 0 , & y & { } = e ^ { - a } \\rho ( b ) ^ { - 1 } \\Bigl ( \\rho ( c ) z - \\frac { 1 } { 2 } \\rho \\Bigl ( \\frac { c } { 2 } \\Bigr ) ^ 2 \\langle S , T \\rangle \\Bigr ) ; \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} \\begin{aligned} \\iint f ( x ) ^ 2 & \\left ( \\tau _ n ( x ) - \\tau _ n ( y ) \\right ) ^ 2 \\frac { \\psi _ 0 ( x ) \\psi _ 0 ( y ) } { | x - y | ^ { d + \\alpha } } d x d y \\\\ & \\leq \\| f \\| _ \\infty ^ 2 \\iint \\left ( \\tau _ n ( x ) - \\tau _ n ( y ) \\right ) ^ 2 \\frac { \\psi _ 0 ( x ) \\psi _ 0 ( y ) } { | x - y | ^ { d + \\alpha } } d x d y \\rightarrow 0 \\end{aligned} \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} \\partial _ x G _ { u , v } g ( x ) = u ' ( x ) \\int _ { x } ^ b v ( y ) g ( y ) \\d y + v ' ( x ) \\int _ a ^ { x } u ( y ) g ( y ) \\d y . \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} P _ { 2 2 } = N ^ \\top P N ; \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} \\langle \\widehat { u } g , \\widehat { u } f _ 0 \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } + \\langle \\widehat { v } g , \\widehat { v } f _ 0 \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = F ^ { ( \\alpha ) } [ f _ 0 ] \\langle g , f _ 0 \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } , \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} \\frac { d } { d t } \\langle g _ n ^ \\lambda | 1 \\rangle = i \\mathcal H _ \\lambda \\frac { \\langle g _ n ^ \\lambda | 1 \\rangle } { \\lambda _ n + \\lambda } \\ . \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} \\check { \\nabla } ^ \\pm _ { \\sigma ' } \\sigma \\oplus \\epsilon : = \\rho ( \\sigma ' ) ( \\sigma ) \\oplus \\epsilon + \\sigma \\oplus X ^ * \\nabla ^ \\pm _ { \\sigma ' } \\epsilon . \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} D ^ k [ f \\circ g ] ( x ) = D ^ k f ( g ( x ) ) ( \\underbrace { D g ( x ) , \\dots , D g ( x ) } _ { k D g ( x ) } ) + f . \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} \\mathcal { V } : = \\{ u \\in \\mathcal { F } _ 1 \\times \\mathcal { F } _ 2 : \\nabla \\cdot u = 0 \\} . \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} \\varphi _ { p , q } ( \\lambda ) - ( \\lambda + 1 ) \\phi _ { p , q } ( \\lambda ) = & \\left ( 5 \\ , p q - 5 \\ , p - 2 \\ , q + 2 \\right ) { \\lambda } ^ { 3 } \\\\ & + \\left ( 1 4 \\ , p q - 1 4 \\ , p - 4 \\ , q + 4 \\right ) { \\lambda } ^ { 2 } \\\\ & + \\left ( 1 8 \\ , p q - 1 8 \\ , p - 8 \\ , q + 8 \\right ) \\lambda + 8 \\ , p q - 8 \\ , p - 4 \\ , q + 4 . \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} ( A _ h ) _ w = ( ( A _ h ) _ { w ' } ) ^ k , w ' : = w / k . \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} \\Delta ^ { I } n _ { 1 } = \\Delta ^ { I } \\Big ( - \\frac { a u } { \\sqrt { g } } \\Big ) = \\lambda _ { 1 1 } \\Big ( - \\frac { a u } { \\sqrt { g } } \\Big ) + \\lambda _ { 1 2 } \\Big ( - \\frac { b v } { \\sqrt { g } } \\Big ) + \\lambda _ { 1 3 } \\Big ( \\frac { 1 } { \\sqrt { g } } \\Big ) , \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} I = I _ 1 - I _ 2 \\ , , \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} \\mho _ { \\frac 1 { \\mathbf { F } } } [ \\varphi ] = \\frac 1 { \\Omega _ \\mathbf { F } \\left [ \\frac 1 \\varphi \\right ] } . \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} ( \\varphi ^ c ) ^ j : = \\sup _ { x \\in X } ( - c ( x , y _ j ) - \\varphi ( x ) ) , ( \\psi ^ { c ^ * } ) ( x ) : = \\max _ { 1 \\leq j \\leq N } ( - c ( x , y _ j ) - \\psi ^ j ) . \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} \\partial _ { \\xi } ^ { 2 } \\left ( \\xi ^ { 2 } \\frac { K _ { 1 } ( \\xi \\lambda ( t ) ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\right ) = \\partial _ { \\xi } ^ { 2 } \\left ( \\frac { \\xi } { \\lambda ( t ) \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\right ) + F _ { v _ { 2 } } ( \\xi , \\lambda ( t ) ) , \\xi \\leq \\frac { 1 } { 4 } \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} C ^ { 2 } . { h } ^ { l } B & \\equiv ( a _ { l } h ^ { l } + a _ { l - 2 } h ^ { l - 2 } + \\cdots + a _ { 0 } ) B \\\\ \\\\ a _ { l } & = - 4 ( l + 1 ) ^ { 2 } \\\\ k & = 2 , 4 , \\ldots , k - 3 \\ ( k - 2 ) \\\\ a _ { l - k } & = - \\frac { 1 } { 2 } \\bigg ( ( 8 - \\mu ) { l \\choose k } 4 ^ { k } + 6 { l \\choose k + 1 } 4 ^ { k + 1 } + { l \\choose k + 2 } 4 ^ { k + 2 } \\bigg ) . \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} \\int _ { M } | f ( x ) | d { \\rm v o l } ( x ) = \\lim _ { r \\to \\infty } \\int _ { B _ { r } ( x _ { 0 } ) } | f ( x ) | d { \\rm v o l } ( x ) , \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} X = Y _ 0 \\supset Y _ 1 = E _ i \\supset \\cdots \\supset Y _ d = \\{ p \\} \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} J ( z ) = \\sqrt { 1 + h _ z ^ 2 } \\ ; h ^ n ( z ) + H h ^ { n + 1 } ( z ) . \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} I = \\langle & E _ { 1 , 2 } ^ { 2 } , E _ { 1 , 3 } ^ { 2 } , E _ { 1 , 4 } ^ { 2 } , E _ { 2 , 3 } ^ { 2 } , E _ { 2 , 4 } ^ { 2 } , E _ { 3 , 4 } ^ { 2 } , E _ { 1 , 2 } E _ { 1 , 3 } , E _ { 1 , 2 } E _ { 1 , 4 } , E _ { 1 , 3 } E _ { 1 , 4 } , E _ { 2 , 3 } E _ { 2 , 4 } , \\\\ & E _ { 1 , 3 } E _ { 2 , 4 } + E _ { 1 , 4 } E _ { 2 , 3 } , E _ { 1 , 3 } E _ { 2 , 3 } , E _ { 1 , 4 } E _ { 2 , 4 } , E _ { 1 , 4 } E _ { 3 , 4 } , E _ { 2 , 4 } E _ { 3 , 4 } \\rangle . \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} \\int _ { \\Omega } F _ k ( D u _ k ) \\ , d x \\rightarrow \\int _ { \\Omega } F ( D u ) \\ , d x = \\int _ { \\Omega } F ( D v ) \\ , d x , \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} \\varphi ( \\hat { x } , \\hat { t } ) - \\varphi ( \\hat { x } , \\hat { t } - h ) & \\le \\xi _ 1 ^ { y , s , \\varepsilon } ( \\hat { x } , \\hat { t } ) - \\xi _ 1 ^ { y , s , \\varepsilon } ( \\hat { x } , \\hat { t } - h ) \\\\ & = - M _ 2 ( | \\hat { t } - s | - | \\hat { t } - h - s | ) \\\\ & \\le M _ 2 h \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} ( - P ) ^ { \\alpha } ( - P ) ^ { \\beta } u = ( - P ) ^ { \\alpha + \\beta } u \\ ; \\ ; u \\in { \\rm d o m } \\ , ( ( - P ) ^ { \\alpha + \\beta } ) \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} \\omega = F _ 0 d x _ 0 + \\dots + F _ n d x _ n \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} R _ { s ; i } ^ 2 & = 0 , \\\\ R _ { s ; i _ 1 } \\cdots R _ { s ; i _ k } & = ( - 1 ) ^ { k ( k - 1 ) / 2 } R _ { s ; i _ 1 , \\dots , i _ k } q _ { s , 0 } ^ { k - 1 } \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} \\hat { T } _ i = \\hat { L } ^ 2 _ i - g _ i ( \\Omega _ i ) , ~ ~ ~ ~ ~ ~ \\hat { T } _ N = \\hat { L } ^ 2 _ N \\\\ \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} \\begin{aligned} & F ( r ) = r ^ { \\kappa } e ^ { - \\sqrt { - E } r } L ^ { ( 2 \\kappa - 1 ) } _ { N _ r } ( 2 \\sqrt { - E } r ) , ~ ~ ~ \\kappa = 2 \\sum _ { s = 1 } ^ { N - 1 } J _ s + N - \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { N } \\sqrt { 1 + 4 \\lambda _ s } , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} \\lim _ { x \\downarrow c _ { 0 } ^ { - 1 } } \\frac { q _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( x ) } { \\left | x - c _ { 0 } ^ { - 1 } \\right | ^ { 1 / 3 } } = \\frac { a _ { 1 } } { \\pi \\sqrt [ 3 ] { \\left | c _ { 3 } \\right | } } \\sin \\frac { \\theta } { 3 } , \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} \\mathrm { G r } ( T ) = \\{ ( T u , u ) | u \\in M \\} \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} ( Q M _ 0 ^ j ) ^ { q } = Q M _ 0 ^ j Q ^ { - 1 } \\cdot Q ^ 2 M _ 0 ^ j Q ^ { - 2 } \\cdot Q ^ 3 M _ 0 ^ j Q ^ { - 3 } \\cdots Q ^ { q - 1 } M _ 0 ^ j Q ^ { - ( q - 1 ) } \\cdot M _ 0 ^ j \\in \\Ref ^ { ( n ) } . \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} \\beta _ { \\Lambda ^ c ; i , j } \\beta _ { \\Lambda , \\Lambda _ { 0 } ; i , j } = \\beta _ { \\Lambda _ { 0 } ^ c ; i , j } . \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\ , f _ i \\ ! \\cdot \\ ! g _ i = 1 . \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} \\mathcal { Q } ( \\Phi , h ) & = \\frac { 1 } { h } \\Phi ( x , F ( h , x ) ) \\\\ & = \\frac { 1 } { h } [ \\Phi ( x , F ( h , x ) ) - \\Phi ( x , x ) ] \\\\ & = \\frac { \\Phi ( x , F ( h , x ) ) - \\Phi ( x , F ( 0 , x ) ) } { h } . \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} ( \\delta g ^ { i l } ) g _ { l k } = - g ^ { i l } ( \\delta g _ { l k } ) = 2 u g ^ { i l } h _ { l k } , \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { r = 1 } ^ { k - 1 } ( - 1 ) ^ { r - i } \\binom { k - i } { r - i } \\binom { m - r } { s - r } = \\sum _ { r = i } ^ { s } ( - 1 ) ^ { r - i } \\binom { k - i } { r - i } \\binom { m - r } { s - r } & = \\sum _ { r = 0 } ^ { s - i } ( - 1 ) ^ r \\binom { k - i } { r } \\binom { m - i - r } { s - i - r } \\\\ & = ( - 1 ) ^ { s - i } \\binom { k - m + s - i - 1 } { s - i } . \\end{aligned} \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} I _ { a _ 2 } = - \\sum _ { i , j \\neq k = 1 } ^ { 2 } \\int _ { \\Omega } \\partial _ { i } u _ i \\partial _ j ^ 2 \\theta \\partial _ k ^ 2 \\theta \\dd x - \\sum _ { i , j \\neq k = 1 } ^ { 2 } \\int _ { \\Omega } u _ i \\partial _ j ^ 2 \\theta \\partial _ { i } \\partial _ k ^ 2 \\theta \\dd x \\ ; . \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} \\det ( T _ { g _ { i j } , k } ) = \\det ( T _ { g _ { 1 1 } , k } ) \\cdot \\ldots \\cdot \\det ( T _ { g _ { l l } , k } ) . \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} \\iff \\sum _ { k , j = 0 } ^ { n } \\left ( \\sum _ { t , s , l , r = 1 } ^ { d } x _ { t } \\overline { a } _ { l , t } ^ { ( j ) } E _ { l , s } ( b _ { j } ^ * b _ { k } ) a _ { s , r } ^ { ( k ) } x ^ { * } _ { r } \\right ) \\geq 0 \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} D _ { k , q } : = 2 H - 2 E _ k - 2 E _ q - \\sum _ { p \\neq i } E _ p \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} a _ i = d _ i + \\binom { m - i } { 1 } d _ { i + 1 } + \\cdots + \\binom { m - i } { k - 1 - i } d _ { k - 1 } + d _ k . \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { \\ell _ R ( I ( n D ) / I ( \\mu _ i ) _ { \\delta n } ) } { n ^ d } = \\mu { \\rm V o l } ( \\Delta ( D ) ) . \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} F _ J ( x , y ; \\mu ) = \\begin{bmatrix} f _ J ( x , y ; \\mu ) \\\\ g _ J ( x , y ; \\mu ) \\end{bmatrix} , \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon \\to 0 } \\int _ { y _ \\varepsilon } ^ { x _ \\varepsilon } \\frac { p _ \\varepsilon } { z ^ { \\alpha + 1 } } d z \\le \\limsup _ { \\varepsilon \\to 0 } \\frac { p _ \\varepsilon } { ( x _ \\varepsilon \\wedge y _ \\varepsilon ) ^ { \\alpha + 1 } } \\int _ { y _ \\varepsilon } ^ { x _ \\varepsilon } d z = \\limsup _ { \\varepsilon \\to 0 } \\frac { ( x _ \\varepsilon - y _ \\varepsilon ) ^ 2 } { \\varepsilon ( x _ \\varepsilon \\wedge y _ \\varepsilon ) ^ { \\alpha + 1 } } = 0 . \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} d \\sigma = \\frac { d k } { 2 \\pi | \\Pi ( k ) | ^ 2 } + d \\sigma _ s , \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} a = \\sum \\limits _ { k \\in \\mathbb Z ^ c } a _ k \\otimes \\varepsilon ^ k , \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\frac { 1 - h ^ { - 1 } ( i t ) } { | 1 - h ^ { - 1 } ( i t ) | } = \\exp \\big ( i \\eta \\tfrac \\pi 2 \\big ) . \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} I _ { \\mathcal { W } ^ p } ( S ^ 2 ( r ) ) ( u , u ) & = \\delta ^ 2 \\int _ { S ^ 2 ( r ) } H ^ p \\ , d S \\geq \\int _ { S ^ 2 ( r ) } \\frac { 2 p ^ 2 - 3 p + 4 } { 2 r ^ 2 } \\ , u ^ 2 \\ , d S \\\\ & \\geq C ( p , r ) \\int _ { S ^ 2 ( r ) } u ^ 2 \\ , d S , \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} i _ Z ^ ! \\circ p _ { 1 3 * } \\circ ( p _ { 1 2 } \\circ p _ { 2 3 } ) ^ ! ( - ) & = q _ { T * } \\left ( e ( Y ) ^ { - 2 } \\cdot _ 2 ( i _ { Z ^ { ( 2 ) } } ) ^ ! _ { i _ { Y ^ 4 } } \\circ ( p _ { 1 2 } \\circ p _ { 2 3 } ) ^ ! ( - ) \\right ) \\\\ & = q _ { T * } \\left ( e ( Y ) ^ { - 2 } \\cdot _ 2 ( p _ T ) ^ ! _ { p _ { 1 2 } \\times p _ { 2 3 } } \\circ i _ { Z ^ 2 } ^ ! ( - ) \\right ) \\\\ & = q _ { T * } \\left ( e ( Y ) ^ { - 1 } \\cdot _ 2 p _ T ^ * \\circ i _ { Z ^ 2 } ^ ! ( - ) \\right ) , \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} \\boldsymbol \\psi _ N ^ { \\beta } ( t _ { n + 1 } ) = \\exp ( i D _ N ^ { \\beta } \\dd { t } ) \\boldsymbol \\psi _ N ^ { \\beta } ( t _ { n } ) . \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} & | \\int _ { 0 } ^ { s - t } \\rho d \\rho \\frac { \\lambda '' ( s ) } { r } \\partial _ { r } \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( - 1 - \\rho ^ { 2 } + r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } \\right ) | \\leq C ( s - t ) ^ { 2 } | \\lambda '' ( s ) | , s - t \\leq \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} & \\| x _ { n + 1 } ^ { \\frac { 1 - 2 s } { 2 } } \\nabla \\tilde { u } \\| _ { L ^ { 2 } ( B _ { r } ^ { + } ) } ^ { 2 } \\\\ \\le & C \\bigg [ r ^ { - 2 } \\| x _ { n + 1 } ^ { \\frac { 1 - 2 s } { 2 } } \\tilde { u } \\| _ { L ^ { 2 } ( B _ { 2 r } ^ { + } ) } ^ { 2 } + \\sum _ { j = 1 } ^ { n } \\| x _ { n + 1 } ^ { \\frac { 1 - 2 s } { 2 } } f _ { j } \\| _ { L ^ { 2 } ( B _ { 2 r } ^ { + } ) } ^ { 2 } + \\| \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { u } \\| _ { L ^ { 2 } ( B _ { 2 r } ' ) } \\| u \\| _ { L ^ { 2 } ( B _ { 2 r } ' ) } \\bigg ] . \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} ( \\partial _ { n + 1 } \\phi ) ^ { 2 } ( \\partial _ { n + 1 } ^ { 2 } \\phi ) = 8 ( x _ { n + 1 } ^ { 1 - 2 s } - x _ { n + 1 } ) ^ { 2 } ( ( 2 s - 1 ) x _ { n + 1 } ^ { - 2 s } + 1 ) \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} \\partial _ { t } \\upsilon - i \\left [ \\Delta \\upsilon + A \\upsilon + \\tilde { V } \\upsilon \\right ] = \\eta _ { r } ^ { \\prime } \\left ( t \\right ) \\theta _ { M } \\left ( x \\right ) \\tilde { u } \\left ( x , t \\right ) - \\left ( 2 \\nabla \\theta _ { M } . \\nabla \\tilde { u } + \\tilde { u } \\Delta \\theta _ { M } \\right ) \\eta _ { r } . \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} \\textnormal { d i a m } ( S _ i ) \\leq 2 ^ { - 8 0 } A ^ { - 1 } \\textnormal { f o r } \\ \\ , i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} \\begin{aligned} n \\norm { f _ { 1 } } { L ^ { \\infty } ( \\Omega _ { 1 } ) } \\leq \\| T ^ { \\Omega _ { 1 } } _ { t ^ { 2 } _ { n } e ^ { i \\phi } } f _ { 1 } \\| _ { L ^ { \\infty } ( \\Omega _ { 1 } ) } = \\| T ^ { \\Omega _ { t _ { n } } } _ { e ^ { i \\phi } } f _ { n } \\| _ { L ^ { \\infty } ( \\Omega _ { t _ { n } } ) } . \\end{aligned} \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{gather*} g _ n = T g _ { s _ { k - 1 } + r } , \\quad \\mbox { w h e n } s _ k + 1 \\le n \\le s _ k + n _ k . \\\\ g _ n = T ^ * g _ { r + 1 - k } \\quad \\mbox { w h e n } s _ k + n _ k + 1 \\le n \\le s _ { k + 1 } - 1 . \\\\ g _ n = e _ k \\quad \\mbox { w h e n } n = s _ { k + 1 } . \\end{gather*}"}
-{"id": "8383.png", "formula": "\\begin{align*} U _ { 2 4 } : = \\{ A _ { 1 } ^ { \\pm 1 } , A _ { 2 } ^ { \\pm 1 } , A _ { 3 } ^ { \\pm 1 } , B _ { 1 } ^ { \\pm 1 } , B _ { 2 } ^ { \\pm 1 } , B _ { 3 } ^ { \\pm 1 } , A _ { 1 } '^ { \\pm 1 } , A _ { 2 } '^ { \\pm 1 } , A _ { 3 } '^ { \\pm 1 } , B _ { 1 } '^ { \\pm 1 } , B _ { 2 } '^ { \\pm 1 } , B _ { 3 } '^ { \\pm 1 } \\} . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\alpha \\in U ^ { r } \\\\ \\alpha \\not = \\beta _ { 0 } } } \\sum _ { \\substack { x < n \\le x + H \\\\ \\gcd ( n , M ) = 1 } } \\chi _ { \\alpha } ( n ) \\ge - 2 ^ { \\omega ( M ) } ( 2 ^ { r } - 1 ) \\left ( \\dfrac { 1 } { 3 \\log 3 } \\sqrt { \\Gamma } \\log \\Gamma + \\dfrac { 1 3 } { 2 } \\sqrt { \\Gamma } \\right ) . \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} M & = \\left ( \\begin{array} { c c c | c c c | c } 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & m _ { 6 , 6 } & m _ { 6 , 7 } \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} \\widehat { H } ^ { ( \\alpha ) } f _ { \\nu } = \\nu f _ { \\nu } \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} u ^ { N } & = N ! \\frac { u ^ { N } } { N ! } . \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} | \\tau - \\alpha | ^ 2 = | \\tau + q ^ { 1 / 3 } / p ^ { 1 / 3 } | | \\Bar { \\tau } + q ^ { 1 / 3 } / p ^ { 1 / 3 } | . \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( F \\left ( ( \\eta ^ N _ n ) _ { n = 1 } ^ { N } \\right ) \\right ) = \\frac { \\mathbf { E } \\left ( \\nu ( \\eta _ 0 ) ^ { - 1 } F \\left ( ( \\eta _ n ) _ { n = 1 } ^ { N } \\right ) \\ : \\vline \\ : \\eta _ N = \\eta _ 0 , \\ : S _ N > 0 \\right ) } { \\mathbf { E } \\left ( \\nu ( \\eta _ 0 ) ^ { - 1 } \\ : \\vline \\ : \\eta _ N = \\eta _ 0 , \\ : S _ N > 0 \\right ) } , \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} A _ i ' : = r ^ { - 1 } ( A _ i ) , 1 \\le i \\le k . \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} & r _ { l } ^ { x , y , z } : = \\min \\big \\{ h _ { l } ( ( x , 0 ) ) ( 2 ) , h _ { l } ( ( y , 0 ) ) ( 2 ) , h _ { l } ( ( z , 0 ) ) ( 2 ) \\big \\} , \\\\ & s _ { l } ^ { x , y , z } : = \\max \\big \\{ h _ { l } ( ( x , 0 ) ) ( 2 ) , h _ { l } ( ( y , 0 ) ) ( 2 ) , h _ { l } ( ( z , 0 ) ) ( 2 ) \\big \\} . \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} A _ { i j } = \\bigcap _ { g \\in D _ { i j } } \\pi _ { i j , g } ^ { - 1 } ( g W ) ( K ) . \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} Q ( D _ p , D _ q ) \\ = \\ Q ( C _ p , C _ q ) \\ p , q \\in [ k ] . \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} Q _ { A , i } = \\begin{cases} R _ { A , i } \\setminus { \\displaystyle \\bigcup _ { ( m , k ) \\in { M } _ { A , i } ' } } ( \\mathcal { R } _ { A , m , i } \\times \\mathcal { T } _ { k } ^ A ) \\ \\textnormal { f o r } \\ A \\geq 2 ^ { 1 0 1 } , \\\\ \\ R _ { 2 ^ { 1 0 0 } , i } \\ , \\textnormal { f o r } \\ A = 2 ^ { 1 0 0 } , \\end{cases} \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} \\tilde { m } _ i = \\begin{cases} m _ i , & i \\neq s , \\\\ m _ s - 1 , & i = s . \\end{cases} \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} p _ { k , 0 } = \\phi _ n \\left ( \\sum _ { i } u _ i ^ k \\right ) , p _ { k , 1 } = \\frac { 1 } { k } \\sum _ i \\left ( ( x _ i + c _ i ) ^ k - x _ i ^ k \\right ) = \\frac { 1 } { k } \\phi _ n \\left ( \\sum _ { i } v _ i ^ k - \\sum _ { i } u _ i ^ k \\right ) , \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} Q ( B _ 1 \\ltimes A _ 1 , B _ 2 \\ltimes A _ 2 ) \\ = \\ Q ( B _ 1 , B _ 2 ) \\cdot | A _ 1 | \\cdot | A _ 2 | . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} F _ { 3 } ( r , \\rho , \\lambda ( s ) ) - F _ { 3 } ( r , \\rho , \\lambda ( t ) ) = \\int _ { 0 } ^ { 1 } \\frac { - 4 r ^ { 2 } z _ { \\sigma } ( 1 + ( r ^ { 2 } - \\rho ^ { 2 } ) z _ { \\sigma } ^ { 2 } ) } { ( 1 + 2 ( \\rho ^ { 2 } + r ^ { 2 } ) z _ { \\sigma } ^ { 2 } + ( \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } z _ { \\sigma } ^ { 4 } ) ^ { 3 / 2 } } \\left ( \\lambda ( s ) ^ { \\alpha - 1 } - \\lambda ( t ) ^ { \\alpha - 1 } \\right ) d \\sigma \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\int _ { p _ 0 } ^ { p ( t ) } \\frac { \\varepsilon ( r ) } { r } \\d r & : = t , \\\\ A ( t ) & : = \\int _ 0 ^ t \\frac { \\gamma ( p ( r ) ) } { \\varepsilon ( p ( r ) ) } \\d r . \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} F ( ( X \\otimes Z _ 2 ) / Y ) = 2 X _ 1 \\oplus X _ 3 \\oplus 2 X _ 4 . \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} & | \\left ( e _ { 1 } ( t ) - e _ { 2 } ( t ) \\right ) \\int _ { 0 } ^ { \\infty } d \\xi \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) \\frac { \\sin ( t \\xi ) } { t ^ { 2 } } F _ { v _ { 2 } } ( \\xi , \\lambda _ { 1 } ( t ) ) | \\leq C | e _ { 1 } ( t ) - e _ { 2 } ( t ) | \\frac { \\lambda _ { 1 } ( t ) | \\log ( \\lambda _ { 1 } ( t ) ) | } { t ^ { 3 } } \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} B _ { k , k } D W ' = \\begin{bmatrix} B _ { k - 1 , k } A _ { k - 1 , m } W _ { k - 1 , m } & B _ { k - 1 , k } \\vec { \\bf 1 } + v \\\\ \\vec { \\bf 0 } ^ \\intercal & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} f _ { t } ( \\hat { \\mathcal { X } } ) = \\big \\{ \\big ( \\varphi ( \\hat { \\pi } ( t ) , t ) , \\psi ( t ) \\big ) : \\hat { \\pi } \\in \\hat { \\mathcal { X } } \\big \\} = \\big \\{ \\varphi ( \\hat { \\pi } ( t ) , t ) : \\hat { \\pi } \\in \\hat { \\mathcal { X } } \\big \\} \\times \\big \\{ \\psi ( t ) \\big \\} . \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} y = \\lambda ( t ) ^ 2 + w ^ 2 - 1 \\end{align*}"}
-{"id": "146.png", "formula": "\\begin{align*} \\| B \\| = \\sup \\{ \\| B ( x , y ) \\| : x \\in X _ 1 , \\ , y \\in Y _ 1 \\} , \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} \\langle X ( m ) , e _ { 1 } \\rangle = \\max \\left \\{ \\langle x , e _ { 1 } \\rangle : \\xi _ { L _ { k } } ^ { m } ( x ) > 0 \\right \\} . \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} I I \\ge c _ 2 ' : = \\inf \\left \\{ \\delta _ 1 ^ { 1 / n } s \\exp ( - c _ 1 / s ) | \\frac { \\delta _ 2 } { 2 \\delta _ 1 } \\le s \\le \\int _ X \\omega _ 0 ^ n \\right \\} , \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} | | P | | _ b ^ 2 : = \\| \\chi P \\| ^ 2 _ 1 + \\| \\phi [ \\mathcal { V } , P ] \\| ^ 2 _ 1 + \\| [ \\mathcal { V } , P ] \\| ^ 2 _ 1 + \\| [ \\phi , P ] \\| ^ 2 + \\| P \\| ^ 2 \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} C : = A + _ G B , \\ , \\ , \\ , \\ , D : = A \\cdot _ G B . \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} \\eta _ { t } ( x ) = \\bar { \\eta } _ { t } ( x ) , ( x , t ) \\in B f ( \\eta ) = f ( \\bar { \\eta } ) . \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} c _ { k + 1 } ( A ) = \\ ; & c _ { k + 1 } ( A + \\lambda v \\otimes \\phi ) = c _ { k + 1 } ( A ) + \\lambda \\sum _ { j = 0 } ^ k { ( - 1 ) ^ j c _ { k - j } ( A ) \\phi A ^ j v } \\\\ = \\ ; & c _ { k + 1 } ( A ) + ( - 1 ) ^ k \\lambda \\phi A ^ k v \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ k \\gamma _ i W _ i = \\sum _ { \\substack { i = 1 \\\\ i \\neq i _ 0 } } ^ s \\gamma _ i W _ i + \\gamma _ { i _ 0 } W _ { i _ 0 } \\longmapsto \\sum _ { \\substack { i = 1 \\\\ i \\neq i _ 0 } } ^ k \\gamma _ i W _ i + \\gamma _ { i _ 0 } \\sum _ { \\substack { j = 1 \\\\ j \\neq h } } ^ s ( - \\alpha _ j \\alpha _ h ^ { - 1 } L a _ j R ) \\\\ & \\textit { ( a n d t h e f u r t h e r c a n c e l l a t i o n s i f t h e r e a r e a n y ) } . \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} | f ^ { ( k ) } ( z ) | & = \\left | \\sum _ { i = 0 } ^ k \\binom { k } { i } f _ 1 ^ { ( i ) } ( z ) f _ 2 ^ { ( k - i ) } ( z ) \\right | \\\\ & \\leq \\sum _ { i = 0 } ^ k \\binom { k } { i } | f _ 1 ^ { ( i ) } ( z ) | | f _ 2 ^ { ( k - i ) } ( z ) | \\\\ & \\leq \\sum _ { i = 0 } ^ k \\binom { k } { i } C ^ { f _ 1 } _ { m - k + i , i } ( 1 + | z | ) ^ { m - k } M ^ { f _ 2 } _ { k - i } = : C ' _ { m , k } ( 1 + | z | ) ^ { m - k } , \\mbox { f o r a l l } ~ m \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} F ( t ; x , y , 1 , u , 1 , v ) & = \\frac { x v t ^ 2 ( 1 - y r ) } { ( 1 - y t u ) ( 1 - t u v ( y - y r + 1 ) ) ( t u x + y ^ { - 1 } - t u ) } \\\\ & \\quad + \\frac { t x } { ( t u x + y ^ { - 1 } - t u ) } F ( t ; x , y - y r + 1 , 1 , u , 1 , v ) \\\\ & \\quad - \\frac { y u ^ 2 v t ^ 2 ( 1 - v ) ( y r - 1 ) } { ( 1 - y t u ) ( 1 - t u v ( y - y r + 1 ) ) } F ( t ; x , y , 1 , u , 1 , v ) , \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} \\mathcal { A } _ t ^ x f ( t , x ) = \\mathcal { L } _ { t , x } f ( t , x ) \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} Q _ \\rho - Q _ { \\tilde { \\rho } } = Q _ \\rho \\left ( P _ { \\tilde { \\rho } } \\circ T - P _ \\rho \\circ T \\right ) Q _ { \\tilde { \\rho } } = Q _ \\rho \\left ( \\tilde { \\rho } \\circ T - \\rho \\circ T \\right ) Q _ { \\tilde { \\rho } } . \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} \\partial _ t \\partial _ k \\omega + u \\cdot \\nabla \\partial _ k \\omega + \\partial _ k u \\cdot \\nabla \\omega - \\partial _ 1 ^ 2 \\partial _ k \\omega = \\partial _ 1 \\partial _ k \\theta . \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde a & = - 2 { \\tilde f _ 1 } ' { \\tilde f _ 1 } ^ { - 1 } - \\tilde f _ 1 \\varphi '' ( \\varphi ' ) ^ { - 1 } { \\tilde f _ 1 } ^ { - 1 } \\\\ & = - 2 ( h ' f _ 1 + h f _ 1 ' ) f _ 1 ^ { - 1 } h ^ { - 1 } - h ( f _ 1 \\varphi '' ( \\varphi ' ) ^ { - 1 } f _ 1 ^ { - 1 } ) h ^ { - 1 } \\\\ & = - 2 h ' h ^ { - 1 } + h ( - 2 f _ 1 ' f _ 1 ^ { - 1 } - f _ 1 \\varphi '' ( \\varphi ' ) ^ { - 1 } f _ 1 ^ { - 1 } ) h ^ { - 1 } . \\end{aligned} \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} \\frac { ~ ^ C d ^ { \\alpha } } { d t ^ { \\alpha } } d _ c f ( t ) = c ^ { \\alpha } \\frac { ~ ^ C d ^ { \\alpha } } { d t ^ { \\alpha } } f ( c t ) , c , t > 0 , \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\frac { \\ln g ( n ) } { n } & = \\inf \\limits _ { n > 0 } \\frac { \\ln g ( n ) } { n } , \\\\ \\lim \\limits _ { n \\to - \\infty } \\frac { \\ln g ( n ) } { n } & = \\sup \\limits _ { n < 0 } \\frac { \\ln g ( n ) } { n } . \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} H _ { L + 1 } - G _ { L + 1 } = c _ 1 > 0 = 2 \\left ( H _ { L } - G _ { L } \\right ) . \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} T _ { \\rm s l i d e } ( 0 ) = \\frac { a _ { 0 R } } { \\gamma } \\ , \\ln \\left ( 1 + \\frac { P _ L ( 0 ) } { \\kappa } \\right ) . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} \\mathcal H ( u ) = \\sum _ { n = 1 } ^ \\infty n ^ 2 \\gamma _ n - \\sum _ { n = 1 } ^ \\infty \\big ( \\sum _ { k = n } ^ \\infty \\gamma _ k \\big ) ^ 2 \\ . \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} [ k ; r _ { k + 1 } , \\dots , r _ s ] : = \\frac { 1 } { k ! } \\left | \\begin{array} { c c c } x _ 1 & \\cdots & x _ s \\\\ \\cdot & \\cdots & \\cdot \\\\ x _ 1 & \\cdots & x _ s \\\\ y _ 1 ^ { p ^ { r _ { k + 1 } } } & \\cdots & y _ s ^ { p ^ { r _ { k + 1 } } } \\\\ \\cdot & \\cdots & \\cdot \\\\ y _ 1 ^ { p ^ { r _ { s } } } & \\cdots & y _ s ^ { p ^ { r _ { s } } } \\end{array} \\right | . \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} \\left ( \\frac d { d t } \\ L _ u + [ L _ u , B _ u ] \\right ) h = - \\Pi \\big ( \\big ( \\partial _ t u + 2 u \\partial _ x u - H \\partial ^ 2 _ { x } u \\big ) h \\big ) \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} \\begin{aligned} f _ 1 ( 0 , 0 ; 0 ) & > 0 , & g _ 1 ( 0 , 0 ; 0 ) & < 0 , \\\\ f _ 2 ( 0 , 0 ; 0 ) & < 0 , & g _ 2 ( 0 , 0 ; 0 ) & < 0 , \\\\ f _ 3 ( 0 , 0 ; 0 ) & < 0 , & g _ 3 ( 0 , 0 ; 0 ) & > 0 , \\\\ f _ 4 ( 0 , 0 ; 0 ) & > 0 , & g _ 4 ( 0 , 0 ; 0 ) & > 0 , \\end{aligned} \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} \\lambda ''' ( t ) = \\partial _ { t } \\left ( \\frac { R H S _ { 2 } ( t ) } { g _ { 2 } ( t ) } \\right ) \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} \\tilde { \\psi } ( z \\otimes w ) = \\psi ( z , w ) ( z , w \\in X ) . \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} \\begin{cases} D ^ { 1 - \\epsilon } _ 0 y ( t ) = w ( t ) , & y ( 0 ) = \\frac { \\gamma _ 1 - b _ 1 s } { a _ 1 } , \\\\ D ^ { \\epsilon } _ 0 w ( t ) = z ( t ) , & w ( 0 ) = 0 , \\\\ D ^ { \\alpha _ 2 - 1 } _ 0 z ( t ) = f ( t , y ( t ) , w ( t ) ) , & z ( 0 ) = s . \\end{cases} \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} L w = f ( t , x ) + R \\Bigl ( t , x , v + w , \\frac { \\partial v } { \\partial x } + \\frac { \\partial w } { \\partial x } \\Bigr ) - R \\Bigl ( t , x , v , \\frac { \\partial v } { \\partial x } \\Bigr ) . \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} \\mathbf { O } _ A ( p , \\varepsilon ) = \\mathbf { O } ( p , \\varepsilon ) \\cap A , . \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} ( - P ) _ { B } ^ { \\alpha } \\phi = ( - P ) _ { B } ^ { \\alpha - n } ( - P ) ^ { n } \\phi . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} \\begin{aligned} \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( M + m - \\frac { \\overline { a } + { { a } _ { i } } } { 2 } \\right ) } & \\le \\sum \\limits _ { i = 1 } ^ { n } { \\frac { { { w } _ { i } } } { \\overline { a } - { { a } _ { i } } } \\int _ { M + m - \\overline { a } } ^ { M + m - { { a } _ { i } } } { f \\left ( t \\right ) d t } } \\\\ & \\le \\frac { f \\left ( M + m - \\overline { a } \\right ) + \\sum \\nolimits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( M + m - { { a } _ { i } } \\right ) } } { 2 } . \\end{aligned} \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} R _ t \\leq \\begin{cases} \\frac { 1 } { 2 } , & t \\leq 0 , \\\\ \\frac { 1 } { 1 + | 1 - \\alpha - \\beta | ^ t } , & t \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} Y = \\bigcup _ { \\ell = 1 } ^ k \\bigcup _ { j = 1 } ^ { d _ \\ell } Y _ j ^ \\ell . \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} \\tilde { F } \\left ( x , t \\right ) = \\left ( \\sqrt { \\alpha \\beta } \\sigma \\left ( t \\right ) \\right ) ^ { \\frac { n } { 2 } + 2 } \\left ( \\sqrt { \\alpha \\beta } x \\sigma \\left ( t \\right ) , \\beta t \\sigma \\left ( t \\right ) \\right ) . \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} \\det \\left ( W ^ { \\top } G \\left ( \\lambda \\right ) W - D + \\lambda I _ { 2 } \\right ) = 0 . \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} \\frac { \\omega _ H ^ { \\wedge \\dim B } } { \\pi ^ * ( \\pi _ * ( \\omega ^ { \\wedge ( \\dim X + 1 ) } ) ) ^ { \\wedge \\dim B } } = \\lambda _ 1 \\cdot \\ldots \\cdot \\lambda _ { \\dim B } . \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} r = A _ c k _ c + g _ c \\circ K { } { } - k _ c \\circ ( A _ c + r ) . \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} K ( r ) : = \\left \\{ \\begin{array} { l c } r ^ { - d - \\alpha } , & \\textrm { i f } r \\le 1 , \\\\ r ^ { - d - \\alpha - \\beta _ 1 - \\beta _ 2 } , & \\textrm { i f } r > 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} i v _ n & = i \\frac 1 2 \\sum _ { k = 1 } ^ { n - 1 } a _ { n - k - 1 } a _ { k - 1 } ( n - k ) ! k ! \\sum _ { P \\in Q ^ { ( n , k ) } } X _ { P } . \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} M _ t ^ g : = g ( Y _ t ) - g ( Y _ 0 ) - \\sum _ { i = 1 } ^ m \\frac { 1 } { 2 } \\int _ 0 ^ t { \\rm H e s s } g ( Y _ s ) ( Z _ s ^ i , Z _ s ^ i ) d s + \\int _ 0 ^ t \\langle \\nabla g ( Y _ s ) , f ( Y _ s , Z _ s ) \\rangle d s , \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} \\mathbb { P } ( D ) & : = \\{ q \\ , : \\ , \\mbox { $ q $ i s p r i m e a n d } ( D / q ) = - 1 \\} , \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} C \\Phi _ { 1 1 } ( q ) = \\sum _ { n = 0 } ^ { \\infty } \\bigg ( p _ { [ 1 ^ 0 1 1 ^ 1 ] } ( n ) + 1 1 \\cdot p ( 1 1 n - 5 ) \\bigg ) q ^ n . \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} \\{ s \\in S : s ^ { - 1 } A \\in q \\} & = \\{ s \\in S : A \\in s \\cdot q \\} \\\\ & = \\{ s \\in S : A \\in \\rho _ q ( s ) \\} \\\\ & = \\{ s \\in S : s \\in \\rho _ q ^ { - 1 } ( A ) \\} \\\\ \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} \\Phi _ { g } : M _ n ( A ) & \\to M _ { n , d } ( E ) \\\\ a & \\mapsto a g . \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} i v _ n & = i \\frac 1 2 \\sum _ { k = 1 } ^ { n - 1 } a _ { n - k - 1 } a _ { k - 1 } ( n - k ) ! k ! \\sum _ { P \\in Q ^ { ( n , k ) } } X _ { P } . \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} H ^ 2 _ { \\lambda , N } ( \\Omega ) = \\left \\{ u \\in H ^ 2 ( \\Omega ) : \\mathcal { Q } _ { \\lambda , N } ( u , \\varphi ) = 0 \\ , , \\ \\forall \\varphi \\in \\mathcal { H } ^ 2 _ { 0 , D } ( \\Omega ) \\right \\} . \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } h ( t ) - h ( t - \\varepsilon ) = 0 , \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\begin{pmatrix} \\lambda & 0 \\\\ 0 & 1 \\end{pmatrix} \\quad \\ , ; \\qquad \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\ , ; \\qquad \\begin{pmatrix} \\lambda & - 1 \\\\ 1 & \\lambda \\end{pmatrix} \\quad . \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } a _ { j k } \\partial _ { j } \\partial _ { k } \\bigg ] w & = f \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ w & = 0 \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} \\sum u _ { i } ' \\sigma _ i = \\phi _ t ^ * \\left ( \\sum u _ { i } \\sigma _ i \\right ) = \\begin{pmatrix} e ^ t & t e ^ t \\\\ & e ^ t \\end{pmatrix} \\ , \\begin{pmatrix} \\sigma _ 1 \\\\ \\sigma _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} E _ { 7 ( - 5 ) } ~ = \\mathcal { N } _ { 1 } ^ { 7 ( - 5 ) - } \\oplus ~ s o ^ { \\ast } ( 1 2 ) \\oplus s o ( 1 , 1 ) \\oplus \\mathcal { N } _ { 1 } ^ { 7 ( - 5 ) } \\newline \\mathbf { , } \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} N \\left ( H _ { k + 1 } - H _ { k } - H _ { k - 1 } - \\dots - H _ 2 - H _ 1 \\right ) + H _ { k + m + 1 } \\leq H _ { L - 1 } + H _ { k + 1 } + \\sum _ { a = 1 } ^ { k } \\sum _ { i = a } ^ { a + m - 1 } H _ { i } . \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} & \\phi _ 1 = ( { \\phi _ 1 } ^ + , { \\phi _ 1 } ^ - ) \\quad \\quad { \\phi _ 1 } ^ \\pm = U ^ \\pm , \\\\ [ 1 m m ] & \\phi _ 2 = ( { \\phi _ 2 } ^ + , { \\phi _ 2 } ^ - ) \\quad \\quad { \\phi _ 2 } ^ \\pm = \\frac { 1 } { 2 } \\Big ( { U ^ \\pm } ' - \\frac { U ^ \\pm } { r } \\Big ) , \\\\ [ 1 m m ] & \\phi _ 0 = ( { \\phi _ 0 } ^ + , { \\phi _ 0 } ^ - ) \\quad \\quad { \\phi _ 0 } ^ \\pm = \\frac { 1 } { 2 } \\Big ( { U ^ \\pm } ' + \\frac { U ^ \\pm } { r } \\Big ) . \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} \\begin{aligned} \\bigwedge _ i a ^ L _ i \\wedge a ^ L & \\le a ^ R \\vee \\bigvee _ j a ^ R _ j & & \\in A , \\\\ b ^ L _ i & \\le f _ i ( a ^ L _ i ) \\vee b ^ R _ i & & \\in B _ i \\ ; \\forall i , \\\\ c ^ L _ j \\wedge g _ j ( a ^ R _ j ) & \\le c ^ R _ j & & \\in C _ j \\ ; \\forall j . \\end{aligned} \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} & i \\partial _ t u ( x , t ) = - \\frac { 1 } { 2 } \\Delta u ( x , t ) + \\frac { 1 } { 2 } x ^ \\top \\ ! \\textbf { A } x \\ , u ( x , t ) , x \\in \\mathbb { R } ^ 2 , t > 0 , \\\\ & u ( x , 0 ) = \\pi ^ { - \\frac { 1 } { 2 } } \\big ( \\frac { 1 } { 2 } x _ 1 ^ 2 + \\frac { 1 } { 2 } ( x _ 2 - 1 ) ^ 2 \\big ) , \\\\ & \\textbf { A } = \\begin{pmatrix} 2 & - 1 \\\\ - 1 & 3 \\end{pmatrix} . \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} \\| D \\| _ { \\mathcal { H } _ p ( X ) } = \\| \\mathfrak { B } _ X ^ { - 1 } ( D ) \\| _ { H _ { p } ( \\mathbb { T } ^ { \\infty } , X ) } \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} P ( x _ 1 , \\cdots , x _ N ) = \\left ( y _ 1 , \\cdots , y _ M \\right ) , y _ l = \\frac { 1 } { K } \\sum _ { i \\in B ( l ) } x _ i . \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} Y _ 3 : = Y _ 2 \\cup S \\cup \\left ( \\bigsqcup _ i \\ , \\overline { \\Delta } _ i \\right ) \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} - \\partial _ { t t } w + \\partial _ { r r } w + \\frac { 1 } { r } \\partial _ { r } w - \\frac { w } { r ^ { 2 } } = 0 \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} \\begin{aligned} a & = - ( f _ 1 '' - f _ 2 '' f _ 2 ^ { - 1 } f _ 1 ) ( f _ 1 ' - f _ 2 ' f _ 2 ^ { - 1 } f _ 1 ) ^ { - 1 } \\\\ b & = - ( f _ 1 '' - f _ 2 '' ( f _ 2 ' ) ^ { - 1 } f _ 1 ' ) ( f _ 1 - f _ 2 ( f _ 2 ' ) ^ { - 1 } f _ 1 ' ) ^ { - 1 } \\end{aligned} \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} L _ I f : = \\lim _ { r \\downarrow 0 } c _ { \\alpha , d } \\int _ { y : | y | > r } \\frac { f ( \\cdot + y ) - f ( \\cdot ) } { | y | ^ { d + \\alpha } } d y \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} \\mathcal { C } _ c ( G : H ) : = \\{ f \\in \\mathcal { C } _ c ( G ) : R _ h f = f \\ \\forall h \\in H \\} . \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\mathcal { P } = \\{ P ^ { \\theta } \\big | \\frac { d P ^ { \\theta } } { d P } = f _ { T } ^ { \\theta } \\ \\ ; \\ \\theta \\in \\Theta \\} \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} T = \\sum _ { m \\in \\mathbb Z ^ c } B _ m S _ m , \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} W = ( w _ i ) : = D [ \\nabla F ( { \\bf 0 } ) ] ^ T \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} \\left \\langle N _ 1 ( v ) , f \\right \\rangle = \\sum _ { j = 1 } ^ { \\infty } \\sqrt { \\lambda _ j ( 0 ) } \\beta _ j \\hat b _ j \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} \\Theta : = ( H _ 1 , H _ 2 , H _ 3 , H _ 4 ) \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} B = M + \\sqrt { \\frac { p \\left ( n + 2 \\right ) } { \\left ( 1 - p \\right ) } } l l ^ { \\top } \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} | \\int _ { 0 } ^ { \\infty } d \\xi \\chi _ { \\leq 1 } ( r \\xi ) \\sin ( t \\xi ) J _ { 1 } ( r \\xi ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } | & \\leq C r \\int _ { 0 } ^ { \\infty } d \\xi \\chi _ { \\leq 1 } ( r \\xi ) \\xi \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\\\ & \\leq \\frac { C } { r \\log ^ { b - 1 } ( r ) } , r \\geq \\frac { t } { 2 } > 4 \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} F ( r ) = \\frac 1 { e ^ { r } - 1 } - \\frac 1 { r } + \\frac 1 2 - \\frac 1 { 1 2 } r \\left ( = \\sum _ { k = 2 } ^ { \\infty } \\frac { B _ { 2 k } } { ( 2 k ) ! } r ^ { 2 k - 1 } \\right ) . \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k ( - 1 ) ^ j \\binom { n } { j } = ( - 1 ) ^ k \\binom { n - 1 } { k } \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} \\mathcal I ( \\bar x ) : = I ^ { 0 + } ( \\bar x ) \\cup I ^ { + 0 } ( \\bar x ) \\cup I ^ { 0 0 } ( \\bar x ) \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} \\forall \\tau \\in [ 1 , \\infty ) \\colon \\rho _ a ( \\tau ) : = \\frac { \\abs { \\{ s \\in \\mathcal S \\ , | \\ , r _ { s , a } \\leq \\tau \\} } } { \\abs { \\mathcal S } } \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} G _ { 0 } ( X ( \\gamma ) , Y ( \\gamma ) ) = \\int _ 0 ^ 1 g ( X ( \\gamma ; t ) , Y ( \\gamma ; t ) ) d t . \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} \\begin{pmatrix} \\omega _ 1 & a _ 1 \\\\ \\omega _ 2 & a _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} x v _ j = \\sum _ { i = 1 } ^ n v _ i \\varphi _ { i j } ( x ) . \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} \\sin ^ 2 \\phi & = \\frac { 4 n ^ 2 ( 1 - m ^ 2 ) ^ 2 } { ( n + m ) ^ 2 ( 1 - m n ) ^ 2 } \\cdot \\frac { m ( 1 - n ^ 2 ) } { n ( 1 - m ^ 2 ) } \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} \\widetilde { \\phi } ( x ) = \\phi ( x ) + \\phi ( x - 1 ) . \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} ( x ^ 2 - ( q / p ) ^ { 1 / 3 } x + { q / p } ^ { 2 / 3 } + y ^ 2 ) ^ 2 = \\big [ | \\tau | ^ 2 - ( q / p ) ^ { 1 / 3 } \\big ( \\frac { \\tau + \\Bar { \\tau } } { 2 } \\big ) + ( q / p ) ^ { 2 / 3 } \\big ] ^ 2 , \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} \\tilde { p } ( z ) = p ( z ) Q \\begin{pmatrix} V \\left ( z \\right ) & 0 \\\\ 0 & I _ m \\end{pmatrix} , \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} d _ W \\left ( \\frac { S _ T ( u ) - T ^ d \\Phi ( u ) } { \\sqrt { T ^ d } } , \\mathcal N ( 0 , \\sigma ^ 2 ( u ) ) \\right ) = O \\left ( 1 / ( \\log T ) ^ { 1 / 4 } \\right ) , \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} e _ R ( \\mathcal I ( i ) ^ { [ 1 ] } , \\mathcal I ( j ) ^ { [ 1 ] } ; R ) = - ( \\Delta _ i \\cdot \\Delta _ j ) \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} \\dim ( C \\cap A ) = \\dim ( C ) - \\dim ( A ^ \\perp ) + \\dim ( C ^ \\perp \\cap A ^ \\perp ) , \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} v _ { 4 } ( t , r ) & = \\frac { - 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } G ( s , r , \\rho ) \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} \\xi = x _ 0 - \\int _ { t _ 0 } ^ T \\frac { G ( \\tau , \\xi ) } { \\tau } d \\tau . \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} | \\mathfrak { A } ( \\xi _ 1 ' , \\eta _ 1 ' , \\xi _ 2 ' , \\eta _ 2 ' ) | & : = | \\xi _ 1 ' \\eta _ 2 ' - \\xi _ 2 ' \\eta _ 1 ' | \\geq A ^ { - 1 } K ^ { - \\frac { 1 } { 2 } } , \\\\ | F ( \\xi _ 1 ' , \\eta _ 1 ' , \\xi _ 2 ' , \\eta _ 2 ' ) | & = | \\xi _ 1 ' \\eta _ 2 ' + \\xi _ 2 ' \\eta _ 1 ' + 2 ( \\xi _ 1 ' \\eta _ 1 ' + \\xi _ 2 ' \\eta _ 2 ' ) | \\geq 2 ^ { - 1 } K ^ { - \\frac { 3 } { 2 } } . \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} V ( z ) & = ( { L } ( x ^ * , \\lambda ^ * ) - { L } ( x ^ * , \\lambda ) ) + ( { L } ( x , \\lambda ^ * ) - { L } ( x ^ * , \\lambda ^ * ) ) \\\\ & ~ ~ ~ + \\frac { 1 } { 2 } \\norm { z - z ^ * } ^ 2 . \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} E ( v _ { 4 } ( t ) , \\partial _ { t } v _ { 4 } ) & \\leq C \\left ( \\int _ { t } ^ { \\infty } | | v _ { 4 , c } ( s ) | | _ { L ^ { 2 } ( r d r ) } d s \\right ) ^ { 2 } \\leq \\frac { C } { t ^ { 2 } \\log ^ { 4 N + 4 b } ( t ) } \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { \\alpha } : = 2 \\langle f , g \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } + \\langle \\widehat { u } f , \\widehat { u } g \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } + \\langle \\widehat { v } f , \\widehat { v } g \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} T = \\begin{pmatrix} 1 & 1 & 1 & 0 & 0 \\\\ 1 & 1 & 1 & 1 & 1 \\\\ 0 & 1 & 1 & 1 & 1 \\\\ 0 & 1 & 1 & 1 & 1 \\\\ 0 & 1 & 1 & 1 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} L ( t ) \\cap \\partial \\left ( ( - \\infty , \\lambda ] \\times ( - \\infty , \\mu ] \\right ) = \\left \\{ \\begin{array} { c } ( \\lambda , \\lambda - t ) ~ \\mbox { i f } ~ t > \\theta , \\\\ ( \\mu + t , \\mu ) ~ \\mbox { i f } ~ t \\leq \\theta , \\end{array} \\right . \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} P _ { 1 , 1 } ( \\eta ) P _ { i , 2 } ( \\eta ) - P _ { 1 , 2 } ( \\eta ) P _ { i , 1 } ( \\eta ) = 0 . \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} \\{ Q _ { i } ( X , Y _ { 1 } , Y _ { 2 } ) = P _ { i , 1 } ( X ) Y _ { 1 } + P _ { i , 2 } ( X ) Y _ { 2 } \\} _ { i = 1 } ^ { q } \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} x \\circ y = ( x _ i y _ i ) \\in \\R ^ m . \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} a * b = R ( a ) b + a R ( b ) - R ( a ) R ( b ) , ~ a , b \\in A . \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} \\pi _ { i , N } = \\frac { c _ { N } w _ { N } ( X _ { i } ) } { \\sum _ { j = 1 } ^ { N } w ( X _ { j } ) } , i = 1 , 2 , \\dots , N . \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} f ! _ { \\lambda } g = f . ( 1 - \\lambda ) \\Box g . \\lambda . \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} U _ t U & = \\frac { 3 ( c ^ 2 - a ^ 2 ) + 6 ( a b + c d ) \\varepsilon _ i } { 8 \\varepsilon _ i ^ 3 } t ^ 2 + \\frac { a b - c d } { 2 \\varepsilon _ i } t - \\frac { 3 ( c ^ 2 - a ^ 2 ) + 2 ( a b + c d ) \\varepsilon _ i } { 8 \\varepsilon _ i } , \\\\ U _ t ^ 2 + U _ { t t } U & = \\frac { 3 ( c ^ 2 - a ^ 2 ) + 6 ( a b + c d ) \\varepsilon _ i } { 4 \\varepsilon _ i ^ 3 } t + \\frac { a b - c d } { 2 \\varepsilon _ i } , \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} u ( x , t ) = \\sum _ { i = 1 } ^ { n } p _ i ( t ) e ^ { - | x - q _ i ( t ) | } . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ { t t } u _ { 1 } + \\Delta u _ { 1 } = 0 \\\\ u _ { 1 } ( s ) = 0 \\\\ \\partial _ { t } u _ { 1 } ( s ) = - \\lambda '' ( s ) \\log ( 1 + | x | ^ { 2 } ) \\end{cases} \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} f ' ( 0 ) = 0 < \\inf \\sigma ( - \\Delta + V ( x ) ) < a , \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} \\frac { d } { d t } \\| \\partial _ k \\omega ( t ) \\| _ { L ^ p } ^ 2 \\leq C \\| \\nabla \\theta \\| _ { L ^ p } ^ 2 + \\| \\nabla u \\| _ { L ^ \\infty } \\| \\nabla \\omega \\| _ { L ^ p } ^ 2 . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} \\aligned \\Delta _ { p } ( A , B ) & = \\min \\{ \\Delta _ { p } ( A ) , \\Delta _ { p } ( B ) \\} , \\\\ \\lambda ( A , B ) & = \\min \\{ \\lambda ( A ) , \\lambda ( B ) \\} , \\\\ \\Lambda ( A , B ) & = \\max \\{ \\Lambda ( A ) , \\Lambda ( B ) \\} . \\endaligned \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} \\int _ { 0 } ^ x \\frac { e ^ { i ( u ^ 2 / ( 4 t ) - k u ) } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) d u = e ^ { - i t k ^ 2 } \\int _ { 0 } ^ x \\frac { e ^ { i ( u / ( 2 \\sqrt t ) - k \\sqrt t ) ^ 2 } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) d u \\ , . \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} t _ 3 = \\begin{cases} \\min \\{ \\kappa , t , r _ 2 , n - r _ 1 + k _ 4 , n - k _ 4 + k _ 5 , n - r _ 2 + k _ 6 \\} & \\mbox { i f $ \\tilde h _ j ( x ) $ i s a u n i t } \\\\ & \\mbox { f o r s o m e $ j $ } , \\\\ \\min \\{ t , r _ 2 , n - r _ 1 + k _ 4 , n - k _ 4 + k _ 5 , n - r _ 2 + k _ 6 \\} & \\mbox { i f $ \\tilde h _ j ( x ) = 0 $ } \\\\ & \\mbox { f o r a l l $ j $ } , \\end{cases} \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ \\infty & \\left [ { a _ 1 \\atop b _ 1 + \\alpha n } \\right ] _ { p _ 1 } \\left [ { a _ 2 \\atop b _ 2 + \\alpha n } \\right ] _ { p _ 2 } \\frac { 1 } { ( - z q ^ n , - q ^ { 1 - n } / z ; q ) _ \\infty } \\\\ & = \\int _ { - \\infty } ^ \\infty \\left [ { a _ 1 \\atop b _ 1 + \\alpha x } \\right ] _ { p _ 1 } \\left [ { a _ 2 \\atop b _ 2 + \\alpha x } \\right ] _ { p _ 2 } \\frac { d x } { \\left ( - z q ^ x , - q ^ { 1 - x } / z ; q \\right ) _ \\infty } . \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} f ^ { 2 } ( r ) = \\int _ { 0 } ^ { r } \\left ( 2 f ( s ) f ' ( s ) \\right ) d s = 2 \\int _ { 0 } ^ { r } \\frac { f ( s ) } { \\sqrt { s } } ( f ' ( s ) \\sqrt { s } ) d s \\leq 2 | | \\frac { f } { r } | | _ { L ^ { 2 } ( r d r ) } | | f ' | | _ { L ^ { 2 } ( r d r ) } \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} d \\ge M : = \\frac { q - p } { p q } r _ * ^ p . \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} v ' = F ( v ) , \\ \\ v ( 0 ) = ( \\max _ { x } u _ i ( 0 , x ) ) \\in [ \\mathbf { 0 } , \\mathbf { \\hat u } ] , \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} \\frac 1 N \\sum _ { j = 1 } ^ N p ( \\mathbf { x } _ j ) = \\frac { 1 } { 4 \\pi } \\int _ { \\mathbb { S } ^ 2 } p ( \\mathbf { x } ) \\omega ( \\mathbf { x } ) \\quad \\forall p \\in \\mathbb { P } _ t . \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} d ^ 2 = 1 + \\frac { 2 } { \\lambda ^ * } \\ , { \\boldsymbol { \\pi } } D D _ Q ^ { \\sharp } D \\mathbf { 1 } . \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} \\hat { E } : b \\in \\mathcal { B } \\to \\hat { E } ( b ) : = ( E _ { i , j } ( b ) ) _ { i , j \\in I } \\in M _ { d _ { I } } ( \\mathbb { C } ) \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( F \\left ( ( \\eta ^ N _ n ) _ { n = 1 } ^ { N } \\right ) \\right ) = \\mathbf { E } \\left ( F \\left ( ( \\eta _ n ) _ { n = 1 } ^ { N } \\right ) \\ : \\vline \\ : \\mathcal { A } _ { N , K } , \\ : S _ N > 0 \\right ) , \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} \\tilde { n } _ i = \\begin{cases} n _ i , & i \\neq s , \\\\ n _ s - 1 , & i = s . \\end{cases} \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} l ( a b ) & = l \\left ( \\Phi ( u ^ \\ast ) \\Phi ( u ) \\right ) = l \\left ( \\Phi ( u ^ \\ast u ) \\right ) = l \\left ( \\Phi ( p ) \\right ) = e . \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} \\Delta ^ { I } n _ { 1 } = \\Delta ^ { I } \\Big ( - \\frac { a u } { \\sqrt { \\Phi } } \\Big ) = \\lambda _ { 1 1 } \\Big ( - \\frac { a u } { \\sqrt { \\Phi } } \\Big ) + \\lambda _ { 1 2 } \\Big ( - \\frac { b v } { \\sqrt { \\Phi } } \\Big ) + \\lambda _ { 1 3 } \\Big ( \\frac { \\sqrt { \\omega } } { \\sqrt { \\Phi } } \\Big ) , \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} \\psi _ { n _ k } ( x , y ) = \\langle M _ { \\hat { x } } ( h _ { n _ k } ) , M _ { \\hat { x } } ( h _ { n _ k } ) \\rangle _ { L ^ 2 ( \\mu ) } \\rightarrow 0 , \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} \\Omega _ { \\rho _ { 1 } } = \\{ x + i y : x \\in \\mathbb { R } , y > f ( x ) \\} , \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} b ^ { i } _ { j } = c _ { i + 2 } + \\dots + c _ { j } - 1 \\leq i \\leq n - 3 , i + 2 \\leq j \\leq n - 1 . \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} Q ( C _ i , C _ j ) \\ & = \\ 3 \\cdot Q ( A _ { i } , A _ j ) \\ \\ i , j \\in J \\ \\ i , j \\in J ' \\\\ Q ( C _ i , C _ j ) \\ & = \\ Q ( A _ { i } , A _ j ) \\ \\ i \\in J \\ \\ j \\in J ' . \\\\ Q ( C _ i , C _ { n + 1 } ) \\ & = \\ N ^ 2 \\ \\ i \\in J ' , \\\\ Q ( C _ { n + 1 } , C _ i ) \\ & = \\ N ^ 2 \\ \\ i \\in J . \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} \\Sigma _ { i j } = \\bigcap _ { k \\geq i } p _ { i j k } ( A _ { k j } ) \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} Z ^ { \\epsilon } _ T = | T | ^ { 1 / 2 } \\left ( C ^ { / T } _ 2 ( f , u ) - C ^ { * } _ 2 ( g , u - \\epsilon X ) \\right ) \\overset { d } { \\underset { T \\nearrow \\R ^ 2 } { \\longrightarrow } } \\Theta _ \\epsilon ( u ) , \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} \\mathcal { H } ( M ) = \\int _ M H \\ , d S . \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} 2 C ^ d { } _ { [ a | g | } \\widetilde { C } ^ { g l } { } _ { b ] } + 2 \\widetilde { C } ^ { d g } { } _ { [ b } C ^ f { } _ { a ] g } - C ^ g { } _ { a b } \\widetilde { C } ^ { d l } { } _ { g } = 0 . \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} w = \\cdots a z b \\cdots c \\cdots \\rightsquigarrow w ^ * = \\cdots z a b \\cdots c \\cdots . \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} \\Phi ^ { \\overline { e } } _ { w v } \\Phi ^ e _ { v w } \\epsilon & = \\Phi ^ { \\overline { e } } _ { w v } w ( \\epsilon ) ^ { \\frac { 1 } { 2 } } \\ , \\overline { \\epsilon } \\\\ & = w ( \\overline { e } ) ^ { \\frac { 1 } { 2 } } w ( \\epsilon ) ^ { \\frac { 1 } { 2 } } \\epsilon \\\\ & = \\epsilon . \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} S _ { \\tau F } [ ( g , \\tau ) ^ { - 1 } \\cdot ( h , \\chi ) , A ^ g ] = S _ { \\tau F } [ ( h , \\chi ) , A ] - S _ { \\tau F } [ ( g , \\tau ) , A ] . \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} K ( \\Psi ; M ; \\gamma ) = K ( \\Psi ; M \\setminus j ; \\gamma ) - K ( \\Psi ; M \\setminus j ; \\gamma - \\epsilon _ j ) \\ , , \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} { A ' } ^ 0 _ n & = - i x _ { n } b ' _ { n - 1 } ( 1 , 2 , \\ldots , n - 1 ) \\big | _ { x _ n = 0 } + . \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} v _ { 4 } ( t , r ) = \\int _ { t } ^ { \\infty } d x \\int _ { 0 } ^ { \\infty } d \\xi J _ { 1 } ( r \\xi ) \\sin ( ( t - x ) \\xi ) \\widehat { v _ { 4 , c } } ( x , \\xi ) \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} ( u _ 1 ' ) ^ 2 + ( u _ 2 ' ) ^ 2 = e ^ { 2 \\lambda \\ , t } \\ , \\left ( u _ 1 ^ 2 + u _ 2 ^ 2 \\right ) \\ , . \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} ( a ; q ) _ \\infty = \\prod _ { n = 0 } ^ \\infty ( 1 - a q ^ n ) . \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} \\beta = \\frac { 1 } { 2 } \\int _ { [ 0 , + \\infty ] } \\left ( \\frac { 1 } { t } + t \\right ) \\ , d \\sigma ( 1 / t ) \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 \\overline { s } } \\partial _ { n + 1 } \\overline { v } \\bigg | _ { C _ { \\overline { s } , 1 / 2 } ' } = \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 \\overline { s } } \\partial _ { n + 1 } \\tilde { v } \\bigg | _ { C _ { \\overline { s } , 1 / 2 } ' } , \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} \\left | g ( x ) \\right | \\chi _ Q ( x ) & \\leq \\sum _ { j \\in \\mathbb { N } } \\left | g _ j ( x ) \\right | \\chi _ { Q _ j } ( x ) + ( c _ \\mu 2 ^ { n _ \\mu } + 1 ) L \\chi _ Q ( x ) . \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} p ^ * ( x ) ^ { \\alpha - 1 } = q ( x ) ^ { \\alpha - 1 } - \\dfrac { \\alpha - 1 } { \\alpha } \\sum \\limits _ { i = 1 } ^ k \\lambda _ i \\big [ f _ i ( x ) - a _ i \\big ] - \\dfrac { \\alpha - 1 } { \\alpha } \\nu . \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} g = \\left [ \\begin{array} { c c } \\frac { 2 } { 1 + e ^ { - 2 s _ 2 } } & 0 \\\\ 0 & \\frac { 2 } { 1 - e ^ { - 2 s _ 2 } } \\end{array} \\right ] \\ ; . \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} 1 = \\mu _ { - } + \\mu _ 0 + \\mu _ { + } , \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} v _ 0 = v , d ^ { s t } v _ i = - d ^ { \\chi } v _ { i + 1 } . \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } v _ { 4 } ( t , r ) | \\leq \\begin{cases} \\frac { C } { t ^ { 4 } \\log ^ { 3 b - 2 + N } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { t ^ { 3 5 / 1 2 } \\log ^ { 2 b - 1 } ( t ) } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} u ( x , t ) = \\exp \\left [ - ( b t + a ) x \\right ] \\cos x , \\ , \\ , x , t \\in \\mathbb { R } ^ + \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} & | - 4 \\int _ { t } ^ { \\infty } \\frac { e ''' ( s ) d s } { ( 1 + s - t ) } \\left ( \\frac { 1 } { \\log ( \\lambda _ { 0 , 0 } ( s ) ) } - \\frac { 1 } { \\log ( \\lambda _ { 0 , 0 } ( t ) ) } \\right ) | \\leq \\frac { C } { t ( \\log ( \\log ( t ) ) ) ^ { 2 } \\log ( t ) } \\int _ { t } ^ { \\infty } | e ''' ( s ) | d s \\\\ & \\leq \\frac { C \\sup _ { x \\geq t } \\left ( | e ''' ( x ) | x ^ { 3 / 2 } \\right ) } { t ^ { 3 / 2 } ( \\log ( \\log ( t ) ) ) ^ { 2 } \\log ( t ) } \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} \\frac { 1 } { \\beta ^ { \\Lambda } \\sum _ { i \\in I _ { 0 } , \\beta _ { i i } = \\beta } 1 } \\sum _ { i \\in I _ { 0 } , \\beta _ { i i } = \\beta } \\prod _ { x \\in \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , i } ^ * b _ { x } \\right ) \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ K & \\int _ { K _ 2 } | x - y | ^ { - ( \\alpha + d - 2 ) } | x | ^ { \\alpha - d } | y | ^ { \\alpha - d } d y d x \\\\ & \\leq \\left ( \\int _ K \\int _ { K _ 2 } \\frac { | x | ^ { 2 ( \\alpha - d ) } } { | x - y | ^ { \\alpha + d - 2 } } d y d x \\right ) ^ { 1 / 2 } \\cdot \\left ( \\int _ K \\int _ { K _ 2 } \\frac { | y | ^ { 2 ( \\alpha - d ) } } { | x - y | ^ { \\alpha + d - 2 } } d y d x \\right ) ^ { 1 / 2 } \\\\ & \\leq \\int _ K | x | ^ { 2 ( \\alpha - d ) } d x \\int _ { | y | \\leq \\varepsilon } | y | ^ { - ( \\alpha + d - 2 ) } d y < \\infty . \\end{aligned} \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} \\Phi ( z ) = \\gamma z \\exp \\left [ \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t z } { z - t } \\ , d \\sigma ( t ) \\right ] , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} \\Phi _ { T + 1 } \\leq \\Phi _ 1 - \\rho \\sum _ { t = 1 } ^ T \\Phi _ t + 2 \\rho ^ 2 \\sigma ^ 2 T + \\frac { L ^ 2 \\gamma ^ 2 } { \\rho } \\sum _ { t = 1 } ^ T \\Delta _ t \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } a _ { j k } \\partial _ { j } \\partial _ { k } \\bigg ] \\overline { w } & = f \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ \\overline { w } & = 0 \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} H = ( H _ { \\rho } \\oplus V _ { \\rho } ) \\oplus ( H _ { \\tau } \\oplus V _ { \\tau } ) \\textfractionsolidus \\{ ( H _ { \\rho } \\oplus V _ { \\rho } ) \\cap ( H _ { \\tau } \\oplus V _ { \\tau } ) \\} \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} \\sup _ { z \\in S ( \\beta , \\ , \\rho ) } \\| \\Delta ( z ) \\| & = \\sup _ { z \\in S ( \\beta , \\ , \\rho ) } \\ , \\max \\{ \\ , | \\Delta ^ x ( z ) | , | \\Delta ^ y ( z ) | \\} \\leq \\ , \\max \\ , \\{ \\tfrac { d } { 2 } \\ , \\rho ^ { n } , \\ , \\tfrac { d } { 2 } \\ , \\rho ^ { m } \\} \\ , < \\frac { d } { 2 } . \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} \\kappa = \\frac { b _ 0 \\omega } { \\lambda ^ 2 + \\omega ^ 2 } . \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} \\phi ( x ) = \\frac { \\psi _ { \\gamma } ^ 2 ( x ) } { \\int \\psi _ { \\gamma } ^ 2 ( x ) d x } \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} \\langle E _ i , v ^ T \\rangle = \\langle E _ i , v \\rangle - \\langle E _ i , \\Pi ( v ) \\rangle = 0 \\ , . \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} V ( x ) = ( v _ i ( x ) ) : = p \\Phi ^ { ( 1 ) } ( x ) - \\Phi ^ { ( 2 ) } ( x ) . \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} H _ { i } ( F \\otimes _ { R } B ) = 0 \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} { } ^ Q \\Omega ^ - _ a = 0 , \\quad { } ^ Q \\Omega ^ + _ a = ( \\rho ^ T E \\rho ) ^ { - 1 } \\rho ^ \\lambda _ a \\partial _ \\lambda ( \\rho ^ T E \\rho ) . \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} \\eta _ { x , \\varphi } ( t ) = \\begin{cases} \\sup _ { { a } \\in \\mathbf { O } _ A ( { x } , t ) } \\varphi ( { a } ) & \\\\ \\inf \\varphi & \\end{cases} \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} \\limsup _ { M \\to \\infty } \\limsup _ { n \\to \\infty } P \\left ( \\| \\hat { K } _ { \\lambda _ n } - K _ Y \\| _ { \\ell _ 2 } > M \\sqrt { s _ n } \\lambda _ n \\right ) \\leq \\limsup _ { L \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\Omega _ { n , L } ^ c ) = 0 \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} P _ { 2 k } ^ - \\lbrace M ( t ) \\le \\beta \\rbrace \\ = \\ P _ { 2 k } ^ + \\lbrace M ( t ) \\le \\beta \\rbrace + \\binom { 2 k } { k } \\frac { ( c ^ 2 t ^ 2 - \\beta ^ 2 ) ^ k } { ( 2 c t ) ^ { 2 k } } . \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} \\sum _ { s = \\tau } ^ { t - 1 } \\beta ( s ) \\theta ^ { t - s } \\leq M \\beta ( t ) , \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} ( P _ n f ) ( z ) = \\langle f , K _ { n , z } \\rangle , ( P _ { ( n ) } f ) ( z ) = \\langle f , K _ { ( n ) , z } \\rangle . \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} \\varphi _ t ( x _ \\varepsilon , t _ \\varepsilon ) \\le & J [ u _ \\varepsilon ^ * , p _ \\varepsilon ] ( x _ \\varepsilon , t _ \\varepsilon ) + K _ { ( 0 , x _ \\varepsilon - \\delta ) } [ \\varphi , p _ \\varepsilon ] ( x _ \\varepsilon , t _ \\varepsilon ) \\\\ & + K _ { ( x _ \\varepsilon - \\delta , x _ \\varepsilon ) } [ u _ \\varepsilon ^ * , p _ \\varepsilon ] ( x _ \\varepsilon , t _ \\varepsilon ) + f ( x _ \\varepsilon , t _ \\varepsilon ) , \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} \\pi _ n \\subset \\pi _ { n - 1 } \\subset \\ldots \\subset \\pi _ 1 \\subset \\pi _ 0 = \\pi \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} c _ { b } = \\begin{cases} \\frac { 4 b } { \\pi ( b - 1 ) } , b \\neq 1 \\\\ \\frac { - 4 } { \\pi } , b = 1 \\end{cases} \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} \\Omega : = \\C ~ \\Big \\backslash \\Big ( \\Gamma \\ , \\cup \\ , \\bigcup \\limits _ { n = 2 } ^ { + \\infty } ( \\Gamma _ { n } \\cup \\Lambda _ { n } ) \\Big ) \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} G _ { u , v } - G _ \\rightarrow & = | u \\rangle \\langle v | , \\\\ G _ { u , v } - G _ \\leftarrow & = | v \\rangle \\langle u | , \\\\ G _ \\rightarrow - G _ \\leftarrow & = | v \\rangle \\langle u | - | u \\rangle \\langle v | = G _ \\leftrightarrow , \\\\ G _ { u , v } - G _ { u _ 1 , v _ 1 } & = | u \\rangle \\langle v | - | u _ 1 \\rangle \\langle v _ 1 | , \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} f & : I \\to I \\\\ f ( x ) & = p ( R ( p ^ { - 1 } ( x ) ) ) . \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} B ( t , g , h ) : = Q ^ { + } ( t , g ( t ) ) - Q ^ { - } ( t , g ( t ) ) + a \\left ( \\left \\| \\Lambda g ( t ) \\right \\| + \\int _ 0 ^ t \\Delta ( s , h ( s ) ) d s \\right ) \\Lambda g ( t ) , \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} \\nabla _ x f ( x ^ k ) + \\left [ \\displaystyle \\sum _ { i = 1 } ^ p [ y ^ I ] ^ k _ i \\nabla ^ 2 c _ i ( x ^ k ) - A A ^ T - { \\cal J } c ( x ^ k ) ^ T { \\cal J } _ { z ^ I } \\Psi _ { \\varepsilon } ( [ y ^ I ] ^ k , [ z ^ I ] ^ k ) { \\cal J } c ( x ^ k ) \\right ] \\lambda ^ k = 0 . \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} \\langle \\mathrm { g r a d } L , v \\rangle _ r = D _ z L ( v ) , \\forall v \\in T _ z \\mathcal { M } . \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} h ( r ) = \\Psi ( r e ^ { i f ( r ) } ) , r \\in ( 0 , + \\infty ) , \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { 2 \\left ( \\alpha ( \\vec { \\bf b } ) + \\frac 1 m \\right ) \\binom { b _ { k - 1 } + b _ k } { b _ k } } \\right ) ^ { \\frac 1 k } \\le f ( m , H ( \\vec { \\bf b } ) ) \\le \\frac { k ( k - 1 ) } { \\alpha ( \\vec { \\bf b } ) } + k - 1 . \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} \\{ e _ { k } = \\left | \\zeta _ { n } ^ { k } - 1 \\right | : 1 \\leq k < n \\} . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} \\varphi ( t ) = \\int _ 0 ^ t \\frac { \\mu ( \\tau ) } { \\tau } d \\tau , 0 < t \\leq T _ 0 , \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} A ^ 1 _ 4 & = - i b _ 3 \\left ( x _ 1 + x _ 2 + x _ 3 + x _ 4 \\right ) . \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{gather*} f _ i ( x , u , 0 ) = f _ i ( x , 0 , v ) = 0 , \\ , \\ , \\ , i = 1 , 2 , \\end{gather*}"}
-{"id": "1409.png", "formula": "\\begin{align*} \\int _ { \\R ^ d _ + } G _ { \\lambda } f ( x ) g ( x ) \\ , d x = \\int _ { \\R ^ d _ + } f ( x ) G _ { \\lambda } g ( x ) \\ , d x \\ , . \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} | v _ { 4 } ( t , r ) | \\leq \\begin{cases} \\frac { C r } { t ^ { 2 } \\log ^ { 3 b + 2 N - 1 } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { \\sqrt { r } t \\log ^ { \\frac { 3 N } { 2 } + 3 b - 1 } ( t ) } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} ( k _ { 1 } * k _ { 2 } ) ( x , y ) = \\int _ { M } k _ { 1 } ( x , z ) k _ { 2 } ( z , y ) d { \\rm v o l } ( z ) \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} Z _ { \\Gamma } ( s ) = \\prod _ { \\{ P _ { 0 } \\} } \\prod _ { k = 0 } ^ { \\infty } ( 1 - N ( P _ { 0 } ) ^ { - s - k } ) , \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} M _ { n } ^ { ( 2 ) } = - \\frac { 1 } { 7 } \\left ( 4 V _ { n + 1 } ^ { ( 2 ) } - V _ { n } ^ { ( 2 ) } \\right ) \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} \\left \\{ ( \\pm \\sqrt { 2 l + 1 } ) \\lambda \\ , \\colon \\ , l = 0 , 1 , 2 , \\ldots \\right \\} , \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l l } d \\bar { x } _ { t } & = ( F _ { t } \\bar { x } _ { t } + f _ { t } ) d t + P _ { t } G _ { t } ^ { \\top } R _ { t } ^ { - 1 } d I _ { t } , \\\\ \\bar { x } ( 0 ) & = x _ { 0 } , \\end{array} \\right . \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} P ( \\varepsilon z ) = \\sum _ { k = 0 } ^ { m / 2 } \\sum _ { A \\in \\mathcal { A } _ k } \\varepsilon _ A P _ A ( z ) . \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} p \\ge p _ n ( k ) : = 2 + \\frac { 6 } { 2 ( n - 1 ) + ( k - 1 ) \\prod _ { i = k } ^ { n - 1 } \\frac { 2 i } { 2 i + 1 } } . \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} \\mathcal { \\tilde { F } } _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) \\Delta \\nabla \\mathbb { K } _ { n } ^ { ( j ) } \\left ( x \\right ) + \\mathcal { \\tilde { G } } _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) \\Delta \\mathbb { K } _ { n } ^ { ( j ) } \\left ( x \\right ) + \\mathcal { \\tilde { H } } _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) \\mathbb { K } _ { n } ^ { ( j ) } \\left ( x \\right ) = 0 , \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} & \\psi _ { j ' , k ' } * f ( x _ { I ' } , x _ { J ' } , x _ { S ' } ) \\\\ & = \\sum \\limits _ { j , k \\in \\mathbb Z } \\sum _ { R = I \\times J \\times S \\in \\mathcal R ^ N _ { \\frak z } ( j , k ) } | R | \\psi _ { j ' , k ' } * \\big ( \\widetilde \\psi _ { j , k } ( \\cdot _ 1 , \\cdot _ 2 , \\cdot _ 3 , x _ { I } , x _ { J } , x _ { S } ) \\big ) ( x _ { I ' } , x _ { J ' } , x _ { S ' } ) \\psi _ { j , k } * f ( x _ I , x _ J , x _ S ) . \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} \\alpha \\in J _ { C , n } : = \\bigcup _ { t = 0 } ^ { c _ n - 1 } \\left ( \\frac { I _ C } { c _ n } + \\frac { t } { c _ n } \\right ) \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} \\mathcal { G } _ 1 = \\ell ^ 2 ( E _ { \\ell } , \\lambda ) \\oplus \\ell ^ 2 ( E _ { r } , \\theta ) \\ ; \\ ; \\mathcal { G } _ 2 = \\ell ^ 2 ( E _ { \\ell } , \\theta ) \\oplus \\ell ^ 2 ( E _ { r } , \\lambda ) \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} & g ( t _ 0 ) < - K _ 1 = - \\underline h ( 0 ) \\ { \\rm a n d } \\ \\underline h ( 0 ) = K _ 1 < h ( t _ 0 ) , \\\\ & U ( t _ 0 , x ) \\succeq ( 1 - \\theta ^ { - \\alpha } ) \\mathbf { u } ^ * \\succeq \\underline U ( 0 , x ) \\ \\mbox { f o r } \\ \\ x \\in [ - \\underline h ( 0 ) , \\underline h ( 0 ) ] . \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} J ^ * ( P ' ( 2 m + 2 , 2 k + 1 ) ) = J ( P ' ( 2 m + 1 , 2 k + 1 ) ) . \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} N _ { i , A } : = | \\{ X \\in C \\mid X _ i ^ 1 = A \\} | , \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} | E _ { v _ { 3 } , i p } | & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 2 } ( t ) } \\int _ { t } ^ { \\infty } \\frac { d x } { ( \\log ^ { ( \\alpha - 1 ) b } ( t ) + x - t ) ( 1 + x - t ) ^ { 3 } } \\\\ & \\leq \\frac { C \\log ( \\log ( t ) ) } { t ^ { 2 } \\log ^ { b + 2 } ( t ) } \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\begin{aligned} z _ t = & \\mathcal { F } _ \\psi [ z , u _ \\psi ] \\doteqdot z z _ { \\psi \\psi } + z _ \\psi \\left ( \\frac { q - 1 } { \\psi } - \\frac { z } { \\psi } \\right ) - \\frac { 1 } { 2 } z _ \\psi ^ 2 + \\frac { 2 ( q - 1 ) } { \\psi ^ 2 } z ( 1 - z ) - 2 p z ^ 2 u _ \\psi ^ 2 \\\\ u _ t = & \\mathcal { E } _ \\psi [ z , u ] \\doteqdot z u _ { \\psi \\psi } + u _ \\psi \\frac { z + q - 1 } { \\psi } - ( p - 1 ) e ^ { - 2 u } \\end{aligned} \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} \\sigma _ k ^ 2 = \\beta _ k ^ 2 + \\gamma _ k ^ 2 . \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} 2 ^ { g + 1 } = 2 ^ { k + 1 } H _ { g + 1 - k } = 2 ^ { k + 1 } 2 ^ { g - k } \\leq \\sum _ { i = 1 } ^ { g + 1 } 2 ^ { i - 1 } + 1 = 2 ^ { g + 1 } . \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} g ( x ) = \\frac { 1 / x } { \\log ( 1 / x ) } , \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} X = Y _ 0 \\supset Y _ 1 \\supset \\cdots \\supset Y _ d = \\{ p \\} \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} \\big ( \\rho _ + ( y ) \\big ) _ k = \\rho ^ k y _ k , \\big ( \\rho _ - ( y ) \\big ) _ k = \\rho ^ { - k } y _ k , \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} \\left \\Vert \\alpha \\beta \\sigma ^ { 2 } \\left ( t \\right ) V \\left ( \\sqrt { \\alpha \\beta } x \\sigma \\left ( t \\right ) , \\beta t \\sigma \\left ( t \\right ) \\right ) \\right \\Vert _ { L \\left ( H \\right ) } \\leq \\alpha ^ { - 1 } \\beta \\left \\Vert V \\right \\Vert _ { B } = \\left ( k a _ { 0 } ^ { - 1 } \\right ) ^ { \\frac { 1 } { p } } \\left \\Vert V \\right \\Vert _ { B } \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} M _ { n p } = \\frac { \\gamma _ { n + 1 } \\langle f _ { n + 1 } | S f _ n \\rangle } { \\langle f _ { n + 1 } | 1 \\rangle } \\frac { \\langle f _ p | 1 \\rangle } { \\lambda _ p - \\lambda _ n - 1 } \\ , . \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } a _ { j k } \\partial _ { j } \\partial _ { k } \\bigg ] \\tilde { u } & = f \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { u } & = V \\tilde { u } \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} ( a _ n ) _ { n = 1 } ^ \\infty \\mapsto \\sum _ { n = 1 } ^ \\infty | a _ n | ^ { p _ n } , \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} C _ 1 \\ & = \\ B _ 1 \\ltimes A _ 1 \\ = \\ \\{ 2 + 2 , 6 + 2 , 1 0 + 2 \\} \\ = \\ \\{ 4 , 8 , 1 2 \\} , \\\\ C _ 2 \\ & = \\ B _ 2 \\ltimes A _ 2 \\ = \\ \\{ 2 + 1 , 4 + 1 , 1 4 + 1 \\} \\ = \\ \\{ 3 , 5 , 1 5 \\} , \\\\ C _ 3 \\ & = \\ B _ 3 \\ltimes [ 1 ] \\ = \\ \\{ 0 + 1 , 8 + 1 , 1 2 + 1 \\} \\ = \\ \\{ 1 , 9 , 1 3 \\} . \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} \\int _ { G / H } T _ H ( f ) ( x H ) d \\mu ( x H ) = \\int _ G f ( x ) d x , \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} i \\to j \\Longleftrightarrow Q ( A _ i , A _ j ) > 0 , \\\\ \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} U _ t = D \\circ \\int _ \\R \\mathbf { J } ( x - y ) \\circ U ( t , y ) { \\rm d } y - D \\circ U + F ( U ) \\mbox { f o r } t > 0 , \\ ; x \\in \\R , \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} \\{ ( x , u , v , w ) \\in X \\times \\C ^ 3 : ( a ( x ) - d ( x ) ) u + b ( x ) v + c ( x ) w = 0 , \\ u ^ 2 + v w = - 1 \\} . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} P _ { 2 k + 1 } ^ - \\lbrace M ( t ) \\le \\beta \\rbrace \\ = \\ \\frac { 2 k + 1 } { 2 k + 2 } P _ { 2 k } ^ - \\lbrace M ( t ) \\le \\beta \\rbrace + \\frac { 1 } { 2 k + 2 } P _ { 2 k + 1 } ^ + \\lbrace M ( t ) \\le \\beta \\rbrace \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} \\nabla G _ k ( \\lambda , v ) = - ( \\lambda + \\lambda _ { k - 1 } ( v ) + 1 ) \\nabla \\chi _ k ( \\lambda , v ) - \\chi _ k ( \\lambda , v ) \\nabla \\lambda _ { k - 1 } ( v ) \\ . \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} \\frac { - c _ { b } r } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\sin ^ { 2 } ( \\theta ) d \\theta \\left ( ( b - 1 ) \\int _ { 0 } ^ { \\frac { t _ { + } } { 2 } } \\frac { d u \\sin ( u ) } { u \\log ^ { b } ( \\frac { t _ { + } } { u } ) t _ { + } ^ { 2 } } \\right ) = \\frac { - b r } { 2 t ^ { 2 } \\log ^ { b } ( t ) } + E _ { v _ { 2 } , 2 } ( t , r ) , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} ( { } ^ Q \\Omega ^ - ) ^ a { } _ { b c } = 0 = ( \\Omega ^ - ) ^ a { } _ { \\mu b } \\rho ^ \\mu _ c + C ^ a { } _ { b c } , \\Longrightarrow ( \\Omega ^ - ) ^ a { } _ { \\mu b } = - ( \\rho ^ + ) ^ c _ \\mu C ^ a { } _ { b c } . \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} \\vline \\begin{aligned} & & & F ( U , V ; \\gamma ) \\coloneqq \\frac { \\gamma } { 2 } \\| U - V \\| _ F ^ 2 + f ( U , V ) . \\end{aligned} \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} \\Big \\Vert \\sum a _ { n } n ^ { - s } \\Big \\Vert _ { \\mathcal { H } _ { p ' } } \\leq C \\Big ( \\sum _ { n = 1 } ^ { \\infty } \\vert a _ { n } \\vert ^ { p } \\Big ) ^ { \\frac { 1 } { p } } \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} N \\sum _ { k = 1 } ^ N \\sum _ { z < 0 } h _ { b _ N } ( k , z ) P \\big [ \\sigma ^ { z } _ { [ R , \\infty ) } < N - k < \\sigma _ { \\{ 0 \\} } ^ { z } , S ^ { z } _ { N - k } = - c _ N \\big ] , \\\\ \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} \\Phi = \\{ ( x , y , z ) : F ( x , y , z ) = 0 , \\ , z ^ E = 0 , \\ , ( y ^ I , z ^ I ) \\in \\Theta \\} \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} c _ x \\leq \\left \\| 2 - \\frac { 2 } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } \\right \\| ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} \\widetilde { C } ^ c { } _ { a b } = ( K ^ { - 1 } ) ^ c { } _ { d } ( K ^ e { } _ { a } K ^ f { } _ { b } & C ^ d { } _ { e f } + K ^ e { } _ { a } ( X ^ { - 1 } _ * \\rho ) _ e ( K ^ d { } _ { b } ) - K ^ e { } _ { b } ( X ^ { - 1 } _ * \\rho ) _ e ( K ^ d { } _ { a } ) ) , \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} & \\sup _ { Q _ { y , x } } N ( \\epsilon \\lVert F \\circ \\phi _ { y } \\rVert _ { L _ { 2 } ( Q _ { y , x } ) } , \\mathcal { F } \\circ \\phi _ { y } , L _ { 2 } ( Q _ { y , x } ) ) = \\\\ & = \\sup _ { R _ { y , x } } N ( \\epsilon \\lVert ( F \\circ \\phi _ { y } ) / ( w _ { \\theta } \\circ \\phi _ { x } ) \\rVert _ { L _ { 2 } ( R _ { y , x } ) } , \\mathcal { F } / w _ { \\theta } , L _ { 2 } ( R _ { y , x } ) ) \\\\ \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} | Q _ { 1 } | \\leq C b _ q 2 ^ { - 2 q s } ( \\| u \\| _ { L ^ 2 } + \\| \\omega \\| _ { L ^ 2 } ^ \\frac 1 2 \\| \\partial _ 1 \\omega \\| _ { L ^ 2 } ^ \\frac 1 2 ) ( \\| f \\| _ { H ^ s } ^ 2 + \\| f \\| _ { H ^ s } \\| \\partial _ 1 f \\| _ { H ^ s } ) . \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} \\alpha _ { 1 } = - \\beta ^ 2 - \\frac { 1 } { 2 } \\log \\left ( 1 - 4 \\beta ^ 2 \\right ) \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} \\begin{aligned} - \\Delta _ r u & = \\lambda | u | ^ { r - 2 } u & & \\Omega , \\\\ u & = 0 & & \\partial \\Omega . \\end{aligned} \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | = m } \\partial ^ \\alpha a _ \\alpha ( x , t , u ) - u _ t \\ge 0 \\mbox { i n } B _ { 2 R } ^ y \\times ( 0 , \\infty ) , \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} P ( z ) = \\sum _ { | \\alpha | = m } x _ \\alpha z ^ \\alpha . \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} ( \\alpha - 1 ) p _ \\theta ( { \\bf { x } } ) ^ { \\alpha - 2 } \\partial _ r [ p _ \\theta ( { \\bf { x } } ) ] & = \\partial _ r [ Z ( \\theta ) ^ { \\alpha - 1 } ] [ h ( { \\bf { x } } ) + w ( \\theta ) ^ \\top f ( { \\bf { x } } ) ] + Z ( \\theta ) ^ { \\alpha - 1 } \\partial _ r [ w ( \\theta ) ] ^ \\top f ( { \\bf { x } } ) . \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} u _ 1 = u \\ , e ^ { \\lambda s } \\ , , u _ 2 = e ^ s \\ , . \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} \\mathcal { D } ( L _ { \\max } ) & : = \\big \\{ f \\in L ^ 2 ] a , b [ \\ , \\cap \\ , A C ^ 1 ] a , b [ \\ \\mid L f \\in L ^ 2 ] a , b [ \\big \\} , \\\\ L _ { \\max } f & : = L f , f \\in \\mathcal { D } ( L _ { \\max } ) . \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} Q ^ s _ { \\Sigma } ( \\phi , \\phi ) = Q _ { \\Sigma } ( \\phi , \\phi ) + s \\int _ { \\Sigma } | A _ { \\Sigma } | ^ 2 \\phi ^ 2 = \\int _ { \\Sigma } | \\nabla ^ { \\Sigma } \\phi | ^ 2 - { \\big ( } ( 1 - s ) | A _ { \\Sigma } | ^ 2 + R i c ( \\nu , \\nu ) { \\big ) } \\phi ^ 2 \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} \\tilde { x } & = \\frac { x } { r } , & \\tilde { y } & = \\frac { y } { r } , \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} C B . { h } ^ { n } & \\equiv a _ { n + 1 } h ^ { n + 1 } + a _ { n - 1 } h ^ { n - 1 } + \\cdots + a _ { 1 } h \\\\ \\\\ a _ { n + 1 } & = 2 ( n + 1 ) \\\\ k & = 1 , 3 , \\ldots , k - 2 \\ ( k - 1 ) \\\\ a _ { n - k } & = \\frac { 1 } { 2 } \\bigg ( - \\mu { n \\choose k } 4 ^ { k } + 2 { n \\choose k + 1 } 4 ^ { k + 1 } + { n \\choose k + 2 } 4 ^ { k + 2 } \\bigg ) . \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} V _ g : L ^ 2 ( G ) & \\to L ^ 2 ( G \\times \\widehat { G } ) \\\\ V _ g f ( \\xi ) & = \\langle f , \\pi ( \\xi ) g \\rangle , \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} \\pi _ { \\Phi [ \\rho ] } ( z ) = \\pi _ \\rho ( K z ) k ( z ) \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} \\gamma = \\sum _ I p ^ I \\gamma _ I + \\sum _ I q _ I \\gamma ^ I , \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} \\phi ( \\infty ) = \\infty \\Longleftrightarrow b > 0 \\textrm { o r } \\int _ { A } ^ { 1 } m ( d r ) = \\infty \\textrm { f o r s o m e } A \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} \\begin{pmatrix} r \\\\ k _ s \\\\ k _ s \\end{pmatrix} = \\Lambda = \\Theta ^ { } ( \\Lambda ) = \\begin{pmatrix*} [ l ] A _ c k _ c + g _ c \\circ K { } { } - k _ c \\circ ( A _ c + r ) \\\\ A _ u ^ { - 1 } k _ u \\circ ( A _ c + r ) - A _ u ^ { - 1 } g _ u \\circ K { } { } \\\\ A _ s k _ s \\circ ( A _ c + r ) ^ { - 1 } + g _ s \\circ K { } { } \\circ ( A _ c + r ) ^ { - 1 } \\end{pmatrix*} . \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} \\hat { M } = I _ m \\otimes M ^ * \\in \\R ^ { m n \\times m n } \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} \\mu _ { r , 7 } ( 0 , 1 ) = \\begin{cases} & 0 \\ ; \\ ; \\\\ & 1 \\ ; \\ ; \\\\ \\end{cases} , \\ ; \\ ; \\ ; \\ ; \\lambda _ r ( 0 , 1 ) = \\begin{cases} & 0 \\ ; \\ ; \\\\ & 1 \\ ; \\ ; \\end{cases} . \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} ( L _ u + & ( \\lambda + 1 ) I d ) ^ { - 1 } - S ^ * ( L _ u + \\lambda I d ) ^ { - 1 } S \\\\ & = - \\frac { \\langle \\ , \\cdot \\ , | S ^ * ( L _ u + \\lambda I d ) ^ { - 1 } 1 \\rangle } { \\langle ( L _ u + \\lambda I d ) ^ { - 1 } 1 | 1 \\rangle } S ^ * ( L _ u + \\lambda I d ) ^ { - 1 } 1 . \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} | \\frac { - 4 ( 1 - \\alpha ) \\lambda ( t ) ^ { - \\alpha } \\lambda ' ( t ) } { \\log ( \\lambda _ { 0 , 0 } ( t ) ) } \\int _ { t } ^ { \\infty } \\frac { e '' ( s ) d s } { ( \\lambda ( t ) ^ { 1 - \\alpha } + s - t ) ^ { 2 } ( 1 + s - t ) ^ { 3 } } | \\leq \\frac { C } { t ^ { 3 } \\log ^ { b + 2 } ( t ) ( \\log ( \\log ( t ) ) ) ^ { 3 / 2 } } \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} \\mathcal { S } : = \\{ [ s _ { 1 } , s _ { 2 } , s _ { 3 } , t _ { 1 } , t _ { 2 } , t _ { 3 } ] : ( 1 , s _ { 1 } , s _ { 2 } , s _ { 3 } ) \\sim ( 1 , t _ { 1 } , t _ { 2 } , t _ { 3 } ) \\mathcal { C } _ { 4 } \\} \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} 0 & = \\nabla f ( \\bar x ) + \\sum \\limits _ { i \\in I ^ g ( \\bar x ) } \\lambda _ i \\nabla g _ i ( \\bar x ) + \\sum \\limits _ { j \\in \\mathcal P } \\rho _ j \\nabla h _ j ( \\bar x ) \\\\ & + \\sum \\limits _ { l \\in I ^ { \\varphi _ \\textup { K S } } ( \\bar x ) } \\xi _ l \\bigl ( H _ l ( \\bar x ) \\nabla G _ l ( \\bar x ) + G _ l ( \\bar x ) \\nabla H _ l ( \\bar x ) \\bigr ) . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} T _ \\mu W ( M ) : = \\overline { \\{ \\nabla \\varphi \\mid \\varphi \\in \\mathcal { C } ^ \\infty _ c ( M ) \\} } ^ { L ^ 2 ( T M , \\mu ) } \\subset L ^ 2 ( T M , \\mu ) . \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} \\gamma = \\left ( \\frac { \\partial f _ L } { \\partial y } g _ R - \\frac { \\partial g _ L } { \\partial y } f _ R \\right ) \\bigg | _ { x = y = \\mu = 0 } . \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} \\{ \\gamma _ p , \\varphi _ n \\} = \\delta _ { p n } \\ , \\ p \\ge 1 \\ , \\forall u \\in L ^ 2 _ { r , 0 } \\setminus Z _ n \\ . \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} \\| S ' \\| _ { L ^ 2 _ \\sigma ( \\R ) } & \\leq \\left \\| \\sum _ { j \\in \\N } | P _ j | \\right \\| _ { L ^ 2 _ \\sigma ( \\R ) } \\leq \\sum _ { j \\in \\N } \\| P _ j \\| _ { L ^ 2 _ \\sigma ( \\R ) } = \\sum _ { j \\in \\N } j ^ { - 1 } j \\| P _ j \\| _ { L ^ 2 _ \\sigma ( \\R ) } \\lesssim L ^ { 1 / 2 } \\ , \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} a _ i = v _ { n - 1 } + \\sum _ { j = i } ^ { n - 1 } ( d - C v ) _ j \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} ( \\Omega ^ - ) ^ a { } _ { \\mu b } = - ( \\rho ^ + ) ^ c _ \\mu C ^ a { } _ { b c } , \\quad ( \\Omega ^ + ) ^ a { } _ { \\mu b } = ( ( E \\rho ) ^ + ) ^ a { } _ { \\lambda } ( \\Psi _ b ) _ { \\lambda \\mu } , \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} \\Psi ( z ) = \\gamma z \\exp \\left [ \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t \\eta _ { \\mu _ { 1 } } ( z ) } { \\eta _ { \\mu _ { 1 } } ( z ) - t } \\ , d \\sigma ( t ) \\right ] , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} x _ 1 : = & z ^ { - 1 } ( \\log ( u _ 2 ) - 2 \\log ( u _ 1 ) ) , & y _ { 1 , d } : = & d z ^ { - 1 } \\log ( u _ 1 ) , \\\\ x _ 2 : = & z ^ { - 1 } ( \\log ( u _ 1 ) - 2 \\log ( u _ 2 ) ) , & y _ { 2 , d } : = & d z ^ { - 1 } \\log ( u _ 2 ) . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} \\sum _ { g _ 0 \\cdots g _ k = 1 } | f _ 0 ( g _ 0 ) f _ 1 ( g _ 1 ) \\cdots f _ k ( g _ k ) c ( g _ 1 , \\dots , g _ k ) | \\leq C \\nu _ m ( f _ 0 ) \\cdots \\nu _ m ( f _ k ) \\ , . \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} \\bar { d } _ { E , \\phi } ( F ) : = \\limsup _ { N \\to \\infty } \\frac { | \\{ n _ i : i \\in \\N \\} \\cap I _ N | } { | I _ N | } \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} \\psi _ { n _ k } ( x , x ) = \\langle M _ { \\hat { x } } ( h _ { n _ k } ) , M _ { \\hat { x } } ( h _ { n _ k } ) \\rangle _ { L ^ 2 ( \\mu ) } = \\| M _ { \\hat { x } } ( h _ { n _ k } ) \\| ^ 2 _ { L ^ 2 ( \\mu ) } . \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} \\Phi ( \\eta _ { \\mu } ( z ) ) = z , z \\in \\mathbb { C } \\backslash \\mathbb { R _ { + } } , \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} v ^ { \\lambda } _ { 3 , 1 } ( t , r ) = \\frac { - 1 } { r } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { ( s - t ) } \\lambda '' ( s ) \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} b _ 1 = 0 \\ \\ b _ i = c _ 1 + \\dots + c _ { i - 1 } \\ \\ 2 \\leq i \\leq n . \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} \\frac { z } { f ( z ) } - z \\left ( \\frac { z } { f ( z ) } \\right ) ' - \\mu = \\lambda \\Omega ( z ) , \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} ( X _ 1 ^ { * * } , X _ 2 ^ { * * } ) = \\{ X ^ * _ 1 , ~ ( X ^ * _ 2 - \\rho _ * X ^ * _ 1 ) / \\sqrt { 1 - \\rho _ * ^ 2 } \\} . \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} \\sigma \\big ( ( \\Phi ^ e _ { v w } ) ^ * \\Phi ^ e _ { v w } \\big ) & = \\sigma \\Big ( \\big ( ( \\Phi ^ e _ { v w } ) ^ * \\Phi ^ e _ { v w } \\big ) ^ { - 1 } \\Big ) ^ { - 1 } \\\\ & = \\sigma \\Big ( \\big ( | \\Phi ^ e _ { v w } | \\big ) ^ { - 1 } \\big ( | \\Phi ^ e _ { v w } | ^ * \\big ) ^ { - 1 } \\Big ) ^ { - 1 } \\\\ & = \\sigma \\big ( ( \\Phi ^ { \\overline { e } } _ { w v } ) ^ * \\Phi ^ { \\overline { e } } _ { w v } \\big ) ^ { - 1 } \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} B ' = B \\setminus \\{ B _ { a , b } ^ { ( s ) } \\mid b = 1 , \\ldots , m _ s \\} . \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} R ^ * _ k u = k \\ - u , R ^ * _ j u = j \\ - u \\ , j , R ^ * _ j u ' = j \\ - u ' , R ^ * _ k u ' = u ' , \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} & \\alpha B + \\beta B ^ { 1 / 2 } W B ^ { 1 / 2 } + B ^ { 1 / 2 } W ^ 2 B ^ { 1 / 2 } \\\\ & \\qquad \\quad + \\int _ { ( 0 , \\infty ) } \\biggl ( { 1 \\over 1 + s } \\ , B ^ { 1 / 2 } W B ^ { 1 / 2 } - B ^ { 1 / 2 } h _ s ( W ) B ^ { 1 / 2 } \\biggr ) \\ , d \\mu ( s ) \\\\ & \\quad = B ^ { 1 / 2 } \\biggl [ \\alpha I + \\beta W + \\gamma W ^ 2 + \\int _ { ( 0 , \\infty ) } \\phi _ s ( W ) \\ , d \\mu ( s ) \\biggr ] B ^ { 1 / 2 } = B ^ { 1 / 2 } f ( W ) B ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} \\vec { d } = W ' \\vec { b } . \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} \\begin{aligned} & \\rho _ t + \\nabla \\cdot ( \\rho Q \\ast \\mathbf { u } ) = 0 , \\ ; x \\in \\mathbb { R } ^ N , \\ ; t > 0 , \\\\ & \\mathbf { u } _ t + ( \\mathbf { u } \\cdot \\nabla ) \\mathbf { u } = \\rho ( Q \\ast \\mathbf { u } - \\mathbf { u } ) , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} | | f | | _ { \\dot { H } ^ { 1 } _ { e } } ^ { 2 } = \\int _ { 0 } ^ { \\infty } \\left ( ( \\partial _ { r } f ) ^ { 2 } + \\frac { f ^ { 2 } } { r ^ { 2 } } \\right ) r d r \\leq C \\left ( | | L f | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } + | | f | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } \\right ) \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} \\alpha _ 1 = \\alpha _ 2 = \\alpha _ 3 = 0 , \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} Q _ 1 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 ) = Q _ 2 ( x _ 0 , x _ 2 , x _ 3 , x _ 4 , x _ 5 ) = Q _ 3 ( x _ 0 , x _ 1 , x _ 3 , x _ 4 , x _ 5 ) = 0 , \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} & | \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { 0 } ^ { 1 } d w \\lambda '' ( w + t ) \\left ( \\partial _ { 2 2 } K _ { 1 } ( w , \\lambda ( t ) ) \\lambda ' ( t ) - \\frac { \\lambda ' ( t ) } { 2 ( 1 + w ) } \\right ) \\lambda ' ( t ) | \\\\ & \\leq \\frac { C } { t ^ { 3 } \\log ^ { 2 } ( t ) } \\int _ { 0 } ^ { 1 } d w \\left ( w + \\frac { 1 } { 1 + w } \\right ) \\frac { 1 } { t \\log ^ { b + 1 } ( t ) } \\leq \\frac { C } { t ^ { 4 } \\log ^ { b + 3 } ( t ) } \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} \\kappa _ g = \\frac { e ^ { - 4 s _ 2 } \\left ( e ^ { 2 s _ 2 } - 2 \\right ) } { e ^ { - 4 s _ 2 } \\left ( e ^ { 2 s _ 2 } + 1 \\right ) ^ 2 } = \\frac { e ^ { 2 s _ 2 } - 2 } { \\left ( e ^ { 2 s _ 2 } + 1 \\right ) ^ 2 } \\ ; . \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} \\beta ^ { \\ast } : = \\frac { n - \\widehat { w } _ { n } } { n ( 1 + \\widehat { w } _ { n } ) } . \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\bar { P } _ { } ^ e = 0 . 1 6 8 \\Phi \\bigg ( 1 . 7 5 2 { e ^ { { \\mu _ \\gamma } + \\frac { { \\sigma _ \\gamma ^ 2 } } { 2 } } } , \\frac { { { \\sigma _ \\gamma } } } { 2 } \\bigg ) + 0 . 1 4 4 \\Phi \\bigg ( 1 . 0 5 { e ^ { { \\mu _ \\gamma } + \\frac { { \\sigma _ \\gamma ^ 2 } } { 2 } } } , \\frac { { { \\sigma _ \\gamma } } } { 2 } \\bigg ) + 0 . 0 0 2 \\Phi \\bigg ( 1 . 2 0 6 { e ^ { { \\mu _ \\gamma } + \\frac { { \\sigma _ \\gamma ^ 2 } } { 2 } } } , \\frac { { { \\sigma _ \\gamma } } } { 2 } \\bigg ) , \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} \\begin{cases} \\ln ( \\ln \\theta ) \\geq - \\ln K _ 1 + 2 \\\\ [ 2 m m ] \\dd K _ 1 \\leq K _ 2 \\sum _ { i = 1 } ^ { m _ 0 } \\frac { \\mu _ i C _ 1 \\theta _ i ( 1 - \\beta ) } { 2 } , \\end{cases} \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} ( R ^ * \\tilde t _ 2 ) ^ { - 1 } \\cdot ( R ^ * \\tilde t _ 1 ) = R ^ * ( ( i ^ { - 1 } ) ^ * t _ { 1 2 } ) = t _ 1 ^ { - 1 } \\cdot t _ 2 { \\rm o n } ( \\mathfrak M ( H ^ \\infty ( U ) ) \\setminus \\overline { O } _ 2 ) \\cap O _ 1 , \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} f ( r ) = 2 \\beta \\Im \\frac { 1 } { 1 - \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) } = 2 \\pi \\beta k ( \\Psi ( r e ^ { i f ( r ) } ) ) . \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} \\psi ( \\hat { y } - z , \\hat { s } ) - \\psi ( \\hat { y } , \\hat { s } ) - \\varphi ( \\hat { y } - z , \\hat { s } ) + \\varphi ( \\hat { y } , \\hat { s } ) & = \\psi ( \\hat { y } - z , \\hat { s } ) - \\varphi ( \\hat { y } - z , \\hat { s } ) - \\lambda ' \\\\ & \\ge \\psi ( \\hat { y } - z , \\hat { s } ) - w ^ * ( \\hat { y } - z , \\hat { s } ) \\\\ & \\ge 0 . \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} & \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } f _ 0 ( x , v ) \\psi ( 0 , x , v ) d x d v \\\\ \\leq & \\ ; \\ ; \\| f _ 0 \\| _ { L ^ \\infty } \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } \\big | \\psi ( 0 , x , v ) \\big | d x d v \\ ; \\leq \\| f _ 0 \\| _ { L ^ \\infty } . \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} \\langle P u , \\phi \\rangle = ( u , P \\phi ) _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } \\le \\| u \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } \\| P \\phi \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } \\le C \\| u \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } \\| \\phi \\| _ { H ^ { 2 } ( \\mathbb { R } ^ { n } ) } \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} D ^ { ( r ) } ( f g ) = \\sum _ { i = 0 } ^ { r } D ^ { ( i ) } ( f ) D ^ { ( r - i ) } ( g ) \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} x ^ 2 + \\frac { b } { x ^ 2 } = B ^ 2 . \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} \\bar { H } ( \\eta _ 0 ^ { \\nu } ) \\leq C ( 1 + \\| \\eta _ 0 ^ { \\nu } \\| _ { L ^ 2 ( Y ) } ^ 2 ) = C ( 1 + \\| \\bar { \\eta } _ 0 ^ { \\nu } \\| _ { L ^ 2 } ^ 2 ) \\leq C . \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} \\frac { d } { d \\sigma } A ( \\sigma ) = m , \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} \\frac { d ^ 2 \\varphi _ i } { d z ^ 2 } + \\frac { n r ^ { n - 2 } + \\lambda _ i ^ { \\rm c y l } } { r ^ n } \\varphi _ i = 0 . \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} | | v _ { 6 } ( t ) | | _ { \\dot { H } ^ { 1 } _ { e } } = | | v _ { 6 } ( t , \\cdot \\lambda ( t ) ) | | _ { \\dot { H } ^ { 1 } _ { e } } \\leq C \\left ( | | v _ { 6 } ( t , \\cdot \\lambda ( t ) ) | | _ { L ^ { 2 } ( R d R ) } + | | L ( v _ { 6 } ( t , \\cdot \\lambda ( t ) ) ) | | _ { L ^ { 2 } ( R d R ) } \\right ) \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} f = \\sum _ { x \\le y } f ( x , y ) e _ { x y } , \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} D _ 1 \\ & = \\ \\{ 3 , 5 , 7 \\} , \\\\ D _ 2 \\ & = \\ \\{ 2 , 4 , 9 \\} , \\\\ D _ 3 \\ & = \\ \\{ 1 , 6 , 8 \\} . \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} \\Lambda ( h ^ * ) = [ 1 , \\infty ) . \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} u _ \\lambda ( x ) = \\int _ 0 ^ \\infty e ^ { - \\lambda t } p ( t , 0 , x ) d t = \\frac { 1 } { ( 2 \\pi ) ^ d } \\int _ { \\mathbb { R } ^ d } \\frac { e ^ { - i \\langle \\xi , x \\rangle } } { \\lambda + | \\xi | ^ \\alpha } d \\xi . \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} \\Psi [ \\varphi ] ( p ) = \\inf _ { a \\in A } \\left [ \\varphi ( a ) + \\frac { d ( a , p ) } { d ( p , A ) } - 1 \\right ] , \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} | \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } \\lambda '' ( s ) r \\left ( \\frac { 1 } { s - t } - \\frac { 1 } { 1 + s - t } \\right ) d s | & \\leq r \\sup _ { x \\geq t } \\left ( | \\lambda '' ( x ) | \\right ) \\log ( 1 + \\frac { 1 } { 2 + 2 r } ) \\\\ & \\leq C \\sup _ { x \\geq t } \\left ( | \\lambda '' ( x ) | \\right ) r \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} & 0 = \\frac { 1 } { \\pi } \\boldsymbol { B } \\Big ( \\frac { \\partial w } { \\partial { x _ 1 } } , \\frac { \\partial w } { \\partial { x _ 1 } } \\Big ) = \\mathbb B _ 1 ( - \\Phi _ 2 , \\Phi _ 0 ) = { \\mathbb D } ( \\Phi _ 2 , \\Phi _ 0 ) , \\\\ [ 1 m m ] & 0 = \\frac { 1 } { \\pi } \\boldsymbol { B } \\Big ( \\frac { \\partial w } { \\partial { x _ 2 } } , \\frac { \\partial w } { \\partial { x _ 2 } } \\Big ) = \\mathbb B _ 1 ( i \\Phi _ 2 , i \\Phi _ 0 ) = { \\mathbb D } ( \\Phi _ 2 , \\Phi _ 0 ) . \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} \\begin{cases} & \\underline h ( t ) : = c _ 0 t + \\delta ( t ) , \\ \\ \\ t \\geq 0 , \\\\ & \\underline U ( t , x ) : = ( 1 - \\epsilon ( t ) ) [ \\Phi ( x - \\underline h ( t ) ) + \\Phi ( - x - \\underline h ( t ) ) - \\mathbf { u } ^ * ] , \\ \\ \\ t \\geq 0 , x \\in [ - \\underline h ( t ) , \\underline h ( t ) ] , \\end{cases} \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} \\tilde { u } \\left ( x , t \\right ) = \\left ( \\sqrt { \\alpha \\beta } \\sigma \\left ( t \\right ) \\right ) ^ { \\frac { n } { 2 } } u \\left ( \\sqrt { \\alpha \\beta } x \\sigma \\left ( t \\right ) , \\beta t \\sigma \\left ( t \\right ) \\right ) e ^ { \\eta } . \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} { } ^ b { \\rm T r } _ { \\chi } \\left ( [ \\Phi _ { A _ { 1 } } , \\Phi _ { A _ { 2 } } ] \\right ) = \\frac { i } { 2 \\pi } \\int _ { \\mathbb { R } } \\int _ G { \\rm T r } _ { \\partial S } \\left ( \\frac { \\partial I ( \\Phi _ { A _ 1 } , h ^ { - 1 } , \\lambda ) } { \\partial \\lambda } \\circ I ( \\Phi _ { A _ 2 } , h , \\lambda ) \\right ) d h d \\lambda . \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} R ^ m ( y ) S ( y ) = 1 + c _ 1 y + c _ 2 y ^ 2 + \\cdots + c _ q y ^ q = 1 + y ^ { k + n - l } U ( y ) , \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} R & = \\nabla Q ^ T ( x ) \\nabla Q ( x ) \\\\ & = \\begin{bmatrix} \\big ( \\frac { \\partial g } { \\partial x } \\big ) ^ T \\frac { \\partial g } { \\partial x } & \\big ( - \\frac { \\partial g } { \\partial x } \\big ) ^ T \\\\ - \\frac { \\partial g } { \\partial x } & I \\end{bmatrix} \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\vec { \\mathbf { v } } ^ { k + 1 } = ( 1 - \\Delta t _ k \\eta ) \\vec { \\mathbf { v } } ^ { k } + \\Delta t _ k \\mathbf { F } ^ { k } \\vec { \\mathbf { u } } ^ { k } , \\\\ \\vec { \\mathbf { u } } ^ { k + 1 } = \\vec { \\mathbf { u } } ^ { k } + \\Delta t _ k \\vec { \\mathbf { v } } ^ { k + 1 } , \\\\ \\vec { \\mathbf { u } } _ { 0 } = \\vec { \\mathbf { u } } ^ d , \\vec { \\mathbf { v } } _ 0 = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} e ^ t & = e + D ^ A t , e ^ \\gamma = \\gamma \\ - e \\gamma , \\\\ A ^ t & = A , A ^ \\gamma = \\gamma \\ - A \\gamma + \\gamma \\ - d \\gamma . \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} \\widetilde V : = \\left \\{ \\xi \\in \\mathfrak M ( H ^ \\infty ) \\ , : \\ , \\hat h ( \\xi ) \\le \\mbox { $ \\frac 1 2 $ } \\right \\} \\subset \\widetilde U _ 1 . \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} \\langle T \\rho ( x ) \\rho ( y ) \\rangle & = \\langle T \\phi ( x ) \\phi ( y ) + \\sum _ { k = 2 } ^ \\infty \\frac { \\left ( b _ k \\right ) ^ 2 } { ( k ! ) ^ 2 } \\langle { T \\phi ^ k ( x ) \\phi ^ k ( y ) } \\rangle . \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} V ^ { N } ( x ) : = N ^ { d \\alpha } V ( N ^ { \\alpha } x ) , \\alpha \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} \\frac { c _ { 1 2 } x ^ p _ d y ^ { p } _ d } { r ^ { d + \\alpha + \\beta _ 1 + \\beta _ 2 } } \\left \\{ \\begin{array} { l l l } y _ d ^ { 2 \\alpha - 2 p + \\beta _ 1 + \\beta _ 2 } \\left ( \\log \\frac { r } { y _ d } \\right ) ^ { \\beta _ 4 } , & 2 \\alpha - 2 p + \\beta _ 1 + \\beta _ 2 < 0 ; \\\\ \\left ( \\log \\frac { r } { y _ d } \\right ) ^ { \\beta _ 4 + 1 } , & 2 \\alpha - 2 p + \\beta _ 1 + \\beta _ 2 = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} \\bar { V } ^ { \\perp } _ { x _ i , x _ j } = x _ i \\partial _ { x _ j } ^ { \\perp } - x _ j \\partial _ { x _ i } ^ { \\perp } \\ , . \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & \\ , \\to \\ , \\min & & & \\\\ g _ i ( x ) & \\ , \\leq \\ , 0 & \\qquad & i \\in \\mathcal M & \\\\ h _ j ( x ) & \\ , = \\ , 0 & & j \\in \\mathcal P , & \\end{aligned} \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} p \\otimes q = \\{ A \\in \\mathfrak { m } \\otimes \\mathfrak { m } : \\{ s \\in X : A _ s \\in q \\} \\in p \\} \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} \\int _ 0 ^ t y ^ { - \\alpha a - 1 } m _ { \\alpha } \\left ( \\frac { y } { t } \\right ) d y = t ^ { \\alpha } \\int _ 0 ^ t y ^ { - ( \\alpha + \\alpha a + 1 ) } m ( y ) d y < \\infty . \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} 0 = \\phi _ 2 \\left ( \\sum _ { i = 1 } ^ n \\alpha _ i A _ i \\right ) = \\sum _ { i = 1 } ^ n \\alpha _ i \\phi _ 1 ( A _ i ) . \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} x _ { \\frac { n } { 4 } } = - 1 / 2 = - x _ { \\frac { n } { 4 } + 1 } . \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} \\partial ( n _ 1 , . . . , n _ { 2 ^ \\ell } ) = \\partial ( \\partial ( n _ 1 , n _ 2 ) , . . . , \\partial ( n _ { 2 ^ \\ell - 1 } , n _ { 2 ^ \\ell } ) ) . \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} [ D , T ] = \\{ S \\in \\mathcal { T } : r _ k ( S ) = D \\mathrm { \\ a n d \\ } S \\le T \\} . \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} A \\triangleq \\left \\{ \\omega \\bigg { | } \\lim _ { s \\to \\infty } \\frac { t _ { s + 1 } - t _ s } { t _ s ^ { \\min \\{ \\tilde { \\beta } - \\beta , 1 \\} } } = 0 \\right \\} . \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} \\Omega _ h = \\operatorname { I n t } \\left ( \\bigcup _ { K \\in \\mathcal { T } _ h } K \\right ) ; \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} B \\ltimes A \\ = \\ \\{ M ( b - 1 ) + a : b \\in B , a \\in A \\} . \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} \\Phi ( P _ f ( A , B ) ) & = \\Phi ( P _ { f _ 0 } ( A , B ) ) + \\alpha \\Phi ( B ) + \\beta \\Phi ( A ) , \\\\ P _ f ( \\Phi ( A ) , \\Phi ( B ) ) & = P _ { f _ 0 } ( \\Phi ( A ) , \\Phi ( B ) ) + \\alpha \\Phi ( B ) + \\beta \\Phi ( A ) , \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} K _ { n } ^ { p , N } \\left ( x \\right ) = p ^ n \\left ( - N \\right ) _ { n } \\ , _ { 2 } F _ { 1 } \\left ( \\begin{array} { c | c } - n , - x & \\\\ & p ^ { - 1 } \\\\ - N & \\end{array} \\right ) , \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} h ( t ) = \\frac { f ( t ) } { 1 - F ( t ) } . \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} \\ , \\ , \\ , \\ , \\ , \\ , & t _ { } \\\\ ~ ~ ~ ~ & { \\rm { P } } _ { } ^ e - h < 0 , \\\\ & t _ { s } = t _ { } + t _ { } . \\\\ \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ { t t } v _ { s } + \\partial _ { r r } v _ { s } + \\frac { 1 } { r } \\partial _ { r } v _ { s } - \\frac { v _ { s } } { r ^ { 2 } } = 0 \\\\ v _ { s } ( s , r ) = 0 \\\\ \\partial _ { t } v _ { s } ( s , r ) = - 2 \\lambda '' ( s ) \\frac { r } { 1 + r ^ { 2 } } \\end{cases} \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} \\underline { \\psi } _ { N , j } = \\liminf _ { Q \\to \\infty } \\psi _ { N , j } ( Q ) , \\qquad \\overline { \\psi } _ { N , j } = \\limsup _ { Q \\to \\infty } \\psi _ { N , j } ( Q ) , \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} N _ { \\Theta ( \\varepsilon ) } ( y ^ I , z ^ I ) = \\widehat N _ { \\Theta ( \\varepsilon ) } ( y ^ I , z ^ I ) = { \\cal J } _ { y ^ I , z ^ I } \\Psi _ { \\varepsilon } ( y ^ I , z ^ I ) ^ T \\Re ^ p . \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to + \\infty } c ( \\gamma ) = + \\infty . \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} | \\partial _ { r } \\left ( \\frac { \\phi ( r , \\xi ) } { \\sqrt { r } } \\right ) | \\leq \\begin{cases} C \\left ( | \\phi _ { 0 } ' ( r ) | + \\xi \\log ( 1 + r ^ { 2 } ) \\right ) , r ^ { 2 } \\xi \\leq 4 \\\\ \\frac { C | a ( \\xi ) | \\xi ^ { 1 / 4 } } { r ^ { 1 / 2 } } , r ^ { 2 } \\xi > 4 \\end{cases} \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} \\nu _ { N , s } ^ { \\sigma } ( d x ) : = \\frac { 1 } { Z } \\exp \\left ( \\sigma \\sum _ { i = 1 } ^ { N } x _ i - H _ s ( x ) \\right ) d x . \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} a _ { 0 R } = f _ R ( 0 , 0 ; 0 ) . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } ( \\| | D | ^ s X ( t ) \\| _ { L ^ 2 } ^ 2 & = - \\int _ { \\R ^ 2 } [ | D | ^ s , u \\cdot \\nabla ] X \\cdot | D | ^ s X ~ d x + \\int _ { \\R ^ 2 } | D | ^ s ( X \\cdot \\nabla u ) \\cdot | D | ^ s X ~ d x \\\\ & \\triangleq M _ 1 + M _ 2 . \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} \\mathcal { S } \\left ( J _ { 3 } ^ { \\mathbb { O } } \\right ) \\supset \\mathcal { S } \\left ( \\mathbb { R } \\oplus J _ { 2 } ^ { \\mathbb { O } } \\right ) \\cong \\left . s o ( 1 , q ) \\right \\vert _ { q = 8 } \\supset \\left . s o ( q - 1 ) \\right \\vert _ { q = 8 } \\oplus s o ( 1 , 1 ) . \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} \\mathfrak { p } _ { N } ^ { R } ( \\mathbf { s } _ { N } ; \\mathbf { p } _ { N } ) : = \\begin{cases} \\frac { \\prod _ { i = 1 } ^ { N } p _ { i , N } ^ { s _ { i } } ( 1 - p _ { i , N } ) ^ { 1 - s _ { i } } } { \\sum _ { \\mathbf { s } _ { N } ' \\in \\Omega _ { N , n } } \\prod _ { i = 1 } ^ { N } p _ { i , N } ^ { s _ { i } ' } ( 1 - p _ { i , N } ) ^ { 1 - s _ { i } ' } } & \\mathbf { s } _ { N } \\in \\Omega _ { N , n } , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} | \\nabla \\varrho _ c | = | \\nabla u | v _ { h , c } ( \\varrho _ c ) ^ { \\frac { 1 } { p - 1 } } \\le | \\nabla u | v _ { h } ( \\varrho ) ^ { \\frac { 1 } { p - 1 } } = | \\nabla \\varrho | \\ , \\Omega _ c . \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} \\gamma _ 1 ( H ^ 2 ( \\Omega ) ) = \\gamma _ 1 ( H ^ 2 _ { \\mu , D } ( \\Omega ) ) = \\mathcal S ^ { \\frac { 1 } { 2 } } ( \\partial \\Omega ) \\ ( = \\mathcal S ^ { \\frac { 1 } { 2 } } _ { \\mu } ( \\partial \\Omega ) ) . \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} \\sigma \\Big ( ( \\Phi ^ e _ { v w } ) ^ * \\Phi ^ e _ { v w } \\Big ) = \\sigma \\Big ( U ^ { e } _ { v w } ( \\Psi ^ e _ { \\varphi ^ a ( v ) \\varphi ^ b ( w ) } ) ^ * \\Psi ^ e _ { \\varphi ^ a ( v ) \\varphi ^ b ( w ) } ( U ^ { e } _ { v w } ) ^ * \\Big ) = \\sigma \\Big ( ( \\Psi ^ e _ { \\varphi ^ a ( v ) \\varphi ^ b ( w ) } ) ^ * \\Psi ^ e _ { \\varphi ^ a ( v ) \\varphi ^ b ( w ) } \\Big ) \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} \\frac 1 2 \\int _ \\Omega | x | ^ { - s } ( u ^ m ( x , \\widetilde { T } ) ) ^ 2 d x - \\frac 1 2 \\int _ \\Omega | x | ^ { - s } ( u ^ m ( x , 0 ) ) ^ 2 d x + \\int _ 0 ^ { \\widetilde { T } } \\| \\nabla u ^ m \\| ^ p _ p d t = \\int _ 0 ^ { \\widetilde { T } } \\int _ \\Omega | u ^ m | ^ q \\ln | u ^ m | d x d t . \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} G ( d , s , t ) = \\frac { d ^ 2 } { 2 s } + \\frac { d } { 2 s } ( 2 P - 2 - s ) + \\frac { d } { s } ( \\beta - 1 ) + 1 . \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} \\begin{cases} c _ 0 t - h ( t ) \\geq \\sigma \\ , t ^ { 3 - { \\gamma } } & { \\rm i f } \\ { \\gamma } \\in ( 2 , 3 ) , \\\\ c _ 0 t - h ( t ) \\geq \\sigma \\ln t & { \\rm i f } \\ { \\gamma } = 3 . \\end{cases} \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} ( \\phi / \\psi _ { \\gamma _ n } ^ 2 , Q ^ n _ t ( \\phi / \\psi _ { \\gamma _ n } ^ 2 ) ) _ { H _ n } = ( Q ^ n _ { t / 2 } ( \\phi / \\psi _ { \\gamma _ n } ^ 2 ) , Q ^ n _ { t / 2 } ( \\phi / \\psi _ { \\gamma _ n } ^ 2 ) ) _ { H _ n } \\leq ( \\phi / \\psi _ { \\gamma _ n } ^ 2 , \\phi / \\psi _ { \\gamma _ n } ^ 2 ) _ { H _ n } . \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} { \\bf L } \\Lambda ^ { \\mathfrak { m } } \\left ( E _ { R } ( k ) \\right ) & \\simeq { \\bf L } \\Lambda ^ { \\mathfrak { m } } \\left ( { \\bf R } \\Gamma _ { \\mathfrak { m } } ( D ) \\right ) \\\\ & \\simeq { \\bf L } \\Lambda ^ { \\mathfrak { m } } ( D ) \\\\ & \\simeq D \\otimes _ { R } ^ { \\bf L } \\widehat { R } ^ { \\mathfrak { m } } \\\\ & \\simeq D \\otimes _ { R } \\widehat { R } ^ { \\mathfrak { m } } . \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} A _ 1 ( K _ 1 ) : = A _ 1 - B _ 1 B _ 1 ^ { \\top } K _ 1 . \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} E ( v _ { 3 } ( t ) , \\partial _ { t } v _ { 3 } ( t ) ) & \\leq C \\left ( \\int _ { t } ^ { \\infty } | | F _ { 0 , 1 } ( s ) | | _ { L ^ { 2 } ( r d r ) } d s \\right ) ^ { 2 } \\leq \\frac { C \\log ( \\log ( t ) ) } { t ^ { 2 } \\log ^ { 2 b + 2 } ( t ) } \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} \\psi ( x _ 1 , x _ 2 ) = \\psi _ { \\nu } ^ { \\pm } ( x _ 1 , x _ 2 ) = e ^ { \\pm i \\nu \\sqrt { 3 } x _ 1 } K _ { i \\nu } \\left ( 4 e ^ { - \\sqrt { 3 } x _ 2 } \\right ) , \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} \\sum _ { y \\in Y } m _ y \\deg _ k ( y ) = \\deg ( d f _ 3 ) = \\deg ( f _ 3 ) + 2 g - 1 . \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty R ^ { p p _ n / | p _ n - p | } < \\infty , \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\sup _ { t \\ge T } \\int _ 0 ^ { \\frac t 2 } \\omega ( t - \\tau , \\lambda _ 1 ) p ( \\tau ) d \\tau = 0 . \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } \\theta ' \\varphi \\ , U = - \\int _ 0 ^ { + \\infty } \\theta \\ , \\varphi \\ , U ' . \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} u \\prec v = u \\cdot T ( v ) , u \\succ v = T ( u ) \\cdot v ~ ~ ~ ~ ~ ~ u \\curlyvee v = H ( T u , T v ) , ~ u , v \\in M . \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} & ( \\rho ( x _ 1 , x _ 2 ) b ) \\rho ( x _ 3 , x _ 4 ) + ( - 1 ) ^ { \\bar x _ 3 ( \\bar x _ 1 + \\bar x _ 2 + \\bar b ) + \\bar b \\bar x _ 2 } ( \\rho ( x _ 3 , x _ 1 ) b ) \\rho ( x _ 2 , x _ 4 ) \\\\ & + ( - 1 ) ^ { \\bar x _ 1 ( \\bar x _ 2 + \\bar x _ 3 + \\bar b ) + \\bar x _ 1 \\bar b } ( \\rho ( x _ 2 , x _ 3 ) b ) \\rho ( x _ 1 , x _ 4 ) = 0 . \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} q _ 1 \\sim q _ 2 \\Leftrightarrow \\ f ^ { ( i ) } ( q _ 1 ) = f ^ { ( i ) } ( q _ 2 ) \\ \\mbox { f o r a l l } \\ f \\in C ^ \\infty ( M ) _ { ( i ) } , \\ 0 \\le i \\le r . \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} \\sigma ^ { i j } : = \\det ( g ) ^ { \\frac { 1 } { N + 1 } } g ^ { i j } \\in S ^ 2 ( M ) \\otimes ( \\Lambda ^ N ( M ) ) ^ { \\frac { 2 } { N + 1 } } \\ , , \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} ( x _ 1 - 1 ) ^ 2 + ( x _ 2 - 2 ) ^ 2 + ( x _ 3 + 2 ) ^ 2 & \\ , \\to \\ , \\min \\limits _ { x , u , v } \\\\ 4 - x _ 1 - u & \\ , \\leq \\ , 0 \\\\ 5 - x _ 1 - ( x _ 2 - 2 ) ^ 2 - ( x _ 3 + 2 ) ^ 2 - u & \\ , \\leq \\ , 0 \\\\ x _ 1 ^ 2 + x _ 2 ^ 2 - x _ 3 - v & \\ , \\leq \\ , 0 \\\\ 1 - ( x _ 1 - 1 ) ^ 2 - x _ 2 ^ 2 - x _ 3 - v & \\ , \\leq \\ , 0 \\\\ x _ 2 - v & \\ , \\leq \\ , 0 \\\\ u \\ , \\leq \\ , 0 \\ , \\lor \\ , v & \\ , \\leq \\ , 0 \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} c _ { 3 } = c _ { 1 } c _ { 2 } \\overline { c _ { 0 } } - 3 ^ { - 1 } c _ { 1 } ^ { 3 } \\overline { c _ { 0 } } ^ { 2 } + 3 ^ { - 1 } c _ { 0 } ( \\overline { \\alpha } ^ { 3 } + 3 b _ { 1 } a _ { 3 } + 6 b _ { 2 } a _ { 1 } a _ { 2 } + 3 b _ { 3 } a _ { 1 } ^ { 3 } ) . \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} K : = \\left [ I - \\gamma ^ 2 ( I - Y _ k Z _ k ) ^ { - 1 } T _ k ( I - Z _ k Y _ k ) ^ { - 1 } S _ k \\right ] ^ { - 1 } , \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} ( A + g ) \\circ K { } { } = K { } { } \\circ ( A _ c + r ) & & K { } { } = \\iota + \\begin{pmatrix} k _ c \\\\ k _ h \\end{pmatrix} . \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} F ' ( z ) = \\beta + \\int _ { [ 0 , + \\infty ) } \\frac { 1 + t ^ { 2 } } { ( t - z ) ^ { 2 } } \\ , d \\rho ( t ) \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} \\gamma _ { k } ^ { o } ( D ) \\leq n - | X ' | \\leq n - \\frac { | X ' | + | Y ' | } { 2 } = n - \\frac { | X | + | Y | - n _ { < k } } { 2 } = \\frac { n + n _ { < k } } { 2 } . \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} y _ i ( t _ 0 ) = \\begin{cases} y ^ { ( 0 ) } & ~ i = 1 , \\\\ y ^ { ( l ) } & ~ \\alpha _ { i - 1 } = l \\in \\N , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\sup _ { \\mu , \\sigma _ + } \\{ \\ell _ { n , 1 } ^ * ( \\mu , \\sigma _ + ) - \\ell _ { n , 1 } ^ * ( 0 , 1 ) \\} = O _ p ( 1 ) . \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} Y = \\bigwedge _ i ( U _ i - > V _ i ) \\subseteq X , \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} \\liminf _ { a \\uparrow a ^ * } & ( a ^ * - a ) ^ { - p _ { j _ 0 } / p } \\int _ { \\R ^ d } K ( x _ { j _ 0 } + ( a ^ * - a ) ^ { 1 / p } x ) | v _ a ( { x + ( a ^ * - a ) ^ { - 1 / p } x _ { j _ 0 } } ) | ^ { 2 + 4 / d } \\\\ & \\ge \\int _ { \\R ^ d } \\lambda _ { j _ 0 } | x | ^ { p _ { j _ 0 } } \\Big ( b ^ { d / 2 } Q _ 0 ( b ( x + x _ 0 ) ) \\Big ) ^ { 2 + 4 / d } \\\\ & = \\lambda _ { j _ 0 } b ^ { 2 - p _ { j _ 0 } } \\int _ { \\R ^ d } | x | ^ { p _ { j _ 0 } } | Q _ 0 ( x + b x _ 0 ) | ^ { 2 + 4 / d } \\d x . \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} F _ { 3 } = N ( u ) + L _ { 1 } ( u ) \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} V ' ( s ) = a _ n \\int _ { z _ 1 } ^ { z _ 2 } h _ s h ^ n \\ ; d z = ( n + 1 ) a _ n H ' ( s ) \\int _ { z _ 1 } ^ { z _ 2 } u h ^ n \\ ; d z \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} p l \\xi & = c m ^ { 1 / 2 } n ^ { 1 / 2 } - \\frac { c ^ 2 m } { 2 } + O ( m ^ { 3 / 2 } n ^ { - 1 / 2 } ) \\\\ & = \\xi \\bigg ( 1 - \\frac { c m ^ { 1 / 2 } } { 2 n ^ { 1 / 2 } } + O ( m n ^ { - 1 } ) \\bigg ) \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} P _ { \\Gamma } ( y ) \\xi = \\xi - ( \\xi \\cdot \\nu ( y ) ) \\nu ( y ) \\quad \\mbox { f o r } \\xi \\in \\R ^ n , \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} \\det ( I d - z M _ N ) = \\det ( I d - z M _ { N - 1 } ) , \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} \\mathcal { E } \\emph { ( } \\theta \\emph { ) } = \\int _ { M } P _ { 0 } \\lambda \\cdot \\lambda d \\mu _ { 0 } + \\int _ { M } Q _ { 0 } \\lambda d \\mu _ { 0 } \\theta \\in \\lbrack \\theta _ { 0 } ] . \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} t \\frac { \\partial u } { \\partial t } = u - \\Bigl ( \\frac { \\partial u } { \\partial x } \\Bigr ) ^ 2 \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} \\Delta _ 2 = \\left ( { { e } ^ { q _ 2 } } - { { e } ^ { q _ 1 } } \\right ) \\left ( { { e } ^ { q _ 3 } } - { { e } ^ { q _ 2 } } \\right ) \\left ( { { e } ^ { q _ 2 } } + { { e } ^ { q _ 1 } } \\right ) \\left ( { { e } ^ { q _ 3 } } + { { e } ^ { q _ 2 } } \\right ) . \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} \\rho [ f ] = \\overline { \\lim } _ { n \\to \\infty } \\left \\{ \\ \\frac { n \\ \\ln n } { | \\ln | \\xi _ n | \\ | } \\ \\right \\} , \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} \\frac { 1 + | x | ^ { 2 } + \\rho ^ { 2 } - \\sqrt { - 4 | x | ^ { 2 } \\rho ^ { 2 } + ( 1 + | x | ^ { 2 } + \\rho ^ { 2 } ) ^ { 2 } } } { 2 | x | \\rho } = \\frac { 2 | x | \\rho } { 1 + | x | ^ { 2 } + \\rho ^ { 2 } + \\sqrt { - 4 | x | ^ { 2 } \\rho ^ { 2 } + ( 1 + | x | ^ { 2 } + \\rho ^ { 2 } ) ^ { 2 } } } \\leq 1 \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { \\infty , \\infty } } = \\sup _ { t > 0 } f ^ { * * } ( t ) = \\sup _ { t > 0 } \\frac { 1 } { t } \\int _ { 0 } ^ { t } f ^ { * } ( s ) d s = f ^ * ( 0 ) = \\| f \\| _ { L ^ { \\infty } } . \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} \\varprojlim _ { i \\in I } ( Z _ i , \\varphi _ { i j } ) = \\varprojlim _ { i \\in I } Z _ i \\subset \\prod _ { i \\in I } Z _ i \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} \\sum y _ j \\log z _ j ^ { - 1 } \\ge - Y \\log ( Z / Y ) + \\sum y _ j \\log y _ j ^ { - 1 } = \\sum y _ j \\log ( y _ j Z / Y ) ^ { - 1 } , \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} \\| f _ 0 \\| _ { L ^ \\infty } + \\kappa - C \\delta & \\leq \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } F ( t _ * , x , v ) \\psi ( t _ * , x , v ) d x d v \\\\ & = \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } f _ 0 ( x , v ) \\psi ( 0 , x , v ) d x d v \\\\ [ 3 p t ] & \\leq \\| f _ 0 \\| _ { L ^ \\infty } . \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} A _ m ( x ) & = B _ { m - 1 } ( x ) + x ^ 2 A _ { m - 1 } ( x ) \\\\ & = x A _ { m - 1 } ( x ) + B _ { m - 2 } ( x ) + x ^ 2 A _ { m - 1 } ( x ) \\\\ & = x A _ { m - 1 } ( x ) + A _ { m - 1 } ( x ) - x ^ 2 A _ { m - 2 } ( x ) + x ^ 2 A _ { m - 1 } ( x ) \\\\ & = ( 1 + x + x ^ 2 ) A _ { m - 1 } ( x ) - x ^ 2 A _ { m - 2 } ( x ) . \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} G = \\pm ( \\zeta _ { 6 } + \\zeta _ { 6 } ^ { 5 } + \\zeta _ { 2 3 } + \\cdots + \\zeta _ { 2 3 } ^ { 2 2 } ) . \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} { f ( C ^ p ) \\over f ( C ) ^ p } = { C ^ p g ( C ^ { - p } ) \\over ( C g ( C ^ { - 1 } ) ) ^ p } = { g ( C ^ { - p } ) \\over g ( C ^ { - 1 } ) ^ p } , \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\mathcal { A } _ \\gamma u . \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} \\sigma _ e ^ { n _ 1 r } = \\rho ^ { n _ 1 r } = L _ { \\rho ( e ) } ^ { n _ 1 } = \\sigma _ { \\sigma _ e ( e ) } ^ { n _ 1 } \\sigma _ e ^ { - n _ 1 } , \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} C _ { a , b } ( t ) & = \\left ( t C _ { a , b } ^ { \\frac a b } ( t ) + 1 \\right ) ^ b . \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} a ^ { 2 } b ( a + 1 ) ^ { 2 } ( b + 1 ) u ^ { 4 } + f ( u , v ) = \\lambda _ { 1 1 } { \\Phi ^ { 3 } } . \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} \\overline { \\lim } _ { n \\to \\infty } \\left \\{ \\frac { n \\ \\ln n } { | \\ln | \\eta _ n | \\ | } \\ \\right \\} = \\rho . \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} ( ( f \\circ _ i g ) \\circ _ { i + j - 1 } h ) ( [ r ] ; a _ 1 , \\ldots , a _ { m + n + p - 2 } ) = ( f \\circ _ i ( g \\circ _ j h ) ) ( [ r ] ; a _ 1 , \\ldots , a _ { m + n + p - 2 } ) \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} d i v _ C \\big ( \\nabla _ C u + b _ 0 ( x , u , \\nabla _ C u ) \\big ) + | A _ C | ^ 2 u + b _ 1 ( x , u , \\nabla _ C u ) = 0 \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} \\widetilde { a } _ { \\mu , \\nu } = \\sum _ { \\mu ^ { \\prime } , \\nu ^ { \\prime } } a _ { \\mu ^ { \\prime } , \\nu ^ { \\prime } } R ( \\mu ^ { \\prime } - \\mu , \\nu ^ { \\prime } - \\nu ) . \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} \\Delta H + 2 H ( H ^ 2 - K + k _ 0 ) = 0 , \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} e ^ { - W } \\hat { H } _ l e ^ W = - \\frac { \\partial ^ 2 } { \\partial r _ l ^ 2 } + \\omega ^ 2 r _ l ^ 2 + \\frac { 1 } { r _ l ^ 2 } \\big [ - \\hat { T } _ l + \\frac { 1 } { 4 } ( d _ l - 1 ) ( d _ l - 3 ) \\big ] . \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{gather*} \\mathcal { Q } ^ X = \\big \\{ u + t e _ x : u \\in X _ 1 \\cap B _ { R _ 1 } , \\ , t \\in [ 0 , 1 ] \\big \\} , \\end{gather*}"}
-{"id": "3454.png", "formula": "\\begin{align*} S _ p ( \\beta _ 1 , \\beta _ 2 ) : = \\big \\{ p + i t e ^ { i \\theta } \\colon t > 0 , \\ ; - \\beta _ 2 < \\theta < \\beta _ 1 \\big \\} , \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} \\kappa _ U ( x ) = \\int _ { { \\R ^ d _ + } \\setminus U } J ( x , y ) \\ , d y + \\kappa ( x ) \\ , , x \\in U \\ , , \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} & | \\partial _ { t } \\left ( - 1 6 \\int _ { t } ^ { \\infty } \\lambda '' ( s ) \\left ( K _ { 3 } ( s - t , \\lambda ( t ) ) - K _ { 3 , 0 } ( s - t , \\lambda ( t ) ) \\right ) d s \\right ) | \\\\ & \\leq C \\sup _ { x \\geq t } | e ''' ( x ) | + \\frac { C } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} & l a _ 1 b _ 1 r + l a _ 2 b _ 1 r + l a _ 3 s _ 1 ^ { - 1 } b _ 1 r = 0 , \\\\ & l a _ 1 s _ 1 b _ 2 r + l a _ 2 s _ 1 b _ 2 r + l a _ 3 b _ 2 r = 0 , \\\\ & l a _ 1 \\cdot r + l a _ 2 \\cdot r + l a _ 3 s _ 1 ^ { - 1 } \\cdot r = 0 , \\\\ & l a _ 1 b _ 1 r + l a _ 1 s _ 1 b _ 2 r + l a _ 1 \\cdot r = 0 , \\\\ & l a _ 2 b _ 1 r + l a _ 2 s _ 1 b _ 2 r + l a _ 2 \\cdot r = 0 , \\\\ & l a _ 3 s _ 1 ^ { - 1 } b _ 1 r + l a _ 3 b _ 2 r + l a _ 3 s _ 1 ^ { - 1 } \\cdot r = 0 . \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} | \\partial _ { t } v _ { 1 } ( t , r ) | & \\leq \\frac { C r } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\int _ { 0 } ^ { 1 } B _ { 3 } ( r ( 1 - \\sigma ) ) d \\sigma + \\frac { C r B _ { 3 } ( r ) } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\leq \\frac { C r } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} L ( A , B ) = : \\int _ 0 ^ 1 A \\sharp _ t B d t = A ^ { 1 / 2 } F \\big ( A ^ { - 1 / 2 } B A ^ { - 1 / 2 } \\big ) A ^ { 1 / 2 } , \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} h _ 0 ( \\partial Q ) \\cap ( \\partial B _ \\rho \\cap X ) = h _ 0 ( \\partial Q ) \\cap \\partial B _ \\rho \\cap X = ( \\bar { B } _ R \\cap V ) \\cap \\partial B _ \\rho \\cap X = \\emptyset . \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} \\ddot { u } + \\left ( \\eta - \\frac { 1 } { 2 } \\partial _ t \\log ( \\abs { \\nabla u } ^ 2 ) \\right ) \\dot { u } = \\abs { \\nabla u } \\operatorname { d i v } \\left ( \\frac { \\nabla u } { \\abs { \\nabla u } } \\right ) . \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} \\dim ( \\nu _ { \\beta } ) = \\min \\left \\{ 1 , \\frac { H ( \\beta ) } { \\log ( \\beta ) } \\right \\} . \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} u _ 1 ' = \\left ( u _ 1 \\cos ( t ) + u _ 2 \\sin ( t ) \\right ) \\ , e ^ { \\lambda t } \\ , , u _ 2 ' = \\left ( u _ 2 \\cos ( t ) - u _ 1 \\sin ( t ) \\right ) \\ , e ^ { \\lambda t } \\ , , \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} I ^ \\alpha = \\{ f \\in S ^ \\alpha \\ , \\vert \\ , \\Lambda ( f g ) = 0 \\ \\mbox { f o r a l l } \\ g \\in S ^ { N - \\alpha } \\} ; \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} T _ i ( x ) : = x + \\alpha _ i ^ { \\vee } ( x ) \\cdot \\alpha _ i . \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} \\delta _ 0 = \\eta \\inf _ { t \\ge 0 } \\left [ \\big ( \\omega ( t , \\lambda _ 1 ) + ( \\ell + \\zeta ) \\omega ( \\cdot , \\lambda _ 1 ) * p ( t ) \\big ) ^ { - 1 } \\big ( 1 - ( \\ell + \\zeta ) \\omega ( \\cdot , \\lambda _ 1 ) * p ( t ) \\big ) \\right ] , \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} D _ { 3 } ( t v , v _ \\lambda ) = \\frac { 2 \\lambda ^ { 3 } } { 3 } + \\frac { 3 t ^ { 3 } } { 8 } - t - \\frac { \\left ( \\frac { \\lambda } { 2 } + \\frac { 1 } { 2 } \\right ) ^ { 3 } } { 3 } - \\left ( \\frac { \\lambda } { 2 } + \\frac { 1 } { 2 } \\right ) ^ { 2 } \\left ( - \\frac { \\lambda } { 2 } + \\frac { t } { 2 } - \\frac { 1 } { 2 } \\right ) + \\frac { 2 } { 3 } \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} \\rho ( L _ i ) = \\frac { - h ( z ) ^ { i + 1 } } { h ' ( z ) } \\partial + ( b ( z ) + i \\cdot c ) \\cdot h ( z ) ^ i \\quad \\forall L _ i \\in { \\mathfrak { g } } \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } J _ { 0 } ( r \\xi ) \\left ( \\partial _ { r } v _ { 4 , c } ( t , r ) + \\frac { v _ { 4 , c } ( t , r ) } { r } \\right ) r d r = - \\int _ { 0 } ^ { \\infty } v _ { 4 , c } ( t , r ) J _ { 0 } ' ( r \\xi ) \\xi r d r = \\xi \\int _ { 0 } ^ { \\infty } v _ { 4 , c } ( t , r ) J _ { 1 } ( r \\xi ) r d r \\\\ & = \\xi \\widehat { v _ { 4 , c } } ( t , \\xi ) \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} \\xi _ { ( g , x ) } . y = x ( y \\gamma ) ^ g , \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} A : = \\{ ( a , b ) \\in \\R ^ 2 \\ , | \\ , a > 0 \\ , \\land \\ , b > 0 \\} , B : = \\{ ( a , b ) \\in \\R ^ 2 \\ , | \\ , a < 0 \\ , \\lor \\ , b < 0 \\} , \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} F ( t , x , v ) & = \\bar { f _ 0 } \\big ( X ( t _ 0 ) , V ( t _ 0 ) \\big ) + \\int _ { t _ 0 } ^ t Q ^ \\varepsilon [ F ] \\big ( s , X ( s ) , V ( s ) \\big ) d s \\\\ & = : \\mathcal { T } [ F ] ( t , x , v ) , \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n - 1 } | a _ j p _ j | & \\leq \\sum _ { j = 1 } ^ { n - 1 } | a _ j U _ j | + 2 C \\sum _ { j = 1 } ^ { n - 1 } | a _ j | \\\\ & \\leq N \\sum _ { j = 1 } ^ { n - 1 } | a _ j | + 2 C \\sum _ { j = 1 } ^ { n - 1 } | a _ j | \\\\ & \\leq ( N + 2 C ) ( n - 1 ) \\frac { C } { N ^ 3 } \\\\ & \\leq ( N + 2 C ) \\frac { C } { N ^ 2 } \\\\ \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} h = h _ { \\b + 1 , \\gamma } , \\ P = e _ { \\gamma } e _ { \\gamma } ^ { \\top } , \\ , \\ , R = \\frac { 1 } { A } = G _ { \\b , \\gamma - 1 } \\left ( E + \\i \\eta \\right ) , S = \\frac { 1 } { A + B } = G _ { \\b , \\gamma } \\left ( E + \\i \\eta \\right ) . \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} F ( U ) = F ( \\Sigma _ U ) - F ( \\Sigma ) \\ , . \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} \\zeta _ { A } ^ { * } ( s ) : = \\sum _ { B * A } | A : B | ^ { - s } = \\sum _ { i = 0 } ^ { n } a _ { q ^ { i } } ^ { * } ( A ) q ^ { - i s } , \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} G ( p ) : = \\chi \\Big ( { \\rm d i s t } ^ 2 _ N ( p ) \\Big ) , \\ p \\in \\R ^ L . \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} \\mathcal { C } [ f , \\mathcal { L } ] ( v , w ) = \\gamma D \\left ( \\Delta ( v ^ 2 w ) - 2 v \\Delta v w - v ^ 2 \\Delta w \\right ) = \\gamma D \\left ( 4 v \\nabla v \\cdot \\nabla w + 2 \\nabla v \\cdot \\nabla v w \\right ) . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} \\Re ( \\Omega ) = \\gamma _ 0 . \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} f ^ { \\prime \\infty ( \\ @ B _ \\kappa ) \\delta } ( g ^ { \\prime * } ( B ) ) = g ^ * ( f ^ { \\infty ( \\ @ B _ \\kappa ) _ \\delta } ( B ) ) . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} \\mathcal { A } ^ \\infty _ G ( M , E ) : = \\left ( H ^ \\infty _ L ( G ) \\hat { \\otimes } \\Psi ^ { - \\infty } ( S , E | _ S ) \\right ) ^ { K \\times K } , \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} p ^ { ( 1 / \\alpha ) } ( { \\bf { x } } ) & = \\frac { Z ( \\theta ) ^ { 1 / \\alpha } } { \\int p ( { \\bf { y } } ) ^ { 1 / \\alpha } d { \\bf { y } } } \\big [ h ( { \\bf { x } } ) + w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\big ] ^ { \\frac { 1 } { 1 - \\alpha } } . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} \\overline { D } ^ 2 = \\overline { \\nabla } ^ * \\overline { \\nabla } + \\frac { 1 } { 2 } ( \\rho - e _ 0 \\cdot j ^ \\sharp \\cdot ) , \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} X _ i ^ { ( 0 ) } & : = X _ i , \\\\ X _ i ^ { ( \\alpha + 1 ) } & : = X _ i ^ { ( \\alpha ) \\prime } , \\\\ X _ i ^ { ( \\alpha ) } & : = \\bigwedge _ { \\beta < \\alpha } X _ i ^ { ( \\beta ) } \\quad . \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} - \\rho _ t ( t , x ) = \\beta K _ 4 K _ 1 ( t + \\theta ) ^ { - \\beta - 1 } \\hat V ^ * \\succeq { \\bf 0 } . \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} b e l { e q : c h i } \\chi ( X , \\mathcal { O } _ X ( D ) ) & = 1 + \\frac { D \\cdot ( D - K _ X ) } { 2 } \\\\ & = \\dim H ^ 0 ( X , \\mathcal { O } _ X ( D ) ) - \\dim H ^ 1 ( X , \\mathcal { O } _ X ( D ) ) . \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} \\dot { B } ( t ) & = - [ \\gamma ( t ) , B ( t ) ] + d _ { A ( t ) } j ( t ) \\\\ \\dot { A } ( t ) & = d _ { A ( t ) } \\gamma ( t ) \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} r _ n : = n ^ { - \\nu } , \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} \\{ x \\in \\mathbb { R } : p _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ( x ) > 0 \\} = \\{ x : F _ { \\rho _ { 1 } } ( x ) \\in \\mathbb { H } \\} , \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} \\lambda ^ { \\epsilon _ 1 } _ { i _ 1 - 1 } \\cdots \\lambda ^ { \\epsilon _ s } _ { i _ s - 1 } \\otimes a ^ \\epsilon b ^ { [ t ] } \\mapsto \\left \\{ \\begin{array} { l l } \\lambda ^ { \\epsilon _ 1 } _ { p i _ 1 - 1 } \\cdots \\lambda ^ { \\epsilon _ s } _ { p i _ s - 1 } \\otimes a b ^ { [ p ( t + 1 ) - 1 ] } , & \\epsilon _ 1 = \\cdots = \\epsilon _ s = \\epsilon = 1 , \\\\ 0 , & \\end{array} \\right . \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} C \\left ( \\mu , \\Psi \\right ) & = \\inf _ { L \\in A _ \\phi } c _ \\mu 2 ^ { n _ \\mu } \\phi \\left [ c \\left ( 1 + \\frac { L [ \\phi ^ { - 1 } ] ' ( L ) } { \\phi ^ { - 1 } ( L ) } \\right ) \\right ] \\left ( \\frac { \\phi ^ { - 1 } ( L ) } { L [ \\phi ^ { - 1 } ] ' ( L ) } + 1 \\right ) \\\\ & \\leq \\inf _ { \\phi ( s ) \\in A _ \\phi } c _ \\mu 2 ^ { n _ \\mu } \\phi \\left [ c \\left ( 1 + \\frac { \\phi ( s ) } { s \\phi ' ( s ) } \\right ) \\right ] \\left ( \\frac { s \\phi ' ( s ) } { \\phi ( s ) } + 1 \\right ) , \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{gather*} \\mathcal { L } : = \\frac { d } { d z } \\left ( \\sigma ( z ) \\frac { d } { d z } \\right ) + \\frac { n h _ 0 ^ { n - 2 } } { \\sqrt { 1 + h _ { 0 z } ^ 2 } } , \\\\ \\sigma ( z ) : = \\frac { h _ 0 ^ n } { \\left ( 1 + h _ { 0 z } ^ 2 \\right ) ^ { 3 / 2 } } . \\end{gather*}"}
-{"id": "577.png", "formula": "\\begin{align*} \\lim _ { y \\uparrow + \\infty } \\frac { F ( i y ) } { i y } = 1 . \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} & \\tfrac { c } { 6 } \\left ( x _ { 4 } ^ { 3 } - 3 x _ { 3 } ^ { 2 } x _ { 4 } + \\cos \\theta ( 3 x _ { 1 } ^ { 2 } x _ { 2 } - x _ { 2 } ^ { 3 } ) + \\sin \\theta ( 3 x _ { 2 } ^ { 2 } x _ { 1 } - x _ { 1 } ^ { 3 } ) \\right ) , \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\bar { H } ( \\eta ^ { \\nu } ( t ) ) \\beta ( t ) d t & = \\int _ 0 ^ T \\left ( \\bar { H } ( \\eta ^ { \\nu } ( t ) ) - \\bar { H } _ { } ( \\eta ^ { \\nu } ( t ) ) \\right ) \\beta ( t ) d t \\\\ & + \\int _ 0 ^ T \\bar { H } _ { } ( \\eta ^ { \\nu } ( t ) ) \\beta ( t ) d t . \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} \\lim _ { d \\to \\infty } \\sup _ { 0 \\le t \\le x } | \\ell _ m ^ { ( d ) } ( t ) | = 0 . \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} \\begin{array} { l } \\partial ^ 2 _ t u = \\partial ^ 2 _ x u + g _ 1 ( u ) + f _ 1 ( u ) \\dot { W } ( t , x ) , t \\geq 0 , \\\\ u ( 0 , x ) = u _ 1 ( x ) , \\partial _ t u ( 0 , x ) = v _ 1 ( x ) , \\\\ \\end{array} \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} h ( A ) = \\begin{cases} \\varphi ( A \\cap Y ) \\cup f ( A \\cap L ) , & A \\in C ( p , L \\cup Y ) ; \\\\ g ^ { - 1 } ( f ^ { - 1 } ( A \\cap K ) , f ( A \\cap L ) ) , & A \\in C ( Y , X ) . \\end{cases} \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} | \\frac { L _ { 1 } ( u _ { 1 } ) ( t , R \\lambda ( t ) ) - L _ { 1 } ( u _ { 2 } ) ( t , R \\lambda ( t ) ) } { R } | & \\leq \\frac { C | \\overline { v } _ { 1 } - \\overline { v } _ { 2 } | ( t , R ) } { R ^ { 3 } \\lambda ( t ) ^ { 2 } } \\left ( | v _ { c o r r } ( t , R \\lambda ( t ) ) | ^ { 2 } + \\frac { R | v _ { c o r r } ( t , R \\lambda ( t ) ) | } { ( R ^ { 2 } + 1 ) } \\right ) \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} & \\frac { - 4 b } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\int _ { t + \\log ^ { ( \\alpha - 1 ) b } ( t ) } ^ { t + \\log ^ { ( \\alpha - 1 ) b } ( t ) + \\frac { 1 } { 2 } } \\frac { d x } { ( \\log ^ { ( \\alpha - 1 ) b } ( t ) + x - t ) } | \\\\ & = \\frac { - 4 b } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\left ( \\log ( 2 \\log ^ { ( \\alpha - 1 ) b } ( t ) + \\frac { 1 } { 2 } ) - \\log ( 2 \\log ^ { ( \\alpha - 1 ) b } ( t ) ) \\right ) \\\\ & = \\frac { 4 b ^ { 2 } ( \\alpha - 1 ) \\log ( \\log ( t ) ) } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } + E _ { v _ { 3 , i p } , 1 b } \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} B _ { r } ^ { + } ( x _ { 0 } ) & : = \\{ x \\in \\mathbb { R } _ { + } ^ { n + 1 } : | x - x _ { 0 } | \\le r \\} , \\\\ B _ { r } ' ( x _ { 0 } ) & : = \\{ ( x ' , 0 ) \\in \\mathbb { R } ^ { n } \\times \\{ 0 \\} : | ( x ' , 0 ) - x _ { 0 } | \\le r \\} , \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} \\| u ( \\cdot , t ) \\| & \\le \\left ( \\frac { \\lambda _ 1 } { \\lambda _ 1 - \\| p \\| _ \\infty ( \\ell + \\theta ) } + 1 \\right ) \\| \\xi \\| _ \\infty , \\ ; \\forall t \\ge 0 , \\\\ \\lim _ { t \\to \\infty } \\| u ( \\cdot , t ) \\| & = 0 , \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} \\begin{aligned} & \\max \\left \\{ \\sup _ { [ 0 , T ^ * ] } | | \\rho ^ { k + 1 } ( t ) - \\rho ^ { k } ( t ) | | _ \\infty , \\ , \\sup _ { [ 0 , T ^ * ] } | | \\mathbf { u } ^ { k + 1 } ( t ) - \\mathbf { u } ^ { k } ( t ) | | _ \\infty \\right \\} \\\\ \\leq & \\ \\frac { 1 } { 2 } \\max \\left \\{ \\sup _ { [ 0 , T ^ * ] } | | \\rho ^ { k } ( t ) - \\rho ^ { k - 1 } ( t ) | | _ \\infty , \\ , \\sup _ { [ 0 , T ^ * ] } | | \\mathbf { u } ^ { k } ( t ) - \\mathbf { u } ^ { k - 1 } ( t ) | | _ \\infty \\right \\} , \\end{aligned} \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & \\ , \\to \\ , \\min & & & \\\\ g _ i ( x ) & \\ , \\leq \\ , 0 & \\qquad & i \\in \\mathcal M \\\\ h _ j ( x ) & \\ , = \\ , 0 & \\qquad & j \\in \\mathcal P \\\\ G _ l ( x ) & \\ , \\leq \\ , 0 & \\qquad & l \\in I ^ { 0 + } ( \\bar x ) \\cup I \\\\ H _ l ( x ) & \\ , \\leq \\ , 0 & \\qquad & l \\in I ^ { + 0 } ( \\bar x ) \\cup ( I ^ { 0 0 } ( \\bar x ) \\setminus I ) \\end{aligned} \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} \\psi _ n \\left ( z , ( \\alpha _ 1 \\star w _ 1 + \\alpha _ 2 \\star w _ 2 ) \\right ) = \\alpha _ 1 \\psi _ n ( z , w _ 1 ) + \\alpha _ 2 \\psi _ n ( z , w _ 2 ) , \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} ( N - 2 ) H _ { n - k } = ( N - 2 ) H _ { n - k - 1 } + ( N - 2 ) . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} Q ( C _ { n ( i _ 1 - 1 ) + j _ 1 } , C _ { n ( i _ 2 - 1 ) + j _ 2 } ) \\ & = \\ Q ( B _ { i _ 1 } , B _ { i _ 2 } ) \\cdot N ^ 2 \\ i _ 1 \\not = i _ 2 , \\\\ Q ( C _ { n ( i - 1 ) + j _ 1 } , C _ { n ( i - 1 ) + j _ 2 } ) \\ & = \\ Q ( A _ { j _ 1 } , A _ { j _ 2 } ) \\cdot K \\ j _ 1 \\not = j _ 2 . \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } \\frac { 1 } { w _ 1 ^ { 1 + s p } } \\mathrm { d } w _ 1 \\int _ { 0 } ^ { \\infty } \\frac { y ^ { n - 2 } } { \\left ( 1 + y ^ 2 \\right ) ^ { \\frac { n + s p } { 2 } } } \\mathrm { d } y = \\frac { 1 } { s p } \\int _ { 0 } ^ { \\infty } \\frac { y ^ { n - 2 } } { \\left ( 1 + y ^ 2 \\right ) ^ { \\frac { n + s p } { 2 } } } \\mathrm { d } y . \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} \\mbox { $ \\Gamma _ 1 \\subset K \\cap \\{ z > 0 \\} $ a n d $ \\Gamma _ 2 \\subset K \\cap \\{ z < 0 \\} $ f o r $ K : = \\{ x ^ 2 + y ^ 2 < z ^ 2 \\sinh ^ 2 \\tau \\} $ } , \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} K _ { 1 , 2 } \\le 2 C _ \\alpha L \\phi ' ( | e | ) \\int _ { \\bar { x } } ^ { \\bar { y } } \\frac { d z } { z ^ { \\alpha + 1 } } = \\frac { 2 C _ \\alpha L \\phi ' ( | e | ) } { \\alpha } \\left ( \\frac { 1 } { \\bar { x } ^ \\alpha } - \\frac { 1 } { \\bar { y } ^ \\alpha } \\right ) . \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} & | | - \\frac { ( \\cos ( 2 ( v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } ) ) - 1 ) } { r ^ { 3 } } \\left ( \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } + 2 v _ { 5 } ) - \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\right ) \\vert _ { r = R \\lambda ( t ) } | | _ { L ^ { 2 } ( R d R ) } \\\\ & \\leq C | | \\frac { v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } } { r } | | _ { L ^ { \\infty } } ^ { 2 } | | \\frac { v _ { 5 } } { r } \\vert _ { r = R \\lambda ( t ) } | | _ { L ^ { 2 } ( R d R ) } \\\\ & \\leq \\frac { C \\log ^ { 6 + b } ( t ) } { t ^ { 5 } } \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} v _ { 3 } ^ { \\lambda } ( t , r ) = - r \\int _ { t + 6 r } ^ { \\infty } d s \\lambda '' ( s ) ( s - t ) \\left ( \\frac { 1 } { 1 + ( s - t ) ^ { 2 } } - \\frac { 1 } { \\lambda ( t ) ^ { 2 - 2 \\alpha } + ( s - t ) ^ { 2 } } \\right ) + E _ { 5 } ^ { \\lambda } ( t , r ) \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} z = \\frac { a } { 2 } \\ , x ^ { 2 } + \\frac { b } { 2 } \\ , y ^ { 2 } , a , b \\in \\mathbb { R } , a , b > 0 . \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} \\lim _ { n } \\left ( ( T x _ n ) _ f , y _ n + N _ f \\right ) _ f = 0 . \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} w ' _ { l ' } = { z ' } _ { l ' } \\overline { { z ' } } _ { \\tau ' \\left ( l ' \\right ) } + { \\lambda ' } _ { l ' } \\left ( { z ' } _ { l ' } { z ' } _ { \\sigma ' \\left ( l ' \\right ) } + \\overline { { z ' } } _ { l ' } \\overline { z } _ { \\sigma ' \\left ( l ' \\right ) } \\right ) + \\displaystyle \\sum _ { k \\geq 3 } { \\varphi ' } _ { k } ^ { \\left ( l ' \\right ) } \\left ( z ' , \\overline { z ' } \\right ) , \\quad \\mbox { f o r a l l $ l ' = 1 , \\dots , N ' $ , } \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} p _ 1 = ( 1 , 0 , 0 , 0 , 0 , 0 ) , \\ p _ 2 = ( 0 , 0 , 1 , 0 , 0 , 0 ) , \\ p _ 3 = ( 0 , 0 , 1 , 0 , 0 , 0 ) . \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} \\Phi ^ { \\overline { e } } _ { w v } & = ( \\Phi ^ e _ { v w } ) ^ { - 1 } \\\\ & = | \\Phi ^ e _ { v w } | ^ { - 1 } ( V ^ e _ { v w } ) ^ { - 1 } \\\\ & = \\underbrace { ( V ^ e _ { v w } ) ^ * } _ { } \\underbrace { V ^ e _ { v w } | \\Phi ^ e _ { v w } | ^ { - 1 } ( V ^ e _ { v w } ) ^ * } _ { } \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\left ( \\log | f _ X / f _ Y | \\right ) } { \\partial X \\partial Y } & = \\frac { \\partial } { \\partial X } \\left ( \\frac { 1 + \\lambda } { - 2 \\lambda X + ( 1 + \\lambda ) Y } + \\frac { 2 } { - 2 Y + ( 1 + \\lambda ) X } \\right ) \\\\ & = \\frac { 2 \\lambda ( 1 + \\lambda ) } { ( - 2 \\lambda X + ( 1 + \\lambda ) Y ) ^ 2 } - \\frac { 2 ( 1 + \\lambda ) } { ( - 2 Y + ( 1 + \\lambda ) X ) ^ 2 } . \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} \\frac { 1 } { h ( 1 / r ) } = h _ { * } ( r ) & \\le \\frac { 1 } { \\gamma } r \\exp ( \\sigma _ { * } ( [ 0 , + \\infty ] ) + 2 ) \\\\ & = \\frac { 1 } { \\gamma } r \\exp ( \\sigma ( [ 0 , + \\infty ] ) + 2 ) \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} K ( Q ) = \\{ ( z _ { 0 } , \\ldots , z _ { N } ) : \\vert z _ { 0 } \\vert \\leq Q , \\vert z _ { 1 } \\vert \\leq Q ^ { - 1 / N } , \\ldots , \\vert z _ { N } \\vert \\leq Q ^ { - 1 / N } \\} , \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ 2 } = \\| \\widehat { f } \\| _ { L ^ 2 } \\approx \\| F \\| _ { \\ell ^ 2 } = 1 , \\| g \\| _ { L ^ 2 } = \\| \\widehat { g } \\| _ { L ^ 2 } \\approx \\| G \\| _ { \\ell ^ 2 } = 1 . \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } ( L ^ { * } L v ( t , R ) ) ^ { 2 } R d R & = \\lambda ( t ) ^ { 2 } | | \\omega \\lambda ( t ) ^ { 2 } y ( t ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } ^ { 2 } \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ z ( - 1 ) ^ t \\binom x t = ( - 1 ) ^ z \\binom { x - 1 } { z } , \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} H ( \\rho ) : = \\min _ { z \\in ( 0 , 1 ] } \\log _ { | \\pi | } \\Big ( \\frac { f ( z ) } { z ^ \\rho } \\Big ) = \\min _ { z \\in ( 0 , 1 ] } \\frac { 1 } { m n } \\log _ q \\Big ( \\frac { f ( z ) } { z ^ \\rho } \\Big ) . \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} X = \\sum _ { k = 1 } ^ { D } ( p _ k L _ { D k } + L _ { D k } p _ k ) + x _ D \\bigg ( \\frac { \\eta } { r } - \\sum _ { i = 1 } ^ { D - 1 } \\frac { 2 \\alpha _ i } { x _ i ^ 2 } \\bigg ) , \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} A _ { i j } = N ^ 2 \\left ( - \\delta _ { i , j - 1 } + 2 \\delta _ { i , j } - \\delta _ { i , j + 1 } \\right ) , \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} \\Xi _ n ( \\tau _ i ) = \\widetilde { \\xi } _ { 1 , \\frac { \\Delta _ { i , i + 1 } } { x _ { i + 1 } - x _ i } } + \\widetilde { \\xi } _ { s _ i , 1 + \\frac { \\Delta _ { i , i + 1 } } { x _ { i + 1 } - x _ i } } . \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} F _ { n } ( \\beta ) : = \\frac { 1 } { n } \\log \\left ( Z _ { n } ( \\beta ) \\right ) . \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} z _ { x y } = - \\frac { e ^ { r + \\alpha } } { ( x - y ) ^ 2 } . \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} Z = \\bigsqcup _ { y \\in Y } ( Z \\wedge [ x | - > y ] ) . \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} L ( H , x ) = \\begin{cases} \\{ ( u ^ 1 ( a ^ * ) , u ^ 2 ( a ^ * ) ) \\} , & x = 0 , 1 , \\ldots , c ( a ^ * ) - 1 \\\\ \\{ ( 0 , 0 ) \\} , & x = c ( a ^ * ) , \\ldots , B \\end{cases} \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} x _ { i _ { 1 } , i _ { 2 } } y _ { i _ { 3 } , i _ { 4 } } & = x _ { i _ { 1 } , k } y _ { i _ { 3 } , k } \\\\ x _ { i _ { 1 } , i _ { 2 } } y _ { i _ { 1 } , i _ { 2 } } & = z _ { 1 } ^ { 2 } \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} \\Delta ^ { I } \\boldsymbol { x } = - \\nabla _ { 1 } \\boldsymbol { e _ { 1 } } - \\nabla _ { 2 } \\boldsymbol { e _ { 2 } } - q _ { 2 } \\boldsymbol { e _ { 1 } } + q _ { 1 } \\boldsymbol { e _ { 2 } } . \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} \\begin{cases} H _ n = n , H _ { ( L - m - 1 ) + n } = L - m - 1 + n + \\frac { n ( n + 1 ) ( n + 2 ) } { 6 } , & \\\\ [ 1 . 5 m m ] G _ n = n , G _ { L - m + n } = L - m + n + \\frac { n ( n + 1 ) ( n + 2 ) } { 6 } , & . \\end{cases} \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} ( x _ 1 - 1 ) ^ 2 & \\ , \\to \\ , \\min \\limits _ { x , y , z } \\\\ y , z & \\ , \\leq \\ , 0 \\\\ ( x _ 1 - y ) ( x _ 2 - z ) & \\ , = \\ , 0 \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} f ( \\alpha ^ * ) = & = \\bigg [ \\bigg ( \\frac { p ( 1 - x ) } { x ( 1 - p ) } \\bigg ) ^ { x } \\bigg ( x \\frac { 1 - p } { 1 - x } + 1 - p \\bigg ) \\bigg ] ^ { m t } \\\\ & = \\bigg [ \\bigg ( x \\frac { 1 - p } { 1 - x } + 1 - p \\bigg ) \\bigg ( \\frac { p } { x } \\bigg ) ^ x \\bigg ( \\frac { 1 - p } { 1 - x } \\bigg ) ^ x \\bigg ] ^ { m t } \\\\ & = \\bigg [ \\bigg ( \\frac { p } { x } \\bigg ) ^ x \\bigg ( \\frac { 1 - p } { 1 - x } \\bigg ) ^ { 1 - x } \\bigg ] ^ { m t } \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} \\lambda _ 2 ( s ) > 0 \\ ; \\ ; \\ ; \\& \\ ; \\ ; \\ ; \\begin{cases} H ' ( s ) V ' ( s ) < 0 \\\\ H ' ( s ) V ' ( s ) \\geq 0 \\end{cases} \\ ; \\ ; \\ ; \\Longrightarrow \\ ; \\ ; \\ ; \\ ; \\begin{cases} \\mbox { u n s t a b l e } \\\\ \\mbox { s t a b l e } \\end{cases} . \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} ( \\varphi \\ast \\psi ) ( h ) = \\sum _ { ( h ) } \\varphi ( h _ { ( 1 ) } ) \\psi ( h _ { ( 2 ) } ) \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} C _ i ^ { / T } ( h , u ) : = \\frac { L _ i ( h , u , T ) } { | T | } , ~ \\mbox { f o r } i = 0 , 1 , 2 , \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 g } { \\partial t ^ 2 } - c _ 1 c _ 2 \\frac { \\partial ^ 2 g } { \\partial x ^ 2 } + ( c _ 1 - c _ 2 ) \\frac { \\partial ^ 2 g } { \\partial x \\partial t } = \\frac { 1 } { t } \\Bigl [ ( m + n ) \\frac { \\partial p } { \\partial t } + ( c _ 1 m - c _ 2 n ) \\frac { \\partial g } { \\partial x } \\Bigr ] \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} F _ { J - 2 r , K - 2 r } ( a - r , b - r ) = \\left ( F ^ { ( 1 ) } _ { J , K } ( a , b ) - r , F ^ { ( 2 ) } _ { J , K } ( a , b ) - r \\right ) . \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} \\dot { z } = \\beta \\{ P ^ r _ \\mathcal { M } [ z - \\alpha G _ r ( z ) ] - z \\} . \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} I ^ { 0 + } _ \\textup { C C } ( \\bar x , \\bar y , \\bar z ) & = I ^ { - 0 } ( \\bar x ) \\cup I ^ { - + } ( \\bar x ) \\cup I ^ { 0 + } ( \\bar x ) , \\\\ I ^ { + 0 } _ \\textup { C C } ( \\bar x , \\bar y , \\bar z ) & = I ^ { 0 - } ( \\bar x ) \\cup I ^ { + - } ( \\bar x ) \\cup I ^ { + 0 } ( \\bar x ) , \\\\ I ^ { 0 0 } _ \\textup { C C } ( \\bar x , \\bar y , \\bar z ) & = I ^ { - - } ( \\bar x ) \\cup I ^ { 0 0 } ( \\bar x ) . \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\mathcal { F } ( f ) ( \\xi ) = \\int _ { 0 } ^ { \\infty } \\phi ( r , \\xi ) f ( r ) d r \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} \\phi ( Q ( j ) , Q ( i ) ) & = ( l _ j + \\Delta ( Q ( j ) , j ) ) - 2 ( l _ { i + 1 } - \\Delta ( i + 1 , Q ( i ) + 1 ) ) - \\Delta ( Q ( j ) , Q ( i ) + 1 ) \\\\ & \\quad - \\Delta ( Q ( i ) + 1 - \\alpha ( Q ( j ) , Q ( i ) ) , Q ( i ) + 1 ) \\\\ & = l _ j - 2 l _ { i + 1 } - [ \\Delta ( j , Q ( j ) ) + \\Delta ( Q ( i ) + 1 , i + 1 ) + \\Delta ( Q ( j ) , Q ( i ) + 1 ) ] \\\\ & \\quad - [ \\Delta ( Q ( i ) + 1 , i + 1 ) + \\Delta ( Q ( i ) + 1 - \\alpha ( Q ( j ) , Q ( i ) ) , Q ( i ) + 1 ) ] \\\\ & = l _ j - 2 l _ { i + 1 } - \\Delta ( j , i + 1 ) - \\Delta ( Q ( i ) + 1 - \\alpha ( Q ( j ) , Q ( i ) ) , i + 1 ) \\ , \\cdot \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} R \\ = \\ ( R | J ' ) \\rhd ( R | J ) . \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} w ( t , x ) = & \\psi ( \\phi ( T ; t , x ) ) \\exp \\Bigl [ - \\int _ { t } ^ { T } \\frac { \\lambda ( s , \\phi ( s ; t , x ) ) } { s } d s \\Bigr ] \\\\ & - \\int _ t ^ T \\exp \\Bigl [ - \\int _ t ^ { \\tau } \\frac { \\lambda ( s , \\phi ( s ; t , x ) ) } { s } d s \\Bigr ] \\frac { g ( \\tau , \\phi ( \\tau ; t , x ) ) } { \\tau } d \\tau . \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} & c _ 1 ( Z ) ^ 2 = 2 1 6 c _ 2 ( Z ) = 3 3 6 \\quad \\\\ & \\kappa = 3 2 8 \\delta = 3 9 2 \\lambda = 6 0 , \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} \\Phi ( u ) ( \\cdot , t ) = S ( t ) \\xi ( \\cdot , 0 ) + \\int _ 0 ^ t S ( t - s ) f ( s , u [ \\xi ] _ \\rho ( \\cdot , s ) ) d s , \\ ; u \\in \\mathcal { B C } _ 0 ^ \\xi . \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} X = X _ { N , m } : = \\left \\{ x \\in \\mathbb { R } ^ { N } : \\ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ i = m \\right \\} \\subset \\mathbb { R } ^ { N } . \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} q _ 1 = a ^ 2 - a ' , q _ 2 = a ^ 2 + a ' \\ , . \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} \\inf _ { s \\in [ 0 , 1 ] } G _ s \\big ( f , g ; \\lambda \\big ) = f ! _ \\lambda g \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\sup _ { s \\in [ 0 , 1 ] } G _ s \\big ( f , g ; \\lambda \\big ) = f \\sharp _ { \\lambda } g , \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} \\deg ( H , ( Q ) , \\rho e ) = \\deg ( H _ 0 , ( Q ) , \\rho e ) , \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} | I | \\leq C _ 1 + \\tilde C _ 1 + C _ 2 + C _ 3 : = C < \\infty \\mbox { f o r a l l } M > 0 . \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} A _ 4 ^ 4 & = - i b _ 3 \\left ( x _ 1 + x _ 2 + x _ 3 + x _ 4 \\right ) - i b _ 2 ^ 2 \\left ( \\frac { ( x _ 1 + x _ 2 ) ( x _ 3 + x _ 4 ) } { x _ { 1 + 2 } } + \\right ) . \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} \\mathcal H _ { \\lambda } ( v ) = - \\frac { \\lambda _ { k - 1 } ( v ) + 1 + \\lambda } { \\lambda _ k ( v ) + \\lambda } \\chi _ k ( \\lambda , v ) \\ . \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} | \\partial _ { 3 } F _ { 3 } ( r , \\rho , z ) | & \\leq \\frac { C r ^ { 2 } \\lambda _ { 0 , 0 } ( s ) ^ { 2 \\alpha - 3 } } { ( 1 + ( - \\rho ^ { 2 } + r ^ { 2 } ) \\lambda _ { 0 , 0 } ( s ) ^ { 2 \\alpha - 2 } ) ^ { 2 } + 4 \\rho ^ { 2 } \\lambda _ { 0 , 0 } ( s ) ^ { 2 \\alpha - 2 } } \\\\ & \\leq \\frac { C r ^ { 2 } \\lambda _ { 0 , 0 } ( s ) ^ { 2 \\alpha - 3 } } { 1 + 2 ( \\rho ^ { 2 } + r ^ { 2 } ) \\lambda _ { 0 , 0 } ( t ) ^ { 2 \\alpha - 2 } + ( \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } \\lambda _ { 0 , 0 } ( t ) ^ { 4 \\alpha - 4 } } \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} \\mathcal { A } ^ y _ { h _ L , \\ , F } ( f _ 0 ) - \\mathcal { A } ^ y _ { h _ L , \\ , F } ( \\tilde { f } _ 0 ) & = \\varphi _ { h _ L } ( f _ 0 , \\ , \\tilde { f } _ 0 ) ( f _ 0 ^ x - \\tilde { f } _ 0 ^ x ) + \\psi _ { h _ L } ( f _ 0 , \\ , \\tilde { f } _ 0 ) ( f _ 0 ^ y - \\tilde { f } _ 0 ^ y ) \\\\ & + \\theta ( f _ 0 , \\ , \\tilde { f } _ 0 ) ( f _ 0 - \\tilde { f } _ 0 ) \\cdot h _ L , \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} & { \\mathcal L } ( \\psi _ 0 ) = h _ 0 , \\\\ [ 1 m m ] & { \\mathcal L } ( \\psi _ j ^ \\ell ) = h _ j ^ \\ell , ~ ~ ~ j \\in \\mathbb { N } ^ + , \\ , \\ell = 1 , 2 . \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\dot { x } & = y + \\frac { x ^ 2 - y ^ 2 } { 2 } , \\\\ \\dot { y } & = - x , \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} H ' ( s ) \\int _ { z _ 1 } ^ { z _ 2 } e h ^ n \\ ; d z = 0 . \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} ( X \\xrightarrow { 1 _ X \\otimes \\omega } X \\otimes \\overline { X } \\otimes X \\xrightarrow { \\overline { \\omega } ^ { \\ , * } \\otimes 1 _ X } X ) = 1 _ X , ( \\overline { X } \\xrightarrow { \\omega \\otimes 1 _ { \\overline { X } } } \\overline { X } \\otimes X \\otimes \\overline { X } \\xrightarrow { 1 _ { \\overline { X } } \\otimes \\overline { \\omega } ^ { \\ , * } } \\overline { X } ) = 1 _ { \\overline { X } } . \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} \\ss _ 2 [ 1 ] = \\beta _ 0 ^ 2 + \\beta _ 1 ^ 2 + ( \\eta ^ 2 - 1 ) = \\tt [ 2 ] + \\tt [ 3 ] / 2 - \\beta _ 1 ^ 2 ( \\sigma ^ 2 _ + - 1 ) = \\tt [ 2 ] + \\tt [ 3 ] / 2 + O _ p ( \\| \\tt \\| ^ 2 ) . \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} S _ { i j } = & R _ { i j } - h R _ { i \\gamma } R _ { \\gamma j } + h ^ 2 R _ { i \\gamma } R _ { \\gamma \\gamma } R _ { \\gamma j } \\dots ( - 1 ) ^ k h ^ k R _ { i \\gamma } R ^ { k - 1 } _ { \\gamma \\gamma } S _ { \\gamma j } \\\\ = & R _ { i j } - h R _ { i \\gamma } R _ { \\gamma j } \\left ( \\sum _ { l = 0 } ^ k \\left ( - h R _ { \\gamma \\gamma } \\right ) ^ l \\right ) + ( - 1 ) ^ k h ^ k R _ { i \\gamma } R ^ { k - 1 } _ { \\gamma \\gamma } S _ { \\gamma j } \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{align*} \\dot { z } ( t ) = L ^ { - 1 } A L ^ { - \\top } z ( t ) + L ^ { - 1 } B u ( t ) \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} ( x _ 1 - 1 ) ^ 2 & \\ , \\to \\ , \\min \\\\ x _ 1 \\ , \\leq \\ , 0 \\ , \\lor \\ , x _ 2 & \\ , \\leq \\ , 0 . \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} \\omega = \\sum _ { j = 0 } ^ n ( x _ { j + 1 } x _ j - x _ { j - 1 } ^ 2 ) d x _ j \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} H _ { T _ u w } = T _ w H _ { \\Pi u } + H _ w T _ u - \\langle \\Pi u | \\ , \\cdot \\rangle w \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} ( \\delta ^ { 2 * } _ 2 ( \\epsilon ) \\circ \\delta ^ { 2 * } _ 0 ( \\epsilon ) ) ( m \\otimes 1 ) & = \\sum _ { I , J } \\nabla ( \\xi _ J ) ( \\nabla ( \\xi _ I ) ( m ) ) \\otimes \\delta ^ 2 _ 2 ( \\Gamma _ J ) \\cdot \\delta ^ 2 _ 0 ( \\Gamma _ I ) \\\\ \\delta ^ { 2 * } _ 1 ( \\epsilon ) ( m \\otimes 1 ) & = \\sum _ K \\nabla ( \\xi _ K ) ( m ) \\otimes \\delta ^ 2 _ 1 ( \\Gamma _ K ) . \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} B ( l ) : = \\{ ( l - 1 ) K + 1 , \\cdots , l K \\} l \\in \\{ 1 , \\cdots , M \\} . \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} D _ p ( \\Pi _ N x _ n , x _ n ) = D _ p ( z , x _ n ) - D _ p ( z , x _ { n + 1 } ) \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} i v _ 4 & = 6 i a _ 2 \\left ( X _ 1 + X _ 2 + X _ 3 + X _ 4 \\right ) + 4 i a _ 1 ^ 2 \\left ( X _ { 1 + 2 } + X _ { 1 + 3 } + X _ { 2 + 3 } \\right ) . \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} H ^ { d - 1 } _ { \\mathfrak { a } } ( M / a M ) \\rightarrow H ^ { d } _ { \\mathfrak { a } } ( M ) \\xrightarrow { a } H ^ { d } _ { \\mathfrak { a } } ( M ) \\rightarrow H ^ { d } _ { \\mathfrak { a } } ( M / a M ) = 0 , \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} \\mathcal O _ { X , p } = \\mathcal O _ { Y _ 0 , p } \\overset { \\sigma _ 1 } { \\rightarrow } \\mathcal O _ { Y _ 1 , p } = \\mathcal O _ { Y _ 0 , p } / ( a _ 1 ) \\overset { \\sigma _ 2 } { \\rightarrow } \\cdots \\overset { \\sigma _ { d - 1 } } { \\rightarrow } \\mathcal O _ { Y _ { d - 1 } , p } = \\mathcal O _ { Y _ { d - 2 } , p } / ( a _ { d - 1 } ) . \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} \\gamma ( \\overrightarrow { s _ 1 } , \\cdots , \\overrightarrow { s _ l } , \\overrightarrow { t _ 1 } , \\cdots , \\overrightarrow { t _ { l } } ) = \\bigcup \\limits _ { i = 1 } ^ l \\gamma _ { C _ i } ( \\overrightarrow { s _ i } , \\overrightarrow { t _ i } ) . \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} \\partial _ { s } ^ 2 \\mathcal { N } ( t , s , \\xi , v , \\overline v ) = e ^ { ( t - s ) \\mathcal { L } } \\mathcal { C } ^ 2 \\left [ f , \\mathcal { L } \\right ] \\left ( e ^ { s \\mathcal { L } } v , e ^ { s \\mathcal { L } } e ^ { \\xi \\mathcal { A } } \\overline v \\right ) \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} F _ I = \\frac { \\partial F } { \\partial X ^ I } . \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} V _ i = U _ { x _ i } \\setminus \\bigcup _ { j < i } ( U _ { x _ j } \\cap U _ { x _ i } ) . \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} u _ { \\sigma } ( z ) = \\int _ { \\mathbb { T } } \\frac { t ( 1 + z ) + z } { t ( 1 + z ) - z } \\ , d \\sigma ( t ) , \\quad \\Re z > - \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} N _ 1 \\leq C b _ q 2 ^ { - 2 q s } \\| \\nabla u \\| _ { L ^ \\infty } \\| \\omega \\| _ { H ^ s } ^ 2 . \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} \\Psi ^ { - \\infty , \\epsilon } _ { G , c } ( M ) = \\{ T _ { \\widetilde { k } } , \\ ; \\widetilde { k } \\in ( C ^ \\infty _ c ( G ) \\hat { \\otimes } \\ , \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } \\} \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ d \\setminus [ - a , a ] ^ d } | \\rho ( t ) | \\ , d t = O \\left ( 1 / \\log a \\right ) , a \\to + \\infty , \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} \\alpha _ i A u ( \\mathcal { O } _ i ) = \\sum _ { j : I _ j \\sim \\mathcal { O } _ i } \\pm \\gamma _ { i j } D _ j u ( \\mathcal { O } _ i ) . \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} \\Xi ( x , y ) \\ ! = \\ ! \\left ( \\ ! \\frac { B ^ r x } { \\mu _ r } \\ ! \\right ) ^ { y } \\ ! \\ ! { \\rm G } _ { 3 , 3 } ^ { 3 , 3 } \\ ! \\ ! \\left [ \\ ! { \\frac { \\kappa m { \\cal C } } { \\kappa _ I m _ I \\bar { \\gamma } } } \\ ! \\ ! \\left | \\begin{array} { c c c c } \\ ! \\ ! \\ ! 1 \\ ! - \\kappa _ I \\ ! , 1 \\ ! - \\ ! L m _ I \\ ! , 1 + y \\ ! \\\\ \\kappa , N m , 0 \\end{array} \\ ! \\ ! \\ ! \\ ! \\right . \\right ] , \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} \\mbox { a d } : \\mathfrak g \\wedge \\mathfrak g \\rightarrow \\mathfrak { g l } ( \\mathfrak g ) , ~ ~ ~ ~ \\mbox { a d } _ { x , y } ( z ) = [ x , y , z ] . ~ \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} \\langle \\tilde { L } G _ D f , \\varphi \\rangle & = \\langle G _ D f , L \\varphi \\rangle = \\lim _ { n \\to \\infty } \\langle G _ { D _ n } f , L \\varphi \\rangle \\\\ & = \\lim _ { n \\to \\infty } - \\langle f \\mathbf { 1 } _ { D _ n } , \\varphi \\rangle = - \\langle f , \\varphi \\rangle . \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} d _ X G _ 0 ( Y , Z ) = G _ 0 ( \\nabla ^ { \\mathcal { H } ^ 0 } _ X Y , Z ) + G _ 0 ( Y , \\nabla ^ { \\mathcal { H } ^ 0 } _ X Z ) . \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} 2 ^ { \\delta } h _ { K _ n } ( \\overline { X } _ n ) = h _ { L _ n } ( \\overline X _ n ) \\leq 4 p ^ e ( 2 g _ { L _ n } - 2 + | V | ) , \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} \\mathcal H ( u ) = - \\frac 1 3 s _ 1 ^ 3 + \\sum _ { n = 1 } ^ \\infty \\gamma _ n \\big ( \\big ( n - s _ n \\big ) ^ 2 + \\gamma _ n \\big ( n - s _ n \\big ) + \\frac 1 3 \\gamma _ n ^ 2 \\big ) \\ , . \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} S ^ { ( s ) } _ n & = - i \\lambda _ s \\sum _ { k = s } ^ n B _ { k , s } ( 1 , 2 ! a _ 1 , 3 ! a _ 2 , 4 ! a _ 3 , \\ldots ) B _ { n , k } \\left ( b _ 1 , b _ 2 , \\ldots \\right ) . \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} \\tilde { \\mathcal { G } } _ 2 & = \\ker ( 1 + \\tilde { A } ^ \\star ) = \\bigoplus _ { e \\in E } \\ker ( 1 - [ \\partial P _ 1 ] ) \\subseteq \\bigoplus _ { e \\in E } \\operatorname { L } ^ 2 ( 0 , 1 ) \\\\ \\tilde { \\mathcal { G } } _ 1 & = \\ker ( 1 - \\tilde { A } ^ \\star ) = \\bigoplus _ { e \\in E } \\ker ( 1 + [ \\partial P _ 1 ] ) \\subseteq \\bigoplus _ { e \\in E } \\operatorname { L } ^ 2 ( 0 , 1 ) \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 1 & 1 \\\\ 0 & 0 & 1 & 0 & 0 & 1 \\\\ 1 & 0 & 0 & 1 & 0 & 0 \\\\ 1 & 1 & 0 & - 1 & - 1 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & - 1 \\end{pmatrix} \\begin{pmatrix} a _ { 3 } \\\\ b _ { 2 } \\\\ b _ { 3 } \\\\ a _ { 3 } ' \\\\ b _ { 2 } ' \\\\ b _ { 3 } ' \\end{pmatrix} = \\begin{pmatrix} 1 / 2 - a _ { 1 } - a _ { 2 } - b _ { 1 } \\\\ 1 / 2 - a _ { 1 } ' - a _ { 2 } ' - b _ { 1 } ' \\\\ - a _ { 2 } - a _ { 2 } ' \\\\ - b _ { 1 } - b _ { 1 } ' \\\\ 0 \\\\ - a _ { 1 } + a _ { 1 } ' \\end{pmatrix} . \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} | N ( u ) ( t , R \\lambda ( t ) ) | & \\leq C \\left ( | | v ( t ) | | ^ { 2 } _ { \\dot { H } ^ { 1 } _ { e } } + | | L ^ { * } L v ( t ) | | _ { L ^ { 2 } ( R d R ) } ^ { 2 } \\right ) \\left ( \\log ^ { 2 } ( \\frac { 1 } { R } ) + 1 \\right ) \\cdot \\frac { | v ( t , R ) | } { \\lambda ( t ) ^ { 2 } } \\\\ & + C | | v ( t ) | | _ { \\dot { H } ^ { 1 } _ { e } } \\frac { | v ( t , R ) | } { R \\lambda ( t ) ^ { 2 } } \\left ( 1 + \\frac { | v _ { c o r r } ( t , R \\lambda ( t ) ) | } { R } \\right ) \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} \\sin \\phi _ 1 = \\frac { 2 ( p + q ) ( 1 - p q ) } { ( 1 + q ^ 2 ) ( 1 + p ^ 2 ) } \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} \\int _ { S ^ n } | X _ * ( x ) | _ { g _ b } d x & = \\int _ { S ^ n } \\frac { | X _ * ( x ) | _ { g ^ n } } { d ( f _ * ( x ) ) } d x \\\\ & = \\int _ { S ^ n } \\frac { \\sqrt { | X ( x _ 1 ) | ^ 2 _ g + \\ldots + | X ( x _ n ) | ^ 2 _ g } } { d ( f _ * ( x ) ) } d x \\\\ & \\leq \\int _ { S ^ n } \\frac { | X ( x _ 1 ) | _ g + \\ldots + | X ( x _ n ) | _ g } { d ( f _ * ( x ) ) } d x \\\\ & = n \\int _ { S ^ n } \\frac { | X ( x _ 1 ) | _ g } { d ( f _ * ( x ) ) } d x . \\\\ \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} S [ X , A ] = \\frac { 1 } { 2 } \\int _ { \\Sigma } G _ { \\mu \\nu } D X ^ \\mu \\wedge \\star D X ^ \\nu + \\int _ { \\Sigma } A _ \\mu \\wedge \\d X ^ \\mu + \\frac { 1 } { 2 } \\pi ^ { \\mu \\nu } A _ \\mu \\wedge A _ \\nu , \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} A _ n ^ 1 = - i C _ n \\cdot \\left ( X _ 1 + \\ldots + X _ n \\right ) . \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} P _ { F \\Sigma K } = P _ { \\cup ^ k _ { i = 1 } S _ { i } } = \\sum ^ k _ { i = 1 } ( - 1 ) ^ { i + 1 } \\sum _ { 1 \\leq j _ 1 < \\cdots < j _ i \\leq k } P _ { S _ { j _ 1 } \\cap \\cdots \\cap S _ { j _ i } } \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\nabla ^ 2 u + u g = 0 , & \\\\ \\frac { \\partial u } { \\partial \\nu } - \\cot \\theta u = 0 , & . \\end{array} \\right . \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} B _ { n , \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } \\mid { \\boldsymbol { \\omega } } ^ { - } [ j ] ) = & - B _ { n , \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } + \\omega _ { j } \\mathbf { 1 } \\mid { \\boldsymbol { \\omega } } ) . \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} q = \\frac { p ^ * } { 1 ^ * } , q ' = \\frac { p ( \\nu - 1 ) } { \\nu ( p - 1 ) } \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} \\begin{array} { r c l } Z _ 0 . d \\phi _ t + Z _ { t h } . d \\phi _ 1 & = & ( 1 - t ) Z _ 0 . d \\phi _ 0 + t [ Z _ 0 . d \\phi _ 1 + Z _ h . d \\phi _ 1 ] \\\\ & \\geq & ( 1 - t ) \\delta ( | Z _ 0 | ^ 2 + | d \\phi _ 0 | ^ 2 ) + t \\delta ' ( | Z _ 0 + Z _ h | ^ 2 + | d \\phi _ 1 | ^ 2 ) \\\\ & \\geq & m i n ( \\delta , \\delta ' ) [ ( 1 - t ) | Z _ 0 | ^ 2 + t | Z _ 0 + Z _ h | ^ 2 + ( 1 - t ) | d \\phi _ 0 | ^ 2 + t | d \\phi _ 1 | ^ 2 ] \\end{array} \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} G ( n _ 1 , \\ldots , n _ r ) = \\sum _ { d _ 1 + \\cdots + d _ r = d } \\frac { 1 } { d _ 1 ! \\cdots d _ r ! } e _ R ( \\mathcal I ( 1 ) ^ { [ d _ 1 ] } , \\ldots , \\mathcal I ( r ) ^ { [ d _ r ] } ; R ) n _ 1 ^ { d _ 1 } \\cdots n _ r ^ { d _ r } . \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} J _ { \\frac { d - 2 } { 2 } } ( x ) = \\frac { \\left ( \\frac { x } { 2 } \\right ) ^ { \\frac { d - 2 } { 2 } } } { \\pi ^ { \\frac { d - 1 } { 2 } } } \\left ( e ^ { - i x } \\Phi _ { \\frac { d - 2 } { 2 } } ( x ) \\right ) , x > 0 \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} & \\sup _ { t > 0 } \\frac { e ^ { - n t } } { \\pi ^ { n / 2 } ( 1 - e ^ { - 2 t } ) ^ { n / 2 } } \\exp \\Big ( - \\frac { | x - e ^ { - t } y | ^ 2 } { 1 - e ^ { - 2 t } } \\Big ) \\\\ & = \\sup _ { s \\in ( 0 , 1 ) } \\frac { ( 1 - s ) ^ n } { \\pi ^ { n / 2 } ( 4 s ) ^ { n / 2 } } \\exp { \\left ( - \\frac { | x ( 1 - s ) + y ( 1 + s ) | ^ 2 } { 4 s } \\right ) } e ^ { - | x | ^ 2 + | y | ^ 2 } \\\\ & \\lesssim e ^ { - | x | ^ 2 + | y | ^ 2 } \\Big [ ( 1 + | x | ) ^ n \\land ( | x | \\sin \\theta ( x , y ) ) ^ { - n } \\Big ] , \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} \\Phi _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } ) = \\sum _ { \\mathbf { n } \\subset \\mathbf { m } } \\frac { 1 } { | \\mathbf { m } | - | \\mathbf { n } | + 1 } \\binom { \\mathbf { m } } { \\mathbf { n } } ^ { ( d ) } B _ { \\mathbf { n } } ^ { ( d ) } ( \\mathbf { z } ) . \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} W \\begin{bmatrix} u _ 1 \\\\ u _ 2 \\end{bmatrix} = \\begin{bmatrix} D & - C \\\\ - B & A \\end{bmatrix} \\begin{bmatrix} u _ 1 \\\\ u _ 2 \\end{bmatrix} = \\begin{bmatrix} v _ 1 \\\\ v _ 2 \\end{bmatrix} , \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} 2 ( u _ 1 ^ 2 + u _ 2 ^ 2 ) = u _ 3 ^ 2 + u _ 4 ^ 2 \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} I = \\left ( i _ { 1 } , \\dots , i _ { k } , \\dots , i _ { N } \\right ) \\in \\mathbb { N } ^ { N } \\quad \\mbox { s u c h t h a t $ i _ { 1 } + \\dots + i _ { k } + \\dots + i _ { N } = p \\geq 3 , $ } \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} I _ 4 ( p ^ 0 , 0 , q _ 0 , 0 ) = - ( p ^ 0 q _ 0 ) ^ 2 , \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} \\begin{aligned} & \\rho _ t + \\nabla \\cdot ( \\rho \\mathbf { u } ) = 0 , \\\\ & \\mathbf { u } _ t + \\left ( ( \\mathbf { u } - \\rho \\boldsymbol \\gamma ) \\cdot \\nabla \\right ) \\mathbf { u } = 0 , \\end{aligned} \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} { \\psi } ^ { \\pm } = e ^ { i \\theta } \\big [ { \\tilde \\psi _ 1 } ^ { \\pm } \\sin { \\theta } + i { \\tilde \\psi _ 2 } ^ { \\pm } \\cos { \\theta } \\big ] . \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} \\sigma ( \\pi ) = ( s _ 1 , s _ 2 , \\ldots , s _ n ) , \\quad \\mbox { w h e r e } s _ i : = | \\{ j : j < i \\ , \\mbox { a n d } \\ , \\pi _ j > \\pi _ i \\} | . \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} \\zeta ' _ B ( 0 ; a , 1 , 1 ) = \\frac 1 { 1 2 } \\left ( a + \\frac 1 a \\right ) ( \\gamma - \\log a ) - \\frac 1 4 \\log a + \\frac 5 { 2 4 } a - \\frac 1 4 \\log ( 2 \\pi ) + J ( a ) , \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} ( \\psi _ \\varepsilon ) _ t ( \\hat { x } , \\hat { t } ) & \\le J [ u ^ * , p _ \\varepsilon ] ( \\hat { x } , \\hat { t } ) + K _ { ( 0 , \\hat { x } - \\delta ) } [ \\psi _ \\varepsilon , p _ \\varepsilon ] ( \\hat { x } , \\hat { t } ) \\\\ & \\quad + K _ { ( \\hat { x } - \\delta , \\hat { x } ) } [ u ^ * , p _ \\varepsilon ] ( \\hat { x } , \\hat { t } ) + f ( \\hat { x } , \\hat { t } ) \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} F ( 1 ) = 0 . \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} \\| \\lambda _ q \\| _ { M ( G ) } = \\| \\lambda \\| _ { M ( G / H ) } . \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} \\Phi ( \\eta _ { \\mu } ( z ) ) = z \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k \\lambda _ i ( A ) \\le \\prod _ { i = 1 } ^ k \\lambda _ i ( B ) , \\qquad 1 \\le k \\le n , \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} \\tilde { \\tau } ^ { 2 - 2 s } e ^ { 2 s t } | \\varphi '' | = \\tau ^ { 2 - 2 s } e ^ { 2 t } | \\varphi '' | \\le \\tau ^ { 2 - 2 s } | \\varphi ' | ^ { 2 } | \\varphi '' | . \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} \\frac { 1 } { k + h ^ \\vee } + \\frac { 1 } { k ' + h ^ \\vee } = n . \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} \\Sigma = \\prod _ { c \\in G / H } \\Sigma _ c . \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k + 1 \\} = \\Bigl ( \\frac { c _ 2 } { c _ 1 + c _ 2 } \\Bigr ) ^ { k + 1 } \\sum _ { j = 0 } ^ k A _ j ^ { ( k ) } \\Bigl ( \\frac { c _ 1 } { c _ 1 + c _ 2 } \\Bigr ) ^ { j } \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} [ \\delta _ { \\varepsilon _ 1 } , \\delta _ { \\varepsilon _ 2 } ] X ^ \\mu = \\delta _ { \\varepsilon _ 3 } X ^ \\mu , [ \\delta _ { \\varepsilon _ 1 } , \\delta _ { \\varepsilon _ 2 } ] A ^ a = \\delta _ { \\varepsilon _ 3 } A ^ a , \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} d \\mu ( t ) = : \\frac { d t } { t ( 1 - t ) \\Big ( \\pi ^ 2 + \\big ( \\log \\frac { t } { 1 - t } \\big ) ^ 2 \\Big ) } . \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} & \\widetilde { M } _ L \\cdot M _ L ^ { ( 1 ) } = M _ L ^ { ( 1 ) } ( v , w ) , \\ M _ R ^ { ( 1 ) } \\cdot \\widetilde { M } _ R = M _ R ^ { ( 1 ) } ( v , w ) , \\\\ & \\widetilde { M } _ L \\cdot M _ L ^ { ( 2 ) } = M _ L ^ { ( 2 ) } ( v , w ) , \\ M _ R ^ { ( 2 ) } \\cdot \\widetilde { M } _ R = M _ R ^ { ( 2 ) } ( v , w ) \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} \\| P _ n y _ \\ell - x \\| _ { X _ { \\mathcal { H } } } ^ 2 = \\langle P _ n y _ \\ell - x , P _ n y _ \\ell - x \\rangle _ { X _ { \\mathcal { H } } } = \\langle P _ n y _ \\ell - x , \\mathcal { H } P _ n y _ \\ell - \\mathcal { H } x \\rangle \\to 0 \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} \\AA _ { m , n } [ \\mathcal { M } _ 1 \\times \\mathcal { M } _ 2 ] ( s ) = \\sum _ { n _ 1 + n _ 2 = n } \\AA _ { m _ 1 , n _ 1 } [ \\mathcal { M } _ 1 ] ( s ) \\cdot \\AA _ { m _ 2 , n _ 2 } [ \\mathcal { M } _ 2 ] ( s ) \\ , . \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} \\ - X _ i ^ { ( 0 ) } & : = X _ i , \\\\ \\ - X _ i ^ { ( \\alpha + 1 ) } & : = \\bigwedge _ { j \\succ i } \\ - f _ { j i } ( \\ - X _ j ^ { ( \\alpha ) } ) , \\\\ \\ - X _ i ^ { ( \\alpha ) } & : = \\bigwedge _ { \\beta < \\alpha } \\ - X _ i ^ { ( \\beta ) } \\quad . \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} \\sup _ { \\frac { 1 } { 8 ( 1 + | x | ) ^ 2 } < s < 1 } K _ { 2 3 } ( t , x , y ) | _ { t = \\log \\left ( \\frac { 1 + s } { 1 - s } \\right ) } & \\lesssim \\sup _ { \\frac { 1 } { 8 ( 1 + | x | ) ^ 2 } < s < 1 } \\frac { | x | } { s ^ { ( n - 1 ) / 2 } } e ^ { | y | ^ 2 - | x | ^ 2 } \\\\ & \\lesssim ( 1 + | x | ) ^ n e ^ { | y | ^ 2 - | x | ^ 2 } , ( x , y ) \\in N ^ c . \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} \\Phi _ \\varepsilon ( r ) : = 2 ( r + \\varepsilon ) ^ { 1 / 2 } - 2 \\varepsilon ^ { 1 / 2 } \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} - L _ { \\Sigma } u _ 0 = \\lambda _ 0 \\alpha \\cdot u _ 0 \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} W _ { r , c } = \\begin{cases} P \\cdot \\frac { \\Lambda + w - 1 } { w } \\ ! \\ ! & \\ c \\leq r \\leq c + w - 1 , \\\\ 0 \\ ! \\ ! & \\end{cases} \\ ! r \\in \\left [ L _ R \\right ] , c \\in \\left [ L _ C \\right ] . \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} \\| x \\| _ 1 = \\left ( 1 - \\frac { c _ x } { 2 } \\right ) \\sqrt { n } \\| x \\| _ 2 . \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} A _ { i j } ' : = r ^ { - 1 } ( A _ { i j } ) . \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} \\nabla _ l | h | ^ 2 = \\nabla _ l ( h ^ { i j } h _ { i j } ) = ( \\nabla _ l h ^ { i j } ) h _ { i j } + h ^ { i j } ( \\nabla _ l h _ { i j } ) = 2 h ^ { i j } ( \\nabla _ l h _ { i j } ) , \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} ( 1 - H ) ( P _ n x + w _ k ) & = P _ n ( 1 - H ) ( P _ n x + w _ k ) + Q _ n ( 1 - H ) ( P _ n x + w _ k ) \\\\ & = P _ n ( 1 - H ) x + Q _ n ( 1 - H ) ( P _ n x + w _ k ) \\phantom { \\mathop { \\longrightarrow } ^ { k \\to \\infty } } \\\\ & = y + Q _ n ( 1 - H ) ( P _ n x + w _ k ) \\mathop { \\longrightarrow } ^ { k \\to \\infty } y . \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} \\forall \\ : z \\in D ( H ) \\cap X _ { n } \\colon \\langle x , z \\rangle = 0 \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} a _ { 0 } ^ { 2 } + \\lambda a _ { 1 } = \\lambda ^ { 2 } \\left ( \\left [ \\int _ { \\mathbb { R } } \\frac { 1 } { \\alpha - t } \\frac { d \\mu _ { 1 } ( t ) } { \\lambda } \\right ] ^ { 2 } - \\int _ { \\mathbb { R } } \\frac { 1 } { ( \\alpha - t ) ^ { 2 } } \\frac { d \\mu _ { 1 } ( t ) } { \\lambda } \\right ) < 0 . \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} ( i d \\times r ) ( r \\times i d ) ( i d \\times r ) = ( r \\times i d ) ( i d \\times r ) ( r \\times i d ) . \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t / 2 } d y \\frac { e ^ { - y } \\sin ( a \\tan ^ { - 1 } ( \\frac { \\pi } { 2 ( \\log ( t ) - \\log ( y ) ) } ) ) } { y ( ( \\log ( t ) - \\log ( y ) ) ^ { 2 } + \\frac { \\pi ^ { 2 } } { 4 } ) ^ { a / 2 } } & = a \\frac { \\pi } { 2 } \\int _ { 0 } ^ { t / 2 } d y \\frac { e ^ { - y } } { y \\log ^ { a + 1 } ( t / y ) } + \\int _ { 0 } ^ { t / 2 } d y \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} \\partial _ t \\omega + u \\cdot \\nabla \\omega - \\partial _ 1 ^ 2 \\omega = \\partial _ 1 \\theta . \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) \\le \\beta \\ | \\ V ( 0 ) = - c _ 2 \\} - P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 \\} = \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { | J _ N \\cap A | } \\sum _ { n \\in J _ N \\cap A } \\psi ( s _ n ) = \\lim _ { N \\to \\infty } \\frac { | J _ N | } { | J _ N \\cap A | } L ( \\psi ) = \\frac { L ( \\psi ) } { d _ { \\N , \\phi ' } ( A ) } \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} \\lim _ { y \\uparrow + \\infty } \\frac { \\Phi ( i y ) } { i y } = 1 \\Im \\Phi ( z ) \\le \\Im z , z \\in \\mathbb { H } . \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} \\mathcal D ( Q ) = \\{ u \\in L ^ 2 ( \\R ^ d _ { 1 + } ) : Q ( u , u ) < \\infty \\} . \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} & \\quad \\frac 1 2 \\frac { d } { d t } ( \\| | D | ^ s \\omega ( t ) \\| _ { L ^ 2 } ^ 2 + \\| | D | ^ s \\theta ( t ) \\| _ { L ^ 2 } ^ 2 ) + \\| \\partial _ 1 | D | ^ s \\omega \\| _ { L ^ 2 } ^ 2 \\\\ & = \\int _ { \\R ^ 2 } \\partial _ 1 | D | ^ s \\theta | D | ^ s \\omega ~ d x - \\int _ { \\R ^ 2 } [ | D | ^ s , u \\cdot \\nabla ] \\omega | D | ^ s \\omega ~ d x - \\int _ { \\R ^ 2 } [ | D | ^ s , u \\cdot \\nabla ] \\theta | D | ^ s \\theta ~ d x \\\\ & \\triangleq K _ 1 + K _ 2 + K _ 3 . \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} \\hat { H } = - \\sum _ { i = 1 } ^ { D } \\frac { \\partial ^ 2 } { \\partial x _ i ^ 2 } + \\omega ^ 2 r ^ 2 + \\frac { f _ 1 ( \\Omega _ 1 ) } { x _ 1 ^ 2 + \\cdots + x _ { n _ 1 } ^ 2 } + \\frac { f _ 2 ( \\Omega _ 2 ) } { x _ { n _ 1 + 1 } ^ 2 + \\cdots + x _ { n _ 2 } ^ 2 } + \\cdots + \\frac { f _ N ( \\Omega _ N ) } { x _ { n _ { N - 1 } + 1 } ^ 2 + \\cdots + x _ { n _ { N } } ^ 2 } , \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} B ( e _ { x y } , e _ { u v } ) = B ( e _ x , e _ { u v } ) = B ( e _ x , e _ u e _ { u v } e _ v ) = 0 \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} - \\Delta u = \\frac { 1 } { \\epsilon ^ 2 } u ( 1 - | u | ^ 2 ) \\quad B _ 1 ( 0 ) , u = g \\partial B _ 1 ( 0 ) , \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} \\lambda ( \\tau ) = \\alpha ( \\tau ) + \\beta ( \\tau ) i . \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ 3 } \\xi ( u ) d u = 1 , \\\\ [ 2 p t ] & \\int _ { \\mathbb { R } ^ 3 } u _ { i } \\xi ( u ) d u = 0 , \\ ( i = 1 , 2 , 3 ) , \\\\ [ 2 p t ] & \\int _ { \\mathbb { R } ^ 3 } u _ i u _ { j } \\xi ( u ) d u = \\delta _ { i j } , \\ ( i , j = 1 , 2 , 3 ) . \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} | v _ { 4 } ^ { \\lambda _ { 1 } } - v _ { 4 } ^ { \\lambda _ { 2 } } | ( t , r ) \\leq \\begin{cases} \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\sqrt { \\log ( \\log ( t ) ) } \\log ^ { N + 3 b - 1 } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { t \\log ^ { 3 b + 2 N } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } , r \\geq \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} H _ l ( x ^ { B ( l ) } | \\bar { x } ^ { B ( l ) } ) = \\sum _ { i \\in B ( l ) } \\psi ( x _ i ) + \\frac { 1 } { 2 } \\sum _ { i , j \\in B ( l ) } M _ { i j } x _ i x _ j + \\sum _ { \\substack { i \\in B ( l ) \\\\ k \\notin B ( l ) } } M _ { i j } x _ i x _ k \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} S _ { } [ X , A ] = \\frac { 1 } { 2 } \\int _ { \\Sigma } G _ { \\mu \\nu } D X ^ \\mu \\wedge \\star D X ^ \\nu + \\int _ { \\Sigma _ 3 } H + \\int _ { \\Sigma } ( A ^ a \\wedge \\alpha _ a + \\frac { 1 } { 2 } \\gamma _ { a b } A ^ a \\wedge A ^ b ) , \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} | \\overline { R } _ { \\varphi _ 2 } | & = [ G _ 1 : G _ 2 ] | \\overline { R } _ { \\phi _ 2 } | = [ G _ 1 : G _ 2 ] [ G _ 2 : _ { G _ 2 } ( I _ 2 ) ] \\\\ & = [ G _ 1 : _ { G _ 1 } ( I _ 1 ) ] = | \\overline { R } _ { \\phi _ 1 } | . \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} \\Phi _ h ( ( a , u ) \\cdot _ H ( b , v ) ) = ~ & \\Phi _ h ( a b , a \\cdot v + u \\cdot b + H ( a , b ) ) \\\\ = ~ & ( a b , a \\cdot v + u \\cdot b + H ( a , b ) - h ( a b ) ) \\\\ = ~ & ( a b , a \\cdot v + u \\cdot b + H ( a , b ) - a \\cdot h ( b ) - h ( a ) \\cdot b + ( \\delta h ) ( a , b ) ) \\\\ = ~ & ( a , u - h ( a ) ) \\cdot _ { H + \\delta h } ( b , v - h ( b ) ) \\\\ = ~ & \\Phi _ h ( a , u ) \\cdot _ { H + \\delta h } \\Phi _ h ( b , v ) . \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} & \\int _ { \\Omega } | u | ^ { \\alpha - 2 p + 2 } | \\nabla u | ^ { 2 p - 2 } \\langle \\nabla u , \\nabla \\Delta _ { p , f } u \\rangle \\ , d \\mu \\\\ = & - ( p - 1 ) \\lambda _ { 1 ; p , f } \\int _ { \\Omega } | u | ^ { \\alpha - p } | \\nabla u | ^ { 2 p } \\ , d \\mu . \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} f ^ { - 1 } ( F _ e , F _ w ) = \\widetilde { Q } _ \\mu \\cap w . \\widetilde { Q } _ \\lambda \\subset Q _ n , \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} \\mathcal { L } _ + ^ \\beta = \\{ p \\in \\mathcal { L } ^ \\beta : \\forall A \\in p \\ , \\ , \\ , \\ , \\lambda ( A ) > 0 \\} \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} & | \\left ( \\frac { \\cos ( 2 ( v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } ) ) - 1 } { 2 r ^ { 2 } } \\right ) \\left ( \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\left ( \\cos ( 2 v _ { 5 } ) - 1 \\right ) + \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\sin ( 2 v _ { 5 } ) \\right ) | \\\\ & \\leq \\frac { C ( v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } ) ^ { 2 } } { r ^ { 2 } } \\left ( \\frac { r \\lambda ( t ) v _ { 5 } ^ { 2 } } { r ^ { 2 } + \\lambda ( t ) ^ { 2 } } + | v _ { 5 } | \\right ) \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} \\gamma = \\frac { \\pi } { 4 } - \\arcsin \\left ( \\frac { 1 } { \\sqrt { 2 } } \\cos \\frac { \\pi } { n } \\right ) = \\frac { \\pi ^ 2 } { 2 n ^ 2 } - \\frac { \\pi ^ 4 } { 6 n ^ 4 } + O \\left ( \\frac { 1 } { n ^ 6 } \\right ) . \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} \\ @ O _ \\kappa ( \\ @ D _ \\kappa ^ \\alpha ( X ) ) : = \\kappa \\Sigma ^ 0 _ { 1 + \\alpha } ( X ) = \\ @ N _ \\kappa ^ \\alpha ( \\ @ O _ \\kappa ( X ) ) . \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} & - 2 p \\lambda _ { 1 ; p , f } \\int _ { \\Omega } | u | ^ { \\alpha - p + 1 } | \\nabla u | ^ { 2 p - 3 } \\langle \\nabla u , \\nabla | \\nabla u | \\rangle \\ , d \\mu \\\\ = & 2 ( \\alpha - p + 1 ) \\lambda _ { 1 ; p , f } \\int _ { \\Omega } | u | ^ { \\alpha - p } | \\nabla u | ^ { 2 p } \\ , d \\mu - 2 \\lambda _ { 1 ; p , f } ^ 2 \\int _ { \\Omega } | u | ^ { \\alpha } | \\nabla u | ^ { p } \\ , d \\mu . \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} \\begin{cases} D ^ { \\alpha _ n } _ a y ( t ) = f ( t , y ( t ) , D ^ { \\alpha _ 1 } _ a y ( t ) , D ^ { \\alpha _ 2 } _ a y ( t ) , \\cdots , D ^ { \\alpha _ { n - 1 } } _ a y ( t ) ) , \\\\ y ^ { ( j ) } ( a ) = y ^ { ( j ) } _ { a } , ~ j = 0 , 1 , \\cdots , \\lfloor \\alpha _ n \\rfloor , \\end{cases} \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} A \\mathsf { q = - } \\left ( \\mathsf { q , \\hat { p } } _ { \\beta , \\mathsf { G i b b s } } \\right ) _ { 2 } \\mathsf { \\hat { p } } _ { \\beta , \\mathsf { G i b b s } } \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} \\min \\{ J , K \\} \\le T ^ { t + 1 } \\eta _ n + T ^ t W _ n = T ^ t \\eta _ n + T ^ t W _ { n - 1 } \\le \\max \\{ J , K \\} , \\forall n \\ge N . \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} J ( x _ 0 , u ) : = \\lim _ { T \\rightarrow \\infty } J _ T ( x _ 0 , u ) . \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} \\hat r = r _ E \\circ p . \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} K _ X = - 2 \\sin ( y ) ^ 3 \\cos ( x ) \\cos ( y ) d x ^ 2 - 2 \\sin ( y ) ^ 2 \\sin ( x ) d x d y \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\partial ^ 2 ( h _ { i j } ) = \\begin{pmatrix} 0 & 0 & \\bar { a _ { i , i } } & \\bar { a _ { i , j } } \\\\ 0 & 0 & \\bar { a _ { j , i } } & \\bar { a _ { j , j } } \\\\ a _ { i , i } & a _ { i , j } & 0 & 0 \\\\ a _ { i , j } & a _ { j , j } & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} q ^ m \\Delta > ( q ^ m - 1 ) t \\mbox { a n d } | H | = \\Big \\lfloor \\frac { q ^ m \\Delta } { q ^ m \\Delta - ( q ^ m - 1 ) t } \\Big \\rfloor . \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} & | | \\left ( \\partial _ { r } + \\frac { 1 } { r } \\right ) N _ { 2 } ( f _ { v _ { 5 } } ) ( t , r ) | | _ { L ^ { 2 } ( r d r ) } \\leq | | \\partial _ { r } N _ { 2 } ( f _ { v _ { 5 } } ) ( t , r ) | | _ { L ^ { 2 } ( r d r ) } + | | \\frac { N _ { 2 } ( f _ { v _ { 5 } } ) ( t , r ) } { r } | | _ { L ^ { 2 } ( r d r ) } \\\\ & \\leq \\frac { C \\log ( t ) } { t ^ { 2 3 / 8 } } \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} S ( Z ) = ( z ' ( 0 ) ) ^ { - 1 } z ''' ( 0 ) - \\frac 3 2 ( ( z ' ( 0 ) ) ^ { - 1 } z '' ( 0 ) ) ^ 2 . \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} g ( R ) \\int \\ , f _ { x } ( R ) \\ , d \\ , \\mu ( x ) = h ( R ) \\int \\ , u _ { y } ( R ) \\ , d \\ , \\nu ( y ) . \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} A ^ p \\sigma _ h B ^ p & \\ge \\lambda _ { \\min } ( A ^ { p ' } \\sigma _ h B ^ { p ' } ) ( A ^ { p ' } \\sigma _ h B ^ { p ' } ) \\\\ & \\ge \\lambda _ { \\min } \\bigl ( \\lambda _ { \\min } ^ { p ' - 1 } ( A \\sigma _ h B ) ( A \\sigma _ h B ) \\bigr ) \\cdot \\lambda _ { \\min } ^ { p ' - 1 } ( A \\sigma _ h B ) ( A \\sigma _ h B ) \\\\ & = \\lambda _ { \\min } ^ { 2 p ' - 1 } ( A \\sigma _ h B ) ( A \\sigma _ h B ) = \\lambda _ { \\min } ^ { p - 1 } ( A \\sigma _ h B ) ( A \\sigma _ h B ) . \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} A _ i & = ( Z _ { 1 i } , { ( Z _ { 1 i } ^ 2 - 1 ) } / { 2 } , ( Z _ { 2 i } ^ 2 - 1 ) / 2 ) ^ \\tau , \\\\ B _ i & = ( ( Z _ { 1 i } ^ 2 - 1 ) ( Z _ { 2 i } ^ 2 - 1 ) / 2 , Z _ { 1 i } ( Z _ { 2 i } ^ 2 - 1 ) / 2 , - ( Z _ { 2 i } ^ 4 - 6 Z _ { 2 i } ^ 2 + 3 ) / 1 2 ) ^ \\tau . \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} j _ { 1 } + j _ { 2 } + \\dots + j _ { N } + i _ { 1 } + i _ { 2 } + \\dots + i _ { N } = p , j _ { 1 } + j _ { 2 } + \\dots + j _ { N } = k - 1 . \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} v _ { 4 } ( t , r ) = \\frac { - 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\int _ { 0 } ^ { 2 \\pi } d \\theta \\frac { v _ { 4 , c } ( s , \\sqrt { r ^ { 2 } + 2 r \\rho \\cos ( \\theta ) + \\rho ^ { 2 } } ) } { \\sqrt { r ^ { 2 } + \\rho ^ { 2 } + 2 r \\rho \\cos ( \\theta ) } } ( r + \\rho \\cos ( \\theta ) ) \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} \\psi _ 0 ( 0 ) = \\psi ( 0 ) - \\log | w _ 0 ' ( 0 ) | = \\log 2 + \\frac 1 { 2 \\beta + 1 } \\log c _ \\beta . \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} f ( p ) = \\inf _ { a \\in A } \\big [ \\varphi ( a ) d ( p , A ) + d ( a , p ) \\big ] , p \\in X . \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } \\chi _ 1 & = H _ 2 - 1 , \\\\ \\chi _ 2 & = ( H _ 1 ) ^ 2 - \\epsilon H _ 2 , \\\\ \\chi _ 3 & = - \\epsilon ( 3 H _ 2 H _ 3 + H _ 3 + 4 H _ 2 ) \\cdot \\chi _ 2 \\\\ & - ( \\epsilon H _ 2 + 3 ( H _ 1 ) ^ 2 ) ( H _ 4 + \\epsilon ) \\cdot \\chi _ 1 . \\end{alignedat} \\right . \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} \\eta ' _ { \\nu } ( 0 ) = \\int _ { \\mathbb { T } } t \\ , d \\nu ( t ) > 0 , \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} e ^ { C \\left ( \\gamma , \\varepsilon \\right ) } \\left \\Vert \\tilde { u } \\right \\Vert _ { L ^ { 2 } \\left ( O \\left ( r , \\varepsilon \\right ) ; H \\right ) } \\leq N _ { \\gamma , \\varepsilon } , O \\left ( r , \\varepsilon \\right ) = O _ { \\frac { r } { 8 } } \\times \\left [ \\frac { 1 - \\varepsilon } { 2 } , \\frac { 1 + \\varepsilon } { 2 } \\right ] \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} & ( \\alpha - p + 1 ) I _ { p , \\alpha } \\int _ { \\Omega } | u | ^ { \\alpha } | \\nabla u | ^ { p } \\ , d \\mu \\\\ & = \\int _ { \\Omega } \\Big ( \\lambda _ { 1 ; p , f } | u | ^ { \\frac { \\alpha } { 2 } } | \\nabla u | ^ { \\frac { p } { 2 } } - p | u | ^ { \\frac { \\alpha } { 2 } - p + 1 } | \\nabla u | ^ { \\frac { 3 } { 2 } p - 3 } \\langle \\nabla u , \\nabla | \\nabla u | \\rangle \\Big ) | u | ^ { \\frac { \\alpha } { 2 } } | \\nabla u | ^ { \\frac { p } { 2 } } \\ , d \\mu . \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} X _ { } = \\left [ v _ 1 \\cdots v _ d \\right ] ^ T X . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} \\mathbf { P } _ { \\beta , T } ( d \\omega ) = \\frac { \\exp \\{ - \\beta H _ T ( \\omega ) \\} } { Z _ { \\beta , T } } \\mathbf { P } _ T ( d \\omega ) = \\frac { \\exp \\{ \\beta \\int _ 0 ^ T v ( \\omega ( t ) ) d t \\} } { Z _ { \\beta , T } } \\mathbf { P } _ T ( d \\omega ) , \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} S _ { m } \\left ( \\cos ( \\theta ) \\right ) = \\cos ( m \\theta ) . \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ N w _ j g ( \\mathbf { x } _ j ) = \\int _ { \\Omega } g \\omega \\quad \\forall g \\in \\mathbb { P } _ { 2 L } . \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} [ s _ { 1 } ^ { 2 } , s _ { 2 } ^ { 2 } , s _ { 3 } ^ { 2 } , t _ { 1 } ^ { 2 } , t _ { 2 } ^ { 2 } , t _ { 3 } ^ { 2 } ] = [ - A _ { 3 } \\overline { B _ { 2 } ' } , - A _ { 1 } \\overline { B _ { 3 } ' } , - A _ { 2 } \\overline { B _ { 1 } ' } , - A _ { 2 } B _ { 3 } ' , - A _ { 3 } B _ { 1 } ' , - A _ { 1 } B _ { 2 } ' ] \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} H ( p , q ) = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ n p _ i p _ j e ^ { - | q _ i - q _ j | } . \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} p ( x , t ) = \\frac { e ^ { - \\lambda t } } { c _ 1 + c _ 2 } \\Biggl [ \\lambda I _ 0 \\Bigl ( \\frac { 2 \\lambda } { c _ 1 + c _ 2 } \\sqrt { ( c _ 1 t - x ) ( c _ 2 t + x ) } \\Bigr ) + \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} D \\varphi = - B ^ { - 1 } D \\psi ( B ^ { - 1 } + \\varphi ) ( B ^ { - 1 } + D \\varphi ) . \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} - \\lambda _ { 1 2 } \\sqrt { 1 + b v ^ { 2 } } + \\lambda _ { 1 3 } v = 0 . \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} F _ { 4 } ( t , r ) & = \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) \\right ) \\left ( F _ { 0 , 2 } ( t , r ) + \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) - 1 } { r ^ { 2 } } \\right ) \\left ( v _ { 1 } + v _ { 2 } + v _ { 3 } \\right ) \\right ) \\\\ & + \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) - 1 } { r ^ { 2 } } \\right ) \\left ( v _ { 4 } + v _ { 5 } \\right ) \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 2 } v _ { 5 } ( t , r ) & = \\frac { - 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\int _ { 0 } ^ { 2 \\pi } d \\theta \\frac { \\partial _ { 1 } ^ { 2 } N _ { 2 } ( f ) ( s , \\sqrt { r ^ { 2 } + \\rho ^ { 2 } + 2 r \\rho \\cos ( \\theta ) } ) } { \\sqrt { r ^ { 2 } + \\rho ^ { 2 } + 2 r \\rho \\cos ( \\theta ) } } ( r + \\rho \\cos ( \\theta ) ) \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} & \\delta \\int _ { M } H ^ p \\ , d S = \\int _ M \\left ( \\frac { p } { 2 } H ^ { p - 1 } \\Delta u + ( 2 H ^ 2 - K + 2 k _ 0 ) p H ^ { p - 1 } u - 2 H ^ { p + 1 } u \\right ) \\ , d S , \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ { t t } v _ { 4 , s } + \\partial _ { r r } v _ { 4 , s } + \\frac { 1 } { r } \\partial _ { r } v _ { 4 , s } - \\frac { v _ { 4 , s } } { r ^ { 2 } } = 0 \\\\ v _ { 4 , s } ( s , r ) = 0 \\\\ \\partial _ { t } v _ { 4 , s } ( s , r ) = v _ { 4 , c } ( s , r ) \\end{cases} \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} X ' ( \\sigma ) = h , { X ' } ^ * A = A ' . \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} Q ( A , B ) \\ = \\ | \\{ ( a , b ) \\in A \\times B : a > b \\} | - | \\{ ( a , b ) \\in A \\times B : b > a \\} | , \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} b _ i \\ge \\sum _ { r = i + 1 } ^ { k - 1 } \\binom { \\left \\lfloor \\frac { \\alpha m } k \\right \\rfloor } { r - i } b _ r \\ge \\sum _ { r = i + 1 } ^ { k - 1 } \\binom { \\left \\lceil \\frac { \\alpha m } k \\right \\rceil - k + r - i - 1 } { r - i } b _ r \\ge \\sum _ { r = i + 1 } ^ { k - 1 } ( - 1 ) ^ { r - i + 1 } \\binom { \\left \\lceil \\frac { \\alpha m } k \\right \\rceil - k + r - i - 1 } { r - i } b _ r , \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} a _ 1 = n _ 1 a _ 1 { a _ 0 ' } - a _ 0 { a ' _ 1 } . \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} ( C , ( Q _ { t = 0 } ) _ { ( 0 ) } ) \\cong ( C , ( Q _ { t } ) _ { ( 0 ) } ) \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} Q ( u ( \\cdot + z _ a ) , u ( \\cdot + z _ a ) ) = \\int _ { \\R ^ d _ { 1 + } } \\int _ { \\R ^ d _ { 1 + } } ( u ( x + z _ a ) - u ( y + z _ a ) ) ^ 2 K ( | x - y | ) \\ , d x \\ , d y \\le Q ( u , u ) . \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} { \\mathbb T } ^ h ( \\omega ) \\vec { u } = \\vec { 0 } , \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} \\dim W = \\dim W _ D + \\dim W ( - D ) . \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} \\beta = \\left ( \\frac { \\partial f _ L } { \\partial \\mu } \\frac { \\partial g _ L } { \\partial y } - \\frac { \\partial g _ L } { \\partial \\mu } \\frac { \\partial f _ L } { \\partial y } \\right ) \\bigg | _ { x = y = \\mu = 0 } , \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} x y = f ( x ) + g ( y ) \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & \\ , \\rightarrow \\ , \\min & & & \\\\ g _ i ( x ) & \\ , \\in \\ , \\R ^ 0 _ - & \\qquad & i \\in \\mathcal M & \\\\ h _ j ( x ) & \\ , \\in \\ , \\{ 0 \\} & \\qquad & j \\in \\mathcal P & \\\\ ( G _ l ( x ) , H _ l ( x ) ) & \\ , \\in \\ , O & \\qquad & l \\in \\mathcal Q . & \\end{aligned} \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} | \\frac { 1 } { r ^ { 3 / 2 } \\langle \\log ( r ) \\rangle } \\phi ( r , \\xi ) | \\leq \\begin{cases} \\frac { C } { \\langle \\log ( r ) \\rangle } \\left ( \\frac { \\phi _ { 0 } ( r ) } { r } + \\frac { \\log ( 1 + r ^ { 2 } ) } { r ^ { 2 } } \\right ) , r ^ { 2 } \\xi \\leq 4 \\\\ \\frac { C | a ( \\xi ) | } { \\xi ^ { 1 / 4 } r ^ { 1 / 2 } \\cdot r \\langle \\log ( r ) \\rangle } , r ^ { 2 } \\xi > 4 \\end{cases} \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} F ( z ) = - \\alpha + \\beta z + \\int _ { ( 0 , + \\infty ) } \\frac { z ( 1 + t ^ { 2 } ) } { t ( t - z ) } \\ , d \\rho ( t ) , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} L _ { k } + L _ { i _ 1 } + L _ { i _ 2 } + \\ldots + L _ { i _ t } = \\textbf { 0 } \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} & u ^ { ( i _ 1 ) } = e m _ v \\left ( u ^ { ( i _ 1 ) } \\right ) f , \\ \\widetilde { u } ^ { ( i _ 1 ) } = \\widetilde { e } m _ v \\left ( u ^ { ( i _ 1 ) } \\right ) \\widetilde { f } , \\textit { a n d } \\\\ & \\widetilde { u } ^ { ( i _ 1 ) } = \\widetilde { e } \\left ( e ^ { - 1 } \\cdot u ^ { ( i _ 1 ) } \\cdot f ^ { - 1 } \\right ) \\widetilde { f } , \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} a ( n ) = 2 a ( n - 1 ) + ( 2 ^ { k + 1 } - 1 ) ( 2 ^ { n - 2 } ) . \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} \\left \\Vert \\tilde { u } _ { \\varepsilon } \\left ( . , t _ { i } \\right ) \\right \\Vert _ { X } \\geq \\frac { 1 } { 4 N _ { 1 } } \\left \\Vert u \\left ( . , 0 \\right ) \\right \\Vert _ { X } , 0 < \\varepsilon \\leq \\varepsilon _ { 0 } , i = 1 , 2 , . . . m \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\int _ { X \\times Y } c d \\gamma _ i = \\int _ { X \\times Y } c d \\gamma . \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} \\varphi ( \\gamma ( t _ { j } \\Lambda , y ) ) & = \\varphi ( \\gamma t _ { j } \\alpha ( \\gamma , t _ { j } \\Lambda ) \\Lambda , \\alpha ( \\gamma , t _ { j } \\Lambda ) ^ { - 1 } y ) \\\\ & = \\varphi ( t _ { i } \\Lambda , y ) = t _ { i } \\Lambda . \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 , \\ N ( t ) = 1 \\} = P \\{ T _ 1 < \\frac { \\beta } { c _ 1 } \\ | \\ N ( t ) = 1 \\} = \\frac { \\beta } { c _ 1 t } \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} \\frac { u _ 1 ' } { u _ 2 ' } = \\frac { u _ 1 + u _ 2 \\tan ( t ) } { u _ 2 - u _ 1 \\tan ( t ) } \\ , , \\quad \\tan ( t ) = \\tan ( s ' - s ) \\ , , \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} \\mu = 4 ( n ^ 2 - n ) , \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} \\begin{cases} D ^ { 1 - \\epsilon } _ 0 y ( t ) = w ( t ) , & y ( 0 ) = y _ 0 , \\\\ D ^ { \\epsilon } _ 0 w ( t ) = z ( t ) , & w ( 0 ) = 0 , \\\\ D ^ { \\alpha _ 2 - 1 } _ 0 z ( t ) = f ( t , y ( t ) , w ( t ) ) , & z ( 0 ) = s . \\end{cases} \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} ( E \\rho ) ^ + E \\rho = I . \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} \\omega _ n = n ^ 2 - 2 \\sum _ { k = 1 } ^ \\infty \\min ( k , n ) \\gamma _ k \\ . \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } P _ 0 & = 0 , \\\\ C _ 1 K & = 0 . \\end{alignedat} \\right . \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} \\left \\vert \\sum _ { j = 1 } ^ { N + 1 } \\psi _ { N , j } ^ { \\ast } ( Q ) \\right \\vert \\leq \\frac { C _ { N } ^ { \\ast } } { q } , \\left \\vert \\sum _ { j = 1 } ^ { N + 1 } L _ { N , j } ^ { \\ast } ( q ) \\right \\vert \\leq C _ { N } ^ { \\ast } , \\qquad q > 0 , \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} \\dot { \\bar { y } } & = \\bar { \\mu } - \\bar { y } - A \\bar { y } | \\bar { y } - 1 | , \\\\ \\dot { \\bar { \\mu } } & = \\delta _ 0 ( \\lambda - \\bar { y } ) . \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} \\ @ N _ \\kappa ^ 0 ( A ) & : = A , \\\\ \\ @ N _ \\kappa ^ { \\alpha + 1 } ( A ) & : = \\ @ N _ \\kappa ( \\ @ N _ \\kappa ^ \\alpha ( A ) ) , \\\\ \\ @ N _ \\kappa ^ \\alpha ( A ) & : = \\injlim _ { \\beta < \\alpha } \\ @ N _ \\kappa ^ \\beta ( A ) \\quad , \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} & t \\ , \\frac { \\partial q } { \\partial t } - ( b ( t , x ) + b _ 1 ( t , x ) ) \\frac { \\partial q } { \\partial x } \\\\ & = ( \\lambda ( t , x ) + \\lambda _ 1 ( t , x ) + c ( t , x ) ) q + \\gamma ( t , x ) u \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} X _ { i } & : = \\frac { \\beta _ { i - 1 } - \\beta _ { i } } { 2 } - \\frac { ( 1 + \\sigma _ { i } ) } { 2 } \\Big ( \\frac { 1 } { 2 } - \\frac { 1 } { p _ 0 } \\Big ) ; \\\\ Y _ { i } & : = \\frac { \\beta _ { i - 1 } } { 2 } - \\big ( 1 + ( n - i ) ( 1 - \\sigma _ { i } ) \\big ) \\Big ( \\frac { 1 } { 2 } - \\frac { 1 } { p _ 0 } \\Big ) . \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} q _ { i j } \\circ p _ { i , j + 1 } ( \\sigma ) = q _ { i j } ( p _ { i , j + 1 } ( \\sigma ) ) = q _ { i j } ( x \\vert _ { M ^ { i + j + 1 } } ) = ( \\tau ( x ) ) \\vert _ { M ^ { i + j } } . \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} ( T _ j ^ { n _ 0 } ) ^ { - 1 } \\{ x \\} = \\{ y _ k : \\ , k < D _ { n _ 0 } \\} \\ , \\ , \\ , \\ , ( T _ j ^ { n _ 0 } ) ^ { - 1 } \\{ x ' \\} = \\{ y ' _ k : \\ , k < D _ { n _ 0 } \\} \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} & \\underline U _ t ( t , x ) = K _ 1 K _ 2 \\frac { ( 1 - \\beta ) \\ln ( t + \\theta ) + 1 } { ( t + \\theta ) ^ { \\beta } } \\Theta + \\frac { K _ 2 \\beta | x | } { ( t + \\theta ) ^ { 1 + \\beta } } \\Theta , \\ \\ \\ \\underline h ( t ) - ( t + \\theta ) ^ \\beta < | x | \\leq \\underline h ( t ) , \\\\ & \\underline U _ t ( t , x ) = { \\bf 0 } , \\ | x | < \\underline h ( t ) - ( t + \\theta ) ^ \\beta . \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} { \\rm V o l } ( D ) = \\lim _ { m \\rightarrow \\infty } \\frac { \\dim _ k \\Gamma ( X , \\mathcal O _ X ( m D ) ) } { m ^ d / d ! } = d ! [ \\mathcal O _ { X , p } / m _ p : k ] { \\rm V o l } ( \\Delta ( D ) ) . \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} \\beta = - \\left ( \\frac { \\partial g } { \\partial \\mu } \\frac { \\partial f } { \\partial y } \\middle ) \\right | _ { x = y = \\mu = 0 } . \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{gather*} \\delta _ { \\lambda ' , \\lambda } : Y _ { \\lambda ' } = Y _ { \\lambda ' } \\times _ { Y _ \\lambda } Y _ \\lambda \\hookrightarrow Y _ { \\lambda ' } \\times _ { Q _ n } Y _ { \\lambda } = Z _ { \\lambda ' , \\lambda } , \\\\ \\delta _ { \\lambda , \\lambda ' } : Y _ { \\lambda ' } = Y _ \\lambda \\times _ { Y _ \\lambda } Y _ { \\lambda ' } \\hookrightarrow Y _ { \\lambda } \\times _ { Q _ n } Y _ { \\lambda ' } = Z _ { \\lambda , \\lambda ' } . \\end{gather*}"}
-{"id": "7047.png", "formula": "\\begin{align*} \\int _ a ^ b \\big ( \\phi ( x ) | A ( x ) \\phi ( x ) \\big ) = \\int | f ( x ) | ^ 2 \\d x , \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} { \\det \\Delta } = 4 \\pi \\det ( { \\Delta } \\ ! \\ ! \\restriction _ { \\overline { \\Bbb C } _ \\epsilon } ) \\cdot \\det ( { \\Delta } \\ ! \\ ! \\restriction _ { | w _ + | \\leq \\epsilon } ) \\cdot \\det ( \\Delta \\ ! \\ ! \\restriction _ { | w _ 0 | \\leq \\epsilon } ) \\cdot \\det ( { \\Delta } \\ ! \\ ! \\restriction _ { | w _ - | \\leq \\epsilon } ) \\cdot \\frac { \\det \\mathcal N \\ ! \\ ! \\restriction _ { \\partial \\Bbb C _ \\epsilon } } { \\ell ( \\partial \\Bbb C _ \\epsilon , m ) } , \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} p '' ( t ) & = \\frac { 1 } { 4 t } h '' ( \\sqrt { t } ) - \\frac { 1 } { 4 t ^ { 3 / 2 } } ( h ' ( \\sqrt { t } ) - h ' ( 0 ) ) \\\\ & = \\frac { 1 } { 4 t } h '' ( \\sqrt { t } ) - \\frac { 1 } { 4 t ^ { 3 / 2 } } \\int _ { 0 } ^ { \\sqrt { t } } h '' ( \\theta ) d \\theta \\\\ & = \\frac { 1 } { 4 t } \\int _ { 0 } ^ { 1 } [ h '' ( \\sqrt { t } ) - h '' ( \\sqrt { t } \\theta ) ] d \\theta \\ , . \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} \\left \\{ ( s _ j ) _ { j = 1 } ^ { \\infty } \\in \\mathbb R ^ { \\infty } : ( j ^ { \\frac { 3 } { 2 ( N - 1 ) } } s _ j ) _ { j = 1 } ^ { \\infty } \\in l ^ 2 \\right \\} \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} u ( t , x ) = S \\ast ( \\frac { \\kappa ^ 2 } { 4 } | u | ^ r - \\frac { c _ 1 ^ 2 + c _ 2 ^ 2 } { 2 } u ) ( t , x ) + I ( t , x ) , \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} | A _ 1 \\cap \\cdots \\cap A _ { k - 2 } | = b _ { k - 2 } + 2 b _ { k - 1 } + b _ k . \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} m = \\frac { d } { d \\sigma } A _ N ( \\sigma ) = \\frac { 1 } { N } \\frac { \\int _ { \\mathbb { R } ^ N } \\sum _ { i = 1 } ^ { N } x _ i \\exp \\left ( \\sigma \\sum _ { i = 1 } ^ { N } x _ i - H ( x ) \\right ) d x } { \\int _ { \\mathbb { R } ^ N } \\exp \\left ( \\sigma \\sum _ { i = 1 } ^ { N } x _ i - H ( x ) \\right ) d x } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N m _ i . \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} \\tilde A _ { h , \\tilde h } = ( \\tilde x ' ) ^ { - m } s ^ { - m } ( s D _ s ) ^ j P _ { m - j } ( h , h \\tilde h s \\tilde x ' , y , D _ y ) \\bigl ( \\delta ( s - 1 ) \\delta ( y - y ' ) \\bigr ) \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} D _ { k } = \\frac { 1 } { a _ { 0 } ^ { ( n + \\gamma ) / \\beta } } \\sum _ { j = 0 } ^ { n } { b _ { n - j } \\sum _ { i = 0 } ^ { j } { - \\frac { n + \\gamma } { \\beta } \\choose i } \\frac { 1 } { a _ { 0 } } { B } _ { j , i } ( a _ { 1 } , a _ { 2 } , . . . a _ { j - i + 1 } ) } , \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} \\int _ a ^ b x ' y = x ( b ) y ( b ) - x ( a ) y ( a ) - \\int _ a ^ b x y ' . \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} \\| f \\| _ { L _ p ( \\mathbb { T } ) } \\asymp \\| f \\| _ { L _ 2 ( \\mathbb { T } ) } , 0 < p < \\infty , \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} \\psi = ( \\psi ^ + , \\psi ^ - ) = ( i w ^ + \\varphi ^ + , i w ^ - \\varphi ^ - ) , \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} \\Psi ^ { - \\infty , \\epsilon } _ { G , c } ( M ) = \\{ T _ { \\widetilde { k } } , \\ ; \\widetilde { k } \\in ( C ^ \\infty _ c ( G ) \\hat { \\otimes } \\ , \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } \\} \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} F ( x , y ; \\mu ) = \\begin{bmatrix} f ( x , y ; \\mu ) \\\\ g ( x , y ; \\mu ) \\end{bmatrix} , \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} { \\mathcal Q } _ A ^ * = { \\mathcal Q } _ { A ^ { - 1 } } . \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta w ^ + + \\big [ A _ + ( | w ^ + | ^ 2 - { t ^ + } ^ 2 ) + B ( | w ^ - | ^ 2 - { t ^ - } ^ 2 ) \\big ] w ^ + = 0 , \\\\ [ 2 m m ] - \\Delta w ^ - + \\big [ A _ - ( | w ^ - | ^ 2 - { t ^ - } ^ 2 ) + B ( | w ^ + | ^ 2 - { t ^ + } ^ 2 ) \\big ] w ^ - = 0 , \\end{cases} \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} \\| \\overline D \\psi \\| ^ 2 _ { L ^ 2 } & = ( \\psi , \\overline D \\overline D \\psi ) = \\| \\overline \\nabla \\psi \\| ^ 2 _ { L ^ 2 } + \\frac 1 2 ( \\psi , \\rho \\psi ) - \\frac 1 2 ( \\psi , e _ 0 \\cdot j ^ \\sharp \\cdot \\psi ) \\\\ & \\geq \\frac 1 2 ( \\psi , \\rho \\psi ) - \\frac 1 2 ( \\psi , \\| j \\| \\psi ) = \\frac 1 2 ( \\psi , ( \\rho - \\| j \\| ) \\psi ) > 0 \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} I _ { N _ 1 , N _ 2 } = \\{ ( i , j ) \\ | \\ i \\in \\{ 1 , \\cdots , N _ 1 \\} , j \\in \\{ N _ 1 + 1 , \\cdots , N _ 1 + N _ 2 \\} , | i - j | \\leq R \\} . \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} \\Psi ( L _ i , L _ j ) \\ , : = \\ , \\frac 1 { 1 2 } ( i ^ 3 - i ) \\delta _ { i + j , 0 } \\ , . \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} { } _ 4 \\phi _ 3 \\ ! \\left [ \\begin{matrix} q ^ { - n } , a , b , c \\\\ d , e , a b c q ^ { 1 - n } / d e \\end{matrix} ; q , q \\right ] = \\frac { ( e / a , d e / b c ; q ) _ n } { ( e , d e / a b c ; q ) _ n } \\ , { } _ 4 \\phi _ 3 \\ ! \\left [ \\begin{matrix} q ^ { - n } , a , d / b , d / c \\\\ d , a q ^ { 1 - n } / e , d e / b c \\end{matrix} ; q , q \\right ] . \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} | x + \\tau \\xi _ x | ^ 2 & = | | x | - \\tau \\langle \\xi , \\hat x \\rangle | ^ 2 + | \\tau \\pi _ { V _ x } ( \\xi ) | ^ 2 , \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} y u _ t x ^ * = d _ \\sigma v _ { \\omega ( \\phi ( \\sigma ) ) ^ { - 1 } } y u _ t x ^ * v _ { \\delta ( \\sigma ) } , \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} \\bigl ( A x \\bigr ) ( t ) & = \\sum \\limits _ { m \\in \\mathbb Z ^ c } \\bigl ( b _ { k m } x _ { k - m } \\bigr ) ( t - k ) \\\\ & = \\sum \\limits _ { m \\in \\mathbb Z ^ c } \\int _ { [ 0 , 1 ) ^ c + k - m } n _ { k m } ( t - k , s - k + m ) \\ , x ( s ) \\ , d s \\\\ & = \\int _ { \\mathbb R ^ c } n ( t , s ) \\ , x ( s ) \\ , d s , \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} ( - 1 ) ^ { j - i } \\sum _ { s = 0 } ^ { j - i } \\binom { m - k + j - s - i - 1 } { j - i - s } \\binom { k - m + s - 1 } { s } . \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} \\partial _ { 2 } ^ { 2 } \\phi ( r , \\xi ) = \\frac { 1 } { \\sqrt { r } } \\sum _ { j = 2 } ^ { \\infty } j ( j - 1 ) \\xi ^ { j - 2 } r ^ { 2 j } \\phi _ { j } ( r ^ { 2 } ) , r \\sqrt { \\xi } \\leq 2 \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} \\Im { m \\left ( E + \\i \\eta \\right ) = \\frac { 1 } { n } \\sum _ { i } \\Im { G _ { i i } \\left ( E + \\i \\eta \\right ) } } . \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\widehat { A } ( M , \\partial M ; \\varphi ) : = \\int _ { M } \\chi \\widehat { A } ( M ) \\wedge \\omega _ \\varphi - \\frac { 1 } { 2 } \\widehat { \\eta } _ \\varphi ( D _ \\partial ) \\ , , \\ ; \\ ; [ \\varphi ] \\in H ^ { 2 * } _ { { \\rm d i f f } } ( G ) . \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} R = \\sigma _ { \\overline { 1 } , 1 } + \\sigma _ { 1 , \\overline { 1 } } - \\sigma _ { 0 } . \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} \\xi ^ 2 = F ( \\mu ) : = 4 \\bigg ( Q ( \\mu ) + \\frac { f ( \\mu ) } { \\mu ^ 2 } \\bigg ) . \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} b _ N \\geq 1 \\binom { b _ N + 2 } { 2 } \\le ( 1 - \\varepsilon _ 0 ) N \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u | _ { \\Gamma _ 1 } = u | _ { \\Gamma _ 2 } , ~ ~ u _ { x _ 1 } | _ { \\Gamma _ 1 } = u _ { x _ 1 } | _ { \\Gamma _ 2 } , \\\\ & u | _ { \\Gamma _ 3 } = u | _ { \\Gamma _ 4 } , ~ ~ u _ { x _ 2 } | _ { \\Gamma _ 3 } = u _ { x _ 2 } | _ { \\Gamma _ 4 } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t ^ 2 - c ^ 2 ( x ) \\Delta ) u ( x , t ) = 0 , & ( x , t ) \\in \\left ( \\mathbb { R } ^ n \\setminus \\Omega \\right ) \\times ( 0 , \\infty ) , \\ , n \\ge 2 , \\\\ u ( x , 0 ) = u _ 0 ( x ) , \\\\ \\partial _ t u ( x , 0 ) = u _ 1 ( x ) , \\\\ u ( t , x ) = 0 , & ( x , t ) \\in \\partial \\Omega \\times ( 0 , \\infty ) , \\end{cases} \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} \\vert \\langle L _ u h , h \\rangle - \\langle L _ v h , h \\rangle \\vert = \\vert \\langle ( u - v ) , | h | ^ 2 \\rangle \\vert \\leq \\| u - v \\| \\| h \\| _ { L ^ 4 } ^ 2 \\ \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} ( V _ 0 ) _ 0 = V _ 0 . \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} H _ { ( M _ { i j } ) } ( z , y ) : = \\frac { 1 } { 2 } \\langle z , ( I d + ( M _ { i j } ) ) z \\rangle + \\langle z , ( M _ { i j } ) N P ^ * y \\rangle + \\langle z , s \\rangle + \\sum _ { i = 1 } ^ N \\delta \\psi ( z _ i + ( N P ^ * y ) _ i ) , \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} G _ 1 - G _ 2 = \\sum _ { j = 1 } ^ n | u _ j \\rangle \\langle \\phi _ j | . \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} ( 1 _ A \\otimes m ) ( m ^ * \\otimes 1 _ A ) = m ^ * m = ( m \\otimes 1 _ A ) ( 1 _ A \\otimes m ^ * ) \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} H ( t ; x , y , 1 , u , z , v ) & = z ( y t u v ( 1 - z ) + z ) H ( t ; x , y , 1 , u , 1 , v ) . \\\\ F ( t ; x , y , 1 , u , z , v ) & = \\frac { t x ( y - y z r + z ) F ( t ; x , y - y r + 1 , 1 , u , z , v ) } { ( t u x + y ^ { - 1 } - t u ) ( y - y r + 1 ) } \\\\ & \\quad + z ( y t u v ( 1 - z ) + z ) H ( t ; x , y , 1 , u , 1 , v ) . \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} \\int _ 0 ^ r f ( \\rho ) P ( \\rho , k ) d \\rho = \\int _ 0 ^ { [ r ] } f ( \\rho ) P ( \\rho , k ) d \\rho + \\int _ { [ r ] } ^ { r } f ( \\rho ) P ( \\rho , k ) d \\rho \\ , . \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} X \\circ Y & = Y \\circ X , \\\\ X ^ { 2 } \\circ ( X \\circ Y ) & = X \\circ ( X ^ { 2 } \\circ Y ) , \\forall \\ X , Y \\in \\mathfrak { J } . \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} c ( \\gamma ) & = \\int _ 0 ^ { + \\infty } \\frac { g ( \\tau ) } { \\tau ^ { 1 + 2 s } } \\ , d \\tau \\\\ & = \\int _ 0 ^ { \\frac { 1 } { 4 \\sqrt { N } } } \\frac { g ( \\tau ) } { \\tau ^ { 1 + 2 s } } \\ , d \\tau + \\int _ { \\frac { 1 } { 4 \\sqrt { N } } } ^ { \\frac { 1 } { 2 \\sqrt { N } } } \\frac { g ( \\tau ) } { \\tau ^ { 1 + 2 s } } \\ , d \\tau + \\int _ { \\frac { 1 } { 2 \\sqrt { N } } } ^ { + \\infty } \\frac { g ( \\tau ) } { \\tau ^ { 1 + 2 s } } \\ , d \\tau \\\\ & = : I _ 1 + I _ 2 + I _ 3 . \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} S _ q ( a , n ) & = \\bigg \\{ \\frac { 1 } { ( 1 - q ) ^ 2 } - \\frac { n ( 1 - a ) a ^ { ( n - 1 ) / 2 } } { ( 1 - q / a ) ( 1 - a q ) ( 1 - a ^ n ) } \\bigg \\} \\\\ [ 5 p t ] & \\quad \\times \\frac { ( 1 - a q ^ n ) ( a - q ^ n ) } { ( 1 - a ) ^ 2 } - \\frac { 1 } { q ( q ^ { n - 1 } ; q ^ 2 ) _ 2 } . \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} \\kappa _ G ( s , y , y ^ \\prime ) = \\int _ { { \\rm I m } \\lambda = r } s ^ { i \\lambda } G ( \\lambda ) ( y , y ^ \\prime ) d \\lambda \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} \\mathcal B _ D ( \\Omega ) : = \\left \\{ u \\in \\mathcal H ^ 2 _ { 0 , D } ( \\Omega ) : \\mathcal Q _ { \\sigma } ( u , \\varphi ) = 0 \\ , , \\forall \\varphi \\in H ^ 2 _ 0 ( \\Omega ) \\right \\} . \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} E _ { t } ( x ) & : = \\int _ 0 ^ t \\nabla \\cdot e ^ { ( t - s ) \\Delta } \\langle \\mu ^ N _ { s } , V ^ N ( x - \\cdot ) \\left ( F \\big ( K \\ast u ^ N _ s ( x ) \\big ) - F \\big ( K \\ast u ^ N _ s ( \\cdot ) \\big ) \\right ) \\rangle \\ d s , \\\\ M ^ N _ { t } ( x ) & : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\nabla V ^ N ( x - X _ s ^ { i , N } ) \\cdot d W ^ i _ s . \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} C ^ 2 . u = ( - 1 6 h - 4 a ) B \\in N , \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty \\frac { \\mu _ + ( x ) } { x } e ^ { - i \\xi x } d x = \\int _ 2 ^ \\infty \\frac { e ^ { - i \\xi x } } { x } d x + O ( 1 ) \\ , . \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} W ( u , v ; x ) = W _ x ( u , v ) = u ( x ) v ' ( x ) - u ' ( x ) v ( x ) . \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} \\int _ { 0 } ^ a ( e ^ { i u ^ 2 } ) ' g ( u \\epsilon ) u ^ { - 1 } d u = e ^ { i a ^ 2 } \\epsilon \\left ( \\frac { g ( a \\epsilon ) } { a \\epsilon } \\right ) - \\epsilon g ' ( 0 ) - \\epsilon \\int _ { 0 } ^ a e ^ { i u ^ 2 } \\left ( \\frac { g ( u \\epsilon ) } { \\epsilon u } \\right ) ' d u \\ , . \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} S ( t ) \\xi = \\sum _ { n = 1 } ^ \\infty \\omega ( t , \\lambda _ n ) \\xi _ n \\varphi _ n . \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} \\Psi | \\partial \\Omega _ { \\rho _ { 1 } } , \\partial \\Omega _ { \\rho _ { 1 } } = \\{ R ( t ) t : t \\in \\mathbb { T } \\} , \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} F ( ( x - 2 k t ) / ( 2 \\sqrt t ) ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { \\mathbb { R } } \\widehat F ( \\xi ) \\exp ( i \\xi ( x - 2 k t ) / ( 2 \\sqrt t ) ) d \\xi \\ , . \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} \\eta ^ \\rightarrow \\left ( \\prod _ { i \\in \\mathcal { I } } U _ i \\right ) = f ^ { - 1 } \\left ( \\prod _ { i \\in \\mathcal { I } } \\eta ^ \\rightarrow _ i ( U _ i ) \\right ) . \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} \\lim _ { s \\to - \\infty } \\mathbf { P } \\left ( M _ { s } > S _ { t } \\right ) = 0 , \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} C _ { n , k } : = \\left ( \\frac { 1 } { \\sqrt { n } } \\right ) ^ { k } \\sum _ { i _ 0 , i _ 1 , \\ldots , i _ { k - 1 } } A _ { i _ { 0 } , i _ { 1 } } A _ { i _ { 1 } , i _ { 2 } } \\ldots A _ { i _ { k - 1 } , i _ { 0 } } . \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} A ( G / H ) : = \\{ \\varphi \\in \\mathcal { C } _ c ( G / H ) : L _ h \\varphi = \\varphi \\ \\forall h \\in H \\} . \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} \\Pr \\left ( Q _ t = A | Q _ { t - 1 } = A B \\right ) & = p _ 0 ( 1 - \\alpha ) \\frac { \\beta } { 1 - \\alpha } + ( 1 - p _ 0 ) \\beta \\\\ & = \\beta . \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} T _ { k + 1 } = T _ k \\cup \\{ t _ \\sigma , t _ \\tau , t _ \\sigma \\wedge t _ \\tau \\} \\cup \\{ t _ s : s \\in E _ k \\} . \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} a _ 1 x _ 1 + \\cdots + a _ i x _ i + \\cdots + a _ j x _ j + \\cdots + a _ k x _ k = 0 \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} \\dim \\mu _ { \\eta } = \\min \\{ 1 , \\frac { h ( \\eta ) } { \\log \\eta ^ { - 1 } } \\} . \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} \\nabla \\zeta _ n = - \\frac { 1 } { 2 } \\frac { \\nabla \\kappa _ n } { \\kappa _ n ^ { 3 / 2 } } \\langle 1 | f _ n \\rangle + \\frac { 1 } { \\sqrt { \\kappa _ n } } \\nabla \\langle 1 | f _ n \\rangle \\ , . \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} n = x _ 1 ^ 2 + y _ 1 ^ 2 + z _ 1 ^ 2 + w _ 1 ^ 2 \\ \\ x _ 1 + 3 y _ 1 = 2 \\eta . \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} \\lambda _ k = \\frac { k ( k + 1 ) } { r ^ 2 } , N _ k = { k + 2 \\choose 2 } . \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} 4 \\langle \\Delta ' u , \\partial _ { n + 1 } ^ { 2 } u \\rangle = 4 \\langle \\nabla ' \\partial _ { n + 1 } u , \\nabla ' \\partial _ { n + 1 } u \\rangle - 4 \\langle \\Delta ' u , \\partial _ { n + 1 } u \\rangle _ { 0 } \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} g _ 2 ^ { - 1 } g _ 1 \\in _ G ( I _ \\eta ) & \\iff I _ \\eta = g _ 2 ^ { - 1 } g _ 1 I _ \\eta g _ 1 ^ { - 1 } g _ 2 \\\\ & \\iff g _ 2 I _ \\eta g _ 2 ^ { - 1 } = g _ 1 I _ \\eta g _ 1 ^ { - 1 } \\\\ & \\iff I _ { g _ 2 } = I _ { g _ 1 } . \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} \\begin{aligned} u ( \\xi ) & = e ^ { \\xi \\mathcal { L } } u _ 0 + \\int _ 0 ^ \\xi e ^ { ( \\xi - s ) \\mathcal { L } } f \\left ( u ( s ) , \\overline u ( s ) \\right ) d s = e ^ { \\xi \\mathcal { L } } u _ 0 + \\xi f ( u _ 0 , \\overline u _ 0 ) + \\mathcal { R } _ { 2 , a } ( 0 , \\xi ) \\end{aligned} \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} F ( z ) = \\gamma \\exp ( H _ { \\sigma } ( z ) ) , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} \\AA _ r = \\R [ \\epsilon ] / ( \\epsilon ^ { r + 1 } ) , \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} \\langle T x , \\tau \\rangle = 0 \\ , . \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} L = \\phi \\left [ c + c \\frac { L [ \\phi ^ { - 1 } ] ' ( L ) } { \\phi ^ { - 1 } ( L ) } \\right ] . \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} ( i d \\otimes r ) ( r \\otimes i d ) ( i d \\otimes r ) = ( r \\otimes i d ) ( i d \\otimes r ) ( r \\otimes i d ) . \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} \\int _ { G / H } \\psi ( x H ) d T _ H ( \\nu ) ( x H ) = \\int _ G \\psi _ q ( x ) d \\nu ( x ) , \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} \\left \\{ \\{ B _ 1 , \\ldots , B _ \\ell \\} : | B _ 1 \\cap \\cdots \\cap B _ \\ell | = a \\right \\} . \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} 2 ^ { s _ t q } \\| \\Delta _ q \\rho ( t ) \\| _ { L ^ 2 } & \\leq d _ q \\| \\rho _ 0 \\| _ { B ^ { s _ t } _ { 2 , r } } + d _ q \\int _ 0 ^ t \\| f ( \\tau ) \\| _ { B ^ { s _ t } _ { 2 , 1 } } ~ d \\tau \\\\ & \\quad + C d _ q \\int _ 0 ^ t 2 ^ { \\big ( - \\eta \\int _ { \\tau } ^ { t } V ( s ) d s \\big ) q } ( \\sqrt { q + 2 } V ( \\tau ) + 1 ) \\| \\rho \\| _ { B ^ { s _ \\tau } _ { 2 , r } } ~ d \\tau . \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} a & = 0 . 5 , & K _ p & = 1 , & K _ d & = 1 , & \\theta ^ * & = 1 . \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} m _ \\beta = c _ \\beta ^ 2 | z ^ 2 - 1 | ^ { 2 \\beta } | z | ^ { - 4 - 4 \\beta } \\ , | d z | ^ 2 , - 1 < \\beta < - 1 / 2 , \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} ( n - 1 ) - \\sum _ { j = 1 } ^ s m _ j = i \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} X = \\frac { \\partial } { \\partial x } , Y = \\frac { \\partial } { \\partial x } + f ( x ) \\frac { \\partial } { \\partial y } , \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} d _ { \\N , \\phi ' } ( A ) = \\lim _ { \\substack { N \\to \\infty \\\\ I _ N \\in \\phi ' } } \\frac { | I _ N \\cap A | } { | I _ N | } \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} S [ X ] = \\int d z d \\bar { z } ( G _ { \\mu \\nu } + B _ { \\mu \\nu } ) \\partial X ^ \\mu \\bar { \\partial } X ^ \\nu = : \\int d z d \\bar { z } E _ { \\mu \\nu } \\partial X ^ \\mu \\bar { \\partial } X ^ \\nu . \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} B ( | \\xi | ; \\rho , \\theta ) = \\frac { 1 } { 2 } \\left ( | \\xi | ^ { 2 \\rho } + | \\xi | ^ { 2 \\theta } \\right ) B _ 0 + i | \\xi | B _ 1 \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} M _ l ( \\lambda , \\mu ) = \\begin{pmatrix} l ( l - 1 - \\lambda ) & ( l + 2 ) ( l + 1 - \\lambda ) \\\\ l ( l + N - 2 ) ( l - 1 ) - \\mu & l ( l ( l - 5 ) + N ( l - 1 ) - 2 ) - \\mu , \\end{pmatrix} \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} { \\mathcal L } ( \\psi ) = h ~ ~ ~ ~ \\mathbb R ^ 2 , \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} \\| \\nabla \\omega \\| _ { L ^ \\infty _ t ( L ^ p ) } ^ 2 + \\| \\nabla \\theta \\| _ { L ^ \\infty _ t ( L ^ p ) } ^ 2 \\leq C . \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} \\partial _ t f ^ { \\varepsilon , a , n } + \\big \\{ \\beta _ \\varepsilon ( v ) + [ v \\ ! - \\ ! \\beta _ \\varepsilon ( v ) ] \\eta _ \\varepsilon ( x ) \\big \\} \\cdot \\nabla _ { \\ ! x } f ^ { \\varepsilon , a , n } - \\mathbf { B } \\cdot \\nabla _ { \\ ! v } f ^ { \\varepsilon , a , n } = Q ^ \\varepsilon [ f ^ { \\varepsilon , a , n } ] \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} d X _ { t } ^ { i , N } = F _ { A } \\left ( \\frac { 1 } { N } \\sum _ { k = 1 } ^ { N } ( K \\ast V ^ { N } ) ( X _ { t } ^ { i , N } - X _ { t } ^ { k , N } ) \\right ) \\ ; d t + \\sqrt { 2 } \\ ; d W _ { t } ^ { i } , t \\leq T , 1 \\leq i \\leq N , \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\left \\| e ^ { i t \\partial ^ 2 _ { x x } } h - \\frac { 1 } { 1 + i } \\frac { e ^ { i x ^ 2 / ( 4 t ) } } { \\sqrt t } \\widehat h ( x / ( 2 t ) ) \\right \\| _ { L ^ 2 ( \\R ) } = 0 , \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} w = \\cdots y a b \\cdots c \\cdots \\rightsquigarrow w ^ * = \\cdots y b a \\cdots c \\cdots . \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { 1 } { T } \\int _ 0 ^ T f ( X _ t ) d t = \\int _ { \\mathcal { E } } f ( y ) \\mu _ x ( d y ) . \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} | | \\gamma ^ X ( t ) | | ^ 2 = ( A _ 0 \\gamma ( t ) , \\gamma ( t ) ) + | | \\gamma ^ V ( t ) | | ^ 2 , \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} N ! \\prod _ { j = 1 } ^ p \\big | f ( \\beta _ j ) \\big | ^ { m _ j } \\le \\prod _ { i = 1 } ^ s \\Big ( n _ i ! \\prod _ { k \\neq i } | \\alpha _ i - \\alpha _ k | ^ { n _ k } \\Big ) . \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} a ( u , v ) = l ( v ) ( \\forall v \\in H ^ s ( \\Omega ) ) . \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} V = V _ 1 \\oplus \\ldots \\oplus V _ k \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} 2 ^ { g + k + 1 } - \\sum _ { i = k + n + 3 } ^ { g + k + 1 } H _ i = 2 ^ { g + k + 1 } - \\sum _ { i = k + n + 2 } ^ { g + k + 1 } H _ i + H _ { k + n + 2 } . \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} h ( x , t ) = \\frac { ( c _ 1 t - x ) ^ m ( c _ 2 t + x ) ^ n } { t ^ { m + n + 1 } } , \\ \\ \\ \\ \\ - c _ 2 t \\le x \\le c _ 1 t \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\infty } \\sup _ { 0 \\leq t \\leq T } \\| \\bar { \\eta } ^ { \\nu } ( t , \\cdot ) - \\zeta ( t , \\cdot ) \\| _ { H ^ { - 1 } } ^ 2 = 0 \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} | A \\cup B | - \\sum _ { \\beta \\in B } ( m ( \\beta ) + 1 ) = s - \\sum _ { \\beta \\in B } m ( \\beta ) = 1 . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} & \\bar { a } _ k ( h ) = \\oint _ { C _ k ( h ) } \\frac { a _ 1 ( x ) \\nabla H ( x ) \\cdot \\nabla H ( x ) } { | \\nabla H ( x ) | } d s , \\\\ & \\bar { \\beta } _ k ( h ) = \\oint _ { C _ k ( h ) } \\frac { L _ 1 H ( x ) } { | \\nabla H ( x ) | } d s , \\\\ & \\beta _ i ( \\O _ k ) = \\oint _ { \\tilde { C } _ i ( \\O _ k ) } \\frac { a _ 1 ( x ) \\nabla H ( x ) \\cdot \\nabla H ( x ) } { | \\nabla H ( x ) | } d s , \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} \\mathcal U _ N = \\big \\{ u = h + \\overline h \\ , : \\ , \\ h ( x ) = - { \\rm e } ^ { i x } \\frac { Q ' ( { \\rm e } ^ { i x } ) } { Q ( { \\rm e } ^ { i x } ) } \\ , , \\ , \\ , Q \\in \\C _ { N } ^ + [ z ] \\big \\} \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} S _ i = 0 ( i = 1 , \\ldots , 4 ) . \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} L _ { } ( y , t ) : = \\max \\lbrace 0 , 1 - y t \\rbrace \\ , \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} & ( \\rho _ { i _ 1 } \\cdots \\rho _ { i _ p } ) \\rho _ 0 ( \\rho _ { i _ 1 } \\cdots \\rho _ { i _ p } ) ^ { - 1 } \\cdot ( \\rho _ { j _ 1 } \\cdots \\rho _ { j _ q } ) \\rho _ 0 ( \\rho _ { j _ 1 } \\cdots \\rho _ { j _ q } ) ^ { - 1 } \\\\ & = ( \\rho _ { j _ 1 } \\cdots \\rho _ { j _ q } ) \\rho _ 0 ( \\rho _ { j _ 1 } \\cdots \\rho _ { j _ q } ) ^ { - 1 } \\cdot ( \\rho _ { i _ 1 } \\cdots \\rho _ { i _ p } ) \\rho _ 0 ( \\rho _ { i _ 1 } \\cdots \\rho _ { i _ p } ) ^ { - 1 } , \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} E _ { d } \\left | \\frac { 1 } { \\sqrt { N } } \\sum _ { i = 1 } ^ { N } \\left ( \\frac { S _ { i , N } } { \\pi _ { i , n } } - 1 \\right ) \\right | ^ { 2 } \\leq \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\frac { 1 - \\pi _ { i , N } } { \\pi _ { i , N } } , \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} \\frac { d ^ a } { d X ^ a } F ( X ) \\Big | _ { X = \\l } = 0 \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\xi _ 4 ^ { 3 n } \\sum _ { a = 0 } ^ n ( - 1 ) ^ { a } \\sum _ { b = 0 } ^ a \\xi _ 4 ^ { - b } ( b + 1 ) . \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} \\psi _ 0 = ( i w ^ + { \\varphi _ 0 } ^ + , i w ^ - { \\varphi _ 0 } ^ - ) , \\psi _ j ^ \\ell = ( i w ^ + { \\varphi _ j ^ \\ell } ^ + , i w ^ - { \\varphi _ j ^ \\ell } ^ - ) , ~ \\ell = 1 , 2 , \\ , j \\in \\mathbb { N } ^ + , \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} K ( \\Psi ; M ; \\gamma ) = \\prod _ { j \\in M } ( 1 - L _ j ) \\prod _ { ( i , j ) \\in \\Delta ^ + \\setminus \\Psi } ( 1 - R _ { i j } ) \\ , k _ \\gamma \\ , . \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} B ( e , f x g ) & = f B ( e , x g ) + B ( e , f ) x g = f B ( e , x g ) \\\\ & = f x B ( e , g ) + f B ( e , x ) g = f B ( e , x ) g . \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} s ( f , \\Psi , \\mathcal { P } ^ 1 _ n ) & = \\sum _ { k } f ( x _ k ^ { n , 1 } ) \\ , | I ^ { n , 1 } _ k | _ \\Psi \\\\ & = \\sum _ { k } \\int _ { I _ k ^ { n , 1 } } f ( x _ k ^ { n , 1 } ) ( \\psi - \\psi ( x _ k ^ { n , 1 } ) ) + S ( f \\psi , \\Lambda , \\mathcal { P } _ n ) , \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} r _ { t } = \\sup \\{ \\langle x , e _ { 1 } \\rangle : \\xi _ { t } ( x ) > 0 \\} . \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} \\Delta ^ { I } n _ { 2 } = \\Delta ^ { I } \\Big ( - \\frac { b v } { \\sqrt { \\Phi } } \\Big ) = \\lambda _ { 2 1 } \\Big ( - \\frac { a u } { \\sqrt { \\Phi } } \\Big ) + \\lambda _ { 2 2 } \\Big ( - \\frac { b v } { \\sqrt { \\Phi } } \\Big ) + \\lambda _ { 2 3 } \\Big ( \\frac { \\sqrt { \\omega } } { \\sqrt { \\Phi } } \\Big ) , \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} \\zeta ' ( - 1 ) = \\frac { \\partial } { \\partial s } \\zeta _ 1 ( s , 1 ) \\big | _ { s = - 1 } , \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} { \\mathcal L } _ \\pm ( \\phi ) = & - \\Delta \\phi ^ \\pm \\ , + \\ , \\Big [ \\ , A _ \\pm \\big ( { U ^ \\pm } ^ 2 - { t ^ \\pm } ^ 2 \\big ) + B \\big ( { U ^ \\mp } ^ 2 - { t ^ \\mp } ^ 2 \\big ) \\ , \\Big ] \\phi ^ \\pm \\\\ & \\ , + \\ , 2 A _ \\pm { \\mathbf { R e } } \\Big ( w ^ \\pm \\overline { \\phi ^ \\pm } \\Big ) w ^ \\pm \\ , + \\ , 2 B { \\mathbf { R e } } \\Big ( w ^ \\mp \\overline { \\phi ^ \\mp } \\Big ) w ^ \\pm . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} ( \\xi _ 2 ^ { y , \\varepsilon } ) _ t ( x , t ) - ( D _ x ^ \\alpha \\xi _ 2 ^ { y , \\varepsilon } ) _ x ( x , t ) - f ( x , t ) = - ( N _ 2 + \\| f \\| _ \\infty ) - N _ 1 \\Gamma ( 2 + \\alpha ) - f ( x , t ) \\le 0 \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} H : = \\mathbb { C } [ E _ { i , j } | 1 \\leq i < j \\leq n ] / \\mathcal { I } , \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} Q ( \\pi ( A _ p ) , \\pi ( A _ q ) ) \\ = \\ & Q ( A _ p , A _ q ) \\ \\{ p , q \\} \\not = \\{ p _ 1 , p _ 2 \\} , \\\\ Q ( \\pi ( A _ { p _ 1 } ) , \\pi ( A _ { p _ 2 } ) ) \\ = \\ & Q ( A _ { p _ 1 } , A _ { p _ 2 } ) + 2 . \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} \\Phi ( z ) = z \\left ( \\frac { z } { \\eta _ { \\nu } ( z ) } \\right ) ^ { k - 1 } , z \\in \\mathbb { D } . \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} V _ { n , k , w } & = \\left ( \\frac { 1 } { n } \\right ) ^ { \\frac { k } { 2 } } \\sum _ { \\emptyset \\subsetneq E _ { f } \\subsetneq E _ { w } } \\prod _ { e \\in E _ { f } } \\sigma _ { e } ( \\frac { 2 \\beta } { \\sqrt { n } } ) \\prod _ { e \\in E _ { w } \\backslash E _ { f } } B _ { e } \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d Y ( t ) & = \\bar { b } ( Y ( t ) ) \\ , d t + \\bar { \\sigma } ( Y ( t ) ) d B ( t ) , \\\\ Y ( 0 ) & = F ( x ) . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} \\partial _ { t } \\upsilon = i \\left ( \\Delta \\upsilon + A \\upsilon + F \\left ( x , t \\right ) \\right ) , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} \\alpha = \\sigma _ { { \\rm f o l d } , L } - \\sigma _ { { \\rm f o l d } , R } \\ , , \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} | | f | | _ { \\dot { H } ^ { 1 } _ { e } } ^ { 2 } = | | \\partial _ { r } f | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } + | | \\frac { f } { r } | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} 2 h = f _ 7 + f _ { 1 0 } = f _ 4 + f _ { 1 2 } = f _ 8 + f _ 9 = f _ 3 + f _ 6 = f _ 5 + f _ { 1 1 } = f _ { 1 } + f _ { 2 } , \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} \\int _ { 0 } ^ x \\frac { e ^ { i ( u / ( 2 \\sqrt t ) - k \\sqrt t ) ^ 2 } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) d u = \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{gather*} \\bar { I } ( u _ 0 , v _ 0 ) = \\min _ { ( u , v ) \\in X } \\bar { I } ( u , v ) \\quad \\quad \\ , \\bar { I } ' ( u _ 0 , v _ 0 ) = 0 . \\end{gather*}"}
-{"id": "5815.png", "formula": "\\begin{align*} g ( z ) = N ( z - \\beta _ 1 ) ^ { m _ 1 } \\cdots ( z - \\beta _ p ) ^ { m _ p } . \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} y ^ 2 - m ^ 3 x y = x ^ 3 + ( - 2 n ^ 6 + n ^ 3 m ^ 3 ) x ^ 2 + ( n ^ { 1 2 } - n ^ 9 m ^ 3 ) x \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} e ^ { z u } \\prod _ { j = 1 } ^ { n } \\frac { u } { e ^ { \\omega _ { j } u } - 1 } = \\sum _ { m \\geq 0 } B _ { n , m } \\left ( z \\mid { \\boldsymbol { \\omega } } \\right ) \\Psi _ { m } ( u ) ( | \\omega _ { j } u | < 2 \\pi , j = 1 , \\ldots , n ) . \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} - \\frac { h ( z ) } { h ' ( z ) } t ' ( z ) \\ , = \\ , - 2 t ( z ) \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\Big \\| | x | ^ { - \\frac s 2 } u _ \\tau \\Big \\| _ 2 ^ 2 d \\tau + J ( u ( x , t ) ) = J ( u _ 0 ) \\quad t \\in ( 0 , T ) . \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} U _ { 1 2 } : = \\{ A _ { 1 } , A _ { 2 } , A _ { 3 } , B _ { 1 } , B _ { 2 } , B _ { 3 } , A _ { 1 } ' , A _ { 2 } ' , A _ { 3 } ' , B _ { 1 } ' , B _ { 2 } ' , B _ { 3 } ' \\} . \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} \\lambda \\ast _ { G / H } \\lambda ' ( \\psi ) = \\int _ { G / H } \\psi ( g H ) d \\lambda \\ast _ { G / H } \\lambda ' ( g H ) . \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} B _ { t } = \\left \\{ \\sum _ { x \\in H ( i ) } \\eta ^ { * } _ { t } ( x ) \\leq \\sum _ { x \\in H ( i ) } \\eta _ { t } ( x ) , i \\in I \\right \\} , \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} \\underline U ( t , \\pm \\underline h ( t ) ) = 0 \\ \\mbox { f o r } \\ t \\geq 0 . \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} a _ v = \\frac { n ^ { g - 1 } B ^ n } { 4 e ^ { | \\alpha | _ v } \\alpha ^ n ( n - 1 ) ! } \\in K _ v ^ \\times . \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} X ( \\epsilon ) = \\sum _ { \\ell = 0 } ^ { \\infty } \\frac { X _ \\ell } { \\ell ! } \\epsilon ^ \\ell , \\ ; \\ ; \\ ; X = h , A , V , \\ ; \\mbox { o r } \\ ; H . \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} P ( x , y ) = x y = \\tfrac { 1 } { 4 } ( x + y ) ^ { 2 } - \\tfrac { 1 } { 4 } ( x - y ) ^ { 2 } = P \\circ \\pi _ { 1 } ( x , y ) + P \\circ \\pi _ { 2 } ( x , y ) . \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} \\alpha ( \\chi ; A ) = d \\beta ( \\chi ; A ) , \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} & P _ { d } \\left \\{ | \\mathbb { H } _ { N } ' f | \\leq M \\right \\} + \\widetilde { B } _ { N } \\geq 1 - \\eta \\quad N = 1 , 2 , \\dots \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} \\dim H ^ 0 ( X _ j , \\mathcal { O } _ { X _ j } ( F _ j ) ) = \\dim H ^ 1 ( X _ j , \\mathcal { O } _ { X _ j } ( F _ j ) ) = 1 . \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} ( X , T ) = S ^ 1 \\times ( Y , K ) , \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} \\alpha _ { k , 0 } Y ^ n + \\displaystyle \\sum _ { j = 1 } ^ { k } \\left ( \\alpha _ { k , j - 1 } + \\alpha _ { k , j } \\right ) Y ^ { n + j } + \\alpha _ { k , k } Y ^ { n + k + 1 } = f ( t _ n , Y ^ n ) \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} 0 & \\le J [ u ^ * , p _ \\varepsilon ] ( x _ \\varepsilon , t _ \\varepsilon ) - J [ v _ * , p _ \\varepsilon ] ( y _ \\varepsilon , s _ \\varepsilon ) \\\\ & \\quad + K _ { ( 0 , x _ \\varepsilon ) } [ u ^ * , p _ \\varepsilon ] ( x _ \\varepsilon , t _ \\varepsilon ) - K _ { ( 0 , y _ \\varepsilon ) } [ v _ * , p _ \\varepsilon ] ( y _ \\varepsilon , s _ \\varepsilon ) \\\\ & \\quad + f ( x _ \\varepsilon , t _ \\varepsilon ) - f ( y _ \\varepsilon , s _ \\varepsilon ) . \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} \\widetilde A ( k ) = 1 - \\Pi ( k ) \\cdot \\chi _ { E _ s ^ c } , \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} X _ \\tau : = \\{ ( x _ 1 , x _ 2 , \\ldots , x _ k ) \\in X : x _ i \\ \\ i \\in C _ j , \\ 1 \\leq j \\leq \\ell ( \\tau ) \\} \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\ddot { u } ( t ) + \\eta \\dot { u } ( t ) & = \\abs { \\nabla u ( t ) } \\operatorname { d i v } \\left ( \\frac { \\nabla u ( t ) } { \\abs { \\nabla u ( t ) } } \\right ) , & \\R ^ 2 \\times ( 0 , \\infty ) , \\\\ u ( 0 ) = u _ 0 , & \\dot { u } ( 0 ) = 0 \\ ; & \\R ^ 2 \\times 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} D _ p ( x _ n , x _ m ) = & D _ p ( x _ n , \\Pi _ M ^ p x _ m ) + D _ p ( \\Pi _ M ^ p x _ m , x _ m ) + \\\\ & + \\langle j _ p ( \\Pi _ M ^ p x _ m ) - j _ p ( x _ m ) , x _ n - \\Pi _ M ^ p x _ m \\rangle \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} \\mathcal B ' = \\{ e _ { t _ 1 } * e _ { t _ 2 } * \\ldots * e _ { t _ { n - r } } * f _ { k _ 1 } * f _ { k _ 2 } * \\ldots * f _ { k _ r } ; ~ r \\in [ 0 ; n ] , ~ 0 \\leqslant t _ 1 \\leqslant t _ 2 \\leqslant \\ldots \\leqslant t _ { n - r } , ~ 0 < k _ 1 \\leqslant k _ 2 \\leqslant \\ldots \\leqslant k _ { r } \\} \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} \\Delta _ p \\psi ( \\varrho ) = \\left [ ( p - 1 ) \\psi '' + \\frac { v _ h ' ( \\varrho ) } { v _ h ( \\varrho ) } \\psi ' \\right ] | \\nabla \\varrho | ^ p . \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} \\sigma _ { A } ( \\xi , \\eta ) = \\sum _ { j , k \\in \\Z } a _ { j , k } \\phi ( 2 ^ { - j } \\xi ) \\phi ( 2 ^ { - k } \\eta ) , \\xi , \\eta \\in \\R ^ n , \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} \\begin{aligned} \\deg ( { } ^ { i } \\ ! t ^ { j ' , h ' } _ { j , h } ) & = \\min ( h - h ' + 1 , h - h ' + 1 + j ' - j ) , \\\\ \\deg ( { } ^ { i } \\ ! s ^ { j ' , h ' } _ { j , h } ) & = \\min ( h - h ' , h - h ' + j ' - j ) . \\end{aligned} \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} \\tilde A _ { h , \\tilde h } = \\tilde h ^ m A _ h - \\tilde \\omega . \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} \\dim ( \\hat { \\sigma } ( X ) ) ^ 2 = \\sigma ( \\dim ( X ) ^ 2 ) \\dim ( \\mathcal { Z } ( \\mathcal { C } ) ) / \\sigma ( \\dim ( \\mathcal { Z } ( \\mathcal { C } ) ) ) . \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} \\lambda ( t ) ^ { \\alpha - 1 } \\leq | z _ { \\sigma } | = | \\sigma \\lambda ( s ) ^ { \\alpha - 1 } + ( 1 - \\sigma ) \\lambda ( t ) ^ { \\alpha - 1 } | \\leq \\lambda ( s ) ^ { \\alpha - 1 } , 0 \\leq \\sigma \\leq 1 , s \\geq t \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} { \\rm S c h } ( h ) _ t = { \\rm S c h } ( h ) ^ { ''' } + 3 { \\rm S c h } ( h ) ' { \\rm S c h } ( h ) . \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 1 & 0 \\\\ 0 & - 1 \\end{bmatrix} = h , B = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} , C = \\begin{bmatrix} 0 & 1 \\\\ - 1 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } [ 4 k - 1 ] \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a , q ^ { - 1 } / b , c q ^ { - 1 } , d q ^ { - 1 } , q ^ { - 1 } ; q ^ 2 ) _ k } { ( q ^ 2 / a , a q ^ 2 , b q ^ 2 , q ^ 2 / c , q ^ 2 / d , q ^ 2 ; q ^ 2 ) _ k } \\bigg ( \\frac { b q ^ 7 } { c d } \\bigg ) ^ k \\equiv 0 \\pmod { \\Phi _ n ( q ) } . \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} h ( u _ h ) = \\rho e = v _ h \\in ( \\partial B _ \\rho \\cap X ) , \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} J = \\left ( j _ { 1 } , \\dots , j _ { k } , \\dots , j _ { N } \\right ) \\in \\mathbb { N } ^ { N } \\quad \\mbox { s u c h t h a t $ j _ { 1 } + \\dots + j _ { k } + \\dots + j _ { N } = p - 1 \\geq 2 $ , } \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} \\tau ( t ) = \\tau _ 0 \\exp ( - \\int _ 0 ^ t K ( s ) d s ) , \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} \\Theta ^ { } _ 1 ( \\Lambda ) ( 0 ) & = A _ c k _ c ( 0 ) + g _ c ( K { } { } ( 0 ) ) - k _ c ( 0 ) . \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} \\begin{array} { r } \\displaystyle 0 \\ne \\hat f ( x _ 2 ) - \\hat f ( x _ 1 ) = \\lim _ \\alpha f ( x _ { 2 \\alpha } ) - \\lim _ \\alpha f ( x _ { 1 \\alpha } ) = \\lim _ \\alpha f ( q _ j ( x _ { 2 \\alpha } ) ) - \\lim _ \\alpha f ( q _ j ( x _ { 1 \\alpha } ) ) \\medskip \\\\ \\displaystyle = \\tilde f ( q _ j ( x _ 2 ) ) - \\tilde f ( q _ j ( x _ 1 ) ) = 0 , \\end{array} \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} \\sum _ { \\alpha } \\Vert x _ { \\alpha } \\Vert ^ { q } \\rho ^ { \\vert \\alpha \\vert q } = \\sum _ { k = 0 } ^ { N } \\rho ^ { q k } \\sum _ { \\substack { \\alpha \\in F \\\\ \\alpha _ { n } = k } } \\Vert x _ { \\alpha } \\Vert ^ { q } \\rho ^ { ( \\vert \\alpha \\vert - \\alpha _ { n } ) q } \\ , . \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} I ^ \\varphi ( \\bar x ) : = \\{ l \\in \\mathcal Q \\ , | \\ , \\varphi ( G _ l ( \\bar x ) , H _ l ( \\bar x ) ) = 0 \\} . \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} \\mathbb { P } _ \\eta \\left ( \\mathcal { G } , \\mathcal { T } _ t ( 0 ) \\leq \\frac { 1 - p } { 4 } \\delta ^ d t \\right ) \\leq | D | d e ^ { - \\frac { c ( 1 - p ) \\delta ^ { d - 1 } } { 8 } t } = ( \\lfloor 2 d \\alpha t \\rfloor + 1 ) ^ d d e ^ { - \\frac { c ( 1 - p ) \\delta ^ { d - 1 } } { 8 } t } \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} \\log \\det \\Delta _ { - 2 / 3 } = \\frac 2 3 \\log \\pi + \\frac 1 3 \\log \\frac 2 3 - 2 \\log \\Gamma \\left ( \\frac 2 3 \\right ) \\approx 0 . 0 2 1 7 ; \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} ( f \\circ _ i g ) \\circ _ { i + j - 1 } h = ~ & f \\circ _ i ( g \\circ _ j h ) , ~ 1 \\leq i \\leq m , ~ 1 \\leq j \\leq n , \\\\ ( f \\circ _ i g ) \\circ _ { j + n - 1 } h = ~ & ( f \\circ _ j h ) \\circ _ i g , ~ 1 \\leq i < j \\leq m , \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} B _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } + \\mathbf { 1 } ) & = \\sum _ { \\mathbf { n } \\subset \\mathbf { m } } \\binom { \\mathbf { m } } { \\mathbf { n } } ^ { ( d ) } B _ { \\mathbf { n } } ^ { ( d ) } ( \\mathbf { z } ) \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} \\partial _ { t } \\left ( \\frac { 1 6 } { \\lambda ( t ) ^ { 3 } } \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) \\left ( K ( s - t , \\lambda ( t ) ) + K _ { 1 } ( s - t , \\lambda ( t ) ) \\right ) \\right ) = I + I I + I I I + I V \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} ( \\mathcal { T } Q ) _ j & : = \\min \\left \\{ \\sum _ { l = 1 } ^ j Q _ l - \\sum _ { l = 1 } ^ { j - 1 } ( \\mathcal { T } Q ) _ l , { E } _ j \\right \\} , \\\\ ( \\mathcal { T } { E } ) _ j & : = Q _ { j + 1 } + { E } _ j - ( \\mathcal { T } Q ) _ j , \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} \\int _ A h d m = \\int A d \\mu = \\mu ( A ) \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} H ( t ; x , y , 1 , u , z , v ) : & = F ( t ; x , y , 1 , u , z , v ) - \\frac { t x ( y - y z r + z ) F ( t ; x , y - y r + 1 , 1 , u , z , v ) } { ( t u x + y ^ { - 1 } - t u ) ( y - y r + 1 ) } . \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} | - \\frac { c _ { b } } { 2 } \\int _ { 0 } ^ { \\infty } d \\xi \\sin ( t \\xi ) \\xi J _ { 2 } ( r \\xi ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } | & \\leq \\frac { C r ^ { 2 } } { t ^ { 4 } \\log ^ { b } ( t ) } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} v ^ - = f ( v ^ + ) = \\begin{pmatrix} 0 \\ & \\ - I _ { l - h } \\ & \\ 0 \\\\ I _ { h - 1 } \\ & \\ 0 \\ & \\ 0 \\\\ 0 \\ & \\ 0 \\ & \\ - I _ { n - l + 2 } \\end{pmatrix} v ^ + + \\begin{pmatrix} \\beta _ { l - h } \\\\ 0 _ { h - 1 } \\\\ 2 \\cdot \\beta _ { n - l + 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} A _ w : = \\frac { i } { 2 \\pi } \\int _ \\gamma \\lambda ^ w ( A - \\lambda ) ^ { - 1 } \\ , \\dd \\lambda , w \\in \\C , \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} | A _ 1 \\cap \\cdots \\cap A _ { k - 2 } \\cap B | = b _ { k - 1 } + b _ k . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} L _ c w = f ( t , x ) + \\sum _ { j + \\alpha \\geq 2 } c _ { j , \\alpha } ( t , x ) w ^ j \\Bigl ( \\frac { \\partial w } { \\partial x } \\Bigr ) ^ { \\alpha } . \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} \\chi ( D ) = \\inf \\{ \\varepsilon > 0 : D \\varepsilon - \\} \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} \\Phi _ A ( g , s _ 1 , s _ 2 ) : = A ( s _ 1 , g s _ 2 ) . \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} G ( z ) = F ( \\eta _ { \\mu _ { 1 } } ( z ) ) , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} \\partial _ t f + v \\cdot \\nabla _ { \\ ! x } f = \\bar { A } _ g f , \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} 0 & = \\det \\left ( M - \\mu I _ { n } + \\sqrt { \\frac { p \\left ( n + 2 \\right ) } { 1 - p } } l l ^ { \\top } \\right ) \\\\ & = \\det \\left ( M - \\mu I _ { n } \\right ) \\det \\left ( I _ { n } + G _ { M } \\left ( \\mu \\right ) \\sqrt { \\frac { p \\left ( n + 2 \\right ) } { 1 - p } } l l ^ { \\top } \\right ) \\\\ & = \\det \\left ( M - \\mu I _ { n } \\right ) \\left ( 1 + l ^ { \\top } G _ { M } \\left ( \\mu \\right ) \\sqrt { \\frac { p \\left ( n + 2 \\right ) } { 1 - p } } l \\right ) , \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} [ V ^ b ( D ) ] - [ e _ 1 ] = \\operatorname { I n d } _ \\infty ( D ) \\ ; \\ ; \\ ; \\ ; K _ 0 ( \\mathcal { I } _ G ( M ) ) = K _ 0 ( C ^ * ( M _ 0 \\subset M ) ^ G ) \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\left ( \\deg f ^ \\ast D _ i - \\frac { 1 } { m _ i } \\deg f ^ \\ast D _ i \\right ) \\leq \\sum _ { i = 1 } ^ r \\left ( a _ i \\cdot \\mathrm { g e n u s } ( C ) + \\frac { \\epsilon } { r } \\deg f ^ \\ast L \\right ) . \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} a _ k ( n ) & = a _ k ( n - 1 ) + a _ k ( n - 2 ) - a _ k ( n - 5 ) - a _ k ( n - 7 ) \\\\ & \\quad \\ , + a _ k ( n - 1 2 ) + a _ k ( n - 1 5 ) - a _ k ( n - 2 2 ) - a _ k ( n - 2 6 ) + \\cdots . \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\left \\langle x - A x , y - A y \\right \\rangle \\right | & = \\left | \\left \\langle x , y \\right \\rangle - \\left \\langle \\left ( 2 A - A ^ 2 \\right ) x , y \\right \\rangle \\right | \\\\ & \\ge \\left | \\left \\langle x , y \\right \\rangle \\right | - \\left | \\left \\langle \\left ( 2 A - A ^ 2 \\right ) x , y \\right \\rangle \\right | . \\end{aligned} \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} T ( u ) f ( v ) + f ( u ) T ( v ) = ~ & T ( u \\cdot f ( v ) + H ( T u , f ( v ) ) ) + T ( f ( u ) \\cdot v + H ( f ( u ) , T v ) ) \\\\ & + f ( u \\cdot T ( v ) + T ( u ) \\cdot v + H ( T u , T v ) ) , ~ u , v \\in M . \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} \\begin{aligned} \\kappa \\Sigma ^ 0 _ { 1 + \\alpha } ( X ) & : = \\ @ N _ \\kappa ^ \\alpha ( \\ @ O _ \\kappa ( X ) ) , \\\\ \\kappa \\Pi ^ 0 _ { 1 + \\alpha } ( X ) & : = \\{ \\neg B \\mid B \\in \\ @ N _ \\kappa ^ \\alpha ( \\ @ O _ \\kappa ( X ) ) \\} . \\end{aligned} \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} | R ( \\eta ) - \\tau _ 0 | = & \\frac { | P _ 1 ( \\eta ) + \\tau _ 0 P _ 2 ( \\eta ) | } { | P _ 2 ( \\eta ) | } \\\\ \\le & \\frac { c ^ { n ' } + | P _ 1 ( \\eta ) - P _ 1 ( \\l _ 0 ) | + | \\tau _ 0 | | P _ 2 ( \\eta ) - P _ 2 ( \\l _ 0 ) | } { | P _ 2 ( \\eta ) | } \\\\ \\le & \\frac { c ^ { n ' } + ( 1 + | \\tau _ 0 | ) L ( n ' ) ^ 2 | \\eta - \\l _ 0 | } { | P _ 2 ( \\eta ) | } . \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} ( x _ 1 - 1 ) ^ 2 & \\ , \\to \\ , \\min \\limits _ { x , y , z } \\\\ x _ 1 - y & \\ , \\leq \\ , 0 \\\\ x _ 2 - z & \\ , \\leq \\ , 0 \\\\ 0 \\ , \\leq y \\ , \\perp \\ , z & \\ , \\geq \\ , 0 . \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} \\begin{cases} & d X _ t = K \\ast u _ t ( X _ t ) \\ , d t + \\sqrt { 2 } d W _ t , t \\leq T , \\\\ & \\mathcal { L } ( X _ t ) = u _ t , ~ \\mathcal { L } ( X _ 0 ) = u _ 0 . \\end{cases} \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } \\varphi \\Phi g d x & = \\iint _ { \\mathbb { R } ^ { 2 n } } \\frac { \\partial } { \\partial y _ j } ( \\varphi \\Phi ) d \\pi \\\\ & = - \\iint _ { \\mathbb { R } ^ { 2 n } } \\varphi ( x ) \\frac { \\partial \\Phi } { \\partial y _ j } ( x , y ) d \\pi ( x , y ) \\\\ & = - \\int _ { \\mathbb { R } ^ { n } } \\varphi ( x ) \\frac { \\partial \\Phi } { \\partial y _ j } ( x , F ( x ) ) g ( x ) d x , \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} ( u , v ) _ { H _ 0 } : = ( u , v ) , ( \\psi , \\phi ) _ { H _ 1 } : = ( \\psi , \\phi ) \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} \\forall s \\in \\mathcal S \\ , \\forall a \\in \\mathcal A \\colon r _ { s , a } : = \\frac { Q _ \\delta ( x ^ a _ s ) } { \\min \\{ Q _ \\delta ( x ^ \\alpha _ s ) \\ , | \\ , \\alpha \\in \\mathcal A \\} } . \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} | | | \\Phi | | | ^ 2 _ m : = \\int _ G | | \\Phi ( g ) | | ^ 2 _ { b } ( 1 + L ( g ) ) ^ { 2 m } d g \\ , . \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} u g _ 1 ( x ) = u ^ 2 ( x - 1 ) ^ { r _ 1 } + u ^ 3 ( x - 1 ) ^ { k _ 4 } p _ 4 ( x ) , \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} T _ { \\eqref { e _ 3 7 _ a u x _ a p p r o x } } & \\overset { \\eqref { d _ a u x _ c g _ h a m } } { = } \\int _ 0 ^ T \\int _ { \\mathbb { T } ^ 1 } \\bar { H } _ K ( \\bar { \\eta } ^ { \\nu } ) \\beta ( t ) d \\theta d t \\\\ & = \\int _ 0 ^ T \\int _ { \\mathbb { T } ^ 1 } \\varphi ( \\bar { \\eta } ^ { \\nu } ) \\beta ( t ) d \\theta d t + \\int _ 0 ^ T \\int _ { \\mathbb { T } ^ 1 } \\left ( \\bar { H } _ K ( \\bar { \\eta } ^ { \\nu } ) - \\varphi ( \\bar { \\eta } ^ { \\nu } ) \\right ) \\beta ( t ) d \\theta d t \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} & | \\frac { \\sin ( 2 ( v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } ) ) } { 2 r ^ { 2 } } \\left ( \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\left ( \\cos ( 2 v _ { 5 } ) - 1 \\right ) - \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\sin ( 2 v _ { 5 } ) \\right ) | \\\\ & \\leq C \\frac { | v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } | } { r ^ { 2 } } \\left ( v _ { 5 } ( t , r ) ^ { 2 } + \\frac { r \\lambda ( t ) } { r ^ { 2 } + \\lambda ( t ) ^ { 2 } } | v _ { 5 } ( t , r ) | \\right ) \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} ( I - P _ { N } ) ( I - P _ M ) x = j _ { p ^ * } ( \\Pi ^ { p ^ * } _ { N ^ \\perp } \\Pi ^ { p ^ * } _ { M ^ \\perp } j _ p ( x ) ) . \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} \\eta _ S ( U \\cap S ) = f ^ { - 1 } ( \\eta ( U ) \\cap \\eta ( S ) ) . \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} h _ 0 ( u ) : = \\gamma ^ X ( t ) + | v | , u = R t e + v \\in Q , \\ ; t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} \\bar { x } ( \\theta ) = x _ j , \\theta \\in \\left ( \\frac { j - 1 } { N } , \\frac { j } { N } \\right ] . \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} \\frac { n } { \\pi } \\Im { m \\left ( E + \\i \\eta \\right ) } = \\sum _ { i \\in \\left [ n \\right ] } \\frac { 1 } { \\pi } \\frac { \\eta } { \\left ( \\lambda _ { i } - E \\right ) ^ { 2 } + \\eta ^ { 2 } } . \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n ^ { 1 / 6 } } Y ( m , k ) p ^ k & \\leq ( m - 1 ) ! ( 1 + C m ^ { 3 / 2 } n ^ { - 1 / 2 } e ^ { m ^ { 3 / 2 } n ^ { - 1 / 2 } } ) \\\\ & \\leq 2 ( m - 1 ) ! C m ^ { 3 / 2 } n ^ { - 1 / 2 } e ^ { m ^ { 3 / 2 } n ^ { - 1 / 2 } } \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} \\underline { \\psi } _ { N , 1 } = - \\overline { \\psi } _ { N , N + 1 } ^ { \\ast } , \\qquad \\overline { \\psi } _ { N , 1 } = - \\underline { \\psi } _ { N , N + 1 } ^ { \\ast } . \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} W = \\begin{bmatrix} W _ { 1 , 1 } & 0 & 0 & \\cdots & 0 \\\\ W _ { 2 , 1 } & W _ { 2 , 2 } & 0 & \\cdots & 0 \\\\ W _ { 3 , 1 } & W _ { 3 , 2 } & W _ { 3 , 3 } & \\cdots & 0 \\\\ 0 & W _ { 4 , 2 } & W _ { 4 , 3 } & \\ddots & 0 \\\\ 0 & 0 & W _ { 5 , 3 } & \\ddots & W _ { L _ { R } - 2 , L _ C } \\\\ \\vdots & \\vdots & \\vdots & \\ddots & W _ { L _ { R } - 1 , L _ C } \\\\ 0 & 0 & 0 & \\cdots & W _ { L _ R , L _ C } \\end{bmatrix} . \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} | v _ { 5 } ( t , r ) | & \\leq \\frac { C r } { t ^ { 7 / 2 } \\log ^ { 3 b - 3 + \\frac { 5 N } { 2 } } ( t ) } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} \\mathcal { S } _ G ^ c ( M , E ) : = \\{ S _ A , A \\in \\mathcal { A } _ { G } ^ { c } ( M , E ) \\} . \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} | f ( \\alpha _ i ) | = \\prod _ { k = 1 } ^ r | \\alpha _ i - \\alpha _ k | ^ { n _ k - 1 } = ( n _ i - 1 ) ! \\prod _ { k \\neq i } | \\alpha _ i - \\alpha _ k | ^ { n _ k - 1 } . \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} A = \\left [ a _ { m j } \\right ] m , j = 1 , 2 , . . . , N , N \\in \\mathbb { N } \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} \\psi _ { \\Lambda } = \\langle \\Psi _ { \\Lambda } , \\ ( \\ \\cdot \\ ) \\Psi _ { \\Lambda } \\rangle = \\hbox { T r } _ { \\mathcal { H } _ { \\Lambda } } ( \\Psi _ { \\Lambda } \\Psi _ { \\Lambda } ^ * \\ ( \\ \\cdot \\ ) ) . \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} \\begin{cases} | x | ^ { - s } u _ { t } - \\mathrm { d i v } ( | \\nabla u | ^ { p - 2 } \\nabla u ) = | u | ^ { q - 2 } u \\ln | u | & ~ \\Omega \\times ( 0 , T ) , \\\\ u ( x , t ) = 0 & ~ \\partial \\Omega \\times ( 0 , T ) , \\\\ u ( x , 0 ) = u _ 0 ( x ) & ~ x \\in \\Omega , \\end{cases} \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} H ( x ) = \\sum _ { i = 1 } ^ { N } \\left ( \\psi ( x _ i ) + \\frac { 1 } { 2 } \\sum _ { j : \\ 1 \\leq | j - i | \\leq R } M _ { i j } x _ i x _ j \\right ) , \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} P = c x _ { n + 1 } ^ { 3 } + x _ { n + 1 } \\sum _ { i = 1 } ^ { n } a _ { i } x _ { i } ^ { 2 } + B ( x _ { 1 } , \\dots , x _ { n } ) , \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} \\sqrt { R _ { \\ell } } \\varphi ' ( U _ { \\ell } ) \\mathbf { T } _ { N , \\ell , j , k } = \\partial _ { j } ( { R _ { \\ell } } U _ { \\ell } \\varphi ' ( U _ { \\ell } ) U _ { \\ell , k } ) - 2 \\sqrt { R _ { \\ell } } U _ { \\ell , } \\partial _ { j } \\sqrt { R _ { \\ell } } + g _ { j , k , \\varphi } , \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} S = | G | , \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} & \\mathrm { H o m } _ { D G _ { T ^ { 2 n } _ { J = T } } } ( E _ { ( r , A , r ' , \\mathcal { U } , p , q ) } , E _ { ( s , B , s ' , \\mathcal { V } , u , v ) } ) \\\\ & : = \\Gamma ( E _ { ( r , A , r ' , \\mathcal { U } , p , q ) } , E _ { ( s , B , s ' , \\mathcal { V } , u , v ) } ) \\bigotimes _ { C ^ { \\infty } ( T ^ { 2 n } _ { J = T } ) } \\Omega ^ { 0 , * } ( T ^ { 2 n } _ { J = T } ) , \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} T = \\mu [ U ] \\left ( a ^ { ( 1 ) } \\otimes \\ldots \\otimes \\widetilde { t } ^ { ( i ) } \\otimes \\ldots \\otimes a ^ { ( m ) } \\right ) & = \\\\ \\sum _ { k = 1 } ^ l \\beta _ k \\mu [ U ] \\left ( a ^ { ( 1 ) } \\right . & \\left . \\otimes \\ldots \\otimes b ^ { ( i ) } _ k \\otimes \\ldots \\otimes a ^ { ( m ) } \\right ) . \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} \\frac { d \\mathbb { Q } _ { n } } { d \\mathbb { P } _ { n } } = \\frac { 1 } { \\tau _ { n } } \\exp \\left \\{ - ( n - 1 ) \\beta ^ 2 + \\beta J \\right \\} \\exp \\left \\{ - \\frac { \\beta } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } A _ { i , i } - \\beta J ' \\right \\} Z _ { n } ( \\beta ) , \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} & H = \\dot \\rho \\left ( \\dot s + s ^ 2 \\dot \\rho \\right ) , & & P = \\dot s + 2 s ^ 2 \\dot \\rho - 2 \\nu s , \\\\ [ 4 p t ] & D = \\rho P - s \\dot \\rho , & & K = \\rho ^ 2 P + ( 1 - 2 s \\rho ) \\dot \\rho + 2 \\nu \\rho . \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} a _ 1 ' & = a _ 1 b _ 2 , \\cr a _ 2 ' & = - a _ { 2 } a _ { 3 } { b _ 4 } , \\cr a _ j ' & = - a _ { 2 j - 2 } a _ { 2 j - 1 } { b _ { 2 j - 4 } } { b _ { 2 j } } ; \\cr b _ 0 ' & = b _ 0 , \\cr b _ 1 ' & = b _ 1 b _ 2 + a _ 2 , \\\\ b _ j ' & = b _ { 2 j - 2 } b _ { 2 j - 1 } b _ { 2 j } + a _ { 2 j } b _ { 2 j - 2 } + a _ { 2 j - 1 } b _ { 2 j } . \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} \\phi ^ { \\pm } = \\sum _ { k \\in \\mathbb { Z } } \\phi _ k ^ { \\pm } ( r ) e ^ { i k \\theta } , \\quad \\mbox { o r } \\phi = \\sum _ { k \\in \\mathbb { Z } } \\phi _ k ( r ) e ^ { i k \\theta } \\ \\mbox { w i t h } \\phi _ k ( r ) = \\big ( \\phi _ k ^ + ( r ) , \\phi _ k ^ - ( r ) \\big ) , \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} K = - \\frac { \\partial ^ 2 } { \\partial x _ 3 ^ 2 } \\left ( \\frac { 1 } { 2 \\phi } \\right ) . \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} N : = - ( 2 a ^ 2 + 2 a \\tilde { \\phi } ( a ^ 2 - x _ 3 ^ 2 ) + 8 a \\tilde { \\phi } x _ 3 ^ 2 ) \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{gather*} \\varepsilon = 1 , u = \\phi = e ^ { - t } x ^ 2 y ^ 2 ( 1 - x ) ^ 2 ( 1 - y ) ^ 2 . \\end{gather*}"}
-{"id": "8893.png", "formula": "\\begin{align*} A _ { k , m } = ( a ^ { ( m ) } _ { i j } ) _ { 1 \\le i , j \\le k } , \\mbox { a n d } B _ { k , m } = ( b ^ { ( m ) } _ { i j } ) _ { 1 \\le i , j \\le k } , \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} \\widehat { { \\boldsymbol { \\omega } } } ( j ) : = & \\ , ( \\omega _ { 1 } , \\cdots , \\omega _ { j - 1 } , \\omega _ { j + 1 } , \\cdots , \\omega _ { r } ) \\in \\mathbb { C } ^ { r - 1 } \\\\ = & \\ , ( \\omega _ { 1 } , \\cdots , \\widehat { \\omega } _ { j } , \\cdots , \\omega _ { r } ) , \\\\ { \\boldsymbol { \\omega } } ^ { - } [ j ] : = & \\ , ( \\omega _ { 1 } , \\cdots , - \\omega _ { j } , \\cdots , \\omega _ { r } ) \\in \\mathbb { C } ^ { r } . \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} \\Gamma = \\gamma _ 1 ( [ 0 , 1 ) ) \\cup \\gamma _ 2 ( [ 0 , 1 ] ) \\cup \\cdots \\cup \\gamma _ { k - 1 } ( [ 0 , 1 ] ) \\cup \\gamma _ k ( [ 0 , 1 ) ) \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} H _ { s _ j } ( F ( x ) ) = ( - 1 ) ^ { \\epsilon _ j } v _ 0 ^ { s _ j } v _ 1 ^ { s _ j - s _ 1 } v _ 2 ^ { s _ j - s _ 2 } \\cdots v _ { j - 1 } ^ { s _ j - s _ { j - 1 } } , \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} \\frac { | t | ^ { n - 1 } } { ( n - 1 ) ! } - 2 \\sum ^ { n - 1 } _ { i = 1 } \\frac { | t | ^ { i - 1 } } { ( i - 1 ) ! } > 0 \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} I ^ g ( \\bar x ) : = \\{ i \\in \\mathcal M \\ , | \\ , g _ i ( \\bar x ) = 0 \\} . \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} \\| \\widehat { u } f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } \\leq \\left ( 2 \\| f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 + \\| \\widehat { u } f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 + \\| \\widehat { v } f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } = \\| f \\| _ { \\alpha } \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} q _ j : = \\frac { \\sum _ { k = j } ^ s m _ k ^ 2 } { \\sum _ { k = j } ^ s m _ k } . \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} Z _ I ( u _ 0 ) & = \\sum _ { \\substack { u _ 1 , \\ldots , u _ { 2 t - 1 } : \\\\ d ( u _ 0 , u _ i ) \\ , = \\ , d ( u _ 0 , u _ { 2 t - i } ) \\ , = \\ , i \\ , \\ , \\forall i \\in [ t ] , \\\\ u _ i \\ , = \\ , u _ { 2 t - i } \\ , \\ , \\forall i \\in I . } } W \\ ! \\left ( u _ 0 , \\ldots , u _ { 2 t - 1 } , u _ 0 \\right ) , \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} d _ { 4 } ( \\mathcal { X } , \\mathcal { Y } ) : = \\mathcal { \\Delta } ( \\mathbf { x } - \\mathbf { y } ) - \\left ( \\Phi - \\Psi + \\left \\{ \\mathbf { x } , \\mathbf { y } \\right \\} \\right ) ^ { 2 } , \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} S _ Q [ X , A ] = \\int _ { \\Sigma } ( X ^ { * * } E ) _ { \\alpha \\beta } D _ - \\sigma ^ \\alpha \\wedge D _ + \\sigma ^ \\beta - \\int _ { \\Sigma } X ^ { * * } C + \\int _ { \\Sigma _ 3 } X ^ { * * } H , \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} \\left \\| B _ n \\left ( T \\right ) \\right \\| = \\sum _ { j , l } \\left | \\sum _ { A \\in \\left [ b \\right ] ^ { \\times d - 2 } } T _ { j , \\ell , A } \\right | \\le \\sum _ { j , l } \\sum _ { A \\in \\left [ b \\right ] ^ { \\times d - 2 } } \\left | T _ { j , \\ell , A } \\right | = \\left \\| T \\right \\| , \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} \\frac { u } { e ^ { u } - 1 } = \\sum _ { m = 0 } ^ { \\infty } \\frac { B _ { m } } { m ! } u ^ { m } , | u | < 2 \\pi , \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} f \\left ( \\eta ^ \\rightarrow \\left ( \\prod _ { i \\in \\mathcal { I } } U _ i \\right ) \\right ) = \\prod _ { i \\in \\mathcal { I } } \\eta ^ \\rightarrow _ i ( U _ i ) . \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} T \\cdot [ z ] : = \\left [ z _ i e ^ { \\langle \\Lambda _ i , T \\rangle } \\right ] _ { i = 1 } ^ n \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sup _ { 0 \\leq t \\leq T } \\int \\| \\bar { x } - \\zeta ( t , \\cdot ) \\| _ { H ^ { - 1 } } ^ 2 f ( t , x ) \\mu _ { N , m } ( d x ) = 0 . \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} g _ { 1 1 } ^ { 2 } \\cdot \\ldots \\cdot g _ { l l } ^ { 2 } = \\det ( f _ { i j } ) . \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} C _ { k , 0 } ( n ) & = \\{ \\lambda = ( \\lambda _ 1 ^ { m _ 1 } , \\lambda _ 2 ^ { m _ 2 } , \\ldots , \\lambda _ \\ell ^ { m _ \\ell } , a ^ b ) \\in \\mathcal { P } ( n ) \\mid m _ i < k \\ , \\forall i , \\ , b \\geq k , \\ , k \\mid a \\} , \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} \\abs { \\l _ k } \\le \\frac { c } { L ^ 2 } k = 0 , \\ldots , d - 1 . \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} \\langle L _ { \\max } f | g \\rangle - \\langle f | L _ { \\max } g \\rangle = \\langle \\psi ( f ) | \\phi ( g ) \\rangle - \\langle \\phi ( f ) | \\psi ( g ) \\rangle . \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} B _ { j } u = \\sum \\limits _ { \\left \\vert \\beta \\right \\vert \\leq m _ { j } } \\ b _ { j \\beta } \\left ( y \\right ) D _ { y } ^ { \\beta } u \\left ( x , y , t \\right ) = 0 x \\in R ^ { n } , y \\in \\partial G , j = 1 , 2 , . . . , m , \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} \\partial _ { j } = e ^ { - t } ( \\omega _ { j } \\partial _ { t } + \\Omega _ { j } ) \\quad j = 1 , \\cdots , n + 1 . \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} & 0 \\leq \\langle I _ { \\sigma } ^ { \\prime } ( U ) , \\left ( U _ { \\epsilon } - U \\right ) \\rangle = \\epsilon \\langle I _ { \\sigma } ^ { \\prime } ( U ) , \\left ( \\varphi , \\psi \\right ) \\rangle - \\langle I _ { \\sigma } ^ { \\prime } ( U ) , \\left ( w ^ { \\epsilon } , z ^ { \\epsilon } \\right ) \\rangle + \\langle I _ { \\sigma } ^ { \\prime } ( U ) , \\left ( w _ { \\epsilon } , z _ { \\epsilon } \\right ) \\rangle , \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} F _ { \\gamma } ( \\gamma ) = 1 - \\prod _ { X \\in \\{ 1 , 2 \\} } F _ { \\gamma _ X } ^ { ( c ) } ( \\gamma ) . \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} V _ { \\alpha ( \\varphi ) } ( t , \\varphi _ t ) = 0 , \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} ( Q _ { 0 } ) _ { \\ker } = 0 . \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} \\| \\nabla _ { x } \\Phi _ a \\| _ { 2 } = \\| \\Phi _ { a } \\| _ { H ^ { 1 } } \\lesssim \\| \\partial _ { t } a ( t ) \\| _ { H ^ { - 1 } _ 0 } \\lesssim \\| b ( t ) \\| _ { 2 } . \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} z _ { x y } = F ( x , y , z , z _ x , z _ y ) Z _ { X Y } = G ( X , Y , Z , Z _ X , Z _ Y ) , \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} u _ { n - 1 } - u _ n \\geq C \\begin{cases} u _ { n - 1 } a _ { n - 1 } ^ { 1 - \\kappa } = u _ { n - 1 } ^ { - \\kappa } & \\\\ u _ { n - 1 } a _ { n - 1 } ^ { - 1 } \\log a _ { n - 1 } = u _ { n - 1 } ^ 2 \\log ( 1 / u _ { n - 1 } ) & \\end{cases} \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} \\inf _ { R \\in ( 0 , \\infty ) , | \\hat \\xi | = 1 } \\int _ { | y | \\leq 1 } \\left | \\hat \\xi \\cdot y \\right | ^ 2 \\tilde \\nu _ R ( d y ) > 0 , \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} A = P D P ^ { - 1 } , \\ \\ D \\coloneqq \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} . \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} Y _ i ( \\Omega _ i ) = \\prod _ { k = 1 } ^ { d _ i - 1 } h ^ i _ k ( \\phi ^ i _ k ) , ~ ~ ~ i = 1 , 2 , \\cdots , N , \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} 2 \\alpha ( N + 1 ) = 2 \\pi . \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} u ( t ) = u _ 2 ( t ) + \\mathcal { R } _ { 2 , 0 } ( t ) \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} \\Big ( \\sum _ { n = 0 } ^ { \\infty } \\Vert x _ n \\Vert ^ { q } \\rho ^ { q n } \\Big ) ^ { 1 / q } \\leq \\sup _ { | z | < 1 } \\| f ( z ) \\| , \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} d _ { - 1 , 1 } d _ { 0 , - 1 } ^ 2 - d _ { 0 , 1 } d _ { - 1 , - 1 } d _ { 0 , - 1 } + d _ { 1 , 1 } d _ { - 1 , - 1 } ^ 2 = 0 . \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} e _ { x y } B ( e _ { z w } , e _ { u v } ) + B ( e _ { x y } , e _ { u v } ) e _ { z w } & = e _ { x y } B ( e _ { y w } , e _ u ) + B ( e _ { x y } , e _ u ) e _ { y w } . \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} \\mu ( f ^ { - 1 } ( B ) ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ B e ^ { - \\frac { x ^ 2 } { 2 } } x \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} \\phi ( r , \\omega \\lambda ( t ) ^ { 2 } ) = \\widetilde { \\phi _ { 0 } } ( r ) + \\frac { 1 } { \\sqrt { r } } \\sum _ { j = 1 } ^ { \\infty } ( r ^ { 2 } \\omega \\lambda ( t ) ^ { 2 } ) ^ { j } \\phi _ { j } ( r ^ { 2 } ) \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\ddot { u } _ { \\lambda } ( t ) + \\eta \\dot { u } _ { \\lambda } ( t ) + A _ { \\lambda } ( u _ { \\lambda } ( t ) ) = 0 , & \\textmd { i n } \\ \\Omega \\times ( 0 , \\infty ) , \\\\ u _ { \\lambda } ( 0 ) = u _ 0 , \\dot { u } _ { \\lambda } ( 0 ) = 0 , & \\textmd { i n } \\ \\Omega \\times 0 , \\\\ \\partial _ \\nu u _ \\lambda ( t ) = 0 , & \\textmd { o n } \\ \\partial \\Omega \\times ( 0 , \\infty ) . \\end{array} \\right . \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} G ( x , y , z ) = \\left ( \\begin{array} { c c } G _ { 1 1 } ( x , y , z ) & G _ { 1 2 } ( x , y , z ) \\\\ G _ { 2 1 } ( x , y , z ) & G _ { 2 2 } ( x , y , z ) \\end{array} \\right ) = \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} d ( ( \\Gamma , p ) , ( \\Delta , q ) ) = \\sum _ { i = 1 } ^ { k } \\Big ( | p _ i - q _ i | + \\| g _ i - h _ i \\| + \\| g _ i ^ { - 1 } - h _ i ^ { - 1 } \\| + \\sup _ { [ 0 , \\beta ] \\cup [ 1 - \\beta , 1 ] } | g _ i ' - h _ i ' | \\Big ) , \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} b _ { 0 } = 2 0 - \\mu . \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} p ( { \\bf { x } } ) & = Z ( \\theta ) \\Big [ h ( { \\bf { x } } ) + w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\Big ] ^ { \\frac { 1 } { \\frac { 1 } { \\alpha } - 1 } } , \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} P = \\begin{bmatrix} 1 - \\alpha & \\alpha & 0 \\\\ \\beta & 1 - \\beta & 0 \\\\ 1 - \\alpha & 1 - \\beta & \\alpha + \\beta - 1 \\end{bmatrix} , \\alpha + \\beta > 1 . \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} V ' ( s ) = a _ n \\int _ { z _ 1 } ^ { z _ 2 } h ^ n h _ s d z . \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} - 3 n \\mu + 8 n ( n - 2 ) + 4 n ( n - 2 ) ( n - 3 ) \\\\ = - 6 n \\mu + 2 4 n ( n - 1 ) + 1 6 n ( n - 1 ) ( n - 2 ) \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} m & = 1 , & b & = 0 . 5 , & k & = 1 , & F & = 1 . \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} \\lambda _ { r } = \\left \\{ \\begin{array} { l l } c & \\mbox { i f $ r $ i s e v e n } , \\\\ d & \\mbox { i f $ r $ i s o d d } . \\end{array} \\right . \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} \\alpha ( x ) \\to \\infty \\ \\ \\ x \\to S i n g ( \\Sigma ) = \\{ p \\} \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} \\lim _ { \\xi \\rightarrow \\overline { c _ { 0 } } ^ { + } } \\frac { p _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( \\xi ) } { \\left | \\xi - \\overline { c _ { 0 } } \\right | ^ { 1 / 2 } } = \\frac { 2 \\left | a _ { 1 } \\right | } { \\sqrt { \\left | c _ { 2 } \\right | } } \\cos \\left ( \\frac { \\theta } { 2 } - \\frac { \\pi } { 4 } \\right ) , \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} W _ a ( \\bar u , u ) = 0 , W _ a ( \\bar v , v ) = 0 . \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} \\left ( K _ { n } ^ { ( 3 ) } \\right ) ^ { 2 } + \\left ( K _ { n + 1 } ^ { ( 3 ) } \\right ) ^ { 2 } + \\left ( K _ { n + 2 } ^ { ( 3 ) } \\right ) ^ { 2 } = 2 1 \\cdot 2 ^ { 2 n } + 2 ^ { n + 1 } \\left ( M _ { n + 1 } ^ { ( 2 ) } + 3 M _ { n + 2 } ^ { ( 2 ) } \\right ) + 6 , \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} \\Delta h _ { B ' } ( n ) = h ( n ) - h ( n - m - 1 ) \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} \\lim _ { \\omega _ { n + 1 } \\rightarrow 0 } \\omega _ { n + 1 } ^ { 1 - 2 s } \\Omega _ { n + 1 } \\overline { u } = \\tilde { V } \\overline { u } , \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} c _ { 1 R } & = \\frac { - 4 b _ { 0 R } } { a _ { 2 R } } , & c _ { 2 R } & = \\frac { - 4 \\sigma _ { { \\rm f o l d } , R } } { 3 } , \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} A _ { R } ^ { ( n ) } = \\sum _ { k = 0 } ^ { n - 1 } T _ { \\xi _ { k } } ( 1 , R ( X ) ) X ^ k \\qquad A _ { R } = \\sum _ { k = 0 } ^ { \\infty } T _ { \\xi _ { k } } ( 1 , R ( X ) ) X ^ k , \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} \\Big ( \\sum _ { \\vert \\alpha \\vert = m } \\Vert \\widehat { f } ( \\alpha ) \\Vert ^ { q } \\Big ) ^ { \\frac { 1 } { q } } \\leq c ^ { m } \\| f _ m \\| _ p \\le c ^ { m } \\| f \\| _ p . \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} \\sqrt { \\tfrac { n + 2 } { n } } P _ { n } ( x ) _ { i j } = ( n + 1 ) x _ { i } x _ { j } - \\tfrac { n + 2 } { n + 1 } h _ { i j } . \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{gather*} x _ i f = f x _ i , \\quad \\tau _ i f = s _ i ( f ) \\tau _ i \\quad f \\in F ^ { \\otimes n } , \\\\ \\tau _ i x _ j = x _ { s _ i ( j ) } \\tau _ i - ( \\delta _ { i , j } - \\delta _ { i + 1 , j } ) \\Delta _ { i , i + 1 } , \\end{gather*}"}
-{"id": "6969.png", "formula": "\\begin{align*} D = - K _ { \\hat { X } } + n \\hat { F } , \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} \\dot { x } = - \\nabla f ( x ) - \\Big ( \\frac { \\partial g } { \\partial x } \\Big ) ^ T g ( x ) . \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} \\Pi = \\frac { 1 } { 2 \\pi i } \\int _ \\Theta { \\mathbb T } ^ h ( \\omega ) ^ { - 1 } d \\omega . \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} R _ 1 = 2 c _ 2 k _ 0 R \\| u \\| _ \\infty . \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} \\langle V _ { g _ 1 } f _ 1 , V _ { g _ 2 } f _ 2 \\rangle _ { L ^ 2 ( \\mathbb { R } ^ { 2 d } ) } = \\langle f _ 1 , f _ 2 \\rangle _ { L ^ 2 ( \\mathbb { R } ^ d ) } \\overline { \\langle g _ 1 , g _ 2 \\rangle } _ { L ^ 2 ( \\mathbb { R } ^ d ) } , \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} R = \\frac { 1 } { \\overline { \\lim } _ { n \\to \\infty } \\sqrt [ n ] { \\sigma _ n } } . \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } \\sum _ { n = 1 } ^ N | a ( r _ n ) - \\psi ( r _ n ) | > 0 . \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} f _ R ( 0 , 0 ; 0 ) = 0 . \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} u _ n \\Phi ( x ) u _ n ^ { - 1 } - \\Phi ( x ) = & \\Phi \\left ( v _ n x v _ n ^ { - 1 } - x \\right ) = - 2 \\Phi ( y _ n ) + \\sum \\limits _ { k = 1 } ^ { n - 1 } \\left ( u _ n \\Phi ( y _ k ) u _ n ^ { - 1 } - \\Phi ( y _ k ) \\right ) . \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } | | \\lambda ^ { \\perp } - \\lambda _ { \\infty } ^ { \\perp } | | _ { 2 } ^ { 2 } = 0 \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} f \\left ( M + m - \\overline { a } \\right ) \\le \\sum \\limits _ { i = 1 } ^ { n } { \\frac { { { w } _ { i } } } { \\overline { a } - { { a } _ { i } } } \\int _ { M + m - \\overline { a } } ^ { M + m - { { a } _ { i } } } { f \\left ( t \\right ) d t } } \\le f \\left ( M \\right ) + f \\left ( m \\right ) - \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( { { a } _ { i } } \\right ) } \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} \\phi ( x _ 1 , x _ 2 , x _ 3 ) = \\phi ^ { ( 1 ) } ( x _ 1 ) \\phi ^ { ( 2 ) } ( x _ 2 , x _ 3 ) \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} V ^ i = \\sum _ { j = k + 2 } ^ { n + 1 } \\ , J ^ { i j } \\ , \\partial _ j \\ , . \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} \\begin{aligned} F _ { k + 1 } & = F _ k ( I _ m - H _ k G _ k ) ^ { - 1 } F _ k , \\ \\ \\ E _ { k + 1 } = E _ k ( I _ n - G _ k H _ k ) ^ { - 1 } E _ k , \\\\ H _ { k + 1 } & = H _ k + F _ k ( I _ m - H _ k G _ k ) ^ { - 1 } H _ k E _ k , \\ \\ \\ G _ { k + 1 } = G _ k + E _ k ( I _ n - G _ k H _ k ) ^ { - 1 } G _ k F _ k . \\end{aligned} \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} \\lambda _ { 1 ; p , f } \\int _ { \\Omega } | u | ^ { \\alpha } | \\nabla u | ^ { p } \\ , d \\mu = & ( \\alpha - p + 1 ) I _ { p , \\alpha } \\int _ { \\Omega } | u | ^ { \\alpha } | \\nabla u | ^ { p } \\ , d \\mu \\\\ & + p \\int _ { \\Omega } | u | ^ { \\alpha - p + 1 } | \\nabla u | ^ { 2 p - 3 } \\langle \\nabla u , \\nabla | \\nabla u | \\rangle \\ , d \\mu , \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} \\ i \\omega + b - K + K ( \\cos \\omega \\tau - i \\sin \\omega \\tau ) = 0 . \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} & \\epsilon ( \\partial _ { t } ^ + e _ h ^ { u ^ n } , e _ h ^ { u ^ n } ) + \\| e _ h ^ { \\phi ^ n } \\| ^ 2 _ { \\mathcal { T } _ h } = - ( \\phi ^ n - \\phi _ { I h } ^ n , e _ h ^ { \\phi ^ n } ) _ { \\mathcal { T } _ h } \\\\ & - \\epsilon ^ { - 1 } ( f ^ n ( u _ h ^ n ) - f ( u ^ n ) , e _ h ^ { \\phi ^ n } ) _ { \\mathcal { T } _ h } + \\epsilon ( \\partial _ { t } ^ + u _ { I h } ^ n - \\partial _ t u ^ n , e _ h ^ { u ^ n } ) _ { \\mathcal { T } _ h } . \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} \\tilde A _ { h , \\tilde h } = \\tilde h ^ m e ^ { - m \\hat h s _ h } D _ { s _ h } ^ j \\hat h ^ { m - j } P _ { m - j } ( \\hat h x ' , e ^ { \\hat h s _ h } x ' , y , \\hat h ^ { - 1 } D _ { Y _ h } ) \\bigl ( \\delta ( s _ h ) \\delta ( Y _ h ) \\bigr ) \\cdot \\hat h ^ { - n } . \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} W _ { \\alpha } = \\left \\{ k \\in \\left ( \\Z \\bigcap \\left [ k _ { m i n } , k _ { m a x } \\right ] \\right ) \\setminus \\left \\{ \\frac { \\ell \\pi } { \\alpha } \\right \\} _ { \\ell \\in \\Z } \\right \\} . \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} W ( \\mu , \\nu ) = \\inf _ { ( \\mu _ t , v _ t ) } \\int _ 0 ^ 1 \\| v _ t \\| _ { L ^ 2 ( \\mu _ t ) } \\ d t , \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} P _ { 2 k + 1 } ^ - \\lbrace M ( t ) = \\mathcal { T } ( t ) \\rbrace = \\binom { 2 k + 1 } { k } \\frac { 1 } { 2 ^ { 2 k + 1 } } = P _ { 2 k + 1 } ^ - \\lbrace M ( t ) = 0 \\rbrace . \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} h ^ { k } B ^ { l } C ^ { 2 } = h ^ { k } B ^ { l } ( h ^ { 2 } + B ^ { 2 } - \\mu ) = h ^ { k } ( B ^ { l } h ^ { 2 } + B ^ { l + 2 } - \\mu B ^ l ) . \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} | \\tilde { \\Phi } ( \\xi _ 2 ' , \\eta _ 2 ' ) | & : = \\Bigl | \\xi _ 1 ' { \\xi _ 2 ' } ^ 2 + \\eta _ 1 ' { \\eta _ 2 ' } ^ 2 - \\frac { { \\xi _ 1 ' } ^ 3 + { \\eta _ 1 ' } ^ 3 } { 4 } \\Bigr | \\leq 2 ^ 4 A ^ { - 1 } N _ 1 ^ 3 , \\\\ | \\tilde { F } ( \\xi _ 2 ' , \\eta _ 2 ' ) | & : = \\Bigl | \\frac { 3 } { 2 } \\ , \\xi _ 1 ' \\ , \\eta _ 1 ' + 2 \\ , \\xi _ 2 ' \\ , \\eta _ 2 ' \\Bigr | \\leq 2 ^ 4 A ^ { - 1 } N _ 1 ^ 2 , \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} \\P \\left [ X \\geq A \\right ] = \\P \\left [ e ^ { X } \\geq e ^ { A } \\right ] \\leq \\exp \\left \\{ - A + \\rho ( e - 1 ) \\right \\} \\leq e ^ { 2 \\rho } e ^ { - A } , \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} K _ { 1 } ( w , \\lambda ( t ) ) = \\int _ { 0 } ^ { \\infty } \\frac { R d R } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\int _ { 0 } ^ { w } \\frac { \\rho d \\rho } { w } \\left ( 1 + \\frac { R ^ { 2 } \\lambda ( t ) ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( R ^ { 2 } \\lambda ( t ) ^ { 2 } - 1 - \\rho ^ { 2 } ) ^ { 2 } + 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } } \\right ) \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} ( d d ^ c u _ t ) ^ n = e ^ { \\dot { u } + \\tilde { F } ( t , z , u _ t ) } \\mu \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} \\mathrm { T r } ( \\mathrm { T r } ( \\alpha ^ i \\beta ^ j \\gamma _ k \\omega _ l ) ) & = \\mathrm { T r } ( \\beta ^ j \\omega _ l \\mathrm { T r } ( \\alpha ^ i \\gamma _ k ) ) \\\\ & = \\begin{cases} 0 , & i \\not = k \\\\ \\mathrm { T r } ( \\beta ^ j \\omega _ l ) , & i = k \\end{cases} \\\\ & = \\begin{cases} 0 , & j \\not = l k \\not = i \\\\ 1 , & j = l k = i \\end{cases} \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\min ( m - 1 , L ) } c _ { i } \\left ( H _ { L + m - i } - G _ { L + m - i } \\right ) \\geq \\sum _ { i = 1 } ^ { \\min ( m - 1 , L ) } 2 c _ { i } \\left ( H _ { L + m - 1 - i } - G _ { L + m - 1 - i } \\right ) . \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} { \\mathcal L } ( \\phi ) = 0 \\quad \\phi = c _ 1 \\frac { \\partial w } { \\partial x _ 1 } + c _ 2 \\frac { \\partial w } { \\partial x _ 2 } + i ( c _ 3 \\ , w ^ + , c _ 4 \\ , w ^ - ) ~ ~ ~ \\phi \\in \\mathcal H . \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} \\begin{array} { l } \\eta _ 0 ^ * \\nabla f ( x ^ * ) + \\left [ \\displaystyle \\sum _ { i = 1 } ^ p [ y ^ I ] ^ * _ j \\nabla ^ 2 c _ j ( x ^ * ) - A ^ T A - { \\cal J } c _ { \\gamma } ( x ^ * ) ^ T { \\cal J } c _ { \\gamma } ( x ^ * ) - { \\cal J } c _ { \\beta } ( x ^ * ) ^ T { \\rm D i a g } ( [ v _ b ] _ { \\beta } ) { \\cal J } c _ { \\beta } ( x ^ * ) \\right ] \\eta ^ * _ 1 = 0 , \\end{array} \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ \\ , \\textbf { d } _ { t } \\ , \\big ] \\ , = \\ , \\frac { 1 } { n } \\ , \\big ( \\ , S _ { t } ^ { 3 } - \\tilde { S } _ { t } ^ { 3 } + \\tilde { S } _ { t } ^ { 2 } - S _ { t } ^ { 2 } \\ , \\big ) \\ , \\leq \\ , 0 . \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} \\beta = \\left ( \\frac { \\frac { \\partial f _ R } { \\partial \\mu } } { \\frac { \\partial f _ R } { \\partial y } } - \\frac { \\frac { \\partial f _ L } { \\partial \\mu } } { \\frac { \\partial f _ L } { \\partial y } } \\right ) \\bigg | _ { x = y = \\mu = 0 } . \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} C _ { 3 , 1 } ( 9 ) & = \\{ ( 6 , 1 ^ 3 ) , \\ , ( 5 , 1 ^ 4 ) , \\ , ( 4 , 2 , 1 ^ 3 ) , \\ , ( 4 , 1 ^ 5 ) , \\ , ( 3 ^ 2 , 1 ^ 3 ) , \\ , ( 3 , 2 ^ 3 ) , \\\\ & \\quad \\ ( 3 , 2 , 1 ^ 4 ) , \\ , ( 3 , 1 ^ 6 ) , \\ , ( 2 ^ 4 , 1 ) , \\ , ( 2 ^ 2 , 1 ^ 5 ) , \\ , ( 2 , 1 ^ 7 ) , \\ , ( 1 ^ 9 ) \\} . \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} g = \\dd x ^ 2 + x ^ 2 k ( x ) . \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} h \\star \\rho _ \\delta ( x , y ) : = \\int _ { \\mathbb { R } ^ d } h ( x - z , y - z ) \\rho _ \\delta ( z ) d z . \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} f \\left ( \\bigcap _ { k \\in \\N } C _ k ( K ) \\right ) = \\bigcap _ { k \\in \\N } f ( C _ k ( K ) ) = \\bigcap _ { k \\in \\N } f ( C _ k ) ( K ) . \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} \\frac { C \\log ( t ) } { t ^ { 2 3 / 8 } } \\geq | | \\partial _ { r } N _ { 2 } ( f _ { v _ { 5 } } ) ( t , r ) + \\frac { N _ { 2 } ( f _ { v _ { 5 } } ) ( t , r ) } { r } | | _ { L ^ { 2 } ( r d r ) } = | | \\xi \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( t , \\xi ) | | _ { L ^ { 2 } ( \\xi d \\xi ) } \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} \\Omega _ q ( n ) = [ n ] ^ 3 \\bigg \\{ \\frac { n ^ 2 ( 1 - q ) ^ 2 - ( 1 + 2 2 q + q ^ 2 ) } { 2 4 } - \\frac { 1 } { q [ n ] ^ 2 [ n - 1 ] [ n + 1 ] } \\bigg \\} \\frac { q ^ { ( n + 5 ) / 2 } } { 1 + q ^ 2 } . \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} G _ { 1 ^ \\ell } ^ { ( k + 1 ) } G _ \\mu ^ { ( k + 1 ) } = \\sum _ \\gamma c _ { \\gamma \\mu } \\ , G _ \\gamma ^ { ( k + 1 ) } \\ , ( - 1 ) ^ { | \\gamma | - \\ell - | \\mu | } c _ { \\gamma \\mu } \\in \\Z _ { \\geq 0 } \\ , . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} & \\tilde { F } ^ { \\leq } ( x , y ) = \\begin{pmatrix} x + \\gamma \\ , c \\ , y \\\\ \\ \\ , y \\end{pmatrix} + \\begin{pmatrix} 0 \\\\ \\gamma ^ { - 1 } \\ , a _ k \\ , x ^ k + b _ l \\ , y \\ , x ^ { l - 1 } + h . o . t . \\end{pmatrix} . \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} p _ { L } ( x ) = x ^ { L } - x ^ { L - 1 } - \\left \\lceil \\frac { L ( L + 1 ) } { 4 } \\right \\rceil - 1 . \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} \\tilde A _ { h , \\tilde h } = \\tilde h ^ m h ^ m ( x ' ) ^ { - m } s ^ { - m } ( s D _ s ) ^ j P _ { m - j } ( h , s x ' , y , D _ y ) \\bigl ( \\delta ( s - 1 ) \\delta ( y - y ' ) \\bigr ) , \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} & r \\int _ { 0 } ^ { 1 } d \\beta \\int _ { t } ^ { \\infty } d s \\int _ { B _ { s - t } ( 0 ) \\cap ( B _ { \\frac { s } { 2 } } ( - \\beta x ) ) ^ { c } } d A ( y ) \\left ( \\frac { 1 } { \\sqrt { ( s - t ) ^ { 2 } - | y | ^ { 2 } } } - \\frac { 1 } { ( s - t ) } \\right ) | \\partial _ { 2 } v _ { 4 , c } ( s , | \\beta x + y | ) | \\\\ & \\leq \\frac { C r } { t ^ { 4 } \\log ^ { 2 b - 2 } ( t ) } \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ( L _ 2 ) = h ( z ) ^ { 2 } ( \\rho ( L _ 0 ) + 2 \\lambda ) ( \\rho ( L _ 0 ) - \\lambda - 2 ) ( \\rho ( L _ 0 ) - \\lambda - 1 ) \\ , = \\\\ = \\ , h ( z ) ^ { 2 } \\left ( L ^ 3 - 3 L ^ 2 + ( 2 - 3 \\lambda - 3 \\lambda ^ 2 ) L + 4 \\lambda + 6 \\lambda ^ 2 + 2 \\lambda ^ 3 \\right ) \\end{aligned} \\end{align*} % \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} \\langle f _ 1 , f _ 2 g \\rangle & = \\langle f _ 1 g ^ * , f _ 2 \\rangle , \\\\ \\langle f _ 1 , f _ 2 g \\rangle & = \\langle f _ 1 f _ 2 ^ * , g \\rangle , \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} ( x + y ) z _ { x y } - \\mathcal { W } ( z _ x , z _ y ) = 0 , \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 _ + } G ( r x ) / \\inf _ { \\partial B _ r } G = G _ 0 ( x ) \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} a b + b a & = \\Phi ( u ^ \\ast ) \\Phi ( u ) + \\Phi ( u ) \\Phi ( u ^ \\ast ) = \\mathbf { 1 } \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} T \\eta ^ \\varepsilon \\buildrel { d } \\over { = } \\eta ^ \\varepsilon \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} \\gamma _ j & : = \\frac { n - m - 1 } { 2 } \\cdot \\frac { 1 } { ( n - j ) ( n - j - 1 ) } \\cdot \\prod _ { i = n - m } ^ { n - j } \\frac { 2 i } { 2 i + 1 } \\textrm { f o r $ 1 \\leq j \\leq m $ , } \\\\ \\gamma _ 0 & : = 1 - \\sum _ { j = 1 } ^ { m } \\gamma _ j , \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} - k _ F = \\langle D , F \\rangle < 0 . \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} \\mathcal { D } ^ \\mu _ { J , K } \\left ( \\mu _ { J , K } \\right ) : = \\mu _ { J , K } ^ { \\otimes \\mathbb { Z } } \\circ W _ 0 ^ { - 1 } . \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} e _ R ( \\mathcal I ; M ) = e _ R ( \\mathcal I ^ { [ d ] } ; M ) \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} Q ( B _ 0 \\ltimes A _ i , B _ 0 \\ltimes A _ j ) \\ = \\ Q ( A _ i , A _ j ) . \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} f _ { \\equiv } ( \\eta _ X ( x _ 1 ) ) = \\eta _ Y ( f ( x _ 1 ) ) = \\eta _ Y ( f ( x _ 2 ) ) = f _ { \\equiv } ( \\eta _ X ( x _ 2 ) ) . \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} Q = ( p ^ 0 , 0 , q _ 0 , 0 ) , \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} | | \\partial _ { t } v _ { 5 } | | _ { L ^ { 2 } ( r d r ) } + | | \\left ( \\partial _ { r } + \\frac { 1 } { r } \\right ) v _ { 5 } | | _ { L ^ { 2 } ( r d r ) } & \\leq \\int _ { t } ^ { \\infty } | | N _ { 2 } ( f _ { v _ { 5 } } ) ( x , r ) | | _ { L ^ { 2 } ( r d r ) } d x \\\\ & \\leq C \\frac { \\log ^ { 3 } ( t ) } { t ^ { 7 / 4 } } \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} [ h _ i , h _ j ] & = 0 \\\\ [ h _ i , x _ { \\alpha _ j } ] & = C ( j , i ) x _ { \\alpha _ j } \\\\ [ h _ i , x _ { - \\alpha _ j } ] & = - C ( j , i ) x _ { - \\alpha _ j } \\\\ [ x _ { \\alpha _ i } , x _ { - \\alpha _ j } ] & = \\delta _ { i , j } h _ i . \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} G ( z ) = \\nabla _ z L = \\begin{bmatrix} - \\nabla _ x L ( x , \\lambda ) \\\\ \\nabla _ \\lambda L ( x , \\lambda ) \\end{bmatrix} \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} \\Phi ( z ) = z F ( z ) , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} \\begin{aligned} & E = 2 \\omega \\sum _ { i = 1 } ^ { N } k _ i + \\omega \\sum _ { i = 1 } ^ { N } \\gamma _ i + \\frac { \\omega N } { 2 } , ~ ~ ~ ~ ~ k _ i = 0 , 1 , 2 , \\cdots \\\\ & \\gamma _ i = \\frac { 1 } { 2 } ( 1 + \\sqrt { 1 + 4 \\lambda _ i + ( d _ i - 1 ) ( d _ i - 3 ) } ) ~ ~ ~ ~ ~ i = 1 , 2 , \\cdots , N . \\end{aligned} \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} \\eta ^ \\rightarrow ( B ) = \\eta ( B ) = \\{ \\eta ( x ) \\mid x \\in B \\} , \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} P _ { 2 k + 1 } ^ - \\lbrace M ( t ) = 0 , \\ \\mathcal { T } ( t ) \\le 0 \\rbrace = \\binom { 2 k + 1 } { k } \\frac { 1 } { 2 ^ { 2 k + 1 } } = P _ { 2 k + 1 } ^ - \\lbrace M ( t ) = 0 \\rbrace , \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} \\exp \\left \\{ \\sum _ { i = 1 } ^ { \\infty } \\frac { \\mu _ { i } ^ { 2 } } { \\sigma _ { i } ^ { 2 } } \\right \\} - \\exp \\left \\{ \\sum _ { i = 1 } ^ { m } \\frac { \\mu _ { i } ^ { 2 } } { \\sigma _ { i } ^ { 2 } } \\right \\} < \\epsilon . \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} v _ { 2 } ( t , r ) & = \\frac { c _ { b } ( 1 - ( t - r ) ) } { 2 \\sqrt { 2 r } \\sqrt { | t - r | } \\log ^ { b - 1 } ( | t - r | ) } + E _ { 2 } ( t , r ) \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial ^ 2 u } { \\partial r ^ 2 } _ { | _ { r = 1 } } = \\lambda \\frac { \\partial u } { \\partial r } _ { | _ { r = 1 } } , \\\\ - \\frac { 1 } { r ^ 2 } { \\Delta _ S } \\Big ( \\frac { \\partial u } { \\partial r } - \\frac { u } { r } \\Big ) - \\frac { \\partial \\Delta u } { \\partial r } _ { | _ { r = 1 } } = \\mu u _ { | _ { r = 1 } } , \\end{cases} \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} \\alpha & = \\ln { r } + \\frac { \\pi \\tau } { \\sqrt { 4 \\delta - \\tau ^ 2 } } , & \\beta & = \\delta , & \\gamma & = r . \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} \\rho ( L _ { 0 } ) \\ , & = \\ , - \\frac { h ( z ) } { h ' ( z ) } \\partial + b ( z ) \\ , , \\\\ \\rho ( L _ i ) \\ , & = \\ , h ( z ) ^ i \\left ( \\rho ( L _ 0 ) + i \\lambda \\right ) P ( \\rho ( L _ 0 ) + \\lambda + 1 , - i ) \\ , , \\ \\ i \\neq 0 , \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} t \\frac { \\partial U } { \\partial t } = \\lambda _ c ( t , x ) U + b _ c ( t , x ) \\frac { \\partial U } { \\partial x } + \\sum _ { j + \\alpha \\geq 2 } c _ { j , \\alpha } ( t , x ) U ^ j \\Bigl ( \\frac { \\partial U } { \\partial x } \\Bigr ) ^ { \\alpha } , \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} & \\bigl ( \\mathcal { T } _ { k _ 1 } ^ { A , d } \\times \\mathcal { T } _ { k } ^ { A , d } \\bigr ) \\cap ( \\mathfrak { D } _ { j _ 1 } ^ A \\times \\mathfrak { D } _ { j } ^ A ) \\not = \\emptyset , \\\\ | \\Phi ( \\xi _ 1 , \\eta _ 1 , \\xi , \\eta ) | & \\geq A ^ { - \\frac { 3 } { 2 } } d ^ { - 1 } N _ 1 ^ { 3 } \\ \\ \\textnormal { f o r a n y } \\ ( \\xi _ 1 , \\eta _ 1 ) \\times ( \\xi , \\eta ) \\in \\mathcal { T } _ { k _ 1 } ^ { A , d } \\times \\mathcal { T } _ { k } ^ { A , d } . \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} \\mathcal { B } ( \\mathbb { R } ^ 2 ) : = \\left \\{ f \\in \\mathcal { S } ^ { \\prime } ( \\mathbb { R } ^ 2 ) : ~ \\| f \\| _ { \\mathcal { B } } < + \\infty \\right \\} \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} ( - P ) _ { B } ^ { \\alpha } \\phi = ( - P ) _ { B } ^ { \\alpha - n } ( - P ) ^ { n } \\phi . \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} \\psi = \\begin{pmatrix} \\pi _ 1 & 0 & \\\\ - \\sigma _ 2 & \\pi _ 2 & \\\\ & & & \\ddots \\\\ & & & & \\pi _ { k - 1 } & 0 \\\\ 0 & & & & - \\sigma _ k & \\pi _ k \\end{pmatrix} . \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} \\Delta ( t , x ) : = & ( 1 - \\epsilon ) \\delta \\big [ \\Phi ' ( x - \\underline h ( t ) ) + \\Phi ' ( - x - \\underline h ( t ) ) \\big ] \\\\ & + ( 1 - \\epsilon ) \\big [ F ( \\Phi ( x - \\underline h ( t ) ) ) + F ( \\Phi ( - x - \\underline h ( t ) ) ) \\big ] - F ( \\underline U ( t , x ) _ + ) . \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} Q _ n ( x ) & = \\sum _ { k \\ge 0 } q _ { n , k } x ^ k = \\sum _ { k \\ge 0 } ( q _ { n - 1 , k } + q _ { n - 2 , k } + q _ { n - 2 , k - 1 } ) x ^ k \\\\ & = \\sum _ { k \\ge 0 } q _ { n - 1 , k } x ^ k + \\sum _ { k \\ge 0 } q _ { n - 2 , k } x ^ k + x \\sum _ { k \\ge 0 } q _ { n - 2 , k - 1 } x ^ { k - 1 } \\\\ & = Q _ { n - 1 } ( x ) + ( 1 + x ) Q _ { n - 2 } ( x ) . \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} \\widetilde { f } _ i ^ { p R / 2 , q R } ( x ) & = \\int _ { D _ { q R } ^ R } \\int _ { D _ { p R / 2 } } G _ { D _ { p R / 2 } } ( x , z ) j ( | z - y | ) d z f _ i ^ * ( d y ) \\\\ & \\le ( 1 + \\varepsilon ) \\Lambda _ { 0 , q R } ( f _ i ^ * ) P _ { D _ { p R / 2 } } ( x , 0 ) , \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} n _ 0 = x _ 0 ^ 2 + y _ 0 ^ 2 + z _ 0 ^ 2 + w _ 0 ^ 2 \\ \\ x _ 0 + 3 y _ 0 \\in \\{ 2 \\times 1 , 2 \\times 2 ^ 2 \\} . \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} \\| \\Gamma \\| : = \\sup _ { \\{ f \\in \\mathcal { C } _ 0 ( X ) : \\| f \\| _ { \\sup } \\le 1 \\} } | \\Gamma ( f ) | . \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} { } ^ b \\mathcal { A } ^ \\epsilon _ G ( M ) : = { } ^ b \\widetilde { A } ^ \\epsilon ( M ) + \\widetilde { A } ^ \\epsilon ( M ) \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} \\phi _ t ^ * \\sigma = \\phi _ t ^ * \\left ( \\sum u _ { i } \\sigma _ i \\right ) = e ^ { - \\frac 5 3 t } u _ 1 \\sigma _ 1 + e ^ { - \\frac 2 3 t } u _ 2 \\sigma _ 2 + e ^ { \\frac 4 3 t } u _ 3 \\sigma _ 3 \\ , . \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} \\begin{array} { l l } { \\rm T r } ( A ) : = \\sum \\limits _ { i = 1 } ^ d a _ { i i } , \\rm { v e c } ( A ) : = [ a _ { 1 1 } , \\dots , a _ { 1 d } , a _ { 2 1 } , \\ldots , a _ { 2 d } , \\ldots , a _ { d 1 } , \\ldots , a _ { d d } ] ^ \\top , \\end{array} \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} a _ N ( \\sigma ) : = \\log \\int _ { \\mathbb { R } ^ N } \\exp \\left ( \\sigma \\sum _ { i = 1 } ^ { N } x _ i - H _ N ( x ) \\right ) d x . \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} = \\left ( \\int _ { 2 a t } ^ { 2 b t } A ( 2 u ) P ( 2 u , k ) d u \\right ) \\mu _ + ( ( b - k ) \\sqrt t ) - \\int _ { 2 a t } ^ { 2 b t } \\left ( \\int _ { 2 a t } ^ x A ( 2 u ) P ( 2 u , k ) d u \\right ) \\frac { \\mu ' _ + ( ( x - 2 k t ) / ( 2 \\sqrt t ) ) } { 2 \\sqrt t } d x \\ , . \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} H _ f = H - \\langle \\nabla f , \\nu \\rangle , \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} \\mathbb E _ { \\delta } ( \\int _ 0 ^ t \\int _ D v ( s , y ; \\omega ) \\dot { W } ( d s d y ) ) ^ 2 = \\mathbb E _ { \\delta } \\int _ 0 ^ t \\int _ D [ v ( s , y ; \\omega ) ] ^ 2 d s d y , 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} & \\rho ' ( t ) = \\rho ( t ) + \\alpha , & & s ' ( t ) = s ( t ) , & & u ' ( t ) = u ( t ) \\\\ [ 4 p t ] & \\rho ' ( t ) = \\rho ( t ) + \\beta \\rho ( t ) , & & s ' ( t ) = s ( t ) - \\beta s ( t ) , & & u ' ( t ) = u ( t ) + \\beta , \\\\ [ 4 p t ] & \\rho ' ( t ) = \\rho ( t ) + \\gamma \\rho ^ 2 ( t ) , & & s ' ( t ) = s ( t ) + \\gamma ( 1 - 2 \\rho ( t ) s ( t ) ) , & & u ' ( t ) = u ( t ) + 2 \\gamma \\rho ( t ) , \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} T _ { \\eqref { e _ d i f f e r e n c e _ c g _ h a m i l t o n i a n } } = \\frac { 1 } { N } \\sum _ { l = 1 } ^ { M } \\log \\left ( \\mathbb { E } _ { \\mu _ K ( d x ^ { B ( l ) } | y _ l ) } \\left [ \\frac { 1 } { 2 } \\sum _ { l = 1 } ^ { M } \\sum _ { n \\neq l } \\sum _ { \\substack { i \\in B ( l ) \\\\ j \\in B ( n ) } } | M _ { i j } | x _ i ^ 2 \\right ] \\right ) . \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} f ( s ) & = \\left ( c _ { 1 } \\lambda _ { 1 } ^ { \\alpha _ 1 } + c _ 2 \\lambda _ { 2 } ^ { \\alpha _ 2 } \\right ) + \\left ( c _ { 1 } \\alpha _ { 1 } \\lambda ^ { \\alpha _ { 1 } - 1 } + c _ { 2 } \\alpha _ { 2 } \\lambda ^ { \\alpha _ { 2 } - 1 } \\right ) s + . . . \\\\ & = f ( 0 ) + \\sum _ { k = 0 } ^ { \\infty } { a _ { k } s ^ { k + \\beta } } , \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} 0 = D ^ 2 [ R \\circ T ] = D ^ 2 R ( T ) ( D T , D T ) + D R ( T ) D ^ 2 T . \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} & | | \\omega \\lambda ( t ) ^ { 2 } ( T ( y _ { 1 } ) - T ( y _ { 2 } ) ) ( t ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } \\\\ & \\leq C | | y _ { 1 } - y _ { 2 } | | _ { Z } \\frac { \\log ^ { - 1 } ( t ) + \\log ^ { - \\epsilon + 2 b } ( t ) + \\log ^ { - \\frac { \\epsilon } { 2 } + b } ( t ) } { t ^ { 3 } \\log ^ { \\frac { \\epsilon } { 2 } + b } ( t ) } \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} p _ { k , 0 } = \\sum _ { i } x _ i ^ k , p _ { k , 1 } = \\sum _ { i } c _ i x _ i ^ k . \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { ( p ^ r + 1 ) / 2 } ( 4 k - 1 ) \\frac { ( - \\frac { 1 } { 2 } ) _ k ^ 6 } { k ! ^ 6 } \\equiv \\frac { 1 5 p ^ r ( p ^ { 2 r } - p ^ { 4 r } - 1 ) } { 6 4 ( p ^ { 2 r } - 1 ) } \\sum _ { k = 0 } ^ { ( p ^ r - 3 ) / 2 } \\frac { ( \\frac { 3 } { 2 } ) _ k ^ 3 ( \\frac { 7 } { 2 } ) _ k } { k ! ( k + 2 ) ! ^ 3 } \\pmod { p ^ { r + 3 } } . \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} & \\frac { \\nabla { G } _ r ( z ) + \\nabla { G } ^ T _ r ( z ) } { 2 } \\\\ = & \\begin{bmatrix} k \\nabla ^ 2 f ( x ) - A ^ T A & \\frac { 1 } { 2 } ( A \\nabla ^ 2 f ( x ) ) ^ T \\\\ \\frac { 1 } { 2 } A \\nabla ^ 2 f ( x ) & A A ^ T \\end{bmatrix} \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} \\begin{aligned} ( \\mathbf { x } ) = 1 0 0 \\cdot \\frac { | | \\mathbf { x _ { 1 } } - \\mathbf { x } | | _ 2 } { | | \\mathbf { x _ 1 } | | _ 2 } \\end{aligned} \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ k [ \\tilde g ( | x | ^ 2 + \\tau ^ 2 + 2 | x | \\tau | \\langle \\hat x , \\xi _ i \\rangle | ) + \\tilde g ( | x | ^ 2 + \\tau ^ 2 - 2 | x | \\tau | \\langle \\hat x , \\xi _ i \\rangle | ) - 2 \\tilde g ( | x | ^ 2 ) ] \\\\ \\leq & \\Big { ( } \\tilde g ( | x | ^ 2 + \\tau ^ 2 + 2 | x | \\tau ) + \\tilde g ( | x | ^ 2 + \\tau ^ 2 - 2 | x | \\tau ) - 2 \\tilde g ( | x | ^ 2 ) \\Big { ) } + 2 ( k - 1 ) \\Big { ( } \\tilde g ( | x | ^ 2 + \\tau ^ 2 ) - \\tilde g ( | x | ^ 2 ) \\Big { ) } , \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} T _ L ( 0 ; 0 ) = \\frac { \\hat { s } ( \\nu ) } { \\omega } . \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} \\gamma = \\left ( \\frac { \\partial f _ R } { \\partial y } \\ , g _ R \\middle ) \\right | _ { x = y = \\mu = 0 } . \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} \\int _ { \\R ^ + } e ^ { - \\rho | x - \\xi | } \\alpha ( \\xi ) e ^ { - \\rho \\xi / 2 } d \\xi = e ^ { - \\rho x / 2 } \\cdot e ^ { - \\rho x / 2 } \\int _ 0 ^ x e ^ { \\rho \\xi / 2 } \\alpha ( \\xi ) d \\xi + e ^ { \\rho x } \\int _ x ^ \\infty \\alpha ( \\xi ) e ^ { - 3 \\rho \\xi / 2 } d \\xi \\ , . \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} \\Delta _ 0 \\ , z _ j & = \\Delta \\ , z _ j - \\sum _ { i = k + 1 } ^ n \\langle \\Pi ( \\partial _ { z _ j } ) , \\nabla _ { E _ i } E _ i \\rangle + \\sum _ { i = 1 } ^ k \\langle \\Pi _ 0 ( \\partial _ { z _ j } ^ T ) , \\nabla _ { E _ i } E _ i \\rangle \\ , . \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} P _ f ( A , B ) = \\alpha B + \\beta A - B \\sigma _ h A , \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\psi _ { \\Lambda _ { 1 } } ( b _ { \\Lambda } ) = \\sum _ { i , j \\in I } \\prod _ { x \\in \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * b _ { x } \\right ) \\prod _ { y \\in \\Lambda _ { 1 } \\setminus \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { y } } \\left ( h _ { y , i } h _ { y , j } ^ * \\right ) . \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} \\frac { \\delta } { { a } ^ { p ^ * - 1 } } & = \\delta ( 2 \\Lambda ) ^ { p ^ * - 1 } = \\delta 2 ^ { \\frac { 1 } { p - 1 } } \\Lambda ^ { \\frac { 1 } { p - 1 } } \\\\ & = 2 ^ { \\frac { 1 } { p - 1 } } \\delta ^ { \\frac { p } { 2 ( p - 1 ) } } \\Big ( \\frac { p - 2 } { p } \\Big ) ^ { \\frac { p - 2 } { 2 ( p - 1 ) } } L ^ { \\frac { p } { 2 ( p - 1 ) } } . \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} \\mathcal S _ N = \\big \\{ \\psi ( x , y , z ) \\in C ^ \\infty ( \\mathbb R ^ 3 ) : \\ \\| \\psi \\| _ { \\mathcal S _ N } < \\infty \\big \\} , \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} \\alpha ^ 2 \\lambda _ j ' ( \\epsilon ) = - ( n + 1 ) H '' ( \\epsilon ) V ' ( \\epsilon ) \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\min _ { j = \\pm 1 } \\Bigl | \\frac { 3 } { 4 } ( \\sqrt { 2 } - 1 ) ^ { \\frac { 4 } { 3 } } \\frac { N ^ 2 } { \\xi _ 2 ' } + ( \\sqrt { 2 } + 1 ) ^ { \\frac { 2 } { 3 } } \\xi _ 2 ' - j \\sqrt { 3 } ( \\sqrt { 2 } - 1 ) ^ { \\frac { 1 } { 3 } } N \\Bigr | \\leq 2 ^ { 2 3 } { A ' } ^ { - 1 } N . \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} Y = \\int _ { - 1 } ^ { 1 } \\frac { ( w ^ { n + 1 } + K ) \\psi _ Y ( w ) } { \\sqrt { ( w ^ n + w ^ { n + 1 } + K ) ( 1 - \\zeta ^ 2 ) g ( w ) } } d \\zeta , \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\ell ( I ) = 2 ^ { n ( I ) } . \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} \\forall Q > 1 \\ \\ \\Rightarrow \\ \\ \\sum _ { k = 1 } ^ { \\infty } T ( Q ^ k ) < \\infty . \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} ( Q _ n H _ 1 Q _ n ) ^ { \\star } = Q _ n H _ 1 ^ { \\star } Q _ n \\ ; \\ ; ( Q _ n H _ 2 Q _ n ) ^ { \\star } = Q _ n H _ 2 ^ { \\star } Q _ n \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} \\langle 1 , g \\rangle & = \\langle 1 , \\theta _ { k _ 1 } \\cdots \\theta _ { k _ t } \\overline { \\theta } _ { l _ 1 } \\cdots \\overline { \\theta } _ { l _ u } g ^ \\prime \\rangle \\\\ & : = w ( k _ 1 , \\dots , k _ t ) \\delta _ { t , u } \\delta _ { k _ 1 , l _ u } \\cdots \\delta _ { k _ t , l _ 1 } \\ , \\langle 1 , g ^ \\prime \\rangle \\\\ & = w ( k ) \\delta _ { t , u } \\delta _ { k , l ^ T } \\ , \\langle 1 , g ^ \\prime \\rangle . \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} D : = 3 H - 2 E _ 1 - \\ldots - 2 E _ 7 . \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\xi _ 8 ^ n \\sum _ { a = 0 } ^ n \\xi _ 8 ^ { 2 a } \\sum _ { b = 0 } ^ a \\xi _ 8 ^ { 2 b } \\sum _ { c = 0 } ^ b \\xi _ 8 ^ { 2 c } . \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} T ( u ) T _ 1 ( v ) + T _ 1 ( u ) T ( v ) = ~ & T _ 1 ( u \\cdot T ( v ) + T ( u ) \\cdot v + H ( T u , T v ) ) \\\\ & + T \\big ( u \\cdot T _ 1 ( v ) + T _ 1 ( u ) \\cdot v + H ( T ( u ) , T _ 1 ( v ) ) + H ( T _ 1 ( u ) , T ( v ) ) \\big ) , \\\\ T _ 1 ( u ) T _ 1 ( v ) = ~ & T ( H ( T _ 1 ( u ) , T _ 1 ( v ) ) ) + T _ 1 \\big ( u \\cdot T _ 1 ( v ) + T _ 1 ( u ) \\cdot v + H ( T ( u ) , T _ 1 ( v ) ) + H ( T _ 1 ( u ) , T ( v ) ) \\big ) , \\\\ 0 = ~ & T _ 1 ( H ( T _ 1 ( u ) , T _ 1 ( v ) ) ) . \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} \\begin{array} { l } ( x - 1 ) ^ { \\epsilon _ 1 } h _ 1 ( x ) \\\\ = \\begin{cases} ( x - 1 ) ^ { n - r + k _ 2 } p _ 2 ( x ) - ( x - 1 ) ^ { n - 2 r + 2 k _ 1 } p _ 1 ^ 2 ( x ) & \\mbox { i f ~ } n - r + k _ 1 > r , \\\\ ( x - 1 ) ^ { r - k _ 1 + k _ 2 } p _ 2 ( x ) - ( x - 1 ) ^ { k _ 1 } p _ 1 ^ 2 ( x ) & \\mbox { i f ~ } n - r + k _ 1 \\le r , \\end{cases} \\end{array} \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} & l ( m _ B \\otimes 1 _ X ) = l ( 1 _ B \\otimes l ) , r ( 1 _ X \\otimes m _ A ) = r ( r \\otimes 1 _ A ) , \\\\ & l ( \\iota _ B \\otimes 1 _ X ) = 1 _ X = r ( 1 _ X \\otimes \\iota _ A ) , r ( l \\otimes 1 _ A ) = l ( 1 _ B \\otimes r ) . \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} A _ { \\varepsilon } ( H , x ) = \\min \\left \\{ \\bigcup _ { k = 0 } ^ x A _ { \\varepsilon } ( H _ 1 , k ) \\odot A _ { \\varepsilon } ( H _ 2 , x - k ) \\right \\} x = 0 , 1 , \\ldots , B , \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} P \\{ M ( t ) = 0 , \\ \\mathcal { T } ( t ) = - c t \\ | \\ V ( 0 ) = - c \\} = e ^ { - \\lambda t } . \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} \\mu x y - \\lambda S ( x , y ) = & \\mu v y ^ 2 - \\lambda ( d _ { - 1 , 0 } y + d _ { 0 , 1 } v y + R ( v y , y ) ) \\\\ = & y \\left ( \\mu v y - \\lambda ( d _ { - 1 , 0 } + d _ { 0 , 1 } v + R ' ( v , y ) ) \\right ) , \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} \\eta ( x ) = \\begin{cases} 1 & C _ { s , 3 / 1 6 } ^ { + } , \\\\ 0 & \\mathbb { R } _ { + } ^ { n + 1 } \\setminus C _ { s , 1 / 4 } ^ { + } , \\end{cases} \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} ( \\Delta f ) _ g = h ^ 2 \\Big [ ( \\Delta f ) _ { g _ \\kappa } - ( n - 2 ) g _ \\kappa ( ( \\nabla f ) _ { g _ \\kappa } , ( \\nabla \\log h ) _ { g _ \\kappa } ) \\Big ] . \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle H S _ { q } ( g r ^ { i , j } ( J _ { \\infty } ( B _ { i . j - 1 } ) ) ) = H S _ { q } ( J _ { \\infty } ( B _ { i . j - 1 } ) ) & \\mbox { i f } j - 1 \\geq 1 \\\\ H S _ { q } ( g r ^ { i . j } ( J _ { \\infty } ( B _ { i - 1 . i - 1 } ) ) ) = H S _ { q } ( J _ { \\infty } ( B _ { i - 1 . i - 1 } ) ) & \\mbox { i f } j = 1 \\\\ \\end{cases} . \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} \\sigma = f _ 1 ( s , u ) \\ , \\sigma _ 1 + f _ 2 ( s , u ) \\ , \\sigma _ 2 \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( F \\left ( ( \\eta ^ { N , b } _ n ) _ { n = 1 } ^ { N } \\right ) \\right ) = \\mathbf { E } \\left ( F \\left ( ( \\eta _ n ) _ { n = 1 } ^ { N } \\right ) \\ : \\vline \\ : \\hat { W } _ N = \\hat { W } _ 0 , \\ : \\max _ { 1 \\leq n \\leq N } \\hat { W } _ n \\leq K , \\ : S _ N > 0 \\right ) . \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} \\| ( - P ) ^ { s } u \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } ^ { 2 } = ( ( - P ) ^ { 2 s } u , u ) _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } . \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} H _ { n } = H _ { n - 1 } + N H _ { n - k - 2 } \\geq H _ { n - k } + N - 2 , \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} s = t ^ { \\gamma - ( \\gamma - 1 ) \\rho _ n ^ { - 1 } } , \\delta = \\frac { 1 + ( \\gamma - 1 ) \\sigma _ m ^ { - 1 } } { \\gamma - ( \\gamma - 1 ) \\rho _ n ^ { - 1 } } . \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} e ^ { f + r } = 2 \\cosh \\alpha + 2 ( \\alpha > 0 ) , \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} & b ^ { ( k ) } = p ^ { ( k ) } t ( b ) , \\textit { w h e r e } p ^ { ( k ) } \\textit { i s a s m a l l p i e c e ( p o s s i b l y e m p t y ) } , \\\\ & U ^ { ( k ) } = A ^ { ( k ) } b ^ { ( k ) } B , \\textit { w h e r e } A ^ { ( k ) } \\textit { i s a p r e f i x o f } U ^ { ( k ) } , \\ B \\textit { i s a s u f f i x o f } U ^ { ( k ) } , \\\\ & k = 1 , \\ldots , k _ 1 . \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} \\sum _ { \\mathsf { j = 1 } } ^ { + \\infty } \\sum _ { \\mathsf { k = 1 } , \\mathsf { k \\neq j } } ^ { + \\infty } \\left \\vert \\left ( \\frac { \\mathsf { \\hat { r } } _ { \\mathsf { j } } } { \\left \\Vert \\mathsf { \\hat { r } } _ { \\mathsf { j } } \\right \\Vert _ { 2 } } , \\frac { \\mathsf { \\hat { r } } _ { \\mathsf { k } } } { \\left \\Vert \\mathsf { \\hat { r } } _ { \\mathsf { k } } \\right \\Vert _ { 2 } } \\right ) _ { 2 } \\right \\vert ^ { 2 } < + \\infty \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} p ^ { e _ l } ( a ) = p _ 0 + a p _ 1 + a ^ 2 p _ 2 & = f ( a ) , \\cr p ^ { e _ l } ( b ) = p _ 0 + b p _ 1 + b ^ 2 p _ 2 & = f ( b ) , \\cr p ^ { e _ l } ( c ) = p _ 0 + c p _ 1 + c ^ 2 p _ 2 & = f ( c ) . \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} \\Phi ( r e ^ { i \\theta } ) = \\gamma \\exp [ u ( r e ^ { i \\theta } ) + i v ( r e ^ { i \\theta } ) ] , \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} \\lambda _ { i _ 1 , \\dots , i _ { n - 1 } } = ( - 1 ) ^ { n + i _ 1 + \\dots + i _ { n - 1 } } \\frac { ( \\alpha \\beta + ( - 1 ) ^ { i _ 1 + \\dots + i _ { n - 1 } } \\prod _ { k = 1 } ^ { n - 1 } \\gamma _ k ^ { i _ k } ) \\prod _ { k = 1 } ^ { n - 1 } \\gamma _ k ^ { 1 - i _ k } } { \\alpha \\beta } . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} Q _ { \\Sigma } ^ s ( \\phi , \\phi ) = \\int _ { \\Sigma } \\nabla \\phi \\cdot \\nabla ( \\epsilon B - v ) - ( 1 - s ) | A _ { \\Sigma } | ^ 2 \\phi ( \\epsilon B - v ) = \\int _ { \\Sigma } - \\phi \\cdot L ^ s _ { \\Sigma } ( \\epsilon B - v ) \\leq 0 \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ K \\int _ { G _ 1 } | x - y | ^ { - ( d + \\alpha ) } & | x | ^ { \\alpha - d } | y | ^ { \\alpha - d } d y d x \\\\ & \\leq \\int _ K \\int _ { G _ 1 } \\left ( \\frac { | y | } { | x - y | } \\right ) ^ { \\alpha + d } | x | ^ { \\alpha - d } | y | ^ { - 2 d } d y d x . \\end{aligned} \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} k ( x , t ) = \\frac { ( c _ 1 t - x ) ^ m ( c _ 2 t + x ) ^ n } { t ^ { m + n + r } } \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} e _ i ^ * = \\begin{cases} 0 , \\ \\ \\ \\ \\ { \\rm i f } \\ \\ h _ i \\geq 0 \\\\ \\frac { a } { \\mu } h _ i , \\ \\ { \\rm i f } \\ \\ - \\mu < h _ i < 0 \\\\ - a , \\ \\ \\ { \\rm i f } \\ \\ h _ i \\leq - \\mu . \\end{cases} \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} - \\nabla \\cdot \\left ( \\frac { 1 } { \\epsilon ( x , \\omega ) } \\nabla \\psi \\right ) = \\left ( \\frac { \\omega } { c } \\right ) ^ 2 \\psi . \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} & \\hat { h } ( c ( s ) z , s ) = c ( s ) h ( z , s ) , \\\\ & c ' ( s ) \\hat { h } _ z ( c ( s ) z , s ) + \\hat { h } _ s ( c ( s ) z , s ) = c ' ( s ) h ( z , s ) + c ( s ) h _ s ( z , s ) . \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } K _ { \\varepsilon , 1 } = K _ { ( \\delta , \\bar { x } ) } [ u , p + q ] ( \\bar { x } , \\bar { t } ) - K _ { ( \\delta , \\bar { y } ) } [ u , p ] ( \\bar { y } , \\bar { t } ) , \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} \\int _ { - M } ^ 0 | x | ^ { \\sigma } \\psi ( x ) d x = \\int _ { \\R } \\int _ 0 ^ M | x | ^ { \\sigma } J ( y ) \\psi ( - x ) d x d y . \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} t ^ I = \\frac { p ^ I + i \\frac { \\partial I _ 1 } { \\partial q _ I } } { p ^ 0 + i \\frac { \\partial I _ 1 } { \\partial q _ 0 } } . \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} \\tilde t _ 2 ^ { - 1 } \\cdot \\tilde t _ 1 = ( i ^ { - 1 } ) ^ * t _ { 1 2 } { \\rm o n } Y _ 1 \\cap Y _ 3 . \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} \\begin{aligned} \\log \\det \\Delta _ \\beta = \\frac 1 6 \\left ( 2 \\beta + \\frac 4 { \\beta + 1 } - \\frac 1 { 2 \\beta + 1 } \\right ) \\log 2 - \\frac 1 6 \\left ( \\frac 2 { \\beta + 1 } - \\frac 1 { 2 \\beta + 1 } - 1 \\right ) \\log c _ \\beta \\\\ - 4 \\zeta ' _ B ( 0 ; \\beta + 1 , 1 , 1 ) - 2 \\zeta ' _ B ( 0 ; - 2 \\beta - 1 , 1 , 1 ) + 2 \\zeta _ R ' ( - 1 ) \\\\ - \\log ( \\beta + 1 ) - \\frac 1 2 \\log ( - 2 \\beta - 1 ) - \\frac { 5 } { 2 } \\log 2 - \\log \\pi . \\end{aligned} \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} \\partial _ s A _ \\nu ( s , x ) = \\nabla ^ \\mu F _ { \\mu \\nu } ( s , x ) . \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} L ( f , g ) = \\int _ 0 ^ 1 f ! _ t g \\ ; d \\mu ( t ) , \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} T _ { \\mu , D } = B _ { \\mu , D } ^ { ( - 1 ) } \\circ J _ 1 . \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} A ^ p ( G : H ) : = \\{ f \\in L ^ p ( G ) : L _ h f = f \\ \\forall h \\in H \\} , \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} b ^ { ( k ) } = p ^ { ( k ) } t ( b ) , & \\textit { w h e r e } p ^ { ( k ) } \\textit { i s a p r e f i x o f } b ^ { ( k ) } , \\\\ & p ^ { ( k ) } \\textit { i s a s m a l l p i e c e ( p o s s i b l y e m p t y ) , } k = 1 , \\ldots , K . \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} \\psi ( B ^ { - 1 } ( x + y ) + \\varphi ( x + y ) ) & - \\psi ( B ^ { - 1 } x - \\varphi ( x ) ) \\\\ & = \\int _ 0 ^ 1 D \\psi ( z ( s , x , y ) ) s ( B ^ { - 1 } y + \\varphi ( x + y ) - \\varphi ( y ) ) , \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} \\mu _ { 2 } = \\mu _ { } = \\frac { \\mu _ { 1 } \\alpha \\xi ^ 2 ( \\xi ^ 2 + 1 ) ^ { - 2 } ( \\xi ^ 2 + 2 ) ( g + \\Omega ) } { ( \\alpha + 1 ) [ 2 g ( g + 2 \\Omega ) + \\Omega ^ 2 ( 1 + \\frac { 1 } { \\beta } ) ] } . \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} | \\frac { 4 \\alpha ( \\log ( \\lambda _ { 0 , 0 } ( t ) ) - \\log ( \\lambda ( t ) ) ) \\lambda ''' ( t ) } { \\log ( \\lambda _ { 0 , 0 } ( t ) ) } | & \\leq \\frac { C } { \\log ( \\log ( t ) ) } | \\log ( 1 + \\frac { e ( t ) } { \\lambda _ { 0 , 0 } ( t ) } ) | \\left ( \\frac { 1 } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } + | e ''' ( t ) | \\right ) \\\\ & \\leq \\frac { C } { ( \\log ( \\log ( t ) ) ) ^ { 3 / 2 } } \\left ( \\frac { 1 } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } + | e ''' ( t ) | \\right ) \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} I _ { r _ { j } } ( \\theta _ { j } ) \\le I _ { r _ { j } } ( f ( r _ { j } ) ) \\le 1 , j = 1 , 2 . \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} \\begin{array} { l l l } - \\frac { 1 } { 2 } ( ( n _ 1 \\Delta _ 1 + \\cdots + n _ r \\Delta _ r ) ^ 2 ) & = & \\lim _ { \\epsilon \\rightarrow 0 } - \\frac { 1 } { 2 } ( A _ { \\epsilon } ^ 2 ) \\\\ & = & \\lim _ { n \\rightarrow \\infty } \\frac { \\ell _ R ( R / I ( n n _ 1 D _ 1 ) \\cdots I ( n n _ r D _ r ) ) } { n ^ 2 } \\\\ & = & G ( n _ 1 , \\ldots , n _ r ) . \\end{array} \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} M _ 0 & = m , \\\\ M _ t & = M _ { t - 1 } - 1 + B _ t t \\geq 1 , \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} h ( x , t ) = C \\cdot \\frac { ( c _ 1 t - x ) ^ m ( c _ 2 t + x ) ^ n } { t ^ { m + n + 1 } } \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} D \\left ( A \\right ) = \\left \\{ u \\in W ^ { 2 , 2 } \\left ( 0 , 1 \\right ) , A u \\left ( j \\right ) = 0 , j = 0 , 1 \\right \\} , \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} - 4 \\int _ { t } ^ { \\infty } \\frac { \\lambda '' ( x ) d x } { 1 + x - t } + \\frac { 4 b } { t ^ { 2 } \\log ^ { b } ( t ) } + 4 \\alpha \\log ( \\lambda ( t ) ) \\lambda '' ( t ) - 4 \\int _ { t } ^ { \\infty } \\frac { \\lambda '' ( s ) d s } { ( \\lambda ( t ) ^ { 1 - \\alpha } + s - t ) ( 1 + s - t ) ^ { 3 } } = G ( t , \\lambda ( t ) ) \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } } \\frac { | 1 - \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } } { | t - \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } } \\ , d \\sigma ( t ) = \\frac { | 1 - \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } \\log | z | } { | \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } - 1 } . \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} \\int _ G f ( x h ) d \\lambda _ q ( x ) = \\int _ G f ( x ) d \\lambda _ q ( x ) . \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} u ( x , t ) = ( c ^ 2 t ^ 2 - x ^ 2 ) ^ { m } \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} N ^ { = } = \\sum _ { i = 1 } ^ t ( - 1 ) ^ { i + 1 } \\binom { t } { i } N _ i . \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} S p e c \\left ( C _ { \\alpha } \\right ) = \\left \\lbrace \\left [ \\alpha ^ 2 + 2 \\alpha \\right ] ^ 1 , \\left [ \\alpha \\right ] ^ { \\frac { \\alpha ^ 3 + 2 \\alpha ^ 2 - \\alpha - 2 } { 2 } } , \\left [ - \\alpha \\right ] ^ { \\frac { \\alpha ^ 3 + 2 \\alpha ^ 2 + \\alpha + 2 } { 2 } } \\right \\rbrace . \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} \\epsilon \\cdot X ^ { ( - i ) } = X ^ { ( - i - 1 ) } . \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} | \\partial _ { r } v _ { 4 , c } ( t , r ) | & \\leq C \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) \\begin{cases} \\frac { 1 } { r ^ { 4 } t ^ { 2 } \\log ^ { 3 b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { \\log ( r ) } { \\log ^ { 2 b } ( t ) r ^ { 4 } ( t - r ) ^ { 2 } } + \\frac { 1 } { \\log ^ { 3 b } ( t ) t ^ { 2 } r ^ { 4 } } , r \\geq \\frac { t } { 2 } \\end{cases} \\\\ & + \\frac { C | \\chi ' ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) | } { \\log ^ { 5 N + 2 b } ( t ) } \\frac { r } { t ^ { 2 } \\log ^ { b } ( t ) } \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} F _ { \\nu } ^ { \\prime } ( \\alpha ) = \\lim _ { \\varepsilon \\downarrow 0 } \\Re \\left [ \\frac { F _ { \\nu } ( \\alpha + i \\varepsilon ) - F _ { \\nu } ( \\alpha ) } { i \\varepsilon } \\right ] = F _ { \\nu } ( \\alpha ) ^ { 2 } \\int _ { \\mathbb { R } } \\frac { d \\nu ( t ) } { ( \\alpha - t ) ^ { 2 } } \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} { \\rm T r } _ \\chi ( A ) = \\lim _ { \\epsilon \\downarrow 0 } \\int _ M \\chi _ \\epsilon ( x ) A ( x , x ) d x = { \\rm T r } _ S ( \\Phi _ A ( e ) ) , \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} I I & = | - 2 r \\int _ { 6 r } ^ { \\infty } d w \\left ( \\lambda _ { 1 } '' ( t + w ) - \\lambda _ { 2 } '' ( t + w ) \\right ) \\left ( \\frac { w } { 2 ( 1 + w ^ { 2 } ) } - \\frac { w } { 2 ( \\lambda _ { 1 } ( t + w ) ^ { 2 - 2 \\alpha } + w ^ { 2 } ) } \\right ) | + E _ { I I } \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} \\mathcal { C } ^ 2 [ T , \\mathcal { L } ] ( v _ { 1 } , v _ { 2 } , v _ { 3 } ) = | \\nabla | ^ 2 ( v _ { 1 } v _ { 2 } v _ { 3 } ) - | \\nabla | ^ 2 v _ { 1 } v _ { 2 } v _ { 3 } - v _ { 1 } | \\nabla | ^ 2 v _ { 2 } v _ { 3 } - v _ { 1 } v _ { 2 } | \\nabla | ^ 2 v _ { 3 } + \\mathcal { R } _ { 1 } + \\mathcal { R } _ { 2 } \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} \\dot { u } ( \\Gamma ( t ) , t ) = - \\langle \\nabla u ( \\Gamma ( t ) , t ) , \\dot { \\Gamma } ( t ) \\rangle . \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} b = \\frac { c _ { \\iota _ { 1 } ( Q ) , n } } { u _ { \\iota _ { 1 } ( Q ) } ^ n } + \\ldots + \\frac { c _ { \\iota _ { 1 } ( Q ) , 2 } } { u _ { \\iota _ { 1 } ( Q ) } ^ 2 } + \\frac { c _ { \\iota _ { 1 } ( Q ) , 1 } } { u _ { \\iota _ { 1 } ( Q ) } } + \\tilde { g } \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} \\delta _ z ( \\sigma ) = \\sum _ { k = 0 } ^ \\infty \\frac { z ^ k } { k ! } D ^ k ( \\sigma ) \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} \\tilde { w } = \\cdots a \\cdots b z c \\cdots \\rightsquigarrow \\tilde { w } ^ * = \\cdots a \\cdots z b c \\cdots \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} A _ { 1 } = \\left \\{ z : = r e ^ { i \\theta } \\in A \\ , : \\ , - \\pi < \\theta < \\frac { \\pi } { 2 } \\right \\} { \\rm a n d } A _ 2 = \\left \\{ z : = r e ^ { i \\theta } \\in A \\ , : \\ , 0 < \\theta < \\frac { 3 \\pi } { 2 } \\right \\} , \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} & - \\frac { 1 } { \\omega } \\int _ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } ^ { \\infty } \\phi ( r , \\omega \\lambda ( t ) ^ { 2 } ) \\sqrt { r } F _ { 4 } ( t , r \\lambda ( t ) ) d r = - \\frac { 2 } { \\omega } \\left ( \\int _ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } ^ { \\infty } a ( \\omega \\lambda ( t ) ^ { 2 } ) \\psi ^ { + } ( r , \\omega \\lambda ( t ) ^ { 2 } ) \\sqrt { r } F _ { 4 } ( t , r \\lambda ( t ) ) d r \\right ) \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} \\Sigma ' ( f , g ) : = E & \\left \\{ \\frac { E w ( X _ { 2 } ) - \\theta w _ { \\theta } ( X _ { 1 } ) } { \\theta w _ { \\theta } ( X _ { 1 } ) } \\left [ f ( Y _ { 1 } ) - R ( f ) \\frac { \\theta w _ { \\theta } ( X _ { 1 } ) } { E w ( X _ { 2 } ) } \\right ] \\times \\right . \\\\ & \\quad \\left . \\times \\left [ g ( Y _ { 1 } ) - R ( g ) \\frac { \\theta w _ { \\theta } ( X _ { 1 } ) } { E w ( X _ { 2 } ) } \\right ] \\right \\} , f , g \\in \\mathcal { F } , \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} & \\Bigl | \\int _ { \\R ^ 3 \\times \\R ^ 3 } { h ( \\tau _ 1 + \\tau _ 2 , \\xi _ 1 + \\xi _ 2 , \\eta _ 1 + \\eta _ 2 ) f ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) g ( \\tau _ 2 , \\xi _ 2 , \\eta _ 2 ) } d \\sigma _ 1 d \\sigma _ 2 \\Bigr | \\\\ & \\lesssim A ^ { \\frac { 1 } { 2 } } N _ 1 ^ { - 2 } ( L _ 0 L _ 1 L _ 2 ) ^ { \\frac { 1 } { 2 } } \\| f \\| _ { L ^ 2 } \\| g \\| _ { L ^ 2 } \\| h \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} Q ( B _ 1 ^ + \\ltimes A _ i , B _ 2 ^ + \\ltimes A _ j ) \\ & = \\ 0 , \\\\ Q ( B _ 1 ^ + \\ltimes A _ i , B _ 0 \\ltimes A _ j ) \\ = \\ Q ( B _ 2 ^ + \\ltimes A _ i , & B _ 0 \\ltimes A _ j ) \\ = \\ 2 N ^ 2 . \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} f | _ x ^ y = f ( x , y ) e _ { x y } + \\sum _ { x \\le v < y } f ( x , v ) e _ { x v } + \\sum _ { x < u \\le y } f ( u , y ) e _ { u y } . \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} 6 P _ { n } ( \\tfrac { x } { | x | } ) = 6 | x | ^ { - 3 } P _ { n } ( x ) = \\tfrac { ( n + 1 ) ^ { 3 } } { n ^ { 3 / 2 } ( n + 1 ) ^ { 3 / 2 } } \\tfrac { ( n - 1 ) n ^ { 3 / 2 } ( n + 2 ) ^ { 3 / 2 } } { ( n + 1 ) ^ { 3 } } = n - 1 . \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} F ( f ) ( x ) : = p _ 1 ( x - a ) + Q _ a f ( x ) + T _ a g ( x ) . \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} & n _ { t } + \\nabla \\cdot ( n Q ^ \\epsilon \\ast \\mathbf { v } ) + O ( \\epsilon ) = 0 , \\\\ & \\mathbf { v } _ { t } + ( \\mathbf { v } \\cdot \\nabla ) \\mathbf { v } = \\frac { n ( Q ^ \\epsilon \\ast \\mathbf { v } - \\mathbf { v } ) } { \\epsilon } + n ( Q ^ \\epsilon \\ast \\mathbf { u _ 1 } - \\mathbf { u _ 1 } ) + \\rho _ 1 ( Q ^ \\epsilon \\ast \\mathbf { v } - \\mathbf { v } ) + O ( \\epsilon ) . \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} A _ i : = \\{ \\beta \\in A \\ , ; \\ , | \\alpha _ i - \\beta | _ v < \\rho \\} ( 0 \\le i \\le k ) \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} q = v + j w , v = t + i x = { 1 \\over 2 } ( q - i q i ) , y - i z = { 1 \\over 2 } ( - j q + k q i ) . \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} & \\alpha _ 1 = \\frac { r } { 2 } , \\alpha _ 2 = \\frac { r } { 2 } , \\beta _ 1 = \\frac { r } { 2 } , \\beta _ 2 = \\frac { r } { 2 } , \\\\ [ 1 m m ] & b _ 1 = \\frac { - 1 } { r } , b _ 2 = \\frac { - 1 } { r } , b _ 3 = B U ^ + U ^ - r . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} d _ \\infty ( D ) & = \\lim _ { T \\to \\infty } \\sup _ { u \\in D } \\sup _ { t \\ge T } \\| u ( \\cdot , t ) \\| , \\\\ \\chi _ \\infty ( D ) & = \\sup _ { T > 0 } \\chi _ T ( \\pi _ T ( D ) ) , \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} \\begin{aligned} 2 A _ + { U ^ + } ^ 2 | \\tilde \\psi _ 1 ^ + | ^ 2 \\ , + \\ , 4 B U ^ + U ^ - \\tilde \\psi _ 1 ^ + \\tilde \\psi _ 1 ^ - \\ , + \\ , 2 A _ - { U ^ - } ^ 2 | \\tilde \\psi _ 1 ^ - | ^ 2 \\ , \\geq \\ , \\Lambda \\big ( | \\tilde \\psi _ 1 ^ + | ^ 2 + | \\tilde \\psi _ 1 ^ - | ^ 2 \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{gather*} A _ 1 = - ( n + 1 ) a _ n \\int _ { z _ 1 } ^ { z _ 2 } H _ 0 h _ 0 ^ n h _ 1 d z + a _ n \\Big [ \\frac { h _ 0 ^ n h _ { 0 z } } { \\sqrt { 1 + h _ { 0 z } ^ 2 } } h _ 1 \\Big ] _ { z _ 1 } ^ { z _ 2 } , \\\\ V _ 1 = a _ n \\int _ { z _ 1 } ^ { z _ 2 } h _ 0 ^ n h _ 1 d z . \\end{gather*}"}
-{"id": "5579.png", "formula": "\\begin{align*} d _ W \\left ( | T | ^ { 1 / 2 } \\ , \\left ( C ^ { / T } _ 2 ( g , u ) - C ^ { * } _ 2 ( g , u ) \\right ) , \\mathcal N ( 0 , \\sigma ^ 2 ( u / \\sigma _ g ) ) \\right ) = O \\left ( ( \\log | T | ) ^ { - 1 / 1 2 } \\right ) , \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} & \\varepsilon ^ * = \\varepsilon _ { \\vec \\gamma } , \\ \\ k ^ * = k _ { \\vec \\gamma } , \\ \\ t _ d = p _ { \\vec \\gamma } ( d ) , \\cr t _ { i , j } & = p _ { \\vec { \\gamma } } ( i , \\delta _ { \\vec { \\gamma } } ( j ) ) \\mathrm { \\ f o r \\ } i < d , \\ j < k ^ * . \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} \\| u \\| _ U ^ 2 = \\frac { \\i } { 2 } W _ b ( \\bar u , u ) , \\| v \\| _ U ^ 2 = \\frac { \\i } { 2 } W _ b ( \\bar v , v ) , \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} x _ 0 = x , x _ { 2 n + 1 } = \\Pi _ { M } ^ p x _ { 2 n } , x _ { 2 n } = \\Pi _ { N } ^ p x _ { 2 n - 1 } , \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} \\rho x _ i = \\sum _ { v _ j \\in V ( G ) } { d _ { i j } x _ j } . \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} | f _ i ( u ) - f _ i ( v ) | \\leq L ( K ) \\sum _ { i = 1 } ^ { m } | u _ i - v _ i | . \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} \\sin ^ 2 \\phi & = 1 - \\cos ^ 2 \\phi \\\\ \\sin ^ 2 \\phi & = \\frac { 4 m n ( 1 - m ^ 2 ) ( 1 - n ^ 2 ) } { ( n + m ) ^ 2 ( 1 - m n ) ^ 2 } \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} H ( x , y ) F ( x , y ) = x ^ q + h _ 1 ( y ) x ^ { q - 1 } + h _ 2 ( y ) x ^ { q - 2 } + \\cdots + h _ q ( y ) , \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} a _ k ( n ) + k c _ { k , 0 } ( n ) = \\ , \\mid A _ k ( n ) \\ , \\dot \\cup \\ , A ' _ k ( n ) \\mid = \\ , \\mid C _ { k , 0 } ( n ) \\ , \\dot \\cup \\ , C _ { k , 1 } ( n ) \\mid = c _ { k , 0 } ( n ) + c _ { k , 1 } ( n ) , \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} q _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( 1 / x ) = \\frac { 1 } { \\pi } \\Im \\psi _ { \\mu _ { 1 } } \\left ( \\eta _ { \\rho _ { 1 } } ( x ) \\right ) , x \\in ( 0 , + \\infty ) . \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} i v _ 3 & = i \\frac 1 2 a _ 1 2 ! 1 ! \\left ( X _ 1 + X _ 2 + X _ 3 \\right ) + i \\frac 1 2 1 ! 2 ! \\left ( X _ { 1 + 2 } + X _ { 1 + 3 } + X _ { 2 + 3 } \\right ) \\\\ & = 2 i a _ 1 \\left ( X _ 1 + X _ 2 + X _ 3 \\right ) . \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} \\int _ { z _ 1 } ^ { z _ 2 } u h ^ n \\ ; d z = \\frac { V ' ( s ) } { ( n + 1 ) a _ n H ' ( s ) } . \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} ( | x | ^ { - s } u _ t , \\phi ) + ( | \\nabla u | ^ { p - 2 } \\nabla u , \\nabla \\phi ) = ( | u | ^ { q - 2 } u \\ln | u | , \\phi ) \\quad t \\in ( 0 , T ) \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\left | \\int _ 0 ^ \\infty f ( x ) \\mathcal { E } ( x , k ) d x \\right | ^ 2 d \\sigma = \\| f \\| _ 2 ^ 2 \\ , , \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} h ^ { 1 + d / 2 } \\omega ( h ) ^ { - 1 / 2 } = h ^ { 1 + d / 2 } h ^ { - d / 2 - \\alpha / 2 } = h ^ { 1 - \\alpha / 2 } \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} \\hat { A } = \\mathbb { E } [ { S } \\otimes A ] , \\hat { B } = \\mathbb { E } [ { s } \\otimes B ] , \\hat { C } = \\mathbb { E } [ { S } \\otimes C ] , \\hat { E } = \\mathbb { E } [ { S } \\otimes E ] \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} A : B = \\mathrm { t r } \\ : A ^ t B = \\sum _ { i , j = 1 } ^ N a _ { i j } b _ { i j } \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} h _ { i j } = \\begin{cases} \\sqrt { \\frac { 1 - p } { p } } \\frac { 1 } { \\sqrt { n } } & p , \\\\ - \\sqrt { \\frac { p } { 1 - p } } \\frac { 1 } { \\sqrt { n } } & 1 - p . \\end{cases} \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} \\delta _ \\varepsilon D _ \\pm X ^ \\mu = & \\varepsilon ^ a ( \\partial _ \\nu \\rho ^ \\mu _ a - \\rho ^ \\mu _ b ( \\Omega ^ \\pm ) ^ b { } _ { \\nu a } ) D _ \\pm X ^ \\nu . \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} \\dot { x } = A x + B u , x ( 0 ) = x _ 0 , \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} \\chi ' = \\chi \\left [ \\chi - \\frac { 1 } { p - 1 } \\frac { v _ h ' } { v _ h } \\right ] . \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} N _ { j } = \\varphi _ { m , ( a , b ) } ( x _ { 0 } , y _ { 0 } ) + 4 c ( m - 2 ) z _ { 0 } ( ( m - 2 ) z _ { 0 } - ( m - 4 ) ) \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} P ( x ; t , s ) = \\sum _ { n \\geq 1 } P _ n ( t , s ) \\frac { x ^ n } { n ! } = \\frac { - 2 + 2 e ^ { u ( t , s ) x } } { ( 1 + s + u ( t , s ) ) - ( 1 + s - u ( t , s ) ) e ^ { u ( t , s ) x } } , \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} \\sigma = \\frac { 1 } { n ( 1 + m ) } \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} \\left ( \\widehat { p } _ 1 ^ 2 - \\widehat { p } _ 2 ^ 2 - 4 8 e ^ { - 2 \\sqrt { 3 } \\widehat { q } _ 2 } \\right ) \\psi = 0 . \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} f _ { 1 } ( t ) & = 0 , f _ { 2 } ( t ) = V ^ t f _ { 0 , 2 } , \\\\ f _ { n } ( t ) & = V ^ t f _ { 0 , n } + S ( t , f _ { n - 1 } , f _ { n - 2 } ) , t \\geq 0 ; n = 3 , 4 , . . . . \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} | \\sin \\angle \\left ( ( \\xi _ 1 , \\ , \\eta _ 1 ) , \\ , ( \\xi _ 2 , \\eta _ 2 ) \\right ) | & = \\frac { | \\xi _ 1 \\eta _ 2 - \\xi _ 2 \\eta _ 1 | } { | ( \\xi _ 1 , \\eta _ 1 ) | | ( \\xi _ 2 , \\eta _ 2 ) | } \\\\ & \\geq \\frac { | \\xi _ 1 \\eta - \\xi \\eta _ 1 | } { 4 | ( \\xi _ 1 , \\eta _ 1 ) | | ( \\xi , \\eta ) | } \\\\ & = 2 ^ { - 2 } | \\sin \\angle \\left ( ( \\xi , \\ , \\eta ) , \\ , ( \\xi _ 1 , \\eta _ 1 ) \\right ) | > 2 ^ { - 2 2 } , \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} \\rho ^ { { y } } ( x ) = \\begin{cases} \\displaystyle - \\frac { 1 } { \\Gamma ( 2 + \\alpha ) } x ^ { 1 + \\alpha } + \\frac { C } { \\Gamma ( 1 + \\alpha ) } x ^ \\alpha \\quad & \\\\ \\displaystyle - \\frac { 1 } { \\Gamma ( 2 + \\alpha ) } ( x ^ { 1 + \\alpha } - L ^ { 1 + \\alpha } ) \\quad & \\end{cases} \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} V = a _ 1 \\Gamma _ 1 + \\hdots + a _ \\ell \\Gamma _ \\ell \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} K _ 2 \\leq C ( \\| \\nabla u \\| _ { L ^ \\infty } + \\| \\nabla \\omega \\| _ { L ^ { p ' } } ) \\times ( \\| | D | ^ s \\omega \\| _ { L ^ 2 } ^ 2 + 1 ) . \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} J _ { 0 , 1 } = \\left ( \\begin{array} { c c c c } \\frac { k + a c d } { c ( k - d ) } & \\theta & 1 & 1 \\\\ - 2 \\theta & - b + K & 0 & 0 \\\\ - 1 & 0 & - c & 0 \\\\ - \\frac { d k ( 1 + a c ) } { c ( k - d ) } & - d \\theta & 0 & - k \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} | \\mathfrak { A } ( \\xi _ 1 ' , \\eta _ 1 ' , \\xi _ 2 ' , \\eta _ 2 ' ) | & = | \\xi _ 1 ' \\eta _ 2 ' - \\xi _ 2 ' \\eta _ 1 ' | \\geq A ^ { - \\frac { 3 } { 2 } } , \\\\ | F ( \\xi _ 1 ' , \\eta _ 1 ' , \\xi _ 2 ' , \\eta _ 2 ' ) | & = | \\xi _ 1 ' \\eta _ 2 ' + \\xi _ 2 ' \\eta _ 1 ' + 2 ( \\xi _ 1 ' \\eta _ 1 ' + \\xi _ 2 ' \\eta _ 2 ' ) | \\geq 2 ^ { - 1 } A ^ { - \\frac { 3 } { 2 } } d ^ { - 1 } , \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} p _ \\varepsilon : = \\frac { 1 - \\varepsilon c } { 2 } . \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} \\tau = x + i y \\mapsto ( x , y ) , \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} \\begin{cases} g ( t ) = 1 - 4 ( v ^ { - 1 } - 1 ) ( v ^ { - 1 } - 2 ) \\lambda ^ { 2 v } t ^ { - 2 } \\to 1 \\ ; \\mbox { a s } \\ ; t \\to \\infty ; \\\\ f ( t ) = 2 \\lambda ^ { v } \\left ( 1 + 2 \\lambda ^ { v } ( v ^ { - 1 } - 1 ) t ^ { - 1 } \\right ) \\ ; \\mbox { w i t h } \\ ; f ^ { \\prime } ( t ) \\to 0 \\ ; \\mbox { a s } \\ ; t \\to \\infty . \\end{cases} \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} X _ { ( i ; j ) } & : = \\frac 1 2 ( X _ { i ; j } + X _ { j ; i } ) = \\frac 1 2 \\left ( X ^ a _ { , j } \\ , g _ { a i } + X ^ a g _ { a i , j } + X ^ a _ { , i } \\ , g _ { a j } + X ^ a g _ { a j , i } - 2 X ^ a g _ { a k } \\Gamma ^ k _ { j i } \\right ) \\\\ \\mathrm { d i v } X & : = X ^ i _ { ; i } = X ^ i _ { , i } + X ^ j \\Gamma ^ i _ { i j } = \\frac { 1 } { \\sqrt { \\det g } } ( \\sqrt { \\det g } \\ , X ^ i ) _ { , i } \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} \\mathfrak { \\widehat D } _ k = - k a ^ { k - 2 } X ^ d + \\frac { k ( k - 3 ) } { 2 } a ^ { k - 4 } X ^ { 2 d } - \\frac { k ( k - 4 ) ( k - 5 ) } { 6 } a ^ { k - 6 } X ^ { 3 d } \\\\ + \\cdots + ( - 1 ) ^ { \\frac { k } { 2 } - 1 } \\ , \\frac { k ^ 2 } { 4 } a ^ 2 X ^ { d ( \\frac { k } { 2 } - 1 ) } + ( - 1 ) ^ { \\frac { k } { 2 } } \\cdot 2 \\cdot X ^ { \\frac { d k } { 2 } } . \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} D [ \\Theta ^ { } _ 1 ( \\Lambda ) ] ( 0 ) & = A _ c D k _ c ( 0 ) + D g _ c ( K { } { } ( 0 ) ) D K { } { } ( 0 ) - D k _ c ( ( A _ c + r ) ( 0 ) ) D ( A _ c + r ) ( 0 ) \\\\ & = A _ c D k _ c ( 0 ) + D g _ c ( 0 ) D K { } { } ( 0 ) - D k _ c ( 0 ) ( A _ c + D r ) ( 0 ) . \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} X = x , Y = y - \\frac { 1 } { b } , Z = z , U = u \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} | \\Phi ( \\xi _ 1 , \\eta _ 1 , \\xi , \\eta ) | = & | \\xi _ 1 \\xi ( \\xi _ 1 + \\xi ) + \\eta _ 1 \\eta ( \\eta _ 1 + \\eta ) | \\\\ \\geq & r _ 1 r ( r _ 1 + r ) | \\cos ^ 3 \\theta _ 1 + \\sin ^ 3 \\theta _ 1 | - 2 ^ { 9 } A ^ { - 1 } r _ 1 r ( r _ 1 + r ) \\\\ = & r _ 1 r ( r _ 1 + r ) ( 1 - 2 ^ { - 1 } \\sin 2 \\theta _ 1 ) | \\cos \\theta _ 1 + \\sin \\theta _ 1 | - 2 ^ { 9 } A ^ { - 1 } r _ 1 r ( r _ 1 + r ) \\\\ \\geq & 2 ^ { - 1 3 } r _ 1 r ( r _ 1 + r ) , \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} v _ { 4 } ( t , r ) = \\int _ { t } ^ { \\infty } d x \\int _ { 0 } ^ { \\infty } d \\xi J _ { 1 } ( r \\xi ) \\sin ( ( t - x ) \\xi ) \\widehat { v _ { 4 , c } } ( x , \\xi ) \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} \\Phi _ \\epsilon ^ { c _ 1 } ( x ) \\succeq P ^ { 2 } [ \\Phi _ \\epsilon ^ { c _ 1 } ] ( x ) \\succeq P ^ { 2 } [ \\Phi _ { n } ^ 2 ] ( x ) = \\Phi _ { n + 1 } ^ 2 ( x ) \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} \\mu _ { r , 5 } ( 0 , 1 ) = \\begin{cases} & 0 \\ ; \\ ; \\\\ & 1 \\ ; \\ ; \\\\ \\end{cases} , \\ ; \\ ; \\ ; \\ ; \\lambda _ r ( 0 , 1 ) = \\begin{cases} & 0 \\ ; \\ ; \\\\ & 1 \\ ; \\ ; \\end{cases} . \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} \\delta = \\frac { ( 2 - c ^ 2 ) m ^ { 1 / 2 } } { 2 c n ^ { 1 / 2 } } + O ( m n ^ { - 1 } ) \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} H ^ { \\ast } ( \\mathbb { P } ( E ) ; \\Z ) \\cong H ^ { \\ast } ( X ; \\Z ) [ x ] \\bigg / \\left \\langle \\sum _ { k = 1 } ^ { n + 1 } ( - 1 ) ^ k x ^ { n + 1 - k } c _ { k } ( E ) \\right \\rangle , \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} A \\ & = \\ \\{ 1 , 6 , 1 0 , 2 2 , 2 4 , 3 0 \\} , \\\\ B \\ & = \\ \\{ 7 , 1 2 , 1 3 , 1 5 , 1 9 , 2 7 \\} , \\\\ C \\ & = \\ \\{ 3 , 4 , 1 7 , 1 8 , 2 3 , 2 8 \\} , \\\\ D \\ & = \\ \\{ 2 , 9 , 1 1 , 1 6 , 2 6 , 2 9 \\} , \\\\ E \\ & = \\ \\{ 5 , 8 , 1 4 , 2 0 , 2 1 , 2 5 \\} , \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} h _ s = ( n + 1 ) H ' ( s ) u + b e , \\exists b \\in { \\mathbb R } . \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } \\leq \\sqrt { n } \\left ( \\sum _ { i = 1 } ^ { n } a _ i a _ i ^ * \\right ) ^ \\frac { 1 } { 2 } , \\forall n \\in \\mathbb { N } , \\forall a _ 1 , \\dots , a _ n \\in \\mathcal { A } . \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} | { \\rm H e s s } \\ , u | _ A ^ 2 & = \\sum _ { i , j } u _ { i j } ^ 2 + 2 ( p - 2 ) \\sum _ { k = 1 } ^ { n } u _ { k n } ^ 2 + ( p - 2 ) ^ { 2 } u _ { n n } ^ 2 \\\\ & = ( p - 1 ) ^ 2 u _ { n n } ^ 2 + 2 ( p - 1 ) \\sum _ { k = 1 } ^ { n - 1 } u _ { k n } ^ 2 + \\sum _ { k , l = 1 } ^ { n - 1 } u _ { k l } ^ 2 \\\\ & \\geq ( p - 1 ) ^ 2 u _ { n n } ^ 2 . \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} \\frac { d \\lambda } { d \\tau } [ R ' ( \\lambda ) + Q ' ( \\lambda ) e ^ { - \\lambda \\tau } - \\tau Q ( \\lambda ) e ^ { - \\lambda \\tau } ] - \\lambda Q ( \\lambda ) e ^ { - \\lambda \\tau } = 0 \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} Q : = - \\frac { 4 } { 3 } \\mathrm { R e } [ \\left ( R , _ { 1 } - i A _ { 1 1 , \\overline { 1 } } \\right ) _ { \\overline { 1 } } ] = - \\frac { 2 } { 3 } ( W _ { 1 \\overline { 1 } } + W _ { \\overline { 1 } 1 } ) . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} \\mbox { $ H ' \\neq 0 $ a t $ s $ } \\ ; & \\ ; \\Longrightarrow \\ ; \\ ; \\mbox { $ \\lambda _ 2 $ d o e s n o t c h a n g e s i g n a t $ s $ } , \\\\ \\mbox { $ H ' = 0 $ \\& $ H '' \\ne 0 $ \\& $ V ' \\ne 0 $ a t $ s $ } \\ ; & \\ ; \\Longrightarrow \\ ; \\ ; \\mbox { $ \\lambda _ 2 = 0 $ \\& $ \\lambda _ 2 $ c h a n g e s s i g n a t $ s $ } . \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} \\sum _ { b = 0 } ^ a \\xi _ 3 ^ { b } ( b + 1 ) & = \\sum _ { b = 0 } ^ { \\frac { a - 2 } { 3 } } \\xi _ 3 ^ { 3 b } ( 3 b + 1 ) + \\xi _ 3 ^ { 3 b + 1 } ( 3 b + 2 ) + \\xi _ 3 ^ { 3 b + 2 } ( 3 b + 3 ) \\\\ & = \\sum _ { b = 0 } ^ { \\frac { a - 2 } { 3 } } \\xi _ 3 ^ { 3 b + 1 } + 2 \\xi _ 3 ^ { 3 b + 2 } = \\frac { a + 1 } { 3 } ( \\xi _ 3 + 2 \\xi _ 3 ^ { 2 } ) . \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} R ^ + | I \\ & = \\ R , \\\\ R ^ + ( u ) \\ & = \\ \\{ v \\} \\cup J ' , \\\\ R ^ + ( v ) \\ & = \\ J , \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} 0 = B ( e f , e x f ) & = e B ( f , e x f ) + B ( e , e x f ) f = e B ( f , x ) f + e B ( e , x ) f \\\\ & = B ( f , e x f ) + B ( e , e x f ) = - B ( e x f , f ) + B ( e , e x f ) . \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} & { \\psi _ 0 } ^ { \\pm } = e ^ { i \\theta } \\Big [ \\ , { \\psi _ { 0 1 } } ^ { \\pm } \\ , + \\ , i { \\psi _ { 0 2 } } ^ { \\pm } \\ , \\Big ] , \\\\ [ 1 m m ] & { \\psi _ j ^ 1 } ^ { \\pm } = e ^ { i \\theta } \\Big [ \\ , { \\psi _ { j 1 } ^ 1 } ^ { \\pm } \\sin { j \\theta } \\ , + \\ , i { \\psi _ { j 2 } ^ 1 } ^ { \\pm } \\cos { j \\theta } \\ , \\Big ] , \\\\ [ 1 m m ] & { \\psi _ j ^ 2 } ^ { \\pm } = e ^ { i \\theta } \\Big [ \\ , { \\psi _ { j 1 } ^ 2 } ^ { \\pm } \\cos { j \\theta } \\ , + \\ , i { \\psi _ { j 2 } ^ 2 } ^ { \\pm } \\sin { j \\theta } \\ , \\Big ] . \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} \\sigma _ { { \\rm f o l d } , J } = \\frac { a _ { 1 J } } { b _ { 0 J } } + \\frac { b _ { 2 J } } { b _ { 0 J } } - \\frac { a _ { 5 J } } { a _ { 2 J } } . \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} H _ T ( \\omega ) = - \\int _ 0 ^ T v ( \\omega ( t ) ) d t , \\omega \\in \\Omega _ T . \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} \\sum _ { { \\bf m } \\in \\mathbb { N } _ { \\geq 0 } ^ { n ( n - 1 ) / 2 } } \\frac { q ^ { C ( { \\bf m } ) } \\cdot F } { \\displaystyle \\prod _ { 1 \\leq i < j \\leq n } ( q ) _ { m _ { i , j } } } = \\sum _ { k _ 1 , \\cdots , k _ { n - 1 } \\geq 0 } \\frac { q ^ { k _ 1 ^ 2 / 2 + \\cdots + k _ { n - 1 } ^ 2 / 2 - k _ 1 k _ 2 - \\cdots - k _ { n - 2 } k _ { n - 1 } } x _ 1 ^ { k _ 1 } \\cdots x _ { n - 1 } ^ { k _ { n - 1 } } } { ( q ) _ { k _ 1 } \\cdots ( q ) _ { k _ { n - 1 } } } , \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} g _ { \\sigma ^ 2 } ( x ) = \\frac { 1 } { ( 2 \\pi \\sigma ^ 2 ) ^ { \\frac { d } { 2 } } } e ^ { - \\frac { | x | ^ 2 } { 2 \\sigma ^ 2 } } . \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} M : = W + \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 \\mu _ { k + 1 } Z _ { k } - \\mu _ { k + 1 } ^ 2 } { 4 ( k + 1 ) } \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} & \\Bigl | \\int _ { \\R ^ 3 \\times \\R ^ 3 } { h ( \\tau _ 1 + \\tau _ 2 , \\xi _ 1 + \\xi _ 2 , \\eta _ 1 + \\eta _ 2 ) f ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) g ( \\tau _ 2 , \\xi _ 2 , \\eta _ 2 ) } d \\sigma _ 1 d \\sigma _ 2 \\Bigr | \\\\ & \\lesssim A d ^ { \\frac { 1 } { 2 } } N _ 1 ^ { - 2 } ( L _ 0 L _ 1 L _ 2 ) ^ { \\frac { 1 } { 2 } } \\| f \\| _ { L ^ 2 } \\| g \\| _ { L ^ 2 } \\| h \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} \\begin{aligned} U _ \\eta \\to U & L ^ 2 ( 0 , T ; L ^ { 2 } ( \\mathbb T ^ d _ { \\ell } ) ) - w , \\\\ \\sqrt { R _ \\eta } U _ \\eta \\to \\sqrt { R } U & C ( [ 0 , T ] ; L ^ 2 ( \\mathbb T ^ d _ { \\ell } ) - w ) , \\\\ R _ \\eta ^ { \\frac 1 4 } U _ \\eta \\to R ^ { \\frac 1 4 } U & L ^ 4 ( 0 , T ; L ^ 4 ( \\mathbb T ^ d _ { \\ell } ) ) - w , \\\\ R _ \\eta U _ \\eta \\to R U & L ^ 2 ( 0 , T ; L ^ 2 ( \\mathbb T ^ d _ { \\ell } ) ) . \\end{aligned} \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} b _ { 0 } = 4 - \\mu . \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} & | | \\sqrt { \\xi } \\overline { y } ( \\xi ) | | _ { L ^ { 2 } ( \\rho ( \\xi ) d \\xi ) } ^ { 2 } = | | L \\overline { v } | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\theta _ 2 ( \\varepsilon ) = \\max \\left \\{ \\theta _ { 2 , 1 } , \\theta _ { 2 , 2 } , \\theta _ { 2 , 3 } \\right \\} < 1 . \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} H \\mathcal { H } P z = H \\mathcal { H } \\tilde { z } = ( P _ n H \\mathcal { H } z ) _ { n \\in \\mathbb { N } } = P H \\mathcal { H } z \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} \\partial _ { | x | } u & = - \\frac { \\lambda '' ( s ) } { 2 \\pi } \\int _ { 0 } ^ { t - s } \\rho d \\rho \\int _ { 0 } ^ { 2 \\pi } d \\theta \\frac { 2 ( | x | + \\rho \\cos ( \\theta ) ) } { ( 1 + | x | ^ { 2 } + 2 | x | \\rho \\cos ( \\theta ) + \\rho ^ { 2 } ) ( ( t - s ) ^ { 2 } - \\rho ^ { 2 } ) ^ { 1 / 2 } } \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} f ( p ) = \\sup _ { a \\in A } \\frac { \\varphi ( a ) } { \\left ( 1 + \\left [ d ( a , p ) \\right ] ^ 2 \\right ) ^ { \\frac 1 { d ( p , A ) } } } , \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } \\frac { 1 } { 2 } \\alpha ' ( t ) & 0 \\\\ 0 & \\frac { 1 } { 2 } \\beta ' ( t ) \\\\ \\end{array} \\right ) = \\left ( \\begin{array} { c c } - b ( t ) \\alpha ( t ) & 0 \\\\ 0 & - \\frac { 1 } { b ( t ) } \\beta ( t ) \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} \\frac { \\Upsilon ( w , \\theta ) } { \\Upsilon ( w , \\theta - 1 ) } = \\Lambda ( w , \\theta \\b 1 ) . \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) v _ { 1 } \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle & = \\frac { - 1 6 } { \\lambda ( t ) ^ { 3 } } \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) K ( s - t , \\lambda ( t ) ) + \\frac { - 1 6 } { \\lambda ( t ) ^ { 3 } } \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) K _ { 1 } ( s - t , \\lambda ( t ) ) \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} \\partial _ { 1 } L ( x , \\lambda ( t ) ) = L _ { a } ( x , \\lambda ( t ) ) + L _ { b } ( x , \\lambda ( t ) ) \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} \\forall ( a , b ) \\in \\R ^ 2 \\colon \\varphi _ \\textup { F B } ( a , b ) : = a + b - \\sqrt { a ^ 2 + b ^ 2 } \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} x K _ { n } ^ { p , N } \\left ( x \\right ) = K _ { n + 1 } ^ { p , N } \\left ( x \\right ) + \\alpha _ { n } ^ { p , N } K _ { n } ^ { p , N } \\left ( x \\right ) + \\beta _ { n } ^ { p , N } K _ { n - 1 } ^ { p , N } \\left ( x \\right ) , n \\geq 0 , \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} \\begin{aligned} \\phi _ 7 ^ 7 ( z ) & = g _ 7 ^ 2 ( 7 z ) \\{ 3 4 3 \\phi _ 7 ^ 6 ( z ) + 3 4 3 \\phi _ 7 ^ 5 ( z ) + 1 4 7 \\phi _ 7 ^ 4 ( z ) + 4 9 \\phi _ 7 ^ 3 ( z ) + 2 1 \\phi _ 7 ^ 2 ( z ) \\\\ & + 7 \\phi _ 7 ( z ) + 1 \\} + g _ 7 ( 7 z ) \\{ 7 \\phi _ 7 ^ 4 ( z ) + 3 5 \\phi _ 7 ^ 5 ( z ) + 4 9 \\phi _ 7 ^ 6 ( z ) \\} . \\end{aligned} \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} \\Phi ^ { \\prime } ( \\alpha ) = \\beta + ( 1 - \\beta ) F _ { \\nu } ^ { \\prime } ( \\alpha ) . \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} m _ { s c } \\left ( z \\right ) = - 1 + O ( \\sqrt { s } ) . \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} B _ { H , n } \\coloneqq 1 + \\sum _ { i = 1 } ^ { n - 1 } H _ i - H _ n . \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} 0 & = a _ 4 \\sqrt { \\tilde { x } } + a _ 2 \\tilde { y } , \\\\ \\dot { \\tilde { y } } & = - \\frac { \\beta } { a _ 2 } + b _ 4 \\sqrt { \\tilde { x } } + b _ 2 \\tilde { y } . \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} \\mu _ N \\left ( d x \\right ) : = \\frac { 1 } { Z } \\exp \\left ( - H ( x ) \\right ) d x . \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} [ \\mathfrak { g } , [ J , \\mathfrak { g } ] ] \\subset J , [ \\mathfrak { g } , [ J , J ] ] = \\{ 0 \\} . \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} ( x y ) v _ j = x ( y v _ j ) = \\sum _ { k = 1 } ^ n ( x v _ k ) \\varphi _ { k j } ( y ) = \\sum _ { i = 1 } ^ n \\sum _ { k = 1 } ^ n v _ i \\varphi _ { i k } ( x ) \\varphi _ { k j } ( y ) . \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { H _ n ( u , x ) \\ , t ^ n } { ( 1 - u ) ^ { n + 1 } ( 1 - x ) ^ { n + 1 } } = \\sum _ { k \\ge 0 } \\sum _ { m \\ge 0 } \\frac { u ^ k x ^ m } { ( 1 - t ) ^ { k m } } , \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} \\left | \\bar { H } _ N ( m ) - \\varphi ( m ) \\right | & \\leq \\left | \\bar { H } _ N ( m ) - \\mathcal { H } _ N ( m ) \\right | + \\left | \\mathcal { H } _ N ( m ) - \\varphi ( m ) \\right | \\\\ & \\leq C \\frac { 1 } { N } + C \\frac { m ^ 2 + 1 } { N } = C \\frac { m ^ 2 + 1 } { N } . \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} u _ t = ( D ^ \\alpha _ x u ) _ x + f \\quad \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} & \\lim _ { a \\to 1 } \\bigg \\{ \\frac { 1 } { ( 1 - q ) ^ 2 } - \\frac { n ( 1 - a ) a ^ { ( n - 1 ) / 2 } } { ( 1 - q / a ) ( 1 - a q ) ( 1 - a ^ n ) } \\bigg \\} \\frac { ( 1 - a q ^ n ) ( a - q ^ n ) } { ( 1 - a ) ^ 2 } \\\\ [ 5 p t ] & = \\lim _ { a \\to 1 } \\frac { \\{ ( 1 - q / a ) ( 1 - a q ) ( 1 - a ^ n ) - n ( 1 - a ) a ^ { ( n - 1 ) / 2 } ( 1 - q ) ^ 2 \\} ( 1 - a q ^ n ) ( a - q ^ n ) } { ( 1 - q ) ^ 2 ( 1 - q / a ) ( 1 - a q ) ( 1 - a ^ n ) ( 1 - a ) ^ 2 } \\\\ [ 5 p t ] & = [ n ] ^ 2 \\frac { n ^ 2 ( 1 - q ) ^ 2 - ( 1 + 2 2 q + q ^ 2 ) } { 2 4 ( 1 - q ) ^ 2 } . \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} f ( \\alpha ) = \\frac { 2 E } { \\lambda } r ^ 3 \\alpha ^ 3 + \\frac { E ^ 2 } { \\lambda ^ 2 } r ^ 4 \\alpha ^ 4 , \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} u _ p ^ { + , e x t } ( 1 , \\omega ) = u _ p ^ + ( 1 , \\omega ) \\ \\ \\ \\omega \\in S _ p \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} 2 ^ { k + 1 } H _ { g + 1 - k } \\leq \\sum _ { i = 1 } ^ { g + 1 } H _ i + 1 . \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} n = x '^ 2 + y '^ 2 + z '^ 2 + w '^ 2 \\ \\ x ' + 3 y ' = 2 . \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} \\frac { c _ { b } } { 2 } \\int _ { 0 } ^ { \\infty } \\sin ( t \\xi ) \\xi J _ { 0 } ( r \\xi ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } d \\xi = \\frac { - b } { t ^ { 2 } \\log ^ { b } ( t ) } + E _ { \\partial _ { r } v _ { 2 } , 1 } ( t , r ) \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} \\alpha _ 0 + 2 \\sum _ { k = 1 } ^ { n / 4 - 1 } \\alpha _ k + \\alpha _ { n / 4 } = \\frac { \\pi } { 2 } , \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} L ^ { [ k ] } ( x ) & = \\sum _ { j } L ^ { [ k ] } _ j x ^ { j } = \\frac { \\int _ 0 ^ x \\lambda ^ { [ k ] } ( z ) d z } { \\int _ 0 ^ 1 \\lambda ^ { [ k ] } ( z ) d z } . \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} F ( z ) = \\sum _ { y \\in S ( z ) } w ( z , y ) g ( y ) = \\sum _ { y \\in S ( z ) } w ( x , y ) F ( y ) . \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} \\norm { \\eta ( x ) + \\eta ( y ) } ^ * = \\norm { \\eta ( x + y ) } ^ * = \\norm { x + y } \\leq \\norm x + \\norm y = \\norm { \\eta ( x ) } ^ * + \\norm { \\eta ( y ) } ^ * . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} h ^ \\Theta ( z ) \\ , = \\ , \\frac 1 { h ( z ) } , b ^ \\Theta ( z ) \\ , = \\ , - b ( z ) , c ^ \\Theta \\ , = \\ , c \\ , . \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} \\kappa _ G ( s , y , y ^ \\prime ) = \\int _ { { \\rm I m } \\lambda = r } s ^ { i \\lambda } G ( \\lambda ) ( y , y ^ \\prime ) d \\lambda \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } & C _ 2 , D _ 4 , K \\ne 0 , \\\\ & S _ i = 0 ( i = 1 , \\ldots , 4 ) , \\end{alignedat} \\right . \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} \\phi _ { n , 1 } ^ c ( x _ { n } ^ c ) = u _ 1 ^ * / 2 , \\ \\ \\ \\phi _ { n , 1 } ^ c ( x ) < u _ 1 ^ * / 2 \\ \\ \\ \\ { \\rm f o r } \\ x > x _ { n } ^ c . \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} \\lim _ n \\sum _ k { \\rm { o s c \\ , } } ( f , I _ k ^ n ) \\ , \\big ( \\Psi ( x _ { k , r } ^ n ) - \\Psi ( x _ { k , l } ^ n ) \\big ) = 0 \\ , . \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} \\limsup _ { M \\to \\infty } \\limsup _ { n \\to \\infty } P \\left ( \\| \\hat { K } _ { \\lambda _ n } ^ - - K _ Y ^ - \\| _ { \\ell _ 1 } > M s _ n \\lambda _ n \\right ) \\leq \\limsup _ { L \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\Omega _ { n , L } ^ c ) = 0 , \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} V ( D ) = \\left ( \\begin{array} { c c } e ^ { - D ^ - D ^ + } & e ^ { - \\frac { 1 } { 2 } D ^ - D ^ + } \\left ( \\frac { I - e ^ { - D ^ - D ^ + } } { D ^ - D ^ + } \\right ) D ^ - \\\\ e ^ { - \\frac { 1 } { 2 } D ^ + D ^ - } D ^ + & I - e ^ { - D ^ + D ^ - } \\end{array} \\right ) \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} \\nu \\ast _ G \\nu ' ( f ) = \\int _ G \\int _ G f ( x y ) d \\nu ( x ) d \\nu ' ( y ) , \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} e ( \\beta Q ^ { 2 p + p - 1 - i } \\beta Q ^ i ) & = 2 ( 2 p ^ 2 + p - 1 - i ) - 2 ( p - 1 ) i = 4 p ^ 2 + 2 p - 2 - 2 p i \\\\ & = 4 p ^ 2 + 2 p - 2 - 4 p ^ 2 - 2 p a = 2 p - 2 p a - 2 < 0 . \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} \\vec { f } _ x ( g ) = f ( x ) g ' ( x ) - f ' ( x ) g ( x ) = W _ x ( f , g ) \\forall g \\in C ^ 1 ] a , b [ . \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} \\frac { \\partial H } { \\partial \\bar \\xi } = G . \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} P ( n _ 1 , \\ldots , n _ r ) = \\lim _ { m \\rightarrow \\infty } \\frac { \\ell _ R ( M / I ( 1 ) _ { m n _ 1 } \\cdots I ( r ) _ { m n _ r } M ) } { m ^ d } \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} Q : = \\frac { I - \\exp ( - \\frac { 1 } { 2 } D ^ - D ^ + ) } { D ^ - D ^ + } D ^ + \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} ( \\theta + d d ^ c q ) ^ n ( t _ 0 , x _ 0 ) & \\geq \\nu ( x _ 0 ) = e ^ { \\partial _ t u ( t _ 0 , x _ 0 ) + u ( t _ 0 , x _ 0 ) + F ( t _ 0 , x _ 0 , u ( t _ 0 , x _ 0 ) ) } \\mu ( t _ 0 , x _ 0 ) \\\\ & \\geq e ^ { \\partial _ t q ( t _ 0 , x _ 0 ) ) + F ( t _ 0 , x _ 0 , q ( t _ 0 , x _ 0 ) ) } \\mu ( t _ 0 , x _ 0 ) . \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} \\frac { g ( r ) } { \\sqrt { r } } = \\int _ { 0 } ^ { \\infty } \\frac { 2 } { f ( u ) ^ { 2 } } \\mathcal { F } _ { H } ( \\frac { g ( \\cdot ) } { \\sqrt { \\cdot } } ) ( u ) \\rho ( u ^ { 2 } ) \\tilde { \\phi } _ { u } ( r ) u d u \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} F _ { \\lambda , t } = \\begin{bmatrix} 0 & G _ { \\lambda , t } ^ { * } \\\\ G _ { \\lambda , t } & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} & f _ n ( t ) = f _ { 0 , n } + \\int _ 0 ^ t [ Q ^ { + } ( s , f _ { n - 1 } ( s ) ) - Q ^ { - } ( s , f _ { n - 1 } ( s ) ) ] d s \\\\ & + \\int _ 0 ^ t \\biggl [ a \\left ( \\left \\| \\Lambda f _ { n - 1 } ( s ) \\right \\| + \\int _ 0 ^ s \\Delta ( \\tau , f _ { n - 2 } ( \\tau ) ) d \\tau \\right ) \\Lambda f _ { n - 1 } ( s ) \\\\ & - a ( \\left \\| \\Lambda f _ 0 \\right \\| ) \\Lambda f _ n ( s ) \\biggr ] d s , \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} e ( N _ { \\{ 1 \\} \\times C ^ n } Y _ \\mu ) ^ { - 1 } e ( N _ { \\{ w \\} \\times C ^ n } Y _ \\lambda ) ^ { - 1 } = \\left ( \\gamma _ \\mu \\gamma _ \\lambda ^ w \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} T _ u ( S h ) = S T _ u h + \\langle u S h | 1 \\rangle 1 \\ , . \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\widetilde { \\varphi } _ 3 ( N ) = - \\beta Q ^ { 2 p ^ 2 } \\beta Q ^ { p - 1 } \\beta Q ^ { 1 } . \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} \\langle g , f _ 0 \\rangle _ { \\alpha } = ( \\gamma + 2 ) \\langle g , f _ 0 \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } . \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} \\dot { B } ( t ) & = - [ \\gamma ( t ) , B ( t ) ] + d _ { A ( t ) } j ( t ) \\\\ \\dot { A } ( t ) & = d _ { A ( t ) } \\gamma ( t ) , \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} \\mathsf { \\hat { t } } _ { \\mathsf { j } } : = \\left \\Vert Q _ { \\mathsf { j } } \\mathsf { \\hat { p } } _ { \\mathsf { j } } \\right \\Vert _ { 2 } ^ { - 2 } Q _ { \\mathsf { j } } \\mathsf { \\hat { p } } _ { \\mathsf { j } } , \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} c _ { s } = \\frac { 4 ^ { s } \\Gamma ( s ) } { 2 s \\Gamma ( - s ) } < 0 ( c _ { 1 / 2 } = - 1 ) , \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ s _ p u = f ( x , u ) , & x \\in \\Omega , \\\\ u = 0 , & x \\in \\mathbb { R } ^ n \\setminus \\Omega , \\end{cases} \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } C _ 2 & = 0 , \\\\ C _ 3 & = - C _ 1 , \\\\ K & = \\frac { P _ 4 - P _ 2 } { 4 } . \\end{alignedat} \\right . \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} { A ' } ^ 4 _ 4 & = - i \\lambda _ 3 \\left ( - i \\lambda _ 3 - i b _ 2 \\left ( x _ 1 + x _ 2 + x _ 3 + x _ 4 \\right ) \\right ) \\left ( \\frac { i } { x _ { 1 + 2 } } + \\frac { i } { x _ { 1 + 3 } } + \\frac { i } { x _ { 1 + 4 } } \\right ) \\\\ & + A _ 4 . \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ { N } a _ k U _ k ^ 2 \\right | \\leq \\frac { C } { N } \\textrm { a n d } | a _ k | \\leq \\frac { C } { N ^ 3 } \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N } \\frac { 1 } { \\log K _ { j } \\log \\log K _ { j } } \\le C \\left ( 1 + \\frac { 1 } { n _ { 1 } } \\sum _ { j = 1 } ^ { N } \\log K _ { j } \\right ) . \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda } \\log \\mathcal H _ \\lambda ( u ) = - \\frac { 1 } { \\lambda _ 0 + \\lambda } + \\sum _ { n = 1 } ^ \\infty \\frac { \\gamma _ n } { ( \\lambda + \\lambda _ { n - 1 } + 1 ) ( \\lambda + \\lambda _ n ) } \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} s ( f , \\Psi , { \\mathcal P } _ n ^ 1 ) = \\sum _ { k } f ( x _ k ^ { n , 1 } & ) \\ , | I _ k ^ { n , 1 } | _ \\Psi \\\\ = \\sum _ { k } f ( x _ k ^ { n , 1 } & ) \\int _ { I _ k ^ { n , 1 } } \\big ( \\psi - \\psi ( x _ k ^ { n , 1 } ) \\big ) + \\sum _ { k } f ( x _ k ^ { n , 1 } ) \\psi ( x _ k ^ { n , 1 } ) \\ , | I _ k ^ { n , 1 } | \\\\ & = A _ n + B _ n , \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} N _ { \\Z / n \\Z } ( a _ 1 , a _ 2 , \\ldots , a _ k ; b ) & = \\sum _ { i = 1 } ^ k \\sum _ { \\tau \\in S _ k : \\ell ( \\tau ) = i } ( - 1 ) ^ { k - i } | X _ \\tau | \\\\ & = \\sum _ { i = 2 } ^ k ( - 1 ) ^ { k - i } c ( k , i ) n ^ { i - 1 } \\\\ & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ k ( - 1 ) ^ { k - i } c ( k , i ) n ^ i - ( - 1 ) ^ { k - 1 } c ( k , 1 ) \\\\ & = \\frac { 1 } { n } ( n ) _ k + ( - 1 ) ^ { k } ( k - 1 ) ! \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} \\langle w , v \\rangle _ E - \\langle w , v \\rangle _ { \\pi ( E ) } = \\int _ E ( w v - ( w \\circ \\pi ) ( v \\circ \\pi ) ) \\ , d \\gamma _ h + \\int _ E ( w \\circ \\pi ) ( v \\circ \\pi ) \\ , d \\gamma _ h - \\int _ { \\pi ( E ) } w v \\ , d \\gamma , \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\sum _ { { \\bf n } \\in \\mathbb { Z } ^ k _ { \\geq 0 } } \\frac { q ^ { \\frac 1 2 { \\bf n } ^ \\top A { \\bf n } } { \\bf x } ^ { { \\bf n } } } { ( q ) _ { n _ 1 } \\cdots ( q ) _ { n _ k } } = \\sum _ { { \\bf m } \\in \\mathbb { Z } ^ { \\ell } _ { \\geq 0 } } \\frac { q ^ { B ( { \\bf m } ) } { \\bf x } ^ { U { \\bf m } } } { ( q ) _ { m _ 1 } \\cdots ( q ) _ { m _ \\ell } } , \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} ( q ; q ^ 2 ) _ \\infty ( q ^ { 2 } ; q ^ { 2 } ) _ \\infty \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { 2 n } } { 1 + q ^ { 2 n } } = ( q ; q ) _ \\infty \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { 2 n } } { 1 + q ^ { 2 n } } . \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} W = \\bigcup _ { x \\in G \\setminus \\{ 1 \\} } U _ x . \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} \\mathcal { W } ( M ) : = \\int _ M H ^ 2 + k _ 0 \\ , \\ , d S . \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle G _ m = H _ m ^ { - 1 } , H _ { m } ^ { - 1 } \\cdot F _ m \\cdot H _ m = { \\rm d i a g } ( 1 , 0 ) { \\rm a n d } \\\\ \\\\ \\displaystyle \\| H _ m \\| _ { { \\scriptstyle H _ { 2 } ^ \\infty \\rightarrow H _ 2 ^ \\infty } } \\le 1 , \\| H _ m ^ { - 1 } \\| _ { { \\scriptstyle H _ { 2 } ^ \\infty \\rightarrow H _ 2 ^ \\infty } } \\le 1 + \\| F _ m \\| _ { { \\scriptstyle H _ { 2 } ^ \\infty \\rightarrow H _ 2 ^ \\infty } } . \\end{array} \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} L _ N = \\{ ( x , y ) \\in G \\times G \\mid x ^ { - 1 } y \\in N \\} . \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } ( ( a , i ) ) = ( a + \\delta _ { ( a , i ) } , i + r \\delta _ { ( a , i ) } + 1 ) , \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} \\begin{pmatrix} D r \\\\ D k _ u \\\\ D k _ s \\end{pmatrix} & = \\begin{pmatrix*} [ l ] A _ c D k _ c + D g _ c ( K { } { } ) D K { } { } - D k _ c \\left ( R \\right ) D R \\\\ - A _ u ^ { - 1 } D g _ u ( K { } { } ) D K { } { } + A _ u ^ { - 1 } D k _ u ( R ) D R \\\\ A _ s D k _ s ( T ) D T + D g _ s ( K { } { } \\circ T ) D K { } { } ( T ) D T \\end{pmatrix*} , \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} \\tilde { \\gamma } = \\frac { \\gamma } { a _ { 2 L } a _ { 2 R } } , \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} \\partial _ { t } u = \\left ( a + i b \\right ) \\left [ \\Delta u + A \\left ( x \\right ) u + V \\left ( x , t \\right ) u \\right ] + F \\left ( x , t \\right ) , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} \\int _ { \\{ 0 , 1 \\} ^ { \\mathbb { Z } ^ d } } \\left | \\mathbb { E } _ \\eta ( f ( \\eta _ t ) ) - \\mu ( f ) \\right | \\mathrm { d } \\nu ( \\eta ) = \\| f - \\mu ( f ) \\| _ \\infty \\int _ { \\{ 0 , 1 \\} ^ { \\mathbb { Z } ^ d } } \\left | \\mathbb { E } _ \\eta ( g ( \\eta _ t ) ) \\right | \\mathrm { d } \\nu ( \\eta ) \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} \\psi _ { \\mu _ { * } } ( z ) = - 1 - \\psi _ { \\mu } ( 1 / z ) , \\quad \\eta _ { \\mu _ { * } } ( z ) = \\frac { 1 } { \\eta _ { \\mu } ( 1 / z ) } , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} \\Delta ( t , g ) : = \\left \\Vert \\Lambda Q ^ { - } ( t , g ) \\right \\Vert - \\left \\Vert \\Lambda Q ^ { + } ( t , g ) \\right \\Vert \\geq 0 , \\forall g \\in { \\mathcal D } _ { + } ( \\Lambda ^ 2 ) , \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} A ^ p ( G / H , \\mu ) : = \\{ \\varphi \\in L ^ p ( G / H , \\mu ) : L _ h \\varphi = \\varphi \\ \\forall h \\in H \\} \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} m ^ 2 + \\varepsilon m n + n ^ 2 = 1 \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} c _ { \\gamma \\mu } = \\big \\langle g _ \\gamma ^ { ( k + 1 ) } , \\sum _ \\beta c _ { \\beta \\mu } \\ , G _ \\beta ^ { ( k + 1 ) } \\big \\rangle = \\big \\langle g _ \\gamma ^ { ( k + 1 ) } , G _ { 1 ^ \\ell } ^ { ( k + 1 ) } G _ \\mu ^ { ( k + 1 ) } \\big \\rangle = \\big \\langle ( G _ { 1 ^ \\ell } ^ { ( k + 1 ) } ) ^ \\perp g _ \\gamma ^ { ( k + 1 ) } , G _ \\mu ^ { ( k + 1 ) } \\big \\rangle \\ , . \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} b _ { l - 2 } = \\frac { ( 8 - \\mu ) l + 2 4 { l \\choose 2 } + 1 6 { l \\choose 3 } } { 2 } . \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} ( A _ { i j } ) _ s = \\Pi _ s ( F _ { i j } ) + \\Pi ( V _ { i j } ) = \\Pi _ s ( A _ { i j } ) + V _ { i j } ^ { \\perp } = V _ { i j } ^ { \\perp } - F _ k \\ , \\langle \\nabla _ { F _ k } ^ { \\perp } V , A _ { i j } \\rangle \\ , . \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} \\gamma = \\left ( \\frac { \\partial f _ L } { \\partial y } \\frac { \\partial g _ R } { \\partial y } - \\frac { \\partial f _ R } { \\partial y } \\frac { \\partial g _ L } { \\partial y } \\middle ) \\right | _ { x = y = \\mu = 0 } . \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} \\omega ( u , v ) = \\langle u \\ , | \\ , \\partial _ x ^ { - 1 } v \\rangle , \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} a \\overline { H } a ^ { - 1 } = ( l _ a \\circ r _ { a ^ { - 1 } } ) ( \\overline { H } ) \\subseteq \\overline { ( l _ a \\circ r _ { a ^ { - 1 } } ) ( H ) } = \\overline { a H a ^ { - 1 } } = \\overline { H } . \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} \\Theta ( t ) : = \\frac { 1 } { 2 N } \\int \\left \\langle \\left ( x - N P ^ * \\eta ( t ) \\right ) , A ^ { - 1 } \\left ( x - N P ^ * \\eta ( t ) \\right ) \\right \\rangle f ( t , x ) \\mu ( d x ) . \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} v _ { 3 } ( t , r ) = - r \\int _ { t + 6 r } ^ { \\infty } d s \\lambda '' ( s ) ( s - t ) \\left ( \\frac { 1 } { 1 + ( s - t ) ^ { 2 } } - \\frac { 1 } { \\lambda ( t ) ^ { 2 - 2 \\alpha } + ( s - t ) ^ { 2 } } \\right ) + E _ { 5 } ( t , r ) \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} \\mathcal { A } ^ \\infty _ G ( M ) : = \\left ( H ^ \\infty _ L ( G ) \\hat { \\otimes } \\Psi ^ { - \\infty } _ c ( S ) \\right ) ^ { K \\times K } \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} L ' ( t ) = & - I ( u ( x , t ) ) = \\frac { q - p } { p } \\| \\nabla u \\| _ p ^ p + \\frac { 1 } { q } \\| u \\| _ q ^ q - q J ( u ( x , t ) ) \\\\ & \\ge \\frac { q - p } { p } \\Big ( \\frac { 2 } { \\widetilde { C } } L ( t ) - 1 \\Big ) - q J ( u ( x , t ) ) \\\\ & = \\frac { q - p } { p } \\frac { 2 } { \\widetilde { C } } \\Big ( L ( t ) - C _ 1 - C _ 2 J ( u ( x , t ) \\Big ) , \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} I _ 1 ( t ) : = [ \\underline h ( t ) - K _ 0 , \\underline h ( t ) ] , \\ I _ 2 ( t ) : = [ - \\underline h ( t ) , - \\underline h ( t ) + K _ 0 ] , \\ I _ 3 ( t ) : = [ - \\underline h ( t ) + K _ 0 , \\underline h ( t ) - K _ 0 ] . \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} { A ' } ^ 4 _ 4 & = \\Big ( - i b ' _ 2 ( 1 + 2 ) \\Big ) \\Big ( - i b ' _ 2 ( 3 + 4 ) \\Big ) \\frac { i } { x _ { 1 + 2 } } + \\\\ & - i b _ 3 \\cdot ( x _ 1 + x _ 2 + x _ 3 + x _ 4 ) . \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} \\{ T ( \\mathbf { x } , \\mathbf { y } , \\mathbf { w } ) , \\mathbf { z } \\} = 2 \\Delta ( \\mathbf { x } , \\mathbf { y } , \\mathbf { w } , \\mathbf { z } ) , \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} \\frac { \\sum _ { w _ 0 = 0 } ^ K \\prod _ { n = 1 } ^ M P _ W ( w ^ { x , w _ 0 } _ { n - 1 } , w ^ { x , w _ 0 } _ { n } ) \\mathbf { P } \\left ( \\hat { W } _ { N - M } = w _ 0 , \\ : \\max _ { 1 \\leq n \\leq { N - M } } \\hat { W } _ n \\leq K \\ : \\vline \\ : \\hat { W } _ { 0 } = w ^ { x , w _ 0 } _ M \\right ) } { \\sum _ { w _ 0 = 0 } ^ K \\mathbf { P } \\left ( \\hat { W } _ N = w _ 0 , \\ : \\max _ { 1 \\leq n \\leq N } \\hat { W } _ n \\leq K \\ : \\vline \\ : \\hat { W } _ { 0 } = w _ 0 \\right ) } . \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to \\infty } \\int _ { \\mathbb { R } } \\left ( \\sup _ { \\rho < r _ 1 < r _ 2 } \\left | \\int _ { r _ 1 } ^ { r _ 2 } A ( r ) P ( r , k ) d x \\right | \\right ) ^ 2 \\frac { d \\sigma } { 1 + k ^ 2 } = 0 \\ , . \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } F ( t , x , v ) \\psi ( t , x , v ) d x d v = \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } f _ 0 ( x , v ) \\psi ( 0 , x , v ) d x d v , \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} 0 \\ , = \\ , E _ 0 \\ , \\subset \\ , E _ 1 \\ , \\subsetneq \\ , E _ 2 \\ , \\subsetneq \\ , \\cdots \\ , \\subsetneq \\ , E _ b \\ , \\subsetneq \\ , E _ { b + 1 } \\ , = \\ , E \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} D _ 0 ( x ) & = 1 \\\\ D _ 1 ( x ) & = 2 x \\\\ D _ 2 ( x ) & = 2 x + x ^ 2 \\\\ D _ 3 ( x ) & = 2 x + 2 x ^ 2 \\\\ D _ 4 ( x ) & = x + 4 x ^ 2 + x ^ 3 \\\\ D _ 5 ( x ) & = 5 x ^ 2 + 4 x ^ 3 + x ^ 4 \\\\ D _ 6 ( x ) & = 3 x ^ 2 + 9 x ^ 3 + 3 x ^ 4 + x ^ 5 \\\\ D _ 7 ( x ) & = x ^ 2 + 1 1 x ^ 3 + 1 0 x ^ 4 + 3 x ^ 5 + x ^ 6 \\\\ \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} \\dim \\mu _ { \\lambda } = \\lim _ n \\frac { H ( \\mu _ { \\lambda } ; \\l ^ n ) } { n \\log \\l ^ { - 1 } } = \\lim _ n \\frac { H ( \\mu _ { \\lambda } ^ { ( n ) } ; \\l ^ n ) } { n \\log \\l ^ { - 1 } } \\le \\lim _ n \\frac { H ( \\mu _ { \\lambda } ^ { ( n ) } ) } { n \\log \\l ^ { - 1 } } , \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} E _ { x } \\left [ \\int _ { 0 } ^ { t } f ( X _ s ) \\ , d L _ s \\right ] = \\int _ { 0 } ^ { t } \\int _ { \\partial D } p _ { s } ( x , y ) f ( y ) \\ , d \\sigma ( y ) \\ , d s \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u + \\left \\Vert u \\right \\Vert ^ { \\frac { 4 } { n } } u = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} e ''' ( t ) = \\frac { R H S _ { 3 } ( t ) } { 4 \\alpha } - \\int _ { t } ^ { \\infty } \\frac { R H S _ { 3 } ( z ) } { 4 \\alpha } r _ { 2 } ( - t , - z ) d z , t \\geq T _ { 0 } \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} P ( \\tilde { r } ; \\mu ) = P _ L \\left ( P _ R ( \\tilde { r } ; \\mu ) ; \\mu \\right ) , \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} ( a ) \\frac { u _ 1 } { u _ 1 ' } > 0 \\quad u _ 2 = u _ 2 ' = 0 \\qquad \\qquad \\qquad ( b ) \\frac { u _ 2 } { u _ 2 ' } > 0 \\quad \\frac { u _ 1 } { | u _ 2 | ^ \\lambda } = \\frac { u _ 1 ' } { | u _ 2 ' | ^ \\lambda } = : u \\ , . \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} G _ 1 ( X ( \\gamma ) , Y ( \\gamma ) ) = \\int _ 0 ^ 1 g ( X ( \\gamma ; t ) , Y ( \\gamma ; t ) ) d t \\\\ + \\int _ 0 ^ 1 g ( \\nabla X ( \\gamma ; t ) , \\nabla Y ( \\gamma ; t ) ) d t . \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} \\frac { 1 } { 2 t _ n } \\left ( u _ n \\Phi ( x ) u _ n ^ { - 1 } - \\Phi ( x ) \\right ) & = - \\Phi ( x _ n ) + \\frac { 1 } { 2 t _ n } \\sum \\limits _ { k = 1 } ^ { n - 1 } \\left ( u _ n \\Phi ( t _ k x _ k ) u _ n ^ { - 1 } - \\Phi ( t _ k x _ k ) \\right ) . \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} \\left ( \\frac { h ' } { h } \\right ) ' + \\left ( \\frac { h ' } { h } \\right ) ^ 2 = H \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} A _ 0 ^ { ( 0 ) } = 1 , \\ \\ \\ A _ k ^ { ( k ) } = A _ { k - 1 } ^ { ( k ) } , \\ \\ \\ A _ j ^ { ( k ) } = \\sum _ { i = 0 } ^ j A _ i ^ { ( k - 1 ) } , \\ \\ \\ \\ k > j \\ge 0 \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb E _ { \\delta } \\int _ 0 ^ t \\int _ D S ( t - s , x - y ) | f ( u ( s , y ) ) | ^ { 2 } d s d y < + \\infty , \\\\ & \\mathbb E _ { \\delta } \\int _ 0 ^ t \\int _ D S ( t - s , x - y ) | u ( s , y ) | ^ { q } d s d y < + \\infty , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} \\mathcal { C } ^ 2 [ H , L ] ( v _ { 1 } , \\cdots , v _ { n } ) = \\mathcal { C } [ \\mathcal { C } [ H , L ] , L ] ( v _ { 1 } , \\cdots , v _ { n } ) . \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} L _ { n , 1 } ( q - s ) = L _ { n , 1 } ( q ) + s = ( - \\alpha + \\delta ) q + s , \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} \\hat { V } _ { \\rm h e m } & = \\frac { \\frac 1 2 v _ { n + 2 } R ^ { n + 2 } } { \\frac 1 2 v _ { n + 2 } L ^ { n + 2 } } = \\left ( \\frac { R } { L } \\right ) ^ { n + 2 } , \\ ; \\ ; \\ ; ( 0 \\leq R \\leq L ) , \\\\ \\hat { A } _ { \\rm h e m } & = 1 , \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} - 1 & = u ^ 2 + v w \\\\ & = u ^ 2 + v \\tfrac { ( d - a ) u - b v } { c } \\\\ & = u ^ 2 + \\tfrac { d - a } { c } u v - \\tfrac { b } { c } v ^ 2 \\\\ & = \\Big ( u + \\tfrac { d - a } { 2 c } v \\Big ) ^ 2 - \\Big ( \\tfrac { ( d - a ) ^ 2 } { 4 c ^ 2 } + \\tfrac { b } { c } \\Big ) v ^ 2 . \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 p } { \\partial t ^ 2 } + 2 \\lambda ( t ) \\frac { \\partial p } { \\partial t } = c ^ 2 \\frac { \\partial ^ 2 p } { \\partial x ^ 2 } \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} \\tilde { \\mathbb B } _ 0 ( \\phi , \\phi ) = \\int _ 0 ^ \\infty ( { U ^ + } ^ 2 { \\xi _ 1 ' } ^ 2 + { U ^ - } ^ 2 { \\xi _ 2 ' } ^ 2 ) r \\ , { \\mathrm d } r . \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} & [ Y ] = - \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { d } \\big ( Y \\times _ j \\textbf { D } \\big ) + Y \\bigtimes _ { i = 1 } ^ d \\textbf { V } _ { c o s } \\in \\R ^ { N \\times \\dots \\times N } , \\\\ & \\textbf { D } = \\texttt { t r i d i a g } ( - 1 , 2 , - 1 ) \\in \\R ^ { N \\times N } , \\\\ & \\textbf { V } _ : = \\{ 1 - \\cos ( \\frac { 2 \\pi j } { N } ) \\} , j = - N / 2 , \\dots , N / 2 - 1 \\ . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ \\infty \\left [ { a \\atop b + \\alpha n } \\right ] _ { p } \\frac { 1 } { ( - z q ^ n , - q ^ { 1 - n } / z ; q ) _ \\infty } = \\int _ { - \\infty } ^ \\infty \\left [ { a \\atop b + \\alpha x } \\right ] _ { p } \\frac { 1 } { \\left ( - z q ^ x , - q ^ { 1 - x } / z ; q \\right ) _ \\infty } \\ , d x , \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} \\limsup _ n | C _ n | = 0 \\ , . \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N } \\frac { 1 } { \\log K _ { j } \\log \\log K _ { j } } \\le C \\left ( 1 + \\frac { 1 } { n _ { 1 } } \\sum _ { j = 1 } ^ { N } \\log K _ { j } \\right ) . \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} \\Theta ^ { } _ 1 ( \\Lambda ) ( 0 ) & = g _ c ( 0 ) . \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} & | \\frac { - c _ { b } r } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\sin ^ { 2 } ( \\theta ) d \\theta \\int _ { 0 } ^ { \\infty } d \\xi \\psi ( \\xi ) \\left ( \\frac { \\sin ( \\xi t _ { + } ) } { t _ { + } ^ { 2 } } + \\frac { \\sin ( \\xi t _ { - } ) } { t _ { - } ^ { 2 } } \\right ) | \\leq C r \\int _ { 0 } ^ { \\pi } d \\theta \\int _ { 0 } ^ { \\infty } \\frac { | \\psi ' ( \\xi ) | d \\xi } { t ^ { 3 } } \\\\ & \\leq \\frac { C r } { t ^ { 3 } } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} \\begin{cases} u ^ 3 ( x - 1 ) ^ { n - r _ 1 + k _ 5 } p _ 5 ( x ) - u ^ 3 ( x - 1 ) ^ { n - r _ 1 - r _ 2 + k _ 4 + k _ 6 } p _ 4 ( x ) p _ 6 ( x ) & \\mbox { i f ~ } n - r _ 1 + k _ 4 > r _ 2 , \\\\ u ^ 3 ( x - 1 ) ^ { r _ 2 - k _ 4 + k _ 5 } p _ 5 ( x ) - u ^ 3 ( x - 1 ) ^ { k _ 6 } p _ 4 ( x ) p _ 6 ( x ) & \\mbox { i f ~ } n - r _ 1 + k _ 4 \\le r _ 2 , \\\\ \\end{cases} \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} \\sqrt { R _ 0 } \\ge 0 \\Omega , ( \\sqrt { R } U ) _ 0 = 0 \\{ \\sqrt { R _ 0 } = 0 \\} . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} f _ { n } ( t ) : = \\begin{cases} n ^ { - \\epsilon } t ^ { p + \\epsilon } , & 0 \\leq t \\leq n , \\\\ \\frac { p + \\epsilon } { 2 } n ^ { p - 2 } t ^ { 2 } + ( 1 - \\frac { p + \\epsilon } { 2 } ) n ^ { p } , & t \\geq n \\end{cases} \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} - ( y ^ { 1 - 2 \\alpha } \\psi _ k ^ { \\prime } ) ^ { \\prime } = \\mu _ k y ^ { 1 - 2 \\alpha } \\psi _ k , \\ \\ 0 < y < \\infty , \\ \\psi _ k ( 0 ) = 1 , \\ \\lim _ { y \\to \\infty } \\psi _ k ( y ) = 0 . \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} B _ { \\epsilon } = \\{ ( \\beta _ 0 , \\beta _ 1 , \\eta ) : \\beta _ 0 ^ 2 + \\beta _ 1 ^ 2 + ( \\eta ^ 2 - 1 ) ^ 2 \\leq \\epsilon ^ 2 \\} \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} - \\Delta u = u ( 1 - | u | ^ 2 ) , \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} \\Psi ( z ) = c _ { 0 } + c _ { 3 } ( z - \\alpha ) ^ { 3 } + \\sum _ { n = 4 } ^ { \\infty } c _ { n } ( z - \\alpha ) ^ { n } . \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} \\phi = c _ 1 \\frac { \\partial w } { \\partial x _ 1 } + c _ 2 \\frac { \\partial w } { \\partial x _ 2 } . \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} \\{ f , g \\} = \\frac { d } { d g } \\left ( \\frac { d ^ 2 f / d g ^ 2 } { d f / d g } \\right ) - \\frac { 1 } { 2 } \\left ( \\frac { d ^ 2 f / d g ^ 2 } { d f / d g } \\right ) ^ 2 . \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} 0 < - h ^ * K _ X \\cdot C = ( C + ( m + 2 - b ) F _ W ) \\cdot C = - m + ( m + 2 - b ) = 2 - b . \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} \\tilde { \\varphi } & = \\tilde { \\varphi } ( f _ 0 ) = p ' \\circ ( K ^ x + f _ 0 ^ x ) + ( K ^ y + f _ 0 ^ y ) \\ , q ' \\circ ( K ^ x + f _ 0 ^ x ) + ( D _ 1 u + D _ 1 g ) \\circ ( K + f _ 0 ) , \\\\ \\tilde { \\psi } & = \\tilde { \\psi } ( f _ 0 ) = q \\circ ( K ^ x + f _ 0 ^ x ) + ( D _ 2 u + D _ 2 g ) \\circ ( K + f _ 0 ) , \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} A \\diamond B _ { i , j } = a _ { i , j } b _ { i , j } . \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} H ^ 0 ( X ^ { ( 1 ) } , B ) = k x , \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} \\nabla \\gamma _ n = - | f _ n | ^ 2 + | f _ { n - 1 } | ^ 2 . \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} f ( d , M ) : = \\dfrac { 2 d M 2 ^ { \\omega ( d M ) } } { \\phi ( d M ) } \\left ( \\dfrac { 1 } { 3 \\log 3 } \\sqrt { d } \\log d + \\dfrac { 1 3 } { 2 } \\sqrt { d } + 1 \\right ) . \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} B _ { H , \\ ; L + m + 1 } - B _ { H , \\ ; L + m } = 2 H _ { L + m } - H _ { L + m + 1 } , \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} \\partial ^ \\textup { M } \\psi ( \\bar x ) : = \\left \\{ y \\in \\R ^ n \\ , \\middle | \\ , \\begin{aligned} & \\exists \\{ x _ k \\} _ { k \\in \\N } \\subset \\R ^ n \\ , \\exists \\{ y _ k \\} _ { k \\in \\N } \\subset \\R ^ n \\colon \\\\ & x _ k \\to \\bar x , \\ , y _ k \\to y , \\ , y _ k \\in \\partial ^ \\textup { F } \\psi ( x _ k ) \\ , \\forall k \\in \\N \\end{aligned} \\right \\} . \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} R ( t _ s ) \\leq \\hat { \\lambda } ^ { s - \\hat { s } } R ( t _ { \\hat { s } } ) + \\sum _ { \\tau = \\hat { s } } ^ { s - 1 } c ( \\tau ) t _ { \\tau + 1 } ^ { \\beta } \\hat { \\lambda } ^ { s - \\tau - 1 } . \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} F _ g = \\frac { D _ g } { \\sqrt { 1 + D _ g ^ 2 } } , \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} \\| b \\| _ { \\mathcal { B M O _ B } } : = \\sup _ { B \\in \\mathcal { B } } \\| b - b _ B \\| _ { { \\rm e x p } \\ , L , B } = \\sup _ { B \\in \\mathcal { B } } \\inf \\left \\{ \\lambda > 0 : \\ , \\frac { 1 } { \\mu ( B ) } \\int _ B e ^ { \\frac { | b ( x ) - b _ B | } { \\lambda } } \\ , d \\mu \\leq 2 \\right \\} . \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} G ^ { - 1 } ( Z _ { t - } + \\tilde \\rho ( Z _ { t - } , y ) ) - G ^ { - 1 } ( Z _ { t - } ) = \\rho ( X _ { t - } , y ) . \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} p = \\frac { \\sin \\phi _ 3 } { 1 + \\cos \\phi _ 3 } = \\frac { 4 A } { ( u _ 1 + u _ 2 ) ^ 2 - u _ 3 ^ 2 } \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\left ( \\langle x , x \\rangle ^ \\frac { - 1 } { 2 } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } - \\frac { 1 } { \\sqrt { n } } \\right ) \\left ( \\langle x , x \\rangle ^ \\frac { - 1 } { 2 } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } - \\frac { 1 } { \\sqrt { n } } \\right ) ^ * = c _ x . \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} \\lim _ { \\tau \\rightarrow + \\infty } \\left \\Vert \\exp \\left [ \\tau A \\right ] \\mathsf { p } ^ { \\ast } - \\left ( \\mathsf { p } ^ { \\ast } \\mathsf { , \\hat { q } } _ { \\mathsf { 1 } } \\right ) _ { 2 } \\mathsf { \\hat { p } } _ { \\mathsf { 1 } } \\right \\Vert _ { 2 } = 0 \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} \\begin{cases} D _ { 0 } ^ { 1 - \\epsilon } y ( t ) = z ( t ) & y ( 0 ) = s , \\\\ D _ { 0 } ^ { \\alpha } z ( t ) = f ( t ) + c ( t ) y ( t ) + b ( t ) z ( t ) & z ( 0 ) = ( \\gamma _ 1 - a _ 1 s ) / b _ 1 . \\end{cases} \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} | | L ( v _ { 6 } ( t , \\cdot \\lambda ( t ) ) ) | | _ { L ^ { 2 } ( R d R ) } = \\lambda ( t ) | | \\sqrt { \\omega } \\lambda ( t ) y _ { 0 } ( t ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} & \\sum _ { \\substack { r _ 1 , r _ 2 , r _ 3 \\geq 0 } } y _ 1 ^ { r _ 1 } y _ 2 ^ { 2 r _ 3 + r _ 2 } \\frac { q ^ { r _ 1 ^ 2 + ( r _ 2 + r _ 3 ) ^ 2 + r _ 3 ^ 2 - r _ 1 ( r _ 2 + 2 r _ 3 ) } } { ( q ) _ { r _ 1 } ( q ) _ { r _ 2 } ( q ) _ { r _ 3 } } = \\sum _ { \\substack { n _ 1 , n _ 2 , n _ 3 , n _ 4 , n _ { 5 } \\geq 0 } } \\\\ & y _ 1 ^ { n _ 1 + n _ 2 + n _ 4 } y _ 2 ^ { 2 n _ 1 + 2 n _ 3 + n _ 4 + n _ 5 } \\frac { q ^ { n _ 1 ^ 2 + n _ 2 ^ 2 + ( n _ 3 + n _ 5 ) ^ 2 + n _ 4 ^ 2 + n _ 3 ^ 2 + ( 2 n _ 3 + n _ 5 ) ( n _ 1 ) + n _ 4 ( n _ 1 + n _ 2 ) + n _ 3 n _ 4 } } { ( q ) _ { n _ 1 } ( q ) _ { n _ 2 } ( q ) _ { n _ 3 } ( q ) _ { n _ 4 } ( q ) _ { n _ 5 } } \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 , \\ N ( t ) = 2 k \\} = \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} c k \\cdot f ( | y _ n | ) & \\leq c f ( | x _ n | ) f ( | y _ n | ) \\leq | f ( B ( x _ n , y _ n ) ) | = | f ( ( T x _ n ) ( y _ n ) ) | \\\\ & = | ( y _ n + N _ f , ( T x _ n ) _ f ) _ f | \\qquad ( \\mbox { b y ~ } \\eqref { P a s 1 } ) \\ , . \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} \\langle \\tilde { L } G _ D f , \\varphi \\rangle & = \\langle G _ D f , L \\varphi \\rangle = \\lim _ { n \\to \\infty } \\langle G _ D f _ n , L \\varphi \\rangle \\\\ & = \\lim _ { n \\to \\infty } - \\langle f _ n , \\varphi \\rangle = - \\langle f , \\varphi \\rangle . \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} d X ( t ) = b ( X ( t ) ) d t + \\sigma ( X ( t ) ) d B ( t ) - d \\eta ( t ) \\ , ; \\ , \\ , \\ , \\ , \\ , X ( 0 ) = x \\in { \\mathcal O } \\ , , \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} \\partial _ { t } v _ { 4 } ( t , r ) = r \\int _ { 0 } ^ { 1 } d \\beta \\partial _ { t r } v _ { 4 } ( t , r \\beta ) \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} E _ { k } ( m ) = \\left \\{ \\xi _ { t } ^ { m } ( x ) = 0 , ( x , t ) \\in R _ { k } ( m ) \\right \\} . \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} \\int ^ T _ 0 \\mathcal { E } ( t ) d t = \\frac { 3 } { 2 \\eta } \\left ( \\mathcal { E } ( 0 ) - \\mathcal { E } ( T ) \\right ) + ( h ( 0 ) - h ( T ) ) \\leq \\left ( \\frac { 3 } { 2 \\eta } \\mathcal { E } ( 0 ) + h ( 0 ) \\right ) - h ( T ) . \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} & g ( t _ 0 ) < - K = - \\underline h ( 0 ) \\ { \\rm a n d } \\ \\underline h ( 0 ) = K < h ( t _ 0 ) , \\\\ & U ( t _ 0 , x ) \\succeq 1 - \\epsilon \\succeq \\underline U ( 0 , x ) \\ \\mbox { f o r } \\ \\ x \\in [ - \\underline h ( 0 ) , \\underline h ( 0 ) ] , \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} ( B _ 1 ( u , \\theta ) , A _ 1 ^ 2 \\theta ) & = \\sum _ { i , j , k = 1 } ^ { 2 } \\int _ { \\Omega } u _ i \\partial _ i \\theta \\partial _ j ^ 2 \\partial _ k ^ 2 \\theta \\dd x \\\\ & = - \\sum _ { i , j , k = 1 } ^ { 2 } \\int _ { \\Omega } u _ i \\partial _ { i j } \\theta \\partial _ j \\partial _ k ^ 2 \\theta \\dd x - \\sum _ { i , j , k = 1 } ^ { 2 } \\int _ { \\Omega } \\partial _ j u _ i \\partial _ i \\theta \\partial _ j \\partial _ k ^ 2 \\theta \\dd x = I + I I \\ ; . \\\\ \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} & \\phi ( - q ^ { 1 / 2 } x _ 3 ) \\phi ( - q ^ { 1 / 2 } x _ 1 ) \\phi ( - q ^ { 1 / 2 } x _ 2 ) \\\\ & = \\sum \\frac { q ^ { \\frac { \\ell ^ 2 } { 2 } } ( x _ { 3 } ) ^ { \\ell } } { ( q ) _ { \\ell } } \\frac { q ^ { \\frac { m ^ 2 } { 2 } } ( x _ { 1 } ) ^ { m } } { ( q ) _ { m } } \\frac { q ^ { \\frac { n ^ 2 } { 2 } } ( x _ { 2 } ) ^ { n } } { ( q ) _ { n } } \\\\ & = \\sum \\frac { q ^ { \\frac { n ^ 2 } { 2 } + \\frac { \\ell ^ 2 } { 2 } + \\frac { m ^ 2 } { 2 } - m n - n \\ell } ( x _ { 2 } ) ^ { n } ( x _ { 3 } ) ^ { \\ell } ( x _ { 1 } ) ^ { m } } { ( q ) _ { n } ( q ) _ { \\ell } ( q ) _ { m } } . \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} F _ t = \\begin{cases} , & t \\leq 0 , \\\\ , & t \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} \\alpha ^ { ( k ) } ( X , \\Delta ; L ) = \\inf _ { m \\ge 1 } \\alpha _ m ^ { ( k ) } ( X , \\Delta ; L ) \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\begin{cases} V _ { 0 } \\in ( ( 1 / 1 2 ) \\mathbb { Z } / \\mathbb { Z } ) ^ { 4 } , & \\gcd ( V _ { 1 } ) = 1 , \\\\ V _ { 0 } \\in ( ( 1 / 6 ) \\mathbb { Z } / \\mathbb { Z } ) ^ { 4 } , & \\gcd ( V _ { 1 } ) = 2 . \\end{cases} \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} a ( n ) - 2 \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { n } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - m ^ 2 ) = a ( n ) - 2 \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { \\frac { n } { 2 } } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - 2 m ^ 2 ) , \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} \\left ( F . _ { \\mathcal { P } } \\xi \\right ) ( k , [ v ] , r , s ) = & \\sum _ { k _ { 1 } , k _ { 2 } } F \\left ( k - k _ { 2 } + k _ { 1 } , [ v + k _ { 1 } \\theta ] , r - k _ { 1 } , s + k _ { 2 } \\right ) \\\\ & \\hphantom { \\sum _ { k _ { 1 } , k _ { 2 } } } \\cdot \\xi \\left ( k _ { 1 } , [ v + k _ { 1 } \\theta ] , k _ { 2 } , [ v + b ( r + s ) + ( k _ { 2 } - k ) \\theta ] \\right ) . \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} & \\left . \\psi ^ { l _ 1 \\cdots l _ n } _ { m _ 1 \\cdots ( m _ k = 3 ) \\cdots m _ n } \\right | _ { O ^ { l _ 1 \\cdots l _ n } _ { m _ 1 \\cdots ( m _ k = 3 ) \\cdots m _ n } \\cap O ^ { l _ 1 \\cdots l _ n } _ { m _ 1 \\cdots ( m _ k = 1 ) \\cdots m _ n } } \\\\ & = U _ k \\left . \\psi ^ { l _ 1 \\cdots l _ n } _ { m _ 1 \\cdots ( m _ k = 1 ) \\cdots m _ n } \\right | _ { O ^ { l _ 1 \\cdots l _ n } _ { m _ 1 \\cdots ( m _ k = 3 ) \\cdots m _ n } \\cap O ^ { l _ 1 \\cdots l _ n } _ { m _ 1 \\cdots ( m _ k = 1 ) \\cdots m _ n } } , \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} \\Delta _ { p _ 1 } ( f , < p _ 1 > _ f ) = \\frac { \\mu } { \\sqrt { a } } \\left [ 1 + \\left ( \\frac { \\eta } { 4 \\mu ^ 2 } \\right ) ^ 2 a b \\right ] ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} \\bigl ( a . _ { \\mathcal { H } _ { - b } } \\Psi \\bigr ) ( n , [ x ] , r ) & = \\sum _ { m \\in \\mathbb { Z } } a ( [ x - r b ] , m ) \\Psi ( n - m , [ x - m \\theta ] , r - m ) . \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} A _ i [ U ] = \\left \\langle \\left \\lbrace a ^ { ( i ) } \\mid u ^ { ( i ) } \\textit { a n d } a ^ { ( i ) } \\textit { a r e } U \\textit { - i n c i d e n t m o n o m i a l s } \\right \\rbrace \\right \\rangle . \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} & w ' = \\left ( { w ' } _ { 1 } , \\dots , { w ' } _ { N } , { w ' } _ { N + 1 } - w _ { 1 } , { w ' } _ { N + 2 } , \\dots , { w ' } _ { N ' } \\right ) \\\\ & { z ' } = \\left ( { z ' } _ { 1 } , \\dots , { z ' } _ { N } , { z ' } _ { N + 1 } , { z ' } _ { N + 2 } , \\dots , { z ' } _ { N ' } \\right ) , \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} B ( e x f , g y h ) & = - B ( g y h , e x f ) = - e B ( g y h , e x f ) h = e B ( e x f , g y h ) h . \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} \\nabla ^ \\pm e _ a : = ( \\Omega ^ \\pm ) ^ b { } _ { a } \\otimes e _ b . \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} x ^ 4 + 4 \\tilde { c } _ 1 x ^ 3 - 6 x ^ 2 y - 1 2 \\tilde { c } _ 1 x y + 9 y ^ 2 - 2 \\tilde { c } _ 2 x + 1 2 \\tilde { c } _ 1 ^ 2 y + \\tilde { c } _ 1 \\tilde { c } _ 2 = 0 \\ , . \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} C _ 0 = \\frac { \\epsilon C _ 1 S _ 1 + \\epsilon C _ 2 S _ 2 - D _ 3 S _ 3 - D _ 4 S _ 4 } { A } . \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\mu = 4 - \\mu \\mbox { h e n c e } \\mu = 8 . \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} F ( \\Phi ^ c ( x - \\delta ) ) - F ( \\Phi ^ c ( x ) ) = V ( x ) E ( x ) \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} Q A = d _ A c ; \\ \\ Q c = \\frac 1 2 [ c , c ] ; \\ \\ Q A ^ \\dag = d _ A \\star F _ A + [ c , A ^ \\dag ] ; \\ \\ Q c ^ \\dag = d _ A A ^ \\dag + [ c , c ^ \\dag ] \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} \\gamma ( \\mu ) = \\frac { \\partial R } { \\partial y } ( 0 ; \\mu ) . \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} \\nabla _ A U = \\nabla U + i A U \\Omega . \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} \\| \\lambda ^ { \\ast ^ { G / H } } \\| _ { M ( G / H ) } & = \\| T _ H ( \\lambda _ q ^ { * ^ G } ) \\| _ { M ( G / H ) } \\\\ & \\le \\| \\lambda _ q ^ { \\ast ^ G } \\| _ { M ( G ) } \\\\ & = \\| \\lambda _ q \\| _ { M ( G ) } = \\| \\lambda \\| _ { M ( G / H ) } . \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} t ^ * = \\bigcup _ { \\xi < \\nu } { t _ { \\xi } } \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\varrho _ p ( A _ 1 , \\ldots , A _ N ) & \\leq \\varrho _ { \\lfloor p \\rfloor } ( A _ 1 , \\ldots , A _ N ) ^ { p - \\lfloor p \\rfloor } \\varrho _ { 1 + \\lfloor p \\rfloor } ( A _ 1 , \\ldots , A _ N ) ^ { 1 + \\lfloor p \\rfloor - p } \\\\ & \\leq \\rho \\left ( \\sum _ { i = 1 } ^ N A _ i ^ { \\otimes \\lfloor p \\rfloor } \\right ) ^ { p - \\lfloor p \\rfloor } \\rho \\left ( \\sum _ { i = 1 } ^ N A _ i ^ { \\otimes ( 1 + \\lfloor p \\rfloor ) } \\right ) ^ { 1 + \\lfloor p \\rfloor - p } \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\ddot { \\Gamma } ( t ) + \\left ( \\eta \\operatorname { I d } + \\nu ( t ) \\otimes \\frac { \\nabla \\dot { u } ( \\Gamma ( t ) , t ) } { \\abs { \\nabla u ( \\Gamma ( t ) , t ) } } \\right ) \\dot { \\Gamma } ( t ) = - \\operatorname { d i v } \\left ( \\nu ( t ) \\right ) \\nu ( t ) \\ ; ; \\\\ \\dot { \\Gamma } ( 0 ) = 0 , \\ ; \\Gamma ( 0 ) = \\Gamma _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} \\widetilde { C } = \\begin{cases} C & \\quad \\frac { 2 N } { N + 2 - s } = p , \\\\ C | \\Omega | ^ { \\frac { N + 2 - s } { N } - \\frac { 2 } { p } } & \\quad \\frac { 2 N } { N + 2 - s } < p . \\end{cases} \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} \\omega _ { \\dot { \\gamma } } = \\sum _ { i = 1 } ^ 3 \\dot { \\gamma } _ i \\left ( d x _ i \\wedge \\eta + \\phi d x _ j \\wedge d x _ k \\right ) , \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} v _ i ^ { \\prime } \\cdot v _ i ^ { - 1 } \\cdot T _ 1 & = v _ i ^ { \\prime } \\cdot v _ i ^ { - 1 } \\cdot ( 1 - v - v w ) = v _ i ^ { \\prime } \\cdot v _ i ^ { - 1 } \\cdot ( 1 - v _ i v _ m v _ f - v _ i v _ m v _ f w ) = \\\\ & = v _ i ^ { \\prime } \\cdot v _ i ^ { - 1 } - v _ i ^ { \\prime } v _ m v _ f - v _ i ^ { \\prime } v _ m v _ f w . \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} \\frac { d } { d t } \\lambda ( t ) = - \\frac { \\delta _ \\lambda } { \\langle t \\rangle ^ { 2 \\tilde { q } } } ( 1 + \\lambda ( t ) ) , t > 1 , \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} \\partial _ t \\Delta _ q \\rho + \\Delta _ q ( v \\cdot \\nabla \\rho ) = \\Delta f . \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} \\delta _ 0 & \\ge \\eta \\left [ 1 + ( \\ell + \\epsilon ) M \\right ] ^ { - 1 } \\inf _ { t \\in [ 0 , T ] } \\big [ 1 - ( \\ell + \\epsilon ) \\omega ( \\cdot , \\lambda _ 1 ) * p ( t ) \\big ] \\\\ & \\ge \\eta \\left [ 1 + ( \\ell + \\epsilon ) M \\right ] ^ { - 1 } \\big [ 1 - ( \\ell + \\epsilon ) \\sup _ { t \\in [ 0 , T ] } \\omega ( \\cdot , \\lambda _ 1 ) * p ( t ) \\big ] \\\\ & = \\eta \\left [ 1 + ( \\ell + \\epsilon ) M \\right ] ^ { - 1 } \\big [ 1 - ( \\ell + \\epsilon ) M \\big ] > 0 . \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} \\Delta _ A ( f _ s , 0 ) \\Delta _ B ( f _ s , 0 ) = | s | ^ { n - m } C _ 1 \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} ( \\alpha - 1 ) p _ \\theta ( { \\bf { x } } ) ^ { \\alpha - 2 } \\partial _ r [ p _ \\theta ( { \\bf { x } } ) ] & = \\partial _ r [ F ( \\theta ) ] + \\partial _ r [ w ( \\theta ) ] ^ \\top f ( { \\bf { x } } ) . \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} D _ p = \\{ i : m _ i \\in C _ p \\} \\ p \\in [ k ] . \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} \\rho ( L _ i ) = \\frac { - h ( z ) ^ { i + 1 } } { h ' ( z ) } \\partial + ( b ( z ) + i \\cdot c ) \\cdot h ( z ) ^ i \\ , . \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} \\forall ( a , b ) \\in \\R ^ 2 \\colon \\varphi _ \\textup { K S } ( a , b ) : = \\begin{cases} a b & a + b \\geq 0 , \\\\ - \\tfrac 1 2 ( a ^ 2 + b ^ 2 ) & a + b < 0 . \\end{cases} \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} [ X ^ { ( - i ) } , Y ^ { ( - j ) } ] = [ X , Y ] ^ { ( - i - j ) } , \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} t & = { 1 \\over 4 } ( q - i q i - j q j - k q k ) , \\cr x & = { 1 \\over 4 i } ( q - i q i + j q j + k q k ) , \\cr y & = { 1 \\over 4 j } ( q + i q i - j q j + k q k ) , \\cr z & = { 1 \\over 4 k } ( q + i q i + j q j - k q k ) . \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} \\int _ { D _ 0 } ( \\nabla + i { \\boldsymbol k } ) u \\cdot \\overline { ( \\nabla + i { \\boldsymbol k } ) v } d x = \\left ( \\frac { \\omega } { c } \\right ) ^ 2 \\int _ { D _ 0 } \\epsilon ( x , \\omega ) u \\overline { v } d x \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} ( \\alpha , A ) + ( \\beta , B ) & = ( \\alpha + \\beta , A + B ) , \\\\ \\lambda ( \\alpha , A ) & = ( \\lambda \\alpha , A ) , \\\\ ( \\alpha , A ) ( \\beta , B ) & = ( \\alpha \\beta , \\alpha B + \\beta A + A B ) . \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } \\partial _ { j } a _ { j k } \\partial _ { k } \\bigg ] u _ { 2 } & = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ u _ { 2 } & = u - \\zeta u \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} . \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} \\widehat { \\mathbb { L } } _ n : = \\Big \\{ p \\in \\mathcal { P } : \\sum \\limits _ { x \\in \\mathbb { S } } p ( x ) f _ i ( x ) = \\bar { f _ i } , i \\in \\{ 1 , \\ldots , k \\} \\Big \\} , \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} \\sup _ { x \\ge 0 } \\int _ \\R \\frac { | P ( x , k ) | ^ 2 } { k ^ 2 + 1 } d \\sigma = \\sup _ { x \\ge 0 } \\int _ \\R \\frac { ( k ^ 2 + \\rho ^ 2 ) | P ( x , k ) | ^ 2 } { ( k ^ 2 + \\rho ^ 2 ) ( k ^ 2 + 1 ) } d \\sigma \\lesssim \\rho \\sup _ { x \\ge 0 } \\int _ \\R \\frac { \\rho | P ( x , k ) | ^ 2 } { k ^ 2 + \\rho ^ 2 } d \\sigma \\ , . \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} A = \\frac { ( n - 1 ) ( k - 1 ) ^ 2 } { n k - k - 2 n + 3 - \\widehat { \\lambda } _ { n } ( \\underline { \\xi } _ { n } ) } , B = \\frac { ( n - 1 ) \\lambda _ { n } ( \\underline { \\xi } _ { n } ) + \\widehat { \\lambda } _ { n } ( \\underline { \\xi } _ { n } ) + n - 2 } { 1 - \\widehat { \\lambda } _ { n } ( \\underline { \\xi } _ { n } ) } . \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} K ( x - t , z _ { 1 } ) - K ( x - t , z _ { 2 } ) = ( z _ { 1 } - z _ { 2 } ) \\int _ { 0 } ^ { 1 } d q \\partial _ { 2 } K ( x - t , z _ { 2 } + q ( z _ { 1 } - z _ { 2 } ) ) \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { Z } _ r + \\frac { \\alpha + \\frac { 1 } { 2 } } { 2 \\alpha r } \\xi \\mathcal { Z } _ \\xi = & \\mathcal { F } _ \\xi [ \\mathcal { Z } , \\mathcal { U } _ \\xi ] \\\\ \\mathcal { U } _ r + \\frac { \\alpha + \\frac { 1 } { 2 } } { 2 \\alpha r } ( \\xi \\mathcal { U } _ \\xi - 1 ) = & \\mathcal { E } _ \\xi [ \\mathcal { Z } , \\mathcal { U } ] \\end{aligned} \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} b _ { n - 1 } & = q _ { n - 1 } a _ { n - 1 , n - 1 } + a _ { n , n - 1 } \\\\ b _ { n - 2 } - q _ { n - 1 } a _ { n - 1 , n - 2 } & = q _ { n - 2 } a _ { n - 2 , n - 2 } + a _ { n , n - 2 } \\\\ & \\vdots \\\\ b _ 1 - q _ { n - 1 } a _ { n - 1 , 1 } - \\cdots - q _ 2 a _ { 2 , 1 } & = q _ 1 a _ { 1 , 1 } + a _ { n , 1 } \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} \\Theta \\nabla F ( \\mathbf { 0 } ) ^ T = \\lambda _ 1 \\Theta , \\ \\ \\tilde \\Theta \\nabla F ( \\mathbf { 0 } ) = \\lambda _ 1 \\tilde \\Theta , \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} A = \\left \\{ \\begin{array} { c } \\eta \\\\ H ^ { * } s _ { k } \\\\ H T \\end{array} \\right \\} \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} \\mathfrak m = \\mathfrak m _ 1 ' + \\ldots + \\mathfrak m _ s ' \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} G _ r ( z ) & = R ^ { - 1 } G ( z ) , \\\\ & = \\begin{bmatrix} k \\nabla f ( x ) - A ^ T A x + k A ^ T \\lambda + A ^ T b \\\\ A \\nabla f ( x ) - k A x + A A ^ T \\lambda + k b \\end{bmatrix} \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} L ( { \\mathcal Q } _ A , { \\mathcal Q } _ B ) = { \\mathcal Q } _ { L ( A , B ) } , \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ s _ p u ( x ) + c ( x ) u ( x ) \\geq ( - \\Delta ) ^ s _ p v ( x ) + c ( x ) v ( x ) , & x \\in \\Omega , \\\\ u ( x ) \\geq v ( x ) , & x \\in \\mathbb { R } ^ n \\setminus \\Omega . \\end{cases} \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} b ^ 2 & = \\frac { x _ 0 ^ 2 \\Delta _ { [ x _ 0 : x _ 1 ] } ^ x } { x _ 1 ^ 2 ( \\sum _ { i = 1 } ^ 2 x _ 0 ^ i x _ 1 ^ { 2 - i } t d _ { i - 1 , 1 } ) ^ 2 } \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} H ^ { * } = \\Big [ - 3 \\rho T , n + 3 \\rho T \\Big ] ^ { d } . \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} r e s _ k ( v ) = \\langle f _ { k , 1 } ( \\cdot \\ , v ) , 1 \\rangle ^ 2 + \\langle f _ { k , 2 } ( \\cdot \\ , v ) , 1 \\rangle ^ 2 \\ . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} \\varphi ( r ) = \\begin{cases} \\exp \\big ( B \\ , r ^ { 1 + \\frac { \\gamma } { 2 } } \\big ) & \\quad \\hbox { i f } \\gamma > - 2 \\\\ r ^ \\delta & \\quad \\hbox { i f } \\gamma = - 2 \\\\ r & \\quad \\hbox { i f } \\gamma < - 2 \\end{cases} \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} \\begin{cases} d _ { n } ^ { * } = b ^ { v } _ { n } + 4 ( v ^ { - 1 } - 1 ) \\lambda ^ { 2 v } b _ { n } ^ { - v } \\\\ c _ { n } ^ { * } = f ( b _ { n } ^ { v } ) = 2 \\lambda ^ { v } + 4 ( v ^ { - 1 } - 1 ) \\lambda ^ { 2 v } b _ { n } ^ { - v } , \\end{cases} \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{cases} \\begin{bmatrix} x + y \\\\ 1 \\end{bmatrix} , & x < 0 , \\\\ \\begin{bmatrix} y \\\\ - 1 + \\eta _ 1 x + \\eta _ 2 y \\end{bmatrix} , & x > 0 , \\end{cases} \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} & \\int _ { \\left [ 0 , 1 \\right ] ^ d } \\left ( p ( x ) - \\hat { p } \\left ( x \\right ) \\right ) ^ 2 d x \\\\ & \\quad = \\int _ { \\left [ 0 , 1 \\right ] ^ d } \\hat { p } \\left ( x \\right ) ^ 2 d x - 2 \\int _ { \\left [ 0 , 1 \\right ] ^ d } p ( y ) \\hat { p } ( y ) d y + \\int _ { \\left [ 0 , 1 \\right ] ^ d } p ( z ) ^ 2 d z . \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ i = m . \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } a _ i a _ i ^ * \\leq \\sqrt { n } \\left ( \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ 2 \\right ) ^ \\frac { 1 } { 2 } , \\forall n \\in \\mathbb { N } , \\forall a _ 1 , \\dots , a _ n \\in \\mathcal { A } . \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} \\mathcal { A } A \\mathcal { B } = \\left ( \\begin{array} { c c c c c c c } \\tilde { a _ 1 } & & & & & \\\\ & \\ddots & & & & \\\\ & & \\tilde { a _ t } & & & \\\\ & & & 0 & & \\\\ & & & & \\ddots & \\\\ & & & & & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} \\langle x , y \\rangle _ Y : = \\frac { 1 } { M } \\sum _ { l = 1 } ^ M x _ l y _ l . \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} & \\int _ { D \\cap V ^ c } M _ { D _ n } ( x , y ) \\eta _ n ( d y ) \\le \\int _ { D _ k } M _ { D _ n } ( x , y ) \\eta _ n ( d y ) \\\\ & \\qquad \\qquad \\qquad = \\int \\limits _ { D _ k } G _ { D _ n } ( x , v ) \\int \\limits _ { D \\setminus D _ n } j ( | v - y | ) f ( y ) d y d v \\\\ & \\qquad \\qquad \\qquad \\le c _ k \\left ( \\int \\limits _ { D _ k } G _ D ( x , v ) d v \\right ) \\left ( \\int \\limits _ { D \\setminus D _ n } f ( y ) ( 1 \\wedge j ( | y | ) ) d y \\right ) \\overset { n \\to \\infty } { \\longrightarrow } 0 , \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} y B & = \\frac { 1 } { 2 } ( B ^ { 2 } - C B ) \\\\ & = \\frac { 1 } { 2 } ( B ^ { 2 } - B C - 2 h ) \\\\ & \\equiv \\frac { 1 } { 2 } ( \\mu - { h } ^ { 2 } - 2 { h } ) . \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} D _ q ( M _ 1 , v ) = \\left ( 1 - \\frac { 1 } { q } \\right ) | z | ^ q , D _ q ( M _ 2 , v ) = \\left ( 1 - \\frac { 1 } { q } \\right ) | y | ^ q \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} \\overline { u } _ \\beta ( x , t ) & = ( \\limsup _ { \\alpha \\to \\beta , \\alpha \\neq \\beta } { } ^ { * } u _ \\alpha ) ( x , t ) \\\\ & = \\lim _ { \\delta \\to 0 ^ + } \\sup \\{ u _ \\alpha ( y , s ) \\mid ( y , s ) \\in Q _ { T , 0 } \\cap \\overline { B _ \\delta ( x , t ) } , \\ 0 < | \\alpha - \\beta | < \\delta \\} \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} e _ 0 ( x _ i ) = 1 , \\ \\ \\ l e _ l ( x _ i ) = \\sum _ { m = 1 } ^ l ( - 1 ) ^ { m - 1 } e _ { l - m } ( x _ i ) p _ m ( x _ i ) , 1 \\le l \\le \\# T _ k \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} & q _ { l } ( z , \\overline { z } ) = \\overline { q _ { l } ( z , \\overline { z } ) } , \\quad \\mbox { f o r a l l $ l = 1 , \\dots , N _ { 0 } $ , } \\\\ & q _ { l } ( z , \\overline { z } ) \\neq \\overline { q _ { l } ( z , \\overline { z } ) } , \\quad \\mbox { f o r a l l $ l = N _ { 0 } + 1 , \\dots , N $ , } \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} \\psi ( y + x ) = \\frac { l _ 2 - ( y + x ) } { l _ 1 } = \\frac { l _ 2 - x } { l _ 1 } - \\frac { y } { l _ 1 } = \\psi ( x ) - \\frac { y } { l _ 1 } . \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} \\mbox { $ \\Psi ( x ) = ( \\psi _ i ( x ) ) : = \\mathbf { u } ^ * - \\Phi ( x ) $ a n d $ G ( u ) = ( g _ i ( u ) ) : = - F ( \\mathbf { u } ^ * - u ) $ . } \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} G = \\sum _ { i = 1 } ^ { m } \\xi _ { i } = 0 , \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} \\frac { p } { 2 } \\Delta ( H ^ { p - 1 } ) + p ( 2 H ^ 2 - K + 2 k _ 0 ) H ^ { p - 1 } - 2 H ^ { p + 1 } = 0 . \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} f \\in S : = \\left \\{ f \\in \\mathcal { B } ^ { \\alpha } ( \\mathbb { R } ^ 2 ) : ~ \\| f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = 1 \\right \\} . \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} \\dot { \\tilde { y } } = - \\frac { \\beta } { a _ 2 } + \\frac { \\gamma } { a _ 4 } \\ , \\tilde { y } , \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} P _ { } = F _ \\gamma ( \\gamma _ { t h } ) , \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} \\alpha ^ { \\ast } = \\frac { n - w _ { n } } { n ( 1 + w _ { n } ) } . \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} L \\models \\phi ( b _ { 1 } , . . . , b _ { M } ) : = \\exists \\overline { a } \\bigwedge _ { i = 1 } ^ { k } f _ { i } ( \\overline { a } ) = 0 \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} \\bigcap _ { j = 0 } ^ k \\{ c _ 1 T _ 1 - c _ 2 ( T _ 2 - T _ 1 ) + \\dots + c _ 1 ( T _ { 2 j + 1 } - T _ { 2 j } ) < \\beta \\} \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } F ( t _ * , x , v ) \\varphi ( x , v ) d x d v = \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } f _ 0 ( x , v ) \\psi ( 0 , x , v ) d x d v . \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( 1 - \\sigma ) D ^ 2 v _ { j , \\mu } : D ^ 2 \\varphi + \\sigma \\Delta v _ { j , \\mu } \\Delta \\varphi d x = \\lambda _ j ( \\mu ) \\left ( v _ { j , \\mu } , \\varphi \\right ) _ { \\partial \\Omega } \\ , , \\ \\ \\ \\forall \\varphi \\in \\mathcal { H } ^ 2 _ { 0 , D } ( \\Omega ) , \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} \\bar { H } _ Y ( y ) & : = - \\frac { 1 } { N } \\log \\int _ { P x = y } \\exp ( - H ( x ) ) \\mathcal { L } ^ { N - M } ( d x ) . \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} \\limsup _ { t _ 2 \\to \\infty } \\int _ K u ( x , t _ 2 ) \\ , d x \\le C \\sum _ { i = 1 } ^ N \\left ( \\varepsilon R _ i ^ n + \\frac { \\varepsilon ^ p R _ i ^ { n - m } } { \\lambda } + \\frac { R _ i ^ { n - m / ( 1 - p ) } } { \\lambda ^ { 1 / ( 1 - p ) } } \\right ) . \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} P ( Z _ 2 ; d ) ( R ( 1 ) - R ( 0 ) ) = P ( Z _ 2 ; d ) ( P ^ { - 1 } ( Z _ 2 ; d ) P ( Z _ 1 ; d ) - I d ) = P ( Z _ 1 ; d ) - P ( Z _ 2 ; d ) . \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} b _ 2 & = - \\frac 1 2 \\frac { 1 } { X _ { 1 + 2 } } \\sum _ { j = 1 } ^ { 2 } a _ { 2 - j } a _ { j - 1 } ( 2 + 1 - j ) ! j ! \\sum _ { Q \\in Q ^ { ( 3 , j ) } } X _ { Q } \\Big | _ { X _ 1 = 0 = X _ 2 } \\\\ & = - \\frac { 1 } { 2 X _ { 1 + 2 } } \\left ( a _ 1 a _ 0 2 ! 1 ! \\left ( X _ 1 + X _ 2 + X _ { 1 + 2 } \\right ) + a _ 0 a _ 1 1 ! 2 ! \\left ( X _ { 1 + 2 } + X _ { 1 } + X _ 2 \\right ) \\right ) \\Big | _ { X _ 1 = 0 = X _ 2 } \\\\ & = - 2 a _ 1 . \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} d X _ t & = \\mu ( X _ { t - } ) d t + \\sigma ( X _ { t - } ) d W _ t + \\int _ { \\R } \\rho ( X _ { t - } , y ) \\nu ( d y , d t ) , t \\in [ 0 , T ] , X _ 0 = x _ 0 \\in \\R ^ d , \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} \\Delta = - 2 7 B ^ 4 + ( A B ) ^ 3 ; \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} T _ 1 = \\gamma _ 1 ( c _ 1 - d _ 1 ) = \\gamma _ 1 ( u _ 1 c v _ 1 - d _ 1 ) , \\\\ T _ 2 = \\gamma _ 2 ( c _ 2 - d _ 2 ) = \\gamma _ 2 ( u _ 2 c v _ 2 - d _ 2 ) . \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} \\cal B : = \\{ P * f _ { k _ 1 } * f _ { k _ 2 } * \\ldots * f _ { k _ r } : r \\in [ 0 ; n ] , P \\in \\widetilde { B } _ { n - r } , 0 < k _ 1 \\leqslant k _ 2 \\leqslant \\ldots \\leqslant k _ r \\} . \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( z ) = \\frac { \\psi _ { \\mu } ( z ) } { 1 + \\psi _ { \\mu } ( z ) } , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} { \\rm c h } [ W ( \\Lambda _ 0 ) ] = \\sum _ { r _ 1 , r _ 2 , r _ 3 \\geq 0 } \\frac { q ^ { r _ 1 ^ 2 + ( r _ 2 + r _ 3 ) ^ 2 + r _ 3 ^ 2 - r _ 1 ( r _ 2 + 2 r _ 3 ) } } { ( q ) _ { r _ 1 } ( q ) _ { r _ 2 } ( q ) _ { r _ 3 } } . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} ( - K _ { F _ { \\alpha , i , \\beta } } ) ^ { r _ { \\alpha , i } } = ( - K _ { F ^ { \\mathrm { g e n } } _ { \\alpha , i } } ) ^ { r _ { \\alpha , i } } \\le C . \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} \\C ^ { 2 ^ n } = W ^ { + } \\oplus W ^ { - } ; W ^ { \\pm } = \\bigoplus _ { ( i _ 1 , \\dots , i _ { n - 1 } ) } \\C f _ { 1 2 ; i _ 1 , \\dots , i _ { n - 1 } } ^ { \\pm } . \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} \\langle F ' ( D u ) , D u \\rangle = F ^ * ( F ' ( D u ) ) + F ( D u ) . \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} Y _ 1 : = i ^ { - 1 } \\bigl ( \\mathfrak M ( H ^ \\infty ( U ) ) \\setminus \\overline { O } _ 2 \\bigr ) { \\rm a n d } Y _ 2 : = i ^ { - 1 } \\bigl ( O _ 1 \\setminus K _ 1 \\bigr ) \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\infty } ( \\Omega _ { y , x } , \\mathcal { A } _ { y , x } , P _ { y , x } ) \\times ( \\Omega _ { d } , \\mathcal { A } _ { d } , P _ { d } ) \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda _ L & = \\frac { \\nu _ 1 - b _ L } { 2 } , \\\\ \\lambda _ R & = \\frac { \\nu _ 1 - b _ L - b _ R } { 2 } , \\\\ \\omega _ L & = \\sqrt { k _ L - \\frac { ( \\nu _ 1 - b _ L ) ^ 2 } { 4 } } , \\\\ \\omega _ R & = \\sqrt { k _ L + k _ R - \\frac { ( \\nu _ 1 - b _ L - b _ R ) ^ 2 } { 4 } } . \\end{aligned} \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} \\mathfrak { J } _ A ^ K & = \\Bigl \\{ j \\in \\N \\ | \\ \\frac { A } { K } \\leq j \\leq 2 \\frac { A } { K } , A - 2 \\frac { A } { K } \\leq j \\leq A - \\frac { A } { K } \\Bigr \\} , \\\\ \\mathfrak { J } _ A & = \\left \\{ j \\in \\N \\ | \\ 0 \\leq j \\leq 2 ^ { 1 0 } , A - 2 ^ { 1 0 } \\leq j \\leq A - 1 \\right \\} . \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} & t \\frac { d u ^ * } { d t } = ( \\lambda ^ * ( t ) + \\lambda _ 1 ^ * ( t ) ) u ^ * \\mbox { o n $ ( t _ { \\xi } , T ] $ } , \\\\ & t \\frac { d q ^ * } { d t } = ( \\lambda ^ * ( t ) + \\lambda _ 1 ^ * ( t ) + c ^ * ( t ) ) q ^ * + \\gamma ^ * ( t ) u ^ * \\mbox { o n $ ( t _ { \\xi } , T ] $ } . \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} \\| \\phi \\| _ { C ^ k _ * ( ( 0 , \\tau ) \\times S ) } : = \\sum _ { 0 \\leq i + j \\leq k } r ^ { i + j - 1 } \\| \\partial ^ i _ r \\nabla _ S ^ j \\phi \\| _ { C ^ 0 } \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} K _ { 1 } ( x , \\lambda ( t ) ) = \\int _ { 0 } ^ { \\infty } \\frac { r d r } { \\lambda ( t ) ^ { 2 } ( 1 + \\frac { r ^ { 2 } } { \\lambda ( t ) ^ { 2 } } ) ^ { 3 } } \\int _ { 0 } ^ { x } \\frac { \\rho d \\rho } { x } \\left ( 1 + \\frac { r ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( r ^ { 2 } - 1 - \\rho ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } \\right ) \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} & \\hat { \\rho } \\Big ( \\big ( \\varphi ( \\hat { \\pi } _ { 1 } ( t _ { 1 } ) , t _ { 1 } ) , \\psi ( t _ { 1 } ) \\big ) , \\big ( \\varphi ( \\hat { \\pi } _ { 2 } ( t _ { 2 } ) , t _ { 2 } ) , \\psi ( t _ { 2 } ) \\big ) \\Big ) \\\\ & ~ = \\big | \\varphi ( \\hat { \\pi } _ { 1 } ( t _ { 1 } ) , t _ { 1 } ) - \\varphi ( \\hat { \\pi } _ { 2 } ( t _ { 2 } ) , t _ { 2 } ) \\big | \\vee | \\psi ( t _ { 1 } ) - \\psi ( t _ { 2 } ) | \\\\ & ~ = \\rho \\big ( ( \\hat { \\pi } _ { 1 } ( t _ { 1 } ) , t _ { 1 } ) , ( \\hat { \\pi } _ { 2 } ( t _ { 2 } ) , t _ { 2 } ) \\big ) , \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} D = ( d _ i ) , \\ ; { \\mathbf J } ( x ) = ( J _ i ( x ) ) . \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} \\begin{matrix} \\displaystyle \\min _ { P _ { \\cal F } , P _ { \\cal R } } & ~ P _ { \\text o u t } = G ( \\mathcal { A } _ F P _ { \\cal F } ^ { - a } + \\mathcal { A } _ R P _ { \\cal R } ^ { - a } ) \\\\ \\\\ \\textrm { s . t . } & P _ { \\cal F } + P _ { \\cal R } \\leq P _ { t o t } \\\\ \\\\ & - P _ { \\cal R } \\leq 0 , ~ P _ { \\cal F } \\leq \\mathcal { S } & \\end{matrix} \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} a ^ L _ i & : = ( \\ - a ^ L _ i \\vee \\bigvee _ { s \\in B _ i \\cap S } \\ - a ^ L _ { s , i } ) \\wedge \\bigwedge _ { s \\in S \\setminus B _ i } \\ - a ^ L _ { s , i } , \\\\ a ^ R _ j & : = ( \\ - a ^ R _ j \\wedge \\bigwedge _ { s \\in C _ j \\cap S } \\ - a ^ R _ { s , j } ) \\vee \\bigvee _ { s \\in S \\setminus C _ j } \\ - a ^ R _ { s , j } . \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} \\liminf _ { n \\to - \\infty } Y _ n = r . \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} ( S _ t ) _ { t \\geq 0 } \\ : \\vline \\ : \\left \\{ \\inf _ { t \\geq 0 } S _ t = 0 \\right \\} \\buildrel { d } \\over { = } ( 2 \\bar { M } _ t - S _ t ) _ { t \\geq 0 } \\ : \\vline \\ : \\left \\{ M _ 0 = 0 \\right \\} , \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} y ^ 2 + x y = x ^ 3 + A ^ 4 x ^ 2 + B ^ 8 . \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} E _ { \\omega } T _ x = \\omega ( x ) T _ x E _ { \\omega } . \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} & n < \\infty \\textit { f o r e v e r y } n \\in \\mathbb { Z } , \\\\ & \\infty + n = n + \\infty = \\infty \\textit { f o r e v e r y } n \\in \\mathbb { Z } , \\\\ & \\infty + \\infty = \\infty . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} \\int _ { G / H } \\psi ( x H ) d \\lambda ^ { \\ast _ { G / H } } ( x H ) = \\int _ { G / H } \\left ( \\int _ H \\psi ( h ^ { - 1 } x ^ { - 1 } H ) d h \\right ) d \\overline { \\lambda } ( x H ) , \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} \\Phi ( x ) = \\phi ( x _ 1 ) \\cdots \\phi ( x _ d ) , x = ( x _ 1 , \\dots , x _ d ) \\in \\R ^ d , \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} \\langle x , y \\rangle _ { X } = 0 . \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} ( f ^ { - 1 } ) '' = - ( f ^ { - 1 } f ' f ^ { - 1 } ) ' = 2 f ^ { - 1 } f ' f ^ { - 1 } f ' f ^ { - 1 } - f ^ { - 1 } f '' f ^ { - 1 } , \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} \\Sigma ^ * _ { i j } = ( q _ { i 0 } \\circ \\dots \\circ q _ { i , j - 1 } ) ^ { - 1 } ( \\Sigma _ { i 0 } ^ * ) . \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} \\dim H ^ 0 ( X , \\mathcal { O } _ X ( D ) ) = \\chi ( X , \\mathcal { O } _ X ( D ) ) + \\sum _ { C \\cdot D < 0 } \\binom { k _ C } { 2 } . \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} b _ 0 ( u , v , v ) = 0 , b _ 1 ( u , \\theta , \\theta ) = 0 , u , v \\in V _ 0 , \\ \\theta \\in V _ 1 . \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} & X ^ { s , \\ , b } ( \\R ^ 3 ) : = \\{ f \\in \\mathcal { S } ' ( \\R ^ 3 ) \\ | \\ \\| f \\| _ { X ^ { s , \\ , b } } < \\infty \\} , \\\\ & \\| f \\| _ { X ^ { s , \\ , b } } : = \\Bigl ( \\sum _ { N , \\ , L } N ^ { 2 s } L ^ { 2 b } \\| P _ N Q _ L f \\| ^ 2 _ { L _ { x , t } ^ { 2 } } \\Bigr ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\gamma _ 1 = \\sum _ { I \\in \\mathcal { I } } \\omega _ I u _ 1 ^ { \\epsilon _ 1 } v _ 1 ^ { ( p - 1 ) i _ 1 - \\epsilon _ 1 } \\cdots u _ s ^ { \\epsilon _ s } v _ s ^ { ( p - 1 ) i _ s - \\epsilon _ s } S _ s ( \\Sigma ^ { - ( s - 1 ) } x _ s y _ s ^ { - 1 } \\otimes m ) \\in ( \\Gamma ^ + \\Sigma ^ { - ( s - 1 ) } \\widehat { P } \\otimes M ) _ s . \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} y ^ { ( m ) } ( t , \\mu ) = \\displaystyle { \\sum _ { i = 1 } ^ m } w _ i ( t ) \\Phi _ i ( \\mu ) \\mbox { a n d } x ^ { ( m ) } ( t , \\mu ) = \\displaystyle { \\sum _ { i = 1 } ^ m } v _ i ( t ) \\Phi _ i ( \\mu ) . \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} \\begin{cases} D _ { a } ^ { \\alpha _ 2 } y ( t ) = f ( t , y , D _ { a } ^ { \\alpha _ 1 } y ( t ) ) , \\ t \\in [ a , b ] , \\\\ a _ 1 y ( a ) + b _ 1 y ' ( a ) = \\gamma _ 1 , \\ a _ 2 y ( b ) + b _ 2 y ' ( b ) = \\gamma _ 2 , \\end{cases} \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} & v _ 0 = w _ 1 = l a _ 1 b _ 1 m _ 1 c _ 1 m _ 2 d _ 1 e _ 1 f _ 1 r , \\\\ & v _ 1 = l a _ 2 s _ 1 ^ { - 1 } b _ 1 m _ 1 c _ 1 m _ 2 d _ 1 e _ 1 f _ 1 r , \\\\ & v _ 2 = l a _ 2 b _ 2 m _ 1 c _ 1 m _ 2 d _ 1 e _ 1 f _ 1 r , \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} \\begin{aligned} ( f ^ j , \\phi ) & = \\lim _ { n \\to \\infty } ( f _ n \\ast \\varphi _ j , \\phi ) = \\lim _ { n \\to \\infty } ( f _ n \\ast \\varphi _ j \\ast ( \\varphi _ { j - 1 } + \\varphi _ j + \\varphi _ { j + 1 } ) , \\phi ) \\\\ & = ( f ^ j \\ast ( \\varphi _ { j - 1 } + \\varphi _ j + \\varphi _ { j + 1 } ) , \\phi ) , \\end{aligned} \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} p _ { \\mu } ( \\xi ) = \\frac { 1 } { 2 \\pi } \\Re \\frac { 1 + \\eta _ { \\mu } ( \\overline { \\xi } ) } { 1 - \\eta _ { \\mu } ( \\overline { \\xi } ) } , \\xi \\in \\mathbb { T } , \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} \\phi _ 5 ^ 5 ( z ) = g _ 5 ( 5 z ) \\left ( 5 ^ 2 \\phi _ 5 ^ 4 ( z ) + 5 ^ 2 \\phi _ 5 ^ 3 ( z ) + 5 \\cdot 3 \\phi _ 5 ^ 2 ( z ) + 5 \\phi _ 5 ( z ) + 1 \\right ) . \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} Y = X + \\sqrt { X } { Z _ 1 } + { Z _ 0 } , \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} \\inf I - \\frac { 1 } { 4 c _ \\mu } { \\mu ( Q ) } \\geq \\frac { 1 } { 2 c _ \\mu } { \\mu ( Q ) } - \\frac { 1 } { 4 c _ \\mu } { \\mu ( Q ) } = \\frac { 1 } { 4 c _ \\mu } { \\mu ( Q ) } , \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} Y _ t = \\xi - \\int _ t ^ T Z _ s d B _ s - \\int _ t ^ T f ( s , Y _ s , Z _ s ) d s , \\ t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} 0 & = \\det ( B ( | \\xi | ; \\rho , \\theta ) - \\lambda I _ { 4 \\times 4 } ) \\\\ & = \\lambda ^ 4 - 2 \\left ( | \\xi | ^ { 2 \\rho } + | \\xi | ^ { 2 \\theta } \\right ) \\lambda ^ 3 + \\left ( \\left ( | \\xi | ^ { 2 \\rho } + | \\xi | ^ { 2 \\theta } \\right ) ^ 2 + \\left ( a ^ 2 + b ^ 2 \\right ) | \\xi | ^ 2 \\right ) \\lambda ^ 2 \\\\ & \\quad - \\left ( a ^ 2 + b ^ 2 \\right ) | \\xi | ^ 2 \\left ( | \\xi | ^ { 2 \\rho } + | \\xi | ^ { 2 \\theta } \\right ) \\lambda + a ^ 2 b ^ 2 | \\xi | ^ 4 . \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} | w | _ { H ^ 2 ( \\Omega ) } = | w - \\overline w | _ { H ^ 2 ( \\Omega ) } \\le \\| w - \\overline w \\| _ { H ^ 2 ( \\Omega ) } \\le C _ { } \\| \\Delta _ h w _ h \\| _ { \\mathcal { T } _ h } , \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } \\sum _ { n = 1 } ^ N | a ( r _ n ) - \\psi ( r _ n ) | > 0 . \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } - \\frac { \\coth \\theta \\cosh t - \\sinh t } { \\cosh t - \\coth \\theta \\sinh t } u = 0 . \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} D ( H \\mathcal { H } ) = \\bigl \\{ x \\in D ( \\mathcal { H } ) \\ : ; \\ : \\mathcal { H } x \\in D ( H ) H \\mathcal { H } x \\in X _ { \\mathcal { H } } \\bigl \\} , \\ ; \\ ; x \\mapsto H \\mathcal { H } x . \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} \\lim _ { x _ { n + 1 } \\rightarrow 0 } \\nabla ' \\overline { w } = 0 , \\quad \\lim _ { x _ { n + 1 } \\rightarrow 0 } \\Delta ' \\overline { w } = 0 \\quad \\quad \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\overline { w } = \\eta \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { w } . \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} f _ a ^ { - 1 } ( x , y ) = \\begin{cases} \\emptyset & a \\neq y , \\\\ \\{ x \\} & a = y , \\end{cases} \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} \\left | \\partial ^ \\alpha _ x \\partial ^ \\beta _ \\xi \\left ( a ( x , \\xi ) - \\varphi ( x , \\xi ) \\sum _ { j = - m } ^ { N - 1 } a _ j ( x , \\xi ) \\right ) \\right | \\le C _ { N , \\alpha , \\beta } ( 1 + \\| x \\| ^ 2 + \\| \\xi \\| ^ 2 ) ^ { - N / 2 } \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} \\nu = 2 4 _ a + 2 1 _ { b ^ 2 } + 1 6 _ a + 1 3 _ { b ^ 2 } + 1 1 _ { a ^ 2 } + 7 _ b + 6 _ a \\ , \\cdot \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} y ^ 2 + x y = x ^ 3 + a x ^ 2 + b \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} \\begin{cases} a \\cdot ( u \\cdot b ) - ( u \\cdot b ) \\cdot a + H ( a , T ( u \\cdot b ) ) - H ( T ( u \\cdot b ) , a ) \\\\ = u \\cdot ( a b - b a ) + ( a \\cdot u - u \\cdot a + H ( a , T u ) - H ( T u , a ) ) \\cdot b , \\\\ ( a \\cdot u - u \\cdot a + H ( a , T u ) - H ( T u , a ) ) \\cdot ( a b - b a ) = 0 , ~ b \\in A , u \\in M . \\end{cases} \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } = \\frac { \\boldsymbol { \\pi } D } { \\boldsymbol { \\pi } D \\mathbf { 1 } } \\boldsymbol { \\pi } = \\frac { \\boldsymbol { \\alpha } ( - C ) ^ { - 1 } } { \\boldsymbol { \\alpha } ( - C ) ^ { - 1 } \\mathbf { 1 } } = \\lambda ^ * \\boldsymbol { \\alpha } ( - C ) ^ { - 1 } , \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} W _ t ^ { ( 2 ) } + \\Lambda _ 1 ( | \\xi | ) W ^ { ( 2 ) } + R _ 2 ( | \\xi | ) W ^ { ( 2 ) } = 0 , \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} P _ G \\left ( x \\right ) & = x ^ { n - t - 1 } \\left ( x + t \\mu \\right ) \\left ( x - \\mu \\right ) ^ t \\\\ & = x ^ { n - t - 1 } \\left ( x - \\mu \\right ) ^ { t - 2 } \\left [ x ^ 3 + \\left ( t - 2 \\right ) \\mu x ^ 2 + \\left ( 1 - 2 t \\right ) \\mu ^ 2 x + t \\mu ^ 3 \\right ] . \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} \\xi _ i ( t ) : = g _ i ( \\mathbf { u ^ * } + t ( u - \\mathbf { u ^ * } ) , \\mathbf { u ^ * } + t ( v - \\mathbf { u ^ * } ) \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} \\varphi _ { p , q } ( \\lambda ) = & { \\lambda } ^ { 5 } - \\left ( q - 5 + p \\right ) { \\lambda } ^ { 4 } \\\\ & - \\left ( 3 \\ , p q + 4 \\ , p + q - 1 0 \\right ) { \\lambda } ^ { 3 } \\\\ & - \\left ( 8 \\ , p q + 6 \\ , p - 4 \\ , q - 8 \\right ) { \\lambda } ^ { 2 } \\\\ & - \\left ( p q + 1 0 \\ , p - 8 \\right ) \\lambda + 3 \\ , p q - 6 \\ , p - 2 \\ , q + 4 . \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} U _ { 1 } = I d , U _ { x } V _ { y , x } = V _ { x , y } U _ { x } , U _ { U _ { x } y } = U _ { x } U _ { y } U _ { x } \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} U _ { \\vec { N } } ( x , y ) : = \\sum \\limits _ { i = 0 } ^ { N _ x } \\sum \\limits _ { j = 0 } ^ { N _ y } u _ { i , j } B _ { i } ( x ) B _ { j } ( y ) \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} d ( \\lambda \\otimes h ) & = d ( \\lambda ) \\otimes h + \\sum _ { i - \\epsilon \\geq 0 } ( - 1 ) ^ { \\deg \\lambda + ( 1 - \\epsilon ) \\deg h } \\lambda \\lambda ^ { \\epsilon } _ { i - 1 } \\otimes h \\beta ^ { 1 - \\epsilon } P ^ i . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} D ^ m [ f _ 1 \\circ f _ 2 ] ( x ) = \\sum _ { i = 1 } ^ m \\sum _ { \\pi \\in P _ m ^ i } D ^ i f _ 1 ( f _ 2 ( x ) ) \\left ( D ^ { \\pi ( 1 ) } f _ 2 ( x ) , \\dots , D ^ { \\pi ( i ) } f _ 2 ( x ) \\right ) , \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} \\theta ( w ) = \\displaystyle \\min _ { z } \\left \\{ \\displaystyle \\frac { 1 } { 2 } \\| z \\| ^ 2 : g ( w ) + z \\in K \\right \\} . \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} v _ { 3 } ^ { \\lambda } ( t , r ) = \\frac { - 1 } { r } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\lambda '' ( s ) \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( - 1 - \\rho ^ { 2 } + r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} \\{ a , b \\cdot c \\} = \\{ a , b \\} \\cdot c + b \\cdot \\{ a , c \\} . \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} \\begin{aligned} = \\ ! { \\sigma L o c } _ \\sigma & \\simeq \\ ! { Q P o l } : = , \\\\ = \\ ! { \\sigma B o r L o c } _ \\sigma & \\simeq \\ ! { S B o r } : = . \\end{aligned} \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} \\varphi ^ { \\ast _ { G / H } } ( x H ) = \\Delta _ G ( x ^ { - 1 } ) \\int _ H \\overline { \\varphi ( h ^ { - 1 } x ^ { - 1 } H ) } d h , \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k - 1 } \\Biggl [ \\binom { 2 k - 1 } { j } - \\binom { 2 k - 1 } { j - 1 } \\Biggr ] \\frac { c _ 1 ^ { j } \\ , c _ 2 ^ { 2 k - 1 - j } } { ( c _ 1 + c _ 2 ) ^ { 2 k - 1 } } = \\sum _ { j = 0 } ^ { k } \\Biggl [ \\binom { 2 k } { j } - \\binom { 2 k } { j - 1 } \\Biggr ] \\frac { c _ 1 ^ { j } \\ , c _ 2 ^ { 2 k - j } } { ( c _ 1 + c _ 2 ) ^ { 2 k } } \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} D _ i Z = \\partial _ i Z + \\partial _ i K \\cdot Z , \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} \\left [ \\displaystyle \\sum _ { i = 1 } ^ p [ y ^ I ] ^ k _ i \\nabla ^ 2 c _ i ( x ^ k ) - A A ^ T - { \\cal J } c ( x ^ k ) ^ T { \\cal J } _ { z ^ I } \\Psi _ { \\varepsilon } ( [ y ^ I ] ^ k , [ z ^ I ] ^ k ) { \\cal J } c ( x ^ k ) \\right ] \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} p _ I = p _ { i _ 1 \\ldots i _ k } : Y ^ { n } \\to Y ^ { k } . \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} \\Omega _ \\Sigma \\Psi ( A ^ { 1 , 0 } ) = 0 \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} \\nabla _ k h _ { i j } - \\nabla _ j h _ { i k } = \\overline { R } _ { i j k l } N ^ l = k _ 0 ( \\delta _ { i k } \\delta _ { j l } - \\delta _ { j k } \\delta _ { i l } ) N ^ l , \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} \\Sigma ^ * _ { i j } = ( q _ { i 0 } \\circ \\dots \\circ q _ { i , j - 1 } ) ^ { - 1 } ( \\Sigma _ { i 0 } ^ * ) \\subset \\Sigma _ { i j } . \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} \\Delta _ { \\lambda , \\epsilon } : = \\Delta - \\lambda \\sum \\beta _ i \\Delta _ i - \\epsilon \\Delta _ 1 , \\bar \\Delta _ { \\lambda , \\epsilon } : = \\varphi _ * \\Delta _ { \\lambda , \\epsilon } . \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} | v _ { 3 , 1 , 2 } ( t , r ) | \\leq \\begin{cases} \\frac { C r | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\sqrt { \\log ( \\log ( t ) ) } \\log ^ { b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { r \\log ^ { b } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} \\alpha ^ k _ l : = 1 - \\frac { G _ l ( x _ k ) } { \\sqrt { G _ l ^ 2 ( x _ k ) + H _ l ^ 2 ( x _ k ) + 2 t _ k } } \\beta ^ k _ l : = 1 - \\frac { H _ l ( x _ k ) } { \\sqrt { G _ l ^ 2 ( x _ k ) + H _ l ^ 2 ( x _ k ) + 2 t _ k } } \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} \\hat { v } = v _ 0 + v _ 1 + v _ 2 + \\dots \\quad \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} | f _ 1 ( z ) | = | e ^ { - z ^ 2 } | \\leq e ^ { W ^ 2 } e ^ { - { \\rm R e } ( z ) ^ 2 } \\leq C ^ { f _ 1 } _ m ( 1 + | z | ) ^ m , \\mbox { f o r a l l } ~ m \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} \\hat { \\Sigma } _ n - \\hat { \\Theta } _ { \\lambda _ n } ^ { - 1 } + \\lambda _ n \\hat { V } _ n \\hat { Z } _ n \\hat { V } _ n = 0 , \\| \\hat { Z } _ n \\| _ { \\ell _ \\infty } \\leq 1 , \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\langle y , \\tilde { y } \\rangle _ Y = \\frac { 1 } { M } \\sum _ { l = 1 } ^ M y _ l \\tilde { y } _ l . \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( F \\left ( ( \\eta ^ L _ t ) _ { t \\in [ 0 , L ] } \\right ) \\right ) = \\frac { \\mathbf { E } \\left ( \\nu ( \\eta _ 0 ) ^ { - 1 } F \\left ( ( \\eta _ t ) _ { t \\in [ 0 , L ] } \\right ) \\ : \\vline \\ : \\eta _ L = \\eta _ 0 , \\ : S _ L > 0 \\right ) } { \\mathbf { E } \\left ( \\nu ( \\eta _ 0 ) ^ { - 1 } \\ : \\vline \\ : \\eta _ L = \\eta _ 0 , \\ : S _ L > 0 \\right ) } , \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} - \\dfrac { \\nu ^ 2 } { \\tau ^ 4 } \\int _ { \\mathbb T ^ d _ { \\ell } } R \\mathbb D U : \\nabla ^ 2 \\log R + \\dfrac { \\nu ^ 2 } { \\tau ^ 4 } \\int _ { \\mathbb T ^ d _ { \\ell } } R \\nabla U \\cdot \\nabla ^ 2 \\log { R } = \\dfrac { \\nu ^ 2 } { \\tau ^ 4 } \\int _ { \\mathbb T ^ d _ { \\ell } } R \\mathbb A U : \\nabla ^ 2 \\log R = 0 , \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta ' _ B ( 0 ; a , 1 , 1 ) = \\left ( \\frac 1 { 1 2 } - \\zeta ' _ R ( - 1 ) \\right ) \\frac 1 a & - \\frac 1 4 \\log ( 2 \\pi ) + \\frac { \\gamma a } { 1 2 } \\\\ & + \\sum _ { k = 2 } ^ \\infty \\frac { B _ { 2 k } \\zeta _ R ( 2 k - 1 ) } { 2 k ( 2 k - 1 ) } a ^ { 2 k - 1 } , a \\to 0 ^ + , \\end{aligned} \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} & - \\frac { 1 } { r } \\int _ { t + 6 r } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { ( s - t ) } \\lambda '' ( s ) \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( - 1 - \\rho ^ { 2 } + r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) \\\\ & = - 2 r \\int _ { 6 r } ^ { \\infty } d w \\lambda '' ( t + w ) w \\left ( \\frac { 1 } { 2 ( 1 + w ^ { 2 } ) } - \\frac { 1 } { 2 ( \\lambda ( t + w ) ^ { 2 - 2 \\alpha } + w ^ { 2 } ) } \\right ) + E _ { 4 } \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} & | - \\lambda ( t ) \\langle \\left ( \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) \\left ( v _ { 1 } + v _ { 2 } + v _ { 3 } \\right ) + F _ { 0 , 2 } \\right ) \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | \\\\ & \\leq C \\lambda ( t ) \\int _ { 0 } ^ { \\infty } | v _ { 4 , c } ( t , R \\lambda ( t ) ) | \\phi _ { 0 } ( R ) R d R \\leq \\frac { C } { t ^ { 2 } \\log ^ { 3 b + 2 N } ( t ) } \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{bmatrix} a _ 1 x + a _ 2 y \\\\ b _ 0 + b _ 1 x + b _ 2 y \\end{bmatrix} , \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} \\lambda ( f ) : = \\int _ X f ( x ) d \\lambda ( x ) . \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} \\mathrm { e } ^ { - \\lambda _ \\gamma t } P _ t \\psi _ \\gamma \\leq \\psi _ \\gamma , \\lim _ { t \\downarrow 0 } \\mathrm { e } ^ { - \\lambda _ \\gamma t } P _ t \\psi _ \\gamma = \\psi _ \\gamma . \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} 0 \\leq \\chi ( c _ i \\leq c _ j ) \\leq \\Delta ( i , j ) \\leq j - i \\ , \\ , , \\ , \\Delta ( j , i ) = - \\Delta ( i , j ) \\ , \\cdot \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} \\psi _ { \\Lambda _ { 1 } } ( b _ { \\Lambda } ) = \\sum _ { i , j \\in I _ { 0 } } \\prod _ { x \\in \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * b _ { x } \\right ) \\beta _ { i j } ^ { | \\Lambda _ { 1 } \\setminus \\Lambda | } . \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} \\left [ \\widehat { q } _ 1 , \\widehat { p } _ 1 \\right ] = \\left [ \\widehat { q } _ 2 , \\widehat { p } _ 2 \\right ] = i , \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} \\langle \\bar { M } y , \\Delta ( y ) \\rangle _ w = \\langle \\rho _ - \\bar { M } y , \\rho _ - \\Delta ( y ) \\rangle \\geq 0 \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} \\lambda _ 1 ( G ^ c ) & = x ^ T D ( G ^ c ) x \\\\ & = x ^ T ( J _ n - I _ n ) x + x ^ T A ( G ) x \\\\ & < x ^ T ( J _ n - I _ n ) x + x ^ T A ( H ( s , t ) ) x \\\\ & \\le x ^ T D ( H ^ c ( s , t ) ) x . \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} & { \\rm R e } \\lambda _ c ( 0 , 0 ) = { \\rm R e } \\lambda ( 0 , 0 ) < a _ 1 , \\\\ & | b _ c ( t , x ) | \\leq ( B + C _ 2 ) ( \\mu ( t ) + t ^ d ) \\mbox { o n $ [ 0 , T _ N ] \\times D _ { R _ N } $ } . \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} H ' ( \\beta ) = 2 \\beta ^ { - 1 / 2 } \\sum _ { k = 1 } ^ \\infty \\frac { k ^ { k - 2 } } { k ! } \\Bigg [ k \\beta ^ { k } - ( k - 1 ) \\beta ^ { k - 1 } \\Bigg ] e ^ { - k \\beta } \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} \\frac { 1 } { \\check { \\ell } + h ^ \\vee } = \\frac { 1 } { k ' + h ^ \\vee } + 1 \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} \\Psi _ * X _ p & = R _ { \\gamma ( p ) * } X _ p + \\left \\{ \\gamma \\ - d \\gamma _ { | p } ( X _ p ) \\right \\} ^ v _ { p \\gamma ( p ) } = R _ { \\gamma ( p ) * } \\left ( X _ p + \\left \\{ d \\gamma { \\gamma \\ - } _ { | p } ( X _ p ) \\right \\} ^ v _ p \\right ) , \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} V ( D ) = \\left ( \\begin{array} { c c } e ^ { - D ^ - D ^ + } & e ^ { - \\frac { 1 } { 2 } D ^ - D ^ + } \\left ( \\frac { I - e ^ { - D ^ - D ^ + } } { D ^ - D ^ + } \\right ) D ^ - \\\\ e ^ { - \\frac { 1 } { 2 } D ^ + D ^ - } D ^ + & I - e ^ { - D ^ + D ^ - } \\end{array} \\right ) \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} q : = ( 1 - C ) ^ \\alpha \\in ( 0 , 1 ) \\qquad C = 2 C ' D _ p ( \\Pi _ { M \\cap N } ^ p x _ 0 , x _ 0 ) ^ \\alpha \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{gather*} \\lim _ { \\alpha \\to - \\infty } \\liminf _ { n \\to \\infty } d ^ { ( n ) } \\big ( t _ { \\textnormal { m i x } } ^ { ( n ) } + \\alpha w _ { n } \\big ) = 1 \\quad \\lim _ { \\alpha \\to \\infty } \\limsup _ { n \\to \\infty } d ^ { ( n ) } \\big ( t _ { \\textnormal { m i x } } ^ { ( n ) } + \\alpha w _ { n } \\big ) = 0 . \\end{gather*}"}
-{"id": "9399.png", "formula": "\\begin{align*} \\Omega _ { j } \\Omega _ { k } - \\omega _ { j } \\Omega _ { k } = \\Omega _ { k } \\Omega _ { j } - \\omega _ { k } \\Omega _ { j } , \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} \\mathbb { G } _ { N } '' f - \\widetilde { \\mathbb { G } } _ { N } '' f = \\left ( \\frac { N } { \\widehat { N } } - 1 \\right ) \\widetilde { \\mathbb { G } } _ { N } '' f , f \\in \\mathcal { F } , \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} \\rho ( L _ i ) ( z ^ n ) \\ , = \\ , ( - z ^ { i + 1 } \\partial + ( \\alpha i + \\beta ) z ^ i ) ( z ^ n ) \\ , = \\ , ( \\alpha i + \\beta - n ) z ^ { n + i } \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} E _ { P ^ { \\theta ^ { \\ast } } } \\Vert { x } _ { t } - \\hat { x } _ { t } \\Vert ^ { 2 } = \\inf _ { \\zeta \\in \\bar { \\mathcal { K } } _ { t } } E _ { P ^ { \\theta ^ { \\ast } } } \\Vert { x } _ { t } - \\zeta \\Vert ^ { 2 } \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} \\sigma _ { i } = \\frac { n } { 2 \\pi } \\textup { A r g } ( s _ { i } ) \\tau _ { i } = \\frac { n } { 2 \\pi } \\textup { A r g } ( t _ { i } ) , \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} d ( \\lambda ^ 0 _ { - 1 } b ^ { [ t ] } ) = \\lambda ^ 0 _ { - 1 } \\lambda ^ 0 _ { - 1 } a b ^ { [ t - 1 ] } \\mod F ^ { 2 ( t - 1 ) } , \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 2 } v _ { 4 } ^ { 0 } ( t , r ) = \\frac { - 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\int _ { 0 } ^ { 2 \\pi } d \\theta \\frac { \\partial _ { 1 } ^ { 2 } v _ { 4 , c } ^ { 0 } ( s , \\sqrt { r ^ { 2 } + \\rho ^ { 2 } + 2 r \\rho \\cos ( \\theta ) } ) \\left ( r + \\rho \\cos ( \\theta ) \\right ) } { \\sqrt { r ^ { 2 } + \\rho ^ { 2 } + 2 r \\rho \\cos ( \\theta ) } } \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} r _ h ( v , \\chi ) = a _ h ( v , \\chi ) - l _ h ( \\chi ) \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} d ( b ^ { [ p + \\ell + 1 ] } ) = \\lambda ^ 0 _ { - 1 } a b ^ { [ p + \\ell ] } + ( \\ell + 2 ) \\lambda ^ 1 _ 0 b ^ { [ \\ell + 2 ] } \\mod F ^ { 2 ( \\ell + 2 ) - 1 } . \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} \\int _ { \\Omega } [ a ( t + \\epsilon ) - a ( t ) ] \\phi ( x ) = \\int _ { t } ^ { t + \\epsilon } \\int _ { \\Omega } ( b \\cdot \\nabla _ { x } ) \\phi ( x ) . \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} { \\rm t r } \\left [ \\mathbf { M } ( \\tau ) \\right ] & = \\big [ \\nabla _ { \\ ! x } \\cdot \\mathbf { W _ { \\ ! \\varepsilon } } - \\nabla _ { \\ ! v } \\cdot \\mathbf { B } \\big ] \\big ( \\tau , X ( \\tau ) , V ( \\tau ) \\big ) \\\\ [ 2 p t ] & = \\big [ V ( \\tau ) - \\beta _ \\varepsilon \\big ( V ( \\tau ) \\big ) \\big ] \\cdot \\nabla _ { \\ ! x } \\eta _ \\varepsilon \\big ( X ( \\tau ) \\big ) \\\\ & \\ ; - \\big ( \\nabla _ { \\ ! v } \\cdot \\mathbf { B } \\big ) \\big ( \\tau , X ( \\tau ) , V ( \\tau ) \\big ) \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} \\# \\left \\{ t \\in [ 0 , L ) : \\ : t \\in L M ( S ^ { * L } ) \\right \\} = J , \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} & \\delta g = - 2 u \\ , h , \\\\ & \\delta g ^ { - 1 } = 2 \\ , u \\ , \\hat { h } \\\\ & \\delta ( d S ) = - 2 H u \\ , d S , \\\\ & \\delta ( 2 H ) = \\Delta { u } + 2 u ( 2 H ^ 2 - K + 2 k _ 0 ) , \\\\ & \\delta K = 2 H \\Delta { u } - \\langle h , \\ , u \\rangle + 2 H K u , \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} 4 z ^ { 3 } + 3 p z ^ { 2 } + 2 q z + u = 0 . \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} i \\frac 1 2 a _ { j - 2 } a _ { k - 2 } ( j - 1 ) ! ( k - 1 ) ! X _ e + i \\frac 1 2 a _ { k - 2 } a _ { j - 2 } ( k - 1 ) ! ( j - 1 ) ! X _ e & = i a _ { j - 2 } a _ { k - 2 } ( j - 1 ) ! ( k - 1 ) ! X _ e . \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { \\ell } v _ i ' = \\sum \\limits _ { i = 3 } ^ { \\ell } v _ i ' \\leq \\sum \\limits _ { i = 3 } ^ { \\ell } ( 5 - v _ i ) = 5 ( \\ell - 2 ) - n + v _ 1 + v _ 2 < n , \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} o ( \\epsilon ) = o ( 1 ) \\epsilon \\mbox { w i t h } o ( 1 ) \\to 0 \\mbox { a s } \\theta _ 0 \\to \\infty . \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) \\le \\beta \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k + 1 \\} = \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} \\frac { d } { d t } \\phi _ t = d _ { \\phi _ t } \\gamma _ t . \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} f ^ n ( u _ h ^ n ) - f ( u ^ n ) & = ( u _ h ^ n ) ^ 3 - ( u _ { I h } ^ n ) ^ 3 + ( u _ { I h } ^ n ) ^ 3 - ( u ^ n ) ^ 3 - \\Delta t \\partial _ t ^ + u _ h ^ n - ( u _ h ^ { n - 1 } - u ^ { n - 1 } ) \\\\ & + \\Delta t \\partial _ t ^ + u ^ n . \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} \\xi _ { R } ( x ) = \\begin{cases} 1 & , R \\le | x | \\le 2 R , \\\\ 0 & , | x | \\le \\frac { R } { 2 } | x | \\ge 2 R , \\end{cases} \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\epsilon _ { i n } ^ { ( 3 ) } = o _ p ( n ^ { 1 / 2 } | \\ss _ 2 | ) = o _ p ( 1 ) + o _ p ( n \\| \\ss _ 2 \\| ^ 2 ) . \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} c _ 1 = \\frac { 1 } { \\sqrt { 2 } } \\sum _ { k = 1 } ^ { \\infty } \\frac { 1 } { k ^ { 3 / 2 } } \\frac { \\Gamma \\left ( k - \\frac { 1 } { 2 } \\right ) } { k ! } . \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty \\lVert a _ i \\rVert _ { L _ q } \\ ; \\lVert y _ i \\rVert _ { L _ p } < \\infty . \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} \\tt = ( \\mu , \\beta _ 0 ^ 2 / 2 + ( \\sigma _ + ^ 2 - 1 ) , \\beta _ 0 ^ 2 , \\beta _ 1 ^ 2 , \\beta _ 0 \\beta _ 1 ) ^ \\tau . \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} G _ { \\left ( l ' \\right ) } ( z , w ) = q _ { l ' } \\left ( F ( z , w ) , F ( z , w ) \\right ) + \\displaystyle \\sum _ { k \\geq 3 } { \\varphi ' } _ { k } ^ { \\left ( l ' \\right ) } \\left ( F ( z , w ) , \\overline { F ( z , w ) } \\right ) , \\quad \\mbox { f o r a l l $ l ' = 1 , \\dots , N ' $ . } \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} \\langle m _ { d + k , k } , b _ { d + q , q } \\rangle = \\begin{cases} \\sqrt { q ! \\ , ( d + q ) ! } , & k = q ; \\\\ [ 0 . 5 e x ] 0 , & k < q . \\end{cases} \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} \\{ ( x , \\ ~ a _ i , \\ ~ b _ j ) _ { i , j } \\in X \\times \\prod _ i \\ ~ A _ i \\times \\prod _ j \\ ~ B _ j \\mid \\bigwedge _ i ( f _ i ( a _ i ) \\le x ) \\wedge \\bigwedge _ j ( x \\le g _ j ( b _ j ) ) \\} _ { \\ ! { \\kappa B o r L o c } _ \\kappa } = \\emptyset . \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} \\begin{aligned} \\bigg ( \\frac { f ( x ) } { \\psi _ \\gamma ( x ) } & - \\frac { f ( y ) } { \\psi _ \\gamma ( y ) } \\bigg ) ( g ( x ) - g ( y ) ) \\psi _ \\gamma ( x ) \\psi _ \\gamma ( y ) \\\\ & = ( f ( x ) - f ( y ) ) \\cdot ( g ( x ) \\psi _ \\gamma ( x ) ) - ( f ( x ) - f ( y ) ) \\cdot ( g ( y ) \\psi _ \\gamma ( y ) ) \\\\ & + ( f ( x ) g ( x ) - f ( y ) g ( y ) ) \\cdot \\psi _ \\gamma ( y ) - ( f ( x ) g ( x ) - f ( y ) g ( y ) ) \\cdot \\psi _ \\gamma ( x ) . \\end{aligned} \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} \\begin{cases} \\big ( a ( t ) \\theta + d d ^ c \\phi _ t \\big ) ^ n = e ^ { \\partial _ t \\phi _ t + \\phi _ t } \\mu \\\\ \\phi ( 0 , x ) = \\phi _ 0 \\end{cases} \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} \\| \\nabla u \\| ^ 2 _ { L ^ 2 ( B _ 1 ( 0 ) ) } = \\int _ 0 ^ 1 r ^ { n - 1 } \\ d r \\int _ S | \\partial _ r u | ^ 2 + \\frac { 1 } { r ^ 2 } | \\nabla _ S u | ^ 2 \\ d \\omega = + \\infty \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} \\begin{aligned} \\log \\det \\Delta _ \\beta = & - \\frac { \\log ( \\beta + 1 ) } { 6 ( \\beta + 1 ) } + \\left ( \\frac 1 6 \\log ( 8 \\pi ) - 4 \\log A \\right ) \\frac 1 { \\beta + 1 } \\\\ & \\qquad \\quad \\ - \\log ( \\beta + 1 ) - \\log 2 + O ( \\beta + 1 ) \\quad \\quad \\beta \\to - 1 ^ + , \\end{aligned} \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} \\chi ^ + _ 1 = - \\int _ r ^ \\infty \\frac { 1 } { s { U ^ + ( s ) } ^ 2 } \\mathrm d s \\int _ 0 ^ s { \\mathfrak h } _ 1 ^ + ( t ) { U ^ + ( t ) } ^ 2 \\ , \\mathrm d t , \\\\ [ 1 m m ] \\chi ^ - _ 1 = - \\int _ r ^ \\infty \\frac { 1 } { s { U ^ - ( s ) } ^ 2 } \\mathrm d s \\int _ 0 ^ s { \\mathfrak h } _ 1 ^ - ( t ) { U ^ - ( t ) } ^ 2 \\ , \\mathrm d t , \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{gather*} x _ r x _ t = x _ t x _ r , x _ r c _ t = c _ t x _ r , c _ r c _ t = c _ t c _ r , c _ r ^ 2 = 0 ; \\\\ \\tau _ k ^ 2 = 1 , \\tau _ k \\tau _ { k + 1 } \\tau _ k = \\tau _ { k + 1 } \\tau _ k \\tau _ { k + 1 } , \\tau _ k \\tau _ l = \\tau _ l \\tau _ k \\mbox { i f } | l - k | > 1 ; \\\\ \\tau _ k c _ k = c _ { k + 1 } \\tau _ k , \\qquad \\tau _ k c _ r = c _ r \\tau _ k \\mbox { i f } r \\ne k , k + 1 ; \\\\ \\tau _ k x _ k - x _ { k + 1 } \\tau _ k = - c _ k - c _ { k + 1 } = x _ { k } \\tau _ k - \\tau _ k x _ { k + 1 } , \\tau _ k x _ r = x _ r \\tau _ k \\mbox { i f } r \\ne k , k + 1 . \\end{gather*}"}
-{"id": "931.png", "formula": "\\begin{align*} \\ 0 \\leq \\xi ( x ) \\leq 1 , \\ \\ \\ \\xi ( x ) = 1 \\ { \\rm f o r } \\ | x | < \\tilde { \\epsilon } , \\ \\xi ( x ) = 0 \\ { \\rm f o r } \\ | x | > 2 \\tilde { \\epsilon } , \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} \\begin{aligned} ( \\Phi ' _ { 1 , - 1 } \\circ \\rho ) ( L _ { - 1 } ) & \\ , = \\ , - \\partial + z , \\\\ ( \\Phi ' _ { 1 , - 1 } \\circ \\rho ) ( L _ { 0 } ) & \\ , = \\ , - \\partial ^ 2 + z \\partial + \\frac { 1 - d } 2 , \\\\ ( \\Phi ' _ { 1 , - 1 } \\circ \\rho ) ( L _ { 1 } ) & \\ , = \\ , - \\partial ^ 3 + z \\partial ^ 2 + ( 1 - d ) \\partial . \\end{aligned} \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} I \\ge c _ 2 : = \\inf \\left \\{ \\delta _ 1 ^ { \\frac 1 n - 1 } s \\exp ( - \\delta _ 1 c _ 1 / s ) | \\delta _ 2 / 2 \\le s \\le \\delta _ 1 \\int _ X \\omega _ 0 ^ n \\right \\} , \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} P _ { } ^ e ( \\gamma _ { } ) = \\frac { 1 } { 2 } ( \\sqrt { \\gamma _ { } } ) , \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} f = \\sum _ { j = 1 } ^ { \\infty } \\hat a _ j \\hat u _ { j , \\lambda } \\ , , \\ \\ \\ g = \\sum _ { j = 1 } ^ { \\infty } \\hat b _ j \\hat v _ { j , \\mu } \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} \\lim _ { | t | \\to + \\infty } \\frac { | f ( x , t ) | } { \\exp ( \\beta | t | ^ { \\frac { n } { n - 1 } + | x | ^ \\alpha } ) } = + \\infty \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} g _ { 5 , 3 } ^ { - 1 } ( ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { 2 } , { 3 } , 7 , { 3 } , { \\bf 4 } , 9 , 9 , { \\bf 4 } ) ) = ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { 2 } , { 3 } , 7 , { 3 } , { 4 } , { \\bf 4 } , { \\bf 4 } ) . \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} m _ { i j } \\leq 0 i \\neq j \\sum \\limits _ { k = 0 } ^ { r } m _ { k j } \\geq 0 , \\sum \\limits _ { k = 0 } ^ { r } m _ { i k } \\geq 0 \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\mathrm { g r a d } _ r L = R ^ { - 1 } \\nabla L ^ T \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( t ) < \\beta \\ | \\ V ( 0 ) = c _ 1 , \\ N ( t ) = 3 \\} = \\frac { \\beta ^ 3 } { ( c _ 1 t ) ^ 3 } + \\frac { 3 \\beta ( c _ 1 t - \\beta ) ( c _ 2 t + \\beta ) } { ( c _ 1 + c _ 2 ) \\ , c _ 1 ^ 2 \\ , t ^ 3 } \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} \\boldsymbol \\xi ( t , \\cdot ) \\leq \\sup \\boldsymbol \\xi ( 0 ) = 0 . \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} a _ { 1 2 } = b _ { 1 2 } = 0 , a _ { 1 1 } = b _ { 1 1 } , c = a _ { 2 2 } = b _ { 2 2 } . \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} c _ 1 ( \\mathcal { O } _ { \\mathbb { P } ( F ^ * ) } ( 1 ) , h ^ { \\mathcal { O } } ) _ { H } = \\frac { 1 } { p } \\pi ^ * c _ 1 ( L _ H , h ^ { L _ H } ) . \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} \\partial _ t h + v \\cdot \\nabla _ { \\ ! x } h = \\bar { A } _ g ^ { \\theta } h . \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} f _ J ( 0 , 0 ; 0 ) = g _ J ( 0 , 0 ; 0 ) = 0 , \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} & \\| T _ { \\sigma _ { A , \\Phi } } ( f , g ) \\| _ { ( L ^ 2 , \\ell ^ { \\infty } ) } \\ge \\| T _ { \\sigma _ { A , \\Phi } } ( f , g ) \\| _ { L ^ 2 ( Q ) } \\gtrsim \\bigg | \\int _ { Q } T _ { \\sigma _ { A , \\Phi } } ( f , g ) ( x ) h ( x ) \\ , d x \\bigg | \\\\ & = \\bigg | \\sum _ { \\mu , \\nu \\in \\Z ^ n } a _ { \\mu , \\nu } F ( \\mu ) G ( \\nu ) H ( \\mu + \\nu ) \\bigg | , \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} L ( H , x ) = \\min \\left \\{ \\bigcup _ { k = 0 } ^ x L ( H _ 1 , k ) \\odot L ( H _ 2 , x - k ) \\right \\} x = 0 , 1 , \\ldots , B , \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} \\sup _ { Q _ T \\cup \\partial _ p Q _ T } ( u ^ * ) ^ + = \\sup _ { \\partial _ p Q _ T } ( u ^ * ) ^ + \\quad \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} s c a l _ g = \\kappa n ( n - 1 ) h ^ 2 + ( n - 1 ) \\Big [ 2 h ( \\Delta h ) _ { g _ \\kappa } - n \\| ( \\nabla h ) _ { g _ \\kappa } \\| ^ 2 \\Big ] , \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} u = g \\quad \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { n } } d \\mathbb { Q } _ { n , \\sigma } = \\exp \\left \\{ \\frac { \\beta J } { n } \\left ( \\sum _ { i = 1 } ^ { n } \\sigma _ { i } \\right ) ^ 2 \\right \\} . \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} \\mathbb { L } : = \\Big \\{ p \\in \\mathcal { P } : \\sum \\limits _ { x \\in \\mathbb { S } } p ( x ) f _ i ( x ) = a _ i , r \\in \\{ 1 , \\ldots , k \\} \\Big \\} . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} \\bar { Y } : = \\left \\{ \\bar { y } : \\mathbb { T } ^ 1 \\to \\mathbb { R } ; \\ \\bar { y } \\left ( \\frac { l - 1 } { M } , \\frac { l } { M } \\right ] l = 1 , \\cdots , M , m \\right \\} . \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} \\left \\langle \\bar f ^ R ( p , u ) , \\bar \\nabla { \\rm d i s t } _ N \\left ( p \\right ) \\right \\rangle = 0 , \\ \\ \\forall \\ p \\in B ( N , 3 \\delta _ 0 ) , u \\in \\R ^ { m L } , \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} \\Lambda _ { 0 , p } ( \\mu ) & = \\int _ { \\overline B _ { p } } \\mu ( d y ) , \\\\ \\Lambda _ { 0 , p , q } ( \\mu ) & = \\int _ { D _ { p , q } } \\mu ( d y ) . \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} \\int _ { \\mathbb T ^ { d } _ { \\ell } } S ^ 0 _ { \\ell , \\iota } | y | ^ 2 & = \\int _ { \\mathbb T ^ d _ { \\ell } } | [ \\sqrt { R _ 0 } \\chi _ \\ell ] \\ast \\zeta _ { \\iota } | ^ 2 | y | ^ 2 \\leq \\int _ { \\mathbb T ^ { d } _ { \\ell } } [ | \\sqrt { R _ 0 } \\chi _ { \\ell } | ^ 2 * \\zeta _ { \\iota } ] | y | ^ 2 \\\\ & \\leq \\int _ { \\mathbb T ^ { d } _ { \\ell } } | \\sqrt { R _ 0 } \\chi _ { \\ell } | ^ 2 ( ( 1 + \\iota ) | y | ^ 2 + C \\iota ) , \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} \\mathcal Q _ { \\sigma } ( u , \\varphi ) = \\xi _ j ( \\gamma _ 0 ( u ) , \\gamma _ 0 ( \\varphi ) ) _ { \\partial \\Omega } \\ , , \\ \\ \\ \\forall \\varphi \\in H ^ 2 ( \\Omega ) , u \\in U _ { \\xi _ j } , \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} \\mathcal H _ \\lambda ( u ) = \\sum _ { n = 0 } ^ \\infty \\frac { | \\langle 1 | f _ n \\rangle | ^ 2 } { \\lambda _ n + \\lambda } \\ . \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} A = ( w , \\overline { w } ) \\begin{pmatrix} \\alpha _ + ^ { - 1 } & 0 \\\\ 0 & \\alpha _ - ^ { - 1 } \\end{pmatrix} ( w , \\overline { w } ) ^ { - 1 } \\in \\mathbb { R } ^ { 2 \\times 2 } . \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} \\gamma ^ \\circ = \\sum _ { g \\in \\Gamma } g \\otimes \\gamma ^ g \\in H _ * ( Y ) , \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathcal { R } _ { 2 , a } ( t _ n , \\xi ) = \\int _ 0 ^ \\xi e ^ { ( \\xi - s ) \\mathcal { L } } f \\left ( u ( t _ n + s ) , \\overline u ( t _ n + s ) \\right ) d s - \\xi f ( u ( t _ n ) , \\overline u ( t _ n ) ) \\\\ & \\Psi _ { \\xi } ( v , w ) = \\mathcal { B } \\left ( F \\left ( v \\right ) \\cdot e ^ { \\xi \\mathcal { A } } \\mathcal { C } [ G , \\mathcal { A } ] \\left ( w \\right ) \\right ) , \\ , \\xi \\in \\mathbb { R } . \\end{aligned} \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} f ( B ) : = f ^ { \\ @ B _ \\infty } ( B ) : = \\min \\{ C \\in \\ @ B _ \\infty ( Y ) \\mid B \\le f ^ * ( C ) \\} , . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} \\mathcal L _ { \\tilde X } ( \\bar x ) : = \\left \\{ d \\in \\R ^ n \\ , \\middle | \\ , \\begin{aligned} \\nabla g _ i ( \\bar x ) \\cdot d & \\ , \\leq \\ , 0 & & i \\in I ^ g ( \\bar x ) \\\\ \\nabla h _ j ( \\bar x ) \\cdot d & \\ , = \\ , 0 & & j \\in \\mathcal P \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} \\beta C _ { n , 1 } + \\sum _ { k = 2 } ^ { m _ { n } } \\frac { 2 \\mu _ { k } \\left ( C _ { n , k } - ( n - 1 ) \\mathbb { I } _ { k = 2 } \\right ) - \\mu _ { k } ^ 2 } { 4 k } \\stackrel { d } { \\to } N \\left ( \\beta ^ 2 + \\frac { 1 } { 4 } \\log ( 1 - 4 \\beta ^ 2 ) , - \\beta ^ 2 - \\frac { 1 } { 2 } \\log ( 1 - 4 \\beta ^ 2 ) \\right ) . \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} \\int _ 2 ^ \\infty \\frac { e ^ { - i \\xi x } } { x } d x = \\int _ { 2 \\xi } ^ \\infty \\frac { e ^ { - i u } } { u } d u = \\int _ { 2 \\xi } ^ 1 \\frac { e ^ { - i u } } { u } d u + \\int _ { 1 } ^ \\infty \\frac { e ^ { - i u } } { u } d u = O ( | \\log \\xi | + 1 ) \\ , . \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} \\phi ( x ) & = \\sum _ { k = 0 } ^ \\infty \\alpha _ k \\left ( - \\partial _ \\mu \\partial ^ \\mu \\right ) ^ k \\left ( \\sum _ { j = 0 } ^ \\infty a _ j \\rho ^ { j + 1 } ( x ) \\right ) \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} \\| u \\| _ { \\lambda , N } ^ 2 = \\mathcal { Q } _ { \\lambda , N } ( u , u ) + b \\| \\gamma _ 0 ( u ) \\| _ { L ^ 2 ( \\Omega ) } ^ 2 , \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} d _ { \\mathbf { m } } \\frac { 1 } { \\left ( \\frac { n } { r } \\right ) _ { \\mathbf { m } } } = \\frac { P _ { \\mathbf { m } } \\left ( \\mathbf { 1 } ; \\frac { d } { 2 } \\right ) } { P _ { \\mathbf { m } } ^ { \\mathrm { i p } } \\left ( \\mathbf { m } + \\frac { d } { 2 } \\delta ; \\frac { d } { 2 } \\right ) } . \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} a _ 1 c _ 2 = a _ 2 c _ 1 , b _ 1 d _ 2 = b _ 2 d _ 1 , a _ 1 d _ 2 = a _ 2 d _ 1 , b _ 1 c _ 2 = b _ 2 c _ 1 . \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} \\alpha _ q ( \\eta ) & \\leq \\limsup _ { t \\rightarrow \\infty } \\frac { \\hat { m } ( N _ t - N _ { z ( t ) } ) } { D _ t } = \\limsup _ { t \\rightarrow \\infty } \\frac { N _ t - N _ { z ( t ) } } { N _ t } \\\\ & = 1 - \\liminf _ { t \\rightarrow \\infty } \\frac { N _ { z ( t ) } } { N _ t } \\leq 1 - \\eta \\Big ( 1 - \\frac { 1 } { n ^ * q ^ { \\hat { m } } } \\Big ) ^ { - 1 } , \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} \\int \\xi ( x ) \\mu _ { N , m } ( d x | y ) = \\int \\left ( \\int \\xi ( x ^ { \\Lambda _ 1 } , x ^ { \\Lambda _ 2 } ) \\mu _ { N , m } ( d x ^ { \\Lambda _ 1 } | x ^ { \\Lambda _ 2 } , y ) \\right ) \\bar { \\mu } _ { N , m } ( d x ^ { \\Lambda _ 2 } ) . \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} k _ u = A _ u ^ { - 1 } \\left ( A _ u k _ u \\right ) = A _ u ^ { - 1 } \\left ( k _ u \\circ ( A _ c + r ) - g _ u \\circ K { } { } \\right ) . \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} & n _ { t } + \\nabla \\cdot ( n \\mathbf { v } ) = 0 , \\\\ & \\mathbf { v } _ { t } + ( \\mathbf { v } \\cdot \\nabla ) \\mathbf { v } = n \\lim _ { \\epsilon \\to 0 } \\frac { ( Q ^ \\epsilon \\ast \\mathbf { v } - \\mathbf { v } ) } { \\epsilon } . \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} Q ( B \\ltimes A _ 1 , B \\ltimes A _ 2 ) \\ = \\ Q ( A _ 1 , A _ 2 ) \\cdot | B | . \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} f ( 0 , 0 ) & = 0 , & \\frac { \\partial f } { \\partial y } ( 0 , 0 ) & > 0 , & g ( 0 , 0 ) & < 0 . \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} a \\sim \\sum _ { j = - m } ^ \\infty a _ j a _ j ( s x , s \\xi ) = s ^ { - j } a _ j ( x , \\xi ) s > 0 , ( x , \\xi ) \\ne ( 0 , 0 ) \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} \\frac { \\partial \\lambda ^ { \\perp } } { \\partial t } = - ( Q _ { 0 } ^ { \\perp } + 2 P _ { 0 } \\lambda ^ { \\perp } ) \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} { S } ( \\mu ) = ( \\Phi _ i ( \\mu ) \\Phi _ j ( \\mu ) ) _ { i , j = 1 , \\ldots , m } \\mbox { a n d } { s } ( \\mu ) = ( \\Phi _ i ( \\mu ) ) _ { i = 1 , \\ldots , m } . \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} & \\int _ { \\mathbb R ^ 2 } \\sum _ { k \\in \\Bbb Z } 2 ^ { 2 k } \\vert \\psi _ { j , k } * f ( y _ 1 , y _ 2 , y _ 3 ) \\vert ^ 2 w ( R _ { 2 ^ { - j } , 2 ^ { - k } } ( y _ 1 , y _ 2 , y _ 3 ) ) { d y _ 2 \\ , d y _ 3 } \\\\ & = \\int _ { \\mathbb R ^ 2 } \\sum _ { k \\in \\Bbb Z } 2 ^ { 2 k } \\big \\vert \\big ( f \\ast _ 1 \\psi ^ { ( 1 ) } _ { j } ( y _ 1 , \\cdot , \\cdot ) \\big ) \\ast _ { 2 , 3 } \\psi ^ { ( 2 ) } _ { j , k } ( y _ 2 , y _ 3 ) \\big \\vert ^ 2 w _ { y _ 1 , 2 ^ { - j } } ( R _ { 2 ^ { - k } , 2 ^ { - j - k } } ( y _ 2 , y _ 3 ) ) { d y _ 2 \\ , d y _ 3 } . \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} \\Theta ( \\l , K ) = \\left ( \\begin{array} { c c c c c } \\l & 0 & 0 & \\cdots & 0 \\\\ 1 & \\l & 0 & \\cdots & 0 \\\\ 0 & 2 & \\l & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & K - 1 & \\l \\end{array} \\right ) , \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) \\le \\beta \\ | \\ V ( 0 ) = - c _ 2 \\} > P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 \\} \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} f _ N ( m ) d m = \\frac { 1 } { Z _ N } e ^ { - N \\bar { H } _ N ( m ) } d m . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} \\mathfrak { a } \\left ( \\mathbb { K } _ { n } ^ { ( j ) } ( x ) \\right ) = \\Lambda _ { n } ^ { \\left ( j \\right ) } \\left ( x ; 1 , 1 \\right ) \\mathbb { K } _ { n - 1 } ^ { ( j ) } ( x ) , \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} \\left \\{ ( s _ j ) _ { j = 1 } ^ { \\infty } \\in \\mathbb R ^ { \\infty } : ( j ^ { \\frac { 1 } { 2 ( N - 1 ) } } s _ j ) _ { j = 1 } ^ { \\infty } \\in l ^ 2 \\right \\} . \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} t ^ I = X ^ I / X ^ 0 \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} \\rho : = \\frac { 1 } { 2 } \\left ( \\rho _ 1 + \\rho _ 2 + \\frac { \\kappa ^ 2 } { \\rho _ 1 } - \\sqrt { \\left ( \\rho _ 1 + \\rho _ 2 + \\frac { \\kappa ^ 2 } { \\rho _ 1 } \\right ) ^ 2 - 4 \\rho _ 1 \\rho _ 2 } \\right ) > 0 . \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} \\nu = \\sum _ { A \\subset \\{ 0 , \\ldots , n - 1 \\} } ( 2 / 3 ) ^ { | A | } ( 1 / 3 ) ^ { n - | A | } \\nu _ A . \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ \\infty \\left ( { a \\atop b + \\alpha n } \\right ) e ^ { i \\theta n } = \\int _ { - \\infty } ^ \\infty \\left ( { a \\atop b + \\alpha x } \\right ) e ^ { i \\theta x } \\ , d x , ~ 0 < \\alpha < 1 . \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} & a ^ I = a + \\sum _ { k = 1 } ^ n i _ k ( 1 - c _ k ) , b ^ I = b + \\sum _ { k = 1 } ^ n i _ k ( 1 - c _ k ) , \\\\ & c ^ I = ( c _ 1 + 2 i _ 1 ( 1 - c _ 1 ) , \\dots , c _ n + 2 i _ n ( 1 - c _ n ) ) . \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\psi ( a _ { \\Lambda } ) \\psi ( J _ t ( b _ { \\Lambda ^ { ' } } ) ) = \\psi ( a _ { \\Lambda } ) \\alpha ^ { ( | \\Lambda ^ { ' } | ) } ( J ^ { - 1 } _ { \\Lambda ^ { ' } } ( b _ { \\Lambda ^ { ' } } ) ) . \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\left \\{ ( 2 l + 1 ) \\abs { \\lambda } \\ , \\colon \\ , l = 1 , 2 , \\ldots \\right \\} . \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} & ( \\xi _ 1 ' , \\eta _ 1 ' ) + ( \\xi _ 2 ' , \\eta _ 2 ' ) = ( \\xi ' , \\eta ' ) , \\\\ \\lambda _ 1 ' = \\phi _ { \\tilde { c _ 1 } } ( \\xi _ 1 ' , \\eta _ 1 ' ) \\in S _ 1 , \\ \\ & \\lambda _ 2 ' = \\phi _ { \\tilde { c _ 2 } } ( \\xi _ 2 ' , \\eta _ 2 ' ) \\in S _ 2 , \\ \\ \\lambda _ 3 ' = ( \\psi ( \\xi ' , \\eta ' ) , \\xi ' , \\eta ' ) \\in S _ 3 . \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} L _ { 0 } = L > 2 ^ { d } L _ { k + 1 } = L _ { k } ^ { d + 7 } . \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} \\omega _ H = \\pi ^ * \\big ( g \\cdot \\pi _ * ( \\omega ^ { \\wedge ( \\dim X + 1 ) } ) \\big ) . \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} & \\frac { 1 } { \\mu ( B ) } \\int _ { B } \\abs { \\varphi ( x ) - \\avg { \\varphi } _ { B } } \\ , d \\mu \\\\ & = \\frac { 1 } { \\mu ( B ) } \\left ( \\int _ { B _ + } \\big ( \\varphi ( x ) - \\avg { \\varphi } _ { B } \\big ) \\ , d \\mu + \\int _ { B _ - } - \\big ( \\varphi ( x ) - \\avg { \\varphi } _ { B } \\big ) \\ , d \\mu \\right ) . \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} ( \\prod _ { s = i _ { 2 } } ^ { j _ { 2 } } x _ { s } ) ^ { m _ { i _ { 2 } , j _ { 2 } + 1 } } \\cdots ( \\prod _ { s = i _ { 1 } } ^ { n } x _ { s } ) ^ { m _ { i _ { 1 } , n + 1 } } = q ^ { - ( j _ { 2 } - i _ { 1 } ) m _ { i _ { 1 } , n + 1 } m _ { i _ { 2 } , j _ { 2 } + 1 } } \\cdots , \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { 2 \\left ( \\frac { b _ 1 } { m b _ 2 } + \\frac 1 m \\right ) \\binom { b _ { 2 } + b _ 3 } { b _ 3 } } \\right ) ^ { \\frac 1 3 } \\le f ( m , H ( \\vec { \\bf b } ) ) \\le \\frac { 6 m b _ 2 } { b _ 1 } + 2 . \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} d ( 1 ) = \\mu _ n ( c ( 1 ) , \\dots , c ( n + 1 ) ) = c ( n + 1 ) + \\nu _ n ( c ( 1 ) , \\dots , c ( n ) ) . \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} \\underset { \\Omega - \\Omega _ { \\delta } } { i n f } \\sigma \\leq \\sigma ( \\omega _ f ) = \\underset { n \\rightarrow + \\infty } { l i m } \\sigma _ n ( \\omega _ f ) \\leq t _ 0 = t _ 0 ( \\Omega _ { \\delta } ) \\leq T . \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} h _ s = c e , \\exists c \\in { \\mathbb R } \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} \\mathcal { C } _ { J , K } ^ { r , a l t } : = \\left \\{ \\eta \\in \\mathcal { C } _ { J , K } : \\ : \\max _ { i \\in \\{ 0 , 1 \\} } \\liminf _ { n \\to - \\infty } \\mathbf { 1 } _ { \\{ \\eta _ { 2 n + i } \\ge K - r , \\ : \\eta _ { 2 n + 1 + i } \\le J - K + r \\} } = 1 \\right \\} \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} T ( x ) = b , \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} R _ 2 = D a ^ { - 1 } 2 c _ 2 k _ 0 R \\| u \\| _ \\infty . \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} H ^ { ( 1 ) } ( \\xi _ { 0 } ) = - 2 \\overline { \\xi _ { 0 } } \\int _ { \\mathbb { T } } \\frac { d \\sigma ( t ) } { \\left | \\xi _ { 0 } - t \\right | ^ { 2 } } . \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} J ^ a ( z ) = { \\pi } _ 1 ( x _ a ) ( z ) + \\sum _ { \\beta , \\gamma \\in \\Delta _ + } c _ { a , \\beta } ^ \\gamma : \\psi _ { \\gamma } ( z ) \\psi _ { \\beta } ^ * ( z ) : \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} \\begin{cases} c _ 1 T _ 1 < \\beta \\\\ c _ 1 T _ 1 - c _ 2 ( T _ 2 - T _ 1 ) + c _ 1 ( T _ 3 - T _ 2 ) < \\beta \\\\ \\cdot \\\\ c _ 1 T _ 1 - c _ 2 ( T _ 2 - T _ 1 ) + \\dots + c _ 1 ( t - T _ { 2 k } ) < \\beta \\end{cases} \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} \\ell _ { t } = \\mathbb { E } _ { Q _ { t } } [ \\ell \\left ( Q _ t \\right ) ] . \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} W _ D [ G _ D f ] = W _ D [ P _ D \\lambda ] = 0 . \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} \\omega _ r \\subset \\omega _ { r - 1 } \\subset \\ldots \\subset \\omega _ 0 = ( \\pi _ 1 \\times \\pi _ 2 ) _ { N _ j } \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} e _ R ( \\mathcal I ( 1 ) \\mathcal I ( 2 ) ; R ) ^ { \\frac { 1 } { 2 } } = e _ R ( \\mathcal I ( 1 ) ; R ) ^ { \\frac { 1 } { 2 } } + e _ R ( \\mathcal I ( 2 ) ; R ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} H _ { g + 2 - k } M = 2 ^ { g + 1 - k } 2 ^ { k + 1 } = 2 ^ { g + 2 } = 2 ( 2 ^ { g + 1 } ) = 2 H _ { g + 2 } . \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} \\Omega _ { \\alpha } = \\{ ( x , x ^ { \\prime } ) \\in \\R \\times \\R ^ { d - 1 } : x > 0 , \\ | x ^ { \\prime } | < c e ^ { - \\alpha x } \\} , \\alpha > 0 , \\ c > 0 . \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} \\hat { h } _ s ( z , s ) = \\hat { h } _ s ( - z , s ) , \\ \\hat { h } _ { s z } ( z , s ) = - \\hat { h } _ { s z } ( - z , s ) , \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} ( \\mu _ \\varphi ) ^ { \\ast ^ { G / H } } = \\mu _ { \\varphi ^ { \\ast ^ { G / H } } } . \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} \\begin{pmatrix} \\mbox { A u x } \\Lambda & \\mbox { O } _ { N ^ { p - 1 } } & \\dots & \\mbox { O } _ { N ^ { p - 1 } } \\\\ \\mbox { O } _ { N ^ { p - 1 } } & \\mbox { A u x } \\Lambda & \\dots & \\mbox { O } _ { N ^ { p - 1 } } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\mbox { O } _ { N ^ { p - 1 } } & \\mbox { O } _ { N ^ { p - 1 } } & \\dots & \\mbox { A u x } \\Lambda \\end{pmatrix} \\tilde { Z } \\left [ J \\right ] . \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} \\frac { \\left ( \\det H \\right ) ^ \\frac { 1 } { 2 } } { \\det K } = \\frac { \\left ( \\det H \\right ) ^ \\frac { 1 } { 2 } } { \\det \\left ( 2 I - \\frac { 1 } { 2 } H - \\frac { 1 } { 2 } J H J \\right ) } < \\left ( \\frac { 1 } { 2 \\ , n } \\right ) ^ n . \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} C _ { \\rm l o w } = - \\frac { 1 } { 2 } \\ln \\left ( { 2 \\pi e { \\sigma ^ 2 } } \\right ) + 1 + m + n \\xi P + { f _ { { \\rm { l o w } } } } \\left ( { \\xi P } \\right ) , \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} P _ { A , l } ( z ) = \\sum _ { \\alpha \\in \\Lambda _ { A , l } } x _ \\alpha z ^ \\alpha = \\sum _ { \\gamma \\in \\Gamma _ { A , l } } \\sum _ { \\beta \\in B _ { A , l } } x _ { 2 \\beta + 2 \\gamma + 1 _ A } z ^ { 2 \\beta + 2 \\gamma + 1 _ A } = \\sum _ { \\gamma \\in \\Gamma _ { A , l } } \\left ( \\sum _ { \\beta \\in B _ { A , l } } x _ { 2 \\beta + 2 \\gamma + 1 _ A } z ^ { 2 \\beta } \\right ) z ^ { 2 \\gamma + 1 _ A } . \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} V _ p ( G ) _ 0 = V _ p ( G ^ \\circ ) , \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} r \\left ( \\frac { - 1 + ( s - t ) ^ { 2 } - r ^ { 2 } } { \\sqrt { ( s - t ) ^ { 4 } - 2 ( s - t ) ^ { 2 } ( - 1 + r ^ { 2 } ) + ( 1 + r ^ { 2 } ) ^ { 2 } } } \\right ) = r \\left ( 1 + E _ { \\partial _ { r } v _ { 3 } , 1 } \\right ) \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} x = ( ( a _ 1 a _ 1 ^ * ) ^ \\frac { 1 } { 2 } , \\dots , ( a _ n a _ n ^ * ) ^ \\frac { 1 } { 2 } ) , y = ( 1 , \\dots , 1 ) \\in \\mathcal { A } ^ n . \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} H _ { \\mu _ { * } } ( z ) = 1 + 2 \\psi _ { \\mu } ( z ) = \\frac { 1 + \\eta _ { \\mu } ( z ) } { 1 - \\eta _ { \\mu } ( z ) } , z \\in \\mathbb { D } . \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} \\Delta _ { q _ 1 } ( f , < q _ 1 > _ f ) = \\left ( \\frac { \\theta ^ 2 } { 4 \\lambda ^ 2 b } + \\frac { a ^ 2 E ^ 2 } { 8 } \\right ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} 2 s _ 1 & = y + 2 a x \\\\ 2 s _ 2 & = x + z \\\\ 2 \\lambda _ 2 s _ 2 & = y \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} ( q ; q ) _ \\infty ( - q ; q ^ 2 ) _ \\infty = \\dfrac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( - q ^ 2 ; q ^ 2 ) _ \\infty } = \\sum _ { m = - \\infty } ^ \\infty ( - 1 ) ^ m ( q ^ 2 ) ^ { m ^ 2 } = \\sum _ { m = - \\infty } ^ \\infty ( - 1 ) ^ m q ^ { 2 m ^ 2 } , \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} \\zeta _ n ( u ( t ) ) = \\zeta _ n ( u _ 0 ) \\ , e ^ { i \\omega _ n ( u _ 0 ) t } \\ , t \\in \\R . \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} L _ { \\phi , \\psi } G _ { u , v } g = g , g \\in L ^ 2 ] a , b [ . \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } ( t ) \\\\ \\dot { y } ( t ) \\end{bmatrix} = \\begin{cases} F _ L ( x ( t ) , y ( t ) ) , & x ( t - \\mu ) < 0 , \\\\ F _ R ( x ( t ) , y ( t ) ) , & x ( t - \\mu ) > 0 , \\end{cases} \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} V ( \\hat { f _ 1 } ^ h , \\ldots , \\hat { f _ n } ^ h ) \\cap & \\{ X _ 0 \\neq 0 \\} \\subset V ( \\hat { f _ 1 } , \\ldots , \\hat { f _ n } ) , \\\\ V ( \\hat { f _ 1 } ^ h , \\ldots , \\hat { f _ n } ^ h ) \\cap & \\{ X _ 0 = 0 \\} \\subset V ( X _ { i _ 1 } , \\ldots , X _ { i _ n } ) . \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} i \\partial _ { t } \\hat { u } \\left ( \\xi , t \\right ) + \\hat { A } _ { \\xi } \\hat { u } \\left ( \\xi , t \\right ) = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} & \\mathcal K ( x _ 1 - y _ 1 , x _ 2 - y _ 2 , x _ 3 - y _ 3 ) \\\\ & = ( ( x _ 1 - y _ 1 ) \\cdot ( x _ 2 - y _ 2 ) ) \\bigg \\{ \\frac { 1 } { | x _ 1 - y _ 1 | ^ { 2 } | x _ 2 - y _ 2 | ^ { 2 } + | x _ 3 - y _ 3 | ^ 2 } \\bigg \\} \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} u ( t ) = u _ 1 ( t ) + \\mathcal { R } _ { 1 , 0 } ( t , u ) \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} S / J ( D ) \\cong \\bigoplus _ { i = 1 } ^ t S _ { m _ i } / \\Gamma ( X _ i , \\mathcal O _ { X _ i } ( - D _ i ) ) \\cong \\bigoplus _ { i = 1 } ^ t S _ { m _ i } / J ( D _ i ) \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} \\left \\| \\Lambda _ { 1 } f _ n ( t ) \\right \\| \\leq \\left \\| \\Lambda _ { 1 } f _ { 0 , n } \\right \\| \\leq \\left \\| \\Lambda _ { 1 } f _ 0 \\right \\| , \\forall t \\geq 0 , n = 2 , 3 , . . . . \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} f & = ( f _ 0 + f _ 1 + f _ 2 + f _ 3 ) ^ 4 + ( f _ 1 + f _ 3 ) ^ 4 ( g + 1 ) + ( f _ 2 + f _ 3 ) ^ 4 ( g + 1 ) ^ 2 + f _ 3 ^ 4 ( g + 1 ) ^ 3 \\\\ & = f _ 0 ^ 4 + ( f _ 3 g ) ^ 4 g ^ { - 1 } + ( f _ 2 g ) ^ 4 g ^ { - 2 } + ( f _ 1 g ) ^ 4 g ^ { - 3 } \\\\ & = f _ 0 ^ 4 + ( t ^ { - 1 } f _ 1 ) ^ 4 t ^ 4 g + ( t ^ { - 2 } f _ 2 ) ^ 4 ( t ^ 4 g ) ^ 2 + ( t ^ { - 3 } f _ 3 ) ^ 4 ( t ^ 4 g ) ^ 3 . \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} \\bar { L } u ( z ) = \\frac { 1 } { 2 } \\bar { a } ( z ) \\frac { d ^ 2 u } { d z ^ 2 } + \\bar { \\beta } ( z ) \\frac { d u } { d z } . \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} \\int _ { r , \\theta } & ( r + 1 ) ^ { - 2 s } ( | u | ^ 2 + | h u ' | ^ 2 ) \\lesssim \\\\ & \\frac { 1 } { h ^ 2 } \\int _ { r , \\theta } ( r + 1 ) ^ { 2 s } | P ^ { \\pm } _ \\varphi ( h ) u | ^ 2 + \\frac { \\varepsilon } { h } \\int _ { r , \\theta } | u | ^ 2 , h \\in ( 0 , h _ 0 ] . \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} \\max _ { \\lambda \\in \\Lambda } \\sum _ { i = 1 } ^ k [ \\tilde g ( a + b \\sqrt { \\lambda _ i } ) + \\tilde g ( a - b \\sqrt { \\lambda _ i } ) ] = \\tilde g ( a + b ) + \\tilde g ( a - b ) + 2 ( k - 1 ) \\tilde g ( a ) . \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} f \\left ( x _ 1 , \\ldots , x _ d \\right ) \\quad = \\sum _ { i _ 1 = 1 } ^ k \\cdots \\sum _ { i _ d = 1 } ^ k W _ { i _ 1 , \\ldots , i _ d } f _ { 1 , i _ 1 } ( x _ 1 ) \\cdots f _ { d , i _ d } ( x _ d ) . \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} | f ^ { ( k ) } ( z ) | & = \\left | \\sum _ { i = 0 } ^ k \\binom { k } { i } f _ 1 ^ { ( i ) } ( z ) f _ 2 ^ { ( k - i ) } ( z ) \\right | \\\\ & \\leq \\sum _ { i = 0 } ^ k \\binom { k } { i } | f _ 1 ^ { ( i ) } ( z ) | | f _ 2 ^ { ( k - i ) } ( z ) | \\\\ & \\leq \\sum _ { i = 0 } ^ k \\binom { k } { i } C ^ { f _ 1 } _ { m - k + i , i } ( 1 + | z | ) ^ { m - k } M ^ { f _ 2 } _ { k - i } = : C ' _ { m , k } ( 1 + | z | ) ^ { m - k } , \\mbox { f o r a l l } ~ m \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} & L _ { \\sigma _ e ( x ) } \\sigma _ e L _ y = L _ { \\sigma _ e ( y ) } \\sigma _ e L _ x . \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} \\frac { d } { d t } \\| \\partial _ k \\theta ( t ) \\| _ { L ^ p } ^ 2 \\leq C \\| \\nabla u \\| _ { L ^ \\infty } \\| \\nabla \\theta \\| _ { L ^ p } ^ 2 . \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} \\Big ( \\sum _ { n = 1 } ^ \\infty \\| a _ n \\| _ X ^ { q } \\Big ) ^ { 1 / { q } } \\leq C \\| D \\| _ { \\mathcal { H } _ { q ' } ( X ) } ; \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} & - 2 \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } d s \\lambda '' ( s ) ( s - t ) \\left ( \\frac { ( r ^ { 2 } - 1 - ( s - t ) ^ { 2 } ) } { \\sqrt { \\beta } ( 1 + r ^ { 2 } + ( s - t ) ^ { 2 } + \\sqrt { \\beta } ) } \\right ) \\\\ & = \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } d s \\frac { \\lambda '' ( s ) } { ( s - t ) } + E _ { 1 , \\partial _ { r } v _ { 1 } } ( t , r ) \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{gather*} w ( x ) F ( x ) = w ( x ) F [ u ] ( x ) = w ( x ) | h u ' ( x ) | ^ 2 - w ( x ) ( V ( x ) - ( \\varphi ' ( x ) ) ^ 2 - E ) | u ( x ) | ^ 2 , \\\\ u \\in \\mathcal { D } ( P ) \\langle x \\rangle ^ s P u \\in L ^ 2 ( \\R ) . \\end{gather*}"}
-{"id": "6431.png", "formula": "\\begin{align*} y _ 0 = j _ p ( x ) , y _ { 2 n + 1 } = \\Pi _ { M ^ \\perp } ^ { p ^ * } y _ { 2 n } , y _ { 2 n } = \\Pi _ { N ^ \\perp } ^ { p ^ * } y _ { 2 n - 1 } , \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} | \\partial _ { r } N _ { 2 } ( f _ { v _ { 5 } } ) ( t , r ) | & \\leq \\begin{cases} \\frac { C \\log ( r ) } { r ^ { 3 } ( t - r ) ^ { 2 } } , t - \\sqrt { t } \\leq r \\leq t - t ^ { 1 / 4 } t + t ^ { 1 / 4 } \\leq r \\leq t + \\sqrt { t } \\\\ \\frac { C } { r ^ { 7 / 2 } } , t - t ^ { 1 / 4 } \\leq r \\leq t + t ^ { 1 / 4 } \\end{cases} \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} \\tilde { f } _ A ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) & = \\tilde { f } ( \\tau _ 1 , \\xi _ 1 , A ^ { \\frac { 1 } { 2 } } \\eta _ 1 ) , \\\\ \\tilde { g } _ A ( \\tau _ 2 , \\xi _ 2 , \\eta _ 2 ) & = \\tilde { g } ( \\tau _ 2 , \\xi _ 2 , A ^ { \\frac { 1 } { 2 } } \\eta _ 2 ) , \\\\ \\tilde { h } _ A ( \\tau , \\xi , \\eta ) & = \\tilde { h } ( \\tau , \\xi , A ^ { \\frac { 1 } { 2 } } \\eta ) . \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} b _ 0 ( u , u , A _ 0 u ) = 0 , \\forall \\ u \\in D ( A _ 0 ) , \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} \\kappa _ \\sigma = \\frac { 1 } { g ( X _ a , X _ a ) g ( X _ b , X _ b ) - g ( X _ a , X _ b ) ^ 2 } \\sum _ { i , j , k , l = 1 } ^ 3 R _ { i j k l } a _ i a _ k b _ j b _ l , \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} x _ { n + 1 } ^ { - 2 s } w ( x ' , x _ { n + 1 } ) = x _ { n + 1 } ^ { 1 - 2 s } \\int _ { 0 } ^ { 1 } \\partial _ { n + 1 } w ( x ' , t x _ { n + 1 } ) \\ , d t = \\int _ { 0 } ^ { 1 } ( t x _ { n + 1 } ) ^ { 1 - 2 s } \\partial _ { n + 1 } w ( x ' , t x _ { n + 1 } ) t ^ { 2 s - 1 } \\ , d t . \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} A ( s ) : = \\Phi ^ * ( s ) , \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} d _ i = | \\mathcal { B } _ i | = n _ i - n _ { i - 1 } \\geq 1 . \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} D _ { k } ( m ) = D _ { k } ( y , j L _ { k } ) = \\left \\{ \\begin{array} { c } \\xi _ { t } ^ { m } ( x ) > 0 , ( x , t ) \\\\ | | x - y | | _ { \\infty } = L _ { k } ^ { d + 6 } , ( t - j L _ { k } ) \\leq L _ { k } \\end{array} \\right \\} , \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} \\sharp \\mathcal { C } ( u _ 1 , \\dots , u _ m , n ) = \\sharp \\mathcal { D } ( u _ 1 , \\dots , u _ m , n ) \\ , \\cdot \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} [ \\omega , \\phi , \\delta , F ] = \\sum _ { i = 1 } ^ { N } \\left ( \\zeta _ { F , \\omega _ { i } ^ { \\phi } } ^ { R ^ { \\delta } } \\circ \\phi \\right ) \\otimes ( d x _ { 1 } ^ { \\phi } \\wedge \\dots \\widehat { d x _ { i } ^ { \\phi } } \\wedge \\dots d x _ { N } ^ { \\phi } ) ; \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} \\frac { d \\mathbb { \\tilde { Q } } _ { n } } { d \\mathbb { P } _ { n } } \\left | \\mathbb { P } _ { n } \\right . \\stackrel { d } { \\to } \\exp \\left \\{ \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 \\mu _ { k + 1 } Z _ { k } - \\mu _ { k + 1 } ^ 2 } { 4 ( k + 1 ) } \\right \\} \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} \\varphi ( t ) = 1 + \\frac { \\eta } { 2 } + \\frac { c ^ 4 } { c ^ 4 + 1 } ( t - 1 ) , t \\ge 1 . \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} \\begin{aligned} & \\log \\det \\Delta _ \\beta = \\frac { \\log ( - 2 \\beta - 1 ) } { 1 2 ( 2 \\beta + 1 ) } - \\left ( \\frac 1 6 + \\frac { \\log \\pi } { 1 2 } - 2 \\log A \\right ) \\frac 1 { 2 \\beta + 1 } \\\\ & \\ \\ - \\frac 3 4 \\log ( - 2 \\beta - 1 ) - \\frac 1 2 \\log 2 - \\frac 1 4 \\log \\pi + O ( 2 \\beta + 1 ) \\quad \\quad \\beta \\to - 1 / 2 ^ - , \\end{aligned} \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} \\langle [ S , A ] _ { 1 } u , u \\rangle = - \\frac { 1 } { 2 } \\tau ^ { 3 } \\| | x ' | u \\| ^ { 2 } - 2 \\tau \\| \\nabla ' u \\| ^ { 2 } . \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } G ( x , t ) & = - c _ { 1 } \\left ( \\lambda _ 1 + \\frac { \\partial } { \\partial x } \\right ) ^ { \\alpha _ 1 } G ( x , t ) - c _ { 2 } \\left ( \\lambda _ 2 + \\frac { \\partial } { \\partial x } \\right ) ^ { \\alpha _ 2 } G ( x , t ) + \\lambda _ 1 ^ { \\alpha _ 1 } { c _ 1 } G ( x , t ) + \\lambda _ 2 ^ { \\alpha _ 1 } { c _ 2 } G ( x , t ) , \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} F ( X , Y , Z ) = \\alpha \\beta X ^ 2 + & Y ^ 2 + Z ^ 2 + ( \\alpha + \\beta ) X Y - ( \\alpha + \\beta ) X Z - 2 Y Z \\\\ & + ( \\alpha ^ 2 \\beta + \\beta ^ 2 \\alpha ) X + ( \\alpha ^ 2 + \\beta ^ 2 ) Y - 2 \\alpha \\beta Z + \\alpha ^ 2 \\beta ^ 2 . \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} \\lambda '' ( t ) = \\frac { R H S _ { 2 } ( t ) } { g _ { 2 } ( t ) } \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} \\delta ( h _ { i j } ) = \\langle \\delta \\mathbf { N } , \\mathbf { r } _ { i j } \\rangle + \\langle \\mathbf { N } , \\delta \\mathbf { r } _ { i j } \\rangle = g _ { i j } k _ 0 u + u _ { i j } - u h _ { i l } h _ j ^ l , \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} \\mathcal { G } = \\mathcal { A } \\cup \\mathcal { B } , \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} g = G ( d , s , t ) - \\frac { d } { s } \\left ( \\beta - 1 \\right ) + O ( 1 ) , \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{gather*} \\varepsilon ^ { ( m ) } _ k = \\begin{cases} 1 k = ( 0 , \\dots , 0 , \\underset { m } { 1 } , 0 , \\dots , 0 ) , \\\\ 0 k \\neq ( 0 , \\dots , 0 , \\underset { m } { 1 } , 0 , \\dots , 0 ) , \\end{cases} \\end{gather*}"}
-{"id": "5147.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\infty } \\frac { \\lambda ( t ) w ^ { 5 } } { ( 3 6 \\lambda ( t ) ^ { 2 } + w ^ { 2 } ) ^ { 3 } } | \\frac { 1 } { 1 + w ^ { 2 } } - \\frac { 1 } { ( \\lambda ( t ) ^ { 2 - 2 \\alpha } + w ^ { 2 } ) } | d w \\\\ & \\leq \\frac { C \\log ( \\log ( t ) ) } { \\lambda ( t ) } \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} C _ i \\ = \\ & B _ 1 \\ltimes A _ i \\ i \\in J ' , \\\\ C _ j \\ = \\ & B _ 2 \\ltimes A _ j \\ j \\in J , \\\\ \\bar U & \\ = \\ B _ 3 \\ltimes [ N ] , \\\\ \\bar V & \\ = \\ B _ 4 \\ltimes [ N ] . \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} \\Theta _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) \\nabla \\mathbb { K } _ { n } ^ { ( j ) } ( x ) + \\Lambda _ { n } ^ { \\left ( j \\right ) } \\left ( x ; 2 , 1 \\right ) \\mathbb { K } _ { n } ^ { ( j ) } ( x ) = \\Lambda _ { n } ^ { \\left ( j \\right ) } \\left ( x ; 1 , 1 \\right ) \\mathbb { K } _ { n - 1 } ^ { ( j ) } ( x ) . \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} ( - P ) ^ { s } u ( x ' ) = c _ { s } \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { u } ( x ) \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} v _ { 5 , 1 , 2 } ( t , r ) & = \\frac { - 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\\\ & \\int _ { 0 } ^ { 2 \\pi } d \\theta \\left ( N _ { 2 } ( f _ { v _ { 5 } } ^ { \\lambda _ { 1 } } ) - N _ { 2 } ( f _ { v _ { 5 } } ^ { \\lambda _ { 2 } } ) \\right ) ( s , \\sqrt { r ^ { 2 } + \\rho ^ { 2 } + 2 \\rho r \\cos ( \\theta ) } ) \\frac { ( r + \\rho \\cos ( \\theta ) ) } { \\sqrt { r ^ { 2 } + 2 r \\rho \\cos ( \\theta ) + \\rho ^ { 2 } } } \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} Q ( P ( i ) ) - P ( i ) & = | \\{ j \\in J / \\ , P ( j ) > P ( i ) , \\psi ( P ( j ) , P ( i ) ) > 0 \\} | \\\\ & = | \\{ j \\in J / \\ , P ( j ) > P ( i ) , j < i \\} | \\\\ & = | \\{ j \\in J / \\ , j < i , \\phi ( j , i ) < 0 \\} | \\\\ & = i - P ( i ) \\ , \\ , \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} K i r _ { T , S } \\Phi ( x ) = \\psi ( x ) = \\int _ { \\mathbb { R } ^ n } \\frac { \\Phi ( x , y ) - \\Phi ( x , x ) } { \\abs { x - y } ^ { n + 2 s } } d y \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} { \\mathcal F } _ h ( y , y ^ * ) = : h ( y ) + h ^ * ( y ^ * ) - \\Re e \\langle y ^ * , y \\rangle \\geq 0 . \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} H _ { L + 1 } \\leq \\sum _ { i = 1 } ^ { L } H _ { i } + 1 , \\ \\ \\ H _ { L + 2 } \\leq \\sum _ { i = 1 } ^ { L + 1 } H _ { i } + 1 . \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ { n + 1 } ( 1 + x _ i ) \\right ) h _ { n - | I | } ( \\underline x _ n ^ I ) \\prod _ { i \\in I } \\frac { 1 } { 1 + x _ i } = h _ { n - | I | } ( \\underline x _ n ^ I ) \\prod _ { i \\in [ 1 , n + 1 ] \\setminus I } ( 1 + x _ i ) \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} \\vec { g } = B _ { k , k } D W ' \\cdot \\vec { b } . \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} & | \\partial _ { t } ^ { 2 } \\left ( \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { t } ^ { \\infty } d x \\lambda '' ( x ) \\left ( K _ { 1 } ( x - t , \\lambda ( t ) ) - \\frac { \\lambda ( t ) ^ { 2 } } { 4 ( 1 + x - t ) } \\right ) \\right ) | + | \\partial _ { t } ^ { 2 } \\left ( \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { t } ^ { \\infty } d x \\lambda '' ( x ) K ( x - t , \\lambda ( t ) ) \\right ) | \\\\ & \\leq \\frac { C } { t ^ { 4 } \\log ^ { b + 1 } ( t ) } + C \\sup _ { x \\geq t } | e '''' ( x ) | \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} \\partial _ { t } v _ { 5 } ( t , r ) = \\int _ { t } ^ { \\infty } d x \\int _ { 0 } ^ { \\infty } d \\xi J _ { 1 } ( r \\xi ) \\cos ( ( t - x ) \\xi ) \\xi \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( x , \\xi ) \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} \\psi \\colon \\Z [ x ] / ( \\Phi _ { m _ 1 } ( x ) \\cdots \\Phi _ { m _ r } ( x ) ) \\to \\prod _ { i = 1 } ^ r \\Z [ x ] / ( \\Phi _ { m _ i } ( x ) ) \\cong \\prod _ { i = 1 } ^ r \\Z [ \\zeta _ { m _ i } ] . \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} \\Phi _ { * } ( z ) = \\frac { 1 } { \\Phi ( 1 / z ) } = \\frac { 1 } { \\gamma } z \\exp \\left [ \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t z } { z - t } \\ , d \\sigma _ { * } ( t ) \\right ] , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} F ^ { ( 1 ) } _ { J , K } ( a , b ) & = a + \\min \\{ b , J - a \\} - \\min \\{ a , K - b \\} , \\\\ F ^ { ( 2 ) } _ { J , K } ( a , b ) & = b - \\min \\{ b , J - a \\} + \\min \\{ a , K - b \\} , \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} f ( x ) = x ^ { L } - x ^ { L - 1 } - N _ { L } - 1 , \\ ; \\ ; \\ ; g ( x ) = x ^ { L + 1 } - x ^ { L } - N _ { L + 1 } - 1 . \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} \\dot { x } ^ 2 = \\frac { k } { H ( x , y , \\gamma ' _ 1 = 1 , \\gamma ' _ 2 = y _ x ) } \\ , . \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} \\frac { P _ { n , 1 } ( X ) } { P _ { n , 2 } ( X ) } = \\frac { P _ { n + 1 , 1 } ( X ) } { P _ { n + 1 , 2 } ( X ) } \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} - 2 u g ^ { i j } g ^ { k l } h _ { j l , i } f _ k + u g ^ { i j } g ^ { k l } h _ { i j , l } f _ k = - u g ^ { i j } g ^ { k l } h _ { i j , l } f _ k , \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} \\hat { u } ( f ) = f ( u ) ( f \\in ( X ^ * ) _ 1 ) . \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} \\bigg ( \\frac { 1 } { n } \\bigg ) ^ 2 - m \\bigg ( \\frac { 1 } { n } \\bigg ) + m ^ 2 = 1 \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} T _ { \\frac { Q - \\epsilon } { 2 } } ( a ) = \\left ( \\frac { 2 ( a + 1 ) } { Q } \\right ) , \\ ; \\ ; U _ { \\frac { Q - \\epsilon } { 2 } - 1 } ( a ) = 0 \\ ; \\ ; \\ ; m o d \\ ; \\ ; Q . \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} \\lambda ^ { 4 } + ( a _ { 1 } + a _ 2 ) \\lambda ^ { 3 } + ( b _ { 1 } + b _ 2 ) \\lambda ^ { 2 } + ( c _ { 1 } + c _ 2 ) \\lambda + d _ { 1 } = 0 . \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} F _ { m } ( z ) = \\alpha + \\frac { e ^ { m \\pi i } } { ( c _ { 2 } ) ^ { 1 / 2 } } ( z - c _ { 0 } ) ^ { 1 / 2 } + e ^ { m \\pi i } \\sum _ { n = 2 } ^ { \\infty } d _ { n , m } ( z - c _ { 0 } ) ^ { n / 2 } , m = 0 , 1 . \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\lim _ { m \\rightarrow \\infty } \\frac { h ^ 0 ( Z , \\mathcal O _ { m A } ) } { m ^ d } & = & \\lim _ { m \\rightarrow \\infty } \\frac { \\chi ( \\mathcal O _ { m A } ) } { m ^ d } \\\\ & = & \\lim _ { m \\rightarrow \\infty } \\frac { \\chi ( \\mathcal O _ Z ) - \\chi ( \\mathcal O _ Z ( - m A ) ) } { m ^ d } \\\\ & = & \\lim _ { m \\rightarrow \\infty } \\frac { - \\chi ( \\mathcal O _ Z ( - m A ) ) } { m ^ d } \\\\ & = & \\frac { - ( ( - A ) ^ d ) } { d ! } , \\end{array} \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} & \\nabla _ { e _ 0 } e _ 0 = \\frac { 1 } { 2 \\phi ^ { 3 / 2 } } \\sum _ { i = 1 } ^ 3 \\frac { \\partial \\phi } { \\partial x _ i } e _ i ; \\\\ & \\nabla _ { e _ i } e _ 0 = - \\frac { 1 } { 2 \\phi ^ { 3 / 2 } } \\sum _ { j , k = 1 } ^ 3 \\epsilon _ { i j k } \\frac { \\partial \\phi } { \\partial x _ j } e _ k ; \\\\ \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} = P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c , \\ N ( t ) = 2 k \\} = \\binom { 2 k } { k } \\frac { 1 } { 2 ^ { 2 k } } \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} r _ k ( T ) = \\bigcup _ { m < k } T ( m ) , \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} g _ { \\rm s l i d e } ( y ; \\mu ) = \\frac { y h ( y ; \\mu ) } { f _ L ( 0 , y ; \\mu ) - f _ R ( 0 , y ; \\mu ) } , \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} \\mathbb { P } _ { n _ { l _ q } } \\left ( \\left | \\log ( \\frac { d \\tilde { \\mathbb { Q } } _ { n _ { l _ { q } } } } { d \\mathbb { P } _ { n _ { l _ { q } } } } ) - \\sum _ { k = 2 } ^ { K } \\frac { 2 \\mu _ { k } ( C _ { n _ { l _ q } , k } - ( n - 1 ) \\mathbb { I } _ { k = 2 } ) - \\mu _ { k } ^ { 2 } } { 4 k } \\right | \\ge \\frac { \\epsilon } { 2 } \\right ) \\le \\frac { \\delta } { 2 } . \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} P ( a ) ( P ( b ) p ( x ) ) = ( P ( a ) P ( b ) ) p ( x ) = P ( P ( a ) b + a P ( b ) + \\lambda a b ) p ( x ) \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} \\underline U _ t ( t , x ) = & K _ 2 \\Theta \\frac { | x | \\underline h ' ( t ) } { \\underline h ^ 2 ( t ) } \\preceq K _ 2 \\Theta \\frac { \\underline h ' ( t ) } { \\underline h ( t ) } = \\frac { K _ 1 K _ 2 \\Theta } { { \\gamma } - 1 } \\underline h ( t ) ^ { 1 - \\gamma } . \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\frac { \\delta ^ + _ { \\Omega , p } ( t _ n ) } { \\delta ^ - _ { \\Omega , p } ( t _ n ) } = + \\infty . \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} f _ L ( 0 , 0 ; 0 ) = g _ L ( 0 , 0 ; 0 ) = f _ R ( 0 , 0 ; 0 ) = g _ R ( 0 , 0 ; 0 ) = 0 , \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} \\tilde f _ 1 | _ { N ' } - \\tilde f _ 2 | _ { N ' } = 0 . \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} \\eta ( x ) = \\overline { \\{ x \\} } \\cap \\bigcap _ { \\substack { U \\in \\tau \\\\ x \\in U } } U . \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} ( \\mu , \\sigma _ + , \\beta _ 0 , \\beta _ 1 , \\eta ) = ( 0 , 1 , 0 , 0 , 1 ) . \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} & \\Pr \\left ( Q _ t = A | Q _ { t - 1 } = A \\right ) = \\Pr \\left ( X _ t = A | X _ { t - 1 } = A \\right ) = 1 - \\alpha , & \\\\ & \\Pr \\left ( Q _ t = B | Q _ { t - 1 } = A \\right ) = \\Pr \\left ( X _ t = B | X _ { t - 1 } = A \\right ) = \\alpha , & \\\\ & \\Pr \\left ( Q _ t = A | Q _ { t - 1 } = B \\right ) = \\Pr \\left ( X _ t = A | X _ { t - 1 } = B \\right ) = \\beta , & \\\\ & \\Pr \\left ( Q _ t = B | Q _ { t - 1 } = B \\right ) = \\Pr \\left ( X _ t = B | X _ { t - 1 } = B \\right ) = 1 - \\beta . & \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} \\Pi \\ , \\Pi _ s \\ , \\Pi = 0 \\ , . \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} \\overline \\partial G = \\omega \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\Delta _ L U _ { 1 , 0 } ( \\gamma ) = - \\int _ 0 ^ 1 U _ { 1 , t } ( \\gamma ) \\nabla ^ \\mu F _ { \\mu \\nu } ( \\gamma ( t ) ) \\dot { \\gamma } ^ \\nu ( t ) U _ { t , 0 } ( \\gamma ) d t . \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } \\left ( - \\lambda ( t ) \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) - 1 } { r ^ { 2 } } \\right ) v _ { 4 } \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle \\right ) | \\leq \\frac { C } { t ^ { 4 } \\log ^ { 2 b + N - 2 } ( t ) } + \\frac { C \\lambda ( t ) ^ { 2 - 2 \\alpha } \\sup _ { x \\geq t } \\left ( \\frac { | e '''' ( x ) | x } { \\lambda ( x ) ^ { 2 - 2 \\alpha } } \\right ) } { t \\log ^ { b - 3 + N } ( t ) } \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} \\begin{aligned} & \\partial _ t K + \\mathbf { u } \\cdot \\nabla _ { z , v } K - \\Delta _ t K = \\mathbf { H } - \\gamma ^ 2 \\gamma _ 1 \\partial _ { z z } \\phi - 2 ( \\partial _ { v } - t \\partial _ z ) \\partial _ z f \\end{aligned} \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} L _ \\beta = \\begin{cases} x _ \\beta & \\\\ x _ \\beta - \\varphi ( \\{ \\alpha , \\beta \\} ) x _ \\alpha & \\end{cases} \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} \\lambda \\ast _ { G / H } \\lambda ' ( \\psi ) = \\int _ { G / H } \\int _ { G / H } \\left ( \\int _ H \\psi ( x h y H ) d h \\right ) d \\lambda ( x H ) d \\lambda ' ( y H ) , \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} \\mathcal B _ N ( \\Omega ) : = \\left \\{ u \\in \\mathcal H ^ 2 _ { 0 , N } ( \\Omega ) : \\mathcal Q _ { \\sigma } ( u , \\varphi ) = 0 \\ , , \\forall \\varphi \\in H ^ 2 _ 0 ( \\Omega ) \\right \\} . \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} & D _ p = D \\cap \\overline { B } ^ c _ p , \\\\ & D _ p ^ R = ( D \\setminus D _ p ) \\cup \\overline B _ R , \\\\ & D _ { p , q } = D _ { q } \\setminus D _ { p } . \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} \\int _ \\Omega \\Big | | \\nabla u ^ m | ^ { p - 2 } \\nabla u ^ m \\Big | ^ { \\frac { p } { p - 1 } } d x = \\| \\nabla u ^ m \\| _ p ^ p < \\frac { d p q } { q - p } . \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} A S ' & = ( A ^ k { T ' } ^ { k + 1 } A ) A ^ k { T ' } ^ { k + 1 } = ( A ^ k { T ' } ^ k ) A ^ k { T ' } ^ { k + 1 } = A ^ k v ^ { k + 1 } B ^ k A ^ { 2 k } { T ' } ^ { k + 1 } = A ^ k { T ' } ^ { k + 1 } ( B ^ k A ^ k ) ^ { k + 1 } \\\\ & = S ^ { k + 1 } = A ^ k { T ' } ^ { k + 1 } = S ' . \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} & \\int _ { \\mathbb R } \\sum _ j 2 ^ { 2 j } \\big \\vert f ( \\cdot , y _ 2 , y _ 3 ) \\ast _ 1 \\psi ^ { ( 1 ) } _ { j } ( y _ 1 ) \\big \\vert ^ 2 w _ { y _ 2 , y _ 3 } ( I _ { y _ 1 , j } ) d y _ 1 \\\\ & = \\int _ { \\mathbb R } S ^ 2 _ { \\psi ^ { ( 1 ) } } \\big ( f ( \\cdot , y _ 2 , y _ 3 ) \\big ) ( y _ 1 ) w _ { y _ 2 , y _ 3 } ( y _ 1 ) d y _ 1 , \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} \\hat { \\mathfrak { C } } _ n : = n \\sum _ { h = 1 } ^ n \\hat { \\chi } _ h \\hat { \\chi } _ h ^ \\top - \\frac { n } { 2 } \\sum _ { h = 1 } ^ { n - 1 } \\left ( \\hat { \\chi } _ h \\hat { \\chi } _ { h + 1 } ^ \\top + \\hat { \\chi } _ { h + 1 } \\hat { \\chi } _ { h } ^ \\top \\right ) . \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} | P _ { U } | & \\leq \\left | P ( 0 ) + P ' ( 0 ) + \\frac { 1 } { 2 } \\ , P '' ( 0 ) \\right | + C \\ , ( 1 + | x | ^ 6 ) \\ , | U | _ 2 ^ 3 = \\frac { 1 } { 2 } \\ , \\left | P '' ( 0 ) \\right | + C \\ , ( 1 + | x | ^ 6 ) \\ , | U | _ 2 ^ 3 \\ , . \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} H : = H ( s , t ) : = \\frac { s ^ 2 } { 2 t } + \\frac { s } { 2 } ( t - 4 ) - \\frac { ( t - 1 - \\beta ) ( 1 + \\beta ) ( t - 1 ) } { 2 t } + 1 . \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} a _ { n } = \\lim _ { \\varepsilon \\downarrow 0 } \\frac { G _ { \\mu _ { 1 } } ^ { ( n ) } ( \\alpha + i \\varepsilon ) } { n ! } = \\lim _ { \\varepsilon \\downarrow 0 } ( - 1 ) ^ { n } \\int _ { \\mathbb { R } } \\frac { d \\mu _ { 1 } ( t ) } { ( \\alpha - t + i \\varepsilon ) ^ { n + 1 } } = ( - 1 ) ^ { n } \\int _ { \\mathbb { R } } \\frac { d \\mu _ { 1 } ( t ) } { ( \\alpha - t ) ^ { n + 1 } } , \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} | E _ { i n t , 1 } | & \\leq \\frac { ( b + 1 ) } { t \\log ^ { b + 2 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } + \\frac { 1 } { 2 t \\log ^ { b + 2 } ( t ) ( \\log ( \\log ( t ) ) ) ^ { 3 / 2 } } \\\\ & \\leq \\frac { 1 } { 1 0 0 } \\frac { 1 } { t \\log ^ { b + 1 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } , t \\geq T _ { 0 } \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} & a _ 1 c _ 2 = a _ 2 c _ 1 , b _ 1 d _ 2 = b _ 2 d _ 1 , \\\\ & a _ 1 d _ 2 - a _ 2 d _ 1 + b _ 1 c _ 2 - b _ 2 c _ 1 = 0 , \\\\ & a _ 1 d _ 1 + a _ 2 d _ 2 - b _ 1 c _ 1 - b _ 2 c _ 2 = 1 . \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} \\gamma ( t ) ( x ) : = \\tilde { u } \\left ( \\dfrac { x } { t L } \\right ) , \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t y ( t , \\omega ) - \\Delta _ \\omega y ( t , \\omega ) - \\tfrac 1 { 1 0 } \\chi _ { \\Omega _ u } ( \\omega ) u ( t ) - \\tfrac 1 { 1 0 } \\chi _ { \\Omega _ v } ( \\omega ) v ( t ) & \\ , = \\ , 0 & \\qquad & I \\times \\Omega & \\\\ \\vec { \\mathbf n } ( \\omega ) \\cdot \\nabla _ \\omega y ( t , \\omega ) & \\ , = \\ , 0 & & I \\times \\Gamma & \\\\ y ( 0 , \\omega ) & \\ , = \\ , 0 & & \\Omega \\end{aligned} \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} V _ { n + 2 } ^ { ( 2 ) } = - V _ { n + 1 } ^ { ( 2 ) } - V _ { n } ^ { ( 2 ) } , \\ V _ { 0 } ^ { ( 2 ) } = 2 \\ \\textrm { a n d } \\ V _ { 1 } ^ { ( 2 ) } = - 3 . \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} \\Delta ^ { I } n _ { 2 } = \\Delta ^ { I } \\Big ( - \\frac { b v } { \\sqrt { g } } \\Big ) = \\lambda _ { 2 1 } \\Big ( - \\frac { a u } { \\sqrt { g } } \\Big ) + \\lambda _ { 2 2 } \\Big ( - \\frac { b v } { \\sqrt { g } } \\Big ) + \\lambda _ { 2 3 } \\Big ( \\frac { 1 } { \\sqrt { g } } \\Big ) , \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} D ^ \\alpha _ x v ( x ) = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\int _ 0 ^ x \\frac { v ' ( y ) } { ( x - y ) ^ \\alpha } d y . \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} f _ L ( 0 , 0 ; 0 ) = f _ R ( 0 , 0 ; 0 ) = 0 , \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( x ) = - C _ { N , s } \\mathrm { P . V . } \\int _ { \\R ^ N } [ u ( x + z ) - u ( x ) ] | z | ^ { - ( N + 2 s ) } d z , \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} \\begin{aligned} D ^ { \\beta _ 1 } _ a y _ 1 ( t ) & = y _ 2 ( t ) , \\\\ D ^ { \\beta _ 2 } _ a y _ 2 ( t ) & = y _ 3 ( t ) , \\\\ \\vdots \\\\ D ^ { \\beta _ { n - 1 } } _ a y _ { n - 1 } & = y _ n ( t ) , \\\\ D ^ { \\beta _ { n } } _ a y _ { n } & = f ( t , y _ 1 , y _ 2 , \\cdots , y _ { n - 2 } , y _ { n - 1 } ) , \\end{aligned} \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} V ^ * ( \\kappa ) = \\frac { \\kappa } { 6 } ( p ^ 0 ) ^ 2 + \\frac { 6 } { \\kappa } ( q _ 0 ) ^ 2 . \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} \\left \\langle \\rho ( p ) \\rho ( - p ) \\right \\rangle & = \\frac { i } { X _ p } . \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to 0 } \\frac { \\rho _ { X } ( \\tau ) } { \\tau } = 0 . \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ { k } b _ { i } ^ { * } \\Phi ( a _ { i } ^ { * } a _ { j } ) b _ { j } \\ge 0 ; a _ 1 , \\cdots , a _ k \\in \\mathcal { A } , b _ 1 , \\cdots , b _ k \\in \\mathcal { B } \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} \\varphi ( x , x ) = \\| x \\| ^ 2 \\mbox { a n d } \\varphi ( x , y ) = 0 . \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} \\begin{aligned} p _ { [ 1 ^ 0 7 ^ 1 ] } \\bigg ( 7 ^ { 2 r } m + \\frac { 7 - 7 ^ { 2 r + 1 } } { 2 4 } \\bigg ) & \\equiv 0 \\pmod { 7 ^ { r } } , \\\\ p _ { [ 1 ^ 4 7 ^ { - 3 } ] } \\bigg ( 7 ^ { 2 r } m + \\frac { 1 7 \\cdot 7 ^ { 2 r } - 1 7 } { 2 4 } \\bigg ) & \\equiv 0 \\pmod { 7 ^ { r } } , \\\\ p _ { [ 1 ^ 8 7 ^ { - 7 } ] } \\bigg ( 7 ^ { 2 r } m + \\frac { 4 1 \\cdot 7 ^ { 2 r } - 4 1 } { 2 4 } \\bigg ) & \\equiv 0 \\pmod { 7 ^ { r } } . \\end{aligned} \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} \\alpha ( \\vec { \\bf b } ) ( m ) : = \\min _ { 1 \\le i \\le k - 2 } \\left ( \\frac { b _ i ( m ) } { m b _ { i + 1 } ( m ) } \\right ) . \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} b = - { \\tilde f _ 1 } '' { \\tilde f _ 1 } ^ { - 1 } = \\mbox { u s i n g \\eqref { e q : i n t e r 3 } } = - \\left ( \\frac { \\tilde f _ 1 } { 4 } \\theta - \\frac { \\tilde f _ 1 } { 2 } \\theta ' \\right ) { \\tilde f _ 1 } ^ { - 1 } \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} \\left . \\begin{array} { @ { } l l @ { } } & \\langle v , j _ 1 \\rangle \\rightsquigarrow \\langle v , j _ 2 \\rangle 2 \\le j _ 1 < j _ 2 \\le \\mu , \\\\ & \\langle v , j \\rangle \\rightsquigarrow \\langle v , 1 \\rangle \\rightsquigarrow \\langle v , 2 \\rangle j = 3 , 4 , \\ldots , \\mu , \\end{array} \\right \\} \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} e _ x f e _ y = \\begin{cases} f ( x , y ) e _ { x y } , & x \\le y , \\\\ 0 , & x \\not \\le y . \\end{cases} \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} \\int _ { \\gamma _ i } ^ 2 \\rho _ { s c } ( x ) \\d x = \\frac { i } { n } . \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} g = e ^ { \\lambda x } \\ , \\begin{pmatrix} q _ 1 ( y ) & q _ 2 ( y ) \\\\ q _ 2 ( y ) & q _ 3 ( y ) \\end{pmatrix} \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} b _ 1 + b _ 2 & = \\tau ( y ) ( \\tau ( x ) - x ) + x ( \\tau ( y ) - y ) + \\tau ( y ) ( \\tau ( x ) - x ) = \\tau ( x y ) - x y . \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} [ \\rho ( L _ i ) , & \\rho ( L _ j ) ] \\ , = \\ , h ( z ) ^ { i + j } \\ , \\cdot \\\\ & \\left ( ( L + i \\lambda - j ) P ( L - \\lambda - i - j , i ) ( L + j \\lambda ) P ( L - \\lambda - j , j ) \\right . \\ , - \\\\ & - \\ , \\left . ( L + j \\lambda - i ) P ( L - \\lambda - i - j , j ) ( L + i \\lambda ) P ( L - \\lambda - i , i ) \\right ) \\ , = \\\\ & = \\ , ( i - j ) h ( z ) ^ { i + j } ( L + ( i + j ) \\lambda ) P ( L - \\lambda - ( i + j ) , i + j ) \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} T ( u ) T ( v ) = T ( T ( u ) \\cdot v + u \\cdot T ( v ) ) , ~ u , v \\in M . \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} u _ 1 ^ 1 ( t , x , y ) = \\sum _ { i = 1 } ^ n \\partial _ { x _ i } u _ 0 ^ 1 ( t , x ) w _ i ^ 1 ( y ) , \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} ( t _ 1 , \\ldots , t _ s ) \\cdot ( x _ 1 , \\ldots , x _ s , y _ 1 , \\ldots , y _ s ) = ( x _ 1 , \\ldots , x _ s , y _ 1 + t _ 1 x _ 1 , \\ldots , y _ s + t _ s x _ s ) \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} B _ { G , L + 1 } - B _ { H , L + 1 } = \\sum _ { i = 1 } ^ { L } G _ { i } - G _ { L + 1 } - \\left ( 1 + \\sum _ { i = 1 } ^ { L - 1 } H _ { L } - H _ { L + 1 } \\right ) = H _ { L + 1 } - G _ { L + 1 } = c _ 1 > 0 . \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} \\bar { H } ( y ) = \\frac { 1 } { 2 } \\langle y , ( I d + P ( M _ { i j } ) N P ^ * y \\rangle _ Y + \\langle P ^ * y , s \\rangle - \\frac { 1 } { N } \\log \\int _ { P x = 0 } \\exp \\left ( - H _ { ( M _ { i j } ) } ( x , y ) \\right ) \\mathcal { L } ( d x ) . \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} | | \\mathcal { F } ( \\sqrt { \\cdot } \\left ( F _ { 5 } + F _ { 6 } \\right ) ( x , \\cdot \\lambda ( x ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } ^ { 2 } & = \\int _ { 0 } ^ { \\infty } \\frac { r } { \\lambda ( x ) ^ { 4 } } \\left ( F _ { 5 } + F _ { 6 } \\right ) ^ { 2 } ( x , r ) d r \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} v _ { 4 } ( t , r ) = \\int _ { t } ^ { \\infty } v _ { 4 , s } ( t , r ) d s \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} a ( n - 1 ) = \\sum _ { i = 1 } ^ { n - 2 } a ( i ) + ( 2 ^ { k + 1 } - 1 ) ( 2 ^ { n - 2 } + 1 ) . \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} \\mu _ { n + 1 } ( p ) = p ( n + 1 ) + \\nu _ { n + 1 } ( p ( 0 ) , \\dots , p ( n ) ) , \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} F ( \\hat K ( t ) ) - \\hat K ( \\hat R ( t ) ) = ( O ( t ^ { n + N } ) , O ( t ^ { n + 2 N - 1 } ) ) , \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} d _ { \\lambda , + } : = \\lambda \\mathsf { M } + \\tfrac { \\partial \\ ; } { \\partial r } \\quad d _ { \\lambda , - } : = \\lambda \\mathsf { M } - \\tfrac { \\partial \\ ; } { \\partial r } \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} Y _ n \\stackrel { d } { \\to } \\exp \\left \\{ \\sum _ { i = 1 } ^ \\infty \\frac { \\mu _ i Z _ i - \\frac { 1 } { 2 } \\mu _ i ^ 2 } { \\sigma _ i ^ 2 } \\right \\} . \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} N _ { \\Phi ( \\varepsilon ) } ( x , y , z ^ I ) = \\widehat N _ { \\Phi _ \\varepsilon } ( x , y , z ^ I ) = { \\cal J } F _ \\varepsilon ( \\vartheta , \\mu , y , z ^ I ) ^ T \\Re ^ { n + q + p + p } . \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} R : = v ( S ) \\cup \\{ ( g , v ( u ( g ) ) ) \\mid g \\in G \\} . \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} I I = \\int _ { \\R ^ 2 } \\Delta _ q f \\Delta _ q \\rho \\leq \\| \\Delta _ q f \\| _ { L ^ 2 } \\| \\Delta _ q \\rho \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} I ' ( u ) v = ( u ^ + , v ^ + ) - ( u ^ - , v ^ - ) - \\int _ { \\mathbb { R } ^ N } f _ 0 ( u ( x ) ) v ( x ) \\ ; d x = 0 , \\forall \\ ; v \\in E . \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} T & = i v _ j \\cdot \\frac { i } { X _ e } \\cdot i v _ k , \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} \\left | \\mathbb { E } _ { \\mu _ { N , m } } \\left [ x _ i \\right ] - \\mathbb { E } _ { \\mu _ N ^ \\sigma } \\left [ x _ i \\right ] \\right | = O ( \\frac { 1 } { N } ) . \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} \\partial _ { t } f = S f + K f + i \\left [ A + e ^ { \\tilde { \\gamma } \\varphi } F \\right ] \\left ( x , t \\right ) \\in R ^ { n } \\times \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} u _ t - u _ { x x t } + 3 u u _ x - 2 u _ x u _ { x x } - u u _ { x x x } = 0 . \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} G _ { \\bar { g } } = r i c _ { \\bar { g } } - \\frac { s c a l _ { \\bar { g } } } { 2 } \\bar { g } , \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} A + P = H \\left ( I + S + H ^ { - 1 } P \\right ) \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} \\sigma _ { \\rm f o l d } = \\frac { a _ 1 } { b _ 0 } + \\frac { b _ 2 } { b _ 0 } - \\frac { a _ 5 } { a _ 2 } , \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} \\partial _ { 3 } F _ { 3 } ( r , \\rho , z ) = \\frac { - 4 ( - 1 + \\alpha ) r ^ { 2 } z ^ { - 3 + 2 \\alpha } ( 1 + ( r ^ { 2 } - \\rho ^ { 2 } ) z ^ { 2 \\alpha - 2 } ) } { ( ( 1 + ( r ^ { 2 } - \\rho ^ { 2 } ) z ^ { 2 \\alpha - 2 } ) ^ { 2 } + 4 \\rho ^ { 2 } z ^ { 2 \\alpha - 2 } ) ^ { 3 / 2 } } \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} { } ^ Q \\nabla ^ \\pm _ { q _ 1 } q _ 2 : = \\nabla ^ \\pm _ { \\rho ( q _ 2 ) } q _ 1 + [ q _ 1 , q _ 2 ] _ Q . \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} { \\rm I n d } _ { c , \\varphi } ( D ) : = ( - 1 ) ^ p \\frac { 2 p ! } { p ! } \\ , \\left < \\tau _ \\varphi ^ M , { \\rm I n d } _ c ( D ) \\right > . \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} \\int _ B \\frac { 1 } { \\ 2 m s + | x | ^ { \\alpha } } | t _ \\varepsilon u _ \\varepsilon ( x ) | ^ { \\ 2 m s + | x | ^ \\alpha } d x & = \\int _ B \\frac { ( 1 + R _ \\varepsilon ) ^ { \\ 2 m s + | x | ^ \\alpha } - 1 } { \\ 2 m s + | x | ^ { \\alpha } } | u _ \\varepsilon ( x ) | ^ { \\ 2 m s + | x | ^ \\alpha } d x \\\\ & + \\int _ B \\frac { 1 } { \\ 2 m s + | x | ^ { \\alpha } } | u _ \\varepsilon ( x ) | ^ { \\ 2 m s + | x | ^ \\alpha } d x \\\\ & = I + I I . \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} F _ { 5 } ( t , r ) & = N _ { 2 } ( v _ { 5 } ) ( t , r ) + \\frac { \\sin ( 2 ( v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } ) ) } { 2 r ^ { 2 } } \\left ( \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } + 2 v _ { 5 } ) - \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\right ) \\\\ & + \\left ( \\frac { \\cos ( 2 ( v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } ) ) - 1 } { 2 r ^ { 2 } } \\right ) \\left ( \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } + 2 v _ { 5 } ) - \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\right ) \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} D ^ k [ f \\circ g ] ( x ) = D ^ k f ( g ( x ) ) \\left ( D g ( x ) \\right ) ^ { \\otimes k } + f , \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} | \\langle \\mu _ { t } ^ { N } , \\phi \\rangle - \\langle u ^ N _ t , \\phi \\rangle | & = | \\langle \\mu _ { t } ^ { N } , ( \\phi - \\phi \\ast V ^ { N } ) \\rangle | \\\\ & \\leq \\left \\langle \\mu _ { t } ^ { N } , \\int _ { \\R ^ { d } } V ( y ) ~ | \\phi ( . ) - \\phi ( \\frac { y } { N ^ { \\alpha } } - . ) | d y \\right \\rangle \\\\ & \\leq \\frac { C \\| \\phi \\| _ { } } { N ^ { \\alpha } } . \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} p _ { n + 1 } ( \\underline x _ { n + 1 } ) = \\sum _ { I \\subseteq [ 2 , n ] } \\left ( h _ { n - | I | + 1 } ( \\underline x _ { n + 1 } ^ I ) + h _ { n - | I | } ( \\underline x _ { n + 1 } ^ { I \\cup \\{ n + 1 \\} } ) \\right ) = \\sum _ { I \\subseteq [ 2 , n + 1 ] } h _ { n + 1 - | I | } ( \\underline x _ { n + 1 } ^ I ) . \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} S _ { g W Z } [ A ^ { h } , h \\cdot g ] & = S _ { g W Z } [ A , g ] - S _ { g W Z } [ A , h ] \\\\ S ^ { 1 , 0 } _ { g W Z W } [ ( A ^ { 1 , 0 } ) ^ { h } , h \\cdot g ] & = S ^ { 1 , 0 } _ { g W Z } [ A ^ { 1 , 0 } , g ] - S ^ { 1 , 0 } _ { g W Z } [ A ^ { 1 , 0 } , h ] . \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} \\prod _ { t = 1 } ^ r \\left ( \\frac { 1 } { x _ { i _ t } - x _ { j _ t } } \\right ) \\cdot w _ { 0 , a , b } , \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } _ { + } } \\frac { t r _ { 0 } } { ( 1 - t r _ { 0 } ) ^ { 2 } } \\ , d \\mu _ { 1 } ( t ) = \\int _ { \\mathbb { R } _ { + } } \\frac { 1 } { ( 1 - t r _ { 0 } ) ^ { 2 } } \\ , d \\mu _ { 1 } ( t ) - \\int _ { \\mathbb { R } _ { + } } \\frac { 1 } { 1 - t r _ { 0 } } \\ , d \\mu _ { 1 } ( t ) \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} & | - \\frac { 1 } { \\omega } \\int _ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } ^ { \\infty } \\phi ( r , \\omega \\lambda ( t ) ^ { 2 } ) \\sqrt { r } F _ { 4 } ( t , r \\lambda ( t ) ) d r | \\leq \\frac { C | a ( \\omega \\lambda ( t ) ^ { 2 } ) | } { \\omega ^ { 5 / 4 } \\lambda ( t ) ^ { 1 / 2 } } \\int _ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } ^ { \\infty } \\sqrt { r } | F _ { 4 } ( t , r \\lambda ( t ) ) | d r \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} g _ { \\alpha , 0 , \\alpha , \\lambda } ( x , t ) = e ^ { - \\lambda x + \\lambda ^ { \\alpha } t } \\frac { 1 } { \\pi } \\int _ { 0 } ^ { \\infty } e ^ { - w x } e ^ { - t w ^ { \\alpha } \\cos ( \\pi \\alpha ) } \\sin ( t w ^ { \\alpha } s i n ( \\pi \\alpha ) ) d w \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 , \\ N ( t ) = 2 k + 1 \\} = \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} \\begin{cases} w _ i ( t , x ) > 0 \\ { \\rm o n } \\ [ 0 , T _ 0 ) \\times \\R \\ \\mbox { f o r } 1 \\leq i \\leq m , \\\\ A : = \\min _ { 1 \\leq i \\leq m } \\inf _ { x \\in \\R } w _ i ( T _ 0 , x ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} \\norm { \\eta ( x ) } ^ * = \\norm x = \\norm { x - \\mathbf { 0 } } = d ( x , \\mathbf { 0 } ) > 0 , \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} \\lambda = - 2 n , 2 n - 2 , \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} W ^ { 1 , \\mathcal { H } } ( \\Omega ) = \\left \\{ u \\in L ^ \\mathcal { H } ( \\Omega ) : | \\nabla u | \\in L ^ { \\mathcal { H } } ( \\Omega ) \\right \\} \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} \\kappa ( y , x , p _ 2 , p _ 1 ) \\kappa ( z , y , p _ 2 , p _ 1 ) = \\kappa ( z , x , p _ 2 , p _ 1 ) , \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} \\left ( \\frac { C _ { n , k _ { 1 } } - ( n - 1 ) \\mathbb { I } _ { k _ { 1 } = 2 } } { \\sqrt { 2 k _ { 1 } } } , \\ldots , \\frac { C _ { n , k _ { l } } } { \\sqrt { 2 k _ { l } } } \\right ) \\stackrel { d } { \\to } N _ { l } ( 0 , I _ { l } ) . \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} ( R _ \\nabla ) ^ d { } _ { a b c } = 2 \\rho ^ \\mu _ { [ a | } \\partial _ \\mu \\Omega ^ d { } _ { | b ] c } + 2 \\Omega ^ d { } _ { [ a | e } \\Omega ^ e { } _ { | b ] c } - C ^ e { } _ { a b } \\Omega ^ d { } _ { e c } . \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} \\widetilde { \\mathbb { G } } _ { N } '' f : = \\frac { 1 } { \\sqrt { N } } \\sum _ { i = 1 } ^ { N } \\left ( \\frac { S _ { i , N } } { \\pi _ { i , N } } - 1 \\right ) [ f ( Y _ { i } ) - \\mathbb { P } _ { y , N } f ] , f \\in \\mathcal { F } , \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} \\mathcal { F } ( t , \\xi , v , \\overline v ) = e ^ { ( t - \\xi ) \\mathcal { L } } f \\left ( e ^ { \\xi \\mathcal { L } } v , e ^ { \\xi \\overline { \\mathcal { L } } } \\overline v \\right ) \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ { N - 1 } a _ { k + 1 } U _ k ^ 2 \\right | \\leq \\frac { C } { N } \\textrm { a n d } | a _ k | \\leq \\frac { C } { N ^ 3 } \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} & \\psi ( a \\cdot u ) = \\phi ( a ) \\cdot \\psi ( u ) ~ ~ ~ ~ ~ ~ \\psi ( u \\cdot a ) = \\psi ( u ) \\cdot \\phi ( a ) , \\\\ & \\psi \\circ H = H ' \\circ ( \\phi \\otimes \\phi ) , \\\\ & \\phi \\circ T = T ' \\circ \\psi , ~ a \\in A , u \\in M . \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} ( - Y - Z ) X ^ 3 + ( 2 Y ^ 2 + Z Y ) X ^ 2 + ( - Y ^ 3 + Z Y ^ 2 - 2 ( Z ^ 2 ) Y + Z ^ 3 ) X + ( 2 Z ^ 2 Y ^ 2 - 3 Z ^ 3 Y ) = 0 . \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} \\widetilde { \\Phi } ( x ) = \\widetilde { \\phi } ( x _ 1 ) \\cdots \\widetilde { \\phi } ( x _ d ) , x = ( x _ 1 , \\dots , x _ d ) \\in \\R ^ d , \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} d \\mu _ { 1 } ( t ) = \\frac { 5 } { 3 3 } t ^ { 4 } \\ , d t , - 2 \\leq t \\leq 1 . \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} | | \\partial _ { r } F _ { 6 } ( t , r ) | | _ { L ^ { 2 } ( r d r ) } & \\leq \\frac { C } { t ^ { 9 / 2 } \\log ^ { 5 b - 1 + \\frac { 5 N } { 2 } } ( t ) } + | | \\partial _ { r } v _ { 5 } | | _ { L ^ { 2 } ( r d r ) } \\cdot | | \\frac { \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\lambda ( t ) ^ { 2 } } { ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) ^ { 2 } } | | _ { L ^ { \\infty } _ { r } } \\\\ & \\leq \\frac { C } { t ^ { 9 / 2 } \\log ^ { 5 b - 1 + \\frac { 5 N } { 2 } } ( t ) } \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} \\tau ( x ) ( g ) = \\mu ( ( g ^ { - 1 } x ) \\vert _ M ) x \\in B ^ G g \\in G . \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} \\Big | \\big [ V ( \\tau ) - \\beta _ \\varepsilon \\big ( V ( \\tau ) \\big ) \\big ] \\cdot \\nabla _ { \\ ! x } \\eta _ \\varepsilon \\big ( X ( \\tau ) \\big ) \\Big | & = \\big [ v \\ ! - \\ ! \\beta _ \\varepsilon ( v ) \\big ] _ { \\perp } \\big ( V ( \\tau ) \\big ) \\left | \\frac { \\partial \\eta _ \\varepsilon } { \\partial x _ { \\ ! \\perp } } \\big ( X ( \\tau ) \\big ) \\right | \\\\ [ 3 p t ] & = O ( \\varepsilon ^ 4 ) O ( 1 / \\varepsilon ) = O ( \\varepsilon ^ 3 ) . \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{gather*} J ( \\bar { u } , 0 ) = J _ 1 ( \\bar { u } ) = c _ 1 \\ , \\ , J ( 0 , \\bar { v } ) = J _ 2 ( \\bar { v } ) = c _ 2 . \\end{gather*}"}
-{"id": "3049.png", "formula": "\\begin{align*} x ^ * = P _ X ( x ^ * - \\alpha F ( x ^ * ) ) \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} { \\bf 1 } _ E ( \\xi ) \\ , \\psi _ n ^ u [ f ] ( \\xi ) = 0 \\ , \\forall \\ , 0 \\le \\xi \\le 1 , \\ , f \\in L ^ 2 ( \\R ) \\ , \\ \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\{ f , h \\} ( d h ) ^ 2 = \\{ f , g \\} ( d g ) ^ 2 + \\{ g , h \\} ( d h ) ^ 2 . \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} \\int _ { a - b i } ^ { a + 1 - b i } F ( z ) d z + \\int _ { a + 1 - b i } ^ { a + 1 + b i } F ( z ) d z + \\int _ { a + 1 + b i } ^ { a + b i } F ( z ) d z + \\int _ { a + b i } ^ { a - b i } F ( z ) d z = 0 . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mathfrak { g } _ { - 2 } \\oplus \\mathfrak { g } _ { - 1 } \\oplus \\mathfrak { g } _ 0 \\oplus \\mathfrak { g } _ 1 \\oplus \\mathfrak { g } _ 2 \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} \\sin ( 2 \\overline { v } _ { 1 } ) \\partial _ { R } \\overline { v } _ { 1 } - \\sin ( 2 \\overline { v } _ { 2 } ) \\partial _ { R } \\overline { v } _ { 2 } & = \\sin ( 2 \\overline { v } _ { 1 } ) \\left ( \\partial _ { R } ( \\overline { v } _ { 1 } - \\overline { v } _ { 2 } ) \\right ) + \\left ( \\sin ( 2 \\overline { v } _ { 1 } ) - \\sin ( 2 \\overline { v } _ { 2 } ) \\right ) \\partial _ { R } \\overline { v } _ { 2 } \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} \\phi _ 1 ( K _ 1 ) : = A _ 1 ^ T K _ 1 + K _ 1 A _ 1 + Q _ 1 - K _ 1 B _ 1 B _ 1 ^ { \\top } K _ 1 & = 0 , \\\\ A _ 1 ( K _ 1 ) ^ { \\top } K _ { 1 2 } + K _ { 1 2 } A _ { 2 } & = - ( Q _ { 1 2 } + K _ 1 A _ { 1 2 } ) , \\\\ A _ 2 ^ { \\top } K _ 2 + K _ 2 A _ 2 & = K _ { 1 2 } ^ { \\top } B _ 1 B _ 1 ^ { \\top } K _ { 1 2 } - K _ { 1 2 } ^ { \\top } A _ { 1 2 } - A _ { 1 2 } ^ { \\top } K _ { 1 2 } - Q _ 2 \\ , . \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} B ( x , r ) = \\{ y \\in X \\mid d ( x , y ) < r \\} \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} | \\partial _ { t } v _ { 4 } ( t , r ) | \\leq \\begin{cases} \\frac { C r } { t ^ { 3 } \\log ^ { 3 b + 2 N - 2 } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { t ^ { 2 } \\log ^ { 3 b + 2 N - 1 } ( t ) } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} H [ \\hat { h } ( \\ast , s ) ] = H _ 0 . \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} 0 = \\nabla ^ 2 _ { e _ i , e _ n } u & = \\nabla _ { e _ i } \\nabla _ { e _ n } u - \\nabla _ { \\nabla _ { e _ i } e _ n } u \\\\ & = a ( t ) u _ i - h ( t ) _ { i j } u _ j . \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { - W } \\bigg ( \\sum _ { i = 1 } ^ { l } \\hat { H } _ i \\bigg ) e ^ W = & - \\frac { \\partial ^ 2 } { \\partial r '^ 2 } - \\frac { \\partial ^ 2 } { \\partial r _ l ^ 2 } + \\omega ^ 2 ( r '^ 2 + r _ l ^ 2 ) + \\frac { 1 } { r '^ 2 } \\big [ - \\hat { Z } _ { l - 1 } + \\frac { 1 } { 4 } ( D _ { l - 1 } - 1 ) ( D _ { l - 1 } - 3 ) \\big ] \\\\ & + \\frac { 1 } { r _ l ^ 2 } \\big [ - \\hat { T } _ l + \\frac { 1 } { 4 } ( d _ l - 1 ) ( d _ l - 3 ) \\big ] , \\end{aligned} \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} \\sum _ { j = \\frac { X - r } { r d } } ^ \\frac { X } { r d } \\frac { 1 } { j ^ 2 } & = \\frac { ( X - r ) / r d } { ( ( X - r ) / r d ) ^ 2 } - \\frac { X / r d } { ( X / r d ) ^ 2 } - \\int _ { ( X - r ) / r d } ^ { X / r d } t \\ ( \\frac { - 2 } { t ^ 3 } \\ ) d t \\\\ & = \\frac { 1 } { ( X - r ) / r d } - \\frac { 1 } { X / r d } + 2 \\int _ { ( X - r ) / r d } ^ { X / r d } \\frac { 1 } { t ^ 2 } d t \\\\ & = - \\frac { r ^ 2 d } { X ( X - r ) } + 2 \\frac { r ^ 2 d } { X ( X - r ) } \\\\ & = \\frac { r ^ 2 d } { X ( X - r ) } . \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} A d m _ { \\Z } ^ { k - 1 } = A d m _ { \\Z } ^ { \\check { \\ell } } = P ^ { p - h ^ \\vee } _ + , A d m _ { \\Z } ^ k = A d m _ { \\Z } ^ { \\ell } = P ^ { p + q - h ^ { \\vee } } _ + \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } - \\Delta u + V ( x ) u = f ( u ) , \\\\ u \\in { H } ^ { 1 } ( \\mathbb { R } ^ { N } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} \\Sigma _ i ( - ) ^ { p ( p _ 1 + \\ldots + p _ i ) } P ( A _ 1 , \\ldots , [ A _ i , \\eta ] , \\ldots , A _ k ) = 0 , \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} \\beta C _ { n , 1 } + \\sum _ { k = 2 } ^ { K } \\frac { 2 \\mu _ { k } \\left ( C _ { n , k } - ( n - 1 ) \\mathbb { I } _ { k = 2 } \\right ) - \\mu _ { k } ^ 2 } { 4 k } \\left | \\mathbb { P } _ { n } \\right . \\stackrel { d } { \\to } W + \\sum _ { k = 1 } ^ { K - 1 } \\frac { 2 \\mu _ { k + 1 } Z _ { k } - \\mu _ { k + 1 } ^ 2 } { 4 ( k + 1 ) } . \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( t ) = R ( h ( t ) ) h ( t ) , \\ \\eta _ { \\mu _ { n } } ( t ) = R _ { n } ( h _ { n } ( t ) ) h _ { n } ( t ) t \\in \\mathbb { T } , n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} \\phi ( x ) & = \\sum _ { k = 0 } ^ \\infty \\alpha _ k \\left ( - \\partial _ \\mu \\partial ^ \\mu \\right ) ^ k \\rho ( x ) . \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} C _ { n , k } & = \\left ( \\frac { 1 } { n } \\right ) ^ { \\frac { k } { 2 } } \\sum _ { w \\in \\mathfrak { W } _ { k + 1 , k } } X _ { w } . \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} \\lim _ { \\Lambda _ { 1 } \\uparrow \\uparrow V } \\prod _ { x \\in \\Lambda _ { 1 } \\setminus \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * \\right ) = \\lim _ { \\Lambda _ { 1 } \\uparrow \\uparrow V } e ^ { \\sum _ { x \\in \\Lambda _ { 1 } \\setminus \\Lambda } H _ { x , i , j } } \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\frac { h ^ i ( Z , \\mathcal O _ Z ( m G ) ) } { m ^ d } = 0 \\mbox { i f } i > 0 \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} | I I _ { r , 2 } | & \\leq C \\int _ { 0 } ^ { \\infty } | \\partial _ { \\xi } ^ { 2 } \\left ( \\left ( 1 - \\chi _ { \\leq 1 } ( r \\xi ) \\right ) \\xi ( r \\xi ) ^ { 2 } \\Phi _ { 2 } ( r \\xi ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\right ) | \\frac { d \\xi } { ( t - r ) ^ { 2 } } \\\\ & \\leq \\frac { C } { ( t - r ) ^ { 2 } \\log ^ { b - 1 } ( r ) } , r \\geq \\frac { t } { 2 } \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} \\left | \\frac t { e ^ t - 1 } - 1 + \\frac 1 2 t - \\frac 1 { 1 2 } t ^ 2 - \\sum _ { k = 2 } ^ { N - 1 } \\frac { B _ { 2 k } } { ( 2 k ) ! } t ^ { 2 k } \\right | \\leq \\frac { | B _ { 2 N } | } { ( 2 N ) ! } t ^ { 2 N } , t \\geq 0 . \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} \\| \\sum _ { j = 1 } ^ { \\ell - 1 } \\binom { 2 \\ell - 1 } { 2 j } ( - \\partial ( m _ { s _ 1 } , . . . , m _ { s _ { 2 ^ { \\ell - 1 } } } ) ) ^ { 2 j } ( \\partial ( m _ { t _ 1 } , . . . , m _ { t _ { 2 ^ { \\ell - 1 } } } ) ) ^ { 2 ( \\ell - j ) - 1 } \\alpha - \\beta _ 1 \\| < \\frac { \\epsilon } { 1 0 } , \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} 0 < \\lim _ { \\varepsilon \\to 0 } \\mu [ Q ( x _ Q , t + \\varepsilon ) ] - \\mu [ Q ( x _ Q , t ) ] = \\mu [ \\overline { Q ( x _ Q , t ) } ] - \\mu [ Q ( x _ Q , t ) ] = \\mu [ \\partial Q ( x _ Q , t ) ] , \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} f ( \\ell , x ) = \\delta _ { \\ell } \\cdot f ( 0 , x ) , \\mbox { f o r a l l $ \\ell \\in \\mathbb { Z } _ + $ } , \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} - \\partial _ { t t } v _ { 3 , 1 , 2 } + \\partial _ { r r } v _ { 3 , 1 , 2 } + \\frac { 1 } { r } \\partial _ { r } v _ { 3 , 1 , 2 } - \\frac { v _ { 3 , 1 , 2 } } { r ^ { 2 } } = F _ { 0 , 1 } ^ { \\lambda _ { 1 } } ( t , r ) - F _ { 0 , 1 } ^ { \\lambda _ { 2 } } ( t , r ) \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} ( & ( W ^ { ( 1 ) } \\mapsto \\ldots \\mapsto W ^ { ( k _ 1 ) } = U ^ { ( k _ 2 ) } ) , \\\\ & ( ( a _ h ^ { ( 1 ) } , a _ j ^ { ( 1 ) } ) , \\ldots , ( a _ h ^ { ( k _ 1 - 1 ) } , a _ j ^ { ( k _ 1 - 1 ) } ) ) , \\\\ & ( d ^ { ( 1 ) } , \\ldots , d ^ { ( k _ 1 ) } = b ^ { ( k _ 2 ) } ) \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} \\frac { C } { 2 } \\log ( M ) + \\frac { C } { 2 } & \\le \\frac { C } { 2 } \\log ( M ) + \\frac { C } { 2 } \\log ( M ) \\\\ & = C \\log ( M ) \\le n . \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { C } e ^ { s x } \\overline { G } ( s , t ) d s = \\sum \\mbox { R e s } \\ e ^ { s x } \\overline { G } ( s , t ) . \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} z _ { x y } = F ( x , y , z , z _ x , z _ y ) . \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} \\tau ( a x ) y = \\tau ( x ) a y \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} a ^ { p ^ * } & = ( 2 \\Lambda ) ^ { p ^ * } = 2 ^ { \\frac { p } { p - 1 } } \\Big ( \\frac { p - 2 } { p \\delta } \\Big ) ^ { \\frac { p - 2 } { 2 } \\cdot \\frac { p } { p - 1 } } L ^ { \\frac { p } { 2 } \\cdot \\frac { p } { p - 1 } } \\\\ & = O _ p \\bigg ( \\Big ( \\frac { 1 } { \\epsilon } \\Big ) ^ { \\frac { p ( p - 2 ) } { p - 1 } } L ^ p \\bigg ) . \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} \\bar { H } ( y ) & = - \\frac { 1 } { N } \\log \\int _ { P x = y } \\exp \\left ( - H ( x ) \\right ) \\mathcal { L } ( d x ) \\\\ & = \\frac { 1 } { 2 } \\langle y , ( I d + P ( M _ { i j } ) N P ^ * y \\rangle _ Y + \\langle P ^ * y , s \\rangle - \\frac { 1 } { N } \\log \\int _ { P z = 0 } \\exp \\left ( - H _ { ( M _ { i j } ) } ( z , y ) \\right ) \\mathcal { L } ( d z ) . \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} { M _ { \\infty } \\ge \\sum _ { x = 0 } ^ { \\infty } \\dfrac { 1 } { w _ { \\infty } ( x ) } = \\sum _ { x = 0 } ^ { \\infty } \\dfrac { 1 } { ( N _ x + 1 ) ^ { \\rho } ( x + 1 ) ^ { \\alpha } } . } \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} ( u , v ) : & = \\int _ { \\Omega } u \\cdot v , \\ u , v \\in L ^ 2 ( \\Omega ) , \\\\ ( ( u , v ) ) : & = { \\sum _ { i , j = 1 } ^ 2 \\int _ { \\Omega } \\frac { \\partial u _ i } { \\partial x _ j } \\frac { \\partial v _ i } { \\partial x _ j } } , \\ u , v \\in H ^ 1 ( \\Omega ) , \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} I I _ { t } & = \\frac { c _ { b } } { 4 \\pi ^ { 3 / 2 } } \\left ( \\int _ { 0 } ^ { \\infty } \\left ( 1 - \\chi _ { \\leq 1 } ( r \\xi ) \\right ) \\xi \\left ( e ^ { i ( t - r ) \\xi } + e ^ { - i ( t + r ) \\xi } \\right ) \\frac { r \\xi \\Phi _ { 1 } ( r \\xi ) \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } d \\xi \\right ) \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} f ^ \\varphi ( x _ 0 , \\ldots , x _ k ) : = \\int _ { G ^ { \\times ( k + 1 ) } } \\chi ( g _ 0 ^ { - 1 } x _ 0 ) \\cdots \\chi ( g _ k ^ { - 1 } x _ k ) \\varphi ( g _ 0 , \\ldots , g _ k ) d \\mu ( g _ 0 ) \\cdots d \\mu ( g _ k ) . \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} & E _ { ( r , A , r ' , \\mathcal { U } , p , q ) } = ( E _ { ( r , A , r ' , \\mathcal { U } , p , q ) } , \\nabla _ { ( r , A , r ' , \\mathcal { U } , p , q ) } ) , \\\\ & E _ { ( s , B , s ' , \\mathcal { V } , u , v ) } = ( E _ { ( s , B , s ' , \\mathcal { V } , u , v ) } , \\nabla _ { ( s , B , s ' , \\mathcal { V } , u , v ) } ) , \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} | \\nabla \\varrho _ c | = | \\nabla u | v _ h ( \\varrho _ c ) ^ { \\frac { 1 } { p - 1 } } \\le | \\nabla u | v _ h ( \\varrho ) ^ { \\frac { 1 } { p - 1 } } = | \\nabla \\varrho | . \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} ( T x ) _ k = \\sum \\limits _ { m \\in \\mathbb Z ^ c } b _ { k m } x _ { k - m } , k \\in \\mathbb Z ^ c . \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} \\gamma = \\left ( k a _ { 0 } \\right ) ^ { \\frac { 1 } { 2 } } \\beta = k ^ { \\frac { 1 } { p } } , \\alpha = a _ { 0 } ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} h ^ 1 _ 1 ( \\phi ^ 1 _ 1 ) = ( 1 - \\sin 3 \\phi ^ 1 _ 1 ) ^ { ( A - B ) / 6 } ( 1 + \\sin 3 \\phi ^ 1 _ 1 ) ^ { ( A + B ) / 6 } \\cdot \\frac { 1 } { 2 A - 3 - 2 B \\sin 3 \\phi ^ 1 _ 1 } \\hat { P } ^ { ( \\alpha , \\beta ) } _ { l + 1 } ( \\sin 3 \\phi ^ 1 _ 1 ) \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} \\int _ X \\left ( n \\mu _ { \\hat \\omega } \\omega _ 0 - R i c ( \\omega _ 0 ) \\right ) ^ n & = \\lim _ { t \\to n \\mu _ { \\hat \\omega } } \\int _ X \\left ( t \\omega _ 0 - R i c ( \\omega _ { 0 } ) + \\sqrt { - 1 } \\partial \\bar \\partial \\Phi ( t ) \\right ) ^ n \\\\ & = \\lim _ { t \\to n \\mu _ { \\hat \\omega } } \\int _ X e ^ { \\Phi ( t ) } e ^ { - t \\psi } \\omega _ { 0 } ^ n \\\\ & \\ge \\liminf _ { t \\to n \\mu _ { \\hat \\omega } } \\int _ X e ^ { \\Phi ( t ) } \\omega _ { 0 } ^ n \\\\ & \\ge c _ 4 , \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} F ^ { [ 1 , i ] } _ { 0 b } \\left ( m \\right ) & { = } \\int _ { \\mathbb { R } ^ 2 } \\mathbb { P } ( y , v s ^ { [ 2 , i ] } | b ^ { [ 2 , i ] } { = } b ) f ^ { [ 1 , i ] } ( y , v s ^ { [ 2 , i ] } ) \\ , d y \\ , d v s ^ { [ 2 , i ] } , \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} \\begin{array} { l l } U _ { x ^ { \\sigma } } V _ { y ^ { - \\sigma } , x ^ { \\sigma } } & = V _ { x ^ { \\sigma } , y ^ { - \\sigma } } U _ { x ^ { \\sigma } } \\\\ V _ { U _ { x ^ { \\sigma } } y ^ { - \\sigma } , y ^ { - \\sigma } } & = V _ { x ^ { \\sigma } , U _ { y ^ { - \\sigma } } x ^ { \\sigma } } \\\\ U _ { U _ { x ^ { \\sigma } } y ^ { - \\sigma } } & = U _ { x ^ { \\sigma } } U _ { y ^ { - \\sigma } } U _ { x ^ { \\sigma } } \\end{array} \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} A _ 2 ^ 4 \\Big | _ { x _ 3 = 0 = x _ 4 } & = - i b _ 3 \\left ( x _ 1 + x _ 2 \\right ) - i b _ 2 ^ 2 \\left ( \\frac { x _ 1 x _ 2 } { x _ { 1 + 3 } } + \\frac { x _ 1 x _ 2 } { x _ { 1 + 4 } } \\right ) . \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} & r _ 1 = \\sup _ { ( 0 , T ] \\times D _ R } | u ( t , x ) | , r _ 2 = \\sup _ { ( 0 , T ] \\times D _ R } | q ( t , x ) | , \\\\ & C = \\sup _ { ( 0 , T ] \\times D _ R } | c ( t , x ) | , L = \\sup _ { ( 0 , T ] \\times D _ R } | \\gamma ( t , x ) | . \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} | C | \\le \\Big \\lfloor \\frac { q ^ m n \\Delta } { q ^ m n \\Delta - ( q ^ m - 1 ) N } \\Big \\rfloor = \\Big \\lfloor \\frac { q ^ m \\Delta } { q ^ m \\Delta - ( q ^ m - 1 ) t } \\Big \\rfloor \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} f ^ { * * } ( t ) = \\frac { 1 } { t } \\int _ { 0 } ^ { t } f ^ { * } ( s ) d s , \\ ; f ^ { * } ( s ) = \\inf \\{ t \\geq 0 : a _ { f } ( t ) \\leq s \\} , \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} Q = e ^ { - 4 \\lambda } ( Q _ { 0 } + 2 P _ { 0 } \\lambda ) , \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} f _ i ( x ) = \\int _ { D ^ c \\cup B ( z , r ) ^ c } f _ i ( y ) \\omega ^ x _ { D \\cap B ( z , r ) } ( d y ) = \\int _ { B ( z , r ) ^ c } P _ { D \\cap B ( z , r ) } ( x , y ) f _ i ( y ) d y . \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} C _ { n , 1 } : = \\left ( \\frac { 1 } { \\sqrt { n } } \\right ) \\sum _ { i } A _ { i , i } . \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} D _ q ( M _ 1 \\cap M _ 2 , v ) = \\left ( 1 - \\frac { 1 } { q } \\right ) \\left ( | y | ^ q + | z | ^ q \\right ) . \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} ( 0 , + \\infty ) \\times S \\to C , \\ \\ \\ ( r , \\omega ) \\mapsto x : = r \\omega \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} & \\Bigl | \\int h ( \\phi _ { c _ 1 } ( \\xi _ 1 , \\eta _ 1 ) + \\phi _ { c _ 2 } ( \\xi _ 2 , \\eta _ 2 ) ) f ( \\phi _ { c _ 1 } ( \\xi _ 1 , \\eta _ 1 ) ) g ( \\phi _ { c _ 2 } ( \\xi _ 2 , \\eta _ 2 ) ) d \\xi _ 1 d \\eta _ 1 d \\xi _ 2 d \\eta _ 2 \\Bigr | \\\\ & \\lesssim A ^ { \\frac { 1 } { 2 } } N _ 1 ^ { - 2 } \\| f \\circ \\phi _ { c _ 1 } \\| _ { L _ { \\xi , \\eta } ^ 2 } \\| g \\circ \\phi _ { c _ 2 } \\| _ { L _ { \\xi , \\eta } ^ 2 } \\| h \\| _ { L _ { \\xi , \\eta , \\tau } ^ 2 } , \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} \\Delta t \\sum _ { n = 1 } ^ { m } ( \\partial _ { t } ^ + f ^ n ( u _ h ^ n ) , \\phi _ h ^ n ) _ { \\mathcal { T } _ h } & \\le C \\Delta t \\sum _ { n = 1 } ^ { m } \\| \\partial _ { t } ^ + u _ h ^ n \\| _ { \\mathcal T _ h } \\| \\phi _ h ^ n \\| _ { L ^ 6 ( \\Omega ) } \\\\ & \\le \\epsilon \\Delta t \\sum _ { n = 1 } ^ { m } \\| \\partial _ { t } ^ + u _ h ^ n \\| _ { \\mathcal T _ h } ^ 2 + C \\Delta t \\sum _ { n = 1 } ^ m \\| \\phi _ h ^ n \\| _ { L ^ 6 ( \\Omega ) } ^ 2 . \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} \\widetilde { w } ( x , y ) = w ( x _ 1 , x _ 2 ) \\cdot w ( x _ 2 , x _ 3 ) \\cdot \\ , \\cdots \\ , \\cdot w ( x _ { m - 1 } , x _ m ) \\in F - \\{ 0 \\} . \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} a _ v = \\frac { p ^ { t _ v } } { \\alpha ^ n ( n - 1 ) ! } \\in K _ v ^ \\times \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} \\int _ { \\Omega } G ( D u ) \\ , d x = [ \\ ! [ G ' ( D u ) , D \\psi ] \\ ! ] _ { u _ 0 } ( \\overline { \\Omega } ) - \\int _ { \\Omega } G ^ * ( G ' ( D u ) ) \\ , d x . \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} T _ { \\Phi } ( x , y , z ) = \\left \\{ d : d _ { z ^ E } = 0 , ( d _ { y ^ I } , d _ { z ^ I } ) \\in T _ { \\Theta } ( y ^ I , z ^ I ) : { \\cal J } F ( x , y , z ) d = 0 \\right \\} . \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} r ^ { 2 } \\partial _ { r r } v _ { 3 } = r ^ { 2 } F _ { 0 , 1 } ( t , r ) + r ^ { 2 } \\partial _ { t t } v _ { 3 } - r \\partial _ { r } v _ { 3 } + v _ { 3 } \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\overline { v } ( r ) = \\int _ { 0 } ^ { \\infty } \\overline { y } ( \\xi ) \\lim _ { r \\rightarrow \\infty } \\left ( \\frac { \\phi ( r , \\xi ) } { \\sqrt { r } } \\right ) \\rho ( \\xi ) d \\xi = 0 \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} \\ker ( f ) : = \\{ ( x , y ) \\in X ^ 2 \\mid f ( x ) = f ( y ) \\} , \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} P _ n H x = P _ n H z _ k . \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} \\mathbb { S } _ { A ^ { - 1 } M N _ 1 } ^ { \\ell _ 1 } \\cap { \\mathfrak { D } } _ { j _ 1 } ^ A \\cap { \\mathcal { T } } _ { k _ 1 } ^ { A ' } \\not = \\emptyset , \\mathbb { S } _ { A ^ { - 1 } M N _ 1 } ^ { \\ell _ 1 } \\cap { \\mathfrak { D } } _ { j _ 1 } ^ A \\subset \\bigcup _ { \\# k _ 1 \\sim M } { \\mathcal { T } } _ { k _ 1 } ^ { A ' } . \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} F _ 0 ( x _ 0 , \\dots , x _ 3 ) G ( x _ 0 , \\dots , x _ 3 ) + F _ 1 ( x _ 0 , \\dots , x _ 3 ) F _ 2 ( x _ 0 , \\dots , x _ 3 ) F _ 3 ( x _ 0 , \\dots , x _ 3 ) F _ 4 ( x _ 0 , \\dots , x _ 3 ) = 0 , \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} & \\displaystyle \\sum _ { { \\bf k } = ( k _ 1 , . . . , k _ { n - 1 } ) \\in \\mathbb { N } ^ { n - 1 } _ { \\geq 0 } } \\frac { { \\bf x ^ { k } } } { \\prod _ { i = 1 } ^ { n - 1 } ( q ) _ { k _ i } } = \\sum _ { \\eta } \\frac { q ^ { { \\rm c o d i m } ( \\eta ) } { \\bf x ^ { \\overline { d i m } ( \\eta ) } } } { \\displaystyle \\prod _ { 1 \\leq i \\leq j \\leq n - 1 } ( q ) _ { m _ { [ i , j ] } ( \\eta ) } } , \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} | \\tilde { F } ( s _ k ) | = | a _ 2 y ^ c _ N ( s _ k ) + b _ 2 z ^ c _ N ( s _ k ) - \\gamma _ 2 | \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} & \\Phi ^ { - 1 } ( L ' ) = L , \\\\ & \\Phi ^ * \\mathcal { L } ' \\cong \\mathcal { L } , \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} [ v ] _ { A _ { \\infty , Y } } : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { Y ( Q ) } \\int _ Q M ( v \\chi _ Q ) ( x ) \\ , \\d x < \\infty , \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} \\sum _ { d \\mid p ^ 2 } \\limits \\dfrac { p ^ 2 } { d } = p ^ 2 + p + 1 \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} \\sum _ { i + j = n } \\Phi _ i \\circ \\pi _ j = \\sum _ { i + j + k = n } \\pi _ i ' \\circ ( \\Phi _ j \\otimes \\Phi _ k ) , ~ n \\geq 0 . \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} \\int _ { 1 / ( N + 1 ) } ^ { N / ( N + 1 ) } & \\frac { | Z _ N ( t ) - \\sigma B _ N ( t ) | | B _ N ( t ) | ^ { p - 1 } } { t ^ { 1 + p / 2 } } d t \\\\ & \\leq \\sup _ { 1 / ( N + 1 ) \\leq t \\leq N / ( N + 1 ) } \\frac { | Z _ N ( t ) - \\sigma B _ N ( t ) | } { ( t ( 1 - t ) ) ^ \\zeta } \\int _ { 1 / ( N + 1 ) } ^ { N / ( N + 1 ) } \\frac { | B _ N ( t ) | ^ { p - 1 } } { t ^ { 1 + p / 2 - \\zeta } } d t \\\\ & = O _ P \\left ( N ^ { - 1 / 2 + \\zeta } \\right ) \\int _ { 1 / ( N + 1 ) } ^ { N / ( N + 1 ) } \\frac { ( t ( 1 - t ) ) ^ { ( p - 1 ) / 2 } } { t ^ { 1 + p / 2 - \\zeta } } d t \\\\ & = O _ P ( 1 ) . \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} \\mathcal B _ D ( \\Omega ) : = \\left \\{ u \\in \\mathcal H ^ 2 _ { 0 , D } ( \\Omega ) : \\int _ { \\Omega } \\Delta u \\Delta \\varphi d x = 0 \\ , , \\forall \\varphi \\in H ^ 2 _ 0 ( \\Omega ) \\right \\} . \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} = h _ { n - | I | } ( \\underline x _ n ^ I ) \\left ( \\prod _ { i \\in [ 1 , n + 1 ] \\setminus I } ( 1 + x _ i ) - 1 \\right ) + h _ { n - | I | } ( \\underline x _ n ^ I ) \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} I _ X = \\left ( X _ { ( i ; j ) } - \\frac { 2 } { n + 1 } \\mathrm { d i v } X \\ , g _ { i j } \\right ) \\dot { x } ^ i \\dot { x } ^ j \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { ( p ^ r + 1 ) / 2 } ( 4 k - 1 ) \\frac { ( - \\frac { 1 } { 2 } ) _ k ^ 5 ( - \\frac { 3 } { 2 } ) _ k } { k ! ^ 5 ( k + 1 ) ! } \\\\ [ 5 p t ] & \\quad \\equiv \\frac { 3 5 p ^ r ( p ^ { 2 r } - p ^ { 4 r } - 1 ) } { 6 4 ( p ^ { 2 r } - 1 ) } \\sum _ { k = 0 } ^ { ( p ^ r - 3 ) / 2 } \\frac { ( \\frac { 3 } { 2 } ) _ k ^ 3 ( \\frac { 9 } { 2 } ) _ k } { k ! ( k + 2 ) ! ^ 2 ( k + 3 ) ! } \\pmod { p ^ { r + 3 } } \\quad p > 3 . \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} \\log _ q ( | C | ) & \\leq \\sum _ { i = j ( t ) } ^ t m _ i n _ i = \\hat { m } ( N _ t - N _ { j ( t ) } + n _ { j ( t ) } ) \\\\ & \\leq \\hat { m } ( N _ t - N _ { j ( t ) } + \\hat { m } ) \\leq \\hat { m } ( N _ t - \\eta N _ t + 1 + \\hat { m } ) . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} g _ X = - 2 \\deg ( f ) + \\sum _ { x \\in X _ { \\overline { k } } ^ \\circ } ( e _ x - 1 ) , \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} \\delta _ i ( \\beta _ 0 , \\beta _ 1 , \\eta ) = \\sum _ { s + t + k = 1 } ^ { 4 } \\frac { \\beta _ 0 ^ s \\beta _ 1 ^ t ( \\eta ^ 2 - 1 ) ^ k } { s ! t ! k ! } \\delta ^ { ( s , t , k ) } _ i ( 0 , 0 , 1 ) + \\epsilon _ { i n } ^ { ( 1 ) } , \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } a _ { j k } \\partial _ { j } \\partial _ { k } \\bigg ] w = f , \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} x ' = x + \\frac { c _ 2 - c _ 1 } { 2 } t \\ , \\ \\ \\ \\ \\ t ' = t \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} \\widetilde { \\Psi } \\left ( \\begin{array} { c c } x _ { 1 1 } & x _ { 1 2 } \\\\ x _ { 2 1 } & x _ { 2 2 } \\\\ \\end{array} \\right ) = \\left ( \\begin{array} { c c } \\Psi ( x _ { 1 1 } ) & \\Psi ( x _ { 1 2 } ) \\\\ \\Psi ( x _ { 2 1 } ) & \\Psi ( x _ { 2 2 } ) \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} I _ 4 : = - I _ { 4 , 1 } + I _ { 4 , 2 } , \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} Q _ { \\Sigma } ( u ^ { ( s ) } , u ^ { ( s ) } ) \\leq Q _ { \\Sigma } ^ s ( u ^ { ( s ) } , u ^ { ( s ) } ) = \\lambda _ s \\leq \\lambda _ { 1 / 2 } \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} e _ { k } = \\prod _ { p i = k } e _ { i } p \\mid n . \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} e ^ { \\xi \\mathcal { A } } = e ^ { - 2 i \\xi \\mathcal { L } } = 1 + \\mathcal { O } ( \\xi \\mathcal { L } ) . \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} \\ddot { y } ^ k + \\Gamma ^ k _ { i j } \\dot { y } ^ i \\dot { y } ^ j = 0 \\ , , k = 1 , \\dots , N \\ , . \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} N _ 0 ( n ) = 4 r + 4 s + N _ 0 ( n - 2 ) . \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} & \\int _ { \\lambda ( t ) ^ { 1 - \\alpha } } ^ { \\frac { 1 } { 2 } } | \\frac { - 1 } { 4 ( \\lambda ( t ) ^ { 1 - \\alpha } + w ) } + \\frac { 1 } { 4 ( \\lambda ( t ) ^ { 1 - \\alpha } + w ) ( 1 + w ) ^ { 3 } } | d w \\\\ & \\leq C \\int _ { \\lambda ( t ) ^ { 1 - \\alpha } } ^ { \\frac { 1 } { 2 } } \\frac { w d w } { ( \\lambda ( t ) ^ { 1 - \\alpha } + w ) } \\\\ & \\leq C \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} Z _ { n } ( \\beta ) : = \\int \\exp \\left \\{ \\beta H _ { n } ( \\sigma ) \\right \\} d \\Psi _ { n } ( \\sigma ) \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} \\partial _ { 2 } ^ { 2 } \\phi ( r , \\xi ) = ( \\partial _ { 2 } ^ { 2 } \\phi ( r , \\xi ) ) _ { 0 } + ( \\partial _ { 2 } ^ { 2 } \\phi ( r , \\xi ) ) _ { 1 } \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} F _ { \\nu } ( z ) = a + z - N _ { \\tau } ( z ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { ( 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } ) ^ { 2 } + 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } + 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } } = \\frac { 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } - \\sqrt { ( 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } ) ^ { 2 } + 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } } { - 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} m _ { 1 , 5 } & = 0 , & m _ { 2 , 5 } & = 0 , & 1 - m _ { 1 , 3 } m _ { 6 , 8 } & = 0 , \\\\ m _ { 6 , 8 } + m _ { 1 , 3 } m _ { 7 , 8 } & = 0 , & 2 m _ { 2 , 3 } m _ { 6 , 8 } + m _ { 7 , 8 } & = 0 , & m _ { 4 , 5 } - m _ { 7 , 8 } & = 0 , \\\\ 1 - m _ { 2 , 3 } m _ { 7 , 8 } & = 0 , & 1 - m _ { 4 , 5 } m _ { 6 , 8 } & = 0 , & 2 m _ { 6 , 8 } + m _ { 4 , 5 } m _ { 7 , 8 } & = 0 . \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} \\sum _ { k = K + 1 } ^ { m _ { n _ { { l _ { q } } _ { m } } } } \\frac { 2 \\mu _ { k } \\left ( C _ { n _ { { l _ { q } } _ { m } } , k } - ( n _ { { l _ { q } } _ { m } } - 1 ) \\mathbb { I } _ { k = 2 } \\right ) - \\mu _ { k } ^ 2 } { 4 k } \\left | \\mathbb { P } _ { n _ { { l _ { q } } _ { m } } } \\right . \\stackrel { d } { \\to } M _ { 1 } - M _ { 2 , K } . \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } = c _ x \\sqrt { n } \\left ( \\sum _ { i = 1 } ^ { n } a _ i a _ i ^ * \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} A : = \\left \\{ \\frac { u \\sqrt v } { \\sqrt w } : u , v , w v , w \\leq n ^ { 1 / 5 } , u \\leq n ^ { 3 / 5 } \\right \\} . \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} S \\leq \\sum _ { i = 1 } ^ t ( n _ i K _ { i , 1 } + ( n _ i - 1 ) K _ { i , 2 } ) = \\sum _ { i = 1 } ^ t \\big ( n _ i | C | ( | C | - 1 ) - K _ { i , 2 } \\big ) = N | C | ( | C | - 1 ) - \\sum _ { i = 1 } ^ t K _ { i , 2 } . \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} a ( l _ T ( u , a ) - r _ T ( a , u ) ) - ( l _ T ( u , a ) - r _ T ( a , u ) ) a = 0 \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} \\widetilde { C } ^ a { } _ { b c } = ( K ^ { - 1 } ) ^ a { } _ { x } C ^ x { } _ { y z } K ^ y { } _ b K ^ z { } _ c . \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} \\psi _ { j , k } * f ( x _ 1 , x _ 2 , x _ 3 ) = \\int _ { \\mathbb R ^ 3 } \\psi _ { j , k } ( x _ 1 - y _ 1 , x _ 2 - y _ 2 , x _ 3 - y _ 3 ) f ( y _ 1 , y _ 2 , y _ 3 ) d y _ 1 d y _ 2 d y _ 3 . \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} \\int \\xi d \\mu _ { N , m } = \\int _ Y \\left ( \\int \\xi ( x ) \\mu _ { N , m } ( d x | y ) \\right ) \\bar { \\mu } _ { N , m } ( d y ) . \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} w ^ { ( 2 ) } ( \\theta ) = [ w _ { i j } ( \\theta ) ] ^ \\top _ { i , j \\in \\{ 1 , \\ldots , d \\} , ~ i \\leq j } , f ^ { ( 2 ) } ( { \\bf { x } } ) = [ f _ { i j } ( { \\bf { x } } ) ] ^ \\top _ { i , j \\in \\{ 1 , \\ldots , d \\} , ~ i \\leq j } , \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} x _ 0 = \\xi + \\int _ { t _ 0 } ^ T \\frac { b ( \\tau , \\phi _ 1 ( \\tau ; T , \\xi ) ) + b _ 1 ( \\tau , \\phi _ 1 ( \\tau ; T , \\xi ) ) } { \\tau } d \\tau \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} [ a _ { ( m ) } , b _ { ( n ) } ] = \\sum _ { j \\geq 0 } \\begin{pmatrix} m \\\\ j \\end{pmatrix} ( a _ { ( j ) } b ) _ { ( m + n - j ) } \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} G _ { \\bar { g } } = T , \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} P _ n ^ + \\lbrace M ( t ) > \\beta , \\ \\mathcal { T } ( t ) \\le \\beta \\rbrace = P _ n ^ - \\lbrace \\mathcal { T } ( t ) > \\beta \\rbrace . \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} | | u ( t , \\cdot \\lambda ( t ) ) | | ^ { 2 } _ { L ^ { 2 } ( R d R ) } = | | v ( t , \\cdot ) | | ^ { 2 } _ { L ^ { 2 } ( R d R ) } = | | w ( t ) | | ^ { 2 } _ { L ^ { 2 } ( d R ) } = | | \\mathcal { F } ( w ) ( t ) | | _ { L ^ { 2 } ( \\rho ( \\xi ) d \\xi ) } ^ { 2 } = \\lambda ( t ) ^ { 2 } | | y ( t ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } ^ { 2 } \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} \\sigma = \\frac { n } { 1 + m } \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} - \\Delta \\Phi _ b = & \\partial _ t b ( t ) , \\ \\ \\Omega , \\\\ \\Phi _ b \\cdot n = & 0 , \\ \\ \\partial \\Omega \\\\ \\partial _ n \\Phi _ b = & ( \\partial _ n \\Phi _ b \\cdot n ) n \\ \\ \\partial \\Omega . \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} \\forall l \\in I ^ { \\tilde G \\tilde H } ( \\bar x ) \\colon \\tilde \\mu _ l = 0 \\ , \\land \\ , \\tilde \\nu _ l = 0 . \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} A _ { n , n } = \\begin{pmatrix} P _ { n - 1 , n } ( 0 ) & P _ { n - 1 , n } ( \\alpha ) \\\\ P _ { n , n - 1 } ( 0 ) & P _ { n , n - 1 } ( \\alpha ) \\end{pmatrix} = ( n - 1 ) ! C _ n C _ { n - 1 } \\cdots C _ 1 \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} W ( f , g ; a ) = - \\int _ { a } ^ { d } \\big ( ( L f ) ( x ) g ( x ) - f ( x ) L g ( x ) \\big ) \\d x + W ( f , g ; d ) . \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{cases} F _ L ( x , y ; \\mu ) , & x \\le 0 , \\\\ F _ R ( x , y ; \\mu ) , & x \\ge 0 . \\end{cases} \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} ( \\widehat { A } - a \\widehat { I } ) f = i c ( \\widehat { B } - b \\widehat { I } ) f \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} P ( R ^ * \\leq x ) = P \\Big ( \\max \\{ w _ 1 ^ 2 + ( w _ 2 ^ + ) ^ 2 , w _ 1 ^ 2 + ( w _ 3 ^ + ) ^ 2 \\} \\leq x \\Big ) = \\int _ { 0 } ^ x \\Phi ^ 2 ( \\sqrt { x - y } ) ( 2 \\pi y ) ^ { - 1 / 2 } \\exp ( - y / 2 ) d y \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} \\frac { \\partial _ r [ w ( \\theta ) ] ^ \\top \\mathbb { E } _ \\theta [ f ( \\textbf { X } ) ] } { \\mathbb { E } _ \\theta [ h ( \\textbf { X } ) + w ( \\theta ) ^ \\top f ( \\textbf { X } ) ] } = \\frac { \\partial _ r [ w ( \\theta ) ] ^ \\top \\bar { f } } { \\bar { h } + w ( \\theta ) ^ \\top \\bar { f } } , \\quad ~ r \\in \\{ 1 , \\ldots , k \\} . \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} \\sum _ { q = 1 } ^ { \\widetilde { l } } \\gamma _ q \\widetilde { T } _ q = \\sum _ { \\substack { r , s = 1 \\\\ r < s } } ^ { \\widetilde { l } } \\nu _ { r , s } ( \\delta _ r ^ { - 1 } \\widetilde { T } _ r - \\delta _ s ^ { - 1 } \\widetilde { T } _ s ) . \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} \\| A ^ { r - \\frac { 1 } { 2 } } & e ^ { \\tau A } \\partial _ t \\widetilde { v } \\| \\leq C _ r \\Big ( \\| e ^ { \\tau A } \\overline { v } \\| _ { H ^ { r + \\frac { 1 } { 2 } } } ^ 2 + \\| e ^ { \\tau A } \\widetilde { v } \\| _ { H ^ { r + \\frac { 1 } { 2 } } } ^ 2 + | \\Omega | \\| A ^ { r } e ^ { \\tau A } \\widetilde { v } \\| \\Big ) . \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} \\mathcal { A } ^ { \\exp } ( G ) : = \\left \\{ f \\in C ^ \\infty ( G ) , ~ \\sup _ { g \\in G } e ^ { q L ( g ) } | D f ( g ) | < C ~ \\forall q , ~ D \\in U ( \\mathfrak { g } ) \\right \\} \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} & \\left ( \\frac { 1 } { n } \\right ) ^ { \\frac { 2 k + 1 } { 2 } } \\sum _ { w ~ | ~ ~ l ( w ) = 2 k + 2 } X _ { w } \\\\ & = \\left ( \\frac { 1 } { n } \\right ) ^ { \\frac { 2 k + 1 } { 2 } } \\sum _ { r = 3 } ^ { 2 k + 1 } \\sum _ { w \\in \\mathfrak { W } _ { 2 k + 2 , r , t } } X _ { w } + \\left ( \\frac { 1 } { n } \\right ) ^ { \\frac { 2 k + 1 } { 2 } } \\sum _ { w \\in \\mathcal { W } _ { 1 , 2 k + 1 } } X _ { w } . \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} \\mathbb P ( B ) = \\underset { B \\subset G , G : o p e n } { i n f } \\mathbb P ( G ) . \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} w = \\frac { 1 } { 2 } ( 1 + p ^ 2 ) ( 1 + q ^ 2 ) v \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} ( L ^ { \\otimes p } , ( h ^ { L } ) ^ { \\otimes p } ) = ( L _ V , h ^ { L _ V } ) \\otimes ( \\pi ^ * L _ H , \\pi ^ * h ^ { L _ H } ) , \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} R ( X ) = - \\frac { P _ { N , 1 } ( X ) } { P _ { N , 2 } ( X ) } . \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} \\langle T \\rho ( x ) \\rho ( y ) \\rangle & = \\sum _ { j = 1 } ^ \\infty \\sum _ { k = 1 } ^ \\infty \\frac { b _ j b _ k } { j ! k ! } \\langle T \\phi ^ j ( x ) \\phi ^ k ( y ) \\rangle . \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} \\sigma = \\kappa \\ , ( \\sigma _ 1 \\pm \\sigma _ 2 ) \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} x = \\pi ( x ) + d ( x ) \\nu ( \\pi ( x ) ) ( x \\in \\Gamma ( \\delta ) , \\ , \\pi ( x ) \\in \\Gamma ) . \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} v _ j ( R , \\l , K ) = ( T _ { j } ( 1 , R ( \\l ) ) , T _ { j } ( 0 , \\frac { d } { d X } R ( \\l ) ) , \\cdots , T _ { j } ( 0 , \\frac { d ^ { K - 1 } } { d X ^ { K - 1 } } R ( \\l ) ) ) ^ T . \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} H \\cdot H & = 1 , \\\\ H \\cdot E _ i & = 0 , \\\\ E _ i \\cdot E _ j & = - \\delta _ { i , j } . \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} \\begin{aligned} & X ^ n : \\Xi \\rightarrow \\Omega , \\varpi \\mapsto X ^ n _ \\cdot ( \\varpi ) , \\\\ & X : \\Xi \\rightarrow \\Omega , \\varpi \\mapsto X _ \\cdot ( \\varpi ) , \\end{aligned} \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} ( a * b ) ( t , s ) & = \\int _ { J } a ( t , u ) b ( u , s ) d u \\\\ ( a * f ) ( t ) & = \\int _ { J } a ( t , u ) f ( u ) d u \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} \\chi ^ * ( D ) = d _ \\infty ( D ) + \\chi _ \\infty ( D ) , \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} \\| \\langle x \\rangle ^ { s } ( P ( h ) - E \\pm i \\varepsilon ) v \\| _ { L ^ 2 } & \\le \\| ( P ( h ) - E \\pm i \\varepsilon ) \\langle x \\rangle ^ { s } v \\| _ { L ^ 2 } + \\| [ P ( h ) , \\langle x \\rangle ^ { s } ] \\langle x \\rangle ^ { - s } \\langle x \\rangle ^ { s } v \\| _ { L ^ 2 } \\\\ & \\le C _ { \\varepsilon , h } \\| \\langle x \\rangle ^ { s } v \\| _ { H ^ 2 } , \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} \\check { \\nabla } ^ \\pm \\check { e } _ i = ( \\check { \\Omega } ^ \\pm ) ^ j { } _ { i } \\otimes \\check { e } _ j . \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } J _ { 1 } ( r \\xi ) r v _ { 4 , c } ( t , r ) d r & = \\sum _ { k = 1 } ^ { 5 } \\int _ { I _ { k } } J _ { 1 } ( r \\xi ) r v _ { 4 , c } ( t , r ) d r \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} Q _ l ( x ^ { \\Lambda _ 2 ^ { ( l ) } } | \\bar { x } ^ { B ( l ) } ) = - \\log \\int _ { \\frac { 1 } { K - R } \\sum _ { i \\in \\Lambda _ 1 ^ { ( l ) } } x _ i = \\tilde { y } _ l } \\exp \\left ( - H ( x ^ { B ( l ) } | \\bar { x } ^ { B ( l ) } ) \\right ) \\mathcal { L } ( d x ^ { \\Lambda _ 1 ^ { ( l ) } } ) . \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} D R ( T ) D ^ m T = - D ^ m r ( T ) \\left ( D T \\right ) ^ { \\otimes m } - \\mathcal { P } _ m ( R , T ) . \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} ( I I ) \\leq \\sum _ { p \\in S i n g ( \\Sigma ) } C ( \\tau ) r _ j ( p ) ^ n \\cdot \\| V _ j \\| ( B ^ M _ \\tau ( p ) ) = o ( c _ j ^ 2 ) \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} p _ { \\mu } ( t ) = \\frac { f ( x ) } { \\pi \\left | F _ { \\mu } ( t ) \\right | ^ { 2 } } , t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} \\psi ( P ( j ) , P ( i ) ) & = ( l _ j + \\Delta ( P ( j ) , j ) ) - ( l _ i + \\Delta ( P ( i ) , i ) ) - \\Delta ( P ( j ) , P ( i ) ) \\\\ & = l _ j - l _ i - \\Delta ( j , i ) \\ , \\cdot \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} x \\circ y & = y \\circ x \\\\ ( x \\circ y ) \\circ x ^ 2 & = x \\circ ( y \\circ ( x ^ 2 ) ) ; \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} Q _ \\delta ( x ^ a _ s ) : = \\begin{cases} f ( x ^ a _ s ) - f _ \\textup { m i n } + \\delta & x ^ a _ s , \\\\ + \\infty & \\end{cases} \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} & | - 4 \\int _ { t } ^ { \\infty } \\frac { e ''' ( s ) d s } { ( 1 + s - t ) ^ { 3 } ( \\lambda _ { 0 , 0 } ( t ) ^ { 1 - \\alpha } + s - t ) } \\left ( \\frac { 1 } { \\log ( \\lambda _ { 0 , 0 } ( s ) ) } - \\frac { 1 } { \\log ( \\lambda _ { 0 , 0 } ( t ) ) } \\right ) | \\\\ & \\leq \\frac { C \\sup _ { x \\geq t } | e ''' ( x ) | } { t \\log ( t ) ( \\log ( \\log ( t ) ) ) ^ { 2 } } \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} G ( x , y ) = G \\left ( \\frac { x } { | x - y | } , \\frac { y } { | x - y | } \\right ) | x - y | ^ { \\alpha - d } \\ , . \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} M _ n = \\max \\{ M _ { n - 1 } , \\tilde { S } _ n \\} , \\forall n \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} F ( \\xi ) = F ( - \\xi ) . \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} \\mathcal { B } _ { 3 , M } & = \\ker ( \\varphi _ { 3 , M } ) = \\langle e _ 1 + 7 e _ 3 , e _ 2 + 5 e _ 3 , 1 3 e _ 3 \\rangle , \\\\ \\mathcal { W } _ { 3 , M } & = \\{ - 3 e _ 1 + \\mathcal { B } _ { 3 , M } , 4 e _ 1 + \\mathcal { B } _ { 3 , M } , - 2 e _ 1 + \\mathcal { B } _ { 3 , M } \\} . \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac d { d t } U ^ a _ { t , s } ( \\gamma ) = - A ^ a _ \\mu ( \\gamma ( t ) ) \\dot { \\gamma } ^ \\mu ( t ) U ^ a _ { t , s } ( \\gamma ) \\\\ \\frac d { d s } U ^ a _ { t , s } ( \\gamma ) = U ^ a _ { t , s } ( \\gamma ) A ^ a _ \\mu ( \\gamma ( s ) ) \\dot { \\gamma } ^ \\mu ( s ) \\\\ \\left . U ^ a _ { t , s } \\right | _ { t = s } = I d . \\end{aligned} \\right . \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} \\int \\limits _ D L \\varphi ( x ) u ( x ) d x + \\int \\limits _ { D ^ c } L \\varphi ( x ) \\lambda ( d x ) = \\int \\limits _ U L \\varphi ( x ) P _ U \\tilde { \\lambda } ( x ) d x + \\int \\limits _ { U ^ c } L \\varphi ( x ) \\tilde \\lambda ( d x ) = 0 . \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} r < r _ c : = \\frac { \\sqrt { n } L } { \\pi } , \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} & \\sup _ { R _ { y , x } } N ( \\epsilon \\lVert ( F \\circ \\phi _ { y } ) / ( w _ { \\theta } \\circ \\phi _ { x } ) \\rVert _ { L _ { 2 } ( R _ { y , x } ) } , \\mathcal { F } / w _ { \\theta } , L _ { 2 } ( R _ { y , x } ) ) \\\\ & = \\sup _ { Q _ { y , x } } N ( \\epsilon \\lVert ( F \\circ \\phi _ { y } ) / ( w _ { \\theta } \\circ \\phi _ { x } ) \\rVert _ { L _ { 2 } ( Q _ { y , x } ) } , \\mathcal { F } / w _ { \\theta } , L _ { 2 } ( Q _ { y , x } ) ) \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ 2 u = 0 \\ , , & { \\rm i n \\ } \\Omega , \\\\ u = f \\ , , & { \\rm o n \\ } \\partial \\Omega , \\\\ \\frac { \\partial u } { \\partial \\nu } = g \\ , , & { \\rm o n \\ } \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty u ^ k P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c , \\ N ( t ) = 2 k + 1 \\} = \\frac { \\beta } { c t } \\sum _ { k = 0 } ^ \\infty u ^ k \\sum _ { j = 0 } ^ { k } \\binom { 2 j } { j } \\Bigl ( \\frac { \\sqrt { c ^ 2 t ^ 2 - \\beta ^ 2 } } { 2 c t } \\Bigr ) ^ { 2 j } = \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} \\begin{array} { c } \\langle \\widehat { q } _ 1 ^ 2 f , f \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = E ^ 2 ( x _ 1 ^ { ( 0 ) } ) ^ 4 + 2 \\lambda E ( x _ 1 ^ { ( 0 ) } ) ^ 3 + \\left ( \\lambda ^ 2 + \\frac { 3 } { 2 } a E ^ 2 \\right ) ( x _ 1 ^ { ( 0 ) } ) ^ 2 + \\frac { 3 } { 2 } \\lambda a E x _ 1 ^ { ( 0 ) } + \\\\ \\\\ + \\frac { \\lambda ^ 2 a } { 4 } + \\frac { \\theta ^ 2 } { 4 \\lambda ^ 2 b } + \\frac { 3 a ^ 2 E ^ 2 } { 1 6 } \\end{array} \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} \\mathcal { E } \\Vert { x } _ { t } - \\hat { x } _ { t } \\Vert ^ { 2 } = \\inf _ { \\zeta \\in \\mathcal { K } _ { t } } \\mathcal { E } \\Vert { x } _ { t } - \\zeta \\Vert ^ { 2 } \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} \\partial _ t ^ + \\xi ^ n & = \\partial _ t ^ + u _ { I h } ^ n ( u _ { I h } ^ n + u _ { I h } ^ { n - 1 } ) + \\partial _ t u _ { I h } ^ n u _ h ^ { n - 1 } + \\Delta t ( \\partial _ t ^ + u _ h ^ n ) ^ 2 \\\\ & \\quad + \\partial _ t ^ + u _ h ^ n ( u _ { I h } ^ n + 2 u _ { I h } ^ { n - 1 } - 2 e _ h ^ { u ^ { n - 1 } } ) . \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} D ^ m r & = A _ c D ^ m k _ c + D g _ c ( K { } { } ) D ^ m K { } { } + D ^ m g _ c ( K { } { } ) \\left ( D K { } { } \\right ) ^ { \\otimes m } + \\mathcal { P } _ m ( g _ c , K { } { } ) \\\\ & \\quad - D k _ c ( R ) D ^ m r - D ^ m k _ c ( R ) \\left ( D R \\right ) ^ { \\otimes m } - \\mathcal { P } _ m ( k _ c , R ) \\\\ & = f _ { m , 1 } + D g _ c ( K { } { } ) D ^ m K { } { } + D k _ c ( R ) D ^ m r , \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} \\mathcal { W } ^ p ( M ) = \\int _ M H ^ p \\ , d S , p \\geq 1 . \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu ( B ) } \\int _ { B _ - } - \\big ( \\varphi ( x ) - \\avg { \\varphi } _ { B } \\big ) \\ , d \\mu & = \\frac { p - 1 } { \\mu ( B ) } \\int _ { B _ - } - \\frac { \\big ( \\varphi ( x ) - \\avg { \\varphi } _ { B } \\big ) } { p - 1 } \\ , d \\mu \\\\ & \\leq \\frac { p - 1 } { \\mu ( B ) } \\int _ { B _ { - } } e ^ { - ( \\varphi ( x ) - \\avg { \\varphi } _ { B } ) / p - 1 } \\ , d \\mu \\\\ & \\leq ( p - 1 ) [ w ] _ { A _ { p , \\mathcal { B } } } ^ { 1 \\over p - 1 } , \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} \\mathcal { F } _ L f = \\sum _ { \\ell = 1 } ^ { d } h \\left ( \\frac { \\deg p _ { \\ell } } { L } \\right ) \\left < f , p _ { \\ell } \\right > _ N p _ { \\ell } . \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} K _ { p , q } ( \\varphi ; \\Omega ) = \\| K _ p \\mid L _ { \\kappa } ( \\Omega ) \\| < \\infty , \\ , \\ , \\ , 1 / \\kappa = 1 / q - 1 / p \\ , \\ , ( \\kappa = \\infty \\ , \\ , \\ , \\ , p = q ) , \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} \\mathcal { C } [ H , L ] ( v _ { 1 } , \\cdots , v _ { n } ) = - L ( H \\left ( v _ { 1 } , \\cdots v _ { n } \\right ) ) + \\sum _ { i = 1 } ^ n \\ , D _ { i } H ( v _ { 1 } , \\cdots , v _ { n } ) \\cdot L v _ { i } \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} V ( x ) = V _ { 0 } \\left ( | x | ^ { - 1 2 } - | x | ^ { - 6 } \\right ) , \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} \\lim _ { \\Lambda _ { 1 } \\uparrow \\uparrow V } \\prod _ { x \\in \\Lambda _ { 1 } \\setminus \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * \\right ) = \\lim _ { \\Lambda _ { 1 } \\uparrow \\uparrow V } e ^ { \\sum _ { x \\in \\Lambda _ { 1 } \\setminus \\Lambda } H _ { x , i , j } } \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} B _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } ) = \\sum _ { \\mathbf { n } \\subset \\mathbf { m } } B _ { | \\mathbf { m } | - | \\mathbf { n } | } \\binom { \\mathbf { m } } { \\mathbf { n } } ^ { ( d ) } \\Phi _ { \\mathbf { n } } ^ { ( d ) } ( \\mathbf { z } ) . \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} F _ { J , K } \\circ F _ { J , K } ( a , b ) = ( a , b ) . \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} x = B ^ 2 , y ^ 2 + B ^ 2 y = B ^ 6 + A ^ 4 B ^ 4 + B ^ 8 . \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} \\norm { H } _ { H S } ^ 2 = \\sum _ { i , j = 1 } ^ n h _ { i j } ^ 2 = \\sum _ { i = 1 } ^ n \\lambda _ i ^ 2 , \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} \\mathrm { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * b _ { x } \\right ) = \\begin{cases} \\delta _ { i , \\bar i } \\delta _ { j , \\bar j } c _ { i j } , i f x = \\overline { x } , \\\\ \\\\ 0 , i f x \\ne \\overline { x } \\end{cases} \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} | s | ^ { n - m } C _ 1 = \\Delta _ A ( f _ s , 0 ) \\Delta _ B ( f _ s , 0 ) \\geq \\Delta _ A ( f _ s , < A > _ { f _ s } ) \\Delta _ B ( f _ s , < B > _ { f _ s } ) \\to 0 , \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } { \\bf P } ( | \\eta _ k | / \\sigma _ k > Q ^ k ) < \\infty . \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} \\Theta ^ { } ( \\Lambda ) = \\Theta ^ { } \\begin{pmatrix} r \\\\ k _ u \\\\ k _ s \\end{pmatrix} = \\begin{pmatrix*} [ l ] A _ c k _ c + g _ c \\circ K { } { } - k _ c \\circ ( A _ c + r ) \\\\ A _ u ^ { - 1 } k _ u \\circ ( A _ c + r ) - A _ u ^ { - 1 } g _ u \\circ K { } { } \\\\ A _ s k _ s \\circ ( A _ c + r ) ^ { - 1 } + g _ s \\circ K { } { } \\circ ( A _ c + r ) ^ { - 1 } \\end{pmatrix*} . \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} A ^ { - 1 } \\big ( R _ y w \\big ) & = A ^ { - 1 } \\langle w _ 1 , w _ 2 , - w _ 3 \\rangle = w _ 1 \\cdot \\partial _ 1 \\vec { \\eta } + w _ 2 \\cdot \\partial _ 2 \\vec { \\eta } - w _ 3 \\cdot \\vec { n } \\\\ & = R _ x \\big ( w _ 1 \\cdot \\partial _ 1 \\vec { \\eta } + w _ 2 \\cdot \\partial _ 2 \\vec { \\eta } + w _ 3 \\cdot \\vec { n } \\big ) = R _ x \\big ( A ^ { - 1 } w \\big ) ; \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} \\hat { J } ^ { \\sigma } _ { p , \\phi } ( r ; Q ) : = \\sup \\big \\{ Q , \\int _ { A ^ { C _ p } _ { r , K r } ( p ) } \\phi ^ 2 ( x ) \\cdot | x | ^ { - n - 2 \\sigma } \\ d v o l _ { C _ p } ( x ) \\big \\} \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} \\Delta ( t , x ) = ( \\Delta _ i ( t , x ) ) \\preceq { \\bf 0 } \\mbox { f o r } t > 0 , \\ x \\in ( - \\underline h ( t ) , \\underline h ( t ) ) . \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} \\exp ( - t D ^ 2 ) = \\frac { 1 } { 2 \\pi i } \\int _ \\gamma e ^ { - t \\lambda } ( D ^ 2 - \\lambda ) ^ { - 1 } d \\lambda \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} \\begin{cases} \\ & h ( R , \\omega ) = ( c _ 1 + c _ 2 \\log R ) R ^ { \\gamma _ 1 ^ + } w _ 1 ( \\omega ) + O _ 2 ( R ^ { \\gamma _ 1 ^ + - \\epsilon } ) \\\\ \\ & h ( R , \\omega ) = c _ 1 R ^ { \\gamma _ 1 ^ - } w _ 1 ( \\omega ) + O _ 2 ( R ^ { \\gamma _ 1 ^ - - \\epsilon } ) \\end{cases} \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} P ( x ) = \\tfrac { 1 } { 6 } \\sum _ { i = 1 } ^ { p } ( x _ { 2 i - 1 } ^ { 3 } - 3 x _ { 2 i - 1 } x _ { 2 i } ^ { 2 } ) + \\sum _ { j = 1 } ^ { q } x _ { 2 p + 3 j - 1 } x _ { 2 p + 3 j - 2 } x _ { 2 p + 3 j } \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} \\alpha ^ { - 1 } _ { \\alpha ^ { - 1 } _ a ( b ) } \\alpha ^ { - 1 } _ a = \\alpha ^ { - 1 } _ { \\alpha ^ { - 1 } _ b ( a ) } \\alpha ^ { - 1 } _ b . \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} C _ { n + 1 } R _ n = R _ { n + 1 } \\begin{pmatrix} 0 & 1 \\\\ 1 & a _ { k ( n + 1 ) } \\end{pmatrix} \\cdots \\begin{pmatrix} 0 & 1 \\\\ 1 & a _ { k ( n ) + 1 } \\end{pmatrix} , \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| \\nabla h \\| ^ { 2 } _ { L ^ { 2 } _ { m / 2 } } = W _ { 1 } + W _ { 2 } + W _ { 3 } + W _ { 4 } , \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} \\left [ \\displaystyle \\sum _ { i = 1 } ^ p [ y ^ I ] ^ * _ j \\nabla ^ 2 c _ j ( x ^ * ) - A ^ T A - { \\cal J } c _ { \\gamma } ( x ^ * ) ^ T { \\cal J } c _ { \\gamma } ( x ^ * ) \\right ] \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} H ^ 0 ( \\mathcal { I } _ Z ( d ) ) \\hookrightarrow H ^ 1 ( F _ 1 ) \\hookrightarrow \\dots \\hookrightarrow H ^ { n - 2 } ( F _ { n - 2 } ) \\hookrightarrow H ^ { n - 1 } ( \\mathcal { O } _ { \\mathbb { P } ^ n } ( ( d + 1 ) ( 1 - n ) ) ) = \\{ 0 \\} . \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} w = \\cdots z a b \\cdots c \\cdots \\rightsquigarrow w ^ * = \\cdots a z b \\cdots c \\cdots . \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} \\nabla _ x f ( x ) + \\left [ \\displaystyle \\sum _ { i = 1 } ^ p [ y ^ I ] _ i \\nabla ^ 2 c _ i ( x ) - A A ^ T - { \\cal J } c ( x ) ^ T { \\cal J } _ { z ^ I } \\Psi _ { \\varepsilon } ( y ^ I , z ^ I ) { \\cal J } c ( x ) \\right ] \\lambda = 0 . \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} \\frac { d } { d s } \\big | _ { s = 0 } \\ , P _ { U ( s ) } = 0 \\ , . \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} \\lim _ n \\frac { 1 } { n } H \\Big ( \\sum _ { k = 0 } ^ { n - 1 } T _ { \\xi _ k } ( 1 , w ) z ^ k \\Big ) . \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} D ^ m [ f _ 1 \\circ f _ 2 \\circ f _ 3 ] ( x ) & = D [ f _ 1 \\circ f _ 2 ] ( f _ 3 ( x ) ) D ^ m f _ 3 ( x ) + D ^ m [ f _ 1 \\circ f _ 2 ] ( f _ 3 ( x ) ) \\left ( D f _ 3 ( x ) \\right ) ^ { \\otimes m } \\\\ & + \\mathcal { P } _ m ( f _ 1 \\circ f _ 2 , f _ 3 ) ( x ) . \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d Y ( t ) & = b ( Y ( t ) ) d t + \\sigma ( Y ( t ) ) d \\tilde { B } ( t ) - d \\eta _ Y ( t ) , \\\\ Y ( 0 ) & = x . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} z : = \\frac { y } { D _ p ( y , 0 ) ^ \\frac { 1 } { p } } \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} \\mu [ \\overline { Q ( x _ Q , t ) } ] & \\leq c _ \\mu \\mu [ Q ( x _ Q , t ) ] = c _ \\mu \\left [ \\mu [ \\overline { Q ( x _ Q , t ) } ] - \\mu [ \\partial Q ( x _ Q , t ) ] \\right ] , \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} \\tau ^ { n } ( d ) ( k ) & = \\mu _ { n } ( d ( k ) , \\dots , d ( n + k ) ) \\\\ & = d ( n + k ) + \\nu _ { n } ( d ( k ) , \\dots , d ( k + n - 1 ) ) \\\\ & = c ( k ) , \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} \\partial _ t f + v \\cdot \\nabla _ { \\ ! x } f = \\bar { A } _ g f + \\bar { K } _ g f , \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} 2 ^ { g + k + 1 } - \\sum _ { i = k + n + 3 } ^ { g + k + 1 } H _ i \\leq 2 ^ g + 2 ^ { k + n + 2 } - 2 ^ { n + 1 } + H _ { k + n + 2 } . \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} \\langle v _ H , v _ V \\rangle _ I = \\begin{bmatrix} \\dot { x } & \\mathrm { 0 } \\end{bmatrix} \\begin{bmatrix} I & \\mathrm { 0 } \\\\ \\mathrm { 0 } & I \\end{bmatrix} \\begin{bmatrix} \\mathrm { 0 } \\\\ \\dot { \\lambda } \\end{bmatrix} \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} ( - 1 ) ^ { j - i } \\sum _ { s = 0 } ^ { j - i } ( - 1 ) ^ s \\binom { m - k } s \\binom { m - k + j - i - 1 - s } { j - i - s } , \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} W _ a ( f , g ) & = \\frac { - W _ a ( f , k ) W _ a ( g , h ) + W _ a ( f , h ) W _ a ( g , k ) } { W _ a ( h , k ) } \\\\ & = \\frac { - \\vec f _ a ( k ) \\vec g _ a ( h ) - \\vec f _ a ( h ) \\vec g _ a ( k ) } { W _ a ( h , k ) } . \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } \\theta \\ , ( \\operatorname { d i v } \\varphi - i A \\cdot \\varphi ) \\ , U ( t ) = - \\int _ 0 ^ { + \\infty } \\theta \\ , \\varphi \\ , \\nabla _ A U ( t ) \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} \\{ ( a + b ) ( c + d ) = \\lambda ~ : ~ a \\in A , \\ , b \\in B , \\ , c \\in C , \\ , d \\in D \\} \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} \\kappa _ g = \\frac { e ^ { ( q _ 1 - q _ 2 ) } - 2 } { e ^ { 2 ( q _ 1 - q _ 2 ) } + 2 e ^ { ( q _ 1 - q _ 2 ) } + 1 } . \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} & | | \\sqrt { \\omega } \\lambda ( x ) \\left ( K \\left ( K ( y ( x , \\frac { \\cdot } { \\lambda ( x ) ^ { 2 } } ) ) \\right ) ( \\omega \\lambda ( x ) ^ { 2 } ) \\right ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } \\\\ & \\leq C \\left ( | | y ( x ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } + | | \\sqrt { \\omega } \\lambda ( x ) y ( x ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } \\right ) \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} c ( f ( A ) , f ( B ) , f ( C ) ; f ( K ) , f ( L ) , f ( M ) ) - 1 = : \\epsilon ^ 2 S _ 3 ( f , \\xi , \\eta ; x ) + o ( \\epsilon ^ 2 ) , \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} r _ { t } = \\sup \\{ \\langle x , e _ { 1 } \\rangle : \\xi _ { t } ( x ) > 0 \\} , \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} G _ \\rightarrow ( x , y ) : = \\begin{cases} 0 , & x < y , \\\\ v ( x ) u ( y ) - u ( x ) v ( y ) , & x > y . \\end{cases} \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} \\Omega _ L & = \\left \\{ ( x , y ) \\ , \\middle | \\ , x < 0 , y \\in \\mathbb { R } \\right \\} , \\\\ \\Omega _ R & = \\left \\{ ( x , y ) \\ , \\middle | \\ , x > 0 , y \\in \\mathbb { R } \\right \\} , \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} A _ { \\frac { n } { 4 } } = \\frac { 1 } { 2 } ( x _ { \\frac { n } { 4 } + 1 } ( y _ { \\frac { n } { 4 } - 1 } - 1 / 2 ) - ( y _ { \\frac { n } { 4 } + 1 } - 1 / 2 ) x _ { \\frac { n } { 4 } - 1 } ) = \\frac { 1 } { 4 } \\sin \\left ( \\frac { \\pi } { n } + \\beta \\right ) . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\nabla _ { 1 } \\boldsymbol { n } = - a \\boldsymbol { e _ { 1 } } , \\ \\ \\ \\nabla _ { 2 } \\boldsymbol { n } = - c \\boldsymbol { e _ { 2 } } . \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} \\beta = \\frac { d \\Lambda } { d \\mu } ( 0 ) . \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} \\mathcal { R } _ { 2 , a } ( t _ n , \\xi ) = \\int _ 0 ^ \\xi e ^ { ( \\xi - s ) \\mathcal { L } } f \\left ( u ( t _ n + s ) , \\overline u ( t _ n + s ) \\right ) d s - \\xi f ( u ( t _ n ) , \\overline u ( t _ n ) ) . \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} \\Omega = \\Omega _ 1 \\times \\Omega _ 2 \\times \\cdots \\times \\Omega _ k \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} \\mathbb { K } _ { \\psi } ^ { F } ( \\Omega ) : = \\left \\{ z \\in u _ 0 + { W ^ { 1 , p } _ 0 ( \\Omega ) } : z \\ge \\psi \\ , \\ , \\textnormal { a . e . i n $ \\Omega $ } , \\ , \\ , \\ , F ( D z ) \\in L ^ 1 ( \\Omega ) \\right \\} , \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} \\frac { d } { d t } f ( t ) = A f + Q ^ { + } ( t , f ( t ) ) - Q ^ { - } ( t , f ( t ) ) , f ( 0 ) = f _ 0 \\in X _ { + } , t \\geq 0 , \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} H ( A _ { R } ^ { ( n ) } ; 4 n ) = H ( A _ { R } ^ { ( n ) } ) \\ge n h ( R ) \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} \\gamma = - \\frac { \\partial \\phi } { \\partial y } ( 0 ; 0 ) , \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { j \\nearrow + \\infty } \\int _ { B \\setminus B _ a } | u _ j | ^ { \\ 2 m s } d x & = \\lim _ { j \\nearrow + \\infty } \\int _ { B \\setminus B _ a } | u _ j | ^ { \\ 2 m s + | x | ^ \\alpha } d x \\\\ & = \\lim _ { j \\nearrow + \\infty } \\int _ { B \\setminus B _ a } \\frac { | u _ j | ^ { \\ 2 m s + | x | ^ \\alpha } } { \\ 2 m s + | x | ^ \\alpha } d x = 0 \\end{aligned} \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} \\Phi ( k , l ) = \\int \\limits _ 0 ^ \\infty { \\frac { 1 } { { x \\sqrt { 2 \\pi } l } } } { e ^ { - k { x ^ 2 } } } { e ^ { - \\frac { { { { ( \\ln ( x ) + { l ^ 2 } ) } ^ 2 } } } { { 2 { l ^ 2 } } } } } d x . \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} \\Delta _ { q _ 1 } ( f , < q _ 1 > _ f ) = \\left [ a \\left ( E x _ 1 ^ { ( 0 ) } + \\frac { \\lambda } { 2 } \\right ) ^ 2 + \\frac { \\theta ^ 2 } { 4 \\lambda ^ 2 b } + \\frac { a ^ 2 E ^ 2 } { 8 } \\right ] ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{gather*} A _ 2 = a _ n \\sum _ { i = 1 } ^ \\infty c _ i ^ 2 \\lambda _ i , \\\\ V _ 1 = a _ n \\sum _ { i = 1 } ^ \\infty c _ i \\int _ { z _ 1 } ^ { z _ 2 } h _ 0 ^ n \\varphi _ i d z , \\end{gather*}"}
-{"id": "7858.png", "formula": "\\begin{align*} b _ 0 ( u , v , w ) : = \\langle B _ 0 ( u , v ) , w \\rangle _ { V _ 0 ' } : = ( ( u \\cdot \\nabla ) v , w ) , u , v , w \\in \\mathcal { V } . \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} | x | \\ , | x e ^ \\alpha - y | & \\ge h ( n - 1 ) c _ 4 n ^ { - g + 1 } \\frac { | \\alpha | ^ n } { 4 ^ n \\ , n ! } \\\\ & \\ge c _ 4 ^ 2 | \\alpha | n ^ { - 2 g } \\binom { 2 n - 2 } { n - 1 } 4 ^ { - n } \\ge c _ 5 n ^ { - 2 g - 1 } \\ge c _ 5 ( \\log | x | ) ^ { - 2 g - 1 } , \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} \\frac { \\langle x , x \\rangle ^ \\frac { - 1 } { 2 } } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } + \\left ( \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } \\right ) \\frac { \\langle x , x \\rangle ^ \\frac { - 1 } { 2 } } { \\sqrt { n } } = 2 - c _ x \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} I ^ { G } ( \\bar x , \\bar y , \\bar z ) & : = \\{ l \\in \\mathcal Q \\ , | \\ , G _ l ( \\bar x ) = \\bar y _ l \\ , \\land \\ , H _ l ( \\bar x ) \\neq \\bar z _ l \\} , \\\\ I ^ { H } ( \\bar x , \\bar y , \\bar z ) & : = \\{ l \\in \\mathcal Q \\ , | \\ , G _ l ( \\bar x ) \\neq \\bar y _ l \\ , \\land \\ , H _ l ( \\bar x ) = \\bar z _ l \\} , \\\\ I ^ { G H } ( \\bar x , \\bar y , \\bar z ) & : = \\{ l \\in \\mathcal Q \\ , | \\ , G _ l ( \\bar x ) = \\bar y _ l \\ , \\land \\ , H _ l ( \\bar x ) = \\bar z _ l \\} . \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} E = - \\frac { \\eta ^ 2 } { ( 2 N _ r + 4 \\sum _ { s = 1 } ^ { N - 1 } J _ s + 2 N - 1 + 2 \\sum _ { s = 1 } ^ { N } \\gamma _ s ) ^ 2 } , ~ ~ ~ N _ r = 0 , 1 , \\cdots . \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} A _ { l } ( z , \\overline { z } ) = \\displaystyle \\sum _ { I , J \\in \\mathbb { N } ^ { N } \\atop \\left | I \\right | + \\left | J \\right | = p - 2 } a _ { I ; J } ^ { ( l ) } z ^ { I } \\overline { z } ^ { J } , C ( z , \\overline { z } ) = \\displaystyle \\sum _ { I , J \\in \\mathbb { N } ^ { N } \\atop \\left | I \\right | + \\left | J \\right | = p } c _ { I ; J } z ^ { I } \\overline { z } ^ { J } , \\quad \\mbox { f o r a l l $ l \\in 1 , \\dots , N $ . } \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} m _ { n + 1 } - k = ( n - 1 ) d - \\displaystyle \\sum _ { i = 1 } ^ { n } m _ i \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} \\frac { p - 1 } { p } \\Delta _ { p , f } | \\nabla u | ^ { p } = & | \\nabla u | ^ { 2 p - 4 } \\big [ | { \\rm H e s s } \\ , u | _ A ^ 2 + { \\rm R i c } _ f ( \\nabla u , \\nabla u ) \\big ] \\\\ & + | \\nabla u | ^ { p - 2 } \\langle \\nabla u , \\nabla \\Delta _ { p , f } u \\rangle , \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} \\liminf _ n \\Vert u _ n \\Vert _ \\infty = \\liminf _ n \\Vert v _ n \\Vert _ \\infty > 0 , \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} \\lambda | \\xi | ^ { 2 } \\le \\sum _ { j , k = 1 } ^ { n } a _ { j k } ( x ) \\xi _ j \\xi _ { k } \\le \\lambda ^ { - 1 } | \\xi | ^ { 2 } \\quad 0 < \\lambda \\le 1 . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} \\gamma = \\frac { \\gamma _ 1 \\gamma _ 2 } { \\gamma _ 1 + \\gamma _ 2 + 1 } . \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} \\rho ( L _ { - 1 } ) \\ , & = \\ , h ( z ) ^ { - 1 } \\left ( \\rho ( L _ 0 ) ^ 2 + \\rho ( L _ 0 ) - c ( c + 1 ) \\right ) \\ , , \\\\ \\rho ( L _ { 0 } ) \\ , & = \\ , - \\frac { h ( z ) } { h ' ( z ) } \\partial + b ( z ) \\ , , \\\\ \\rho ( L _ 1 ) \\ , & = \\ , h ( z ) \\ , . \\end{align*} % \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} b _ { n + 1 } & = \\sum _ { k = 1 } ^ { n } \\frac { ( n + k ) ! } { n ! } B _ { n , k } \\left ( - 1 ! a _ 1 , - 2 ! a _ 2 , \\ldots , - n ! a _ n \\right ) . \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} \\beta = \\frac { d \\Lambda } { d \\mu } ( 0 ) . \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} L = \\sqrt { g _ { 1 1 } + 2 g _ { 1 i } y ^ i _ x + g _ { i j } y ^ i _ x y ^ j _ x } \\ , d x \\ , = : \\mathsf { L } \\ , d x \\ , . \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} R ( a ) R ( b ) = R ( a R ( b ) + R ( a ) b - R ( a ) R ( b ) ) , ~ a , b \\in A . \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} \\psi _ \\pm ( 0 ) = \\psi ( \\pm 1 ) - \\log | w ' _ \\pm ( \\pm 1 ) | = - \\frac \\beta { \\beta + 1 } \\log 2 - \\frac 1 { \\beta + 1 } \\log c _ \\beta \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} R _ \\tau b _ { p , q } = \\tau ^ { q - p } b _ { p , q } = \\tau ^ { - d } b _ { p , q } , \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} \\tilde A _ { h , \\tilde h } = \\tilde h ^ m ( \\hat x ' ) ^ { - m } s ^ { - m } ( s D _ s ) ^ j P _ { m - j } ( h , h s \\hat x ' , y , D _ y ) \\bigl ( \\delta ( s - 1 ) \\delta ( y - y ' ) \\bigr ) . \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} g _ { n - 1 } & = 1 , \\\\ ( w _ + + w _ - ) g _ { n - 1 } - g _ { n - 2 } & = 1 , \\\\ w _ + w _ - g _ { p + 2 } - ( w _ + + w _ - ) g _ { p + 1 } + g _ p & = 0 , \\ ; \\ ; \\ ; ( 0 \\leq p \\leq n - 3 ) , \\\\ w _ + w _ - g _ 1 - ( w _ + + w _ - ) g _ 0 & = 0 , \\\\ w _ + w _ - g _ 0 & = K . \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} M ^ { ( c ) } _ { \\gamma _ 2 } ( x ) = \\frac { { \\rm \\mathcal { H } } _ { 3 , 4 } ^ { 3 , 3 } \\Biggl [ \\frac { \\kappa _ I m _ I \\bar { \\gamma } } { \\kappa m } s \\Bigg \\vert \\ { ( \\chi _ 2 , X _ 2 ) \\atop ( \\upsilon _ 2 , \\Upsilon _ 2 ) } \\Biggr ] } { s \\Gamma ( N m ) \\Gamma ( \\kappa ) \\Gamma ( L m _ I ) \\Gamma ( \\kappa _ I ) } . \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 } \\frac { u _ { \\lambda + \\mu } ( x ) } { u _ \\lambda ( x ) } = 1 . \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\underline U ( t , \\pm \\underline h ( t ) ) = ( 1 - \\epsilon ) [ \\Phi ( - 2 \\underline h ( t ) ) - \\mathbf { u } ^ * ] \\preceq \\mathbf { 0 } \\ \\mbox { f o r } \\ \\ t \\geq 0 . \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} \\bar { H } ( y _ l | \\bar { x } ^ { B ( l ) } ) = - \\frac { 1 } { K } \\log \\int _ { \\frac { 1 } { K } \\sum _ { i \\in B ( l ) } x _ i = y _ l } \\exp \\left ( - H ( x ^ { B ( l ) } | \\bar { x } ^ { B ( l ) } ) \\right ) \\mathcal { L } ^ { K - 1 } ( d x ^ { B ( l ) } ) . \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & - 1 & 0 & - 1 \\\\ 0 & 1 & 0 & 4 & 5 & 8 \\\\ 0 & 0 & 1 & - 3 & - 5 & - 6 \\\\ 0 & 0 & 0 & - 3 & - 6 & - 7 \\\\ 0 & 0 & 0 & - 1 & - 1 & - 2 \\\\ 0 & 0 & 0 & 2 & 3 & 4 \\end{pmatrix} ^ { - 1 } \\begin{pmatrix} x _ { 1 } \\\\ x _ { 2 } \\\\ x _ { 3 } \\\\ x _ { 4 } \\\\ x _ { 5 } \\\\ x _ { 6 } \\end{pmatrix} = \\begin{pmatrix} 1 / 2 \\\\ 0 \\\\ 1 / 2 \\\\ 0 \\\\ 0 \\\\ x \\end{pmatrix} \\begin{pmatrix} 1 / 2 \\\\ 0 \\\\ 1 / 2 \\\\ 0 \\\\ 1 / 2 \\\\ x \\end{pmatrix} , \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} - L ^ s _ { \\Sigma } u ^ { ( s ) } = \\lambda _ s \\alpha \\cdot u ^ { ( s ) } \\ \\ \\ \\ B _ { r _ 2 } \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} \\left ( \\widehat { u } ^ 2 + \\widehat { v } ^ 2 \\right ) f _ 0 = \\gamma f _ 0 . \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} \\mu _ { m } ^ { k } | _ { I } = \\ell ( I ) ^ { d } \\frac { \\mu _ { m } ^ { k + 1 } | _ { I } } { \\mu _ { m } ^ { k + 1 } ( I ) } < \\frac { 1 } { 2 } \\mu _ { m } ^ { k + 1 } | _ { I } . \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} u ^ 3 ( x - 1 ) ^ { \\tau _ 4 } \\tilde h _ 4 ( x ) = u ^ 3 ( x - 1 ) ^ { k _ 4 } p _ 4 ( x ) - u ^ 3 ( x - 1 ) ^ { r _ 1 - r _ 2 + k _ 6 } p _ 6 ( x ) , \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} Q _ { \\Sigma } ( v , v ) + \\mu ^ 2 \\sum _ { i = 1 } ^ I ( \\int _ { \\Sigma } v \\cdot \\phi _ i \\ ) ^ 2 \\geq \\lambda \\| v \\| ^ 2 _ { L ^ 2 ( \\Sigma ) } \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} I I = \\frac { 1 6 } { \\lambda ( t ) ^ { 3 } } \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) \\partial _ { 2 } K _ { 1 } ( s - t , \\lambda ( t ) ) \\lambda ' ( t ) \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} \\partial _ t \\partial _ k \\theta + u \\cdot \\nabla \\partial _ k \\theta + \\partial _ k u \\cdot \\nabla \\theta = 0 . \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} \\inf _ { r \\in ( 0 , \\infty ) , | \\xi | = c } s ( r ) \\int _ { | y | \\leq N _ \\nu } ( 1 - \\cos ( y \\cdot \\xi ) ) \\nu ( r \\ , d y ) > 0 , \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} \\left ( I _ { N ^ { p } } - \\mbox { A u x } _ { p } A \\right ) Z + B \\overline { Z } = V \\left ( z _ { 1 } , z _ { 2 } , \\dots , z _ { N } \\right ) , \\quad \\mbox { w h e r e $ A $ , $ B \\in \\mathcal { M } _ { N ^ { p } \\times N ^ { p } } \\left ( \\mathbb { C } \\right ) $ } , \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} \\mu = \\mu _ { 1 } \\boxplus \\mu _ { 2 } \\quad \\quad \\mu _ { 1 } = \\mu _ { 2 } = \\nu _ { \\boxplus } ^ { \\frac { \\gamma } { 2 } , \\frac { \\sigma } { 2 } } . \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} T ( y ) ( t , \\omega ) = - \\int _ { t } ^ { \\infty } \\frac { \\sin ( ( t - x ) \\sqrt { \\omega } ) } { \\sqrt { \\omega } } \\left ( F _ { 2 } ( y ) ( x , \\omega ) - \\mathcal { F } ( \\sqrt { \\cdot } F ( x , \\cdot \\lambda ( x ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) - \\mathcal { F } ( \\sqrt { \\cdot } F _ { 3 } ( u ( y ) ) ( x , \\cdot \\lambda ( x ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) \\right ) d x \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} c _ 1 \\Big ( \\langle L _ 0 , \\ldots , L _ m \\rangle , \\| \\cdot \\| ^ 2 \\Big ) = \\pi _ * \\Big ( c _ 1 ( L _ 0 , h ^ { L _ 0 } ) \\wedge \\cdots \\wedge c _ 1 ( L _ m , h ^ { L _ m } ) \\Big ) . \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} \\dim H ( p ) & = \\dim X \\\\ H ( p ) \\cap \\mathcal { M } ( p ) & = \\Phi \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} \\eta _ { \\nu ^ { \\boxtimes k } } ( z ) = \\eta _ { \\mu } ( z ) \\left ( \\frac { \\eta _ { \\mu } ( z ) } { z } \\right ) ^ { 1 / ( k - 1 ) } , z \\in \\mathbb { D } . \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } - \\frac { 1 } { \\theta - t } u = 0 . \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} G ( z ) = \\frac { z + 0 . 8 1 1 1 } { z ^ 4 + 1 . 5 5 2 z ^ 3 + 0 . 6 9 9 5 z ^ 2 + 0 . 0 6 0 4 2 z - 0 . 0 1 2 4 1 } \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} \\xi _ 1 ^ { y , s , \\varepsilon } ( x , t ) & \\le g ( y , s ) - 2 \\varepsilon - M _ 1 \\rho ( x ) - M _ 2 | t - s | \\\\ & \\le g ( x , t ) + \\omega ( | x - y | ) + \\omega ( | t - s | ) - 2 \\varepsilon - M _ 1 \\rho ( x ) - M _ 2 | t - s | \\\\ & \\le g ( x , t ) . \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} \\gamma D _ \\alpha ^ { - 1 } C u _ 2 + \\gamma D _ \\alpha ^ { - 1 } v _ 1 & = u _ 1 - D _ \\alpha ^ { - 1 } D _ { - \\beta } u _ 1 , \\\\ \\gamma A _ \\beta ^ { - 1 } B u _ 1 + \\gamma A _ \\beta ^ { - 1 } v _ 2 & = u _ 2 - A _ \\beta ^ { - 1 } A _ { - \\alpha } u _ 2 . \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} \\mathcal { F } ( f ( x ) ) '' ( \\xi ) = \\int _ { 0 } ^ { \\frac { 2 } { \\sqrt { \\xi } } } f ( x , r ) \\partial _ { 2 } ^ { 2 } \\phi ( r , \\xi ) d r + \\int _ { \\frac { 2 } { \\sqrt { \\xi } } } ^ { \\infty } f ( x , r ) \\partial _ { 2 } ^ { 2 } \\phi ( r , \\xi ) d r \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} F ( x , y ) = \\begin{pmatrix} x + c \\ , y + f _ 1 ( x , y ) \\\\ y + f _ 2 ( x , y ) \\end{pmatrix} , \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = F ( x , y ; S ( x ) ; \\mu ) , \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} b ^ { ( k ) } = i ( b ) s ^ { ( k ) } , & \\textit { w h e r e } s ^ { ( k ) } \\textit { i s a s u f f i x o f } b ^ { ( k ) } , \\\\ & s ^ { ( k ) } \\textit { i s a s m a l l p i e c e ( p o s s i b l y e m p t y ) , } k = 1 , \\ldots , K . \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} x ^ s = x _ 1 ^ { s _ 1 } \\cdots x _ n ^ { s _ n } , \\ \\ \\ s = ( s _ 1 , \\ldots , s _ n ) \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} \\inf _ { Q _ T \\cup \\partial _ p Q _ T } ( v _ * ) ^ - = \\inf _ { \\partial _ p Q _ T } ( v _ * ) ^ - \\quad \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} \\beta Q ^ { 2 p ^ 2 } \\beta Q ^ { p - 1 } = \\sum _ { i } ( - 1 ) ^ { 2 p ^ 2 + i + 1 } \\binom { ( p - 1 ) ( i + 1 - p ) - 1 } { p i - 2 p ^ 2 - 1 } \\beta Q ^ { 2 p + p - 1 - i } \\beta Q ^ i . \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} \\pm \\left ( \\zeta _ { 3 } + \\zeta _ { 3 } ^ { 2 } + \\zeta _ { 1 0 } + \\zeta _ { 1 0 } ^ { 9 } + \\sum _ { i = 1 } ^ { 1 0 } ( \\zeta _ { 5 } + \\zeta _ { 5 } ^ { 4 } ) \\zeta _ { 1 1 } ^ { i } \\right ) . \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} - \\Delta \\Phi _ a = & \\partial _ t a ( t ) , \\ \\ \\Omega , \\\\ \\frac { \\partial \\Phi _ a } { \\partial n } = & 0 , \\ \\ \\partial \\Omega \\\\ \\int _ \\Omega \\Phi _ a d x = & 0 . \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} d - 2 = \\sum _ { i = 1 } ^ { j - 1 } n _ i + \\delta - 1 = \\sum _ { i = 1 } ^ { j - 1 } \\tilde { n } _ i + \\delta \\ \\ \\dim ( C ' ) = \\dim ( C ) = \\sum _ { i = j } ^ t \\tilde { n } _ i m _ i - m _ j \\delta , \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} - \\langle u | 1 \\rangle = \\lambda _ 0 + \\sum _ { n = 1 } ^ \\infty \\gamma _ n \\ . \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | = m } \\partial ^ \\alpha a _ \\alpha ( x , t , u ) - u _ t = f ( x , t ) g ( u ) \\mbox { i n } \\Omega \\times ( 0 , \\infty ) \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} \\left | \\mathbb { E } _ \\eta \\left ( \\mathbb { 1 } _ { \\{ \\mathcal { T } _ t ( 0 ) > \\frac { 1 - p } { 4 } \\delta ^ d t \\} } g ( \\eta _ t ) \\right ) \\right | = \\left | \\mathbb { E } _ \\eta \\left ( \\mathbb { 1 } _ { \\{ \\mathcal { T } _ t ( 0 ) > \\frac { 1 - p } { 4 } \\delta ^ d t \\} } \\mathbb { E } _ \\eta \\left ( g ( \\eta _ t ) | \\mathcal { F } _ { t , \\Lambda ^ c } \\right ) \\right ) \\right | , \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} L f = \\frac { L _ 0 ( f v ) } { v } . \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } \\| \\partial _ k \\theta ( t ) \\| _ { L ^ 2 } ^ 2 \\leq C \\| \\nabla u \\| _ { L ^ \\infty } \\| \\nabla \\theta \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} A = \\begin{bmatrix} A _ 1 & A _ { 1 2 } \\\\ 0 & A _ 2 \\end{bmatrix} , B = \\begin{bmatrix} B _ 1 \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} & ( \\xi _ 1 ' , \\eta _ 1 ' ) + ( \\xi _ 2 ' , \\eta _ 2 ' ) = ( \\xi ' , \\eta ' ) , \\\\ \\phi _ { \\tilde { c _ 1 } } ^ K ( \\xi _ 1 ' , \\eta _ 1 ' ) & \\in S _ 1 ^ K , \\ \\ \\phi _ { \\tilde { c _ 2 } } ^ K ( \\xi _ 2 ' , \\eta _ 2 ' ) \\in S _ 2 ^ K , \\ \\ ( \\psi ^ K ( \\xi ' , \\eta ' ) , \\xi ' , \\eta ' ) \\in S _ 3 ^ K . \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} I ( \\nu ) _ n = I _ R ( \\nu ) = \\{ f \\in R \\mid \\nu ( f ) \\ge n \\} \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} ( x + y ) z _ { x y } - ( { W _ 0 } ( e ^ { z _ x } ) + 1 ) ( { W _ { - 1 } } ( - e ^ { z _ y } ) + 1 ) = 0 , \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} a _ 0 = \\sum _ { k = 1 } ^ { \\infty } \\frac { 1 } { k ^ { 3 / 2 } } \\frac { \\Gamma \\left ( k - \\frac { 1 } { 2 } \\right ) } { k ! } . \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} D _ n = \\phi ' _ + ( \\alpha ) \\ , S ( f \\ , \\psi , { \\mathcal P } _ n ) . \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} \\sigma _ \\alpha ( X , T , \\gamma ) = \\rho _ \\alpha ( V , \\gamma ) . \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} \\partial _ { r } f - \\frac { f } { r } = L f + \\left ( \\frac { \\cos ( Q _ { 1 } ( r ) ) - 1 } { r } \\right ) f \\in L ^ { 2 } ( ( 0 , \\infty ) , r d r ) \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} h _ { x , j } = J _ { x } ^ { * } h _ { j } \\qquad ; \\qquad \\forall j \\in I _ { 0 } \\ , \\ \\forall x \\in V . \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} Q ( B _ 2 , B _ 3 ) \\ = \\ Q ( B _ 3 , B _ 1 ) \\ = \\ 2 \\cdot 5 - 9 \\ = \\ 1 . \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} \\| | D | ^ s \\omega \\| _ { L ^ \\infty _ t ( L ^ 2 ) } ^ 2 + \\| | D | ^ s \\theta \\| _ { L ^ \\infty _ t ( L ^ 2 ) } ^ 2 + \\| \\partial _ 1 | D | ^ s \\omega \\| _ { L ^ 2 _ t ( L ^ 2 ) } ^ 2 \\leq C . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} \\Big \\Vert \\sum _ { n = 1 } ^ { N } a _ { n } n ^ { - s } \\Big \\Vert _ { \\mathcal { H } _ { p ' } ( X ) } \\leq C \\left ( \\sum _ { n = 1 } ^ { N } \\| a _ n \\| _ X ^ p \\right ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} F _ { \\varepsilon } \\left ( x , t \\right ) = \\frac { i } { \\varepsilon + i } e ^ { \\varepsilon t W } V _ { 2 } \\left ( x , t \\right ) u \\left ( x , t \\right ) \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} p _ 1 - p _ 2 = - \\sqrt { 2 H _ 1 - 2 p _ 1 p _ 2 ( 1 + e ^ { - z } ) } . \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} H _ { L - m + k } = L - m + \\frac { 1 } { 6 } k ( k + 1 ) ( k + 2 ) + k . \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} p = \\lambda \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} \\langle \\rho ( k ) \\rho ( - k ) \\rangle & = \\frac { i } { X _ k } = \\frac { i } { \\beta _ 0 + \\beta _ 1 k ^ 2 + \\beta _ 2 ( k ^ 2 ) ^ 2 + \\ldots } \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { 1 } { p } \\right ) \\| x _ n \\| ^ p = D _ p ( 0 , x _ n ) \\leq D _ p ( 0 , x _ 0 ) \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} p ( x , t ) = \\frac { 1 } { B \\bigl ( \\alpha , \\frac { 1 } { 2 } \\bigr ) } \\frac { ( c ^ 2 t ^ 2 - x ^ 2 ) ^ { \\alpha - 1 } } { ( c t ) ^ { 2 \\alpha - 1 } } \\ , \\ \\ \\ \\ \\ | x | < c t \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ( L _ { - 1 } ) \\rho ( L _ { k + 1 } ) & = h ( z ) ^ k ( L - c - k - 1 ) ( L + ( k + 1 ) c ) \\\\ \\rho ( L _ { 0 } ) \\rho ( L _ { k } ) & = h ( z ) ^ k ( L - k ) ( L + k c ) \\\\ \\rho ( L _ { 1 } ) \\rho ( L _ { k - 1 } ) & = h ( z ) ^ k ( L + c - k + 1 ) ( L + ( k - 1 ) c ) \\end{aligned} \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} 0 = \\nabla f ( \\bar x ) + \\sum \\limits _ { i \\in I ^ g ( \\bar x ) } \\lambda _ i \\nabla g _ i ( \\bar x ) + \\sum \\limits _ { j \\in \\mathcal P } \\rho _ j \\nabla h _ j ( \\bar x ) . \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} \\Phi ^ { \\overline { e } } _ { w v } \\Phi ^ e _ { v w } ( \\epsilon ) & = \\Phi ^ { \\overline { e } } _ { w v } w ( \\epsilon ) ^ { \\frac { 1 } { 2 } } \\ , ( \\overline { \\epsilon } ) \\\\ & = w ( \\overline { e } ) ^ { \\frac { 1 } { 2 } } w ( \\epsilon ) ^ { \\frac { 1 } { 2 } } \\cdot \\epsilon \\\\ & = \\epsilon . \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} _ c ( N ) = _ c ( G _ { N } ) , \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} R ( r _ 1 , \\cdots , r _ { N } ) = r ^ { - \\frac { N - 1 } { 2 } } F ( r ) \\prod _ { i = 1 } ^ { N - 1 } y _ i ( \\theta _ i ) , \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} \\Phi ^ e _ { v w } & = U ^ { \\overline { e } } _ { w v } \\circ \\Psi ^ { e } _ { \\varphi ^ a ( v ) \\varphi ^ b ( w ) } \\circ ( U ^ { e } _ { v w } ) ^ * . \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} p _ { \\mathsf { m } } ^ { \\ast } \\geq 0 \\sum _ { \\mathsf { m = 1 } } ^ { + \\infty } p _ { \\mathsf { m } } ^ { \\ast } = 1 . \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} T ( e ) ( t ) = \\int _ { t } ^ { \\infty } d x _ { 1 } \\int _ { x _ { 1 } } ^ { \\infty } d x _ { 2 } T ( f ) '' ( x _ { 2 } ) \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} \\lambda ( t ) = \\frac { 3 } { 4 } \\lambda _ 0 + \\frac { 1 } { 4 } \\lambda ^ \\prime , t \\leq 1 \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} G _ \\bullet ( y - 0 , y ) - G _ \\bullet ( y + 0 , y ) & = 0 , \\\\ \\partial _ 1 G _ \\bullet ( y - 0 , y ) - \\partial _ 1 G _ \\bullet ( y + 0 , y ) & = 1 ; \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} M : = \\dd \\max _ { 1 \\leq i \\leq m } \\sup _ { x \\leq 0 } | \\phi _ i ' ( x ) | . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} t \\mu p _ 1 p _ { t + 1 } = t \\mu ^ 3 . \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 ^ + } ( \\alpha _ \\varepsilon , x _ \\varepsilon , t _ \\varepsilon , u _ { \\alpha _ \\varepsilon } ^ * ( x _ \\varepsilon , t _ \\varepsilon ) ) = ( \\beta , \\hat { x } , \\hat { t } , \\overline { u } _ \\beta ( \\hat { x } , \\hat { t } ) ) . \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} M & = \\left ( \\begin{array} { c c c | c c c | c } 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & - \\frac { 1 } { a } & m _ { 5 , 7 } \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 p } { \\partial t ^ 2 } = - \\frac { ( c _ 1 - c _ 2 ) } { 2 } \\frac { \\partial ^ 2 p } { \\partial x \\partial t } - \\frac { ( c _ 1 + c _ 2 ) } { 2 } \\frac { \\partial ^ 2 w } { \\partial x \\partial t } = \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} r \\left ( \\frac { 1 + ( r ^ { 2 } - ( s - t ) ^ { 2 } ) \\lambda ( s ) ^ { 2 \\alpha - 2 } } { \\sqrt { 1 + 2 ( ( s - t ) ^ { 2 } + r ^ { 2 } ) \\lambda ( s ) ^ { 2 \\alpha - 2 } + ( ( s - t ) ^ { 2 } - r ^ { 2 } ) ^ { 2 } \\lambda ( s ) ^ { 4 \\alpha - 4 } } } \\right ) = r \\left ( - 1 + E _ { \\partial _ { r } v _ { 3 } , 2 } \\right ) \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} { } ^ Q \\Omega ^ + = E ^ { - 1 } \\partial _ x E \\d x - E ( { } ^ Q \\Omega ^ - ) E ^ { - 1 } , \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} \\varphi ( x _ 0 , x _ 1 , x _ 2 ) = ( F _ 1 , F _ 2 , F _ 3 ) . \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} f ( r ) = \\Im \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t ^ { 2 } } { | t - \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) | ^ { 2 } } \\ , d \\sigma ( t ) . \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} & p + i R \\ , \\ , \\exp \\big ( i ( \\beta _ 1 - \\beta _ 2 ) / 2 \\big ) \\in \\Omega , \\\\ \\intertext { a n d i f $ \\beta _ 1 = \\pi \\neq \\beta _ 2 $ o r $ \\beta _ 1 \\neq \\pi = \\beta _ 2 $ , t h e n w e a d d i t i o n a l l y r e q u i r e t h a t } & p - i R \\ , \\exp \\big ( i ( \\beta _ 1 - \\beta _ 2 ) / 2 \\big ) \\in \\C \\setminus \\Omega \\quad . \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} \\mathcal R _ A ( v , \\epsilon ) = \\{ n \\in \\N \\ , | \\ , \\mu ( A \\cap T ^ { - v ( n ) } A ) > \\mu ^ 2 ( A ) - \\epsilon \\} \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} C _ { s , r } ^ { + } & : = \\bigg \\{ ( x ' , x _ { n + 1 } ) \\in \\mathbb { R } _ { + } ^ { n + 1 } : x _ { n + 1 } \\le \\bigg [ ( 1 - s ) \\bigg ( r - \\frac { | x ' | ^ { 2 } } { 4 } \\bigg ) \\bigg ] ^ { \\frac { 1 } { 2 - 2 s } } \\bigg \\} \\\\ C _ { s , r } ' & : = \\bigg \\{ ( x ' , 0 ) \\in \\mathbb { R } ^ { n } \\times \\{ 0 \\} : 0 \\le \\bigg [ ( 1 - s ) \\bigg ( r - \\frac { | x ' | ^ { 2 } } { 4 } \\bigg ) \\bigg ] ^ { \\frac { 1 } { 2 - 2 s } } \\bigg \\} . \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} N ^ s : = \\bigcup _ { j \\in J _ N ^ { s } } A _ j F ^ s : = \\bigcup _ { j \\in J _ F ^ { s } } A _ j . \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} - \\lambda _ { i _ 1 , \\dots , i _ { n - 1 } } = ( - 1 ) ^ { n - 1 + i _ 1 + \\dots + i _ { n - 1 } } \\frac { ( \\alpha \\beta + ( - 1 ) ^ { i _ 1 + \\dots + i _ { n - 1 } } \\prod _ { k = 1 } ^ { n - 1 } \\gamma _ k ^ { i _ k } ) \\prod _ { k = 1 } ^ { n - 1 } \\gamma _ k ^ { 1 - i _ k } } { \\alpha \\beta } . \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} \\max ( L _ 1 , L _ 2 ) & \\gtrsim | ( \\tau _ 1 - \\xi _ 1 ^ 3 - \\eta _ 1 ^ 3 ) + ( \\tau - \\tau _ 1 ) - ( \\xi - \\xi _ 1 ) ^ 3 - ( \\eta - \\eta _ 1 ) ^ 3 | \\\\ & = | ( \\tau - \\xi ^ 3 - \\eta ^ 3 ) + 3 ( \\xi \\xi _ 1 ( \\xi - \\xi _ 1 ) + \\eta \\eta _ 1 ( \\eta - \\eta _ 1 ) ) | \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} \\frac { \\nabla G + \\nabla G ^ T } { 2 } & = \\begin{bmatrix} \\nabla ^ 2 f ( x ) + \\lambda ^ T \\nabla ^ 2 g ( x ) & \\mathbf { 0 } \\\\ \\mathbf { 0 } & \\mathbf { 0 } \\end{bmatrix} \\\\ & \\geq { 0 } , \\forall z \\in \\Omega , \\forall t \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} \\frac { p _ t ( z ) } { t } = t ^ { - \\frac { d + \\alpha } { \\alpha } } p _ 1 \\left ( \\frac { z } { t ^ { 1 / \\alpha } } \\right ) \\lesssim | z | ^ { - d - \\alpha } , \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{cases} F _ L ( x , y ) , & { \\rm u n t i l ~ } x = \\mu , \\\\ F _ R ( x , y ) , & { \\rm u n t i l ~ } x = - \\mu , \\end{cases} \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} x l \\xi = \\xi \\bigg ( 1 - \\frac { m ^ { 1 / 2 } } { c n ^ { 1 / 2 } } \\bigg ) \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} \\gamma _ 1 T _ 1 + \\gamma _ 2 T _ 2 & = \\gamma _ 1 \\widetilde { M } _ L ^ { - 1 } \\cdot Q _ 1 ( v , w ) \\cdot \\widetilde { M } _ R ^ { - 1 } + \\gamma _ 2 \\widetilde { M } _ L ^ { - 1 } \\cdot Q _ 2 ( v , w ) \\cdot \\widetilde { M } _ R ^ { - 1 } = \\\\ & = \\widetilde { M } _ L ^ { - 1 } \\cdot Q ( v , w ) \\cdot \\widetilde { M } _ R ^ { - 1 } . \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} f \\in \\mathcal { D } ( L _ { \\max } ) \\quad \\Rightarrow \\lim _ { x \\to \\infty } f ( x ) = 0 , \\ \\lim _ { x \\to \\infty } f ' ( x ) = 0 . \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} & \\partial _ { x _ 1 } u ( x _ 1 , x _ 2 ) = 4 ^ { - \\frac { 1 } { 3 } } ( \\partial _ x + \\partial _ y ) v ( x , y ) , \\\\ & \\partial _ { x _ 2 } u ( x _ 1 , x _ 2 ) = \\sqrt { 3 } 4 ^ { - \\frac { 1 } { 3 } } ( \\partial _ x - \\partial _ y ) v ( x , y ) , \\\\ & \\partial _ { x _ 1 } ( \\partial _ { x _ 1 } ^ 2 + \\partial _ { x _ 2 } ^ 2 ) u ( x _ 1 , x _ 2 ) = ( \\partial _ x ^ 3 + \\partial _ y ^ 3 ) v ( x , y ) . \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} h ( R , \\l , K ) = \\lim _ { n \\to \\infty } \\frac { H ( B _ { R , \\l , K } ^ { ( n ) } ) } { n } , \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} G ( t ; x , q , u , 1 , v ) & = G ( t ; x , v , u , 1 , q ) \\\\ & = \\frac { q v t } { 1 - q v t u } + \\sum _ { m = 0 } ^ { \\infty } \\frac { r v ( 1 - q r ) ( 1 - r ) ^ m } { ( x - x u + u ( 1 - q r ) ( 1 - r ) ^ m ) ( 1 - q t u v ) } \\\\ & \\quad \\times \\prod _ { i = 0 } ^ { m } \\frac { x ( 1 - ( 1 - q r ) ( 1 - r ) ^ i ) ( x - x u + u ( 1 - q r ) ( 1 - r ) ^ i ) } { ( x - u ( x - 1 ) ( 1 - q r ) ( 1 - r ) ^ i ) ( x - x u + u ( 1 - r v ) ( 1 - q r ) ( 1 - r ) ^ i ) } \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} 0 < \\frac { \\bar { u } _ { \\bar { t } } } { \\bar { u } } & = \\frac { 1 } { ( 1 - C _ 3 ) \\bar { t } + \\frac { N } { l _ i } - ( 1 - C _ 3 ) b _ i } \\\\ & \\le \\frac { 1 } { ( 1 - C _ 3 ) \\bar { t } + \\frac { N } { l _ i } - \\frac { 2 ( 1 - C _ 3 ) } { l _ i } } \\\\ & \\le \\frac 1 { ( 1 - C _ 3 ) \\bar { t } } . \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} K _ { 1 } ( x , \\lambda ( t ) ) & = \\int _ { 0 } ^ { \\infty } \\frac { r d r } { \\lambda ( t ) ^ { 2 } ( 1 + \\frac { r ^ { 2 } } { \\lambda ( t ) ^ { 2 } } ) ^ { 3 } } \\int _ { 0 } ^ { x } \\frac { \\rho d \\rho } { x } \\left ( 1 + \\frac { r ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( r ^ { 2 } - 1 - \\rho ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } \\right ) \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} \\frac { 1 } { \\beta ^ { \\Lambda } \\sum _ { i , j \\in I _ { 0 } , \\beta _ { i j } = \\beta } 1 } \\sum _ { i , j \\in I _ { 0 } , \\beta _ { i j } = \\beta } \\prod _ { x \\in \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * b _ { x } \\right ) \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} ( t , x , y ^ i , \\dot { x } , \\dot { y } ^ i , \\ddot { x } , \\ddot { y } ^ i ) \\in J ^ 2 \\pi \\to \\left ( x , y ^ i , \\frac { \\dot { y } ^ i } { \\dot { x } } , \\frac { \\ddot { y } ^ i \\dot { x } - \\ddot { x } \\dot { y } ^ i } { \\dot { x } ^ 3 } \\right ) = ( x , y ^ i , y ^ i _ { x } , y ^ i _ { x x } ) \\in J ^ 2 ( M , 1 ) \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} \\begin{cases} \\mbox { $ \\widehat D = ( \\hat d _ i ) $ w i t h $ \\hat d _ i : = d _ i | | K _ i | | _ { L ^ 1 ( \\R ) } $ , } \\\\ \\mbox { $ \\mathbf { \\widehat K } = ( \\widehat K _ i ) $ w i t h $ \\widehat K _ i : = \\frac { 1 } { \\| K _ i \\| _ { L ^ 1 ( \\R ) } } K _ i $ , } \\\\ \\mbox { $ \\widehat F ( u ) : = F ( u ) - ( \\widehat D - D ) \\circ u $ f o r $ u \\in \\R ^ m $ . } \\end{cases} \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} x ^ m A _ h = \\sum _ { k + | \\alpha | \\leq m } h ^ m a _ { m k \\alpha } ( h , x , y ) ( x D _ x ) ^ k D _ y ^ \\alpha + \\sum _ { k + | \\alpha | \\leq j \\leq m - 1 } x ^ { m - j } h ^ j a _ { j k \\alpha } ( h , x , y ) ( x D _ x ) ^ k D _ y ^ \\alpha . \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} - \\int _ { \\R ^ 2 } \\Delta _ q ( u \\cdot \\nabla X ) \\Delta _ q X ~ d x \\leq & C b _ q 2 ^ { - 2 q s } \\| \\nabla u \\| _ { L ^ \\infty } \\| X \\| _ { H ^ s } ^ 2 . \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} \\{ F _ { \\rho _ { 1 } } ( x ) : x \\in I \\} = \\{ s + i f ( s ) : s \\in J \\} , \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} q = \\frac { \\sin \\phi _ 2 } { 1 + \\cos \\phi _ 2 } = \\frac { 4 A } { ( u _ 1 + u _ 3 ) ^ 2 - u _ 2 ^ 2 } \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} \\bigcup _ { 1 \\leq i \\ne j \\leq k - 2 } D ( V _ { l - 1 + i } , \\widetilde { V } _ { l - 1 + j } ) = & \\bigcup _ { 1 \\leq i \\ne j \\leq k - 2 } D ( \\{ ( i , 0 ) \\} , k \\boxtimes \\{ ( j , 0 ) \\} ) \\\\ = & k \\boxtimes \\left ( \\bigcup _ { 1 \\leq i \\ne j \\leq k - 2 } D ( \\{ ( i , 0 ) \\} , \\{ ( j , 0 ) \\} ) \\right ) \\\\ = & k ( k - 3 ) \\boxtimes ( ( \\Z _ { k - 1 } \\backslash \\{ 0 \\} ) \\times \\{ 0 \\} ) . \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} E _ { \\alpha , i } \\cdot ( \\mathfrak { A } | _ { X _ { \\alpha } } ) ^ { n - 1 } \\le \\mathfrak { D } \\cdot X _ { \\alpha } \\cdot \\mathfrak { A } ^ { n - 1 } = ( - m K _ { \\mathfrak { X } } + m \\mathfrak { h } ^ * \\mathfrak { L } - \\mathfrak { A } ) \\cdot X _ { \\alpha } \\cdot \\mathfrak { A } ^ { n - 1 } . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} \\eta = \\begin{cases} 1 & C _ { \\overline { s } , 1 } ^ { + } , \\\\ 0 & C _ { \\overline { s } , 2 } ^ { + } , \\end{cases} \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} \\mathcal { W ' } _ { k } \\left [ I ; J \\right ] , \\quad \\mbox { f o r a l l $ k = 3 , \\dots , p + 1 $ . } \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d x _ { t } & = ( F _ { t } x _ { t } + f _ { t } ) d t + d w _ { t } , \\\\ x ( 0 ) & = x _ { 0 } , \\\\ d m _ { t } & = ( G _ { t } { x } _ { t } + g _ { t } ) d t + d v _ { t } , \\\\ m ( 0 ) & = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} \\partial _ { t } u = \\left ( a + i b \\right ) \\left [ \\Delta u + A \\left ( x \\right ) u + V \\left ( x , t \\right ) u + F \\left ( x , t \\right ) \\right ] , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "9456.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } \\frac { 1 } { 2 } \\alpha ' ( t ) & 0 \\\\ 0 & \\frac { 1 } { 2 } \\beta ' ( t ) \\\\ \\end{array} \\right ) = \\left ( \\begin{array} { c c } - \\frac { 1 } { c ( t ) } \\alpha ( t ) & 0 \\\\ 0 & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} \\begin{aligned} f _ { 1 , i } ( t ) & = 0 , f _ { 2 , i } ( t ) = V ^ t f _ { 0 , i } , \\\\ f _ { n , i } ( t ) & = V ^ t f _ { 0 , i } + \\int _ 0 ^ t V ^ { t - s } B ( s , f _ { n - 1 , i } , f _ { n - 2 , i } ) d s , ( t \\geq 0 ) ; n = 3 , 4 , . . . , \\end{aligned} \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} \\partial _ { t } u - i \\left [ \\Delta u + A u + V \\left ( x , t \\right ) u \\right ] = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} z ( \\tau + 1 ) = \\beta P ^ r _ \\mathcal { M } \\{ z ( \\tau ) - \\alpha G _ r ( \\tau ) \\} . \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} \\beta & = a _ { 3 L } b _ { 2 L } - a _ { 2 L } b _ { 3 L } \\ , , \\\\ \\gamma & = a _ { 2 L } b _ { 0 R } - a _ { 0 R } b _ { 2 L } \\ , , \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} { } ^ b \\Psi ^ { - \\infty , \\epsilon } _ { G , c } ( M ) = \\{ T _ { \\widetilde { k } } , \\ ; \\widetilde { k } \\in ( C ^ \\infty _ c ( G ) \\hat { \\otimes } \\ , { } ^ b \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } \\} \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} \\Theta : N \\rightarrow \\mathcal { S } : = \\{ ( h _ i ) \\in \\R ^ 4 : h _ 1 > 0 , ( h _ 1 ) ^ 2 \\ne \\epsilon , h _ 2 \\ne 0 \\} \\subset \\R ^ 4 . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} ( Q _ t ) _ { ( 0 ) } = d _ t ^ { s t } + d ^ { \\chi } , ( d _ t ^ { s t } ) ^ 2 = ( d ^ { \\chi } ) ^ 2 = \\{ d _ t ^ { s t } , d ^ { \\chi } \\} = 0 , \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} \\forall l \\in I ^ { 0 0 } ( \\bar x ) \\colon \\mu _ l \\nu _ l = 0 , \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} \\varepsilon = \\varepsilon ^ a X ^ * e _ a = \\tilde { \\varepsilon } ^ a X ^ * \\tilde { e } _ a = \\tilde { \\varepsilon } ^ a K ( \\sigma ) ^ b { } _ { a } X ^ * e _ b , \\quad \\Longrightarrow \\tilde { \\varepsilon } ^ a = ( K ^ { - 1 } ) ^ a { } _ b \\varepsilon ^ b . \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} \\phi ^ * _ t ( u _ 1 \\sigma _ 1 + u _ 2 \\sigma _ 2 ) = e ^ { \\lambda _ 1 t } u _ 1 \\sigma _ 1 + e ^ { \\lambda _ 2 t } u _ 2 \\sigma _ 2 \\ , , \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} | | | P | | | ^ 2 : = \\| \\chi P \\| ^ 2 _ 1 + \\| \\phi [ \\mathcal { V } , P ] \\| ^ 2 _ 1 + \\| [ \\mathcal { V } , P ] \\| ^ 2 _ 1 + \\| [ \\phi , P ] \\| ^ 2 + \\| P \\| ^ 2 \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} \\mathfrak S _ f ( \\mathfrak R ) = \\mathfrak S _ f ( \\mathfrak { R } _ { n i l } ^ { ( 2 ) } ) . \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} | z ' ( t ) | = \\sqrt { R ( t ) ^ { 2 } + R ' ( t ) ^ { 2 } . } \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} \\mathcal G = \\bigcap _ n \\mathcal G _ { C _ n , I _ n } . \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} \\check C _ n ( \\mathcal R ^ 2 ; \\omega ) = \\coprod _ { a _ 0 , \\dots , a _ n } \\tilde { \\mathcal R } ^ 2 _ { a _ 0 } \\bigcap \\tilde { \\mathcal R } ^ 2 _ { a _ 1 } \\bigcap \\dots \\bigcap \\tilde { \\mathcal R } ^ 2 _ { a _ n } , \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} t _ 1 ^ { \\kappa - p / 2 + 1 } \\int _ { t _ 1 } ^ { 1 / 2 } \\frac { \\displaystyle | Z _ N ( t ) - \\sigma B _ N ( t ) | | B _ N ( t ) | ^ { p - 1 } } { t ^ { \\kappa } } d t = O _ P \\left ( ( N t _ 1 ) ^ { - 1 / 2 + \\zeta } \\right ) = o _ P ( 1 ) \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} Q ( A _ j , A _ { n + p } ) \\ = \\ & 1 \\ 1 \\leq p \\leq j , \\\\ Q ( A _ { n + j } , A _ { q } ) \\ = \\ & 1 \\ 0 \\leq q \\leq j - 1 . \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow 0 } \\frac { \\overline { v } ( r ) } { r \\langle \\log ( r ) \\rangle } = \\int _ { 0 } ^ { \\infty } \\overline { y } ( \\xi ) \\lim _ { r \\rightarrow 0 } \\left ( \\frac { \\phi ( r , \\xi ) } { r ^ { 3 / 2 } \\langle \\log ( r ) \\rangle } \\right ) \\rho ( \\xi ) d \\xi = 0 \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} i \\partial _ { t } \\upsilon _ { n } + \\Delta \\upsilon _ { n } + A \\upsilon _ { n } = V _ { n } \\left ( x , t \\right ) \\upsilon _ { n } + \\phi _ { n } \\left ( x _ { 1 } \\right ) F \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} \\frac { N ! } { n _ 1 ! \\cdots n _ s ! } \\Big ( \\frac { n _ 1 } { N } \\Big ) ^ { n _ 1 } \\cdots \\Big ( \\frac { n _ s } { N } \\Big ) ^ { n _ s } \\le \\Big ( \\frac { n _ 1 } { N } + \\cdots + \\frac { n _ s } { N } \\Big ) ^ N = 1 . \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} \\mathfrak m ^ { ( i _ 1 , \\ldots , i _ r ) } = \\widetilde { \\mathfrak n } _ 1 + \\mathfrak m _ 1 ^ - + \\ldots + \\widetilde { \\mathfrak n } _ k + \\mathfrak m _ k ^ - + \\widetilde { \\mathfrak n } _ { k + 1 } . \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} \\sum _ { s = i } ^ { n - 1 } m _ { [ s , s ] } m _ { [ s + 1 , s + 1 ] } = ( n - i ) m ^ { 2 } _ { [ i , n ] } + \\cdots . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} \\alpha ^ { [ \\mathbb { Q } ( \\alpha ) : \\mathbb { Q } ] } = N _ { \\mathbb { Q } ( \\alpha ) / \\mathbb { Q } } ( \\alpha ) u \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} \\rho ( L _ 0 ) ^ 2 \\ , = \\ , \\rho ( L _ 0 ) + \\rho ( L _ { - 1 } ) \\rho ( L _ 1 ) + \\frac 1 4 ( \\rho ( \\mathcal C ) - 1 ) \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( z ) = \\frac { \\psi _ { \\mu } ( z ) } { 1 + \\psi _ { \\mu } ( z ) } , z \\in \\mathbb { D } . \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} \\Lambda _ { j , t } ( r \\ , d y ) = \\frac { c _ j ( t , r y ) } { r ^ { \\alpha } | y ^ j | ^ { 1 + \\alpha } } d y ^ j \\epsilon _ 0 ( d y ^ 1 , \\cdots , d y ^ { j - 1 } , d y ^ { j + 1 } , \\cdots , d y ^ d ) . \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} k _ s = A _ s k _ s \\circ ( A _ c + r ) ^ { - 1 } + g _ s \\circ K { } { } \\circ ( A _ c + r ) ^ { - 1 } . \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} \\varlimsup _ { R \\to + 0 } \\biggl [ \\ , \\lim _ { T \\to + 0 } \\Bigl ( \\frac { 1 } { R ^ 2 } \\sup _ { ( 0 , T ] \\times D _ R } | u ( t , x ) | \\Bigr ) \\biggr ] = 0 . \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} c _ k ' ( 0 ^ + ) = & - C _ s \\left ( \\int _ { 0 } ^ { + \\infty } \\ln | 1 - \\tau ^ 4 | \\tau ^ { - ( 1 + 2 s ) } d \\tau \\right . \\\\ & \\qquad \\quad \\left . + \\ , ( k - 2 ) \\int _ { 0 } ^ { \\infty } \\ln ( 1 + \\tau ^ 2 ) \\tau ^ { - ( 1 + 2 s ) } d \\tau \\right ) \\mbox { f o r } \\ k \\geq 2 . \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} C \\Phi _ { 1 3 } ( q ) = \\frac { 1 } { ( q ^ { 1 3 } ; q ^ { 1 3 } ) _ { \\infty } } + 1 3 \\sum _ { n = 0 } ^ { \\infty } p ( 1 3 n - 7 ) q ^ n + 2 6 \\cdot q \\prod _ { n = 1 } ^ { \\infty } \\frac { ( 1 - q ^ { 1 3 n } ) } { ( 1 - q ^ n ) ^ 2 } . \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} \\partial ^ v _ { n - 1 } \\left [ Q ( u , v ) ( v _ { n - 1 } - u _ n ) ^ 2 ( v _ n - u _ n ) u _ n ^ { k - 1 } \\right ] & = Q ( u , v ) ( v _ { n - 1 } - u _ n ) ( v _ n - u _ n ) u _ n ^ { k - 1 } \\partial _ { n - 1 } ^ v ( v _ { n - 1 } - u _ n ) \\\\ & = Q ( u , v ) ( v _ { n - 1 } - u _ n ) ( v _ n - u _ n ) u _ n ^ { k - 1 } . \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} \\mu _ { ( T , q ) } ( O ( Z _ n , r _ n ) ) > \\mu _ n ( Z _ n ) - \\frac { 1 } { 2 n } > 1 - \\frac { 1 } { 2 n } - \\frac { 1 } { 2 n } = 1 - \\frac { 1 } { n } . \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} F ^ n ( \\Lambda \\otimes H ) : = \\{ \\lambda \\otimes h \\in \\Lambda \\otimes H : | h | \\leq n \\} . \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} \\Delta ^ x _ { [ x _ 0 : x _ 1 ] } = & t ^ 2 \\Big [ ( d _ { - 1 , 0 } x _ 1 ^ 2 - \\frac { 1 } { t } x _ 0 x _ 1 + d _ { 1 , 0 } x _ 0 ^ 2 ) ^ 2 \\\\ & - 4 ( d _ { - 1 , 1 } x _ 1 ^ 2 + d _ { 0 , 1 } x _ 0 x _ 1 + d _ { 1 , 1 } x _ 0 ^ 2 ) ( d _ { - 1 , - 1 } x _ 1 ^ 2 + d _ { 0 , - 1 } x _ 0 x _ 1 + d _ { 1 , - 1 } x _ 0 ^ 2 ) \\Big ] . \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} \\left ( \\begin{array} { l } \\nabla f ( x ^ * ) + \\displaystyle \\sum _ { j = 1 } ^ p [ y ^ I ] ^ * _ j \\nabla ^ 2 c _ j ( x ^ * ) \\eta _ 1 ^ * + - A ^ T A \\eta _ 1 ^ * + { \\cal J } c ( x ^ * ) ^ T [ \\xi _ b ] ^ * \\\\ { \\cal J } c ( x ^ * ) \\eta _ 1 ^ * + [ \\xi _ a ] ^ * + [ \\xi _ b ] ^ * \\\\ \\end{array} \\right ) = 0 . \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k - 1 \\} = \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} & ( m _ B \\otimes 1 _ X ) ( 1 _ B \\otimes l ^ * ) = l ^ * l = ( 1 _ B \\otimes l ) ( m _ B ^ * \\otimes 1 _ X ) , \\\\ & ( 1 _ X \\otimes m _ A ) ( r ^ * \\otimes 1 _ A ) = r ^ * r = ( r \\otimes 1 _ A ) ( 1 _ X \\otimes m _ A ^ * ) . \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} Q _ { \\ker } = 0 . \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} a _ { N _ 1 + N _ 2 } ( \\sigma ) - a _ { N _ 1 } ( \\sigma ) - a _ { N _ 2 } ( \\sigma ) = \\log \\left ( \\mathbb { E } _ { \\mu _ { N _ 1 } ^ { \\sigma } \\bigotimes \\mu _ { N _ 2 } ^ { \\sigma } } \\left [ \\exp \\left ( - \\sum _ { ( i , j ) \\in I _ { N _ 1 , N _ 2 } } M _ { i j } u _ i v _ j \\right ) \\right ] \\right ) . \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} \\partial ^ 2 ( h _ { i j } ) = \\begin{pmatrix} 0 & 0 & \\bar { a _ { i , i } } & \\bar { a _ { i , j } } \\\\ 0 & 0 & \\bar { a _ { j , i } } & \\bar { a _ { j , j } } \\\\ a _ { i , i } & a _ { i , j } & * & * \\\\ a _ { i , j } & a _ { j , j } & * & * \\end{pmatrix} \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} J ^ { \\sigma } _ u ( r , K r ) = \\int _ r ^ { K r } t ^ { n - 1 - n - 2 \\sigma } \\ d t \\int _ S u ^ 2 \\ d \\omega = \\sum _ { k \\geq 1 } \\int _ r ^ { K r } t ^ { - 1 - 2 \\sigma } v _ k ( t ; c _ k ^ + , c _ k ^ - ) ^ 2 \\ d t \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} \\int _ X \\varphi \\psi d \\mu = \\iint _ { X \\times X } \\varphi \\Phi d \\pi \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} \\{ x \\mapsto ( \\Delta ^ { ( n ) } ) ^ { - 1 } \\Theta ( \\l _ n , K ) \\Delta ^ { ( n ) } x + ( \\Delta ^ { ( n ) } ) ^ { - 1 } v _ j ( R _ n , \\l _ n , K ) : j = 1 , \\ldots , m \\} . \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} \\sum _ { m \\in \\mathbb { Z } } \\mu ( T _ m ( \\phi _ 1 ) ) x ^ m & = \\left ( \\frac { 1 + x ^ { - 1 } } { 2 } \\right ) \\left ( \\frac { x } { 6 } + \\frac { 5 } { 6 } \\right ) \\left ( \\frac { 2 7 } { 4 8 } + \\frac { 2 1 } { 4 8 } x ^ { - 1 } \\right ) \\\\ & = \\frac { 3 5 } { 1 9 2 } x ^ { - 2 } + \\frac { 2 9 } { 6 4 } x ^ { - 1 } + \\frac { 6 1 } { 1 9 2 } + \\frac { 3 } { 6 4 } x . \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} Q ( x ^ { \\Lambda _ 2 } | y ) = - \\log \\int _ { \\substack { \\frac { 1 } { K - R } \\sum _ { i \\in \\Lambda _ 1 ^ { ( k ) } } x _ i = \\tilde { y } _ k \\\\ k \\in [ M ] \\setminus \\{ l \\} } } \\exp \\left ( - H ( \\bar { x } ^ { B ( l ) } ) \\right ) \\exp \\left ( - Q _ l ( x ^ { \\Lambda _ 2 ^ { ( l ) } } | \\bar { x } ^ { B ( l ) } ) \\right ) \\mathcal { L } ( d \\bar { x } ^ { \\Lambda _ 1 ^ { ( l ) } } ) , \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} \\begin{aligned} z ( t _ i ) - z ( t _ j ) & = z ' ( t _ i - t _ j ) + \\frac 1 2 z '' ( t _ i ^ 2 - t _ j ^ 2 ) + \\frac 1 6 z ''' ( t _ i ^ 3 - t _ j ^ 3 ) + . . . \\\\ & = ( t _ i - t _ j ) ( z ' + \\frac 1 2 z '' ( t _ i + t _ j ) + \\frac 1 6 z ''' ( t _ i ^ 2 + t _ i t _ j + t _ j ^ 2 ) ) + . . . \\end{aligned} \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} c _ i ( \\cal E ) = c _ { i , 0 } \\otimes 1 + c _ { i , 1 } \\otimes p , \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} \\int _ D P _ D \\lambda ( x ) L \\varphi ( x ) d x = - \\int _ D \\left ( \\int _ { D ^ c } j ( | x - y | ) \\lambda ( d y ) \\right ) \\varphi ( x ) d x . \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { \\dd } { \\dd t } \\left [ \\frac { 1 } { 2 \\tau ^ 2 } R | \\nabla \\log R | ^ 2 \\right ] + \\dfrac { \\dot { \\tau } } { \\tau ^ 3 } \\int _ { \\mathbb T ^ d _ { \\ell } } 4 | \\nabla \\sqrt { R } | ^ 2 + \\dfrac { \\delta _ 1 } { \\tau ^ 4 } \\int _ { \\mathbb T ^ d _ \\ell } R | \\nabla ^ 2 \\log R | ^ 2 \\\\ & = \\dfrac { 1 } { \\tau ^ 4 } \\int _ { \\mathbb T ^ d _ { \\ell } } R \\nabla U : \\nabla ^ { 2 } \\log R . \\end{aligned} \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} K i r _ \\varepsilon \\Phi ( x ) = \\frac { 1 } { h ^ \\varepsilon ( x ) } \\int _ { \\{ y : \\abs { x - y } > \\varepsilon \\} } \\Phi ( x , y ) \\frac { 1 } { x - y } d y \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} R \\setminus \\bar S = \\bigsqcup _ { i = 1 } ^ k D _ i , \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} = \\left ( \\prod _ { i = 1 } ^ { n + 1 } ( 1 + x _ i ) \\right ) \\sum _ { J \\subseteq [ 2 , n ] } ( - 1 ) ^ { | J | } \\left ( \\prod _ { j \\in J } \\frac { x _ j } { 1 + x _ j } \\right ) p _ { n - | J | } ( \\underline x _ n ^ J ) . \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} w t _ { s p } ( C ) = \\begin{cases} 2 & \\mbox { i f ~ } t _ 3 = 0 , \\\\ 3 & \\mbox { i f ~ } t _ 3 = 1 , \\\\ 4 & \\mbox { i f ~ } 2 \\le t _ 3 \\le 2 5 , \\\\ 6 & \\mbox { i f ~ } 2 6 \\le t _ 3 \\le 5 0 , \\\\ 8 & \\mbox { i f ~ } 5 1 \\le t _ 3 \\le 6 2 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} ( \\lambda \\ast _ { G / H } \\lambda ' ) ^ { \\ast ^ { G / H } } = \\lambda '^ { \\ast ^ { G / H } } \\ast _ { G / H } \\lambda ^ { \\ast ^ { G / H } } \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} \\lim _ { \\xi \\rightarrow \\overline { c _ { 0 } } ^ { + } } \\frac { p _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( \\xi ) } { \\left | \\xi - \\overline { c _ { 0 } } \\right | ^ { 1 / 3 } } = \\frac { - 2 \\left | a _ { 1 } \\right | } { \\sqrt [ 3 ] { \\left | c _ { 3 } \\right | } } \\cos \\left ( \\frac { \\theta } { 3 } - \\frac { 5 \\pi } { 6 } \\right ) , \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} \\int _ { \\Sigma } | \\nabla \\hat { v } _ \\infty | ^ 2 - ( | A _ { \\Sigma } | ^ 2 + R i c _ M ( \\nu , \\nu ) ) \\hat { v } _ \\infty ^ 2 \\ d x = \\int _ { \\Sigma } - \\hat { v } _ \\infty \\cdot L _ { \\Sigma } \\hat { v } _ \\infty \\geq 0 \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} \\langle M y , \\Delta _ { \\rho } ( y ) \\rangle = \\langle M y , \\rho _ - \\Delta ( \\rho _ + y ) \\rangle = \\langle M \\rho _ - \\rho _ + y , \\rho _ - \\Delta ( \\rho _ + y ) \\rangle \\geq 0 \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} T ( \\omega ) u = 0 . \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} \\tau ^ { 3 ^ { n + 1 } } ( d ) ( k ) = \\tau ^ { 3 ^ { n + 1 } } ( d _ { n + 1 } ) ( k ) = d _ n ( k ) = d ( k ) . \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} \\langle u ^ - , v ^ - \\rangle _ { * \\mathcal R ^ - } = \\int _ 0 ^ \\infty \\Big [ \\ , A _ - \\big ( { t ^ - } ^ 2 - { U ^ - } ^ 2 \\big ) - B \\big ( { t ^ + } ^ 2 - { U ^ + } ^ 2 \\big ) \\ , \\Big ] ( u ^ - _ 1 v ^ - _ 1 + u ^ - _ 2 v ^ - _ 2 ) r \\ , { \\mathrm d } r . \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} \\tilde { F } _ { \\varepsilon } \\left ( x , t \\right ) = \\frac { i } { \\varepsilon + i } e ^ { \\varepsilon t \\left ( \\Delta + A \\right ) } \\tilde { V } \\left ( x , t \\right ) \\tilde { u } \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} m & = \\frac { 1 } { 2 } \\left [ J ^ { - 1 } \\left ( I ^ { [ 2 ] } _ { V S } \\right ) \\right ] ^ 2 . \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} \\widetilde { t } ^ { ( i ) } = \\sum _ { k = 1 } ^ l \\beta _ k b ^ { ( i ) } _ k , \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} \\mathcal { H } _ p ( X ) = H _ { p } ( \\mathbb { T } ^ { \\infty } , X ) \\ , 1 \\le p < \\infty . \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} \\kappa _ g > 0 \\ ; \\mbox { i f } \\ ; q _ 1 - q _ 2 > \\ln 2 , \\ ; \\kappa _ g = 0 \\ ; \\mbox { i f } \\ ; q _ 1 - q _ 2 = \\ln 2 , \\ ; \\mbox { a n d } \\ ; \\kappa _ g < 0 \\ ; \\mbox { i f } \\ ; q _ 1 - q _ 2 < \\ln 2 . \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} \\int _ { z _ 1 } ^ { z _ 2 } e h ^ n \\ ; d z = 0 . \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} \\delta _ 0 = \\eta \\inf _ { t \\in [ 0 , T ] } \\big \\{ \\big [ \\omega ( t , \\lambda _ 1 ) + ( \\ell + \\epsilon ) \\omega ( \\cdot , \\lambda _ 1 ) * p ( t ) \\big ] ^ { - 1 } \\big [ 1 - ( \\ell + \\epsilon ) \\omega ( \\cdot , \\lambda _ 1 ) * p ( t ) \\big ] \\big \\} , \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} v _ { s } ( t , r ) \\sim \\frac { 2 \\lambda '' ( s ) \\left ( 1 - \\sqrt { 1 - a ^ { 2 } } \\right ) } { a } , 0 < a = \\frac { r } { ( s - t ) } < 1 \\ll s - t \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\phi _ { j } ^ { ( 2 ) } ( x _ n ) } { \\phi _ { j } ^ { ( 1 ) } ( x _ n ) } = \\frac { ( \\phi _ j ^ 2 ) '' ( 0 ^ - ) } { ( \\phi _ j ^ 1 ) '' ( 0 ^ - ) } < p \\mbox { i f } j \\in \\{ m _ 0 + 1 , . . . , m \\} \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} \\dot { h } = \\frac { 1 } { 2 } \\left ( a ^ 2 - h ^ 2 \\right ) . \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} a \\prec b = a N ( b ) , a \\succ b = N ( a ) b ~ ~ ~ ~ ~ ~ a \\curlyvee b = - N ( a b ) \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} ( f B ( g , h ) + B ( f , h ) g ) ( x , x ) = f ( x , x ) B ( g , h ) ( x , x ) + B ( f , h ) ( x , x ) g ( x , x ) = 0 , \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} \\sqrt { R _ { 0 , \\ell } } : = S ^ { 0 } _ { \\ell , \\theta _ { \\ell } , \\iota _ { \\ell } } , \\forall \\ , \\ell \\in \\mathbb N ^ * , \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} & { } _ 4 \\phi _ 3 \\ ! \\left [ \\begin{matrix} ( 1 - r ) ^ j , 1 - a , b , c \\\\ d , e , ( 1 - r ) ^ { j + 1 } ( 1 - a ) b c / d e \\end{matrix} ; 1 - r , 1 - r \\right ] \\\\ & = \\frac { ( ( 1 - r ) / e , ( 1 - r ) ( 1 - a ) b c / d e ; 1 - r ) _ j } { ( ( 1 - r ) ( 1 - a ) / e , ( 1 - r ) b c / d e ; 1 - r ) _ j } \\\\ & \\quad \\ ; \\times { } _ 4 \\phi _ 3 \\ ! \\left [ \\begin{matrix} ( 1 - r ) ^ j , 1 - a , d / b , d / c \\\\ d , d e / b c , ( 1 - r ) ^ { j + 1 } ( 1 - a ) / e \\end{matrix} ; 1 - r , 1 - r \\right ] . \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} & \\lambda _ { 1 } = m _ { 1 2 } + m _ { 1 3 } + m _ { 1 4 } + n _ { 1 4 } + n _ { 1 3 } + n _ { 1 2 } , \\\\ & \\lambda _ { 2 } = m _ { 2 3 } + m _ { 1 4 } + m _ { 1 3 } + m _ { 2 4 } + n _ { 2 4 } + n _ { 2 3 } + n _ { 1 3 } + 2 n _ { 1 2 } + n _ { 1 4 } , \\\\ & \\lambda _ { 3 } = m _ { 3 4 } + m _ { 1 4 } + m _ { 2 4 } + n _ { 2 3 } + n _ { 1 3 } + n _ { 1 2 } , \\\\ & \\lambda _ { 4 } = n _ { 3 4 } + n _ { 1 4 } + n _ { 2 4 } + n _ { 2 3 } + n _ { 1 3 } + n _ { 1 2 } , \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} \\nu ( M , N ) = F _ { - r } \\supset \\cdots F _ { - 1 } \\supset F _ 0 = N , \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} & \\frac { - c _ { b } r } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\sin ^ { 2 } ( \\theta ) d \\theta \\int _ { 0 } ^ { \\frac { 1 } { 2 } } d \\xi \\left ( \\frac { b - 1 } { \\xi \\log ^ { b } ( \\frac { 1 } { \\xi } ) } + \\frac { b ( b - 1 ) } { \\xi \\log ^ { b + 1 } ( \\frac { 1 } { \\xi } ) } \\right ) \\left ( \\frac { \\sin ( \\xi t _ { + } ) } { t _ { + } ^ { 2 } } + \\frac { \\sin ( \\xi t _ { - } ) } { t _ { - } ^ { 2 } } \\right ) \\\\ & = \\frac { - b r } { t ^ { 2 } \\log ^ { b } ( t ) } + E _ { v _ { 2 } , 3 } ( t , r ) \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} \\langle \\rho ( p ) \\rho ( - p ) \\rangle & = \\langle \\phi ( - p ) \\phi ( p ) \\rangle + \\sum _ { k = 2 } ^ \\infty \\frac { \\left ( b _ k \\right ) ^ 2 } { k ! } M ^ { ( k - 1 ) } ( p ) . \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { r = 1 } ^ { k - 1 } \\sum _ { s = 1 } ^ { k - 1 } b ^ { ( k ) } _ { i r } a ^ { ( m ) } _ { r s } w ^ { ( m ) } _ { s j } & = \\sum _ { r = 1 } ^ { k - 1 } \\sum _ { s = 1 } ^ { k - 1 } ( - 1 ) ^ { r - i + j - s } \\binom { k - i } { r - i } \\binom { m - r } { s - r } \\binom { m - k + j - s - 1 } { j - s } \\\\ & = \\sum _ { s = 1 } ^ { k - 1 } ( - 1 ) ^ { j - s } \\binom { m - k + j - s - 1 } { j - s } \\sum _ { r = 1 } ^ { k - 1 } ( - 1 ) ^ { r - i } \\binom { k - i } { r - i } \\binom { m - r } { s - r } . \\end{aligned} \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} \\left \\vert \\left ( \\nu _ { \\mathsf { k } } - a _ { \\mathsf { m } ^ { \\ast } , \\mathsf { m } ^ { \\ast } } \\right ) \\right \\vert \\leqslant \\sum _ { \\mathsf { n = 1 } , \\mathsf { n \\neq m } ^ { \\ast } } ^ { + \\infty } r _ { \\mathsf { n , m } ^ { \\ast } } = \\left \\vert a _ { \\mathsf { m } ^ { \\ast } , \\mathsf { m } ^ { \\ast } } \\right \\vert \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} p _ a ( \\Sigma ) = P ( s , t ) = \\sum _ { i = 1 } ^ { + \\infty } s - h ( i ) = t ^ 3 - 5 t ^ 2 + ( \\beta + 7 ) t + \\frac { \\beta ^ 2 - 7 \\beta } { 2 } - 1 . \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} \\mathcal S ^ { \\frac { 3 } { 2 } } ( \\partial \\Omega ) = \\mathcal S ^ { \\frac { 3 } { 2 } } _ { \\lambda } ( \\partial \\Omega ) : = \\left \\{ f \\in L ^ 2 ( \\partial \\Omega ) : f = \\sum _ { j = 1 } ^ { \\infty } \\hat a _ j \\hat u _ { j , \\lambda } { \\rm \\ w i t h \\ } \\left ( \\sqrt { | \\mu _ j ( \\lambda ) | } \\hat a _ j \\right ) _ { j = 1 } ^ { \\infty } \\in l ^ 2 \\right \\} , \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} I I _ { r r } = \\frac { c _ { b } } { 4 } \\int _ { 0 } ^ { \\infty } \\sin ( t \\xi ) \\xi ^ { 2 } \\left ( 1 - \\chi _ { \\leq 1 } ( r \\xi ) \\right ) \\left ( - 3 J _ { 1 } ( r \\xi ) + J _ { 3 } ( r \\xi ) \\right ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } d \\xi \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} y ^ 1 \\leq y ^ 2 & \\Leftrightarrow y ^ 1 _ k \\leq y ^ 2 _ k k = 1 , 2 y ^ 1 \\neq y ^ 2 , \\\\ y ^ 1 \\leqq y ^ 2 & \\Leftrightarrow y ^ 1 _ k \\leq y ^ 2 _ k k = 1 , 2 . \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} \\sum _ { e } \\frac { d u _ e } { d x _ e } ( v ) = 0 \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} \\lim _ n \\frac { H ( \\mu _ { \\lambda } ^ { ( n ) } ; \\lambda ^ { 1 0 n } ) } { n \\log \\lambda ^ { - 1 } } = \\dim \\mu _ { \\lambda } . \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} - g _ { 1 1 } \\kappa _ g = \\left ( \\frac { \\partial \\Gamma _ { 1 2 } ^ 2 } { \\partial s _ 1 } - \\frac { \\partial \\Gamma _ { 1 1 } ^ 2 } { \\partial s _ 2 } + \\Gamma _ { 1 2 } ^ 1 \\Gamma _ { 1 1 } ^ 2 + \\Gamma _ { 1 2 } ^ 2 \\Gamma _ { 1 2 } ^ 2 - \\Gamma _ { 1 1 } ^ 1 \\Gamma _ { 1 2 } ^ 2 - \\Gamma _ { 1 1 } ^ 2 \\Gamma _ { 2 2 } ^ 2 \\right ) , \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} x _ 0 : = x , x _ { 2 n + 1 } : = \\Pi ^ 2 _ { M } x _ { 2 n } \\quad x _ { 2 n } : = \\Pi ^ 2 _ { N } x _ { 2 n - 1 } \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} W _ { 1 } = 2 P _ { 1 } u + i \\left ( A _ { 1 1 } \\gamma _ { \\overline { 1 } } - \\gamma _ { 1 , 0 } \\right ) \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} \\det ( z I _ n - ( D + P ) ) = \\det ( z I _ n - D ) \\cdot \\det ( I _ n - ( z I _ { n } - D ) ^ { - 1 } P ) . \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} \\int _ H \\psi ( x h y H ) d h & = \\int _ H \\psi ( x y y ^ { - 1 } h y H ) d h \\\\ & = \\int _ H \\psi ( x y H ) d h = \\psi ( x y H ) . \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} \\mathbf { A } = ( A ( 1 ) , \\dotsc , A ( \\nu ) ) , \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} \\forall r \\in [ s , t ] , \\widetilde { I } _ { r } ^ N & : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ s ^ r \\nabla e ^ { ( t - u ) \\Delta } V ^ N ( X _ u ^ { i , N } - x ) \\cdot d W ^ i _ u \\ , , \\\\ \\forall r \\in [ 0 , s ] , \\widetilde { I I } _ { r } ^ N & : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ 0 ^ r \\nabla e ^ { ( s - u ) \\Delta } \\left [ e ^ { ( t - s ) \\Delta } V ^ N ( X _ u ^ { i , N } - x ) - V ^ N ( X _ u ^ { i , N } - x ) \\right ] \\cdot d W ^ i _ u \\ , . \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} & \\partial _ { t } \\left ( - 1 6 \\int _ { t } ^ { \\infty } \\lambda '' ( s ) \\left ( K _ { 3 } ( s - t , \\lambda ( t ) ) - K _ { 3 , 0 } ( s - t , \\lambda ( t ) ) \\right ) d s \\right ) \\\\ & = - 1 6 \\int _ { t } ^ { \\infty } \\lambda ''' ( s ) \\left ( K _ { 3 } ( s - t , \\lambda ( t ) ) - K _ { 3 , 0 } ( s - t , \\lambda ( t ) ) \\right ) d s \\\\ & - 1 6 \\int _ { t } ^ { \\infty } \\lambda '' ( s ) \\left ( \\partial _ { 2 } ( K _ { 3 } - K _ { 3 , 0 } ) ( s - t , \\lambda ( t ) ) \\right ) \\lambda ' ( t ) d s \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} b _ { l - 2 } = \\frac { ( 8 - \\mu ) { l \\choose 2 } + 2 4 { l \\choose 3 } + 1 6 { l \\choose 4 } } { 2 ( l ) } . \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} \\lim _ { c \\nearrow C _ * } \\Phi ^ c ( x ) = 0 \\mbox { l o c a l l y u n i f o r m l y i n } ( - \\infty , 0 ] . \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} \\Delta t \\sum _ { n = 1 } ^ m \\big ( 1 + \\| \\partial _ t ^ + u _ { I h } ^ n \\| _ { \\mathcal { T } _ h } ^ 2 + \\| \\partial _ t ^ + u _ { h } ^ n \\| _ { \\mathcal { T } _ h } ^ 2 \\big ) \\leq C . \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} S = | G | \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} s & = \\ , ( 0 , 1 , 2 , 0 , 1 , 4 , 4 , { \\bf 1 } , 5 , 2 , 1 , 3 , 1 ) \\mbox { b y s u b s t i t u t i o n } ( 6 ) \\mbox { o f } \\R _ 2 , \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , 4 , 4 , 5 , 4 , 2 , { \\bf 1 } , 3 , 1 ) \\mbox { b y s u b s t i t u t i o n } ( 5 ) \\mbox { o f } \\R _ 2 , \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , 4 , 4 , 5 , 4 , 2 , 4 , 3 , 1 ) . \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} K _ { n + 3 } ^ { ( 3 ) } = K _ { n + 2 } ^ { ( 3 ) } + K _ { n + 1 } ^ { ( 3 ) } + 2 K _ { n } ^ { ( 3 ) } , \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} f _ k ( \\bar { x } ) = f _ 1 \\left ( \\left ( \\mathbf { 1 } _ { \\{ - \\overleftarrow { H ^ { k - 1 } _ W ( \\Delta ( { w } ) ) } _ n = 1 \\} } \\right ) _ { n = 1 } ^ l \\right ) , \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} b \\vee ( a \\wedge \\neg b ) \\vee ( c \\wedge \\neg d ) & = b \\vee a \\vee ( c \\wedge \\neg d ) \\\\ & \\ge ( c \\wedge d ) \\vee ( c \\wedge \\neg d ) = c , \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\sum _ { i + j = n } & \\binom { i + j } { i } \\lambda _ { i - 1 + p m } \\lambda _ { j - 1 + m } = 0 , \\\\ \\sum _ { i + j = n } & \\binom { i + j } { i } ( \\lambda _ { i - 1 + p m } \\mu _ { j - 1 + m } - \\mu _ { i - 1 + p m } \\lambda _ { j - 1 + m } ) = 0 , \\\\ \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} & { \\rm C o e f f } _ { m _ { 1 } , \\cdots m _ { n - 1 } } ( \\sum _ { ( \\lambda _ { 1 } \\cdots \\lambda _ { n - 1 } ) \\in \\mathbb { N } ^ { n - 1 } } M _ { 2 } x _ 1 ^ { \\lambda _ 1 } \\cdots x _ { n - 1 } ^ { \\lambda _ { n - 1 } } ) \\\\ & = { \\rm C o e f f } _ { m _ { 1 } , \\cdots m _ { n - 1 } } ( \\sum _ { ( k _ { 1 } \\cdots k _ { n - 1 } ) \\in \\mathbb { N } ^ { n - 1 } } N _ { 2 } x _ 1 ^ { k _ 1 } \\cdots x _ { n - 1 } ^ { k _ { n - 1 } } ) . \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} 3 J _ { n } ^ { ( 3 ) } + j _ { n } ^ { ( 3 ) } = 2 ^ { n + 1 } , \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } \\left ( \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) - 1 } { r ^ { 2 } } \\right ) v _ { 4 } \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle \\right ) | \\leq \\frac { C } { t ^ { 3 } \\log ^ { 2 N - 2 - b } ( t ) } \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} \\eta _ { m } ( z ) = 1 + \\frac { e ^ { i 2 m \\pi / 3 } } { \\sqrt [ 3 ] { \\left | c _ { 3 } \\right | } e ^ { i \\theta / 3 } } ( z - 1 ) ^ { 1 / 3 } + e ^ { i 2 m \\pi / 3 } \\sum _ { n = 2 } ^ { \\infty } d _ { n , m } ( z - 1 ) ^ { n / 3 } , m = 0 , 1 , 2 , \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} \\psi _ { k , k + 1 } ^ { \\ast } ( Q ^ { \\prime } ) \\geq - \\frac { \\sum _ { j = 1 } ^ { k - 1 } \\psi _ { k , j } ^ { \\ast } ( Q ^ { \\prime } ) } { 2 } - O ( q ^ { - 1 } ) \\geq - \\frac { \\tau } { 2 } - \\epsilon _ { 1 1 } - O ( q ^ { - 1 } ) . \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} & d ' = - d ( d - \\rho ) , \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} \\lim _ { t \\longrightarrow \\infty } \\frac { 1 } { t } \\log _ { | \\pi | } \\Big ( V _ { t \\rho } ( \\pi ^ t ) \\Big ) = \\min _ { z \\in ( 0 , 1 ] } \\log _ { | \\pi | } \\Big ( \\frac { f ( z ) } { z ^ \\rho } \\Big ) . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\phi _ { 0 } ( r ) \\partial _ { x } ( F _ { 4 } ( x , r \\lambda ( x ) ) ) r d r = 0 , x \\geq T _ { 0 } \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} \\lim _ { x \\downarrow c _ { 0 } } \\frac { p _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ( x ) } { \\left | x - c _ { 0 } \\right | ^ { 1 / 2 } } = \\frac { - a _ { 1 } } { \\pi \\sqrt { \\left | c _ { 2 } \\right | } } \\sin \\frac { \\theta } { 2 } , \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} \\widehat f ( \\xi ) \\ , : = \\ , \\int _ { - \\infty } ^ { \\infty } f ( z ) e ^ { - 2 \\pi { \\mathbf i } z \\xi } d z \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} ( q _ 1 , \\dots , q _ J ) & = \\left ( a \\delta ^ Q _ 1 , \\dots , a \\delta ^ Q _ { J - 1 } , a \\left ( 1 - \\sum _ { j = 1 } ^ { J - 1 } \\delta ^ Q _ j \\right ) \\right ) , \\\\ ( e _ 1 , \\dots , e _ { J - 1 } ) & = \\left ( ( L - a ) \\delta ^ { E } _ 1 , \\dots , ( L - a ) \\delta ^ { E } _ { J - 1 } \\right ) , \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} L ( A , B ) = \\int _ 0 ^ 1 A ! _ t B \\ ; d \\mu ( t ) , \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} ( B _ 1 ( u , \\theta ) , A _ 1 \\theta ) & = - \\sum _ { i , j = 1 } ^ { 2 } \\int _ { \\Omega } u _ i \\partial _ i \\theta \\partial _ j ^ 2 \\theta \\dd x \\\\ & = \\sum _ { i , j = 1 } ^ { 2 } \\int _ { \\Omega } u _ i \\partial _ { i j } \\theta \\partial _ j \\theta \\dd x + \\sum _ { i , j = 1 } ^ { 2 } \\int _ { \\Omega } \\partial _ j u _ i \\partial _ i \\theta \\partial _ j \\theta \\dd x \\ ; . \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} \\Psi ( z ) = \\gamma z \\exp \\left [ \\int _ { \\mathbb { T } } \\frac { t + \\eta _ { \\mu _ { 1 } } ( z ) } { t - \\eta _ { \\mu _ { 1 } } ( z ) } \\ , d \\sigma ( t ) \\right ] , z \\in \\mathbb { D } . \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} \\xi _ { n , h } : = \\sqrt { n } \\left \\{ \\int _ { t _ { h - 1 } } ^ { t _ h } ( M _ t - M _ { t _ { h - 1 } } ) d N _ t + \\int _ { t _ { h - 1 } } ^ { t _ h } ( N _ t - N _ { t _ { h - 1 } } ) d M _ t \\right \\} . \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} Y _ t ^ k = \\xi ^ k - \\sum _ { l = 1 } ^ m \\int _ t ^ T Z _ s ^ { k , l } d B _ s ^ l + \\frac { 1 } { 2 } \\sum _ { l = 1 } ^ m \\int _ t ^ T \\sum _ { i , j = 1 } ^ n \\Gamma _ { i j } ^ k ( Y _ s ) Z _ s ^ { i , l } Z _ s ^ { j , l } d s . \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} ( k _ { 1 } * k _ { 2 } ) ( x , y ) = \\int _ { M } k _ { 1 } ( x , z ) k _ { 2 } ( z , y ) d { \\rm v o l } ( z ) \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} \\gamma _ { l ' i _ { 1 } } \\gamma _ { \\tau ' \\left ( l ' \\right ) i _ { 2 } } = 0 , \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} k ' = k [ \\alpha ] / ( \\alpha ^ 2 + \\alpha + A ^ 4 ) , k '' = k ' [ \\beta ] / ( \\beta ^ 2 + \\beta + B ^ 2 ) . \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} B _ \\kappa : = \\{ b \\in B \\mid b \\} . \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\Sigma '' ( f , g ) : = \\Sigma ' ( f , g ) - \\frac { P _ { y } f } { C } \\Sigma ' ( C , g ) - \\frac { P _ { y } g } { C } \\Sigma ' ( f , C ) + \\frac { P _ { y } f P _ { y } g } { C ^ { 2 } } \\Sigma ' ( C , C ) , f , g \\in \\mathcal { F } , \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} \\bar { c } _ i = ( \\widetilde { S q } ^ 0 ) ^ i ( \\lambda _ 3 ^ 2 b ^ { [ 2 ] } ) , i \\geq 0 . \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\int _ { \\mathcal { M } } f ( \\mu ) g ( \\mu ) \\rho ( \\mu ) \\ ; { \\rm d } \\mu \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} S t _ s ( x _ 1 , y _ 1 , \\dots , x _ s , y _ s ; m ) : = ( - 1 ) ^ { s \\left [ \\frac { | m | } { 2 } \\right ] } L _ s ^ { \\frac { p - 1 } { 2 } | m | } S _ s ( m ) . \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} \\xi = \\sum _ { 1 \\leq i _ 1 \\leq \\ldots \\leq i _ \\ell \\leq n + k } \\xi _ { i _ 1 \\ldots i _ \\ell } d x ^ { i _ 1 } \\wedge \\ldots \\wedge d x ^ { i _ \\ell } . \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} w = ( w ^ + , w ^ - ) \\quad \\mbox { w i t h } w ^ \\pm = U ^ \\pm e ^ { i \\theta } , \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} \\eta _ { \\mu _ { * } } ^ { \\langle - 1 \\rangle } ( z ) = \\frac { 1 } { \\eta _ { \\mu } ^ { \\langle - 1 \\rangle } ( 1 / z ) } \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} \\inf _ { 0 \\le x _ { n + 1 } \\le 1 / 2 } ( x _ { n + 1 } ^ { 1 - 2 s } - x _ { n + 1 } ) = \\inf _ { 0 \\le x _ { n + 1 } \\le 1 / 2 } \\psi _ { s } ( x _ { n + 1 } ) = \\psi _ { s } \\bigg ( \\frac { 1 } { 2 } \\bigg ) = \\frac { 1 } { 2 } ( 4 ^ { s } - 1 ) \\ge \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} \\Delta '' = \\prod _ { \\substack { \\alpha \\in A \\\\ \\gamma \\in D _ G ( \\alpha ) } } | \\alpha - \\gamma | _ v ^ { n _ \\gamma } \\prod _ { \\substack { \\beta \\notin R \\\\ \\gamma \\notin D _ G ( \\beta ) \\cup \\{ \\beta \\} } } | \\beta - \\gamma | _ v ^ { n _ \\gamma } \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} \\Delta ^ c = \\bigcup _ { ( x , y ) \\in \\Delta ^ c } U _ { x y } \\times V _ { x y } . \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} \\sigma _ { ( a ' , i ' ) } ^ { ( s + 1 ) n _ 1 } = \\sigma _ { \\sigma _ { ( a ' , i ' ) } ( ( a ' , i ' ) ) } ^ { n _ 1 } = \\sigma _ { \\sigma _ { \\varphi ( ( a , i ) ) } ( \\varphi ( ( a , i ) ) ) } ^ { n _ 1 } = \\varphi \\sigma _ { \\sigma _ { ( a , i ) } ( ( a , i ) ) } ^ { n _ 1 } \\varphi ^ { - 1 } = \\varphi \\sigma _ { ( a , i ) } ^ { ( r + 1 ) n _ 1 } \\varphi ^ { - 1 } = \\sigma _ { ( a ' , i ' ) } ^ { ( r + 1 ) n _ 1 } \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} E ( \\mu ) \\dot { \\tilde { x } } ( t , \\mu ) & = \\tilde { f } ( \\tilde { x } ( t , \\mu ) , \\mu ) = f ( \\tilde { x } ( t , \\mu ) + x ^ * ( \\mu ) , \\mu ) \\\\ [ 1 e x ] y ( t , \\mu ) & = \\tilde { g } ( \\tilde { x } ( t , \\mu ) , \\mu ) = g ( \\tilde { x } ( t , \\mu ) + x ^ * ( \\mu ) , \\mu ) , \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} u ^ N _ { t } ( x ) - u _ { t } ( x ) & = e ^ { t \\Delta } ( u ^ N _ { 0 } - u _ { 0 } ) ( x ) - \\int _ { 0 } ^ t \\nabla \\cdot e ^ { ( t - s ) \\Delta } \\left ( \\langle \\mu _ s ^ N , V ^ N ( x - \\cdot ) F \\big ( K \\ast u ^ N _ s ( \\cdot ) \\big ) \\rangle - u _ { s } F ( K \\ast u _ s ) ( x ) \\right ) \\ , d s \\\\ & - \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\nabla V ^ N ( x - X _ s ^ { i , N } ) \\cdot d W ^ i _ s . \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} | g | _ { H S } = \\sqrt { n } \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} c _ { \\mathsf { p , } \\alpha , \\beta } : = \\sum _ { \\mathsf { n = 1 } } ^ { \\mathsf { + \\infty } } \\exp \\left [ - \\frac { \\alpha - 1 } { 2 } \\beta \\lambda _ { \\mathsf { n } } \\right ] p _ { \\mathsf { n } } . \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} D T ( x ) = D ( A _ c + r ) ^ { - 1 } ( x ) = \\left ( D R ( R ^ { - 1 } ( x ) ) \\right ) ^ { - 1 } = \\left ( D R ( T ( x ) ) \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} B _ { k , 1 } ( n ) & = \\{ \\lambda = ( \\lambda _ 1 ^ { m _ 1 } , \\lambda _ 2 ^ { m _ 2 } , \\ldots , \\lambda _ \\ell ^ { m _ \\ell } , a ^ b ) \\in \\mathcal { P } ( n ) \\mid k \\nmid \\lambda _ i \\ , \\forall i , \\ , k \\mid a , \\ , k \\nmid b \\} . \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} & \\mathcal { F } ( \\sqrt { \\cdot } F _ { 4 } ( x , \\cdot \\lambda ( x ) ) ) '' ( \\omega \\lambda ( x ) ^ { 2 } ) \\\\ & = \\int _ { 0 } ^ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( x ) } } \\sqrt { r } F _ { 4 } ( x , r \\lambda ( x ) ) \\partial _ { 2 } ^ { 2 } \\phi ( r , \\omega \\lambda ( x ) ^ { 2 } ) d r + \\int _ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( x ) } } ^ { \\infty } \\sqrt { r } F _ { 4 } ( x , r \\lambda ( x ) ) \\partial _ { 2 } ^ { 2 } \\phi ( r , \\omega \\lambda ( x ) ^ { 2 } ) d r \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} H ( t ; x , y , 1 , u , 1 , v ) & = \\frac { x v t ^ 2 ( 1 - y r ) } { ( 1 - y t u ) ( 1 - t u v ( y - y r + 1 ) ) ( t u x + y ^ { - 1 } - t u ) } \\\\ & \\quad + \\frac { y u ^ 2 v t ^ 2 ( 1 - v ) ( 1 - y r ) } { ( 1 - y t u ) ( 1 - t u v ( y - y r + 1 ) ) } F ( t ; x , y , 1 , u , 1 , v ) . \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} ( 1 + R _ \\varepsilon ) ^ { \\ 2 m s + | x | ^ \\alpha } & = 1 + ( \\ 2 m s + | x | ^ \\alpha ) R _ \\varepsilon \\\\ & + ( \\ 2 m s + | x | ^ \\alpha ) ( \\ 2 m s + | x | ^ \\alpha - 1 ) R _ \\varepsilon ^ 2 \\int _ 0 ^ 1 ( 1 + s R _ \\varepsilon ) ^ { \\ 2 m s + | x | ^ \\alpha - 2 } ( 1 - s ) d s \\\\ & = 1 + ( \\ 2 m s + | x | ^ \\alpha ) R _ \\varepsilon + O ( R _ \\varepsilon ^ 2 ) , \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\| A \\| _ { \\mathrm E } = \\sqrt { \\sum _ { i , j } A _ { i j } ^ 2 } . \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} \\lVert x \\rVert _ { L _ p } & = \\sup \\Bigl \\{ \\ , \\Bigl | \\int _ E x ( t ) y ( t ) \\ , d t \\Bigr | : \\ , \\lVert y \\rVert _ { L _ q } \\le 1 \\ , \\Bigr \\} , \\\\ \\lVert y \\rVert _ { L _ q } & = \\sup \\Bigl \\{ \\ , \\Bigl | \\int _ E x ( t ) y ( t ) \\ , d t \\Bigr | : \\ , \\lVert x \\rVert _ { L _ p } \\le 1 \\ , \\Bigr \\} . \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} R = - \\frac { ( x - y ) S e ^ f } { ( x - y ) e ^ f + S + 1 } . \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} A ( G : H ) : = \\{ f \\in \\mathcal { C } _ c ( G ) : L _ h f = f \\ \\forall h \\in H \\} , \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} H _ { L + k + 1 } = H _ { L + k } + H _ { k + m + 1 } + \\dots + H _ { k + 2 } + N H _ { k + 1 } \\leq \\sum _ { i = 1 } ^ { L + k } H _ { i } + 1 . \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} Z = \\left ( \\frac { 1 } { 1 - \\lambda } \\right ) ^ 2 ( X - Y ) ( Y - \\lambda X ) , \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} \\partial _ { t } u = \\left ( a + i b \\right ) \\left [ \\Delta u + A \\left ( \\left ( x \\right ) \\right ) u + V \\left ( x , t \\right ) u + F \\left ( x , t \\right ) \\right ] , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} \\gamma _ { \\mu ' } ^ { - 1 } E ( \\gamma _ { \\lambda ' } ^ { - 1 } ) ^ g = \\zeta _ l ^ { - 1 } = \\beta _ g ( \\gamma _ \\lambda \\gamma _ \\mu ) ^ { - 1 } . \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} c _ 2 ( E ) = \\sum _ i ( \\Omega ^ { 4 } ) _ { i i } = \\sum _ { i , j } d e t ( \\partial ^ { 2 } h _ { i j } ) d z ^ { 1 } \\wedge d \\bar { z } ^ { 1 } \\wedge d z ^ { 2 } \\wedge d \\bar { z } ^ { 2 } \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} \\mathbb { P } _ \\eta \\left ( \\left . \\mathcal { T } _ t ( x ) < \\frac { 1 - p } { 4 } t \\right | \\mathcal { F } _ { k } \\right ) \\leq \\mathbb { P } _ \\eta \\left ( \\left . \\int _ { t / 2 } ^ { t } \\mathbb { 1 } _ { \\{ \\eta _ s ( x ) = 0 \\} } \\mathrm { d } s < \\frac { 1 - p } { 4 } t \\right | \\mathcal { F } _ { k } \\right ) \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} ( p _ T ) ^ ! _ { p _ { 1 2 } \\times p _ { 2 3 } } = e ( Y ) \\cdot _ 2 p _ T ^ * . \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} \\cos { { \\psi } _ { x , y } } = \\frac { \\left | \\left \\langle x , y \\right \\rangle \\right | } { \\left \\| x \\right \\| \\left \\| y \\right \\| } ; 0 \\leq \\psi _ { x , y } \\leq \\frac { \\pi } { 2 } . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 s ^ { \\alpha - 1 } ( 1 - s ) ^ { \\beta - 1 } d s = \\frac { \\Gamma ( \\alpha ) \\Gamma ( \\beta ) } { \\Gamma ( \\alpha + \\beta ) } \\ , , \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} - \\partial _ { t t } v _ { 3 } + \\partial _ { r r } v _ { 3 } + \\frac { 1 } { r } \\partial _ { r } v _ { 3 } - \\frac { v _ { 3 } } { r ^ { 2 } } = F _ { 0 , 1 } ( t , r ) \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} C \\Phi _ { 1 3 } ( q ) \\equiv \\sum _ { n = 0 } ^ { \\infty } p _ { [ 1 ^ 0 1 3 ^ 1 ] } ( n ) q ^ n \\pmod { 1 3 } . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} \\begin{array} { l } \\partial ^ 2 _ t u = \\partial ^ 2 _ x u + \\frac { \\kappa ^ 2 } { 4 } | u | ^ r - \\frac { c _ 1 ^ 2 + c _ 2 ^ 2 } { 2 } u , t > 0 , x \\in D , \\\\ u ( t , x ) = 0 , \\mbox { f o r } ~ x \\in \\partial D , ~ t \\geq 0 , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\partial _ t u ( 0 , x ) = v _ 0 ( x ) , \\end{array} \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} \\mathfrak h ^ \\perp = \\mathfrak p _ u \\oplus \\bigoplus \\limits _ { \\mu \\in \\Psi } \\mathfrak g ( \\mu ) . \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} \\langle T , \\eta _ 1 \\otimes \\eta _ 2 \\rangle = \\langle T , \\eta _ 2 \\otimes \\eta _ 1 \\rangle , \\langle T , \\eta _ 1 \\otimes \\eta _ 1 \\rangle \\ge 0 \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} \\| ( x , y ) \\| _ { \\infty } = \\max \\{ \\| x \\| _ X , \\| y \\| _ Y \\} , \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} P _ L ( \\tilde { r } ; 0 ) & = \\tilde { \\gamma } - P _ { \\rm a f f i n e } \\left ( \\tilde { \\gamma } - \\tilde { r } ; \\lambda _ L , \\omega _ L , \\frac { \\beta _ L } { a _ { 2 L } } \\right ) , \\\\ T _ L ( \\tilde { r } ; 0 ) & = T _ { \\rm a f f i n e } \\left ( \\tilde { \\gamma } - \\tilde { r } ; \\lambda _ L , \\omega _ L , \\frac { \\beta _ L } { a _ { 2 L } } \\right ) , \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} \\max _ { O \\in \\O } \\| f _ { O } \\| _ 2 ^ 2 \\lesssim _ { \\varepsilon } \\Big ( \\prod _ { i = 1 } ^ { m } r _ { i } ^ { - 1 / 2 } D _ { i } ^ { \\delta } \\Big ) \\Big ( \\prod _ { i = 1 } ^ { \\ell } r _ { i } ^ { - 1 / 2 } \\Big ) \\Big ( \\prod _ { i = \\ell + 1 } ^ { m + 1 } D _ { i } ^ { - ( n - i ) } \\Big ) R ^ { O ( \\varepsilon _ { \\circ } ) } \\| f \\| _ { L ^ { \\infty } ( B ^ { n - 1 } ) } ^ 2 \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} k _ \\gamma = k _ { 0 ^ r } k _ { \\gamma _ 1 } ^ { ( 0 ) } \\cdots k _ { \\gamma _ s } ^ { ( s - 1 ) } = k _ { 0 ^ r } \\left ( \\sum _ { i _ 1 = 0 } ^ r ( - 1 ) ^ { i _ 1 } \\binom { r } { i _ 1 } k _ { \\gamma _ 1 - i _ 1 } ^ { ( r ) } \\right ) \\cdots \\left ( \\sum _ { i _ s = 0 } ^ r ( - 1 ) ^ { i _ s } \\binom { r } { i _ s } k _ { \\gamma _ s - i _ s } ^ { ( r + s - 1 ) } \\right ) = \\prod _ { j = r + 1 } ^ { r + s } ( 1 - L _ j ) ^ r k _ { ( 0 ^ r , \\gamma ) } \\ , . \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} D _ { k , j } : = 2 H - 2 E _ k - 2 E _ j - \\sum _ { p \\neq j } E _ p \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} \\eta _ t ( x ) = \\begin{cases} 1 , & r ( x ) \\ge t \\\\ ( 1 / \\log ( t ) ) ( \\log t ^ 2 - \\log r ( x ) ) , & t ^ 2 \\le r ( x ) \\le t \\\\ 0 & r ( x ) \\le t ^ 2 . \\end{cases} \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} \\mu ( d x ) : = \\frac { 1 } { Z } \\exp \\left ( - H ( x ) \\right ) d x . \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} T ( f ) : = \\mathcal K \\ast f : = \\lim _ { \\epsilon _ 1 , \\epsilon _ 2 , \\epsilon _ 3 \\to 0 \\atop N _ 1 , N _ 2 , N _ 3 \\to \\infty } \\mathcal K _ { \\epsilon } ^ N \\ast f , \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} W ^ { ( 4 ) } _ { t } + ( \\Lambda _ 1 ( | \\xi | ) + \\Lambda _ 2 ( | \\xi | ) + \\Lambda _ 3 ( | \\xi | ) ) W ^ { ( 4 ) } + R _ 4 ( | \\xi | ) W ^ { ( 4 ) } = 0 , \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} H _ { n , \\mathcal { H } } ^ { \\star } \\tilde { P } _ n \\tilde { x } = ( H \\mathcal { H } ) ^ { \\star } \\tilde { P } _ n \\tilde { x } . \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} f ( a , b , z ) = f ( a p , b , z ) - a f ( a p , b , z / q ) . \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} \\quad \\hat Y _ s ^ x ( \\omega ) : = \\lim _ { s \\to t ; s \\in \\mathbb { Q } } & \\Bigg ( h ( B _ T + x ) - \\sum _ { i = 1 } ^ m \\Big ( \\int _ s ^ T Z _ r ^ { x , i } d B _ r + \\frac { 1 } { 2 } \\int _ s ^ T \\bar A ( Y _ r ^ x ) \\left ( Z _ r ^ { x , i } , Z _ r ^ { x , i } \\right ) d r \\Big ) \\\\ & + \\int _ s ^ T \\bar f \\left ( Y _ r ^ x , Z _ r ^ x \\right ) d r \\Bigg ) . \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} ( \\lambda - \\psi ^ { \\mu } ( D ) ) ^ { \\alpha / 2 } & = ( \\lambda - \\psi ^ { \\mu } ( D ) ) ^ { ( \\alpha - 2 n ) / 2 } ( \\lambda - \\psi ^ { \\mu } ( D ) ) ^ { n } \\\\ & = ( \\lambda - \\psi ^ { \\mu } ( D ) ) ^ { ( \\alpha - 2 n ) / 2 } ( \\lambda - \\psi ^ { \\mu } ( D ) ) \\cdots ( \\lambda - \\psi ^ { \\mu } ( D ) ) . \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} Z _ i = \\bigcap _ { k = 0 } ^ { \\infty } Z _ { i k } \\neq \\varnothing , f _ { i j } ( Z _ j ) = \\bigcap _ { k = 0 } ^ { \\infty } f _ { i j } ( Z _ { j k } ) . \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} e ^ * F = f . \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} B ( e _ { x y } , e _ { u v } ) = B ( e _ x , e _ { x v } ) = B ( e _ x , e _ x e _ { x v } e _ v ) = e _ x B ( e _ x , e _ { x v } ) e _ v = B ( e _ x , e _ { x v } ) ( x , v ) e _ { x v } \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} \\| \\Pi u \\| ^ 2 = \\sum _ { n = 0 } ^ \\infty \\lambda _ n ^ 2 \\ , | \\langle 1 | f _ n \\rangle | ^ 2 \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} \\dim H ^ 1 ( X , \\mathcal { O } _ { X } ( D ) ) = 0 . \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} H ^ 2 ( \\Omega ) = U _ j \\oplus U _ j ^ { \\perp } . \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} | \\Theta ^ n x | = r _ \\Theta ^ { n + o ( n ) } n \\rightarrow \\infty . \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} | v _ { 4 , c } ^ { \\lambda _ { 1 } } - v _ { 4 , c } ^ { \\lambda _ { 2 } } | \\leq C \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) | | e _ { 1 } - e _ { 2 } | | _ { X } \\begin{cases} \\frac { 1 } { r ^ { 3 } t ^ { 2 } \\log ^ { 3 b } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } , r \\leq \\frac { t } { 2 } \\\\ \\frac { \\log ( r ) } { r ^ { 4 } \\log ^ { 2 b } ( t ) \\sqrt { \\log ( \\log ( t ) ) } | t - r | } + \\frac { 1 } { t ^ { 2 } \\log ^ { 3 b + 1 - 2 \\alpha b } ( t ) \\sqrt { \\log ( \\log ( t ) ) } r ^ { 3 } } , t > r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} | | \\frac { | \\partial _ { r } v _ { 5 } ( t , r ) | v _ { 5 } ( t , r ) ^ { 2 } } { r ^ { 2 } } \\vert _ { r = R \\lambda ( t ) } | | _ { L ^ { 2 } ( R d R ) } & \\leq | | ( \\partial _ { 2 } v _ { 5 } ) ( t , \\cdot \\lambda ( t ) ) | | _ { L ^ { 2 } ( R d R ) } \\cdot | | \\frac { v _ { 5 } ^ { 2 } } { r ^ { 2 } } | | _ { L ^ { \\infty } } \\\\ & \\leq \\frac { C \\log ^ { 1 1 + b } ( t ) } { t ^ { 3 1 / 4 } } \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} a \\times b : = \\iota _ 1 ( a ) \\wedge \\iota _ 2 ( b ) \\in A \\otimes B \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} p _ { L } ( x ) = x ^ { L } - x ^ { L - 1 } - \\left \\lceil \\frac { L ( L + 1 ) } { 4 } \\right \\rceil - 1 . \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} ( q ; q ) _ \\infty & ( - q ^ 2 ; q ^ 2 ) _ \\infty \\sum _ { n = 0 } ^ \\infty a _ k ( n ) q ^ n \\\\ & = ( - q ^ { 2 } ; q ^ { 2 } ) _ \\infty ( q ^ { k } ; q ^ { k } ) _ \\infty \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { k n } + 2 q ^ { 2 k n } + \\dots + ( k - 1 ) q ^ { ( k - 1 ) k n } } { 1 + q ^ { k n } + q ^ { 2 k n } + \\dots + q ^ { ( k - 1 ) k n } } . \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} g ( w ) : = \\frac { w ^ n - w ^ { n + 1 } - K } { ( w _ + - w ) ( w - w _ - ) } = \\sum _ { p = 0 } ^ { n - 1 } g _ p w ^ p . \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} W _ t ^ { ( 3 ) } + ( \\Lambda _ 1 ( | \\xi | ) + \\Lambda _ 2 ( | \\xi | ) ) W ^ { ( 3 ) } + R _ 3 ( | \\xi | ) W ^ { ( 3 ) } = 0 , \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\zeta ' _ B ( 0 ; a , 1 , 1 ) - \\left ( \\frac 1 { 1 2 } - \\zeta ' _ R ( - 1 ) \\right ) \\frac 1 a + \\frac 1 4 \\log ( 2 \\pi ) - \\frac { \\gamma a } { 1 2 } - \\sum _ { k = 2 } ^ { N - 1 } \\frac { B _ { 2 k } \\zeta _ R ( 2 k - 1 ) } { 2 k ( 2 k - 1 ) } a ^ { 2 k - 1 } \\right | \\\\ \\leq \\frac { | B _ { 2 N } \\zeta _ R ( 2 N - 1 ) | } { 2 N ( 2 N - 1 ) } a ^ { 2 N - 1 } , \\end{aligned} \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} \\dot { \\tilde { \\tilde { x } } } & = a _ { 1 L } \\tilde { \\tilde { x } } + a _ { 2 L } \\tilde { \\tilde { y } } , \\\\ \\dot { \\tilde { \\tilde { y } } } & = \\frac { \\beta _ L } { a _ { 2 L } } + b _ { 1 L } \\tilde { \\tilde { x } } + b _ { 2 L } \\tilde { \\tilde { y } } . \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} \\partial _ { t } \\left ( \\int _ { 0 } ^ { \\frac { t } { 2 } } \\frac { \\sin ( u ) b ( b - 1 ) d u } { t ^ { 2 } u \\log ^ { b + 1 } ( \\frac { t } { u } ) } \\right ) & = \\frac { \\sin ( \\frac { t } { 2 } ) b ( b - 1 ) } { t ^ { 3 } \\log ^ { b + 1 } ( 2 ) } + \\int _ { 0 } ^ { \\frac { t } { 2 } } \\frac { \\sin ( u ) b ( b - 1 ) } { u } \\left ( \\frac { - 2 } { t ^ { 3 } \\log ^ { b + 1 } ( \\frac { t } { u } ) } - \\frac { ( b + 1 ) } { \\log ^ { b + 2 } ( \\frac { t } { u } ) t ^ { 3 } } \\right ) d u \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} f ( 0 , x , v ) = f _ { 0 } ( x , v ) , \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} \\hat H = p _ 1 p _ 2 p _ 3 \\left ( 1 + e ^ { - ( q _ 1 - q _ 3 ) } - e ^ { - ( q _ 1 - q _ 2 ) } - e ^ { - ( q _ 2 - q _ 3 ) } \\right ) \\ ; \\ ; \\mbox { f o r } \\ ; \\ ; n = 3 . \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} N ^ { \\prime } ( \\eta _ { \\mu _ { 1 } } ( \\alpha ) ) = N ^ { ( 1 ) } ( \\eta _ { \\mu _ { 1 } } ( \\alpha ) ) = u _ { \\sigma } ^ { ( 1 ) } ( a _ { 0 } ) \\cdot \\frac { 1 } { ( 1 - \\eta _ { \\mu _ { 1 } } ( \\alpha ) ) ^ { 2 } } = \\frac { b _ { 1 } } { ( 1 - \\eta _ { \\mu _ { 1 } } ( \\alpha ) ) ^ { 2 } } , \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} \\hat { H } _ { c o u l } = - \\sum _ { i = 1 } ^ { D } \\frac { \\partial ^ 2 } { \\partial x _ i ^ 2 } - \\frac { \\eta } { r } + \\frac { g _ 1 ( \\Omega _ 1 ) } { x _ 1 ^ 2 + \\cdots + x _ { n _ 1 } ^ 2 } + \\frac { g _ 2 ( \\Omega _ 2 ) } { x _ { n _ 1 + 1 } ^ 2 + \\cdots + x _ { n _ 2 } ^ 2 } + \\cdots + \\frac { g _ { N - 1 } ( \\Omega _ { N - 1 } ) } { x _ { n _ { N - 2 } + 1 } ^ 2 + \\cdots + x _ { n _ { N - 1 } } ^ 2 } , \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} \\frac { 1 } { T } \\sum _ { t = 1 } ^ T \\Delta _ t \\leq \\frac { \\rho \\tau \\Theta _ 1 + ( 1 - \\rho ) \\Theta _ 1 + 2 \\sigma ^ 2 \\rho ^ 2 T } { \\gamma C _ { \\mu } T } \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} \\eta ( z _ 1 ) = \\eta ( z _ 2 ) \\iff \\eta _ i ( z _ 1 ( i ) ) = \\eta _ i ( z _ 2 ( i ) ) i \\in \\mathcal { I } . \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} \\big ( \\mathbb { E } \\| P _ k ( \\varepsilon ) \\| _ X ^ q \\big ) ^ { 1 / q } \\leq \\int _ { 0 } ^ 1 \\big ( \\mathbb { E } \\| P ( t \\varepsilon ) p _ { k + 1 } ^ { ( m ) } ( t ) \\| _ X ^ q \\big ) ^ { 1 / q } d t \\leq B ^ m \\int _ { 0 } ^ 1 \\big ( \\mathbb { E } \\| P ( t \\varepsilon ) \\| _ X ^ q \\big ) ^ { 1 / q } d t . \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} ( v a r ^ n [ y _ n ( t , x ) ] ) ^ { \\frac { 1 } { 2 } } & \\leq \\dfrac { 1 } { \\sqrt { M } } ( v a r ^ { n - 1 } [ y _ n ( t , x ) ] ) ^ { \\frac { 1 } { 2 } } + \\dfrac { C _ f } { \\sqrt { M } } \\sum _ { j = 1 } ^ Q w _ { n , j } \\left ( E ^ n [ ( y _ { n - 1 } - y _ { n - 2 } ) ^ 2 ( t _ { n , j } , x + W _ { t _ { n , j } - t } ^ { ( n ) } ) ] \\right ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} M _ 1 = \\langle \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix} , \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\end{pmatrix} \\rangle \\qquad M _ 2 = \\langle \\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\end{pmatrix} , \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\end{pmatrix} \\rangle \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} 3 t \\geq \\sum _ { i = 1 } ^ { n } | N ^ { - } _ { S ' \\cap \\{ u _ 1 , \\dots , u _ n \\} } | \\geq n . \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } x _ 1 = r \\cos ( \\phi _ 1 ) \\\\ \\\\ x _ 2 = r \\sin ( \\phi _ 1 ) \\cos ( \\phi _ 2 ) \\\\ \\\\ \\omega _ 1 = r \\sin ( \\phi _ 1 ) \\sin ( \\phi _ 2 ) \\cos ( \\phi _ 3 ) \\\\ \\\\ \\omega _ 2 = r \\sin ( \\phi _ 1 ) \\sin ( \\phi _ 2 ) \\sin ( \\phi _ 3 ) \\end{array} \\right . \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} G ^ { 1 , 1 } _ { 0 } = { \\rm s p a n } \\left \\{ ( E _ { 1 , 3 } ) ^ { l } _ { ( k ) } | l \\in \\mathbb { N } , k \\in \\mathbb { Z } _ { < 0 } \\right \\} . \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} \\eta _ { \\mu _ { 1 } } ( \\eta _ { \\rho _ { 1 } } ( z ) ) = \\eta _ { \\mu _ { 2 } } ( \\eta _ { \\rho _ { 2 } } ( z ) ) = \\frac { \\eta _ { \\rho _ { 1 } } ( z ) \\eta _ { \\rho _ { 2 } } ( z ) } { z } , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} P _ L ( \\tilde { r } ; 0 ) & = - P _ { \\rm a f f i n e } \\left ( - \\tilde { r } ; \\lambda , \\omega , b _ 3 \\right ) , \\\\ T _ L ( \\tilde { r } ; 0 ) & = T _ { \\rm a f f i n e } \\left ( - \\tilde { r } ; \\lambda , \\omega , b _ 3 \\right ) , \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p , q } } : = \\left \\{ \\begin{array} { l } \\bigg ( \\int _ 0 ^ { \\infty } \\big ( t ^ { 1 / p } f ^ { * * } ( t ) \\big ) ^ { q } \\frac { d t } { t } \\bigg ) ^ { 1 / q } , \\ ; 1 \\leq q < \\infty \\\\ \\sup \\limits _ { t > 0 } \\ ; t ^ { 1 / p } f ^ { * * } ( t ) , \\ ; q = \\infty , \\end{array} \\right . \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} \\int _ a ^ b | m u + v | ^ 2 \\Im ( \\lambda - V ) = \\Im m , \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} \\lim _ { k } { | \\psi _ { k } ( ( u z ) ^ * x u z - x ) | } = \\lim _ { k } { | \\phi _ k ( \\Phi ( ( u z ) ^ * x u z ) - \\Phi ( x ) ) | } = \\lim _ { k } { | \\phi _ { k } ( u \\Phi ( x ) u ^ * - \\Phi ( x ) ) | } = 0 \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} p ( t ) = \\left \\{ \\begin{array} { l l } \\frac { \\nabla u ( t ) } { \\abs { \\nabla u ( t ) } } , & \\nabla u ( t ) \\neq 0 ; \\\\ \\gamma ( t ) \\abs { \\gamma ( t ) } \\leq 1 , & \\nabla u ( t ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} p _ { L } \\left ( 1 + \\varepsilon \\right ) = \\sum _ { n = 1 } ^ { L } \\varepsilon ^ { n } \\binom { L - 1 } { n - 1 } - \\left \\lceil L ( L + 1 ) / 4 \\right \\rceil - 1 . \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} H \\left ( W _ { x , t } \\right ) = L . \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} 2 \\alpha _ n = 3 \\alpha _ { n - 1 } + ( n - 1 ) \\alpha _ { n - 2 } - ( n - 1 ) \\alpha _ { n - 3 } , \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} \\| x _ { n + 1 } ^ { \\frac { 1 - 2 s } { 2 } } u _ { 1 } \\| _ { L ^ { 2 } ( B _ { 1 0 } ^ { + } ) } ^ { 2 } & \\le \\int _ { 0 } ^ { 1 0 } \\int _ { \\mathbb { R } ^ { n } } x _ { n + 1 } ^ { 1 - 2 s } | u _ { 1 } ( x ' , x _ { n + 1 } ) | ^ { 2 } \\ , d x ' \\ , d x _ { n + 1 } \\\\ & \\le \\bigg ( \\int _ { 0 } ^ { 1 0 } x _ { n + 1 } ^ { 1 - 2 s } \\ , d x _ { n + 1 } \\bigg ) \\| u \\| _ { L ^ { 2 } ( B _ { 1 6 } ' ) } ^ { 2 } = C \\| u \\| _ { L ^ { 2 } ( B _ { 1 6 } ' ) } ^ { 2 } . \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} B = \\mathbb { C } [ E _ { i , j } | 1 \\leq i < j \\leq n ] / I , \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} ( \\theta ^ t S ) _ s = S _ { t + s } - S _ t , \\qquad \\forall s \\in \\mathbb { R } , \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} B _ { \\theta } ( t ) \\triangleq \\left \\{ \\omega \\bigg { | } \\frac { \\hat { t } - t } { t ^ { \\beta } } > \\theta \\mbox { w h e r e } \\hat { t } = \\inf \\{ \\tau > t | a ( \\tau ) \\leq \\lambda \\} \\right \\} . \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} K i r _ k \\Phi ( x ) & = \\frac { 1 } { P _ k ( x ) } \\int _ { y \\in \\mathbb { R } } \\Phi ( x , y ) \\mathcal { X } _ { [ - \\tfrac { 1 } { k } , \\tfrac { 1 } { k } ] } ( x - y ) d y \\\\ & = \\frac { \\pi } { k } ( 1 + k ^ 2 x ^ 2 ) \\int _ { x - \\tfrac { 1 } { k } } ^ { x + \\tfrac { 1 } { k } } \\Phi ( x , y ) d y \\\\ & = 2 \\pi \\left ( \\frac { 1 } { k ^ 2 } + x ^ 2 \\right ) \\frac { k } { 2 } \\int _ { x - \\tfrac { 1 } { k } } ^ { x + \\tfrac { 1 } { k } } \\Phi ( x , y ) d y . \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} \\frac { z } { f ( z ) } = 1 + \\sum _ { k = 1 } ^ { \\infty } b _ k z ^ k = 1 - a _ 2 z + ( a _ 2 ^ 2 - a _ 3 ) z ^ 2 + \\cdots . \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} \\left < v , z \\right > _ { N } : = \\sum _ { j = 1 } ^ N w _ j v ( \\mathbf { x } _ j ) z ( \\mathbf { x } _ j ) , \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} \\begin{cases} D ^ { \\alpha _ 2 } _ 0 y ( t ) = f ( t , y ( t ) , D ^ { 1 - \\varepsilon } _ 0 y ( t ) ) , ~ ~ t \\in [ 0 , b ] , \\\\ y ( 0 ) = y _ 0 , \\ y ( b ) = y _ b , \\end{cases} \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} r ( \\mathcal { L } \\otimes \\sigma + \\mathcal { L } - \\sigma - 1 ) & = \\frac { r ( \\mathcal { L } \\otimes \\sigma ) r ( \\mathcal { L } ) } { r ( \\sigma ) } \\\\ & = \\frac { r ( t + z ) r ( t ) } { r ( z ) } . \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} \\kappa ( x , y , z , t ) = \\kappa ( w , y , z , t ) \\kappa ( x , w , z , t ) \\\\ \\kappa ( x , y , z , t ) = 1 - \\kappa ( t , y , z , x ) , \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} \\mathcal { B } ^ { \\alpha } ( \\mathbb { R } ^ 2 ) : = \\left \\{ f \\in \\mathcal { S } ^ { \\prime } ( \\mathbb { R } ^ 2 ) : ~ \\| f \\| _ { \\alpha } < + \\infty \\right \\} \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\R _ + ^ m : = \\{ x = ( x _ 1 , . . . , x _ m ) \\in \\R ^ m : x _ i \\geq 0 \\mbox { f o r } i = 1 , . . . , m \\} , \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} & \\mathcal { T } _ k ^ { A , d } : = \\{ ( \\xi , \\eta ) \\in \\R ^ 2 \\ , | \\ , \\xi \\in A ^ { - \\frac { 3 } { 2 } } d ^ { - 1 } N _ 1 [ k _ { ( 1 ) } , k _ { ( 1 ) } + 1 ) , \\ \\eta \\in A ^ { - 1 } d ^ { - 1 } N _ 1 [ k _ { ( 2 ) } , k _ { ( 2 ) } + 1 ) \\} , \\\\ & \\tilde { \\mathcal { T } } _ k ^ { A , d } : = \\R \\times \\mathcal { T } _ k ^ { A , d } . \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} \\mathbb B _ 1 ( \\phi _ { 2 } , \\phi _ { 0 } ) = \\mathbb B _ 1 ( \\phi _ { 2 } ^ \\mathbb { R } , \\phi _ { 0 } ^ \\mathbb { R } ) + \\mathbb B _ 1 ( i \\phi _ { 2 } ^ \\mathbb { I } , i \\phi _ { 0 } ^ \\mathbb { I } ) = { \\mathbb D } ( - \\phi _ { 2 } ^ \\mathbb { R } , \\phi _ { 0 } ^ \\mathbb { R } ) + { \\mathbb D } ( \\phi _ { 2 } ^ \\mathbb { I } , \\phi _ { 0 } ^ \\mathbb { I } ) , \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} & | - \\int _ { t } ^ { t + \\frac { 1 } { 2 } } \\frac { d s } { ( s - t ) } \\int _ { 0 } ^ { s - t } \\rho d \\rho \\frac { \\lambda '' ( s ) } { r } \\partial _ { r } \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( - 1 - \\rho ^ { 2 } + r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } \\right ) | \\\\ & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\mathcal { E } } F = \\mathrm { g h } ( F ) F . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} \\sigma _ p ( G ) & = \\bigcup _ { I \\in \\mathcal { I } _ p ( G ) } \\sigma _ p ( I ) . \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} \\Big ( \\sum _ { k = 1 } ^ N \\big \\| x _ k \\big \\| ^ q \\Big ) ^ { \\frac { 1 } { q } } \\leq C \\Big ( \\int _ { \\mathbb { T } } \\Big \\| \\sum _ { k = 1 } ^ N x _ k z ^ { k } \\Big \\| ^ { q ' } d z \\Big ) ^ { \\frac { 1 } { q ' } } \\ , . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} d _ { \\mathcal { P } , \\pm } = \\left [ b ( \\mathsf { M } ^ { \\mathbb { R } } _ { 1 } + \\mathsf { M } ^ { \\mathbb { Z } } ) \\mp \\frac { 1 } { 2 \\pi } \\ , \\partial _ { 1 } \\right ] + \\left [ b \\ , \\mathsf { M } ^ { \\mathbb { R } } _ { 2 } \\pm \\frac { 1 } { 2 \\pi } \\ , \\partial _ { 2 } \\right ] \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} \\beta _ { m a x } = \\frac { c _ 1 ( c _ 1 - c _ 2 ) t } { 2 c _ 1 - c _ 2 } \\ \\ \\ \\ \\ \\ \\ i f \\ \\ c _ 1 > c _ 2 \\ \\ o r \\ \\ c _ 1 < \\frac { c _ 2 } { 2 } \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} \\Psi ( z ) = c _ { 0 } + c _ { 2 } ( z - \\alpha ) ^ { 2 } + \\sum _ { n = 3 } ^ { \\infty } c _ { n } ( z - \\alpha ) ^ { n } , \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} \\kappa = \\frac { b _ { 0 L } } { a _ { 2 L } } - \\frac { b _ { 0 R } } { a _ { 2 R } } . \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} | - \\frac { 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { B _ { s - t } ( 0 ) } \\frac { d A ( y ) } { \\sqrt { ( s - t ) ^ { 2 } - | y | ^ { 2 } } } _ { v _ { 4 , 2 } ^ { 1 } } | & \\leq \\frac { C } { t ^ { 3 } \\log ^ { 3 b - 1 + 2 N - 2 \\alpha b } ( t ) } \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} \\beta [ f ] = \\overline { \\lim } _ { n \\to \\infty } \\left \\{ \\ n ^ { 1 / \\rho [ f ] } \\ \\sqrt [ n ] { | \\xi _ n | } \\ \\right \\} . \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} \\deg \\tau = 0 , \\deg x _ i = 2 , \\deg f = \\deg _ F f . \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} \\{ l \\in I ^ G ( \\bar x , \\bar y , \\bar z ) \\cup I ^ { G H } ( \\bar x , \\bar y , \\bar z ) \\ , | \\ , \\bar y _ l = 0 \\} = \\{ l \\in \\mathcal Q \\ , | \\ , G _ l ( \\bar x ) = \\bar y _ l = 0 \\} = I ^ { 0 + } ( \\bar x ) \\cup I ^ { 0 0 } ( \\bar x ) \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} \\begin{cases} H _ { i - 1 } = G _ i - 1 , & \\\\ H _ { i - 1 } = G _ { i } , & \\\\ H _ { i - 1 } > G _ i , & \\\\ \\end{cases} \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} & \\left ( \\cdots \\left ( \\varphi _ { k _ 1 l _ 1 } ^ { r _ 2 ' \\cdots r _ s ' } \\right ) \\cdots \\right ) _ { k _ s l _ s } ( X _ 1 , \\cdots , X _ j + 2 \\pi r _ j ' , \\cdots , X _ n , Y _ 1 , \\cdots , Y _ n ) \\\\ & = \\left ( \\cdots \\left ( \\varphi _ { k _ 1 l _ 1 } ^ { r _ 2 ' \\cdots r _ s ' } \\right ) \\cdots \\right ) _ { k _ s l _ s } ( X _ 1 , \\cdots , X _ n , Y _ 1 , \\cdots , Y _ n ) , \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} \\mathrm { S l o p e } [ t \\mapsto \\varphi _ t ( z _ 1 ) , \\tau ] = \\mathrm { S l o p e } [ t \\mapsto \\varphi _ t ( z _ 2 ) , \\tau ] . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\nabla \\times \\frac { 1 } { \\epsilon ( x , \\omega ) } \\nabla \\times \\mathbf { H } = \\left ( \\frac { \\omega } { c } \\right ) ^ 2 \\mathbf { H } , \\\\ & \\nabla \\cdot \\mathbf { H } = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} \\sigma = \\sqrt { 2 \\pi } . \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} f \\mapsto \\Gamma _ \\lambda ( f ) : = \\int _ { G / H } T _ H ( f ) ( x H ) d \\lambda ( x H ) , \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} J : = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , A : = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} , B ( x ) : = \\begin{pmatrix} - V ( x ) & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} S [ g , A ] : = S _ { t o t } ^ + [ \\mathcal { A } ] - S _ { C S } [ \\mathcal { A } ] = \\int _ { \\partial M } A _ + \\partial _ - \\phi + \\frac 1 2 A _ + A _ - + \\frac 1 2 \\partial _ + \\phi \\partial _ - \\phi \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} \\Psi _ { n } ( \\theta ) = \\Phi _ { n } ( r e ^ { i \\theta + f _ { n } ( r ) } ) , \\qquad 0 < \\theta \\leq \\frac { \\pi } { 2 } - 1 , \\ ; n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} \\tau _ { \\Sigma } ( \\epsilon ) : = m i n \\{ \\tau _ p ( \\epsilon ) : p \\in S i n g ( \\Sigma ) \\} \\cup \\{ d i s t _ M ( p , q ) / 2 : p \\neq q \\in S i n g ( \\Sigma ) \\} , \\ \\ \\ \\epsilon \\in ( 0 , 1 ] \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & \\frac { \\partial } { \\partial \\Theta _ 1 } - i \\frac { \\partial } { \\partial \\Theta _ 2 } \\\\ \\frac { \\partial } { \\partial \\Theta _ 1 } + i \\frac { \\partial } { \\partial \\Theta _ 2 } & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} B ( x y , z ) & = B ( x , z ) y + x B ( y , z ) , \\\\ B ( x , y z ) & = B ( x , y ) z + y B ( x , z ) . \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} W \\begin{bmatrix} u _ 1 \\\\ u _ 2 \\end{bmatrix} = \\begin{bmatrix} v _ 1 \\\\ v _ 2 \\end{bmatrix} \\ge 0 , \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} { } ^ { i } \\ ! t ^ { j , h ' } _ { j , h } = \\begin{cases} 0 , & h ' = 1 \\\\ ( - 1 ) ^ { h - h ' + 1 } \\left ( \\begin{matrix} h - 1 \\\\ h ' - 2 \\end{matrix} \\right ) c _ { i + 1 } ^ { h - h ' + 1 } , & 2 \\leq h ' \\leq h + 1 \\end{cases} \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} \\lim _ { \\beta \\to 0 ^ + } F ( \\beta ) = 0 . \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} \\sum _ { m = - \\infty } ^ \\infty ( - 1 ) ^ m a _ k ( n - 2 m ^ 2 ) = a ( n ) - 2 \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { \\frac { n } { 2 } } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - 2 m ^ 2 ) . \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} 2 b = b - \\iota _ 1 ( b ) = \\tau ( g + \\iota _ 2 ( g ) ) - ( g + \\iota _ 2 ( g ) ) + ( h - \\iota _ 1 ( h ) ) \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} 1 = \\theta ( h ) g h ^ { - 1 } g ^ { - 1 } = \\theta ( h ) g _ 0 t ^ { \\varepsilon _ 1 } g _ 1 t ^ { \\varepsilon _ 2 } . . . g _ { n - 1 } t ^ { \\varepsilon _ n } g _ n h ^ { - 1 } g _ n ^ { - 1 } t ^ { - \\varepsilon _ n } g ^ { - 1 } _ { n - 1 } t ^ { - \\varepsilon _ { n - 1 } } . . . t ^ { - \\varepsilon _ 1 } g _ 0 ^ { - 1 } h \\in \\Sigma _ 1 . \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} A ^ { ( \\rho ) } = \\{ \\alpha \\in A \\ , ; \\ , | \\alpha - \\rho | _ v < \\delta \\} , \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} 0 \\longrightarrow \\Omega ^ 2 _ { \\mathbb { P } ^ n } ( d + 2 ) \\otimes \\mathcal { I } _ Z \\longrightarrow \\mathcal { I } _ Z ( d ) ^ { \\oplus \\binom { n + 1 } { 2 } } \\longrightarrow \\Omega ^ 1 _ { \\mathbb { P } ^ n } ( d + 2 ) \\otimes \\mathcal { I } _ Z \\longrightarrow 0 \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} j ( E ) = \\frac { ( t + 6 ) ^ 3 ( t ^ 3 + 1 8 t ^ 2 + 8 4 t + 2 4 ) ^ 3 } { t ( t + 8 ) ^ 3 ( t + 9 ) ^ 2 } , t \\neq 0 , - 8 , - 9 . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} K ( \\Psi ; M ; \\gamma ) = g \\left ( \\prod _ { ( i , j ) \\in \\Psi } \\left ( 1 - \\frac { z _ i } { z _ j } \\right ) ^ { - 1 } \\prod _ { j \\in M } \\left ( 1 - \\frac { 1 } { z _ j } \\right ) \\mathbf { z } ^ \\gamma \\right ) \\ , , \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} h _ i : = | \\{ ( x , B ) : \\ x \\in A _ 1 \\cap \\cdots \\cap A _ i \\setminus ( A _ { i + 1 } \\cup \\cdots \\cup A _ { k - 1 } ) , \\ B \\in F _ 3 , \\ x \\in B \\} | . \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} | z | = | z - \\beta x + \\beta x | \\leq | z - \\beta x | + r < s - t + r \\leq s - \\frac { t } { 2 } \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} \\phi _ t ^ * \\left ( \\sum _ 1 ^ 2 u _ { i } \\sigma _ i \\right ) = u _ 1 e ^ t \\sigma _ 1 + u _ 1 t e ^ t \\sigma _ 2 + u _ 2 e ^ t \\sigma _ 2 = u _ 1 ' \\sigma _ 1 + u _ 2 ' \\sigma _ 2 \\ , , \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} P _ { D _ { \\bar p R / 2 } } ( x , y ) = \\int _ { D _ { \\bar p R / 2 } } G _ { D _ { \\bar p R / 2 } } ( x , z ) j ( | z - y | ) d z \\le c P _ { D _ { \\bar p R / 2 } } ( x , 0 ) , \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} K _ { 1 } ( w , \\lambda ( t ) ) - \\frac { \\lambda ( t ) ^ { 2 } } { 4 w } = - \\int _ { 0 } ^ { \\infty } \\frac { R d R } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\frac { 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } { 2 w \\left ( \\sqrt { ( 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } ) ^ { 2 } + 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } + 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } \\right ) } \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} H _ { \\mu } ( z ) = \\int _ { \\mathbb { T } } \\frac { t + z } { t - z } \\ , d \\mu ( t ) , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} q _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( 1 / x ) = \\begin{cases} \\frac { 1 } { \\pi } \\Im \\frac { 1 } { 1 - \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) } , & x = h ( r ) f ( r ) > 0 , \\\\ 0 , & x = h ( r ) f ( r ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} u _ 2 ' = u _ 1 \\ , \\frac { u _ 1 ' } { u _ 1 } \\ln \\left ( \\frac { u _ 1 ' } { u _ 1 } \\right ) + u _ 2 \\ , \\frac { u _ 1 ' } { u _ 1 } \\ , . \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} \\lim _ { x \\uparrow c _ { 0 } ^ { - 1 } } \\frac { q _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( x ) } { \\left | x - c _ { 0 } ^ { - 1 } \\right | ^ { 1 / 3 } } = \\frac { a _ { 1 } } { \\pi \\sqrt [ 3 ] { \\left | c _ { 3 } \\right | } } \\cos \\left ( \\frac { \\theta } { 3 } - \\frac { \\pi } { 6 } \\right ) , \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle H _ { 1 m } \\cdot G _ { 1 m } = ( I _ 2 - F _ m ) = ( I _ 2 - F _ m ) ^ 2 = H _ { 1 m } \\cdot ( G _ { 1 m } \\cdot H _ { 1 m } ) \\cdot G _ { 1 m } , \\\\ \\\\ H _ { 2 m } \\cdot G _ { 2 m } = F _ m = F _ m ^ 2 = H _ { 2 m } \\cdot ( G _ { 2 m } \\cdot H _ { 2 m } ) \\cdot G _ { 2 m } , \\\\ \\\\ ( H _ { 1 m } \\cdot G _ { 1 m } ) ( H _ { 2 m } \\cdot G _ { 2 m } ) = ( H _ { 2 m } \\cdot G _ { 2 m } ) ( H _ { 1 m } \\cdot G _ { 1 m } ) = 0 . \\end{array} \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} \\norm { ( x _ n ) } _ p = \\left ( \\sum _ { n = 0 } ^ \\infty { \\abs { x _ n } ^ p } \\right ) ^ \\frac { 1 } { p } . \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} C ^ \\infty ( U ) _ { ( i ) } = \\{ f \\in C ^ \\infty ( U ) | \\ \\forall j < i \\colon f ^ { ( j ) } \\mbox { v a n i s h e s o n } \\ Q \\cap T _ r U \\} ; \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} p _ { n + 1 } ( \\underline x _ { n + 1 } ) = \\left ( \\prod _ { i = 1 } ^ { n + 1 } ( 1 + x _ i ) \\right ) \\sum _ { I \\subseteq [ 2 , n ] } k _ { n - | I | } ( \\underline x _ n ^ I ) \\cdot \\prod _ { i \\in I } \\frac { 1 } { 1 + x _ i } . \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} \\frac { 1 } { h ( r ) } p _ { \\mu } \\left ( \\frac { 1 } { h ( r ) } \\right ) = \\frac { 1 } { \\pi } \\frac { r \\sin f ( r ) } { | 1 - r e ^ { i f ( r ) } | ^ { 2 } } , r > 0 , \\ ; h ( r ) \\notin A _ { \\mu } . \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} i w = i \\phi _ 1 e ^ { i \\theta } , \\frac { \\partial w } { \\partial { x _ 1 } } = \\phi _ 2 e ^ { 2 i \\theta } + \\phi _ 0 , \\frac { \\partial w } { \\partial { x _ 2 } } = - i \\phi _ 2 e ^ { 2 i \\theta } + i \\phi _ 0 , \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} \\beta = \\frac { d \\Lambda } { d \\mu } ( 0 ) . \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} \\begin{cases} D ^ { \\alpha } _ 0 y ( t ) = f ( t , y ( t ) ) , \\ t \\in [ 0 , b ] , \\\\ D ^ { ( i ) } y ( 0 ) = c _ i , \\ i = 0 , \\ldots \\lfloor \\alpha \\rfloor . \\end{cases} \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} \\begin{cases} \\bigl | { \\xi _ 2 } + \\frac { 1 } { 2 } N + \\frac { \\sqrt { 3 } } { 2 } ( \\sqrt { 2 } - 1 ) N \\bigr | \\leq 2 ^ { 1 2 } { A ' } ^ { - \\frac { 1 } { 2 } } N , \\\\ | \\eta _ 2 - ( \\sqrt { 2 } + 1 ) ^ { \\frac { 2 } { 3 } } \\xi _ 2 + ( \\sqrt { 3 } + 1 ) ( \\sqrt { 2 } - 1 ) ^ { \\frac { 1 } { 3 } } N | \\leq 2 ^ { 2 5 } { A ' } ^ { - 1 } N , \\end{cases} \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} v _ { 3 , 2 , a } ^ { \\lambda } ( t , r ) : = \\frac { - 1 } { r } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\rho d \\rho \\left ( \\frac { 1 } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } - \\frac { 1 } { ( s - t ) } \\right ) \\lambda '' ( s ) \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + 1 \\right ) \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} G : = [ 0 , \\mu ( Q ) ] \\backslash h \\left ( [ 0 , \\ell ( Q ) ] \\right ) . \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} 0 & \\equiv A + B = \\varphi ( f + D ^ 2 u ( \\nu , \\nu ) ) + \\bar { \\nabla } f \\cdot \\bar { \\nabla } \\varphi - h ( \\bar { \\nabla } f , \\bar { \\nabla } f ) \\\\ & = \\varphi ^ 2 ( - H + ( k - 1 ) \\cot \\theta - ( n - k ) \\tan \\theta ) + \\cot \\theta | \\bar { \\nabla } f | ^ 2 - h ( \\bar { \\nabla } f , \\bar { \\nabla } f ) . \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} \\begin{cases} d ( x , z ) - d ( y , z ) \\leq d ( x , y ) , \\\\ d ( y , z ) - d ( x , z ) \\leq d ( x , y ) . \\end{cases} \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} D ^ { 2 } _ i = \\{ \\alpha ^ { i + 2 j } \\ , : \\ , 0 \\leq j \\leq 2 k - 1 \\} , \\ , \\ , \\ , \\ , i = 0 , 1 \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} \\partial _ { t } T ( y ) ( t , \\omega ) = - \\int _ { t } ^ { \\infty } \\cos ( ( t - x ) \\sqrt { \\omega } ) \\left ( F _ { 2 } ( y ) ( x , \\omega ) - \\mathcal { F } ( \\sqrt { \\cdot } F ( x , \\cdot \\lambda ( x ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) - \\mathcal { F } ( \\sqrt { \\cdot } F _ { 3 } ( x , \\cdot \\lambda ( x ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) \\right ) d x \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} B _ { j } u = \\sum \\limits _ { \\left \\vert \\beta \\right \\vert \\leq m _ { j } } \\ b _ { j \\beta } D _ { y } ^ { \\beta } u \\left ( x , y , t \\right ) = 0 x \\in R ^ { n } , y \\in \\partial G , j = 1 , 2 , . . . , m , \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} \\psi ^ t _ { 0 , \\lambda } ( r ) = \\left ( \\frac { \\lambda } { \\sqrt { \\pi } } \\right ) ^ { \\frac { 1 } { 2 } } \\cdot e ^ { - \\frac { \\lambda ( r - t ) ^ { 2 } } { 2 } } , \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} \\beta _ J = \\left ( \\frac { \\partial f _ J } { \\partial \\mu } \\frac { \\partial g _ J } { \\partial y } - \\frac { \\partial g _ J } { \\partial \\mu } \\frac { \\partial f _ J } { \\partial y } \\right ) \\bigg | _ { x = y = \\mu = 0 } . \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} \\psi ( a , f ) \\ , ( [ x ] , k ) : = \\sum _ { n \\in \\mathbb { Z } } a ( [ x - k \\theta ] , - n ) \\ , f ( [ x ] , n - k ) , \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} t \\leq \\frac { ( q - 1 ) ( q ^ m + 1 ) - q ^ { n - \\delta } + 1 } { q ^ n - 1 } + \\ell + 1 = \\ell + \\frac { ( q - 1 ) ( q ^ m + 1 ) + q ^ n - q ^ { n - \\delta } } { q ^ n - 1 } \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} \\langle F _ k , \\nabla _ { F _ j } ^ T V \\rangle = \\langle F _ k , \\nabla _ { F _ j } V \\rangle = - \\langle F _ { j k } , V \\rangle \\ , , \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} e l ( e ) = e ( e i ( e ) ) = e ^ 2 i ( e ) = e i ( e ) = l ( e ) . \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} ( w t _ { s p } ( C ) , w t _ { R T } ( C ) ) = \\begin{cases} ( 3 , 2 ) & \\mbox { $ C $ i s g i v e n i n E x a m p l e \\ref { 0 9 _ 1 0 _ 2 0 2 0 _ e x _ 2 } ( i ) } , \\\\ ( 2 , 1 ) & \\mbox { $ C $ i s g i v e n i n E x a m p l e s \\ref { 0 9 _ 1 0 _ 2 0 2 0 _ e x _ 2 } ( i i ) a n d \\ref { 0 9 _ 1 0 _ 2 0 2 0 _ e x _ 3 } ( i ) } , \\\\ ( 4 , 4 ) & \\mbox { $ C $ i s g i v e n i n E x a m p l e \\ref { 0 9 _ 1 0 _ 2 0 2 0 _ e x _ 3 } ( i i ) } . \\end{cases} \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} T _ j ( x _ { j } , x _ { j + 1 } , x _ { j + 2 } , . . ) = ( x _ { j + 1 } , x _ { j + 2 } , . . . ) . \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} \\Psi _ Z ( t , \\xi / c ) = \\Psi _ { Z ^ c } ( t , \\xi ) . \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} \\inf _ { r \\in ( 0 , \\infty ) , | \\xi | = c } s ( r ) \\int _ { | y | \\leq N _ \\nu } ( 1 - \\cos ( y \\cdot \\xi ) ) \\nu ( r \\ , d y ) > 0 , \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} \\lambda _ { 1 ; p , f } I _ { p , \\alpha } = ( \\alpha + 1 ) \\frac { \\int _ { \\Omega } | u | ^ { \\alpha - p } | \\nabla u | ^ { 2 p } \\ , d \\mu } { \\int _ { \\Omega } | u | ^ { \\alpha + p } \\ , d \\mu } , \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { \\ell _ R ( R / I _ n ) } { n ^ 2 } = \\lim _ { n \\rightarrow \\infty } \\frac { \\ell _ { \\hat R } ( \\hat R / I _ n \\hat R ) } { n ^ 2 } = \\lim _ { n \\rightarrow \\infty } \\frac { \\ell _ { R _ 1 } ( R _ 1 / I _ n R _ 1 ) } { n ^ 2 } + \\lim _ { n \\rightarrow \\infty } \\frac { \\ell _ { R _ 2 } ( R _ 2 / I _ n R _ 2 ) } { n ^ 2 } . \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} \\sum _ { i + j = n } & \\binom { i + j } { i } \\lambda _ { i + p m } \\mu _ { j - 1 + m } = 0 , \\\\ \\sum _ { i + j = n } & \\binom { i + j } { i } \\mu _ { i + p m } \\mu _ { i - 1 + m } = 0 , \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} e ^ { \\tau ( \\phi _ { + } - \\phi _ { - } ) } = \\frac { \\lim _ { x _ { n + 1 } \\rightarrow 0 } \\| x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { w } \\| _ { L ^ { 2 } ( C _ { s , 1 / 2 } ' ) } ^ { 1 - \\alpha } } { \\| x _ { n + 1 } ^ { \\frac { 1 - 2 s } { 2 } } \\tilde { w } \\| _ { L ^ { 2 } ( C _ { s , 1 / 2 } ^ { + } ) } ^ { 1 - \\alpha } } \\le \\frac { 1 } { c _ { 0 } } \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} \\delta g _ { i j } & = \\frac { d } { d t } \\big \\langle \\ * { r } _ i , \\ * { r } _ j \\big \\rangle \\bigg | _ { t = 0 } = 2 \\big \\langle ( \\delta \\ * { r } ) _ i , \\ * { r } _ j \\big \\rangle = 2 \\langle u _ i \\ * { N } + u \\ * { N } _ i , \\ * { r } _ j \\rangle \\\\ & = 2 u _ i \\langle \\ * { N } , \\ * { r } _ j \\rangle + 2 u \\langle \\ * { N } _ i , \\ * { r } _ j \\rangle = - 2 u h _ { i j } . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} K _ n \\subset E _ n , \\mathbb { P } ( K _ n ) \\geq \\mathbb { P } ( E _ { n + 1 } ) > 0 \\mbox { a n d } \\overset { n } { \\underset { m = 1 } { \\cap } } K _ m \\subset \\overset { n } { \\underset { m = 1 } { \\cap } } E _ m , \\mbox { f o r a l l } n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} \\begin{aligned} f _ L ( 0 , 0 ; \\mu ) & = 0 , & f _ R ( 0 , 0 ; \\mu ) & = 0 , \\\\ g _ L ( 0 , 0 ; \\mu ) & = 0 , & g _ R ( 0 , 0 ; \\mu ) & = 0 , \\end{aligned} \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} C ^ { 2 } . u - ( - 4 n ^ { 2 } u ) = g ^ { * } ( h ) B \\in W , \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} \\beta _ q ( k ) = [ 4 k - 1 ] \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a , q ^ { - 1 } / b , c q ^ { - 1 } , d q ^ { - 1 } , q ^ { - 1 } ; q ^ 2 ) _ k } { ( q ^ 2 / a , a q ^ 2 , b q ^ 2 , q ^ 2 / c , q ^ 2 / d , q ^ 2 ; q ^ 2 ) _ k } \\bigg ( \\frac { b q ^ 7 } { c d } \\bigg ) ^ k . \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} \\sum _ { m \\in \\mathbb { Z } } \\mu ( T _ m ( \\phi _ 1 ) ) x ^ m = \\left ( \\frac { 1 + x ^ { - 1 } } { 2 } \\right ) \\left ( \\sum _ { U \\in \\mathbb { Q } _ 2 ^ \\times / ( \\mathbb { Q } _ 2 ^ \\times ) ^ 2 } x ^ { \\mathrm { o r d } _ 3 c _ 2 ( \\phi _ { 1 , s } ) } \\mu _ 2 ( U ) \\right ) \\left ( \\sum _ { U \\in \\mathbb { Q } _ 7 ^ \\times / ( \\mathbb { Q } _ 7 ^ \\times ) ^ 2 } x ^ { \\mathrm { o r d } _ 3 c _ 7 ( \\phi _ { 1 , s } ) } \\mu _ 7 ( U ) \\right ) , \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} & ( 1 + \\epsilon ) F ( v ) - F ( ( 1 + \\epsilon ) v + \\rho ) \\\\ = & ( 1 + \\epsilon ) F ( v ) - F ( ( 1 + \\epsilon ) v ) + F ( ( 1 + \\epsilon ) v ) - F ( ( 1 + \\epsilon ) v + \\rho ) \\\\ \\succeq & C _ l \\epsilon \\mathbf { 1 } - C _ f K _ 4 \\epsilon \\mathbf { 1 } \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} \\Delta ^ { I } \\boldsymbol { n } = - \\nabla _ { 1 } \\nabla _ { 1 } \\boldsymbol { n } - \\nabla _ { 2 } \\nabla _ { 2 } \\boldsymbol { n } - q _ { 2 } \\nabla _ { 1 } \\boldsymbol { n } + q _ { 1 } \\nabla _ { 2 } \\boldsymbol { n } . \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} = 2 e ^ { - i t k ^ 2 } \\int _ { \\sqrt t ( 0 . 5 \\alpha + k ) } ^ { \\sqrt t ( 0 . 5 \\beta + k ) } \\exp ( i \\xi ^ 2 ) \\widehat h ( - k + \\xi / \\sqrt t ) d \\xi \\ , . \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} \\frac { d v } { d t } & = i , \\\\ \\frac { d i } { d t } & = - v + k _ 1 i - k _ 2 i ^ 3 , \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} \\int _ { t } ^ { \\infty } | e _ { 1 } '' ( s ) - e _ { 2 } '' ( s ) | | \\frac { 1 } { \\log ( \\lambda _ { 0 } ( t ) ) } - \\frac { 1 } { \\log ( \\lambda _ { 0 } ( s ) ) } | \\frac { d s } { 1 + s - t } \\leq \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\log ^ { b + 2 } ( t ) ( \\log ( \\log ( t ) ) ) ^ { 5 / 2 } } \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} \\Gamma _ { 2 , } ^ { - 1 } ( k ) = \\langle \\phi ( k ) \\phi ( - k ) \\rangle & = \\frac { i } { X _ k } . \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} \\mu ( B ) = \\P ( g _ { 1 } g _ { 2 } \\in B ) . \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi _ { n , \\alpha } ( z ) & \\ , = \\ , z \\\\ \\Phi _ { n , \\alpha } ( \\partial ) & \\ , = \\ , \\alpha z ^ n + \\partial \\end{aligned} \\begin{aligned} \\Phi ' _ { n , \\alpha } ( z ) & \\ , = \\ , z + \\alpha \\partial ^ n \\\\ \\Phi ' _ { n , \\alpha } ( \\partial ) & \\ , = \\ , \\partial \\end{aligned} \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} H ( \\xi ) = \\frac { 1 } { \\pi } \\iint _ { A _ { i j } } \\ , \\frac { G ( z ) } { \\xi - z } \\ , d x \\ , d y , \\xi \\in A _ { i } \\cup Z _ \\rho , x = { \\rm R e } \\ , z , y = { \\rm I m } \\ , z , \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} \\varphi ( m ) : = \\lim _ { N \\rightarrow \\infty } - \\frac { 1 } { N } \\log \\int _ { \\{ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ i = m \\} } \\exp \\left ( - H ( x ) \\right ) \\mathcal { L } ^ { N - 1 } ( d x ) \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} \\lim _ { x \\downarrow c _ { 0 } } \\frac { p _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ( x ) } { \\left | x - c _ { 0 } \\right | ^ { 1 / 3 } } = \\frac { - a _ { 1 } } { \\pi \\sqrt [ 3 ] { \\left | c _ { 3 } \\right | } } \\cos \\left ( \\frac { \\theta } { 3 } - \\frac { \\pi } { 6 } \\right ) , \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} \\mathcal { I } _ N u ( x ) = U _ N ( x ) = \\sum _ { i = 0 } ^ N u _ { i } B _ { i } ( x ) \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} \\partial _ { j } \\partial _ { k } = e ^ { - 2 t } ( \\omega _ { j } \\omega _ { k } \\partial _ { t } ^ { 2 } + \\omega _ { j } \\Omega _ { k } \\partial _ { t } + \\omega _ { k } \\Omega _ { j } \\partial _ { t } + ( \\delta _ { j k } - 2 \\omega _ { j } \\omega _ { k } ) \\partial _ { t } + \\Omega _ { j } \\Omega _ { k } - \\omega _ { j } \\Omega _ { k } ) . \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| \\Phi _ n f \\| _ { H _ n } = \\| f \\| _ H \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} \\top _ { A _ i } \\wedge \\top _ { A _ j } = \\bot \\quad . \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} m \\ddot { x } + b \\dot { x } + k x = F _ { \\rm a p p l y } \\ , . \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} A _ { 2 r } ( 0 , 1 ) = 2 r - 1 , \\ ; \\ ; A _ { 2 r } ( 1 , 0 ) = 2 r . \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} H _ 1 ( z ) = \\frac { 1 } { ( n + 1 ) h _ 0 ^ n } \\mathcal { L } h _ 1 , \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} \\Delta ^ { I } \\boldsymbol { n } = \\Lambda \\boldsymbol { n } , \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} \\left ( \\eta - \\overline { \\partial } _ { b } \\varphi \\right ) = \\gamma \\in \\ker \\left ( \\square _ { b } \\right ) , \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} \\sigma ^ { y } ( x ) = \\frac { 1 } { \\alpha } y ^ { 1 + \\alpha } - \\frac { 1 + \\alpha } { \\alpha } y x ^ { \\alpha } + x ^ { 1 + \\alpha } . \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} \\langle u \\ , | \\ , 1 \\rangle = - \\sum _ { n = 0 } ^ \\infty \\lambda _ n | \\langle 1 | f _ n \\rangle | ^ 2 \\ . \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} \\rho = \\chi \\otimes \\psi \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t g _ B & = - 2 R i c _ B + 2 q \\frac { \\nabla ^ 2 \\psi } { \\psi } \\\\ \\partial _ t \\psi & = \\Delta \\psi - ( q - 1 ) \\frac { 1 - | \\nabla \\psi | ^ 2 } { \\psi } \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} ( h - 2 n ) ( h + ( 2 n - 2 ) ) r ( h - 4 ) = ( h - 2 n ) ( h - ( 2 n + 2 ) ) r ( h ) , \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} \\bar f ( p , u ) : = \\begin{cases} & \\phi \\big ( { \\rm d i s t } _ N ( p ) \\big ) f \\big ( P _ N ( p ) , \\Pi _ N ( P _ N ( p ) ) u \\big ) , \\ p \\in B ( N , 2 \\delta _ 0 ) , \\\\ & 0 , \\ \\ p \\in \\R ^ L / B ( N , 2 \\delta _ 0 ) . \\end{cases} \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} S = \\cosh \\alpha \\pm \\sinh \\alpha . \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} ( h ' ) ^ { - 1 } h ''' = ( 3 / 2 ) ( h ' ) ^ { - 2 } h ''^ 2 - 2 F _ 2 . \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} { } ^ { i } \\ ! s ^ { j , h ' } _ { j , h } = \\begin{cases} \\left ( \\begin{matrix} h - 1 \\\\ h ' - 1 \\end{matrix} \\right ) c _ { i + 1 } ^ { h - h ' } , & h \\neq j - i \\\\ P ( i , h ' , j ) + \\left ( \\begin{matrix} h - 1 \\\\ h ' - 1 \\end{matrix} \\right ) c _ { i + 1 } ^ { h - h ' } , & 1 \\leq h ' \\leq h , h = j - i \\end{cases} \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} g ( e ( x ) ) & = \\{ A \\in \\mathfrak { m } _ 2 : \\varphi ^ { - 1 } ( A ) \\in e ( x ) \\} \\\\ & = \\{ A \\in \\mathfrak { m } _ 2 : x \\in \\widehat { \\varphi ^ { - 1 } ( A ) } \\} \\\\ & = \\{ A \\in \\mathfrak { m } _ 2 : \\varphi ( x ) \\in A \\} \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} \\hat { \\Sigma } _ n \\hat { \\Theta } _ { \\lambda _ n } - \\mathsf { E } _ d + \\lambda _ n \\hat { V } _ n \\hat { Z } _ n \\hat { V } _ n \\hat { \\Theta } _ { \\lambda _ n } = 0 . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} \\partial _ { z } e ^ { z u } & = e ^ { z u } u . \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} E ( \\mu ) = - \\int _ { \\mathbb { R } } \\mu ( x ) \\log \\left ( \\mu ( x ) \\right ) d x , \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} \\lambda _ i = \\exp \\left ( \\log ( \\lambda _ { \\min } ) + \\frac { i - 1 } { m - 1 } \\log ( \\lambda _ { \\max } / \\lambda _ { \\min } ) \\right ) , i = 1 , \\dots , m . \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} D _ p ( x , y ) = D _ { p ^ { * } } ( j _ p ( y ) , j _ p ( x ) ) \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} I _ { r } ( 0 ) = \\lim _ { \\theta \\downarrow 0 } I _ { r } ( \\theta ) = r \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t ^ { 2 } } { ( r - t ) ^ { 2 } } \\ , d \\sigma ( t ) \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} z & = s _ 2 + \\lambda _ 2 \\frac { a + \\lambda _ 2 } { 1 + a \\lambda _ 2 } s _ 2 \\\\ z & = \\frac { s _ 2 } { 1 + a \\lambda _ 2 } ( 1 + 2 a \\lambda _ 2 + \\lambda _ 2 ^ 2 ) \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} F ( x , y ) = - 2 \\ln | x - y | ( x \\ne y ) . \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} E _ \\ast ( V / T ) = E _ \\ast ( V ) / T . \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} e ^ { \\rho x } \\int _ x ^ \\infty \\alpha ( \\xi ) e ^ { - 2 \\rho \\xi } d \\xi = e ^ { - \\rho x } \\int _ 0 ^ \\infty e ^ { - 2 \\rho \\eta } \\alpha ( x + \\eta ) d \\eta \\ , . \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} \\mathfrak { A } : = \\left \\{ \\sum _ { n , m \\in \\mathbb { Z } } a _ { n , m } V ^ n U ^ m \\ ; | \\ ; ( a _ { n , m } ) _ { n , m } \\in S ( \\Z ^ { 2 } ) \\right \\} \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} ( t ^ { - 1 } x , t \\bar { x } , p \\rho _ { - w _ { 0 } ( \\lambda ) } ( t ) ^ { - 1 } , \\rho _ { - w _ { 0 } ( \\lambda ) } ( t ) q ) = ( \\rho _ { V } ( t ) x \\rho _ { V } ( t ) ^ { - 1 } , \\rho _ { V } ( t ) \\bar { x } \\rho _ { V } ( t ) ^ { - 1 } , \\rho _ { V } ( t ) p , q \\rho _ { V } ( t ) ^ { - 1 } ) . \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & a \\ , = \\ , \\alpha - \\gamma \\\\ & b = \\alpha + \\beta + \\gamma \\end{aligned} \\right . \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} I ( u ) = - \\dfrac { 1 } { 2 } | | u | | ^ 2 - \\int _ { \\mathbb { R } ^ N } F _ 0 ( u ) \\ ; d x \\leq 0 . \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} & F ( z , w ) = \\left ( z _ { 1 } + \\mbox { O } ( 2 ) , z _ { 2 } + \\mbox { O } ( 2 ) , \\dots , z _ { N } + \\mbox { O } ( 2 ) , \\mbox { O } ( 2 ) , \\dots , \\mbox { O } ( 2 ) \\right ) , \\\\ & G ( z , w ) = \\left ( w _ { 1 } + \\mbox { O } ( 2 ) , w _ { 2 } + \\mbox { O } ( 2 ) , \\dots , w _ { N } + \\mbox { O } ( 2 ) , \\mbox { O } ( 2 ) , \\dots , \\mbox { O } ( 2 ) \\right ) . \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} \\partial _ { t } u _ { m } = i \\left [ \\Delta u _ { m } + \\sum \\limits _ { j = 1 } ^ { \\infty } \\left ( a _ { m j } \\left ( x \\right ) + b _ { m j } \\left ( x , t \\right ) \\right ) u _ { j } \\right ] , x \\in R ^ { n } , t \\in \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} \\dot { y } = g _ { \\rm s l i d e } ( y ) . \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} a _ { n } = \\int _ { \\mathbb { R } _ { + } } \\frac { t ^ { n } } { ( 1 - \\alpha t ) ^ { n + 1 } } \\ , d \\mu _ { 1 } ( t ) , n = 2 , \\cdots , 2 k - 1 , \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} ( k _ { 1 } * k _ { 2 } ) ( x , y ) : = \\int _ { G } \\int _ { M } k _ { 1 } ( g x , z ) c ( z ) k _ { 2 } ( z , g y ) d { \\rm v o l } ( z ) d \\mu ( g ) . \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k \\norm [ 1 ] { x _ i } _ 2 & \\le n ^ { k / 2 } \\prod _ { \\ell = 0 } ^ L ( 1 + 2 ^ { - \\ell / 4 } u ) \\le n ^ { k / 2 } \\exp \\Big ( u \\sum _ { \\ell = 0 } ^ L 2 ^ { - \\ell / 4 } \\Big ) \\\\ & \\le n ^ { k / 2 } \\exp ( C u ) \\le n ^ { k / 2 } ( 1 + e ^ { 2 C } u ) . \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} \\deg V = d _ 1 \\cdots d _ m . \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} g _ i N _ { \\lambda } \\cup N _ { \\mu , i } = g _ i N _ { \\lambda } \\cup \\left ( ( s _ i N _ { \\mu , i - 1 } \\setminus M ^ i _ { t + 1 , t } ) \\sqcup M ^ i _ { t , t + 1 } \\right ) . \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} \\mathfrak { \\widehat E } _ k = ( 1 - k ) a ^ { k - 2 } X ^ d + \\frac { ( k - 2 ) ( k - 3 ) } { 2 ! } a ^ { k - 4 } X ^ { 2 d } - \\frac { ( k - 3 ) ( k - 4 ) ( k - 5 ) } { 3 ! } a ^ { k - 6 } X ^ { 3 d } + \\cdots . \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} ( \\Phi , \\rho ) \\ , \\mapsto \\ , \\rho ^ \\Phi \\qquad \\rho ^ \\Phi ( L _ k ) : = \\Phi \\circ \\rho ( L _ k ) \\circ \\Phi ^ { - 1 } \\forall k \\ , . \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} P ( z ) = \\sum _ { \\alpha \\in F } x _ { \\alpha } z ^ { \\alpha } \\ , , \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} \\Sigma = \\tau ' ( \\Sigma ' ) = \\tau ' \\left ( \\prod _ { c \\in G / H } \\Sigma ' _ c \\right ) = \\prod _ { c \\in G / H } \\tau ' _ c ( \\Sigma ' _ c ) = \\prod _ { c \\in G / H } \\Sigma _ c . \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} D _ p ( x , y ) = D _ p ( x , z ) + D _ p ( z , y ) + \\langle j _ p ( z ) - j _ p ( y ) , x - z \\rangle , \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} \\psi _ n ( z , w ) = \\langle M _ { \\hat { z } } ( h _ n ) , M _ { \\hat { w } } ( h _ n ) \\rangle _ { L ^ 2 ( \\mu ) } ( z \\in X , w \\in \\overline { X } ) . \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} ( X , \\N [ X ] \\to M ) \\otimes ( X , \\mathbb { N } [ X ] \\to M ) : = ( X \\wedge X , \\N [ X \\wedge X ] \\to M \\otimes M ) \\ , . \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} \\mathcal { F } ^ n & = \\mathcal { C } _ { K _ n } \\| \\cdot \\| _ { H ^ { 1 } ( D ) } , \\\\ \\mathcal { E } ^ n & = \\mathcal { E } | _ { \\mathcal { F } ^ n \\times \\mathcal { F } ^ n } . \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} K _ 3 \\leq C ( \\| \\nabla u \\| _ { L ^ \\infty } + \\| \\nabla \\theta \\| _ { L ^ { p ' } } ) \\times ( \\| | D | ^ s \\omega \\| _ { L ^ 2 } ^ 2 + \\| | D | ^ s \\theta \\| _ { L ^ 2 } ^ 2 + 1 ) . \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} \\liminf _ { R \\to \\infty } \\frac { \\log \\| h ( \\varrho ) ^ \\mu \\| _ { L ^ \\infty ( \\Omega \\cap B _ R ) } } { R ^ 2 } = 0 , \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} a f _ 1 ( f _ 1 ^ { - 1 } f _ 1 ' f _ 1 ^ { - 1 } f _ 2 - f _ 1 ^ { - 1 } f _ 2 ' ) = f _ 1 ( f _ 1 ^ { - 1 } f _ 2 '' - f _ 1 ^ { - 1 } f _ 1 '' f _ 1 ^ { - 1 } f _ 2 ) . \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} & \\partial _ { R } \\left ( \\frac { \\cos ( 2 Q _ { 1 } ( R ) ) } { 2 R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\right ) \\left ( n _ { 1 } ( \\overline { v } _ { 1 } ) - n _ { 2 } ( \\overline { v } _ { 2 } ) \\right ) \\leq \\frac { C } { \\lambda ( t ) ^ { 2 } } \\left ( \\frac { \\overline { v } _ { 1 } ^ { 2 } } { R ^ { 2 } } + \\frac { \\overline { v } _ { 2 } ^ { 2 } } { R ^ { 2 } } \\right ) \\frac { | \\overline { v } _ { 1 } - \\overline { v } _ { 2 } | } { R } \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } P _ * ( r , k ) = \\Pi ( k ) \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} A \\sigma B = \\int _ 0 ^ 1 A ! _ t B \\ ; d p ( t ) , \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} S _ s ( x _ 1 , y _ 1 , & \\dots , x _ s , y _ s ; m ) : = \\\\ & \\sum _ { \\tiny \\begin{array} { c } \\epsilon _ j = 0 , 1 , \\\\ i _ j \\geq 0 \\end{array} } ( - 1 ) ^ { \\epsilon _ 1 + i _ 1 + \\cdots + \\epsilon _ s + i _ s } u _ s ^ { \\epsilon _ s } \\cdots u _ 1 ^ { \\epsilon _ 1 } v _ 1 ^ { - ( p - 1 ) i _ 1 - \\epsilon _ 1 } \\cdots v _ s ^ { - ( p - 1 ) i _ s - \\epsilon _ s } \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\otimes ( \\beta ^ { \\epsilon _ 1 } P ^ { i _ 1 } \\cdots \\beta ^ { \\epsilon _ s } P ^ { i _ s } ) ( m ) . \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} \\begin{array} { r c l l } \\partial _ r ^ 2 & = & 0 & \\mbox { \\rm f o r } r \\in [ 1 , n - 1 ] , \\\\ \\partial _ r \\partial _ t & = & \\partial _ t \\partial _ r & \\mbox { \\rm f o r } r , t \\in [ 1 , n - 1 ] , ~ | r - t | > 1 , \\\\ \\partial _ r \\partial _ { r + 1 } \\partial _ r & = & \\partial _ { r + 1 } \\partial _ r \\partial _ { r + 1 } & \\mbox { \\rm f o r } r \\in [ 1 , n - 2 ] . \\\\ \\end{array} \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} 0 & { } = \\langle J ( B + X ) , Y + Z \\rangle = \\langle Z + J X , Y + Z \\rangle = 1 + \\langle J X , Y \\rangle \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} B ( f , g ) ( x , y ) = B ( f | _ x ^ y , g | _ x ^ y ) ( x , y ) . \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\left ( 1 + | x | \\right ) ^ { d + 1 } \\ , & \\left | \\nabla e ^ { ( s - u ) \\Delta } \\left [ e ^ { ( t - s ) \\Delta } V ^ N ( x ) - V ^ N ( x ) \\right ] \\right | ^ 2 d x \\\\ & \\leq \\int _ { \\R ^ d } \\left ( 1 + | x | \\right ) ^ { d + 1 } \\left [ \\left | \\nabla e ^ { ( t - u ) \\Delta } V ^ N ( x ) \\right | ^ 2 + \\left | \\nabla e ^ { ( s - u ) \\Delta } V ^ N ( x ) \\right | ^ 2 \\right ] d x \\\\ & = : A + B . \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} C _ { 0 } ^ { \\infty } ( ( - 1 , 1 ) ) \\ni \\theta \\mapsto \\int _ { - 1 } ^ { 1 } \\theta ( x ) \\phi ( x - j ) \\ , d x \\in \\C , j = - 1 , 0 , 1 , \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} A / _ G B : = \\{ a / b : ( a , b ) \\in G \\} . \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & \\ , \\to \\ , \\min \\limits _ { x , y , z } & & & \\\\ g _ i ( x ) & \\ , \\leq \\ , 0 & \\qquad & i \\in \\mathcal M & \\\\ h _ j ( x ) & \\ , = \\ , 0 & & j \\in \\mathcal P & \\\\ G _ l ( x ) - y _ l & \\ , \\leq \\ , 0 & & l \\in \\mathcal Q & \\\\ H _ l ( x ) - z _ l & \\ , \\leq \\ , 0 & & l \\in \\mathcal Q & \\\\ 0 \\ , \\leq \\ , y _ l \\ , \\perp \\ , z _ l & \\ , \\geq \\ , 0 & & l \\in \\mathcal Q & \\end{aligned} \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} \\Delta \\phi & = \\Delta ( u ^ 2 + | \\nabla u | ^ 2 ) \\\\ & = 2 \\left ( u \\Delta u + | \\nabla u | ^ 2 + | \\nabla ^ 2 u | ^ 2 + \\nabla \\Delta u \\cdot \\nabla u + R i c ( \\nabla u , \\nabla u ) \\right ) \\\\ & \\geq 2 \\left ( - n u ^ 2 + | \\nabla u | ^ 2 + \\frac { ( \\Delta u ) ^ 2 } { n } - n | \\nabla u | ^ 2 + ( n - 1 ) | \\nabla u | ^ 2 \\right ) \\\\ & = 0 . \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} \\begin{array} { l } ( x - 1 ) ^ { \\epsilon _ 4 } h _ 2 ( x ) \\\\ = \\begin{cases} ( x - 1 ) ^ { n - r + k _ 2 } p _ 2 ( x ) - ( x - 1 ) ^ { n - r + k _ 1 - r _ 1 + k _ 4 } p _ 1 ( x ) p _ 4 ( x ) & \\mbox { i f ~ } n - r + k _ 1 > r _ 1 , \\\\ ( x - 1 ) ^ { r _ 1 - k _ 1 + k _ 2 } p _ 2 ( x ) - ( x - 1 ) ^ { k _ 4 } p _ 1 ( x ) p _ 4 ( x ) & \\mbox { i f ~ } n - r + k _ 1 \\le r _ 1 , \\end{cases} \\end{array} \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} U _ \\delta = \\Big \\{ | \\nabla \\varrho | ^ 2 > 1 + \\delta \\Big \\} \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} y ^ k _ { x x } = f _ { 1 1 } ^ k + f _ { 1 i } ^ k y ^ i _ x + f _ { i j } ^ k y ^ i _ x y ^ j _ x + f ^ 1 _ { i j } \\ , y ^ i _ x y ^ j _ x y ^ k _ x \\ , , k = 2 , \\dots , N \\ , , \\ , \\ , \\ , i , j \\geq 2 \\ , , f ^ k _ { i j } = f ^ k _ { j i } \\ , . \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} ( - P ) ^ { s } u ( x ) : = \\int _ { 0 } ^ { \\infty } \\lambda ^ { s } \\ , \\mathsf { d } E _ { \\lambda } = \\frac { 1 } { \\Gamma ( - s ) } \\int _ { 0 } ^ { \\infty } ( e ^ { t P } - 1 ) u ( x ) \\ , \\frac { \\mathsf { d } t } { t ^ { 1 + s } } \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} \\Psi _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } ) = \\frac { 1 } { | \\mathbf { m } | ! } C _ { \\mathbf { m } } ^ { \\left ( \\frac { 2 } { d } \\right ) } ( \\mathbf { z } ) . \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} B _ i = V \\circ U = U \\circ V . \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} Q = \\begin{bmatrix} Q _ 1 & Q _ { 1 2 } \\\\ Q _ { 1 2 } ^ { \\top } & Q _ 2 \\end{bmatrix} \\ , , K = \\begin{bmatrix} K _ 1 & K _ { 1 2 } \\\\ K _ { 1 2 } ^ { \\top } & K _ 2 \\end{bmatrix} \\ , , \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} w _ \\ell & : = \\left ( \\sum _ { q \\in \\left [ b \\right ] } p _ { n } \\left ( q , \\ell \\right ) \\right ) \\\\ f _ { \\ell , 1 } & : = \\left ( \\sum _ { j \\in \\left [ b \\right ] } \\frac { p _ { n } \\left ( j , \\ell \\right ) } { \\sum _ { q \\in \\left [ b \\right ] } p _ { n } \\left ( q , \\ell \\right ) } h _ { 1 , b , j } \\right ) \\\\ f _ { \\ell , 2 } & : = h _ { 1 , b , \\ell } \\\\ f _ { i , j } & : = h _ { 1 , b , 1 } , \\forall j > 2 , \\forall i . \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} \\langle j _ p ( x ^ * ) - j _ p ( x ) , z - x ^ * \\rangle = 0 \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} 0 = { } ^ h A = h A h ^ { - 1 } + \\d h h ^ { - 1 } \\Longrightarrow A = - h ^ { - 1 } \\d h , \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} \\psi _ { n , 1 } ^ { \\ast } ( Q ) \\leq \\frac { \\sum _ { j = 1 } ^ { n - 1 } \\psi _ { n , j } ^ { \\ast } ( Q ) } { n - 1 } \\leq \\frac { - \\alpha - \\gamma } { n - 1 } + \\epsilon _ { 4 } + O ( q ^ { - 1 } ) . \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} \\Pi = \\Pi _ q \\big ( n _ 1 \\times m _ 1 \\mid \\cdots \\mid n _ { t _ 1 } \\times m _ { t _ 1 } \\mid \\underbrace { 1 \\times 1 \\mid \\cdots \\mid 1 \\times 1 } _ { t _ 2 } \\big ) , \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} b ' _ n & \\overset = 0 \\forall n . \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} \\bigl | \\triangle _ 2 ^ { \\ell } R _ n ( \\cos \\theta ) \\bigr | \\le C \\begin{cases} \\theta ^ \\ell ( 1 + n \\theta ) ^ { - \\frac { d - 1 } 2 } , & \\theta \\in [ 0 , \\pi / 2 ] , \\\\ ( \\pi - \\theta ) ^ { \\ell } ( 1 + n ( \\pi - \\theta ) ) ^ { - \\frac { d - 1 } 2 } , & \\theta \\in [ \\pi / 2 , \\pi ] . \\end{cases} \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} C C ^ k ( A , B ) : = C C ^ k ( A ) \\oplus C C ^ { k + 1 } ( B ) , \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} \\omega ( x , u ) : = x ^ { \\top } Q x + u ^ { \\top } R u = \\begin{bmatrix} x ^ { \\top } & u ^ { \\top } \\end{bmatrix} W \\begin{bmatrix} x \\\\ u \\end{bmatrix} \\ , , W : = \\begin{bmatrix} Q & 0 \\\\ 0 & R \\end{bmatrix} \\ , , R = I _ m . \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} R _ l ^ { } = \\log _ 2 ( 1 + \\gamma _ l ) ~ ~ , \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} \\mbox { $ P _ 1 $ , $ { \\rm G L } _ { 1 } ( q ) \\wr S _ 2 $ , $ { \\rm G L } _ { 1 } ( q ^ 2 ) $ , $ { \\rm G L } _ { 2 } ( q _ 0 ) $ { \\rm ( } $ q = q _ 0 ^ k $ , $ k $ p r i m e { \\rm ) } , } \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} v _ { 4 } ( t , r ) & = \\frac { - r } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\int _ { 0 } ^ { 1 } \\partial _ { 2 } G ( s , r \\beta , \\rho ) d \\beta \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} f ( x ) & = \\int _ { D ^ c } P _ { B ( x , \\varepsilon ) } ( x , y ) \\lambda ( d y ) + \\int _ { D \\setminus B ( x , \\varepsilon ) } f ( y ) P _ { B ( x , \\varepsilon ) } ( x , y ) d y , \\\\ f ( x _ n ) & = \\int _ { D ^ c } P _ { B ( x , \\varepsilon ) } ( x _ n , y ) \\lambda ( d y ) + \\int _ { D \\setminus B ( x , \\varepsilon ) } f ( y ) P _ { B ( x , \\varepsilon ) } ( x _ n , y ) d y . \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} F ( \\underline U ( t , x ) _ + ) \\geq & ( 1 - \\epsilon ) \\Big [ F \\big ( [ \\Phi ( x - \\underline h ( t ) ) + \\Phi ( - x - \\underline h ( t ) ) - \\mathbf { u } ^ * ] _ + \\big ) \\Big ] \\\\ \\succeq & ( 1 - \\epsilon ) \\big [ F ( \\Phi ( x - \\underline h ( t ) ) ) - m L \\epsilon _ 2 \\mathbf { 1 } \\big ] . \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} \\begin{aligned} K ( a , b , c ) q _ { 0 } ^ { 2 } q _ { i + 1 } < ( m - 3 ) q _ { 1 } q _ { 2 } \\cdots q _ { i } \\end{aligned} \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} Q ( i ) - P ( Q ( i ) ) & = | \\{ j \\in J / \\ , Q ( j ) < Q ( i ) , \\phi ( Q ( j ) , Q ( i ) ) < 0 \\} | \\\\ & = | \\{ j \\in J / \\ , Q ( j ) < Q ( i ) , j > i \\} | \\\\ & = | \\{ j \\in J / \\ , j > i , \\psi ( j , i ) > 0 \\} | \\\\ & = Q ( i ) - i \\ , \\ , , \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} u ( a ) = 1 , & u ' ( a ) = 0 , \\\\ v ( a ) = 0 , & v ' ( a ) = 1 . \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} u ^ 2 & = ( e - l ( e ) ) ^ 2 = e ^ 2 - e l ( e ) - l ( e ) e + l ( e ) ^ 2 = e - l ( e ) - e + l ( e ) = 0 . \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} ( \\gamma , \\rho ) \\ , \\mapsto \\ , \\rho ^ \\gamma \\qquad \\rho ^ \\gamma ( L _ k ) : = \\gamma \\circ \\rho ( L _ k ) \\circ \\gamma ^ { - 1 } \\forall k \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} \\begin{cases} \\Phi _ { \\epsilon _ 1 } ^ c \\preceq \\Phi _ { \\epsilon _ 2 } ^ c \\ \\ & { \\rm i f \\ } 0 < \\epsilon _ 1 \\leq \\epsilon _ 2 \\ll 1 , \\\\ \\Phi _ { \\epsilon } ^ { c _ 1 } \\succeq \\Phi _ { \\epsilon } ^ { c _ 2 } \\ \\ & { \\rm i f \\ } 0 < c _ 1 \\leq c _ 2 . \\end{cases} \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} f _ k ( x ) = f _ k ( \\theta _ { P e r } x ) , \\forall k \\geq 0 , \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} T ( r \\zeta ) = \\frac { r ^ { 2 } - 1 } { \\log r } \\int _ { \\mathbb { T } } \\frac { d \\sigma ( t ) } { | t - r \\zeta | ^ { 2 } } , r \\in ( 0 , 1 ) , \\zeta \\in \\mathbb { T } . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} \\P _ { \\rho } [ \\eta \\in A ] \\leq \\sum _ { k = 1 } ^ { \\infty } \\P _ { \\rho } [ \\eta \\in A ( k ) ] \\leq c ( n ^ { d } + T ^ { d } ) e ^ { - T } . \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} \\mu ( [ x , y ] ) = \\mu ( x ) \\mu ( y ) - ( - 1 ) ^ { \\bar x \\bar y } \\mu ( y ) \\mu ( x ) . \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} & \\max _ { \\boldsymbol { Q } } \\textrm { t r } ( ( \\boldsymbol { R } + \\boldsymbol { V } ) \\boldsymbol { Q } ) \\\\ & ~ ~ \\textrm { s . t . } \\quad ~ \\boldsymbol { Q } \\succcurlyeq 0 ; Q _ { n , n } = 1 , \\forall n = 1 , \\ldots , N + 1 . \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} w - H w = x - y \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} f ( x ) = x ^ { L } - \\sum _ { i = 1 } ^ { L } c _ { i } x ^ { L - i } g ( x ) = x ^ { L + 1 } - \\sum _ { i = 1 } ^ { L } c _ { i } x ^ { L + 1 - i } - c _ { L + 1 } . \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} Q _ { P S M } \\beta = d \\Pi \\beta \\beta ; \\ \\ \\ Q _ { P S M } \\eta ^ \\dag = d X + \\eta ^ \\dag d \\Pi \\beta + \\Pi \\eta . \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} \\nabla ^ { \\mathcal H ^ 0 } _ Y X ( \\gamma ; t ) = d _ Y X ( \\gamma , t ) + \\Gamma _ { \\sigma ( t ) } ( \\gamma ( t ) ) ( X ( t ) , Y ( t ) ) . \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} \\langle \\bar \\nabla \\bar g ( p ) , \\bar f ( p , u ) \\rangle = \\langle \\nabla g ( p ) , f ( p , u ) \\rangle . \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} \\nabla _ { x _ i , q } \\circ \\phi = \\dd \\cdot x _ i ^ { p - 1 } \\cdot \\phi \\circ \\nabla _ { x _ i , q } \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} ( t \\omega _ 0 - R i c ( \\omega _ 0 ) + \\sqrt { - 1 } \\partial \\bar \\partial \\Phi ( t ) ) ^ n = e ^ { \\Phi ( t ) - t \\psi } \\omega _ 0 ^ n , \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} \\frac { d \\mathbb { \\tilde { Q } } _ { n } } { d \\mathbb { P } _ { n } } \\left | \\mathbb { P } _ { n } \\right . \\stackrel { d } { \\to } \\exp \\left \\{ \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 \\mu _ { k + 1 } Z _ { k } - \\mu _ { k + 1 } ^ 2 } { 4 ( k + 1 ) } \\right \\} . \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} z ^ i = x ^ i - i \\lambda ^ i , \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\sum _ { j = 1 } ^ { k } \\left ( N _ j - \\binom { \\deg h _ j + 2 } { 2 } \\right ) & \\ge \\frac { N - ( \\sum _ { j = 1 } ^ k \\binom { \\deg h _ j + 2 } { 2 } ) } { k } \\ge \\frac { N - \\binom { b _ N + 2 } { 2 } } { k } \\\\ & \\geq \\frac { \\varepsilon _ 0 N } { b _ N } \\ge \\varepsilon _ 0 \\sqrt { N } \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} F ( \\eta _ { \\mu _ { 1 } } ( z ) ) z = \\eta _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ^ { \\langle - 1 \\rangle } ( \\eta _ { \\mu _ { 1 } } ( z ) ) = \\eta _ { \\rho _ { 1 } } ^ { \\langle - 1 \\rangle } ( z ) \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} a x + b y & = d \\\\ a ' x + b ' y & = e \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} \\lVert A \\rVert _ { \\mathfrak S _ 1 } = \\inf \\sum _ { i = 1 } ^ \\infty \\lVert a _ i \\rVert \\cdot \\lVert y _ i \\rVert , \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} H _ { 2 g + 3 } = H _ { 2 g + 2 } + H _ { 2 g + 1 } + \\dots + H _ { g + 3 } + 2 ^ { k + 1 } H _ { g + 2 - k } . \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} \\mathbf { x } = \\begin{pmatrix} \\alpha & A \\\\ B & \\beta \\end{pmatrix} , ~ \\mathbf { y } = \\begin{pmatrix} \\gamma & C \\\\ D & \\delta \\end{pmatrix} . \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} \\left ( R , _ { 1 } - i A _ { 1 1 , \\overline { 1 } } \\right ) u _ { \\overline { 1 } } = W _ { 1 } u _ { \\overline { 1 } } = \\left [ 2 P _ { 1 } u + i \\left ( A _ { 1 1 } \\gamma _ { \\overline { 1 } } - \\gamma _ { 1 , 0 } \\right ) \\right ] u _ { \\overline { 1 } } . \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} \\mathrm { M } & = \\begin{bmatrix} 2 k \\nabla ^ 2 f ( x ) - 2 A ^ T A - q _ 1 \\mathrm { I } & ( A \\nabla ^ 2 f ( x ) ) ^ T \\\\ A \\nabla ^ 2 f ( x ) & 2 A A ^ T - q _ 1 \\mathrm { I } \\end{bmatrix} \\\\ & \\geq \\begin{bmatrix} 2 k \\nabla ^ 2 f ( x ) - 2 A ^ T A - q _ 1 \\mathrm { I } & ( A \\nabla ^ 2 f ( x ) ) ^ T \\\\ A \\nabla ^ 2 f ( x ) & A A ^ T \\end{bmatrix} . \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} m ( \\iota \\otimes 1 _ A ) = 1 _ A = m ( 1 _ A \\otimes \\iota ) , m ( m \\otimes 1 _ A ) = m ( 1 _ A \\otimes m ) . \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} \\theta ( x ) u _ t - c ^ \\alpha \\lambda \\mathrm { d i v } ( | \\nabla u ^ m | ^ { p - 2 } \\nabla u ^ m ) = f ( u ) , \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} F ( \\Psi ^ { \\beta ^ * } ( x ) ) - F ( k \\Phi ( x ) ) = [ W ^ { \\beta ^ * } ( x ) ] E ( x ) \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} \\mu _ { ( l ) } ( \\lambda ) \\left ( \\eta _ { ( l ) } - \\lambda \\right ) = \\left ( 3 l ^ 4 + 2 ( N - 2 ) l ^ 3 - ( N + 1 ) l ^ 2 - ( N - 2 ) l - \\xi _ { ( l ) } \\lambda \\right ) . \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} \\epsilon ^ { - 1 } \\left ( f ^ n ( u _ { h , 1 } ^ n ) - f ^ n ( u _ { h , 2 } ^ n ) , U _ h ^ n \\right ) _ { \\mathcal T _ h } & = \\epsilon ^ { - 1 } \\left ( ( u _ { h , 1 } ^ n ) ^ 3 - ( u _ { h , 2 } ^ n ) ^ 3 , U _ h ^ n \\right ) _ { \\mathcal T _ h } - \\epsilon ^ { - 1 } \\| U _ h ^ n \\| _ { \\mathcal { T } _ h } ^ 2 . \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} T = \\begin{pmatrix} C _ 1 & A _ 1 & 0 \\\\ B _ 1 & C _ 2 & A _ 2 \\\\ 0 & B _ 2 & C _ 3 & \\ddots \\\\ & \\ , & \\ddots & \\ddots \\end{pmatrix} , \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} M _ { s ; i _ 1 , \\dots , i _ k } & : = [ k ; 0 , \\dots , \\hat { i } _ 1 , \\dots , \\hat { i } _ k , \\dots , s - 1 ] , \\\\ R _ { s ; i _ 1 , \\dots , i _ k } & : = M _ { s ; i _ 1 , \\dots , i _ k } L _ { s } ^ { p - 2 } . \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} C ( { \\bf m } ) = \\sum _ { \\substack { m _ { i , j } \\geq 0 \\\\ 1 \\leq i < j \\leq n } } \\frac { ( 2 - ( j - i ) ) m _ { i , j } ^ { 2 } } { 2 } \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} \\gamma = \\oplus _ i ( 1 _ { \\overline { X } _ i } \\otimes \\overline { \\gamma } _ { X _ i } \\otimes 1 _ { X _ i } ) \\gamma _ { X _ i } \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} \\square _ { b } \\gamma = 0 \\Longrightarrow \\overline { \\partial } _ { b } \\gamma = 0 = \\overline { \\partial } _ { b } ^ { \\ast } \\gamma \\Longrightarrow \\gamma _ { \\overline { 1 } , 1 } = 0 \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} c ' ( s _ 0 ) ( h _ 0 ) _ z + \\hat { h } _ s ( z , s _ 0 ) = c ' ( s _ 0 ) h _ 0 + h _ s ( z , s _ 0 ) . \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} p ^ * : = \\frac { N p } { N - p } , \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} & \\sum _ { \\substack { r = 0 \\\\ } } ^ { 2 n - 1 } x _ { r / 2 } x ^ * _ { r / 2 + t \\bmod n } + \\sum _ { \\substack { r = 0 \\\\ } } ^ { 2 n - 1 } y _ { ( r - 1 ) / 2 } y ^ * _ { ( r - 1 ) / 2 + t \\bmod n } \\\\ & = R _ X ( t ) + R _ Y ( t ) . \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} \\sum _ { { \\bf n } \\in \\mathbb { Z } ^ k _ { \\geq 0 } } \\frac { q ^ { \\frac 1 2 { \\bf n } ^ \\top \\tilde { A } { \\bf n } } { \\bf x } ^ { { \\bf n } } } { ( q ) _ { n _ 1 } \\cdots ( q ) _ { n _ k } } = \\sum _ { { \\bf m } \\in \\mathbb { Z } ^ \\ell _ { \\geq 0 } } \\frac { q ^ { \\tilde { B } ( { \\bf m } ) } { \\bf x } ^ { U { \\bf m } } } { ( q ) _ { m _ 1 } \\cdots ( q ) _ { m _ \\ell } } , \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( f ( ( \\eta ^ { * N } _ n ) _ { n = 1 } ^ N ) \\right ) = \\mathbf { E } \\left ( f ( ( \\eta _ n ) _ { n = 1 } ^ N ) \\ : \\vline \\ : \\eta _ 1 = 1 , \\ : \\eta _ N = 0 , \\ : S _ N > 0 \\right ) , \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta w ^ + + \\Big [ A _ + \\big ( | w ^ + | ^ 2 - { t ^ + } ^ 2 \\big ) + B \\big ( | w ^ - | ^ 2 - { t ^ - } ^ 2 \\big ) \\Big ] w ^ + = 0 , \\\\ [ 3 m m ] - \\Delta w ^ - + \\Big [ A _ - \\big ( | w ^ - | ^ 2 - { t ^ - } ^ 2 \\big ) + B \\big ( | w ^ + | ^ 2 - { t ^ + } ^ 2 \\big ) \\Big ] w ^ - = 0 , \\end{cases} \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} \\lim _ { x \\downarrow c _ { 0 } ^ { - 1 } } \\frac { q _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( x ) } { \\left | x - c _ { 0 } ^ { - 1 } \\right | ^ { 1 / 2 } } = \\frac { - a _ { 1 } } { \\pi \\sqrt { \\left | c _ { 2 } \\right | } } \\cos \\frac { \\theta } { 2 } , \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} s = t ^ { \\gamma - ( \\gamma - 1 ) \\rho _ n ^ { - 1 } } , \\delta = \\frac { 1 + ( \\gamma - 1 ) \\sigma _ m ^ { - 1 } } { \\gamma - ( \\gamma - 1 ) \\rho _ n ^ { - 1 } } . \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} Q ( x , y , z ) = P ( x + y + z ) - P ( x + y ) - P ( y + z ) - P ( z + x ) + P ( x ) + P ( y ) + P ( z ) \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} \\overline D = \\sum _ { k = 0 } ^ \\infty \\sqrt { \\lambda _ k ^ 2 + \\frac 1 4 \\tau ^ 2 } \\ ; \\pi _ { F _ k } + \\sum _ { k = 0 } ^ \\infty \\left ( - \\sqrt { \\lambda _ k ^ 2 + \\frac 1 4 \\tau ^ 2 } \\right ) \\pi _ { a ( F _ k ) } + \\frac 1 2 \\tau \\pi _ k - \\frac 1 2 \\tau \\pi _ { a ( K ) } \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} a ( t ) & : = \\mathrm { e x p } ( t x ) = 1 _ n + \\displaystyle \\sum _ { i < j } \\frac { t ^ { j - i } } { ( j - i ) ! } e _ { i , j } , \\\\ b ( s ) & : = \\mathrm { e x p } ( s y ) = 1 _ n + s e _ { n , 1 } , \\\\ c ( r ) & : = \\mathrm { e x p } ( r z ) = 1 _ n + \\sum _ { j = 2 } ^ n \\sum _ { i = 1 } ^ { j - 1 } c _ { j - i , j } \\frac { r ^ { j - i } } { ( j - i ) ! } e _ { j , i } c _ { j - i , j } = \\prod _ { k = 1 } ^ { j - i } b _ { j - k } . \\\\ \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } P \\bigg ] \\tilde { u } & = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ \\tilde { u } & = u \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\\\ \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } \\tilde { u } ( x ) & = V u \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} \\mathcal Q _ { \\sigma } ( w , \\varphi ) = \\left \\langle F , \\gamma _ 1 ( \\varphi ) \\right \\rangle + \\left \\langle G , \\gamma _ 0 ( \\varphi ) \\right \\rangle \\ , , \\ \\ \\ \\forall \\varphi \\in H ^ 2 ( \\Omega ) . \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} 0 \\geqslant \\Re \\langle P _ n x , H _ n x \\rangle = \\Re \\langle P _ n x , P _ n H x \\rangle = \\Re \\langle P _ n x , H x \\rangle \\mathop { \\longrightarrow } ^ { n \\to \\infty } \\Re \\langle x , H x \\rangle \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} G ( n _ 1 , \\ldots , n _ r ) = \\sum _ { d _ 1 + \\cdots + d _ r = d } \\frac { 1 } { d _ 1 ! \\cdots d _ r ! } e _ R ( \\mathcal I ( 1 ) ^ { [ d _ 1 ] } , \\ldots , \\mathcal I ( r ) ^ { [ d _ r ] } ; M ) n _ 1 ^ { d _ 1 } \\cdots n _ r ^ { d _ r } . \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} \\phi _ { r } \\left ( x \\right ) = \\left \\{ \\begin{array} { c } \\left \\vert x \\right \\vert ^ { 2 } , \\left \\vert x \\right \\vert \\leq r \\\\ r ^ { 2 } , \\left \\vert x \\right \\vert > r \\end{array} \\right . \\varphi _ { \\rho , r } = d \\left ( t \\right ) \\theta _ { \\rho } \\ast \\phi _ { r } \\left ( x \\right ) \\upsilon _ { \\rho , r } = e ^ { \\varphi _ { \\rho , r } } u , \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} | \\frac { - 4 \\alpha \\lambda _ { 0 } '' ( t ) \\left ( \\log ( \\lambda _ { 1 } ( t ) ) - \\log ( \\lambda _ { 2 } ( t ) ) \\right ) } { \\log ( \\lambda _ { 0 } ( t ) ) } | & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) \\log ( \\log ( t ) ) } \\frac { | \\lambda _ { 1 } ( t ) - \\lambda _ { 2 } ( t ) | } { \\lambda _ { 0 , 0 } ( t ) } \\\\ & \\leq \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\log ^ { b + 1 } ( t ) ( \\log ( \\log ( t ) ) ) ^ { 3 / 2 } } \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} \\eta _ t = \\frac { h _ t } { \\| \\nabla _ t \\| ^ 2 } \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} f ' ( 0 ) < \\inf \\sigma ( - \\Delta + V ( x ) ) < a = + \\infty . \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} \\nabla F ( x ) = \\Bigg ( \\frac { \\partial F _ i ( x ) } { \\partial x _ j } \\Bigg ) _ { i , j = 1 , 2 , \\ldots , n } , \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} \\tilde C _ { p , \\nu } = \\left [ 3 \\over k - 1 \\right ] ^ p . \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} \\overleftarrow { \\eta } \\buildrel { d } \\over { = } \\eta , \\bar { W } \\buildrel { d } \\over { = } W , T \\eta \\buildrel { d } \\over { = } \\eta . \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} D = \\sum _ { | s | \\le p } f _ s V ^ s , \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} g _ { \\alpha , 0 , \\alpha , 0 } ( x , t ) = \\frac { 1 } { \\pi } \\int _ { 0 } ^ { \\infty } e ^ { - w x } e ^ { - t w ^ { \\alpha } \\cos ( \\pi \\alpha ) } \\sin ( t w ^ { \\alpha } \\sin ( \\pi \\alpha ) ) d w , \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} \\frac { d ( \\mu _ { 1 } ) _ { * } } { d t } ( x ) = \\frac { 1 } { x ^ { 2 } } \\frac { d \\mu _ { 1 } } { d t } \\left ( \\frac { 1 } { x } \\right ) = A ( x - \\alpha ) ^ { 2 k } \\frac { h ( 1 / x ) } { x ^ { 2 k + 2 } } , x \\in J , \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} \\mathcal R ( v , \\epsilon ) \\index { $ \\mathcal R ( v , \\epsilon ) $ } = \\{ n \\in \\N = \\{ 1 , 2 , . . . \\} \\ , | \\ , \\| v ( n ) \\| < \\epsilon \\} , \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} r i c _ g - \\frac { s c a l _ g } { n } g = \\frac { n - 2 } { h } \\left [ ( \\nabla ^ 2 h ) _ { g _ \\kappa } - \\frac { ( \\Delta h ) _ { g _ \\kappa } } { n } g _ \\kappa \\right ] , \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} \\langle x , \\tilde { x } \\rangle _ { X } : = \\sum _ { i = 1 } ^ N x _ i \\tilde { x } _ i . \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} & ( q ; q ) _ \\infty ( q ; q ^ 2 ) _ \\infty \\sum _ { n = 0 } ^ \\infty p ( n ) q ^ n \\ ! = ( q ; q ^ 2 ) _ \\infty \\mbox { a n d } ( q ; q ) _ \\infty ( - q ; q ^ 2 ) _ \\infty \\sum _ { n = 0 } ^ \\infty p ( n ) q ^ n \\ ! = ( - q ; q ^ 2 ) _ \\infty . \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} d ( \\gamma _ t , \\gamma _ s ) = | t - s | d ( \\gamma _ 0 , \\gamma _ 1 ) \\ \\ \\forall t , s \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} \\hat { h } _ { s } ( z , s _ 0 ) = h _ { s } ( z , s _ 0 ) . \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} 1 + \\zeta _ { p } + \\cdots + \\zeta _ { p } ^ { p - 1 } = 0 . \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} f ( z ) & = ( z - \\alpha _ 1 ) ^ { n _ 1 } \\cdots ( z - \\alpha _ s ) ^ { n _ s } , \\\\ f ' ( z ) & = N ( z - \\alpha _ 1 ) ^ { n _ 1 - 1 } \\cdots ( z - \\alpha _ s ) ^ { n _ s - 1 } ( z - \\beta _ 1 ) ^ { m _ 1 } \\cdots ( z - \\beta _ p ) ^ { m _ p } , \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} t \\frac { \\partial w } { \\partial t } = \\lambda w + f ( t ) \\frac { \\partial w } { \\partial x } + \\Bigl ( \\frac { \\partial w } { \\partial x } \\Bigr ) ^ 2 \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} Y _ \\lambda ^ * ( x ) A ( x ) Y _ \\lambda ( x ) = Z _ \\lambda ^ * ( x ) Y _ \\mu ^ * ( x ) A ( x ) Y _ \\mu ( x ) Z _ \\lambda ( x ) \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} V & = \\frac { v _ { n + 1 } } { ( - H ) ^ { n + 2 } } \\int _ { w _ - } ^ { w _ + } \\frac { w ^ { n + 1 } } { \\sqrt { - U ( w ) } } d w , \\\\ A & = \\frac { a _ n } { ( - H ) ^ { n + 1 } } \\int _ { w _ - } ^ { w _ + } \\frac { w ^ { n } \\sqrt { 1 - U ( w ) } } { \\sqrt { - U ( w ) } } d w , \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} | ( \\theta , w ) | _ 0 = | A ( \\theta ) | + \\sum _ { \\alpha \\in A ( \\theta ) } \\log _ 2 ( w ( \\alpha ) + 1 ) . \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} 2 g _ { L _ n } - 2 = 2 ^ { \\delta } ( 2 g _ { K _ n } - 2 ) + 2 ^ { \\delta } \\sum _ { v \\in M _ { K _ n } } \\frac { e _ v - 1 } { e _ v } \\leq 2 ^ { \\delta } ( g _ { K _ n } - 2 ) + 2 ^ { \\delta } | R | , \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} h _ { \\pm , i } ^ { ( d ) } ( \\mathbf { m } ) & : = \\prod _ { 1 \\leq k \\not = i \\leq r } \\frac { m _ { i } - m _ { k } - \\frac { d } { 2 } ( i - k ) \\pm \\frac { d } { 2 } } { m _ { i } - m _ { k } - \\frac { d } { 2 } ( i - k ) } . \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} \\frac { d G } { d \\tilde { r } } = \\left ( \\frac { \\tilde { \\gamma } } { \\tilde { r } ^ * \\left ( \\tilde { \\gamma } - \\tilde { r } ^ * \\right ) } - \\frac { \\tilde { \\gamma } \\ , \\frac { \\partial P _ R } { \\partial \\tilde { r } } } { P _ R \\left ( \\tilde { \\gamma } - P _ R \\right ) } \\right ) G ( \\tilde { r } ) , \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} { } ^ { ( i ) } \\pi : = \\theta _ { n - i } ( \\theta _ n ( \\pi ) ^ { ( i ) } ) , \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} J ( D ) = \\Gamma ( X , \\mathcal O _ X ( - D ) ) , J ( D _ i ) = \\Gamma ( X _ i , \\mathcal O _ { X _ i } ( - D _ i ) ) , I ( D ) = J ( D ) \\cap R , I ( D _ i ) = J ( D _ i ) \\cap R . \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} \\Delta _ p ( e ^ { i \\phi } I ) = \\cos \\phi - \\mod { 1 - 2 / p } = \\cos \\phi - \\cos \\phi _ { p } . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} \\mathfrak h ^ \\perp = \\bigoplus \\limits _ { \\alpha \\in Y ( \\delta ) } ( \\mathfrak h ^ \\perp \\cap V ( \\alpha ) ) . \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} P _ 1 \\doteq \\prod _ { k = 0 } ^ { \\infty } ( t _ k \\theta _ k ) ^ { p ^ { - { k } } } = t ^ { \\frac { p } { p - 1 } } ( 1 - \\xi ^ { \\sigma _ 1 } ) \\prod _ { k = 1 } ^ \\infty \\left [ \\xi ^ { \\sigma _ k } - \\xi ^ { \\sigma _ { k + 1 } } \\right ] ^ { \\frac { 1 } { p ^ k } } \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} c \\| x \\| \\leq \\sup _ { \\| v \\| = 1 } \\| B ( x , v ) \\| = \\sup _ { \\| v \\| = 1 } \\| \\tau ( v ) \\| = \\| \\tau \\| \\ , . \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\Re \\eta _ { m } ( r ) = 1 + \\frac { \\sqrt [ 3 ] { 1 - r } } { \\sqrt [ 3 ] { \\left | c _ { 3 } \\right | } } \\cos \\left ( \\frac { 2 m \\pi } { 3 } - \\frac { \\theta } { 3 } + \\frac { \\pi } { 3 } \\right ) + o ( 1 ) \\quad r \\rightarrow 1 ^ { - } , \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\Psi : S _ { n } ^ { K } ( k ) & \\longrightarrow & \\mathrm { S p e c } ( K [ X _ { 1 } , . . . , X _ { n } ] ) \\\\ p & \\longmapsto & I _ { p } : = \\{ f \\in F [ X _ { 1 } , . . . , X _ { n } ] : f ( \\overline { x } ) = 0 \\in p \\} \\end{array} \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} t ^ * = \\frac { 2 \\sqrt { 1 - y ( 0 ) } } { y ( 0 ) \\sqrt { ( 1 + y ( 0 ) ) ( 2 H _ 1 - H _ 0 ^ 2 ) } } , \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} f ( t ) = \\boldsymbol { \\eta } e ^ { C t } D { \\mathbf 1 } , h ( t ) = \\frac { \\boldsymbol { \\eta } e ^ { C t } D \\mathbf { 1 } } { \\boldsymbol { \\eta } e ^ { C t } \\mathbf { 1 } } . \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} S ^ * S = I d \\ , \\ S S ^ * = I d - \\langle \\ , \\cdot \\ , | 1 \\rangle 1 \\ . \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} f ( \\eta _ S ( U \\cap S ) ) & = \\{ f ( \\eta _ S ( x ) ) \\mid x \\in U \\cap S \\} \\\\ & = \\{ \\eta ( x ) \\mid x \\in U \\cap S \\} \\\\ & = \\{ \\eta ( x ) \\mid x \\in U \\} \\cap \\{ \\eta ( x ) \\mid x \\in S \\} \\\\ & = \\eta ( U ) \\cap \\eta ( S ) . \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } H _ 2 & = 1 , \\\\ H _ 3 & = - 1 . \\end{alignedat} \\right . \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} M _ 1 = \\langle \\begin{pmatrix} 1 \\\\ 0 \\\\ \\frac { 1 } { 2 } \\end{pmatrix} , \\begin{pmatrix} 1 \\\\ 1 \\\\ \\frac { 9 9 } { 1 0 0 } \\end{pmatrix} \\rangle \\qquad M _ 2 = \\langle \\begin{pmatrix} 1 \\\\ 0 \\\\ \\frac { 1 } { 2 } \\end{pmatrix} , \\begin{pmatrix} 1 \\\\ 1 \\\\ \\frac { 1 0 1 } { 1 0 0 } \\end{pmatrix} \\rangle \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} \\left \\| \\Lambda ^ 2 f _ n ( t ) \\right \\| \\leq \\left \\| \\Lambda ^ 2 f _ 0 \\right \\| \\exp ( \\rho ( \\left \\| \\Lambda _ 1 f _ 0 \\right \\| ) t ) , \\forall t \\geq 0 , n = 1 , 2 , . . . . \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} \\ss _ 1 = ( \\mu , ~ \\sigma _ + ^ 2 - 1 , \\beta _ 0 ^ 2 + \\beta _ 1 ^ 2 + ( \\eta ^ 2 - 1 ) ) ^ \\tau ; ~ ~ \\ss _ 2 = ( \\beta _ 1 ^ 2 , ~ \\beta _ 0 \\beta _ 1 , ~ \\beta _ 0 ^ 4 ) ^ \\tau \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} \\Delta ^ { I } f = - \\nabla _ { 1 } \\nabla _ { 1 } f - \\nabla _ { 2 } \\nabla _ { 2 } f - q _ { 2 } \\nabla _ { 1 } f + q _ { 1 } \\nabla _ { 2 } f . \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } R _ n ( x ) y ^ n & = \\sum _ { m \\ge 0 } A _ m ( x ) y ^ { 2 m } + \\sum _ { m \\ge 0 } B _ m ( x ) y ^ { 2 m + 1 } \\\\ & = \\sum _ { m \\ge 0 } A _ m ( x ) y ^ { 2 m } + y \\sum _ { m \\ge 0 } B _ m ( x ) y ^ { 2 m } \\\\ & = \\frac { 1 - x y ^ 4 - x ^ 2 y ^ 4 + x ^ 2 y ^ 6 } { 1 - ( 1 + x + x ^ 2 ) y ^ 2 + x ^ 2 y ^ 4 } + y \\frac { 1 + x - x y ^ 2 - x ^ 2 y ^ 2 } { 1 - ( 1 + x + x ^ 2 ) y ^ 2 + x ^ 2 y ^ 4 } \\\\ & = \\frac { 1 + ( 1 + x ) y - ( x + x ^ 2 ) y ^ 3 - ( x + x ^ 2 ) y ^ 4 + x ^ 2 y ^ 6 } { 1 - ( 1 + x + x ^ 2 ) y ^ 2 + x ^ 2 y ^ 4 } . \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} \\hat { \\Sigma } _ { Z , n } : = \\hat { \\Sigma } _ { Y , n } - \\hat { \\beta } _ n \\hat { \\Sigma } _ { X , n } \\hat { \\beta } _ n ^ \\top . \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & \\ , \\rightarrow \\ , \\min & & & \\\\ g _ i ( x ) & \\ , \\leq \\ , 0 & \\qquad & i \\in \\mathcal M & \\\\ h _ j ( x ) & \\ , = \\ , 0 & \\qquad & j \\in \\mathcal P & \\\\ G _ l ( x ) \\ , \\leq 0 \\ , \\lor \\ , H _ l ( x ) & \\ , \\leq \\ , 0 & \\qquad & l \\in \\mathcal Q . & \\end{aligned} \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} \\frac { p _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( \\xi ) } { p _ { \\nu _ { 1 } \\boxtimes \\nu _ { 2 } } ( \\xi ) } = \\beta \\frac { | \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } - 1 } { | 1 - \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } \\log | z | } , \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} \\int _ { z _ 1 } ^ { z _ 2 } e { \\cal \\ ; L } h _ s d z = ( n + 1 ) H ' ( s ) \\int _ { z _ 1 } ^ { z _ 2 } e h ^ n \\ ; d z . \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} \\begin{array} { l } \\bullet ~ ( x - 1 ) ^ { \\epsilon _ 1 } h _ 1 ( x ) = ( x - 1 ) ( 1 + ( x - 1 ) ) , \\\\ \\bullet ~ ( x - 1 ) ^ { \\epsilon _ 4 } h _ 2 ( x ) = - 1 + ( x - 1 ) , \\\\ \\bullet ~ ( x - 1 ) ^ { \\min \\{ k _ 1 , r - r _ 1 + k _ 4 \\} + \\hat e _ 3 } h _ 3 ( x ) = 0 , \\\\ \\bullet ~ ( \\epsilon _ 2 , \\epsilon _ 3 ) = ( r - k _ 1 + k _ 3 , k _ 2 ) = ( 2 , 0 ) , \\\\ \\bullet ~ ( \\epsilon _ 5 , \\epsilon _ 6 ) = ( r _ 1 - k _ 1 + k _ 3 , k _ 5 ) = ( 1 , 1 ) , \\end{array} \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\tau _ { k } ^ { ( j ) } = \\infty , k = 1 , 2 , 3 , 4 . \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} U _ p \\left ( f ( z ) g ( p z ) \\right ) = g ( z ) U _ p \\left ( f ( z ) \\right ) . \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} H ( \\mu ^ { I } _ { \\l , \\tau } ; r _ 1 | r _ 2 ) = H ( \\mu ^ { \\l ^ k I } _ { \\l , \\tau } ; \\l ^ k r _ 1 | \\l ^ k r _ 2 ) \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} \\{ \\gamma _ n , \\gamma _ p \\} = 0 \\ , \\ \\{ \\gamma _ n , \\varphi _ p \\} = \\delta _ { p n } \\ , \\ 1 \\leq n , p \\leq N \\ . \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} e ^ { n + 1 } = e ^ { \\tau \\mathcal { L } } \\Big [ e ^ n + \\tau \\left ( \\Psi ^ \\tau _ { } ( u ( t _ { n } ) ) - \\Psi ^ \\tau _ { } ( u ^ n ) \\right ) \\Big ] + \\mathcal { E } ( \\tau , t _ { n } ) . \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} \\lambda _ j ( 0 ) = \\min _ { \\substack { V \\subset H _ c ^ 2 ( \\Omega ) \\\\ { \\rm d i m } V = j } } \\max _ { \\substack { v \\in V \\\\ \\frac { \\partial v } { \\partial \\nu } \\ne 0 } } \\frac { \\mathcal { Q } _ { \\sigma } ( v , v ) } { \\int _ { \\partial \\Omega } \\left ( \\frac { \\partial v } { \\partial \\nu } \\right ) ^ 2 d \\sigma } , \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} H _ { L + 1 } = c _ 1 H _ { L } + \\dots + c _ { L } H _ 1 + 1 = c _ 1 G _ { L } + \\dots + c _ { L } G _ 1 + 1 = G _ { L + 1 } + 1 . \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} & | | \\sqrt { \\omega } \\lambda ( x ) \\left ( [ \\xi \\partial _ { \\xi } , K ] ( y ( x , \\frac { \\cdot } { \\lambda ( x ) ^ { 2 } } ) ) \\right ) ( \\omega \\lambda ( x ) ^ { 2 } ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } \\\\ & \\leq C \\left ( | | y ( x ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } + | | \\sqrt { \\omega } \\lambda ( x ) y ( x ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } \\right ) \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} H _ { k + m } + H _ { k - L + 2 m + 1 } + \\cdots + H _ { k - L + 1 } \\leq H _ { L - 1 } + H _ { k + 1 } + \\sum _ { a = 1 } ^ { k } \\sum _ { i = a } ^ { a + m - 1 } H _ { i } . \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } \\left ( \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) - 1 } { r ^ { 2 } } \\right ) \\left ( \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) ( v _ { 1 } + v _ { 2 } + v _ { 3 } ) \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle \\right ) | \\leq \\frac { C } { t ^ { 3 } \\log ^ { 1 + 3 N + 2 \\alpha b } ( t ) } \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} \\big ( A _ { 1 , i } , \\ldots , A _ { t _ 1 , i } , ( g _ { i 1 } ) , \\ldots , ( g _ { i t _ 2 } ) \\big ) \\ i = 1 , \\ldots , m _ { t _ 1 } . \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} \\chi ( X _ { 4 , 1 4 } , \\mathcal { O } _ { X _ { 4 , 1 4 } } ( D _ 2 ) ) = \\binom { 6 } { 2 } - 1 4 = 1 . \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} b _ { n } = \\int _ { \\mathbb { R } } \\frac { t ^ { n - 1 } ( 1 + t ^ { 2 } ) } { ( 1 - t a _ { 0 } ) ^ { n + 1 } } \\ , d \\sigma ( t ) , n \\geq 2 . \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} x _ { \\pm } = \\pm \\sqrt { \\dfrac { 2 ( 8 ^ { \\delta } v + w _ { 0 } ) } { a ( m - 2 ) } + \\left ( \\dfrac { m - 4 } { 2 ( m - 2 ) } \\right ) ^ { 2 } } + \\dfrac { m - 4 } { 2 ( m - 2 ) } . \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} \\left \\langle N _ 0 ( u ) , g \\right \\rangle = \\sum _ { j = 1 } ^ { \\infty } \\sqrt { \\mu _ j ( 0 ) } \\alpha _ j \\hat a _ j \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} \\langle 1 \\ , | \\ , f _ n ( \\cdot , Q _ \\tau u ) \\rangle = e ^ { i n \\tau } \\langle 1 \\ , | \\ , f _ n ( \\cdot , u ) \\rangle \\ , , \\forall \\ , n \\ge 0 \\ , . \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} [ S , A ] _ { 1 } = & [ \\Delta ' + \\tau ^ { 2 } | \\nabla ' \\phi | ^ { 2 } , - 2 \\tau \\nabla ' \\phi \\cdot \\nabla ' - \\tau \\Delta ' \\phi ] , \\\\ { } [ S , A ] _ { 2 } = & \\bigg [ \\partial _ { n + 1 } ^ { 2 } + \\tau ^ { 2 } ( \\partial _ { n + 1 } \\phi ) ^ { 2 } + \\frac { 1 - 4 s ^ { 2 } } { 4 } x _ { n + 1 } ^ { - 2 } , - 2 \\tau \\partial _ { n + 1 } \\phi \\partial _ { n + 1 } - \\tau \\partial _ { n + 1 } ^ { 2 } \\phi \\bigg ] . \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} \\mu ( \\mathbb { R } ^ n ) = \\mu \\left ( \\bigcup _ { k \\in \\mathbb { N } } k Q \\right ) = \\lim _ { k \\to \\infty } \\mu ( k Q ) \\leq \\lim _ { k \\to \\infty } c _ \\mu k ^ { n _ \\mu } \\mu ( Q ) = \\lim _ { k \\to \\infty } c _ \\mu k ^ { n _ \\mu } \\cdot 0 = 0 , \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} a x = a _ 1 ( s _ 2 - \\lambda _ 2 s _ 1 ) = s _ 1 - \\lambda _ 2 s _ 2 \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} 0 = B ( e f , f ) = e B ( f , f ) + B ( e , f ) f = B ( e , f ) f . \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} T ^ { ( 0 ) } = \\frac { 1 } { \\omega } \\left ( \\phi + \\frac { 3 \\pi } { 2 } \\right ) . \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} \\mathcal { P } _ { 3 } = \\{ ( 2 4 , 0 , 0 ) , ( 1 6 , 4 , 4 ) , ( 1 4 , 5 , 5 ) , ( 1 2 , 6 , 6 ) , ( 1 0 , 7 , 7 ) , ( 8 , 8 , 8 ) , ( 6 , 9 , 9 ) , ( 4 , 1 0 , 1 0 ) , ( 0 , 1 2 , 1 2 ) \\} . \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} \\nu _ { N _ 1 + N _ 2 } ^ { \\sigma } ( s ) ( d x ) : = \\frac { 1 } { Z } \\exp \\left ( \\sigma \\sum _ { k = 1 } ^ { N _ 1 + N _ 2 } w _ k - H _ { N _ 1 + N _ 2 } ( w ) + s \\sum _ { ( i , j ) \\in I _ { N _ 1 , N _ 2 } } M _ { i j } u _ i v _ j \\right ) . \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} \\partial _ { t } h + g \\partial _ { v } h = - \\frac { 1 } { t } \\left ( f _ 0 + h \\right ) \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} H _ e ( \\alpha y + u ) = H ( u ) - \\alpha y , u \\in D ( H ) , \\alpha \\in { \\mathbb C } . \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} | f ( x ) - f _ { Q , \\mu } | & = f ( x ) - f _ { Q , \\mu } \\geq f ( x ) - \\log \\beta _ \\mu = \\log \\left ( \\frac { v ( x ) } { \\beta _ \\mu } \\right ) \\\\ & = \\frac { 1 } { 2 } \\log \\left [ \\frac { M ( w \\chi _ Q ) ( x ) } { \\beta _ \\mu ^ 2 w _ { Q , \\mu } } \\right ] \\geq \\frac { 1 } { 2 } \\log \\left [ \\frac { w ( x ) \\chi _ Q ( x ) } { \\beta _ \\mu ^ 2 w _ { Q , \\mu } } \\right ] . \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { | | \\mathbf { x _ { k + 1 } } - \\mathbf { x _ k } | | _ 2 } { | | \\mathbf { x _ { k + 1 } } | | _ 2 } \\leq 1 0 ^ { - 5 } . \\end{aligned} \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} D _ s f ( x ) = \\frac { 1 } { \\sqrt { | s | } } f \\left ( \\frac { x } { s } \\right ) \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} w _ { i j } ( t ) = \\begin{cases} \\frac { 1 } { 2 } , & i = j = \\ell ( t ) \\\\ \\frac { 1 } { 2 } , & i = \\ell ( t ) , j \\in \\{ s ( t ) , \\ell ( t - 1 ) \\} \\setminus \\{ \\ell ( t ) \\} \\\\ 1 , & i = j \\not = \\ell ( t ) \\\\ 0 , & \\mbox { o t h e r w i s e } \\end{cases} , \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 2 ( q _ 0 ) ^ 2 / \\kappa ^ 3 & 0 \\\\ 0 & ( g _ { i j } ) \\end{pmatrix} , \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} 0 \\ne ( d \\omega , d \\omega ) = ( \\omega , d ^ * d \\omega ) = \\lim ( d \\varphi _ n , d ^ * d \\omega ) = \\lim ( d d \\varphi _ n , d \\omega ) = \\lim 0 = 0 \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A \\left ( x \\right ) u + V \\left ( x , t \\right ) u + F \\left ( x , t \\right ) = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} | S _ d ^ + ( x ) | = X ^ { 1 / 2 } + O ( 1 ) . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} P ( x ) & = x _ { 1 } x _ { 3 } x _ { 5 } - x _ { 1 } x _ { 4 } x _ { 6 } - x _ { 2 } x _ { 3 } x _ { 6 } - x _ { 2 } x _ { 4 } x _ { 5 } , \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} I = \\left ( \\sum _ { i = 1 } ^ { r _ { \\alpha - 1 } } f _ i R _ { k ' } \\right ) + \\left ( \\sum _ { i = 1 } ^ \\beta \\sum _ { j = 1 } ^ \\gamma g _ i ( f _ { 1 , j } ) R _ { k ' } \\right ) + \\left ( \\sum _ { i = r _ \\alpha + 1 } ^ { s } f _ i R _ { k ' } \\right ) . \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} \\tilde { x } & = \\frac { x } { \\mu } , & \\tilde { y } & = \\frac { y - \\zeta _ R ( \\mu ) } { \\mu } . \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} m = 4 ( 1 + | z | ^ 2 ) ^ { - 2 } | d z | ^ 2 \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\hat { p } _ t ( \\xi ) : = \\int _ { \\mathbb { R } ^ d } e ^ { i \\langle x , \\xi \\rangle } p _ t ( x ) d x = e ^ { - t | \\xi | ^ \\alpha } , \\xi \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} P _ A ( z ) = \\sum _ { l = k } ^ { m / 2 } \\sum _ { \\alpha \\in \\Lambda _ { A , l } } x _ \\alpha z ^ \\alpha = \\sum _ { l = k } ^ { m / 2 } P _ { A , l } ( z ) . \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\partial _ { r } \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) \\rho d \\rho = 0 \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} \\frac { \\partial g _ { \\rm s l i d e } } { \\partial y } \\left ( y _ { \\rm e q } ( \\mu ) ; \\mu \\right ) = - \\frac { a _ { 2 R } b _ { 0 L } - a _ { 2 L } b _ { 0 R } } { f _ L \\left ( 0 , y _ { \\rm e q } ( \\mu ) ; \\mu \\right ) - f _ R \\left ( 0 , y _ { \\rm e q } ( \\mu ) ; \\mu \\right ) } \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { \\nabla } _ { a B + U + x Z } ( b B + V + y Z ) = { } & \\Bigl ( \\frac { 1 } { 2 } \\langle U , V \\rangle + x y \\Bigr ) B - \\frac { 1 } { 2 } \\bigl ( b U + y J U + x J V \\bigr ) \\\\ & { } + \\Bigl ( \\frac { 1 } { 2 } \\langle J U , V \\rangle - b x \\Bigr ) Z . \\end{aligned} \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} G = \\zeta _ { 1 0 } + \\zeta _ { 1 0 } ^ { 3 } + \\zeta _ { 1 0 } ^ { 7 } + \\zeta _ { 1 0 } ^ { 9 } + \\cdots + ( \\zeta _ { 6 } + \\zeta _ { 6 } ^ { 5 } ) \\zeta _ { 1 7 } ^ { j } + \\cdots + ( \\zeta _ { 6 } + \\zeta _ { 6 } ^ { 5 } ) \\zeta _ { 1 7 } ^ { k } + \\cdots + ( \\zeta _ { 6 } + \\zeta _ { 6 } ^ { 5 } ) \\zeta _ { 1 7 } ^ { 1 7 - k } + \\cdots + ( \\zeta _ { 6 } + \\zeta _ { 6 } ^ { 5 } ) \\zeta _ { 1 7 } ^ { 1 7 - j } + \\cdots , \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} \\| b \\| _ { { \\rm b m o } _ { \\mathcal { B } } } : = \\sup _ { B \\in \\mathcal { B } } \\frac { 1 } { \\mu ( B ) } \\int _ B | b ( x ) - b _ B | \\ , d \\mu , \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} u _ 2 , \\theta , \\frac { \\partial u _ 1 } { \\partial x _ 2 } = 0 , \\textrm { f o r } x _ 2 = - 1 , 0 , 1 \\ , . \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} \\ker P _ { 1 } = \\ker P _ { 0 } . \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } w = \\lim _ { x _ { n + 1 } \\rightarrow 0 } x _ { n + 1 } ^ { 1 - 2 s } \\eta _ { R } \\partial _ { n + 1 } \\tilde { u } = c _ { n , s } ^ { - 1 } q \\eta _ { R } u = c _ { n , s } ^ { - 1 } q w \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} . \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} & | | - \\frac { \\sin ( 2 ( v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } ) ) } { r ^ { 3 } } \\left ( \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } + 2 v _ { 5 } ) - \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) \\right ) \\vert _ { r = R \\lambda ( t ) } | | _ { L ^ { 2 } ( R d R ) } \\\\ & \\leq \\frac { C } { t ^ { 1 1 / 2 } \\log ^ { 3 b - 3 + \\frac { 5 N } { 2 } } ( t ) } \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} \\frac { d ^ a } { d X ^ a } ( P _ 1 ( X ) + P _ 2 ( X ) R ( X ) ) \\Big | _ { X = \\l } = 0 \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} \\tilde { f } = - K ^ { 2 } \\tau | \\varphi ' | ^ { 2 } \\omega _ { n + 1 } ^ { 1 - 2 s } u _ { 1 } \\quad \\quad \\tilde { V } \\equiv 0 \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} \\begin{cases} \\alpha ^ { * } _ { n } = p \\alpha _ { n } \\beta _ { n } ^ { p - 1 } ; \\\\ \\beta ^ { * } _ { n } = \\beta _ { n } ^ { p } . \\end{cases} \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} A = u _ 1 u _ 2 \\sin \\phi \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} e ^ { - i t H _ 0 } f = \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } e ^ { - i t \\xi ^ 2 } \\left ( \\int _ { \\mathbb { R } ^ + } f ( u ) \\sin ( \\xi u ) d u \\right ) \\sin ( \\xi x ) d \\xi = \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } } e ^ { - i t \\xi ^ 2 } \\left ( \\int _ { \\mathbb { R } } f _ o ( u ) e ^ { - i \\xi u } d u \\right ) e ^ { i \\xi x } d \\xi \\ , . \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} \\sum ^ { 2 g + 2 } _ { i = 1 } H _ i + 1 = 2 H _ { 2 g + 2 } . \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} w _ \\lambda = \\left ( 1 - \\frac { 5 0 } { 4 9 } \\lambda \\right ) \\begin{pmatrix} 1 \\\\ 0 \\\\ \\frac { 1 } { 2 } \\end{pmatrix} + \\frac { 5 0 } { 4 9 } \\lambda \\begin{pmatrix} 1 \\\\ 1 \\\\ \\frac { 9 9 } { 1 0 0 } \\end{pmatrix} \\in M _ 1 \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} K = - \\gamma ^ 2 \\partial _ z \\rho + \\Delta _ t f . \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ 0 \\int _ 0 ^ \\infty J _ i ( x - y ) d y d x = \\int _ 0 ^ \\infty x J _ i ( x ) d x < \\infty , \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} \\begin{cases} c _ 1 t _ 1 < \\beta \\\\ \\beta - [ c _ 1 t _ 1 - c _ 2 ( t _ 2 - t _ 1 ) ] \\le c _ 1 ( t - t _ 2 ) \\end{cases} \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} \\lim _ { U \\uparrow D } \\eta _ U | u | ( A ) = 0 . \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} ( f \\Box g ) ^ * = f ^ * + g ^ * . \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} \\psi = \\hat \\psi ^ * + c _ 1 \\frac { \\partial w } { \\partial x _ 1 } + c _ 2 \\frac { \\partial w } { \\partial x _ 2 } , \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} \\| u _ n \\| _ { 0 , s _ 1 , p _ 1 } ^ { p _ 1 } + \\| v _ n \\| _ { 0 , s _ 2 , p _ 2 } ^ { p _ 2 } = 1 . \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} f ( - 1 ) = \\frac { E ^ 2 } { \\lambda ^ 2 } r ^ 4 - \\frac { 2 E } { \\lambda } r ^ 3 , \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} \\mathbf { f } = \\rho _ 0 h _ z ( \\Omega ) \\mathbf { e } _ z , \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} ( x + y ) z _ { x y } + 2 \\sqrt { z _ x z _ y } = 0 . \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} ( T x ) _ k = \\sum \\limits _ { m \\in \\mathbb Z ^ c } b _ { k m } x _ { k - m } , k \\in \\mathbb Z ^ c , \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} \\Delta ' \\Delta '' = \\delta ^ N \\prod _ { \\beta \\in A } \\prod _ { \\gamma \\in A \\setminus \\{ \\beta \\} } | \\beta - \\gamma | _ v ^ { n _ \\gamma } . \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} | \\partial _ { t } \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) v _ { 4 } \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | & \\leq \\frac { C } { t ^ { 3 } \\log ^ { b + N - 2 } ( t ) } + \\frac { C \\sup _ { x \\geq t } \\left ( \\frac { x | e ''' ( x ) | } { \\lambda ( x ) ^ { 2 - 2 \\alpha } } \\right ) } { t \\log ^ { N - 3 + 2 b - 2 b \\alpha } ( t ) } \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} ( S + 1 ) ^ 2 = | S | e ^ { f + r } , \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} \\mu _ { [ r , 1 1 ] } ( 0 , 1 ) = \\begin{cases} & 0 \\ ; \\ ; \\\\ & 1 \\ ; \\ ; \\\\ \\end{cases} , \\ ; \\ ; \\ ; \\ ; \\lambda _ r ( 0 , 1 ) = \\begin{cases} & 0 \\ ; \\ ; \\\\ & 1 \\ ; \\ ; \\end{cases} . \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} & M _ n ( A ) = \\mathbb { K } _ { M _ d ( B ) } ( M _ { n , d } ( E ) ) , \\\\ & M _ d ( B ) = \\mathbb { K } _ { M _ n ( A ) } ( M _ { n , d } ( E ) ) . \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} u _ 3 ^ 2 = u _ 1 ^ 2 + u _ 2 ^ 2 - 2 u _ 1 u _ 2 \\cos \\phi _ 3 \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} \\overline { \\lim } _ { n \\to \\infty } \\frac { n \\ \\ln n } { | \\ln \\ | \\eta _ n | \\ | \\ } = \\rho \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} \\dot { z } = h \\left ( 1 - e ^ { - z } \\right ) , \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} f | _ { V _ i } \\in \\sum _ { \\substack { f ' \\in I \\\\ } } f ' R [ g _ i ^ { - 1 } ] = I R [ g _ i ^ { - 1 } ] \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = \\langle \\widehat { A } f , g \\rangle _ { \\alpha } , \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} F _ J ( x , y ) = \\begin{bmatrix} f _ J ( x , y ) \\\\ g _ J ( x , y ) \\end{bmatrix} , \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} d \\theta _ { 1 } { } ^ { 1 } = R \\theta ^ { 1 } \\wedge \\theta ^ { \\bar { 1 } } + 2 i \\mathrm { I m } ( A ^ { \\bar { 1 } } { } _ { 1 , \\bar { 1 } } \\theta ^ { 1 } \\wedge \\theta ) , \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} R = \\frac { \\mathbb { C } [ x _ 1 , x _ 2 , \\ldots , x _ n ] } { ( f _ { 1 } , f _ { 2 } , \\ldots , f _ { n } ) } . \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} - \\frac 1 { h ' ( z ) } c _ { - 1 } ' ( z ) \\ , = \\ , 0 \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} \\Lambda ^ k f _ { n } ( t ) \\nearrow \\Lambda ^ k f ( t ) , a s n \\rightarrow \\infty ; k = 0 , 1 , 2 . \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} 0 & = \\nabla f ( \\bar x ) + \\sum \\limits _ { i \\in I ^ g ( \\bar x ) } \\lambda _ i \\nabla g _ i ( \\bar x ) + \\sum \\limits _ { j \\in \\mathcal P } \\rho _ j \\nabla h _ j ( \\bar x ) \\\\ & + \\sum \\limits _ { l \\in I ^ { 0 + } ( \\bar x ) } \\xi _ l \\nabla G _ l ( \\bar x ) + \\sum \\limits _ { l \\in I ^ { + 0 } ( \\bar x ) } \\xi _ l \\nabla H _ l ( \\bar x ) + \\sum \\limits _ { l \\in I ^ { 0 0 } ( \\bar x ) } \\xi _ l ( \\alpha _ l \\nabla G _ l ( \\bar x ) + \\beta _ l \\nabla H _ l ( \\bar x ) ) . \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} \\hat { \\Sigma } _ { Y , \\lambda } : = \\hat { \\beta } _ n \\hat { \\Sigma } _ { X , n } \\hat { \\beta } _ n ^ \\top + \\hat { \\Theta } _ { Z , \\lambda } ^ { - 1 } . \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} \\psi _ { \\Lambda } = \\langle \\Psi _ { \\Lambda } , \\ ( \\ \\cdot \\ ) \\Psi _ { \\Lambda } \\rangle = \\hbox { T r } _ { \\mathcal { H } _ { \\Lambda } } ( \\Psi _ { \\Lambda } \\Psi _ { \\Lambda } ^ * \\ ( \\ \\cdot \\ ) ) . \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} E _ { P ^ { \\theta ^ { \\ast } } } \\Vert { x } _ { t } - \\hat { x } _ { t } \\Vert ^ { 2 } = \\inf _ { \\zeta \\in \\mathcal { K } _ { t } } E _ { P ^ { \\theta ^ { \\ast } } } \\Vert { x } _ { t } - \\zeta \\Vert ^ { 2 } . \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} v _ { 3 , 2 , b } ^ { \\lambda } ( t , r ) : = \\frac { - 1 } { r } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\rho d \\rho \\left ( \\frac { 1 } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } - \\frac { 1 } { ( s - t ) } \\right ) \\lambda '' ( s ) \\left ( - 1 + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} \\sum _ { s = \\tau } ^ { t - 1 } \\sum _ { \\ell = \\tau } ^ { t - 1 } \\alpha ( s ) \\alpha ( \\ell ) \\lambda ^ { t - s } \\lambda ^ { t - \\ell } = \\left ( \\sum _ { s = \\tau } ^ { t - 1 } \\alpha ( s ) \\lambda ^ { t - s } \\right ) ^ 2 . \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} \\mu _ \\varphi \\ast _ { G / H } \\mu _ { \\varphi ' } = \\mu _ { \\varphi \\ast _ { G / H } \\varphi ' } . \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} \\left < \\tau ^ M _ \\varphi , [ V ( D ) ] - [ e _ 1 ] \\right > = \\lim _ { t \\downarrow 0 } \\left < \\tau ^ M _ \\varphi , [ V ( t D ) ] - [ e _ 1 ] \\right > = \\frac { p ! } { 2 p ! } \\frac { ( - 1 ) ^ p } { ( 2 \\pi i ) ^ p } \\ , \\int _ M \\chi \\hat { A } ( M ) \\wedge \\omega _ \\varphi . \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} B _ { n , \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } \\mid { \\boldsymbol { \\omega } } ) + B _ { n , \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } \\mid { \\boldsymbol { \\omega } } ^ { - } [ j ] ) = & - \\sum _ { i = 1 } ^ { r } B _ { n - 1 , \\mathbf { m } _ { i } } ^ { ( d ) } ( z \\mid \\widehat { { \\boldsymbol { \\omega } } } ( j ) ) \\left ( m _ { i } + \\frac { d } { 2 } ( r - i ) \\right ) h _ { - , i } ^ { ( d ) } ( \\mathbf { m } ) . \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} x ^ c : = - \\lim _ { n \\to \\infty } x _ n ^ c \\in ( 0 , \\infty ] \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} \\gamma _ { l ' i } \\overline { \\gamma _ { \\tau ' \\left ( l ' \\right ) j } } = \\left \\{ \\begin{matrix} G ^ { \\left ( l ' \\right ) } _ { 0 ; 0 , \\dots , 1 , \\dots , 0 } ( z ) , & \\mbox { f o r a l l $ i = 1 , \\dots , N $ a n d $ j = N + 1 , \\dots , N ' $ w i t h $ j = \\tau \\left ( i \\right ) $ , } \\\\ \\quad \\quad \\quad \\quad \\quad 0 , & \\quad \\mbox { f o r a l l $ i = 1 , \\dots , N $ a n d $ j = N + 1 , \\dots , N ' $ w i t h $ j \\neq \\tau \\left ( i \\right ) $ , } \\end{matrix} \\right . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} B _ { \\eta } : = \\inf \\left \\{ B _ { c , i } ^ { a l t } : \\ : c \\in \\{ 0 , 1 , \\dots , J \\} , \\ : i \\in \\{ 0 , 1 \\} , \\ : B _ { c , i } ^ { a l t } > - \\infty \\right \\} . \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} S _ 0 ( G ) : = \\{ f \\in L ^ 2 ( G ) \\mid V _ f f \\in L ^ 1 ( G \\times \\widehat { G } ) \\} . \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { ( n + 1 ) / 2 } \\beta _ { \\zeta } ( k ) = \\sum _ { j = 0 } ^ { ( n / t - 3 ) / 2 } \\beta _ { \\zeta } ( j t ) \\sum _ { k = 0 } ^ { t - 1 } \\beta _ { \\zeta } ( k ) + \\sum _ { k = 0 } ^ { ( t + 1 ) / 2 } \\beta _ { \\zeta } ( ( n - t ) / 2 + k ) = 0 . \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} F _ { x y } = - 2 e ^ F . \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} G : = \\frac { d ^ 2 } { 2 s } + \\frac { d } { 2 } \\left ( \\frac { 2 \\pi - 2 } { s } - 1 \\right ) + R , \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} \\frac { \\phi ( 2 ^ \\delta q _ 1 ^ { e _ 1 } \\cdots q _ k ^ { e _ k } ) } 2 - 1 \\ge \\sum _ { i = 1 } ^ k ( \\frac { \\phi ( 2 ^ { \\delta } q _ i ^ { e _ i } ) } 2 - 1 ) \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} ( A - A ^ 2 S ' ) ^ k B ^ k ( A - A ^ 2 S ' ) ^ k & = ( A ^ k - A ^ { k + 1 } S ' ) B ^ k ( A ^ k - A ^ { k + 1 } S ' ) \\\\ & = A ^ k B ^ k A ^ k - A ^ { k + 1 } S ' B ^ k A ^ k - A ^ k B ^ k A ^ k B ^ k A ^ k S ' + A ^ { k + 1 } ( B ^ k A ^ k ) ^ 2 { S ' } ^ 2 \\\\ & = A ^ { k + 1 } - A ^ { k + 2 } S ' = ( A - A ^ 2 S ' ) ^ { k + 1 } . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} 0 & = \\varphi ^ R _ { T _ R ( r ; \\mu ) } \\left ( G _ L ( r ; \\mu ) , H _ L ( r ; \\mu ) \\right ) , \\\\ P _ R ( r ; \\mu ) & = \\psi ^ R _ { T _ R ( r ; \\mu ) } \\left ( G _ L ( r ; \\mu ) , H _ L ( r ; \\mu ) \\right ) . \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } D ( t ) t ^ \\beta = 0 \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} \\begin{aligned} p _ { [ 1 ^ 0 5 ^ 1 ] } \\bigg ( 5 ^ { 2 r } m + \\frac { 5 - 5 ^ { 2 r + 1 } } { 2 4 } \\bigg ) \\equiv 0 \\pmod { 5 ^ { 2 r - 1 } } , \\\\ p _ { [ 1 ^ 6 5 ^ { - 5 } ] } \\bigg ( 5 ^ { 2 r } m + \\frac { 1 9 \\cdot 5 ^ { 2 r } - 1 9 } { 2 4 } \\bigg ) \\equiv 0 \\pmod { 5 ^ { 2 r } } . \\end{aligned} \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} \\max & \\left ( \\sum _ { i = 1 } ^ n p ^ 1 _ i x _ i , \\sum _ { i = 1 } ^ { n } p ^ 2 _ i x _ i \\right ) \\\\ & \\sum _ { i = 1 } ^ n w _ i x _ i \\leq b \\\\ & x _ i \\in \\{ 0 , 1 \\} , & i = 1 , \\ldots , n \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} \\varphi ( x , y ) = \\psi ( x , y ) ( x , y \\in X ) . \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\alpha } ( - \\Delta ) ^ { \\beta } = ( - \\Delta ) ^ { \\alpha + \\beta } ( - \\Delta ) ^ { s } \\in \\mathcal { L } ( \\dot { H } ^ { \\beta + s } ( \\mathbb { R } ^ { n } ) , \\dot { H } ^ { \\beta - s } ( \\mathbb { R } ^ { n } ) ) . \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} Z _ j & = \\big ( A _ { 1 } , A _ { j } , \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots , A _ { j } \\ , , \\ , ( 0 ) , \\ldots , ( 0 ) , ( 1 ) , ( 0 ) , \\ldots , ( 0 ) , ( 0 ) \\big ) \\ j \\in [ m ] , \\\\ Z _ { m + 1 } & = \\big ( A _ { 2 } , A _ { 1 } , \\ldots , A _ { m } , B _ 1 , B _ 1 , B _ 2 , \\ldots , B _ { s - m - 1 } , ( 0 ) , \\ldots , ( 0 ) , ( 0 ) , ( 0 ) , \\ldots , ( 0 ) , ( 1 ) \\big ) , \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} \\widehat { | D | f } = | \\xi | \\widehat { f } , \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} \\sigma = \\sigma ( n ) = \\sqrt { \\frac { 8 \\pi } { n - 2 } } \\mbox { i f } \\ n \\geq 4 , \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} - \\Delta \\phi _ c = & c ( t ) \\ \\ \\Omega , \\\\ \\frac { \\partial \\phi _ c } { \\partial n } = & 0 \\ \\ \\partial \\Omega . \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} v _ { i j } ( h ) = \\begin{cases} \\frac { 3 } { 4 } , & i = j = h \\\\ \\frac { 1 } { 4 \\delta _ i } , & i = h , ( i , j ) \\in E \\\\ \\frac { 1 } { 4 \\delta _ i } , & j = h , ( i , j ) \\in E \\\\ 1 , & i = j \\not = h \\\\ 0 , & \\mbox { o t h e r w i s e } \\end{cases} , \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} \\sigma _ { \\pi ( ( a , i ) ) } \\pi \\sigma _ { ( b , j ) } ( c , k ) = \\sigma _ { ( - a , 1 - i ) } \\pi ( c + j , k ) = \\sigma _ { ( - a , 1 - i ) } ( - c - j , 1 - k ) = ( - c - j + 1 - i , 1 - k ) \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} v '' ( t ) + v ( t ) | v ( t ) | ^ { p - 2 } = \\lambda v ( t ) \\forall t \\in ( j , j + 1 ) , j \\in \\{ 0 , 1 , 2 , \\ldots \\} , \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} \\mathbb E _ { \\delta } \\int _ 0 ^ t \\int _ D v ( s , y ; \\omega ) \\dot { W } ( d s d y ) = 0 , 0 \\leq t \\leq T , \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} ( { h } ^ { 2 } - 2 { h } - 4 ( n ^ 2 - n ) ) r ( { h } - 4 ) = ( h ^ 2 - ( 4 n + 2 ) h + ( 4 n ^ 2 + 4 n ) ) r ( h ) \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} \\sigma _ { l s b w } ( T ) & : = \\{ \\lambda \\in \\mathbb { C } : \\lambda I - T \\thinspace \\mbox { i s n o t l o w e r s e m i B - W e y l } \\} , \\\\ \\sigma _ { u s b w } ( T ) & : = \\{ \\lambda \\in \\mathbb { C } : \\lambda I - T \\thinspace \\mbox { i s n o t u p p e r s e m i B - W e y l } \\} , \\\\ \\sigma _ { b w } ( T ) & : = \\{ \\lambda \\in \\mathbb { C } : \\lambda I - T \\thinspace \\mbox { i s n o t B - W e y l } \\} , \\thinspace \\mbox { r e s p e c t i v e l y . } \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} F _ m ( a , b ) & = \\frac { b } { m a + b } \\binom { m a + b } { m } = b \\frac { ( m a + b - 1 ) ! } { ( m a + b - m ) ! m ! } . \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} & ( \\xi _ 1 ' , \\eta _ 1 ' ) + ( \\xi _ 2 ' , \\eta _ 2 ' ) = ( \\xi ' , \\eta ' ) , \\\\ \\phi _ { \\tilde { c _ 1 } } ^ A ( \\xi _ 1 ' , \\eta _ 1 ' ) & \\in S _ 1 ^ A , \\ \\ \\phi _ { \\tilde { c _ 2 } } ^ A ( \\xi _ 2 ' , \\eta _ 2 ' ) \\in S _ 2 ^ A , \\ \\ ( \\psi ^ A ( \\xi ' , \\eta ' ) , \\xi ' , \\eta ' ) \\in S _ 3 ^ A . \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} \\begin{aligned} & Q \\in W ^ { 1 , 1 } ( \\mathbb { R } ^ N ) , \\\\ & Q = Q ( | x - y | ) \\ ; \\ x , y \\in \\mathbb { R } ^ N \\ \\frac { \\partial Q } { \\partial r } \\leq 0 , \\ \\ r > 0 , \\\\ & \\int _ { \\mathbb { R } ^ N } \\ ! \\ ! Q ( x ) \\ , d x = 1 . \\end{aligned} \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { n } j _ { s } ^ { ( 3 ) } = \\frac { 1 } { 4 9 } \\left ( 1 6 K _ { n + 3 } ^ { ( 3 ) } - 5 K _ { n + 2 } ^ { ( 3 ) } + 2 K _ { n + 1 } ^ { ( 3 ) } \\right ) - 1 . \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} \\begin{cases} | \\vec x | _ { \\pmb \\sigma } \\leq t \\\\ | \\Theta \\vec x - \\vec y | _ { \\pmb \\rho } \\leq t ^ { - \\gamma } \\end{cases} . \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} \\Gamma _ { 1 ; f } & \\leq \\frac { \\beta ^ 2 } { 2 \\beta - 1 } \\Big [ 1 + \\frac { ( \\beta - 1 ) ^ 2 } { \\Big ( 2 \\beta - \\frac { 7 } { 3 } \\Big ) - \\frac { 2 } { 3 } \\sqrt { 2 ( 3 \\beta - 4 ) } } \\Big ] \\lambda _ { 1 ; f } ^ 2 \\\\ & = \\frac { \\beta ^ 2 } { 2 \\beta - 1 } \\Big [ 1 + \\frac { 1 } { 3 } \\Big ( \\frac { ( 3 \\beta - 4 ) + 1 } { \\sqrt { 2 ( 3 \\beta - 4 ) } - 1 } \\Big ) ^ 2 \\Big ] \\lambda _ { 1 ; f } ^ 2 . \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} \\langle f _ { 1 } , T _ { g } f _ { 2 } \\rangle & = \\langle f _ { 1 } , P ( f _ { 2 } g ) \\rangle = \\langle P f _ { 1 } , f _ { 2 } g \\rangle = \\langle f _ { 1 } , f _ { 2 } g \\rangle = \\langle f _ { 1 } g ^ { * } , f _ { 2 } \\rangle \\\\ & = \\langle f _ { 1 } g ^ { * } , P f _ { 2 } \\rangle = \\langle P ( f _ { 1 } g ^ { * } ) , f _ { 2 } \\rangle = \\langle T _ { g ^ { * } } f _ { 1 } , f _ { 2 } \\rangle . \\blacksquare \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} L _ { 2 r , \\ell } ( \\tau ) & = \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ n ) ^ c ( 1 - q ^ { \\ell \\cdot n } ) ^ d \\sum _ { m \\geq \\mu _ { 2 r } } p _ { [ 1 ^ c \\ell ^ d ] } ( \\ell ^ { 2 r } m + n _ { 2 r } ) q ^ m , \\\\ L _ { 2 r - 1 , \\ell } ( \\tau ) & = \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ { \\ell \\cdot n } ) ^ c ( 1 - q ^ n ) ^ d \\sum _ { m \\geq \\mu _ { 2 r - 1 } } p _ { [ 1 ^ c \\ell ^ d ] } ( \\ell ^ { 2 r - 1 } m + n _ { 2 r - 1 } ) q ^ m . \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} F = \\omega ( E _ k ) \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} ( a f ) _ { i , j } = \\sum _ { k \\in \\Z _ n } a _ { i , k } f _ { k , j } , \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} \\{ \\xi _ n : = \\left ( g _ 1 ( X ^ n ) , \\cdots , g _ m ( X ^ n ) \\right ) : n \\geq 1 \\} \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} D \\left ( A \\right ) = \\left \\{ u \\in W ^ { 2 , 2 } \\left ( 0 , 1 \\right ) , B _ { j } u = A u \\left ( j \\right ) = 0 j = 0 , 1 \\right \\} , \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} f ( y \\cdot ( \\xi _ 1 + \\xi _ 2 ) ) - f ( y \\cdot \\xi _ 1 ) = \\int _ 0 ^ 1 f ' ( y \\cdot ( \\xi _ 1 + \\theta \\xi _ 2 ) ) ( y \\cdot \\xi _ 2 ) d \\theta \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} K ( \\Psi ; M ; ( \\gamma , 0 ) ) = K ( \\hat { \\Psi } ; \\hat { M } ; \\gamma ) \\ , , \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} \\zeta _ B ( s ; a , b , x ) = \\sum _ { m , n = 0 } ^ \\infty ( a m + b n + x ) ^ { - s } , \\Re s > 2 , a > 0 , b > 0 , x \\in \\Bbb R , \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} \\int _ 0 ^ s ( \\ddot { v } ( t ) + \\eta \\dot { v } ( t ) , \\phi ^ s _ v ( t ) ) d t = \\int _ 0 ^ s ( \\operatorname { d i v } ( p ) - \\operatorname { d i v } ( \\bar { p } ) , \\phi ^ s _ v ( t ) ) d t . \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} \\mathbf { F } ^ { k } = \\mathbf { B } ^ { k } \\mathbf { G } ^ { k } , \\mathbf { B } ^ { k } = \\operatorname { d i a g } ( b ^ k _ { 1 , 1 } , b ^ k _ { 2 , 1 } , \\cdots , b ^ k _ { M , 1 } , b ^ k _ { 1 , 2 } , \\cdots , b ^ k _ { M , N } ) , \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} T _ { \\bar { \\nabla } } ( f _ i \\otimes q ^ i , f ' _ j \\otimes q '^ j ) : = f _ i f ' _ j X ^ * T _ \\nabla ( q _ i , q _ j ) , \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} D _ p ( x , y ) = \\frac 1 p \\| x \\| ^ p - \\frac 1 p \\| y \\| ^ p - \\langle j _ p ( y ) , x - y \\rangle \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} ( C , ( Q _ { t = 0 } ) _ { ( 0 ) } ) \\cong ( C , ( Q _ { t = 1 } ) _ { ( 0 ) } ) \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\lambda _ { \\ker } ( t , p ) - \\lambda _ { \\ker } ( 0 , p ) = \\displaystyle \\int _ { 0 } ^ { t } r d t . \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\beta = \\frac { \\partial f _ R } { \\partial \\mu } ( 0 , 0 ; 0 ) . \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} \\| f \\| \\leq \\| f \\| _ \\infty + L ( f ) \\leq ( c _ 2 + b ) m ( f ) = ( c _ 2 + b ) \\int f d m \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} & a = \\widehat { a } _ 1 c \\widehat { a } _ 2 , \\\\ & b = \\widehat { b } _ 1 c \\widehat { b } _ 2 , \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} \\nu ( [ a , b ] ) = [ a , \\nu ( b ) ] , \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} k ' ( x ( r ) ) = \\frac { ( d / d r ) k ( x ( r ) ) } { ( d / d r ) x ( r ) } = \\frac { ( 1 / 2 \\pi \\beta ) f ' ( r ) } { [ x ' ( r ) / x ( r ) ] x ( r ) } , \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} a _ { 1 } & = 1 , & b _ { 0 } & = 0 , \\\\ a _ { 3 k + 0 } & = - 2 ( 4 k - 1 ) ^ 2 k x ^ 3 , & b _ { 3 k + 0 } & = - 1 6 k ^ 2 x ^ 2 - 4 k x + 1 , \\\\ a _ { 3 k + 1 } & = - 2 ( 4 k + 1 ) ^ 2 k x ^ 3 , & b _ { 3 k + 1 } & = - 2 ( 3 k + 1 ) x + 1 , \\\\ a _ { 3 k + 2 } & = - ( 2 k + 1 ) ^ 2 x ^ 2 ; & b _ { 3 k + 2 } & = - 2 ( 3 k + 2 ) x + 1 . \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} ( 1 + \\varepsilon ) ^ { - 1 } \\le \\frac { M _ D ( x , z _ j ) } { M _ { D _ n } ( x , z _ j ) } \\le ( 1 + \\varepsilon ) , j = 1 , \\dots , m . \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} & | - \\frac { 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { B _ { s - t } ( 0 ) } \\frac { d A ( y ) } { \\sqrt { ( s - t ) ^ { 2 } - | y | ^ { 2 } } } _ { v _ { 4 , 1 } } | \\leq \\frac { C } { t ^ { 2 } \\log ^ { 3 b + 2 N - 1 } ( t ) } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} P ^ + _ { 2 k + 1 } \\{ M ( t ) \\le \\beta \\} = P ^ + _ { 2 k + 2 } \\{ M ( t ) \\le \\beta \\} = \\frac { \\beta } { c t } \\sum _ { j = 0 } ^ k \\binom { 2 j } { j } \\frac { ( c ^ 2 t ^ 2 - \\beta ^ 2 ) ^ j } { ( 2 c t ) ^ { 2 j } } \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} A ^ 1 _ { n + 1 } : = - i \\left ( x _ 1 + x _ 2 + \\ldots + x _ n + x _ { 1 + 2 + \\ldots + n } \\right ) b _ n . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} W _ \\phi ( z + 1 ) = \\phi ( z ) W _ \\phi ( z ) , W _ \\phi ( 1 ) = 1 . \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} u _ k ( r , f ) : = \\begin{cases} \\frac { - 1 } { 2 b _ k } { \\big ( } r ^ { \\gamma ^ + _ k } \\int _ r ^ 1 s ^ { - \\gamma _ k ^ + + 1 } f _ k ( s ) \\ d s - r ^ { \\gamma ^ - _ k } \\int _ r ^ 1 s ^ { - \\gamma _ k ^ - + 1 } f _ k ( s ) \\ d s { \\big ) } , & b _ k \\neq 0 ; \\\\ r ^ { - \\frac { n - 2 } { 2 } } { \\big ( } \\log r \\int _ 0 ^ r s ^ { \\frac { n } { 2 } } f _ k ( s ) \\ d s - \\int _ 0 ^ r s ^ { \\frac { n } { 2 } } \\log s f _ k ( s ) \\ d s { \\big ) } , & b _ k = 0 . \\end{cases} \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} 1 - \\theta _ { n , p } ( x ) = \\frac { ( 1 - v ^ { - 1 } ) ^ 3 ( \\log \\log n ) ^ { 2 } } { 2 \\log n } - \\frac { ( 1 - v ^ { - 1 } ) ^ { 2 } ( 1 + x - \\log \\{ 2 \\Gamma ( \\frac { 1 } { v } ) \\} ) \\log \\log n } { \\log n } + o \\big ( \\frac { \\log \\log n } { \\log n } \\big ) \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} p _ { \\beta , \\mathsf { m } , \\mathsf { G i b b s } } : = Z _ { \\beta } ^ { - 1 } \\exp \\left [ - \\beta \\lambda _ { \\mathsf { m } } \\right ] \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} \\tilde { x } & = \\frac { x } { \\mu ^ 2 } , & \\tilde { y } & = \\frac { y - \\zeta ( \\mu ) } { | \\mu | } , \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} q _ { s , i } = q _ { s - 1 , i - 1 } ^ p + q _ { s - 1 , i } V _ s ^ { p - 1 } . \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} z ^ \\top \\tilde { E } z & = \\sum _ { \\ell = 1 } ^ k \\gamma _ { \\ell } z ^ \\top ( S ( \\mu _ { \\ell } ) \\otimes { E } ' ( \\mu _ { \\ell } ) ) z = \\sum _ { \\ell = 1 } ^ k \\gamma _ { \\ell } \\sum _ { i , j = 1 } ^ m \\Phi _ i ( \\mu _ { \\ell } ) \\Phi _ j ( \\mu _ { \\ell } ) z _ i ^ \\top { E } ' ( \\mu _ { \\ell } ) z _ j \\\\ & = \\sum _ { \\ell = 1 } ^ k \\gamma _ { \\ell } v _ { \\ell } ^ \\top { E } ' ( \\mu _ { \\ell } ) v _ { \\ell } . \\\\ \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} P _ f ( A , B + L _ n ) & = P _ { f _ 0 } ( A , B + L _ n ) + \\alpha ( B + L _ n ) + \\beta A , \\\\ B ^ { 1 / 2 } f ( W ) B ^ { 1 / 2 } & = B ^ { 1 / 2 } f _ 0 ( W ) B ^ { 1 / 2 } + \\alpha B + \\beta A , \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} | \\gamma | _ T = \\sum _ i \\log _ 2 ( | \\langle \\gamma , e _ i \\rangle | + 1 ) . \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} F _ { 6 } ( t , r ) = \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) \\left ( v _ { 4 } ( t , r ) + v _ { 5 } ( t , r ) \\right ) \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} A ( b , \\lambda ) & = - 3 b ^ 2 + b ^ 3 + 4 \\lambda + 6 \\lambda ^ 2 + 2 \\lambda ^ 3 + b \\left ( - 3 \\lambda ^ 2 - 3 \\lambda + 2 \\right ) , \\\\ B ( b , \\lambda ) & = 3 ( b ^ 2 - b - \\lambda ^ 2 - \\lambda ) . \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} X ( t ) = N ( S _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( t ) ) , \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} \\{ ( x , t ) : x \\in \\Omega _ 0 , u _ j ^ - ( x ) < t < u _ j ^ + ( x ) \\} \\setminus B ^ M _ { 2 \\tau } ( S i n g ( \\Sigma ) ) = U _ j \\setminus B ^ M _ { 2 \\tau } ( S i n g ( \\Sigma ) ) \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} = \\frac { 1 } { \\bigl [ ( c _ 1 + c _ 2 ) t \\bigr ] ^ { 2 k } } \\sum _ { j = 1 } ^ { k - 1 } \\binom { 2 k } { j } \\Bigl [ ( c _ 1 t - \\beta ) ^ { j } ( c _ 2 t + \\beta ) ^ { 2 k - j } - \\Bigl ( \\frac { c _ 2 } { c _ 1 } \\Bigr ) ^ { 2 k - 2 j } ( c _ 1 t - \\beta ) ^ { 2 k - j } ( c _ 2 t + \\beta ) ^ { j } \\Bigr ] \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} = e ^ { - \\lambda t } \\sum _ { k = 0 } ^ \\infty I _ { 2 k } \\Bigl ( \\frac { 2 \\lambda } { c _ 1 + c _ 2 } \\sqrt { ( c _ 1 t - \\beta ) ( c _ 2 t + \\beta ) } \\Bigr ) \\Biggl ( \\sqrt { \\frac { c _ 2 t + \\beta } { c _ 1 t - \\beta } } \\Biggr ) ^ { 2 k } \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} \\psi _ { \\Lambda } ( b _ { \\Lambda } ) = \\langle \\hat { e } _ { I } , \\left ( \\diamond _ { x \\in \\Lambda } \\hat { E } _ { x } ( b _ { x } ) \\right ) \\hat { e } _ { I } \\rangle \\quad ; b _ { \\Lambda } : = \\bigotimes _ { x \\in \\Lambda } b _ { x } \\ , \\ b _ { x } \\in \\mathcal { B } _ { x } \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} \\alpha = \\lambda _ L \\ , , \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\underline U ^ n ( 0 , x ) = ( 1 - \\epsilon ) \\big [ \\Phi ^ n ( x - K ) + \\Phi ^ n ( - x - K ) - \\mathbf { u ^ * _ n } \\big ] \\prec ( 1 - \\epsilon ) \\mathbf { u ^ * _ n } \\to ( 1 - \\epsilon ) \\mathbf { u ^ * } \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} L \\bar { v } = \\frac { G } { \\rho ^ 3 T } \\cdot f ( 1 - \\sum _ { p \\in S i n g ( \\Sigma ) } \\eta _ p ) - \\sum _ { p \\in S i n g ( \\Sigma ) } ( 2 \\nabla \\eta _ p \\cdot \\nabla v _ p + v _ p \\Delta \\eta _ p ) \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} \\mathsf { \\hat { r } } _ { \\mathsf { k } } : = \\left \\{ \\begin{array} { c } \\mathsf { \\hat { p } } _ { 1 } \\mathsf { k = 1 , } \\\\ \\\\ \\mathsf { \\hat { p } } _ { \\mathsf { k } } - \\mathsf { \\hat { p } } _ { 1 } , \\mathsf { k } \\in \\left \\{ 2 , 3 , . . . \\right \\} , \\end{array} \\right . \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} \\Big ( \\int _ B | \\nabla ^ m \\bar u _ \\varepsilon | ^ 2 d x \\Big ) ^ { 1 / 2 } = 1 + O ( \\varepsilon ^ { n - 2 m } ) _ { \\varepsilon \\searrow 0 } , \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\inf _ { ( U , V ) \\in \\mathcal { N } } \\| ( U ^ { ( p ) } , V ^ { ( p ) } ) - ( U , V ) \\| _ F = 0 . \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} { \\rm R i c } ^ m _ { V } : = { \\rm R i c } - \\frac { 1 } { 2 } L _ V g - \\frac { 1 } { m - n } V ^ * \\oplus V ^ * , \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} K _ { - n } ^ { ( 3 ) } & = 2 ^ { - n } + M _ { - n } ^ { ( 2 ) } = 2 ^ { - n } + M _ { n } ^ { ( 2 ) } \\\\ & = 2 ^ { - n } + 2 ^ { n } + M _ { n } ^ { ( 2 ) } - 2 ^ { n } \\\\ & = K _ { n } ^ { ( 3 ) } + 2 ^ { - n } - 2 ^ { n } . \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } F ( t _ * , x , v ) \\varphi ( x , v ) d x d v \\leq & \\ ; \\ ; \\| \\psi \\| _ { L ^ \\infty ( Q _ { t _ * } ) } \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } \\big | f _ 0 ( x , v ) \\big | d x d v \\\\ \\leq & \\ ; \\ ; \\| f _ 0 \\| _ { L ^ 1 ( \\Omega \\times \\mathbb { R } ^ 3 ) } . \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} \\tilde { V } \\left ( x , t \\right ) = \\alpha \\beta \\sigma ^ { 2 } \\left ( t \\right ) V \\left ( \\sqrt { \\alpha \\beta } x \\sigma \\left ( t \\right ) , \\beta t \\sigma \\left ( t \\right ) \\right ) , \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} 0 = \\langle \\widehat { A } g , f \\rangle _ { \\alpha } = \\langle g , f \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } , \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{align*} a \\cdot H ( b , c ) - H ( a b , c ) + H ( a , b c ) - H ( a , b ) \\cdot c = 0 , ~ a , b , c \\in A . \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} M _ { i j } = h ( | i - j | ) i , j \\in \\mathbb { N } . \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} \\Bigl | \\Bigl ( \\frac { d G } { d \\xi _ 2 ' } \\Bigr ) ( \\xi _ 2 ' ) \\Bigr | = \\Bigl | \\frac { 2 \\xi _ 1 ' } { { \\xi _ 2 ' } ^ 3 } \\Bigl ( { \\xi _ 2 ' } ^ 4 - \\frac { 9 { \\xi _ 1 ' } { \\eta _ 1 ' } ^ 3 } { 1 6 } \\Bigr ) \\Bigr | \\geq 2 ^ { - 1 0 0 } N _ 1 ^ 2 , \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} \\Gamma _ { 2 , } ^ { - 1 } = \\langle \\rho ( p ) \\rho ( - p ) \\rangle & = \\frac { i } { p ^ 2 - m ^ 2 } = \\frac { i } { x _ p } . \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} y _ { x x } = \\sin ( y ) \\cos ( y ) + 2 \\cot ( y ) y _ x ^ 2 \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} P ^ { ( 1 ) } = - \\frac { b _ 2 } { a _ 2 } \\ , \\Phi ^ { ( 1 ) } _ { \\frac { \\pi } { \\omega } } + \\Psi ^ { ( 1 ) } _ { \\frac { \\pi } { \\omega } } . \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} g _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( x , t ) = \\frac { 1 } { 2 \\pi i } \\int _ { x _ { 0 } - i \\infty } ^ { x _ { 0 } + i \\infty } e ^ { s x } \\overline { G } ( s , t ) d s . \\\\ \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} u _ t = ( D _ x ^ \\alpha u ) _ x + f _ 1 ( x , t ) \\quad \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} { A ' } ^ 2 _ 2 & = - i x _ p \\\\ { A ' } ^ j _ n & = 0 \\ \\forall n > 2 , j > 0 . \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\langle s _ 0 , \\ldots , f s _ i , \\ldots , s _ m \\rangle = { \\rm { N o r m } } _ { Y / B } ( f ) \\cdot \\langle s _ 0 , \\ldots , s _ m \\rangle , \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\int _ { \\mathbb { R } ^ N } \\dfrac { f _ 0 ( u _ n ( x ) ) } { u _ n ( x ) } \\left [ ( v _ n ^ + ( x ) ) ^ 2 - ( v _ n ^ - ( x ) ) ^ 2 \\right ] d x = 1 . \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} \\tilde { u } _ { \\varepsilon } \\left ( x , t \\right ) = e ^ { \\varepsilon t \\left ( \\Delta + A \\right ) } \\tilde { u } \\left ( x , t \\right ) t \\in \\left [ 0 , 1 \\right ] . \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} | v _ { 5 , 1 } ( t , r ) | & \\leq C \\frac { r } { t ^ { 4 } \\log ^ { b - 2 } ( t ) } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} A r e a _ { g ' } ( g r a p h _ { \\Sigma , g } ( \\phi ) ) = \\int _ { \\Sigma } F ^ { g ' } ( x , \\phi ( x ) , \\nabla _ { \\Sigma } \\phi ( x ) ) \\ d v o l _ { g | _ { \\Sigma } } ( x ) \\ \\ \\ \\ \\forall \\phi \\in C ^ 1 ( \\Sigma ) \\| \\phi \\| ^ * _ { 1 ; \\Sigma } \\leq \\delta _ 1 \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} & \\sqrt { R } \\varphi ' _ i ( U ) [ \\mathbf T _ { N } ] _ { j k } = \\partial _ j ( R \\varphi ' _ i ( U ) U _ k ) - 2 \\sqrt { R } U _ k \\partial _ j \\sqrt { R } + g _ \\varphi \\ , , \\forall i , j , k \\in \\{ 1 , \\cdots , d \\} , \\\\ [ 6 p t ] & \\mathbf S _ K = \\sqrt { R } \\nabla ^ 2 \\sqrt { R } - \\nabla \\sqrt { R } \\otimes \\nabla \\sqrt { R } \\ , . \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} F ^ 1 ( x , t ) & : = - K ( x , 0 , t ) Q ( x , 0 , t ) \\ \\ \\ \\ \\ \\mbox { a n d } & F ^ 2 ( y , t ) : = - K ( 0 , y , t ) Q ( 0 , y , t ) . \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} \\mu _ { m } ^ { k } | _ { I } = \\mu _ { m } ^ { k + 1 } | _ { I } . \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} p _ { t } ^ { n } ( x , x ) = \\sum _ { k = 1 } ^ { \\infty } e ^ { - \\lambda _ k t } \\varphi _ k ( x ) ^ 2 > 0 \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} f ( X ) = p X ^ 3 + q . \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} \\rho [ f ] = \\overline { \\lim } _ { n \\to \\infty } \\left \\{ \\ \\frac { n \\ \\ln n } { | \\ln \\sigma _ n \\ | } \\ \\right \\} , \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} \\nabla f = ( D _ 1 f , D _ 2 f , \\cdots , D _ d f ) . \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} E ( X ) : = \\det \\left ( ( P _ { i , j } ( X ) ) _ { i , j = 1 } ^ { 2 } \\right ) \\ne 0 . \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\psi ( a _ { \\Lambda } ) = \\sum _ { i , j \\in I } \\prod _ { x \\in \\Lambda } \\left ( \\hbox { T r } ( h _ { x , i } h ^ { * } _ { x , i } ) \\hbox { T r } ( h _ { x , j } h ^ { * } _ { x , j } ) \\right ) \\beta _ { \\Lambda ^ c , i , j } \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} & ( \\xi _ 1 ' , \\eta _ 1 ' ) + ( \\xi _ 2 ' , \\eta _ 2 ' ) = ( \\xi ' , \\eta ' ) , \\\\ \\phi _ { \\tilde { c _ 1 } } ( \\xi _ 1 ' , \\eta _ 1 ' ) & \\in S _ 1 , \\ \\ \\phi _ { \\tilde { c _ 2 } } ( \\xi _ 2 ' , \\eta _ 2 ' ) \\in S _ 2 , \\ \\ ( \\psi ( \\xi ' , \\eta ' ) , \\xi ' , \\eta ' ) \\in S _ 3 , \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} & | | \\frac { | v _ { 1 } + v _ { 2 } + v _ { 3 } + v _ { 4 } | } { r ^ { 2 } } \\left ( \\frac { | \\partial _ { r } v _ { 5 } | r \\lambda ( t ) } { r ^ { 2 } + \\lambda ( t ) ^ { 2 } } \\right ) \\vert _ { r = R \\lambda ( t ) } | | _ { L ^ { 2 } ( R d R ) } \\leq \\frac { C \\log ^ { 4 } ( t ) } { t ^ { 2 1 / 4 } } \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} \\mathbb { L } _ { C S } ^ \\bullet = \\mathcal { P } _ { f ^ \\bullet _ { B F - C S } } ( \\mathbb { L } _ { B F } ^ \\bullet ) . \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} c = \\int _ { \\mathbb { R } } \\frac { 1 + t ^ { 2 } } { t ^ { 2 } } \\ , d \\sigma ( t ) \\leq 1 , \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} a f _ 1 = - 2 f _ 1 ' - f _ 1 \\varphi '' ( \\varphi ' ) ^ { - 1 } , \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} I : = \\sum _ { k = 1 } ^ n \\sum _ { z \\leq | y | / 2 } , J _ { \\leq n / 2 } : = \\sum _ { k = 1 } ^ { n / 2 } \\sum _ { z > | y | / 2 } \\mbox { a n d } J _ { > n / 2 } : = \\sum _ { k = 1 + n / 2 } ^ n \\sum _ { z > | y | / 2 } \\mbox { ( s a y ) } . \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} v _ \\lambda : = ( 1 - \\lambda ) \\begin{pmatrix} 1 \\\\ 0 \\\\ \\frac { 1 } { 2 } \\end{pmatrix} + \\lambda \\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\end{pmatrix} \\qquad v : = \\begin{pmatrix} 1 \\\\ 0 \\\\ \\frac { 1 } { 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} D _ 1 & : = ( ( x _ i , y _ i ) , \\ldots , ( x _ l , y _ l ) ) , \\\\ D _ 2 & : = ( ( x _ { l + 1 } , y _ { l + 1 } ) , \\ldots , ( x _ n , y _ n ) ) , \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} x ( 1 - x ) ^ { p + 1 } G ' ( x ) - b ( 1 - x ) ^ { p + 1 } G ( x ) - ( 1 - x ) ^ { p } + x ^ { b } ( 1 - x ) = 0 . \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} \\lambda ^ { 4 } + a _ { 1 } \\lambda ^ { 3 } + b _ { 1 } \\lambda ^ { 2 } + c _ { 1 } \\lambda + d _ { 1 } + ( a _ { 2 } \\lambda ^ { 3 } + b _ { 2 } \\lambda ^ { 2 } + c _ { 2 } \\lambda ) e ^ { - \\lambda \\tau } = 0 \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} A _ { 2 r } ( 0 , 1 ) = r , \\ ; \\ ; A _ { 2 r } ( 4 , - 3 ) = r , \\ ; \\ ; A _ { 2 r } ( 8 , - 7 ) = r . \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} \\binom { k + 1 + 2 d } { d } & = d + 1 + ( d + 1 ) \\binom { k + d } { d - 1 } + \\binom { k + d } { d } . \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u + F \\left ( u , \\bar { u } \\right ) u = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} g ( p ) = \\{ A \\in \\mathfrak { m } _ 2 : \\varphi ^ { - 1 } ( A ) \\in p \\} . \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} & - 4 \\int _ { t } ^ { \\infty } \\frac { \\lambda _ { 0 , 1 } '' ( s ) } { 1 + s - t } d s + 4 \\alpha \\log ( \\lambda _ { 0 , 0 } ( t ) ) \\lambda _ { 0 , 0 } '' ( t ) - 4 \\int _ { t } ^ { \\infty } \\frac { \\lambda _ { 0 , 0 } '' ( s ) d s } { ( \\lambda _ { 0 , 0 } ( t ) ^ { 1 - \\alpha } + s - t ) ( 1 + s - t ) ^ { 3 } } \\\\ & = E _ { v _ { 3 } , i p , 0 1 } + E _ { v _ { 3 } , i p , f } + E _ { \\lambda _ { 0 , 1 } } \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} \\begin{aligned} ( - \\Delta ) ^ s _ p u ( x ) & = \\frac { ( 2 x ) ^ { s p } } { ( 1 - x ^ 2 ) ^ s } \\lim _ { \\epsilon \\rightarrow 0 } \\int \\limits _ { \\{ w \\in \\mathbb { R } ^ n : | w | \\geq \\frac { 2 x \\epsilon } { 1 - x ^ 2 } \\} } \\frac { \\left | 1 - \\left ( 1 + w _ 1 - \\frac { 1 - x ^ 2 } { 4 x ^ 2 } | w | ^ 2 \\right ) ^ s _ + \\right | ^ { p - 2 } \\left [ 1 - \\left ( 1 + w _ 1 - \\frac { 1 - x ^ 2 } { 4 x ^ 2 } | w | ^ 2 \\right ) ^ s _ + \\right ] } { | w | ^ { n + s p } } \\mathrm { d } w . \\end{aligned} \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} \\begin{bmatrix} E _ 0 & G _ 0 \\\\ H _ 0 & F _ 0 \\end{bmatrix} = \\begin{bmatrix} D _ \\alpha & - \\beta C \\\\ - \\alpha B & A _ \\beta \\end{bmatrix} ^ { - 1 } \\begin{bmatrix} D _ { - \\beta } & \\alpha C \\\\ \\beta B & A _ { - \\alpha } \\end{bmatrix} , \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\pi \\circ F _ { J , K } = F _ { K , J } \\circ \\pi , \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} \\lambda _ i \\lambda _ j = \\sum _ { t } \\binom { t - j - 1 } { 2 t - i } \\lambda _ { i + j - t } \\lambda _ t , i > 2 j . \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} \\ell _ { n , 1 } ^ * ( \\bar \\mu , \\bar \\sigma _ + ) - \\ell _ { n , 1 } ^ * ( 0 , 1 ) = O _ p ( 1 ) = o _ p ( n ) . \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} p _ { \\nu _ { \\beta } } ( t ) = \\frac { \\beta f ( x ) } { \\pi ( \\beta - 1 ) \\left | F _ { \\nu } \\left ( F _ { \\mu } ( t ) \\right ) \\right | ^ { 2 } } , \\quad t \\in \\mathbb { R } , \\ ; F _ { \\nu } \\left ( F _ { \\mu } ( t ) \\right ) \\neq 0 . \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} d ( b ^ { [ k p ^ i ] } ) = \\lambda ^ 0 _ { - 1 } a b ^ { [ k p ^ i - 1 ] } + \\lambda ^ 1 _ 0 b ^ { [ k p ^ i - p + 1 ] } \\mod F ^ { 2 ( k p ^ i - p + 1 ) - 1 } ; \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} \\| D \\| _ { \\mathcal { H } _ q ( X ) } \\leq C \\left ( \\sum _ { n = 1 } ^ \\infty R ^ { \\Omega ( n ) } \\| a _ n \\| ^ p \\right ) ^ { \\frac { 1 } { p } } ; \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} a ^ { c } _ { \\Lambda } ( t ) : = \\frac { 1 } { c } \\left ( a ( t ) - 1 _ { c \\in ( 0 , 1 ) } \\int _ { c < | y | \\leq 1 } y \\Lambda _ t ( d y ) + 1 _ { c \\in ( 1 , \\infty ) } \\int _ { 1 < | y | \\leq c } y \\Lambda _ t ( d y ) \\right ) , ~ ~ A ^ c ( t ) : = \\frac { 1 } { c ^ 2 } A ( t ) . \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} I _ 1 ( t ) : = [ \\underline h ( t ) - K _ 0 , \\underline h ( t ) ] , \\ I _ 2 ( t ) : = [ - \\underline h ( t ) , - \\underline h ( t ) + K _ 0 ] , \\ I _ 3 ( t ) : = [ - \\underline h ( t ) + K _ 0 , \\underline h ( t ) - K _ 0 ] . \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} \\eta & = \\sum \\limits _ { i \\in I ^ g ( \\bar x ) } \\lambda ^ 1 _ i \\nabla g _ i ( \\bar x ) + \\sum \\limits _ { j \\in \\mathcal P } \\rho ^ 1 _ j \\nabla h _ j ( \\bar x ) \\\\ & + \\sum \\limits _ { l \\in I ^ { 0 + } ( \\bar x ) } \\mu ^ 1 _ l \\nabla G _ l ( \\bar x ) + \\sum \\limits _ { l \\in I ^ { + 0 } ( \\bar x ) } \\nu ^ 1 _ l \\nabla H _ l ( \\bar x ) \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} \\mathfrak { L } ^ { ( \\alpha ) } \\left [ f , \\gamma \\right ] = F ^ { ( \\alpha ) } \\left [ f \\right ] + \\gamma \\left ( 1 - \\| f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 \\right ) , \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} Q _ t = \\phi _ t \\left ( X _ { ( t ) } , S _ { ( t ) } \\right ) . \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} \\sigma _ \\alpha ( S ^ 1 \\times Y , S ^ 1 \\times K ) = \\sigma _ \\alpha ( Y , K ) \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} Q _ t ( z ) = & \\sum _ { \\alpha \\in \\Delta _ { > 0 } } ( \\pi _ 1 ( x _ \\alpha ) ( z ) + t ^ { 2 \\alpha ( x _ 0 ) } \\pi _ 2 ( x _ \\alpha ) ( z ) ) \\psi _ { \\alpha } ^ * ( z ) \\\\ & + \\sum _ { \\alpha \\in \\Delta _ { > 0 } } \\Phi _ { \\alpha } ( z ) \\psi _ { \\alpha } ^ * ( z ) - \\frac { 1 } { 2 } \\sum _ { \\alpha , \\beta , \\gamma \\in \\Delta _ { > 0 } } c _ { \\alpha , \\beta } ^ \\gamma \\psi _ { \\alpha } ^ * ( z ) \\psi _ { \\beta } ^ * ( z ) \\psi _ { \\gamma } ( z ) , \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} \\widetilde { \\Theta } ^ E ( \\tau ) : = \\sum _ { 1 \\leq j , k \\leq n } \\sum _ { 1 \\leq \\lambda , \\mu \\leq r } c _ { j k \\lambda \\mu } \\tau _ { j \\lambda } \\overline { \\tau } _ { k \\mu } \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} \\sum _ \\mu c _ { \\gamma \\mu } g _ \\mu ^ { ( k + 1 ) } = ( G _ { 1 ^ \\ell } ^ { ( k + 1 ) } ) ^ \\perp g _ \\gamma ^ { ( k + 1 ) } = g _ \\lambda ^ { ( k ) } , \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} \\Big ( \\sum _ { n = 1 } ^ \\infty \\frac { \\| a _ n \\| ^ q } { n ^ \\delta } \\Big ) ^ { 1 / q } \\leq C \\| D \\| _ { \\mathcal { H } _ { p } ( X ) } . \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} f _ L ( 0 , 0 ; 0 ) = g _ L ( 0 , 0 ; 0 ) = 0 , \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ i = m . \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} D : = 3 m H - m E _ 1 - \\ldots - m E _ 9 . \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} \\mathcal { R } _ { 1 , 1 } ( t ) = \\int _ 0 ^ t \\int _ 0 ^ \\xi e ^ { ( t - s ) \\mathcal { L } } \\mathcal { C } [ f , \\mathcal { L } ] \\left ( e ^ { s \\mathcal { L } } v , e ^ { s \\mathcal { L } } e ^ { \\xi \\mathcal { A } } \\overline v \\right ) d s d \\xi \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} \\{ l \\in I ^ H ( \\bar x , \\bar y , \\bar z ) \\cup I ^ { G H } ( \\bar x , \\bar y , \\bar z ) \\ , | \\ , \\bar z _ l = 0 \\} = \\{ l \\in \\mathcal Q \\ , | \\ , H _ l ( \\bar x ) = \\bar z _ l = 0 \\} = I ^ { + 0 } ( \\bar x ) \\cup I ^ { 0 0 } ( \\bar x ) \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} { \\rm v a r } ( \\mu _ { 2 } ) = \\int _ { \\mathbb { R } } t ^ { 2 } \\ , d \\mu _ { 2 } ( t ) - \\left [ \\int _ { \\mathbb { R } } t \\ , d \\mu _ { 2 } ( t ) \\right ] ^ { 2 } \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} & c \\ , K _ { k + 1 } ^ y - 2 R _ k = 0 , a _ k - ( k + 1 ) K _ { k + 1 } ^ y R _ k = 0 , \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a _ 1 & a _ 2 \\\\ b _ 1 & b _ 2 \\end{bmatrix} , \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } \\leq \\sqrt { n } \\left ( \\sum _ { i = 1 } ^ { n } a _ i a _ i ^ * \\right ) ^ \\frac { 1 } { 2 } , \\forall n \\in \\mathbb { N } , \\forall a _ 1 , \\dots , a _ n \\in \\mathcal { A } . \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} & ~ [ x ^ k ] [ y ^ n ] \\frac { 1 + y - x ^ 2 y ^ 2 } { ( 1 - x y ) ( 1 - x y ^ 2 ) - x y ^ 3 } \\\\ & = [ x ^ k ] [ y ^ n ] f ( x , y ) + [ x ^ k ] [ y ^ { n - 1 } ] f ( x , y ) - [ x ^ { k - 2 } ] [ y ^ { n - 2 } ] f ( x , y ) \\\\ & = \\sum _ { j = 0 } ^ k \\binom { n - 2 j } { k - j } \\binom { j } { n - k - j } + \\sum _ { j = 0 } ^ k \\binom { n - 2 j - 1 } { k - j } \\binom { j } { n - k - j - 1 } \\\\ & ~ - \\sum _ { j = 0 } ^ k \\binom { n - 2 j - 2 } { k - j - 2 } \\binom { j } { n - k - j } . \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} \\left [ a b ^ { 2 } ( a + 1 ) ( b + 1 ) ^ { 2 } v ^ { 4 } + g ( u , v ) \\right ] = \\lambda _ { 2 2 } { \\Phi ^ { 3 } } . \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} \\int _ { 2 a t } ^ { 2 b t } { P _ * } ( 2 x , k ) \\left ( \\int _ { 2 a t } ^ x \\frac { e ^ { i u ^ 2 / ( 4 t ) } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) e ^ { - i k u } d u \\right ) ' d x = \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} | F ( \\xi _ 1 , \\eta _ 1 , \\xi _ 2 , \\eta _ 2 ) | & = | \\xi _ 1 \\eta _ 2 + \\xi _ 2 \\eta _ 1 + 2 ( \\xi _ 1 \\eta _ 1 + \\xi _ 2 \\eta _ 2 ) | \\\\ & \\geq 2 | \\xi _ 2 \\eta _ 1 + \\xi _ 1 \\eta _ 1 + \\xi _ 2 \\eta _ 2 | - | \\xi _ 1 \\eta _ 2 - \\xi _ 2 \\eta _ 1 | \\\\ & \\geq 2 | \\xi _ 2 \\eta _ 2 | - 2 | \\xi _ 1 + \\xi _ 2 | | \\eta _ 2 | - 2 ^ 7 A ^ { - 1 } N _ 1 ^ 2 \\\\ & \\geq K ^ { - 1 } N _ 1 ^ 2 - 2 ^ 8 A ^ { - 1 } N _ 1 ^ 2 \\\\ & \\geq 2 ^ { - 1 } K ^ { - 1 } N _ 1 ^ 2 . \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( f ( \\eta ^ { * N } ) \\right ) = \\mathbf { E } \\left ( f ( \\eta ^ N ) \\ : \\vline \\ : \\eta ^ N _ 0 = 0 , \\ : \\eta ^ N _ 1 = 1 \\right ) , \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} q _ { \\ell , \\theta } : = - \\frac { 2 \\ , \\ell ^ 2 - 2 5 \\ , \\ell + 5 7 } { 1 8 \\ , ( l - 2 ) } \\ , \\left ( 1 - \\frac { \\theta } { \\ell } \\right ) + \\frac { \\theta } { \\ell } , \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} f _ 2 ( x ) r ' _ k ( x ) - k f _ 1 ( x ) r _ k ( x ) = c ( x ) r _ k ( x ) . \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} \\left ( \\frac { C _ { n , k _ { 1 } } - ( n - 1 ) \\mathbb { I } _ { k _ { 1 } = 2 } - \\mu _ { k _ 1 } } { \\sqrt { 2 k _ { 1 } } } , \\ldots , \\frac { C _ { n , k _ { l } } - \\mu _ { k _ { l } } } { \\sqrt { 2 k _ { l } } } \\right ) \\stackrel { d } { \\to } N _ { l } ( 0 , I _ { l } ) . \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} \\gamma ( t ) ( x ) : = \\tilde { u } \\left ( \\frac { x - y } { t L } \\right ) , \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} \\left \\| \\sum _ { k = 1 } ^ n \\alpha _ k \\cdot b _ k \\right \\| _ X \\coloneqq & \\sum _ { k = 1 } ^ n \\nu ( \\alpha _ k ) , \\\\ \\left \\| \\sum _ { k = 1 } ^ n \\hat { \\alpha } _ k \\cdot I ( b _ k ) \\right \\| _ { \\hat { X } } \\coloneqq & \\sum _ { k = 1 } ^ n \\hat { \\nu } ( \\hat { \\alpha } _ k ) . \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} \\frac { P _ 1 ( X ) } { P _ 2 ( X ) } + R ( X ) = 0 . \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} Q ( x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } ) & = \\tfrac { 1 } { 6 } \\left ( x _ { 1 } ^ { 3 } - 3 x _ { 1 } y _ { 1 } ^ { 2 } - 3 x _ { 1 } ( x _ { 2 } ^ { 2 } - y _ { 2 } ^ { 2 } ) + 6 y _ { 1 } x _ { 2 } y _ { 2 } \\right ) \\\\ & = \\tfrac { 1 } { 6 } x _ { 1 } ^ { 3 } - \\tfrac { 1 } { 2 } x _ { 1 } ( x _ { 2 } ^ { 2 } + y _ { 1 } ^ { 2 } - y _ { 2 } ^ { 2 } ) + y _ { 1 } x _ { 2 } y _ { 2 } , \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} \\frac { 1 } { t } \\left ( p _ t \\ast u _ \\lambda - u _ \\lambda \\right ) = \\frac { \\mathrm { e } ^ { \\lambda t } - 1 } { t } \\int _ t ^ \\infty \\mathrm { e } ^ { - \\lambda s } s ^ { - \\frac { d } { \\alpha } } p _ 1 \\left ( \\frac { x } { s ^ { 1 / \\alpha } } \\right ) d s - \\frac { 1 } { t } \\int _ 0 ^ t \\mathrm { e } ^ { - \\lambda s } s ^ { - \\frac { d } { \\alpha } } p _ 1 \\left ( \\frac { x } { s ^ { 1 / \\alpha } } \\right ) d s . \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} H _ 1 ( q , p ) = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ n p _ i p _ j e ^ { - | q _ i - q _ j | } . \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} P ( \\varepsilon z ) = \\sum _ { k = 0 } ^ { ( m - 1 ) / 2 } \\sum _ { A \\in \\mathcal { A } ' _ k } \\sum _ { l = k } ^ { ( m - 1 ) / 2 } \\sum _ { \\gamma \\in \\Gamma ' _ { A , l } } \\sum _ { \\beta \\in B ' _ { A , l } } x _ { 2 \\beta + 2 \\gamma + 1 _ A } \\varepsilon _ A z ^ { 2 \\beta + 2 \\gamma + 1 _ A } , \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} \\Phi ( z ) = \\gamma z \\exp \\left [ \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t z } { z - t } \\ , d \\sigma ( t ) \\right ] , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} \\Gamma : = \\{ h \\in C ( Q , E ) : h | _ { Q _ 0 } \\in \\Gamma _ 0 \\} . \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} \\mathbf { T } _ { N , \\ell } = \\sqrt { R _ { \\ell } } \\nabla U _ { \\ell } . \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} \\chi ( \\mathcal { O } _ { \\P } ( n ) ) : = \\dim H ( \\mathcal { O } _ { \\P } ( n ) ) = { N + n \\choose n } , \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} \\widehat { C } _ { 0 , T } ( f , u ) & = C ^ { / T } _ 0 ( f , u ) - \\frac { | \\partial T | _ 1 } { \\pi | T | } C ^ { / T } _ 1 ( f , u ) + \\left ( \\frac { 1 } { 2 \\pi } \\left ( \\frac { | \\partial T | _ 1 } { | T | } \\right ) ^ 2 - \\frac { 1 } { | T | } \\right ) C ^ { / T } _ 2 ( f , u ) , \\\\ \\widehat { C } _ { 1 , T } ( f , u ) & = C ^ { / T } _ 1 ( f , u ) - \\frac { | \\partial T | _ 1 } { 2 | T | } C ^ { / T } _ 2 ( f , u ) , \\\\ \\widehat { C } _ { 2 , T } ( f , u ) & = C ^ { / T } _ 2 ( f , u ) . \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 , \\ N ( t ) = 2 k + 1 \\} = \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} T _ 0 \\ , \\subsetneq \\ , T _ 1 \\ , \\subsetneq \\ , \\cdots \\ , \\subsetneq \\ , T _ { \\ell } \\ , \\subsetneq \\ , T _ { \\ell + 1 } \\ , = \\ , T X \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} \\hat { P } ( x , r ) = \\tfrac { 1 } { 6 } r ^ { 3 } + \\tfrac { 1 } { 2 } r | x | ^ { 2 } + P ( x ) . \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} \\bigg ( \\int _ { \\mathbb { T } ^ { N } } \\Big \\Vert \\sum _ { n = 1 } ^ { N } x _ { n } z _ { n } \\Big \\Vert ^ { p } d z \\bigg ) ^ { \\frac { 1 } { p } } \\leq C \\Big ( \\sum _ { n = 1 } ^ { N } \\Vert x _ { n } \\Vert ^ { p } \\Big ) ^ { \\frac { 1 } { p } } \\ , . \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( 1 - \\sigma ) D ^ 2 v : D ^ 2 \\varphi + \\sigma \\Delta v \\Delta \\varphi d x = \\lambda \\int _ { \\partial \\Omega } \\frac { \\partial v } { \\partial \\nu } \\frac { \\partial \\varphi } { \\partial \\nu } d \\sigma \\ , , \\ \\ \\ \\forall \\varphi \\in H ^ 2 ( \\Omega ) , \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} \\sum _ { i + j = n } \\pi _ i \\circ _ 1 \\pi _ j = \\sum _ { i + j = n } \\pi _ i \\circ _ 2 \\pi _ j . \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} \\left ( e ^ { t \\Delta } f \\right ) \\left ( x \\right ) = \\int _ { \\mathbb { R } ^ { d } } g _ { 2 t } ( x - y ) \\ , f \\left ( y \\right ) d y , \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} & | \\frac { - 3 2 \\lambda ' ( t ) } { \\lambda ( t ) ^ { 3 } } \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) \\left ( K _ { 1 } ( s - t , \\lambda ( t ) ) - \\frac { \\lambda ( t ) ^ { 2 } } { 4 ( 1 + s - t ) } \\right ) | \\\\ & \\leq \\frac { C } { t ^ { 3 } \\log ^ { 2 - b } ( t ) } \\int _ { t } ^ { \\infty } | K _ { 1 } ( s - t , \\lambda ( t ) ) - \\frac { \\lambda ( t ) ^ { 2 } } { 4 ( 1 + s - t ) } | d s \\\\ & \\leq \\frac { C } { t ^ { 3 } \\log ^ { b + 2 } ( t ) } \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} & \\omega \\lambda ( t ) ^ { 2 } T ( y ) ( t , \\omega ) \\\\ & = - \\int _ { t } ^ { \\infty } \\sin ( ( t - x ) \\sqrt { \\omega } ) \\frac { \\lambda ( t ) ^ { 2 } } { \\lambda ( x ) } \\lambda ( x ) \\sqrt { \\omega } \\left ( F _ { 2 } ( y ) ( x , \\omega ) - \\mathcal { F } ( \\sqrt { \\cdot } F ( x , \\cdot \\lambda ( x ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) - \\mathcal { F } ( \\sqrt { \\cdot } F _ { 3 } ( x , \\cdot \\lambda ( x ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) \\right ) d x \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} A = A _ 0 \\le \\ldots \\le A _ { i _ 0 } \\le \\ldots \\le A _ { i _ 1 } \\le \\ldots \\le A _ m = D \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} w ( \\epsilon ) w ( \\overline { \\epsilon } ) & = 1 , \\ a n d \\\\ \\pi ( \\overline { \\epsilon } ) & = \\overline { \\pi ( \\epsilon ) } . \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { 1 - x ^ 2 } } = \\sum _ { j = 0 } ^ \\infty \\frac { \\Gamma \\bigl ( \\frac { 1 } { 2 } \\bigr ) } { j ! \\Gamma \\bigl ( \\frac { 1 } { 2 } - j \\bigr ) } \\bigl ( - x ^ 2 \\bigr ) ^ j = \\sum _ { j = 0 } ^ \\infty \\binom { 2 j } { j } \\Bigl ( \\frac { x } { 2 } \\Bigr ) ^ { 2 j } \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} \\psi _ { j , k } ( x _ 1 , x _ 2 , x _ 3 ) : = 2 ^ { - 2 ( j + k ) } \\psi ^ { ( 1 ) } ( 2 ^ { - j } x _ 2 ) \\psi ^ { ( 2 ) } ( 2 ^ { - k } x _ 1 , 2 ^ { - ( j + k ) } x _ 3 ) . \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { 3 } v _ i ' \\leq \\sum \\limits _ { i = 1 } ^ { 3 } ( 5 - v _ i ) = 1 5 - \\sum \\limits _ { i = 1 } ^ { 3 } v _ i = 6 < 9 , \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} S _ { t } + \\left [ S , K \\right ] = \\gamma \\partial _ { t } ^ { 2 } \\varphi + 2 \\gamma ^ { 2 } a \\nabla \\varphi . \\nabla \\varphi _ { t } - 2 i b \\gamma \\left ( 2 \\nabla \\varphi _ { t } . \\nabla + \\Delta \\varphi _ { t } \\right ) - \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} u _ { \\sigma } ( z ) = \\int _ { [ 0 , + \\infty ] } \\frac { ( 1 + z ) + t z } { z - ( 1 + z ) t } \\ , d \\sigma ( t ) , z \\in \\mathbb { C } \\setminus \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} M _ { k } = \\left ( \\Z ^ d \\times L _ { k } \\N _ { 0 } \\right ) \\cap B _ { k + 1 } , \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} R i c ( T , T ) & = - m \\frac { \\bar { u } _ { \\bar { t } \\bar { t } } } { \\bar { u } } - ( n - 1 ) \\frac { \\bar { g } _ { \\bar { t } \\bar { t } } } { \\bar { g } } \\\\ & \\ge - m \\frac { C _ 3 } { ( 1 - C _ 3 ) ^ 2 \\bar { t } ^ 2 } + 2 ( n - 1 ) \\frac { C _ 3 } { ( 1 - C _ 3 ) ^ 2 \\bar { t } ^ 2 } \\\\ & > 0 , \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} \\Phi _ { p _ n , \\widetilde { p _ n } } ( x _ 1 ) & = \\Phi ( p _ n x p _ n ) \\widetilde { p _ n } = \\Phi ( p _ n ) \\Phi ( x ) \\Phi ( p _ n ) \\widetilde { p _ n } = \\Phi ( p _ n ) \\Phi ( x ) \\widetilde { p _ n } \\\\ & = \\Phi ( p _ n ) y \\widetilde { p _ n } = \\Phi ( p _ n ) \\widetilde { p _ n } y \\widetilde { p _ n } = \\widetilde { p _ n } y \\widetilde { p _ n } = y . \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} \\widehat { w } _ { k } ( \\underline { \\xi } _ { k } ) \\geq \\frac { k - 1 } { 1 - \\frac { \\widehat { \\lambda } _ { n } ( \\underline { \\xi } _ { n } ) + n - 2 } { ( n - 1 ) ( k - 1 ) } } = \\frac { ( n - 1 ) ( k - 1 ) ^ 2 } { n k - k - 2 n + 3 - \\widehat { \\lambda } _ { n } ( \\underline { \\xi } _ { n } ) } = A , \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\sqrt { R _ 0 } \\ge 0 \\R ^ d , ( \\sqrt { R } U ) _ 0 = 0 \\{ \\sqrt { R _ 0 } = 0 \\} . \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ \\infty f ( n ) = \\int _ { - \\infty } ^ \\infty f ( x ) d x . \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} M & = \\left ( \\begin{array} { c c c c c } 1 & m _ { 1 , 2 } & \\cdots & m _ { 1 , c } & m _ { 1 , c + 1 } \\\\ 0 & 0 & \\cdots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\cdots & 0 & 0 \\\\ 0 & 0 & \\cdots & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} \\mathbb { Z } \\setminus \\{ 0 \\} = T _ { - 2 } ( \\phi _ 1 ) \\cup T _ { - 1 } ( \\phi _ 1 ) \\cup T _ 0 ( \\phi _ 1 ) \\cup T _ 1 ( \\phi _ 1 ) . \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} \\sigma _ { ( 0 , 1 ) } ^ a \\sigma _ { ( 0 , 0 ) } ^ { \\delta _ { ( a , i ) } - a } ( ( b , j ) ) = ( b + a , j + i ) . \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} a _ n = \\max \\{ a _ 1 , \\dots , a _ n \\} . \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} g _ { \\rm s l i d e } ( y ; \\mu ) = \\frac { \\left ( y - \\xi _ L ( \\mu ) \\right ) \\left ( y - \\xi _ R ( \\mu ) \\right ) h ( y ; \\mu ) } { f _ L ( 0 , y ; \\mu ) - f _ R ( 0 , y ; \\mu ) } , \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} & \\int _ { \\Omega } | \\nabla u | ^ { 3 p - 4 } | { \\rm H e s s } \\ , u | _ A ^ 2 \\ , d \\mu \\\\ = & \\frac { p - 1 } { p } \\int _ { \\Omega } | \\nabla u | ^ { p } \\Delta _ { p , f } ( | \\nabla u | ^ { p } ) \\ , d \\mu - \\int _ { \\Omega } | \\nabla u | ^ { 3 p - 4 } { \\rm R i c } _ f ( \\nabla u , \\nabla u ) \\ , d \\mu \\\\ & - \\int _ { \\Omega } | \\nabla u | ^ { 2 p - 2 } \\langle \\nabla u , \\nabla \\Delta _ { p , f } u \\rangle \\ , d \\mu . \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} \\pi _ 1 - \\pi _ 1 ' = \\pi \\circ ( \\Phi _ 1 \\otimes \\mathrm { i d } ) + \\pi ( \\mathrm { i d } \\otimes \\Phi _ 1 ) - \\Phi _ 1 \\circ \\pi = \\delta _ \\pi ( \\Phi _ 1 ) . \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} \\bar { f } ( t ) = U ( t ) \\bar { f } _ { 0 } + \\int _ { 0 } ^ { t } U ( t - s ) \\bar K _ { g } \\bar { f } ( s ) d s . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} { \\mathcal S } ( ( \\ref { 6 . 0 . 1 } ) , X _ 1 ^ + ) = \\{ U ( \\psi ) \\ , ; \\ , \\psi ( x ) \\in { \\mathcal O } _ 0 \\} . \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} ( \\Phi _ 1 * \\Phi _ 2 ) ( g ) = \\int _ G \\left ( \\Phi _ 1 ( g h ^ { - 1 } ) \\circ \\Phi _ 2 ( h ) \\right ) d h . \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} \\begin{aligned} \\nu ( \\xi ) & = \\frac { ( n - 1 ) \\cos ^ 2 ( \\xi ) } { n } \\Big \\{ ( 1 - \\xi ^ 2 ) [ ( \\tan ( \\xi ) - \\sqrt { n - 1 } ) ^ 2 + \\sec ^ 2 ( \\xi ) ] + \\\\ & - n \\xi ( \\tan ( \\xi ) - \\sqrt { n - 1 } ) + ( n - 2 ) ( 1 - \\xi ^ 2 ) \\tan ( \\xi ) ( \\tan ( \\xi ) - \\sqrt { n - 1 } ) \\Big \\} + \\\\ & - \\frac { n - 2 } { n } \\mu ( \\xi ) , \\end{aligned} \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} \\mathcal { L } _ L ^ { \\lambda } f : = \\sum _ { \\ell = 0 } ^ L \\sum _ { k = 1 } ^ { 2 \\ell + 1 } \\mathcal { S } _ { \\lambda \\mu _ { \\ell } } \\left ( \\sum _ { j = 1 } ^ N w _ j f ( \\mathbf { x } _ j ) Y _ { \\ell , k } ( \\mathbf { x } _ j ) \\right ) Y _ { \\ell , k } . \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { j = 0 } ^ k b _ j d ^ j : b _ j \\in \\{ a _ 1 , \\ldots , a _ d \\} \\right \\} \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} \\tilde { w } = \\cdots a \\cdots z b c \\cdots \\rightsquigarrow \\tilde { w } ^ * = \\cdots a \\cdots b z c \\cdots \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} H _ 0 ( q , p ) = p _ 1 + \\cdots + p _ n \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} Q _ { 1 2 1 2 } = \\frac { 3 e ^ { 3 q _ 3 - q _ 2 - 2 q _ 1 } + 2 e ^ { 2 q _ 3 - 2 q _ 1 } - 2 e ^ { q _ 3 + q _ 2 - 2 q _ 1 } - 2 e ^ { 2 q _ 2 - 2 q _ 1 } - e ^ { 2 q _ 3 - q _ 2 - q _ 1 } + e ^ { q _ 3 - q _ 1 } + e ^ { q _ 2 - q _ 1 } } { ( 1 + e ^ { q _ 2 - q _ 1 } ) \\Delta _ 3 } , \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} | ( T ( e _ { 1 } ) - T ( e _ { 2 } ) ) '' ( t ) | & \\leq \\frac { | R H S ( e _ { 1 } , t ) - R H S ( e _ { 2 } , t ) | } { \\alpha } + \\frac { 2 } { \\alpha } \\sup _ { z \\geq t } \\left ( | R H S ( e _ { 1 } , z ) - R H S ( e _ { 2 } , z ) | \\right ) \\\\ & \\leq \\frac { 3 C _ { l i p } } { \\alpha } \\frac { | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\log ^ { b + 1 } ( t ) ( \\log ( \\log ( t ) ) ) ^ { 3 / 2 } } \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} u _ { t } = e ^ { t \\Delta } u _ { 0 } - B ( u , u ) ( t ) , \\forall t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} S _ { i j } = R _ { i j } + h R _ { i \\gamma } R _ { j \\gamma } + h ^ { 2 } R _ { i \\gamma } R _ { \\gamma \\gamma } R _ { j \\gamma } + h ^ { 3 } R _ { i \\gamma } R _ { \\gamma \\gamma } ^ { 2 } S _ { j \\gamma } , \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} \\nabla _ a \\sigma ^ { b c } - \\frac { 1 } { N + 1 } ( \\delta ^ c _ a \\nabla _ i \\sigma ^ { i b } + \\delta ^ b _ a \\nabla _ i \\sigma ^ { i c } ) = 0 \\ , . \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} | \\mathbb { E } _ \\eta ( \\eta _ t ( x ) ) - p | = | \\eta ( x ) - p | \\mathbb { P } _ \\eta ( \\tau _ x > t ) \\geq ( p \\wedge ( 1 - p ) ) \\mathbb { P } _ \\eta ( \\tau _ x > t ) , \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d \\eta } { d t } = - \\bar { A } \\nabla _ Y \\bar { H } _ Y ( \\eta ) \\\\ \\eta ( 0 ) = \\eta _ 0 . \\end{cases} \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} H P ( G ) _ d = G _ d ^ { \\Z _ + } \\cap H P ( G ) , \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{gather*} B = \\Gamma _ 0 ^ { - 1 } ( A ' ) ^ { - 1 } \\Gamma _ 0 T ' T = \\Gamma _ 0 - B ' \\Gamma _ 0 B \\\\ \\Gamma _ 0 = ( A ' ) ^ { - 1 } \\Gamma _ 0 A ^ { - 1 } + ( A ' ) ^ { - 1 } A ^ { - 1 } . \\end{gather*}"}
-{"id": "8863.png", "formula": "\\begin{align*} c ^ 2 + 1 = \\frac { M _ 2 } { M _ 1 ^ 2 } = \\frac { - 2 \\ , ( 1 / \\lambda ^ * ) \\ , \\boldsymbol { \\pi } \\ , C ^ { - 1 } { \\mathbf 1 } } { ( 1 / \\lambda ^ * ) ^ 2 } = 2 \\boldsymbol { \\pi } C { \\mathbf 1 } \\boldsymbol { \\pi } C ^ { - 1 } { \\mathbf 1 } . \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { k = 1 } ^ { \\infty } ( \\sum _ { j = 0 } ^ { N - 1 } q _ j ^ k ) e ^ { i k x } + \\sum _ { k = 1 } ^ { \\infty } ( \\sum _ { j = 0 } ^ { N - 1 } \\overline { q _ j } ^ k ) e ^ { - i k x } \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} ( f \\circ _ i g ) ( [ r ] ; a _ 1 , \\ldots , a _ { m + n - 1 } ) = \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} \\Pr { \\prod _ { i = 1 } ^ d \\norm [ 1 ] { x _ i } _ 2 > n ^ { d / 2 } ( 1 + u ) } \\le 2 \\exp \\Big ( - \\frac { c n u ^ 2 } { 4 d } \\Big ) . \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} P _ { n } ^ \\pm \\{ \\ \\cdot \\ \\} \\coloneqq P \\{ \\ \\cdot \\ | \\ V ( 0 ) = \\pm c , \\ N ( t ) = n \\} . \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { r } } \\phi ( r , \\xi ) = \\frac { 1 } { 2 } \\phi _ { 0 } ( r ) + \\frac { 1 } { r } \\sum _ { j = 1 } ^ { \\infty } ( r ^ { 2 } \\xi ) ^ { j } \\phi _ { j } ( r ^ { 2 } ) , r ^ { 2 } \\xi \\leq 4 \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} X _ { P _ \\lambda } & = X _ { P _ { \\lambda _ 1 } } \\cdots X _ { P _ { \\lambda _ { \\ell ( \\lambda ) } } } \\\\ & = \\prod _ { i = 1 } ^ { \\ell ( \\lambda ) } \\left ( \\sum _ { \\beta \\vDash \\lambda _ i } ( - 1 ) ^ { \\lambda _ i - \\ell ( \\beta ) } p _ { \\widetilde \\beta } \\right ) \\\\ & = \\sum _ { \\alpha \\preccurlyeq \\lambda } ( - 1 ) ^ { | \\lambda | - \\ell ( \\alpha ) } p _ { \\widetilde \\alpha } . \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} ( \\varphi \\ast _ { G / H } \\psi ) ^ { \\ast ^ { G / H } } = \\psi ^ { \\ast ^ { G / H } } \\ast _ { G / H } \\varphi ^ { \\ast ^ { G / H } } . \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} \\int _ { \\Omega } | u | ^ { \\alpha } | \\nabla u | ^ { p } \\ , d \\mu = \\frac { 1 } { \\alpha + 1 } \\lambda _ { 1 ; p , f } \\int _ { \\Omega } | u | ^ { \\alpha + p } \\ , d \\mu . \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} g = - [ G : H ] + 1 + \\frac { [ G : H ] } { 2 } \\sum _ { i = 1 } ^ { 3 } \\frac { | I _ i | - 1 } { | I _ i | } - \\frac { [ N _ G ( H ) : H ] } { 2 } \\sum _ { i = 1 } ^ { 3 } \\frac { | N _ G ( I _ i ) | } { | N _ H ( I _ i ) | } \\frac { | H \\cap I _ i | - 1 } { | H \\cap I _ i | } . \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} \\| w \\| _ U ^ 2 = \\Im m . \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} a ( x ) = a ( 0 ) + \\int _ { 0 } ^ { x } a ' ( \\xi ) d \\xi , x \\in { \\mathbb R } _ { + } . \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} \\Psi ' _ a = E \\rho ( E \\rho ) ^ + \\Psi ' _ a , \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} Z = \\frac { 1 } { 2 } \\left ( ( \\alpha + \\beta ) X + 2 Y + 2 \\alpha \\beta \\pm ( \\alpha - \\beta ) \\sqrt { X ^ 2 - 4 Y } \\right ) \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} | 1 - F _ { 3 } ( r , \\rho , \\lambda ( s ) ) | = 1 - F _ { 3 } ( r , \\rho , \\lambda ( s ) ) = F _ { 3 } ( 0 , \\rho , \\lambda ( s ) ) - F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} \\nu _ { j \\pm } = i \\left ( \\frac { n - 2 } { 2 } \\pm \\sqrt { \\left ( \\frac { n - 2 } { 2 } \\right ) ^ 2 + \\lambda _ j ^ 2 } \\ , \\right ) . \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} I _ 1 : = - I _ { 1 , 1 } + I _ { 1 , 2 } , \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\log \\big | \\det { } \\big ( T _ { \\omega _ H ( \\partial z _ i , \\overline { \\partial z } _ j ) , k } \\big ) \\big | ^ { 1 / N _ k } = \\frac { 1 } { N _ k } \\sum \\log ( \\lambda _ { i , k } ) , \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} K ^ { \\ast } ( Q ) = \\{ ( y _ { 0 } , \\ldots , y _ { N } ) \\in \\mathbb { R } ^ { N + 1 } : Q \\vert y _ { 0 } \\vert + Q ^ { - N } \\vert y _ { 1 } \\vert + \\cdots + Q ^ { - N } \\vert y _ { N } \\vert \\leq 1 \\} , \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} w _ n ( x ) = \\frac { \\sqrt \\mu } { \\| v _ n \\| _ 2 } \\ , v _ n ( x ) \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} \\varphi \\sigma _ x = \\sigma ' _ { \\varphi ( x ) } \\varphi . \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} \\| \\omega ( t ) \\| _ { H ^ s } ^ 2 + \\| \\theta ( t ) \\| _ { H ^ { 1 + s } } ^ 2 + \\int _ 0 ^ t \\| \\partial _ 1 \\theta ( \\tau ) \\| _ { H ^ { 1 + s } } ^ 2 ~ d \\tau \\leq C ( t ) e ^ { \\int _ 0 ^ t \\| \\nabla u ( \\tau ) \\| _ { L ^ \\infty } ~ d \\tau } . \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} S _ u ( x ) = \\frac { ( u + 1 ) ^ 2 x ^ 4 S _ { u + 1 } ( x ) } { \\left ( ( u + 1 ) ( x - 1 ) x ^ 2 F _ { u + 1 } ( x ) + ( 1 - x - ( u + 1 ) x ^ 2 ) \\right ) ^ 2 } . \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} w = i \\frac { 1 + z } { 1 - z } \\quad a + b w - N _ { \\sigma ^ { \\prime } } ( w ) = i H _ { ( \\mu _ { 1 } ) _ { * } } ( z ) , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} \\phi _ { 0 } ( r ) = \\frac { d } { d \\lambda } \\Bigr | _ { \\lambda = 1 } Q _ { \\lambda } ( r ) = \\frac { 2 r } { 1 + r ^ { 2 } } \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} \\Gamma _ { 1 ; f } & \\leq \\frac { \\int _ { \\Omega } ( \\Delta _ f \\varphi ) ^ 2 \\ , d \\mu } { \\int _ { \\Omega } \\varphi ^ 2 \\ , d \\mu } \\\\ & = \\frac { 4 \\int _ { \\Omega } \\Big [ u ^ 2 ( \\Delta _ f u ) ^ 2 + 2 u | \\nabla u | ^ 2 \\Delta _ f u + | \\nabla u | ^ 4 \\Big ] \\ , d \\mu } { \\int _ { \\Omega } u ^ { 4 } \\ , d \\mu } \\\\ & = 4 \\Big ( \\lambda _ { 1 ; f } ^ 2 - \\frac { 2 } { 3 } \\lambda _ { 1 ; f } ^ 2 + \\frac { 1 } { 3 } \\lambda _ { 1 ; f } I _ { 2 , 2 } \\Big ) \\\\ & \\leq \\frac { 1 6 } { 3 } \\lambda _ { 1 ; f } ^ 2 , \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} \\begin{pmatrix} \\tilde C _ 1 & \\tilde A _ 1 & 0 & \\dots \\\\ \\tilde B _ 1 & \\tilde C _ 2 & \\tilde A _ 2 & \\ddots \\\\ 0 & \\tilde B _ 2 & \\tilde C _ 3 & \\ddots \\\\ \\vdots & \\ddots & \\ddots & \\ddots \\end{pmatrix} \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} p _ k = \\P _ { \\rho _ { k } } \\left [ E _ { k } ( m ) \\right ] , \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} \\inf I & = \\inf I - 0 \\geq \\mu ( Q ) - \\sup I \\geq \\mu ( Q ) - ( \\inf I + | I | ) \\\\ & \\geq \\mu ( Q ) - \\inf I - \\frac { c _ \\mu - 1 } { c \\mu } \\mu ( Q ) = \\frac { 1 } { c _ \\mu } \\mu ( Q ) - \\inf I , \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} q _ i \\mapsto q _ i + c , i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} \\mu _ \\theta ( \\omega ) = \\mu ^ * + \\theta \\Delta \\mu ( \\omega ) \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} u ( r e ^ { i \\theta } ) = \\log r + \\int _ { [ 0 , + \\infty ] } \\frac { ( 1 - t ^ { 2 } ) r \\cos \\theta + t ( r ^ { 2 } - 1 ) } { | r e ^ { i \\theta } - t | ^ { 2 } } \\ , d \\sigma ( t ) , \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} G _ m \\leq 1 + \\sum ^ { m - 1 } _ { i = 1 } G _ i . \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} \\eta = 2 l + 1 \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} e ( N _ { \\{ 1 \\} \\times C ^ n } Z _ { \\mu , \\lambda } ^ w ) = e ( T _ { ( F _ e , F _ w ) } \\Omega _ w ) e \\left ( N _ { C ^ n } \\ ! \\left ( \\widetilde { Q } _ \\mu \\cap w . \\widetilde { Q } _ \\lambda \\right ) \\right ) = \\prod _ { ( i , j ) \\in N _ \\mu \\cup g N _ \\lambda } ( x _ j - x _ i ) \\prod _ { ( i , j ) \\in I _ \\mu \\cap g I _ \\lambda } ( x _ i - x _ j + \\Delta _ { i j } ) . \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} e ^ { - \\rho x / 2 } \\int _ 0 ^ x e ^ { \\rho \\xi / 2 } \\alpha ( \\xi ) d \\xi = \\int _ 0 ^ x e ^ { - \\rho \\eta / 2 } \\alpha ( x - \\eta ) d \\eta \\le \\int _ 0 ^ 1 e ^ { - \\rho \\eta / 2 } \\alpha ( x - \\eta ) d \\eta + \\sum _ { j = 1 } ^ { \\infty } \\int _ j ^ { j + 1 } e ^ { - \\rho \\eta / 2 } \\alpha ( x - \\eta ) d \\eta \\le \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} v ^ 1 _ k ( x ) = \\eta ^ 2 ( x ) [ ( u _ k - \\psi ) ( x + h ) - ( u _ k - \\psi ) ( x ) ] . \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\Lambda ( \\mu ) = \\frac { \\lambda _ L ( \\mu ) } { \\omega _ L ( \\mu ) } + \\frac { \\lambda _ R ( \\mu ) } { \\omega _ R ( \\mu ) } . \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} = ( 1 + x _ 1 ) ( 1 + x _ { n + 1 } ) \\cdot \\sum _ { J \\subseteq [ 2 , n ] } \\left ( \\prod _ { j \\in J } x _ j \\right ) ( - 1 ) ^ { | J | } p _ { n - | J | } ( \\underline x _ n ^ J ) \\cdot \\left ( \\prod _ { i \\in [ 2 , n ] \\setminus J } ( 1 + x _ i ) \\right ) . \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} c = \\liminf _ { z \\to t } \\frac { 1 - | \\eta _ { \\mu _ { 1 } } ( z ) | } { 1 - | z | } \\in ( 0 , + \\infty ) , \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} m _ { 1 , 5 } m _ { 7 , 9 } + m _ { 1 , 6 } m _ { 8 , 9 } & = 0 , & m _ { 1 , 5 } & = 0 , & m _ { 1 , 6 } m _ { 7 , 9 } & = 0 , \\\\ 1 - m _ { 1 , 2 } m _ { 7 , 9 } - m _ { 1 , 3 } m _ { 8 , 9 } & = 0 , & m _ { 1 , 2 } - m _ { 7 , 9 } & = 0 , & - m _ { 1 , 3 } m _ { 7 , 9 } - m _ { 8 , 9 } & = 0 , \\\\ 1 + m _ { 4 , 5 } m _ { 7 , 9 } + m _ { 4 , 6 } m _ { 8 , 9 } & = 0 , & - m _ { 4 , 5 } + m _ { 7 , 9 } & = 0 , & m _ { 4 , 6 } m _ { 7 , 9 } + m _ { 8 , 9 } & = 0 . \\\\ \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} ( a \\prec b ) \\prec c = ~ & a \\prec ( b * c ) , \\\\ ( a \\succ b ) \\prec c = ~ & a \\succ ( b \\prec c ) , \\\\ ( a * b ) \\succ c = ~ & a \\succ ( b \\succ c ) , \\\\ ( a \\curlyvee b ) \\prec c + ( a * b ) \\curlyvee c = ~ & a \\succ ( b \\curlyvee c ) + a \\curlyvee ( b * c ) , \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} ( A - A ^ 2 S ' ) ^ { p + 1 } & = ( A - A ^ 2 S ' ) ( A - A ^ 2 S ' ) ^ p \\\\ & = ( A - A ^ 2 S ' ) ( A ^ p - A ^ { p + 1 } S ' ) \\\\ & = A ^ { P + 1 } - A ^ { P + 2 } S ' - A ^ { P + 2 } S ' + A ^ { P + 3 } { S ' } ^ 2 \\\\ & = A ^ { p + 1 } - A ^ { p + 2 } S ' . \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} | \\partial _ { t } \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) E _ { 5 } \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | & \\leq \\frac { C } { t ^ { 3 } \\log ( t ) } + \\frac { C \\sup _ { x \\geq t } \\left ( \\frac { x | e ''' ( x ) | } { \\lambda ( x ) ^ { 3 - 2 \\alpha } } \\right ) } { \\log ^ { ( 2 - 2 \\alpha ) b } ( t ) t } \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} F ( t , x , v ) = \\mu ( v ) + \\mu ^ { 1 / 2 } ( v ) f ( t , x , v ) , \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} [ T ^ { p ^ 2 } \\textbf { a } ] _ i \\equiv \\sum _ { j = 0 } ^ { p ^ 2 } \\binom { p ^ 2 } { j } \\textbf { a } _ { i + j } \\equiv \\sum _ { j = 0 } ^ { p } \\binom { p ^ 2 } { j p } \\textbf { a } _ { i + j p } \\equiv \\sum _ { j = 0 } ^ { p } \\binom { p } { j } \\textbf { a } _ { i + j p } \\pmod { p ^ 2 } , \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} c w _ { i _ 0 } ' ( x ) = \\sum _ { j = 1 } ^ m e _ { j i _ 0 } ( x ) w _ j ( x ) . \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} \\overline { \\{ x \\} } = \\bigcap _ { \\substack { x \\in F \\\\ F } } F = \\bigcap _ { \\substack { y \\in F \\\\ F } } F = \\overline { \\{ y \\} } . \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} p _ { i } = \\min \\{ p ( e _ { i } ) , p ( - e _ { i } ) \\} , \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} \\psi _ n ^ u [ L _ u ^ \\R f ] ( \\xi ) = ( \\xi + \\lambda _ n ( u ) ) \\psi _ n ^ u [ f ] ( \\xi ) \\ , \\ 0 \\le \\xi < 1 \\ . \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} H _ { n + 1 } = c _ 1 H _ n + \\cdots + c _ n H _ 1 + 1 = H _ n + \\cdots + H _ 1 + 1 , \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { N \\to \\infty } P \\left \\{ \\int _ { 0 } ^ \\delta \\frac { | Z _ N ( t ) - \\sigma B _ N ( t ) | ^ p } { w ( t ) } d t > \\varepsilon \\right \\} = 0 . \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} \\tilde { w } = \\cdots a \\cdots b c y \\cdots \\rightsquigarrow \\tilde { w } ^ * = \\cdots a \\cdots b y c \\cdots \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\xi _ 6 ^ { 5 n } \\sum _ { a = 0 } ^ n \\xi _ 6 ^ { - a } \\sum _ { b = 0 } ^ a \\xi _ 6 ^ { - 2 b } \\sum _ { c = 0 } ^ b \\xi _ 6 ^ { - c } . \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} \\| \\sum _ { j = 1 } ^ { \\ell - 1 } \\binom { 2 \\ell - 1 } { 2 ( \\ell - j ) - 1 } ( - \\partial ( m _ { s _ 1 } , . . . , m _ { s _ { 2 ^ { \\ell - 1 } } } ) ) ^ { 2 ( \\ell - j ) - 1 } ( \\partial ( m _ { t _ 1 } , . . . , m _ { t _ { 2 ^ { \\ell - 1 } } } ) ) ^ { 2 j } \\alpha - \\beta _ 2 \\| < \\frac { \\epsilon } { 1 0 } . \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} \\left ( A + P \\right ) \\mathsf { \\hat { p } } _ { \\beta , \\mathsf { G i b b s } } = \\mathsf { \\hat { p } } _ { \\beta , \\mathsf { G i b b s } } . \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} y ^ 2 + y & = x ^ 3 - x ^ 2 - 7 x + 1 0 , \\\\ t ^ 2 & = - ( 4 x ^ 3 + 7 x ^ 2 - 6 x + 1 9 ) . \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} \\partial _ t f _ { 0 } + g \\partial _ v f _ 0 - ( v ^ \\prime ) ^ 2 \\partial _ { v v } f _ 0 - v ^ { \\prime \\prime } \\partial _ { v } f _ { 0 } + v ^ \\prime < \\nabla ^ \\perp \\phi _ { \\neq } \\cdot \\nabla _ { z , \\upsilon } f _ { \\neq } > = 0 . \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} f ( X , Y , Z ) = g \\ ( \\frac { X - Y - Z } { 2 } , \\ - Y + Z , \\ Y + Z \\ ) . \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} \\alpha = \\ln ( \\gamma ) + \\frac { \\lambda \\pi } { \\omega } , \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} - \\Delta \\varphi _ \\beta = - e ^ { - 2 \\psi } ( - 4 \\partial _ z \\partial _ { \\bar z } \\varphi _ \\beta ) = e ^ { - 2 \\psi } ( - 4 \\partial _ z \\partial _ { \\bar z } \\psi ) = 1 \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} B _ { 3 , 1 } ( 9 ) & = \\{ ( 9 ) , \\ , ( 6 , 2 , 1 ) , \\ , ( 6 , 1 ^ 3 ) , \\ , ( 5 , 3 , 1 ) , \\ , ( 4 , 3 , 2 ) , \\ , ( 4 , 3 , 1 ^ 2 ) , \\ , \\\\ & \\\\ & \\quad \\ ( 3 ^ 2 , 2 , 1 ) , \\ , ( 3 ^ 2 , 1 ^ 3 ) , \\ , ( 3 , 2 ^ 3 ) , \\ , ( 3 , 2 ^ 2 , 1 ^ 2 ) , \\ , ( 3 , 2 , 1 ^ 4 ) , \\ , ( 3 , 1 ^ 6 ) \\} . \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} \\Big [ \\lambda - ( - b + K - K e ^ { - \\lambda \\tau } ) \\Big ] ( \\lambda ^ { 2 } + P \\lambda + Q ) = 0 . \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} \\| f \\| _ { T , \\mathcal { X } } = \\sup _ { s \\in [ 0 , T ] } \\| f _ { s } \\| _ { \\mathcal { X } } . \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} \\bar { X } _ { t } ^ { t _ n , x } = x + \\int _ { t _ n } ^ { t } \\bar { b } ( s , \\bar { X } _ { s } ^ { t _ n , x } ) \\ , d s + \\int _ { t _ n } ^ { t } \\bar { b } ( s , \\bar { X } _ { s } ^ { t _ n , x } ) \\ , d W _ s . \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} g = E ^ { - 1 } = \\left [ \\begin{array} { c c c } \\frac { 1 } { 1 - e ^ { - 2 ( q _ 1 - q _ 2 ) } } & - \\frac { e ^ { - ( q _ 1 - q _ 2 ) } } { 1 - e ^ { - 2 ( q _ 1 - q _ 2 ) } } & 0 \\\\ - \\frac { e ^ { - ( q _ 1 - q _ 2 ) } } { 1 - e ^ { - 2 ( q _ 1 - q _ 2 ) } } & \\frac { 1 - e ^ { - 2 ( q _ 1 - q _ 3 ) } } { ( 1 - e ^ { - 2 ( q _ 1 - q _ 2 ) } ) ( 1 - e ^ { - 2 ( q _ 2 - q _ 3 ) } ) } & - \\frac { e ^ { - ( q _ 2 - q _ 3 ) } } { 1 - e ^ { - 2 ( q _ 2 - q _ 3 ) } } \\\\ 0 & - \\frac { e ^ { - ( q _ 2 - q _ 3 ) } } { 1 - e ^ { - 2 ( q _ 2 - q _ 3 ) } } & \\frac { 1 } { 1 - e ^ { - 2 ( q _ 2 - q _ 3 ) } } \\end{array} \\right ] \\ ; , \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} a \\wedge \\bigvee _ i b _ i = \\bigvee _ i ( a \\wedge b _ i ) \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} | | L v ( t ) | | _ { L ^ { 2 } ( R d R ) } = \\lambda ( t ) | | \\sqrt { \\omega } \\lambda ( t ) y ( t ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} X _ { ( P _ { \\ell ( \\alpha ) } , \\alpha ) } = \\sum _ { \\beta \\succcurlyeq \\alpha ^ c } ( - 1 ) ^ { \\ell ( \\alpha ^ c ) - \\ell ( \\beta ) } X _ { P _ { \\widetilde \\beta } } . \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} P ( x ; 1 , - 1 ) & = \\frac { - 2 + 2 e ^ { 2 x i } } { 2 i + 2 i e ^ { 2 x i } } = \\tan ( x ) . \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} u ^ d ( x ) = u ( x + d ( x ) ) \\approx u ( x ) + \\abs { d ( x ) } \\left \\langle \\frac { \\nabla u ( x ) } { \\abs { \\nabla u ( x ) } } , \\nabla u ( x ) \\right \\rangle , \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} v _ F ( j ) \\leq 1 2 v _ F ( A ) - 3 ( v _ F ( B ) + v _ F ( A ) ) = 3 ( 3 v _ F ( A ) - v _ F ( B ) ) \\leq 0 , \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} \\frac { \\delta S } { \\delta A ( t ) } = 0 \\iff d _ { A ( t ) } \\gamma ( t ) = \\frac { d } { d t } A ( t ) \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} | g _ i ( \\Psi ( x ) ) | \\leq L \\sum _ { j = 1 } ^ m \\psi _ j ( x ) : = L \\hat \\psi ( x ) \\mbox { f o r s o m e $ L > 0 $ a n d a l l } x < 0 , \\ ; i \\in \\{ 1 , . . . , m \\} . \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} A _ { 0 } = \\frac { b _ { 2 } } { \\left ( 2 - \\bar { t } \\right ) ^ { p } } A _ { 1 } = \\frac { b _ { 2 } } { \\left ( 1 - \\bar { t } \\right ) ^ { p } } . \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} { _ { 0 } \\mathcal { F } _ 0 } ^ { ( d ) } \\left ( \\begin{matrix} \\\\ \\end{matrix} ; \\mathbf { z } , \\mathbf { u } \\right ) \\prod _ { i = 1 } ^ { n } \\frac { | \\mathbf { u } | } { e ^ { \\omega _ { i } | \\mathbf { u } | } - 1 } = \\sum _ { \\mathbf { m } \\in \\mathcal { P } } B _ { n , \\mathbf { m } } ^ { ( d ) } \\left ( \\mathbf { z } \\mid { \\boldsymbol { \\omega } } \\right ) \\Psi _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { u } ) \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} \\underset { t \\rightarrow t _ 0 - } { l i m } \\underset { x \\in \\overline { D } } { m a x } \\underset { \\Omega - \\Omega _ { \\delta } } { e s s s u p } | u ( t , x ) | = + \\infty , \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} \\mathfrak { a } _ { \\phi } = \\sup \\{ u \\leq 0 ; | \\phi ( u ) | = \\infty \\} \\in ( - \\infty , 0 ] . \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } R d R \\left ( \\frac { \\cos ( 2 Q _ { 1 } ( R ) ) - 1 } { R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\right ) \\phi _ { 0 } ( R ) v _ { 2 } ( t , R \\lambda ( t ) ) = \\frac { 4 b } { \\lambda ( t ) t ^ { 2 } \\log ^ { b } ( t ) } + E _ { v _ { 2 } , i p } ( t , \\lambda ( t ) ) \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} \\mathcal { B } ^ { n + 1 , n } & = \\mathcal { C } ' _ { K _ n } \\| \\cdot \\| _ { H ^ { 1 } ( I _ { n + 1 } ) } , \\\\ \\mathcal { A } ^ { n + 1 , n } & = \\mathcal { A } ^ { n + 1 } | _ { \\mathcal { B } ^ { n + 1 , n } \\times \\mathcal { B } ^ { n + 1 , n } } , \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} Z ( u _ 0 ) = \\sum _ { I \\subseteq [ t - 1 ] , \\ , I \\neq \\emptyset } ( - 1 ) ^ { | I | } Z _ I ( u _ 0 ) . \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} \\left | \\sigma _ { \\infty } - \\sigma _ N \\right | \\leq C \\left | A ' ( \\sigma _ { \\infty } ) - A ' ( \\sigma _ N ) \\right | = C \\left | A _ N ' ( \\sigma _ N ) - A ' ( \\sigma _ N ) \\right | & \\overset { \\eqref { e _ q u a n t i t a t i v e _ c o n v _ a n _ d e r } } { \\leq } C \\frac { ( | \\sigma _ N | + 1 ) } { N } \\overset { \\eqref { h 1 } } { \\leq } C \\frac { | m | + 1 } { N } . \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c , \\ N ( t ) = 2 k - 1 \\} = \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} \\int _ \\R \\sup _ { t \\in \\R ^ + } \\left | \\int _ 0 ^ t f ( r ) P ( r , k ) d r \\right | ^ 2 \\frac { d \\sigma ( k ) } { \\kappa ( k ) } = \\int _ \\R \\sup _ { t \\in \\R ^ + } \\left | \\int _ 0 ^ { [ t ] } f ( r ) P ( r , k ) d r + \\int _ { [ t ] } ^ t f ( r ) P ( r , k ) d r \\right | ^ 2 \\frac { d \\sigma ( k ) } { \\kappa ( k ) } \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} P _ { n + 1 } ( \\hat { x } - \\hat { y } ) = P _ { n + 1 } ( x - y , 0 ) = \\sqrt { \\tfrac { n + 2 } { n } } P _ { n } ( x - y ) = 0 \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} P _ { 2 m + 1 } ( A ^ p , B ^ p ) & = A ^ p B ^ { - p } P _ { 2 m - 1 } ( A ^ p , B ^ p ) B ^ { - p } A ^ p \\\\ & \\le A ^ p B ^ { - p } f ( A ) ^ { 2 ( n - m ) p } B ^ { - p } A ^ p \\\\ & \\le A ^ p B ^ { - p } f ( X ) ^ { 2 ( n - m ) p } B ^ { - p } A ^ p \\\\ & = A ^ p \\left ( { { f ( X ) ^ { n - m - 1 } } \\over { X } } \\right ) ^ { 2 p } A ^ p \\\\ & \\le A ^ p \\left ( { { f ( A ) ^ { n - m - 1 } } \\over { A } } \\right ) ^ { 2 p } A ^ p = f ( A ) ^ { 2 ( n - m - 1 ) p } . \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} R [ A ] \\ = \\ \\{ ( i , j ) : j - i \\in A \\} . \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} ( I ) \\quad \\phi ^ * _ t \\left ( \\sum u _ { i } \\sigma _ i \\right ) & = e ^ { \\lambda t } u _ 1 \\sigma _ 1 + e ^ t u _ 2 \\sigma _ 2 \\\\ ( I I ) \\quad \\phi ^ * _ t \\left ( \\sum u _ { i } \\sigma _ i \\right ) & = e ^ { \\lambda t } u _ 1 \\sigma _ 1 + ( t e ^ t u _ 1 + e ^ t u _ 2 ) \\sigma _ 2 \\\\ ( I I I ) \\quad \\phi ^ * _ t \\left ( \\sum u _ { i } \\sigma _ i \\right ) & = e ^ { \\lambda t } \\left ( ( \\cos ( t ) u _ 1 - \\sin ( t ) u _ 2 ) \\sigma _ 1 + ( \\sin ( t ) u _ 1 + \\cos ( t ) u _ 2 ) \\sigma _ 2 \\right ) \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} t ^ k \\ge 2 t \\binom { t } { k - 1 } \\ge \\frac { m b _ { i + 1 } } { 2 ( b _ i + b _ { i + 1 } ) \\binom { b _ { k - 1 } + b _ k } { b _ k } } . \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} b _ { n - 2 } = \\frac { - \\mu { n \\choose 2 } + 8 { n \\choose 3 } + 1 6 { n \\choose 4 } } { 2 ( n - 1 ) } . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} H _ { ( \\mu _ { 1 } ) _ { * } } ( z ) = \\int _ { \\mathbb { T } } \\frac { t + z } { t - z } \\ , d ( \\mu _ { 1 } ) _ { * } ( t ) \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} C _ 1 = & \\ [ \\emptyset , 1 , \\emptyset , 3 , \\emptyset , \\emptyset , \\emptyset , \\emptyset , \\emptyset , 1 0 ] , \\\\ C _ 2 ' = & \\ [ 6 , \\emptyset , \\emptyset , \\emptyset , \\emptyset , \\emptyset , 1 2 , 1 5 , \\emptyset , \\emptyset ] , \\\\ C _ 2 = & \\ [ \\emptyset , \\emptyset , \\emptyset , \\emptyset , \\emptyset , \\emptyset , 1 2 , 1 5 , 1 7 , \\emptyset ] , \\\\ C _ 3 = & \\ [ \\emptyset , \\emptyset , 1 8 , \\emptyset , 1 9 , 2 4 , \\emptyset , \\emptyset , \\emptyset , \\emptyset ] . \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} t \\frac { \\partial u } { \\partial t } = x t ^ { \\mu } + \\lambda u + \\Bigl ( \\frac { \\partial u } { \\partial x } \\Bigr ) ^ 2 . \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} \\sum _ { m \\in \\mathbb { Z } } \\mu ( T _ m ( \\phi _ 2 ) ) y ^ m & = \\left ( \\frac { 1 + y ^ { - 1 } } { 2 } \\right ) \\left ( \\frac { y ^ { - 1 } } { 6 } + \\frac { 5 } { 6 } \\right ) \\left ( \\frac { 2 7 } { 4 8 } + \\frac { 2 1 } { 4 8 } y ^ { - 1 } \\right ) \\\\ & = \\frac { 7 } { 1 9 2 } y ^ { - 2 } + \\frac { 1 7 } { 6 4 } y ^ { - 1 } + \\frac { 8 9 } { 1 9 2 } + \\frac { 1 5 } { 6 4 } y . \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} \\psi _ \\alpha ( \\bar { x } _ 0 , \\ldots , \\bar { x } _ \\lambda ) = \\bigwedge \\limits _ { W \\in { \\mathcal W } ( \\bar { x } _ 0 , \\ldots , \\bar { x } _ \\lambda ) } \\psi ' _ \\alpha ( W ( \\bar { x } _ 0 , \\ldots , \\bar { x } _ \\lambda ) ) , \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} \\exp \\left \\{ \\sum _ { i = 1 } ^ { \\infty } \\frac { \\mu _ { i } ^ 2 } { \\sigma _ { i } ^ 2 } \\right \\} - \\exp \\left \\{ \\sum _ { i = 1 } ^ { k } \\frac { \\mu _ { i } ^ 2 } { \\sigma _ { i } ^ 2 } \\right \\} < \\frac { \\delta \\tilde { \\epsilon } ^ 2 } { 1 0 0 } . \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} \\sup _ { A _ { t , r _ 2 } } G / G ' = \\sup _ { \\partial B _ t } G / G ' \\ \\ \\ \\ \\inf _ { A _ { t , r _ 2 } } G / G ' = \\inf _ { \\partial B _ t } G / G ' \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\varepsilon _ { A } = \\prod _ { n \\in A } \\varepsilon _ { n } . \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} y _ { x x } = 2 \\frac { x } { x ^ 2 + y } y _ x - \\frac { 1 } { x ^ 2 + y } y _ x ^ 2 \\ , . \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} \\Phi ( \\xi , \\eta ) = \\sum _ { k , \\ell \\in \\Z ^ n } b _ { k , \\ell } e ^ { 2 \\pi i k \\cdot \\xi / T } e ^ { 2 \\pi i \\ell \\cdot \\eta / T } , ( \\xi , \\eta ) \\in T Q \\times T Q , \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} f \\in \\mathcal { S } ^ { m } _ W \\Longrightarrow f ( D ) = f ( D ) ^ \\# + f ( D ) ^ b , \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( q ^ { - 1 } ; q ^ 2 ) _ k ^ 4 ( q ^ { - 3 } ; q ^ 2 ) _ k ^ 2 } { ( q ^ 2 ; q ^ 2 ) _ k ^ 4 ( q ^ { 4 } ; q ^ 2 ) _ k ^ 2 } q ^ { 1 1 k } \\equiv \\frac { \\Omega _ q ( n ) [ 7 ] [ 9 ] } { [ 2 ] ^ 2 [ 4 ] ^ 2 [ 6 ] ^ 2 } \\sum _ { k = 0 } ^ { ( n - 3 ) / 2 } \\frac { ( q ^ 3 ; q ^ 2 ) _ k ^ 3 ( q ^ { 1 1 } ; q ^ 2 ) _ k } { ( q ^ 2 , q ^ 6 , q ^ 8 , q ^ 8 ; q ^ 2 ) _ k } q ^ { 2 k } . \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} \\mathrm { A v g } _ { s \\in \\Sigma } \\ , | \\mathrm { S e l } _ { \\phi _ s } ( A _ s ) | = 1 + \\mathrm { A v g } _ { s \\in \\Sigma } \\ , c ( \\phi _ s ) \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} & \\lambda _ { 1 } ( t ) E _ { 0 , 1 } ( \\lambda _ { 1 } ( t ) , \\lambda _ { 1 } ' ( t ) , \\lambda _ { 1 } '' ( t ) ) - \\lambda _ { 2 } ( t ) E _ { 0 , 1 } ( \\lambda _ { 2 } ( t ) , \\lambda _ { 2 } ' ( t ) , \\lambda _ { 2 } '' ( t ) ) \\\\ & = \\int _ { 0 } ^ { 1 } D F _ { 0 , 0 , 1 } ( \\lambda _ { \\sigma } ) \\cdot ( \\lambda _ { 1 } ( t ) - \\lambda _ { 2 } ( t ) , \\lambda _ { 1 } ' ( t ) - \\lambda _ { 2 } ' ( t ) , \\lambda _ { 1 } '' ( t ) - \\lambda _ { 2 } '' ( t ) ) d \\sigma \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d X _ \\varepsilon ( t ) & = b ( X _ \\varepsilon ( t ) ) d t + \\sigma ( X _ \\varepsilon ( t ) ) d B ( t ) - \\frac { 1 } { \\varepsilon } \\beta ( X _ \\varepsilon ( t ) ) d t , \\\\ X ( 0 ) & = x . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} \\Gamma \\int _ { \\mathfrak S } g ( s ) d s = \\int _ { \\mathfrak S } \\Gamma g ( s ) d s . \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} A _ n & = p _ { n + 1 } - p _ { n } , C _ n = q _ { n } - q _ { n + 1 } \\\\ B _ n & = - p _ { n } , D _ n = q _ { n } . \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} 0 \\le \\mu _ 0 \\le 1 , \\mu _ 0 = 1 \\ , \\ , { \\rm i f } \\ , \\ , | x | < 1 \\ , . \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} d | \\hat { X } ( t ) | ^ 2 = \\Big [ 2 < \\hat { X } ( t ) , \\hat { b } ( t ) + \\sigma ( X ( t ) ) \\rho ( t ) > + t r ( \\hat { \\sigma } ( t ) \\hat { \\sigma } ( t ) ^ t ) \\Big ] d t + 2 < \\hat { X } ( t ) , \\hat { \\sigma } ( t ) d \\tilde { B } ( t ) > - 2 < \\hat { X } ( t ) , d \\hat { \\eta } ( t ) > \\ , . \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} Q ( B , A ) \\ = \\ - \\ Q ( A , B ) . \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} s _ n = \\left \\{ \\begin{array} { c l } ( a n - b ) ^ { - 1 } & a n - b \\not \\equiv 0 \\mod p \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} ( T _ j ^ { n } ) ^ { - 1 } \\{ x \\} = \\{ y _ { k , n } : \\ , k < D _ { n } \\} \\ , \\ , \\ , \\ , ( T _ j ^ { n } ) ^ { - 1 } \\{ x ' \\} = \\{ y ' _ { k , n } : \\ , k < D _ { n } \\} \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} e ^ t & = e + D ^ A t , e ^ \\gamma = \\gamma \\ - e \\gamma , \\\\ * A ^ t & = A , A ^ \\gamma = \\gamma \\ - A \\gamma + \\gamma \\ - d \\gamma . \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} | \\int _ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( x ) } } ^ { \\infty } \\sqrt { r } F _ { 4 } ( x , r \\lambda ( x ) ) ( \\partial _ { 2 } ^ { 2 } \\phi ( r , \\omega \\lambda ( x ) ^ { 2 } ) ) _ { 2 } d r | & \\leq \\frac { C | a ( \\omega \\lambda ( x ) ^ { 2 } ) | } { \\omega ^ { 9 / 4 } \\lambda ( x ) ^ { 9 / 2 } } \\int _ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( x ) } } ^ { \\infty } \\sqrt { r } | F _ { 4 } ( x , r \\lambda ( x ) ) | d r \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { ( p ^ r + 1 ) / 2 } ( 4 k - 1 ) \\frac { ( - \\frac { 1 } { 2 } ) _ k ^ 4 ( - \\frac { 3 } { 2 } ) _ k ^ 2 } { k ! ^ 4 ( k + 1 ) ! ^ 2 } \\\\ [ 5 p t ] & \\quad \\equiv \\frac { 6 3 p ^ r ( p ^ { 2 r } - p ^ { 4 r } - 1 ) } { 6 4 ( p ^ { 2 r } - 1 ) } \\sum _ { k = 0 } ^ { ( p ^ r - 3 ) / 2 } \\frac { ( \\frac { 3 } { 2 } ) _ k ^ 3 ( \\frac { 1 1 } { 2 } ) _ k } { k ! ( k + 2 ) ! ( k + 3 ) ! ^ 2 } \\pmod { p ^ { r + 3 } } \\quad p > 3 . \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} H ( A ; r ) = \\int _ { 0 } ^ 1 H ( \\lfloor r ^ { - 1 } A + t \\rfloor ) d t , \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} \\mathcal { P } = \\mathcal { O } \\cup \\mathcal { E } \\ , \\cdot \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} \\beta _ { E , A , p } ( R ) : = \\ell ( R ) ^ { d } + \\sum _ { Q \\subseteq R } \\beta _ { E } ^ { d , p } ( A B _ Q ) ^ { 2 } \\ell ( Q ) ^ { d } , \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} \\nabla \\theta ( x ) = { \\rm D } g ( x ) ^ * \\Pi _ { K ^ { \\circ } } ( g ( x ) ) . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\phi _ { \\alpha } } ( z ) = - \\frac { 1 } { \\phi _ { \\alpha } ' ( 0 ^ + ) } \\frac { 1 } { 2 \\pi i } \\int _ { a - i \\infty } ^ { a + i \\infty } \\frac { \\phi _ { \\alpha } ( - \\xi ) } { \\xi } \\frac { \\Gamma ( - \\xi ) \\Gamma ( 1 - \\xi ) } { W _ { \\phi _ { \\alpha } } ( 1 - \\xi ) } ( - z ) ^ { - \\xi } d \\xi \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} \\rho ( x ) & = \\phi ( x ) - \\frac { \\lambda _ s } { ( s - 1 ) ! x _ p } \\phi ^ { s - 1 } ( x ) \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} h _ \\epsilon ( y ) = f _ { \\tiny \\mbox { B E P } } ^ { \\delta = 0 } ( y ) ( 1 + \\gamma _ 1 - \\gamma _ 2 ) + f _ { \\tiny \\mbox { B E P } } ^ { \\delta = 2 } ( y ) ( \\gamma _ 2 - 2 \\gamma _ 1 ) + f _ { \\tiny \\mbox { B E P } } ^ { \\delta = 4 } ( y ) \\gamma _ 1 + O ( \\epsilon ^ 3 ) , \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} ( \\partial _ { 2 } ^ { 2 } \\phi ( r , \\xi ) ) _ { 0 } & = 2 \\left ( a '' ( \\xi ) \\psi ^ { + } ( r , \\xi ) + 2 a ' ( \\xi ) \\partial _ { 2 } \\psi ^ { + } ( r , \\xi ) \\right ) \\\\ & + 2 \\left ( a ( \\xi ) \\left ( \\partial _ { \\xi } ^ { 2 } \\left ( \\frac { e ^ { i r \\sqrt { \\xi } } } { \\xi ^ { 1 / 4 } } \\sigma ( r \\sqrt { \\xi } , r ) \\right ) + \\frac { r ^ { 2 } } { 4 \\xi ^ { 5 / 4 } } e ^ { i r \\sqrt { \\xi } } \\sigma ( r \\sqrt { \\xi } , r ) \\right ) \\right ) \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} \\int _ { r } ^ { \\infty } \\frac { 1 } { \\varphi ( t ) ^ { n - 1 } } d t = C \\ , r ^ { 2 - n } \\ , . \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} \\beta _ J = a _ { 3 J } b _ { 2 J } - a _ { 2 J } b _ { 3 J } \\ , , \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} \\int _ { 1 / ( N + 1 ) } ^ \\delta & \\frac { | Z _ N ( t ) - \\sigma B _ N ( t ) | ^ p } { w ( t ) } d t \\\\ & \\leq \\sup _ { 1 / ( N + 1 ) < t < N / ( N + 1 ) } \\left ( \\frac { | Z _ N ( t ) - \\sigma B _ N ( t ) | } { t ^ \\zeta } \\right ) ^ p \\int _ { 1 / ( N + 1 ) } ^ \\delta \\frac { t ^ { p \\zeta } } { w ( t ) } d t \\\\ & = O _ P ( N ^ { ( - 1 / 2 + \\zeta ) p } ) ( N + 1 ) ^ { - p \\zeta + p / 2 } \\int _ { 0 } ^ \\delta \\frac { t ^ { p / 2 } } { w ( t ) } d t . \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} \\partial _ r [ w ( \\theta ) ] ^ \\top \\mathbb { E } _ \\theta [ f ( \\textbf { X } ) ] = \\partial _ r [ w ( \\theta ) ] ^ \\top \\bar { f } , \\quad ~ r \\in \\{ 1 , \\ldots , k \\} . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} \\frac d { d t } f _ n ( t ) = B ( t , f _ { n - 1 } , f _ { n - 2 } ) - a ( \\left \\| \\Lambda f _ 0 \\right \\| ) \\Lambda f _ n ( t ) , t > 0 a . e . , n \\geq 3 . \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} \\| R \\| = \\| Q \\| = | Q ( w _ L , w _ R ) | \\leq \\frac { \\eta ^ 3 } { 2 ^ 4 } + | R ( w _ L , w _ R ) | \\leq \\frac { \\eta ^ 3 } { 2 ^ 4 } + 1 + \\eta \\bigl | \\hat { T } ( w _ L , \\hat { x } _ R ) \\bigr | \\bigl | \\hat { T } ( \\hat { x } _ L , w _ R ) \\bigr | . \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} \\omega _ { 3 1 } = - a \\omega _ { 1 } , \\ \\ \\ \\ \\omega _ { 3 2 } = - c \\omega _ { 2 } . \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} { \\tilde f _ 1 } ' = - \\frac { \\tilde f _ 1 } { 2 } \\theta . \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} \\Phi ( \\xi , \\eta ) = \\sum _ { k , \\ell \\in \\Z ^ n } b _ { k , \\ell } e ^ { 2 \\pi i k \\cdot \\xi / T } e ^ { 2 \\pi i \\ell \\cdot \\eta / T } \\phi ( \\xi ) \\phi ( \\eta ) , ( \\xi , \\eta ) \\in \\R ^ n \\times \\R ^ n . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} U ^ { a _ { n } , a _ { 1 } } _ { d , c } ( \\gamma ) = U _ { t _ { n } , t _ { n - 1 } } ^ { a _ { n - 1 } } ( \\gamma ) \\psi _ { a _ n a _ { n - 1 } } ( \\gamma ( t _ { n - 1 } ) ) \\ldots U _ { t _ { 3 } , t _ { 2 } } ^ { a _ { 2 } } ( \\gamma ) \\psi _ { a _ { 2 } a _ { 1 } } ( \\gamma ( t _ { 2 } ) ) U _ { t _ { 2 } , t _ { 1 } } ^ { a _ { 1 } } ( \\gamma ) . \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} w _ { k } ( \\underline { \\xi } _ { k } ) \\geq \\frac { ( n - 1 ) \\lambda _ { n } ( \\underline { \\xi } _ { n } ) + \\widehat { \\lambda } _ { n } ( \\underline { \\xi } _ { n } ) + n - 2 } { 1 - \\widehat { \\lambda } _ { n } ( \\underline { \\xi } _ { n } ) } = B , \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} n _ { 2 r , 5 } ( 0 , 1 ) = \\frac { 5 - 5 ^ { 2 r + 1 } } { 2 4 } , \\ ; \\ ; \\ ; n _ { 2 r , 5 } ( 6 , - 5 ) = \\frac { 1 9 \\cdot 5 ^ { 2 r } - 1 9 } { 2 4 } . \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} \\psi ( t ) = \\int _ 0 ^ t \\left [ a ( \\left \\| \\Lambda f _ 0 \\right \\| ) - a \\left ( \\left \\| \\Lambda f ( s ) \\right \\| + \\int _ 0 ^ s \\Delta ( \\tau , f ( \\tau ) ) d \\tau \\right ) \\right ] \\left \\| \\Lambda ^ 2 f ( s ) \\right \\| d s , \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} m ( p , R ) : = \\sup _ { r \\in ( 0 , R ] } \\frac { 1 } { r } \\int _ { B ( p , r ) } | H | , \\kappa ( p , R ) : = \\inf _ { r \\in ( 0 , R ] } \\frac { 1 } { r ^ 2 } V ( p , r ) , \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} N ( \\alpha , x ) & = \\alpha B ( x ) \\ ( \\alpha , x ) \\in V , \\\\ \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} \\mathcal { I } _ { J , K } ^ { i n v } : = \\left \\{ \\mu _ { J , K } \\in \\mathcal { M } ^ { r e v } _ { J , K } : \\ : \\mu _ { J , K } ^ { \\otimes \\mathbb { Z } } \\circ T _ { J , K } ^ { - 1 } = \\mu _ { J , K } ^ { \\otimes \\mathbb { Z } } \\right \\} . \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} \\xi ( x , t ) & \\ge \\sup _ { \\varepsilon > 0 } ( g ( x , 0 ) - 2 \\varepsilon - ( L _ g + N _ 2 + \\| f \\| _ \\infty ) t ) \\\\ & = g ( x , 0 ) - ( L _ g + \\| f \\| _ \\infty ) t - \\inf _ { \\varepsilon > 0 } 2 \\varepsilon - N _ 2 t \\\\ & = g ( x , 0 ) - ( L _ g + \\| f \\| _ \\infty + N _ 2 ) t . \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} H _ p ^ 1 ( D _ 0 ) = \\{ v \\in H ^ 1 ( D _ 0 ) | ~ v | _ { \\Gamma _ 1 } = v | _ { \\Gamma _ 2 } , v | _ { \\Gamma _ 3 } = v | _ { \\Gamma _ 4 } \\} . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} S = \\cosh \\alpha + \\sinh \\alpha ( \\alpha \\ne 0 ) . \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} \\tfrac { \\partial ( f \\bullet \\phi ) } { \\partial r } - f \\bullet \\tfrac { \\partial \\phi } { \\partial r } = \\tfrac { \\partial f _ { b } } { \\partial r } \\cdot \\phi . \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} \\widehat \\Omega : = \\{ \\Gamma \\in [ C ( \\R ) ] ^ m : 0 \\preceq \\Gamma \\preceq \\mathbf { \\hat u } ^ * \\} , \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} \\Omega _ n ( \\gamma ) \\neq 0 \\implies \\sigma ( \\gamma ) = ( - 1 ) ^ { n + 1 } . \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} C _ { N , s } = \\Big { ( } \\int _ { { \\R ^ N } } \\frac { 1 - \\cos ( z _ 1 ) } { | z | ^ { N + 2 s } } d z \\Big { ) } ^ { - 1 } . \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} ( 1 + z ) ^ { - \\alpha } = \\frac { P _ m ( z ) } { Q _ m ( z ) } + \\epsilon _ m ( z ) : = r _ m ( z ) + \\epsilon _ m ( z ) , \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} \\varrho _ p ( A _ 1 , \\ldots , A _ N ) = \\rho \\left ( \\sum _ { i = 1 } ^ N A _ i ^ { \\otimes p } \\right ) \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\varphi d ( - { \\rm d i v } \\sigma ) = \\int _ { \\Omega } \\langle \\sigma , D \\varphi \\rangle \\ , d x , \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} & R ( \\lambda ) = \\lambda ^ { 4 } + a _ { 1 } \\lambda ^ { 3 } + b _ { 1 } \\lambda ^ { 2 } + c _ { 1 } \\lambda + d _ { 1 } , \\\\ & Q ( \\lambda ) = a _ { 2 } \\lambda ^ { 3 } + b _ { 2 } \\lambda ^ { 2 } + c _ { 2 } \\lambda . \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} \\hat H = H _ 1 - \\frac { 1 } { 2 } H _ 0 ^ 2 \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} w _ { i j } ( \\theta ) = b _ \\alpha N ^ { \\alpha - 1 } _ { \\theta , \\alpha } \\sigma ^ { i j } , i , j \\in \\{ 1 , \\ldots , d \\} , ~ i \\leq j , f _ { i i } ( { \\bf { x } } ) = x _ i ^ 2 , ~ f _ { i j } ( { \\bf { x } } ) = 2 x _ i x _ j , i , j \\in \\{ 1 , \\ldots , d \\} , ~ i < j . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} F _ 1 = \\frac { ( | x - \\alpha | ^ 2 + y ^ 2 ) ( | x - \\beta | ^ 2 + y ^ 2 ) ( | x - \\gamma | ^ 2 + y ^ 2 ) } { y ^ 3 } , \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} \\phi ( j , i ' ) - \\phi ( j , i ) & = 2 ( l _ { i + 1 } - l _ { i ' + 1 } ) - \\Delta ( i + 1 , i ' + 1 ) + \\Delta ( i + 1 - \\alpha ( j , i ) , i + 1 ) \\\\ & + \\Delta ( i ' + 1 , i ' + 1 - \\alpha ( j , i ' ) ) \\\\ & = 2 ( l _ { i + 1 } - l _ { i ' + 1 } - \\Delta ( i + 1 , i ' + 1 ) ) + \\Delta ( i + 1 - \\alpha ( j , i ) , i ' + 1 - \\alpha ( j , i ' ) ) \\\\ & \\geq 2 \\alpha ( i + 1 , i ' + 1 ) + \\Delta ( i + 1 - \\alpha ( j , i ) , i ' + 1 - \\alpha ( j , i ' ) ) \\ , \\cdot \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } \\partial _ { j } a _ { j k } \\partial _ { k } \\bigg ] \\tilde { w } & = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ \\tilde { w } & = w \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} G _ { 1 m } \\cdot H _ { 1 m } = 1 , G _ { 2 m } \\cdot H _ { 2 m } = 1 , G _ { 1 m } \\cdot H _ { 2 m } = G _ { 2 m } \\cdot H _ { 1 m } = 0 . \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { N \\to \\infty } P \\left \\{ \\int ^ { 1 } _ { 1 - \\delta } \\frac { | Z _ N ( t ) - \\sigma B _ N ( t ) | | \\sigma B _ N ( t ) | ^ { p - 1 } } { w ( t ) } d t > \\varepsilon \\right \\} = 0 . \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} \\lambda ^ 4 + a \\lambda ^ 3 + b \\lambda ^ 2 + c \\lambda + d + r e ^ { - \\lambda \\tau } = 0 , \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} \\sum _ { i + j = n } \\pi _ i \\circ _ 1 \\pi _ j = \\sum _ { i + j = n } \\pi _ i \\circ _ 2 \\pi _ j , ~ n = 0 , 1 , \\ldots , N . \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} ( | x | ^ { - s } u ^ m _ t , \\phi _ j ) + ( | \\nabla u ^ m | ^ { p - 2 } \\nabla u ^ m , \\nabla \\phi _ j ) = ( | u ^ m | ^ { q - 2 } u ^ m \\ln | u ^ m | , \\phi _ j ) , \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} \\left \\langle \\xi , \\frac { \\partial \\zeta } { \\partial t } \\right \\rangle _ { H ^ { - 1 } } = - \\int _ { \\mathbb { T } ^ 1 } \\xi \\varphi ' ( \\zeta ) d \\theta \\xi \\in L ^ 2 , t \\in [ 0 , T ] \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} \\sum _ { u , v , n \\geq 0 } \\sharp \\mathcal { D } ( u , v , n ) a ^ u b ^ v q ^ n = \\sum _ { u , v , n \\geq 0 } \\sharp \\mathcal { C } ( u , v , n ) a ^ u b ^ v q ^ n = ( - a q ; q ) _ { \\infty } ( - b q ; q ) _ { \\infty } \\ , \\cdot \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} \\partial _ x W ( u , v ; x ) = - ( L u ) ( x ) v ( x ) + u ( x ) ( L v ) ( x ) . \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} \\omega ( z ) = \\omega ( r ( z ) ) : = \\int _ a ^ { r ( z ) } \\sqrt { Q _ { \\frac { r ( ( \\gamma ( s ) ) } { 4 } } ( r ( \\gamma ( s ) ) } \\ , d s , \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} Q = ( p ^ 0 , p ^ i , q _ 0 , q _ i ) . \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} Q _ { \\Sigma } ( \\phi , \\phi ) : = \\int _ { \\Sigma } | \\nabla \\phi | ^ 2 - ( | A _ { \\Sigma } | ^ 2 + R i c _ M ( \\nu , \\nu ) ) \\phi ^ 2 \\geq 0 \\ \\ \\ \\ \\forall \\phi \\in C ^ 1 _ c ( \\Sigma \\cap U ) \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} c - c _ D \\leq \\binom { l _ n } { n - 1 } + \\dots + \\binom { l _ { \\delta } } { \\delta - 1 } . \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} - ( \\sum _ { i = 1 } ^ p \\alpha _ i c _ i ^ 2 u _ i ) \\ , ( \\sum _ { i = 1 } ^ p \\alpha _ i u _ i ) + ( \\sum _ { i = 1 } ^ p \\alpha _ i c _ i u _ i ) ^ 2 . \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} \\begin{aligned} f ( t , t _ 1 , t _ 2 , t _ 3 ) & = \\kappa ( Z ( t _ 3 ) , Z ( t _ 1 ) , Z ( t ) , Z ( t _ 2 ) ) \\\\ & = ( z ( t _ 2 ) - z ( t _ 1 ) ) ^ { - 1 } ( z ( t _ 1 ) - z ( t ) ) ( z ( t ) - z ( t _ 3 ) ) ^ { - 1 } ( z ( t _ 3 ) - z ( t _ 2 ) ) , \\end{aligned} \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} \\forall \\delta > 0 \\ \\Rightarrow \\sum _ { k = 1 } ^ { \\infty } { \\bf P } \\left ( \\ \\frac { \\sigma _ k } { | \\eta _ k | } > k ^ { \\delta \\ k } \\ \\right ) < \\infty . \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} & v ^ * ( t ) = v ( t , \\phi ( t ; 0 , x _ 0 ) ) , \\\\ & \\lambda ^ * ( t ) = \\lambda ( t , \\phi ( t ; 0 , x _ 0 ) ) . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} S _ { } [ X , V , W ] = \\int _ { \\Sigma } \\Big ( \\frac { 1 } { 2 } g _ { \\mu \\nu } D X ^ \\mu \\wedge \\star D X ^ \\nu + W _ \\mu \\wedge ( \\d X ^ \\mu - \\frac { 1 } { 2 } V ^ \\mu ) \\Big ) + \\int _ { \\Sigma _ 3 } H , \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} \\vec { g } = B _ { k , k } \\vec { a } . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow 0 } \\left < \\tau _ \\varphi , [ V ( t D ) ] - [ e _ 1 ] \\right > = \\lim _ { t \\downarrow 0 } { \\rm T r } ^ \\chi _ s ( \\Pi ( t ) ) = \\frac { 1 } { ( 2 \\pi ) ^ { \\dim M } } \\int _ { T ^ * M } \\chi ( x ) { \\rm t r } _ s ( \\sigma _ { t ^ { - 1 } } ( \\Pi ( t ) ) ) ( x , \\xi ) d x d \\xi , \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} \\| \\mu - \\nu \\| _ { 0 } = \\sup \\left \\{ \\int _ { \\R ^ d } \\phi \\ , d ( \\mu - \\nu ) \\ , ; ~ \\phi \\| \\phi \\| _ { L ^ \\infty ( \\R ^ d ) } \\leq 1 \\| \\phi \\| _ { } \\leq 1 \\right \\} . \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} \\begin{bmatrix} K _ 1 ^ - \\\\ W _ 1 \\end{bmatrix} = \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & K _ { 1 , 2 2 } ^ - & K _ { 1 , 2 3 } ^ - \\\\ 0 & K _ { 1 , 2 3 } ^ { - \\top } & K _ { 1 , 3 3 } ^ - \\\\ W _ { 1 1 } & W _ { 1 2 } & W _ { 1 3 } \\end{bmatrix} = \\begin{bmatrix} 0 & 0 & D _ 3 \\end{bmatrix} \\ , . \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} \\bold { m } _ { i } C _ { j } = \\left ( Y ^ { ( 1 ) } _ { i , j } , \\ldots , Y ^ { ( n ) } _ { i , j } \\right ) M _ { \\underline { a } , \\underline { b } } \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} f ^ { j ' , h ' } _ { j , h } = \\begin{cases} \\operatorname { I d } _ { D _ j } ~ ~ h ' = h + 1 , \\ , j ' = j \\\\ q _ j x _ { j - 1 } \\ldots x _ { h ' + 1 } x _ { h ' } \\bar { x } _ { h ' } \\bar { x } _ { h ' + 1 } \\ldots \\bar { x } _ { j ' - 1 } p _ { j ' } ~ ~ h = j , \\\\ 0 , ~ . \\end{cases} \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } \\psi _ { k , j } ^ { \\ast } ( Q ^ { \\prime } ) \\leq ( k - n ) A + B , \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} | I _ 2 | & \\leq \\int _ { - \\infty } ^ 0 \\int _ { M } ^ { M - y } M _ 1 J ( y ) d x d y + \\int _ { - \\infty } ^ 0 \\int _ { 0 } ^ { - y } M _ 1 J ( y ) d x d y \\\\ & = 2 M _ 1 \\int _ 0 ^ \\infty y J ( y ) d y : = C _ 2 < \\infty , \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} a = - ( f _ 1 '' f _ 1 ^ { - 1 } - f _ 2 '' f _ 2 ^ { - 1 } ) ( f _ 1 ' f _ 1 ^ { - 1 } - f _ 2 ' f _ 2 ^ { - 1 } ) ^ { - 1 } \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} G _ { 1 ^ \\ell } ^ \\perp \\ , g _ { \\lambda + 1 ^ \\ell } ^ { ( k + 1 ) } = g _ \\lambda ^ { ( k ) } \\ , . \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} \\varphi _ { m , ( a ^ { \\prime } , b ^ { \\prime } , c ^ { \\prime } ) } ( x , y , z ) = & \\dfrac { \\mathfrak { c } ( m - 4 ) } { 2 } \\left ( a ^ { \\prime } \\left ( x - \\dfrac { d } { \\mathfrak { c } } \\right ) ^ { 2 } + b ^ { \\prime } \\left ( y - \\dfrac { d } { \\mathfrak { c } } \\right ) ^ { 2 } + c ^ { \\prime } \\left ( z - \\dfrac { d } { \\mathfrak { c } } \\right ) ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} \\{ ( g _ 1 , g _ 2 , g _ 3 ) \\in C : g _ i \\in C _ i g _ i \\not = 1 g _ 1 g _ 2 g _ 3 = 1 \\} \\not = \\oslash . \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} F ( \\underline U ( t , x ) ) = K _ 2 \\frac { \\underline h ( t ) - | x | } { \\underline h ( t ) } \\Theta \\Big ( [ \\nabla F ( { \\bf 0 } ) ] ^ T + o ( 1 ) { \\bf I } _ m \\Big ) \\succeq K _ 2 \\frac { \\underline h ( t ) - | x | } { \\underline h ( t ) } \\frac { 3 } { 4 } \\lambda _ 1 \\Theta = \\frac { 3 } { 4 } \\lambda _ 1 \\underline U ( t , x ) , \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} \\partial _ r [ F ( \\theta ) ] = c _ 1 \\partial _ r [ w _ 1 ( \\theta ) ] + \\cdots + c _ k \\partial _ r [ w _ k ( \\theta ) ] . \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} p ( 2 ) = 2 ^ { L } - \\sum _ { i = 1 } ^ { L } c _ { i } 2 ^ { L - i } > 0 \\iff \\sum _ { i = 1 } ^ { L } c _ { i } 2 ^ { L - i } < 2 ^ { L } . \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} L : = \\exp \\left \\{ \\sum _ { i = 1 } ^ { \\infty } \\frac { 2 \\mu _ { i } Z _ { i } - \\mu _ { i } ^ { 2 } } { 2 \\sigma _ { i } ^ { 2 } } \\right \\} , L ^ { ( m ) } : = \\exp \\left \\{ \\sum _ { i = 1 } ^ { m } \\frac { 2 \\mu _ { i } Z _ { i } - \\mu _ { i } ^ { 2 } } { 2 \\sigma _ { i } ^ { 2 } } \\right \\} , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } \\frac { 1 } { 2 } \\alpha ' ( t ) & 0 \\\\ 0 & \\frac { 1 } { 2 } \\beta ' ( t ) \\\\ \\end{array} \\right ) = \\left ( \\begin{array} { c c } - a ( t ) \\alpha ( t ) & 0 \\\\ 0 & \\frac { 1 } { a ( t ) } \\beta ( t ) \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} G _ 1 - G _ 2 = K , \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} \\nu ^ { \\boxtimes k } = ( \\nu \\boxtimes \\nu ) ^ { k / 2 } \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} I ( u ) = \\| \\nabla u \\| _ p ^ p - \\int _ \\Omega | u | ^ q \\ln | u | d x . \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} a _ 0 a ' _ i - n _ i a _ i a ' _ 0 \\ , = \\ , ( - i ) a _ i , \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} \\mathcal { H } _ N ( m ) = \\sup _ { \\sigma \\in \\mathbb { R } } \\left ( \\sigma m - A _ N ( \\sigma ) \\right ) = \\sigma _ N m - A _ N ( \\sigma _ N ) . \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} \\bigl ( b _ { k m } x _ { k - m } \\bigr ) ( t - k ) & = \\int _ { [ 0 , 1 ) ^ c + k - m } n _ { k m } ( t - k , \\sigma - k + m ) \\ , x _ { k - m } ( \\sigma - k + m ) \\ , d \\sigma \\\\ & = \\int _ { [ 0 , 1 ) ^ c + k - m } n _ { k m } ( t - k , \\sigma - k + m ) \\ , x ( \\sigma ) \\ , d \\sigma , t \\in [ 0 , 1 ) ^ c + k . \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} M _ D ( x , z _ 0 ) = \\int _ { D \\setminus U } M _ D ( y , z _ 0 ) \\omega _ U ^ x ( d y ) , x \\in U . \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} \\big \\{ f \\in \\mathcal { D } ( L _ { \\max } ) \\ \\mid \\ W ( f , g ; a ) = 0 W ( f , g ; b ) = 0 g \\in \\mathcal { D } ( L _ { \\max } ) \\big \\} . \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} K _ 1 = \\bigsqcup _ n R ^ { - 1 } ( \\partial \\Delta _ n ) \\sqcup R ^ { - 1 } ( S ) ; \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} \\partial _ { t } u _ { m } = i \\left [ \\Delta u _ { m } + \\sum \\limits _ { j = 1 } ^ { N } \\left ( a _ { m j } + b _ { m j } \\left ( x , t \\right ) \\right ) u _ { j } \\right ] , x \\in R ^ { n } , t \\in \\left ( 0 , 1 \\right ) . \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\phi \\| ^ 2 _ { \\mathcal H } = & \\int _ { \\mathbb { R } ^ 2 } | \\nabla \\phi | ^ 2 \\ , + \\ , \\int _ { \\mathbb { R } ^ 2 } \\Big [ \\ , A _ + \\big ( { t ^ + } ^ 2 - { U ^ + } ^ 2 \\big ) - B \\big ( { t ^ - } ^ 2 - { U ^ - } ^ 2 \\big ) \\ , \\Big ] | \\phi ^ + | ^ 2 \\\\ & \\ , + \\ , \\int _ { \\mathbb { R } ^ 2 } \\Big [ \\ , A _ - \\big ( { t ^ - } ^ 2 - { U ^ - } ^ 2 \\big ) - B \\big ( { t ^ + } ^ 2 - { U ^ + } ^ 2 \\big ) \\ , \\Big ] | \\phi ^ - | ^ 2 , \\end{aligned} \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} d \\mu _ { 1 } ( t ) = \\frac { 5 } { 2 9 \\sqrt { 2 } - 4 0 } ( 1 - t ) ^ { 4 } \\ , d t , 0 \\leq t \\leq \\sqrt { 2 } , \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} B _ { H , L + m + 1 } - B _ { H , L + m } = 2 H _ { L + m } - H _ { L + m + 1 } , \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} j _ { n } ^ { ( 3 ) } = \\frac { 8 } { 7 } 2 ^ { n } + \\frac { 3 + 2 i \\sqrt { 3 } } { 7 } \\omega _ { 1 } ^ { n } + \\frac { 3 - 2 i \\sqrt { 3 } } { 7 } \\omega _ { 2 } ^ { n } , \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} \\lambda _ 1 ( G '^ c ) & = y ^ T D ( G '^ c ) y \\\\ & = y ^ T ( J _ n - I _ n ) y + y ^ T A ( G ' ) y \\\\ & \\le y ^ T ( J _ n - I _ n ) y + y ^ T A ( G ) y \\\\ & = y ^ T D ( G ^ c ) y . \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} \\begin{aligned} C _ { \\phi , n } ^ 2 & = ( \\phi / \\psi _ { \\gamma _ n } ^ 2 , R ^ n _ 1 ( \\phi / \\psi _ { \\gamma _ n } ^ 2 ) ) _ { H _ n } = \\int _ 0 ^ \\infty \\mathrm { e } ^ { - t } \\cdot ( \\phi / \\psi _ { \\gamma _ n } ^ 2 , Q ^ n _ t ( \\phi / \\psi _ { \\gamma _ n } ^ 2 ) ) _ { H _ n } d t \\\\ & \\leq ( \\phi / \\psi _ { \\gamma _ n } ^ 2 , \\phi / \\psi _ { \\gamma _ n } ^ 2 ) _ { H _ n } = \\int \\left ( \\phi / \\psi _ { \\gamma _ n } \\right ) ^ 2 ( x ) d x \\leq \\int \\left ( \\phi / \\psi _ { \\gamma _ 1 } \\right ) ^ 2 ( x ) d x = C _ \\phi ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} \\psi _ { \\mu _ { 1 } } ( z ) = z \\int _ { \\Gamma } \\frac { g ( t ) } { t - z } \\ , d m ( t ) + z \\int _ { \\mathbb { T } \\setminus \\Gamma } \\frac { 1 } { t - z } \\ , d ( \\mu _ { 1 } ) _ { * } ( t ) , \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} B ( x , v ) = \\tau ( v ) \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} U = L a _ h R \\longmapsto \\sum \\limits _ { \\substack { j = 0 \\\\ j \\neq h } } ^ { n } ( - \\alpha _ { h } ^ { - 1 } \\alpha _ { j } L \\cdot a _ { j } \\cdot R ) , \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} M & = \\left ( \\begin{array} { c c | c | c c } m _ { 1 , 1 } & m _ { 1 , 2 } & m _ { 1 , 3 } & 0 & 0 \\\\ 0 & m _ { 2 , 2 } & m _ { 2 , 3 } & 0 & 0 \\\\ \\hline 0 & 0 & m _ { 3 , 3 } & 0 & 0 \\\\ \\hline 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} V ( z ) & = ( I _ 2 - A z ) ^ { - 1 } ( I - B z ) T ^ { - 1 } \\\\ & = ( \\underbrace { \\left ( z - \\alpha _ + \\right ) \\left ( z - \\alpha _ - \\right ) } _ { = a ( z ) } ) ^ { - 1 } \\underbrace { ( w , \\overline { w } ) \\begin{pmatrix} z - \\alpha _ - & 0 \\\\ 0 & z - \\alpha _ + \\end{pmatrix} ( w , \\overline { w } ) ^ { - 1 } ( I - B z ) T ^ { - 1 } } _ { = b ( z ) } = a ^ { - 1 } ( z ) b ( z ) . \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} Q ( z ) = \\sum _ { \\alpha \\in \\Delta _ { > 0 } } x _ \\alpha ( z ) \\psi _ { \\alpha } ^ * ( z ) + \\sum _ { \\alpha \\in \\Delta _ { > 0 } } \\Phi _ { \\alpha } ( z ) \\psi _ { \\alpha } ^ * ( z ) - \\frac { 1 } { 2 } \\sum _ { \\alpha , \\beta , \\gamma \\in \\Delta _ { > 0 } } c _ { \\alpha , \\beta } ^ \\gamma \\psi _ { \\alpha } ^ * ( z ) \\psi _ { \\beta } ^ * ( z ) \\psi _ { \\gamma } ( z ) , \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} [ T ^ { p ^ 4 } \\textbf { a } ] _ i & \\equiv \\sum _ { j = 0 } ^ { p ^ 4 } \\binom { p ^ 4 } { j } \\textbf { a } _ { i + j } \\equiv \\sum _ { j = 0 } ^ { p ^ 2 } \\binom { p ^ 4 } { j p ^ 2 } \\textbf { a } _ { i + j p ^ 2 } \\equiv \\sum _ { j = 0 } ^ { p ^ 2 } \\binom { p ^ 2 } { j } \\textbf { a } _ { i + j p ^ 2 } \\pmod { p ^ 3 } , \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } T _ { \\xi _ j } ( 1 , \\tau ) \\l ^ j , \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} \\mathcal { A } ^ x _ { h _ L , \\ , F } ( f _ 0 ) = & \\ , c \\ , h _ L ^ y , \\\\ \\mathcal { A } ^ x _ { h _ L , \\ , F } ( f _ 0 ) = & \\ , p ' \\circ ( K ^ x + f _ 0 ^ x ) \\cdot h _ L ^ x + ( K ^ y + f _ 0 ^ y ) \\cdot q ' \\circ ( K ^ x + f _ 0 ^ x ) \\ , h _ L ^ x \\\\ & + q \\circ ( K ^ x + f _ 0 ^ x ) \\cdot h _ L ^ y + ( D u + D g ) \\circ ( K + f _ 0 ) \\cdot h _ L , \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} \\partial _ { r } v _ { 4 } ( t , r ) = \\frac { - 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\partial _ { 2 } G ( s , r , \\rho ) \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} J ^ { \\sigma } _ u ( r , s ) : = \\int _ { A _ { r , s } } u ^ 2 ( x ) | x | ^ { - n - 2 \\sigma } \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} x _ g = g ^ { - 1 } x _ { 1 _ G } \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} \\| T \\| _ j : = \\| \\delta ^ j ( T ) \\| \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} \\rho _ { ( n , m , a ) } = \\min _ { K ' } \\left \\{ \\frac { \\lambda ' } { \\widetilde { k ' } m } \\ , : \\ , \\exists \\ , ( n , m , K ' , a , \\lambda ' ) \\ , s . t . \\ , \\sum _ { 1 \\leq i \\leq m } k ' _ i = a \\right \\} . \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} t ^ { j ' , h ' } _ { j , h } + \\sum _ { \\substack { h ' - 1 < h '' < h \\\\ h ' - 1 - j ' < h '' - j '' < h - j } } s ^ { j '' , h '' } _ { j , h } t ^ { j ' , h ' } _ { j '' , h '' } + s ^ { j ' , h ' - 1 } _ { j , h } = 0 \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} \\sigma \\circ ( 1 \\otimes \\sigma ) = \\sigma \\circ ( m \\otimes 1 ) . \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} \\begin{cases} D ^ { \\alpha _ 2 } _ 0 y ( t ) = f ( t , y ( t ) , D ^ { \\alpha _ 1 } _ 0 y ( t ) ) , \\ t \\in [ 0 , b ] \\\\ y ( 0 ) = y _ { 0 } , \\ y ( T ) = y _ b . \\end{cases} \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} \\partial _ { t r } v _ { 1 } ( t , r ) = \\int _ { t } ^ { \\infty } \\frac { \\lambda ''' ( s ) d s } { 1 + s - t } + E _ { \\partial _ { t r } v _ { 1 } } ( t , r ) \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} t _ { n + 1 } - y _ n \\ge t _ { n + 1 } - t _ { n + 1 } \\Big ( 1 - \\frac { ( t _ { n } \\sin \\alpha _ n ) ^ 2 } { 2 t _ { n + 1 } ^ { 2 } } \\Big ) = \\frac { ( n - 1 ) ! } { 2 n ( n + 1 ) } , \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} u _ { m - 1 } ' = \\frac { u _ m ' } { u _ m } \\ , u _ { m - 1 } + \\ln \\left ( \\frac { u _ m ' } { u _ m } \\right ) \\ , u _ 1 ' \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} | \\sigma | ^ 2 \\le \\| f _ 1 \\| ^ 2 _ 2 \\cdot \\sum _ x \\left | \\sum _ { g \\in G } f _ 2 ( g x ) \\right | ^ 2 = \\| f _ 1 \\| ^ 2 _ 2 \\cdot \\sum _ g r _ { G G ^ { - 1 } } ( g ) \\sum _ x f _ 2 ( x ) \\overline { f _ 2 ( g x ) } \\ , . \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = F ( x , y ; \\mu ) = \\begin{bmatrix} f ( x , y ; \\mu ) \\\\ g ( x , y ; \\mu ) \\end{bmatrix} , \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} \\mathrm { H e s s } ( f \\circ \\psi ) = f ' ( \\psi ) \\mathrm { H e s s } \\psi + f '' ( \\psi ) \\nabla \\psi \\circ \\nabla \\psi . \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} ( a , u ) \\cdot _ H ( b , v ) = ( a b , a \\cdot v + u \\cdot b + H ( a , b ) ) , ~ ( a , u ) , ( b , v ) \\in A \\oplus M . \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} \\Lambda ( w , \\eta \\b 1 ) : = \\frac { \\Gamma ( w + \\eta ) \\cdot e ^ w } { \\sqrt { 2 \\pi } \\cdot w ^ { \\eta + w - \\frac { 1 } { 2 } } } , \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} H ( x ) = N \\left ( \\frac { 1 } { 2 } \\langle y , ( I d + P ( M _ { i j } ) N P ^ * y \\rangle _ Y + \\langle P ^ * y , s \\rangle \\right ) + H _ { M _ { i j } } ( z , y ) , \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} & 0 = \\nabla f ( \\bar x ) + \\sum \\limits _ { i \\in I ^ g ( \\bar x ) } \\lambda _ i \\nabla g _ i ( \\bar x ) + \\sum \\limits _ { j \\in \\mathcal P } \\rho _ j \\nabla h _ j ( \\bar x ) \\\\ & + \\sum \\limits _ { l \\in I ^ { 0 + } ( \\bar x ) \\cup I ^ { 0 0 } ( \\bar x ) } \\mu _ l \\nabla G _ l ( \\bar x ) + \\sum \\limits _ { l \\in I ^ { + 0 } ( \\bar x ) \\cup I ^ { 0 0 } ( \\bar x ) } \\nu _ l \\nabla H _ l ( \\bar x ) , \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} D = \\{ a _ { i } - a _ { j } : 1 \\le i , j \\le m \\} , \\qquad L _ { 0 } = \\max ( D ) . \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} b _ 3 & = - \\frac { 1 } { 2 X _ { 1 + 2 + 3 } } \\left ( 3 b _ 2 \\sum _ { j = 1 } ^ 2 a _ { 2 - j } a _ { j - 1 } ( 3 - j ) ! j ! \\sum _ { Q \\in Q ^ { ( 3 , j ) } } X _ { Q } + \\sum _ { j = 1 } ^ 3 a _ { 3 - j } a _ { j - 1 } ( 4 - j ) ! j ! \\sum _ { Q \\in Q ^ { ( 4 , j ) } } X _ { Q } \\right ) . \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} \\sigma _ { j D } \\circ \\hat { X } _ D \\circ \\sigma ^ { - 1 } _ { j D } = \\hat { X } _ j , ~ ~ ~ \\sigma _ { j D } \\circ \\hat { Y } _ 1 \\circ \\sigma ^ { - 1 } _ { j D } = \\hat { Y } _ 1 , ~ ~ ~ \\sigma _ { j D } \\circ \\hat { H } _ { c o u l } \\circ \\sigma ^ { - 1 } _ { j D } = \\hat { H } _ { c o u l } . \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} \\Delta ^ x \\circ R - \\Delta ^ x & = c \\ , \\Delta ^ y , \\\\ \\Delta ^ y \\circ R - \\Delta ^ y & = p \\circ ( K ^ x + \\Delta ^ x ) - p \\circ K ^ x + K ^ y \\cdot ( q \\circ ( K ^ x + \\Delta ^ x ) - q \\circ K ^ x ) \\\\ & \\quad + \\ , \\Delta ^ y \\cdot q \\circ ( K ^ x + \\Delta ^ x ) + u \\circ ( K + \\Delta ) - u \\circ K + g \\circ ( K + \\Delta ) . \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} \\chi _ { \\alpha } ( n ) & = \\prod _ { i = 1 } ^ { r } ( \\eta _ { i } \\chi _ { i } ) ^ { \\alpha _ { i } } ( n ) = \\prod _ { i = 1 } ^ { r } ( \\chi _ { i } ( n ) ) ^ { 2 \\alpha _ { i } } = 1 \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} \\rho _ 2 = \\rho _ 1 \\exp \\left ( - \\delta \\psi \\right ) , \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} B _ { \\mathbf { m } } ^ { ( d ) } \\left ( \\mathbf { z } + \\mathbf { 1 } \\right ) - B _ { \\mathbf { m } } ^ { ( d ) } \\left ( \\mathbf { z } \\right ) = \\sum _ { i = 1 } ^ { r } \\Phi _ { \\mathbf { m } _ { i } } ^ { ( d ) } ( \\mathbf { z } ) \\left ( m _ { i } + \\frac { d } { 2 } ( r - i ) \\right ) h _ { - , i } ^ { ( d ) } ( \\mathbf { m } ) . \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} Q _ { w _ { 0 , a , b } } = w _ { 0 , a , b } \\left ( \\frac { 1 } { \\prod _ { 1 \\leqslant i \\leqslant a } \\prod _ { a + 1 \\leqslant j \\leqslant n } ( x _ i - x _ j ) } \\right ) = \\frac { 1 } { \\prod _ { b + 1 \\leqslant i \\leqslant n } \\prod _ { 1 \\leqslant j \\leqslant b } ( x _ i - x _ j ) } . \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} | \\partial _ { t } \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) v _ { 5 } \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | & \\leq \\frac { C } { t ^ { 9 / 2 } \\log ^ { 2 b - 2 + \\frac { 5 N } { 2 } } ( t ) } + \\frac { C } { t ^ { 7 / 2 } \\log ^ { b - 3 + \\frac { 5 N } { 2 } } ( t ) } \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} ( f ) = \\frac { a ^ 4 f + b ^ 4 } { c ^ 4 f + d ^ 4 } . \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} a _ 1 x _ 1 + a _ 2 x _ 2 + \\cdots + a _ k x _ k = b , \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} p ^ { 2 f } - p ^ f + 1 = 3 ( 6 f + 1 ) \\mbox { o r } p ^ { 2 f } - p ^ f + 1 = 3 ( 6 f + 1 ) ^ 2 . \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} T ( r ; \\mu ) = T _ R ( r ; \\mu ) + T _ L \\left ( P _ R ( r ; \\mu ) ; \\mu \\right ) . \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} \\Phi _ m ^ + ( \\phi ) = \\cos ( m \\phi ) , \\Phi _ m ^ - ( \\phi ) = \\sin ( m \\phi ) \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} \\tilde { y } _ l : = \\frac { K y _ l - \\sum _ { j \\in \\Lambda _ 2 ^ { ( l ) } } x _ j } { K - R } . \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} S = \\frac { 1 } { A + B } = R - h R P R + h ^ { 2 } \\left ( R P \\right ) ^ { 2 } R + \\dots h ^ { k } \\left ( R P \\right ) ^ { k } S . \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} h _ { \\mathsf { m , n } } : = - b _ { \\mathsf { m } } \\delta _ { \\mathsf { m , n } } \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} X _ { ( G _ \\lambda , w _ \\lambda ) } = \\prod _ { i = 1 } ^ { \\ell ( \\lambda ) } X _ { ( G _ { \\lambda _ i } , w _ { \\lambda _ i } ) } \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} F ( X ) = { \\sum _ { j \\ge n } } ^ * ( T _ { \\omega _ { 1 , j } } - T _ { \\omega ' _ { 1 , j } } - T _ { \\omega _ { 2 , j } } + T _ { \\omega ' _ { 2 , j } } ) ( 1 , R ( X ) ) X ^ j , \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\infty } \\int _ 0 ^ T \\bar { H } ( \\eta ^ { \\nu } ( t ) + \\xi ^ { \\nu } ( t ) ) \\beta ( t ) d t = \\int _ 0 ^ T \\int _ { \\mathbb { T } ^ 1 } \\varphi ( \\eta _ * ( t , \\theta ) + \\xi ( \\theta ) ) \\beta ( t ) d \\theta d t . \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} F ( z ) = \\gamma + z - N _ { \\sigma } ( z ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} \\frac { v _ i } { \\mathrm { d } t _ { \\widetilde { Q _ a } } } ~ \\Big | _ { t _ { \\widetilde { Q _ a } } = 0 } ~ \\neq 0 , \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} F ( f ) ( x ) : = p _ 0 + p _ 1 ( x - d ) + Q _ d f ( x ) + T _ d g ( x ) . \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} \\begin{aligned} & R ( r _ 1 , \\cdots , r _ N ) = \\prod _ { i = 1 } ^ { N } r _ i ^ { \\gamma _ i } \\exp \\big \\{ - \\frac { \\omega } { 2 } r _ i ^ 2 \\big \\} L ^ { ( \\gamma _ i - 1 / 2 ) } _ { k _ i } ( \\omega r _ i ^ 2 ) , \\\\ & \\gamma _ i = \\frac { 1 } { 2 } ( 1 + \\sqrt { 1 + 4 l _ i ( l _ i + d _ i - 2 ) + 4 \\beta _ i + ( d _ i - 1 ) ( d _ i - 3 ) } ) , \\end{aligned} \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} \\begin{array} { c } m ( x , \\omega ) = \\sqrt { 2 + \\left ( x _ 1 + \\frac { E x _ 1 ^ 2 } { \\lambda } \\right ) ^ 2 + x _ 2 ^ 2 + 4 \\pi ^ 2 \\omega _ 1 ^ 2 + 4 \\pi ^ 2 \\omega _ 2 ^ 2 } \\\\ \\\\ \\geq \\sqrt { 2 + \\left ( x _ 1 + \\frac { E x _ 1 ^ 2 } { \\lambda } \\right ) ^ 2 + x _ 2 ^ 2 + \\omega _ 1 ^ 2 + \\omega _ 2 ^ 2 } \\end{array} \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} \\tilde { \\omega } ^ a { } _ { \\mu b } = ( K ^ { - 1 } ) ^ a { } _ c \\omega ^ c { } _ { \\mu d } K ^ d { } _ { b } - K ^ c { } _ { b } \\partial _ \\mu ( K ^ { - 1 } ) ^ a { } _ { c } , \\tilde { \\phi } ^ a { } _ { \\mu b } = ( K ^ { - 1 } ) ^ a { } _ { c } \\phi ^ c { } _ { \\mu d } K ^ d { } _ b . \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( q ^ { - 1 } ; q ^ 2 ) _ k ^ 6 } { ( q ^ 2 ; q ^ 2 ) _ k ^ 6 } q ^ { 7 k } \\equiv \\frac { \\Omega _ q ( n ) [ 3 ] [ 5 ] } { [ 2 ] ^ 4 [ 4 ] ^ 2 } \\sum _ { k = 0 } ^ { ( n - 3 ) / 2 } \\frac { ( q ^ 3 ; q ^ 2 ) _ k ^ 3 ( q ^ 7 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k ( q ^ 6 ; q ^ 2 ) _ k ^ 3 } q ^ { 2 k } . \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\lim _ { | t | \\to + \\infty } \\frac { | f ( x , t ) | } { \\exp ( \\beta | t | ^ { \\frac { n } { n - 1 } + | x | ^ \\alpha } ) } = 0 \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} ( h ^ { 2 } + & 2 h - \\mu ) ( h + 4 ) ^ { n } = h ^ { n + 2 } + 2 ( 2 n + 1 ) h ^ { n + 1 } + \\\\ & \\sum _ { k = 0 } ^ { n - 2 } \\bigg ( - \\mu { n \\choose k } ( 4 ) ^ { k } + 2 { n \\choose k + 1 } ( 4 ) ^ { k + 1 } + { n \\choose k + 2 } ( 4 ) ^ { k + 2 } \\bigg ) h ^ { n - k } \\\\ & + \\big ( - \\mu n ( 4 ) ^ { n - 1 } + 2 ( 4 ) ^ { n } \\big ) h - \\mu ( 4 ) ^ { n } . \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\eta ^ { ( \\beta ) } } [ \\textbf { X } ] = \\mathbb { E } _ \\theta [ \\textbf { X } ] = \\mu ~ ~ \\mathbb { E } _ { \\eta ^ { ( \\beta ) } } [ X _ i X _ j ] = \\mathbb { E } _ \\theta [ X _ i X _ j ] = k _ { i j } + \\mu _ i \\mu _ j , \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} c ( \\Gamma ( \\cal E ( - 1 ) ) ) = c ( \\Gamma ( \\cal E ) ) ^ 2 / c ( \\Gamma ( \\cal E ( 1 ) ) ) . \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} w = \\frac { w _ + + w _ - } { 2 } + \\frac { w _ + - w _ - } { 2 } \\zeta . \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} \\left ( \\frac { d h } { d z } \\right ) ^ 2 + 1 - \\left ( \\frac { h ^ n } { C - H h ^ { n + 1 } } \\right ) ^ 2 = 0 . \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} \\lim _ n \\frac { 1 } { 3 n } H ( A _ { R } ^ { ( 4 n ) } ; 4 n | n ) = & \\lim _ n \\frac { 1 } { 3 n } \\left ( H ( A _ { R } ^ { ( 4 n ) } ; 4 n ) - H ( A _ { R } ^ { ( n ) } ; n ) \\right ) \\\\ = & \\dim \\mu _ R . \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} e _ R ( \\mathcal I ( j ) ^ { [ 2 ] } ; R ) = - ( \\Delta _ j ^ 2 ) \\mbox { f o r a l l $ j $ } \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} \\ell _ A = \\max \\{ | t | : t \\in A \\} \\mathrm { \\ \\ a n d \\ \\ } \\max ( A ) = \\{ s \\in A : | s | = \\ell _ A \\} . \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} & \\frac { 1 6 } { ( \\lambda _ { 0 } ( t ) + e _ { 1 } ( t ) ) ^ { 2 } } \\int _ { t } ^ { \\infty } d x ( \\lambda _ { 0 } '' ( x ) + e _ { 1 } '' ( x ) ) K ( x - t , \\lambda _ { 0 } ( t ) + e _ { 1 } ( t ) ) \\\\ & - \\frac { 1 6 } { ( \\lambda _ { 0 } ( t ) + e _ { 2 } ( t ) ) ^ { 2 } } \\int _ { t } ^ { \\infty } d x ( \\lambda _ { 0 } '' ( x ) + e _ { 2 } '' ( x ) ) K ( x - t , \\lambda _ { 0 } ( t ) + e _ { 2 } ( t ) ) \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} \\delta \\int _ { S ^ 2 ( r ) } H ^ p \\ , d S = \\frac { p - 2 } { r ^ { p + 1 } } \\int _ { S ^ 2 ( r ) } u \\ , d S = 0 . \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} & \\left | \\bar A ( y _ 1 ) ( z _ 1 , z _ 1 ) - \\bar A ( y _ 2 ) ( z _ 2 , z _ 2 ) \\right | \\le c _ 1 ( 1 + | z _ 1 | ^ 2 + | z _ 2 | ^ 2 ) \\left ( | y _ 1 - y _ 2 | + | z _ 1 - z _ 2 | \\right ) , \\\\ & \\left | \\bar f ( y _ 1 , z _ 1 ) - \\bar f ( y _ 2 , z _ 2 ) \\right | \\le c _ 1 ( 1 + | z _ 1 | + | z _ 2 | ) \\left ( | y _ 1 - y _ 2 | + | z _ 1 - z _ 2 | \\right ) . \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} \\int \\left ( \\sum _ { j \\in B ( l ) } \\left ( \\sum _ { i = 1 } ^ N M _ { i j } x _ i \\right ) + \\psi _ b ' ( x _ j ) \\right ) \\mu _ { N , m } ( d x ^ { B ( l ) } | \\bar { x } ^ { B ( l ) } , y ) \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} \\tau : & = \\Phi _ { k , n } ( - \\alpha - \\gamma ) + ( k - n ) \\Phi _ { k , n } ( - \\frac { \\alpha + \\gamma } { n - 1 } ) + ( n - 2 ) \\frac { k - n } { k ( n + 1 ) } \\\\ & = - ( \\alpha + \\gamma ) \\frac { ( k + 1 ) ( k - 1 ) n } { ( n + 1 ) ( n - 1 ) k } + \\frac { ( k - 1 ) ( k - n ) } { k ( n + 1 ) } . \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} U _ r = \\{ ( x , y ) \\in X \\times X \\mid d ( x , y ) < r \\} . \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} \\langle \\overline { D } _ t \\rangle = \\langle \\partial _ t + \\dot { y } ^ k \\partial _ { y ^ k } - \\Gamma ^ k _ { i j } \\dot { y } ^ i \\dot { y } ^ j \\partial _ { \\dot { y } ^ k } \\rangle \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} & \\Sigma _ { 1 } + \\dots + \\Sigma _ { k } = - \\tfrac { m } { n } | x | ^ { 2 } , & & c _ { 1 } \\Sigma _ { 1 } + \\dots + c _ { k } \\Sigma _ { k } = \\tfrac { m } { n } \\left ( \\tfrac { m } { n } - 1 \\right ) | x | ^ { 2 } . \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} & \\max _ { 1 \\le i \\le m } \\left \\{ \\sum _ { j \\in E _ { \\mathrm { s s } } [ i ] } \\alpha _ { i j } ^ * + \\sum _ { k \\in E _ { \\mathrm { s a } } [ i ] } \\alpha _ { i k } ^ * + \\sum _ { j \\in E _ { \\mathrm { s s } } [ i ] } | \\alpha _ { i j } ^ * | \\right \\} \\\\ & \\quad \\le \\max _ { 1 \\le i \\le m } v ( i ) = \\sqrt { 2 f ( U ^ { ( 0 ) } , V ^ { ( 0 ) } ) } \\max _ { 1 \\le i \\le m } \\sqrt { 4 | E _ { \\mathrm { s s } } [ i ] | + | E _ { \\mathrm { s a } } [ i ] | } , \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} U = \\bigsqcup _ { g \\in \\pi _ 1 ( \\Omega ) } g ( U ' ) \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} F ( z ) = \\gamma \\exp \\left [ \\beta \\frac { 1 + z } { 1 - z } \\right ] , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} ( \\Psi . _ { \\mathcal { H } _ { - b } } a ) ( n , [ x ] , r ) & = \\sum _ { m \\in \\mathbb { Z } } \\Psi ( m , [ x ] , r ) a ( [ x - m \\theta ] , n - m ) . \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} H _ * ( X ) = H _ * ( Y ) ^ \\Gamma \\hookrightarrow H _ * ( Y ) , x \\mapsto x ^ \\circ . \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} \\widehat { G } _ { \\epsilon } : = \\mathrm { i n t } \\bigcup _ { k \\in K _ { \\epsilon } } \\epsilon \\left ( \\overline { G } + k \\right ) , \\quad \\Lambda _ { \\epsilon } : = \\Omega \\setminus \\overline { \\widehat { G } _ { \\epsilon } } . \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} { \\rm V o l } ( D ) = - ( \\Delta ^ 2 ) \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} c b _ \\alpha ^ { - 1 } N ^ { 1 - \\alpha } _ { \\theta , \\alpha } + \\sum \\limits _ { i = 1 } ^ d \\sum \\limits _ { j = i } ^ d c _ { i j } \\sigma ^ { i j } = 0 . \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} \\dim W = \\dim \\ker r + \\dim W _ D . \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 1 } z & : = s + e ^ { - h } , \\\\ p & : = e ^ { f - h } , \\\\ q & : = - e ^ { F - f - h } , \\end{alignedat} \\right . \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} \\chi ( X , \\mathcal { O } _ { X } ( D ) ) = \\dim H ^ 0 ( X , \\mathcal { O } _ { X } ( D ) ) - \\dim H ^ 1 ( X , \\mathcal { O } _ { X } ( D ) ) = 1 . \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} H ' ( s _ 0 ) = V ' ( s _ 1 ) & = V ' ( s _ 2 ) = H ' ( s _ 3 ) = 0 , \\\\ 0 < s _ 0 < s _ 1 & < s _ 2 < s _ 3 < 1 . \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} B ( p ) & = \\cot ^ 2 \\theta \\frac { f ( p ) } { \\varphi ( p ) } | \\bar { \\nabla } f | ^ 2 - h ( \\bar { \\nabla } f , \\bar { \\nabla } f ) \\\\ & \\leq \\cot \\theta | \\bar { \\nabla } f | ^ 2 ( \\cot \\theta \\frac { f ( p ) } { \\varphi ( p ) } - 1 ) \\leq 0 . \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} s _ { j , j + 1 } = \\eta _ j ^ { - 1 } t _ j - \\eta _ { j + 1 } ^ { - 1 } t _ { j + 1 } . \\\\ \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} \\frac { u } { e ^ { u } - 1 } e ^ { z u } = \\sum _ { m = 0 } ^ { \\infty } \\frac { B _ { m } ( z ) } { m ! } u ^ { m } , | u | < 2 \\pi . \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} C _ { \\beta , 2 } : = \\{ \\psi \\in h _ \\alpha ^ \\beta ( \\tau ( \\kappa _ \\alpha ) ) ( d _ \\beta ^ \\tau ) : ( H _ \\beta ^ 2 ) ^ \\tau ( \\psi ( \\kappa _ \\beta ) ) = ( H _ \\beta ^ 2 ) ^ r ( \\psi ( \\kappa _ \\beta ) ) \\} \\in h _ \\alpha ^ \\beta ( \\tau ( \\kappa _ \\alpha ) ) ( d _ \\beta ^ \\tau ) . \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} \\Lambda ^ i _ { 1 , \\ , R } ( f _ 0 ^ i ) & = 0 , \\\\ \\Lambda ^ i _ { 2 , \\ , R } ( f _ 0 ^ i , \\ , f _ 1 ^ i ) & = - f _ 1 ^ i \\circ R \\ , D ^ 2 R , \\\\ \\Lambda ^ i _ { L , \\ , R } ( f _ 0 ^ i , \\ , \\dots , \\ , f _ { L - 1 } ^ i ) & = D [ \\Lambda ^ i _ { L - 1 , \\ , R } ( f _ 0 ^ i , \\ , \\dots , \\ , f _ { L - 2 } ^ i ) ] \\\\ & - ( L - 1 ) \\ , f _ { L - 1 } ^ i \\circ R \\ , ( D R ) ^ { L - 2 } \\ , D ^ 2 R , L \\in \\{ 3 , \\ , \\dots , \\ , r \\} , \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} c ' ( 0 ) = - \\frac 1 2 \\int _ 0 ^ { + \\infty } \\frac { \\ln \\left ( 1 + 2 \\left ( 1 - \\frac 2 N \\right ) \\tau ^ 2 + \\tau ^ 4 \\right ) } { \\tau ^ { 1 + 2 s } } < 0 . \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} \\begin{cases} \\dot { X } _ t = 0 , & ( X _ 0 , Y _ 0 ) = ( x , y ) , \\\\ \\dot { Y } _ t = \\dot { W } _ t ^ 2 + n _ 2 ( X _ t , Y _ t ) \\dot { \\phi } _ t , & \\phi _ 0 = 0 . \\end{cases} \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} \\nu _ { S } ( d x ) = \\frac { 1 } { \\pi } \\int _ { 0 } ^ { \\infty } { e ^ { - x \\lambda - w x } \\left ( c _ 1 w ^ { \\alpha _ { 1 } } \\sin { ( \\pi \\alpha _ { 1 } ) } + c _ { 2 } w ^ { \\alpha _ { 2 } } \\sin { ( \\pi \\alpha _ { 2 } ) } \\right ) } d w . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} A _ K : = K \\otimes _ k A \\ , . \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} L v = 0 \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} \\lim _ { k } { \\| \\beta _ k - ( u z ) ^ * \\beta _ k u z \\| _ 1 } = 0 \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} { \\lambda } \\sum _ { k = 1 } ^ { n } \\frac { | u ( k ) | ^ 2 } { z - \\lambda _ k } = 1 . \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} \\int _ { X } | f ( x ) | \\ , d \\mu ( x ) = \\left ( 1 - \\frac { c _ f } { 2 } \\right ) \\| f \\| _ 2 \\sqrt { \\mu ( X ) } . \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} | I _ 3 | & \\leq \\int _ 0 ^ M M _ 1 y J ( y ) d y + \\int _ M ^ { \\infty } M _ 1 M J ( y ) d y \\\\ & \\leq M _ 1 \\int _ 0 ^ \\infty y J ( y ) d y + M _ 1 M _ 3 : = C _ 3 < \\infty . \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } R d R \\left ( \\frac { \\cos ( 2 Q _ { 1 } ( R ) ) - 1 } { R ^ { 2 } \\lambda ( t ) ^ { 2 } } \\right ) \\phi _ { 0 } ( R ) v _ { 2 } ( t , R \\lambda ( t ) ) = - 2 c _ { b } \\int _ { 0 } ^ { \\infty } \\xi ^ { 2 } d \\xi \\sin ( t \\xi ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } K _ { 1 } ( \\xi \\lambda ( t ) ) \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} ( X , \\N [ X ] \\to M ) \\otimes ( X ' , \\N [ X ' ] \\to M ' ) = ( X \\wedge X ' , \\N [ X \\wedge X ' ] \\to M \\otimes M ' ) \\ , , \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + a \\frac { \\partial ^ { 2 } u } { \\partial y ^ { 2 } } + b \\frac { \\partial u } { \\partial y } = F \\left ( u , \\bar { u } \\right ) , \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ N ( t ) = 2 k , \\ V ( 0 ) = - c \\} = \\binom { 2 k } { k } \\frac { 1 } { 2 ^ { 2 k } } \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} \\frac { d ( \\mathrm { R e } \\lambda ( \\tau ) ) } { d \\tau } \\Big \\vert _ { \\tau = \\tau _ { 0 } } \\neq 0 \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} \\| \\varphi ^ * - \\mathcal { L } ^ { \\lambda } _ L \\varphi ^ * \\| _ 2 = \\left ( \\sum _ { \\ell = 1 , ~ | c _ { \\ell } | \\leq \\lambda \\mu _ { \\ell } } ^ d | c _ { \\ell } | ^ 2 + \\sum _ { \\ell = 1 , ~ | c _ { \\ell } | > \\lambda \\mu _ { \\ell } } ^ d | \\lambda \\mu _ { \\ell } | ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} \\tilde { B } ( x \\otimes y ) = B ( x , y ) ( x \\in X , y \\in Y ) . \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} \\| \\partial _ { t } ^ + u _ h ^ n \\| ^ 2 _ { - 1 , h } + ( \\partial _ { t } ^ + u _ h ^ n , \\phi _ h ^ n ) _ { \\mathcal { T } _ h } = 0 . \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} \\pi ( A _ { p _ 1 } ) \\ = \\ & ( A _ { p _ 1 } \\setminus \\{ k \\} ) \\cup \\{ k + 1 \\} , \\\\ \\pi ( A _ { p _ 2 } ) \\ = \\ & ( A _ { p _ 2 } \\setminus \\{ k + 1 \\} ) \\cup \\{ k \\} , \\\\ \\pi ( A _ p ) \\ = \\ & A _ p \\ p \\not = p _ 1 , p _ 2 . \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} \\sum _ { m \\geq 0 } B _ { m } ( 1 - z ) \\frac { u ^ { m } } { m ! } & = \\frac { u } { e ^ { u } - 1 } e ^ { ( 1 - z ) u } \\\\ & = \\frac { u e ^ { u } } { e ^ { u } - 1 } e ^ { - z u } \\\\ & = \\frac { - u } { e ^ { - u } - 1 } e ^ { - z u } \\\\ & = \\sum _ { m \\geq 0 } B _ { m } ( z ) \\frac { ( - u ) ^ { m } } { m ! } \\\\ & = \\sum _ { m \\geq 0 } ( - 1 ) ^ { m } B _ { m } ( z ) \\frac { u ^ { m } } { m ! } . \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} \\Gamma _ I : = \\prod _ { \\substack { j = 1 \\\\ i _ j \\neq 0 } } ^ { n } \\gamma _ { i _ j } ( \\epsilon _ j ) , \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} A _ i = \\{ x _ 1 ^ { ( i ) } , \\ldots , x _ { a _ i } ^ { ( i ) } \\} \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} & \\left ( \\cdots \\left ( \\varphi _ { k _ 1 l _ 1 } ^ { r _ 2 ' \\cdots r _ s ' } \\right ) \\cdots \\right ) _ { k _ j l _ j } ^ { r _ { j + 1 } ' \\cdots r _ s ' } ( X _ 1 , \\cdots , X _ j + 2 \\pi , \\cdots , X _ n , Y _ 1 , \\cdots , Y _ n ) \\\\ & = \\left ( \\cdots \\left ( \\varphi _ { k _ 1 l _ 1 } ^ { r _ 2 ' \\cdots r _ s ' } \\right ) \\cdots \\right ) _ { ( k _ j + 1 ) ( l _ j + 1 ) } ^ { r _ { j + 1 } ' \\cdots r _ s ' } ( X _ 1 , \\cdots , X _ n , Y _ 1 , \\cdots , Y _ n ) , \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} \\ ! { L o c } _ \\kappa = \\ ! { \\kappa L o c } _ \\kappa = \\ ! { \\lambda L o c } _ \\kappa \\quad . \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} C C ^ k ( A , B ) : = C C ^ k ( A ) \\oplus C C ^ { k + 1 } ( B ) , \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} I ^ { G H } ( \\bar x , \\bar y , \\bar z ) = \\{ l \\in \\mathcal Q \\ , | \\ , G _ l ( \\bar x ) = \\bar y _ l \\ , \\land \\ , H _ l ( \\bar x ) = \\bar z _ l \\} = I ^ { 0 0 } ( \\bar x ) \\cup I ^ { - - } ( \\bar x ) . \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} \\left < p _ i , p _ j \\right > _ N = \\delta _ { i j } , \\quad 1 \\leq i , j \\leq d . \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} \\| v \\| _ { W ^ { 1 , p } ( \\omega , E ) } : = \\left ( \\| v \\| _ { L ^ p ( \\omega , E ) } ^ p + \\| \\nabla v \\| _ { L ^ p ( \\omega , E ) } ^ p \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} \\Delta _ 0 w & = \\Delta w - 2 ( n - k ) - \\sum _ { i = k + 1 } ^ n \\langle \\Pi ( \\nabla w ) , \\nabla _ { E _ i } E _ i \\rangle + \\sum _ { i = 1 } ^ k \\langle \\Pi _ 0 ( \\nabla ^ T w ) , \\nabla _ { E _ i } E _ i \\rangle \\ , . \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ { \\R } \\rho ( X _ { s - } , y ) \\nu ( d y , d s ) = \\sum _ { k = 1 } ^ { N _ t } \\xi _ k \\cdot \\rho ( X _ { \\tau _ { k } - } ) = \\sum _ { 0 < s \\leq t } \\rho ( X _ { s - } ) \\Delta L _ s = \\int \\limits _ 0 ^ t \\rho ( X _ { s - } ) d L _ s . \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} \\pi ( \\xi _ 1 ) \\pi ( \\xi _ 2 ) = c ( \\xi _ 1 , \\xi _ 2 ) \\pi ( \\xi _ 1 + \\xi _ 2 ) , \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} [ P ^ b _ { Q } ] - [ e _ 1 ] \\in K _ 0 ( \\mathcal { I } _ G ( M ) ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; e _ 1 : = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} X _ { N , m } : = \\left \\{ x \\in \\mathbb { R } ^ { N } : \\ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ i = m \\right \\} \\subset \\mathbb { R } ^ { N } . \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} \\int _ 0 ^ a e ^ { i u ^ 2 } G ( u \\epsilon ) d u = G ( 0 ) \\int _ 0 ^ a e ^ { i u ^ 2 } d u + \\int _ 0 ^ a e ^ { i u ^ 2 } G _ 1 ( \\epsilon u ) d u \\ , . \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} t \\frac { \\partial U } { \\partial t } = & \\lambda ( t , x ) U + b ( t , x ) \\frac { \\partial U } { \\partial x } \\\\ & + R \\Bigl ( t , x , u _ 0 + U , \\frac { \\partial u _ 0 } { \\partial x } + \\frac { \\partial U } { \\partial x } \\Bigr ) - R \\Bigl ( t , x , u _ 0 , \\frac { \\partial u _ 0 } { \\partial x } \\Bigr ) . \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} 2 V = \\bigg [ \\frac { \\kappa } { 6 } ( 1 + 4 g ) + \\frac { h ^ 2 } { 6 \\kappa } + \\frac { 3 } { 8 \\kappa } g ^ { i j } h _ i h _ j \\bigg ] ( p ^ 0 ) ^ 2 + \\frac { 6 } { \\kappa } ( q _ 0 ) ^ 2 + \\frac { 2 } { \\kappa } h p ^ 0 q _ 0 . \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} \\inf _ { | \\xi | = 1 } \\int _ { | y | \\leq N _ \\nu } | y \\cdot \\xi | ^ 2 \\nu ( d y ) > 0 \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 p } { \\partial t ^ 2 } = - ( c _ 1 - c _ 2 ) \\frac { \\partial ^ 2 p } { \\partial x \\partial t } + c _ 1 c _ 2 \\frac { \\partial ^ 2 p } { \\partial x ^ 2 } - 2 \\lambda ( t ) \\Bigl [ \\frac { \\partial p } { \\partial t } + \\frac { ( c _ 1 - c _ 2 ) } { 2 } \\frac { \\partial p } { \\partial x } \\Bigr ] \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} s ( \\lambda ) = \\min \\Big \\{ 1 , \\frac { H ( p ) } { \\log \\lambda ^ { - 1 } } \\Big \\} , h ( \\lambda ) = \\lim _ { n \\to \\infty } \\frac { 1 } { n } H ( \\mu _ { \\lambda } ^ { ( n ) } ) . \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} g _ i ( u ) \\geq & - f _ i ( ( 1 + \\epsilon ) \\mathbf { u } ^ * ) - ( 1 + \\epsilon ) b _ 1 \\sum _ { j = 1 } ^ m ( u _ j ^ * - u _ j ) \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} L w = g ( t , x ) , w \\bigr | _ { t = T } = \\psi ( x ) . \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} \\mathbf { P } \\left ( W _ n = b \\ : \\vline \\ : W _ { n - 1 } = a \\right ) = \\mu _ { J , K } \\left ( \\{ x : \\ : F ^ { ( 2 ) } ( x , a ) = b \\} \\right ) , \\qquad \\forall a , b \\in \\{ 0 , 1 , \\dots , K \\} . \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} \\mu = \\frac { s c a l _ g } { 2 } \\quad \\mbox { a n d } \\quad \\nu = \\frac { n - 1 } { n } \\frac { ( \\Delta f ) _ g } { f } - \\frac { n - 2 } { n } \\mu , \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} K _ { 3 } ( w , \\lambda ( t ) ) - K _ { 3 , 0 } ( w , \\lambda ( t ) ) = \\left ( \\frac { w } { 1 + w ^ { 2 } } - \\frac { w } { \\lambda ( t ) ^ { 2 - 2 \\alpha } + w ^ { 2 } } \\right ) \\frac { w ^ { 4 } } { 4 ( w ^ { 2 } + 3 6 \\lambda ( t ) ^ { 2 } ) ^ { 2 } } + \\frac { 1 } { 4 ( \\lambda ( t ) ^ { 1 - \\alpha } + w ) ( 1 + w ) ^ { 3 } } \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} h _ t = ( h ' ) { \\rm S c h } ( h ) \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} \\sigma ( u _ 1 ) = - u _ 1 ^ { - 1 } u _ 2 , \\qquad \\qquad \\sigma ( u _ 2 ) = u _ 1 ^ { - 1 } . \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} = \\hbox { T r } _ { \\mathcal { H } } \\left ( h _ { i } h _ { j } ^ * \\right ) = \\langle h _ { j } , h _ { i } \\rangle = : \\beta _ { i j } . \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} \\begin{aligned} ( z & ( t _ 2 ) - z ( t _ 1 ) ) ^ { - 1 } ( z ( t _ 1 ) - z ( t ) ) = \\\\ & = \\frac { t _ 1 - t } { t _ 2 - t _ 1 } \\Bigl ( 1 + ( t - t _ 2 ) \\bigl ( \\frac 1 2 ( z ' ) ^ { - 1 } z '' + \\frac 1 6 ( z ' ) ^ { - 1 } z ''' ( t + t _ 1 + t _ 2 ) - \\frac 1 4 ( ( z ' ) ^ { - 1 } z '' ) ^ 2 ( t _ 2 + t _ 1 ) \\bigr ) \\Bigr ) + . . . \\end{aligned} \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } - \\frac { \\sin t + \\cot \\theta \\cos t } { \\cos t - \\cot \\theta \\sin t } u = 0 . \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} g ( n ) & = 1 , \\\\ g ( n ) & = ( 1 + | n | ) ^ s , \\\\ g ( n ) & = e ^ { a | n | ^ b } ( 1 + | n | ) ^ s , \\\\ g ( n ) & = e ^ { a | n | ^ b } ( 1 + | n | ) ^ s \\ln ^ t ( e + | n | ) \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ f ^ 2 u = \\Gamma _ f \\ , u , \\ \\ { \\rm i n } \\ \\Omega , \\\\ u | _ { \\partial \\Omega } = \\frac { \\partial u } { \\partial \\nu } | _ { \\partial \\Omega } = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} \\Delta ^ { \\{ e _ n \\} } _ L f ( x ) = \\lim _ { n \\to \\infty } \\frac 1 n \\sum _ { k = 1 } ^ n < f '' ( x ) e _ k , e _ k > . \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} ( t \\hat \\omega _ i - R i c ( \\omega _ 0 ) + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi _ i ( t ) ) ^ n = e ^ { \\varphi _ i ( t ) } \\omega _ 0 ^ n , \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} G _ s \\big ( f , g ; \\lambda \\big ) = \\int _ 0 ^ 1 f ! _ { s t + ( 1 - s ) \\lambda } g \\ ; d \\nu _ { \\lambda } ( t ) , \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} \\left ( \\sum _ { n = 1 } ^ { N } \\| a _ n \\| _ X ^ { q } \\right ) ^ { 1 / { q } } \\leq C \\Big \\Vert \\sum _ { n = 1 } ^ { N } a _ { n } n ^ { - s } \\Big \\Vert _ { \\mathcal { H } _ { q ' } ( X ) } \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} \\tilde { I } _ i = \\frac { I _ i ( \\dot \\gamma , \\dot \\gamma ) } { g ( \\dot \\gamma , \\dot \\gamma ) } = \\tilde { c } _ i \\in \\mathbb { R } \\ , . \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} T _ t ( u ) T _ t ( v ) = T _ t ( u \\cdot T _ t ( v ) + T _ t ( u ) \\cdot v + H ( T _ t ( u ) , T _ t ( v ) ) ) , \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} A ( X ) = ( \\d L _ { h ^ { - 1 } } ) _ h ( X ) , \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} \\delta _ \\pi ( f ) : = ( - 1 ) ^ { n - 1 } \\big [ \\sum _ { i = 1 } ^ 2 ( - 1 ) ^ { ( i - 1 ) ( n - 1 ) } ~ \\pi \\circ _ i f - ( - 1 ) ^ { n - 1 } \\sum _ { i = 1 } ^ n ( - 1 ) ^ { ( i - 1 ) } ~ f \\circ _ i \\pi \\big ] . \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ N ( t ) = n , \\ V ( 0 ) = c _ 1 \\} = \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} B _ 0 & = \\{ \\mu \\in A ^ \\prime : ( \\langle f ^ \\prime \\oplus \\mu , \\lambda ( \\mu ) , h ^ 0 ( \\mu ) , h ^ 1 ( \\mu ) , ( H ^ 2 ) ^ \\prime ( \\mu ( \\kappa ) ) \\rangle , a ^ \\prime ) \\Vdash b \\} . \\\\ B _ 1 & = \\{ \\mu \\in A ^ \\prime : ( \\langle f ^ \\prime \\oplus \\mu , \\lambda ( \\mu ) , h ^ 0 ( \\mu ) , h ^ 1 ( \\mu ) , ( H ^ 2 ) ^ \\prime ( \\mu ( \\kappa ) ) \\rangle , a ^ \\prime ) \\Vdash \\neg b \\} . \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} \\begin{pmatrix} \\bar \\alpha & 0 \\\\ 0 & \\alpha \\end{pmatrix} . \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} \\psi _ { \\Lambda _ { 1 } } ( b _ { \\Lambda } ) = \\sum _ { i , j \\in I _ { 0 } } \\prod _ { x \\in \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , j } ^ * b _ { x } \\right ) \\beta _ { i j } ^ { | \\Lambda _ { 1 } \\setminus \\Lambda | } . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} ( \\phi \\ast f ) ( [ v ] , r ) = \\phi ( [ v ] , r ) f ( [ a v + r b ] ) , \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} ( \\lambda \\ast _ { G / H } \\lambda ' ) ^ { \\ast ^ { G / H } } & = T _ H \\left ( ( \\lambda \\ast _ { G / H } \\lambda ' ) _ q ^ { \\ast ^ { G } } \\right ) \\\\ & = T _ H \\left ( ( \\lambda _ q \\ast _ { G } \\lambda ' _ q ) ^ { \\ast ^ { G } } \\right ) = T _ H \\left ( { \\lambda ' } _ q ^ { \\ast ^ { G } } \\ast _ { G } \\lambda _ q ^ { \\ast ^ { G } } \\right ) . \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} & | \\partial _ { r } \\left ( - \\frac { 1 } { r } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\lambda '' ( s ) \\partial _ { 3 } F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\lambda ' ( s ) \\right ) | \\leq \\frac { C } { t ^ { 3 } \\log ^ { 1 + b } ( t ) } \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} b _ { \\mathsf { m } } : = Z _ { \\frac { \\alpha + 1 } { 2 } \\beta } \\exp \\left [ - \\frac { \\alpha - 1 } { 2 } \\beta \\lambda _ { \\mathsf { m } } \\right ] . \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} \\boldsymbol { Y } = ( \\beta \\boldsymbol { G } \\boldsymbol { \\Theta } \\boldsymbol { S } \\boldsymbol { h } _ r + \\boldsymbol { h } _ d ) \\boldsymbol { x } ^ { \\rm T } + \\boldsymbol { W } , \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} x _ k = \\sum \\limits _ { i = 0 } ^ { 2 ^ { k } - 1 } ( - 1 ) ^ i v _ k ^ { i } a _ k v _ k ^ { - i } \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} \\gamma _ - f ^ { \\varepsilon , a , n } = ( 1 \\ ! - \\ ! a ) \\mathcal { R } \\left [ \\gamma _ + f ^ { \\varepsilon , a , n - 1 } \\right ] , \\ \\ n \\geq 2 , \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} \\sqrt { n } & \\sum _ { k = 2 } ^ { \\log n } \\frac { k ^ { k - 2 } } { k ! } \\int _ { 1 0 0 0 \\log n } ^ \\infty x ^ { k - 1 } e ^ { - x } e ^ { - ( k - 1 ) \\cdot 1 0 0 0 \\log n } \\dd x \\\\ & \\leq O ( n ^ { 1 0 0 1 } ) \\sum _ { k = 2 } ^ { \\log n } \\frac { k ^ { k - 2 } } { k ! } ( k - 1 ) ! n ^ { - 1 0 0 0 k } \\\\ & \\leq O ( n ^ { 1 0 0 1 } ) \\sum _ { k = 2 } ^ { \\log n } \\left ( \\frac { k } { n ^ { 1 0 0 0 } } \\right ) ^ k = O ( n ^ { - 5 0 0 } ) . \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} \\begin{aligned} ( z & ( t ) - z ( t _ 3 ) ) ^ { - 1 } ( z ( t _ 3 ) - z ( t _ 2 ) ) = \\\\ & = \\frac { t _ 3 - t _ 2 } { t - t _ 3 } \\Bigl ( 1 + ( t _ 2 - t ) \\bigl ( \\frac 1 2 ( z ' ) ^ { - 1 } z '' + \\frac 1 6 ( z ' ) ^ { - 1 } z ''' ( t + t _ 2 + t _ 3 ) - \\frac 1 4 ( ( z ' ) ^ { - 1 } z '' ) ^ 2 ( t + t _ 3 ) \\bigr ) \\Bigr ) + . . . \\end{aligned} \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} \\begin{aligned} \\theta _ { 1 7 } ( \\lambda , \\mu ) & = \\theta _ { 1 7 } ( \\lambda - 1 7 , \\mu ) , \\\\ \\theta _ { 1 7 } ( \\lambda , \\mu ) & = \\theta _ { 1 7 } ( \\lambda + 6 , \\mu - 4 ) . \\end{aligned} \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} \\bar { X } : = \\left \\{ \\bar { x } : \\mathbb { T } ^ 1 \\to \\mathbb { R } ; \\ \\bar { x } \\left ( \\frac { j - 1 } { N } , \\frac { j } { N } \\right ] j = 1 , \\cdots , N , m \\right \\} . \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } a _ n = \\lim _ { n \\to \\infty } \\left ( \\sup _ { m \\geq n } a _ m \\right ) . \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} \\Phi _ { p , \\widetilde { p } } ( x ) = d \\Psi ( x ) d ^ { - 1 } \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\begin{aligned} & \\bigg [ - \\hat { L } ^ 2 _ i + g _ i ( \\Omega _ i ) + \\frac { 1 } { 4 } ( d _ i - 1 ) ( d _ i - 3 ) \\bigg ] Y _ i ( \\Omega _ i ) = \\lambda _ i Y _ i ( \\Omega _ i ) , ~ ~ ~ i = 1 , 2 , \\cdots , N - 1 , \\\\ & \\bigg [ - \\hat { L } ^ 2 _ N + \\frac { 1 } { 4 } ( d _ N - 1 ) ( d _ N - 3 ) \\bigg ] Y _ N ( \\Omega _ N ) = \\lambda _ N Y _ N ( \\Omega _ N ) , \\end{aligned} \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} \\min \\limits _ { p \\in \\mathbb { P } _ L } ~ ~ \\left \\{ \\frac 1 2 \\sum _ { j = 1 } ^ N w _ j \\left ( p ( \\mathbf { x } _ j ) - f ( \\mathbf { x } _ j ) \\right ) ^ 2 + \\lambda \\sum _ { \\ell = 1 } ^ { d } | \\mu _ { \\ell } \\beta _ { \\ell } | \\right \\} \\quad p = \\sum _ { \\ell = 1 } ^ { d } \\beta _ { \\ell } p _ { \\ell } \\in \\mathbb { P } _ L , \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} \\partial _ { t t } y + \\omega y = - \\mathcal { F } ( \\sqrt { \\cdot } F ( t , \\cdot \\lambda ( t ) ) ) ( \\omega \\lambda ( t ) ^ { 2 } ) + F _ { 2 } ( y ) ( t , \\omega ) - \\mathcal { F } ( \\sqrt { \\cdot } F _ { 3 } ( u ( y ) ) ( t , \\cdot \\lambda ( t ) ) ) ( \\omega \\lambda ( t ) ^ { 2 } ) \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} F _ n ^ { ( 1 ) } ( z ) = \\frac { P _ n ^ { ( 1 ) } ( z ) } { \\prod _ { u \\in U _ n } ( z - u ) } , F _ n ^ { ( 2 ) } ( z ) = \\frac { P _ n ^ { ( 2 ) } ( z ) } { \\prod _ { w \\in W _ n } ( z - w ) } . \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} Q ^ h _ { \\Sigma } ( \\phi , \\phi ) : = Q _ { \\Sigma } ( \\phi , \\phi ) + \\int _ { \\Sigma } h \\phi ^ 2 \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} H _ 1 : = X _ 1 , H _ 2 : = X _ 1 X _ 2 , H _ 3 : = X _ 1 X _ 3 , H _ 4 : = \\frac { X _ 1 X _ 4 } { X _ 2 } \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ M \\left ( \\frac { \\partial } { \\partial t } \\varphi ( x , t ) + h ( \\nabla \\varphi ( x , t ) , v _ t ( x ) ) \\right ) \\ d \\mu _ t ( x ) d t = 0 \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} & c = s _ 1 \\cdots s _ { n _ 1 } , \\\\ & d = t _ 1 \\cdots t _ { n _ 2 } , \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} \\Psi _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } ) = d _ { \\mathbf { m } } \\frac { 1 } { \\left ( \\frac { n } { r } \\right ) _ { \\mathbf { m } } } \\Phi _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } ) , \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} \\zeta _ 1 ( s , x \\b a ) = a ^ { - s } \\cdot \\zeta _ H \\left ( \\frac { x } { a } \\right ) , \\Gamma _ 1 ( x \\b a ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\cdot \\Gamma \\left ( \\frac { x } { a } \\right ) \\cdot a ^ { \\frac { x } { a } - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} ( { } ^ Q \\Omega ^ \\pm ) ^ a { } _ { b c } = \\rho ^ \\mu _ c ( \\Omega ^ \\pm ) ^ a { } _ { \\mu b } + C ^ a { } _ { b c } . \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} \\begin{aligned} f ( Y _ t ) = f ( y _ 0 ) & + \\sum _ { k = 1 } ^ d \\int _ 0 ^ t \\frac { \\partial f } { \\partial y _ k } ( Y _ { s - } ) d ( Y ^ c _ s ) ^ k + \\frac { 1 } { 2 } \\sum _ { k , j = 1 } ^ d \\int _ 0 ^ t \\frac { \\partial ^ 2 f } { \\partial y _ k \\partial y _ j } ( Y _ { s - } ) d [ ( Y ^ c _ s ) ^ k , ( Y ^ c _ s ) ^ j ] \\\\ & + \\sum _ { 0 < s \\le t } \\big ( f ( Y _ s ) - f ( Y _ { s - } ) \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 \\pi ) ^ { \\dim M } } \\int _ { T ^ * M } \\chi ( x ) { \\rm t r } _ s ( \\sigma _ { t ^ { - 1 } } ( \\Pi ( t ) ) ) ( x , \\xi ) d x d \\xi & = \\frac { ( - 1 ) ^ p } { ( 2 \\pi i ) ^ p } \\frac { p ! } { 2 p ! } \\ , \\int _ M \\chi \\hat { A } ( M ) f _ 0 ^ \\varphi d f ^ \\varphi _ 1 \\wedge d f ^ \\varphi _ k \\\\ & = \\frac { ( - 1 ) ^ p } { ( 2 \\pi i ) ^ p } \\frac { p ! } { 2 p ! } \\ , \\int _ M \\chi \\hat { A } ( M ) \\wedge \\omega _ \\varphi . \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} h ( z ) = h ( z , \\omega ) = h [ \\{ \\eta _ k \\} ] ( z ) \\stackrel { d e f } { = } \\sum _ { k = 0 } ^ { \\infty } \\eta _ k \\ z ^ k . \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} F _ C ( a , b , c ; x ) = \\sum _ { m _ 1 , \\ldots , m _ n = 0 } ^ { \\infty } \\frac { ( a , m _ 1 + \\cdots + m _ n ) ( b , m _ 1 + \\cdots + m _ n ) } { ( c _ 1 , m _ 1 ) \\cdots ( c _ n , m _ n ) m _ 1 ! \\cdots m _ n ! } x _ 1 ^ { m _ 1 } \\cdots x _ n ^ { m _ n } , \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} \\begin{array} { l } u ( x - 1 ) ^ { r } + u ^ 2 ( x - 1 ) ^ { k _ 1 } p _ 1 ( x ) + u ^ 3 ( x - 1 ) ^ { k _ 2 } p _ 2 ( x ) \\\\ - ( x - 1 ) ^ { r - r _ 1 } ( u ( x - 1 ) ^ { r _ 1 } + u ^ 2 ( x - 1 ) ^ { k _ 4 } p _ 4 ( x ) + u ^ 3 ( x - 1 ) ^ { k _ 5 } p _ 5 ( x ) ) \\\\ = u ^ 2 ( x - 1 ) ^ { k _ 1 } p _ 1 ( x ) - u ^ 2 ( x - 1 ) ^ { r - r _ 1 + k _ 4 } p _ 4 ( x ) \\\\ + u ^ 3 ( x - 1 ) ^ { k _ 2 } p _ 2 ( x ) - u ^ 3 ( x - 1 ) ^ { r - r _ 1 + k _ 5 } p _ 5 ( x ) , \\\\ = u ^ 2 ( x - 1 ) ^ { \\min \\{ k _ 1 , r - r _ 1 + k _ 4 \\} + \\hat e _ 3 } h _ 3 ( x ) + u ^ 3 ( x - 1 ) ^ { k _ 2 } p _ 2 ( x ) - u ^ 3 ( x - 1 ) ^ { r - r _ 1 + k _ 5 } p _ 5 ( x ) , \\end{array} \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} \\pi _ { S _ { \\alpha , \\lambda } } ( x ) = \\frac { \\alpha } { \\Gamma ( 1 - \\alpha ) } \\frac { e ^ { - \\lambda x } } { x ^ { \\alpha + 1 } } , ~ x > 0 . \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} P f = \\sum _ { k = 0 } ^ { 4 } \\langle f , \\chi _ k \\rangle \\chi _ k , \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} \\begin{aligned} 0 \\to X _ 1 \\to & M _ 1 \\to X _ 0 \\to 0 , \\\\ 0 \\to X _ 0 \\to & M _ 0 \\to N \\to 0 . \\end{aligned} \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} \\mathfrak { \\widehat H } _ k = ( 4 - k ) a ^ { k - 2 } X ^ d + \\frac { ( k - 3 ) ( k - 8 ) } { 2 } a ^ { k - 4 } X ^ { 2 d } - \\frac { ( k - 4 ) ( k - 5 ) ( k - 1 2 ) } { 3 ! } a ^ { k - 6 } X ^ { 3 d } + \\cdots . \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} \\gamma _ j = \\frac { \\sqrt { 1 + 4 \\lambda _ j } } { 2 } , ~ ~ ~ j = 1 , 2 , \\cdots , N . \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} \\dot { x } & = y , \\\\ \\dot { y } & = \\begin{cases} - \\frac { k } { m } \\ , x - \\frac { b } { m } \\ , y + \\frac { F } { m } , & { \\rm u n t i l ~ } x = \\mu , \\\\ - \\frac { k } { m } \\ , x - \\frac { b } { m } \\ , y - \\frac { F } { m } , & { \\rm u n t i l ~ } x = - \\mu , \\end{cases} \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} y ^ { 3 } + p _ { 1 } y + q _ { 1 } = 0 , \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} P \\{ M ( s ) \\le \\beta \\ | \\ V ( 0 ) = c , \\ N ( t ) = 2 k \\} & = \\sum _ { j = 0 } ^ { 2 k } P \\{ M ( s ) \\le \\beta , \\ N ( s ) = j \\ | \\ V ( 0 ) = c , \\ N ( t ) = 2 k \\} \\\\ & = \\sum _ { j = 0 } ^ { 2 k } P _ j ^ + \\{ M ( s ) \\le \\beta \\} \\ , \\frac { P \\{ N ( s ) = j \\} \\ , P \\{ N ( t - s ) = 2 k - j \\} } { P \\{ N ( t ) = 2 k \\} } \\\\ & = \\frac { 1 } { t ^ { 2 k } } \\sum _ { j = 0 } ^ { 2 k } P _ j ^ + \\{ M ( s ) \\le \\beta \\} \\binom { 2 k } { j } s ^ j ( t - s ) ^ { 2 k - j } , \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} f _ { n , 1 } ( \\cdot \\ , u ) : = \\big ( f _ n ( \\cdot \\ , u ) + f _ n ^ - ( \\cdot \\ , u ) \\big ) / 2 \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} R ( y ) & = y \\sqrt { 1 - \\nu + \\frac { \\nu ^ 2 } { 2 } } , \\\\ \\Theta ( y ) & = \\tan ^ { - 1 } \\left ( \\frac { 2 } { \\nu } - 1 \\right ) . \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ * = \\langle u ^ + , \\ , v ^ + \\rangle _ { * \\mathcal R ^ + } \\ , + \\ , \\langle { u ^ - } , \\ , v ^ - \\rangle _ { * \\mathcal R ^ - } , \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Big ( \\P \\{ | M _ { n , r } | ^ { p } \\le c _ { n } x + d _ { n } \\} - \\Lambda _ { r } ( x ) \\Big ) = 0 , \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\limsup _ { n \\to \\infty } P \\left ( \\left \\| \\mathbb { I } _ n \\right \\| _ { \\ell _ \\infty } > L \\sqrt { \\frac { \\log ( d + r ) } { n } } \\right ) = 0 . \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} \\varphi \\ast _ { G / H } \\psi ( x H ) = \\int _ { G / H } \\varphi ( y H ) J \\psi ( y ^ { - 1 } x H ) d \\mu ( y H ) , \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\left \\Vert e ^ { \\gamma \\left \\vert x \\right \\vert ^ { 2 } } \\tilde { u } \\left ( . , t \\right ) \\right \\Vert _ { X } = \\left \\Vert e ^ { \\mu ^ { 2 } \\left ( t \\right ) \\left \\vert x \\right \\vert ^ { 2 } } u \\left ( . , s \\right ) \\right \\Vert _ { X } , \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} - \\alpha = \\frac { 1 - n \\lambda _ { n } } { n ( 1 + \\lambda _ { n } ) } . \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} \\begin{aligned} ( \\nabla ^ 2 f ) _ g & = ( \\nabla ^ 2 f ) _ { g _ \\kappa } + \\\\ & + d f \\otimes d ( \\log h ) + d ( \\log h ) \\otimes d f - g _ { \\kappa } ( ( \\nabla f ) _ { g _ \\kappa } , ( \\nabla \\log h ) _ { g _ \\kappa } ) . \\end{aligned} \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\frac { 2 \\mu _ { k + 1 } Z _ { k } - \\mu _ { k + 1 } ^ 2 } { 4 ( k + 1 ) } . \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } \\psi ( 0 , x , v ) d x d v \\leq \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } \\psi ( t _ * , x , v ) d x d v = 1 . \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} \\tau ( t ) = \\tau _ 0 - 2 t C _ r ( 1 + \\| e ^ { \\tau _ 0 A } \\overline { v } _ 0 \\| _ { H ^ r } ^ 2 + \\| e ^ { \\tau _ 0 A } \\widetilde { v } _ 0 \\| _ { H ^ r } ^ 2 ) , \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} | \\{ ( a + b ) ( b + d ) = 1 ~ : ~ a \\in A , \\ , d \\in D , \\ , b \\in \\o \\cdot [ N ] \\} | \\ll \\sqrt { | A | | D | } N \\max \\{ | D | ^ { - 1 / 2 } , N ^ { - 1 / 5 } \\} \\ , . \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\Big ( \\sum _ { n = 1 } ^ \\infty \\frac { \\| a _ n \\| ^ q _ X } { n ^ \\delta } \\Big ) ^ { 1 / q } \\leq C \\| D \\| _ { \\mathcal { H } _ { p } ( X ) } , \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} P ( r , k ) = e ^ { i r k } \\overline { P _ * ( r , k ) } , \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} V [ v ^ { ( i _ 0 + 1 ) } \\leadsto x ^ { ( i _ 0 + 1 ) } ] = W [ w ^ { ( i _ 0 + 1 ) } \\leadsto y ^ { ( i _ 0 + 1 ) } ] . \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} \\partial _ { 2 } \\phi ( r , \\xi ) = ( \\partial _ { 2 } \\phi ( r , \\xi ) ) _ { 0 } + ( \\partial _ { 2 } \\phi ( r , \\xi ) ) _ { 1 } \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} | P _ { 1 } | \\leq C b _ q 2 ^ { 2 q s } ( \\| u \\| _ { L ^ 2 } + \\| \\omega \\| _ { L ^ 2 } ) ( \\| f \\| _ { H ^ s } ^ 2 + \\| f \\| _ { H ^ s } \\| \\partial _ 1 f \\| _ { H ^ s } ) . \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} \\forall ( a , b ) \\in \\R ^ 2 \\colon \\varphi ^ t _ \\textup { F B } ( a , b ) : = a + b - \\sqrt { a ^ 2 + b ^ 2 + 2 t } . \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow 0 } \\mathbf { Q } ^ \\phi \\big [ h _ 1 ( X _ { t _ 1 } ^ n ) \\cdots h _ k ( X _ { t _ k } ^ n ) \\big ] = \\mathbf { Q } ^ \\phi \\big [ h _ 1 ( X _ { t _ 1 } ) \\cdots h _ k ( X _ { t _ k } ) \\big ] , \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} \\sum \\limits _ { x \\in \\mathbb { S } } \\big [ p ( x ) - p ^ * ( x ) \\big ] \\big [ p ^ * ( x ) ^ { \\alpha - 1 } - q ( x ) ^ { \\alpha - 1 } \\big ] = 0 \\quad \\forall p \\in \\mathbb { L } . \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} B : = \\frac { \\mathbb { C } [ x _ { \\epsilon _ 1 + \\epsilon _ 2 } , x _ { \\epsilon _ 1 - \\epsilon _ 2 } , x _ { \\epsilon _ 2 } , x _ { \\epsilon _ 1 } ] } { ( x _ { \\epsilon _ 1 + \\epsilon _ 2 } ^ 2 , x _ { \\epsilon _ 1 - \\epsilon _ 2 } ^ 2 , x _ { \\epsilon _ 1 } ^ 2 , x _ { \\epsilon _ 2 } x _ { \\epsilon _ 1 + \\epsilon _ 2 } , x _ { \\epsilon _ 1 } x _ { \\epsilon _ 1 - \\epsilon _ 2 } , x _ { \\epsilon _ 1 } x _ { \\epsilon _ 1 + \\epsilon _ 2 } , x _ { \\epsilon _ 2 } ^ 3 , x _ { \\epsilon _ 2 } ^ 2 x _ { \\epsilon _ 1 } ) } , \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} & ( U _ l ^ * - V _ l ^ * ) ^ \\top \\nabla _ { U _ l } F ( U ^ * , V ^ * ; \\gamma ) + ( V _ l ^ * - U _ l ^ * ) ^ \\top \\nabla _ { V _ l } F ( U ^ * , V ^ * ; \\gamma ) \\\\ & = ( U _ l ^ * - V _ l ^ * ) ^ \\top \\left \\{ 2 \\gamma I _ m - \\left ( \\sum _ { i j \\in E _ { \\mathrm { s s } } } \\alpha _ { i j } ^ * \\bar { A } _ { i j } + \\sum _ { i k \\in E _ { \\mathrm { s a } } } \\alpha _ { i k } ^ * \\bar { A } _ { i k } \\right ) \\right \\} ( U _ l ^ * - V _ l ^ * ) = 0 . \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} \\bar A ( p ) ( u , u ) : = \\begin{cases} & \\phi \\big ( { \\rm d i s t } _ N ( p ) \\big ) \\sum _ { i , j = 1 } ^ L \\frac { \\partial ^ 2 P _ N } { \\partial p _ i \\partial p _ j } ( P _ N ( p ) ) u _ i u _ j , \\ \\ \\ p \\in B ( N , 2 \\delta _ 0 ) , \\\\ & 0 , \\ \\ \\ p \\in \\R ^ L / B ( N , 2 \\delta _ 0 ) \\end{cases} \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} \\delta \\left ( r \\right ) \\geq C _ { \\mu } \\left \\Vert u \\left ( . , 0 \\right ) \\right \\Vert _ { X } e ^ { - \\left ( M + 1 0 \\varkappa \\right ) } = C _ { \\mu } \\left \\Vert u \\left ( . , 0 \\right ) \\right \\Vert _ { X } e ^ { - \\left ( M + 2 0 \\right ) a _ { 1 } k ^ { \\frac { 1 } { 2 - p } } } \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} \\partial _ { t } \\rho - \\partial _ { v } \\phi _ { \\neq } \\partial _ { z } \\rho _ { l o } = 0 . \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} S = S ^ { c l } + \\iota _ { Q _ { B R S T } } \\alpha \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} \\beta _ n & = \\sum _ { k = 0 } ^ { n - 1 } \\alpha _ { n - 1 - k } \\alpha _ k - m ^ 2 \\sum _ { k = 0 } ^ n \\alpha _ { n - k } \\alpha _ k . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} B V ( \\Omega ) : = \\left \\{ u \\in L ^ 1 ( \\Omega ) : ~ \\norm { u } _ { B V } < + \\infty \\right \\} , \\ ; \\norm { u } _ { B V } : = \\norm { u } _ { L ^ 1 } + \\norm { D u } . \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\lim _ { m \\rightarrow \\infty } \\frac { \\ell _ R ( H ^ 1 ( X , \\mathcal O _ X ( - m A ) ) } { m ^ d } & = & \\lim _ { m \\rightarrow \\infty } \\frac { 1 } { [ R / m _ R : k _ 1 ] } \\frac { \\dim _ { k _ 1 } H ^ 1 ( X , \\mathcal O _ X ( - m A ) ) } { m ^ d } \\\\ & = & \\lim _ { m \\rightarrow \\infty } \\frac { 1 } { [ R / m _ R : k _ 1 ] } \\frac { h ^ 0 ( Y , R ^ 1 \\Psi _ * \\mathcal O _ Z ( m ( a L - A ) ) } { m ^ d } \\\\ & = & 0 \\end{array} \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} \\lim _ { n } \\big ( u ( f , \\Psi , \\mathcal { P } _ n ^ 1 ) - l ( f , \\Psi , \\mathcal { P } _ n ^ 1 ) \\big ) = 0 \\ , , \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} \\lambda & = 0 . 5 , & \\eta & = - 1 , & \\nu & = - 1 . \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} - \\left ( b ^ 2 - a ^ 2 \\right ) ^ 2 | \\xi | ^ 2 = 2 \\left ( a ^ 2 + b ^ 2 \\right ) \\left ( | \\xi | ^ { 2 \\rho } + | \\xi | ^ { 2 \\theta } \\right ) ^ 2 . \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} y ^ 2 + y = x ^ 3 + a x + b \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} 2 N \\leq \\sum _ { i = m + 3 } ^ L G _ i + 4 . \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} Y = \\int _ { w _ - } ^ { w _ + } \\frac { \\psi _ Y ( w ) } { \\sqrt { - U ( w ) } } d w , \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} \\begin{array} { l @ { { } \\iff { } } l } g \\left ( r _ 1 \\right ) > f \\left ( r _ 1 \\right ) & r _ 1 ^ { L } - \\sum _ { i = 1 } ^ { L } c _ { i } r _ 1 ^ { L - i } + r _ 1 ^ { m } - 1 > r _ 1 ^ { L } - \\sum _ { i = 1 } ^ { L } c _ { i } r _ 1 ^ { L - 1 } \\\\ & r _ 1 ^ { m } - 1 > 0 \\\\ & r _ 1 > 1 . \\\\ \\end{array} \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} \\begin{pmatrix} S _ { n + 1 } ( a ) \\cr V _ { n } ( a ) \\end{pmatrix} = \\begin{pmatrix} a & a ^ 2 + 1 \\cr 1 & a \\end{pmatrix} ^ { n } \\begin{pmatrix} a \\cr 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} | \\partial _ { r } v _ { 4 } ( t , r ) | \\leq \\begin{cases} \\frac { C } { t ^ { 2 } \\log ^ { 3 b + 2 N - 1 } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { \\sqrt { r } t \\log ^ { 3 b - 1 + \\frac { 5 N } { 2 } } ( t ) } , r \\geq \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} \\mathfrak h _ \\infty ^ \\perp = \\mathfrak p _ u \\oplus ( \\mathfrak d ^ \\perp \\cap \\mathfrak c ) \\oplus \\bigoplus \\limits _ { \\Omega \\in \\widetilde \\Psi } \\mathfrak u _ \\infty ( \\Omega ) \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { k = 0 } ^ { n - 1 } \\sum _ { j = 0 } ^ \\infty \\alpha _ { j , k } m _ { j , k } ( z ) , \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} \\int _ { D \\cap V } M _ D ( x , y ) \\eta _ n ( d y ) \\to \\int _ { D ^ * \\cap V } M _ D ( x , y ) \\mu ^ * ( d y ) = \\int _ { \\partial ^ * D } M _ D ( x , y ) \\mu ^ * ( d y ) . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} X _ { ( P _ 3 , ( 2 , 1 , 2 ) ) } & = X _ { ( P _ { ( 3 , 1 ) } , ( 2 , 1 , 1 , 1 ) ) } - X _ { ( P _ 4 , ( 2 , 1 , 1 , 1 ) ) } \\\\ & = X _ { P _ { ( 1 , 3 , 1 ) } } - X _ { P _ { ( 4 , 1 ) } } - X _ { P _ { ( 1 , 4 ) } } + X _ { P _ 5 } \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} f ( s ) + \\int _ { \\mathbb { R } } \\frac { \\Im G _ { \\mu _ { 1 } } ( s + i f ( s ) ) } { | 1 - t G _ { \\mu _ { 1 } } ( s + i f ( s ) ) | ^ { 2 } } \\ , ( 1 + t ^ { 2 } ) d \\sigma ( t ) = 0 , s \\in J . \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} \\lambda ^ { \\ast ^ { G / H } \\ast ^ { G / H } } = \\lambda , \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} \\ast _ { \\mathbb { R } ^ 3 } d \\phi = \\alpha . \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} \\mathbb { P } _ \\eta ( \\mathcal { G } ^ c ) \\leq e ^ { - c _ 3 ' } \\mathbb { E } _ \\eta \\left ( \\prod _ { k = d \\lfloor \\alpha t \\rfloor + 1 } ^ { \\lfloor \\beta t \\rfloor } ( \\mathbb { 1 } _ { \\mathcal { U } _ k } \\mathbb { 1 } _ { \\mathcal { G } _ { k } ^ c } ) \\right ) , \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = - c _ 2 \\} = \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} \\mathsf { r o w s u m } _ 1 ( M ) & : = \\mbox { t h e s u m o f e n t r i e s i n t h e f i r s t r o w o f } M , \\\\ \\mathsf { n e } ( M ) & : = \\mbox { t h e n u m b e r o f w e a k l y n o r t h - e a s t c e l l s o f } M , \\\\ \\mathsf { t r } ( M ) & : = \\mbox { t h e n u m b e r o f n o n - z e r o c e l l s o n t h e m a i n d i a g o n a l o f } M . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} x = P _ M x + j _ { p ^ * } \\left ( \\Pi _ { M ^ \\perp } ^ { p ^ * } j _ p ( x ) \\right ) . \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} \\sigma _ { 1 } = \\left ( \\overline { \\varphi } \\right ) _ { 1 } + \\gamma _ { 1 } . \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} W = M E V . \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} ( w , v ) _ { \\mathcal { T } _ h } & : = \\sum _ { K \\in \\mathcal { T } _ h } ( w , v ) _ K = \\sum _ { K \\in \\mathcal { T } _ h } \\int _ K w v , \\\\ \\left \\langle \\zeta , \\rho \\right \\rangle _ { \\partial \\mathcal { T } _ h } & : = \\sum _ { K \\in \\mathcal { T } _ h } \\left \\langle \\zeta , \\rho \\right \\rangle _ { \\partial K } = \\sum _ { K \\in \\mathcal { T } _ h } \\int _ { \\partial K } \\zeta \\rho . \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} \\partial _ t \\mu _ t + \\nabla \\cdot ( v _ t \\mu _ t ) = 0 , \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} \\frac { 1 6 - \\mu } { 2 } = \\frac { - 3 \\mu + 1 1 2 } { 2 } , \\mbox { h e n c e } \\mu = 4 8 , \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda _ D & = ( b _ i - d _ i ) ^ { - 1 } ( c _ i - d _ i ) , \\ i = 1 , 2 , \\\\ \\lambda _ E & = ( c _ i - e _ i ) ^ { - 1 } ( a _ i - e _ i ) , \\ i = 1 , 2 , \\\\ \\lambda _ F & = ( a _ i - f _ i ) ^ { - 1 } ( b _ i - f _ i ) , \\ i = 1 , 2 . \\end{aligned} \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} \\gamma _ { - 1 } ( B ( x , R ) ) & = \\int _ { | u | < R } e ^ { | u + x | ^ 2 } \\ , d u \\geq e ^ { | x | ^ 2 + R ^ 2 / 4 } \\int _ { \\varphi _ 0 } ^ { \\pi / 2 } \\int _ { R / 2 } ^ R r ^ { n - 1 } e ^ { 2 r | x | \\cos \\varphi _ 1 } \\sin \\varphi _ 1 \\ , d r \\ , d \\varphi _ 1 \\\\ & = C \\frac { e ^ { | x | ^ 2 + R ^ 2 / 4 } } { | x | } \\int _ { R / 2 } ^ R r ^ { n - 2 } \\int _ 0 ^ { 2 r | x | \\cos \\varphi _ 0 } e ^ u \\ , d u \\ , d r , \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} f _ 1 \\\\ f _ 2 \\end{matrix} \\right ) \\rightarrow \\cal { F } = \\int _ 0 ^ \\infty f _ 1 ( x ) \\phi ( x , k ) d x + \\int _ 0 ^ \\infty f _ 2 ( x ) \\psi ( x , k ) d x \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} \\lambda _ { 1 } = { \\lambda ' } _ { N + 1 } , \\quad \\tau ( 1 ) \\equiv \\sigma ( 1 ) , 0 \\neq \\gamma _ { N + 1 , 1 } \\overline { \\gamma _ { \\tau ' \\left ( N + 1 \\right ) \\tau ( 1 ) } } = \\gamma _ { N + 1 , 1 } \\gamma _ { \\sigma ' \\left ( N + 1 \\right ) \\tau ( 1 ) } = G ^ { \\left ( N + 1 \\right ) } _ { 0 ; 1 , 0 , \\dots , 0 , \\dots , 0 } ( z ) , \\quad \\gamma _ { N + 1 , 1 } = \\overline { \\gamma _ { N + 1 , 1 } } . \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} \\frac { 1 } { r ^ { 2 } } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho \\lambda '' ( s ) } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) = \\frac { - v _ { 3 } ( t , r ) } { r } \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} \\Psi ( z ) = \\frac { \\Phi ( \\eta _ { \\mu _ { 1 } } ( z ) ) } { \\eta _ { \\mu _ { 1 } } ( z ) } z , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 , \\ N ( t ) = 2 k \\} = \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} e _ { x y } B ( e _ { z w } , e _ { u v } ) + B ( e _ { x y } , e _ { u v } ) e _ { z w } & = e _ { x y } B ( e _ { y w } , e _ x ) + B ( e _ { x y } , e _ x ) e _ { y w } . \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} ( G ^ { ( k ) } ) = ( G _ k ) + 1 = m = ( G ) . \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} | d H _ { \\sigma } / d z | & = \\left | \\int _ { \\mathbb { T } } \\frac { 2 t } { ( t - z ) ^ { 2 } } \\ , d \\sigma ( t ) \\right | \\le 2 \\int _ { \\mathbb { T } } \\frac { d \\sigma ( t ) } { | t - z | ^ { 2 } } . \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\nu _ t } - a ( t ) u = 0 . \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} x _ 0 : = x , x _ { 2 n + 1 } : = ( I - P _ M ) x _ { 2 n } \\quad x _ { 2 n } : = ( I - P _ N ) x _ { 2 n - 1 } \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} f _ { * } ( r ) = f ( 1 / r ) , h _ { * } ( r ) = \\frac { 1 } { h ( 1 / r ) } , r \\in ( 0 , + \\infty ) . \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } & \\partial _ t u + u \\cdot \\nabla u - \\nu \\partial ^ 2 _ 1 u = - \\nabla p + \\theta e _ 2 , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ x \\in { \\mathbb { R } ^ 2 } , ~ t > 0 \\\\ & \\partial _ t \\theta + u \\cdot \\nabla \\theta = 0 , \\\\ & \\nabla \\cdot u = 0 , \\\\ & u ( 0 , x ) = u _ 0 ( x ) , \\theta ( 0 , x ) = \\theta _ 0 ( x ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} T ( K , U ) = T ( K , \\psi ^ { - 1 } ( U ^ \\prime ) ) = \\pi ^ { - 1 } ( T ( K , U ^ \\prime ) ) \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} v _ { 1 } ( t , r ) = r \\int _ { t } ^ { \\infty } \\frac { \\lambda '' ( s ) } { 1 + s - t } d s + \\emph { E r r } ( t , r ) \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} \\varphi = \\varphi _ 0 + \\sum _ { j = 1 } ^ \\infty { \\varphi _ j ^ 1 } + \\sum _ { j = 1 } ^ \\infty { \\varphi _ j ^ 2 } , \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} M _ { G , H , I } = \\left ( \\begin{array} { c c c c } \\frac { n _ 1 + 2 } { 2 } & 0 & - \\frac { n _ 1 + 2 } { 2 } & 0 \\\\ 0 & \\frac { n _ 2 + 1 } { 2 } & 0 & - \\frac { n _ 2 + 1 } { 2 } \\\\ - \\frac { n _ 1 + 2 } { 2 } & 0 & \\frac { n _ 1 + 2 } { 2 } & 0 \\\\ 0 & - \\frac { n _ 2 + 1 } { 2 } & 0 & \\frac { n _ 2 + 1 } { 2 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} \\begin{aligned} G \\left ( u ( x _ 0 ) - u ( y ) \\right ) - G \\left ( v ( x _ 0 ) - v ( y ) \\right ) & \\leq 0 , \\forall y \\in \\mathbb { R } ^ n , \\\\ G \\left ( u ( x _ 0 ) - u ( y ) \\right ) - G \\left ( v ( x _ 0 ) - v ( y ) \\right ) & < 0 , \\forall y \\in \\mathbb { R } ^ n \\setminus \\Omega . \\end{aligned} \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} \\tilde { \\tilde { x } } & = - \\tilde { x } , \\\\ \\tilde { \\tilde { y } } & = - \\left ( \\tilde { y } - \\tilde { \\gamma } \\right ) , \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} d - \\frac { 2 d } { p + 1 } - \\frac { d } { 2 } = \\frac { d } { 2 } \\left ( 1 - \\frac { 4 } { p + 1 } \\right ) , \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} \\mathbb { T } ( > \\kappa _ n ) = \\prod _ { m > n } { \\mathbb { T } ( \\kappa _ m ) } \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} Y _ n \\coloneqq \\bigcap _ { j = i } ^ n \\bigcap _ { k = 0 } ^ n f _ { i j } ( Z _ { j k } ) \\subset X _ i ( K ) . \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} s _ { T , a } : = \\int _ 0 ^ T u _ { g , h , T } ( s ) d s . \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 , \\ N ( t ) = 4 \\} = \\frac { 1 } { \\bigl [ ( c _ 1 + c _ 2 ) t \\bigr ] ^ { 4 } } \\cdot \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} H _ { k + m + 1 } \\leq H _ { L - 1 } + H _ { k + 1 } + \\sum _ { a = 1 } ^ { k } \\sum _ { i = a } ^ { a + m - 1 } H _ { i } + N H _ { k - 1 } . \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} & d ( 0 , 0 ) = d ( 1 , 1 ) = d ( * , * ) = 0 , \\\\ & d ( 0 , * ) = d ( 1 , * ) = 1 , ~ ~ ~ ~ d ( 0 , 1 ) = \\infty . \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} \\frac { d } { d t } r ( k , t ) = - \\sum _ { i = 1 } ^ { 2 } c _ { i } \\{ ( \\lambda _ { i } + \\mu ( 1 - B ) ) ^ { \\alpha _ { i } } - \\lambda _ { i } ^ { \\alpha _ { i } } \\} r ( k , t ) , \\ ; \\alpha _ { i } \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N } \\pi _ { i , N } : = \\sum _ { i = 1 } ^ { N } \\min \\left \\{ c _ { N } \\frac { w ( X _ { i } ) } { \\sum _ { j = 1 } ^ { N } w ( X _ { j } ) } ; 1 \\right \\} = n _ { N } . \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} U _ { N \\pm 1 } ( x _ { i } , t ) = U _ N ( x _ { i } , t ) , i = 0 , . . . , N \\pm 1 , \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta | \\nabla \\psi | ^ 2 & \\ge | \\mathrm { H e s s } _ { \\psi } | ^ 2 + \\langle \\nabla \\Delta \\psi , \\nabla \\psi \\rangle + K | \\nabla \\psi | ^ 2 \\\\ & \\ge \\frac { ( \\mathrm { t r } ( \\mathrm { H e s s } _ { \\psi } ) ) ^ 2 } { n } + \\langle \\nabla \\Delta \\psi , \\nabla \\psi \\rangle + K | \\nabla \\psi | ^ 2 \\\\ & = \\frac { ( \\Delta \\psi ) ^ 2 } { n } + \\langle \\nabla \\Delta \\psi , \\nabla \\psi \\rangle + K | \\nabla \\psi | ^ 2 . \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} P _ { n + 1 } & ( x , x _ { n + 1 } ) = \\tfrac { 1 } { 6 } \\sum _ { i = 1 } ^ { n } ( x _ { n + 1 } ^ { 3 } - 3 x _ { n + 1 } x _ { i } ^ { 2 } ) + \\sqrt { \\tfrac { n + 2 } { n } } P _ { n } ( x ) \\\\ & = \\tfrac { n + 3 } { 6 } \\left ( x _ { n + 1 } ^ { 3 } - \\tfrac { 3 } { n + 3 } x _ { n + 1 } E _ { n + 1 } ( x , x _ { n + 1 } ) \\right ) + \\sqrt { \\tfrac { n + 2 } { n } } P _ { n } ( x ) . \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{gather*} \\mathcal { Q } ^ Y = \\big \\{ v + s e _ y : v \\in Y _ 1 \\cap B _ { R _ 2 } , \\ , s \\in [ 0 , 1 ] \\big \\} , \\end{gather*}"}
-{"id": "1347.png", "formula": "\\begin{align*} X _ { ( P _ { \\ell ( \\alpha ) } , \\alpha ) } & = \\sum _ { \\beta \\succcurlyeq \\alpha ^ c } ( - 1 ) ^ { \\ell ( \\alpha ^ c ) - \\ell ( \\beta ) } X _ { P _ { \\beta } } \\\\ & = \\sum _ { \\beta \\succcurlyeq \\alpha ^ c } ( - 1 ) ^ { \\ell ( \\alpha ^ c ) - \\ell ( \\beta ) } X _ { P _ { \\widetilde \\beta } } \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} s _ { 0 , 0 } s _ { 1 , 1 } - s _ { 0 , 1 } s _ { 1 , 0 } = s _ { 0 , 0 } s _ { 1 , 2 } - s _ { 1 , 0 } s _ { 0 , 2 } = 0 \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} \\alpha _ 0 \\widehat { h } _ 0 = 0 ; \\alpha _ 0 \\widehat { h } _ 0 ( k ) = 0 , 1 \\leq k < p - 1 . \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} p _ { \\nu _ { k } } ( \\xi ) = \\frac { 1 - R ( t ) ^ { \\frac { 2 k } { k - 1 } } } { 2 \\pi \\left | 1 - \\eta _ { \\nu } \\left ( R ( t ) t \\right ) \\right | ^ { 2 } } , \\quad t \\in \\mathbb { T } , \\ ; \\eta _ { \\nu } \\left ( R ( t ) t \\right ) \\neq 1 . \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} u _ { x y } = \\frac { \\sqrt { 1 - { u _ x } ^ 2 } \\sqrt { 1 - { u _ y } ^ 2 } } { \\sin u } v _ { x y } = 0 , \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} K ( \\Psi ; M ; \\gamma ) = K ( \\Psi ' ; M ' ; \\gamma ) \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} \\sigma _ k : = F ' _ k ( D u _ k ) \\in \\mathcal { S } _ - ^ { p ' } ( \\Omega ) \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} N ( \\mathbf { g } ( X ) ) = \\lambda N ( X ) , \\lambda \\in \\mathbb { R } , \\forall \\ \\mathbf { g } \\in S t r ( \\mathfrak { J } ) ; \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} \\sup _ { \\beta _ 0 , \\beta _ 1 , \\eta } \\{ \\ell _ { n , 2 } ^ * ( \\beta _ 0 , \\beta _ 1 , \\eta ) - \\ell _ { n , 2 } ^ * ( 0 , 0 , 1 ) \\} = O _ p ( 1 ) . \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} & \\iint _ { \\mathbb { R } ^ 3 \\ ! \\times \\mathbb { R } ^ 3 } \\left [ ( \\bar { f } \\varphi ) ( t ) - ( \\bar { f } \\varphi ) ( 0 ) \\right ] d y ' d w ' \\\\ & = \\int _ 0 ^ t \\ ! \\iint _ { \\mathbb { R } ^ 3 \\ ! \\times \\mathbb { R } ^ 3 } \\Big \\{ \\bar { f } \\big [ ( \\partial _ s + w ' \\ ! \\cdot \\ ! \\nabla _ { \\ ! y ' } ) \\varphi - \\mathbb { B } \\cdot \\nabla _ { \\ ! w ' } \\varphi \\big ] - \\nabla _ { \\ ! w ' } \\bar { f } \\cdot ( \\mathbb { A } \\nabla _ { \\ ! w ' } \\varphi ) \\Big \\} d y ' d w ' d s \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} ( \\mathsf { r o w s u m } _ 1 , \\mathsf { n e } , \\mathsf { t r } ) M = ( \\mathsf { r o w s u m } _ 1 , \\mathsf { t r } , \\mathsf { n e } ) ( \\phi \\circ \\Phi \\circ \\phi ^ { - 1 } ) M , \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} H _ { } ( x ) & : = H ( x ) - \\frac { 1 } { 2 } \\sum _ { l = 1 } ^ { M } \\sum _ { n \\neq l } \\sum _ { \\substack { i \\in B ( l ) \\\\ j \\in B ( n ) } } M _ { i j } x _ i x _ j . \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} \\boldsymbol { B } ( \\psi , \\psi ) = \\boldsymbol { B } ( \\psi _ 0 , \\psi _ 0 ) + \\sum _ { j = 1 } ^ { \\infty } \\boldsymbol { B } ( \\psi _ j ^ 1 , \\psi _ j ^ 1 ) + \\sum _ { j = 1 } ^ { \\infty } \\boldsymbol { B } ( \\psi _ j ^ 2 , \\psi _ j ^ 2 ) . \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} { \\mathbb T } _ { T M } ^ h ( \\omega ) = S _ { \\epsilon ^ { - 1 } } ^ h + i A _ { \\epsilon ^ { - 1 } } ^ { 1 , h } - i A _ { \\epsilon ^ { - 1 } } ^ { 2 , h } + | { \\boldsymbol k } | ^ 2 M _ { \\epsilon ^ { - 1 } } ^ h - \\left ( \\frac { \\omega } { c } \\right ) ^ 2 M ^ h . \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} \\partial _ { w _ { 0 , a , b } } ( P ) = \\partial _ { w _ { 0 , a , b } } [ \\partial _ { w _ { 0 , a } } ( Q _ 0 ) \\partial _ { w _ { 0 , b } } ( R _ 0 ) ] = \\partial _ { w _ { 0 , n } } ( Q _ 0 R _ 0 ) . \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} \\bar { d } _ E ( F ) = \\limsup _ { N - M \\to \\infty } \\frac { | \\{ n _ i : i \\in \\N \\} \\cap \\{ M + 1 , M + 2 , \\ldots , N \\} | } { N - M } \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} \\mathcal { H } _ N ( m ) & : = \\sup _ { \\sigma \\in \\mathbb { R } } \\left ( \\sigma m - A _ N ( \\sigma ) \\right ) , \\\\ \\varphi ( m ) & : = \\sup _ { \\sigma \\in \\mathbb { R } } \\left ( \\sigma m - A ( \\sigma ) \\right ) . \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} H ^ \\bullet _ { D S , f } ( M ) : = H ^ { \\bullet } ( C ( M ) , Q _ { ( 0 ) } ) \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c \\} = e ^ { - \\lambda t } \\Bigl [ I _ 0 ( \\lambda t ) + I _ 1 ( \\lambda t ) \\Bigr ] \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} \\mathcal { R } _ { 1 , 2 } ( t ) = \\int _ 0 ^ t \\int _ 0 ^ \\xi \\mathcal { B } \\left ( F ( v ) \\cdot e ^ { ( \\xi - s ) \\mathcal { A } } \\mathcal { C } [ G , \\mathcal { A } ] \\left ( e ^ { s \\mathcal { A } } \\overline v \\right ) \\right ) d s d \\xi . \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} \\beta _ { E , A , p } ( R ) : = \\ell ( R ) ^ { d } + \\sum _ { Q \\subseteq R } \\beta _ { E } ^ { d , p } ( A B _ Q ) ^ { 2 } \\ell ( Q ) ^ { d } . \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} { } ^ b { \\rm T r } _ { \\chi } \\left ( [ \\Phi _ { A _ { 1 } } , \\Phi _ { A _ { 2 } } ] \\right ) = \\frac { i } { 2 \\pi } \\int _ { \\mathbb { R } } \\int _ G { \\rm T r } _ { \\partial S } \\left ( \\frac { \\partial I ( \\Phi _ { A _ 1 } , h ^ { - 1 } , \\lambda ) } { \\partial \\lambda } \\circ I ( \\Phi _ { A _ 2 } , h , \\lambda ) \\right ) d h d \\lambda . \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix*} [ r ] A _ c k _ c + g _ c \\circ K { } { } - r - k _ c \\circ ( A _ c + r ) = 0 , \\\\ A _ u k _ u + g _ u \\circ K { } { } - k _ u \\circ ( A _ c + r ) = 0 , \\\\ A _ s k _ s + g _ s \\circ K { } { } - k _ s \\circ ( A _ c + r ) = 0 . \\end{matrix*} \\right . \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} H ( x , 0 ) F ( x , 0 ) = F ( x , 0 ) \\prod _ { j = 1 } ^ n ( x - b _ j ) ^ m = x ^ q + c _ 1 x ^ { q - 1 } + c _ 2 x ^ { q - 2 } + \\cdots + c _ q . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} \\hat { A } _ { \\rm c y l } = \\left ( \\frac { 2 a _ n } { a _ { n + 1 } } \\right ) ^ { \\frac { 1 } { n + 2 } } \\left ( \\frac { n + 1 } { n + 2 } \\right ) ^ { \\frac { n + 1 } { n + 2 } } \\left ( \\frac { L } { r } \\right ) ^ { \\frac { 1 } { n + 2 } } . \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} A _ { n + p } \\ = \\ \\{ N n + j : j \\in B _ p \\} \\ p \\in [ m ] . \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} \\pi \\left ( f ( \\tau ) \\right ) : = \\mbox { m i n } _ { n \\geq n _ 0 } \\left \\{ \\pi ( a ( n ) \\right \\} . \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} \\langle \\bullet , \\bullet \\rangle = \\langle \\bullet , \\bullet \\rangle _ { L ^ { 2 } ( \\mathbb { R } _ { + } ^ { n } , x _ { n + 1 } ^ { 1 - 2 s } \\tilde { A } ) } \\quad \\quad \\| \\bullet \\| = \\| \\bullet \\| _ { L ^ { 2 } ( \\mathbb { R } _ { + } ^ { n } , x _ { n + 1 } ^ { 1 - 2 s } \\tilde { A } ) } . \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} & | | \\sqrt { \\omega } \\lambda ( x ) K ( \\partial _ { 1 } y ( x , \\frac { \\cdot } { \\lambda ( x ) ^ { 2 } } ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } \\\\ & \\leq C \\left ( | | \\partial _ { 1 } y ( x ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } + | | \\sqrt { \\omega } \\lambda ( x ) \\partial _ { 1 } y ( x ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } \\right ) \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} f _ j - ( x - 1 ) ^ { \\omega _ j - \\omega _ i } f _ i ( x ) h _ { i 1 } ^ { - 1 } ( x ) h _ { j 1 } ( x ) = u ^ 3 ( x - 1 ) ^ { \\tau _ k } \\tilde h _ k ( x ) , \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} & \\frac { 1 6 } { \\lambda ( t ) } \\int _ { 0 } ^ { \\infty } \\frac { R ^ { 3 } } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\int _ { t + 6 R \\lambda ( t ) } ^ { \\infty } d s \\lambda '' ( s ) \\left ( \\frac { s - t } { 1 + ( s - t ) ^ { 2 } } - \\frac { ( s - t ) } { \\lambda ( t ) ^ { 2 - 2 \\alpha } + ( s - t ) ^ { 2 } } \\right ) \\\\ & = \\frac { 1 6 } { \\lambda ( t ) } \\int _ { t } ^ { \\infty } K _ { 3 } ( s - t , \\lambda ( t ) ) \\lambda '' ( s ) d s \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} t = \\frac { - 2 } { b _ { 0 R } } \\ , r + S , \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} | p _ N | & = \\frac { 1 } { \\sin ( \\theta ) } | a _ 1 p _ 1 \\sin ( \\theta ) + a _ 2 p _ 2 \\sin ( 2 \\theta ) + \\ldots + a _ { N - 1 } p _ { N - 1 } \\sin ( ( N - 1 ) \\theta ) | \\\\ & = | a _ 1 p _ 1 U _ 1 + a _ 2 p _ 2 U _ 2 + \\ldots + a _ { N - 1 } p _ { N - 1 } U _ { N - 1 } | \\\\ & \\leq | a _ 1 U _ 1 ^ 2 + a _ 2 U _ 2 ^ 2 + \\ldots + a _ { N - 1 } U _ { N - 1 } ^ 2 | + 2 C \\sum _ { i = 1 } ^ { N - 1 } | a _ i U _ i | \\\\ & \\leq \\frac { C } { N } + 2 C \\frac { C } { N ^ 3 } N ^ 2 = \\frac { C ' } { N } . \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} \\sigma _ { \\overline { 1 } } = \\varphi _ { \\overline { 1 } } + \\gamma _ { \\overline { 1 } } . \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} J _ T ( x _ 0 , u ) = \\int _ 0 ^ T \\omega ( x ( t ; x _ 0 , u ) ) \\ , d t \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} \\rho = \\int _ { \\R ^ 3 } f ( v ) \\ , d v , u = \\frac { 1 } { \\rho } \\int _ { \\R ^ 3 } v \\ , f ( v ) \\ , d v , T = \\frac { 1 } { 3 \\rho } \\int _ { \\R ^ 3 } | v - u | ^ 2 \\ , f ( v ) \\ , d v . \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} 2 ^ { g + k + 1 } - \\sum _ { i = k + n + 2 } ^ { g + k + 1 } H _ i \\leq 2 ^ g + 2 ^ { k + n + 2 } - 2 ^ { n + 1 } \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} c _ { 1 } = 1 + b _ { 1 } a _ { 1 } = \\frac { ( a _ { 0 } ^ { 2 } + \\lambda a _ { 1 } ) \\beta ^ { 2 } - 2 a _ { 0 } \\beta + 1 } { ( 1 - \\beta a _ { 0 } ) ^ { 2 } } . \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} C _ { p , \\nu } = 2 ^ { \\nu } \\max \\left \\{ [ 1 + p ] ^ p , \\frac { 3 ^ p \\nu ^ \\nu } { p ^ p ( \\nu - p ) ^ { \\nu - p } } \\right \\} . \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( \\overline { \\xi } ) = R ( t ) t \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} \\mathbb { P } _ { n _ { l } } \\left [ - M \\le \\log \\left ( Y _ { n _ { l } } \\right ) \\le M \\cap - M \\le \\left \\{ \\sum _ { i = 1 } ^ { k } \\frac { 2 \\mu _ { i } W _ { n _ { l } , i } - \\mu _ { i } ^ 2 } { 2 \\sigma _ { i } ^ 2 } \\right \\} \\le M \\right ] \\ge 1 - \\frac { \\delta } { 5 0 } . \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\dot { v } & = q ( v ) - w + I , \\\\ \\dot { w } & = b ( v - c w ) , \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} \\| \\lambda \\| _ { M ( X ) } : = | \\lambda | ( X ) , \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} \\left ( \\prod _ { j = 1 } ^ { m - 1 } Z _ j , T _ m \\right ) \\stackrel { \\cong } { \\longrightarrow } \\left ( \\prod _ { j = 1 } ^ { m - 1 } ( Z _ j , T _ m ) \\right ) , \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} H \\left ( \\Phi \\left [ \\gamma ( L _ n ) \\right ] \\right ) - H \\left ( \\gamma ( L _ n ) \\right ) = \\log | \\det K | _ p . \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} \\sum _ { i + j = n + 1 } ^ { } \\sum _ { \\sigma } ^ { } ( - 1 ) ^ \\sigma \\epsilon ( \\sigma ) ~ ( - 1 ) ^ { i ( j - 1 ) } ~ l _ j \\big ( l _ i ( x _ { \\sigma ( 1 ) } , \\ldots , x _ { \\sigma ( i ) } ) , x _ { \\sigma ( i + 1 ) } , \\ldots , x _ { \\sigma ( n ) } \\big ) = 0 , \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} g ^ { i j } \\ , \\Pi \\left ( F _ { s i j } \\right ) & = g ^ { i j } \\ , \\nabla _ { F _ i } ^ { \\perp } \\nabla _ { F _ j } ^ { \\perp } V - g ^ { i j } \\ , \\nabla _ { F _ i } ^ { \\perp } \\left ( F _ { \\ell } \\ , g ^ { \\ell k } \\langle F _ { j k } , V \\rangle \\right ) \\\\ & = g ^ { i j } \\ , \\nabla _ { F _ i } ^ { \\perp } \\nabla _ { F _ j } ^ { \\perp } V - g ^ { i j } \\ , A _ { i \\ell } \\ , g ^ { \\ell k } A ^ V _ { j k } \\ , . \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} F ( z ) = - \\alpha + \\beta z + \\int _ { ( 0 , + \\infty ) } \\left [ \\frac { 1 + z t } { t - z } - \\frac { 1 } { t } \\right ] \\ , d \\rho ( t ) , \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} u ^ { ( s ) } ( r , \\cdot ) \\begin{cases} = 0 & r \\geq r _ 2 \\\\ > 0 & r < r _ 2 \\end{cases} ; \\ \\ \\ \\ \\ \\int _ { \\Sigma } ( u ^ { ( s ) } ) ^ 2 \\cdot \\alpha = 1 \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} \\theta = [ \\mu _ i , \\sigma _ { i j } ] ^ \\top _ { i , j \\in \\{ 1 , \\ldots , d \\} , i \\le j } , Z ( \\theta ) = S ( \\theta ) ^ { \\frac { 1 } { \\alpha - 1 } } N _ { \\theta , \\alpha } , h ( { \\bf { x } } ) \\equiv 1 , w ( \\theta ) = \\big [ w ^ { ( 1 ) } ( \\theta ) , w ^ { ( 2 ) } ( \\theta ) \\big ] ^ \\top , f ( { \\bf { x } } ) = \\big [ f ^ { ( 1 ) } ( { \\bf { x } } ) , f ^ { ( 2 ) } ( { \\bf { x } } ) \\big ] ^ \\top , \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} u ^ { \\ell + 1 } = e ^ { \\tau \\mathcal { L } } \\Big ( u ^ \\ell + \\tau f ( u ^ \\ell , \\overline u ^ \\ell ) \\Big ) . \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} L G _ \\leftrightarrow = 0 , G _ \\leftrightarrow L = 0 , G _ \\leftrightarrow ( x , y ) = - G _ \\leftrightarrow ( y , x ) . \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} \\int _ { \\mathbb T ^ d _ { \\ell } } R | \\mathbb D U | ^ 2 - \\int _ { \\mathbb T ^ d _ { \\ell } } R \\nabla U : \\nabla ^ { \\top } U = \\int _ { \\mathbb T ^ d _ { \\ell } } R | \\mathbb A U | ^ 2 . \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} \\Big \\| | x | ^ { - \\frac s 2 } u ^ m \\Big \\| _ 2 ^ 2 = \\int _ \\Omega | x | ^ { - s } ( u ^ m ) ^ { 2 } d x \\le C \\Big ( \\int _ \\Omega | \\nabla u ^ m | ^ { \\frac { 2 N } { N + 2 - s } } d x \\Big ) ^ { \\frac { N + 2 - s } { N } } . \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} I _ \\Omega : = \\left | \\Pi \\vec { g } _ h \\right | \\approx \\bigg | \\frac { 1 } { 2 \\pi i } \\sum \\limits _ { j = 1 } ^ { m _ 0 } w _ j \\vec { x } _ h ( \\omega _ j ) \\bigg | , \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} \\lambda ^ { \\ast ^ { G / H } \\ast ^ { G / H } } ( \\psi ) & = \\overline { \\lambda ^ { \\ast ^ { G / H } } ( \\psi ^ { \\ast ^ { G / H } } ) } \\\\ & = \\lambda ( \\psi ^ { \\ast ^ { G / H } \\ast ^ { G / H } } ) \\\\ & = \\lambda ( J \\psi ) = \\lambda ( J _ 0 \\psi ) . \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} D _ j ^ { ( 1 ) } & : = \\left ( ( x _ i , y _ i ) _ { i \\in \\lbrace 1 , \\ldots , l \\rbrace } : x _ i \\in A _ j \\right ) , j \\in \\lbrace 1 , \\ldots , m _ n \\rbrace , \\\\ D _ j ^ { ( 2 ) } & : = \\left ( ( x _ i , y _ i ) _ { i \\in \\lbrace l + 1 , \\ldots , n \\rbrace } : x _ i \\in A _ j \\right ) , j \\in \\lbrace 1 , \\ldots , m _ n \\rbrace , \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} \\beta _ { 0 } = \\left ( a _ { 0 } + \\sqrt { \\lambda \\left | a _ { 1 } \\right | } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} \\| g \\| = \\| L _ { \\phi , \\psi } G _ { u , v } g \\| \\geq c \\| G _ { u , v } g \\| \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} \\sigma _ { A , \\Phi } ( \\xi , \\eta ) = \\sum _ { \\mu , \\nu \\in \\Z ^ n } a _ { \\mu , \\nu } \\Phi ( \\xi - \\mu , \\eta - \\nu ) , \\xi , \\eta \\in \\R ^ n . \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} S _ 1 = & \\bigl \\{ \\phi _ { \\tilde { c _ 1 } } ( \\xi _ 1 , \\eta _ 1 ) \\in \\R ^ 3 \\ | \\ ( \\xi _ 1 , \\eta _ 1 ) \\in \\mathcal { T } _ { \\tilde { k _ 1 } } ^ { N _ 1 ^ { - 1 } A ' } \\bigr \\} , \\\\ S _ 2 = & \\bigl \\{ \\phi _ { \\tilde { c _ 2 } } ( \\xi _ 2 , \\eta _ 2 ) \\in \\R ^ 3 \\ | \\ ( \\xi _ 2 , \\eta _ 2 ) \\in \\mathcal { T } _ { \\tilde { k _ 2 } } ^ { N _ 1 ^ { - 1 } A ' } \\bigr \\} . \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} \\Omega : = \\{ \\Gamma \\in [ C ( \\R ) ] ^ m : 0 \\preceq \\Gamma \\preceq \\mathbf { u } ^ * \\} . \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{align*} \\begin{cases} a \\cdot ( b \\cdot u ) - ( b \\cdot u ) \\cdot a + H ( a , T ( b \\cdot u ) ) - H ( T ( b \\cdot u ) , a ) \\\\ = ( a b - b a ) \\cdot u + b \\cdot ( a \\cdot u - u \\cdot a + H ( a , T u ) - H ( T u , a ) ) , \\\\ ( a b - b a ) \\cdot ( a \\cdot u - u \\cdot a + H ( a , T u ) - H ( T u , a ) ) = 0 , ~ b \\in A , u \\in M . \\end{cases} \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} \\mathcal { K } ( x _ 1 , x _ 2 , x _ 3 ) = \\sum _ { k , j \\in { \\Bbb Z } } 2 ^ { - 2 ( k + j ) } \\phi \\Big ( { { x _ 1 } \\over { 2 ^ j } } , { { x _ 2 } \\over { 2 ^ k } } , { { x _ 3 } \\over { 2 ^ { j + k } } } \\Big ) \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} \\Omega _ { \\mu } = \\{ r t : t \\in \\mathbb { T } , r \\in [ 0 , R ( t ) ) \\} , \\ \\Omega _ { \\mu _ { n } } = \\{ r t : t \\in \\mathbb { T } , r \\in [ 0 , R _ { n } ( t ) ) \\} . \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} \\left ( t - 1 \\right ) \\mu \\left ( p _ 1 + p _ { t + 1 } \\right ) - p _ 1 p _ { t + 1 } = \\left ( 1 - 2 t \\right ) \\mu ^ 2 \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} m = \\begin{cases} 1 & { \\rm i f } \\ G \\ { \\rm b e l o n g s \\ t o \\ } \\Phi _ { 1 6 } , \\Phi _ { 3 1 } , \\Phi _ { 3 7 } , \\Phi _ { 3 9 } , \\Phi _ { 4 3 } , \\Phi _ { 5 8 } , \\Phi _ { 6 0 } \\ { \\rm o r } \\ \\Phi _ { 8 0 } , \\\\ 2 & { \\rm i f } \\ G \\ { \\rm b e l o n g s \\ t o \\ } \\Phi _ { 1 0 6 } \\ { \\rm o r } \\ \\Phi _ { 1 1 4 } , \\\\ 3 & { \\rm i f } \\ G \\ { \\rm b e l o n g s \\ t o \\ } \\Phi _ { 3 0 } . \\end{cases} \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 ) = \\sum _ { j , k \\in \\Bbb Z } \\sum _ { R \\in \\R ^ { j , k } _ { \\mathfrak z } } | R | \\psi _ { j , k } ( x _ 1 - x _ I , x _ 2 - x _ J , x _ 3 - x _ S ) ( \\psi _ { j , k } * f ) ( x _ I , x _ J , x _ S ) , \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} J _ 1 = J _ k z ^ 1 _ i = z _ i ^ k i \\in J _ 1 . \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} L ^ \\bullet _ { Y M } & = \\mathrm { T r } \\left [ \\frac 1 2 F _ A \\star F _ A + A ^ \\dag d _ A c + \\frac 1 2 c ^ \\dag [ c , c ] + c d _ A \\star F _ A + \\frac 1 2 A ^ \\dag [ c , c ] + \\frac 1 2 [ c , c ] \\star F _ A \\right ] \\\\ \\theta ^ \\bullet _ { Y M } & = \\mathrm { T r } \\left [ A ^ \\dag \\delta A + c ^ \\dag \\delta c + \\delta A \\star F _ A + A ^ \\dag \\delta c + c \\delta ( \\star F _ A ) \\right ] , \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} x = \\begin{pmatrix} a & z & \\bar { y } \\\\ \\bar { z } & b & x \\\\ y & \\bar { x } & c \\end{pmatrix} ; \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\partial _ { t } u = \\left ( a + i b \\right ) \\left [ \\Delta u + A \\left ( x \\right ) u + V _ { 1 } \\left ( x \\right ) u \\right ] , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} \\begin{cases} \\big ( \\theta + d d ^ c \\tilde \\phi _ t \\big ) ^ n = e ^ { a ( t ) \\partial _ t \\tilde \\phi _ t + \\tilde \\phi _ t } \\mu \\\\ \\tilde \\phi ( 0 , x ) = \\phi _ 0 \\end{cases} . \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} { \\cal L } \\hat { h } _ s \\Big \\vert _ { s = s _ 0 } = h _ 0 ^ n ( n + 1 ) \\frac { \\partial H [ \\hat { h } ( \\ast , s ) ] } { \\partial s } \\Big \\vert _ { s = s _ 0 } = 0 . \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} \\begin{aligned} \\bigg | \\int & \\left ( \\bar { f } ( x , y ) - \\tilde { f } ( x , y ) \\right ) g ( x , y ) d x d y \\bigg | \\\\ & \\leq \\left | \\int \\left ( \\bar { f } ( x , y ) - \\bar { f } _ n ( x , y ) \\right ) g ( x , y ) d x d y \\right | + \\left | \\int \\left ( \\bar { f } _ n ( x , y ) - \\tilde { f } ( x , y ) \\right ) g ( x , y ) d x d y \\right | . \\end{aligned} \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} A x = \\sum _ { i = 1 } ^ \\infty a _ i ( x ) y _ i , \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\frac { B _ { 2 k } } { ( 2 k ) ! } t ^ { 2 k - 2 } \\frac 1 { e ^ { t / a } - 1 } \\ , d t = \\frac { B _ { 2 k } } { ( 2 k ) ! } a ^ { 2 k - 1 } \\int _ 0 ^ \\infty \\frac { t ^ { 2 k - 2 } } { e ^ { t } - 1 } \\ , d t = \\frac { B _ { 2 k } \\zeta _ R ( 2 k - 1 ) } { 2 k ( 2 k - 1 ) } a ^ { 2 k - 1 } . \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow - \\infty } \\mu _ j ( \\lambda ) = \\xi _ j , \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} \\varrho ( \\hat { s } ; \\nu ) = 0 . \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} \\bar { \\Delta } f + ( k - 1 ) \\csc ^ 2 \\theta f & = 0 , \\\\ \\sin ^ 2 \\theta | \\bar { \\nabla } f | ^ 2 + f ^ 2 & = \\sin ^ 2 \\theta . \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} - \\int _ { \\R ^ 2 } \\Delta _ q ( u \\cdot \\nabla f ) \\Delta _ q f ~ d x \\leq & C b _ q 2 ^ { - 2 q s } \\| \\nabla u \\| _ { L ^ \\infty } \\| f \\| _ { H ^ s } ^ 2 , \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} v _ { 4 } ( t , r ) = \\frac { - 1 } { 2 \\pi } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\int _ { 0 } ^ { 2 \\pi } d \\theta \\frac { v _ { 4 , c } ( s , \\sqrt { r ^ { 2 } + 2 r \\rho \\cos ( \\theta ) + \\rho ^ { 2 } } ) } { \\sqrt { r ^ { 2 } + 2 r \\rho \\cos ( \\theta ) + \\rho ^ { 2 } } } \\left ( r + \\rho \\cos ( \\theta ) \\right ) \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} g ( r ) = \\int _ { 0 } ^ { \\infty } \\phi ( r , \\xi ) \\mathcal { F } ( g ) ( \\xi ) \\rho ( \\xi ) d \\xi = \\int _ { 0 } ^ { \\infty } \\frac { \\sqrt { r } } { f ( \\sqrt { \\xi } ) ^ { 2 } } \\mathcal { F } _ { H } ( \\frac { g ( \\cdot ) } { \\sqrt { \\cdot } } ) ( \\sqrt { \\xi } ) \\rho ( \\xi ) \\tilde { \\phi } _ { \\sqrt { \\xi } } ( r ) d \\xi \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} ( I - P _ M ) x = j _ { p ^ * } ( \\Pi ^ { p ^ * } _ { M ^ \\perp } j _ p ( x ) ) . \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} ( c _ { k _ 2 } - c _ { k _ 1 } ) - ( c _ 2 - c _ 1 ) = \\sum _ { j \\in \\alpha ( k _ 1 , k _ 2 ) } n _ j . \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ N ( t ) = n , \\ V ( 0 ) = c _ 1 \\} = \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} u _ 1 ( x _ 1 , x _ 2 ) & = u _ 1 ( x _ 1 , - x _ 2 ) , u _ 2 ( x _ 1 , x _ 2 ) = - u _ 2 ( x _ 1 , - x _ 2 ) , \\\\ p ( x _ 1 , x _ 2 ) & = p ( x _ 1 , - x _ 2 ) , \\quad \\quad \\theta ( x _ 1 , x _ 2 ) = - \\theta ( x _ 1 , - x _ 2 ) \\ , , \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} | \\Phi [ \\varphi ] ( x ) - \\Phi [ \\psi ] ( x ) | & \\leq \\frac 1 { \\rho ( { x } ) } \\int _ { \\rho ( { x } ) } ^ { 2 \\rho ( { x } ) } \\left | \\eta _ { x , \\varphi } ( t ) - \\eta _ { x , \\psi } ( t ) \\right | d t \\\\ & \\leq \\frac 1 { \\rho ( { x } ) } \\int _ { \\rho ( { x } ) } ^ { 2 \\rho ( { x } ) } \\| \\varphi - \\psi \\| d t = \\| \\varphi - \\psi \\| . \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} \\hat { A } : = \\left ( \\begin{array} { c c c c c c c } a _ { 1 i _ 1 } & \\cdots & & \\cdots & & \\cdots & a _ { 1 t } \\\\ 0 & a _ { 2 i _ 2 } & & & & \\cdots & a _ { 2 t } \\\\ \\vdots & & \\ddots & & \\vdots & & \\vdots \\\\ 0 & \\cdots & 0 & \\cdots & a _ { n i _ n } & \\cdots & a _ { n t } \\end{array} \\right ) , \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} g & = g _ 0 ^ 4 + r ^ 2 h = g _ 0 ^ 4 + ( g _ 1 ^ 2 + g _ 3 ^ 2 h ) ^ 2 h \\\\ f & = ( f _ 1 ^ 2 + f _ 3 ^ 2 g ) ^ 2 g \\\\ & = ( f _ 1 ^ 2 + f _ 3 ^ 2 ( g _ 0 ^ 4 + r ^ 2 h ) ) ^ 2 ( g _ 0 ^ 4 + g _ 1 ^ 4 h + g _ 3 ^ 4 h ^ 3 ) \\\\ & = ( ( f _ 1 + f _ 3 g _ 0 ^ 2 ) ^ 4 + ( f _ 3 r ) ^ 4 h ^ 2 ) ( g _ 0 ^ 4 + g _ 1 ^ 4 h + g _ 3 ^ 4 h ^ 3 ) \\\\ & = ( f _ 1 g _ 0 + f _ 3 g _ 0 ^ 3 ) ^ 4 + ( f _ 1 g _ 1 + f _ 3 g _ 0 ^ 2 g _ 1 + f _ 3 g _ 3 r h ) ^ 4 h + ( f _ 3 g _ 0 r ) ^ 4 h ^ 2 + ( f _ 1 g _ 3 + f _ 3 g _ 0 ^ 2 g _ 3 + f _ 3 g _ 1 r ) h ^ 3 . \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} \\left [ \\theta _ k ( \\theta _ k + c _ k - 1 ) - x _ k ( \\theta + a ) ( \\theta + b ) \\right ] f ( x ) = 0 ( k = 1 , \\dots , n ) . \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} F ( \\Phi ^ { ( k ) } ( x ) ) = \\Phi ^ { ( k ) } ( x ) E ^ { ( k ) } ( x ) \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} V \\cong V ( 2 n - 2 ) = \\begin{cases} V ( \\lambda ) & \\lambda \\geq 0 , \\\\ V ( - ( \\lambda + 2 ) ) & \\lambda < 0 . \\end{cases} \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} \\partial ^ 2 ( h _ { i j } ) = \\begin{pmatrix} 0 & 0 & \\bar { a _ { i , i } } & \\bar { a _ { i , j } } \\\\ 0 & 0 & \\bar { a _ { j , i } } & \\bar { a _ { j , j } } \\\\ a _ { i , i } & a _ { i , j } & * & * \\\\ a _ { i , j } & a _ { j , j } & * & * \\end{pmatrix} \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} ( \\mathcal { T } Q ) _ j & : = \\min \\left \\{ Q _ j - { D } _ j , { E } _ j \\right \\} , \\\\ ( \\mathcal { T } { E } ) _ j & : = Q _ { j + 1 } + { E } _ j - ( \\mathcal { T } Q ) _ j , , \\\\ { D } _ j & : = \\min _ { 0 \\leq k \\leq J - 1 } \\sum _ { l = 1 } ^ k \\left ( { E } _ { j - l } - Q _ { j - l } \\right ) . \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} \\delta ( g ^ { i j } \\Gamma _ { i j } ^ k f _ k ) & = - u g ^ { i j } g ^ { k l } h _ { i j , l } f _ k - g ^ { i j } g ^ { k l } u _ i h _ { j l } f _ k - g ^ { i j } g ^ { k l } u _ j h _ { i l } f _ k + g ^ { i j } g ^ { k l } u _ l h _ { i j } f _ k \\\\ & = - 2 u \\langle \\nabla H , \\nabla f \\rangle - 2 \\ , h ( \\nabla { u } , \\nabla { f } ) + 2 H \\langle \\nabla u , \\nabla f \\rangle , \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} ( ( r + w ) ^ { 2 } + 1 ) ( ( r - w ) ^ { 2 } + 1 ) = ( r ^ { 2 } + w ^ { 2 } + 1 ) ^ { 2 } - 4 r ^ { 2 } w ^ { 2 } \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} | E _ { 1 } ( t , r ) | & \\leq \\sup _ { x \\geq t } \\left ( | \\lambda '' ( x ) | \\right ) r ( 1 + r ) ^ { 2 } \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } \\frac { d s } { ( s - t ) ^ { 3 } } \\\\ & \\leq C r \\sup _ { x \\geq t } \\left ( | \\lambda '' ( x ) | \\right ) \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} & \\lim _ { T \\to 0 } r _ 1 = o ( R ^ 2 ) \\mbox { ( a s $ R \\longrightarrow + 0 $ ) } , \\\\ & \\lim _ { T \\to 0 } r _ 2 = o ( R ) \\mbox { ( a s $ R \\longrightarrow + 0 $ ) } \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} & s = k r , \\\\ & B = k A . \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} \\lim _ n \\frac { H ( \\mu _ { \\lambda , \\tau } ^ { ( n ) } ; \\lambda ^ { 1 0 n } ) } { n \\log \\lambda ^ { - 1 } } = \\dim \\mu _ { \\lambda , \\tau } . \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} \\Gamma _ { 1 2 } ^ 1 = \\frac { e ^ { - 2 s _ 2 } } { 1 + e ^ { - 2 s _ 2 } } \\ ; , \\Gamma _ { 1 1 } ^ 2 = - \\frac { e ^ { - 2 s _ 2 } \\left ( 1 - e ^ { - 2 s _ 2 } \\right ) } { \\left ( 1 + e ^ { - 2 s _ 2 } \\right ) ^ 2 } \\ ; , \\ ; \\ ; \\Gamma _ { 2 2 } ^ 2 = - \\frac { e ^ { - 2 s _ 2 } } { 1 - e ^ { - 2 s _ 2 } } . \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} u \\circ ( \\gamma _ x ( t ) ) = u ( 0 ) \\cos t - \\cot \\theta u ( 0 ) \\sin t . \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} U ^ { ( k ) } = A ^ { ( k _ 1 ) } b ^ { ( k ) } B ^ { ( k ) } = A ^ { ( k _ 1 ) } p ^ { ( k _ 1 ) } m ( b ) s ^ { ( k ) } B ^ { ( k ) } , \\ k = k _ 1 , \\ldots , k _ 2 , \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} - \\frac { r _ 0 \\nu } { \\tau ^ 4 } \\int _ { \\mathbb T ^ d _ { \\ell } } U \\cdot \\nabla \\log R = \\frac { \\dd } { \\dd t } \\left [ \\frac { r _ 0 \\nu } { \\tau ^ 2 } \\int _ { \\mathbb T ^ d _ { \\ell } } \\log R \\right ] + \\dfrac { 2 r _ 0 \\nu \\dot { \\tau } } { \\tau ^ 3 } \\int _ { \\mathbb T ^ d _ { \\ell } } \\log R - \\frac { r _ 0 \\nu \\delta _ 1 } { \\tau ^ 4 } \\int _ { \\mathbb T ^ d _ { \\ell } } \\frac { \\Delta R } { R } . \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} ( \\omega _ { 2 } \\rtimes v ) ( \\xi ) \\left ( f \\right ) = \\lambda ^ { - k _ { 1 } l _ { 2 } } z ^ { l _ { 2 } + l _ { 1 } } \\otimes \\varepsilon _ { k _ { 2 } - k _ { 1 } } , \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} Q ( x , y , t ) = \\sum _ { i , j , n \\geq 0 } q _ { i , j } ( n ) x ^ i y ^ j t ^ n \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} z _ { x y } = F ( x , y , z , z _ x , z _ y ) \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} u _ 1 & = ( p + q ) ( 1 - p q ) v \\\\ u _ 2 & = q ( 1 + p ^ 2 ) v \\\\ u _ 3 & = p ( 1 + q ^ 2 ) v \\\\ A & = p q ( p + q ) ( 1 - p q ) v ^ 2 \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} \\{ \\varphi _ p , \\varphi _ n \\} = 0 \\ . \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} \\chi _ 0 & = \\mu ^ { 1 / 2 } , \\\\ \\chi _ j & = v ^ j \\mu ^ { 1 / 2 } , \\ j = 1 , 2 , 3 \\\\ \\chi _ 4 & = \\frac { | v | ^ 2 - 3 } { \\sqrt { 6 } } \\mu ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} \\frac { 1 } { \\psi } + \\frac { 1 } { \\phi _ n } = n . \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} \\langle \\gamma ' , e _ k \\rangle & = \\langle e _ k , \\gamma ' \\rangle = \\langle e _ k , g _ 2 ( e _ j ) \\rangle = \\langle T , g _ 2 ( T ) \\rangle _ { k j } . \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} = \\sum _ { j = 1 } ^ { k } \\binom { 2 k + 1 } { j } \\frac { \\Bigl [ ( c _ 1 t - \\beta ) ^ { j } ( c _ 2 t + \\beta ) ^ { 2 k + 1 - j } - \\Bigl ( \\frac { c _ 2 } { c _ 1 } \\Bigr ) ^ { 2 k + 1 - 2 j } ( c _ 1 t - \\beta ) ^ { 2 k + 1 - j } ( c _ 2 t + \\beta ) ^ { j } \\Bigr ] } { \\bigl [ ( c _ 1 + c _ 2 ) t \\bigr ] ^ { 2 k + 1 } } \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} h _ * : \\pi _ * ^ S X = \\pi _ * Q X \\to H _ * Q X \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} \\widetilde { D _ s f } ( \\xi ) = D _ { \\frac { 1 } { s } } \\widetilde { f } ( \\xi ) \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} G ( x , y , i \\rho ) = G ^ 0 ( x , y , i \\rho ) - k _ 0 ( x , y ) + k _ 1 ( x , y ) + \\ldots \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\hat { F } _ 0 ( u _ n ( x ) ) \\ ; d x = 2 I ( u _ n ) - I ' ( u _ n ) u _ n \\to 2 c n \\to + \\infty . \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} \\widetilde { U } _ i = ( \\widetilde { S q } ^ 0 ) ^ i ( \\lambda _ { 1 9 1 } ( \\lambda _ { 1 5 } ^ 2 \\lambda _ { 3 9 } + \\lambda _ { 3 9 } \\lambda _ { 1 5 } ^ 2 ) \\lambda _ 0 + \\lambda _ { 6 3 } ^ 2 \\lambda _ { 4 7 } \\lambda _ { 8 7 } \\lambda _ 0 + \\lambda _ { 1 2 7 } \\lambda _ { 3 1 } \\lambda _ { 6 3 } \\lambda _ { 3 9 } \\lambda _ 0 ) , i \\geq 0 . \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} \\overline { B } _ R ^ { ( \\alpha ) } : = \\left \\{ f \\in \\mathcal { B } ^ { \\alpha } ( \\mathbb { R } ^ 2 ) : ~ \\| f \\| _ { \\alpha } \\leq R \\right \\} . \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} D = \\left \\{ t \\in \\mathbb { R } : F _ { \\nu } ( F _ { \\mu } ( t ) ) = 0 \\right \\} . \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} D _ x & : = \\bigvee _ { y \\in Y } [ x | - > y ] \\in \\ @ O ( \\# S ^ { X \\times Y } ) , & E _ y & : = \\bigvee _ { x \\in X } [ x | - > y ] \\in \\ @ O ( \\# S ^ { X \\times Y } ) . \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} \\widetilde { w } _ + : = w _ + - \\sum \\limits _ { m = - l } ^ n w _ { 2 m } \\widetilde { w } _ - : = w _ - - \\sum \\limits _ { m = - l } ^ n ( - 1 ) ^ m w _ { 2 m } \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} \\dot { \\mathcal { L } } _ \\lambda ( t ) = - \\eta \\| \\dot { u } _ \\lambda ( t ) \\| ^ 2 \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} f ^ { [ - 1 ] } ( t ) h ( t ^ { - 1 } ) = ( t ^ { 2 n - 2 } h ) ^ * ( t ) h ( t ^ { - 1 } ) = t ^ { 2 n - 2 } . \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} d ^ * ( \\eta ( x ) , \\eta ( y ) ) = \\norm { \\eta ( x ) - \\eta ( y ) } ^ * = \\norm { \\eta ( x - y ) } ^ * = \\norm { x - y } = d ( x , y ) , \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} f _ { \\phi } ( \\xi + 1 ) = \\frac { \\xi } { \\phi _ { \\alpha } ( \\xi ) } f _ { \\phi } ( \\xi ) . \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} - \\frac { 1 } { 4 n } ( h + 2 n ) & ( r ( h - 2 ) - r ( h + 2 ) ) \\\\ & = - \\frac { 1 } { 4 n } ( ( h + 2 n ) r ( h - 2 ) - ( h + 2 n ) r ( h + 2 ) ) \\\\ & \\overset { b y ( \\ref { r h - 2 & r h + 2 } ) } { { = \\joinrel = \\joinrel = } } - \\frac { 1 } { 4 n } ( ( h - 2 n ) r ( h + 2 ) - ( h + 2 n ) r ( h + 2 ) ) \\\\ & = r ( h + 2 ) . \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} h ''' = ( 3 / 2 ) h '' ( h ' ) ^ { - 1 } h '' - 2 h ' F _ 2 \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} a = 0 . 2 5 , b = 0 . 5 , c = 0 . 5 , \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} ( - 1 ) ^ { j - i } \\cdot ( - 1 ) ^ { j - i } \\cdot \\binom { m - k - m + k - j + i + 1 + j - i - 1 } { j - i } = \\binom { 0 } { j - i } . \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} \\sup _ { x \\geq t } \\left ( x ^ { 3 / 2 } | e ''' ( x ) | \\right ) = y ( t ) ^ { 3 / 2 } | e ''' ( y ( t ) ) | & \\leq \\frac { C } { y ( t ) ^ { 3 / 2 } \\log ^ { b + 1 } ( y ( t ) ) } + \\frac { C \\sup _ { x \\geq y ( t ) } \\left ( x ^ { 3 / 2 } | e ''' ( x ) | \\right ) } { \\sqrt { \\log ( \\log ( y ( t ) ) } } \\\\ & \\leq \\frac { C } { t ^ { 3 / 2 } \\log ^ { b + 1 } ( t ) } + \\frac { C \\sup _ { x \\geq t } \\left ( x ^ { 3 / 2 } | e ''' ( x ) | \\right ) } { \\sqrt { \\log ( \\log ( t ) ) } } \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} x \\ > = \\ > \\begin{pmatrix} u & w \\\\ w ^ t & v \\end{pmatrix} \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} \\forall q \\in ( 0 , 1 ) \\ \\Rightarrow \\sum _ { k = 1 } ^ { \\infty } { \\bf P } \\left ( \\ \\frac { | \\eta _ k | } { \\sigma _ k } < q ^ k \\ \\right ) < \\infty , \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { i = 1 } ^ p [ y ^ I ] _ i \\nabla ^ 2 c _ i ( x ) - A A ^ T - { \\cal J } c ( x ) ^ T { \\cal J } _ { z ^ I } \\Psi _ { \\varepsilon } ( y ^ I , z ^ I ) { \\cal J } c ( x ) \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} \\gamma _ n ( u ) : = \\lambda _ n ( u ) - \\lambda _ { n - 1 } ( u ) - 1 \\ , \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} \\forall ( x , y ) \\in X \\times X \\colon \\bigl [ \\ , \\forall \\ ; u \\in D ( H ) : \\Re \\langle x - u , y - H u \\rangle \\leqslant 0 \\ ; \\Longrightarrow x \\in D ( H ) y = H x \\ , \\bigr ] \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} \\sum _ { i = 2 ( L - m ) } ^ L H _ i \\geq \\sum _ { i = 2 ( L - m ) + 1 } ^ L G _ i + H _ { m + 3 } + H _ { m + 2 } + ( 2 L - 3 ( m + 1 ) ) . \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\varrho ( \\hat { s } ( \\nu ) ; - \\nu ) = \\left ( 1 + \\nu ^ 2 \\right ) \\sin ^ 2 ( \\hat { s } ( \\nu ) ) , \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} \\dot { x } ( t ) & = y ( t ) , \\\\ \\dot { y } ( t ) & = \\begin{cases} - \\frac { k } { m } \\ , x ( t ) - \\frac { b } { m } \\ , y ( t ) + \\frac { F } { m } , & x ( t - \\mu ) < 0 , \\\\ - \\frac { k } { m } \\ , x ( t ) - \\frac { b } { m } \\ , y ( t ) - \\frac { F } { m } , & x ( t - \\mu ) > 0 . \\end{cases} \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} \\bar C _ { p , \\nu } = 2 ^ { \\nu } [ 1 + p ] ^ p . \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} ( D ^ \\alpha u ) _ x ( x ) = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } & \\left ( \\frac { \\alpha ( u ( 0 ) - u ( x ) ) + ( \\alpha + 1 ) u ' ( x ) x } { x ^ { \\alpha + 1 } } \\right . \\\\ & \\quad \\left . + \\alpha ( \\alpha + 1 ) \\int _ 0 ^ x [ u ( x - z ) - u ( x ) + u ' ( x ) z ] \\frac { d z } { z ^ { \\alpha + 2 } } \\right ) \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} \\begin{array} { l } \\nabla f ( x ^ * ) + \\left [ \\displaystyle \\sum _ { i = 1 } ^ p [ y ^ I ] ^ * _ j \\nabla ^ 2 c _ j ( x ^ * ) - A ^ T A - { \\cal J } c _ { \\gamma } ( x ^ * ) ^ T { \\cal J } c _ { \\gamma } ( x ^ * ) - { \\cal J } c _ { \\beta } ( x ^ * ) ^ T { \\rm D i a g } ( [ v _ b ] _ { \\beta } ) { \\cal J } c _ { \\beta } ( x ^ * ) \\right ] \\eta ^ * _ 1 = 0 . \\end{array} \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} \\gamma z \\exp \\left [ \\int _ { \\mathbb { T } } \\frac { t + \\eta _ { \\mu _ { 1 } } ( z ) } { t - \\eta _ { \\mu _ { 1 } } ( z ) } \\ , d \\sigma ( t ) \\right ] = \\Psi ( z ) = \\gamma z \\exp \\left [ \\beta \\frac { 1 + \\eta _ { \\nu _ { 1 } } ( z ) } { 1 - \\eta _ { \\nu _ { 1 } } ( z ) } \\right ] . \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} q ( \\rho , \\theta ) : = \\left \\{ \\begin{aligned} & \\frac { 2 \\theta - 1 } { 2 - 2 \\rho } , & & \\rho + \\theta < 1 , \\\\ & \\frac { 1 - 2 \\rho } { 2 - 2 \\rho } , & & \\rho + \\theta \\geq 1 , \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} \\widetilde { \\varphi } _ 3 ( N ) = \\sum _ { j = 1 } ^ { p - 1 } \\frac { ( - 1 ) ^ { j } } { j } \\beta Q ^ { 2 p ^ 2 } \\beta Q ^ { j } \\beta Q ^ { p - j } . \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = 1 . \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} r _ j : = m _ j - q _ j \\geq 0 . \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} \\sigma _ p ( I ) & = \\left \\{ \\frac { j } { m _ I } \\in \\mathbb { Q } \\mid j > m _ I p \\nmid j \\gcd ( j , m _ I ) = \\frac { | ( I ) | } { p } \\right \\} . \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} \\begin{aligned} \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( M + m - \\left ( \\left ( 1 - t \\right ) \\overline { a } + t { { a } _ { i } } \\right ) \\right ) } & \\ge f \\left ( \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } \\left ( M + m - \\left ( \\left ( 1 - t \\right ) \\overline { a } + t { { a } _ { i } } \\right ) \\right ) } \\right ) \\\\ & = f \\left ( M + m - \\overline { a } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} Q ( C _ i , C _ j ) \\ & = \\ 3 \\cdot Q ( A _ { i } , A _ j ) \\ \\ i , j \\in J \\ \\ i , j \\in J ' \\\\ Q ( C _ i , C _ j ) \\ & = \\ Q ( A _ { i } , A _ j ) \\ \\ j \\in J , \\ i \\in J ' . \\\\ Q ( C _ j , \\bar U ) \\ & = \\ Q ( \\bar U , \\bar V ) \\ = \\ Q ( \\bar U , C _ i ) \\ = \\ N ^ 2 \\ \\ j \\in J , \\ i \\in J ' . \\\\ Q ( \\bar V , C _ j ) \\ & = \\ Q ( C _ i , \\bar V ) \\ = \\ N ^ 2 \\ \\ j \\in J , \\ i \\in J ' . \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} ( X ^ { \\sharp } ) ^ { \\sharp } = N ( X ) X , \\forall X \\in \\mathfrak { J } . \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} & ( L ^ 2 , \\ell ^ 2 ) = L ^ 2 , \\\\ & ( L ^ { 2 } , \\ell ^ { q } ) \\hookrightarrow L ^ { q } \\ ; \\ ; \\ ; \\ ; 1 < q < 2 , \\\\ & ( L ^ 2 , \\ell ^ 1 ) \\hookrightarrow h ^ 1 \\hookrightarrow L ^ 1 , \\\\ & L ^ { q } \\hookrightarrow ( L ^ { 2 } , \\ell ^ { q } ) \\ ; \\ ; \\ ; \\ ; 2 < q < \\infty , \\\\ & L ^ { \\infty } \\hookrightarrow b m o \\hookrightarrow ( L ^ 2 , \\ell ^ { \\infty } ) , \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} ( x - 1 ) ^ { n - r _ 1 } g _ 1 ( x ) = u ^ 2 ( x - 1 ) ^ { n - r _ 1 + k _ 4 } p _ 4 ( x ) + u ^ 3 ( x - 1 ) ^ { n - r _ 1 + k _ 5 } p _ 5 ( x ) . \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} \\phi ( t ) & = j C _ { D , E } ( t ) - j C _ { \\tilde E , \\tilde D } ( t ) + j C _ { D , E } ( t ) - j C _ { \\tilde E , \\tilde D } ( t ) = 0 , \\\\ \\psi ( t ) & = - i C _ { \\tilde D , D } ( t ) - C _ { E , \\tilde E } ( t ) + i C _ { \\tilde D , D } ( t ) = - C _ { E , \\tilde E } ( t ) . \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} \\ell ( t + 1 ) = \\begin{cases} \\ell ( t ) , & \\mbox { w i t h p r o b a b i l i t y $ \\frac { 1 } { 2 } $ } \\\\ s ( t + 1 ) , & \\mbox { w i t h p r o b a b i l i t y $ \\frac { 1 } { 2 } $ } \\end{cases} . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 h } { \\partial t ^ 2 } - c _ 1 c _ 2 \\frac { \\partial ^ 2 h } { \\partial x ^ 2 } + ( c _ 1 - c _ 2 ) \\frac { \\partial ^ 2 h } { \\partial x \\partial t } = - \\frac { 1 } { t } \\Biggl [ ( m + n + 2 ) \\frac { \\partial h } { \\partial t } + \\Bigl [ ( c _ 1 - c _ 2 ) ( m + n + 1 ) - ( c _ 1 m - c _ 2 n ) \\Bigr ] \\frac { \\partial h } { \\partial x } \\Biggr ] \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} \\frac { \\phi ( 2 ^ \\delta q _ 1 ^ { e _ 1 } \\cdots q _ k ^ { e _ k } ) } 2 - 1 \\ge \\sum _ { i = 1 } ^ k ( \\frac { \\phi ( 2 ^ { \\delta - 1 } q _ i ^ { e _ i } ) } 2 - 1 ) + \\frac { \\phi ( 2 ^ { \\delta } ) } 2 - 1 . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} \\gamma . \\mathbf { z } . \\nu = \\left ( [ v - l _ { 1 } \\theta ] , \\tfrac { k _ { 1 } + k _ { 2 } \\theta + w - v } { b } + l _ { 1 } , k _ { 2 } + l _ { 2 } - l _ { 1 } \\right ) . \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{gather*} H _ 0 ( z ) = { \\rm c o n s t . } , \\\\ h _ { 0 z } ( z _ 1 ) = h _ { 0 z } ( z _ 2 ) = 0 . \\end{gather*}"}
-{"id": "8379.png", "formula": "\\begin{align*} \\frac { ( s _ { 3 } / s _ { 1 } - 1 ) ( s _ { 2 } - 1 ) } { ( s _ { 2 } / s _ { 1 } - 1 ) ( s _ { 3 } - 1 ) } = \\frac { ( t _ { 3 } / t _ { 1 } - 1 ) ( t _ { 2 } - 1 ) } { ( t _ { 2 } / t _ { 1 } - 1 ) ( t _ { 3 } - 1 ) } \\in \\mathbb { R } . \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} \\Gamma _ v = \\{ q \\in \\Gamma \\ , | \\ , ( x _ { k + 2 } ( q ) , \\dots , x _ { n + 1 } ( q ) ) = v \\} \\ , . \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} D ^ u _ n D ^ v _ n & [ ( v _ 1 - u _ n ) ^ 2 \\ldots ( v _ { n - 1 } - u _ n ) ^ 2 ( v _ n - u _ n ) u _ n ^ k ] \\\\ & = D ^ u _ n D ^ v _ { n - r } [ ( v _ 1 - u _ n ) ^ 2 \\ldots ( v _ { n - r - 1 } - u _ n ) ^ 2 ( v _ { n - r } - u _ n ) \\cdots ( v _ n - u _ n ) u _ n ^ k ] \\\\ & \\equiv D ^ u _ { n - r } D ^ v _ { n - r } [ ( v _ 1 - u _ { n - r } ) ^ 2 \\ldots ( v _ { n - r - 1 } - u _ { n - r } ) ^ 2 ( v _ { n - r } - u _ { n - r } ) u _ { n - r } ^ k ] . \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } | f ( x ) | ^ 2 \\ , d x = \\frac { 1 } { 4 \\pi ^ 2 } \\int _ { 0 } ^ { 2 \\pi } \\Big ( \\int _ 0 ^ 1 | \\psi [ f ] ( x , \\xi ) | ^ 2 \\ , d \\xi \\Big ) \\ , d x \\ , \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} \\left ( \\varphi ^ { - 1 } \\right ) ^ { \\ast } : L ^ 1 _ { \\tilde { q } } ( \\Omega ) \\to L ^ 1 _ { 1 } ( \\widetilde { \\Omega } ) , \\ , \\ , \\ , \\tilde { q } = q / ( q - n + 1 ) . \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty { \\abs { x _ n } ^ p } < \\infty , \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} & \\lambda _ c ( t , x ) = \\lambda ( t , x ) + c _ 1 ( t , x ) , \\\\ & b _ c ( t , x ) = b ( t , x ) + c _ 2 ( t , x ) , \\\\ & L _ c = t \\frac { \\partial } { \\partial t } - \\lambda _ c ( t , x ) - b _ c ( t , x ) \\frac { \\partial } { \\partial x } \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} | a _ \\alpha ( x , t , \\zeta ) | \\le A \\zeta ^ p , | \\alpha | = m , \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} L G _ \\bullet g = g , g \\in L _ { \\mathrm { c } } ^ 1 ] a , b [ . \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} | N _ { 2 } ( v _ { 5 } ) | ( t , r ) \\leq \\begin{cases} \\frac { C r } { ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) t ^ { 7 } \\log ^ { 7 b - 6 + 5 N } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C \\log ^ { 8 } ( t ) } { r ^ { 7 / 2 } \\log ^ { b } ( t ) t ^ { 7 / 2 } } , r \\geq \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} \\int _ { 2 a t } ^ { 2 b t } A ( 2 x ) P ( 2 x , k ) \\mu _ + ( ( x - 2 k t ) / ( 2 \\sqrt t ) ) d x = \\int _ { 2 a t } ^ { 2 b t } \\left ( \\int _ { 2 a t } ^ x A ( 2 u ) P ( 2 u , k ) d u \\right ) ' \\mu _ + ( ( x - 2 k t ) / ( 2 \\sqrt t ) ) d x \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} P & = \\big \\lbrace \\emptyset \\neq P _ i \\subseteq \\left \\lbrace 1 , \\ldots , n \\right \\rbrace ~ \\forall i , P _ i \\cap P _ j = \\emptyset ~ \\forall i \\neq j , \\\\ & P _ 1 \\cup \\ldots \\cup P _ k = \\left \\lbrace 1 , \\ldots , n \\right \\rbrace \\big \\rbrace . \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} \\{ \\xi \\in \\mathbb { T } : p _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( \\xi ) > 0 \\} & = \\{ \\overline { \\Psi ( R ( t ) t ) } : R ( t ) < 1 \\} \\\\ & = \\{ \\xi \\in \\mathbb { T } : p _ { \\nu _ { 1 } \\boxtimes \\nu _ { 2 } } ( \\xi ) > 0 \\} . \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} \\lim _ { \\| u \\| _ { 1 , \\mathcal { H } , 0 } \\to \\infty } \\frac { \\langle \\mathcal { A } u , u \\rangle _ { \\mathcal { H } } } { \\| u \\| _ { 1 , \\mathcal { H } , 0 } } = + \\infty . \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} \\Vert u _ n - u \\Vert _ 2 ^ 2 = \\Vert u _ n \\Vert _ 2 ^ 2 + \\Vert u \\Vert _ 2 ^ 2 - 2 \\langle u _ n , u \\rangle _ 2 \\to \\mu - m . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} | E _ { 2 } ( t , r ) | & \\leq C \\left ( \\frac { 1 } { \\sqrt { r } \\sqrt { | t - r | } \\log ^ { b } ( | t - r | ) } \\right ) + \\frac { C } { \\sqrt { r } \\sqrt { t + r } \\log ^ { b - 1 } ( t + r ) } + \\frac { C } { r \\log ^ { b - 1 } ( r ) } , r > \\frac { t } { 2 } , | t - r | > 5 \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} x _ l \\leq \\ell _ l \\ , \\lor \\ , x _ l \\geq u _ l l = 1 , \\ldots , n . \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} m _ { 7 , 8 } & = - \\sqrt [ 3 ] { 2 } , & m _ { 1 , 5 } = m _ { 2 , 5 } & = 0 , \\\\ m _ { 4 , 5 } & = m _ { 7 , 8 } , & m _ { 2 , 3 } = m _ { 6 , 8 } & = \\frac { 1 } { m _ { 7 , 8 } } , \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} H _ 0 & = 1 , \\\\ H _ { 4 k + 1 } & = ( - 1 ) ^ { k } \\frac { ( 2 k ) ! ^ 2 } { 2 ^ { 4 k ( 2 k - 1 ) } } \\prod _ { j = 1 } ^ { 2 k - 1 } ( 2 j + 1 ) ! ^ 4 , \\\\ H _ { 4 k + 2 } & = 0 , \\\\ H _ { 4 k + 3 } & = ( - 1 ) ^ { k + 1 } \\frac { ( 2 k + 1 ) ! ^ 2 } { 2 ^ { 4 k ( 2 k + 1 ) } } \\prod _ { j = 1 } ^ { 2 k } ( 2 j + 1 ) ! ^ 4 , \\\\ H _ { 4 k + 4 } & = ( - 1 ) ^ { k + 1 } \\frac { ( 2 k + 1 ) ! ^ 2 ( 4 k + 3 ) ! ^ 2 } { 2 ^ { 2 ( 2 k + 1 ) ^ 2 } } \\prod _ { j = 1 } ^ { 2 k } ( 2 j + 1 ) ! ^ 4 . \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} D \\cdot D = D \\cdot K _ X = - 1 . \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} \\begin{cases} ( \\theta + d d ^ c \\psi _ t ) ^ n = e ^ { \\partial _ t \\psi + \\psi _ t } \\mu \\\\ \\phi ( 0 , x ) = \\phi _ 0 , \\end{cases} \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} \\begin{aligned} H _ * ( & Z ) \\otimes H _ * ( Y ) \\to H _ { * - \\deg \\gamma } ( Y ) , \\\\ a & \\otimes x \\mapsto ( p _ { 1 } ) _ * ( ( \\gamma x ) \\cdot _ 2 a ) . \\end{aligned} \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} \\psi _ \\varepsilon ( \\hat { x } - z , \\hat { t } ) - \\psi _ \\varepsilon ( \\hat { x } , \\hat { t } ) + p z & \\le \\psi ( \\hat { x } - z , \\hat { t } ) - \\psi ( \\hat { x } , \\hat { t } ) + p z \\\\ & = \\varphi ( \\hat { x } - z , \\hat { t } ) - \\varphi ( \\hat { x } , \\hat { t } ) + p z \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} & \\sigma _ { ( a , i ) } ^ { ( r + 1 ) n _ 1 } ( ( b , j ) ) = ( b + ( r + 1 ) n _ 1 \\delta _ { ( a , i ) } , j + ( r + 1 ) n _ 1 ( r \\delta _ { ( a , i ) } + 1 ) ) = \\\\ & ( b , j + r n _ 1 + r n _ 1 \\delta _ { ( a , i ) } + n _ 1 ) = ( b + n _ 1 ( \\delta _ { ( a , i ) } + 1 ) , j + n _ 1 ( r ( \\delta _ { ( a , i ) } + 1 ) + 1 ) ) = \\\\ & \\sigma _ { \\sigma _ { ( a , i ) } ( ( a , i ) ) } ^ { n _ 1 } ( ( b , j ) ) , \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} Z \\coloneqq F _ { ( i , j ) , ( k , l ) } ( \\Sigma ^ * _ { k l } ) = F _ { ( i , j ) , ( k , l ) } ( \\Sigma _ { k l } ) \\cap \\Sigma ^ * _ { i j } , \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} \\limsup _ { k \\rightarrow + \\infty } \\int _ { \\Omega } F _ k ( D u _ k ) \\ , d x \\le \\ , \\limsup _ { k \\rightarrow + \\infty } \\int _ { \\Omega } F _ k ( D u ) \\ , d x = \\int _ { \\Omega } F ( D u ) \\ , d x . \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} ( P _ H f ) ( z ) = \\langle f , K _ z \\rangle _ { L ^ 2 ( \\Omega ) } . \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} \\delta _ f ^ { - 1 } c L = \\delta _ f '^ { - 1 } c L \\quad \\Leftrightarrow \\quad \\Gamma _ { c , L } \\delta = \\Gamma _ { c , L } \\delta ' . \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} \\mathbf { D } ( s , t ) = \\frac s t , s \\geq t > 0 . \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} a _ { ( x , t , s ) } ( \\lambda ) & = \\frac { s } { \\sqrt { 1 + \\lambda ^ 2 } } + \\frac { 1 - s } { \\sqrt { 1 + \\lambda ^ 2 + \\frac 1 4 t ^ 2 \\tau ( g ( x ) ) } } \\\\ b _ { ( x , t , s ) } ( \\lambda ) & = s + \\frac { ( 1 - s ) \\frac 1 2 \\tau ( g ( x ) ) } { \\sqrt { 1 + \\lambda ^ 2 + \\frac 1 4 t ^ 2 \\tau ( g ( x ) ) } } . \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} \\hat { E } \\dot { \\hat { v } } ( t ) & = \\hat { A } \\hat { v } ( t ) + \\hat { B } u ( t ) \\\\ [ 1 e x ] \\hat { w } ( t ) & = \\hat { C } \\hat { v } ( t ) \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} h _ n ( y ) = n \\left ( f ( y , \\frac { 1 } { n } ) - f ( y , 0 ) \\right ) . \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} J _ { n } ^ { ( 3 ) } = \\frac { 2 } { 7 } 2 ^ { n } - \\frac { 3 + 2 i \\sqrt { 3 } } { 2 1 } \\omega _ { 1 } ^ { n } - \\frac { 3 - 2 i \\sqrt { 3 } } { 2 1 } \\omega _ { 2 } ^ { n } \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} e _ R ( \\mathcal I ( 1 ) ^ { [ d _ 1 ] } , \\ldots , \\mathcal I ( r ) ^ { [ d _ r ] } ; M ) = 0 \\mbox { i f } d _ { s + 1 } + \\cdot + d _ r > 0 \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} f ( p ) = \\inf _ { a \\in A } \\left [ \\varphi ( a ) + \\frac { d ( a , p ) } { d ( p , A ) } - 1 \\right ] , \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{cases} \\begin{bmatrix} \\lambda _ L x + y \\\\ - x + \\lambda _ L y \\end{bmatrix} , & x < 0 , \\\\ \\begin{bmatrix} y + \\mu \\\\ - 1 \\end{bmatrix} , & x > 0 , \\end{cases} \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} \\ddot { u } ( \\Gamma ( t ) , t ) = - \\langle ( \\partial _ t \\nabla u ( \\Gamma ( t ) , t ) ) , \\dot { \\Gamma } ( t ) \\rangle - \\langle \\nabla u ( \\Gamma ( t ) , t ) , \\ddot { \\Gamma } ( t ) \\rangle . \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} K _ { K S } ( x ) = - \\chi \\frac { x } { | x | ^ d } . \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k \\} \\ = \\ \\binom { 2 k } { k } \\frac { \\alpha ^ k } { ( 1 + \\alpha ) ^ { 2 k } } + ( 1 - \\alpha ) \\sum _ { j = 0 } ^ { k - 1 } \\binom { 2 k } { j } \\frac { \\alpha ^ { j } } { ( 1 + \\alpha ) ^ { 2 k } } \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} & D _ H = \\{ ( x , y ) \\in \\R \\times \\R ^ { d } \\mid x > 1 , \\ | y | < H ( x ) \\} , \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} & | \\frac { 1 } { r ^ { 2 } } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho \\lambda '' ( s ) } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) \\right ) | \\\\ & \\leq C \\frac { \\log ( \\log ( t ) ) } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} f ( t ) & = \\sum _ { n = 1 } ^ \\infty f _ n t ^ n g ( t ) = \\sum _ { n = 0 } ^ \\infty g _ n t ^ n \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} J _ j ^ \\ell ( \\psi _ j ^ \\ell ) = \\frac { 1 } { 2 } \\boldsymbol { B } ( \\psi _ j ^ \\ell , \\psi _ j ^ \\ell ) - \\langle \\psi _ j ^ \\ell , h _ j ^ \\ell \\rangle , ~ ~ ~ j \\in \\mathbb { N } ^ + , \\ , \\ell = 1 , 2 . \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\to + \\infty } \\sup _ { y \\in \\mathbb { R } ^ N } \\int _ { B _ r ( y ) } v _ n ^ 2 ( x ) \\ ; d x = 0 ; \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { p ^ k } \\log \\left [ \\xi ^ { \\sigma _ k } - \\xi ^ { \\sigma _ { k + 1 } } \\right ] \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} d _ { 4 } ( \\mathcal { X } , \\mathcal { Y } ) = 0 , ~ \\forall \\left ( \\mathcal { X } , \\mathcal { Y } \\right ) \\in \\left ( \\mathbf { E F } \\left ( \\mathfrak { J } \\right ) \\right ) ^ { 2 } , \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} n _ { [ 2 r , 1 1 ] } ( 0 , 1 ) = \\frac { 1 1 - 1 1 ^ { 2 r + 1 } } { 2 4 } , \\ ; \\ ; \\ ; n _ { [ 2 r + 1 , 1 1 ] } ( 1 , 0 ) = \\frac { - 1 1 ^ { 2 r + 1 } + 1 } { 2 4 } \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} A _ { k , i } = \\left \\{ \\begin{array} { c } k + \\frac { i } { 8 } k + \\frac { i + 1 } { 8 } , \\\\ ( k + i ) e _ { 1 } ( k + i + 1 ) e _ { 1 } , \\\\ ( k + i ) e _ { 1 } ( k + i + 1 ) e _ { 1 } \\end{array} \\right \\} , \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} \\gamma ^ 2 + ( n - 2 ) \\gamma - \\mu _ k = 0 \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} x \\nabla K _ { n } ^ { p , N } \\left ( x \\right ) = n K _ { n } ^ { p , N } \\left ( x \\right ) + n p ( N - n + 1 ) K _ { n - 1 } ^ { p , N } \\left ( x \\right ) . \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} \\mathcal { L } _ L f : = \\sum _ { \\ell = 1 } ^ { d } \\left < f , p _ { \\ell } \\right > _ N p _ { \\ell } . \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} \\Big ( \\sum _ { \\alpha \\in \\N _ 0 ^ { ( \\N ) } } r ^ { | \\alpha | } \\| \\widehat { f } ( \\alpha ) \\| ^ q \\Big ) ^ { \\frac { 1 } { q } } = \\Big ( \\sum _ { m = 1 } ^ { \\infty } r ^ { m } \\sum _ { | \\alpha | = m } \\| \\widehat { f } ( \\alpha ) \\| ^ q \\Big ) ^ { \\frac { 1 } { q } } \\le \\Big ( \\sum _ { m = 1 } ^ { \\infty } ( r c ^ q ) ^ { m } \\Big ) ^ { \\frac { 1 } { q } } \\| f \\| _ p \\le C \\| f \\| _ p , \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} f _ { S _ 4 } ( z ) = - \\frac { 1 } { 4 3 2 } \\frac { ( 1 6 z ^ 8 - 5 6 z ^ 4 + 1 ) ^ 3 } { z ^ 4 ( 4 z ^ 4 + 1 ) 4 } . \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} \\sup _ \\Omega | \\nabla \\varrho | ^ 2 > \\max \\Big \\{ 1 , \\limsup _ { y \\to \\partial \\Omega } | \\nabla \\varrho ( y ) | ^ 2 \\Big \\} , \\Big ( \\sup _ \\Omega | \\nabla \\varrho | ^ 2 > 1 \\ \\ \\ \\Omega = M \\Big ) . \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} \\phi _ n - \\phi _ { n + 1 } = \\phi _ n \\phi _ { n + 1 } = \\psi ( n \\phi _ n - ( n + 1 ) \\phi _ { n + 1 } ) \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} ( \\beta + 1 ) \\| \\zeta - \\frac { 1 } \\beta \\phi _ n \\| \\leqslant \\| \\zeta - \\frac 1 \\beta \\phi _ n + \\beta ( 1 - H ) ( \\zeta - \\frac 1 \\beta \\phi _ n ) \\| = \\| \\zeta - \\frac 1 \\beta \\phi _ n + \\beta ( 1 - H ) \\zeta - ( 1 - H ) \\phi _ n \\| . \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} K _ { ( j , h ) , ( i , k ) } : = E _ { j , i } \\left ( b _ { h } ^ * b _ { k } \\right ) \\quad ; ( j , h ) , ( i , k ) \\in I \\times \\{ 1 , \\dots , n \\} \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} u _ 1 ( T _ 1 , x _ 1 ) \\leq \\psi _ 1 ( x _ 1 ) \\leq \\psi _ 1 ( 0 ) = u _ 1 ^ * / 2 , \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} & F ( \\Phi ( - x - \\underline h ( t ) ) ) = F ( \\Phi ( - x - \\underline h ( t ) ) ) - F ( \\mathbf { u } ^ * ) \\preceq m L \\epsilon _ 2 \\mathbf { 1 } \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} A ( \\mu ) ^ \\top M ( \\mu ) E ( \\mu ) + E ( \\mu ) ^ \\top M ( \\mu ) A ( \\mu ) + F = 0 \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} \\langle h _ l - g , u \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = \\langle \\widehat { A } ( h _ l - g ) , u \\rangle _ { \\alpha } . \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} W _ t ^ { ( 3 ) } + ( \\Lambda _ 1 ( | \\xi | ) + \\Lambda _ 2 ( | \\xi | ) ) W ^ { ( 3 ) } + R _ 3 ( | \\xi | ) W ^ { ( 3 ) } = 0 , \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} \\Delta _ L \\varphi = d i v _ L ( g r a d _ { H ^ 0 } \\varphi ) . \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} x _ { e + f } \\Big | _ { x _ e = 0 } = \\Big ( \\left ( p _ e + p _ f \\right ) ^ 2 - m ^ 2 \\Big ) _ { p _ e ^ 2 = m ^ 2 } = m ^ 2 + 2 p _ e p _ f + p _ f ^ 2 - m ^ 2 \\neq p _ f ^ 2 - m ^ 2 = x _ f . \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} r ^ { 2 } \\partial _ { r } ^ { 2 } v _ { 4 } = r ^ { 2 } v _ { 4 , c } ( t , r ) + r ^ { 2 } \\partial _ { t t } v _ { 4 } - r \\partial _ { r } v _ { 4 } + v _ { 4 } \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} S _ { B F } ^ { c l } \\left [ ( A , B ) ^ { ( g , \\tau ) } \\right ] - S _ { B F } ^ { c l } [ ( A , B ) ] = S _ { \\tau F } [ ( g , \\tau ) , A ] . \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} f B ( e x f , y ) g & = f e B ( x f , y ) g + f B ( e , y ) x f g = 0 . \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} M ( G : H ) : = \\{ \\nu \\in M ( G ) : \\nu ^ h = \\nu \\ \\forall h \\in H \\} , \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} \\Psi _ a = E \\rho ( E \\rho ) ^ + \\Psi _ a . \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} \\operatorname { I n d } _ \\infty ( D ) : = [ P ^ b _ Q ] - [ e _ 1 ] \\in K _ 0 ( \\mathcal { A } ^ \\infty _ G ( M ) ) \\equiv K _ 0 ( C ^ * ( M _ 0 \\subset M , E ) ^ G ) \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} v _ { 5 } ( t , r ) = \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { \\infty } d \\xi J _ { 1 } ( r \\xi ) \\sin ( ( t - s ) \\xi ) \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( s , \\xi ) \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} u _ 1 = u e ^ { \\lambda s } \\sin ( s ) \\qquad u _ 2 = u e ^ { \\lambda s } \\cos ( s ) \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} U = X _ 1 \\leadsto X _ 2 \\leadsto \\ldots \\leadsto X _ { n + 1 } = Z _ { t } \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} q = \\begin{cases} 2 , & p \\in ( 1 , 2 ] , \\\\ p ^ * = \\frac { p } { p - 1 } , & p \\in ( 2 , \\infty ) \\end{cases} \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} - ( \\nabla + i { \\boldsymbol k } ) \\cdot \\frac { 1 } { \\epsilon ( x , \\omega ) } ( \\nabla + i { \\boldsymbol k } ) u ( x ) = \\left ( \\frac { \\omega } { c } \\right ) ^ 2 u ( x ) ~ ~ { \\rm i n } ~ D _ 0 . \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\iint \\eta _ n ( y ) ^ 2 \\left ( f ( x ) - f ( y ) \\right ) ^ 2 \\frac { \\psi _ 0 ( x ) \\psi _ 0 ( y ) } { | x - y | ^ { d + \\alpha } } d x d y = 0 . \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} - 2 c _ { b } \\lambda ( t ) \\int _ { 0 } ^ { \\infty } d \\xi \\frac { \\xi ^ { 2 } \\sin ( t \\xi ) } { t ^ { 2 } } \\psi _ { v _ { 2 } } ( \\xi , \\lambda ( t ) ) & = \\frac { 2 c _ { b } \\lambda ( t ) } { t ^ { 5 } } \\int _ { 0 } ^ { \\infty } \\cos ( t \\xi ) \\partial _ { \\xi } ^ { 3 } \\left ( \\xi ^ { 2 } \\psi _ { v _ { 2 } } ( \\xi , \\lambda ( t ) ) \\right ) d \\xi \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} \\Psi ' ( \\lambda ) = d e t ( I _ 5 - D _ { L ''^ c } ) = & { \\lambda } ^ { 5 } - \\left ( 2 \\ , n - 1 0 \\right ) { \\lambda } ^ { 4 } - \\left ( - 2 8 + 1 0 \\ , n \\right ) { \\lambda } ^ { 3 } \\\\ & - \\left ( 1 0 + 8 \\ , n \\right ) { \\lambda } ^ { 2 } - \\left ( 1 0 3 - 1 5 \\ , n \\right ) \\lambda + 1 4 \\ , n - 7 0 . \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} \\lambda ''' ( t ) = \\frac { R H S _ { 2 } ' ( t ) } { g _ { 2 } ( t ) } - \\frac { g _ { 2 } ' ( t ) } { g _ { 2 } ( t ) ^ { 2 } } R H S _ { 2 } ( t ) \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} ( V _ 2 ^ { ( 1 ) } - V _ 2 ^ { ( 2 ) } ) v ^ { ( 3 ) } = 0 \\Omega . \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} k _ { m - 1 } ^ { ( r ) } + k _ m ^ { ( r - 1 ) } = \\sum _ { i = 0 } ^ { m } \\left [ \\binom { r + i - 2 } { i - 1 } + \\binom { r + i - 2 } { i } \\right ] h _ { m - i } = \\sum _ { i = 0 } ^ m \\binom { r + i - 1 } { i } h _ { m - i } = k _ m ^ { ( r ) } \\ , . \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} S _ n & = - i \\lambda _ n \\forall n \\geq 3 . \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} l _ 1 ' ( x ) = l _ 2 ( \\alpha , x ) - \\frac { 1 } { 2 } ~ l _ 3 ( \\alpha , \\alpha , x ) , ~ l _ 2 ' ( x , y ) = l _ 2 ( x , y ) - l _ 3 ( \\alpha , x , y ) ~ ~ ~ ~ l _ 3 ' ( x , y , z ) = l _ 3 ( x , y , z ) . \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} \\pi _ { \\mathbf { 0 } } ^ { ( n ) } ( \\sigma , \\gamma ) ( t ) \\stackrel { d } { = } \\hat { \\pi } _ { n } ( N ( n ^ { 2 } \\gamma t ) / n ^ { 2 } ) + \\dfrac { 1 } { n } \\Big [ \\dfrac { n ^ { 2 } \\gamma t - R _ { N ( n ^ { 2 } \\gamma t ) } } { R _ { N ( n ^ { 2 } \\gamma t ) + 1 } - R _ { N ( n ^ { 2 } \\gamma t ) } } Z _ { N ( n ^ { 2 } \\gamma t ) + 1 } \\Big ] . \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} ( \\chi _ t + d d ^ c \\phi ) ^ n = e ^ { \\partial _ t \\phi + \\tilde { F } ( t , x , \\phi ) } \\mu . \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m e _ { j i _ 0 } ( x ) w _ j ( x ) > e _ { i _ 0 i _ 0 } ( x ) w _ { i _ 0 } ( x ) \\geq - M w _ { i _ 0 } ( x ) \\mbox { f o r } x < 0 \\mbox { a n d s o m e c o n s t a n t } M . \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} \\mu ^ { I _ 1 \\dot \\cup \\cdots \\dot \\cup I _ N } _ { \\l , \\tau } = \\mu ^ { I _ 1 } _ { \\l , \\tau } * \\ldots * \\mu ^ { I _ N } _ { \\l , \\tau } \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} \\eta _ { R } ( x ) = \\begin{cases} 1 & , 2 \\le | x | \\le R , \\\\ 0 & , | x | \\le 1 | x | \\ge 2 R , \\end{cases} \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mu _ t + \\nabla \\cdot ( v _ t ^ \\top \\mu _ t ) = \\frac { d } { d t } \\mu _ t + \\nabla \\cdot ( ( v _ t ^ \\top + v _ t ^ \\perp ) \\mu _ t ) = 0 . \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} P x = ( y _ 1 , \\cdots , y _ M ) , \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} e ^ { - M } \\left \\Vert u \\left ( . , 0 \\right ) \\right \\Vert _ { X } \\leq \\left \\Vert u \\left ( . , t \\right ) \\right \\Vert _ { X } = \\left \\Vert \\tilde { u } \\left ( . , s \\right ) \\right \\Vert _ { X } \\leq e ^ { M } \\left \\Vert u \\left ( . , 0 \\right ) \\right \\Vert _ { X } t , s \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{gather*} - \\frac { z } { \\psi } u _ \\psi = - \\frac { \\psi _ s \\phi _ s } { \\psi \\phi } \\end{gather*}"}
-{"id": "1990.png", "formula": "\\begin{align*} ( x + y ) z _ { x y } + 2 \\sqrt { z _ x z _ y } = 0 \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\mathcal { B } = \\frac { \\varphi } { 2 \\Gamma ( p ) } \\sum _ { j = 1 } ^ { n } q _ j ^ p \\int _ { 0 } ^ { \\infty } e ^ { - q _ j x } x ^ { p - 1 } F _ { \\gamma } ( x ) \\mathrm { d } x , \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} F = \\sin ( y ) ^ 3 \\cos ( x ) \\cos ( y ) \\dot { x } ^ 2 + \\sin ( y ) ^ 2 \\sin ( x ) \\dot { x } \\dot { y } \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} | \\widehat { N _ { 2 } ( f _ { v _ { 5 } } ) } ( t , \\xi ) | & \\leq \\frac { C } { \\xi t ^ { 4 } \\log ^ { 3 b } ( t ) } + \\frac { C } { \\xi ^ { 3 } t ^ { 6 } \\log ^ { 3 b } ( t ) } + \\frac { C \\log ^ { 3 } ( t ) } { \\sqrt { \\xi } t ^ { 5 / 2 } } , 1 \\leq \\frac { 1 } { \\xi } \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\langle A ( u _ n ) , u _ n - u \\rangle _ { \\mathcal { H } } = \\limsup _ { n \\to \\infty } \\langle \\mathcal { A } ( u _ n ) , u _ n - u \\rangle _ { \\mathcal { H } } \\leq 0 . \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} g ^ { i j } \\ , \\Pi \\left ( F _ { s i j } \\right ) & = g ^ { i j } \\ , \\Pi \\left ( \\nabla _ { F _ i } \\nabla _ { F _ j } V \\right ) = g ^ { i j } \\ , \\nabla _ { F _ i } ^ { \\perp } \\nabla _ { F _ j } ^ { \\perp } V + g ^ { i j } \\ , \\nabla _ { F _ i } ^ { \\perp } \\nabla _ { F _ j } ^ T V \\ , . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} U _ p \\left ( f ( z ) \\right ) = \\sum _ { p n \\geq N } a ( p n ) q ^ n . \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} \\begin{cases} 0 > x _ { n } ^ { c } \\geq x _ { m } ^ { c } \\ \\ & { \\rm i f } \\ n \\leq m , \\\\ 0 > x _ { n } ^ { c _ 1 } \\geq x _ { n } ^ { c _ 2 } \\ \\ & { \\rm i f } \\ c _ 1 \\leq c _ 2 . \\end{cases} \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} T ( f ( u ) \\cdot v ~ + ~ & u \\cdot f ( v ) ) + f ( T ( u ) \\cdot v + u \\cdot T ( v ) ) - T ( u ) f ( v ) - f ( u ) T ( v ) \\\\ & + T \\big ( H ( T ( u ) , f ( v ) ) + H ( f ( u ) , T ( v ) ) \\big ) + f ( H ( T ( u ) , T ( v ) ) ) = 0 , ~ u , v \\in M . \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} \\partial _ t ^ h v ( t ) : = \\frac { v ( t + h ) - v ( t ) } { h } . \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} C ^ { 2 } . f ( { h } ) = & \\frac { 1 } { 2 } ( x - y ) ( f ( { h } - 2 ) - f ( { h } + 2 ) ) { B } \\\\ = & \\frac { 1 } { 2 } ( f ( h - 4 ) x B - f ( h ) x B - f ( h ) y B + f ( h + 4 ) y B ) \\\\ \\equiv & \\frac { 1 } { 4 } \\big ( ( - { h } ^ { 2 } + 2 { h } + \\mu ) f ( { h } - 4 ) - 2 ( \\mu - { h } ^ { 2 } ) f ( { h } ) \\\\ & + ( - { h } ^ { 2 } - 2 { h } + \\mu ) f ( { h } + 4 ) \\big ) \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} 0 = \\langle y , x \\rangle _ \\mathcal { H } = \\langle y , \\mathcal { H } x \\rangle = \\langle \\mathcal { H } y , x \\rangle . \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} a _ k ( n ) = 2 \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { n } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - m ^ 2 ) , a _ k ( n ) = 2 \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { \\frac { n } { 2 } } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - 2 m ^ 2 ) . \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} 1 & = 2 c d + 2 d ^ 2 + c ^ 2 + b ^ 2 + d b + a ^ 2 + c a + 2 d a \\\\ & = \\left ( \\frac { a } { 2 } + d + c \\right ) ^ 2 + \\left ( \\frac { d } { 2 } + b \\right ) ^ 2 + \\left ( \\frac { d } { \\sqrt { 2 } } + \\frac { a } { \\sqrt { 2 } } \\right ) ^ 2 + \\left ( \\frac { d } { 2 } \\right ) ^ 2 + \\left ( \\frac { a } { 2 } \\right ) ^ 2 . \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 \\overline { s } } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 \\overline { s } } \\sum _ { j , k = 1 } ^ { n } \\partial _ { j } a _ { j k } \\partial _ { k } \\bigg ] v & = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ v & = f \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} . \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} Q & = Q ( y , z ) \\\\ & = \\min \\left \\{ \\frac { \\sigma L - k z - \\left ( \\alpha ( \\sigma y \\pm k z ) + \\widehat { \\alpha } z \\right ) } { \\sigma } , \\frac { \\sigma \\alpha y \\pm \\alpha k z + \\widehat { \\alpha } z - 2 k z + \\sqrt { ( 2 \\sigma L ) ^ 2 - 3 ( \\sigma \\alpha y \\pm \\alpha k z - \\widehat { \\alpha } z ) ^ 2 } } { 2 \\sigma } \\right \\} . \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} T _ j ^ n = T _ { j + n - 1 } \\circ \\cdots \\circ T _ { j + 1 } \\circ T _ j . \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( 1 - \\sigma ) D ^ 2 v : D ^ 2 \\varphi + \\sigma \\Delta v \\Delta \\varphi d x = \\lambda \\int _ { \\partial \\Omega } \\frac { \\partial v } { \\partial \\nu } \\frac { \\partial \\varphi } { \\partial \\nu } d \\sigma \\ , , \\ \\ \\ \\forall \\varphi \\in H _ c ^ 2 ( \\Omega ) . \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} \\boldsymbol { a } _ { n } = \\left [ 1 , e ^ { j \\frac { 2 \\pi D } { \\lambda _ r } \\sin \\vartheta } , \\ldots , e ^ { j \\frac { 2 \\pi D } { \\lambda _ r } ( n - 1 ) \\sin \\vartheta } \\right ] ^ { T } , \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} \\Psi = \\Psi _ 1 \\cup \\ldots \\cup \\Psi _ p \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} \\mu _ { 2 r - 1 , \\ell } & = \\left \\lceil \\frac { \\ell \\cdot c + d } { 2 4 } - \\dfrac { c + \\ell \\cdot d } { 2 4 \\cdot \\ell ^ { 2 r - 1 } } \\right \\rceil , \\\\ \\mu _ { 2 r , \\ell } & = \\left \\lceil \\frac { c + \\ell \\cdot d } { 2 4 } - \\dfrac { c + \\ell \\cdot d } { 2 4 \\cdot { \\ell } ^ { 2 r } } \\right \\rceil . \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} X _ t = \\begin{cases} X _ { t - 1 } + \\eta _ t - 1 & X _ { t - 1 } > 0 , \\\\ \\eta _ t & \\end{cases} \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} W ^ { ( 3 ) } _ { t } + ( \\Lambda _ 1 ( | \\xi | ) + \\Lambda _ 2 ( | \\xi | ) ) W ^ { ( 3 ) } + R _ 3 ( | \\xi | ) W ^ { ( 3 ) } = 0 , \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} G _ \\rightarrow - G _ \\leftarrow = | v \\rangle \\langle u | - | u \\rangle \\langle v | . \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} I _ 1 + I _ 2 = \\frac { 1 } { 2 } \\partial _ { x x } ^ 2 \\left ( | \\partial _ x v | ^ 2 \\right ) - | ( \\partial _ { x x } ^ 2 v ) ^ T | ^ 2 . \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} u ^ { \\ell + 1 } = \\Phi _ { , 1 } ^ \\tau ( { u ^ \\ell } ) = e ^ { \\tau \\mathcal { L } } \\Big ( u ^ \\ell + \\tau \\mathcal { B } \\left ( F ( u ^ \\ell ) \\cdot \\varphi _ 1 \\big ( \\tau \\mathcal { A } \\big ) G ( \\overline { u ^ \\ell } ) \\right ) \\Big ) \\quad \\varphi _ 1 ( z ) = \\frac { e ^ { z } - 1 } { z } \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} \\sqrt { R } \\ge 0 ( 0 , \\infty ) \\times \\Omega , \\sqrt { R } U = 0 \\{ \\sqrt { R } = 0 \\} . \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} \\tau ( t ) = \\tau _ 0 - 2 t C _ r ( 1 + \\| e ^ { \\tau _ 0 A } \\overline { v } _ 0 \\| _ { H ^ r } ^ 2 + \\| e ^ { \\tau _ 0 A } \\widetilde { v } _ 0 \\| _ { H ^ r } ^ 2 ) . \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} l l ^ * = 1 _ X = r r ^ * . \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} ( - L _ { \\Sigma } + h ) \\phi = \\lambda _ j \\phi \\ \\ \\ \\ \\forall \\phi \\in E _ j \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} & x _ { i _ { 1 } , i _ { 2 } } x _ { i _ { 3 } , i _ { 4 } } = x _ { i _ { \\sigma _ { 1 } } , i _ { \\sigma _ { 2 } } } x _ { i _ { \\sigma _ { 3 } } , i _ { \\sigma _ { 4 } } } , \\\\ & y _ { i _ { 1 } , i _ { 2 } } y _ { i _ { 3 } , i _ { 4 } } = y _ { i _ { \\sigma _ { 1 } } , i _ { \\sigma _ { 2 } } } y _ { i _ { \\sigma _ { 3 } } , i _ { \\sigma _ { 4 } } } , \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} \\begin{cases} D ^ { \\alpha _ 1 } _ a y _ { s s } ( t ) = w _ { s s } ( t ) , & y _ { s s } ( 0 ) = 0 , \\\\ D ^ { 1 - \\alpha _ 1 } _ a w _ { s s } ( t ) = z _ { s s } ( t ) , & w _ { s s } ( 0 ) = 0 , \\\\ D ^ { \\alpha _ 2 - 1 } _ a z _ { s s } ( t ) = f _ { s s } ( t , y ( t ) , w ( t ) ) , & z _ { s s } ( 0 ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} \\frac { \\delta } { { a } ^ { p ^ * - 1 } } & = \\delta ( 2 \\Lambda ) ^ { p ^ * - 1 } = \\delta 2 ^ { \\frac { 1 } { p - 1 } } \\Lambda ^ { \\frac { 1 } { p - 1 } } \\\\ & = 2 ^ { \\frac { 1 } { p - 1 } } \\delta ^ { \\frac { p } { 2 ( p - 1 ) } } \\Big ( \\frac { p - 2 } { p } \\Big ) ^ { \\frac { p - 2 } { 2 ( p - 1 ) } } L ^ { \\frac { p } { 2 ( p - 1 ) } } . \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} y _ { x x } = f _ 0 + f _ 1 \\ , y _ x + f _ 2 \\ , y _ x ^ 2 + f _ 3 \\ , y _ x ^ 3 \\ , , f _ i = f _ i ( x , y ) \\ , , \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} & p ( n - 1 ) - p ( n - 4 ) + p ( n - 9 ) - p ( n - 1 6 ) + \\cdots \\\\ & \\qquad \\ , = p ( n - 2 ) - p ( n - 8 ) + p ( n - 1 8 ) - p ( n - 3 2 ) + \\cdots = [ p ( n ) - p _ { \\mathcal { D O } } ( n ) ] / 2 , \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} G ( u , v ) = ( g _ i ( u , v ) ) : = ( 1 - \\epsilon ) [ F ( u ) + F ( v ) ] - F ( ( 1 - \\epsilon ) ( u + v - \\mathbf { u } ^ * ) ) , \\ \\ u , v \\in \\R ^ m . \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} & \\tilde { F } ( x , y ) = \\begin{pmatrix} x + \\gamma c y \\\\ \\ \\ , y \\end{pmatrix} + \\begin{pmatrix} 0 \\\\ \\gamma ^ { - 1 } \\ , a _ k \\ , x ^ k + b _ l \\ , y \\ , x ^ { l - 1 } + \\cdots \\end{pmatrix} , \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} { } ^ Q \\Omega ^ + _ a = ( \\rho ^ T E \\rho ) ^ + \\rho _ a ( \\rho ^ T E \\rho ) - ( E \\rho ) ^ + E ^ T \\rho { } ^ Q \\Omega ^ - _ a . \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} | f _ { v _ { 5 } } ^ { \\lambda } | \\leq \\begin{cases} \\frac { C r } { t ^ { 2 } \\log ^ { b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C \\log ( r ) } { | t - r | } , t > r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} \\nabla _ { 1 } c = q _ { 2 } ( a - c ) , \\ \\ \\ \\nabla _ { 2 } a = q _ { 1 } ( a - c ) . \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} & | \\partial _ { t } ^ { 2 } \\left ( \\lambda ( t ) \\langle \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) F _ { 0 , 2 } \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle \\right ) | \\leq \\frac { C } { t ^ { 4 } \\log ^ { 3 b + 1 + 2 N - 2 b \\alpha } ( t ) } + \\frac { C | e '''' ( t ) | } { \\log ^ { 2 b + 2 N - 2 b \\alpha } ( t ) } \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} P _ * ( r , k ) = 1 - \\int _ 0 ^ r A ( \\rho ) P ( \\rho , k ) d \\rho \\ , . \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\boldsymbol { B } ( \\phi , \\phi ) = 2 \\mathbb B _ 0 ( \\phi _ 1 , \\phi _ 1 ) + \\sum _ { k = 1 } ^ { \\infty } { \\mathbb B } _ k ( \\phi _ { 1 + k } , \\phi _ { 1 - k } ) , \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( 1 - \\sigma ) D ^ 2 u : D ^ 2 \\varphi + \\sigma \\Delta u \\Delta \\varphi d x = \\xi \\int _ { \\partial \\Omega } u \\varphi d \\sigma \\ , , \\ \\ \\ \\forall \\varphi \\in \\mathcal { H } ^ 2 _ { 0 , N } ( \\Omega ) , \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} \\begin{cases} u = ( h \\bar { x } - z ) ^ { - 1 } ( y - t x ) ^ { - 1 } p ( w ) \\\\ \\tilde { w } = w - t h q ( u ) \\end{cases} \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } \\right ) \\langle x , x \\rangle ^ \\frac { 1 } { 2 } + \\langle x , x \\rangle ^ \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } = \\sqrt { n } \\langle x , x \\rangle ^ \\frac { 1 } { 2 } ( 2 - c _ x ) \\langle x , x \\rangle ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} \\mathcal { W } _ { k } \\left [ I ; J \\right ] , \\quad \\mbox { f o r a l l $ k = 2 , \\dots , p + 1 $ , } \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} S ^ { ( s ) } _ { n , n } & = - i w ^ { ( s ) } _ n \\underbrace { b _ 1 b _ 1 \\cdots b _ 1 } _ { n } . \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} b _ { j + 1 } - b _ j = ( j + 1 ) ^ q - j ^ q \\leq c \\ , j ^ { q - 1 } \\ , , \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} \\Omega _ 1 & = \\{ ( x , y ) \\ , | \\ , x > 0 , \\ , y > 0 \\} , \\\\ \\Omega _ 2 & = \\{ ( x , y ) \\ , | \\ , x > 0 , \\ , y < 0 \\} , \\\\ \\Omega _ 3 & = \\{ ( x , y ) \\ , | \\ , x < 0 , \\ , y < 0 \\} , \\\\ \\Omega _ 4 & = \\{ ( x , y ) \\ , | \\ , x < 0 , \\ , y > 0 \\} . \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} { \\mathbf E } ( w ) = \\int _ { \\mathbb { R } ^ 2 } \\frac { 1 } { 2 } | \\nabla w | ^ 2 + & \\int _ { \\mathbb { R } ^ 2 } \\frac { 1 } { 4 } \\Big [ \\ , A _ + ( | w ^ + | ^ 2 - { t ^ + } ^ 2 ) ^ 2 + A _ - ( | w ^ - | ^ 2 - { t ^ - } ^ 2 ) ^ 2 \\\\ [ 1 m m ] & \\qquad \\quad + 2 B ( | w ^ + | ^ 2 - { t ^ + } ^ 2 ) ( | w ^ - | ^ 2 - { t ^ - } ^ 2 ) \\ , \\Big ] , \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} W = \\begin{bmatrix} D & - C \\\\ - B & A \\end{bmatrix} \\in \\mathbb { R } ^ { ( m + n ) \\times ( m + n ) } \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} \\bigl ( A x \\bigr ) ( t ) & = \\bigl ( \\varphi ^ { - 1 } \\bigl ( \\{ ( T z ) _ i \\} \\bigr ) \\bigr ) ( t ) = \\bigl [ ( T z ) _ k \\bigr ] ( t - k ) \\\\ & = \\sum \\limits _ { m \\in \\mathbb Z ^ c } \\bigl ( b _ { k m } x _ { k - m } \\bigr ) ( t - k ) , t \\in [ 0 , 1 ) ^ c + k . \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { u } = i \\left [ \\Delta \\tilde { u } + A \\tilde { u } + \\tilde { V } \\left ( x , t \\right ) \\tilde { u } + \\tilde { F } \\left ( x , t \\right ) \\right ] , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} R _ { 2 3 2 3 } = \\frac { \\left ( 3 { { e } ^ { 5 q _ 2 } + 2 { { e } ^ { 4 q _ 2 + q _ 1 } } - 2 { { e } ^ { 3 q _ 2 + 2 q _ 1 } } - 2 { { e } ^ { 2 q _ 2 + 3 q _ 1 } } } \\right ) { { e } ^ { 2 q _ 3 } } + \\left ( - { { e } ^ { 5 q _ 2 + q _ 1 } } + { { e } ^ { 4 q _ 2 + 2 q _ 1 } } + { { e } ^ { 3 q _ 2 + 3 q _ 1 } } \\right ) { { e } ^ { q _ 3 } } } { ( e ^ { q _ 2 } + e ^ { q _ 3 } ) \\Delta _ 1 } , \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} \\lim _ { \\norm { y } \\to 0 } \\frac { \\norm { \\varphi ( x + y ) - \\varphi ( y ) - \\mathcal { A } ( x ) y } } { \\norm { y } } = 0 \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} \\mathcal { N } ( t , 0 , \\xi , v , \\overline v ) = e ^ { t \\mathcal { L } } f ( v , \\overline v ) \\quad \\mathcal { N } ( t , t , \\xi , v , \\overline v ) = f \\left ( e ^ { t \\mathcal { L } } v , e ^ { t \\mathcal { L } } \\overline v \\right ) \\xi . \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} \\gamma ^ { \\prime } = \\gamma \\exp \\left [ \\frac { 1 } { 2 } \\int _ { [ 0 , + \\infty ] } \\left ( \\frac { 1 } { t } - t \\right ) \\ , d \\sigma ( 1 / t ) \\right ] . \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{align*} \\lVert k _ i \\rVert _ { L _ 1 } & = \\lVert y _ i ( \\cdot ) a _ i ( \\cdot \\cdot ) \\rVert _ { L _ 1 } = \\lVert y _ i ( \\cdot ) \\rVert _ { L _ 1 } \\cdot \\lVert a _ i ( \\cdot \\cdot ) \\rVert _ { L _ 1 } \\\\ & \\le M ^ { \\frac 1 { p } } \\lVert a _ i \\rVert _ { L _ q } \\cdot M ^ { \\frac 1 { q } } \\lVert y _ i \\rVert _ { L _ p } = M \\lVert a _ i \\rVert _ { L _ q } \\cdot \\lVert y _ i \\rVert _ { L _ p } . \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 { \\rm d i s t } _ N ^ 2 } { \\partial p _ i \\partial p _ j } ( p ) = 2 \\sum _ { k = 1 } ^ L \\left ( \\left ( \\delta _ { i k } - \\frac { \\partial P _ N ^ k } { \\partial p _ i } ( p ) \\right ) \\left ( \\delta _ { j k } - \\frac { \\partial P _ N ^ k } { \\partial p _ j } ( p ) \\right ) - \\left ( p _ k - P _ N ^ k ( p ) \\right ) \\frac { \\partial ^ 2 P _ N ^ k } { \\partial p _ i \\partial p _ j } ( p ) \\right ) . \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty | \\widehat { \\phi ^ { ( 1 ) } } ( s \\xi _ 1 ) | ^ 2 { d s \\over s } = 1 \\mbox { f o r a l l } \\xi _ 1 \\in \\Bbb { R } \\backslash \\{ 0 \\} , \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} \\sigma _ { p B f } ( T ) & : = \\{ \\lambda \\in \\mathbb { C } : \\lambda I - T \\thinspace \\mbox { i s n o t p s e u d o B - F r e d h o l m } \\} , \\\\ \\sigma _ { p B w } ( T ) & : = \\{ \\lambda \\in \\mathbb { C } : \\lambda I - T \\thinspace \\mbox { i s n o t p s e u d o B - W e y l } \\} , \\thinspace \\mbox { r e s p e c t i v e l y . } \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} p _ \\mu ( x ) = N _ \\alpha [ 1 + { b } _ \\alpha ( x - \\mu ) ^ 2 ] ^ { \\frac { 1 } { \\alpha - 1 } } , \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} - \\partial _ { t t } u + \\partial _ { r r } u + \\frac { 1 } { r } \\partial _ { r } u - \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) } { r ^ { 2 } } u & = F ( t , r ) + F _ { 3 } ( t , r ) \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} M = \\{ ( x , y ) \\in X : - 2 \\leq x < 1 \\} \\cup \\{ q \\} . \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} X _ 0 ' & = m , \\\\ X _ t ' & = X _ { t - 1 } ' - 1 + W _ t t \\geq 1 , \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} ( \\lambda _ k + \\lambda ) \\mathcal H _ \\lambda ( v ) = - ( \\lambda + \\lambda _ { k - 1 } ( v ) + 1 ) \\chi _ k ( \\lambda , v ) \\ , \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} \\langle D , D \\rangle = \\langle m F , m F \\rangle = m ^ 2 \\cdot \\langle F , F \\rangle = i . \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } \\partial _ { j } a _ { j k } \\partial _ { k } \\bigg ] u _ { 1 } & = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ u _ { 1 } & = \\zeta u \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} h _ { i j } = \\begin{cases} \\sqrt { \\frac { 1 - p } { p } } \\frac { 1 } { \\sqrt { n } } & p , \\\\ - \\sqrt { \\frac { p } { 1 - p } } \\frac { 1 } { \\sqrt { n } } & 1 - p , \\end{cases} \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} A _ k ( n ) & = \\{ ( \\lambda = ( \\lambda _ 1 ^ { m _ 1 } , \\lambda _ 2 ^ { m _ 2 } , \\ldots , \\lambda _ \\ell ^ { m _ \\ell } ) , \\lambda _ { i _ 0 } , m ) \\mid \\\\ & \\qquad \\lambda \\in \\mathcal { P } ( n ) , \\ , m _ i < k \\ , \\forall i , \\ , 1 \\leq i _ 0 \\leq \\ell , \\ , k \\mid \\lambda _ { i _ 0 } , \\ , 0 \\leq m < m _ { i _ 0 } \\} . \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} a \\left ( j \\right ) u _ { y y } \\left ( x , j , t \\right ) + b \\left ( j \\right ) u _ { y } \\left ( x , j , t \\right ) = 0 j = 0 , 1 x \\in R ^ { n } , t \\in \\left ( 0 , 1 \\right ) . \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} q ( \\beta ) : = d i v _ { \\Sigma } ( \\beta ( \\nu , \\cdot ) ) - \\frac { 1 } { 2 } t r _ { \\Sigma } ( \\nabla ^ M _ { \\nu } \\beta ) \\ 0 \\Sigma \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} s ( f , \\Psi , \\mathcal { P } ^ 1 ) = \\sum _ { k } f ( x _ k ^ 1 ) \\ , | I ^ 1 _ k | _ \\Psi \\ , , x _ k ^ 1 \\in I ^ 1 _ k , \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} H ( u ) : = | | ( 1 - P ) h ( u ) | | e + P ( h ( u ) ) , \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 w } { \\partial { } ^ 2 t } ( \\xi , t ) = - \\frac { 1 } { \\rho ( \\xi ) } \\frac { \\partial } { \\partial \\xi } \\Bigl ( T ( \\xi ) \\frac { \\partial { } w } { \\partial \\xi } ( \\xi , t ) \\Bigr ) \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} d _ { 0 , 1 } d _ { 0 , - 1 } - d _ { 1 , 1 } d _ { - 1 , - 1 } = 0 . \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} u _ { \\sigma } ( z ) = I \\left ( \\frac { 1 + z } { z } \\right ) , \\quad I ( w ) = \\int _ { [ 0 , + \\infty ] } \\frac { w + t } { 1 - t w } \\ , d \\sigma ( t ) , w \\in \\mathbb { C } \\setminus \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} h ( z ) = z ^ { 4 } + p z ^ { 3 } + q z ^ { 2 } + u z + v . \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} \\Pi u ( z ) = \\sum _ { j = 0 } ^ { N - 1 } \\frac { q _ j z } { 1 - q _ j z } \\ , . \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} & \\frac { d X ( s ) } { d s } = \\beta _ \\varepsilon \\big ( V ( s ) \\big ) + \\big [ V ( s ) - \\beta _ \\varepsilon \\big ( V ( s ) \\big ) \\big ] \\eta _ \\varepsilon \\big ( X ( s ) \\big ) , X ( t ) = x \\\\ [ 5 p t ] & \\frac { d V ( s ) } { d s } = - \\mathbf { B } \\big ( s , X ( s ) , V ( s ) \\big ) , V ( t ) = v \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} \\lim _ { x \\uparrow c _ { 0 } } \\frac { p _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ( x ) } { \\left | x - c _ { 0 } \\right | ^ { 1 / 3 } } = \\frac { - a _ { 1 } } { \\pi \\sqrt [ 3 ] { \\left | c _ { 3 } \\right | } } \\sin \\frac { \\theta } { 3 } , \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} f _ \\epsilon ( t ) = \\begin{cases} 0 & 0 \\leq t < \\epsilon , \\\\ \\frac { 1 } { \\epsilon } ( t - \\epsilon ) & \\epsilon \\leq t \\leq 2 \\epsilon , \\\\ 1 & \\end{cases} \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} s & = \\ , ( 0 , 1 , 2 , 0 , 1 , 4 , { \\bf 1 } , 2 , 1 , 1 ) \\mbox { b y s u b s t i t u t i o n } ( 3 ) \\mbox { o f } \\R _ 1 , \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , 4 , 2 , 4 , { \\bf 1 } , 1 ) \\mbox { b y s u b s t i t u t i o n } ( 1 ) \\mbox { o f } \\R _ 1 , \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , 1 , 4 , 2 , 4 , 4 , 1 ) . \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} p _ { \\nu _ { \\beta } } \\left ( t \\right ) = \\left ( t r \\right ) ^ { \\frac { 1 } { \\beta - 1 } } \\frac { \\sin \\left ( \\frac { \\beta } { \\beta - 1 } f ( r ) \\right ) } { \\sin ( f ( r ) ) } \\frac { \\left | 1 - r e ^ { i f ( r ) } \\right | ^ { 2 } } { \\left | 1 - \\eta _ { \\nu } \\left ( r e ^ { i f ( r ) } \\right ) \\right | ^ { 2 } } \\ , p _ { \\mu } \\left ( t \\right ) . \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} 0 \\leq \\abs { \\norm x - \\norm y } \\leq \\norm { x - y } = 0 , \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} u \\left ( x , t \\right ) = F ^ { - 1 } \\left [ e ^ { i \\hat { A } _ { \\xi } t } \\hat { f } \\left ( \\xi \\right ) \\right ] , \\hat { A } _ { \\xi } = \\hat { A } \\left ( \\xi \\right ) - \\left \\vert \\xi \\right \\vert ^ { 2 } , \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} g _ { i j } = \\langle F _ i , F _ j \\rangle \\ , . \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} M _ { K } = \\left ( \\begin{array} { r r r r r r r r r r r r r r r r r r r r } 1 & 0 & 0 & - 1 & 0 \\\\ - 1 & 0 & 0 & 1 & 0 \\\\ 0 & 1 & - 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 \\\\ 0 & - 1 & 1 & 0 & 0 \\end{array} \\right ) , M _ { L , M } = \\left ( \\begin{array} { r r r r r r r r r r r r } 1 & 0 & 0 & 0 & - 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 \\\\ - 1 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & - 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & - 1 & 0 & 1 & 0 & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} K ( \\Psi ' ; M ' ; ( \\lambda , 0 ^ { k + 1 } , 1 ^ r ) ) = \\sum _ { a = 0 } ^ r \\sum _ { \\substack { \\mu = \\lambda + \\epsilon _ { S } + \\epsilon _ { S ' } \\\\ S \\subset \\{ \\ell + r - k + 1 - a , \\ldots , \\ell \\} \\\\ | S | = a \\\\ S ' = \\{ \\ell + 1 , \\ldots , \\ell + r - a \\} } } K ( \\Psi '' ; ( M ' \\setminus L ( E ) ) \\sqcup S ; \\mu ) \\ , . \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} ( a b ) \\cdot u = a \\cdot ( b \\cdot u ) , ( a \\cdot u ) \\cdot b = a \\cdot ( u \\cdot b ) , ( u \\cdot a ) \\cdot b = u \\cdot ( a b ) , \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} A ^ \\dagger _ n f = \\sum _ { j = 0 } ^ \\infty \\langle f , b _ { j , n - 1 } \\rangle \\ , A ^ { \\dagger } _ n b _ { j , n - 1 } = \\sum _ { j = 0 } ^ \\infty \\langle f , b _ { j , n - 1 } \\rangle \\ , b _ { j , n } , \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} & 4 \\tau ^ { 2 } \\langle \\Delta ' u , | \\nabla \\phi | ^ { 2 } u \\rangle \\\\ = & \\tau ^ { 2 } \\langle \\Delta ' u , | x ' | ^ { 2 } u \\rangle + 1 6 \\tau ^ { 2 } \\langle \\Delta ' u , ( x _ { n + 1 } ^ { 1 - 2 s } - x _ { n + 1 } ) ^ { 2 } u \\rangle \\\\ \\ge & - \\tau ^ { 2 } ( 1 + \\epsilon _ { 0 } ) \\| \\nabla ' u \\| ^ { 2 } - \\tau ^ { 2 } C \\epsilon _ { 0 } ^ { - 1 } \\| u \\| ^ { 2 } - 1 6 \\tau ^ { 2 } \\| ( x _ { n + 1 } ^ { 1 - 2 s } - x _ { n + 1 } ) \\nabla ' u \\| ^ { 2 } . \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} P _ { J , K } \\circ T _ { J , K } ^ { - 1 } & = P _ { J , K } \\circ \\mathcal { D } _ { J , K } ^ { - 1 } \\circ \\theta ^ { - 1 } \\circ \\mathcal { D } _ { J , K } = P _ { K , J } \\circ \\theta ^ { - 2 } \\circ \\mathcal { D } _ { J , K } \\\\ & = P _ { K , J } \\circ \\theta ^ { - 1 } \\circ \\mathcal { D } _ { J , K } = P _ { K , J } \\circ \\mathcal { D } _ { J , K } ^ { - 1 } \\circ \\theta = P _ { J , K } , \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} P ( x ) & = \\tfrac { 1 } { 6 } x _ { 4 } ^ { 3 } - \\tfrac { 1 } { 2 } x _ { 4 } ( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } - x _ { 3 } ^ { 2 } ) + x _ { 1 } x _ { 2 } x _ { 3 } \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} \\Phi ( t ) = B \\mu ( t ) + B _ 1 r _ 1 \\Bigl ( \\frac { t } { T } \\Bigr ) ^ a + B _ 2 r _ 2 \\Bigl ( \\frac { t } { T } \\Bigr ) ^ a + B _ 2 L r _ 1 \\Bigl ( \\frac { t } { T } \\Bigr ) ^ a \\log \\Bigl ( \\frac { T } { t } \\Bigr ) . \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} & b ^ { ( k ) } = i ( b ^ { ( k _ 1 ) } ) s ^ { ( k ) } , \\textit { w h e r e } s ^ { ( k ) } \\textit { i s a s m a l l p i e c e ( p o s s i b l y e m p t y ) } , \\\\ & U ^ { ( k ) } = A ^ { ( k _ 1 ) } b ^ { ( k ) } B ^ { ( k ) } , \\textit { w h e r e } A ^ { ( k _ 1 ) } \\textit { i s a p r e f i x o f } U ^ { ( k ) } , \\ B ^ { ( k ) } \\textit { i s a s u f f i x o f } U ^ { ( k ) } , \\\\ & k = k _ 1 , \\ldots , k _ 2 . \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} P _ 2 = \\begin{bmatrix} 0 & 0 & I & 0 \\\\ I & 0 & 0 & 0 \\\\ 0 & I & 0 & 0 \\\\ 0 & 0 & 0 & I \\end{bmatrix} \\in \\mathbb { R } ^ { 2 ^ { k + 1 } ( p + q ) \\times 2 ^ { k + 1 } ( p + q ) } , \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} | \\beta _ 3 | \\leq 2 ^ { 1 0 } A ^ { - 1 } N _ 1 , \\ ( \\mathfrak { R } _ 1 ^ { A } + \\mathfrak { R } _ 2 ^ { A } ) \\cap \\mathfrak { R } _ 3 ^ { A } \\not = \\emptyset , \\mathfrak { R } _ 1 ^ { A } \\cap \\mathcal { T } _ { k _ 1 } ^ { A , d } \\not = \\emptyset , \\mathfrak { R } _ 2 ^ { A } \\cap \\mathcal { T } _ { k } ^ { A , d } \\not = \\emptyset . \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} \\omega _ { n + 1 } ^ { \\frac { 2 s - 1 } { 2 } } e ^ { \\tau \\varphi } \\tilde { f } = L ^ { + } \\overline { v } = ( S - A + ( I ) + ( I I ) + ( I I I ) ) \\overline { v } \\quad \\mathcal { S } _ { + } ^ { n } \\times \\mathbb { R } , \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} \\Gamma _ { 1 1 } ^ 1 = \\Gamma _ { 2 2 } ^ 1 = \\Gamma _ { 1 2 } ^ 2 = 0 . \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} h _ \\rho ( z , \\rho ) = 0 , \\forall z \\in [ z _ 1 , z _ 2 ] \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} | \\beta | & = \\sum _ { i = 1 } ^ n \\beta _ i = \\sum _ { i \\in A ^ c } \\frac { \\alpha _ i } { 2 } = \\frac { | \\alpha | } { 2 } - \\sum _ { i \\in A } \\frac { \\alpha _ i } { 2 } = \\frac { m } { 2 } - l , \\intertext { a n d } | \\gamma | & = \\sum _ { i = 1 } ^ n \\gamma _ i = \\sum _ { i \\in A } \\frac { \\alpha _ i - 1 } { 2 } = \\frac { 2 l - | A | } { 2 } = l - k . \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} F \\models \\phi ( b _ { 1 } , . . . , b _ { M } ) : = \\exists \\overline { a } \\bigwedge _ { i = 1 } ^ { k } f _ { i } ( \\overline { a } ) = 0 \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} B ( e x f , f y g ) & = B ( e x f , f ) y g + f B ( e x f , y g ) = e B ( e , x ) f y g + f B ( e x f , y g ) . \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} U ^ { \\overline { e } } _ { w v } = ( U ^ e _ { v w } ) ^ * , \\\\ | \\Phi ^ { \\overline { e } } _ { w v } | = U ^ e _ { v w } | \\Phi ^ e _ { v w } | ^ { - 1 } ( U ^ e _ { v w } ) ^ * . \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} \\xi _ { ( g , x ) } * \\xi _ { ( h , y ) } & = \\sum _ { f _ 1 , f _ 2 \\in \\Gamma } \\delta _ { f _ 1 g , f _ 2 } ( f _ 1 \\otimes f _ 2 h \\otimes x ^ { f _ 1 } ( y \\gamma ) ^ { f _ 2 } ) = \\sum _ { f \\in \\Gamma } f \\otimes f g h \\otimes x ^ { f } ( y \\gamma ) ^ { f g } \\\\ & = \\xi _ { ( g h , x ( y \\gamma ) ^ g ) } . \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} \\mathrm { d i s } _ \\Delta R _ n = \\mathrm { d i s } _ { \\overline { \\Delta } } R _ n \\leq \\frac { 1 } { 2 } \\left ( 2 \\frac { r } { n } \\right ) = \\frac { r } { n } , \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} C ^ \\infty _ M = C ^ \\infty _ { M , ( 0 ) } \\supset C ^ \\infty _ { M , ( 1 ) } \\supset \\cdots \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} Q \\left ( x \\right ) = b _ { 1 } ^ { - \\frac { n } { 2 } } e ^ { - b _ { 2 } \\left \\vert x \\right \\vert ^ { p } } b _ { 1 } b _ { 2 } > 0 p > 1 ; \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} = e ^ { - \\lambda t } \\sum _ { k = 0 } ^ \\infty I _ { 2 k + 2 } \\Bigl ( \\frac { 2 \\lambda } { c _ 1 + c _ 2 } \\sqrt { ( c _ 1 t - \\beta ) ( c _ 2 t + \\beta ) } \\Bigr ) \\Biggl [ \\Biggl ( \\sqrt { \\frac { c _ 2 t + \\beta } { c _ 1 t - \\beta } } \\Biggr ) ^ { 2 k + 2 } - \\Biggl ( \\frac { c _ 2 } { c _ 1 } \\sqrt { \\frac { c _ 1 t - \\beta } { c _ 2 t + \\beta } } \\Biggr ) ^ { 2 k + 2 } \\ : \\Biggr ] \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} K _ { X _ { n , s } } : = - ( n + 1 ) H + ( n - 1 ) E _ 1 + \\dots + ( n - 1 ) E _ s . \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} F \\circ H = H \\circ R , \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} \\frac { 1 } { \\displaystyle \\prod _ { i = 1 } ^ { n - 1 } ( q ) _ { k _ i } } = \\sum _ { \\eta } q ^ { { \\rm c o d i m } ( \\eta ) } \\frac { 1 } { \\displaystyle \\prod _ { 1 \\leq i \\leq j \\leq n - 1 } ( q ) _ { m _ { [ i , j ] } ( \\eta ) } } \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} \\widehat { d F } _ { \\mu } ( v ) _ y : = \\int _ { f ^ { - 1 } ( { y } ) } d f ( v _ { x } ) d \\mu ^ y ( x ) , \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} S _ 3 ( f , \\xi , \\eta ; x ) = \\frac 5 6 \\frac { \\kappa '' _ { \\eta \\eta } ( x ) - \\kappa '' _ { \\xi \\xi } ( x ) } { \\kappa ( x ) } , \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} ( x _ { j - 1 } \\cdots x _ { i } ) ^ { m } = q ^ { - ( j - i - 1 ) \\frac { m ( m + 1 ) } { 2 } } x _ { i } ^ { m } x _ { i + 1 } ^ { m } \\cdots x _ { j - 1 } ^ { m } . \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} G _ { L + 2 } = G _ { L + 1 } + \\sum _ { i = 3 } ^ { m + 2 } G _ i + N G _ 2 \\leq 1 + \\sum _ { i = 1 } ^ { L + 1 } G _ i , \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} b _ { 0 } = - \\frac { 1 } { 2 } \\mu . \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } & \\partial _ t | D | ^ s \\omega + u \\cdot \\nabla | D | ^ s \\omega - \\partial _ 1 ^ 2 | D | ^ s \\omega = \\partial _ 1 | D | ^ s \\theta - [ | D | ^ s , u \\cdot \\nabla ] \\omega , \\\\ & \\partial _ t | D | ^ s \\theta + u \\cdot \\nabla | D | ^ s \\theta = - [ | D | ^ s , u \\cdot \\nabla ] \\theta . \\end{array} \\right . \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} a _ i = d _ i + \\binom { m - i } { 1 } d _ { i + 1 } + \\cdots + \\binom { m - i } { k - 1 - i } d _ { k - 1 } + d _ k \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} k _ { I , \\sigma _ t } : = t ( \\sum _ { i = 1 } ^ { n + 3 } m _ i ) + \\sum _ { p \\in I } m _ p - { ( ( n + 1 ) t + | I | - 1 ) d . } \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} K ( \\Psi ; M ; \\gamma ) = K ( \\Psi \\setminus ( i , j ) ; M ; \\gamma ) + K ( \\Psi ; M ; \\gamma + \\epsilon _ { i } - \\epsilon _ { j } ) \\ , , \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} \\epsilon ( X / k ) : = \\log _ p ( { \\rm l e n g t h } _ R R ) . \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} \\phi ( y ) \\phi ( x ) = \\phi ( x ) \\phi ( - y x ) \\phi ( y ) . \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} \\lambda _ L - 1 \\geq \\frac { L + 2 } { L ^ 2 - 4 } = \\frac { 1 } { L - 2 } \\quad \\iff \\lambda _ L \\geq \\frac { L - 1 } { L - 2 } . \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} F _ { m } ( z ) = \\alpha + \\frac { e ^ { 2 m \\pi i / 3 } } { ( c _ { 3 } ) ^ { 1 / 3 } } ( z - c _ { 0 } ) ^ { 1 / 3 } + e ^ { 2 m \\pi i / 3 } \\sum _ { n = 2 } ^ { \\infty } d _ { n , m } ( z - c _ { 0 } ) ^ { n / 3 } , m = 0 , 1 , 2 , \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} \\{ ( I - P _ { M _ k } ) ( I - P _ { M _ { k - 1 } } ) \\cdots ( I - P _ { M _ { 1 } } ) ^ { n } x \\} _ { n = 0 } ^ \\infty \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} \\tilde { u } \\left ( x , t \\right ) = \\left ( \\sqrt { \\alpha \\beta } \\rho ^ { - 1 } \\left ( t \\right ) \\right ) ^ { \\frac { n } { 2 } } u \\left ( \\sqrt { \\alpha \\beta } x \\rho ^ { - 1 } \\left ( t \\right ) , \\beta t \\rho ^ { - 1 } \\left ( t \\right ) \\right ) e ^ { \\varphi } . \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} \\kappa \\Sigma ^ 0 _ { 1 + \\alpha } ( X ) & : = \\ @ N _ \\kappa ^ \\alpha ( \\ @ O _ \\kappa ( X ) ) \\subseteq \\ @ B _ \\kappa ( X ) , \\\\ \\kappa \\Pi ^ 0 _ { 1 + \\alpha } ( X ) & : = \\neg \\ @ N _ \\kappa ^ \\alpha ( \\ @ O _ \\kappa ( X ) ) \\subseteq \\ @ B _ \\kappa ( X ) , \\\\ \\kappa \\Delta ^ 0 _ { 1 + \\alpha } ( X ) & : = \\kappa \\Sigma ^ 0 _ { 1 + \\alpha } ( X ) \\cap \\kappa \\Pi ^ 0 _ { 1 + \\alpha } ( X ) = \\ @ N _ \\kappa ^ \\alpha ( \\ @ O _ \\kappa ( X ) ) _ \\neg \\subseteq \\ @ B _ \\kappa ( X ) . \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} T _ k ( u ) : = { \\bf P } ( | \\eta _ k | > u ) , \\ \\ \\ S _ k ( u ) : = { \\bf P } ( | \\eta _ k | \\le u ) = 1 - T _ k ( u ) , \\ \\ u \\ge 0 . \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n y _ j \\log z _ j ^ { - 1 } \\ge \\sum _ { j = 1 } ^ { n } y _ j \\log ( y _ j Z / Y ) ^ { - 1 } . \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} x _ { n + 1 } ^ { \\frac { 2 s - 1 } { 2 } } f = \\Delta \\tilde { u } + \\mathring { c } _ { s } x _ { n + 1 } ^ { - 2 } \\tilde { u } + \\sum _ { j , k = 1 } ^ { n } ( a _ { j k } - \\delta _ { j k } ) \\partial _ { j } \\partial _ { k } \\tilde { u } , \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} \\psi : = k + h ^ \\vee \\phi _ n : = \\frac { \\psi } { n \\psi - 1 } , \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} \\Big | \\frac { P _ { 1 , 1 } } { P _ { 1 , 2 } } ( \\l ) + \\tau \\Big | = \\frac { | Q _ 1 ( \\l , 1 , \\tau ) | } { | P _ { 1 , 2 } ( \\l ) | } \\le r ^ { 1 / 2 } , \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} | C | \\leq q ^ { \\sum _ { i = j } ^ t m _ i n _ i - m _ j \\delta } . \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} O : = \\{ ( a , b ) \\in \\mathbb R ^ 2 \\ , | \\ , a \\leq 0 \\ , \\lor \\ , b \\leq 0 \\} , \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} & \\left ( \\frac { \\lambda ( t ) } { \\lambda ( x ) } \\right ) ^ { 2 } \\left ( \\int _ { \\frac { 4 } { \\lambda ( x ) ^ { 2 } } } ^ { \\infty } \\frac { \\omega \\lambda ( x ) ^ { 2 } d \\omega } { \\omega ^ { 5 } } \\right ) ^ { 1 / 2 } \\frac { 1 } { \\lambda ( x ) ^ { 3 } x ^ { 4 } \\log ^ { 1 - 2 \\alpha b } ( x ) } \\leq C \\left ( \\frac { \\lambda ( t ) } { \\lambda ( x ) } \\right ) ^ { 2 } \\frac { 1 } { x ^ { 4 } \\log ^ { b + 1 - 2 \\alpha b } ( x ) } \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} f ' ( \\theta ) = 2 \\cos ( \\theta ) | x | \\int _ { { \\R ^ k } } \\left [ \\tilde g ' ( | x | ^ 2 + \\tau ^ 2 + 2 | x | \\tau _ 1 \\sin ( \\theta ) ) - \\tilde g ' ( | x | ^ 2 + \\tau ^ 2 - 2 | x | \\tau _ 1 \\sin ( \\theta ) ) \\right ] \\tau _ 1 K ( \\tau ) d \\tau \\ , . \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } & \\frac { d P _ { t } } { d t } = F _ { t } P _ { t } + P _ { t } F _ { t } ^ { \\top } - P _ { t } G _ { t } ^ { \\top } R _ { t } ^ { - 1 } G _ { t } P _ { t } + Q _ { t } , \\\\ & P ( 0 ) = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} \\forall a \\in \\mathcal H ( A ) , x , y \\in \\mathcal H ( L ) : [ x , a y ] = \\mu ( x ) a y + ( - 1 ) ^ { \\bar a \\bar x } a [ x , y ] . \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} \\sigma ( m \\alpha + n \\alpha ^ \\vee ) = ( - 1 ) ^ { m ( n + 1 ) } . \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} f = \\sum _ { j = 1 } ^ { \\infty } \\hat a _ j \\hat u _ { j , \\lambda } = \\sum _ { j = 1 } ^ { \\infty } \\hat a _ j \\sqrt { \\mu _ j ( \\lambda ) + b } \\cdot \\frac { \\hat u _ { j , \\lambda } } { \\sqrt { \\mu _ j ( \\lambda ) + b } } = \\sum _ { j = 1 } ^ { \\infty } \\hat a _ j \\sqrt { \\mu _ j ( \\lambda ) + b } \\cdot \\gamma _ 0 ( u _ { j , \\lambda } ) , \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} ( f \\# g ) ( v ) = \\frac { 1 } { ( 2 \\pi ) ^ { 2 n } } \\int _ { H _ p } e ^ { 2 i \\ , \\omega ( x , y ) } f ( v + x ) g ( v + y ) d x \\ , d y \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} \\| f \\| _ 1 = \\int _ { X } | f ( x ) | \\ , d \\mu ( x ) \\leq \\left ( 1 - \\frac { c _ f } { 2 } \\right ) \\sqrt { \\mu ( X ) } . \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} [ g ( t ) , h ( t ) ] \\supset [ - \\underline h ( t - t _ 0 ) , \\underline h ( t - t _ 0 ) ] \\supset [ - c _ 0 t + C , c _ 0 t - C ] \\mbox { w i t h } C : = C _ 0 + c _ 0 t _ 0 , \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} ( x + y ) z _ { x y } + 2 \\sqrt { z _ x z _ y } = 0 , \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} A = \\frac { 1 } { 2 } u _ 1 u _ 2 \\sin \\phi _ 3 \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} [ \\rho _ a , \\rho _ b ] = : C ^ c { } _ { a b } \\rho _ c , C ^ c { } _ { a b } \\in C ^ \\infty ( M ) , \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { Z } _ r + \\frac { \\alpha + \\frac { 1 } { 2 } } { 2 \\alpha r } \\xi \\mathcal { Z } _ \\xi = & \\mathcal { F } _ \\xi [ \\mathcal { Z } , \\mathcal { U } _ \\xi ] \\\\ \\mathcal { U } _ r + \\frac { \\alpha + \\frac { 1 } { 2 } } { 2 \\alpha r } ( \\xi \\mathcal { U } _ \\xi - 1 ) = & \\mathcal { E } _ \\xi [ \\mathcal { Z } , \\mathcal { U } ] \\end{aligned} \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} T _ { ( f _ 1 , \\tau _ 1 ) ^ * , ( f _ 2 , \\tau _ 2 ) ^ * } \\circ T _ { ( f _ 0 , \\tau _ 0 ) ^ * , ( f _ 1 , \\tau _ 1 ) ^ * } = T _ { ( f _ 0 , \\tau _ 0 ) ^ * , ( f _ 2 , \\tau _ 2 ) ^ * } . \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} \\bigl ( \\ell _ { \\min \\{ p , n - 1 \\} } ^ { ( | p - n + 1 | ) } ( t ) \\bigr ) ^ 2 = \\bigl ( \\ell _ { \\min \\{ q , n - 1 \\} } ^ { ( | q - n + 1 | ) } ( t ) \\bigr ) ^ 2 , \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} \\partial _ { n + 1 } u & = e ^ { \\tau \\phi } \\bigg ( x _ { n + 1 } ^ { \\frac { 1 - 2 s } { 2 } } \\partial _ { n + 1 } w - \\frac { 2 s - 1 } { 2 } x _ { n + 1 } ^ { - \\frac { 1 + 2 s } { 2 } } w \\bigg ) + x _ { n + 1 } ^ { \\frac { 3 - 2 s } { 2 } } R \\\\ \\nabla ' u & = e ^ { \\tau \\phi } x _ { n + 1 } ^ { \\frac { 1 - 2 s } { 2 } } \\nabla ' w + x _ { n + 1 } ^ { s + \\frac { 1 } { 2 } } R ' , \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} G ( s ) = 0 . \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} C ^ \\infty ( U ) = C ^ \\infty ( U ) _ { ( 0 ) } \\supset C ^ \\infty ( U ) _ { ( 1 ) } \\supset \\cdots , \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} \\sum _ i \\frac { x _ i } { 1 - x _ i } \\le \\sum _ i \\frac { x _ i } { 1 - \\sum _ j x _ j } = \\frac { \\sum _ i x _ i } { 1 - \\sum _ i x _ i } , \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} \\delta K _ E = 2 H \\ , \\delta ( 2 H ) - \\frac { 1 } { 2 } \\delta | h | ^ 2 = 2 H \\Delta u - \\langle h , \\ , u \\rangle + 2 H K u . \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} M _ 2 \\leq C ( \\| | D | ^ { s } \\omega \\| _ { L ^ 2 } + \\| \\nabla u \\| _ { L ^ { \\infty } } ) \\| | D | ^ s X \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} \\nabla ^ \\mu F _ { \\mu \\nu } = 0 . \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} \\Phi ( z ) = \\gamma + z + N _ { \\sigma } , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} \\dot { x } & = y - y ^ 4 , \\\\ \\dot { y } & = - 1 , \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} P \\mu = \\sum _ { i = 1 } ^ { k } p _ i \\ , g _ i \\mu , \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} \\langle 1 \\ , | \\ , f _ N \\rangle = \\frac { 1 } { \\sqrt { N + \\gamma _ N } } \\zeta _ N \\ . \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} F ( x , y ; z ; \\mu ) = \\begin{bmatrix} f ( x , y ; z ; \\mu ) \\\\ g ( x , y ; z ; \\mu ) \\end{bmatrix} . \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} B _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { 1 } - \\mathbf { z } ) = ( - 1 ) ^ { | \\mathbf { m } | } B _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } ) \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} \\mu _ j ( \\lambda ) = \\min _ { \\substack { U \\subset H ^ 2 ( \\Omega ) \\\\ { \\rm d i m } U = j } } \\max _ { \\substack { u \\in U \\\\ \\gamma _ 0 ( u ) \\ne 0 } } \\frac { \\mathcal { Q } _ { \\lambda , N } ( u , u ) } { \\int _ { \\partial \\Omega } u ^ 2 d \\sigma } , \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} P ( x ) & = \\det \\begin{pmatrix} x _ { 1 1 } & x _ { 1 2 } / \\sqrt { 2 } & x _ { 1 3 } / \\sqrt { 2 } \\\\ x _ { 1 2 } / \\sqrt { 2 } & x _ { 2 2 } & x _ { 2 3 } / \\sqrt { 2 } \\\\ x _ { 1 3 } / \\sqrt { 2 } & x _ { 2 3 } / \\sqrt { 2 } & x _ { 3 3 } \\end{pmatrix} \\\\ & = x _ { 1 1 } x _ { 2 2 } x _ { 3 3 } - \\tfrac { 1 } { 2 } \\left ( x _ { 1 1 } x _ { 2 3 } ^ { 2 } + x _ { 2 2 } x _ { 1 3 } ^ { 2 } + x _ { 3 3 } x _ { 1 2 } ^ { 2 } \\right ) + \\tfrac { 1 } { \\sqrt { 2 } } x _ { 1 2 } x _ { 2 3 } x _ { 1 3 } , \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} d \\mu _ y ^ { c , \\ , t } \\left ( z \\right ) = t \\ , d \\mu ^ c _ y \\left ( z \\right ) + \\left ( 1 - t \\right ) \\delta _ y \\left ( z \\right ) \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} P _ { 2 k + 1 } ^ - \\{ M ( s ) \\le \\beta \\} - P _ { 2 k + 2 } ^ - \\{ M ( s ) \\le \\beta \\} = - 2 \\beta \\binom { 2 k + 1 } { k } \\frac { ( c ^ 2 s ^ 2 - \\beta ^ 2 ) ^ k } { ( 2 c s ) ^ { 2 k + 2 } } ( c s - \\beta ) , \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} \\Phi _ e : X _ e \\to X , X _ e : = X , \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} Y ( u _ 0 ) = \\sum _ { \\substack { d _ 1 , \\ldots , d _ { 2 t } : \\\\ d _ 1 = 1 , \\ , \\ , | d _ { i + 1 } - d _ i | \\le 1 } } \\ , \\ , \\sum _ { a _ 1 , \\ldots , a _ { 2 t } \\in [ 2 t ] } \\ , \\ , \\sum _ { \\substack { u _ 1 , \\ldots , u _ { 2 t } : \\\\ u _ i \\in B ( d _ i , a _ i ) } } W ( u _ 0 , u _ 1 , \\ldots , u _ { 2 t } , u _ 0 ) . \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} J \\partial _ x \\phi ( x ) = \\lambda A ( x ) \\phi ( x ) + B ( x ) \\phi ( x ) \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} 1 + \\sum _ { i = 1 } ^ { N } { a _ i } \\leq n < 1 + \\sum _ { i = 1 } ^ { N + 1 } { a _ i } . \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} T ^ { \\prime } - \\sum _ { s = 1 } ^ l \\gamma _ s T _ s ^ { \\prime } = \\sum _ { j _ 1 , \\ldots , j _ m } \\alpha _ { j _ 1 , \\ldots , j _ m } a ^ { ( 1 ) } _ { j _ 1 } \\otimes \\ldots \\otimes a ^ { ( m ) } _ { j _ m } , \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} d ( k ) = c ( k ) \\mbox { i f } k \\leq 0 , \\\\ \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} f _ L ( 0 , 0 ) = f _ R ( 0 , 0 ) = 0 , \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\Delta _ q \\partial _ X u \\cdot \\Delta _ q X ~ d \\tau \\leq C b _ q 2 ^ { - 2 q s } ( \\| X \\| _ { H ^ s } + \\| \\nabla u \\| _ { L ^ \\infty } \\| X \\| _ { H ^ s } ^ 2 ) . \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} - \\Delta \\phi _ a = & a ( t ) \\ \\ \\Omega , \\\\ \\frac { \\partial \\phi _ a } { \\partial n } = & 0 \\ \\ \\partial \\Omega . \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} \\begin{array} { l } \\nabla f ( x ^ * ) + \\left [ \\displaystyle \\sum _ { i = 1 } ^ p [ y ^ I ] ^ * _ j \\nabla ^ 2 c _ j ( x ^ * ) - A ^ T A - { \\cal J } c _ { \\gamma } ( x ^ * ) ^ T { \\cal J } c _ { \\gamma } ( x ^ * ) - { \\cal J } c _ { \\beta } ( x ^ * ) ^ T { \\rm D i a g } ( \\eta _ { \\beta } ) { \\cal J } c _ { \\beta } ( x ^ * ) \\right ] \\lambda ^ * = 0 \\end{array} \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { j = 1 } ^ { k - 1 } b _ { i j } ^ { ( m ) } = \\sum _ { j = i } ^ { k - 1 } ( - 1 ) ^ { j - i } \\binom { k - i } { j - i } = \\sum _ { j = 0 } ^ { k - i - 1 } ( - 1 ) ^ { j } \\binom { k - i } { j } = 0 - ( - 1 ) ^ { k - i } = - v _ i , \\end{aligned} \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} \\psi _ 0 = ( { \\psi _ 0 } ^ + , { \\psi _ 0 } ^ - ) , \\psi _ j ^ 1 = ( { \\psi _ j ^ 1 } ^ + , { \\psi _ j ^ 1 } ^ - ) , \\psi _ j ^ 2 = ( { \\psi _ j ^ 2 } ^ + , { \\psi _ j ^ 2 } ^ - ) , \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} \\widetilde { h } _ \\epsilon ( y ) = f _ { \\tiny \\mbox { B E P } } ^ { \\delta = 0 } ( y ) ( 1 + \\gamma _ 1 - \\gamma _ 2 ) + f _ { \\tiny \\mbox { B E P } } ^ { \\delta = 2 } ( y ) ( \\gamma _ 2 - 2 \\gamma _ 1 ) + f _ { \\tiny \\mbox { B E P } } ^ { \\delta = 4 } ( y ) \\gamma _ 1 , \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} \\eta _ { \\nu _ { \\beta } } ( z ) = \\eta _ { \\nu } \\left ( \\eta _ { \\mu } ( z ) \\right ) , z \\in \\mathbb { C } \\setminus \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} A x = \\sum _ { i = 1 } ^ \\infty a _ i ( x ) y _ i , \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} f ( d ) = p _ 0 , f ' ( d ) = p _ 1 . \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} a ( n ) = ( 2 ^ { k + 1 } - 1 ) ( 2 ^ n + 2 ^ { n - 2 } ( n - 1 ) ) . \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} ( V _ a \\ , y _ b ) | _ { N \\cap U } = \\delta ^ a _ b \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} \\varphi ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) = ( F _ 1 , F _ 2 , F _ 3 , F _ 4 ) . \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} P ( z ) = \\sum _ { \\alpha \\in \\Lambda \\subseteq \\mathbb { N } _ { 0 } ^ { n } } x _ { \\alpha } z ^ { \\alpha } \\ , , \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} F _ I = \\frac { \\partial F } { \\partial X ^ I } : \\ I = 0 , \\cdots , n . \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} V = \\frac { \\kappa } { 6 } ( p ^ 0 ) ^ 2 + \\frac { 6 } { \\kappa } ( q _ 0 ) ^ 2 , \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} \\rho _ 1 ( c ( x , a ) , ( y , b ) ) & = \\rho _ 1 ( ( c x , c a ) , ( y , b ) ) = \\rho ( c x , y ) \\\\ & = ( - 1 ) ^ { \\bar c \\bar x } \\rho ( x , c y ) = ( - 1 ) ^ { \\bar c \\bar x } \\rho _ 1 ( ( x , a ) , ( c y , c b ) ) = ( - 1 ) ^ { \\bar c \\bar x } \\rho _ 1 ( ( x , a ) , c ( y , b ) ) . \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} & | 2 c _ { b } ( e _ { 1 } ( t ) - e _ { 2 } ( t ) ) \\int _ { 0 } ^ { \\infty } d \\xi \\frac { \\sin ( t \\xi ) } { t ^ { 2 } } \\psi _ { v _ { 2 } } ( \\xi , \\lambda _ { 1 } ( t ) ) | \\\\ & \\leq C | e _ { 1 } ( t ) - e _ { 2 } ( t ) | | \\int _ { 0 } ^ { \\infty } d \\xi \\frac { \\sin ( t \\xi ) } { t ^ { 2 } } \\psi _ { v _ { 2 } } ( \\xi , \\lambda _ { 1 } ( t ) ) | \\\\ & \\leq C \\frac { | e _ { 1 } ( t ) - e _ { 2 } ( t ) | } { t ^ { 3 } \\lambda _ { 1 } ( t ) } \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} \\Bigl [ \\{ \\nabla g _ i ( \\bar x ) \\ , | \\ , i \\in I ^ g ( \\bar x ) \\} & \\cup \\{ \\nabla G _ l ( \\bar x ) \\ , | \\ , l \\in I ^ { 0 + } ( \\bar x ) \\cup I ^ { 0 0 } ( \\bar x ) \\} \\\\ & \\cup \\{ \\nabla H _ l ( \\bar x ) \\ , | \\ , l \\in I ^ { + 0 } ( \\bar x ) \\cup I ^ { 0 0 } ( \\bar x ) \\} \\Bigr ] \\cup \\{ \\nabla h _ j ( \\bar x ) \\ , | \\ , j \\in \\mathcal P \\} \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} \\| \\psi \\| _ { L ^ \\infty ( Q _ { t _ * } ) } \\leq \\| \\psi _ { t _ * } \\| _ { L ^ \\infty ( \\Omega \\times \\mathbb { R } ^ 3 ) } = \\| \\varphi \\| _ { L ^ \\infty ( \\Omega \\times \\mathbb { R } ^ 3 ) } \\leq 1 . \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} \\tau _ { j } = \\frac { 1 } { \\omega _ { + } } \\arccos \\frac { K - b } { K } + \\frac { 2 j \\pi } { \\omega _ { + } } , j = 0 , 1 , 2 , \\dots \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} X & = \\{ 2 ^ { i } + \\lambda 2 ^ j : i , j \\in \\mathbb N , i , j \\leq n \\} , \\\\ Y & = \\{ - 2 ^ { k } - 2 ^ l : k , l \\in \\mathbb N , k , l \\leq n \\} , \\\\ G & = \\{ ( 2 ^ { i } + \\lambda 2 ^ j , - 2 ^ { i } - 2 ^ j ) : i , j \\in \\mathbb N , i , j \\leq n \\} . \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} \\begin{cases} D _ { 0 } ^ { \\alpha _ 2 } y ( t ) = f ( t ) + c ( t ) y ( t ) + b ( t ) y ' ( t ) , \\\\ a _ 1 y ( 0 ) + b _ 1 y ' ( 0 ) = \\gamma _ 1 , ~ ~ a _ 2 y ( 1 ) + b _ 2 y ' ( 1 ) = \\gamma _ 2 , \\end{cases} \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} A ^ { \\shortparallel } ( x ) = ( A _ 1 , \\dotsc , A _ d ) ( x , 0 ) . \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} f = \\sum _ { j = 0 } ^ m f _ { n _ j } , f _ { n _ j } \\in \\mathcal { H } _ { n _ j } ^ d , j = 0 , 1 , \\dots , m , \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{gather*} \\alpha ( e _ 1 ) = x y \\frac { \\partial } { \\partial y } , \\alpha ( e _ 2 ) = y \\frac { \\partial } { \\partial y } , \\\\ \\alpha ( e _ 3 ) = y \\frac { \\partial } { \\partial x } , \\alpha ( e _ 4 ) = y \\frac { \\partial } { \\partial y } - x \\frac { \\partial } { \\partial x } \\end{gather*}"}
-{"id": "3789.png", "formula": "\\begin{align*} a _ { n + 1 } \\leq 1 + \\sum _ { i = 1 } ^ { n } { a _ i } \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} B ( e _ { x y } e _ { z w } , e _ { u v } ) = e _ { x y } B ( e _ { z w } , e _ { u v } ) + B ( e _ { x y } , e _ { u v } ) e _ { z w } . \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} R _ q ^ { - 1 } ( A \\times B ) & = \\{ s \\in S : s \\otimes q \\in \\widehat { A \\times B } \\} \\\\ & = \\{ s \\in S : A \\times B \\in s \\otimes q \\} \\\\ & = \\{ s \\in S : \\{ u : \\{ t : ( u , t ) \\in A \\times B \\} \\in q \\} \\in s \\} \\\\ & = \\{ s \\in S : \\{ t : ( s , t ) \\in A \\times B \\} \\in q \\} \\\\ & = \\{ s \\in S : s \\in A , B \\in q \\} = A \\in \\mathfrak { m } \\\\ \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} e ^ { \\beta } + \\beta - \\frac { 1 } { 2 } > \\sum _ { k = 3 } ^ \\infty \\frac { k ^ { k - 2 } } { k ! } \\Bigg [ k - 1 - k \\beta \\Bigg ] \\big ( \\beta e ^ { - \\beta } \\big ) ^ { k - 2 } . \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} \\xi = \\sum _ { 1 \\leq i _ 1 < \\ldots < i _ { \\ell - 1 } \\leq n + k } \\ , \\xi _ { i _ 1 \\ldots i _ { \\ell - 1 } } d x ^ { i _ 1 } \\wedge \\ldots \\wedge d x ^ { i _ { \\ell - 1 } } \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} c o n f ( \\mathfrak { J } ) = g ^ { - 1 } \\oplus s t r \\left ( \\mathfrak { J } \\right ) \\oplus g ^ { 1 } . \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} \\deg ( h _ 4 ) & \\leq 2 g + \\sum _ y ( \\lfloor m _ y / 4 \\rfloor + 1 ) \\deg _ k ( y ) \\\\ & \\leq 2 g + \\sum _ y ( m _ y / 2 ) \\deg _ k ( y ) = 3 g + \\frac { 1 } { 2 } ( \\deg ( f _ 3 ) - 1 ) \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} \\lambda _ { 1 ; p , f } \\int _ { \\Omega } | u | ^ { \\alpha + p } \\ , d \\mu & = - \\int _ { \\Omega } | u | ^ { \\alpha + 1 } \\Delta _ { p , f } u \\ , d \\mu \\\\ & = \\int _ { \\Omega } | \\nabla u | ^ { p - 2 } \\langle \\nabla u , \\nabla | u | ^ { \\alpha + 1 } \\rangle \\ , d \\mu \\\\ & = ( \\alpha + 1 ) \\int _ { \\Omega } | u | ^ { \\alpha } | \\nabla u | ^ { p } \\ , d \\mu , \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} X ^ I & = p ^ I + i \\phi ^ I \\\\ F _ I & = q _ I + i \\psi _ I . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} p _ { { \\boldsymbol { \\mu } } , \\boldsymbol { \\Sigma } } ( { \\bf { x } } ) = N _ { \\boldsymbol { \\Sigma } , \\nu } \\Big [ 1 + \\frac { 1 } { \\nu } ( { \\bf { x } } - { \\boldsymbol { \\mu } } ) ^ \\top \\boldsymbol { \\Sigma } ^ { - 1 } ( { \\bf { x } } - { \\boldsymbol { \\mu } } ) \\Big ] _ + ^ { - \\frac { \\nu + d } { 2 } } , \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} | \\partial _ { t r } N _ { 2 } ( f ) | ( t , r ) \\leq \\begin{cases} \\frac { C } { t ^ { 5 } \\log ^ { 3 b } ( t ) ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C \\log ^ { 3 } ( r ) } { r ^ { 2 } | t - r | ^ { 5 } } + \\frac { C \\log ^ { 3 } ( r ) } { r ^ { 5 / 2 } \\sqrt { t } ( t - r ) ^ { 4 } } + \\frac { C \\log ^ { 2 } ( r ) } { r ^ { 2 } t ^ { 3 / 2 } | t - r | ^ { 3 } \\log ^ { 3 b - 1 + \\frac { 5 N } { 2 } } ( t ) } , t > r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} q & = \\pm R _ { s ; i _ 1 , \\cdots , i _ k } q _ { s , 0 } ^ { j _ 1 + k - 1 } \\cdots q _ { s , s - 1 } ^ { j _ s } S _ s ( m ) \\\\ & = \\pm R _ { s ; i _ 1 , \\cdots , i _ k } q _ { s , 0 } ^ { j _ 1 + k - 1 - n } \\cdots q _ { s , s - 1 } ^ { j _ s } \\frac { 1 } { L _ s ^ { \\frac { p - 1 } { 2 } \\delta } } S t _ s ( m ) . \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} 0 & \\leq \\rho \\tau \\Theta _ 1 + ( 1 - \\rho ) \\Theta _ 1 + 2 \\sigma ^ 2 \\rho ^ 2 T + \\frac { L ^ 2 \\gamma ^ 2 } { \\rho } \\sum _ { t = 1 } ^ T \\Delta _ t + \\frac { \\gamma ^ 3 \\tau ^ 2 \\eta C _ L } { 2 } \\sum _ { t = 1 } ^ T \\Delta _ t \\\\ & ~ ~ ~ ~ ~ - \\gamma \\left ( \\mu - \\frac { \\gamma L } { 2 } - \\frac { 2 } { \\eta } \\right ) \\sum _ { t = 1 } ^ T \\Delta _ t + ( \\frac { \\gamma \\eta } { 2 } - \\rho ) \\sum _ { t = 1 } ^ { T + 1 } \\Phi _ t \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} E _ H ( M ) : = \\int _ M k _ c ( 2 H + c _ 0 ) ^ 2 + \\overline { k } K \\ , d S , \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} ( A + g ) \\circ K { } { } = K { } { } \\circ ( A _ c + r ) . \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { n } \\right \\rfloor } ( - 1 ) ^ { m - 1 } p ( n - m ^ 2 ) = \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { \\frac { n } { 2 } } \\right \\rfloor } ( - 1 ) ^ { m - 1 } p ( n - 2 m ^ 2 ) = \\frac { p ( n ) - p _ { \\mathcal { D O } } ( n ) } 2 , \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} M _ { n + 1 } - M _ n = \\begin{cases} \\dfrac { 1 } { w _ n ( x ) } & \\mbox { i f $ ( X _ n , X _ { n + 1 } ) = ( x , x + 1 ) $ } , \\\\ - \\dfrac { 1 } { w _ n ( x ) } & \\mbox { i f $ ( X _ n , X _ { n + 1 } ) = ( x + 1 , x ) $ } \\\\ \\end{cases} \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } \\frac { d P _ { t } } { d t } & = F _ { t } P _ { t } + P _ { t } F ^ { \\intercal } _ { t } + 2 E _ { P ^ { \\theta ^ { \\ast } } } [ \\widehat { x _ { t } \\theta ^ { \\ast \\intercal } _ { t } } - \\hat { x } _ { t } \\widehat { \\theta ^ { \\ast \\intercal } _ { t } } ] - P _ { t } G ^ { \\intercal } _ { t } R ^ { - 1 } _ { t } G _ { t } P _ { t } + Q _ { t } , \\\\ P ( 0 ) & = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} \\phi = ( \\phi _ i ) : X \\to \\bigoplus _ { i = 1 } ^ k X ( s _ i , t _ i ) , \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} ( 8 - \\mu ) { l \\choose 2 } 4 ^ { 2 } + 6 { l \\choose 3 } 4 ^ { 3 } + { l \\choose 4 } 4 ^ { 4 } + 8 b _ { l - 2 } ( l - 1 ) ^ { 2 } = 8 b _ { l - 2 } ( l + 1 ) ^ { 2 } , \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} \\lambda _ s : = Q _ { \\Sigma } ^ s ( u ^ { ( s ) } , u ^ { ( s ) } ) = \\inf \\{ Q ^ s _ { \\Sigma } ( \\phi , \\phi ) : \\phi \\in C _ c ^ 1 ( B _ { r _ 2 } ) , \\ \\int _ { \\Sigma } \\phi ^ 2 \\cdot \\alpha = 1 \\} \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} \\frac { - \\mu { n \\choose 2 } + 8 { n \\choose 3 } + 1 6 { n \\choose 4 } } { 2 ( n - 1 ) } = \\frac { - n \\mu + 8 { n \\choose 2 } + 1 6 { n \\choose 3 } } { 2 } . \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} \\| \\widehat { q _ 2 } f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 = \\lambda ^ 2 \\| x _ 2 f \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 + \\frac { \\theta ^ 2 } { 4 \\lambda ^ 2 } \\| \\xi _ 1 \\widetilde { f } \\| _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } ^ 2 . \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} \\omega \\mapsto \\Big ( \\sum _ { j = n } ^ { n + N - 1 } T _ { \\omega _ j } ( 1 , R ( X ) ) X ^ j \\Big ) ^ { [ n , n + N ) } . \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} \\mathcal { M } - \\Theta ^ { } ( \\mathcal { M } ) = \\begin{pmatrix} - \\mathcal { F } _ 1 \\\\ A _ u ^ { - 1 } \\mathcal { F } _ 2 \\\\ - \\mathcal { F } _ 3 \\circ ( A _ c + r _ 0 ) ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} S ( \\alpha , \\beta , \\gamma , k ) = \\prod _ { i = 0 } ^ { k - 1 } \\frac { \\Gamma ( \\alpha + i \\gamma ) \\Gamma ( \\beta + i \\gamma ) \\Gamma ( 1 + ( i + 1 ) \\gamma ) } { \\Gamma ( \\alpha + \\beta + ( k + i - 1 ) \\gamma ) \\Gamma ( 1 + \\gamma ) } . \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} c : ( G \\times \\widehat { G } ) \\times ( G \\times \\widehat { G } ) & \\to \\mathbb { T } \\\\ ( \\xi _ 1 , \\xi _ 2 ) & \\mapsto \\overline { \\omega _ 2 ( x _ 1 ) } \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} \\Delta _ p w _ p = | \\nabla w _ p | ^ p \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} \\begin{array} { l } \\partial ^ 2 _ t u = \\partial ^ 2 _ x u + \\frac { \\kappa ^ 2 } { 4 } | u | ^ r - \\frac { c _ 1 ^ 2 + c _ 2 ^ 2 } { 2 } u , t > 0 , x \\in D , \\\\ u ( t , x ) = 0 , \\mbox { f o r } ~ x \\in \\partial D , ~ t \\geq 0 , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\partial _ t u ( 0 , x ) = v _ 0 ( x ) , \\end{array} \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} \\sigma _ { y } ( x ) = \\left ( | x - y | ^ 2 + \\left ( \\kappa h \\right ) ^ 2 \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} ( M ( b _ 1 - 1 ) + a _ 1 ) > ( M ( b _ 2 - 1 ) + a _ 2 ) \\Longleftrightarrow \\begin{cases} \\ b _ 1 > b _ 2 \\ \\\\ b _ 1 = b _ 2 \\ \\ a _ 1 > a _ 2 . \\end{cases} \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} i \\ne j \\Rightarrow | \\langle h _ { j } , h _ { i } \\rangle | = \\beta _ { i j } < \\| h _ { j } \\| \\ \\| h _ { i } \\| = \\sqrt { \\beta _ { i i } } \\sqrt { \\beta _ { j j } } . \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} \\langle u , \\varphi \\rangle _ { \\lambda , N } = \\mathcal { Q } _ { \\lambda , N } ( u , \\varphi ) + b ( \\gamma _ 0 ( u ) , \\gamma _ 0 ( \\varphi ) ) _ { \\partial \\Omega } , \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} Q ( B \\ltimes A _ 1 , B \\ltimes A _ 2 ) \\ = \\ \\sum _ { b \\in B } Q ( \\{ b \\} \\ltimes A _ 1 , \\{ b \\} \\ltimes A _ 2 ) . \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} \\{ x \\in \\mathbb { R } : p _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ( x ) > 0 \\} = \\{ x \\in \\mathbb { R } : p _ { \\nu _ { 1 } \\boxplus \\nu _ { 2 } } ( x ) > 0 \\} . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} P _ G \\left ( x \\right ) & = x ^ { n - ( t + 1 ) } \\left [ \\prod _ { i = 1 } ^ { t + 1 } \\left ( x - \\mu \\right ) - \\sum _ { i = 1 } ^ { t + 1 } \\left ( - \\mu \\right ) \\prod _ { \\substack { j = 1 \\\\ j \\neq i } } ^ { t + 1 } \\left ( x - \\mu \\right ) \\right ] \\\\ & = x ^ { n - t - 1 } \\left [ \\left ( x - \\mu \\right ) ^ { t + 1 } + \\left ( t + 1 \\right ) \\mu \\left ( x - \\mu \\right ) ^ { t } \\right ] \\\\ & = x ^ { n - t - 1 } \\left ( x - \\mu \\right ) ^ { t } \\left ( x + t \\mu \\right ) . \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} E _ { 7 ( - 2 5 ) } ~ = \\mathcal { N } _ { 3 } ^ { 7 ( - 2 5 ) - } \\oplus ~ E _ { 6 ( - 2 6 ) } \\oplus s o ( 1 , 1 ) \\oplus \\mathcal { N } _ { 3 } ^ { 7 ( - 2 5 ) } \\newline \\mathbf { , } \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} D ( j _ 0 , \\ldots , j _ { n - 1 } ) = \\{ ( \\o _ 0 , \\o _ 1 , \\ldots ) \\in \\Omega : \\o _ l = j _ l \\} . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} \\partial _ t f = Q ( f , f ) ( v ) , \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} b & = \\tau ( g ) - g + h . \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} \\begin{array} { l } \\int _ { M } ( R - \\frac { 1 } { 2 } T o r ) \\left ( \\gamma , \\gamma \\right ) d \\mu + \\int _ { M } \\left \\vert \\gamma _ { 1 , 1 } \\right \\vert ^ { 2 } d \\mu \\\\ - \\frac { 1 } { 2 } \\int _ { M } T o r ^ { \\prime } \\left ( \\gamma , \\gamma \\right ) d \\mu + \\frac { 1 } { 4 } \\int _ { M } ( 3 Q + 2 P u ) u d \\mu \\\\ = 0 . \\end{array} \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} Q ( x , y , z _ { 1 } , z _ { 2 } , z _ { 3 } ) & = \\tfrac { 3 \\sqrt { 3 } } { 2 } \\det \\begin{pmatrix} - \\tfrac { 1 } { \\sqrt { 3 } } x + y & z _ { 3 } & \\bar { z } _ { 1 } \\\\ \\bar { z } _ { 3 } & - \\tfrac { 1 } { \\sqrt { 3 } } x - y & z _ { 2 } \\\\ z _ { 1 } & \\bar { z } _ { 2 } & \\tfrac { 2 } { \\sqrt { 3 } } x \\end{pmatrix} . \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} N ( u ) ( t , r ) = \\left ( \\frac { \\sin ( 2 u ( t , r ) ) - 2 u ( t , r ) } { 2 r ^ { 2 } } \\right ) \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) + \\left ( \\frac { \\cos ( 2 u ( t , r ) ) - 1 } { 2 r ^ { 2 } } \\right ) \\sin ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) + 2 v _ { c o r r } ) \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} \\begin{cases} | \\xi _ 2 + \\frac { N } { 2 } - \\alpha | \\leq 2 ^ { 3 0 } A ^ { - 1 } M _ 0 ^ { \\frac { 1 } { 2 } } N , \\\\ | \\eta _ 2 - ( \\sqrt { 2 } + 1 ) ^ { \\frac { 2 } { 3 } } \\xi _ 2 + C ' ( \\alpha , M _ 0 , N ) | \\leq 2 ^ { 5 5 } A ^ { - 1 } N , \\end{cases} \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} \\nu = F ^ { ( \\alpha ) } \\left [ f _ { \\nu } \\right ] \\geq F ^ { ( \\alpha ) } \\left [ f _ 0 \\right ] \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} C \\Phi _ { 1 1 } ( q ) = \\prod _ { n = 1 } ^ { \\infty } \\frac { 1 } { 1 - q ^ { 1 1 n } } + 1 1 \\sum _ { j = 1 } ^ { \\infty } p ( 1 1 j - 5 ) q ^ j . \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} & P ^ 1 = \\langle 1 , 2 \\rangle \\langle 2 , 2 \\rangle \\ldots \\langle \\lambda - 1 , 2 \\rangle \\langle \\lambda - 1 , 1 \\rangle . \\\\ & P ^ 2 = \\langle 1 , 2 \\rangle \\langle 2 , 2 \\rangle \\ldots \\langle \\lambda - 1 , 2 \\rangle \\langle \\lambda , 2 \\rangle . \\\\ & P ^ 3 = P ^ 2 \\langle \\lambda , 1 \\rangle . \\\\ & P ^ 4 = \\langle 2 , 2 \\rangle \\langle 2 , 1 \\rangle \\langle 3 , 1 \\rangle \\ldots \\langle \\lambda , 1 \\rangle . \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} N \\prod _ { j = 1 } ^ p ( \\alpha _ k - \\beta _ j ) ^ { m _ j } = n _ k \\prod _ { i \\neq k } ( \\alpha _ k - \\alpha _ i ) ( 1 \\le k \\le s ) . \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} H _ n = a _ 1 r _ 1 ^ { n } + q _ 2 ( n ) r _ 2 ^ n + \\cdots + q _ r ( n ) r _ k ^ n . \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} | v _ { 4 , c } ( t , r ) | \\leq C | \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) | \\begin{cases} \\frac { 1 } { r ^ { 3 } t ^ { 2 } \\log ^ { 3 b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { \\log ( r ) } { \\log ^ { 2 b } ( t ) r ^ { 4 } | t - r | } + \\frac { \\log ^ { 2 b \\alpha } ( t ) } { t ^ { 2 } r ^ { 3 } \\log ^ { 3 b + 1 } ( t ) } , \\frac { t } { 2 } \\leq r \\leq t - \\sqrt { t } , r > t + \\sqrt { t } \\\\ \\frac { 1 } { \\log ^ { 2 b } ( t ) r ^ { 9 / 2 } } , t - \\sqrt { t } \\leq r \\leq t + \\sqrt { t } \\end{cases} \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} \\tau ^ r _ c ( A _ 0 , \\ldots , A _ k ) : = \\int _ { G ^ { \\times k } } { } ^ b { \\rm T r } _ S \\left ( \\Phi _ { A _ 0 } ( ( g _ 1 \\cdots g _ k ) ^ { - 1 } ) \\circ \\Phi _ { A _ 1 } ( g _ 1 ) \\circ \\ldots \\circ \\Phi _ { A _ k } ( g _ k ) \\right ) c ( e , g _ 1 , g _ 1 g _ 2 , \\ldots , g _ 1 \\cdots g _ k ) d g _ 1 \\cdots d g _ k \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} Y = \\int _ { w _ - } ^ { w _ + } \\frac { ( w ^ { n + 1 } + K ) \\psi _ Y ( w ) } { \\sqrt { ( w ^ n + w ^ { n + 1 } + K ) ( w _ + - w ) ( w - w _ - ) g ( w ) } } d w , \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} | w - w ( 0 ) | \\geq \\min _ { \\nu \\in T _ q \\Gamma _ 0 } \\ , \\ , | \\nu - w ( 0 ) | = \\left | v _ i ( q ) - E _ i ( p ) \\right | \\ , . \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leq x \\\\ p \\equiv b \\pmod a } } e \\bigg ( \\sum _ { j = 1 } ^ m \\beta _ j \\gamma _ j ( p - h ) ^ { \\theta _ j } \\bigg ) = o \\bigg ( \\frac { \\pi ( x ) } { \\phi ( a ) } \\bigg ) = o \\big ( \\pi ( x ) \\big ) , x \\to \\infty . \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } d s \\frac { \\lambda '' ( s ) } { 1 + s - t } & = \\int _ { t } ^ { \\infty } d s \\frac { \\lambda '' ( s ) } { 1 + s - t } - \\int _ { t } ^ { t + 2 ( r + 1 ) } d s \\frac { \\lambda '' ( s ) } { 1 + s - t } \\\\ & = \\int _ { t } ^ { \\infty } d s \\frac { \\lambda '' ( s ) } { 1 + s - t } + E _ { 3 , \\partial _ { r } v _ { 1 } } ( t , r ) \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} \\dim \\mu _ { \\eta } = \\min \\{ 1 , \\frac { h ( \\eta ) } { \\log \\eta ^ { - 1 } } \\} . \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} R e g ( \\Sigma ) \\ & : = \\{ x \\in C l o s ( \\Sigma ) \\cap U : C l o s ( \\Sigma ) x \\} \\\\ S i n g ( \\Sigma ) & : = U \\cap C l o s ( \\Sigma ) \\setminus R e g ( \\Sigma ) \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} \\xi _ { ( g , x ) } = \\sum _ { h \\in \\Gamma } h \\otimes h g \\otimes x ^ h . \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} \\textup { P r } _ c ( G ) = \\textup { P r } _ { a n n } ( R _ G ) . \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} D _ \\alpha \\rho _ t & = - D _ \\alpha ( \\nabla \\cdot ( \\rho Q \\ast \\mathbf { u } ) ) \\\\ & = - [ D _ \\alpha ( \\nabla \\cdot ( \\rho Q \\ast \\mathbf { u } ) ) - Q \\ast \\mathbf { u } \\cdot \\nabla ( D _ \\alpha \\rho ) ] - Q \\ast \\mathbf { u } \\cdot \\nabla ( D _ \\alpha \\rho ) . \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} \\begin{pmatrix} T _ { n + 1 } ( a ) \\cr U _ { n } ( a ) \\end{pmatrix} = \\begin{pmatrix} a & a ^ 2 - 1 \\cr 1 & a \\end{pmatrix} ^ { n } \\begin{pmatrix} a \\cr 1 \\end{pmatrix} \\ ; \\ ; m o d \\ ; \\ ; \\ ; Q . \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} \\frac { d \\eta ^ { v } } { d t } = - \\bar { A } \\nabla _ Y \\bar { H } ( \\eta ^ v ) \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} \\mathcal { B } ( \\mathbb { R } ^ 2 ) = M _ m ^ 2 ( \\mathbb { R } ^ 2 ) \\subset \\subset M ^ 2 ( \\mathbb { R } ^ 2 ) = L ^ 2 ( \\mathbb { R } ^ 2 ) . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} \\textrm { i n t h e $ x _ 2 $ - d i r e c t i o n : } & u _ 2 , \\theta = 0 \\ \\\\ & \\frac { \\partial u _ 1 } { \\partial x _ 2 } = 0 \\ \\\\ \\textrm { i n t h e $ x _ 1 $ - d i r e c t i o n : } & u , \\theta , p \\ L \\ ; , \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} R _ { m } = \\frac { \\mathbb { C } [ x _ { j , ( - 1 - i ) } \\ ; | \\ ; 0 \\leq i \\leq m , \\ ; 1 \\leq j \\leq n ] } { ( T ^ { j } f _ { i } | i = 1 , \\ldots n , \\ ; j \\in \\mathbb { N } ) } , \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} I _ 2 : = - I _ { 2 , 1 } + I _ { 2 , 2 } , \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} w = \\cdots a b y \\cdots c \\cdots \\rightsquigarrow w ^ * = \\cdots b a y \\cdots c \\cdots . \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} \\Delta _ 3 = ( 1 + e ^ { q _ 2 - q _ 1 } ) ( 1 + e ^ { q _ 3 - q _ 2 } ) . \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} \\Gamma _ k ( \\alpha ) = \\Gamma ( \\alpha ) \\ , \\Gamma ( \\alpha - \\dfrac { 1 } { 2 } ) \\cdots \\Gamma ( \\alpha - \\dfrac { k - 1 } { 2 } ) = \\prod \\nolimits _ { j = 1 } ^ k \\Gamma ( \\alpha - \\dfrac { j - 1 } { 2 } ) , \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} g _ i ( u ) \\geq & - \\epsilon \\nabla f _ i ( \\mathbf { u } ^ * ) \\cdot \\mathbf { u } ^ * + o ( \\epsilon ) - 2 C _ 2 \\epsilon \\sum _ { j = 1 } ^ m ( u _ j ^ * - u _ j ) . \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} \\sum _ { \\mathsf { m = 1 } } ^ { + \\infty } \\left ( A \\mathsf { h } _ { \\mathsf { m } } , \\mathsf { h } _ { \\mathsf { m } } \\right ) _ { 2 } \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} D Q _ { \\tilde { \\rho } } = - Q _ { \\tilde { \\rho } } D P _ { \\tilde { \\rho } } ( T ) \\left ( D T , Q _ { \\tilde { \\rho } } \\right ) . \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} h ( 2 t - 5 ) = s - 1 , h ( 2 t - 4 ) = s . \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} \\mathcal { L } _ L ^ { \\lambda } f : = \\sum _ { \\ell = 1 } ^ { d } \\mathcal { S } _ { \\lambda \\mu _ { \\ell } } \\left ( \\left < f , p _ { \\ell } \\right > _ N \\right ) p _ { \\ell } , \\lambda > 0 . \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} h ( m ) = \\max \\{ h _ F ( m ) , h _ C ( m ) \\} , h _ F ( m ) = \\frac { m L } { n _ F } \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} \\tilde { x } & = - \\frac { x } { \\mu } , & \\tilde { y } & = - \\frac { y } { \\mu } , \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} Q = ( p ^ 0 , p , q , q _ 0 ) \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} \\varphi ( b ) = \\min \\limits _ { \\begin{array} { c } \\phi _ 1 , \\dots , \\phi _ { T } \\in \\mathbb { R } ^ { \\widehat { r } _ u } \\end{array} } \\left | \\left ( Y - \\sum _ { k = 1 } ^ K b _ k X _ { k } - \\left ( \\widehat { \\lambda } _ 1 , \\dots , \\widehat { \\lambda } _ N \\right ) ^ \\top \\left ( \\phi _ 1 , \\dots , \\phi _ T \\right ) \\right ) \\widehat { M } _ v \\right | ^ 2 _ 2 . \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 , \\ N ( t ) = 2 \\} = \\frac { ( c _ 2 t + \\beta ) ^ 2 - \\bigl ( \\frac { c _ 2 } { c _ 1 } \\bigr ) ^ 2 ( c _ 1 t - \\beta ) ^ 2 } { ( c _ 1 + c _ 2 ) ^ 2 t ^ 2 } = \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} g r a p h _ { \\Sigma , g } ( u ) : = \\{ e x p ^ M _ x ( u ( x ) \\cdot \\nu ( x ) ) : x \\in E \\} \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} h ^ { i j } ( \\nabla _ i h _ { j l } ) u ^ l & = h ^ { i j } \\nabla _ i ( h _ { l j } ) u ^ l = h ^ { i j } ( \\nabla _ l h _ { i j } - R _ { i l j k } N ^ k ) u ^ l \\\\ & = \\frac { 1 } { 2 } \\langle \\nabla | h | ^ 2 , \\nabla u \\rangle - 2 H k _ 0 \\langle \\mathbf { N } , \\nabla u \\rangle - k _ 0 h ( \\mathbf { N } , \\nabla u ) = \\frac { 1 } { 2 } \\langle \\nabla | h | ^ 2 , \\nabla u \\rangle , \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} ( x _ 3 + G _ 2 ) ( x _ 3 - G _ 2 ) = F _ 1 F _ 2 F _ 3 G _ 1 , \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} \\begin{cases} B _ { H , n } \\geq 0 , & \\\\ B _ { H , n } > 0 , & \\end{cases} \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} \\{ H , F \\} = 2 H \\left ( \\frac { \\partial \\mathrm { t r a c e } ( L ) } { \\partial x ^ i } g ^ { i j } p _ j \\right ) \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} e ( \\lambda ^ 1 _ { p i _ 1 - 1 } \\cdots \\lambda ^ 1 _ { p i _ s - 1 } ) & = 2 p i _ 1 - \\sum _ { k = 2 } ^ s 2 p ( p - 1 ) i _ k + ( s - 2 ) \\\\ & = p e ( \\lambda ^ 1 _ { i _ 1 - 1 } \\cdots \\lambda ^ 1 _ { i _ s - 1 } ) - ( p - 1 ) ( s - 2 ) . \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( \\xi ' , \\eta ' ) & : = \\Bigl | \\xi _ 1 ' { \\xi ' } ^ 2 + \\eta _ 1 ' { \\eta ' } ^ 2 - \\frac { { \\xi _ 1 ' } ^ 3 + { \\eta _ 1 ' } ^ 3 } { 4 } \\Bigr | \\leq 2 ^ 5 A ^ { - \\frac { 3 } { 2 } } d ^ { - 1 } N _ 1 ^ 3 , \\\\ \\tilde { F } ( \\xi ' , \\eta ' ) & : = \\Bigl | \\frac { 3 } { 2 } \\ , \\xi _ 1 ' \\ , \\eta _ 1 ' + 2 \\ , \\xi ' \\ , \\eta ' \\Bigr | \\leq 2 ^ 5 A ^ { - 1 } d ^ { - 1 } N _ 1 ^ 2 , \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} _ { r } F _ { s } \\left ( \\begin{array} { c | c } a _ { 1 } , \\ldots , a _ { r } & \\\\ & x \\\\ b _ { 1 } , \\ldots , b _ { s } & \\end{array} \\right ) = \\sum _ { k \\geq 0 } \\frac { \\left ( a _ { 1 } , \\ldots , a _ { r } \\right ) _ { k } } { \\left ( b _ { 1 } , \\ldots , b _ { s } \\right ) _ { k } } \\frac { x ^ { k } } { k ! } , \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} \\sum - \\frac { y _ j } { Y } \\log ( z _ j / y _ j ) = \\sum - v _ j \\log u _ j \\ge - \\log ( Z / Y ) . \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} \\binom { k + 1 + 2 d } { d } & = d + 1 + d ( d - 1 ) \\binom { k + d } { 1 } + ( d + 1 ) \\binom { k + d } { d - 1 } + \\binom { k + d } { d } ; \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} c _ { 1 R } = \\frac { a _ { 0 R } b _ { 2 L } - a _ { 2 L } b _ { 0 R } } { a _ { 0 R } } + \\frac { a _ { 0 L } \\left ( a _ { 2 R } b _ { 0 R } - a _ { 0 R } b _ { 2 R } \\right ) } { a _ { 0 R } ^ 2 } , \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} \\begin{cases} | \\vec x | _ { \\pmb \\sigma } \\leq t \\\\ | \\Theta \\vec x - \\vec y - \\pmb \\eta | _ { \\pmb \\rho } \\leq t ^ { - \\gamma } \\end{cases} . \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} A _ 0 ( G : H ) : = \\{ f \\in \\mathcal { C } _ 0 ( G ) : L _ h f = f \\ \\forall h \\in H \\} , \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} 0 \\le ( x _ { n + 1 } ^ { 1 - 2 s } - x _ { n + 1 } ) x _ { n + 1 } ^ { s } = x _ { n + 1 } ^ { 1 - s } - x _ { n + 1 } ^ { 1 + s } \\le x _ { n + 1 } ^ { 1 - s } \\le 1 , \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} \\Gamma = \\Gamma _ e \\oplus \\Gamma _ m , \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} U ( r _ \\alpha ) = X _ { ( P _ { \\ell ( \\alpha ) } , \\alpha ) } \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} 1 - \\| \\mu ^ { ( i ) } \\| = \\frac { \\big ( 1 - \\| \\mu ^ { ( i - 1 ) } \\| \\big ) e ^ { c ( \\beta ) } } { D _ { i } } . \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} A d m _ { \\Z } ^ k = P ^ { p - h ^ { \\vee } } _ + , \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} ( I - \\Delta ) ^ \\frac { \\beta } { 2 } f : = \\mathcal { F } ^ { - 1 } \\left ( ( 1 + | \\cdot | ^ 2 ) ^ { \\frac { \\beta } { 2 } } \\mathcal { F } f \\right ) . \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} & \\delta \\mathcal { A } = \\delta \\int _ M d S = \\int _ U \\delta \\sqrt { \\det ( \\ * g ) } \\ , d A = \\int _ U \\frac { 1 } { 2 \\sqrt { \\det ( \\ * g ) } } \\ , \\delta ( \\det ( \\ * g ) ) \\ , d A \\\\ & = \\int _ U 2 H u \\sqrt { \\det ( \\ * g ) } \\ , d A = \\int _ M - 2 H u \\ , d S . \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} J \\eta & = - X - W + ( \\| X \\| ^ 2 + \\| W \\| ^ 2 ) B \\\\ & = \\left ( \\frac { \\| W \\| ^ 2 } { 1 + \\| X \\| ^ 2 } ( B + X ) - W \\right ) + \\frac { 1 + \\| X \\| ^ 2 + \\| W \\| ^ 2 } { 1 + \\| X \\| ^ 2 } ( \\| X \\| ^ 2 B - X ) , \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} \\lim _ { R \\rightarrow \\infty } ( R ^ { - 4 } \\| e ^ { \\tau \\phi } x _ { n + 1 } ^ { \\frac { 1 - 2 s } { 2 } } | x | \\tilde { u } \\| _ { L ^ { 2 } ( A _ { R , 2 R } ^ { + } ) } ^ { 2 } + R ^ { - 2 } \\| e ^ { \\tau \\phi } x _ { n + 1 } ^ { \\frac { 1 - 2 s } { 2 } } | x | \\nabla \\tilde { u } \\| _ { L ^ { 2 } ( A _ { R , 2 R } ^ { + } ) } ^ { 2 } ) = 0 . \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} & \\max _ { \\boldsymbol { \\theta } } \\mathbb { E } \\| \\boldsymbol { G } \\boldsymbol { \\Theta } \\boldsymbol { S } \\boldsymbol { h } _ r + \\boldsymbol { h } _ d \\| _ 2 ^ 2 \\\\ & ~ ~ \\textrm { s . t . } \\quad ~ | \\theta _ n | = 1 , ~ n = 1 , \\ldots , N . \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} & \\int _ { - M } ^ 0 \\int _ { - \\infty } ^ 0 | x | ^ \\sigma J ( x - y ) \\psi ( y ) d y d x \\\\ = & \\int _ { - \\infty } ^ 0 \\int _ { - y } ^ { M - y } ( x + y ) ^ { \\sigma } J ( y ) \\psi ( - x ) d x d y + \\int _ 0 ^ M \\int _ { 0 } ^ { M - y } ( x + y ) ^ { \\sigma } J ( y ) \\psi ( - x ) d x d y . \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} E \\colon y ^ 2 + x y + y = x ^ 3 + 4 x - 6 \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} U ^ { - 1 } T U = \\begin{pmatrix} * & * & * & 0 & \\cdots \\\\ * & * & * & * & * & * & 0 & \\cdots \\\\ * & * & \\cdots \\\\ 0 & * & \\\\ 0 & * & \\\\ 0 & * & \\\\ \\vdots & 0 & \\end{pmatrix} , \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} \\forall l \\in I ^ { G H } ( \\bar x , \\bar y , \\bar z ) \\colon \\mu _ l \\nu _ l = 0 . \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} \\mathfrak { L } = \\mathfrak { J } ^ { \\prime } \\oplus s t r ( \\mathfrak { J } ) \\oplus \\mathfrak { J } , \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} \\delta _ { i j } = \\int _ { 0 } ^ 1 t ^ { i - 1 } p _ j ^ { ( m ) } ( t ) d t = \\sum _ { k = 1 } ^ { m + 1 } a _ { k j } ^ { ( m ) } \\int _ { 0 } ^ 1 t ^ { i + k - 2 } d t = \\sum _ { k = 1 } ^ { m + 1 } \\frac { 1 } { i + k - 1 } a _ { k j } ^ { ( m ) } , \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N } \\frac { 1 } { \\log K _ { j } \\log \\log K _ { j } } \\le C \\left ( 1 + \\frac { 1 } { n _ { 1 } } \\sum _ { j = 1 } ^ { N } \\log K _ { j } \\right ) . \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} q _ { t z } ^ x \\kappa ( x , y , z , t ) q _ { z t } ^ x = q _ { t z } ^ y \\kappa ( x , y , z , t ) q _ { z t } ^ y = \\kappa ( y , x , t , z ) ; \\\\ q _ { x z } ^ y \\kappa ( x , y , z , t ) q _ { z x } ^ y = q _ { x z } ^ t \\kappa ( x , y , z , t ) q _ { z x } ^ t = \\kappa ( z , t , x , y ) ; \\\\ q _ { y z } ^ x \\kappa ( x , y , z , t ) q _ { z y } ^ x = q _ { y z } ^ t \\kappa ( x , y , z , t ) q _ { z y } ^ t = \\kappa ( t , z , x , y ) ; \\\\ \\kappa ( x , y , z , t ) ^ { - 1 } = \\kappa ( y , x , z , t ) . \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} | \\det Y _ \\lambda ( x ) | = | \\det Y _ \\lambda ( a ) | = 1 . \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} \\left \\langle h ^ { k } B ^ { l } C ^ { m } \\mid k + l + m \\leq n m = 0 , 1 \\right \\rangle . \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} \\Omega _ { 0 , F } & = p \\circ ( K ^ x + \\Delta ^ x ) - p \\circ K ^ x + K ^ y \\cdot ( q \\circ ( K ^ x + \\Delta ^ x ) - q \\circ K ^ x ) \\\\ & \\quad + \\ , \\Delta ^ y \\cdot q \\circ ( K ^ x + \\Delta ^ x ) + u \\circ ( K + \\Delta ) - u \\circ K + g \\circ ( K + \\Delta ) \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} T = T _ 1 + T _ 2 \\circ P _ 1 + T _ 3 \\circ P _ 2 \\circ P _ 1 + T _ 4 \\circ P _ 3 \\circ P _ 2 \\circ P _ 1 \\ , . \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} a _ 2 \\leq 2 a _ 1 = 2 = a _ 1 + 1 . \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu ( B ) } \\int _ B e ^ { \\gamma \\frac { | b ( x ) - b _ B | } { \\lambda \\| b \\| _ { { \\rm b m o } _ { \\mathcal { B } } } } } \\ , d \\mu \\leq \\left ( \\frac { 1 } { \\mu ( B ) } \\int _ B e ^ { \\gamma \\frac { | b ( x ) - b _ B | } { \\| b \\| _ { { \\rm b m o } _ { \\mathcal { B } } } } } \\ , d \\mu \\right ) ^ { \\frac { 1 } { \\lambda } } \\leq C ^ { \\frac { 1 } { \\lambda } } = 2 , \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} \\mathcal { D } ^ 2 = \\left ( \\begin{array} { c c } H _ 1 & 0 \\\\ 0 & H _ 2 \\end{array} \\right ) \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} \\hat c ( \\gamma ) & : = C _ s \\mathrm { P . V . } \\int _ { - \\infty } ^ { + \\infty } [ | 1 + \\tau | ^ { - \\gamma } - 1 ] | \\tau | ^ { - ( 1 + 2 s ) } d \\tau \\\\ c ^ \\perp ( \\gamma ) & : = 2 C _ s \\int _ { 0 } ^ { + \\infty } [ ( 1 + \\tau ^ 2 ) ^ { - \\gamma / 2 } - 1 ] \\tau ^ { - ( 1 + 2 s ) } d \\tau . \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} { \\Pr } \\bigg ( { { x } _ R } = 1 ~ \\bigg | ~ { x _ T } = { s } \\bigg ) = { \\Pr } \\bigg ( g _ { T , R } ^ \\theta \\ge { \\tau _ R } ~ \\bigg | ~ { x _ T } = { s } \\bigg ) = \\frac { 1 } { 2 } \\left ( 1 - \\left ( \\frac { { { \\tau _ R } - { { \\mu } _ { { { s } _ { T , R } } } } } } { { \\sqrt { 2 { { \\sigma } _ { { { s } _ { T , R } } } } ^ 2 } } } \\right ) \\right ) , \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} S _ { } [ g ] = \\frac { 1 } { 2 } \\int _ { \\Sigma } \\d { } ^ 2 z ( g ^ { - 1 } \\partial g ) ^ \\mu E _ { \\mu \\nu } ( g ^ { - 1 } \\bar { \\partial } g ) ^ \\nu , \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} G _ { u , v } L _ { \\phi , \\psi } f ( x ) & = \\int _ a ^ b \\Big ( \\big ( - \\partial _ y ^ 2 + V ( y ) \\big ) G _ { u , v } ( x , y ) \\Big ) f ( y ) \\d y \\\\ & + \\lim _ { y \\to a } \\big ( G _ { u , v } ( x , y ) f ' ( y ) - \\partial _ y G _ { u , v } ( x , y ) f ( y ) \\big ) \\\\ & - \\lim _ { y \\to b } \\big ( G _ { u , v } ( x , y ) f ' ( y ) - \\partial _ y G _ { u , v } ( x , y ) f ( y ) \\big ) \\\\ & = f ( x ) + v ( x ) W ( u , f ; a ) - u ( x ) W ( v , f ; b ) = f ( x ) . \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} u = \\frac { h _ s } { ( n + 1 ) H ' ( s ) } . \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} a _ { 2 J } & = \\frac { \\partial f _ J } { \\partial y } ( 0 , 0 ) , & b _ { 0 J } & = g _ J ( 0 , 0 ) , \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} g _ { i j } ^ { ( 2 k + 2 ) } = \\begin{cases} 0 & \\mbox { i f } j \\not \\in \\{ i , k + 1 \\} \\\\ 1 & \\mbox { i f } i = j \\\\ - a ^ { ( 2 k + 1 ) } _ { i k + 1 } \\medskip & \\mbox { i f } i \\neq k + 1 , j = k + 1 \\end{cases} . \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} b ' _ 4 ( 1 , 2 , 3 , 4 ) & = b _ 4 + b _ 3 \\cdot ( - i \\lambda _ 3 ) \\left ( \\frac { i } { x _ { 1 + 2 } } + \\right ) \\\\ & + b ' _ 2 ( 1 + 2 + 3 + 4 ) \\cdot ( - i \\lambda _ 3 ) ^ 2 \\left ( \\frac { i ^ 2 } { x _ { 1 + 2 } x _ { 3 + 4 } } + \\right ) \\\\ & + b ' _ 2 ( 1 + 2 + 3 + 4 ) \\cdot ( - i \\lambda _ 3 ) ^ 2 \\left ( \\frac { i ^ 2 } { x _ { 1 + 2 } x _ { 1 + 2 + 3 } } + \\right ) . \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} \\Psi _ { \\Lambda } : = \\sum _ { i \\in I } \\bigotimes _ { x \\in \\Lambda } h _ { x , i } \\in \\bigotimes _ { x \\in \\Lambda } \\mathcal { H } _ { x } = : \\mathcal { H } _ { \\Lambda } \\quad ; \\Lambda \\subset _ { f i n } V \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} x ^ { \\top } K x = - ( H x ) ^ { \\top } ( H x ) = 0 \\ ; \\ ; \\Leftrightarrow \\ ; \\ ; H x = 0 \\ ; \\ ; \\Rightarrow \\ ; \\ ; - H ^ { \\top } ( H x ) = K x = 0 . \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} \\beta : = \\max \\{ | \\beta _ { i j } | \\ : \\ i , j \\in I _ { 0 } \\} = \\max \\{ | \\beta _ { i i } | \\ : \\ i \\in I _ { 0 } \\} \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} K ( x , \\lambda ( t ) ) = \\int _ { 0 } ^ { \\infty } d R \\frac { R } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\int _ { 0 } ^ { x } \\rho d \\rho \\left ( \\frac { 1 } { \\sqrt { x ^ { 2 } - \\rho ^ { 2 } } } - \\frac { 1 } { x } \\right ) \\left ( 1 + \\frac { R ^ { 2 } \\lambda ( t ) ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( R ^ { 2 } \\lambda ( t ) ^ { 2 } - 1 - \\rho ^ { 2 } ) ^ { 2 } + 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } } \\right ) \\geq 0 \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( F \\left ( ( \\eta ^ { * L } _ t ) _ { t \\in [ 0 , L ] } \\right ) \\right ) = \\mathbf { E } \\left ( F \\left ( ( \\eta _ t ) _ { t \\in [ 0 , L ] } \\right ) \\ : \\vline \\ : \\eta _ 0 = 1 , \\ : \\eta _ L = 0 , \\ : S _ L > 0 \\right ) , \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} S _ d ( X ) = S _ d ^ - ( X ) \\cup S _ d ^ + ( X ) , \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} A = \\mathbb { C } [ E _ { 1 , 2 } , E _ { 1 , 3 } , E _ { 1 , 4 } , E _ { 2 , 3 } , E _ { 2 , 4 } , E _ { 3 , 4 } ] / I , \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & \\ , \\to \\ , \\min \\limits _ { x , y , z } & & & \\\\ g _ i ( x ) & \\ , \\leq \\ , 0 & \\qquad & i \\in \\mathcal M & \\\\ h _ j ( x ) & \\ , = \\ , 0 & & j \\in \\mathcal P & \\\\ y _ l , z _ l & \\ , \\leq \\ , 0 & & l \\in \\mathcal Q & \\\\ ( G _ l ( x ) - y _ l ) ( H _ l ( x ) - z _ l ) & \\ , = \\ , 0 & & l \\in \\mathcal Q & \\end{aligned} \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ t ( n _ i - r _ i ) - \\sum _ { i : r _ i < n _ i } q ^ { - m _ i } + 1 \\leq \\eta N _ t . \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} U ^ a _ { t , s } ( \\gamma ) = \\psi _ { a b } ( \\gamma ( t ) ) U ^ b _ { t , s } ( \\gamma ) \\psi _ { b a } ( \\gamma ( s ) ) . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} G ^ { \\mathcal { G } } _ n = \\bigcup _ { p \\in \\mathcal { G } , | p | > n } \\sigma ^ p _ n \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} \\alpha ^ + _ { \\Omega , p } ( t ) : = \\min \\big \\{ \\pi , \\ , \\sup \\{ \\alpha > 0 : p + i t e ^ { - i \\theta } \\in \\Omega \\mathrm { \\ f o r \\ a l l \\ } \\theta \\in [ 0 , \\alpha ] \\} \\big \\} \\in ( 0 , \\pi ] . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} p _ { \\nu _ { k } } ( \\xi ) = \\frac { \\left | 1 - R ( t ) t \\right | ^ { 2 } } { \\left | 1 - \\eta _ { \\nu } \\left ( R ( t ) t \\right ) \\right | ^ { 2 } } \\frac { 1 - R ( t ) ^ { \\frac { 2 k } { k - 1 } } } { 1 - R ( t ) ^ { 2 } } \\ , p _ { \\mu } ( \\xi ) \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} F = ( x _ { 1 } ^ { m _ { 1 , 2 } } ) ( x _ { 1 } ^ { m _ { 1 , 3 } } x _ { 2 } ^ { m _ { 1 , 3 } + m _ { 2 , 3 } } ) \\cdots ( x _ { 1 } ^ { m _ { 1 , n } } \\cdots x _ { n - 1 } ^ { m _ { 1 , n } } x _ { 2 } ^ { m _ { 2 , n } } \\cdots x _ { n - 1 } ^ { m _ { 2 , n } } \\cdots x _ { n - 1 } ^ { m _ { n - 1 , n } } ) . \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} & g _ k ( q ) = ( q ; q ) _ \\infty ( - q ; q ^ 2 ) _ \\infty \\sum _ { n = 0 } ^ \\infty a _ k ( n ) q ^ n \\\\ & \\qquad \\ = ( - q ; q ^ 2 ) _ \\infty ( q ^ { k } ; q ^ { k } ) _ \\infty \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { k n } + 2 q ^ { 2 k n } + \\dots + ( k - 1 ) q ^ { ( k - 1 ) k n } } { 1 + q ^ { k n } + q ^ { 2 k n } + \\dots + q ^ { ( k - 1 ) k n } } . \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} & T _ { a } b = \\sum _ { q } S _ { q - 1 } a \\Delta _ { q } b , S _ { q - 1 } a = \\sum _ { k \\leq q - 2 } \\Delta _ { k } a , R ( u , v ) = \\sum _ { | p - q | \\leq 2 } \\Delta _ { p } a \\Delta _ { q } b . \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} f = \\sum _ { j = 0 } ^ \\infty \\sum _ { k = 0 } ^ { n - 1 } \\langle f , b _ { j , k } \\rangle b _ { j , k } , \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} d X ( t ) = - A \\nabla H ( X ( t ) ) d t + \\sqrt { 2 A } d B ( t ) , \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} \\mu _ { N _ 1 } ^ { \\sigma } ( s ) ( d x ) : = \\frac { 1 } { Z } \\exp \\left ( \\sigma \\sum _ { i = 1 } ^ { N _ 1 } x _ i - H _ { N _ 1 } ( x ) + s \\cdot \\frac { 1 } { 2 } \\sum _ { ( i , j ) \\in I _ { N _ 1 , N _ 2 } } | M _ { i j } | x _ i ^ 2 \\right ) d x . \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} \\| u \\| _ \\pi = \\inf \\Big \\{ \\sum _ { i = 1 } ^ { n } \\| x _ i \\| \\| y _ i \\| : u = \\sum _ { i = 1 } ^ { n } x _ i \\otimes y _ i \\Big \\} . \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} \\begin{aligned} D _ { 3 } ( \\Pi ^ { 3 } _ { M _ 1 \\cap M _ 2 } v _ \\lambda , v _ \\lambda ) & = \\frac { 3 \\lambda ^ { 3 } } { 4 } - \\frac { \\lambda ^ { 2 } \\sqrt { \\lambda ^ { 2 } + 2 \\lambda + 9 } } { 3 6 } + \\frac { \\lambda ^ { 2 } } { 4 } - \\frac { \\lambda \\sqrt { \\lambda ^ { 2 } + 2 \\lambda + 9 } } { 1 8 } + \\frac { \\lambda } { 4 } \\\\ & \\qquad - \\frac { \\sqrt { \\lambda ^ { 2 } + 2 \\lambda + 9 } } { 4 } + \\frac { 3 } { 4 } \\end{aligned} \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} ( q _ { i 0 } \\circ \\dots \\circ q _ { i , j - 1 } ) ( x ) ( 1 _ G ) = 0 . \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} K _ { 1 } ' ( y ) & = - \\left ( \\frac { y K _ { 0 } ( y ) + K _ { 1 } ( y ) } { y } \\right ) \\\\ K _ { 1 } '' ( y ) & = K _ { 1 } ( y ) + \\frac { K _ { 2 } ( y ) } { y } \\\\ K _ { 1 } ''' ( y ) & = \\frac { - ( 3 + y ^ { 2 } ) } { y ^ { 2 } } K _ { 2 } ( y ) \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} G ^ { i , j } _ { 0 } = { \\rm s p a n } \\left \\{ ( E _ { i , i + 2 } ) ^ { l _ { 1 } } _ { ( k _ { 1 } ) } ( E _ { i - 1 , i + 2 } ) ^ { l _ { 2 } } _ { ( k _ { 2 } ) } \\cdots ( E _ { j , i + 2 } ) ^ { l _ { i - j + 1 } } _ { ( k _ { i - j + 1 } ) } | l _ { r } \\in \\mathbb { N } , k _ { s } \\in \\mathbb { Z } _ { < 0 } \\right \\} . \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} w & : = - H h , \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} \\mathcal { R } _ n ( t ) = \\begin{cases} t + R _ l t ^ l & \\ 1 \\leq n \\leq l - 1 , \\\\ t + R _ l t ^ l + R _ { 2 l - 1 } t ^ { 2 l - 1 } & \\ n \\ge l , \\\\ \\end{cases} \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} \\phi _ t : = \\mathrm { i d } _ A + t ( \\mathrm { a d } ^ l _ a - \\mathrm { a d } ^ r _ a ) ~ ~ ~ ~ ~ ~ \\psi _ t = \\mathrm { i d } _ M + t ( l _ a - r _ a + H ( a , T - ) - H ( T - , a ) ) \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} \\mathcal { F } ( g ) ( \\xi ) = \\int _ { 0 } ^ { \\infty } \\phi ( r , \\xi ) g ( r ) d r = \\int _ { 0 } ^ { \\infty } \\frac { \\sqrt { r } \\tilde { \\phi } _ { \\sqrt { \\xi } } ( r ) } { f ( \\sqrt { \\xi } ) } g ( r ) d r = \\frac { 1 } { f ( \\sqrt { \\xi } ) } \\mathcal { F } _ { H } ( \\frac { g ( \\cdot ) } { \\sqrt { \\cdot } } ) ( \\sqrt { \\xi } ) \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} M ^ { ( c ) } _ { \\gamma _ 1 } ( s ) = \\frac { \\xi ^ 2 A r \\mu _ r } { \\Gamma ( \\alpha ) B ^ r } \\sum _ { k = 1 } ^ { \\beta } \\frac { b _ k } { \\Gamma ( k ) } { \\rm \\mathcal { H } } _ { 4 , 3 } ^ { 1 , 4 } \\Biggl [ \\ ! \\frac { \\mu _ r } { B ^ r } s \\Bigg \\vert \\ { ( \\delta _ 2 , \\Delta _ 2 ) \\atop ( \\lambda _ 2 , \\Lambda _ 2 ) } \\Biggr ] . \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( e ^ { - s S _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( t ) } \\right ) = e ^ { - t \\left ( c _ 1 \\left ( ( s + \\lambda _ 1 ) ^ { \\alpha _ 1 } - \\lambda _ 1 ^ { \\alpha _ 1 } \\right ) + c _ 2 \\left ( ( s + \\lambda _ 2 ) ^ { \\alpha _ 2 } - \\lambda _ 2 ^ { \\alpha _ 2 } \\right ) \\right ) } , \\ ; s > 0 , \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\nabla v ( x ) \\cdot \\nabla \\varphi ( x ) \\ ; d x + \\int _ { \\mathbb { R } ^ N } V _ 0 ( x ) v ( x ) \\varphi ( x ) \\ ; d x = a _ 0 \\int _ { \\mathbb { R } ^ N } v ( x ) \\varphi ( x ) \\ ; d x , \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} \\Gamma _ 2 ( x \\b 1 , 1 ) ^ { - 1 } = \\rho \\cdot G ( x ) \\cdot ( 2 \\pi ) ^ { - \\frac { x } { 2 } } = \\rho \\cdot G ( x + 1 ) \\cdot \\Gamma ( x ) \\cdot ( 2 \\pi ) ^ { - \\frac { x } { 2 } } . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} \\mathrm { E R e r r } _ k : = \\max _ { i , j } \\frac { | ( H _ k - X ) _ { ( i , j ) } | } { X _ { ( i , j ) } } \\le \\varepsilon , \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} L _ u S = S L _ u + S - \\langle \\ , u S \\cdot \\ , | 1 \\rangle 1 \\ . \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} \\frac { d \\eta } { d t } = C ( | | \\tilde { \\mathbf { v } } ( t ) | | _ \\infty + | | \\tilde { w } ( t ) | | _ \\infty ) , \\eta ( 0 ) = 0 . \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} \\phi ( 2 ^ \\delta q _ 1 ^ { e _ 1 } \\ ! \\cdots \\ ! q _ k ^ { e _ k } ) = 2 ^ { \\delta - 1 } \\phi ( q _ 1 ^ { e _ 1 } \\ ! \\cdots \\ ! q _ k ^ { e _ k } ) \\ge \\sum _ { i = 1 } ^ k \\ ! 2 ^ { \\delta - 1 } \\phi ( q _ i ^ { e _ i } ) = \\sum _ { i = 1 } ^ k \\ ! \\phi ( 2 ^ { \\delta } q _ i ^ { e _ i } ) \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} \\mu ( d x ) : = \\frac { 1 } { Z } \\exp \\left ( - H ( x ) \\right ) d x . \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial \\zeta } { \\partial t } = \\frac { \\partial ^ 2 } { \\partial \\theta ^ 2 } \\varphi ' ( \\zeta ) , \\\\ \\zeta ( 0 , \\cdot ) = \\zeta _ 0 , \\end{cases} \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} c _ 1 ( Z ) ^ 2 = & ( c _ 1 ( M ) - c _ 1 ) ^ 2 c _ 2 - 2 ( c _ 1 ( M ) - c _ 1 ) c _ 3 + c _ 4 \\\\ c _ 2 ( Z ) = & ( c _ 2 ( M ) - c _ 1 ( M ) c _ 1 + c _ 2 ( A ) - c _ 2 ( B ) + c _ 1 ( B ) ^ 2 - c _ 1 ( A ) c _ 1 ( B ) ) c _ 2 + \\\\ & + ( - c _ 1 ( M ) + 2 c _ 1 ) c _ 3 + c _ 4 . \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} \\sigma _ { \\sigma _ x ( y ) } = \\sigma _ y . \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} G ( d , s , \\tau ) = G ( d , s , t ) + \\frac { d } { 2 } ( A ' - A ) + ( \\rho ' - \\rho ) . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} G _ { \\mu } ( z ) = \\int _ { \\mathbb { R } } \\frac { d \\mu ( t ) } { z - t } , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} \\gamma ^ + _ j ( t ) = h ^ { - 1 } \\left ( \\rho ^ { 2 j } ( t - 1 ) ^ { 1 / \\ell } \\right ) ( 1 \\le t < 1 + \\delta ) . \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} x _ 1 ( 0 ) = \\textstyle \\alpha x _ 3 ( 1 ) + \\beta x _ 4 ( 1 ) , \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} d X ( t ) = b ( X ( t ) ) d t + \\sigma ( X ( t ) ) d { B } ( t ) + \\int _ { \\mathbb { R } } \\nu ( d x ) \\ , d L _ t ^ x ( X ) \\ , , \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} \\widehat { g } _ { \\sigma _ { 1 } , \\sigma _ { 2 } , \\sigma _ { 3 } } : = \\left ( \\frac { \\widehat { e } _ { \\sigma _ { 3 } - \\sigma _ { 1 } } \\widehat { e } _ { \\sigma _ { 2 } } } { \\widehat { e } _ { \\sigma _ { 2 } - \\sigma _ { 1 } } \\widehat { e } _ { \\sigma _ { 3 } } } \\cdot \\widehat { f } _ { 1 } , \\cdots , \\frac { \\widehat { e } _ { \\sigma _ { 3 } - \\sigma _ { 1 } } \\widehat { e } _ { \\sigma _ { 2 } } } { \\widehat { e } _ { \\sigma _ { 2 } - \\sigma _ { 1 } } \\widehat { e } _ { \\sigma _ { 3 } } } \\cdot \\widehat { f } _ { j } \\right ) . \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} H ^ 2 ( \\Omega ) = \\mathcal H ^ 2 _ { 0 , D } ( \\Omega ) + \\mathcal H ^ 2 _ { 0 , N } ( \\Omega ) . \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} \\alpha \\star x = \\overline { \\alpha } x , \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} M _ g \\rho ( \\tau ) = \\rho ( \\tau ) M _ { g \\circ \\alpha ( \\tau ) } . \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} d y = ( x ^ 2 + a ) \\ , d x . \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} \\| y - H w \\| ^ 2 = \\Re \\langle y - H w , y - H w \\rangle = \\Re \\langle x - w , y - H w \\rangle \\leqslant 0 \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} f ( 0 , 0 ; \\mu ) = g ( 0 , 0 ; \\mu ) = 0 , \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta \\ | \\ V ( 0 ) = c _ 1 , \\ N ( t ) = 2 k \\} = \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} Y _ t ^ k = \\xi ^ k - \\sum _ { l = 1 } ^ m \\int _ t ^ T Z _ s ^ { k , l } d B _ s ^ l + \\frac { 1 } { 2 } \\sum _ { l = 1 } ^ m \\int _ t ^ T \\sum _ { i , j = 1 } ^ n \\Gamma _ { i j } ^ k ( Y _ s ) Z _ s ^ { i , l } Z _ s ^ { j , l } d s + \\int _ t ^ T f ^ k ( Y _ s , Z _ s ) d s , \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} u ^ m ( x , 0 ) = \\sum _ { j = 1 } ^ { m } b _ j ^ m \\phi _ j ( x ) \\to u _ 0 ( x ) W _ 0 ^ { 1 , p } ( \\Omega ) . \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} \\bar { R } _ { k } = \\left \\{ ( x , t ) \\in B _ { k } : | | x | | _ { \\infty } = L _ { k } ^ { d + 6 } x _ { 1 } \\geq 7 L _ { k } t = L _ { k } \\right \\} , \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} \\psi _ \\alpha ( \\bar { x } _ 0 , \\ldots , \\bar { x } _ \\lambda ) = \\bigwedge \\limits _ { V \\in \\mathcal { V } } \\psi ' _ \\alpha ( V ( \\bar { x } _ 0 , \\ldots , \\bar { x } _ \\lambda ) ) \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} | \\Theta ( \\l , K ) ^ n x | = \\l ^ { n + o ( n ) } n \\rightarrow \\infty . \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } X _ t = X _ 0 + \\int _ 0 ^ t a ( s , X _ s , Y _ s , Z _ s ) \\ , d s + \\int _ 0 ^ t b ( s , X _ s , Y _ s , Z _ s ) \\ , d W _ s , \\\\ Y _ t = \\xi + \\int _ t ^ T f ( s , X _ s , Y _ s , Z _ s ) \\ , d s - \\int _ t ^ T Z _ s \\ , d W _ s . \\end{array} \\right . \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } | x | ^ { d + 1 } \\ , \\left \\vert \\nabla e ^ { ( t - u ) \\Delta } V ^ { N } ( x ) \\right \\vert ^ { 2 } d x & = \\int _ { \\R ^ d } | x | ^ { d + 1 } \\ , \\left \\vert \\int _ { \\R ^ d } g _ { 2 ( t - u ) } ( x - y ) \\ , \\nabla V ^ { N } ( y ) \\ , d y \\right \\vert ^ { 2 } d x \\\\ & \\leq \\int _ { \\R ^ d } | x | ^ { d + 1 } \\int _ { \\R ^ d } \\left | \\nabla V ^ { N } ( y ) \\right | ^ 2 \\ , g _ { 2 ( t - u ) } ( x - y ) \\ , d y \\ , d x , \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} b _ i = a _ i - \\binom { k - i } { 1 } a _ { i + 1 } + \\binom { k - i } { 2 } a _ { i + 2 } - \\cdots + ( - 1 ) ^ { k - i } \\binom { k - i } { k - i } a _ k . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} \\Big ( { \\sum _ { j \\ge n } } ^ * T _ { \\omega _ { 2 , j } } ( 1 , R ( X ) ) X ^ j \\Big ) ^ { [ n , n + N ) } = \\Big ( { \\sum _ { j \\ge n } } ^ * T _ { \\omega ' _ { 2 , j } } ( 1 , R ( X ) ) X ^ j \\Big ) ^ { [ n , n + N ) } , \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} | E ( \\l ) | = | P _ { 1 , 1 } ( \\l ) | = | Q _ 1 ( \\l , 1 , \\tau ) | \\le r , \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} H _ \\pm ( z , t , \\theta ) = \\lim _ { \\tau \\to 0 } \\Big ( ( 2 \\pi i \\tau ) \\cdot \\log \\psi _ { \\pm } ( t ) \\Big ) . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} - 2 \\log c _ \\beta = 2 \\log \\Gamma ( \\beta + 1 ) + 2 \\log \\Gamma ( - \\beta - 1 / 2 ) + \\log \\sin \\bigl ( \\pi ( - 2 \\beta - 1 ) \\bigr ) - \\log ( 4 \\pi ) . \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} & w _ 1 = l a _ 1 b _ 1 m _ 1 c _ 1 m _ 2 d _ 1 e _ 1 f _ 1 r , \\\\ & w _ 2 = l a _ 2 b _ 2 m _ 1 c _ 2 m _ 2 d _ 2 e _ 2 f _ 2 r . \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} \\mathcal { L } _ \\lambda ( t ) = T V ( J _ { \\lambda } ( u _ \\lambda ( t ) ) ) + \\frac { 1 } { 2 } \\| \\dot { u } _ \\lambda ( t ) \\| ^ 2 , \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} Q _ { x } ^ { N + 1 } ( T _ { B ( x , r ) } ^ { N + 1 , N } \\le t ) = P _ { x } ( \\tau _ { B ( x , r ) } ^ { N } \\le t ) . \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} \\| u \\| _ { 1 , \\mathcal { H } } = \\| \\nabla u \\| _ { \\mathcal { H } } + \\| u \\| _ { \\mathcal { H } } \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} & { \\varphi _ 0 } ^ { \\pm } = { \\varphi _ { 0 1 } } ^ { \\pm } + i { \\varphi _ { 0 2 } } ^ { \\pm } , \\\\ [ 1 m m ] & { \\varphi _ j ^ 1 } ^ { \\pm } = { \\varphi _ { j 1 } ^ { 1 } } ^ { \\pm } \\cos { j \\theta } + i { \\varphi _ { j 2 } ^ { 1 } } ^ { \\pm } \\sin { j \\theta } , \\\\ [ 1 m m ] & { \\varphi _ j ^ 2 } ^ { \\pm } = { \\varphi _ { j 1 } ^ 2 } ^ { \\pm } \\sin { j \\theta } + i { \\varphi _ { j 2 } ^ 2 } ^ { \\pm } \\cos { j \\theta } , \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} \\left | e _ 2 ^ \\perp - \\left ( ( 1 - \\dot x _ k ^ 2 ) e _ 2 - \\dot x _ k \\dot y _ k e _ 3 \\right ) \\right | = O ( \\varepsilon ) , \\left | e _ 3 ^ \\perp - \\left ( ( 1 - \\dot y _ k ^ 2 ) e _ 3 - \\dot x _ k \\dot y _ k e _ 2 \\right ) \\right | = O ( \\varepsilon ) . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} \\Psi _ { \\mathrm { G F } } ( f ( z ) \\partial ^ m , g ( z ) \\partial ^ n ) \\ , : = \\ , \\frac { m ! n ! } { ( m + n + 1 ) ! } \\operatorname { R e s } _ { z = 0 } ( \\partial ^ { n + 1 } f ( z ) ) ( \\partial ^ m g ( z ) ) d z \\ , . \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} h ^ \\Theta ( z ) \\ , = \\ , - \\frac 1 { h ( z ) } , b ^ \\Theta ( z ) \\ , = \\ , - b ( z ) , c ^ \\Theta \\ , = \\ , c \\ , . \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} x \\sim y \\Leftrightarrow \\sigma _ x = \\sigma _ y . \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} C _ n = \\sum _ { \\ell = 0 } ^ { p - 1 } \\binom { p + k + n - \\ell } { n } \\lambda ^ 1 _ { n p + \\ell } b ^ { [ s - ( n + 1 ) p ^ 2 + ( k + n - \\ell ) p + \\ell + 1 ] } , \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} k \\log ( 4 ) + k \\log ( k ) \\le \\sqrt { l o g ( n ) } \\log ( \\sqrt { \\log n } ) = o ( l o g ( n ) ) \\Leftrightarrow ( 4 ) ^ { k } k ^ { k } = o ( n ) . \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} \\frac { 1 - \\left | \\eta _ { \\mu _ { 1 } } ( z ) \\right | } { 1 - \\left | z \\right | } = \\frac { 1 + 2 \\Re \\psi _ { \\mu _ { 1 } } ( z ) } { \\left ( 1 + \\left | \\eta _ { \\mu _ { 1 } } ( z ) \\right | \\right ) \\left | 1 + \\psi _ { \\mu _ { 1 } } ( z ) \\right | ^ { 2 } \\left ( 1 - \\left | z \\right | \\right ) } , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} \\phi ( u ) = \\begin{cases} ( 1 + u ) ^ 4 ( 1 - u ) ^ 4 , & | u | \\le 1 , \\\\ 0 , & | u | > 1 . \\end{cases} \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} I ( { \\rm c l } _ { T _ K ^ { \\max } ( X ) } [ A ] ) = & I ( X ) \\cap { \\rm c l } _ { T _ { \\hat { K } } ^ { \\max } ( \\hat { X } ) } [ I ( A ) ] , \\\\ I ( { \\rm i n t } _ { T _ K ^ { \\max } ( X ) } [ { \\rm c l } _ { T _ K ^ { \\max } ( X ) } [ A ] ] ) = & I ( X ) \\cap { \\rm i n t } _ { T _ { \\hat { K } } ^ { \\max } ( \\hat { X } ) } [ { \\rm c l } _ { T _ { \\hat { K } } ^ { \\max } ( \\hat { X } ) } [ I ( A ) ] ] . \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} \\int _ { M } | G ( p , y ) | | f ( y ) | \\ , d y & = \\int _ { M \\setminus B ^ \\rho _ { R } ( p ) } | G ( p , y ) | | f ( y ) | \\ , d y + \\int _ { B ^ \\rho _ { R } ( p ) } | G ( p , y ) | | f ( y ) | \\ , d y \\\\ & \\leq \\int _ { M \\setminus B ^ \\rho _ { R } ( p ) } | G ( p , y ) | | f ( y ) | \\ , d y + C , \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} | Z ( F ) \\cap ( A \\times B \\times C ) | = O _ d ( | A | ^ { 1 / 2 } | B | ^ { 2 / 3 } | C | ^ { 2 / 3 } + | A | ^ { 1 / 2 } ( | A | ^ { 1 / 2 } + | B | + | C | ) ) . \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\displaystyle \\int _ { M } ( P _ { 0 } ^ { k } ( \\lambda ^ { \\perp } - \\lambda _ { \\infty } ^ { \\perp } ) ) ^ { 2 } d \\mu _ { 0 } = 0 . \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} \\partial Q \\cap ( \\partial B _ \\rho \\cap X ) = ( \\partial Q \\cap \\partial B _ \\rho ) \\cap X = ( V \\cap \\partial B _ \\rho ) \\cap X = \\emptyset . \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} h _ 0 ( \\partial Q ) = ( \\partial B _ R \\cap V ) \\oplus \\{ \\gamma ^ X ( t ) : 0 \\leq t \\leq 1 \\} \\cup ( \\bar { B } _ R \\cap V ) \\cup ( \\bar { B } _ R \\cap V ) \\oplus \\{ \\gamma ^ X ( 1 ) \\} \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} \\begin{aligned} & \\partial ^ 2 _ t u = \\partial ^ 2 _ x u + c _ 1 u + ( c _ 2 u + f ( u ) ) \\dot { W } ( t , x ) , t > 0 , x \\in \\overline { D } , \\\\ & u ( 0 , x ) = ( J + T + 1 ) ( 1 + u _ 0 ( x ) ) , \\partial _ t u ( 0 , x ) = ( J + T + 1 ) v _ 0 ( x ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} H _ { L - m + 1 } = H _ { L - m } + H _ 1 + 1 = L - m + 2 = L - m + \\sum _ { a = 1 } ^ { 1 } \\frac { a ( a + 1 ) } { 2 } + 1 . \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } R = \\sigma _ { \\overline { 1 } , 1 } + \\sigma _ { 1 , \\overline { 1 } } - \\sigma _ { 0 } \\\\ A _ { 1 1 , } { } ^ { 1 } = \\sigma _ { 1 , 0 } + i \\sigma _ { 0 , 1 } - A _ { 1 1 } \\sigma ^ { 1 } \\end{array} \\right . . \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ( L _ { - 1 } ) \\rho ( L _ { k + 1 } ) & = h ( z ) ^ k ( L - \\lambda - k - 1 ) ( L + ( k + 1 ) \\lambda ) P ( L - \\lambda - k , k ) \\\\ \\rho ( L _ { 0 } ) \\rho ( L _ { k } ) & = h ( z ) ^ k ( L - k ) ( L + k \\lambda ) P ( L - \\lambda - k , k ) \\\\ \\rho ( L _ { 1 } ) \\rho ( L _ { k - 1 } ) & = h ( z ) ^ k ( L + \\lambda - k + 1 ) ( L + ( k - 1 ) \\lambda ) P ( L - \\lambda - k , k ) \\end{aligned} \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} D _ p ( M , x ) : = D _ p ( \\Pi _ M ^ p x , x ) \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} \\langle \\mathbf { N } , \\delta \\mathbf { r } _ { i j } \\rangle & = \\langle \\mathbf { N } , ( u \\mathbf { N } ) _ { i j } \\rangle = u _ { i j } + u \\langle \\mathbf { N } , \\mathbf { N } _ { i j } \\rangle = u _ { i j } - u \\langle h _ i ^ l \\mathbf { r } _ l , h _ j ^ k \\mathbf { r } _ k \\rangle \\\\ & = u _ { i j } - u h _ { i l } h ^ l _ j . \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} \\sum _ { j = i } ^ { k - 1 } ( - 1 ) ^ { j - i } \\binom { m - k + j - i - 1 } { j - i } b _ j \\ge 0 . \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} \\widehat { R } = \\epsilon \\left ( \\widehat { q } _ 1 + \\frac { \\theta } { 1 + \\sqrt { 1 - \\xi } } \\widehat { p } _ 2 \\right ) \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y _ l } \\left ( \\frac { 1 } { 2 } \\langle y , ( I d + P ( M _ { i j } ) N P ^ * ) y \\rangle _ Y \\right ) = \\frac { 1 } { M } y _ l + \\frac { 1 } { N } \\sum _ { n = 1 } ^ M \\sum _ { i \\in B ( l ) , j \\in B ( n ) } M _ { i j } y _ n . \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( 1 - \\sigma ) D ^ 2 v : D ^ 2 \\varphi + \\sigma \\Delta v \\Delta \\varphi d x = \\eta \\int _ { \\partial \\Omega } \\frac { \\partial v } { \\partial \\nu } \\frac { \\partial \\varphi } { \\partial \\nu } d \\sigma \\ , , \\ \\ \\ \\forall \\varphi \\in \\mathcal { H } ^ 2 _ { 0 , D } ( \\Omega ) , \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} K _ { n } ^ { ( 3 ) } K _ { m } ^ { ( 3 ) } + K _ { n + 1 } ^ { ( 3 ) } K _ { m + 1 } ^ { ( 3 ) } + K _ { n + 2 } ^ { ( 3 ) } K _ { m + 2 } ^ { ( 3 ) } = \\left \\lbrace \\begin{array} { c } 2 1 \\cdot 2 ^ { n + m } \\\\ + 2 ^ { n } \\left ( M _ { m + 1 } ^ { ( 2 ) } + 3 M _ { m + 2 } ^ { ( 2 ) } \\right ) \\\\ + 2 ^ { m } \\left ( M _ { n + 1 } ^ { ( 2 ) } + 3 M _ { n + 2 } ^ { ( 2 ) } \\right ) \\\\ + 3 ( \\omega _ { 1 } ^ { n } \\omega _ { 2 } ^ { m } + \\omega _ { 1 } ^ { m } \\omega _ { 2 } ^ { n } ) \\end{array} \\right \\rbrace , \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{align*} \\log \\det \\Delta _ \\beta ^ { S } = \\log \\det \\Delta _ \\beta - \\zeta _ \\beta ( 0 ) \\log S . \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} | \\int _ { t } ^ { \\infty } d s \\frac { - \\lambda '' ( s ) } { r ^ { 2 } } \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\left ( 1 + \\frac { r ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( r ^ { 2 } - 1 - \\rho ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } \\right ) | & \\leq \\frac { C } { r ^ { 2 } } \\int _ { t } ^ { \\infty } d s | \\lambda '' ( s ) | ( s - t ) \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} u g ^ { i j } g ^ { k l } \\nabla _ j h _ { i l } f _ k & = u ( \\nabla ^ i h ^ j _ i ) f _ j = u ( \\nabla ^ j h ^ i _ i + \\overline { R } ^ { \\ , i j } _ { \\ , \\ , \\ , \\ , l i } N ^ l ) f _ j = 2 u H ^ j f _ j - u f _ j \\overline { R } ^ { \\ , j } _ l N ^ l \\\\ & = 2 u H ^ j f _ j - u k _ 0 f _ j g ^ j _ l N ^ l = 2 u \\langle \\nabla H , \\nabla f \\rangle - u k _ 0 \\langle \\nabla f , \\mathbf { N } \\rangle \\\\ & = 2 u \\langle \\nabla H , \\nabla f \\rangle , \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} \\gamma _ J = \\left ( \\frac { \\partial f _ J } { \\partial y } \\ , g _ J \\middle ) \\right | _ { x = y = \\mu = 0 } , \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} \\ddot { \\theta } = \\begin{cases} a \\theta , & | \\theta - b \\dot { \\theta } | < \\theta ^ * , \\\\ ( a - K _ p ) \\theta - K _ d \\dot { \\theta } , & | \\theta - b \\dot { \\theta } | > \\theta ^ * . \\end{cases} \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} u ( x ) = \\begin{cases} \\frac { 1 } { | x | ^ t } , & x \\in B ^ + _ 1 ( 0 ) : = \\{ x \\in B _ 1 ( 0 ) \\rvert x _ n > 0 \\} ; \\\\ 0 , & x \\in \\mathbb { R } ^ n \\setminus B ^ + _ 1 ( 0 ) . \\end{cases} \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} p \\cdot \\pi _ * ( c _ 1 ( L ) ^ { \\dim X + 1 } ) = ( \\dim X + 1 ) \\int _ X c _ 1 ( L ) ^ { \\dim X } \\cdot c _ 1 ( L _ H ) . \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} z \\eta _ { \\mu } ^ { \\langle - 1 \\rangle } ( z ) = \\eta _ { \\mu _ { 1 } } ^ { \\langle - 1 \\rangle } ( z ) \\eta _ { \\mu _ { 2 } } ^ { \\langle - 1 \\rangle } ( z ) \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} \\vartheta _ { 0 } = 8 \\vartheta _ { k + 1 } = \\vartheta _ { k } - \\frac { 6 } { \\pi ^ { 2 } ( k + 1 ) ^ { 2 } } . \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} \\beta \\int _ { \\mathbb { T } } \\frac { d \\mu _ { 1 } ( \\overline { \\xi } ) } { | \\xi - z ( t ) | ^ { 2 } } = \\frac { \\log R ( t ) } { R ( t ) ^ { 2 } - 1 } \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } D ^ 2 u = 0 , & \\\\ \\frac { \\partial u } { \\partial \\nu } - \\frac { 1 } { \\theta } u = 0 , & . \\end{array} \\right . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} \\left \\| g + h \\right \\| = \\left \\| g \\right \\| + \\left \\| h \\right \\| g , h \\in X _ { + } . \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} \\phi _ s = L \\ , V - F _ j \\ , g ^ { i j } \\ , \\langle \\nabla _ { F _ i } ^ { \\perp } V , \\phi \\rangle \\ , . \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta _ { x } u + \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert \\leq 2 m } a _ { \\alpha } \\left ( x , y \\right ) D _ { y } ^ { \\alpha } u \\left ( x , y , t \\right ) + F \\left ( u , \\bar { u } \\right ) u = 0 , \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} \\frac { d u } { d t } ( t _ 0 ) = \\displaystyle \\sum _ { i = 0 } ^ { k } \\alpha _ { k , i } \\left ( u ( t _ i ) + u ( t _ { i + 1 } ) \\right ) + \\mathcal { O } \\left ( \\displaystyle \\sum _ { i = 0 } ^ { k } \\alpha _ { k , i } \\left ( ( \\Delta t _ i ) ^ { k + 1 } + ( \\Delta t _ { i + 1 } ) ^ { k + 1 } \\right ) \\right ) \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} \\theta = \\sqrt { \\frac { k b ( 1 + a c ) } { c ( d - k ) } + 1 } . \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} E ( i ) & \\coloneqq \\{ ( x , z ) \\in X \\times \\C ^ 2 : A ( x ) z = i z \\} , \\\\ E ( - i ) & \\coloneqq \\{ ( x , z ) \\in X \\times \\C ^ 2 : A ( x ) z = - i z \\} . \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} \\mathcal { Z } ^ n _ T ( M , A ) = \\{ f \\in \\mathrm { H o m } ( M ^ { \\otimes n } , A ) | ~ d _ T ( f ) = 0 \\} ~ ~ ~ ~ ~ ~ ~ ~ \\mathcal { B } ^ n _ T ( M , A ) = \\{ d _ T g | ~ g \\in \\mathrm { H o m } ( M ^ { \\otimes n - 1 } , A ) \\} \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} f ( p ) = \\sup _ { a \\in A } \\varphi ( a ) \\cdot \\frac { d ( p , A ) } { d ( a , p ) } , \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} & P _ { d } \\left \\{ \\sup _ { 1 \\leq i \\leq k _ { m } } \\sup _ { f , g \\in \\mathcal { F } _ { m , i } } | \\mathbb { H } _ { N } ' f - \\mathbb { H } _ { N } ' g | > \\epsilon _ { m } \\right \\} < 2 ^ { - m } \\eta + \\widetilde { D } _ { m , N } \\\\ & N = 1 , 2 , \\dots \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} Y ( t ) = N ( E _ { \\alpha _ 1 , \\lambda _ 1 , \\alpha _ 2 , \\lambda _ 2 } ( t ) ) . \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{gather*} \\beta _ i = \\begin{cases} 0 & i \\in A \\\\ \\frac { \\alpha _ i } { 2 } & i \\in A ^ c \\end{cases} , \\gamma _ i = \\begin{cases} \\frac { \\alpha _ i - 1 } { 2 } & i \\in A \\\\ 0 & i \\in A ^ c \\end{cases} 1 _ { A , i } = \\begin{cases} 1 & i \\in A \\\\ 0 & i \\in A ^ c \\end{cases} , \\end{gather*}"}
-{"id": "6301.png", "formula": "\\begin{align*} \\rho ^ + = ( \\rho ^ T \\rho ) ^ { - 1 } \\rho ^ T . \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} & x _ 1 = e _ { 2 3 } + e _ { 5 6 } , ~ y _ 1 = e _ { 3 2 } + e _ { 6 5 } , ~ h _ 1 = [ x _ 1 , y _ 1 ] \\\\ & x _ 2 = e _ { 1 2 } + e _ { 3 4 } + e _ { 4 5 } + e _ { 6 7 } , ~ y _ 2 = e _ { 2 1 } + 2 e _ { 4 3 } + 2 e _ { 5 4 } + e _ { 7 6 } , ~ h _ 2 = [ x _ 2 , y _ 2 ] . \\\\ \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} \\partial _ { t } f = S f + K f + \\left ( a + i b \\right ) \\left ( V f + e ^ { \\gamma \\varphi } F \\right ) R ^ { n } \\times \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} \\int _ { D E } e ^ { s x } \\overline { G } ( s , t ) d s + \\int _ { F G } e ^ { s x } \\overline { G } ( s , t ) d s & = - \\int _ { r } ^ { - r + \\lambda _ 2 - \\lambda _ 1 } e ^ { - x \\lambda _ 1 } e ^ { - w x } e ^ { t ( c _ 1 \\lambda _ 1 ^ { \\alpha _ 1 } + c _ 2 \\lambda _ 2 ^ { \\alpha _ 2 } ) } \\\\ & \\times \\left [ e ^ { - t ( c _ 1 w ^ { \\alpha _ 1 } \\cos ( \\pi \\alpha _ 1 ) + c _ 2 ( \\lambda _ 2 - \\lambda _ 1 + w e ^ { - i \\pi } ) ^ { \\alpha _ 2 } ) } 2 i \\sin ( t c _ 1 w ^ { \\alpha _ 1 } \\sin ( \\pi \\alpha _ 1 ) ) \\right ] d w . \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} \\mathbb { E } [ f ] = \\displaystyle \\int _ { \\Omega } f ( \\mu ( \\omega ) ) \\ ; { \\rm d } P ( \\omega ) = \\int _ { \\mathcal { M } } f ( \\mu ) \\rho ( \\mu ) \\ ; { \\rm d } \\mu \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} ( 2 i + 1 ) \\nu ( h ( z ) ) - ( i + 1 ) \\nu ( h ' ( z ) ) \\geq 0 i = - 1 , 0 , 1 \\ , , \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} U _ { y } ( z ) = J F ^ k ( y ) ^ { - 1 } e ^ { z S _ { j , k } u ( y ) } f ( y ) \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} \\lambda ^ { \\ast ^ { G / H } \\ast ^ { G / H } } ( \\psi ) = \\lambda \\circ J _ 0 ( \\psi ) , \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} & D _ k : = 2 H - 2 E _ k - \\sum _ { p \\neq k } E _ p \\\\ & D _ j : = 2 H - 2 E _ j - \\sum _ { p \\neq j } E _ p . \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t / 2 } d y \\frac { e ^ { - y } \\sin ( a \\tan ^ { - 1 } ( \\frac { \\pi } { 2 ( \\log ( t ) - \\log ( y ) ) } ) ) } { y ( ( \\log ( t ) - \\log ( y ) ) ^ { 2 } + \\frac { \\pi ^ { 2 } } { 4 } ) ^ { a / 2 } } & = a \\frac { \\pi } { 2 } \\left ( \\frac { 1 } { a \\log ^ { a } ( t ) } + O \\left ( \\frac { 1 } { \\log ^ { a + 1 } ( t ) } \\right ) \\right ) + O \\left ( \\frac { 1 } { \\log ^ { a + 2 } ( t ) } \\right ) \\\\ & = \\frac { \\pi } { 2 \\log ^ { a } ( t ) } + O \\left ( \\frac { 1 } { \\log ^ { a + 1 } ( t ) } \\right ) \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} I _ { \\Omega ^ i _ k } : = \\bigg | \\frac { 1 } { 2 \\pi i } \\sum \\limits _ { j = 1 } ^ { m _ 0 } w _ j \\vec { x } _ h ( \\omega _ j ) \\bigg | . \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} d ( m , n ) & = \\sum _ { r = 1 } ^ m u ( m - r , n + r ) + f ( m - r , n + r ) , \\\\ f ( m , n ) & = \\sum _ { r = 1 } ^ m u ( m - r , n ) + d ( m - r , n ) , \\\\ u ( m , n ) & = \\sum _ { r = 1 } ^ m f ( m - r , n - r ) + d ( m - r , n - r ) , \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} p ( x , y , t ) = \\sum _ { i \\ge 0 } e ^ { - \\lambda _ i t } \\phi _ i ( x ) \\phi _ i ( y ) \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} 0 = f ( x ) - \\int \\limits _ { D \\setminus U } f ( z ) \\omega _ U ^ x ( d z ) = \\int \\limits _ { \\partial ^ * D } \\left ( M _ D ( x , y ) - \\int _ { D \\setminus U } M _ D ( z , y ) \\omega _ U ^ x ( d z ) \\right ) \\mu ^ * ( d y ) , \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} P _ 1 ( T ) U ( T ) = P _ 2 ( T ) V ( T ) \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} Q _ { i } ( \\eta , 1 , \\sigma ) = P _ { i , 1 } ( \\eta ) - \\frac { P _ { 1 , 1 } ( \\eta ) } { P _ { 1 , 2 } ( \\eta ) } P _ { i , 2 } ( \\eta ) = 0 . \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{align*} f _ { ( \\Gamma , p ) } ( x ) = \\frac { \\| x , A _ { ( \\Gamma , p ) } \\| } { \\| x , A _ { ( \\Gamma , p ) } \\| + \\| x , Z _ { ( \\Gamma , p ) } \\| } , \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} Q = \\left [ \\begin{array} { c c c } Q _ { 2 3 2 3 } & - Q _ { 1 3 2 3 } & Q _ { 1 2 2 3 } \\\\ - Q _ { 1 3 2 3 } & Q _ { 1 3 1 3 } & - Q _ { 1 2 1 3 } \\\\ Q _ { 1 2 2 3 } & - Q _ { 1 2 1 3 } & Q _ { 1 2 1 2 } \\end{array} \\right ] . \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} \\left ( \\frac { C _ { n , k _ { 1 } } - ( n - 1 ) \\mathbb { I } _ { k _ { 1 } = 2 } - \\mu _ { k _ { 1 } } } { \\sqrt { 2 k _ { 1 } } } , \\ldots , \\frac { C _ { n , k _ { l } } - \\mu _ { k _ { l } } } { \\sqrt { 2 k _ { l } } } \\right ) \\stackrel { d } { \\to } N _ { l } ( 0 , I _ { l } ) \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} z \\rightsquigarrow \\sum _ { k = 0 } ^ { n } \\Big ( \\sum _ { \\substack { \\alpha \\in F \\\\ \\alpha _ { n } = k } } c _ { \\alpha } z _ { 1 } ^ { \\alpha _ { 1 } } \\cdots z _ { n - 1 } ^ { \\alpha _ { n - 1 } } \\Big ) z ^ { k } \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} \\{ x \\mapsto \\Theta ( \\l , K ) x + v _ j ( R , \\l , K ) : j = 1 , \\ldots , m \\} \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} F ^ a _ { \\nabla ^ \\omega } = \\d A ^ a + \\omega ^ a { } _ { \\mu b } A ^ b \\wedge D X ^ \\mu + \\tfrac { 1 } { 2 } C ^ a { } _ { b c } A ^ b \\wedge A ^ c , \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} { \\rm T r } _ \\chi ( A ) : = \\int _ M \\chi ( x ) A ( x , x ) d x . \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} P * Q = \\partial ^ u _ { w _ { 0 , a , b } } \\partial ^ v _ { w _ { 0 , a , b } } \\left ( ( P \\otimes Q ) \\prod _ { i = 1 } ^ a \\prod _ { j = a + 1 } ^ b ( v _ i - u _ j ) ^ 2 \\right ) , \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} \\forall l \\in I ^ { 0 0 } _ \\textup { C C } ( \\bar x ) \\colon \\bar \\mu _ l \\bar \\nu _ l = 0 \\ , \\lor \\ , ( \\bar \\mu _ l > 0 \\ , \\land \\ , \\bar \\nu _ l > 0 ) , \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ 3 \\left ( \\frac { b _ 2 c _ n } { a _ 2 } - d _ n \\right ) \\frac { q _ n } { \\omega ^ n } & = 2 a _ 3 k _ 1 + a _ 4 k _ 2 + 2 a _ 5 k _ 3 \\\\ & \\quad + 2 b _ 3 \\ell _ 1 + b _ 4 \\ell _ 2 + 2 b _ 5 \\ell _ 3 \\ , , \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} H _ { D S , f , - } ^ \\bullet ( M ) = H _ { D S , f , ( w _ 0 , - x _ 0 ) } ^ { \\bullet } ( M ) \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} \\mu _ { m } ^ { n ( I ) } ( J ) & = \\mu _ { m } ^ { n ( I _ { b ( I ) } ) } ( J ) \\stackrel { \\eqref { e : f r o s t - c a s e - 1 } } { < } \\frac { 1 } { 2 } \\mu _ { m } ^ { n ( I _ { b ( I ) } ) + 1 } ( J ) \\stackrel { \\eqref { e : f r o s t - c a s e - 2 } } { = } \\frac { 1 } { 2 } \\mu _ { m } ^ { n ( I _ { b ( I ) + 1 } ) } ( J ) < \\cdots \\\\ & \\cdots < \\frac { 1 } { 2 ^ { b ( J ) - b ( I ) } } \\mu _ { m } ^ { n ( I _ { b ( J ) } ) } ( J ) = \\frac { 1 } { 2 ^ { b ( J ) - b ( I ) } } \\mu _ { m } ^ { m } ( J ) = \\frac { \\ell ( J ) ^ { d } } { 2 ^ { b ( J ) - b ( I ) } } . \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} G = ( \\d x ) ^ 2 + ( \\d y ) ^ 2 + ( 1 + x ^ 2 ) ( \\d z ) ^ 2 , C = 2 x \\d x \\wedge \\d z , \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} \\| | D | ^ s X \\| _ { L ^ \\infty _ t ( L ^ 2 ) } ^ 2 \\leq C . \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} T _ { \\eqref { e _ d i f f e r e n c e _ c g _ h a m i l t o n i a n } } \\lesssim \\frac { 1 } { N } \\sum _ { l = 1 } ^ { M } \\left ( 1 + y _ l ^ 2 \\right ) = \\frac { 1 } { K } \\left ( 1 + \\| y \\| _ { L ^ 2 ( Y ) } ^ 2 \\right ) \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} 2 = [ A _ 1 \\otimes B , A _ 2 \\otimes B ] = [ A _ 1 \\otimes A _ 2 , B \\otimes B ] = [ A _ 1 \\oplus C , B \\otimes B ] . \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} \\hat { H } _ { c o u l } = - \\sum _ { i = 1 } ^ { D } \\frac { \\partial ^ 2 } { \\partial x _ i ^ 2 } - \\frac { \\eta } { r } + \\sum _ { i = 1 } ^ { N - 1 } \\frac { 1 } { r _ i ^ 2 } g _ i ( \\Omega _ i ) . \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} \\begin{aligned} \\dd \\partial _ t w _ i - d _ i \\mathcal { L } _ i [ w _ i ] - \\sum _ { j = 1 } ^ m q _ { i j } w _ j & = \\dd \\partial _ t v _ i - d _ i \\mathcal { L } _ i [ v _ i ] - \\sum _ { j = 1 } ^ m q _ { i j } v _ j + \\epsilon A e ^ { A t } - \\epsilon e ^ { A t } \\sum _ { j = 1 } ^ n q _ { i j } \\\\ & \\geq \\Big ( A - \\sum _ { j = 1 } ^ n q _ { i j } \\Big ) \\epsilon e ^ { A t } \\geq \\epsilon e ^ { A t } \\ \\ { \\rm f o r } \\ ( t , x ) \\in ( 0 , T ] \\times \\R . \\end{aligned} \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} C _ { n ( i - 1 ) + j } \\ = \\ B _ i \\ltimes A _ j \\ \\ M = N n . \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} \\varepsilon _ 3 = \\varepsilon ^ b _ 1 \\varepsilon ^ c _ 2 C ^ a { } _ { b c } X ^ * e _ a = \\tilde { \\varepsilon } ^ b _ 1 \\tilde { \\varepsilon } ^ c _ 2 \\widetilde { C } ^ a { } _ { b c } X ^ * \\tilde { e } _ a = \\varepsilon ^ y _ 1 \\varepsilon ^ z _ 2 ( K ^ { - 1 } ) ^ b { } _ { y } ( K ^ { - 1 } ) ^ c { } _ { z } \\widetilde { C } ^ x _ { y z } K ^ a { } _ x X ^ * e _ a , \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} H = - \\sum _ { i = 1 } ^ { D } \\frac { \\partial ^ 2 } { \\partial x _ i ^ 2 } - \\frac { \\eta } { r } + \\sum _ { i = 1 } ^ { D - 1 } \\frac { \\alpha _ i } { x _ i ^ 2 } \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} a _ k ( n ) & = a _ k ( n - 1 ) + a _ k ( n - 2 ) - a _ k ( n - 5 ) - a _ k ( n - 7 ) \\\\ & \\quad \\ , + a _ k ( n - 1 2 ) + a _ k ( n - 1 5 ) - a _ k ( n - 2 2 ) - a _ k ( n - 2 6 ) + \\cdots . \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} e _ { x y } e _ { u v } = \\delta _ { y u } e _ { x v } , \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} [ T ^ { p ^ v } \\textbf { a } ] _ i & \\equiv \\sum _ { j = 0 } ^ { p ^ v } \\binom { p ^ v } { j } \\textbf { a } _ { i + j } \\equiv \\sum _ { j = 0 } ^ { p ^ 2 } \\binom { p ^ v } { j p ^ { v - 2 } } \\textbf { a } _ { i + j p ^ { v - 2 } } \\equiv \\sum _ { j = 0 } ^ { p ^ 2 } \\binom { p ^ 2 } { j } \\textbf { a } _ { i + j p ^ { v - 2 } } \\pmod { p ^ 3 } , \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} | \\partial _ { t } N _ { 2 } ( f ) ( t , r ) | \\leq \\begin{cases} \\frac { C r } { t ^ { 5 } \\log ^ { 3 b } ( t ) ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { r ^ { 3 } t ^ { 7 / 2 } \\log ^ { \\frac { 3 N } { 2 } + 5 b } ( t ) } + \\frac { C \\log ^ { 3 } ( r ) } { r ^ { 2 } ( t - r ) ^ { 4 } } , t > r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} \\mathcal { P } _ L f : = \\sum _ { \\ell = 1 } ^ { d } \\left < f , p _ { \\ell } \\right > p _ { \\ell } . \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} E = \\left ( \\begin{matrix} 1 & 0 & x \\\\ 0 & 1 & 0 \\\\ - x & 0 & 1 + x ^ 2 \\end{matrix} \\right ) , H = \\d C = 0 . \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} \\beta ( F , A _ 1 , A _ 2 ) \\cdot _ { \\overline { M _ { i + 1 } } } \\delta _ j & = m _ 1 m _ 2 F \\cdot \\delta _ j j \\neq 0 , 1 , i \\\\ \\beta ( F , A _ 1 , A _ 2 ) \\cdot _ { \\overline { M _ { i + 1 } } } \\delta _ 0 & = m _ 1 m _ 2 F \\cdot \\delta _ 0 + \\sum _ { \\ell = 1 } ^ { 2 } \\Big ( m _ { 2 - \\ell } ( m _ \\ell ( 2 h - 2 ) - ( 2 g ( A _ \\ell ) - 2 ) + A _ \\ell \\cdot A _ \\ell ) - A _ 1 \\cdot A _ 2 \\Big ) . \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } t } D _ { 3 } ( t v , v _ \\lambda ) = \\frac { 9 t ^ { 2 } } { 8 } - \\frac { \\left ( \\frac { \\lambda } { 2 } + \\frac { 1 } { 2 } \\right ) ^ { 2 } } { 2 } - 1 \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} B ^ 1 = J \\oplus \\pi ^ * ( I ^ 1 ) . \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} K _ { \\tilde X } + \\tilde \\Delta = p ^ * ( K _ X + \\Delta ) + E . \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} ( \\sum _ { s = i _ { 1 } } ^ { j _ { 2 } } m _ { s , s + 1 } ) m _ { i _ { 2 } , j _ { 2 } + 1 } = - ( j _ { 2 } - i _ { 1 } + 1 ) m _ { i _ { 2 } , j _ { 2 } + 1 } m _ { i _ { 1 } , n + 1 } + \\cdots , \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} ( \\omega _ 1 \\otimes \\omega _ 2 ) : ( X _ 1 ^ \\flat \\otimes X _ 2 ^ \\flat ) & : = \\omega _ 1 ( X _ 1 ) \\ , \\omega _ 2 ( X _ 2 ) , \\\\ \\langle \\omega _ 1 \\wedge \\omega _ 2 , X _ 1 ^ \\flat \\wedge X _ 2 ^ \\flat \\rangle & : = \\det \\omega _ i ( X _ j ) . \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} V ( q ) = Z ^ 2 + \\frac { 3 } { 2 } \\sum _ { x , y } g ^ { x y } \\partial _ x Z \\partial _ y Z , \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} & \\psi ( X ^ 1 , \\cdots , X ^ i + 2 \\pi r _ i ' , \\cdots , X ^ n ) = \\psi ( X ^ 1 , \\cdots , X ^ i , \\cdots , X ^ n ) \\ ( i = 1 , \\cdots , s ) , \\\\ & \\psi ( X ^ 1 , \\cdots , X ^ i + 2 \\pi , \\cdots , X ^ n ) = \\psi ( X ^ 1 , \\cdots , X ^ i , \\cdots , X ^ n ) \\ ( i = s + 1 , \\cdots , n ) . \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} \\Sigma _ { n , \\ , N } ^ \\alpha & = \\{ f \\in \\Sigma _ { n , \\ , N } \\ | \\ \\| f \\| _ { \\Sigma _ { n , \\ , N } } \\leq \\alpha \\} . \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} \\int _ M r ( y ) ^ \\gamma \\ , u ( y ) ^ 2 \\ , d y & = \\int _ M r ( y ) ^ \\gamma \\ , \\left ( \\sum _ k \\varphi _ k ( y ) u ( y ) \\right ) ^ 2 \\ , d y \\\\ & \\leq 2 \\sum _ k \\int _ M r ( y ) ^ \\gamma \\ , \\varphi _ k ( y ) ^ 2 u ( y ) ^ 2 \\ , d y \\\\ & \\leq C \\sum _ k ( k - 1 ) ^ \\gamma \\int _ M \\varphi _ k ( y ) ^ 2 u ( y ) ^ 2 \\ , d y \\\\ & \\leq C \\sum _ k \\int _ M | \\nabla \\left ( \\varphi _ k ( y ) u ( y ) \\right ) | ^ 2 \\ , d y , \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} \\Vert u \\Vert _ p ^ p \\leq \\Vert u \\Vert _ \\infty ^ { p - 2 } \\Vert u \\Vert _ 2 ^ 2 = \\Vert u \\Vert _ \\infty ^ { p - 2 } \\ , \\mu , \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} \\frac { \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) } { \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) } & \\leq C \\frac { \\omega \\lambda ( x ) ^ { 2 } \\log ^ { 2 } ( \\omega \\lambda ( x ) ^ { 2 } ) } { \\omega \\lambda ( t ) ^ { 2 } \\log ^ { 2 } ( \\omega \\lambda ( t ) ^ { 2 } ) } \\\\ & \\leq C \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} F _ j ( x , y ; \\mu ) = \\begin{bmatrix} f _ j ( x , y ; \\mu ) \\\\ g _ j ( x , y ; \\mu ) \\end{bmatrix} , \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} N = \\left \\lceil \\exp ^ { ( 2 ) } ( \\log ^ { ( 2 ) } ( j _ { 0 } + 1 ) + C ^ { 2 } ) \\right \\rceil , \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} T = \\sum _ { j = 1 } ^ k \\delta _ j T _ j , \\textit { w h e r e } T _ j = \\mu [ U ] \\left ( a ^ { ( 1 ) } \\otimes \\ldots \\otimes t _ j \\otimes \\ldots \\otimes a ^ { ( m ) } \\right ) . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} h _ n ( \\underline x _ n ) = \\prod _ { r = 1 } ^ n \\left ( \\prod _ { s = 1 } ^ r ( 1 + x _ s ) - 1 \\right ) \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} { \\rm s g n } \\left ( \\phi ( y ; \\mu ) \\right ) = - { \\rm s g n } ( y ) , ~ y \\ge 0 , \\ , ~ \\mu \\in \\mathbb { R } . \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} \\omega ^ { 2 } = 2 K b - b ^ { 2 } \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} X ^ G _ t = \\left \\lbrace \\begin{aligned} & X _ t , t < \\sigma _ F : = \\{ s > 0 : X _ s = F \\} , \\\\ & \\partial , \\quad \\ ; \\ ; t \\geq \\sigma _ F , \\end{aligned} \\right . \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} H _ { L - 1 } + H _ { k + 1 } + \\sum _ { a = 1 } ^ { k } \\sum _ { i = a } ^ { a + m - 1 } H _ { i } \\geq \\left ( \\sum _ { i = 1 } ^ { k + m - 1 } H _ { i } + 1 \\right ) + \\left ( \\sum _ { i = 1 } ^ { k + m - 2 } H _ { i } + 1 \\right ) . \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} x _ j = \\bar { x } \\left ( \\frac { j } { N } \\right ) , j = 1 , \\cdots , N . \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} \\operatorname { I d } + \\tilde { k } _ c = ( \\operatorname { I d } + k _ c ) \\circ ( \\operatorname { I d } + \\psi ) \\circ ( \\operatorname { I d } + \\tilde { k } _ c ) = ( \\operatorname { I d } + k _ c ) \\circ ( \\operatorname { I d } + \\varphi ) , \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} \\sup _ { \\zeta \\in [ 0 , t ] } \\| \\Phi ( u ) ( \\cdot , \\zeta ) \\| & \\le \\| \\xi \\| _ \\infty + \\int _ 0 ^ t p ( s ) ( 1 + \\| \\xi \\| _ \\infty + \\sup _ { \\zeta \\in [ 0 , s ] } \\| u ( \\zeta ) \\| ) d s \\\\ & \\le \\| \\xi \\| _ \\infty + \\int _ 0 ^ t p ( s ) ( 1 + \\| \\xi \\| _ \\infty + \\psi ( s ) ) d s = \\psi ( t ) , \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} \\langle \\nabla u , \\nabla | \\nabla u | \\rangle = u _ n u _ { n n } \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} \\mathsf { M } ( f \\bullet \\phi ) = f \\bullet \\left ( \\mathsf { M } \\phi \\right ) , \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} m _ j ( \\mu ) = \\frac { g _ j ( 0 , 0 ; \\mu ) } { f _ j ( 0 , 0 ; \\mu ) } , \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } { d x } _ { t } & = ( F _ { t } { x } _ { t } + f _ { t } + \\theta _ { t } ^ { \\ast } ) d t + d w _ { t } ^ { \\theta ^ { \\ast } } , \\\\ { x } ( 0 ) & = x _ { 0 } , \\\\ { d m } _ { t } & = ( G _ { t } x _ { t } + g _ { t } ) d t + d v _ { t } , \\\\ { m } ( 0 ) & = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} H _ 3 = - 1 . \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} \\partial _ t \\partial _ k X + u \\cdot \\nabla \\partial _ k X + \\partial _ k u \\cdot \\nabla X = \\partial _ k \\partial _ X u \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} - \\partial _ { t t } u + \\partial _ { r r } u + \\frac { 1 } { r } \\partial _ { r } u - \\frac { \\sin ( 2 u ) } { 2 r ^ { 2 } } = - \\left ( F _ { 4 } + F _ { 5 } + F _ { 6 } \\right ) \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} J _ { n + 3 } ^ { ( 3 ) } = J _ { n + 2 } ^ { ( 3 ) } + J _ { n + 1 } ^ { ( 3 ) } + 2 J _ { n } ^ { ( 3 ) } , \\ J _ { 0 } ^ { ( 3 ) } = 0 , \\ J _ { 1 } ^ { ( 3 ) } = J _ { 2 } ^ { ( 3 ) } = 1 , \\ n \\geq 0 , \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} \\Phi ( \\widetilde { A } ) = \\sup \\limits _ { f \\in L ^ 1 _ p ( \\widetilde { A } ) \\cap C _ 0 ( \\widetilde { A } ) } \\left ( \\frac { \\| \\varphi ^ { \\ast } ( f ) \\mid L ^ 1 _ q ( \\Omega ) \\| } { \\| f \\mid L ^ 1 _ p ( \\widetilde { A } ) \\| } \\right ) ^ { \\kappa } , \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} & \\mu [ U ] \\left ( b ^ { ( 1 ) } \\otimes \\ldots \\otimes b ^ { ( m ) } \\right ) = V [ v ^ { ( i _ 0 + 1 ) } \\leadsto x ^ { ( i _ 0 + 1 ) } ] , \\\\ & \\mu [ U ] \\left ( c ^ { ( 1 ) } \\otimes \\ldots \\otimes c ^ { ( m ) } \\right ) = W [ w ^ { ( i _ 0 + 1 ) } \\leadsto y ^ { ( i _ 0 + 1 ) } ] . \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} \\lbrace \\omega \\in \\Omega \\ : \\ V ( \\omega , 0 ) = + c , \\ N ( \\omega , t ) = n , \\max _ { 0 \\le s \\le t } \\mathcal { T } ( \\omega , s ) > \\beta , \\ \\mathcal { T } ( \\omega , t ) \\le \\beta \\rbrace , \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} G _ { \\nu _ { 1 } } ( z ) = \\frac { 1 } { \\beta } \\int _ { \\mathbb { R } } \\frac { 1 + t ^ { 2 } } { F _ { \\mu _ { 1 } } ( z ) - t } \\ , d \\sigma ( t ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} b _ { s - 1 } & = - \\frac { \\lambda _ s } { x _ { 1 + 2 + \\ldots + ( s - 1 ) } } \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} P _ 1 ( A ^ p , B ^ p ) & = P _ { t h } ( A ^ p , B ^ p ) = P _ { h ( 1 / t ) } ( B ^ p , A ^ p ) \\\\ & \\le P _ { h ( 1 / t ) } ( g ( A ) ^ p , A ^ p ) = A ^ p h \\left ( \\left ( { A \\over g ( A ) } \\right ) ^ p \\right ) \\\\ & = A ^ p h \\left ( \\left ( { 1 \\over { f ( A ) } } \\right ) ^ p \\right ) \\\\ & \\le \\left ( A h \\left ( { 1 \\over { f ( A ) } } \\right ) \\right ) ^ p = f ( A ) ^ { 2 ( n - 1 ) p } . \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { k } P _ { 2 k } [ 2 r ] r \\psi _ { 2 r } = 0 . \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} u _ 1 ( t ) = e ^ { t \\mathcal { L } } u _ 0 + \\int _ 0 ^ t e ^ { ( t - \\xi ) \\mathcal { L } } f \\left ( e ^ { \\xi \\mathcal { L } } u _ 0 , e ^ { \\xi \\overline { \\mathcal { L } } } \\overline u _ 0 \\right ) d \\xi \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} S ( B ^ k A ^ k ) S & = B ^ k T ^ 2 A ^ k ( B ^ k A ^ k ) B ^ k T ^ 2 A ^ k = ( B ^ k T ^ 2 A ^ k ) ( B ^ k A ^ k T ) = B ^ k T ^ 2 A ^ { k + 1 } T = B ^ k T ^ 2 A ^ k = S . \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} \\alpha \\in J _ { D , n } : = \\bigcup _ { t = 0 } ^ { d _ n - 1 } \\left ( \\frac { I _ D } { d _ n } + \\frac { t } { d _ n } \\right ) \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} { \\phi } _ { p , \\alpha } \\left ( 1 + \\frac { 1 } { [ { \\phi } _ { p , \\alpha } ] _ { 1 } } \\right ) \\left ( [ { \\phi } _ { p , \\alpha } ] _ { 2 } + 1 \\right ) = { \\phi } _ { p , \\alpha } \\left ( 1 + \\frac { 1 } { p } \\right ) \\left ( p + \\alpha + 1 \\right ) , \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} \\mathbf { E } \\left ( f ( \\eta ^ * ) \\right ) = \\mathbf { E } \\left ( f ( \\eta ) \\ : \\vline \\ : \\eta _ 0 = 0 , \\ : \\eta _ 1 = 1 \\right ) \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} \\nabla \\cdot \\overline { v } = \\int _ 0 ^ 1 \\nabla \\cdot v ( \\boldsymbol { x } ' , z ) d z = - \\int _ 0 ^ 1 \\partial _ z w ( \\boldsymbol { x } ' , z ) d z = 0 . \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} \\begin{aligned} { } ^ { i } \\ ! t ^ { j ' , h ' } _ { j , h } & = 0 & & ( j ' , h ' ) \\neq ( j , h + 1 ) \\\\ { } ^ { i } \\ ! s ^ { j ' , h ' } _ { j , h } & = 0 & & ( j ' , h ' ) \\neq ( j , h ) h \\neq j - i \\end{aligned} \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} \\boldsymbol { x } ( u , v ) = \\left ( u , v , \\frac { a } { 2 } u ^ { 2 } + \\frac { b } { 2 } v ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} q _ \\gamma ( t , x , y ) = \\frac { \\mathrm { e } ^ { - \\lambda _ \\gamma t } p _ \\gamma ( t , x , y ) \\psi _ \\gamma ( y ) } { \\psi _ \\gamma ( x ) } , \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} < D P ( Z ; d ) , \\delta Z > = - P ( Z ; d ) \\int _ { c } ^ { d } P ^ { - 1 } ( Z ; t ) \\delta Z ( t ) P ( Z ; t ) d t . \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} \\frac { { \\lambda ' } _ { l } } { \\lambda _ { l } } \\gamma _ { l i } \\gamma _ { \\sigma ' \\left ( l \\right ) \\sigma ( i ) } = G ^ { \\left ( 1 \\right ) } _ { 0 ; 0 , \\dots , 1 , \\dots , 0 } ( z ) , \\quad \\mbox { f o r a l l $ l = 1 , \\dots , N $ . } \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} u = \\begin{cases} ( c ^ + r ^ { \\gamma _ 1 ^ + } + c ^ - r ^ { \\gamma _ 1 ^ - } ) w _ 1 , & \\gamma _ 1 ^ + > \\gamma _ 1 ^ - \\\\ c r ^ { - \\frac { n - 2 } { 2 } } w _ 1 , & \\gamma _ 1 ^ + = \\gamma _ 1 ^ - \\end{cases} \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} f \\nabla _ { \\lambda } g = g \\nabla _ { 1 - \\lambda } f , \\ ; f ! _ { \\lambda } g = g ! _ { 1 - \\lambda } f , \\ ; f \\sharp _ { \\lambda } g = g \\sharp _ { 1 - \\lambda } f , \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} \\eta ( x ) = \\bigcap _ { U \\in \\mathcal { U } } U ( x ) . \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} & | \\partial _ { t } ^ { 2 } \\left ( \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { t } ^ { \\infty } \\lambda '' ( x ) K _ { 1 } ( x - t , \\lambda ( t ) ) + \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { t } ^ { \\infty } \\lambda '' ( x ) K ( x - t , \\lambda ( t ) ) \\right ) | \\\\ & \\leq \\frac { C } { t ^ { 3 } \\log ^ { 1 - 2 b } ( t ) } \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} J _ t ( \\Lambda ^ { ' } ) \\subset D _ { t } ^ { c } ; | J _ t ( \\Lambda ^ { ' } ) | = | \\Lambda ^ { ' } | . \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} \\Pi u ( z ) = \\langle ( I d - z M ( u ) ) ^ { - 1 } X ( u ) | Y ( u ) \\rangle _ { \\ell ^ 2 } \\ , \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} e ^ { \\beta } u ' ( \\beta ) = e ^ { 2 \\beta } + ( 1 - e ) e ^ { \\beta } + e \\beta ^ 2 + \\left ( \\frac { 2 e ^ 3 } { 3 ^ { 5 / 2 } \\sqrt { 2 \\pi } } - \\frac { 5 e } { 2 } \\right ) \\beta + \\frac { e } { 2 } - \\frac { 2 e ^ 3 } { 3 ^ { 5 / 2 } \\sqrt { 2 \\pi } } . \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} \\lambda _ 2 = \\frac { e ^ { q _ 3 - q _ 1 } } { 1 + e ^ { q _ 3 - q _ 2 } + e ^ { q _ 2 - q _ 1 } + e ^ { q _ 3 - q _ 1 } } = \\frac { 1 } { e ^ { q _ 1 - q _ 3 } + e ^ { q _ 1 - q _ 2 } + e ^ { q _ 2 - q _ 3 } + 1 } < 1 / 4 , \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} \\gamma _ { - 1 } ( B ( x , R ) ) & = \\int _ { | u | < R } e ^ { x ^ 2 + u ^ 2 + 2 x u } \\ , d u \\geq e ^ { x ^ 2 + R ^ 2 / 4 } \\int _ { R / 2 < u < R } e ^ { 2 x u } \\ , d u \\\\ & \\gtrsim \\frac { e ^ { x ^ 2 + R ^ 2 / 4 + R x } } { x } , \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} \\{ \\gamma _ k , \\langle 1 | f _ n \\rangle \\} = i \\langle 1 , f _ n \\rangle \\delta _ { k n } \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} 2 H _ l = \\nabla _ l ( g ^ { i j } h _ { i j } ) = ( \\nabla _ l g ^ { i j } ) h _ { i j } + g ^ { i j } ( \\nabla _ l h _ { i j } ) = 0 + g ^ { i j } \\nabla _ l h _ { i j } . \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} I I I & = - \\frac { 1 6 } { \\lambda ( t ) ^ { 3 } } \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) \\left ( \\partial _ { 1 } K ( s - t , \\lambda ( t ) ) + \\partial _ { 1 } K _ { 1 } ( s - t , \\lambda ( t ) ) \\right ) \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\displaystyle { \\sum _ { u _ 1 , \\dots , u _ m , n \\geq 0 } } \\sharp \\mathcal { D } ( u _ 1 , \\dots , u _ m , n ) a _ 1 ^ { u _ 1 } \\cdots a _ m ^ { u _ m } q ^ n & = & \\displaystyle { \\sum _ { u _ 1 , \\dots , u _ m , n \\geq 0 } } \\sharp \\mathcal { C } ( u _ 1 , \\dots , u _ m , n ) a _ 1 ^ { u _ 1 } \\cdots a _ m ^ { u _ m } q ^ n \\\\ \\\\ & = & ( - a _ 1 q ; q ) _ { \\infty } \\cdots ( - a _ m q ; q ) _ { \\infty } \\ , \\cdot \\end{array} \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\frac { w } { 2 \\lambda ( t ) } } \\frac { R d R } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\frac { 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } { 2 w \\left ( \\sqrt { ( 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } ) ^ { 2 } + 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } + 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } \\right ) } \\\\ & \\leq C \\int _ { 0 } ^ { \\infty } d R \\frac { R } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\frac { R ^ { 2 } \\lambda ( t ) ^ { 2 } } { w ( 1 + w ^ { 2 } ) } \\leq \\frac { C \\lambda ( t ) ^ { 2 } } { w ( 1 + w ^ { 2 } ) } , w \\geq 1 \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} p _ { \\mu _ { 1 } ^ { \\prime } \\boxtimes \\mu _ { 2 } ^ { \\prime } } \\left ( \\overline { z } \\right ) = 1 + 2 \\Re \\psi _ { \\mu _ { 1 } ^ { \\prime } \\boxtimes \\mu _ { 2 } ^ { \\prime } } ( z ) = 1 + 2 \\Re \\psi _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } \\left ( c _ { 0 } z \\right ) = p _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } \\left ( \\overline { c _ { 0 } z } \\right ) , \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} f ( x ) = \\int f ( y ) \\omega _ { D _ i } ^ x ( d y ) . \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} g _ 1 \\dots g _ { 2 k } = g ' _ 1 \\dots g ' _ { 2 k } + s p \\ , , \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} \\tau ^ M _ \\varphi ( A _ 0 , \\ldots , A _ k ) & = \\int _ { G ^ { \\times k } } { \\rm T r } _ S \\left ( \\Phi _ { A _ 0 } ( ( g _ 1 \\cdots g _ k ) ^ { - 1 } ) \\circ \\Phi _ { A _ 1 } ( g _ 1 ) \\circ \\ldots \\circ \\Phi _ { A _ k } ( g _ k ) \\right ) \\varphi ( e , g _ 1 , g _ 1 g _ 2 , \\ldots , g _ 1 \\cdots g _ k ) d g _ 1 \\cdots d g _ k \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} \\mathcal Q _ { \\mu , D } ( u , \\varphi ) = \\mathcal Q _ { \\mu , D } ( w , \\varphi ) \\ , , \\ \\ \\ \\forall \\varphi \\in U _ j , \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} [ f ' ( 0 ) , a ] \\cap \\sigma _ { e s s } ( A ) \\subset \\left [ \\inf _ { s \\in \\mathbb { R } \\setminus \\{ 0 \\} } \\dfrac { f ( s ) } { s } , \\sup _ { s \\in \\mathbb { R } \\setminus \\{ 0 \\} } \\dfrac { f ( s ) } { s } \\right ] \\cap \\sigma _ { e s s } ( A ) = \\emptyset a \\notin \\sigma ( A ) . \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\Omega _ { n , L } ^ c ) = 0 . \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} a ^ { p ^ * } & = ( 2 \\Lambda ) ^ { p ^ * } = 2 ^ { \\frac { p } { p - 1 } } \\Big ( \\frac { p - 2 } { p \\delta } \\Big ) ^ { \\frac { p - 2 } { 2 } \\cdot \\frac { p } { p - 1 } } L ^ { \\frac { p } { 2 } \\cdot \\frac { p } { p - 1 } } \\\\ & = O _ p \\bigg ( \\Big ( \\frac { 1 } { \\epsilon } \\Big ) ^ { \\frac { p ( p - 2 ) } { p - 1 } } L ^ p \\bigg ) . \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} E ( t ^ n , \\omega ^ { n k } ) = \\bigoplus _ { j = 0 } ^ { n - 1 } E ( t , \\omega ^ { d j + k } ) \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} \\big ( T ^ * f \\big ) ( x ) & = \\int _ a ^ b \\overline { T ( y , x ) } f ( y ) \\d y , \\\\ \\big ( T ^ \\# f \\big ) ( x ) & = \\int _ a ^ b T ( y , x ) f ( y ) \\d y , \\\\ \\big ( \\overline { T } f \\big ) ( x ) & = \\int _ a ^ b \\overline { T ( x , y ) } f ( y ) \\d y . \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} w _ + & = \\frac { 1 - \\rho ^ n } { 1 - \\rho ^ { n + 1 } } = \\frac { \\sum _ { m = 0 } ^ { n - 1 } \\rho ^ m } { \\sum _ { m = 0 } ^ n \\rho ^ m } , \\\\ w _ - & = \\frac { \\rho ( 1 - \\rho ^ n ) } { 1 - \\rho ^ { n + 1 } } = \\frac { \\sum _ { m = 0 } ^ { n - 1 } \\rho ^ { m + 1 } } { \\sum _ { m = 0 } ^ n \\rho ^ m } , \\\\ K & = \\frac { ( 1 - \\rho ) \\rho ^ n ( 1 - \\rho ^ n ) ^ n } { ( 1 - \\rho ^ { n + 1 } ) ^ { n + 1 } } = \\frac { ( \\sum _ { m = 0 } ^ { n - 1 } \\rho ^ { m + 1 } ) ^ n } { ( \\sum _ { m = 0 } ^ n \\rho ^ m ) ^ { n + 1 } } . \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} H _ m = G _ m \\leq 1 + \\sum ^ { m - 1 } _ { i = 1 } G _ i = 1 + \\sum ^ { m - 1 } _ { i = 1 } H _ i , \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} \\frac { - 1 } { 2 \\omega } \\int _ { 0 } ^ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } r \\phi _ { 0 } ( r ) F _ { 4 } ( t , r \\lambda ( t ) ) d r = \\frac { 1 } { 2 \\omega } \\int _ { \\frac { 2 } { \\sqrt { \\omega } \\lambda ( t ) } } ^ { \\infty } \\phi _ { 0 } ( r ) F _ { 4 } ( t , r \\lambda ( t ) ) r d r \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} \\| [ b _ 0 , T ] \\| _ { L ^ 2 ( \\mathbb R ^ 3 ) \\to L ^ 2 ( \\mathbb R ^ 3 ) } = C _ 0 < \\infty . \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} f ^ { e _ l } ( c ) = ( c - b ) ( a - b ) ^ { - 1 } f ^ { e _ l } ( a ) + ( c - a ) ( b - a ) ^ { - 1 } f ^ { e _ l } ( b ) , \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} \\mathcal { P } _ n ^ - = \\lbrace \\omega \\in \\Omega \\ : s \\mapsto X ( \\omega , s ) \\ s . t . \\ V _ 0 ( \\omega ) = - c , \\ N ( \\omega , t ) = n , \\ X ( \\omega , t ) = 2 \\beta - x \\rbrace \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} N _ 0 ( n - 1 ) = 2 s + N _ 0 ( n - 2 ) . \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} \\frac { 1 } { A + B } = \\frac { 1 } { A } - \\frac { 1 } { A } B \\frac { 1 } { A + B } . \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} R _ t \\leq \\begin{cases} \\frac { 1 } { 2 } , & F _ t = , \\\\ \\frac { 1 } { 1 + | 1 - \\alpha - \\beta | ^ { t - F ^ { - } ( t ) } } , & F _ t = , \\end{cases} \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} j _ { n + 3 } ^ { ( 3 ) } = j _ { n + 2 } ^ { ( 3 ) } + j _ { n + 1 } ^ { ( 3 ) } + 2 j _ { n } ^ { ( 3 ) } , \\ j _ { 0 } ^ { ( 3 ) } = 2 , \\ j _ { 1 } ^ { ( 3 ) } = 1 , \\ j _ { 2 } ^ { ( 3 ) } = 5 , \\ n \\geq 0 , \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} D _ { j } : = \\left ( ( x _ i , y _ i ) \\in D : i \\in I _ j \\right ) . \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} I \\left ( X _ 0 ; Q _ { t } \\right ) = 0 , \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} \\delta _ { \\varepsilon } ( F _ { \\nabla ^ \\omega } ) ^ a = ( C ^ a { } _ { b c } - \\rho ^ \\mu _ b \\omega ^ a { } _ { c \\mu } ) \\varepsilon ^ c ( F _ { \\nabla ^ \\omega } ) ^ b + ( R _ { \\nabla ^ \\omega } ) ^ a { } _ { b \\mu \\nu } \\varepsilon ^ b D X ^ \\mu \\wedge D X ^ \\nu + D ^ a { } _ { b c \\mu } \\varepsilon ^ c D X ^ \\mu \\wedge A ^ b , \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} L f ( x ) : = \\sum _ j \\ell _ j ( x ) f ( x _ j ) , \\ell _ j ( x ) : = \\prod _ { j \\ne k } { x - x _ k \\over x _ j - x _ k } . \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} T ^ { Y } \\eta _ n = J - \\eta _ n , \\qquad \\forall n \\le N , \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} \\frac 1 2 \\int _ \\Omega | x | ^ { - s } ( u ( x , \\widetilde { T } ) ) ^ 2 d x - \\frac 1 2 \\int _ \\Omega | x | ^ { - s } ( u _ 0 ( x ) ) ^ 2 d x + \\int _ 0 ^ { \\widetilde { T } } ( \\xi , \\nabla u ) d t = \\int _ 0 ^ { \\widetilde { T } } ( | u | ^ { q - 2 } u \\ln | u | , u ) d t . \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} { } ^ Q \\widetilde { \\Omega } ^ + _ a = ( \\tilde { \\rho } ^ T E \\tilde { \\rho } ) ^ + \\tilde { \\rho } _ a ( \\tilde { \\rho } ^ T E \\tilde { \\rho } ) . \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} \\phi ( x ) = \\phi ( x ' , x _ { n + 1 } ) : = - \\frac { | x ' | ^ { 2 } } { 4 } + 2 \\bigg ( - \\frac { 1 } { 2 - 2 s } x _ { n + 1 } ^ { 2 - 2 s } + \\frac { 1 } { 2 } x _ { n + 1 } ^ { 2 } \\bigg ) . \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} \\dot { V } _ 1 ( z ) & = \\nabla V _ 1 ( z ) \\dot { z } \\\\ & = - \\langle \\nabla L ( x , \\lambda ^ * ) - \\nabla L ( x ^ * , \\lambda ) + z - z ^ * , z - \\tilde { z } \\rangle _ r \\\\ & = - \\langle G _ r ( z ) + z - z ^ * , z - \\tilde { z } \\rangle _ r \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} \\Phi ( u ) ( \\cdot , t ) = S ( t ) \\xi ( \\cdot , 0 ) + \\int _ 0 ^ t S ( t - s ) f ( s , u [ \\xi ] _ \\rho ( \\cdot , s ) ) d s , \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} { \\tilde f _ 1 } ' = ( h f _ 1 ) ' = \\frac 1 2 h a f _ 1 + h f _ 1 ' = \\mbox { u s i n g e q u a t i o n \\eqref { e q : e q 2 } } = - h f _ 1 ' - \\frac 1 2 h f _ 1 \\theta + h f _ 1 ' = - \\frac { \\tilde f _ 1 } { 2 } \\theta . \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} u _ h \\in V _ h , a _ h ( u _ h , \\chi ) = l _ h ( \\chi ) ( \\forall \\chi \\in V _ h ) . \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} L = - \\partial _ x ^ 2 + V ( x ) \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} l _ T ( u , a ) = T ( u ) a - T ( u \\cdot a + H ( T u , a ) ) , r _ T ( a , u ) = a T ( u ) - T ( a \\cdot u + H ( a , T u ) ) , ~ ~ ~ u \\in M , a \\in A \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} G ( x , y ) = \\lim _ { R \\to \\infty } G _ { R } ( x , y ) , \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} Y _ j ^ \\ell = \\left \\{ \\begin{array} { c c } \\{ v _ j ^ S : S \\in { [ m ] \\choose \\ell } \\} , & \\ \\ell < k \\\\ \\{ w _ j \\} , & \\ \\ell = k \\end{array} \\right \\} \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} \\rho ( L _ { - 1 } ) \\ , & = \\ , h ( z ) ^ { - 1 } \\ , , \\\\ \\rho ( L _ { 0 } ) \\ , & = \\ , - \\frac { h ( z ) } { h ' ( z ) } \\partial + b ( z ) \\ , , \\\\ \\rho ( L _ 1 ) \\ , & = \\ , h ( z ) \\left ( \\rho ( L _ 0 ) ^ 2 - \\rho ( L _ 0 ) - c ( c + 1 ) \\right ) \\ , . \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} \\eta _ { \\mu } ( x ) = \\lim _ { y \\downarrow 0 } \\eta _ { \\mu } ( x + i y ) . \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} | \\{ ( \\xi _ 1 , \\eta _ 1 ) \\ | \\ ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) \\in E ( \\tau , \\xi , \\eta ) \\} | = & \\int _ { \\theta _ 1 } \\int _ { r _ 1 } { \\chi } _ { E ( \\tau , \\xi , \\eta ) } ( | \\xi _ 1 | , \\theta _ 1 ) r _ 1 d r _ 1 d \\theta _ 1 \\\\ \\lesssim & ( N _ 1 A ) ^ { - 1 } \\max ( L _ 1 , L _ 2 ) . \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} \\lambda & = \\frac { a _ 1 + b _ 2 } { 2 } , & \\omega & = \\sqrt { - a _ 2 b _ 1 - \\frac { ( a _ 1 - b _ 2 ) ^ 2 } { 4 } } . \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} N ( x , d y ) = c _ { \\alpha , d } \\frac { \\psi _ \\gamma ( y ) } { \\psi _ \\gamma ( x ) } \\frac { d y } { | x - y | ^ { d + \\alpha } } , H _ t = t . \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} N _ { \\sigma } ( z ) = \\int _ { \\mathbb { R } } \\frac { 1 + t z } { z - t } \\ , d \\sigma ( t ) \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} x ^ 2 + 2 a x y + y ^ 2 = z ^ 2 \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c \\} = e ^ { - \\lambda t } \\Bigl [ I _ { 0 } \\bigl ( \\lambda t \\bigr ) + I _ { 1 } \\bigl ( \\lambda t \\bigr ) \\Bigr ] \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} \\tau _ k ( \\chi ) = \\Big \\{ \\begin{aligned} \\bar { \\chi } ( k ) \\tau _ 1 ( \\chi ) , & \\ ( k , q ) = 1 \\\\ 0 , & \\ , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & \\ , \\to \\ , \\min & & & \\\\ g _ i ( x ) & \\ , \\leq \\ , 0 & \\qquad & i \\in \\mathcal M & \\\\ h _ j ( x ) & \\ , = \\ , 0 & & j \\in \\mathcal P & \\\\ \\tilde G _ l ( x ) \\tilde H _ l ( x ) & \\ , = \\ , 0 & & l \\in \\mathcal Q & \\end{aligned} \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} \\begin{aligned} J _ 0 ( B ( r ) \\times B ( 2 r ) ^ c ) & \\lesssim \\int _ { B ( r ) } | x | ^ { \\alpha - d } d x \\int _ { B ( 2 r ) ^ c } \\frac { | y | ^ { \\alpha - d } } { | x - y | ^ { d + \\alpha } } d y \\\\ & \\lesssim \\int _ { B ( r ) } | x | ^ { \\alpha - d } d x \\int _ { B ( 2 r ) ^ c } | y | ^ { - 2 d } d y < \\infty . \\end{aligned} \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} t = \\frac { \\sqrt { \\lambda ^ { 2 } + 2 \\lambda + 9 } } { 3 } \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) \\le \\beta , \\ \\bigcup _ { k = 0 } ^ \\infty \\{ N ( t ) = 2 k + 1 \\} \\ | \\ V ( 0 ) = - c _ 2 \\} = \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} j _ { n + 1 } ^ { ( 3 ) } + j _ { n } ^ { ( 3 ) } = 3 J _ { n + 2 } ^ { ( 3 ) } , \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} \\Pi _ { \\Theta , v } \\omega = \\lim _ n \\psi _ { \\Theta , v ; \\omega _ 0 } \\circ \\ldots \\circ \\psi _ { \\Theta , v ; \\omega _ n } ( 0 ) . \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} \\{ a ^ \\epsilon b ^ { [ t ] } : \\epsilon = 0 , 1 , s \\geq 0 , t + \\epsilon > 0 \\} , \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} \\left \\Vert K _ { n } ^ { p , N } \\right \\Vert ^ { 2 } = n ! ( - N ) _ n p ^ n ( p - 1 ) ^ n . \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} D _ { M , N } ^ { } : = \\left \\{ [ x , y , x , y ] \\ , \\vert \\ , [ x , y ] \\in \\mathbb { T } ^ 2 \\right \\} \\subseteq \\mathbb { T } ^ { 2 } \\times \\mathbb { T } ^ { 2 } = ( \\mathcal { B } _ { M ( \\theta ) } \\times \\mathcal { B } _ { N ( \\theta ) } ) ^ { ( 0 ) } \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} \\delta ( 2 H ) & = \\delta ( g ^ { i j } h _ { i j } ) = ( \\delta g ^ { i j } ) h _ { i j } + g ^ { i j } ( \\delta h _ { i j } ) \\\\ & = 2 u h ^ { i j } h _ { i j } + g ^ { i j } ( u _ { i j } - u h _ { i l } h _ j ^ l + g _ { i j } k _ 0 u ) \\\\ & = \\Delta { u } + u ( 4 H ^ 2 - 2 K + 4 k _ 0 ) . \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} \\begin{array} { l } g _ 0 ( x ) F _ 1 ( x ) + g _ 1 ( x ) F _ 2 ( x ) \\\\ = ( ( x - 1 ) ^ r + u ( x - 1 ) ^ { k _ 1 } p _ 1 ( x ) + u ^ 2 ( x - 1 ) ^ { k _ 2 } p _ 2 ( x ) + u ^ 3 ( x - 1 ) ^ { k _ 3 } p _ 3 ( x ) ) F _ 1 ( x ) \\\\ + ( u ( x - 1 ) ^ { r _ 1 } + u ^ 2 ( x - 1 ) ^ { k _ 4 } p _ 4 ( x ) + u ^ 3 ( x - 1 ) ^ { k _ 5 } p _ 5 ( x ) ) F _ 2 ( x ) , \\\\ \\end{array} \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} & u ^ * ( t ) = u ( t , x ( t ) ) , q ^ * ( t ) = q ( t , x ( t ) ) , \\\\ & \\lambda ^ * ( t ) = \\lambda ( t , x ( t ) ) , \\lambda _ 1 ^ * ( t ) = \\lambda _ 1 ( t , x ( t ) ) , \\\\ & c ^ * ( t ) = c ( t , x ( t ) ) , \\gamma ^ * ( t ) = \\gamma ( t , x ( t ) ) : \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} c _ { 1 } = c _ { 0 } \\overline { \\alpha } \\left ( 1 - 2 c \\int _ { \\mathbb { T } } \\frac { d \\sigma ( t ) } { \\left | \\eta _ { \\mu _ { 1 } } ( \\alpha ) - t \\right | ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} A = \\{ \\sqrt i \\pm \\sqrt j : i , j \\in [ n ] \\} , \\ , \\ , \\ , \\ , G = \\{ ( \\sqrt i + \\sqrt j , \\sqrt i - \\sqrt j ) : i , j \\in [ n ] \\} . \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} e _ { x y } B ( e _ { z w } , e _ { u v } ) + B ( e _ { x y } , e _ { u v } ) e _ { z w } & = e _ { x y } B ( e _ { y u } , e _ u ) + B ( e _ { x y } , e _ u ) e _ { y u } . \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ t v ( t , x ) - \\frac { 1 } { 2 } \\partial _ { x x } ^ 2 v ( t , x ) = - \\frac { 1 } { 2 } \\bar A \\left ( v ( t , x ) \\right ) \\left ( \\partial _ x v ( t , x ) , \\partial _ x v ( t , x ) \\right ) + \\bar f \\left ( v ( t , x ) , \\partial _ x v ( t , x ) \\right ) , \\\\ & v ( 0 , x ) = h ( x ) , \\ \\ \\ \\ t \\in ( 0 , T _ 1 ) . \\end{cases} \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } \\leq \\sqrt { s } \\langle x , x \\rangle ^ \\frac { 1 } { 2 } \\iff \\left ( 1 - \\frac { c _ x } { 2 } \\right ) \\sqrt { n } \\leq \\sqrt { s } \\iff 1 - \\frac { c _ x } { 2 } \\leq \\sqrt { \\frac { s } { n } } . \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} f _ { 5 , 4 } ^ { - 1 } ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , 2 , { \\bf 3 } , 7 , { \\bf 3 } , 7 , 7 , 4 ) = ( 0 , 1 , 2 , 0 , 1 , 2 , 5 , { \\bf 2 } , { 3 } , { \\bf 2 } , { 3 } , 8 , 8 , 4 ) = s ^ { \\bullet } . \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} 0 \\neq w ( { h } ) = \\frac { 1 } { \\gamma } g ( h ) B . v - v = \\frac { 1 } { \\gamma } g ( { h } ) q ( { h } ) - \\gamma \\in N , \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} a _ { 0 } = \\int _ { \\mathbb { R } _ { + } } \\frac { \\alpha t } { 1 - \\alpha t } \\ , d \\mu _ { 1 } ( t ) , a _ { 1 } = \\begin{cases} \\int _ { \\mathbb { R } _ { + } } \\frac { t } { ( 1 - \\alpha t ) ^ { 2 } } \\ , d \\mu _ { 1 } ( t ) & a _ { 0 } \\neq - 1 , \\\\ \\frac { 1 } { \\alpha } \\int _ { \\mathbb { R } _ { + } } \\frac { 1 } { ( 1 - \\alpha t ) ^ { 2 } } \\ , d \\mu _ { 1 } ( t ) & a _ { 0 } = - 1 , \\end{cases} \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} \\Omega _ { N } = \\Bigg \\{ \\sup _ { \\substack { t \\in [ 0 , T ] \\\\ i \\in \\{ 1 , \\dots , N \\} } } \\frac { 1 } { N } \\bigg | \\sum _ { k = 1 } ^ { N } ( K \\ast V ^ { N } ) ( X _ { t } ^ { i , N , ( A ) } - X _ { t } ^ { k , N , ( A ) } ) \\bigg | \\leq A \\Bigg \\} \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} p = \\chi _ { E _ s ^ c } \\cdot \\sqrt { 2 \\pi } \\frac { \\overline { \\Pi ( k ) } \\widehat f _ + ( - k ) - \\Pi ( k ) \\widehat f _ + ( k ) } { 2 i } \\ , . \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} \\nabla G = \\begin{bmatrix} \\nabla ^ 2 f ( x ) + \\lambda ^ T \\nabla ^ 2 g ( x ) & \\nabla g ( x ) ^ T \\\\ - \\nabla g ( x ) & \\mathbf { 0 } \\end{bmatrix} \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\frac 1 2 \\int R | \\nabla ^ 2 \\log R | ^ 2 = \\int \\frac { \\Delta \\sqrt { R } } { \\sqrt { R } } \\Delta R . \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} \\rho ( a x , y ) = ( - 1 ) ^ { \\bar a \\bar x } \\rho ( x , a y ) = a \\rho ( x , y ) . \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} 1 / \\eta _ { \\rho _ { 1 } } ( 1 / x _ { 0 } ) = 1 / \\eta _ { \\rho _ { 1 } ' } ( \\eta _ { \\rho _ { 1 } '' } ( 1 / x _ { 0 } ) ) \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} c _ 1 : = \\sup \\{ - ( b _ 0 + c _ 0 ) ^ n \\int _ X u \\omega _ 0 ^ n | u \\in P S H ( X , ( b _ 0 + c _ 0 ) \\omega _ 0 ) , \\sup _ X u = 0 \\} < + \\infty . \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} \\mathcal { D } ^ { P } _ { J , K } \\left ( P _ { J , K } \\right ) : = P _ { J , K } \\circ \\mathcal { D } _ { J , K } ^ { - 1 } \\circ \\theta \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} U ^ * = \\{ t ^ * _ i : i \\le d \\} \\cup \\{ u ^ * : u \\in \\max ( D ) ^ + \\setminus B \\} \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} \\gamma _ { N + 1 , 1 } = \\gamma _ { N + 1 , 2 } = \\dots = \\gamma _ { N + 1 , N } = 0 . \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} a _ { I , l , l ' } = \\overline { b _ { I , l , l ' } } , \\quad \\mbox { f o r a l l $ I \\in \\mathbb { N } ^ { N } $ h a v i n g l e n g t h $ p \\geq 3 $ a n d $ l , l ' = 1 , \\dots , N $ . } \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} a \\partial _ { y } ^ { 2 } u \\left ( x , j , t \\right ) + b \\partial _ { y } u \\left ( x , j , t \\right ) = 0 j = 0 , 1 . \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} \\int _ { \\R ^ d _ + \\setminus D ( r , r ) } \\frac { \\big ( 1 + { \\bf 1 } _ { | w | \\ge 1 } ( \\log | w | ) ^ { \\beta _ 3 } \\big ) } { | w - y | ^ { d - \\alpha } | w | ^ { d + \\alpha + \\beta _ 1 } } d w = \\int _ { \\R ^ d _ + \\cap B ( y , r ) } + \\int _ { \\R ^ d _ + \\setminus ( D ( r , r ) \\cup B ( y , r ) ) } = : I + I I . \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} f \\left ( E \\right ) : = \\frac { \\left ( w _ { 1 } ^ { \\top } G \\left ( E \\right ) w _ { 1 } - d _ { 1 1 } + E \\right ) \\left ( w _ { 2 } ^ { \\top } G \\left ( E \\right ) w _ { 2 } - d _ { 2 2 } + E \\right ) } { \\left ( w _ { 1 } ^ { \\top } G \\left ( E \\right ) w _ { 2 } - d _ { 1 2 } \\right ) ^ { 2 } } . \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} t _ { s } = t _ { } + t _ { } , \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} \\{ x _ 2 ' \\phi ' _ 2 + x _ 3 ' \\phi ' _ 3 + \\phi ' _ 4 = 0 \\} \\subset \\mathbb C ^ 4 _ { x _ 1 ' , \\dots , x _ 4 ' } , \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} a ( E _ k ) = \\tilde R ( e _ { n + 1 } ) a ( E _ k ) = L ( e _ 0 ) ( E _ k ) \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} \\pi _ i ( [ 1 ] ; a , b ) = a \\prec _ i b , \\pi _ i ( [ 2 ] ; a , b ) = a \\succ _ i b ~ ~ ~ ~ ~ ~ \\pi _ i ( [ 3 ] ; a , b ) = a \\curlyvee _ i b . \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} \\dot { y } = - e ^ { - z } \\dot { z } \\geq - y \\left ( - \\sqrt { 1 - y } \\sqrt { - ( 1 + y ) ( H _ 0 ^ 2 - 2 H _ 1 ) } \\right ) . \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} H _ { 2 g + 2 } = \\sum ^ { 2 g + 1 } _ { i = 1 } H _ i + 1 . \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} ( u - \\varphi ) ( x _ 0 ) = 0 \\leq ( u - \\varphi ) ( x ) \\quad \\forall x \\in B _ \\delta ( x _ 0 ) \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} \\int _ { M } | f ( x ) | d { \\rm v o l } ( x ) = \\lim _ { r \\to \\infty } \\int _ { B _ { r } ( x _ { 0 } ) } | f ( x ) | d { \\rm v o l } ( x ) , \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} S _ { r , \\ell } ( z ) = \\ell \\cdot U _ { \\ell } \\bigg ( \\phi _ { \\ell } ^ r ( z ) \\bigg ) . \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} \\frac { q _ { \\nu _ { 1 } \\boxtimes \\nu _ { 2 } } ( x ) } { q _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( x ) } = \\frac { | 1 - \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) | ^ { 2 } } { 2 \\beta } \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t ^ { 2 } } { | t - \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) | ^ { 2 } } \\ , d \\sigma ( t ) . \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} T r ( T ^ { - 1 } , S y m ^ n ( V ) ) & = \\sum _ { a = 0 } ^ { n } \\sum _ { b = 0 } ^ a ( - 1 ) ^ { b } ( b + 1 ) = \\sum _ { b = 0 } ^ n ( - 1 ) ^ { b } ( b + 1 ) = \\frac { n } { 2 } + 1 . \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} = \\sum _ { ( j , h ) , ( i , k ) \\in I \\times \\{ 1 , \\dots , n \\} } \\sum _ { m ' , m \\in F } \\bar { z } _ { j , h } E _ { \\mathcal { B } ; j , i } \\left ( b ^ * _ { h , m ' } b _ { k , m } \\right ) E _ { \\mathcal { C } ; j , i } \\left ( c ^ * _ { h , m ' } c _ { k , m } \\right ) z _ { i , k } . \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} \\bar C _ { p , \\nu } = 2 ^ { p + \\nu } . \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} \\dfrac { \\partial } { \\partial p ( x ) } B _ \\alpha ( p , q ) = \\dfrac { \\alpha } { \\alpha - 1 } \\big [ p ( x ) ^ { \\alpha - 1 } - q ( x ) ^ { \\alpha - 1 } \\big ] , \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} & c = p _ 1 \\cdots p _ l ^ { \\prime } , \\\\ & d = p _ l ^ { \\prime \\prime } \\cdots p _ n , \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} ( V _ { N , x _ L } ^ { \\beta } ( t ) ) _ { \\ell j } = \\int _ { - \\infty } ^ { \\infty } ( V ( x ) + V _ { \\rm e x } ( x , t ) ) \\hat { \\mathcal { H } } _ { \\ell - 1 } ^ { \\beta } ( x - x _ L ) \\hat { \\mathcal { H } } _ { j - 1 } ^ { \\beta } ( x - x _ L ) \\dd { x } . \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} R ^ m ( x ) S ( x ) = 1 + x ^ { \\operatorname { d e g } R + \\operatorname { d e g } S + 1 } P ( x ) . \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} \\partial _ { r } v _ { 2 } ( t , r ) = \\frac { c _ { b } } { 2 } \\int _ { 0 } ^ { \\infty } \\sin ( t \\xi ) \\xi \\left ( J _ { 0 } ( r \\xi ) - J _ { 2 } ( r \\xi ) \\right ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } d \\xi \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\sum _ { j = k } ^ N \\delta _ j ^ 2 j ^ q & = \\sum _ { j = k } ^ N b _ j ( a _ j - a _ { j + 1 } ) = b _ k a _ k - b _ { N } a _ { N + 1 } + \\sum _ { j = k } ^ { N - 1 } a _ { j + 1 } ( b _ { j + 1 } - b _ j ) \\\\ & \\leq k ^ q \\ , \\sum _ { j = k } ^ { \\infty } \\delta _ j ^ 2 + C \\ , \\sum _ { j = k } ^ { \\infty } j ^ { - \\rho } \\ , j ^ { q - 1 } \\ , . \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} N _ 2 \\leq \\| \\nabla u \\| _ { L ^ \\infty } \\| \\nabla \\omega \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} \\Psi = \\prod _ { i = 1 } ^ { N } r _ i ^ { - ( d _ i - 1 ) / 2 } R ( r _ 1 , \\cdots , r _ N ) \\prod _ { i = 1 } ^ { N } Y _ i ( \\Omega _ i ) , \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} \\phi = x _ 2 \\phi _ 2 + x _ 3 \\phi _ 3 + x _ 4 \\phi _ 4 = 0 , \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} f ( x , \\xi ) = \\sum _ { i = 1 } ^ N \\beta _ i \\xi _ i + \\rho ( x ) \\quad x \\in \\Omega \\xi \\in \\R ^ N , \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} \\left \\{ F = \\sum _ { j = 1 } ^ { \\infty } \\hat \\beta _ j \\hat v _ { j , 0 } : \\left ( \\frac { \\hat \\beta _ j } { \\sqrt { \\lambda _ j ( 0 ) } } \\right ) _ { j \\geq N + 1 } \\in l ^ 2 \\right \\} , \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} \\{ \\theta \\in ( 0 , \\pi ) : I _ { r } ( \\theta ) < 1 \\} = ( f ( r ) , \\pi ) . \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { t - 1 } \\beta _ { \\zeta } ( k ) = \\sum _ { k = 0 } ^ { ( t + 1 ) / 2 } \\beta _ { \\zeta } ( k ) = 0 . \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} Y = \\left \\{ ( y _ 1 , \\cdots , y _ M ) ; \\ \\frac { 1 } { M } \\sum _ { l = 1 } ^ M \\alpha _ l y _ l = m , \\alpha _ l = \\frac { M K _ l } { N } \\right \\} . \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} S _ n & : = \\sum _ { s = 3 } ^ \\infty S _ n ^ { ( s ) } = - i \\sum _ { s = 3 } ^ n \\lambda _ s \\sum _ { k = s } ^ n B _ { n , k } \\left ( b _ 1 , b _ 2 , \\ldots \\right ) B _ { k , s } ( 1 , 2 ! a _ 1 , 3 ! a _ 2 , 4 ! a _ 3 , \\ldots ) . \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} T ^ 2 _ { 1 2 } = T ^ 2 _ { 1 3 } = T ^ 2 _ { 1 4 } = T ^ 2 _ 1 = \\bar T ^ 2 _ 1 = R ^ 2 _ 1 = 0 ; \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} \\mathcal { D } _ { J , K } \\circ T _ { J , K } = \\theta \\circ \\mathcal { D } _ { J , K } \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} Q : = \\frac { I - \\exp ( - \\frac { 1 } { 2 } D ^ - D ^ + ) } { D ^ - D ^ + } D ^ + \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} & \\mathbf { B } ( i , j ) = \\mathbf { A } ( i , j ) ^ { \\mathsf { T } } \\circ \\mathbf { A } ( i , j ) = 0 , \\ a n d \\\\ & \\mathbf { B } ( j , i ) = \\mathbf { A } ( j , i ) ^ { \\mathsf { T } } \\circ \\mathbf { A } ( j , i ) = 0 . \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} V ' ( s ) = a _ n \\int _ { z _ 1 } ^ { z _ 2 } h _ s h ^ n \\ ; d z = c a _ n \\int _ { z _ 1 } ^ { z _ 2 } e h ^ n \\ ; d z . \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} J _ { i j } = \\langle \\partial _ i ^ T , \\partial ^ T _ j \\rangle = \\langle \\partial _ i - \\Pi ( \\partial _ i ) , \\partial _ j - \\Pi ( \\partial _ j ) \\rangle \\ , . \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} F ( t ; x , y , 1 , u , z , v ) & = \\frac { t x ( y - y z r + z ) F ( t ; x , y - y r + 1 , 1 , u , z , v ) } { ( t u x + y ^ { - 1 } - t u ) ( y - y r + 1 ) } \\\\ & \\quad - \\frac { t x z ( y t u v ( 1 - z ) + z ) } { t u x + y ^ { - 1 } - t u } F ( t ; x , y - y r + 1 , 1 , u , 1 , v ) \\\\ & \\quad + z ( y t u v ( 1 - z ) + z ) F ( t ; x , y , 1 , u , 1 , v ) . \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} \\dot { x } & = r x \\left ( 1 - \\frac { x } { K } \\right ) - q ( x ) y , \\\\ \\dot { y } & = \\left ( k q ( x ) - \\delta \\right ) y , \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} \\| \\tilde { f } _ m - f \\| _ H \\rightarrow 0 , \\lim _ m \\limsup _ n \\| \\Phi _ n \\tilde { f } _ m - f _ n \\| _ { H _ n } = 0 . \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { N } ( t , s , \\xi , v , \\overline v ) & = e ^ { ( t - s ) \\mathcal { L } } f \\left ( e ^ { s \\mathcal { L } } v , e ^ { s \\mathcal { L } } e ^ { \\xi \\mathcal { A } } \\overline v \\right ) = e ^ { ( t - s ) \\mathcal { L } } \\mathcal { B } \\left ( F ( e ^ { s \\mathcal { L } } v ) \\cdot G \\left ( e ^ { s \\mathcal { L } } e ^ { \\xi \\mathcal { A } } \\overline v \\right ) \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} S ( x ) = \\begin{cases} 0 , & x \\le 0 , \\\\ \\sqrt { x } , & x \\ge 0 . \\end{cases} \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} \\Sigma _ Y = \\beta \\Sigma _ X \\beta ^ \\top + \\Sigma _ Z . \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} \\epsilon \\| \\Delta _ h u _ h ^ n \\| ^ 2 _ { \\mathcal { T } _ h } & = ( \\epsilon ^ { - 1 } f ^ n ( u ^ n _ h ) , \\Delta _ h u ^ n _ h ) _ { \\mathcal { T } _ h } - ( \\phi _ h ^ n , \\Delta _ h u ^ n _ h ) _ { \\mathcal { T } _ h } \\\\ & \\le \\epsilon ^ { - 1 } ( \\| u _ h \\| _ { 0 , 6 } ^ 3 + \\| u _ h ^ n \\| _ { \\mathcal T _ h } ) \\| \\Delta _ h u _ h ^ n \\| _ { \\mathcal { T } _ h } + \\| \\phi _ h ^ n \\| _ { \\mathcal T _ h } \\| \\Delta _ h u _ h ^ n \\| _ { \\mathcal { T } _ h } . \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} \\psi _ { \\mu _ { 1 } } ( z ) = \\int _ { \\mathbb { R } _ { + } \\setminus I } \\frac { t z } { 1 - t z } \\ , d \\mu _ { 1 } ( t ) + z \\int _ { J } \\frac { 1 } { t - z } \\ , d ( \\mu _ { 1 } ) _ { * } ( t ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} f = h _ 1 + h { \\rm o n } A _ { i j } ' \\cap Z _ \\rho ' , Z _ \\rho ' : = r ^ { - 1 } ( Z _ \\rho ) ; \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} \\phi _ { p , q } ( \\lambda ) - \\psi ( \\lambda ) = & \\left ( - 8 \\ , p q + 8 \\ , p + 8 \\ , q \\right ) { \\lambda } ^ { 2 } \\\\ & + \\left ( - 1 4 \\ , p q + 1 2 \\ , p + 1 2 \\ , q + 8 \\right ) \\lambda - 5 \\ , p q + 2 \\ , p + 2 \\ , q + 1 2 . \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} \\mathcal { P } _ { 2 } = \\{ ( 2 0 , 4 ) , ( 1 2 , 1 2 ) , ( 4 , 2 0 ) \\} \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} \\frac { 1 } { 6 } ( p ^ 0 ) ^ 2 - \\frac { 6 } { \\kappa ^ 2 } ( q _ 0 ) ^ 2 = 0 , \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} S _ 1 & = \\sum _ { i = 1 } ^ k ( - 1 ) ^ { k - i } \\sum _ { \\substack { 2 \\leq j \\leq k \\\\ j \\equiv k \\ ( { \\rm m o d } \\ p ) } } p ( k , i , j ) q ^ { i - 1 } , \\\\ S _ 2 & = \\sum _ { i = 1 } ^ k ( - 1 ) ^ { k - i } \\sum _ { \\substack { 1 \\leq j _ 1 , j _ 2 \\leq k \\\\ j _ 1 + j _ 2 \\leq k \\\\ j _ 1 \\equiv 1 - a _ 1 \\ ( { \\rm m o d } \\ p ) \\\\ j _ 2 \\equiv 1 - a _ 2 \\ ( { \\rm m o d } \\ p ) } } p ( k , i , j _ 1 , j _ 2 ) q ^ { i - 1 } . \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} A _ 1 \\ & = \\ \\{ 3 , 5 , 7 \\} , \\\\ A _ 2 \\ & = \\ \\{ 2 , 4 , 9 \\} , \\\\ A _ 3 \\ & = \\ \\{ 1 , 6 , 8 \\} . \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} f = \\alpha \\chi u + \\beta \\chi v + f _ a , f _ a \\in \\mathcal { D } ( L _ a ) . \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} \\alpha ( X , \\Delta ; L ) = \\frac { A ( E ; X , \\Delta ) } { \\tau ( L ; E ) } \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ { n } \\frac { \\prod _ { k = 1 } ^ n ( q ^ k - 1 ) } { \\prod _ { k = 1 } ^ d ( q ^ k - 1 ) \\prod _ { k = 1 } ^ { n - d } ( q ^ k - 1 ) } , \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} F ( X \\otimes Z _ 2 ) = F ( X ) \\otimes F ( Z _ 2 ) = X _ 1 \\otimes ( X _ 0 \\oplus X _ 2 \\oplus X _ 3 ) = 3 X _ 1 \\oplus X _ 3 \\oplus 2 X _ 4 . \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} \\begin{cases} A _ { 1 } A _ { 2 } A _ { 3 } B _ { 1 } B _ { 2 } B _ { 3 } A _ { 1 } ' A _ { 2 } ' A _ { 3 } ' B _ { 1 } ' B _ { 2 } ' B _ { 3 } ' = 1 , \\\\ A _ { 1 } A _ { 2 } A _ { 3 } B _ { 1 } B _ { 2 } B _ { 3 } = A _ { 1 } ' A _ { 2 } ' A _ { 3 } ' B _ { 1 } ' B _ { 2 } ' B _ { 3 } ' = - 1 , \\\\ A _ { 2 } B _ { 3 } A _ { 2 } ' B _ { 3 } ' = A _ { 3 } B _ { 1 } A _ { 3 } ' B _ { 1 } ' = A _ { 1 } B _ { 2 } A _ { 1 } ' B _ { 2 } ' = 1 , \\\\ A _ { 3 } B _ { 2 } \\overline { A _ { 3 } ' B _ { 2 } ' } = A _ { 1 } B _ { 3 } \\overline { A _ { 1 } ' B _ { 3 } ' } = A _ { 2 } B _ { 1 } \\overline { A _ { 2 } ' B _ { 1 } ' } = 1 . \\end{cases} \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} \\Pi u = - \\sum _ { n = 0 } ^ \\infty \\lambda _ n \\langle 1 | f _ n \\rangle f _ n \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} K ( q ) \\le K ( q ) + L ( q ) = h ( q ) \\le c _ 3 \\phi ( \\frac { 1 } { q ^ 2 } ) \\le c _ 4 j ( q ) q ^ d . \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} J _ 0 ( \\psi _ 0 ) = \\frac { 1 } { 2 } \\boldsymbol { B } ( \\psi _ 0 , \\psi _ 0 ) - \\langle \\psi _ 0 , h _ 0 \\rangle , \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} { \\rm R i c } _ f = { \\rm R i c } + \\nabla ^ 2 f , \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} v _ { 1 } ( t , r ) + v _ { 2 } ( t , r ) = r \\left ( \\int _ { t } ^ { \\infty } \\frac { \\lambda '' ( x ) } { 1 + x - t } d x - \\frac { b } { t ^ { 2 } \\log ^ { b } ( t ) } \\right ) + ( t , r ) + E _ { v _ { 2 } } ( t , r ) , 0 \\leq r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} L = - A - K , \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} & u ^ { ( i _ 0 ) } = e m _ v ( u ^ { ( i _ 0 ) } ) f , \\ \\widetilde { u } ^ { ( i _ 0 ) } = \\widetilde { e } m _ v ( u ^ { ( i _ 0 ) } ) \\widetilde { f } , \\textit { a n d } \\\\ & \\widetilde { u } ^ { ( i _ 0 ) } = \\widetilde { e } \\left ( e ^ { - 1 } \\cdot u ^ { ( i _ 0 ) } \\cdot f ^ { - 1 } \\right ) \\widetilde { f } , \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} F _ n ^ { ( k ) } = F _ { n - 1 } ^ { ( k ) } + F _ { n - 2 } ^ { ( k ) } + \\cdots + F _ { n - k } ^ { ( k ) } , \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} \\lim _ { x _ { n + 1 } \\rightarrow 0 } \\| x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } u _ { 1 } \\| _ { H ^ { - 2 s } ( \\mathbb { R } ^ { n } \\times \\{ 0 \\} ) } & = \\| ( - P ) ^ { s } u _ { 1 } \\| _ { H ^ { - 2 s } ( \\mathbb { R } ^ { n } \\times \\{ 0 \\} ) } \\\\ & \\le C \\| u _ { 1 } \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { n } \\times \\{ 0 \\} ) } \\le C \\| \\tilde { u } \\| _ { L ^ { 2 } ( B _ { 1 6 } ' ) } , \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} C _ { L D } ( n ) : = \\inf _ { M ' } \\frac { \\ell ( \\partial M ' ) } { d ( M ' ) } = 2 , \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} \\Omega _ { \\mu } = \\{ z \\in \\mathbb { H } : \\Phi ( z ) \\in \\mathbb { H } \\} . \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} \\ss _ 1 ^ \\tau A _ i + \\ss _ 2 ^ \\tau B _ i = \\{ \\tt ^ \\tau + O _ p ( \\| \\tt \\| ^ 2 ) \\} D _ i . \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} T ^ { ( 0 ) } = \\frac { \\pi } { \\omega } . \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} | \\Phi ( \\xi _ 1 , \\eta _ 1 , \\xi , \\eta ) | & = | \\xi _ 1 \\xi ( \\xi _ 1 + \\xi ) + \\eta _ 1 \\eta ( \\eta _ 1 + \\eta ) | \\\\ & \\geq 2 ^ { - 1 } r _ 1 r | r _ 1 - r | - 2 ^ { 1 0 } A ^ { - 1 } K ^ { - 1 } N _ 1 ^ 3 . \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} \\Phi _ 2 = ( \\Phi _ 2 ^ + , \\Phi _ 2 ^ - ) , ~ ~ ~ \\Phi _ 0 = ( \\Phi _ 0 ^ + , \\Phi _ 0 ^ - ) , \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} z = g ^ { - 1 } \\mathbf { i } \\ \\ \\implies \\ \\ E _ i ( g , s ) = \\left \\{ \\begin{matrix} E _ i ( z , s ) & \\Gamma \\mathrm { I d } \\\\ 2 E _ i ( z , s ) & \\end{matrix} \\right . . \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} \\sup _ { s } \\ , \\langle X _ { s } , e _ { 1 } \\rangle & \\ = \\sup _ { s } \\left \\{ \\langle X _ { s } - \\vec { v } s , e _ { 1 } \\rangle + v _ { 1 } s \\right \\} \\\\ & \\leq \\max \\left \\{ \\sup _ { s } \\left \\{ v _ { 1 } s + \\delta s \\right \\} , \\sup _ { s } \\left \\{ v _ { 1 } s + \\mathcal { R } _ { \\delta } \\right \\} \\right \\} \\\\ & \\leq \\mathcal { R } _ { \\delta } , \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} \\phi v = 0 \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} \\beta [ g ] = \\overline { \\lim } _ { n \\to \\infty } \\left \\{ \\ n ^ { 1 / \\rho [ f ] } \\ \\sqrt [ n ] { | c _ n | } \\ \\right \\} . \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} \\rho ( x ) & = \\sum _ { n = 1 } ^ \\infty \\frac { b _ n } { n ! } \\phi ^ n ( x ) \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} \\mathfrak u = \\bigoplus \\limits _ { \\Omega \\in \\widetilde \\Psi } \\mathfrak u ( \\Omega ) \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} \\partial _ t \\overline { v } + \\mathbb { P } _ h \\Big ( \\overline { v } \\cdot \\nabla \\overline { v } \\Big ) + \\mathbb { P } _ h P _ 0 \\Big ( ( \\nabla \\cdot \\widetilde { v } ) \\widetilde { v } + \\widetilde { v } \\cdot \\nabla \\widetilde { v } \\Big ) + \\Omega \\int _ { \\mathbb { T } ^ 2 } \\overline { v } ^ \\perp ( \\boldsymbol { x } ' ) d \\boldsymbol { x } ' = 0 . \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} U _ s ( f , g ) = \\int _ 0 ^ 1 \\Big ( f ! _ { s t + ( 1 - s ) \\frac { 1 } { 2 } } g \\Big ) d \\mu ( t ) , \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} \\sup _ { \\lambda > 0 } \\| A _ { \\lambda } ( w ) \\| \\leq \\left | ( \\partial T V ) ^ 0 ( w ) \\right | : = \\min _ { v \\in \\partial T V ( w ) } \\| v \\| . \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} \\frac { \\partial P _ R } { \\partial \\tilde { r } } = - \\frac { \\varrho _ { \\rm n o d e } \\left ( \\eta _ R T _ R ; \\frac { \\lambda _ R } { \\eta _ R } \\right ) } { \\varrho _ { \\rm n o d e } \\left ( \\eta _ R T _ R ; - \\frac { \\lambda _ R } { \\eta _ R } \\right ) } , \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} d s ^ 2 = \\sin ( y ) ^ 2 d x ^ 2 + d y ^ 2 \\ , . \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} \\eta _ j ^ { - 1 } T _ j - \\eta _ { j + 1 } ^ { - 1 } T _ { j + 1 } & = \\mu [ U ] \\left ( a ^ { ( 1 ) } \\otimes \\ldots \\otimes ( \\eta _ j ^ { - 1 } t _ j - \\eta _ { j + 1 } ^ { - 1 } t _ { j + 1 } ) \\otimes \\ldots \\otimes a ^ { ( m ) } \\right ) = \\\\ & = \\eta _ j ^ { - 1 } L \\widetilde { t } _ j R - \\eta _ { j + 1 } ^ { - 1 } L \\widetilde { t } _ { j + 1 } R . \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} F ( x , t ) = \\begin{cases} x & 0 \\leq t < 1 , \\\\ \\pi ( x ) & t = 1 . \\end{cases} \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } \\varphi ( x ) K i r _ { T , S } \\Phi ( x ) d x & = \\langle T , \\varphi K i r _ { T , S } \\Phi \\rangle \\\\ & = \\langle \\ ! \\langle S , \\varphi \\Phi \\rangle \\ ! \\rangle \\\\ & = \\iint _ { \\mathbb { R } ^ { 2 n } } \\frac { \\varphi ( x ) \\Phi ( x , y ) - \\varphi ( x ) \\Phi ( x , x ) } { \\abs { x - y } ^ { n + 2 s } } d x d y \\\\ & = \\int _ { \\mathbb { R } ^ { n } } \\varphi ( x ) \\left ( \\int _ { \\mathbb { R } ^ { n } } \\frac { \\Phi ( x , y ) - \\Phi ( x , x ) } { \\abs { x - y } ^ { n + 2 s } } d y \\right ) d x \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} \\begin{cases} G ( x , 0 ) = \\delta ( x ) \\\\ G ( 0 , t ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} \\delta _ { \\Omega , p } ^ + ( t ) : = \\min \\big \\{ t , ~ \\inf \\{ | z - ( p + i t ) | \\colon \\Re z \\geq \\Re p , ~ z \\in \\C \\setminus \\Omega \\} \\} \\in [ 0 , t ] . \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} \\begin{aligned} x _ { k } & = \\frac { \\sin \\frac { 2 k \\pi } { n } \\sin \\left ( \\beta - ( - 1 ) ^ k \\frac { \\pi } { n } \\right ) } { \\sin \\frac { 2 \\pi } { n } } , & x _ { k + \\frac { n } { 2 } + 1 } & = x _ { k } + ( - 1 ) ^ k \\sin \\frac { 2 k \\pi } { n } , \\\\ y _ { k } & = \\frac { 1 } { 2 } + \\frac { \\cos \\frac { 2 k \\pi } { n } \\sin \\left ( \\beta - ( - 1 ) ^ k \\frac { \\pi } { n } \\right ) } { \\sin \\frac { 2 \\pi } { n } } , & y _ { k + \\frac { n } { 2 } + 1 } & = y _ { k } + ( - 1 ) ^ k \\cos \\frac { 2 k \\pi } { n } . \\end{aligned} \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} u _ 1 ' = \\frac { u _ 2 ' } { u _ 2 } \\ , u _ 1 + \\ln \\left ( \\frac { u _ 2 ' } { u _ 2 } \\right ) \\ , u _ 1 ' \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} \\| \\mathcal { T } [ F ] ( t ) \\| _ { L ^ 1 } & \\leq \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } \\big | \\bar { f _ 0 } \\big ( X ( t _ 0 ) , V ( t _ 0 ) \\big ) \\big | d x d v \\\\ & + \\iint _ { \\Omega \\times \\mathbb { R } ^ 3 } \\int _ { t _ 0 } ^ t \\big | Q ^ \\varepsilon [ F ] \\big ( s , X ( s ) , V ( s ) \\big ) \\big | d s d x d v \\\\ [ 3 p t ] & = : I + I I , \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} \\tau _ { m , m - 1 } ( D ) = \\circ _ { n = 1 } ^ { m - 1 } \\left ( \\circ _ { \\alpha \\in L _ n ' } \\left ( \\psi ^ { n , m - 1 } _ \\alpha \\bullet \\widetilde { N ^ { n , \\alpha } } \\right ) \\right ) . \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} \\lambda _ \\gamma = \\left ( \\frac { \\gamma } { c ( \\alpha , d ) } \\right ) ^ { \\alpha / ( d - \\alpha ) } , \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} \\Delta _ A ( f _ 1 , 0 ) \\Delta _ B ( f _ 1 , 0 ) = C _ 1 \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} e _ 3 ( \\left \\lbrace z _ 1 , \\ldots , z _ 5 \\right \\rbrace ) & = z _ 1 z _ 2 z _ 3 + z _ 1 z _ 2 z _ 4 + + z _ 2 z _ 4 z _ 5 + z _ 3 z _ 4 z _ 5 . \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} \\tau & = - 0 . 5 , & \\delta & = 1 , & b _ 1 & = 1 , & b _ 2 & = 1 . \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} H _ m > 1 + \\sum ^ { m - 1 } _ { i = 1 } H _ i = 1 + \\sum ^ { L } _ { i = 1 } H _ i + \\sum _ { i = L + 1 } ^ { m - 1 } H _ i \\geq G _ m - \\sum _ { i = L + 1 } ^ { m - 1 } G _ i + \\sum _ { i = L + 1 } ^ { m - 1 } H _ i , \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} u & = \\sum _ { j = 1 } ^ { r + 2 } ( - 1 ) ^ { n - r - 3 + j } C ( 2 n - 4 - r , n - r - 3 + j ) e _ { j , n - r - 2 + j } \\\\ v & = \\sum _ { j = 1 } ^ { r + 2 } ( - 1 ) ^ { j - 1 } C ( r , j - 1 ) e _ { n - r + j - 2 , j } . \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\frac { d s } { d t } = \\alpha \\beta \\sigma ^ { 2 } \\left ( t \\right ) d t = \\left ( \\alpha \\beta \\right ) ^ { - 1 } \\sigma ^ { - 2 } \\left ( t \\right ) d s . \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ { n + 1 } ( 1 + x _ i ) \\right ) \\sum _ { J \\subseteq [ 2 , n ] } ( - 1 ) ^ { | J | } \\left ( \\prod _ { j \\in J } \\frac { x _ j } { 1 + x _ j } \\right ) \\sum _ { K \\subseteq [ 2 , n ] \\setminus J } k _ { n - | J | - | K | } ( \\underline x _ n ^ { J \\cup K } ) . \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} \\begin{cases} T _ 1 ( u ) + a T ( u ) - T ( u ) a = T ( a \\cdot u - u \\cdot a + H ( a , T u ) - H ( T u , a ) ) + T _ 1 ' ( u ) , \\\\ a T _ 1 ( u ) - T _ 1 ( u ) a = T _ 1 ' ( a \\cdot u - u \\cdot a + H ( a , T u ) - H ( T u , a ) ) . \\end{cases} \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} ( f g ) ' = f ' g + f g ' . \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} R _ { i \\rho } = R ^ 0 _ { i \\rho } - R ^ 0 _ { i \\rho } V R ^ 0 _ { i \\rho } + R ^ 0 _ { i \\rho } V R ^ 0 _ { i \\rho } V R ^ 0 _ { i \\rho } + \\ldots \\ , . \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} \\dim ( \\Delta ^ { ( n ) } ) ^ { - 1 } \\nu _ { R _ n , \\l _ n , K } = \\dim \\nu _ { R _ n , \\l _ n , K } . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} I = i _ 0 + i _ 1 + \\cdots + i _ { 2 l + 1 } \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} A _ { i j } = \\begin{cases} \\rho _ i , & i = j \\\\ - \\kappa _ { i j } , & i \\neq j \\end{cases} . \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { \\nabla } _ { S } S & { } = \\frac { 1 } { 2 } B , & \\bar { \\nabla } _ { X } X & { } = \\frac { 1 + 2 \\lVert T \\rVert ^ { 2 } } { 2 ( 1 + \\lVert T \\rVert ^ { 2 } ) } B - \\frac { 1 } { 1 + \\lVert T \\rVert ^ { 2 } } J T . \\end{aligned} \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} & \\iota _ { Q ^ { ( k ) } } \\Omega ^ { ( k ) } = \\delta S ^ { ( k ) } + \\pi _ { ( k ) } ^ * \\alpha ^ { ( k + 1 ) } \\\\ & \\frac 1 2 \\iota _ { Q ^ { ( k ) } } \\iota _ { Q ^ { ( k ) } } \\Omega ^ { ( k ) } = \\pi _ { ( k ) } ^ * S ^ { ( k + 1 ) } , \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} \\mathcal { S } ^ { m } _ W : = \\{ f \\in C ^ \\infty ( \\bar { \\Omega } _ W ) ~ \\mbox { e v e n a n d h o l o m o r p h i c o n } ~ \\Omega _ W , ~ | f ^ { ( k ) } ( z ) | \\leq C _ k ( 1 + | z | ) ^ { m - k } , ~ \\forall k . \\} , \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} C ^ k _ { b _ 1 , \\ldots , b _ k } ( T ) ( f ) : = [ b _ k , \\cdots , [ b _ 2 , [ b _ 1 , T ] ] \\cdots ] ( f ) \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} \\partial _ { s } u = \\left ( a + i b \\right ) \\left [ \\Delta u + A u + Q \\left ( y , s \\right ) \\right ] , y \\in R ^ { n } , s \\in \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} \\Psi _ { n } ( 0 ) = h _ { n } ( r ) , \\ ; \\Phi ( r ) = h ( r ) . \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\frac { d \\mu _ { 1 } ( t ) } { ( x - t ) ^ { 2 } } \\geq \\liminf _ { \\delta \\downarrow 0 } \\int _ { ( x - \\delta , x + \\delta ) } \\frac { A ( \\alpha - t ) ^ { 2 k } h ( t ) } { ( x - t ) ^ { 2 } } \\ , d t = + \\infty . \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} \\left \\| \\left ( \\sum _ { i = 1 } ^ { N } \\left ( 1 + \\sup _ { u \\in [ s , t ] } | X ^ { i , N } _ { s } | \\right ) ^ { d + 1 } \\right ) ^ { \\frac { 1 } { 2 } } \\right \\| _ { L ^ m ( \\Omega ) } \\leq C \\ , N ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} B _ { G , L } - B _ { H , L } = 1 + \\sum _ { i = 1 } ^ { L - 1 } G _ { i } - G _ { L } - \\left ( 1 + \\sum _ { i = 1 } ^ { L - 1 } H _ { i } - H _ { L } \\right ) = H _ { L } - G _ { L } = 1 > 0 . \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} \\begin{aligned} P _ \\rho : \\ & X _ c \\to \\mathcal { L } ( X _ c , X _ c ) \\\\ & x \\mapsto A _ c + \\rho ( x ) \\end{aligned} & & & & \\begin{aligned} Q _ \\rho : \\ & X _ c \\mapsto \\mathcal { L } ( X _ c , X _ c ) \\\\ & x \\mapsto ( P _ \\rho ( T ( x ) ) ) ^ { - 1 } \\end{aligned} \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} \\frac { d } { d t } \\det A = \\det A \\ ; \\mathrm { t r } \\left ( A ^ { - 1 } \\frac { d A } { d t } \\right ) . \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} I = I _ M : = \\int _ { - M } ^ 0 | x | ^ { \\sigma } \\left [ \\int _ { - \\infty } ^ 0 J ( x - y ) \\psi ( y ) d y - \\psi ( x ) \\right ] d x \\in [ - C , C ] \\mbox { f o r a l l } M > 0 . \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} \\| J \\psi \\| _ { \\sup } = \\sup _ { x H \\in G / H } | J \\psi ( x H ) | \\le \\| \\psi \\| _ { \\sup } . \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} \\eta _ { \\nu _ { k } } ( z ) = \\eta _ { \\mu } ( z ) \\left ( \\frac { \\eta _ { \\mu } ( z ) } { z } \\right ) ^ { \\frac { 1 } { k - 1 } } , z \\in \\mathbb { T } . \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } P \\bigg ] \\tilde { u } = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} \\Phi ( z ) = \\gamma z \\exp \\left [ \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t z } { z - t } \\ , d \\sigma ( t ) \\right ] , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } , \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} ( U _ + \\cdot \\nabla ) ( \\overline { \\phi } + i \\overline { \\phi } ^ \\perp ) = ( U _ + \\cdot \\nabla \\overline { \\phi } - U ^ \\perp _ + \\cdot \\nabla \\overline { \\phi } ^ \\perp ) = U ^ \\perp _ + ( \\nabla ^ \\perp \\cdot \\overline { \\phi } ) , \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ n \\left ( a _ { i j } \\zeta _ j \\langle B \\zeta , \\zeta \\rangle - b _ { i j } \\zeta _ j \\langle A \\zeta , \\zeta \\rangle \\right ) = 0 , i = 1 , \\ldots , n , \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} | \\frac { \\phi ( r , \\xi ) } { r ^ { 3 / 2 } } | \\leq \\begin{cases} C , r ^ { 2 } \\xi \\leq 4 \\\\ C | a ( \\xi ) | \\sqrt { \\xi } , r ^ { 2 } \\xi > 4 \\end{cases} \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} \\left \\| \\Lambda ^ 2 f _ n ( t ) \\right \\| \\leq \\left \\| \\Lambda ^ 2 f _ { 0 } \\right \\| + \\rho ( ( \\left \\Vert \\Lambda _ { 1 } f _ 0 \\right \\Vert ) \\int _ 0 ^ t \\left \\Vert \\Lambda ^ { 2 } f _ n ( s ) \\right \\Vert d s \\ ; \\forall t \\geq 0 ; \\ ; n = 2 , 3 , . . . , \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} D ( H ) = \\{ x \\in D ( H _ 0 ^ \\star ) \\ : ; \\ : F _ 1 ( x ) = T F _ 2 ( x ) \\} . \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} A _ n = R _ n \\begin{pmatrix} 0 & 1 \\\\ 1 & a _ { k ( n ) } \\end{pmatrix} \\cdots \\begin{pmatrix} 0 & 1 \\\\ 1 & a _ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} u ^ N _ t ( x ) = & u ^ N _ 0 ( x ) - \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ 0 ^ t \\nabla V ^ N ( x - X _ s ^ { i , N } ) \\cdot F \\big ( K \\ast u ^ N _ s ( X _ { s } ^ { i , N } ) \\big ) \\ d s \\\\ & - \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ 0 ^ t \\nabla V ^ N ( x - X _ s ^ { i , N } ) \\cdot d W ^ i _ s + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ 0 ^ t \\Delta V ^ N ( x - X _ s ^ { i , N } ) \\ d s . \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} \\mathbb { I } _ { \\gamma } = \\mathbb { I } _ { \\bar { \\gamma } } \\ast \\dot { \\mathbb { B } } _ { \\mathbb { I } _ { \\bar { \\gamma } } ( < \\kappa _ n ) } ( \\dot { T } _ { \\bar { \\gamma } } ) \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} \\tilde { w } = \\cdots a \\cdots z b c \\cdots \\rightsquigarrow \\tilde { w } ^ * = \\cdots a \\cdots z c b \\cdots \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} & \\rho : = \\rho ^ { k + 1 } , \\mathbf { u } : = \\mathbf { u } ^ { k + 1 } , w : = \\rho ^ k , \\mathbf { v } : = \\mathbf { u } ^ k , \\\\ & \\tilde { \\rho } : = \\rho ^ { k + 1 } - \\rho ^ k , \\tilde { \\mathbf { u } } : = \\mathbf { u } ^ { k + 1 } - \\mathbf { u } ^ k , \\tilde { w } : = \\rho ^ { k } - \\rho ^ { k - 1 } , \\tilde { \\mathbf { v } } : = \\mathbf { u } ^ { k } - \\mathbf { u } ^ { k - 1 } , \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} H ( s ) = C ( s E - A ) ^ { - 1 } B , \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\| R \\| ^ 2 \\leq \\| R \\| ^ 2 _ { { \\rm H S } } = \\int _ { M \\times M } | K _ R | ^ 2 d { \\rm v o l } _ { M \\times M } \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} T _ t ( u ) T _ t ( v ) = T _ t ( u \\cdot T _ t ( v ) + T _ t ( u ) \\cdot v + H ( T _ t ( u ) , T _ t ( v ) ) ) , ~ u , v \\in M . \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} ( q ; q ) _ \\infty ( q ; q ^ 2 ) _ \\infty = \\dfrac { ( q ; q ) _ \\infty } { ( - q ; q ) _ \\infty } = \\sum _ { m = - \\infty } ^ \\infty ( - 1 ) ^ m q ^ { m ^ 2 } . \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\int _ { S _ { \\varepsilon , k , i } } | H | = \\frac { 1 } { 2 } \\ell ( \\partial M _ i ) \\int _ { \\gamma _ k } | \\kappa | d s . \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} \\frac { C } { | t - r | } \\int _ { 0 } ^ { \\infty } d \\xi | \\partial _ { \\xi } \\left ( \\left ( 1 - \\chi _ { \\leq 1 } ( r \\xi ) \\right ) r \\xi \\phi ( r \\xi ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\right ) | \\leq \\frac { C } { | t - r | \\log ^ { b - 1 } ( r ) } \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} \\boldsymbol { B } ( \\phi , \\phi ) = \\langle { \\mathcal L } ( \\phi ) , \\phi \\rangle = \\langle { \\mathcal L } _ + ( \\phi ) , \\phi ^ + \\rangle _ { \\mathcal R } \\ , + \\ , \\langle { \\mathcal L } _ - ( \\phi ) , \\phi ^ - \\rangle _ { \\mathcal R } . \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} ( b + B ) & - I ^ * \\\\ 0 & - ( b + B ) \\end{matrix} \\right ) \\left ( \\begin{matrix} \\tau ^ r _ \\varphi \\\\ \\sigma _ \\varphi \\end{matrix} \\right ) = \\left ( \\begin{matrix} \\tau ^ r _ { \\delta \\varphi } \\\\ \\sigma _ { \\delta \\varphi } \\end{matrix} \\right ) \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} N ( \\mathbf { g } ( X ) ) = N ( X ) , \\forall \\ \\mathbf { g } \\in S t r _ { 0 } ( \\mathfrak { J } ) . \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} \\zeta _ n ( u ) : = \\frac { \\langle 1 | f _ n \\rangle } { \\sqrt { \\kappa _ n } } \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} | E ( \\l ) | = | \\det ( M ) | \\le C r \\le r ^ { 1 / 2 } \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} C = \\big \\langle ( A _ { 1 , 1 } , \\ldots , A _ { t , 1 } ) , \\ldots , ( A _ { 1 , m _ t } , \\ldots , A _ { t , m _ t } ) \\big \\rangle . \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} \\partial _ t u - ( 1 + \\gamma \\partial ^ \\alpha _ t ) \\Delta u & = f ( t , u _ \\rho ) \\ ; \\Omega , t > 0 , \\\\ u & = 0 \\ ; \\partial \\Omega , \\ ; t \\ge 0 , \\\\ u ( x , s ) & = \\xi ( x , s ) , \\ ; x \\in \\Omega , s \\in [ - \\tau , 0 ] , \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} P _ { p } ( \\zeta , \\eta ) : = F _ { p } ( \\zeta , \\eta ) + K _ p \\left ( F _ p ( \\zeta ) + F _ p ( \\eta ) \\right ) , \\zeta , \\eta \\in \\R ^ { 2 } . \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} \\partial ^ \\textup { F } \\psi ( \\bar x ) : = \\left \\{ y \\in \\R ^ n \\ , \\middle | \\ , \\liminf \\limits _ { x \\to \\bar x } \\frac { \\psi ( x ) - \\psi ( \\bar x ) - y \\cdot ( x - \\bar x ) } { \\norm { x - \\bar x } { 2 } } \\geq 0 \\right \\} \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} \\rho = \\mathbf { P } \\left ( \\eta _ 0 = 1 \\right ) = \\frac { p _ 0 } { 1 - p _ 1 + p _ 0 } , \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} \\theta _ T : = \\frac { 2 k } { ( \\sqrt { p } - 1 ) \\lambda } \\in ( 0 , 2 ) \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} \\langle h _ { j } , h _ { i } \\rangle = c \\qquad ; \\qquad \\forall i , j \\in \\{ 1 , 2 \\} \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } = \\left ( 1 - \\frac { c _ x } { 2 } \\right ) \\sqrt { n } \\langle x , x \\rangle ^ \\frac { 1 } { 2 } = \\sqrt { n } \\langle x , x \\rangle ^ \\frac { 1 } { 2 } \\left ( 1 - \\frac { c _ x } { 2 } \\right ) . \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} \\frac { T ' _ 1 T ' _ 3 + ( T ' _ 2 ) ^ 2 } { ( T ' _ 1 ) ^ 2 + h ( T ' _ 3 ) ^ 2 } = b ' . \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} a ^ { 2 } \\left ( x _ { 1 } \\right ) \\geq 0 , \\left \\Vert \\partial _ { x _ { 1 } } ^ { j } a \\left ( x _ { 1 } \\right ) \\right \\Vert _ { L ^ { \\infty } \\left ( \\mathbb { R } \\right ) } \\leq C _ { j } n ^ { - j } , j = 1 , 2 , . . . , \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} G _ { \\mu } ( x ) = \\lim _ { y \\downarrow 0 } G _ { \\mu } ( x + i y ) , x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} \\Psi _ { \\Lambda } : = \\sum _ { i \\in I } \\bigotimes _ { x \\in \\Lambda } h _ { x , i } \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} S = S _ M : = \\int _ { - M } ^ 0 | x | ^ { \\alpha - 1 } \\left [ \\int _ { - \\infty } ^ 0 J ( x - y ) \\psi ( y ) d y - \\psi ( x ) \\right ] d x \\leq C \\mbox { f o r a l l } M > 0 . \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} K ( \\Psi ; M ; \\gamma ) = \\sum _ { A \\subset M } ( - 1 ) ^ { | A | } H ( \\Psi ; \\gamma - \\epsilon _ A ) \\ , , \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} P u = \\sum _ { j , k = 1 } ^ { n } a _ { j k } \\partial _ { j } \\partial _ { k } u + \\sum _ { j , k = 1 } ^ { n } ( \\partial _ { j } a _ { j k } ) \\partial _ { k } u . \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} \\omega ^ n = T _ n ( a ) + U _ { n - 1 } ( a ) \\sqrt { a ^ 2 - 1 } , \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} - 2 \\int _ { 0 } ^ { \\infty } \\sin ( t \\xi ) \\xi \\widehat { v _ { 2 , 0 } } ( \\xi ) d \\xi = F ( t ) , t \\geq 0 \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} \\frac { d } { d t } ( q _ 1 - q _ 2 ) = p _ 2 ( 0 ) \\left ( e ^ { q _ 2 - q _ 1 } - 1 \\right ) , \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\mathfrak { g } ( \\varphi ) = \\psi \\iff T \\vdash \\mathcal { G } ( \\ulcorner \\varphi \\urcorner , \\ulcorner \\psi \\urcorner ) \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} \\begin{array} { l l } \\eta = [ \\mu _ i , \\sigma _ { i j } ] ^ \\top _ { i , j \\in \\{ 1 , \\ldots , d \\} , i \\le j } , Z ( \\eta ) = S ( \\eta ) ^ { \\frac { 1 } { 1 - \\beta } } M _ { \\eta , \\beta } , h ( { \\bf { x } } ) \\equiv 1 , \\\\ w ( \\eta ) = \\big [ w ^ { ( 1 ) } ( \\eta ) , w ^ { ( 2 ) } ( \\eta ) \\big ] ^ \\top , f ( { \\bf { x } } ) = \\big [ f ^ { ( 1 ) } ( { \\bf { x } } ) , f ^ { ( 2 ) } ( { \\bf { x } } ) \\big ] ^ \\top , \\end{array} \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} \\left ( I - \\mbox { A u x } _ { p } \\tilde { A } \\right ) \\tilde { Z } + \\tilde { B } \\overline { \\tilde { Z } } = \\tilde { V } \\left ( z _ { 1 } , z _ { 2 } , \\dots , z _ { N } \\right ) , \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} \\sigma ( u ) & = \\sigma ( ( - 1 ) ^ \\delta u _ 1 ^ x u _ 2 ^ y ) = ( - 1 ) ^ { \\delta + x } u _ 1 ^ { - x - y } u _ 2 ^ x , \\\\ \\sigma ^ 2 ( u ) & = \\sigma ( ( - 1 ) ^ { \\delta + x } u _ 1 ^ { - x - y } u _ 2 ^ x ) = ( - 1 ) ^ { \\delta - y } u _ 1 ^ y u _ 2 ^ { - x - y } . \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} u ( \\cdot , t ) = S ( t ) \\xi + \\int _ 0 ^ t S ( t - s ) F ( \\cdot , s ) d s , \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} \\mathcal S ^ { \\frac { 1 } { 2 } } ( \\partial \\Omega ) = \\mathcal S ^ { \\frac { 1 } { 2 } } _ { \\mu } ( \\partial \\Omega ) : = \\left \\{ f \\in L ^ 2 ( \\partial \\Omega ) : f = \\sum _ { j = 1 } ^ { \\infty } \\hat b _ j \\hat v _ { j , \\mu } { \\rm \\ w i t h \\ } \\left ( \\sqrt { \\lambda _ j ( \\mu ) } \\hat b _ j \\right ) _ { j = 1 } ^ { \\infty } \\in l ^ 2 \\right \\} . \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} | n _ { 1 } ( x _ { 1 } ) - n _ { 1 } ( x _ { 2 } ) | & \\leq | x _ { 1 } - x _ { 2 } | _ { \\theta \\in [ 0 , 1 ] } | n _ { 1 } ' ( \\theta x _ { 1 } + ( 1 - \\theta ) x _ { 2 } ) | \\\\ & \\leq | x _ { 1 } - x _ { 2 } | _ { \\theta \\in [ 0 , 1 ] } | 4 \\sin ^ { 2 } ( \\theta x _ { 1 } + ( 1 - \\theta ) x _ { 2 } ) | \\\\ & \\leq C | x _ { 1 } - x _ { 2 } | \\left ( | x _ { 1 } | ^ { 2 } + | x _ { 2 } | ^ { 2 } \\right ) \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} H _ { k - L + 2 m + 1 } + \\cdots + H _ { k - L + 1 } \\leq \\sum _ { i = 1 } ^ { k + m - 2 } H _ { i } + 1 , \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} \\nu _ { N _ 1 + N _ 2 } ^ { \\sigma } ( 0 ) = \\mu _ { N _ 1 + N _ 2 } ^ { \\sigma } , \\nu _ { N _ 1 + N _ 2 } ^ { \\sigma } ( 1 ) = \\mu _ { N _ 1 } ^ { \\sigma } \\otimes \\mu _ { N _ 2 } ^ { \\sigma } . \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} \\C ^ { 2 ^ n } = W ^ { + } \\oplus W ^ { - } ; W ^ { \\pm } = \\bigoplus _ { ( i _ 1 , \\dots , i _ { n - 2 } ) } \\C f _ { 1 4 ; i _ 1 , \\dots , i _ { n - 2 } } ^ { \\pm } \\oplus \\bigoplus _ { ( i _ 1 , \\dots , i _ { n - 2 } ) } \\C f _ { 2 3 ; i _ 1 , \\dots , i _ { n - 2 } } ^ { \\pm } . \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} \\lim _ { \\mu \\to 0 } T _ R ( \\tilde { r } ; \\mu ) = \\kappa \\ln \\left ( 1 + \\frac { a _ 2 } { \\kappa \\beta } \\ , \\tilde { r } \\right ) . \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } d \\xi | \\chi _ { \\leq 1 } ' ( r \\xi ) | r | r \\xi \\Phi _ { 1 } ( r \\xi ) | \\frac { | \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) | } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } & \\leq C r \\int _ { 0 } ^ { 1 / r } \\frac { d \\xi } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\\\ & \\leq \\frac { C } { \\log ^ { b - 1 } ( r ) } , r \\geq \\frac { t } { 2 } \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} & w ^ * ( t ) = w ( t , \\phi ( t ; T , x _ 0 ) ) , \\\\ & \\lambda ^ * ( t ) = \\lambda ( t , \\phi ( t ; T , x _ 0 ) ) , \\\\ & g ^ * ( t ) = g ( t , \\phi ( t ; T , x _ 0 ) ) . \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} X = ( X _ 1 , \\ldots , X _ { s + 1 } , ( x _ 1 ) , \\ldots , ( x _ m ) , ( 1 ) ) . \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} \\mathsf { p = } c _ { \\mathsf { p , } \\alpha , \\beta } \\mathsf { \\hat { p } } _ { \\mathsf { k } } , \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} \\boldsymbol \\psi _ { N , x _ L } ^ { \\beta } ( t _ { n + 1 } ) = \\exp \\left [ i \\int _ { t _ n } ^ { t _ { n + 1 } } ( D _ N ^ { \\beta } + V _ { N , x _ L } ^ { \\beta } ( t ) ) \\dd { t } \\right ] \\boldsymbol \\psi _ { N , x _ L } ^ { \\beta } ( t _ { n } ) \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} l _ 1 ( \\alpha ) + \\frac { 1 } { 2 ! } l _ 2 ( \\alpha , \\alpha ) - \\frac { 1 } { 3 ! } l _ 3 ( \\alpha , \\alpha , \\alpha ) - \\cdots = 0 . \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} \\mathcal { A } ^ { \\exp } _ G ( M , E ) : = \\left \\{ A \\in C ^ { \\infty } ( M \\times M , E \\boxtimes E ^ * ) ^ { G } , ~ \\sup _ { x , y \\in M } \\left | e ^ { q d _ M ( x , y ) } \\nabla ^ { m } _ { x } \\nabla ^ { n } _ { y } A ( x , y ) \\right | < C _ q , ~ \\mbox { f o r a l l } ~ q , m , n \\in \\mathbb { N } \\right \\} . \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} \\phi ( - q ^ { \\frac { j } { 2 } } x ) = \\sum _ { n \\geq 0 } \\frac { q ^ { \\frac 1 2 n ^ 2 + \\frac { j - 1 } { 2 } n } x ^ n } { ( q ) _ n } . \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} \\| ( I - P _ { M + N } ) x - x _ n \\| & \\leq C _ 1 D _ p ( x _ n , ( I - ( P _ { M + N } ) ) x ) ^ { 1 / \\rho } = C _ 1 D _ { p ^ * } ( \\Pi ^ { p ^ * } _ { M \\perp \\cap N ^ \\perp } y _ 0 , y _ n ) ^ { 1 / \\rho } \\\\ & \\leq C _ 1 C _ 2 ( q ' ) ^ { n / \\rho } = : C q ^ { n } \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} { \\psi _ h } = \\{ x \\in { \\rm { \\textbf { d o m } } } ~ f ~ | ~ f ( x ) \\le h \\} , \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} \\rho ^ \\lambda _ a \\partial _ \\lambda E _ { \\mu \\nu } = \\widetilde { C } ^ { k l } _ a \\rho ^ \\lambda _ k \\rho ^ \\tau _ l E _ { \\lambda \\nu } E _ { \\mu \\tau } - ( \\partial _ \\mu \\rho ^ \\lambda _ a ) E _ { \\lambda \\nu } - ( \\partial _ \\nu \\rho ^ \\lambda _ a ) E _ { \\mu \\lambda } . \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} F ( z ) = \\int _ \\gamma f ( \\zeta , z ) d \\zeta , \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} \\Phi _ { , 1 } ^ \\tau ( { u ^ \\ell } ) = e ^ { \\tau \\mathcal { L } } \\Big ( u ^ \\ell + \\tau \\mathcal { B } \\left ( F ( u ^ \\ell ) \\cdot \\varphi _ 1 \\big ( \\tau \\mathcal { A } \\big ) G ( \\overline { u ^ \\ell } ) \\right ) \\Big ) = e ^ { \\tau \\mathcal { L } } \\left ( u ^ \\ell + \\tau \\Psi ^ \\tau _ { , 1 } ( u ^ l ) \\right ) . \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} \\mu \\left ( \\{ x \\in B : \\ | b ( x ) - b _ B | > t \\} \\right ) & \\leq \\mu \\left ( \\{ x \\in B : \\ | b ( x ) - b _ B | > C k \\alpha \\} \\right ) \\\\ & \\leq \\sum _ j \\mu ( B ^ { ( k ) } _ j ) \\leq { 1 \\over \\alpha ^ k } \\mu ( B ^ { ( 0 ) } ) = \\mu ( B ) e ^ { - k \\log \\alpha } \\\\ & \\leq \\mu ( B ) \\alpha e ^ { - { t \\log ( \\alpha ) \\over C \\alpha } } , \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} L ' ( t ) = - I ( u ( x , t ) ) = \\frac { q - p } { p } \\| \\nabla u \\| _ p ^ p + \\frac { 1 } { q } \\| u \\| _ q ^ q - q J ( u ( x , t ) ) \\ge q K ( t ) . \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} M _ \\nu \\uparrow \\infty , N _ \\nu \\uparrow \\infty , K _ \\nu = \\frac { N _ \\nu } { M _ \\nu } \\uparrow \\infty . \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ N h ( \\nabla \\phi ( x , t ) , u _ t ( x ) ) d \\alpha _ t \\ , d t = \\int _ 0 ^ T ( \\int _ N h ( \\nabla \\phi ( x , t ) , v _ t ( x ) ) \\lambda d \\mu _ t + h ( \\nabla \\phi ( x , t ) , w _ t ( x ) ) ( 1 - \\lambda ) d \\nu _ t ) \\ , d t \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left \\| \\frac { e ^ { i t k ^ 2 } \\Pi ( k ) } { 2 i ( 1 + i ) } \\cdot \\chi _ { E _ s ^ c } \\cdot \\int _ { 2 a t } ^ { 2 b t } \\frac { e ^ { i u ^ 2 / ( 4 t ) } } { \\sqrt t } \\widehat f _ + ( u / ( 2 t ) ) e ^ { - i k u } d u - \\frac { \\sqrt { 2 \\pi } \\Pi ( k ) } { 2 i } \\cdot \\chi _ { E _ s ^ c } \\widehat f _ + ( k ) \\right \\| _ { 2 , \\sigma } = 0 \\ , . \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\int _ \\mathcal C g ( F ( \\omega ) ) e ^ { 2 \\pi i ( n _ k ^ 3 + M _ 1 n _ k ^ 2 + M _ 2 n _ k ) F ( \\omega ) } \\nu ( \\omega ) = \\int _ \\mathcal C g ( F ( \\omega ) ) \\nu ( \\omega ) . \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} F ( z ) = \\alpha _ { 0 } + \\beta z + \\int _ { \\mathbb { R } } \\frac { 1 + z t } { t - z } \\ , d \\rho ( t ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} k _ E = D \\cdot E = 9 m - 9 m = 0 . \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} + \\infty > c : = \\inf _ { h \\in \\Gamma } \\sup _ { u \\in Q } I ( h ( u ) ) > \\omega : = \\sup _ { h _ 0 \\in \\Gamma _ 0 } \\sup _ { u \\in Q _ 0 } I ( h _ 0 ( u ) ) , \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} \\mathrm { i } \\begin{bmatrix} 0 & - 1 \\\\ 1 & 0 \\end{bmatrix} \\begin{bmatrix} u ' \\\\ v ' \\end{bmatrix} + \\begin{bmatrix} 0 & 0 \\\\ 0 & \\mathrm { i } \\end{bmatrix} \\begin{bmatrix} u \\\\ v \\end{bmatrix} = \\mathrm { i } \\begin{bmatrix} f \\\\ 0 \\end{bmatrix} \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} & \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } \\lambda '' ( s ) \\frac { 2 r ( s - t ) } { ( 1 + r ^ { 2 } + ( s - t ) ^ { 2 } + \\sqrt { ( 1 + ( r + s - t ) ^ { 2 } ) ( 1 + ( r - ( s - t ) ) ^ { 2 } ) } } d s \\\\ & = \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } \\lambda '' ( s ) \\frac { 2 r } { ( s - t ) } \\left ( \\frac { 1 } { 2 } + O \\left ( \\frac { ( 1 + r ) ^ { 2 } } { ( s - t ) ^ { 2 } } \\right ) \\right ) d s \\\\ & = \\int _ { t + 2 ( r + 1 ) } ^ { \\infty } \\lambda '' ( s ) \\frac { r } { ( s - t ) } d s + E _ { 1 } ( t , r ) \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} N _ k : = \\dim H ^ 0 ( X , L ^ { \\otimes k } \\otimes G ) . \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { r r } 1 & 0 \\\\ 1 & - 1 \\end{array} \\right ) , W = \\left ( \\begin{array} { c c } 0 & - 1 \\\\ 1 & - 1 \\end{array} \\right ) , J _ 2 = \\left ( \\begin{array} { r r } 0 & - 1 \\\\ 1 & 0 \\end{array} \\right ) , \\\\ \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} M _ { B , C } = \\left ( \\begin{array} { c c } \\frac { ( n _ 1 + 2 ) ( n _ 2 + 1 ) } { 2 } & 0 \\\\ 0 & - \\frac { ( n _ 1 + 2 ) ( n _ 2 + 1 ) } { 2 } \\\\ \\end{array} \\right ) , M _ { D , E } = \\left ( \\begin{array} { c c c c } \\frac { n _ 1 + n _ 2 + 3 } { 3 } & \\frac { n _ 2 - n _ 1 - 1 } { 3 } & - \\frac { 2 n _ 2 + 2 } { 3 } \\\\ - \\frac { 2 n _ 1 + 4 } { 3 } & \\frac { 2 n _ 1 + 4 } { 3 } & 0 \\\\ \\frac { n _ 1 - n _ 2 + 1 } { 3 } & - \\frac { n _ 1 + n _ 2 + 3 } { 3 } & \\frac { 2 n _ 2 + 2 } { 3 } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} - i w _ n & : = - i \\sum _ { s = 3 } ^ n w _ n ^ { ( s ) } = - i \\sum _ { s = 3 } ^ n \\lambda _ s B _ { n , s } \\left ( 1 ! , 2 ! a _ 1 , 3 ! a _ 2 , \\ldots \\right ) . \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} S = - i \\tilde { \\gamma } \\left ( 2 \\nabla \\varphi . \\nabla + \\Delta \\varphi \\right ) K = i \\left ( \\Delta + A + \\tilde { \\gamma } ^ { 2 } \\left \\vert \\nabla \\varphi \\right \\vert ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} \\delta _ j = \\sqrt { F ( \\Sigma _ { j - 1 } ) - F ( \\Sigma _ { j + 2 } ) } = \\left ( \\int _ { j - 1 } ^ { j + 2 } \\| \\phi \\| _ { L ^ 2 } ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } \\ , . \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} M _ { s ; i } = M _ { s - 1 ; i } V _ s + q _ { s - 1 , i } M _ { s ; s - 1 } . \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} z _ 2 = \\frac { 2 5 } { 6 4 } z _ 1 \\ ; . \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} \\psi _ \\gamma : = u _ { \\lambda _ \\gamma } , \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} & ( n - 1 ) d ^ 2 - \\sum _ { i = 1 } ^ { s } m _ i ^ 2 = i , \\\\ & d ( n + 1 ) - \\sum _ { i = 1 } ^ s m _ i = 2 + i . \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} A ( K ) : = A - B B ^ T K . \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} i _ Z ^ ! p _ { 1 3 * } ( - ) & = q _ { T * } ( e ( N _ { Y ^ { 4 T } } Y ^ 4 ) ^ { - 1 } f _ T ^ * ( e ( N _ { Y ^ { 2 T } } Y ^ 2 ) ) \\cdot ( i _ { Z ^ { ( 2 ) } } ) ^ ! _ { i _ { Y ^ 4 } } ( - ) ) \\\\ & = q _ { T * } \\left ( e ( Y ) ^ { - 2 } \\cdot _ 2 ( i _ { Z ^ { ( 2 ) } } ) ^ ! _ { i _ { Y ^ 4 } } ( - ) \\right ) \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} F _ 1 ( u ) - u [ \\nabla F _ 1 ( u ) ] ^ T = ( a _ 1 u _ 1 u _ 2 , a _ 2 u _ 1 u _ 2 ) . \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} \\hat { \\pi } ( t ) : = \\left \\{ \\begin{array} { r l } \\pi ( \\varsigma _ { \\pi } ) & t \\leq \\varsigma _ { \\pi } , \\\\ \\pi ( t ) ~ & t > \\varsigma _ { \\pi } . \\end{array} \\right . \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} \\theta _ { j + k d ^ h + 1 } \\triangleq \\begin{cases} \\exp \\left ( \\frac { \\pi i k \\left ( 2 j + k d ^ h \\right ) } { d ^ { h + 1 } } \\right ) & d \\\\ \\exp \\left ( \\frac { \\pi i k \\left ( 2 j + \\left ( k + 1 \\right ) d ^ h \\right ) } { d ^ { h + 1 } } \\right ) & d \\end{cases} , \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} \\prec _ t = \\sum _ { i = 0 } ^ N t ^ i \\prec _ i \\succ _ t = \\sum _ { i = 0 } t ^ i \\succ _ i ~ ~ ~ \\curlyvee _ t = \\sum _ { i = 0 } ^ N t ^ i \\curlyvee _ i \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} | F _ 3 | \\ge \\frac { \\frac m 2 - t } { \\binom t { k - 1 } \\binom { b _ { k - 1 } + b _ { k } } { b _ k } } . \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} \\mathfrak { a } ^ { \\dagger } \\left ( \\mathbb { K } _ { n - 1 } ^ { ( j ) } ( x ) \\right ) = \\Lambda _ { n } ^ { \\left ( j \\right ) } \\left ( x ; 2 , 2 \\right ) \\mathbb { K } _ { n } ^ { ( j ) } ( x ) , \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} | \\Phi ( \\xi _ 1 ' , \\eta _ 1 ' , \\xi , \\eta ) | & = | \\xi _ 1 ' \\xi ( \\xi _ 1 ' + \\xi ) + \\eta _ 1 ' \\eta ( \\eta _ 1 ' + \\eta ) | \\leq 2 ^ 5 A ^ { - \\frac { 3 } { 2 } } d ^ { - 1 } N _ 1 ^ 3 , \\\\ | F ( \\xi _ 1 ' , \\eta _ 1 ' , \\xi , \\eta ) | & = | \\xi _ 1 ' \\eta + \\xi \\eta _ 1 ' + 2 ( \\xi _ 1 ' \\eta _ 1 ' + \\xi \\eta ) | \\leq 2 ^ 5 A ^ { - 1 } d ^ { - 1 } N _ 1 ^ 2 , \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} & | \\int _ { t } ^ { t + \\lambda ( t ) ^ { 1 - \\alpha } } \\frac { d s } { ( s - t ) } \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho \\lambda '' ( s ) } { r } \\partial _ { r } F _ { 3 } ( r , \\rho , \\lambda ( s ) ) | \\leq C \\int _ { t } ^ { t + \\lambda ( t ) ^ { 1 - \\alpha } } \\frac { d s } { ( s - t ) } \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho } { r } | \\lambda '' ( s ) | r \\lambda ( s ) ^ { 2 \\alpha - 2 } \\\\ & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} L ( g , \\bar { g } ) : = \\left ( \\frac { \\det ( \\bar { g } ) } { \\det ( g ) } \\right ) ^ { \\frac { 1 } { N + 1 } } \\ , \\bar { g } ^ { - 1 } g \\ , . \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} \\phi ( K ) = \\begin{bmatrix} \\phi _ 1 ( K _ 1 ) & A _ 1 ( K _ 1 ) ^ { \\top } K _ { 1 2 } + K _ { 1 2 } A _ 2 + K _ 1 A _ { 1 2 } + Q _ { 1 2 } \\\\ * & A _ 2 ^ { \\top } K _ 2 + K _ 2 A _ 2 + K _ { 1 2 } ^ { \\top } A _ { 1 2 } + A _ { 1 2 } ^ { \\top } K _ { 1 2 } + Q _ 2 - K _ { 1 2 } ^ { \\top } B _ 1 B _ 1 ^ { \\top } K _ { 1 2 } \\end{bmatrix} \\ , . \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} \\eta ^ \\rightarrow ( B _ 1 \\cap B _ 2 ) & = \\eta ^ \\rightarrow ( ( B _ 1 ^ c \\cup B _ 2 ^ c ) ^ c ) \\\\ & = \\{ [ \\eta ^ \\rightarrow ( B _ 1 ) ] ^ c \\cup [ \\eta ^ \\rightarrow ( B _ 2 ) ] ^ c \\} ^ c \\\\ & = \\eta ^ \\rightarrow ( B _ 1 ) \\cap \\eta ^ \\rightarrow ( B _ 2 ) . \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} J \\partial _ x \\phi ( x ) = \\lambda A ( x ) \\phi ( x ) + B ( x ) \\phi ( x ) , \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} f ' ( z ) = ( z - \\alpha _ 1 ) ^ { n _ 1 - 1 } \\cdots ( z - \\alpha _ s ) ^ { n _ s - 1 } g ( z ) \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} < f '' ( x ) u , v > = \\int _ 0 ^ 1 \\int _ 0 ^ 1 K _ V ( x ; t , s ) u ( t ) v ( s ) d t d s + \\int _ 0 ^ 1 K _ L ( x ; t ) u ( t ) v ( t ) d t , \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} { f _ { { \\rm { l o w } } } } \\left ( { \\xi P } \\right ) = \\frac { 1 } { 2 } \\ln \\left ( { 1 + \\frac { { 2 { \\varsigma ^ 2 } { \\sigma ^ 2 } } } { { \\xi P } } } \\right ) - \\frac { { \\xi P + { \\varsigma ^ 2 } { \\sigma ^ 2 } } } { { { \\varsigma ^ 2 } { \\sigma ^ 2 } } } + \\frac { { \\sqrt { \\xi P \\left ( { \\xi P + 2 { \\varsigma ^ 2 } { \\sigma ^ 2 } } \\right ) } } } { { { \\varsigma ^ 2 } { \\sigma ^ 2 } } } . \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} C _ { a , b } ( t ) & = \\sum _ { m = 0 } ^ \\infty t ^ m F _ m ( a , b ) , \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} \\Upsilon \\left ( \\frac { \\pm z } { 2 \\pi i t } , \\mp \\theta \\right ) = \\left ( \\frac { \\pm z } { 2 \\pi i t } \\right ) ^ { \\frac { 1 } { 1 2 } } \\cdot \\lim _ { \\tau \\to 1 } \\psi _ \\pm ( t ) \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} \\beta = \\left ( \\frac { \\partial f _ L } { \\partial \\mu } \\frac { \\partial g _ L } { \\partial y } - \\frac { \\partial g _ L } { \\partial \\mu } \\frac { \\partial f _ L } { \\partial y } \\right ) \\bigg | _ { x = y = \\mu = 0 } . \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} \\left | \\frac { R ' ( t ) } { R ( t ) } \\right | = | ( \\log R ) ' ( t ) | , \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} \\mathcal { S } ( T , X ) = \\sum _ { t _ { j } \\leqslant T } X ^ { i t _ { j } } \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} \\lambda + b = 0 . \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} w _ { l } = z _ { l } \\overline { z } _ { \\tau \\left ( l \\right ) } + \\lambda _ { l } \\left ( z _ { l } z _ { \\sigma \\left ( l \\right ) } + \\overline { z } _ { l } \\overline { z } _ { \\sigma \\left ( l \\right ) } \\right ) + \\displaystyle \\sum _ { k \\geq 3 } \\varphi _ { k } ^ { \\left ( l \\right ) } ( z , \\overline { z } ) , \\quad \\mbox { f o r a l l $ l = 1 , \\dots , N \\leq N ' $ , } \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} c b _ \\alpha ^ { - 1 } N _ { \\theta , \\alpha } ^ { 1 - \\alpha } - 2 \\sum \\limits _ { i = 1 } ^ { d } c _ i \\Big [ \\sum \\limits _ { j = 1 } ^ d \\sigma ^ { i j } \\mu _ j \\Big ] + \\sum \\limits _ { i = 1 } ^ d \\sum \\limits _ { j = i } ^ d c _ { i j } \\sigma ^ { i j } = 0 . \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} \\partial _ x u ( x ) = A ( x ) u ( x ) , u ( a ) = u _ 0 . \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} M _ 0 \\cdot f _ { 1 2 ; i _ 1 , \\dots , i _ { n - 1 } } ^ { + } = f _ { 1 2 ; i _ 1 , \\dots , i _ { n - 1 } } ^ { + } + \\lambda _ { i _ 1 , \\dots , i _ { n - 1 } } f _ { 1 2 ; 1 , \\dots , 1 } ^ { + } , \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { P \\to \\infty } { C _ { \\rm g a p } } = \\ln \\left ( { \\frac { 1 } { { 1 - \\beta } } } \\right ) . \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} ( \\sigma _ - ) _ { 2 j } = ( - 1 ) ^ j ( \\sigma _ + ) _ { 2 j } \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} 1 - \\frac { y r - 1 } { y ( 1 - w ) } = 0 , \\mbox { t h a t i s } , \\ , w = 1 + y ^ { - 1 } - r \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } \\frac { 3 + 3 \\tau _ i + \\tau _ i ^ 2 } { \\sqrt { 1 + ( \\varepsilon _ i / \\tau _ i ) ^ 2 } } = 0 . \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} \\pi ( \\xi ) = \\pi ( x , \\omega ) = E _ { \\omega } T _ x . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} \\lim _ { N - M \\to \\infty } \\frac { 1 } { N - M } \\sum _ { n = M } ^ { N - 1 } | a ( n ) - \\psi ( n ) | = 0 \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} \\left ( \\frac { 1 - \\sqrt { 1 - 4 z ^ 2 } } { 2 z } \\right ) ^ r = \\sum _ { m = r } ^ { \\infty } f ( m , r ) z ^ { m } . \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} | \\partial _ { r } N _ { 2 } ( f _ { v _ { 5 } } ) ( t , r ) | & \\leq \\frac { C } { r ^ { 3 } t ^ { 3 } \\log ^ { 4 N + 7 b - 2 } ( t ) } + \\frac { C \\log ( r ) } { r ^ { 3 } | t - r | t ^ { 3 / 2 } \\log ^ { 4 b - 1 + \\frac { 5 N } { 2 } } ( t ) } + \\frac { C \\log ^ { 2 } ( r ) } { r ^ { 2 } t ^ { 3 / 2 } ( t - r ) ^ { 2 } \\log ^ { 3 b - 1 + \\frac { 5 N } { 2 } } ( t ) } \\\\ & + \\frac { C \\log ^ { 3 } ( r ) } { r ^ { 2 } ( t - r ) ^ { 4 } } , t - \\sqrt { t } > r > \\frac { t } { 2 } r > t + \\sqrt { t } \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} 0 < a _ - : = \\inf _ { B _ { r _ 2 } } G / G ' \\leq a _ + : = \\sup _ { B _ { r _ 2 } } G / G ' < + \\infty \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta u = 0 , & ; \\\\ u = f , & , \\end{array} \\right . \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} \\frac { d ^ a } { d X ^ a } \\sum _ { j = 0 } ^ { M - 1 } T _ { \\omega _ { 1 , j } } ( 1 , R ( X ) ) X ^ j \\Big | _ { X = \\l } = & \\frac { d ^ a } { d X ^ a } \\sum _ { j = 0 } ^ { M - 1 } T _ { \\omega _ { 2 , j } } ( 1 , R ( X ) ) X ^ j \\Big | _ { X = \\l } \\\\ \\frac { d ^ a } { d X ^ a } \\sum _ { j = 0 } ^ { M ' - 1 } T _ { \\omega ' _ { 1 , j } } ( 1 , R ( X ) ) X ^ j \\Big | _ { X = \\l } = & \\frac { d ^ a } { d X ^ a } \\sum _ { j = 0 } ^ { M ' - 1 } T _ { \\omega ' _ { 2 , j } } ( 1 , R ( X ) ) X ^ j \\Big | _ { X = \\l } \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} H _ { L + 1 } = c _ 1 H _ L + c _ 2 H _ { L - 1 } + \\cdots + c _ L H _ 1 = H _ L + H _ { L - 1 } + \\cdots + H _ 2 + c _ L H _ 1 . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} \\| F \\| _ { \\ell ^ 2 } = \\bigg ( \\sum _ { \\mu \\in \\Z ^ n } | F ( \\mu ) | ^ 2 \\bigg ) ^ { 1 / 2 } < \\infty . \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} \\tilde { u } ' ( \\beta ) & = 2 e ^ { 2 \\beta } + ( 1 - e ) e ^ { \\beta } + 2 e \\beta + \\frac { 2 e ^ 3 } { 3 ^ { 5 / 2 } \\sqrt { 2 \\pi } } - \\frac { 5 e } { 2 } \\\\ & < 2 e ^ { 4 / 5 } + 1 - e + \\frac { 4 e } { 5 } + \\frac { 2 e ^ 3 } { 3 ^ { 5 / 2 } \\sqrt { 2 \\pi } } - \\frac { 5 e } { 2 } < - 0 . 8 \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} \\phi ( | z - e _ \\varepsilon | ) + \\phi ( | z + e _ \\varepsilon | ) - 2 \\phi ( | e _ \\varepsilon | ) & = - \\alpha ( 1 + \\alpha ) | e _ \\varepsilon | ^ { \\alpha - 1 } z ^ 2 \\int _ 0 ^ 1 d t \\int _ { - 1 } ^ 1 \\left ( 1 + \\frac { z t \\tau } { | e _ \\varepsilon | } \\right ) ^ { \\alpha - 1 } t \\ , d \\tau \\\\ & \\le - \\alpha ( 1 + \\alpha ) 2 ^ { \\alpha - 1 } | e _ \\varepsilon | ^ { \\alpha - 1 } . \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} \\begin{aligned} & R _ { \\eta } ( 1 + | y | ^ 2 + | \\log R _ \\eta | ) L ^ \\infty ( 0 , T ; L ^ 1 ( \\mathbb T ^ d _ { \\ell } ) ) , & \\nabla \\sqrt { R _ \\eta } L ^ \\infty ( 0 , T ; L ^ 2 ( \\mathbb T ^ d _ { \\ell } ) ) , \\\\ & \\sqrt { R _ \\eta } U _ { \\eta } L ^ \\infty ( 0 , T ; L ^ 2 ( \\mathbb T ^ d _ { \\ell } ) ) , & \\sqrt { R _ \\eta } \\nabla U _ \\eta L ^ 2 ( 0 , T ; L ^ 2 ( \\mathbb T ^ d _ { \\ell } ) ) . \\end{aligned} \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} & \\psi ( y + l _ 2 - l _ 1 + B ) = \\frac { l _ 2 - ( y + l _ 2 - l _ 1 + B ) } { l _ 1 } = 1 - \\frac { y + B } { l _ 1 } , \\ \\ \\ y \\in [ - B , B ] . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} \\mathcal { H } _ \\beta = \\frac { 1 } { 2 } \\Delta + \\beta \\cdot v : L ^ 2 ( \\mathbb { R } ^ d ) \\rightarrow L ^ 2 ( \\mathbb { R } ^ d ) , \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { r _ 0 } | u ( r ) | ^ { q ( r ) } r ^ { \\beta - 1 } d r & = \\int _ 0 ^ { r _ 0 } | u ( r ) | ^ { \\frac { 2 \\beta } { \\beta - 2 } } | u ( r ) | ^ { f ( r ) } r ^ { \\beta - 1 } d r \\\\ & \\leq \\int _ 0 ^ { r _ 0 } | u ( r ) | ^ { \\frac { 2 \\beta } { \\beta - 2 } } \\Big ( \\frac { a r ^ { 2 - \\beta } } { \\beta - 2 } \\Big ) ^ { f ( r ) / 2 } r ^ { \\beta - 1 } d r . \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} \\partial _ t | D | ^ s X + u \\cdot \\nabla | D | ^ s X = - [ | D | ^ s , u \\cdot \\nabla ] X + | D | ^ s ( X \\cdot \\nabla u ) . \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} q _ i ( d _ { m j k } p ^ j ) = 0 . \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } a _ { \\ell } ( n ) q ^ n = \\prod _ { n = 1 } ^ { \\infty } \\frac { ( 1 - q ^ { \\ell n } ) ^ { \\ell } } { ( 1 - q ^ n ) } . \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} B _ { k } = \\left [ - L _ { k } ^ { d + 6 } , L _ { k } ^ { d + 6 } \\right ] ^ { d } \\times [ 0 , L _ { k } ] , \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} d \\nu _ { \\lambda } ( t ) = : \\frac { \\sin ( \\pi \\lambda ) } { \\pi } \\frac { t ^ { \\lambda - 1 } } { ( 1 - t ) ^ { \\lambda } } d t , \\ ; \\ ; \\lambda \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} \\Big [ \\lambda - ( - b + K - K e ^ { - \\lambda \\tau } ) \\Big ] ( \\lambda ^ { 3 } + p _ { 1 } \\lambda ^ { 2 } + p _ { 2 } \\lambda + p _ { 3 } ) = 0 \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} F _ { m } ( c _ { 0 } + \\varepsilon e ^ { i \\tau } ) = \\alpha + \\sqrt [ 3 ] { \\frac { \\varepsilon } { \\left | c _ { 3 } \\right | } } \\exp \\left ( \\frac { 2 m \\pi i - \\theta i + \\tau i } { 3 } \\right ) + o ( 1 ) \\quad ( \\varepsilon \\rightarrow 0 ^ { + } ) . \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} \\Big ( 2 D _ 4 u ( x _ 1 , x _ 2 ) & - ( 1 + z _ 1 - z _ 2 ) D _ 1 u ( x _ 1 , x _ 2 ) - ( z _ 3 - z _ 1 ) D _ 2 u ( x _ 1 , x _ 2 ) \\\\ & - ( 1 + z _ 2 - z _ 3 ) D _ 3 u ( x _ 1 , x _ 2 ) \\Big ) \\Big | _ { x _ 1 = 0 } = 0 . \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} [ X , [ X , [ X , [ X , Y _ { - 2 } ] ] ] = I _ 4 ( X ) Y _ 2 . \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} K _ 2 = R ^ { - 1 } \\bigl ( U \\cup \\bigl ( \\mathfrak M ( H ^ \\infty ( V ) ) \\setminus ( V \\cup S ) \\bigr ) \\bigr ) . \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} F \\circ K = K \\circ R . \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } \\partial _ { j } a _ { j k } \\partial _ { k } \\bigg ] \\tilde { w } & = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ \\tilde { w } & = w \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} p : = \\inf \\{ \\rho \\geq 1 : \\rho \\Phi ^ { ( 1 ) } ( x ) \\succeq \\Phi ^ { ( 2 ) } ( x ) \\ { \\rm f o r } \\ x \\leq 0 \\} . \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} C _ 0 ^ { - 1 } 2 ^ { j m _ s \\theta _ 0 } = C _ 0 ^ { - 1 } c _ s ^ { j \\theta _ 0 } \\leq s ( c _ s ^ { - j } ) ^ { - 1 } \\leq C _ 0 c _ s ^ { j \\theta _ 1 } = C _ 0 2 ^ { j m _ s \\theta _ 1 } , \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} \\frac { d P } { d r } = - \\frac { \\varrho \\left ( \\omega T ; \\frac { \\lambda } { \\omega } \\right ) } { \\varrho \\left ( \\omega T ; - \\frac { \\lambda } { \\omega } \\right ) } , \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} q = ( 0 , 1 , 0 ) , \\ , p _ { 1 } = ( 1 , 0 , 0 ) , \\ , p _ { 2 } = ( - 1 , 1 , 0 ) , \\ , r _ { 1 } = ( 0 , 0 , 1 ) , \\ , r _ { 2 } = ( 0 , 1 , - 1 ) , \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} \\int _ X c ( x , T ( x ) ) d \\mu = \\min _ { S _ \\# \\mu = \\nu } \\int _ X c ( x , S ( x ) ) d \\mu . \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} \\operatorname { I n d } _ \\infty ( D ) : = [ P ^ b _ Q ] - [ e _ 1 ] \\in K _ 0 ( \\mathcal { I } _ G ( M ) ) \\equiv K _ 0 ( C ^ * ( M _ 0 \\subset M , E ) ^ G ) \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} \\gamma _ C ( \\overrightarrow { s } , \\overrightarrow { t } ) = \\bigcup \\limits _ { i = 1 } ^ { m ( C ) - 1 } \\gamma _ i ( s _ i , t _ i ) , \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} \\partial _ { t } u = i \\left ( \\Delta u + L u \\right ) + V \\left ( x , t \\right ) u , x \\in R ^ { n } , y \\in \\Omega , t \\in \\left [ 0 , T \\right ] . \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d X ( t ) & = b ( X ( t ) ) d t + \\sigma ( X ( t ) ) d { B } ( t ) + \\int _ { \\mathbb { R } } \\nu ( d a ) \\ , d L _ t ^ a ( X ) \\\\ X ( 0 ) & = x \\ , , \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} \\Re ( g \\cdot \\Omega ) = \\lambda \\gamma , \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} \\nabla ( a b ) = \\nabla ( a ) b + a \\nabla ( b ) a , b \\in A \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} & \\frac { C } { ( \\log ( \\log ( t ) ) ) ^ { 2 } t ^ { 2 } \\log ^ { b + 1 } ( t ) } + \\frac { C } { ( \\log ( \\log ( t ) ) ) ^ { 2 } \\log ^ { b + 1 } ( t ) t ^ { 2 } } + \\frac { C \\log ( \\log ( t ) ) } { t ^ { 2 } \\log ^ { b + 2 } ( t ) } + \\frac { C } { ( \\log ( \\log ( t ) ) ) ^ { 3 / 2 } t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\\\ & + \\frac { C } { t ^ { 2 } \\log ^ { b + 2 } ( t ) ( \\log ( \\log ( t ) ) ^ { 5 / 2 } } + \\frac { C } { ( \\log ( \\log ( t ) ) ^ { 5 / 2 } t ^ { 3 } \\log ^ { b + 2 } ( t ) } \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} s _ { T , b } : = \\int _ 0 ^ T V ^ { s } u _ { g , h , T } d s . \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} S & = \\{ ( a _ i , d _ i ) \\in \\Z ^ 2 : 1 \\leq i \\leq r \\} , \\\\ S ' & = \\{ ( - a _ i + ( d _ i - 1 ) k , d _ i ) \\in \\Z ^ 2 : 1 \\leq i \\leq r \\} , \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} \\sharp \\mathcal { C } ( u _ 1 , \\dots , u _ m , n ) = \\sharp \\mathcal { D } ( u _ 1 , \\dots , u _ m , n ) \\ , \\cdot \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} a _ 1 x _ 1 + \\cdots + a _ i x _ j + \\cdots + a _ j x _ i + \\cdots + a _ k x _ k = 0 \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} I _ 3 : = - I _ { 3 , 1 } + I _ { 3 , 2 } , \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} H _ { n + 1 } \\ = \\ c _ 1 H _ n + c _ 2 H _ { n - 1 } + \\cdots + c _ n H _ { 1 } + 1 . \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} \\hat X \\phi ( x ) & = 0 \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} W ( f , g ) W ( h , k ) + W ( g , h ) W ( f , k ) + W ( h , f ) W ( g , k ) = 0 , \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} \\varphi ( A + B ) & = \\varphi ( A ) + \\varphi ( B ) , \\\\ \\varphi ( \\alpha A ) & = \\alpha \\varphi ( A ) , \\\\ \\varphi ( A B ) & = \\varphi ( A ) \\varphi ( B ) \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} V _ p ( \\xi ) = ( 1 + | \\xi | ^ 2 ) ^ { \\frac { p - 2 } { 4 } } \\xi . \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} K _ { ( n + 1 ) , z } ( w ) = \\frac { 1 } { n } \\left ( z - \\frac { \\partial } { \\partial \\overline { z } } \\right ) \\left ( \\overline { w } - \\frac { \\partial } { \\partial w } \\right ) K _ { ( n ) , z } ( w ) . \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} \\frac { 1 } { x } \\frac { d \\mu } { d t } \\left ( \\frac { 1 } { x } \\right ) = \\frac { 1 } { \\pi } \\Im \\frac { 1 } { 1 - \\eta _ { \\mu } ( x ) } \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } \\left ( - \\lambda ( t ) \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ( r ) ) - 1 } { r ^ { 2 } } \\right ) v _ { 5 } \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle \\right ) | \\leq \\frac { C } { t ^ { 4 } \\log ^ { 3 N - 2 } ( t ) } \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} ( a * _ \\gamma b ) * _ \\gamma c = p _ { 1 4 * } ( p _ 2 \\times p _ 3 \\times p _ { 1 2 } \\times p _ { 2 3 } \\times p _ { 3 4 } ) ^ ! ( \\gamma \\otimes \\gamma \\otimes a \\otimes b \\otimes c ) . \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} k _ n ( x _ 1 , \\dots , x _ n ) = \\sum _ M \\prod _ { r = 1 } ^ n x _ r ^ { t _ r ( M ) } \\in \\Z [ x _ 1 , \\dots , x _ n ] \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} \\Phi ( \\varepsilon ) = \\left \\{ ( x , y ^ E , y ^ I , z ^ I ) \\in \\Re ^ n \\times \\Re ^ q \\times \\Theta ( \\varepsilon ) : G ( x , y , z ^ I ) = 0 \\right \\} . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} \\sigma _ { ( a , i ) } ^ { - 1 } ( ( b , j ) ) = ( b + a r - i , j - i r + a r ^ 2 - 1 ) . \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} X = M _ + \\cup _ H M _ - \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} H ( \\nu ) \\ge \\sum _ { A \\subset \\{ 0 , \\ldots , n - 1 \\} } ( 2 / 3 ) ^ { | A | } ( 1 / 3 ) ^ { n - | A | } ( h n - \\log ( 2 ) ( n - | A | ) ) = ( h - \\frac { \\log 2 } { 3 } ) n , \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} F ( \\xi ) + F ^ * ( F ' ( \\xi ) ) = \\langle F ' ( \\xi ) , \\xi \\rangle , \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} \\lambda _ { ( l ) } ( \\mu ) \\left ( \\xi _ { ( l ) } - \\mu \\right ) = \\left ( 3 l ^ 4 + 2 ( N - 2 ) l ^ 3 - ( N + 1 ) l ^ 2 - ( N - 2 ) l - \\eta _ { ( l ) } \\mu \\right ) \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} ( 1 - t _ 2 ) ^ { \\kappa - p / 2 + 1 } \\int _ { 1 / 2 } ^ { t _ 2 } \\left | \\frac { \\displaystyle | Z _ N ( t ) | ^ p - | \\sigma B _ N ( t ) | ^ p } { ( 1 - t ) t ^ { \\kappa } } \\right | d t = o _ P ( 1 ) . \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} \\int _ { r } ^ { \\infty } \\frac { 1 } { \\varphi ( t ) ^ { n - 1 } } d t = C \\ , r ^ { - \\delta ( n - 1 ) + 1 } \\ , . \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\epsilon _ { i n } ^ { ( 2 ) } = o _ p ( n ^ { 1 / 2 } | \\ss _ 2 | ) . \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} K _ { 2 3 } ( t , x , y ) | _ { t = \\log \\left ( \\frac { 1 + s } { 1 - s } \\right ) } & \\lesssim \\frac { ( 1 - s ) ^ n | ( 1 + s ) y - ( 1 - s ) x | } { s ^ { ( n - 1 ) / 2 } } \\exp \\Big ( - \\frac { | ( 1 + s ) y - ( 1 - s ) x | ^ 2 } { 8 s } \\Big ) e ^ { | y | ^ 2 - | x | ^ 2 } \\\\ & \\lesssim \\frac { ( 1 - s ) ^ n } { s ^ { n / 2 } } \\exp \\Big ( - \\frac { | ( 1 + s ) y - ( 1 - s ) x | ^ 2 } { 1 6 s } \\Big ) e ^ { | y | ^ 2 - | x | ^ 2 } , ( x , y ) \\in N ^ c . \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} \\mathbb { H } = \\{ x + i y : x , y \\in \\mathbb { R } , y > 0 \\} , \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} \\frac { \\partial u _ { m } } { \\partial t } = i \\left [ \\Delta u _ { m } + \\sum \\limits _ { j = 1 } ^ { N } a _ { m j } \\left ( x \\right ) u _ { j } + \\sum \\limits _ { j = 1 } ^ { N } b _ { m j } \\left ( x \\right ) u _ { j } \\right ] , x \\in R ^ { n } , t \\in \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} \\langle D _ 1 , D _ 1 \\rangle = - 1 \\chi ( X _ { 3 , 9 } , \\mathcal { O } _ { X _ { 3 , 9 } } ( D _ 1 ) ) = 1 \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} \\rho _ t ( t , x ) = & - \\xi ' \\bar h ' K _ 4 \\epsilon ( t ) V ^ * + \\xi K _ 4 \\epsilon ' ( t ) V ^ * \\\\ \\succeq & - \\xi _ * K _ 4 \\epsilon ( t ) V ^ * - K _ 4 \\beta ( t + \\theta ) ^ { - 1 } \\epsilon ( t ) V ^ * \\\\ \\succeq & - ( \\xi _ * + \\beta \\theta ^ { - 1 } ) K _ 4 \\epsilon ( t ) V ^ * , \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} \\| g _ { \\frak z } ^ d ( \\mathcal K \\ast f ) \\| _ { L ^ 2 _ w ( \\mathbb R ^ 3 ) } & \\leq C \\bigg \\| \\bigg \\{ \\sum _ { j ' , k ' } | \\psi _ { j ' , k ' } \\ast f | ^ 2 \\bigg \\} ^ { \\frac { 1 } { 2 } } \\bigg \\| _ { L ^ 2 _ w ( \\mathbb R ^ 3 ) } \\\\ & = C \\| g _ { \\frak z } ^ d ( f ) \\| _ { L ^ 2 _ w ( \\mathbb R ^ 3 ) } \\\\ & \\leq C \\| f \\| _ { L ^ 2 _ w ( \\mathbb R ^ 3 ) } , \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} \\psi _ { n , 2 } ^ { \\ast } ( Q ^ { \\ast } ) = \\frac { L _ { n , 2 } ^ { \\ast } ( q + s ^ { \\ast } ) } { q + s ^ { \\ast } } \\leq \\frac { n - \\widehat { w } _ { n } } { n ( 1 + \\widehat { w } _ { n } ) } + \\epsilon _ { 1 } . \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} 2 \\ell ( J ) ^ { d } \\geq \\mu ^ { I } ( J ) = \\mu _ { m } ^ { n ( I ) } ( J ) = \\mu _ { m } ^ { n ( I ) + 1 } ( J ) \\frac { \\ell ( I ) ^ { d } } { \\mu _ { m } ^ { n ( I ) + 1 } ( I ) } \\geq 2 ^ { d - n - 1 } \\ell ( J ) ^ { d } . \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} D _ k \\cdot D _ j = L _ { k j } + C . \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} v _ { 3 } ( t , r ) = v _ { 3 , 1 } ( t , r ) + v _ { 3 , 2 } ( t , r ) \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} R ^ { - 1 } ( Y _ 3 ) = O _ 1 . \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial t _ 2 } ( 0 , t _ 1 , 0 , t _ 3 ) = - ( z ( 0 ) - z ( t _ 1 ) ) ^ { - 1 } z ' ( 0 ) + ( z ( 0 ) - z ( t _ 3 ) ) ^ { - 1 } z ' ( 0 ) . \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} E _ { d } e ^ { \\lambda \\sum _ { i = 1 } ^ { N } Z _ { i , N } ( f , g ) } = E _ { d } \\prod _ { i = 1 } ^ { N } e ^ { \\lambda Z _ { i , N } ( f , g ) } \\leq \\prod _ { i = 1 } ^ { N } E _ { d } e ^ { \\lambda Z _ { i , N } ( f , g ) } . \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} k _ { n } & = ( - 1 ) ^ { n } t [ c _ { 1 } \\alpha _ { 1 } ( \\alpha _ { 1 } - 1 ) ( \\alpha _ { 1 } - 2 ) \\cdots ( \\alpha _ { 1 } - n + 1 ) { \\lambda _ { 1 } } ^ { \\alpha _ { 1 } - n } + c _ { 2 } \\alpha _ { 2 } ( \\alpha _ { 2 } - 1 ) ( \\alpha _ { 2 } - 2 ) \\\\ & \\cdots ( \\alpha _ { 2 } - n + 1 ) { \\lambda _ { 2 } } ^ { \\alpha _ { 1 } - n } ] . \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ l g _ { j } ^ { p _ j ( n _ 1 , \\ldots , n _ r ) } = \\prod _ { I } z _ { l _ 1 , \\ldots , l _ r } ^ { \\binom { n _ 1 } { l _ 1 } \\ldots \\binom { n _ r } { l _ r } } \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} v ^ 2 _ k ( x ) = \\eta ^ 2 ( x - h ) [ ( u _ k - \\psi ) ( x - h ) - ( u _ k - \\psi ) ( x ) ] \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| p _ t \\| _ { 2 , \\mu } = 1 , \\lim _ { t \\to \\infty } \\int _ { \\Delta } | p - p _ t | ^ 2 d \\mu = 0 \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} S _ { W Z } [ g ] = \\frac { 1 } { 1 2 } \\int _ M \\langle \\tilde { g } ^ { - 1 } d \\tilde { g } , [ \\tilde { g } ^ { - 1 } d \\tilde { g } , \\tilde { g } ^ { - 1 } d \\tilde { g } ] \\rangle , \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} p _ { i j } \\circ q _ { i + 1 , j } ( \\sigma ) = p _ { i j } ( q _ { i + 1 , j } ( \\sigma ) ) = p _ { i j } ( ( \\tau ( x ) ) \\vert _ { M ^ { i + j + 1 } } ) = ( \\tau ( x ) ) \\vert _ { M ^ { i + j } } . \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} \\Theta ^ { [ 2 ] } ( \\mathcal { M } ) = \\Theta ^ { [ 2 ] } \\begin{pmatrix} \\rho \\\\ \\kappa _ u \\\\ \\kappa _ s \\end{pmatrix} = \\begin{pmatrix*} [ l ] A _ c D k _ c + D g _ c ( K { } { } ) \\kappa - D k _ c \\left ( R \\right ) P _ \\rho \\\\ - A _ u ^ { - 1 } D g _ u ( K { } { } ) \\kappa + A _ u ^ { - 1 } \\kappa _ u ( R ) P _ \\rho \\\\ A _ s \\kappa _ s ( T ) Q _ \\rho + D g _ s ( K { } { } \\circ T ) \\kappa ( T ) Q _ \\rho \\end{pmatrix*} , \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} Q ( i ) = i + | \\{ j \\in J / j > i \\ , , \\psi ( j , i ) > 0 \\} | \\ , \\cdot \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} \\Phi _ \\tau ( r ) : = ( r + \\tau ) ^ { p / 2 - 1 } \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} & | 2 c _ { b } \\lambda _ { 2 } ( t ) \\int _ { 0 } ^ { \\infty } d \\xi \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) \\frac { \\sin ( t \\xi ) } { t ^ { 2 } } \\left ( F _ { v _ { 2 } } ( \\xi , \\lambda _ { 1 } ( t ) ) - F _ { v _ { 2 } } ( \\xi , \\lambda _ { 2 } ( t ) ) \\right ) | \\leq C \\frac { | \\lambda _ { 1 } ( t ) - \\lambda _ { 2 } ( t ) | \\lambda _ { 0 } ( t ) | \\log ( \\lambda _ { 0 } ( t ) ) | } { t ^ { 3 } } \\\\ & \\leq C \\frac { | | e _ { 1 } - e _ { 2 } | | _ { X } \\sqrt { \\log ( \\log ( t ) ) } } { t ^ { 3 } \\log ^ { 2 b } ( t ) } \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} \\psi _ { \\mu _ { 1 } } ' ( r e ^ { i f ( r ) } ) = \\int _ { \\mathbb { R } _ { + } } \\frac { t } { ( 1 - t r e ^ { i f ( r ) } ) ^ { 2 } } \\ , d \\mu _ { 1 } ( t ) , \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { x _ { k + 1 } } = \\mathbf { x _ k } - \\alpha [ \\mathbf { x _ k ^ T ( x _ k x _ k ^ T - X ) } + \\gamma ( \\mathbf { x _ k ^ T } \\mathbf { x _ k } ) \\mathbf { 1 ^ T } ] ^ { \\mathbf { T } } \\\\ \\end{aligned} \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} \\Big ( \\sum _ { n = 1 } ^ { N } \\Vert x _ { n } \\Vert ^ { q } \\Big ) ^ { \\frac { 1 } { q } } \\leq C \\bigg ( \\int _ { \\mathbb { T } ^ { N } } \\Big \\Vert \\sum _ { n = 1 } ^ { N } x _ { n } z _ { n } \\Big \\Vert ^ { q } d z \\bigg ) ^ { \\frac { 1 } { q } } \\ , , \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} b _ R & : = { 1 \\over | R | } \\int _ R b ( x _ 1 , x _ 2 , x _ 3 ) d x _ 1 d x _ 2 d x _ 3 = { 1 \\over a ^ 4 } \\int _ { a } ^ { a + a ^ 2 } \\int _ { a } ^ { 2 a } \\int _ { a } ^ { 2 a } x _ 1 \\ , d x _ 1 d x _ 2 d x _ 3 \\\\ & = { 1 \\over a } { x _ 1 ^ 2 \\over 2 } \\bigg | _ a ^ { 2 a } = { 3 a \\over 2 } . \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} \\inf _ { c \\in \\mathbb { R } } \\left ( \\frac { 1 } { \\mu ( E ) } \\int _ E | f ( x ) - c | ^ p d \\mu ( x ) \\right ) ^ { 1 / p } & \\leq \\left ( \\frac { 1 } { \\mu ( E ) } \\int _ E | f ( x ) - f _ E | ^ p d \\mu ( x ) \\right ) ^ { 1 / p } \\\\ & \\leq 2 \\inf _ { c \\in \\mathbb { R } } \\left ( \\frac { 1 } { \\mu ( E ) } \\int _ E | f ( x ) - c | ^ p d \\mu ( x ) \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ { 2 } \\int _ { \\Omega } u _ i \\partial _ { i j } \\theta \\partial _ j \\theta \\dd x & = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ { 2 } \\int _ { \\Omega } u _ i \\partial _ { i } ( \\partial _ j \\theta ) ^ 2 \\dd x \\\\ & = - \\frac { 1 } { 2 } \\int _ { \\Omega } ( \\partial _ 1 u _ 1 + \\partial _ 2 u _ 2 ) \\left [ ( \\partial _ 1 \\theta ) ^ 2 + ( \\partial _ 2 \\theta ) ^ 2 \\right ] \\dd x = 0 \\ ; . \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} I _ 4 = \\frac { 1 } { 2 } \\partial _ t ^ + \\| e _ h ^ { u ^ n } \\| _ { \\mathcal { T } _ h } ^ 2 + \\frac { \\Delta t } { 2 } \\| \\partial _ t ^ + e _ h ^ { u ^ n } \\| _ { \\mathcal { T } _ h } ^ 2 . \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} g = L _ { \\phi , \\psi } G _ { u , v } g , g \\in L _ \\mathrm { c } ^ 2 ] a , b [ . \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = \\xi ^ { \\prime \\prime } 4 x _ i x _ j + \\xi ^ \\prime 2 \\delta _ { i j } - \\frac { \\xi ^ { \\prime \\prime } 4 r + \\xi ^ \\prime 2 n } { n } \\delta _ { i j } + \\xi ^ \\prime \\frac { 4 x _ i x _ j } { 1 + r } + \\xi ^ \\prime \\frac { 4 x _ i x _ j } { 1 + r } - \\frac { 2 } { n } \\xi ^ \\prime \\frac { 4 r } { 1 + r } \\delta _ { i j } \\\\ & = 4 \\left ( \\xi ^ { \\prime \\prime } + \\frac { 2 } { 1 + r } \\xi ^ \\prime \\right ) \\left ( x _ i x _ j - \\frac { r } { n } \\delta _ { i j } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} B _ { n + k , k } \\left ( 0 , - 2 ! a _ 1 , - 3 ! a _ 2 , \\ldots \\right ) & = \\frac { ( n + k ) ! } { n ! } B _ { n , k } \\left ( - 1 ! a _ 1 , - 2 ! a _ 2 , - 3 ! a _ 3 , \\ldots \\right ) . \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} \\mathcal { D } _ { J , K } ( \\eta ) = ( ( T ^ { t } W ) _ 0 ) _ { t \\in \\mathbb { Z } } . \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} ( x H ) ( y H ) ^ { - 1 } \\in \\phi ( U ) [ \\phi ( V ) ] ^ { - 1 } = \\phi ( U ) \\phi ( V ^ { - 1 } ) = \\phi ( U V ^ { - 1 } ) \\subseteq W . \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} a f _ 1 ' f _ 1 ^ { - 1 } f _ 2 + f _ 1 '' f _ 1 ^ { - 1 } f _ 2 = a f _ 2 ' + f _ 2 '' . \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} \\forall ( a , b ) \\in \\R ^ 2 \\colon \\varphi _ \\textup { m i n } ( a , b ) : = \\min \\{ a ; b \\} , \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} & [ - \\underline h ^ n ( 0 ) , \\underline h ^ n ( 0 ) ] = [ - K , K ] \\subset [ g ( T ^ n ) , h ( T ^ n ) ] , \\\\ & \\underline U ^ n ( 0 , x ) \\preceq U ( T ^ n , x ) \\ \\mbox { f o r } \\ x \\in [ - K , K ] . \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} | \\varphi ^ { - 1 } ( \\widetilde { A } ) | ^ { \\frac { 1 } { s } - \\frac { 1 } { n } } \\leq \\| H _ s \\mid L _ { \\kappa } ( \\widetilde { A } ) \\| | \\widetilde { A } | ^ { \\frac { 1 } { q } - \\frac { 1 } { n } } , \\ , \\ , { 1 } / { \\kappa } = { 1 } / { s } - { 1 } / { q } , \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} \\frac { | \\mathbf { u } | ^ { N } } { N ! } \\Psi _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { u } ) & = \\sum _ { \\substack { | \\mathbf { n } | - | \\mathbf { m } | = N , \\\\ \\mathbf { n } \\in \\mathcal { P } } } \\binom { \\mathbf { n } } { \\mathbf { m } } ^ { ( d ) } \\Psi _ { \\mathbf { n } } ^ { ( d ) } ( \\mathbf { u } ) . \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} \\nu ( a ) \\nu ( b ) = \\left \\{ \\begin{array} { l l } \\nu ( J - a ) \\nu ( 2 a + b - J ) , & \\hbox { w h e n } J \\leq a + b \\leq K , \\\\ \\nu ( b + J - K ) \\nu ( a + K - J ) , & \\hbox { w h e n } a + b \\geq K . \\end{array} \\right . \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} \\phi _ { s , t } ( \\lambda ) = & \\lambda ^ 4 + ( - 2 s - 2 t + 4 ) \\lambda ^ 3 + ( - 9 s - 9 t - 5 s t + 3 ) \\lambda ^ 2 \\\\ & + ( - 1 2 s - 1 2 t - 4 s t - 4 ) \\lambda + ( - 4 s - 4 t - 4 ) . \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & a _ { 1 , 3 } \\\\ 0 & 1 & a _ { 2 , 3 } \\end{pmatrix} , \\begin{pmatrix} 1 & a _ { 1 , 2 } & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , \\textrm { o r } \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} \\Big ( \\sum _ { n = 0 } ^ { \\infty } \\Vert x _ n \\Vert ^ { q } \\rho ^ { q n } \\Big ) ^ { 1 / q } \\le \\Big ( \\int _ { \\mathbb { T } } \\Vert f ( z ) \\Vert ^ { q } d z \\Big ) ^ { 1 / q } , \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial ^ 2 } { \\partial x _ 2 ^ 2 } - \\frac { \\partial ^ 2 } { \\partial x _ 1 ^ 2 } - 4 8 e ^ { - 2 \\sqrt { 3 } x _ 2 } \\right ) \\psi ( x _ 1 , x _ 2 ) = 0 , \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} { } ^ Q \\Omega ^ \\pm _ a = ( K _ \\pm ) ^ { - 1 } \\rho _ a ( K _ \\pm ) = ( K _ \\pm ) ^ { - 1 } \\rho ^ \\mu _ a \\partial _ \\mu K _ \\pm , \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} \\begin{aligned} G _ l ( x ) & \\ , \\leq \\ , M y _ l & \\quad & l \\in \\mathcal Q & \\\\ H _ l ( x ) & \\ , \\leq \\ , M ( 1 - y _ l ) & & l \\in \\mathcal Q & \\\\ y _ l & \\ , \\in \\ , \\{ 0 , 1 \\} & & l \\in \\mathcal Q & \\end{aligned} \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} \\mathcal U _ N : = \\{ u \\in L ^ 2 _ { r , 0 } \\ , : \\ , \\gamma _ N ( u ) > 0 \\ , \\ , \\ , \\gamma _ j ( u ) = 0 \\ , \\ , \\forall \\ , j > N \\} . \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\beta _ { \\zeta } ( k ) = \\sum _ { j = 0 } ^ { n / t - 1 } \\sum _ { k = 0 } ^ { t - 1 } \\beta _ { \\zeta } ( j t + k ) = \\sum _ { j = 0 } ^ { n / t - 1 } \\beta _ { \\zeta } ( j t ) \\sum _ { k = 0 } ^ { t - 1 } \\beta _ { \\zeta } ( k ) = 0 , \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} \\forall \\ : \\tilde { y } \\in D ( H \\mathcal { H } ) \\cap X _ { n , \\mathcal { H } } \\colon \\langle \\tilde { x } , \\tilde { y } \\rangle _ { X _ { n , \\mathcal { H } } } = 0 \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} r ^ { 2 } \\partial _ { r r } ( v _ { 1 } + v _ { 2 } ) = r ^ { 2 } \\left ( \\frac { - 2 r \\lambda '' ( t ) } { 1 + r ^ { 2 } } \\right ) + r ^ { 2 } \\partial _ { t t } ( v _ { 1 } + v _ { 2 } ) - r \\partial _ { r } ( v _ { 1 } + v _ { 2 } ) + ( v _ { 1 } + v _ { 2 } ) \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} L _ { n , 1 } ( q - s ) = L _ { n , 2 } ( q - s ) \\leq \\frac { 1 - n \\widehat { \\lambda } _ { n } } { n ( 1 + \\widehat { \\lambda } _ { n } ) } ( q - s ) + \\epsilon _ { 1 } q . \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} \\Psi ( x , t ) = \\frac { 1 } { \\sqrt { \\zeta + i t } } \\exp \\left [ i k ( x - k t ) - \\frac { ( x - 2 k t ) ^ 2 } { 4 ( \\zeta + i t ) } \\right ] , \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} ( u _ h ^ n , 1 ) _ { \\mathcal T _ h } = ( u _ h ^ { n - 1 } , 1 ) _ { \\mathcal T _ h } = \\cdots = ( u ^ 0 , 1 ) _ { \\mathcal T _ h } , ( \\phi _ h ^ n , 1 ) _ { \\mathcal T _ h } = ( \\epsilon ^ { - 1 } f ^ n ( u ^ n _ h ) , 1 ) _ { \\mathcal T _ h } . \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} a _ j = \\sqrt { \\mu _ j ( \\lambda ) + b } \\hat a _ j \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} \\chi _ \\beta ( z ) = \\log c _ \\beta + \\beta \\log | z + 1 | + \\beta \\log | z - 1 | - 2 ( 1 + \\beta ) \\log | z | , \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} \\omega _ n t ^ { ( n + 2 ) / 2 } g _ t = c _ n g - \\frac { c _ n } { 3 } \\left ( \\mathrm { R i c } _ g - \\frac { 1 } { 2 } \\mathrm { S c a l } _ g g \\right ) t + O ( t ^ 2 ) , \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} \\langle u ^ + , v ^ + \\rangle _ { * \\mathcal R ^ + } = \\int _ 0 ^ \\infty \\Big [ \\ , A _ + \\big ( { t ^ + } ^ 2 - { U ^ + } ^ 2 \\big ) - B \\big ( { t ^ - } ^ 2 - { U ^ - } ^ 2 \\big ) \\ , \\Big ] ( u ^ + _ 1 v ^ + _ 1 + u ^ + _ 2 v ^ + _ 2 ) r \\ , { \\mathrm d } r , \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} A ' _ 3 ( 9 ) = & \\{ ( ( 3 ^ 3 ) , 3 , 0 ) , ( ( 3 ^ 3 ) , 3 , 1 ) , ( ( 3 ^ 3 ) , 3 , 2 ) \\} . \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} \\partial _ t F + v \\cdot \\nabla _ { \\ ! x } F = Q ( F , F ) , \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} \\sigma _ x = L _ { \\rho ( e ) } ^ k \\rho = ( L _ { \\rho ( e ) } \\rho ^ { - r } ) ^ k \\rho ^ { k r + 1 } = \\lambda ^ l \\rho ^ { k r + 1 } , \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} c ( \\gamma ) : = \\int _ 0 ^ { + \\infty } \\frac { \\left ( 1 + \\tau ^ 2 + \\frac { 2 } { \\sqrt { N } } \\tau \\right ) ^ { - \\gamma / 2 } + \\left ( 1 + \\tau ^ 2 - \\frac { 2 } { \\sqrt { N } } \\tau \\right ) ^ { - \\gamma / 2 } - 2 } { \\tau ^ { 1 + 2 s } } \\ , d \\tau \\ , . \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} \\mathfrak { \\widehat D } _ { k } = - k X ^ { d } a ^ { k - 2 } + \\frac { ( k - 3 ) k } { 2 ! } X ^ { 2 d } a ^ { k - 4 } - \\frac { ( k - 4 ) ( k - 5 ) k } { 3 ! } X ^ { 3 d } a ^ { k - 6 } + \\\\ \\cdots + ( - 1 ) ^ { \\frac { k - 3 } { 2 } } \\ , \\frac { ( k - 1 ) k ( k + 1 ) } { 2 4 } \\ , X ^ { \\frac { d ( k - 3 ) } { 2 } } a ^ 3 + ( - 1 ) ^ { \\frac { k - 1 } { 2 } } \\ , k \\ , X ^ { \\frac { d ( k - 1 ) } { 2 } } a . \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } P ' ( r , k ) = i k P ( r , k ) - \\overline { A ( r ) } P _ * ( r , k ) , & P ( 0 , k ) = 1 \\\\ P _ * ' ( r , k ) = - { A } ( r ) P ( r , k ) , & P _ * ( 0 , k ) = 1 \\end{array} , k \\in \\mathbb { C } , r \\ge 0 \\ , . \\right . \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} \\Theta _ 2 ( \\pi ) = * . \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} & C r \\sup _ { x \\geq t } | \\lambda ''' ( x ) | + C r \\sup _ { x \\geq t } \\left ( | \\lambda ''' ( x ) | \\lambda ( x ) ^ { \\alpha - 1 } \\left ( \\lambda ( x ) ^ { \\alpha - 1 } - \\lambda ( t ) ^ { \\alpha - 1 } \\right ) \\right ) \\lambda ( t ) ^ { 2 - 2 \\alpha } \\\\ & \\leq \\frac { C r } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } + \\frac { C r } { t \\log ^ { b - b \\alpha } ( t ) } \\sup _ { x \\geq t } \\left ( \\frac { | e ''' ( x ) | x } { \\lambda ( x ) ^ { 1 - \\alpha } } \\right ) \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} | \\alpha ( n ) - \\alpha ( m ) | & = \\int _ { m \\leq d ( x , x _ { 0 } ) \\leq n } | f ( x ) | d { \\rm v o l } ( x ) \\\\ & \\leq C \\int _ { m \\leq d ( x , x _ { 0 } ) \\leq n } e ^ { - q d ( x , x _ { 0 } ) } d { \\rm v o l } ( x ) \\quad \\mbox { f o r a l l } ~ q \\in \\mathbb { N } , \\\\ & \\leq C ' e ^ { - q m + n p } , \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} Q _ w = w \\left ( \\frac { 1 } { \\prod _ { 1 \\leqslant i \\leqslant a } \\prod _ { a + 1 \\leqslant j \\leqslant n } ( x _ i - x _ j ) } \\right ) . \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} \\frac { d x } { d t } = - \\frac { b ( t , x ) + b _ 1 ( t , x ) } { t } , x \\bigr | _ { t = T } = \\xi . \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} L _ { w } u ( t ) : = \\ddot { u } ( t ) + \\eta \\dot { u } ( t ) - \\sum _ { i , j } \\left ( \\delta _ { i j } - \\frac { w _ { y _ i } ( t ) w _ { y _ j } ( t ) } { \\abs { \\nabla w ( t ) } ^ 2 + \\epsilon ^ 2 } \\right ) u _ { x _ i x _ j } ( t ) . \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} \\left ( f ^ { - 1 } \\right ) _ n & = \\frac 1 { n ! } \\sum _ { k = 1 } ^ { n - 1 } \\frac { 1 } { f _ 1 ^ { n + k } } B _ { n - 1 + k , k } \\left ( 0 , - 2 ! f _ 2 , - 3 ! f _ 3 , \\ldots \\right ) , \\left ( f ^ { - 1 } \\right ) _ 1 = \\frac { 1 } { f _ 1 } . \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} \\int _ { M } \\left | \\nabla \\left ( \\sqrt { G ( p , y ) } \\phi ( y ) \\right ) \\right | ^ 2 \\ , d y & \\leq \\frac { 1 } { 2 } \\int _ { \\mathcal { L } _ { p } ( \\frac { \\delta \\varepsilon } { 2 } , 2 \\varepsilon ) } \\frac { | \\nabla G ( p , y ) | ^ 2 } { G ( p , y ) } \\ , d y + 2 \\int _ M G ( p , y ) | \\nabla \\phi | ^ 2 \\ , d y \\\\ & = C ( - \\log \\delta + 1 ) + 2 \\int _ M G ( p , y ) | \\nabla \\phi | ^ 2 \\ , d y \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} Q ( u , u ) : = \\int _ { \\R ^ d _ { 1 + } } \\int _ { \\R ^ d _ { 1 + } } ( u ( x ) - u ( y ) ) ^ 2 K ( | x - y | ) \\ , d x \\ , d y . \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( 1 - \\sigma ) D ^ 2 v : D ^ 2 \\varphi + \\sigma \\Delta v \\Delta \\varphi d x - \\mu \\int _ { \\partial \\Omega } v \\varphi d \\sigma = \\lambda ( \\mu ) \\int _ { \\partial \\Omega } \\frac { \\partial v } { \\partial \\nu } \\frac { \\partial \\varphi } { \\partial \\nu } d \\sigma \\ , , \\ \\ \\ \\forall \\varphi \\in H ^ 2 ( \\Omega ) , \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} F ( \\overline { z } ) = \\overline { F ( z ) } , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} 4 \\left ( Q ( \\mu ( \\xi ) ) + \\frac { \\varphi ( \\mu ( \\xi ) ) } { \\mu ^ 2 ( \\xi ) } \\right ) = \\xi ^ 2 , \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} \\forall \\ , x , y \\in L _ q , \\| \\mathfrak { s } ( x ) - \\mathfrak { s } ( y ) \\| _ Q = \\| x - y \\| _ q ^ { \\frac { q } { Q } } . \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} b _ { k , m } ( n ) = | B _ { k , m } ( n ) | = | C _ { k , m } ( n ) | = c _ { k , m } ( n ) , \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} u ^ N _ t ( x ) = e ^ { t \\Delta } u ^ N _ 0 ( x ) - \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\langle \\mu _ s ^ N , \\nabla V ^ N ( x - \\cdot ) \\cdot F \\big ( K \\ast u ^ N _ s ( \\cdot ) \\big ) \\rangle \\ d s \\\\ - \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\nabla V ^ N ( x - X _ s ^ { i , N } ) \\cdot d W ^ i _ s . \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} \\langle F _ i , \\Pi _ s \\Pi ( W ) \\rangle = \\langle \\Pi _ s ( F _ i ) , \\Pi ( W ) \\rangle = - \\langle \\Pi ( \\nabla _ { F _ i } F _ s ) , \\Pi ( W ) \\rangle = - \\langle \\Pi ( \\nabla _ { F _ i } F _ s ) , W \\rangle \\ , . \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} \\mbox { $ W = W ^ 1 + W ^ 2 = ( w _ i ^ 1 ) + ( w _ i ^ 2 ) $ w i t h } \\begin{cases} w _ i ^ 1 = 0 \\mbox { f o r } i = m _ 0 + 1 , . . . , m , \\\\ w _ i ^ 2 = 0 \\mbox { f o r } i = 1 , . . . , m _ 0 . \\end{cases} \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} A _ N ( \\sigma ) : = \\frac { 1 } { N } \\log \\int _ { \\mathbb { R } ^ N } \\exp \\left ( \\sigma \\sum _ { i = 1 } ^ N x _ i - H ( x ) \\right ) d x . \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} \\frac 1 2 \\frac { d } { d t } \\| \\Delta _ q \\omega ( t ) \\| _ { L ^ 2 } ^ 2 & = - \\int _ { \\R ^ 2 } \\Delta _ q ( u \\cdot \\nabla \\omega ) \\Delta _ q \\omega ~ d x + \\int _ { \\R ^ 2 } \\partial _ 1 \\Delta _ q \\theta \\Delta _ q \\omega ~ d x \\\\ & \\triangleq N _ 1 + N _ 2 . \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} T _ j ^ { - 1 } \\{ x \\} = \\{ y _ k : \\ , k < D \\} \\ , \\ , \\ , \\ , T _ j ^ { - 1 } \\{ x ' \\} = \\{ y ' _ k : \\ , k < D \\} \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} \\varepsilon ( p ) & : = \\frac { \\beta p } { 2 ( p - 1 ) } , \\\\ \\gamma ( p ) & : = \\frac { 2 \\chi } { p } - \\frac { K \\beta ( p - 2 ) ^ 2 } { 2 p ( p - 1 ) } . \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} \\partial _ { t } ( T ( y _ { 1 } ) - T ( y _ { 2 } ) ) & = - \\int _ { t } ^ { \\infty } \\cos ( ( t - x ) \\sqrt { \\omega } ) F _ { 2 } ( y _ { 1 } - y _ { 2 } ) d x \\\\ & + \\int _ { t } ^ { \\infty } \\cos ( ( t - x ) \\sqrt { \\omega } ) \\mathcal { F } ( \\sqrt { \\cdot } ( F _ { 3 } ( y _ { 1 } ) - F _ { 3 } ( y _ { 2 } ) ( x , \\cdot \\lambda ( x ) ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) d x \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} \\hat { A } _ { i j } ' = \\mathbb { E } [ \\Phi _ i \\Phi _ j A ' ] \\approx \\sum _ { \\ell = 1 } ^ k \\gamma _ { \\ell } \\Phi _ i ( \\mu _ { \\ell } ) \\Phi _ j ( \\mu _ { \\ell } ) E ( \\mu _ { \\ell } ) ^ \\top M ( \\mu _ { \\ell } ) A ( \\mu _ { \\ell } ) \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} I ^ { j , [ k ] } _ { V C } & = J \\left ( \\sqrt { ( j - 1 ) \\left [ J ^ { - 1 } ( I ^ { [ k ] } _ { C V } ) \\right ] ^ 2 + \\left [ J ^ { - 1 } ( I ^ { [ k ] } _ { S V } ) \\right ] ^ 2 } \\right ) , \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} h ^ 1 _ 1 ( \\phi ^ 1 _ 1 ) = ( 1 - \\sin 3 \\phi ^ 1 _ 1 ) ^ { ( A - B ) / 6 } ( 1 + \\sin 3 \\phi ^ 1 _ 1 ) ^ { ( A + B ) / 6 } \\cdot \\frac { 1 } { 2 A - 3 - 2 B \\sin 3 \\phi ^ 1 _ 1 } \\hat { P } ^ { ( \\alpha , \\beta ) } _ { l + 1 } ( \\sin 3 \\phi ^ 1 _ 1 ) \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} | f ( B ( u , v ) ) | & = | f ( \\overline { u } v ) | = f ( | u | ) f ( | v | ) \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} I _ { r r } = \\frac { c _ { b } } { 4 } \\int _ { 0 } ^ { \\infty } \\sin ( t \\xi ) \\xi ^ { 2 } \\chi _ { \\leq 1 } ( r \\xi ) \\left ( - 3 J _ { 1 } ( r \\xi ) + J _ { 3 } ( r \\xi ) \\right ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } d \\xi \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} \\psi ^ { + } ( r , \\xi ) = \\frac { e ^ { i r \\sqrt { \\xi } } } { \\xi ^ { 1 / 4 } } \\sigma ( r \\sqrt { \\xi } , r ) , r \\sqrt { \\xi } \\geq 2 \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} S ' \\rhd S \\ = \\ S ' \\cup ( J ' \\times J ) \\cup S , \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} D ^ { \\alpha } _ { x } I ^ { \\Lambda } f ( t , x ) = I ^ { \\Lambda } D ^ { \\alpha } _ { x } f ( t , x ) , \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} \\int ^ { \\infty } _ { 0 } \\| \\dot { u } _ \\lambda ( t ) \\| ^ 2 d t \\leq \\mathcal { L } _ \\lambda ( 0 ) / \\eta = T V ( J _ { \\lambda } ( u _ 0 ) ) / \\eta < \\infty , \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} \\widetilde { f } ( \\xi ) = ( 2 \\pi ) ^ { - 1 } \\mathcal { F } f \\left ( \\frac { \\xi } { 2 \\pi } \\right ) . \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} F _ k ( z ) = \\frac { A _ k z + C _ k } { B _ k z + D _ k } , \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} & P _ + = \\frac { f ( X _ - ^ * ( y ^ * ) , y ^ * ) } { f ( X _ - ^ * ( y ^ * ) , y ^ * ) + f ( X _ + ^ * ( y ^ * ) , y ^ * ) } , \\\\ & P _ - = \\frac { f ( X _ + ^ * ( y ^ * ) , y ^ * ) } { f ( X _ - ^ * ( y ^ * ) , y ^ * ) + f ( X _ + ^ * ( y ^ * ) , y ^ * ) } . \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} D _ { 1 , \\pm } = \\mathsf { M } \\pm \\tfrac { \\partial \\ ; } { \\partial r } , \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} \\Delta : = & d _ { 1 , 0 } ^ 2 d _ { 1 , - 1 } ^ 2 d _ { - 1 , 1 } ^ 2 d _ { - 1 , 0 } ^ 2 t ^ 8 ( 1 6 t ^ 4 d _ { 1 , 0 } ^ 2 d _ { - 1 , 0 } ^ 2 - 3 2 t ^ 4 d _ { 1 , 0 } d _ { - 1 , 0 } d _ { 1 , - 1 } d _ { - 1 , 1 } \\\\ & + 1 6 t ^ 4 d _ { 1 , - 1 } ^ 2 d _ { - 1 , 1 } ^ 2 - 8 t ^ 2 d _ { 1 , 0 } d _ { - 1 , 0 } - 8 t ^ 2 d _ { 1 , - 1 } d _ { - 1 , 1 } + 1 ) \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} \\begin{cases} x ' = x + \\frac { c _ 2 - c _ 1 } { 2 } t \\\\ t ' = t \\end{cases} \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} \\frac { d \\eta } { d t } = - \\bar { A } \\nabla _ Y \\bar { H } ( \\eta ) , \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} \\delta ( g ^ { i j } f _ { i j } ) & = ( \\delta g ^ { i j } ) f _ { i j } + g ^ { i j } ( \\delta f _ { i j } ) = 2 u h ^ { i j } f _ { i j } + g ^ { i j } \\nabla _ i \\nabla _ j \\dot f \\\\ & = 2 u \\langle h , f \\rangle + \\Delta \\dot f . \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} \\left ( f _ L g _ R - f _ R g _ L \\middle ) \\right | _ { x = y = 0 } = 0 . \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} \\left [ A , B \\right ] _ { \\rho } = \\left [ A , B \\right ] _ 0 + \\sum _ { k = 1 } ^ { \\infty } B _ k ( A , B ) \\rho ^ k , \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} \\xi = \\sum _ { \\star = 1 } ^ \\ell \\ , \\ , \\sum _ { 1 \\leq i _ 1 \\leq \\ldots \\leq i _ \\star \\leq n } \\ , \\ , \\sum _ { n + 1 \\leq i _ { \\star + 1 } \\leq \\ldots \\leq i _ \\ell \\leq n + k } \\xi _ { i _ 1 \\ldots i _ \\star \\ldots i _ \\ell } d x ^ { i _ 1 } \\wedge \\ldots \\wedge d x ^ { i _ \\star } \\wedge \\ldots \\wedge d x ^ { i _ \\ell } . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} \\Psi _ { \\Gamma } ( X ) = X + \\sum _ { 1 / 2 < s _ { j } < 1 } \\frac { X ^ { s _ { j } } } { s _ { j } } + \\sum _ { | t _ { j } | \\leqslant T } \\frac { X ^ { s _ { j } } } { s _ { j } } + O \\left ( \\frac { X } { T } ( \\log X ) ^ { 2 } \\right ) \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} u _ t = ( D ^ \\alpha _ x u ) _ x + f , \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} \\begin{array} { l } ( \\epsilon _ 1 , \\epsilon _ 2 , \\epsilon _ 3 ) \\\\ = \\begin{cases} ( \\min \\{ n - r + k _ 2 , n - 2 r + 2 k _ 1 \\} + \\hat e _ 1 , n - r + k _ 3 , & \\mbox { i f ~ } n - r + k _ 1 > r , \\\\ ~ ~ ~ ~ ~ ~ ~ n - 2 r + k _ 1 + k _ 2 ) & \\\\ ( \\min \\{ k _ 1 , r - k _ 1 + k _ 2 \\} + \\hat e _ 1 , r - k _ 1 + k _ 3 , k _ 2 ) & \\mbox { i f ~ } n - r + k _ 1 \\le r , \\end{cases} \\end{array} \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} f ( U ^ { ( 0 ) } , V ^ { ( 0 ) } ) & = F ( U ^ { ( 0 ) } , V ^ { ( 0 ) } ; \\gamma ) \\ge F ( U ^ { ( p ) } , V ^ { ( p ) } ; \\gamma ) \\\\ & \\ge \\frac { 1 } { 2 } \\sum _ { i j \\in E _ { \\mathrm { s s } } } ( \\alpha _ { i j } ^ { ( p ) } ) ^ 2 + \\frac { 1 } { 2 } \\sum _ { i k \\in E _ { \\mathrm { s a } } } ( \\alpha _ { i k } ^ { ( p ) } ) ^ 2 . \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} u _ l ( r , \\theta ) = ( A r ^ l + B r ^ { l + 2 } ) H _ l ( \\theta ) , \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} \\phi = \\vec u _ a , \\psi = \\vec v _ b , \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} \\alpha = \\left ( \\frac { \\xi _ 1 - \\xi _ 2 } { m _ 1 } + \\frac { \\xi _ 3 - \\xi _ 4 } { m _ 4 } \\middle ) \\right | _ { \\mu = 0 } \\ , . \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} \\tilde A _ { h , \\tilde h } = \\tilde h ^ m A _ h - \\tilde \\omega \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} g ( z ) = \\sum _ { k = 1 } ^ s n _ k ( z - \\alpha _ 1 ) \\cdots \\widehat { ( z - \\alpha _ k ) } \\cdots ( z - \\alpha _ s ) . \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } d w w \\frac { | \\lambda '' ( t + w ) | } { ( \\lambda ( t + w ) ^ { 2 - 2 \\alpha } + w ^ { 2 } ) } & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\frac { 1 } { \\log ^ { ( 2 \\alpha - 2 ) b } ( t ) } \\int _ { 0 } ^ { 1 } \\frac { w d w } { 1 + w ^ { 2 } \\lambda ( t ) ^ { 2 \\alpha - 2 } } \\\\ & \\leq \\frac { C \\log ( \\log ( t ) ) } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } X _ t ^ { s , x } = x + \\int _ s ^ t a ( r , X ^ { s , x } _ r ) \\ , d s + \\int _ s ^ t b ( r , X ^ { s , x } _ r ) \\ , d W _ r , \\\\ Y ^ { s , x } _ t = g ( X ^ { s , x } _ T ) + \\int _ t ^ T f ( r , X ^ { s , x } _ r , Y ^ { s , x } _ r , Z ^ { s , x } _ r ) \\ , d r - \\int _ t ^ T Z ^ { s , x } _ r \\ , d W _ r , \\end{array} \\right . \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} \\phi \\left [ c + c \\frac { L [ \\phi ^ { - 1 } ] ' ( L ) } { \\phi ^ { - 1 } ( L ) } \\right ] = \\phi \\left [ c + c \\frac { \\phi ( s ) } { s \\phi ' ( s ) } \\right ] \\leq \\phi \\left [ c + c \\frac { 1 } { [ \\phi ] _ 1 } \\right ] \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} \\begin{aligned} { F } _ { i } ( t ) & = f _ { 0 , i } + \\int _ 0 ^ t Q ( s , { F } _ { i } ( s ) ) d s \\\\ + & \\int _ 0 ^ t \\left [ a \\left ( \\left \\| \\Lambda { F } _ { i } ( s ) \\right \\| + \\int _ 0 ^ s \\Delta ( \\tau , { F } _ { i } ( \\tau ) ) d \\tau \\right ) - a ( \\left \\| \\Lambda f _ 0 \\right \\| ) \\right ] \\Lambda { F } _ { i } ( s ) d s , t \\geq 0 . \\end{aligned} \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} S [ X ] = \\frac { 1 } { 2 } \\int ( X ^ * G ) _ { \\mu \\nu } \\d X ^ \\mu \\wedge \\star \\d X ^ \\nu + \\int _ { \\Sigma _ 3 } X ^ * H , \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} \\frac { { \\lambda ' } _ { l ' } } { \\lambda _ { l ' } } \\gamma _ { l ' i } \\gamma _ { \\sigma ' \\left ( l ' \\right ) j } = \\left \\{ \\begin{matrix} G ^ { \\left ( l ' \\right ) } _ { 0 ; 0 , \\dots , 1 , \\dots , 0 } ( z ) , & \\mbox { f o r a l l $ i , j = 1 , \\dots , N $ w i t h $ j = \\sigma \\left ( i \\right ) $ , } \\\\ \\quad \\quad \\quad \\quad \\quad 0 , & \\quad \\mbox { f o r a l l $ i , j = 1 , \\dots , N $ w i t h $ j \\neq \\sigma \\left ( i \\right ) $ , } \\end{matrix} \\right . \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} & \\sum _ { \\mu \\equiv j } \\sum _ { \\nu \\equiv k } u ( \\xi - \\mu ) v ( \\eta - \\nu ) \\sigma _ { \\mu , \\nu } ( \\xi , \\eta ) \\\\ & = \\bigg ( \\sum _ { \\mu \\equiv j } \\sum _ { \\nu \\equiv k } \\sigma _ { \\mu , \\nu } ( \\xi , \\eta ) \\bigg ) \\bigg ( \\sum _ { \\mu ^ { \\prime } \\equiv j } u ( \\xi - \\mu ^ { \\prime } ) \\bigg ) \\bigg ( \\sum _ { \\nu ^ { \\prime } \\equiv k } v ( \\eta - \\nu ^ { \\prime } ) \\bigg ) \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} F _ { n } ^ { ( k ) } = 2 ^ { n - 2 } { } 2 \\leq n \\leq k + 1 . \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} T _ H ^ * ( \\lambda ) = \\lambda _ q , \\ \\forall \\lambda \\in M ( G / H ) . \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} | \\tilde { \\Phi } ( \\xi _ 2 ' , \\eta _ 2 ' ) | & : = \\Bigl | \\xi _ 1 ' { \\xi _ 2 ' } ^ 2 + \\eta _ 1 ' { \\eta _ 2 ' } ^ 2 - \\frac { { \\xi _ 1 ' } ^ 3 + { \\eta _ 1 ' } ^ 3 } { 4 } \\Bigr | \\leq 2 ^ 4 A ^ { - 1 } N _ 1 ^ 3 , \\\\ | \\tilde { F } ( \\xi _ 2 ' , \\eta _ 2 ' ) | & : = \\Bigl | \\frac { 3 } { 2 } \\ , \\xi _ 1 ' \\ , \\eta _ 1 ' + 2 \\ , \\xi _ 2 ' \\ , \\eta _ 2 ' \\Bigr | \\leq 2 ^ 4 A ^ { - 1 } N _ 1 ^ 2 , \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ 6 \\phi _ 2 ( \\| \\mathbf { z } _ i - \\mathbf { x } \\| _ 2 ) , \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} \\left \\Vert e ^ { \\gamma \\left \\vert x \\right \\vert ^ { p } } \\tilde { u } _ { k } \\left ( x , 0 \\right ) \\right \\Vert _ { X } = \\left \\Vert e ^ { \\gamma \\left ( \\frac { \\alpha } { \\beta } \\right ) ^ { p / 2 } \\left \\vert x \\right \\vert ^ { p } } \\upsilon \\left ( x , 0 \\right ) \\right \\Vert _ { X } = a _ { 0 } , \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} W = Z \\overline { Z } ^ { t } + \\rm { O } \\left ( 3 \\right ) , \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T \\Phi _ t \\leq \\frac { \\Phi _ 1 } { \\rho } + 2 \\rho \\sigma ^ 2 T + \\frac { L ^ 2 \\gamma ^ 2 } { \\rho ^ 2 } \\sum _ { t = 1 } ^ T \\Delta _ t \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 ^ + } \\sup _ { x \\in [ 0 , l ] } | J ^ { 1 - \\alpha } f '' ( x ) - J ^ 1 f '' ( x ) | = 0 \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} j = \\dfrac { A ^ 3 ( A ^ 3 - 2 4 B ) ^ 3 } { B ^ 3 ( A ^ 3 - 2 7 B ) } \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} x _ { 3 } ^ { 3 } - \\tfrac { 3 } { 2 } x _ { 3 } ( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } ) + \\tfrac { 1 } { \\sqrt { 2 } } \\cos \\theta \\left ( x _ { 1 } ^ { 3 } - 3 x _ { 1 } x _ { 2 } ^ { 2 } \\right ) + \\tfrac { 1 } { \\sqrt { 2 } } \\sin \\theta \\left ( x _ { 2 } ^ { 3 } - 3 x _ { 2 } x _ { 1 } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} \\psi _ a ( x _ 1 , x _ 2 ) = \\mathcal { R } _ a ( x _ 2 ) \\exp \\left [ \\frac { i x _ 1 } { \\mu } \\left ( a - \\frac { \\eta } { 2 \\mu } x _ 2 \\right ) \\right ] , \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} | ( \\phi _ h ^ n , 1 ) _ { \\mathcal { T } _ h } | = \\frac { 1 } { \\epsilon } | \\big ( f ^ n ( u _ h ^ n ) , 1 \\big ) _ { \\mathcal { T } _ h } | \\leq \\frac { 1 } { \\epsilon } ( | | u _ h ^ n | | _ { L ^ 3 } ^ 3 + | | u _ h ^ n | | _ { L ^ 1 } ) \\leq \\frac { C } { \\epsilon } , \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { F _ n ^ { ( 1 ) } ( z ) } { F _ n ^ { ( 2 ) } ( z ) } = R _ 0 ( z ) ( z - \\l _ 0 ) ^ { k _ 2 - k _ 1 } \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} q _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( x ) = \\frac { 1 } { \\pi } \\Im \\frac { 1 } { 1 - \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) } = \\frac { 1 } { \\pi } \\frac { \\Im \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) } { | 1 - \\eta _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) | ^ { 2 } } , \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} \\bar { \\nabla } _ \\sigma ( f _ i \\otimes q ^ i ) = \\sigma ( f _ i ) \\otimes q ^ i + f _ i \\otimes \\nabla _ { X _ * ( \\sigma ) } q ^ i , \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} \\varphi ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) = ( F _ 1 , F _ 2 , F _ 3 , F _ 4 ) . \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} & | - \\int _ { t } ^ { t + \\frac { 1 } { 2 } } \\frac { d s } { ( s - t ) } \\int _ { 0 } ^ { s - t } \\rho d \\rho \\frac { \\lambda '' ( s ) } { r } \\partial _ { r } \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } \\right ) | \\\\ & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} | | v ( t ) | | _ { L ^ { 2 } ( R d R ) } = \\lambda ( t ) | | y ( t ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} & a _ k ( n - 1 ) - a _ k ( n - 4 ) + a _ k ( n - 9 ) - a _ k ( n - 1 6 ) + \\cdots \\\\ & \\qquad \\ , = a _ k ( n - 2 ) - a _ k ( n - 8 ) + a _ k ( n - 1 8 ) - a _ k ( n - 3 2 ) + \\cdots . \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} \\sum \\limits _ { \\textbf { { x } } \\in \\mathbb { S } } p _ n ( \\textbf { { x } } ) s ( \\textbf { { x } } ; \\theta ) = 0 , \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} ( - h ^ { 2 } + & 2 h + \\mu ) ( h - 4 ) ^ { n } = - h ^ { n + 2 } + 2 ( 2 n + 1 ) h ^ { n + 1 } \\\\ & + \\sum _ { k = 0 } ^ { n - 2 } \\bigg ( \\mu { n \\choose k } ( - 4 ) ^ { k } + 2 { n \\choose k + 1 } ( - 4 ) ^ { k + 1 } \\\\ & - { n \\choose k + 2 } ( - 4 ) ^ { k + 2 } \\bigg ) h ^ { n - k } \\\\ & + \\big ( \\mu n ( - 4 ) ^ { n - 1 } + 2 ( - 4 ) ^ { n } \\big ) h + \\mu ( - 4 ) ^ { n } , \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} & c ^ { ( k ) } = p ^ { ( 1 ) } m ( b ) s ^ { ( k ) } , \\ k = k _ 1 , \\ldots , k _ 2 , \\\\ & d ^ { ( k ) } = p ^ { ( k ) } m ( b ) s ^ { ( k _ 2 ) } , \\ k = 1 , \\ldots , k _ 1 . \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } N _ { 2 } ( f ) ( t , r ) | \\leq C \\begin{cases} \\frac { 1 } { t ^ { 5 } \\log ^ { b } ( t ) ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { \\log ( r ) } { r ^ { 3 } t ^ { 2 } | t - r | \\log ^ { 3 N + 2 b } ( t ) } + \\frac { \\log ^ { 2 } ( r ) } { r ^ { 2 } ( t - r ) ^ { 2 } t ^ { 2 } \\log ^ { 3 N + b } ( t ) } + \\frac { \\log ^ { 3 } ( r ) } { t ^ { 2 } r ^ { 2 } \\log ( t ) | t - r | ^ { 3 } } + \\frac { \\log ^ { 5 / 2 } ( r ) } { r ^ { 9 / 4 } ( t - r ) ^ { 4 } } , \\frac { t } { 2 } \\leq r < t \\end{cases} \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} F ( X ^ I ) = \\sum _ { i , j , k } N _ { i j k } \\frac { X ^ i X ^ j X ^ k } { X ^ 0 } , \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} \\rho _ \\tau ( x , y ) = \\tau ( x ) \\mu ( y ) - ( - 1 ) ^ { \\bar x \\bar y } \\tau ( y ) \\mu ( x ) , \\forall x , y \\in \\mathcal H ( \\mathfrak g ) . \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} \\mathfrak { a } = \\Theta _ { n } ^ { \\left ( j \\right ) } \\left ( x \\right ) \\nabla + \\Lambda _ { n } ^ { \\left ( j \\right ) } \\left ( x ; 2 , 1 \\right ) I , \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} b ( n ) = \\sum _ { i = 1 } ^ { g } b \\left ( n - i \\right ) + \\left ( 2 ^ { k + 1 } - 1 \\right ) H _ { n - ( g + k + 1 ) } . \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} \\eta ^ \\rightarrow ( U _ x ) = \\eta ^ \\rightarrow \\left ( \\bigcap _ { \\substack { U \\in \\tau \\\\ x \\in U } } U \\right ) \\subseteq \\bigcap _ { \\substack { U \\in \\tau \\\\ x \\in U } } \\eta ^ \\rightarrow ( U ) = U _ { \\eta ( x ) } . \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} \\int _ { r , \\theta } h \\varphi '' | \\psi u | ^ 2 = - \\real \\int _ { r , \\theta } 2 \\varphi ' h ( \\psi u ) ' \\overline { \\psi u } . \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} L _ { } ( y , t ) : = \\mathbf { 1 } _ { ( - \\infty , 0 ] } ( y \\ , t ) , \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} H _ { 2 k + 3 } = H _ { 2 k + 2 } + \\dots + H _ { k + 3 } + 4 H _ { k + 1 } . \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} \\sum _ { i + j = n } T _ i ( u ) T _ j ( v ) = \\sum _ { i + j = n } T _ i ( u \\cdot T _ j ( v ) + T _ j ( u ) \\cdot v ) + \\sum _ { i + j + k = n } T _ i ( H ( T _ j ( u ) , T _ k ( v ) ) ) . \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} \\widetilde U _ 1 : = \\left \\{ \\xi \\in \\mathfrak M ( H ^ \\infty ) \\ , : \\ , \\hat h ( \\xi ) < \\mbox { $ \\frac 3 4 $ } \\right \\} . \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} i ^ ! _ Y f _ * ( - ) = e ( N _ { ( Y ' ) ^ T } Y ' ) \\cdot f _ { T * } \\left ( e ( N _ { ( X ' ) ^ T } X ' ) ^ { - 1 } \\cdot i ^ ! _ X ( - ) \\right ) . \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} \\Sigma _ G ( G / N _ 1 ) = \\lbrace \\alpha _ 2 , \\alpha _ 3 \\rbrace \\ \\ \\Sigma _ G ( G / N _ 2 ) = \\lbrace \\alpha _ 1 , \\alpha _ 2 \\rbrace . \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} [ t ^ n ] \\left ( f \\left ( g ( t ) \\right ) \\right ) & = \\frac { 1 } { n ! } \\sum _ { k = 1 } ^ n k ! f _ k \\cdot B _ { n , k } \\left ( 1 ! g _ 1 , 2 ! g _ 2 \\ldots , ( n + 1 - k ) ! g _ { n + 1 - k } \\right ) . \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } \\left ( \\frac { - 4 \\lambda ' ( t ) ^ { 2 } } { \\lambda ( t ) ^ { 2 } } \\int _ { 0 } ^ { \\infty } \\chi _ { \\geq 1 } ( \\frac { 2 R \\lambda ( t ) } { \\log ^ { N } ( t ) } ) \\frac { R ^ { 2 } \\phi _ { 0 } ( R ) d R } { ( R ^ { 2 } + 1 ) ^ { 2 } } \\right ) | \\leq \\frac { C } { t ^ { 4 } \\log ^ { 2 + 2 b + 2 N } ( t ) } \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} - \\Delta u + q ( x ) u ^ m = 0 \\Omega , \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} p \\cdot \\omega _ H = \\pi ^ * c _ 1 ( L _ H , h ^ { L _ H } ) . \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\omega \\to \\sum _ { j = 0 } ^ { n ' - 1 } T _ { \\omega _ j } ( 1 , \\tau _ 0 ) \\l _ 0 ^ { j } . \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} | | \\mathcal { F } ( \\sqrt { \\cdot } L _ { 1 } ( u ) ( t , \\cdot \\lambda ( t ) ) ) ( \\omega \\lambda ( t ) ^ { 2 } ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) d \\omega ) } ^ { 2 } & = \\frac { 1 } { \\lambda ( t ) ^ { 2 } } \\int _ { 0 } ^ { \\infty } R ( L _ { 1 } ( u ) ( t , R \\lambda ( t ) ) ) ^ { 2 } d R \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} K _ { ( n ) , z } ( w ) = e ^ { \\overline { z } w } L _ { n - 1 } ( | w - z | ^ 2 ) . \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} ( D ^ { 2 } \\Gamma ) ( \\zeta ) = \\gamma ^ { \\prime \\prime } ( | \\zeta | ) \\frac { \\zeta } { | \\zeta | } \\otimes \\frac { \\zeta } { | \\zeta | } + \\frac { \\gamma ^ { \\prime } ( | \\zeta | ) } { | \\zeta | } \\left [ I _ { \\R ^ { 2 } } - \\frac { \\zeta } { | \\zeta | } \\otimes \\frac { \\zeta } { | \\zeta | } \\right ] , \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} \\sum _ { g \\in \\Gamma } g \\cdot h = 1 , \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} \\ss _ 1 = ( \\mu , ~ \\sigma _ + ^ 2 - 1 ) ^ \\tau ; ~ ~ \\ss _ 2 = ( \\beta _ 0 ^ 2 + \\beta _ 1 ^ 2 + ( \\eta ^ 2 - 1 ) , ~ \\beta _ 1 ^ 2 , ~ \\beta _ 0 \\beta _ 1 , ~ \\beta _ 0 ^ 4 ) ^ \\tau . \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} p ^ C _ { C \\otimes Y } = l _ Y ^ * l _ Y , p ^ A _ { X \\otimes A } = r _ X ^ * r _ X , ( p ^ C _ { Z \\otimes Y } \\otimes 1 _ X ) ( 1 _ { Z } \\otimes p ^ B _ { Y \\otimes X } ) = ( 1 _ { Z } \\otimes p ^ B _ { Y \\otimes X } ) ( p ^ C _ { Z \\otimes Y } \\otimes 1 _ X ) . \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} \\pi \\circ _ 1 \\pi _ { N + 1 } - \\pi \\circ _ 2 \\pi _ { N + 1 } + \\pi _ { N + 1 } \\circ _ 1 \\pi - \\pi _ { N + 1 } \\circ _ 2 \\pi = - \\sum _ { i + j = N + 1 , i , j \\geq 1 } \\big ( \\pi _ i \\circ _ 1 \\pi _ j - \\pi _ i \\circ _ 2 \\pi _ j \\big ) . \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} \\varphi ( s ) = ( - 1 ) ^ { ( h - h _ { 0 } ) / 2 } \\left ( \\frac { \\Gamma ( 1 - s ) } { \\Gamma ( s ) } \\right ) ^ { h } \\left ( \\frac { A } { \\pi ^ { h } } \\right ) ^ { 1 - 2 s } \\sideset { } { ^ { \\star } } \\prod _ { \\psi } \\frac { L ( 2 - 2 s , \\overline { \\psi } ) } { L ( 2 s , \\psi ) } . \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} s _ { k + 1 } = s _ k - \\frac { F ( s _ k ) } { F _ s ( s _ k ) } , ~ ~ ~ k = 0 , 1 , 2 , \\ldots , m , \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} \\| \\mathcal { T } [ F _ 1 ] ( t ) - \\mathcal { T } [ F _ 2 ] ( t ) \\| _ { L ^ \\infty } & = \\| \\mathcal { T } [ F _ 1 \\ ! - \\ ! F _ 2 ] ( t ) \\| _ { L ^ \\infty } \\\\ [ 5 p t ] & \\leq 0 + T _ 1 \\big | Q ^ \\varepsilon [ F _ 1 \\ ! - \\ ! F _ 2 ] \\big | \\\\ [ 1 p t ] & \\leq \\frac { 4 T _ 1 } { \\varepsilon ^ 2 } \\sup _ { t \\in [ 0 , T _ 1 ] } \\| F _ 1 ( t ) - F _ 2 ( t ) \\| _ { L ^ \\infty } , \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} ( \\Gamma ^ { \\sim } _ \\sigma ( \\gamma ) ( X , Y ) ) ( t ) = \\Gamma _ { \\sigma ( t ) } ( \\gamma ( t ) ) ( X ( t ) , Y ( t ) ) . \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} F ( \\tau ) = p ^ 0 + p ^ 1 \\tau + q _ 1 \\tau ^ 2 - \\frac { q _ 0 } { 3 } \\tau ^ 3 . \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} L _ { \\Sigma } : = \\Delta + | A _ { \\Sigma } | ^ 2 + R i c _ M ( \\nu , \\nu ) \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} ( - P ) _ { B } ^ { \\alpha } \\phi = \\frac { \\sin \\alpha \\pi } { \\pi } \\int _ { 0 } ^ { \\infty } \\lambda ^ { \\alpha - 1 } ( \\lambda - P ) ^ { - 1 } ( - P ) \\phi \\ , d \\lambda . \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} \\partial Q = ( \\partial B _ R \\cap V ) \\oplus \\{ r e : 0 \\leq r \\leq R \\} \\cup ( \\bar { B } _ R \\cap V ) \\cup ( \\bar { B } _ R \\cap V ) \\oplus \\{ R e \\} , \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} \\hat X & = \\beta _ 0 + \\beta _ 1 \\left ( - \\partial _ \\mu \\partial ^ \\mu \\right ) + \\beta _ 2 \\left ( - \\partial _ \\mu \\partial ^ \\mu \\right ) ^ 2 + \\ldots = \\sum _ { k = 0 } ^ \\infty \\beta _ k \\left ( - \\partial _ \\mu \\partial ^ \\mu \\right ) ^ k . \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} m _ p = \\max \\{ p - 1 , 1 \\} . \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} \\det ( \\Delta _ g ) : = e ^ { - \\zeta ' _ { g } ( 0 ) } . \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} [ h , L _ 1 ] = L _ 1 , [ h , f ] = - f , [ L _ 1 , f ] = 2 h . \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} f _ 1 & = x ^ 9 - 3 x ^ 7 - 1 2 x ^ 6 + 3 x ^ 5 + 6 x ^ 4 - 6 1 x ^ 3 - 1 2 x ^ 2 + 2 4 x - 6 2 , \\\\ f _ 2 & = x ^ 9 - 3 x ^ 7 - 1 8 x ^ 6 + 3 x ^ 5 + 1 8 x ^ 4 - 1 0 9 x ^ 3 - 1 8 x ^ 2 - 2 1 4 \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} \\dot { w } = - \\partial \\Phi ( w ) , \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} & | - \\int _ { t + \\frac { 1 } { 2 } } ^ { \\infty } \\frac { d s } { ( s - t ) } \\int _ { 0 } ^ { s - t } \\rho d \\rho \\frac { \\lambda '' ( s ) } { r } \\partial _ { r } \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } \\right ) | \\\\ & \\leq \\int _ { t + \\frac { 1 } { 2 } } ^ { \\infty } \\frac { d s } { ( s - t ) } | \\lambda '' ( s ) | \\cdot 2 \\\\ & \\leq C \\sup _ { x \\geq t } \\left ( x | \\lambda '' ( x ) | \\right ) \\frac { \\log ( t ) } { t } \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} \\sqrt { \\omega } \\lambda ( t ) ( T ( y _ { 1 } ) - T ( y _ { 2 } ) ) & = - \\int _ { t } ^ { \\infty } \\lambda ( t ) \\sin ( ( t - x ) \\sqrt { \\omega } ) F _ { 2 } ( y _ { 1 } - y _ { 2 } ) d x \\\\ & + \\lambda ( t ) \\int _ { t } ^ { \\infty } \\sin ( ( t - x ) \\sqrt { \\omega } ) \\mathcal { F } ( \\sqrt { \\cdot } ( F _ { 3 } ( y _ { 1 } ) - F _ { 3 } ( y _ { 2 } ) ( x , \\cdot \\lambda ( x ) ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) d x \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} I _ { r } = \\frac { c _ { b } } { 2 } \\int _ { 0 } ^ { \\infty } d \\xi \\sin ( t \\xi ) \\xi \\chi _ { \\leq 1 } ( r \\xi ) \\left ( J _ { 0 } ( r \\xi ) - J _ { 2 } ( r \\xi ) \\right ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} & | \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { t } ^ { \\infty } d x \\lambda ''' ( x ) \\left ( K _ { 1 } ( x - t , \\lambda ( t ) ) - \\frac { \\lambda ( t ) ^ { 2 } } { 4 ( 1 + x - t ) } \\right ) | \\\\ & \\leq \\frac { C } { \\lambda ( t ) ^ { 2 } } \\sup _ { x \\geq t } ( | \\lambda ''' ( x ) | ) \\int _ { t } ^ { \\infty } | K _ { 1 } ( x - t , \\lambda ( t ) ) - \\frac { \\lambda ( t ) ^ { 2 } } { 4 ( 1 + x - t ) } | d x \\\\ & \\leq C \\sup _ { x \\geq t } | \\lambda ''' ( x ) | \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} 2 m - l + k + r & = 2 m - l + k + ( n - k - m ) \\\\ & = m + n - l \\geq n . \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} | f _ 1 ( z ) | = | e ^ { - z ^ 2 } | \\leq e ^ { W ^ 2 } e ^ { - { \\rm R e } ( z ) ^ 2 } \\leq C ^ { f _ 1 } _ m ( 1 + | z | ) ^ m , \\mbox { f o r a l l } ~ m \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { r / 2 } d \\rho \\frac { ( \\rho ^ { 2 } + r ^ { 2 } + 1 ) } { ( ( \\rho ^ { 2 } - r ^ { 2 } + 1 ) ^ { 2 } + 4 r ^ { 2 } ) ^ { 3 / 2 } } \\frac { 1 } { \\sqrt { 1 + \\rho ^ { 2 } } } \\leq \\frac { C } { r ^ { 4 } } \\sinh ^ { - 1 } \\left ( \\frac { r } { 2 } \\right ) \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} \\zeta ( x , y ) = \\begin{cases} - 1 & x \\in E , y \\in E ^ c \\\\ 1 & x \\in E ^ c , y \\in E . \\end{cases} \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} \\sum _ { O \\in \\O } \\| f _ { O } \\| _ 2 ^ { 2 } \\lesssim _ { \\varepsilon } \\Big ( \\prod _ { i = 1 } ^ { m + 1 } D _ { i } ^ { 1 + \\delta } \\Big ) R ^ { O ( \\varepsilon _ { \\circ } ) } \\| f \\| _ { L ^ 2 ( B ^ { n - 1 } ) } ^ { 2 } . \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} Y _ { i t } = \\sum _ { k = 1 } ^ K \\beta _ k X _ { k i t } + \\sum _ { j = 1 } ^ { r _ N } \\lambda _ { i j } f _ { t j } \\delta _ { j } + E _ { i t } , , \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} c ( \\phi _ i ^ { ( k ) } ) ' ( x ) = - f _ i ( \\Phi ^ { ( k ) } ( x ) ) \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} T _ { \\frac { Q - \\epsilon } { 2 } } ( a ) = \\delta , \\ ; \\ ; \\ ; U _ { \\frac { Q - \\epsilon } { 2 } - 1 } ( a ) = 0 \\ ; \\ ; m o d \\ ; \\ ; Q , \\ ; \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} H _ { n } ^ { S K } ( \\sigma ) : = \\frac { 1 } { \\sqrt { n } } \\langle \\sigma , A \\sigma \\rangle = \\frac { 1 } { \\sqrt { n } } \\sum _ { i , j } A _ { i , j } \\sigma _ { i } \\sigma _ { j } = \\frac { 2 } { \\sqrt { n } } \\sum _ { 1 \\le i < j \\le n } A _ { i , j } \\sigma _ { i } \\sigma _ { j } . \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} { \\rm m a x b a n d } ( \\Psi , \\gamma ) = \\max \\{ \\gamma _ i + { \\rm n r } ( \\Psi ) _ i : i \\in [ \\ell ] \\} , \\ , { \\rm n r } ( \\Psi ) _ i : = \\big | \\big \\{ j \\in \\{ i + 1 , \\dots , \\ell \\} : ( i , j ) \\notin \\Psi \\big \\} \\big | . \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} \\big ( h \\eta - ( h ' \\psi ) ' \\big ) ^ { \\wedge } = 0 \\ h \\eta - ( h ' \\psi ) ' = 0 \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} [ s _ 1 \\oplus \\beta _ 1 , s _ 2 \\oplus \\beta _ 2 ] _ A : = [ s _ 1 , s _ 2 ] _ { T \\Sigma } \\oplus \\Big ( \\bar { \\nabla } _ { s _ 1 } \\beta _ 2 - \\bar { \\nabla } _ { s _ 2 } \\beta _ 1 - T _ { \\bar { \\nabla } } ( \\beta _ 1 , \\beta _ 2 ) \\Big ) , \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} ( q ; q ) _ \\infty & ( - q ^ m ; q ^ m ) _ \\infty \\sum _ { n = 0 } ^ \\infty a _ k ( n ) q ^ n \\\\ & = ( - q ^ { m } ; q ^ { m } ) _ \\infty ( q ^ { k } ; q ^ { k } ) _ \\infty \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { k n } + 2 q ^ { 2 k n } + \\dots + ( k - 1 ) q ^ { ( k - 1 ) k n } } { 1 + q ^ { k n } + q ^ { 2 k n } + \\dots + q ^ { ( k - 1 ) k n } } , \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} A \\cdot _ G B = C , \\ , \\ , \\ , \\ , \\ , \\ , \\ , ( A + \\alpha ) \\cdot _ G ( B + \\beta ) = D , \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} [ V ^ b ( D ) ] - [ e _ 1 ] = \\operatorname { I n d } _ \\infty ( D ) \\ ; \\ ; \\ ; \\ ; K _ 0 ( \\mathcal { A } ^ { \\infty , \\delta } _ G ( M ) ) = K _ 0 ( C ^ * ( M _ 0 \\subset M ) ^ G ) \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { p ^ k } \\log \\left [ \\xi ^ { \\sigma _ k } - \\xi ^ { \\sigma _ { k + 1 } } \\right ] = \\tau \\log \\xi \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { p ^ k } = \\frac { \\tau \\log \\xi } { p - 1 } \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} & | - \\lambda ( t ) \\langle \\left ( \\frac { \\cos ( 2 Q _ { \\frac { 1 } { \\lambda ( t ) } } ) - 1 } { r ^ { 2 } } \\right ) v _ { 4 } \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) \\vert _ { r = R \\lambda ( t ) } , \\phi _ { 0 } \\rangle | \\leq \\frac { C } { \\lambda ( t ) } \\int _ { 0 } ^ { \\frac { t } { 2 \\lambda ( t ) } } \\frac { R ^ { 2 } } { ( 1 + R ^ { 2 } ) ^ { 3 } } | v _ { 4 } ( t , R \\lambda ( t ) ) | d R \\\\ & \\leq \\frac { C } { t ^ { 2 } \\log ^ { 3 b + 2 N - 1 } ( t ) } \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow 0 } \\frac { 1 } { t } \\left ( p _ t \\ast u _ \\lambda - u _ \\lambda \\right ) = \\lambda u _ \\lambda . \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} \\sum _ { i + j = N + 1 } \\pi _ i \\circ _ 1 \\pi _ j = \\sum _ { i + j = N + 1 } \\pi _ i \\circ _ 2 \\pi _ j , , \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} f ' * h _ { \\alpha } ( t ) = \\frac { t ^ { - \\alpha } } { \\Gamma ( 1 - \\alpha ) } \\int _ 0 ^ t f ' ( y ) \\left ( 1 - \\frac { y } { t } \\right ) ^ { - \\alpha } d y = t ^ { - \\alpha } f ' \\star g _ { \\alpha } ( t ) \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} \\mathbb { E } _ \\theta [ f ( \\textbf { X } ) ] = \\bar { f } . \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} \\begin{aligned} \\limsup _ { j \\nearrow + \\infty } \\int _ B \\Big ( | u _ j ( x ) | ^ { \\ 2 m s + | x | ^ \\alpha } & - ( \\eta _ \\delta | u _ j ( x ) | ) ^ { \\ 2 m s + | x | ^ \\alpha } \\Big ) d x \\\\ & \\leq \\lim _ { j \\nearrow + \\infty } \\int _ { B \\setminus B _ { \\delta / 2 } } | u _ j ( x ) | ^ { \\ 2 m s + | x | ^ \\alpha } d x = 0 . \\end{aligned} \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} \\rho _ 1 : E \\wedge E \\rightarrow D e r ( A ) , \\rho _ 1 ( ( x , a ) , ( y , b ) ) = \\rho ( x , y ) . \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} \\frac { d } { d t } \\det ( \\mathbf { g } ) = \\det ( \\mathbf { g } ) \\ ; \\mathrm { t r } \\left ( \\mathbf { g } ^ { - 1 } \\frac { d \\mathbf { g } } { d t } \\right ) = \\det ( \\mathbf { g } ) ( - 2 u g ^ { i j } h _ { i j } ) = - 4 H u \\det ( \\mathbf { g } ) . \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} { } ^ b \\widetilde { A } ^ \\epsilon ( M ) : = ( \\mathcal { A } ( G ) \\hat { \\otimes } \\ , { } ^ b \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } \\ , , \\widetilde { A } ^ \\epsilon ( M ) : = ( \\mathcal { A } ( G ) \\hat { \\otimes } \\ , \\Psi ^ { - \\infty , \\epsilon } ( S ) ) ^ { K \\times K } \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { r l } d x _ { t } & = ( F _ { t } x _ { t } + f _ { t } + \\theta _ { t } ) d t + d w _ { t } ^ { \\theta } , \\\\ x ( 0 ) & = x _ { 0 } , \\\\ d m _ { t } & = ( G _ { t } { x } _ { t } + g _ { t } ) d t + d v _ { t } , \\\\ m ( 0 ) & = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} \\max _ { 0 \\le t \\le 1 } | f ( \\gamma ( t ) ) | = | f ( \\beta ) | . \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} \\| a \\| _ { l _ { 1 , g } } = \\sum \\limits _ { k \\in \\mathbb Z ^ c } g ( k ) \\| a _ k \\| < \\infty . \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} ( k _ { 1 } * k _ { 2 } ) ( x , y ) : = \\int _ { G } \\int _ { M } k _ { 1 } ( g x , z ) c ( z ) k _ { 2 } ( z , g y ) d { \\rm v o l } ( z ) d \\mu ( g ) . \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} \\mathcal { L } _ L ^ { \\lambda } f : = \\sum _ { \\ell = 0 } ^ L \\mathcal { S } _ { \\lambda \\mu _ { \\ell } } \\left ( \\sum _ { j = 1 } ^ N w _ j f ( x _ j ) p _ { \\ell } ( x _ j ) \\right ) p _ { \\ell } . \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} \\nu ^ { \\boxtimes k } = \\underbrace { \\nu \\boxtimes \\cdots \\boxtimes \\nu } _ { k } \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} c v _ i '' ( 0 ^ - ) = - \\sum _ { j = 1 } ^ m \\partial _ j f _ i ( \\mathbf { 0 } ) v _ j ' ( 0 ^ - ) > 0 . \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} \\bar { E } \\dot { \\bar { v } } ( t ) & = \\bar { f } ( \\bar { v } ( t ) ) = V ^ \\top \\hat { f } ( V \\bar { v } ( t ) ) \\\\ [ 1 e x ] \\bar { w } ( t ) & = \\bar { g } ( \\bar { v } ( t ) ) = \\hat { g } ( V \\bar { v } ( t ) ) \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} B ( f , B ( g , h ) ) + B ( g , B ( h , f ) ) + B ( h , B ( f , g ) ) = 0 . \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} \\tau ^ M _ \\varphi ( A _ 0 , \\ldots , A _ k ) : = { \\rm T r } _ \\chi ( f ^ \\varphi _ 0 \\cdot A _ 0 \\cdots f ^ \\varphi _ k \\cdot A _ k ) , \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq | j - i | \\leq R } | M _ { i j } | + \\delta \\leq M _ { i i } = 1 . \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} & ( i _ 1 , \\dots , i _ n ) \\prec ( i ' _ 1 , \\dots , i ' _ n ) \\\\ & \\Longleftrightarrow \\exists r ~ \\textrm { s . t . } ~ i _ j = i ' _ j \\ ( j = r + 1 , \\ldots , n ) , \\ i _ r = 0 , \\ i ' _ r = 1 . \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} \\int _ { G / H } \\psi ( x H ) d \\mu _ \\varphi ( x H ) = \\int _ { G / H } \\psi ( x H ) \\varphi ( x H ) d \\mu ( x H ) , \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} & \\eta _ { \\Gamma } ( t _ { 0 } , t ; a , b ) : = \\# \\big \\{ \\pi ( t _ { 0 } + t ) : \\pi \\in \\Gamma , \\varsigma _ { \\pi } \\leq t _ { 0 } , \\pi ( t _ { 0 } ) \\in [ a , b ] \\big \\} , \\\\ & \\hat { \\eta } _ { \\Gamma } ( t _ { 0 } , t ; a , b ) : = \\# \\big \\{ \\pi ( t _ { 0 } + t ) : \\pi \\in \\Gamma , \\varsigma _ { \\pi } \\leq t _ { 0 } , \\pi ( t _ { 0 } + t ) \\in [ a , b ] \\big \\} . \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} a = \\sum \\limits _ { n \\in \\mathbb Z ^ c } a _ n \\varepsilon ^ { n } , \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} f ^ { S _ k } ( x ) + \\sum _ { j = 0 } ^ { k - 1 } C ( S _ j , S _ { j + 1 } ) = f ^ { \\tilde { S } _ \\ell } ( x ) + \\sum _ { j = 0 } ^ { \\ell - 1 } C ( \\tilde { S } _ j , \\tilde { S } _ { j + 1 } ) . \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} \\Gamma _ { i j } ^ k = 1 / 2 \\sum _ { r = 1 } ^ 2 \\left ( \\frac { \\partial g _ { i r } } { \\partial s _ j } + \\frac { \\partial g _ { j r } } { \\partial s _ i } - \\frac { \\partial g _ { i j } } { \\partial s _ r } \\right ) g ^ { k r } , \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} Z _ { t - } + \\tilde \\rho ( Z _ { t - } , y ) = G ( G ^ { - 1 } ( Z _ { t - } ) + \\rho ( X _ { t - } , y ) ) . \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} p _ { k + 1 } = t _ { k } p _ k - p _ { k - 1 } \\\\ q _ { k + 1 } = t _ { k } q _ k - q _ { k - 1 } . \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} \\begin{array} { l } \\mbox { e q u a t i o n ~ } \\eqref { 0 9 _ 0 7 _ 2 0 2 0 _ e q _ 1 2 } = g _ 0 ( x ) ( x - 1 ) ^ { n - k _ 1 } \\hat f _ { 0 0 } ( x ) , \\\\ \\mbox { e q u a t i o n ~ } \\eqref { 0 9 _ 0 7 _ 2 0 2 0 _ e q _ 1 3 } = g _ 0 ( x ) ( x - 1 ) ^ { n - r } ( u \\tilde f _ { 0 1 } ( x ) ) , \\\\ \\mbox { e q u a t i o n ~ } \\eqref { 0 9 _ 0 6 _ 2 0 2 0 _ e q _ 2 } = g _ 1 ( x ) ( x - 1 ) ^ { n - r _ 1 } \\tilde f _ { 1 0 } ( x ) , \\\\ \\mbox { e q u a t i o n ~ } \\eqref { 0 9 _ 0 6 _ 2 0 2 0 _ e q _ 1 } = g _ 0 ( x ) ( u ^ 2 f _ { 0 2 } ( x ) ) , \\\\ \\mbox { e q u a t i o n ~ } \\eqref { 0 9 _ 0 6 _ 2 0 2 0 _ e q _ 3 } = g _ 1 ( x ) ( u f _ { 1 1 } ( x ) ) , \\end{array} \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} \\frac { C _ 1 } { | x | ^ { { \\gamma } } + 1 } \\leq \\sum _ { i = 1 } ^ { m _ 0 } \\mu _ i J _ i ( x ) \\leq \\frac { C _ 2 } { | x | ^ { { \\gamma } } + 1 } \\ \\mbox { f o r } \\ x \\in \\R \\ \\mbox { a n d s o m e } { \\gamma } \\in ( 1 , 2 ] . \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} ( \\varphi _ { k i } ' ) ^ { - 1 } ( z ) & = \\varphi _ { k i } ^ { - 1 } ( z ) \\cap Z ' _ i = \\bigcap _ { n = i } ^ \\infty \\varphi _ { k i } ^ { - 1 } ( z ) \\cap Y _ n = \\bigcap _ { n = i } ^ \\infty T _ n \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} \\begin{aligned} a _ { 2 L } & > 0 , & a _ { 2 R } & > 0 , \\\\ b _ { 0 L } & > 0 , & b _ { 0 R } & < 0 . \\end{aligned} \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} \\left | { \\int \\limits _ { t } ^ T E \\left [ f ( s , y ( s , x + W _ { s - t } ) ) \\right ] d s - \\sum _ { j = 1 } ^ Q w _ { n , j } E \\left [ f ( t _ j , y ( t _ j , x + W _ { t _ j - t } ) ) \\right ] } \\right | & = \\dfrac { [ Q ! ] ^ 4 ( T - t ) ^ { 2 Q + 1 } } { ( 2 Q + 1 ) [ ( 2 Q ) ! ] ^ 3 } \\left | F ^ { 2 Q } ( \\xi ) \\right | \\\\ & \\leq \\dfrac { [ Q ! ] ^ 4 ( T - t ) ^ { 2 Q + 1 } } { ( 2 Q + 1 ) [ ( 2 Q ) ! ] ^ 3 } C _ d = \\epsilon _ n ( t ) \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} | ( v _ { 3 } ^ { \\lambda _ { 1 } } - v _ { 3 } ^ { \\lambda _ { 2 } } ) ( t , r ) | & \\leq \\frac { C r | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\sqrt { \\log ( \\log ( t ) ) } \\log ^ { b } ( t ) } \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} f ( z ) : = e ^ { - z ^ 2 / 2 } \\frac { ( 1 - e ^ { - z ^ 2 } ) } { z ^ 2 } . \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{gather*} Z _ m ' = \\{ ( x _ { i + 1 } , w ( x _ i ) ) \\mid i \\in [ 1 , k - 1 ] \\} \\sqcup \\{ ( x _ 1 , S ( y ) ) \\} \\sqcup \\{ ( y , w ( x _ k ) ) \\} , \\\\ Z _ m '' = \\{ ( x _ i , w ( x _ { i } ) ) \\mid i \\in [ 1 , k ] \\} , Z _ m ''' = \\{ ( y , S ( y ) ) \\} . \\end{gather*}"}
-{"id": "7347.png", "formula": "\\begin{align*} \\pi _ { i , N } : = \\min \\left \\{ c _ { N } ( X _ { 1 } , X _ { 2 } , \\dots , X _ { N } ) \\frac { w ( X _ { i } ) } { \\sum _ { j = 1 } ^ { N } w ( X _ { j } ) } ; 1 \\right \\} , i = 1 , 2 , \\dots , N , \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} \\dim \\mu _ { \\l _ 0 , \\tau } = \\min \\Big \\{ 1 , \\frac { h ( \\l _ 0 , \\tau ) } { \\log \\l _ 0 ^ { - 1 } } \\Big \\} , \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} L : = - \\partial ^ 2 + V . \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} \\inf _ { x \\in K } R ^ { ( \\gamma ) , 0 } _ 1 \\varphi ( x ) = 1 - \\sup _ { x \\in K } w _ { 1 , \\lambda _ \\gamma } ( x ) > 0 , \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c l } a _ { 1 1 } X _ 1 ^ { d _ 1 } & + a _ { 1 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { 1 t } X _ t ^ { d _ t } = b _ 1 \\\\ a _ { 2 1 } X _ 1 ^ { d _ 1 } & + a _ { 2 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { 2 t } X _ t ^ { d _ t } = b _ 2 \\\\ \\ ; \\vdots & & \\quad \\vdots \\\\ a _ { n 1 } X _ 1 ^ { d _ 1 } & + a _ { n 2 } X _ 2 ^ { d _ 2 } + \\cdots + & a _ { n t } X _ t ^ { d _ t } = b _ n , \\end{array} \\right . \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} \\sigma _ f ( g ) = \\int _ G g ( x ) f ( x ) d \\sigma ( x ) , \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ \\ , \\textbf { d } _ { t } \\ , \\big ] = - \\frac { 1 } { n } \\ , \\big ( \\ , S _ { t } ^ { 2 } - \\tilde { S } _ { t } ^ { 2 } \\ , \\big ) \\leq 0 . \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } } \\frac { 1 - | \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } } { | t - \\eta _ { \\mu _ { 1 } } ( z ) | ^ { 2 } } \\ , d \\sigma ( t ) = \\beta \\frac { 1 - | \\eta _ { \\nu _ { 1 } } ( z ) | ^ { 2 } } { | 1 - \\eta _ { \\nu _ { 1 } } ( z ) | ^ { 2 } } , \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} \\lambda _ n \\langle 1 | f _ n \\rangle = \\langle 1 | L _ u f _ n \\rangle = \\langle L _ u 1 | f _ n \\rangle = - \\langle \\Pi u | f _ n \\rangle = - \\langle u | f _ n \\rangle \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} & \\mu [ U ] \\left ( b ^ { ( 1 ) } \\right ) = U [ u ^ { ( 1 ) } \\leadsto b ^ { ( 1 ) } ] , \\\\ & \\mu [ U ] \\left ( c ^ { ( 1 ) } \\right ) = U [ u ^ { ( 1 ) } \\leadsto c ^ { ( 1 ) } ] . \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} & L v = 0 , \\\\ & L w = R \\Bigl ( t , x , v + w , \\frac { \\partial v } { \\partial x } + \\frac { \\partial w } { \\partial x } \\Bigr ) . \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ f ^ 2 u = - \\Lambda _ f \\Delta _ f u , \\ \\ { \\rm i n } \\ \\Omega , \\\\ u | _ { \\partial \\Omega } = \\frac { \\partial u } { \\partial \\nu } | _ { \\partial \\Omega } = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} A _ 0 ( G / H ) : = \\{ \\varphi \\in \\mathcal { C } _ 0 ( G / H ) : L _ h \\varphi = \\varphi \\ \\forall h \\in H \\} . \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} \\psi ( \\lambda ) = { \\lambda } ^ { 4 } + \\left ( - n + 4 \\right ) { \\lambda } ^ { 3 } + \\left ( 4 - 6 \\ , n \\right ) { \\lambda } ^ { 2 } + \\left ( - 6 \\ , n - 8 \\right ) \\lambda - 1 2 . \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} F { } { } \\circ \\begin{pmatrix} \\operatorname { I d } + k _ c \\\\ k _ u \\\\ k _ s \\end{pmatrix} = \\begin{pmatrix} \\operatorname { I d } + k _ c \\\\ k _ u \\\\ k _ s \\end{pmatrix} \\circ ( A _ c + r ) \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} \\overline { K } ( a _ { i , 0 } , a _ { i , 1 } , y _ 0 , y _ 1 ) = & ( a _ { i , 0 } a _ { i , 1 } - t \\sum _ { \\ell = 0 } ^ { 2 } d _ { \\ell - 1 , 0 } a _ { i , 0 } ^ { \\ell } a _ { i , 1 } ^ { 2 - \\ell } ) y _ 0 y _ 1 \\\\ & + t ( \\sum _ { \\ell = 0 } ^ 2 d _ { \\ell - 1 , - 1 } a _ { i , 0 } ^ { \\ell } a _ { i , 1 } ^ { 2 - \\ell } ) y _ 1 ^ 2 . \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} N _ 2 \\leq & C \\| \\partial _ 1 \\Delta _ q \\theta \\| _ { L ^ 2 } \\| \\Delta _ q \\omega \\| _ { L ^ 2 } \\leq C b _ q 2 ^ { - 2 q s } \\| \\partial _ 1 \\theta \\| _ { H ^ s } \\| \\omega \\| _ { H ^ s } \\leq C b _ q 2 ^ { - 2 q s } ( \\| \\theta \\| _ { H ^ { 1 + s } } ^ 2 + \\| \\omega \\| _ { H ^ s } ^ 2 ) . \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} \\Omega _ { \\rho _ { 1 } } = \\eta _ { \\rho _ { 1 } } ( \\mathbb { D } ) \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} g _ { i j } ^ { ( 2 k + 1 ) } = \\begin{cases} 1 & \\mbox { i f } i = j \\not = k + 1 \\\\ 0 & \\mbox { i f } i \\not = j \\\\ \\dfrac { 1 } { a ^ { ( 2 k ) } _ { k + 1 k + 1 } } & \\mbox { i f } i = j = k + 1 \\end{cases} \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} ( \\theta + d d ^ c \\phi ) ^ n = e ^ { \\phi } \\mu \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} \\| F \\| _ { p , \\ell _ 2 } : = \\left \\| \\sqrt { \\sum _ { a _ 1 , \\dots , a _ k = 1 } ^ { d ' } F ( a _ 1 , \\dots , a _ k ) ^ 2 } \\right \\| _ p . \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} a \\prec b = \\pi ( [ 1 ] ; a , b ) , a \\succ b = \\pi ( [ 2 ] ; a , b ) ~ ~ ~ ~ ~ ~ a \\curlyvee b = \\pi ( [ 3 ] ; a , b ) , ~ a , b \\in A . \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} & \\delta \\int _ M \\mathcal { E } ( H , K ) \\ , d S \\\\ & = \\int _ M \\bigg ( \\frac { 1 } { 2 } \\mathcal { E } _ H + 2 H \\mathcal { E } _ K \\bigg ) \\Delta u + \\bigg ( ( 2 H ^ 2 - K + 2 k _ 0 ) \\mathcal { E } _ H + 2 H K \\mathcal { E } _ K - 2 H \\mathcal { E } \\bigg ) u \\\\ & \\phantom { \\int _ M h } - \\mathcal { E } _ K \\langle h , \\ , u \\rangle \\ , \\ , d S , \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} \\ - f ( B ) : = \\ - f ^ 1 ( B ) \\ge \\ - f ^ 2 ( B ) \\ge \\dotsb \\ge \\ - f ^ \\alpha ( B ) \\ge \\dotsb ( \\ge f ( B ) , ) . \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} F _ { \\mu } ( z ) = \\frac { 1 } { G _ { \\mu } ( z ) } , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} T _ { \\Theta ( \\varepsilon ) } ( y ^ I , z ^ I ) = \\left \\{ ( \\triangle y ^ I , \\triangle z ^ I ) \\in \\Re ^ { 2 m } : { \\cal J } _ { y ^ I , z ^ I } \\Psi _ { \\varepsilon } ( y ^ I , z ^ I ) ( \\triangle y ^ I , \\triangle z ^ I ) = 0 \\right \\} , \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a , q ^ { - 1 } / b , c q ^ { - 1 } , d q ^ { - 1 } , q ^ { - 1 } ; q ^ 2 ) _ k } { ( q ^ 2 / a , a q ^ 2 , b q ^ 2 , q ^ 2 / c , q ^ 2 / d , q ^ 2 ; q ^ 2 ) _ k } \\bigg ( \\frac { b q ^ 7 } { c d } \\bigg ) ^ k \\equiv 0 \\pmod { \\Phi _ n ( q ) } . \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} ( f ^ { k - i } ( x _ { i } ) , f ^ { k - i - 1 } ( x _ { i + 1 } ) ) = ( f ^ { k - i - 1 } ( f ( x _ { i } ) ) , f ^ { k - i - 1 } ( x _ { i + 1 } ) ) ) \\in S \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} \\mathcal M _ { C , I } ( g ) = \\{ \\Sigma _ 1 , \\hdots , \\Sigma _ N \\} . \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} ( \\partial _ { | x | } u ) ( t , r ) & = - \\frac { \\lambda '' ( s ) } { | x | } \\int _ { 0 } ^ { t - s } \\frac { \\rho d \\rho } { \\sqrt { ( t - s ) ^ { 2 } - \\rho ^ { 2 } } } \\left ( 1 + \\frac { | x | ^ { 2 } - 1 - \\rho ^ { 2 } } { \\sqrt { ( 1 + | x | ^ { 2 } + \\rho ^ { 2 } ) ^ { 2 } - 4 | x | ^ { 2 } \\rho ^ { 2 } } } \\right ) \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} c _ { 1 L } = \\frac { a _ { 0 L } b _ { 2 R } - a _ { 2 R } b _ { 0 L } } { a _ { 0 L } } + \\frac { a _ { 0 R } \\left ( a _ { 2 L } b _ { 0 L } - a _ { 0 L } b _ { 2 L } \\right ) } { a _ { 0 L } ^ 2 } , \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} - g _ { 1 1 } \\kappa _ g = \\left ( - \\frac { \\partial \\Gamma _ { 1 1 } ^ 2 } { \\partial s _ 2 } - \\frac { e ^ { - 4 s _ 2 } \\left ( 1 - e ^ { - 2 s _ 2 } \\right ) } { \\left ( 1 + e ^ { - 2 s _ 2 } \\right ) ^ 3 } - \\frac { e ^ { - 4 s _ 2 } } { \\left ( 1 + e ^ { - 2 s _ 2 } \\right ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} Q _ C ( v _ { s , t } , v _ { s , t } ) = ( - \\log t ) + \\frac { ( - \\log t ) ^ 2 } { \\log t - \\log s } \\geq ( - \\log t ) \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} { } ^ b \\Psi ^ { * , \\epsilon } _ { G , \\mathcal { A } } ( M ) : = { } ^ b \\Psi ^ { * } _ { G , c } ( M ) + { } ^ b \\mathcal { A } ^ \\epsilon _ G ( M ) \\ , . \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | = m } \\partial ^ \\alpha a _ \\alpha ( x , t , u ) - u _ t \\ge f ( x , t ) g ( u ) \\mbox { i n } \\Omega \\times ( 0 , \\infty ) , \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} a ( p ) = \\int _ { - \\infty } ^ \\infty \\ ! \\int _ { - \\infty } ^ \\infty \\ ! \\int _ { - \\infty } ^ \\infty & | x y | ^ p \\Biggl \\{ \\frac { 1 } { 2 \\pi ( 1 - \\exp ( - 2 | u | ) ) ^ { 1 / 2 } } \\exp \\biggl ( - \\frac { 1 } { 2 ( 1 - \\exp ( - 2 | u | ) ) } ( x ^ 2 + y ^ 2 \\\\ & - 2 \\exp ( - | u | ) | x y | ) \\biggl ) - \\phi ( x ) \\phi ( y ) \\Biggl \\} d x d y d u , \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} \\begin{array} { c l } \\displaystyle \\min _ { x , y , z ^ I } & f ( x ) \\\\ { \\rm s . t . } & G ( x , y , z ^ I ) = 0 , \\\\ & ( y ^ I , z ^ I ) \\in \\Theta ( \\varepsilon ) , \\end{array} \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} v ' ( \\beta ) & = e ^ \\beta + 1 - \\frac { 3 e ^ 3 } { 5 \\sqrt { 2 \\pi } } ( 1 - \\beta ) e ^ { - \\beta } \\\\ & > e ^ { 2 / 5 } + 1 - \\frac { 3 e ^ 3 } { 5 \\sqrt { 2 \\pi } } \\frac { 3 } { 5 } e ^ { - 2 / 5 } > 0 . 5 \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} \\begin{array} { c } \\Delta _ A ( f _ s , 0 ) = \\left ( \\int _ { \\mathbb { R } } x ^ { 2 n } | f _ s ( x ) | ^ 2 d x \\right ) ^ { \\frac { 1 } { 2 } } = | s | ^ { - 1 / 2 } \\left ( \\int _ { \\mathbb { R } } x ^ { 2 n } | f _ 1 \\left ( s ^ { - 1 } x \\right ) | ^ 2 d x \\right ) ^ { \\frac { 1 } { 2 } } = \\\\ \\\\ = | s | ^ { - 1 / 2 } \\left ( \\int _ { \\mathbb { R } } ( s y ) ^ { 2 n } | f _ 1 \\left ( y \\right ) | ^ 2 | s | d y \\right ) ^ { \\frac { 1 } { 2 } } = | s | ^ n \\Delta _ A ( f _ 1 , 0 ) \\end{array} \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} b _ i \\ge \\alpha m b _ { i + 1 } \\ge \\sum _ { r = i + 1 } ^ { k - 1 } \\frac { \\alpha m } { k } \\cdot b _ { i + 1 } \\ge \\sum _ { r = i + 1 } ^ { k - 1 } \\frac { \\alpha ^ { r - i } m ^ { r - i } } { k } \\cdot b _ r \\ge \\sum _ { r = i + 1 } ^ { k - 1 } \\left ( \\frac { \\alpha m } { k } \\right ) ^ { r - i } b _ r \\ge \\sum _ { r = i + 1 } ^ { k - 1 } \\binom { \\left \\lfloor \\frac { \\alpha m } { k } \\right \\rfloor } { r - i } b _ r . \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} \\xi \\widehat { v _ { 2 , 0 } } ( \\xi ) = \\frac { - 1 } { \\pi } \\int _ { 0 } ^ { \\infty } F ( \\frac { \\sigma } { \\xi } ) \\sin ( \\sigma ) \\frac { d \\sigma } { \\xi } \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} \\mathcal { A } ( u ) = A ( u ) - N _ f ( u ) . \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} \\psi _ { \\nu _ { 1 } } ( z ) = \\frac { 1 } { \\beta } \\int _ { \\mathbb { T } } \\frac { t \\eta _ { \\mu _ { 1 } } ( z ) } { 1 - t \\eta _ { \\mu _ { 1 } } ( z ) } \\ , d \\sigma ( \\overline { t } ) , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} \\int _ { z _ 1 } ^ { z _ 2 } e { \\cal \\ ; L } h _ s d z = \\int _ { z _ 1 } ^ { z _ 2 } h _ s { \\cal L } e \\ ; d z = 0 . \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} { \\bf h } _ { X D } = \\sqrt { P _ X \\psi ( d _ { X D } ) } \\widetilde { { \\bf h } } _ { X D } , \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} \\left ( \\sum _ { n = 1 } ^ \\infty r ^ { \\Omega ( n ) } \\| a _ n \\| ^ q \\right ) ^ { \\frac { 1 } { q } } \\leq C \\| D \\| _ { \\mathcal { H } _ p ( X ) } . \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} p _ \\theta ( { \\bf { x } } ) ^ { - 1 } & = Z ( \\theta ) ^ { - 1 } \\exp \\big [ - h ( { \\bf { x } } ) - w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\big ] \\\\ p _ \\theta ( { \\bf { x } } ) ^ { - 1 } & = \\exp \\big [ - h ( { \\bf { x } } ) - Z ' ( \\theta ) - w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\big ] , \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} T \\cong \\prod _ { i = 1 } ^ s C _ { p _ i ^ { a _ i } } \\times \\prod _ { \\iota = 1 } ^ \\rho C _ { 2 ^ { \\epsilon _ \\iota } } \\times C _ { 2 ^ { \\epsilon } } ^ \\sigma \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} \\varphi \\big ( Z ( A \\otimes B ) \\big ) & = \\varphi \\big ( Z ( A ) \\otimes Z ( B ) \\big ) = \\pi ( Z ( A ) ) \\otimes Z ( B ) \\\\ & = Z ( A / I ) \\otimes Z ( B ) = Z \\big ( ( A / I ) \\otimes B \\big ) . \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} & \\langle \\mathcal { L } _ g f , \\psi \\rangle _ { Q _ t } + \\langle \\mathcal { L } _ g ^ { * } \\psi , f \\rangle _ { Q _ t } \\\\ [ 2 p t ] = \\ ; & \\int _ { \\Omega \\times \\mathbb { R } ^ 3 } \\ ! \\big [ ( f \\psi ) ( t , x , v ) - ( f \\psi ) ( 0 , x , v ) \\big ] d x d v + \\int _ { \\Sigma ^ { t } } \\ ! \\gamma f \\gamma \\psi ( v \\ ! \\cdot \\ ! n _ x ) d S _ x d v d \\tau , \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} \\lim _ { D \\rightarrow \\infty } \\frac { \\sum _ { x \\in \\Gamma _ D } f ( x ) } { | \\Gamma _ D | } = \\int _ F f \\rho , \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} \\partial _ { r } ^ { 2 } v _ { 2 } ( t , r ) = \\frac { c _ { b } } { 4 } \\int _ { 0 } ^ { \\infty } \\sin ( t \\xi ) \\xi ^ { 2 } \\left ( - 3 J _ { 1 } ( r \\xi ) + J _ { 3 } ( r \\xi ) \\right ) \\frac { \\chi _ { \\leq \\frac { 1 } { 4 } } ( \\xi ) } { \\log ^ { b - 1 } ( \\frac { 1 } { \\xi } ) } d \\xi \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} \\frac { 1 } { n } A A ^ { \\mathrm { T } } = \\left ( \\begin{array} { l l } \\beta J & 2 \\beta ^ 3 J \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sigma _ { i } \\right ) \\\\ 2 \\beta ^ 3 J \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sigma _ { i } \\right ) & 2 \\beta ^ 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} r = - e ^ h + G ( s ) , \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} a \\wedge \\neg a = \\bot , & & a \\vee \\neg a = \\top _ { A _ i } \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} V ^ { l _ { 1 } } U ^ { k _ { 1 } } . \\left ( \\Phi \\otimes ( z ^ { 0 } \\otimes \\varepsilon _ { 0 } ) \\right ) = \\lambda ^ { l _ { 1 } k _ { 1 } } \\ , \\Phi \\otimes ( z ^ { l _ { 1 } } \\otimes \\varepsilon _ { k _ { 1 } } ) . \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} \\frac d { d t } f ( t ) + a ( \\left \\| \\Lambda f _ 0 \\right \\| ) \\Lambda f = B ( t , f , f ) , f ( 0 ) = f _ 0 \\in X _ { + } , t \\geq 0 , \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} \\iint _ { B _ \\varepsilon ^ c } \\frac { \\abs { \\Phi ( x , y ) - \\Phi ( x , x ) } } { \\abs { x - y } ^ { n + 2 s } } d x d y & = \\int _ { x \\in K } \\left ( \\int _ { \\abs { x - y } \\geq \\varepsilon } \\frac { \\abs { \\Phi ( x , y ) - \\Phi ( x , x ) } } { \\abs { x - y } ^ { n + 2 s } } d y \\right ) d x \\\\ & \\leq c _ { n , s } \\norm { \\Phi } _ { \\infty } \\abs { K } \\varepsilon ^ { - 2 s } . \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} T _ { { } ^ Q \\nabla ^ \\pm } ( q _ 1 , q _ 2 ) = & - T _ { \\overline { \\nabla } ^ \\pm } ( q _ 1 , q _ 2 ) , \\\\ R _ { { } ^ Q \\nabla ^ \\pm } ( q _ 1 , q _ 2 ) ( q _ 3 ) = & R _ { \\overline { \\nabla } ^ \\pm } ( q _ 1 , q _ 3 ) q _ 2 - R _ { \\overline { \\nabla } ^ \\pm } ( q _ 2 , q _ 3 ) q _ 1 \\\\ & + \\overline { \\nabla } ^ \\pm _ { q _ 3 } T _ { \\nabla ^ \\pm } ( q _ 1 , q _ 2 ) - T _ { \\overline { \\nabla } ^ \\pm } ( \\overline { \\nabla } ^ \\pm _ { q _ 3 } q _ 1 , q _ 2 ) - T _ { \\overline { \\nabla } ^ \\pm } ( q _ 1 , \\overline { \\nabla } ^ \\pm _ { q _ 3 } q _ 2 ) . \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} Q : = \\frac { I - \\exp ( - \\frac { 1 } { 2 } D ^ - D ^ + ) } { D ^ - D ^ + } D ^ + \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} \\lambda + b - K + K e ^ { - \\lambda \\tau } = 0 , \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\left \\Vert e ^ { \\gamma \\left \\vert x \\right \\vert ^ { p } } \\tilde { u } _ { k } \\left ( x , 0 \\right ) \\right \\Vert _ { X } = \\left \\Vert e ^ { \\gamma \\left ( \\frac { \\alpha } { \\beta } \\right ) ^ { p / 2 } \\left \\vert x \\right \\vert ^ { p } } u _ { k } \\left ( x , 0 \\right ) \\right \\Vert _ { X } = b _ { 0 } , \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} A ^ { \\pi } _ i \\ = \\ A _ { \\pi ^ { - 1 } i } , \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} & P _ { d } \\left \\{ \\lVert \\mathbb { H } _ { N } ' \\rVert _ { \\mathcal { F } } \\leq M \\right \\} + \\widetilde { E } _ { N } \\geq 1 - \\eta \\quad N = 1 , 2 , \\dots \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} D _ Q ^ { \\sharp } = \\lim _ { t \\rightarrow \\infty } D ^ { \\sharp } _ Q ( t ) = \\int _ 0 ^ { \\infty } ( e ^ { Q u } - \\mathbf { 1 } { \\boldsymbol { \\pi } } ) \\ , d u . \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} P ( \\tilde { y } ; \\mu ) = P _ L \\left ( P _ R ( \\tilde { y } ; \\mu ) ; \\mu \\right ) , \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} f ( 1 + p t ) = f ( 1 ) + p t f ' ( 1 ) + p t \\phi ( p , t ) , \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} & | \\frac { - 1 } { r } \\int _ { 6 r } ^ { \\infty } \\frac { d w \\lambda '' ( t + w ) r ^ { 2 } w ( 2 - 2 \\alpha ) \\left ( ( 1 - 2 \\alpha ) \\lambda ( t + w ) ^ { - 2 \\alpha } ( \\lambda ' ( t + w ) ) ^ { 2 } + \\lambda ( t + w ) ^ { 1 - 2 \\alpha } \\lambda '' ( t + w ) \\right ) } { ( \\lambda ( t + w ) ^ { 2 - 2 \\alpha } + w ^ { 2 } ) ^ { 2 } } | \\\\ & \\leq \\frac { C r } { t ^ { 4 } \\log ^ { b + 2 } ( t ) } \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} & \\int _ { \\Omega } | u | ^ { \\alpha - 2 p + 2 } | \\nabla u | ^ { p } \\Delta _ { p , f } ( | \\nabla u | ^ { p } ) \\ , d \\mu \\\\ = & - p ( \\alpha - 2 p + 2 ) \\int _ { \\Omega } | u | ^ { \\alpha - 2 p + 1 } | \\nabla u | ^ { 3 p - 3 } \\langle \\nabla u , \\nabla | \\nabla u | \\rangle \\ , d \\mu \\\\ & - p ^ 2 \\int _ { \\Omega } | u | ^ { \\alpha - 2 p + 2 } | \\nabla u | ^ { 3 p - 4 } | \\nabla | \\nabla u | | ^ 2 \\ , d \\mu \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} n _ i \\ , = \\ , n _ 0 + i ( 1 - n _ { - 1 } ) \\forall i \\geq 0 \\ , . \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} f ( x ) = f ^ { S _ k } ( x ) + \\sum _ { j = 0 } ^ { k - 1 } C ( S _ j , S _ { j + 1 } ) . \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} \\begin{pmatrix} \\ell _ { 1 } ( x ) & & \\\\ & \\ddots & \\\\ & & \\ell _ { n } ( x ) \\end{pmatrix} , \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} r i c _ { \\bar { g } } - \\frac { s c a l _ { \\bar { g } } } { 2 } \\bar { g } = ( \\mu + \\nu ) \\bar { g } ( \\cdot , X ) \\otimes \\bar { g } ( \\cdot , X ) + \\nu \\bar { g } , \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } A _ n \\begin{pmatrix} e ^ 3 \\\\ - 1 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} \\int _ { D _ { 2 R } } g ( z ) \\omega _ { D \\setminus D _ { 2 R } } ^ x ( d z ) & = \\int _ { | y | < R } \\left ( \\int _ { D _ { 2 R } } M _ D ( z , y ) \\omega _ { D \\setminus D _ { 2 R } } ^ x ( d z ) \\right ) \\mu ( d y ) \\\\ & = \\int _ { | y | < R } M _ D ( x , y ) \\mu ( d y ) = g ( x ) . \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} r _ \\theta ( g a _ t g ^ { - 1 } ) ( g n _ x g ^ { - 1 } ) ( v ) = r _ \\theta g a _ t \\mathbf { e } _ 1 = e ^ { t / 2 } r _ \\theta v = e ^ { \\frac 1 2 ( t + t _ v ) } r _ { \\theta + \\theta _ v } \\mathbf { e } _ 1 \\ , . \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} \\max \\{ | z | : z \\in E _ k \\} = | \\tau _ 0 | + 2 k \\ge 1 , \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\frac { d t } { t \\Big ( \\pi ^ 2 + \\big ( \\log \\frac { t } { 1 - t } \\big ) ^ 2 \\Big ) } = \\int _ 0 ^ 1 \\frac { d t } { ( 1 - t ) \\Big ( \\pi ^ 2 + \\big ( \\log \\frac { t } { 1 - t } \\big ) ^ 2 \\Big ) } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} F ( z ) = \\frac { \\beta } { \\int _ { \\mathbb { R } } \\frac { 1 + t ^ { 2 } } { F _ { \\mu _ { 1 } } ( z ) - t } \\ , d \\sigma ( t ) } \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} \\P _ { \\rho } \\left [ \\sum _ { i = 1 } ^ { k } \\mathcal { T } _ { i } + \\sum _ { i : j _ { i } < 0 } \\sum _ { \\ell = 1 } ^ { | j _ { i } | } \\mathcal { W } _ { i , \\ell } \\leq t \\right ] \\leq \\P [ X \\geq k + J ] , \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} & \\left ( \\frac { \\lambda ( t ) } { \\lambda ( x ) } \\right ) ^ { 2 } \\left ( \\int _ { \\frac { 4 } { \\lambda ( x ) ^ { 2 } } } ^ { \\infty } \\frac { d \\omega } { \\omega ^ { 3 / 2 } x ^ { 8 } \\log ^ { b + 2 - 4 \\alpha b } ( x ) } \\right ) ^ { 1 / 2 } \\leq C \\left ( \\frac { \\lambda ( t ) } { \\lambda ( x ) } \\right ) ^ { 2 } \\frac { 1 } { x ^ { 4 } \\log ^ { b + 1 - 2 \\alpha b } ( x ) } \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} L ( t ) : = \\frac 1 2 \\Big \\| | x | ^ { - \\frac s 2 } u \\Big \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} G ^ { 1 , 1 } _ { 1 } = { \\rm s p a n } \\left \\{ \\right . & ( E _ { 1 , 2 } ) _ { ( k _ { 1 } ) } ( E _ { 1 , 3 } ) ^ { l } _ { ( k _ { 2 } ) } , \\ , ( E _ { 1 , 4 } ) _ { ( k _ { 1 } ) } ( E _ { 1 , 3 } ) ^ { l } _ { ( k _ { 2 } ) } , \\ , ( E _ { 2 , 3 } ) _ { ( k _ { 1 } ) } ( E _ { 1 , 3 } ) ^ { l } _ { ( k _ { 2 } ) } , \\ , \\\\ & ( E _ { 2 , 4 } ) _ { ( k _ { 1 } ) } ( E _ { 1 , 3 } ) ^ { l } _ { ( k _ { 2 } ) } , \\ , ( E _ { 3 , 4 } ) _ { ( k _ { 1 } ) } ( E _ { 1 , 3 } ) ^ { l } _ { ( k _ { 2 } ) } , \\ , | l \\in \\mathbb { N } , k _ { 1 } , k _ { 2 } \\in \\mathbb { Z } _ { < 0 } \\left . \\right \\} + G ^ { 1 , 1 } _ { 0 } , \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} \\Phi ( z ) = \\gamma z \\exp H _ { \\sigma } ( z ) , z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} \\lambda \\ast _ { G / H } \\lambda ' ( \\psi ) & = \\int _ { G / H } \\int _ { G / H } \\left ( \\int _ H \\psi ( x h y H ) d h \\right ) d \\lambda ( x H ) d \\lambda ' ( y H ) \\\\ & = \\int _ { G / H } \\int _ { G / H } \\psi ( x y H ) d \\lambda ( x H ) d \\lambda ' ( y H ) \\\\ & = \\int _ { G / H } \\int _ { G / H } \\psi ( x H y H ) d \\lambda ( x H ) d \\lambda ' ( y H ) . \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} G ( d , s , t ) = \\frac { d ^ 2 } { 2 s } + \\frac { d } { 2 s } ( 2 P - 2 - s ) + \\frac { d } { s } ( \\beta - 1 ) + \\rho + 1 . \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} f \\left ( \\frac { L + 2 } { L ^ 2 + L + 4 } + 1 \\right ) = - \\frac { 6 1 } { 2 5 } < 0 , \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} \\frac { ~ ^ C d ^ { \\alpha } } { d t ^ { \\alpha } } f ( t ) = f ' * h _ { \\alpha } ( t ) = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\int _ 0 ^ t \\frac { f ' ( y ) } { ( t - y ) ^ { \\alpha } } d y \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} \\underline { x } _ n ^ J = ( x _ 1 , \\dots , \\hat x _ { j _ 1 } , \\dots , \\hat x _ { j _ m } , \\dots , x _ n ) \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} \\dfrac { \\alpha } { \\alpha - 1 } \\big [ p ^ * ( x ) ^ { \\alpha - 1 } - q ( x ) ^ { \\alpha - 1 } \\big ] = - \\sum \\limits _ { i = 1 } ^ k \\lambda _ i \\big [ f _ i ( x ) - a _ i \\big ] + \\mu ( x ) - \\nu x \\in \\mathbb { S } . \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} F _ { \\mu _ { 1 } } ( z ) = c + z - N _ { \\lambda } ( z ) , z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} p _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ( f ( t ) ) & = \\frac { 1 } { 2 \\pi } \\Re \\frac { 1 + \\eta _ { \\mu _ { 1 } } ( z ( t ) ) } { 1 - \\eta _ { \\mu _ { 1 } } ( z ( t ) ) } \\\\ & = \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { T } } \\frac { 1 - | z ( t ) | ^ { 2 } } { | \\xi - z ( t ) | ^ { 2 } } \\ , d \\mu _ { 1 } ( \\overline { \\xi } ) \\\\ & = \\frac { 1 } { 2 \\pi \\beta } \\beta \\int _ { \\mathbb { T } } \\frac { 1 - | z ( t ) | ^ { 2 } } { | \\xi - z ( t ) | ^ { 2 } } \\ , d \\mu _ { 1 } ( \\overline { \\xi } ) = \\frac { - \\log R ( t ) } { 2 \\pi \\beta } . \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} \\psi _ { k , k + 1 } ^ { \\ast } ( Q ^ { \\prime } ) = \\frac { L _ { k , k + 1 } ^ { \\ast } ( q ^ { \\prime } ) } { q ^ { \\prime } } \\geq - \\frac { ( k - n + 1 ) \\Phi _ { k , n } ( \\psi _ { n , 1 } ^ { \\ast } ( Q ) ) + \\Phi _ { k , n } ( \\psi _ { n , 2 } ^ { \\ast } ( Q ) ) } { n - 1 } - O ( q ^ { \\prime - 1 } ) , \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} u = Q _ { \\frac { 1 } { \\lambda ( t ) } } + \\phi _ { L } + \\epsilon \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} \\sum _ { q = 1 } ^ { \\widetilde { l } } \\gamma _ q \\widetilde { T } _ q = \\sum _ { p } \\gamma ^ { \\prime } _ p \\widetilde { T } _ p ^ { \\prime } + Q , \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} | G ( \\xi _ 2 ' ) | \\leq 2 ^ { 7 0 } A ^ { - 1 } N _ 1 ^ 3 \\textnormal { a n d } | G ' ( \\xi _ 2 ' ) | = \\Bigl | \\Bigl ( \\frac { d G } { d \\xi _ 2 ' } \\Bigr ) ( \\xi _ 2 ' ) \\Bigr | \\leq 2 ^ { - 1 0 0 } N _ 1 ^ 2 . \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} \\Phi ( 0 ) = 0 , \\ ; | \\Phi ( z ) | \\ge | z | , z \\in \\mathbb { D } . \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial t } = \\frac { \\nu _ t } { \\mathcal { H } _ t } , \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} - \\widetilde { \\Theta } ^ { F * } ( \\tau ) : = \\sum _ { 1 \\leq j , k \\leq n } \\sum _ { 1 \\leq \\lambda , \\mu \\leq r } c _ { j k \\mu \\lambda } \\tau _ { j \\lambda } \\overline { \\tau } _ { k \\mu } \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} Z _ { \\mu , \\lambda } ^ { \\prec w } = \\bigsqcup _ { w ' \\prec w } Z _ { \\mu , \\lambda } ^ { w ' } , Z _ { \\mu , \\lambda } ^ { \\preccurlyeq w } = Z _ { \\mu , \\lambda } ^ { \\prec w } \\sqcup Z _ { \\mu , \\lambda } ^ w . \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} x & { } = \\rho ( a ) ^ { - 1 } \\Bigl ( \\rho ( c ) z - \\frac { 1 } { 2 } \\rho \\Bigl ( \\frac { c } { 2 } \\Bigr ) ^ 2 \\langle S , T \\rangle \\Bigr ) , & y & { } = 0 . \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} f _ 1 \\left ( x \\right ) = f _ 1 \\left ( \\overleftarrow { x } \\right ) . \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} g _ { i j } ^ { ( 2 k + 1 ) } = \\begin{cases} 1 & \\mbox { i f } i = j \\not = k + 1 \\\\ 0 & \\mbox { i f } i \\not = j \\\\ \\dfrac { m _ { k } } { m _ { k + 1 } } & \\mbox { i f } i = j = k + 1 \\end{cases} . \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} \\int \\widetilde { p } _ n ( \\textbf { { x } } ) ^ \\alpha p _ \\theta ( \\textbf { { x } } ) ^ { 1 - \\alpha } s ( \\textbf { { x } } ; \\theta ) d \\textbf { { x } } = 0 , \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} 1 & = 1 + [ 1 + ( - 1 ) ^ p ] a u + [ 1 + ( - 1 ) ^ q ] b v + [ 4 ( - 1 ) ^ { p + q } + ( - 1 ) ^ q a b + ( - 1 ) ^ p a b + 4 ] u v . \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} ( F { } { } \\circ K { } { } ^ { 1 / \\varepsilon } ) ^ \\varepsilon = F { } { } ^ \\varepsilon \\circ K { } { } = K { } { } \\circ ( A _ c + r ) = ( K { } { } ^ { 1 / \\varepsilon } \\circ ( A _ c + r ^ { 1 / \\varepsilon } ) ) ^ \\varepsilon , \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} W _ { \\b , n } = W _ { \\b + 1 , 0 } . \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} [ \\delta _ { \\varepsilon _ 1 } , \\delta _ { \\varepsilon _ 2 } ] X = \\delta _ { [ \\varepsilon _ 1 , \\varepsilon _ 2 ] } X \\Longrightarrow [ \\check { \\rho } ( \\varepsilon _ 1 ) , \\check { \\rho } ( \\varepsilon _ 2 ) ] _ { T \\Sigma } X = \\check { \\rho } ( [ \\varepsilon _ 1 , \\varepsilon _ 2 ] _ { X ^ { * * } Q } ) X . \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} \\tau ^ G _ \\varphi ( a _ 0 , \\ldots , a _ k ) : = \\int _ { G ^ { \\times k } } a _ 0 ( ( g _ 1 \\cdots g _ k ) ^ { - 1 } ) a _ 1 ( g _ 1 ) \\cdots a _ k ( g _ k ) \\varphi ( e , g _ 1 , g _ 1 g _ 2 , \\ldots , g _ 1 \\cdots g _ k ) d g _ 1 \\cdots d g _ k . \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} p _ 1 p _ 2 ^ { - 1 } a _ 2 a _ 1 ^ { - 1 } = - ( y - c ) ( y b ^ { - 1 } - c x ^ { - 1 } ) ^ { - 1 } ( x - b ) ^ { - 1 } x c ^ { - 1 } ( y b ^ { - 1 } - c x ^ { - 1 } ) b x ^ { - 1 } ( x - a _ 1 ) a _ 1 ^ { - 1 } . \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} \\Phi ( \\xi _ 1 , \\eta _ 1 , \\xi , \\eta ) & = \\xi _ 1 \\xi ( \\xi _ 1 + \\xi ) + \\eta _ 1 \\eta ( \\eta _ 1 + \\eta ) , \\\\ F ( \\xi _ 1 , \\eta _ 1 , \\xi , \\eta ) & = \\xi _ 1 \\eta + \\xi \\eta _ 1 + 2 ( \\xi _ 1 \\eta _ 1 + \\xi \\eta ) . \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} H ( x ) = \\int _ { x } ^ \\infty e ^ { i t ^ 2 } d t \\ , . \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} 2 c _ 1 c _ 2 = c _ { 1 , 0 } ^ 2 , 2 x _ 1 x _ 2 = 6 c _ { 1 , 0 } ^ 2 - 5 c _ { 1 , 0 } c _ { 2 , 1 } + c _ { 2 , 1 } ^ 2 - c _ { 3 , 1 } + 4 c _ { 2 , 0 } . \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} \\forall t \\in ( 0 , 1 ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\Psi ( t ) = \\frac { 1 } { t ( 1 - t ) \\Big ( \\pi ^ 2 + \\big ( \\log \\frac { t } { 1 - t } \\big ) ^ 2 \\Big ) } . \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} ( T ' ) ^ { - 1 } ( I - B ' z ^ { - 1 } ) ( I - A ' z ^ { - 1 } ) ^ { - 1 } ( I - A z ) ^ { - 1 } ( I - B z ) T ^ { - 1 } = I _ 2 \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} \\tt = n ^ { - 1 } \\Big ( \\sum _ { i = 1 } ^ n D _ i [ 1 ] , ~ ~ \\sum _ { i = 1 } ^ n D _ i [ 2 ] , ~ ~ 4 \\big \\{ \\sum _ { i = 1 } ^ n D _ i [ 3 ] \\big \\} ^ { + } , 0 , 0 \\Big ) ^ \\tau . \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\eta _ j \\vert A _ j ( p _ j ) \\vert = \\infty . \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} Z _ { \\mu , \\lambda } : = Y _ \\mu \\times _ { Q _ n } Y _ \\lambda \\subset Y _ \\mu \\times Y _ \\lambda \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} \\frac { \\rho ( \\omega \\lambda ( t ) ^ { 2 } ) } { \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) } & \\leq C \\omega \\lambda ( t ) ^ { 2 } \\omega \\lambda ( x ) ^ { 2 } \\log ^ { 2 } ( \\omega \\lambda ( x ) ^ { 2 } ) \\\\ & \\leq C \\frac { \\lambda ( t ) ^ { 2 } } { \\lambda ( x ) ^ { 2 } } \\left ( \\omega \\lambda ( x ) ^ { 2 } \\right ) ^ { 2 } \\log ^ { 2 } ( \\omega \\lambda ( x ) ^ { 2 } ) \\\\ & \\leq \\frac { C \\lambda ( t ) ^ { 2 } } { \\lambda ( x ) ^ { 2 } } \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} P _ D ( x _ 0 , y ) = \\int _ D G _ D ( x _ 0 , z ) j ( | z - y | ) d z = \\infty , . \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} \\begin{cases} a \\cdot H ( b , c ) - H ( b , c ) \\cdot a + H ( a , T H ( b , c ) ) - H ( T H ( b , c ) , a ) = H ( a b - b a , c ) + H ( b , a c - c a ) , \\\\ H ( a b - b a , a c - c a ) = 0 , ~ b , c \\in A . \\end{cases} \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} B x = ( x ' _ e ) _ { e \\in E } , \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} H _ 1 = \\bigoplus _ { j \\in J _ 1 } ( W _ j \\cap H _ 1 ) , \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} \\frac { d \\zeta _ R } { d \\mu } ( 0 ) = - \\frac { \\frac { \\partial f _ R } { \\partial \\mu } ( 0 , 0 ; 0 ) } { \\frac { \\partial f _ R } { \\partial y } ( 0 , 0 ; 0 ) } . \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} \\lim _ { s \\to t } I _ 2 ( s , t , \\omega ) = 0 , \\ \\ \\omega \\notin \\Pi _ 0 , \\ t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} A = R \\begin{pmatrix} 0 & 1 \\\\ 1 & a _ k \\end{pmatrix} \\cdots \\begin{pmatrix} 0 & 1 \\\\ 1 & a _ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} z _ { x y } = F ( x , y , z , z _ x , z _ y ) Z _ { X Y } = G ( X , Y , Z , Z _ X , Z _ Y ) , \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} \\alpha = \\left ( \\chi _ { { \\rm f o c u s } , L } - \\chi _ { { \\rm f o c u s } , R } \\middle ) \\right | _ { \\mu = 0 } \\ , , \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} | x - \\beta | ^ 2 = ( x - \\beta ) ( x - \\Bar { \\beta } ) = x ^ 2 - ( q / p ) ^ { 1 / 3 } x + { q / p } ^ { 2 / 3 } \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} d ( m , n ) & = \\sum _ { \\underset { r \\not \\in D } { 1 \\leq r \\leq m } } u _ d ( m - r , n + r ) + f _ d ( m - r , n + r ) , \\\\ f ( m , n ) & = \\sum _ { \\underset { r \\not \\in E } { 1 \\leq r \\leq m } } d ( m - r , n ) + u ( m - r , n ) , \\\\ u ( m , n ) & = \\sum _ { \\underset { r \\not \\in C } { 1 \\leq r \\leq m } } d _ u ( m - r , n - r ) + f _ u ( m - r , n - r ) . \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} \\Phi _ { \\alpha } ( z ) = \\chi ( x _ \\alpha ) \\alpha \\in \\Delta _ { \\geq 1 } \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} \\inf _ { \\zeta \\in \\mathcal { K } _ { t } } \\sup _ { P ^ { \\theta } \\in \\mathcal { P } } E _ { P ^ { \\theta } } ( { x } _ { t } - \\zeta ) ^ { 2 } = \\inf _ { \\zeta \\in \\mathcal { K } _ { t } } E _ { P ^ { \\theta ^ { \\ast } } } ( { x } _ { t } - \\zeta ) ^ { 2 } \\geq \\inf _ { \\zeta \\in \\bar { \\mathcal { K } } _ { t } } E _ { P ^ { \\theta ^ { \\ast } } } ( { x } _ { t } - \\zeta ) ^ { 2 } \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} \\Omega _ { ( r , A , r ' , \\mathcal { U } , p , q ) } = - \\frac { \\mathbf { i } } { 2 \\pi r } d x ^ t A ^ t d y \\cdot I _ { r ' } , \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} y _ i y _ { i + 1 } = 1 + y _ { i + 3 } , { \\rm m o d } \\mathbb Z _ 5 . \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} h h ^ * = T \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} D _ x ^ \\alpha x ^ \\beta = \\frac { \\Gamma ( \\beta + 1 ) } { \\Gamma ( \\beta - \\alpha + 1 ) } x ^ { \\beta - \\alpha } \\quad \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} \\tilde { \\chi } _ A ( n _ 1 , n _ 2 ) & = \\chi _ A M _ A ( n _ 1 , n _ 2 ) , & \\tilde { \\chi } _ { B , C } ( n _ 1 , n _ 2 ) & = \\chi _ { B , C } M _ { B , C } ( n _ 1 , n _ 2 ) , \\\\ \\tilde { \\chi } _ K ( n _ 1 , n _ 2 ) & = \\chi _ K M _ K ( n _ 1 , n _ 2 ) , & \\tilde { \\chi } _ N ( n _ 1 , n _ 2 ) & = \\chi _ N M _ N ( n _ 1 , n _ 2 ) . \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} P ^ * ( y _ 1 , \\cdots , y _ M ) = \\frac { 1 } { N } ( \\underbrace { y _ 1 , \\cdots , y _ 1 } _ { K } , \\cdots , \\underbrace { y _ M , \\cdots , y _ M } _ { K } ) \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} & \\int _ { t } ^ { \\infty } d x _ { 1 } \\int _ { x _ { 1 } } ^ { \\infty } \\frac { d x _ { 2 } } { \\sqrt { \\log ( \\log ( x _ { 2 } ) ) } x _ { 2 } ^ { 2 } \\log ^ { b + 1 } ( x _ { 2 } ) } = \\frac { 1 } { b \\log ^ { b } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } + E _ { i n t , 2 } \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } H _ 2 & = 1 \\\\ H _ 3 & = - 1 \\end{alignedat} \\right . \\left \\{ \\begin{alignedat} { 2 } H _ 2 & = \\epsilon ( H _ 1 ) ^ 2 \\\\ H _ 4 & = - \\epsilon \\end{alignedat} \\right . ; \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} C ^ \\infty ( M ) _ { ( i ) } / C ^ \\infty ( M ) _ { ( i + 1 ) } . \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} D = \\sum _ { k = 0 } ^ \\infty \\lambda _ k \\pi _ { E _ k } + \\sum _ { k = 0 } ^ \\infty ( - \\lambda _ k ) \\pi _ { a ( E _ k ) } \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} d _ { \\mathcal { P } , \\pm } = \\left [ b ( \\mathsf { M } ^ { \\mathbb { R } } _ { 1 } + \\mathsf { M } ^ { \\mathbb { Z } } ) \\mp \\frac { 1 } { 2 \\pi } \\ , \\partial _ { 1 } \\right ] + \\left [ b \\ , \\mathsf { M } ^ { \\mathbb { R } } _ { 2 } \\pm \\frac { 1 } { 2 \\pi } \\ , \\partial _ { 2 } \\right ] . \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} \\phi ( x ) = : \\int _ 0 ^ 1 x ^ v \\sin ( \\pi v ) d v = \\frac { ( x + 1 ) \\pi } { \\pi ^ 2 + \\big ( \\log x \\big ) ^ 2 } . \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} \\Pi _ s ( W ^ T ) = - \\Pi \\ , \\left ( \\nabla _ { W ^ T } F _ s \\right ) \\ , . \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} \\widehat \\rho ( L _ i ) = \\left ( - q ^ 2 \\partial _ q \\right ) ^ { i + 1 } \\circ \\left ( \\frac { 1 } { q } \\right ) + \\left ( b \\left ( - q ^ 2 \\partial _ q \\right ) + i c \\right ) \\circ \\left ( - q ^ 2 \\partial _ q \\right ) ^ i \\ , . \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\frac { \\sin ^ { 2 } ( u - Q _ { \\frac { 1 } { \\lambda ( t ) } } + Q _ { \\frac { 1 } { \\lambda ( t ) } } ) } { r ^ { 2 } } r d r \\leq C \\left ( \\int _ { 0 } ^ { \\infty } \\frac { \\sin ^ { 2 } ( Q _ { \\frac { 1 } { \\lambda ( t ) } } ) } { r ^ { 2 } } r d r + \\int _ { 0 } ^ { \\infty } \\frac { | u - Q _ { \\frac { 1 } { \\lambda ( t ) } } | ^ { 2 } } { r ^ { 2 } } r d r \\right ) \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} \\limsup _ { t \\rightarrow t ^ { + } _ { 0 } } \\frac { \\gamma ^ { \\prime \\prime } ( t ) } { \\gamma ^ { \\prime } ( t ) } = + \\infty . \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} \\phi _ t ^ * \\left ( \\sum _ 1 ^ 2 u _ { i } \\sigma _ i \\right ) = e ^ { t + r } \\ , \\sigma _ 1 + e ^ { t + r } \\ , ( t + r + \\kappa ) \\ , \\sigma _ 2 \\ , , \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } a _ 1 & b _ 1 \\\\ c _ 1 & d _ 1 \\end{array} \\right ) \\circ \\left ( \\begin{array} { c c } a _ 2 & b _ 2 \\\\ c _ 2 & d _ 2 \\end{array} \\right ) = \\left ( \\begin{array} { c c c c } 0 & a _ 1 & 0 & b _ 1 \\\\ a _ 2 & 0 & b _ 2 & 0 \\\\ 0 & c _ 1 & 0 & d _ 1 \\\\ c _ 2 & 0 & d _ 2 & 0 \\end{array} \\right ) \\ , . \\\\ \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ K & \\left ( \\int _ { K } \\frac { ( f ( y ) - f ( x ) ) ( \\tilde { \\psi } ( y ) - \\tilde { \\psi } ( x ) ) } { | x - y | ^ { d + \\alpha } } d y \\right ) ^ 2 d x \\\\ & \\leq \\| \\nabla f \\| _ \\infty ^ 2 \\| \\nabla \\tilde \\psi \\| _ \\infty ^ 2 \\int _ K \\left ( \\int _ K \\frac { d y } { | x - y | ^ { d + \\alpha - 2 } } \\right ) ^ 2 d x < \\infty \\end{aligned} \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} \\partial _ t ( \\partial _ k h ) = Q ( f , \\partial _ k h ) + Q ( \\partial _ k f , h ) + Q ( \\partial _ k h , \\mu ) + Q ( h , \\partial _ k \\mu ) . \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} \\left ( \\begin{array} { l l l l l } 1 & 0 & \\ldots & 0 & 0 \\\\ \\binom { 3 } { 2 } & 1 & \\ldots & 0 & 0 \\\\ \\vdots & \\ddots & \\ldots & 1 & 0 \\\\ \\binom { 2 k + 1 } { k + 1 } & \\binom { 2 k + 1 } { k + 2 } & \\ldots & \\binom { 2 k + 1 } { 2 k } & 1 \\end{array} \\right ) ^ { - 1 } = D . \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} \\ < J _ i ' ( t e _ i ) , e _ i \\ > = \\begin{cases} 0 , \\ , t = t _ i , \\\\ > 0 , 0 < t < t _ i , \\\\ < 0 , \\ , t _ i < t < \\infty . \\end{cases} \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} L _ u = D - T _ u \\ , , B _ u : = i D ^ 2 + 2 i T _ { D ( \\Pi u ) } - 2 i D T _ u \\ , . \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} H ^ 2 ( Y , \\mathcal O _ Y ( m a L ) \\otimes \\Psi _ * \\mathcal O _ Z ( - m A ) ) = 0 \\mbox { f o r } m \\gg 0 . \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} R _ q ( a , n ) & = \\bigg \\{ \\frac { q ^ 5 ( q ^ { - 1 } ; q ^ 2 ) _ { ( n + 1 ) / 2 } ^ 2 } { ( q ^ { - 1 } ; q ^ 2 ) _ 2 ( q ^ 2 / a , a q ^ { 2 } ; q ^ 2 ) _ { ( n + 1 ) / 2 } } - \\frac { q ^ { ( n + 7 ) / 2 } } { ( q ^ { n - 1 } ; q ^ 2 ) _ 2 } \\bigg \\} \\\\ [ 5 p t ] & \\quad \\times \\frac { ( 1 - a q ^ n ) ( a - q ^ n ) } { ( 1 - a ) ^ 2 } - \\frac { q ^ { ( n + 7 ) / 2 } } { ( q ^ { n - 1 } ; q ^ 2 ) _ 2 } . \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} B ( e _ { x y } , e _ { u v } ) & = B ( e _ x , e _ { u x } ) = - B ( e _ { u x } , e _ x ) = - B ( e _ u e _ { u x } e _ x , e _ x ) = - B ( e _ u , e _ u e _ { u x } e _ x ) \\\\ & = - e _ u B ( e _ u , e _ { u x } ) e _ x = - B ( e _ u , e _ { u x } ) ( u , x ) e _ { u x } \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} x _ 0 : = x , x _ { 2 n + 1 } = ( I - P _ M ) x _ { 2 n } , x _ { 2 n } = ( I - P _ N ) x _ { 2 n - 1 } \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} d _ n ( f ) ( g _ 1 , \\ldots , g _ { n + 1 } ) & = g _ 1 \\cdot f ( g _ 2 , \\ldots , g _ { n + 1 } ) \\\\ & + \\sum _ { i = 1 } ^ n ( - 1 ) ^ i f ( g _ 1 , \\ldots , g _ i g _ { i + 1 } , \\ldots , g _ { n + 1 } ) \\\\ & + ( - 1 ) ^ { n + 1 } f ( g _ 1 , \\ldots , g _ n ) , \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} \\left \\| \\int _ { { \\mathfrak S } } g ( s ) d s \\right \\| = \\int _ { { \\mathfrak S } } \\left \\| g ( s ) \\right \\| d s , \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} e ( \\lambda _ I ) = e ( I ) = 2 i _ 1 - \\epsilon _ 1 - \\sum _ { k = 2 } ^ { s } 2 ( p - 1 ) i _ k + \\sum _ { k = 2 } ^ s \\epsilon _ s . \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} \\nabla _ { 1 } \\boldsymbol { e _ { 1 } } = q _ { 1 } \\boldsymbol { e _ { 2 } } + a \\boldsymbol { n } , \\ \\ \\ \\nabla _ { 2 } \\boldsymbol { e _ { 1 } } = q _ { 2 } \\boldsymbol { e _ { 2 } } + b \\boldsymbol { n } , \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} h ( ( S _ { \\beta } \\times S _ { \\beta } ^ 2 ) ^ n ( 0 , 0 , 0 , 0 ) ) & = ( n \\alpha , n ^ 2 \\alpha + n \\beta , n \\beta ) \\\\ & = U ^ n ( 0 , 0 , 0 ) \\\\ & = U ^ n ( h ( 0 , 0 , 0 , 0 ) ) , \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} \\begin{aligned} & \\rho _ t + \\nabla \\cdot ( \\rho Q ^ \\epsilon \\ast \\mathbf { u } ) = 0 , \\\\ & \\mathbf { u } _ t + ( \\mathbf { u } \\cdot \\nabla ) \\mathbf { u } = \\frac { \\rho } { \\epsilon } ( Q ^ \\epsilon \\ast \\mathbf { u } - \\mathbf { u } ) , \\end{aligned} \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} u _ { \\varepsilon } \\left ( x , t \\right ) = e ^ { \\varepsilon t W } u \\left ( x , t \\right ) t \\in \\left [ 0 , 1 \\right ] . \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} & \\sigma _ x = L _ { \\lambda ^ i \\rho ^ j ( e ) } \\rho = L _ { ( L _ { \\rho ( e ) } \\rho ^ { - r } ) ^ i \\rho ^ j ( e ) } \\rho = L _ { \\rho ^ { j - i r } ( e ) } \\rho = L _ { \\rho ( e ) } ^ { j - i r } \\rho \\\\ & = ( \\lambda \\rho ^ { r } ) ^ { j - i r } \\rho = \\lambda ^ { j - i r } \\rho ^ { 1 + r ( j - i r ) } = \\lambda ^ { j - i r } \\rho ^ { 1 + j r - i r ^ 2 } \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} \\mathcal { O } _ A ( n ) = \\mathrm { H o m } ( \\mathbb { K } [ C _ n ] \\otimes A ^ { \\otimes n } , A ) , ~ n \\geq 1 . \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} \\bigcup \\limits _ { i = 3 } ^ { 5 } C N _ { C _ { 1 2 } ( 1 , 3 ) } ( V _ i ) \\subseteq V _ o , \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} \\Theta _ \\mathbf { F } [ \\varphi ] = \\Omega _ \\mathbf { F } [ \\varphi ^ + ] - \\Omega _ \\mathbf { F } [ \\varphi ^ - ] , \\varphi \\in \\ell _ \\infty ( A ) . \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} \\left | \\frac { ( d / d r ) \\Psi ( r e ^ { i f ( r ) } ) } { \\Psi ( r e ^ { i f ( r ) } ) } \\right | & = \\left | \\frac { \\Psi ' ( r e ^ { i f ( r ) } ) } { \\Psi ( r e ^ { i f ( r ) } ) } \\right | \\left | \\frac { d ( r e ^ { i f ( r ) } ) } { d r } \\right | \\\\ & = \\left | \\frac { 1 } { r e ^ { i f ( r ) } } - 2 \\beta \\psi ' _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) \\right | \\left | e ^ { i f ( r ) } ( 1 + i r f ' ( r ) ) \\right | \\\\ & = \\frac { 1 } { r } \\left | 1 - 2 \\beta r e ^ { i f ( r ) } \\psi ' _ { \\mu _ { 1 } } ( r e ^ { i f ( r ) } ) \\right | \\sqrt { 1 + r ^ { 2 } f ' ( r ) ^ { 2 } } . \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { N } } \\frac { w _ r ( Q _ j ) } { w _ r ( Q ) } \\leq \\frac { w _ r \\left ( \\bigcup _ { j \\in \\mathbb { N } } Q _ j \\right ) } { w _ r ( Q ) } \\leq \\left ( \\frac { \\mu \\left ( \\bigcup _ { j \\in \\mathbb { N } } Q _ j \\right ) } { \\mu ( Q ) } \\right ) ^ { 1 / r ' } , \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} b _ { 1 , 0 } ( s ) & = \\prod _ { j = 1 } ^ d ( s _ 1 + \\dfrac { d + 1 } { 2 } - \\dfrac { j - 1 } { 2 } ) \\prod _ { k = 1 } ^ { n - d } ( s _ 1 + s _ 2 + \\dfrac { n + 1 } { 2 } - \\dfrac { k - 1 } { 2 } ) , \\\\ b _ { 0 , 1 } ( s ) & = \\prod _ { j = 1 } ^ d ( s _ 2 + \\dfrac { d + 1 } { 2 } - \\dfrac { j - 1 } { 2 } ) ( s _ 2 + \\dfrac { n } { 2 } - \\dfrac { j - 1 } { 2 } ) \\prod _ { k = 1 } ^ { n - d } ( s _ 1 + s _ 2 + \\dfrac { n + 1 } { 2 } - \\dfrac { k - 1 } { 2 } ) . \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} w _ n ( x ) = f ( \\phi _ n ( x ) , x ) \\mbox { f o r $ x \\in \\mathbb { Z } _ + $ } , \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} z ^ { I } - A _ { I , l } ( z , \\overline { z } ) q _ { l } ( z , \\overline { z } ) = C _ { I , l } ( z , \\overline { z } ) , \\quad \\mbox { f o r a l l $ I \\in \\mathbb { N } $ h a v i n g l e n g t h $ p $ , f o r a l l $ l = 1 , \\dots , N $ . } \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} R i c ( \\omega ( t ) ) = - \\omega ( t ) + t \\hat \\omega . \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} \\| \\psi \\| _ { \\mathcal S _ N } : = \\sup _ { ( x , y , z ) \\in \\mathbb R ^ 3 } \\Big ( 1 + | ( x , y , z ) | ^ N \\Big ) \\sum _ { \\alpha , \\beta , \\gamma = 0 } ^ N \\Big | \\partial _ x ^ \\alpha \\partial _ y ^ \\beta \\partial _ z ^ \\gamma \\psi ( x , y , z ) \\Big | . \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} [ v _ a , v _ b ] = C ^ c { } _ { a b } v _ c , \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} \\mathbb { P } _ { n _ { l } } \\left [ \\left | \\sum _ { i = 1 } ^ { k } \\frac { 2 \\mu _ { i } W _ { n _ { l } , i } - \\mu _ { i } ^ 2 } { 2 \\sigma _ { i } ^ 2 } \\right | > M \\right ] \\le \\frac { C ' } { M ^ 2 } \\le \\frac { \\delta } { 1 0 0 } \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} \\langle h _ { 1 } , h _ { 2 } \\rangle ^ 2 = c ^ 2 = \\langle h _ { 1 } , h _ { 1 } \\rangle \\langle h _ { 2 } , h _ { 2 } \\rangle \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} Y _ k = \\begin{bmatrix} 0 & Y _ { k - 1 } \\\\ Y _ { k - 1 } & \\gamma T _ { k - 1 } \\end{bmatrix} , Z _ k = \\begin{bmatrix} 0 & Z _ { k - 1 } \\\\ Z _ { k - 1 } & \\gamma S _ { k - 1 } \\end{bmatrix} \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} \\widehat { \\beta } _ 1 = \\xi _ 2 + \\xi _ 3 , \\widehat { \\beta } _ 2 = \\xi _ 3 + \\xi _ 1 , \\widehat { \\beta } _ 3 = \\xi _ 1 + \\xi _ 2 . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} \\frac { Y ( t ) } { d t } = f ( t , Y ( t ) ) , t \\in [ 0 , T ) \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} \\Theta ( u ) : = \\begin{cases} 0 , & u ( x , y ) \\geq - c ( x , y ) , \\forall ( x , y ) \\in X \\times Y \\\\ + \\infty , & \\end{cases} \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} X ^ I : \\ I = 0 , \\cdots , n , \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} [ v _ a , v _ b ] = C ^ c { } _ { a b } v _ c , C ^ c { } _ { a b } \\in C ^ \\infty ( M ) , \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} \\delta ( 2 H ) = \\Delta u + 2 ( 2 H ^ 2 - K ) u = - \\frac { 2 } { r ^ 2 } u + \\frac { 2 } { r ^ 2 } u = 0 . \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} | 1 - F _ { 3 } ( r , \\rho , \\lambda ( s ) ) | & \\leq C r \\int _ { 0 } ^ { 1 } \\frac { r _ { \\sigma } \\lambda ( s ) ^ { 2 \\alpha - 2 } ( 1 + \\lambda ( s ) ^ { 2 \\alpha - 2 } ( \\rho ^ { 2 } + r _ { \\sigma } ^ { 2 } ) ) } { ( 1 + 2 ( \\rho ^ { 2 } + r _ { \\sigma } ^ { 2 } ) \\lambda ( s ) ^ { 2 \\alpha - 2 } + ( \\rho ^ { 2 } - r _ { \\sigma } ^ { 2 } ) ^ { 2 } \\lambda ( s ) ^ { 4 \\alpha - 4 } ) ^ { 3 / 2 } } d \\sigma \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in \\mathfrak { S } _ n } W _ 2 ( \\sigma ) = \\sum _ { \\sigma \\in \\mathfrak { S } _ n } W _ 3 ( \\sigma ) = \\sum _ { \\sigma \\in \\mathfrak { S } _ n } W _ 4 ( \\sigma ) = E _ n , \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} | \\partial _ { r } ^ { 2 } v _ { 2 } ( t , r ) | \\leq \\begin{cases} \\frac { C r } { t ^ { 4 } \\log ^ { b } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C \\log ( r ) } { | t - r | ^ { 3 } } , \\frac { t } { 2 } < r \\neq t \\\\ \\end{cases} \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} ( i ) \\ a _ 1 < b _ 1 < b _ 2 < a _ 2 \\Longrightarrow & Q ( A , B ) = 2 \\cdot 2 - 4 = 0 , \\\\ ( i i ) \\ b _ 1 < a _ 1 < b _ 2 < a _ 2 \\Longrightarrow & Q ( A , B ) = 2 \\cdot 3 - 4 = 2 . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} P _ { \\mathbf { m } } \\left ( \\mathbf { 1 } ; \\frac { d } { 2 } \\right ) & = \\prod _ { ( i , j ) \\in \\mathbf { m } } \\frac { j - 1 + \\frac { d } { 2 } ( r - i + 1 ) } { m _ { i } - j + \\frac { d } { 2 } ( m _ { j } ^ { \\prime } - i + 1 ) } = \\prod _ { 1 \\leq i < j \\leq r } \\frac { \\left ( \\frac { d } { 2 } ( j - i + 1 ) \\right ) _ { m _ { i } - m _ { j } } } { \\left ( \\frac { d } { 2 } ( j - i ) \\right ) _ { m _ { i } - m _ { j } } } . \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} S _ 2 = ( - 1 ) ^ { k - 2 + \\lfloor \\frac { k - 2 } { p } \\rfloor } q ( k - 2 ) ! \\left ( ( \\lfloor k / p \\rfloor + \\varepsilon ) \\binom { q / p - 1 } { \\lfloor ( k - 2 ) / p \\rfloor } - q / p \\binom { q / p - 2 } { \\lfloor ( k - 2 ) / p \\rfloor - 1 } \\right ) , \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} \\tau _ B ( \\mathrm { G r } ( T ) ) : = \\{ ( T u , u + B ( T u ) ) | ~ u \\in M \\} . \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} \\{ P _ { i , 1 } ( X ) Y _ { 1 } + P _ { i , 2 } ( X ) Y _ { 2 } = Q _ { i } ( X , Y _ { 1 } , Y _ { 2 } ) \\} _ { i = 1 } ^ { q } = \\mathcal { A } \\subset \\mathcal { Q } ^ { ( n ) } \\setminus \\{ 0 \\} , \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} \\sigma _ { \\alpha } ( X , T _ { L , n } ) = \\sum _ { j = 0 } ^ { n - 1 } \\sigma _ { L } ( e ^ { 2 \\pi i j / n } , \\alpha ) - \\sum _ { j = 1 } ^ { n - 1 } \\sigma _ { L } ( e ^ { 2 \\pi i j / n } , 1 ) . \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} - \\int _ { \\R ^ 2 } \\Delta _ q ( u ^ 2 \\partial _ 2 f ) \\Delta _ q f ~ d x = Q _ 1 + Q _ 2 + Q _ 3 , \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} \\bigoplus _ { i = 1 } ^ 3 H ^ 0 ( X , D - E _ i ) \\to H ^ 0 ( X , D ) , \\ ( g _ 1 , g _ 2 , g _ 3 ) \\mapsto f _ 1 g _ 1 + f _ 2 g _ 2 + f _ 3 g _ 3 . \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} \\Xi _ n ( \\tau _ i ) = \\Xi _ n ( [ Z ^ i _ n ] ) - \\Xi _ n ( [ Y _ { 1 ^ n } ] ) = \\Xi _ n ( [ Z ^ i _ n ] ) - \\widetilde { \\xi } _ { ( 1 , 1 ) } . \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} \\ell | I _ { 1 } | \\sim \\sum _ { i = 1 } ^ { \\ell } | I _ { i } | = \\left | \\bigcup _ { i = 1 } ^ { \\ell } I _ { i } \\right | \\leq | 9 I _ { 1 } | \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} & ( f ^ { m - i } ( x _ { n - m + i } ) , f ^ { m - i - 1 } ( x _ { n - m + i + 1 } ) ) \\\\ & = ( f ^ { m - i - 1 } ( f ( x _ { n - m + i } ) ) , f ^ { m - i - 1 } ( x _ { n - m + i + 1 } ) ) \\in S \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} \\frac { p _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ( x ) } { p _ { \\nu _ { 1 } \\boxplus \\nu _ { 2 } } ( x ) } = \\beta \\frac { - \\Im G _ { \\mu _ { 1 } } ( s + i f ( s ) ) } { f ( s ) } = \\beta \\int _ { \\mathbb { R } } \\frac { d \\mu _ { 1 } ( t ) } { ( t - s ) ^ { 2 } + f ( s ) ^ { 2 } } . \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} \\dot { V } & = \\dot { x } \\Big ( - \\nabla ^ 2 f ( x ) \\dot { x } - \\Big [ \\frac { \\partial ^ 2 g } { \\partial ^ 2 x } \\Big ] ^ T g ( x ) \\dot { x } - \\Big [ \\frac { \\partial g } { \\partial x } \\Big ] ^ T \\Big [ \\frac { \\partial g } { \\partial x } \\Big ] \\dot { x } \\Big ) \\\\ & \\leq \\dot { x } ^ T [ - \\nabla ^ 2 f ( x ) - k ] \\dot { x } \\\\ & \\leq - \\gamma V ( x ) \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} { \\mathcal D } ( G ^ \\infty ) \\ni { g } _ n : = n \\int _ 0 ^ \\infty \\varphi ( n t ) S ^ t g d t , n = 1 , 2 , . . . . \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} \\frac d { d t } F ( t ) = Q ^ { + } _ { U } ( t , F ) - Q ^ { - } _ { U } ( t , F ( t ) ) , F ( 0 ) = f _ 0 , t \\geq 0 , a . e . , \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\int _ { \\Delta } | J _ 2 | ^ 2 d \\sigma = 0 \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} \\tau ( G \\setminus p _ j ) \\leq | M _ j | = m , \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} | \\psi _ { n , 1 } ^ { \\ast } ( Q ) - \\frac { n - w _ { n } } { n ( 1 + w _ { n } ) } | = | \\frac { L _ { n , 1 } ^ { \\ast } ( q ) } { q } - \\frac { n - w _ { n } } { n ( 1 + w _ { n } ) } | < \\epsilon . \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} & | \\alpha _ 3 | \\leq 2 ^ { 1 0 } A ^ { - 1 } K ^ { - \\frac { 1 } { 2 } } N _ 1 , | \\beta _ 3 | \\sim K ^ { - 1 } N _ 1 , \\\\ ( \\mathfrak { R } _ 1 ^ { A } + \\mathfrak { R } _ 2 ^ { A } & ) \\cap \\mathfrak { R } _ 3 ^ { A } \\not = \\emptyset , \\mathfrak { R } _ 1 ^ { A } \\cap \\mathfrak { D } _ { j _ 1 } ^ A \\not = \\emptyset , \\mathfrak { R } _ 2 ^ { A } \\cap \\mathfrak { D } _ { j } ^ A \\not = \\emptyset . \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} \\omega _ X ^ { - 3 q } \\otimes N = \\omega _ X \\otimes ( \\omega _ X ^ { - q } ) ^ { \\dim X + 1 } \\otimes ( \\omega _ X ^ { - 1 } \\otimes N ) \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} \\Delta f = \\mathrm { t r } ( \\mathrm { H e s s } _ f ) \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} d = \\inf _ { u \\in \\mathcal { N } } J ( u ) , \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} w _ i ^ \\beta ( x ) \\geq 0 \\mbox { f o r } x \\leq 0 , \\beta \\leq \\beta _ i ^ * , \\ ; \\mbox { a n d } w _ i ^ { \\beta _ i ^ * } ( x _ i ^ 0 ) = 0 . \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} Y _ { 0 } = 0 Y _ { \\ell + 1 } = X ( ( Y _ { \\ell } , \\ell L _ { k } ) ) . \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} t \\leq ( G ) & \\leq ( G ) = ( G ^ \\prime ) \\\\ & \\leq ( G _ R ) \\leq t . \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} P _ { n } ^ \\pm \\{ \\ \\cdot \\ \\} = P \\{ \\ \\cdot \\ | \\ V ( 0 ) = \\pm c , \\ N ( s ) = n \\} . \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} - h ^ * K _ X \\cdot C = ( 2 C + ( m + 2 - b ) F _ W ) \\cdot C = - m + 2 - b \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} \\begin{array} { l l } \\int _ 0 ^ s \\frac { d \\norm { v ( t ) } ^ 2 } { 2 d t } + \\eta \\norm { v ( t ) } ^ 2 d t & = \\int _ 0 ^ s \\left ( \\operatorname { d i v } ( p ( t ) ) - \\operatorname { d i v } ( \\bar { p } ( t ) ) , \\int _ t ^ s v ( r ) d r \\right ) d t \\\\ & = \\int _ 0 ^ s \\left ( \\operatorname { d i v } ( p ( t ) ) - \\operatorname { d i v } ( \\bar { p } ( t ) ) , ( s - t ) v ( t + h _ s ) \\right ) d t , \\end{array} \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} B _ { n , \\mathbf { m } } ^ { ( d ) } ( c \\mathbf { z } \\mid c { \\boldsymbol { \\omega } } ) = & c ^ { | \\mathbf { m } | - n } B _ { n , \\mathbf { m } } ^ { ( d ) } ( \\mathbf { z } \\mid { \\boldsymbol { \\omega } } ) \\ , \\ , \\ , ( c \\in \\mathbb { C } ^ { * } ) . \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\ddot { u } ( t ) + \\eta \\dot { u } ( t ) - \\operatorname { d i v } \\left ( \\frac { \\nabla u ( t ) } { \\abs { \\nabla u ( t ) } } \\right ) = 0 , & \\textmd { i n } \\ \\Omega \\times ( 0 , \\infty ) , \\\\ u ( 0 ) = u _ 0 , \\dot { u } ( 0 ) = 0 , & \\textmd { i n } \\ \\Omega \\times 0 , \\\\ \\partial _ \\nu u ( t ) = 0 , & \\textmd { o n } \\ \\partial \\Omega \\times ( 0 , \\infty ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} \\begin{aligned} & Y _ 1 ( \\Omega _ 1 ) = h ^ 1 _ 1 ( \\phi ^ 1 _ 1 ) \\cdot \\prod _ { a = 2 } ^ { d _ 1 - 1 } ( \\sin \\phi ^ 1 _ a ) ^ { c _ { a - 1 } + \\frac { 1 } { 2 } - \\frac { a - 1 } { 2 } } P ^ { ( c _ { a - 1 } , - 1 / 2 ) } _ { J _ a } ( \\cos 2 \\phi ^ 1 _ a ) \\\\ & c _ a = \\sum _ { s = 1 } ^ { a } J _ s + \\frac { a - 1 } { 2 } + A + J _ 1 , \\end{aligned} \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} & | - \\lambda _ { 1 } ( t ) E _ { v _ { 2 } , i p } ( t , \\lambda _ { 1 } ( t ) ) + \\lambda _ { 2 } ( t ) E _ { v _ { 2 } , i p } ( t , \\lambda _ { 2 } ( t ) ) | \\leq \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 3 } \\sqrt { \\log ( \\log ( t ) ) } } \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} \\mu _ 2 ( d x ) : = \\frac { 1 } { Z } \\exp \\left ( - \\delta \\psi ( x ) \\right ) \\mu _ 1 ( d x ) . \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} \\begin{gathered} a _ { 1 } = r x , a _ { k } = - 2 ( 2 k + r - 2 ) ( 2 k + r - 3 ) ( 2 k - 1 ) ( k - 1 ) x ^ 2 ; \\\\ b _ { 0 } = 1 , b _ { k } = 1 - ( 8 k ^ 2 + 4 k r - 8 k - r + 2 ) x . \\end{gathered} \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} | \\O | = q _ 0 ^ { k - 1 } \\left ( \\frac { q _ 0 ^ { 2 k } - 1 } { q _ 0 ^ 2 - 1 } \\right ) = q _ 0 ^ { k - 1 } \\left ( \\frac { q _ 0 ^ { k } - 1 } { q _ 0 - 1 } \\right ) \\left ( \\frac { q _ 0 ^ { k } + 1 } { q _ 0 + 1 } \\right ) . \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} 6 \\sqrt { n + 2 } { n } P _ { n } ( \\tfrac { x } { | x | } ) = n + \\tfrac { n + 3 } { ( n + 1 ) ^ { 3 } } - \\tfrac { 3 } { n + 1 } = \\tfrac { ( n - 1 ) n ( n + 2 ) ^ { 2 } } { ( n + 1 ) ^ { 3 } } , \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} g _ { i j } = \\frac { 1 } { \\left ( 1 - e ^ { - 2 ( q _ 1 - q _ 2 ) } \\right ) } ( - 1 ) ^ { i + j } e ^ { - | q _ i - q _ j | } . \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} \\| \\nabla u ( t _ 0 ) \\| _ p = \\lim \\limits _ { t \\rightarrow t _ { 0 } } \\| \\nabla u \\| _ p \\geq r _ * > 0 , \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} \\bigcup _ { \\{ i \\ , : \\ , | B _ i | = w _ t \\} } \\bigcup _ { \\{ j : 1 \\leq j \\leq m , \\ , j \\ne i \\} } D ( B _ i , B _ j ) = \\lambda _ t \\boxtimes ( G \\backslash \\{ 0 \\} ) , \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} D : = 4 H - 3 E _ 1 - 3 E _ 2 - 3 E _ 3 - E _ 4 - \\ldots - E _ { 8 - i } \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} 0 < n - a _ { N + 1 } < 1 + \\sum _ { i = 1 } ^ { N } { a _ i } \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} P _ { \\widetilde f } ( A , B ) & = P _ f ( B , A ) , \\\\ P _ { f ^ * } ( A , B ) & = P _ f ( A ^ { - 1 } , B ^ { - 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} \\| \\nabla u \\| _ p ^ p \\ge r _ * ^ p = \\frac { p q } { q - p } M \\quad t \\in [ t _ 0 , T ) . \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} \\varphi ( x _ 0 , x _ 1 , x _ 2 ) = ( F _ 1 , F _ 2 , s _ 9 ) . \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} | \\{ ( a + b ) ( c + d ) = \\lambda ~ : ~ a \\in A , \\ , b \\in B , \\ , c \\in C , \\ , d \\in D \\} | - \\frac { | A | | B | | C | | D | } { p } \\lesssim \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} \\hat { V } _ { \\rm u n d } & = \\frac { V ( s ) } { \\frac 1 2 v _ { n + 2 } L ^ { n + 2 } } , \\\\ \\hat { A } _ { \\rm u n d } & = \\frac { A ( s ) } { \\frac 1 2 a _ { n + 1 } R ^ { n + 1 } } . \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} ( \\partial _ { 2 } ^ { 2 } \\phi ( r , \\xi ) ) _ { 1 } = - 2 \\left ( a ( \\xi ) \\frac { r ^ { 2 } } { 4 \\xi ^ { 5 / 4 } } e ^ { i r \\sqrt { \\xi } } \\sigma ( r \\sqrt { \\xi } , r ) \\right ) \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} u _ { i 0 } \\in C ( [ - h _ 0 , h _ 0 ] ) , \\ \\ u _ { i 0 } ( - h _ 0 ) = u _ { i 0 } ( h _ 0 ) = 0 , \\ u _ { i 0 } ( x ) > 0 \\ \\mbox { i n } ( - h _ 0 , h _ 0 ) , \\ ; 1 \\leq i \\leq m . \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} h _ { 4 } ( x _ { 1 } , x _ { 2 } ) & = ( x _ { 1 } - x _ { 2 } ) ^ { 2 } , \\\\ h _ { 1 2 } ( x _ { 1 } , x _ { 2 } ) & = x _ { 1 } ^ { 2 } x _ { 2 } ^ { 2 } ( x _ { 1 } + x _ { 2 } ) ^ { 2 } , \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} \\beta = \\frac { d \\Lambda } { d \\mu } ( 0 ) . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} Q _ { 1 3 1 3 } = \\frac { { { e } ^ { q _ 3 - q _ 1 } } } { \\Delta _ 3 } , Q _ { 1 2 1 3 } = - \\frac { { { e } ^ { 2 q _ 3 - q _ 2 - q _ 1 } } } { \\Delta _ 3 } , Q _ { 1 2 2 3 } = \\frac { { { e } ^ { 2 q _ 3 - 2 q _ 1 } } } { \\Delta _ 3 } , Q _ { 1 3 2 3 } = - \\frac { { { e } ^ { q _ 3 + q _ 2 - 2 q _ 1 } } } { \\Delta _ 3 } , \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} z _ { x y } = F ( x , y , z , z _ x , z _ y ) ; \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} & & - 3 n \\mu + 4 n ^ 3 - 1 2 n ^ 2 + 8 n & = - 6 n \\mu + 1 6 n ^ 3 - 2 4 n ^ 2 + 8 n \\\\ & & 3 n \\mu & = 1 2 n ^ 3 - 1 2 n , \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} X _ 1 & : = \\{ x \\in \\R ^ 3 \\ , | \\ , x _ 1 \\geq 4 , \\ , x _ 1 + ( x _ 2 - 2 ) ^ 2 + ( x _ 3 + 2 ) ^ 2 \\geq 5 \\} , \\\\ X _ 2 & : = \\{ x \\in \\R ^ 3 \\ , | \\ , x _ 1 ^ 2 + x _ 2 ^ 2 \\leq x _ 3 , \\ , ( x _ 1 - 1 ) ^ 2 + x _ 2 ^ 2 + x _ 3 \\geq 1 , \\ , x _ 2 \\leq 0 \\} . \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} N F _ n : [ e _ i , e _ 1 ] = e _ { i + 1 } , 1 \\leq i \\leq n - 1 \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} \\hat { \\psi } ( b _ { \\Lambda } ) : = \\lim _ { \\Lambda _ { 1 } \\uparrow \\uparrow V } \\left \\langle \\Psi _ { \\Lambda _ 1 } , b _ { \\Lambda } \\Psi _ { \\Lambda _ 1 } \\right \\rangle \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} T ( r ; \\mu ) = T _ R ( r ; \\mu ) + T _ L \\left ( P _ R ( r ; \\mu ) ; \\mu \\right ) , \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} \\langle f | g \\rangle : = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } f ( x ) \\overline { g ( x ) } \\ , d x \\ \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} I = H A , \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} \\begin{aligned} | X | & = | S \\cap X | + | X \\setminus S | \\\\ & = | S \\cap A _ 1 \\cap \\cdots \\cap A _ i \\setminus ( A _ { i + 1 } \\cup \\cdots \\cup A _ { k - 1 } ) | + | A _ 1 \\cap \\cdots \\cap A _ i \\setminus ( A _ { i + 1 } \\cup \\cdots \\cup A _ { k - 1 } \\cup S ) | \\\\ & = b _ { i + 1 } + b _ { i } , \\end{aligned} \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} H _ { j + 1 } = H _ j + \\dots + H _ { j - k + 1 } + H _ { j - k } + H _ { j - k - 1 } + H _ { j - k - 1 } + H _ { j - k - 2 } + \\dots + H _ 1 + 1 . \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { n } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - m ^ 2 ) = \\sum _ { m = 1 } ^ { \\left \\lfloor \\sqrt { \\frac { n } { 2 } } \\right \\rfloor } ( - 1 ) ^ { m - 1 } a _ k ( n - 2 m ^ 2 ) . \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} ( \\partial B _ R \\cap V ) \\oplus \\{ \\gamma ( t ) : 0 \\leq t \\leq 1 \\} \\cap X = \\emptyset . \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} c _ 1 ( L _ H , h ^ { L _ H } ) = g \\pi _ * ( \\omega ^ { \\wedge ( \\dim X + 1 ) } ) \\cdot ( \\dim X + 1 ) \\int _ X c _ 1 ( L ) ^ { \\dim X } . \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\frac { 1 0 0 0 \\log n } { n } } \\frac { n ^ k } { k ! } k ^ { k - 2 } q ^ { k - 1 } ( 1 - q ) ^ { k n } \\frac { \\dd q } { \\sqrt { q } } = \\sqrt { n } \\frac { k ^ { k - 2 } } { k ! } \\int _ 0 ^ { 2 0 0 0 \\log n } x ^ { k - 1 } \\left ( 1 - \\frac { x } { n } \\right ) ^ { k n } \\frac { \\dd x } { \\sqrt { x } } . \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} \\phi _ t ^ * \\left ( \\sum u _ { i } \\sigma _ i \\right ) & = e ^ { \\lambda _ 1 ( t - r ) } \\cos ( \\varphi ) \\ , \\sigma _ 1 + e ^ { \\lambda _ 2 ( t - r ) } \\sin ( \\varphi ) \\ , \\sigma _ 2 \\\\ & = e ^ { \\lambda _ 1 ( t ' ) } \\cos ( \\varphi ) \\ , \\sigma _ 1 + e ^ { \\lambda _ 2 ( t ' ) } \\sin ( \\varphi ) \\ , \\sigma _ 2 \\stackrel { t = r } { = } \\left ( \\cos ( \\varphi ) \\ , \\sigma _ 1 + \\sin ( \\varphi ) \\ , \\sigma _ 2 \\right ) \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} P \\varphi = 2 \\left ( \\delta _ { b } ( ( P _ { 1 } \\varphi ) \\theta ^ { 1 } ) + \\overline { \\delta } _ { b } ( ( \\overline { P } _ { 1 } \\varphi ) \\theta ^ { 1 } ) \\right ) , \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} \\bar \\rho _ { \\phi } ( M ) = \\frac 1 2 \\left ( \\lim _ { \\beta \\to \\alpha ^ { - } } \\rho _ { \\phi _ \\beta } ( M ) + \\lim _ { \\beta \\to \\alpha ^ { + } } \\rho _ { \\phi _ \\beta } ( M ) \\right ) . \\end{align*}"}
-{"id": "8935.png", "formula": "\\begin{align*} A _ 2 = - ( n + 1 ) a _ n \\int _ { z _ 1 } ^ { z _ 2 } H _ 1 ( z ) h _ 0 ^ n h _ 1 d z + a _ n \\left [ h _ 0 ^ n h _ 1 h _ { 1 z } \\right ] _ { z _ 1 } ^ { z _ 2 } . \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} C = \\big \\langle \\big ( ( 1 , 0 ) , \\ldots , ( 1 , 0 ) , \\ , ( 1 ) , \\ , ( 1 ) , \\ , ( 0 ) \\big ) , \\ \\big ( ( 0 , 1 ) , \\ldots , ( 0 , 1 ) , \\ , ( 0 ) , \\ , ( 1 ) , \\ , ( 1 ) \\big ) \\big \\rangle \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} T ( \\tilde { r } ; \\mu ) = T _ R ( \\tilde { r } ; \\mu ) + T _ L \\left ( P _ R ( \\tilde { r } ; \\mu ) ; \\mu \\right ) . \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} \\lambda _ n ( G ^ c ) & = x ^ T D ( G ^ c ) x \\\\ & = x ^ T ( J _ n - I _ n ) x + x ^ T A ( G ) x \\\\ & \\ge x ^ T ( J _ n - I _ n ) x + x ^ T A ( G ' ) x \\\\ & = x ^ T D ( G '^ c ) x . \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} X ( u ) : = - ( \\lambda _ p \\langle 1 | f _ p \\rangle ) _ { p \\ge 0 } \\ , \\ Y ( u ) : = ( \\langle 1 | f _ n \\rangle ) _ { n \\ge 0 } \\ . \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} \\begin{aligned} \\frac 1 2 \\norm { S ( u , v ) - y _ { \\textup d } } { L ^ 2 ( I ; L ^ 2 ( \\Omega ) ) } ^ 2 + \\frac { \\alpha } { 2 } \\left ( \\norm { u } { L ^ 2 ( I ) } ^ 2 + \\norm { v } { L ^ 2 ( I ) } ^ 2 \\right ) & & & \\\\ + \\frac { \\beta } { 2 } \\left ( \\norm { \\partial _ t u } { L ^ 2 ( I ) } ^ 2 + \\norm { \\partial _ t v } { L ^ 2 ( I ) } ^ 2 \\right ) & \\ , \\to \\ , \\min \\limits _ { u , v } & & & \\\\ u ( t ) \\ , \\geq \\ , 0 \\ , \\lor \\ , v ( t ) & \\ , \\geq \\ , 0 & \\qquad & I . \\end{aligned} \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} D _ { \\mathcal { H } , \\pm } = b \\ , \\mathsf { M } ^ { \\mathbb { R } } \\pm \\frac { 1 } { 2 \\pi } \\frac { \\partial \\ ; } { \\partial r } \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} q _ \\eta ( { \\bf { x } } ) : = p _ \\theta ^ { ( \\alpha ) } ( { \\bf { x } } ) = \\widetilde { N } _ { \\eta , \\alpha } [ 1 + b _ \\alpha ( { \\bf { x } } - \\boldsymbol { \\mu } ) ^ \\top \\boldsymbol { \\Sigma } ^ { - 1 } ( { \\bf { x } } - \\boldsymbol { \\mu } ) ] _ + ^ { \\frac { \\alpha } { \\alpha - 1 } } , \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} \\mathcal { S } ^ { m } _ W : = \\{ f \\in C ^ \\infty ( \\bar { \\Omega } _ W ) ~ \\mbox { e v e n a n d h o l o m o r p h i c o n } ~ \\Omega _ W , ~ | f ^ { ( k ) } ( z ) | \\leq C _ k ( 1 + | z | ) ^ { m - k } , ~ \\forall k . \\} , \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} & \\bigg ( - 2 4 , - \\frac { 1 } { 2 4 5 } \\bigg ) , \\bigg ( \\frac { 1 } { 2 } , - \\frac { 2 } { 4 9 } \\bigg ) , \\bigg ( \\frac { m n ( 4 m + 2 n - 3 ) } { ( m + n - 1 ) ( 2 m + 2 n - 1 ) } , \\ \\lambda _ 1 \\bigg ) , \\\\ & \\bigg ( - \\frac { 2 m n ( 3 - 4 m - 2 n + 4 m n ) } { 2 m - 2 n - 1 } , \\ \\lambda _ 2 \\bigg ) , \\bigg ( - \\frac { ( 2 m n + m - 2 n ) ( 2 m n - m - n ) } { m - n } , \\ \\lambda _ 3 \\bigg ) . \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\theta _ 1 ( \\varepsilon ) = \\max \\left \\{ \\theta _ { 1 , 1 } , \\theta _ { 1 , 2 } , \\theta _ { 1 , 3 } \\right \\} < 1 . \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} T _ { \\lambda , N } = B _ { \\lambda , N } ^ { ( - 1 ) } \\circ J _ 0 . \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} Y _ { n _ { l _ { m } } } - \\exp \\left \\{ \\sum _ { i = 1 } ^ { k } \\frac { 2 \\mu _ { i } W _ { n _ { l _ { m } } , i } - \\mu _ { i } ^ 2 } { 2 \\sigma _ { i } ^ 2 } \\right \\} \\stackrel { d } { \\to } H _ { 1 } - \\exp \\left \\{ \\sum _ { i = 1 } ^ { k } \\frac { 2 \\mu _ { i } H _ { i + 1 } - \\mu _ { i } ^ 2 } { 2 \\sigma _ { i } ^ 2 } \\right \\} . \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} \\kappa = \\mu ^ + \\land \\square _ { \\mu , 2 } \\land \\mathsf { S A T P } ( \\kappa ) \\land 2 ^ { \\mu } \\geq \\delta \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} \\begin{pmatrix} a _ 1 & x _ 1 \\\\ a _ 2 & x _ 2 \\end{pmatrix} \\ \\mbox { a n d } \\ \\begin{pmatrix} \\omega _ 1 & x _ 1 \\\\ \\omega _ 2 & x _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} K ( t ) = \\left \\{ \\begin{array} { l l } ( t ^ 2 , K ^ y _ { k + 1 } t ^ { k + 1 } ) + ( O ( t ^ 3 ) , O ( t ^ { k + 2 } ) ) & \\ ; 1 , \\\\ ( t , K ^ y _ { l } t ^ { l } ) + ( O ( t ^ 2 ) , O ( t ^ { l + 1 } ) ) & \\ ; 2 , 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} \\dim H ^ 0 ( X , \\mathcal { O } _ { X } ( D ) ) = 1 . \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} \\begin{cases} \\bigcup _ { n \\in \\N } \\delta _ { n } = \\sigma ( R ) \\\\ n \\geq m \\Rightarrow \\delta _ { n } \\supseteq \\delta _ { m } \\\\ ( \\forall n \\in \\N ) ( g ( \\delta _ { n } ) ) \\end{cases} \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } \\frac { 1 } { z - \\lambda _ k } = \\frac { 1 } { \\lambda } . \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} & \\int _ { \\frac { w } { 2 \\lambda ( t ) } } ^ { \\infty } \\frac { R d R } { ( 1 + R ^ { 2 } ) ^ { 3 } } \\frac { 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } { 2 w \\left ( \\sqrt { ( 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } ) ^ { 2 } + 4 R ^ { 2 } \\lambda ( t ) ^ { 2 } } + 1 + w ^ { 2 } - R ^ { 2 } \\lambda ( t ) ^ { 2 } \\right ) } \\\\ & \\leq \\int _ { \\frac { w } { 2 \\lambda ( t ) } } ^ { \\infty } \\frac { d R } { R ^ { 5 } } \\frac { R ^ { 2 } \\lambda ( t ) ^ { 2 } } { w } \\leq \\frac { C \\lambda ( t ) ^ { 4 } } { w ^ { 3 } } , w \\geq 1 \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} x \\eta _ { \\mu } ( 1 / x ) ^ { k } = \\eta _ { \\nu ^ { \\boxtimes k } } ( 1 / x ) ^ { k - 1 } , x \\in ( 0 , + \\infty ) . \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} & r \\int _ { 0 } ^ { 1 } d \\beta \\int _ { t } ^ { \\infty } d s \\int _ { B _ { s - t } ( 0 ) \\cap ( B _ { \\frac { s } { 2 } } ( - \\beta x ) ) ^ { c } } \\frac { d A ( y ) } { ( s - t ) } | \\frac { v _ { 4 , c } ( s , | \\beta x + y | ) } { | \\beta x + y | } | \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} \\alpha ^ 1 _ 1 = ( A + 3 J _ 1 ) ^ 2 , ~ ~ ~ J _ 1 = 0 , 1 , 2 \\cdots . \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} d \\omega _ i = - d x _ i \\wedge d \\eta + d \\phi \\wedge d x _ j \\wedge d x _ k = 0 . \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } m ( r ) = 0 , ( r + 1 ) ^ { - 1 } m ( r ) \\in L ^ 1 ( 0 , \\infty ) . \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} \\phi ( K ) : = A ^ { \\top } K + K A + Q - K B B ^ { \\top } K = 0 \\ , . \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} h ( n ) = \\binom { n + 2 } { 2 } - 2 \\binom { n + 2 - t } { 2 } + \\binom { n - 2 t + 4 } { 2 } - \\binom { n + 1 - t } { 1 } - \\binom { n + 1 - t - \\nu } { 1 } \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} \\mu ^ { k } _ { i , \\pm } = 1 - \\frac { \\Delta t _ k } { 2 } \\left [ \\left ( \\Delta t _ k b ^ { k } _ i \\lambda ^ { k } _ i + \\eta \\right ) \\pm \\sqrt { \\left ( \\Delta t _ k b ^ { k } _ i \\lambda ^ { k } _ i + \\eta \\right ) ^ 2 - 4 b ^ { k } _ i \\lambda ^ { k } _ i } \\right ] i = 1 , 2 , \\cdots , M N , \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} \\big | P ( \\alpha _ i ) \\big | _ v \\le p ^ 2 ( N - 1 ) \\frac { 1 } { \\delta } \\prod _ { k = 1 } ^ s \\max \\{ | \\alpha _ i - \\alpha _ k | _ v , \\ , \\delta \\} ^ { n _ k } \\le p ^ 3 N \\prod _ { k = 1 } ^ s \\max \\{ | \\alpha _ i - \\alpha _ k | _ v , \\ , \\delta \\} ^ { n _ k } . \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} b & = \\tau ( g ) - g + h . \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} H _ { k + 4 } = H _ { k + 3 } + N H _ 2 = H _ { k + 3 } + 2 N . \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} H _ { L + 1 } \\leq 1 + \\sum _ { i = 1 } ^ L H _ i \\qquad H _ { L + 2 } \\leq 1 + \\sum _ { i = 1 } ^ { L + 1 } H _ i , \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} x _ k = \\lambda _ k \\sum \\limits _ { i = 0 } ^ { 2 ^ { k } - 1 } ( - 1 ) ^ i v _ k ^ { i } a _ k v _ k ^ { - i } \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} \\lim _ n \\frac { 1 } { n } H ( \\mu _ { \\lambda } ^ { ( n ) } ) = \\inf _ n \\frac { 1 } { n } H ( \\mu _ { \\lambda } ^ { ( n ) } ) . \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} \\frac { u ^ { N } } { N ! } \\frac { u ^ { m } } { m ! } & = \\binom { m + N } { m } \\frac { u ^ { m + N } } { ( m + N ) ! } . \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} \\langle F _ 1 , D _ 1 \\rangle = - 2 \\langle F _ 2 , D _ 2 \\rangle = - 2 . \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} \\begin{array} { c } \\langle \\widehat { q } _ 1 f , f \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = \\int _ { \\mathbb { R } ^ 2 } \\left [ \\left ( \\lambda x _ 1 + \\frac { i \\theta } { 2 \\lambda } \\frac { \\partial } { \\partial x _ 2 } + E x _ 1 ^ 2 \\right ) f ( x _ 1 , x _ 2 ) \\right ] \\overline { f ( x _ 1 , x _ 2 ) } d x _ 1 d x _ 2 = \\\\ \\\\ = \\lambda x _ 1 ^ { ( 0 ) } + E \\left ( ( x _ 1 ^ { ( 0 ) } ) ^ 2 + \\frac { a } { 4 } \\right ) \\end{array} \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} & \\Bigl | \\int _ { \\R ^ 3 \\times \\R ^ 3 } { h ( \\tau _ 1 + \\tau _ 2 , \\xi _ 1 + \\xi _ 2 , \\eta _ 1 + \\eta _ 2 ) f ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) g ( \\tau _ 2 , \\xi _ 2 , \\eta _ 2 ) } d \\sigma _ 1 d \\sigma _ 2 \\Bigr | \\\\ & \\lesssim ( A K ) ^ { \\frac { 1 } { 2 } } N _ 1 ^ { - 2 } ( L _ 0 L _ 1 L _ 2 ) ^ { \\frac { 1 } { 2 } } \\| f \\| _ { L ^ 2 } \\| g \\| _ { L ^ 2 } \\| h \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} I ( m D ) = I _ R ( m D ) = \\Gamma ( X , \\mathcal O _ X ( - m D ) ) \\cap R . \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} m ( 1 - m ^ 2 ) & = n ( 1 - n ^ 2 ) \\\\ m - n & = m ^ 3 - n ^ 3 \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} \\int _ { D ^ c } L \\varphi ( x ) \\lambda ( d x ) & = \\int _ { D ^ c } \\left ( \\int _ { D } \\varphi ( y ) j ( | x - y | ) d y \\right ) \\lambda ( d x ) \\\\ & = \\int _ { D } \\varphi ( y ) \\left ( \\int _ { D ^ c } j ( | x - y | ) \\lambda ( d x ) \\right ) d y , \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} t \\ , \\frac { \\partial u } { \\partial t } - ( b ( t , x ) + b _ 1 ( t , x ) ) \\frac { \\partial u } { \\partial x } = ( \\lambda ( t , x ) + \\lambda _ 1 ( t , x ) ) u \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} c _ { 1 J } = \\frac { 2 \\left ( a _ { 2 J } b _ { 0 J } - a _ { 0 J } b _ { 2 J } \\right ) } { a _ { 0 J } ^ 2 } , \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} h ^ { \\prime } ( t ) = \\frac { \\boldsymbol { \\eta } C e ^ { C t } ( - C ) \\mathbf { 1 } \\ , \\ , \\boldsymbol { \\eta } e ^ { C t } \\mathbf { 1 } - \\boldsymbol { \\eta } C e ^ { C t } \\mathbf { 1 } \\ , \\ , \\boldsymbol { \\eta } e ^ { C t } ( - C ) \\mathbf { 1 } } { ( \\boldsymbol { \\eta } e ^ { C t } \\mathbf { 1 } ) ^ 2 } . \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} g = \\phi ^ { - 1 } \\eta ^ 2 + \\phi g _ { \\mathbb { R } ^ 3 } , \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } \\sum _ { j , k = 1 } ^ { n } \\partial _ { j } a _ { j k } \\partial _ { k } \\bigg ] \\tilde { w } & = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ \\tilde { w } & = w \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} c : = \\inf _ { h \\in \\Gamma } \\max _ { u \\in Q } I ( h ( u ) ) , \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} \\| a \\| ^ 2 & = | \\psi ( a , ( a + \\lambda b ) ^ * | \\\\ & \\leq \\| \\psi \\| \\| a \\| \\| ( a + \\lambda b ) ^ * \\| \\\\ & = \\| a \\| \\| a + \\lambda b \\| , \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} u \\frac { u ^ { m } } { m ! } = \\binom { m + 1 } { m } \\frac { u ^ { m + 1 } } { ( m + 1 ) ! } = ( m + 1 ) \\frac { u ^ { m + 1 } } { ( m + 1 ) ! } . \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} B _ { k l } ( v ) = \\left ( v ^ k v ^ l - \\frac { \\delta _ { k l } } { 3 } | v | ^ 2 \\right ) \\sqrt { \\mu } , \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} \\deg ( D ) : = \\langle D , H \\rangle \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} \\| a \\| _ { l _ { 1 , g } } = \\sum \\limits _ { m \\in \\mathbb Z ^ c } g ( m ) | a _ m | . \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} L _ { U \\cap V } \\circ L _ { U \\cap V } = \\{ ( x , z ) \\in G \\times G \\mid x ^ { - 1 } y , y ^ { - 1 } z \\in U \\cap V y \\in G \\} \\subseteq L _ W . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} \\mathcal { W '' } _ { k } \\left [ I ; J \\right ] \\in \\mathcal { M } _ { N ^ { p } \\times N ^ { p } } \\left ( \\mathbb { C } \\right ) , \\quad \\mbox { f o r a l l $ k = 1 , \\dots , p + 1 $ } . \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} 4 N \\leq \\sum _ { i = 1 } ^ { k + 3 } ( f _ { i + 1 } - 1 ) + f _ 1 = \\sum _ { i = 1 } ^ { k + 4 } f _ i + ( k + 3 ) = f _ { k + 6 } + ( k + 5 ) . \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} \\frac { b _ { 0 L } } { a _ { 0 L } } = \\frac { b _ { 0 R } } { a _ { 0 R } } . \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} s ( \\check { x } ) = \\biggl \\{ & \\left ( \\frac { 1 } { r } A \\check { x } + \\frac { 1 } { r } p + \\frac { 2 \\pi } { r } A \\mathcal { B } M _ s \\right ) \\in \\check { \\pi } ^ { - 1 } ( \\check { x } ) \\approx \\mathbb { R } ^ n / 2 \\pi \\mathbb { Z } ^ n \\ | \\\\ & M _ s = ( m _ 1 , \\cdots , m _ s , 0 , \\cdots , 0 ) ^ t \\in \\mathbb { Z } ^ n , \\ 0 \\leq m _ i \\leq r _ i ' - 1 , \\ i = 1 , \\cdots , s \\biggr \\} , \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} \\mathcal R _ N : = \\left \\{ \\frac { P ( z ) } { Q ( z ) } \\ : \\ , P \\in \\C _ { \\le N } [ z ] \\right \\} , Q ( z ) : = \\det ( I d - z M _ N ) , \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} u = u _ 0 \\mbox { o n } \\ \\Gamma \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} A _ \\top : = A \\sqcup \\{ \\top \\} \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} a _ 1 ' & = a _ 1 b _ 2 = 1 , \\cr a _ 2 ' & = - a _ { 2 } a _ { 3 } { b _ 4 } = - x ^ 3 , \\cr a _ { 3 k } ' & = - a _ { 6 k - 2 } a _ { 6 k - 1 } { b _ { 6 k - 4 } } { b _ { 6 k } } = - ( 4 k - 1 ) ^ 2 ( 2 k - 1 ) x ^ 3 , \\\\ a _ { 3 k + 1 } ' & = - a _ { 6 k } a _ { 6 k + 1 } { b _ { 6 k - 2 } } { b _ { 6 k + 2 } } = - 4 k ^ 2 x ^ 2 , \\\\ a _ { 3 k + 2 } ' & = - a _ { 6 k + 2 } a _ { 6 k + 3 } { b _ { 6 k } } { b _ { 6 k + 4 } } = - ( 4 k + 1 ) ^ 2 ( 2 k + 1 ) x ^ 3 . \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} W _ \\alpha = \\{ ( u , \\theta ) \\in V _ 0 \\times V _ 1 : \\int _ \\Omega u _ 1 ( x , t ) \\dd x = \\alpha \\} \\ ; . \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} & \\lim _ { t \\to 0 } \\int _ { \\R ^ d } \\sqrt { R } ( t , y ) \\psi ( y ) \\ , \\dd y = \\int _ { \\R ^ d } \\sqrt { R _ 0 } ( y ) \\psi ( y ) \\ , \\dd y , \\\\ & \\lim _ { t \\to 0 } \\int _ { \\R ^ d } \\sqrt { R } ( t , y ) ( \\sqrt { R } U ) ( t , y ) \\psi ( y ) \\ , \\dd y = \\int _ { \\R ^ d } \\sqrt { R _ 0 } ( y ) \\Lambda _ 0 ( y ) \\psi ( y ) \\ , \\dd y . \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} A = P D P ^ { - 1 } = P e ^ { \\tilde D } P ^ { - 1 } = e ^ { P \\tilde D P ^ { - 1 } } = e ^ B , \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} F ^ * \\rho ' _ { H ' } = a _ { H ' } \\prod _ { H \\in M _ 1 ( X ) } \\rho _ H ^ { e ( H , H ' ) } \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} x ( g ) = ( g ^ { - 1 } x ) ( 1 _ G ) = x _ g ( 1 _ G ) = g ^ { - 1 } x _ { 1 _ G } ( 1 _ G ) = x _ { 1 _ G } ( g ) . \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} \\begin{cases} [ 1 / 2 + x , 1 / 2 + 3 x , 1 / 2 + 4 x , 4 x , 1 2 x , 1 / 2 + 6 x ] , \\\\ { } [ x , 3 x , 1 / 2 + 4 x , 4 x , 1 2 x , 1 / 2 + 6 x ] . \\end{cases} \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } | | \\lambda _ { \\ker } - ( \\lambda _ { \\infty } ) _ { \\ker } | | _ { S ^ { k , 2 } } = 0 \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} | | \\partial _ { t } ( u - v _ { 2 } ) | | _ { L ^ { 2 } ( r d r ) } ^ { 2 } + | | u - Q _ { \\frac { 1 } { \\lambda ( t ) } } - v _ { 2 } | | _ { \\dot { H } ^ { 1 } _ { e } } ^ { 2 } \\leq \\frac { C } { t ^ { 2 } \\log ^ { 2 b } ( t ) } \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} s _ 1 : = \\frac { q _ 1 + q _ 2 } { 2 } , \\ ; \\ ; \\ ; s _ 2 : = \\frac { q _ 1 - q _ 2 } { 2 } \\ ; . \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} & \\psi ( X ^ 1 , \\cdots , X ^ i + 2 \\pi r _ i ' , \\cdots , X ^ n ) = e ^ { \\frac { r _ i ' } { r } \\mathbf { i } ( q ' - q ) ^ t \\mathcal { B } _ i } \\psi ( X ^ 1 , \\cdots , X ^ i , \\cdots , X ^ n ) \\ ( i = 1 , \\cdots , s ) , \\\\ & \\psi ( X ^ 1 , \\cdots , X ^ i + 2 \\pi , \\cdots , X ^ n ) = e ^ { \\frac { \\mathbf { i } } { r } ( q ' - q ) ^ t \\mathcal { B } _ i } \\psi ( X ^ 1 , \\cdots , X ^ i , \\cdots , X ^ n ) \\ ( i = s + 1 , \\cdots , n ) \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} \\Pi u ( z ) = \\langle ( I d - z S ^ * ) ^ { - 1 } \\Pi u | 1 \\rangle \\ , \\forall \\ , | z | < 1 . \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & \\ , \\rightarrow \\ , \\min & & & \\\\ g _ i ( x ) & \\ , \\leq \\ , 0 & \\qquad & i \\in \\mathcal M & \\\\ h _ j ( x ) & \\ , = \\ , 0 & \\qquad & j \\in \\mathcal P & \\\\ \\varphi ^ t ( G _ l ( x ) , H _ l ( x ) ) & \\ , \\leq \\ , 0 & \\qquad & l \\in \\mathcal Q & \\end{aligned} \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} \\mathbf { P } \\left ( ( \\eta ^ N _ n ) _ { n = 1 } ^ { N } = ( x _ n ) _ { n = 1 } ^ { N } \\right ) = \\mathbf { P } \\left ( ( \\eta _ n ) _ { n = 1 } ^ { N } = ( x _ n ) _ { n = 1 } ^ { N } \\ : \\vline \\ : S _ N > 0 \\right ) , \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} Z \\times X & = \\bigsqcup _ { x \\in X } ( Z \\times \\{ x \\} ) = \\bigsqcup _ { x \\in X } \\bigsqcup _ { y \\in Y } ( ( Z \\wedge [ x | - > y ] ) \\times \\{ x \\} ) . \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\delta ^ { ( s , t , k ) } _ i = O _ p ( n ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} \\Lambda = \\begin{pmatrix} \\lambda _ { 1 } & 0 & \\dots & 0 \\\\ 0 & \\lambda _ { 2 } & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & \\lambda _ { N } \\end{pmatrix} , \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} m m ^ * = 1 _ A . \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} \\sin \\phi _ 1 & = \\sin ( \\phi _ 2 + \\phi _ 3 ) = \\sin \\phi _ 2 \\cos \\phi _ 3 + \\cos \\phi _ 2 \\sin \\phi _ 3 \\\\ \\sin \\phi _ 1 & = \\frac { 2 q ( 1 - p ^ 2 ) } { ( 1 + q ^ 2 ) ( 1 + p ^ 2 ) } + \\frac { 2 p ( 1 - q ^ 2 ) } { ( 1 + q ^ 2 ) ( 1 + p ^ 2 ) } \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} | \\partial _ { R } ( v _ { c o r r } ( x , R \\lambda ( x ) ) ) | \\leq C \\lambda ( x ) \\begin{cases} \\frac { \\log ( \\log ( x ) ) } { x ^ { 2 } \\log ^ { b + 1 } ( x ) } , R \\lambda ( x ) \\leq \\log ^ { N } ( x ) \\\\ \\frac { 1 } { x ^ { 2 } \\log ^ { b } ( x ) } , \\log ^ { N } ( x ) \\leq R \\lambda ( x ) \\leq \\frac { x } { 2 } \\\\ \\frac { 1 } { \\sqrt { x } } , R \\lambda ( x ) > \\frac { x } { 2 } \\end{cases} \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} \\sum _ { k = 3 } ^ \\infty \\frac { k ^ { k - 2 } } { k ! } \\Bigg [ k - 1 - k \\beta \\Bigg ] \\big ( \\beta e ^ { - \\beta } \\big ) ^ { k - 2 } & < \\left ( \\sum _ { k = 3 } ^ \\infty \\frac { \\frac { 3 } { 5 } k - 1 } { k ^ { 5 / 2 } } \\right ) \\frac { e ^ 2 } { \\sqrt { 2 \\pi } } \\beta e ^ { 1 - \\beta } \\\\ & < \\frac { 3 } { 5 } \\frac { e ^ 2 } { \\sqrt { 2 \\pi } } \\beta e ^ { 1 - \\beta } , \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{gather*} { \\inf _ { u \\in X _ { 0 , s _ i , p _ i } \\setminus \\{ 0 \\} } \\max _ { t \\geq 0 } J _ i ( t u ) = c _ i ^ { * * } = c _ i = J _ i ( \\bar u ) \\leq \\max _ { t \\in [ 0 , 1 ] } J _ i ( t L \\bar u ) . } \\end{gather*}"}
-{"id": "5645.png", "formula": "\\begin{align*} Z _ p = \\mathcal { S } _ { 1 , p } ( \\alpha _ 1 , \\cdots , \\alpha _ p ) , ~ ~ ~ Y _ p = \\mathcal { S } _ { p , D } ( \\alpha _ p , \\cdots , \\alpha _ D = 0 ) . \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} \\mathcal { P } _ 1 = \\{ p : p \\mid u - 1 \\} \\mathcal { P } _ 2 = \\{ p : p \\mid u ^ 2 + u + 1 \\} . \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} D ( H ) = \\bigl \\{ x \\in D ( A ^ \\star ) ; T \\mathbf { S } _ { - , P _ 1 / \\alpha } { F } _ 2 ( \\mathbf { S } _ { a , b } ^ { - 1 } x ) = \\mathbf { S } _ { + , P _ 1 / \\alpha } { F } _ 1 ( \\mathbf { S } _ { a , b } ^ { - 1 } x ) \\bigr \\} \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} u ( r , \\omega ) & = v _ 1 ( r ) w _ 1 ( \\omega ) + \\sum _ { k \\geq 2 } c _ k r ^ { \\gamma _ k ^ + } w _ k ( \\omega ) \\\\ ( \\ u ( r , \\omega ) & = v _ 1 ( r ) w _ 1 ( \\omega ) + \\sum _ { k \\geq 2 } c _ k r ^ { \\gamma _ k ^ - } w _ k ( \\omega ) \\ ) \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in \\{ 0 , 1 \\} ^ n } x _ { \\alpha } z ^ { \\alpha } = \\sum _ { A \\subseteq [ n ] } x _ A z _ A \\leftrightsquigarrow \\sum _ { A \\subseteq [ n ] } x _ A \\varepsilon _ A \\ , . \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} i ^ { - 1 } \\bigl ( ( \\mathfrak M ( H ^ \\infty ( U ) ) \\setminus \\overline { O } _ 2 ) \\cap O _ 1 \\bigr ) = Y _ 1 \\cap Y _ 3 . \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} \\sum _ { j \\in \\Bbb { Z } } | \\widehat { \\psi ^ { ( 1 ) } } ( 2 ^ { j } \\xi _ 1 ) | ^ 2 = 1 \\mbox { f o r a l l } \\xi _ 1 \\in \\Bbb { R } \\backslash \\{ 0 \\} , \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} { } ^ Q \\Omega ^ + _ a = ( \\rho ^ T E \\rho ) ^ { - 1 } \\rho _ a ( \\rho ^ T E \\rho ) - ( E \\rho ) ^ + E ^ T \\rho { } ^ Q \\Omega ^ - _ a , \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} \\mathcal T _ A ( \\bar x ) : = \\left \\{ d \\in \\R ^ n \\ , \\middle | \\ , \\begin{aligned} & \\exists \\{ x _ k \\} _ { k \\in \\N } \\subset A \\ , \\exists \\{ \\tau _ k \\} _ { k \\in \\N } \\subset \\R _ + \\colon \\\\ & x _ k \\to \\bar x , \\ , \\tau _ k \\downarrow 0 , \\ , ( x _ k - \\bar x ) / \\tau _ k \\to d \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} 2 m - L + \\sum _ { i = L - m + 2 } ^ { m + 1 } i \\geq 2 L - 3 ( m + 1 ) + 2 . \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} & R \\Bigl ( t , x , v + w , \\frac { \\partial v } { \\partial x } + \\frac { \\partial w } { \\partial x } \\Bigr ) - R \\Bigl ( t , x , v , \\frac { \\partial v } { \\partial x } \\Bigr ) \\\\ & = c _ 1 ( t , x ) w + c _ 2 ( t , x ) \\frac { \\partial w } { \\partial x } + \\sum _ { j + \\alpha \\geq 2 } c _ { j , \\alpha } ( t , x ) w ^ j \\Bigl ( \\frac { \\partial w } { \\partial x } \\Bigr ) ^ { \\alpha } , \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} p _ n ( x _ 1 , \\dots , x _ n ) = \\sum _ M \\prod _ { r = 1 } ^ n x _ r ^ { t _ r ( M ) } \\in \\Z [ x _ 1 , \\dots , x _ n ] \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} & \\langle M y , \\Delta _ { \\rho } ( y ) \\rangle \\\\ = ~ & \\langle M y , \\rho _ - \\Delta ( \\rho _ + y ) \\rangle \\\\ = ~ & \\langle M \\rho _ - \\rho _ + y , \\rho _ - \\Delta ( \\rho _ + y ) \\rangle \\geq 0 \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} H _ { P _ { p } } ^ { ( A , B ) } [ \\omega ; ( X , Y ) ] = H _ { F _ { p } } ^ { ( A , B ) } [ \\omega ; ( X , Y ) ] + ( 2 \\kappa _ p ) ^ { p - 1 } \\left ( H _ { F _ p } ^ A [ \\zeta ; X ] + H _ { F _ p } ^ B [ \\eta ; Y ] \\right ) . \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} \\int _ B \\exp ( \\alpha _ 0 | t _ j M _ j | ^ { \\frac n { n - 1 } } ) & \\geq \\int _ B \\exp ( \\alpha _ n | M _ j | ^ { \\frac n { n - 1 } } ) d x \\\\ & = \\int _ { B \\setminus B _ { \\log j / j } ( 0 ) } \\exp ( \\alpha _ n | M _ j | ^ { \\frac n { n - 1 } } ) d x + \\int _ { B _ { \\log j / i } ( 0 ) } \\exp ( \\alpha _ n | M _ j | ^ { \\frac n { n - 1 } } ) d x . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} F { } { } \\circ K { } { } ^ { 1 / \\varepsilon } = \\left ( ( F { } { } \\circ K { } { } ^ { 1 / \\varepsilon } ) ^ \\varepsilon \\right ) ^ { 1 / \\varepsilon } = \\left ( \\left ( K { } { } ^ { 1 / \\varepsilon } \\circ ( A _ c + r ^ { 1 / \\varepsilon } ) \\right ) ^ \\varepsilon \\right ) ^ { 1 / \\varepsilon } = K { } { } ^ { 1 / \\varepsilon } \\circ ( A _ c + r ^ { 1 / \\varepsilon } ) . \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} e _ { k } = e _ { - k } \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} D _ { k , m } ( X , a ) : = \\sum _ { i = 0 } ^ { \\lfloor \\frac k 2 \\rfloor } \\frac { k - m i } { k - i } \\dbinom { k - i } { i } ( - a ) ^ { i } X ^ { k - 2 i } , \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} \\widehat { A } = \\{ p \\in \\mathfrak { m } ^ \\beta : A \\in p \\} . \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n - 1 } \\psi _ { n , j } ^ { \\ast } ( Q ) \\leq - \\psi _ { n , n } ^ { \\ast } ( Q ) - \\psi _ { n , n + 1 } ^ { \\ast } ( Q ) + O ( q ^ { - 1 } ) \\leq - \\alpha - \\gamma + \\epsilon _ { 7 } + O ( q ^ { - 1 } ) . \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} B ( f , B ( g , h ) ) ( x , x ) = B ( g , B ( h , f ) ) ( x , x ) = B ( h , B ( f , g ) ) ( x , x ) = 0 . \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} ( A u ) ( x ) = \\frac { 1 } { ( 2 \\pi ) ^ n } \\iint e ^ { i ( x - y ) \\cdot \\xi } a \\left ( \\frac { x + y } { 2 } , \\xi \\right ) u ( y ) \\ , d y \\ , d \\xi \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} \\mathbb { G } _ { N } '' f : = \\sqrt { N } \\left ( \\frac { 1 } { \\widehat { N } } \\sum _ { i = 1 } ^ { N } \\frac { S _ { i , N } } { \\pi _ { i , N } } f ( Y _ { i } ) - \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } f ( Y _ { i } ) \\right ) , f \\in \\mathcal { F } , \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} & | r \\int _ { 6 r } ^ { \\infty } d w \\lambda '' ( t + w ) w \\partial _ { t } \\left ( \\frac { 1 } { ( \\lambda ( t ) ^ { 2 - 2 \\alpha } + w ^ { 2 } ) } - \\frac { 1 } { \\lambda ( t + w ) ^ { 2 - 2 \\alpha } + w ^ { 2 } } \\right ) | \\\\ & \\leq \\frac { C r } { t ^ { 3 } \\log ^ { b + 2 } ( t ) } \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} L : = - \\partial ^ 2 + V . \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} | q _ 0 k ^ { j - 1 } - p + 1 | = \\frac { k ^ j a - k ^ { j - 1 } a } { 2 } \\ge \\frac { ( k - 1 ) a } { 2 } . \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} \\mathcal { A } ^ \\delta ( X ) : = \\{ u \\in \\rho ^ \\delta L ^ \\infty ( X ) \\ , | \\ , { \\rm D i f f } ^ * _ b ( X ) ( u ) \\subset \\rho ^ \\delta L ^ \\infty ( X ) \\} \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} \\bar { H } _ { } ( \\mathbf { 0 } ) \\overset { \\eqref { d _ a u x _ c g _ h a m } } { = } \\frac { 1 } { M } \\sum _ { l = 1 } ^ { M } \\bar { H } _ K ( 0 ) = \\bar { H } _ K ( 0 ) \\to \\varphi ( 0 ) \\nu \\to \\infty . \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} Y _ n \\supset \\bigcap _ { j = i } ^ n \\bigcap _ { k = 0 } ^ \\infty f _ { i j } ( Z _ { j k } ) = \\bigcap _ { j = i } ^ n f _ { i j } ( Z _ j ) \\supset f _ { i n } ( Z _ n ) = \\varphi _ { i n } ( Z _ n ) \\neq \\varnothing . \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} S ( u _ 0 , u _ { \\ell } , u _ k ) & = \\left ( \\sum _ { \\substack { u _ { \\ell + 1 } , \\ldots , u _ { k - 1 } : \\\\ d ( u _ 0 , u _ i ) \\ , = \\ , i \\ , \\ , \\ , \\forall \\ell \\ , < \\ , i \\ , < \\ , k . } } W \\ ! \\left ( u _ \\ell , \\ldots , u _ { k } \\right ) \\right ) ^ 2 . \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} \\{ x \\mapsto \\l x + a _ j : j = 1 , \\ldots , m \\} \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{gather*} \\mathbb { E } _ { \\mu _ { n } } S ( \\sigma ) = \\big ( \\ , \\mathbb { E } _ { \\mu _ { n } } S ^ { 1 } ( \\sigma ) , \\ , \\mathbb { E } _ { \\mu _ { n } } S ^ { 2 } ( \\sigma ) , \\ , \\mathbb { E } _ { \\mu _ { n } } S ^ { 3 } ( \\sigma ) \\ , \\big ) = \\Big ( \\ , \\frac { 1 } { 3 } , \\ , \\frac { 1 } { 3 } , \\ , \\frac { 1 } { 3 } \\ , \\Big ) \\\\ \\mathbb { V } a r _ { \\mu _ { n } } S ( \\sigma ) = \\mathbb { V } a r _ { \\mu _ { n } } S ^ { 1 } ( \\sigma ) + \\mathbb { V } a r _ { \\mu _ { n } } S ^ { 2 } ( \\sigma ) + \\mathbb { V } a r _ { \\mu _ { n } } S ^ { 3 } ( \\sigma ) = \\frac { 2 } { 3 n } . \\end{gather*}"}
-{"id": "3609.png", "formula": "\\begin{align*} f _ n : = \\# S ^ { g ^ 2 _ Z } _ Z \\times _ Z g ^ n _ Z : Z \\times \\# S ^ { X ^ 2 } \\times X ^ n \\cong Z \\times \\# S ^ { Y ^ 2 } \\times Y ^ n . \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} f = \\theta _ { i _ 1 } \\cdots \\theta _ { i _ r } \\overline { \\theta } _ { j _ 1 } \\cdots \\overline { \\theta } _ { j _ s } f ^ \\prime , \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} \\frac { 1 } { \\epsilon } \\int \\frac { 1 } { \\epsilon ^ N } Q \\left ( \\frac { x - y } { \\epsilon } \\right ) ( \\mathbf { v } ( y ) - \\mathbf { v } ( x ) ) \\ , d y = & \\int Q ( z ) \\frac { \\mathbf { v } ( x + \\epsilon z ) - \\mathbf { v } ( x ) } { \\epsilon } \\ , d z \\\\ = & \\int Q ( z ) ( z \\cdot \\nabla _ x ) \\mathbf { v } ( \\zeta ) \\ , d z , \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } & \\partial _ t \\rho + u \\cdot \\nabla \\rho = f , \\\\ & \\rho ( 0 , x ) = \\rho _ 0 ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} \\Upsilon \\left ( \\frac { \\pm z } { 2 \\pi i t } , \\mp \\theta \\right ) = \\left ( \\frac { \\pm z } { 2 \\pi i t } \\right ) ^ { \\frac { 1 } { 1 2 } } \\cdot \\lim _ { \\tau \\to 1 } \\psi _ \\pm ( t ) , \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} \\Delta ^ t f : = \\langle \\mathrm { H e s s } _ f , g _ t \\rangle + \\frac { 1 } { 4 } \\langle \\nabla _ x \\Delta _ x p ( x , x , 2 t ) , \\nabla f \\rangle . \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} & | \\frac { 1 } { r } \\int _ { t } ^ { \\infty } d s \\int _ { 0 } ^ { s - t } \\frac { \\rho d \\rho \\lambda '' ( s ) } { ( s - t ) \\sqrt { ( s - t ) ^ { 2 } - \\rho ^ { 2 } } } \\left ( \\frac { - 1 - \\rho ^ { 2 } + r ^ { 2 } } { \\sqrt { ( 1 + \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } + 4 r ^ { 2 } } } + 1 + F _ { 3 } ( r , \\rho , \\lambda ( s ) ) - 1 \\right ) | \\\\ & \\leq C \\frac { r } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } + C \\frac { r } { t ^ { 2 } \\log ^ { 1 + b \\alpha } ( t ) } \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} p _ \\theta ( { \\bf { x } } ) & = Z ( \\theta ) e _ \\alpha \\big [ h ( { \\bf { x } } ) + w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\big ] , \\\\ p _ \\theta ( { \\bf { x } } ) ^ { - 1 } & = Z ( \\theta ) ^ { - 1 } ~ e _ \\alpha \\big [ - h ( { \\bf { x } } ) - w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\big ] , \\\\ p _ \\theta ( { \\bf { x } } ) ^ { - 1 } & = e _ \\alpha \\big [ - h ( { \\bf { x } } ) - Z ' ( \\theta ) - w ( \\theta ) ^ \\top f ( { \\bf { x } } ) \\big ] , \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} | \\Phi ( \\xi _ 1 , \\eta _ 1 , \\xi , \\eta ) | & = r _ 1 r | \\cos \\theta _ 1 \\cos \\theta ( r _ 1 \\cos \\theta _ 1 + r \\cos \\theta ) + \\sin \\theta _ 1 \\sin \\theta ( r _ 1 \\sin \\theta _ 1 + r \\sin \\theta ) | \\\\ & \\geq r _ 1 r \\ , ( r _ 1 + r ) \\ , ( 1 - 2 ^ { - 1 } \\sin 2 \\theta _ 1 ) | ( \\cos \\theta _ 1 + \\sin \\theta _ 1 ) | \\\\ & \\geq 2 ^ { - 1 0 } M ^ { - 1 } N _ 1 ^ 3 , \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} \\| \\partial _ X \\omega \\| _ { C ^ { s - 1 } } \\leq C \\| \\partial _ X \\omega \\| _ { L ^ p } \\leq C \\| \\partial _ X \\omega _ 0 \\| _ { L ^ p } + C ( t ) e ^ { 2 \\int _ { 0 } ^ { t } \\| \\nabla u ( \\tau ) \\| _ { L ^ \\infty } ~ d \\tau } . \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} \\int _ B \\Big | \\frac { u ( x ) } { \\lambda _ * \\| \\nabla ^ m u \\| _ { L ^ 2 ( B ) } } \\Big | ^ { \\ 2 m s + f ( | x | ) } d x & = \\lambda _ * ^ { - \\ 2 m s } \\int _ B \\Big | \\frac { u ( x ) } { \\| \\nabla ^ m u \\| _ { L ^ 2 ( B ) } } \\Big | ^ { \\ 2 m s + f ( | x | ) } \\lambda _ * ^ { - f ( | x | ) } d x \\\\ & \\leq \\lambda _ * ^ { - \\ 2 m s } \\int _ B \\Big | \\frac { u ( x ) } { \\| \\nabla ^ m u \\| _ { L ^ 2 ( B ) } } \\Big | ^ { \\ 2 m s + f ( | x | ) } d x \\\\ & \\leq C \\lambda _ * ^ { - \\ 2 m s } \\\\ & \\leq 1 . \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} \\Delta u + V _ { 0 , \\infty } u = f _ 0 ( u ) , \\mathbb { R } ^ N , N \\geq 3 , \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} G _ { 3 , 1 } ( w , r , \\lambda ) : = \\int _ { 0 } ^ { w } \\rho \\left ( F _ { 3 } ( r , \\rho , \\lambda ( s ) ) + \\frac { 2 r ^ { 2 } \\lambda ( s ) ^ { 2 - 2 \\alpha } } { ( \\lambda ( s ) ^ { 2 - 2 \\alpha } + \\rho ^ { 2 } ) ^ { 2 } } \\right ) d \\rho \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} \\begin{alignedat} { 2 } \\epsilon ( 3 H _ 2 H _ 3 + H _ 3 & + 4 H _ 2 ) \\cdot ( ( H _ 1 ) ^ 2 - \\epsilon H _ 2 ) \\\\ & = - ( \\epsilon H _ 2 + 3 ( H _ 1 ) ^ 2 ) ( H _ 4 + \\epsilon ) \\cdot ( H _ 2 - 1 ) \\end{alignedat} \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} { } _ n \\pi \\subset { } _ { n - 1 } \\pi \\subset \\ldots \\subset { } _ 1 \\pi \\subset { } _ 0 \\pi = \\pi \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} \\begin{aligned} W = - \\frac { D _ { l - 1 } - 1 } { 2 } \\log r ' - \\frac { d _ l - 1 } { 2 } \\log r _ l , ~ ~ ~ D _ { l - 1 } = \\sum _ { i = 1 } ^ { l - 1 } d _ i , ~ ~ ~ r ' = \\sqrt { \\sum _ { i = 1 } ^ { l - 1 } r _ i ^ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} ( f | g ) _ { L } : = ( L f | L g ) + ( f | g ) = \\langle \\bar L \\bar f | L g \\rangle + \\langle \\bar f | g \\rangle . \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} C _ i \\ = \\ & B _ 1 \\ltimes A _ i \\ i \\in J , \\\\ C _ i \\ = \\ & B _ 2 \\ltimes A _ i \\ i \\in J ' , \\\\ C _ { n + 1 } & \\ = \\ B _ 3 \\ltimes [ N ] . \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} \\int _ { \\Omega _ 0 } \\beta ( \\nu , \\nabla _ { \\Sigma } \\phi ) + \\frac { 1 } { 2 } t r _ { \\Sigma } ( \\nabla ^ M _ { \\nu } \\beta ) \\cdot \\phi = 0 \\ \\ \\ \\ \\forall \\phi \\in C ^ 2 _ c ( \\Omega _ 0 ) \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} \\left [ \\displaystyle \\sum _ { i = 1 } ^ p [ y ^ I ] ^ * _ j \\nabla ^ 2 c _ j ( x ^ * ) - A ^ T A - { \\cal J } c _ { \\gamma } ( x ^ * ) ^ T { \\cal J } c _ { \\gamma } ( x ^ * ) \\right ] \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} \\lambda _ n ( u ) = n + \\lambda _ 0 ( u ) + \\sum _ { k = 1 } ^ n \\gamma _ k ( u ) \\ge n + \\lambda _ 0 ( u ) \\ . \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} \\begin{aligned} a _ 1 & = \\frac { 1 } { 2 } , & a _ 2 & = 0 , & a _ 3 & = - \\frac { 1 } { 2 } , \\\\ b _ 1 & = 0 , & b _ 2 & = 0 , & b _ 3 & = 0 , \\end{aligned} \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} \\begin{cases} L v = \\frac { G } { \\rho ^ 3 T } \\cdot f \\ & \\Sigma _ T \\\\ v = \\phi \\ & \\partial \\Sigma _ T \\end{cases} \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} & \\iota _ { Q } \\varpi ^ { \\bullet } = \\delta L ^ { \\bullet } + d \\theta ^ \\bullet \\\\ & \\frac 1 2 \\iota _ Q \\iota _ Q \\varpi ^ { \\bullet } = d L ^ { \\bullet } , \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} 1 4 7 J _ { n + 1 } ^ { ( 3 ) } & = 1 4 7 \\left ( J _ { n } ^ { ( 3 ) } + J _ { n - 1 } ^ { ( 3 ) } + 2 J _ { n - 2 } ^ { ( 3 ) } \\right ) \\\\ & = 1 3 K _ { n } ^ { ( 3 ) } + 4 8 K _ { n - 1 } ^ { ( 3 ) } + 2 0 K _ { n - 2 } ^ { ( 3 ) } \\\\ & \\ \\ + 1 3 K _ { n - 1 } ^ { ( 3 ) } + 4 8 K _ { n - 2 } ^ { ( 3 ) } + 2 0 K _ { n - 3 } ^ { ( 3 ) } \\\\ & \\ \\ + 2 6 K _ { n - 2 } ^ { ( 3 ) } + 9 6 K _ { n - 3 } ^ { ( 3 ) } + 4 0 K _ { n - 4 } ^ { ( 3 ) } \\\\ & = 1 3 K _ { n + 1 } ^ { ( 3 ) } + 4 8 K _ { n } ^ { ( 3 ) } + 2 0 K _ { n - 1 } ^ { ( 3 ) } . \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} \\sum _ { m , n \\geq 0 } \\frac { q ^ { m ^ 2 + n ^ 2 - m n } x ^ m y ^ n } { ( q ) _ m ( q ) _ n } = \\sum _ { n _ 1 , n _ 2 , n _ 3 \\geq 0 } \\frac { x ^ { n _ 1 + n _ 3 } y ^ { n _ 2 + n _ 3 } q ^ { n _ 1 ^ 2 + n _ 2 ^ 2 + n _ 3 ^ 2 + n _ 1 n _ 3 + n _ 2 n _ 3 } } { ( q ) _ { n _ 1 } ( q ) _ { n _ 2 } ( q ) _ { n _ 3 } } . \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} \\sigma _ { l b } ( T ) : & = \\{ \\lambda \\in \\mathbb { C } : \\lambda I - T \\thinspace \\mbox { i s n o t l o w e r s e m i - B r o w d e r } \\} , \\\\ \\sigma _ { u b } ( T ) : & = \\{ \\lambda \\in \\mathbb { C } : \\lambda I - T \\thinspace \\mbox { i s n o t u p p e r s e m i - B r o w d e r } \\} , \\\\ \\sigma _ b ( T ) : & = \\{ \\lambda \\in \\mathbb { C } : \\lambda I - T \\thinspace \\mbox { i s n o t B r o w d e r } \\} , \\thinspace \\mbox { r e s p e c t i v e l y } . \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} X _ 1 ^ * = \\sqrt { 2 } ( X _ 1 - \\mu _ * ) / \\sigma _ * , ~ ~ X _ 2 ^ * = \\sqrt { 2 } ( X _ 2 - \\mu _ * ) / \\sigma _ * . \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} \\left | a ^ { ( 2 k ) } _ { i j } \\right | = \\left | \\dfrac { m ^ { 1 \\dots k i } _ { 1 \\dots k j } } { m _ { k } } \\right | \\leq \\left | \\dfrac { m _ { k + 1 } } { m _ k } \\right | = \\left | a ^ { ( 2 k ) } _ { k + 1 k + 1 } \\right | . \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} C \\Phi _ 5 ( q ) = \\prod _ { n = 1 } ^ { \\infty } \\frac { 1 } { 1 - q ^ { 5 n } } + 2 5 q \\prod _ { n = 1 } ^ { \\infty } \\frac { ( 1 - q ^ { 5 n } ) ^ 5 } { ( 1 - q ^ n ) ^ 6 } . \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} ( [ \\delta _ { \\varepsilon _ 1 } , \\delta _ { \\varepsilon _ 2 } ] - \\delta _ { [ \\varepsilon _ 1 , \\varepsilon _ 2 ] _ Q } ) A ^ a = 0 \\Longleftrightarrow \\nabla ^ \\pm _ \\mu ( T _ { \\nabla ^ \\pm } ) ^ a { } _ { b c } = 2 \\rho ^ \\nu _ { [ b | } ( R _ { \\nabla ^ \\pm } ) ^ a { } _ { \\nu \\mu | c ] } . \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) < \\beta , \\ \\bigcup _ { k = 0 } ^ \\infty \\{ N ( t ) = 2 k + 1 \\} \\ | \\ V ( 0 ) = c _ 1 \\} = \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} \\mu _ { N , m } ( d y ) = \\mu _ { N , m } ( d x | y ) \\bar { \\mu } _ { N , m } ( d y ) . \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} h ( R , \\omega ) = \\begin{cases} c R ^ { \\gamma _ 1 ^ + } w _ 1 ( \\omega ) + O ( R ^ { \\gamma _ 1 ^ + - \\epsilon } ) \\ & \\mu _ 1 > ( \\frac { n - 2 } { 2 } ) ^ 2 \\\\ c R ^ { \\gamma _ 1 ^ + } \\log R \\ w _ 1 ( \\omega ) + O ( R ^ { \\gamma _ 1 ^ + - \\epsilon } ) \\ & \\mu _ 1 = ( \\frac { n - 2 } { 2 } ) ^ 2 \\end{cases} \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} p ^ { \\epsilon ( X / k ) } \\leq p ^ { \\ell _ F ( X / k ) ( \\log _ p [ k : k ^ p ] - 1 ) } = ( p ^ { - 1 } \\cdot [ k : k ^ p ] ) ^ { \\ell _ F ( X / k ) } . \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} B ( x _ { \\lambda } , v ) = \\tau _ { \\lambda } ( v ) = \\tau ( v ) \\ , . \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} \\pi ( \\xi ) ^ * = c ( \\xi , \\xi ) \\pi ( - \\xi ) \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} \\eta ^ x _ { U _ n } | u | ( D ) & = \\int _ { D \\setminus U _ n } P _ { U _ n } ( x , y ) | u ( y ) | d y \\\\ & \\le c _ 2 \\int _ { D \\setminus U _ n } P _ { U _ n } ( x _ 0 , y ) | u ( y ) | d y \\le c _ 2 \\cdot M , \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { M } [ 4 k - 1 ] \\frac { ( a q ^ { - 1 } , q ^ { - 1 } / a , q ^ { - 1 } / b , q ^ { - 1 } ; q ^ 2 ) _ k } { ( q ^ { 2 } / a , a q ^ 2 , b q ^ 2 , q ^ 2 ; q ^ 2 ) _ k } ( b q ^ 4 ) ^ k \\\\ [ 5 p t ] \\ : & \\ : \\equiv [ n ] ( b q ) ^ { ( n + 1 ) / 2 } \\frac { ( q ^ { - 2 } / b ; q ^ 2 ) _ { ( n + 1 ) / 2 } } { ( b q ^ 2 ; q ^ 2 ) _ { ( n + 1 ) / 2 } } \\pmod { [ n ] ( 1 - a q ^ n ) ( a - q ^ n ) } , \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} \\rho ( R ) = \\sup \\bigg \\{ \\frac { \\sup L ( x ) } { \\min L ( x ) } \\ \\bigg { | } \\ x \\in R ^ * \\setminus U ( R ) \\bigg \\} \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} { \\rm s g n } ( a ) = \\begin{cases} - 1 , & a < 0 , \\\\ 0 , & a = 0 , \\\\ 1 , & a > 0 . \\end{cases} \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} v _ x ( f _ 1 f _ 3 + f _ 2 ^ 2 ) = 0 , v _ x ( f _ 1 ^ 2 + f _ 3 ^ 2 g ) \\geq 1 , \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} { \\cal L } \\varphi _ i ( z ) & = - \\lambda _ i \\varphi _ i ( z ) , \\\\ \\varphi _ { i z } ( z _ 1 ) & = \\varphi _ { i z } ( z _ 2 ) = 0 , \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} u _ 0 ( x ) : = \\int _ 0 ^ \\infty p ( t , 0 , x ) d t = \\uparrow \\lim _ { \\lambda \\downarrow 0 } u _ \\lambda ( x ) = c _ { - \\alpha , d } \\cdot | x | ^ { \\alpha - d } , \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} f ^ { ( k ) } _ 2 ( z ) = \\frac { a _ 0 - e ^ { - z ^ 2 } ( a _ 0 + a _ 2 z ^ 2 + \\ldots + a _ { 2 k } z ^ { 2 k } ) } { z ^ { k + 2 } } , \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} r _ \\alpha r _ \\beta = r _ { \\alpha \\cdot \\beta } + r _ { \\alpha \\odot \\beta } . \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} \\beta _ q ( f ) : = \\beta ( T _ H ( f ) ) , \\ \\forall f \\in \\mathcal { C } _ 0 ( G ) . \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} { \\phi } _ { p , \\alpha } \\left ( 1 + \\frac { 1 } { p } \\right ) = \\left ( 1 + \\frac { 1 } { p } \\right ) ^ { p } \\left ( 1 + \\log \\left ( 1 + \\frac { 1 } { p } \\right ) \\right ) ^ { \\alpha } \\leq e 2 ^ { \\alpha } . \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} c _ 1 ( \\mathcal { O } _ { \\mathbb { P } ( F ^ * ) } ( 1 ) , h ^ { \\mathcal { O } } ) _ H = \\frac { \\pi ^ * \\omega } { 2 \\pi } . \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} \\lim _ { v \\uparrow \\infty } \\frac { 1 } { N _ v } \\int ( x - N _ v P _ v ^ t \\eta _ { 0 , v } ) \\cdot A _ v ^ { - 1 } ( x - N _ v P _ v ^ t \\eta _ { 0 , v } ) f _ { 0 , v } ( x ) \\mu _ v ( d x ) = 0 . \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} q _ \\eta ( { \\bf { x } } ) = S ( \\eta ) ^ { \\frac { 1 } { 1 - \\beta } } M _ { \\eta , \\beta } \\big [ 1 + c _ \\beta S ( \\eta ) ^ { - 1 } \\lbrace \\rm { v e c } ^ \\top ( \\boldsymbol { \\Sigma } ^ { - 1 } ) \\rm { v e c } ( { \\bf { x } } { \\bf { x } } ^ \\top ) - 2 ( \\boldsymbol { \\Sigma } ^ { - 1 } \\boldsymbol { \\mu } ) ^ \\top { \\bf { x } } \\rbrace \\big ] ^ { \\frac { 1 } { 1 - \\beta } } , \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} | W ' ( \\zeta ) | \\leq C _ 1 d ( x , x ' ) \\| f \\| \\leq d ( x , x ' ) C _ 1 ( b + c _ 2 ) \\int f d m = d ( x , x ' ) C _ 1 b ^ { - 1 } ( b + c _ 2 ) B \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} \\left ( \\frac { d ^ n } { d z ^ n } f ( z ) \\right ) ^ { \\widehat { \\ , } } \\ , & = \\ , ( 2 \\pi { \\mathbf i } \\xi ) ^ n \\widehat f ( \\xi ) \\\\ \\left ( z ^ n f ( z ) \\right ) ^ { \\widehat { \\ , } } \\ , & = \\ , \\left ( \\frac { { \\mathbf i } } { 2 \\pi } \\right ) ^ n \\frac { d ^ n \\widehat f ( \\xi ) } { d \\xi ^ n } \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n - 1 } \\psi _ { n , j } ^ { \\ast } ( Q ) \\leq - \\psi _ { n , n } ^ { \\ast } ( Q ) - \\psi _ { n , n + 1 } ^ { \\ast } ( Q ) + O ( q ^ { - 1 } ) \\leq - \\alpha - \\gamma + \\epsilon + \\epsilon _ { 3 } + O ( q ^ { - 1 } ) . \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} \\sqrt { \\alpha \\beta ^ { - 1 } } \\left \\vert x \\right \\vert \\leq \\left \\vert y \\right \\vert \\sqrt { \\alpha ^ { - 1 } \\beta } \\left \\vert x \\right \\vert = \\left ( k a _ { 0 } ^ { - 1 } \\right ) ^ { \\frac { 1 } { 2 p } } \\left \\vert x \\right \\vert \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} \\frac { d } { d \\sigma } A ( 0 ) = \\frac { 1 } { N } \\mathbb { E } _ { \\mu _ N ^ { 0 } } \\left [ \\sum _ { i = 1 } ^ N X _ i \\right ] . \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} H = \\frac 1 2 \\ , g ^ { i j } p _ i p _ j \\ , , I = \\left ( \\frac { \\det ( g ) } { \\det ( \\bar { g } ) } \\right ) ^ { \\frac 2 3 } \\ , \\bar { g } ^ { i j } p _ i p _ j \\ , , J = \\left ( \\frac { \\det ( g ) } { \\det ( \\hat { g } ) } \\right ) ^ { \\frac 2 3 } \\ , \\hat { g } ^ { i j } p _ i p _ j \\ , , \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} \\mathbb { V = } \\tilde { V } _ { \\chi \\left ( \\left \\vert x > R \\right \\vert \\right ) } \\left ( x , t \\right ) , \\mathbb { F } = \\tilde { V } _ { \\chi \\left ( \\left \\vert x < r \\right \\vert \\right ) } \\left ( x , t \\right ) \\tilde { u } \\left ( x , t \\right ) . \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} | v _ { c o r r } ( t , r ) | \\leq \\begin{cases} \\frac { C r \\log ( \\log ( t ) ) } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } , r \\leq \\log ^ { N } ( t ) \\\\ \\frac { C r } { t ^ { 2 } \\log ^ { b } ( t ) } , \\log ^ { N } ( t ) \\leq r \\leq \\frac { t } { 2 } \\\\ \\frac { C } { \\sqrt { r } } , \\frac { t } { 2 } < r \\end{cases} \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ z ( - 1 ) ^ t \\binom { x } { t } \\binom { y - t } { z - t } = ( - 1 ) ^ z \\binom { x - y + z - 1 } { z } . \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} \\Phi _ { ( n ) } ( S ) = \\bigl ( \\langle S b _ { p , n - 1 } , b _ { p , n - 1 } \\rangle \\bigr ) _ { p = 0 } ^ \\infty , \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} F _ 1 : = 4 H - 2 E _ 1 - \\ldots - 2 E _ 9 , F _ 2 : = 4 H - 2 E _ 1 - \\ldots - 2 E _ { 1 4 } \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} H _ 1 : = \\{ x y = c : c \\in C \\} , \\ , \\ , \\ , \\ , \\ , \\ , H _ 2 : = \\{ ( x + \\alpha ) ( y + \\beta ) = d : d \\in D \\} . \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} ( h ^ { 2 } - & 2 h - \\mu ) ( h - 4 ) ^ { n } = h ^ { n + 2 } - 2 ( 2 n + 1 ) h ^ { n + 1 } \\\\ & + \\sum _ { k = 0 } ^ { n - 2 } \\bigg ( - \\mu { n \\choose k } ( - 4 ) ^ { k } - 2 { n \\choose k + 1 } ( - 4 ) ^ { k + 1 } \\\\ & + { n \\choose k + 2 } ( - 4 ) ^ { k + 2 } \\bigg ) h ^ { n - k } \\\\ & + ( - \\mu n ( - 4 ) ^ { n - 1 } - 2 ( - 4 ) ^ { n } ) h - \\mu ( - 4 ) ^ { n } . \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} D _ 2 = - \\frac { \\epsilon } { A } C _ 2 , C _ 4 = - \\frac { 1 } { A } D _ 4 , D _ 1 = - \\frac { \\epsilon } { A } C _ 1 , C _ 3 = - \\frac { 1 } { A } D _ 3 \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} 1 + 2 ( \\rho ^ { 2 } + r ^ { 2 } ) z ^ { 2 } + ( \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } z ^ { 4 } = 4 \\rho ^ { 2 } z ^ { 2 } + ( 1 + ( r ^ { 2 } - \\rho ^ { 2 } ) z ^ { 2 } ) ^ { 2 } \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} H ^ 2 ( \\Omega ) = V _ j \\oplus V _ j ^ { \\perp } . \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} y ^ 2 + A x y + B y = x ^ 3 , \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} | F _ { 4 } ( \\lambda ( s ) ^ { 1 - \\alpha } , s - t ) - F _ { 4 } ( \\lambda ( t ) ^ { 1 - \\alpha } , s - t ) | & \\leq C \\frac { \\lambda ( t ) ^ { 1 - \\alpha } } { ( s - t ) ^ { 4 } } \\cdot | \\lambda ( t ) ^ { 1 - \\alpha } - \\lambda ( s ) ^ { 1 - \\alpha } | \\\\ & \\leq C \\frac { \\lambda ( t ) ^ { 1 - 2 \\alpha } } { ( s - t ) ^ { 3 } } \\cdot \\sup _ { x \\geq t } \\left ( \\frac { | \\lambda ' ( x ) | \\lambda ( t ) ^ { \\alpha } } { \\lambda ( x ) ^ { \\alpha } } \\right ) \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} n _ { t + 1 } \\in \\bigcap _ { \\ell = 0 } ^ d \\left [ B _ 1 ^ \\ell \\cap \\bigcap _ { s \\in \\{ 1 , . . . , \\min \\{ 2 ^ \\ell , t \\} \\} \\setminus \\{ 2 ^ \\ell \\} } \\bigcap _ { 1 \\leq j _ 1 < \\cdots < j _ s \\leq t } B _ { s + 1 } ^ \\ell ( n _ { j _ 1 } , . . . , n _ { j _ s } ) \\right ] \\in p . \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} B _ \\lambda ( x _ \\lambda , y ) = \\tau _ \\lambda ( y ) ( y \\in Y _ \\lambda ) \\ , . \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} \\Sigma \\supset \\tau ( \\Sigma ) \\supset \\tau ^ 2 ( \\Sigma ) \\supset \\cdots \\supset \\Omega ( \\tau ) = \\bigcap _ { n \\in \\N } \\tau ^ n ( \\Sigma ) , \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} \\Phi ( x + y ) ^ \\ast \\Phi ( x + y ) = \\Phi ( x ) ^ \\ast \\Phi ( x ) + \\Phi ( y ) ^ \\ast \\Phi ( y ) . \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} \\gamma = \\frac { \\gamma _ 1 \\gamma _ 2 } { \\gamma _ 2 + { \\cal C } } , \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} \\partial _ t u = \\partial _ x | w _ \\lambda ( \\cdot , u ) | ^ 2 \\ . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} \\Pr { \\frac { 1 } { d } \\sum _ { i = 1 } ^ d \\big ( \\norm [ 1 ] { x _ i } _ 2 - \\mu _ i \\big ) > s } , \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} b = \\frac { - 1 + \\sqrt { 4 X + 1 } } { 2 } , \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} ( a _ { 1 } , a _ { 2 } , b _ { 1 } , b _ { 3 } , a _ { 1 } ' , a _ { 2 } ' ) = ( 2 / 1 1 , 5 / 6 6 , 0 , 7 / 1 1 , 3 / 1 1 , 4 9 / 6 6 ) . \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} ( q ; q ) _ \\infty ( - q ^ 2 ; q ^ 2 ) _ \\infty = \\dfrac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( - q ; q ^ 2 ) _ \\infty } = \\sum _ { n = 0 } ^ \\infty ( - q ) ^ { n ( n + 1 ) / 2 } , \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} F ( z ) \\eta _ { \\mu _ { 1 } } ^ { - 1 } ( z ) = \\eta _ { \\mu _ { 1 } \\boxtimes \\mu _ { 2 } } ^ { \\langle - 1 \\rangle } ( z ) , \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} \\limsup _ n | A _ n | \\le M _ f \\limsup _ n \\sum _ k { { \\rm o s c } \\ , ( \\psi , I _ k ^ n ) } \\ , | I _ k ^ n | = 0 \\ , . \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} \\dot { u } & = v , \\\\ \\dot { v } & = - \\delta u + \\tau v , \\\\ v & \\mapsto - r v , { \\rm ~ w h e n ~ } u = \\xi . \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\frac { \\partial \\lambda } { \\partial t } = - ( Q _ { 0 } ^ { \\bot } + 2 P _ { 0 } \\lambda ) + r \\\\ \\theta ( \\cdot , t ) = e ^ { 2 \\lambda ( \\cdot , t ) } \\theta _ { 0 } \\\\ \\lambda ( \\cdot , 0 ) = \\lambda _ { 0 } ( \\cdot ) \\int _ { M } e ^ { 4 \\lambda _ { 0 } } d \\mu _ { 0 } = \\int _ { M } d \\mu _ { 0 } \\end{array} \\right . , \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} p _ 2 = \\frac { 1 } { 2 } H _ 0 + \\frac { 1 } { 2 } \\sqrt { \\frac { 4 } { 1 - e ^ { q _ 3 ( t ) - q _ 2 ( t ) } } H _ 1 - \\frac { 1 + e ^ { q _ 3 ( t ) - q _ 2 ( t ) } } { 1 - e ^ { q _ 3 ( t ) - q _ 2 ( t ) } } H _ 0 ^ 2 } \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} \\ell _ { n , 2 } ^ * ( \\beta _ 0 , \\beta _ 1 , \\eta ) - \\ell _ { n , 2 } ^ * ( 0 , 0 , 1 ) \\leq \\ss _ 2 ^ \\tau \\sum _ { i = 1 } ^ n B _ i - ( n / 2 ) \\ss _ 2 ^ \\tau \\Sigma _ B \\ss _ 2 \\{ 1 + o _ p ( 1 ) \\} + o _ p ( 1 ) . \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} & \\int _ { t } ^ { \\infty } \\frac { \\rho d \\rho ( \\rho ^ { 2 } + r ^ { 2 } + \\lambda ( t ) ^ { 2 - 2 \\alpha } ) } { ( 1 + 2 ( \\rho ^ { 2 } + r ^ { 2 } ) \\lambda ( t ) ^ { 2 \\alpha - 2 } + ( \\rho ^ { 2 } - r ^ { 2 } ) ^ { 2 } \\lambda ( t ) ^ { 4 \\alpha - 4 } ) ^ { 3 / 2 } } \\frac { 1 } { \\log ^ { ( 4 \\alpha - 4 ) b } ( t ) t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\\\ & \\leq \\frac { C } { t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) \\le \\beta , \\ \\bigcup _ { k = 0 } ^ \\infty \\{ N ( t ) = 2 k \\} \\ | \\ V ( 0 ) = - c _ 2 \\} = \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} \\top < | C \\implies \\bigvee C = \\top \\in B . \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} \\hat { \\sigma } ( n ) = \\prod _ { \\epsilon _ j \\neq 0 } \\frac { c _ j } { 2 } . \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} \\Big ( { \\sum _ { j \\ge n } } ^ * T _ { \\omega _ { 1 , j } } ( 1 , R ( X ) ) X ^ j \\Big ) ^ { [ n , n + N ) } = \\Big ( { \\sum _ { j \\ge n } } ^ * T _ { \\omega ' _ { 1 , j } } ( 1 , R ( X ) ) X ^ j \\Big ) ^ { [ n , n + N ) } . \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} \\mathcal H ( u ) = \\sum _ { n = 1 } ^ \\infty n ^ 2 \\gamma _ n - \\sum _ { n = 1 } ^ \\infty s _ n ^ 2 \\ . \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} \\widetilde { m } _ { n _ 2 } ^ { ( i _ 2 ) } = e _ 2 ^ { \\prime } \\left ( e _ 2 ^ { - 1 } \\cdot m _ { n _ 2 } ^ { ( i _ 2 ) } \\right ) , \\textit { w h e r e } e _ 2 ^ { \\prime } \\textit { i s t h e o v e r l a p o f } b ^ { ( i _ 1 ) } \\textit { a n d } \\widetilde { m } _ { n _ 2 } ^ { ( i _ 2 ) } . \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} \\frac { 1 } { k + h ^ \\vee } + \\frac { 1 } { k ' + h ^ \\vee } = n \\in \\Z , \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c l } a _ { 1 1 } X _ 1 ^ { d } & + a _ { 1 2 } X _ 2 ^ { d } + \\cdots + & a _ { 1 t } X _ t ^ { d } = 0 \\\\ a _ { 2 1 } X _ 1 ^ { d } & + a _ { 2 2 } X _ 2 ^ { d } + \\cdots + & a _ { 2 t } X _ t ^ { d } = 0 \\\\ \\ ; \\vdots & & \\quad \\vdots \\\\ a _ { n 1 } X _ 1 ^ { d } & + a _ { n 2 } X _ 2 ^ { d } + \\cdots + & a _ { n t } X _ t ^ { d } = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} K _ { ( n + 1 ) , z } ( w ) = \\frac { 1 } { n } \\left ( z - \\frac { \\partial } { \\partial \\overline { z } } \\right ) \\left ( \\overline { w } - \\frac { \\partial } { \\partial w } \\right ) \\sum _ { p = 0 } ^ \\infty \\overline { b _ { p , n - 1 } ( z ) } b _ { p , n - 1 } ( w ) . \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} \\tilde { \\mbox { A u x } } _ { p } = \\tilde { \\mbox { A u x } } \\left [ p - 2 , \\dots , 0 \\right ] \\tilde { \\mbox { A u x } } \\left [ p - 3 , 1 , \\dots , 0 \\right ] \\dots \\tilde { \\mbox { A u x } } \\left [ 0 , \\dots , 1 , p - 3 \\right ] \\tilde { \\mbox { A u x } } \\left [ 0 , \\dots , p - 2 \\right ] , \\quad \\mbox { f o r a l l $ p \\geq 3 $ . } \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} | - 1 + x K _ { 1 } ( x ) | & \\leq C x ^ { 2 } ( | \\log ( x ) | + 1 ) \\\\ | K _ { 0 } ( x ) | & \\leq C ( | \\log ( x ) | + 1 ) \\\\ | x K _ { 1 } ( x ) | & \\leq C \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} B . u = & B \\big ( f ( h ) \\big ( r ( h ) + \\frac { i } { 2 n } ( C . r ( h ) ) \\big ) \\big ) \\\\ \\equiv & \\big ( f ^ { \\prime } ( h ) B + f ^ { \\prime \\prime } ( h ) C \\big ) \\big ( r ( h ) + \\frac { i } { 2 n } ( C . r ( h ) ) \\big ) \\\\ \\equiv & i h f ^ { \\prime } ( h ) ( r ( h ) + \\frac { i } { 2 n } C . r ( h ) ) - 2 n i f ^ { \\prime \\prime } ( h ) ( r ( h ) + \\frac { i } { 2 n } C . r ( h ) ) \\\\ \\equiv & \\big ( i h f ^ { \\prime } ( h ) - 2 n i f ^ { \\prime \\prime } ( h ) \\big ) \\big ( r ( h ) + \\frac { i } { 2 n } C . r ( h ) \\big ) \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} R _ { k } = \\left \\{ ( x , t ) \\in B _ { k } : | | x | | _ { \\infty } = L _ { k } ^ { d + 6 } x _ { 1 } \\geq \\vartheta _ { k } L _ { k } t = L _ { k } \\right \\} , \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} \\int _ { t } ^ { t + 1 } d s | K _ { 1 } ( s - t , \\lambda ( t ) ) - \\frac { \\lambda ( t ) ^ { 2 } } { 4 ( 1 + s - t ) } | & \\leq \\int _ { t } ^ { t + 1 } d s K _ { 1 } ( s - t , \\lambda ( t ) + \\int _ { t } ^ { t + 1 } d s \\frac { \\lambda ( t ) ^ { 2 } } { 4 ( 1 + s - t ) } \\\\ & \\leq C \\lambda ( t ) ^ { 2 } \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} \\| f \\| ^ { * } _ { L ^ { p , q } } : = \\left \\{ \\begin{array} { l } \\bigg ( \\int _ 0 ^ { \\infty } \\big ( t ^ { 1 / p } f ^ { * } ( t ) \\big ) ^ { q } \\frac { d t } { t } \\bigg ) ^ { 1 / q } , \\ ; 1 \\leq q < \\infty \\\\ \\sup \\limits _ { t > 0 } \\ ; t ^ { 1 / p } f ^ { * } ( t ) , \\ ; q = \\infty . \\end{array} \\right . \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} \\begin{cases} & U ^ \\pm \\in C ^ \\infty ( 0 , \\infty ) , 0 < U ^ \\pm < t ^ \\pm ~ ~ r > 0 , \\\\ [ 2 m m ] & U ^ \\pm \\sim r ~ ~ r \\rightarrow 0 , U ^ \\pm \\sim t ^ \\pm - \\frac { c _ \\pm } { 2 r ^ 2 } ~ ~ r \\rightarrow \\infty , \\\\ [ 2 m m ] & { U ^ \\pm } ' > 0 ~ ~ B < 0 , { U ^ \\pm } ' \\sim \\frac { c } { r ^ 3 } ~ ~ r \\rightarrow \\infty , \\end{cases} \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} \\begin{pmatrix} u _ 1 \\\\ u _ 2 \\\\ u _ 3 \\end{pmatrix} = \\begin{pmatrix} e ^ { - \\frac 5 3 r } \\ , \\sin ( \\theta ) \\ , \\cos ( \\varphi ) \\\\ e ^ { - \\frac 2 3 r } \\ , \\sin ( \\theta ) \\ , \\sin ( \\varphi ) \\\\ e ^ { \\frac 4 3 r } \\ , \\cos ( \\theta ) \\end{pmatrix} \\ , , \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} x _ { 0 } = \\sup \\{ x \\in ( 0 , 2 ) : \\ , h ' ( x ) > 0 \\} . \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} \\frac { d } { d \\tau } [ R ( \\lambda ) + Q ( \\lambda ) e ^ { - \\lambda \\tau } ] = 0 \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} K _ { j } = \\frac { C ( \\log n _ { j } ) ^ { 2 } } { \\log \\lambda ^ { - 1 } } . \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} F _ { \\nabla ^ \\omega } = & ( \\d A ^ a + \\omega ^ a { } _ { \\mu b } A ^ b \\wedge d X ^ \\mu - \\omega ^ a _ { \\mu b } A ^ b \\wedge \\rho ^ \\mu _ c A ^ c + \\tfrac { 1 } { 2 } C ^ a { } _ { b c } A ^ b \\wedge A ^ c ) e _ a \\\\ = & ( \\d A ^ a + \\omega ^ a { } _ { \\mu b } A ^ b \\wedge D X ^ \\mu + \\tfrac { 1 } { 2 } C ^ a { } _ { b c } A ^ b \\wedge A ^ c ) e _ a . \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} \\sum _ i ( \\Omega ^ { 4 } ) _ { i i } = \\sum _ { i , j } d e t ( \\partial ^ { 2 } h _ { i j } ) d z ^ { 1 } \\wedge d \\bar { z } ^ { 1 } \\wedge d z ^ { 2 } \\wedge d \\bar { z } ^ { 2 } \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\xi \\frac { \\delta _ { \\zeta } \\mathcal { N } ( t , \\zeta , \\xi , v , \\overline v ) } { t } \\vert _ { \\zeta = t } d \\xi = \\frac { 1 } { t } \\int _ 0 ^ t \\xi \\Big ( \\mathcal { N } ( t , t , \\xi , v , \\overline v ) - \\mathcal { N } ( t , 0 , \\xi , v , \\overline v ) \\Big ) d \\xi . \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} L _ { 0 } ( T ) = 1 , L _ { 1 } ( T ) = \\frac 1 2 | \\partial T | _ 1 , L _ { 2 } ( T ) = | T | , \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} \\sum _ { N = 0 } ^ { \\infty } \\frac { B _ { N } } { N ! } | \\mathbf { u } | ^ { N } = \\sum _ { N = 0 } ^ { \\infty } B _ { N } \\sum _ { | \\mathbf { m } | = N } \\Psi _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { u } ) = \\sum _ { \\mathbf { m } \\in \\mathcal { P } } B _ { | \\mathbf { m } | } \\Psi _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { u } ) . \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} u ^ * T _ { ( f _ 0 , \\tau _ 0 ) ^ * , ( f _ 1 , \\tau _ 1 ) ^ * } = T _ { u ^ * \\circ ( f _ 0 , \\tau _ 0 ) ^ * , u ^ * \\circ ( f _ 1 , \\tau _ 1 ) ^ * } \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} \\Re ( \\Omega ' ) & = \\Re ( X ^ I \\gamma _ I + F _ I \\gamma ^ I ) \\\\ & = \\Re ( \\sqrt { D } X ^ I ) \\gamma _ I + \\Re ( \\sqrt { D } F _ I ) \\gamma ^ I \\\\ \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} \\norm { d g _ \\# } = \\sup _ { \\| v \\| _ { T _ \\mu W ( M ) } = 1 } \\| d g ( v ) \\| _ { T _ { g _ \\# \\mu } W ( M ) } = \\sup _ { \\| v \\| _ { T _ \\mu W ( M ) } = 1 } \\| v \\| _ { T _ { \\mu } W ( M ) } = 1 . \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\sigma _ { 0 } } \\Vert \\hat { S } _ { t } \\Vert _ { 2 } ^ { 2 } = \\Big ( \\ , 1 - \\frac { 3 } { 2 n } \\ , \\Big ) ^ { t } \\ , \\Vert \\hat { S } _ { 0 } \\Vert _ { 2 } ^ { 2 } + \\frac { 2 } { 3 n } \\Big [ \\ , 1 - \\Big ( \\ , 1 - \\frac { 3 } { 2 n } \\ , \\Big ) ^ { t } \\ , \\Big ] . \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} | \\partial _ { r } v _ { 5 } ( t , r ) | \\leq \\begin{cases} \\frac { C } { t ^ { 7 / 2 } \\log ^ { 3 b - 3 + \\frac { 5 N } { 2 } } ( t ) } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C \\log ^ { 3 } ( t ) } { \\sqrt { r } t ^ { 3 / 2 } } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} C _ T ( n ) \\geq \\begin{cases} \\pi / 1 6 & ( n = 3 , 4 ) , \\\\ \\pi / 2 4 & ( n = 5 ) , \\\\ \\pi / 3 2 & ( n \\geq 6 ) . \\end{cases} \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} = \\sum _ { j = 0 } ^ k \\Biggl [ \\binom { 2 k } { j } - \\binom { 2 k } { j - 1 } \\Biggr ] \\Bigl ( \\frac { c _ 1 } { c _ 1 + c _ 2 } \\Bigr ) ^ { j } \\Bigl ( \\frac { c _ 2 } { c _ 1 + c _ 2 } \\Bigr ) ^ { 2 k - j } \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} \\mathcal S _ 0 ^ { \\frac { 1 } { 2 } } ( \\partial \\Omega ) : = \\left \\{ f \\in L ^ 2 ( \\partial \\Omega ) : f = \\sum _ { j = 1 } ^ { \\infty } \\hat b _ j \\hat v _ { j , 0 } { \\rm \\ w i t h \\ } \\left ( \\sqrt { \\lambda _ j ( 0 ) } \\hat b _ j \\right ) _ { j = 1 } ^ { \\infty } \\in l ^ 2 \\right \\} . \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} \\alpha ( n ) : = \\int _ { B _ { n } ( x _ { 0 } ) } | f ( x ) | d { \\rm v o l } ( x ) \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\bar { \\mu } _ { N , m } ( d y ) = \\exp ( - N \\bar { H } _ Y ( y ) ) d y . \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\left | H _ { 1 } - \\exp \\left \\{ \\sum _ { i = 1 } ^ { k } \\frac { 2 \\mu _ { i } H _ { i + 1 } - \\mu _ { i } ^ 2 } { 2 \\sigma _ { i } ^ 2 } \\right \\} \\right | \\ge \\frac { \\tilde { \\epsilon } } { 2 } \\right ] \\le \\frac { \\delta \\tilde { \\epsilon } ^ 2 } { 2 5 \\tilde { \\epsilon } ^ 2 } = \\frac { \\delta } { 2 5 } . \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} B _ N ( v ) [ \\varphi ] = \\mathcal Q _ { \\sigma } ( v , \\varphi ) + b ( \\gamma _ 0 ( v ) , \\gamma _ 0 ( \\varphi ) ) _ { \\partial \\Omega } \\ , , \\ \\ \\ \\forall v , \\varphi \\in \\mathcal { H } ^ 2 _ { 0 , N } ( \\Omega ) , \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} \\sigma \\circ ( 1 \\otimes \\sigma ) \\circ ( P \\otimes P \\otimes p ) = \\sigma \\circ ( P \\otimes p ) \\circ ( m \\otimes 1 ) \\circ ( ( P \\otimes 1 + 1 \\otimes P + \\lambda ( 1 \\otimes 1 ) ) \\otimes 1 ) \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} 0 & < \\psi ( j ' , i ' ) \\\\ & = l _ { j ' } - l _ { i ' } - \\Delta ( j ' , i ' ) \\\\ & = l _ { j ' } - l _ { i ' + 1 } - \\Delta ( i ' , i ' + 1 ) - \\Delta ( j ' , Q ( j ' ) ) - \\Delta ( Q ( j ' ) , Q ( i ' ) + 1 ) - \\Delta ( Q ( i ' ) + 1 , i ' ) \\\\ & = ( l _ { j ' } + \\Delta ( Q ( j ' ) , j ' ) ) - ( l _ { i ' + 1 } + \\Delta ( Q ( i ' ) + 1 , i ' + 1 ) ) - \\Delta ( Q ( j ' ) , Q ( i ' ) + 1 ) \\ , , \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} | \\lambda ^ { ( 2 + k ) } ( t ) | \\leq \\frac { C } { t ^ { 2 + k } \\log ^ { b + 1 } ( t ) } , t \\geq T _ { 0 } , k = 1 , 2 \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} V ( D ) = \\left ( \\begin{array} { c c } e ^ { - D ^ - D ^ + } & e ^ { - \\frac { 1 } { 2 } D ^ - D ^ + } \\left ( \\frac { I - e ^ { - D ^ - D ^ + } } { D ^ - D ^ + } \\right ) D ^ - \\\\ e ^ { - \\frac { 1 } { 2 } D ^ + D ^ - } D ^ + & I - e ^ { - D ^ + D ^ - } \\end{array} \\right ) \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} { \\cal G } _ d = \\min \\left ( N m , \\kappa , \\frac { \\xi ^ 2 } { r } , \\frac { \\alpha } { r } , \\frac { \\beta } { r } \\right ) , \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} \\mathcal { W } ^ p ( M ) : = \\int _ M H ^ p \\ , d S , p \\geq 1 , \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { \\Phi } ^ { ( 1 ) } _ t \\\\ \\dot { \\Psi } ^ { ( 1 ) } _ t \\end{bmatrix} & = A \\begin{bmatrix} \\Phi ^ { ( 1 ) } _ t \\\\ \\Psi ^ { ( 1 ) } _ t \\end{bmatrix} + h ( t ) , & \\begin{bmatrix} \\Phi ^ { ( 1 ) } _ 0 \\\\ \\Psi ^ { ( 1 ) } _ 0 \\end{bmatrix} & = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} { \\rm s p e c } _ { L ^ 2 } ( D _ { \\partial } ) \\cap [ - \\alpha , \\alpha ] = \\emptyset \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} \\Delta ' = \\delta ^ N \\prod _ { \\substack { \\beta \\in R \\\\ \\gamma \\notin D _ G ( \\beta ) \\cup \\{ \\beta \\} } } | \\beta - \\gamma | _ v ^ { n _ \\gamma } . \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & h \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} 1 & - h \\\\ 0 & 1 \\end{pmatrix} = \\begin{pmatrix} a + h c & \\ast \\\\ \\ast & \\ast \\end{pmatrix} \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} \\int _ { S } | H | = \\frac { 1 } { 2 } \\int _ 0 ^ { L _ i } \\int _ 0 ^ { L _ \\varepsilon } | g ^ { j l } A _ { j l } | \\sqrt { \\det { ( g _ { j l } ) } } d s d \\sigma . \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} & | \\partial _ { t } \\left ( \\frac { 1 6 } { \\lambda ( t ) ^ { 2 } } \\int _ { t } ^ { \\infty } d s \\lambda '' ( s ) \\left ( K _ { 1 } ( s - t , \\lambda ( t ) ) - \\frac { \\lambda ( t ) ^ { 2 } } { 4 ( 1 + s - t ) } \\right ) \\right ) | \\\\ & \\leq \\frac { C } { t ^ { 3 } \\log ^ { b + 1 } ( t ) } + C \\sup _ { x \\geq t } | e ''' ( x ) | \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} f ^ * ( t ) = \\inf \\left \\{ s > 0 : \\alpha _ { f } ( s ) < t \\right \\} , \\ ; \\mbox { f o r } t > 0 . \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} \\Xi _ n ( b _ { w , P } ) = \\xi ^ { \\mu \\lambda } _ { ( w , \\beta _ w P ) } + \\sum _ { w ' \\prec w } \\xi _ { ( w ' , a _ { w ' } ) } \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} ( { } ^ Q \\Omega ^ - ) ^ a { } _ { b c } = N ^ a { } _ x \\rho ^ \\mu _ b \\partial _ \\mu ( N ^ { - 1 } ) ^ x { } _ c , \\quad ( { } ^ Q \\Omega ^ + ) ^ a { } _ { b c } = ( ( K N ^ { - 1 } ) ^ { - 1 } ) ^ a { } _ x \\partial ^ \\mu _ b ( K N ^ { - 1 } ) ^ x { } _ c , \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} { \\tilde f _ 1 } '' = - \\left ( \\frac { \\tilde f _ 1 } { 2 } \\theta \\right ) ' = \\frac { \\tilde f _ 1 } { 4 } \\theta - \\frac { \\tilde f _ 1 } { 2 } \\theta ' . \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} \\limsup _ { s \\to \\infty } R ( t _ s ) & \\leq \\limsup _ { s \\to \\infty } \\hat { \\lambda } ^ { s - \\hat { s } } R ( t _ { \\hat { s } } ) + \\sum _ { \\tau = \\hat { s } } ^ { s - 1 } c ( \\tau ) t _ { \\tau + 1 } ^ { \\beta } \\hat { \\lambda } ^ { s - \\tau - 1 } \\cr & = \\lim _ { s \\to \\infty } c ( s ) t _ { s + 1 } ^ { \\beta } , \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} \\begin{aligned} I ( u , v ) & \\geq \\frac { M } { p _ 1 \\vee p _ 2 } \\bigg ( \\| u \\| _ { 0 , s _ 1 , p _ 2 } ^ { p _ 2 } + \\| v \\| _ { 0 , s _ 2 , p _ 2 } ^ { p _ 2 } \\bigg ) - | l | _ 1 , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} \\partial _ x Z _ \\lambda = ( \\lambda - \\mu ) Y _ \\mu ^ { - 1 } J ^ { - 1 } A Y _ \\lambda = ( \\lambda - \\mu ) \\big ( Y ^ * _ \\mu J Y _ \\mu ) ^ { - 1 } Y _ \\mu ^ * A Y _ \\mu Z _ \\lambda . \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} \\sum _ { \\mathbf { m } \\in \\mathcal { P } } B _ { \\mathbf { m } } ^ { ( d ) } \\left ( \\mathbf { 0 } \\right ) \\Psi _ { \\mathbf { m } } ^ { ( d ) } ( \\mathbf { u } ) = \\frac { | \\mathbf { u } | } { e ^ { | \\mathbf { u } | } - 1 } = \\sum _ { N = 0 } ^ { \\infty } \\frac { B _ { N } } { N ! } | \\mathbf { u } | ^ { N } . \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} J _ { 1 } '' ( x ) = \\frac { 1 } { 4 } ( J _ { 3 } ( x ) - 3 J _ { 1 } ( x ) ) \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} \\delta ( \\Delta { f } ) & = \\delta ( g ^ { i j } f _ { ; i j } ) = \\delta \\big ( g ^ { i j } ( f _ { i j } - \\Gamma _ { i j } ^ k f _ { k } ) \\big ) = \\delta ( g ^ { i j } f _ { , i j } ) - \\delta ( g ^ { i j } \\Gamma _ { i j } ^ k f _ { k } ) . \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } e ( t g ( n ) ) \\Lambda ( n ) = o ( x ) , x \\to \\infty . \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} I _ 2 & : = \\int _ { D ( R , a _ 2 ) \\setminus D ( R , a _ 3 ) } f ( y ; \\gamma , \\beta , q , \\delta , x ) \\ , d y \\\\ & \\asymp R ^ { \\gamma + \\alpha - q } \\left ( F \\left ( \\frac { a _ 3 } { R } ; \\gamma + \\alpha - q - 1 , \\beta \\right ) - F \\left ( \\frac { a _ 2 } { R } ; \\gamma + \\alpha - q - 1 , \\beta \\right ) \\right ) . \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} ( x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } , x _ { 5 } , x _ { 6 } ) = \\begin{cases} ( 1 / 2 - x , 8 x , 1 / 2 - 6 x , - 7 x , - 2 x , 4 x ) , \\\\ ( 1 / 2 - x , 1 / 2 + 8 x , - 6 x , - 7 x , 1 / 2 - 2 x , 1 / 2 + 4 x ) . \\end{cases} \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} l \\bullet \\left ( \\mathsf { M } ( - l ) \\bullet \\phi \\right ) ( [ x ] , r ) & = \\left ( \\mathsf { M } ( - l ) . \\phi \\right ) \\left ( [ x - l \\theta ] , r - l \\right ) \\\\ & = \\left ( r - l \\right ) \\cdot \\phi ( [ x ] , r ) . \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} \\int _ { D _ 0 } \\frac { 1 } { \\epsilon ( x , \\omega ) } ( \\nabla + i { \\boldsymbol k } ) u \\cdot \\overline { ( \\nabla + i { \\boldsymbol k } ) v } d x = \\left ( \\frac { \\omega } { c } \\right ) ^ 2 \\int _ { D _ 0 } u \\overline { v } d x \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} \\dot { u } & = - u + H ( u - a v - b ) , \\\\ \\tau \\dot { v } & = - v + H ( u - c v - d ) . \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} Q \\cap ( \\partial B _ \\rho \\cap X ) = \\{ r e : 0 \\leq r \\leq R \\} \\cap \\partial B _ \\rho = \\{ \\rho e \\} \\rho e \\in ( Q ) , \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} T = e ^ { H } = e ^ { U ^ * D _ { H } U } \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} \\rho ^ \\kappa _ a \\partial _ \\kappa ( E _ { \\lambda \\mu } \\rho ^ \\mu _ c ) = & \\rho ^ \\kappa _ a \\partial _ \\kappa ( ( \\rho ^ + ) ^ d _ \\lambda \\rho ^ \\nu _ d E _ { \\nu \\mu } \\rho ^ \\mu _ c ) = \\rho ^ \\kappa _ a \\partial _ \\kappa ( ( \\rho ^ + ) ^ d _ \\lambda ) \\rho ^ \\nu _ d E _ { \\nu \\mu } \\rho ^ \\mu _ c + \\rho ^ \\kappa _ a ( \\rho ^ + ) ^ d _ \\lambda \\partial _ \\kappa ( \\rho ^ \\nu _ d E _ { \\nu \\mu } \\rho ^ \\mu _ c ) . \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & a _ { 1 , 3 } \\\\ 0 & 1 & a _ { 2 , 3 } \\\\ 0 & 0 & 0 \\end{pmatrix} , \\begin{pmatrix} 1 & a _ { 1 , 2 } & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , \\textrm { o r } \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} \\begin{aligned} f \\left ( M + m - \\bar { a } \\right ) & = f \\left ( \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } \\left ( M + m - \\frac { \\bar { a } + { { a } _ { i } } } { 2 } \\right ) } \\right ) \\\\ & \\le \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( M + m - \\frac { \\bar { a } + { { a } _ { i } } } { 2 } \\right ) } \\end{aligned} \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} 0 = - c ' ( s _ 0 ) h _ { z z } ( 0 , s _ 0 ) . \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} J ( u ) = \\frac 1 p \\| \\nabla u \\| _ p ^ p - \\frac 1 q \\int _ \\Omega | u | ^ q \\ln | u | d x + \\frac { 1 } { q ^ 2 } \\| u \\| _ q ^ q , \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} F _ I = \\frac { \\partial F } { \\partial X ^ I } , \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} b _ 3 & = - \\frac { 1 } { 2 X _ { 1 + 2 + 3 } } \\left ( - 8 a _ 1 ^ 2 \\left ( X _ { 1 + 2 } + X _ { 2 + 3 } + X _ { 1 + 3 } + 3 X _ { 1 + 2 + 3 } \\right ) \\right . \\\\ & \\left . + 1 2 a _ 2 \\left ( X _ { 1 + 2 + 3 } \\right ) + 8 a _ 1 ^ 2 \\left ( X _ { 1 + 2 } + X _ { 2 + 3 } + X _ { 1 + 3 } \\right ) \\right ) \\\\ & = 1 2 a _ 1 ^ 2 - 6 a _ 2 . \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} \\tilde { A } = \\sum _ { \\ell = 1 } ^ k \\gamma _ { \\ell } \\ ; S ( \\mu _ { \\ell } ) \\otimes { A } ' ( \\mu _ { \\ell } ) . \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} q ^ { \\prime } \\left ( t \\right ) = - 4 \\left ( a + \\frac { b ^ { 2 } } { a } \\right ) q ^ { 2 } \\left ( t \\right ) , q \\left ( 0 \\right ) = \\gamma , \\gamma \\geq 0 . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} \\hat { L } ^ 2 _ i = \\frac { 1 } { 2 } \\sum _ { x , y \\in \\mathcal { B } _ i } \\bigg ( x \\frac { \\partial } { \\partial y } - y \\frac { \\partial } { \\partial x } \\bigg ) ^ 2 , \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} R ( r ) = \\int _ 0 ^ r U _ { 1 , t } ( s , \\gamma ) ( \\partial _ s A _ \\nu ( s , \\gamma ( t ) ) \\dot { \\gamma } ^ \\nu ( t ) - \\nabla _ \\mu F ^ \\mu _ { \\ \\nu } ( s , \\gamma ( t ) ) \\dot { \\gamma } ^ \\nu ( t ) ) U _ { t , 0 } ( s , \\gamma ) d t . \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} \\dot { x } & = y , \\\\ \\dot { y } & = \\begin{cases} - x - 2 h _ 1 y , & x < a , \\\\ - x + 2 h _ 2 y , & x > a . \\end{cases} \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} \\mathsf { \\hat { q } } _ { \\mathsf { 1 } } : = \\left \\Vert Q \\mathsf { \\hat { p } } _ { 1 } \\right \\Vert _ { 2 } ^ { - 2 } Q \\mathsf { \\hat { p } } _ { 1 } . \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} \\widehat { \\mathcal { K } } _ 1 & = \\bigl \\{ ( \\xi , \\eta ) \\in \\R ^ 2 \\ , | \\ , \\bigl | \\eta - ( \\sqrt { 2 } + 1 ) ^ { \\frac { 2 } { 3 } } ( \\sqrt { 2 } + \\sqrt { 3 } ) \\xi \\bigr | \\leq 2 ^ { - 1 5 } N _ 1 \\bigr \\} , \\\\ \\widehat { \\mathcal { K } } _ 2 & = \\bigl \\{ ( \\xi , \\eta ) \\in \\R ^ 2 \\ , | \\ , \\bigl | \\eta + ( \\sqrt { 2 } + 1 ) ^ { \\frac { 2 } { 3 } } ( \\sqrt { 3 } - \\sqrt { 2 } ) \\xi \\bigr | \\leq 2 ^ { - 1 5 } N _ 1 \\bigr \\} . \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} \\mathbb { E } ( \\| \\mathcal { L } _ L ^ { \\lambda } f ^ { \\epsilon } - f \\| _ 2 ) \\leq & c \\tau _ 3 V ^ { 1 / 2 } G \\sqrt { \\log { N } } + \\left [ ( 1 + \\tau _ 3 ) V ^ { 1 / 2 } C \\| f \\| _ { k , \\zeta } \\right ] L ^ { - k - \\zeta } \\\\ & + \\left [ V \\left ( \\| f \\| _ { \\infty } + C \\| f \\| _ { k , \\zeta } L ^ { - k - \\zeta } \\right ) ^ 2 - \\chi \\right ] ^ { 1 / 2 } , \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} \\sum _ { i = m + 3 } ^ L G _ i + 4 \\leq 2 \\sum _ { i = m + 4 } ^ L H _ i . \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} \\overline { S } _ { - \\infty } : = \\limsup _ { n \\to - \\infty } S _ n , \\underline { S } _ { - \\infty } : = \\liminf _ { n \\to - \\infty } S _ n , \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} \\frac { d } { d t } S _ { \\tau F } [ ( g _ t , \\tau _ t ) , A ] = S _ { \\tau F } [ ( g _ t , \\beta _ t ) , A ] . \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} \\bigg \\| { f ( C ^ p ) \\over f ( C ) ^ p } \\bigg \\| _ \\infty = \\bigg \\| { g ( ( B ^ { 1 / 2 } A ^ { - 1 } B ^ { 1 / 2 } ) ^ p ) \\over g ( B ^ { 1 / 2 } A ^ { - 1 } B ^ { 1 / 2 } ) ^ p } \\bigg \\| _ \\infty = \\bigg \\| { g ( ( A ^ { - 1 / 2 } B A ^ { - 1 / 2 } ) ^ p ) \\over g ( A ^ { - 1 / 2 } B A ^ { - 1 / 2 } ) ^ p } \\bigg \\| _ \\infty \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} \\int _ X e ^ { \\Phi ( t ) ^ * } e ^ { - t \\psi } \\omega _ 0 ^ n & \\le \\int _ X e ^ { - t \\psi } \\omega _ 0 ^ n = c _ 0 ^ { - n } \\int _ X e ^ { - t \\psi } ( c _ 0 \\omega _ 0 ) ^ n \\le c _ 0 ^ { - n } C _ { c _ 0 \\omega _ 0 } , \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} J _ { n } ^ { ( 3 ) } = \\frac { 1 } { 7 } \\left ( 2 ^ { n + 1 } - V _ { n } ^ { ( 2 ) } \\right ) \\ \\textrm { a n d } \\ j _ { n } ^ { ( 3 ) } = \\frac { 1 } { 7 } \\left ( 2 ^ { n + 3 } + 3 V _ { n } ^ { ( 2 ) } \\right ) . \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} \\begin{aligned} { } [ B , Z ] & { } = Z , & [ B , U ] & { } = \\frac { 1 } { 2 } \\ , U , & [ U , V ] & { } = \\langle J U , V \\rangle Z , & [ U , Z ] & { } = 0 . & \\end{aligned} \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} Q _ { \\Sigma } ( v , v ) + \\mu ^ 2 \\sum _ { i = 1 } ^ I ( \\int _ { \\Sigma } v \\cdot \\phi _ i \\ ) ^ 2 \\geq \\lambda \\int _ { \\Sigma } | \\nabla v | ^ 2 + v ^ 2 / \\rho ^ 2 \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ { \\R } \\rho ( X _ { s - } , y ) \\nu ( d y , d s ) = \\sum _ { 0 < s \\leq t } \\rho ( X _ { s - } , \\Delta L _ s ) \\mathbf { 1 } _ D ( s ) = \\sum _ { k = 1 } ^ { N _ t } \\rho ( X _ { \\tau _ { k } - } , \\xi _ k ) , \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} R i c ( \\Sigma _ l , \\Sigma _ l ) & = ( n - 2 ) ( \\frac 1 { g ^ 2 } - \\frac { g _ t ^ 2 } { g ^ 2 } ) - \\frac { g _ { t t } } g - m \\frac { u _ t } u \\frac { g _ t } g \\\\ & \\ge ( n - 2 ) \\left ( \\frac 1 { \\alpha ^ { \\gamma } t ^ { \\gamma } } - \\frac { 4 ( 1 + \\frac r 6 ) ^ 2 \\gamma ^ 2 } { t ^ 2 } \\right ) - \\frac { 1 2 \\gamma ( 1 + 2 r ) ( 1 + \\frac { r \\gamma } 3 ) } { r t ^ 2 } \\\\ & \\quad - m \\frac { ( 1 + 3 c ) ( 1 + \\frac r 6 ) \\gamma } { t ^ 2 \\cos \\Delta } \\\\ & > 0 , \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} \\iota ( a \\star b ) & = a b + \\sum _ { k = 1 } ^ \\infty \\left ( \\frac { i } { 2 } \\right ) ^ k ( - 1 ) ^ k \\rho ^ { 2 k } P _ k ( \\iota a , \\iota b ) \\\\ & = a b + \\sum _ { k = 1 } ^ \\infty \\left ( \\frac { i } { 2 } \\right ) ^ k \\rho ^ { 2 k } P _ k ( \\iota b , \\iota a ) = \\iota ( b ) \\star \\iota ( a ) \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} \\bigsqcup _ { n = 0 } ^ { \\infty } \\lbrace V _ { i } ^ { ( n ) } \\rbrace _ { i \\in I ^ { ( n ) } } = \\lbrace V _ { i } \\rbrace _ { i \\in I } . \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} s & \\ , = ( 0 , 1 , 2 , 0 , { \\bf 1 } , 2 , { \\bf 1 } , 2 , 6 , 3 , 6 , 6 , 4 ) \\\\ & \\rightarrow ( 0 , 1 , 2 , 0 , { 1 } , { 2 } , { 2 } , { \\bf 3 } , 6 , { \\bf 3 } , 6 , 6 , 4 ) ; \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} \\left | \\mathbb { E } _ { \\nu _ { N , 1 } ^ { \\sigma } } \\left [ X _ i \\right ] - \\mathbb { E } _ { \\nu _ { N , 0 } ^ { \\sigma } } \\left [ X _ i \\right ] \\right | \\leq C \\sum _ { j = 1 } ^ { N } \\exp \\left ( - C | i - j | \\right ) \\lesssim 1 . \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} \\left ( \\mathsf { \\hat { p } } _ { \\mathsf { j } } \\mathsf { , \\hat { q } } _ { \\mathsf { k } } \\right ) _ { 2 } = 0 . \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} e _ s ^ \\perp K ( \\Psi ; M ; \\gamma ) = \\sum _ { S \\subset [ \\ell ] , \\ | S | = s } K ( \\Psi ; M ; \\gamma - \\epsilon _ { S } ) \\ , , \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} = \\frac { 1 } { 2 } \\left ( \\frac { 1 } { 2 } s ^ 2 - s + \\frac { 1 } { 2 } + w _ 1 - \\frac { 1 } { 2 } w _ 1 ^ 2 \\right ) + w \\delta + \\frac { 1 } { 2 } w ( w + 1 ) k . \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} \\nu _ 1 ( f ) = n \\mbox { i f a n d o n l y i f } \\nu _ 2 ( f ) = \\frac { b } { a } n . \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} \\pi _ * \\Big ( \\big ( \\lambda _ 1 + \\cdots + \\lambda _ { \\dim B } \\big ) \\omega ^ { \\wedge \\dim X } \\Big ) = \\frac { { \\rm { T r } ' } \\ , \\big [ \\pi _ * ( \\omega ^ { \\wedge ( \\dim X + 1 ) } ) \\big ] } { \\dim X + 1 } = \\frac { \\dim B } { \\dim X + 1 } . \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { \\phi _ n ( x ) - 1 } \\dfrac { ( - 1 ) ^ { \\ell } } { \\delta _ { \\ell } } & = \\sum _ { m = 1 } ^ { k - 1 } \\left ( \\dfrac { 1 } { \\delta _ { 2 m } } - \\dfrac { 1 } { \\delta _ { 2 m + 1 } } \\right ) + \\dfrac { 1 } { \\delta _ { 2 k } } . \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} \\left | \\sum _ { i = k + 1 } ^ n \\langle \\Pi ( z ) , \\nabla _ { E _ i } E _ i \\rangle \\right | \\leq ( n - k ) \\ , | z | \\sup _ { \\eta \\in W } \\ , \\frac { | A ( \\eta , \\eta ) | } { | \\eta | ^ 2 } \\ , . \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} F _ J ( x , y ) = \\begin{bmatrix} f _ J ( x , y ) \\\\ g _ J ( x , y ) \\end{bmatrix} , \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} \\frac 1 2 \\Delta w & = \\frac 1 2 \\eta ^ { 2 } \\Delta | \\nabla v | ^ { 2 } + \\frac 1 2 | \\nabla v | ^ { 2 } \\Delta \\eta ^ { 2 } + \\langle \\nabla | \\nabla v | ^ { 2 } , \\nabla \\eta ^ { 2 } \\rangle . \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} S = \\begin{psmallmatrix} H _ y & I _ y & J _ y \\\\ H _ { p _ x } & I _ { p _ x } & J _ { p _ x } \\\\ H _ { p _ y } & I _ { p _ y } & J _ { p _ y } \\end{psmallmatrix} \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} f ( t , t _ 1 , t _ 2 , t _ 3 ) = 1 - ( t _ 2 - t ) \\left ( ( z ( t ) - z ( t _ 1 ) ) ^ { - 1 } z ' ( t ) - ( z ( t ) - z ( t _ 3 ) ) ^ { - 1 } z ' ( t ) \\right ) + o ( t _ 2 - t ) \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} \\lim _ { \\omega _ { n + 1 } \\rightarrow 0 } \\omega _ { n + 1 } ^ { 1 - 2 s } \\Omega _ { n + 1 } \\omega _ { n + 1 } ^ { \\frac { 2 s - 1 } { 2 } } \\overline { v } = \\tilde { V } \\omega _ { n + 1 } ^ { \\frac { 2 s - 1 } { 2 } } \\overline { v } \\quad \\partial \\mathcal { S } _ { + } ^ { n } \\times \\mathbb { R } . \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} P ^ b _ { Q } : = \\left ( \\begin{array} { c c } { } ^ b S _ { + } ^ 2 & { } ^ b S _ { + } ( I + { } ^ b S _ { + } ) Q ^ b \\\\ { } ^ b S _ { - } D ^ + & I - { } ^ b S _ { - } ^ 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} D ^ { 4 } _ i = \\{ \\alpha ^ { i + 4 j } \\ , : \\ , 0 \\leq j \\leq k - 1 \\} , \\ , \\ , \\ , \\ , 0 \\leq i \\leq 3 . \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} \\gamma + z + N _ { \\sigma } ( F _ { \\mu _ { 1 } } ( z ) ) = F _ { \\mu _ { 1 } \\boxplus \\mu _ { 2 } } ^ { \\langle - 1 \\rangle } ( F _ { \\mu _ { 1 } } ( z ) ) , \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} H _ { j + 1 } = H _ j + \\dots + H _ { j - k + 1 } + 4 H _ { j - k - 1 } . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} | D ^ n f ( \\lambda ) | \\leq \\frac { \\lambda ^ 2 } { 2 } 1 _ { n = 0 } + \\lambda 1 _ { n = 1 } + 1 _ { n \\geq 2 } \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} & \\rho _ t + ( \\rho u ) _ x = 0 , \\ ; x \\in \\mathbb { R } , \\ ; t > 0 , \\\\ & u _ t + u u _ x = Q \\ast ( \\rho u ) - u Q \\ast \\rho , \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} B _ { G , L + 1 } - B _ { H , L + 1 } = 1 + \\sum _ { i = 1 } ^ { L } G _ { i } - G _ { L + 1 } - \\left ( 1 + \\sum _ { i = 1 } ^ { L } H _ { i } - H _ { L + 1 } \\right ) = H _ { L + 1 } - G _ { L + 1 } = 1 > 0 . \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} \\sup _ { t > 0 } | K _ 1 ( t , x , y ) | & \\lesssim \\begin{cases} e ^ { - | x | ^ 2 } , & \\mbox { i f $ \\langle x , y \\rangle \\le 0 $ } , \\\\ { | x + y | } ^ { n } \\exp { \\left ( \\frac { | y | ^ 2 - | x | ^ 2 } { 2 } - \\frac { | x - y | | x + y | } { 2 } \\right ) } , & \\mbox { i f $ \\langle x , y \\rangle > 0 $ } . \\end{cases} \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} \\varphi _ T ^ + ( r _ p ( T ) , \\omega ) : = r _ p ( T ) u _ p ^ + ( 1 , \\omega ) \\ \\ \\ \\forall \\omega \\in S _ p \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} \\begin{aligned} f \\left ( M + m - \\overline { a } \\right ) & \\le \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( M + m - \\left ( \\left ( 1 - t \\right ) \\overline { a } + t { { a } _ { i } } \\right ) \\right ) } \\\\ & \\le f \\left ( M \\right ) + f \\left ( m \\right ) - \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( { { a } _ { i } } \\right ) } \\end{aligned} \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} P = \\begin{bmatrix} 1 - \\alpha & \\alpha & 0 \\\\ \\beta & 1 - \\beta & 0 \\\\ \\beta & \\alpha & 1 - \\alpha - \\beta \\end{bmatrix} , \\alpha + \\beta \\leq 1 , \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} F _ n ^ { ( k ) } - 3 ^ m = F _ { n _ 1 } ^ { ( k ) } - 3 ^ { m _ 1 } ~ ~ ~ ~ ( = c ) \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} d _ { m j k } p ^ j p ^ k = 0 . \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} \\tau = \\underset { E _ 0 \\subset \\Omega , \\mathbb P ( E _ 0 ) = 0 } { s u p } ( \\underset { \\Omega - E _ 0 } { i n f } \\sigma ) \\leq \\underset { \\Omega _ { \\delta } : o p e n , 0 < \\mathbb { P } ( \\Omega _ { \\delta } ) \\leq \\delta \\leq \\frac { 1 } { 3 } } { s u p } ( \\underset { \\Omega - \\Omega _ { \\delta } } { i n f } \\sigma ) . \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} U _ h ( x , 0 ) = \\Psi ^ T r ( \\Lambda ) \\Psi f _ h , \\ \\mbox { w h e r e } \\ r ( z ) = d _ \\alpha \\sum _ { k = 1 } ^ M \\frac { \\psi _ { k , h } ( 0 ) ^ 2 } { \\mu _ { k , h } + z } \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} A = { \\rm O p } ( a ) + K , \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} Q _ { n + 1 } ( x , x _ { n + 1 } ) = c _ { n + 1 } \\left ( \\tfrac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( x _ { n + 1 } ^ { 3 } - 3 x _ { n + 1 } x _ { i } ^ { 2 } ) + b _ { n } ^ { - 1 } Q _ { n } ( x ) \\right ) \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} v _ { 1 } ( t , r ) = \\int _ { t } ^ { \\infty } v _ { s } ( t , r ) d s \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} y _ { \\rm e q } ( \\mu ) = \\left ( 1 - s ( \\mu ) \\right ) \\zeta _ L ( \\mu ) + s ( \\mu ) \\zeta _ R ( \\mu ) , \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} \\Phi _ { p , q } ( \\lambda ) - \\Phi _ { p - 1 , q + 1 } ( \\lambda ) = & \\left ( 3 \\ , p q - 6 \\ , p - 6 \\ , q + 1 2 \\right ) { \\lambda } ^ { 3 } + \\left ( 1 8 \\ , p q - 3 4 \\ , p - 4 0 \\ , q + 7 4 \\right ) { \\lambda } ^ { 2 } \\\\ & + \\left ( 3 0 \\ , p q - 4 9 \\ , p - 6 7 \\ , q + 1 0 1 \\right ) \\lambda + 1 2 \\ , p q - 1 0 \\ , p - 2 2 \\ , q + 2 . \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} \\mathbb { E } [ e ^ { - \\lambda S _ t } ] = e ^ { - t \\phi ( \\lambda ) } , \\lambda , t \\ge 0 . \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} \\mbox { F o r w h i c h s e q u e n c e o f } k \\mbox { - e d g e h y p e r g r a p h s } \\{ H _ m \\} _ { m = 1 } ^ \\infty \\mbox { i s } f ( m , H _ m ) \\mbox { b o u n d e d ( a s } m \\to \\infty \\mbox { ) ? } \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} \\hat { f } ( \\alpha ) = \\int _ { \\mathbb { T } ^ { \\infty } } f ( z ) z ^ { - \\alpha } d z \\ , . \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} f ( B ( x _ \\lambda , v ) ) = f ( \\langle x _ \\lambda , u \\rangle ) \\to f ( \\langle z , u \\rangle ) \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} A \\otimes B = \\begin{pmatrix} A \\ , b _ { 1 1 } & A \\ , b _ { 1 2 } & \\cdots & A \\ , b _ { 1 s } \\\\ A \\ , b _ { 2 1 } & A \\ , b _ { 2 2 } & \\cdots & A \\ , b _ { 2 s } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A \\ , b _ { r 1 } & A \\ , b _ { r 2 } & \\cdots & A \\ , b _ { r s } \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} \\frac { 3 } { 4 } \\int _ { M } Q u d \\mu + \\frac { 1 } { 2 } \\int _ { M } \\left ( P u \\right ) u d \\mu = - \\int _ { M } T o r \\left ( d _ { b } u , \\gamma \\right ) d \\mu . \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} | \\partial _ { t } ^ { 2 } v _ { 5 } ( t , r ) | & \\leq \\frac { C } { t ^ { 4 } \\log ^ { 3 N + b - 2 } ( t ) } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} L _ { r , \\ell } : = U _ { \\ell } \\left ( \\phi _ { \\ell } ^ { \\lambda _ { r - 1 } } ( \\tau ) L _ { r - 1 , \\ell } \\right ) , \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} R _ { n , 2 } ^ * = \\sup _ { \\ss _ 2 } \\{ 2 \\ss _ 2 ^ \\tau \\sum _ { i = 1 } ^ n B _ i - n \\ss _ 2 ^ \\tau \\Sigma _ B \\ss _ 2 \\} + o _ p ( 1 ) . \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} \\partial _ t \\partial _ X \\omega + u \\cdot \\nabla \\partial _ X \\omega = \\partial _ X ( \\partial _ 1 \\theta ) = X \\cdot \\nabla \\partial _ 1 \\theta . \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} L _ h w = 0 , t ^ { - \\lambda ( 0 , x ) } w ( t , x ) \\longrightarrow 0 \\mbox { { \\rm ( } a s $ t \\longrightarrow + 0 $ { \\rm ) } } . \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} A & = \\varphi ( - H \\varphi + ( k - 1 ) \\cot ^ 2 \\theta f - ( n - k ) f ) \\\\ & \\leq ( ( k - 1 ) \\cot \\theta - ( n - k ) \\tan \\theta ) \\varphi [ \\cot \\theta f - \\varphi ] , \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} L _ { p j } = H _ { p j k } \\cdot H _ { p j l } \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} ( X - - - > { \\delta } X ^ 2 ) ^ * ( { \\le } ) & = X , \\\\ ( X ^ 3 - - - > { \\pi _ { 1 2 } } X ^ 2 ) ^ * ( { \\le } ) \\cap ( X ^ 3 - - - > { \\pi _ { 2 3 } } X ^ 2 ) ^ * ( { \\le } ) & \\subseteq ( X ^ 3 - - - > { \\pi _ { 1 3 } } X ^ 2 ) ^ * ( { \\le } ) \\subseteq X ^ 3 , \\\\ ( { \\le } ) \\cap ( X ^ 2 - - - > { \\sigma } X ^ 2 ) ^ * ( { \\le } ) & = ( { = _ X } ) \\subseteq X ^ 2 \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} S _ { t } = \\begin{cases} t & \\mbox { f o r } t < 0 , \\\\ - t + 2 \\sum _ { l = 1 } ^ j { E } _ l , & \\mbox { f o r } \\sum _ { l = 1 } ^ j Q _ l + \\sum _ { l = 1 } ^ j { E } _ l \\leq t \\leq \\sum _ { l = 1 } ^ { j + 1 } Q _ l + \\sum _ { l = 1 } ^ j { E } _ l , \\\\ t - 2 \\sum _ { l = 1 } ^ { j + 1 } Q _ l , & \\mbox { f o r } \\sum _ { l = 1 } ^ { j + 1 } Q _ l + \\sum _ { l = 1 } ^ j { E } _ l \\leq t \\leq \\sum _ { l = 1 } ^ { j + 1 } Q _ l + \\sum _ { l = 1 } ^ { j + 1 } { E } _ l , \\end{cases} \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} J = \\{ d \\in D e r _ { \\alpha ^ { k } } ( \\mathfrak { g } ) | \\tilde { \\phi } ( d , a d _ { k } { \\mathfrak { g } } ) = \\{ 0 \\} \\} . \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} Q = \\left ( \\begin{array} { c c c c } - 1 & 1 & 0 & 0 \\\\ 0 & - 1 & 1 & 0 \\\\ 0 & 0 & - 1 & 1 \\\\ 1 & 0 & 0 & - 1 \\end{array} \\right ) \\mbox { a n d } D = \\left ( \\begin{array} { c c c c } \\displaystyle 0 . 0 1 & 0 & \\ , \\ , 0 & \\ , \\ , \\ , \\ , 0 \\\\ 0 & 0 . 0 1 & \\ , \\ , 0 & \\ , \\ , \\ , \\ , 0 \\\\ 0 & 0 & \\ , \\ , 1 & \\ , \\ , \\ , \\ , 0 \\\\ 0 & 0 & \\ , \\ , 0 & \\ , \\ , \\ , \\ , 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} \\Delta _ B ( f _ s , 0 ) = | s | ^ { - m } \\Delta _ B ( f _ 1 , 0 ) \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} & \\sqrt { R } \\mathbf T _ { N } = \\nabla ( \\sqrt { R } \\sqrt { R } U ) - 2 \\sqrt { R } U \\otimes \\nabla \\sqrt { R } \\ , , \\\\ [ 6 p t ] & \\mathbf S _ K = \\sqrt { R } \\nabla ^ 2 \\sqrt { R } - \\nabla \\sqrt { R } \\otimes \\nabla \\sqrt { R } \\ , . \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} f _ k ( T x ) = f _ k ( x ) , \\forall k \\geq 0 . \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} \\Delta _ { \\rho , j } f ( x ^ j _ k ) & = \\sum _ { i : \\rho ( x ^ j _ k , x ^ j _ i ) \\leq 2 ^ { - j + 2 } } \\left ( f ( x ^ j _ i ) - f ( x ^ j _ k ) \\right ) \\\\ & = \\Bigl ( \\sum _ { i : \\rho ( x ^ j _ k , x ^ j _ i ) \\leq 2 ^ { - j + 2 } } f ( x ^ j _ i ) \\Bigr ) - 3 f ( x ^ j _ k ) . \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} g ( x ) = \\int _ a ^ b \\phi _ x ( y ) ( L _ \\bullet g ) ( y ) \\d y . \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} \\tilde A _ { h , \\tilde h } = e ^ { - m \\hat h \\tilde h s _ { \\tilde h } } D _ { s _ { \\tilde h } } ^ j ( \\hat h \\tilde h ) ^ { m - j } P _ { m - j } ( \\hat h x ' , e ^ { \\hat h \\tilde h s _ { \\tilde h } } x ' , y , ( \\hat h \\tilde h ) ^ { - 1 } D _ { Y _ { \\tilde h } } ) \\bigl ( \\delta ( s _ { \\tilde h } ) \\delta ( Y _ { \\tilde h } ) \\bigr ) \\cdot ( \\hat h \\tilde h ) ^ { - n } . \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} f ( t , x ) = R \\Bigl ( t , x , u , \\frac { \\partial u } { \\partial x } \\Bigr ) : \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} \\Phi ^ * ( v ) = f ^ i \\otimes q _ i , \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} \\widetilde { p } ( 0 ) = \\phi _ \\Omega , \\ \\ \\widetilde { q _ \\Gamma } ( 0 ) = \\phi _ \\Gamma \\hbox { i n $ \\Omega $ } , \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} \\lim _ { \\theta \\rightarrow 0 } \\hat { E } ^ \\top \\hat { M } \\hat { A } = I _ m \\otimes ( E ( \\mu ^ * ) ^ \\top M ^ * A ( \\mu ^ * ) ) \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} E _ 0 = 1 ; \\ E _ 1 = 1 ; E _ n = \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ n \\binom { n - 1 } { k - 1 } E _ { k - 1 } E _ { n - k } . \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} G ( n _ 1 , \\ldots , n _ r ) = \\lim _ { n \\rightarrow \\infty } \\frac { \\ell _ R ( R / I ( \\nu _ 1 ) _ { n n _ 1 } \\cdots I ( \\nu _ r ) _ { n n _ r } ) } { n ^ d } \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} \\ss _ 2 [ 1 ] \\sum _ { i = 1 } ^ n B _ i [ 1 ] - n \\{ \\ss _ 2 [ 1 ] \\} ^ 2 = o _ p ( 1 ) . \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} g _ { \\mathcal H ^ 0 \\otimes \\mathcal { E } } ( \\nabla ^ { \\mathcal H ^ 0 } _ { X } \\psi ( \\gamma ) , Y ( \\gamma ) \\otimes \\Phi ( \\gamma ) ) = g _ { \\mathcal E } ( K _ \\psi ( \\gamma ) < X ( \\gamma ) , Y ( \\gamma ) > , \\Phi ( \\gamma ) ) \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} D ^ m \\Theta ^ { } ( \\mathcal { T } ) = \\mathcal { G } ( \\mathcal { T } , D \\mathcal { T } , \\dots , D ^ m \\mathcal { T } ) . \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} \\mathrm { i n d e x } ( S ^ 7 \\times \\R , \\hat { h } ) & = \\mathrm { i n d e x } ( D ( V _ k ) \\cup ( S ^ 7 \\times [ - 1 , \\infty ) ) , \\hat { g } _ k \\cup \\hat { h } ) \\\\ & = \\mathrm { i n d e x } ( D ( V _ k ) \\cup ( S ^ 7 \\times [ - 1 , 1 ] ) \\cup ( - D ( V _ l ) ) , \\hat { g } _ k \\cup \\hat { h } \\cup \\hat { g } _ l ) . \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} ( f g ) ^ { ( i ) } = \\sum _ { j = 0 } ^ i f ^ { ( j ) } g ^ { ( i - j ) } . \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} | T ( e ) '' ( t ) | & \\leq \\frac { 3 D _ { 2 } } { \\alpha \\log ( \\log ( t ) ) t ^ { 2 } \\log ^ { b + 1 } ( t ) } \\\\ & \\leq \\frac { 1 } { 1 0 0 \\sqrt { \\log ( \\log ( t ) ) } t ^ { 2 } \\log ^ { b + 1 } ( t ) } , e \\in \\overline { B } _ { 1 } ( 0 ) \\subset X \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} \\langle \\tilde \\psi , Z _ 0 \\rangle _ * = \\langle { \\tilde \\psi } ^ + , Z _ 0 ^ + \\rangle _ { * \\mathcal R ^ + } + \\langle { \\tilde \\psi } ^ - , Z _ 0 ^ - \\rangle _ { * \\mathcal R ^ - } = 0 , \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} \\langle \\widehat { u } g , \\widehat { u } f _ 0 \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } + \\langle \\widehat { v } g , \\widehat { v } f _ 0 \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } = \\gamma \\langle g , f _ 0 \\rangle _ { L ^ 2 ( \\mathbb { R } ^ 2 ) } , \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} F ( X \\otimes Z _ 1 ) = ( X _ 1 \\oplus X _ 3 ) \\otimes ( X _ 0 \\oplus X _ 1 \\oplus X _ 3 ) = 2 X _ 0 \\oplus 4 X _ 1 \\oplus 2 X _ 2 \\oplus 5 X _ 3 \\oplus 2 X _ 4 \\oplus X _ 5 . \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} u ( z ) = \\int _ { [ 0 , + \\infty ] } \\frac { 1 + t z } { z - t } \\ , d \\sigma ( t ) , z \\in \\mathbb { C } \\backslash \\mathbb { R } _ { + } . \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} \\frac { C } { t ^ { 2 } \\log ^ { 3 b + 3 N } ( t ) } \\geq | | \\partial _ { r } v _ { 4 , c } ( t , r ) + \\frac { v _ { 4 , c } ( t , r ) } { r } | | _ { L ^ { 2 } ( r d r ) } = | | \\xi \\widehat { v _ { 4 , c } } ( t , \\xi ) | | _ { L ^ { 2 } ( \\xi d \\xi ) } \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} & ( 1 + \\epsilon ( 0 ) ) \\Phi ( - K ) - ( 1 + \\epsilon ( 0 ) / 2 ) \\mathbf { u } ^ * \\\\ \\succeq & ( 1 + \\epsilon ( 0 ) ) \\big [ 1 - C _ 2 K ^ { - \\omega _ * } \\big ] \\mathbf { u } ^ * - ( 1 + \\epsilon ( 0 ) / 2 ) \\mathbf { u } ^ * \\\\ = & \\big [ K _ 1 \\theta ^ { - \\beta } / 2 - C _ 2 K ^ { - \\omega _ * } ( 1 + K _ 1 \\theta ^ { - \\beta } ) \\big ] \\mathbf { u } ^ * \\\\ \\succeq & { \\bf 0 } \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} { } ^ Q \\Omega ^ \\pm = K _ \\pm ^ { - 1 } \\d { } _ Q K _ \\pm , ( { } ^ Q \\Omega ^ \\pm ) ^ a { } _ { b c } = ( K ^ { - 1 } _ \\pm ) ^ a { } _ { d } \\rho ^ \\mu _ b \\partial _ \\mu ( K _ \\pm ) ^ d { } _ { c } , \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} \\tt = ( \\mu , \\beta _ 0 ^ 2 / 2 + \\sigma _ + ^ 2 - 1 , \\beta _ 0 ^ 2 , \\beta _ 1 ^ 2 , \\beta _ 0 \\beta _ 1 ) ^ \\tau . \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} \\eta _ { \\mu _ { n } } ( 1 / x ) \\left [ x \\eta _ { \\mu _ { n } } ( 1 / x ) \\right ] ^ { 1 / ( k _ { n } - 1 ) } = \\eta _ { \\nu _ { n } ^ { \\boxtimes k _ { n } } } ( 1 / x ) , x \\in ( 0 , + \\infty ) , \\ ; n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} ( a - d ) u + b v + c w = 0 , \\ \\ - u ^ 2 - v w = 1 . \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} \\mathcal { L } ^ { - 1 } [ F ( s ) ] = t ^ { q - 1 } M ^ { r } _ { p , q } ( - a t ^ { p } ) . \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} | \\mu ^ { k } _ { i , \\pm } | = | 1 - \\Delta t _ k \\sqrt { b ^ { k } _ i \\lambda ^ { k } _ { i } } | = \\sqrt { 1 - \\eta \\Delta t _ k } , \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} x _ 3 ^ 2 = F _ 1 ( x _ 0 , x _ 1 , x _ 2 ) F _ 2 ( x _ 0 , x _ 1 , x _ 2 ) F _ 3 ( x _ 0 , x _ 1 , x _ 2 ) G _ 1 ( x _ 0 , x _ 1 , x _ 2 ) + G _ 2 ( x _ 0 , x _ 1 , x _ 2 ) ^ 2 , \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} \\bigg [ \\partial _ { n + 1 } x _ { n + 1 } ^ { 1 - 2 s } \\partial _ { n + 1 } + x _ { n + 1 } ^ { 1 - 2 s } P \\bigg ] \\tilde { u } & = 0 \\quad \\mathbb { R } _ { + } ^ { n + 1 } , \\\\ \\tilde { u } & = u \\quad \\mathbb { R } ^ { n } \\times \\{ 0 \\} . \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} \\ ! { \\kappa \\Sigma ^ 1 _ 1 B o r L o c } | = \\forall ( y _ 0 , y _ 1 , \\dotsc ) \\in Y ^ \\# N \\ , ( R ( y _ 1 , y _ 0 ) \\wedge R ( y _ 2 , y _ 1 ) \\wedge \\dotsb \\implies \\bot ) . \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} \\delta & = 1 , & r = 0 . 5 , \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} I ^ { 0 + } _ \\textup { C C } ( \\bar x ) & : = \\{ l \\in \\mathcal Q \\ , | \\ , \\bar G _ l ( \\bar x ) = 0 \\ , \\land \\ , \\bar H _ l ( \\bar x ) > 0 \\} , \\\\ I ^ { + 0 } _ \\textup { C C } ( \\bar x ) & : = \\{ l \\in \\mathcal Q \\ , | \\ , \\bar G _ l ( \\bar x ) > 0 \\ , \\land \\ , \\bar H _ l ( \\bar x ) = 0 \\} , \\\\ I ^ { 0 0 } _ \\textup { C C } ( \\bar x ) & : = \\{ l \\in \\mathcal Q \\ , | \\ , \\bar G _ l ( \\bar x ) = 0 \\ , \\land \\ , \\bar H _ l ( \\bar x ) = 0 \\} \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} | I I | & \\leq C r \\log ( \\log ( t ) ) \\sup _ { x \\geq t } | \\lambda _ { 1 } '' ( x ) - \\lambda _ { 2 } '' ( x ) | + \\frac { C r | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\log ^ { b + 1 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } \\\\ & \\leq \\frac { C r \\log ( \\log ( t ) ) | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\log ^ { b + 1 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } + \\frac { C r | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\log ^ { b + 1 } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} \\Vert f ( 0 ) \\Vert ^ { q } + \\lambda \\Vert f ' ( 0 ) \\Vert ^ { q } \\leq \\int _ { \\mathbb { T } } \\Vert f ( z ) \\Vert ^ { q } d z \\ , . \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} h _ { x ; i } : = \\sum _ { m \\in I } h _ { x ; i , m } e _ { x ; m } \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} | | \\sqrt { \\omega } \\lambda ( x ) \\mathcal { F } ( \\sqrt { \\cdot } \\left ( F _ { 5 } + F _ { 6 } \\right ) ( x , \\cdot \\lambda ( x ) ) ) ( \\omega \\lambda ( x ) ^ { 2 } ) | | _ { L ^ { 2 } ( \\rho ( \\omega \\lambda ( x ) ^ { 2 } ) d \\omega ) } ^ { 2 } = \\frac { 1 } { \\lambda ( x ) ^ { 2 } } \\int _ { 0 } ^ { \\infty } ( L ( \\left ( F _ { 5 } + F _ { 6 } \\right ) ( x , \\cdot \\lambda ( x ) ) ) ) ^ { 2 } ( R ) R d R \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} ( \\delta _ { \\mathrm { H o c h } } f ) ( a _ 1 , \\ldots , a _ { n + 1 } ) = ~ & a _ 1 \\cdot f ( a _ 2 , \\ldots , a _ { n + 1 } ) + \\sum _ { i = 1 } ^ n ( - 1 ) ^ i f ( a _ 1 , \\ldots , a _ { i - 1 } , a _ i a _ { i + 1 } , \\ldots , a _ { n + 1 } ) \\\\ ~ & + ( - 1 ) ^ { n + 1 } f ( a _ 1 , \\ldots , a _ n ) \\cdot a _ { n + 1 } , f \\in C ^ n _ { \\mathrm { H o c h } } ( A , M ) . \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} { \\rm d i a g } ( 1 , 0 ) : = \\left [ \\begin{array} { c c } 1 & 0 \\\\ 0 & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} f ( t ) = V ^ t f _ 0 + \\int _ 0 ^ t V ^ { t - s } B ( s , f , f ) d s , t \\geq 0 \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } \\dots \\int _ { - \\infty } ^ { \\infty } g ( { \\bf x } ) \\exp ( - { \\bf x } ^ { \\top } { \\bf x } ) \\ , d { \\bf x } \\approx \\displaystyle \\sum _ { { \\bf j } = 1 } ^ { L } w _ { { \\bf j } } g ( { \\bf a _ j } ) , \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} \\Theta _ 1 \\leq C b _ q 2 ^ { - 2 q ( 1 + s ) } \\| \\nabla u \\| _ { L ^ \\infty } \\| \\theta \\| _ { H ^ { 1 + s } } ^ 2 . \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} z _ a \\bar { z } _ c , z _ b \\bar { z } _ d \\in \\mathbb { Z } \\mbox { a n d } z _ a \\bar { z } _ d - \\bar { z } _ b z _ c = 1 . \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} ( [ \\delta _ { \\varepsilon _ 1 } , \\delta _ { \\varepsilon _ 2 } ] - \\delta _ { [ \\varepsilon _ 1 , \\varepsilon _ 2 ] _ Q } ) A ^ a = & \\varepsilon ^ b _ 1 \\varepsilon ^ c _ 2 ( - \\nabla ^ - _ \\mu ( T _ { \\nabla ^ - } ) ^ a { } _ { b c } + 2 \\rho ^ \\nu _ { [ b | } ( R _ { \\nabla ^ - } ) ^ a { } _ { \\nu \\mu | c ] } ) D _ - X ^ \\mu \\\\ + & \\varepsilon ^ b _ 1 \\varepsilon ^ c _ 2 ( - \\nabla ^ + _ \\mu ( T _ { \\nabla ^ + } ) ^ a { } _ { b c } + 2 \\rho ^ \\nu _ { [ b | } ( R _ { \\nabla ^ + } ) ^ a { } _ { \\nu \\mu | c ] } ) D _ + X ^ \\mu , \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} J _ i ( u ) = \\begin{cases} 0 , \\ , \\ , u = 0 , \\\\ > 0 , \\ , u = e _ i . \\end{cases} \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} P \\Big \\{ F _ \\beta = \\frac { \\beta } { c } \\ \\Big | \\ V ( 0 ) = c \\Big \\} = e ^ { - \\lambda \\frac { \\beta } { c } } . \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| X ( t ) \\| _ { H ^ s } ^ 2 & \\leq C ( \\| X \\| _ { H ^ s } + \\| \\nabla u \\| _ { L ^ \\infty } \\| X \\| _ { H ^ s } ^ 2 ) . \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} S ^ { ( 3 ) } _ { 4 , 3 , } & = - i \\lambda _ 3 \\cdot \\left ( - 2 a _ 1 \\right ) = 2 i \\lambda _ 3 a _ 1 . \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} \\rho ( f , g ) = \\sum ^ \\infty _ { n = 0 } \\frac { 1 } { 2 ^ n } \\frac { d ( f ( x _ n ) , g ( x _ n ) ) } { 1 + d ( f ( x _ n ) , g ( x _ n ) ) } , \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} R _ { \\nabla ^ + } = \\frac { 1 } { 2 ( 1 + 2 x ^ 2 ) ^ 2 } \\left ( \\begin{matrix} - 2 x ( 5 + x ^ 2 ) & 0 \\\\ 5 - 8 x ^ 2 & 0 \\end{matrix} \\right ) \\d x \\wedge \\d z . \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k - 1 } ( { \\pi _ i } - { \\alpha _ i } ) e ^ { - c _ i t } < \\sum _ { i = k } ^ p ( { \\alpha _ i } - { \\pi _ i } ) e ^ { - c _ i t } . \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} a _ k ( n ) & = a _ k ( n - 1 ) + a _ k ( n - 3 ) - a _ k ( n - 6 ) - a _ k ( n - 1 0 ) \\\\ & \\quad \\ , + a _ k ( n - 1 5 ) + a _ k ( n - 2 1 ) - a _ k ( n - 2 8 ) - a _ k ( n - 3 6 ) + \\cdots , \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} t \\eta _ { \\mu } ( 1 / t ) ^ { \\beta } = \\eta _ { \\nu _ { \\beta } } ( 1 / t ) ^ { \\beta - 1 } , t \\in ( 0 , + \\infty ) , \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p ( \\cdot ) } } : = \\inf \\left \\{ \\lambda > 0 : \\int _ { \\mathbb { R } ^ n } \\left ( \\frac { | f ( x ) | } { \\lambda } \\right ) ^ { p ( x ) } \\ , \\d x \\leq 1 \\right \\} < \\infty . \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} \\theta + d d ^ c \\rho \\geq C ^ { - 1 } \\omega _ 0 \\sup _ X \\rho = 0 , \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} X \\xrightarrow { \\phi } \\bigoplus _ { i = 1 } ^ k X ( s _ i , t _ i ) \\xrightarrow { \\psi } \\bigoplus _ { i = 1 } ^ k C _ i \\to 0 \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} \\mathbb { P } _ \\eta \\left ( \\left . \\mathcal { T } _ t ( x ) < \\frac { 1 - p } { 4 } t \\right | \\mathcal { F } _ { k } \\right ) \\leq \\frac { \\int _ { t / 2 } ^ { t } p \\mathrm { d } s } { \\left ( 1 - \\frac { 1 - p } { 2 } \\right ) \\frac { t } { 2 } } = \\frac { p } { 1 - \\frac { 1 - p } { 2 } } . \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} b _ { 1 } = \\frac { 8 - 3 \\mu } { 4 } . \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} P ( n _ 1 , \\ldots , n _ r ) = \\lim _ { m \\rightarrow \\infty } \\frac { \\ell _ R ( M / I ( 1 ) _ { m n _ 1 } \\cdots I ( r ) _ { m n _ r } M ) } { m ^ d } \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} & \\boldsymbol { Y } = ( \\boldsymbol { A } \\boldsymbol { s } + \\boldsymbol { h } _ d ) \\boldsymbol { x } ^ { \\rm T } + \\boldsymbol { W } = \\boldsymbol { z } \\boldsymbol { x } ^ { \\rm T } + \\boldsymbol { W } , \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} \\varepsilon ( p ) & : = \\frac { \\beta p } { 2 ( p - 1 ) } , \\\\ \\gamma ( p ) & : = - \\frac { K \\beta p } { 2 ( p - 1 ) } . \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} K _ { ( j , h ) , ( i , k ) } : = E _ { x ; j , i } \\left ( X _ { h } ^ * X _ { k } \\right ) \\quad ; ( j , h ) , ( i , k ) \\in I \\times \\{ 1 , \\dots , n \\} \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} \\hat { V } & : = \\frac { V } { \\frac 1 2 v _ { n + 2 } L ^ { n + 2 } } , \\\\ \\hat { A } & : = \\frac { A } { \\frac 1 2 a _ { n + 1 } R ^ { n + 1 } } . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} F ( x , \\ell ) = \\begin{cases} ( x , \\ell + 1 ) & R ( x ) > \\ell + 1 \\\\ ( f _ 0 ( x ) , 0 ) & R ( x ) = \\ell + 1 \\end{cases} . \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} f \\nabla _ 0 g = f ! _ 0 g = f \\sharp _ 0 g = f \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; f \\nabla _ 1 g = f ! _ 1 g = f \\sharp _ 1 g = g . \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} E ^ \\top M E \\dot { x } ( t ) & = E ^ \\top M A x ( t ) + E ^ \\top M B u ( t ) \\\\ [ 1 e x ] y ( t ) & = C x ( t ) . \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} \\sup _ U \\Phi _ { L , C , \\eta } & = u ( l , \\bar { t } ) - u ( \\bar { y } , \\bar { t } ) - L \\phi ( | l - \\bar { y } | ) - C | l - \\hat { x } | ^ 2 - \\frac { 2 } { \\eta ( T - \\bar { t } ) } \\\\ & \\le u ( l , \\bar { t } ) - u ( \\bar { y } , \\bar { t } ) - L \\phi ( | l - \\bar { y } | ) \\\\ & \\le ( L _ 2 - L ( 1 - \\hat { r } ^ \\alpha ) ) | l - \\bar { y } | , \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} A = ( \\psi _ { n , 1 } ^ { \\ast } ( Q ) - 1 ) \\frac { ( k + 1 ) n } { k ( n + 1 ) } + 1 , B = \\sum _ { j = 1 } ^ { n } \\left [ ( \\psi _ { n , j } ^ { \\ast } ( Q ) - 1 ) \\frac { ( k + 1 ) n } { k ( n + 1 ) } + 1 \\right ] . \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} P & = P ( y , z ) \\\\ & = \\max \\left \\{ \\frac { - \\sigma L - k z - \\left ( \\alpha ( \\sigma y \\pm k z ) + \\widehat { \\alpha } z \\right ) } { \\sigma } , \\frac { \\sigma \\alpha y \\pm \\alpha k z + \\widehat { \\alpha } z - 2 k z - \\sqrt { ( 2 \\sigma L ) ^ 2 - 3 ( \\sigma \\alpha y \\pm \\alpha k z - \\widehat { \\alpha } z ) ^ 2 } } { 2 \\sigma } \\right \\} , \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} g ( t ) = A ( t - \\alpha ) ^ { 2 k } h ( t ) , t \\in I , \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} & c ( t , x ) = \\frac { \\partial b } { \\partial x } ( t , x ) + \\frac { \\partial b _ 1 } { \\partial x } ( t , x ) \\in X _ 0 ( ( 0 , T _ 1 ] \\times D _ { R _ 1 } ) , \\\\ & \\gamma ( t , x ) = \\frac { \\partial \\lambda } { \\partial x } ( t , x ) + \\frac { \\partial \\lambda _ 1 } { \\partial x } ( t , x ) \\in X _ 0 ( ( 0 , T _ 1 ] \\times D _ { R _ 1 } ) . \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} & \\int _ { \\mathcal { L } _ { p } \\left ( 0 , ( 2 A ) ^ { - 1 } \\exp ( - B \\omega ( m _ { 0 } ) ) \\right ) } G ( x , y ) \\ , | f ( y ) | \\ , d y \\\\ & = \\sum _ { m \\geq m _ { 0 } } \\int _ { \\mathcal { L } _ { p } \\left ( ( 2 A ) ^ { - 1 } \\exp ( - B \\omega ( m + 1 ) ) , ( 2 A ) ^ { - 1 } \\exp ( - B \\omega ( m ) ) \\right ) } G ( x , y ) \\ , | f ( y ) | \\ , d y \\\\ & \\leq C \\sum _ { m \\geq m _ { 0 } } ^ { \\infty } \\left ( \\omega ( m + 1 ) - \\omega ( m ) \\right ) \\sup _ { M \\setminus B _ m ( p ) } \\left | \\frac { f } { \\rho } \\right | < \\infty \\ , , \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\phi _ { j } ^ { ( 2 ) } ( x _ n ) } { \\phi _ { j } ^ { ( 1 ) } ( x _ n ) } = \\frac { ( \\phi _ j ^ 2 ) ' ( 0 ^ - ) } { ( \\phi _ j ^ 1 ) ' ( 0 ^ - ) } < p \\mbox { i f } j \\in \\{ 1 , . . . , m _ 0 \\} \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} \\nu _ s ( u _ 1 ^ { \\epsilon _ 1 } v _ 1 ^ { ( p - 1 ) i _ 1 - \\epsilon _ 1 } \\cdots u _ s ^ { \\epsilon _ s } v _ s ^ { ( p - 1 ) i _ s - \\epsilon _ s } ) = ( - 1 ) ^ { i _ 1 + \\cdots + i _ s + \\sum _ { \\ell < k } \\epsilon _ \\ell \\epsilon _ k } ( \\lambda _ { i _ 1 - 1 } ^ { \\epsilon _ 1 } \\cdots \\lambda _ { i _ s - 1 } ^ { \\epsilon _ s } ) ^ * , \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} \\mathbb { \\beta } = \\beta ( t ) + p ( t ) d t = q ( t ) + x ( t ) d t \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} \\begin{bmatrix} K _ { 1 , 2 2 } ^ - - K _ { 1 , 2 2 } & K _ { 1 , 2 3 } ^ - - K _ { 1 , 2 3 } \\\\ ( K _ { 1 , 2 3 } ^ - - K _ { 1 , 2 3 } ) ^ { \\top } & K _ { 1 , 3 3 } ^ + - K _ { 1 , 3 3 } \\end{bmatrix} \\geq 0 . \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} \\widetilde { f } _ i ^ { \\bar { p } R / 2 , \\bar { q } R } ( x ) = \\int _ { D _ { \\bar q R } ^ R } P _ { D _ { \\bar p R / 2 } } ( x , y ) f _ i ^ * ( d y ) & \\le c P _ { D _ { \\bar p R / 2 } } ( x , 0 ) \\Lambda _ { 0 , \\bar q R } ( f _ i ^ * ) . \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} \\Lambda : = \\gamma _ 1 ^ { b _ 1 } \\cdots \\gamma _ t ^ { b _ t } - 1 \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} p _ { i j } ( \\sigma ) = \\sigma \\vert _ { M ^ { i + j } } . \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} \\lesssim \\sqrt { \\sigma ( [ - a , a ] ) } \\left ( \\int _ { \\R ^ + } | f ( r ) | ^ 2 \\log ^ 2 ( 2 + r ) d r \\right ) ^ { 1 / 2 } = \\sqrt { \\sigma ( [ - a , a ] ) } L ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} \\partial _ 1 S = \\varepsilon ( \\dot x _ k e _ 2 + \\dot y _ k e _ 3 ) , \\partial _ 2 S = e _ 1 + \\varepsilon ( x _ k \\dot e _ 2 + y _ k \\dot e _ 3 ) , \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} B . v = \\gamma B + q ( { h } ) \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} \\Phi _ { p , q } ( \\lambda ) = & { \\lambda } ^ { 5 } - \\left ( 2 \\ , q - 1 0 + 2 \\ , p \\right ) { \\lambda } ^ { 4 } \\\\ & - \\left ( - 3 \\ , p q + 1 6 \\ , p + 1 6 \\ , q - 4 0 \\right ) { \\lambda } ^ { 3 } \\\\ & - \\left ( - 1 8 \\ , p q + 4 4 \\ , p + 5 0 \\ , q - 7 4 \\right ) { \\lambda } ^ { 2 } \\\\ & - \\left ( - 3 0 \\ , p q + 4 5 \\ , p + 6 3 \\ , q - 5 3 \\right ) \\lambda + 1 2 \\ , p q - 1 0 \\ , p - 2 2 \\ , q + 2 . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} P _ 1 = \\begin{bmatrix} 0 & 0 & I & 0 \\\\ I & 0 & 0 & 0 \\\\ 0 & I & 0 & 0 \\\\ 0 & 0 & 0 & I \\end{bmatrix} \\in \\mathbb { R } ^ { 2 ( p + q ) \\times 2 ( p + q ) } . \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} p _ { t } ^ { n } f ( x ) = \\int _ { K _ n } p _ { t } ^ { n } ( x , y ) f ( y ) \\ , d m ( y ) , m x \\in K _ n . \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} i \\partial _ t u + \\vert \\nabla \\vert u = \\pm | u | ^ 2 u , \\ , \\mathcal { L } = i \\vert \\nabla \\vert , u _ { / t = 0 } = u _ { 0 } . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} \\mathfrak h ^ \\perp = \\mathfrak p _ u \\oplus ( \\mathfrak d ^ \\perp \\cap \\mathfrak c ) \\oplus \\bigoplus \\limits _ { \\Omega \\in \\widetilde \\Psi } \\mathfrak u ( \\Omega ) . \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} [ q _ 1 , [ q _ 2 , q _ 3 ] _ Q ] _ Q = & [ [ q _ 1 , q _ 2 ] _ Q , q _ 3 ] _ Q + [ q _ 2 , [ q _ 1 , q _ 3 ] _ Q ] _ Q , \\\\ [ q _ 1 , f q _ 2 ] _ Q = & f [ q _ 1 , q _ 2 ] _ Q + ( ( \\rho ( q _ 1 ) f ) ) q _ 2 , \\\\ [ q _ 1 , q _ 2 ] _ Q = & - [ q _ 2 , q _ 1 ] _ Q , \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} L _ 0 f ( x ) = \\sum _ { y \\in F ^ { - 1 } \\{ x \\} } J F ( y ) ^ { - 1 } f ( y ) \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} \\widehat { \\eta } _ \\varphi ( D _ \\partial ) : = ( 2 \\pi i ) ^ p \\eta _ \\varphi ( D _ \\partial ) , [ \\varphi ] \\in H ^ { 2 p } _ { { \\rm d i f f } } ( G ) . \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ { \\gamma _ k } | \\kappa | d s = \\pi , \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} H _ { n } ( \\sigma ) : = H _ { n } ^ { S K } ( \\sigma ) + H _ { n } ^ { C W } ( \\sigma ) \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} \\ell ( \\gamma ) = \\sup _ { n \\in \\mathbb { N } , \\ , 0 \\le t _ 1 < \\ldots < t _ n \\le 1 } \\sum _ { k = 1 } ^ n \\| \\gamma ( t _ k ) - \\gamma ( t _ { k - 1 } ) \\| . \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } \\sigma _ { L } ( e ^ { 2 \\pi i j / n } , \\alpha ) \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} \\int _ { t + 1 } ^ { \\infty } d s | K _ { 1 } ( s - t , \\lambda ( t ) ) - \\frac { \\lambda ( t ) ^ { 2 } } { 4 ( 1 + s - t ) } | & \\leq C \\int _ { t + 1 } ^ { \\infty } d s \\frac { \\lambda ( t ) ^ { 2 } } { 4 } \\left ( \\frac { 1 } { s - t } - \\frac { 1 } { 1 + s - t } \\right ) + C \\int _ { t + 1 } ^ { \\infty } d s \\frac { C \\lambda ( t ) ^ { 2 } ( 1 + \\lambda ( t ) ^ { 2 } ) } { ( s - t ) ( 1 + ( s - t ) ^ { 2 } ) } \\\\ & \\leq C \\lambda ( t ) ^ { 2 } ( 1 + \\lambda ( t ) ^ { 2 } ) \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} H _ { L + 2 } - G _ { L + 2 } = c _ 1 + c _ { L + 1 } \\geq 2 = 2 ( 1 ) = 2 \\left ( H _ { L + 1 } - G _ { L + 1 } \\right ) . \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} F ( X \\otimes Z _ 5 ) = F ( X ) \\otimes F ( Z _ 5 ) = 2 X _ 1 \\oplus X _ 3 \\oplus 2 X _ 4 \\oplus X _ 5 . \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} X ^ 0 = p ^ 0 + i \\frac { \\partial I _ 1 } { \\partial q _ 0 } , \\ X ^ I = p ^ I + i \\frac { \\partial I _ 1 } { \\partial q _ I } \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} \\deg ( H _ 0 , ( Q ) , \\rho e ) = \\deg ( h _ 0 , ( Q ) , w ) \\not = 0 . \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} \\delta = & 1 6 7 7 7 2 1 6 ( d _ { 1 , 0 } ^ 2 d _ { - 1 , 0 } ^ 2 - 2 d _ { 1 , 0 } d _ { - 1 , 0 } d _ { 1 , - 1 } d _ { - 1 , 1 } + d _ { 1 , - 1 } ^ 2 d _ { - 1 , 1 } ^ 2 ) d _ { 1 , 0 } ^ 2 d _ { - 1 , 0 } ^ 2 d _ { 1 , - 1 } ^ 2 d _ { - 1 , 1 } ^ 2 . \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial f } { \\partial t } = - c _ 1 \\frac { \\partial f } { \\partial x } - \\lambda ( t ) ( f - b ) \\\\ \\frac { \\partial b } { \\partial t } = c _ 2 \\frac { \\partial b } { \\partial x } + \\lambda ( t ) ( f - b ) \\end{cases} \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} a _ k ( n ) & = a _ k ( n - 1 ) + a _ k ( n - 2 ) - a _ k ( n - 4 ) - a _ k ( n - 8 ) \\\\ & \\quad \\ , + a _ k ( n - 9 ) + a _ k ( n - 1 8 ) - a _ k ( n - 1 6 ) - a _ k ( n - 3 2 ) + \\cdots . \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} V _ { i j } : = \\inf \\{ S _ { 0 T } ( \\varphi ) : \\varphi \\in C _ { 0 T } , \\ \\varphi _ 0 \\in K _ i , \\ \\varphi _ T \\in K _ j , \\ T \\geq 0 \\} . \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} Q ^ { ( n , k ) } : = \\left \\lbrace \\left \\lbrace j _ 1 , \\ldots , j _ k \\right \\rbrace \\subseteq \\left \\lbrace 1 , \\ldots , n \\right \\rbrace \\right \\rbrace . \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} \\begin{aligned} & \\rho ( L _ 1 ^ { \\alpha } L _ 0 ^ { \\beta } L _ { - 1 } ^ { \\gamma } ) \\ , = \\ , \\\\ & = h ( z ) ^ { \\alpha - \\gamma } P ( L + c - \\alpha + \\gamma + 1 , \\alpha ) ( L + \\gamma ) ^ { \\beta } P ( L - c , \\gamma ) \\ , = \\\\ & = \\ , h ( z ) ^ { \\alpha - \\gamma } ( L ^ { \\alpha + \\beta + \\gamma } + \\cdots ) \\end{aligned} \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} \\left | \\mathbb { E } _ { \\mu _ N ^ { \\sigma } } \\left [ X _ i \\right ] \\right | = \\left | \\mathbb { E } _ { \\nu _ { N , 1 } ^ { \\sigma } } \\left [ X _ i \\right ] \\right | \\lesssim \\left | \\mathbb { E } _ { \\nu _ { N , 0 } ^ { \\sigma } } \\left [ X _ i \\right ] \\right | + 1 \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} \\begin{array} { l } < x > _ f = \\langle \\widehat { x } f , f \\rangle _ { L ^ 2 ( \\mathbb { R } ) } = \\int _ { \\mathbb { R } } x | f ( x ) | ^ 2 d x \\\\ \\\\ < \\xi > _ f = \\langle \\widehat { \\xi } f , f \\rangle _ { L ^ 2 ( \\mathbb { R } ) } = \\int _ { \\mathbb { R } } \\xi | \\widetilde { f } ( \\xi ) | ^ 2 d \\xi . \\end{array} \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} W ^ { ( 1 ) } _ { t } + \\Lambda _ 1 ( | \\xi | ) W ^ { ( 1 ) } + \\left ( B _ 0 ^ { ( 1 ) } ( | \\xi | ) + B _ 1 ^ { ( 1 ) } ( | \\xi | ) \\right ) W ^ { ( 1 ) } = 0 , \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} c _ x \\coloneqq 2 - \\frac { \\langle x , x \\rangle ^ \\frac { - 1 } { 2 } } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } - \\left ( \\sum _ { i = 1 } ^ { n } ( a _ i a _ i ^ * ) ^ \\frac { 1 } { 2 } \\right ) \\frac { \\langle x , x \\rangle ^ \\frac { - 1 } { 2 } } { \\sqrt { n } } . \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} u \\prec _ t v = \\sum _ { i \\geq 0 } u \\cdot T _ i ( v ) u \\succ _ t v = \\sum _ { i \\geq 0 } T _ i ( u ) \\cdot v u \\curlyvee _ t v = \\sum _ { i \\geq 0 } t ^ i ( \\sum _ { j + k = i } H ( T _ j ( u ) , T _ k ( v ) ) ) . \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } G ( x - y ) f ( y ) d y = \\mathbb { E } _ x \\left [ \\int _ { 0 } ^ { \\infty } f ( X _ t ) d t \\right ] , x \\in \\R ^ d . \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} \\int _ { M } G ( p , y ) \\ , | f ( y ) | \\ , d y & = \\int _ { \\mathcal { L } _ { p } \\left ( 0 , \\ , A \\exp \\left ( B \\ , \\omega ( a ) \\right ) \\right ) } G ( p , y ) \\ , | f ( y ) | \\ , d y \\\\ & \\ , \\ , \\ , + \\int _ { \\mathcal { L } _ { p } \\left ( A \\exp \\left ( B \\ , \\omega ( a ) \\right ) , \\infty \\right ) } G ( p , y ) \\ , | f ( y ) | \\ , d y \\ , . \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} \\| M \\| ^ 2 _ { L ^ 2 _ \\sigma ( \\R ) } & \\lesssim \\int _ 0 ^ 1 | f ( r ) | ^ 2 d r + \\int _ \\R \\sup _ { j \\in \\N } \\left | \\int _ 1 ^ { 2 ^ j } f ( r ) P ( r , k ) d r \\right | ^ 2 d \\sigma ( k ) + & \\\\ \\int _ \\R \\sup _ { j \\in \\N } \\sup _ { 2 ^ j \\leq n \\leq 2 ^ { j + 1 } } \\left | \\int _ { 2 ^ j } ^ n f ( r ) P ( r , k ) d r \\right | ^ 2 d \\sigma ( k ) & = \\int _ 0 ^ 1 | f ( r ) | ^ 2 d r + \\| S ' \\| ^ 2 + \\| S '' \\| ^ 2 \\lesssim L \\ , . \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} \\begin{dcases} p \\in ( 2 , \\infty ) & \\implies \\omega \\in A _ 1 , \\\\ p = 2 & \\implies \\omega \\in A _ 1 , \\omega ^ { - 1 } \\in A _ 2 ( \\Omega ) \\cap A _ 1 , \\\\ p \\in ( P ' , 2 ] & \\implies \\omega ' \\in A _ { p ' } ( \\Omega ) \\cap A _ 1 , \\end{dcases} \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} ( A \\cup B ) \\cap ( D \\cap C ) & = ( A \\cap ( D \\cap C ) ) \\cup ( B \\cap ( D \\cap C ) ) \\\\ & \\subseteq ( A \\cap C ) \\cup ( B \\cap D ) = \\emptyset . \\\\ \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} \\dot { x } _ 1 & = A _ 1 x _ 1 + A _ { 1 2 } x _ 2 + B _ 1 u , \\ ; x _ 1 ( 0 ) = x _ { 1 , 0 } \\\\ \\dot { x } _ 2 & = A _ 2 x _ 2 , \\ ; x _ 2 ( 0 ) = x _ { 2 , 0 } . \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} \\lim _ { \\gamma _ n \\downarrow 0 } \\frac { w _ { \\gamma _ n , 0 } ( y ) } { w _ { \\gamma _ n , 0 } ( x ) } = 1 . \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} \\begin{aligned} & Y _ 1 ( \\Omega _ 1 ) = h ^ 1 _ 1 ( \\phi ^ 1 _ 1 ) \\cdot \\prod _ { a = 2 } ^ { d _ 1 - 1 } ( \\sin \\phi ^ 1 _ a ) ^ { c _ { a - 1 } + \\frac { 1 } { 2 } - \\frac { a - 1 } { 2 } } P ^ { ( c _ { a - 1 } , - 1 / 2 ) } _ { J _ a } ( \\cos 2 \\phi ^ 1 _ a ) \\\\ & c _ a = \\sum _ { s = 1 } ^ { a } J _ s + \\frac { a - 1 } { 2 } + A + J _ 1 , \\end{aligned} \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\nabla \\tilde { v } ( x ) \\cdot \\nabla \\varphi ( x ) \\ ; d x + \\int _ { \\mathbb { R } ^ N } V _ { 0 , \\infty } \\tilde { v } ( x ) \\varphi ( x ) \\ ; d x = a _ 0 \\int _ { \\mathbb { R } ^ N } \\tilde { v } ( x ) \\varphi ( x ) \\ ; d x , \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} M _ { i j } = M _ { j i } . \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} \\langle y , P ( M _ { i j } ) N P ^ * y \\rangle _ Y = \\frac { 1 } { N } \\sum _ { l , n = 1 } ^ { M } \\sum _ { i \\in B ( l ) , j \\in B ( n ) } M _ { i j } y _ l y _ n , \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} \\norm f _ { u , - n , c , L } = \\norm { f _ c } _ { u , - n , \\Gamma _ { c , L } } \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} u ( r , \\omega ) = \\sum _ { k \\geq 1 } v _ k ( r ; c _ k ^ + , c _ k ^ - ) w _ k ( \\omega ) \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} \\phi ( \\rho ) = \\rho \\cdot C \\left ( \\alpha \\rho ^ { s - 2 } \\right ) . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} c = \\frac { k n ( 2 k + 2 n - 3 ) } { ( k + 2 n - 2 ) ( k + 2 n - 1 ) } . \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} H _ { N _ 1 + N _ 2 } ( w ) - H _ { N _ 1 } ( u ) - H _ { N _ 2 } ( v ) = \\sum _ { ( i , j ) \\in I _ { N _ 1 , N _ 2 } } M _ { i j } u _ i v _ j . \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} \\delta g ^ { i j } = 2 u g ^ { i l } g ^ { j k } h _ { l k } = 2 u h ^ { i j } . \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} \\begin{aligned} \\theta _ 7 ( \\lambda , \\mu ) & = \\theta _ 7 ( \\lambda - 7 , \\mu ) , \\\\ \\theta _ 7 ( \\lambda , \\mu + 1 ) & = \\theta _ 7 ( \\lambda + 4 , \\mu ) . \\end{aligned} \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} c _ n \\| P _ n x \\| ^ 2 \\leqslant \\langle P _ n x , \\mathcal { H } P _ n x \\rangle = \\langle P _ n x , P _ n \\mathcal { H } x \\rangle = 0 \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} C \\left ( \\mu , \\Psi \\right ) : = \\inf _ { L > \\max \\left \\{ 1 , \\left [ { \\Psi } \\left ( ( { C _ Y } \\cdot K ) ^ { - 1 } \\right ) \\right ] ^ { - 1 } \\right \\} } c _ \\mu 2 ^ { n _ \\mu } \\frac { L } { 1 - { C _ Y } \\cdot K \\cdot { \\Psi } ^ { - 1 } \\left ( \\frac { 1 } { L } \\right ) } , \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} i n d ( \\Sigma , U ) = \\sum _ { \\lambda _ j < 0 } \\dim E _ j \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} G _ \\rightarrow ^ \\# = G _ \\leftarrow , L _ b ^ { - 1 } = G _ \\leftarrow . \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} { \\phi } _ { p , \\alpha } ' ( t ) = \\begin{cases} p t ^ { p - 1 } & t < 1 , \\\\ p t ^ { p - 1 } ( 1 + \\log ( t ) ) ^ { \\alpha } + \\alpha t ^ { p - 1 } ( 1 + \\log ( t ) ) ^ { \\alpha - 1 } & t > 1 . \\end{cases} \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} P = \\tfrac { c } { 6 } \\left ( x _ { 4 } ^ { 3 } - x _ { 4 } ( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } ) + \\sqrt { 5 } ( x _ { 1 } ^ { 2 } - x _ { 2 } ^ { 2 } ) x _ { 3 } \\right ) . \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta _ B ' ( 0 ; a , 1 , 1 ) - \\left ( \\frac 1 { 1 2 } - \\zeta ' _ R ( - 1 ) \\right ) \\frac 1 a + \\frac 1 4 \\log ( 2 \\pi ) - \\frac { \\gamma a } { 1 2 } \\\\ = \\int _ 0 ^ \\infty \\left ( \\frac t { e ^ t - 1 } - 1 + \\frac 1 2 t - \\frac 1 { 1 2 } t ^ 2 \\right ) \\frac 1 { t ^ 2 ( e ^ { t / a } - 1 ) } \\ , d t \\\\ = \\int _ 0 ^ \\infty \\frac { F ( a t ) } { t ( e ^ { t } - 1 ) } \\ , d t , \\end{aligned} \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} W = Z \\overline { Z } ^ { t } , \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} { } ^ C D ^ { \\gamma } _ { + } f ( x ) = \\int _ 0 ^ x \\frac { f ^ { ( [ \\gamma ] + 1 ) } ( y ) } { ( x - y ) ^ { \\gamma - [ \\gamma ] } } \\frac { d y } { \\Gamma ( [ \\gamma ] + 1 - \\gamma ) } , \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} P \\{ \\max _ { 0 \\le s \\le t } \\mathcal { T } ( s ) = 0 \\ | \\ V ( 0 ) = - c _ 2 , \\ N ( t ) = 2 k \\} \\ = \\ \\binom { 2 k } { k } \\frac { ( c _ 1 c _ 2 ) ^ k } { ( c _ 1 + c _ 2 ) ^ { 2 k } } + \\Bigl ( 1 - \\frac { c _ 1 } { c _ 2 } \\Bigr ) \\sum _ { j = 0 } ^ { k - 1 } \\binom { 2 k } { j } \\frac { c _ 1 ^ { j } \\ , c _ 2 ^ { 2 k - j } } { ( c _ 1 + c _ 2 ) ^ { 2 k } } \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} \\frac { d } { d t } f ( t ) = Q ^ { + } ( t , f ( t ) ) - Q ^ { - } ( t , f ( t ) ) , f ( 0 ) = f _ 0 \\in X _ { + } , t \\geq 0 , \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\langle x , y \\rangle \\coloneqq \\sum _ { i = 1 } ^ { n } a _ i b _ i ^ * , \\forall x = ( a _ 1 , \\dots , a _ n ) , y = ( b _ 1 , \\dots , b _ n ) \\in \\mathcal { A } ^ n \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k ( - 1 ) ^ j j \\binom { n } { j } = ( - 1 ) ^ k n \\binom { n - 2 } { k - 1 } . \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} | | \\overline { v } ( r ) | | _ { L ^ { 2 } ( r d r ) } = | | \\mathcal { F } ^ { - 1 } ( \\overline { y } ) ( r ) | | _ { L ^ { 2 } ( d r ) } = | | \\overline { y } | | _ { L ^ { 2 } ( \\rho ( \\xi ) d \\xi ) } \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} \\tilde { \\mu } _ { J , K } \\times \\tilde { \\mu } _ { K , J } ' \\circ F _ { \\tilde { J } , \\tilde { K } } ^ { - 1 } = \\tilde { \\mu } _ { J , K } \\times \\tilde { \\mu } _ { K , J } ' \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} \\bigg [ - \\sum _ { i = 1 } ^ { N } \\frac { \\partial ^ 2 } { \\partial r _ i ^ 2 } - \\frac { \\eta } { r } + \\sum _ { i = 1 } ^ { N } \\frac { \\lambda _ i } { r _ i ^ 2 } \\bigg ] R ( r _ 1 , \\cdots , r _ N ) = E R ( r _ 1 , \\cdots , r _ N ) . \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { 2 } \\int _ { \\frac { - 3 \\pi } { 2 } } ^ 0 \\cos ^ 3 ( \\theta ) - \\cos ( \\theta ) \\sin ^ 2 ( \\theta ) \\ , d \\theta = - \\frac { 1 } { 6 } . \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} \\alpha _ l : = 1 - \\frac { G _ l ( x ) } { \\sqrt { G _ l ^ 2 ( x ) + H _ l ^ 2 ( x ) + 2 t } } \\beta _ l : = 1 - \\frac { H _ l ( x ) } { \\sqrt { G _ l ^ 2 ( x ) + H _ l ^ 2 ( x ) + 2 t } } \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} X ^ \\flat ( Y ) : = \\langle X , Y \\rangle \\quad \\quad \\langle \\omega ^ \\sharp , Y \\rangle : = \\omega ( Y ) . \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { | x _ \\varepsilon - y _ \\varepsilon | ^ 2 } { \\varepsilon } = 0 . \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} \\mu ( \\xi ) = 2 \\sqrt { \\frac { \\varphi ( \\mu ( \\xi ) ) } { \\xi ^ 2 - 4 Q ( \\mu ( \\xi ) ) } } , \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} L _ D ( u , v ) = \\left \\{ \\begin{array} { c l } - d ^ + ( v ) & u = v , \\\\ d ( v , u ) & u \\neq v . \\end{array} \\right . \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} \\left | \\frac { P _ { n , 1 } ( X ) } { P _ { n , 2 } ( X ) } + R ( X ) \\right | = \\left | \\frac { Q _ { n } ( X , 1 , R ( X ) ) } { P _ { n , 2 } ( X ) } \\right | \\le 2 ^ { - 3 n } . \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} \\Pi \\colon ( Y , E ) = ( X _ n , D _ n ) \\to \\cdots \\to ( X _ 0 , D _ 0 ) = ( X , D ) \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} \\begin{array} { l } \\lim _ { n \\rightarrow \\infty } \\frac { \\ell _ R ( R / I ( n n _ 1 D ) \\cdots I ( n n _ r D _ r ) ) } { n ^ 2 } = \\sum _ { i = 1 } ^ t - \\frac { 1 } { 2 } [ S / m _ i : R / m _ R ] ( ( n _ 1 \\Delta ( 1 ) _ i + \\cdots + n _ r \\Delta ( r ) _ i ) ^ 2 ) \\\\ = \\sum _ { k _ 1 + \\ldots + k _ r = 2 } \\frac { 1 } { k _ 1 ! \\cdots k _ r ! } \\left ( \\sum _ { i = 1 } ^ t - [ S / m _ i : R / m _ R ] ( \\Delta ( 1 ) _ i ^ { k _ 1 } \\cdot \\ldots \\cdot \\Delta ( r ) _ i ^ { k _ r } ) \\right ) n _ 1 ^ { k _ 1 } \\cdots n _ r ^ { k _ r } \\end{array} \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\mathcal K ( x _ 1 , x _ 2 , x _ 3 ) = ( x _ 1 x _ 2 ) \\bigg \\{ \\frac { 1 } { x _ 1 ^ { 2 } x _ 2 ^ { 2 } + x _ 3 ^ 2 } \\bigg \\} . \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} B _ { G , \\ ; L + m + 1 } - B _ { G , \\ ; L + m } = 2 G _ { L + m } - G _ { L + m + 1 } . \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 h } { \\partial t ^ 2 } - c ^ 2 \\frac { \\partial ^ 2 h } { \\partial x ^ 2 } = - \\frac { m + n + 2 } { t } \\frac { \\partial h } { \\partial t } + \\frac { c ( m - n ) } { t } \\frac { \\partial h } { \\partial x } \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} \\partial _ { t } v _ { 2 } ( t , r ) = I _ { t } ( t , r ) + I I _ { t } ( t , r ) \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{cases} F _ L ( x , y ) , & x < 0 , \\\\ F _ R ( x , y ) , & x > 0 . \\end{cases} \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} \\begin{aligned} T & = ( \\mu + \\nu ) \\bar { g } ( \\cdot , X ) \\otimes \\bar { g } ( \\cdot , X ) + \\nu \\bar { g } \\\\ & = \\sec ^ 2 ( \\xi ( x ) ) \\left [ \\nu ( \\xi ( x ) ) x ^ * g _ 1 + \\mu ( \\xi ( x ) ) \\left ( e ^ { - \\xi ( x ) \\sqrt { n - 1 } } \\right ) ^ 2 d t ^ 2 \\right ] , \\end{aligned} \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} \\left \\{ P _ { k } ( z , \\overline { z } ) \\right \\} _ { k = 1 , \\dots , \\left [ \\frac { p - 1 } { 2 } \\right ] } , \\left \\{ R _ { k } ( z , \\overline { z } ) \\right \\} _ { k = 1 , \\dots , \\left [ \\frac { p - 1 } { 2 } \\right ] } , \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { | s | \\to + \\infty } \\dfrac { f _ 0 ( s ) } { s } = a _ 0 ; \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} p ( \\boldsymbol { s } ) = \\prod _ { n = 1 } ^ N p ( s _ n ) = \\prod _ { n = 1 } ^ N ( 1 - \\rho ) ^ { 1 - s _ n } \\rho ^ { s _ n } . \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} - \\int _ { \\R ^ 2 } \\Delta _ q ( u \\cdot \\nabla f ) \\Delta _ q f ~ d x \\leq & C b _ q 2 ^ { - 2 q s } ( \\| u \\| _ { L ^ 2 } + \\| \\omega \\| _ { L ^ 2 } + \\| \\partial _ 1 \\omega \\| _ { L ^ 2 } ) \\\\ & \\quad \\times ( \\| f \\| _ { H ^ s } ^ 2 + \\| f \\| _ { H ^ s } ^ \\frac 1 2 \\| \\partial _ 1 f \\| _ { H ^ s } ^ \\frac 1 2 + \\| f \\| _ { H ^ s } ^ \\frac 3 2 \\| \\partial _ 1 f \\| _ { H ^ s } ^ \\frac 1 2 ) , \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} K _ { n } ^ { ( 3 ) } K _ { m } ^ { ( 3 ) } & + K _ { n + 1 } ^ { ( 3 ) } K _ { m + 1 } ^ { ( 3 ) } + K _ { n + 2 } ^ { ( 3 ) } K _ { m + 2 } ^ { ( 3 ) } \\\\ & = \\left \\lbrace \\begin{array} { c } \\left ( 2 ^ { n } + M _ { n } ^ { ( 2 ) } \\right ) \\left ( 2 ^ { m } + M _ { m } ^ { ( 2 ) } \\right ) \\\\ + \\left ( 2 ^ { n + 1 } + M _ { n + 1 } ^ { ( 2 ) } \\right ) \\left ( 2 ^ { m + 1 } + M _ { m + 1 } ^ { ( 2 ) } \\right ) \\\\ + \\left ( 2 ^ { n + 2 } + M _ { n + 2 } ^ { ( 2 ) } \\right ) \\left ( 2 ^ { m + 2 } + M _ { m + 2 } ^ { ( 2 ) } \\right ) \\end{array} \\right \\rbrace . \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} [ h _ 0 , e _ { r , r + 1 } + e _ { n - r , n - r + 1 } ] & = ( - 1 ) ^ { r - 1 } \\left ( C ( n - 1 , r ) e _ { r , r + 1 } + C ( n - 1 , n - r ) e _ { n - r , n - r + 1 } \\right ) \\\\ & = ( - 1 ) ^ { r - 1 } C ( n - 1 , r ) ( e _ { r , r + 1 } - e _ { n - r , n - r + 1 } ) . \\\\ \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} \\langle e _ { a , 1 } ( t , R \\lambda ( t ) ) , \\phi _ { 0 } ( R ) \\rangle _ { L ^ { 2 } ( R d R ) } & = - \\frac { 4 } { \\lambda ( t ) } \\int _ { t } ^ { \\infty } \\frac { \\lambda '' ( s ) } { 1 + s - t } d s + \\frac { 4 b } { \\lambda ( t ) t ^ { 2 } \\log ^ { b } ( t ) } + \\frac { 4 \\lambda '' ( t ) \\log ( \\lambda ( t ) ) } { \\lambda ( t ) } + f _ { 2 } ( \\lambda ( t ) , \\lambda ' ( t ) , \\lambda '' ( t ) ) \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} \\{ a ^ i _ s : i \\in [ n ] \\} \\ = \\ \\{ n ( s - 1 ) + j : j \\in [ n ] \\} , \\ s = 1 , 2 , 3 . \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} \\Phi _ 2 ^ \\pm = \\frac { 1 } { 2 } \\Big ( { - U ^ \\pm } ' + \\frac { U ^ \\pm } { r } \\Big ) , \\Phi _ 0 ^ \\pm = \\frac { 1 } { 2 } \\Big ( { U ^ \\pm } ' + \\frac { U ^ \\pm } { r } \\Big ) . \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} & f ( x _ 1 , x _ 2 , x _ 3 ) \\\\ & = \\sum \\limits _ { j , k \\in \\mathbb Z } \\sum _ { R = I \\times J \\times S \\in \\mathcal R ^ N _ { \\frak z } ( j , k ) } | R | \\widetilde \\psi _ { j , k } ( x _ 1 , x _ 2 , x _ 3 , x _ I , x _ J , x _ S ) \\psi _ { j , k } * f ( x _ I , x _ J , x _ S ) , \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} h _ t = ( h ' ) { \\rm N S c h } ( h ) = h ''' - ( 3 / 2 ) h '' ( h ' ) ^ { - 1 } h '' \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} S f = \\sum _ { k = 1 } ^ m \\xi _ k \\langle f , u _ k \\rangle _ H v _ k , \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} \\overline { \\partial } _ { b } ( u + i v ) = 0 . \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} A _ 0 \\to A _ 1 , A _ 2 , A _ 4 ; & A _ 1 \\to A _ 2 , A _ 3 , A _ 5 ; \\\\ A _ 2 \\to A _ 3 , A _ 4 , A _ 6 ; & A _ 3 \\to A _ 0 , A _ 4 , A _ 5 ; \\\\ A _ 4 \\to A _ 1 , A _ 5 , A _ 6 ; & A _ 5 \\to A _ 0 , A _ 2 , A _ 6 ; \\\\ A _ 6 \\to A _ 0 , & A _ 1 , A _ 3 . \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} 2 l _ { i + 1 } - l ' _ s & \\geq \\beta ( i + 1 , s ) + \\Delta ( i + 1 , s ) \\\\ & = s - ( i + 1 ) + \\Delta ( i + 1 , s ) \\\\ & \\geq 2 \\Delta ( i + 1 , s ) \\ , , \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} \\bar { H } _ N ( m ) & : = - \\frac { 1 } { N } \\log \\int _ { \\{ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } x _ i = m \\} } \\exp \\left ( - H ( x ) \\right ) \\mathcal { L } ^ { N - 1 } ( d x ) . \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} & \\langle 1 , j \\rangle \\overset { 1 } \\twoheadrightarrow \\langle 2 , j \\rangle \\overset { 2 } \\twoheadleftarrow \\langle 3 , j \\rangle j = 1 , 2 , \\langle i , 1 \\rangle \\rightsquigarrow \\langle i , 2 \\rangle i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} H _ { 2 \\alpha } ( r , \\phi , r ' , \\phi ' , t ) = \\frac { e ^ { - ( r ^ 2 + r '^ 2 ) / 4 t } } { 8 \\pi \\alpha t } \\int _ { A _ \\phi } e ^ { r r ' \\cos ( z - \\phi ) / 2 t } \\frac { e ^ { i \\pi z / \\alpha } } { e ^ { i \\pi z / \\alpha } - e ^ { i \\pi \\phi ' / \\alpha } } d z . \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} f ^ { \\prime } ( \\xi _ { 0 } ) = \\overline { \\xi _ { 0 } } f ( \\xi _ { 0 } ) \\left [ 1 + 2 \\int _ { \\mathbb { T } } \\frac { d \\sigma ( t ) } { \\left | \\xi _ { 0 } - t \\right | ^ { 2 } } \\right ] . \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} H _ { 2 g + 2 } = H _ { 2 g + 1 } + \\dots + H _ { g + 2 } + 2 ^ { k + 1 } H _ { g + 1 - k } . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} \\langle z , N P ^ * y \\rangle = N \\langle P z , y \\rangle _ Y = 0 . \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { s p } + \\int _ { 0 } ^ { 1 } \\frac { \\left [ 1 - ( 1 - k ) ^ s \\right ] ^ { p - 1 } - \\left [ ( 1 + k ) ^ s - 1 \\right ] ^ { p - 1 } } { k ^ { 1 + s p } } \\mathrm { d } k - \\int _ { 1 } ^ { \\infty } \\frac { \\left [ ( 1 + k ) ^ s - 1 \\right ] ^ { p - 1 } } { k ^ { 1 + s p } } \\mathrm { d } k = 0 . \\end{aligned} \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} \\nabla ' : = \\nabla + [ \\kappa , \\cdot ] , \\ \\theta ' : = \\theta + \\nabla \\kappa + \\kappa ^ 2 \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} & \\frac { i } { x _ { 1 + 2 + 3 } } ( - i \\lambda _ 3 ) \\left ( - i \\lambda _ 3 - i b _ 2 x _ { 1 + 2 + 3 } \\right ) \\left ( \\frac { i } { x _ { 1 + 2 } } + \\frac { i } { x _ { 1 + 3 } } + \\frac { i } { x _ { 1 + 4 } } \\right ) + b _ 3 \\\\ & = ( - i \\lambda _ 3 ) b ' _ 2 ( 1 + 2 + 3 ) \\left ( \\frac { i } { x _ { 1 + 2 } } + \\frac { i } { x _ { 1 + 3 } } + \\frac { i } { x _ { 1 + 4 } } \\right ) + b _ 3 . \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} \\Phi _ A ( k _ 1 g k _ 2 ^ { - 1 } ) = \\rho ( k _ 1 ) \\circ \\Phi ( g ) \\circ \\rho ( k _ 2 ^ { - 1 } ) , k _ 1 , k _ 2 \\in K , ~ g \\in G . \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} \\begin{aligned} ( \\rho ( 0 , x ) , \\mathbf { u } ( 0 , x ) ) = ( \\rho _ 0 ( x ) , \\mathbf { u _ 0 } ( x ) ) , \\end{aligned} \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} \\| F \\| _ { L _ { x , \\omega } ^ { r , s } ( \\mathbb { R } ^ { 2 d } ) } = \\left ( \\int _ { \\mathbb { R } ^ d } \\left ( \\int _ { \\mathbb { R } ^ d } | F ( x , \\omega ) | ^ r d x \\right ) ^ { \\frac { s } { r } } d \\omega \\right ) ^ { \\frac { 1 } { s } } , \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} & ( 1 + \\epsilon ) F ( v ) - F ( ( 1 + \\epsilon ) v - \\rho ) \\\\ = & ( 1 + \\epsilon ) F ( v ) - F ( ( 1 + \\epsilon ) v ) + F ( ( 1 + \\epsilon ) v ) - F ( ( 1 + \\epsilon ) v - \\rho ) \\\\ \\succeq & C _ l \\epsilon \\mathbf { 1 } - C _ f \\rho \\succeq C _ l \\epsilon \\mathbf { 1 } - C _ f K _ 4 \\epsilon \\hat V ^ * \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} N = n _ 1 + \\cdots + n _ s \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} V _ h ( t , x ) & : = \\sup \\Big \\{ \\int _ 0 ^ t ( c ( \\psi _ s , h _ s ) - \\frac { 1 } { 2 } | \\dot { \\psi } _ s | ^ 2 ) d s : \\\\ & \\psi : [ 0 , T ] \\to \\R ^ n , \\psi _ 0 \\in G _ 0 , \\psi _ t = x \\Big \\} , \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} \\Psi _ { \\Lambda } \\Psi _ { \\Lambda } ^ * : = \\sum _ { i , j \\in I } \\bigotimes _ { x \\in \\Lambda } h _ { x , i } h _ { x , j } ^ * \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} \\frac { \\partial \\omega _ t } { \\partial t } = - R i c ( \\omega _ t ) - \\omega _ t , { \\omega _ t } _ { | _ { t = 0 } } = \\hat \\omega . \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\partial _ { r } \\overline { v _ { M } } ( r ) = \\int _ { 0 } ^ { \\infty } \\overline { y } ( \\xi ) \\lim _ { r \\rightarrow \\infty } \\partial _ { r } \\left ( \\frac { \\phi ( r , \\xi ) } { \\sqrt { r } } \\right ) \\chi _ { \\leq 1 } \\left ( \\frac { \\xi } { M } \\right ) \\rho ( \\xi ) d \\xi = 0 \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} \\bar P _ { _ { { \\rm { } } } } ^ e = \\frac { 1 } { 2 } \\bigg [ 1 - \\sqrt { \\frac { { { { \\bar \\gamma } _ { } } } } { { 1 + { { \\bar \\gamma } _ { } } } } } \\bigg ] , \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} \\Re \\langle x , \\overline { H } x \\rangle = \\lim _ { n \\to \\infty } \\Re \\langle x _ n , H x _ n \\rangle \\leqslant 0 . \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} & \\int _ { t } ^ { \\infty } | e _ { 1 } '' ( s ) - e _ { 2 } '' ( s ) | | \\frac { 1 } { \\log ( \\lambda _ { 0 } ( t ) ) } - \\frac { 1 } { \\log ( \\lambda _ { 0 } ( s ) ) } | \\frac { d s } { ( \\lambda _ { 0 } ( t ) ^ { 1 - \\alpha } + s - t ) ( 1 + s - t ) ^ { 3 } } \\\\ & \\leq \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 3 } \\log ^ { b + 2 } ( t ) ( \\log ( \\log ( t ) ) ) ^ { 5 / 2 } } \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} { \\rm N C S c h } ( h ) : = ( h ' ) ^ { - 1 } h ''' - ( 3 / 2 ) ( h ' ) ^ { - 1 } h '' ( h ' ) ^ { - 1 } h '' \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} h ( \\overrightarrow { x } , \\overrightarrow { y } , \\overrightarrow { z } ) = \\gamma \\circ h _ { 2 m - l } ( \\overrightarrow { x } ) \\cup \\alpha \\circ h _ k ( \\overrightarrow { y } ) \\cup \\beta \\circ h _ r ( \\overrightarrow { z } ) \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} \\norm { x } = \\max _ { 1 \\le i \\le n } \\left \\{ \\norm { x _ i } _ i \\right \\} , & & x = \\oplus _ { i = 1 } ^ n x _ i \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} \\Delta _ { b } \\varphi = T r \\left ( ( \\nabla ^ { H } ) ^ { 2 } \\varphi \\right ) = ( \\varphi _ { 1 \\bar { 1 } } + \\varphi _ { \\bar { 1 } 1 } ) . \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} f B ( e x f , y g ) & = f y B ( e x f , g ) + f B ( e x f , y ) g = - f y B ( g , e x f ) + f B ( e x f , y ) g , \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} | F _ { 4 } ( t , r ) | & \\leq C \\frac { \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 2 r } { \\log ^ { N } ( t ) } ) \\right ) } { ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) ^ { 2 } } \\left ( \\frac { r } { t ^ { 2 } \\log ^ { 3 b + 1 - 2 b \\alpha } ( t ) } \\right ) \\\\ & + C \\frac { \\left ( 1 - \\chi _ { \\geq 1 } ( \\frac { 4 r } { t } ) \\right ) } { ( r ^ { 2 } + \\lambda ( t ) ^ { 2 } ) ^ { 2 } } \\frac { r } { t ^ { 2 } \\log ^ { 5 b + 2 N - 1 } ( t ) } \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} \\sigma ^ k _ { \\sigma _ { ( a , i ) } ( ( a , i ) ) } ( ( b , j ) ) = ( b + k ( \\delta _ { ( a , i ) } + 1 ) , j + k ( r ( \\delta _ { ( a , i ) } + 1 ) + 1 ) ) . \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} M _ g ( h ) = g h ( h \\in L ^ 2 ( \\mu ) ) . \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} Y _ \\mu ^ * ( x ) J Y _ \\mu ( x ) = J + ( \\mu - \\bar \\mu ) \\int _ a ^ x Y _ \\mu ^ * ( y ) A ( y ) Y _ \\mu ( y ) \\d y . \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} E _ { 7 ( - 2 5 ) } ~ = ~ \\mathcal { N } _ { 1 } ^ { 7 ( - 2 5 ) - } \\oplus s o ( 1 0 , 2 ) \\oplus s o ( 1 , 1 ) \\oplus \\mathcal { N } _ { 1 } ^ { 7 ( - 2 5 ) + } \\newline \\mathbf { , } \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} | r ^ { 2 } \\partial _ { r } ^ { 2 } v _ { 4 } ( t , r ) | & \\leq \\frac { C r } { t ^ { 2 } \\log ^ { 3 b + 2 N } ( t ) } + \\frac { C r ^ { 2 } } { t ^ { 4 } \\log ^ { 3 b - 2 + N } ( t ) } + \\frac { C r } { t ^ { 2 } \\log ^ { 3 b + 2 N - 1 } ( t ) } + \\frac { C r } { t ^ { 2 } \\log ^ { 3 b + 2 N - 1 } ( t ) } \\\\ & \\leq \\frac { C r } { t ^ { 2 } \\log ^ { 3 b + 2 N - 1 } ( t ) } , r \\leq \\frac { t } { 2 } \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} 1 - F _ { \\lambda , t } ^ { 2 } = ( 1 - \\chi ^ { 2 } ) ( d _ { \\lambda } ^ { t } ) \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} | v _ { 1 } ^ { \\lambda _ { 1 } } - v _ { 1 } ^ { \\lambda _ { 2 } } | \\leq \\begin{cases} \\frac { C r | | e _ { 1 } - e _ { 2 } | | _ { X } } { t ^ { 2 } \\log ^ { b } ( t ) \\sqrt { \\log ( \\log ( t ) ) } } , r \\leq \\frac { t } { 2 } \\\\ \\frac { C | | e _ { 1 } - e _ { 2 } | | _ { X } } { r \\sqrt { \\log ( \\log ( t ) ) } \\log ^ { b } ( t ) } , r > \\frac { t } { 2 } \\end{cases} \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} U _ { t , s } ( \\gamma ) U _ { s , r } ( \\gamma ) = U _ { t , r } ( \\gamma ) . \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} c _ { 3 } = c _ { 1 } c _ { 2 } c _ { 0 } ^ { - 1 } - 3 ^ { - 1 } c _ { 1 } ^ { 3 } c _ { 0 } ^ { - 2 } + 3 ^ { - 1 } c _ { 0 } ( \\alpha ^ { - 3 } + 3 b _ { 1 } a _ { 3 } + 6 b _ { 2 } a _ { 1 } a _ { 2 } + 3 b _ { 3 } a _ { 1 } ^ { 3 } ) . \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} \\lim _ { \\Lambda _ { 1 } \\uparrow \\uparrow V } \\psi _ { \\Lambda _ { 1 } } ( b _ { \\Lambda } ) = \\sum _ { i \\in I } \\prod _ { x \\in \\Lambda } \\hbox { T r } _ { \\mathcal { H } _ { x } } \\left ( h _ { x , i } h _ { x , i } ^ * b _ { x } \\right ) = : \\hat { \\psi } _ { \\Lambda } ( b _ { \\Lambda } ) . \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} { \\mathcal L } \\Big ( \\frac { \\partial w } { \\partial { x _ 1 } } \\Big ) = { \\mathcal L } \\Big ( \\frac { \\partial w } { \\partial { x _ 2 } } \\Big ) = { \\mathcal L } ( i w ) = 0 . \\end{align*}"}