diff --git "a/process_6/tokenized_finally.jsonl" "b/process_6/tokenized_finally.jsonl"
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+++ "b/process_6/tokenized_finally.jsonl"
@@ -0,0 +1,9336 @@
+{"id": "6391.png", "formula": "\\begin{align*} \\| p - p _ h \\| ^ 2 = \\inf _ { q _ h \\in Q _ h } \\| p - q _ h \\| ^ 2 + \\| \\pi _ { Q _ h } p - p _ h \\| ^ 2 . \\end{align*}"}
+{"id": "3707.png", "formula": "\\begin{align*} ( h + L _ X \\bar g ) _ { i j } & = \\tfrac { 2 } { n - 2 } c \\delta _ { i j } | x | ^ { 2 - n } + O ^ { 2 , \\alpha } ( | x | ^ { \\max \\{ - 2 q , 1 - n \\} } ) \\\\ v + X ( \\bar u ) & = - c | x | ^ { 2 - n } + O ^ { 2 , \\alpha } ( | x | ^ { \\max \\{ - 2 - q , 1 - n \\} } ) \\end{align*}"}
+{"id": "8612.png", "formula": "\\begin{align*} h _ { K \\oplus _ 2 L } ( x ) = \\sqrt { h _ K ( x ) ^ 2 + h _ L ( x ) ^ 2 } \\ge \\sqrt { \\rho _ K ( u ) ^ 2 + \\rho _ L ( u ) ^ 2 } | \\langle x , u \\rangle | = \\sqrt { \\rho _ K ( u ) ^ 2 + \\rho _ L ( u ) ^ 2 } h _ { [ - u , u ] } ( x ) . \\end{align*}"}
+{"id": "8916.png", "formula": "\\begin{align*} \\lambda _ { 1 2 } + \\lambda _ { 1 2 3 ' } & = \\log \\left ( \\frac { 8 1 } { \\zeta } \\frac { \\zeta } { 3 6 } \\right ) = \\log \\frac { 9 } { 4 } \\end{align*}"}
+{"id": "77.png", "formula": "\\begin{align*} \\ell ( x , \\alpha ) = \\langle \\mu , v \\beta \\rangle + \\Phi ^ + ( v \\beta ) - \\Phi ^ + ( w \\alpha ) \\geq 1 - \\Phi ^ + ( w \\alpha ) . \\end{align*}"}
+{"id": "4042.png", "formula": "\\begin{align*} Z _ n = M _ n + r _ n N _ n , \\end{align*}"}
+{"id": "3417.png", "formula": "\\begin{align*} e ^ { L _ { \\lambda } ( E ) } + e ^ { - L _ { \\lambda } ( E ) } = \\frac { \\sqrt { ( 2 + E ) ^ 2 + \\lambda ^ 2 } + \\sqrt { ( 2 - E ) ^ 2 + \\lambda ^ 2 } } { 2 } . \\end{align*}"}
+{"id": "5815.png", "formula": "\\begin{align*} \\rho = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { f _ i } } , \\ ; \\ ; \\ ; \\ ; \\ ; \\rho { { \\bf { u } } ^ { e q } } = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { { \\bf { c } } _ i } { f _ i } } . \\end{align*}"}
+{"id": "8416.png", "formula": "\\begin{align*} \\Psi ^ - _ { 1 1 , x } ( x ; z ) = \\frac { 1 } { 2 i } | u _ x | ^ { 2 } \\Psi ^ - _ { 1 1 } ( x ; z ) + \\frac { 1 } { 2 i } u _ x ( x ) \\Psi ^ - _ { 2 1 } ( x ; z ) . \\end{align*}"}
+{"id": "1195.png", "formula": "\\begin{align*} h _ t ( n , k ) \\le \\frac { n } { n - k } \\binom { n } { k } ^ { - 1 } \\binom { n - 1 } { k } h _ t ( n - 1 , k ) = h _ t ( n - 1 , k ) \\enspace . \\end{align*}"}
+{"id": "6482.png", "formula": "\\begin{align*} \\forall \\theta > 0 \\ \\Rightarrow \\sup _ { p \\ge 1 } \\left [ \\ \\frac { L ( p ^ { \\theta } ) } { L ( p ) } \\ \\right ] = C ( \\theta ) < \\infty . \\end{align*}"}
+{"id": "7292.png", "formula": "\\begin{align*} G : = \\{ ( x , y ) : \\ \\ x \\in ( - b , b ) ^ { d } , \\ \\ g ( x ) - 1 \\le y \\leq g ( x ) \\} , \\end{align*}"}
+{"id": "6320.png", "formula": "\\begin{align*} S _ 2 : 2 + 3 X ^ { p - 1 } Y = 0 \\end{align*}"}
+{"id": "5446.png", "formula": "\\begin{align*} a _ { ( n ) } \\cdot v = 0 , n \\geq N , \\end{align*}"}
+{"id": "4268.png", "formula": "\\begin{align*} N = n ^ { 3 } + \\left ( n + 1 \\right ) ^ { 3 } = \\left ( n + 3 \\right ) ^ { 3 } + \\left ( n + \\alpha \\right ) ^ { 3 } \\end{align*}"}
+{"id": "3486.png", "formula": "\\begin{align*} \\sum _ { l \\neq j } \\ln | \\sin \\pi ( \\theta _ j - \\theta _ l ) | = \\sum _ { l \\in I _ 1 , l \\neq j } \\ln { | \\sin \\pi ( j - l ) \\alpha | } + \\sum _ { l \\in I _ 2 } \\ln { | \\sin \\pi ( j - l ) \\alpha | } \\triangleq \\sum _ { 1 } + \\sum _ { 2 } . \\end{align*}"}
+{"id": "7879.png", "formula": "\\begin{align*} Z ^ i ( v ^ \\mathrm { a p p } ( s ) , s ) - Z ^ i ( v ( s ) , s ) = \\mathcal O ( s ^ { \\min ( \\frac 3 2 , \\frac { i + 1 } 2 ) } ) . \\end{align*}"}
+{"id": "6600.png", "formula": "\\begin{align*} [ X + \\xi , Y + \\eta ] _ H = [ X , Y ] - \\mathrm { a d } _ X ^ * \\eta - \\iota _ Y d \\xi + H ( X , Y , \\cdot ) \\in \\mathfrak { g } \\oplus \\mathfrak { g } ^ * , \\end{align*}"}
+{"id": "3089.png", "formula": "\\begin{align*} \\tilde { \\lambda } ^ { k } = \\lambda ^ k - \\beta ( \\tilde { x } ^ { k } - \\tilde { y } ^ { k } ) . \\end{align*}"}
+{"id": "7685.png", "formula": "\\begin{align*} \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) ) } ^ { N , i } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } h \\Big ( \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) ) } ^ { N , j } \\Big ) = - R ^ { - 1 } B \\Big [ P _ t \\nu ^ { N , i } _ { \\boldsymbol { x } } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } \\phi ^ { N , j } ( t , \\boldsymbol { x } ) \\Big ] . \\end{align*}"}
+{"id": "4329.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t _ 0 \\} } | \\tilde { F } _ { t _ 1 } - F _ { t _ 0 } | ^ 2 _ h c ( - \\Psi ) \\\\ = & \\int _ { \\{ \\Psi < - t _ 0 \\} } | \\tilde { F } _ { t _ 1 } - F _ { t _ 0 } | ^ 2 _ h e ^ { - \\Psi - t _ 0 } c ( t _ 0 ) . \\\\ \\end{align*}"}
+{"id": "4733.png", "formula": "\\begin{align*} & ( i ) \\ \\ \\ \\nabla ( J + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k f _ k ) ( x ^ * ) = 0 \\\\ & ( i i ) \\ \\ \\gamma _ k f _ k ( x ^ * ) = 0 \\ \\ k \\in \\{ 1 , . . . , m \\} \\\\ & ( i i i ) \\ A _ J + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k A _ k \\succeq 0 \\end{align*}"}
+{"id": "3412.png", "formula": "\\begin{align*} A ( \\theta , E ) = \\left ( \\begin{matrix} E - \\lambda \\tan { \\pi \\theta } \\ & - 1 \\\\ 1 & 0 \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "590.png", "formula": "\\begin{align*} \\Phi _ { \\rho } ( w ) : = \\int _ { 1 } ^ { w } \\frac { 1 } { \\rho ( v ) } \\ , d v \\end{align*}"}
+{"id": "6544.png", "formula": "\\begin{align*} x ^ { \\pi ( a ) } + d _ { \\pi ( a ) - 1 } x ^ { \\pi ( a ) - 1 } + d _ { \\pi ( a ) - 2 } x ^ { \\pi ( a ) - 2 } + \\cdots d _ 1 x + d _ 0 = ( x - r _ 1 ) ( x - r _ 2 ) \\cdots ( x - r _ { \\pi ( a ) } ) \\end{align*}"}
+{"id": "3300.png", "formula": "\\begin{align*} \\bar q _ t ^ n ( x , y ) : = \\int _ 0 ^ t \\langle \\Sigma _ s \\delta _ { x _ { j _ 1 } } , \\delta _ { x _ { j _ 2 } } \\rangle d s . \\end{align*}"}
+{"id": "8494.png", "formula": "\\begin{align*} M _ { + , 2 } ( x ; z ) - e _ { 2 } = \\mathcal { P } ^ + \\left ( \\bar { r } _ 1 ( z ) \\mathrm { e } ^ { - 2 i z x } M _ { - , 1 } ( x ; z ) \\right ) ( z ) , z \\in \\mathbb { R } . \\end{align*}"}
+{"id": "5445.png", "formula": "\\begin{align*} a _ \\lambda v = \\sum _ { j \\in \\mathbb { Z _ { + } } } ( a _ { ( j ) } v ) \\frac { \\lambda ^ { j } } { j ! } . \\end{align*}"}
+{"id": "2315.png", "formula": "\\begin{align*} g ( N ( X , Y ) , Z ) + g ( N ( Y , Z ) , X ) + g ( N ( Z , X ) , Y ) = 0 , \\end{align*}"}
+{"id": "7235.png", "formula": "\\begin{align*} S _ \\sigma D = \\sigma ( D ) , \\end{align*}"}
+{"id": "6424.png", "formula": "\\begin{align*} \\log \\det ( u _ { , i j } ) = - v _ j x ^ j + u _ { , i } \\xi ^ i + c , \\xi \\in \\R ^ n , v \\in ( \\R ^ n ) ^ * . \\end{align*}"}
+{"id": "5314.png", "formula": "\\begin{align*} \\int _ { \\C ^ n } | g ( z ) | ^ 2 d z = ( 2 \\pi ) ^ { - 2 n } \\ , | \\lambda | ^ { 2 n } \\ , \\sum _ { k = 0 } ^ \\infty \\int _ { \\C ^ n } | g \\ast _ \\lambda \\varphi _ { k , \\lambda } ^ { n - 1 } ( z ) | ^ 2 d z . \\end{align*}"}
+{"id": "3645.png", "formula": "\\begin{align*} \\alpha _ x = \\mu _ x , \\end{align*}"}
+{"id": "3227.png", "formula": "\\begin{align*} Y _ t = \\mathcal S ( t ) Y _ 0 + \\int _ 0 ^ t \\mathcal S ( t - s ) \\alpha _ s d s + \\int _ 0 ^ t \\mathcal S ( t - s ) \\sigma _ s d W _ s , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "8027.png", "formula": "\\begin{align*} \\sum _ { u = 1 } ^ t d _ u = 2 j , \\end{align*}"}
+{"id": "905.png", "formula": "\\begin{align*} \\Sigma _ 0 = \\sum \\limits _ { n \\leq X } \\sum \\limits _ { D \\leq d < 2 D \\atop { n ^ 2 + n + 1 \\equiv 0 \\ , ( d ^ 2 ) } } 1 \\ , , \\end{align*}"}
+{"id": "4901.png", "formula": "\\begin{align*} h ( x ) & = \\frac { \\left [ \\Phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] ^ { \\alpha - 1 } \\ , \\left [ 1 - \\Phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] ^ { \\beta - 1 } \\ , \\phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) } { \\sigma \\ , \\mathrm { B } ( \\alpha , \\beta ) \\ , \\left [ 1 - I _ { \\Phi ( \\frac { x - \\mu } { \\sigma } ) } ( \\alpha , \\beta ) \\right ] } , \\end{align*}"}
+{"id": "3759.png", "formula": "\\begin{align*} \\# Q ^ i _ { n _ 0 ; l _ 1 , \\ldots , l _ { n _ 0 } } = q ^ { n _ 0 - i } ( \\# P _ { n _ 0 ; l _ 1 , \\ldots , l _ { i - 1 } , l _ i - 1 , l _ { i + 1 } , \\ldots , l _ { n _ 0 } } - \\# R ^ { i + 1 } _ { n _ 0 ; l _ 1 , \\ldots , l _ { i - 1 } , l _ i - 1 , l _ { i + 1 } , \\ldots , l _ { n _ 0 } } ) . \\end{align*}"}
+{"id": "7118.png", "formula": "\\begin{align*} g \\circ L _ f = f \\bullet _ \\delta g \\ \\mbox { a n d } \\ g \\circ R _ f = g \\bullet _ \\delta f . \\end{align*}"}
+{"id": "6486.png", "formula": "\\begin{align*} T _ { \\xi } ( t ) = \\exp \\left ( \\ - 0 . 5 \\ \\gamma ^ { - 1 } \\ ( \\ln ^ 2 t ) \\ \\right ) , \\ t \\ge e , \\end{align*}"}
+{"id": "3584.png", "formula": "\\begin{align*} p ^ { 0 } _ { 1 } & = p ^ { 0 } _ { 2 } = \\frac { 1 } { 4 } \\cdot \\dfrac { d _ { 1 } + d _ { 2 } } { \\frac { 2 } { 4 } } = \\frac { 1 } { 2 } ( d _ { 1 } + d _ { 2 } ) , \\\\ p ^ { 0 } _ { 3 } & = p ^ { 0 } _ { 4 } = \\frac { 1 } { 4 } \\cdot \\dfrac { d _ { 3 } + d _ { 4 } } { \\frac { 2 } { 4 } } = \\frac { 1 } { 2 } ( d _ { 3 } + d _ { 4 } ) \\end{align*}"}
+{"id": "9078.png", "formula": "\\begin{align*} \\tilde { \\sigma } ( u _ 1 , \\dots , u _ n ) : = ( \\tilde { \\sigma } _ 1 ( u _ 1 , u _ 2 ) , \\dots , \\tilde { \\sigma } _ { n - 1 } ( u _ { n - 1 } , u _ n ) ) . \\end{align*}"}
+{"id": "2724.png", "formula": "\\begin{align*} \\| P \\| _ K = \\max _ { x \\in K } \\sum _ { j = 1 } ^ { n + 1 } | \\lambda _ j ( x ) | = \\max _ { x \\in K } \\sum _ { j = 1 } ^ { n + 1 } \\frac { | \\Delta _ j ( x ) | } { | \\Delta | } . \\end{align*}"}
+{"id": "8661.png", "formula": "\\begin{align*} m = \\left \\lfloor { \\frac { { i + { N _ { { \\rm { C P } } } } } } { { K + { N _ { { \\rm { C P } } } } } } } \\right \\rfloor , \\end{align*}"}
+{"id": "6613.png", "formula": "\\begin{align*} [ v _ a , v _ b ] = \\alpha _ c \\varepsilon _ c v _ c , \\forall \\quad \\mbox { c y c l i c } \\quad ( a , b , c ) \\in \\mathfrak { S } _ 3 , H = h \\mathrm { v o l } _ g , \\end{align*}"}
+{"id": "6654.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\sup _ { t \\in [ 0 , 1 ] } | \\hat { X } ^ { \\epsilon } ( t ) - \\varphi ( t ) | < \\delta , \\chi _ 1 < 1 \\Big ) = - \\infty . \\end{align*}"}
+{"id": "6889.png", "formula": "\\begin{align*} f ( x ) & = \\sum _ { i = 1 } ^ n V ( x _ i ) + \\sum _ { 1 \\le i < j \\le n } J _ { i j } K ( x _ i - x _ j ) , \\end{align*}"}
+{"id": "1084.png", "formula": "\\begin{align*} \\prescript J { } \\ell ( w ) : = \\# \\{ \\beta \\in \\Phi ^ + _ J \\mid w ^ { - 1 } \\beta \\in \\Phi ^ - \\} . \\end{align*}"}
+{"id": "1649.png", "formula": "\\begin{align*} \\theta _ n ( f ) ( \\gamma _ 1 , \\dots , \\gamma _ n ) = f ( \\rho ( \\gamma _ 1 ) , \\dots , \\rho ( \\gamma _ n ) ) . \\end{align*}"}
+{"id": "5117.png", "formula": "\\begin{align*} \\int _ { \\{ | r - r ' | < r / 2 \\} } \\frac { ( r r ' ) ^ { 2 \\tau + 1 } } { | r - r ' | ^ { 4 \\tau + 2 - 2 / q } r ' } \\dd r ' = C r ^ { 2 / q } . \\end{align*}"}
+{"id": "6053.png", "formula": "\\begin{align*} 2 H = \\eta z ^ m + \\lambda . \\end{align*}"}
+{"id": "5817.png", "formula": "\\begin{align*} \\rho { \\bf { u } } = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { { \\bf { c } } _ i } { f _ i } } + \\frac { 1 } { 2 } { \\bf { F } } \\Delta t . \\end{align*}"}
+{"id": "7389.png", "formula": "\\begin{align*} \\Delta _ + = \\Delta _ - ^ 0 \\cup \\{ \\alpha + n \\delta , n \\delta | n \\in \\mathbb Z _ { < 0 } , \\alpha \\in \\Delta ^ 0 \\} \\end{align*}"}
+{"id": "6066.png", "formula": "\\begin{align*} \\kappa = \\frac { \\alpha } { f \\sqrt { 1 + { f ' } ^ 2 } } + \\varpi \\end{align*}"}
+{"id": "4078.png", "formula": "\\begin{align*} A '' _ n ( j ) = A _ n ( j ) A ' _ n ( j ) \\end{align*}"}
+{"id": "190.png", "formula": "\\begin{align*} B _ { G ' , \\infty } = ( B _ { G ' , n } \\widehat { \\otimes } _ { A _ n } A ) ^ { \\wedge - u } , \\end{align*}"}
+{"id": "4510.png", "formula": "\\begin{align*} \\sum _ { i \\in I } x _ i & \\ge \\sum _ { i \\in I } ( t ' - 1 + \\abs { S \\cap S _ i } + p _ i ) \\\\ & = \\abs { I } ( t ' - 1 ) + \\abs { S } + p \\ge ( k + 1 ) ( t ' - 1 ) + ( 2 k - t ' - p ) + p = k t ' + k - 1 \\ge k t ' , \\end{align*}"}
+{"id": "5168.png", "formula": "\\begin{align*} b _ n - b _ { 1 , n } - b _ { 2 , n } = \\nabla \\times ( \\phi _ n ( \\chi _ { R _ n / 2 } - \\chi _ { R _ 0 } ) \\nabla \\theta ) + G _ n ( \\chi _ { R _ n / 2 } - \\chi _ { R _ 0 } ) \\nabla \\theta , \\end{align*}"}
+{"id": "2938.png", "formula": "\\begin{align*} \\left \\Vert \\varphi \\right \\Vert _ { p , \\beta } = \\max _ { 0 \\leq q \\leq p } | \\sup _ { x \\in \\mathbb { R } } \\left ( e ^ { p \\left \\vert x \\right \\vert ^ { \\beta } } \\varphi ^ { ( q ) } \\left ( x \\right ) \\right ) | , \\ ; \\ ; \\ ; \\beta > 1 \\end{align*}"}
+{"id": "1613.png", "formula": "\\begin{align*} \\begin{aligned} \\theta _ { i , m } ^ \\star = & \\begin{cases} \\angle \\chi _ { i , m } + \\pi , & \\angle \\chi _ { i , m } \\in [ 0 , \\pi ) , \\\\ \\angle \\chi _ { i , m } - \\pi , & \\angle \\chi _ { i , m } \\in [ \\pi , 2 \\pi ) , \\end{cases} \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\forall i \\in \\{ \\mathrm { t , r } \\} , \\forall m \\in \\mathcal { M } . \\end{aligned} \\end{align*}"}
+{"id": "7924.png", "formula": "\\begin{align*} 3 2 \\gamma _ 1 & = 1 6 \\left ( c _ 5 ( q , 1 , p _ 0 ) - \\frac { c _ 2 ( q , 1 , p _ 0 ) c _ 3 ( q , 2 , 1 , p _ 0 ) } { c _ 1 ( q , 2 , p _ 0 ) } \\right ) \\\\ & = q _ 1 - \\frac { q _ 2 ^ 2 } { q _ 1 } \\\\ & = \\frac { 1 } { q _ 1 } ( q _ 1 + q _ 2 ) ( q _ 1 - q _ 2 ) \\end{align*}"}
+{"id": "3027.png", "formula": "\\begin{align*} \\left ( d L _ { \\left ( x , p \\right ) } \\right ) _ { \\left ( 0 , 1 \\right ) } \\left ( \\xi , \\eta \\right ) = \\left . \\frac { d } { d t } \\right \\vert _ { 0 } \\left ( x + p t \\xi \\bar { p } , p e ^ { t \\eta } \\right ) = \\left ( p \\xi \\bar { p } , p \\eta \\right ) \\end{align*}"}
+{"id": "3954.png", "formula": "\\begin{align*} d _ I ( \\C ) = \\min _ { \\mathbf { a } , \\mathbf { b } \\in \\C , \\mathbf { a } \\neq \\mathbf { b } } \\left \\{ \\ , \\ , d _ I ( \\mathbf { a } , \\mathbf { b } ) \\ , \\ , \\right \\} . \\end{align*}"}
+{"id": "7501.png", "formula": "\\begin{align*} \\| \\varphi ^ { + } \\| _ { L ^ { m _ { k } } ( \\partial B _ 1 ) } & \\leq ( C m _ 0 ) ^ { \\frac { 1 } { m _ { 0 } } \\sum _ { j = k - l } ^ { k - 1 } \\left ( \\frac { 2 } { 2 ^ { \\star } } \\right ) ^ { j } } \\left ( \\frac { 2 ^ { \\star } } { 2 } \\right ) ^ { \\frac { 1 } { m _ { 0 } } \\sum _ { j = k - l } ^ { k - 1 } j \\left ( \\frac { 2 } { 2 ^ { \\star } } \\right ) ^ { j } } . \\end{align*}"}
+{"id": "42.png", "formula": "\\begin{align*} B _ { r } = \\{ g : \\rho ( g ) < r \\} . \\end{align*}"}
+{"id": "6229.png", "formula": "\\begin{align*} \\mathcal { M _ { \\mathbb { N } } } & = \\{ \\omega = \\frac { \\sum _ { h = 1 } ^ { H } c _ h \\delta _ { a _ h } } { \\sum _ { h = 1 } ^ { H } c _ h } : \\ ; c _ h \\in \\mathbb { N } ^ + \\} , \\\\ \\mathcal { M _ { \\mathbb { Q } } } & = \\{ \\omega = \\frac { \\sum _ { h = 1 } ^ { H } c _ h \\delta _ { a _ h } } { \\sum _ { h = 1 } ^ { H } c _ h } : \\ ; c _ h \\in \\mathbb { Q } ^ + \\} , \\\\ \\mathcal { M _ { \\mathbb { R } } } & = \\{ \\omega = \\frac { \\sum _ { h = 1 } ^ { H } c _ h \\delta _ { a _ h } } { \\sum _ { h = 1 } ^ { H } c _ h } : \\ ; c _ h \\in \\mathbb { R } ^ + \\} , \\end{align*}"}
+{"id": "5395.png", "formula": "\\begin{align*} v : = - d + \\frac { 1 } { 8 b _ { n - 1 } } d ^ 2 . \\end{align*}"}
+{"id": "8386.png", "formula": "\\begin{align*} & h _ { 1 } = \\frac { 1 } { 2 i } \\int _ { - \\infty } ^ { x } ( | u _ y | ^ 2 f _ 1 + u _ y f _ 2 ) d y , \\\\ & h _ { 2 } = - \\frac { 1 } { 2 i } \\int _ { - \\infty } ^ { x } ( ( 2 i \\bar { u } _ { y y } + \\bar { u } _ y | u _ y | ^ 2 ) f _ 1 + | u _ y | ^ 2 f _ 2 ) e ^ { 2 i z ( x - y ) } d y . \\end{align*}"}
+{"id": "2437.png", "formula": "\\begin{align*} F _ { q } ( s ) = \\int _ { 0 } ^ { \\infty } \\exp _ { q } ( - s t ) f ( t ) d t , \\end{align*}"}
+{"id": "3734.png", "formula": "\\begin{align*} \\mathbf { g } = \\pm u ^ 2 d t ^ 2 + g . \\end{align*}"}
+{"id": "4547.png", "formula": "\\begin{align*} | \\Gamma _ { \\ell , m , n } | = ( q - 1 ) ^ 3 q ^ { 3 \\ell + m - n + 6 } \\end{align*}"}
+{"id": "1869.png", "formula": "\\begin{align*} \\mathcal W ( E ) = \\bigcap _ { p \\ge 1 } \\mathcal L _ 1 ^ p ( E ) \\cap \\{ \\nabla : \\mathcal { Y M } _ e ^ 0 ( \\nabla ) < \\infty \\} \\ , . \\end{align*}"}
+{"id": "2696.png", "formula": "\\begin{align*} \\beta ^ m _ k \\stackrel { \\rm d e f } { = } \\max \\left \\{ \\| g _ k \\| , - \\lambda _ { \\rm m i n } ( H _ k ) \\right \\} . \\end{align*}"}
+{"id": "5677.png", "formula": "\\begin{align*} & ( p _ { 1 } , \\ldots , p _ { n - 3 } , f _ { n - 2 } , f _ { n - 1 } , f _ { n } ) = \\\\ & \\Big [ p _ { 1 } , \\ldots , p _ { n - 2 } , f _ { n - 1 } \\circ ( p _ { 1 } , \\ldots , \\widetilde { f } _ { n - 2 } , p _ { n - 1 } , p _ { n } ) , f _ { n } \\circ ( p _ { 1 } , \\ldots , \\widetilde { f } _ { n - 2 } , p _ { n - 1 } , p _ { n } ) \\Big ] \\circ \\\\ & ( p _ { 1 } , \\ldots , f _ { n - 2 } , p _ { n - 1 } , p _ { n } ) . \\end{align*}"}
+{"id": "2103.png", "formula": "\\begin{align*} \\theta \\mapsto P _ \\theta ( A ) = \\sum _ { i = 1 } ^ k w _ i F _ { v _ i } ( A ) \\end{align*}"}
+{"id": "3463.png", "formula": "\\begin{align*} | \\phi ( k ) | \\leq e ^ { 2 4 \\varepsilon q _ n } \\frac { e ^ { \\beta _ n q _ n } } { \\max ( | \\ell | , 1 ) } \\max \\begin{cases} c _ { n , \\ell - 1 } g _ { | x _ 2 - k | , \\ell } e ^ { - q _ n L } r _ { \\ell - 1 } \\\\ g _ { | x _ 2 - k | , \\ell } e ^ { - q _ n L } r _ { \\ell } \\\\ c _ { n , \\ell } ( c _ { n , \\ell + 1 } ) ^ 2 e ^ { - q _ n L } r _ { \\ell + 1 } \\\\ c _ { n , \\ell } c _ { n , \\ell + 1 } e ^ { - 2 q _ n L } r _ { \\ell + 2 } \\end{cases} \\end{align*}"}
+{"id": "3807.png", "formula": "\\begin{align*} ( 3 m ) ^ 2 z ^ { 3 m } = z ^ 2 \\frac { d } { d z ^ 2 } ( z ^ { 3 m } ) 3 m z ^ { 3 m } . \\end{align*}"}
+{"id": "8256.png", "formula": "\\begin{align*} \\kappa ^ { n } _ m ( t ) : = \\sum _ { i = m } ^ { 2 m } i \\psi ^ { n } _ i + 2 m \\sum _ { i = 2 m + 1 } ^ { n } \\psi ^ { n } _ i , \\end{align*}"}
+{"id": "3107.png", "formula": "\\begin{align*} G = \\{ ( \\omega , x ) \\in \\Omega \\times R ^ n \\mid x \\in C ( \\omega ) , ~ \\langle \\mathcal { F } ( x ^ * ) ( \\omega ) , x - x ^ * ( \\omega ) \\rangle < 0 \\} . \\end{align*}"}
+{"id": "3572.png", "formula": "\\begin{align*} \\varphi ( - t ) ^ 2 G ( t ) H ( - t ) + \\varphi ( t ) ^ 2 G ( t ) H ( t ) = 2 \\varphi ( t ^ 4 ) ^ 2 G ( t ^ 4 ) G ( t ) . \\end{align*}"}
+{"id": "1770.png", "formula": "\\begin{align*} \\bar { w } _ 0 ( 0 ) = \\int _ Y w ^ 0 d y . \\end{align*}"}
+{"id": "8846.png", "formula": "\\begin{align*} B ( x , y ) = \\begin{pmatrix} x _ 1 y _ 3 \\\\ - x _ 2 y _ 3 \\\\ ( x _ 2 - x _ 1 ) ( y _ 2 + y _ 1 ) \\end{pmatrix} . \\end{align*}"}
+{"id": "5409.png", "formula": "\\begin{align*} u _ { n n } \\sigma _ 1 ( b ) - \\sum _ { \\alpha \\leq n - 1 } u _ { n \\alpha } ^ 2 + \\sigma _ 2 ( b ) = f . \\end{align*}"}
+{"id": "4658.png", "formula": "\\begin{align*} C \\log ( 2 N + 1 ) \\leq \\Vert M _ { H _ 0 } : S _ { 1 } ^ { 2 N + 1 } \\rightarrow S _ { 1 } ^ { 2 N + 1 } \\Vert . \\end{align*}"}
+{"id": "7108.png", "formula": "\\begin{align*} - \\ln \\frac { | \\xi - z | } { | \\xi - z ^ * | | z | } \\leq \\frac { 1 } { 2 } \\left ( \\frac { \\frac { ( 1 - | \\xi | ^ 2 ) ( 1 - | z | ^ 2 ) } { | \\xi - z ^ * | ^ 2 | z | ^ 2 } } { \\frac { | \\xi - z | ^ 2 } { | \\xi - z ^ * | ^ 2 | z | ^ 2 } } \\right ) ^ { \\frac { 1 } { 2 } } \\leq \\frac { ( 1 - | \\xi | ) ^ { \\frac { 1 } { 2 } } ( 1 - | z | ) ^ { \\frac { 1 } { 2 } } } { | \\xi - z | } . \\end{align*}"}
+{"id": "3737.png", "formula": "\\begin{align*} { \\bf g } | _ { \\partial \\bf N } & = \\pm u ^ 2 d t ^ 2 + g ^ \\intercal \\\\ A _ { \\bf g } & = \\pm u \\nu ( u ) d t ^ 2 + A _ g . \\end{align*}"}
+{"id": "1564.png", "formula": "\\begin{align*} \\omega _ { 1 1 } = \\partial _ { z _ 1 } \\bar \\partial _ { z _ 1 } \\left ( \\frac { i } { z _ 1 - \\bar z _ 1 } + \\vert z _ 2 \\vert ^ 2 \\right ) = \\frac { 1 } { 4 } \\left ( \\frac { \\partial } { \\partial x _ 1 } - i \\frac { \\partial } { \\partial y _ 1 } \\right ) \\left ( \\frac { \\partial } { \\partial x _ 1 } + i \\frac { \\partial } { \\partial y _ 1 } \\right ) \\frac { 1 } { 2 y _ 1 } = \\frac { 1 } { 4 } \\frac { 1 } { y _ 1 ^ 3 } \\end{align*}"}
+{"id": "2286.png", "formula": "\\begin{align*} \\langle T ^ { ( \\lambda ) } _ a f , g \\rangle _ \\lambda = \\langle a f , g \\rangle _ \\lambda , \\end{align*}"}
+{"id": "6111.png", "formula": "\\begin{align*} ( \\mathcal { F } _ N ) = \\begin{pmatrix} N + 2 \\\\ 2 \\end{pmatrix} \\ , . \\end{align*}"}
+{"id": "1484.png", "formula": "\\begin{align*} g ( u _ 1 ) = \\begin{cases} e ^ { - \\frac { 1 } { u _ 1 } } & \\mbox { i f } u _ 1 > 0 ; \\\\ 0 & \\mbox { i f } u _ 1 \\leq 0 , \\end{cases} \\end{align*}"}
+{"id": "7353.png", "formula": "\\begin{gather*} \\inf _ { Q \\in \\Gamma } Q ( c ) = \\sup _ { f _ 1 , \\ldots , f _ n } \\ , \\sum _ { i = 1 } ^ n \\mu _ i ( f _ i ) \\end{gather*}"}
+{"id": "3080.png", "formula": "\\begin{align*} \\deg T _ { \\varphi _ A ^ * ( f ) } = \\deg T _ f . \\end{align*}"}
+{"id": "3480.png", "formula": "\\begin{align*} \\begin{cases} | \\phi ( x _ 1 - 1 ) | \\leq e ^ { 3 0 \\varepsilon q _ n } \\max \\{ e ^ { - ( ( \\ell - 1 ) q _ n - x _ 1 ) L } r _ { \\ell - 1 } ^ - , e ^ { - ( x _ 1 - ( \\ell - 2 ) q _ n ) L } r _ { \\ell - 2 } ^ + \\} \\\\ | \\phi ( x _ 2 + 1 ) | \\leq e ^ { 3 0 \\varepsilon q _ n } \\max \\{ e ^ { - ( ( \\ell + 1 ) q _ n - x _ 2 ) L } r _ { \\ell + 1 } ^ - , e ^ { - ( x _ 2 - \\ell q _ n ) L } r _ { \\ell } ^ + \\} \\end{cases} \\end{align*}"}
+{"id": "8985.png", "formula": "\\begin{align*} i _ { j _ 1 \\ell _ 1 } = - x _ { 1 , j _ 1 } D _ { 2 , \\ell _ 2 - 1 } , i _ { j _ 1 \\ell _ 2 } = - D _ { 1 , \\ell _ 1 - 1 } x _ { 2 , j _ 2 } , i _ { j _ 1 j _ 2 } = x _ { j _ 1 } x _ { j _ 2 } , \\end{align*}"}
+{"id": "1244.png", "formula": "\\begin{align*} \\abs { \\hat { c } _ \\Delta ( \\eta , \\xi _ \\Delta ) } = \\abs { c _ \\Delta ( \\xi _ \\Delta \\eta _ { \\Delta ^ c } , \\eta _ \\Delta ) \\frac { \\gamma _ \\Delta ( \\xi _ { \\Delta } | \\eta _ { \\Delta ^ c } ) } { \\gamma _ \\Delta ( \\eta _ \\Delta | \\eta _ { \\Delta ^ c } ) } } \\leq \\frac { 1 } { \\delta } e ^ { \\abs { \\Delta } } \\norm { c _ \\Delta ( \\cdot , \\eta _ { \\Delta } ) } _ \\infty \\leq \\frac { 1 } { \\delta } e ^ R \\norm { c _ \\Delta ( \\cdot , \\eta _ \\Delta ) } _ \\infty . \\end{align*}"}
+{"id": "9159.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { \\zeta } = & \\ , A \\zeta + B z \\\\ x = & \\ , C \\zeta \\end{aligned} \\end{align*}"}
+{"id": "3770.png", "formula": "\\begin{align*} \\begin{aligned} & w ( j ) < w ( i _ 2 ) w ( j ) > w ( i _ 1 ) = w ( l + 1 ) i _ 1 < j < i _ 2 , \\\\ & w ( l + 1 ) = w ( i _ 1 ) > w ( l ) > w ( i _ 2 ) . \\end{aligned} \\end{align*}"}
+{"id": "5215.png", "formula": "\\begin{align*} \\int _ 0 ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ j ( \\partial _ t \\varphi + ( v _ j + u _ { \\infty } ) \\cdot \\nabla \\varphi ) \\dd x \\dd s + \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ 0 \\varphi _ 0 \\dd x = - \\mu _ j \\int _ 0 ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( \\Phi _ j \\Delta \\varphi - \\frac { 2 } { r } \\partial _ r \\Phi _ j \\varphi \\right ) \\dd x \\dd s . \\end{align*}"}
+{"id": "79.png", "formula": "\\begin{align*} v ^ { - 1 } \\alpha = \\beta _ 1 > \\cdots > \\beta _ \\ell \\in \\Phi ^ + \\end{align*}"}
+{"id": "4237.png", "formula": "\\begin{align*} \\dot { s } \\varepsilon ( z ) \\sum _ { x \\in \\bar { \\mathbb { F } } _ q } a _ x \\varepsilon ( x ) \\xi _ 1 = \\sum _ { x \\in \\bar { \\mathbb { F } } _ q } a _ x ( \\varepsilon ( x ( z ^ 2 - z ) ^ { - 1 } - z ^ { - 1 } ) + \\varepsilon ( x z ^ { - 1 } ) - \\varepsilon ( x ( z - 1 ) ^ { - 1 } ) ) \\xi _ 1 \\end{align*}"}
+{"id": "2333.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 4 D ^ g \\theta ( J e _ i , \\left ( D ^ g _ { e _ i } J \\right ) X ) & = \\frac { 1 } { 2 } g ( D ^ g _ { J X } \\theta , J \\theta ) + \\frac { 1 } { 2 } g ( D ^ g _ X \\theta , \\theta ) + g ( d \\theta , N _ X ) . \\end{align*}"}
+{"id": "3862.png", "formula": "\\begin{align*} x _ 2 v _ 1 - x _ 1 v _ 2 + k v _ 3 = 0 . \\end{align*}"}
+{"id": "7580.png", "formula": "\\begin{align*} N \\mapsto \\mu _ N = \\frac { 1 } { | \\Phi _ N | } \\sum _ { n \\in \\Phi _ N } \\delta _ { T ^ n a } \\end{align*}"}
+{"id": "8351.png", "formula": "\\begin{align*} & a ( k ) = e ^ { - i c } + \\mathcal { O } \\left ( k ^ { - 1 } \\right ) , \\ \\ b ( k ) = \\mathcal { O } \\left ( k ^ { - 1 } \\right ) , k \\rightarrow \\infty , \\\\ & a ( k ) = e ^ { - i c } \\left ( 1 + \\mathcal { O } \\left ( k ^ 2 \\right ) \\right ) , \\ \\ b ( k ) = \\mathcal { O } \\left ( k ^ 3 \\right ) , k \\rightarrow 0 , \\end{align*}"}
+{"id": "3710.png", "formula": "\\begin{align*} \\Delta v + \\big ( \\Delta ' ( h ) \\big ) \\bar u = 0 \\mbox { i n } M \\setminus \\Omega . \\end{align*}"}
+{"id": "5017.png", "formula": "\\begin{align*} S t _ { D N D C } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( 1 , z ) - 1 | | _ { \\infty } = 0 , \\end{align*}"}
+{"id": "7106.png", "formula": "\\begin{align*} - \\ln \\frac { | \\xi - z | } { | \\xi - z ^ * | | z | } = - \\frac { 1 } { 2 } \\ln \\left ( 1 - \\frac { ( 1 - | \\xi | ^ 2 ) ( 1 - | z | ^ 2 ) } { | \\xi - z ^ * | ^ 2 | z | ^ 2 } \\right ) \\geq \\frac { 1 } { 2 } \\frac { ( 1 - | \\xi | ^ 2 ) ( 1 - | z | ^ 2 ) } { | \\xi - z ^ * | ^ 2 | z | ^ 2 } , \\end{align*}"}
+{"id": "6978.png", "formula": "\\begin{align*} x l _ 1 + y k _ 1 = 1 \\end{align*}"}
+{"id": "2792.png", "formula": "\\begin{align*} h ^ { D _ { N } } ( x ) = S _ { k } ( x ) + ( S _ { k - l } ( x ) - S _ { k } ( x ) ) + ( S _ { 0 } ( x ) - S _ { k - l } ( x ) ) \\end{align*}"}
+{"id": "8161.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\mu \\xi ( \\mu , t ) \\ d \\mu = \\int _ { 0 } ^ { \\infty } \\mu \\xi ^ { \\mathrm { i n } } ( \\mu ) \\ d \\mu = \\varrho _ { 0 } , \\forall ~ t \\geq 0 . \\end{align*}"}
+{"id": "7774.png", "formula": "\\begin{align*} \\gamma _ { w , k , m } \\le \\epsilon = 1 / 2 ^ b ~ { \\rm f o r } ~ k \\ge ( b + 2 ) \\log _ { \\Delta _ { v , m } } ( 2 ) ~ { \\rm i f } ~ 2 \\Delta _ { v , m } ^ k \\le 1 . \\end{align*}"}
+{"id": "59.png", "formula": "\\begin{align*} I : = \\{ \\beta \\in \\Phi ^ + \\setminus \\{ \\alpha \\} \\mid s _ \\alpha ( \\beta ) \\in \\Phi ^ - \\} . \\end{align*}"}
+{"id": "1493.png", "formula": "\\begin{align*} \\phi _ 4 ' = - \\frac { 1 } { \\epsilon } [ c \\phi _ 4 - \\kappa \\phi _ 2 g ( \\phi _ 1 ) ] . \\end{align*}"}
+{"id": "1471.png", "formula": "\\begin{align*} 0 = \\varphi ' ( s ) = \\beta \\left [ ( M s - A ) ^ { 2 } - ( B - M s ) ^ { 2 } \\right ] \\end{align*}"}
+{"id": "2758.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ m \\ell ( D F _ j ( h _ { j - 1 } ( u ) ) ) \\leq \\ell ( D h _ m ( u ) ) \\leq | D h _ m ( u ) | \\leq \\prod _ { j = 1 } ^ m | D F _ j ( h _ { j - 1 } ( u ) ) | , \\end{align*}"}
+{"id": "6033.png", "formula": "\\begin{align*} c _ { b } = \\mathop { \\inf } \\limits _ { ( u , v ) \\in E \\setminus \\{ ( 0 , 0 ) \\} } \\mathop { \\max } \\limits _ { t > 0 } I ( \\gamma _ { u , v } ( t ) ) . \\end{align*}"}
+{"id": "5851.png", "formula": "\\begin{align*} \\delta x = - \\frac { 1 } { 2 } \\left ( { { f _ 1 } - { f _ 2 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) - \\frac { 1 } { 4 } { F _ x } , \\end{align*}"}
+{"id": "6065.png", "formula": "\\begin{align*} - \\frac { 1 } { \\sqrt { 1 + { f ' } ^ 2 } } = \\frac { \\eta } { m + 1 } f ^ { m + 1 } + \\lambda f + c , \\end{align*}"}
+{"id": "6684.png", "formula": "\\begin{align*} 2 c ( \\gamma , x ) = h ( \\gamma x , \\gamma y , \\gamma z ) - h ( x , y , z ) . \\end{align*}"}
+{"id": "6216.png", "formula": "\\begin{align*} W _ + ( r ) & = ( 2 L + 3 ) \\biggl ( - \\frac { f } { r } + \\Q \\frac { \\Q ^ 2 + 4 \\kappa } { \\Q ^ 2 + 1 0 \\kappa } + 3 \\kappa \\frac { \\Q ^ 2 + 2 \\kappa } { \\Q ^ 2 + 1 0 \\kappa } \\frac { r } { f } + \\frac { 6 \\kappa \\Q } { \\Q ^ 2 + 1 0 \\kappa } \\frac { 1 } { f ^ 2 } \\\\ & \\quad { } + \\frac { 6 \\kappa ^ 2 } { \\Q ^ 2 + 1 0 \\kappa } \\frac { r } { f ^ 3 } \\biggr ) , \\\\ W _ - ( r ) & = - \\frac { f } { r } - \\Q + 3 \\kappa \\frac { r } { f } , \\end{align*}"}
+{"id": "715.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { y \\in \\R ^ N } \\int _ { B _ r ( y ) } u _ { \\lambda _ n } ^ 2 d x = 0 , \\end{align*}"}
+{"id": "7208.png", "formula": "\\begin{align*} \\overline { W ^ { k , i } } = W ^ { - k , i } , \\quad \\big [ W ^ { k , i } , W ^ { l , j } \\big ] _ { t } = 2 t \\delta _ { k , - l } \\delta _ { i , j } . \\end{align*}"}
+{"id": "210.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\rightarrow 0 ^ + } \\left ( \\lim _ { n \\rightarrow \\infty } \\sqrt [ n ] { \\binom { n + \\lceil \\alpha n \\rceil } { \\lceil \\alpha n \\rceil } } \\right ) = \\lim _ { \\alpha \\rightarrow 0 ^ + } \\left ( \\lim _ { n \\rightarrow \\infty } \\sqrt [ n ] { \\binom { n } { \\lceil \\alpha n \\rceil } } \\right ) = 1 . \\end{align*}"}
+{"id": "6279.png", "formula": "\\begin{align*} 0 & = \\nabla _ { X _ { k + 1 } } ( \\Omega _ { k - 1 } ( X _ 0 , \\ldots , X _ k ) \\varphi ) = \\nabla _ { X _ { k + 1 } } \\Omega _ { k - 1 } ( X _ 0 , \\ldots , X _ k ) \\varphi - \\Omega _ { k - 1 } ( X _ 0 , \\ldots , X _ k ) \\circ \\omega ( X _ { k + 1 } ) \\varphi \\\\ & = \\Omega _ k ( X _ 0 , \\ldots , X _ { k + 1 } ) \\varphi , \\quad \\forall X _ i \\in \\Gamma ( T L ) , \\end{align*}"}
+{"id": "1867.png", "formula": "\\begin{align*} C _ 0 = \\frac { n ( n - 1 ) } { 2 ( n + 2 ) \\omega _ n ^ { \\frac 2 n } } \\ , . \\end{align*}"}
+{"id": "387.png", "formula": "\\begin{align*} 2 ( J _ { n - 1 } - I _ { n - 1 } ) u _ k = 2 \\ 1 _ { n - 1 } \\ 1 _ { n - 1 } ' u _ k - 2 u _ k = - 2 u _ k . \\end{align*}"}
+{"id": "243.png", "formula": "\\begin{align*} g & = t ^ { - \\sigma ^ - } \\cdot \\prod _ { i = \\sigma ^ - } ^ { \\sigma ^ + - 1 } \\big ( w _ i ( a _ 1 , \\dots , a _ d ) \\ , t ^ { - 1 } \\big ) \\cdot w _ { \\sigma ^ + } ( a _ 1 , \\ldots , a _ d ) \\cdot t ^ { \\sigma ^ + } , \\\\ g & = t ^ { - \\sigma ^ + } \\cdot \\prod _ { j = \\sigma ^ - } ^ { \\sigma ^ + - 1 } \\big ( w _ { \\sigma ^ + + \\sigma ^ - - j } ( a _ 1 , \\dots , a _ d ) \\ , t \\big ) \\cdot w _ { \\sigma ^ - } ( a _ 1 , \\ldots , a _ d ) \\cdot t ^ { \\sigma ^ - } , \\end{align*}"}
+{"id": "5932.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\lim _ { n \\rightarrow \\infty } I I I \\leq \\lim _ { M \\rightarrow \\infty } \\Big [ C _ M \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ { T } \\Vert X _ n ( s ) - X ( s ) \\Vert _ H ^ 2 d s \\Big ] = 0 . \\end{align*}"}
+{"id": "2029.png", "formula": "\\begin{align*} d X _ t ^ i & = b ( t , X _ t ^ i , \\overline \\mu _ t ^ N , u _ t ^ i ) d t + \\sigma d W _ t , \\end{align*}"}
+{"id": "8379.png", "formula": "\\begin{align*} \\widetilde { Q } = \\frac { 1 } { 2 i } \\begin{pmatrix} | u _ x | ^ 2 & u _ x \\\\ - 2 i \\bar { u } _ { x x } - \\bar { u } _ x | u _ x | ^ 2 & - | u _ x | ^ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "3871.png", "formula": "\\begin{align*} \\begin{cases} - \\delta ^ 2 ( K ( x ) \\nabla w ) = ( w - q ) ^ { p } _ + , \\ \\ & x \\in \\Omega , \\\\ w = 0 , \\ \\ & x \\in \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "7855.png", "formula": "\\begin{align*} \\hat \\Phi ^ \\dagger _ { a a } ( 0 , 0 ) & = \\langle z _ 1 ^ * , \\Phi ^ \\dagger _ { v v } ( 0 , p _ 0 ) [ u _ \\ell , u _ \\ell ] \\rangle = \\langle z _ 1 ^ * , Q ^ \\dagger F ^ { q , \\dagger } _ { v v } ( 0 , p _ 0 ) [ u _ \\ell , u _ \\ell ] \\rangle = \\langle z _ 1 ^ * , 0 \\rangle = 0 . \\end{align*}"}
+{"id": "2065.png", "formula": "\\begin{align*} [ M , v \\wedge \\partial _ v ] = 2 \\sum _ { 1 \\leq i \\leq 2 } \\big ( [ \\Lambda _ i , v ] \\wedge \\partial _ v \\big ) \\Lambda _ i = 2 \\sum _ { 1 \\leq i \\leq 2 } \\Lambda _ i \\big ( [ \\Lambda _ i , v ] \\wedge \\partial _ v \\big ) , \\textrm { w i t h } [ \\Lambda _ i , v ] = c _ i \\big ( ( t - t _ 0 ) ^ { 1 \\over 2 } , 0 , 0 \\big ) , \\end{align*}"}
+{"id": "8686.png", "formula": "\\begin{align*} S \\subset \\bigcup _ { i = 1 } ^ { C _ { B C } } \\bigcup _ { B \\in \\mathcal { B } ^ { ( i ) } } B . \\end{align*}"}
+{"id": "5738.png", "formula": "\\begin{align*} a _ 1 x ^ q + a _ 1 x = c _ 1 y ^ q + b _ 1 y . \\end{align*}"}
+{"id": "7604.png", "formula": "\\begin{align*} \\rho ( x ) = ( \\rho _ k ( x ) : k \\in \\N ) \\end{align*}"}
+{"id": "8617.png", "formula": "\\begin{align*} g ( t ) = \\frac { | A \\oplus _ 2 \\sqrt { t } B | ^ 2 } { | \\partial ( A \\oplus _ 2 \\sqrt { t } B ) | ^ 2 _ { n - 1 } } \\end{align*}"}
+{"id": "8445.png", "formula": "\\begin{align*} & \\| z ^ { - 2 } k ^ { - 1 } b ( k ) \\| _ { L ^ 2 ( - \\delta , \\delta ) } = \\| z ^ { - 1 } ( z ^ { - 1 } k ^ { - 1 } b ( k ) ) \\| _ { L ^ 2 ( - \\delta , \\delta ) } \\\\ & \\leq \\| z ^ { - 1 } \\| _ { L ^ { 1 / 2 } ( - \\delta , \\delta ) } ^ { 1 / 5 } \\| z ^ { - 1 } k ^ { - 1 } b ( k ) \\| _ { L ^ 8 } ^ { 4 / 5 } ( - \\delta , \\delta ) \\leq c . \\end{align*}"}
+{"id": "5525.png", "formula": "\\begin{align*} f ( x ) = \\left ( \\log \\left ( 1 + \\frac { 1 } { 2 x } \\right ) + \\frac { 1 } { 4 x } \\right ) - \\left ( \\psi \\left ( x + \\frac 1 2 \\right ) - \\psi ( x ) \\right ) . \\end{align*}"}
+{"id": "737.png", "formula": "\\begin{align*} z : = \\zeta ^ { 2 } | \\tilde w ^ \\varepsilon | ^ 2 - \\lambda \\tilde w ^ \\varepsilon , \\end{align*}"}
+{"id": "9292.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u _ 1 ^ t \\wedge \\dots \\wedge \\Delta u _ k ^ t & \\wedge \\beta _ n ^ { n - m } \\mapsto \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ k \\wedge \\beta _ n ^ { n - m } , k = 1 \\dots , m . \\end{aligned} \\end{align*}"}
+{"id": "7219.png", "formula": "\\begin{align*} d u = \\Delta _ p u \\ , d t + \\sum _ { n = 0 } ^ { + \\infty } \\xi _ { n } \\cdot \\nabla u \\circ d B _ t ^ { n } , \\end{align*}"}
+{"id": "8742.png", "formula": "\\begin{align*} g ^ { - 1 } \\alpha ( g ) = n ^ { - 1 } a ^ { - 1 } \\alpha ( a n ) K = n ^ { - 1 } a ^ { - 1 } a \\alpha ( n ) K = n ^ { - 1 } \\alpha ( n ) K . \\end{align*}"}
+{"id": "4852.png", "formula": "\\begin{align*} K ( c _ 1 , \\dots , c _ n , 0 ) = & p _ i \\circ H ( c _ 1 , \\dots , c _ n , m ) \\\\ = & p _ i \\circ \\iota _ { U _ k } ( c 1 , \\dots , c _ n ) = p _ i ( c _ 1 , \\dots , c _ n ) , \\end{align*}"}
+{"id": "6261.png", "formula": "\\begin{align*} \\gamma ( X _ i , X _ j ) = 0 \\quad \\forall i \\neq j , \\end{align*}"}
+{"id": "3004.png", "formula": "\\begin{align*} F ^ \\eta ( x , y ) = \\big \\{ l \\in \\mathbb R ^ n \\colon \\| l \\| \\leq c _ H ( 1 + \\| x \\| + \\| y \\| ) + \\eta \\big \\} , \\ \\ x , y \\in \\mathbb R ^ n , \\ \\ \\eta \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "1077.png", "formula": "\\begin{align*} \\beta _ i : = \\beta + i w ^ { - 1 } \\gamma \\in \\Phi , i = 0 , \\dotsc , k . \\end{align*}"}
+{"id": "1456.png", "formula": "\\begin{align*} \\operatorname { \\mathbb { J } } _ { 1 / 2 } ( \\Omega , u ) : = \\sum _ { s \\in S _ { u } \\cap \\Omega } | J ( s ) | ^ { 1 / 2 } , \\end{align*}"}
+{"id": "6822.png", "formula": "\\begin{align*} J _ { l , 0 } \\left ( \\rho \\right ) = \\int _ { 0 } ^ { \\infty } \\left ( q ( s ) I _ { l } ^ { \\left ( 1 \\right ) } ( s , \\rho ) + Q ^ { \\prime } ( s ) I _ { l } ^ { \\left ( 2 \\right ) } ( s , \\rho ) \\right ) d s , \\end{align*}"}
+{"id": "3299.png", "formula": "\\begin{align*} \\mathcal S ( t + \\Delta _ n ) e _ i - \\mathcal S ( t ) e _ i = \\int _ t ^ { t + \\Delta _ n } \\mathcal S ( u ) \\mathcal A e _ i d u \\end{align*}"}
+{"id": "5743.png", "formula": "\\begin{align*} h _ { a , b , c } ( x , y ) = c ( ( a + 1 ) y / b + a x ) ^ { q + 1 } + ( b x + a y ) ^ { q + 1 } ) . \\end{align*}"}
+{"id": "8873.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\prod _ { k = 1 } ^ { j - 1 } f _ k ( T _ { g ( k ) } ^ { p _ k ( a _ n ) } x ) ( f _ j - \\mathbb { E } ( f _ j | \\mathcal { A } _ { g _ { j _ 0 } } ) ) ( T _ { g ( j ) } ^ { p _ j ( a _ n ) } x ) \\prod _ { l = j + 1 } ^ { m } \\mathbb { E } ( f _ { l } | \\mathcal { A } ) ( T _ { g ( l ) } ^ { p _ { l } ( a _ n ) } x ) \\rightarrow 0 \\end{align*}"}
+{"id": "5361.png", "formula": "\\begin{align*} b - b _ 1 = l ( p - q ) + ( p - q r ) + 1 \\leq l ( p - q ) \\leq ( l q - 1 ) ( r - 1 ) = b _ 1 ( r - 1 ) . \\end{align*}"}
+{"id": "3188.png", "formula": "\\begin{align*} 0 = \\lambda ( - x ^ 0 x ^ 0 + x ^ 1 x ^ 1 + x ^ 2 x ^ 2 + x ^ 3 x ^ 3 ) + \\mu { ( x ^ 0 - x ^ 3 ) } ^ 2 . \\end{align*}"}
+{"id": "1614.png", "formula": "\\begin{align*} \\begin{aligned} \\phi _ { \\mathrm { t } , m } ^ \\star = \\sqrt { \\alpha _ { \\mathrm { t } , m } ^ \\star } e ^ { \\jmath \\theta _ { \\mathrm { t } , m } ^ \\star } , \\phi _ { \\mathrm { r } , m } ^ \\star = \\sqrt { 1 - \\alpha _ { \\mathrm { t } , m } ^ \\star } e ^ { \\jmath \\theta _ { \\mathrm { r } , m } ^ \\star } , \\forall m \\in \\mathcal { M } . \\end{aligned} \\end{align*}"}
+{"id": "6350.png", "formula": "\\begin{align*} \\frac { d } { d x } f ( x ) = & \\frac { \\theta ^ { 2 } } { B ( a , b ) \\ , x ^ 6 } \\exp \\left ( - \\frac { a \\theta } { x ^ 2 } \\right ) \\left [ 1 - \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) \\right ] ^ { b - 1 } \\\\ & \\times \\left [ 4 a - \\frac { 6 x ^ 2 } { \\theta } + \\frac { 4 ( b - 1 ) } { 1 - \\exp \\left ( \\frac { \\theta } { x ^ 2 } \\right ) } \\right ] , x > 0 . \\end{align*}"}
+{"id": "9010.png", "formula": "\\begin{align*} U _ { r } ^ { \\tau } = U _ { r } ^ { \\tau _ 1 } = \\cdots = U _ { r } ^ { \\tau _ { s + 1 } } . \\end{align*}"}
+{"id": "306.png", "formula": "\\begin{align*} i ( x ) = d ( x ) \\end{align*}"}
+{"id": "8857.png", "formula": "\\begin{align*} | x _ t | ^ 2 - | x _ 0 | ^ 2 = 2 \\int _ 0 ^ t x _ s \\cdot \\sigma d W _ s - 2 \\int _ 0 ^ t A x _ s \\cdot x _ s d s + t \\sum _ { i , j = 1 } ^ n | \\sigma _ { i j } | ^ 2 . \\end{align*}"}
+{"id": "5754.png", "formula": "\\begin{align*} \\langle \\langle Z \\cdot \\varphi , \\varphi ' \\rangle \\rangle = \\langle \\langle \\varphi , Z \\cdot \\varphi ' \\rangle \\rangle . \\end{align*}"}
+{"id": "3825.png", "formula": "\\begin{align*} \\mathcal { H } ( \\boldsymbol { \\nu } , U , \\bar { U } ; x , t ) = \\int \\eta ( \\lambda | \\bar { U } ; x , t ) d \\boldsymbol { \\nu } ( \\lambda ) \\end{align*}"}
+{"id": "4652.png", "formula": "\\begin{align*} \\langle i _ p ( T _ \\phi ( x _ 1 , \\ldots , x _ n ) ) , i _ { p ' } ( y ^ * ) \\rangle _ { p , p ' } = \\left \\langle \\left ( M _ { \\widetilde { \\phi } } ( i _ { p _ 1 , \\alpha } ( x _ 1 ) , \\ldots , i _ { p _ n , \\alpha } ( x _ n ) ) \\right ) _ { \\alpha \\in I } , i _ { p ' } ( y ^ * ) \\right \\rangle _ { p , p ' } , \\end{align*}"}
+{"id": "1801.png", "formula": "\\begin{align*} c ^ { 3 } ( q ) = 2 7 \\sum _ { n , s = 1 } ^ { \\infty } \\chi _ { - 3 } ( n ) s ^ { 2 } q ^ { n s } , \\end{align*}"}
+{"id": "2440.png", "formula": "\\begin{align*} F _ { q } ( s - s _ { 0 } ) = L _ { q } \\bigg [ f ( t ) \\exp _ { q } \\bigg ( \\frac { s _ { 0 } t } { 1 - ( 1 - q ) s t } \\bigg ) \\bigg ] . \\end{align*}"}
+{"id": "9016.png", "formula": "\\begin{align*} \\dfrac { \\hat { k } _ 2 { ( \\Delta ) } } { \\hat { k } _ 2 ( \\Psi ) } = \\dfrac { D _ { 2 } D _ { 3 } D _ { 5 } } { D _ { 1 } D _ { 2 } D _ { 4 } } = \\dfrac { D _ { 3 } D _ { 5 } } { D _ { 1 } D _ { 4 } } . \\end{align*}"}
+{"id": "2466.png", "formula": "\\begin{align*} f ( t ) = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ { n } \\frac { \\left ( \\alpha t \\right ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! } = \\sin \\alpha t . \\end{align*}"}
+{"id": "7632.png", "formula": "\\begin{align*} \\mathcal { J } ( \\mu , \\nu ; \\alpha ) : = \\frac { 1 } { 2 } \\mathbb { E } \\Big \\{ \\int _ 0 ^ T \\Big [ Q \\left ( x _ t ^ { \\xi , \\alpha } + l ( \\nu _ t ) \\right ) ^ 2 + R \\left ( \\alpha _ t + h ( \\mu _ t ) \\right ) ^ 2 \\Big ] d t + G \\left ( x _ T ^ { \\xi , \\alpha } + g ( \\nu _ T ) \\right ) ^ 2 \\Big \\} , \\end{align*}"}
+{"id": "3272.png", "formula": "\\begin{align*} 1 - \\mathbb { P } _ { Y _ n } [ K _ { \\epsilon } ] \\leq \\sum _ { k = 1 } ^ { \\infty } \\mathbb P _ { Y _ n } [ ( A _ k ^ { \\epsilon } ) ^ c ] \\leq \\epsilon , \\end{align*}"}
+{"id": "1051.png", "formula": "\\begin{align*} x _ n : = \\prescript { \\sigma ^ { 1 - n } } { } { } \\left ( \\left ( x ^ { \\ast , \\sigma , n - 1 } \\right ) ^ { - 1 } x ^ { \\ast , \\sigma , n } \\right ) , \\end{align*}"}
+{"id": "6933.png", "formula": "\\begin{align*} \\partial _ { i i } f ( x ) & = V '' ( x _ i ) + \\sum _ { j \\neq i } J _ { i j } K '' ( x _ i - x _ j ) , \\partial _ { i j } f ( x ) = - J _ { i j } K '' ( x _ i - x _ j ) . \\end{align*}"}
+{"id": "4598.png", "formula": "\\begin{align*} F _ j \\subseteq \\bigcup _ { n = 0 } ^ \\infty [ x _ { n , j } , y _ { n , j } ] \\end{align*}"}
+{"id": "6695.png", "formula": "\\begin{align*} | \\delta ( \\theta , \\gamma ^ + ) | = | f ( \\theta \\gamma ^ + , \\gamma ^ - ) - f ( \\gamma ^ + , \\gamma ^ - ) | \\leq 2 M . \\end{align*}"}
+{"id": "611.png", "formula": "\\begin{align*} \\ell _ \\eta : = \\sup _ { \\substack { g , q \\in \\phi ( Y ) \\\\ \\pi ( g ) = \\pi ( q ) } } \\eta \\left ( \\frac { d ( g , \\pi ^ { - 1 } ( \\bar y ) ) } { d ( q , \\pi ^ { - 1 } ( \\bar y ) ) } \\right ) < \\infty , \\end{align*}"}
+{"id": "4666.png", "formula": "\\begin{align*} \\int _ X | | \\xi ( x ) | | G \\ \\mu ( d x ) = \\int _ X \\beta ( x ) \\ \\mu ( d x ) , \\end{align*}"}
+{"id": "5560.png", "formula": "\\begin{align*} \\left [ A ( y _ 1 x ) - A ( y _ 1 x ' ) \\right ] + \\sum _ { n = 2 } ^ \\infty \\left [ A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) \\right ] \\end{align*}"}
+{"id": "7607.png", "formula": "\\begin{align*} X _ \\beta = \\{ ( x , y , w , z ) \\in X : w - x = \\beta \\} \\end{align*}"}
+{"id": "3217.png", "formula": "\\begin{align*} x \\in X = p _ X [ \\mu \\cdot ( \\nu \\cdot \\gamma ) ] \\end{align*}"}
+{"id": "2391.png", "formula": "\\begin{align*} \\sum _ h a _ h v ( \\sigma _ h ) \\sigma _ h = \\sum _ h \\left [ \\sum _ { \\sigma _ h \\subset \\tau _ j } c _ { h , j } b _ j \\frac { v ( \\sigma _ h ) } { v ( \\tau _ j ) } \\right ] \\sigma _ h , \\end{align*}"}
+{"id": "602.png", "formula": "\\begin{align*} \\null _ { j } \\eta = \\eta _ { j , 1 } \\ , \\null _ { j , 2 } \\eta \\quad \\quad \\null _ { k } \\hat { \\eta } = \\hat { \\eta } _ { k , 1 } \\ , \\null _ { k , 2 } \\hat { \\eta } \\end{align*}"}
+{"id": "6419.png", "formula": "\\begin{align*} \\Gamma _ { i k } ^ k = \\left ( \\log \\det g ^ { 1 / 2 } \\right ) _ { , i } = \\frac { 1 } { 2 } ( - v _ i + u _ { , i q } \\xi ^ q ) . \\end{align*}"}
+{"id": "7183.png", "formula": "\\begin{align*} \\Lambda _ { g } = A Q - D \\end{align*}"}
+{"id": "6022.png", "formula": "\\begin{align*} c + o _ { n } ( 1 ) \\| ( u _ { n } , v _ { n } ) \\| _ { E } & = I ( ( u _ { n } , v _ { n } ) ) - \\frac { 1 } { 2 } \\langle I ' ( u _ { n } , v _ { n } ) , ( u _ { n } , v _ { n } ) \\rangle \\\\ & = - \\big ( B ( u + B ( v ) \\big ) + \\Big ( \\frac { 1 } { 2 } - \\frac { 1 } { 2 p } \\Big ) F ( u , v ) + o _ { n } ( 1 ) \\\\ & = o _ { n } ( 1 ) , \\end{align*}"}
+{"id": "7442.png", "formula": "\\begin{align*} \\widetilde { a } _ k ( x ) h _ k '' ( x ) + \\widetilde { b } _ k ( x ) h _ k ' ( x ) + ( 2 k - 1 ) \\widetilde { c } _ k ( x ) h _ k ( x ) = 0 , \\end{align*}"}
+{"id": "2890.png", "formula": "\\begin{align*} \\partial _ t T ( t , u ) & = \\frac { D } { 4 \\gamma } \\partial _ u ^ 2 T ( t , u ) u \\in ( 0 , 1 ) \\\\ T ( t , 0 ) & = T _ - , \\partial _ u T ( t , 1 ) = - \\frac { 4 \\gamma J } { D } , T ( 0 , u ) = T _ 0 ( u ) , \\end{align*}"}
+{"id": "8471.png", "formula": "\\begin{align*} \\rho _ - \\left \\| Q _ { j , - } \\right \\| _ { L ^ 2 } ^ 2 \\leq \\operatorname { R e } \\int _ { \\mathbb { R } } Q _ { j , - } ( I + J ) Q _ { j , - } ^ H d z = \\operatorname { R e } \\int _ { \\mathbb { R } } \\mathcal { P } ^ - ( D _ j ) Q _ { j , - } ^ H d z \\leq \\| D _ j \\| _ { L ^ 2 } \\left \\| Q _ { j , + } \\right \\| _ { L ^ 2 } . \\end{align*}"}
+{"id": "8339.png", "formula": "\\begin{align*} \\varphi ^ { \\pm } ( x , t ; k ) = I + k \\int _ { \\pm \\infty } ^ { x } e ^ { - 2 i k ^ 2 ( x - y ) \\widehat { \\sigma } _ 3 } P _ y ( y ) \\varphi ^ { \\pm } ( y , t ; k ) d y . \\end{align*}"}
+{"id": "63.png", "formula": "\\begin{align*} \\beta ' : = - s _ \\alpha ( \\beta ) = \\langle \\alpha ^ \\vee , \\beta \\rangle \\alpha - \\beta . \\end{align*}"}
+{"id": "5464.png", "formula": "\\begin{align*} d = 2 n = 4 l + 2 \\equiv 2 \\mod 3 \\end{align*}"}
+{"id": "5142.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 5 } } \\nabla _ y \\varphi \\cdot \\nabla _ y \\tilde { \\varphi } \\dd y = \\mu ^ { 2 } \\int _ { \\mathbb { R } ^ { 5 } } \\left ( \\varphi - 1 - \\frac { \\gamma } { r ^ { 2 } } \\right ) _ { + } \\tilde { \\varphi } \\dd y , \\end{align*}"}
+{"id": "5541.png", "formula": "\\begin{align*} a \\wedge b : = ( a \\cdot b ) a = ( b \\cdot a ) b . \\end{align*}"}
+{"id": "6941.png", "formula": "\\begin{align*} M _ n = n \\sup _ { Q \\in \\P ( \\R ) } \\left ( \\int _ { \\R } V \\ , d Q + \\frac 1 2 \\int _ { \\R } \\int _ { \\R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \\right ) , \\end{align*}"}
+{"id": "287.png", "formula": "\\begin{align*} \\nabla _ H \\psi ( q _ 0 ) = \\nabla _ H \\varphi ( q _ 0 ) . \\end{align*}"}
+{"id": "613.png", "formula": "\\begin{align*} \\psi _ * \\mu \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) = \\mu ( \\psi ^ { - 1 } \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) ) = \\mu ( \\pi \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) ) , \\end{align*}"}
+{"id": "4210.png", "formula": "\\begin{align*} \\frac { 1 } { \\varepsilon ^ 2 } \\int _ { A _ \\varepsilon } \\left ( u _ \\varepsilon - q \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ { p } q d x = \\frac { 2 c _ \\varepsilon } { \\ln \\frac { 1 } { \\varepsilon } } + O \\left ( \\frac { 1 } { \\ln \\frac { 1 } { \\varepsilon } } \\right ) \\leq C . \\end{align*}"}
+{"id": "7125.png", "formula": "\\begin{align*} \\begin{array} { l l l } f _ L ( L \\circ s ) & = & f ^ M _ L ( \\mathcal { M } _ { \\mathbb { C } , \\mathsf { S } } ( L \\circ s ) ) \\\\ & = & f ^ M _ L ( ( L , R ) \\star \\mathcal { M } _ { \\mathbb { C } , \\mathsf { S } } ( s ) ) \\\\ & = & f ^ M _ L ( L , R ) \\circ f ^ M _ L ( \\mathcal { M } _ { \\mathbb { C } , \\mathsf { S } } ( s ) ) \\\\ & = & f ^ M _ { L } ( L , R ) \\circ f _ L ( s ) . \\end{array} \\end{align*}"}
+{"id": "2337.png", "formula": "\\begin{align*} \\left ( d J \\theta \\right ) ^ { ( 2 , 0 ) + ( 0 , 2 ) } = \\left ( \\mathcal { L } _ { \\theta ^ \\sharp } J \\right ) ^ { a n t i - s y m } . \\end{align*}"}
+{"id": "3064.png", "formula": "\\begin{align*} \\alpha _ w = \\sum _ { i = 0 } ^ { n } w _ i \\frac { d x _ i } { x _ i } . \\end{align*}"}
+{"id": "1765.png", "formula": "\\begin{align*} \\det ( A + u \\otimes v ) = ( 1 + v ^ T A ^ { - 1 } u ) \\det ( A ) , \\end{align*}"}
+{"id": "5384.png", "formula": "\\begin{align*} m : = \\inf _ { \\partial \\Omega } \\frac { G ( ( D ^ 2 u ) ' ) } { \\widetilde { f } } , \\end{align*}"}
+{"id": "3197.png", "formula": "\\begin{align*} _ { A _ 1 A _ 0 } = \\left ( \\begin{array} { c c c c } 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & - 1 \\\\ 0 & 0 & - 1 & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
+{"id": "2267.png", "formula": "\\begin{align*} a ( U Z ) = a ( Z ) , \\end{align*}"}
+{"id": "4426.png", "formula": "\\begin{align*} X = \\{ \\sum _ { n = 0 } ^ { \\infty } a _ n z ^ n \\mid \\left ( a _ n / w _ n \\right ) _ { n \\in \\mathbb { N } } \\in \\ell ^ 2 \\} \\end{align*}"}
+{"id": "5427.png", "formula": "\\begin{align*} \\left | \\frac { \\partial ^ 2 \\varphi } { \\partial x _ \\beta \\partial x _ \\mu } ( x ' , \\rho ( x ' ) ) \\right | \\leq C \\delta \\sqrt { \\tilde { b } _ \\beta \\tilde { b } _ \\mu } , \\ \\beta , \\mu = 1 , \\ldots , n - 1 ; \\beta \\neq \\mu \\end{align*}"}
+{"id": "6881.png", "formula": "\\begin{align*} u ( \\infty ) = \\Re f ( \\infty ) = \\Psi _ 1 ^ { - 1 } ( \\Psi _ 2 ^ { - 1 } ( \\Phi ( \\infty ) ) ) = \\Psi _ 1 ^ { - 1 } ( \\Psi _ 2 ^ { - 1 } ( 0 ) ) . \\end{align*}"}
+{"id": "4261.png", "formula": "\\begin{align*} \\alpha ( g , Z ) = ( \\xi + z _ 1 \\psi _ 1 ) \\alpha ( g , Z ) - \\alpha ( g , Z ) \\cdot \\left ( \\sum _ { \\begin{smallmatrix} 1 \\in I \\subseteq [ 1 , n ] \\\\ | I | > 1 \\end{smallmatrix} } ( z _ { I ' } + 1 ) \\cdot \\delta _ { I ' } \\right ) \\end{align*}"}
+{"id": "9329.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\sum _ { i = 1 } ^ m U _ i < \\left ( \\frac { \\pi } { 4 } + \\frac { 1 9 9 \\pi } { 3 2 0 0 k } \\right ) \\| \\mathbf { x } \\| . \\end{align*}"}
+{"id": "2657.png", "formula": "\\begin{align*} d _ { 4 } ( n ) = d _ { e } ( n ) . \\end{align*}"}
+{"id": "1312.png", "formula": "\\begin{align*} I ( u _ 0 ) = m | | u _ 0 | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } + | | \\nabla _ { H } u _ 0 | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } - { \\rm R e } \\int _ { \\mathbb { H } ^ n } \\overline { u } _ 0 f ( u _ 0 ) d x < 0 , \\end{align*}"}
+{"id": "6913.png", "formula": "\\begin{align*} H ( Q _ k \\ , | \\ , P ) = \\frac { 1 } { Q ( B _ k ) } \\int _ { B _ k } \\log \\frac { d Q } { d P } \\ , d Q - \\log Q ( B _ k ) \\end{align*}"}
+{"id": "1122.png", "formula": "\\begin{align*} R _ { \\max } ^ { \\rm { N } } = W { \\log _ 2 } \\left ( { 1 + \\frac { { \\left ( { P - { p _ K } \\left ( { \\overline \\varepsilon , W } \\right ) } \\right ) { { \\left | { { { \\widetilde h } _ b } } \\right | } ^ 2 } } } { { { p _ K } \\left ( { \\overline \\varepsilon , W } \\right ) { { \\left | { { { \\widetilde h } _ b } } \\right | } ^ 2 } + W { N _ 0 } } } } \\right ) . \\end{align*}"}
+{"id": "534.png", "formula": "\\begin{align*} A _ { \\mathcal N } ( a ) : = \\bigoplus _ { \\lambda \\in { \\mathcal N } } A _ \\lambda ( a ) . \\end{align*}"}
+{"id": "970.png", "formula": "\\begin{align*} F ( P , \\mathcal { U } ) : = \\{ ( \\pi , \\tau ) \\in X ( \\Gamma _ d ( q ) ) \\mid P \\in \\pi \\tau \\in \\mathcal { U } \\} . \\end{align*}"}
+{"id": "3963.png", "formula": "\\begin{align*} ( \\mathbf { u } G _ 2 , \\mathbf { u } G _ 3 , \\cdots , \\mathbf { u } G _ { 2 k } ) = ( \\mathbf { v } G _ 2 , \\mathbf { v } G _ 3 , \\cdots , \\mathbf { v } G _ { 2 k } ) = ( 1 , 1 , \\cdots , 1 ) , \\end{align*}"}
+{"id": "9103.png", "formula": "\\begin{align*} ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + r ( j - 1 ) < \\chi _ j < ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + r j , \\end{align*}"}
+{"id": "7042.png", "formula": "\\begin{align*} \\aligned F _ { 0 , 9 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 1 , 8 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 2 , 7 } \\ , : = \\ , 5 0 4 0 , \\ \\ \\ \\ \\ F _ { 3 , 6 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 4 , 5 } & \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 5 , 4 } \\ , : = \\ , 8 4 0 , \\ \\ \\ \\ \\ F _ { 6 , 3 } \\ , : = \\ , 0 , \\\\ F _ { 7 , 2 } & \\ , : = \\ , 4 2 \\ , \\theta , \\ \\ \\ \\ \\ F _ { 8 , 1 } \\ , : = \\ , \\tfrac { 2 4 5 } { 2 } , \\ \\ \\ \\ \\ F _ { 9 , 0 } \\ , : = \\ , 4 \\ , \\theta ^ 2 . \\endaligned \\end{align*}"}
+{"id": "763.png", "formula": "\\begin{align*} B & \\leq \\int _ { | u - t | > 2 ^ { 2 s } \\ell ( Q ) ^ { 2 s } } \\frac { | P _ s * ( \\phi \\mu ) ( x , t ) - P _ s * ( \\phi \\mu ) ( y , t ) | } { | u - t | ^ { 2 - \\frac 1 { 2 s } } } \\ , d u \\\\ & + \\int _ { | u - t | > 2 ^ { 2 s } \\ell ( Q ) ^ { 2 s } } \\frac { | P _ s * ( \\phi \\mu ) ( x , u ) - P _ s * ( \\phi \\mu ) ( y , u ) | } { | u - t | ^ { 2 - \\frac 1 { 2 s } } } \\ , d u = B _ 1 + B _ 2 . \\end{align*}"}
+{"id": "4848.png", "formula": "\\begin{align*} \\bigl ( T \\xRightarrow { Q _ 1 ^ 0 } Q _ 1 \\circ A \\xRightarrow { \\beta * 1 _ A } Q _ 2 \\circ A \\bigr ) = \\bigl ( T \\xRightarrow { Q _ 2 ^ 0 } Q _ 2 \\circ A \\bigr ) \\end{align*}"}
+{"id": "5011.png", "formula": "\\begin{align*} S t _ { D N P } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( z ^ { 2 } , z ) - z ^ { 2 } | | _ { \\infty } = 0 . \\\\ \\end{align*}"}
+{"id": "6918.png", "formula": "\\begin{align*} G _ 1 ( M ( t ) ) & = \\int _ { \\R ^ n } f \\Big ( ( 1 - t ) x _ 1 + t T _ 1 ( x _ 1 ) , \\dots , ( 1 - t ) x _ n + t T _ n ( x _ n ) \\Big ) \\prod _ { i = 1 } ^ { n } Q ^ * _ i ( d x _ i ) , \\end{align*}"}
+{"id": "7375.png", "formula": "\\begin{gather*} \\mathcal { U } = \\bigl \\{ Q \\in \\mathbb { P } : Q \\ll P ^ * ( \\pi _ 1 , \\ldots , \\pi _ n ) Q \\bigr \\} . \\end{gather*}"}
+{"id": "8691.png", "formula": "\\begin{align*} d ( \\bar x , y ) & \\le d ( \\bar x , x ) + d ( x , y ) \\le \\bar K r _ { m ' } + ( 6 \\lambda r _ i + 3 \\lambda r _ m ) \\le \\left ( 6 \\lambda + \\frac 1 { \\bar K } + \\frac { 3 \\lambda } { \\bar K ^ 2 } \\right ) r _ i \\le 7 \\lambda r _ i . \\end{align*}"}
+{"id": "1265.png", "formula": "\\begin{align*} \\int _ \\Omega \\varphi _ n ( \\eta ) \\mu ( d \\eta ) = 0 . \\end{align*}"}
+{"id": "3680.png", "formula": "\\begin{align*} \\omega _ { i j } V _ i V _ j = \\tfrac { 1 } { 2 } h _ { i j } V _ i V _ j . \\end{align*}"}
+{"id": "4863.png", "formula": "\\begin{align*} \\frac { 1 } { e ^ { - \\phi ( x ) } } \\int _ { - \\infty } ^ { T ( x , y ) } e ^ { - \\psi ( x , u ) } d u = \\frac { 1 } { e ^ { - \\phi ( x _ 0 ) } } \\int _ { - \\infty } ^ y e ^ { - \\psi ( x _ 0 , u ) } d u . \\end{align*}"}
+{"id": "6303.png", "formula": "\\begin{align*} \\alpha _ t ^ { - 2 } \\| x ^ { t + 1 } - y ^ t \\| ^ 2 \\overset { \\eqref { l 2 - e 3 } } { = } \\| z ^ { t + 1 } - \\beta _ t y ^ t - ( 1 - \\beta _ t ) z ^ t \\| ^ 2 \\le \\beta _ t \\| z ^ { t + 1 } - y ^ t \\| ^ 2 + ( 1 - \\beta _ t ) \\| z ^ { t + 1 } - z ^ t \\| ^ 2 . \\end{align*}"}
+{"id": "886.png", "formula": "\\begin{align*} r a + 1 = k q \\ , , \\end{align*}"}
+{"id": "8598.png", "formula": "\\begin{align*} | A | ^ { m - 1 } V ( A [ n - m ] , [ 0 , u _ 1 ] , \\dots , [ 0 , u _ m ] ) \\le \\frac { n ^ m ( n - m ) ! } { n ! } \\prod _ { i = 1 } ^ m V ( A [ n - 1 ] , [ 0 , u _ i ] ) . \\end{align*}"}
+{"id": "2954.png", "formula": "\\begin{align*} U \\backslash \\partial A = ( U \\cap A ) \\cup ( U \\backslash \\bar A ) , \\end{align*}"}
+{"id": "8203.png", "formula": "\\begin{align*} x _ { 2 i - 1 , j } + x _ { 2 i - 1 , j + 1 } + x _ { 2 i , j } + x _ { 2 i , j + 1 } & = x _ { 2 i - 1 , j } + x _ { 2 i - 1 , j + 1 } \\\\ & + ( \\tfrac { 1 } { 2 } S - x _ { 2 i , n + 1 - j } ) + ( \\tfrac { 1 } { 2 } S - x _ { 2 i , n - j } ) \\\\ & = a _ j + S - a _ j = S . \\end{align*}"}
+{"id": "5313.png", "formula": "\\begin{align*} g ( z ) = ( 2 \\pi ) ^ { - n } \\ , | \\lambda | ^ n \\ , \\sum _ { k = 0 } ^ \\infty g \\ast _ \\lambda \\varphi _ { k , \\lambda } ^ { n - 1 } ( z ) . \\end{align*}"}
+{"id": "5146.png", "formula": "\\begin{align*} \\begin{aligned} & S _ h \\subset T _ h \\subset \\left \\{ b \\in L ^ { 2 } _ { \\sigma , \\textrm { a x i } } ( \\mathbb { R } ^ { 3 } ) \\ \\middle | \\ H [ b ] = h \\right \\} , \\\\ & I _ h = \\inf \\left \\{ E [ b ] \\ \\middle | \\ b \\in T _ h \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "3883.png", "formula": "\\begin{align*} \\hat { q } _ i = q ( z _ i ) + \\frac { \\hat { q } _ i } { \\ln \\frac { R } { \\varepsilon } } g _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ i ) - \\Sigma _ { j \\neq i } \\frac { \\hat { q } _ j } { \\ln \\frac { R } { \\varepsilon } } \\bar { G } _ { z _ j } ( T _ { z _ j } z _ i , T _ { z _ j } z _ j ) , \\end{align*}"}
+{"id": "1798.png", "formula": "\\begin{align*} L ( f , 3 ) & = \\frac { 4 \\pi ^ { 3 } } { 3 } \\int _ { 0 } ^ { \\infty } b ^ { 2 } ( e ^ { - 2 \\pi u } ) c ( e ^ { - 6 \\pi u } ) u ^ { 2 } d u \\\\ & = \\frac { 4 \\pi ^ { 3 } } { 3 \\sqrt { 3 } } \\int _ { 0 } ^ { \\infty } b ( e ^ { - 2 \\pi u } ) c ( e ^ { - 6 \\pi u } ) \\cdot c ( e ^ { - \\frac { 2 \\pi } { 3 u } } ) u d u . \\end{align*}"}
+{"id": "8874.png", "formula": "\\begin{align*} \\mathbb { B } _ { k , S ; ( n , g ) } ( \\tau , z ) = \\mathbb { B } _ { k , S [ u ] ; ( n , g ) } ( \\tau , ( u ^ t ) ^ { - 1 } z ) , \\end{align*}"}
+{"id": "4735.png", "formula": "\\begin{align*} \\gamma _ 0 J ( x ) + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k f _ k ( x ) \\ge \\gamma _ 0 J ( x ^ * ) \\end{align*}"}
+{"id": "7107.png", "formula": "\\begin{align*} | \\xi - z ^ * | \\ , | z | \\leq | \\xi - z | + | z - z ^ * | \\ , | z | = | z - \\xi | + 1 - | z | ^ 2 \\leq | z - \\xi | + 2 ( 1 - | z | ) \\leq 5 M \\end{align*}"}
+{"id": "230.png", "formula": "\\begin{align*} \\dot g ( j _ 1 ) _ { | i ( g ) } = \\prod _ { k = 1 } ^ { \\kappa ( j _ 1 ) } \\left ( s _ { \\iota ( k ) } \\right ) _ { | \\ , i ( g ) - \\sigma ( k - 1 ) } \\ne \\prod _ { k = 1 } ^ { \\kappa ( j _ 2 ) } \\left ( s _ { \\iota ( k ) } \\right ) _ { | \\ , i ( g ) - \\sigma ( k - 1 ) } = \\dot g ( j _ 2 ) _ { | i ( g ) } , \\end{align*}"}
+{"id": "619.png", "formula": "\\begin{align*} \\begin{aligned} & \\omega _ { S ^ 1 } = ( x - i y ) d x + ( y + i x ) d y \\\\ & = ( \\cos \\theta - i \\sin \\theta ) d ( \\cos \\theta ) + ( \\sin \\theta + i \\cos \\theta ) d ( \\sin \\theta ) \\\\ & = - \\cos \\theta \\sin \\theta d \\theta + \\sin \\theta \\cos \\theta d \\theta + i ( \\cos ^ 2 \\theta d \\theta + \\sin ^ 2 \\theta d \\theta ) = i d \\theta . \\end{aligned} \\end{align*}"}
+{"id": "8949.png", "formula": "\\begin{align*} \\biggl ( \\int _ { a ( t ) } ^ { b ( t ) } v _ 1 ^ { - p ' } \\biggr ) ^ { 1 / p ' } \\biggl ( \\int _ { a ( t ) } ^ { b ( t ) } v _ 0 ^ p \\biggr ) ^ { 1 / p } = 1 , t > 0 . \\end{align*}"}
+{"id": "2912.png", "formula": "\\begin{align*} \\sum _ { x , y = 0 } ^ n ( \\delta _ { x , y } - M _ { x , y } ( m ) ) f _ y ^ \\star f _ x = \\sum _ { j , j ' = 0 } ^ n \\left ( 1 - \\Theta _ m ( \\mu _ j , \\mu _ { j ' } ) \\right ) \\left | \\sum _ { x = 0 } ^ n \\psi _ j ( x ) f _ x \\psi _ { j ' } ( x ) \\right | ^ 2 . \\end{align*}"}
+{"id": "7227.png", "formula": "\\begin{align*} L = a _ 0 x + a _ 1 x ^ q + a _ 2 x ^ { q ^ 2 } + \\cdots + a _ n x ^ { q ^ n } \\in F [ x ] . \\end{align*}"}
+{"id": "9060.png", "formula": "\\begin{align*} = H _ { \\beta _ { m - k + 1 } } v _ { k - 1 } + \\sum _ { l = 1 } ^ { k - 1 } ( - ) ^ { k - l } X _ { \\beta _ { m - k + 1 } } \\cdots X _ { \\beta _ { m - l } } X _ { e _ { m - l + 1 } - e _ { m - k + 1 } } v _ { l - 1 } , \\end{align*}"}
+{"id": "6081.png", "formula": "\\begin{align*} \\frac { p ^ { 2 n + 1 } - p } { p ^ 2 - p } = 1 + p + p ^ 2 + \\cdot + p ^ { 2 n - 1 } . \\end{align*}"}
+{"id": "9251.png", "formula": "\\begin{align*} \\omega = \\eta _ 1 ^ { * } \\widetilde \\omega ^ 0 \\wedge \\eta _ 1 ^ { * } \\widetilde \\omega ^ 1 \\wedge \\cdots \\wedge \\eta _ k ^ { * } \\widetilde \\omega ^ 0 \\wedge \\eta _ k ^ { * } \\widetilde \\omega ^ 1 , \\end{align*}"}
+{"id": "5974.png", "formula": "\\begin{align*} \\sigma ( \\sqrt { a } ) = - \\sqrt { a } , \\ & \\sigma ( \\sqrt { d } ) = \\sqrt { d } \\\\ \\tau ( \\sqrt { a } ) = \\sqrt { a } , \\ & \\tau ( \\sqrt { d } ) = - \\sqrt { d } . \\end{align*}"}
+{"id": "1297.png", "formula": "\\begin{align*} \\frac { a ( x _ a ; x _ c ; \\beta ) } { a ( x _ a ; x _ b ; \\alpha ) a ( x _ a ; x _ d ; { \\beta - \\alpha } ) } = 1 , \\\\ \\frac { c ( x _ a ; x _ c ; \\beta ) } { c ( x _ a ; x _ b ; \\alpha ) a ( x _ a ; x _ d ; { \\beta - \\alpha } ) } = 1 , \\end{align*}"}
+{"id": "5024.png", "formula": "\\begin{align*} h ( T ) & = x y \\log ( x y ) - \\sum _ { k = 1 } ^ { r } \\left ( \\Delta ( i _ k ) + \\Delta ( j _ k ) \\right ) \\\\ & \\le x y \\log ( x y ) - \\sum _ { k = 1 } ^ { r } \\left ( \\Delta ( x ) + \\Delta ( y + 1 - k ) \\right ) \\\\ & = h ( T ' ) , \\end{align*}"}
+{"id": "658.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | e ^ { i k \\phi } \\sin ( n \\theta ) \\left | l _ 2 , m _ 2 \\right \\rangle = \\end{align*}"}
+{"id": "3382.png", "formula": "\\begin{align*} C ( 1 + S ( p _ 1 ; N ) + S ( p _ 2 ; K ) ) = C Z = 1 , Z = Z ( p _ 1 , p _ 2 ; N , K ) . \\end{align*}"}
+{"id": "6995.png", "formula": "\\begin{align*} q _ { i j } ( x ) & = 0 | i - j | > m , \\\\ q _ { i j } ( x ) & \\in ( 0 , M ] 0 < | i - j | \\leq m , \\end{align*}"}
+{"id": "8456.png", "formula": "\\begin{align*} M _ { \\pm } ( x ; z ) = I + \\mathcal { P } ^ { \\pm } \\left ( M _ { - } ( x ; \\cdot ) R ( x ; \\cdot ) \\right ) ( z ) , z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "6569.png", "formula": "\\begin{align*} \\int _ { \\Omega } n \\xi ( \\cdot , t _ { 2 } ) - \\int _ { t _ { 1 } } ^ { t _ { 2 } } \\int _ { \\Omega } n \\xi _ { t } + \\int _ { t _ { 1 } } ^ { t _ { 2 } } \\int _ { \\Omega } ( \\nabla n - n S ( x , n , c ) \\cdot \\nabla c ) \\cdot \\nabla \\xi = \\int _ { \\Omega } n \\xi ( \\cdot , t _ { 1 } ) , \\end{align*}"}
+{"id": "7530.png", "formula": "\\begin{align*} \\ell \\binom { k } { 2 } m ^ { k - 2 } < s \\binom { k } { 2 } m ^ { k - 2 } \\leq \\frac { d m ^ { k - 1 } } { 4 } \\leq \\frac { m ^ k } { 4 } \\end{align*}"}
+{"id": "453.png", "formula": "\\begin{align*} g _ - ( x ) = \\sum _ { k = 1 } ^ \\infty { \\bf 1 } _ { \\{ x \\in ( - 2 b ^ { n _ k } \\delta , \\ , - b ^ { n _ k } \\delta ] \\} } b ^ { - n _ k } \\delta ^ { - 1 } n ^ { - 1 / 2 } _ k . \\end{align*}"}
+{"id": "1781.png", "formula": "\\begin{align*} T ( z , t ) = \\left ( \\begin{matrix} 1 & - \\bar z & \\frac { - | z | ^ 2 + i t } { 2 } \\\\ 0 & 1 & z \\\\ 0 & 0 & 1 \\end{matrix} \\right ) \\end{align*}"}
+{"id": "5792.png", "formula": "\\begin{align*} \\nu _ z = - \\frac { 1 } { 2 } h _ z - ( \\alpha - I \\beta ) h _ { \\overline { z } } . \\end{align*}"}
+{"id": "8623.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\lim _ { N \\to \\infty } \\sum _ { m = 1 } ^ { n } \\sum _ { i = N } ^ { \\infty } \\mathbb { E } _ { n , m } [ \\langle \\chi _ { n , m } , e _ i \\rangle ^ 2 ] = 0 \\end{align*}"}
+{"id": "8988.png", "formula": "\\begin{align*} \\hat { k } _ d ( \\Lambda ) = \\prod _ { \\sigma \\in \\Lambda _ d } \\prod _ { v _ { q , i } \\in \\sigma } x _ { q , i } = \\prod _ { q , i } x _ { q , i } ^ { f ( q , i ) } = \\prod _ { q , i } x _ { q , i } ^ { e ( q , i ) } \\prod _ { \\rho \\in \\Gamma _ { d - 1 } } x _ { m ( \\rho ) , 1 } . \\end{align*}"}
+{"id": "3055.png", "formula": "\\begin{align*} ( x _ 0 f _ { x _ 0 } ( x ) : \\cdots : x _ n f _ { x _ n } ( x ) ) = ( \\lambda _ 0 : \\cdots : \\lambda _ n ) . \\end{align*}"}
+{"id": "432.png", "formula": "\\begin{align*} \\delta _ { r j } = \\sum _ { h = 1 } ^ { n } p _ { r j h } \\theta ^ { h - 1 } , \\end{align*}"}
+{"id": "7762.png", "formula": "\\begin{align*} x \\Longleftrightarrow y = \\frac { x - c } { \\rho } , \\end{align*}"}
+{"id": "1722.png", "formula": "\\begin{align*} | | y ( t ) - y ( s ) | | _ { L ^ 2 } = & \\left | \\left | \\int _ s ^ t ( \\partial _ t y ) ( \\tau ) d \\tau \\right | \\right | _ { L ^ 2 } \\le \\left | \\int _ s ^ t | | ( \\partial _ t y ) ( \\tau ) | | _ { L ^ 2 } d \\tau \\right | \\\\ \\le & | t - s | | | \\partial _ t y | | _ { L ^ { \\infty } ( 0 , \\tau _ 1 ; L ^ 2 ) } \\le M _ 1 | t - s | , \\end{align*}"}
+{"id": "6282.png", "formula": "\\begin{align*} \\partial _ v \\Gamma _ { u v } ^ v - \\Gamma _ { v u } ^ u \\Gamma _ { u v } ^ v + g _ { u v } = 0 . \\end{align*}"}
+{"id": "802.png", "formula": "\\begin{align*} m = \\int _ { \\Gamma ( t ) } c ( t ) \\ , \\mathrm { d } \\mathcal { H } ^ d = c ( t ) \\mathcal { H } ^ d \\big ( \\Gamma ( t ) \\big ) = \\alpha _ d c ( t ) R ( t ) ^ d \\phantom { x x } \\Leftrightarrow \\phantom { x x } c ( t ) = \\frac { m } { \\alpha _ d } R ( t ) ^ { - d } \\end{align*}"}
+{"id": "5060.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } | \\Phi _ { + } | ^ { 2 } \\dd x = \\int _ { \\mathbb { R } ^ { 3 } } | \\Phi _ { 0 , + } | ^ { 2 } \\dd x . \\end{align*}"}
+{"id": "4975.png", "formula": "\\begin{align*} \\begin{aligned} \\xi _ 1 ^ { \\mathsf { L } ^ { k } } ( u ) = \\xi _ 1 ^ { \\mathsf { L } ^ { k + 1 } } ( u ) = & \\frac { \\xi _ 0 f _ 1 ( u ) } { \\xi _ 0 f _ 1 ( u ) + ( 1 - \\xi _ 0 ) f _ 2 ( u ) } . \\end{aligned} \\end{align*}"}
+{"id": "4209.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { c _ \\varepsilon } { \\ln \\frac { 1 } { \\varepsilon } } = \\pi \\min \\limits _ { \\overline { \\Omega } } q ^ 2 \\sqrt { d e t ( K _ H ) } . \\end{align*}"}
+{"id": "9226.png", "formula": "\\begin{align*} \\begin{aligned} & \\| x ( t ) - z ( t ) \\| \\le \\| x ( 0 ) - z ( 0 ) \\| + \\gamma ^ 2 3 L _ \\mathcal { D } M _ \\mathcal { D } T t \\\\ & + \\gamma L _ \\mathcal { D } \\int _ 0 ^ t ( \\| x ( 0 ) - z ( 0 ) \\| + \\gamma ^ 2 3 L _ \\mathcal { D } M _ \\mathcal { D } T \\tau ) \\\\ & \\cdot e ^ { \\gamma L _ \\mathcal { D } ( t - \\tau ) } \\ , d \\tau \\\\ & = \\| x ( 0 ) - z ( 0 ) \\| e ^ { \\gamma L _ \\mathcal { D } t } + \\gamma 3 M _ \\mathcal { D } T \\left ( e ^ { \\gamma L _ \\mathcal { D } t } - 1 \\right ) \\end{aligned} \\end{align*}"}
+{"id": "2768.png", "formula": "\\begin{align*} \\frac { 1 } { 6 4 } \\left ( \\frac { \\sqrt { 2 } t } { 5 } \\right ) ^ { - 2 \\beta } \\frac { t ^ 2 } { 3 1 ^ 2 } = \\frac { 2 ^ { - \\beta } } { 6 4 \\cdot 5 ^ { - 2 \\beta } \\cdot 3 1 ^ 2 } \\cdot t ^ { 2 - 2 \\beta } , \\end{align*}"}
+{"id": "5470.png", "formula": "\\begin{align*} \\Lambda _ C : = \\{ ( x , y ) \\in \\Lambda : x \\in C \\} \\end{align*}"}
+{"id": "3394.png", "formula": "\\begin{align*} 3 \\left ( { k \\choose 2 } + { l + 1 \\choose 2 } + { k + 1 \\choose 2 } + { l \\choose 2 } \\right ) = 3 ( k ^ 2 + l ^ 2 ) \\geq ( k + l ) ^ 2 . \\end{align*}"}
+{"id": "9144.png", "formula": "\\begin{align*} \\chi ( s ) : = \\beta ^ { - 1 } ( s , 0 ) . \\end{align*}"}
+{"id": "9032.png", "formula": "\\begin{align*} [ F , G ] ( x ) = \\sum _ { { \\bf k , n } \\in \\Gamma _ { r } } [ \\widehat { F } _ { \\bf k } , \\widehat { G } _ { \\bf n } ] { \\rm e } ^ { { \\rm i } \\langle { \\bf k + n } , x \\rangle } . \\end{align*}"}
+{"id": "2598.png", "formula": "\\begin{align*} \\tilde { z } = z + \\varepsilon ( x \\wedge _ g y ) z + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
+{"id": "4253.png", "formula": "\\begin{align*} ( I - N _ j ( \\varphi ) + P ) f _ l & = ( I - M _ j ( \\varphi ) + P ) f _ l \\\\ & = \\dfrac { 1 } { n } \\sum _ { i = 1 } ^ j ( I - C _ { \\varphi _ i } ) f _ l + P f _ l \\\\ & = \\dfrac { 1 } { n } \\sum _ { i = 1 } ^ j ( I + C _ \\varphi + . . . + C _ { \\varphi _ { i - 1 } } ) ( I - C _ { \\varphi } ) f _ l + P f _ l \\\\ & = \\dfrac { 1 } { n } \\sum _ { i = 1 } ^ j ( I + T _ 1 + . . . + T _ { i - 1 } ) ( I - T _ 1 ) f _ l + P f _ l \\rightarrow 0 , \\\\ \\end{align*}"}
+{"id": "2591.png", "formula": "\\begin{align*} T _ 1 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , v , J v ) = 0 \\textrm { f o r e v e r y } x _ 1 , x _ 2 , x _ 3 , x _ 4 , v \\in V . \\end{align*}"}
+{"id": "8509.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ x ^ 2 \\left ( e ^ { i c _ + ( x ) } u _ x ( x ) \\right ) & = I _ 3 ' ( x ) + I _ 4 ' ( x ) , \\end{aligned} \\end{align*}"}
+{"id": "8677.png", "formula": "\\begin{align*} \\begin{aligned} R & = \\frac { { { n _ c } - { n _ { \\rm { O F D M } } } { N _ { \\rm { C P } } } } } { { { n _ c } } } \\frac { 1 } { { K } } \\sum \\limits _ { k = 0 } ^ { K - 1 } { { { \\log } _ 2 } \\left ( { 1 + { \\gamma } _ k } \\right ) } \\\\ & = \\frac { 1 } { { K + { N _ { { \\rm { C P } } } } } } \\sum \\limits _ { k = 0 } ^ { K - 1 } { { { \\log } _ 2 } \\left ( { 1 + { \\gamma _ k } } \\right ) } . \\end{aligned} \\end{align*}"}
+{"id": "836.png", "formula": "\\begin{align*} \\Gamma _ 0 ^ \\varepsilon = \\big \\{ \\theta _ { \\rho ^ \\varepsilon } ( 0 , z ) \\ , \\big | \\ , z \\in M \\big \\} \\end{align*}"}
+{"id": "635.png", "formula": "\\begin{align*} { \\bf e } = \\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix} , \\ \\ \\ \\ { \\bf f } = \\begin{bmatrix} 0 & 0 \\\\ 1 & 0 \\end{bmatrix} , \\ \\ \\ \\ { \\bf h } = \\begin{bmatrix} 1 & 0 \\\\ 0 & - 1 \\end{bmatrix} . \\end{align*}"}
+{"id": "9263.png", "formula": "\\begin{align*} \\int _ \\Omega u \\Delta v _ 1 \\wedge \\dots \\wedge \\Delta v _ { m - 1 } \\wedge \\beta _ n ^ { n - m } \\wedge \\Delta \\omega = \\int _ \\Omega \\omega \\Delta u \\wedge \\Delta v _ 1 \\wedge \\dots \\wedge \\Delta v _ { m - 1 } \\wedge \\beta _ n ^ { n - m } \\end{align*}"}
+{"id": "9092.png", "formula": "\\begin{align*} \\chi _ j = \\chi - \\sum _ { \\substack { i = 1 , \\\\ i \\neq j } } ^ { n } \\chi _ i + ( n - 1 ) r . \\end{align*}"}
+{"id": "6490.png", "formula": "\\begin{align*} | \\{ j \\le 2 : j \\not = i , m _ j > 0 \\} | = \\begin{cases} 0 \\\\ 1 \\end{cases} \\end{align*}"}
+{"id": "8875.png", "formula": "\\begin{align*} \\lim \\nolimits _ { k \\to \\infty } C ( l ' , r ' ; P _ { k , S } ^ { l , r } ) = \\delta _ S ( l , r ; \\ , l ' , r ' ) , \\end{align*}"}
+{"id": "3229.png", "formula": "\\begin{align*} \\int _ U ^ T \\Sigma _ s ^ T d s : = \\int _ U ^ T \\mathcal S ( T - s ) \\Sigma _ s \\mathcal S ( T - s ) ^ * d s . \\end{align*}"}
+{"id": "6590.png", "formula": "\\begin{align*} W ( r ) : = M _ { 0 } r ^ { d } . \\end{align*}"}
+{"id": "1657.png", "formula": "\\begin{align*} T ( F r o b _ v ) = T _ v \\end{align*}"}
+{"id": "3428.png", "formula": "\\begin{align*} \\beta _ n : = \\frac { \\ln q _ { n + 1 } } { q _ n } , \\end{align*}"}
+{"id": "7799.png", "formula": "\\begin{align*} f \\cdot g : = \\int _ \\S f ( x ) g ( x ) \\ \\d x + \\int _ \\S D f ( x ) D g ( x ) \\ \\d x \\end{align*}"}
+{"id": "1596.png", "formula": "\\begin{align*} \\mathbf { \\Phi } _ { \\mathrm { r } } ^ { H } \\mathbf { \\Phi } _ { \\mathrm { r } } + \\mathbf { \\Phi } _ { \\mathrm { t } } ^ { H } \\mathbf { \\Phi } _ { \\mathrm { t } } = \\mathbf { I } _ { M } . \\end{align*}"}
+{"id": "3244.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ] } t ^ { - \\frac r 2 } \\| ( \\mathcal S ( t ) - I ) Q ^ { \\frac 1 2 } \\| _ { L _ { } ( U , H ) } = & \\sup _ { t \\in [ 0 , T ] } t ^ { - \\frac r 2 } \\| A ^ { - \\frac r 2 } ( \\mathcal S ( t ) - I ) A ^ { \\frac r 2 } Q ^ { \\frac 1 2 } \\| _ { L _ { } ( U , H ) } \\\\ \\leq & C \\| A ^ { \\frac r 2 } Q ^ { \\frac 1 2 } \\| _ { L _ { } ( U , H ) } \\\\ = & C \\| Q ^ { \\frac r 2 } \\| _ { L _ { } ( U , \\dot H ^ r ) } < \\infty . \\end{align*}"}
+{"id": "1882.png", "formula": "\\begin{align*} \\textstyle \\Pi _ \\lambda f = \\sum _ { \\alpha : \\ , d + 2 | \\alpha | = \\lambda } \\ , \\langle f , \\Phi _ \\alpha \\rangle \\Phi _ \\alpha . \\end{align*}"}
+{"id": "2421.png", "formula": "\\begin{align*} L _ { q } [ L _ { q } ^ { - 1 } [ F _ { q } ( s ) ] ] = \\frac { 1 } { 2 \\pi i } \\ ; \\int _ { c - i \\infty } ^ { c + i \\infty } d s ^ { \\prime } \\frac { F _ { q } ( s ^ { \\prime } ) } { ( s - s ^ { \\prime } ) } . \\end{align*}"}
+{"id": "1020.png", "formula": "\\begin{align*} \\chi _ K ( \\alpha ) + \\chi _ K ( - \\alpha ) = 1 - \\Phi _ K ^ + ( \\alpha ) - \\Phi _ K ^ + ( - \\alpha ) = \\begin{cases} 1 , & \\alpha \\in \\Phi \\setminus \\Phi _ K , \\\\ 0 , & \\alpha \\in \\Phi _ K . \\end{cases} \\end{align*}"}
+{"id": "3041.png", "formula": "\\begin{align*} \\left ( d b \\right ) _ { p } \\left ( y \\right ) = \\left ( d \\mathcal { L } _ { g } \\right ) _ { \\iota } \\left ( d \\bar { b } _ { o } \\left ( x \\right ) \\right ) y ^ { b } = \\left ( d g \\right ) _ { o } \\left ( x ^ { \\bar { b } } \\right ) \\end{align*}"}
+{"id": "4559.png", "formula": "\\begin{align*} & \\sqrt { q } \\sigma _ 1 ( \\textbf { z } ) a _ { \\ell + 1 , 0 , 0 } = ( q ^ 3 + q ^ 2 + q ) a _ { \\ell + 1 , 1 , 0 } + a _ { \\ell + 2 , 0 , 0 } \\\\ & { q } \\sigma _ 2 ( \\textbf { z } ) a _ { \\ell , 0 , 0 } = ( q ^ 5 + q ^ 4 + q ^ 3 ) a _ { \\ell , 1 , 1 } + ( q ^ 3 + q ^ 2 + q ) a _ { \\ell + 1 , 1 , 0 } \\\\ & q ^ { 3 / 2 } \\sigma _ 3 ( \\textbf { z } ) a _ { \\ell - 1 , 0 , 0 } = q ^ 2 a _ { \\ell - 2 , 0 , 0 } + ( q ^ 5 + q ^ 4 + q ^ 3 ) a _ { \\ell , 1 , 1 } . \\end{align*}"}
+{"id": "4609.png", "formula": "\\begin{align*} \\frac { { x _ 1 } ^ 2 } { g } ( x _ { k } { x _ { k + 1 } } ^ 2 + { x _ { k + 1 } } ^ 2 x _ { k + 2 } ) = \\begin{cases} x _ 1 & ( k = 1 ) \\\\ 0 & ( k : ) , \\end{cases} \\end{align*}"}
+{"id": "7046.png", "formula": "\\begin{align*} 0 \\ , = \\ , G _ { 0 , 5 } \\ , = \\ , G _ { 1 , 4 } , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ G _ { 2 , 3 } \\ , = \\ , 6 , \\ \\ \\ \\ \\ \\ \\ G _ { 3 , 2 } \\ , = \\ , 6 , \\end{align*}"}
+{"id": "8454.png", "formula": "\\begin{align*} r ( k ) = \\begin{cases} - 2 i k r _ 1 ( z ) & | k | \\leq 1 \\\\ ( 2 i k ) ^ { - 1 } r _ 2 ( z ) & | k | \\geq 1 . \\end{cases} \\end{align*}"}
+{"id": "8335.png", "formula": "\\begin{align*} u _ x ( x , t ) = 2 i m ( x , t ) e ^ { 4 i \\int _ { - \\infty } ^ x | m ( s , t ) | ^ 2 d s } , \\end{align*}"}
+{"id": "23.png", "formula": "\\begin{align*} P ( \\lambda ) = \\left \\{ ( x _ \\alpha ) \\in \\mathbb { R } ^ { \\sharp R ^ + } _ { \\geq 0 } \\mid \\forall \\ ; \\mathbf { p } , \\sum _ { \\alpha _ \\in \\mathbf { p } } x _ \\alpha \\leq m _ { e ( p ) } - m _ { s ( p ) } \\right \\} \\end{align*}"}
+{"id": "4569.png", "formula": "\\begin{align*} a _ { \\ell , m , 0 } = A _ { 1 , m , 0 } ( \\sqrt { q } z _ 1 ) ^ \\ell + A _ { 2 , m , 0 } ( \\sqrt { q } z _ 2 ) ^ \\ell + A _ { 3 , m , 0 } ( \\sqrt { q } z _ 3 ) ^ \\ell + A _ { 4 , m , 0 } ( \\sqrt { q } z _ 4 ) ^ \\ell . \\end{align*}"}
+{"id": "511.png", "formula": "\\begin{align*} \\Phi _ { \\nu } ( x ) = \\phi _ { \\nu _ { 1 } } ( x _ { 1 } ) \\phi _ { \\nu _ { 2 } } ( x _ { 2 } ) . . . \\phi _ { \\nu _ { d } } ( x _ { d } ) , \\ \\ x = ( x _ { 1 } , . . . , x _ { d } ) . \\end{align*}"}
+{"id": "5118.png", "formula": "\\begin{align*} \\int _ { \\{ | r - r ' | \\geq r / 2 \\} } \\frac { ( r r ' ) ^ { 2 \\tau + 1 } } { | r - r ' | ^ { 4 \\tau + 2 - 2 / q } r ' } \\dd r ' = C r ^ { 2 / q } . \\end{align*}"}
+{"id": "2651.png", "formula": "\\begin{align*} ( q ; q ) _ { \\infty } \\sum \\limits _ { n = 0 } ^ { \\infty } d _ { e } ( n ) q ^ { n } & = \\left ( 1 + \\sum \\limits _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n } q ^ { \\frac { n ( 3 n \\pm 1 ) } { 2 } } \\right ) \\sum \\limits _ { n = 0 } ^ { \\infty } d _ { e } ( n ) q ^ { n } \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { k = 0 } ^ { n } c _ { k } d _ { e } ( n - k ) q ^ { n } \\end{align*}"}
+{"id": "9191.png", "formula": "\\begin{align*} \\begin{aligned} | x ( t ) | _ { \\mathcal { A } _ \\delta } & \\le | x _ 1 ( t ) | _ { \\mathcal { A } _ \\delta } + | x ( t ) - x _ 1 ( t ) | \\\\ & \\le \\kappa \\bar c _ 0 e ^ { - \\gamma \\lambda ( t - \\bar { t } / \\gamma ) } + \\bar c \\\\ & \\le \\kappa \\bar c _ 0 + \\bar c \\leq d & & \\forall \\ , t \\in I _ 1 \\end{aligned} \\end{align*}"}
+{"id": "3921.png", "formula": "\\begin{align*} \\phi ( u _ 0 e _ 1 u _ 1 ) = - \\tau ( u _ 0 , e _ 1 ) \\phi ( e _ 1 ) . \\end{align*}"}
+{"id": "8356.png", "formula": "\\begin{align*} N _ + ( x ; k ) = N _ - ( x ; k ) V ( x ; k ) , \\end{align*}"}
+{"id": "3606.png", "formula": "\\begin{align*} \\mu ( A _ { h , j } ) \\geq c h ^ d , j = 1 , \\ldots , N _ n , \\end{align*}"}
+{"id": "4446.png", "formula": "\\begin{align*} x _ i : = ( - 1 ) ^ { \\ell _ i - 1 } \\sum _ { j = 1 } ^ K y _ { i , j } . \\end{align*}"}
+{"id": "1516.png", "formula": "\\begin{align*} \\varDelta : = \\sqrt { \\sum _ { i = 1 } ^ n \\| F _ i ( x ) - \\tilde { F } _ i ( x ) \\| ^ 2 } \\end{align*}"}
+{"id": "6924.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( T _ i ( x ^ \\pm _ n ) - x ^ \\pm _ n ) Q _ i ( x ^ \\pm _ n ) = 0 . \\end{align*}"}
+{"id": "7770.png", "formula": "\\begin{align*} 1 + \\gamma _ { w , k , m } = \\frac { 1 + \\Delta _ { w , k + 1 , m } } { 1 + \\Delta _ { w , k , m } } , ~ | \\Delta _ { w , i , m } | \\le \\frac { w - m } { m } \\Big | \\frac { z _ { w } } { z _ { w - m } } \\Big | ^ i , ~ i = k , k + 1 , \\end{align*}"}
+{"id": "5426.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\varphi } { \\partial x _ \\beta ^ 2 } ( x ' , \\rho ( x ' ) ) \\geq b _ \\beta - C \\delta \\tilde { b } _ \\beta , \\ \\beta = 1 , \\ldots , n - 1 \\end{align*}"}
+{"id": "5640.png", "formula": "\\begin{align*} C _ { * } ( n ) : = ( C _ { * } ( R ^ { 2 n } ) , d ) = \\cdots \\rightarrow C _ { 2 } ( R ^ { 2 n } ) \\rightarrow C _ { 1 } ( R ^ { 2 n } ) \\xrightarrow { \\varepsilon } \\mathbb { Z } \\rightarrow 0 \\end{align*}"}
+{"id": "3534.png", "formula": "\\begin{align*} A _ k \\ = \\ \\sum _ { j = 0 } ^ { k - 1 } \\binom { k } { j } B _ j . \\end{align*}"}
+{"id": "8658.png", "formula": "\\begin{align*} y \\left [ i \\right ] = \\sum \\limits _ { t = 0 } ^ { { n ' _ { { \\rm { s p a n } } } } } { \\left ( { \\sum \\limits _ { l ' = 1 } ^ { L ' } { { \\bf { g } } _ { l ' } ^ H \\left [ t \\right ] { { \\bf { \\bar H } } _ { l ' } ^ \\bot { \\bf { \\bar X } } _ { l ' } } } } \\right ) { { \\bf { d } } \\left [ { i - t } \\right ] } } + z \\left [ { i } \\right ] . \\end{align*}"}
+{"id": "5035.png", "formula": "\\begin{align*} \\left | \\dfrac { - \\gamma \\omega } { - \\gamma \\alpha \\omega } \\right | & = \\left | 1 + \\dfrac { \\gamma \\alpha \\omega - \\gamma \\omega } { - \\gamma \\alpha \\omega } \\right | \\leq \\max \\left ( 1 , \\left | \\dfrac { \\gamma \\alpha \\omega - \\gamma \\omega } { - \\gamma \\alpha \\omega } \\right | \\right ) , \\end{align*}"}
+{"id": "7521.png", "formula": "\\begin{align*} r ^ { n - 1 } | u ( r \\sigma ) | ^ p & \\leq 2 ^ { p - 1 } r ^ { n - 1 } | u ( \\sigma ) | ^ p + 2 ^ { p - 1 } r ^ { n - 1 } \\int _ { r } ^ { 1 } | \\nabla u ( t \\sigma ) | ^ p \\d t \\\\ & \\leq 2 ^ { p - 1 } | u ( \\sigma ) | ^ p + 2 ^ { p - 1 } \\int _ { 0 } ^ { 1 } t ^ { n - 1 } | \\nabla u ( t \\sigma ) | ^ p \\d t . \\end{align*}"}
+{"id": "3255.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sup _ { n \\in \\mathbb N } \\sum _ { k \\geq N } \\mathbb E \\left [ \\langle Y _ n , e _ k \\rangle ^ 2 \\right ] = 0 , \\end{align*}"}
+{"id": "5376.png", "formula": "\\begin{align*} x _ n = \\rho ( x ' ) = \\frac { 1 } { 2 } \\sum _ { \\alpha , \\beta < n } B _ { \\alpha \\beta } x _ \\alpha x _ \\beta + O ( | x ' | ^ 3 ) , \\end{align*}"}
+{"id": "8714.png", "formula": "\\begin{align*} \\varphi ( [ m , n ] ) = \\int _ { \\widehat { N } } \\chi ( [ m , n ] ) \\ , \\mathrm { d } \\mu _ \\varphi ( \\chi ) = \\int _ { \\widehat { N } } \\chi ^ m ( n ) \\chi ^ { - 1 } ( n ) \\ , \\mathrm { d } \\mu _ \\varphi ( \\chi ) = 1 . \\end{align*}"}
+{"id": "2457.png", "formula": "\\begin{align*} f ( t ) = \\exp \\left ( \\pm \\sigma _ { \\epsilon } \\ ; t \\right ) = { } _ { 0 } F _ { 0 } \\left ( \\ ; ; \\ ; ; \\pm \\sigma _ { \\epsilon } \\ ; t \\right ) . \\end{align*}"}
+{"id": "4014.png", "formula": "\\begin{align*} E ^ L _ Y ( u ) \\supset E ^ L _ Y ( s \\sigma ( v ' ) w ) = \\varphi _ { \\sigma , s } ^ L ( E ^ L _ X ( v ' ) ) = \\varphi _ { \\sigma , s } ^ L ( E ^ L _ X ( v ) ) . \\end{align*}"}
+{"id": "2366.png", "formula": "\\begin{align*} C _ p ( C _ { \\ast } ) = B _ { p } ( C _ { \\ast } ) \\oplus \\ell _ p ( H _ p ( C _ { \\ast } ) ) \\oplus s _ p ( B _ { p - 1 } ( C _ { \\ast } ) ) . \\end{align*}"}
+{"id": "8270.png", "formula": "\\begin{align*} \\int _ { t _ 1 } ^ { t _ 2 } \\sum _ { i = m } ^ n \\sum _ { j = 1 } ^ i ( j g _ { i + 1 } - j g _ i - g _ j ) V _ { i , j } \\psi _ i ( s ) \\psi _ j ( s ) d s \\to \\int _ { t _ 1 } ^ { t _ 2 } \\sum _ { i = m } ^ { \\infty } \\sum _ { j = 1 } ^ i ( j g _ { i + 1 } - j g _ i - g _ j ) V _ { i , j } \\psi _ i ( s ) \\psi _ j ( s ) d s \\end{align*}"}
+{"id": "5444.png", "formula": "\\begin{align*} [ L _ \\lambda L ] = ( \\partial + 2 \\lambda ) L , [ L _ \\lambda H ] = ( \\partial + \\lambda ) H , [ H _ \\lambda L ] = \\lambda H , [ H _ \\lambda H ] = 0 . \\end{align*}"}
+{"id": "6734.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } s } \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) & = \\frac { 1 } { 2 \\Gamma ( 1 / s ) } \\begin{cases} s ^ 2 \\tilde \\psi \\left ( s , ( \\frac { \\mu - x } { \\sigma } ) ^ s \\right ) + \\psi ( 1 / s ) \\Gamma ( 1 / s , ( \\frac { \\mu - x } { \\sigma } ) ^ s ) , & x \\leq \\mu , \\\\ - [ s ^ 2 \\tilde \\psi \\left ( s , ( \\frac { x - \\mu } { \\sigma } ) ^ s \\right ) + \\psi ( 1 / s ) \\Gamma ( 1 / s , ( \\frac { x - \\mu } { \\sigma } ) ^ s ) ] , & x > \\mu . \\end{cases} \\end{align*}"}
+{"id": "7695.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\bar { x } ^ i _ t = & ~ \\Big [ ( A - B ^ 2 R ^ { - 1 } P _ t ) \\bar { x } ^ i _ t - B ^ 2 R ^ { - 1 } \\Phi ( t , \\bar { \\nu } _ t ^ i ) - B h \\Big ( k \\big ( t , \\bar { \\nu } _ t ^ i , \\Phi ( t , \\bar { \\nu } _ t ^ i ) \\big ) \\Big ) \\Big ] d t \\\\ & + \\sigma d W _ t ^ i + \\sigma _ 0 d W _ t ^ 0 , \\\\ \\bar { x } ^ i _ { t _ 0 } = & ~ \\xi ^ i , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7015.png", "formula": "\\begin{align*} \\aligned L & \\ , = \\ , \\ \\ \\big ( T _ 1 + A _ { 1 , 1 } \\ , x + A _ { 1 , 2 } \\ , y + B _ 1 \\ , u \\big ) \\ , \\frac { \\partial } { \\partial x } \\\\ & \\ \\ \\ \\ \\ + \\big ( T _ 2 + A _ { 2 , 1 } \\ , x + A _ { 2 , 2 } \\ , y + B _ 2 \\ , u \\big ) \\ , \\frac { \\partial } { \\partial y } \\\\ & \\ \\ \\ \\ \\ + \\big ( T _ 0 + C _ 1 \\ , x + C _ 2 \\ , y + D \\ , u \\big ) \\ , \\frac { \\partial } { \\partial u } . \\endaligned \\end{align*}"}
+{"id": "5102.png", "formula": "\\begin{align*} - L \\eta = G \\textrm { i n } \\ \\mathbb { R } ^ { 2 } _ { + } , \\eta = 0 \\textrm { o n } \\ \\partial \\mathbb { R } ^ { 2 } _ { + } , \\end{align*}"}
+{"id": "5398.png", "formula": "\\begin{align*} F ^ { i j } v _ { i j } \\geq \\delta _ 1 \\left ( \\frac { 1 } { b _ { n - 1 } ^ { 1 / k } } + \\sum _ { i = 1 } ^ n F ^ { i i } \\right ) \\mbox { i n } \\omega . \\end{align*}"}
+{"id": "933.png", "formula": "\\begin{align*} u _ 2 ( t ) = U ( t - t _ 0 ) u _ 2 ( t _ 0 ) - i \\lambda _ 6 \\int ^ t _ { t _ 0 } U ( t - \\tau ) \\mathcal { N } _ 2 ( u _ 1 ( \\tau ) , u _ 2 ( \\tau ) ) \\ d \\tau \\end{align*}"}
+{"id": "6595.png", "formula": "\\begin{align*} \\min _ { r \\in [ 0 , R ] } c ( r , t ) = c ( 0 , t ) . \\end{align*}"}
+{"id": "1842.png", "formula": "\\begin{align*} \\begin{aligned} \\Rightarrow & B _ 0 r ^ q \\big ( g - d r \\otimes d r \\big ) \\leq H e s s ( r ) \\leq ( \\sqrt { A } \\coth \\sqrt { A } ) r ^ q \\big ( g - d r \\otimes d r \\big ) \\ , , \\operatorname { f o r } \\ , r \\geq 1 \\ , , \\end{aligned} \\end{align*}"}
+{"id": "7579.png", "formula": "\\begin{align*} A = \\{ n \\in \\N : T ^ n a \\in E \\} . \\end{align*}"}
+{"id": "323.png", "formula": "\\begin{align*} | q _ { t , \\gamma } ( x , y ) | \\leq | q _ t ( x , y ) | \\leq \\frac { C _ N } { t ^ { n / 2 + 1 } } e ^ { - \\frac { c | x - y | ^ 2 } { t } } \\Big ( 1 + \\frac { \\sqrt { t } } { \\rho ( x ) } + \\frac { \\sqrt { t } } { \\rho ( y ) } \\Big ) ^ { - N } \\end{align*}"}
+{"id": "6284.png", "formula": "\\begin{align*} \\ker ( D _ { \\varphi } ) = \\bigg \\{ \\sum _ j \\varphi _ j e _ j \\bigg \\} . \\end{align*}"}
+{"id": "1145.png", "formula": "\\begin{align*} { S } ^ { \\rm P R } _ { i , \\ , j , \\ , \\ell } ( { \\cal A } ) : = \\kappa ( \\rho ) \\ , { y _ { N ( \\ell - 1 ) + i } \\ , x _ { N ( \\ell - 1 ) + j } } , \\end{align*}"}
+{"id": "1610.png", "formula": "\\begin{align*} \\min _ { \\overline { \\mathbf { \\Phi } } } ~ ~ & \\mathsf { T r } ( \\overline { \\mathbf { \\Phi } } \\mathbf { Y } \\overline { \\mathbf { \\Phi } } ^ H \\mathbf { Z } ) - 2 \\Re \\{ \\mathsf { T r } ( \\overline { \\mathbf { \\Phi } } \\mathbf { X } ) \\} \\\\ \\mathrm { s . t . } ~ ~ & \\overline { \\mathbf { \\Phi } } ^ H \\overline { \\mathbf { \\Phi } } = \\mathbf { I } _ M , \\end{align*}"}
+{"id": "2270.png", "formula": "\\begin{align*} a ( Z ) = a \\big ( ( Z ^ * Z ) ^ \\frac { 1 } { 2 } \\big ) \\end{align*}"}
+{"id": "302.png", "formula": "\\begin{align*} G _ { \\tilde u } = T G _ u { } ^ t T \\end{align*}"}
+{"id": "331.png", "formula": "\\begin{align*} \\norm { f } ^ 2 - \\norm { ( 1 - H ) f } ^ 2 & \\ = \\sum _ { i , j = 0 } ^ n a _ i \\overline { a _ j } \\left ( \\frac { 1 } { i + j + 1 } - \\frac { i j } { ( i + 1 ) ( j + 1 ) ( i + j + 1 ) } \\right ) \\\\ & \\ = \\sum _ { i , j = 0 } ^ n a _ i \\overline { a _ j } \\left ( \\frac { 1 } { ( i + 1 ) ( j + 1 ) } \\right ) \\\\ & \\ = | \\int _ 0 ^ 1 f ( x ) d x | ^ 2 . \\end{align*}"}
+{"id": "2280.png", "formula": "\\begin{align*} a \\circ \\widehat { \\varphi } ( [ U , V ] , x ) = a ( U D ( x ) V ) = a ( D ( x ) V ) = a \\circ \\widehat { \\varphi } ( [ I _ n , V ] , x ) , \\end{align*}"}
+{"id": "3556.png", "formula": "\\begin{align*} b ^ \\Lambda _ \\lambda = \\eta ( \\tau ) ^ \\ell c ^ \\Lambda _ \\lambda \\end{align*}"}
+{"id": "4474.png", "formula": "\\begin{align*} \\frac { 3 } { 4 p } = \\frac { p ^ { 3 } } { 1 6 ( p ^ { 2 } / 1 2 ) } \\leq \\frac { p ^ { 3 } } { 1 6 a _ { 1 } ^ { * } a _ { 2 } ^ { * } } = a _ { 3 } ^ { * } a _ { 4 } ^ { * } \\leq \\frac { 4 } { 3 } ( a _ { 3 } ^ { * } ) ^ { 2 } \\end{align*}"}
+{"id": "440.png", "formula": "\\begin{align*} ( \\omega \\kappa ) ^ { - 1 } \\alpha \\omega \\kappa \\begin{pmatrix} 1 \\\\ \\vdots \\\\ \\theta ^ { n - 1 } \\end{pmatrix} = \\theta \\begin{pmatrix} 1 \\\\ \\vdots \\\\ \\theta ^ { n - 1 } \\end{pmatrix} . \\end{align*}"}
+{"id": "4429.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ n \\frac { \\mu ( k ) } { k } ( I - S ) h _ k ( z ) = \\sum _ { k = 1 } ^ n \\frac { \\mu ( k ) } { k } \\left [ \\log ( 1 - z ^ k ) - \\log ( 1 - z ) - \\log k \\right ] \\end{align*}"}
+{"id": "3713.png", "formula": "\\begin{align*} ( \\nabla ^ { \\ell + 2 } _ \\nu h ) ( e _ a , e _ b ) = 0 , \\mbox { o r i n t h e l o c a l f r a m e } h _ { a b ; \\tiny \\underbrace { 0 \\cdots 0 } _ { \\ell + 2 - \\mbox { t i m e s } } } = 0 . \\end{align*}"}
+{"id": "910.png", "formula": "\\begin{align*} M = X ^ { \\frac { 1 } { 2 } } \\end{align*}"}
+{"id": "8974.png", "formula": "\\begin{align*} \\dfrac { \\hat { k } _ d ( \\Delta ) } { \\hat { k } _ d ( \\Psi ) } = x _ \\sigma . \\end{align*}"}
+{"id": "8396.png", "formula": "\\begin{align*} a ( z ) = 1 + \\frac { 1 } { 2 i } \\int _ { \\mathbb { R } } \\left ( | u _ y | ^ 2 \\Psi ^ - _ { 1 1 } ( y ; z ) + u _ y \\Psi ^ - _ { 2 1 } ( y ; z ) \\right ) \\mathrm { d } y , \\end{align*}"}
+{"id": "5641.png", "formula": "\\begin{align*} \\mathcal { I U } _ { k } ( R ^ { 2 n } ) : = \\{ ( v _ { 1 } , \\dots , v _ { k } ) : v _ { i } \\in R ^ { 2 n } , ( v _ { 1 } , \\dots , v _ { k } ) \\ , \\ , \\} . \\end{align*}"}
+{"id": "7186.png", "formula": "\\begin{align*} \\frac { \\partial \\textbf { \\textit { U } } } { \\partial x _ n } + Q \\textbf { \\textit { U } } = \\textbf { \\textit { W } } \\end{align*}"}
+{"id": "8059.png", "formula": "\\begin{align*} \\bar { R } _ { c , k } & = \\mathbb { E } \\left [ R _ { c , k } | \\mathbf { \\hat { G } } \\right ] & \\bar { R } _ { k } = \\mathbb { E } \\left [ R _ { k } | \\mathbf { \\hat { G } } \\right ] , \\end{align*}"}
+{"id": "654.png", "formula": "\\begin{align*} \\max \\{ | X ( t ) - x ( t / n ) n | , | R ( t ) - r ( t / n ) n | , | C _ { k _ 1 , k _ 2 } ( t ) - c _ { k _ 1 , k _ 2 } ( t / n ) n | | \\} = o ( n ) . \\end{align*}"}
+{"id": "1571.png", "formula": "\\begin{align*} \\pi \\circ \\gamma ( a , y ) = \\pi ( a , y ) = \\pi ( \\gamma ' ( a , y ) ) = \\pi ( a ' , y ' ) . \\end{align*}"}
+{"id": "8748.png", "formula": "\\begin{align*} \\mathcal D _ { \\psi } : = \\psi _ 0 \\frac { \\partial } { \\partial x } + \\psi _ 1 \\frac { \\partial } { \\partial y } . \\end{align*}"}
+{"id": "1091.png", "formula": "\\begin{align*} \\prescript J { } \\pi ( s _ \\beta w ) = ( w ' , \\lambda + \\Phi ^ + ( - \\beta ) w ^ { - 1 } \\beta ^ \\vee - \\lambda ' ) \\end{align*}"}
+{"id": "831.png", "formula": "\\begin{align*} w ( t , y ) & \\geq w _ 0 ( y ) + \\left ( - g \\big ( c _ 0 ( y ) \\big ) \\frac { 1 } { w _ 0 ( y ) } - \\varepsilon \\right ) \\cdot t = 1 + ( - g _ O - \\varepsilon ) \\cdot t \\\\ & > 1 + ( - g _ I + \\varepsilon ) \\cdot t = w _ 0 ( x ) + \\left ( - g \\big ( c _ 0 ( x ) \\big ) \\frac { 1 } { w _ 0 ( x ) } + \\varepsilon \\right ) \\cdot t \\geq w ( t , x ) \\end{align*}"}
+{"id": "6741.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } \\beta } \\ell = & \\psi ( \\alpha + \\beta ) - \\psi ( \\beta ) + \\log \\left [ 1 - \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] . \\end{align*}"}
+{"id": "5719.png", "formula": "\\begin{align*} f '' ( x ) = a + b ' x + c ' x ^ q , \\end{align*}"}
+{"id": "649.png", "formula": "\\begin{align*} G _ { u } : = c \\left ( R ^ { \\frac { 6 } { r } } \\left ( \\frac { 1 } { R ^ 3 } \\int _ { \\mathcal { C } ( R / 2 , R ) } \\vert \\nabla u \\vert ^ { \\frac { p } { 2 } } d x \\right ) ^ { \\frac { 2 } { p } } , S _ { u } : = R ^ { \\frac { 6 } { r } } \\left ( \\frac { 1 } { R ^ 3 } \\int _ { \\mathcal { C } ( R / 2 , R ) } \\vert u \\vert ^ { p } d x \\right ) ^ { \\frac { 2 } { p } } \\right ) , \\end{align*}"}
+{"id": "451.png", "formula": "\\begin{align*} \\alpha ( \\log x ) : = \\begin{cases} \\ 0 & \\ x = x _ 0 \\\\ \\frac { - 1 } { \\log | x - x _ 0 | } & \\ | x - x _ 0 | < 2 \\delta . \\end{cases} \\end{align*}"}
+{"id": "4746.png", "formula": "\\begin{align*} \\| ( \\lambda x _ v + ( 1 - \\lambda ) x _ w ) \\| ^ 2 = \\lambda \\| x _ v \\| ^ 2 + ( 1 - \\lambda ) \\| x _ w \\| ^ 2 - \\lambda ( 1 - \\lambda ) \\| x _ v - x _ w \\| ^ 2 . \\end{align*}"}
+{"id": "1688.png", "formula": "\\begin{align*} \\tilde { V } _ n ( \\varphi ) = V ( \\varphi ) - V _ n ( \\varphi ) , \\forall \\varphi \\in \\mathcal { C } _ { p w } ( - h , 0 ; \\mathbb { R } ^ m ) . \\end{align*}"}
+{"id": "6089.png", "formula": "\\begin{align*} 2 C _ j \\ , [ C _ { i k } , C _ { k \\ell } ] + ( C _ { i j } - C _ i - C _ j ) [ C _ { k \\ell } , C _ { j k } ] + ( C _ { j \\ell } - C _ j - C _ \\ell ) [ C _ { j k } , C _ { i k } ] - ( C _ { j k } - C _ j - C _ k ) [ C _ { i j } , C _ { j \\ell } ] = 0 \\ , . \\end{align*}"}
+{"id": "6036.png", "formula": "\\begin{align*} c _ { b _ { k } } & = \\frac { 1 } { 2 } \\| ( \\widetilde { u } _ { k } , \\widetilde { v } _ { k } ) \\| _ { E } ^ { 2 } + \\frac { 1 } { 2 } \\Big ( B ( \\widetilde { u } _ { k } ) + B ( \\widetilde { v } _ { k } ) \\Big ) - \\frac { 1 } { 2 p } F ( \\widetilde { u } _ { k } , \\widetilde { v } _ { k } ) \\\\ & \\geq \\Big ( \\frac { 1 } { 2 } - \\frac { \\alpha } { 2 ( p \\alpha - 1 ) } \\Big ) \\| ( \\widetilde { u } _ { k } , \\widetilde { v } _ { k } ) \\| _ { E } ^ { 2 } , \\end{align*}"}
+{"id": "3069.png", "formula": "\\begin{align*} v _ i = \\deg \\left ( \\overline { \\gamma _ Z ^ { - 1 } ( \\P ^ { n - i } ) } \\right ) \\end{align*}"}
+{"id": "3636.png", "formula": "\\begin{align*} 2 \\frac { F ^ { 1 1 } h _ { 1 1 i } ^ 2 } { \\kappa _ 1 ( \\kappa _ 1 - \\tilde { \\kappa } _ i ) } - \\frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \\kappa _ 1 ^ 2 } = ( 2 \\kappa _ 1 - 1 ) \\frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \\kappa _ 1 ^ 2 } \\geq \\frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \\kappa _ 1 ^ 2 } . \\end{align*}"}
+{"id": "4206.png", "formula": "\\begin{align*} \\frac { 1 } { \\varepsilon ^ 2 \\ln \\frac { 1 } { \\varepsilon } } \\int _ \\Omega \\left ( v ^ \\tau _ \\varepsilon - q \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ { p } _ + q \\ln \\frac { 1 } { \\varepsilon } d x \\rightrightarrows q ^ { p + 1 } ( \\bar { x } ) \\int _ { \\hat { B } _ \\tau ( 0 ) } ( \\hat { U } ( y ) + \\ln \\tau ) ^ { p } _ + d y . \\end{align*}"}
+{"id": "1342.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { 3 m } \\alpha _ j + \\sum _ { k = 1 } ^ l \\beta _ k \\leq \\sum _ { i = 1 } ^ { n + 1 } \\lambda _ i + \\varepsilon . \\end{align*}"}
+{"id": "4309.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi _ 1 \\ge - t _ 1 \\} \\cap D _ 3 } | \\tilde { F } | ^ 2 _ h e ^ { - \\varphi _ 1 - \\Psi _ 1 } \\le \\frac { 1 } { C _ 0 } \\int _ { \\{ \\Psi _ 1 \\ge - t _ 1 \\} \\cap D _ 3 } | \\tilde { F } | ^ 2 _ h e ^ { - \\varphi _ 1 } c ( - \\Psi _ 1 ) < + \\infty . \\end{align*}"}
+{"id": "8938.png", "formula": "\\begin{align*} E ( X - Y ) ^ { n } \\allowbreak = \\allowbreak \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ \\frac { n ! } { ( n / 2 ) ! } \\left ( \\beta \\right ) ^ { \\left ( n / 2 \\right ) } ( 1 - \\rho ) ^ { n / 2 } & & n \\end{array} \\right . . \\end{align*}"}
+{"id": "7618.png", "formula": "\\begin{align*} a _ i = - \\frac { 1 } { \\norm { C _ j ^ { \\top } b _ i } ^ 2 } S _ j ^ { \\top } R _ j C _ j ^ { \\top } b _ i i = 1 , \\dots , r . \\end{align*}"}
+{"id": "7113.png", "formula": "\\begin{align*} \\begin{array} { l l l } L _ f \\circ g & = & \\mu \\circ ( f \\otimes i d _ S ) \\circ \\lambda _ S ^ { - 1 } \\circ g \\\\ & = & \\mu \\circ ( f \\otimes i d _ S ) \\circ ( i d _ I \\otimes g ) \\circ \\lambda _ I ^ { - 1 } \\\\ & = & f \\bullet _ \\mu g \\end{array} \\end{align*}"}
+{"id": "5283.png", "formula": "\\begin{align*} g g _ 1 = n _ u a _ t k _ \\theta n _ { u _ 1 } a _ { t _ 1 } k _ { \\theta _ 1 } = n _ u a _ t n _ { u _ 1 } a _ { t _ 1 } ( ( n _ { u _ 1 } a _ { t _ 1 } ) ^ { - 1 } k _ \\theta n _ { u _ 1 } a _ { t _ 1 } ) k _ { \\theta _ 1 } . \\end{align*}"}
+{"id": "8673.png", "formula": "\\begin{align*} { \\bf { u } } _ k ^ { \\star } = { \\left [ { \\bf { \\bar U } } ^ { \\star } \\right ] _ { : , k + 1 } } = { \\left [ \\begin{array} { l } { { \\bf { I } } _ K } \\\\ { { \\bf { 0 } } _ { \\left ( { { M _ t } - K } \\right ) \\times K } } \\end{array} \\right ] _ { : , k + 1 } } , \\ \\forall k , \\end{align*}"}
+{"id": "1126.png", "formula": "\\begin{align*} P _ { \\min } ^ { \\rm { O } } \\left ( { \\overline S , \\overline \\varepsilon , \\overline R } \\right ) = \\mathop { \\arg \\min } \\limits _ { { W _ s } \\in \\left [ { { W ^ { l o w } } , { W ^ { u p } } } \\right ] } P _ { \\min } ^ { \\rm { O } } \\left ( { \\overline S , \\overline \\varepsilon , \\overline R | { W _ s } } \\right ) . \\end{align*}"}
+{"id": "1704.png", "formula": "\\begin{align*} A ( t ) y ( t , \\xi ) & = - i e ^ { - W } \\Delta ( e ^ W y ) - ( \\mu + \\tilde { \\mu } ) y \\\\ & = - i ( \\Delta + b ( t , \\xi ) \\cdot \\nabla + c ( t , \\xi ) ) y ( t , \\xi ) , \\end{align*}"}
+{"id": "2327.png", "formula": "\\begin{align*} \\delta ^ g D ^ g _ { \\theta ^ \\sharp } \\theta = \\frac { 1 } { 2 } \\delta ^ g \\left ( d \\theta ( \\theta ^ \\sharp , \\cdot ) \\right ) = - \\frac { 1 } { 2 } \\delta ^ g d ( \\| \\theta \\| ^ 2 ) = - \\frac { 1 } { 2 } \\Delta ^ g ( \\| \\theta \\| ^ 2 ) . \\end{align*}"}
+{"id": "7719.png", "formula": "\\begin{align*} \\hat M ^ { ( k ) } _ { j , [ k + 1 , n ] } = ( M + \\tilde E ^ { ( k ) } ) _ { j , [ k + 1 , n ] } - ( M + \\tilde E ^ { ( k ) } ) _ { j , [ k ] } \\big ( ( M + \\tilde E ^ { ( k ) } ) _ { [ k ] , [ k ] } \\big ) ^ { - 1 } ( M + \\tilde E ^ { ( k ) } ) _ { [ k ] , [ k + 1 , n ] } , \\end{align*}"}
+{"id": "8940.png", "formula": "\\begin{gather*} H _ { j , n } = \\frac { \\sqrt { n ! } } { \\Gamma ( \\beta ) \\sqrt { \\Gamma ( n + \\beta ) } } \\sum _ { k = 0 } ^ { n } \\frac { ( - 1 ) ^ { k } } { k ! } \\frac { ( \\beta ) ^ { ( n ) } } { ( n - k ) ! ( \\beta ) ^ { ( k ) } } \\int _ { 0 } ^ { \\infty } x ^ { k + j } x ^ { \\beta - 1 } \\exp ( - x ) d x \\\\ = \\frac { \\sqrt { n ! } } { \\sqrt { \\Gamma ( n + \\beta ) } } \\sum _ { k = 0 } ^ { n } \\frac { ( - 1 ) ^ { k } } { k ! } \\frac { ( \\beta ) ^ { ( n ) } ( \\beta ) ^ { ( k + j ) } } { ( n - k ) ! ( \\beta ) ^ { ( k ) } } . \\end{gather*}"}
+{"id": "9196.png", "formula": "\\begin{align*} \\begin{aligned} F _ a ( \\eta _ a ) & : = \\left ( \\begin{array} { c } - \\dfrac { b _ { 1 , \\delta } ( x _ a ) } { 2 } \\\\ - \\bar { y } _ a + \\dfrac { a _ { 0 , \\delta } ( x _ a ) } { 2 } \\end{array} \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "8667.png", "formula": "\\begin{align*} { y _ { \\rm f } } \\left [ { m , k } \\right ] = { { { \\bf { \\tilde h } } } ^ H } \\left [ k \\right ] { { \\bf { u } } _ k } s \\left [ { m , k } \\right ] + z \\left [ m , k \\right ] , \\ \\forall k \\in \\left [ 0 , { K } - 1 \\right ] . \\end{align*}"}
+{"id": "8472.png", "formula": "\\begin{align*} \\begin{aligned} & \\rho _ - \\left \\| Q _ { j , + } \\right \\| _ { L ^ 2 } ^ 2 \\leq \\operatorname { R e } \\int _ { \\mathbb { R } } Q _ { j , + } ( I + J ) ^ H Q _ { j , + } ^ { H } d z \\\\ & = \\operatorname { R e } \\int _ { \\mathbb { R } } \\mathcal { P } ^ + ( D _ j ) ( I + J ) ^ { H } Q _ { j , + } ^ { H } d z \\leq \\rho _ + \\| D _ j \\| _ { L ^ 2 } \\left \\| Q _ { j , + } \\right \\| _ { L ^ 2 } . \\end{aligned} \\end{align*}"}
+{"id": "5948.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ { d } [ a _ i ( t , x , u , z ) - a _ i ( t , x , \\tilde { u } , \\tilde { z } ) ] ( z _ i - \\tilde { z } _ i ) \\\\ + & [ a _ 0 ( t , x , u , z ) - a _ 0 ( t , x , \\tilde { u } , \\tilde { z } ) ] ( u - \\tilde { u } ) \\geq - c ( f _ 3 ( t ) + | u | ^ { \\gamma } + | \\tilde { u } | ^ { \\gamma } ) | u - \\tilde { u } | ^ { 2 } . \\end{align*}"}
+{"id": "4443.png", "formula": "\\begin{align*} W '' = e ^ { - \\lambda ^ * T } \\big ( W ' + W ^ D \\big ) . \\end{align*}"}
+{"id": "787.png", "formula": "\\begin{align*} \\partial _ t c = \\Delta _ \\Gamma \\big ( G ' ( c ) \\big ) \\end{align*}"}
+{"id": "8221.png", "formula": "\\begin{align*} N K B L ( m , n ) & = \\sum _ { A \\in M N A P P ( m , n ) } N ( A ) \\cdot ( m - 1 ) ! \\\\ & \\ge \\sum _ { i = 1 } ^ { 6 } N ( A _ i ) \\cdot ( m - 1 ) ! \\ge ( 6 \\cdot 2 ^ m ) \\cdot ( m - 1 ) ! . \\end{align*}"}
+{"id": "5044.png", "formula": "\\begin{align*} { \\mathcal { E } } = \\frac { 1 } { 2 } \\int _ { \\Omega } \\left ( | u | ^ { 2 } + | B | ^ { 2 } \\right ) \\dd x , { \\mathcal { H } } = \\int _ { \\Omega } A \\cdot B \\dd x , \\end{align*}"}
+{"id": "7135.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} F _ 2 ^ { X , D } ( \\alpha ) _ x & = H ^ 0 \\left ( P ^ D _ X ( \\alpha ) _ x , T _ { P ^ D _ X ( \\alpha ) _ x } ( - [ s ( x ) ] ) \\right ) \\\\ F _ 1 ^ { X , D } ( \\alpha ) _ x & = H ^ 0 \\left ( P ^ D _ X ( \\alpha ) _ x , T _ { P ^ D _ X ( \\alpha ) _ x } ( - 2 [ s ( x ) ] ) \\right ) \\end{aligned} \\right . \\end{align*}"}
+{"id": "89.png", "formula": "\\begin{align*} v _ n ^ { - 1 } \\lambda = \\sigma ^ n v ^ { - 1 } ( \\sigma \\circ w ) ^ { - n } ( \\lambda ) . \\end{align*}"}
+{"id": "3872.png", "formula": "\\begin{align*} - \\varepsilon ^ 2 ( K ( x ) \\nabla u ) + V ( x ) u = u ^ { p } , \\ \\ \\ \\ x \\in \\mathbb { R } ^ n , \\end{align*}"}
+{"id": "2301.png", "formula": "\\begin{align*} W ^ { \\mu * } \\otimes W ^ \\mu = \\bigoplus _ { j = 1 } ^ { \\dim W ^ \\mu } W ^ { \\mu * } \\otimes w _ j \\simeq \\bigoplus _ { j = 1 } ^ { \\dim W ^ \\mu } W ^ { \\mu * } \\end{align*}"}
+{"id": "6972.png", "formula": "\\begin{align*} \\abs { t _ n } _ p & = \\abs { t _ \\ell } _ p \\abs { k } _ p , \\intertext { w h i l e f o r $ p = 2 $ , } \\abs { t _ n } _ 2 & = \\begin{cases} \\abs { t _ \\ell } _ 2 & k \\\\ 2 \\abs { t _ { 2 \\ell } } _ 2 \\abs { k } _ 2 & k \\end{cases} \\end{align*}"}
+{"id": "5331.png", "formula": "\\begin{align*} \\widehat { f _ \\delta } ( a , z ) = ( 2 \\pi ) ^ { - p - q } | \\lambda | ^ { p + q } \\ , \\Big ( \\sum _ { j = 1 } ^ { d ( p , q ) } P _ { p , q } ^ j ( z ) \\ , \\mathcal { G } _ { n + p + q } ( g _ { \\delta , j } ) ( a ( p , q ) ) \\Big ) \\ , \\varphi _ { k - q , \\lambda } ^ { n + p + q - 1 } ( z ) . \\end{align*}"}
+{"id": "8662.png", "formula": "\\begin{align*} n = { \\rm m o d } \\left ( { i + { N _ { { \\rm { C P } } } } , K + { N _ { { \\rm { C P } } } } } \\right ) - { N _ { { \\rm { C P } } } } . \\end{align*}"}
+{"id": "3076.png", "formula": "\\begin{align*} \\sum \\mu _ p ( C ) = 2 , \\mathfrak { i } = 2 , \\quad \\mathfrak { t } = 3 \\end{align*}"}
+{"id": "1721.png", "formula": "\\begin{align*} | | \\Delta U ( \\cdot , 0 ) x | | _ { L ^ { \\infty } ( 0 , \\tau _ 1 ; L ^ 2 ) \\cap L ^ { q } ( 0 , \\tau _ 1 ; L ^ { p } ) } & \\lesssim | | x | | _ { H ^ 2 } , \\\\ | | \\mu x | | _ { L ^ 2 } \\lesssim | | x | | _ { L ^ 2 } \\lesssim | | x | | _ { H ^ 2 } , | | g ( x ) | | _ { L ^ 2 } & = | | x | | ^ { \\alpha } _ { L ^ { 2 \\alpha } } \\lesssim | | x | | ^ { \\alpha } _ { H ^ 2 } , \\end{align*}"}
+{"id": "6222.png", "formula": "\\begin{align*} E _ 0 = \\frac { 1 } { 4 } \\left \\{ - \\Q ^ 2 [ ( 2 L + 3 ) a _ 0 + 1 ] ^ 2 + ( m + 1 ) ^ 2 \\kappa [ ( 2 L + 3 ) a _ 0 - 1 ] ^ 2 \\right \\} . \\end{align*}"}
+{"id": "104.png", "formula": "\\begin{align*} n = \\dim ( X _ \\ast ( T ) _ { \\Gamma _ 0 } \\otimes \\mathbb Q ) ^ { \\sigma } - ( X _ \\ast ( T ) _ { \\Gamma _ 0 } \\otimes \\mathbb Q ) ^ { \\sigma w } . \\end{align*}"}
+{"id": "4270.png", "formula": "\\begin{align*} N = n ^ { 3 } + \\left ( n + 1 \\right ) ^ { 3 } = \\left ( n + a \\right ) ^ { 3 } + \\left ( n + b \\right ) ^ { 3 } \\end{align*}"}
+{"id": "5553.png", "formula": "\\begin{align*} \\theta ( . . . , y _ 2 , y _ 1 | x _ 1 , x _ 2 , . . . ) = ( . . . , x _ 2 , x _ 1 | y _ 1 , y _ 2 , . . . ) . \\end{align*}"}
+{"id": "5354.png", "formula": "\\begin{align*} N + E _ N ^ + - E _ N ^ - = f ^ * ( f _ * N ) . \\end{align*}"}
+{"id": "8579.png", "formula": "\\begin{align*} \\kappa ( t ) = h _ { \\alpha } ( t ) \\cdot \\ , f _ 1 ( t ) , \\ f _ 1 ( t ) = \\sum _ { k = 0 } ^ { + \\infty } \\ , a _ k t ^ k , \\ a _ 0 \\not = 0 , \\ 0 < \\alpha < 1 , \\end{align*}"}
+{"id": "2064.png", "formula": "\\begin{align*} \\begin{aligned} \\norm { M ^ { i } ( \\phi f ) } _ { ( 2 , 0 ) } \\leq \\norm { \\phi M ^ { i } f } _ { ( 2 , 0 ) } + \\sum _ { 1 \\leq j \\leq 2 i } \\frac { 1 } { j ! } \\norm { ( \\partial _ { v _ 1 } ^ j \\phi ) P _ j f } _ { ( 2 , 0 ) } , \\end{aligned} \\end{align*}"}
+{"id": "4029.png", "formula": "\\begin{align*} K _ { \\lambda , \\beta } = \\sum _ { J \\subset \\varrho ^ { - 1 } ( \\beta ) } \\tilde \\kappa _ { \\lambda , J } , \\end{align*}"}
+{"id": "7818.png", "formula": "\\begin{align*} \\hat \\Phi \\left ( A _ \\phi \\begin{pmatrix} a \\\\ b \\end{pmatrix} , s \\right ) = A _ \\phi \\hat \\Phi \\left ( \\begin{pmatrix} a \\\\ b \\end{pmatrix} , s \\right ) , \\end{align*}"}
+{"id": "4944.png", "formula": "\\begin{align*} W ( \\xi _ 0 , n , x , \\theta ) = E _ P \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\biggr ] , \\end{align*}"}
+{"id": "1714.png", "formula": "\\begin{align*} C _ t = & \\sup \\{ | | U ( \\cdot , 0 ) u _ 0 | | _ { L ^ { q _ 1 } ( 0 , t ; W ^ { 2 , p _ 1 } ) } ; | | u _ 0 | | _ { H ^ 2 } \\le 1 \\} \\\\ & + \\sup \\left \\{ \\left | \\left | \\int _ 0 ^ { \\cdot } U ( \\cdot , s ) f ( s ) d s \\right | \\right | _ { L ^ { q _ 1 } ( 0 , t ; W ^ { 2 , p _ 1 } ) } ; | | f | | _ { L ^ { q ' _ 2 } ( 0 , t ; W ^ { 2 , p ' _ 2 } ) } = 1 \\right \\} . \\end{align*}"}
+{"id": "4205.png", "formula": "\\begin{align*} & \\frac { 1 } { \\varepsilon ^ 2 } \\int _ \\Omega \\left ( v ^ \\tau _ \\varepsilon - q \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ { p + 1 } _ + d x = \\int _ { \\hat { B } _ \\tau ( 0 ) } q ^ { p + 1 } ( \\bar { x } + \\varepsilon y ) ( \\hat { U } ( y ) + \\ln \\tau ) ^ { p + 1 } _ + d y \\\\ \\rightrightarrows & q ^ { p + 1 } ( \\bar { x } ) \\int _ { \\hat { B } _ \\tau ( 0 ) } ( \\hat { U } ( y ) + \\ln \\tau ) ^ { p + 1 } _ + d y , \\end{align*}"}
+{"id": "3356.png", "formula": "\\begin{align*} N e w t o n ( F ) = C o n v ( \\bigcup \\limits _ { \\mu } S u p p ( s _ \\mu ) ) . \\end{align*}"}
+{"id": "1483.png", "formula": "\\begin{align*} \\mathbf { u } _ { t } ( t , \\mathbf { x } ) = D \\Delta _ { \\mathbf { x } } \\mathbf { u } ( t , \\mathbf { x } ) + f ( \\mathbf { u } ( t , \\mathbf { x } ) ) , \\mathbf { u } \\in \\mathbb { R } ^ { n } , d \\geq 2 , t \\geq 0 , n \\geq 2 , \\end{align*}"}
+{"id": "5596.png", "formula": "\\begin{align*} W ( 0 1 ^ l | 0 ^ k 1 ) & = A ( 1 0 ^ k 1 . . . ) - A ( 1 0 ^ \\alpha 1 . . . ) + \\sum _ { j = 2 } ^ l \\left [ A ( 1 ^ j 0 ^ k 1 . . . ) - A ( 1 ^ j 0 ^ \\alpha 1 . . . ) \\right ] \\\\ & = A ( 1 0 ^ k 1 . . . ) - A ( 1 0 ^ \\alpha 1 . . . ) \\\\ & = d _ { k } - d _ \\alpha \\ . \\end{align*}"}
+{"id": "5642.png", "formula": "\\begin{align*} d ( v _ { 1 } , \\dots , v _ { k } ) : = \\sum _ { i = 1 } ^ { k } ( - 1 ) ^ { i + 1 } d _ { i } ( v _ { 1 } , \\dots , v _ { k } ) , \\end{align*}"}
+{"id": "1848.png", "formula": "\\begin{align*} X = r \\nabla r , X ^ { \\flat } = r d r \\nabla X ^ { \\flat } = ( \\frac 1 2 r ^ 2 ) = d r \\otimes d r + r ( r ) \\ , . \\end{align*}"}
+{"id": "7997.png", "formula": "\\begin{align*} - \\frac { 1 } { 4 } V _ 1 ( u _ n ) + \\frac { 1 } { 4 } V _ 2 ( u _ n ) + \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { p } \\right ) | u | ^ p _ p = I ( u _ n ) - \\frac { 1 } { 2 } I ' ( u _ n ) [ u _ n ] = c + o ( 1 ) , \\end{align*}"}
+{"id": "6697.png", "formula": "\\begin{align*} \\delta ( \\rho , \\varrho \\gamma ^ + ) = \\delta ( \\rho \\varrho , \\gamma ^ + ) - \\delta ( \\varrho , \\gamma ^ + ) \\end{align*}"}
+{"id": "8369.png", "formula": "\\begin{align*} N _ - ( x ; k ) = I + \\mathcal { P } ^ { - } \\left ( N _ { - } J \\right ) ( z ) , z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "1119.png", "formula": "\\begin{align*} { R _ { \\max } } = W { \\log _ 2 } \\left ( { 1 + \\frac { { P { { \\left | { { h _ b } } \\right | } ^ 2 } } } { { W { N _ 0 } } } } \\right ) . \\end{align*}"}
+{"id": "3468.png", "formula": "\\begin{align*} r _ { \\ell } ^ + : = \\max _ { y \\in R _ { \\ell } ^ + } | \\phi ( y ) | r _ { \\ell } ^ - : = \\max _ { y \\in R _ { \\ell } ^ - } | \\phi ( y ) | \\end{align*}"}
+{"id": "7076.png", "formula": "\\begin{gather*} \\mathcal { A } ( x , \\xi ) = D _ { \\xi } F ( x , \\xi ) . \\end{gather*}"}
+{"id": "488.png", "formula": "\\begin{align*} e ^ { \\gamma x } f _ r ( x ) = \\int _ { - \\infty } ^ \\infty e ^ { \\gamma ( x - y ) } f _ { r + } ( x - y ) e ^ { \\gamma y } \\mu _ - ( d y ) \\le c \\int _ { - \\infty } ^ \\infty e ^ { \\gamma y } \\mu _ - ( d x ) < \\infty . \\end{align*}"}
+{"id": "4352.png", "formula": "\\begin{align*} \\frac { 1 } { r _ 1 ^ 2 } \\int _ { \\{ p \\Psi < 2 \\log r _ 1 \\} } | f | ^ 2 _ h & \\ge \\frac { 1 } { r _ 1 ^ 2 } G _ { p } ( - 2 \\log r _ 1 ) \\\\ & \\ge G _ { p } ( 0 ) \\\\ & \\ge G ( 0 ; c \\equiv 1 , \\Psi , h , I _ + ( h , 2 a _ { z _ 0 } ^ f ( \\Psi ; h ) \\Psi ) _ { z _ 0 } , f ) , \\end{align*}"}
+{"id": "4073.png", "formula": "\\begin{align*} e ( 4 H - 3 , G ^ { M ( 2 , 0 ) } _ { 2 , 1 } ) & = 4 H - 3 + 2 ( 2 - 1 - 1 ) = 4 H - 3 \\\\ e ( 4 H - 2 , G ^ { M ( 2 , 0 ) } _ { 2 , 3 } ) & = 4 H - 2 + ( 2 - 2 - 1 ) = 4 H - 3 \\end{align*}"}
+{"id": "5414.png", "formula": "\\begin{align*} \\begin{aligned} u ( x ) \\geq u ( x _ 0 ) - C b _ \\alpha | x - x _ 0 | = \\ , & u ( x _ 0 ) - C b _ \\alpha ( x _ n - \\rho ( 0 , \\tilde { x } ) ) \\\\ \\geq \\ , & u ( x _ 0 ) - C \\epsilon _ 0 \\delta ^ 2 b _ \\alpha ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "3655.png", "formula": "\\begin{align*} D _ { y , 1 } = \\frac { 3 } { 2 } + \\frac { L _ { x y } } { 2 \\mu _ { y } } D _ { x , 1 } + \\frac { L _ { l x } } { 2 L _ { x y } } + \\frac { L _ { l y } } { 2 \\mu _ { y } } , \\end{align*}"}
+{"id": "8523.png", "formula": "\\begin{align*} \\begin{aligned} & \\begin{pmatrix} \\delta _ - ( z ) & 0 \\\\ 0 & \\delta _ { - } ^ { - 1 } ( z ) \\end{pmatrix} ( I + R ( x ; z ) ) \\begin{pmatrix} \\delta _ + ^ { - 1 } ( z ) & 0 \\\\ 0 & \\delta _ + ( z ) \\end{pmatrix} \\\\ & = \\begin{pmatrix} 1 & \\delta _ - ( z ) \\delta _ + ( z ) \\bar { r } _ 1 ( z ) e ^ { - 2 i z x } \\\\ \\bar { \\delta } _ + ( z ) \\bar { \\delta } _ - ( z ) r _ 2 ( z ) e ^ { 2 i z x } & 1 + \\bar { r } _ 1 ( z ) r _ 2 ( z ) \\end{pmatrix} , \\end{aligned} \\end{align*}"}
+{"id": "5505.png", "formula": "\\begin{align*} \\tilde H ( p , s ) = \\tilde G ( p , s ) - F ( p , s ) . \\end{align*}"}
+{"id": "6683.png", "formula": "\\begin{align*} d k ( \\gamma , x ) = k ( \\gamma x ) - k ( x ) \\qquad ( \\gamma \\in \\Gamma , \\ x \\in X ) . \\end{align*}"}
+{"id": "6251.png", "formula": "\\begin{align*} d _ { i j } : = \\langle \\eta _ i , \\eta _ j \\rangle = 1 + \\frac { \\delta _ { i j } } { \\varphi _ i } , \\end{align*}"}
+{"id": "2901.png", "formula": "\\begin{align*} E : = \\begin{bmatrix} 1 & 0 & \\dots & 0 \\\\ 0 & 0 & \\dots & 0 \\\\ \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & \\dots & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "5048.png", "formula": "\\begin{align*} \\begin{aligned} v _ t + ( v + u _ { \\infty } ) \\cdot \\nabla v + \\nabla p & = ( b + B _ { \\infty } ) \\cdot \\nabla b + \\nu \\Delta v , \\\\ b _ t + ( v + u _ { \\infty } ) \\cdot \\nabla b & = ( b + B _ { \\infty } ) \\cdot \\nabla v + \\mu \\Delta b , \\\\ \\nabla \\cdot v = \\nabla \\cdot b & = 0 . \\end{aligned} \\end{align*}"}
+{"id": "7848.png", "formula": "\\begin{align*} & ( I - Q ^ \\dagger ) F ^ { q , \\dagger } _ { v v } ( v + \\psi ^ \\dagger ( v , p ) , p ) [ v _ 1 + \\psi ^ \\dagger _ v ( v , p ) v _ 1 , v _ 2 + \\psi ^ \\dagger _ v ( v , p ) v _ 2 ] \\\\ & \\quad + ( I - Q ^ \\dagger ) F ^ { q , \\dagger } _ v ( v + \\psi ^ \\dagger ( v , p ) , p ) \\psi ^ \\dagger _ { v v } ( v , p ) [ v _ 1 , v _ 2 ] = 0 . \\end{align*}"}
+{"id": "4302.png", "formula": "\\begin{align*} G ( t _ 0 ; \\tilde { c } ) = \\int _ { \\{ \\Psi < - t _ 0 \\} } | \\tilde { F } | ^ 2 _ h \\tilde { c } ( - \\Psi ) = \\frac { G ( T _ 1 ; c ) } { \\int _ { T _ 1 } ^ { + \\infty } c ( t ) e ^ { - t } d t } \\int _ { t _ 0 } ^ { + \\infty } \\tilde { c } ( s ) e ^ { - s } d s , \\end{align*}"}
+{"id": "1803.png", "formula": "\\begin{align*} L ( f , 3 ) = \\frac { 2 \\pi ^ { 2 } } { 2 7 } \\int _ { 0 } ^ { 1 } b ^ { 3 } ( q ) \\sum _ { k , r = 1 } ^ { \\infty } \\frac { \\chi _ { - 3 } ( k r ) } { k } \\left ( q ^ { \\frac { k r } { 3 } } - q ^ { k r } \\right ) \\frac { d q } { q } . \\end{align*}"}
+{"id": "8929.png", "formula": "\\begin{align*} E X ^ { m } Y ^ { n - m } = \\sum _ { j = 0 } ^ { \\min ( m , n - m ) } \\rho ^ { j } H _ { m , j } H _ { n - m , j } . \\end{align*}"}
+{"id": "1130.png", "formula": "\\begin{align*} 1 \\leq \\sum _ { j = 1 } ^ { N L } x _ j \\leq \\sqrt { N L } , 1 \\leq \\sum _ { j = 1 } ^ { N L } y _ j \\leq \\sqrt { N L } . \\end{align*}"}
+{"id": "4965.png", "formula": "\\begin{align*} W ( \\xi _ 0 , n , x , \\bar { \\theta } ) = & E _ { \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ n Z ^ { \\bar { \\theta } } \\biggr ) \\biggr ] \\\\ = & E _ { \\xi _ 0 } \\biggl [ V \\biggl ( \\frac { \\xi _ 0 f _ 1 ( X _ 1 ) } { \\xi _ 0 f _ 1 ( X _ 1 ) + ( 1 - \\xi _ 0 ) f _ 2 ( X _ 1 ) } , n - 1 , x + X _ 1 \\biggr ) \\biggr ] . \\end{align*}"}
+{"id": "1800.png", "formula": "\\begin{align*} 4 \\sqrt { 3 } \\pi ^ { 3 } \\int _ { 0 } ^ { \\infty } \\left ( \\sum _ { n , s = 1 } ^ { \\infty } \\chi _ { - 3 } ( n ) s ^ { 2 } e ^ { - 2 \\pi u n s } \\right ) \\left ( \\sum _ { k , r = 1 } ^ { \\infty } \\frac { \\chi _ { - 3 } ( k r ) } { k } \\left ( e ^ { - \\frac { 2 \\pi k r } { 9 u } } - e ^ { - \\frac { 2 \\pi k r } { 3 u } } \\right ) \\right ) u d u . \\end{align*}"}
+{"id": "7806.png", "formula": "\\begin{align*} \\sigma ( F ^ q _ v ( 0 , p ) ) = \\operatorname { c l } ( \\{ \\xi _ k , k \\in \\N \\} ) . \\end{align*}"}
+{"id": "7584.png", "formula": "\\begin{align*} x _ { 0 0 } & = a \\\\ x _ { 0 1 } & = \\lim _ { m \\to \\infty } T ^ { c _ 2 ( m ) } a \\\\ x _ { 1 0 } & = \\lim _ { j \\to \\infty } T ^ { c _ 1 ( j ) } a \\\\ x _ { 1 1 } & = \\lim _ { j \\to \\infty } \\lim _ { m \\to \\infty } T ^ { c _ 1 ( j ) + c _ 2 ( m ) } a = \\lim _ { m \\to \\infty } \\lim _ { j \\to \\infty } T ^ { c _ 1 ( j ) + c _ 2 ( m ) } a . \\end{align*}"}
+{"id": "2655.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } d _ { p } ( n , k , r ) q ^ { n } & = \\frac { ( q ^ { r } ; q ^ { p } ) _ { \\infty } } { ( q ; q ) _ { \\infty } } \\sum \\limits _ { n = 0 } ^ { \\infty } \\frac { q ^ { k ( p n + r ) } } { 1 - q ^ { p n + r } } \\prod \\limits _ { j \\neq n , j = 0 } ^ { \\infty } ( 1 + q ^ { p j + r } + q ^ { 2 ( p j + r ) } + \\cdots + q ^ { ( k - 1 ) ( p j + r ) } ) \\end{align*}"}
+{"id": "7948.png", "formula": "\\begin{align*} \\eta ( z _ 1 , z _ 2 ) ^ 2 & = 1 - ( z _ 1 \\bar { z _ 2 } ) \\leq 2 - ( z _ 1 ) - ( z _ 2 ) - ( z _ 1 ) ( z _ 2 ) \\\\ & \\leq \\eta ( 1 , z _ 1 ) ^ 2 + \\eta ( 1 , z _ 2 ) ^ 2 + | ( z _ 1 ) | | ( z _ 2 ) | \\leq \\eta ( 1 , z _ 1 ) ^ 2 + \\eta ( 1 , z _ 2 ) ^ 2 + 2 \\eta ( 1 , z _ 1 ) \\eta ( 1 , z _ 2 ) \\\\ & = ( \\eta ( 1 , z _ 1 ) + \\eta ( 1 , z _ 2 ) ) ^ 2 . \\end{align*}"}
+{"id": "6283.png", "formula": "\\begin{align*} \\langle \\overline { \\alpha } ' _ 1 , \\overline { \\alpha } ' _ 1 \\rangle \\langle \\overline { \\alpha } ' _ 2 , \\overline { \\alpha } ' _ 2 \\rangle < \\langle \\overline { \\alpha } ' _ 1 , \\overline { \\alpha } ' _ 2 \\rangle ^ 2 = \\langle \\alpha ' _ 1 , \\alpha ' _ 2 \\rangle ^ 2 = \\cosh ( \\theta ) ^ 2 = 1 - s . \\end{align*}"}
+{"id": "9152.png", "formula": "\\begin{align*} h ( x ) = h _ 0 + \\left \\{ \\begin{array} { c c } ( x - \\pi ) ^ 2 - 1 & x < \\pi \\\\ \\cos ( x - \\pi ) - 2 & x \\in [ \\pi , \\ , 2 \\pi ) \\\\ ( x - 2 \\pi ) ^ 2 - 3 & x \\ge 2 \\pi \\end{array} \\right . \\end{align*}"}
+{"id": "1835.png", "formula": "\\begin{align*} \\aligned 0 & = \\delta ^ \\nabla \\big ( \\exp \\ , ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) R ^ \\nabla \\big ) \\\\ & = - \\sum _ { i = 1 } ^ m \\big ( \\nabla _ { e _ i } \\exp \\ , ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) R ^ \\nabla \\big ) ( e _ i , \\cdot ) \\\\ & = \\exp \\ , ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) \\delta ^ \\nabla R ^ \\nabla - i _ { \\big ( \\exp \\ , ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) \\big ) } R ^ \\nabla \\ , . \\endaligned \\end{align*}"}
+{"id": "7885.png", "formula": "\\begin{align*} \\dot \\theta _ k = \\frac 1 M \\sum _ { j = 1 } ^ M a _ { k - j } \\sin ( \\theta _ j - \\theta _ k ) - \\frac 1 M \\sum _ { j = 1 } ^ M a _ { 1 - j } \\sin ( \\theta _ j ) . \\end{align*}"}
+{"id": "6967.png", "formula": "\\begin{align*} \\prod ^ { n e } _ { \\substack { j = 1 \\\\ j \\equiv i \\bmod e } } ( x - \\zeta ^ j _ { n e } y ) = x ^ n - \\zeta ^ i _ e y ^ n \\end{align*}"}
+{"id": "1785.png", "formula": "\\begin{align*} \\begin{array} { c } \\frac { v _ 0 } { v _ 2 } = \\frac { - | z | ^ 2 + i \\tilde { t } \\sqrt { d } } { 2 } \\\\ \\frac { v _ 1 } { v _ 2 } = z \\end{array} . \\end{align*}"}
+{"id": "251.png", "formula": "\\begin{align*} u _ \\alpha - 1 & = \\sum _ { i = 1 } ^ k ( \\sum _ { j = 1 } ^ s a _ j w _ { j i } + \\sum _ { j = 1 } ^ t b _ j v _ { j i } ) X _ i \\\\ & = \\sum _ { j = 1 } ^ s ( \\sum _ { i = 1 } ^ k a _ j w _ { j i } X _ i ) + \\sum _ { j = 1 } ^ t ( \\sum _ { i = 1 } ^ k b _ j v _ { j i } X _ i ) \\\\ & = \\sum _ { j = 1 } ^ s a _ j ( \\sum _ { i = 1 } ^ k w _ { j i } X _ i ) + \\sum _ { j = 1 } ^ t b _ j ( \\sum _ { i = 1 } ^ k v _ { j i } X _ i ) . \\end{align*}"}
+{"id": "8362.png", "formula": "\\begin{align*} 1 - | r ( k ) | ^ 2 = \\frac { 1 } { | a ( k ) | ^ 2 } \\geq c _ { 0 } ^ 2 > 0 k \\in i \\mathbb { R } , \\end{align*}"}
+{"id": "7973.png", "formula": "\\begin{align*} \\partial _ { t } P = \\Theta \\partial _ { t } \\sigma _ { k } + \\sigma _ { k } \\partial _ { t } \\Theta , \\end{align*}"}
+{"id": "6088.png", "formula": "\\begin{align*} & [ C _ { I J } , C _ { J K } ] = [ C _ { J K } , C _ { I K } ] = [ C _ { I K } , C _ { I J } ] \\ , , \\\\ & \\big [ C _ { J K } , [ C _ { I J } , C _ { J K } ] \\big ] = 2 C _ { I K } C _ { J K } - 2 C _ { J K } C _ { I J } + 2 ( C _ J - C _ K ) ( C _ { I J K } - C _ I ) \\ , , \\end{align*}"}
+{"id": "4431.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\frac { \\mu ( k ) } { k } \\log ( 1 - z ^ k ) \\to - z \\ \\ell ^ q \\end{align*}"}
+{"id": "141.png", "formula": "\\begin{align*} u ( t ) = U _ \\alpha ( t ) u _ 0 + \\int _ 0 ^ t U _ \\alpha ( t - s ) F ( u ( s ) ) d s . \\end{align*}"}
+{"id": "1724.png", "formula": "\\begin{align*} | | \\nabla ( y ( t ) - y ( s ) ) | | _ { L ^ { 2 \\alpha } } \\le & | | \\nabla ( y ( t ) - y ( s ) ) | | ^ { \\alpha } _ { L ^ 2 } | | \\nabla ( y ( t ) - y ( s ) ) | | ^ { 1 - \\alpha } _ { H ^ 1 } \\\\ \\lesssim & | | y ( t ) - y ( s ) | | ^ { \\alpha ' } _ { L ^ 2 } | | y ( t ) - y ( s ) | | ^ { 1 - \\alpha ' } _ { H ^ 2 } \\\\ \\lesssim & M _ 1 ^ { \\alpha ' } | t - s | ^ { \\alpha ' } M _ 1 ^ { 1 - \\alpha ' } = M _ 1 | t - s | ^ { \\alpha ' } , \\alpha ' \\le \\alpha , \\end{align*}"}
+{"id": "3043.png", "formula": "\\begin{align*} \\sigma ^ { - 1 } \\left ( r \\right ) - \\sigma ^ { - 1 } \\left ( s \\right ) = \\int _ { s } ^ { r } \\tfrac { 1 } { \\sqrt { 2 } } \\left ( \\ell ^ { \\prime } \\left ( t \\right ) \\right ) ^ { 1 / 2 } ~ d t \\end{align*}"}
+{"id": "5270.png", "formula": "\\begin{align*} \\int _ a ^ b e ^ { i \\lambda \\phi ( x ) } \\psi ( x ) d x = \\int _ a ^ b e ^ { i \\lambda \\phi ( x ) } ( D ^ t ) ^ N ( \\psi ) ( x ) d x . \\end{align*}"}
+{"id": "3435.png", "formula": "\\begin{align*} \\| ( \\ell _ n ^ { ( 1 ) } - \\ell ) q _ n \\alpha \\| = | \\ell _ n ^ { ( 1 ) } - \\ell | \\cdot \\| q _ n \\alpha \\| . \\end{align*}"}
+{"id": "6837.png", "formula": "\\begin{align*} \\left \\Vert \\mathcal { L } _ { 1 } ^ { + } \\left ( \\rho \\right ) \\right \\Vert _ { L _ { 2 } \\left ( \\mathbb { R } \\right ) } ^ { 2 } = \\int _ { - \\infty } ^ { \\infty } \\left ( \\int _ { M ^ { + } } \\left \\vert Q ^ { \\prime } ( s ) \\right \\vert \\beta ^ { k - 2 } ( s ) d s \\right ) \\left ( \\int _ { M ^ { + } } \\left \\vert Q ^ { \\prime } ( s ^ { \\prime } ) \\right \\vert \\overline { \\beta } ^ { k - 2 } ( s ^ { \\prime } ) d s ^ { \\prime } \\right ) d \\rho . \\end{align*}"}
+{"id": "1113.png", "formula": "\\begin{align*} R _ { b , m \\to b } ^ { \\rm { S - N } } = { W _ m } { \\log _ 2 } \\left ( { 1 + \\frac { { { p _ { b , m } } { { \\left | { { h _ b } } \\right | } ^ 2 } } } { { { p _ s } { { \\left | { { h _ b } } \\right | } ^ 2 } + { W _ m } { N _ 0 } } } } \\right ) . \\end{align*}"}
+{"id": "6077.png", "formula": "\\begin{align*} X = X _ 1 \\boxtimes X _ 2 \\boxtimes \\cdots \\boxtimes X _ k , \\end{align*}"}
+{"id": "6547.png", "formula": "\\begin{align*} ( 2 a ) ^ { \\pi ( a ) } + d _ { \\pi ( a ) - 1 } ( 2 a ) ^ { \\pi ( a ) - 1 } + d _ { \\pi ( a ) - 2 } ( 2 a ) ^ { \\pi ( a ) - 2 } + \\cdots d _ 1 ( 2 a ) = - d _ 0 \\end{align*}"}
+{"id": "3831.png", "formula": "\\begin{align*} \\begin{aligned} \\Big | \\langle \\boldsymbol { \\nu } , \\eta _ t \\rangle - \\bar \\eta _ t + \\langle \\boldsymbol { \\nu } , q _ { \\alpha , x _ \\alpha } \\rangle - \\bar q _ { \\alpha , x _ \\alpha } \\Big | & \\le c _ 3 \\langle \\boldsymbol { \\nu } , \\eta ( \\lambda | \\bar U ; x , t ) \\rangle = c _ 3 \\mathcal { H } ( \\boldsymbol { \\nu } , U , \\bar { U } ; x , t ) \\ ; , \\end{aligned} \\end{align*}"}
+{"id": "1271.png", "formula": "\\begin{align*} \\delta + x = \\lambda . \\end{align*}"}
+{"id": "1498.png", "formula": "\\begin{align*} | | \\mathbf { v } ( T + t , \\mathbf { v } ^ 0 ) | | _ { \\mathcal { E } } = | | \\mathbf { v } ( t , \\mathbf { v } ^ 1 ) | | _ { \\mathcal { E } } \\leq \\delta . \\end{align*}"}
+{"id": "7907.png", "formula": "\\begin{align*} \\hat W _ r ( 0 ) = k \\hat W _ \\frac { r } { k } ( 0 ) . \\end{align*}"}
+{"id": "8156.png", "formula": "\\begin{align*} g ^ * f _ ! = f ' _ ! g '^ * \\end{align*}"}
+{"id": "8310.png", "formula": "\\begin{align*} R _ c = \\pi \\sqrt { 5 0 / 2 } \\approx 1 5 . 7 \\ , , \\end{align*}"}
+{"id": "5952.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathbb { T } ^ d } \\langle a ( u _ n ( x ) ) \\nabla u ( x ) , \\nabla u _ n ( x ) - \\nabla u ( x ) \\rangle d x = 0 . \\end{align*}"}
+{"id": "2801.png", "formula": "\\begin{align*} Y _ { m } ^ { S _ { n } } ( \\mathrm { d } x ) : = \\frac { \\sqrt { \\pi } } { 4 } \\sum _ { j = 1 } ^ { 1 6 ^ { m } } e ^ { \\pi \\lambda \\Phi ^ { S _ { n } , \\tilde { S } _ { n , m } } ( x ) + \\frac { 4 \\lambda ^ { 2 } } { \\gamma } s _ { S _ { n + m , j } } ( x ) } 1 _ { S _ { n + m , j } } ( x ) \\mathrm { d } x \\end{align*}"}
+{"id": "706.png", "formula": "\\begin{align*} \\| P _ 0 \\Psi _ { R ( \\lambda ) } \\| _ { H ^ 1 } = O ( b - \\lambda ) ~ ~ \\lambda \\to b . \\end{align*}"}
+{"id": "5328.png", "formula": "\\begin{align*} f ^ \\lambda \\ast _ \\lambda \\varphi _ { k , \\lambda } ^ { n - 1 } ( z ) = ( 2 \\pi ) ^ { - p - q } \\lambda ^ { p + q } \\ , P ( z ) \\ , g ^ \\lambda \\ast _ \\lambda \\varphi _ { k - p , \\lambda } ^ { n + p + q - 1 } ( z ) \\end{align*}"}
+{"id": "397.png", "formula": "\\begin{align*} \\Omega _ { N } : = \\{ \\omega \\in \\Omega : | \\omega ( t ) | \\leq N e ^ { | t | } , t \\in \\R \\} , \\ N \\in \\N . \\end{align*}"}
+{"id": "4588.png", "formula": "\\begin{align*} ( B ^ { \\circ } S _ { T - ( \\cdot ) } ^ { \\circ } x ^ { \\circ } \\odot \\lambda ) ( \\boldsymbol { \\cdot } ) u = \\int _ { ( \\boldsymbol { \\cdot } ) } \\langle B ^ { \\circ } S _ { T - t } ^ { \\circ } x ^ { \\circ } , u \\rangle \\d t \\qquad ( u \\in U ) \\end{align*}"}
+{"id": "4306.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t _ 1 \\} } | \\tilde { F } _ 1 - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f F | ^ 2 _ { \\tilde { h } } e ^ { v _ { t _ 0 , B } ( \\Psi ) } c ( - v _ { t _ 0 , B } ( \\Psi ) ) \\\\ \\le & \\left ( \\int _ { t _ 1 } ^ { t _ 0 + B } c ( s ) e ^ { - s } d s \\right ) \\int _ { M } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } { | f F | } ^ 2 _ { \\tilde { h } } . \\end{align*}"}
+{"id": "7258.png", "formula": "\\begin{align*} t = r _ 0 + r _ 1 q + \\cdots + r _ { n - 1 } q ^ { n - 1 } , \\end{align*}"}
+{"id": "6150.png", "formula": "\\begin{align*} \\mathcal { X } ^ { 1 } _ t = \\frac { t _ { k + 1 } - t } { t _ { k + 1 } - t _ k } { X } ^ { 1 } _ { t _ k } + \\frac { t - t _ k } { t _ { k + 1 } - t _ k } { X } ^ { 1 } _ { t _ { k + 1 } } , \\end{align*}"}
+{"id": "76.png", "formula": "\\begin{align*} X ( x , \\alpha ) = \\begin{cases} + , & \\ell ( x , \\alpha ) > 0 , \\\\ \\bigcirc , & \\ell ( x , \\alpha ) = 0 , \\\\ - , & \\ell ( x , \\alpha ) < 0 . \\end{cases} \\end{align*}"}
+{"id": "6936.png", "formula": "\\begin{align*} z ^ \\top \\nabla ^ 2 f ( x ) z & \\le \\sum _ { i = 1 } ^ n z _ i ^ 2 V '' ( x _ i ) \\le - \\kappa | z | ^ 2 , \\end{align*}"}
+{"id": "4983.png", "formula": "\\begin{align*} \\Delta _ { 1 } ( x , \\xi _ 0 ) = E _ { \\xi _ 0 } \\left [ \\varphi \\left ( x + X _ { 1 } \\right ) \\right ] - E _ { \\xi _ { 0 } } \\left [ \\varphi \\left ( x + Y _ { 1 } \\right ) \\right ] \\end{align*}"}
+{"id": "5898.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ { T } \\Vert P _ n B ( t , X _ n ( t ) ) Q _ n - B ( t , X ( t ) ) \\Vert _ { L _ 2 } ^ 2 d t = 0 . \\end{align*}"}
+{"id": "3581.png", "formula": "\\begin{align*} L ( p \\mid u ) = \\prod _ { i = 1 } ^ m p _ j ^ { u _ j } . \\end{align*}"}
+{"id": "7044.png", "formula": "\\begin{align*} 0 \\ , = \\ , F _ { 0 , 5 } \\ , = \\ , F _ { 1 , 4 } , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ F _ { 2 , 3 } \\ , = \\ , 6 , \\ \\ \\ \\ \\ \\ \\ F _ { 3 , 2 } \\ , = \\ , 6 , \\end{align*}"}
+{"id": "8498.png", "formula": "\\begin{align*} & u ( x ) e ^ { 2 i ( c _ - ( x ) + c ) } = \\lim _ { k \\rightarrow 0 } ( k ^ { - 1 } \\psi ^ + ( x ; k ) ) _ { 1 2 } , \\\\ & u ( x ) e ^ { - i ( 2 c _ - ( x ) + c ) } = \\lim _ { k \\rightarrow 0 } ( k ^ { - 1 } \\psi ^ - ( x ; k ) ) _ { 1 2 } , \\\\ & \\partial _ x \\left ( \\bar { u } _ x ( x ) e ^ { i c _ \\pm } \\right ) = 2 i \\lim _ { | z | \\rightarrow \\infty } ( z \\Psi ^ { \\pm } ( x ; z ) ) _ { 2 1 } , \\end{align*}"}
+{"id": "1194.png", "formula": "\\begin{align*} h ( k , k ) = 1 \\ \\mbox { a n d } \\ h _ t ( n , k ) \\le h _ t ( n - 1 , k ) \\le h _ t ( k + 1 , k ) = \\frac { t } { k + 1 } \\enspace . \\end{align*}"}
+{"id": "2454.png", "formula": "\\begin{align*} L _ { q } \\left [ t ^ { m - 1 } \\right ] = \\displaystyle \\int _ { 0 } ^ { \\infty } d t \\exp _ { q } ( - s t ) t ^ { m - 1 } . \\end{align*}"}
+{"id": "6241.png", "formula": "\\begin{align*} \\partial ^ 2 _ { j k } v _ i - \\Gamma _ { k j } ^ j \\partial _ j v _ i - \\Gamma _ { j k } ^ k \\partial _ k v _ i = 0 . \\end{align*}"}
+{"id": "7757.png", "formula": "\\begin{align*} \\frac { 1 } { N ( f ( x ) ) } : = - \\frac { f ' ( x ) } { f ( x ) } \\end{align*}"}
+{"id": "6407.png", "formula": "\\begin{align*} \\Phi = y _ i \\xi ^ i + c , \\end{align*}"}
+{"id": "5258.png", "formula": "\\begin{align*} T ( t ) x = \\tau \\lim _ { n \\to \\infty } \\left ( \\frac { n } { t } R \\left ( \\frac { n } { t } , A \\right ) \\right ) ^ { n } x \\end{align*}"}
+{"id": "8402.png", "formula": "\\begin{align*} \\lim _ { | z | \\rightarrow \\infty } \\Psi ^ - _ 2 ( x ; z ) = e ^ { i c _ - ( x ) } e _ 2 . \\end{align*}"}
+{"id": "9040.png", "formula": "\\begin{align*} \\omega ( E , \\epsilon ) = 0 , L ( E , \\epsilon ) = \\log \\Big | \\frac { E } { 2 } + \\frac { \\sqrt { E ^ { 2 } - 4 } } { 2 } \\Big | . \\end{align*}"}
+{"id": "587.png", "formula": "\\begin{align*} d e t ^ { S ^ 2 } ( \\otimes _ { 1 \\leq i < j \\leq 2 d } ( v _ { i , j } ) ) = \\sum _ { ( \\Gamma _ 1 , . . . , \\Gamma _ d ) \\in \\mathcal { P } ^ { h , c f } _ d ( K _ { 2 d } ) } \\varepsilon _ d ^ { S ^ 2 } ( ( \\Gamma _ 1 , . . . , \\Gamma _ d ) ) M _ { ( \\Gamma _ 1 , . . . , \\Gamma _ d ) } ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) , \\end{align*}"}
+{"id": "1627.png", "formula": "\\begin{align*} F _ p C ^ \\ast ( \\C { O } _ h ( G ) , M ) : = h ^ p C ^ \\ast ( \\C { O } _ h ( G ) , M ) , p \\geq 0 , \\end{align*}"}
+{"id": "9300.png", "formula": "\\begin{align*} \\left ( \\Delta K _ m \\right ) ^ m \\wedge \\beta _ n ^ { n - m } = 0 \\mathbb { H } ^ n \\setminus \\{ 0 \\} . \\end{align*}"}
+{"id": "3295.png", "formula": "\\begin{align*} \\int _ 0 ^ { 2 - i \\Delta _ n + s } | x - y - i \\Delta _ n + s | ^ { 2 \\mathfrak H - 1 } d y = \\int _ { i \\Delta _ n - s } ^ 2 | x - y | ^ { 2 \\mathfrak H - 1 } d y \\leq \\int _ 0 ^ 2 | x - y | ^ { 2 \\mathfrak H - 1 } d y \\leq \\frac { 2 ^ { 2 \\mathfrak H } } { \\mathfrak H } , \\end{align*}"}
+{"id": "8911.png", "formula": "\\begin{align*} \\mathbf { g } = & [ \\log 9 , \\log 9 , \\log 6 , \\log \\zeta , \\log 5 4 , \\log 5 4 , \\log 2 1 6 ] ^ \\intercal \\end{align*}"}
+{"id": "8776.png", "formula": "\\begin{align*} x \\cdot B ( x , x ) = 0 . \\end{align*}"}
+{"id": "46.png", "formula": "\\begin{align*} & \\int _ { B _ r } d i v _ { \\mathbb { G } } F | \\nabla _ { H } v | ^ 2 - 2 \\sum _ { i = 1 } ^ { m } \\int _ { B _ r } X _ i v [ X _ i , F ] v - 2 \\int _ { B _ r } F v \\Delta _ { H } v \\\\ & = \\int _ { \\partial B _ r } | \\nabla _ { H } v | ^ 2 < F , \\nu > - 2 \\sum _ { i = 1 } ^ { m } \\int _ { \\partial B _ r } F v X _ i v < X _ i , \\nu > . \\end{align*}"}
+{"id": "8801.png", "formula": "\\begin{align*} | \\tilde { Y } _ t | = | P ^ { - 1 } Y _ t | \\le \\| P ^ { - 1 } \\| _ { \\C ^ n \\to \\C ^ n } | Y _ t | , \\end{align*}"}
+{"id": "3137.png", "formula": "\\begin{align*} \\overbrace { [ x + a , y + b ] } & = ( x + a ) \\cdot _ { \\ltimes } ( y + b ) - ( y + b ) \\cdot _ { \\ltimes } ( x + a ) \\\\ & = x \\cdot y + \\ell ( x ) b + r ( y ) a - y \\cdot x - \\ell ( y ) a - r ( x ) b \\\\ & = [ x , y ] + ( \\ell - r ) ( x ) b - ( \\ell - r ) ( y ) a . \\end{align*}"}
+{"id": "5793.png", "formula": "\\begin{align*} g _ 2 = - \\frac { 1 } { \\nu } I \\widehat { g _ 1 } I + \\frac { 2 i } { \\sqrt { \\tau _ 0 } } h _ z J g _ 1 . \\end{align*}"}
+{"id": "356.png", "formula": "\\begin{align*} u _ k = \\frac { - 1 } { 8 \\cos ^ { 2 } ( \\frac { \\pi k } { n - 1 } ) } S v _ k , \\end{align*}"}
+{"id": "4982.png", "formula": "\\begin{align*} D _ n ( x , t _ X , t _ Y ) = \\begin{cases} ( t _ X + t _ Y ) \\Delta _ n ( x , \\frac { t _ X } { t _ X + t _ Y } ) & \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "4416.png", "formula": "\\begin{align*} \\Big \\| \\sum \\limits _ { k = 2 } ^ { n } \\mu ( k ) ( I - S ) h _ k - ( 1 - z ) \\Big \\| _ { H ^ 2 ( \\mathbb { D } ) } \\longrightarrow 0 n \\to \\infty \\end{align*}"}
+{"id": "3809.png", "formula": "\\begin{align*} \\mathcal { A } _ q ( p , x ) = \\mathcal { A } _ q ^ p ( x ) : = \\displaystyle \\sum _ { n = 0 } ^ \\infty \\frac { \\overline { p } ^ n } { \\sqrt { \\Gamma ( q n + 1 ) } } \\psi _ n ( x ) , p \\in \\mathbb { H } , x \\in \\mathbb { R } \\end{align*}"}
+{"id": "745.png", "formula": "\\begin{align*} \\partial _ t P _ s ( x , t ) = - ( - \\Delta ) ^ { s } P _ s ( x , t ) = - t ^ { - \\frac { N } { 2 s } - 1 } ( - \\Delta ) ^ { s } \\phi ( z ) , z = | x | t ^ { - \\frac 1 { 2 s } } , t > 0 , \\end{align*}"}
+{"id": "6349.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { n = 0 } ^ \\infty c _ n ( a , b ) \\ , g ( x ; ( a + n ) \\theta ) , \\end{align*}"}
+{"id": "3213.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu \\left ( \\left \\{ \\tau \\in \\partial _ e ( T ( A ) ) \\mid \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 0 } ^ { n - 1 } \\tau ( \\alpha ^ i ( a ) ) \\le z \\right \\} \\right ) = \\frac { 1 } { \\sigma \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ z \\exp \\left ( - \\frac { t ^ 2 } { 2 \\sigma ^ 2 } \\right ) d t \\end{align*}"}
+{"id": "8286.png", "formula": "\\begin{align*} \\gamma ^ i ( x ) = \\frac 1 { p - 1 } \\sum _ { j \\neq i , j \\in \\C } K ( x ^ i - x ^ j ) , ~ ~ ~ ~ \\forall x \\in \\mathbb R ^ { N d } , \\end{align*}"}
+{"id": "4154.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n - 1 } } \\widetilde { \\Phi } ( x ' ) \\ , d x ' & = \\bigg ( \\int _ { \\R ^ { n - 1 } } \\Phi ( - x ' ) \\ , d x ' \\bigg ) ^ { \\top } = \\bigg ( \\int _ { \\R ^ { n - 1 } } \\Phi ( x ' ) \\ , d x ' \\bigg ) ^ { \\top } \\\\ & = \\bigg ( \\int _ { \\R ^ { n - 1 } } \\partial _ n K ^ L ( x ' , 1 ) \\ , d x ' \\bigg ) ^ { \\top } = 0 , \\end{align*}"}
+{"id": "2936.png", "formula": "\\begin{align*} \\overline { \\nu _ { \\alpha } ^ { \\prime } } \\left ( x \\right ) = \\nu _ { \\alpha } ^ { \\prime } \\left ( x , \\infty \\right ) = \\frac { 1 } { \\alpha \\log ^ { \\alpha } \\left ( 1 + x \\right ) } \\end{align*}"}
+{"id": "7576.png", "formula": "\\begin{align*} \\Phi _ N = \\{ L _ N , L _ N + 1 , \\dots , M _ N - 1 \\} \\end{align*}"}
+{"id": "2616.png", "formula": "\\begin{align*} & ( x \\cdot y ) \\cdot \\alpha ( z ) - \\alpha ( x ) \\cdot ( y \\cdot z ) = \\varepsilon ( x , y ) \\big ( ( y \\cdot x ) \\cdot \\alpha ( z ) - \\alpha ( y ) \\cdot ( x \\cdot z ) \\big ) , \\\\ & ( x \\cdot y ) \\cdot \\alpha ( z ) = \\varepsilon ( y , z ) ( x \\cdot z ) \\cdot \\alpha ( y ) . \\end{align*}"}
+{"id": "7021.png", "formula": "\\begin{align*} 0 \\ , = \\ , F _ { 0 , 4 } \\ , = \\ , F _ { 1 , 3 } \\ , = \\ , F _ { 2 , 2 } \\ , = \\ , F _ { 3 , 1 } . \\end{align*}"}
+{"id": "3337.png", "formula": "\\begin{align*} \\overline { \\chi } = ( T \\cap \\overline { \\psi } ) \\cup ( T \\setminus \\overline { \\psi } ) = T . \\end{align*}"}
+{"id": "2861.png", "formula": "\\begin{align*} \\tilde S ^ { ( p ) } _ { j , j ' } = \\Theta ( \\mu _ j , \\mu _ { j ' } ) \\tilde F _ { j , j ' } , \\Theta ( \\mu _ j , \\mu _ { j ' } ) = \\left [ 1 + \\frac { ( \\mu _ j - \\mu _ { j ' } ) ^ 2 } { 8 \\gamma ^ 2 ( \\mu _ j + \\mu _ { j ' } ) } \\right ] ^ { - 1 } , \\end{align*}"}
+{"id": "1457.png", "formula": "\\begin{align*} \\alpha _ { 0 } : = \\frac { 1 6 } { \\sqrt { 3 } } . \\end{align*}"}
+{"id": "639.png", "formula": "\\begin{align*} \\rho ^ * \\omega & = \\rho ^ * \\left ( \\frac { 4 d x \\wedge d z } { x } \\right ) = \\frac { 4 t ( 1 + \\cos \\theta ) } { t ( 1 + \\cos \\theta ) } d t \\wedge d \\theta \\\\ & = 4 \\ d t \\wedge d \\theta . \\end{align*}"}
+{"id": "3794.png", "formula": "\\begin{align*} \\displaystyle T ( f ) ( x ) = \\int _ { \\mathbb { C } } { A _ q ( z , x ) } f ( z ) | z | ^ { \\frac { 2 } { q } - 2 } e ^ { - \\frac { | z | ^ 2 } { q } } d A ( z ) , \\end{align*}"}
+{"id": "2034.png", "formula": "\\begin{align*} q = x _ { 0 } + x _ { 1 } i + x _ { 2 } j + x _ { 3 } k , \\end{align*}"}
+{"id": "2420.png", "formula": "\\begin{align*} \\mathcal { I } _ { q } ( s , s ^ { \\prime } ) = \\int _ { 0 } ^ { \\infty } d t \\exp _ { q } ( - s t ) \\ ; [ \\exp _ { q } ( - s ^ { \\prime } t ) ] ^ { 2 q - 3 } = \\frac { 1 } { ( 2 - q ) } \\ ; \\frac { 1 } { ( s - s ^ { \\prime } ) } . \\end{align*}"}
+{"id": "7687.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\underset { j = 1 } { \\overset { N } { \\sum } } \\partial _ { x ^ j x ^ j } \\phi ^ { N , i } ( t , \\boldsymbol { x } ) \\sigma ^ 2 + \\frac { 1 } { 2 } \\underset { j , \\tau = 1 } { \\overset { N } { \\sum } } \\partial _ { x ^ j } \\partial _ { x ^ \\tau } \\phi ^ { N , i } ( t , \\boldsymbol { x } ) \\sigma _ 0 ^ 2 = \\frac { 1 } { 2 } \\partial _ { \\nu \\nu } \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } ) \\sigma _ 0 ^ 2 + O ( \\frac { 1 } { N } ) . \\end{align*}"}
+{"id": "3291.png", "formula": "\\begin{align*} \\left | \\mathbb E \\left [ \\langle B _ 1 , B _ 2 \\rangle \\sum _ { i = 1 } ^ n \\int _ { ( i - 1 ) \\Delta _ n } ^ { i \\Delta _ n } \\Delta _ n ^ { - ( \\mathfrak H + \\epsilon ) } \\langle ( \\mathcal S ( i \\Delta _ n - s ) - I ) B _ 1 , B _ 2 \\rangle d s \\right ] \\right | > c , \\end{align*}"}
+{"id": "8762.png", "formula": "\\begin{align*} \\langle \\vartheta _ l , \\varphi _ l \\rangle _ { \\C } = 0 \\ \\ \\textrm { o n } \\ \\C \\ \\textrm { f o r } \\ l = 1 , 2 . \\end{align*}"}
+{"id": "1148.png", "formula": "\\begin{align*} { S } ^ { \\rm P R } ( { \\cal A } ) = \\kappa ( \\rho ) \\| W | _ { \\mathcal D } \\| _ F \\frac { W | _ { \\mathcal D } } { \\| W | _ { \\mathcal D } \\| _ F } . \\end{align*}"}
+{"id": "1887.png", "formula": "\\begin{align*} \\mathcal D ( x , y ) = - | x | ^ 2 | y | ^ 2 \\sin ^ 2 \\ ! \\ ! \\measuredangle ( x , y ) + ( 1 - | x | ^ 2 ) ( 1 - | y | ^ 2 ) , \\end{align*}"}
+{"id": "7368.png", "formula": "\\begin{gather*} \\abs { P ( f ) } \\le P ( \\abs { f } ) \\le \\sup \\abs { g } \\ , P ( \\abs { \\pi _ 2 - \\pi _ 1 } ) \\\\ = \\sup \\abs { g } \\ , P ( \\pi _ 1 - \\pi _ 2 ) = \\sup \\abs { g } \\ , \\bigl \\{ P ( \\pi _ 1 ) - P ( \\pi _ 2 ) \\bigr \\} = 0 . \\end{gather*}"}
+{"id": "3326.png", "formula": "\\begin{align*} \\dim K = | \\sigma | - 1 > | \\tau | - 1 = \\dim L . \\end{align*}"}
+{"id": "4176.png", "formula": "\\begin{align*} \\partial _ t w + ( \\mathbf { v } \\cdot \\nabla ) w = 0 . \\end{align*}"}
+{"id": "800.png", "formula": "\\begin{align*} ( V , \\partial ^ \\square c ) = - \\mathrm { E } ( \\Sigma , c ) = - ( V _ g , w _ g ) \\end{align*}"}
+{"id": "296.png", "formula": "\\begin{align*} \\tilde { d } _ H ( \\xi , \\eta ) = \\tilde { d } _ H ( p _ 0 , q _ { \\delta , \\beta } ) . \\end{align*}"}
+{"id": "1237.png", "formula": "\\begin{align*} \\abs { \\log \\left ( \\frac { \\rho ( \\xi _ { \\Delta } \\eta _ { \\Lambda \\setminus \\Delta } ) } { \\rho ( \\eta _ { \\Lambda } ) } \\right ) } = \\abs { \\log \\left ( \\frac { \\rho ( \\xi _ { \\Delta } \\lvert \\eta _ { \\Lambda \\setminus \\Delta } ) } { \\rho ( \\eta _ { \\Delta } \\lvert \\eta _ { \\Lambda \\setminus \\Delta } ) } \\right ) } \\leq \\abs { \\Delta } \\log \\left ( \\frac { 1 } { \\delta ( \\rho ) } \\right ) . \\end{align*}"}
+{"id": "6901.png", "formula": "\\begin{align*} g ( x ) = G \\bigg ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { x _ i } \\bigg ) , \\ \\ G : \\P ( \\R ) \\to \\R , \\end{align*}"}
+{"id": "3969.png", "formula": "\\begin{align*} M _ { I J } ^ \\prime = \\begin{pmatrix} G _ { I _ 1 } ^ \\prime & G _ { I _ 2 } ^ \\prime & \\cdots & G _ { I _ { 2 k } } ^ \\prime \\\\ G _ { J _ 1 } ^ \\prime & G _ { J _ 2 } ^ \\prime & \\cdots & G _ { J _ { 2 k } } ^ \\prime \\end{pmatrix} \\end{align*}"}
+{"id": "7811.png", "formula": "\\begin{align*} \\Phi ( v , p ) = Q F ^ q ( v + \\psi ( v , p ) , p ) , \\end{align*}"}
+{"id": "6548.png", "formula": "\\begin{align*} ( \\langle z _ 1 , \\ldots , z _ n \\rangle \\in \\mathfrak { C } ) \\implies ( z _ 1 = z _ n ) z _ 1 \\ldots , z _ n \\in X . \\end{align*}"}
+{"id": "3474.png", "formula": "\\begin{align*} | \\phi ( k ) | \\leq & \\frac { | \\tilde { P } _ { x _ 2 - k } ( \\theta _ { k + 1 } ) | } { | \\tilde { P } _ { 2 q _ n - 1 } ( \\theta _ { x _ 1 } ) | } \\prod _ { j = x _ 1 } ^ k | \\cos ( \\pi \\theta _ j ) | \\ , | \\phi ( x _ 1 - 1 ) | + \\frac { | \\tilde { P } _ { k - x _ 1 } ( \\theta _ { x _ 1 } ) | } { | \\tilde { P } _ { 2 q _ n - 1 } ( \\theta _ { x _ 1 } ) | } \\prod _ { j = k } ^ { x _ 2 } | \\cos ( \\pi \\theta _ j ) | \\ , | \\phi ( x _ 2 + 1 ) | \\end{align*}"}
+{"id": "8057.png", "formula": "\\begin{align*} R _ { c , k } = \\log _ 2 \\left ( 1 + \\gamma _ { c , k } \\right ) . \\end{align*}"}
+{"id": "1778.png", "formula": "\\begin{align*} P = \\sum _ { k = 0 } ^ d p _ { k } x _ 1 ^ { d - k } x _ 2 ^ k , \\end{align*}"}
+{"id": "4342.png", "formula": "\\begin{align*} \\int _ { \\{ z \\in M : - \\Psi ( z ) = t \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h = \\int _ { \\{ z \\in M : - \\Psi ( z ) = t \\} } | F _ { t _ 0 , \\tilde { c } } - \\tilde { F } | ^ 2 _ h \\end{align*}"}
+{"id": "1100.png", "formula": "\\begin{align*} x = \\sqrt { { p _ s } } { x _ s } + \\sqrt { { p _ b } } { x _ b } , \\end{align*}"}
+{"id": "3709.png", "formula": "\\begin{align*} Q ( h , v ) = \\Big ( v A + \\bar u A ' ( h ) - ( \\nu ( u ) ) ' \\bar g ^ \\intercal , \\ , 2 v \\Big ) \\mbox { o n } \\hat \\Sigma . \\end{align*}"}
+{"id": "2521.png", "formula": "\\begin{align*} \\mathbf { d } _ { h y } = 0 . \\end{align*}"}
+{"id": "6354.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { h ( x ) } { 1 / x } = & \\frac { 2 \\theta } { \\operatorname { B } ( a , b ) } \\lim _ { x \\to \\infty } \\frac { \\left [ 1 - \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) \\right ] ^ { b - 1 } / x ^ 2 } { \\operatorname { I } _ { 1 - \\exp ( - \\theta / x ^ 2 ) } ( b , a ) } . \\end{align*}"}
+{"id": "732.png", "formula": "\\begin{align*} \\begin{aligned} & ( i ) & \\left | \\tilde w ^ \\varepsilon ( x , t ) \\right | & \\leq K , \\medskip \\\\ & ( i i ) & \\left | \\partial _ x \\tilde w ^ \\varepsilon ( x , t ) \\right | & \\leq K , \\medskip \\\\ & ( i i i ) & \\left | \\tilde w ^ \\varepsilon ( x , t ) - \\tilde w ^ \\varepsilon ( x , s ) \\right | & \\leq K \\left ( \\left | t - s \\right | ^ { 1 / 2 } + | t - s | \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "4599.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ d y _ { n , j } < \\min _ { k = 1 , 2 , \\ldots , d } \\left ( x _ { n + 1 , k } + \\sum _ { j \\neq k } x _ { 0 , j } \\right ) \\forall n \\geq N _ 0 . \\end{align*}"}
+{"id": "8739.png", "formula": "\\begin{align*} \\psi _ n ( k h ) = \\psi ( h ) \\forall k \\in K _ n , \\forall h \\in H . \\end{align*}"}
+{"id": "8482.png", "formula": "\\begin{align*} \\mathcal { P } ^ - ( D _ j ) = \\mathcal { P } ^ - \\left ( R ( x ; z ) V _ { j } ( k ) \\right ) ( k ) , j = 1 , 2 , \\end{align*}"}
+{"id": "7302.png", "formula": "\\begin{align*} R _ { k , j } ^ m = B _ { \\rm R F } \\log _ 2 \\left ( 1 + \\frac { p _ { k , j } ^ m \\left | G _ { k , j } ^ m \\right | ^ 2 } { { \\sum \\limits _ { j ' \\ne j } \\sum \\limits _ { k ' \\ne { { k } } } { p _ { k ' , j ' } ^ { { m } } { { { \\left | G _ { k ' , j } ^ m \\right | ^ 2 } } } } } + N _ { \\rm R F } B _ { \\rm R F } } \\right ) , k \\ne 0 , \\end{align*}"}
+{"id": "6074.png", "formula": "\\begin{align*} | C _ G ( x ) | = 1 + \\deg ( x ) = { \\frac { | G | } { | x ^ G | } } , \\textrm { f o r a l l } x \\in G . \\end{align*}"}
+{"id": "2748.png", "formula": "\\begin{align*} | y - s | \\geq | s | - | y | \\geq | s | - \\frac 3 2 | x | > | s | - \\frac 3 4 | s | = \\frac 1 4 | s | . \\end{align*}"}
+{"id": "2921.png", "formula": "\\begin{align*} v ( x ) \\left \\{ \\begin{array} { c l } \\ge 0 & \\bar u ( x ) = u _ a , \\\\ \\le 0 & \\bar u ( x ) = u _ b . \\end{array} \\right . \\end{align*}"}
+{"id": "2958.png", "formula": "\\begin{align*} R _ { \\bar A } ( c ( x _ n ) ) = \\Phi ( t _ n , c ( x _ n ) ) \\in \\partial A . \\end{align*}"}
+{"id": "7716.png", "formula": "\\begin{align*} s _ { \\min } ( Q ) & = \\min _ { v \\in S ^ { k _ 0 - 1 } } \\| \\tilde Q v \\| \\ge \\min _ { v \\in S ^ { k _ 0 - 1 } } \\max _ { i \\in [ k _ 0 ] } \\| P _ i \\tilde Q v \\| _ 2 = \\min _ { v \\in S ^ { k _ 0 - 1 } } \\max _ { i \\in [ k _ 0 ] } \\| P _ i ( \\tilde Q _ { [ 2 k _ 0 ] , i } ) \\| _ 2 | v _ i | \\\\ & \\geq \\frac { \\min _ { j \\in [ k _ 0 ] } \\| P _ j ( \\tilde Q _ { [ 2 k _ 0 ] , j } ) \\| _ 2 } { \\sqrt { k _ 0 } } . \\end{align*}"}
+{"id": "3788.png", "formula": "\\begin{align*} B _ q ( \\psi _ m ) ( z ) = \\displaystyle \\frac { z ^ m } { \\sqrt { \\Gamma ( q m + 1 ) } } , \\end{align*}"}
+{"id": "6630.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { I } ( x ) = & \\inf \\left \\{ \\int ^ { 1 } _ { 0 } \\Gamma ( \\phi ( r ) ) d r ; ~ \\phi \\in L ^ 2 ( [ 0 , 1 ] ) \\ { \\rm i s \\ s u c h \\ t h a t \\ } \\right . \\\\ & \\left . x ( t ) = x _ { 0 } + \\int ^ { t } _ { 0 } b ( x ( s ) ) d s + \\int ^ { t } _ { 0 } \\sigma ( x ( s ) ) \\phi ( s ) d s \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "8639.png", "formula": "\\begin{align*} { \\bf { h } } \\left [ n \\right ] = \\sum \\limits _ { l = 1 } ^ L { { \\bf { h } } _ l \\delta \\left [ { n - { n _ l } } \\right ] } , \\end{align*}"}
+{"id": "2682.png", "formula": "\\begin{align*} R & = ( a _ 1 , b _ 1 ) \\times \\cdots \\times ( a _ d , b _ d ) , \\\\ R ' & = ( a ' _ 1 , b ' _ 1 ) \\times \\cdots \\times ( a ' _ d , b ' _ d ) . \\end{align*}"}
+{"id": "2824.png", "formula": "\\begin{align*} \\mathbf { r } ^ { ( k ) } = g _ { \\Phi , \\rho ^ { ( k ) } } \\left ( \\mathbf { x } ^ { ( k - 1 ) } , \\mathbf { y } \\right ) = \\mathbf { x } ^ { ( k - 1 ) } - \\rho ^ { ( k ) } \\Phi ^ { \\top } \\left ( \\Phi \\mathbf { x } ^ { ( k - 1 ) } - \\mathbf { y } \\right ) . \\end{align*}"}
+{"id": "4890.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 f ( z ) } { \\partial z ^ 2 } = & \\frac { 1 } { \\mathrm { B } ( \\alpha , \\beta ) } \\ , \\phi ( z ) \\ , [ \\Phi ( z ) ] ^ { \\alpha - 2 } \\ , [ 1 - \\Phi ( z ) ] ^ { \\beta - 2 } \\ , \\frac { \\partial s ( z ) } { \\partial z } . \\end{align*}"}
+{"id": "7366.png", "formula": "\\begin{gather*} T ( A ) = 1 \\quad A \\in \\mathcal { R } A \\supset \\{ \\pi _ 1 > \\pi _ 2 \\} . \\end{gather*}"}
+{"id": "2654.png", "formula": "\\begin{align*} \\sum \\limits _ { n = 0 } ^ { \\infty } d _ { o } ( n ) q ^ { n } \\sum _ { n = 0 } ^ { \\infty } q ^ { n ( n + 1 ) / 2 } \\equiv \\sum \\limits _ { n = 1 } ^ { \\infty } \\frac { q ^ { 2 n } } { 1 - q ^ { 2 n } } - \\sum \\limits _ { n = 1 } ^ { \\infty } \\frac { q ^ { 4 n } } { 1 - q ^ { 4 n } } \\pmod { 2 } \\end{align*}"}
+{"id": "1383.png", "formula": "\\begin{align*} g _ { x _ d } ( x ) & = 0 , x _ d = x _ { d - 1 } \\\\ g _ { x _ { d - 1 } } ( x ) & = 0 , x _ { d - 1 } = x _ { d - 2 } \\\\ & \\ldots \\\\ g _ { x _ 2 } ( x ) & = 0 , x _ { 2 } = x _ 1 . \\end{align*}"}
+{"id": "907.png", "formula": "\\begin{align*} & \\Sigma _ 1 = \\sum \\limits _ { D \\leq d < 2 D } \\sum \\limits _ { n \\in \\mathcal { N } ' ( d ) } \\psi \\left ( \\frac { - n } { d ^ 2 } \\right ) \\ , , \\\\ & \\Sigma _ 2 = \\sum \\limits _ { D \\leq d < 2 D } \\sum \\limits _ { n \\in \\mathcal { N } ' ( d ) } \\psi \\left ( \\frac { X - n } { d ^ 2 } \\right ) \\ , . \\end{align*}"}
+{"id": "6421.png", "formula": "\\begin{align*} \\phi = \\frac { 1 } { 2 } \\left ( u _ { , p } \\xi ^ p + v _ q x ^ q \\right ) . \\end{align*}"}
+{"id": "1804.png", "formula": "\\begin{align*} q \\frac { d } { d q } E _ { 0 } ( q ) = \\sum _ { k , r = 1 } ^ { \\infty } \\chi _ { - 3 } ( k r ) r \\left ( \\frac { 1 } { 3 } q ^ { \\frac { k r } { 3 } } - q ^ { k r } \\right ) & = \\frac { 1 } { 9 } b ( q ^ { \\frac { 1 } { 3 } } ) c ( q ) - \\frac { 1 } { 3 } b ( q ) c ( q ^ { 3 } ) \\\\ & = \\frac { 1 } { 9 } \\left ( a ( q ) c ( q ) - c ^ { 2 } ( q ) - a ( q ) b ( q ) + b ^ { 2 } ( q ) \\right ) . \\end{align*}"}
+{"id": "4139.png", "formula": "\\begin{align*} \\Theta f ( x ' , t ) : = \\int _ { \\R ^ { n - 1 } } \\theta ( x ' , t ; y ' ) f ( y ' ) \\ , d y ' , ( x ' , t ) \\in \\R ^ n _ + . \\end{align*}"}
+{"id": "6054.png", "formula": "\\begin{align*} \\kappa _ 1 = a \\kappa _ 2 + b , \\end{align*}"}
+{"id": "7207.png", "formula": "\\begin{align*} W ( t , x ) = \\sqrt { C _ { d } \\kappa } \\ , \\sum _ { k \\in \\Z ^ d _ 0 } \\sum _ { i = 1 } ^ { d - 1 } \\theta _ { k } \\sigma _ { k , i } ( x ) W ^ { k , i } _ { t } , \\end{align*}"}
+{"id": "4897.png", "formula": "\\begin{align*} \\left . \\frac { \\partial s ( z ) } { \\partial z } \\right | _ { z = 0 } = - \\frac { 1 } { 4 } + \\frac { 1 - \\alpha } { \\pi } . \\end{align*}"}
+{"id": "168.png", "formula": "\\begin{align*} ( F ^ { \\bullet , R l a } ) ^ { R l a } & = \\varinjlim _ { h , h ' \\to \\infty } ( F ^ { R G _ { ( h ' ) } - a n } ) ^ { R G _ { ( h ) } - a n } \\\\ & = \\varinjlim _ { h \\to \\infty } ( F ^ { R G _ { ( h ) } - a n } ) ^ { R G _ { ( h ) } - a n } \\\\ & = \\varinjlim _ { h \\to \\infty } F ^ { R G _ { ( h ) } - a n } \\\\ & = F ^ { R l a } , \\end{align*}"}
+{"id": "6653.png", "formula": "\\begin{align*} \\hat { Y } ^ { \\epsilon } ( t ) & = - \\big ( 3 \\delta ^ { 1 / 2 } + \\sup _ { t \\in [ 0 , 1 ] } | \\varphi ( t ) - \\varphi _ m ( t ) | ^ { 1 / 2 } \\big ) + \\int ^ { t } _ { 0 } \\big ( b ( \\hat { X } ^ { \\epsilon } ( s ) ) - b ( X ^ { \\epsilon } ( s ) ) \\big ) d s \\\\ & \\quad - \\lambda ( \\epsilon ) \\epsilon \\int ^ { t } _ { 0 } \\big ( \\sigma ( \\hat { X } ^ { \\epsilon } ( s - ) ) - \\sigma ( X ^ { \\epsilon } ( s - ) ) \\big ) \\epsilon \\theta ^ { \\epsilon } ( s - ) \\tilde { N } ^ { \\epsilon ^ { - 2 } } ( d s ) . \\end{align*}"}
+{"id": "9132.png", "formula": "\\begin{align*} \\Sigma _ 0 = \\Sigma \\setminus ( S _ p ( F ) \\cup S _ { \\infty } ( F ) ) \\supset S . \\end{align*}"}
+{"id": "7400.png", "formula": "\\begin{align*} d ( z ) & = \\sum _ { \\alpha \\in S _ + } : X ^ \\alpha \\varphi ^ \\alpha : - \\frac { 1 } { 2 } \\sum _ { \\alpha , \\beta , \\gamma \\in S _ + } { f ^ { \\alpha \\beta } } _ \\gamma : \\varphi _ \\gamma \\varphi ^ \\alpha \\varphi ^ \\beta : + \\sum _ { \\alpha \\in S _ + } ( f | q ^ \\alpha ) \\varphi ^ \\alpha + \\sum _ { \\alpha \\in S _ { \\frac { 1 } { 2 } } } : \\varphi ^ \\alpha \\Phi _ \\alpha : . \\end{align*}"}
+{"id": "2355.png", "formula": "\\begin{align*} & y _ { n } ( x , 0 ) \\leq y _ { m } ( x , 0 ) , \\forall x \\in [ 0 , m ] . \\\\ & y _ n ( 0 , t ) = 0 = y _ m ( 0 , t ) , \\ \\ y _ n ( m , t ) \\leq 1 = y _ m ( m , t ) , \\forall t > 0 . \\end{align*}"}
+{"id": "4630.png", "formula": "\\begin{align*} \\widetilde { \\tau _ 2 } { \\widetilde { \\sigma } } ^ { - 1 } { \\widetilde { \\tau _ 2 } } ^ { - 1 } { \\widetilde { \\tau _ 1 } } ^ { - 1 } = { \\widetilde \\sigma } ^ { - 1 } { \\widetilde { \\tau _ 1 } } ^ { - 1 } [ \\widetilde \\sigma , \\widetilde { \\tau _ 2 } ] . \\end{align*}"}
+{"id": "4115.png", "formula": "\\begin{align*} H _ r ( E ^ { \\vee } ) = H _ r ( E ) ^ { - 1 } \\end{align*}"}
+{"id": "2066.png", "formula": "\\begin{align*} \\big [ M ^ k , v \\wedge \\partial _ v \\big ] = k M ^ { k - 1 } \\big [ M , v \\wedge \\partial _ v \\big ] = 2 k M ^ { k - 1 } \\sum _ { 1 \\leq i \\leq 2 } \\Lambda _ i \\big ( \\big [ \\Lambda _ i , v \\big ] \\wedge \\partial _ v \\big ) , \\end{align*}"}
+{"id": "779.png", "formula": "\\begin{align*} \\delta = \\frac { \\gamma } { 1 + 2 \\gamma } , \\end{align*}"}
+{"id": "2624.png", "formula": "\\begin{align*} \\circlearrowleft _ { x , y , z } \\varepsilon ( z , x ) \\Big ( 2 \\alpha ^ 2 ( h ) [ [ x , y ] , \\alpha ( z ) ] \\Big ) & = \\circlearrowleft _ { x , y , z } \\varepsilon ( z , x ) \\Big ( [ \\alpha ( h ) , [ x , y ] , \\alpha ^ 2 ( z ) ] \\\\ & + \\varepsilon ( h , x + y ) [ \\alpha ( [ x , y ] ) , \\alpha ( h \\cdot z ) ] \\Big ) . \\end{align*}"}
+{"id": "9294.png", "formula": "\\begin{align*} \\mathcal { E } : = u _ 1 ^ t \\Delta u _ 2 ^ t \\wedge \\dots \\wedge \\Delta u _ { k } ^ t \\wedge \\beta _ n ^ { n - m } - u _ 1 \\Delta u _ 2 \\wedge \\dots \\wedge \\Delta u _ { k } \\wedge \\beta _ n ^ { n - m } \\longrightarrow 0 , t \\to \\infty , \\end{align*}"}
+{"id": "8532.png", "formula": "\\begin{align*} T _ { \\delta } - \\mathcal { P } ^ - \\left ( T _ { \\delta } R _ { \\delta } \\right ) = F _ { \\delta } , \\end{align*}"}
+{"id": "3502.png", "formula": "\\begin{align*} \\| ( j - \\ell _ w ) \\alpha \\| \\geq \\| q _ n \\alpha \\| \\geq \\frac { 1 } { 2 q _ { n + 1 } } = \\frac { 1 } { 2 } e ^ { - \\beta _ n q _ n } \\geq \\frac { 1 } { 2 } e ^ { - 3 0 0 \\varepsilon q _ n } . \\end{align*}"}
+{"id": "7935.png", "formula": "\\begin{align*} D _ x ( A ^ q \\eta ) ( x ) & = D _ x \\left ( \\int _ { x - r } ^ { x + r } ( \\eta ( y ) - \\eta ( x ) ) \\cos ( 2 \\pi q ( y - x ) ) \\ \\d y \\right ) \\\\ & = ( \\eta ( x + r ) - \\eta ( x ) ) \\cos ( 2 \\pi q r ) - ( \\eta ( x - r ) - \\eta ( x ) ) \\cos ( 2 \\pi q r ) \\\\ & + \\int _ { x - r } ^ { x + r } - \\partial \\eta ( x ) \\ \\cos ( 2 \\pi q ( y - x ) ) + 2 \\pi q ( \\eta ( y ) - \\eta ( x ) ) \\sin ( 2 \\pi q ( y - x ) ) \\ \\d y . \\end{align*}"}
+{"id": "7736.png", "formula": "\\begin{align*} 0 = u c \\lim _ i \\| \\varpi \\otimes \\varpi ( \\cdot ) \\xi _ i - \\xi _ i \\| \\end{align*}"}
+{"id": "4015.png", "formula": "\\begin{align*} \\begin{cases} a \\mapsto a b , \\\\ c \\mapsto c , & c \\neq a , \\end{cases} \\begin{cases} a \\mapsto b a , \\\\ c \\mapsto c , & c \\neq a . \\end{cases} \\end{align*}"}
+{"id": "8744.png", "formula": "\\begin{align*} ( D _ { b ^ - } ^ { \\alpha } f ) ( x ) : = ( - 1 ) \\frac { d } { d x } \\left [ ( { \\bf I } _ { b ^ - } ^ { 1 - \\alpha } f ) ( x ) \\right ] . \\end{align*}"}
+{"id": "7268.png", "formula": "\\begin{align*} \\ell ( x ) = a _ n x ^ n + a _ { n - 1 } x ^ { n - 1 } + \\cdots + a _ 0 \\end{align*}"}
+{"id": "457.png", "formula": "\\begin{align*} 1 \\le \\frac { \\sup _ { t \\ge x } f ( t ) } { f ( x ) } = \\sup _ { t \\ge x } \\frac { f ( t ) } { \\alpha ( x ) } \\frac { \\alpha ( x ) } { f ( x ) } \\le \\sup _ { t \\ge x } \\frac { f ( t ) } { \\alpha ( t ) } \\cdot \\frac { \\alpha ( x ) } { f ( x ) } \\to 1 \\end{align*}"}
+{"id": "7473.png", "formula": "\\begin{align*} | p | _ { A ( x ) } : = \\left ( a _ { i j } ( x ) p _ i p _ j \\right ) ^ { 1 / 2 } \\end{align*}"}
+{"id": "7537.png", "formula": "\\begin{align*} E ( G ) = \\{ V _ i \\cup V _ j : \\ : \\{ i , j \\} \\in F \\} . \\end{align*}"}
+{"id": "1079.png", "formula": "\\begin{align*} s _ \\alpha ( - w ^ { - 1 } \\gamma ) = \\langle \\alpha ^ \\vee , w ^ { - 1 } \\gamma \\rangle \\alpha - w ^ { - 1 } \\gamma \\in \\Phi ^ - . \\end{align*}"}
+{"id": "2211.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } i . ~ ~ \\phi = 2 \\pi N , ~ N ~ i s ~ i n t e g e r \\\\ i i . ~ ~ \\phi = \\frac { 2 \\pi Z } { N } , \\frac { Z } { N } ~ i s ~ n o t ~ i n t e g e r . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "6864.png", "formula": "\\begin{align*} \\hat z = \\Phi ( z ) = \\frac { 1 } { \\Phi _ 1 ( z ) } \\end{align*}"}
+{"id": "9073.png", "formula": "\\begin{align*} G r _ w ( E ) = G r _ w ( F ) . \\end{align*}"}
+{"id": "5321.png", "formula": "\\begin{align*} f ( z , t ) = \\int _ \\Omega \\int _ { \\C ^ n } \\widehat { f } ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \\ , d \\nu ( a ) . \\end{align*}"}
+{"id": "8655.png", "formula": "\\begin{align*} y \\left [ i \\right ] = \\left ( { \\sum \\limits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { \\bf { H } } _ l ^ \\bot { \\bf { X } } _ l } } \\right ) { \\bf { d } } \\left [ { i - { n _ { \\max } } } \\right ] + z \\left [ i \\right ] . \\end{align*}"}
+{"id": "4721.png", "formula": "\\begin{align*} & { M i n i m i z e \\ , } _ { x \\in \\mathbb { R } ^ n } \\ \\ J ( x ) : = a _ J x ^ T x + 2 b _ J ^ T x + c _ J \\\\ & s . t . \\ \\ f _ k ( x ) : = a _ k x ^ T x + 2 b _ k ^ T x + c _ k \\le 0 , \\ \\ k = 1 , . . . , m \\end{align*}"}
+{"id": "238.png", "formula": "\\begin{align*} \\dot g ( J ) \\in A t ^ { \\rho ( \\dot g ( J ) ) } , \\end{align*}"}
+{"id": "1947.png", "formula": "\\begin{align*} = z ^ { - \\mu } 2 ^ { \\mu - 1 } G _ { { 0 } , { 2 } } ^ { { 2 } , { 0 } } \\left ( \\frac { z ^ { 2 } } { 4 } \\bigg { | } \\begin{array} { l l l } \\\\ \\frac { \\mu + \\nu } { 2 } ~ , ~ \\frac { \\mu - \\nu } { 2 } \\end{array} \\right ) , \\end{align*}"}
+{"id": "7327.png", "formula": "\\begin{align*} | A | ^ m | A ' + B _ 1 + \\ldots + B _ m | \\le ( 1 + \\varepsilon ) | A ' | \\prod _ { i = 1 } ^ m | A + B _ i | . \\end{align*}"}
+{"id": "5484.png", "formula": "\\begin{align*} h _ q '' ( t ) = - 2 ^ q \\Gamma ( 2 - q ) q ( q + 1 ) ( 3 - t ) ^ { - q - 2 } + \\frac { q ^ 2 ( 1 - q ^ 2 ) } { 6 } \\end{align*}"}
+{"id": "7326.png", "formula": "\\begin{align*} { \\rho ^ { \\circ } ( K ) } = { \\sup _ { x \\in K } \\rho ( \\{ x \\} ) } < { \\infty } , \\end{align*}"}
+{"id": "4070.png", "formula": "\\begin{align*} e ( 0 , G ^ { N ( 1 , 0 ) } _ { 1 } ) & = 0 + ( 2 - 1 - 1 ) = 0 , & e ( - 1 , G ^ { N ( 1 , 0 ) } _ { 2 } ) & = - 1 + ( 2 - 1 - 0 ) = 0 . \\end{align*}"}
+{"id": "3843.png", "formula": "\\begin{align*} \\| X _ { n + 1 } \\| ^ 2 = & \\ \\| L _ n ( X _ n , Y _ { n + 1 } ) \\| ^ 2 + \\| F _ n ( X _ n , Y _ { n + 1 } ) \\| ^ 2 + \\| \\hat G _ n ( X _ n , Y _ { n + 1 } ) \\| ^ 2 + \\| \\tilde G ( X _ n , Y _ { n + 1 } , \\xi _ { n + 1 } ) \\| ^ 2 \\\\ & \\ + 2 \\ < L _ n ( X _ n , Y _ { n + 1 } ) , F _ n ( X _ n , Y _ { n + 1 } ) \\ > + 2 \\ < ( L _ n + F _ n + \\hat G _ n ) ( X _ n , Y _ { n + 1 } ) , \\tilde G ( X _ n , Y _ { n + 1 } , \\xi _ { n + 1 } ) \\ > \\\\ & + 2 \\ < ( L _ n + F _ n ) ( X _ n , Y _ { n + 1 } ) , \\hat G _ n ( X _ n , Y _ { n + 1 } ) \\ > . \\end{align*}"}
+{"id": "8717.png", "formula": "\\begin{align*} \\chi ^ g = \\chi \\Rightarrow \\chi ( n ^ g ) = \\chi ( n ) \\forall n \\in N \\Rightarrow n ^ g = n \\forall n \\in N . \\end{align*}"}
+{"id": "7293.png", "formula": "\\begin{align*} F ( z ) ( 0 ) = \\lambda _ { \\Theta } [ ( \\Theta \\circ h ) ( z ) , \\alpha _ z ] ( 0 ) = ( \\Theta \\circ h ) ( z ) \\end{align*}"}
+{"id": "60.png", "formula": "\\begin{align*} \\# \\{ \\beta \\in \\Phi ^ + \\setminus I \\mid \\varphi ( v ' \\beta ) < 0 \\} = & \\# \\{ \\beta \\in \\Phi ^ + \\setminus ( I \\cup \\{ \\alpha \\} ) \\mid \\varphi ( v s _ \\alpha \\beta ) < 0 \\} \\\\ = & \\# \\{ \\beta \\in \\Phi ^ + \\setminus ( I \\cup \\{ \\alpha \\} ) \\mid \\varphi ( v \\beta ) < 0 \\} \\\\ = & \\# \\{ \\beta \\in \\Phi ^ + \\setminus I \\mid \\varphi ( v \\beta ) < 0 \\} - 1 . \\end{align*}"}
+{"id": "4021.png", "formula": "\\begin{align*} \\varphi ^ L _ { \\sigma , s ' } ( D ) = \\varphi ^ L _ { \\sigma , s ' } ( \\{ c \\} ) \\varphi ^ L _ { \\sigma , s ' } ( E ) = \\varphi ^ L _ { \\sigma , s ' } ( C _ i ) . \\end{align*}"}
+{"id": "377.png", "formula": "\\begin{align*} U ^ { \\dagger } = A + \\sum _ { k = 1 } ^ { n - 2 } \\Delta _ k . \\end{align*}"}
+{"id": "4668.png", "formula": "\\begin{align*} \\psi _ { \\tau } ( p ) \\stackrel { d e f } { = } \\left [ \\ p \\int _ 0 ^ { \\infty } \\ t ^ { p - 1 } \\ T _ { \\tau } ( t ) \\ d t \\ \\right ] ^ { 1 / p } = | | \\tau | | L _ p ( \\Omega ) , \\end{align*}"}
+{"id": "5855.png", "formula": "\\begin{align*} { f _ { 1 3 } } = { f _ { 1 4 } } - \\frac { 1 } { 2 } \\left ( { { f _ 1 } - { f _ 2 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) - \\frac { 1 } { 8 } \\left ( { 2 { F _ x } - { F _ z } } \\right ) - \\frac { 1 } { 6 } \\rho { u _ z } , \\end{align*}"}
+{"id": "3054.png", "formula": "\\begin{align*} u = z \\cdot ( n _ 1 , \\dotsc , n _ l ) \\in \\C ^ l \\end{align*}"}
+{"id": "9083.png", "formula": "\\begin{align*} \\chi _ 2 \\geq ( 1 - \\sum \\limits _ { j = 3 } ^ n w _ j ) \\chi - \\chi _ 1 - \\sum \\limits _ { j = 3 } ^ n k _ j + ( n - 2 ) r \\end{align*}"}
+{"id": "1541.png", "formula": "\\begin{align*} \\frac { d L } { d t _ { k } } = \\left [ \\epsilon ^ { k - 1 } ( L ^ { k } ) _ { D } , L \\right ] _ \\epsilon = \\epsilon ^ { k } \\left [ ( L ^ { k } ) _ { D } , L \\right ] \\ ; , k \\geq 1 \\ ; , \\end{align*}"}
+{"id": "4109.png", "formula": "\\begin{align*} H ( E ) = \\left ( \\N \\left ( \\prod _ { i = 1 } ^ { \\ell } \\mathfrak a _ i \\right ) \\prod _ { v \\in V _ \\infty } \\det ( h _ { v } ( b _ i , b _ j ) ) ^ { e _ { v } / 2 } \\right ) ^ { 1 / [ K : \\mathbb { Q } ] } \\end{align*}"}
+{"id": "6351.png", "formula": "\\begin{align*} t ( y ) = \\frac { 6 } { y ^ 2 } + 4 ( b - 1 ) \\frac { \\exp ( y ) } { \\left [ 1 - \\exp ( y ) \\right ] ^ 2 } . \\end{align*}"}
+{"id": "9334.png", "formula": "\\begin{align*} \\lambda ^ * = - 3 \\sqrt 2 \\frac { \\sqrt { 1 + a ^ 2 } } { 2 + \\frac { 1 9 7 } { 1 9 9 } a ^ 2 } \\eta . \\end{align*}"}
+{"id": "8548.png", "formula": "\\begin{align*} f ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\ , \\int _ 0 ^ t ( t - \\tau ) ^ { \\alpha - 1 } \\phi ( \\tau ) \\ , d \\tau , \\ t > 0 , \\ 0 < \\alpha < 1 . \\end{align*}"}
+{"id": "7712.png", "formula": "\\begin{align*} h _ s ( i ) : = & \\exp \\Big ( - \\max \\Big \\{ \\frac { 1 2 8 \\alpha \\ , \\log n } { \\tilde \\varepsilon ^ 2 ( 1 + \\tilde \\varepsilon ) ^ { - i } m _ s } , \\ , C _ h \\Big \\} \\Big ) , \\end{align*}"}
+{"id": "589.png", "formula": "\\begin{align*} p x E \\big [ X Y ^ { p - \\gamma } ] E \\big [ Y ^ { p } \\big ] ^ { \\frac { \\beta } { p } } & \\leq p x E \\big [ X \\big ] _ { \\frac { p } { \\gamma } } E \\big [ Y ^ { p } \\big ] ^ { 1 - \\frac { \\gamma - \\beta } { p } } \\\\ & \\leq \\big [ X \\big ] _ { \\frac { p } { \\gamma } } \\big ( ( \\gamma - \\beta ) x ^ { \\frac { p } { \\gamma - \\beta } } + ( p + \\beta - \\gamma ) E \\big [ Y ^ { p } \\big ] \\big ) \\end{align*}"}
+{"id": "3953.png", "formula": "\\begin{align*} ( x _ 1 , x _ 2 , \\cdots , x _ k ) \\bar { G } = { \\bf 0 } , \\end{align*}"}
+{"id": "6977.png", "formula": "\\begin{align*} ( a / b ) ^ { k _ 1 } = ( \\alpha / \\beta ) ^ { l _ 1 } \\zeta \\end{align*}"}
+{"id": "1017.png", "formula": "\\begin{align*} \\prescript { L _ i } { } { } x { } ^ { R _ j } = \\prescript { L _ i } { } { } \\left ( \\prescript L { } { } x { } ^ R \\right ) { } ^ { R _ j } \\underset { } \\leq \\prescript { L _ i } { } { } \\left ( \\prescript L { } { } y { } ^ R \\right ) { } ^ { R _ j } = \\prescript { L _ i } { } { } y { } ^ { R _ j } . \\end{align*}"}
+{"id": "3548.png", "formula": "\\begin{align*} q ^ { 1 / 2 4 } ( - q ^ 2 ; q ^ 3 ) _ \\infty ( - q ; q ^ 3 ) _ \\infty ( q ^ 3 ; q ^ 3 ) _ \\infty & = q ^ { 1 / 2 4 } \\frac { ( q ^ 4 ; q ^ 6 ) _ \\infty } { ( q ^ 2 ; q ^ 3 ) _ \\infty } \\frac { ( q ^ 2 ; q ^ 6 ) _ \\infty } { ( q ; q ^ 3 ) _ \\infty } ( q ^ 3 ; q ^ 3 ) _ \\infty \\\\ & = q ^ { 1 / 2 4 } \\frac { \\varphi _ 2 / \\varphi _ 6 } { \\varphi _ 1 / \\varphi _ 3 } \\varphi _ 3 \\\\ & = q ^ { 1 / 2 4 } \\frac { \\varphi _ 2 \\varphi _ 3 ^ 2 } { \\varphi _ 1 \\varphi _ 6 } = \\frac { \\eta _ 2 \\eta _ 3 ^ 2 } { \\eta _ 1 \\eta _ 6 } . \\end{align*}"}
+{"id": "6919.png", "formula": "\\begin{align*} \\hat { f } _ i ( x _ i ) = \\int _ { \\R ^ { n - 1 } } f ( x _ 1 , \\ldots , x _ n ) \\ , \\prod _ { j \\neq i } Q _ j ( d x _ j ) \\ge - c _ 1 e ^ { c _ 2 x _ i ^ 2 } \\prod _ { j \\neq i } \\int _ { \\R } e ^ { c _ 2 x _ j ^ 2 } Q _ j ( x _ j ) \\ , d x _ j \\end{align*}"}
+{"id": "4943.png", "formula": "\\begin{align*} Z _ { i } ^ { \\theta } : = \\theta _ { i } X _ { i } + ( 1 - \\theta _ { i } ) Y _ { i } , ~ 1 \\leq i \\leq n . \\end{align*}"}
+{"id": "2063.png", "formula": "\\begin{align*} \\| \\Lambda ^ \\ell g \\| _ { ( 2 , 0 ) } ^ 2 \\leq \\left \\{ \\begin{aligned} & \\| M ^ { \\ell / 2 } g \\| _ { ( 2 , 0 ) } ^ 2 \\mbox { f o r e v e n n u m b e r } \\ , \\ , \\ell , \\\\ & \\| M ^ { \\ell / 2 } g \\| _ { ( 2 , 0 ) } ^ 2 \\leq \\norm { M ^ { ( \\ell + 1 ) / 2 } g } _ { ( 2 , 0 ) } \\norm { M ^ { ( \\ell - 1 ) / 2 } g } _ { ( 2 , 0 ) } \\mbox { f o r o d d n u m b e r } \\ , \\ , \\ell . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5530.png", "formula": "\\begin{align*} X _ n = ( \\mathfrak X , \\mathcal O _ { \\mathfrak X } \\otimes _ R R _ n ) , \\end{align*}"}
+{"id": "5121.png", "formula": "\\begin{align*} \\begin{aligned} B & = \\nabla \\times ( \\Phi \\nabla \\theta ) + G \\nabla \\theta , \\\\ A & = \\nabla \\times ( \\eta \\nabla \\theta ) + \\Phi \\nabla \\theta , \\\\ \\Phi & = \\phi - \\phi _ { \\infty } . \\end{aligned} \\end{align*}"}
+{"id": "1579.png", "formula": "\\begin{align*} [ x , e ^ f ( x ) ] = s ( \\pi _ \\mathcal Z ( x ) ) = s ( \\pi _ \\mathcal Z ( \\gamma x ) ) = [ \\gamma x , e ^ f ( \\gamma x ) ] = [ x , \\rho ( \\gamma ) ^ { - 1 } e ^ f ( \\gamma x ) ] , \\end{align*}"}
+{"id": "6954.png", "formula": "\\begin{align*} T _ { W , \\psi } ( \\mu _ \\infty ) - I ( \\mu _ \\infty ) & \\ge \\limsup _ { n \\to \\infty } \\big ( T _ { W _ { n J } , \\psi } ( \\mu _ n ( Q ^ n ) ) - I ( \\mu _ n ( Q ^ n ) ) \\big ) = \\limsup _ { n \\to \\infty } M _ n ^ \\psi ( Q ^ n ) / n . \\end{align*}"}
+{"id": "9165.png", "formula": "\\begin{align*} \\begin{aligned} & \\dfrac { b _ { 1 , \\delta } ( x ) } { 2 } = \\ , \\int _ 0 ^ 1 h ( x + \\delta \\sin ( 2 \\pi t ) ) \\sin ( 2 \\pi t ) \\ , d t \\\\ = & \\ , \\int _ 0 ^ 1 [ \\tilde { h } ( x + \\delta \\sin ( 2 \\pi t ) ) + m ( x + \\delta \\sin ( 2 \\pi t ) ) ] \\sin ( 2 \\pi t ) \\ , d t \\\\ = & \\ , \\int _ 0 ^ 1 \\tilde { h } ( x + \\delta \\sin ( 2 \\pi t ) ) \\sin ( 2 \\pi t ) \\ , d t \\\\ & \\ , + \\int _ 0 ^ 1 [ m ( x + \\delta \\sin ( 2 \\pi t ) ) - m ( x ) ] \\sin ( 2 \\pi t ) \\ , d t . \\end{aligned} \\end{align*}"}
+{"id": "1817.png", "formula": "\\begin{align*} \\begin{aligned} S _ { e , \\mathcal { Y M } ^ 0 } ( X , Y ) & = \\big ( \\exp ( \\frac { | | R ^ { \\nabla } | | ^ 2 } 2 ) - 1 \\big ) g ( X , Y ) - \\exp ( \\frac { | | R ^ { \\nabla } | | ^ 2 } 2 ) \\langle i _ X R ^ { \\nabla } , i _ Y R ^ { \\nabla } \\rangle \\ , , \\\\ \\end{aligned} \\end{align*}"}
+{"id": "5233.png", "formula": "\\begin{align*} \\ddot { \\kappa } + \\frac { 1 } { \\rho } \\dot { \\kappa } + \\left ( 1 - \\frac { ( 3 / 2 ) ^ { 2 } } { \\rho ^ { 2 } } \\right ) \\kappa = 0 , \\kappa & > 0 , 0 < \\rho < R _ 0 , \\\\ \\kappa ( R _ 0 ) & = 0 . \\end{align*}"}
+{"id": "6640.png", "formula": "\\begin{align*} & \\frac { | \\mathbb { G } _ { \\epsilon } ( t ) - \\mathbb { G } _ { \\epsilon } ( s ) | ^ { 2 } } { d ^ 2 ( t , s ) } \\\\ & = \\frac { | \\mathbb { G } _ { \\epsilon } ( t ) - \\mathbb { G } _ { \\epsilon } ( s ) | ^ { 2 } } { \\int ^ { 1 } _ { 0 } ( K ( \\mathcal { G } ^ { - 1 } ( t ) , r ) - K ( \\mathcal { G } ^ { - 1 } ( s ) , r ) ) ^ { 2 } d r } \\cdot \\frac { \\int ^ { 1 } _ { 0 } ( K ( \\mathcal { G } ^ { - 1 } ( t ) , r ) - K ( \\mathcal { G } ^ { - 1 } ( s ) , r ) ) ^ { 2 } d r } { d ^ 2 ( t , s ) } . \\end{align*}"}
+{"id": "9218.png", "formula": "\\begin{align*} \\dot { x } = \\gamma f ( x , t ) x ( 0 ) = x _ 0 . \\end{align*}"}
+{"id": "2989.png", "formula": "\\begin{align*} \\widetilde \\mu _ { ( x , t ) } = ( \\phi _ t ) _ * \\widetilde \\mu _ { ( x , 0 ) } = ( \\phi _ t ) _ * ( \\widehat \\mu _ { x } \\times \\delta _ 0 ) = \\widehat \\mu _ { x } \\times \\delta _ t \\end{align*}"}
+{"id": "7179.png", "formula": "\\begin{align*} E _ 1 = i \\sum _ \\alpha \\frac { \\partial ( q _ 1 - b _ 1 ) } { \\partial \\xi _ \\alpha } \\frac { \\partial q _ 1 } { \\partial x _ \\alpha } + b _ 0 q _ 1 + \\frac { \\partial q _ 1 } { \\partial x _ n } - c _ 1 . \\end{align*}"}
+{"id": "7013.png", "formula": "\\begin{align*} 0 \\ , \\neq \\ , \\left \\vert \\ ! \\begin{array} { c c c } a _ { 1 , 1 } & 0 & b _ 1 \\\\ a _ { 2 , 1 } & a _ { 2 , 2 } & b _ 2 \\\\ 0 & 0 & a _ { 1 , 1 } ^ 2 \\end{array} \\ ! \\right \\vert \\ , = \\ , a _ { 1 , 1 } \\ , a _ { 2 , 2 } \\ , a _ { 1 , 1 } ^ 2 , \\end{align*}"}
+{"id": "995.png", "formula": "\\begin{align*} g ( \\delta _ 1 ) & = - q ^ { d + 1 } ( q ^ 3 - ( \\alpha d + 8 d + 4 \\alpha + 2 2 ) q ^ 2 - ( 8 d + 4 ) q - 4 d ) \\\\ & \\hphantom { = { } } \\mathrel { + } ( d + 5 ) q ^ 6 + ( d + 4 ) q ^ 5 + ( d ^ 2 + 8 d + 1 6 ) q ^ 4 , \\\\ g ( \\delta _ 2 ) & = - q ^ { 2 d + 2 } ( ( \\alpha + 3 ) q - 2 \\alpha ^ 2 - 1 2 \\alpha - 1 8 ) \\\\ & \\hphantom { = { } } \\mathrel { + } ( \\alpha + 3 ) q ^ { d + 5 } + ( d + \\alpha + 6 ) q ^ { d + 4 } \\\\ & \\hphantom { = { } } \\mathrel { + } ( 5 d + 1 0 ) q ^ { d + 3 } + ( 8 d + 4 ) q ^ { d + 2 } + 4 d q ^ { d + 1 } + q ^ 6 . \\end{align*}"}
+{"id": "4171.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\mathbf { w } + ( \\mathbf { v } \\cdot \\nabla ) \\mathbf { w } = ( \\mathbf { w } \\cdot \\nabla ) \\mathbf { v } , \\ \\ & D \\times ( 0 , T ) , \\\\ \\nabla \\cdot \\mathbf { v } = 0 , \\ \\ & D \\times ( 0 , T ) , \\\\ \\mathbf { v } \\cdot \\mathbf { n } = v _ n , \\ \\ & \\partial D \\times ( 0 , T ) , \\\\ \\mathbf { w } ( x , 0 ) = \\nabla \\times \\mathbf { v } _ 0 ( x ) , \\ \\ & D . \\end{cases} \\end{align*}"}
+{"id": "1708.png", "formula": "\\begin{align*} i \\frac { \\partial u } { \\partial t } & = \\Delta u + c u + b \\cdot \\nabla u + f \\ ( s , T ) \\times \\mathbb { R } ^ d , \\\\ u ( s ) & = x , \\end{align*}"}
+{"id": "9266.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { \\Omega } \\Delta u _ { \\epsilon } ( x ) \\wedge \\Delta v _ 1 ( x ) \\wedge \\dots \\wedge \\Delta v _ { m - 1 } ( x ) \\wedge \\beta _ n ^ { n - m } \\wedge \\omega ( x ) \\\\ = & \\int _ { B ( 0 , \\epsilon ) } \\chi _ { \\epsilon } ( y ) d V ( y ) \\int _ { \\Omega } u ( z ) \\Delta v _ 1 ( z + y ) \\wedge \\dots \\wedge \\Delta v _ { m - 1 } ( z + y ) \\wedge \\beta _ n ^ { n - m } \\wedge \\Delta \\omega ( z + y ) \\geq 0 , \\end{aligned} \\end{align*}"}
+{"id": "4727.png", "formula": "\\begin{align*} \\Omega _ 0 : = \\{ ( f _ 0 ( x ) , f _ 1 ( x ) , . . . , f _ m ( x ) ) \\ | \\ x \\in \\mathbb { R } ^ n \\} + i n t \\mathbb { R } ^ { m + 1 } _ + \\end{align*}"}
+{"id": "4107.png", "formula": "\\begin{align*} \\forall x \\in E _ v \\ , , \\ \\Vert x \\Vert _ v = \\begin{cases} \\sqrt { h _ v ( x , x ) } v \\in V _ { \\infty } \\\\ \\inf \\left \\lbrace \\vert \\alpha \\vert _ v \\ , , \\ \\alpha \\in K _ v \\ , , \\ x \\in \\alpha L \\right \\rbrace v \\in V _ f . \\end{cases} \\end{align*}"}
+{"id": "2159.png", "formula": "\\begin{align*} z & = \\frac { 1 } { \\sqrt { 1 - a + \\tau ^ 2 ( 1 + a ) } } \\left ( \\frac { \\sqrt { 1 - a } \\ , \\cos ( t ) } { \\sqrt { 1 + a } } + \\frac { i \\ , \\tau \\ , \\sqrt { 1 + a } \\ , \\sin ( t ) } { \\sqrt { 1 - a } } \\right ) , \\\\ w & = \\frac { 1 } { \\sqrt { 1 - a + \\tau ^ 2 ( 1 + a ) } } \\left ( \\frac { - \\sqrt { 1 - a } \\ , \\sin ( t ) } { \\sqrt { 1 + a } } + \\frac { i \\ , \\tau \\ , \\sqrt { 1 + a } \\ , \\cos ( t ) } { \\sqrt { 1 - a } } \\right ) . \\end{align*}"}
+{"id": "594.png", "formula": "\\begin{align*} \\Sigma ( x , \\mu ) = \\hat { \\kappa } + \\mathrm { d i a g } ( x ) \\hat { \\eta } + \\sum _ { n = 1 } ^ { N } \\null _ { n } \\eta \\int _ { \\R ^ { m } } g _ { n } ( x , y ) \\ , \\mu ( d y ) \\end{align*}"}
+{"id": "7771.png", "formula": "\\begin{align*} | z _ { w } | = | z _ { w - 1 } | = \\cdots = | z _ { w - m + 1 } | < | z _ { w - m } | = \\min _ { j \\ge w - m } | z _ j | . \\end{align*}"}
+{"id": "5688.png", "formula": "\\begin{align*} & x ^ { t + 1 } ( \\gamma ) = ( I + \\gamma B ) ^ { - 1 } \\left ( x ^ t + \\alpha ( \\gamma ) ( x ^ t - x ^ { t - 1 } ) - \\gamma \\left ( F ( x ^ t ) + \\beta ( \\gamma ) ( F ( x ^ t ) - F ( x ^ { t - 1 } ) ) \\right ) \\right ) \\forall \\gamma > 0 , \\\\ & L _ { t + 1 } ( \\gamma ) = \\| F ( x ^ { t + 1 } ( \\gamma ) ) - F ( x ^ { t } ) - \\kappa _ t \\gamma ^ { - 1 } ( x ^ { t + 1 } ( \\gamma ) - x ^ t ) \\| / \\| x ^ { t + 1 } ( \\gamma ) - x ^ t \\| \\forall \\gamma > 0 , \\end{align*}"}
+{"id": "3430.png", "formula": "\\begin{align*} \\delta = \\limsup \\max ( 0 , \\delta _ n ) . \\end{align*}"}
+{"id": "4101.png", "formula": "\\begin{align*} \\{ f , g \\} . \\{ f ' , g ' \\} ( \\xi , \\eta ) & = \\{ f , g \\} ( f ' ( \\xi ) , g ' ( \\xi ) + D f ' ( \\xi ) \\eta ) \\\\ * & = ( f \\circ f ' ( \\xi ) , g \\circ f ' ( \\xi ) + D f ( f ' ( \\xi ) ) ( g ' ( \\xi ) + D f ' ( \\xi ) \\eta ) ) \\\\ * & = ( f \\circ f ' ( \\xi ) , h ( \\xi ) + D ( f \\circ f ' ) ( \\xi ) \\eta ) , \\end{align*}"}
+{"id": "5753.png", "formula": "\\begin{align*} \\langle \\langle \\varphi , \\varphi ' \\rangle \\rangle = \\tau \\langle \\langle \\varphi ' , \\varphi \\rangle \\rangle \\end{align*}"}
+{"id": "7304.png", "formula": "\\begin{align*} R _ j = \\sum \\limits _ { \\forall v } \\sum \\limits _ { \\forall q } R _ { v , j } ^ q + \\sum \\limits _ { \\forall k , k \\ne 0 } \\sum \\limits _ { \\forall m } R _ { k , j } ^ m + \\sum \\limits _ { \\forall n } R _ { 0 , j } ^ n , \\end{align*}"}
+{"id": "9130.png", "formula": "\\begin{align*} E ( \\Q _ { p ^ f , n } ) _ p = \\begin{cases} ( d _ { f , n } , d _ { f , n - 1 } ) _ { R _ { f , n } } & ( n \\geq 0 ) \\\\ ( d _ { f , - 1 } ) _ { R _ { f , - 1 } } & ( n = - 1 ) . \\end{cases} \\end{align*}"}
+{"id": "7529.png", "formula": "\\begin{align*} \\sum _ { x \\in V _ i } \\deg _ { F _ \\ell } ( x ) = \\ell < s = \\frac { d m } { 2 k ^ 2 } . \\end{align*}"}
+{"id": "2800.png", "formula": "\\begin{align*} \\eta ^ { D } ( A \\times [ b , \\infty ) ) = e ^ { - \\pi \\lambda b } \\langle \\eta ^ { D } , f _ { b } \\rangle & = e ^ { - \\pi \\lambda b } \\langle \\eta ^ { D } , f \\rangle \\\\ & = ( \\pi \\lambda ) ^ { - 1 } e ^ { - \\pi \\lambda b } Z _ { \\lambda } ^ { D } ( A ) = \\int _ { A \\times [ b , \\infty ) } Z _ { \\lambda } ^ { D } ( \\mathrm { d } x ) e ^ { - \\pi \\lambda h } \\mathrm { d } h \\end{align*}"}
+{"id": "8885.png", "formula": "\\begin{align*} \\sum _ { \\lambda , \\mu } \\exp \\Big ( \\frac { - \\pi ( u ^ 2 + v ^ 2 ) } { v } S [ \\lambda ] - \\frac { \\pi } { v } S [ \\mu ] - \\frac { 2 \\pi } { v } u \\lambda ^ t S \\mu \\Big ) = \\sum _ { \\lambda , \\mu } \\exp \\Big ( - \\pi v S [ \\lambda ] - \\frac { \\pi } { v } S [ u \\lambda + \\mu ] \\Big ) . \\end{align*}"}
+{"id": "2909.png", "formula": "\\begin{align*} \\sum _ { y = 0 } ^ n M _ { x , y } ( m ) g _ y = S ^ { p } _ { x , x } ( m ) . \\end{align*}"}
+{"id": "2810.png", "formula": "\\begin{align*} \\mathbf { D } _ { i l } = \\begin{cases} \\mathbf { I } _ { N _ { } } , & l \\in \\mathcal { M } _ { i } , \\\\ \\mathbf { O } _ { N _ { } } , & l \\notin \\mathcal { M } _ { i } . \\end{cases} \\end{align*}"}
+{"id": "7611.png", "formula": "\\begin{align*} \\xi _ z = \\frac { 1 } { r ! } \\sum _ { j = 1 } ^ { r ! } T ^ j ( \\xi _ { i _ z , z } ) \\end{align*}"}
+{"id": "8121.png", "formula": "\\begin{align*} Q ( G , x ) = 1 + ( n - 1 ) x + ( m - n + 1 ) x ^ 2 , \\end{align*}"}
+{"id": "8005.png", "formula": "\\begin{align*} & \\rho _ L ( \\theta ) = \\sum _ { | l | \\leq L } \\widehat { \\rho _ L } ( l ) e ( l \\theta ) = \\widehat { \\rho _ L } ( 0 ) + \\sum _ { 1 \\leq l \\leq L } \\widehat { \\rho _ L } ( l ) 2 \\cos ( 2 \\pi l \\theta ) , \\end{align*}"}
+{"id": "4404.png", "formula": "\\begin{align*} \\frac { d } { d s } \\left ( \\frac { \\zeta ' ( s ) } { \\zeta ( s ) } \\right ) = \\frac { 1 } { ( s - 1 ) ^ 2 } - \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { ( s + 2 n ) ^ 2 } - \\sum _ \\rho \\frac { 1 } { ( s - \\rho ) ^ 2 } . \\end{align*}"}
+{"id": "1026.png", "formula": "\\begin{align*} ( x = w \\varepsilon ^ \\mu , x ' = w ' \\varepsilon ^ { \\mu ' } , v , v ' , L , R , J ) \\end{align*}"}
+{"id": "5671.png", "formula": "\\begin{align*} A = \\begin{pmatrix} \\sigma & 0 \\\\ 0 & 1 _ { 2 ( n - p ) } \\end{pmatrix} \\end{align*}"}
+{"id": "2087.png", "formula": "\\begin{align*} H = \\bigcup _ { n \\in \\mathbb { N } } Q _ n \\end{align*}"}
+{"id": "584.png", "formula": "\\begin{align*} M _ 2 ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 4 } ) = \\begin{pmatrix} \\alpha _ { 1 , 2 } & 0 & - \\alpha _ { 2 , 3 } & 0 & \\alpha _ { 2 , 4 } & 0 \\\\ \\beta _ { 1 , 2 } & 0 & - \\beta _ { 2 , 3 } & 0 & \\beta _ { 2 , 4 } & 0 \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "5579.png", "formula": "\\begin{align*} W ( 1 ^ \\infty | 1 ^ \\infty ) = c - d + \\sum _ { n = 2 } ^ { \\infty } ( c - c _ n ) \\end{align*}"}
+{"id": "818.png", "formula": "\\begin{align*} \\partial ^ \\square H = g ( c ) \\Delta _ \\Gamma H + 2 \\nabla _ \\Gamma g ( c ) \\cdot \\nabla _ \\Gamma H + \\big ( \\Delta _ \\Gamma g ( c ) + \\big | \\nabla _ \\Gamma \\nu \\big | ^ 2 g ( c ) \\big ) H \\end{align*}"}
+{"id": "3579.png", "formula": "\\begin{align*} A ^ { 1 } A ^ { 2 } : = \\begin{bmatrix} A ^ { 1 } \\\\ \\hline A ^ { 2 } \\end{bmatrix} . \\end{align*}"}
+{"id": "176.png", "formula": "\\begin{align*} 0 = W _ j + \\gamma _ j ( W _ i ) - W _ i - \\gamma _ i ( W _ j ) + ( 1 - R _ { H ' , n } ) ( Q _ 2 ) \\end{align*}"}
+{"id": "4167.png", "formula": "\\begin{align*} F ( X ( s ) ) = F ( X ( t _ j ) ) + F ' ( X ( t _ j ) ) ( X ( s ) - X ( t _ j ) ) + R _ { F , j } ( s ) \\end{align*}"}
+{"id": "4902.png", "formula": "\\begin{align*} h ( \\mu ) = \\frac { 2 ^ { 2 - \\alpha - \\beta } } { \\sqrt { 2 \\pi } \\ , \\sigma \\ , \\mathrm { B } ( \\alpha , \\beta ) \\left [ 1 - I _ { \\frac { 1 } { 2 } } ( \\alpha , \\beta ) \\right ] } . \\end{align*}"}
+{"id": "2610.png", "formula": "\\begin{align*} R \\cdot R ( u , J u , x , J x ; x , J x ) = \\frac { 1 } { 2 } \\ , L ( p , \\pi ^ h ) \\ , Q ^ c ( g , R ) ( u , J u , x , J x ; x , J x ) . \\end{align*}"}
+{"id": "8699.png", "formula": "\\begin{align*} i x , x ' \\in E _ i , ~ \\frac { q ( x ) } { p ( x ) } = \\frac { q ( x ' ) } { p ( x ' ) } , \\end{align*}"}
+{"id": "7709.png", "formula": "\\begin{align*} L : = \\max \\bigg ( \\frac { 1 } { \\tilde \\varepsilon } , 8 0 p \\ , \\frac { C ( \\tilde { \\varepsilon } ) } { c ( \\tilde { \\varepsilon } ) } \\bigg ) . \\end{align*}"}
+{"id": "8694.png", "formula": "\\begin{align*} [ a _ 1 , a _ 2 , \\cdots , a _ m ] : = \\cfrac { 1 } { a _ 1 - \\cfrac { 1 } { a _ 2 - \\cfrac { 1 } { \\ddots - \\cfrac { 1 } { a _ { m } } } } } \\end{align*}"}
+{"id": "4169.png", "formula": "\\begin{align*} J _ 3 \\leq \\begin{cases} C t _ m ^ { \\alpha - 1 - \\frac { \\alpha \\zeta } { 2 \\beta } } h ^ { \\min \\{ \\frac { 1 } { 2 } + \\frac { \\alpha } { 2 \\beta } ( r + \\lambda - \\kappa ) , 1 \\} } , & \\mbox { i f } \\alpha + \\gamma = 1 , \\\\ C t _ m ^ { \\alpha - 1 - \\frac { \\alpha \\zeta } { 2 \\beta } } h ^ { \\min \\{ \\frac { \\alpha } { 2 \\beta } \\min \\{ \\kappa , r + \\lambda \\} + ( \\gamma - \\frac { 1 } { 2 } ) ^ { + } , 1 - \\varepsilon \\} } , & \\mbox { i f } \\alpha + \\gamma \\neq 1 , \\end{cases} \\end{align*}"}
+{"id": "3892.png", "formula": "\\begin{align*} Q _ \\delta u : = u - \\sum _ { j = 1 } ^ m \\sum _ { h = 1 } ^ 2 C _ { j , h } \\frac { \\partial } { \\partial z _ { j , h } } ( - \\delta ^ 2 ( K ( z _ j ) \\nabla V _ { \\delta , Z , j } ) ) , \\end{align*}"}
+{"id": "7270.png", "formula": "\\begin{align*} \\lambda _ 1 \\alpha ^ { m - 1 } + \\cdots + \\lambda _ m \\beta ^ { m - 1 } = 0 , \\end{align*}"}
+{"id": "4531.png", "formula": "\\begin{align*} v _ 1 ( x , t ) = v _ 0 ( x - \\lambda t ) + \\int _ { t _ { 1 , e x } ( x , t ) } ^ t v ( s , x - \\lambda t + \\lambda s ) \\dd s + \\int _ { 0 } ^ { t _ { 1 , e n } ( x , t ) } v ( s , x - \\lambda t + \\lambda s ) \\dd s . \\end{align*}"}
+{"id": "6496.png", "formula": "\\begin{align*} \\mu | _ { E _ { 2 , m + e _ i - e _ j } } = \\lambda ^ j = \\lambda _ x | _ { E _ { 2 , m + e _ i - e _ j } } \\end{align*}"}
+{"id": "7826.png", "formula": "\\begin{align*} \\Phi ( [ a \\cos ( 2 \\pi \\ell \\phi ) + b \\sin ( 2 \\pi \\ell \\phi ) ] u _ \\ell + [ - a \\sin ( 2 \\pi \\ell \\phi ) + b \\cos ( 2 \\pi \\ell \\phi ) ] w _ \\ell , p ) \\\\ = [ c \\cos ( 2 \\pi \\ell \\phi ) + d \\sin ( 2 \\pi \\ell \\phi ) ] u _ \\ell + [ - c \\sin ( 2 \\pi \\ell \\phi ) + d \\cos ( 2 \\pi \\ell \\phi ) ] w _ \\ell . \\end{align*}"}
+{"id": "374.png", "formula": "\\begin{align*} \\phi ^ 2 _ { n - 1 - k } & = \\cos ^ 2 ( \\frac { \\pi } { n - 1 } ( n - 1 - k ) ) \\\\ & = \\cos ^ 2 ( \\pi - \\frac { \\pi } { n - 1 } k ) \\\\ & = \\cos ^ 2 ( \\frac { \\pi } { n - 1 } k ) = \\phi ^ 2 _ { k } . \\end{align*}"}
+{"id": "1845.png", "formula": "\\begin{align*} X = \\nabla f , X ^ { \\flat } = d f \\nabla X ^ { \\flat } = ( f ) \\ , . \\end{align*}"}
+{"id": "3885.png", "formula": "\\begin{align*} V _ { \\delta , Z } ( x ) - q ( x ) = V _ { \\delta , z _ i , \\hat { q } _ { \\delta , i } , z _ i } ( x ) - \\hat { q } _ { \\delta , i } + O \\left ( \\frac { \\ln | \\ln \\varepsilon | } { | \\ln \\varepsilon | ^ 2 } \\right ) . \\end{align*}"}
+{"id": "8637.png", "formula": "\\begin{align*} \\forall \\varepsilon > 0 , \\lim _ { n \\to \\infty } \\sum _ { m = 1 } ^ n \\mathbb { E } _ { n , m } ' [ \\| \\chi _ { n , m } ' \\| _ { L ^ 2 ( \\mu ) } ^ 2 : \\| \\chi _ { n , m } ' \\| _ { L ^ 2 ( \\mu ) } > \\varepsilon ] = 0 \\end{align*}"}
+{"id": "2024.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d y _ t = & \\big [ f ^ \\alpha ( t , \\zeta _ t , \\mu _ t ) + f _ 0 ( t ) \\big ] d t + \\big [ g ^ \\alpha ( t , \\zeta _ t , \\mu _ t ) + g _ 0 ( t ) \\big ] d \\overleftarrow B _ t - z _ t d W _ t , \\\\ d p _ t = & \\big [ F ^ \\alpha ( t , \\zeta _ t , \\mu _ t ) + F _ 0 ( t ) \\big ] d t + \\big [ G ^ \\alpha ( t , \\zeta _ t , \\mu _ t ) + G _ 0 ( t ) \\big ] d W _ t - q _ t d \\overleftarrow B _ t , \\\\ y _ T = & \\xi , \\ , \\ , p _ 0 = \\Psi ^ \\alpha ( y _ 0 , \\mathcal L ( y _ 0 ) ) + \\Psi _ 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "3460.png", "formula": "\\begin{align*} r _ { \\ell } \\leq e ^ { 4 0 \\varepsilon q _ n } \\frac { e ^ { - q _ n L } } { \\max ( | \\ell | , 1 ) } \\max ( r _ { \\ell - 1 } , r _ { \\ell + 1 } ) \\times \\begin{cases} \\max ( | \\ell | , e ^ { \\delta _ n q _ n } , 1 ) , & \\beta _ n \\geq \\delta _ n + 2 0 0 \\varepsilon \\\\ e ^ { \\beta _ n q _ n } , & \\beta _ n < \\delta _ n + 2 0 0 \\varepsilon \\end{cases} . \\end{align*}"}
+{"id": "8891.png", "formula": "\\begin{align*} S = \\frac { ( 4 \\pi ) ^ { 2 k - 3 } } { \\Gamma ( 2 k - 3 ) } \\Big ( 1 + \\frac { p } { p + 1 } C ( p ) + O ( k ^ { - \\alpha } p ^ { - \\alpha / 2 } D ^ { - \\alpha } + p ^ { - 9 / 1 6 + \\epsilon } k ^ { - 5 / 2 4 } D ^ { 7 / 8 + \\epsilon } ) \\Big ) . \\end{align*}"}
+{"id": "3562.png", "formula": "\\begin{align*} q ^ { - 1 3 / 6 0 } & \\varphi _ 1 ^ { - 6 } \\varphi _ \\frac { 1 } { 2 } ^ 2 G _ \\frac { 1 } { 2 } + q ^ { 1 7 / 6 0 } \\varphi _ 1 ^ { - 6 } \\varphi _ 2 ^ 2 H _ 2 \\\\ & = q ^ { - 1 3 / 6 0 } \\varphi _ 1 ^ { - 6 } ( \\varphi _ \\frac { 1 } { 2 } ^ 2 G _ \\frac { 1 } { 2 } + q ^ { 1 / 2 } \\varphi _ 2 ^ 2 H _ 2 ) . \\end{align*}"}
+{"id": "6923.png", "formula": "\\begin{align*} - \\int _ { \\R } \\left ( T _ i ' ( x _ i ) - 1 \\right ) Q _ i ( x _ i ) d x _ i = \\int _ \\R \\left ( T _ i ( x _ i ) - x _ i \\right ) Q _ i ' ( x _ i ) \\ , d x _ i . \\end{align*}"}
+{"id": "2365.png", "formula": "\\begin{align*} C _ { \\ast } ( K ; \\mathfrak { g } _ { \\mathrm { A d } _ \\rho } ) : = \\displaystyle C _ { \\ast } ( \\widetilde { K } ; \\mathbb { Z } ) \\displaystyle \\otimes \\mathfrak { g } / \\sim , \\end{align*}"}
+{"id": "7111.png", "formula": "\\begin{align*} \\int _ { t _ 0 } ^ { t _ 1 } \\int _ { \\Omega } \\omega ( t , x ) f _ h ' ( \\phi ( x ) ) \\ , u ( t , x ) \\cdot \\psi ( x ) d x d t + \\int _ { \\Omega } \\omega ( t _ 0 , x ) f _ h ( \\phi ( x ) ) d x - \\int _ { \\Omega } \\omega ( t _ 1 , x ) f _ h ( \\phi ( x ) ) d x = 0 , \\end{align*}"}
+{"id": "1985.png", "formula": "\\begin{align*} f ( - x - T ( x ) ) = f ( z - 2 g ^ { - 1 } ( z ) ) . \\end{align*}"}
+{"id": "8175.png", "formula": "\\begin{align*} \\int _ { R _ { 0 } / 2 } ^ { R _ { 0 } } \\nu \\xi ( \\nu , s ) \\ d \\nu = 0 , s \\in ( 0 , t _ { 0 } ) . \\end{align*}"}
+{"id": "381.png", "formula": "\\begin{align*} \\begin{aligned} ( n - 1 ) ( a + 2 b ) & = 1 \\\\ c + 2 a ( n - 2 ) + b ( 3 n - 7 ) & = 1 \\\\ 2 c + a ( 3 n - 7 ) + 4 b ( n - 3 ) & = 1 . \\end{aligned} \\end{align*}"}
+{"id": "4479.png", "formula": "\\begin{align*} { \\rm I m } ( M ( w ) ) = \\frac { { \\rm I m } ( w ) } { | c w + d | ^ { 2 } } . \\end{align*}"}
+{"id": "3573.png", "formula": "\\begin{align*} A ( t ) + B ( t ) & = \\varphi ( - t ) ^ 2 ( G ( - t ) H ( t ) + G ( t ) H ( - t ) ) \\\\ & = \\varphi ( - t ) ^ 2 \\frac { 2 \\psi ( t ^ 2 ) } { \\varphi ( t ^ 2 ) } \\\\ & = \\varphi ( t ^ 2 ) \\phi ( t ) \\frac { 2 \\varphi ( t ^ 4 ) ^ 2 / \\varphi ( t ^ 2 ) } { \\varphi ( t ^ 2 ) } \\\\ & = \\frac { \\phi ( t ) } { \\varphi ( t ^ 2 ) } 2 \\varphi ( t ^ 4 ) ^ 2 \\\\ & = 2 \\varphi ( t ^ 4 ) ^ 2 ( G ( t ) G ( t ^ 4 ) + t H ( t ) H ( t ^ 4 ) ) \\end{align*}"}
+{"id": "5604.png", "formula": "\\begin{align*} d = d _ \\alpha + b - a _ { \\alpha + 1 } + \\sum _ { j = 2 } ^ \\infty ( a _ j - a _ { \\alpha + j } ) \\ . \\end{align*}"}
+{"id": "696.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\aligned & - \\Delta u + V ( x ) u = \\left ( I _ \\alpha \\ast | u | ^ p \\right ) | u | ^ { p - 2 } u + \\lambda u ~ ~ x \\in \\R ^ N , \\\\ & u ( x ) \\to 0 ~ ~ ~ ~ | x | \\to \\infty , \\endaligned \\end{array} \\right . \\end{align*}"}
+{"id": "7677.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } _ t ^ * : = ( \\alpha _ t ^ { * , i } ) _ { 1 \\leq i \\leq N } = \\left ( - R ^ { - 1 } B ( P _ t x _ t ^ { * , i } + \\varphi _ t ^ { * , i } ) - h \\Big ( k _ i ^ N ( t , \\nu ^ { N , 1 } _ { \\boldsymbol { x } ^ * _ t } , \\ldots , \\nu ^ { N , N } _ { \\boldsymbol { x } ^ * _ t } , \\lambda ^ { N , 1 } _ { \\boldsymbol { \\varphi } ^ * _ t } , \\ldots , \\lambda ^ { N , N } _ { \\boldsymbol { \\varphi } ^ * _ t } ) \\Big ) \\right ) _ { 1 \\leq i \\leq N } . \\end{align*}"}
+{"id": "9326.png", "formula": "\\begin{align*} \\P \\left ( \\frac { 1 } { m } \\sum _ { i = 1 } ^ m V _ i \\leq \\left ( \\frac { \\pi } { 4 } + \\frac { 1 9 9 \\pi } { 3 2 0 0 k } \\right ) \\| \\mathbf { x } \\| \\right ) \\leq 2 \\exp \\left ( - \\frac { c _ 5 m } { k } \\right ) . \\end{align*}"}
+{"id": "6361.png", "formula": "\\begin{align*} S _ r ( a , b ) = \\Gamma ( b ) \\Gamma \\left ( 1 - \\frac { r } { 2 } \\right ) \\sum _ { n = 0 } ^ \\infty \\frac { ( - 1 ) ^ n } { ( a + n ) ^ { 1 - r / 2 } \\Gamma ( b - n ) n ! } , r < 2 . \\end{align*}"}
+{"id": "4497.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 \\arctan \\lambda _ i = \\hat \\theta _ 0 \\in \\left ( \\pi , \\frac { 3 \\pi } { 2 } \\right ) . \\end{align*}"}
+{"id": "2504.png", "formula": "\\begin{align*} \\mu = \\mu ( \\mathbf { x } , \\mathbf { s } ) = \\mathbf { x } ^ T \\mathbf { s } / k , \\end{align*}"}
+{"id": "9222.png", "formula": "\\begin{align*} \\epsilon ( x , t ) = \\int _ 0 ^ t f ( x , \\tau ) - f _ a ( x ) \\ , d \\tau = \\int _ { N T } ^ t f ( x , \\tau ) - f _ a ( x ) \\ , d \\tau , \\end{align*}"}
+{"id": "2997.png", "formula": "\\begin{align*} \\mathbb G = [ 0 , \\vartheta ] \\times \\mathbb R ^ n \\times \\mathrm { P L i p } , \\mathbb G _ * = \\cup _ { i = 0 } ^ { I - 1 } ( i h , ( i + 1 ) h ) \\times \\mathbb R ^ n \\times \\mathrm { P L i p } _ * . \\end{align*}"}
+{"id": "5397.png", "formula": "\\begin{align*} F ^ { i j } = \\frac { \\partial F ( D ^ 2 u ) } { \\partial u _ { i j } } . \\end{align*}"}
+{"id": "1376.png", "formula": "\\begin{align*} S _ j ( n ) = \\inf \\{ t \\geq 0 : N _ { d - j + 1 } ( t ) \\geq n \\} , n \\geq 0 . \\end{align*}"}
+{"id": "2677.png", "formula": "\\begin{align*} \\bigl ( | u _ t | ^ { \\ell - 2 } u _ t \\bigr ) _ t - a \\nabla \\cdot \\bigl ( | \\nabla u | ^ { q - 2 } \\nabla u \\bigr ) + b ( t , x ) | u _ t | ^ { m - 2 } u _ t = 0 , \\end{align*}"}
+{"id": "5209.png", "formula": "\\begin{align*} \\partial _ t a _ j + \\mathbb { P } ( B _ j \\times u _ j ) = - \\mu _ j \\nabla \\times b _ j \\textrm { o n } \\ L ^ { 2 } _ { \\sigma } ( \\mathbb { R } ^ { 3 } ) , \\end{align*}"}
+{"id": "4260.png", "formula": "\\begin{align*} \\alpha ( g , ( z _ 1 + 1 , z _ 2 , \\ldots ) ) = ( \\xi + ( z _ 1 + 1 ) \\psi _ 1 ) \\alpha ( g , Z ) - \\sum _ { \\overline { \\Gamma } \\in { \\rm B i c } ( { g } , { Z } ) _ 1 } m ( \\overline { \\Gamma } ) \\alpha ( \\overline { \\Gamma } ) . \\end{align*}"}
+{"id": "7554.png", "formula": "\\begin{align*} | f ( 0 ) | \\leq { | a _ k | } ^ { - 1 } \\sup _ { | z | = 1 } | f ( z ) p ( z ) | . \\end{align*}"}
+{"id": "382.png", "formula": "\\begin{align*} A : = \\frac { 9 ( n - 1 ) } { ( n + 4 ) ^ 2 } \\left [ \\begin{array} { c c c c c c c c c } 1 & \\frac { ( n - 2 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } ' & \\frac { - ( n + 1 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } ' \\\\ \\frac { ( n - 2 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } & \\frac { ( n - 2 ) ^ 2 } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } & \\frac { - ( n - 2 ) ( n + 1 ) } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } \\\\ \\frac { - ( n + 1 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } & \\frac { - ( n - 2 ) ( n + 1 ) } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } & \\frac { ( n + 1 ) ^ 2 } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } \\end{array} \\right ] , \\end{align*}"}
+{"id": "4950.png", "formula": "\\begin{align*} V ( \\xi _ 0 , n , x ) = \\sup _ { \\theta \\in \\Theta _ n } W ( \\xi _ 0 , n , x , \\theta ) = \\sup _ { \\theta \\in \\Theta _ n } E _ { \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\biggr ] . \\end{align*}"}
+{"id": "8616.png", "formula": "\\begin{align*} f ( t ) = \\frac { | A \\oplus _ 2 \\sqrt { t } B | ^ { \\frac { 2 } { k } } } { | V ( ( A \\oplus _ 2 \\sqrt { t } B ) [ n - k ] , Z _ 1 , \\dots , Z _ k ) | ^ { \\frac { 2 } { k } } _ { n - k } } \\end{align*}"}
+{"id": "2579.png", "formula": "\\begin{align*} R ( X , Y ) Z = c ( X \\wedge _ g Y ) Z , \\end{align*}"}
+{"id": "7122.png", "formula": "\\begin{align*} \\begin{array} { l l l } f _ R ( R ( s ) ) ( t ) \\cdot f _ L ( u ) ( v ) & = & f _ R ( u ) ( f _ R ( R ( s ) ) ) ( t ) \\cdot v \\\\ & = & f _ R ( R ( s ) * u ) ( t ) \\cdot v \\\\ & = & f _ R ( s * L ( u ) ) ( t ) \\cdot v \\\\ & = & f _ R ( L ( u ) ) ( f _ R ( s ) ( t ) ) \\cdot v \\\\ & = & f _ R ( L ( u ) ) ( f _ R ( s ' ) ( t ' ) ) \\cdot v \\\\ & = & f _ R ( s ' * L ( u ) ) ( t ' ) \\cdot v \\\\ & = & f _ R ( R ( s ' ) * u ) ( t ' ) \\cdot v \\\\ & = & f _ R ( u ) ( f _ R ( R ( s ' ) ) ( t ' ) ) \\cdot v \\\\ & = & f _ R ( R ( s ' ) ) ( t ' ) \\cdot f _ L ( u ) ( v ) . \\end{array} \\end{align*}"}
+{"id": "9063.png", "formula": "\\begin{align*} v _ 0 = \\sum Y ^ { + } Y _ { \\ 1 } ^ { - } Y _ { \\ 0 } ^ { - } v _ k , \\end{align*}"}
+{"id": "6678.png", "formula": "\\begin{align*} B _ { x , y } ( \\xi ) = \\lim _ { t \\rightarrow \\infty } ( d _ X ( y , \\gamma _ \\xi ( t ) ) - t ) \\end{align*}"}
+{"id": "3166.png", "formula": "\\begin{align*} & \\sum \\limits _ { y \\neq 0 , x } [ ( 2 + h \\chi _ 4 ( y ) + \\overline { h } \\overline { \\chi _ 4 } ( y ) ) ( 2 + h \\chi _ 4 ( x - y ) + \\overline { h } \\overline { \\chi _ 4 } ( x - y ) ) ] \\\\ & = 4 ( q - 3 ) - 4 h \\chi _ 4 ( x ) - 4 \\overline { h } \\overline { \\chi _ 4 } ( x ) - 2 \\chi _ 4 ( g ) \\varphi ( x ) J ( \\chi _ 4 , \\chi _ 4 ) + 2 \\chi _ 4 ( g ) \\varphi ( x ) \\overline { J ( \\chi _ 4 , \\chi _ 4 ) } . \\end{align*}"}
+{"id": "9272.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u _ 1 ^ { t } \\wedge \\dots \\wedge \\Delta u _ k ^ { t } \\wedge \\beta _ n ^ { n - m } \\rightarrow \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ k \\wedge \\beta _ n ^ { n - m } . \\end{aligned} \\end{align*}"}
+{"id": "6657.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { m \\to \\infty } \\lim _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\sup _ { t \\in [ 0 , 1 ] } | \\hat { X } ^ { \\epsilon } ( t ) - \\varphi ( t ) | < \\delta , \\chi _ 2 < 1 \\Big ) = - \\infty . \\end{align*}"}
+{"id": "1007.png", "formula": "\\begin{align*} x '' : = w '' \\varepsilon ^ { \\mu '' } : = x ' r _ { a } = w ' s _ \\alpha \\varepsilon ^ { \\mu ' - ( 1 + \\langle \\mu ' , \\alpha \\rangle ) \\alpha ^ \\vee } < x ' . \\end{align*}"}
+{"id": "4271.png", "formula": "\\begin{align*} 3 n ^ { 2 } \\left ( a + b - 1 \\right ) + 3 n \\left ( a ^ { 2 } + b ^ { 2 } - 1 \\right ) + \\left ( a ^ { 3 } + b ^ { 3 } - 1 \\right ) = 0 \\end{align*}"}
+{"id": "6227.png", "formula": "\\begin{align*} \\psi _ \\alpha ( f ) : & = \\nu ( \\{ x \\in X : | f ( x ) | > \\alpha \\} ) \\end{align*}"}
+{"id": "9071.png", "formula": "\\begin{align*} 0 = E ^ 0 \\subset E ^ 1 \\subset \\cdots \\subset E ^ { k - 1 } \\subset E ^ { k } = E , \\end{align*}"}
+{"id": "7525.png", "formula": "\\begin{align*} I _ j = W _ j \\setminus \\bigcup _ { e \\in G [ W _ j ] } e . \\end{align*}"}
+{"id": "7430.png", "formula": "\\begin{align*} \\mathbb { P } ( S _ k = 0 ) ^ { 1 / p _ k } \\leq \\mathbb { P } ( S _ k = - k ) ^ { 1 / p _ k } + \\mathbb { P } ( S _ k = k ) ^ { 1 / p _ k } \\end{align*}"}
+{"id": "1738.png", "formula": "\\begin{align*} \\psi _ n ( x ) = \\sum _ { j \\leq n , x \\geq \\sup I ^ + _ j } \\lambda ( I ^ + _ j ) \\ ; \\ ; - \\sum _ { j \\leq n , x \\leq \\inf I ^ - _ j } \\lambda ( I ^ - _ j ) . \\end{align*}"}
+{"id": "1343.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ l \\beta _ k & = \\lambda _ { n + 1 } \\sum _ { k = 1 } ^ l \\frac { d ( \\gamma ( t _ { k - 1 } ) , \\gamma ( t _ { k } ) ) } { d ( p _ { n + 1 } , q _ { n + 1 } ) } \\leq \\frac { \\lambda _ { n + 1 } L ( \\gamma ) } { d ( p _ { n + 1 } , q _ { n + 1 } ) } \\\\ & < \\lambda _ { n + 1 } + \\frac { \\lambda _ { n + 1 } \\varepsilon \\delta } { d ( p _ { n + 1 } , q _ { n + 1 } ) } \\leq \\lambda _ { n + 1 } + \\frac { \\varepsilon } { 2 } . \\end{align*}"}
+{"id": "1537.png", "formula": "\\begin{align*} \\overline { \\Psi } ( A _ t ) = \\left \\{ \\sum _ { \\alpha \\in { \\mathbb { Z } } } a _ { \\alpha } \\ , \\dd ^ { \\alpha } \\in \\widehat { \\Psi } ( A _ t ) \\ ; | \\ ; v a l _ t ( a _ \\alpha ) \\geq \\alpha \\right \\} \\end{align*}"}
+{"id": "8195.png", "formula": "\\begin{align*} x _ { i , j } & + x _ { i + 1 , j } + x _ { i , j + 1 } + x _ { i + 1 , j + 1 } \\\\ & = \\bigl ( \\tfrac { 1 } { 2 } S - x _ { i , n + 1 - j } \\bigr ) + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { i + 1 , n + 1 - j } \\bigr ) + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { i , n - j } \\bigr ) + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { i + 1 , n - j } \\bigr ) \\\\ & = 2 S - \\bigl ( x _ { i , n + 1 - j } + x _ { i + 1 , n + 1 - j } + x _ { i , n - j } + x _ { i + 1 , n - j } \\bigr ) = S . \\end{align*}"}
+{"id": "7347.png", "formula": "\\begin{align*} \\epsilon ' = \\sum _ { i \\in I } \\bar { \\nu } _ { j ! } & \\ & ( \\delta _ k ^ n ) _ ! = \\sum _ { J = \\{ i _ 0 < \\hdots < \\widehat { i _ k } < \\hdots < i _ n \\} \\subset K = \\{ i _ 0 < \\hdots < i _ n \\} } ( \\nu _ { K } ^ J ) _ ! \\end{align*}"}
+{"id": "7941.png", "formula": "\\begin{align*} \\lim _ { \\norm { \\eta _ n } _ { H ^ 1 } \\to 0 } \\frac { 1 } { \\norm { \\eta _ n } _ { H ^ 1 } } \\norm { A ^ { \\tilde \\Psi + \\eta _ n } [ \\eta _ 1 , \\dots , \\eta _ { n - 1 } ] - A ^ { \\tilde \\Psi } [ \\eta _ 1 , \\dots , \\eta _ { n - 1 } ] - A ^ { \\tilde \\Psi } [ \\eta _ 1 , \\dots , \\eta _ n ] } _ { H ^ 1 } = 0 . \\end{align*}"}
+{"id": "2322.png", "formula": "\\begin{align*} \\left ( R ^ W _ { X , Y } \\right ) ^ { J , - } = \\frac { 1 } { 2 } \\left ( D ^ W _ X \\left ( D ^ W _ Y J \\right ) - D ^ W _ Y \\left ( D ^ W _ X J \\right ) - D ^ W _ { [ X , Y ] } J \\right ) . \\end{align*}"}
+{"id": "4316.png", "formula": "\\begin{align*} \\lim _ { j \\to + \\infty } \\int _ { \\{ \\Psi < - t _ 0 \\} } | f _ j | ^ 2 _ h c ( - \\Psi ) = 0 , \\end{align*}"}
+{"id": "6166.png", "formula": "\\begin{align*} \\hat { H } _ 1 = \\hat { A } ^ + \\hat { A } ^ - + E _ 0 , \\hat { H } _ 2 = \\hat { A } ^ - \\hat { A } ^ + + E _ 0 , \\end{align*}"}
+{"id": "5628.png", "formula": "\\begin{align*} f _ \\xi ( z ) = \\begin{cases} \\frac { f ( z ) } { \\xi - z } & , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "9219.png", "formula": "\\begin{align*} \\dot { x } _ a = \\gamma f _ a ( x _ a ) x _ a ( 0 ) = x _ 0 \\end{align*}"}
+{"id": "3575.png", "formula": "\\begin{align*} A ( t ) & = \\frac { 1 } { 2 } ( ( A ( t ) + B ( t ) ) - ( B ( t ) - A ( t ) ) ) \\\\ & = \\varphi ( t ^ 4 ) ^ 2 ( 2 t H ( t ) H ( t ^ 4 ) ) \\\\ & = 2 t \\varphi ( t ^ 4 ) ^ 2 H ( t ^ 4 ) H ( t ) \\end{align*}"}
+{"id": "909.png", "formula": "\\begin{align*} \\Sigma _ 3 = \\sum \\limits _ { D \\leq d < 2 D } \\sum \\limits _ { n \\in \\mathcal { N } ( d ) } \\psi \\left ( \\frac { \\sqrt { X } - n } { d } \\right ) \\ , . \\end{align*}"}
+{"id": "219.png", "formula": "\\begin{align*} I = ( \\iota , \\sigma ) = \\big ( ( 2 , 4 , 1 , 5 , 5 , 4 , 5 , 4 , 5 ) , ( 0 , 2 , 2 , 5 , 3 , 1 , 1 , - 1 , - 1 , - 3 ) \\big ) , \\end{align*}"}
+{"id": "9163.png", "formula": "\\begin{align*} \\begin{aligned} \\dfrac { b _ { 1 , \\delta } ( x ) } { 2 } & \\geq \\underline { \\delta } ^ \\star ( \\delta ) & \\forall \\ , x \\geq x ^ \\star + \\delta \\\\ \\dfrac { b _ { 1 , \\delta } ( x ) } { 2 } & \\leq - \\underline { \\delta } ^ \\star ( \\delta ) & \\forall \\ , x \\leq x ^ \\star - \\delta \\ , . \\end{aligned} \\end{align*}"}
+{"id": "4628.png", "formula": "\\begin{align*} C _ { B _ n } ( \\sigma ) = \\{ \\tau \\in B _ n : \\sigma \\tau = \\tau \\sigma \\} \\end{align*}"}
+{"id": "5487.png", "formula": "\\begin{align*} f ( q ) = \\log \\Gamma ( 2 - q ) + \\frac { q ( 1 - q ) } { 2 } + \\frac { q ^ 2 ( 1 - q ^ 2 ) } { 1 2 } \\end{align*}"}
+{"id": "1987.png", "formula": "\\begin{align*} F ( z ) = - F ( z + F ( z ) ) . \\end{align*}"}
+{"id": "6534.png", "formula": "\\begin{align*} ( 2 a ) ^ { \\pi ( a ) } + c _ { \\pi ( a ) - 1 } ( 2 a ) ^ { \\pi ( a ) - 1 } + c _ { \\pi ( a ) - 2 } ( 2 a ) ^ { \\pi ( a ) - 2 } + \\cdots c _ 1 ( 2 a ) = - c _ 0 . \\end{align*}"}
+{"id": "9335.png", "formula": "\\begin{align*} \\phi ( \\lambda ^ * , \\eta ) & = \\frac { 5 9 7 } { 5 0 } \\eta ^ 2 \\frac { - 1 - \\frac { 4 0 0 } { 1 9 9 } a ^ 2 } { 2 + \\frac { 1 9 7 } { 1 9 9 } a ^ 2 } + \\frac { 1 9 9 } { 5 0 } \\frac { 1 + a ^ 2 } { 2 } - \\frac { 2 0 1 } { 1 0 0 } \\\\ \\ & \\ge \\frac { 5 9 7 } { 3 5 0 } \\frac { - 1 - \\frac { 4 0 0 } { 1 9 9 } a ^ 2 } { 2 + \\frac { 1 9 7 } { 1 9 9 } a ^ 2 } + \\frac { 1 9 9 } { 5 0 } \\frac { 1 + a ^ 2 } { 2 } - \\frac { 2 0 1 } { 1 0 0 } . \\end{align*}"}
+{"id": "773.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } u _ i - d _ i \\partial _ { x x } u _ i = f _ i ( x , t , u ) , & x \\in \\Omega , ~ t > 0 , \\\\ \\partial _ x u _ i ( 0 , t ) = \\partial _ x u _ i ( L , t ) = 0 , & t > 0 , \\\\ u _ i ( x , 0 ) = u _ { i , 0 } ( x ) , & x \\in \\Omega , \\end{cases} \\end{align*}"}
+{"id": "4125.png", "formula": "\\begin{align*} H _ { m i n } ( E \\otimes ( F _ 1 [ t ] \\times F _ 1 [ t ] ^ { \\vee } ) ) & = H _ { m i n } ( E \\otimes F _ 1 [ t ] \\times E \\otimes F _ 1 [ t ] ^ { \\vee } ) \\\\ & = \\min H _ { m i n } ( E \\otimes F _ 1 [ t ] , E \\otimes F _ 1 [ t ] ^ { \\vee } ) \\\\ & \\leq H _ { m i n } ( E \\otimes F _ 1 [ t ] ) = t H _ { m i n } ( E \\otimes F _ 1 ) \\leq t H _ { m i n } ( E ) H _ { m i n } ( F _ 1 ) \\end{align*}"}
+{"id": "4613.png", "formula": "\\begin{align*} & | \\o ( t , x ) - \\o ^ { \\beta ^ \\prime } ( t , x ) | \\\\ & = | \\o _ 0 ( X ( 0 ; t , x ) ) - \\o ^ { \\beta ^ \\prime } _ 0 ( X ^ { \\beta ^ \\prime } ( 0 ; t , x ) ) | \\\\ & \\leq | \\o _ 0 ( X ( 0 ; t , x ) ) - \\o ^ \\ell _ 0 ( X ( 0 ; t , x ) ) | + | \\o _ 0 ^ \\ell ( X ^ { \\beta ^ \\prime } ( 0 ; t , x ) ) - \\o _ 0 ^ { \\beta ^ \\prime } ( X ^ { \\beta ^ \\prime } ( 0 ; t , x ) ) | \\\\ & + | \\o ^ \\ell _ 0 ( X ( 0 ; t , x ) ) - \\o ^ \\ell _ 0 ( X ^ { \\beta ^ \\prime } ( 0 ; t , x ) ) | . \\end{align*}"}
+{"id": "3786.png", "formula": "\\begin{align*} \\langle A _ q ^ w , A _ q ^ z \\rangle _ { L ^ 2 ( \\mathbb { R } ) } = K _ q ( w , z ) . \\end{align*}"}
+{"id": "630.png", "formula": "\\begin{align*} \\chi _ K ( 2 ) = \\begin{cases} 1 & d \\equiv 1 \\mod 8 \\\\ - 1 & d \\equiv 5 \\mod 8 \\\\ 0 & \\end{cases} . \\end{align*}"}
+{"id": "8746.png", "formula": "\\begin{align*} ( D _ { a ^ + } ^ { \\alpha } 1 ) ( x ) = \\frac { ( x - a ) ^ { - \\alpha } } { \\Gamma [ 1 - \\alpha ] } , \\ \\forall x \\in [ a , b ] . \\end{align*}"}
+{"id": "3116.png", "formula": "\\begin{align*} \\begin{array} { l l } [ \\alpha ( [ x , z ] ) , \\alpha ( [ y , t ] ) ] & = [ [ [ x , y ] , \\alpha ( z ) ] , \\alpha ^ { 2 } ( t ) ] + [ [ [ y , z ] , \\alpha ( t ) ] , \\alpha ^ { 2 } ( x ) ] \\\\ & + [ [ [ z , t ] , \\alpha ( x ) ] , \\alpha ^ { 2 } ( y ) ] + [ [ [ t , x ] , \\alpha ( y ) ] , \\alpha ^ { 2 } ( z ) ] . \\end{array} \\end{align*}"}
+{"id": "1204.png", "formula": "\\begin{align*} \\left \\| g ( x ) - h ( x ) \\right \\| & = \\lim _ { n \\rightarrow \\infty } | 2 | ^ { n } \\left \\| g ( 2 ^ { n + 1 } x ) - h ( 2 ^ { n + 1 } x ) \\right \\| \\\\ & \\leq \\lim _ { k \\rightarrow \\infty } | 2 | ^ { n } \\max \\{ \\left \\| g ( 2 ^ { n + 1 } x ) - h ( 2 ^ { n + 1 } x ) \\right \\| , \\left \\| g ( 2 ^ { n + 1 } x ) - h ( 2 ^ { n + 1 } x ) \\right \\| \\} \\\\ & \\leq \\lim _ { j \\rightarrow \\infty } \\lim _ { n \\rightarrow \\infty } \\max \\{ | 2 | ^ { k } \\mu ( 0 , 2 ^ { k + 1 } x ) ; j \\leq k < n + j \\} \\\\ & = 0 \\end{align*}"}
+{"id": "4889.png", "formula": "\\begin{align*} z = ( 2 - \\alpha - \\beta ) \\frac { \\phi ( z ) } { [ 1 - \\Phi ( z ) ] } + ( \\alpha - 1 ) \\frac { \\phi ( z ) } { \\Phi ( z ) [ 1 - \\Phi ( z ) ] } . \\end{align*}"}
+{"id": "282.png", "formula": "\\begin{align*} \\lambda _ 0 = \\sup \\{ \\lambda \\geq 0 : p \\cdot \\lambda ( p ^ { - 1 } \\cdot q ) \\notin \\Omega \\} . \\end{align*}"}
+{"id": "5886.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\sup _ { n \\in \\mathbb { N } } I I _ n = 0 . \\end{align*}"}
+{"id": "6870.png", "formula": "\\begin{align*} \\Psi _ 1 ( \\hat \\alpha ) = 0 , \\Psi _ 1 ' ( \\hat \\alpha ) > 0 \\end{align*}"}
+{"id": "4234.png", "formula": "\\begin{align*} \\dot { s } \\varepsilon ( z ) \\varpi = \\varepsilon ( - z ^ { - 1 } ) h ( ( z ^ 2 - z ) ^ { - \\frac { 1 } { 2 } } ) ) \\varpi + ( h ( z ^ { - \\frac { 1 } { 2 } } ) - h ( ( z - 1 ) ^ { - \\frac { 1 } { 2 } } ) ) \\varpi \\end{align*}"}
+{"id": "2474.png", "formula": "\\begin{align*} g ( E ) = \\displaystyle \\lim _ { k \\rightarrow \\infty } \\frac { ( - 1 ) ^ { k } } { \\Gamma ( k + 1 ) } Z _ { q } ^ { ( k ) } ( \\beta ) \\beta ^ { k + 1 } Q _ { 1 } ( 2 - q ) \\Biggr | _ { \\beta = \\frac { k \\xi _ { m } } { E } } . \\end{align*}"}
+{"id": "3598.png", "formula": "\\begin{align*} \\omega _ { u v } = \\begin{cases} x _ u y _ v & t ( u , \\bullet ) = t ( \\bullet , v ) \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "1479.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } | D z _ { n } | ( ( a , b ) ) = \\lim _ { n \\to + \\infty } \\sum _ { i = 1 } ^ { \\infty } | J _ { n } ( i ) | = \\sum _ { i = 1 } ^ { \\infty } | J _ { \\infty } ( i ) | . \\end{align*}"}
+{"id": "6953.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } T _ { W _ { n J } , \\psi } ( \\mu _ n ( Q ^ n ) ) & \\le \\limsup _ { n \\to \\infty } T _ { W _ { n J } , \\psi _ m } ( \\mu _ n ( Q ^ n ) ) \\le \\limsup _ { n \\to \\infty } T _ { W , \\psi _ m } ( \\mu _ n ( Q ^ n ) ) = T _ { W , \\psi _ m } ( \\mu _ \\infty ) , \\end{align*}"}
+{"id": "1629.png", "formula": "\\begin{align*} E _ 0 ^ { i , j } ( \\C { O } _ h ( G ) , M ) : = \\frac { F ^ i C ^ { i + j } ( \\C { O } _ h ( G ) , M ) } { F ^ { i + 1 } C ^ { i + j } ( \\C { O } _ h ( G ) , M ) } \\cong h ^ i C ^ { j + i } ( \\C { O } ( G ) , M _ 0 ) . \\end{align*}"}
+{"id": "6096.png", "formula": "\\begin{align*} & \\big ( \\mu _ n ^ { ( 1 2 ) } - \\mu _ { n + 1 } ^ { ( 1 2 ) } - 1 \\big ) Y _ { n + 1 , p } - \\big ( \\mu _ { n - 1 } ^ { ( 1 2 ) } - \\mu _ n ^ { ( 1 2 ) } + 1 \\big ) Y _ { n , p } = \\\\ & \\big ( \\psi ^ { 0 0 } _ { n , p } \\big ) ^ 2 + ( 2 \\mu _ n ^ { ( 1 2 ) } - \\mu ^ { ( 1 ) } - \\mu ^ { ( 2 ) } - \\mu ^ { ( 3 ) } - \\mu _ p ^ { ( 1 2 3 ) } ) \\ , \\psi ^ { 0 0 } _ { n , p } - ( \\mu ^ { ( 1 ) } - \\mu _ p ^ { ( 1 2 3 ) } ) ( \\mu ^ { ( 3 ) } - \\mu ^ { ( 2 ) } ) \\ , , \\end{align*}"}
+{"id": "7782.png", "formula": "\\begin{align*} s _ { h , q } ( p , c , \\rho ) : = \\frac { \\rho } { q } \\sum _ { g = 0 } ^ { q - 1 } \\zeta ^ { ( h + 1 ) g } ~ \\frac { p ' ( c + \\rho \\zeta ^ g ) } { p ( c + \\rho \\zeta ^ g ) } ~ { \\rm f o r } ~ \\zeta ~ { \\rm o f ~ ( \\ref { e q z t } ) ~ a n d } ~ h = 0 , 1 , \\dots , q - 1 , \\end{align*}"}
+{"id": "5090.png", "formula": "\\begin{align*} \\phi _ t + u \\cdot \\nabla \\phi = \\mu \\Delta \\phi . \\end{align*}"}
+{"id": "6071.png", "formula": "\\begin{align*} A u t \\left ( \\mathcal { R P } ( G ) \\right ) = \\mathbb Z _ 2 \\times { \\prod _ { i = 1 } ^ { t } S _ { \\widehat { u _ i } } } . \\end{align*}"}
+{"id": "9346.png", "formula": "\\begin{align*} E ( r ) & \\leq C \\left ( 1 + \\frac { 1 } { 4 r ^ 2 } \\| v \\| ^ 3 _ { L ^ 3 ( Q _ { 2 r } ( z _ 0 ) ) } + \\frac { 1 } { 4 r ^ 2 } \\| \\pi | ^ \\frac { 3 } { 2 } _ { L ^ \\frac { 3 } { 2 } ( Q _ { 2 r } ( z _ 0 ) ) } \\right ) = : C ( 1 + A ( 2 r ) + B ( 2 r ) ) , \\end{align*}"}
+{"id": "7242.png", "formula": "\\begin{align*} ( \\alpha + \\beta ) ^ { q ^ i } = \\alpha ^ { q ^ i } + \\beta ^ { q ^ i } \\end{align*}"}
+{"id": "5862.png", "formula": "\\begin{align*} J a = \\frac { { { c _ p } \\left ( { { T _ w } - { T _ { s a t } } } \\right ) } } { { { h _ { l v } } } } , \\end{align*}"}
+{"id": "5174.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi _ 1 + \\phi _ 2 - \\phi _ \\infty ) _ { + } G _ 2 \\frac { 1 } { r ^ { 2 } } \\dd x = \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi _ 2 - \\phi _ \\infty ) _ { + } G _ 2 \\frac { 1 } { r ^ { 2 } } \\dd x . \\end{align*}"}
+{"id": "711.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { y \\in \\R ^ N } \\int _ { B _ r ( y ) } u _ n ^ 2 d x = 0 , \\end{align*}"}
+{"id": "4968.png", "formula": "\\begin{align*} \\mathsf { M } ^ n = & \\{ \\mathsf { m } ^ n _ 1 , \\cdots , \\mathsf { m } ^ n _ n \\} \\\\ = & \\left \\{ g \\bigl ( \\xi _ 0 \\bigr ) , g \\bigl ( \\xi _ 1 ^ { \\mathsf { M } ^ n } \\bigr ) , \\cdots , g \\bigl ( \\xi _ { n - 1 } ^ { \\mathsf { M } ^ n } \\bigr ) \\right \\} \\in \\Theta _ n , \\end{align*}"}
+{"id": "3778.png", "formula": "\\begin{align*} S H ^ { * } _ { M } ( K ; \\Lambda _ { 0 } ) : = H ^ { * } ( \\widehat { t e l } ( \\mathcal { C } ; \\Lambda _ { 0 } ) ) . \\end{align*}"}
+{"id": "8041.png", "formula": "\\begin{align*} f \\circ \\gamma ( t ) = \\gamma ( t + b _ \\gamma ( f ) ) . \\end{align*}"}
+{"id": "4265.png", "formula": "\\begin{align*} 3 ^ { 3 } + 4 ^ { 3 } + 5 ^ { 3 } = 6 ^ { 3 } \\end{align*}"}
+{"id": "8517.png", "formula": "\\begin{align*} \\delta ( z ) = \\exp [ { \\mathcal { C } \\log \\left ( 1 + \\bar { r } _ 1 r _ 2 \\right ) } ] , z \\in \\mathbb { C } ^ { \\pm } , \\end{align*}"}
+{"id": "8344.png", "formula": "\\begin{align*} & u ( x , t ) e ^ { 2 i ( c _ - ( x ) + c ) } = \\lim _ { k \\rightarrow 0 } ( k ^ { - 1 } \\psi ^ + ( x , t ; k ) ) _ { 1 2 } , \\\\ & u ( x , t ) e ^ { - i ( 2 c _ - ( x ) + c ) } = \\lim _ { k \\rightarrow 0 } ( k ^ { - 1 } \\psi ^ - ( x , t ; k ) ) _ { 1 2 } . \\end{align*}"}
+{"id": "3235.png", "formula": "\\begin{align*} \\int _ 0 ^ { ( i - 1 ) \\Delta _ n } ( \\mathcal S ( \\Delta _ n ) - I ) \\Sigma _ s ^ { \\mathcal S _ n } ( \\mathcal S ( \\Delta _ n ) - I ) ^ * d s = & ( i - 1 ) \\Delta _ n ( e \\otimes ( \\Delta _ i \\mathcal S f ) ) ( e \\otimes ( \\Delta _ i \\mathcal S f ) ) ^ * \\\\ = & ( i - 1 ) \\Delta _ n ( \\Delta _ i \\mathcal S f ) ^ { \\otimes 2 } . \\end{align*}"}
+{"id": "155.png", "formula": "\\begin{align*} A _ 0 ( \\Lambda ) = \\frac { \\Lambda - 2 \\pm \\sqrt { \\Lambda ( \\Lambda - 4 ) } } { 2 } , \\ B _ 0 ( \\Lambda ) = \\frac { \\Lambda - 2 \\mp \\sqrt { \\Lambda ( \\Lambda - 4 ) } } { 2 } . \\end{align*}"}
+{"id": "624.png", "formula": "\\begin{align*} O _ { \\gamma } ( f _ 0 ) = 1 + ( q + 1 ) \\frac { q ^ { d _ \\gamma } - 1 } { q - 1 } . \\end{align*}"}
+{"id": "6819.png", "formula": "\\begin{align*} h _ { p } ^ { \\left ( 0 \\right ) } = \\frac { \\left ( - 1 \\right ) ^ { l } } { 2 \\pi i } \\frac { 1 } { \\left ( p + 1 \\right ) \\left ( p + 2 \\right ) \\cdots \\left ( p + l \\right ) } \\int _ { \\mathbb { T } } \\left ( \\widetilde { \\varphi } _ { 0 } ^ { - } \\left ( u \\right ) \\right ) ^ { \\left ( l \\right ) } u ^ { p + l } d u . \\end{align*}"}
+{"id": "1215.png", "formula": "\\begin{align*} \\gamma _ \\Lambda ( \\eta _ { \\Lambda } \\eta _ \\Delta | \\eta _ { \\Lambda ^ c } ) = \\gamma _ { \\Lambda } ( \\eta _ { \\Lambda } | \\eta _ { \\Lambda ^ c } ) \\mathbf { 1 } _ { \\eta _ \\Delta } ( \\eta _ { \\Lambda ^ c } ) . \\end{align*}"}
+{"id": "4622.png", "formula": "\\begin{align*} x _ i x _ j = \\epsilon x _ j x _ i , 1 \\leq i , j \\leq n , \\ i \\neq j , \\end{align*}"}
+{"id": "128.png", "formula": "\\begin{align*} \\sup _ { t _ N \\in [ - T , T ] } \\| P _ N u ( t _ N ) \\| _ { L ^ 2 } ^ 2 & \\leqslant \\| P _ N u _ 0 \\| _ { L ^ 2 } ^ 2 \\\\ & + \\sup _ { t _ N \\in [ - T , T ] } \\Big | \\int _ { [ 0 , t _ N ] \\times \\R ^ 2 } P _ N u P _ N ( u \\partial _ x u ) ~ d t d x d y \\Big | . \\end{align*}"}
+{"id": "5439.png", "formula": "\\begin{align*} [ \\partial a _ \\lambda b ] & = - \\lambda [ a _ \\lambda b ] , \\ \\ [ a _ \\lambda \\partial b ] = ( \\partial + \\lambda ) [ a _ \\lambda b ] \\ \\ \\mbox { ( c o n f o r m a l \\ s e s q u i l i n e a r i t y ) } , \\\\ { [ a _ \\lambda b ] } & = - [ b _ { - \\lambda - \\partial } a ] \\ \\ \\mbox { ( s k e w - s y m m e t r y ) } , \\\\ { [ a _ \\lambda [ b _ \\mu c ] ] } & = [ [ a _ \\lambda b ] _ { \\lambda + \\mu } c ] + [ b _ \\mu [ a _ \\lambda c ] ] \\ \\ \\mbox { ( J a c o b i \\ i d e n t i t y ) } . \\end{align*}"}
+{"id": "4360.png", "formula": "\\begin{align*} G _ { \\alpha \\beta } ^ T H ^ { \\beta } _ i \\overline { G _ { \\alpha \\beta } } & = H ^ { \\alpha } _ i \\\\ & = C ^ { \\alpha } _ { i } H ^ { \\alpha } _ { 1 } ( \\overline { C ^ { \\alpha } _ { i } } ^ { T } ) = C ^ { \\alpha } _ { i } G _ { \\alpha \\beta } ^ T H ^ { \\beta } _ 1 \\overline { G _ { \\alpha \\beta } } ( \\overline { C ^ { \\alpha } _ { i } } ^ { T } ) . \\end{align*}"}
+{"id": "6395.png", "formula": "\\begin{align*} g \\big | _ { U _ \\alpha } = D ^ 2 u _ \\alpha . \\end{align*}"}
+{"id": "8236.png", "formula": "\\begin{align*} \\left ( \\mathfrak { M } ^ \\prime ( f ) \\eta \\right ) ( x , m ) = f \\left ( O ^ m x \\right ) \\eta ( x , m ) , \\ ; \\left ( \\mathcal { F } ' ( O ^ k ) \\eta \\right ) ( x , m ) = \\eta ( x , m - k ) . \\end{align*}"}
+{"id": "5802.png", "formula": "\\begin{align*} H ( d G , d G ) = Q _ 0 d z ^ 2 + \\left ( \\frac { \\tau _ 0 } { 4 } + \\frac { 4 | Q _ 0 | ^ 2 } { \\tau _ 0 } \\right ) d z d \\overline { z } + \\overline { Q _ 0 } d \\overline { z } ^ 2 . \\end{align*}"}
+{"id": "890.png", "formula": "\\begin{align*} x = - r , y = - q . \\end{align*}"}
+{"id": "1972.png", "formula": "\\begin{align*} & \\sum _ q \\sum _ p \\sum _ { u \\in \\mathcal { O } ^ * _ F } \\int _ { \\mathbb { A } ^ n } f _ { B } ( x ) f _ { B } ( \\gamma u x ) d \\alpha ^ n \\\\ & = \\sum _ q \\sum _ p \\sum _ { u \\in \\mathcal { O } ^ * _ F } \\sum _ { 1 \\leq i , j \\leq h _ F } \\frac { B ^ n } { w _ F ^ 2 } \\int _ { \\mathbb { A } ^ n } \\phi ( a _ i x ) \\phi ( a _ j \\gamma u x ) d \\alpha ^ n , \\end{align*}"}
+{"id": "958.png", "formula": "\\begin{align*} \\begin{aligned} & \\Phi _ { j } [ ( v _ { 1 } , v _ { 2 } ) ] ( t ) \\\\ & = i \\int _ { t } ^ { \\infty } U ( t - \\tau ) \\left \\{ \\tilde { { \\mathcal N } } _ { j } ( v _ { 1 } + u _ { \\mathrm { a p } , 1 } , v _ { 2 } + u _ { \\mathrm { a p } , 2 } ) - { \\tilde { \\mathcal N } } _ { j } ( u _ { \\mathrm { a p } , 1 } , u _ { \\mathrm { a p } , 2 } ) \\right \\} ( \\tau ) d \\tau \\\\ & + i \\int _ { t } ^ { \\infty } U ( t - \\tau ) \\mathcal { E } _ { j } ( \\tau ) d \\tau + \\mathcal { R } _ j \\end{aligned} \\end{align*}"}
+{"id": "3959.png", "formula": "\\begin{align*} ( f _ 1 ( \\mathbf { x } ) , f _ 2 ( \\mathbf { x } ) , \\cdots , f _ { n - 1 } ( \\mathbf { x } ) ) = ( f _ 2 ( \\mathbf { y } ) , f _ 3 ( \\mathbf { y } ) , \\cdots , f _ n ( \\mathbf { y } ) ) = ( f _ 2 ( \\mathbf { x } ) , f _ 3 ( \\mathbf { x } ) , \\cdots , f _ n ( \\mathbf { x } ) ) . \\end{align*}"}
+{"id": "7062.png", "formula": "\\begin{align*} \\aligned e _ 1 & \\ , : = \\ , ( 1 - y ) \\ , \\partial _ x - z \\ , \\partial _ y + x \\ , \\partial _ u , \\\\ e _ 2 & \\ , : = \\ , ( 1 - y ) \\ , \\partial _ y - 2 z \\ , \\partial _ z + u \\ , \\partial _ u , \\\\ e _ 3 & \\ , : = \\ , u \\ , \\partial _ x - \\tfrac { 4 } { 3 } \\ , x \\ , \\partial _ y + ( 1 - y ) \\ , \\partial _ z , \\\\ e _ 4 & \\ , : = \\ , x \\ , \\partial _ x - z \\ , \\partial _ z + 2 \\ , u \\ , \\partial _ u . \\endaligned \\end{align*}"}
+{"id": "6574.png", "formula": "\\begin{align*} \\int _ { \\Omega \\cap B _ { \\eta } } n c ^ { 2 } \\psi ^ { 2 } + \\int _ { \\Omega \\cap B _ { \\eta } } | \\nabla c | ^ { 2 } \\psi ^ { 2 } = \\int _ { \\partial \\Omega } \\nabla c \\cdot \\nu c \\psi ^ { 2 } - 2 \\int _ { \\Omega \\cap B _ { \\eta } } \\nabla c \\cdot \\nabla \\psi c \\psi . \\end{align*}"}
+{"id": "1259.png", "formula": "\\begin{align*} \\sum _ { \\xi _ { \\Theta } } \\int _ { \\Omega } h _ { \\Delta } ( \\eta ) ( \\hat { c } _ { \\Theta } ( \\eta , \\xi _ { \\Theta } ) - c _ { \\Theta } ( \\eta , \\xi _ { \\Theta } ) ) \\mu ( d \\eta ) = \\sum _ { \\xi _ { \\Theta } } \\int _ { \\Omega } c _ { \\Theta } ( \\eta , \\xi _ { \\Theta } ) \\left ( h _ { \\Delta } ( \\xi _ { \\Theta } \\eta _ { \\Theta ^ c } ) - h _ { \\Delta } ( \\eta ) \\right ) \\mu ( d \\eta ) . \\end{align*}"}
+{"id": "3493.png", "formula": "\\begin{align*} | G _ { I ( z _ j ' ) } ( z _ j ' , z _ { j + 1 } ) | = \\frac { | P _ { | y _ { j + 1 } - z _ j ' | } ( \\theta + ( z _ j ' + 1 ) \\alpha ) | } { | P _ { | I ( z _ j ' ) | } ( \\theta + z _ { j + 1 } \\alpha ) | } = & \\frac { | \\tilde { P } _ { | y _ { j + 1 } - z _ j ' | } ( \\theta + ( z _ j ' + 1 ) \\alpha ) | } { | \\tilde { P } _ { | I ( z _ j ' ) | } ( \\theta + z _ { j + 1 } \\alpha ) | } \\prod _ { \\ell = z _ { j + 1 } } ^ { z _ j ' } | \\cos ( \\pi ( \\theta + \\ell \\alpha ) ) | . \\end{align*}"}
+{"id": "5638.png", "formula": "\\begin{align*} O _ { n , n } ( R ) : = \\{ A \\in G L _ { 2 n } ( R ) | \\ , \\ , ^ { t } A \\psi _ { 2 n } A = \\psi _ { 2 n } \\} . \\end{align*}"}
+{"id": "2848.png", "formula": "\\begin{align*} T ( u ) = T _ - - \\frac { 4 \\gamma J u } { D } , u \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "8942.png", "formula": "\\begin{gather*} = \\sum _ { s = 0 } ^ { n } \\binom { n } { s } y ^ { n - s } ( 1 - \\rho ) ^ { s } \\sum _ { t = s } ^ { n } ( - 1 ) ^ { n - t } \\frac { ( n - s ) ! } { ( t - s ) ! t ! } \\rho ^ { t - s } ( \\beta + t - s ) ^ { ( s ) } \\\\ = \\sum _ { s = 0 } ^ { n } \\binom { n } { s } y ^ { n - s } ( 1 - \\rho ) ^ { s } \\sum _ { m = 0 } ^ { n - s } ( - 1 ) ^ { n - m - s } \\binom { n - s } { m } \\rho ^ { m } ( \\beta + m ) ^ { ( s ) } . \\end{gather*}"}
+{"id": "3712.png", "formula": "\\begin{align*} h _ { 0 j ; k _ 1 k _ 2 } = 0 , h _ { i j ; k _ 1 a } = 0 , h _ { i j ; a k _ 1 } = 0 \\mbox { o n } \\hat \\Sigma . \\end{align*}"}
+{"id": "7488.png", "formula": "\\begin{align*} \\nabla ( \\eta ^ 2 ) = - a \\zeta ^ 2 | x | _ { A ^ { - 1 } ( 0 ) } ^ { - ( a + 2 ) } A ^ { - 1 } ( 0 ) x + | x | _ { A ^ { - 1 } ( 0 ) } ^ { - a } \\nabla ( \\zeta ^ 2 ) , \\end{align*}"}
+{"id": "1678.png", "formula": "\\begin{align*} v _ { 1 2 } v _ { 2 3 } - u _ 2 v _ { 1 3 } = \\sum _ { i < j } g _ { i j } f _ { i j } \\end{align*}"}
+{"id": "6917.png", "formula": "\\begin{align*} M _ i ( t ) = Q ^ * _ i \\circ ( ( 1 - t ) \\mathrm { I d } + t T _ i ) ^ { - 1 } , \\end{align*}"}
+{"id": "8856.png", "formula": "\\begin{align*} \\bar { y } _ t = - \\int _ 0 ^ t \\exp \\left ( - \\int _ s ^ t Z _ { t ' , 3 } d t ' \\right ) \\bar { z } _ { s , 3 } y _ s d s . \\end{align*}"}
+{"id": "4866.png", "formula": "\\begin{align*} d _ x T ( x _ 0 , y ) & = - \\frac { 1 } { e ^ { - { \\psi ( x _ 0 , y ) } } } \\left ( d \\phi _ { x _ 0 } \\int _ { - \\infty } ^ y e ^ { - { \\psi ( x _ 0 , v ) } } d v - \\int _ { - \\infty } ^ y e ^ { - { \\psi ( x , v ) } } d _ x { \\psi } ( x _ 0 , v ) d v \\right ) \\\\ & = \\Gamma ( x _ 0 , y ) \\end{align*}"}
+{"id": "6573.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\nabla \\psi _ { \\eta } \\| _ { L ^ { 2 } ( \\R ^ { d } ) } ^ { 2 } & = \\sigma _ { d } \\int _ { 0 } ^ { \\eta } \\frac { 1 } { | r \\ln r | ^ { 2 } } r ^ { d - 1 } d r \\\\ & = \\sigma _ { d } \\int _ { \\ln \\frac { 1 } { \\eta } } ^ { \\infty } \\frac { 1 } { \\rho ^ { 2 } } e ^ { - ( d - 2 ) \\rho } d \\rho . \\end{aligned} \\end{align*}"}
+{"id": "2573.png", "formula": "\\begin{align*} ( \\frac { p } { q } ) ^ { u _ { s , t } } = \\frac { B _ { n _ { s , t } } } { m _ { k } N _ { n _ { s , t } } } . \\end{align*}"}
+{"id": "4471.png", "formula": "\\begin{align*} \\dfrac { p } { p + 1 } \\displaystyle \\sum _ { n = 1 } ^ \\infty \\dfrac { 2 ^ { \\omega ( \\gcd ( n , p ) ) } \\cdot 2 r _ { Q ^ \\ast } ( n ) ^ 2 } { n } \\displaystyle \\sum _ { d = 1 } ^ \\infty \\psi \\left ( d \\sqrt { \\dfrac { n } { p } } \\right ) \\leq \\frac { 1 } { \\min Q ^ { * } } + 3 2 1 6 . 6 5 2 4 \\frac { M ( 2 5 . 0 9 p ^ { 3 5 / 6 } ) } { p ^ { 1 / 4 } } . \\end{align*}"}
+{"id": "8450.png", "formula": "\\begin{align*} & r _ 2 ( z ) = 4 z r _ 1 ( z ) , z \\in \\mathbb { R } , \\\\ & \\bar { r } _ 1 ( z ) r _ 2 ( z ) = | r ( k ) | ^ 2 , z \\in \\mathbb { R } ^ { + } , k \\in \\mathbb { R } , \\\\ & \\bar { r } _ 1 ( z ) r _ 2 ( z ) = - | r ( k ) | ^ { 2 } , z \\in \\mathbb { R } ^ { - } , k \\in i \\mathbb { R } . \\end{align*}"}
+{"id": "2464.png", "formula": "\\begin{align*} f ( t ) = ( \\alpha t ) ^ { \\delta } { } _ { 0 } F _ { 1 } \\left ( \\ ; ; \\frac { 1 } { 2 } + \\delta ; - \\frac { 1 } { 4 } \\alpha ^ { 2 } t ^ { 2 } \\right ) = \\left \\{ \\begin{array} { l l } \\cos \\alpha t & \\mbox { f o r } \\ ; \\delta = 0 \\\\ \\sin \\alpha t & \\mbox { f o r } \\ ; \\delta = 1 \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "371.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 2 } \\Delta _ k = \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ { n - 2 } B _ k . \\end{align*}"}
+{"id": "7504.png", "formula": "\\begin{align*} \\div ( A ( x ) \\nabla v ) + h ( x ) = 0 B _ 1 . \\end{align*}"}
+{"id": "6592.png", "formula": "\\begin{align*} ( 1 - d ) r _ { 1 } ^ { - 1 } F _ { r } ( r _ { 1 } , t _ { 1 } ) = 0 , - F ( r _ { 1 } , t _ { 1 } ) = - \\varepsilon < 0 , \\end{align*}"}
+{"id": "5545.png", "formula": "\\begin{align*} x y = y x = x \\wedge y \\end{align*}"}
+{"id": "5080.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( U - f \\textrm { c u r l } ^ { - 1 } U ) \\cdot \\tilde { U } \\dd x = 0 , f = \\frac { \\int _ { \\Omega } U _ 0 \\cdot U \\dd x } { \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } U _ 0 \\cdot U \\dd x } . \\end{align*}"}
+{"id": "3607.png", "formula": "\\begin{align*} \\lambda _ n ( A ) = \\frac { \\lambda ( A \\cap A _ n ) } { \\lambda ( A _ n ) } , \\end{align*}"}
+{"id": "8001.png", "formula": "\\begin{align*} G _ x ( \\theta ) : = \\sum _ { n \\in \\Z } g \\left ( \\pi _ N ( x ) ( \\theta + n ) \\right ) , \\end{align*}"}
+{"id": "2978.png", "formula": "\\begin{align*} \\limsup _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\min _ { 1 \\leq i < j \\leq k } \\widetilde { d } ( T _ { \\vec { v } } ^ n ( x _ i , t _ i ) , T _ { \\vec { v } } ^ n ( x _ j , t _ j ) ) \\geq \\eta > 0 . \\end{align*}"}
+{"id": "2353.png", "formula": "\\begin{align*} v _ t = \\mu ( v ) v _ { x x } + \\phi ' ( x ) ( 1 + \\mu ( v ) ) v _ x + \\phi '' ( x ) ( 1 + \\mu ( v ) ) v - 2 \\mu ( v ) ^ 2 e ^ { 2 \\phi } ( v _ x + \\phi ' ( x ) v ) ^ 2 v . \\end{align*}"}
+{"id": "9256.png", "formula": "\\begin{align*} ( \\Delta u ) ^ { m } \\wedge \\beta _ n ^ { n - m } = m ! ( n - m ) ! \\mathcal { H } _ m ( u ) \\Omega _ { 2 n } . \\end{align*}"}
+{"id": "3639.png", "formula": "\\begin{align*} \\sum _ { k \\leq j < m } \\delta _ { 0 , j } + \\sum _ { 0 < i < k } \\delta _ { i , k } - \\sum _ { k < j \\leq m } \\delta _ { k , j } - \\sum _ { 0 < i < k } \\delta _ { i , m } = 1 + \\sum _ { 0 \\leq i < m } ( \\delta _ { 0 , i } + \\delta _ { i , k } - 1 ) , \\end{align*}"}
+{"id": "6513.png", "formula": "\\begin{align*} & \\lambda _ x | _ { E _ { d ^ * ( y ) } } = \\lambda _ y \\\\ & \\lambda _ x | _ { E _ { [ w b ^ { - n } , w b ^ { - n + 1 } a ] } } = \\phi _ n \\end{align*}"}
+{"id": "6625.png", "formula": "\\begin{align*} G _ { \\epsilon } ( t ) = \\int _ 0 ^ 1 K ( t , r ) \\theta _ { \\epsilon } ( r ) d r , 0 \\leq t \\leq 1 , \\end{align*}"}
+{"id": "7965.png", "formula": "\\begin{align*} h _ { i } = e _ { i } \\cdot F ( x ) . \\end{align*}"}
+{"id": "380.png", "formula": "\\begin{align*} \\frac { 2 n ^ 2 - 1 5 n + 3 3 } { 2 ( n + 4 ) ( n - 1 ) } = \\frac { n - 1 } { 2 } ( c ^ 2 + ( n - 1 ) a ^ 2 + ( n - 1 ) b ^ 2 ) . \\end{align*}"}
+{"id": "6641.png", "formula": "\\begin{align*} X ^ { \\epsilon } ( t ) & = x _ { 0 } + \\int ^ { t } _ { 0 } b ( X ^ { \\epsilon } ( s ) ) d s + \\lambda ( \\epsilon ) \\int ^ { t } _ { 0 } \\sigma ( X ^ { \\epsilon } ( s ) ) \\theta _ { \\epsilon } ( s ) d s , \\end{align*}"}
+{"id": "6915.png", "formula": "\\begin{align*} S _ i : = \\sup _ { Q _ i \\in \\P ( \\R ) } \\left ( \\int _ { \\R } \\hat { f } _ i \\ , d Q _ i - H ( Q _ i ) \\right ) . \\end{align*}"}
+{"id": "3072.png", "formula": "\\begin{align*} \\deg T _ f = 1 + \\binom { n } { 1 } ( k - 1 ) + \\dotsb + \\binom { n } { n - 1 } ( k - 1 ) ^ { n - 1 } + ( k - 1 ) ^ n - \\sum _ { p \\in D } \\mu _ p ( D ) \\end{align*}"}
+{"id": "5289.png", "formula": "\\begin{align*} \\Sigma ^ { \\pm } ( s - w , \\phi , F _ { \\pm } ) = & \\lim _ { z \\downarrow 0 } 2 \\int _ 0 ^ \\infty \\lambda ^ { 4 ( 1 - s + w ) - 1 } \\int _ { - \\infty } ^ \\infty \\int _ 0 ^ \\infty \\sum _ { \\ell , m = 1 } ^ \\infty \\rho _ \\phi ( 3 \\ell m ) K _ \\nu ( 6 \\pi \\ell m t ^ 2 ) \\\\ & \\times \\cos \\left ( \\frac { 6 \\pi \\ell m t ^ 3 u } { \\lambda } \\right ) \\hat { F } \\left ( 0 , 0 , \\frac { 3 m \\lambda } { t } , u \\right ) \\frac { d \\lambda } { \\lambda } t d t d u . \\end{align*}"}
+{"id": "1171.png", "formula": "\\begin{align*} H \\left ( \\rho _ n ^ { k } ( t , \\cdot ) \\mid \\bar { \\rho } ^ { \\otimes k } ( \\cdot ) \\right ) = \\sup _ { g \\in L ^ \\infty ( \\mathbb { T } ^ { k d } ) } \\mathbb { E } _ { \\rho _ n ^ { k } } ( g ) - \\mathbb { E } _ { \\bar { \\rho } ^ { \\otimes k } } ( e ^ g ) + 1 . \\end{align*}"}
+{"id": "5824.png", "formula": "\\begin{align*} { \\bar F _ i } = - { { w } _ i } { \\bar F } , \\end{align*}"}
+{"id": "9248.png", "formula": "\\begin{align*} \\begin{aligned} \\det \\left ( \\frac { \\partial ^ 2 u } { \\partial \\overline { q _ l } \\partial { q _ k } } \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "4085.png", "formula": "\\begin{align*} ( \\widehat { h } ^ i , \\widehat { h } ^ i { } _ j , \\widehat { h } ^ i { } _ { j k } ) & = ( \\phi ( h ^ a ) ^ i , D \\phi ( h ^ a ) ^ i { } _ \\alpha h ^ \\alpha { } _ j , H \\phi ( h ^ a ) ^ i { } _ { \\alpha \\beta } h ^ \\alpha { } _ j h ^ \\beta { } _ k + D \\phi ( h ^ a ) ^ i { } _ \\alpha h ^ \\alpha { } _ { j k } ) \\\\ * & = ( \\phi ( h ^ a ) ^ i , ( D \\phi ( h ^ a ) ^ i { } _ j , H \\phi ( h ^ a ) ^ i { } _ { j k } ) ( h ^ i { } _ j , h ^ i { } _ { j k } ) ) , \\end{align*}"}
+{"id": "5518.png", "formula": "\\begin{align*} \\max _ { k \\leq n + 1 } a _ k = 1 \\leq \\frac { 1 } { \\sqrt { 2 } } \\sqrt { 1 + x } = \\frac { 1 } { \\sqrt { 2 } } \\left ( \\sum _ { k = 1 } ^ { n + 1 } a _ k ^ 2 \\right ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "1593.png", "formula": "\\begin{align*} \\mathbf { \\Phi } _ { \\mathrm { r } } ^ { H } \\mathbf { \\Phi } _ { \\mathrm { r } } + \\mathbf { \\Phi } _ { \\mathrm { t } } ^ { H } \\mathbf { \\Phi } _ { \\mathrm { t } } = \\mathbf { I } _ { M } . \\end{align*}"}
+{"id": "8271.png", "formula": "\\begin{align*} u ( t ) : = \\sum _ { i = 1 } ^ { \\infty } \\big | \\psi _ i ( t ) - \\rho _ i ( t ) \\big | = \\sum _ { i = 1 } ^ { \\infty } | u _ i ( t ) | . \\end{align*}"}
+{"id": "7308.png", "formula": "\\begin{align*} \\begin{array} { l } \\mathop { \\max } \\limits _ { \\bf p } \\eta = \\frac { R _ T } { P _ T } \\\\ { \\rm { s } } { \\rm { . t } } { \\rm { . } } \\\\ C 2 , \\ , C 3 , \\ , C 8 , \\ , { \\rm a n d } \\ , \\ , C 9 . \\end{array} \\end{align*}"}
+{"id": "4984.png", "formula": "\\begin{align*} W ( \\xi _ 0 , k + 1 , x , \\mathsf { M } ^ { k + 1 } ) = V ( \\xi _ 0 , k + 1 , x ) . \\end{align*}"}
+{"id": "5489.png", "formula": "\\begin{align*} G ( p , s ) = \\int _ 0 ^ \\infty e ^ { - s t ^ 2 / 8 } t ^ { p - 1 } \\dd t = s ^ { - p / 2 } 2 ^ { 3 p / 2 - 1 } \\Gamma ( p / 2 ) . \\end{align*}"}
+{"id": "1665.png", "formula": "\\begin{align*} z = ( \\pi \\otimes ( \\chi \\circ \\det ) , \\alpha _ z ) \\end{align*}"}
+{"id": "7351.png", "formula": "\\begin{gather*} \\Omega = \\Omega _ 1 \\times \\ldots \\times \\Omega _ n \\quad \\quad \\mathcal { A } = \\mathcal { A } _ 1 \\otimes \\ldots \\otimes \\mathcal { A } _ n \\end{gather*}"}
+{"id": "7205.png", "formula": "\\begin{align*} d u = \\Delta _ p u \\ , d t + \\nabla u \\circ d W , u ( 0 , \\cdot ) = u _ 0 , \\end{align*}"}
+{"id": "3385.png", "formula": "\\begin{align*} s _ l ( k ) = \\left \\{ \\begin{array} { l l } { k + l - 1 \\choose l - 1 } & k \\in [ 0 , N ] \\\\ { k + l - 1 \\choose l - 1 } - l { k + l - N - 2 \\choose l - 1 } & k \\in [ N + 1 , 2 N ] \\end{array} \\right . . \\end{align*}"}
+{"id": "7702.png", "formula": "\\begin{align*} \\dim H & \\geq x - \\lfloor \\tilde \\varepsilon x \\rfloor + \\ell _ { i + 1 } '' - \\lfloor \\tilde \\varepsilon x \\rfloor + \\sum _ { j = 1 } ^ i \\big ( \\ell _ { j } - \\lfloor 2 ^ { j - i - 1 } \\tilde \\varepsilon x \\rfloor \\big ) \\\\ & \\geq r + x - \\ell _ { i + 1 } ' - 3 \\tilde \\varepsilon x \\\\ & \\geq r + x - ( 1 + \\tilde \\varepsilon ) ^ { - i - 1 } ( r + x ) . \\end{align*}"}
+{"id": "5888.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 + } \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } \\left ( \\int _ 0 ^ { T - \\delta } \\Vert Y _ n ( t + \\delta ) - Y _ n ( t ) \\Vert _ { H } ^ { \\alpha } d t > \\epsilon \\right ) = 0 . \\end{align*}"}
+{"id": "7449.png", "formula": "\\begin{align*} \\cos ^ 2 \\theta \\delta _ { m , k } + \\sin ^ 2 \\theta \\sqrt { m } = \\cos ^ 2 \\theta \\sqrt { m } + \\sin ^ 2 \\theta \\delta _ { m , l } = \\gamma _ { m , k , l } . \\end{align*}"}
+{"id": "3735.png", "formula": "\\begin{align*} \\beta _ { \\bar { \\mathbf { g } } } \\mathbf { g } = \\beta _ { \\bar g } g + \\bar u ^ { - 2 } u d u - \\bar u ^ { - 1 } g ( \\nabla _ { \\bar g } \\bar u , \\cdot ) . \\end{align*}"}
+{"id": "6692.png", "formula": "\\begin{align*} \\delta ( \\eta , \\theta \\gamma ^ + ) = f ( \\eta \\theta \\gamma ^ + , \\theta \\gamma ^ - ) - f ( \\theta \\gamma ^ + , \\theta \\gamma ^ - ) . \\end{align*}"}
+{"id": "8866.png", "formula": "\\begin{align*} \\tilde p : = p _ \\beta ( \\beta ) , \\end{align*}"}
+{"id": "1989.png", "formula": "\\begin{align*} g ^ { - 1 } ( z - 2 g ^ { - 1 } ( z ) ) = - g ^ { - 1 } ( z ) . \\end{align*}"}
+{"id": "4358.png", "formula": "\\begin{align*} & \\int _ M ( \\eta + g ^ { - 1 } ) | D ^ { '' * } v | ^ 2 _ Q d V _ M + \\int _ M \\eta | D ^ { '' } v | ^ 2 _ Q d V _ M \\\\ \\ge & \\int _ M \\langle [ \\eta \\sqrt { - 1 } \\Theta _ Q - \\sqrt { - 1 } \\partial \\bar { \\partial } \\eta - \\sqrt { - 1 } g \\partial \\eta \\wedge \\bar { \\partial } \\eta , \\Lambda _ { \\omega } ] v , v \\rangle _ Q d V _ M . \\end{align*}"}
+{"id": "1246.png", "formula": "\\begin{align*} \\sup _ { y \\in \\Z ^ d } \\sum _ { \\Delta \\ni y } \\sum _ { z \\neq y } \\sum _ { \\xi _ { \\Delta } } \\sum _ { i = 1 } ^ q \\norm { \\nabla ^ i _ z \\hat { c } _ \\Delta ( \\cdot , \\xi _ \\Delta ) } _ \\infty < \\infty . \\end{align*}"}
+{"id": "1954.png", "formula": "\\begin{align*} r ' ( X \\cup z ) = r ( X ) + 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
+{"id": "1796.png", "formula": "\\begin{align*} b ( e ^ { - 2 \\pi u } ) = \\frac { 1 } { \\sqrt { 3 } u } c ( e ^ { - \\frac { 2 \\pi } { 3 u } } ) , \\end{align*}"}
+{"id": "5720.png", "formula": "\\begin{align*} | Z ' _ { f '' } | + 1 = | Z ' _ { f ' } | = | Z ' _ f | \\in \\{ 0 , 1 , 2 , p ^ \\delta + 1 \\} . \\end{align*}"}
+{"id": "424.png", "formula": "\\begin{align*} a _ { i } = \\sum _ { j = 1 } ^ { n } k _ { i j } \\theta ^ { j - 1 } . \\end{align*}"}
+{"id": "5277.png", "formula": "\\begin{align*} \\partial _ t \\left [ t ^ 2 \\left [ \\left ( \\frac { \\alpha } { t ^ 4 } + \\beta + \\gamma \\frac { u ^ 2 } { t ^ 4 } \\right ) ^ 2 + \\delta ^ 2 \\frac { u ^ 2 } { t ^ 4 } \\right ] \\right ] = 0 . \\end{align*}"}
+{"id": "8352.png", "formula": "\\begin{align*} c = c _ - ( x ) - c _ + ( x ) = \\frac { 1 } { 2 } \\int _ { - \\infty } ^ \\infty | u _ x | ^ 2 d x , \\end{align*}"}
+{"id": "2727.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n + 1 } \\lambda _ j ( x ) ^ 2 = \\frac { | | x | | ^ 2 + 1 } { n + 1 } . \\end{align*}"}
+{"id": "508.png", "formula": "\\begin{align*} \\Omega ^ 1 _ { O ( \\Delta ( \\rho _ 1 , \\ldots , \\rho _ n ) ) } \\cong \\bigoplus _ { i = 1 } ^ n O ( \\Delta ( \\rho _ 1 , \\ldots , \\rho _ n ) ) . d x _ i , \\end{align*}"}
+{"id": "542.png", "formula": "\\begin{align*} v _ i v _ j & = ( c _ i - 4 s _ { i } - 4 z _ { i } ) ( c _ j - 4 s _ { j } - 4 z _ { j } ) \\\\ & = c _ i c _ j - 4 \\left ( { c _ i ( s _ { j } + z _ i ) + c _ j ( s _ { i } + z _ { i } ) } \\right ) \\\\ & \\phantom { { } = { } } + 1 6 ( s _ { i } s _ { j } + s _ i z _ { j } + s _ j z _ { i } + z _ { i } z _ { j } ) \\\\ & = - 3 c _ { i , j } { - 6 t _ { i , j } } + c _ i c _ j + 1 6 ( s _ i z _ { j } + s _ j z _ { i } + z _ { i } z _ { j } ) \\end{align*}"}
+{"id": "3315.png", "formula": "\\begin{align*} & \\Phi _ S ( \\sigma , \\tau ) _ i = \\left \\{ \\begin{aligned} & \\sigma & & ( i \\in S ) \\\\ & \\tau & & ( i \\notin S ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5566.png", "formula": "\\begin{align*} W ( y | x ) = b _ k - a + \\sum _ { n = 2 } ^ \\infty ( a _ n - a ) \\ , \\end{align*}"}
+{"id": "6744.png", "formula": "\\begin{align*} x = \\mu - \\sigma \\left [ F ^ { - 1 } _ \\Gamma ( 1 - 2 u ) \\right ] ^ { 1 / s } , \\end{align*}"}
+{"id": "5226.png", "formula": "\\begin{align*} & { \\mathcal { E } } [ v _ n , b _ n ] \\leq { \\mathcal { E } } [ v _ { 0 , n } , b _ { 0 , n } ] , \\\\ & H _ { \\gamma } [ b _ n ] = H _ { \\gamma } [ b _ { 0 , n } ] . \\end{align*}"}
+{"id": "5254.png", "formula": "\\begin{align*} X _ { \\gamma } ' = \\{ x ' \\in X ' \\ ; | \\ ; x ' \\ ; \\tau \\| \\cdot \\| \\} \\eqqcolon X ^ { \\circ } \\end{align*}"}
+{"id": "4899.png", "formula": "\\begin{align*} \\beta _ \\alpha ( z ) = \\frac { \\alpha - 1 } { \\Phi ( z ) } - z \\frac { 1 - \\Phi ( z ) } { \\phi ( z ) } + 2 - \\alpha . \\end{align*}"}
+{"id": "6213.png", "formula": "\\begin{align*} W _ - ( r ) = \\frac { f ( r ) d W _ + ( r ) / d r + E _ 0 - E _ 1 } { W _ + ( r ) } . \\end{align*}"}
+{"id": "6740.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } \\alpha } \\ell = & \\psi ( \\alpha + \\beta ) - \\psi ( \\alpha ) + \\log \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\end{align*}"}
+{"id": "7552.png", "formula": "\\begin{align*} D ( f \\circ l _ g ) = ( D f ) \\circ l _ g , \\ ; \\ ; \\ ; f \\in C ^ \\infty ( G ) , \\forall g \\in G , \\end{align*}"}
+{"id": "989.png", "formula": "\\begin{align*} f ( \\delta _ 2 ) & = - ( \\alpha + 3 1 ) q ^ { 2 d + 1 } + ( 2 \\alpha ^ 2 + 3 7 \\alpha + 2 5 6 ) q ^ { 2 d } + q ^ { d + 4 } + 3 q ^ { d + 3 } , \\end{align*}"}
+{"id": "4187.png", "formula": "\\begin{align*} \\begin{cases} \\nabla w \\cdot \\nabla ^ \\perp ( u - q \\ln \\frac { 1 } { \\varepsilon } ) = 0 , \\\\ w = \\mathcal { L } _ H u , \\\\ \\nabla ^ \\perp u \\cdot \\nu | _ { \\partial \\Omega } = 0 . \\end{cases} \\end{align*}"}
+{"id": "8308.png", "formula": "\\begin{align*} M ^ 2 ( t ) = 2 a t . \\end{align*}"}
+{"id": "4545.png", "formula": "\\begin{align*} & | \\Gamma _ { \\ell , 0 , 0 } | = ( q - 1 ) ^ 3 ( q ^ 2 + q + 1 ) ( q + 1 ) q ^ { 3 \\ell + 6 } \\\\ & | \\Gamma _ { \\ell , \\ell , 0 } | = ( q - 1 ) ^ 3 ( q + 1 ) ^ 2 q ^ { 4 \\ell + 6 } \\\\ & | \\Gamma _ { \\ell , \\ell , \\ell } | = ( q - 1 ) ^ 3 ( q ^ 2 + q + 1 ) ( q + 1 ) q ^ { 3 \\ell + 6 } . \\end{align*}"}
+{"id": "3801.png", "formula": "\\begin{align*} [ D _ * ^ q , M _ { z ^ q } ] ( f ) ( z ) = a _ 0 \\Gamma ( q + 1 ) + \\sum _ { n = 1 } ^ { \\infty } a _ n \\beta _ { m , q } \\ ; z ^ { q n } , \\end{align*}"}
+{"id": "627.png", "formula": "\\begin{align*} \\chi _ K ( p ) = \\begin{cases} 1 & p K \\\\ - 1 & p K \\\\ 0 & p K \\end{cases} . \\end{align*}"}
+{"id": "2250.png", "formula": "\\begin{align*} K _ \\lambda ( Z , W ) = \\det ( I _ n - Z W ^ * ) ^ { - \\lambda } . \\end{align*}"}
+{"id": "7149.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } e ^ { - t \\tau _ { k } } = t ^ { 1 - n } \\int _ { \\partial \\Omega } a _ 0 ( x ) \\ , d S + t ^ { 2 - n } \\int _ { \\partial \\Omega } a _ 1 ( x ) \\ , d S + \\begin{cases} O ( t \\log t ) , & n = 2 , \\\\ O ( t ^ { 3 - n } ) , & n \\geqslant 3 , \\end{cases} \\ t \\to 0 ^ + . \\end{align*}"}
+{"id": "6179.png", "formula": "\\begin{align*} E _ 0 = \\kappa ( L + 1 ) ^ 2 - \\frac { Q ^ 2 } { 4 ( L + 1 ) ^ 2 } + \\frac { \\kappa ( L + 1 ) B _ 1 } { Q } \\left ( \\frac { ( L + 1 ) B _ 1 } { Q } + 2 L + 2 \\right ) . \\end{align*}"}
+{"id": "7461.png", "formula": "\\begin{align*} N ( f ) & = \\frac { 1 } { d } \\sum _ { i = 0 } ^ { d - 1 } \\lvert ( 1 - m ) \\det ( I - A ^ i M ) \\rvert \\\\ & = \\frac { 1 } { d } \\sum _ { i = 0 } ^ { d - 1 } \\lvert ( 1 - m ) \\det ( I - B ^ i C ^ i M ) \\rvert \\\\ & = \\frac { 1 } { d } \\sum _ { i = 0 } ^ { d - 1 } \\lvert ( 1 - m ) \\det ( I - C ^ i M ) \\rvert . \\end{align*}"}
+{"id": "3821.png", "formula": "\\begin{align*} \\eta ( U | \\bar U ; x , t ) \\ge \\begin{cases} c _ 1 ' \\big | U - \\bar U \\big | ^ 2 & \\ ; | U | \\le r _ 2 , \\\\ c _ 2 ' \\ , \\big | U - \\bar U \\big | ^ p & \\ ; | U | \\ge r _ 2 , \\end{cases} \\end{align*}"}
+{"id": "1874.png", "formula": "\\begin{align*} E ( u ) = \\int _ M \\frac 1 2 \\sum _ { i = 1 } ^ m h \\big ( d u ( e _ i ) , d u ( e _ i ) \\big ) \\ , d v = \\int _ M { \\frac 1 2 } { \\big ( \\sigma _ 1 ( u ^ { \\ast } ) \\big ) } \\ , d v . \\end{align*}"}
+{"id": "7645.png", "formula": "\\begin{align*} \\mu _ t ^ { * , \\xi } = \\rho ( - R ^ { - 1 } B P _ t \\nu _ t ^ { * , \\xi } - R ^ { - 1 } B \\varphi _ t ^ { * , \\xi } ) \\end{align*}"}
+{"id": "2232.png", "formula": "\\begin{align*} x ^ { ( k ) } _ { i } = x ^ { ( \\hat { k } ) } _ { i } ( \\forall i \\in \\{ 1 , \\dots , D \\} ) ( \\forall k > \\hat { k } ) . \\end{align*}"}
+{"id": "1146.png", "formula": "\\begin{align*} { S } ^ { \\rm P R \\ , s y m } _ { i , \\ , j , \\ , \\ell } ( { \\cal A } ) : = 2 \\ , { x _ { N ( \\ell - 1 ) + i } \\ , x _ { N ( \\ell - 1 ) + j } } . \\end{align*}"}
+{"id": "2273.png", "formula": "\\begin{align*} a ( Z ) = a \\big ( ( Z Z ^ * ) ^ \\frac { 1 } { 2 } \\big ) \\end{align*}"}
+{"id": "4996.png", "formula": "\\begin{align*} [ - T , T ] = \\bigcup _ { j = - [ T / \\tau ] - 1 } ^ { [ T \\tau ] } [ j \\tau , ( j + 1 ) \\tau ] \\cap [ - T , T ] . \\end{align*}"}
+{"id": "1687.png", "formula": "\\begin{align*} \\mathbf { P } _ n = \\begin{bmatrix} U ( 0 ) & \\mathbf { Q } _ n \\\\ \\ast & \\mathbf { T } _ n + \\mathbf { I } _ n ^ { - 1 } \\end{bmatrix} . \\end{align*}"}
+{"id": "1430.png", "formula": "\\begin{align*} D ( m , a ) = \\det ( \\Gamma ( a + i + j ) ) _ { i , j = 1 } ^ { m } . \\end{align*}"}
+{"id": "7877.png", "formula": "\\begin{align*} Z ^ i ( v ^ \\mathrm { a p p } ( s ) , s ) - Z ^ i ( v ( s ) , s ) = \\mathcal O ( s ^ \\frac 3 2 ) . \\end{align*}"}
+{"id": "1853.png", "formula": "\\begin{align*} \\frac { d } { d \\rho } \\int _ { B _ \\rho ( x _ 0 ) } \\big ( \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) - 1 \\big ) \\ , d v = \\int _ { \\partial B _ \\rho ( x _ 0 ) } \\big ( \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) - 1 \\big ) \\ , d s \\ , . \\end{align*}"}
+{"id": "1658.png", "formula": "\\begin{align*} D ( F r o b _ v ) = q _ v S _ v \\end{align*}"}
+{"id": "8307.png", "formula": "\\begin{align*} \\begin{aligned} & A y '' _ { b e n } + M _ { b e n } y _ { b e n } = E y _ { b e n } \\\\ & a A y '' _ { n b } + M _ { n b } y _ { n b } = E y _ { n b } , \\end{aligned} \\end{align*}"}
+{"id": "3306.png", "formula": "\\begin{align*} \\frac { \\sqrt { p } } { \\log p + 1 } > \\frac { ( s + 1 ) 2 ^ { l + 1 } } { \\left ( 1 - \\sum _ { i = 1 } ^ s \\frac 1 { p _ i } \\right ) \\prod _ { j = 1 } ^ l \\left ( 1 - \\frac 1 { q _ j } \\right ) } . \\end{align*}"}
+{"id": "6070.png", "formula": "\\begin{align*} \\int _ \\Omega \\left ( \\eta x _ { n + 1 } ^ m + \\lambda \\right ) d V = \\int _ \\Sigma \\langle Y , \\nu \\rangle \\ , d A . \\end{align*}"}
+{"id": "1783.png", "formula": "\\begin{align*} d _ { C y } ( ( z , t , u ) , ( z ' , t ' , u ' ) ) = \\big \\lvert ( \\lvert z - z ' \\rvert ^ { 2 } + \\lvert u - u ' \\rvert ) ^ { 2 } + \\lvert t - t ' + 2 \\Im ( z \\overline { z ' } ) \\rvert ^ { 2 } \\big \\rvert ^ { 1 / 4 } . \\end{align*}"}
+{"id": "4710.png", "formula": "\\begin{align*} w _ { r } & = 2 b x _ n ^ a ( 1 - r ^ 2 ) ^ { b - 1 } r \\\\ w _ { r r } & = 2 b x _ n ^ a ( 1 - r ^ 2 ) ^ { b - 2 } [ 1 - ( 2 b - 1 ) r ^ 2 ] \\\\ w _ { x _ n } & = 1 - a x _ n ^ { a - 1 } ( 1 - r ^ 2 ) ^ b \\\\ w _ { x _ n x _ n } & = a ( 1 - a ) x _ n ^ { a - 2 } ( 1 - r ^ 2 ) ^ b \\\\ w _ { x _ n r } & = 2 a b x _ n ^ { a - 1 } ( 1 - r ^ 2 ) ^ { b - 1 } r . \\end{align*}"}
+{"id": "3324.png", "formula": "\\begin{align*} | T | = | T ' | \\end{align*}"}
+{"id": "5876.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\Big [ \\sup _ { t \\leq T } \\Vert X ( t , x _ n ) - X ( t , x ) \\Vert _ H ^ p \\Big ] = 0 . \\end{align*}"}
+{"id": "8753.png", "formula": "\\begin{align*} f ( z ) - f ( 0 ) & = \\int _ { 0 } ^ 1 \\frac { d f ( t z ) } { d t } d t \\\\ & = \\int _ { 0 } ^ 1 \\left ( \\frac { \\partial f } { \\partial x } ( t z ) x + \\frac { \\partial f } { \\partial y } ( t z ) y \\right ) d t . \\end{align*}"}
+{"id": "6808.png", "formula": "\\begin{align*} R ( \\rho ) = T _ { + } ( \\rho ) G _ { - } ( \\rho ) \\end{align*}"}
+{"id": "6080.png", "formula": "\\begin{align*} \\bar { N } ( v ) = \\{ u \\in V \\mid ( u , v ) \\in E \\} \\cup \\{ v \\} . \\end{align*}"}
+{"id": "2878.png", "formula": "\\begin{align*} \\bar K ^ { ( n , \\ell ) } ( { x , y } ) : = \\frac { 1 } { 4 ( n + 1 ) ^ 2 } \\sum _ { j , j ' = - n - 1 } ^ n \\Phi \\left ( \\frac { j } { n + 1 } , \\frac { j ' } { n + 1 } \\right ) \\cos \\left ( \\frac { \\pi j ( y - x ) } { n + 1 } \\right ) \\cos \\left ( \\frac { \\pi j ' ( y - x - \\ell ) } { n + 1 } \\right ) . \\end{align*}"}
+{"id": "6676.png", "formula": "\\begin{align*} \\ell _ \\beta ( \\langle \\gamma \\rangle ) = \\beta ( \\gamma , \\gamma ^ + ) \\end{align*}"}
+{"id": "3091.png", "formula": "\\begin{align*} T ( \\theta ) : = \\begin{pmatrix} F ( x ) - \\lambda \\\\ \\lambda \\\\ x - y \\end{pmatrix} , \\forall \\ \\theta : = \\begin{pmatrix} x \\\\ y \\\\ \\lambda \\end{pmatrix} \\in \\mathcal { K } . \\end{align*}"}
+{"id": "506.png", "formula": "\\begin{align*} \\lim _ { \\sum m _ i \\to \\infty } | \\lambda _ { m _ 1 , \\ldots , m _ n } | ^ { 1 / \\sum m _ i } = 0 . \\end{align*}"}
+{"id": "7086.png", "formula": "\\begin{gather*} \\Phi ( r ) = \\displaystyle \\int _ { B _ r } | D u | ^ { p + 2 \\beta } d x , \\end{gather*}"}
+{"id": "5373.png", "formula": "\\begin{align*} S _ k ^ { i j } ( \\Delta u ) _ { i j } + \\sum _ i S _ k ^ { p q , r s } u _ { p q i } u _ { r s i } = \\Delta f , \\end{align*}"}
+{"id": "3669.png", "formula": "\\begin{align*} ( T _ h ) ^ i _ { j k } = \\tfrac { 1 } { 2 } ( h ^ i _ { j ; k } + h ^ i _ { k ; j } - h _ { j k ; } ^ { \\ ; \\ ; \\ ; \\ ; \\ ; i } ) \\end{align*}"}
+{"id": "6397.png", "formula": "\\begin{align*} \\omega ( X _ v , X _ w ) = 0 \\end{align*}"}
+{"id": "8957.png", "formula": "\\begin{align*} I _ 1 ^ p : = \\sum _ { | k | \\le N } \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { p } ( x ) \\frac { \\Bigl | \\bigl [ V _ 1 ^ - ( x ) \\bigr ] ' \\Bigr | ^ p } { \\bigl [ V _ 1 ^ - ( x ) \\bigr ] ^ { 2 p } } \\biggl ( \\int _ { \\eta _ { k - 1 } } ^ x v _ 1 ^ { - p ' } ( t ) \\biggl ( \\int _ { a ( x ) } ^ t v _ 1 ^ { - p ' } \\biggr ) ^ { 1 - \\delta } [ V _ 1 ( t ) ] ^ { \\delta } \\bigl | G ^ { ( \\delta ) } _ { 1 , k } ( t ) \\bigr | ^ { p ' - 1 } \\ , d t \\biggr ) ^ p \\ , d x \\end{align*}"}
+{"id": "5380.png", "formula": "\\begin{align*} \\begin{aligned} \\Psi \\leq & - \\frac { \\theta } { 2 } | x ' | ^ 2 , \\mbox { o n } \\partial _ 1 \\omega _ \\delta \\\\ \\Psi \\leq & - \\frac { \\delta ^ 2 } { 2 } , \\mbox { o n } \\partial _ 2 \\omega _ \\delta \\\\ \\Psi \\leq & - \\frac { \\theta \\delta ^ 2 } { 2 } , \\mbox { o n } \\partial _ 3 \\omega _ \\delta . \\end{aligned} \\end{align*}"}
+{"id": "8994.png", "formula": "\\begin{align*} m = \\max \\{ q \\in [ M - 1 ] : \\ r _ q \\neq \\ell _ q \\} , \\end{align*}"}
+{"id": "1418.png", "formula": "\\begin{align*} P _ { 1 1 } & : = \\int _ { W ^ d \\cap \\{ \\max _ j ( y _ j - y _ { j - 1 } ) \\le \\sqrt n , | y _ 1 - x _ 1 - n | \\le \\sqrt n \\} } \\P _ x ( \\rho > [ n / 2 ] , S _ { [ n / 2 ] } \\in d y ) \\P _ y ( \\rho > n - [ n / 2 ] ) \\\\ P _ { 1 2 } & : = \\int _ { W ^ d \\cap \\{ \\max _ j ( y _ j - y _ { j - 1 } ) \\le \\sqrt n , | y _ 1 - x _ 1 - n | > \\sqrt n \\} } \\P _ x ( \\rho > [ n / 2 ] , S _ { [ n / 2 ] } \\in d y ) \\P _ y ( \\rho > n - [ n / 2 ] ) \\end{align*}"}
+{"id": "7681.png", "formula": "\\begin{align*} \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) ) } ^ { N , i } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } h \\Big ( \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) ) } ^ { N , j } \\Big ) = - R ^ { - 1 } B \\Big [ P _ t \\nu ^ { N , i } _ { \\boldsymbol { x } } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } \\Phi ( t , \\nu ^ { N , j } _ { \\boldsymbol { x } } ) \\Big ] \\end{align*}"}
+{"id": "6296.png", "formula": "\\begin{align*} \\begin{aligned} \\min & \\ \\frac { 1 } { 2 } x ^ T A x + b ^ T x \\\\ \\mbox { s . t . } & \\ \\frac { 1 } { 2 } x ^ T B _ i x + c _ i ^ T x + d _ i \\leq 0 , i = 1 , \\dots , m , \\end{aligned} \\end{align*}"}
+{"id": "7259.png", "formula": "\\begin{align*} t = r _ 0 + r _ 1 q + \\cdots + r _ { n - 1 } q ^ { n - 1 } , \\end{align*}"}
+{"id": "4786.png", "formula": "\\begin{align*} F ( \\eta ) : = T - \\frac { 1 } { \\eta } I , \\eta \\in \\Omega . \\end{align*}"}
+{"id": "2793.png", "formula": "\\begin{align*} h ^ { D _ { N } } ( y ) = S _ { k } ( x ) + ( S _ { k - l } ( y ) - S _ { k } ( x ) ) + ( S _ { 0 } ( y ) - S _ { k - l } ( y ) ) , \\end{align*}"}
+{"id": "8022.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ j a _ f ( p _ i ^ { 2 m _ i } ) \\prod _ { i = j + 1 } ^ r a _ f ( p _ i ^ { l _ i } ) \\prod _ { i = 1 } ^ j a _ f ( q _ i ^ { m _ i ' } ) \\prod _ { i = j + 1 } ^ r a _ f ( q _ i ^ { l ' _ i } ) = \\prod _ { u = 1 } ^ t \\left ( \\prod _ { i \\in \\mathcal I ( s _ u ) } a _ f ( s _ u ^ { 2 i } ) \\right ) , \\end{align*}"}
+{"id": "405.png", "formula": "\\begin{align*} \\phi ( Y ) = f ^ { - 1 } ( z _ 0 ) . \\end{align*}"}
+{"id": "4351.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi < - t _ 0 \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h \\tilde { c } ( - \\Psi ) \\ge \\int _ { \\{ \\Psi < - t _ 0 \\} } | \\tilde F | ^ 2 _ h \\tilde { c } ( - \\Psi ) . \\end{align*}"}
+{"id": "6527.png", "formula": "\\begin{align*} 2 0 = 2 2 - 2 = 2 3 - 3 = 2 5 - 5 = 2 7 - 7 . \\end{align*}"}
+{"id": "8886.png", "formula": "\\begin{align*} \\lambda _ { f } ( n ) = \\lambda _ t ( n ) = \\lambda _ s ( n ) \\end{align*}"}
+{"id": "985.png", "formula": "\\begin{align*} | T \\cap M ' | \\le \\begin{cases} \\theta _ { d - 1 } q ^ { d ^ 2 - d + 1 } & P \\in \\tau _ 1 \\cup \\tau _ 2 , \\\\ ( q ^ { d - 1 } + \\theta _ { d - 1 } ) q ^ { d ^ 2 - d } & \\end{cases} \\end{align*}"}
+{"id": "3321.png", "formula": "\\begin{align*} \\dim K = | \\tau ^ 0 | - 1 = \\dim L . \\end{align*}"}
+{"id": "6184.png", "formula": "\\begin{align*} R = - E _ 0 . \\end{align*}"}
+{"id": "1619.png", "formula": "\\begin{align*} \\psi : C ^ { n - 1 } _ c ( G , V ) \\to C ^ n _ c ( G , V ) ^ G , \\psi ( c ) ( x _ 0 , \\ldots , x _ n ) : = x _ 0 \\cdot c ( x _ 0 ^ { - 1 } x _ 1 , x _ 1 ^ { - 1 } x _ 2 , \\ldots , x _ { n - 1 } ^ { - 1 } x _ n ) . \\end{align*}"}
+{"id": "7856.png", "formula": "\\begin{align*} \\hat \\Phi ^ \\dagger _ s ( 0 , 0 ) = 0 , \\qquad \\hat \\Phi ^ \\dagger _ { a s } ( 0 , 0 ) = \\gamma _ 2 , \\end{align*}"}
+{"id": "3516.png", "formula": "\\begin{align*} ( - 1 ) ^ p F _ n - n ^ p \\ \\ = \\ \\ \\mathcal { C } _ n ^ { ( p ) } \\ - \\ 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } \\mathcal { C } _ n ^ { ( p - 2 j - 1 ) } . \\end{align*}"}
+{"id": "1600.png", "formula": "\\begin{align*} \\iota _ { k } ^ \\star = \\gamma _ { k } , \\forall k \\in \\mathcal { K } , \\end{align*}"}
+{"id": "4166.png", "formula": "\\begin{align*} \\textup { S t o C } : = \\min \\{ \\alpha + \\gamma - \\frac { 1 } { 2 } - \\frac { \\alpha } { 2 \\beta } ( \\kappa - r ) , 1 - \\varepsilon \\} \\leq \\min \\Big \\{ \\alpha + \\frac { 1 } { 2 } , 1 - \\varepsilon \\Big \\} . \\end{align*}"}
+{"id": "114.png", "formula": "\\begin{align*} \\omega _ \\alpha ( \\xi , \\eta ) = | \\xi | ^ { \\alpha } \\xi + \\frac { \\eta ^ 2 } { \\xi } , \\xi \\in \\R \\backslash \\{ 0 \\} , \\ ; \\eta \\in \\R , \\end{align*}"}
+{"id": "1053.png", "formula": "\\begin{align*} \\rho _ { \\prescript { \\sigma ^ n } { } x } ( \\prescript { \\sigma ^ n } { } ( u ) ) = \\prescript { \\sigma ^ n } { } ( \\rho _ x ( u ) ) . \\end{align*}"}
+{"id": "8457.png", "formula": "\\begin{align*} T ( x ; z ) = \\left ( M _ { - , 1 } ( x ; z ) - e _ 1 , M _ { + , 2 } ( x ; z ) - e _ 2 \\right ) , \\end{align*}"}
+{"id": "7828.png", "formula": "\\begin{align*} F ^ { q , \\dagger } \\colon X ^ \\dagger \\times \\mathcal P \\to Z ^ \\dagger , F ^ { q , \\dagger } ( v , p ) : = F ^ q ( v , p ) , \\end{align*}"}
+{"id": "5978.png", "formula": "\\begin{align*} \\# \\{ \\tau _ { s _ k } \\in { \\bf { S } } _ { s _ k } : f _ { \\tau _ { s _ k } } \\not \\equiv 0 \\} \\prod _ { i = 1 } ^ { k - 1 } \\max _ { \\tau _ { s _ { i + 1 } } \\in { \\bf { S } } _ { s _ { i + 1 } } } \\# \\{ \\tau _ { s _ i } \\in { \\bf { S } } _ { s _ i } : \\tau _ { s _ i } \\subset \\tau _ { s _ { i + 1 } } , f _ { \\tau _ { s _ i } } \\not \\equiv 0 \\} . \\end{align*}"}
+{"id": "4423.png", "formula": "\\begin{align*} \\Big \\| \\sum \\limits _ { k = 2 } ^ { n } \\mu ( k ) ( I - S ) h _ k - ( 1 - z ) \\Big \\| _ { X } \\longrightarrow 0 n \\to \\infty . \\end{align*}"}
+{"id": "434.png", "formula": "\\begin{align*} k _ { r s } = \\sum _ { j , h , l = 1 } ^ { n } p _ { r j h } k _ { j l } c _ { h l s } , \\end{align*}"}
+{"id": "4873.png", "formula": "\\begin{align*} - \\Phi ^ * ( x , u ) = { \\sup } \\{ f \\in U S C ( \\overline { \\Omega } ) : f \\in F ( \\Omega ) f ( \\tau ) \\le - { \\phi ^ * _ \\tau } ( u ) \\tau \\in \\partial \\Omega \\} \\end{align*}"}
+{"id": "1481.png", "formula": "\\begin{align*} | D z _ { \\infty } | ( ( a , b ) ) = \\sum _ { s \\in S _ { z _ { \\infty } } } | J _ { z _ { \\infty } } ( s ) | \\leq \\sum _ { i = 1 } ^ { \\infty } | J _ { \\infty } ( i ) | , \\end{align*}"}
+{"id": "1346.png", "formula": "\\begin{align*} \\| y \\| & \\leq \\sum _ { j = 1 } ^ m \\alpha _ j \\| { y _ j } \\| = \\sum _ { j = 1 } ^ m \\frac { \\alpha _ j } { 2 } \\| m _ { a _ j , b _ j } + y _ j - m _ { a _ j , b _ j } + y _ j \\| \\\\ & \\leq \\sum _ { j = 1 } ^ m \\frac { \\alpha _ j } { 2 } \\bigl ( \\| m _ { a _ j , b _ j } + y _ j \\| + \\| m _ { a _ j , b _ j } - y _ j \\| \\bigr ) \\\\ & \\leq \\sum _ { j = 1 } ^ m \\alpha _ j \\bigl ( 1 + \\frac { \\varepsilon } { 5 m \\alpha _ j } \\bigr ) < 1 + \\frac { 2 \\varepsilon } { 5 } , \\end{align*}"}
+{"id": "1019.png", "formula": "\\begin{align*} \\chi _ K ( \\alpha ) + \\chi _ K ( - \\alpha ) = \\begin{cases} 1 , & \\alpha \\in \\Phi \\setminus \\Phi _ K , \\\\ 0 , & \\alpha \\in \\Phi _ K . \\end{cases} \\end{align*}"}
+{"id": "8975.png", "formula": "\\begin{align*} \\dfrac { \\hat { k } _ 2 ( \\Delta ) } { \\hat { k } _ 2 ( \\Psi ) } & = \\dfrac { ( x _ { 1 , 1 } x _ { 1 , 2 } x _ { 2 , 1 } x _ { 2 , 2 } x _ { 3 , 1 } x _ { 3 , 2 } ) ^ 3 ( x _ { 1 , 1 } + x _ { 1 , 2 } ) ( x _ { 2 , 1 } + x _ { 2 , 2 } ) ( x _ { 3 , 1 } + x _ { 3 , 2 } ) } { ( x _ { 1 , 1 } x _ { 1 , 2 } x _ { 2 , 1 } x _ { 2 , 2 } x _ { 3 , 1 } x _ { 3 , 2 } ) ^ 4 / ( x _ { 1 , 2 } x _ { 2 , 2 } x _ { 3 , 2 } ) } \\\\ & = \\dfrac { ( x _ { 1 , 1 } + x _ { 1 , 2 } ) ( x _ { 2 , 1 } + x _ { 2 , 2 } ) ( x _ { 3 , 1 } + x _ { 3 , 2 } ) } { x _ { 1 , 1 } x _ { 2 , 1 } x _ { 3 , 1 } } . \\end{align*}"}
+{"id": "637.png", "formula": "\\begin{align*} \\rho ( t , \\theta ) = ( t ( \\cos \\theta + 1 ) , t ( \\cos \\theta - 1 ) , t \\sin \\theta ) . \\end{align*}"}
+{"id": "6604.png", "formula": "\\begin{align*} \\mathcal { B } ( u , v , w ) = \\langle [ u , v ] _ H , w \\rangle , u , v , w \\in E . \\end{align*}"}
+{"id": "1262.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu ( \\eta _ { \\Lambda _ n } ) } \\int \\mathbf { 1 } _ { \\eta _ { \\Lambda _ n } } ( \\omega ) \\left ( \\sum _ { \\Theta \\Subset \\Z ^ d } h _ { \\Theta } ( \\omega ) \\right ) ^ 2 \\mu ( d \\omega ) = 0 . \\end{align*}"}
+{"id": "7857.png", "formula": "\\begin{align*} \\hat \\Phi _ s ( 0 , 0 ) = 0 . \\end{align*}"}
+{"id": "3832.png", "formula": "\\begin{align*} U \\cdot V = U _ { i _ 1 \\cdots i _ n } V _ { i _ 1 \\cdots i _ n } , \\quad | U | ^ 2 = U \\cdot U . \\end{align*}"}
+{"id": "8692.png", "formula": "\\begin{align*} \\abs { \\mathcal { R } } \\le C _ { V B } = C _ { V B } ( M , g ) . \\end{align*}"}
+{"id": "3904.png", "formula": "\\begin{align*} Q _ \\delta L _ \\delta \\omega = Q _ \\delta l _ { 1 , \\delta } + Q _ \\delta l _ { 2 , \\delta } + Q _ \\delta R _ \\delta ( \\omega ) , \\end{align*}"}
+{"id": "3377.png", "formula": "\\begin{align*} h _ \\mu ( t ) = t - \\log ( 1 + \\mu t ) . \\end{align*}"}
+{"id": "979.png", "formula": "\\begin{align*} \\R : = \\{ R _ i : i \\in \\{ 1 , \\dots , j - \\theta _ { d - 1 } - 1 \\} R _ i \\notin \\tau \\} . \\end{align*}"}
+{"id": "3453.png", "formula": "\\begin{align*} F _ k ( \\theta _ y ) = F _ { k - q _ n } ( \\theta _ { ( \\ell + 1 ) q _ n + m _ n + 1 } ) F ( \\theta _ { ( \\ell + 1 ) q _ n + m _ n } ) F _ { q _ n - 1 } ( \\theta _ { \\ell q _ n + m _ n + 1 } ) \\end{align*}"}
+{"id": "9276.png", "formula": "\\begin{align*} \\mathcal { U } ( E , \\Omega ) : = \\{ u \\in Q S H _ m ( \\Omega ) , u \\vert _ { \\Omega } \\leq 0 , u \\vert _ { E } \\leq - 1 \\} , \\end{align*}"}
+{"id": "4646.png", "formula": "\\begin{align*} m ^ a \\ell ^ { t _ { \\widetilde K } } = m ^ a ( \\ell ' v ) ^ { t _ { \\widetilde K } } = m ^ a { \\ell ' } ^ { t _ { \\widetilde K } } \\cdot ( v + \\ell ' ( v ) + \\dots + \\ell '^ { t _ { \\widetilde K } - 1 } ( v ) ) . \\end{align*}"}
+{"id": "3568.png", "formula": "\\begin{align*} \\varphi ( - q ^ { 1 / 2 } ) ^ 2 H ( - q ^ { 1 / 2 } ) + \\varphi ( q ^ { 1 / 2 } ) ^ 2 H ( q ^ { 1 / 2 } ) = 2 \\varphi ( q ^ 2 ) ^ 2 G ( q ^ 2 ) . \\end{align*}"}
+{"id": "9205.png", "formula": "\\begin{align*} \\begin{aligned} x ( t ) - z _ 1 ( t ) = & \\ , x ( 0 ) - z _ 1 ( 0 ) + \\int _ 0 ^ t \\dot { x } ( \\tau ) - \\dot { x } _ a ( \\tau ) d \\tau \\\\ & - \\gamma \\int _ 0 ^ t \\dfrac { \\partial \\epsilon _ 1 ( \\eta , t ) } { \\partial \\eta } \\Big | _ { \\eta = \\eta _ a ( \\tau ) } \\dot { \\eta } _ a ( \\tau ) d \\tau \\\\ & - \\gamma \\int _ 0 ^ t \\dfrac { \\partial } { \\partial \\tau } \\epsilon _ 1 ( \\eta _ a ( \\tau ) , \\tau ) d \\tau . \\end{aligned} \\end{align*}"}
+{"id": "1041.png", "formula": "\\begin{align*} \\ell ( r _ { ( \\alpha , k ) } , \\beta ) = \\begin{cases} 1 , & \\beta = \\alpha , \\\\ - 1 , & \\beta = - \\alpha , \\\\ 0 , & \\beta \\neq \\pm \\alpha . \\end{cases} \\end{align*}"}
+{"id": "6060.png", "formula": "\\begin{align*} H ( s , t ) = \\frac { \\kappa ( s ) } { n } . \\end{align*}"}
+{"id": "9084.png", "formula": "\\begin{align*} - \\sum \\limits _ { j = 3 } ^ { n - 1 } k _ j \\geq - ( n - 3 ) r \\end{align*}"}
+{"id": "2032.png", "formula": "\\begin{align*} x _ 0 ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 = \\eta ^ m \\end{align*}"}
+{"id": "2721.png", "formula": "\\begin{align*} g ( x ) = \\sum _ { i = 1 } ^ n \\frac { f ( x + \\sigma u _ i ) - f ( x ) } { \\sigma } \\tilde u _ i . \\end{align*}"}
+{"id": "291.png", "formula": "\\begin{align*} u ^ K _ n ( p ) = \\begin{cases} u _ n ( p ) & \\\\ K & \\end{cases} \\end{align*}"}
+{"id": "4174.png", "formula": "\\begin{align*} \\mathbf { v } \\cdot \\overrightarrow { \\zeta } = 0 , \\end{align*}"}
+{"id": "7657.png", "formula": "\\begin{align*} \\mu _ t ^ { * , t _ 0 , \\xi } : = \\rho \\big ( - R ^ { - 1 } B \\mathbb E \\big [ U ( t , x _ t ^ { * , t _ 0 , \\xi } , \\nu _ t ^ { * , t _ 0 , \\xi } ) | \\mathcal { F } _ t ^ { W ^ 0 } \\big ] \\big ) \\quad \\nu _ t ^ { * , t _ 0 , \\xi } : = \\mathbb E \\big [ x _ t ^ { * , t _ 0 , \\xi } | \\mathcal { F } _ t ^ { W ^ 0 } \\big ] . \\end{align*}"}
+{"id": "2225.png", "formula": "\\begin{align*} \\mathbf { v } ^ { 0 } = \\mathbf { e } _ { k } = [ \\cdots , 0 , 0 , \\cdots , 1 , \\cdots ] ^ { T } , k = 0 , \\cdots , N - 1 . \\end{align*}"}
+{"id": "8551.png", "formula": "\\begin{align*} ( \\kappa \\ , * \\ , k ) ( t ) = \\int _ 0 ^ t \\ , \\kappa ( t - \\tau ) \\ , k ( \\tau ) \\ , d \\tau \\ , = \\ , h _ 1 ( t ) \\ , = \\ , \\{ 1 \\} , \\ t > 0 . \\end{align*}"}
+{"id": "7911.png", "formula": "\\begin{align*} g ( \\upsilon ) = \\frac 1 4 \\left ( f ( \\upsilon ) + \\frac 1 3 f ( 3 \\upsilon ) - 2 f ( \\upsilon ) - u ( \\upsilon ) f ( \\upsilon ) \\right ) . \\end{align*}"}
+{"id": "5843.png", "formula": "\\begin{align*} \\rho { u _ y } = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { c _ { i y } } { f _ i } + \\frac { { \\Delta t } } { 2 } { F _ y } } , \\end{align*}"}
+{"id": "9270.png", "formula": "\\begin{align*} \\sum _ { A = 0 } ^ { 2 n - 1 } \\nabla _ { A \\alpha } \\varrho \\left ( d _ 1 u _ k \\wedge \\Theta \\wedge ( \\Delta \\varrho ) ^ { n - k } \\right ) _ A \\Omega _ { 2 n } = d _ \\alpha \\varrho \\wedge d _ 1 u _ k \\wedge \\Theta \\wedge ( \\Delta \\varrho ) ^ { n - k } , \\end{align*}"}
+{"id": "2494.png", "formula": "\\begin{align*} \\mathbf { \\bar { x } } = ( \\Theta \\mathbf { G } ) ^ T \\mathbf { x } , \\mathbf { \\bar { s } } = { { ( \\Theta \\mathbf { G } ) } ^ { - 1 } } \\mathbf { s } , \\end{align*}"}
+{"id": "8116.png", "formula": "\\begin{align*} C ( G , x ) = \\sum _ { i = 1 } ^ { r } ( x + 1 ) ^ { n _ i } - \\sum _ { j = 1 } ^ { r - 1 } ( x + 1 ) ^ { l _ j } \\end{align*}"}
+{"id": "4995.png", "formula": "\\begin{align*} d \\geq 2 : 0 < s < s _ s \\leq - \\frac d 2 , d = 1 : s < \\frac 1 6 s \\ne - \\frac 1 2 , \\end{align*}"}
+{"id": "2268.png", "formula": "\\begin{align*} a ( Z V ) = a ( Z ) , \\end{align*}"}
+{"id": "8867.png", "formula": "\\begin{align*} ( b ^ { [ j ] } \\wedge a ) ( \\alpha ) ( \\xi ) = \\begin{cases} b ( \\alpha ) ( \\xi ) & \\alpha < \\beta , \\\\ x _ j ( \\beta ) ( \\xi ) & \\alpha = \\beta \\xi \\le \\zeta , \\\\ b ( \\beta ) ( \\xi ) & \\alpha = \\beta \\xi > \\zeta , \\\\ a ( \\alpha ) ( \\xi ) & \\end{cases} \\end{align*}"}
+{"id": "2075.png", "formula": "\\begin{align*} m ( D _ { t _ 0 } \\setminus P ) = 0 \\end{align*}"}
+{"id": "6505.png", "formula": "\\begin{align*} d ^ * ( \\lceil x \\rceil ) & = q ( c ( \\lceil x \\rceil ) ) \\\\ & = q ( c _ a c _ a c _ b c _ b c _ b c _ b c _ b c _ b c _ b c _ b ) \\\\ & = a ^ 2 b ^ 8 \\end{align*}"}
+{"id": "4462.png", "formula": "\\begin{align*} \\frac { p } { p + 1 } \\sum _ { n = 1 } ^ { \\infty } \\frac { 2 ^ { \\omega ( \\gcd ( n , p ) ) } \\cdot 2 r _ { Q ^ { \\ast } } ( n ) ^ { 2 } } { n } \\sum _ { d = 1 } ^ { \\infty } \\psi \\left ( d \\sqrt { \\frac { n } { p } } \\right ) \\end{align*}"}
+{"id": "6852.png", "formula": "\\begin{align*} I _ { 2 , n } ( x ) = I _ { 2 , n - 1 } ( x ) + e ^ { \\frac { x } { 2 } } \\xi ( x ) b _ { n - 1 } ( x ) - \\int _ { - \\infty } ^ { x } \\left ( \\xi ( t ) e ^ { \\frac { t } { 2 } } \\right ) ^ { \\prime } b _ { n - 1 } ( t ) d t . \\end{align*}"}
+{"id": "5611.png", "formula": "\\begin{align*} B : = p _ { 1 1 } \\chi _ { ( 0 0 ] } + p _ { 1 2 } \\chi _ { ( 1 0 ] } + p _ { 2 1 } \\chi _ { ( 0 1 ] } + p _ { 2 2 } \\chi _ { ( 1 1 ] } \\end{align*}"}
+{"id": "7773.png", "formula": "\\begin{align*} | z _ 1 | = | z _ 2 | = \\cdots = | z _ m | > | z _ { m + 1 } | = \\max _ { j > m } | z _ j | . \\end{align*}"}
+{"id": "285.png", "formula": "\\begin{align*} ( u _ \\ast - \\varphi ) ( p ) > ( u _ \\ast - \\varphi ) ( p _ 0 ) = 0 \\end{align*}"}
+{"id": "626.png", "formula": "\\begin{align*} \\lim _ { s \\to 1 ^ + } ( s - 1 ) \\zeta _ K ( s ) = \\frac { 2 ^ { r + t } \\pi ^ t R _ K h _ K } { w _ K | \\Delta _ K | ^ { 1 / 2 } } , \\end{align*}"}
+{"id": "8055.png", "formula": "\\begin{align*} \\gamma _ { c , k } & = \\frac { a _ c ^ 2 \\lvert \\mathbf { \\hat { h } } _ k ^ { \\textrm { T } } \\mathbf { p } _ c \\rvert ^ 2 } { \\sum \\limits _ { i = 1 } ^ K a _ i ^ 2 \\lvert \\mathbf { h } _ k ^ { \\textrm { T } } \\mathbf { p } _ i \\rvert ^ 2 + \\sigma _ w ^ 2 } . \\end{align*}"}
+{"id": "4738.png", "formula": "\\begin{align*} J ( x ) \\ge J ( x ) + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k f _ k ( x ) \\end{align*}"}
+{"id": "2042.png", "formula": "\\begin{align*} v \\wedge \\eta = ( v _ 2 \\eta _ 3 - v _ 3 \\eta _ 2 , v _ 3 \\eta _ 1 - v _ 1 \\eta _ 3 , v _ 1 \\eta _ 2 - v _ 2 \\eta _ 1 ) . \\end{align*}"}
+{"id": "7891.png", "formula": "\\begin{align*} c _ 1 ( q , k , ( r _ 0 , \\lambda , 0 ) ) = c _ 1 ( q , k , ( r _ 0 , 0 , 0 ) ) + \\lambda h ( q , r _ 0 ) . \\end{align*}"}
+{"id": "6723.png", "formula": "\\begin{align*} F ( x ) = \\frac { 1 } { \\mathrm { B } ( \\alpha , \\beta ) } \\int _ 0 ^ { \\Phi _ s ( \\frac { x - \\mu } { \\sigma } ) } t ^ { \\alpha - 1 } \\ , ( 1 - t ) ^ { \\beta - 1 } \\ , \\mathrm { d } t = I _ { \\Phi _ s ( \\frac { x - \\mu } { \\sigma } ) } ( \\alpha , \\beta ) . \\end{align*}"}
+{"id": "5789.png", "formula": "\\begin{align*} d g _ 1 ' = ( \\log \\sqrt { \\tau _ 0 } ) _ z d z \\ g ' _ 1 + \\frac { 1 } { \\sqrt { \\tau _ 0 } } \\left ( Q _ 0 d z + \\frac { \\tau _ 0 } { 4 } d \\overline { z } \\right ) \\ J \\widehat { g _ 1 ' } I , \\end{align*}"}
+{"id": "6775.png", "formula": "\\begin{align*} L a _ { 0 } - a _ { 0 } ^ { \\prime } = q , \\end{align*}"}
+{"id": "9012.png", "formula": "\\begin{align*} ( - 1 ) ^ { | \\pi ( \\tau ) | } v _ { \\tau _ { s + 1 } } & = ( - 1 ) ^ { | \\pi ( \\tau _ { s + 1 } ) | + s } i _ { \\tau _ { s + 1 } } / x _ { \\tau _ { s + 1 } } \\\\ & = ( - 1 ) ^ { s } \\frac { ( - x _ 1 ) \\left ( \\prod _ { r = 2 } ^ { s + 1 } D _ { j _ { r - 1 } } \\right ) U } { x _ { \\tau _ { s + 1 } } } \\\\ & = ( - 1 ) ^ { s } \\frac { ( - x _ { j _ { s + 1 } } ) \\left ( \\prod _ { r = 2 } ^ { s + 1 } D _ { j _ { r - 1 } } \\right ) U } { x _ \\tau } . \\end{align*}"}
+{"id": "6617.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c } - \\varepsilon _ 1 \\mu & \\varepsilon _ 1 \\nu \\\\ \\varepsilon _ 3 \\mu & - \\varepsilon _ { 3 } \\nu \\end{array} \\right ) \\end{align*}"}
+{"id": "1602.png", "formula": "\\begin{align*} \\mathbf { w } _ { k } ^ \\star = \\left ( \\sum _ { p \\in \\mathcal { K } } \\bar { \\mathbf { h } } _ { p } \\bar { \\mathbf { h } } _ { p } ^ H + \\lambda ^ \\star \\mathbf { I } _ N \\right ) ^ { - 1 } \\times \\sqrt { 1 + \\iota _ { k } } \\bar { \\mathbf { h } } _ { k } , \\forall k \\in \\mathcal { K } , \\end{align*}"}
+{"id": "3683.png", "formula": "\\begin{align*} h ( \\nu , \\cdot ) = 0 , ( \\nabla _ { \\nu } h ) ( \\nu , \\cdot ) = 0 . \\end{align*}"}
+{"id": "6976.png", "formula": "\\begin{align*} u _ n = a ( \\theta _ 1 ^ n - \\zeta \\theta _ 2 ^ n ) \\end{align*}"}
+{"id": "1428.png", "formula": "\\begin{align*} q _ { y , z } ( n ) = ( - 1 ) ^ n \\sum _ { k = 0 } ^ { N - 1 } \\sum _ { j = 0 } ^ { N - 1 } a _ j ^ { ( k ) } n ^ { - k - 1 / 2 - j } \\frac { ( z - y ) ^ { 2 k + 1 } } { ( 2 k + 1 ) ! } . \\end{align*}"}
+{"id": "7915.png", "formula": "\\begin{align*} \\iota ( \\upsilon ) = - \\frac { 1 } { 4 } f ( \\upsilon ) + \\frac { h ( \\upsilon ) } { 4 g ( \\upsilon ) } \\left ( - 4 \\upsilon + \\frac 1 2 f ( 2 \\upsilon ) \\right ) . \\end{align*}"}
+{"id": "8858.png", "formula": "\\begin{align*} d \\tilde { x } _ t = \\lambda _ A \\tilde { x } _ t d t + e ^ { \\lambda _ A t } B ( x _ t , x _ t ) d t - A \\tilde { x } _ t d t + e ^ { \\lambda _ A t } \\sigma d W _ t . \\end{align*}"}
+{"id": "4214.png", "formula": "\\begin{align*} \\begin{cases} - ( K _ H ( x ) \\nabla v ) = \\frac { 1 } { \\varepsilon ^ 2 } \\left ( v - q _ { \\hat { x } } \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ { p } _ + , \\ \\ \\ \\ & \\ \\Omega _ { \\hat { x } } . \\\\ v = 0 , \\ \\ \\ \\ & \\ \\partial \\Omega _ { \\hat { x } } . \\end{cases} \\end{align*}"}
+{"id": "2380.png", "formula": "\\begin{align*} [ \\mathbf { h } ' _ { 3 i } , \\mathbf { h } _ { 3 i } ] = 1 . \\end{align*}"}
+{"id": "8782.png", "formula": "\\begin{align*} B ( x , x ) = 0 , \\lim _ { t \\to \\infty } \\norm { e ^ { t L ^ \\perp _ { x } } } = \\infty . \\end{align*}"}
+{"id": "3702.png", "formula": "\\begin{align*} \\hat M = ( M \\setminus \\Omega ) \\cup \\overline U \\end{align*}"}
+{"id": "2662.png", "formula": "\\begin{align*} [ P ( u ^ \\prime ( t ) ) ] ^ \\prime + A ( u ( t ) ) + Q ( t , u ^ \\prime ( t ) ) + F ( u ( t ) ) = 0 , \\end{align*}"}
+{"id": "8864.png", "formula": "\\begin{align*} \\pi ^ * ( \\eta ^ { - 1 } C ) = ^ * \\nu ^ { - 1 } C . \\end{align*}"}
+{"id": "6753.png", "formula": "\\begin{align*} J _ { i , k } ^ { ( s ) } ( l _ 1 , l _ 2 ) & = \\frac { 1 } { [ 2 \\Gamma ( 1 / s ) ] ^ { k + 1 } } \\int _ { l _ 1 } ^ { l _ 2 } y ^ { \\frac { i + 1 } { s } - 1 } \\ , \\exp ( - y ) \\left [ 2 \\Gamma ( 1 / s ) - \\Gamma ( 1 / s , y ) \\right ] ^ k \\mathrm { d } y . \\end{align*}"}
+{"id": "7187.png", "formula": "\\begin{align*} \\frac { \\partial \\textbf { \\textit { U } } } { \\partial x _ n } \\bigg | _ { \\partial \\Omega } = - Q \\textbf { \\textit { U } } | _ { \\partial \\Omega } + \\mathcal { R } \\textbf { \\textit { V } } . \\end{align*}"}
+{"id": "6884.png", "formula": "\\begin{align*} \\frac { \\alpha } { z _ 0 } + \\frac { \\beta } { z _ 0 - 1 } + \\frac { \\gamma } { z _ 0 - t } = \\frac { 1 } { \\i y _ 0 } \\ , . \\end{align*}"}
+{"id": "2299.png", "formula": "\\begin{align*} \\chi _ { - 2 | \\mu _ 1 | } ( z ) = z ^ { - 2 | \\mu _ 1 | } , \\chi _ { - 2 | \\mu _ 2 | } ( z ) = z ^ { - 2 | \\mu _ 2 | } , \\end{align*}"}
+{"id": "2668.png", "formula": "\\begin{align*} \\mathcal { G } ( t ) = \\mathcal { H } ( t ) + \\nu \\alpha ( t ) \\delta ^ { \\frac { 1 } { \\ell } } ( t ) \\mathcal { E } ^ { r + \\frac { 1 } { q } + \\frac { 1 } { \\ell ^ { \\prime } } } ( t ) , \\end{align*}"}
+{"id": "4835.png", "formula": "\\begin{align*} d x = b ( x ) d \\tau + \\sigma ( x ) d \\beta ( \\tau ) , \\ ; \\ ; \\ ; x \\in \\mathbb { R } ^ l , \\ ; \\tau \\geqslant 0 , \\end{align*}"}
+{"id": "6503.png", "formula": "\\begin{align*} \\lceil x \\rceil = & \\lambda ( e , b ) \\lambda ( b , b ^ 2 ) . . . \\lambda ( b ^ { m - 1 } , b ^ m ) \\\\ \\lceil x \\rceil = & \\lambda ( e , a ) \\lambda ( a , a ^ 2 ) . . . \\lambda ( a ^ { n - 1 } , a ^ n ) \\\\ \\lceil x \\rceil = & \\lambda ( e , a ) \\lambda ( a , a ^ 2 ) . . . \\lambda ( a ^ n , a ^ n b ) . . . \\lambda ( b ^ { m - 1 } , b ^ m ) \\end{align*}"}
+{"id": "7119.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\tilde { F } ( \\mathsf { g } _ { \\mathbb { C } , \\mathsf { S } } ( L ) ) \\circ \\Phi _ { S , S } & = & F ( \\mu ) \\circ F ( i d _ S \\otimes _ C L ) \\circ \\Phi _ { S , S } \\\\ & = & F ( \\mu ) \\circ \\Phi _ { S , S } \\circ ( i d _ { F ( S ) } \\otimes _ D F ( L ) ) \\\\ & = & \\mathsf { g } _ { \\mathbb { D } , \\widetilde { \\mathbb { F } } ( \\mathsf { S } ) } ( \\tilde { F } ( L ) ) . \\end{array} \\end{align*}"}
+{"id": "2375.png", "formula": "\\begin{align*} C _ 1 ( \\mathcal { H } _ { \\ast } ) = \\mathrm { I m } \\ , \\partial ' _ 0 \\oplus s _ { _ { 1 } } ( \\mathrm { I m } \\ , \\partial _ 0 ) . \\end{align*}"}
+{"id": "5419.png", "formula": "\\begin{align*} \\bar { x } _ n - x _ n ^ 0 = \\frac { - \\varepsilon _ 1 \\bar { \\xi } _ 1 + | \\hat { x } | \\bar { \\xi } _ 2 } { \\sqrt { \\varepsilon _ 1 ^ 2 + | \\hat { x } | ^ 2 } } \\leq | D \\rho ( 0 , \\tilde { x } ) | | \\bar { \\xi } _ 1 | + | \\bar { \\xi } _ 2 | \\leq C \\epsilon _ 0 \\delta ^ 2 b _ \\alpha \\end{align*}"}
+{"id": "2068.png", "formula": "\\begin{align*} K _ n = [ x - \\frac { 1 } { 2 n + 1 } , x + \\frac { 1 } { 2 n + 1 } ] \\ \\textrm { f o r } \\ n \\neq m ^ 2 \\ \\textrm { w h e r e } \\ m \\in \\mathbb { N } \\\\ K _ n = [ x - n , x + n ] \\ \\ \\ \\textrm { f o r } \\ n = m ^ 2 \\ \\textrm { w h e r e } \\ m \\in \\mathbb { N } \\\\ \\end{align*}"}
+{"id": "9131.png", "formula": "\\begin{align*} D _ g ^ 1 = D _ g ^ + \\oplus D _ g ^ - . \\end{align*}"}
+{"id": "3308.png", "formula": "\\begin{align*} & \\mathcal { Z } ^ * _ K ( \\underline { \\Delta ^ { J - 1 } } , \\underline { \\partial \\Delta ^ { J - 1 } } ) ( \\underline { \\Delta ^ { J - 1 } } , \\underline { \\partial \\Delta ^ { J - 1 } } ) = \\{ ( \\Delta ^ { j _ i - 1 } , \\partial \\Delta ^ { j _ i - 1 } ) \\} _ { i \\in \\{ 1 , \\ldots , m \\} } . \\end{align*}"}
+{"id": "3785.png", "formula": "\\begin{align*} B _ q ( \\phi ) ( z ) : = \\displaystyle \\int _ { \\mathbb { R } } \\overline { A _ q ( z , x ) } \\phi ( x ) d x . \\end{align*}"}
+{"id": "7029.png", "formula": "\\begin{align*} F _ { 2 , 4 } \\ , = \\ , 4 ! \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0 \\ , = \\ , F _ { 0 , 6 } \\ , = \\ , F _ { 1 , 5 } \\ , = \\ , F _ { 3 , 3 } \\ , = \\ , F _ { 4 , 2 } \\ , = \\ , F _ { 5 , 1 } \\ , = \\ , F _ { 6 , 0 } . \\end{align*}"}
+{"id": "1022.png", "formula": "\\begin{align*} \\prescript L { } { } \\ell { } ^ R ( x , \\alpha ) + \\prescript L { } { } \\ell { } ^ R ( x , - \\alpha ) = & \\langle \\mu , \\alpha \\rangle + \\langle \\mu , - \\alpha \\rangle + \\underbrace { \\chi _ R ( \\alpha ) + \\chi _ R ( - \\alpha ) } _ { \\in \\{ 0 , 1 \\} } \\\\ & \\qquad - \\underbrace { ( \\chi _ L ( w \\alpha ) + \\chi _ L ( - w \\alpha ) ) } _ { \\in \\{ 0 , 1 \\} } \\\\ \\in & \\{ - 1 , 0 , 1 \\} . \\end{align*}"}
+{"id": "4049.png", "formula": "\\begin{align*} u _ n ( A ' ) = n ^ { 2 H - 1 / 2 } \\sum _ { j \\in [ n ] } A ' _ n ( j ) I _ 1 ( 1 _ j ) 1 _ j , \\end{align*}"}
+{"id": "2562.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( \\hat { \\mu } \\right ) = \\bigcup _ { n = 1 } ^ { \\infty } \\mathcal { Z } \\left ( M _ { \\mathcal { D } _ { n } } ( \\rho ^ { n } \\xi ) \\right ) = \\bigcup _ { n = 1 } ^ { \\infty } \\frac { \\rho ^ { - n } a _ { n } } { N _ { n } } , \\ a _ { n } \\in \\mathbb { Z } \\backslash N _ { n } \\mathbb { Z } . \\end{align*}"}
+{"id": "337.png", "formula": "\\begin{align*} f ( z ) - P _ 0 ( z ) / Q _ 0 ( z ) = f ( z ) - P _ d ( z ) / Q _ d ( z ) = \\frac { 1 } { Q _ d ( z ) } O ( z ^ { 2 m + 2 d + 1 } ) = O ( z ^ { m + d + 1 } ) . \\end{align*}"}
+{"id": "5469.png", "formula": "\\begin{align*} [ i , j ] : = ( | \\cdot | ^ i , | \\cdot | ^ { i + 1 } , \\ldots , | \\cdot | ^ j ) \\end{align*}"}
+{"id": "3182.png", "formula": "\\begin{align*} & \\mid 2 ^ f h ^ { i ' } \\overline { h } ^ { j ' } \\sum \\limits _ { a _ m \\neq a _ i } \\chi _ 4 ( g ( a _ m ) ) \\mid \\\\ & = \\mid 2 ^ f h ^ { i ' } \\overline { h } ^ { j ' } \\left \\lbrace \\sum \\limits _ { a _ m } \\chi _ 4 ( g ( a _ m ) ) - \\chi _ 4 ( g ( a _ 1 ) ) - \\cdots - \\chi _ 4 ( g ( a _ { m - 1 } ) ) \\right \\rbrace \\mid \\\\ & \\leq 2 ^ { 3 m } \\cdot 3 m \\sqrt { q } + 2 ^ { 2 m - 3 } \\\\ & \\leq 2 ^ { 2 m } ( 1 + 2 ^ m \\cdot 3 m \\sqrt { q } ) , \\end{align*}"}
+{"id": "4389.png", "formula": "\\begin{align*} & \\bigg ( \\sup _ { X _ j } e ^ { - u ( - v ( \\Psi ) ) } \\bigg ) \\int _ { X _ j } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f F | ^ 2 _ h \\\\ \\le & \\bigg ( \\sup _ { X } e ^ { - u ( - v ( \\Psi ) ) } \\bigg ) \\int _ { X } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f F | ^ 2 _ h < + \\infty . \\end{align*}"}
+{"id": "1449.png", "formula": "\\begin{align*} \\mathrm { M S E } = \\sum _ { m = 1 } ^ { M } \\left | P _ { \\mathrm { d } } \\left ( \\theta _ { m } \\right ) - \\mathbf { a } ^ { H } \\left ( \\theta _ { m } \\right ) \\mathbf { R } _ { X } \\mathbf { a } \\left ( \\theta _ { m } \\right ) \\right | ^ { 2 } , \\end{align*}"}
+{"id": "6670.png", "formula": "\\begin{align*} & \\limsup _ { \\delta \\to 0 } \\limsup _ { \\epsilon \\to 0 } \\epsilon ^ { 2 } \\log \\mathbb P \\Big ( \\sup _ { | t - s | < \\delta } \\big | \\epsilon \\Theta _ { \\epsilon } ( t ) - \\epsilon \\Theta _ { \\epsilon } ( s ) \\big | > a \\Big ) = - \\infty , \\end{align*}"}
+{"id": "7188.png", "formula": "\\begin{align*} & \\Phi ( \\tau ) ( \\Lambda _ g - \\tau I ) = I \\mod O P S ^ { - \\infty } , \\\\ & ( \\Lambda _ g - \\tau I ) \\Phi ( \\tau ) = I \\mod O P S ^ { - \\infty } . \\end{align*}"}
+{"id": "799.png", "formula": "\\begin{align*} V _ g & = u _ g c H - G ( c ) H = \\big ( G ' ( c ) c - G ( c ) \\big ) H , \\\\ w _ g & = - \\Delta _ \\Sigma u _ g + c H V _ g = - \\Delta _ \\Sigma \\big ( G ' ( c ) \\big ) + c H V _ g . \\end{align*}"}
+{"id": "9207.png", "formula": "\\begin{align*} \\begin{aligned} | \\varepsilon _ \\delta ( z _ 1 , \\bar { y } & , t ) - \\varepsilon _ \\delta ( x _ a , \\bar { y } _ a , t ) | \\\\ \\le & L _ r \\gamma | \\epsilon _ 1 ( \\eta _ a , t ) | + | \\bar { y } ( t ) - \\bar { y } _ a ( t ) | \\\\ \\le & \\gamma \\delta L _ r \\left ( 2 + L _ r \\ ( 3 + L _ r ) \\right ) + | \\bar { y } - z _ 2 | . \\end{aligned} \\end{align*}"}
+{"id": "5443.png", "formula": "\\begin{align*} [ a _ { ( m ) } , b _ { ( n ) } ] = \\sum _ { j \\in \\mathbb { Z _ { + } } } \\left ( \\begin{array} { c c c } m \\\\ j \\end{array} \\right ) ( a _ { ( j ) } b ) _ { ( m + n - j ) } . \\end{align*}"}
+{"id": "310.png", "formula": "\\begin{align*} \\tau ( n ) : = \\min \\left \\{ 1 , \\frac 1 4 C ^ { - 1 } ( n , 1 / 2 ) , ( A ( n ) \\sqrt { C ' ( n ) } ) ^ { - 1 } \\right \\} \\end{align*}"}
+{"id": "5259.png", "formula": "\\begin{align*} p ( T ( t ) x ) = \\lim _ { n \\to \\infty } \\left ( \\frac { n } { t } \\right ) ^ { n } p \\left ( R \\left ( \\frac { n } { t } , A \\right ) ^ { n } x \\right ) \\underset { \\eqref { e q : r e s _ c o n t } } { \\leq } p ( x ) \\end{align*}"}
+{"id": "2749.png", "formula": "\\begin{align*} F _ m = \\begin{cases} H & \\ k \\ , \\\\ G & \\ k \\ , \\end{cases} \\end{align*}"}
+{"id": "6137.png", "formula": "\\begin{align*} \\left ( \\psi ^ L _ { a ; S } \\right ) _ { 0 j } \\colon & f ( x , y , z ) \\mapsto \\eta ^ { - a \\vert S \\vert j } f ( x , \\eta ^ { - a } x , z ) \\\\ \\left ( \\psi ^ R _ { S ; b } \\right ) _ { i 0 } \\colon & f ( x , y , z ) \\mapsto f ( x , \\eta ^ { b } z , z ) . \\end{align*}"}
+{"id": "1504.png", "formula": "\\begin{align*} I _ { a + } ^ { \\alpha } D _ { a + } ^ { \\alpha } u ( t ) = u ( t ) + \\sum _ { i = 1 } ^ { n } c _ { i } ( t - a ) ^ { \\alpha - n + ( i - 1 ) } , \\end{align*}"}
+{"id": "5222.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { + } \\frac { G } { r ^ { 2 } } \\dd x = \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } \\frac { G _ 0 } { r ^ { 2 } } \\dd x , \\end{align*}"}
+{"id": "6755.png", "formula": "\\begin{align*} \\int _ { l _ 1 } ^ { l _ 2 } y ^ { \\frac { i + j + 1 } { s } - 1 } \\ , \\exp ( - y ) & \\left [ \\sum _ { m = 0 } ^ \\infty \\frac { ( - y ) ^ m } { ( 1 / s + m ) m ! } \\right ] ^ j \\mathrm { d } y = \\sum _ { m = 0 } ^ \\infty c _ { m , j } \\int _ { l _ 1 } ^ { l _ 2 } y ^ { m + \\frac { i + j + 1 } { s } - 1 } \\exp ( - y ) \\mathrm { d } y \\\\ & = \\sum _ { m = 0 } ^ \\infty c _ { m , j } \\left [ \\Gamma \\left ( m + \\frac { i + j + 1 } { s } , l _ 1 \\right ) - \\Gamma \\left ( m + \\frac { i + j + 1 } { s } , l _ 2 \\right ) \\right ] , \\end{align*}"}
+{"id": "5345.png", "formula": "\\begin{align*} \\alpha = \\alpha _ { ( 0 ) } + \\alpha _ { ( 1 ) } \\end{align*}"}
+{"id": "49.png", "formula": "\\begin{align*} I _ 7 = 4 \\beta \\int \\rho ^ { 2 \\beta - 2 \\alpha - 2 } Z v \\ \\Delta _ { H } v = 4 \\beta \\int \\rho ^ { - Q + 2 } Z v \\ ; \\Delta _ { H } v . \\end{align*}"}
+{"id": "7037.png", "formula": "\\begin{align*} a _ { 2 , 1 } \\ , : = \\ , 0 . \\end{align*}"}
+{"id": "5684.png", "formula": "\\begin{align*} \\hat { u } _ { 2 } \\circ p _ { 1 2 } \\circ u _ { 2 } & = \\hat { u } _ { 2 } \\circ u ' _ { 2 } \\circ p _ { 1 2 } ~ ~ ~ ~ ~ ~ ( \\because p _ { 1 2 } \\circ u _ { 2 } = u ' _ { 2 } \\circ p _ { 1 2 } ) \\\\ & = \\hat { u } _ { 2 } \\circ p _ { 1 2 } ~ ~ ~ ( \\because \\hat { u } = ( \\hat { u } _ { 1 } , \\hat { u } _ { 2 } ) \\in U ^ { G } ( X _ { 1 } , X _ { 2 } ) ) . \\end{align*}"}
+{"id": "8972.png", "formula": "\\begin{align*} \\hat { k } _ d ( \\Delta ) = \\sum _ { \\Upsilon \\in \\mathcal { T } _ d ( \\Delta ) } \\prod _ { \\tau \\in \\Upsilon _ d } \\left ( \\prod _ { v \\in \\tau } x _ { \\tau } \\right ) { \\mathbf { t } _ { d - 1 } ( \\Upsilon ) ^ 2 } = \\sum _ { \\Upsilon \\in \\mathcal { T } _ d ( \\Delta ) } \\left ( \\prod _ { v \\in V } x _ { v } ^ { \\deg _ \\Upsilon v } \\right ) { \\mathbf { t } _ { d - 1 } ( \\Upsilon ) ^ 2 } . \\end{align*}"}
+{"id": "3297.png", "formula": "\\begin{align*} \\mathbb E \\left [ \\| \\Delta _ j \\mathcal S B ^ { \\mathfrak H } \\| _ H ^ { 8 } \\right ] \\leq & \\int _ 0 ^ 2 \\mathbb E \\left [ \\left ( B ^ { \\mathfrak H } _ { x + i \\Delta _ n } - B ^ { \\mathfrak H } _ { x + ( i - 1 ) \\Delta _ n } \\right ) ^ 8 \\right ] d x = 2 1 0 \\Delta _ n ^ { 8 H } = 2 1 0 \\Delta _ n ^ 2 , \\end{align*}"}
+{"id": "4724.png", "formula": "\\begin{align*} r i \\ , C : = \\{ x \\in \\textit { a f f } \\ , C \\ , | \\ , ( \\exists \\ , \\epsilon > 0 ) \\ , ( x + \\epsilon B ) \\cap ( \\textit { a f f } \\ , C ) \\subset C \\} \\end{align*}"}
+{"id": "7674.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { * , i } = & ~ \\Big [ A x _ t ^ { * , i } - B ^ 2 R ^ { - 1 } y _ t ^ { * , i } - B h \\big ( \\rho _ i ^ N ( \\Delta ^ { N , 1 } _ { * , t } , \\ldots , \\Delta ^ { N , N } _ { * , t } ) \\big ) \\Big ] d t + \\sigma d W _ t ^ i + \\sigma _ 0 d W ^ 0 _ t , \\\\ x ^ { * , i } _ 0 = & ~ \\xi ^ i . \\end{aligned} \\right . \\end{align*}"}
+{"id": "4189.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { L } _ H u = - ( K _ H ( x ) \\nabla u ) = \\frac { 1 } { \\varepsilon ^ 2 } \\left ( u - q \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ { p } _ + , \\ \\ & x \\in \\Omega , \\\\ u = 0 , \\ \\ & x \\in \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "7139.png", "formula": "\\begin{align*} h ^ 0 \\left ( X , K _ { X _ s } ^ { \\otimes 2 } ( - D _ s ) \\right ) - h ^ 1 \\left ( X , K _ { X _ s } ^ { \\otimes 2 } ( - D _ s ) \\right ) = 3 g - 3 - n \\end{align*}"}
+{"id": "3982.png", "formula": "\\begin{align*} \\mathbf { a } + \\mathbf { b } = ( a _ 1 + \\beta _ 1 , a _ 2 + b _ 1 , a _ 3 + b _ 2 , a _ 4 + b _ 3 , \\cdots , a _ { 2 k - 1 } + b _ { 2 k - 2 } , \\alpha _ 1 + b _ { 2 k - 1 } , \\alpha _ 2 + \\beta _ 2 , \\alpha _ 3 + \\beta _ 3 , \\alpha _ 4 + \\beta _ 4 ) , \\end{align*}"}
+{"id": "9097.png", "formula": "\\begin{align*} ( \\sum \\limits _ { i = 1 } ^ { j } w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + ( j - 1 ) r \\leq \\chi _ j \\leq ( \\sum \\limits _ { i = 1 } ^ { j } w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + j r . \\end{align*}"}
+{"id": "3980.png", "formula": "\\begin{align*} \\alpha _ i + \\bar { \\alpha } _ j = 1 . \\end{align*}"}
+{"id": "2305.png", "formula": "\\begin{align*} S T = T S \\end{align*}"}
+{"id": "8767.png", "formula": "\\begin{align*} { } _ a D _ g ^ { \\alpha } \\circ { } _ { a } I _ g ^ { \\alpha } = I , \\end{align*}"}
+{"id": "2626.png", "formula": "\\begin{align*} \\varepsilon ( h , x + y ) [ [ \\alpha ( x ) , \\alpha ( y ) ] , \\alpha ( h \\cdot z ) ] & + \\varepsilon ( x + y , z ) [ [ h \\cdot z , \\alpha ( x ) ] , \\alpha ^ 2 ( y ) ] \\\\ & + \\varepsilon ( h , y ) \\varepsilon ( x , y + z ) [ [ \\alpha ( y ) , h \\cdot z ] , \\alpha ^ 2 ( x ) ] = 0 , \\end{align*}"}
+{"id": "7568.png", "formula": "\\begin{align*} F = \\{ x \\in [ 0 , 1 ) : E ( x ) \\} . \\end{align*}"}
+{"id": "8584.png", "formula": "\\begin{align*} \\phi ( t ) = f ( t ) + C ( \\kappa \\ , * \\ , k _ 1 ) ( t ) , \\ t > 0 . \\end{align*}"}
+{"id": "1030.png", "formula": "\\begin{align*} \\ell ( x , \\alpha ) = \\prescript L { } { } \\ell { } ^ R ( x , \\alpha ) + \\Phi ^ + _ R ( \\alpha ) \\geq 0 . \\end{align*}"}
+{"id": "267.png", "formula": "\\begin{align*} \\tilde { d } _ H ( \\xi , p ) = | \\xi \\cdot p ^ { - 1 } | _ G = | \\zeta \\cdot w ^ { - 1 } | _ G \\leq - \\lambda , \\end{align*}"}
+{"id": "6444.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { \\begin{subarray} { c } r _ 1 + \\cdots + r _ { n + 1 } = r \\\\ r _ i \\in \\mathbb { Z } _ { \\ge 0 } \\end{subarray} } Z _ { I } ( 1 + r _ 1 , \\{ 2 + r _ i \\} _ { i = 2 } ^ { n + 1 } ; ( \\alpha , \\beta ) ) \\\\ = & \\sum _ { \\begin{subarray} { c } r _ 1 + \\cdots + r _ { n } = r \\\\ r _ i \\in \\mathbb { Z } _ { \\ge 0 } \\end{subarray} } Z _ { I I } ( \\{ 2 + r _ i \\} _ { i = 1 } ^ { n - 1 } , 3 + r _ n ; ( \\alpha , \\beta ) ) \\end{aligned} \\end{align*}"}
+{"id": "8319.png", "formula": "\\begin{align*} g _ 1 ( x ) = \\frac { x ^ 2 } { 1 + \\cos x } , g _ 2 ( x ) = \\frac { x \\sinh x } { 1 + \\cosh x } . \\end{align*}"}
+{"id": "2019.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d y _ t = & f ( t , y _ t , y _ t , u _ t , \\mathcal L ( y _ t , z _ t , u _ t ) ) d t + g ( t , y _ t , z _ t , u _ t , \\mathcal L ( y _ t , z _ t , u _ t ) ) d \\overleftarrow B _ t \\\\ & - z _ t d W _ t , \\ , t \\in [ 0 , T ] , \\\\ y _ T = & \\xi , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4672.png", "formula": "\\begin{align*} \\psi _ { \\tau } ( p ) = | | \\tau | | _ p \\le C _ 1 ( \\beta , \\gamma , L ) \\ ( \\beta - p ) ^ { - ( \\gamma + 1 ) / \\beta } \\ L ^ { 1 / \\beta } ( 1 / ( \\beta - p ) ) , \\end{align*}"}
+{"id": "4480.png", "formula": "\\begin{align*} \\sum _ { r _ { Q } ( n ) = 0 } n \\ll p ^ { 1 + \\epsilon } \\frac { m } { \\min Q ^ { * } } \\ll \\frac { p ^ { 3 + \\epsilon } } { \\left ( \\min Q ^ { * } \\right ) ^ { 2 } } . \\end{align*}"}
+{"id": "5966.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 + } \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } \\left ( \\int _ 0 ^ { T - \\delta } \\Vert X _ n ( t + \\delta ) - X _ n ( t ) \\Vert _ Y ^ p d t > \\epsilon \\right ) = 0 . \\end{align*}"}
+{"id": "3469.png", "formula": "\\begin{align*} r _ { \\ell } \\leq & e ^ { 5 0 \\varepsilon q _ n } \\frac { e ^ { - q _ n L } } { \\max ( | \\ell | , 1 ) } \\max ( r _ { \\ell - 1 } , r _ { \\ell + 1 } ) \\times \\begin{cases} \\max ( | \\ell | , e ^ { \\delta _ n q _ n } ) , \\beta _ n \\geq \\delta _ n + 2 0 0 \\varepsilon , \\\\ e ^ { \\beta _ n q _ n } , \\beta _ n < \\delta _ n + 2 0 0 \\varepsilon \\end{cases} \\end{align*}"}
+{"id": "7121.png", "formula": "\\begin{align*} \\mathbb { x } _ c ^ { \\mathcal { X } ' , \\mathbb { F } } \\circ \\alpha ^ { m } ( \\mathbb { C } ) _ c = \\alpha ^ m ( \\mathbb { D } ) _ { F c } \\circ \\mathbb { x } ^ { \\mathcal { X } , \\mathbb { F } } _ c . \\end{align*}"}
+{"id": "1382.png", "formula": "\\begin{align*} g _ { x _ 1 x _ 2 \\ldots x _ d } = { ( - 1 ) } ^ d \\prod _ { j = 1 } ^ d \\lambda _ j g ( x _ 1 , \\ldots , x _ d ) \\end{align*}"}
+{"id": "8968.png", "formula": "\\begin{align*} c = \\sum _ { \\tau \\in \\Delta _ i } c _ { \\tau } [ \\tau ] , \\end{align*}"}
+{"id": "369.png", "formula": "\\begin{align*} T _ { k } ( r , s ) = v _ { k } ( r ) v _ { k } ^ { \\ast } ( s ) = \\omega ^ { k ( r - 1 ) } \\omega ^ { - k ( s - 1 ) } = \\omega ^ { ( r - s ) k } . \\end{align*}"}
+{"id": "4194.png", "formula": "\\begin{align*} j ' ( t ) = & - \\frac { 1 } { t ^ 2 \\varepsilon ^ 2 } \\int _ { \\Omega } \\left ( p ( t u - q \\ln \\frac { 1 } { \\varepsilon } \\right ) _ + ^ { p - 1 } t u ^ 2 - \\left ( t u - q \\ln \\frac { 1 } { \\varepsilon } \\right ) _ + ^ { p } u ) d x \\\\ = & \\frac { 1 } { t ^ 2 \\varepsilon ^ 2 } \\int _ { \\Omega } \\left ( t u - q \\ln \\frac { 1 } { \\varepsilon } \\right ) _ + ^ { p - 1 } u [ ( 1 - p ) ( t u - q \\ln \\frac { 1 } { \\varepsilon } ) - p q \\ln \\frac { 1 } { \\varepsilon } ] d x \\\\ \\leq & 0 . \\end{align*}"}
+{"id": "634.png", "formula": "\\begin{align*} \\begin{aligned} & x _ 0 = 1 , & y _ 0 = 0 , \\\\ & x _ 1 , & y _ 1 = 0 , \\\\ & x _ 2 = x _ 1 , & y _ 2 , \\\\ \\end{aligned} \\end{align*}"}
+{"id": "5049.png", "formula": "\\begin{align*} b = \\nabla \\times ( \\phi \\nabla \\theta ) + G \\nabla \\theta , \\end{align*}"}
+{"id": "1325.png", "formula": "\\begin{align*} \\widetilde { L } : = \\bigcup _ i \\widetilde { U _ i } , \\end{align*}"}
+{"id": "2852.png", "formula": "\\begin{align*} q ' _ x ( t ) : = q _ x ( t ) - \\bar q _ x ( t ) \\quad \\mbox { a n d } p ' _ x ( t ) : = p _ x ( t ) - \\bar p _ x ( t ) \\end{align*}"}
+{"id": "6705.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } l _ n = g . \\end{align*}"}
+{"id": "7573.png", "formula": "\\begin{align*} X ^ { \\Delta } ( \\beta ) = \\left \\{ \\sum _ { \\ell = 0 } ^ { m - 1 } a _ \\ell \\beta ^ { m - 1 - \\ell } : m \\in \\N , \\ a _ 0 , \\ldots , a _ { m - 1 } \\in \\Delta \\right \\} \\end{align*}"}
+{"id": "6028.png", "formula": "\\begin{align*} J ( u , v ) = & \\alpha \\big ( \\| \\nabla u \\| _ { 2 } ^ { 2 } + \\| \\nabla v \\| _ { 2 } ^ { 2 } \\big ) + ( \\alpha - 1 ) \\big ( \\| u \\| _ { 2 } ^ { 2 } + \\omega \\| v \\| _ { 2 } ^ { 2 } \\big ) \\\\ & + ( 3 \\alpha - 2 ) \\Big ( B ( u ) + B ( v ) \\Big ) - \\frac { p \\alpha - 1 } { p } F ( u , v ) . \\end{align*}"}
+{"id": "1732.png", "formula": "\\begin{align*} \\ < \\big ( \\widehat { f } \\ ; \\big ) ^ * ( g ) , \\delta ( x ) \\ > = \\ < g , \\widehat { f } ( \\delta ( x ) ) \\ > = \\ < g , \\delta ( f ( x ) ) \\ > = g \\circ f ( x ) = \\ < C _ f ( g ) , \\delta ( x ) \\ > . \\end{align*}"}
+{"id": "3489.png", "formula": "\\begin{align*} \\ln \\left \\lbrace \\prod _ { j \\neq k } \\frac { | \\sin \\pi ( \\theta - \\theta _ j ) | } { | \\sin \\pi ( \\theta _ k - \\theta _ j ) | } \\right \\rbrace = \\sum _ { j \\neq k } \\ln | \\sin \\pi ( \\theta - \\theta _ j ) | - \\sum _ { j \\neq k } \\ln | \\sin \\pi ( \\theta _ k - \\theta _ j ) | < ( 2 q _ n - 1 ) ( \\frac { \\ln { q _ { n + 1 } / | \\ell | } } { 2 q _ n - 1 } + \\epsilon ) . \\end{align*}"}
+{"id": "6148.png", "formula": "\\begin{align*} X _ { t } = X _ 0 + \\int _ 0 ^ t a _ s \\ , d s + \\int _ 0 ^ t b _ s \\ , d W _ s , \\end{align*}"}
+{"id": "5299.png", "formula": "\\begin{align*} l ( y _ p ) \\sim r ( y _ p ) = p = p _ 1 + \\cdots + p _ n = r ( y _ { p _ 1 } ) + \\cdots + r ( y _ { p _ n } ) \\succ l ( y _ { p _ 1 } ) \\vee \\cdots \\vee l ( y _ { p _ n } ) . \\end{align*}"}
+{"id": "8587.png", "formula": "\\begin{align*} V ( Z _ 1 , \\dots , Z _ n ) = \\sum V ( [ 0 , w _ { i _ 1 1 } ] , \\dots [ 0 , w _ { i _ n n } ] ) = \\frac { 1 } { n ! } \\sum | \\det ( \\{ w _ { i _ j j } \\} _ { j = 1 } ^ n ) | , \\end{align*}"}
+{"id": "3989.png", "formula": "\\begin{align*} \\beta _ i + \\bar { \\beta _ j } = 1 \\ , \\ , { \\rm a n d } \\ , \\ , \\beta _ u + \\bar { \\beta } _ 4 = 1 . \\end{align*}"}
+{"id": "5850.png", "formula": "\\begin{align*} { f _ { 1 7 } } = { f _ { 1 8 } } - f _ { 1 8 } ^ { \\left ( { e q } \\right ) } + f _ { 1 7 } ^ { \\left ( { e q } \\right ) } + \\delta y - \\delta z \\end{align*}"}
+{"id": "6942.png", "formula": "\\begin{align*} M _ n & \\ge \\sup _ { Q \\in \\P ( \\R ) } \\bigg ( n \\int _ { \\R } V ( x ) \\ , Q ( d x ) + \\frac 1 2 \\sum _ { i , j = 1 } ^ n J _ { i j } \\int _ { \\R } \\int _ { \\R } K ( x - y ) \\ , Q ( d x ) Q ( d y ) - n H ( Q ) \\bigg ) \\\\ & = n \\sup _ { Q \\in \\P ( \\R ) } \\bigg ( \\int _ { \\R } V \\ , d Q + \\frac 1 2 \\int _ { \\R } \\int _ { \\R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \\bigg ) . \\end{align*}"}
+{"id": "8304.png", "formula": "\\begin{align*} \\begin{aligned} & \\dot y _ { b e n } = A _ { b e n } y '' _ { b e n } + M _ { b e n } y _ { b e n } , y \\in ( - R / 2 , R / 2 ) , \\\\ & \\dot y _ { n b } = A _ { n b } y '' _ { n b } + M _ { n b } y _ { n b } , y \\in ( R / 2 , r + R / 2 ) , \\end{aligned} \\end{align*}"}
+{"id": "7012.png", "formula": "\\begin{align*} \\aligned u & \\ , = \\ , \\tfrac { x ^ 2 } { 2 } + F _ { 3 , 0 } \\ , \\tfrac { x ^ 3 } { 6 } + F _ { 2 , 1 } \\ , \\tfrac { x ^ 2 y } { 2 } + F _ { 1 , 2 } \\ , \\tfrac { x y ^ 2 } { 2 } + F _ { 0 , 3 } \\ , \\tfrac { y ^ 3 } { 6 } + { \\rm O } _ { x , y } ( 4 ) , \\\\ v & \\ , = \\ , \\tfrac { r ^ 2 } { 2 } + G _ { 3 , 0 } \\ , \\tfrac { r ^ 3 } { 6 } + G _ { 2 , 1 } \\ , \\tfrac { r ^ 2 s } { 2 } + G _ { 1 , 2 } \\ , \\tfrac { r s ^ 2 } { 2 } + G _ { 0 , 3 } \\ , \\tfrac { s ^ 3 } { 6 } + { \\rm O } _ { r , s } ( 4 ) . \\endaligned \\end{align*}"}
+{"id": "3704.png", "formula": "\\begin{align*} ( \\nu ( u ) ) ' = \\nu ' ( h ) ( \\bar u ) + \\nu ( v ) \\end{align*}"}
+{"id": "1396.png", "formula": "\\begin{align*} \\lim _ { \\lambda _ 1 , \\ldots , \\lambda _ d \\rightarrow 1 } \\frac { \\det ( \\varphi _ i ( \\lambda _ j ) ) _ { i , j = 1 } ^ d } { \\Delta ( \\lambda _ 1 , \\ldots , \\lambda _ d ) } = \\frac { \\det ( \\varphi _ i ^ { ( j - 1 ) } ( 1 ) ) _ { i , j = 1 } ^ d } { \\prod _ { j = 1 } ^ { d - 1 } j ! } . \\end{align*}"}
+{"id": "9343.png", "formula": "\\begin{align*} & \\| v ( \\cdot , t ) \\| _ { L ^ \\infty ( \\mathbb { R } ^ 3 ) } \\leq \\frac { C } { \\sqrt { - t } } t \\in ( - 1 , 0 ) , \\\\ & v _ 3 \\in L ^ p ( - 1 , 0 ; L ^ q ( \\mathbb { R } ^ 3 ) ) \\frac { 2 } { p } + \\frac { 3 } { q } = 1 , q \\in ( 3 , \\infty ] . \\end{align*}"}
+{"id": "6492.png", "formula": "\\begin{align*} E _ { 2 , m + e _ i } = E _ { 2 , m } \\cup ( E _ { 2 , m + e _ i - e _ j } ) \\cup ( E _ { 2 , [ m - e _ j , m + e _ i ] } ) \\end{align*}"}
+{"id": "8040.png", "formula": "\\begin{align*} \\abs { f } = \\inf _ { x \\in X } d _ f ( x ) . \\end{align*}"}
+{"id": "3374.png", "formula": "\\begin{align*} u _ i & = a _ 1 + B _ 1 ( u _ i , u _ i ) + B _ 2 ( b _ i , b _ i ) \\\\ b _ i & = a _ 2 + B _ 3 ( u _ i , b _ i ) . \\end{align*}"}
+{"id": "490.png", "formula": "\\begin{align*} g _ s ( x ) = \\left \\{ \\begin{array} { l l } g ( x ) / \\lambda _ s & \\ x > c _ 1 \\\\ g ( c _ 1 ) / \\lambda _ s & \\ - c _ 1 \\le x \\le c _ 1 \\\\ g ( - x ) / \\lambda _ s & \\ x < - c _ 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "8345.png", "formula": "\\begin{align*} \\psi ^ - ( x , t ; k ) = \\psi ^ + ( x , t ; k ) e ^ { - i k ^ 2 x \\widehat { \\sigma } _ 3 } S ( k ) , \\end{align*}"}
+{"id": "5756.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } B ( X ) : = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ p e _ j \\cdot B ( X , e _ j ) \\ \\in C l ( T M \\oplus E \\oplus \\mathcal { E } _ 2 ) \\end{align*}"}
+{"id": "5736.png", "formula": "\\begin{align*} y ^ q ( x ^ { q + 1 } + d ) = ( x + d ) . \\end{align*}"}
+{"id": "6748.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty x ^ n \\ , h _ j ^ { ( s ) } \\mathrm { d } x & = \\sigma \\ , \\int _ { - \\infty } ^ \\infty ( \\sigma z + \\mu ) ^ n \\ , [ \\Phi _ s ( z ) ] ^ { j + \\alpha - 1 } \\ , \\phi _ s ( z ) \\mathrm { d } z \\\\ & = \\sigma \\mu ^ n \\ , \\sum _ { i = 0 } ^ n \\ , \\binom { n } { i } \\ , \\left ( \\frac { \\sigma } { \\mu } \\right ) ^ i \\ , \\int _ { - \\infty } ^ \\infty z ^ i \\ , [ \\Phi _ s ( z ) ] ^ { j + \\alpha - 1 } \\ , \\phi _ s ( z ) \\mathrm { d } z . \\end{align*}"}
+{"id": "2788.png", "formula": "\\begin{align*} \\mathrm { v a r } [ S _ { k } ( x ) - S _ { m } ( x ) ] = ( m - k ) \\gamma + o ( 1 ) \\end{align*}"}
+{"id": "8440.png", "formula": "\\begin{align*} \\psi ^ - _ { 1 1 } - 1 = - z \\int _ { - \\infty } ^ x u _ y ( y ) e ^ { 2 i z y } d y \\int _ { - \\infty } ^ { y } \\bar { u } _ s ( s ) e ^ { - 2 i z s } \\psi ^ - _ { 1 1 } d s : = - z K \\psi ^ - _ { 1 1 } , \\end{align*}"}
+{"id": "588.png", "formula": "\\begin{align*} \\hat { Y } [ i , k ] = H [ k ] \\bar { Y } [ i , k ] + W [ i , k ] , \\end{align*}"}
+{"id": "3751.png", "formula": "\\begin{align*} & M = \\begin{pmatrix} D & * \\\\ 0 & * \\end{pmatrix} , D = d i a g ( d _ 1 , d _ 2 , \\ldots , d _ { l _ 0 - 1 } ) \\\\ & \\deg ( \\det ( D ) ) \\geq \\deg ( \\det ( D _ 0 ) ) , \\deg ( \\det ( M ) ) = \\deg ( \\det ( M _ 0 ) ) \\end{align*}"}
+{"id": "4731.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { m } \\gamma _ { k } f _ { k } ( x ) \\ge 0 \\ \\ \\forall \\ x \\in \\mathbb { R } ^ { n } . \\end{align*}"}
+{"id": "8073.png", "formula": "\\begin{align*} \\lambda _ c ^ { \\left ( \\right ) } = 2 \\sum _ { j = 1 } ^ { M } \\lvert \\hat { \\phi } ^ { \\left ( j , c \\right ) } \\rvert ^ 2 + \\sigma _ { e , i } ^ 2 \\lVert \\mathbf { p } _ j \\rVert ^ 2 - f ^ { \\left ( c \\right ) } \\end{align*}"}
+{"id": "7814.png", "formula": "\\begin{align*} \\hat \\Phi \\left ( \\begin{pmatrix} a \\\\ b \\end{pmatrix} , s \\right ) : = \\begin{pmatrix} \\langle z _ 1 ^ * , \\Phi ( a u _ \\ell + b w _ \\ell , p ( s ) ) \\rangle \\\\ \\langle z _ 2 ^ * , \\Phi ( a u _ \\ell + b w _ \\ell , p ( s ) ) \\rangle \\end{pmatrix} . \\end{align*}"}
+{"id": "5025.png", "formula": "\\begin{align*} h ( T ' ) & = x y \\log ( x y ) - r \\Delta ( x ) - \\sum _ { k = 1 } ^ { r } \\Delta ( y + 1 - k ) \\\\ & = m \\log ( m + r ) - 2 r - o ( 1 ) \\\\ & = m \\log ( m ) - r - o ( 1 ) , \\end{align*}"}
+{"id": "3743.png", "formula": "\\begin{align*} & \\Delta x _ j - \\bar u ^ { - 1 } \\nabla \\bar u \\cdot \\nabla x _ j = 0 \\\\ & x _ j ( p ) = 0 \\ ; \\mbox { a n d } \\ ; \\frac { \\partial x _ j } { \\partial y _ i } ( p ) = \\delta _ { i j } . \\end{align*}"}
+{"id": "1096.png", "formula": "\\begin{align*} \\chi _ J ( \\beta ) = \\Phi ^ + ( \\beta ) - \\Phi ^ + _ J ( \\beta ) . \\end{align*}"}
+{"id": "6371.png", "formula": "\\begin{gather*} \\delta _ 1 ( X ) = 2 \\mu F ( \\mu ) - 2 \\mu + 2 \\int _ { \\mu } ^ { \\infty } x f ( x ) \\mathrm { d } x \\delta _ 2 ( X ) = 2 \\int _ { m } ^ { \\infty } x f ( x ) \\mathrm { d } x - \\mu , \\end{gather*}"}
+{"id": "1942.png", "formula": "\\begin{align*} B ( \\alpha , \\beta ) = \\left \\{ \\begin{array} { l l } \\displaystyle { \\int _ { 0 } ^ { 1 } t ^ { \\alpha - 1 } ( 1 - t ) ^ { \\beta - 1 } d t } , & ( \\Re ( \\alpha ) > 0 , \\Re ( \\beta ) > 0 ) \\\\ \\\\ \\displaystyle { \\frac { \\Gamma ( \\alpha ) \\Gamma ( \\beta ) } { \\Gamma ( \\alpha + \\beta ) } } , & ( ( \\alpha , \\beta ) \\notin { \\bf Z } _ { 0 } ^ { - } ) . \\end{array} \\right . \\end{align*}"}
+{"id": "5859.png", "formula": "\\begin{align*} a = \\frac { { 0 . 4 5 7 2 4 { R ^ 2 } T _ c ^ 2 } } { { { p _ c } } } , b = \\frac { { 0 . 0 7 7 8 R { T _ c } } } { { { p _ c } } } , \\end{align*}"}
+{"id": "6882.png", "formula": "\\begin{align*} ( 1 - \\alpha ) \\pi , \\ ; ( 1 - \\beta ) \\pi , \\ ; ( 1 - \\gamma ) \\pi , \\ ; ( 1 - \\delta ) \\pi , \\alpha , \\beta , \\gamma , \\delta \\in ( 0 , 1 ) , \\alpha + \\beta + \\gamma + \\delta = 2 . \\end{align*}"}
+{"id": "562.png", "formula": "\\begin{align*} f ( z ) = \\int _ { \\Gamma } g ( \\omega ) e ^ { - \\omega z } d \\omega . \\end{align*}"}
+{"id": "203.png", "formula": "\\begin{align*} f ( g ) = \\sum _ k r _ k g \\cdot v _ { k } . \\end{align*}"}
+{"id": "7732.png", "formula": "\\begin{align*} ( x , & y , z , \\zeta ) ( x ' , y ' , z ' , \\zeta ' ) \\\\ & = ( x + x ' , y + y ' , z + z ' + x y ' - x ' y , \\zeta + \\zeta ' + \\xi ( x y ' - x ' y ) ) . \\end{align*}"}
+{"id": "3838.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\gamma _ 1 = \\min \\{ K _ 1 , K _ 4 , K _ 7 , K _ { 1 0 } \\} > 0 , ~ \\gamma _ 2 = \\min \\{ K _ 2 , K _ 5 , K _ 8 , K _ { 1 1 } \\} > 0 , \\\\ & \\gamma _ 3 = \\min \\{ K _ 3 , K _ 6 , K _ 9 , K _ { 1 2 } \\} > 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4437.png", "formula": "\\begin{align*} \\frac { d } { d t } E \\big [ [ M _ v ] ( t ) \\big ] = \\sum _ { k = 1 } ^ N e ^ { - 2 \\lambda t } \\lambda ^ 2 v _ { k } ^ 2 E | Z _ k ( t ) | \\le 4 | v | ^ 2 e ^ { - 2 \\lambda t } E \\bigg [ \\sum _ { k = 1 } ^ N | Z _ k ( t ) | \\bigg ] . \\end{align*}"}
+{"id": "719.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { c } , k } = \\frac { { \\lvert \\mathbf { h } _ { k } ^ { H } \\mathbf { p } _ { \\textrm { c } } \\rvert } ^ { 2 } } { \\sum _ { i = 1 } ^ { K } { \\lvert \\mathbf { h } _ { k } ^ { H } \\mathbf { p } _ { \\textrm { p } , i } \\rvert } ^ { 2 } + \\sigma _ { z } ^ { 2 } } . \\end{align*}"}
+{"id": "8076.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\mathbf { y ' } ^ H \\mathbf { s } ^ { \\left ( \\right ) } \\right . & \\left . | \\hat { \\mathbf { H } } \\right ] = a _ c \\sum _ { i = 1 } ^ M \\hat { \\phi } ^ { \\left ( i , c \\right ) ^ * } + \\sum _ { j = 1 } ^ { M } a _ j \\hat { \\phi } ^ { \\left ( j , j \\right ) ^ * } . \\end{align*}"}
+{"id": "2311.png", "formula": "\\begin{align*} d _ n ^ T ( \\{ e ( k x ) \\} _ { k = - N } ^ N , L _ p ) \\le \\log ^ { - c _ 1 ( p ) } N , \\quad \\mbox { i f $ n \\ge N \\exp ( - \\log ^ { c _ 2 } N ) $ } . \\end{align*}"}
+{"id": "1332.png", "formula": "\\begin{align*} x ' = \\sum _ { i = 1 } ^ n \\lambda _ i \\ , m _ { p _ i , q _ i } . \\end{align*}"}
+{"id": "2970.png", "formula": "\\begin{align*} x = \\Phi \\left ( - t , \\Phi ( t , x ) \\right ) = \\Phi \\left ( - t , \\Phi ( s , y ) \\right ) . \\end{align*}"}
+{"id": "4062.png", "formula": "\\begin{align*} a _ { n , k } & = \\begin{cases} O ( 1 ) \\quad & H < H ( k ) \\\\ O ( \\log n ) \\quad & H = H ( k ) \\\\ O ( n ^ { ( 2 H - 1 ) k - 1 } ) \\quad & H > H ( k ) \\end{cases} \\end{align*}"}
+{"id": "961.png", "formula": "\\begin{align*} \\mathcal { E } _ { j } & = - U ( t ) { { \\mathcal F } } ^ { - 1 } [ t ^ { - 1 } { \\tilde { \\mathcal N } } _ { j } ( F _ { 1 } , F _ { 2 } ) ] + { \\tilde { \\mathcal N } } _ { j } ( M ( t ) D ( t ) F _ 1 , M ( t ) D ( t ) F _ 2 ) \\\\ & = - t ^ { - 1 } M ( t ) D ( t ) \\mathcal { F } M ( t ) \\mathcal { F } ^ { - 1 } \\tilde { \\mathcal { N } } _ j ( F _ 1 , F _ 2 ) + t ^ { - 1 } M ( t ) D ( t ) \\tilde { \\mathcal { N } } _ j ( F _ 1 , F _ 2 ) \\\\ & = - t ^ { - 1 } M ( t ) D ( t ) \\mathcal { F } ( M ( t ) - 1 ) \\mathcal { F } ^ { - 1 } \\tilde { \\mathcal { N } } _ j ( F _ 1 , F _ 2 ) . \\end{align*}"}
+{"id": "5750.png", "formula": "\\begin{align*} \\varphi _ { c , d } ( u ) = \\varphi _ { c , d } ( v ) , \\end{align*}"}
+{"id": "3699.png", "formula": "\\begin{align*} v = 0 , \\nu ( v ) = 0 \\mbox { o n } \\hat \\Sigma . \\end{align*}"}
+{"id": "6225.png", "formula": "\\begin{align*} E _ 0 & = 1 6 \\kappa \\left ( \\frac { ( L + 1 ) \\Q ^ 2 + \\kappa ( 4 L + 1 ) } { \\Q ^ 2 + 1 0 \\kappa } \\right ) ^ 2 - \\Q ^ 2 \\left ( \\frac { ( L + 2 ) \\Q ^ 2 + \\kappa ( 4 L + 1 1 ) } { \\Q ^ 2 + 1 0 \\kappa } \\right ) ^ 2 , \\\\ E _ 1 & = 1 6 \\kappa \\left ( \\frac { ( L + 2 ) \\Q ^ 2 + \\kappa ( 4 L + 1 1 ) } { \\Q ^ 2 + 1 0 \\kappa } \\right ) ^ 2 - \\Q ^ 2 \\left ( \\frac { ( L + 1 ) \\Q ^ 2 + \\kappa ( 4 L + 1 ) } { \\Q ^ 2 + 1 0 \\kappa } \\right ) ^ 2 , \\end{align*}"}
+{"id": "3690.png", "formula": "\\begin{align*} Z ^ { ( i ) } = \\partial _ i \\mbox { o r } Z ^ { ( i , j ) } = x _ i \\partial _ j - x _ j \\partial _ i \\mbox { f o r } i , j = 1 , \\dots , n . \\end{align*}"}
+{"id": "3891.png", "formula": "\\begin{align*} E _ { \\delta , Z } = \\{ u \\in W ^ { 2 , p } \\cap H ^ 1 _ 0 ( \\Omega ) \\mid \\int _ { \\Omega } ( K ( x ) \\nabla \\frac { \\partial V _ { \\delta , Z , j } } { \\partial z _ { j , h } } ) u = 0 , \\ \\ \\forall j = 1 , \\cdots , m , \\ h = 1 , 2 \\} . \\end{align*}"}
+{"id": "5084.png", "formula": "\\begin{align*} B _ n & \\rightharpoonup B \\textrm { i n } \\ L ^ { 2 } ( \\Omega ) , \\\\ A _ n = \\textrm { c u r l } ^ { - 1 } B _ n & \\to A = \\textrm { c u r l } ^ { - 1 } B \\textrm { i n } \\ L ^ { 2 } ( \\Omega ) . \\end{align*}"}
+{"id": "2814.png", "formula": "\\begin{align*} R _ { k } = \\log _ { 2 } ( 1 + _ { k } ^ ) . \\end{align*}"}
+{"id": "2197.png", "formula": "\\begin{align*} \\hat { \\theta } _ i = \\arcsin \\left ( \\frac { \\lambda \\arg z _ i } { 2 \\pi d } \\right ) \\end{align*}"}
+{"id": "5920.png", "formula": "\\begin{align*} \\chi = \\begin{cases} \\max \\{ 1 + \\beta , 1 + \\lambda , 1 + \\gamma + \\frac { 2 \\theta } { \\alpha } \\} , & \\alpha \\leq 2 , \\\\ \\max \\{ 1 + \\beta , 3 + \\lambda - \\alpha , 3 + \\gamma + \\theta - \\alpha \\} , & \\alpha > 2 . \\end{cases} \\end{align*}"}
+{"id": "7713.png", "formula": "\\begin{align*} \\forall j \\in [ i ] , g ( \\lfloor ( 1 - ( 1 + \\tilde \\varepsilon ) ^ { - j } ) r \\rfloor ) = g _ s ( j ) , \\end{align*}"}
+{"id": "1458.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\left ( A _ { 0 , n } , A _ { 1 , n } , B _ { 0 , n } , B _ { 1 , n } \\right ) = ( u ( a ) , 0 , u ( b ) , 0 ) , \\end{align*}"}
+{"id": "768.png", "formula": "\\begin{align*} K = \\frac 1 { | t | ^ { \\frac 1 M ( M + 1 - \\frac 1 { 2 s } ) } } * \\overset { M ) } { \\cdots } * \\frac 1 { | t | ^ { \\frac 1 M ( M + 1 - \\frac 1 { 2 s } ) } } , \\end{align*}"}
+{"id": "6754.png", "formula": "\\begin{align*} J _ { i , k } ^ { ( s ) } ( l _ 1 , l _ 2 ) = & \\frac { 1 } { [ 2 \\Gamma ( 1 / s ) ] ^ { k + 1 } } \\sum _ { j = 0 } ^ k \\binom { k } { j } [ \\Gamma ( 1 / s ) ] ^ { k - j } \\\\ & \\times \\int _ { l _ 1 } ^ { l _ 2 } y ^ { \\frac { i + 1 } { s } - 1 } \\exp ( - y ) \\left [ y ^ { 1 / s } \\sum _ { m = 0 } ^ \\infty \\frac { ( - y ) ^ m } { ( 1 / s + m ) m ! } \\right ] ^ j \\mathrm { d } y . \\end{align*}"}
+{"id": "9208.png", "formula": "\\begin{align*} \\begin{aligned} \\Bigg | & \\dfrac { \\partial \\epsilon _ 1 ( \\eta , t ) } { \\partial \\eta } \\Bigg | _ { \\eta = \\eta _ a } \\dot { \\eta } _ a \\Bigg | \\le \\ , \\gamma | a _ { 0 , \\delta } ( x _ a ) / 2 - \\bar { y } _ a | \\\\ & + \\gamma \\left | \\left . \\dfrac { \\partial \\varepsilon _ \\delta ( x , \\bar { y } _ a , t ) - b _ { 1 , \\delta } ( x ) / 2 } { \\partial x } \\right | _ { x = x _ a } \\right | \\dfrac { | b _ { 1 , \\delta } ( x _ a ) | } { 2 } . \\end{aligned} \\end{align*}"}
+{"id": "4920.png", "formula": "\\begin{align*} M ( - t ) = \\frac { 1 } { \\sqrt 2 \\pi } \\sum _ { r = 0 } ^ \\infty \\pi _ r \\ , \\ , \\int _ { - \\infty } ^ { \\infty } \\Phi ( x ) ^ r \\ , \\exp \\left ( - t \\ , x - \\frac { x ^ 2 } { 2 } \\right ) \\mathrm { d } x . \\end{align*}"}
+{"id": "4380.png", "formula": "\\begin{align*} D '' u _ { m ' , \\epsilon , j } + P _ { m ' } \\big ( \\sqrt { N _ 1 \\tilde { \\lambda } _ { m ' } } h _ { m ' , \\epsilon , j } \\big ) = D '' \\left ( ( 1 - v ' _ { \\epsilon } ( \\Psi ) ) f F ^ { 1 + \\delta } \\right ) . \\end{align*}"}
+{"id": "2219.png", "formula": "\\begin{align*} \\mathbf { v } ^ { 0 } = [ \\cdots , 1 , 0 , 0 , 0 , \\cdots , 1 , \\cdots ] ^ { T } , m , n = 1 , \\cdots , N . \\end{align*}"}
+{"id": "1180.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\mathbb { P } } \\left [ \\left | \\frac { 1 } { t ^ { \\frac { 1 } { 2 } - H } } \\sum _ { j = 2 } ^ n \\triangle k _ t ^ { 1 , j , n } \\right | ^ { 2 p } \\right ] \\leq p ! \\left ( ( n - 1 ) \\beta \\right ) ^ { p } . \\end{align*}"}
+{"id": "3806.png", "formula": "\\begin{align*} [ D ^ { * } _ { q } , M _ { z ^ 4 } ] ( f ) ( z ) = 2 4 f ( z ^ 4 ) + 1 2 0 z \\frac { d } { d z } ( f ( z ^ 4 ) ) + 5 6 z ^ 2 \\frac { d ^ 2 } { d z ^ 2 } ( f ( z ^ 4 ) ) + 1 6 z ^ 3 \\frac { d ^ 3 } { d z ^ 3 } ( f ( z ^ 4 ) ) . \\\\ \\end{align*}"}
+{"id": "4213.png", "formula": "\\begin{align*} c _ \\varepsilon = I _ \\varepsilon ( u _ \\varepsilon ) = \\pi \\min \\limits _ { \\overline { \\Omega } } q ^ 2 \\sqrt { d e t ( K _ H ) } \\ln \\frac { 1 } { \\varepsilon } + O ( 1 ) . \\end{align*}"}
+{"id": "5845.png", "formula": "\\begin{align*} { u _ z } = \\frac { 1 } { \\rho } \\left [ { { f _ 0 } + { f _ 1 } + { f _ 2 } + { f _ 3 } + { f _ 4 } + { f _ 7 } + { f _ 8 } + { f _ 9 } + { f _ { 1 0 } } + 2 \\left ( { { f _ 5 } + { f _ { 1 1 } } + { f _ { 1 4 } } + { f _ { 1 5 } } + { f _ { 1 8 } } } \\right ) + \\frac { 1 } { 2 } { F _ z } } \\right ] - 1 , \\end{align*}"}
+{"id": "2999.png", "formula": "\\begin{align*} \\partial ^ { c i } _ { \\tau , w } g _ * ( \\tau , z , w ( \\cdot ) ) = G ( \\tau , w ( - h ) , \\mathrm { d } ^ + w ( - h ) / \\mathrm { d } \\xi ) , \\nabla _ z g _ * ( \\tau , z , w ( \\cdot ) ) = 0 \\end{align*}"}
+{"id": "1538.png", "formula": "\\begin{align*} G ( \\overline { \\Psi } ( A _ t ) ) = \\{ P \\in \\overline { \\Psi } ( A _ t ) \\ ; | \\ ; P \\vert _ { t = 0 } \\in G _ { A _ t } \\} \\end{align*}"}
+{"id": "896.png", "formula": "\\begin{align*} \\Omega ( X , q , n ) = \\sum \\limits _ { m \\leq X \\atop { m \\equiv n \\ , ( q ) } } 1 \\ , . \\end{align*}"}
+{"id": "8575.png", "formula": "\\begin{align*} ( I _ { 0 + } ^ \\alpha \\ , _ * D _ { 0 + } ^ \\alpha \\ , f ) ( t ) \\ , = \\ , f ( t ) - f ( 0 ) , \\ t > 0 . \\end{align*}"}
+{"id": "4438.png", "formula": "\\begin{align*} W ( a u + b v , x ) = a W ( u , x ) + b W ( v , x ) . \\end{align*}"}
+{"id": "6948.png", "formula": "\\begin{align*} M _ n ^ \\psi ( Q ) & : = \\sum _ { i = 1 } ^ n \\int _ { \\R } V ( x ) \\ , Q _ i ( d x ) + \\sum _ { i , j = 1 } ^ n J _ { i j } \\int _ { \\R } \\int _ { \\R } \\psi ( x , y ) \\ , Q _ i ( d x ) Q _ j ( d y ) - \\sum _ { i = 1 } ^ n H ( Q _ i ) \\\\ & = \\sum _ { i , j = 1 } ^ n J _ { i j } \\int _ { \\R } \\int _ { \\R } \\psi ( x , y ) \\ , Q _ i ( d x ) Q _ j ( d y ) - \\sum _ { i = 1 } ^ n H ( Q _ i \\ , | \\ , \\rho ) . \\end{align*}"}
+{"id": "129.png", "formula": "\\begin{align*} f _ 1 & = \\mathcal { F } ( \\gamma ( N _ { \\max } ^ { ( 5 - 2 \\alpha ) + \\varepsilon } t - n ) \\mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ { N _ 1 } u ) , \\\\ \\tilde { f } _ 2 & = \\mathcal { F } ( \\gamma ( N _ { \\max } ^ { ( 5 - 2 \\alpha ) + \\varepsilon } t - n ) \\mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ { N _ 2 } \\partial _ x u ) , \\\\ f _ 3 & = \\mathcal { F } ( \\gamma ( N _ { \\max } ^ { ( 5 - 2 \\alpha ) + \\varepsilon } t - n ) \\mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ N u ) , \\end{align*}"}
+{"id": "5096.png", "formula": "\\begin{align*} b ( x _ 1 , x _ 2 ) = \\nabla \\times ( \\phi ( x _ 1 , x _ 2 ) \\nabla z ) + G ( x _ 1 , x _ 2 ) \\nabla z . \\end{align*}"}
+{"id": "2991.png", "formula": "\\begin{align*} \\bigvee _ { ( s , t ) \\in \\Lambda _ N ^ { \\vec { v } } ( 1 ) } T ^ { - ( s , t ) } \\{ A , A ^ c \\} \\bigvee _ { n = 0 } ^ { N - 1 } \\mathcal { A } ^ n . \\end{align*}"}
+{"id": "4274.png", "formula": "\\begin{align*} n _ { i } & = 3 n _ { i - 1 } - 3 n _ { i - 2 } + n _ { i - 3 } + \\kappa \\\\ \\left ( n + a \\right ) _ { i } & = 3 \\left ( n + a \\right ) _ { i - 1 } - 3 \\left ( n + a \\right ) _ { i - 2 } + \\left ( n + a \\right ) _ { i - 3 } + \\lambda \\\\ \\left ( n + b \\right ) _ { i } & = 3 \\left ( n + b \\right ) _ { i - 1 } - 3 \\left ( n + b \\right ) _ { i - 2 } + \\left ( n + b \\right ) _ { i - 3 } - \\lambda \\end{align*}"}
+{"id": "5877.png", "formula": "\\begin{align*} P _ n u : = \\sum _ { i = 1 } ^ { n } \\langle u , e _ i \\rangle e _ i . \\end{align*}"}
+{"id": "1371.png", "formula": "\\begin{align*} \\P _ x ( Z _ d ( n _ 1 ) \\geq \\xi _ 1 , \\ldots , Z _ d ( n _ m ) \\geq \\xi _ m ) = \\det ( I - \\chi _ \\xi K \\chi _ { \\xi } ) _ { l ^ 2 ( \\{ n _ 1 , \\ldots n _ k \\} \\times \\mathbb { N } } \\end{align*}"}
+{"id": "6655.png", "formula": "\\begin{align*} \\begin{aligned} | \\mathcal { R } _ { 2 , \\epsilon } ( t ) | & \\leq \\vartheta \\lambda ^ { 2 } ( \\epsilon ) \\int ^ { t } _ { 0 } 1 + | \\varphi ( s ) | + | \\varphi ' _ m ( s ) | d s \\\\ & + \\lambda ( \\epsilon ) \\epsilon ^ { 2 } \\Big | \\int ^ { t } _ { 0 } \\big ( \\sigma ( \\varphi _ { m } ( s ) ) - \\sigma ( \\hat { X } ^ { \\epsilon } ( s - ) ) \\big ) \\theta _ { \\epsilon } ( s - ) \\tilde { N } ^ { \\epsilon ^ { - 2 } } ( d s ) \\Big | , \\end{aligned} \\end{align*}"}
+{"id": "2469.png", "formula": "\\begin{align*} f ( t ) = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( \\ ; \\alpha ^ { 2 } t ^ { 2 } \\right ) ^ { n } } { ( 2 n ) ! } = \\cosh \\alpha t . \\end{align*}"}
+{"id": "3470.png", "formula": "\\begin{align*} \\gamma : = \\begin{cases} \\max ( e ^ { \\delta _ n q _ n } , | \\ell | , 1 ) e ^ { - \\beta _ n q _ n } , \\beta _ n \\geq \\delta _ n + 2 0 0 \\varepsilon \\\\ 1 , \\end{cases} \\end{align*}"}
+{"id": "3018.png", "formula": "\\begin{align*} \\gamma _ k ( t _ k , x _ k , \\xi _ k , y _ k ) \\leq \\gamma _ k ( \\tau , z , \\tau , z ) = \\varphi ( \\tau , z , w ( \\cdot ) ) . \\end{align*}"}
+{"id": "8689.png", "formula": "\\begin{align*} \\Sigma \\cap T \\cap \\bigcap _ { j = 1 } ^ J B ( p _ j , s _ \\Sigma ( p _ j ) ) \\partial \\Sigma \\cap T \\cap \\bigcap _ { j = 1 } ^ J B ( p _ j , s _ \\Sigma ( p _ j ) ) \\end{align*}"}
+{"id": "8965.png", "formula": "\\begin{align*} - \\int _ 0 ^ \\infty u \\phi = \\int _ 0 ^ \\infty f \\phi ' , \\qquad \\phi \\in C _ 0 ^ \\infty ( 0 , \\infty ) , \\end{align*}"}
+{"id": "7039.png", "formula": "\\begin{align*} A _ { 2 , 1 } \\ , : = \\ , - \\ , \\tfrac { 1 } { 3 } \\ , F _ { 7 , 0 } \\ , T _ 1 . \\end{align*}"}
+{"id": "3445.png", "formula": "\\begin{align*} \\tilde { m } _ n = \\begin{cases} m _ n - q _ { n - 1 } , \\ell _ n = 1 \\\\ m _ n + q _ { n - 1 } , \\ell _ n = - 1 \\end{cases} . \\end{align*}"}
+{"id": "5591.png", "formula": "\\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) = A ( 1 ^ { n + k } 0 . . . ) - A ( 1 ^ n 0 ^ \\infty ) = c _ { n + k } - c _ n \\ . \\end{align*}"}
+{"id": "2316.png", "formula": "\\begin{align*} R i c ^ B = \\rho ^ \\nabla + d J \\theta , \\end{align*}"}
+{"id": "5993.png", "formula": "\\begin{align*} I ( u , v ) = \\frac { 1 } { 2 } \\| ( u , v ) \\| _ { E } ^ { 2 } + \\frac { 1 } { 2 } \\Big ( B ( u ) + B ( v ) \\Big ) - \\frac { 1 } { 2 p } F ( u , v ) , \\end{align*}"}
+{"id": "7412.png", "formula": "\\begin{align*} _ { \\rm r e f } [ V ] ( y , z , q ) = _ { ^ F V } \\left ( q ^ { L _ 0 - \\frac { c } { 2 4 } } z ^ { h } y ^ { - 2 g } \\right ) . \\end{align*}"}
+{"id": "6328.png", "formula": "\\begin{align*} w ^ { q ^ 2 } - w = x ^ { q ^ 3 + 1 } - x ^ { q ^ 2 + q } . \\end{align*}"}
+{"id": "5317.png", "formula": "\\begin{align*} \\frac { ( k + n - 1 ) } { k ! ( n - 1 ) ! } \\| \\rho _ k ^ \\lambda ( f ) \\| _ { H S } ^ 2 = ( 2 \\pi ) ^ { - n } | \\lambda | ^ n \\ , \\int _ { \\C ^ n } | f ^ { - \\lambda } \\ast _ { - \\lambda } \\varphi ^ { n - 1 } _ { k , \\lambda } ( z ) | ^ 2 d z . \\end{align*}"}
+{"id": "2787.png", "formula": "\\begin{align*} \\hat { \\eta } _ { N } ^ { D , M } : = \\frac { 1 } { K _ { N } } \\sum _ { x \\in D _ { N } } 1 _ { T _ { N , M } ( x ) } \\delta _ { x / N } \\otimes \\delta _ { h ^ { D _ { N } } ( x ) - a _ { N } } . \\end{align*}"}
+{"id": "6178.png", "formula": "\\begin{align*} \\xi = - L - 1 , \\eta = \\frac { Q } { 2 ( L + 1 ) } , \\zeta = \\frac { \\kappa ( L + 1 ) B _ 1 } { Q } . \\end{align*}"}
+{"id": "2413.png", "formula": "\\begin{align*} F ( s ) = \\int _ { 0 } ^ { \\infty } \\exp ( - s t ) f ( t ) d t . \\end{align*}"}
+{"id": "9096.png", "formula": "\\begin{align*} \\tilde { \\sigma } | _ { \\oplus _ { i = 1 } ^ n j _ { i _ * } ( F _ i ) } : \\oplus _ { i = 1 } ^ n j _ { i _ * } ( F _ i ) \\rightarrow \\oplus _ { i = 1 } ^ { n - 1 } j _ { p _ { i _ * } } ( j _ { p _ i } ^ * ( j _ { p _ { { i + 1 } _ * } } E _ { i + 1 } ) ) . \\end{align*}"}
+{"id": "3682.png", "formula": "\\begin{align*} ( X _ { i ; j } + X _ { j ; i } - h _ { i j } ) V _ i V _ j & = ( 2 \\omega _ { i j } - h _ { i j } ) V _ i V _ j = 0 \\\\ ( X _ { a ; j } + X _ { j ; a } - h _ { a j } ) V _ j & = ( \\omega _ { a j } - h _ { a j } ) V _ n = 0 \\mbox { f o r $ a = 1 , \\dots , n - 1 $ } \\end{align*}"}
+{"id": "8364.png", "formula": "\\begin{align*} V = b _ - ^ { - 1 } b _ + , \\ \\ \\ b _ - = I , \\ \\ \\ b _ + = V , \\end{align*}"}
+{"id": "5923.png", "formula": "\\begin{align*} X ( t ) = x + \\int _ 0 ^ t \\mathcal { A } ( s ) d s + \\int _ 0 ^ t \\mathcal { B } ( s ) d W ( s ) , \\mathbb { P } \\otimes d t , \\end{align*}"}
+{"id": "5413.png", "formula": "\\begin{align*} \\begin{aligned} | u _ { \\tilde { \\nu } ( y ' ) } ( y ) | \\ , & \\leq | u _ { \\tilde { \\nu } ( 0 ) } ( 0 ) | \\\\ & + \\sum _ { \\beta = \\alpha + 1 } ^ { n - 1 } \\sup _ { \\xi \\in \\partial \\omega \\cap \\partial \\Omega } \\left ( | \\nabla _ { \\tilde { \\nu } \\tau _ \\beta } u ( \\xi ) | + \\sum _ { \\gamma = 1 } ^ { n - 1 } | \\rho _ { \\gamma \\beta } ( \\xi ) u _ \\gamma ( \\xi ) | \\right ) \\cdot | y _ \\beta | \\leq C b _ \\alpha \\end{aligned} \\end{align*}"}
+{"id": "1995.png", "formula": "\\begin{align*} - T ( x ) + x = \\varphi \\varphi ( - T ( x ) + x ) = \\varphi ( - T ( x ) - x ) . \\end{align*}"}
+{"id": "6483.png", "formula": "\\begin{align*} \\sup _ { p \\ge 1 } \\left \\{ \\ \\frac { | | \\tau | | _ { p , \\Omega } } { \\psi _ { m , L } ( p ) } \\ \\right \\} = C ( m , L ) < \\infty \\end{align*}"}
+{"id": "7972.png", "formula": "\\begin{align*} & \\partial _ { t } Q - \\Theta \\sigma ^ { i j } _ { k } \\nabla _ { i j } Q \\\\ & = \\frac { 1 } { P } ( \\partial _ { t } P - \\Theta \\sigma ^ { i j } _ { k } \\nabla _ { i j } P ) - A \\left [ \\partial _ { t } \\left ( \\frac { \\rho ^ { 2 } } { 2 } \\right ) - \\Theta \\sigma ^ { i j } _ { k } \\nabla _ { i j } \\left ( \\frac { \\rho ^ { 2 } } { 2 } \\right ) \\right ] + \\frac { \\Theta } { P ^ { 2 } } \\sigma ^ { i j } _ { k } \\nabla _ { i } P \\nabla _ { j } P . \\end{align*}"}
+{"id": "2204.png", "formula": "\\begin{align*} \\mathbf { v } ^ { 0 } = [ b _ 0 , b _ 1 , \\cdots , b _ { N - 1 } ] ^ { T } , \\end{align*}"}
+{"id": "2905.png", "formula": "\\begin{align*} & H _ { x , x ' } ^ { ( n ) } = \\frac { 1 } { ( n + 1 ) ^ 2 } \\sum _ { j , j ' = 0 } ^ n \\Phi \\left ( \\frac { j } { n + 1 } , \\frac { j ' } { n + 1 } \\right ) \\frac { \\exp \\left \\{ \\frac { 2 i \\pi x } { n + 1 } \\right \\} - 1 } { \\exp \\left \\{ \\frac { 2 i \\pi x } { n + 1 } \\right \\} - 1 } \\exp \\left \\{ \\frac { 2 i \\pi j x } { n + 1 } \\right \\} \\exp \\left \\{ \\frac { 2 i \\pi j ' x ' } { n + 1 } \\right \\} \\end{align*}"}
+{"id": "7919.png", "formula": "\\begin{align*} \\hat W _ r ( k ) = \\frac 1 k f ( r k ) \\end{align*}"}
+{"id": "828.png", "formula": "\\begin{align*} g \\big ( c ( t , x ) \\big ) \\left ( \\frac { \\partial _ { x x } w ( t , x ) } { 1 + | \\partial _ x w ( t , x ) | ^ 2 } - \\frac { 1 } { w ( t , x ) } \\right ) \\begin{cases} \\leq - g \\big ( c _ 0 ( x ) \\big ) \\frac { 1 } { w _ 0 ( x ) } + \\varepsilon , \\\\ \\geq - g \\big ( c _ 0 ( x ) \\big ) \\frac { 1 } { w _ 0 ( x ) } - \\varepsilon \\end{cases} \\end{align*}"}
+{"id": "2005.png", "formula": "\\begin{align*} f _ \\alpha ^ 2 e _ \\alpha & = f _ \\alpha e _ \\alpha f _ \\alpha + f _ \\alpha [ f _ \\alpha , e _ \\alpha ] \\\\ & = f _ \\alpha e _ \\alpha f _ \\alpha + f _ \\alpha ( - h _ \\alpha ) \\\\ & = f _ \\alpha e _ \\alpha f _ \\alpha + ( - h _ \\alpha - 2 ) f _ \\alpha . \\end{align*}"}
+{"id": "7673.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } _ t ^ * = ( \\alpha _ t ^ { * , i } ) _ { 1 \\leq i \\leq N } = \\left ( - R ^ { - 1 } B y _ t ^ { * , i } - h \\big ( \\rho _ i ^ N ( \\Delta ^ { N , 1 } _ { * , t } , \\ldots , \\Delta ^ { N , N } _ { * , t } ) \\big ) \\right ) _ { 1 \\leq i \\leq N } . \\end{align*}"}
+{"id": "278.png", "formula": "\\begin{align*} F : = \\lim _ { n \\to \\infty } T ^ n [ f ] . \\end{align*}"}
+{"id": "17.png", "formula": "\\begin{align*} \\alpha _ { i _ k , j _ \\ell } \\oplus \\alpha _ { i _ r , j _ t } = \\emptyset \\end{align*}"}
+{"id": "3398.png", "formula": "\\begin{align*} f ( x ) ( x f ' ( x ) + x ^ 2 f '' ( x ) ) & = \\left ( \\sum _ { i = k } ^ l \\frac { x ^ i } { i ! } \\right ) \\left ( \\sum _ { i = k } ^ l \\frac { x ^ i } { ( i - 1 ) ! } + \\frac { x ^ i } { ( i - 2 ) ! } \\right ) = \\left ( \\sum _ { i = k } ^ l \\frac { x ^ i } { i ! } \\right ) \\left ( \\sum _ { i = k } ^ l \\frac { i x ^ i } { ( i - 1 ) ! } \\right ) \\\\ & \\geq \\left ( \\sum _ { i = k } ^ l \\frac { x ^ i } { ( i - 1 ) ! } \\right ) ^ 2 = ( x f ' ( x ) ) ^ 2 . \\end{align*}"}
+{"id": "4325.png", "formula": "\\begin{align*} v _ { t _ 0 } ( t ) : = \\lim _ { j \\to + \\infty } v _ { t _ 0 , B _ j } ( t ) = \\left \\{ \\begin{aligned} & - t _ 0 & & x < - t _ 0 , \\\\ & \\ t & & x \\ge t _ 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7184.png", "formula": "\\begin{align*} \\Bigl ( I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } + Q \\Bigr ) \\textbf { \\textit { U } } & = \\textbf { \\textit { W } } , \\textbf { \\textit { U } } \\big | _ { x _ n = 0 } = \\textbf { \\textit { V } } , \\\\ \\Bigl ( I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } + B - Q \\Bigr ) \\textbf { \\textit { W } } & = \\textbf { \\textit { Y } } \\in ( C ^ \\infty ( [ 0 , T ] ; \\mathfrak { D } ^ \\prime ( \\mathbb { R } ^ { n } ) ) ) ^ { n + 1 } . \\end{align*}"}
+{"id": "4489.png", "formula": "\\begin{align*} \\frac { \\mathit { n } ( v , t ) } { \\mathit { g l } ( v , t ) } = \\sum _ { v ^ * } t ^ { - \\sum _ { k < l } \\langle v ^ { ( k ) } , v ^ { ( l ) } \\rangle } \\prod _ { k \\ge 1 } \\frac { H ( v ^ { ( k ) } , v ^ { ( k + 1 ) } , t ^ { - 1 } ) } { \\mathit { g l } ( v ^ { ( k ) } , t ) } , \\end{align*}"}
+{"id": "6791.png", "formula": "\\begin{align*} G ( z ) = - \\frac { z - 1 } { 2 ( z + 1 ) } + \\frac { 1 } { 2 } \\int _ { - \\infty } ^ { 0 } q ( t ) d t + ( z + 1 ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } z ^ { n } c _ { n } ( 0 ) . \\end{align*}"}
+{"id": "6537.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\pi ( a ) } q _ i \\neq \\prod _ { i = 1 } ^ { \\pi ( a ) } p _ i ^ { \\alpha _ i } . \\end{align*}"}
+{"id": "9193.png", "formula": "\\begin{align*} \\hat { b } _ { 1 , \\delta } ( x ) = \\left \\{ \\begin{array} { c c } b _ { 1 , \\delta } ( x ) & x \\in \\bar \\Delta \\\\ 0 & x \\not \\in \\hat \\Delta \\ , . \\end{array} \\right . \\end{align*}"}
+{"id": "623.png", "formula": "\\begin{align*} O _ { \\gamma } ( f _ 0 ) = q ^ { d _ \\gamma } = | D ( \\gamma ) | ^ { - 1 / 2 } \\end{align*}"}
+{"id": "1189.png", "formula": "\\begin{align*} j _ 1 = \\min \\{ k _ 1 , i _ A , \\max \\{ i _ + , i _ - \\} \\} \\ , , \\end{align*}"}
+{"id": "647.png", "formula": "\\begin{align*} b \\in \\mbox { I m } ( f ) & \\mbox { i f f } b + I m ( f ) = 0 \\\\ & \\mbox { i f f } \\pi ( b ) = 0 \\\\ & \\mbox { i f f } b \\in \\mbox { K e r } ( \\pi ) \\end{align*}"}
+{"id": "7036.png", "formula": "\\begin{align*} G _ { 6 , 0 } \\ , : = \\ , 0 \\ , = : \\ , F _ { 6 , 0 } , \\end{align*}"}
+{"id": "2422.png", "formula": "\\begin{align*} \\int _ { c - i \\infty } ^ { c + i \\infty } d s ^ { \\prime } \\frac { F _ { q } ( s ^ { \\prime } ) } { ( s - s ^ { \\prime } ) } = 2 \\pi i F _ { q } ( s ) . \\end{align*}"}
+{"id": "8521.png", "formula": "\\begin{align*} \\delta _ { + } \\delta _ { - } = \\mathrm { e } ^ { - \\mathrm { i } \\mathcal { H } \\log \\left ( 1 + \\bar { r } _ 1 r _ 2 \\right ) } . \\end{align*}"}
+{"id": "8288.png", "formula": "\\begin{align*} m = \\left \\lceil \\frac { \\log C _ 0 + 1 } { \\beta \\tau } \\right \\rceil . \\end{align*}"}
+{"id": "7474.png", "formula": "\\begin{align*} r = | x | , u _ r = \\dfrac { x } { | x | } \\cdot \\nabla u \\end{align*}"}
+{"id": "1653.png", "formula": "\\begin{align*} H _ K \\times H ^ n ( Y _ K , N ) \\rightarrow H ^ n ( Y _ K , N ) , \\ , \\ , \\ , ( \\chi , c ) \\mapsto \\chi \\cdot c : = \\chi \\cup c \\end{align*}"}
+{"id": "1895.png", "formula": "\\begin{align*} \\mathfrak M ^ 0 _ { i , i } ( x , y ) & = 1 - \\frac { ( x _ i - y _ i ) ^ 2 } { | x - y | ^ 2 } , & & 1 \\le i \\le d , \\\\ \\mathfrak M _ { 1 , j } ^ 0 ( x , y ) & = - \\frac { x _ j - y _ j } { 2 | x - y | ^ 2 } \\big ( 2 x _ 1 - ( 1 + \\cos S _ c ) y _ 1 \\big ) , & & j \\ge 2 , \\\\ [ 2 p t ] \\mathfrak M _ { i , 1 } ^ 0 ( x , y ) & = \\mathfrak M _ { 1 , i } ^ 0 ( y , x ) , & & i \\ge 2 , \\\\ [ 2 p t ] \\mathfrak M _ { i , j } ^ 0 ( x , y ) & = - \\frac { ( x _ i - y _ i ) ( x _ j - y _ j ) } { | x - y | ^ 2 } , & & i , j \\ge 2 , \\ , \\ , i \\neq j . \\end{align*}"}
+{"id": "6647.png", "formula": "\\begin{align*} \\hat { X } ^ { \\epsilon } ( t ) & = x _ { 0 } + \\int ^ { t } _ { 0 } b ( \\varphi ( s ) ) d s + \\lambda ( \\epsilon ) \\int ^ { t } _ { 0 } \\sigma ( \\varphi ( s ) ) \\theta _ { \\epsilon } ( s ) d s . \\end{align*}"}
+{"id": "4182.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t w + \\nabla w \\cdot \\nabla ^ \\perp \\varphi = 0 , \\\\ w = \\mathcal { L } _ H \\varphi , \\\\ \\varphi | _ { \\partial \\Omega } = 0 . \\end{cases} \\end{align*}"}
+{"id": "5173.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi _ 1 + \\phi _ 2 - \\phi _ \\infty ) _ { + } G _ 1 \\frac { 1 } { r ^ { 2 } } \\dd x = \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi _ 1 - \\phi _ \\infty ) _ { + } G _ 1 \\frac { 1 } { r ^ { 2 } } \\dd x . \\end{align*}"}
+{"id": "6965.png", "formula": "\\begin{align*} \\Phi _ { n , e } ^ { ( i ) } ( x , y ) = \\prod ^ { n e } _ { \\substack { j = 1 \\\\ ( j , n e ) = 1 \\\\ j \\equiv i \\bmod e } } ( x - \\zeta ^ j _ { n e } y ) . \\end{align*}"}
+{"id": "5693.png", "formula": "\\begin{align*} 0 \\le 2 \\langle z ^ k - z ^ k _ * , z ^ k _ * - x ^ * \\rangle = \\| z ^ k - x ^ * \\| ^ 2 - \\| z ^ k - z ^ k _ * \\| ^ 2 - \\| z ^ k _ * - x ^ * \\| ^ 2 . \\end{align*}"}
+{"id": "4647.png", "formula": "\\begin{align*} \\widetilde { \\phi } ( s _ 0 , \\ldots , s _ n ) = \\phi ( s _ 0 s _ 1 ^ { - 1 } , s _ 1 s _ 2 ^ { - 1 } , \\ldots , s _ { n - 1 } s _ n ^ { - 1 } ) , s _ i \\in G . \\end{align*}"}
+{"id": "5150.png", "formula": "\\begin{align*} I _ h \\leq \\inf \\left \\{ E [ b ] \\ \\middle | \\ b \\in \\tilde { T } _ h \\right \\} \\leq \\inf \\left \\{ E [ b ] \\ \\middle | \\ b \\in T _ h \\right \\} = I _ h , \\end{align*}"}
+{"id": "7427.png", "formula": "\\begin{align*} \\exists C _ 3 = C _ 3 ( m ) \\in ( 0 , \\infty ) \\ \\Rightarrow T [ M ^ * n ] ( t ) \\le \\exp ( - C _ 3 ( m ) \\ t ^ m ) , \\ t \\ge 0 , \\end{align*}"}
+{"id": "6287.png", "formula": "\\begin{align*} ( R ( X , T ) S ) ^ h = 0 \\quad \\forall T , S \\in \\Delta , \\ , X \\in T M . \\end{align*}"}
+{"id": "2819.png", "formula": "\\begin{align*} \\mathbf { z } _ { k l } ^ { \\mathrm { o p t } } = { \\alpha _ { k } { \\mu } _ { k } { \\mathbf { Z } } ^ { - 1 } \\overline { \\mathbf { h } } _ { k l } } , \\end{align*}"}
+{"id": "1786.png", "formula": "\\begin{align*} \\begin{array} { c } a _ 0 + b _ 0 \\Re \\tau _ d = - \\frac { 1 } { 2 } | z _ 0 | ^ 2 \\Re v _ 2 - \\frac { 1 } { 2 } \\tilde { t } _ 0 \\sqrt { d } \\Im v _ 2 \\\\ b _ 0 \\Im \\tau _ d = - \\frac { 1 } { 2 } | z _ 0 | ^ 2 \\Im v _ 2 + \\frac { 1 } { 2 } \\tilde { t } _ 0 \\sqrt { d } \\Re v _ 2 \\\\ \\end{array} \\end{align*}"}
+{"id": "225.png", "formula": "\\begin{align*} g _ { | i ( g ) } = \\prod _ { j = 1 } ^ { \\ell } ( s _ { \\iota ( \\kappa ( j ) ) } ) _ { | \\ , i ( g ) - \\sigma ( \\kappa ( j ) - 1 ) } , \\end{align*}"}
+{"id": "7234.png", "formula": "\\begin{align*} \\sigma ( \\delta ) = ( \\det \\sigma ) \\ \\delta . \\end{align*}"}
+{"id": "8632.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\sum _ { m = 1 } ^ n \\| z _ { n , m } - w _ { n , m } \\| _ { L ^ 2 ( \\mu ) } \\leq \\limsup _ { n \\to \\infty } \\varepsilon \\| g \\| _ { L ^ 2 ( \\mu ) } ^ 3 \\sum _ { m = 1 } ^ n \\mathbb { E } _ { n , m } \\big [ \\| \\chi _ { n , m } \\| _ { L ^ 2 ( \\mu ) } ^ 2 \\big ] \\end{align*}"}
+{"id": "7369.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ n \\mu _ i ( f _ i ) = P \\Bigl ( \\oplus _ { i = 1 } ^ n f _ i \\Bigr ) \\ge \\inf _ { Q \\in \\mathcal { U } } Q \\Bigl ( \\oplus _ { i = 1 } ^ n f _ i \\Bigr ) \\end{gather*}"}
+{"id": "6063.png", "formula": "\\begin{align*} \\left ( \\alpha + 1 \\right ) \\kappa \\dot { P } - \\varpi \\dot { P } = \\alpha P . \\end{align*}"}
+{"id": "6195.png", "formula": "\\begin{align*} r _ 0 = \\frac { 1 } { \\sqrt { 2 \\kappa } } \\left ( 1 - \\frac { Q } { \\sqrt { Q ^ 2 + 1 6 \\kappa ( L + 1 ) ^ 2 ( L + 2 ) ^ 2 } } \\right ) ^ { 1 / 2 } \\end{align*}"}
+{"id": "4817.png", "formula": "\\begin{align*} \\| u \\| _ { 2 , h _ n } = \\sum _ { K \\in \\mathcal { T } _ n } ( u , u ) _ { H ^ 2 ( K ) } , \\end{align*}"}
+{"id": "5402.png", "formula": "\\begin{align*} w ( x ) : = x _ n - y _ n - L ( x ) + \\kappa | x ' - y ' | ^ 2 \\geq 0 \\mbox { o n } \\partial \\omega \\cap \\partial \\Omega . \\end{align*}"}
+{"id": "8436.png", "formula": "\\begin{align*} \\begin{aligned} 2 i k z b ( k ) = & \\Psi ^ + _ { 1 1 } ( 0 ; z ) \\left ( z \\Psi ^ - _ { 2 1 } ( 0 ; z ) - \\widehat { \\Psi } ^ - _ { 2 1 } ( 0 ) \\right ) - \\Psi ^ - _ { 1 1 } ( 0 ; z ) \\left ( z \\Psi ^ + _ { 2 1 } ( 0 ; z ) - \\widehat { \\Psi } ^ + _ { 2 1 } ( 0 ) \\right ) \\\\ & + \\widehat { \\Psi } ^ - _ { 2 1 } ( 0 ) \\left ( \\Psi ^ + _ { 1 1 } ( 0 ; z ) - e ^ { i c _ + ( 0 ) } \\right ) - \\widehat { \\Psi } ^ + _ { 2 1 } ( 0 ) \\left ( \\Psi ^ - _ { 1 1 } ( 0 ; z ) - e ^ { i c _ - ( 0 ) } \\right ) \\end{aligned} \\end{align*}"}
+{"id": "4339.png", "formula": "\\begin{align*} \\int _ { \\{ - t ' _ 1 \\le \\Psi < - t ' _ 2 \\} } | \\tilde { F } | ^ 2 _ h \\mathbb { I } _ E ( - \\Psi ) \\ge & \\liminf _ { j \\to + \\infty } \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in V _ j \\} } | \\tilde { F } | ^ 2 _ h \\\\ \\ge & \\liminf _ { j \\to + \\infty } \\frac { G ( T _ 1 ) } { \\int _ { T _ 1 } ^ { + \\infty } c ( s ) e ^ { - s } d s } \\int _ { V _ j } e ^ { - s } d s \\\\ = & \\frac { G ( T _ 1 ) } { \\int _ { T _ 1 } ^ { + \\infty } c ( s ) e ^ { - s } d s } \\int _ { t _ 2 } ^ { t _ 1 } e ^ { - s } \\mathbb { I } _ E ( s ) d s , \\end{align*}"}
+{"id": "9037.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { { \\bf k } \\in \\Lambda _ { 1 } } \\begin{pmatrix} a _ { \\bf k } & 0 \\\\ 0 & - a _ { \\bf k } \\end{pmatrix} { \\rm e } ^ { { \\rm i } \\langle { \\bf k } , x \\rangle } + \\sum _ { { \\bf k } \\in \\Lambda _ { 2 } } \\begin{pmatrix} 0 & b _ { \\bf k } \\\\ c _ { \\bf k } & 0 \\end{pmatrix} { \\rm e } ^ { { \\rm i } \\langle { \\bf k } , x \\rangle } , \\end{align*}"}
+{"id": "7215.png", "formula": "\\begin{align*} \\| u ( t _ { n + 1 } ) \\| _ { L ^ 2 } ^ 2 = \\| u ( t _ n ) \\| _ { L ^ 2 } ^ 2 - 2 \\int _ { t _ n } ^ { t _ { n + 1 } } \\| \\nabla u ( s ) \\| _ { L ^ p } ^ p \\ , d s . \\end{align*}"}
+{"id": "8432.png", "formula": "\\begin{align*} b ( k ) = \\psi ^ + _ { 1 1 } ( 0 ; k ) \\psi ^ - _ { 2 1 } ( 0 ; k ) - \\psi ^ + _ { 2 1 } ( 0 ; k ) \\psi ^ - _ { 1 1 } ( 0 ; k ) . \\end{align*}"}
+{"id": "7417.png", "formula": "\\begin{align*} \\{ 1 , \\ldots , \\ell \\} \\simeq \\bigcup _ { i = 1 } ^ L \\{ m _ 1 + \\cdots + m _ { i - 1 } + 1 , \\ldots , m _ 1 + \\cdots + m _ i \\} \\end{align*}"}
+{"id": "4038.png", "formula": "\\begin{align*} i _ { \\chi \\theta } ( f ) ( I _ \\bullet ) = \\sum _ { J _ \\bullet \\in \\partial _ { \\chi \\theta } ^ { - 1 } ( I _ \\bullet ) } f ( J _ \\bullet ) \\end{align*}"}
+{"id": "5536.png", "formula": "\\begin{align*} \\int _ { \\mathfrak U } | \\omega | & = { \\pi ' _ s } _ ! \\int _ { \\mathfrak U ' } | \\omega | = { \\pi ' _ s } _ ! \\Big ( { i ' _ s } ^ * \\int _ { \\mathfrak X ' } | \\omega | \\Big ) \\\\ & = ( \\mathfrak i _ 0 ^ * \\circ { \\pi _ s } _ ! ) \\int _ { \\mathfrak X ' } | \\omega | \\\\ & = \\mathfrak i _ 0 ^ * \\int _ { \\mathfrak X } | \\omega | , \\end{align*}"}
+{"id": "472.png", "formula": "\\begin{align*} \\limsup _ { x \\to \\infty } \\frac { g ^ { \\ast 2 } ( x ) } { g ( x ) } & \\le ( 2 / \\lambda ^ 2 ) e ^ \\lambda \\Big ( \\lambda - e ^ { - \\lambda } \\sum _ { n \\neq 2 } ^ \\infty \\liminf _ { x \\to \\infty } \\frac { \\lambda ^ n } { n ! } \\frac { g ^ { \\ast n } ( x ) } { g ( x ) } \\Big ) = 2 , \\end{align*}"}
+{"id": "8390.png", "formula": "\\begin{align*} & w _ 0 = e _ 1 , \\ \\ w _ { n + 1 } ( x ; z ) = F w _ n = \\int _ { - \\infty } ^ x F ( x , y ; z ) w _ n ( y ) d y . \\end{align*}"}
+{"id": "4931.png", "formula": "\\begin{align*} \\l _ c ^ \\infty ( \\Z ^ d ) & = \\big \\{ \\{ b _ k \\} _ { k \\in \\Z ^ d } : b _ k = 0 \\} , \\\\ c _ 0 ( \\Z ^ d ) & = \\big \\{ \\{ b _ k \\} _ { k \\in \\Z ^ d } : b _ k \\to 0 | k | \\to \\infty \\big \\} . \\end{align*}"}
+{"id": "3279.png", "formula": "\\begin{align*} & \\lim _ { N \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\sup _ { n \\in \\mathbb N } \\sum _ { m , k \\geq N } \\mathbb E \\left [ \\langle \\tilde Z _ t ^ { 2 , n } , e _ k \\otimes e _ m \\rangle _ { \\mathcal H } ^ 2 \\right ] \\\\ = & \\lim _ { N \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\sup _ { n \\in \\mathbb N } \\mathbb E \\left [ \\| p _ N \\tilde Z _ t ^ { 2 , n } \\| _ { \\mathcal H } ^ 2 \\right ] = 0 , \\end{align*}"}
+{"id": "1544.png", "formula": "\\begin{align*} P _ 1 ( f , g ) & { } = \\left ( f - \\frac { 1 } { 2 } ( ( f - g ) \\vee 0 ) , g + \\frac { 1 } { 2 } ( ( f - g ) \\vee 0 ) \\right ) , \\\\ P _ { 2 , \\alpha } ( f , g ) & { } = \\left ( g + \\frac { 1 } { 2 } \\varphi _ \\alpha \\circ ( f - g ) , f - \\frac { 1 } { 2 } \\varphi _ \\alpha \\circ ( f - g ) \\right ) , \\end{align*}"}
+{"id": "1089.png", "formula": "\\begin{align*} \\prescript J { } \\pi ( s _ \\beta w ) = ( w ' , \\lambda - \\lambda ' + \\Phi ^ + ( - \\beta ) w ^ { - 1 } \\beta ^ \\vee ) . \\end{align*}"}
+{"id": "18.png", "formula": "\\begin{align*} \\alpha _ { i _ k , j _ \\ell } \\oplus \\alpha _ { i _ r , j _ t } = \\{ \\alpha _ { i _ k , j _ t } , \\alpha _ { j _ \\ell , i _ r } \\} \\end{align*}"}
+{"id": "2844.png", "formula": "\\begin{align*} { \\rm e } _ { q , \\ell , \\ell ' } = \\delta _ { \\ell , \\ell ' } \\quad \\mbox { a n d } { \\rm e } _ { p , \\ell , \\ell ' } = \\delta _ { n + 1 + \\ell , \\ell ' } , \\ell ' = 1 , \\ldots , 2 n + 2 . \\end{align*}"}
+{"id": "4463.png", "formula": "\\begin{align*} Q = a _ { 1 } ( x _ { 1 } + m _ { 1 2 } x _ { 2 } + m _ { 1 3 } x _ { 3 } + m _ { 1 4 } x _ { 4 } ) ^ { 2 } + a _ { 2 } ( x _ { 2 } + m _ { 2 3 } x _ { 3 } + m _ { 2 4 } x _ { 4 } ) ^ { 2 } + a _ { 3 } ( x _ { 3 } + m _ { 3 4 } x _ { 4 } ) ^ { 2 } + a _ { 4 } x _ { 4 } ^ { 2 } . \\end{align*}"}
+{"id": "2196.png", "formula": "\\begin{align*} \\hat { \\Theta } _ { R M } = \\left \\{ \\hat { \\theta } _ i , i \\in \\{ 1 , 2 , \\cdots , 2 N - 2 \\} \\right \\} , \\end{align*}"}
+{"id": "1285.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { \\abs { \\Lambda _ { n } } } s _ { n } ( \\nu | \\mu ) = \\lim _ { n \\to \\infty } \\frac { G _ { n } } { \\abs { \\Lambda _ { n } } } s _ { n } ( \\nu | \\mu ) , \\end{align*}"}
+{"id": "7788.png", "formula": "\\begin{align*} \\dot \\phi _ i = \\frac 1 M \\sum _ { j = 1 } ^ M a _ { i - j } \\sin ( \\phi _ j - \\phi _ i ) , i = 1 , \\dots , M , \\end{align*}"}
+{"id": "3187.png", "formula": "\\begin{align*} \\cosh { { d } } = \\frac { - \\langle ~ \\mathbf { x } , ~ \\mathbf { y } \\rangle } { \\sqrt { \\langle ~ \\mathbf { x } , ~ \\mathbf { x } \\rangle \\langle ~ \\mathbf { y } , ~ \\mathbf { y } \\rangle } } . \\end{align*}"}
+{"id": "6867.png", "formula": "\\begin{align*} w = \\Psi ( \\hat z ) \\end{align*}"}
+{"id": "973.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { 2 c - 2 } a _ i q ^ i & \\le \\sum _ { i = 2 } ^ { 2 c } c ^ i q ^ { 2 c - i } = q ^ { 2 c - 2 } c ^ 2 \\sum _ { i = 0 } ^ { 2 c - 2 } \\frac { c ^ i } { q ^ i } \\le \\frac { q ^ { 2 c - 2 } c ^ 2 } { 1 - c / q } < q ^ { 2 c - 1 } \\end{align*}"}
+{"id": "8483.png", "formula": "\\begin{align*} & \\mathcal { P } ^ + \\left ( f ( z ) e ^ { - 2 i z x } \\right ) = \\int _ { 2 x } ^ { + \\infty } \\widehat { f } ( \\xi ) e ^ { i z ( \\xi - 2 x ) } \\mathrm { d } \\xi , \\\\ & \\mathcal { P } ^ - \\left ( f ( z ) e ^ { 2 i z x } \\right ) = - \\int ^ { \\infty } _ { 2 x } \\widehat { f } ( \\xi ) e ^ { - i z ( \\xi - 2 x ) } \\mathrm { d } \\xi , \\end{align*}"}
+{"id": "930.png", "formula": "\\begin{align*} \\Phi _ 1 ( t ) & = | \\alpha | ^ 2 \\log t + \\theta _ 1 + O ( t ^ { - 1 / 4 + C _ 1 \\varepsilon ^ 2 _ 1 } ) \\end{align*}"}
+{"id": "6777.png", "formula": "\\begin{align*} z : = \\frac { \\frac { 1 } { 2 } + i \\rho } { \\frac { 1 } { 2 } - i \\rho } . \\end{align*}"}
+{"id": "2120.png", "formula": "\\begin{align*} \\rho _ { Z Z } ( L , L ) = a \\rho _ w ^ 2 + b \\rho _ z ^ 2 . \\end{align*}"}
+{"id": "752.png", "formula": "\\begin{align*} \\int _ { 4 I _ Q } \\| R _ j \\partial _ j h ( \\cdot , t ) \\| _ { L ^ 1 ( 8 Q _ 1 ) } d t \\lesssim \\ell ( Q ) ^ N , \\ ; \\ ; j = 1 , 2 , \\cdots , N . \\end{align*}"}
+{"id": "6488.png", "formula": "\\begin{align*} & E _ { k , [ p , q ] } ^ 0 = \\{ n \\in \\mathbb { N } ^ k : p \\le n \\le q \\} \\\\ & E _ { k , [ p , q ] } ^ 1 = \\{ x \\in E _ { k , q } ^ 1 : s ( x ) , r ( x ) \\in E _ { k , [ p , q ] } ^ 0 \\} \\end{align*}"}
+{"id": "9126.png", "formula": "\\begin{align*} E ^ + ( \\Q _ { p ^ g , n } ) _ p \\cap E ^ - ( \\Q _ { p ^ g , n } ) _ p = E ( \\Q _ { p ^ g , - 1 } ) _ p , E ^ + ( \\Q _ { p ^ g , n } ) _ p + E ^ - ( \\Q _ { p ^ g , n } ) _ p = E ( \\Q _ { p ^ g , n } ) _ p . \\end{align*}"}
+{"id": "6124.png", "formula": "\\begin{align*} \\widetilde { \\varphi } ^ { H } _ { n , p } & = \\varphi ^ { H } _ { n , p } - \\psi _ { n , p } \\ , , \\\\ \\widetilde { \\varphi } ^ { V } _ { n , p } & = \\varphi ^ { V } _ { n , p } - \\rho _ { n , p } \\ , . \\end{align*}"}
+{"id": "3245.png", "formula": "\\begin{align*} & \\sqrt n \\left ( \\hat { \\Sigma } _ t ^ { n , m } - t Q \\right ) \\\\ = & \\sqrt n \\left ( R V _ t ^ n - t Q \\right ) + \\left ( \\sqrt n \\left ( \\hat { \\Sigma } _ t ^ { n , m } - \\Pi _ m t Q \\Pi _ m \\right ) - \\sqrt n \\left ( R V _ t ^ n - t Q \\right ) \\right ) + \\sqrt n \\left ( \\Pi _ m t Q \\Pi _ m - t Q \\right ) . \\end{align*}"}
+{"id": "5212.png", "formula": "\\begin{align*} \\Phi _ { t } + u \\cdot \\nabla \\Phi & = 0 \\textrm { i n } \\ \\mathbb { R } ^ { 3 } \\times ( 0 , T ) . \\end{align*}"}
+{"id": "4273.png", "formula": "\\begin{align*} n = \\frac { - 3 \\left ( a ^ { 2 } + b ^ { 2 } - 1 \\right ) \\pm \\sqrt { 3 \\left ( \\left ( a - b \\right ) ^ { 4 } - \\left ( a ^ { 4 } + b ^ { 4 } + \\left ( a - 1 \\right ) ^ { 4 } + \\left ( b - 1 \\right ) ^ { 4 } \\right ) + 1 \\right ) } } { 6 \\left ( a + b - 1 \\right ) } \\end{align*}"}
+{"id": "5840.png", "formula": "\\begin{align*} g _ i ^ { \\left ( { n e q } \\right ) } \\left ( { { { \\bf { x } } _ b } , t } \\right ) = { g _ i } \\left ( { { { \\bf { x } } _ f } , t } \\right ) - g _ i ^ { \\left ( { e q } \\right ) } \\left ( { { { \\bf { x } } _ f } , t } \\right ) . \\end{align*}"}
+{"id": "9160.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } h ( s ) \\sin ( 2 \\pi t ) \\ , d t = 0 \\end{align*}"}
+{"id": "3861.png", "formula": "\\begin{align*} x _ 2 \\partial _ { x _ 1 } v _ 3 - x _ 1 \\partial _ { x _ 2 } v _ 3 + k \\partial _ { x _ 3 } v _ 3 = 0 . \\end{align*}"}
+{"id": "6981.png", "formula": "\\begin{align*} \\theta _ 2 ^ { l _ 1 + k _ 1 n } u _ n = b \\beta ^ n \\zeta ^ { y - x n } ( \\theta _ 1 ^ { l _ 1 + k _ 1 n } - ( - \\zeta ^ { x n - y } ) \\theta _ 2 ^ { l _ 1 + k _ 1 n } ) \\end{align*}"}
+{"id": "4033.png", "formula": "\\begin{align*} M _ { I } = \\bigoplus _ { J \\subset I } R _ { J } , \\end{align*}"}
+{"id": "3098.png", "formula": "\\begin{align*} e _ { G } ( \\tilde { \\theta } ^ k , T , \\mathcal { K } ) = \\tilde { \\theta } ^ k - \\Pi _ { \\mathcal { K } , G } ( \\tilde { \\theta } ^ k - G ^ { - 1 } T ( \\tilde { \\theta } ^ k ) ) . \\end{align*}"}
+{"id": "2010.png", "formula": "\\begin{align*} ( G _ \\alpha T _ \\alpha f \\circ \\psi _ M ( m ) ) ( s ) & = G _ \\alpha T _ \\alpha f ( s \\otimes m ) \\\\ & = T _ \\alpha f ( s \\otimes m ) \\\\ & = s \\otimes f ( m ) \\\\ & = \\psi _ M ( f ( m ) ) ( s ) \\\\ & = ( \\psi _ M \\circ f ) ( m ) ( s ) . \\end{align*}"}
+{"id": "7723.png", "formula": "\\begin{align*} 2 x ^ { \\tilde \\varepsilon x / 2 } h _ s ( i ) ^ { ( \\tilde \\varepsilon x ) ^ 2 / 6 4 } = 2 \\big ( x h _ s ( i ) ^ { \\tilde \\varepsilon x / 3 2 } \\big ) ^ { \\tilde \\varepsilon x / 2 } \\underbrace { \\le } _ { \\mbox { b y \\eqref { e q : P r o b h x 0 } } } 2 n ^ { - \\tilde \\varepsilon x / 2 } h _ s ( i ) ^ { ( \\tilde \\varepsilon x ) ^ 2 / 1 2 8 } . \\end{align*}"}
+{"id": "7331.png", "formula": "\\begin{align*} \\left | \\Delta _ C \\Delta _ B ( A ) \\right | & = \\left | \\big ( ( A + B ) \\setminus A \\big ) + C \\right | - \\left | ( A + B ) \\setminus A \\right | \\\\ & \\geq \\left | ( A + B + C ) \\setminus ( A + C ) \\right | - \\left | A + B \\right | + \\left | A \\right | \\\\ & = | A + B + C | - | A + C | - | A + B | + | A | , \\end{align*}"}
+{"id": "1924.png", "formula": "\\begin{align*} z _ \\star ^ { ( i ) } = \\begin{cases} 2 T - i + 1 , & \\textnormal { i f } 1 \\leq i \\leq 2 T , \\\\ 0 & \\textnormal { o t h e r w i s e } . \\end{cases} y _ \\star ^ { ( i ) } = \\begin{cases} 1 , & \\textnormal { i f } 1 \\leq i \\leq 2 T , \\\\ 0 & \\textnormal { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "7141.png", "formula": "\\begin{align*} \\xi _ p ( V ) = \\left ( \\sum _ { l \\ge n } ( l + 2 ) ( l + n + 3 ) ( l - n + 1 ) a _ { l + 3 } z ^ l \\right ) d z ^ { \\otimes 2 } \\end{align*}"}
+{"id": "7204.png", "formula": "\\begin{align*} a _ 2 ( x ) = \\frac { \\Gamma ( n - 2 ) \\operatorname { v o l } ( \\mathbb { S } ^ { n - 2 } ) } { ( 2 \\pi ) ^ { n - 1 } } \\bigg ( a _ { 2 , 1 } H ^ 2 + a _ { 2 , 2 } \\sum _ { \\alpha } \\kappa _ \\alpha ^ 2 + a _ { 2 , 3 } R + a _ { 2 , 4 } R _ { n n } \\bigg ) \\end{align*}"}
+{"id": "5493.png", "formula": "\\begin{align*} A ( p , s ) = 2 ^ { - p / 2 } \\left ( \\frac { 3 s } { 2 } - p \\right ) \\frac { 1 2 s - \\left ( \\frac { p } { 2 } + 2 \\right ) \\left ( \\frac { p } { 2 } + 3 \\right ) } { 1 4 4 } \\end{align*}"}
+{"id": "2449.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } d _ { q } s \\ , F _ { q } ( s ) = \\int _ { s } ^ { \\infty } d _ { q } s \\int _ { 0 } ^ { \\infty } \\exp _ { q } ( - s t ) f ( t ) d t . \\end{align*}"}
+{"id": "8581.png", "formula": "\\begin{align*} k _ 2 ( t ) = h _ { 1 - \\alpha - \\gamma } ( t ) \\cdot \\ , f _ 3 ( t ) , \\ f _ 3 ( t ) = \\sum _ { k = 0 } ^ { + \\infty } \\ , c _ k t ^ k \\end{align*}"}
+{"id": "276.png", "formula": "\\begin{align*} T [ f ] ( w ) = 1 - t ^ 2 > 0 = \\max \\{ T [ f ] ( p ) , T [ f ] ( q ) \\} . \\end{align*}"}
+{"id": "5779.png", "formula": "\\begin{align*} \\nabla _ X \\psi _ 1 = \\frac { 1 } { 2 } X _ 1 \\cdot \\psi _ 1 - \\frac { 1 } { 2 } S ( X ) \\cdot N \\cdot \\psi _ 1 \\end{align*}"}
+{"id": "959.png", "formula": "\\begin{align*} { { { \\bf X } } } _ { T } & = \\{ ( v _ { 1 } , v _ { 2 } ) \\in C ( [ T , \\infty ) ; L ^ 2 ( \\R ) ; \\| ( v _ { 1 } , v _ { 2 } ) \\| _ { { { \\bf X } _ T } } \\le 1 \\} , \\\\ \\| ( v _ { 1 } , v _ { 2 } ) \\| _ { { { \\bf X } _ T } } & = \\sup _ { t \\ge T } ( t ^ { \\tilde { \\nu } + \\frac 1 2 } \\| v _ { 1 } \\| _ { L _ x ^ { 2 } } + t ^ { \\tilde { \\nu } } \\| J v _ { 1 } \\| _ { L _ x ^ { 2 } } + t ^ { \\tilde { \\nu } + \\frac 1 2 - \\delta } \\| v _ { 2 } \\| _ { L _ x ^ { 2 } } + t ^ { \\tilde { \\nu } - \\delta } \\| J v _ { 2 } \\| _ { L _ x ^ { 2 } } ) \\end{align*}"}
+{"id": "6707.png", "formula": "\\begin{align*} \\Omega ( n , m ) : = [ h ( l _ n \\xi , l _ n \\eta , l _ n \\omega ) - h ( g \\xi , g \\eta , g \\omega ) ] - [ h ( l _ m \\xi , l _ m \\eta , l _ m \\omega ) - h ( g \\xi , g \\eta , g \\omega ) ] . \\end{align*}"}
+{"id": "5068.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\bold { A } _ j \\cdot \\bold { B } _ k \\dd x = \\frac { 1 } { \\lambda _ k } \\int _ { \\Omega } \\bold { A } _ j \\cdot \\nabla \\times \\bold { B } _ k \\dd x = \\frac { 1 } { \\lambda _ k } \\int _ { \\Omega } \\bold { B } _ j \\cdot \\bold { B } _ k \\dd x = \\frac { 1 } { \\lambda _ k } \\delta _ { j , k } . \\end{align*}"}
+{"id": "4979.png", "formula": "\\begin{align*} W ( \\xi _ 0 , n , x , \\mathsf { U } ^ n ) = W ( \\xi _ 0 , n , x , \\mathsf { V } ^ n ) . \\end{align*}"}
+{"id": "5109.png", "formula": "\\begin{align*} \\int _ { D ( 0 , 2 ) \\backslash D ( 0 , 1 ) } | \\phi _ n | ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r = 1 , \\int _ { D ( 0 , 2 ) \\backslash D ( 0 , 1 ) } | \\nabla \\phi _ n | ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r \\to 0 . \\end{align*}"}
+{"id": "581.png", "formula": "\\begin{align*} e _ { i , j } = \\begin{cases} e _ t ~ ~ { \\rm i f } ~ ~ i < 2 t - 1 , ~ ~ i ~ ~ { \\rm i s ~ o d d } ~ ~ { \\rm a n d } ~ ~ j = 2 t - 1 , \\\\ e _ t ~ ~ { \\rm i f } ~ ~ i < 2 t , ~ ~ i ~ ~ { \\rm i s ~ e v e n } ~ ~ { \\rm a n d } ~ ~ j = 2 t , \\\\ e _ t ~ ~ { \\rm i f } ~ ~ i = 2 t - 1 , ~ j > 2 t - 1 ~ { \\rm a n d } ~ ~ j ~ ~ { \\rm i s ~ e v e n } , \\\\ e _ t ~ ~ { \\rm i f } ~ ~ i = 2 t , ~ j > 2 t ~ { \\rm a n d } ~ ~ j ~ ~ { \\rm i s ~ o d d } . \\\\ \\end{cases} \\end{align*}"}
+{"id": "9128.png", "formula": "\\begin{align*} \\varepsilon _ { f , n } = \\sum _ { j = 1 } ^ { \\infty } ( - 1 ) ^ { j - 1 } \\zeta _ { ( f ) } ^ { \\varphi ^ { - ( n + 1 + 2 j ) } } p ^ j \\end{align*}"}
+{"id": "5769.png", "formula": "\\begin{align*} ( \\partial - \\nabla ) _ X Y = - \\sum _ { i = 1 , 2 } \\sqrt { c _ i } \\ \\langle X _ i , Y \\rangle \\nu _ i + B ( X , Y _ T ) - B ^ * ( X , Y _ N ) \\end{align*}"}
+{"id": "2447.png", "formula": "\\begin{align*} D _ { q } ^ { n } \\{ L _ { q } [ f ( t ) ] \\} ( s ) = L _ { q } [ ( - t ) ^ { n } f ( t ) ] . \\end{align*}"}
+{"id": "3111.png", "formula": "\\begin{align*} A = \\bigcup _ { r = 1 } ^ { \\infty } \\bigcup _ { k = 1 } ^ { \\infty } A _ { k , r } . \\end{align*}"}
+{"id": "3844.png", "formula": "\\begin{align*} X _ { n + 1 } = L ( X _ n ) + F ( X _ n ) + G ( X _ n ) \\xi _ { n + 1 } , \\end{align*}"}
+{"id": "7823.png", "formula": "\\begin{align*} \\psi ( B _ \\phi u , p ) = B _ \\phi \\psi ( u , p ) \\end{align*}"}
+{"id": "4815.png", "formula": "\\begin{align*} a _ n ( u _ n , v _ n ) = ( p _ n f , v _ n ) \\quad \\ , v _ n \\in X _ n , \\end{align*}"}
+{"id": "6949.png", "formula": "\\begin{align*} M _ n ^ \\psi ( Q ) = & \\sum _ { i = 1 } ^ n \\int _ { \\R } V ( x ) \\tilde { Q } _ { i } ( d x ) + \\sum _ { i , j = 1 } ^ n J _ { \\pi _ n ( i ) \\pi _ n ( j ) } \\int _ { \\R } \\int _ { \\R } \\psi ( x , y ) \\tilde { Q } _ { i } ( d x ) \\tilde { Q } _ { j } ( d y ) - \\sum _ { i = 1 } ^ n H ( \\tilde { Q } _ { i } ) , \\end{align*}"}
+{"id": "340.png", "formula": "\\begin{align*} \\vert S \\cap [ 1 , n ] \\vert = o ( \\alpha ( n ) \\cdot n / \\log n ) \\end{align*}"}
+{"id": "4111.png", "formula": "\\begin{align*} H ^ { ( k ) } ( E ) = \\min \\limits _ { \\substack { 0 \\neq F \\subset E \\\\ \\dim F = k } } H _ r ( F ) \\end{align*}"}
+{"id": "6535.png", "formula": "\\begin{align*} 2 a = ( a + b ) + ( a - b ) \\end{align*}"}
+{"id": "7170.png", "formula": "\\begin{align*} A ^ { - 1 } \\mathcal { L } _ g = I _ { n + 1 } \\frac { \\partial ^ 2 } { \\partial x _ n ^ 2 } + B \\frac { \\partial } { \\partial x _ n } + C , \\end{align*}"}
+{"id": "6944.png", "formula": "\\begin{align*} M _ n ( Q ) / n & \\le \\int _ \\R V \\ , d \\overline { Q } + \\frac 1 2 \\int _ \\R \\int _ \\R K ( x - y ) \\ , \\overline { Q } ( d x ) \\overline { Q } ( d y ) - H ( \\overline { Q } ) = M _ n ( \\overline { Q } ^ { \\otimes n } ) / n . \\end{align*}"}
+{"id": "8602.png", "formula": "\\begin{align*} h ^ 2 _ { [ - u _ 1 , u _ 1 ] \\oplus _ 2 \\cdots \\oplus _ 2 [ - u _ m , u _ m ] } ( x ) & = \\sum _ { i = 1 } ^ m \\langle u _ i , x \\rangle ^ 2 = \\sum _ { i = 1 } ^ m \\langle U e _ i , x \\rangle ^ 2 = \\sum _ { i = 1 } ^ m \\langle e _ i , U ^ * x \\rangle ^ 2 = | U ^ * x | ^ 2 = h ^ 2 _ { U B _ 2 ^ m } ( x ) \\\\ & = \\langle U ^ * x , U ^ * x \\rangle = \\langle \\sqrt { U U ^ * } x , \\sqrt { U U ^ * } x \\rangle = | \\sqrt { U U ^ * } x | ^ 2 = h ^ 2 _ { \\sqrt { U U ^ * } B _ 2 ^ n } ( x ) . \\end{align*}"}
+{"id": "7433.png", "formula": "\\begin{align*} \\phi _ k \\Big ( \\frac { 1 } { k } \\Big ) & = \\frac { \\log ( 2 \\pi ) } { 2 } - \\frac { ( 2 k - p _ k ) \\log k } { k } + \\Big ( 2 k - p _ k + \\frac { p _ k } { 2 } - \\frac { 3 } { 2 } \\Big ) \\log \\Big ( 1 - \\frac { 1 } { k } \\Big ) \\\\ & > \\frac { \\log ( 2 \\pi ) } { 2 } - \\frac { \\log k } { 2 k } \\Big ( \\log _ 2 ( \\pi k ) + 1 \\Big ) - \\frac { 1 0 1 } { 2 0 0 k } \\Big ( \\log _ 2 ( \\pi k ) + 1 \\Big ) \\\\ & > \\frac { \\log ( 2 \\pi ) } { 2 } - \\frac { 6 0 1 } { 2 0 0 0 } > 0 , \\end{align*}"}
+{"id": "3123.png", "formula": "\\begin{align*} & [ N _ T ( x + a ) , N _ T ( y + b ) ] _ { \\varrho } = [ T ( a ) + 0 , T ( b ) + 0 ] _ { \\varrho } = [ T ( a ) , T ( b ) ] , \\\\ & N _ T \\big ( [ N _ T ( x + a ) , y + b ] _ { \\varrho } - [ N _ T ( y + b ) , x + a ] _ { \\varrho } - N _ T ( [ x + a , y + b ] _ { \\varrho } ) \\big ) \\\\ & = N _ T \\big ( ( [ T ( a ) , y ] + \\varrho ( T ( a ) ) b ) - ( [ T ( b ) , x ] + \\varrho ( T ( b ) ) a ) - ( 0 + T ( \\varrho ( x ) b - \\varrho ( y ) a ) ) \\big ) \\\\ & = T ( \\varrho ( T ( a ) ) b - \\varrho ( T ( b ) ) a ) \\end{align*}"}
+{"id": "4871.png", "formula": "\\begin{align*} \\partial ^ 2 _ { x _ i x _ j } \\phi = \\partial ^ 2 _ { x _ i x _ j } \\psi + \\partial ^ 2 _ { x _ i y } \\psi \\Gamma _ j + \\partial ^ 2 _ { x _ j y } \\psi \\Gamma _ i + \\partial ^ 2 _ y \\psi \\Gamma _ j \\Gamma _ i + \\partial _ y \\psi \\partial ^ 2 _ { x _ j x _ i } T - \\partial _ y T \\partial ^ 3 _ { x _ j x _ i y } T + \\partial _ { y } \\Gamma _ i \\partial _ { y } \\Gamma _ j \\end{align*}"}
+{"id": "9188.png", "formula": "\\begin{align*} \\begin{aligned} \\gamma _ 2 ^ \\star : = \\dfrac { \\bar c } { 3 e ^ { L _ r \\bar { t } } } \\gamma _ 3 ^ \\star : = \\dfrac { \\bar c L _ r } { 3 \\bar k ( L _ r , M _ r , \\delta ) ( e ^ { L _ r \\bar { t } } - 1 ) } \\end{aligned} \\end{align*}"}
+{"id": "4543.png", "formula": "\\begin{align*} \\tau ( [ L ] ) : = \\log _ q [ \\mathcal { O } _ F ^ d : \\pi ^ i L ] , \\end{align*}"}
+{"id": "2438.png", "formula": "\\begin{align*} F _ { q } ( s - s _ { 0 } ) = \\int _ { 0 } ^ { \\infty } \\exp _ { q } ( - ( s - s _ { 0 } ) t ) f ( t ) d t . \\end{align*}"}
+{"id": "4655.png", "formula": "\\begin{align*} \\Vert T _ { H \\vert _ { \\mathbb { Z } \\times \\mathbb { Z } } } ^ { ( 2 N + 1 ) } : L _ { p _ 1 } ( \\mathbb { T } , S _ { p _ 1 } ^ { 2 N + 1 } ) \\times L _ { p _ 2 } ( \\mathbb { T } , S _ { p _ 2 } ^ { 2 N + 1 } ) \\rightarrow L _ { 1 } ( \\mathbb { T } , S _ { 1 } ^ { 2 N + 1 } ) \\Vert \\leq A _ { p _ 1 , p _ 2 , 2 N + 1 } . \\end{align*}"}
+{"id": "4292.png", "formula": "\\begin{align*} \\tilde { u } ^ { [ n ] } ( x ) = \\sum _ i w ^ { [ n ] } _ i \\sigma _ i ( x ) , \\end{align*}"}
+{"id": "1841.png", "formula": "\\begin{align*} \\Rightarrow \\quad \\frac 1 r \\big ( g - d r \\otimes d r \\big ) = H e s s ( r ) ; \\end{align*}"}
+{"id": "8991.png", "formula": "\\begin{align*} i _ { \\eta \\cup j } = ( - 1 ) ^ { p ( \\eta , j ) - 1 } \\left ( \\prod _ { r = 1 } ^ { q - 1 } { D _ { { \\ell _ { r } } - 1 } } \\right ) \\left ( - x _ j \\right ) \\left ( \\prod _ { r = q + 1 } ^ { s } { D _ { \\ell _ { r - 1 } } } \\right ) \\left ( \\prod _ { r = s + 1 } ^ { d + 1 } { D _ { \\ell _ { r } - 1 } } \\right ) . \\end{align*}"}
+{"id": "8349.png", "formula": "\\begin{align*} & a ( k ) = \\det \\left ( \\psi ^ - _ 1 ( 0 ; k ) , \\psi ^ + _ 2 ( 0 ; k ) \\right ) , \\\\ & b ( k ) = \\det \\left ( \\psi _ 1 ^ + ( 0 ; k ) , \\psi _ 1 ^ - ( 0 ; k ) \\right ) , \\end{align*}"}
+{"id": "2213.png", "formula": "\\begin{align*} \\mathbf { v } ^ { 0 } = [ 1 , 1 , 1 , 1 , \\cdots , - 1 , - 1 , \\cdots , - 1 ] ^ { T } . \\end{align*}"}
+{"id": "3626.png", "formula": "\\begin{align*} \\tilde { h } _ { i j } = h _ { i j } - B _ { i j } . \\end{align*}"}
+{"id": "9050.png", "formula": "\\begin{align*} \\tilde { \\rho } ( X _ { \\alpha } ) = \\sum _ { k = 1 } ^ L \\frac { \\partial } { \\partial z _ { s , k } } r _ { t , k } ( 1 \\pm \\alpha _ { t , k } ) . \\end{align*}"}
+{"id": "8249.png", "formula": "\\begin{align*} \\Psi ^ { n } _ 1 ( \\psi ) & : = - V _ { 1 , 1 } \\psi _ 1 ^ 2 - \\psi _ 1 \\sum _ { j = 1 } ^ { n - 1 } V _ { 1 , j } \\psi _ j \\\\ \\Psi ^ { n } _ i ( \\psi ) & : = \\psi _ { i - 1 } \\sum _ { j = 1 } ^ { i - 1 } j V _ { i - 1 , j } \\psi _ j - \\psi _ i \\sum _ { j = 1 } ^ { i } j V _ { i , j } \\psi _ j - \\psi _ i \\sum _ { j = i } ^ { n - 1 } V _ { i , j } \\psi _ j , \\\\ \\Psi ^ { n } _ n ( \\psi ) & : = \\psi _ { n - 1 } \\sum _ { j = 1 } ^ { n - 1 } j V _ { n - 1 , j } \\psi _ j . \\end{align*}"}
+{"id": "6997.png", "formula": "\\begin{align*} V & = \\Big ( Z ( T ' , M _ n ) + e _ 1 M _ n ( F ) e _ 3 + e _ 3 M _ n ( F ) e _ 2 \\Big ) \\cap \\Lambda ^ \\perp \\\\ & = \\left ( Z ( T ' , M _ n ) \\cap \\Lambda ^ \\perp \\right ) + e _ 1 M _ n ( F ) e _ 3 + e _ 3 M _ n ( F ) e _ 2 \\end{align*}"}
+{"id": "7842.png", "formula": "\\begin{align*} F ^ { q , \\dagger } _ v ( 0 , p ) [ u _ k ] = c _ 1 ( q , k , p ) u _ k . \\end{align*}"}
+{"id": "8882.png", "formula": "\\begin{align*} p _ { k , S } ( n , r ) = \\frac { 2 ^ { g / 2 } ( 2 \\pi D ) ^ \\ell } { \\Gamma ( \\ell ) ( \\det 2 S ) ^ { \\ell - 1 / 2 } } \\big ( 2 + O \\big ( \\frac { D ^ { g / 2 + \\epsilon } } { \\ell ^ { g / 2 + 1 / 3 } ( \\det 2 S ) ^ { g / 2 + 1 / 2 + \\epsilon } } \\big ) \\big ) . \\end{align*}"}
+{"id": "915.png", "formula": "\\begin{align*} K _ { v , m } ( t ) = \\sum \\limits _ { \\eta _ 1 ( v ) \\leq u \\leq t \\atop { ( u , v ) = 1 } } e \\left ( \\frac { m \\overline { u } _ { | v | } } { v } \\right ) \\ , . \\end{align*}"}
+{"id": "4378.png", "formula": "\\begin{align*} & \\int _ { X _ j } | u _ { m , m ' , \\epsilon , j } | ^ 2 _ { h _ { m ' } } e ^ { v _ \\epsilon ( \\Psi _ m ) - \\delta M _ { \\eta _ m } } c ( - v _ \\epsilon ( \\Psi _ m ) ) + \\int _ { X _ j } | h _ { m , m ' , \\epsilon , j } | ^ 2 _ { h _ { m ' } } e ^ { - \\phi - \\delta M _ { \\eta _ m } } \\\\ \\le & \\int _ { X _ j } v '' _ { \\epsilon } ( \\Psi _ m ) | f F ^ { 1 + \\delta } | ^ 2 _ { h _ { m ' } } e ^ { - \\phi - \\delta M _ { \\eta _ m } } < + \\infty . \\end{align*}"}
+{"id": "352.png", "formula": "\\begin{align*} A : = \\frac { 9 ( n - 1 ) } { ( n + 4 ) ^ 2 } \\left [ \\begin{array} { c c c c c c c c c } 1 & \\frac { ( n - 2 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } ' & \\frac { - ( n + 1 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } ' \\\\ \\frac { ( n - 2 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } & \\frac { ( n - 2 ) ^ 2 } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } & \\frac { - ( n - 2 ) ( n + 1 ) } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } \\\\ \\frac { - ( n + 1 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } & \\frac { - ( n - 2 ) ( n + 1 ) } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } & \\frac { ( n + 1 ) ^ 2 } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } \\end{array} \\right ] , \\end{align*}"}
+{"id": "7599.png", "formula": "\\begin{align*} ( \\phi x ) _ \\epsilon = x _ { \\phi ( \\epsilon ) } \\end{align*}"}
+{"id": "3997.png", "formula": "\\begin{align*} \\frac { u ^ k - u ^ { k - 1 } } { \\tau _ k } + \\mathcal { L } u ^ k = F ^ k , \\end{align*}"}
+{"id": "3529.png", "formula": "\\begin{align*} \\mathcal { C } _ n ^ { ( k ) } \\ = \\ A _ k F _ n + B _ k F _ { n + 1 } - \\sum _ { j = 0 } ^ k \\binom { k } { j } B _ j n ^ { k - j } . \\end{align*}"}
+{"id": "3920.png", "formula": "\\begin{align*} \\hat { q } _ i = q ( z _ i ) + \\frac { \\hat { q } _ i } { \\ln \\frac { R } { \\varepsilon } } g _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ i ) - \\Sigma _ { j \\neq i } \\frac { \\hat { q } _ j } { \\ln \\frac { R } { \\varepsilon } } \\bar { G } _ { z _ j } ( T _ { z _ j } z _ i , T _ { z _ j } z _ j ) , \\end{align*}"}
+{"id": "8326.png", "formula": "\\begin{align*} & \\psi _ x + i k ^ 2 [ \\sigma _ 3 , \\psi ] = k P _ x \\psi , \\\\ & \\psi _ t + i \\eta ^ 2 [ \\sigma _ 3 , \\psi ] = H \\psi . \\end{align*}"}
+{"id": "8747.png", "formula": "\\begin{align*} \\langle z , w \\rangle _ { \\mathbb C } : = \\frac { 1 } { 2 } \\left ( \\overline z w + \\overline w z \\right ) = \\frac { 1 } { 2 } \\left ( z \\overline w + w \\overline z \\right ) . \\end{align*}"}
+{"id": "8357.png", "formula": "\\begin{align*} J ( x ; k ) = \\begin{cases} \\begin{pmatrix} | r ( k ) | ^ 2 & \\overline { r ( k ) } \\mathrm { e } ^ { - 2 i k ^ 2 x } \\\\ r ( k ) e ^ { 2 i k ^ 2 x } & 0 \\end{pmatrix} , k \\in \\mathbb { R } , \\\\ \\\\ \\begin{pmatrix} - | r ( k ) | ^ 2 & - \\overline { r ( k ) } e ^ { - 2 i k ^ 2 x } \\\\ r ( k ) e ^ { 2 i k ^ 2 x } & 0 \\end{pmatrix} , k \\in i \\mathbb { R } . \\end{cases} \\end{align*}"}
+{"id": "8229.png", "formula": "\\begin{align*} \\frac { 2 \\pi } { N } = \\int _ { \\alpha _ k } ^ { \\beta _ k } \\frac { s b ' ( s ) } { b ( s ) } d m ( s ) \\geq | b ' ( s _ k ) | | \\beta _ k - \\alpha _ k | \\ , k = 1 , \\ldots , N . \\end{align*}"}
+{"id": "5984.png", "formula": "\\begin{align*} f _ { j , \\theta } ( x ' , x _ 3 ) = \\int _ { I _ { K ^ { - 1 } } } \\int _ { \\pi _ 3 ( \\theta ) } \\widehat { f } _ { j , \\theta } ( \\nu ' , \\nu _ 3 ) e ^ { 2 \\pi i x ' \\cdot \\xi ' } e ^ { 2 \\pi i x _ 3 \\xi _ 3 } d \\nu ' d \\nu _ 3 . \\end{align*}"}
+{"id": "2687.png", "formula": "\\begin{align*} a _ d ( 0 ) & = 1 , \\\\ a _ d ( 1 ) & = d , \\\\ a _ d ( 2 ) & = d ^ 2 + d , \\\\ a _ d ( 3 ) & = d ^ 3 + \\frac { 3 } { 2 } d ^ 2 + \\frac { 1 } { 2 } d , \\\\ a _ d ( 4 ) & = d ^ 4 + 2 d ^ 3 + 2 d ^ 2 + d , \\\\ a _ d ( 5 ) & = d ^ 5 + \\frac { 5 } { 2 } d ^ 4 + \\frac { 7 } { 2 } d ^ 3 + 2 d ^ 2 , \\\\ a _ d ( 6 ) & = d ^ 6 + 3 d ^ 5 + \\frac { 2 1 } { 4 } d ^ 4 + \\frac { 9 } { 2 } d ^ 3 + \\frac { 9 } { 4 } d ^ 2 + d . \\end{align*}"}
+{"id": "1570.png", "formula": "\\begin{align*} \\alpha _ 2 : = \\underset { x \\in K _ 2 } { \\inf } d ^ \\mathcal Z ( x , \\partial \\mathcal Z ) \\beta _ 2 : = \\underset { x \\in K _ 2 } { \\sup } \\ ; d ^ \\mathcal Z ( x , \\partial \\mathcal Z ) . \\end{align*}"}
+{"id": "4003.png", "formula": "\\begin{align*} u ^ k = \\arg \\min _ { w \\in H ^ 1 ( \\Omega ) } \\Big ( \\frac { 1 } { 2 } \\| w - u ^ { k - 1 } \\| _ { L _ 2 ( \\Omega ) } ^ 2 + \\tau _ k N ( w ) \\Big ) , \\end{align*}"}
+{"id": "1249.png", "formula": "\\begin{align*} \\frac { d ( \\mu \\circ G _ { \\xi _ { \\Delta } } ^ { - 1 } ) } { d \\mu } ( \\eta ) = \\mathbf { 1 } _ { [ \\xi _ { \\Delta } ] } ( \\eta ) \\sum _ { \\zeta _ { \\Delta } } \\frac { \\gamma _ { \\Delta } ( \\zeta _ { \\Delta } | \\eta _ { \\Delta ^ c } ) } { \\gamma _ { \\Delta } ( \\xi _ { \\Delta } | \\eta _ { \\Delta ^ c } ) } , \\eta \\in \\Omega . \\end{align*}"}
+{"id": "550.png", "formula": "\\begin{align*} \\d ( x , y ) = \\left \\{ \\begin{array} { l l } \\mbox { $ 0 $ ( i . e . $ x = y $ ) } & \\mbox { i f $ | D | = 1 $ } \\\\ \\frac { n } { 2 } & \\mbox { i f $ | D | = 2 $ , } \\end{array} \\right . \\end{align*}"}
+{"id": "4448.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\\\ r _ { Q } ( n ) = 0 } } n \\ll \\max \\left \\{ \\frac { p ^ { 3 + \\epsilon } } { ( \\min Q ^ { * } ) ^ { 2 } } , p ^ { 5 / 2 + \\epsilon } \\right \\} . \\end{align*}"}
+{"id": "7928.png", "formula": "\\begin{align*} ( A ^ q \\eta ) ( x ) = \\int _ \\S W _ r ( x - y ) ( \\eta ( y ) - \\eta ( x ) ) \\cos ( 2 \\pi q ( y - x ) ) \\ \\d y - \\int _ \\S W _ r ( y ) \\eta ( y ) \\cos ( 2 \\pi q y ) ) \\ \\d y . \\end{align*}"}
+{"id": "2277.png", "formula": "\\begin{align*} \\widehat { a } ( Z ) = \\widehat { a } \\big ( ( Z ^ * Z ) ^ \\frac { 1 } { 2 } \\big ) \\end{align*}"}
+{"id": "2726.png", "formula": "\\begin{align*} { \\bf S } ^ { - 1 } = \\frac { 1 } { n + 1 } \\ , { \\bf S } ^ { T } . \\end{align*}"}
+{"id": "7768.png", "formula": "\\begin{align*} z : = x , ~ k = 2 ^ h , ~ { \\rm a n d } ~ f _ k ( z ) : = p _ h ( x ) , \\end{align*}"}
+{"id": "7291.png", "formula": "\\begin{align*} x ( t , s ) : = t + s A ( t ) \\nabla g ( t ) , \\ \\ \\ A ( t ) = ( \\sqrt { 1 + \\| \\nabla g ( t ) \\| ^ 2 } ) ^ { - 1 } , \\end{align*}"}
+{"id": "4130.png", "formula": "\\begin{align*} \\| u \\| _ { \\mathcal { C } ( \\R ^ n _ + ) } : = \\sup _ { Q \\subset \\R ^ { n - 1 } } \\bigg ( \\frac { 1 } { | Q | } \\iint _ { T _ Q } | \\nabla u ( x ) | ^ 2 \\ , t \\ , d x ' d t \\bigg ) ^ { \\frac 1 2 } , \\end{align*}"}
+{"id": "4506.png", "formula": "\\begin{align*} \\abs { c _ 1 ( S ^ 1 \\setminus S _ i ) } \\le \\abs { c ( S \\setminus S _ i ) } + \\sum _ { h \\in [ \\chi ] \\setminus \\{ i \\} } \\abs { c _ 1 ( P _ i ' ) } \\le \\abs { S \\setminus S _ i } + \\sum _ { h \\in [ \\chi ] \\setminus \\{ i \\} } q _ h = \\abs { S } - \\abs { S \\cap S _ i } + q - q _ i . \\end{align*}"}
+{"id": "5622.png", "formula": "\\begin{align*} \\hat { \\mu } _ { A ^ * } = \\hat { \\mu } _ A \\ . \\end{align*}"}
+{"id": "6186.png", "formula": "\\begin{align*} \\xi ' = - L - 2 , \\eta ' = \\frac { Q } { 2 ( L + 2 ) } , \\zeta ' = \\frac { \\kappa ( L + 2 ) B _ 1 } { Q } , \\end{align*}"}
+{"id": "2314.png", "formula": "\\begin{align*} g ( \\left ( D ^ W _ Z J \\right ) X , J Y ) = - 2 g ( N ( X , Y ) , Z ) , \\end{align*}"}
+{"id": "4472.png", "formula": "\\begin{align*} \\frac { p ^ { 3 } } { 1 6 } = a _ { 1 } ^ { * } a _ { 2 } ^ { * } a _ { 3 } ^ { * } a _ { 4 } ^ { * } \\leq \\left ( \\frac { 4 } { 3 } \\right ) ^ { 6 } ( a _ { 1 } ^ { * } ) ^ { 4 } \\end{align*}"}
+{"id": "7345.png", "formula": "\\begin{align*} \\mu _ { n } = \\left ( \\begin{array} { c c c c c c c c c } 0 & 1 \\\\ 1 & 0 & 1 \\\\ & 1 & - h & 1 & 0 & & 1 \\\\ & & 1 & - h & 1 \\\\ & & 0 & 1 & \\ddots & 1 \\\\ & & & & 1 & - h & 0 \\\\ & & 1 & & & 0 & - h & 1 \\\\ & & & & & & 1 & \\ddots & 1 \\\\ & & & & & & & 1 & - h \\end{array} \\right ) \\in \\mathrm { M } _ { 2 n + 1 , 2 n + 1 } ( \\mathrm { G W } ( k ) ) \\end{align*}"}
+{"id": "3028.png", "formula": "\\begin{align*} \\left \\langle \\left ( x , \\xi \\right ) , \\left ( y , \\eta \\right ) \\right \\rangle = \\tfrac { 1 } { 2 } \\left ( \\left \\langle x , \\eta \\right \\rangle + \\left \\langle y , \\xi \\right \\rangle \\right ) \\end{align*}"}
+{"id": "174.png", "formula": "\\begin{align*} R S _ { H ' } ( F ) = S _ { H ' } ( F ) = \\varinjlim _ n S _ { H ' , n } ( F ) , \\end{align*}"}
+{"id": "3075.png", "formula": "\\begin{align*} f = 4 x _ 1 ^ 3 - x _ 0 x _ 1 ^ 2 - 1 8 x _ 0 x _ 1 x _ 2 + 2 7 x _ 0 x _ 2 ^ 2 + 4 x _ 0 ^ 2 x _ 2 . \\end{align*}"}
+{"id": "3948.png", "formula": "\\begin{align*} \\langle - \\Delta z + \\zeta z , p \\rangle = \\langle \\nabla z , \\nabla p \\rangle + \\langle \\zeta z , p \\rangle = \\langle - \\Delta p , z \\rangle + \\langle \\zeta p , z \\rangle = \\langle \\overline { y } - g , z \\rangle , \\ ; \\ ; z \\in Y . \\end{align*}"}
+{"id": "8798.png", "formula": "\\begin{align*} \\Pi _ V J _ { z / | z | } ^ \\perp x = \\lambda \\Pi _ V x \\forall x \\in \\C ^ n . \\end{align*}"}
+{"id": "3288.png", "formula": "\\begin{align*} \\mathbb E \\left [ B _ i ( t ) B _ j ( s ) \\right ] = \\frac { \\rho _ { i , j } } 2 ( | s | ^ { \\mathfrak H _ { i , j } } + | t | ^ { \\mathfrak H _ { i , j } } - | t - s | ^ { \\mathfrak H _ { i , j } } ) , \\end{align*}"}
+{"id": "2236.png", "formula": "\\begin{align*} K _ { \\beta , f } : = \\{ ( | \\cdot | ^ { - \\beta } \\chi _ { ( 0 , 1 ] } ) \\ast f \\} , \\end{align*}"}
+{"id": "5899.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\langle K _ t ( R ) u _ n , u _ n - v \\rangle = \\langle K _ t ( R ) u , u - v \\rangle , \\end{align*}"}
+{"id": "4408.png", "formula": "\\begin{align*} \\theta _ 0 ( x ) = \\frac { 1 } { 2 \\pi i } \\int _ { 2 - i \\infty } ^ { 2 + i \\infty } \\frac { x ^ s } { s } \\bigg \\{ \\sum _ p \\frac { \\log p } { p ^ s } \\bigg \\} d s . \\end{align*}"}
+{"id": "1437.png", "formula": "\\begin{align*} \\det ( e ^ { x _ i z _ j / \\sqrt { n } } ) _ { i , j = 1 } ^ d & \\sim C n ^ { - d ( d - 1 ) / 2 - d / 2 } \\Delta ( x ) \\Delta ( z ) . \\end{align*}"}
+{"id": "1660.png", "formula": "\\begin{align*} \\chi ( a ) = \\prod _ \\tau \\tau ( a ) ^ w \\end{align*}"}
+{"id": "6041.png", "formula": "\\begin{align*} J _ { 1 } ( t _ { 0 } ^ { \\alpha } u ( t _ { 0 } \\cdot ) ) & < \\mathop { \\lim } \\limits _ { n \\rightarrow \\infty } \\Big ( \\frac { 1 } { 2 } \\| \\nabla \\overline { u } _ { k } \\| _ { 2 } ^ { 2 } + \\frac { 1 } { 2 } \\| \\overline { u } _ { k } \\| _ { 2 } ^ { 2 } + \\frac { 1 } { 2 } B ( \\overline { u } _ { k } ) - \\frac { 1 } { 2 p } \\| \\overline { u } _ { k } \\| _ { 2 p } ^ { 2 p } \\Big ) \\\\ & = \\widetilde { E } _ { 1 } , \\end{align*}"}
+{"id": "756.png", "formula": "\\begin{align*} \\gamma _ \\Theta ^ { 1 / 2 } ( E ) = \\sup | \\langle \\nu , 1 \\rangle | , \\end{align*}"}
+{"id": "7936.png", "formula": "\\begin{align*} h _ 1 ( x ) = \\prod _ { i = 1 } ^ n ( \\eta _ i ( x + r ) - \\eta _ i ( x ) ) \\sin ^ { [ n ] } ( \\tilde \\Psi ( x + r ) - \\tilde \\Psi ( x ) ) \\\\ h _ 2 ( x ) = - \\prod _ { i = 1 } ^ n ( \\eta _ i ( x - r ) - \\eta _ i ( x ) ) \\sin ^ { [ n ] } ( \\tilde \\Psi ( x - r ) - \\tilde \\Psi ( x ) ) \\end{align*}"}
+{"id": "4507.png", "formula": "\\begin{align*} k t ' = k ( t - p ) > \\sum _ { i \\in [ \\chi ] } x _ i . \\end{align*}"}
+{"id": "2955.png", "formula": "\\begin{align*} \\begin{aligned} T _ 1 & : = T ^ e _ A ( x ) \\\\ T _ 2 & : = T ^ r _ { \\bar A } ( x ) . \\end{aligned} \\end{align*}"}
+{"id": "7524.png", "formula": "\\begin{align*} b ( n , d , k ) = \\max \\{ \\vartheta ( G ) : \\ : \\} \\end{align*}"}
+{"id": "3988.png", "formula": "\\begin{align*} \\alpha _ i + \\bar { \\alpha } _ j = 1 . \\end{align*}"}
+{"id": "4845.png", "formula": "\\begin{align*} ( r + \\nu ) V ( s , i ) = H \\left ( s , i , p , q \\right ) . \\end{align*}"}
+{"id": "2835.png", "formula": "\\begin{align*} \\| \\nabla u \\| _ 6 \\le C \\| \\nabla u \\| ^ { 1 - \\kappa } \\| \\Lambda ^ { s + 1 } u \\| ^ \\kappa = C \\| \\nabla u \\| ^ { \\frac { s - 1 } { s } } \\| \\Lambda ^ { s + 1 } u \\| ^ { \\frac { 1 } { s } } . \\end{align*}"}
+{"id": "1676.png", "formula": "\\begin{align*} \\pi = \\pi ^ \\infty \\otimes \\pi _ \\infty \\end{align*}"}
+{"id": "1199.png", "formula": "\\begin{align*} f ( 2 x + y ) + f \\left ( \\frac { x + y } { 2 } \\right ) = \\frac { 2 f ( x ) f ( y ) } { f ( x ) + f ( y ) } + \\frac { 2 f ( x + y ) f ( y - x ) } { 3 f ( y - x ) - f ( x + y ) } \\end{align*}"}
+{"id": "7472.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l l } - L \\varphi & \\geq & f ' ( u ) \\varphi & \\Omega , \\\\ \\varphi & > & 0 & \\Omega , \\\\ \\varphi & = & 0 & \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "2371.png", "formula": "\\begin{align*} C _ p ( \\mathcal { H } _ { \\ast } ) = B _ p ( \\mathcal { H } _ { \\ast } ) \\oplus s _ { _ { p } } ( B _ { p - 1 } ( \\mathcal { H } _ { \\ast } ) ) . \\end{align*}"}
+{"id": "456.png", "formula": "\\begin{align*} M _ n ( \\theta ) = n ^ { - 1 } \\sum _ { i = 1 } ^ n \\log f ( X _ i ; \\theta ) . \\end{align*}"}
+{"id": "5230.png", "formula": "\\begin{align*} \\varphi ( y ) = \\frac { 1 } { 8 \\pi ^ { 2 } } \\int _ { \\{ \\varphi > W / 2 \\} } \\frac { \\mu ^ { 2 } } { | y - w | ^ { 3 } } \\left ( \\varphi - \\frac { W } { 2 } \\right ) _ { + } \\dd w . \\end{align*}"}
+{"id": "7562.png", "formula": "\\begin{align*} x = \\sum _ { m = 0 } ^ \\infty \\sum _ { i = 0 } ^ { p - 1 } \\frac { a _ { m p + i } } { ( \\beta _ 0 \\cdots \\beta _ { p - 1 } ) ^ m \\beta _ 0 \\cdots \\beta _ i } . \\end{align*}"}
+{"id": "1252.png", "formula": "\\begin{align*} \\lim _ { \\Lambda \\uparrow \\Z ^ d } \\inf _ { \\xi : \\xi _ { \\Lambda } = \\eta _ { \\Lambda } } f ( \\xi ) = f ( \\eta ) , \\lim _ { \\Lambda \\uparrow \\Z ^ d } \\sup _ { \\xi : \\xi _ { \\Lambda } = \\eta _ { \\Lambda } } f ( \\xi ) = f ( \\eta ) . \\end{align*}"}
+{"id": "3457.png", "formula": "\\begin{align*} g _ { k , \\ell } : = \\begin{cases} \\max ( e ^ { \\delta _ n q _ n } , | \\ell | , 1 ) e ^ { - ( \\beta _ n - 6 \\varepsilon ) q _ n } \\ & \\beta _ n \\geq \\delta _ n + 2 0 0 \\varepsilon \\\\ e ^ { 2 \\varepsilon k } & \\beta _ n < \\delta _ n + 2 0 0 \\varepsilon \\end{cases} \\end{align*}"}
+{"id": "6114.png", "formula": "\\begin{align*} \\varphi ^ { + + } _ { n , p } = \\varphi ^ { - - } _ { n , p } = : \\varphi ^ { D } _ { n , p } , \\varphi ^ { + - } _ { n , p } = \\varphi ^ { - + } _ { n , p } = : \\varphi ^ { A } _ { n , p } , \\varphi ^ { 0 + } _ { n , p } = \\varphi ^ { 0 - } _ { n , p } = : \\varphi ^ { V } _ { n , p } , \\varphi ^ { + 0 } _ { n , p } = \\varphi ^ { - 0 } _ { n , p } = : \\varphi ^ { H } _ { n , p } . \\end{align*}"}
+{"id": "8757.png", "formula": "\\begin{align*} f ( z ) = a _ 0 + \\left ( x - \\frac { \\psi _ 0 } { \\psi _ 1 } y \\right ) a _ 1 + \\left ( x - \\frac { \\psi _ 0 } { \\psi _ 1 } y \\right ) ^ 2 h _ 2 ( z ) , \\\\ \\end{align*}"}
+{"id": "9123.png", "formula": "\\begin{align*} ( n - ( j + 2 ) ) r < \\sum \\limits _ { i = j + 2 } ^ n \\chi _ i , \\end{align*}"}
+{"id": "5857.png", "formula": "\\begin{align*} { f _ { 1 7 } } = { f _ { 1 8 } } - \\frac { 1 } { 2 } \\left ( { { f _ 3 } - { f _ 4 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) + \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) - \\frac { 1 } { 8 } \\left ( { 2 { F _ y } - { F _ z } } \\right ) - \\frac { 1 } { 6 } \\rho { u _ z } . \\end{align*}"}
+{"id": "8624.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { m = 1 } ^ n \\mathbb { E } _ { n , m } [ \\| \\chi _ { n , m } \\| _ { L ^ 2 ( \\mu ) } ^ 2 : \\| \\chi _ { n , m } \\| _ { L ^ 2 ( \\mu ) } > \\varepsilon ] = 0 \\end{align*}"}
+{"id": "4519.png", "formula": "\\begin{align*} X _ i \\left ( t , x _ { 0 } , t _ { 0 } \\right ) : = \\lambda _ i ( t - t _ { 0 } ) + x _ { 0 } , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "8908.png", "formula": "\\begin{align*} \\mathbf { h } = \\lambda _ { 1 } \\mathbf { e } _ 1 + \\lambda _ { 2 } \\mathbf { e } _ 2 + \\lambda _ { 3 } \\mathbf { e } _ 3 + \\lambda _ { 1 2 } \\mathbf { e } _ { 1 2 } + \\lambda _ { 1 2 3 ' } \\mathbf { e } _ { 1 2 3 ' } , \\end{align*}"}
+{"id": "5360.png", "formula": "\\begin{align*} b - b _ 1 = \\sum _ { i = 2 } ^ r b _ i = l ( p - q ) + ( p - q r ) + 1 . \\end{align*}"}
+{"id": "530.png", "formula": "\\begin{align*} K _ { t } ( x , y ) = \\int _ { 0 } ^ { 1 } \\lambda ^ { - 1 / 2 } K _ { t } ^ { * } ( x , y , \\lambda ) d \\lambda . \\end{align*}"}
+{"id": "5697.png", "formula": "\\begin{align*} \\deg ( F ) = \\max \\{ i \\ : \\ A _ i \\ne 0 \\} \\end{align*}"}
+{"id": "297.png", "formula": "\\begin{align*} d _ H ( D , E ) = \\max \\left \\{ \\sup _ { p \\in D } d _ H ( p , E ) , \\ \\sup _ { p \\in E } d _ H ( p , D ) \\right \\} . \\end{align*}"}
+{"id": "3113.png", "formula": "\\begin{align*} J ( x , y , [ x , z ] ) = [ J ( x , y , z ) , x ] , \\end{align*}"}
+{"id": "26.png", "formula": "\\begin{align*} P _ A ( \\lambda ) : = \\pi _ A ( P ( \\lambda ) ) S _ A ( \\lambda ) = \\pi _ A ( S ( \\lambda ) ) . \\end{align*}"}
+{"id": "6982.png", "formula": "\\begin{align*} ( \\theta _ 1 , \\theta _ 2 ) = ( a ^ x \\alpha ^ y , b ^ x \\beta ^ y ) \\end{align*}"}
+{"id": "1350.png", "formula": "\\begin{align*} t _ { 2 K - 1 } = \\min \\bigl \\{ t \\colon d ( \\gamma _ { a _ 1 , b _ 1 } ( t ) , b _ 1 ) = R \\bigr \\} , \\end{align*}"}
+{"id": "7830.png", "formula": "\\begin{align*} F ^ { q , \\dagger } ( 0 , p ) = 0 \\end{align*}"}
+{"id": "5308.png", "formula": "\\begin{align*} \\widetilde { f } ( \\lambda , b ) = \\int _ { G / K } f ( x ) e ^ { ( - i \\lambda + \\rho ) A ( x , b ) } d x \\end{align*}"}
+{"id": "4670.png", "formula": "\\begin{align*} \\forall \\theta > 0 \\ \\Rightarrow \\sup _ { p \\ge 1 } \\left [ \\ \\frac { L ( p ^ { \\theta } ) } { L ( p ) } \\ \\right ] = C ( \\theta ) < \\infty . \\end{align*}"}
+{"id": "4818.png", "formula": "\\begin{align*} F _ n ( \\eta ) = T _ n - \\frac { 1 } { \\eta } I _ n , \\eta \\in \\Omega , \\end{align*}"}
+{"id": "8340.png", "formula": "\\begin{align*} \\varphi ( x , t ; k ) = I + k \\begin{pmatrix} 0 & u \\\\ v & 0 \\end{pmatrix} + \\mathcal { O } \\left ( k ^ 2 \\right ) , k \\rightarrow 0 , \\end{align*}"}
+{"id": "2811.png", "formula": "\\begin{align*} \\mathbf { h } _ { k l } = \\beta _ { k l , 0 } \\mathbf { a } ( \\theta _ { k l , 0 } ) + \\sum _ { p = 1 } ^ { P } \\beta _ { k l , p } \\mathbf { a } ( \\theta _ { k l , p } ) , \\end{align*}"}
+{"id": "1477.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\int _ { a } ^ { b } | u _ { n } ' ( x ) - w _ { n } ' ( x ) | \\ , d x = 0 , \\end{align*}"}
+{"id": "7699.png", "formula": "\\begin{align*} B = \\begin{bmatrix} F & M \\\\ W & Q \\end{bmatrix} , \\end{align*}"}
+{"id": "8649.png", "formula": "\\begin{align*} y \\left [ n \\right ] = \\left ( { \\sum \\limits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { \\bf { H } } _ l ^ \\bot { { \\bf { b } } _ l } } } \\right ) s \\left [ { n - { n _ { \\max } } } \\right ] + z \\left [ n \\right ] . \\end{align*}"}
+{"id": "1943.png", "formula": "\\begin{align*} F _ { 1 } ( b _ { 1 } , b _ { 2 } , b _ { 3 } ; c _ { 1 } ; x , y ) = \\frac { \\Gamma ( c _ { 1 } ) } { \\Gamma ( b _ { 1 } ) \\Gamma ( c _ { 1 } - b _ { 1 } ) } \\int _ { 0 } ^ { 1 } t ^ { b _ { 1 } - 1 } ( 1 - t ) ^ { c _ { 1 } - b _ { 1 } - 1 } ( 1 - x t ) ^ { - b _ { 2 } } ( 1 - y t ) ^ { - b _ { 3 } } d t , \\end{align*}"}
+{"id": "4650.png", "formula": "\\begin{align*} & \\phi _ k ( s _ 1 , \\ldots , s _ n ) \\\\ : = & \\int _ { G ^ { \\times n } } \\phi ( t _ 1 ^ { - 1 } s _ 1 t _ 2 , t _ 2 ^ { - 1 } s _ 2 t _ 3 , \\ldots , t _ { n - 2 } ^ { - 1 } s _ { n - 2 } t _ { n - 1 } , t _ { n - 1 } ^ { - 1 } s _ { n - 1 } , s _ n t _ { n } ^ { - 1 } ) \\left ( \\prod _ { j = 1 } ^ { n } \\varphi _ k ( t _ j ) \\right ) d t _ 1 \\ldots d t _ n \\end{align*}"}
+{"id": "3277.png", "formula": "\\begin{align*} \\mathbb E [ Z f ( Y _ n ) ] \\to \\tilde { \\mathbb E } [ Z f ( Y ) ] = \\int _ { \\Omega } Z ( \\omega ) \\int _ { \\mathcal D ( [ 0 , T ] , H ) } f ( x ) K ( \\omega , d x ) \\mathbb P [ d \\omega ] , n \\to \\infty \\end{align*}"}
+{"id": "8051.png", "formula": "\\begin{align*} a _ c \\left [ t + 1 \\right ] & = a _ c \\left [ t \\right ] - \\mu \\frac { \\partial \\mathbb { E } \\left [ \\lvert \\varepsilon \\rvert ^ 2 | \\hat { \\mathbf { H } } ^ { } \\right ] } { \\partial a _ c } , \\\\ a _ i \\left [ t + 1 \\right ] & = a _ i \\left [ t \\right ] - \\mu \\frac { \\partial \\mathbb { E } \\left [ \\lvert \\varepsilon \\rvert ^ 2 | \\hat { \\mathbf { H } } ^ { } \\right ] } { \\partial a _ i } , \\end{align*}"}
+{"id": "7662.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\bar x _ t ^ { * , t _ 0 , \\xi } = & ~ [ A \\bar x _ t ^ { * , t _ 0 , \\xi } - B ^ 2 R ^ { - 1 } \\bar U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) - B h ( \\bar \\mu _ t ^ { * , t _ 0 , \\xi } ) + f ( \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) + b ( \\bar \\mu _ t ^ { * , t _ 0 , \\xi } ) ] d t \\\\ & + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ \\bar x _ { t _ 0 } ^ { * , t _ 0 , \\xi } = & ~ \\xi \\end{aligned} \\right . \\end{align*}"}
+{"id": "5742.png", "formula": "\\begin{align*} I _ 0 & = \\{ \\omega , \\omega ^ 2 , \\omega ^ 4 \\} , \\\\ I _ 1 & = \\{ \\omega ^ 3 , \\omega ^ 5 , \\omega ^ 6 \\} , \\\\ I _ 2 & = \\{ 0 \\} , \\textrm { a n d } \\\\ I _ 3 & = \\{ 1 \\} . \\end{align*}"}
+{"id": "1709.png", "formula": "\\begin{align*} | | u | | _ { X _ k ( T ) } & = | | u | | _ { L ^ 2 ( A _ 0 ) } + \\sup _ { j > 0 } | | \\langle \\xi \\rangle ^ { - \\frac { 1 } { 2 } } u | | _ { L ^ 2 ( A _ j ) } , \\ k \\ge 0 , \\\\ | | u | | _ { X _ k ( T ) } & = 2 ^ { \\frac { k } { 2 } } | | u | | _ { L ^ 2 ( A _ { < - k } ) } + \\sup _ { j \\ge - k } | | ( | \\xi | + 2 ^ { - k } ) ^ { - \\frac { 1 } { 2 } } u | | _ { L ^ 2 ( A _ j ) } , \\ k < 0 , \\end{align*}"}
+{"id": "645.png", "formula": "\\begin{align*} g \\in 1 \\ast f & \\mbox { i f f } g ( x ) \\in 1 ( x ) \\ast f ( x ) = 1 \\ast f ( x ) \\\\ & \\mbox { i f f } g ( x ) = f ( x ) \\\\ & \\mbox { i f f } f = g \\end{align*}"}
+{"id": "4798.png", "formula": "\\begin{align*} u _ n = T _ n f _ n . \\end{align*}"}
+{"id": "4037.png", "formula": "\\begin{align*} M _ I = \\bigoplus _ { J \\subset I } R _ J . \\end{align*}"}
+{"id": "3320.png", "formula": "\\begin{align*} | \\tau | = | \\tau ' | . \\end{align*}"}
+{"id": "7451.png", "formula": "\\begin{align*} X X ^ \\top = l \\ , V ^ \\top V \\mbox { a n d } X ^ \\top X = k \\ , W ^ \\top W . \\end{align*}"}
+{"id": "7028.png", "formula": "\\begin{align*} u \\ , = \\ , \\tfrac { x ^ 2 } { 2 } + \\tfrac { x ^ 2 y } { 2 } + \\tfrac { x ^ 2 y ^ 2 } { 2 } + 0 + 0 + \\tfrac { x ^ 2 y ^ 3 } { 2 } + { \\rm O } _ { x , y } ( 6 ) , \\end{align*}"}
+{"id": "2913.png", "formula": "\\begin{align*} \\lim _ { m \\to + \\infty } \\Big ( 1 - \\Theta _ m ( c , c ' ) \\Big ) = 1 \\end{align*}"}
+{"id": "9135.png", "formula": "\\begin{align*} \\varrho ( x , z ) = ( z + 1 ) f ( x ) - z b ( x ) \\end{align*}"}
+{"id": "5907.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow 0 } g _ { n } ^ { + } ( t , \\omega ) = 0 , ( t , \\omega ) . \\end{align*}"}
+{"id": "5689.png", "formula": "\\begin{align*} & \\gamma \\beta ( \\gamma ) = \\frac { \\gamma _ { t - 1 } ( 1 - \\kappa _ t ) } { ( 1 - \\kappa _ { t - 1 } ) } \\left ( 1 + \\frac { 2 \\mu \\gamma _ { t - 1 } } { 1 - \\kappa _ { t - 1 } } \\right ) ^ { - 1 } \\leq \\frac { 3 } { 2 } \\gamma _ 0 , \\\\ & \\alpha ( \\gamma ) = \\frac { \\kappa _ { t - 1 } \\gamma \\beta ( \\gamma ) } { \\gamma _ { t - 1 } } = \\frac { \\kappa _ { t - 1 } ( 1 - \\kappa _ { t } ) } { 1 - \\kappa _ { t - 1 } } \\left ( 1 + \\frac { 2 \\mu \\gamma _ { t - 1 } } { 1 - \\kappa _ { t - 1 } } \\right ) ^ { - 1 } \\leq \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "4516.png", "formula": "\\begin{align*} \\hat U ( \\xi , t ) = \\exp ( E ( \\xi ) t ) \\hat w _ 0 ( \\xi ) . \\end{align*}"}
+{"id": "816.png", "formula": "\\begin{align*} V = g ( c ) H , \\end{align*}"}
+{"id": "2642.png", "formula": "\\begin{align*} [ x , y ] = x \\diamond y - \\varepsilon ( x , y ) y \\diamond x , \\quad \\forall x , y \\in \\mathcal { H } ( A ) . \\end{align*}"}
+{"id": "846.png", "formula": "\\begin{align*} c ( t ) = - w ( t ) = 0 M t \\in ( 0 , t _ 0 ] \\phantom { x } \\phantom { x } c ( t ) = - w ( t ) > 0 M t \\in ( t _ 0 , T ] . \\end{align*}"}
+{"id": "5091.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\phi | ^ { 2 } \\dd x + 2 \\mu \\int _ { 0 } ^ { t } \\int _ { \\Omega } | \\nabla \\phi | ^ { 2 } \\dd x \\dd s = \\int _ { \\Omega } | \\phi _ 0 | ^ { 2 } \\dd x , \\end{align*}"}
+{"id": "5103.png", "formula": "\\begin{align*} \\begin{aligned} & \\eta ( z , r ) = \\int _ { \\mathbb { R } ^ { 2 } _ { + } } { \\mathcal { G } } ( z , r , z ' , r ' ) \\frac { G ( z ' , r ' ) } { r ' } \\dd z ' \\dd r ' , \\\\ & { \\mathcal { G } } ( z , r , z ' , r ' ) = \\frac { r r ' } { 2 \\pi } \\int _ { 0 } ^ { \\pi } \\frac { \\cos \\theta \\dd \\theta } { \\sqrt { | z - z ' | ^ { 2 } + r ^ { 2 } + r '^ { 2 } - 2 r r ' \\cos \\theta } } . \\end{aligned} \\end{align*}"}
+{"id": "5335.png", "formula": "\\begin{align*} P _ { n , k } ( z ) = z ^ k A _ n ( z ) . \\end{align*}"}
+{"id": "3153.png", "formula": "\\begin{align*} & \\sum _ { y \\in \\mathbb { F } _ q } \\chi _ 4 ( y ) \\overline { \\chi _ 4 } ( a - y ) = \\sum _ { y ' \\in \\mathbb { F } _ q } \\chi _ 4 ( a y ' ) \\overline { \\chi _ 4 } ( a - a y ' ) \\\\ & = \\sum _ { y ' \\in \\mathbb { F } _ q } \\chi _ 4 ( y ' ) \\overline { \\chi _ 4 } ( 1 - y ' ) \\\\ & = \\sum _ { y ' \\in \\mathbb { F } _ q } \\chi _ 4 \\left ( y ' ( 1 - y ' ) ^ { - 1 } \\right ) \\\\ & = \\sum _ { y '' \\in \\mathbb { F } _ q , y '' \\neq - 1 } \\chi _ 4 ( y '' ) = - 1 , \\end{align*}"}
+{"id": "6539.png", "formula": "\\begin{align*} \\beta ( a + 1 ) = \\begin{cases} 1 , & a + 1 \\in \\mathbb { P } , \\\\ 0 , & a + 1 \\not \\in \\mathbb { P } . \\end{cases} \\end{align*}"}
+{"id": "3062.png", "formula": "\\begin{align*} \\rho _ { E _ 1 } = \\sum _ { Z \\in \\mathcal I } m _ Z \\rho _ { Z } \\end{align*}"}
+{"id": "5909.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ T | g _ n ( t ) | d t = 0 . \\end{align*}"}
+{"id": "2338.png", "formula": "\\begin{align*} \\left ( d \\theta \\right ) ^ { ( 2 , 0 ) + ( 0 , 2 ) } _ { J \\cdot , \\cdot } = g \\left ( J \\left ( \\mathcal { L } _ { J \\theta ^ \\sharp } J \\right ) ^ { a n t i - s y m } \\cdot , \\cdot \\right ) . \\end{align*}"}
+{"id": "1587.png", "formula": "\\begin{align*} G : = D \\rtimes \\langle ( A _ 1 , ( \\alpha , 1 ) ) , ( A _ 2 , ( 1 , \\sigma ( \\alpha ) ) ) \\rangle \\end{align*}"}
+{"id": "8687.png", "formula": "\\begin{align*} \\Theta _ { { \\mathrm { E u c l } } } ^ { \\Sigma _ \\infty } ( 0 , 1 0 \\lambda K ) - \\Theta _ { { \\mathrm { E u c l } } } ^ { \\Sigma _ \\infty } ( 0 , 1 / ( 3 K ) ) = 0 . \\end{align*}"}
+{"id": "4875.png", "formula": "\\begin{align*} & = \\sup _ { ( x ' , y ' ) \\in X \\times \\mathbb R } \\{ \\psi ( x ' , y ) - \\frac { 1 } { 2 \\epsilon } \\left ( \\| x - x ' \\| ^ 2 - \\| x \\| ^ 2 \\right ) - \\frac { 1 } { 2 } \\left ( | y - y ' | ^ 2 - \\| y \\| ^ 2 \\right ) \\} \\\\ & = \\sup _ { ( x ' , y ' ) \\in X \\times \\mathbb R } \\{ \\psi ( x ' , y ) + \\frac { 1 } { \\epsilon } x \\cdot x ' + \\frac { 1 } { \\epsilon } y \\cdot y ' - \\frac { 1 } { 2 \\epsilon } \\| x ' \\| ^ 2 - \\frac { 1 } { 2 \\epsilon } \\| y ' \\| ^ 2 \\} \\end{align*}"}
+{"id": "1274.png", "formula": "\\begin{align*} \\log \\leq \\log ^ + : = \\max \\left \\{ 0 , \\log ( \\cdot ) \\right \\} \\end{align*}"}
+{"id": "3600.png", "formula": "\\begin{align*} a ^ i _ S ( T ) = \\begin{cases} 1 , & T \\cap F _ i = S \\\\ 0 , & \\end{cases} \\end{align*}"}
+{"id": "1858.png", "formula": "\\begin{align*} \\int _ { B _ \\rho ( x _ 0 ) } \\big ( \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) - 1 \\big ) \\ , d v = o ( \\rho ^ \\lambda ) \\rho \\rightarrow \\infty \\ , , \\end{align*}"}
+{"id": "2410.png", "formula": "\\begin{align*} d ( J ) \\leq d ( \\mathcal { G } ) = n ^ { 2 } . \\end{align*}"}
+{"id": "6330.png", "formula": "\\begin{align*} ( y - y ^ { q ^ 4 } ) x ^ { q ^ 3 } + ( x - x ^ { q ^ 4 } ) x ^ { 2 q ^ 3 } = ( y - y ^ { q ^ 2 } ) x ^ { q } + ( x - x ^ { q ^ 2 } ) x ^ { 2 q } + ( y - y ^ { q ^ 2 } ) ^ { q ^ 2 } x ^ { q ^ 3 } + ( x - x ^ { q ^ 2 } ) ^ { q ^ 2 } x ^ { 2 q ^ 3 } . \\end{align*}"}
+{"id": "3907.png", "formula": "\\begin{align*} \\begin{cases} - \\varepsilon ^ 2 ( K _ H ( x ) \\nabla u ) = \\left ( u - \\left ( \\frac { \\alpha | x | ^ 2 } { 2 } + \\beta \\right ) \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ { p } _ + , \\ \\ & x \\in B _ { R ^ * } ( 0 ) , \\\\ u = 0 , \\ \\ & x \\in \\partial B _ { R ^ * } ( 0 ) , \\end{cases} \\end{align*}"}
+{"id": "8910.png", "formula": "\\begin{align*} & \\log 4 \\mathbf { e } _ 1 + \\log 4 \\mathbf { e } _ 2 + \\log 4 \\mathbf { e } _ 3 + \\log \\frac { 3 } { 2 } \\mathbf { e } _ { 1 2 } + \\log \\frac { 3 } { 2 } \\mathbf { e } _ { 1 2 3 ' } \\\\ & = [ \\log 9 , \\log 9 , \\log 6 , \\log 5 4 , \\log 5 4 , \\log 5 4 , \\log 2 1 6 ] ^ \\intercal \\end{align*}"}
+{"id": "2054.png", "formula": "\\begin{align*} \\Lambda _ 1 = \\frac { 1 } { 2 \\sqrt { - 1 } } \\Big ( ( t - t _ 0 ) ^ { 1 / 2 } \\partial _ { v _ 1 } + ( t - t _ 0 ) ^ { 3 / 2 } \\partial _ { x _ 1 } \\Big ) , \\Lambda _ 2 = \\frac { \\sqrt 3 } { 6 \\sqrt { - 1 } } \\Big ( 3 ( t - t _ 0 ) ^ { 1 / 2 } \\partial _ { v _ 1 } + ( t - t _ 0 ) ^ { 3 / 2 } \\partial _ { x _ 1 } \\Big ) . \\end{align*}"}
+{"id": "4892.png", "formula": "\\begin{align*} \\frac { \\partial z } { \\partial \\alpha } = & \\frac { \\phi ( z ) \\ , [ 1 - \\Phi ( z ) ] } { \\Phi ( z ) \\ , [ 1 - \\Phi ( z ) ] + z \\ , \\phi ( z ) \\left [ \\alpha - ( \\alpha + \\beta ) \\ , \\Phi ( z ) \\right ] - ( 2 - \\alpha - \\beta ) \\ , \\phi ^ 2 ( z ) } . \\end{align*}"}
+{"id": "8144.png", "formula": "\\begin{align*} d _ n ( f ) ( g _ 0 , \\dots , g _ { n + 1 } ) ) = \\sum _ { i = 0 } ^ n ( - 1 ) ^ i f ( g _ 0 , \\dots , \\hat g _ i , \\dots , g _ { n + 1 } ) , . \\end{align*}"}
+{"id": "1185.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ H \\left ( \\mu [ t ] \\mid P _ { Y ^ 1 , \\cdots , Y ^ k } ^ { ( k + 1 \\mid k ) } [ t ] \\right ) \\right ] = H \\left ( \\mu ^ { \\otimes k + 1 } [ t ] \\mid P ^ { ( k + 1 ) } [ t ] \\right ) - H \\left ( \\mu ^ { \\otimes k } [ t ] \\mid P ^ { ( k ) } [ t ] \\right ) = H _ t ^ { k + 1 } - H _ t ^ k . \\end{align*}"}
+{"id": "7661.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { t _ 0 , x _ 0 , \\xi , \\alpha } = & ~ [ A x _ t ^ { t _ 0 , x _ 0 , \\xi , \\alpha } + B \\alpha _ t + f ( \\nu _ t ^ { * , t _ 0 , \\xi } ) + b ( \\mu _ t ^ { * , t _ 0 , \\xi } ) ] d t + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ { t _ 0 } ^ { t _ 0 , x _ 0 , \\xi , \\alpha } = & ~ x _ 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "4105.png", "formula": "\\begin{align*} & \\hphantom { { } = { } } \\{ f , g ; F , G \\} ( \\xi , \\eta ; \\dot { \\xi } , \\dot { \\eta } ) \\\\ * & = ( f ( \\xi ) , D f ( \\xi ) \\eta + g ( \\xi ) ; F ( \\xi ) \\dot { \\xi } , ( D F ( \\xi ) ^ i { } _ { \\alpha j } \\eta ^ \\alpha + G ( \\xi ) ) \\dot { \\xi } + F ( \\xi ) \\dot { \\eta } ) , \\end{align*}"}
+{"id": "107.png", "formula": "\\begin{align*} x = x _ 1 , \\dotsc , x _ n \\in \\widetilde W \\end{align*}"}
+{"id": "5306.png", "formula": "\\begin{align*} \\widehat { f } ( \\lambda , \\omega ) = ( 2 \\pi ) ^ { - n / 2 } \\int _ { \\R ^ n } f ( x ) e ^ { - i \\lambda x \\cdot \\omega } d x , \\ , \\ , \\ , \\lambda \\in \\R , \\omega \\in S ^ { n - 1 } . \\end{align*}"}
+{"id": "8160.png", "formula": "\\begin{align*} \\xi ( \\mu , 0 ) = \\xi ^ { \\mathrm { i n } } ( \\mu ) \\ge 0 , \\ \\mu \\in \\mathbb { R } _ { > 0 } . \\end{align*}"}
+{"id": "1466.png", "formula": "\\begin{align*} g ' ( \\theta ) = ( \\cos \\theta - \\sin \\theta ) \\left ( C _ { 0 } - 6 C _ { 1 } \\cos \\theta \\sin \\theta ( \\cos \\theta + \\sin \\theta ) \\right ) . \\end{align*}"}
+{"id": "3995.png", "formula": "\\begin{align*} \\C = \\left \\{ ( \\lambda + \\mu \\theta ^ { i _ 1 } , \\lambda + \\mu \\theta ^ { i _ 2 } , \\cdots , \\lambda + \\mu \\theta ^ { i _ n } ) \\ , | \\ , \\lambda , \\mu \\in \\mathbb { F } _ { p ^ e } \\right \\} \\end{align*}"}
+{"id": "7827.png", "formula": "\\begin{align*} O : = \\{ f \\in X : f ( x ) = - f ( - x ) \\} . \\end{align*}"}
+{"id": "8020.png", "formula": "\\begin{align*} & \\sum _ { a = 0 } ^ t \\pi _ N ( x ) ^ a ( \\log \\log x ) ^ { t - a } L ^ { 2 r - a } L ^ { 2 r + x ' - 2 t } \\sum _ { ( b _ 1 , b _ 2 , \\dots , b _ { t } ) \\atop { b _ u \\in \\mathcal J _ { \\mathcal I ( s _ u ) } \\atop { a ( \\underline { b } ) = a } } } L ^ { - a _ 3 ( \\underline { b } ) } \\\\ & \\ll _ r \\sum _ { a = 0 } ^ t \\pi _ N ( x ) ^ a ( \\log \\log x ) ^ { t - a } L ^ { 2 r - a } L ^ { 2 r + x ' - 2 t } , \\end{align*}"}
+{"id": "5210.png", "formula": "\\begin{align*} \\int _ { B ( 0 , R ) } \\partial _ t a _ { j } \\cdot \\xi \\dd x + \\int _ { B ( 0 , R ) } B _ j \\times u _ j \\cdot \\mathbb { P } \\xi \\dd x = - \\mu _ j \\int _ { B ( 0 , R ) } b _ j \\cdot \\nabla \\times \\xi \\dd x . \\end{align*}"}
+{"id": "6101.png", "formula": "\\begin{align*} \\frac { \\psi ^ { + 0 } _ { n , p - 1 } } { \\psi ^ { + 0 } _ { n , p } } \\ \\frac { \\rho ^ { 0 + } _ { n , p } } { \\rho ^ { 0 + } _ { n - 1 , p } } = \\frac { ( n - p + a _ { 1 2 } - a _ { 1 2 3 } + j ^ { ( 3 ) } + 1 ) ( n - p + a _ { 1 2 } - a _ { 1 2 3 } - j ^ { ( 3 ) } ) } { ( n - p + a _ { 1 2 } - a _ { 1 2 3 } + j ^ { ( 3 ) } ) ( n - p + a _ { 1 2 } - a _ { 1 2 3 } - j ^ { ( 3 ) } - 1 ) } \\ . \\end{align*}"}
+{"id": "4916.png", "formula": "\\begin{align*} e _ { k , i } = ( i \\ , d _ 0 ) ^ { - 1 } \\sum _ { m = 1 } ^ { i } [ m \\ , ( k + 1 ) - i ] \\ , d _ m \\ , e _ { k , i - m } . \\end{align*}"}
+{"id": "2328.png", "formula": "\\begin{align*} R i c ^ W & = R i c ^ g - \\frac { 1 } { 2 } \\left ( \\| \\theta \\| ^ 2 g - \\theta \\otimes \\theta \\right ) , \\\\ & = \\frac { s ^ W } { 4 } g + ( R i c ^ g ) ^ { J , - } + \\frac { 1 } { 2 } ( \\theta \\otimes \\theta ) ^ { J , - } . \\end{align*}"}
+{"id": "6899.png", "formula": "\\begin{align*} f _ { y , \\mathbf { X } } ( \\beta ) : = \\sum _ { i = 1 } ^ p V ( \\beta _ i ) - \\frac { 1 } { 2 \\sigma ^ 2 } \\left | y - \\mathbf { X } \\beta \\right | ^ 2 . \\end{align*}"}
+{"id": "3007.png", "formula": "\\begin{align*} H ( t , z , r , s ) = \\min \\limits _ { u \\in \\mathbb U } \\big ( \\langle f ( t , z , r , u ) , s \\rangle + f ^ 0 ( t , z , r , u ) \\big ) , \\ t \\in [ 0 , \\vartheta ] , \\ z , r , s \\in \\mathbb R ^ n , \\end{align*}"}
+{"id": "2104.png", "formula": "\\begin{align*} h _ c ( G ) & = h _ c ( G \\backslash v ) + f _ c ( 1 ) - f _ c ( 0 ) + \\Delta _ c ( d _ 1 ) \\\\ & \\le h _ c ( K _ { 1 , m - 1 } ) + f _ c ( 1 ) - f _ c ( 0 ) + \\Delta _ c ( m ) \\\\ & = h _ c ( K _ { 1 , m } ) \\end{align*}"}
+{"id": "7367.png", "formula": "\\begin{gather*} m ( f _ 1 ) + m ( f _ 2 ) = \\lim _ { \\epsilon \\rightarrow 0 } \\ , \\Bigl \\{ \\int _ \\epsilon ^ 1 f _ 1 ( x ) \\ , d x + \\int _ 0 ^ { 1 - \\epsilon } f _ 2 ( x ) \\ , d x \\Bigr \\} \\\\ = \\lim _ { \\epsilon \\rightarrow 0 } \\int _ 0 ^ { 1 - \\epsilon } \\bigl \\{ f _ 1 ( x + \\epsilon ) + f _ 2 ( x ) \\bigr \\} \\ , d x \\le \\lim _ { \\epsilon \\rightarrow 0 } \\int _ 0 ^ { 1 - \\epsilon } c ( x + \\epsilon , x ) \\ , d x = 0 . \\end{gather*}"}
+{"id": "2181.png", "formula": "\\begin{align*} \\mathbf { R } _ y = \\frac { 1 } { K } \\sum \\limits _ { n = 1 } ^ { K } \\mathbf { y } ( n ) \\mathbf { y } ^ { H } ( n ) , \\end{align*}"}
+{"id": "5706.png", "formula": "\\begin{align*} R = ( ( a _ 0 , b _ 0 , c _ 0 , d _ 0 ) _ q , ( a _ 1 , b _ 1 , c _ 1 , d _ 1 ) _ q ) = ( f , g ) . \\end{align*}"}
+{"id": "5647.png", "formula": "\\begin{align*} d ( u _ { i } , g _ { i } ) & = ( d u _ { i } , g _ { i } ) + ( - 1 ) ^ { p + 1 } ( u _ { i } , d g _ { i } ) = ( d u _ { i } , g _ { i } ) + ( - 1 ) ^ { p + 1 } ( u _ { i } , f _ { i } ) . \\end{align*}"}
+{"id": "2090.png", "formula": "\\begin{align*} p _ { S } ( x ) : = \\int _ S \\frac { 1 _ { B _ y } ( x ) } { U ( B _ y ) } d U ( y ) , \\end{align*}"}
+{"id": "3736.png", "formula": "\\begin{align*} \\beta _ { \\bar { \\mathbf { g } } } \\mathbf { g } ( \\partial _ t ) & = 0 \\\\ \\beta _ { \\bar { \\mathbf { g } } } \\mathbf { g } ( \\partial _ a ) & = \\beta _ { \\bar g } g ( \\partial _ a ) + \\bar u ^ { - 2 } u \\partial _ a u - \\bar u ^ { - 1 } g ( \\nabla _ { \\bar g } \\bar u , \\partial _ a ) \\mbox { f o r $ a = 1 , \\dots , n $ } . \\end{align*}"}
+{"id": "6594.png", "formula": "\\begin{align*} \\gamma - m _ { 0 } < \\gamma - c ( R ) = c _ { r } ( R ) < \\frac { M _ { 0 } m _ { 0 } } { \\sigma _ { d } } R . \\end{align*}"}
+{"id": "1684.png", "formula": "\\begin{align*} \\ell _ n ( \\theta ) \\ ! = \\ ! \\begin{bmatrix} l _ 0 ( \\theta ) I _ m & l _ 1 ( \\theta ) I _ m & \\dots & l _ { n - 1 } ( \\theta ) I _ m \\end{bmatrix} ^ { \\ ! \\top } \\ ! \\ ! \\ ! , \\ ; \\forall \\theta \\in [ - h , 0 ] . \\end{align*}"}
+{"id": "1634.png", "formula": "\\begin{align*} x ( t ) = \\int ^ { T _ 2 } _ { T _ 1 } G ( t , \\tau ) f ( x ( \\tau ) ) \\ ; d \\tau , t \\in [ T _ 1 , T _ 2 ] , \\end{align*}"}
+{"id": "5813.png", "formula": "\\begin{align*} { f _ i } \\left ( { { \\bf { x } } + { { \\bf { c } } _ i } \\Delta t , t + \\Delta t } \\right ) - { f _ i } \\left ( { { \\bf { x } } , t } \\right ) = { \\left ( { { { { \\bf { \\hat M } } } ^ { - 1 } } { \\bf { \\hat \\Lambda \\hat M } } } \\right ) _ { i j } } \\left [ { { f _ i } \\left ( { { \\bf { x } } , t } \\right ) - f _ i ^ { \\left ( { e q } \\right ) } \\left ( { { \\bf { x } } , t } \\right ) } \\right ] + { { \\bf { F } } _ i } \\left ( { { \\bf { x } } , t } \\right ) , \\end{align*}"}
+{"id": "632.png", "formula": "\\begin{align*} \\begin{aligned} x ^ 2 & = ( x _ 0 + 2 x _ 1 + 4 x _ 2 + 8 x _ 3 + 1 6 x _ 4 + \\ldots ) ^ 2 \\\\ { } & = x _ 0 ^ 2 + 4 ( x _ 0 x _ 1 + x _ 1 ^ 2 ) + 8 ( x _ 0 x _ 2 ) + 1 6 ( x _ 2 ^ 2 + x _ 1 x _ 2 + x _ 0 x _ 3 ) \\\\ { } & + 2 ^ 5 ( x _ 0 x _ 4 + x _ 1 x _ 3 ) + 2 ^ 6 ( x _ 3 ^ 2 + x _ 0 x _ 5 + x _ 1 x _ 4 + x _ 2 x _ 3 ) + \\ldots . \\end{aligned} \\end{align*}"}
+{"id": "2057.png", "formula": "\\begin{align*} \\Lambda ^ \\ell g \\cdot \\ , \\Lambda ^ \\ell h : = \\sum ^ { 2 } _ { j _ 1 = 1 } \\cdot \\cdot \\cdot \\sum ^ { 2 } _ { j _ \\ell = 1 } ( \\Lambda _ { j _ 1 } \\cdots \\Lambda _ { j _ \\ell } g ) \\ , ( \\Lambda _ { j _ 1 } \\cdots \\Lambda _ { j _ \\ell } h ) . \\end{align*}"}
+{"id": "2556.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { i = 1 } ^ { n } d _ i ^ 2 \\leq \\frac { 4 m ^ 2 } { n } + \\frac { n } { 4 } { ( \\triangle ( G ) - \\delta ( G ) ) } ^ 2 . \\end{align*}"}
+{"id": "2879.png", "formula": "\\begin{align*} | \\sum _ { y = 0 } ^ n \\bar K ^ { ( n , \\ell ) } ( { y , x } ) - \\sum _ { y = x - n - 1 } ^ { x + n } \\bar K ^ { ( n , \\ell ) } ( { y , x } ) | \\le \\frac { C } { n ^ 2 } , \\delta n \\le x \\le ( 1 - \\delta ) n . \\end{align*}"}
+{"id": "1191.png", "formula": "\\begin{align*} \\forall s \\in S : C ( s ) = b ( s ) \\enspace . \\end{align*}"}
+{"id": "6834.png", "formula": "\\begin{align*} \\frac { \\partial ^ { k } \\varphi _ { 0 } ^ { - } \\left ( \\rho , x \\right ) } { \\partial x ^ { k } } = \\frac { \\left ( 2 i \\right ) ^ { k } } { 2 \\pi i } \\int _ { 0 } ^ { \\infty } \\left ( q ( s ) L _ { k } ^ { \\left ( 1 \\right ) } \\left ( s , \\rho \\right ) + Q ^ { \\prime } ( s ) L _ { k } ^ { \\left ( 2 \\right ) } \\left ( s , \\rho \\right ) \\right ) d s \\end{align*}"}
+{"id": "7999.png", "formula": "\\begin{align*} V ' _ 1 ( u _ n ) [ u _ n - \\bar { u } ] & = 4 B _ 1 ( u _ n ^ 2 , u _ n ( u _ n - \\bar { u } ) ) = 4 B _ 1 ( u _ n ^ 2 , ( u _ n - \\bar { u } ) ^ 2 ) + 4 B _ 1 ( u _ n ^ 2 , \\bar { u } ( u _ n - \\bar { u } ) ) \\\\ & = 4 B _ 1 ( u _ n ^ 2 , ( u _ n - \\bar { u } ) ^ 2 ) + o ( 1 ) . \\end{align*}"}
+{"id": "591.png", "formula": "\\begin{align*} \\Psi _ { \\rho } ( v , w ) : = \\Phi _ { \\rho } ^ { - 1 } \\big ( \\Phi _ { \\rho } ( v ) + w \\big ) \\end{align*}"}
+{"id": "5423.png", "formula": "\\begin{align*} \\omega _ \\epsilon : = \\{ x \\in \\Omega : | x _ \\beta | < \\epsilon \\delta \\frac { b _ \\alpha } { \\sqrt { b _ \\beta } } , \\beta = \\alpha + 1 , \\ldots , n - 1 , x _ n < \\epsilon ^ 2 \\delta ^ 2 b _ \\alpha \\} \\end{align*}"}
+{"id": "4433.png", "formula": "\\begin{align*} \\begin{aligned} v ^ { ( 1 ) } = \\textup { R e } \\ , u ^ { ( 1 ) } & = \\big ( 1 , \\cos ( \\pi / N ) , \\cos ( 2 \\pi / N ) , \\ldots , \\cos ( ( N - 1 ) \\pi / N ) \\big ) \\\\ v ^ { ( 2 ) } = \\textup { I m } \\ , u ^ { ( 1 ) } & = \\big ( 0 , \\sin ( \\pi / N ) , \\sin ( 2 \\pi / N ) , \\ldots , \\sin ( ( N - 1 ) \\pi / N ) \\big ) \\end{aligned} \\end{align*}"}
+{"id": "5208.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { + } ^ { 2 } \\chi _ R \\dd x - \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } ^ { 2 } \\chi _ R \\dd x + 2 \\mu \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { 3 } } | \\nabla \\Phi _ { + } | ^ { 2 } \\chi _ R \\dd x \\dd s \\\\ & = \\mu \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { + } ^ { 2 } \\left ( \\Delta + \\frac { 2 } { r } \\partial _ r \\right ) \\chi _ R \\dd x \\dd s + \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { 3 } } u \\cdot \\nabla \\chi _ R \\Phi _ { + } ^ { 2 } \\dd x \\dd s . \\end{align*}"}
+{"id": "4764.png", "formula": "\\begin{align*} & f _ 1 ( x ^ * ) = a _ 1 \\| x ^ * \\| ^ 2 + b _ 1 ^ T { x ^ * } + c _ 1 = 0 \\\\ & \\| x ^ * \\| ^ 2 = - \\frac { b _ 1 ^ T { x ^ * } } { a _ 1 } - \\frac { c _ 1 } { a _ 1 } \\\\ & \\| x ^ * \\| ^ 2 = \\frac { 1 } { a _ 1 w } b _ 1 ^ T ( \\gamma _ 1 b _ 1 + \\gamma _ 2 b _ 2 ) - \\frac { c _ 1 } { a _ 1 } \\end{align*}"}
+{"id": "6348.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { n = 0 } ^ \\infty c _ n ( a , b ) \\ , G ( x ; ( a + n ) \\theta ) . \\end{align*}"}
+{"id": "8106.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ( r ) & = u ( 0 ) + \\int _ { 0 } ^ { r } t ^ { 1 - N } \\left ( \\int _ { 0 } ^ { t } s ^ { a } v ^ { p } ( s ) d s \\right ) d t , & & r > 0 , \\\\ v ( r ) & = v ( 0 ) + \\int _ { 0 } ^ { r } t ^ { 1 - N } \\left ( \\int _ { 0 } ^ { t } s ^ { b } v ^ { q } ( s ) f ( | u ' ( s ) | ) d s \\right ) d t , & & r > 0 , \\\\ u ( 0 ) & > 0 , v ( 0 ) > 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2750.png", "formula": "\\begin{align*} h _ m = F _ m \\circ F _ { m - 1 } \\circ \\dots \\circ F _ 1 , \\end{align*}"}
+{"id": "5476.png", "formula": "\\begin{align*} K _ { p , d } = 2 ^ { - p } \\pi ^ { - d / 2 } \\frac { \\Gamma \\left ( \\frac { d - p } { 2 } \\right ) } { \\Gamma \\left ( \\frac { p } { 2 } \\right ) } . \\end{align*}"}
+{"id": "8953.png", "formula": "\\begin{align*} V _ 1 ( t ) = 2 V _ 1 ^ + ( t ) \\le 2 \\int _ { \\eta _ { k - 1 } } ^ { b ( t ) } v _ 1 ^ { - p ' } \\le 2 \\int _ { \\eta _ { k - 1 } } ^ { b ( x ) } v _ 1 ^ { - p ' } \\le 2 V _ 1 ( x ) = 4 V _ 1 ^ - ( x ) , \\eta _ { k - 1 } \\le t \\le x , \\end{align*}"}
+{"id": "1756.png", "formula": "\\begin{align*} \\bar { J } _ 0 ( t , x ) : = \\int _ { Y ^ { \\ast } } J _ 0 ( t , x , y ) d y . \\end{align*}"}
+{"id": "3878.png", "formula": "\\begin{align*} \\begin{cases} - \\delta ^ 2 ( K ( \\hat { x } ) \\nabla v ) = ( V _ { \\delta , \\hat { x } , \\hat { q } , z } - \\hat { q } ) ^ { p } _ + , \\ \\ & \\ \\Omega , \\\\ v = 0 , \\ \\ & \\ \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "545.png", "formula": "\\begin{align*} x = x _ a + x _ s + x _ p \\end{align*}"}
+{"id": "7101.png", "formula": "\\begin{align*} \\omega : = \\nabla \\times u = \\partial _ { x _ 1 } u _ { 2 } - \\partial _ { x _ 2 } u _ { 1 } \\end{align*}"}
+{"id": "7766.png", "formula": "\\begin{align*} r _ d \\le d \\Big | \\frac { p _ 0 } { p _ 1 } \\Big | = d \\Big | \\frac { p ( 0 ) } { p ' ( 0 ) } \\Big | . \\end{align*}"}
+{"id": "4451.png", "formula": "\\begin{align*} \\beta _ { q } ( n ) = \\lim _ { v \\to \\infty } \\frac { \\# \\{ \\vec { x } \\in ( \\Z / p ^ { v } \\Z ) ^ { 4 } | Q ( \\vec { x } ) \\equiv n \\pmod { p ^ { v } } \\} } { p ^ { 3 v } } , \\end{align*}"}
+{"id": "1304.png", "formula": "\\begin{align*} \\overline { x } = x + \\sqrt { x ^ 2 - 1 } . \\end{align*}"}
+{"id": "8164.png", "formula": "\\begin{align*} \\varpi _ 1 ( \\mu , \\nu ) = \\nu \\varpi ' ( \\mu ) - \\varpi ( \\nu ) , \\end{align*}"}
+{"id": "8129.png", "formula": "\\begin{align*} M ' ( S ) = \\varinjlim _ i N ( T _ i ) = \\varinjlim _ i \\varprojlim _ { n \\in \\Delta } N ( T _ { i , n } ) = \\varprojlim _ { n \\in \\Delta } \\varinjlim _ i N ( T _ { i , n } ) = \\varprojlim _ { n \\in \\Delta } M ' ( S _ n ) , \\end{align*}"}
+{"id": "9145.png", "formula": "\\begin{align*} \\dot { { x } } _ a = \\ , - \\gamma c _ 1 \\delta \\left . \\dfrac { \\partial h } { \\partial x } \\right | _ { { x } _ a } - \\gamma \\sum _ { k = 2 } ^ { \\infty } c _ k \\ , { \\delta ^ { 2 k - 1 } } \\ , \\left . \\dfrac { \\partial ^ { 2 k - 1 } h } { \\partial x ^ { 2 k - 1 } } \\right | _ { { x } _ a } \\end{align*}"}
+{"id": "6118.png", "formula": "\\begin{align*} \\begin{aligned} & ( C _ { a j } - C _ j - C _ a ) \\ , [ C _ { i k } , C _ { k \\ell } ] + ( C _ { a i } - C _ i - C _ a ) [ C _ { k \\ell } , C _ { j k } ] + ( C _ { a \\ell } - C _ \\ell - C _ a ) [ C _ { i j } , C _ { j k } ] \\\\ & \\qquad + ( C _ { a k } - C _ k - C _ a ) [ C _ { i \\ell } , C _ { j \\ell } ] = 0 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "3246.png", "formula": "\\begin{align*} \\langle \\Pi _ m Q \\Pi _ m h - Q h , h \\rangle \\leq \\| ( I - \\Pi _ m ) Q h \\| + \\| ( I - \\Pi _ m ) Q h _ m \\| = ( 1 ) _ m + ( 2 ) _ m . \\end{align*}"}
+{"id": "5062.png", "formula": "\\begin{align*} g ( \\Phi ) _ { t } + u \\cdot \\nabla g ( \\Phi ) = 0 , \\end{align*}"}
+{"id": "683.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | \\sin ( n \\theta ) \\left | l _ 2 , m _ 2 \\right \\rangle = 0 \\ , , \\end{align*}"}
+{"id": "4095.png", "formula": "\\begin{align*} \\Theta ^ { 2 ( 1 ) } & = d \\theta ^ i { } _ j + \\theta ^ i { } _ { \\alpha j } \\wedge \\theta ^ \\alpha + \\theta ^ i { } _ \\alpha \\wedge \\theta ^ \\alpha _ j \\\\ * & = \\Omega ^ i { } _ j + \\theta ^ i { } _ { \\alpha j } \\wedge \\theta ^ \\alpha , \\\\ * \\Theta ^ { 2 ( 2 ) } & = d \\theta ^ i { } _ j + \\theta ^ i { } _ { j \\alpha } \\wedge \\theta ^ \\alpha + \\theta ^ i { } _ \\alpha \\wedge \\theta ^ \\alpha { } _ j \\\\ * & = \\Omega ^ i { } _ j + \\theta ^ i { } _ { j \\alpha } \\wedge \\theta ^ \\alpha . \\end{align*}"}
+{"id": "3020.png", "formula": "\\begin{align*} p _ 0 = - 2 k ( t _ k - \\xi _ k ) - \\lambda _ \\varphi \\| x _ k - z \\| , p = - 2 k ( x _ k - y _ k ) . \\end{align*}"}
+{"id": "3795.png", "formula": "\\begin{align*} ( B _ q ) ^ { - 1 } = ( B _ q ) ^ { * } = T . \\end{align*}"}
+{"id": "2209.png", "formula": "\\begin{align*} \\phi = 2 \\pi \\frac { d } { \\lambda } { s i n } \\theta _ 0 = 2 \\pi Z \\end{align*}"}
+{"id": "8849.png", "formula": "\\begin{align*} \\chi _ D ( x _ 1 ) = \\varphi \\left ( \\frac { \\sqrt { K } | x _ 1 | } { 4 R } \\right ) , \\chi _ T ( x _ 1 ) = 1 - \\varphi \\left ( \\frac { \\sqrt { K } | x _ 1 | } { R } \\right ) . \\end{align*}"}
+{"id": "2926.png", "formula": "\\begin{align*} \\left \\Vert \\varphi \\right \\Vert _ { p } = \\max _ { 0 \\leq q \\leq p } | \\sup _ { x \\in \\mathbb { R } } \\left ( e ^ { p \\left \\vert x \\right \\vert } \\varphi ^ { ( q ) } \\left ( x \\right ) \\right ) | \\end{align*}"}
+{"id": "9181.png", "formula": "\\begin{align*} x ( t ) - z ( t ) = x ( 0 ) - z ( 0 ) + \\int _ 0 ^ t \\dot { x } ( \\tau ) - \\dot { z } ( \\tau ) \\ , d \\tau \\end{align*}"}
+{"id": "8625.png", "formula": "\\begin{align*} \\chi _ { n , m } ' ( w , M , \\cdot ) : = \\chi _ { n , m } ( w , \\cdot ) \\mathbb { 1 } _ { X _ 1 ( M ) } + ( \\mathbb { E } _ { n , m } [ \\chi _ { n , m } ( w ' , \\cdot ) | O ( \\chi _ { n , m } ' ( w ' , M , \\cdot ) ) = \\\\ O ( \\chi _ { n , m } ' ( w , M , \\cdot ) ) ] + \\mathcal { G } ( 0 , \\mathcal { K } _ { n , m } ^ { w , M } ) ( \\cdot ) ) \\mathbb { 1 } _ { X _ 0 ( M ) } \\end{align*}"}
+{"id": "5416.png", "formula": "\\begin{align*} \\begin{aligned} x _ n ^ * - x _ n ^ 0 = \\ , & x _ n - \\sum _ { \\beta = 1 } ^ \\alpha x _ \\beta \\rho _ \\beta ( 0 , \\tilde { x } ) - \\rho ( 0 , \\tilde { x } ) \\\\ > \\ , & \\rho ( \\hat { x } , \\tilde { x } ) - \\sum _ { \\beta = 1 } ^ \\alpha x _ \\beta \\rho _ \\beta ( 0 , \\tilde { x } ) - \\rho ( 0 , \\tilde { x } ) \\geq 0 \\end{aligned} \\end{align*}"}
+{"id": "1072.png", "formula": "\\begin{align*} p : w = w _ 1 \\rightarrow w _ 2 \\rightarrow \\cdots \\rightarrow w _ { \\ell + 1 } = w ' . \\end{align*}"}
+{"id": "4200.png", "formula": "\\begin{align*} - ( K _ H ( x ) \\nabla u _ { \\bar { x } } ) = \\frac { 1 } { \\varepsilon ^ 2 } \\left ( u _ { \\bar { x } } - q _ { \\bar { x } } \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ { p } _ + , \\ \\ \\ \\ \\ \\Omega _ { \\bar { x } } . \\end{align*}"}
+{"id": "1815.png", "formula": "\\begin{align*} \\mathcal { Y M } _ e ( \\nabla ) = \\int _ M \\exp ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) \\ , d v \\ , , \\end{align*}"}
+{"id": "942.png", "formula": "\\begin{align*} i \\partial _ t v _ 2 & = \\lambda _ 6 t ^ { - 1 } \\mathcal { F } M ( t ) ^ { - 1 } \\mathcal { F } ^ { - 1 } \\mathcal { N } _ 2 ( \\mathcal { F } M ( t ) { { \\mathcal F } } ^ { - 1 } v _ 1 , \\mathcal { F } M ( t ) { { \\mathcal F } } ^ { - 1 } v _ 2 ) \\\\ & = \\lambda _ 6 t ^ { - 1 } \\mathcal { N } _ 2 ( v _ 1 , v _ 2 ) + i R _ 2 ( s ) . \\end{align*}"}
+{"id": "5586.png", "formula": "\\begin{align*} A ( y _ 1 x ) - A ( y _ 1 x ' ) = A ( 1 ^ \\infty ) - A ( 1 0 ^ \\infty ) = c - d \\ . \\end{align*}"}
+{"id": "6779.png", "formula": "\\begin{align*} g ( \\rho , x ) = e ^ { - i \\rho x } \\left ( 1 + \\left ( z + 1 \\right ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } z ^ { n } b _ { n } ( x ) \\right ) \\end{align*}"}
+{"id": "1561.png", "formula": "\\begin{align*} h _ { x , t } : = b \\vert _ { E ^ s } + \\varphi ( t ) b \\vert _ { E ^ u } + d t ^ 2 \\end{align*}"}
+{"id": "5761.png", "formula": "\\begin{align*} \\Phi ( \\mathcal { P } ( Z ) ) = \\mathcal { P } ' ( \\Phi ( Z ) ) \\end{align*}"}
+{"id": "5513.png", "formula": "\\begin{align*} L ( p ) = \\log F ( p , 8 / 3 ) , R ( p ) = \\log \\left ( e ^ { - p / 6 } \\tilde G ( p , 2 ) \\right ) \\end{align*}"}
+{"id": "8244.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\deg _ p ( N _ k ' ) = \\lim _ { k \\to \\infty } \\deg _ p ( N _ k '' ) ; \\end{align*}"}
+{"id": "8473.png", "formula": "\\begin{align*} M _ { \\pm } ( x ; z ) = \\left ( M _ { \\pm , 1 } ( x ; z ) , M _ { \\pm , 2 } ( x ; z ) \\right ) . \\end{align*}"}
+{"id": "1060.png", "formula": "\\begin{align*} \\overline { I x I } = \\bigsqcup _ { y \\leq x } I y I \\subseteq G ( \\breve F ) . \\end{align*}"}
+{"id": "2979.png", "formula": "\\begin{align*} \\widetilde { C } = C \\times \\{ t \\} . \\end{align*}"}
+{"id": "2485.png", "formula": "\\begin{align*} \\mathbf { x } ^ { ( i ) } \\circ \\mathbf { s } ^ { ( i ) } = \\left ( { \\left ( \\mathbf { x } ^ { ( i ) } \\right ) } ^ T \\mathbf { s } ^ { ( i ) } , \\mathbf { x } _ 1 ^ { ( i ) } \\mathbf { s } _ { 2 : n _ i } ^ { ( i ) } + \\mathbf { s } _ 1 ^ { ( i ) } \\mathbf { x } _ { 2 : n _ i } ^ { ( i ) } \\right ) , \\forall \\mathbf { x } ^ { ( i ) } , \\mathbf { s } ^ { ( i ) } \\in \\mathcal { R } ^ { n _ i } \\end{align*}"}
+{"id": "3663.png", "formula": "\\begin{align*} v = 0 , ( \\nu ' ) ( h ) ( \\bar u ) + \\nu ( v ) = 0 , A ' ( h ) = 0 \\mbox { o n } \\hat \\Sigma . \\end{align*}"}
+{"id": "1475.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\| u _ { n } - z _ { n } \\| _ { L ^ { p } ( ( a , b ) ) } = 0 \\forall p \\in [ 1 , + \\infty ) . \\end{align*}"}
+{"id": "8094.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = | x | ^ { a } v ^ { p } & & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta v & = | x | ^ { b } v ^ { q } f ( | \\nabla u | ) & & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4151.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n - 1 } } \\langle f ( x ' ) , h ( x ' ) \\rangle d x ' = \\lim _ { \\varepsilon \\to 0 ^ + } \\int _ { \\R ^ { n - 1 } } \\langle f ( x ' ) , \\widetilde { \\Psi } _ { \\varepsilon , \\varepsilon ^ { - 1 } } * h ( x ' ) \\rangle d x ' . \\end{align*}"}
+{"id": "4287.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\mathcal { L } ( u ) + f ( t , x ) , \\mathcal { B } ( u ) = g ( t , x ) , \\end{align*}"}
+{"id": "4842.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} S ^ \\prime _ t & = - \\beta ( 1 - \\theta L _ t ) S _ t ( 1 - \\theta L _ t ) I _ t , & S _ 0 & = s _ 0 \\\\ I ^ \\prime _ t & = \\beta ( 1 - \\theta L _ t ) S _ t ( 1 - \\theta L _ t ) I _ t - \\gamma I _ t - I _ t \\phi \\left ( I _ t \\right ) , & I _ 0 & = i _ 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1723.png", "formula": "\\begin{align*} | | y ( t ) - y ( s ) | | _ { L ^ { 2 \\alpha } } \\le & | | y ( t ) - y ( s ) | | ^ { \\alpha } _ { L ^ 2 } | | y ( t ) - y ( s ) | | ^ { 1 - \\alpha } _ { H ^ 2 } \\\\ \\lesssim & M _ 1 ^ { \\alpha } | t - s | ^ { \\alpha } M _ 1 ^ { 1 - \\alpha } = M _ 1 | t - s | ^ { \\alpha } , \\end{align*}"}
+{"id": "8180.png", "formula": "\\begin{align*} \\frac { d } { d \\tau } \\mathcal { U } _ { \\lambda } ( \\tau ) = C _ { 1 } \\Theta _ { \\lambda } ( \\tau ) \\geq C _ { 1 } \\mathcal { U } _ { \\lambda } ( \\tau ) . \\end{align*}"}
+{"id": "8805.png", "formula": "\\begin{align*} \\tau = \\frac { ( 1 / 2 + r + \\epsilon ) \\log ( | z _ 0 | ) } { \\lambda _ R | z _ 0 | } \\end{align*}"}
+{"id": "3619.png", "formula": "\\begin{align*} \\sigma _ { k } ^ { i i } = \\frac { \\partial \\sigma _ k } { \\partial \\kappa _ i } , \\sigma _ { k } ^ { i i , j j } = \\frac { \\partial ^ 2 \\sigma _ { k } } { \\partial \\kappa _ i \\partial \\kappa _ j } . \\end{align*}"}
+{"id": "39.png", "formula": "\\begin{align*} \\Gamma ( g , g ' ) = \\tilde { \\Gamma } ( g ^ { - 1 } \\circ g ' ) \\end{align*}"}
+{"id": "1690.png", "formula": "\\begin{align*} \\mathcal { E } ( \\eta ) = - \\frac { \\kappa _ 1 + \\kappa _ 2 } { \\kappa _ 2 + 1 } + \\sqrt { \\left ( \\frac { \\kappa _ 1 + \\kappa _ 2 } { \\kappa _ 2 + 1 } \\right ) ^ 2 + \\frac { \\eta } { h ( \\kappa _ 2 + 1 ) } } , \\end{align*}"}
+{"id": "5682.png", "formula": "\\begin{align*} f _ { 2 } = f _ { 2 } \\circ u _ { 2 } = f _ { 2 } \\circ \\iota _ { 2 } \\circ p _ { 2 } = f _ { 2 2 } \\circ p _ { 2 } = p _ { 2 } ~ ~ ~ ~ ~ ~ ( \\because u _ { 2 } = \\iota _ { 2 } \\circ p _ { 2 } ) . \\end{align*}"}
+{"id": "8318.png", "formula": "\\begin{align*} \\begin{aligned} & z '' _ { b e n } + ( M ^ S _ { b e n } - P ) z _ { b e n } = 0 \\\\ & z '' _ { n b } + ( M ^ S _ { n b } - P ) z _ { n b } = 0 , \\end{aligned} \\end{align*}"}
+{"id": "1582.png", "formula": "\\begin{align*} h : = \\left ( \\prod \\limits _ { j = 1 } ^ s \\frac { 1 } { y _ j } \\right ) \\left ( \\sum \\limits _ { k , k ' = 1 } ^ s \\frac { 1 } { y _ k y _ { k ' } } d x _ k \\otimes d x _ { k ' } + d y _ k \\otimes d y _ { k ' } \\right ) + d x _ { s + 1 } ^ 2 + d y _ { s + 1 } ^ 2 , \\end{align*}"}
+{"id": "937.png", "formula": "\\begin{align*} u _ 2 ( t ) = M ( t ) D ( t ) \\mathcal { F } M ( t ) \\mathcal { F } ^ { - 1 } v _ 2 ( t ) . \\end{align*}"}
+{"id": "6562.png", "formula": "\\begin{align*} X _ { T } : = \\{ f \\in \\mathcal { C } ( [ 0 , T ] ; L ^ { p } ( \\Omega ) ) \\ , | \\ , \\| f \\| _ { L ^ { \\infty } ( 0 , T ; L ^ { p } ( \\Omega ) ) } \\le M , \\ , \\ , \\ , f \\ge 0 \\ , \\mbox { f o r } \\ , t \\le T \\} . \\end{align*}"}
+{"id": "2166.png", "formula": "\\begin{align*} \\min _ { \\substack { \\omega _ { n , k } , \\ , { { \\bf { u } } } _ k \\\\ { \\alpha _ z , \\ss _ { z } , \\gamma ( . ) } } } & \\mathcal { L } \\\\ & \\omega _ { n , k } \\in \\{ 0 , 1 \\} , \\forall n , \\forall k . \\end{align*}"}
+{"id": "3230.png", "formula": "\\begin{align*} Y ^ T _ t : = & \\mathcal S ( T ) Y _ 0 + \\int _ 0 ^ t \\mathcal S ( T - s ) \\alpha _ s d s + \\int _ 0 ^ t \\mathcal S ( T - s ) \\sigma _ s d W _ s \\\\ = & \\tilde Y _ 0 + \\int _ 0 ^ t \\tilde { \\alpha } _ s d s + \\int _ 0 ^ t \\tilde { \\sigma } _ s d W _ s , t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "4951.png", "formula": "\\begin{align*} E _ { \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\biggr ] = E _ { \\xi _ 0 } \\left [ h \\bigl ( x , Z _ 1 ^ \\theta \\bigr ) \\right ] , \\forall x \\in \\mathbb R , \\end{align*}"}
+{"id": "8365.png", "formula": "\\begin{align*} w _ - = 0 , \\ w _ + = J , \\ w = w _ - + w _ + = J . \\end{align*}"}
+{"id": "923.png", "formula": "\\begin{align*} F _ { 1 } ( t , \\xi ) = W _ 1 ( \\xi ) e ^ { - 3 i \\lambda _ { 1 } | W _ 1 ( \\xi ) | ^ { 2 } \\log t } , \\end{align*}"}
+{"id": "668.png", "formula": "\\begin{align*} \\cos ( n \\theta ) = \\sum \\limits _ { l = 0 } ^ n a ^ n _ l P _ l ^ 0 ( \\cos \\theta ) \\ , , \\qquad \\sin ( n \\theta ) = \\sum \\limits _ { l = 0 } ^ n b ^ n _ l P _ l ^ 1 ( \\cos \\theta ) \\ , . \\end{align*}"}
+{"id": "9041.png", "formula": "\\begin{align*} \\omega ( E , \\epsilon ) = - 1 , L ( E , \\epsilon ) = - \\epsilon + \\log | \\lambda | . \\end{align*}"}
+{"id": "2929.png", "formula": "\\begin{align*} K _ { \\alpha } \\left ( t \\right ) = L _ { \\alpha } ^ { M } \\left ( t \\right ) + L _ { \\alpha } ^ { P } \\left ( t \\right ) \\end{align*}"}
+{"id": "3154.png", "formula": "\\begin{align*} \\sum \\limits _ { x , y \\in \\mathbb { F } _ q , x \\neq 1 } \\overline { \\chi _ 4 } ( x ) \\chi _ 4 ( y ) \\chi _ 4 ( 1 - y ) \\chi _ 4 ( x - y ) = - 2 \\rho \\end{align*}"}
+{"id": "8862.png", "formula": "\\begin{align*} M _ p = \\max \\{ d ( x _ 0 , p ) , d ( u , p ) \\} . \\end{align*}"}
+{"id": "1429.png", "formula": "\\begin{align*} \\P _ x ( \\rho > n ) & = \\mathfrak { X } \\Delta ( x ) n ^ { - d ( d - 1 ) / 4 } \\bigg ( 1 + \\sum _ { r = 1 } ^ { \\infty } U _ r ( x ) n ^ { - r } \\bigg ) + O \\left ( \\frac { 1 + ( x _ d - x _ 1 ) ^ { 2 N + 1 } } { n ^ { N + 1 / 2 } } \\right ) \\end{align*}"}
+{"id": "734.png", "formula": "\\begin{align*} v ( x , t ) = \\frac { 1 } { R ^ { 2 } - | x | ^ { 2 } } + \\mu t + M \\end{align*}"}
+{"id": "8285.png", "formula": "\\begin{align*} \\d X _ t ^ i = b ^ i ( X _ t ) \\d t + \\sigma \\d W _ t ^ i , ~ ~ \\tilde X _ { n + 1 } ^ i = \\tilde X _ n ^ i + b ^ i ( \\tilde X _ n ) \\tau + \\sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , ~ ~ i = 1 , \\cdots , N . \\end{align*}"}
+{"id": "1136.png", "formula": "\\begin{align*} \\| \\mathbf { c } _ X \\| _ 2 \\| \\mathbf { c } _ Y \\| _ 2 \\leq N L \\| \\mathbf { x } \\| _ 2 \\| \\mathbf { y } \\| _ 2 = N L , \\end{align*}"}
+{"id": "8583.png", "formula": "\\begin{align*} h _ 0 \\ , * \\ , k _ 1 \\ , * \\ , k _ 2 \\ , = \\ , h _ 1 \\ , \\Rightarrow \\ , k _ 1 \\ , * \\ , k _ 2 \\ , = \\ , h _ 1 \\end{align*}"}
+{"id": "1647.png", "formula": "\\begin{align*} \\theta _ s ( f ^ \\zeta ) ( \\gamma _ 1 , \\dots , \\gamma _ s ) = \\theta _ r ( f ) ( \\gamma _ { \\zeta ( 1 ) } , \\dots , \\gamma _ { \\zeta ( r ) } ) , \\end{align*}"}
+{"id": "1213.png", "formula": "\\begin{align*} h _ \\Lambda ( \\nu | \\mu ) : = \\begin{cases} \\sum _ { \\omega _ \\Lambda \\in \\Omega _ \\Lambda } \\nu ( \\omega _ { \\Lambda } ) \\log \\frac { \\nu ( \\omega _ { \\Lambda } ) } { \\mu ( \\omega _ \\Lambda ) } , & \\nu \\ll \\mu , \\\\ \\ \\infty , & \\end{cases} \\end{align*}"}
+{"id": "6304.png", "formula": "\\begin{align*} 2 \\alpha _ t ^ { - 1 } ( Q _ t - R _ t ) = \\ & - \\beta _ t \\| y ^ t - z ^ { t + 1 } \\| ^ 2 - ( 1 - \\beta _ t ) \\| z ^ t - z ^ { t + 1 } \\| ^ 2 + \\alpha _ t ^ { - 2 } \\| x ^ { t + 1 } - y ^ t \\| ^ 2 - \\mu \\gamma _ t \\alpha _ t ^ { - 2 } ( 1 - \\alpha _ t ) \\| x ^ t - y ^ t \\| ^ 2 \\\\ \\le \\ & - \\beta _ t \\| y ^ t - z ^ { t + 1 } \\| ^ 2 - ( 1 - \\beta _ t ) \\| z ^ t - z ^ { t + 1 } \\| ^ 2 + \\alpha _ t ^ { - 2 } \\| x ^ { t + 1 } - y ^ t \\| ^ 2 \\leq 0 , \\end{align*}"}
+{"id": "3580.png", "formula": "\\begin{align*} A ^ { 1 , \\dots , k } : = A ^ { 1 } A ^ { 2 } \\dots A ^ { k } . \\end{align*}"}
+{"id": "852.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } t } \\int _ { \\Gamma ( t ) } f \\ , \\mathrm { d } \\mathcal { H } ^ d = \\int _ { \\Gamma ( t ) } \\partial ^ \\square f - f H V \\ , \\mathrm { d } \\mathcal { H } ^ d . \\end{align*}"}
+{"id": "5288.png", "formula": "\\begin{align*} \\Sigma ^ { \\pm } ( s - w , \\phi , F _ { \\pm } ) = & \\lim _ { z \\downarrow 0 } \\int _ 0 ^ \\infty w _ z ( \\lambda ) \\lambda ^ { 4 ( 1 - s + w ) } \\int _ { - \\infty } ^ \\infty \\int _ { 0 } ^ \\infty \\phi ( n _ { u } ^ t a _ { \\frac { 1 } { t } } ) \\\\ & \\times \\sum _ { m = 1 } ^ \\infty \\sum _ { n = 0 } ^ { 3 m - 1 } \\hat { F } ( \\lambda a _ { \\frac { 1 } { t } } n _ u \\cdot ( 0 , 0 , 3 m , n ) ) \\frac { d \\lambda } { \\lambda } \\frac { d t } { t ^ 3 } d u . \\end{align*}"}
+{"id": "1401.png", "formula": "\\begin{align*} \\frac { d ^ k } { d x ^ k } q _ n ( x ) & = q _ { n - k } ( x ) , x > 0 , \\\\ \\int _ 0 ^ x \\frac { ( x - u ) ^ { k - 1 } } { ( k - 1 ) ! } q _ n ( u ) d u & = q _ { n + k } ( x ) , x > 0 . \\end{align*}"}
+{"id": "6735.png", "formula": "\\begin{align*} \\log \\phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) & = \\log \\left [ \\frac { s } { 2 \\Gamma ( 1 / s ) } \\exp \\left ( - \\left | \\frac { x - \\mu } { \\sigma } \\right | ^ s \\right ) \\right ] \\\\ & = \\log s - \\log ( 2 \\Gamma ( 1 / s ) ) - \\left | \\frac { x - \\mu } { \\sigma } \\right | ^ s . \\end{align*}"}
+{"id": "3173.png", "formula": "\\begin{align*} k _ 3 ( \\langle H \\rangle ) = \\frac { q - 1 } { 1 2 } [ k _ 3 ( \\langle H \\rangle , 1 ) + k _ 3 ( \\langle H \\rangle , g ) ] . \\end{align*}"}
+{"id": "536.png", "formula": "\\begin{align*} c _ i : = 2 a _ 0 - ( a _ { - i } + a _ i ) \\end{align*}"}
+{"id": "885.png", "formula": "\\begin{align*} r ^ 2 + r q + q ^ 2 = n \\ , . \\end{align*}"}
+{"id": "2498.png", "formula": "\\begin{align*} \\begin{aligned} & d _ 2 ( \\mathbf { x } , \\mathbf { s } , \\kappa , \\tau ) = \\sqrt { 2 } \\left \\| \\mathbf { w } _ { x s \\kappa \\tau } - \\mu \\mathbf { e } \\right \\| , \\\\ & d _ \\infty ( \\mathbf { x } , \\mathbf { s } , \\kappa , \\tau ) = \\max \\left ( d _ \\infty ( \\mathbf { x } , \\mathbf { s } ) , \\left | \\kappa \\tau - \\mu \\right | \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "856.png", "formula": "\\begin{align*} N _ f ( X ) = C _ f X + \\Delta \\end{align*}"}
+{"id": "7228.png", "formula": "\\begin{align*} L = a _ 0 x + a _ 1 x ^ q + a _ 2 x ^ { q ^ 2 } + \\cdots + a _ n x ^ { q ^ n } . \\end{align*}"}
+{"id": "8629.png", "formula": "\\begin{align*} \\forall \\varepsilon > 0 , \\lim _ { n \\to \\infty } \\sum _ { m = 1 } ^ n \\mathbb { E } _ { n , m } ' [ \\| \\chi _ { n , m } ' \\| _ { L ^ 2 ( \\mu ) } ^ 2 : \\| \\chi _ { n , m } ' \\| _ { L ^ 2 ( \\mu ) } > \\varepsilon ] = 0 \\end{align*}"}
+{"id": "3110.png", "formula": "\\begin{align*} \\Phi _ { k , r } ( \\omega ) = \\{ x \\in C ( \\omega ) \\cap \\bar { B } ( 0 , r ) \\mid \\langle \\mathcal { F } ( x ^ * ) ( \\omega ) , x - x ^ * ( \\omega ) \\rangle \\le - \\frac { 1 } { k } \\} . \\end{align*}"}
+{"id": "4264.png", "formula": "\\begin{align*} i \\leq N _ v / 2 , \\ \\beta _ i ^ v = \\beta _ i ^ { v _ l } \\oplus \\beta _ i ^ { v _ r } , i > N _ v / 2 , \\ \\beta _ i ^ v = \\beta ^ { v _ r } _ { i - N _ v / 2 } . \\end{align*}"}
+{"id": "5653.png", "formula": "\\begin{align*} D _ { a , k } : = \\begin{pmatrix} D _ { a } \\\\ & \\ddots \\\\ & & D _ { a } \\\\ & & & 1 _ { 2 n - 2 k } \\end{pmatrix} = \\left ( \\bigoplus _ { 1 } ^ { k } D _ { a } \\right ) \\bigoplus 1 _ { 2 ( n - k ) } , D _ { a } : = \\begin{pmatrix} a & 0 \\\\ 0 & a ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
+{"id": "2506.png", "formula": "\\begin{align*} \\begin{aligned} & d _ 2 ( \\mathbf { x } , \\mathbf { s } ) = \\sqrt { 2 } \\left \\| \\mathbf { w } _ { x s } - \\mu \\mathbf { e } \\right \\| , \\\\ & d _ \\infty ( \\mathbf { x } , \\mathbf { s } ) = \\max _ { i = 1 , \\cdots , k , j = 0 , 1 } \\left ( \\left | ( \\mathbf { w } _ { x s } ) _ 0 ^ { ( i ) } + \\| ( \\mathbf { w } _ { x s } ) _ 1 ^ { ( i ) } \\| - \\mu \\right | , \\left | ( \\mathbf { w } _ { x s } ) _ 0 ^ { ( i ) } - \\| ( \\mathbf { w } _ { x s } ) _ 1 ^ { ( i ) } \\| - \\mu \\right | \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "5202.png", "formula": "\\begin{align*} A _ { t } \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) + \\nabla Q \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) = - \\mu \\nabla \\times B \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) \\textrm { o n } \\ L ^ { 1 } ( \\mathbb { R } ^ { 3 } ) , \\end{align*}"}
+{"id": "1336.png", "formula": "\\begin{align*} L ( \\gamma _ { u _ { k - 1 } , u _ k } ) & < d ( u _ { k - 1 } , u _ k ) + \\varepsilon \\delta = d ( u _ { k - 1 } , u _ k ) + \\varepsilon \\min \\bigl \\{ 1 , \\frac { d ( u _ { k - 1 } , u _ k ) } { C _ { n + 1 } \\beta _ k } \\bigr \\} . \\end{align*}"}
+{"id": "7026.png", "formula": "\\begin{align*} u \\ , = \\ , \\tfrac { x ^ 2 } { 2 } + \\tfrac { x ^ 2 y } { 2 } + \\tfrac { x ^ 2 y ^ 2 } { 2 } + F _ { 5 , 0 } \\ , \\tfrac { x ^ 5 } { 1 2 0 } + F _ { 4 , 1 } \\ , \\tfrac { x ^ 4 y } { 2 4 } + \\tfrac { x ^ 2 y ^ 3 } { 2 } + { \\rm O } _ { x , y } ( 6 ) . \\end{align*}"}
+{"id": "3458.png", "formula": "\\begin{align*} \\begin{cases} I ^ - : = [ \\ell q _ n + b _ n , \\ell q _ n + m _ n - 1 ] , \\\\ I ^ + : = [ \\ell q _ n + m _ n + 1 , ( \\ell + 1 ) q _ n - b _ n ] , \\\\ | \\phi ( x _ 0 ^ - ) | : = \\max _ { y \\in I ^ - } | \\phi ( y ) | , \\\\ | \\phi ( x _ 0 ^ + ) | : = \\max _ { y \\in I ^ + } | \\phi ( y ) | , \\\\ R _ { \\ell } : = [ \\ell q _ n - b _ n , \\ell q _ n + b _ n ] , \\\\ r _ { \\ell } : = \\max _ { k \\in R _ { \\ell } } | \\phi ( k ) | . \\end{cases} \\end{align*}"}
+{"id": "9075.png", "formula": "\\begin{align*} \\mathcal { F } _ i : = \\mathcal { H } o m ( \\pi ^ * _ i ( \\iota ^ * _ i ( \\mathcal { P } _ i ) ) , \\pi ^ * _ { i + 1 } ( { \\iota ' } ^ * _ i ( \\mathcal { P } _ { i + 1 } ) ) ) . \\end{align*}"}
+{"id": "6381.png", "formula": "\\begin{align*} f _ { i : n } ( x ) = \\frac { f ( x ) } { \\operatorname { B } ( i , n - i + 1 ) } \\sum _ { l = 0 } ^ { n - i } \\binom { n - i } { l } ( - 1 ) ^ l F ( x ) ^ { i + l - 1 } . \\end{align*}"}
+{"id": "3923.png", "formula": "\\begin{align*} 0 = \\phi ^ \\theta ( e ) = \\tau ( u _ { l - 1 } , e ) \\theta ( u _ { l - 1 } ) + \\phi ( e ) + \\tau ( u _ l , e ) \\theta ( u _ l ) , \\end{align*}"}
+{"id": "7058.png", "formula": "\\begin{align*} A _ { 1 , 1 } \\ , = \\ , \\mp \\ , \\tfrac { 1 } { 2 } \\ , F _ { 5 , 1 , 0 } \\ , T _ 1 - \\tfrac { 3 } { 2 } \\ , T _ 2 , \\end{align*}"}
+{"id": "7612.png", "formula": "\\begin{align*} \\phi _ m ( x ) = \\begin{cases} 1 & x \\in A _ m ^ { ( 1 ) } \\\\ 0 & x \\notin A _ m ^ { ( 2 ) } . \\end{cases} \\end{align*}"}
+{"id": "3073.png", "formula": "\\begin{align*} \\mathfrak { t } : = \\sum _ { j = 0 } ^ 2 \\sum _ { p \\in C \\cap H _ j } ( I _ p ( C , H _ { j } ) - 1 ) . \\end{align*}"}
+{"id": "5234.png", "formula": "\\begin{align*} \\kappa & = C _ 1 J _ { 3 / 2 } ( \\rho ) , \\\\ R _ 0 & = c _ { 3 / 2 } . \\end{align*}"}
+{"id": "741.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } p ^ { \\varepsilon } ( x , t ) = Y _ t h ' ( x ) \\partial _ { x } p ^ { \\varepsilon } ( x , t ) + \\frac { p ^ { \\varepsilon } ( x , t ) } { \\varepsilon } \\left ( - \\frac { h ^ 2 ( x ) } { 2 } + \\frac { \\left ( Y _ t h ' ( x ) \\right ) ^ 2 } { 2 } + \\frac { \\varepsilon } { 2 } Y _ t h '' ( x ) \\right ) + \\frac { \\varepsilon } { 2 } \\partial ^ 2 _ { x x } p ^ { \\varepsilon } ( x , t ) , \\end{align*}"}
+{"id": "6888.png", "formula": "\\begin{align*} \\log Z = \\log \\int _ { \\R ^ n } e ^ f \\ , d \\mu ^ { \\otimes n } = \\sup _ { Q \\in \\P ( \\R ^ n ) } \\left ( \\int _ { \\R ^ n } f \\ , d Q - H ( Q \\ , | \\ , \\mu ^ { \\otimes n } ) \\right ) , \\end{align*}"}
+{"id": "4113.png", "formula": "\\begin{align*} \\forall t > 0 , \\ H _ { m i n } ( E [ t ] ) = t H _ { m i n } ( E ) . \\end{align*}"}
+{"id": "2452.png", "formula": "\\begin{align*} F _ { q } ^ { ( k ) } \\left ( \\frac { k } { x } \\right ) = ( - 1 ) ^ { k } f ( x \\xi _ { m } ) x ^ { k + 1 } \\frac { \\Gamma ( k + 1 ) } { k ^ { k + 1 } ( 2 - q ) } . \\end{align*}"}
+{"id": "3301.png", "formula": "\\begin{align*} & \\| \\hat { \\Sigma } _ t ^ { n } - \\bar { \\Sigma } _ t ^ n \\| _ { L _ { } ( L ^ 2 ( [ 0 , 1 ] ) ) } ^ 2 \\\\ = & \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\left ( \\hat { q } _ t ^ n ( x , y ) - \\bar { q } _ t ^ n ( x , y ) \\right ) ^ 2 d x d y \\\\ = & \\sum _ { j _ 1 , j _ 2 = 0 } ^ n \\Delta _ n ^ 2 \\left ( \\langle \\left ( S A R C V _ t ^ n - \\int _ 0 ^ t \\Sigma _ s d s \\right ) \\delta _ { x _ { j _ 1 } } , \\delta _ { x _ { j _ 2 } } \\right ) ^ 2 \\\\ \\leq & 4 \\| S A R C V _ t ^ n - \\int _ 0 ^ t \\Sigma _ s \\| _ { } ^ 2 , \\end{align*}"}
+{"id": "6443.png", "formula": "\\begin{align*} \\begin{aligned} Z _ { I } ( 1 , \\{ 2 \\} ^ { n } ; ( \\alpha , \\beta ) ) = Z _ { I I } ( \\{ 2 \\} ^ { n - 1 } , 3 ; ( \\alpha , \\beta ) ) \\end{aligned} \\end{align*}"}
+{"id": "9307.png", "formula": "\\begin{align*} \\int _ { \\hat { h } _ { P } ^ { - 1 } ( b ) ( F ) } f _ { \\hat { \\alpha } } \\abs { \\omega _ { \\hat { h } _ { P } } } = \\int _ { \\check { h } _ { P } ^ { - 1 } ( b ) ( F ) } f _ { \\check { \\alpha } } \\abs { \\omega _ { \\check { h } _ { P } } } . \\end{align*}"}
+{"id": "1616.png", "formula": "\\begin{align*} C ^ { n + 1 } _ c ( G , V ) : = C ^ 0 _ c ( G , C ^ n _ c ( G , V ) ) , n \\geq 0 . \\end{align*}"}
+{"id": "7397.png", "formula": "\\begin{align*} [ M ] & = \\frac { \\prod \\limits _ { i = 0 } ^ N v _ i ^ { x _ i } } { \\prod \\limits _ { n = 1 } ^ \\infty ( 1 - z ^ n ) ^ { N } \\prod \\limits _ { n = 1 } ^ \\infty \\prod \\limits _ { 1 \\leq i < j \\leq N } ( 1 - v _ j v _ i ^ { - 1 } z ^ { n - 1 } ) ( 1 - v _ i v _ j ^ { - 1 } z ^ n ) } . \\end{align*}"}
+{"id": "7752.png", "formula": "\\begin{align*} p _ h ( x ) : = \\sum _ { i = 0 } ^ d p _ i ^ { ( h ) } x ^ i = \\prod _ { j = 1 } ^ d ( x - x _ j ^ { ( h ) } ) , ~ x _ j ^ { ( h ) } = x _ j ^ { 2 ^ h } , ~ j = 1 , \\dots , d ; ~ h = 0 , 1 , \\dots , \\end{align*}"}
+{"id": "2386.png", "formula": "\\begin{align*} \\mathbb { T } _ { \\varrho } ( M , \\{ \\mathbf { h } _ p ^ { M } \\} _ { p = 0 } ^ 3 ) = \\frac { \\mathbb { T } _ { { \\psi _ { _ 1 } } } ( M _ 1 , \\{ \\mathbf { h } _ p ^ { M _ 1 } \\} _ { p = 0 } ^ { 3 } ) \\ ; \\mathbb { T } _ { { \\psi _ { _ 2 } } } ( M _ 2 , \\{ \\mathbf { h } _ p ^ { M _ 2 } \\} _ { p = 0 } ^ { 3 } ) } { \\mathbb { T } _ { { \\varrho _ { | _ { \\mathbb { D } ^ 2 } } } } ( \\mathbb { D } ^ 2 , \\{ \\mathbf { h } _ 0 ^ { \\mathbb { D } ^ 2 } \\} ) } . \\end{align*}"}
+{"id": "6392.png", "formula": "\\begin{align*} \\begin{pmatrix} A _ { \\mathrm { c } \\mathrm { c } } - A _ { \\mathrm { R } \\mathrm { c } } ^ { \\top } R + R ^ { \\top } A _ { \\mathrm { R } \\mathrm { c } } & B _ 0 ^ { \\top } \\\\ B _ 0 & 0 \\\\ \\end{pmatrix} \\begin{pmatrix} U _ \\mathrm { c } \\\\ P _ 0 \\end{pmatrix} = \\begin{pmatrix} F _ \\mathrm { c } - R ^ \\top F _ \\mathrm { R } \\\\ 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "3112.png", "formula": "\\begin{align*} \\langle \\mathcal { F } ( x ^ * ) , \\tilde { \\eta } - x ^ * \\rangle _ { \\mathcal { L } ^ 2 } = & \\int _ { \\Omega } \\langle \\mathcal { F } ( x ^ * ) ( \\omega ) , \\tilde { \\eta } ( \\omega ) - x ^ * ( \\omega ) \\rangle P ( d ( \\omega ) ) \\\\ = & \\int _ { A _ { k , r } } \\langle \\mathcal { F } ( x ^ * ) ( \\omega ) , \\eta ( \\omega ) - x ^ * ( \\omega ) \\rangle P ( d ( \\omega ) ) \\\\ < & - \\frac { 1 } { k } P ( A _ { k , r } ) , \\end{align*}"}
+{"id": "4941.png", "formula": "\\begin{align*} \\frac { \\zeta ' } { \\zeta } ( s ) = - \\sum _ { n \\leq x y } \\frac { \\Lambda _ { x , y } ( n ) } { n ^ s } + \\frac { 1 } { \\log { y } } \\sum _ { \\rho } \\frac { x ^ { \\rho - s } - ( x y ) ^ { \\rho - s } } { \\left ( \\rho - s \\right ) ^ { 2 } } + \\frac { 1 } { \\log { y } } \\sum _ { n = 1 } ^ { \\infty } \\frac { x ^ { - 2 n - s } - ( x y ) ^ { - 2 n - s } } { \\left ( 2 n + s \\right ) ^ { 2 } } - \\frac { x ^ { 1 - s } - ( x y ) ^ { 1 - s } } { \\log y \\left ( 1 - s \\right ) ^ { 2 } } . \\end{align*}"}
+{"id": "4313.png", "formula": "\\begin{align*} & \\int _ { U _ 0 } | \\tilde { F } _ 0 - ( 1 - b _ { 1 } ( \\Psi ) ) f _ 0 F ^ { 2 } | ^ 2 _ h e ^ { - \\varphi _ 1 + v _ { 1 } ( \\Psi _ 1 ) - \\Psi _ 1 } c ( - v _ { 1 } ( \\Psi _ 1 ) ) \\\\ \\le & \\liminf _ { j \\to + \\infty } \\int _ { U _ 0 } | \\tilde { F } _ j - ( 1 - b _ { 1 } ( \\Psi ) ) f _ j F ^ { 2 } | ^ 2 _ h e ^ { - \\varphi _ 1 + v _ { 1 } ( \\Psi ) - \\Psi } c ( - v _ { 1 } ( \\Psi _ 1 ) ) \\\\ < & + \\infty , \\end{align*}"}
+{"id": "4517.png", "formula": "\\begin{align*} \\{ A ( \\xi ) \\} \\cap \\mathrm { K e r } ( B ) = \\{ 0 \\} , \\forall \\xi \\in \\R ^ * . \\end{align*}"}
+{"id": "3975.png", "formula": "\\begin{align*} c _ { k + 1 } = \\sum _ { i = 1 } ^ { t - 1 } c _ i a _ i + c _ { k } \\sum _ { i \\in \\Omega _ t ^ e } a _ i + c _ { k + 1 } \\sum _ { i \\in \\Omega _ t ^ o } a _ i , \\end{align*}"}
+{"id": "1958.png", "formula": "\\begin{align*} \\varphi _ { + } ( F ( z ) ) = T _ 1 ( \\varphi _ { + } ( z ) ) \\end{align*}"}
+{"id": "1633.png", "formula": "\\begin{align*} ( \\varphi \\psi ) ( h ) = \\varphi ( h _ { ( 1 ) } ) \\psi ( h _ { ( 2 ) } ) = \\left < h _ { ( 1 ) } , \\varphi \\right > \\left < h _ { ( 2 ) } , \\psi \\right > \\end{align*}"}
+{"id": "6998.png", "formula": "\\begin{align*} \\alpha _ 1 : = \\{ \\{ 1 , 2 \\} , \\{ 3 , 4 \\} , \\{ 5 , 6 \\} , \\{ 7 , 8 \\} , \\ldots , \\{ 2 b - 1 , 2 b \\} \\} , \\\\ \\alpha _ 2 : = \\{ \\{ 1 , 4 \\} , \\{ 3 , 2 \\} , \\{ 5 , 6 \\} , \\{ 7 , 8 \\} , \\ldots , \\{ 2 b - 1 , 2 b \\} \\} , \\\\ \\alpha _ 3 : = \\{ \\{ 1 , 6 \\} , \\{ 3 , 4 \\} , \\{ 5 , 2 \\} , \\{ 7 , 8 \\} , \\ldots , \\{ 2 b - 1 , 2 b \\} \\} , \\\\ \\alpha _ 4 : = \\{ \\{ 1 , 8 \\} , \\{ 3 , 4 \\} , \\{ 5 , 6 \\} , \\{ 7 , 8 \\} , \\ldots , \\{ 2 b - 1 , 2 b \\} \\} . \\end{align*}"}
+{"id": "2698.png", "formula": "\\begin{align*} \\mathcal { F } ^ \\prime _ { k - 1 } = \\sigma \\left ( \\left ( \\xi ^ { ( 0 ) } _ 0 , { \\xi ^ { ( 0 + ) } _ { 0 } } , \\xi ^ { ( 1 ) } _ 0 , \\xi ^ { ( 2 ) } _ 0 \\right ) , \\dots , \\left ( \\xi ^ { ( 0 ) } _ { k - 1 } , { \\xi ^ { ( 0 + ) } _ { k - 1 } } , \\xi ^ { ( 1 ) } _ { k - 1 } , \\xi ^ { ( 2 ) } _ { k - 1 } \\right ) , \\left ( \\xi ^ { ( 1 ) } _ k , \\xi ^ { ( 2 ) } _ k \\right ) \\right ) \\end{align*}"}
+{"id": "6205.png", "formula": "\\begin{align*} \\xi ' = - L - 2 , \\eta ' = \\frac { Q } { 2 ( L + 2 ) } - \\sqrt { \\kappa B _ 4 } , \\zeta ' = \\kappa \\left ( \\frac { B _ 3 } { 2 \\sqrt { \\kappa B _ 4 } } + 3 \\right ) , \\sigma ' = \\sqrt { \\kappa B _ 4 } , \\end{align*}"}
+{"id": "7429.png", "formula": "\\begin{align*} p _ k = \\log _ 2 \\binom { 2 k } { k } . \\end{align*}"}
+{"id": "8927.png", "formula": "\\begin{align*} L _ { n } ( x | \\beta ) = \\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \\frac { ( \\beta ) ^ { ( n ) } } { ( n - k ) ! ( \\beta ) ^ { ( k ) } } x ^ { k } / k ! . \\end{align*}"}
+{"id": "900.png", "formula": "\\begin{align*} \\Sigma \\big ( X , q \\big ) = X \\frac { \\lambda ( q ) } { q } + \\mathcal { O } \\big ( X ^ \\varepsilon \\big ) \\ , . \\end{align*}"}
+{"id": "6921.png", "formula": "\\begin{align*} g _ 1 ^ { \\prime + } ( 0 ) & = \\sum _ { i = 1 } ^ n \\int _ { \\R ^ n } \\partial _ i f ( x ) \\big ( T _ i ( x _ i ) - x _ i \\big ) Q ( x ) d x , \\\\ g _ 2 ^ { \\prime + } ( 0 ) & = - \\sum _ { i = 1 } ^ n \\int _ { \\R } \\left ( T _ i ' ( x _ i ) - 1 \\right ) Q _ i ( x _ i ) d x _ i . \\end{align*}"}
+{"id": "6174.png", "formula": "\\begin{align*} V ( r ) = \\frac { L ( L + 1 ) } { r ^ 2 } - \\frac { Q } { r } f + \\kappa \\sum _ { k = 1 } ^ m \\left ( B _ { 2 k - 1 } \\frac { r } { f ^ { 2 k - 1 } } + B _ { 2 k } \\frac { 1 } { f ^ { 2 k } } \\right ) , \\end{align*}"}
+{"id": "6377.png", "formula": "\\begin{align*} H _ \\alpha ( X ) = \\frac { 1 } { 1 - \\alpha } \\log \\left ( \\int ^ \\infty _ 0 f ( x ) ^ { \\alpha } \\mathrm { d } x \\right ) , \\alpha > 0 , \\ ; \\alpha \\neq 1 , \\end{align*}"}
+{"id": "1306.png", "formula": "\\begin{align*} E ( t ) & = \\frac { 1 } { 2 } | | u _ t | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } + \\frac { m } { 2 } | | u | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } + \\frac { 1 } { 2 } | | \\nabla _ { H } u | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } - \\int _ { \\mathbb { H } ^ n } F ( u ) d x , \\end{align*}"}
+{"id": "2969.png", "formula": "\\begin{align*} \\mathcal { O } ( x ) : = \\{ \\Phi ( t , x ) \\mid ( t , x ) \\in \\mathrm { d o m } \\ , \\Phi \\} . \\end{align*}"}
+{"id": "4810.png", "formula": "\\begin{align*} F _ n ( \\eta ) : = T _ n - \\frac { 1 } { \\eta } I _ n . \\end{align*}"}
+{"id": "1162.png", "formula": "\\begin{align*} d X _ t = \\left ( b _ 0 ( t , X ) + \\langle b ( t , X , \\cdot ) , \\mu \\rangle \\right ) d t + d B ^ H _ t , \\quad \\operatorname { L a w } ( X ) = \\mu , \\quad \\operatorname { L a w } ( X _ 0 ) = \\mu _ 0 . \\end{align*}"}
+{"id": "8087.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\mathbf { y ' } ^ H \\mathbf { y } ' \\right ] = \\sum _ { l = 1 } ^ M \\sum \\limits _ { \\substack { j = 1 \\\\ j \\neq l } } ^ M \\mathbb { E } \\left [ y _ l ^ * y _ j \\right ] + 2 \\sum _ { i = 1 } ^ { M } \\mathbb { E } \\left [ y _ i ^ * y _ i \\right ] , \\end{align*}"}
+{"id": "4945.png", "formula": "\\begin{align*} \\xi _ 1 ^ \\theta ( s ) = \\left \\{ \\begin{aligned} & \\frac { \\xi _ 0 f _ 1 ( s ) } { \\xi _ 0 f _ 1 ( s ) + ( 1 - \\xi _ 0 ) f _ 2 ( s ) } , & & \\\\ & \\frac { \\xi _ 0 f _ 2 ( s ) } { \\xi _ 0 f _ 2 ( s ) + ( 1 - \\xi _ 0 ) f _ 1 ( s ) } , & & \\end{aligned} \\right . \\end{align*}"}
+{"id": "5983.png", "formula": "\\begin{align*} | B _ r | \\| S _ { B _ r } f \\| _ { L ^ 2 _ { a v g } } ^ 4 & = | B _ r | ^ { - 1 } \\left ( \\int \\sum _ { \\tau } | f _ \\tau | ^ 2 W _ { B _ r } \\right ) ^ 2 \\\\ & \\lesssim \\sum _ { j = 1 } ^ { 1 0 0 0 } | B _ r | ^ { - 1 } \\left ( \\int \\sum _ { \\tau } | f _ { j , \\tau } | ^ 2 W _ { B _ r } \\right ) ^ 2 \\\\ & = \\sum _ { j = 1 } ^ { 1 0 0 0 } | B _ r | ^ { - 1 } \\left ( \\int | f _ { j } | ^ 2 W _ { B _ r } \\right ) ^ 2 . \\end{align*}"}
+{"id": "5666.png", "formula": "\\begin{align*} ( e _ { 1 } . \\dots , e _ { k } ) = D _ { a , k } ( a ^ { - 1 } e _ { 1 } , \\dots , a ^ { - 1 } e _ { k } ) = \\kappa ( a ^ { - 1 } e _ { 1 } , \\dots , a ^ { - 1 } e _ { k } ) . \\end{align*}"}
+{"id": "5052.png", "formula": "\\begin{align*} h _ C = 2 \\lambda ^ { 1 / 2 } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { C , + } ^ { 2 } \\frac { 1 } { r ^ { 2 } } \\dd x > 0 . \\end{align*}"}
+{"id": "4247.png", "formula": "\\begin{align*} | z - w | & \\leq | z - w _ R | + | w _ R - w | \\\\ & < R \\sqrt { s } + R ^ 2 \\Big ( \\dfrac { 1 - | w | ^ 2 } { 1 - R ^ 2 | w | ^ 2 } \\Big ) \\\\ & < 2 R \\leq 2 r = 2 \\beta ( z , w ) . \\end{align*}"}
+{"id": "8284.png", "formula": "\\begin{align*} \\gamma ^ i ( x ) : = \\frac 1 { N - 1 } \\sum _ { j \\neq i } K ( x ^ i - x ^ j ) , ~ ~ ~ ~ \\forall x \\in \\mathbb R ^ { N d } , \\end{align*}"}
+{"id": "6091.png", "formula": "\\begin{align*} w _ { 1 , 2 , 3 } = 0 \\ , w _ { 1 , 2 , 4 } = 0 \\ , w _ { 1 , 3 , 4 } = 0 \\ , w _ { 2 , 3 , 4 } = 0 \\ , x _ { 1 2 3 4 } = 0 \\ , . \\end{align*}"}
+{"id": "847.png", "formula": "\\begin{align*} \\partial _ t c _ { \\min \\ , | t } = \\partial _ t c _ { | ( t , p ) } \\end{align*}"}
+{"id": "6307.png", "formula": "\\begin{align*} \\gamma _ { t - 1 } \\alpha ( \\gamma ) ^ 2 = ( 1 - \\alpha ( \\gamma ) ) \\alpha _ { t - 1 } ^ 2 \\gamma + \\mu \\gamma \\gamma _ { t - 1 } \\alpha ( \\gamma ) . \\end{align*}"}
+{"id": "5305.png", "formula": "\\begin{align*} f ( z , t ) = \\int _ \\Omega \\int _ { \\C ^ n } \\widehat { f } ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \\ , d \\nu ( a ) . \\end{align*}"}
+{"id": "5927.png", "formula": "\\begin{align*} & \\lim _ { M \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ T \\psi ( t ) \\Big [ \\Vert X ( t \\wedge \\tau _ u ^ M ) \\Vert _ H ^ 2 - \\Vert x \\Vert _ H ^ 2 \\Big ] d t \\\\ = & \\lim _ { M \\rightarrow \\infty } \\liminf _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ T \\psi ( t ) \\Big [ \\Vert X _ n ( t \\wedge \\tau _ u ^ M ) \\Vert _ H ^ 2 - \\Vert P _ n x \\Vert _ H ^ 2 \\Big ] d t . \\end{align*}"}
+{"id": "5634.png", "formula": "\\begin{align*} \\psi _ { 2 n } = \\begin{pmatrix} \\psi _ { 2 } \\\\ & \\psi _ { 2 } \\\\ & & \\ddots \\\\ & & & \\psi _ { 2 } \\end{pmatrix} = \\bigoplus _ { 1 } ^ { n } \\psi _ { 2 } , \\psi _ { 2 } = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "3085.png", "formula": "\\begin{align*} \\zeta _ { x } ^ k : = F ( x ^ k ) - F ( \\tilde { x } ^ k ) + \\beta ( x ^ k - \\tilde { x } ^ k ) , \\end{align*}"}
+{"id": "5466.png", "formula": "\\begin{align*} ( h _ k , \\ldots , h _ 1 , h _ 0 ) \\cdot ( x _ { k k - 1 } , \\ldots , x _ { 2 1 } , x _ { 1 0 } ) = ( h _ { k - 1 } x _ { k k - 1 } h _ k ^ { - 1 } , \\ldots , h _ 1 x _ { 2 1 } h _ 2 ^ { - 1 } , h _ 0 x _ { 1 0 } h _ 1 ^ { - 1 } ) . \\end{align*}"}
+{"id": "6465.png", "formula": "\\begin{align*} M _ 1 = & \\bigcup _ { i = 1 } ^ { \\infty } \\Big ( [ \\epsilon _ i \\mathcal { H } _ { n _ i } , \\mathcal { H } _ { n _ i } - L _ i ] \\cup [ \\mathcal { H } _ { n _ i } + L _ i , \\mathcal { H } _ { n _ i + 1 } ] \\cap \\N \\Big ) , \\\\ M _ 2 = & M \\setminus M _ 1 . \\end{align*}"}
+{"id": "7890.png", "formula": "\\begin{align*} h ( q , r _ 0 ) = - \\hat W _ { r _ 0 } ( q ) = - \\frac { 2 } { \\pi q } \\sin ( 2 \\pi q r _ 0 ) . \\end{align*}"}
+{"id": "1623.png", "formula": "\\begin{align*} g \\cdot \\overline { h } : = \\overline { g h } . \\end{align*}"}
+{"id": "3464.png", "formula": "\\begin{align*} & \\prod _ { j = k } ^ { x _ 2 } | \\cos ( \\pi \\theta _ j ) ) | \\leq C ( \\varepsilon ) c _ { n , \\ell } e ^ { ( - \\ln { 2 } + \\varepsilon ) | x _ 2 - k | } , \\ \\ \\\\ & \\prod _ { j = x _ 1 } ^ k | \\cos ( \\pi \\theta _ j ) ) | \\leq C ( \\varepsilon ) c _ { n , \\ell - 1 } e ^ { ( - \\ln 2 + \\varepsilon ) | k - x _ 1 | } . \\end{align*}"}
+{"id": "9338.png", "formula": "\\begin{align*} \\xi ^ * ( \\eta ) \\ge \\xi ^ * \\left . \\left ( \\frac { 9 9 } { 1 0 0 } \\right ) \\right | _ { a ^ 2 = \\frac { 9 } { 1 0 } } \\geq 0 . 9 4 4 4 3 7 > \\frac { 4 7 } { 5 0 } . \\end{align*}"}
+{"id": "7246.png", "formula": "\\begin{align*} L ( x ) = a _ 0 x + a _ 1 x ^ { q ^ s } + a _ 2 x ^ { q ^ { 2 s } } + \\cdots + a _ m x ^ { q ^ { m s } } \\in F [ x ] , \\end{align*}"}
+{"id": "4963.png", "formula": "\\begin{align*} \\bar \\theta _ 1 & = 1 , \\\\ \\bar \\theta _ i & = \\pi ^ { [ n - 1 ] } _ { i - 1 } ( \\xi ^ { \\bar \\theta } _ 1 ( X _ 1 ) , x + X _ 1 , X _ 2 , Y _ 2 , \\dots , X _ { i - 1 } , Y _ { i - 1 } ) , \\end{align*}"}
+{"id": "7643.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\varphi _ t ^ { * , \\xi } = & - [ ( A - B ^ 2 R ^ { - 1 } P _ t ) \\varphi _ t ^ { * , \\xi } + P _ t f ( \\nu _ t ^ { * , \\xi } ) + P _ t b ( \\mu _ t ^ { * , \\xi } ) \\\\ & + Q l ( \\nu _ t ^ { * , \\xi } ) - P _ t B h ( \\mu _ t ^ { * , \\xi } ) ] d t + \\Lambda _ t ^ { 0 , * , \\xi } d W ^ 0 _ t , \\\\ \\varphi _ T ^ { * , \\xi } = & ~ G g ( \\nu _ T ^ { * , \\xi } ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "9055.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ L z _ { s , k } \\frac { \\partial } { \\partial z _ { s , k } } \\Lambda _ a = \\left \\{ \\begin{array} { l l } 0 & \\mbox { f o r $ 1 \\leq s \\leq m - a $ , } \\\\ 1 & \\mbox { f o r $ m - a < s \\leq m $ . } \\end{array} \\right . \\end{align*}"}
+{"id": "5551.png", "formula": "\\begin{align*} \\hat { A } ( y | x ) = A ( x ) \\ , \\ \\forall \\ ( y | x ) \\in \\hat { \\Omega } \\ . \\end{align*}"}
+{"id": "6245.png", "formula": "\\begin{align*} \\tilde { Q } _ { i j } ( H ) = \\partial ^ 2 _ { i j } H - \\tilde { \\Gamma } ^ i _ { j i } \\partial _ i H - \\tilde { \\Gamma } ^ j _ { i j } \\partial _ j H = 0 , \\quad \\forall i < j , \\end{align*}"}
+{"id": "8101.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = | x | ^ { a } v ^ { p } & & \\quad \\mbox { i n } B _ R , \\\\ \\Delta v & = | x | ^ { b } v ^ { q } f ( | \\nabla u | ) & & \\quad \\mbox { i n } B _ R , \\end{aligned} \\right . \\end{align*}"}
+{"id": "6087.png", "formula": "\\begin{align*} & [ C _ I , C _ J ] = 0 I \\cap J = \\emptyset \\ \\ \\ \\ I \\subseteq J \\ , , \\\\ & C _ I = \\frac { 1 } { 2 } \\sum _ { i , j \\in I \\atop i \\neq j } C _ { i j } - ( | I | - 2 ) \\sum _ { i \\in I } C _ i \\ , , \\end{align*}"}
+{"id": "5453.png", "formula": "\\begin{align*} [ \\phi _ { \\lambda } \\psi ] _ { \\mu } a = \\phi _ { \\lambda } ( \\psi _ { \\mu - \\lambda } a ) - \\psi _ { \\mu - \\lambda } ( \\phi _ { \\lambda } a ) , \\forall \\ a \\in \\mathcal { A } , \\end{align*}"}
+{"id": "4201.png", "formula": "\\begin{align*} \\int _ { \\hat { B } _ { \\delta } ( 0 ) \\setminus \\hat { B } _ { \\varepsilon } ( 0 ) } q ^ 2 ( K _ { H } ) _ { 2 2 } ( x + \\bar { x } ) \\partial _ 2 \\hat { U } \\left ( \\frac { x } { \\varepsilon } \\right ) \\partial _ 2 \\hat { U } \\left ( \\frac { x } { \\varepsilon } \\right ) d x = \\frac { l _ 1 } { l _ 2 } \\pi q ^ 2 ( K _ { H } ) _ { 2 2 } ( \\bar { x } ) \\ln \\frac { \\delta } { \\varepsilon } + O ( 1 ) . \\end{align*}"}
+{"id": "7951.png", "formula": "\\begin{align*} \\partial _ s \\sigma _ k + \\mathbf l ' _ k H ( \\mathbf r _ k , \\mathbf r _ k ) \\big ( m \\cdot \\nabla _ u \\big ) \\big ( 2 ^ { - 1 } \\sigma _ k ^ 2 \\big ) = 0 , \\forall \\ , k = 1 , \\ldots , n . \\end{align*}"}
+{"id": "3771.png", "formula": "\\begin{align*} \\begin{aligned} & w ( j ) < w ( i _ 2 ) = w ( l ) w ( j ) > w ( i _ 1 ) i _ 1 < j < i _ 2 , \\\\ & w ( i _ 1 ) > w ( l + 1 ) > w ( i _ 2 ) = w ( l ) . \\end{aligned} \\end{align*}"}
+{"id": "8759.png", "formula": "\\begin{align*} K ( w - z ) = & \\frac { 1 } { 2 \\pi i } \\frac { 1 } { w - z } = \\frac { 1 } { 2 \\pi i } \\frac { 1 } { w - z } = \\frac { 1 } { 2 \\pi i ( w - a ) } \\frac { 1 } { 1 - \\frac { z - a } { w - a } } \\\\ \\\\ = & \\frac { 1 } { 2 \\pi i } \\sum _ { n = 0 } ^ { \\infty } ( z - a ) ^ n ( w - a ) ^ { - ( n + 1 ) } = \\frac { 1 } { 2 \\pi i } \\sum _ { n = 0 } ^ { \\infty } u _ n ( z ) v _ n ( w ) , \\end{align*}"}
+{"id": "1983.png", "formula": "\\begin{align*} f ( x - T ( x ) ) & = f ( z ) . \\end{align*}"}
+{"id": "5016.png", "formula": "\\begin{align*} S t _ { D N D C } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( f , z ) - f ( z ) | | _ { \\infty } = 0 \\end{align*}"}
+{"id": "1753.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ { \\Omega } \\int _ { Y ^ { \\ast } } D _ 0 ( t , x , y ) \\big [ \\nabla _ x u _ 0 ( t , x ) + \\nabla _ y u _ 1 ( x , y ) \\big ] \\cdot \\nabla _ y \\phi ( t , x , y ) d y d x d t = 0 . \\end{align*}"}
+{"id": "2215.png", "formula": "\\begin{align*} \\mathbf { v } ^ { 0 } = \\mathbf { f } & = [ e ^ { - j \\frac { 2 \\pi } { N } / \\sqrt { N } } , e ^ { - j \\frac { 2 \\pi } { N } 2 / \\sqrt { N } } , \\cdots , e ^ { - j \\frac { 2 \\pi } { N } N / \\sqrt { N } } ] ^ { T } \\\\ & = [ e ^ { - j A } , e ^ { - j 2 A } , \\cdots , e ^ { - j N A } ] ^ { T } . \\end{align*}"}
+{"id": "1339.png", "formula": "\\begin{align*} y = \\sum _ { j = 1 } ^ { m } \\bigl ( \\alpha _ j \\ , m _ { a _ j , c _ j } + \\alpha _ { m + j } \\ , m _ { c _ j , d _ j } + \\alpha _ { 2 m + j } \\ , m _ { d _ j , b _ j } \\bigr ) + \\sum _ { k = 1 } ^ l \\beta _ k \\ , m _ { u _ { k - 1 } , u _ k } . \\end{align*}"}
+{"id": "6128.png", "formula": "\\begin{align*} \\widetilde { \\lambda } \\ , ' ( m ) \\ r _ n ( m ) = E ' _ { n } \\ \\widetilde { r } \\ , ' _ { n + 1 } ( m ) + F ' _ n \\ \\widetilde { r } \\ , ' _ { n } ( m ) + G \\ , ' _ n \\ \\widetilde { r } \\ , ' _ { n - 1 } ( m ) \\ , , \\end{align*}"}
+{"id": "3284.png", "formula": "\\begin{align*} \\| ( I - P _ N ) A \\| _ { \\mathcal H } & = \\| \\sum _ { k , l \\leq N - 1 } \\langle A , e _ k \\otimes e _ l \\rangle _ { \\mathcal H } e _ k \\otimes e _ l \\| _ { \\mathcal H } \\\\ & \\leq \\sum _ { k , l \\leq N - 1 } | \\langle A , e _ k \\otimes e _ l \\rangle _ { \\mathcal H } | \\| e _ k \\otimes e _ l \\| _ { \\mathcal H } \\\\ & = \\sum _ { k , l \\leq N - 1 } | \\langle A , e _ k \\otimes e _ l \\rangle _ { \\mathcal H } | \\\\ & = \\sum _ { k , l \\leq N - 1 } | \\langle A e _ k , e _ l \\rangle | . \\end{align*}"}
+{"id": "8410.png", "formula": "\\begin{align*} \\lim _ { | z | \\rightarrow \\infty } z ( \\Psi ^ \\pm _ 1 ( x ; z ) - e ^ { - i c _ \\pm ( x ) } e _ 1 ) = \\widehat { \\Psi } ^ \\pm _ { 1 1 } ( x ) e _ 1 + \\widehat { \\Psi } ^ \\pm _ { 2 1 } ( x ) e _ 2 . \\end{align*}"}
+{"id": "3688.png", "formula": "\\begin{align*} \\Gamma _ { ( g , u ) } ( X ) = - \\Delta _ g X - u ^ { - 1 } \\nabla X ( \\nabla _ g u , \\cdot ) + u ^ { - 2 } X ( u ) d u . \\end{align*}"}
+{"id": "1347.png", "formula": "\\begin{align*} \\bigl \\| x \\pm \\frac { y } { \\| y \\| } \\bigr \\| & \\leq \\bigl \\| x - \\sum _ { j = 1 } ^ m \\alpha _ j \\ , m _ { a _ j , b _ j } \\bigr \\| + \\bigl \\| \\sum _ { j = 1 } ^ m \\alpha _ j \\ , m _ { a _ j , b _ j } \\pm y \\bigr \\| + \\bigl | \\| y \\| - 1 \\bigr | \\\\ & < \\frac { \\varepsilon } { 5 } + \\sum _ { j = 1 } ^ m \\alpha _ j \\| m _ { a _ j , b _ j } \\pm y _ j \\| + \\frac { 2 \\varepsilon } { 5 } \\leq 1 + \\varepsilon . \\end{align*}"}
+{"id": "3574.png", "formula": "\\begin{align*} B ( t ) - A ( t ) & = \\varphi ( - t ) ^ 2 ( G ( t ) H ( - t ) - G ( - t ) H ( t ) ) + 2 \\varphi ( t ) ^ 2 G ( t ) H ( t ) \\\\ & = \\varphi ( - t ^ 2 ) \\dfrac { 2 t \\psi ( t ^ { 1 0 } ) } { \\varphi ( t ^ 2 ) } + 2 \\varphi ( t ) \\varphi ( t ^ 5 ) \\\\ & = 2 t \\phi ( t ) \\psi ( t ^ { 1 0 } ) + 2 \\varphi ( t ) \\varphi ( t ^ 5 ) \\\\ & = 2 \\psi ( t ^ 2 ) \\phi ( t ^ 5 ) \\\\ & = 2 \\varphi ( t ^ 4 ) ^ 2 \\frac { \\phi ( t ^ 5 ) } { \\varphi ( t ^ 2 ) } \\\\ & = 2 \\varphi ( t ^ 4 ) ^ 2 ( G ( t ) G ( t ^ 4 ) - t H ( t ) H ( t ^ 4 ) ) \\end{align*}"}
+{"id": "5917.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\sup _ { t \\leq T \\wedge \\sigma _ n ^ M } \\Big [ \\varphi _ n ( t ) \\Vert X ( t , x _ n ) - X ( t , x ) \\Vert _ H ^ 2 \\Big ] = 0 . \\end{align*}"}
+{"id": "1048.png", "formula": "\\begin{align*} & d ( v _ 1 ' \\Rightarrow w _ 2 v _ 2 ' ) = d ( v _ 1 ' \\Rightarrow \\rho ^ \\vee _ { x _ 1 } ( w _ 2 v _ 2 ' ) ) + d ( \\rho ^ \\vee _ { x _ 1 } ( w _ 2 v _ 2 ' ) \\Rightarrow w _ 2 v _ 2 ' ) , \\\\ & d ( v _ 1 ' \\Rightarrow w _ 2 v _ 2 ' ) = d ( v _ 1 ' \\Rightarrow w _ 2 \\rho _ { x _ 2 } ( v _ 1 ) ) + d ( w _ 2 \\rho _ { x _ 2 } ( v _ 1 ) \\Rightarrow w _ 2 v _ 2 ' ) . \\end{align*}"}
+{"id": "1462.png", "formula": "\\begin{align*} c _ { 1 } : = \\frac { 5 } { 4 } \\left ( \\frac { \\alpha _ { 0 } ^ { 4 } \\beta } { 3 } \\right ) ^ { 1 / 5 } = 1 0 \\left ( \\frac { 2 \\beta } { 2 7 } \\right ) ^ { 1 / 5 } . \\end{align*}"}
+{"id": "4990.png", "formula": "\\begin{align*} P _ { \\mu , f _ { 1 , \\infty } } ( X , \\psi ) : = \\int P _ { \\mu , f _ { 1 , \\infty } } ( x , \\psi ) \\ d \\mu ( x ) . \\end{align*}"}
+{"id": "6807.png", "formula": "\\begin{align*} \\frac { \\partial ^ { j } } { \\partial x ^ { j } } \\varphi \\left ( \\rho , x \\right ) = \\left ( 2 i \\rho \\right ) ^ { j } R ( \\rho ) e ^ { 8 i t \\rho ^ { 3 } + 2 i x \\rho } . \\end{align*}"}
+{"id": "8928.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } l _ { n } ( x | \\beta ) l _ { m } ( x | \\beta ) f _ { g } ( x | \\beta ) d x = \\delta _ { n , m } . \\end{align*}"}
+{"id": "4202.png", "formula": "\\begin{align*} A _ 2 = \\frac { l _ 2 } { l _ 1 } \\pi q ^ 2 ( K _ { H } ) _ { 1 1 } ( \\bar { x } ) \\ln \\frac { 1 } { \\varepsilon } + \\frac { l _ 1 } { l _ 2 } \\pi q ^ 2 ( K _ { H } ) _ { 2 2 } ( \\bar { x } ) \\ln \\frac { 1 } { \\varepsilon } + O ( 1 ) , \\end{align*}"}
+{"id": "6105.png", "formula": "\\begin{align*} 1 - 3 Z ^ { + + } _ { n - 1 , p } + 3 Z ^ { + + } _ { n - 1 , p } Z ^ { + + } _ { n , p } - Z ^ { + + } _ { n - 1 , p } Z ^ { + + } _ { n , p } Z ^ { + + } _ { n + 1 , p } = 0 \\ , , \\quad Z ^ { + + } _ { n , p } = \\frac { \\psi ^ { + 0 } _ { n , p - 1 } \\ \\rho ^ { 0 + } _ { n , p } } { \\psi ^ { + 0 } _ { n , p } \\ \\rho ^ { 0 + } _ { n - 1 , p } } \\ . \\end{align*}"}
+{"id": "8680.png", "formula": "\\begin{align*} \\begin{aligned} \\mathop { \\max } \\limits _ { \\{ { \\mu } _ k \\} _ { k = 0 } ^ { K - 1 } } & \\ \\frac { 1 } { { K } } \\sum \\limits _ { k = 0 } ^ { K - 1 } { { { \\log } _ 2 } } \\left ( { 1 + { \\mu _ k } { { \\left \\| { { { { \\bf { \\bar e } } } _ k } } \\right \\| } ^ 2 } } \\right ) \\\\ { \\rm { s . t . } } & \\ \\ \\sum \\limits _ { k = 0 } ^ { K - 1 } { { \\mu _ k } } \\le K P . \\end{aligned} \\end{align*}"}
+{"id": "8983.png", "formula": "\\begin{align*} i _ { 2 2 1 } & = - D _ { 1 , 1 } D _ { 2 , 1 } x _ { 3 , 1 } , & i _ { 2 2 2 } & = - D _ { 1 , 1 } D _ { 2 , 1 } x _ { 3 , 2 } , & i _ { 1 2 3 } & = - x _ { 1 , 1 } D _ { 2 , 1 } D _ { 3 , 2 } & i _ { 2 1 3 } & = - D _ { 1 , 1 } x _ { 2 , 1 } D _ { 3 , 2 } , \\\\ i _ { 2 1 1 } & = D _ { 1 , 1 } x _ { 2 , 1 } x _ { 3 , 1 } , & i _ { 2 1 2 } & = D _ { 1 , 1 } x _ { 2 , 1 } x _ { 3 , 2 } , & i _ { 1 2 1 } & = x _ { 1 , 1 } D _ { 2 , 1 } x _ { 3 , 1 } , & i _ { 1 2 2 } & = x _ { 1 , 2 } D _ { 2 , 1 } x _ { 3 , 2 } , \\\\ i _ { 1 1 2 } & = - x _ { 1 , 1 } x _ { 2 , 1 } x _ { 3 , 2 } , & i _ { 1 1 1 } & = - x _ { 1 , 1 } x _ { 2 , 1 } x _ { 3 , 2 } . \\end{align*}"}
+{"id": "6756.png", "formula": "\\begin{align*} J _ { i , k } ^ { ( s ) } & = \\frac { 1 } { [ 2 \\Gamma ( 1 / s ) ] ^ { k + 1 } } \\sum _ { j = 0 } ^ k \\binom { k } { j } \\ , [ \\Gamma ( 1 / s ) ] ^ { k - j } \\sum _ { m = 0 } ^ \\infty c _ { m , j } \\ , \\Gamma \\left ( m + \\frac { i + j + 1 } { s } \\right ) . \\end{align*}"}
+{"id": "7914.png", "formula": "\\begin{align*} X ( q , r ) & = - \\frac { 1 } { 4 q } f ( q r ) + \\frac { h ( q r ) } { 4 g ( q r ) } \\left ( - 4 r + \\frac { 1 } { 2 q } f ( 2 q r ) \\right ) \\\\ & = \\frac { 1 } { q } \\iota ( q r ) , \\end{align*}"}
+{"id": "9036.png", "formula": "\\begin{align*} \\varepsilon _ { n + 1 } = 2 \\varepsilon _ { n } ^ { 3 } , \\ h _ { n + 1 } = h _ { n } - \\frac { h - h ' } { ( n + 2 ) ^ { 2 } } , \\ r _ { n + 1 } = r _ { n } - \\frac { r - r ' } { ( n + 2 ) ^ { 2 } } , \\ N _ { n } = \\frac { 2 | \\log \\varepsilon _ { n } | } { h _ { n } - h _ { n + 1 } } . \\end{align*}"}
+{"id": "4876.png", "formula": "\\begin{align*} \\psi _ \\epsilon ( x , y ) = A _ { \\epsilon } \\| y \\| ^ 2 + \\tilde { v } ( x ) \\| y \\| \\ge R + 1 \\end{align*}"}
+{"id": "5138.png", "formula": "\\begin{align*} 0 = \\frac { \\dd } { \\dd \\tau } E [ b + \\tau \\tilde { b } + s ( \\tau ) b _ 0 ] \\Bigg | _ { \\tau = 0 } & = \\int _ { \\mathbb { R } ^ { 3 } } b \\cdot ( \\tilde { b } + \\dot { s } ( 0 ) b _ 0 ) \\dd x \\\\ & = \\int _ { \\mathbb { R } ^ { 3 } } b \\cdot \\tilde { b } \\dd x - \\frac { 1 } { \\partial _ { s } j ( 0 , 0 ) } \\left ( \\int _ { \\mathbb { R } ^ { 3 } } b \\cdot b _ 0 \\dd x \\right ) \\partial _ { \\tau } j ( 0 , 0 ) , \\\\ & = \\int _ { \\mathbb { R } ^ { 3 } } b \\cdot \\tilde { b } \\dd x - \\frac { \\mu } { 2 } \\partial _ { \\tau } j ( 0 , 0 ) . \\end{align*}"}
+{"id": "8725.png", "formula": "\\begin{align*} \\pi _ 1 : G \\to M , \\pi _ 1 ( s ( q ) z ) = \\overline { \\pi } ( q ) \\chi _ 1 ( z ) \\forall q \\in Q , z \\in Z . \\end{align*}"}
+{"id": "1751.png", "formula": "\\begin{align*} \\partial _ t R _ 0 = g ( u _ 0 , R _ 0 ) \\mbox { a . e . i n } ( 0 , T ) \\times \\Omega . \\end{align*}"}
+{"id": "3720.png", "formula": "\\begin{align*} 0 & = \\lim _ { t _ j \\to 0 } \\frac { 1 } { t _ j } \\left ( S ' | _ { g _ { t _ j } } \\big ( h ( t _ j ) , v ( t _ j ) \\big ) - S ' | _ { g _ { t _ j } } \\big ( \\psi _ { t _ j } ^ * ( h , v ) \\big ) \\right ) \\\\ & = \\lim _ { t _ j \\to 0 } \\frac { 1 } { t _ j } \\left ( S ' | _ { g _ { t _ j } } \\big ( h ( t _ j ) - h , v ( t _ j ) - v \\big ) - S ' | _ { g _ { t _ j } } \\big ( \\psi _ { t _ j } ^ * ( h , v ) - ( h , v ) \\big ) \\right ) \\\\ & = S ' \\big ( p - L _ X h , z - X ( v ) \\big ) . \\end{align*}"}
+{"id": "2331.png", "formula": "\\begin{align*} \\rho ^ \\star ( \\theta ^ \\sharp , X ) = D ^ g \\theta ( J \\theta ^ \\sharp , J X ) + \\sum _ { i = 1 } ^ 4 D ^ g \\theta ( J e _ i , \\left ( D ^ g _ { e _ i } J \\right ) X ) . \\end{align*}"}
+{"id": "4116.png", "formula": "\\begin{align*} E [ t ] ^ { \\vee } = E ^ { \\vee } [ t ^ { - 1 } ] t > 0 . \\end{align*}"}
+{"id": "7264.png", "formula": "\\begin{align*} \\lambda _ 0 \\alpha ^ s + \\lambda _ 1 \\alpha ^ { s - 1 } \\beta + \\cdots + \\lambda _ s \\beta ^ s = 0 , \\end{align*}"}
+{"id": "2736.png", "formula": "\\begin{align*} Z ( x ) = e ^ { x _ 3 } h ( x _ 1 , x _ 2 ) \\quad \\ | x _ 1 | \\leq 1 \\ \\ | x _ 2 | \\leq 1 , \\end{align*}"}
+{"id": "1239.png", "formula": "\\begin{align*} \\rho ( \\eta _ { i _ j } | \\eta _ { [ i _ { j + 1 } , i _ k ] } \\eta _ { \\Lambda \\setminus \\Delta } \\omega _ { \\Lambda ^ c \\cup [ i _ 1 , i _ { j - 1 } ] } ) = \\gamma _ { i _ j } ( \\eta _ { i _ j } | \\eta _ { [ i _ { j + 1 } , i _ k ] } \\eta _ { \\Lambda \\setminus \\Delta } \\omega _ { \\Lambda ^ c \\cup [ i _ 1 , i _ { j - 1 } ] } ) . \\end{align*}"}
+{"id": "483.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } e ^ { \\Lambda _ u } \\frac { 2 } { \\Lambda _ 2 ^ 2 } \\cdot \\frac { f ( x ) } { g _ u ( x ) } = \\lim _ { x \\to \\infty } e ^ { \\Lambda _ u } \\frac { 2 } { \\Lambda _ 2 ^ 2 } \\cdot \\frac { f ( x ) } { g _ 1 ( x ) } \\frac { g _ 1 ( x ) } { g ( x ) } \\frac { g ( x ) } { g _ u ( x ) } = e ^ { \\Lambda _ u } \\frac { 2 } { \\Lambda _ u } . \\end{align*}"}
+{"id": "8769.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } { \\texttt f } \\sigma ^ \\psi _ x \\mathfrak { f } = & \\int _ { \\Omega } \\left ( { \\texttt f } { } ^ \\psi \\mathcal D [ \\mathfrak { f } ] + { } ^ { { \\psi } } \\mathcal D _ r [ { \\texttt f } ] \\mathfrak { f } \\right ) d x , \\end{align*}"}
+{"id": "4149.png", "formula": "\\begin{align*} \\Psi _ { a , b } ( x ' ) : = 4 \\int _ a ^ b \\Phi _ t * \\Phi _ t ( x ' ) \\ , \\frac { d t } { t } , x \\in \\R ^ { n - 1 } . \\end{align*}"}
+{"id": "3351.png", "formula": "\\begin{align*} \\phi _ u & = \\{ v \\in V ( H ) \\ | \\ ( u , v ) \\in \\phi \\} , \\\\ \\overline { \\phi } & = \\{ u \\in V ( G ) \\ | \\ \\phi _ u \\cap U \\neq \\emptyset \\} . \\end{align*}"}
+{"id": "2133.png", "formula": "\\begin{align*} L L \\left ( | \\varrho _ { Z Z } ( L , L ) | ^ 2 \\right ) = - 4 \\varrho _ { Z Z Z } ( L , L , L ) \\overline { \\varrho _ { Z Z } ( N , L ) } - 2 \\varrho _ { Z Z } ( L , L ) ( L \\overline { \\varrho _ { Z Z } ( N , L ) } ) \\\\ + \\overline { \\varrho _ { Z Z } ( L , L ) } \\left ( \\varrho _ { Z Z Z Z } ( L , L , L , L ) - 3 \\varrho _ { Z Z Z } ( L , L , \\varrho _ { Z Z } \\cdot L ) \\right ) . \\end{align*}"}
+{"id": "6894.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { X _ i } \\to Q , . \\end{align*}"}
+{"id": "204.png", "formula": "\\begin{align*} \\theta _ { j , ( 2 ) } ( f ) = \\sum _ { k } r _ k g \\cdot \\theta _ { j , ( 2 ) } ( v _ k ) . \\end{align*}"}
+{"id": "7696.png", "formula": "\\begin{align*} | \\phi ^ { N , i } ( t , \\boldsymbol { x } _ t ) - \\Phi ( t , \\bar { \\nu } _ t ^ i ) | = | \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } _ t } ) - \\Phi ( t , \\bar { \\nu } _ t ^ i ) | \\leq C | \\nu ^ { N , i } _ { \\boldsymbol { x } _ t } - \\bar { \\nu } _ t ^ i | . \\end{align*}"}
+{"id": "9297.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta K _ { m , \\epsilon } = d _ 0 \\left ( \\frac { \\kappa _ m d _ 1 | q | ^ 2 } { ( | q | ^ 2 + \\epsilon ) ^ { \\kappa _ m + 1 } } \\right ) = - \\frac { \\kappa _ m ( \\kappa _ m + 1 ) } { ( | q | ^ 2 + \\epsilon ) ^ { \\kappa _ m + 2 } } d _ 0 | q | ^ 2 \\wedge d _ 1 | q | ^ 2 + \\frac { 8 \\kappa _ m \\beta _ n } { ( | q | ^ 2 + \\epsilon ) ^ { \\kappa _ m + 1 } } = : A + B . \\end{aligned} \\end{align*}"}
+{"id": "3038.png", "formula": "\\begin{align*} \\left [ Z _ { \\kappa } \\left ( x , \\xi \\right ) , Z _ { \\kappa } \\left ( y , \\eta \\right ) \\right ] = Z _ { \\kappa } \\left ( C _ { \\xi } y - C _ { \\eta } x , \\kappa x \\times y + \\xi \\times \\eta \\right ) \\end{align*}"}
+{"id": "3236.png", "formula": "\\begin{align*} \\mathbb E \\left [ \\| \\Delta _ i \\mathcal S B ^ { \\mathfrak H } \\| ^ 4 \\right ] \\geq \\left ( \\int _ 0 ^ { 1 - i \\Delta _ n } \\mathbb E \\left [ \\left ( B ^ { \\mathfrak H } _ { i \\Delta _ n + y } - B ^ { \\mathfrak H } _ { ( i - 1 ) \\Delta _ n + y } \\right ) ^ 2 \\right ] d y \\right ) ^ 2 = & ( n - i ) ^ 2 \\Delta _ n ^ { 2 + 4 \\mathfrak H } . \\end{align*}"}
+{"id": "7212.png", "formula": "\\begin{align*} B _ t ^ { k , i } = \\left \\{ \\begin{array} { l l } { \\rm R e } ( W ^ { k , i } _ { t } ) , & k \\in \\Z _ { + } ^ d , \\\\ - { \\rm I m } ( W ^ { k , i } _ { t } ) , & k \\in \\Z _ { - } ^ d . \\end{array} \\right . \\end{align*}"}
+{"id": "4530.png", "formula": "\\begin{align*} \\begin{cases} v _ i ( x , t ) = S _ { c , i } ( t , t _ { i , e n } ( x , t ) ) S _ { d , i } ( t _ { i , e n } ( x , t ) , 0 ) v _ { i , 0 } ( x ) & \\forall \\ : i \\in [ 1 , p ] , \\\\ v _ i ( x , t ) = S _ { c , i } ( t , t _ { i , e n } ( x , t ) ) S _ { d , i } ( t _ { i , e n } ( x , t ) , 0 ) v _ { i , 0 } ( x ) & \\forall \\ : i \\in [ p + 1 , n ] . \\end{cases} \\end{align*}"}
+{"id": "1192.png", "formula": "\\begin{align*} \\Pr [ \\exists ( S , b ) : Z _ { S , b } < 2 ^ { - ( k + 1 ) } ( n - k ) ] \\le \\binom { n } { k } 2 ^ k \\cdot \\exp \\left ( - 2 ^ { - ( k + 3 ) } ( n - k ) \\right ) \\enspace . \\end{align*}"}
+{"id": "1359.png", "formula": "\\begin{align*} g ( y ) = f ( y ) - ( f - g ) ( y ) \\geq \\| y \\| - \\frac { 4 ( 1 + \\theta ) } { n } , \\end{align*}"}
+{"id": "3732.png", "formula": "\\begin{align*} & \\nabla _ V \\left ( \\nabla _ Y \\hat X - \\omega ( Y ) \\right ) \\\\ & = \\nabla _ Y \\nabla _ V \\hat X + R ( V , Y ) \\hat X - ( \\nabla _ V \\omega ) ( Y ) - \\omega ( \\nabla _ V Y ) \\\\ & = \\nabla _ Y ( \\omega ( V ) ) + R ( V , Y ) \\hat X + R ( \\hat X , V ) Y - T _ h ( V , Y ) - \\omega ( \\nabla _ V Y ) \\mbox { ( b y \\eqref { e q : O D E } ) } \\\\ & = ( \\nabla _ Y \\omega ) ( V ) - R ( Y , \\hat X ) V - T _ h ( V , Y ) \\mbox { ( b y B i a n c h i i d e n t i t y ) } \\end{align*}"}
+{"id": "6079.png", "formula": "\\begin{align*} \\frac { ( p ^ 3 - p ^ 2 ) } { ( p ^ 2 - p ) } = p \\end{align*}"}
+{"id": "3590.png", "formula": "\\begin{align*} \\Theta = \\{ ( s _ 0 , s _ 1 , t _ 0 , t _ 1 , r _ 0 , r _ 1 , r _ 2 ) \\in ( 0 , 1 ] ^ 7 \\ , | \\ , s _ 0 + s _ 1 = 1 , & t _ 0 + t _ 1 = 1 , \\\\ & r _ 0 + r _ 1 + r _ 2 = 1 , r _ 3 + r _ 3 + r _ 4 = 1 \\} . \\end{align*}"}
+{"id": "8510.png", "formula": "\\begin{align*} \\begin{aligned} I _ 3 ' ( x ) = 4 \\pi ^ { - 1 } \\int _ { \\mathbb { R } } z r _ 2 ( z ) e ^ { 2 i z x } d z = 4 \\pi ^ { - 2 } \\widehat { z r _ 2 ( z ) } ( - 2 x ) , \\end{aligned} \\end{align*}"}
+{"id": "3934.png", "formula": "\\begin{align*} f _ { k } ( x , y ) : = \\begin{cases} f ( x , - k ) , & y < - k , \\\\ f ( x , y ) , & | y | \\le k , \\\\ f ( x , k ) , & y > k . \\end{cases} \\end{align*}"}
+{"id": "9344.png", "formula": "\\begin{align*} \\pi ( x , t ) - c _ { r , x _ 0 } ( t ) & = \\frac { 1 } { 3 } | v ( x , t ) | ^ 2 + \\textnormal { p . v . } \\int _ { B _ { 3 r } ( x _ 0 ) } K ( x - y ) : ( v \\otimes v ) ( y , t ) \\ , d y \\\\ & + \\int _ { \\mathbb { R } ^ 3 \\setminus B _ { 3 r } ( x _ 0 ) } ( K ( x - y ) - K ( x _ 0 - y ) ) : ( v \\otimes v ) ( y , t ) \\ , d y \\end{align*}"}
+{"id": "1764.png", "formula": "\\begin{align*} \\chi _ 0 : Y \\rightarrow \\R , \\chi _ 0 ( y ) = \\chi ( d _ { \\Gamma } ( y ) ) , \\end{align*}"}
+{"id": "9070.png", "formula": "\\begin{align*} \\chi ( E ) = \\sum \\limits _ { j = 1 } ^ n \\chi ( E _ j ) - r ( n - 1 ) . \\end{align*}"}
+{"id": "8989.png", "formula": "\\begin{align*} \\hat { k } _ d ( \\Delta ) & = \\prod _ { q , i } x _ { q , i } ^ { e ( q , i ) } \\prod _ { q = 1 } ^ { d + 1 } \\prod _ { \\rho \\in V ' _ 1 \\times \\cdots \\times \\widehat { V ' _ q } \\times \\cdots \\times V ' _ { d + 1 } } D _ { q , n _ q } \\\\ & = \\prod _ { q = 1 } ^ { d + 1 } \\left [ \\left ( \\prod _ { i = 1 } ^ { n _ i } x _ { q , i } \\right ) ^ { e ( q , i ) } ( x _ { q , 1 } + \\cdots + x _ { q , n _ q } ) ^ { \\prod _ { r \\neq q } ( n _ { r } - 1 ) } \\right ] \\end{align*}"}
+{"id": "742.png", "formula": "\\begin{align*} \\Theta ( f ) = \\Delta f + \\partial _ t f = 0 , \\end{align*}"}
+{"id": "4971.png", "formula": "\\begin{align*} W ( \\xi _ 0 , n , x , \\mathsf { L } ^ n ) = W ( 1 - \\xi _ 0 , n , x , \\mathsf { R } ^ n ) , ~ \\forall \\xi _ 0 \\in [ 0 , 1 ] , \\forall x \\in \\mathbb { R } . \\end{align*}"}
+{"id": "3068.png", "formula": "\\begin{align*} \\gamma _ Z ( y , \\varphi ( y ) ) = \\left ( - y _ 1 \\frac { \\partial \\varphi ( y ) } { \\partial y _ 1 } : \\cdots : - y _ { n } \\frac { \\partial \\varphi ( y ) } { \\partial y _ { n } } : \\varphi ( y ) \\right ) \\end{align*}"}
+{"id": "218.png", "formula": "\\begin{align*} g = t ^ { - 2 } \\cdot a _ 0 \\cdot a _ 4 t ^ { - 3 } \\cdot t ^ 2 \\cdot t ^ 2 \\cdot a _ 0 \\cdot t ^ { 2 } \\cdot a _ 0 \\cdot t ^ { 2 } = s _ 2 \\ , s _ 4 \\ , s _ 1 \\ , s _ 5 \\ , s _ 5 \\ , s _ 4 \\ , s _ 5 \\ , s _ 4 \\ , s _ 5 \\end{align*}"}
+{"id": "2144.png", "formula": "\\begin{align*} w = 0 , \\ z = \\sqrt { \\frac { 1 } { 1 + a } } \\cos ( t ) + i \\sqrt { \\frac { 1 } { 1 - a } } \\sin ( t ) , t \\in [ 0 , 2 \\pi ) . \\end{align*}"}
+{"id": "256.png", "formula": "\\begin{align*} X _ 1 & \\mapsto \\begin{pmatrix} 0 & 0 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{pmatrix} , & X _ 2 & \\mapsto \\begin{pmatrix} 0 & 0 & 0 \\\\ \\lambda & 0 & 0 \\\\ \\mu & \\lambda & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "6154.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 } \\frac { m } { 1 + \\lambda r ^ 2 } \\left ( \\sum _ i v _ i ^ 2 + \\lambda \\sum _ { i < j } ( x _ i v _ j - x _ j v _ i ) ^ 2 \\right ) - { \\cal V } ( r ) , { \\cal V } ( r ) = \\frac { 1 } { 2 } m \\alpha ^ 2 \\frac { r ^ 2 } { 1 + \\lambda r ^ 2 } . \\end{align*}"}
+{"id": "1711.png", "formula": "\\begin{align*} u ( t ) = U ( t , 0 ) u _ 0 + \\int _ 0 ^ t U ( t , s ) f ( s ) d s , 0 \\le t \\le T , \\end{align*}"}
+{"id": "8609.png", "formula": "\\begin{align*} \\frac { \\left | \\sum _ { i = 1 } ^ n [ 0 , z _ i ] \\right | } { \\left | \\sum _ { i = 1 } ^ { n - 1 } [ 0 , z _ i ] \\right | } = d ( z _ n , H _ n ) . \\end{align*}"}
+{"id": "4216.png", "formula": "\\begin{align*} h ( \\lambda ) = \\frac { ( \\lambda , \\lambda + 2 \\rho ) } { 2 ( \\kappa + h ^ \\vee ) } + \\frac { ( \\lambda , \\lambda + 2 \\rho ) } { 2 ( \\kappa ^ * + h ^ \\vee ) } - ( \\lambda , \\rho ^ \\vee ) = \\frac { ( \\lambda , \\lambda ) r ^ \\vee n } { 2 } + ( \\lambda , n r ^ \\vee \\rho - \\rho ^ \\vee ) . \\end{align*}"}
+{"id": "1364.png", "formula": "\\begin{align*} \\rho & : = \\inf \\{ n \\geq 1 : S ( n - 1 ) \\nprec S ( n ) \\} \\\\ \\tau & : = \\inf \\{ n \\geq 1 : S ( n ) \\notin W ^ d \\} . \\end{align*}"}
+{"id": "6402.png", "formula": "\\begin{align*} d ( J ^ * d \\theta _ i ) = 0 , \\mbox { l o c a l l y } J ^ * d \\theta _ i = d y _ i . \\end{align*}"}
+{"id": "2067.png", "formula": "\\begin{align*} - [ M ^ k , \\ , \\mathcal { L } ] f & = \\Big ( M ^ k \\Gamma ( \\sqrt \\mu , f ) - \\Gamma ( \\sqrt \\mu , M ^ k f ) \\Big ) + \\Big ( M ^ k \\Gamma ( f , \\sqrt \\mu ) - \\Gamma ( M ^ k f , \\sqrt \\mu ) \\Big ) \\\\ & = \\sum _ { 1 \\leq j \\leq 6 } \\Big ( M ^ k L _ j ( \\sqrt \\mu , f ) - L _ j ( \\sqrt \\mu , M ^ k f ) \\Big ) + \\sum _ { 1 \\leq j \\leq 6 } \\Big ( M ^ k L _ j ( f , \\sqrt \\mu ) - L _ j ( M ^ k f , \\sqrt \\mu ) \\Big ) \\end{align*}"}
+{"id": "7350.png", "formula": "\\begin{align*} \\Pi _ S ( X ) \\rightarrow \\bigoplus _ { i \\in I } \\Pi _ S ( Z _ i , N _ i ) \\rightrightarrows \\cdots \\rightrightarrows \\bigoplus _ { J \\subset I , \\sharp J = n } \\Pi _ S ( Z _ J , N _ J ) \\rightrightarrows \\cdots \\end{align*}"}
+{"id": "7175.png", "formula": "\\begin{align*} \\sum _ { J } \\frac { ( - i ) ^ { | J | } } { J ! } \\partial _ { \\xi } ^ { J } q \\ , \\partial _ { x ^ \\prime } ^ { J } q - \\sum _ { J } \\frac { ( - i ) ^ { | J | } } { J ! } \\partial _ { \\xi } ^ { J } b \\ , \\partial _ { x ^ \\prime } ^ { J } q - \\frac { \\partial q } { \\partial x _ n } + c = 0 . \\end{align*}"}
+{"id": "3124.png", "formula": "\\begin{align*} & [ \\alpha ( y ) , \\alpha ( z ) ] \\cdot \\alpha ( x \\cdot t ) + [ [ x , y ] , \\alpha ( z ) ] \\cdot \\alpha ^ { 2 } ( t ) \\\\ & + \\alpha ^ { 2 } ( y ) \\cdot ( [ x , z ] \\cdot \\alpha ( t ) ) - \\alpha ^ { 2 } ( x ) \\cdot ( \\alpha ( y ) \\cdot ( z \\cdot t ) ) + \\alpha ^ { 2 } ( z ) \\cdot ( \\alpha ( x ) \\cdot ( y \\cdot t ) ) = 0 . \\end{align*}"}
+{"id": "7646.png", "formula": "\\begin{align*} | \\rho ^ \\prime | = \\left | \\frac { 1 } { 1 + h ^ \\prime } \\right | \\leq \\frac { 1 } { \\varepsilon _ 0 } . \\end{align*}"}
+{"id": "8976.png", "formula": "\\begin{align*} x _ { j _ 1 j _ 2 j _ 3 } & = x _ { 1 , j _ 1 } x _ { 2 , j _ 2 } x _ { 3 , j _ 3 } j _ 1 , j _ 2 , j _ 3 \\in \\{ 1 , 2 \\} , \\\\ x _ { 2 2 2 } ^ * & = ( x _ { 1 , 1 } + x _ { 1 , 2 } ) ( x _ { 2 , 1 } + x _ { 2 , 2 } ) ( x _ { 3 , 1 } + x _ { 3 , 2 } ) . \\end{align*}"}
+{"id": "9230.png", "formula": "\\begin{align*} I _ n : = \\left [ n \\dfrac { \\bar { t } } { \\gamma } , \\ , ( n + 1 ) \\dfrac { \\bar { t } } { \\gamma } \\right ) n \\in { \\mathbb { N } } \\end{align*}"}
+{"id": "8565.png", "formula": "\\begin{align*} k ( t ) = h _ { 1 - \\alpha } ( t ) \\cdot k _ 1 ( t ) , \\ k _ 1 ( t ) = \\sum _ { k = 0 } ^ { + \\infty } \\ , b _ k t ^ k , \\end{align*}"}
+{"id": "2107.png", "formula": "\\begin{align*} h _ c ( G ) & = h _ c ( G \\backslash v ) + f _ c ( b ) - f _ c ( 0 ) + \\sum _ { i = 1 } ^ b \\Delta _ c ( d _ i ) \\\\ & \\le h _ c ( G \\backslash v ) + f _ c ( b ) - f _ c ( 0 ) + b \\cdot \\Delta _ c \\left ( \\frac { m + \\binom { b } { 2 } } { b } \\right ) \\\\ & < h _ c ( K _ { 1 , m - b } ) + f _ c ( m ) - f _ c ( m - b ) + b \\Delta _ c \\left ( 1 \\right ) \\\\ & = h _ c ( K _ { 1 , m } ) . \\end{align*}"}
+{"id": "9005.png", "formula": "\\begin{align*} i _ { \\sigma \\setminus \\ell _ { s } \\cup 1 } & = ( - 1 ) ^ { s - 1 } ( - x _ 1 ) \\prod _ { q = 2 } ^ { s } { D _ { \\ell _ { q - 1 } } } \\prod _ { q = s + 1 } ^ { d + 1 } { D _ { \\ell _ q - 1 } } \\\\ & = ( - 1 ) ^ { s } x _ 1 \\prod _ { q = 2 } ^ { s } { D _ { \\ell _ { q - 1 } } } \\prod _ { q = s + 1 } ^ { d + 1 } { D _ { \\ell _ q - 1 } } . \\end{align*}"}
+{"id": "7146.png", "formula": "\\begin{align*} \\Lambda _ { g } ( \\textbf { \\textit { V } } ) : = \\begin{pmatrix} \\lambda \\nu \\operatorname { d i v } + \\mu \\nu S & - \\beta \\nu \\\\ 0 & \\alpha \\partial _ \\nu \\end{pmatrix} \\textbf { \\textit { U } } \\ \\partial \\Omega , \\end{align*}"}
+{"id": "7072.png", "formula": "\\begin{align*} { } [ e _ 1 , e _ 2 ] \\ , = \\ , 0 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ [ e _ 1 , e _ 3 ] \\ , = \\ , \\mp \\ , \\tfrac { 4 } { 1 5 } \\ , e _ 4 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ [ e _ 1 , e _ 4 ] & \\ , = \\ , \\pm \\ , \\tfrac { 5 } { 4 } \\ , e _ 1 , \\\\ { } [ e _ 2 , e _ 3 ] \\ , = \\ , 0 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ [ e _ 2 , e _ 4 ] & \\ , = \\ , 0 , \\\\ { } [ e _ 3 , e _ 4 ] & \\ , = \\ , \\mp \\ , \\tfrac { 5 } { 4 } \\ , e _ 3 . \\end{align*}"}
+{"id": "4653.png", "formula": "\\begin{align*} M _ { \\widetilde { \\phi } } ( i _ { p _ 1 , \\alpha } ( x _ 1 ) , \\ldots , i _ { p _ n , \\alpha } ( x _ n ) ) ( t _ 0 , t _ { n } ) & \\\\ = \\frac { 1 } { \\mu _ G ( F _ \\alpha ) ^ { 1 / p } } 1 _ { F _ \\alpha } ( t _ 0 ) 1 _ { F _ \\alpha } ( t _ n ) \\int _ { F _ \\alpha ^ { \\times n - 1 } } & \\phi ( t _ 0 t _ 1 ^ { - 1 } , \\ldots , t _ { n - 1 } t _ n ^ { - 1 } ) f _ 1 ( t _ 0 t _ 1 ^ { - 1 } ) \\ldots f _ n ( t _ { n - 1 } t _ n ^ { - 1 } ) d t _ 1 \\ldots d t _ { n - 1 } . \\end{align*}"}
+{"id": "823.png", "formula": "\\begin{align*} V _ { | ( t , x , \\varphi ) } = \\frac { 1 } { \\sqrt { 1 + | \\partial _ x w ( t , x ) | ^ 2 } } \\partial _ t w ( t , x ) . \\end{align*}"}
+{"id": "7348.png", "formula": "\\begin{align*} ( \\delta _ n ^ k ) _ ! = \\sum _ { J = \\{ i _ 0 < \\hdots < \\widehat { i _ k } < \\hdots < i _ n \\} \\subset K = \\{ i _ 0 < \\hdots < i _ n \\} } ( \\nu _ { K } ^ J ) _ ! \\end{align*}"}
+{"id": "5007.png", "formula": "\\begin{align*} \\mathcal { E } _ { m } = \\displaystyle \\sum _ { n = x _ { m } + 1 } ^ { y _ { m } } e _ { n } \\ ; \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\ ; \\mathcal { F } _ { m } = \\displaystyle \\sum _ { n = x _ { m } + 1 } ^ { y _ { m } } g _ { n } . \\\\ \\end{align*}"}
+{"id": "7249.png", "formula": "\\begin{align*} \\sigma ( \\lambda \\alpha ) = \\sigma ( \\lambda ) \\sigma ( \\alpha ) \\end{align*}"}
+{"id": "788.png", "formula": "\\begin{align*} \\partial ^ \\square c & = \\Delta _ \\Gamma \\big ( G ' ( c ) \\big ) + c H V . \\end{align*}"}
+{"id": "738.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ x z & = 0 , \\\\ 0 & \\leq \\partial _ t z - \\frac { \\varepsilon } { 2 } \\partial _ { x } ^ 2 z . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6789.png", "formula": "\\begin{align*} g ( z ) = 1 + \\left ( z + 1 \\right ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } z ^ { n } b _ { n } ( 0 ) , \\end{align*}"}
+{"id": "5127.png", "formula": "\\begin{align*} H [ \\nabla \\times ( \\phi \\nabla \\theta ) + G \\nabla \\theta ] = - H [ \\nabla \\times ( \\phi \\nabla \\theta ) - G \\nabla \\theta ] \\end{align*}"}
+{"id": "4539.png", "formula": "\\begin{align*} \\mathcal { I } ( x , t ) = \\bigcup _ { i = 1 } ^ { n } [ t _ { i , e n } ( x , t ) , t ] . \\end{align*}"}
+{"id": "4895.png", "formula": "\\begin{align*} \\alpha = 1 + z \\frac { \\Phi ( z ) } { \\phi ( z ) } \\frac { 1 - \\Phi ( z ) } { 1 - 2 \\Phi ( z ) } , . \\end{align*}"}
+{"id": "8861.png", "formula": "\\begin{align*} x _ { n + 1 } & = ( 1 - \\beta _ n ) U y _ n + \\beta _ n y _ n , \\\\ y _ { n } & = ( 1 - \\alpha _ n ) T x _ n + \\alpha _ n u , \\end{align*}"}
+{"id": "8570.png", "formula": "\\begin{align*} \\kappa ( t ) = h _ { 1 - \\beta + \\alpha } ( t ) \\ , + \\ , h _ { 1 - \\beta } ( t ) , \\ 0 < \\alpha < \\beta < 1 , \\end{align*}"}
+{"id": "4100.png", "formula": "\\begin{align*} & \\hphantom { { } = { } } ( x ^ i , ( a ^ i { } _ j , a ^ i { } _ { j \\alpha } a ^ \\alpha { } _ k ) , ( b ^ i { } _ j , b ^ i { } _ { j k } ) ) . ( g , X ) \\\\ * & = ( x ^ i , ( a ^ i { } _ \\alpha g ^ \\alpha { } _ j , a ^ i { } _ { \\beta \\alpha } g ^ \\beta { } _ j a ^ \\alpha { } _ \\beta g ^ \\beta { } _ k ) , ( h ^ i { } _ \\alpha b ^ \\alpha { } _ \\beta g ^ \\beta { } _ j + X ^ i { } _ j , h ^ i { } _ \\alpha b ^ \\alpha { } _ { \\beta k } g ^ \\beta { } _ j ) ) . \\end{align*}"}
+{"id": "3074.png", "formula": "\\begin{align*} k = \\sum _ { p \\in C } I _ p ( C , H _ { j } ) = \\# ( C \\cap H _ { j } ) + \\sum _ { p \\in C \\cap H _ { j } } ( I _ p ( C , H _ { j } ) - 1 ) \\end{align*}"}
+{"id": "97.png", "formula": "\\begin{align*} \\frac 1 N \\sum _ { k = 1 } ^ N ( \\sigma \\circ w ) ^ k \\mu . \\end{align*}"}
+{"id": "111.png", "formula": "\\begin{align*} \\phi _ { \\lambda } = \\lambda ^ { - \\alpha } \\phi ( \\lambda ^ { - 1 } x , \\lambda ^ { - \\frac { \\alpha + 2 } { 2 } } y ) . \\end{align*}"}
+{"id": "6596.png", "formula": "\\begin{align*} \\log \\frac { c ( r _ { t } , t ) } { c ( 0 , t ) } \\le \\int _ { 0 } ^ { r _ { t } } \\frac { M _ { 0 } } { \\sigma _ { d } } \\rho \\ , d \\rho = \\frac { M _ { 0 } } { 2 \\sigma _ { d } } r _ { t } ^ { 2 } . \\end{align*}"}
+{"id": "4236.png", "formula": "\\begin{align*} \\dot { s } \\varepsilon ( z ) \\dot { s } \\varepsilon ( 1 ) \\xi _ 2 = \\dot { s } ( \\varepsilon ( z - 1 ) - \\varepsilon ( z ) ) \\xi _ 2 + \\dot { s } \\varepsilon ( z ) \\varpi . \\end{align*}"}
+{"id": "1407.png", "formula": "\\begin{align*} \\P _ x ( S ( n ) \\in d z , \\tau > n ) & = \\P _ x ( \\widehat S _ i ( n + d - i ) \\in d z _ i , i = 1 , \\ldots , d , \\widehat \\tau = \\infty ) \\\\ & = \\sum _ { \\pi \\in \\mathcal S _ d } ( \\pi ) \\P _ x ( \\widehat S _ i ( n + d - \\pi ( i ) ) \\in d z _ { \\pi ( i ) } , i = 1 , \\ldots , d , \\widehat \\tau = \\infty ) . \\end{align*}"}
+{"id": "3120.png", "formula": "\\begin{align*} a d ^ \\star ( x ) \\xi = a d ^ * ( \\alpha ( x ) ) ( \\alpha ^ { - 2 } ) ^ * ( \\xi ) , \\end{align*}"}
+{"id": "8684.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 ( I _ N ^ i ( p ) - J _ N ^ i ( p ) ) = O ( p ^ { - \\nu } ) \\quad \\mbox { f o r l a r g e } p > 0 , \\end{align*}"}
+{"id": "2379.png", "formula": "\\begin{align*} [ \\mathbf { h } ' _ 3 , \\mathbf { h } _ 3 ] = 1 . \\end{align*}"}
+{"id": "5359.png", "formula": "\\begin{align*} e m + \\gamma = \\widehat { \\iota } ( D ) c \\geq \\widehat { \\iota } ( D ) ( b _ 1 d + 1 ) . \\end{align*}"}
+{"id": "1896.png", "formula": "\\begin{align*} \\mathfrak M ( x , y ) = - \\sin S _ c ^ l ( x , y ) \\partial _ x \\partial _ y ^ { \\intercal } \\Phi _ s ( x , y ) . \\end{align*}"}
+{"id": "5556.png", "formula": "\\begin{align*} \\mu _ A ( [ x _ 1 , x _ 2 , . . . , x _ { n - 1 } , x _ n ] ) = \\mu _ A ( [ x _ n , x _ { n - 1 } , . . . , x _ 2 , x _ 1 ] ) . \\end{align*}"}
+{"id": "7386.png", "formula": "\\begin{align*} \\mathfrak g = \\mathfrak g _ + \\oplus \\mathfrak h \\oplus \\mathfrak g _ - \\end{align*}"}
+{"id": "2876.png", "formula": "\\begin{align*} | \\sum _ { y = 0 } ^ n \\bar K ^ { ( n ) } _ 1 ( { y , x } ) - \\sum _ { y = x - n - 1 } ^ { x + n } \\bar K ^ { ( n ) } _ 1 ( { y , x } ) | \\le \\frac { C } { n ^ 2 } , \\delta n \\le x \\le ( 1 - \\delta ) n , \\end{align*}"}
+{"id": "8330.png", "formula": "\\begin{align*} & \\psi ^ { \\pm } ( x , t ; k ) = \\mathrm { e } ^ { - i c _ \\pm ( x ) \\sigma _ 3 } + \\mathcal { O } ( k ^ { - 1 } ) , k \\rightarrow \\infty , \\end{align*}"}
+{"id": "2037.png", "formula": "\\begin{align*} N ( \\alpha ) & = w ^ 4 + 2 w ^ 3 x - w ^ 2 x ^ 2 - 2 w x ^ 3 + x ^ 4 + 2 w ^ 2 y ^ 2 + 2 w x y ^ 2 + 3 x ^ 2 y ^ 2 + y ^ 4 + 2 w ^ 2 y z \\\\ & - 8 w x y z - 2 x ^ 2 y z + 2 y ^ 3 z + 3 w ^ 2 z ^ 2 - 2 w x z ^ 2 + 2 x ^ 2 z ^ 2 - y ^ 2 z ^ 2 - 2 y z ^ 3 + z ^ 4 . \\end{align*}"}
+{"id": "7836.png", "formula": "\\begin{align*} \\psi ( v , p ) = \\psi ^ \\dagger ( v , p ) \\in O \\end{align*}"}
+{"id": "4468.png", "formula": "\\begin{align*} P ( x , y ) = \\frac { ( 2 a ( x + \\xi ) + b ( y + \\eta ) ) ^ { 2 } - \\Delta ( y + \\eta ) ^ { 2 } } { 4 a } + P ( - \\xi , - \\eta ) . \\end{align*}"}
+{"id": "5644.png", "formula": "\\begin{align*} A \\cdot ( v _ { 1 } , \\dots , v _ { k } ) : = ( A v _ { 1 } , \\dots , A v _ { k } ) . \\end{align*}"}
+{"id": "4504.png", "formula": "\\begin{align*} \\| \\psi _ { n + m } - \\psi _ n \\| _ 1 \\leq \\sum _ { i = 1 } ^ m \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\left | \\arg \\frac { \\hat { b } _ { n + i } \\left ( e ^ { i \\theta } \\right ) } { e ^ { i \\theta } } \\right | \\ , d \\theta . \\end{align*}"}
+{"id": "7589.png", "formula": "\\begin{align*} \\phi ( x _ { 0 0 } , x _ { 0 1 } , x _ { 1 0 } , x _ { 1 1 } ) = ( x _ { 0 0 } , x _ { 1 0 } , x _ { 1 0 } , x _ { 1 1 } ) . \\end{align*}"}
+{"id": "4575.png", "formula": "\\begin{align*} f ( v _ { \\ell , m , n } ) = \\sqrt { q } ^ { 3 \\ell + m - n } & \\sum _ { \\substack { ( i , j ) \\in \\{ 1 , 3 \\} ^ 2 \\\\ i \\neq j } } \\{ F _ { i i j } z _ i ^ { \\ell + m } z _ j ^ n + F _ { i j i } z _ i ^ { \\ell + n } z _ j ^ m + F _ { i j j } z _ i ^ \\ell z _ j ^ { m + n } \\} , \\\\ \\end{align*}"}
+{"id": "402.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } \\int _ { \\tau } ^ { \\tau + T } Q _ 1 ( t ) \\d t = 0 . \\end{align*}"}
+{"id": "4914.png", "formula": "\\begin{align*} b _ { k + 1 } = \\frac { 1 } { 2 ( 2 k + 3 ) } \\ , \\sum _ { r = 0 } ^ { k } \\frac { ( 2 r + 1 ) \\ , ( 2 k - 2 r + 1 ) \\ , b _ r \\ , b _ { k - r } } { ( r + 1 ) \\ , ( 2 r + 1 ) } . \\end{align*}"}
+{"id": "4667.png", "formula": "\\begin{align*} | | \\tau | | G \\psi \\stackrel { d e f } { = } \\sup _ { p \\in ( 1 , b ) } \\left \\{ \\ \\frac { | | \\tau | | L _ p ( \\Omega ) } { \\psi ( p ) } \\ \\right \\} . \\end{align*}"}
+{"id": "8299.png", "formula": "\\begin{align*} \\dot y = a y '' + M y , y ( 0 , x ) = y _ 0 ( x ) , \\end{align*}"}
+{"id": "1834.png", "formula": "\\begin{align*} \\aligned \\frac d { d t } \\mathcal { Y M } _ e ^ 0 ( \\nabla ^ t ) _ { \\big { | } _ { s = 0 } } & = \\frac d { d t } \\mathcal { Y M } _ e ( \\nabla ^ t ) _ { \\big { | } _ { s = 0 } } \\\\ & = \\int _ M \\ e x p \\ , ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) \\langle R ^ \\nabla , d ^ \\nabla A \\rangle \\ , d v \\\\ & = \\int _ M \\langle \\delta ^ \\nabla \\big ( \\exp \\ , ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) R ^ \\nabla \\big ) , A \\rangle \\ , d v \\ , . \\endaligned \\end{align*}"}
+{"id": "2864.png", "formula": "\\begin{align*} & \\sum _ { x , y = 0 } ^ n ( \\delta _ { x , y } - M _ { x , y } ) f _ y f _ x = \\sum _ { x , y = 0 } ^ n \\sum _ { j , j ' = 0 } ^ n \\left ( 1 - \\Theta ( \\mu _ j , \\mu _ { j ' } ) \\right ) \\psi _ j ( x ) \\psi _ { j ' } ( x ) \\psi _ j ( y ) \\psi _ { j ' } ( y ) f _ y f _ x \\\\ & = \\sum _ { j , j ' = 0 } ^ n \\left ( 1 - \\Theta ( \\mu _ j , \\mu _ { j ' } ) \\right ) \\left ( \\sum _ { x = 0 } ^ n \\psi _ j ( x ) f _ x \\psi _ { j ' } ( x ) \\right ) ^ 2 . \\end{align*}"}
+{"id": "2341.png", "formula": "\\begin{align*} \\begin{cases} 2 | v | ^ 2 + b \\cdot v = & 0 , \\\\ A ^ 1 _ 2 \\ , ( b _ 1 + 2 v _ 1 ) = & 0 , \\\\ A ^ 1 _ 2 \\ , ( b _ 2 + 2 v _ 2 ) = & 0 . \\end{cases} \\end{align*}"}
+{"id": "2968.png", "formula": "\\begin{align*} \\begin{aligned} E _ { D ^ 2 } ( e ^ { 2 \\pi i t } ) & = E _ { D ^ 2 } \\left ( M ^ { - 1 } \\circ M ( e ^ { 2 \\pi i t } ) \\right ) \\\\ & = M ^ { - 1 } \\circ Q \\left ( M ( e ^ { 2 \\pi i t } ) \\right ) \\\\ & = \\hat { P } ( e ^ { 2 \\pi i t } ) = e ^ { 2 \\pi i P ( t ) } , \\end{aligned} \\end{align*}"}
+{"id": "7464.png", "formula": "\\begin{align*} N _ f ( z ) & = \\dfrac { 1 - z } { 1 - m z } . \\end{align*}"}
+{"id": "8997.png", "formula": "\\begin{align*} p ( \\eta , j ) & = p ( \\eta , \\ell _ { m } ) + r - m - 1 \\quad \\\\ | \\pi ( \\eta \\cup j ) | & = | \\pi ( \\eta \\cup \\ell _ { m } ) | + r - m , \\end{align*}"}
+{"id": "5032.png", "formula": "\\begin{align*} f \\left ( \\left ( a + 1 + \\frac { b } { y } \\right ) ( y - 1 ) \\right ) - f \\left ( \\left ( a + \\frac { b } { y } \\right ) ( y - 1 ) \\right ) & = y - 1 + ( y - 1 ) \\int _ { u = a + \\frac b y } ^ { a + 1 + \\frac b y } \\left ( \\log ( u ) + \\log ( y - 1 ) \\right ) \\d u . \\end{align*}"}
+{"id": "2896.png", "formula": "\\begin{align*} { \\cal L } { \\bf T } : = ( { \\frak G } _ 0 { \\bf T } , \\ldots , { \\frak G } _ n { \\bf T } ) , \\end{align*}"}
+{"id": "8372.png", "formula": "\\begin{align*} \\left ( I - \\mathcal { P } ^ - \\right ) s = 0 \\end{align*}"}
+{"id": "5933.png", "formula": "\\begin{align*} \\begin{cases} \\lambda + 2 < p , & \\alpha \\leq 2 , \\\\ \\lambda + 2 < \\alpha + p - 2 , & \\alpha > 2 . \\end{cases} \\end{align*}"}
+{"id": "6131.png", "formula": "\\begin{align*} [ l , r ] \\otimes [ l ' , r ' ] = \\bigoplus \\limits _ { \\nu = \\vert l - l ' \\vert \\mbox { s t e p } 2 } ^ { \\mathrm { m i n \\left ( l + l ' , 2 d - 4 - l - l ' \\right ) } } [ \\nu , r + r ' ] . \\end{align*}"}
+{"id": "8199.png", "formula": "\\begin{align*} x _ { 2 i - 1 , j } + x _ { 2 i - 1 , j + 1 } = a _ j ; \\end{align*}"}
+{"id": "2858.png", "formula": "\\begin{align*} { S ( t ) = \\int _ { 0 } ^ { + \\infty } e ^ { - A s } \\Sigma _ 2 \\big ( \\overline { \\frak p ^ 2 } ( t - s ) \\big ) e ^ { - A ^ T s } \\dd s } , t \\ge 0 \\end{align*}"}
+{"id": "6577.png", "formula": "\\begin{align*} \\begin{aligned} \\biggr { | } \\int _ { \\partial \\Omega \\cap B _ { \\eta } } \\nabla c \\cdot \\nu c \\psi ^ { 2 } \\biggr { | } & = \\biggr { | } \\int _ { \\partial \\Omega \\cap B _ { \\eta } } ( \\gamma - c ) c \\psi ^ { 2 } \\biggr { | } \\\\ & \\le \\gamma ^ { 2 } \\int _ { \\partial \\Omega \\cap B _ { \\eta } } \\psi ^ { 2 } \\\\ & \\le \\gamma ^ { 2 } \\int _ { \\partial \\Omega } \\psi ^ { 2 } . \\end{aligned} \\end{align*}"}
+{"id": "3100.png", "formula": "\\begin{align*} ( \\alpha - \\alpha ^ 2 ) ( 1 - K _ { 2 } ) ^ 2 \\sum \\limits _ { k = 0 } ^ { m } \\norm { \\theta ^ k - \\tilde { \\theta } ^ k } _ { \\mathcal { L } ^ 2 , G } ^ 2 & \\leq \\norm { \\theta ^ 0 - \\theta ^ * } _ { \\mathcal { L } ^ 2 , G } ^ 2 - \\norm { \\theta ^ { m + 1 } - \\theta ^ * } _ { \\mathcal { L } ^ 2 , G } ^ 2 \\\\ & \\leq \\norm { \\theta ^ 0 - \\theta ^ * } _ { \\mathcal { L } ^ 2 , G } ^ 2 , \\forall \\ m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "4126.png", "formula": "\\begin{align*} \\rho ( g ) & = g & g \\in G \\\\ \\rho ^ { \\vee } ( g ) & = ( g ^ { - 1 } ) ^ { \\vee } ( \\varphi ) & g \\in G . \\end{align*}"}
+{"id": "7841.png", "formula": "\\begin{align*} \\hat \\Phi ^ \\dagger ( a , s ) = h ( a , s ) . \\end{align*}"}
+{"id": "3919.png", "formula": "\\begin{align*} \\bigg | \\frac { \\partial G _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) } { \\partial z _ { i , \\hbar } } \\bigg | \\leq C , \\ \\ 1 \\leq i \\neq j \\leq m , \\hbar = 1 , 2 . \\end{align*}"}
+{"id": "7988.png", "formula": "\\begin{align*} \\rho _ { t } & = \\frac { h h _ { t } + \\sum h _ { k } h _ { k t } } { \\rho } , \\\\ \\rho _ { i } & = \\frac { h h _ { i } + \\sum h _ { k } h _ { k i } } { \\rho } = \\frac { h _ { i } w _ { i i } } { \\rho } , \\\\ \\rho _ { i j } & = \\frac { h h _ { i j } + h _ { i } h _ { j } + \\sum h _ { k } h _ { k i j } + \\sum h _ { k i } h _ { k j } } { \\rho } - \\frac { h _ { i } h _ { j } w _ { i i } w _ { j j } } { \\rho ^ { 3 } } . \\end{align*}"}
+{"id": "3161.png", "formula": "\\begin{align*} { _ { 3 } } F _ { 2 } \\left ( \\begin{array} { c c c } \\chi _ 4 , & \\chi _ 4 , & \\chi _ 4 \\\\ & \\varepsilon , & \\varepsilon \\end{array} \\mid 1 \\right ) & = \\frac { \\chi _ 8 ( - 1 ) } { q ^ 2 } [ J ( \\chi _ 8 , \\chi _ 8 ) J ( \\chi _ 8 , \\chi _ 8 ^ 2 ) + \\overline { J ( \\chi _ 8 , \\chi _ 8 ) } J ( \\chi _ 8 , \\chi _ 8 ^ 2 ) ] \\\\ & = \\frac { 1 } { q ^ 2 } J ( \\chi _ 8 , \\chi _ 8 ^ 2 ) \\times 2 R e ( J ( \\chi _ 8 , \\chi _ 8 ) ) \\times \\chi _ 8 ( - 1 ) \\\\ & = \\frac { 1 } { q ^ 2 } [ - 2 u ( - p ) ^ t ] . \\end{align*}"}
+{"id": "1812.png", "formula": "\\begin{align*} \\aligned \\mathcal { Y M } _ p ( \\nabla ) & = \\int _ M \\frac 1 p | | R ^ \\nabla | | ^ p \\ , d v \\ , \\\\ \\big ( r e s p . \\mathcal { Y M } _ F ( \\nabla ) & = \\int _ M F ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) \\ , d v \\ , \\big ) , \\endaligned \\end{align*}"}
+{"id": "1074.png", "formula": "\\begin{align*} w \\varepsilon ^ \\mu = x > w \\varepsilon ^ \\mu s _ \\alpha \\varepsilon ^ { ( \\langle \\mu , \\alpha \\rangle - 1 ) \\alpha ^ \\vee } = w s _ \\alpha \\varepsilon ^ { \\mu - \\alpha ^ \\vee } . \\end{align*}"}
+{"id": "5980.png", "formula": "\\begin{align*} \\widetilde { W } _ { B _ { r _ 1 } } ( x ) = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { ( 2 ^ k ) ^ { 1 0 } } \\int H ( x - y ) \\chi _ { A _ k } ( y ) d y . \\end{align*}"}
+{"id": "1948.png", "formula": "\\begin{align*} = \\cos ( \\pi \\nu ) \\frac { ( 2 z ) ^ { - \\mu } e ^ { z } } { \\sqrt { \\pi } } ~ G _ { { 1 } , { 2 } } ^ { { 2 } , { 1 } } \\left ( 2 z \\bigg { | } \\begin{array} { l l l } \\mu + \\frac { 1 } { 2 } \\\\ \\mu + \\nu , ~ \\mu - \\nu \\end{array} \\right ) , \\end{align*}"}
+{"id": "6832.png", "formula": "\\begin{align*} T ( T _ { + } ) = \\left ( \\begin{array} [ c ] { c c c c c c } t _ { 0 } & 0 & 0 & \\ldots & 0 & \\ldots \\\\ t _ { 1 } & t _ { 0 } & 0 & \\ldots & 0 & \\ldots \\\\ t _ { 2 } & t _ { 1 } & t _ { 0 } & \\ldots & 0 & \\ldots \\\\ & \\ldots & & \\ddots & & \\\\ t _ { n } & t _ { n - 1 } & t _ { n - 2 } & \\ldots & t _ { 0 } & \\ldots \\\\ & \\ldots & & \\ldots & & \\ddots \\end{array} \\right ) \\end{align*}"}
+{"id": "1968.png", "formula": "\\begin{align*} & q ^ { l m ' ( n - m ' ) + ( l - 1 ) ( n - m ' ) ^ 2 - ( l - 1 ) n ^ 2 } \\prod _ { i = 2 } ^ { m ' } q ^ { n ( l _ 1 - l _ i ) } \\cdot q ^ { - m ' ( n - m ' ) - n m ' } \\\\ & = q ^ { - n ( l _ 1 + \\ldots + l _ { m ' } ) } q ^ { m ' n l } \\cdot q ^ { l m ' ( n - m ' ) + ( l - 1 ) ( n - m ' ) ^ 2 - ( l - 1 ) n ^ 2 - m ' ( n - m ' ) - n m ' } \\\\ & = q ^ { - n ( l _ 1 + \\ldots + l _ { m ' } ) } q ^ { l m ' ( 2 n - m ' ) + ( l - 1 ) ( - 2 n m ' + { m ' } ^ 2 ) - 2 n m ' + { m ' } ^ 2 } \\\\ & = q ^ { - n ( l _ 1 + \\ldots + l _ { m ' } ) } , \\end{align*}"}
+{"id": "5690.png", "formula": "\\begin{align*} \\gamma _ t \\beta _ t = \\frac { \\gamma _ { t - 1 } ( 1 - \\kappa _ t ) } { ( 1 - \\kappa _ { t - 1 } ) } \\left ( 1 + \\frac { 2 \\mu \\gamma _ { t - 1 } } { 1 - \\kappa _ { t - 1 } } \\right ) ^ { - 1 } \\leq \\frac { 3 } { 2 } \\gamma _ 0 , \\quad \\alpha _ t = \\frac { \\kappa _ { t - 1 } \\gamma _ t \\beta _ t } { \\gamma _ { t - 1 } } = \\frac { \\kappa _ { t - 1 } ( 1 - \\kappa _ { t } ) } { 1 - \\kappa _ { t - 1 } } \\left ( 1 + \\frac { 2 \\mu \\gamma _ { t - 1 } } { 1 - \\kappa _ { t - 1 } } \\right ) ^ { - 1 } \\leq \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "2973.png", "formula": "\\begin{align*} h ( x ) : = \\begin{cases} x \\sin ( 1 / x ) & x \\neq 0 \\\\ 0 & x = 0 . \\end{cases} \\end{align*}"}
+{"id": "8392.png", "formula": "\\begin{align*} \\| \\widetilde { Q } \\| _ { L ^ 1 } : = \\sum _ { i = 1 } ^ 2 \\sum _ { j = 1 } ^ 2 \\left \\| \\widetilde { Q } _ { i j } \\right \\| _ { L ^ 1 } . \\end{align*}"}
+{"id": "8893.png", "formula": "\\begin{align*} A ^ * ( Z ) = & \\pi ^ k c _ k \\sum _ { r , s } \\sum _ { D _ 1 , D _ 2 , t _ 1 , t _ 2 } \\sum _ { d _ 1 | c ( T _ 1 ) } \\sum _ { d _ 2 | c ( T _ 2 ) } ( d _ 1 d _ 2 ) ^ { k - 1 } S ( D _ 1 d _ 1 ^ { - 2 } , D _ 2 d _ 2 ^ { - 2 } ) e ( \\cdots ) \\\\ & = \\pi ^ k c _ k \\sum _ { T _ 1 , T _ 2 > 0 } \\sum _ { d _ 1 | c ( T _ 1 ) } \\sum _ { d _ 2 | c ( T _ 2 ) } ( d _ 1 d _ 2 ) ^ { k - 1 } S ( D _ 1 d _ 1 ^ { - 2 } , D _ 2 d _ 2 ^ { - 2 } ) e ( T r ( T _ 1 Z + T _ 2 Z ) ) . \\end{align*}"}
+{"id": "8071.png", "formula": "\\begin{align*} e _ { a _ j } \\left [ t + 1 \\right ] = & e _ { a _ j } \\left [ t \\right ] + 2 \\mu \\Re { \\left \\{ \\hat { \\phi } ^ { \\left ( j , j \\right ) } \\right \\} } - 4 \\mu \\left ( \\sum _ { l = 1 } ^ M \\lvert \\hat { \\phi } ^ { \\left ( l , j \\right ) } \\rvert ^ 2 + M \\sigma _ { e , i } ^ 2 \\lVert \\mathbf { p } _ j \\rVert ^ 2 \\right . \\left . + \\sum _ { q = 1 } ^ { M - 1 } \\sum _ { r = q + 1 } ^ M f _ { q , r } ^ { \\left ( j \\right ) } \\right ) a _ j \\left [ t \\right ] . \\end{align*}"}
+{"id": "3765.png", "formula": "\\begin{align*} \\begin{aligned} \\iota ( C ' _ w ) & = C ' _ w , \\\\ q ^ { \\frac { \\ell ( w ) } { 2 } } C ' _ w & = \\sum _ { z \\leq w } P _ { z , w } ( q ) T _ z , \\end{aligned} \\end{align*}"}
+{"id": "3109.png", "formula": "\\begin{align*} A _ { k , r } = \\{ \\o \\in \\O \\mid \\exists ~ x \\in C ( \\omega ) \\cap \\bar { B } ( 0 , r ) ~ s . t . ~ \\langle \\mathcal { F } ( x ^ * ) ( \\omega ) , x - x ^ * ( \\omega ) \\rangle \\le - \\frac { 1 } { k } \\} \\end{align*}"}
+{"id": "2871.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n J _ n \\varphi \\left ( 1 - \\frac 1 n \\right ) = J \\varphi ( 1 ) . \\end{align*}"}
+{"id": "7563.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { a _ { i , n } } { \\prod _ { k = 0 } ^ n \\beta _ { i + k } } = \\frac { 1 } { q _ i } , \\end{align*}"}
+{"id": "2966.png", "formula": "\\begin{align*} F ^ - ( t , x ) : = \\left ( \\begin{array} { c } - R ^ - ( t , x ) \\cos \\pi t , \\\\ - R ^ - ( t , x ) \\sin \\pi t \\end{array} \\right ) , \\end{align*}"}
+{"id": "2224.png", "formula": "\\begin{align*} \\mathbf { a } ( \\theta _ { 0 } ) ^ { T } \\mathbf { v } ^ { 0 } & = e ^ { - j m \\phi } + e ^ { - j n \\phi } + e ^ { - j k \\phi } + e ^ { - j l \\phi } \\\\ & = e ^ { - j m \\phi } . \\\\ & ( 1 + e ^ { - j ( n - m ) \\phi } ( 1 + e ^ { - j ( k - n ) \\phi } ( 1 + e ^ { - j ( l - k ) \\phi } ) ) ) \\end{align*}"}
+{"id": "9129.png", "formula": "\\begin{align*} \\pi _ { f , n } = \\zeta _ { ( f ) } ^ { \\varphi ^ { - ( n + 1 ) } } ( \\zeta _ { p ^ { n + 1 } } - 1 ) . \\end{align*}"}
+{"id": "1335.png", "formula": "\\begin{align*} L ( \\gamma _ { u _ { k - 1 } , u _ k } ) & < d ( u _ { k - 1 } , u _ k ) + \\varepsilon \\delta = d ( u _ { k - 1 } , u _ k ) + \\varepsilon \\min \\bigl \\{ 1 , \\frac { d ( u _ { k - 1 } , u _ k ) } { C _ { n + 1 } \\beta _ k } \\bigr \\} . \\end{align*}"}
+{"id": "3733.png", "formula": "\\begin{align*} \\nu ( H ' | _ g ( h ) ) & = - \\tfrac { 1 } { 2 } R ' ( h ) + \\tfrac { 1 } { 2 } R '^ \\Sigma | _ { g ^ \\intercal } ( h ^ \\intercal ) + A _ g \\cdot ( A _ g \\circ h ) \\\\ & - A _ g \\cdot A ' ( h ) - H _ g H ' ( h ) \\\\ & + \\tfrac { 1 } { 4 } \\big ( - R _ g + R ^ \\Sigma _ g - | A _ g | ^ 2 - H _ { g } ^ 2 \\big ) h ( \\nu , \\nu ) \\\\ & - \\tfrac { 1 } { 2 } \\Delta _ \\Sigma h ( \\nu , \\nu ) + g ^ { a b } \\omega ( e _ a ) e _ b ( H _ g ) . \\end{align*}"}
+{"id": "7651.png", "formula": "\\begin{align*} \\widetilde { \\nu } _ t ^ { * , \\xi } = \\eta _ t \\nu _ t ^ { * , \\xi } , \\widetilde { \\varphi } _ t ^ { * , \\xi } = \\eta ^ { - 1 } _ t \\varphi _ t ^ { * , \\xi } , \\quad \\widetilde { \\Lambda } _ t ^ { 0 , * , \\xi } = \\eta ^ { - 1 } _ t \\Lambda _ t ^ { 0 , * , \\xi } , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "3752.png", "formula": "\\begin{align*} \\deg ( f _ { l _ 0 } m ' _ { l _ 0 , l _ 0 } ) = \\deg ( f _ { l _ 0 } m '' _ { l _ 0 , l _ 0 } ) > k + \\deg ( m ' _ { l _ 0 , l _ 0 } ) - \\deg ( d _ 1 ) - 1 > k . \\end{align*}"}
+{"id": "8247.png", "formula": "\\begin{align*} | | z | | : = \\sum _ { k = 1 } ^ { \\infty } k | z _ k | , \\end{align*}"}
+{"id": "1669.png", "formula": "\\begin{align*} \\rho ( g ) = X \\rho ( g ) \\psi ( g ) X ^ { - 1 } \\end{align*}"}
+{"id": "6999.png", "formula": "\\begin{align*} { \\mathcal M } = \\left ( \\begin{array} { c c c c c c c c } b _ 0 & d _ 0 & 0 & 0 & 0 & \\cdots & 0 & 0 \\\\ c _ 0 & b _ 1 & d _ 1 & 0 & 0 & \\cdots & 0 & 0 \\\\ 0 & c _ 1 & b _ 2 & d _ 2 & 0 & \\cdots & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\cdots & b _ { l - 2 } & d _ { l - 2 } \\\\ 0 & 0 & 0 & 0 & 0 & \\cdots & c _ { l - 2 } & b _ { l - 1 } \\end{array} \\right ) \\end{align*}"}
+{"id": "5606.png", "formula": "\\begin{align*} \\begin{cases} c _ { \\alpha + n + 1 } - c _ { n + 1 } = b _ { \\alpha + n } - b _ { \\alpha + n + 1 } \\\\ d _ n = c _ { \\alpha + 1 } + ( b _ { \\alpha + 1 } - b _ \\alpha ) = d \\\\ b _ { \\alpha + n } = b _ \\alpha + ( d _ 1 - c _ { \\alpha + 1 } ) + \\sum _ { j = 2 } ^ n ( c _ j - c _ { \\alpha + j } ) \\\\ b = b _ \\alpha + d - c _ { \\alpha + 1 } + \\sum _ { j = 2 } ^ \\infty ( c _ j - c _ { \\alpha + j } ) \\end{cases} \\ , \\end{align*}"}
+{"id": "8470.png", "formula": "\\begin{align*} 0 = \\oint q _ 1 ( z ) q _ 2 ( z ) d z = \\int _ { \\mathbb { R } } \\left [ Q _ { j , - } \\mathcal { P } ^ - ( D _ j ) + Q _ { j , - } J \\right ] Q _ { j , - } ^ H \\mathrm { d } z , \\end{align*}"}
+{"id": "8630.png", "formula": "\\begin{align*} \\forall \\varepsilon > 0 , \\lim _ { n \\to \\infty } \\sum _ { m = 1 } ^ n \\mathbb { E } _ { n , m } [ \\| \\chi _ { n , m } \\| _ { L ^ 2 ( \\mu ) } ^ 2 : \\| \\chi _ { n , m } \\| _ { L ^ 2 ( \\mu ) } > \\varepsilon ] = 0 \\end{align*}"}
+{"id": "1354.png", "formula": "\\begin{align*} g ( y _ 1 ) & = \\frac { 1 } { d ( a _ 1 , b _ 1 ) } \\sum _ { k = 1 } ^ { K } \\big ( g ( u _ { 2 k - 2 } ) - g ( u _ { 2 k - 1 } ) - g ( u _ { 2 k - 1 } ) + g ( u _ { 2 k } ) \\big ) \\\\ & = \\frac { 1 } { d ( a _ 1 , b _ 1 ) } ( 2 R + 2 ( K - 2 ) s ) = 1 . \\end{align*}"}
+{"id": "6796.png", "formula": "\\begin{align*} r _ { m } ( x ) : = ( - 1 ) ^ { m + 1 } \\left ( \\sum _ { k = 1 } ^ { N } \\alpha _ { k } ^ { + } e ^ { - 2 \\tau _ { k } x } \\frac { \\left ( \\frac { 1 } { 2 } - \\tau _ { k } \\right ) ^ { m } } { \\left ( \\frac { 1 } { 2 } + \\tau _ { k } \\right ) ^ { m + 1 } } + \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } s ^ { + } \\left ( \\rho \\right ) e ^ { 2 i \\rho x } \\frac { \\left ( \\frac { 1 } { 2 } + i \\rho \\right ) ^ { m } } { \\left ( \\frac { 1 } { 2 } - i \\rho \\right ) ^ { m + 1 } } d \\rho \\right ) , \\end{align*}"}
+{"id": "8868.png", "formula": "\\begin{align*} A ^ U _ \\xi : = I ^ * ( { \\ge } \\xi , { < } \\nu ) \\end{align*}"}
+{"id": "2109.png", "formula": "\\begin{align*} L L ( b , c ) & = b \\left ( f _ c ( b ) - f _ c ( b - 1 ) \\right ) + f _ c ( b ) - f _ c ( 0 ) \\\\ & = b \\Delta _ c ( b ) + f _ c ( b ) - f _ c ( 0 ) \\\\ & = b \\Delta _ c ( b ) + \\sum _ { i = 1 } ^ b \\left [ f _ c ( i ) - f _ c ( i - 1 ) \\right ] \\\\ & = b \\Delta _ c ( b ) + \\sum _ { i = 1 } ^ b \\Delta _ c ( i ) . \\end{align*}"}
+{"id": "7420.png", "formula": "\\begin{align*} | { \\sf N } _ 1 ^ { \\geq 0 } ( i ; c ) | = | { \\sf N } _ 2 ^ { \\geq 0 } ( i ; c ) | \\end{align*}"}
+{"id": "6236.png", "formula": "\\begin{align*} \\psi ( x , w ) = ( \\gamma h + \\nabla \\gamma ) ( x ) + w . \\end{align*}"}
+{"id": "3623.png", "formula": "\\begin{align*} Q = \\ln \\kappa _ 1 - N \\ln \\nu ^ { n + 1 } , \\end{align*}"}
+{"id": "445.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } e ^ { \\gamma x } f _ 0 ( x ) = 0 . \\end{align*}"}
+{"id": "2868.png", "formula": "\\begin{align*} & \\Upsilon _ { 0 , j ' } ( \\ell ) = 0 , j ' = 1 , \\ldots , n , \\\\ & \\Upsilon _ { j , j ' } ( \\ell ) = \\frac { 1 - ( - 1 ) ^ { j + j ' } } { n + 1 } \\cos \\left ( \\frac { \\pi j ' \\ell } { 2 ( n + 1 ) } \\right ) \\cos \\left ( \\frac { \\pi j \\ell } { 2 ( n + 1 ) } \\right ) , j , j ' = 1 , \\ldots , n , \\ , j \\not = j ' \\\\ & \\Upsilon _ { j , j } ( \\ell ) = \\cos \\left ( \\frac { \\pi \\ell j } { n + 1 } \\right ) , j = 0 , \\ldots , n . \\end{align*}"}
+{"id": "7746.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 1 & - 1 \\\\ 0 & 1 \\\\ 0 & - 1 \\end{pmatrix} B = \\begin{pmatrix} 1 & 1 & - 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "9327.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\sum _ { i = 1 } ^ m V _ i > \\left ( \\frac { \\pi } { 4 } + \\frac { 1 9 9 \\pi } { 3 2 0 0 k } \\right ) \\| \\mathbf { x } \\| . \\end{align*}"}
+{"id": "7126.png", "formula": "\\begin{align*} \\begin{array} { l l l } f _ L ^ M ( \\mathcal { M } _ { \\mathbb { C } , \\mathsf { S } } ( a ) ) \\circ u & = & G ( L _ a , R _ a ) ( f _ L ( s ) \\circ t ) \\\\ & = & f _ L ( L _ a \\circ s ) \\circ t \\\\ & = & f _ L ( a \\bullet _ { \\mu _ S } s ) \\circ t \\\\ & = & f _ L ( a ) \\circ f _ L ( s ) \\circ t \\\\ & = & f _ L ( a ) \\circ u . \\end{array} \\end{align*}"}
+{"id": "492.png", "formula": "\\begin{align*} \\mu \\ast \\mu _ a = \\mu _ s \\ast \\mu _ r , \\end{align*}"}
+{"id": "24.png", "formula": "\\begin{align*} P ( \\lambda ) + P ( \\mu ) = P ( \\lambda + \\mu ) S ( \\lambda ) + S ( \\mu ) = S ( \\lambda + \\mu ) , \\end{align*}"}
+{"id": "5179.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { D } ( \\phi _ n - \\phi _ \\infty ) _ + G _ n \\frac { 1 } { r } \\dd z \\dd r = \\int _ { D } ( \\phi - \\phi _ \\infty ) _ + G \\frac { 1 } { r } \\dd z \\dd r . \\end{align*}"}
+{"id": "5796.png", "formula": "\\begin{align*} [ S ] = \\mu \\left ( \\frac { 1 } { 2 } d \\overline { z } + ( \\alpha - I \\beta ) d z \\right ) J \\end{align*}"}
+{"id": "4088.png", "formula": "\\begin{align*} \\theta ^ 0 { } _ u ( X ) & = v ^ i { } _ \\alpha d u ^ \\alpha , \\\\ * \\theta ^ 1 { } _ u ( X ) & = v ^ i { } _ \\alpha d u ^ \\alpha { } _ j - v ^ i { } _ \\alpha u ^ \\alpha { } _ { j \\beta } v ^ \\beta { } _ \\gamma d u ^ \\gamma . \\end{align*}"}
+{"id": "8461.png", "formula": "\\begin{align*} V _ 1 ( k ) : = \\begin{pmatrix} 1 & 0 \\\\ 0 & 2 i k \\end{pmatrix} , V _ 2 ( k ) : = \\begin{pmatrix} ( 2 i k ) ^ { - 1 } & 0 \\\\ 0 & 1 \\end{pmatrix} , \\end{align*}"}
+{"id": "8783.png", "formula": "\\begin{align*} V _ 1 \\oplus V _ 2 = \\mathrm { s p a n } ( \\cos \\ell x _ 1 , \\sin \\ell x _ 1 ) \\oplus \\mathrm { s p a n } ( \\cos k x _ 2 , \\sin k x _ 2 ) , \\end{align*}"}
+{"id": "3263.png", "formula": "\\begin{align*} \\Delta _ n ^ t a ^ j & : = \\int _ t ^ { t + \\Delta _ n } \\int _ t ^ u \\frac d { d u } \\langle \\alpha _ s , \\mathcal S ( u - s ) ^ * e _ j \\rangle d s d u \\\\ & \\qquad \\qquad + \\int _ t ^ { t + \\Delta _ n } \\int _ t ^ u \\langle \\sigma _ s , \\mathcal A ^ * \\mathcal S ( u - s ) ^ * e _ j \\rangle d W _ s d u . \\end{align*}"}
+{"id": "3264.png", "formula": "\\begin{align*} \\langle \\tilde { \\Delta } _ { i } ^ n Y , e _ { j } \\rangle = \\langle \\Delta _ { i } ^ n S , e _ { j } \\rangle + \\Delta _ n ^ { ( i - 1 ) \\Delta _ n } a ^ { j } . \\end{align*}"}
+{"id": "4364.png", "formula": "\\begin{align*} v = D '' u _ 0 + P _ { \\omega , h } ( \\sqrt { \\lambda } \\tau _ 0 ) . \\end{align*}"}
+{"id": "7995.png", "formula": "\\begin{align*} - \\Delta u + V ( x ) u + \\frac { \\gamma } { 2 \\pi } \\left ( \\log ( | \\cdot | ) \\ast u ^ 2 \\right ) u = b | u | ^ { p - 2 } u \\qquad \\ \\mathbb { R } ^ 2 . \\end{align*}"}
+{"id": "8292.png", "formula": "\\begin{align*} ( K * \\mu ) ( x ) = \\int _ { \\mathbb R ^ d } K ( x - z ) \\mu ( \\d z ) . \\end{align*}"}
+{"id": "5523.png", "formula": "\\begin{align*} \\tilde { h } _ d ' ( x ) = \\log d + \\left ( \\psi ( x ) - \\psi \\left ( x + \\frac 1 2 \\right ) \\right ) - \\psi \\left ( x + \\frac { d - 1 } { 2 } \\right ) \\end{align*}"}
+{"id": "1609.png", "formula": "\\begin{align*} \\mathbf { \\Phi } _ { \\mathrm { t } , g } ^ \\star = [ \\mathbf { \\Phi } _ { g } ^ \\star ] _ { 1 : \\bar { M } , : } , ~ ~ ~ \\mathbf { \\Phi } _ { \\mathrm { r } , g } ^ \\star = [ \\mathbf { \\Phi } _ { g } ^ \\star ] _ { \\bar { M } + 1 : 2 \\bar { M } , : } , \\forall g \\in \\mathcal { G } . \\end{align*}"}
+{"id": "6112.png", "formula": "\\begin{align*} \\begin{aligned} & j ^ { ( 1 ) } \\geq j ^ { ( 2 ) } \\geq j ^ { ( 4 ) } \\ , , j ^ { ( 0 ) } \\geq j ^ { ( 4 ) } \\ , , j ^ { ( i ) } \\geq 0 \\ , , i = 0 , 1 , . . . , 4 \\ , , \\\\ & j ^ { ( 1 ) } + j ^ { ( 2 ) } + j ^ { ( 0 ) } + j ^ { ( 4 ) } \\geq j ^ { ( 3 ) } \\geq j ^ { ( 1 ) } + j ^ { ( 2 ) } + j ^ { ( 0 ) } - j ^ { ( 4 ) } \\ , . \\end{aligned} \\end{align*}"}
+{"id": "476.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } e ^ { \\gamma x } f _ r ( x ) = 0 . \\end{align*}"}
+{"id": "825.png", "formula": "\\begin{align*} w ( t , x ) = 1 + \\rho \\big ( t , \\gamma ( x , \\varphi ) \\big ) \\end{align*}"}
+{"id": "8822.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } X _ t = B ( X _ t , X _ t ) \\\\ X _ 0 = x _ 0 \\end{cases} \\end{align*}"}
+{"id": "821.png", "formula": "\\begin{align*} g ( c _ 0 ) ( x ) = \\begin{cases} g _ O , & x \\in [ x _ 0 - 1 , x _ 1 ] , \\\\ g _ I , & x \\in [ x _ 2 , x _ 3 ] , \\\\ g _ O , & x \\in [ x _ 4 , x _ 5 + 1 ] , \\end{cases} \\end{align*}"}
+{"id": "102.png", "formula": "\\begin{align*} J _ b ( F ) = \\{ g \\in G ( L ) \\mid g ^ { - 1 } b \\sigma ( g ) = b \\} . \\end{align*}"}
+{"id": "1472.png", "formula": "\\begin{align*} 0 \\leq \\varphi '' ( s ) = 2 \\beta M [ ( M s - A ) + ( B - M s ) ] . \\end{align*}"}
+{"id": "400.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to - \\infty } \\mathrm { d i s t } _ { \\mathbb { U } } ( \\mathcal { A } ( \\tau , \\omega ) , \\mathcal { A } _ { \\infty } ( \\omega ) ) = 0 \\ \\emph { f o r a l l } \\ \\omega \\in \\Omega , \\end{align*}"}
+{"id": "6565.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } & n ^ { p } ( \\cdot , t ) \\\\ & \\le \\int _ { \\Omega } n _ { 0 } ^ { p } + \\frac { p ( p - 1 ) } { 4 } \\| S _ { 0 } \\| _ { \\mathcal { C } ( [ 0 , \\gamma ] ) } ^ { 2 } \\| \\nabla c \\| _ { L ^ { \\infty } ( 0 , T ; L ^ { \\infty } ( \\Omega ) ) } ^ { 2 } \\int _ { 0 } ^ { t } \\int _ { \\Omega } n ^ { p } \\quad \\mbox { f o r } t \\le T . \\end{aligned} \\end{align*}"}
+{"id": "92.png", "formula": "\\begin{align*} \\nu ^ G ( [ b ] ) = & \\nu ^ { \\tilde G } ( [ b \\gamma ] ) - \\frac 1 { \\# W } \\sum _ { u \\in W } u \\mu _ \\sigma , \\\\ \\implies \\nu _ x = & \\nu ^ { \\tilde G } ( [ b _ { x \\gamma } ] ) - \\frac 1 { \\# W } \\sum _ { u \\in W } u \\mu _ \\sigma . \\end{align*}"}
+{"id": "78.png", "formula": "\\begin{align*} \\ell ( x , \\alpha ) = \\langle \\mu , v \\beta \\rangle + \\Phi ^ + ( \\alpha ) - \\Phi ^ + ( w v \\beta ) \\geq \\Phi ^ + ( \\alpha ) . \\end{align*}"}
+{"id": "9324.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ Z _ { i , j } \\right ] = \\frac { \\pi } { 4 } \\| \\mathbf { x } \\| . \\end{align*}"}
+{"id": "1519.png", "formula": "\\begin{align*} \\begin{pmatrix} \\begin{array} { c | c c c c | c } J L ( 2 ) & J M ( 2 ) & O & \\dots & O & \\\\ \\vdots & O & \\ddots & \\ddots & \\vdots & J R \\\\ \\vdots & \\vdots & \\ddots & \\ddots & O & \\\\ J L ( n ) & O & \\dots & O & J M ( n ) & \\\\ \\end{array} \\end{pmatrix} , \\end{align*}"}
+{"id": "5867.png", "formula": "\\begin{align*} C \\left ( X , Y , Z \\right ) \\equiv \\left ( \\nabla _ { Z } g \\right ) \\left ( X , Y \\right ) = Z g \\left ( X , Y \\right ) - g \\left ( \\nabla _ { Z } X , Y \\right ) - g \\left ( X , \\nabla _ { Z } Y \\right ) \\end{align*}"}
+{"id": "2244.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\frac { e ^ { - \\frac { \\pi | \\xi | ^ 2 } { t ^ 2 } } } { | \\xi | ^ { 2 ( 1 - \\beta ) } } \\ , d \\xi = \\pi ^ { 1 / 2 - \\beta } \\Gamma \\biggr ( \\beta - \\frac { 1 } { 2 } \\biggr ) t ^ { 2 \\beta - 1 } \\leq \\Gamma \\biggr ( \\beta - \\frac { 1 } { 2 } \\biggr ) t ^ { 2 \\beta - 1 } , \\end{align*}"}
+{"id": "552.png", "formula": "\\begin{align*} h g ( h g e _ n h g ) \\cdots ( h g ^ { \\frac { n } { 2 } - 1 } e _ n h g ^ { \\frac { n } { 2 } - 1 } ) ( h g ^ { \\frac { n } { 2 } + 1 } e _ n h g ^ { \\frac { n } { 2 } + 1 } ) \\cdots ( h g ^ { n - 1 } e _ n h g ^ { n - 1 } ) = \\\\ ( h g e _ n h g ) \\cdots ( h g ^ { \\frac { n } { 2 } - 1 } e _ n h g ^ { \\frac { n } { 2 } - 1 } ) ( h g ^ { \\frac { n } { 2 } + 1 } e _ n h g ^ { \\frac { n } { 2 } + 1 } ) \\cdots ( h g ^ { n - 1 } e _ n h g ^ { n - 1 } ) , \\end{align*}"}
+{"id": "6293.png", "formula": "\\begin{align*} \\gamma _ { t - 1 } \\alpha _ t ^ 2 = ( 1 - \\alpha _ t ) \\alpha _ { t - 1 } ^ 2 \\gamma _ t + \\mu \\alpha _ t \\gamma _ t \\gamma _ { t - 1 } , \\end{align*}"}
+{"id": "6090.png", "formula": "\\begin{align*} 2 x _ { 1 2 3 4 } : = w _ { 1 2 , 3 , 4 } - w _ { 1 , 3 , 4 } - w _ { 2 , 3 , 4 } = w _ { 1 , 2 3 , 4 } - w _ { 1 , 2 , 4 } - w _ { 1 , 3 , 4 } = w _ { 1 , 2 , 3 4 } - w _ { 1 , 2 , 3 } - w _ { 1 , 2 , 4 } \\ , . \\end{align*}"}
+{"id": "9236.png", "formula": "\\begin{align*} \\begin{aligned} \\det M = \\sum \\limits _ { \\sigma \\in S _ n } \\sigma { M } _ { n _ { 1 1 } n _ { 1 2 } } \\cdots { M } _ { n _ { 1 l _ 1 } n _ { 1 1 } } M _ { n _ { 2 1 } M _ { 2 2 } } \\cdots { M } _ { n _ { r l _ r } n _ { r 1 } } . \\end{aligned} \\end{align*}"}
+{"id": "6242.png", "formula": "\\begin{align*} ( Q ( \\xi ) ) _ { i j } = Q _ { i j } ( \\xi ) = \\partial ^ 2 _ { i j } \\xi + a _ { i j } ^ j \\partial _ j \\xi + a _ { j i } ^ i \\partial _ i \\xi + b _ { i j } \\xi = 0 \\quad \\forall \\ , 0 \\leq i < j \\leq p , \\end{align*}"}
+{"id": "539.png", "formula": "\\begin{align*} u _ { i , j } & = - 2 ( 3 c _ { i } + 4 s _ { i } + 4 z _ i ) - 2 ( 3 c _ { j } + 4 s _ { j } + 4 z _ j ) \\\\ & \\phantom { { } = { } } + ( 3 c _ { | i - j | } + 4 s _ { | i - j | } + 4 z _ { | i - j | } ) + ( 3 c _ { i + j } + 4 s _ { i + j } + 4 z _ { i + j } ) \\\\ & = 3 c _ { i , j } + 4 t _ { i , j } + 4 z _ { i , j } . \\end{align*}"}
+{"id": "2621.png", "formula": "\\begin{align*} \\rho ( [ x , y ] ) \\beta ( v ) = \\rho ( \\alpha ( x ) ) \\rho ( y ) v - \\varepsilon ( x , y ) \\rho ( \\alpha ( y ) ) \\rho ( x ) v . \\end{align*}"}
+{"id": "6523.png", "formula": "\\begin{align*} R = \\frac { 1 } { 2 J ( X _ { J } ) } \\left [ 2 \\delta ( X _ { J } ) g \\wedge S + 2 \\omega ( X _ { J } ) g \\wedge g + E \\wedge g + F \\wedge S \\right ] , \\end{align*}"}
+{"id": "4686.png", "formula": "\\begin{align*} \\mathfrak h = \\frac { _ { n - 1 } ( \\Sigma ) } { \\min \\{ _ n ( A ) , _ n ( B ) \\} } . \\end{align*}"}
+{"id": "5272.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d \\lambda } \\right ) ^ r \\left [ I ( \\lambda ) - \\lambda ^ { - \\frac { n } { 2 } } \\sum _ { j = 0 } ^ N a _ j \\lambda ^ { - j } \\right ] = O \\left ( \\lambda ^ { - \\frac { n } { 2 } - r - N - 1 } \\right ) . \\end{align*}"}
+{"id": "1735.png", "formula": "\\begin{align*} \\ < g , \\gamma \\ > = \\ < M _ { \\omega \\circ f } ( g ) , \\gamma \\ > = \\lim \\limits _ { \\alpha } \\ < M _ { \\omega \\circ f } ( h _ \\alpha \\circ f ) , \\gamma \\ > = \\lim \\limits _ { \\alpha } \\ < M _ { \\omega } ( h _ \\alpha ) \\circ f , \\gamma \\ > . \\end{align*}"}
+{"id": "6460.png", "formula": "\\begin{align*} h _ { j _ p + 1 } = \\rho _ p h _ { j _ p } + \\frac { \\rho _ p ( \\rho _ p - 1 ) } { 2 } \\end{align*}"}
+{"id": "1143.png", "formula": "\\begin{align*} i _ { \\hat { q } } = N ( \\hat { k } - 1 ) + \\hat { \\imath } , j _ { \\hat { q } } = N ( \\hat { \\ell } - 1 ) + \\hat { \\jmath } \\end{align*}"}
+{"id": "7517.png", "formula": "\\begin{align*} \\int _ { B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x & \\leq C \\int _ { B _ { 8 \\rho } \\setminus B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x \\rho \\leq 1 / 8 \\varepsilon \\leq \\varepsilon _ 0 . \\end{align*}"}
+{"id": "5921.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\Big [ \\sup _ { t \\leq T } \\Vert X ( t , x _ n ) - X ( t , x ) \\Vert _ H ^ p \\Big ] = 0 , \\end{align*}"}
+{"id": "6458.png", "formula": "\\begin{align*} K = \\{ k _ n : n \\in \\N \\} \\subset M ^ c . \\end{align*}"}
+{"id": "4986.png", "formula": "\\begin{align*} M _ { f _ { 1 , \\infty } } ( \\alpha , \\epsilon , Z , \\psi ) : = \\lim _ { n \\to \\infty } M _ { f _ { 1 , \\infty } } ( n , \\alpha , \\epsilon , Z , \\psi ) . \\end{align*}"}
+{"id": "195.png", "formula": "\\begin{align*} Y ' _ { k , ( 2 ) } = Y _ { k , ( 2 ) } - z _ { k } t \\mbox { w i t h } z _ k \\in B _ { \\infty , \\infty } . \\end{align*}"}
+{"id": "1097.png", "formula": "\\begin{align*} \\prescript J { } \\pi ( w _ 1 ) = ( w _ 1 ' , \\mu _ 1 ) , \\prescript J { } \\pi ( w _ 2 ) = ( w _ 2 ' , \\mu _ 2 ) . \\end{align*}"}
+{"id": "514.png", "formula": "\\begin{align*} S _ { R } ^ { \\alpha } f ( x ) = \\sum _ { n = 0 } ^ { \\infty } \\Big ( 1 - \\frac { 2 n + d } { R } \\Big ) _ { + } ^ { \\alpha } P _ { n } f ( x ) , \\end{align*}"}
+{"id": "9325.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m U _ i \\geq \\sum _ { i = 1 } ^ m V _ i . \\end{align*}"}
+{"id": "685.png", "formula": "\\begin{align*} N _ n ( v ) : = \\{ v ' \\in V : d ( v , v ' ) \\leq n \\} \\end{align*}"}
+{"id": "4577.png", "formula": "\\begin{align*} p _ { K } ^ { \\Omega } ( C f ) = \\sup _ { x \\in K } | f | _ \\Omega ( x ) | = \\sup _ { x \\in K } | f ( x ) | = p _ { K } ^ { \\R ^ { d } } ( f ) ( f \\in C _ b ( \\R ^ { d } ) ) , \\end{align*}"}
+{"id": "5842.png", "formula": "\\begin{align*} \\rho { u _ x } = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { c _ { i x } } { f _ i } + \\frac { { \\Delta t } } { 2 } { F _ x } } , \\end{align*}"}
+{"id": "8289.png", "formula": "\\begin{align*} \\lambda : = \\frac { \\beta } { \\log C _ 0 + \\beta / ( 2 \\alpha ) + 1 } , \\end{align*}"}
+{"id": "1980.png", "formula": "\\begin{align*} \\sigma ( x , t ) & = f ( x - t ) + f ( - x - t ) , \\\\ \\mu ( x , t ) & = f ( x - t ) - f ( - x - t ) , \\end{align*}"}
+{"id": "9241.png", "formula": "\\begin{align*} S _ m ( \\lambda ) = \\sum \\limits _ { 1 \\leq j _ 1 < \\cdots < j _ m \\leq n } \\lambda _ { j _ 1 } \\dots \\lambda _ { j _ m } , \\end{align*}"}
+{"id": "1042.png", "formula": "\\begin{align*} \\rho _ { r _ { ( \\alpha , k ) } } ( v ) = \\begin{cases} v , & v ^ { - 1 } \\alpha \\in \\Phi ^ + , \\\\ s _ \\alpha v , & v ^ { - 1 } \\alpha \\in \\Phi ^ - . \\end{cases} \\end{align*}"}
+{"id": "644.png", "formula": "\\begin{align*} f ( x ) & \\in h ( x ) \\ast ( g ( x ) ) ^ { - 1 } = h ( x ) \\ast r ( g ) ( x ) \\mbox { i m p l i e s } f \\in h \\ast r ( g ) \\\\ g ( x ) & \\in ( f ( x ) ) ^ { - 1 } \\ast h ( x ) = r ( f ) ( x ) \\ast h ( x ) \\mbox { i m p l i e s } g \\in r ( f ) \\ast h \\end{align*}"}
+{"id": "8903.png", "formula": "\\begin{align*} \\lambda _ { 1 2 3 ' } & = \\log m , m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "4557.png", "formula": "\\begin{align*} \\lambda _ 1 = q \\sqrt { q } \\sigma _ 1 ( \\textbf { z } ) , \\lambda _ 2 = q ^ 2 \\sigma _ 2 ( \\textbf { z } ) , \\lambda _ 3 = q \\sqrt { q } \\sigma _ 3 ( \\textbf { z } ) . \\end{align*}"}
+{"id": "517.png", "formula": "\\begin{align*} \\int ^ 1 _ 0 | ( 1 - t ) ^ { \\beta - 1 } t ^ \\delta | d t = \\int ^ 1 _ 0 ( 1 - t ) ^ { R e \\beta - 1 } t ^ \\delta d t < \\infty . \\end{align*}"}
+{"id": "894.png", "formula": "\\begin{align*} \\Gamma ( X ) = \\sum \\limits _ { 1 \\leq d \\leq \\sqrt { X ^ 2 + X + 1 } } \\mu ( d ) \\sum \\limits _ { 1 \\leq n \\leq X \\atop { n ^ 2 + n + 1 \\equiv 0 \\ , ( d ^ 2 ) } } 1 = \\Gamma _ 1 ( X ) + \\Gamma _ 2 ( X ) \\ , , \\end{align*}"}
+{"id": "3689.png", "formula": "\\begin{align*} \\left \\langle X , \\Gamma _ { ( g , u ) } ( X ) \\right \\rangle = - \\tfrac { 1 } { 2 } \\Delta | X | ^ 2 - \\tfrac { 1 } { 2 } u ^ { - 1 } { \\nabla u } \\cdot \\nabla | X | _ g ^ 2 + | \\nabla X | ^ 2 + u ^ { - 2 } ( X ( u ) ) ^ 2 . \\end{align*}"}
+{"id": "2536.png", "formula": "\\begin{align*} \\begin{aligned} & \\| \\mathbf { w } _ { x s } - \\nu \\mu \\mathbf { e } \\| ^ 2 = \\| \\mathbf { w } _ { x s } - \\mu \\mathbf { e } \\| ^ 2 + 2 \\| \\mu \\mathbf { e } - \\nu \\mu \\mathbf { e } \\| ^ 2 + 2 ( 1 - \\nu ) \\mu ( \\mathbf { w } _ { x s } - \\mu \\mathbf { e } ) ^ T \\mathbf { e } \\\\ & \\le \\left ( \\gamma ^ 2 / 2 + ( 1 - \\nu ) ^ 2 k \\right ) \\mu ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "448.png", "formula": "\\begin{align*} \\ \\alpha \\ \\lim _ { x \\to \\infty } f ( x ) / \\alpha ( x ) = 1 . \\end{align*}"}
+{"id": "3799.png", "formula": "\\begin{align*} D ^ q _ * ( f ) ( z ) = \\sum _ { n = 1 } ^ { \\infty } a _ n \\frac { \\Gamma ( q n + 1 ) } { \\Gamma ( q ( n - 1 ) + 1 ) } z ^ { q ( n - 1 ) } . \\end{align*}"}
+{"id": "5639.png", "formula": "\\begin{align*} O _ { n , n } ( R ) \\subseteq O _ { n + 1 , n + 1 } ( R ) : A \\mapsto \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & A \\end{pmatrix} . \\end{align*}"}
+{"id": "6294.png", "formula": "\\begin{align*} f _ k ( x ) = f ( x ) + \\frac { 1 } { 2 \\rho _ k } \\| x - x ^ k \\| ^ 2 , F _ k ( x ) = f _ k ( x ) + P ( x ) . \\end{align*}"}
+{"id": "3596.png", "formula": "\\begin{align*} I ( \\tilde { A } ^ { 1 , \\dots , \\ell } ) = \\langle q _ { ( u , s ) } - q _ { ( u , r ) } \\mid u \\in [ \\beta ] , s , r \\in \\{ 0 , \\dots , x _ u - 1 \\} \\rangle + \\tilde { J } . \\end{align*}"}
+{"id": "136.png", "formula": "\\begin{align*} P _ N v \\cdot \\partial _ x P _ N ( P _ { \\gtrsim N } v \\cdot P _ { \\gtrsim N } u _ 1 ) = \\sum _ { N _ 2 \\sim N _ 1 \\gtrsim N } P _ N v \\cdot \\partial _ x P _ N ( P _ { N _ 1 } v \\cdot P _ { N _ 2 } u _ 1 ) . \\end{align*}"}
+{"id": "2022.png", "formula": "\\begin{align*} \\frac { d } { d \\varepsilon } J ( u + \\varepsilon v ) \\Big | _ { \\varepsilon = 0 } \\geq 0 . \\end{align*}"}
+{"id": "5589.png", "formula": "\\begin{align*} W ( [ 1 ^ l 0 ] | [ 0 ^ k 1 ] ) = c - d + \\sum _ { n = 2 } ^ l ( c - c _ n ) + b - b _ l \\ . \\end{align*}"}
+{"id": "6540.png", "formula": "\\begin{align*} x = q _ i ' - p _ i \\end{align*}"}
+{"id": "1699.png", "formula": "\\begin{align*} W ( t , \\xi ) & = \\sum _ { j = 1 } ^ N \\mu _ j e _ j ( \\xi ) \\beta _ j ( t ) , t \\ge 0 , \\ \\xi \\in \\mathbb { R } ^ d , \\\\ \\mu ( \\xi ) & = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ N | \\mu _ j | ^ 2 e _ j ^ 2 ( \\xi ) , \\xi \\in \\mathbb { R } ^ d . \\end{align*}"}
+{"id": "726.png", "formula": "\\begin{align*} p ^ { \\kappa , \\varepsilon } ( x , t ) = \\mathrm { e x p } \\Big ( - \\frac { 1 } { \\varepsilon } y ( t ) h ( x ) \\Big ) q ^ { \\kappa , \\varepsilon } ( x , t ) , \\end{align*}"}
+{"id": "5250.png", "formula": "\\begin{align*} \\tau \\lim _ { n \\to \\infty } T ( t ) ( x _ n - x ) = 0 \\end{align*}"}
+{"id": "1937.png", "formula": "\\begin{align*} f ( m ; z ) = \\sum _ { n = 0 } ^ \\infty b _ 5 ( n ) q ^ { \\frac { 2 4 n + m ( 5 a + b ) + 4 } { 2 4 } } \\cdot \\prod _ { n = 1 } ^ \\infty ( 1 - q ^ { 5 m n } ) ^ a ( 1 - q ^ { m n } ) ^ b . \\end{align*}"}
+{"id": "4018.png", "formula": "\\begin{align*} \\varphi ^ L _ { \\sigma , s ' } ( D ) = \\{ a , b \\} \\varphi ^ L _ { \\sigma , s ' } ( E ) = \\varphi ^ L _ { \\sigma , s ' } ( C _ i ) \\setminus \\{ a \\} = C \\setminus \\{ a \\} . \\end{align*}"}
+{"id": "8150.png", "formula": "\\begin{align*} & \\frac { \\varpi _ n } { \\varpi _ { n + 1 } } x ( \\varpi _ { n + 1 } ) = \\frac { \\varpi _ n } { \\varpi _ { n + 1 } } \\varphi _ M ^ { - n - 1 } ( \\frac \\varpi \\pi x ' ) = \\varphi _ M ^ { - n - 1 } ( \\frac { \\varpi ^ p } { \\varpi } \\frac \\varpi \\pi x ' ) = \\varphi _ M ^ { - n } ( \\varphi _ M ^ { - 1 } ( \\frac { \\varpi ^ p } \\pi x ' ) ) = \\\\ & \\qquad = \\varphi _ M ^ { - n } ( \\frac \\varpi \\pi x ' ) = x ( \\varpi _ n ) . \\end{align*}"}
+{"id": "1642.png", "formula": "\\begin{align*} \\int ^ { 1 } _ 0 G \\left ( \\frac { 1 } { 8 } , \\tau \\right ) d \\tau & = 2 \\int ^ { 1 / 2 } _ 0 G \\left ( \\frac { 1 } { 8 } , \\tau \\right ) d \\tau \\\\ & = 2 \\int ^ { 1 / 8 } _ 0 \\left ( - \\frac { 7 } { 4 8 } \\tau ^ 3 + \\frac { 3 5 } { 1 0 2 4 } \\tau \\right ) d \\tau \\\\ & \\phantom { = } + 2 \\int ^ { 1 / 2 } _ { 1 / 8 } \\left ( \\frac { 1 } { 4 8 } \\tau ^ 3 - \\frac { 1 } { 1 6 } \\tau ^ 2 + \\frac { 4 3 } { 1 0 2 4 } \\tau - \\frac { 1 } { 3 0 7 2 } \\right ) d \\tau \\\\ & = \\frac { 2 7 7 } { 4 9 1 5 2 } . \\end{align*}"}
+{"id": "7892.png", "formula": "\\begin{align*} \\lambda _ 0 = - \\frac { m } { h ( q , r _ 0 ) } = \\frac { \\hat W _ { r _ 0 } ( 2 q ) + \\hat W _ { r _ 0 } ( 0 ) - 2 \\hat W _ { r _ 0 } ( q ) } { 4 \\hat W _ { r _ 0 } ( q ) } . \\end{align*}"}
+{"id": "8663.png", "formula": "\\begin{align*} { \\bf { \\bar q } } \\left [ i \\right ] = \\sum \\limits _ { l = 1 } ^ { L ' } { { { \\bf { \\bar H } } _ { l } ^ \\bot { { { \\bf { \\bar X } } } _ { l } } } { \\bf { d } } \\left [ { i - { \\kappa _ l } } \\right ] } , \\end{align*}"}
+{"id": "262.png", "formula": "\\begin{align*} X _ 1 = \\frac { \\partial } { \\partial x } - \\frac { y } { 2 } \\frac { \\partial } { \\partial z } , X _ 2 = \\frac { \\partial } { \\partial y } + \\frac { x } { 2 } \\frac { \\partial } { \\partial z } , X _ 3 = \\frac { \\partial } { \\partial z } . \\end{align*}"}
+{"id": "3293.png", "formula": "\\begin{align*} \\sup _ { r _ { 1 , 1 } , . . . , r _ { 1 , d _ 1 } , r _ { 2 , 1 } , . . . , r _ { 2 , d _ 2 } \\in [ 0 , 2 ] } \\left | \\mathbb E \\left [ \\prod _ { k = 1 } ^ { d _ 1 } B _ 1 ( r _ { i , 1 } ) \\prod _ { k = 1 } ^ { d _ 2 } B _ 2 ( r _ { 2 , k } ) \\right ] \\right | < \\infty , \\end{align*}"}
+{"id": "1254.png", "formula": "\\begin{align*} \\sum _ { \\Delta \\Subset \\Z ^ d } \\sum _ { \\xi _ { \\Delta } } \\nabla _ { \\Lambda } \\left ( c _ { \\Delta } ( \\cdot , \\xi _ { \\Delta } ) - \\hat { c } ( \\cdot , \\xi _ { \\Delta } ) \\right ) ( \\eta ) = 0 , \\end{align*}"}
+{"id": "6520.png", "formula": "\\begin{align*} d ( \\lambda _ v ) = e = \\iota _ { \\star } = \\lambda _ { d ^ 0 ( v ) } \\end{align*}"}
+{"id": "7401.png", "formula": "\\begin{align*} I ^ \\alpha : = X ^ \\alpha + \\sum _ { \\beta , \\gamma \\in S _ + } { f ^ { \\alpha \\beta } } _ \\gamma : \\varphi _ \\gamma \\varphi ^ \\beta : + \\frac { 1 } { 2 } \\sum _ { \\beta \\in S _ { \\frac { 1 } { 2 } } } { f ^ { \\beta \\alpha } } _ \\gamma \\Phi ^ \\beta \\Phi _ \\gamma \\end{align*}"}
+{"id": "7087.png", "formula": "\\begin{gather*} \\mathcal { A } _ j ( x , \\xi ) = D _ { \\xi } F _ j ( x , \\xi ) , \\end{gather*}"}
+{"id": "950.png", "formula": "\\begin{align*} P \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} \\beta _ 1 ( \\xi _ 0 ) \\\\ \\beta _ 2 ( \\xi _ 0 ) \\end{pmatrix} & { } = P \\lim _ { t \\to \\infty } t ^ { - \\mu | W _ 1 ( \\xi _ 0 ) | ^ 2 } P ^ { - 1 } \\begin{pmatrix} w ( \\xi _ 0 ) \\\\ \\overline { w ( \\xi _ 0 ) } \\end{pmatrix} \\\\ & { } = \\lim _ { t \\to \\infty } t ^ { - \\mu | W _ 1 ( \\xi _ 0 ) | ^ 2 } \\begin{pmatrix} w ( \\xi _ 0 ) \\\\ \\overline { w ( \\xi _ 0 ) } \\end{pmatrix} \\in Z , \\end{align*}"}
+{"id": "8540.png", "formula": "\\begin{align*} & r _ 1 ( t ; z ) = - \\frac { b ( k ) } { 2 i k a ( k ) } e ^ { - 2 i \\eta ^ 2 t } = r _ { 1 } ( z ) e ^ { - 2 i \\eta ^ 2 t } , \\\\ & r _ 2 ( t ; z ) = \\frac { 2 i k b ( k ) } { a ( k ) } e ^ { - 2 i \\eta ^ 2 t } = r _ { 2 } ( z ) e ^ { - 2 i \\eta ^ 2 t } . \\end{align*}"}
+{"id": "3730.png", "formula": "\\begin{align*} \\nabla \\hat X = \\omega \\quad \\mbox { i n } \\Omega . \\end{align*}"}
+{"id": "4348.png", "formula": "\\begin{align*} & \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in U \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h \\\\ = & \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in U \\} } | \\tilde F | ^ 2 _ h + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in U \\backslash N \\} } | F _ { t _ 0 , \\tilde { c } } - \\tilde F | ^ 2 _ h \\\\ & + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in U \\cap N \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h \\end{align*}"}
+{"id": "924.png", "formula": "\\begin{align*} F _ { 2 } ( t , \\xi ) = \\tilde { W } _ 2 ( t , \\xi ) e ^ { - 3 i \\lambda _ { 1 } | W _ 1 ( \\xi ) | ^ { 2 } \\log t } \\end{align*}"}
+{"id": "8081.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ M \\sum \\limits _ { \\substack { j = 1 \\\\ j \\neq l } } ^ M \\mathbb { E } \\left [ y _ l ^ * y _ j | \\hat { \\mathbf { H } } \\right ] = \\sum _ { l = 1 } ^ M \\sum \\limits _ { \\substack { j = 1 \\\\ j \\neq l } } ^ M \\left ( a _ c ^ 2 \\hat { \\phi } ^ { \\left ( l , c \\right ) ^ * } \\hat { \\phi } ^ { \\left ( j , c \\right ) } + \\sum _ { q = 1 } ^ M a _ q ^ 2 \\hat { \\phi } ^ { \\left ( l , q \\right ) ^ * } \\hat { \\phi } ^ { \\left ( j , q \\right ) } \\right ) \\end{align*}"}
+{"id": "1863.png", "formula": "\\begin{align*} \\begin{aligned} \\sqrt { a \\cdot b } & \\le \\frac { a + b } { 2 } \\ , , \\operatorname { f o r } a , b > 0 \\\\ \\operatorname { w i t h } \\ , ` ` & = \" \\ , \\operatorname { h o l d s } \\ , \\operatorname { i f } \\ , \\operatorname { a n d } \\ , \\operatorname { o n l y } \\ , \\operatorname { i f } \\ , a = b \\ , . \\end{aligned} \\end{align*}"}
+{"id": "4640.png", "formula": "\\begin{align*} \\tau = \\tau ' _ 1 v _ 1 \\cdots \\tau ' _ z v _ z \\end{align*}"}
+{"id": "6478.png", "formula": "\\begin{align*} U ( 1 , 1 , \\Delta ) = 1 . \\end{align*}"}
+{"id": "8098.png", "formula": "\\begin{align*} \\int _ { 1 } ^ \\infty \\frac { d s } { \\Big ( \\displaystyle \\int _ 0 ^ s F ( t ) d t \\Big ) ^ { p / ( 2 p - q + 1 ) } } < \\infty \\quad \\mbox { w h e r e } \\ ; F ( t ) = \\int _ 0 ^ t f ( k ) d k . \\end{align*}"}
+{"id": "2797.png", "formula": "\\begin{align*} \\sum _ { x , y \\in A _ { N } } \\mathbb { E } & [ ( 1 _ { \\{ h ^ { D _ { N } } ( x ) \\geq a _ { N } \\} } - e ^ { \\pi \\lambda b } 1 _ { \\{ h ^ { D _ { N } } ( x ) \\geq a _ { N } + b \\} } ) \\\\ & ( 1 _ { \\{ h ^ { D _ { N } } ( y ) \\geq a _ { N } \\} } - e ^ { \\pi \\lambda b } 1 _ { \\{ h ^ { D _ { N } } ( y ) \\geq a _ { N } + b \\} } ) 1 _ { T _ { N , M } ( x ) } 1 _ { T _ { N , M } ( y ) } ] . \\end{align*}"}
+{"id": "4005.png", "formula": "\\begin{align*} \\gamma ( x _ n ^ D ; w ) \\ge \\tau ( x _ n ^ D ) : = 8 \\frac { | \\Gamma _ D | \\mathrm { N } _ I \\lambda _ d ^ 2 } { | \\Omega | \\mathrm { N } _ D \\lambda _ 1 } \\frac { | \\nabla w ( x _ n ^ D ) | ^ 2 } { | \\nabla w ( y _ n ) | ^ 2 } , \\end{align*}"}
+{"id": "5848.png", "formula": "\\begin{align*} { f _ { 1 3 } } = { f _ { 1 4 } } - f _ { 1 4 } ^ { \\left ( { e q } \\right ) } + f _ { 1 3 } ^ { \\left ( { e q } \\right ) } + \\delta x - \\delta z , \\end{align*}"}
+{"id": "6405.png", "formula": "\\begin{align*} \\sigma = ( y _ 1 + \\i \\theta _ 1 ) \\wedge \\dots \\wedge ( y _ n + \\i \\theta _ n ) . \\end{align*}"}
+{"id": "3618.png", "formula": "\\begin{align*} h _ { k l i j } = & \\ h _ { i j k l } - h _ { m l } ( h _ { i m } h _ { k j } - h _ { i j } h _ { m k } ) - h _ { m j } ( h _ { m i } h _ { k l } - h _ { i l } h _ { m k } ) \\\\ & - h _ { m l } ( \\delta _ { i j } \\delta _ { k m } - \\delta _ { i k } \\delta _ { j m } ) - h _ { m j } ( \\delta _ { i l } \\delta _ { k m } - \\delta _ { i k } \\delta _ { l m } ) . \\end{align*}"}
+{"id": "6009.png", "formula": "\\begin{align*} E _ { \\omega } : = \\inf _ { u \\in \\mathcal { N } _ { \\omega } } J _ { \\omega } ( u ) , \\end{align*}"}
+{"id": "4645.png", "formula": "\\begin{align*} \\phi ( m , \\ell ) = \\sum _ { \\widetilde K \\ \\ \\pi ^ { - 1 } ( K ) } \\phi ( m , \\ell _ { \\widetilde K } ) \\equiv \\sum _ { \\widetilde K \\ \\ \\pi ^ { - 1 } ( K ) } ( e _ { \\widetilde K } - 1 ) d _ { \\widetilde K } \\mod 2 . \\end{align*}"}
+{"id": "6342.png", "formula": "\\begin{align*} F ( x ) = \\frac { 1 } { \\operatorname { B } ( a , b ) } \\int ^ { x } _ { 0 } \\frac { 2 \\theta } { y ^ { 3 } } \\exp \\left ( - \\frac { a \\theta } { y ^ { 2 } } \\right ) \\left \\{ 1 - \\exp \\left ( - \\frac { \\theta } { y ^ { 2 } } \\right ) \\right \\} ^ { b - 1 } \\mathrm { d } y . \\end{align*}"}
+{"id": "820.png", "formula": "\\begin{align*} c _ 0 ( x ) = \\begin{cases} c _ { 0 , O } , & x \\in [ x _ 0 - 1 , x _ 1 ] , \\\\ c _ { 0 , I } , & x \\in [ x _ 2 , x _ 3 ] , \\\\ c _ { 0 , O } , & x \\in [ x _ 4 , x _ 5 + 1 ] \\end{cases} \\end{align*}"}
+{"id": "3693.png", "formula": "\\begin{align*} v = 0 , \\nu ' ( h ) ( \\bar u ) + \\nu ( v ) = 0 , A ' ( h ) = 0 \\mbox { o n } \\hat \\Sigma . \\end{align*}"}
+{"id": "7017.png", "formula": "\\begin{align*} T _ 0 \\ , : = \\ , 0 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ C _ 1 \\ , : = \\ , T _ 1 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ C _ 2 \\ , : = \\ , 0 . \\end{align*}"}
+{"id": "1644.png", "formula": "\\begin{align*} k _ i ( x ) = \\prod _ { \\sigma \\in \\Sigma _ \\infty } \\sigma ( x ) ^ { e _ i ( \\sigma ) } \\end{align*}"}
+{"id": "1979.png", "formula": "\\begin{align*} T ( x ) = x - g ( x ) = x - G ^ { - 1 } ( x ) . \\end{align*}"}
+{"id": "7897.png", "formula": "\\begin{align*} X ( q , r ) = - \\frac 1 4 \\hat W _ r ( q ) + \\frac { c _ 3 ( q , 2 q , q , p ^ 0 ( 0 ) ) } { 4 c _ 1 ( q , 2 q , p ^ 0 ( 0 ) ) } ( - \\hat W _ r ( 0 ) + \\hat W _ r ( 2 q ) ) . \\end{align*}"}
+{"id": "1029.png", "formula": "\\begin{align*} \\prescript L { } { } \\ell { } ^ R ( x , \\alpha ) = & \\langle \\mu , \\alpha \\rangle + \\chi _ R ( \\alpha ) - \\chi _ L ( w \\alpha ) \\\\ = & \\langle \\mu , \\alpha \\rangle + \\Phi ^ + ( \\alpha ) - \\Phi ^ + _ R ( \\alpha ) - \\Phi ^ + ( w \\alpha ) + \\Phi _ L ^ + ( w \\alpha ) \\\\ = & \\ell ( x , \\alpha ) - \\Phi ^ + _ R ( \\alpha ) + \\Phi _ L ^ + ( w \\alpha ) \\end{align*}"}
+{"id": "5831.png", "formula": "\\begin{align*} { \\rm O } ( \\epsilon ^ 1 ) : \\rho c _ v { \\bf { I } } \\partial _ { t _ 1 } { { { \\bf { m } } } } ^ { ( 0 ) } + { \\bf { d } } _ { 1 } { { { \\bf { m } } } } ^ { ( 0 ) } = - \\frac { 1 } { \\Delta t } { \\Lambda } { { { \\bf { m } } } } ^ { ( 1 ) } + { { { \\bf { M } } } } { { { \\bf { \\bar F } } } } ^ { ( 1 ) } , \\end{align*}"}
+{"id": "4567.png", "formula": "\\begin{align*} & \\frac { \\lambda _ 1 } { q } a _ { \\ell + 1 , m + 1 , 0 } = ( q ^ 3 + q ^ 2 ) a _ { \\ell + 1 , m + 1 , 1 } + q a _ { \\ell + 1 , m + 2 , 0 } + a _ { \\ell + 2 , m + 1 , 0 } \\\\ & \\frac { \\lambda _ 3 } { q } a _ { \\ell , m , 0 } = q a _ { \\ell - 1 , m , } + a _ { \\ell , m - 1 , 0 } + ( q ^ 3 + q ^ 2 ) a _ { \\ell + 1 , m + 1 , 1 } . \\end{align*}"}
+{"id": "7316.png", "formula": "\\begin{align*} \\delta _ { j k } X _ { i j \\ell } + \\delta _ { i \\ell } X _ { i j k } = \\delta _ { i j } X _ { i k \\ell } + \\delta _ { k \\ell } X _ { i j k } . \\end{align*}"}
+{"id": "4696.png", "formula": "\\begin{align*} { \\rm V o l } ( \\tilde A \\setminus { \\tilde \\Sigma } ^ { r } ) & \\geq { \\rm V o l } ( A \\cap ( \\tilde A \\setminus { \\tilde \\Sigma } ^ { r } ) \\ , ) = { \\rm V o l } ( A ) - { \\rm V o l } ( A \\cap ( \\tilde B \\cup { \\tilde \\Sigma } ^ { r } ) ) \\\\ & \\geq { \\rm V o l } ( A ) - { \\rm V o l } ( { \\tilde \\Sigma } ^ { r } ) - \\sum _ { i = s + 1 } ^ m { \\rm V o l } ( A \\cap B _ { p _ i } ( r ) ) . \\\\ \\end{align*}"}
+{"id": "9313.png", "formula": "\\begin{align*} b _ { 1 } u _ { 1 } ^ { 2 } l ^ { 2 ( k _ { 1 } - k _ { 3 } ) } - b _ { 1 } b _ { 2 } u _ { 3 } ^ { 2 } = - p l ^ { - 2 k _ { 3 } } , \\end{align*}"}
+{"id": "5575.png", "formula": "\\begin{align*} A ( y _ { l + 1 } . . . y _ 1 x ) - A ( y _ { l + 1 } . . . y _ 1 x ' ) = A ( 1 0 ^ { l } 1 ^ k 0 . . . ) - A ( 1 0 ^ \\infty ) = d _ l - d \\ . \\end{align*}"}
+{"id": "2547.png", "formula": "\\begin{align*} \\mathbf { w } _ { x _ w , s _ w } = \\mathbf { T } _ { x _ w } \\mathbf { s } _ w . \\end{align*}"}
+{"id": "4999.png", "formula": "\\begin{align*} \\vec { u } _ { 0 , n } = ( u _ { 0 , n } , u _ { 1 , n } ) & = ( u _ 0 , u _ 1 ) + ( \\phi _ { 0 , n } , \\phi _ { 1 , n } ) \\\\ & = \\vec u _ 0 + \\vec \\phi _ { n } . \\end{align*}"}
+{"id": "6628.png", "formula": "\\begin{align*} \\Lambda ^ { * } ( x ) = \\left \\{ \\begin{array} { l l } 1 - \\sqrt { 1 - x ^ { 2 } } , & ~ | x | \\leq 1 ; \\\\ + \\infty , & | x | > 1 . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "4793.png", "formula": "\\begin{align*} ( p _ n f , v _ n ) = ( f , v _ n ) v _ n \\in X _ n . \\end{align*}"}
+{"id": "3928.png", "formula": "\\begin{align*} { \\bf E } \\left ( { \\bf A } , { \\mathbb V } \\right ) = \\left \\{ { \\bf A } _ { \\theta } : \\theta \\left ( x , y \\right ) = \\sum _ { i = 1 } ^ { s } \\theta _ { i } \\left ( x , y \\right ) e _ { i } \\ \\ \\ \\ \\left \\langle \\left [ \\theta _ { 1 } \\right ] , \\left [ \\theta _ { 2 } \\right ] , \\dots , \\left [ \\theta _ { s } \\right ] \\right \\rangle \\in { \\bf T } _ { s } ( { \\bf A } ) \\right \\} . \\end{align*}"}
+{"id": "8174.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t _ { 0 } } \\int _ { R _ { 0 } / 2 } ^ { R _ { 0 } } \\nu \\xi ( \\nu , s ) \\ d \\nu d s \\leq \\int _ { 0 } ^ { t _ { 0 } } \\int _ { 0 } ^ { R _ { 0 } } \\nu \\xi ( \\nu , s ) \\ d \\nu d s = 0 . \\end{align*}"}
+{"id": "8711.png", "formula": "\\begin{align*} \\varphi _ 1 ( n ) = \\frac { \\tau ( p _ 1 \\pi ( n ) p _ 1 ) } { \\tau ( p _ 1 ) } = \\frac { ( \\varphi _ { | N } ) _ { p _ 1 } ( n ) } { \\tau ( p _ 1 ) } = \\psi _ 1 ( n ) \\end{align*}"}
+{"id": "6255.png", "formula": "\\begin{align*} D _ { \\eta } = A ^ { - 1 } A _ { \\eta } : \\Gamma _ { \\mathbb { C } } ^ \\perp \\rightarrow \\Gamma _ { \\mathbb { C } } ^ \\perp , \\end{align*}"}
+{"id": "1206.png", "formula": "\\begin{align*} A _ { 3 } = D _ { 3 } + M _ { 3 } \\end{align*}"}
+{"id": "4936.png", "formula": "\\begin{align*} B _ { \\sigma , \\lambda } = \\dfrac { ( 2 \\sigma - 1 ) \\left ( 1 - \\exp \\left ( - \\frac { 2 \\lambda ( 1 - \\sigma ) } { 2 \\sigma - 1 } \\right ) \\right ) } { 2 \\lambda ( 1 - \\sigma ) ^ 2 } . \\end{align*}"}
+{"id": "7819.png", "formula": "\\begin{align*} F ^ q ( B _ \\phi \\eta , p ) = B _ \\phi F ^ q ( \\eta , p ) \\end{align*}"}
+{"id": "6183.png", "formula": "\\begin{align*} L ' = L + 1 , Q ' = Q , B ' _ 1 = B _ 1 , B ' _ 2 = B _ 2 + 2 \\frac { ( L + 1 ) B _ 1 } { Q } , \\end{align*}"}
+{"id": "4400.png", "formula": "\\begin{align*} | Z _ 1 | & = \\left | \\frac { x ^ { 1 - s } - x ^ { 2 ( 1 - s ) } } { ( 1 - s ) ^ 2 \\log x } \\right | \\leq \\frac { x ^ { 2 - 2 \\sigma } + x ^ { 1 - \\sigma } } { t ^ 2 \\log x } , \\end{align*}"}
+{"id": "8982.png", "formula": "\\begin{align*} \\dfrac { \\hat { k } _ 2 { ( \\Delta ) } } { \\hat { k } _ 2 ( \\Delta \\setminus \\sigma ) } = \\dfrac { D _ { 1 , 2 } D _ { 2 , 2 } D _ { 3 , 2 } } { D _ { 1 , 1 } D _ { 2 , 1 } D _ { 3 , 1 } } = \\dfrac { ( x _ { 1 , 1 } + x _ { 1 , 2 } ) ( x _ { 2 , 1 } + x _ { 2 , 2 } ) ( x _ { 3 , 1 } + x _ { 3 , 2 } ) } { x _ { 1 , 1 } x _ { 2 , 1 } x _ { 3 , 1 } } . \\end{align*}"}
+{"id": "6144.png", "formula": "\\begin{align*} X _ t \\leq \\int _ 0 ^ t \\alpha _ s \\sup _ { r \\in [ 0 , s ] } X _ r \\ , d s + M _ s + H _ s , \\end{align*}"}
+{"id": "1055.png", "formula": "\\begin{align*} x ^ { \\ast , \\sigma , N + n } = x ^ { \\ast , \\sigma , N } \\cdot \\prescript { \\sigma ^ N } { } x _ \\infty \\cdots \\prescript { \\sigma ^ { N + n - 1 } } { } x _ \\infty \\end{align*}"}
+{"id": "5989.png", "formula": "\\begin{align*} \\binom { n } { k } \\sup _ { g \\in \\Pi _ 1 ^ { - 1 } ( U ) } ( \\phi ( g ) ) \\geq \\# \\Pi _ 1 ^ { - 1 } ( U ) \\sup _ { g \\in \\Pi _ 1 ^ { - 1 } ( U ) } ( \\phi ( g ) ) \\geq \\sum _ { g \\in \\Pi _ 1 ^ { - 1 } ( U ) } \\phi ( g ) . \\end{align*}"}
+{"id": "8467.png", "formula": "\\begin{align*} Q _ { j , + } ( x ; k ) = \\mathcal { P } ^ { + } \\left ( Q _ { j , - } ( x ; k ) J + D _ j \\right ) ( z ) , z \\in \\mathbb { R } . \\end{align*}"}
+{"id": "2770.png", "formula": "\\begin{align*} N _ 2 ( s ) \\leq N _ 2 : = \\frac { 4 ( \\tau + C _ 2 ) } { \\eta ( t ) } + 1 . \\end{align*}"}
+{"id": "2058.png", "formula": "\\begin{align*} c ^ { k + 1 , j } _ { \\ell , p , q } = c ^ { k , j - 2 } _ { \\ell , p - 1 , q } + c ^ { k , j } _ { \\ell , p , q - 1 } + 2 c ^ { k , j - 1 } _ { \\ell - 1 , p , q } \\end{align*}"}
+{"id": "4250.png", "formula": "\\begin{align*} \\| \\varphi _ { 2 k j _ 0 } - \\rho \\| _ \\infty & = \\max \\Big { \\{ } \\sup \\{ | \\varphi _ { 2 k j } ( z ) - \\rho ( z ) | ; \\ z \\in \\mathbb { B } _ n , \\ | \\varphi _ { k j } ( z ) | > r \\} , \\\\ & \\sup \\{ | \\varphi _ { 2 k j } ( z ) - \\rho ( z ) | ; \\ z \\in \\mathbb { B } _ n , \\ | \\varphi _ { k j } ( z ) | \\leq r \\} \\Big { \\} } . \\end{align*}"}
+{"id": "8511.png", "formula": "\\begin{align*} \\| I _ 3 ' ( x ) \\| _ { L ^ 2 _ x } = 4 \\pi ^ { - 2 } \\| \\widehat { z r _ 2 ( z ) } ( - 2 x ) \\| _ { L ^ 2 _ x } \\leq c \\left \\| r _ 2 \\right \\| _ { H ^ { 1 } \\cap L ^ { 2 , 1 } } . \\end{align*}"}
+{"id": "2638.png", "formula": "\\begin{align*} y \\cdot [ x , z ] = \\varepsilon ( y , x ) [ x , y \\cdot z ] - \\varepsilon ( x + y , z ) [ z , y \\cdot x ] + [ y , x ] \\cdot z - \\varepsilon ( x , z ) [ y , z ] \\cdot x . \\end{align*}"}
+{"id": "9277.png", "formula": "\\begin{align*} ( \\Delta \\omega ^ * ( q , K , \\Omega ) ) ^ m \\wedge \\beta _ n ^ { n - m } = 0 \\Omega \\setminus K . \\end{align*}"}
+{"id": "8777.png", "formula": "\\begin{align*} \\frac { d } { d t } z _ t = B ( z _ t , z _ t ) \\end{align*}"}
+{"id": "2389.png", "formula": "\\begin{align*} u _ { n + k } + \\alpha _ { 1 , n } u _ { n + k - 1 } + \\alpha _ { 2 , n } u _ { n + k - 2 } + \\cdots + \\alpha _ { k , n } u _ { n } = 0 \\end{align*}"}
+{"id": "926.png", "formula": "\\begin{align*} ( \\mu + i \\eta ) W _ 2 ( \\xi ) - i \\lambda _ 6 \\tfrac { W _ 1 ( \\xi ) ^ 2 } { | W _ 1 ( \\xi ) | ^ 2 } \\overline { W _ 2 ( \\xi ) } = 0 \\end{align*}"}
+{"id": "3341.png", "formula": "\\begin{align*} K _ i = \\langle \\{ a _ i , b _ i \\} , \\{ b _ i , c _ i \\} \\rangle , \\ L _ i = \\langle \\{ a _ i \\} , \\{ c _ i \\} \\rangle . \\end{align*}"}
+{"id": "887.png", "formula": "\\begin{align*} k = \\frac { z ^ 2 + z + 1 } { n } q - a z - a \\ , . \\end{align*}"}
+{"id": "7947.png", "formula": "\\begin{align*} & ( z _ 1 \\bar { z _ 2 } ) = ( z _ 1 ) ( z _ 2 ) + ( z _ 1 ) ( z _ 2 ) \\geq ( z _ 1 ) + ( z _ 2 ) - 1 + ( z _ 1 ) ( z _ 2 ) . \\end{align*}"}
+{"id": "1818.png", "formula": "\\begin{align*} \\begin{aligned} S _ { e , \\mathcal { Y M } } ( X , Y ) & = \\exp ( \\frac { | | R ^ { \\nabla } | | ^ 2 } 2 ) \\big ( g ( X , Y ) - \\langle i _ X R ^ { \\nabla } , i _ Y R ^ { \\nabla } \\rangle \\big ) \\end{aligned} \\end{align*}"}
+{"id": "1906.png", "formula": "\\begin{align*} { \\mathcal { Q } } _ { i j } ( n ) : = \\textsf { P } \\left \\{ { W ( { n + k } ) = j } \\bigm | { W ( k ) = i } \\right \\} = { \\frac { j q ^ { j - i } } { i \\beta ^ n } } P _ { i j } ( n ) i , j \\in { \\mathcal { E } } , \\end{align*}"}
+{"id": "7640.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { \\xi } = & ~ [ ( A - B ^ 2 R ^ { - 1 } P _ t ) x _ t ^ { \\xi } - B ^ 2 R ^ { - 1 } \\varphi _ t ^ \\xi - B h ( \\mu _ t ) \\\\ & + f ( \\nu _ t ) + b ( \\mu _ t ) ] d t + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ 0 ^ { \\xi } = & ~ \\xi ; \\end{aligned} \\right . \\end{align*}"}
+{"id": "3290.png", "formula": "\\begin{align*} \\| ( \\mathcal S ( t ) - I ) B ^ { \\mathfrak H } ( \\omega ) \\| ^ 2 = & \\int _ 0 ^ { 2 - t } ( B _ { t + x } ^ { \\mathfrak H } ( \\omega ) - B _ x ^ { \\mathfrak H } ( \\omega ) ) ^ 2 d x + \\int _ { 2 - t } ^ 2 ( B _ x ^ { \\mathfrak H } ( \\omega ) ) ^ 2 d x \\leq \\bar C _ { \\omega } t ^ { 2 \\mathfrak H } . \\end{align*}"}
+{"id": "3231.png", "formula": "\\begin{align*} \\hat { \\Gamma } _ t ^ n : = \\Delta _ n ^ { - 1 } \\bigl ( S A M P V _ t ^ n ( 4 ) - S A M P V _ t ^ n ( 2 , 2 ) \\bigr ) . \\end{align*}"}
+{"id": "7503.png", "formula": "\\begin{align*} \\| \\varphi ^ { + } \\| _ { L ^ { m _ { k } } ( \\partial B _ 1 ) } \\leq C ( C m _ 0 ) ^ { \\frac { 1 } { m _ { 0 } } \\sum _ { j = 0 } ^ { k - 1 } \\left ( \\frac { 2 } { 2 ^ { \\star } } \\right ) ^ { j } } \\left ( \\frac { 2 ^ { \\star } } { 2 } \\right ) ^ { \\frac { 1 } { m _ { 0 } } \\sum _ { j = 0 } ^ { k - 1 } j \\left ( \\frac { 2 } { 2 ^ { \\star } } \\right ) ^ { j } } , \\end{align*}"}
+{"id": "6321.png", "formula": "\\begin{align*} z _ 0 ^ { p ^ r } x _ 0 + z _ 1 ^ { p ^ r } x _ 1 \\cdots + z _ n ^ { p ^ r } x _ n = 0 . \\end{align*}"}
+{"id": "7523.png", "formula": "\\begin{align*} \\{ v _ 1 , \\ldots , v _ k \\} \\in G \\bigcap \\limits _ { i = 1 } ^ k S _ { v _ i } = \\emptyset . \\end{align*}"}
+{"id": "4136.png", "formula": "\\begin{align*} K ^ L ( x ' , t ) : = P _ t ^ L ( x ' ) = t ^ { 1 - n } P ^ L ( x ' / t ) , \\forall ( x ' , t ) \\in \\R ^ n _ + , \\end{align*}"}
+{"id": "2608.png", "formula": "\\begin{align*} Q ^ c ( g , R ) ( x , J x , J x , x ; u , J u ) = 2 \\ , Q ^ c ( g , R ) ( u , J u , x , J x ; x , J x ) \\neq 0 . \\end{align*}"}
+{"id": "6824.png", "formula": "\\begin{align*} I _ { l } ^ { \\left ( 2 \\right ) } ( s , \\rho ) = \\frac { 1 } { \\left ( 2 i \\right ) ^ { 2 } } \\int _ { \\mathbb { R } } \\frac { e ^ { i S ( \\tau , x - s ) } \\tau ^ { l - 3 } } { \\left ( \\tau - \\rho \\right ) ^ { l + 1 } } d \\tau . \\end{align*}"}
+{"id": "9267.png", "formula": "\\begin{align*} \\alpha : = \\Delta v _ 1 \\wedge \\dots \\wedge \\Delta v _ { m - 1 } \\wedge \\beta _ n ^ { n - m } . \\end{align*}"}
+{"id": "3685.png", "formula": "\\begin{align*} L _ X g ( \\nu , \\cdot ) = h ( \\nu , \\cdot ) \\mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \\Sigma . \\end{align*}"}
+{"id": "4291.png", "formula": "\\begin{align*} G \\frac { d } { d t } w ( t ) = L w ( t ) + f ( t ) \\end{align*}"}
+{"id": "6773.png", "formula": "\\begin{align*} a _ { 0 } ( x ) = e ( \\frac { i } { 2 } , x ) e ^ { \\frac { x } { 2 } } - 1 \\end{align*}"}
+{"id": "6147.png", "formula": "\\begin{align*} Y ^ n _ 0 = \\xi _ 0 , \\quad Y _ { k + 1 } ^ n = Y _ k ^ n + \\mu ( \\tfrac { k T } { n } , \\mathcal { Y } ^ n ) \\tfrac { T } { n } + \\sigma ( \\tfrac { k T } { n } , \\mathcal { Y } ^ n ) ( W _ { \\frac { ( k + 1 ) T } { n } } - W _ { \\frac { k T } { n } } ) . \\end{align*}"}
+{"id": "4684.png", "formula": "\\begin{align*} \\Delta \\rho _ 1 ( x ) \\leq \\psi _ { K , - H } ( \\rho _ 1 ( x ) ) = ( n - 1 ) \\sqrt K \\cdot \\frac { ( n - 1 ) \\sqrt K \\sinh \\sqrt K \\rho _ 1 - H \\cosh \\sqrt K \\rho _ 1 } { ( n - 1 ) \\sqrt K \\cosh \\sqrt K \\rho _ 1 - H \\sinh \\sqrt K \\rho _ 1 } . \\end{align*}"}
+{"id": "6055.png", "formula": "\\begin{align*} p \\frac { d ^ 2 } { d s ^ 2 } \\left ( \\left ( \\kappa - \\mu \\right ) ^ { p - 1 } \\right ) + p \\kappa ^ 2 \\left ( \\kappa - \\mu \\right ) ^ { p - 1 } - \\kappa \\left ( \\left ( \\kappa - \\mu \\right ) ^ p + \\sigma \\right ) = 0 . \\end{align*}"}
+{"id": "4657.png", "formula": "\\begin{align*} \\max _ { - N \\leq j \\leq N } \\Vert M _ { H _ j } : S _ { 1 } ^ { 2 N + 1 } \\rightarrow S _ { 1 } ^ { 2 N + 1 } \\Vert \\leq \\Vert M _ { \\widetilde { H } \\vert _ { \\mathbb { Z } _ N \\times \\mathbb { Z } _ N } } ^ { ( N ) } : S _ { p _ 1 } ^ { 2 N + 1 } \\times S _ { p _ 2 } ^ { 2 N + 1 } \\rightarrow S _ { 1 } ^ { 2 N + 1 } \\Vert . \\end{align*}"}
+{"id": "266.png", "formula": "\\begin{align*} E _ \\lambda : = \\{ p \\in \\Omega : \\psi _ E ( p ) < \\lambda \\} \\end{align*}"}
+{"id": "1190.png", "formula": "\\begin{align*} j _ 1 = \\min \\{ k _ 1 , i _ A , \\max \\{ i _ + , i _ - \\} \\} \\ , . \\end{align*}"}
+{"id": "9030.png", "formula": "\\begin{align*} L ( E ) = \\max \\{ 0 , \\log | \\lambda | \\} , \\forall \\ E \\in \\Sigma _ { \\lambda \\exp , \\alpha } . \\end{align*}"}
+{"id": "3095.png", "formula": "\\begin{align*} \\varphi ( \\theta ^ { k } , \\tilde { \\theta } ^ { k } , \\zeta ^ k ) = \\langle \\lambda ^ { k } - \\tilde { \\lambda } ^ { k } , \\tilde { y } ^ { k } - y ^ { k } \\rangle _ { \\mathcal { L } ^ 2 } + \\langle \\theta ^ { k } - \\tilde \\theta ^ { k } , G d ( \\theta ^ k , \\tilde { \\theta } ^ { k } , \\zeta ^ k ) \\rangle _ { \\mathcal { L } ^ 2 } , \\end{align*}"}
+{"id": "2967.png", "formula": "\\begin{align*} F ^ + ( t , x ) : = \\left ( \\begin{array} { c } R ^ + ( t , x ) \\cos \\pi t , \\\\ R ^ + ( t , x ) \\sin \\pi t \\end{array} \\right ) , \\end{align*}"}
+{"id": "1115.png", "formula": "\\begin{align*} R _ { b , o } ^ { \\rm { S - N } } = { W _ b } { \\log _ 2 } \\left ( { 1 + \\frac { { { p _ { b , o } } { { \\left | { { h _ b } } \\right | } ^ 2 } } } { { { W _ b } { N _ 0 } } } } \\right ) . \\end{align*}"}
+{"id": "5356.png", "formula": "\\begin{align*} \\nu ^ * A = \\mu ^ * h ^ * D + E . \\end{align*}"}
+{"id": "5990.png", "formula": "\\begin{align*} \\phi _ \\mu ( g ) = \\frac { J _ \\lambda ^ { ( 2 ) } ( a _ 1 , \\ldots , a _ N ) } { J _ \\lambda ^ { ( 2 ) } ( 1 , \\ldots , 1 ) } , \\mu = 2 \\lambda \\end{align*}"}
+{"id": "5597.png", "formula": "\\begin{align*} W ( 0 1 ^ l | 1 ^ k 0 ) & = A ( 1 ^ { k + 1 } 0 1 . . . ) - A ( 1 0 ^ \\alpha 1 . . . ) + \\sum _ { j = 2 } ^ l \\left [ A ( 1 ^ { k + j } 0 . . . ) - A ( 1 ^ j 0 ^ \\alpha 1 . . . ) \\right ] + \\\\ & + A ( 0 1 ^ { k + l } 0 . . . ) - A ( 0 1 ^ l 0 ^ \\alpha 1 . . . ) \\\\ & = c _ { k + 1 } - d _ \\alpha + \\sum _ { j = 2 } ^ l ( c _ { k + j } - c _ j ) + ( b _ { k + l } - b _ l ) . \\end{align*}"}
+{"id": "5279.png", "formula": "\\begin{align*} \\partial _ { t } ^ n \\sqrt { P _ { \\pm , \\pm } \\left ( t , u + \\left ( \\frac { t _ 1 } { t } \\right ) ^ 2 u _ 1 \\right ) } \\Bigg | _ { t = t _ 0 } \\asymp _ n \\frac { 1 } { t _ 0 ^ { n - 1 } } , \\end{align*}"}
+{"id": "953.png", "formula": "\\begin{align*} u _ 2 ( t ) & = M ( t ) D ( t ) e ^ { - 3 i \\lambda _ 1 | W _ 1 | ^ 2 \\log t } \\begin{pmatrix} 1 & 0 \\end{pmatrix} P Q ( t ) P ^ { - 1 } \\begin{pmatrix} W _ 2 \\\\ \\overline { W _ 2 } \\end{pmatrix} \\\\ & + O ( t ^ { - 3 / 4 + C _ 1 \\varepsilon ^ 2 _ 1 + 2 \\mu C _ 2 \\varepsilon ^ 2 _ 1 + C _ 4 \\varepsilon _ 1 ^ 2 } ) . \\end{align*}"}
+{"id": "5972.png", "formula": "\\begin{align*} g ( 0 ) g ( \\infty ) = & \\frac { ( x _ 0 + \\sqrt { a } x _ 1 ) ( \\sqrt { b } x _ 2 + \\sqrt { a b } x _ 3 ) } { ( \\sqrt { b } x _ 2 - \\sqrt { a b } x _ 3 ) ( x _ 0 - \\sqrt { a } x _ 1 ) } > 0 \\\\ \\Leftrightarrow \\ & ( x _ 0 ^ 2 - a x _ 1 ^ 2 ) ( b x _ 2 ^ 2 - a b x _ 3 ^ 2 ) > 0 \\\\ \\Leftrightarrow \\ & ( x _ 0 ^ 2 - a x _ 1 ^ 2 ) ( x _ 0 ^ 2 - a x _ 1 ^ 2 - 1 ) > 0 \\end{align*}"}
+{"id": "5873.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\langle A ( u _ n ) , u _ n \\rangle = \\langle A ( u ) , u \\rangle . \\end{align*}"}
+{"id": "3269.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sup _ { n \\in \\mathbb N } \\mathbb P \\left [ \\sum _ { k \\geq N } \\langle Y _ n , e _ k \\rangle ^ 2 > \\delta \\right ] = 0 , \\end{align*}"}
+{"id": "2448.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } d _ { q } s \\ , F _ { q } ( s ) = L _ { q } \\left \\{ \\frac { f ( t ) } { t } \\right \\} . \\end{align*}"}
+{"id": "9164.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } m ( s ) \\sin ( 2 \\pi t ) \\ , d t = 0 \\end{align*}"}
+{"id": "8558.png", "formula": "\\begin{align*} h _ 0 \\ , * \\ h _ 1 = h _ 1 . \\end{align*}"}
+{"id": "6509.png", "formula": "\\begin{align*} & \\lambda ( q ( c ( z _ 1 z _ 2 . . . z _ { m - 1 } ) ) , q ( c ( z _ 1 z _ 2 . . . z _ { m - 1 } ) ) l _ m ) = z _ m \\\\ & \\lambda ( q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } ) ) , q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } ) ) a ) = z _ { n - 1 } ' \\\\ & \\lambda ( q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } z _ { n - 1 } ' ) ) , q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } z _ { n - 1 } ' ) ) b ) = z _ n ' \\\\ & \\lambda ( q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } z _ { n - 1 } ' z _ n ' ) ) , q ( c ( z _ 1 z _ 2 . . . z _ { n - 2 } z _ { n - 1 } ' z _ n ' ) ) b ) = z _ { n + 1 } ' \\end{align*}"}
+{"id": "8146.png", "formula": "\\begin{align*} g ^ * f _ * \\mathcal M = \\varprojlim _ k g ^ * f _ * \\tau _ { \\le k } \\mathcal M = \\varprojlim _ k f ' _ * g '^ * \\tau _ { \\le k } \\mathcal M = f ' _ * \\varprojlim _ k g '^ * \\tau _ { \\le k } \\mathcal M = f ' _ * g '^ * \\mathcal M , \\end{align*}"}
+{"id": "3597.png", "formula": "\\begin{align*} \\tilde { f } = \\prod _ { n = 1 } ^ N q _ { ( u _ n , 0 ) } - \\prod _ { n = 1 } ^ N q _ { ( v _ n , 0 ) } \\end{align*}"}
+{"id": "6252.png", "formula": "\\begin{align*} \\varphi _ { * } : = - ( \\varphi _ 0 + \\ldots + \\varphi _ p + 1 ) = 0 . \\end{align*}"}
+{"id": "4625.png", "formula": "\\begin{align*} G _ n / \\langle \\epsilon \\rangle = G _ n / \\{ 1 , \\epsilon \\} \\cong ( \\mathbb Z / 2 \\mathbb Z ) ^ n \\rtimes S _ n = B _ n . \\end{align*}"}
+{"id": "7689.png", "formula": "\\begin{align*} \\mathbb { E } \\Big [ \\underset { t _ 0 \\leq t \\leq T } { \\sup } | \\phi ^ { N , i } ( t , \\boldsymbol { x } _ t ^ * ) - \\Psi ^ { N , i } ( t , \\boldsymbol { x } _ t ^ * ) | ^ 2 \\Big ] \\leq \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } \\mathbb { E } [ | \\xi ^ i - \\xi ^ j | ^ 2 ] + \\frac { 1 } { N ^ 2 } \\underset { i , j = 1 } { \\overset { N } { \\sum } } \\mathbb { E } [ | \\xi ^ { i } - \\xi ^ { j } | ^ 2 ] \\Big ) . \\end{align*}"}
+{"id": "8297.png", "formula": "\\begin{align*} \\tilde Y _ { n + 1 } ^ i = \\tilde Y _ n ^ i + \\bigg ( b ( \\tilde Y _ n ^ i ) + \\frac 1 { p - 1 } \\sum _ { j \\neq i , j \\in \\C } K ( \\tilde Y _ n ^ i - \\tilde Y _ n ^ j ) \\bigg ) \\tau + \\sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , ~ ~ i \\in \\C . \\end{align*}"}
+{"id": "4537.png", "formula": "\\begin{align*} \\mathcal { I } ( x , t ) = \\bigcup _ { i = 1 } ^ p [ t _ { i , e n } ( x , t ) , t _ { i , e x } ( x , t ) ] . \\end{align*}"}
+{"id": "7077.png", "formula": "\\begin{align*} \\tau ^ 1 _ { s , h } F ( x ) : = & \\tau _ { s , h } F ( x ) = F ( x + h e _ { s } ) - F ( x ) , \\\\ \\tau ^ r _ { s , h } F ( x ) : = & \\tau _ { s , h } ( \\tau ^ { r - 1 } _ { s , h } F ( x ) ) , r \\in \\N , r \\geq 1 , \\end{align*}"}
+{"id": "9238.png", "formula": "\\begin{align*} h ( a , x ) : = \\min _ k \\mu _ k ( a , x ) > 0 . \\end{align*}"}
+{"id": "1994.png", "formula": "\\begin{align*} \\varphi ( - T ( x ) + x ) = \\varphi ( z ) = z - 2 g ^ { - 1 } ( z ) = x - T ( x ) - 2 x = - T ( x ) - x . \\end{align*}"}
+{"id": "1871.png", "formula": "\\begin{align*} \\frac { 1 } { ( M ) } \\mathcal { Y M } _ p ( { \\nabla } ) = \\frac { | | R ^ \\nabla | | ^ p } { p } = \\frac 1 p \\bigg ( \\frac { 2 } { ( M ) } \\mathcal { Y M } ( \\nabla ) \\bigg ) ^ { \\frac p 2 } \\end{align*}"}
+{"id": "7832.png", "formula": "\\begin{align*} \\Phi ^ \\dagger ( v , p ) : = Q ^ \\dagger F ^ { q , \\dagger } ( v + \\psi ^ \\dagger ( v , p ) , p ) , \\end{align*}"}
+{"id": "6468.png", "formula": "\\begin{align*} \\mu \\big ( T ^ { ( r + 1 ) r _ n } A \\cap A ^ c \\big ) \\leq & \\sum _ { i = 1 } ^ { r + 1 } \\mu \\big ( T ^ { i r _ n } A \\cap T ^ { ( i - 1 ) r _ n } A ^ c \\big ) \\\\ < & \\big ( r + 1 \\big ) \\big ( 1 - \\beta \\big ) \\mu ( A ) \\\\ = & \\Big ( \\big ( r + 1 \\big ) - \\big ( r + 1 \\big ) \\beta \\Big ) \\mu ( A ) \\\\ = & \\Big ( 1 - \\big ( ( r + 1 ) \\beta - r \\big ) \\Big ) \\mu ( A ) \\\\ < & \\Big ( 1 - \\mu ( A ) \\Big ) \\mu ( A ) . \\end{align*}"}
+{"id": "5442.png", "formula": "\\begin{align*} [ a _ \\lambda b ] = \\sum _ { j \\in \\mathbb { Z _ { + } } } ( a _ { ( j ) } b ) \\frac { \\lambda ^ { j } } { j ! } . \\end{align*}"}
+{"id": "236.png", "formula": "\\begin{align*} \\dot g ( J ) = \\dot x ( J ) \\ , t ^ { - 2 C } \\ , \\dot y ( J ) ^ { - 1 } \\ , t ^ { - 2 C } \\dot z ( J ) ; \\end{align*}"}
+{"id": "5835.png", "formula": "\\begin{align*} c _ s ^ 2 \\partial _ { y 1 } T = - \\frac { 1 } { \\Delta t } { \\varsigma } _ 2 { m } _ 2 ^ { ( 1 ) } , \\end{align*}"}
+{"id": "9121.png", "formula": "\\begin{align*} \\frac { \\sum \\limits _ { i = 1 } ^ { j + 1 } \\chi _ i - r ( j + 1 ) } { \\chi } < 1 \\end{align*}"}
+{"id": "3714.png", "formula": "\\begin{align*} h ^ \\intercal = 0 , A ' ( h ) = 0 , ( \\nabla ^ k _ \\nu A ) ' ( h ) = 0 \\mbox { o n } \\hat \\Sigma \\end{align*}"}
+{"id": "839.png", "formula": "\\begin{align*} \\theta _ { \\rho ^ \\varepsilon } ( t , z _ l ) & = F ( z _ l ) + \\rho ^ \\varepsilon ( t , z _ l ) \\nu _ \\Sigma ( z _ l ) = ( 0 , 0 ) + \\rho ^ \\varepsilon ( t , z _ l ) ( - 1 , 0 ) = \\big ( - \\rho ^ \\varepsilon ( t , z _ l ) , 0 \\big ) , \\\\ \\theta _ { \\rho ^ \\varepsilon } ( t , z _ r ) & = F ( z _ r ) + \\rho ^ \\varepsilon ( t , z _ r ) \\nu _ \\Sigma ( z _ r ) = ( 0 , 0 ) + \\rho ^ \\varepsilon ( t , z _ r ) ( + 1 , 0 ) = \\big ( \\rho ^ \\varepsilon ( t , z _ r ) , 0 \\big ) \\end{align*}"}
+{"id": "4765.png", "formula": "\\begin{align*} A = \\mathcal { N } ( A ) - \\mathcal { R } ( A ) . \\end{align*}"}
+{"id": "765.png", "formula": "\\begin{align*} I _ 2 & = | c | | \\langle f , g \\rangle | = \\left | c \\int ( f - m _ Q f ) \\ ; g \\ ; d m \\right | \\\\ & \\lesssim \\int _ { 2 Q } | f - m _ Q f | \\ ; | g | \\ ; d m + \\int _ { \\R ^ { N + 1 } \\setminus 2 Q } ( f - m _ Q f ) \\ ; g \\ ; d m \\\\ & = I _ { 2 1 } + I _ { 2 2 } . \\end{align*}"}
+{"id": "9117.png", "formula": "\\begin{align*} \\chi _ 1 - r < \\chi = \\sum \\limits _ { i = 1 } ^ n \\chi _ i - ( n - 1 ) r \\end{align*}"}
+{"id": "1512.png", "formula": "\\begin{align*} f ( t ) : = \\frac { ( t - a ) ^ { \\alpha - 1 } } { \\Gamma ( \\alpha + 1 ) } \\left [ \\frac { \\alpha } { \\alpha - \\beta } ( b - a ) - ( t - a ) \\right ] , t \\in [ a , b ] . \\end{align*}"}
+{"id": "2304.png", "formula": "\\begin{align*} \\pi _ L ( h ) ( u \\otimes v ) = ( \\pi ( h ) ( u ) ) \\otimes v \\end{align*}"}
+{"id": "6375.png", "formula": "\\begin{align*} B ( p ) = \\frac { 1 } { p \\mu } \\int _ { 0 } ^ { Q ( p ) } x f ( x ) \\mathrm { d } x = \\frac { 1 } { p \\mu } J ( Q ( p ) ) L ( p ) = \\frac { 1 } { \\mu } \\int _ { 0 } ^ { Q ( p ) } x f ( x ) \\mathrm { d } x = \\frac { 1 } { \\mu } J ( Q ( p ) ) , \\end{align*}"}
+{"id": "7269.png", "formula": "\\begin{align*} \\lambda _ 0 \\alpha ^ m + \\lambda _ 1 \\alpha ^ { m - 1 } \\beta + \\cdots + \\lambda _ m \\beta ^ m = 0 . \\end{align*}"}
+{"id": "4753.png", "formula": "\\begin{align*} & M i n i m i z e _ { \\mathbf { x } \\in \\mathbb { R } ^ { 3 n } } \\ \\ \\sum \\limits _ { i = 1 } ^ n ( a _ J , a _ J , b _ { J i } ) ^ T \\mathbf { x } _ i \\\\ & \\sum \\limits _ { i = 1 } ^ n ( a _ k , a _ k , b _ { k i } ) ^ T \\mathbf { x } _ i + c _ k \\le 0 \\ \\ \\forall \\ , k \\in { 1 , . . . , m } \\\\ & \\mathbf { x } _ i \\in \\mathcal { L } _ i \\end{align*}"}
+{"id": "6331.png", "formula": "\\begin{align*} z - z ^ { q ^ 2 } + ( y - y ^ { q ^ 2 } ) x ^ q + ( x - x ^ { q ^ 2 } ) x ^ { 2 q } = 0 \\end{align*}"}
+{"id": "6062.png", "formula": "\\begin{align*} \\kappa ( s ) = \\alpha \\ , \\frac { \\nu _ { n + 1 } ( s ) } { x _ { n + 1 } ( s ) } + \\varpi . \\end{align*}"}
+{"id": "4819.png", "formula": "\\begin{align*} s _ i ( A ) : = \\sqrt { \\lambda _ i ( A ^ * A ) } , i = 1 , 2 , \\dots , n , \\end{align*}"}
+{"id": "6168.png", "formula": "\\begin{align*} \\hat { A } ^ - \\hat { H } _ 1 = \\hat { H } _ 2 \\hat { A } ^ - , \\hat { A } ^ + \\hat { H } _ 2 = \\hat { H } _ 1 \\hat { A } ^ + . \\end{align*}"}
+{"id": "2742.png", "formula": "\\begin{align*} G ( H ( x ) ) = H ( x ) \\quad \\ x \\in \\partial T ( s ) . \\end{align*}"}
+{"id": "1334.png", "formula": "\\begin{align*} \\beta _ k = \\lambda _ { n + 1 } \\frac { d ( u _ { k - 1 } , u _ k ) } { d ( p _ { n + 1 } , q _ { n + 1 } ) } . \\end{align*}"}
+{"id": "2602.png", "formula": "\\begin{align*} R \\cdot R ( x , J x , J x , x ; u , J u ) = L ( p ) \\ , \\ , Q ( g , R ) ( x , J x , J x , x ; u , J u ) , \\end{align*}"}
+{"id": "7593.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } T ^ { c ( n ) } ( x _ { 0 0 } , x _ { 0 1 } ) = ( x _ { 1 0 } , x _ { 1 1 } ) , \\end{align*}"}
+{"id": "7469.png", "formula": "\\begin{align*} f _ { \\tau } = f ^ { M - 1 } _ { \\tau } + ( f _ { \\tau } ^ M - f ^ { M - 1 } _ { \\tau } ) = \\cdots = f ^ { k + 1 } _ { \\tau } ( x ) + \\sum _ { m = k + 1 } ^ { M - 1 } ( f ^ { m + 1 } _ { \\tau } - f ^ { m } _ { \\tau } ) \\end{align*}"}
+{"id": "4515.png", "formula": "\\begin{align*} & \\lambda _ p < \\lambda _ q \\equiv 0 < \\lambda _ r , \\\\ & p = 1 , \\dots , l , q = l + 1 , \\dots , m , r = m + 1 , \\dots , n , \\end{align*}"}
+{"id": "1188.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ r \\frac { t ^ { d _ i p ^ { j _ i } } - 1 } { t ^ { d _ i } - 1 } \\in \\Z [ t ] . \\end{align*}"}
+{"id": "1515.png", "formula": "\\begin{align*} \\tilde { F } _ i ( x ) = \\bar { F } _ i ( x ) \\times \\tilde { H } ( x ) , i = 1 , \\dots , n , \\end{align*}"}
+{"id": "2320.png", "formula": "\\begin{align*} \\tilde { r } = r + \\left ( \\Delta ^ g f + g ( \\theta , d f ) \\right ) F , \\end{align*}"}
+{"id": "4784.png", "formula": "\\begin{align*} T : L ^ 2 ( D ) \\rightarrow L ^ 2 ( D ) T f = u . \\end{align*}"}
+{"id": "561.png", "formula": "\\begin{align*} g ( \\omega ) : = g _ { \\theta } ( \\omega ) , \\omega \\in \\Omega _ { \\theta } \\subset \\Omega . \\end{align*}"}
+{"id": "1494.png", "formula": "\\begin{align*} \\phi _ 1 '' + c \\phi _ 1 ' + \\frac { \\epsilon } { \\kappa } \\phi _ 2 '' + \\frac { c } { \\kappa } \\phi _ 2 ' = 0 . \\end{align*}"}
+{"id": "7110.png", "formula": "\\begin{align*} \\int _ { 0 } ^ \\infty \\int _ { \\Omega } \\omega ( t , x ) g _ r ( t ) f _ h ' ( \\phi ( x ) ) \\ , u ( t , x ) \\cdot \\psi ( x ) d x d t + \\int _ { 0 } ^ \\infty \\int _ { \\Omega } \\omega ( t , x ) g _ r ' ( t ) f _ h ( \\phi ( x ) ) d x d t = 0 . \\end{align*}"}
+{"id": "2376.png", "formula": "\\begin{align*} [ \\mathbf { h } ' _ 1 , \\mathbf { h } _ 1 ] = 1 . \\end{align*}"}
+{"id": "7560.png", "formula": "\\begin{align*} p _ l ( s \\cdot z ) = s ^ l p _ l ( z ) , \\ ; \\ ; \\forall l = 0 , \\dots , k . \\end{align*}"}
+{"id": "8430.png", "formula": "\\begin{align*} a ( z ) - \\hat { a } = \\frac { 1 } { 2 i } \\int _ { \\mathbb { R } } ( | u _ y ( y ) | ^ 2 ( \\Psi ^ - _ { 1 1 } ( y ; z ) - e ^ { - i c _ - } ) + u _ y ( y ) \\Psi ^ - _ { 2 1 } ( y ; z ) ) \\mathrm { d } y . \\end{align*}"}
+{"id": "4240.png", "formula": "\\begin{align*} & \\Big ( P _ { n - s } ( z ) + F _ k ( z ) + \\sum _ { j = k + 1 } ^ \\infty F _ j ( z ) \\Big ) ^ m \\\\ & \\ \\ \\ = \\Big ( F _ k ^ 1 ( z ) + \\sum _ { j = k + 1 } ^ \\infty F _ j ^ 1 ( z ) \\Big ) ^ { m _ 1 } . . . \\Big ( F _ k ^ s ( z ) + \\sum _ { j = k + 1 } ^ \\infty F _ j ^ s ( z ) \\Big ) ^ { m _ s } \\\\ & \\times \\Big ( z _ { s + 1 } + F _ k ^ { s + 1 } ( z ) + \\sum _ { j = k + 1 } ^ \\infty F _ j ^ { s + 1 } ( z ) \\Big ) ^ { m _ { s + 1 } } . . . \\Big ( z _ n + F _ k ^ n ( z ) + \\sum _ { j = k + 1 } ^ \\infty F _ j ^ n ( z ) \\Big ) ^ { m _ n } \\end{align*}"}
+{"id": "9273.png", "formula": "\\begin{align*} \\begin{aligned} ( \\Delta ( u + v ) ) ^ m \\wedge \\beta _ n ^ { n - m } & = ( \\Delta u ) ^ m \\wedge \\beta _ n ^ { n - m } + ( \\Delta v ) ^ m \\wedge \\beta _ n ^ { n - m } + \\sum \\limits _ { p = 1 } ^ { m - 1 } \\binom { m } { p } ( \\Delta u ) ^ p \\wedge ( \\Delta v ) ^ { m - p } \\wedge \\beta _ n ^ { n - m } \\\\ & \\geq ( \\Delta u ) ^ m \\wedge \\beta _ n ^ { n - m } + ( \\Delta v ) ^ m \\wedge \\beta _ n ^ { n - m } . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "7595.png", "formula": "\\begin{align*} x = \\bigl ( \\pi _ 1 ( \\tilde { x } _ { 0 0 } ) , \\pi _ 1 ( \\tilde { x } _ { 0 1 } ) , \\pi _ 1 ( \\tilde { x } _ { 1 0 } ) , \\pi _ 1 ( \\tilde { x } _ { 1 1 } ) \\bigr ) \\end{align*}"}
+{"id": "8646.png", "formula": "\\begin{align*} \\begin{aligned} & \\xi y \\left [ n \\right ] = \\sqrt P s \\left [ { n - { n _ { \\max } } } \\right ] + \\\\ & \\sqrt P \\sum \\limits _ { l = 1 } ^ L { \\sum \\limits _ { l ' \\ne l } ^ L { \\frac { { { \\bf { h } } _ l ^ H { { \\bf { h } } _ { l ' } } } } { { \\sum \\nolimits _ { k = 1 } ^ L { { { \\left \\| { { { \\bf { h } } _ k } } \\right \\| } ^ 2 } } } } s \\left [ { n - { n _ { \\max } } + { n _ { l ' } } - { n _ l } } \\right ] } } + \\xi z \\left [ n \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "7114.png", "formula": "\\begin{align*} R _ f \\circ g = g \\bullet _ \\mu f . \\end{align*}"}
+{"id": "8750.png", "formula": "\\begin{align*} f ( z ) c _ { \\psi } = \\int _ { \\partial \\Omega } f ( w ) E _ { \\psi } ( w , z ) d \\sigma _ { \\psi } ( w ) - \\int _ \\Omega E _ { \\psi } ( w , z ) { \\mathcal D } _ { \\psi } f ( w ) d x d y , \\end{align*}"}
+{"id": "7816.png", "formula": "\\begin{align*} ( B _ \\phi f ) ( x ) = f ( x + \\phi ) - f ( \\phi ) . \\end{align*}"}
+{"id": "2357.png", "formula": "\\begin{align*} g _ { c ^ * } ( 0 , t ) & = c ^ * ( t ) > 0 = x _ n ( 0 , t + ( t ^ * - \\tilde { \\tau } ) ) , \\\\ g _ { c ^ * } ( 1 , t ) & = c ^ * < n = x _ n ( 1 , t + ( t ^ * - \\tilde { \\tau } ) ) , \\end{align*}"}
+{"id": "6677.png", "formula": "\\begin{align*} \\frac { d \\nu _ x } { d \\nu _ y } ( \\xi ) = e ^ { h ( g ) B _ { x , y } ( \\xi ) } , \\end{align*}"}
+{"id": "8378.png", "formula": "\\begin{align*} \\Psi _ x + i z \\left [ \\sigma _ 3 , \\Psi \\right ] = \\widetilde { Q } \\Psi , \\end{align*}"}
+{"id": "4232.png", "formula": "\\begin{align*} \\dot { s } \\varepsilon ( z ) \\dot { s } \\varepsilon ( 1 ) \\eta = \\varepsilon ( - z ^ { - 1 } ) \\dot { s } h ( - z ) \\varepsilon ( - z ^ { - 1 } ) \\varepsilon ( 1 ) \\eta = \\varepsilon ( - z ^ { - 1 } ) \\dot { s } \\varepsilon ( z ^ 2 - z ) \\eta . \\end{align*}"}
+{"id": "6955.png", "formula": "\\begin{align*} I ( \\mu _ m ) & = H ( \\mu ^ m \\ , | \\ , \\overline { \\mu } ) = \\int _ { [ 0 , 1 ] \\times \\R } \\log \\frac { d \\mu ^ m } { d \\overline { \\mu } } \\ , d \\mu ^ m \\\\ & = \\int _ { [ 0 , 1 ] \\times \\R } \\log \\frac { d \\mu ^ m } { d \\mu } \\ , d \\mu ^ m + \\int _ { [ 0 , 1 ] \\times \\R } \\log \\frac { d \\mu } { d \\overline { \\mu } } \\ , d \\mu ^ m . \\end{align*}"}
+{"id": "4440.png", "formula": "\\begin{align*} A _ n & : = \\big \\{ M ^ * _ v ( \\delta n ) \\in ( - \\delta , \\delta ) \\big \\} , \\\\ B _ n & : = \\big \\{ | M _ v ( t ) - M _ v ( \\delta n ) | < \\delta e ^ { ( \\lambda ^ * - \\lambda ) \\delta n } t \\in [ \\delta n , \\delta ( n + 1 ) ] \\big \\} . \\end{align*}"}
+{"id": "755.png", "formula": "\\begin{align*} \\int _ { 2 ^ { k + 2 } I _ Q } \\| R _ j \\partial _ j g _ k ( \\cdot , t ) \\| _ { L ^ 1 ( 2 ^ { k + 3 } Q _ 1 ) } d t & \\lesssim \\int _ { 2 ^ { k + 2 } I } ( 2 ^ { k + 3 } \\ell ( Q ) ) ^ { N / p } \\| R _ j \\partial _ j g _ k ( \\cdot , t ) \\| _ { L ^ q ( 2 ^ { k + 3 } Q _ 1 ) } d t \\\\ & \\lesssim \\int _ { 2 ^ { k + 2 } I _ Q } ( 2 ^ { k + 3 } \\ell ( Q ) ) ^ { N / p } \\| \\partial _ j g _ k ( \\cdot , t ) \\| _ { L ^ q ( 2 ^ { k + 3 } Q _ 1 ) } d t \\\\ & \\lesssim ( 2 ^ k \\ell ( Q ) ) ^ { N / p } 2 ^ k \\ell ( Q ) ( 2 ^ k \\ell ( Q ) ) ^ { N / q } \\frac 1 { 2 ^ k \\ell ( Q ) } \\\\ & = ( 2 ^ k \\ell ( Q ) ) ^ N , \\end{align*}"}
+{"id": "401.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } \\mathrm { d i s t } _ { \\mathbb { U } } ( \\mathcal { A } ( \\tau _ { n _ k } , \\theta _ { \\tau _ { n _ k } } \\omega ) , \\mathcal { A } _ { \\infty } ( \\theta _ { \\tau _ { n _ k } } \\omega ) ) = 0 \\ \\emph { f o r a l l } \\ \\omega \\in \\Omega . \\end{align*}"}
+{"id": "1184.png", "formula": "\\begin{align*} \\begin{aligned} | \\langle \\mu _ t - \\nu , b ( t , x , \\cdot ) \\rangle \\leq \\gamma H \\left ( \\nu \\mid \\mu _ t \\right ) \\quad t \\in [ 0 , T ] , x \\in \\mathbb { R } ^ d , \\\\ \\nu \\in \\mathcal { P } ( \\mathbb { R } ^ d ) b ( t , x , \\cdot ) \\in L ^ 1 ( \\nu ) . \\end{aligned} \\end{align*}"}
+{"id": "8453.png", "formula": "\\begin{align*} & r _ 2 ( z ) - \\widetilde { r } _ 2 ( z ) = \\frac { 2 i k b } { a } - \\frac { 2 i k \\tilde { b } } { \\tilde { a } } , \\\\ & z ^ { - 2 } ( r _ 2 ( z ) - \\widetilde { r } _ 2 ( z ) ) = \\frac { 2 i k z ^ { - 2 } b } { a } - \\frac { 2 i k z ^ { - 2 } \\tilde { b } } { \\tilde { a } } . \\end{align*}"}
+{"id": "8906.png", "formula": "\\begin{align*} \\mathbf { h } = & \\lambda _ { 1 } \\mathbf { e } _ 1 + \\lambda _ { 2 } \\mathbf { e } _ 2 + \\lambda _ { 1 2 3 ' } \\mathbf { e } _ { 1 2 3 ' } \\\\ = & [ \\lambda _ 1 + \\lambda _ { 1 2 3 ' } , \\lambda _ 2 + \\lambda _ { 1 2 3 ' } , \\lambda _ { 1 2 3 ' } , \\lambda _ 1 + \\lambda _ 2 + 2 \\lambda _ { 1 2 3 ' } , \\\\ & \\lambda _ 1 + 2 \\lambda _ { 1 2 3 ' } , \\lambda _ 2 + 2 \\lambda _ { 1 2 3 ' } , \\lambda _ 1 + \\lambda _ 2 + 2 \\lambda _ { 1 2 3 ' } ] ^ \\intercal \\end{align*}"}
+{"id": "2330.png", "formula": "\\begin{align*} \\left ( \\rho ^ \\star \\right ) ^ { s y m } - \\left ( R i c ^ g \\right ) ^ { J , + } = - \\frac { 1 } { 4 } \\left ( 2 \\delta ^ g \\theta + \\| \\theta \\| ^ 2 - 2 \\| N \\| ^ 2 \\right ) g . \\end{align*}"}
+{"id": "2676.png", "formula": "\\begin{align*} q \\mathcal { A } ( u ) = \\Vert \\nabla u \\Vert ^ q _ q = \\langle A ( u ) , u \\rangle , \\\\ \\ell \\mathcal { P } ( v ) = \\Vert v \\Vert ^ \\ell _ { \\ell } = \\langle P ( v ) , v \\rangle \\end{align*}"}
+{"id": "1625.png", "formula": "\\begin{align*} m [ X _ 1 , X _ 2 ] = ( m X _ 1 ) X _ 2 - ( m X _ 2 ) X _ 1 \\end{align*}"}
+{"id": "3416.png", "formula": "\\begin{align*} F ( \\theta , E ) = \\cos { \\pi \\theta } \\cdot A ( \\theta , E ) = \\left ( \\begin{matrix} E \\cos { \\pi \\theta } - \\lambda \\sin { \\pi \\theta } \\ & - \\cos { \\pi \\theta } \\\\ \\cos { \\pi \\theta } & 0 \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "9089.png", "formula": "\\begin{align*} \\bigoplus _ { j = l } ^ { n - 1 } K _ j ' \\subset ( \\bigoplus _ { j = l } ^ { n - 1 } \\sigma _ j \\circ \\rho _ j ' : \\bigoplus _ { j = l } ^ { n - 1 } j _ { j * } E _ j \\rightarrow \\bigoplus _ { j = l } ^ { n - 1 } ( \\sigma _ j ) ) \\end{align*}"}
+{"id": "1193.png", "formula": "\\begin{align*} 2 ^ { - ( k + 2 ) } n \\ge 2 \\ \\mbox { a n d } \\binom { n } { k } 2 ^ k \\cdot \\exp \\left ( - 2 ^ { - ( k + 4 ) } n \\right ) < \\frac { 1 } { ( 2 n ) ^ { \\log ( 2 n ) } } \\end{align*}"}
+{"id": "6365.png", "formula": "\\begin{align*} \\operatorname { E } ( X ^ { - 1 } ) = \\frac { \\sqrt { \\pi / \\theta } } { 2 \\operatorname { B } ( a , b ) } \\sum _ { n = 0 } ^ b ( - 1 ) ^ n \\binom { b - 1 } { n } \\frac { 1 } { \\sqrt { ( a + n ) ^ 3 } } . \\end{align*}"}
+{"id": "6737.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } \\mu } \\ell = & - \\frac { \\alpha - 1 } { \\sigma } \\frac { \\phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) } { \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) } + \\frac { \\beta - 1 } { \\sigma } \\frac { \\phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) } { 1 - \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) } + \\frac { s } { \\sigma } \\left | \\frac { x - \\mu } { \\sigma } \\right | ^ { s - 1 } \\mathrm { s i g n } ( x - \\mu ) \\end{align*}"}
+{"id": "2312.png", "formula": "\\begin{align*} \\nabla _ X Y = D ^ g _ X Y - \\frac { 1 } { 2 } \\theta ( J X ) J Y - \\frac { 1 } { 2 } \\theta ( Y ) X + \\frac { 1 } { 2 } g ( X , Y ) \\theta ^ \\sharp + g ( X , N ( \\cdot , Y ) ) , \\end{align*}"}
+{"id": "4526.png", "formula": "\\begin{align*} v _ n ( x , t ) & = S _ { d , n } ( t , 0 ) v _ 1 ( x , 0 ) . \\end{align*}"}
+{"id": "541.png", "formula": "\\begin{align*} u _ i v _ j & = ( 3 c _ i + 4 s _ { i } + 4 z _ { i } ) ( c _ j - 4 s _ { j } - 4 z _ { j } ) \\\\ & = 3 c _ i c _ j - 1 2 { c _ i ( s _ { j } + z _ { j } ) + 4 c _ j ( s _ { i } + z _ { i } ) } - 1 6 s _ { i } s _ { j } - 1 6 s _ i z _ { j } \\\\ & \\phantom { { } = { } } - 1 6 s _ j z _ { i } - 1 6 z _ { i } z _ { j } \\\\ & = 6 t _ { i , j } - 3 c _ { i , j } + 3 c _ i c _ j - 1 6 ( s _ i z _ { j } + s _ j z _ { i } + z _ { i } z _ { j } ) \\end{align*}"}
+{"id": "229.png", "formula": "\\begin{align*} g _ { | i } = \\begin{cases} \\dot g ( j ) _ { | i } \\ , \\dot g ( j ) _ { | \\ , i + 3 q ( n ) + 4 C } & \\\\ 1 & \\end{cases} \\end{align*}"}
+{"id": "2426.png", "formula": "\\begin{align*} L _ { q } [ f ( t ) ] = \\int _ { 0 } ^ { \\infty } \\left ( \\sum _ { n = 0 } ^ { \\infty } a _ { n } t ^ { n } \\right ) [ 1 - ( 1 - q ) s t ] ^ { \\frac { 1 } { 1 - q } } \\ , d t . \\end{align*}"}
+{"id": "5712.png", "formula": "\\begin{align*} L _ 1 \\circ F \\circ L _ 2 = F ' , \\end{align*}"}
+{"id": "8397.png", "formula": "\\begin{align*} | a ( z ) | \\geq 1 / 2 , \\operatorname { I m } z = 0 . \\end{align*}"}
+{"id": "6498.png", "formula": "\\begin{align*} s ( \\mu \\nu ) = ( \\mu \\nu ) ( d ( \\mu \\nu ) ) = ( \\mu \\nu ) | ^ * _ { E _ { 2 , [ d ( \\mu ) , d ( \\mu ) + d ( \\nu ) ] } } ( d ( \\nu ) ) = \\nu ( d ( \\nu ) ) = s ( \\nu ) \\end{align*}"}
+{"id": "3876.png", "formula": "\\begin{align*} \\int _ { B _ 1 ( 0 ) } \\phi ^ { p + 1 } = \\frac { \\pi ( p + 1 ) } { 2 } | \\phi ' ( 1 ) | ^ 2 , \\ \\ \\int _ { B _ 1 ( 0 ) } \\phi ^ { p } = 2 \\pi | \\phi ' ( 1 ) | . \\end{align*}"}
+{"id": "4632.png", "formula": "\\begin{align*} \\tau = e _ { i _ 1 } + \\dots + e _ { i _ k } \\in ( \\mathbb Z / 2 \\mathbb Z ) ^ n \\leq B _ n \\end{align*}"}
+{"id": "3617.png", "formula": "\\begin{align*} h _ { i j k } = h _ { i k j } . \\end{align*}"}
+{"id": "8934.png", "formula": "\\begin{align*} x ^ { j } \\allowbreak = \\allowbreak j ! \\sum _ { k = 0 } ^ { j } ( - 1 ) ^ { k } \\frac { ( \\beta ) ^ { ( j ) } } { ( j - k ) ! ( \\beta ) ^ { ( k ) } } L _ { k } ^ { ( \\beta ) } ( x ) , \\end{align*}"}
+{"id": "919.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathcal { L } u _ 1 = 3 | u _ 1 | ^ 2 u _ 1 - 3 ( 2 u _ 1 | u _ 2 | ^ 2 + \\overline { u _ 1 } u _ 2 ^ 2 ) , \\\\ & \\mathcal { L } u _ 2 = 3 ( 2 | u _ 1 | ^ 2 u _ 2 + u _ 1 ^ 2 \\overline { u _ 2 } ) - 3 | u _ 2 | ^ 2 u _ 2 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7559.png", "formula": "\\begin{align*} \\varphi ( \\lambda _ 1 , \\lambda _ 2 ) : = h ( \\lambda _ 1 , \\lambda _ 2 ) \\cdot q _ { l , n } ( \\lambda _ 1 , \\lambda _ 2 ) . \\end{align*}"}
+{"id": "9216.png", "formula": "\\begin{align*} \\begin{aligned} f ( 0 ) = & \\ , | x ( 0 ) - z _ 1 ( 0 ) | + | \\bar { y } ( 0 ) - z _ 2 ( 0 ) | \\\\ = & \\ , | x ( 0 ) - x _ a ( 0 ) + \\gamma \\epsilon _ 1 ( \\eta _ a ( 0 ) , 0 ) | \\\\ & \\ , + | \\bar { y } ( 0 ) - \\bar { y } _ a ( 0 ) + \\gamma \\epsilon _ 2 ( \\eta _ a ( 0 ) , 0 ) | \\\\ = & \\ , | x ( 0 ) - x _ a ( 0 ) | + | \\bar { y } ( 0 ) - \\bar { y } _ a ( 0 ) | . \\end{aligned} \\end{align*}"}
+{"id": "5854.png", "formula": "\\begin{align*} { f _ { 1 2 } } = { f _ { 1 1 } } + \\frac { 1 } { 2 } \\left ( { { f _ 1 } - { f _ 2 } } \\right ) + \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) + \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) + \\frac { 1 } { 8 } \\left ( { 2 { F _ x } + { F _ z } } \\right ) - \\frac { 1 } { 6 } \\rho { u _ z } , \\end{align*}"}
+{"id": "1531.png", "formula": "\\begin{align*} \\frac { d L } { d t _ { k } } = \\left [ ( L ^ { k } ) _ { D } , L \\right ] \\ ; , k \\geq 1 \\ ; , \\end{align*}"}
+{"id": "8832.png", "formula": "\\begin{align*} \\sum _ { j = 3 } ^ n | X _ { 0 , j } | ^ 2 & \\le \\delta ^ 2 ( | X _ { 0 , 1 } | ^ 2 + | X _ { 0 , 2 } | ^ 2 ) ^ r \\le 4 \\delta ^ 2 K ^ { 2 r } , \\\\ | X _ { 0 , 1 } | & \\le \\delta _ 1 K ^ r . \\end{align*}"}
+{"id": "882.png", "formula": "\\begin{align*} \\left | \\frac { z } { n } - \\frac { a } { q } \\right | < \\frac { 1 } { q \\sqrt { n } } \\ , , \\quad 1 \\leq q \\leq \\sqrt { n } , \\quad ( a , \\ , q ) = 1 \\ , . \\end{align*}"}
+{"id": "5672.png", "formula": "\\begin{align*} \\varepsilon \\circ i ( B ) = A \\varepsilon ( B ) A ^ { - 1 } . \\end{align*}"}
+{"id": "3361.png", "formula": "\\begin{align*} N e w t o n ( s _ { p ^ { \\downarrow } } ) \\cap \\mathbb { Z } ^ m \\subseteq N e w t o n ( s _ { \\lambda ^ j } ) \\cap \\mathbb { Z } ^ m = S u p p ( s _ { \\lambda ^ j } ) \\subseteq S u p p ( F ) . \\end{align*}"}
+{"id": "6262.png", "formula": "\\begin{align*} f _ { * } \\nabla _ X A D _ i Y & = \\widetilde { \\nabla } _ X f _ { * } A D _ i Y - \\langle A X , A D _ i Y \\rangle \\rho = - \\widetilde { \\nabla } _ X h _ { * } \\pi _ { * } D _ i Y - \\langle h _ { * } \\pi _ { * } X , h _ { * } \\pi _ { * } D _ i Y \\rangle h \\circ \\pi \\\\ & = - h _ { * } \\nabla ' _ { \\hat { X } } \\hat { D } _ i \\hat { Y } - \\alpha ^ h ( \\hat { X } , \\hat { D } _ i \\hat { Y } ) . \\end{align*}"}
+{"id": "7570.png", "formula": "\\begin{align*} \\frac { 1 } { q } = \\sum _ { n = 0 } ^ { s - 1 } \\frac { a _ n } { \\beta ^ { n + 1 } } + \\left ( \\sum _ { n = s } ^ { s + t - 1 } \\frac { a _ n } { \\beta ^ { n + 1 } } \\right ) \\frac { \\beta ^ t } { \\beta ^ t - 1 } \\end{align*}"}
+{"id": "4594.png", "formula": "\\begin{align*} \\tau ( \\widetilde K _ n ) \\ge [ \\Lambda ( g , 2 A , M ) ] ^ { - 1 } ( 1 + \\varepsilon ) & \\geq [ \\Lambda ( g , A , M ) ] ^ { - 2 } ( 1 + \\varepsilon ) \\\\ & > \\left [ \\frac { 1 + \\varepsilon } { 1 + \\frac { \\varepsilon } { 2 } } \\right ] ^ { - 1 } ( 1 + \\varepsilon ) \\\\ & = 1 + \\frac { \\varepsilon } { 2 } , \\end{align*}"}
+{"id": "7791.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Theta ^ M ( t , x ) = \\int _ \\S W _ r ^ M ( x - y ) \\sin ( \\Theta ^ M ( t , y ) - \\Theta ^ M ( t , x ) ) \\ \\d y , \\end{align*}"}
+{"id": "7540.png", "formula": "\\begin{align*} \\left ( \\left ( e ^ { 1 - \\binom { k } { \\ell } } + o ( 1 ) \\right ) m ^ { k - \\ell } \\right ) ^ { m ^ \\ell } \\end{align*}"}
+{"id": "6626.png", "formula": "\\begin{align*} G ( t ) = \\int _ 0 ^ 1 K ( t , r ) d W ( r ) , \\ \\ \\ 0 \\le t \\le 1 . \\end{align*}"}
+{"id": "1566.png", "formula": "\\begin{align*} g = ( w _ 1 ^ 6 \\dd v _ 1 ^ 2 + \\dd w _ 1 ^ 2 ) + ( \\dd x _ 2 ^ 2 + \\dd y _ 2 ^ 2 ) . \\end{align*}"}
+{"id": "8925.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ { \\infty } h _ { n } ( x ) h _ { m } ( x ) \\exp ( - x ^ { 2 } / 2 ) d x = \\delta _ { n m } , \\end{align*}"}
+{"id": "7250.png", "formula": "\\begin{align*} L ( x ) = \\sum _ { i = 0 } ^ d b _ i x ^ { q ^ i } \\end{align*}"}
+{"id": "5727.png", "formula": "\\begin{align*} \\frac { 1 } { A } = h ^ q r \\iff \\frac { 1 } { A ^ { q - 1 } } = h ^ { q ( q - 1 ) } r ^ { q - 1 } = \\frac { ( r + 1 ) ^ q } { r } . \\end{align*}"}
+{"id": "8562.png", "formula": "\\begin{align*} ( D ^ { \\alpha , 1 } _ { 0 + } \\ , f ) ( t ) \\ , = \\ , ( I _ { 0 + } ^ { 1 - \\alpha } \\ , \\frac { d } { d t } \\ , I _ { 0 + } ^ { 0 } \\ , f ) ( t ) \\ , = \\ , ( I _ { 0 + } ^ { 1 - \\alpha } \\ , \\frac { d } { d t } \\ , f ) ( t ) \\ , = \\ , ( _ * D ^ \\alpha _ { 0 + } \\ , f ) ( t ) , \\ 0 \\le \\alpha < 1 . \\end{align*}"}
+{"id": "9304.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\lambda ^ { N _ { i - 1 } } + \\sum _ { \\ell = 1 } ^ { r } c _ \\ell u _ { i - 1 } ^ { \\gamma _ \\ell } \\lambda ^ { ( i - 1 ) \\gamma _ \\ell + N _ { i - 1 } - \\ell } = 0 \\\\ \\lambda = u _ { i - 1 } v _ { i } \\end{array} \\right . \\end{align*}"}
+{"id": "3343.png", "formula": "\\begin{align*} \\phi _ i = \\left \\{ \\begin{aligned} & \\sigma & & ( i = k ) \\\\ & \\tau & & ( i \\neq k ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "5037.png", "formula": "\\begin{align*} | a \\alpha \\omega + b | & = \\dfrac { | ( a ( 1 + T + T ^ 2 ) + b T ^ 3 ) \\omega + b ( 1 + T ) + a | } { | T ^ 3 \\omega + 1 + T | } \\\\ & \\geq 2 ^ { - 1 } \\left | ( a ( 1 + T + T ^ 2 ) + b T ^ 3 ) \\omega + b ( 1 + T ) + a \\right | \\\\ & \\geq 2 ^ { - 1 } . \\end{align*}"}
+{"id": "8237.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 / d & \\sqrt { 1 - 1 / d ^ 2 } \\\\ \\sqrt { 1 - 1 / d ^ 2 } & - 1 / d \\end{pmatrix} , \\end{align*}"}
+{"id": "5214.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { + } ^ { 2 } \\dd x = \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } ^ { 2 } \\dd x , \\end{align*}"}
+{"id": "4934.png", "formula": "\\begin{align*} \\frac { L ' } { L } ( s , \\chi ) = - \\sum _ { n \\leq x y } \\frac { \\Lambda _ { x , y } ( n ) \\chi ( n ) } { n ^ s } + \\frac { 1 } { \\log { y } } \\sum _ { \\rho _ { \\chi } } \\frac { x ^ { \\rho _ { \\chi } - s } - ( x y ) ^ { \\rho _ { \\chi } - s } } { \\left ( \\rho _ { \\chi } - s \\right ) ^ { 2 } } + \\frac { 1 } { \\log { y } } \\sum _ { n = 0 } ^ { \\infty } \\frac { x ^ { - 2 n - \\mathfrak { a } - s } - ( x y ) ^ { - 2 n - \\mathfrak { a } - s } } { \\left ( 2 n + \\mathfrak { a } + s \\right ) ^ { 2 } } \\end{align*}"}
+{"id": "3325.png", "formula": "\\begin{align*} | \\phi | = \\sum _ { i \\in [ m ] } | \\phi _ i | = m ( \\dim K + 1 ) . \\end{align*}"}
+{"id": "2460.png", "formula": "\\begin{align*} F _ { q } ^ { ( k ) } ( s , q ^ { \\prime } ) = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( - \\frac { 1 } { 1 - q ^ { \\prime } } \\right ) _ { n } \\left ( \\mp ( 1 - q ^ { \\prime } ) \\sigma _ { \\epsilon } \\right ) ^ { n } } { Q _ { n + 1 } ( 2 - q ) \\ ; \\ ; n ! } \\frac { ( - 1 ) ^ { k } \\Gamma ( n + k + 1 ) } { s ^ { n + k + 1 } } . \\end{align*}"}
+{"id": "3910.png", "formula": "\\begin{align*} \\alpha = \\frac { c } { 4 \\pi k \\sqrt { k ^ 2 + r _ * ^ 2 } } , \\ \\ \\beta = \\frac { \\alpha } { 2 } ( 3 r _ * ^ 2 + 4 k ^ 2 ) . \\end{align*}"}
+{"id": "6098.png", "formula": "\\begin{align*} j _ p ^ { ( 1 2 3 ) } = p + a _ { 1 2 3 } , \\end{align*}"}
+{"id": "775.png", "formula": "\\begin{align*} | f _ i ( u ) | \\lesssim 1 + | u | ^ p , \\forall u \\in \\R _ + ^ m , \\ ; \\forall i = 1 , \\ldots , m . \\end{align*}"}
+{"id": "817.png", "formula": "\\begin{align*} & \\Delta _ \\Gamma V + V \\big | \\nabla _ \\Gamma \\nu \\big | ^ 2 = A \\colon D _ \\Gamma ^ 2 H + B \\cdot \\nabla _ \\Gamma H + C H \\end{align*}"}
+{"id": "714.png", "formula": "\\begin{align*} c _ { \\lambda _ n } \\geq \\kappa ( \\lambda _ n ) \\geq \\left ( \\frac { 1 } { 8 } - \\frac { 1 } { 2 ^ { 2 p + 1 } } \\right ) \\left ( \\frac { \\beta _ { \\lambda _ n } ^ p } { 2 C ( N , \\alpha , p ) } \\right ) ^ { \\frac { 1 } { p - 1 } } = \\left ( \\frac { 1 } { 8 } - \\frac { 1 } { 2 ^ { 2 p + 1 } } \\right ) \\left ( \\frac { \\beta _ { 0 } ^ p } { 2 C ( N , \\alpha , p ) } \\right ) ^ { \\frac { 1 } { p - 1 } } > 0 . \\end{align*}"}
+{"id": "6825.png", "formula": "\\begin{align*} \\tau = \\beta \\left ( s \\right ) u , \\quad \\rho = \\beta \\left ( s \\right ) \\xi \\quad \\beta \\left ( s \\right ) = \\left ( \\frac { s - x } { 1 2 t } \\right ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "7772.png", "formula": "\\begin{align*} 1 + \\gamma _ { 1 , k , m } = \\frac { 1 + \\Delta _ { 1 , k + 1 , m } } { 1 + \\Delta _ { 1 , k , m } } , ~ | \\Delta _ { 1 , i , m } | \\le \\frac { w - m } { m } \\Big | \\frac { z _ { m + 1 } } { z _ { 1 } } \\Big | ^ i , ~ i = k , k + 1 \\end{align*}"}
+{"id": "1296.png", "formula": "\\begin{align*} x _ a = f _ 0 ( x _ 0 ) , x _ b = f _ 1 ( x _ 1 ) , x _ c = f _ 2 ( x _ 2 ) , x _ d = f _ 3 ( x _ 3 ) , \\alpha = g ( \\alpha _ 1 ) , \\beta = g ( \\alpha _ 3 ) , \\end{align*}"}
+{"id": "5159.png", "formula": "\\begin{align*} I _ { \\theta h } \\leq E [ t ( \\theta ) b ] = t ( \\theta ) ^ { 2 } E [ b ] = t ( \\theta ) ^ { 2 } I _ { h } < \\theta I _ h , \\end{align*}"}
+{"id": "1881.png", "formula": "\\begin{align*} \\pi _ 1 ( M ) = \\cdots = \\pi _ { 6 } ( M ) = 0 . \\end{align*}"}
+{"id": "2553.png", "formula": "\\begin{align*} { { \\xi } _ { o } } = \\left \\| \\left ( ( { { \\mathbf { x } } _ { o } } + { { \\mathbf { s } } _ { o } } ) - { { \\psi } _ { o } } \\mathbf { e } \\right ) \\right \\| + \\rho \\left \\| { { \\mathbf { s } } _ { o } } \\right \\| - \\gamma { { \\psi } _ { o } } . \\end{align*}"}
+{"id": "5621.png", "formula": "\\begin{align*} h _ { \\hat { \\mu } } ( \\hat { \\sigma } ) = h _ { \\hat { \\mu } } ( \\hat { \\sigma } , \\hat { \\P } ) \\ \\ h _ \\mu ( \\sigma ) = h _ \\mu ( \\sigma , \\P ) \\ . \\end{align*}"}
+{"id": "8869.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } 1 _ B ( T _ { 1 } ^ { 3 n ^ 2 } T _ { 2 } ^ { 8 n ^ 2 } x ) ( 1 _ { A } - \\mathbb { E } ( 1 _ A | \\mathcal { A } ) ) ( T _ { 1 } ^ { n ^ 2 } T _ { 2 } ^ { - n ^ 2 } x ) = 0 . \\end{align*}"}
+{"id": "44.png", "formula": "\\begin{align*} E _ { u } = < \\nabla _ { H } u , \\nabla _ { H } \\rho > - \\frac { Z u } { \\rho } | \\nabla _ { H } \\rho | ^ 2 . \\end{align*}"}
+{"id": "927.png", "formula": "\\begin{align*} \\mathcal { L } u _ { 1 } = 3 \\lambda _ 1 | u _ 1 | ^ 2 u _ 1 . \\end{align*}"}
+{"id": "436.png", "formula": "\\begin{align*} \\kappa ' \\begin{pmatrix} 1 \\\\ \\vdots \\\\ \\theta ^ { n - 1 } \\end{pmatrix} = \\begin{pmatrix} a _ { 1 } \\\\ \\vdots \\\\ a _ { n } \\end{pmatrix} . \\end{align*}"}
+{"id": "3409.png", "formula": "\\begin{align*} q _ { k + 1 } = a _ { k + 1 } q _ { k } + q _ { k - 1 } , \\end{align*}"}
+{"id": "7733.png", "formula": "\\begin{align*} s \\cdot ( x , y , z , \\zeta ) = ( a x + b y , c x + d y , ( \\det s ) z , ( \\det s ) \\zeta ) . \\end{align*}"}
+{"id": "2201.png", "formula": "\\begin{align*} \\mathbf { R } \\mathbf { v } ^ { 0 } & = \\mathbf { R } \\frac { ( \\sum \\limits _ { j = 1 } ^ { N } \\mathbf { R } _ { . j } ) ^ { H } } { \\mathbf { S ( R ) } } = \\mathbf { R } ( \\mathbf { 1 } ^ { H } _ { N \\times 1 } \\mathbf { R } ) ^ { H } / \\mathbf { S ( R ) } \\\\ & = \\mathbf { R } \\mathbf { R } ^ { H } \\mathbf { 1 } _ { N \\times 1 } = \\mathbf { 1 } _ { N \\times 1 } \\end{align*}"}
+{"id": "8666.png", "formula": "\\begin{align*} { { { \\bf { \\tilde h } } } ^ H } \\left [ k \\right ] = \\frac { 1 } { { \\sqrt K } } \\sum \\limits _ { t = 0 } ^ { { n ' _ { { \\rm { s p a n } } } } } { \\left ( { \\sum \\limits _ { l ' = 1 } ^ { L ' } { { \\bf { g } } _ { l ' } ^ H \\left [ t \\right ] { \\bf { \\bar H } } _ { l ' } ^ \\bot { { { \\bf { \\bar X } } } _ { l ' } } } } \\right ) { e ^ { - j \\frac { { 2 \\pi } } { K } k t } } } , \\ \\forall k , \\end{align*}"}
+{"id": "441.png", "formula": "\\begin{align*} \\alpha = \\omega \\kappa \\theta \\kappa ^ { - 1 } \\omega ^ { - 1 } , \\end{align*}"}
+{"id": "3697.png", "formula": "\\begin{align*} v = 0 , \\nu ' ( h ) ( \\bar u ) + \\nu ( v ) = 0 , A ' ( h ) = 0 \\mbox { o n } \\hat \\Sigma , \\end{align*}"}
+{"id": "3387.png", "formula": "\\begin{align*} S _ 0 ( x ) S _ 2 ( x ) - S _ 1 ( x ) ^ 2 = \\frac 1 2 \\sum _ { i , j = 0 } ^ N ( i - j ) ^ 2 x ^ i x ^ j \\end{align*}"}
+{"id": "8818.png", "formula": "\\begin{align*} B ( x , x ) = 0 , \\lim _ { t \\to \\infty } \\norm { e ^ { t L ^ \\perp _ { x } } } = \\infty . \\end{align*}"}
+{"id": "5153.png", "formula": "\\begin{align*} H [ b _ n ] & = 4 \\pi \\int _ { \\mathbb { R } ^ { 2 } _ { + } } ( \\phi _ n - \\phi _ \\infty ) _ { + } G _ n \\frac { 1 } { r } \\dd z \\dd r \\\\ & = 4 \\pi \\int _ { Q } ( \\phi _ n - \\phi _ \\infty ) _ { + } G _ n \\frac { 1 } { r } \\dd z \\dd r + 4 \\pi \\int _ { \\mathbb { R } ^ { 2 } _ { + } \\backslash Q } ( \\phi _ n - \\phi _ \\infty ) _ { + } G _ n \\frac { 1 } { r } \\dd z \\dd r . \\end{align*}"}
+{"id": "5934.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\lim _ { n \\rightarrow \\infty } I I _ 1 \\leq C \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ { T } ( 1 + \\Vert X _ n ( s ) \\Vert _ H ^ { \\lambda } ) \\Vert X _ n ( s ) - X ( s ) \\Vert _ H ^ 2 d s = 0 . \\end{align*}"}
+{"id": "6336.png", "formula": "\\begin{align*} \\operatorname { I } _ { y } ( a , b ) = \\frac { 1 } { \\operatorname { B } ( a , b ) } \\int ^ { y } _ { 0 } \\omega ^ { a - 1 } ( 1 - \\omega ) ^ { b - 1 } \\mathrm { d } \\omega , \\end{align*}"}
+{"id": "6011.png", "formula": "\\begin{align*} c _ { \\N } : = \\mathop { \\inf } \\limits _ { ( u , v ) \\in \\N } I ( u , v ) . \\end{align*}"}
+{"id": "9237.png", "formula": "\\begin{align*} M ( x ^ 1 , \\dots , x ^ m ) = \\frac { 1 } { m ! } \\prod _ k \\left ( \\sum \\limits _ i x _ i ^ k \\frac { \\partial } { \\partial x _ i } \\right ) P ( x ) , \\end{align*}"}
+{"id": "8995.png", "formula": "\\begin{align*} U = ( - 1 ) ^ { p ( \\eta , \\ell _ { m } ) } ( - 1 ) ^ { | \\pi ( \\eta \\cup \\ell _ m ) | } \\prod _ { r \\notin [ m + 1 , M ] } U ^ { \\eta \\cup j } _ r . \\end{align*}"}
+{"id": "6799.png", "formula": "\\begin{align*} a _ { m } ( x ) + \\sum _ { n = 0 } ^ { \\infty } a _ { n } ( x ) A _ { m n } ( x ) = r _ { m } ( x ) , m = 0 , 1 , \\ldots \\end{align*}"}
+{"id": "7069.png", "formula": "\\begin{align*} F _ { 7 , 0 , 1 , 0 } \\ , = \\ , - \\ , \\tfrac { 5 0 4 0 } { 5 4 0 0 } \\ , = \\ , - \\ , \\tfrac { 1 4 } { 1 5 } , \\end{align*}"}
+{"id": "5110.png", "formula": "\\begin{align*} \\int _ { B ( 0 , 2 ) \\backslash B ( 0 , 1 ) } | \\nabla \\varphi _ n | ^ { 2 } \\dd y = \\int _ { D ( 0 , 2 ) \\backslash D ( 0 , 1 ) } | \\nabla \\phi _ n | ^ { 2 } \\frac { 2 \\pi ^ { 2 } } { r } \\dd z \\dd r , \\end{align*}"}
+{"id": "1219.png", "formula": "\\begin{align*} h ( \\nu | \\Phi ) : = \\limsup _ { n \\to \\infty } \\frac { 1 } { \\abs { \\Lambda _ n } } h _ { \\Lambda _ n } ( \\nu | \\mu ) . \\end{align*}"}
+{"id": "8786.png", "formula": "\\begin{align*} \\tilde { V } ( x ) = \\frac { 1 } { T } \\int _ 0 ^ T \\mathcal { P } _ t V ( x ) d t \\end{align*}"}
+{"id": "3220.png", "formula": "\\begin{align*} \\gamma ( x ) * ( \\mu \\circ \\nu ) = \\big ( ( \\nu \\cdot \\gamma ) ( x ) * \\mu \\big ) \\circ \\big ( \\gamma ( x ) * \\nu \\big ) \\end{align*}"}
+{"id": "7484.png", "formula": "\\begin{align*} x _ { k } a _ { i j } ( x ) u _ { i k } u _ { j } = \\frac { 1 } { 2 } \\partial _ k \\big ( a _ { i j } ( x ) u _ i u _ j x _ { k } \\big ) - \\frac { n } { 2 } a _ { i j } ( x ) u _ i u _ j - \\frac { 1 } { 2 } x _ { k } \\partial _ k a _ { i j } ( x ) u _ i u _ j , \\end{align*}"}
+{"id": "566.png", "formula": "\\begin{align*} f ( z ) e ^ { p z } = \\frac { 1 } { 2 \\pi i } \\int _ { \\partial \\Delta } \\frac { f ( \\zeta ) e ^ { p \\zeta } } { \\zeta - z } d \\zeta , z \\in \\Delta , \\end{align*}"}
+{"id": "2164.png", "formula": "\\begin{align*} \\phi _ { z } ( { \\bf { v } } , { \\bf { u } } ) \\triangleq \\phi _ { z } ( d ) = \\ss _ { z } - 1 0 \\ , \\alpha _ { z } \\log _ { 1 0 } \\left ( d \\right ) , \\end{align*}"}
+{"id": "8208.png", "formula": "\\begin{align*} x _ { 2 i - 1 , j } = ( - 1 ) ^ { j + 1 } k _ { \\alpha ( i ) } + \\sum _ { \\ell = 1 } ^ { j - 1 } ( - 1 ) ^ { \\ell + 1 } a _ { j - \\ell } . \\end{align*}"}
+{"id": "3129.png", "formula": "\\begin{align*} \\mathcal { R } ( e _ 1 ) = e _ 1 + \\frac { a _ { 4 } } { 2 } e _ 4 , \\ \\mathcal { R } ( e _ 2 ) = \\lambda _ 1 e _ 3 , \\ \\mathcal { R } ( e _ 3 ) = \\mathcal { R } ( e _ 4 ) = 0 , \\end{align*}"}
+{"id": "5028.png", "formula": "\\begin{align*} T & = \\left ( [ x ] \\times [ y ] \\right ) \\backslash \\left ( [ x - r + 1 . . x ] \\times y \\right ) \\backslash \\left ( x \\times [ y - s + 1 . . y - 1 ] \\right ) \\mbox { a n d } \\\\ T ' & = \\left ( [ x - w ] \\times [ y ] \\right ) \\backslash \\left ( ( x - w ) \\times [ y - r ' + 1 . . y ] \\right ) , \\end{align*}"}
+{"id": "7985.png", "formula": "\\begin{align*} \\log ( h _ { t } + h ) = \\log \\sigma _ { n - 1 } + \\chi ( x , t ) , \\end{align*}"}
+{"id": "3203.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu ( h ^ { - n } ( A ) \\cap B ) = \\mu ( A ) \\mu ( B ) . \\end{align*}"}
+{"id": "8860.png", "formula": "\\begin{align*} \\begin{cases} x _ { 2 n + 1 } & = ( 1 - \\alpha _ n ) T x _ { 2 n } + \\alpha _ n u , \\\\ x _ { 2 n + 2 } & = ( 1 - \\beta _ n ) U x _ { 2 n + 1 } + \\beta _ n x _ { 2 n + 1 } . \\end{cases} \\end{align*}"}
+{"id": "7463.png", "formula": "\\begin{align*} N _ f ( z ) & = \\exp \\left ( \\textstyle \\sum _ { k = 1 } ^ { + \\infty } \\ , - \\frac { ( - 1 ) ^ { k + 1 } } { k } ( - z ) ^ k + \\frac { ( - 1 ) ^ { k + 1 } } { k } ( - m z ) ^ k \\right ) \\\\ & = e ^ { - \\ln ( 1 - z ) + \\ln ( 1 - m z ) } \\\\ & = \\dfrac { 1 - m z } { 1 - z } . \\end{align*}"}
+{"id": "5051.png", "formula": "\\begin{align*} H = 2 \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi _ { \\infty } ) _ { + } \\frac { G } { r ^ { 2 } } \\dd x . \\end{align*}"}
+{"id": "5067.png", "formula": "\\begin{align*} < n \\times u , v > _ { \\partial \\Omega } = \\int _ { \\partial \\Omega } u \\times v \\cdot n \\dd H = \\int _ { \\partial \\Omega } \\alpha \\wedge \\beta = \\int _ { \\partial \\Omega } d ( \\phi \\wedge \\beta ) = 0 . \\end{align*}"}
+{"id": "3670.png", "formula": "\\begin{align*} \\nabla _ V ( \\nabla X ) = - R ( X , V ) + T _ h ( V , \\cdot ) \\end{align*}"}
+{"id": "1961.png", "formula": "\\begin{align*} \\det \\rho ( I ) = | \\Delta _ F | ^ { \\frac { 1 } { 2 } } N _ F ( I ) . \\end{align*}"}
+{"id": "3403.png", "formula": "\\begin{align*} b _ n = b _ 0 + \\sum _ { k = 1 } ^ n ( b _ k - b _ { k - 1 } ) = 1 + \\sum _ { k = 0 } ^ { n - 1 } x _ k = \\sum _ { k = 0 } ^ n \\frac { ( e ^ t - 1 ) ^ k } { k ! } \\xrightarrow [ n \\to \\infty ] { } e ^ { e ^ t - 1 } . \\end{align*}"}
+{"id": "3513.png", "formula": "\\begin{align*} B _ { p } \\ = \\ 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } B _ { p - 2 j - 1 } . \\end{align*}"}
+{"id": "4584.png", "formula": "\\begin{align*} p _ { \\gamma _ Z } ( F v + G w ) = p _ { \\gamma _ Z } ( - \\norm { v } _ { V } z ) = \\norm { v } _ { V } p _ { \\gamma _ Z } ( z ) \\leq \\norm { v } _ { V } \\frac { \\varepsilon } { \\norm { v } _ { V } } = \\varepsilon , \\end{align*}"}
+{"id": "8064.png", "formula": "\\begin{align*} _ { \\rm m i n } \\left ( a _ j ^ { \\left ( o \\right ) } \\right ) = 2 a _ j ^ { \\left ( o \\right ) } \\left ( a _ j ^ { \\left ( o \\right ) } \\sum _ { i = 1 } ^ { M } \\lvert \\phi ^ { \\left ( i , j \\right ) } \\rvert ^ 2 - \\Re \\left \\{ \\phi ^ { \\left ( j , j \\right ) } \\right \\} \\right . \\left . + \\sum _ { q = 1 } ^ { M - 1 } \\sum _ { r = q + 1 } ^ { M } f _ { q , r } ^ { \\left ( j \\right ) } \\right ) . \\end{align*}"}
+{"id": "5111.png", "formula": "\\begin{align*} \\int _ { D ( 0 , 2 ) \\backslash D ( 0 , 1 ) } | \\phi _ { n _ { k } } - \\phi | ^ { 2 } \\frac { 2 \\pi ^ { 2 } } { r } \\dd z \\dd r = \\int _ { B ( 0 , 2 ) \\backslash B ( 0 , 1 ) } | \\varphi _ { n _ { k } } - \\varphi | ^ { 2 } \\dd y \\to 0 . \\end{align*}"}
+{"id": "411.png", "formula": "\\begin{align*} \\psi _ * \\mu \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) = \\mu ( \\psi ^ { - 1 } \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) ) = \\mu ( \\pi \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) ) , \\end{align*}"}
+{"id": "5197.png", "formula": "\\begin{align*} \\partial _ t \\Phi _ { + } ^ { 2 } + u \\cdot \\nabla \\Phi _ { + } ^ { 2 } = \\mu \\left ( \\Delta - \\frac { 2 } { r } \\partial _ r \\right ) \\Phi _ { + } ^ { 2 } - 2 \\mu | \\nabla \\Phi _ { + } | ^ { 2 } \\textrm { o n } \\ L ^ { 1 } _ { \\textrm { l o c } } ( \\mathbb { R } ^ { 3 } ) . \\end{align*}"}
+{"id": "9134.png", "formula": "\\begin{align*} M ^ d \\times ( - 1 , 0 ) \\ni ( x , z ) \\mapsto \\mathcal { S } ( x , z ) = ( x , \\varrho ( x , z ) ) \\in \\Omega , \\end{align*}"}
+{"id": "2446.png", "formula": "\\begin{align*} D _ { q } ^ { 2 } \\{ L _ { q } [ f ( t ) ] \\} ( s ) = L _ { q } [ ( - t ) ^ { 2 } f ( t ) ] . \\end{align*}"}
+{"id": "3373.png", "formula": "\\begin{align*} u _ \\lambda ( t , x ) & = \\lambda u ( \\lambda ^ 2 t , \\lambda x ) \\\\ b _ \\lambda ( t , x ) & = \\lambda b ( \\lambda ^ 2 t , \\lambda x ) \\\\ \\pi _ \\lambda ( t , x ) & = \\lambda ^ 2 \\pi ( \\lambda ^ 2 t , \\lambda x ) , \\end{align*}"}
+{"id": "4225.png", "formula": "\\begin{align*} \\overline { \\mathfrak { D } _ { q ^ n } } \\ \\overline { U _ { q ^ n } } \\zeta = ( A + B ) \\overline { U _ { q ^ n } } \\dot { s } \\overline { U ^ * _ { q ^ n } } { \\bf 1 } _ { - } + ( q ^ n - 1 ) C \\overline { U _ { q ^ n } } { \\bf 1 } _ { - } \\in \\Bbbk { \\bf G } \\zeta . \\end{align*}"}
+{"id": "7497.png", "formula": "\\begin{align*} N \\varphi : = A ( x ) \\nabla \\varphi \\cdot \\frac { x } { | x | } . \\end{align*}"}
+{"id": "6256.png", "formula": "\\begin{align*} \\nabla _ T D _ i = [ D _ i , C _ T ] = 0 \\quad \\forall T \\in \\Gamma \\forall i . \\end{align*}"}
+{"id": "4333.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\sum _ { i = 0 } ^ { n - 1 } \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in I _ { i , n } \\backslash U _ k \\} } | \\tilde { F } | ^ 2 _ h \\\\ \\le & \\limsup _ { n \\to + \\infty } \\sum _ { i = 0 } ^ { n - 1 } \\frac { 1 } { \\inf _ { I _ { i , n } \\backslash U _ k } c ( t ) } \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in I _ { i , n } \\backslash U _ k \\} } | \\tilde F | ^ 2 _ h c ( - \\Psi ) . \\end{align*}"}
+{"id": "522.png", "formula": "\\begin{align*} T _ { \\mu } f ( x ) = \\sum _ { n = 0 } ^ { \\infty } \\mu ( 2 n + d ) P _ { n } f ( x ) , \\end{align*}"}
+{"id": "1047.png", "formula": "\\begin{align*} d ( v _ 1 \\Rightarrow w _ 2 v _ 2 ) = d ( v _ 1 \\Rightarrow v _ 1 ' ) + d ( v _ 1 ' \\Rightarrow w _ 2 v _ 2 ' ) + d ( w _ 2 v _ 2 ' \\Rightarrow w _ 2 v _ 2 ) . \\end{align*}"}
+{"id": "5675.png", "formula": "\\begin{align*} { } _ { m } \\tilde { E } ^ { 1 } _ { p , q } = { } _ { m } E ^ { 1 } _ { p , q } ( n - 2 ) [ - 2 , 0 ] = { } _ { m } E ^ { 1 } _ { p - 2 , q } ( n - 2 ) \\cong H _ { q } ( O _ { n - p , n - p } ) \\end{align*}"}
+{"id": "8316.png", "formula": "\\begin{align*} y '' + ( A ^ { - 1 } M - A ^ { - 1 } E ) y = 0 . \\end{align*}"}
+{"id": "3620.png", "formula": "\\begin{align*} \\sum _ i \\frac { u _ i ^ 2 } { u ^ 2 } = 1 - ( \\nu ^ { n + 1 } ) ^ 2 \\leq 1 , ( \\nu ^ { n + 1 } ) _ i = \\frac { u _ i } { u } ( \\nu ^ { n + 1 } - \\kappa _ i ) , \\end{align*}"}
+{"id": "2114.png", "formula": "\\begin{align*} \\pi \\circ \\phi = \\mbox { i d } _ Y , \\end{align*}"}
+{"id": "1718.png", "formula": "\\begin{align*} d ( e ^ { ( \\alpha - 1 ) W ( t - s ) } ) = ( \\alpha - 1 ) e ^ { ( \\alpha - 1 ) W ( t - s ) } d W ( t ) + ( \\alpha - 1 ) ^ 2 \\mu e ^ { ( \\alpha - 1 ) W ( t - s ) } d t . \\end{align*}"}
+{"id": "7802.png", "formula": "\\begin{align*} W _ r ( x ) = \\frac 1 2 \\hat W _ r ( 0 ) + \\sum _ { k = 1 } ^ \\infty \\hat W _ r ( k ) \\cos ( 2 \\pi k x ) , \\end{align*}"}
+{"id": "1730.png", "formula": "\\begin{align*} F ( y ) ( t ) = U ( t , 0 ) x - \\lambda i \\int _ 0 ^ t U ( t , s ) ( e ^ { ( \\alpha - 1 ) W ( s ) } g ( y ( s ) ) ) d s , \\ t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "4811.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 v } { \\partial n _ e ^ 2 } = n _ e \\cdot ( \\triangle v ) n _ e \\end{align*}"}
+{"id": "3204.png", "formula": "\\begin{align*} \\{ d _ F ( \\alpha , \\beta ) = \\max _ { a \\in F } \\| \\alpha ( a ) - \\beta ( a ) \\| \\mid F \\subseteq A \\} . \\end{align*}"}
+{"id": "4550.png", "formula": "\\begin{align*} & | \\Gamma _ { 0 , 0 , 0 } \\cap \\Gamma _ { 1 , 0 , 0 } | = ( q - 1 ) ^ 3 ( q ^ 2 + q + 1 ) ( q + 1 ) q ^ 6 \\\\ & | \\Gamma _ { 0 , 0 , 0 } \\cap \\Gamma _ { 1 , 1 , 0 } | = ( q - 1 ) ^ 3 ( q + 1 ) ^ 2 q ^ 6 \\\\ & | \\Gamma _ { 0 , 0 , 0 } \\cap \\Gamma _ { 1 , 1 , 1 } | = ( q - 1 ) ^ 3 ( q ^ 2 + q + 1 ) ( q + 1 ) q ^ 6 . \\end{align*}"}
+{"id": "3622.png", "formula": "\\begin{align*} \\sum _ i \\sigma _ { k } ^ { i i } \\kappa _ i ^ 2 = \\sigma _ 1 \\sigma _ k - ( k + 1 ) \\sigma _ { k + 1 } . \\end{align*}"}
+{"id": "4785.png", "formula": "\\begin{align*} T ( \\lambda u ) = u . \\end{align*}"}
+{"id": "6181.png", "formula": "\\begin{align*} V _ 2 ( r ) = \\frac { L ' ( L ' + 1 ) } { r ^ 2 } - \\frac { Q ' } { r } f + \\kappa B ' _ 1 \\frac { r } { f } + \\kappa B ' _ 2 \\frac { 1 } { f ^ 2 } + R , \\end{align*}"}
+{"id": "1862.png", "formula": "\\begin{align*} \\sqrt { a \\cdot b } = \\overset { ` ` f ( a v e r a g e ) \" \\swarrow } { \\exp ( \\frac { A + B } { 2 } ) } \\overset { ( \\ , ) } \\le \\overset { ` ` a v e r a g e ( f ) \" \\swarrow } { \\frac { \\exp A + \\exp B } { 2 } } = \\frac { a + b } { 2 } \\ , . \\end{align*}"}
+{"id": "1012.png", "formula": "\\begin{align*} x = x _ L \\cdot \\prescript L { } { } x { } ^ R \\cdot x _ R & \\ell ( x ) = \\ell ( x _ L ) + \\ell \\left ( \\prescript L { } { } x { } ^ R \\right ) + \\ell ( x _ R ) , \\\\ \\prescript { - L } { } { } x { } ^ { - R } = x _ L ' \\cdot x \\cdot x _ R ' & \\ell \\left ( \\prescript { - L } { } { } x { } ^ { - R } \\right ) = \\ell ( x _ L ' ) + \\ell \\left ( x \\right ) + \\ell ( x _ R ' ) . \\end{align*}"}
+{"id": "2180.png", "formula": "\\begin{align*} \\mathbf { a } ( \\theta _ { 0 } ) = [ e ^ { j 2 \\pi d _ { 1 } { s i n } \\theta _ 0 / \\lambda } , e ^ { j 2 \\pi d _ { 2 } { s i n } \\theta _ 0 / \\lambda } , \\cdots , e ^ { j 2 \\pi d _ { N } { s i n } \\theta _ 0 / \\lambda } ] ^ { T } , \\end{align*}"}
+{"id": "6024.png", "formula": "\\begin{align*} \\int _ { \\R ^ { 2 } } ( | \\nabla v _ { n } | ^ { 2 } + v _ { n } ^ { 2 } ) d x + 3 B ( v _ { n } ) = \\int _ { \\R ^ { 2 } } v _ { n } ^ { 2 p } d x + o _ { n } ( 1 ) . \\end{align*}"}
+{"id": "1464.png", "formula": "\\begin{gather*} \\Gamma _ { a } : = \\max \\left \\{ | \\phi ( x , s , \\arctan p ) - \\phi ( x , f ( x ) , 0 ) | : ( x , s , p ) \\in I _ { a } \\right \\} , \\\\ [ 1 e x ] \\Gamma _ { a } ^ { + } : = \\max \\left \\{ | \\phi ( x , s , \\arctan p ) - \\phi ( x , f ( x ) , \\pi / 2 ) | : ( x , s , p ) \\in I _ { a } ^ { + } \\right \\} , \\\\ [ 1 e x ] \\Gamma _ { a } ^ { - } : = \\max \\left \\{ | \\phi ( x , s , \\arctan p ) - \\phi ( x , f ( x ) , - \\pi / 2 ) | : ( x , s , p ) \\in I _ { a } ^ { - } \\right \\} . \\end{gather*}"}
+{"id": "838.png", "formula": "\\begin{align*} a ( \\rho ^ \\varepsilon ) ( 0 , z _ l ) = 1 \\phantom { x x x } \\phantom { x x x } a ( \\rho ^ \\varepsilon ) ( 0 , z _ r ) = 1 \\end{align*}"}
+{"id": "6910.png", "formula": "\\begin{align*} - H ( R \\ , | \\ , P ) + \\log Z & = \\int _ { \\R ^ n } f \\ , d R - H ( R ) \\le \\int _ { \\R ^ n } f \\ , d Q - H ( Q ) = - H ( Q \\ , | \\ , P ) + \\log Z . \\end{align*}"}
+{"id": "6811.png", "formula": "\\begin{align*} & a _ { m } ( x ) + \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( - 1 ) ^ { k + m + 1 } } { 2 \\pi } a _ { k } ( x ) \\int _ { - \\infty } ^ { \\infty } \\frac { \\varphi \\left ( \\tau \\right ) z ^ { k + m + 1 } ( \\tau ) d \\tau } { \\frac { 1 } { 4 } + \\tau ^ { 2 } } \\\\ & = \\frac { ( - 1 ) ^ { m } } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\frac { \\varphi \\left ( \\tau \\right ) z ^ { m + 1 } ( \\tau ) d \\tau } { \\frac { 1 } { 2 } + i \\tau } , m = 0 , 1 , \\ldots , n - 1 . \\end{align*}"}
+{"id": "5619.png", "formula": "\\begin{align*} \\L ^ n _ { \\overline { A } } f \\to \\int f \\ d \\mu \\ , \\ \\ \\ \\L ^ n _ { \\overline { A } ^ * } f \\to \\int f \\circ \\theta ^ { - 1 } \\ d \\mu ^ * = \\int f \\ d \\left ( \\theta ^ { - 1 } _ * \\mu ^ * \\right ) \\ , \\end{align*}"}
+{"id": "2623.png", "formula": "\\begin{align*} 2 \\alpha ( z ) \\cdot [ x , y ] = [ z \\cdot x , \\alpha ( y ) ] + \\varepsilon ( z , x ) [ \\alpha ( x ) , z \\cdot y ] . \\end{align*}"}
+{"id": "1719.png", "formula": "\\begin{align*} I _ 2 = & ( \\alpha - 1 ) \\int _ 0 ^ t U ( t , t - s ) e ^ { ( \\alpha - 1 ) W ( t - s ) } g ( y ( t - s ) ) d W ( s ) \\\\ & + ( \\alpha - 1 ) ^ 2 \\int _ 0 ^ t U ( t , t - s ) \\mu e ^ { ( \\alpha - 1 ) W ( t - s ) } g ( y ( t - s ) ) d s . \\end{align*}"}
+{"id": "2195.png", "formula": "\\begin{align*} f ( \\theta ) & = S ^ { - 1 } ( \\theta ) = \\mathbf { a } ^ H ( \\theta ) \\mathbf { U } _ V \\mathbf { U } ^ H _ V \\mathbf { a } ( \\theta ) \\\\ & = \\mathbf { a } ^ T ( \\frac { 1 } { z } ) \\mathbf { U } _ V \\mathbf { U } ^ H _ V \\mathbf { a } ( z ) \\triangleq f ( z ) \\end{align*}"}
+{"id": "241.png", "formula": "\\begin{align*} \\left | \\big ( A t ^ { - 2 \\ell - 4 C } \\cup A t ^ { 2 \\ell + 4 C } \\big ) \\cap B _ S ( n + D ' ) \\right | \\ge \\frac { 1 } { \\ell + 1 } \\binom { \\ell + 1 } { 2 } \\ , | R _ { q , \\ell } ^ \\flat ( n ) | > \\frac { q ( n ) } { 2 } \\ , | R _ { q , \\ell } ^ \\flat ( n ) | . \\end{align*}"}
+{"id": "3184.png", "formula": "\\begin{align*} \\lim \\limits _ { q \\to \\infty } { _ { 3 } } F _ { 2 } \\left ( \\begin{array} { c c c } \\chi _ 4 , & \\chi _ 4 , & \\chi _ 4 ^ 3 \\\\ & \\varepsilon , & \\varepsilon \\end{array} | 1 \\right ) = 0 . \\end{align*}"}
+{"id": "2571.png", "formula": "\\begin{align*} l _ { k _ { 1 } } - p ^ { l _ { k } - n } b _ { k } q ^ { n } = p ^ { l _ { t } - u } b _ { t } q ^ u . \\end{align*}"}
+{"id": "9244.png", "formula": "\\begin{align*} \\mathcal { H } _ m ( s I + A ) = \\sum \\limits _ { 1 \\leq i _ 1 < \\dots < i _ m \\leq n } ( s + \\lambda _ { i _ 1 } ) \\dots ( s + \\lambda _ { i _ m } ) = \\sum \\limits _ { p = 0 } ^ m \\binom { n - p } { m - p } \\mathcal { H } _ p ( A ) s ^ { m - p } . \\end{align*}"}
+{"id": "466.png", "formula": "\\begin{align*} \\lambda \\Lambda _ 1 L _ { g _ 1 } ( z ) = - \\sum _ { n = 1 } ^ \\infty n ^ { - 1 } ( 1 - e ^ { \\lambda \\Lambda _ 1 } ) ^ n L ^ n _ { f _ 1 } ( z ) \\end{align*}"}
+{"id": "3726.png", "formula": "\\begin{align*} \\nabla _ V ( \\nabla X ) = - R ( X , V ) + T _ h ( V ) \\end{align*}"}
+{"id": "1584.png", "formula": "\\begin{align*} A _ 1 : = \\left ( \\begin{matrix} 0 & 0 & 1 \\\\ 1 & 0 & 2 \\\\ 0 & 1 & - 1 \\end{matrix} \\right ) , \\end{align*}"}
+{"id": "308.png", "formula": "\\begin{align*} \\{ x \\in E : i ( x ) < d ( x ) \\} = \\bigcup _ { 1 \\le k \\le n } \\mathfrak { J } _ k \\end{align*}"}
+{"id": "6285.png", "formula": "\\begin{align*} \\langle [ e _ i ] , [ e _ j ] \\rangle = d _ { i j } = 1 + \\frac { \\delta _ { i j } } { \\varphi _ i } \\quad \\forall i , j \\in I , \\end{align*}"}
+{"id": "6156.png", "formula": "\\begin{align*} { \\cal V } ( r ) = - \\frac { Q } { r } \\sqrt { 1 + \\lambda r ^ 2 } , Q > 0 . \\end{align*}"}
+{"id": "5516.png", "formula": "\\begin{align*} \\Phi _ p ( x ) = \\begin{cases} \\phi _ p ( x ) , & x \\geq 1 , \\\\ 2 \\phi _ p ( 1 ) - \\phi _ p ( 2 - x ) , & 0 \\leq x \\leq 1 . \\end{cases} \\end{align*}"}
+{"id": "6045.png", "formula": "\\begin{align*} & J ( u _ { t } , v _ { t } ) = t \\frac { d } { d t } I ( u _ { t } , v _ { t } ) , \\\\ & \\frac { d } { d t } J ( u _ { t } , v _ { t } ) = \\frac { d } { d t } I ( u _ { t } , v _ { t } ) + t \\cdot \\frac { d ^ { 2 } } { d t ^ { 2 } } I ( u _ { t } , v _ { t } ) . \\end{align*}"}
+{"id": "4794.png", "formula": "\\begin{align*} a _ n ( u _ n , v _ n ) = ( f _ n , v _ n ) v _ n \\in X _ n \\end{align*}"}
+{"id": "2945.png", "formula": "\\begin{align*} \\begin{aligned} E _ A & = R _ { X \\backslash A } , \\\\ R _ B & = E _ { X \\backslash B } . \\end{aligned} \\end{align*}"}
+{"id": "1140.png", "formula": "\\begin{align*} S ^ { \\rm P R } _ { i , \\ , j , \\ , k , \\ , \\ell } ( B ) = \\kappa ( \\rho ) W _ { N ( k - 1 ) + i , \\ , N ( \\ell - 1 ) + j } , \\end{align*}"}
+{"id": "5777.png", "formula": "\\begin{align*} \\nabla _ X \\psi _ 1 = \\frac { 1 } { 2 } X _ 1 \\cdot \\psi _ 1 - \\frac { 1 } { 2 } B ( X ) \\cdot \\psi _ 1 \\end{align*}"}
+{"id": "2523.png", "formula": "\\begin{align*} \\Delta { { \\mathbf { x } } ^ { T } } \\Delta \\mathbf { s } = \\left ( 1 - \\nu \\right ) \\left ( \\Delta { { \\mathbf { y } } ^ { T } } \\mathbf { A x } - \\Delta { { \\mathbf { x } } ^ { T } } \\left ( { { \\mathbf { A } } ^ { T } } \\mathbf { y } + \\mathbf { C } \\mathbf { x } + \\mathbf { s } \\right ) \\right ) . \\end{align*}"}
+{"id": "7158.png", "formula": "\\begin{align*} R ^ { l } _ { i j k } = \\frac { \\partial \\Gamma ^ { l } _ { j k } } { \\partial x _ i } - \\frac { \\partial \\Gamma ^ { l } _ { i k } } { \\partial x _ j } + \\Gamma ^ { h } _ { j k } \\Gamma ^ { l } _ { i h } - \\Gamma ^ { h } _ { i k } \\Gamma ^ { l } _ { j h } , \\end{align*}"}
+{"id": "5580.png", "formula": "\\begin{align*} A ( y _ 1 x ) - A ( y _ 1 x ' ) = A ( 1 ^ { 1 + k } 0 . . . ) - A ( 1 0 ^ \\infty ) = c _ { k + 1 } - d \\ , \\end{align*}"}
+{"id": "4091.png", "formula": "\\begin{align*} j ^ 1 _ q ( F _ \\varphi \\circ f _ \\varphi { } ^ { - 1 } ) & = ( q , F _ \\varphi ( o ) , D F _ \\varphi ( o ) ( D f _ \\varphi ( o ) ) ^ { - 1 } ) \\\\ * & = ( q , F _ \\varphi ( o ) , D F _ \\varphi ( o ) ^ i { } _ { j \\alpha } ( ( F _ \\varphi ( o ) ) ^ { - 1 } ) ^ \\alpha { } _ k ) \\end{align*}"}
+{"id": "1307.png", "formula": "\\begin{align*} & g : [ 0 , \\infty ] \\rightarrow \\mathbb { R } , \\\\ & F ( z ) = g ( | z | ) \\ , \\ , \\ , \\ , z \\in \\mathbb { C } ^ n , \\\\ & f ( z ) = \\frac { g ' ( | z | ) z } { | z | } . \\end{align*}"}
+{"id": "1221.png", "formula": "\\begin{align*} D ( \\Omega ) : = \\Big \\{ f \\in C ( \\Omega ) : \\sum _ { x \\in \\Z ^ d } \\delta _ x ( f ) < \\infty \\Big \\} . \\end{align*}"}
+{"id": "188.png", "formula": "\\begin{align*} ( \\widetilde { B } _ { \\infty } ) ^ { \\Gamma _ n } = \\widetilde { B } _ n \\mbox { a n d } ( \\widetilde { B } _ { \\infty } ) ^ { G ' } = B _ { G ' , \\infty } . \\end{align*}"}
+{"id": "1021.png", "formula": "\\begin{align*} c : = ( \\alpha + \\beta , - \\chi _ K ( \\alpha ) - \\chi _ K ( \\beta ) ) \\in R . \\end{align*}"}
+{"id": "1810.png", "formula": "\\begin{align*} A d ( P ) = P \\times _ { A d } \\mathcal { G } \\ , , \\end{align*}"}
+{"id": "3936.png", "formula": "\\begin{align*} F ' ( y ; h ) ( x ) = f _ { x } ' ( y ( x ) ; h ( x ) ) . \\end{align*}"}
+{"id": "417.png", "formula": "\\begin{align*} k = \\dim ( \\Delta ) = \\dim ( \\pi ( \\Delta ) ) + \\dim ( \\ker ( \\pi | _ \\Delta ) ) = m + \\dim ( \\ker ( \\pi | _ \\Delta ) ) . \\end{align*}"}
+{"id": "9269.png", "formula": "\\begin{align*} \\int _ \\sigma ^ r d t \\int _ { \\varrho \\leq t } ( \\Delta \\varrho ) ^ { n - k } \\wedge \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ k = - \\int _ { \\varrho \\leq r } u _ k ( \\Delta \\varrho ) ^ { n - k + 1 } \\wedge \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ { k - 1 } . \\end{align*}"}
+{"id": "9184.png", "formula": "\\begin{align*} \\begin{aligned} | h ( x ( t ) + \\delta u ( t ) ) - h ( z ( t ) + \\delta u ( t ) ) | \\le & \\ , L _ r | x ( t ) - z ( t ) | \\\\ | h ( z ( t ) + \\delta u ( t ) ) - h ( x _ a ( t ) + \\delta u ( t ) ) | \\le & \\ , \\gamma L _ r | \\epsilon ( x _ a ( t ) , t ) | . \\end{aligned} \\end{align*}"}
+{"id": "1843.png", "formula": "\\begin{align*} B _ 0 = \\min \\{ 1 , - \\frac { q + 1 } 2 + \\big ( B + ( \\frac { q + 1 } 2 ) ^ 2 \\big ) ^ { \\frac { 1 } { 2 } } \\} \\ , . \\end{align*}"}
+{"id": "70.png", "formula": "\\begin{align*} \\eta _ \\sigma ( x ) = \\prescript { \\sigma ^ { - 1 } } { } ( v ) ^ { - 1 } w v \\in W . \\end{align*}"}
+{"id": "8531.png", "formula": "\\begin{align*} \\begin{aligned} & M _ { \\delta , + , 1 } ( x ; z ) = e _ 1 + \\mathcal { P } ^ + \\left ( r _ { \\delta , - , 2 } e ^ { 2 i z x } M _ { \\delta , - , 2 } ( x ; \\cdot ) \\right ) , \\\\ & M _ { \\delta , - , 2 } ( x ; z ) = e _ 2 + \\mathcal { P } ^ { - } \\left ( \\bar { r } _ { \\delta , - , 1 } e ^ { - 2 i z x } M _ { \\delta , + , 1 } ( x ; \\cdot ) \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "6106.png", "formula": "\\begin{align*} 1 - 3 Z ^ { - + } _ { n - 1 , p } + 3 Z ^ { - + } _ { n - 1 , p } Z ^ { - + } _ { n , p } - Z ^ { - + } _ { n - 1 , p } Z ^ { - + } _ { n , p } Z ^ { - + } _ { n + 1 , p } = 0 \\ , , \\quad Z ^ { - + } _ { n , p } = \\frac { \\psi ^ { - 0 } _ { n , p } \\rho ^ { 0 + } _ { n , p } } { \\psi ^ { - 0 } _ { n , p - 1 } \\rho ^ { 0 + } _ { n - 1 , p } } \\ . \\end{align*}"}
+{"id": "3357.png", "formula": "\\begin{align*} \\bigcup \\limits _ { \\mu } S u p p ( s _ \\mu ) = \\bigcup \\limits _ { i = 0 } ^ l S u p p ( s _ { \\lambda ^ i } ) . \\end{align*}"}
+{"id": "5368.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ n u ^ 2 _ { 1 l } = 4 B ^ 2 \\sum _ l x ^ 2 _ l = 4 B ^ 2 | x | ^ 2 \\geq 4 B ^ 2 \\delta _ 0 . \\end{align*}"}
+{"id": "1208.png", "formula": "\\begin{align*} h ( \\nu | \\mu ) : = \\sum _ { x \\in E } \\nu ( x ) \\log \\left ( \\frac { \\nu ( x ) } { \\mu ( x ) } \\right ) , \\end{align*}"}
+{"id": "1829.png", "formula": "\\begin{align*} \\delta ^ \\nabla \\big ( \\exp ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) R ^ \\nabla \\big ) = 0 \\ , . \\end{align*}"}
+{"id": "4396.png", "formula": "\\begin{align*} J ( x , \\delta ) = \\int _ 1 ^ { x } | \\theta ( y + \\delta y ) - \\theta ( y ) - \\delta y | ^ 2 d y \\end{align*}"}
+{"id": "4328.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t _ 1 \\} } | \\tilde { F } _ { t _ 1 } - \\mathbb { I } _ { \\{ \\Psi < - t _ 0 \\} } F _ { t _ 0 } | ^ 2 _ h c ( - \\Psi ) \\\\ = & \\int _ { \\{ \\Psi < - t _ 1 \\} } | \\tilde { F } _ { t _ 1 } - \\mathbb { I } _ { \\{ \\Psi < - t _ 0 \\} } F _ { t _ 0 } | ^ 2 _ h e ^ { - \\Psi + v _ { t _ 0 } ( \\Psi ) } c ( - v _ { t _ 0 } ( \\Psi ) ) \\\\ \\end{align*}"}
+{"id": "4532.png", "formula": "\\begin{align*} \\begin{cases} v _ i ( x , t ) = S _ { d , i } ( t , t _ { 1 , e x } ( x , t ) ) S _ { c , i } ( t _ { i , e x } ( x , t ) , t _ { 1 , e n } ( x , t ) ) S _ { d , i } ( t _ { i , e n } ( x , t ) , 0 ) v _ { i , 0 } ( x ) \\forall \\ : i \\in [ 1 , p ] , \\\\ v _ i ( x , t ) = S _ { d , i } ( t , 0 ) v _ { i , 0 } ( x ) \\forall \\ : i \\in [ p + 1 , n ] . \\end{cases} \\end{align*}"}
+{"id": "4505.png", "formula": "\\begin{align*} \\sum _ { j = n _ r } ^ { n _ { r + 1 } } \\abs { L _ j } & \\ge \\sum _ { j = n _ r } ^ { n _ { r + 1 } } ( \\abs { L } - w ( Q _ j ^ r ) ) \\\\ & > ( n _ { r + 1 } - n _ r + 1 ) \\abs { L } - ( n _ { r + 1 } - n _ r ) k = ( n _ { r + 1 } - n _ r ) ( \\abs { L } - k ) + \\abs { L } \\ge \\abs { L } . \\end{align*}"}
+{"id": "7854.png", "formula": "\\begin{align*} \\hat \\Phi ^ \\dagger _ a ( 0 , 0 ) & = \\langle z _ 1 ^ * , \\Phi _ v ( 0 , p _ 0 ) u _ \\ell \\rangle = 0 . \\end{align*}"}
+{"id": "6030.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\omega } : = \\Bigg \\{ u \\in H _ { r } ^ { 1 } ( \\R ^ { 2 } ) \\setminus \\{ 0 \\} : \\int _ { \\R ^ { 2 } } ( \\alpha | \\nabla u | ^ { 2 } + ( \\alpha - 1 ) \\omega u ^ { 2 } ) d x + ( 3 \\alpha - 2 ) B ( u ) = \\frac { p \\alpha - 1 } { p } \\int _ { \\R ^ { 2 } } u ^ { 2 p } d x \\Bigg \\} . \\end{align*}"}
+{"id": "3034.png", "formula": "\\begin{align*} \\ell ^ { \\prime } \\left ( r \\right ) = \\frac { c } { 3 } \\frac { 1 - \\cos r } { \\left ( r - \\sin r \\right ) ^ { 2 / 3 } } \\end{align*}"}
+{"id": "4190.png", "formula": "\\begin{align*} I _ { \\varepsilon } ( u ) = \\frac { 1 } { 2 } \\int _ { \\Omega } ( K _ H ( x ) \\nabla u | \\nabla u ) d x - \\frac { 1 } { ( p + 1 ) \\varepsilon ^ 2 } \\int _ { \\Omega } \\left ( u - q \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ { p + 1 } _ + d x , \\ \\ \\forall u \\in \\mathcal { H } ( \\Omega ) . \\end{align*}"}
+{"id": "3289.png", "formula": "\\begin{align*} ( f _ 1 ( x ) , f _ 2 ( x ) ) = ( B _ 1 ( x ) ( \\omega ) , B _ 2 ( x ) ( \\omega ) ) , x \\in [ 0 , 2 ] , \\end{align*}"}
+{"id": "2442.png", "formula": "\\begin{align*} L _ { q } [ a _ { 1 } f _ { 1 } ( t ) + a _ { 2 } f _ { 2 } ( t ) ] = a _ { 1 } L _ { q } [ f _ { 1 } ( t ) ] + a _ { 2 } L _ { q } [ f _ { 2 } ( t ) ] . \\end{align*}"}
+{"id": "6681.png", "formula": "\\begin{align*} B _ { x , \\eta x } ( \\xi ) = 0 . \\end{align*}"}
+{"id": "1411.png", "formula": "\\begin{align*} \\widetilde G _ n ( x , y ) & = \\left ( \\frac { 1 } { 2 \\pi } \\right ) ^ { d / 2 } \\int _ { \\mathbb { R } ^ d } \\mbox { d e t } \\left ( e ^ { - i \\theta _ i ( y _ { j } - x _ { i } ) } \\right ) _ { i , j = 1 } ^ d \\prod _ { i = 1 } ^ d ( \\varphi ( \\theta _ i ) ) ^ n d \\theta _ i . \\end{align*}"}
+{"id": "9213.png", "formula": "\\begin{align*} \\begin{aligned} \\Bigg | & \\left . \\dfrac { \\partial \\epsilon _ 2 ( \\eta , \\tau ) } { \\partial \\eta } \\right | _ { \\eta = \\eta _ a ( \\tau ) } \\dot { \\eta } _ a ( \\tau ) \\Bigg | \\le \\\\ & \\gamma \\Bigg | \\left . \\dfrac { \\partial \\epsilon _ 2 ( \\eta , \\tau ) } { \\partial x _ a } \\right | _ { \\eta = \\eta _ a ( \\tau ) } \\Bigg | \\dfrac { | b _ { 1 , \\delta } ( x _ a ) | } { 2 } \\le \\gamma 2 L _ r ^ 2 \\delta ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "743.png", "formula": "\\begin{align*} \\mu ( B _ p ( \\bar x , r ) ) \\leq C r ^ { N + 2 s - 1 } \\quad \\mbox { f o r a l l $ \\bar x \\in \\R ^ { N + 1 } , \\ , r > 0 $ . } \\end{align*}"}
+{"id": "8274.png", "formula": "\\begin{align*} \\d | X _ t ^ i | ^ 2 = 2 X _ t ^ i \\cdot ( b ^ i ( X _ t ^ i ) + \\sigma \\d W _ t ^ i ) + d \\sigma ^ 2 \\d t . \\end{align*}"}
+{"id": "3168.png", "formula": "\\begin{align*} k _ 4 ( P ^ \\ast ( q ) ) & = \\frac { q } { 4 } \\times 0 \\\\ & = \\frac { q } { 4 } \\times k _ 3 ( \\langle H \\rangle ) . \\end{align*}"}
+{"id": "6702.png", "formula": "\\begin{align*} f ( s \\xi , s \\eta ) - f ( \\xi , \\eta ) = \\hat \\beta ( s , \\xi ) + \\hat \\beta ( s , \\eta ) . \\end{align*}"}
+{"id": "1772.png", "formula": "\\begin{align*} Y ^ { t , \\Upsilon ; v } ( s ) = \\Phi ( X _ T ^ { t , \\Upsilon ; v } ) + \\int _ { s } ^ { T } g ( r , X _ r ^ { t , \\Upsilon ; v } , Y ^ { t , \\Upsilon ; v } ( r ) , Z ^ { t , \\Upsilon ; v } ( r ) , v ( r ) ) d r - \\int _ { s } ^ { T } Z ^ { t , \\Upsilon ; v } ( r ) d W ( r ) . \\end{align*}"}
+{"id": "1322.png", "formula": "\\begin{align*} & M '' ( t ) M ( t ) - \\frac { \\omega + 3 } { 4 } ( M ' ( t ) ) ^ 2 \\geq M ( t ) \\left ( M '' ( t ) - ( \\omega + 3 ) \\left ( | | u _ t | | ^ 2 + b \\int _ { 0 } ^ t | | u _ s ( s ) | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } d s \\right ) \\right ) \\\\ & = M ( t ) \\left ( - ( \\omega + 1 ) | | u _ t | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } - ( \\omega + 3 ) b \\int _ { 0 } ^ t | | u _ s ( s ) | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } d s - 2 I ( u ) \\right ) , \\end{align*}"}
+{"id": "3456.png", "formula": "\\begin{align*} \\begin{cases} | b _ 1 | \\leq C \\max ( e ^ { \\delta _ n q _ n } , | \\ell | , 1 ) e ^ { ( \\tilde { L } - \\beta _ n + 4 \\varepsilon ) q _ n } \\\\ | b _ 3 | \\leq \\max ( e ^ { \\delta _ n q _ n } , | \\ell | , 1 ) c _ { n , \\ell + 1 } e ^ { ( \\tilde { L } - \\beta _ n + 4 \\varepsilon ) q _ n } \\\\ | b _ 4 | \\leq c _ { n , \\ell + 1 } e ^ { ( \\tilde { L } + \\varepsilon ) q _ n } \\leq C \\max ( e ^ { \\delta _ n q _ n } , | \\ell | , 1 ) e ^ { ( \\tilde { L } - \\beta _ n + \\varepsilon ) q _ n } \\end{cases} \\end{align*}"}
+{"id": "2783.png", "formula": "\\begin{align*} \\varphi ^ { V , U } \\overset { \\mathrm { l a w } } { = } \\mathbb { E } [ h ^ { V } | \\sigma ( h _ { z } ^ { V } : z \\in V \\backslash U ) ] . \\end{align*}"}
+{"id": "6001.png", "formula": "\\begin{align*} & \\frac { R } { 2 } \\int _ { \\partial B _ { R } } | \\nabla u | ^ { 2 } d S _ { x } + \\int _ { B _ { R } } u ^ { 2 } d x - \\pi h ^ { 2 } ( R ) u ^ { 2 } ( R ) - \\pi \\bigg ( \\int _ { R } ^ { + \\infty } \\frac { h ( s ) } { s } u ^ { 2 } ( s ) d s \\bigg ) u ^ { 2 } ( R ) R ^ { 2 } \\\\ & \\ \\ \\ \\ \\ + 2 \\int _ { \\R ^ { 2 } } \\frac { h ^ { 2 } ( | x | ) } { | x | ^ { 2 } } u ^ { 2 } d x = \\frac { 1 } { p } \\int _ { B _ { R } } u ^ { 2 p } d x - b \\int _ { B _ { R } } | v | ^ { p } | u | ^ { p - 2 } u x \\cdot \\nabla u d x . \\end{align*}"}
+{"id": "4179.png", "formula": "\\begin{align*} \\begin{pmatrix} v _ 1 \\\\ v _ 2 \\end{pmatrix} = - \\frac { 1 } { k ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 } \\begin{pmatrix} x _ 1 x _ 2 & - k ^ 2 - x _ 1 ^ 2 \\\\ k ^ 2 + x _ 2 ^ 2 & - x _ 1 x _ 2 \\end{pmatrix} \\begin{pmatrix} \\partial _ { x _ 1 } \\varphi \\\\ \\partial _ { x _ 2 } \\varphi \\end{pmatrix} . \\end{align*}"}
+{"id": "1455.png", "formula": "\\begin{align*} J ( s ) = \\lim _ { x \\to s ^ { + } } u ( x ) - \\lim _ { x \\to s ^ { - } } u ( x ) \\forall s \\in S _ { u } . \\end{align*}"}
+{"id": "9278.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { U } ^ * ( K , \\Omega ) = \\left \\{ u \\in Q S H _ m ( \\Omega ) \\cap C ( \\Omega ) , u \\vert _ { K } \\leq - 1 , \\varliminf _ { q \\to \\partial \\Omega } u ( q ) \\geq 0 \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "5615.png", "formula": "\\begin{align*} 1 - e ^ { c _ 2 } = \\frac { 1 - e ^ { a _ { \\alpha + n + 1 } + a _ { \\alpha + n + 2 } } } { \\left ( e ^ { - a _ { \\alpha + 1 } } - 1 \\right ) \\left ( 1 - e ^ { a _ { n + 1 } } \\right ) \\exp \\left [ \\sum _ { j = 2 } ^ n ( a _ j - a _ { \\alpha + j } ) \\right ] } \\ , \\ \\forall n \\in \\N \\ . \\end{align*}"}
+{"id": "48.png", "formula": "\\begin{align*} I _ 2 = 0 . \\end{align*}"}
+{"id": "4204.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { 1 } { \\ln \\frac { 1 } { \\varepsilon } } \\int _ { \\Omega } ( K _ H ( x ) \\nabla v ^ \\tau _ \\varepsilon | \\nabla v ^ \\tau _ \\varepsilon ) d x = 2 \\pi q ^ 2 \\sqrt { d e t ( K _ { H } ) } ( \\bar { x } ) \\end{align*}"}
+{"id": "339.png", "formula": "\\begin{align*} \\log \\binom { C n } { 2 L } & < C n \\log C n - \\epsilon n \\log \\epsilon n - ( C - \\epsilon ) n \\log ( C - \\epsilon ) n + O ( \\log n ) \\\\ & = ( C \\log C - \\epsilon \\log \\epsilon - ( C - \\epsilon ) \\log ( C - \\epsilon ) ) n + O ( \\log n ) . \\end{align*}"}
+{"id": "8665.png", "formula": "\\begin{align*} { \\mathbb E } \\left [ { { { \\left \\| { { \\bf { \\bar q } } \\left [ i \\right ] } \\right \\| } ^ 2 } } \\right ] = \\frac { 1 } { K } \\sum \\limits _ { k = 0 } ^ { K - 1 } { { { \\Big \\| { \\sum \\limits _ { { l } = 1 } ^ { L ' } { { { \\bf { \\bar H } } _ { l } ^ \\bot { \\bf { \\bar X } } _ { l } } { { \\bf { u } } _ k } { e ^ { - j \\frac { { 2 \\pi } } { { K } } k { \\kappa _ { l } } } } } } \\Big \\| ^ 2 } } } . \\end{align*}"}
+{"id": "7329.png", "formula": "\\begin{align*} f ( x \\vee y \\vee z ) + f ( x ) - f ( x \\vee y ) - f ( x \\vee z ) = f ( a \\wedge b ) + f ( a \\vee b ) - f ( a ) - f ( b ) \\ge 0 . \\end{align*}"}
+{"id": "3541.png", "formula": "\\begin{align*} \\lambda ' _ 1 & = \\frac { 1 } { 3 } ( 4 , 5 , 6 , 3 , 4 , 2 ) , & \\lambda ' _ 2 & = \\frac { 1 } { 3 } ( 5 , 1 0 , 1 2 , 6 , 8 , 4 ) , & \\lambda ' _ 3 & = ( 2 , 4 , 6 , 3 , 4 , 2 ) \\\\ \\lambda ' _ 4 & = ( 1 , 2 , 3 , 2 , 2 , 1 ) , & \\lambda ' _ 5 & = \\frac { 1 } { 3 } ( 4 , 8 , 1 2 , 6 , 1 0 , 5 ) , & \\lambda ' _ 6 & = \\frac { 1 } { 3 } ( 2 , 4 , 6 , 3 , 5 , 4 ) . \\end{align*}"}
+{"id": "4814.png", "formula": "\\begin{align*} F _ n ( \\eta ) = T _ n - \\frac { 1 } { \\eta } I _ n . \\end{align*}"}
+{"id": "6813.png", "formula": "\\begin{align*} P _ { n } \\left ( I + \\mathbf { H } ( \\varphi ) \\right ) P _ { n } \\cdot P _ { n } A ^ { - 1 } P _ { n } & = P _ { n } \\left ( I + \\mathbf { H } ( \\varphi ) \\right ) A ^ { - 1 } P _ { n } - P _ { n } \\left ( I + \\mathbf { H } ( \\varphi ) \\right ) Q _ { n } A ^ { - 1 } P _ { n } \\\\ & = P _ { n } - P _ { n } \\mathbf { H } ( \\varphi ) Q _ { n } A ^ { - 1 } P _ { n } . \\end{align*}"}
+{"id": "6874.png", "formula": "\\begin{align*} w = ( \\Psi _ 1 ^ { - 1 } \\circ \\Psi _ 2 ^ { - 1 } ) ( \\hat z ) \\end{align*}"}
+{"id": "7981.png", "formula": "\\begin{align*} \\widetilde { E } ( x , t ) = \\log \\lambda _ { m a x } ( \\{ w ^ { i j } \\} ) - d \\log h + \\frac { l } { 2 } \\rho ^ { 2 } , \\end{align*}"}
+{"id": "2709.png", "formula": "\\begin{align*} h ( \\delta ) = C _ 8 \\min \\{ 1 , \\delta ^ { 3 } \\} C _ 8 \\stackrel { \\rm d e f } { = } \\frac { \\eta _ 1 \\eta _ 2 \\kappa _ { \\rm f o d } } { 2 } \\min \\left \\{ \\frac { \\eta _ 2 } { \\kappa _ { \\rm b h m } } , 1 \\right \\} . \\end{align*}"}
+{"id": "1363.png", "formula": "\\begin{align*} \\psi _ * \\mu \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) = \\mu ( \\psi ^ { - 1 } \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) ) = \\mu ( \\pi \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) ) , \\end{align*}"}
+{"id": "385.png", "formula": "\\begin{align*} L _ { o d d } : = A + H + \\sum _ { k = 1 } ^ { \\frac { n - 3 } { 2 } } B _ k . \\end{align*}"}
+{"id": "6610.png", "formula": "\\begin{align*} g ( u \\times v , w ) = \\mathrm { v o l } _ g ( u , v , w ) , \\end{align*}"}
+{"id": "1611.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { V } _ i & \\triangleq \\sum _ { k \\in \\mathcal { K } _ i } | \\tau _ { k } | ^ 2 \\sum _ { p \\in \\mathcal { K } } \\mathbf { v } _ { k , p } \\mathbf { v } _ { k , p } ^ H , \\\\ \\tilde { \\mathbf { v } } _ i & \\triangleq \\sum _ { k \\in \\mathcal { K } _ i } \\sqrt { 1 + \\iota _ { k } } \\mathbf { v } _ { k , k } \\tau _ { k } , \\forall i \\in \\{ \\mathrm { t , r } \\} . \\end{aligned} \\end{align*}"}
+{"id": "1585.png", "formula": "\\begin{align*} A _ 2 : = A _ 1 ^ 2 - 2 \\mathrm { I } _ 3 = \\left ( \\begin{matrix} - 2 & 1 & - 1 \\\\ 0 & 0 & - 1 \\\\ 1 & - 1 & 1 \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "5223.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { j , + } \\frac { G _ j } { r ^ { 2 } } \\dd x = \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } \\frac { G _ 0 } { r ^ { 2 } } \\dd x \\textrm { u n i f o r m l y i n } \\ [ 0 , T ] . \\end{align*}"}
+{"id": "3787.png", "formula": "\\begin{align*} | | A _ q ^ z | | ^ 2 _ { L ^ 2 ( \\mathbb R ) } = E _ q \\left ( | z | ^ 2 \\right ) . \\end{align*}"}
+{"id": "5284.png", "formula": "\\begin{align*} O _ A ( \\tau ^ { - A } ) + \\int _ { \\tau ^ { 1 - O ( \\epsilon ) } } ^ { \\tau ^ { 1 + O ( \\epsilon ) } } d u _ 1 \\int _ { | u _ 1 | ^ { \\frac { 1 } { 2 } } \\tau ^ { - \\epsilon } } ^ { | u _ 1 | ^ { \\frac { 1 } { 2 } } \\tau ^ \\epsilon } \\frac { d t _ 1 } { t _ 1 ^ 3 } y ^ { - 2 } \\tau ^ { - \\frac { 3 } { 2 } + O ( \\epsilon ) } = O _ A ( \\tau ^ { - A } ) + O \\left ( y ^ { - 2 } \\tau ^ { - \\frac { 3 } { 2 } + O ( \\epsilon ) } \\right ) . \\end{align*}"}
+{"id": "1392.png", "formula": "\\begin{align*} h _ 2 ( x ) = \\prod _ { j = 1 } ^ { d - 1 } j ! \\ : \\det ( ( x _ j ) ^ { i + j - 1 - d } L _ { d - j } ^ { ( j + i - 1 - d ) } ( x _ j ) ) _ { i , j = 1 } ^ d . \\end{align*}"}
+{"id": "4503.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\int _ { V } \\Re ( \\alpha _ { t } + \\sqrt { - 1 } \\omega ) ^ { p } - \\cot \\Theta _ { 0 } \\cdot \\Im ( \\alpha _ { t } + \\sqrt { - 1 } \\omega ) ^ { p } \\\\ = { } & p \\int _ { V } \\left ( \\Re ( \\alpha _ { t } + \\sqrt { - 1 } \\omega ) ^ { p - 1 } - \\cot \\Theta _ { 0 } \\cdot \\Im ( \\alpha _ { t } + \\sqrt { - 1 } \\omega ) ^ { p - 1 } \\right ) \\wedge \\omega > 0 . \\end{align*}"}
+{"id": "9249.png", "formula": "\\begin{align*} \\mathcal { H } _ m ( u ) : = \\mathcal { H } _ m \\left ( \\frac { \\partial ^ 2 u } { \\partial \\overline { q _ l } \\partial { q _ k } } \\right ) . \\end{align*}"}
+{"id": "5315.png", "formula": "\\begin{align*} ( 2 \\pi ) ^ { - n } | \\lambda | ^ n W _ \\lambda ( g \\ast _ \\lambda \\varphi _ { k , \\lambda } ^ { n - 1 } ) = W _ \\lambda ( g ) P _ k ( \\lambda ) . \\ , \\ , \\end{align*}"}
+{"id": "2914.png", "formula": "\\begin{align*} \\sum _ { x = 0 } ^ n | M _ { x , 0 } ( m ) | ^ 2 = \\sum _ { j _ 1 , \\ldots , j _ 4 = 0 } ^ n \\Theta _ m ( \\mu _ { j _ 1 } , \\mu _ { j _ 2 } ) \\Theta _ m ( \\mu _ { j _ 3 } , \\mu _ { j _ 4 } ) \\prod _ { k = 1 } ^ 4 \\psi _ { j _ k } ( 0 ) \\sum _ { x = 0 } ^ n \\prod _ { k = 1 } ^ 4 \\psi _ { j _ k } ( x ) . \\end{align*}"}
+{"id": "1605.png", "formula": "\\begin{align*} \\mathbf { \\Phi } _ { g } ^ \\star = \\arg \\min _ { \\mathbf { \\Phi } _ { g } \\in \\mathcal { M } _ g } \\tilde { f } _ g ( \\mathbf { \\Phi } _ { g } ) , \\forall g \\in \\mathcal { G } . \\end{align*}"}
+{"id": "7979.png", "formula": "\\begin{align*} G _ { \\iota \\iota } = ( ( \\nabla ^ { 2 } G \\cdot e _ { \\iota } ) \\cdot e _ { \\iota } ) w ^ { 2 } _ { \\iota \\iota } - ( \\nabla G \\cdot x ) w _ { \\iota \\iota } + ( \\nabla G \\cdot e _ { i } ) w _ { i \\iota \\iota } . \\end{align*}"}
+{"id": "1522.png", "formula": "\\begin{gather*} F _ { \\textrm { B } i } ( x ) = \\bar { F } _ i ( x ) \\tilde { H } ( x ) + F _ { \\textrm { B R } i } ( x ) , \\deg ( F _ { \\textrm { B R } i } ) < d , i = 1 , \\dots , n , \\\\ F _ { \\textrm { S } i } ( x ) = \\bar { F } _ i ( x ) \\tilde { H } ( x ) + F _ { \\textrm { S R } i } ( x ) , \\deg ( F _ { \\textrm { S R } i } ) < d , i = 1 , \\dots , n . \\end{gather*}"}
+{"id": "2630.png", "formula": "\\begin{align*} \\begin{array} { l } \\circlearrowleft _ { x , y , z } \\varepsilon ( z , x ) \\Big ( \\varepsilon ( h , x + y ) [ [ \\alpha ( x ) , \\alpha ( y ) ] , \\alpha ( h \\cdot z ) ] \\Big ) \\\\ + \\circlearrowleft _ { x , y , z } \\varepsilon ( z , x ) \\Big ( 2 [ \\alpha ( h ) \\cdot [ x , y ] , \\alpha ^ 2 ( x ) ] \\Big ) = 0 . \\end{array} \\end{align*}"}
+{"id": "2776.png", "formula": "\\begin{align*} \\eta _ { N } ^ { D } : = \\frac { 1 } { K _ { N } } \\sum _ { x \\in D _ { N } } \\delta _ { x / N } \\otimes \\delta _ { h _ { x } ^ { D _ { N } } - a _ { N } } , \\end{align*}"}
+{"id": "641.png", "formula": "\\begin{align*} \\begin{aligned} \\omega ( x , y , z ) & = - 2 x \\ d H ^ * \\wedge d Y ^ * + 2 y \\ d X ^ * \\wedge d Y ^ * - 2 z \\ d X ^ * \\wedge d H ^ * \\\\ & = - 2 ( x \\ d H ^ * \\wedge d Y ^ * - y \\ d X ^ * \\wedge d Y ^ * + z \\ d X ^ * \\wedge d H ^ * ) . \\end{aligned} \\end{align*}"}
+{"id": "5462.png", "formula": "\\begin{align*} d = k ^ 2 + 2 n k = 4 m k + 2 k - k ^ 2 \\equiv 2 k - k ^ 2 \\mod 4 k \\end{align*}"}
+{"id": "675.png", "formula": "\\begin{align*} I _ { n , l } = \\int _ 0 ^ \\pi d \\theta \\sin \\theta \\cos ( n \\theta ) P _ l ( \\cos \\theta ) = \\end{align*}"}
+{"id": "6426.png", "formula": "\\begin{align*} \\log \\det u _ { , i j } = - v _ p x ^ p + u _ { , q } \\xi ^ q + c . \\end{align*}"}
+{"id": "7290.png", "formula": "\\begin{align*} G : = \\{ ( x , y ) : \\ \\ x \\in [ - b , b ] ^ { d } , \\ \\ g ( x ) - 1 \\leq y \\leq g ( x ) \\} , \\end{align*}"}
+{"id": "6172.png", "formula": "\\begin{align*} V _ 1 ( r ; l , Q ) = \\frac { L ( L + 1 ) } { r ^ 2 } - \\frac { Q } { r } f ( r ) - E _ 0 \\end{align*}"}
+{"id": "7352.png", "formula": "\\begin{gather*} P ( c ) = \\inf _ { Q \\in \\Gamma } Q ( c ) ; \\end{gather*}"}
+{"id": "5389.png", "formula": "\\begin{align*} \\Omega _ \\delta : = \\{ x \\in \\Omega : | x - x _ 0 | < \\delta \\} \\end{align*}"}
+{"id": "376.png", "formula": "\\begin{align*} U ^ { \\dagger } = A - \\sum _ { k = 1 } ^ { n - 2 } { ( \\frac { 2 } { \\theta _ { k } } ) } \\frac { { q _ { k } } { q _ { k } } ^ { \\ast } } { { { \\langle { q _ { k } } } , { q _ { k } } } \\rangle } . \\end{align*}"}
+{"id": "4290.png", "formula": "\\begin{align*} \\mathcal { L } ( u ^ h ( t , x ) ) = \\mathcal { L } \\left ( \\sum _ i w _ i ( t ) \\sigma _ i ( x ) \\right ) = \\sum _ i w _ i ( t ) \\mathcal { L } ( \\sigma _ i ( x ) ) . \\end{align*}"}
+{"id": "5182.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } | \\tilde { b } | ^ { 2 } \\dd x & = \\left ( \\frac { W } { 2 } \\right ) ^ { - 2 } \\int _ { \\mathbb { R } ^ { 3 } } | b | ^ { 2 } \\dd x , \\\\ \\int _ { \\mathbb { R } ^ { 3 } } \\left ( \\tilde { \\phi } - r ^ { 2 } - \\tilde { \\gamma } \\right ) _ + \\frac { \\tilde { G } } { r ^ { 2 } } \\dd x & = \\left ( \\frac { W } { 2 } \\right ) ^ { - 2 } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( \\phi - \\frac { W } { 2 } r ^ { 2 } - \\gamma \\right ) _ + \\frac { G } { r ^ { 2 } } \\dd x . \\end{align*}"}
+{"id": "5582.png", "formula": "\\begin{align*} W ( 1 ^ \\infty | [ 1 ^ k 0 ] ) = c _ { k + 1 } - d + \\sum _ { n = 2 } ^ \\infty ( c _ { n + k } - c _ n ) \\ . \\end{align*}"}
+{"id": "2563.png", "formula": "\\begin{align*} \\rho ^ { - n _ { 2 } } \\frac { a _ { n _ { 2 } } } { N _ { n _ { 2 } } } - \\rho ^ { - n _ { 1 } } \\frac { a _ { n _ { 1 } } } { N _ { n _ { 1 } } } = \\rho ^ { - n _ { s } } \\frac { a _ { n _ { s } } } { N _ { n _ { s } } } . \\end{align*}"}
+{"id": "9074.png", "formula": "\\begin{align*} ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + r ( j - 1 ) \\leq \\chi _ j \\leq ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + r j , \\end{align*}"}
+{"id": "322.png", "formula": "\\begin{align*} q _ { t , \\gamma } ( x , y ) = q _ t ( x , y ) ( 1 - \\varphi ( \\frac { \\sqrt { t } } { \\gamma } ) ) , \\end{align*}"}
+{"id": "6519.png", "formula": "\\begin{align*} & s _ { B S } ( d ( \\lambda ) ) = \\star = d ^ 0 ( s _ { \\Lambda } ) \\\\ & r _ { B S } ( d ( \\lambda ) ) = \\star = d ^ 0 ( r _ { \\Lambda } ) \\end{align*}"}
+{"id": "132.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } \\partial _ t v - D ^ { \\alpha } _ x \\partial _ { x } v - \\partial _ { x } ^ { - 1 } \\partial _ y ^ 2 v & = \\partial _ x ( v ( u _ 1 + u _ 2 ) ) / 2 , ( t , x , y ) \\in \\R \\times \\R \\times \\R , \\\\ v ( 0 ) & = \\phi _ 1 - \\phi _ 2 = : \\phi . \\end{array} \\right . \\end{align*}"}
+{"id": "7923.png", "formula": "\\begin{align*} c _ 2 ( q , 1 , p _ 0 ) = c _ 3 ( q , 2 , 1 , p _ 0 ) = - \\hat W _ { r _ 0 } ( q - 2 ) + 2 \\hat W _ { r _ 0 } ( q - 1 ) - 2 \\hat W _ { r _ 0 } ( q + 1 ) + \\hat W _ { r _ 0 } ( q + 2 ) = : q _ 2 . \\end{align*}"}
+{"id": "4770.png", "formula": "\\begin{align*} F ( \\lambda ) x _ j + \\frac { 1 } { 1 ! } F ^ { ( 1 ) } ( \\lambda ) x _ { j - 1 } + \\ldots + \\frac { 1 } { j ! } F ^ { ( j ) } ( \\lambda ) x _ 0 = 0 , j = 0 , 1 , \\ldots , k , \\end{align*}"}
+{"id": "6812.png", "formula": "\\begin{align*} \\left ( \\left ( P _ { n } + P _ { n } \\mathbf { H } ( \\varphi ) \\right ) P _ { n } y \\right ) \\left ( \\rho \\right ) = - P _ { n } \\mathbf { H } ( \\varphi ) \\left ( 1 \\right ) \\left ( \\rho \\right ) . \\end{align*}"}
+{"id": "2594.png", "formula": "\\begin{align*} Q ^ c ( g , R ) ( v , w , w , v ; x , y ) = & Q ( g , R ) ( v , w , w , v ; x , y ) + Q ( g , R ) ( v , w , w , v ; J x , J y ) \\\\ & - 4 g ( J x , y ) R ( J v , w , w , v ) - 4 g ( J x , y ) R ( v , J w , w , v ) \\\\ = & Q ( g , R ) ( v , w , w , v ; x , y ) + Q ( g , R ) ( v , w , w , v ; J x , J y ) . \\end{align*}"}
+{"id": "4883.png", "formula": "\\begin{align*} D & : = P + Q \\intertext { a n d } \\overline { D } & : = \\overline { P } + \\overline { Q } \\end{align*}"}
+{"id": "841.png", "formula": "\\begin{align*} g \\big ( u _ 0 ( z _ l ) \\big ) = g \\big ( u _ 0 ( z _ r ) \\big ) + K . \\end{align*}"}
+{"id": "1034.png", "formula": "\\begin{align*} x ( b ) = & ( \\alpha , \\Phi ^ + ( - w ^ { - 1 } \\alpha ) - \\langle \\mu , w ^ { - 1 } \\alpha \\rangle ) \\\\ = & ( \\alpha , \\Phi ^ + ( - \\alpha ) + \\ell ( x , - w ^ { - 1 } \\alpha ) \\rangle ) = ( \\alpha , \\Phi ^ + ( - \\alpha ) ) = a . \\end{align*}"}
+{"id": "525.png", "formula": "\\begin{align*} \\| g _ { * , k } ^ { c } ( f ) \\| ^ { 2 } _ { p } & = \\tau \\int _ { \\R ^ { d } } g _ { * } ^ { c } ( f ) ( x ) ^ { 2 } h ( x ) d x \\\\ & = \\tau \\int _ { \\R ^ { d } } \\int _ { \\R ^ { d } } \\int _ { 0 } ^ { \\infty } t ^ { \\frac { 2 - d } { 2 } } ( 1 + t ^ { - 1 } | x - y | ^ { 2 } ) ^ { - k } | \\partial _ { t } H ^ { t } f ( y ) | ^ { 2 } h ( x ) d t d y d x \\\\ & = \\tau \\int _ { \\R ^ { d } } \\int _ { 0 } ^ { \\infty } t | \\partial _ { t } H ^ { t } f ( y ) | ^ { 2 } \\int _ { \\R ^ { d } } t ^ { - \\frac { d } { 2 } } ( 1 + t ^ { - 1 } | x - y | ^ { 2 } ) ^ { - k } h ( x ) d x d t d y . \\end{align*}"}
+{"id": "9102.png", "formula": "\\begin{align*} \\chi _ j \\leq \\chi ( \\sum \\limits _ { i = 1 } ^ j w _ i ) - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + j r . \\end{align*}"}
+{"id": "3503.png", "formula": "\\begin{align*} | W _ i \\cap C ^ { ( i ) } | = \\prod _ { l = 1 } ^ { i } \\frac { \\beta _ { j + l - 1 } } { c _ { l } } . \\end{align*}"}
+{"id": "1505.png", "formula": "\\begin{align*} D _ { a + } ^ { \\nu _ { 2 } } ( t - a ) ^ { \\nu _ { 1 } } = \\frac { \\Gamma ( \\nu _ { 1 } + 1 ) } { \\Gamma ( \\nu _ { 1 } + 1 - \\nu _ { 2 } ) } ( t - a ) ^ { \\nu _ { 1 } - \\nu _ { 2 } } , \\nu _ { 1 } > - 1 , \\nu _ { 2 } \\ge 0 , \\end{align*}"}
+{"id": "8202.png", "formula": "\\begin{align*} x _ { 2 i , j } = \\tfrac { 1 } { 2 } S - x _ { 2 i , n + 1 - j } . \\end{align*}"}
+{"id": "5474.png", "formula": "\\begin{align*} c _ 1 ( q ) = \\begin{cases} c _ { 1 , 2 } ( q ) , & 0 < q \\leq q _ 1 ^ * , \\\\ c _ { 1 , \\infty } ( q ) , & q _ 1 ^ * \\leq q < 2 , \\end{cases} \\end{align*}"}
+{"id": "6559.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } | \\nabla v _ { l } | ^ { 2 } + \\int _ { \\partial \\Omega } | v _ { l } | ^ { 2 } + \\int _ { \\Omega } u _ { l } v _ { l } ^ { 2 } & = \\int _ { \\Omega } f v _ { l } + \\int _ { \\partial \\Omega } \\eta v _ { l } \\\\ & \\le C ( \\| f \\| _ { L ^ { p } ( \\Omega ) } \\| v _ { l } \\| _ { H ^ { 1 } ( \\Omega ) } ) + \\int _ { \\partial \\Omega } \\eta v _ { l } . \\end{aligned} \\end{align*}"}
+{"id": "8354.png", "formula": "\\begin{align*} r ( k ) : = \\frac { b ( k ) } { a ( k ) } , k \\in \\mathbb { R } \\cup i \\mathbb { R } , \\end{align*}"}
+{"id": "1318.png", "formula": "\\begin{align*} M ' ( t ) & = 2 { \\rm R e } ( u , u _ t ) + b | | u ( t ) | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } - b | | u _ 0 | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } \\\\ & = 2 { \\rm R e } ( u , u _ t ) + 2 b \\int _ { 0 } ^ t { \\rm R e } ( u ( s ) , u _ s ( s ) ) d s , \\end{align*}"}
+{"id": "4238.png", "formula": "\\begin{align*} \\varphi _ a ( z ) = \\dfrac { a - P _ a ( z ) - s _ a Q _ a ( z ) } { 1 - \\langle z , a \\rangle } , z \\in \\mathbb { B } _ n , \\end{align*}"}
+{"id": "2059.png", "formula": "\\begin{align*} c ^ { k , j } _ { \\ell , p , q } = 0 \\ \\textrm { i f } j > 2 k \\textrm { o r a n y o n e e n t r y i n t h e i n d e x } ( j , \\ell , p , q ) \\textrm { i s } n e g a t i v e . \\end{align*}"}
+{"id": "1326.png", "formula": "\\begin{align*} \\{ f , g \\} = P ( d f \\wedge d g ) , \\end{align*}"}
+{"id": "2587.png", "formula": "\\begin{align*} R ( J X , J Y ) Z = R ( X , Y ) Z \\textrm { a n d } R ( X , Y ) J Z = J ( R ( X , Y ) Z ) , \\end{align*}"}
+{"id": "7899.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Theta ( t , x ) = \\int _ \\S \\int _ \\S W _ r ( z + y - 2 x ) \\sin ( \\Theta ( t , z ) + \\Theta ( t , y ) - 2 \\Theta ( t , x ) ) \\ \\d y \\d z , \\end{align*}"}
+{"id": "9148.png", "formula": "\\begin{align*} \\dot { { x } } _ a & = - \\gamma \\int _ { 0 } ^ { 1 } \\left ( y _ \\delta ( { x _ a } , \\tau ) - \\bar { y } _ a \\right ) \\ , u ( \\tau ) \\ , d \\tau \\\\ \\dot { \\bar { y } } _ a & = - \\gamma \\ , \\bar { y } _ a + \\gamma \\int _ { 0 } ^ { 1 } y _ \\delta ( { x } _ a , \\tau ) \\ , d \\tau . \\end{align*}"}
+{"id": "3695.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } X ( \\gamma ( t ) ) = B ( t ) X ( \\gamma ( t ) ) + D ( t ) \\end{align*}"}
+{"id": "330.png", "formula": "\\begin{align*} \\lim _ { A \\rightarrow \\infty } \\int _ { | x | > A } | \\mathcal { T } ^ * _ { b , \\gamma } f ( x ) | ^ p w ( x ) d x = 0 , \\end{align*}"}
+{"id": "1111.png", "formula": "\\begin{align*} R _ { b , m \\to s } ^ { \\rm { S - N } } = { W _ m } { \\log _ 2 } \\left ( { 1 + \\frac { { { p _ { b , m } } { { \\left | { { h _ s } } \\right | } ^ 2 } } } { { { p _ s } { { \\left | { { h _ s } } \\right | } ^ 2 } + { W _ m } { N _ 0 } } } } \\right ) . \\end{align*}"}
+{"id": "2987.png", "formula": "\\begin{align*} W ^ { s } ( ( x , t ) , T _ { \\vec { v } } ) = W ^ s ( x , T _ { \\vec { v } } ) \\times \\{ t \\} . \\end{align*}"}
+{"id": "2561.png", "formula": "\\begin{align*} \\hat { \\mu } ( \\xi ) = \\hat { \\mu } _ { \\rho , \\{ \\mathcal { D } _ { n } \\} } ( \\xi ) = \\prod _ { n = 1 } ^ { \\infty } M _ { \\mathcal { D } _ { n } } ( \\rho ^ { n } \\xi ) , \\xi \\in \\mathbb { R } , \\end{align*}"}
+{"id": "778.png", "formula": "\\begin{align*} \\limsup _ { t \\nearrow T _ { \\max } } \\| u _ i ( t ) \\| _ { L ^ \\infty ( \\Omega ) } < \\infty ~ i = 1 , 2 , \\cdots , m , \\end{align*}"}
+{"id": "3694.png", "formula": "\\begin{align*} X _ { i ; j k } = - R _ { \\ell k j i } X ^ \\ell + \\tfrac { 1 } { 2 } ( h _ { i j ; k } + h _ { i k ; j } - h _ { j k ; i } ) . \\end{align*}"}
+{"id": "5593.png", "formula": "\\begin{align*} W ( [ 1 ^ l 0 ] | [ 1 ^ k 0 ] ) = c _ { k + 1 } - d + \\sum _ { n = 2 } ^ { l } ( c _ { n + k } - c _ n ) + b _ { l + k } - b _ l \\end{align*}"}
+{"id": "3545.png", "formula": "\\begin{align*} \\sigma ( \\gamma _ 1 ) & = \\gamma _ 3 , & \\sigma ( \\gamma _ 2 ) & = \\gamma _ 2 , & \\sigma ( \\gamma _ 3 ) & = \\gamma _ 4 , & \\sigma ( \\gamma _ 4 ) & = \\gamma _ 1 . \\end{align*}"}
+{"id": "3889.png", "formula": "\\begin{align*} - \\Delta v - p w ^ { p - 1 } _ + v = 0 , \\ \\ v \\in L ^ { \\infty } ( \\mathbb { R } ^ 2 ) . \\end{align*}"}
+{"id": "8196.png", "formula": "\\begin{align*} x _ { 1 , j } & + x _ { 1 , j + 1 } + x _ { m , n + 1 - j } + x _ { m , n - j } \\\\ & = x _ { 1 , j } + x _ { 1 , j + 1 } + ( 2 m n + 2 - x _ { 1 , n + 1 - j } - x _ { 1 , n - j } ) \\\\ & = 2 m n + 2 + ( x _ { 1 , j } - x _ { 1 , n + 1 - j } ) + ( x _ { 1 , j + 1 } - x _ { 1 , n - j } ) = 2 m n + 2 . \\end{align*}"}
+{"id": "569.png", "formula": "\\begin{align*} \\frac { e ^ { p ( \\zeta - z ) } } { \\zeta - z } = - \\int _ { p + i e ^ { - i \\alpha } [ 0 , + \\infty ) } e ^ { \\omega ( \\zeta - z ) } d \\omega , \\zeta \\in e ^ { i \\alpha } [ 0 , + \\infty ) , \\ ; z \\in \\Delta . \\end{align*}"}
+{"id": "6742.png", "formula": "\\begin{align*} G ( x ) = \\begin{cases} \\frac { 1 } { 2 } \\left \\{ 1 - F _ \\Gamma \\left [ \\left ( - \\frac { x - \\mu } { \\sigma } \\right ) ^ s \\right ] \\right \\} , & x \\leq \\mu , \\\\ \\frac { 1 } { 2 } \\left \\{ 1 + F _ \\Gamma \\left [ \\left ( \\frac { x - \\mu } { \\sigma } \\right ) ^ s \\right ] \\right \\} , & x > \\mu , \\end{cases} \\end{align*}"}
+{"id": "608.png", "formula": "\\begin{align*} \\phi _ { p ^ { - 1 } } ( n _ 1 ) = ( \\pi _ H ( p n _ 1 ) ) ^ { - 1 } \\phi ( \\pi _ N ( p n _ 1 ) ) = \\phi ( m ) ^ { - 1 } \\phi \\left ( m C _ { \\phi ( m ) } ( n _ 1 ) \\right ) , \\forall n _ 1 \\in N \\end{align*}"}
+{"id": "3165.png", "formula": "\\begin{align*} \\sum \\limits _ { y \\neq 0 , x } \\chi _ 4 ( y ( x - y ) ) = \\sum \\limits _ { y \\neq 0 , 1 } \\chi _ 4 ( x y ) \\chi _ 4 ( x - x y ) = \\varphi ( x ) J ( \\chi _ 4 , \\chi _ 4 ) . \\end{align*}"}
+{"id": "7638.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\varphi _ t ^ \\xi = & - [ ( A - B ^ 2 R ^ { - 1 } P _ t ) \\varphi _ t ^ \\xi + P _ t f ( \\nu _ t ) + P _ t b ( \\mu _ t ) \\\\ & + Q l ( \\nu _ t ) - P _ t B h ( \\mu _ t ) ] d t + \\Lambda _ t ^ { 0 , \\xi } d W ^ 0 _ t , \\\\ \\varphi _ T ^ \\xi = & ~ G g ( \\nu _ T ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "848.png", "formula": "\\begin{align*} \\partial ^ \\square c _ { | ( t , p ) } = \\partial ^ \\circ c _ { | ( t , p ) } - V ^ { } _ { \\Gamma \\ , | ( t , p ) } \\cdot \\nabla _ \\Gamma c _ { | ( t , p ) } = \\partial ^ \\circ c _ { | ( t , p ) } = \\partial _ t c _ { | ( t , p ) } \\end{align*}"}
+{"id": "4930.png", "formula": "\\begin{align*} [ T , b ] _ 1 ( f , g ) & = T ( b f , g ) - b T ( f , g ) , \\\\ [ T , b ] _ 2 ( f , g ) & = T ( f , b g ) - b T ( f , g ) . \\end{align*}"}
+{"id": "9093.png", "formula": "\\begin{align*} \\sum \\limits _ { i = j + 1 } ^ n \\chi _ i \\leq ( \\sum \\limits _ { i = j + 1 } ^ n w _ i ) \\chi + \\sum \\limits _ { i = j + 1 } ^ { n - 1 } k _ i + ( n - j ) r . \\end{align*}"}
+{"id": "1432.png", "formula": "\\begin{align*} \\P _ { ( x _ 1 , \\ldots , x _ d ) } ( \\rho > n ) = \\sum _ { l = 1 } ^ d ( - 1 ) ^ { l + 1 } \\P _ { ( x _ 1 , \\ldots , x _ { l - 1 } , x _ { l + 1 } , \\ldots , x _ d ) } ( \\rho > n ) . \\end{align*}"}
+{"id": "8593.png", "formula": "\\begin{align*} \\left | \\det \\begin{pmatrix} x _ j & x _ k \\\\ y _ j & y _ k \\end{pmatrix} \\right | \\le | x _ j y _ k | + | x _ k y _ j | . \\end{align*}"}
+{"id": "1137.png", "formula": "\\begin{align*} \\delta \\rho = \\varepsilon \\kappa ( \\rho ) \\ , { y _ { N ( k - 1 ) + i } \\ , x _ { N ( \\ell - 1 ) + j } } + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
+{"id": "3446.png", "formula": "\\begin{align*} | m _ n + \\ell _ n q _ n - \\tilde { m } _ n | = q _ { n + 1 } = q _ n + q _ { n - 1 } , \\end{align*}"}
+{"id": "8524.png", "formula": "\\begin{align*} \\hat { R } _ { \\delta } ( x ; z ) : = \\begin{pmatrix} 0 & \\bar { r } _ { \\delta , 1 } ( z ) e ^ { - 2 i x z } \\\\ r _ { \\delta , 2 } ( z ) e ^ { 2 i x z } & \\bar { r } _ { \\delta , 1 } ( z ) r _ { \\delta , 2 } ( z ) \\end{pmatrix} , \\end{align*}"}
+{"id": "6805.png", "formula": "\\begin{align*} y \\left ( \\rho \\right ) : = y \\left ( \\rho , x \\right ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { a _ { n } ( x ) z ^ { n } \\left ( \\rho \\right ) } { \\frac { 1 } { 2 } - i \\rho } \\end{align*}"}
+{"id": "631.png", "formula": "\\begin{align*} \\begin{aligned} & \\omega _ T = \\frac { d ( x + \\varphi y ) } { x + \\varphi y } \\wedge \\frac { d ( x + \\varphi ' y ) } { x + \\varphi ' y } = \\frac { ( d x + \\varphi d y ) \\wedge ( d x + \\varphi ' d y ) } { x ^ 2 + x y - y ^ 2 } \\\\ & = \\frac { \\varphi ' - \\varphi } { N _ { E / F } ( x + \\varphi y ) } d x \\wedge d y = \\frac { - \\sqrt { 5 } } { N _ { E / F } ( x + \\varphi y ) } d x \\wedge d y . \\end{aligned} \\end{align*}"}
+{"id": "6673.png", "formula": "\\begin{align*} \\begin{aligned} \\psi ( Y ^ { \\epsilon } ( t ) ) & = \\psi ( y ^ { \\epsilon } _ { 0 } ) + \\int _ 0 ^ t \\mathcal { B } ^ { \\epsilon } ( s ) \\tilde { N } ^ { \\epsilon ^ { - 2 } } ( d s ) \\\\ & \\quad + \\int _ 0 ^ t \\langle \\nabla \\psi ( Y ^ { \\epsilon } ( s - ) ) , h ^ { \\epsilon } ( s ) \\rangle d s + \\int _ 0 ^ t \\mathcal { A } ^ { \\epsilon } ( s ) d s , \\end{aligned} \\end{align*}"}
+{"id": "407.png", "formula": "\\begin{align*} \\psi ( n ) : = n ^ { - 1 } \\phi ( n H ) , \\forall n \\in N \\end{align*}"}
+{"id": "7439.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { j = 0 } ^ { k } \\dbinom { k } { j } ( 1 - q ) ^ { k - j } q ^ j \\dbinom { k } { j } \\Big ( \\frac { 1 } { 2 } \\Big ) ^ k \\bigg ) ^ { 1 / p _ k } \\leq \\Big ( \\frac { 1 - q } { 2 } \\Big ) ^ { k / p _ k } + \\Big ( \\frac { q } { 2 } \\Big ) ^ { k / p _ k } \\end{align*}"}
+{"id": "4602.png", "formula": "\\begin{align*} \\widetilde y _ { n + 1 } + \\widetilde y _ n & = e ^ { r ( n + 1 ) ( A + a ) + r A } + e ^ { r n ( A + a ) + r A } \\\\ & = e ^ { r ( n + 1 ) ( A + a ) } ( e ^ { r A } + e ^ { - r a } ) \\\\ & > 2 e ^ { r ( n + 1 ) ( A + a ) } \\\\ & = 2 \\widetilde x _ { n + 1 } . \\end{align*}"}
+{"id": "670.png", "formula": "\\begin{align*} T _ n ( x ) = \\lim \\limits _ { \\gamma \\rightarrow 0 } \\frac { n + 2 \\gamma } { 2 \\gamma } C ^ \\gamma _ n ( x ) \\ , , P _ l ( x ) = C ^ { 1 / 2 } _ l ( x ) \\ , , \\end{align*}"}
+{"id": "9048.png", "formula": "\\begin{align*} \\Delta _ c ^ + = \\{ e _ i - e _ j | 1 \\leq i < j \\leq m \\} \\cup \\{ f _ i \\pm f _ j | 1 \\leq i < j \\leq n \\} \\cup \\{ f _ j | 1 \\leq j \\leq n \\} , \\end{align*}"}
+{"id": "3595.png", "formula": "\\begin{align*} I ( A ^ { 1 , \\dots , \\ell } ) = \\langle p _ { ( u , s ) } - p _ { ( u , r ) } \\mid u \\in [ \\beta ] , s , r \\in \\{ 0 , \\dots , x _ u Y _ u - 1 \\} \\rangle + J . \\end{align*}"}
+{"id": "2226.png", "formula": "\\begin{align*} | \\sum \\limits _ { k = 2 } ^ { N } \\alpha _ { k } ( { \\frac { { \\lambda } _ { k } } { { \\lambda } _ { 1 } } } ) ^ { n } | \\leq | \\sum \\limits _ { k = 2 } ^ { N } \\alpha _ { 2 } ( { \\frac { { \\lambda } _ { 2 } } { { \\lambda } _ { 1 } } } ) ^ { n } | \\end{align*}"}
+{"id": "9158.png", "formula": "\\begin{align*} x _ 1 ( t ) = H ^ { - 1 } ( H ( x _ { 1 0 } ) - \\gamma ( \\cos ( t ) - 1 ) ) \\end{align*}"}
+{"id": "1928.png", "formula": "\\begin{align*} \\eta ( z ) = q ^ \\frac { 1 } { 2 4 } \\prod _ { n = 1 } ^ \\infty ( 1 - q ^ n ) , \\end{align*}"}
+{"id": "2192.png", "formula": "\\begin{align*} \\hat { \\theta } = \\arcsin ( \\frac { \\phi \\lambda } { 2 \\pi d } ) , \\end{align*}"}
+{"id": "7748.png", "formula": "\\begin{align*} \\tilde { F } _ k ( x , y ) = \\begin{pmatrix} \\max _ { i = 1 , \\ldots , 2 ^ k } x _ i \\\\ \\max _ { i = 1 , \\ldots , 2 ^ k } y _ i \\end{pmatrix} . \\end{align*}"}
+{"id": "3852.png", "formula": "\\begin{align*} c = \\oint _ l \\mathbf { v } \\cdot \\mathbf { t } d l = \\iint _ { \\sigma } \\mathbf { w } \\cdot \\mathbf { n } d \\sigma , \\end{align*}"}
+{"id": "5937.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\lim _ { n \\rightarrow \\infty } I I _ 2 = 0 . \\end{align*}"}
+{"id": "2248.png", "formula": "\\begin{align*} a ( A ^ { - 1 } Z B ) = a ( Z ) \\end{align*}"}
+{"id": "1200.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | 2 | ^ { n } \\mu ( 2 ^ { n + 1 } x , 2 ^ { n + 1 } y ) = 0 \\end{align*}"}
+{"id": "4595.png", "formula": "\\begin{align*} J _ n & = [ 2 g ( x _ n ) , 2 g ( y _ n ) ] , \\\\ J _ { n + 1 } & = [ 2 g ( x _ { n + 1 } ) , 2 g ( y _ { n + 1 } ) ] , \\\\ J _ n ' & = [ g ( x _ n ) + g ( x _ { n + 1 } ) , g ( y _ n ) + g ( y _ { n + 1 } ) ] . \\end{align*}"}
+{"id": "1859.png", "formula": "\\begin{align*} \\int _ { B _ \\rho ( x _ 0 ) - B _ 1 ( x _ 0 ) } \\big ( \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) - 1 \\big ) \\ , d v = o ( \\rho ^ { 1 + \\delta } ) \\rho \\rightarrow \\infty \\end{align*}"}
+{"id": "4744.png", "formula": "\\begin{align*} & \\Tilde { x } = \\sum \\limits _ { i = 1 } ^ p \\alpha _ i u _ i + \\lambda x _ v + ( 1 - \\lambda ) x _ w \\in \\mathcal { S } ^ n \\\\ & \\| \\sum \\limits _ { i = 1 } ^ p \\alpha _ i u _ i + \\lambda x _ v + ( 1 - \\lambda ) x _ w \\| ^ 2 = \\lambda \\| x _ v \\| ^ 2 + ( 1 - \\lambda ) \\| x _ w \\| ^ 2 \\end{align*}"}
+{"id": "5054.png", "formula": "\\begin{align*} \\tilde { { \\mathcal { I } } } ( B _ 0 ) = \\inf \\left \\{ \\frac { 1 } { 2 } \\int _ { \\Omega } | B | ^ { 2 } \\dd x \\ \\middle | \\ B \\in \\overline { M } ^ { w } ( \\Omega , B _ 0 ) \\right \\} , \\end{align*}"}
+{"id": "8083.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ M \\sum \\limits _ { \\substack { j = 1 \\\\ j \\neq l } } ^ M \\mathbb { E } \\left [ y _ l ^ * y _ j | \\hat { \\mathbf { H } } \\right ] = & 2 \\sum _ { l = 1 } ^ { M - 1 } \\sum _ { j = i + 1 } ^ { M } \\sum _ { q = 1 } ^ M a _ r ^ 2 \\Re \\left [ \\hat { \\phi } ^ { \\left ( l , q \\right ) ^ * } \\hat { \\phi } ^ { \\left ( j , q \\right ) } \\right ] + \\sum _ { l = 1 } ^ M \\sum \\limits _ { \\substack { j = 1 \\\\ j \\neq l } } ^ M a _ c ^ 2 \\hat { \\phi } ^ { \\left ( l , c \\right ) } \\hat { \\phi } ^ { \\left ( j , c \\right ) } . \\end{align*}"}
+{"id": "5542.png", "formula": "\\begin{align*} ( x \\wedge y ) \\cdot z = ( x \\cdot y ) x \\cdot z = ( x \\cdot y ) \\cdot ( x \\cdot z ) , \\end{align*}"}
+{"id": "5424.png", "formula": "\\begin{align*} \\bar { y } _ n : = y _ n - \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( x ' _ 0 ) y _ \\beta . \\end{align*}"}
+{"id": "4087.png", "formula": "\\begin{align*} \\Theta & = d \\theta ^ 0 + \\theta ^ 1 \\wedge \\theta ^ 0 , \\\\ * \\Omega & = d \\theta ^ 1 + \\theta ^ 1 \\wedge \\theta ^ 1 , \\end{align*}"}
+{"id": "8075.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\mathbf { s } ^ { \\left ( \\right ) ^ H } \\mathbf { y } ' | \\hat { \\mathbf { H } } \\right ] = a _ c \\sum _ { i = 1 } ^ M \\hat { \\phi } ^ { \\left ( i , c \\right ) } + \\sum _ { j = 1 } ^ { M } a _ j \\hat { \\phi } ^ { \\left ( j , j \\right ) } . \\end{align*}"}
+{"id": "5713.png", "formula": "\\begin{align*} F ' ( x , y ) = L _ 1 \\circ F \\circ L _ 2 ( x , y ) . \\end{align*}"}
+{"id": "4065.png", "formula": "\\begin{align*} u _ n = n ^ { 2 H - 1 / 2 } \\sum _ { j \\in [ n ] } a _ { t _ { j - 1 } } I _ 1 ( 1 _ j ) 1 _ j \\end{align*}"}
+{"id": "7323.png", "formula": "\\begin{align*} \\alpha ( A ) & = \\inf \\{ t > 0 : A \\mbox { i s i n c l u d e d i n s o m e b a l l o f r a d i u s $ t $ } \\} , \\\\ \\beta ( A ) & = \\inf \\{ t > 0 : A \\mbox { h a s a d i a m e t e r l e s s t h a n $ t $ } \\} , \\end{align*}"}
+{"id": "4112.png", "formula": "\\begin{align*} H _ { m i n } ( E ) = \\min \\limits _ { 0 \\neq F \\subset E } H _ r ( F ) . \\end{align*}"}
+{"id": "8314.png", "formula": "\\begin{align*} R _ c ^ { s y m } = \\pi / 0 . 8 6 3 \\approx 3 . 6 4 . \\end{align*}"}
+{"id": "6872.png", "formula": "\\begin{align*} \\tilde z _ k = { \\tt t z e t ( ( k - 1 ) n / 4 + 1 ) } , k = 1 , 2 , 3 , 4 , \\end{align*}"}
+{"id": "8688.png", "formula": "\\begin{align*} T = \\bigcup _ { i = 1 } ^ I A ( p _ i , s _ i , 2 s _ i ) \\end{align*}"}
+{"id": "7083.png", "formula": "\\begin{gather*} v _ { 2 } ( x ) = \\eta ^ { 2 } ( x - h ) [ ( u - \\psi ) ( x - h ) - ( u - \\psi ) ( x ) ] . \\end{gather*}"}
+{"id": "4606.png", "formula": "\\begin{align*} f _ k ( x _ k , x _ { k + 1 } ) = x _ { k + 1 } g _ k ( x _ k , x _ { k + 1 } ) + x _ k g _ { k - 1 } ( 0 , x _ k ) . \\ ; \\ ; ( \\ ; 2 \\leq k \\leq n . ) \\end{align*}"}
+{"id": "8716.png", "formula": "\\begin{align*} \\mu _ \\varphi ( \\{ \\chi \\in \\widehat { H } \\ : : \\ : \\chi ^ g = \\chi \\} ) > 0 . \\end{align*}"}
+{"id": "7513.png", "formula": "\\begin{align*} \\int _ { B _ { 1 / 2 } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x \\leq C \\int _ { B _ { 1 } } r ^ { 2 - n } u _ r ^ 2 \\d x + C \\varepsilon ^ 2 \\int _ { B _ { 1 } } r ^ { 3 - n } | \\nabla u | ^ 2 \\d x \\varepsilon \\leq \\varepsilon _ 0 , \\end{align*}"}
+{"id": "2962.png", "formula": "\\begin{align*} t < t ' = F _ E ( t ' ) \\leq F _ E ( t ) , \\end{align*}"}
+{"id": "2744.png", "formula": "\\begin{align*} T = \\bigcup _ { s \\in S } T ( s ) . \\end{align*}"}
+{"id": "1696.png", "formula": "\\begin{align*} \\dot { W } ( x _ t ) = - \\begin{bsmallmatrix} x ( t ) \\\\ x ( t - h ) \\end{bsmallmatrix} ^ \\top \\begin{bsmallmatrix} \\eta _ 0 \\mathcal { H } ( A ) + \\frac { 1 } { 2 } I _ m & \\eta _ 0 A _ d \\\\ \\ast & \\frac { 1 } { 2 } I _ m \\end{bsmallmatrix} \\begin{bsmallmatrix} x ( t ) \\\\ x ( t - h ) \\end{bsmallmatrix} , \\end{align*}"}
+{"id": "2590.png", "formula": "\\begin{align*} T ^ v ( u , J u , J u , u ) = 0 \\textrm { f o r e v e r y } u \\in V . \\end{align*}"}
+{"id": "1964.png", "formula": "\\begin{align*} & = \\frac { 1 } { q ^ { m - 1 } - 1 } \\left ( q ^ { m - 1 } \\zeta _ \\nu ( n - 1 ) \\cdots \\zeta _ \\nu ( n - m + 1 ) - \\zeta _ \\nu ( n ) \\cdots \\zeta _ \\nu ( n - m + 2 ) \\right ) \\\\ & = \\frac { 1 } { q ^ { m - 1 } - 1 } \\zeta _ \\nu ( n - 1 ) \\cdots \\zeta _ \\nu ( n - m + 2 ) \\left ( q ^ { m - 1 } \\zeta _ \\nu ( n - m + 1 ) - \\zeta _ \\nu ( n ) \\right ) , \\end{align*}"}
+{"id": "8546.png", "formula": "\\begin{align*} h _ \\alpha ( t ) : = \\frac { t ^ { \\alpha - 1 } } { \\Gamma ( \\alpha ) } , \\ \\alpha > 0 . \\end{align*}"}
+{"id": "1759.png", "formula": "\\begin{align*} \\int _ { \\Gamma } J _ 0 d \\sigma _ y = \\int _ { \\Gamma ( t , x ) } d \\sigma = \\vert \\Gamma ( t , x ) \\vert \\end{align*}"}
+{"id": "1784.png", "formula": "\\begin{align*} d _ { C y } ( ( z , t , u ) , ( z ' , t ' , u ' ) ) = \\lvert 2 \\langle \\psi ( z , t , u ) , \\psi ( z ' , t ' , u ' ) \\rangle \\rvert ^ { 1 / 2 } . \\end{align*}"}
+{"id": "4041.png", "formula": "\\begin{align*} G _ \\infty = 2 c _ H ^ 2 T ^ { 4 H - 1 } \\int ^ T _ 0 ( V ^ { [ 1 ] } ( X _ { t } ) ) ^ 4 d t , \\end{align*}"}
+{"id": "1604.png", "formula": "\\begin{align*} \\mathcal { M } _ g = \\{ \\mathbf { \\Phi } _ { g } \\in \\mathbb { C } ^ { 2 \\bar { M } \\times \\bar { M } } : \\mathbf { \\Phi } _ { g } ^ H \\mathbf { \\Phi } _ { g } = \\mathbf { I } _ { \\bar { M } } \\} , \\forall g \\in \\mathcal { G } , \\end{align*}"}
+{"id": "911.png", "formula": "\\begin{align*} \\Sigma _ 3 & = \\sum \\limits _ { D \\leq d < 2 D } \\sum \\limits _ { n \\in \\mathcal { N } ( d ) } \\Bigg ( - \\sum \\limits _ { 1 \\leq | m | \\leq M } \\frac { e \\left ( m \\left ( \\frac { \\sqrt { X } - n } { d } \\right ) \\right ) } { 2 \\pi i m } + \\mathcal { O } \\left ( f _ { M } \\left ( \\frac { \\sqrt { X } - n } { d } \\right ) \\right ) \\Bigg ) \\\\ & = \\Sigma _ 4 + \\Sigma _ 5 \\ , , \\end{align*}"}
+{"id": "5562.png", "formula": "\\begin{align*} A ^ * ( y ) : = A ^ * ( y | x ) = A ( \\tau _ y ( x ) ) + W \\circ \\hat { \\sigma } ^ { - 1 } ( y | x ) - W ( y | x ) \\ , \\end{align*}"}
+{"id": "1495.png", "formula": "\\begin{align*} \\phi _ 3 + c \\phi _ 1 + \\frac { \\epsilon } { \\kappa } \\phi _ 4 + \\frac { c } { \\kappa } \\phi _ 2 = c o n s t a n t : = k . \\end{align*}"}
+{"id": "3768.png", "formula": "\\begin{align*} P _ { e , w } ( q ) = \\sum _ { 1 \\leq i \\leq k } q ^ { \\frac { \\ell ( w ) - \\ell ( w _ i ) } { 2 } } . \\end{align*}"}
+{"id": "4521.png", "formula": "\\begin{align*} \\begin{cases} t _ { i , e x } ( x _ 0 , t _ 0 ) = \\max \\left ( 0 , t _ 0 - \\dfrac { x _ 0 - R } { \\lambda _ i } \\right ) \\forall \\ : i \\in [ 1 , p ] , \\\\ t _ { i , e x } ( x _ 0 , t _ 0 ) = \\max \\left ( 0 , t _ 0 - \\dfrac { x _ 0 + R } { \\lambda _ i } \\right ) \\forall \\ : i \\in [ p + 1 , n ] . \\end{cases} \\end{align*}"}
+{"id": "6301.png", "formula": "\\begin{align*} Q _ t = - \\frac { 1 } { 2 } \\alpha _ t \\beta _ t \\| y ^ t - z ^ { t + 1 } \\| ^ 2 - \\frac { 1 } { 2 } \\alpha _ t ( 1 - \\beta _ t ) \\| z ^ t - z ^ { t + 1 } \\| ^ 2 . \\end{align*}"}
+{"id": "5834.png", "formula": "\\begin{align*} c _ s ^ 2 \\partial _ { x 1 } T = - \\frac { 1 } { \\Delta t } { \\varsigma } _ 1 { m } _ 1 ^ { ( 1 ) } , \\end{align*}"}
+{"id": "753.png", "formula": "\\begin{align*} \\int _ { 4 I _ Q } \\| R _ j \\partial _ j h ( \\cdot , t ) \\| _ { L ^ 1 ( 8 Q _ 1 ) } d t & = \\int _ { 4 I _ Q } \\int _ { 8 Q _ 1 } | R _ j \\partial _ j h ( x , t ) | d x d t \\\\ & \\leq \\int _ { 4 I _ Q } \\ell ( Q ) ^ { N / p } \\| R _ j ( \\partial _ j h ) ( \\cdot , t ) \\| _ { L ^ q ( \\R ^ N ) } d t \\\\ & \\lesssim \\ell ( Q ) ^ { N / p } \\int _ { 4 I _ Q } \\| \\partial _ j h ( \\cdot , t ) \\| _ { L ^ q ( \\R ^ N ) } d t . \\end{align*}"}
+{"id": "9081.png", "formula": "\\begin{align*} ( \\sum \\limits _ { i = 1 } ^ { j } w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i - \\sum \\limits _ { i = j + 1 } ^ { n - 1 } k _ i + ( j - 1 ) r \\leq \\chi _ j \\leq ( \\sum \\limits _ { i = 1 } ^ { j } w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + \\sum \\limits _ { i = 1 } ^ { j } k _ i + ( j - 1 ) r , \\end{align*}"}
+{"id": "4459.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } n \\psi ' ( n x ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { - 1 8 n ^ { 2 } x ( n ^ { 2 } - 6 x ^ { 2 } ) } { \\pi ^ { 2 } ( n ^ { 2 } + 4 x ^ { 2 } ) ^ { 7 / 2 } } . \\end{align*}"}
+{"id": "6368.png", "formula": "\\begin{align*} \\operatorname { I } ^ { - 1 } _ { u } ( a , b ) = \\sum _ { i = 1 } ^ { \\infty } q _ i \\ , [ a \\operatorname { B } ( a , b ) u ] ^ { i / a } , \\end{align*}"}
+{"id": "1174.png", "formula": "\\begin{align*} \\sup _ { n } \\mathbb { E } _ { \\mathbb { P } } \\left [ \\exp \\left \\{ \\kappa \\sum _ { i = 1 } ^ { n } \\int _ { T _ { 0 } } ^ { T _ { 0 } + \\delta } \\triangle b _ { t } ^ { i , n , \\infty } \\cdot d W _ { t } ^ { i } - \\frac { \\kappa } { 2 } \\sum _ { i = 1 } ^ n \\int _ { T _ { 0 } } ^ { T _ { 0 } + \\delta } \\left | \\triangle b _ { t } ^ { i , n , \\infty } \\right | ^ { 2 } d t \\right \\} \\right ] \\end{align*}"}
+{"id": "2390.png", "formula": "\\begin{align*} \\sum _ h a _ h v ( \\sigma _ h ) \\sigma _ h = \\sum _ h \\left [ \\sum _ { \\sigma _ h \\subset \\tau _ j } c _ { h , j } b _ j \\frac { v ( \\sigma _ h ) } { v ( \\tau _ j ) } \\right ] \\sigma _ h , \\end{align*}"}
+{"id": "3864.png", "formula": "\\begin{align*} K _ H ( x _ 1 , x _ 2 ) = \\frac { 1 } { k ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 } \\begin{pmatrix} k ^ 2 + x _ 2 ^ 2 & - x _ 1 x _ 2 \\\\ - x _ 1 x _ 2 & k ^ 2 + x _ 1 ^ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "3389.png", "formula": "\\begin{align*} c _ k = \\frac 1 2 \\sum _ { i = 0 } ^ k ( 2 i - k ) ^ 2 = { k + 2 \\choose 3 } , \\end{align*}"}
+{"id": "5884.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } \\Big ( \\sup _ { t \\leq T } \\Vert Y _ n ( t ) \\Vert _ H > \\sqrt { M } \\Big ) \\leq \\lim _ { M \\rightarrow \\infty } \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } ( \\tau _ { n } ^ M < T ) = 0 , \\end{align*}"}
+{"id": "239.png", "formula": "\\begin{align*} R _ { q , \\ell } ^ \\flat ( n ) = \\{ g \\in A \\cap B _ S ( n ) \\mid \\sigma _ g ^ + - \\sigma _ g ^ - = \\ell \\} , \\ell > q ( n ) , \\end{align*}"}
+{"id": "5240.png", "formula": "\\begin{align*} \\inf \\left \\{ | | b _ 0 - \\tilde { b } | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } \\ | \\ \\tilde { b } \\in S _ { h _ C , W , 0 } \\ \\right \\} & = \\inf \\left \\{ | | b _ 0 - \\tilde { b } | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } \\ \\middle | \\ \\tilde { b } = U _ C ( \\cdot + z e _ z ) - B _ { \\infty } , \\ z \\in \\mathbb { R } \\right \\} \\\\ & = \\inf \\left \\{ | | b _ 0 + B _ { \\infty } - U _ C ( \\cdot + z e _ z ) | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } \\ \\middle | \\ z \\in \\mathbb { R } \\right \\} . \\end{align*}"}
+{"id": "6153.png", "formula": "\\begin{align*} X _ { r } ^ i = \\xi _ r \\quad { X } ^ { i } _ t = \\xi _ 0 + \\int _ { 0 } ^ { t } \\mu ( { \\delta } _ i ( s ) , \\mathcal { X } ^ { i } ) \\ , d s + \\int _ { 0 } ^ { t } \\sigma ( { \\delta } _ i ( s ) , \\mathcal { X } ^ { i } ) \\ , d W _ s . \\end{align*}"}
+{"id": "38.png", "formula": "\\begin{align*} ( d \\circ \\delta _ { \\lambda } ) ( g ) = \\lambda ^ { Q } d g , \\end{align*}"}
+{"id": "265.png", "formula": "\\begin{align*} \\pi ( 0 , \\rho ) : = \\{ ( x , y ) \\in \\R ^ 2 : x ^ 2 + y ^ 2 < \\rho ^ 2 \\} . \\end{align*}"}
+{"id": "555.png", "formula": "\\begin{align*} \\Delta : = \\lbrace z \\in \\mathbb { C } \\setminus \\lbrace 0 \\rbrace \\colon - \\alpha < \\arg ( z ) < \\alpha \\rbrace , \\end{align*}"}
+{"id": "359.png", "formula": "\\begin{align*} \\ 1 _ { n - 1 } ' u _ k = 0 . \\end{align*}"}
+{"id": "1023.png", "formula": "\\begin{align*} \\chi _ K ( s _ \\alpha ( \\beta ) ) = \\chi _ K ( \\beta ) - \\langle \\alpha ^ \\vee , \\beta \\rangle \\chi _ K ( \\alpha ) . \\end{align*}"}
+{"id": "7764.png", "formula": "\\begin{align*} - \\frac { p ' ( 0 ) } { p ( 0 ) } = \\sum _ { j = 1 } ^ d \\frac { 1 } { x _ j } ~ { \\rm a n d } ~ - \\frac { p ' _ { \\rm r e v } ( 0 ) } { p _ { \\rm r e v } ( 0 ) } = \\sum _ { j = 1 } ^ d x _ j . \\end{align*}"}
+{"id": "1289.png", "formula": "\\begin{align*} s _ n ( \\nu | \\mu ) = 0 \\end{align*}"}
+{"id": "9079.png", "formula": "\\begin{align*} \\tilde { K _ i } : = K _ i \\cap { K _ i } ' , \\end{align*}"}
+{"id": "7878.png", "formula": "\\begin{align*} Z ^ i ( v ( s ) , s ) - Z ( v ( s ) , s ) = \\mathcal O ( \\norm { ( v ( s ) , s ) } ^ { i + 1 } ) = \\mathcal O ( s ^ \\frac { i + 1 } { 2 } ) . \\end{align*}"}
+{"id": "3949.png", "formula": "\\begin{align*} \\langle - \\Delta z , p \\rangle = - \\langle F ' ( \\overline { y } ; z ) , p \\rangle + \\langle h , p \\rangle . \\end{align*}"}
+{"id": "8322.png", "formula": "\\begin{align*} & u _ { t x } + \\alpha \\beta ^ 2 u - 2 i \\alpha \\beta u _ x - \\alpha u _ { x x } + \\sigma i \\alpha \\beta ^ 2 | u | ^ 2 u _ x = 0 \\\\ & u ( x , t ) | _ { t = 0 } = u _ 0 ( x ) , \\end{align*}"}
+{"id": "228.png", "formula": "\\begin{align*} \\tau = \\underbrace { \\min \\ ! \\big ( \\mathcal { H } _ 2 - \\rho ( h _ { j , 1 } ) + 3 q ( n ) + 4 C \\big ) } _ { \\ge \\ , - q ( n ) - C + 3 q ( n ) + 4 C \\ , = \\ , 2 q ( n ) + 3 C } - \\underbrace { \\max \\ ! \\big ( \\mathcal { H } _ 1 \\big ) } _ { \\le \\ , q ( n ) + C } \\ge q ( n ) + 2 C , \\end{align*}"}
+{"id": "2708.png", "formula": "\\begin{align*} \\delta _ k \\le \\beta ( x _ k ) \\stackrel { \\rm d e f } { = } \\max \\left \\{ C _ 4 \\| \\nabla \\phi ( x _ k ) \\| - C _ 5 \\epsilon _ g , - C _ 6 \\lambda _ { \\rm m i n } ( \\nabla ^ 2 \\phi ( x _ k ) ) - C _ 7 \\epsilon _ H \\right \\} , \\end{align*}"}
+{"id": "3666.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\int _ { | x | = r } \\sum _ { i , j = 1 } ^ n \\left ( \\frac { \\partial h _ { i j } } { \\partial x _ i } - \\frac { \\partial h _ { i j } } { \\partial x _ j } \\right ) \\frac { x _ j } { | x | } \\ , d \\sigma _ { \\bar g } = 0 . \\end{align*}"}
+{"id": "3221.png", "formula": "\\begin{align*} \\frac { \\gamma _ { i , j } ( x ) ( \\mu _ i \\nu _ { i , j } ) } { \\sum _ { i ' , j ' } \\gamma _ { i ' , j ' } ( x ) ( \\mu _ { i ' } \\nu _ { i ' , j ' } ) } = \\left ( \\frac { \\sum _ { j ' } ( \\nu _ { i , j ' } \\gamma _ { i , j ' } ( x ) ) \\mu _ i } { \\sum _ { i ' } \\sum _ { j ' } ( \\nu _ { i ' , j ' } \\gamma _ { i ' , j ' } ( x ) ) \\mu _ { i ' } } \\right ) \\left ( \\frac { \\gamma _ { i , j } ( x ) \\nu _ { i , j } } { \\sum _ { j ' } \\nu _ { i , j ' } \\gamma _ { i , j ' } ( x ) } \\right ) , \\end{align*}"}
+{"id": "3156.png", "formula": "\\begin{align*} \\sum _ { y \\neq 0 , 1 } \\chi _ 4 ( y ) \\chi _ 4 ( 1 - y ) \\sum _ { x \\neq 0 , 1 , y } \\overline { \\chi _ 4 } ( x ) \\chi _ 4 ( x - y ) & = \\sum _ { y \\neq 0 , 1 } \\chi _ 4 ( y ) \\chi _ 4 ( 1 - y ) \\left [ - 1 - \\chi _ 4 ( y - 1 ) \\right ] \\\\ & = - \\rho - \\sum _ y \\chi _ 4 ( y ) \\varphi ( 1 - y ) = - 2 \\rho , \\end{align*}"}
+{"id": "777.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m f _ i ( u ) \\log u _ i \\leq 0 , \\end{align*}"}
+{"id": "7214.png", "formula": "\\begin{align*} u ( t ) = e ^ { \\kappa t \\Delta } u _ 0 + \\int _ 0 ^ t e ^ { \\kappa ( t - s ) \\Delta } \\Delta _ p u ( s ) \\ , \\ , d s + \\sum _ { k \\in \\Z ^ d } \\sum _ { i = 1 } ^ { d - 1 } \\int _ 0 ^ t e ^ { \\kappa ( t - s ) \\Delta } \\xi _ { k , i } \\cdot \\nabla u ( s ) \\ , d B _ s ^ { k , i } . \\end{align*}"}
+{"id": "6925.png", "formula": "\\begin{align*} \\int _ \\R ( | T _ i ( x _ i ) | + | x _ i | ) Q _ i ( x _ i ) d x _ i = \\int _ \\R | x _ i | Q ^ * _ i ( x _ i ) d x _ i + \\int _ \\R | x _ i | Q _ i ( x _ i ) d x _ i < \\infty . \\end{align*}"}
+{"id": "2915.png", "formula": "\\begin{align*} [ e ^ { - A t } ] _ { x + n + 1 , x ' + n + 1 } = \\sum _ { j = 0 } ^ n E _ j ( t ) \\psi _ j ( x ) \\psi _ j ( x ' ) , \\end{align*}"}
+{"id": "1134.png", "formula": "\\begin{align*} \\| \\mathbf { c } _ X \\| _ 1 = \\sum _ { j = 1 } ^ { N L } x _ j = \\| \\mathbf { x } \\| _ 1 , \\| \\mathbf { c } _ Y \\| _ 1 = \\sum _ { j = 1 } ^ { N L } y _ j = \\| \\mathbf { y } \\| _ 1 . \\end{align*}"}
+{"id": "2188.png", "formula": "\\begin{align*} \\hat { \\lambda } _ { 1 } = \\frac { ( \\mathbf { v } ^ { n + 1 } ) _ i } { ( \\mathbf { v } ^ { n } ) _ i } . \\end{align*}"}
+{"id": "8166.png", "formula": "\\begin{align*} \\mu \\Lambda ( \\mu , \\nu ) \\xi ( \\mu , s ) \\xi ( \\nu , s ) \\chi _ { ( 0 , \\mu ) } ( \\nu ) = 0 a . e . ~ \\mbox { i n } ( 0 , t _ { 0 } ) \\times ( R _ { 0 } , \\infty ) \\times ( 0 , \\infty ) . \\end{align*} % \\end{align*}"}
+{"id": "7873.png", "formula": "\\begin{align*} H = \\begin{pmatrix} \\psi _ { v v } ( 0 , p _ 0 ) [ v , v ] & 0 \\\\ 0 & 0 \\end{pmatrix} \\end{align*}"}
+{"id": "7094.png", "formula": "\\begin{align*} \\mbox { $ ( ( X _ { 2 , 6 } \\cap N ( c _ 2 ) ) \\cup ( X _ { 2 , 7 } \\cap N ( c _ 2 ) ) \\cup \\{ c _ 2 \\} ) \\cap { \\cal N } = \\emptyset $ . } \\end{align*}"}
+{"id": "211.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sqrt [ n ] { \\binom { n } { \\lceil \\alpha n \\rceil } } \\le \\frac { 1 } { \\alpha ^ { \\alpha } ( 1 - \\alpha ) ^ { 1 - \\alpha } } . \\end{align*}"}
+{"id": "3196.png", "formula": "\\begin{align*} v o l ( \\widehat { A V } _ 3 ) = v o l ( [ 3 , 3 , 6 ] ) + v o l ( [ 3 , 4 , 4 ] ) + v o l ( [ 4 , 4 , 3 ] ) + v o l ( [ 3 , 6 , 3 ] ) \\\\ \\end{align*}"}
+{"id": "9322.png", "formula": "\\begin{align*} \\mathbf x ^ { t } \\gets \\underset { \\mathbf z : \\operatorname { s u p p } ( \\mathbf z ) = \\hat { S } ^ { t } } { \\arg \\min } f ( \\mathbf z ) , \\end{align*}"}
+{"id": "7295.png", "formula": "\\begin{align*} \\pi _ { 2 } \\circ \\lambda _ { f \\times i d _ { X } } [ ( f \\times i d _ { X } ) ( r , x ) , \\alpha ] ( t ) = \\pi _ { 2 } ( f ( r ) , \\alpha ( t ) ) = \\alpha ( t ) . \\end{align*}"}
+{"id": "7160.png", "formula": "\\begin{align*} \\Delta _ { B } \\textbf { \\textit { u } } = ( \\nabla ^ j \\nabla _ j u ^ i ) \\frac { \\partial } { \\partial x _ i } , \\textbf { \\textit { u } } = u ^ i \\frac { \\partial } { \\partial x _ i } \\in \\mathfrak { X } ( \\Omega ) . \\end{align*}"}
+{"id": "2515.png", "formula": "\\begin{align*} \\mathbf { S } \\mathbf { u } + \\mathbf { X } \\mathbf { v } = \\mathbf { h } , \\mathbf { u } ^ T \\mathbf { v } \\ge 0 , \\end{align*}"}
+{"id": "2497.png", "formula": "\\begin{align*} \\mu ( \\mathbf { x } , \\mathbf { s } , \\kappa , \\tau ) = ( \\mathbf { x } ^ T \\mathbf { s } + \\kappa \\tau ) / ( k + 1 ) . \\end{align*}"}
+{"id": "7465.png", "formula": "\\begin{align*} N _ f ( z ) = \\dfrac { ( 1 - z ) ( 1 - \\det ( M ) z ) } { ( 1 + z ) ( 1 + \\det ( M ) z ) } . \\end{align*}"}
+{"id": "8527.png", "formula": "\\begin{align*} M _ { \\delta , + } ( x ; z ) = M _ { \\delta , - } ( x ; z ) ( I + \\hat { R } _ { \\delta } ( x ; z ) ) , z \\in \\mathbb { R } . \\end{align*}"}
+{"id": "4900.png", "formula": "\\begin{align*} \\beta _ \\alpha ( - z ) = \\frac { \\alpha - 1 } { 1 - \\Phi ( z ) } + z \\frac { \\Phi ( z ) } { \\phi ( z ) } + 2 - \\alpha . \\end{align*}"}
+{"id": "3648.png", "formula": "\\begin{align*} \\langle \\nabla f _ { \\delta } ( x ^ g _ k ) - \\nabla f ( x ^ * ) , x _ f ^ k - x _ g ^ k \\rangle & \\leq f ( x _ f ^ k ) - f _ { \\delta } ( x _ g ^ k ) - \\langle \\nabla f ( x ^ * ) , x _ f ^ k - x _ g ^ k \\rangle \\\\ & \\leq f ( x _ f ^ k ) - f ( x _ g ^ k ) - \\langle \\nabla f ( x ^ * ) , x _ f ^ k - x _ g ^ k \\rangle + \\delta _ x \\\\ & = B _ f ( x _ f ^ k , x ^ * ) - B _ f ( x _ g ^ k ) + \\delta _ x . \\end{align*}"}
+{"id": "8276.png", "formula": "\\begin{align*} X _ t ^ i - X _ { t _ n } ^ i = \\int _ { t _ n } ^ t b ^ i ( X _ s ) \\d s + \\sigma ( W _ t ^ i - W _ { t _ n } ^ i ) . \\end{align*}"}
+{"id": "4780.png", "formula": "\\begin{align*} - \\Delta u = f & D , \\\\ [ 1 m m ] u = 0 & \\partial D . \\end{align*}"}
+{"id": "8273.png", "formula": "\\begin{align*} u ( t ) & \\leq \\int _ { 0 } ^ { t } \\bigg ( 2 \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j = 1 } ^ { i } C _ { V } j ( i + j ) | u _ j | \\ , \\psi _ i + \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j = i } ^ { \\infty } C _ V i ^ { \\eta } | u _ j | \\ , \\psi _ i \\bigg ) ( h ) d h . \\\\ \\end{align*}"}
+{"id": "672.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | e ^ { i k \\phi } \\sin ( n \\theta ) \\left | l _ 2 , m _ 2 \\right \\rangle = ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
+{"id": "2480.png", "formula": "\\begin{align*} \\Delta + \\Gamma & = \\{ x + y \\mid x \\in \\Delta , y \\in \\Gamma \\} , \\\\ \\Delta - \\Gamma & = \\{ x - y \\mid x \\in \\Delta , y \\in \\Gamma \\} , \\\\ - \\Delta & = \\{ - x \\mid x \\in \\Delta \\} . \\end{align*}"}
+{"id": "2943.png", "formula": "\\begin{align*} T ^ e _ A ( x ) : = \\inf \\{ t > 0 \\mid \\Phi ( t , x ) \\not \\in A \\} . \\end{align*}"}
+{"id": "2150.png", "formula": "\\begin{align*} z & = \\pm \\sqrt { \\frac { s _ 0 } { ( 1 + a ) ( 1 + a + s _ 0 ) } } , \\\\ w & = \\sqrt { \\frac { 1 + a } { 1 + a + s _ 0 } } \\left ( \\cos ( t ) + i \\sin ( t ) \\right ) . \\end{align*}"}
+{"id": "7303.png", "formula": "\\begin{align*} R _ { v , j } ^ q = { \\rho _ { v , j } ^ q B _ { \\rm V L C } \\log _ 2 \\left ( { 1 + { \\frac { \\exp ( 1 ) } { 2 \\pi } } \\frac { p _ { v , j } ^ { q } { \\left ( { { R _ { \\rm { P D } } } { G _ { v , j } ^ { q } } } \\right ) } ^ 2 } { \\sum \\limits _ { j ' \\ne j } { \\sum \\limits _ { v ' \\ne v } p _ { v ' , j ' } ^ { q } { \\left ( { { R _ { \\rm { P D } } } { G _ { v ' , j } ^ { q } } } \\right ) } ^ 2 } + N _ { \\rm V L C } B _ { \\rm V L C } } } \\right ) } , \\end{align*}"}
+{"id": "2146.png", "formula": "\\begin{align*} \\Im \\mathcal { P } [ \\varrho ] = a \\Re ( \\varrho _ z ) \\Im ( \\varrho _ z ) ( ( 5 a ^ 2 - 4 ) | w | ^ 4 - 4 ( | \\varrho _ z | ^ 4 + 2 | w | ^ 2 | \\varrho _ z | ^ 2 ) ) . \\end{align*}"}
+{"id": "3053.png", "formula": "\\begin{align*} d \\log \\varphi _ { \\lambda , u } ( x ) = 0 . \\end{align*}"}
+{"id": "8072.png", "formula": "\\begin{align*} \\left \\lvert 1 - 4 \\mu \\left ( \\sum _ { l = 1 } ^ { M } \\lvert \\hat { \\phi } ^ { \\left ( l , j \\right ) } \\rvert ^ 2 + \\sigma _ { e , i } ^ 2 \\lVert \\mathbf { p } _ j \\rVert ^ 2 + \\sum _ { q = 1 } ^ { M - 1 } \\sum _ { r = q + 1 } ^ { M } f _ { q , r } ^ { \\left ( j \\right ) } \\right ) \\right \\rvert < 1 \\end{align*}"}
+{"id": "3811.png", "formula": "\\begin{align*} \\langle \\mathcal { A } _ q ^ s , \\mathcal { A } _ q ^ p \\rangle _ { L ^ 2 ( \\mathbb { R , \\mathbb { H } } ) } = K _ q ( p , s ) . \\end{align*}"}
+{"id": "2392.png", "formula": "\\begin{align*} \\partial _ n ^ v ( \\sum _ \\ell r _ \\ell \\mu _ \\ell + r _ { m + 1 } \\sigma _ { m + 1 } ) = 0 , \\end{align*}"}
+{"id": "2378.png", "formula": "\\begin{align*} [ \\mathbf { h } ' _ 2 , \\mathbf { h } _ 2 ] = 1 . \\end{align*}"}
+{"id": "6806.png", "formula": "\\begin{align*} \\varphi \\left ( \\rho \\right ) = R ( \\rho ) e ^ { 8 i t \\rho ^ { 3 } + 2 i x \\rho } . \\end{align*}"}
+{"id": "2860.png", "formula": "\\begin{align*} \\tilde S ^ { ( q ) } _ { j , j ' } = \\frac { 2 } { \\mu _ { j } + \\mu _ { j ' } } \\tilde F _ { j , j ' } - \\frac { ( \\mu _ { j } - \\mu _ { j ' } ) ^ 2 } { 8 \\gamma ^ 2 ( \\mu _ { j } + \\mu _ { j ' } ) } \\tilde S ^ { ( q ) } _ { j , j ' } . \\end{align*}"}
+{"id": "4420.png", "formula": "\\begin{align*} \\Lambda ^ { ( s ) } ( 1 ) = - \\frac { 1 } { s } , \\\\ \\ \\ \\Lambda ^ { ( s ) } ( z ^ k ) = f _ k ( s ) : = - \\frac { 1 } { s } \\left ( ( k + 1 ) ^ { 1 - s } - k ^ { 1 - s } \\right ) . \\end{align*}"}
+{"id": "6497.png", "formula": "\\begin{align*} \\Lambda = \\cup _ { m \\in \\mathbb { N } ^ 2 } \\Lambda ^ m . \\end{align*}"}
+{"id": "5503.png", "formula": "\\begin{align*} \\tilde G ( p , s ) = s ^ { - p / 2 } 2 ^ { 3 p / 2 - 1 } \\Gamma ( p / 2 ) D ( p ) \\end{align*}"}
+{"id": "3987.png", "formula": "\\begin{align*} ( \\mathbf { x } , - \\mathbf { y } ) \\underbrace { \\begin{pmatrix} G _ { 4 } & G _ { 5 } & \\cdots & G _ { 2 k + 2 } \\\\ G _ { 5 } & G _ { 6 } & \\cdots & G _ { 2 k + 3 } \\end{pmatrix} } _ M = 0 . \\end{align*}"}
+{"id": "9039.png", "formula": "\\begin{align*} \\| 2 \\rho _ { n _ { j } } - \\langle { \\bf k } ^ { * } _ { n _ { j } } , \\alpha \\rangle \\| _ { \\mathbb { T } } & = \\| 2 \\rho - \\langle { \\bf d } _ j , \\alpha \\rangle \\| _ { \\mathbb { T } } \\\\ & \\geq \\min \\{ \\kappa , \\gamma \\} ( 1 + | 2 { \\bf d } _ j | _ { \\eta } ) ^ { - C _ { 1 } | 2 { \\bf d } _ j | _ { \\eta } ^ { \\frac { 1 } { \\eta + 1 } } } . \\end{align*}"}
+{"id": "1078.png", "formula": "\\begin{align*} \\ell ( w s _ { - w ^ { - 1 } \\gamma } ) = \\ell ( w ) - \\ell ( s _ { - w ^ { - 1 } \\gamma } ) . \\end{align*}"}
+{"id": "396.png", "formula": "\\begin{align*} \\theta _ t \\omega ( \\cdot ) = \\omega ( \\cdot + t ) - \\omega ( t ) , \\ \\ t \\in \\R , \\ \\omega \\in \\Omega . \\end{align*}"}
+{"id": "294.png", "formula": "\\begin{align*} \\psi ( q ) = \\varphi ( p _ 0 ) + \\frac { \\tilde { d } _ H ( p _ 0 , q ) ^ \\beta } { \\delta ^ \\beta } . \\end{align*}"}
+{"id": "7845.png", "formula": "\\begin{align*} F ^ { q , \\dagger } _ { v v v } ( 0 , p ) [ u _ k , u _ k , u _ k ] & = 3 c _ 5 ( q , k , p ) u _ k + c _ 6 ( q , k , p ) u _ { 3 k } , \\end{align*}"}
+{"id": "5278.png", "formula": "\\begin{align*} & E _ { * } ( g _ 1 , w ) = \\\\ & \\int _ { - \\infty } ^ \\infty \\int _ 0 ^ 1 \\int _ 0 ^ \\infty \\int _ 0 ^ \\infty \\Psi _ * \\left ( \\lambda , \\theta , t , u + \\left ( \\frac { t _ 1 } { t } \\right ) ^ 2 u _ 1 \\right ) e ^ { i \\Phi \\left ( \\lambda , \\theta , t , u + \\left ( \\frac { t _ 1 } { t } \\right ) ^ 2 u _ 1 \\right ) } W ( \\lambda , t , u ) d \\lambda d \\theta d t d u . \\end{align*}"}
+{"id": "67.png", "formula": "\\begin{align*} & \\ell ( x , \\alpha ) + \\ell ( x , - \\alpha ) \\\\ = & \\langle \\mu , \\alpha \\rangle + \\Phi ^ + ( \\alpha ) - \\Phi ^ + ( w \\alpha ) + \\langle \\mu , - \\alpha \\rangle + \\Phi ^ + ( - \\alpha ) - \\Phi ^ + ( - w \\alpha ) \\\\ = & \\Phi ^ + ( \\alpha ) + \\Phi ^ + ( - \\alpha ) - ( \\Phi ^ + ( w \\alpha ) + \\Phi ^ + ( - w \\alpha ) ) = 1 - 1 = 0 . \\end{align*}"}
+{"id": "1039.png", "formula": "\\begin{align*} \\rho _ { x _ 1 \\ast x _ 2 } = \\rho _ { x _ 2 } \\circ \\rho _ { x _ 1 } . \\end{align*}"}
+{"id": "4409.png", "formula": "\\begin{align*} \\theta _ 0 ( x ) - x = - \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\frac { x ^ { s } } { s } \\bigg \\{ \\frac { \\zeta ' ( s ) } { \\zeta ( s ) } + g ( s ) \\bigg \\} d t . \\end{align*}"}
+{"id": "8149.png", "formula": "\\begin{align*} x ( \\varpi _ n ) = \\varphi _ M ^ { - n } ( \\frac \\varpi \\pi x ' ) \\end{align*}"}
+{"id": "4092.png", "formula": "\\begin{align*} \\Phi ( u ^ i , u ^ i { } _ j , u ^ i { } _ { j k } ) & = ( u ^ i , u ^ i { } _ j , u ^ i { } _ { j l } v ^ l { } _ k ) , \\\\ * \\Psi ( x ^ i , y ^ i { } _ j , z ^ i { } _ { j k } ) & = ( x ^ i , y ^ i { } _ j , z ^ i { } _ { j l } y ^ l { } _ k ) , \\end{align*}"}
+{"id": "5837.png", "formula": "\\begin{align*} \\rho { c _ v } { \\partial _ { { t } } } T = \\nabla \\cdot \\left ( { \\lambda \\nabla T } \\right ) + { \\bar F } + \\left ( { \\vartheta - 1 } \\right ) \\frac { { \\Delta t } } { 2 } \\rho { c _ v } \\partial _ { { t } } ^ 2 T , \\end{align*}"}
+{"id": "3311.png", "formula": "\\begin{align*} K _ 1 * \\cdots * K _ m = \\left \\{ \\sigma \\subset \\bigsqcup _ { i \\in [ m ] } V ( K _ i ) \\ \\middle | \\ \\sigma \\cap V ( K _ i ) \\in K _ i \\right \\} . \\end{align*}"}
+{"id": "6025.png", "formula": "\\begin{align*} \\frac { 1 } { t _ { n } ^ { 2 } } \\int _ { \\R ^ { 2 } } ( | \\nabla v _ { n } | ^ { 2 } + v _ { n } ^ { 2 } ) d x + 3 B ( v _ { n } ) = t _ { n } ^ { 2 p - 6 } \\int _ { \\R ^ { 2 } } v _ { n } ^ { 2 p } d x . \\end{align*}"}
+{"id": "2944.png", "formula": "\\begin{align*} T ^ r _ B ( x ) : = \\inf \\{ t > 0 \\mid \\Phi ( t , x ) \\in B \\} . \\end{align*}"}
+{"id": "3459.png", "formula": "\\begin{align*} r _ { \\ell } \\leq e ^ { 4 0 \\varepsilon q _ n } \\frac { e ^ { - q _ n L } } { \\max ( | \\ell | , 1 ) } \\max ( r _ { \\ell - 1 } , r _ { \\ell + 1 } ) \\times \\begin{cases} \\max ( | \\ell | , e ^ { \\delta _ n q _ n } ) , & \\beta _ n \\geq \\delta _ n + 2 0 0 \\varepsilon \\\\ e ^ { \\beta _ n q _ n } , & \\beta _ n < \\delta _ n + 2 0 0 \\varepsilon \\end{cases} . \\end{align*}"}
+{"id": "6859.png", "formula": "\\begin{align*} \\sum _ { w \\in E } w t ( w ) = \\sum _ { w \\in O } w t ( w ) . \\end{align*}"}
+{"id": "4157.png", "formula": "\\begin{align*} u _ { \\varepsilon } ( x ' , t ) : = u ( x ' , t + \\varepsilon ) \\quad f _ { \\varepsilon } ( x ' ) : = u ( x ' , \\varepsilon ) , x ' \\in \\R ^ n , \\ , t > 0 . \\end{align*}"}
+{"id": "5511.png", "formula": "\\begin{align*} L ( p ) = b \\frac { \\left ( 1 6 / 3 \\right ) ^ { p / 2 } } { 4 - p } + \\frac { 1 } { 3 6 } ( p / 2 + 2 ) ( p / 2 + 3 ) \\end{align*}"}
+{"id": "1094.png", "formula": "\\begin{align*} \\prescript J { } \\pi ( s _ \\alpha w _ 2 ) = ( w ' _ 2 , \\mu _ 2 + \\Phi ^ + ( - \\alpha ) w _ 2 ^ { - 1 } \\alpha ^ \\vee ) . \\end{align*}"}
+{"id": "1073.png", "formula": "\\begin{align*} w _ 0 w _ 1 = s _ 1 \\cdots s _ q . \\end{align*}"}
+{"id": "5357.png", "formula": "\\begin{align*} \\widehat { \\iota } ( D ) = \\frac { \\beta m + \\gamma } { m + b _ 1 + 1 } . \\end{align*}"}
+{"id": "7509.png", "formula": "\\begin{align*} \\int _ { B _ { 1 / 2 } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x & = \\sum _ { j = 0 } ^ { \\infty } \\int _ { B _ { r _ { j + 1 } } \\setminus B _ { r _ { j + 2 } } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x \\\\ & \\leq C \\sum _ { j = 0 } ^ { \\infty } r _ j ^ { 2 - n } \\int _ { B _ { r _ { j + 1 } } \\setminus B _ { r _ { j + 2 } } } | \\nabla u | ^ 2 \\d x . \\end{align*}"}
+{"id": "6524.png", "formula": "\\begin{align*} d s ^ { 2 } = ( d x _ { 1 } ) ^ { 2 } + x _ 1 ( d x _ { 2 } ) ^ { 2 } + x _ 4 ( d x _ { 3 } ) ^ { 2 } + x _ 3 ( d x _ { 4 } ) ^ { 2 } . \\end{align*}"}
+{"id": "2200.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ { N } \\mathbf { R } _ { . j } = \\mathbf { 1 } ^ { H } _ { N \\times 1 } \\mathbf { R } \\end{align*}"}
+{"id": "5935.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\lim _ { n \\rightarrow \\infty } I I _ 2 \\leq C \\bigg \\{ \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ { T } \\Vert X _ n ( s ) - X ( s ) \\Vert _ H d s \\bigg \\} ^ { \\frac { \\alpha - \\theta } { \\alpha } } = 0 . \\end{align*}"}
+{"id": "503.png", "formula": "\\begin{align*} \\bigcup _ { \\rho _ i > r _ i } K \\ < \\frac { x _ 1 } { \\rho _ 1 } , \\ldots , \\frac { x _ 1 } { \\rho _ n } \\ > = \\bigcup _ { \\rho _ i > r _ i } T ( \\rho _ 1 , \\ldots , \\rho _ n ) \\end{align*}"}
+{"id": "7954.png", "formula": "\\begin{align*} \\partial _ s \\rho ( s , u ) - m \\cdot \\nabla \\big ( \\rho ^ 2 ( s , u ) \\big ) = 0 , \\rho ( 0 , u ) = \\rho ^ { \\rm i n i } ( u ) , \\end{align*}"}
+{"id": "1811.png", "formula": "\\begin{align*} \\mathcal { Y M } ( \\nabla ) = \\int _ M \\frac 1 2 | | R ^ \\nabla | | ^ 2 \\ , d v \\ , , \\end{align*}"}
+{"id": "3925.png", "formula": "\\begin{align*} \\xi ( F ) = u _ \\S ( F ) + 2 . \\end{align*}"}
+{"id": "6248.png", "formula": "\\begin{align*} \\partial _ i \\circ \\pi : = \\partial _ { u _ i } \\circ \\pi = \\pi _ { * } X _ i . \\end{align*}"}
+{"id": "4252.png", "formula": "\\begin{align*} \\dfrac { \\varepsilon } { 2 } \\leq \\beta ( \\varphi _ { 2 k } ( a _ 1 ) , \\rho ( a _ 1 ) ) = \\beta ( \\varphi _ { 2 k } ( a _ 1 ) , \\rho \\circ \\varphi _ { 2 k } ( a _ 1 ) ) \\leq \\beta ( \\varphi _ i ( a _ 1 ) , \\rho \\circ \\varphi _ i ( a _ 1 ) ) . \\end{align*}"}
+{"id": "2873.png", "formula": "\\begin{align*} K _ { j , * } : = \\sup _ { x , n } \\sum _ { y = 0 } ^ n | K ^ { ( n ) } _ j ( x , y ) | < + \\infty , j = 1 , 2 . \\end{align*}"}
+{"id": "7095.png", "formula": "\\begin{align*} \\mbox { $ ( ( X _ { 2 , 7 } \\setminus N ( c _ 2 ) ) \\cup ( X _ { 3 , 7 } \\setminus N ( c _ 3 ) ) \\cup \\{ c _ 7 \\} ) \\cap { \\cal N } = \\emptyset $ . } \\end{align*}"}
+{"id": "4277.png", "formula": "\\begin{align*} \\beta = 2 i - 1 \\end{align*}"}
+{"id": "3582.png", "formula": "\\begin{align*} p ^ { \\ell } = p ^ { \\ell - 1 } \\ast \\frac { A ^ { i } d } { A ^ { i } p ^ { \\ell - 1 } } \\end{align*}"}
+{"id": "4549.png", "formula": "\\begin{align*} & A _ { w , 1 } f ( v _ { 0 , 0 , 0 } ) = ( q ^ 3 + q ^ 2 + q + 1 ) f ( v _ { 1 , 0 , 0 } ) \\\\ & A _ { w , 2 } f ( v _ { 0 , 0 , 0 } ) = ( q ^ 4 + q ^ 3 + 2 q ^ 2 + q + 1 ) f ( v _ { 1 , 1 , 0 } ) \\\\ & A _ { w , 3 } f ( v _ { 0 , 0 , 0 } ) = ( q ^ 3 + q ^ 2 + q + 1 ) f ( v _ { 1 , 1 , 1 } ) \\end{align*}"}
+{"id": "6374.png", "formula": "\\begin{align*} J ( z ) = & \\frac { \\sqrt \\theta } { B ( a , b ) } \\sum _ { n = 0 } ^ \\infty \\frac { ( - 1 ) ^ n } { \\Gamma ( b - n ) n ! } \\int _ { \\theta / z ^ 2 } ^ { \\infty } y ^ { - 1 / 2 } \\exp \\{ - ( a + n ) y \\} \\mathrm { d } y \\\\ = & \\sqrt { \\pi \\theta } \\frac { \\Gamma ( b ) } { B ( a , b ) } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ n } { \\Gamma ( b - n ) n ! } \\frac { 1 } { \\sqrt { a + n } } \\operatorname { e r f c } ( \\sqrt { \\theta / z ^ 2 } \\sqrt { a + n } ) , \\end{align*}"}
+{"id": "3499.png", "formula": "\\begin{align*} \\phi ( k ) = \\sum _ { z \\in \\partial I ( k ) } G _ { I ( k ) } ( z , k ) \\phi ( z ' ) . \\end{align*}"}
+{"id": "6877.png", "formula": "\\begin{align*} { \\tt \\hat z = 1 . / e v a l u ( P h i 1 , z , { ' d ' } ) } \\end{align*}"}
+{"id": "259.png", "formula": "\\begin{align*} N : = \\bigwedge ^ { p - 1 } ( J _ { p - 2 } \\oplus J _ p ^ { \\oplus k - 1 } ) \\cong \\bigoplus ^ { p - 1 } _ { i = 0 } \\bigwedge ^ { p - 1 - i } J _ { p - 2 } \\otimes \\bigwedge ^ { i } ( J _ p ^ { \\oplus k - 1 } ) . \\end{align*}"}
+{"id": "6511.png", "formula": "\\begin{align*} & \\lambda _ x | _ { E _ { d ^ * ( y ' f ' ) } } = \\lambda _ { y ' f ' } \\\\ & \\lambda _ x | _ { E _ { d ^ * ( y ) } } = \\lambda _ y \\\\ & \\lambda _ x | _ { E _ { [ w b ^ { - 1 } , w a ] } } = \\phi ' \\end{align*}"}
+{"id": "4854.png", "formula": "\\begin{align*} A _ t : = t A _ 1 + ( 1 - t ) A _ 0 t \\in [ 0 , 1 ] \\end{align*}"}
+{"id": "5602.png", "formula": "\\begin{align*} a _ { n + 1 } = a _ { \\alpha + n + 1 } + d _ { \\alpha + n + 1 } - d _ { \\alpha + n } \\iff a _ { \\alpha + n + 1 } - a _ { n + 1 } = d _ { \\alpha + n } - d _ { \\alpha + n + 1 } \\ , \\ \\forall \\ n \\in \\N \\ . \\end{align*}"}
+{"id": "4898.png", "formula": "\\begin{align*} \\alpha _ \\beta ( z ) = z \\frac { \\Phi ( z ) } { \\phi ( z ) } - \\frac { ( 2 - \\beta ) \\Phi ( z ) - 1 } { 1 - \\Phi ( z ) } . \\end{align*}"}
+{"id": "1828.png", "formula": "\\begin{align*} \\exp ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) \\delta ^ \\nabla R ^ \\nabla - i _ { \\big ( \\exp ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) \\big ) } R ^ \\nabla = 0 \\ , , \\end{align*}"}
+{"id": "7575.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\dfrac { | ( \\Phi _ N - t ) \\cap \\Phi _ N | } { | \\Phi _ N | } = 1 \\end{align*}"}
+{"id": "5565.png", "formula": "\\begin{align*} W ( 0 ^ \\infty | 1 ^ \\infty ) = b - a + \\sum _ { n = 2 } ^ \\infty ( a _ n - a ) \\ . \\end{align*}"}
+{"id": "3449.png", "formula": "\\begin{align*} | \\phi ( 0 ) | \\geq 1 \\end{align*}"}
+{"id": "4743.png", "formula": "\\begin{align*} \\Tilde { x } = \\sum \\limits _ { i = 1 } ^ p \\alpha _ i u _ i + \\lambda x _ v + ( 1 - \\lambda ) x _ w \\in \\mathcal { S } ^ n , \\end{align*}"}
+{"id": "5089.png", "formula": "\\begin{align*} A _ j = \\textrm { c u r l } ^ { - 1 } B _ j \\to A = \\textrm { c u r l } ^ { - 1 } B \\textrm { i n } \\ L ^ { 2 } _ { \\textrm { l o c } } ( 0 , T ; L ^ { 2 } ( \\Omega ) ) , \\end{align*}"}
+{"id": "687.png", "formula": "\\begin{align*} x \\cdot y : = \\sum _ { g \\in G } \\underline g \\left ( \\sum _ { h _ 1 h _ 2 = g } x _ { h _ 1 } y _ { h _ 2 } \\right ) . \\end{align*}"}
+{"id": "3942.png", "formula": "\\begin{align*} \\mathcal { J } ' ( \\overline { u } ; h ) = \\langle \\overline { y } - g , S ' ( \\overline { u } ; h ) \\rangle + \\alpha \\langle \\overline { u } , h \\rangle \\ge 0 , h \\in T _ { \\mathcal { C } _ { a d } } ( \\overline { u } ) . \\end{align*}"}
+{"id": "1808.png", "formula": "\\begin{align*} ( x , v ) \\sim ( y , w ) \\ : \\Longleftrightarrow \\ x = y \\ \\ w = \\varphi _ { \\beta \\alpha } ( x ) v \\ ( x \\in U _ { \\alpha } , y \\in U _ { \\beta } , v , w \\in \\mathbb R ^ n ) \\ , . \\end{align*}"}
+{"id": "3937.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta z _ { h } + F ' ( y ; z _ { h } ) & = h , \\ ; \\ ; \\Omega , \\\\ z _ { h } & = 0 , \\ ; \\ ; \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "5304.png", "formula": "\\begin{align*} \\hat { f } ( \\chi ) = \\int _ G f ( g ) \\chi ( g ) d g \\end{align*}"}
+{"id": "6257.png", "formula": "\\begin{align*} \\phi _ { i j } ( X ) = \\langle \\nabla ^ \\perp _ X \\eta _ i , \\eta _ j \\rangle . \\end{align*}"}
+{"id": "4713.png", "formula": "\\begin{align*} \\det ( D ^ 2 ( a _ k \\hat C _ { q _ k , N } ) ) = c _ k \\delta _ { q _ k } . \\end{align*}"}
+{"id": "8915.png", "formula": "\\begin{align*} \\log \\lceil 2 ^ { \\lambda _ { 1 2 3 ' } } \\rceil & = \\log \\left \\lceil \\frac { \\zeta } { 3 6 } \\right \\rceil = \\log 3 \\end{align*}"}
+{"id": "6665.png", "formula": "\\begin{align*} L ( \\nu ) = ( \\sqrt { \\nu ( 1 ) } - \\sqrt { \\nu ( - 1 ) } ) ^ { 2 } \\end{align*}"}
+{"id": "7022.png", "formula": "\\begin{align*} 0 \\ , = \\ , F _ { 0 , 5 } \\ , = \\ , F _ { 1 , 4 } \\ , = \\ , F _ { 2 , 3 } \\ , = \\ , F _ { 3 , 2 } \\ , = \\ , F _ { 4 , 1 } . \\end{align*}"}
+{"id": "6543.png", "formula": "\\begin{align*} P ( x ) = x ^ { \\pi ( a ) } + d _ { \\pi ( a ) - 1 } x ^ { \\pi ( a ) - 1 } + d _ { \\pi ( a ) - 2 } x ^ { \\pi ( a ) - 2 } + \\cdots d _ 1 x + d _ 0 . \\end{align*}"}
+{"id": "3815.png", "formula": "\\begin{align*} \\nabla G ( U , x , t ) ^ T \\nabla A ( U , x , t ) & = \\nabla A ( U , x , t ) ^ T \\nabla G ( U , x , t ) \\\\ \\nabla f _ \\alpha ( U , x , t ) ^ T \\nabla A ( U , x , t ) & = \\nabla f _ \\alpha ( U , x , t ) ^ T \\nabla G ( U , x , t ) \\qquad \\alpha = 1 , \\dots , d , \\end{align*}"}
+{"id": "2284.png", "formula": "\\begin{align*} a ( U _ 1 Z ) = a ( U _ 1 U D ( x ) V ) = a ( D ( x ) V ) = a ( U D ( x ) V ) = a ( Z ) . \\end{align*}"}
+{"id": "1663.png", "formula": "\\begin{align*} k : a \\mapsto a ^ { k } : = \\prod _ { \\sigma \\in \\Sigma _ v } \\sigma ( a ) ^ { k _ \\sigma } \\end{align*}"}
+{"id": "6771.png", "formula": "\\begin{align*} A ( x , t ) = \\sum _ { n = 0 } ^ { \\infty } a _ { n } ( x ) L _ { n } ( t - x ) e ^ { \\frac { x - t } { 2 } } \\end{align*}"}
+{"id": "2278.png", "formula": "\\begin{align*} a ( Z ) = a \\big ( ( Z ^ * Z ) ^ \\frac { 1 } { 2 } \\big ) \\end{align*}"}
+{"id": "468.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\breve g _ 1 ( x ) / f _ 1 ( x ) = ( 1 - e ^ { - \\lambda \\Lambda _ 1 } ) / ( \\lambda \\Lambda _ 1 ) , \\end{align*}"}
+{"id": "8597.png", "formula": "\\begin{align*} | A | ^ { m - 1 } V ( A [ n - m ] , B _ 1 , \\dots , B _ m ) \\le \\frac { n ^ m ( n - m ) ! } { n ! } \\prod _ { i = 1 } ^ m V ( A [ n - 1 ] , B _ i ) . \\end{align*}"}
+{"id": "7765.png", "formula": "\\begin{align*} r _ - : = \\min _ { i \\ge 1 } \\Big | \\frac { p _ 0 } { p _ i } \\Big | ^ { \\frac { 1 } { i } } ~ { \\rm a n d } ~ r _ + : = \\max _ { i \\ge 1 } \\Big | \\frac { p _ { d - i } } { p _ d } \\Big | ^ { \\frac { 1 } { i } } . \\end{align*}"}
+{"id": "8338.png", "formula": "\\begin{align*} & \\varphi _ x + i k ^ 2 [ \\sigma _ 3 , \\varphi ] = k P _ x \\varphi , \\\\ & \\varphi _ t + i \\eta ^ 2 [ \\sigma _ 3 , \\varphi ] = H \\varphi , \\end{align*}"}
+{"id": "2588.png", "formula": "\\begin{align*} T _ 1 ( u , J u , J u , u , v , J v ) = T _ 2 ( u , J u , J u , u , v , J v ) , \\end{align*}"}
+{"id": "1454.png", "formula": "\\begin{align*} V _ { 0 } ^ { + } : = \\left \\{ x \\in [ 0 , 1 ] : f ' ( x ) > 0 \\right \\} , V _ { 0 } ^ { - } : = \\left \\{ x \\in [ 0 , 1 ] : f ' ( x ) < 0 \\right \\} . \\end{align*}"}
+{"id": "3834.png", "formula": "\\begin{align*} & \\chi ( \\xi ) + \\sum _ { j \\geq 0 } \\varphi ( 2 ^ { - j } \\xi ) = 1 , ~ ~ \\forall \\xi \\in \\mathbb { R } ^ d , \\\\ & | j - j ' | \\geq 2 \\Rightarrow ~ \\varphi ( 2 ^ { - j } \\xi ) \\cap ~ \\varphi ( 2 ^ { - j ' } \\xi ) = \\emptyset . \\end{align*}"}
+{"id": "8455.png", "formula": "\\begin{align*} & | k r _ 2 ( z ) | ^ 2 = | z r _ 2 ^ 2 ( z ) | = \\bigg | \\int _ 0 ^ z \\left ( r _ 2 ( s ) ^ 2 + 2 s r _ 2 ( s ) r _ 2 ' ( s ) \\right ) d s \\bigg | \\\\ & \\leq \\| r _ 2 ( z ) \\| _ { L ^ 2 } + 2 \\| r _ 2 ' ( z ) \\| _ { L ^ 2 } \\| z r _ 2 ( z ) \\| _ { L ^ 2 } \\leq \\| r _ 2 ( z ) \\| _ { H ^ 1 \\cap L ^ { 2 , 1 } } , \\end{align*}"}
+{"id": "7613.png", "formula": "\\begin{gather*} \\lim _ { N \\to \\infty } \\dfrac { | \\Phi _ N g \\cap \\Phi _ N | } { | \\Phi _ N | } = 1 \\\\ \\lim _ { N \\to \\infty } \\dfrac { | \\Phi _ N \\cap g \\Phi _ N | } { | \\Phi _ N | } = 1 \\end{gather*}"}
+{"id": "7050.png", "formula": "\\begin{align*} \\aligned F _ { 5 , 1 } & \\ , : = \\ , 7 \\ , F _ { 5 , 0 } - \\tfrac { 8 0 } { 9 } , \\\\ F _ { 6 , 0 } & \\ , : = \\ , - \\ , \\tfrac { 3 4 } { 3 } \\ , F _ { 5 , 0 } + \\tfrac { 8 0 0 } { 2 7 } . \\endaligned \\end{align*}"}
+{"id": "324.png", "formula": "\\begin{align*} | q _ { t , \\gamma } ( x + h , y ) - q _ { t , \\gamma } ( x , y ) | & \\leq | q _ { t } ( x + h , y ) - q _ { t } ( x , y ) | \\\\ & \\leq \\frac { C _ N } { t ^ { n / 2 + 1 } } \\Big ( \\frac { | h | } { \\sqrt { t } } \\Big ) ^ \\delta e ^ { - \\frac { c | x - y | ^ 2 } { t } } \\Big ( 1 + \\frac { \\sqrt { t } } { \\rho ( x ) } + \\frac { \\sqrt { t } } { \\rho ( y ) } \\Big ) ^ { - N } . \\end{align*}"}
+{"id": "8808.png", "formula": "\\begin{align*} Y _ t - y _ t = \\int _ 0 ^ t e ^ { L _ { z _ 0 } ^ \\perp ( t - s ) } \\Pi _ { \\mathrm { k e r } A ^ \\perp } ( B ( y _ s , z _ 0 - z _ s ) + B ( z _ 0 - z _ s , y _ s ) - B ( y _ s , y _ s ) + A y _ s ) d s . \\end{align*}"}
+{"id": "9199.png", "formula": "\\begin{align*} \\dot { e } _ { y _ a } = - \\gamma e _ { y _ a } - \\dfrac { \\gamma } { 4 } \\dfrac { \\partial a _ { 0 , \\delta } ( x _ a ) } { \\partial x _ a } b _ { 1 , \\delta } ( x _ a ) e _ { y _ a } ( 0 ) = 0 \\end{align*}"}
+{"id": "3323.png", "formula": "\\begin{align*} | T | | \\sigma ^ 0 | + ( m - | T | ) | \\tau ^ 0 | = | T ' | | \\sigma ^ 0 | + ( m - | T ' | ) | \\tau ^ 0 | . \\end{align*}"}
+{"id": "4752.png", "formula": "\\begin{align*} & M i n i m i z e _ { x , y \\in \\mathbb { R } ^ { n } } \\ \\ \\alpha _ J ^ T y + 2 b _ J ^ T x \\\\ & s . t . \\ \\ \\alpha _ k ^ T y + 2 b _ k ^ T x + c _ k \\le 0 \\ \\ ( \\forall \\ , k \\in \\{ 1 , . . . , m \\} ) \\\\ & s . t . \\ \\ x _ i ^ 2 - y _ i \\le 0 , \\ \\ \\forall \\ , i \\end{align*}"}
+{"id": "200.png", "formula": "\\begin{align*} \\theta _ { i , ( 2 ) } ( z _ j ) = { \\delta } _ { i , j } \\mbox { a n d } \\theta _ { i , ( 1 ) } ( z _ j ) = a _ { i , j } . \\end{align*}"}
+{"id": "5521.png", "formula": "\\begin{align*} \\tilde { h } _ d '' ( x ) = \\sum _ { n = 0 } ^ \\infty \\left ( \\frac { 1 } { ( x + n ) ^ 2 } - \\frac { 1 } { \\left ( x + n + \\frac { 1 } { 2 } \\right ) ^ 2 } - \\frac { 1 } { \\left ( x + n + \\frac { d - 1 } { 2 } \\right ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "4966.png", "formula": "\\begin{align*} \\widetilde \\theta _ 1 & = 0 , \\\\ \\widetilde \\theta _ i & = \\pi ^ { [ n - 1 ] } _ { i - 1 } ( \\xi ^ { \\widetilde \\theta } _ 1 ( Y _ 1 ) , x + Y _ 1 , X _ 2 , Y _ 2 , \\dots , X _ { i - 1 } , Y _ { i - 1 } ) , \\end{align*}"}
+{"id": "5245.png", "formula": "\\begin{align*} S ( t ) = \\begin{bmatrix} e ^ { t \\nu \\Delta } & 0 \\\\ 0 & e ^ { t \\mu \\Delta } \\\\ \\end{bmatrix} . \\end{align*}"}
+{"id": "8279.png", "formula": "\\begin{align*} \\tilde X _ { n + 1 } ^ i = \\tilde X _ n ^ i + b ( \\tilde X _ n ^ i ) \\tau + \\gamma ^ i ( \\tilde X _ n ) \\tau + \\sigma W _ \\tau ^ i , \\end{align*}"}
+{"id": "2539.png", "formula": "\\begin{align*} \\begin{aligned} & \\hat { \\mathbf { r } } _ p ( \\alpha ) = ( 1 - \\alpha \\delta / \\sqrt { 2 k } ) \\hat { \\mathbf { r } } _ p , \\\\ & \\hat { \\mathbf { r } } _ d ( \\alpha ) = ( 1 - \\alpha \\delta / \\sqrt { 2 k } ) \\hat { \\mathbf { r } } _ d , \\\\ & \\mu ( \\alpha ) = ( 1 - \\alpha \\delta / \\sqrt { 2 k } ) \\mu . \\end{aligned} \\end{align*}"}
+{"id": "183.png", "formula": "\\begin{align*} F = \\varinjlim _ n ( \\varprojlim _ { k } X _ { H , n } \\widehat { \\otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V _ k ) \\oplus \\varprojlim _ k A _ { H ' , n } \\widehat { \\otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V _ k ) ) . \\end{align*}"}
+{"id": "8173.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t _ { 0 } } \\int _ { 0 } ^ { R _ { 0 } } \\nu \\xi ( \\nu , s ) \\Lambda ( R _ { 0 } , \\nu ) \\ d \\nu d s = 0 , \\end{align*}"}
+{"id": "1728.png", "formula": "\\begin{align*} F _ n ( z ) ( t ) = U ( \\tau _ n + t , \\tau _ n ) y _ n ( \\tau _ n ) - \\lambda i \\int _ 0 ^ t U ( \\tau _ n + t , \\tau _ n + s ) ( e ^ { ( \\alpha - 1 ) W ( \\tau _ n + s ) } g ( z ( s ) ) ) d s . \\end{align*}"}
+{"id": "7300.png", "formula": "\\begin{align*} L \\left ( d _ { k , j } \\right ) = 1 4 0 . 7 + 3 6 . 7 { \\log _ { 1 0 } } \\left ( { { d _ { { { k , j } } } } } \\right ) , \\end{align*}"}
+{"id": "4031.png", "formula": "\\begin{align*} R _ { \\beta } : = L _ { T ' } , \\end{align*}"}
+{"id": "921.png", "formula": "\\begin{align*} H ^ { s , m } & = \\{ f \\in \\mathcal { S } ' \\ ; \\| f \\| _ { H ^ { s , m } } < \\infty \\} , \\\\ \\| f \\| _ { H ^ { s , m } } & = \\| \\langle x \\rangle ^ m \\mathcal { F } ^ { - 1 } \\langle \\xi \\rangle ^ s { { \\mathcal F } } f \\| _ { L _ { x } ^ { 2 } } , \\end{align*}"}
+{"id": "8159.png", "formula": "\\begin{align*} \\frac { \\partial \\xi ( \\mu , t ) } { \\partial t } = & - \\frac { \\partial } { \\partial \\mu } \\bigg ( \\xi ( \\mu , t ) \\int _ 0 ^ { \\mu } \\nu \\Lambda ( \\mu , \\nu ) \\xi ( \\nu , t ) \\ d \\nu \\bigg ) \\\\ & - \\int _ { \\mu } ^ { \\infty } \\Lambda ( \\mu , \\nu ) \\xi ( \\mu , t ) \\xi ( \\nu , t ) \\ d \\nu , \\ ( \\mu , t ) \\in \\mathbb { R } _ { > 0 } ^ 2 , \\ \\end{align*}"}
+{"id": "8573.png", "formula": "\\begin{align*} C _ { - 1 } ^ 1 ( 0 , + \\infty ) = \\{ f \\in C _ { - 1 } ( 0 , + \\infty ) : \\ , f ^ \\prime \\in C _ { - 1 } ( 0 , + \\infty ) \\} . \\end{align*}"}
+{"id": "7756.png", "formula": "\\begin{align*} q ( x ) : = x p ' ( x ) ~ { \\rm a n d } ~ q _ h ( x ) : = \\sum _ { i = 0 } ^ d q _ i ^ { ( h ) } x ^ i , ~ h = 0 , 1 , \\dots ~ { \\rm ( c f . ~ ( \\ref { e q d n d 2 } ) ) } , \\end{align*}"}
+{"id": "4823.png", "formula": "\\begin{align*} ( A + F ) y = ( b + f ) . \\end{align*}"}
+{"id": "5969.png", "formula": "\\begin{align*} \\Upsilon : = K _ M \\bigcap \\bigcap _ { k = 1 } ^ { \\infty } \\Gamma _ k \\end{align*}"}
+{"id": "6529.png", "formula": "\\begin{align*} x = q _ i + p _ i \\end{align*}"}
+{"id": "3051.png", "formula": "\\begin{align*} \\varphi _ { \\lambda , u } = x _ { 0 } ^ { \\lambda _ { 0 } } \\cdots x _ { n } ^ { \\lambda _ { n } } f _ 1 ^ { u _ 1 } \\cdots f _ { l } ^ { u _ l } \\end{align*}"}
+{"id": "3174.png", "formula": "\\begin{align*} B _ 1 & = 1 6 ( q - 9 ) + 6 B + \\overline { B } h ^ 2 , \\\\ D _ 1 & = 2 \\overline { B } - 8 \\overline { h } + B h ^ 2 - 4 h ^ 3 , \\\\ E _ 1 & = 8 ( q - 9 ) + 4 B h , \\\\ F _ 1 & = 1 6 ( q - 9 ) + 4 R e ( B \\overline { h } ) . \\end{align*}"}
+{"id": "6467.png", "formula": "\\begin{align*} K = \\bigcup _ { n = 1 } ^ { \\infty } [ ( r + 1 ) ^ { ( r + 1 ) n } , ( r + 1 ) ^ { ( r + 1 ) n + 1 } - 1 ] \\cap \\N . \\end{align*}"}
+{"id": "7151.png", "formula": "\\begin{align*} N ( \\tau ) & = \\frac { \\operatorname { v o l } ( \\mathbb { S } ^ { n - 2 } ) \\operatorname { v o l } ( \\partial \\Omega ) } { n - 1 } \\biggl [ \\biggl ( \\frac { \\lambda + 3 \\mu } { 2 \\mu ( \\lambda + \\mu ) } \\biggr ) ^ { n - 1 } + \\frac { 1 } { ( 2 \\mu ) ^ { n - 1 } } + \\frac { n - 2 } { \\mu ^ { n - 1 } } + \\frac { 1 } { \\alpha ^ { n - 1 } } \\biggr ] \\tau ^ { n - 1 } \\\\ & + o ( \\tau ^ { n - 1 } ) \\ \\tau \\to + \\infty . \\end{align*}"}
+{"id": "6985.png", "formula": "\\begin{align*} u _ m - \\beta ^ r u _ { m - r } = a ' \\alpha ^ { m - r } t _ r . \\end{align*}"}
+{"id": "9033.png", "formula": "\\begin{align*} & \\mathcal { B } ^ { \\rm n r e } _ { h , r } ( \\sigma ) = \\Big \\{ F \\in \\mathcal { B } _ { h , r } : F ( x ) = \\mathcal { T } _ { N } F ( x ) \\Big \\} , \\\\ & \\mathcal { B } ^ { \\rm r e } _ { h , r } ( \\sigma ) = \\Big \\{ F \\in \\mathcal { B } _ { h , r } : F ( x ) = \\mathcal { R } _ { N } F ( x ) \\Big \\} . \\end{align*}"}
+{"id": "7319.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ n { T _ { i j r } T _ { k \\ell r } } = 0 \\forall i , j , k , \\ell \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
+{"id": "1369.png", "formula": "\\begin{align*} \\P _ x ( Z _ d ( n _ 1 ) \\leq \\xi _ 1 , \\ldots , Z _ d ( n _ m ) \\leq \\xi _ m ) = \\det ( I - \\bar { \\chi } _ \\xi K \\bar { \\chi } _ { \\xi } ) _ { l ^ 2 ( \\{ n _ 1 , \\ldots n _ k \\} \\times \\mathbb { N } } \\end{align*}"}
+{"id": "1697.png", "formula": "\\begin{align*} \\dot { V } ( \\bar { x } _ t ) = - \\norm { \\bar { x } ( t - h ) } ^ 2 . \\end{align*}"}
+{"id": "7831.png", "formula": "\\begin{align*} \\Phi ^ \\dagger ( v , p ) = 0 , \\end{align*}"}
+{"id": "3256.png", "formula": "\\begin{align*} \\tilde Z _ n ^ N ( i ) : = \\Delta _ n ^ { - \\frac 1 2 } \\left ( ( p _ N \\tilde { \\Delta } _ i ^ n Y ) ^ { \\otimes 2 } - \\int _ { t _ { i - 1 } } ^ { t _ i } p _ N \\mathcal S ( t _ i - s ) \\Sigma _ s \\mathcal S ( t _ i - s ) ^ * p _ N d s \\right ) . \\end{align*}"}
+{"id": "8011.png", "formula": "\\begin{align*} \\sum _ { f \\in \\mathcal F _ { N , k } } a _ f ( n ) = \\begin{cases} \\frac { | \\mathcal F _ { N , k } | } { n ^ { 1 / 2 } } + \\O \\left ( n \\sigma _ 0 ( n ) 4 ^ { \\nu ( N ) } \\right ) , & n \\\\ \\O \\left ( n \\sigma _ 0 ( n ) 4 ^ { \\nu ( N ) } \\right ) , & . \\end{cases} \\end{align*}"}
+{"id": "338.png", "formula": "\\begin{align*} f ^ { ( p + j ) } ( z ) = \\sum _ { i = 0 } ^ { p - 1 } ( A _ { p - i , j } ( z ) / A _ 0 ( z ) ^ { j + 1 } ) f ^ { ( i ) } ( z ) , \\end{align*}"}
+{"id": "5581.png", "formula": "\\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) = A ( 1 ^ { n + k } 0 . . . ) - A ( 1 ^ n 0 ^ \\infty ) = c _ { n + k } - c _ n \\ , \\end{align*}"}
+{"id": "1372.png", "formula": "\\begin{align*} K ( n _ i , y ; n _ j , z ) & = - f _ { n _ j - n _ i } ( z - y ) 1 _ { \\{ i < j \\} } \\\\ & + \\sum _ { k , l = 1 } ^ d \\int _ { y } ^ { \\infty } f _ { n _ m - n _ i } ( u - y ) u ^ { k - 1 } d u ( B ^ { - 1 } ) _ { l k } f _ { n _ j - 1 + l } ( z - x _ l ) \\end{align*}"}
+{"id": "2532.png", "formula": "\\begin{align*} \\widehat { \\Delta \\mathbf { s } } = \\mathbf { T } _ x \\Delta \\mathbf { s } = \\mathbf { T } _ x \\mathbf { X } ^ { - 1 } ( \\nu \\mu \\mathbf { e } - \\mathbf { X } \\mathbf { s } - \\mathbf { S } \\Delta \\mathbf { x } ) = \\nu \\mu \\mathbf { e } - \\mathbf { w } _ { x s } - \\mathbf { R } _ { x s } \\widehat { \\Delta \\mathbf { x } } . \\end{align*}"}
+{"id": "6927.png", "formula": "\\begin{align*} \\log \\int _ { \\R ^ n } e ^ { f } \\ , d \\gamma _ t - \\left ( \\int _ { \\R ^ n } f \\ , d \\gamma _ { y , t } - H ( \\gamma _ { y , t } \\ , | \\ , \\gamma _ t ) \\right ) = H ( \\gamma _ { y , t } \\ , | \\ , P ) , \\end{align*}"}
+{"id": "7261.png", "formula": "\\begin{align*} P ( x _ 1 , \\ldots , x _ n ) = x _ 2 ^ { q + 1 } - x _ 1 ^ q x _ 3 . \\end{align*}"}
+{"id": "5651.png", "formula": "\\begin{align*} \\langle e _ { i } , A f _ { j } \\rangle = \\langle A e _ { i } , A f _ { j } \\rangle = \\langle e _ { i } , f _ { j } \\rangle = \\delta _ { i j } . \\end{align*}"}
+{"id": "4221.png", "formula": "\\begin{align*} \\dot { s } \\varepsilon ( x ) \\dot { s } = \\displaystyle \\varepsilon ( - x ^ { - 1 } ) \\dot { s } h ( - x ) \\varepsilon ( - x ^ { - 1 } ) . \\end{align*}"}
+{"id": "3947.png", "formula": "\\begin{align*} - \\Delta p _ { \\epsilon } + F _ { \\epsilon } ' ( y _ { \\epsilon } ) p _ { \\epsilon } & = y _ { \\epsilon } - g , \\\\ \\langle p _ { \\epsilon } + \\alpha u _ { \\epsilon } , u - u _ { \\epsilon } \\rangle + \\langle | u _ { \\epsilon } - \\overline { u } | ^ { p - 2 } ( u _ { \\epsilon } - \\overline { u } ) , u - u _ { \\epsilon } \\rangle & \\ge 0 u \\in \\mathcal { C } _ { a d } . \\end{align*}"}
+{"id": "690.png", "formula": "\\begin{align*} \\lambda _ g ' \\circ \\bar \\phi \\circ \\eta _ X & = \\lambda _ g ' \\circ \\phi & \\\\ & = \\phi \\circ \\lambda _ g & \\\\ & = \\bar \\phi \\circ \\eta _ X \\circ \\lambda _ g & \\\\ & = \\bar \\phi \\circ \\Q _ \\tau \\lambda _ g \\circ \\eta _ X & \\end{align*}"}
+{"id": "4076.png", "formula": "\\begin{align*} A '' _ n ( k ) = A _ n ( k _ V ) A ' _ n ( k _ { V ' } ) \\end{align*}"}
+{"id": "5951.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathbb { T } ^ d } \\langle b ( u _ n ( x ) ) , \\nabla u _ n ( x ) - \\nabla u ( x ) \\rangle d x = 0 . \\end{align*}"}
+{"id": "5745.png", "formula": "\\begin{align*} A _ { \\alpha , \\beta , u , v } = \\begin{pmatrix} \\alpha & u \\alpha \\\\ \\beta & v \\beta \\end{pmatrix} \\end{align*}"}
+{"id": "2415.png", "formula": "\\begin{align*} \\frac { { \\rm d } ^ { n } } { { \\rm d } s ^ { n } } F ( s ) \\equiv F ^ { ( n ) } ( s ) = ( - 1 ) ^ { n } \\int _ { 0 } ^ { \\infty } t ^ { n } \\exp ( - s t ) f ( t ) d t . \\end{align*}"}
+{"id": "1568.png", "formula": "\\begin{align*} \\frac { \\partial \\Phi _ { i , j } ^ l } { \\partial x _ k } = 0 \\textrm { f o r a l l $ p + 1 \\le l \\le n $ , a n d $ 1 \\le k \\le p $ } \\end{align*}"}
+{"id": "1138.png", "formula": "\\begin{align*} S ^ { \\rm P R } _ { i , \\ , j , \\ , k , \\ , \\ell } ( B ) : = \\kappa ( \\rho ) \\ , y _ { N ( k - 1 ) + i } \\ , x _ { N ( \\ell - 1 ) + j } , \\end{align*}"}
+{"id": "6271.png", "formula": "\\begin{align*} \\sum _ s \\phi _ { i s } ( \\partial _ r ) \\varphi _ s & = \\phi _ { i i } ( \\partial _ r ) \\varphi _ i + \\phi _ { i r } ( \\partial _ r ) \\varphi _ r = \\frac { 1 } { 2 } \\partial _ r ( \\varphi ^ { - 1 } _ i ) \\varphi _ i + \\Gamma _ { r i } ^ i = 0 , \\end{align*}"}
+{"id": "991.png", "formula": "\\begin{align*} 0 & \\le \\sum _ { l \\in \\Omega _ 1 } ( | l \\cap W | - 1 ) ( | l \\cap W | - 2 ) \\\\ & = \\sum _ { l \\in \\Omega _ 1 } | l \\cap W | ( | l \\cap W | - 1 ) - 2 \\sum _ { l \\in \\Omega _ 1 } | l \\cap W | + 2 | \\Omega _ 1 | \\\\ & \\le | W | ( | W | - 1 ) - 2 \\sum _ { l \\in \\Omega _ 1 } | l \\cap W | + 2 | \\Omega _ 1 | , \\end{align*}"}
+{"id": "8479.png", "formula": "\\begin{align*} M _ { \\pm } ( x ; z ) = I + \\mathcal { P } ^ { \\pm } \\left ( M _ { - } ( x ; \\cdot ) R ( x ; \\cdot ) \\right ) , z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "3121.png", "formula": "\\begin{align*} \\alpha \\circ T = T \\circ \\beta , [ T ( a ) , T ( b ) ] = T \\big ( \\varrho ( T ( a ) ) b - \\varrho ( T ( b ) ) a \\big ) . \\end{align*}"}
+{"id": "3427.png", "formula": "\\begin{align*} \\tilde { P } _ { 2 q _ n - 1 } ( \\theta ) = \\sum _ { x _ 1 \\in I _ 0 \\cup I _ { \\ell } } \\tilde { P } _ { 2 q _ n - 1 } ( \\theta _ { x _ 1 } ) \\prod _ { \\substack { j \\in I _ 0 \\cup I _ { \\ell } \\\\ j \\neq x _ 1 } } \\frac { \\sin \\pi ( \\theta - \\theta _ j ) } { \\sin \\pi ( \\theta _ { x _ 1 } - \\theta _ j ) } . \\end{align*}"}
+{"id": "4619.png", "formula": "\\begin{align*} \\mathcal D _ \\pi = \\sum _ { \\widetilde { K } \\ \\ \\widetilde { L } } ( e _ { \\widetilde { K } } - 1 ) [ \\widetilde { K } ] \\in H _ 1 ( \\widetilde M , \\mathbb Z / 2 \\mathbb Z ) \\end{align*}"}
+{"id": "3528.png", "formula": "\\begin{align*} \\mathcal { S } _ n ^ { ( p ) } \\ = \\ \\sum _ { k = 0 } ^ { p } \\binom { p } { k } ( - 1 ) ^ k \\ , \\mathcal { C } _ n ^ { ( k ) } \\ , n ^ { p - k } . \\end{align*}"}
+{"id": "8067.png", "formula": "\\begin{align*} e _ c \\left [ i \\right ] = a _ c { \\left [ i \\right ] } - a _ c ^ { \\left ( o \\right ) } . \\end{align*}"}
+{"id": "5976.png", "formula": "\\begin{align*} W _ { U _ { \\tau _ k , R } } ( x ) = W \\circ R _ { \\tau _ k } ( x ) \\end{align*}"}
+{"id": "1543.png", "formula": "\\begin{align*} C _ 1 & { } = \\left \\{ ( f , g ) \\in \\mathrm { L } ^ 2 ( X , m ; \\mathbb { R } ^ 2 ) : f \\leq g \\right \\} , \\\\ C _ { 2 , \\alpha } & { } = \\left \\{ ( f , g ) \\in \\mathrm { L } ^ 2 ( X , m ; \\mathbb { R } ^ 2 ) : | f - g | \\leq \\alpha \\right \\} . \\end{align*}"}
+{"id": "8415.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { | k | \\rightarrow \\infty } ( z \\Psi ^ - ) _ { 2 1 } ( x ; k ) = - \\frac { 1 } { 2 i } \\lim _ { | z | \\rightarrow \\infty } \\nu ( x ; z ) \\\\ & = - \\frac { 1 } { 4 } \\left ( 2 i \\bar { u } _ { x x } ( x ) + | u _ x ( x ) | ^ { 2 } \\bar { u } _ x ( x ) \\right ) e ^ { i c _ \\pm } ( x ) = \\widehat { \\Psi } ^ - _ { 2 1 } ( x ) , \\end{aligned} \\end{align*}"}
+{"id": "3065.png", "formula": "\\begin{align*} \\mathfrak X ^ 0 ( Z ) = \\{ ( x , w ) \\in Z _ \\mathrm { r e g } \\times \\P ^ { n } \\ \\mid \\ x \\ , \\ , \\ , \\ , \\alpha _ w | _ { Z _ \\mathrm { r e g } } \\} \\end{align*}"}
+{"id": "8516.png", "formula": "\\begin{align*} \\begin{cases} 1 + \\bar { r } _ 1 ( z ) r _ 2 ( z ) = 1 + | r ( k ) | ^ { 2 } \\geq 1 , & k \\in \\mathbb { R } ^ { + } \\\\ 1 + \\bar { r } _ 1 ( z ) r _ 2 ( z ) = 1 - | r ( k ) | ^ { 2 } \\geq c _ { 0 } ^ { 2 } > 0 , & k \\in \\mathbb { R } ^ { - } \\end{cases} \\end{align*}"}
+{"id": "8337.png", "formula": "\\begin{align*} \\varphi ( x , t , k ) = \\phi ( x , t ; k ) e ^ { i ( k ^ 2 x + \\eta ^ 2 t ) \\sigma _ 3 } , \\end{align*}"}
+{"id": "9284.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ B \\varphi _ 0 \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ m \\wedge \\beta _ n ^ { n - m } \\leq \\gamma \\left ( \\| \\varphi _ 0 \\| _ S + \\int _ B \\varphi _ 0 ( \\Delta \\varphi _ 0 ^ { + } ) ^ m \\wedge \\beta _ n ^ { n - m } \\right ) ^ \\kappa , \\end{aligned} \\end{align*}"}
+{"id": "557.png", "formula": "\\begin{align*} I _ f ( \\theta ) : = \\limsup _ { s \\rightarrow + \\infty } \\frac { \\ln \\left | f \\left ( s e ^ { i \\theta } \\right ) \\right | } { s } , \\end{align*}"}
+{"id": "201.png", "formula": "\\begin{align*} \\theta _ i ^ { ( 2 ) } ( z _ j ) = \\delta _ { i , j } \\mbox { a n d } \\theta _ { i } ^ { ( 1 ) } ( z _ j ) = a _ { i , j } . \\end{align*}"}
+{"id": "2127.png", "formula": "\\begin{align*} z = \\frac { 1 } { \\sqrt { 1 + a } } \\cos ( t ) + \\frac { i } { \\sqrt { 1 - a } } \\sin ( t ) , w = 0 . \\end{align*}"}
+{"id": "4308.png", "formula": "\\begin{align*} & \\int _ { D _ 0 } | \\tilde { F } - ( 1 - b _ { t _ 0 } ( \\Psi _ 1 ) ) f F ^ { 2 } | ^ 2 _ { \\tilde { h } } e ^ { v _ { t _ 0 } ( \\Psi _ 1 ) - \\tilde { M } } c ( - v _ { t _ 0 } ( \\Psi _ 1 ) ) \\\\ \\le & \\left ( c ( T _ 1 ) e ^ { - T _ 1 } + \\int _ { T _ 1 } ^ { t _ 0 + 1 } c ( s ) e ^ { - s } d s \\right ) \\int _ { D _ 0 } \\mathbb { I } _ { \\{ - t _ 0 - 1 < \\Psi _ 1 < - t _ 0 \\} } | f F | ^ 2 _ { \\tilde { h } } , \\end{align*}"}
+{"id": "6306.png", "formula": "\\begin{align*} F ( x ^ { t + 1 } ) - F ( x ) + \\frac { \\alpha _ t ^ 2 } { 2 \\gamma _ t } \\| x - z ^ { t + 1 } \\| ^ 2 \\overset { \\eqref { F - i n e q } } { \\leq } \\ & ( 1 - \\alpha _ t ) \\left ( F ( x ^ t ) - F ( x ) \\right ) + \\frac { \\alpha _ { t } ^ 2 ( 1 - \\beta _ t ) } { 2 \\gamma _ { t } } \\| x - z ^ t \\| ^ 2 \\\\ \\overset { \\eqref { a l p h a - e q n } } { = } \\ & ( 1 - \\alpha _ t ) \\left ( F ( x ^ t ) - F ( x ) + \\frac { \\alpha _ { t - 1 } ^ 2 } { 2 \\gamma _ { t - 1 } } \\| x - z ^ t \\| ^ 2 \\right ) . \\end{align*}"}
+{"id": "1317.png", "formula": "\\begin{align*} M ( t ) = | | u | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } + b \\int _ { 0 } ^ t | | u ( s ) | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } d s + b ( T _ 0 - t ) | | u _ 0 | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } , \\end{align*}"}
+{"id": "4245.png", "formula": "\\begin{align*} \\sup _ { z \\in \\mathbb { B } _ n } \\beta ( \\varphi _ { k j } ( z ) , \\rho ( z ) ) & \\leq \\sup _ { z \\in V ^ { - 1 } \\mathbb { B } _ n } \\omega ( \\Big { | } ( ( V ^ { - 1 } \\varphi _ { k j } V ) ^ 1 , . . . , ( V ^ { - 1 } \\varphi _ { k j } V ) ^ s ) ( z ) \\Big { | } , 0 ) \\\\ & = \\dfrac { 1 } { 2 } \\sup _ { z \\in V ^ { - 1 } \\mathbb { B } _ n } \\tanh ^ { - 1 } ( \\Big { | } ( ( V ^ { - 1 } \\varphi _ { k j } V ) ^ 1 , . . . , ( V ^ { - 1 } \\varphi _ { k j } V ) ^ s ) ( z ) \\Big { | } ) \\rightarrow 0 , \\end{align*}"}
+{"id": "4523.png", "formula": "\\begin{align*} v _ 1 ( x , t _ { 1 , e x } ( x , t ) ) = S _ { c , 1 } ( t _ { 1 , e x } ( x , t ) , t _ { 1 , e n } ( x , t ) ) v _ 1 ( x , t _ { 1 , e n } ( x , t ) ) . \\end{align*}"}
+{"id": "880.png", "formula": "\\begin{align*} x x ' + x y ' + y y ' = n \\ , . \\end{align*}"}
+{"id": "1082.png", "formula": "\\begin{align*} \\ell ( s _ \\gamma w s _ \\alpha ) = & \\ , \\ell ( w s _ \\alpha ) + 1 - \\langle - ( w s _ \\alpha ) ^ { - 1 } \\gamma ^ \\vee , 2 \\rho \\rangle \\\\ = & \\ , \\ell ( w ) + 2 - \\langle \\alpha ^ \\vee , 2 \\rho \\rangle - \\langle - w ^ { - 1 } \\gamma ^ \\vee - \\langle - w ^ { - 1 } \\gamma ^ \\vee , \\alpha \\rangle \\alpha ^ \\vee , 2 \\rho \\rangle \\\\ = & \\ , \\ell ( s _ \\gamma w ) + 1 + ( \\langle - w ^ { - 1 } \\gamma ^ \\vee , \\alpha \\rangle - 1 ) \\langle \\alpha ^ \\vee , 2 \\rho \\rangle . \\end{align*}"}
+{"id": "1975.png", "formula": "\\begin{align*} u & = x + y = x + \\psi ( x ) , \\\\ \\psi ( x ) & = u - x . \\end{align*}"}
+{"id": "8186.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\lambda } \\mu \\big [ \\xi ( \\mu , t ) - \\xi ( \\mu , 0 ) ] \\\\ b i g d \\mu = - & \\int _ { 0 } ^ { t } \\int _ { \\lambda } ^ { \\infty } \\int _ { 0 } ^ { \\lambda } \\nu \\Lambda ( \\mu , \\nu ) \\zeta ( \\mu , s ) \\xi ( \\nu , s ) \\ d \\nu d \\mu d s . \\end{align*}"}
+{"id": "2.png", "formula": "\\begin{align*} \\mathcal { P } ( \\delta ) : = \\begin{cases} ( \\delta , 0 ) \\quad - \\delta \\\\ \\{ ( \\beta _ 1 , \\beta _ 2 ) \\in ( R ^ - \\cup \\{ 0 \\} ) ^ 2 \\mid \\beta _ 1 + \\beta _ 2 = \\delta \\} \\ , \\ , . \\end{cases} \\end{align*}"}
+{"id": "4435.png", "formula": "\\begin{align*} E [ M _ v ( t ) ^ 2 ] = E \\big [ [ M _ v ] ( t ) \\big ] . \\end{align*}"}
+{"id": "8889.png", "formula": "\\begin{align*} \\Delta ^ * _ p ( 1 , n ) = \\sum _ { L M = p } \\frac { \\mu ( L ) } { \\nu ( L ) } \\sum _ { \\ell | L ^ \\infty } \\frac { \\ell } { \\nu ( \\ell ) ^ 2 } \\sum _ { d _ 1 , d _ 2 | \\ell } c _ \\ell ( d _ 1 ) c _ \\ell ( d _ 1 ) \\sum _ { v | ( n , L ) } \\frac { v \\ \\mu ( v ) } { \\nu ( v ) } \\sum _ { b | ( \\frac { n } { v } , v ) } \\sum _ { e | ( d _ 2 , \\frac { n } { b ^ 2 } ) } \\Delta _ M ( d _ 1 , \\tfrac { n d _ 2 } { e ^ 2 b ^ 2 } ) , \\end{align*}"}
+{"id": "6558.png", "formula": "\\begin{align*} \\| v _ { l } \\| _ { W ^ { 1 - \\frac { 1 } { p } , p } ( \\partial \\Omega ) } \\le C \\| v _ { l } \\| _ { L ^ { 2 } ( \\Omega ) } ^ { \\theta _ { 2 } } \\| v _ { l } \\| _ { W ^ { 2 , p } ( \\Omega ) } ^ { 1 - \\theta _ { 2 } } , \\qquad \\theta _ { 2 } = \\frac { \\frac { 1 } { d } } { \\frac { 2 } { d } - \\frac { 1 } { p } + \\frac { 1 } { 2 } } \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "248.png", "formula": "\\begin{align*} G ( X , Y ) H ( X , Y ) = P _ 3 ( X , Y ) & = X ^ { p - 1 } Y ^ { p - 1 } + X ^ { p - 1 } + Y ^ { p - 1 } \\\\ & = X ^ { p - 1 } ( 1 + Y ^ { p - 1 } ) + Y ^ { p - 1 } . \\end{align*}"}
+{"id": "6857.png", "formula": "\\begin{align*} b ( 0 , x ) \\equiv b _ 0 , \\xi ( 0 , x ) = \\xi _ 0 , S ( 0 , x ) = x \\cdot \\xi _ 0 . \\end{align*}"}
+{"id": "4832.png", "formula": "\\begin{align*} \\| \\mu _ 1 - \\mu _ 2 \\| _ { L , M } ^ * : = \\sup _ { f \\in C _ b ( M ) , \\ , | f | _ L \\leqslant 1 } \\Big | \\langle f , \\mu _ 1 \\rangle - \\langle f , \\mu _ 2 \\rangle \\Big | \\le 2 , \\end{align*}"}
+{"id": "4736.png", "formula": "\\begin{align*} L ( x , \\gamma ) : = J ( x ) + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k f _ k ( x ) = x ^ T A ( \\gamma ) x + 2 b ( \\gamma ) ^ T x + c ( \\gamma ) \\end{align*}"}
+{"id": "1260.png", "formula": "\\begin{align*} g _ { \\Delta } ( \\omega ) : = \\sum _ { \\zeta _ { \\Lambda } } \\sum _ { \\eta _ { \\Lambda } } \\mathbf { 1 } _ { \\zeta _ { \\Lambda } } ( \\omega ) h _ { \\Delta } ( \\eta _ { \\Lambda } \\omega _ { \\Lambda ^ c } ) \\frac { \\gamma _ { \\Lambda } ( \\eta _ { \\Lambda } | \\omega _ { \\Lambda ^ c } ) } { \\gamma _ { \\Lambda } ( \\zeta _ { \\Lambda } | \\omega _ { \\Lambda ^ c } ) } , \\omega \\in \\Omega , \\end{align*}"}
+{"id": "7807.png", "formula": "\\begin{align*} N : = \\mathcal N ( F ^ q _ v ( 0 , p _ 0 ) ) = \\operatorname { s p a n } \\{ u _ \\ell , w _ \\ell \\} . \\end{align*}"}
+{"id": "4548.png", "formula": "\\begin{align*} \\Gamma _ { \\ell , m , n } = \\left \\{ A = ( a _ { i j } ) \\in \\Gamma : \\substack { a _ { 2 1 } = a _ { 3 1 } = a _ { 3 2 } = a _ { 4 1 } = a _ { 4 2 } = a _ { 4 3 } = 0 , a _ { 1 2 } \\in P ^ { \\ell - m } , \\\\ a _ { 1 3 } \\in P ^ { \\ell - n } ( t ) , a _ { 1 4 } \\in P ^ \\ell ( t ) , a _ { 2 3 } \\in P ^ { m - n } , a _ { 2 4 } \\in P ^ m ( t ) , a _ { 3 4 } \\in P ^ n ( t ) , \\\\ a _ { 1 1 } , a _ { 2 2 } , a _ { 3 3 } , a _ { 4 4 } \\in \\mathbb { F } _ q } \\right \\} / \\{ \\lambda I : \\lambda \\in \\mathbb { F } _ q ^ \\times \\} , \\end{align*}"}
+{"id": "2781.png", "formula": "\\begin{align*} \\left | G ^ { D _ { N } } ( x , y ) - \\gamma \\log \\left ( 2 + N \\frac { \\max ( d ( x / N ) , d ( y / N ) ) } { 1 + | x - y | } \\right ) \\right | = O ( 1 ) . \\end{align*}"}
+{"id": "6470.png", "formula": "\\begin{align*} \\mu \\Big ( \\bigcap _ { j = 0 } ^ { t _ n } T ^ { j h _ n } J \\Big ) \\geq ( 1 - \\epsilon _ n ) \\mu ( J ) . \\end{align*}"}
+{"id": "2917.png", "formula": "\\begin{align*} { \\cal I } = 2 A _ - A _ + \\left ( \\frac { 1 } { 2 \\pi i } \\int _ { C } \\frac { 1 } { \\zeta - \\zeta _ + } \\cdot \\frac { \\dd \\zeta } { \\zeta - \\zeta _ - } \\right ) = \\frac { 2 A _ - A _ + } { \\zeta _ - - \\zeta _ + } = \\frac { 2 ( \\zeta _ - + 1 ) ( \\zeta _ + + 1 ) } { ( \\zeta _ + - \\zeta _ - ) ^ 3 } \\end{align*}"}
+{"id": "612.png", "formula": "\\begin{align*} d ( p , g ) = \\frac { d ( p , g ) } { d ( p , q ) } d ( p , q ) \\leq \\eta \\left ( \\frac { d ( g , \\pi ^ { - 1 } ( y ) ) } { d ( q , \\pi ^ { - 1 } ( y ) ) } \\right ) r \\leq \\ell _ \\eta r , \\end{align*}"}
+{"id": "8696.png", "formula": "\\begin{align*} \\chi ( S ' _ b ) + \\chi ( F ' _ b ) + \\chi ( h ( F ' _ b ) ) & \\ge \\chi ( S ' _ b ) + \\chi ( F ' _ b ) + \\chi ( h ( F ' _ b ) ) + \\chi ( \\Sigma ) \\\\ & = \\chi ( S ) + \\chi ( F ) + \\chi ( h ( F ) ) . \\end{align*}"}
+{"id": "5595.png", "formula": "\\begin{align*} W ( 1 0 ^ l | 1 ^ k 0 ) & = \\left [ A ( 0 1 ^ k 0 . . . ) - A ( 0 ^ { \\alpha + 1 } 1 . . . ) \\right ] + \\sum _ { j = 2 } ^ l \\left [ A ( 0 ^ { j } 1 ^ k 0 ) - A ( 0 ^ { j + \\alpha } 1 . . . ) \\right ] + \\\\ & + \\left [ A ( 1 0 ^ l 1 ^ k 0 . . . ) - A ( 1 0 ^ { l + \\alpha } 1 . . . ) \\right ] \\\\ & = b _ k - a _ { \\alpha + 1 } + \\sum _ { j = 2 } ^ l ( a _ j - a _ { j + \\alpha } ) + d _ l - d _ { l + \\alpha } \\ . \\end{align*}"}
+{"id": "3026.png", "formula": "\\begin{align*} \\epsilon ^ { 1 } \\wedge \\dots \\wedge \\epsilon ^ { m } \\left ( \\xi \\right ) = C ^ { 2 } \\ , e ^ { 1 } \\wedge \\dots \\wedge e ^ { m } \\left ( \\xi \\right ) \\end{align*}"}
+{"id": "2902.png", "formula": "\\begin{align*} & V ( t ; v ) = e ^ { - A _ E t } v + \\int _ 0 ^ t e ^ { - A _ E ( t - s ) } \\Sigma \\dd W ( s ) + \\int _ 0 ^ t e ^ { - A _ E ( t - s ) } { \\cal F } ( s ) { \\rm e } _ { p , n + 1 } \\dd s \\\\ & = U ( t ; v ) + \\int _ 0 ^ t e ^ { - A _ E ( t - s ) } { \\cal F } ( s ) { \\rm e } _ { p , n + 1 } \\dd s . \\end{align*}"}
+{"id": "7047.png", "formula": "\\begin{align*} F _ { 0 , 6 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ \\ \\ F _ { 1 , 5 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ \\ \\ F _ { 2 , 4 } \\ , : = \\ , 2 4 , \\ \\ \\ \\ \\ \\ \\ F _ { 3 , 3 } \\ , : = \\ , 3 6 , \\ \\ \\ \\ \\ \\ \\ F _ { 4 , 2 } \\ , : = \\ , 6 , \\end{align*}"}
+{"id": "3918.png", "formula": "\\begin{align*} \\bigg | \\frac { \\partial h _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) } { \\partial z _ { i , \\hbar } } \\bigg | \\leq C , \\ \\ 1 \\leq i , j \\leq m , \\hbar = 1 , 2 , \\end{align*}"}
+{"id": "1035.png", "formula": "\\begin{align*} ( x _ 1 \\ast x _ 2 ) ^ { - 1 } = x _ 2 ^ { - 1 } \\ast x _ 1 ^ { - 1 } \\end{align*}"}
+{"id": "1827.png", "formula": "\\begin{align*} \\aligned \\frac d { d t } \\mathcal { Y M } _ e ^ 0 ( \\nabla ^ t ) _ { \\big { | } _ { s = 0 } } = \\frac d { d t } \\mathcal { Y M } _ e ^ 0 ( \\nabla ^ t ) _ { \\big { | } _ { s = 0 } } = \\int _ M \\langle \\delta ^ \\nabla \\big ( \\exp ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) R ^ \\nabla \\big ) , A \\rangle \\ , d v \\ , . \\endaligned \\end{align*}"}
+{"id": "4935.png", "formula": "\\begin{align*} y = \\exp { \\left ( \\frac { \\lambda } { \\sigma - \\frac { 1 } { 2 } } \\right ) } , x = y ^ { - 1 } \\log ^ { 2 } { q } , \\end{align*}"}
+{"id": "8992.png", "formula": "\\begin{align*} A ( q ) = A ( q , s ) = \\left ( \\prod _ { r = 1 } ^ { q } D _ { \\ell _ { r } - 1 } \\right ) \\left ( \\prod _ { r = q + 1 } ^ { s } D _ { \\ell _ { r - 1 } } \\right ) \\end{align*}"}
+{"id": "88.png", "formula": "\\begin{align*} \\beta _ i = v s _ { \\alpha _ 1 } \\cdots s _ { \\alpha _ { i - 1 } } ( \\alpha _ i ) \\in \\Phi , i = 1 , \\dotsc , k . \\end{align*}"}
+{"id": "7865.png", "formula": "\\begin{align*} Z ( v , s ) = \\Psi ^ q + \\mathcal O ( \\norm { ( v , s ) } ) . \\end{align*}"}
+{"id": "8952.png", "formula": "\\begin{align*} A _ { 1 } : = \\sup _ { \\eta _ { k - 1 } < t < \\eta _ k } \\biggl ( \\int _ t ^ { \\eta _ k } v _ 0 ^ { p } \\biggr ) ^ { 1 / p } \\biggl ( \\int _ { \\eta _ { k - 1 } } ^ t v _ 1 ^ { - p ' } \\biggr ) ^ { 1 / p ' } \\le \\biggl ( \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 0 ^ { p } \\biggr ) ^ { 1 / p } \\biggl ( \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ { k } } v _ 1 ^ { - p ' } \\biggr ) ^ { 1 / p ' } \\le 1 . \\end{align*}"}
+{"id": "8780.png", "formula": "\\begin{align*} \\exists j \\leq J , \\ ; \\Pi _ { \\mathrm { k e r } A ^ { \\perp } } \\frac { d ^ j } { d t ^ j } X _ t | _ { t = 0 } \\neq 0 . \\end{align*}"}
+{"id": "6689.png", "formula": "\\begin{align*} 0 = c ( 1 , \\gamma ^ - ) = c ( \\gamma ^ { - 1 } \\gamma , \\gamma ^ - ) = c ( \\gamma ^ { - 1 } , \\gamma \\gamma ^ - ) + c ( \\gamma , \\gamma ^ - ) = c ( \\gamma ^ { - 1 } , \\gamma ^ - ) + c ( \\gamma , \\gamma ^ - ) , \\end{align*}"}
+{"id": "4825.png", "formula": "\\begin{align*} \\rho _ d ( z ) : = \\frac { 1 } { \\pi } \\frac { d ^ 2 ( d - 1 ) } { ( d ^ 2 - | z | ^ 2 ) ^ 2 } \\ , \\textbf { 1 } _ { \\{ | z | < \\sqrt { d } \\} } . \\end{align*}"}
+{"id": "2656.png", "formula": "\\begin{align*} o _ { k } ( n ) = d _ { k } ( n ) . \\end{align*}"}
+{"id": "8260.png", "formula": "\\begin{align*} \\xi ^ { n } _ m ( t ) : = e ^ { - t } \\left [ \\mu ^ { n } _ 1 ( t ) - \\sum _ { i = 1 } ^ { m - 1 } i \\psi ^ { n } _ i ( t ) + ( 2 m + 2 ) \\mu ^ { n } _ 1 ( 0 ) ^ 2 \\right ] . \\end{align*}"}
+{"id": "6959.png", "formula": "\\begin{align*} \\Phi _ 2 ( \\mu ^ t ) = \\int _ 0 ^ 1 H \\big ( \\mu ^ 0 _ u \\circ ( ( 1 - t ) \\mathrm { I d } + t T _ u ) ^ { - 1 } \\big ) \\ , d u \\end{align*}"}
+{"id": "5526.png", "formula": "\\begin{align*} q _ d ^ * = - ( d - 1 ) + O ( d ) \\left ( - \\frac { 1 - \\log 2 } { 2 } d \\right ) , d \\to \\infty . \\end{align*}"}
+{"id": "4424.png", "formula": "\\begin{align*} H ^ 2 ( \\mathbb { D } _ \\epsilon ) = \\{ \\sum _ { n = 0 } ^ { \\infty } a _ n z ^ n \\mid ( a _ n \\epsilon ^ n ) _ { n \\in \\mathbb { N } } \\in \\ell ^ 2 \\} \\end{align*}"}
+{"id": "6415.png", "formula": "\\begin{align*} g ^ { q p } u _ { , p q i } = - v _ { i } + u _ { , i q } \\xi ^ q \\end{align*}"}
+{"id": "3869.png", "formula": "\\begin{align*} \\begin{cases} - \\cdot ( K _ H ( x ) \\nabla \\varPhi ) = \\frac { 1 } { \\varepsilon ^ 2 } \\left ( \\varPhi - \\left ( \\frac { \\alpha } { 2 } | x | ^ 2 + \\beta \\right ) | \\ln \\varepsilon | \\right ) ^ p _ + , \\ & x \\in B _ { R ^ * } ( 0 ) , \\\\ \\varPhi ( x ) = 0 , \\ & x \\in \\partial B _ { R ^ * } ( 0 ) , \\end{cases} \\end{align*}"}
+{"id": "2961.png", "formula": "\\begin{align*} \\tilde c ( t ) : = \\begin{cases} c ( 0 ) & ( t = 0 ) \\\\ c ( 1 - t ) & ( 0 < t < 1 ) . \\end{cases} \\end{align*}"}
+{"id": "312.png", "formula": "\\begin{align*} H = \\mathbb { P } \\end{align*}"}
+{"id": "8668.png", "formula": "\\begin{align*} { { \\bf { e } } _ k } = \\frac { 1 } { { \\sigma } } \\sum \\limits _ { l = 1 } ^ L { { { \\bf { h } } _ l } { e ^ { j \\frac { { 2 \\pi } } { K } k \\left ( { { n _ l } - { n _ { \\max } } + { n ' _ { { \\rm { s p a n } } } } } \\right ) } } } . \\end{align*}"}
+{"id": "2214.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } i . ~ ~ \\phi = 2 \\pi N , ~ N ~ i s ~ i n t e g e r \\\\ i i . ~ ~ \\phi = \\frac { 4 \\pi Z } { N } , \\frac { 2 Z } { N } ~ i s ~ n o t ~ i n t e g e r . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "1056.png", "formula": "\\begin{align*} \\ell ( x _ \\infty \\cdots \\prescript { \\sigma ^ { n - 1 } } { } x _ \\infty ) = n \\ell ( x _ \\infty ) . \\end{align*}"}
+{"id": "7582.png", "formula": "\\begin{align*} \\{ n \\in \\Z : T ^ n a \\in E \\} = A . \\end{align*}"}
+{"id": "2287.png", "formula": "\\begin{align*} \\langle T f , g \\rangle _ \\lambda = \\langle T ^ { ( \\lambda ) } _ a f , g \\rangle _ \\lambda \\end{align*}"}
+{"id": "6581.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { \\Omega } \\nabla n \\cdot ( S ( x , n , c ) \\cdot \\nabla c ) \\ , \\varphi ^ { 3 } \\\\ & \\le \\| S _ { 0 } \\| _ { \\mathcal { C } ( [ 0 , \\gamma ] ) } C _ { 2 } ^ { \\frac { 1 } { 4 } } C _ { 3 } ^ { \\frac { 1 } { 4 } } \\| \\nabla c \\| _ { L ^ { 2 } ( \\Omega \\cap B _ { \\delta } ) } ^ { \\frac { 1 } { 4 } } \\int _ { \\Omega } \\frac { | \\nabla n | ^ { 2 } } { n } \\varphi ^ { 3 } + \\frac { 1 } { 5 } \\int _ { \\Omega } \\frac { | \\nabla n | ^ { 2 } } { n } \\varphi ^ { 3 } + M . \\end{aligned} \\end{align*}"}
+{"id": "8536.png", "formula": "\\begin{align*} & \\sup _ { x \\in \\left ( - \\infty , x _ { 0 } \\right ) } \\left \\| \\langle x \\rangle \\mathcal { P } ^ { + } \\left ( z ^ { - i } \\bar { r } _ 1 ( z ) e ^ { 2 i z x } \\right ) \\right \\| _ { L _ z ^ 2 } \\leq \\left \\| z ^ { - i } \\bar r _ 1 ( z ) \\right \\| _ { H ^ 1 } , i = 0 , 1 , \\\\ & \\sup _ { x \\in \\left ( - \\infty , x _ { 0 } \\right ) } \\left \\| \\langle x \\rangle \\mathcal { P } ^ - \\left ( r _ 2 ( z ) e ^ { - 2 i z x } \\right ) \\right \\| _ { L _ { z } ^ 2 } \\leq \\left \\| r _ 2 ( z ) \\right \\| _ { H ^ 1 } . \\end{align*}"}
+{"id": "7835.png", "formula": "\\begin{align*} \\hat h \\left ( A _ \\phi \\begin{pmatrix} a \\\\ b \\end{pmatrix} , s \\right ) = \\hat h \\left ( \\begin{pmatrix} a \\\\ b \\end{pmatrix} , s \\right ) \\end{align*}"}
+{"id": "543.png", "formula": "\\begin{align*} t _ { 3 , i } + z _ { 3 , i } = - \\tfrac { 1 } { 6 } s _ { \\ 0 , 3 } u _ i - \\tfrac { 1 } { 8 } ( a _ 3 u _ i + a _ { - 3 } u _ i ) . \\end{align*}"}
+{"id": "2897.png", "formula": "\\begin{align*} { \\frak G } _ x { \\bf T } ( t ) = \\sum _ { y = 0 } ^ n \\int _ 0 ^ { \\theta } { { \\frak G } _ { x , y } ( s ) } T _ y \\left ( t - s \\right ) \\dd s . \\end{align*}"}
+{"id": "8653.png", "formula": "\\begin{align*} y \\left [ i \\right ] = \\sum \\limits _ { \\left ( { l , l ' } \\right ) \\in { \\cal S } } { { \\bf { h } } _ l ^ H { { \\bf { F } } _ { l ' } } { \\bf { d } } \\left [ { i - { n _ l } - { \\kappa _ { l ' } } } \\right ] } + z \\left [ { i } \\right ] . \\end{align*}"}
+{"id": "2789.png", "formula": "\\begin{align*} \\mathrm { v a r } [ S _ { m } ( y ) - S _ { k } ( x ) ] & = \\mathrm { v a r } [ \\varphi ^ { D _ { N } , \\Delta ^ { k } ( x ) } ( y ) - \\varphi ^ { D _ { N } , \\Delta ^ { k } ( x ) } ( x ) + \\varphi ^ { \\Delta ^ { k } ( x ) , \\Delta ^ { m } ( x ) } ( y ) ] \\\\ & \\geq \\mathrm { v a r } [ \\varphi ^ { \\Delta ^ { k } ( x ) , \\Delta ^ { m } ( x ) } ( y ) ] \\geq \\mathrm { v a r } [ \\varphi ^ { \\Delta ^ { m + 1 } ( x ) , \\Delta ^ { m } ( x ) } ( y ) ] \\geq \\gamma + o ( 1 ) . \\end{align*}"}
+{"id": "5700.png", "formula": "\\begin{align*} D ( X ) = \\sum _ { 0 \\le i \\ne j \\le n - 1 } B _ { i j } X ^ { 2 ^ i + 2 ^ j } . \\end{align*}"}
+{"id": "1301.png", "formula": "\\begin{align*} \\begin{gathered} \\bigl ( C 3 _ { ( 1 / 2 ; \\ , 1 / 2 ; \\ , 0 ) } , C 3 _ { ( 1 / 2 ; \\ , 0 ; \\ , 1 / 2 ) } \\bigr ) , \\ ; \\bigl ( C 3 _ { ( 1 ; \\ , 0 ; \\ , 0 ) } , C 3 _ { ( 1 ; \\ , 0 ; \\ , 0 ) } \\bigr ) , \\ ; \\bigl ( C 3 _ { ( 0 ; \\ , 0 ; \\ , 0 ) } , C 3 _ { ( 0 ; \\ , 0 ; \\ , 0 ) } \\bigr ) , \\\\ \\bigl ( C 2 _ { ( 1 ; \\ , 1 ; \\ , 0 ) } , C 2 _ { ( 1 ; \\ , 0 ; \\ , 1 ) } \\bigr ) , \\ ; \\bigl ( C 2 _ { ( 1 ; \\ , 0 ; \\ , 0 ) } , C 2 _ { ( 1 ; \\ , 0 ; \\ , 0 ) } \\bigr ) , \\ ; \\bigl ( C 2 _ { ( 0 ; \\ , 0 ; \\ , 0 ) } , C 1 _ { ( 1 ) } \\bigr ) , \\ ; \\bigl ( C 1 _ { ( 0 ) } , C 1 _ { ( 0 ) } \\bigr ) . \\end{gathered} \\end{align*}"}
+{"id": "7313.png", "formula": "\\begin{align*} \\mu _ T ( x , \\mu _ T ( y , z ) ) = & \\mu _ T ( x , \\sum _ { i = 1 } ^ k ( y | v _ i ) ( z | v _ i ) v _ i ) = \\sum _ { j = 1 } ^ k \\sum _ { i = 1 } ^ k ( y | v _ i ) ( z | v _ i ) ( x | v _ j ) ( v _ i | v _ j ) v _ j \\\\ = & \\sum _ { i = 1 } ^ k ( y | v _ i ) ( z | v _ i ) ( x | v _ i ) ( v _ i | v _ i ) v _ i \\end{align*}"}
+{"id": "7491.png", "formula": "\\begin{align*} & \\lambda ^ { a / 2 + 1 } \\left ( ( n - 2 - a ) \\int _ { B _ { \\rho } } r ^ { - a } | \\nabla u | ^ 2 \\d x + \\frac { a ( 8 - a ) } { 4 } \\int _ { B _ { \\rho } } r ^ { - a } u _ r ^ 2 \\d x \\right ) \\\\ & \\leq ( n - 2 - a ) \\int _ { B _ 1 } | x | _ { A ^ { - 1 } ( 0 ) } ^ { - a } | \\nabla u | _ { A ( 0 ) } ^ 2 \\zeta ^ 2 \\d x + \\dfrac { a ( 8 - a ) } { 4 } \\int _ { B _ 1 } | x | _ { A ^ { - 1 } ( 0 ) } ^ { - a - 2 } | x \\cdot \\nabla u | ^ 2 \\zeta ^ 2 \\d x . \\end{align*}"}
+{"id": "3118.png", "formula": "\\begin{align*} \\varrho ^ \\star ( x ) ( \\xi ) = \\varrho ^ * ( \\alpha ( x ) ) \\big ( ( \\beta ^ { - 2 } ) ^ * ( \\xi ) \\big ) , \\quad \\forall \\ x \\in A , \\xi \\in V ^ * . \\end{align*}"}
+{"id": "353.png", "formula": "\\begin{align*} B _ { k } : = \\frac { 2 } { ( n - 1 ) ( 2 \\phi _ k + \\frac { 1 } { 2 \\phi _ k } ) ^ { 2 } } \\left [ \\begin{array} { c c c } 0 & \\ 0 & \\ 0 \\\\ \\ 0 & \\frac { C _ k } { 4 \\phi _ { k } ^ { 2 } } & \\frac { C _ k + \\widetilde { C _ { k } } } { 4 \\phi _ k ^ { 2 } } \\\\ \\ 0 & \\frac { C _ k + \\widetilde { C _ { k } } ^ { ' } } { 4 \\phi _ { k } ^ { 2 } } & C _ k \\\\ \\end{array} \\right ] ~ ~ ~ k = 1 , \\dotsc , n - 2 , \\end{align*}"}
+{"id": "7311.png", "formula": "\\begin{align*} \\begin{array} { l } { \\rm S I N R } _ { v , j } ^ q = 2 y _ { v , j } ^ q \\sqrt { p _ { v , j } ^ { q } { \\left ( { { R _ { \\rm { P D } } } { G _ { v , j } ^ { q } } } \\right ) } ^ 2 } - y _ { v , j } ^ q \\left ( { \\sum \\limits _ { j ' \\ne j } { \\sum \\limits _ { v ' \\ne v } p _ { v ' , j ' } ^ { q } { \\left ( { { R _ { \\rm { P D } } } { G _ { v ' , j } ^ { q } } } \\right ) } ^ 2 } + N _ { \\rm V L C } B _ { \\rm V L C } } \\right ) . \\end{array} \\end{align*}"}
+{"id": "6477.png", "formula": "\\begin{align*} U = U [ C , \\beta , \\Delta , p ] = \\stackrel { d e f } { = } \\sup _ { p > \\Delta } \\ \\sup _ { \\xi \\in L _ p } \\ \\sup _ { X \\in L _ p } Y [ C , \\beta , \\Delta ] ( \\xi , X , p ) , \\end{align*}"}
+{"id": "3024.png", "formula": "\\begin{align*} \\ell : \\mathbb { R } \\rightarrow \\mathbb { R } \\ell \\left ( r \\right ) = c \\left ( r - \\sin r \\right ) ^ { 1 / 3 } \\end{align*}"}
+{"id": "8061.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\mathbb { E } \\left [ \\varepsilon | \\hat { \\mathbf { H } } \\right ] } { \\partial a _ c \\partial a _ c } > 0 ~ { \\rm a n d } ~ \\frac { \\partial ^ 2 \\mathbb { E } \\left [ \\varepsilon | \\hat { \\mathbf { H } } \\right ] } { \\partial a _ i \\partial a _ i } > 0 , ~ i = 1 , 2 , \\ldots , K \\end{align*}"}
+{"id": "2303.png", "formula": "\\begin{align*} T ^ { ( \\lambda ) } _ a | _ W = T = T ^ { ( \\lambda ) } _ b | _ W . \\end{align*}"}
+{"id": "7332.png", "formula": "\\begin{align*} h ( X ) = - \\int f ( x ) \\log f ( x ) d x . \\end{align*}"}
+{"id": "7173.png", "formula": "\\begin{align*} Q ^ 2 - B Q - \\Bigl [ I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } , Q \\Bigr ] + C = 0 , \\end{align*}"}
+{"id": "8100.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty \\frac { d s } { \\sqrt { F ( s ) } } < \\infty \\quad \\mbox { w h e r e } \\ ; F ( s ) = \\int _ 0 ^ s f ( t ) d t . \\end{align*}"}
+{"id": "8596.png", "formula": "\\begin{align*} | A | ^ { m - 1 } \\left | P _ { [ u _ 1 , \\dots , u _ m ] ^ \\bot } A \\right | _ { n - m } \\le \\prod _ { i = 1 } ^ m \\left | P _ { u _ i ^ \\bot } A \\right | _ { n - 1 } . \\end{align*}"}
+{"id": "2688.png", "formula": "\\begin{align*} a _ d ( n ) = { } & d a _ d ( n - 1 ) + d a _ d ( n - 2 ) - \\binom { d } { 2 } a _ d ( n - 3 ) + d a _ d ( n - 4 ) - d ^ 2 a _ d ( n - 5 ) \\\\ & + d a _ d ( n - 6 ) - \\sum _ { k \\geq 8 } ^ { n + 1 } \\mu _ d ( k ) a _ d ( n + 1 - k ) . \\end{align*}"}
+{"id": "2455.png", "formula": "\\begin{align*} F _ { q } ( s ) = \\frac { \\Gamma ( m ) } { Q _ { m } ( 2 - q ) } \\frac { 1 } { s ^ { m } } \\ ; , \\mbox { f o r } \\ ; m \\geq 2 . \\end{align*}"}
+{"id": "9314.png", "formula": "\\begin{align*} b _ { 1 } u _ { 1 } ^ { 2 } - b _ { 2 } u _ { 2 } ^ { 2 } = 4 ^ { m } p \\cdot 2 ^ { 2 k } \\implies b _ { 2 } \\equiv 0 \\pmod 2 , \\end{align*}"}
+{"id": "8870.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } ( 1 _ { B } - \\mathbb { E } ( 1 _ B | \\mathcal { A } ) ) ( T _ { 1 } ^ { 3 n ^ 2 } T _ { 2 } ^ { 8 n ^ 2 } x ) \\mathbb { E } ( 1 _ A | \\mathcal { A } ) ( T _ { 1 } ^ { n ^ 2 } T _ { 2 } ^ { - n ^ 2 } x ) = 0 . \\end{align*}"}
+{"id": "5318.png", "formula": "\\begin{align*} f ( z , t ) = ( 2 \\pi ) ^ { - n - 1 } \\ , \\int _ { - \\infty } ^ \\infty e ^ { i \\lambda t } \\ , \\big ( \\sum _ { k = 0 } ^ \\infty \\rho _ k ^ \\lambda ( f ) e ^ { n - 1 } _ { k , \\lambda } ( z , 0 ) \\big ) | \\lambda | ^ n d \\lambda . \\end{align*}"}
+{"id": "6092.png", "formula": "\\begin{align*} \\mu ^ { ( i ) } = j ^ { ( i ) } ( j ^ { ( i ) } + 1 ) \\ , , \\ i = 0 , 1 , \\dots , 4 \\ , , \\mu _ n ^ { ( 1 2 ) } = j _ n ^ { ( 1 2 ) } ( j _ n ^ { ( 1 2 ) } + 1 ) \\quad \\mu _ p ^ { ( 1 2 3 ) } = j _ p ^ { ( 1 2 3 ) } ( j _ p ^ { ( 1 2 3 ) } + 1 ) . \\end{align*}"}
+{"id": "7066.png", "formula": "\\begin{align*} G _ { 6 , 0 , 0 , 1 } \\ , : = \\ , 1 \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ G _ { 6 , 0 , 0 , 1 } \\ , : = \\ , - \\ , 1 , \\end{align*}"}
+{"id": "5977.png", "formula": "\\begin{align*} w _ { R / W _ k } ( x ) = \\frac { c ' } { ( 1 + | x | ^ 2 / ( R / W _ k ) ^ 2 ) ^ { 1 0 } } , \\qquad \\| w _ { R / W _ k } \\| _ 1 = 1 . \\end{align*}"}
+{"id": "4853.png", "formula": "\\begin{align*} K ( c _ 1 , \\dots , c _ n , 2 m ) = & p _ j \\circ H ( c _ 1 , \\dots , c _ n , m ) \\\\ = & p _ j \\circ i _ { U _ k } ( c 1 , \\dots , c _ n ) = p _ j ( c _ 1 , \\dots , c _ n ) . \\end{align*}"}
+{"id": "8500.png", "formula": "\\begin{align*} & u ( x ) e ^ { - i ( 2 c _ - ( x ) + c ) } = 2 i \\lim _ { z \\rightarrow 0 } M _ { + , 1 2 } ( x ; z ) , \\\\ & u ( x ) e ^ { 2 i ( c _ - ( x ) + c ) } = 2 i \\lim _ { z \\rightarrow 0 } M _ { - , 1 2 } ( x ; z ) . \\\\ & \\partial _ x \\left ( \\bar { u } _ x ( x ) e ^ { i c _ + ( x ) } \\right ) = 2 i e ^ { - i c _ + ( x ) } \\lim _ { | z | \\rightarrow \\infty } z M _ { \\pm , 2 1 } ( x ; z ) , \\end{align*}"}
+{"id": "6751.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty z ^ i \\ , \\phi _ s ( z ) \\ , [ \\Phi _ s ( z ) ] ^ { j + \\alpha - 1 } \\ , \\mathrm { d } z = \\sum _ { k = 0 } ^ \\infty \\left [ v _ k ( j + \\alpha - 1 ) + ( - 1 ) ^ { i + k } \\ , \\binom { j + \\alpha - 1 } { k } \\right ] \\ , J _ { i , k } ^ { ( s ) } . \\end{align*}"}
+{"id": "6253.png", "formula": "\\begin{align*} \\varphi _ 0 \\eta _ 0 + \\ldots + \\varphi _ p \\eta _ p = 0 , \\end{align*}"}
+{"id": "633.png", "formula": "\\begin{align*} \\begin{aligned} & x _ 0 = 1 , & y _ 0 = 0 , \\\\ & x _ 1 , & y _ 1 , \\\\ & x _ 2 = y _ 1 , & y _ 2 , \\\\ & x _ 3 = x _ 2 ^ 2 + x _ 1 x _ 2 , & y _ 3 , \\\\ & \\ldots \\end{aligned} \\end{align*}"}
+{"id": "3025.png", "formula": "\\begin{align*} \\Phi : \\mathbb { R } ^ { 3 } \\rightarrow \\mathbb { R } ^ { 3 } \\rtimes S O _ { 3 } \\Phi \\left ( r u \\right ) = \\left ( \\ell \\left ( r \\right ) u , \\exp \\left ( C _ { r u } \\right ) \\right ) \\end{align*}"}
+{"id": "1991.png", "formula": "\\begin{align*} f ( x - T ( x ) ) = - f ( - x - T ( x ) ) , \\end{align*}"}
+{"id": "8947.png", "formula": "\\begin{align*} \\| v _ 1 ^ { - 1 } \\| _ { { L ^ { p ' } ( 0 , c ) } } \\| v _ 0 \\| _ { { L ^ p } ( 0 , c ) } = \\| v _ 1 ^ { - 1 } \\| _ { L ^ { p ' } ( c , \\infty ) } \\| v _ 0 \\| _ { L ^ p ( c , \\infty ) } = \\infty . \\end{align*}"}
+{"id": "2661.png", "formula": "\\begin{align*} u ^ { \\prime \\prime } + B u ^ \\prime ( t ) + A ( u ) = f ( t ) \\end{align*}"}
+{"id": "3366.png", "formula": "\\begin{align*} \\zeta _ k = \\frac { 1 } { k } \\sum _ { j = 0 } ^ { k - 1 } \\delta _ { u _ j } , \\end{align*}"}
+{"id": "7834.png", "formula": "\\begin{align*} \\hat \\Phi \\left ( \\begin{pmatrix} a \\\\ b \\end{pmatrix} , s \\right ) = \\hat h ( ( a , b ) ^ \\top , s ) \\begin{pmatrix} a \\\\ b \\end{pmatrix} , \\end{align*}"}
+{"id": "6907.png", "formula": "\\begin{align*} \\sup _ { Q \\in \\P ( \\R ^ n ) } \\left ( \\int _ { \\R ^ n } f \\ , d Q - H ( Q ) \\right ) = \\log Z \\in ( - \\infty , \\infty ) , \\end{align*}"}
+{"id": "409.png", "formula": "\\begin{align*} d _ { \\mathcal { F } } ( g _ 1 , g _ 2 ) : = \\frac 1 2 \\left ( d ( g _ 1 , \\mathcal { F } _ { g _ 2 } ) + d ( g _ 2 , \\mathcal { F } _ { g _ 1 } ) \\right ) , \\mbox { f o r a l l } g _ 1 , g _ 2 \\in X , \\end{align*}"}
+{"id": "1085.png", "formula": "\\begin{align*} I : = \\{ \\beta \\neq \\gamma \\in \\Phi ^ + _ J \\mid s _ \\beta ( \\gamma ) \\notin \\Phi ^ + _ J \\} . \\end{align*}"}
+{"id": "7498.png", "formula": "\\begin{align*} 2 ^ { \\star } : = \\frac { n - 1 } { n - 2 } 2 \\end{align*}"}
+{"id": "5757.png", "formula": "\\begin{align*} \\partial _ X \\varphi = \\nabla _ X \\varphi - \\frac { 1 } { 2 } ( \\sqrt { c _ 1 } \\ X _ 1 \\cdot \\nu _ 1 + \\sqrt { c _ 2 } \\ X _ 2 \\cdot \\nu _ 2 ) \\cdot \\varphi + \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi \\end{align*}"}
+{"id": "6622.png", "formula": "\\begin{align*} 0 = R _ { 4 1 } ^ \\delta - R _ { 6 3 } ^ \\delta = h ^ 2 + \\varepsilon _ 2 \\delta _ 2 \\left ( \\lambda - \\rho \\right ) . \\end{align*}"}
+{"id": "1255.png", "formula": "\\begin{align*} \\hat { c } _ { \\Delta } ( \\eta , \\xi _ { \\Delta } ) : = c _ { \\Delta } ( \\xi _ { \\Delta } \\eta _ { \\Delta ^ c } , \\eta _ { \\Delta } ) \\frac { \\gamma _ { \\Delta } ( \\xi _ { \\Delta } | \\eta _ { \\Delta ^ c } ) } { \\gamma _ { \\Delta } ( \\eta _ { \\Delta } | \\eta _ { \\Delta ^ c } ) } . \\end{align*}"}
+{"id": "3846.png", "formula": "\\begin{align*} X ^ \\Delta ( t _ { n + 1 } ) = X ^ \\Delta ( t _ { n } ) + F ( X ^ \\Delta ( t _ { n } ) ) \\Delta + \\Delta ^ { 1 / 2 } G ( X ^ \\Delta ( t _ { n } ) ) \\xi _ { n + 1 } . \\end{align*}"}
+{"id": "6237.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } \\partial _ u \\tau = 2 \\Gamma _ { u v } ^ v \\tau ( 1 - \\tau ) \\\\ \\partial _ v \\tau = 2 \\Gamma _ { v u } ^ u ( 1 - \\tau ) , \\end{array} \\right . \\end{align*}"}
+{"id": "4133.png", "formula": "\\begin{align*} { \\rm L M O } ( \\R ^ n _ + ) : = \\{ u \\in \\C ^ { \\infty } ( \\R ^ n _ + , \\mathbb { C } ^ N ) : L u = 0 \\R ^ n _ + , \\ , \\| u \\| _ { \\mathcal { C } ( \\R ^ n _ + ) } < \\infty \\} . \\end{align*}"}
+{"id": "7784.png", "formula": "\\begin{align*} \\theta ^ h = 2 ~ { \\rm a n d } ~ q \\ge h \\log _ 2 ( 1 + d 2 ^ { b _ 0 + 1 } ) . \\end{align*}"}
+{"id": "1976.png", "formula": "\\begin{align*} G ( x ) = - \\frac { 2 ( a x ^ 2 + b x + c ) } { 2 a x + b } , x \\ne - b / ( 2 a ) , \\end{align*}"}
+{"id": "8958.png", "formula": "\\begin{align*} I I _ 1 ^ p : = \\sum _ { | k | \\le N } \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { p } ( x ) \\bigl [ V _ 1 ^ - ( x ) \\bigr ] ^ { - p } \\bigl [ v _ 1 ^ { - p ' } ( a ( x ) ) \\ , a ' ( x ) \\bigr ] ^ p \\biggl ( \\int _ { \\eta _ { k - 1 } } ^ x v _ 1 ^ { - p ' } ( t ) \\bigl | G ^ { ( 0 ) } _ { 1 , k } ( t ) \\bigr | ^ { p ' - 1 } \\ , d t \\biggr ) ^ p \\ , d x . \\end{align*}"}
+{"id": "163.png", "formula": "\\begin{align*} V \\widehat { \\otimes } _ K W = \\varprojlim _ { n , m } V _ { n } \\widehat { \\otimes } _ K W _ m . \\end{align*}"}
+{"id": "2126.png", "formula": "\\begin{align*} ( a ^ 2 + 2 a + 4 ) s ^ 3 + ( 4 - a ( 1 9 a + 3 4 ) ) s ^ 2 + ( a ( 1 9 a - 3 4 ) - 4 ) s - a ^ 2 + 2 a - 4 = 0 . \\end{align*}"}
+{"id": "7983.png", "formula": "\\begin{align*} 0 \\geq \\nabla _ { i i } E & = - w ^ { 1 1 } \\nabla _ { i i } w _ { 1 1 } + 2 w ^ { 1 1 } \\sum w ^ { k k } ( \\nabla _ { 1 } w _ { i k } ) ^ { 2 } - ( w ^ { 1 1 } ) ^ { 2 } ( \\nabla _ { i } w _ { 1 1 } ) ^ { 2 } - d \\left ( \\frac { h _ { i i } } { h } - \\frac { h ^ { 2 } _ { i } } { h ^ { 2 } } \\right ) + l \\rho ^ { 2 } _ { i } + l \\rho \\rho _ { i i } . \\end{align*}"}
+{"id": "6445.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { 0 { \\le } m _ 1 { \\le } \\cdots { \\le } m _ n < \\infty } \\frac { ( \\alpha ) _ { m _ 1 } } { { m _ 1 } ! } \\frac { ( \\beta ) _ { m _ 1 } } { ( \\gamma ) _ { m _ 1 } } \\frac { { m _ n } ! } { ( \\alpha ) _ { m _ n } } \\frac { ( \\gamma ) _ { m _ { n } } } { ( \\beta ) _ { m _ n } } \\\\ & \\times \\left \\{ \\prod _ { i = 1 } ^ { n } \\frac { 1 } { ( m _ i + \\alpha ) ^ { a _ i } ( m _ i + \\beta ) ^ { b _ i } ( m _ i + \\gamma ) ^ { c _ i } } \\right \\} , \\end{aligned} \\end{align*}"}
+{"id": "6376.png", "formula": "\\begin{align*} \\int ^ { \\infty } _ { 0 } \\frac { 1 } { x ^ 2 } f ( x ) \\mathrm { d } x = & \\frac { 2 \\theta } { \\operatorname { B } ( a , b ) } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ n \\Gamma ( b ) } { \\Gamma ( b - n ) n ! } \\int ^ { \\infty } _ { 0 } \\frac { \\exp ( - t ) } { [ 2 ( a + n ) \\theta ] ^ 2 } \\mathrm { d } t \\\\ = & \\frac { 1 } { \\theta \\operatorname { B } ( a , b ) } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ n \\Gamma ( b ) } { \\Gamma ( b - n ) n ! } \\frac { 1 } { ( a + n ) ^ 2 } . \\end{align*}"}
+{"id": "6188.png", "formula": "\\begin{align*} B _ 2 = \\frac { ( L + 2 ) B _ 1 } { Q } \\left ( \\frac { ( L + 2 ) B _ 1 } { Q } - 1 \\right ) - 2 \\frac { ( L + 1 ) B _ 1 } { Q } , \\end{align*}"}
+{"id": "1102.png", "formula": "\\begin{align*} { S ^ { \\rm { O } } } = \\frac { { { W _ s } I } } { { K L } } { \\widetilde \\varepsilon _ K } \\left ( { { \\gamma ^ { \\rm { O } } } } \\right ) . \\end{align*}"}
+{"id": "8791.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } X _ t = B ( X _ t , X _ t ) \\\\ X _ t | _ { t = 0 } = x _ 0 \\end{cases} \\end{align*}"}
+{"id": "3370.png", "formula": "\\begin{align*} & \\| \\boldsymbol { u } _ k - \\boldsymbol { u } _ k ' \\| = \\sum _ { i = 0 } ^ { m - 1 } \\| { u } _ { k + i } - { u } _ { k + i } ' \\| \\leq \\sum _ { i = 0 } ^ { m - 1 } \\| { u } _ { m - 1 } - { u } _ { m - 1 } ' \\| q ^ { k + i - m + 1 } \\\\ & \\leq C _ { m , q } q ^ k \\| \\boldsymbol { u } _ 0 - \\boldsymbol { u } _ 0 ' \\| , 1 \\leq k \\leq n . \\end{align*}"}
+{"id": "2293.png", "formula": "\\begin{align*} \\Phi _ + = \\{ e _ j - e _ k \\mid 1 \\leq j < k \\leq n \\} . \\end{align*}"}
+{"id": "8594.png", "formula": "\\begin{align*} ( x _ 1 , z _ 1 ) = \\lambda _ 1 ( x _ m , z _ m ) , \\dots , ( x _ { m - 1 } , z _ { m - 1 } ) = \\lambda _ { m - 1 } ( x _ m , z _ m ) , \\end{align*}"}
+{"id": "7278.png", "formula": "\\begin{align*} a _ { n - k } ( \\gamma ^ { ( q ^ { n - k } - q ^ n ) } - 1 ) \\beta ^ { q ^ { n - k } } + \\cdots + a _ 0 ( \\gamma ^ { 1 - q ^ n } - 1 ) \\beta = 0 . \\end{align*}"}
+{"id": "8804.png", "formula": "\\begin{align*} \\left | \\Pi _ V e ^ { | z | J ^ \\perp t } \\tilde { Y } _ 0 \\right | ^ 2 & = \\left | e ^ { | z | J ^ \\perp t } \\Pi _ { V } \\tilde { Y } _ 0 \\right | ^ 2 = e ^ { 2 | z | \\lambda _ R t } \\left | \\Pi _ { V } \\tilde { Y } _ 0 \\right | ^ 2 . \\end{align*}"}
+{"id": "1886.png", "formula": "\\begin{align*} \\partial _ s \\mathcal P ( x , y , s ) = - \\frac { \\mathcal { Q } ( x , y , \\cos s ) } { 2 \\sin ^ 2 s } , \\end{align*}"}
+{"id": "7987.png", "formula": "\\begin{align*} & \\frac { h _ { 1 1 t } + h _ { 1 1 } } { h _ { t } + h } - \\frac { ( h _ { 1 } + h _ { 1 t } ) ^ { 2 } } { ( h _ { t } + h ) ^ { 2 } } \\\\ & = \\sum w ^ { i i } \\nabla _ { 1 1 } w _ { i i } - \\sum w ^ { i i } w ^ { k k } ( \\nabla _ { 1 } w _ { i k } ) ^ { 2 } + \\nabla _ { 1 1 } \\chi . \\end{align*}"}
+{"id": "4801.png", "formula": "\\begin{align*} \\Delta ^ 2 u = f \\qquad & D , \\\\ [ 1 m m ] u = \\frac { \\partial u } { \\partial \\nu } = 0 & \\partial D . \\end{align*}"}
+{"id": "1734.png", "formula": "\\begin{align*} \\begin{aligned} \\abs { \\omega ( f ( a ) ) g ( a ) - \\omega ( f ( b ) ) g ( b ) } & \\leq \\abs { \\omega ( f ( a ) ) - \\omega ( f ( b ) ) } \\abs { g ( a ) } + \\abs { \\omega ( f ( b ) ) } \\abs { g ( a ) - g ( b ) } \\\\ & \\leq \\norm { \\omega \\circ f } _ L d ( a , b ) \\norm { g } _ L d ( 0 , a ) + 1 \\cdot \\norm { g } _ L d ( a , b ) \\\\ & \\leq C \\norm { g } _ L d ( a , b ) \\end{aligned} \\end{align*}"}
+{"id": "7374.png", "formula": "\\begin{gather*} \\mathcal { U } = \\bigl \\{ Q \\in \\mathbb { P } : Q \\ll P ^ * ( \\pi _ 1 , \\ldots , \\pi _ n ) Q \\bigr \\} . \\end{gather*}"}
+{"id": "957.png", "formula": "\\begin{align*} \\mathcal { R } _ j : = \\widetilde { u } _ { \\mathrm { a p } , j } - u _ { \\mathrm { a p } , j } . \\end{align*}"}
+{"id": "5908.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ T g _ n ^ { + } ( t ) d t = 0 . \\end{align*}"}
+{"id": "6973.png", "formula": "\\begin{align*} M ( \\lambda _ 1 / \\lambda _ 2 ) = M ( \\lambda _ 2 / \\lambda _ 1 ) = | \\lambda _ 1 \\lambda _ 2 | \\max ( 1 , | \\lambda _ 1 / \\lambda _ 2 | ) \\max ( 1 , | \\lambda _ 2 / \\lambda _ 1 | ) = | \\lambda _ 1 | ^ 2 . \\end{align*}"}
+{"id": "9268.png", "formula": "\\begin{align*} \\int _ \\Omega \\Phi ( x ) \\ , | \\operatorname { g r a d } f ( x ) | \\ , d V ( x ) = \\int _ 0 ^ \\infty d s \\int _ { \\Omega \\cap \\{ | f | = s \\} } \\Phi ( x ) \\ , d S ( x ) , \\end{align*}"}
+{"id": "1267.png", "formula": "\\begin{align*} ( r _ m \\eta ) _ x = \\begin{cases} 1 , & \\abs { x } _ { \\infty } < m , \\\\ \\ \\eta _ x & \\abs { x } _ { \\infty } \\geq m . \\end{cases} \\end{align*}"}
+{"id": "5246.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi & = S ( t ) \\varphi _ 0 + \\int _ { 0 } ^ { t } S ( t - s ) \\mathbb { P } \\nabla \\cdot F ( s ) \\dd s . \\end{aligned} \\end{align*}"}
+{"id": "944.png", "formula": "\\begin{align*} & \\partial _ t Q ( t ) ^ { - 1 } P ^ { - 1 } \\begin{pmatrix} w \\\\ \\overline { w } \\end{pmatrix} = Q ( t ) ^ { - 1 } P ^ { - 1 } \\begin{pmatrix} S _ 2 ( t ) + e ^ { 3 i \\lambda _ 1 ( \\Phi _ 1 - \\theta _ 1 ) } R _ 2 ( t ) \\\\ \\overline { S _ 2 ( t ) + e ^ { 3 i \\lambda _ 1 ( \\Phi _ 1 - \\theta _ 1 ) } R _ 2 ( t ) } \\end{pmatrix} . \\end{align*}"}
+{"id": "5263.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty K _ \\mu ( \\alpha x ) K _ \\nu ( \\beta x ) x ^ { s - 1 } d x & = 2 ^ { s - 3 } \\alpha ^ { - s - \\nu } \\beta ^ \\nu \\Gamma \\left ( \\frac { 1 } { 2 } ( s + \\mu + \\nu ) \\right ) \\Gamma \\left ( \\frac { 1 } { 2 } ( s - \\mu - \\nu ) \\right ) \\\\ & \\int _ 0 ^ 1 t ^ { \\frac { s - \\mu + \\nu } { 2 } - 1 } ( 1 - t ) ^ { \\frac { s + \\mu - \\nu } { 2 } - 1 } \\left ( 1 - \\left ( 1 - \\frac { \\beta ^ 2 } { \\alpha ^ 2 } \\right ) t \\right ) ^ { - \\frac { s + \\mu + \\nu } { 2 } } d t . \\end{align*}"}
+{"id": "8585.png", "formula": "\\begin{align*} V ( L , K [ n - 1 ] ) = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } h _ L ( u ) d S _ K ( u ) , \\end{align*}"}
+{"id": "2882.png", "formula": "\\begin{align*} V _ x ( t ) = \\sum _ { x ' = 1 } ^ n \\int _ 0 ^ \\theta { \\frak g } _ { x , x ' } ( s ) V _ { x ' } ( t - s ) \\dd s + v _ x ( t ) . \\end{align*}"}
+{"id": "3418.png", "formula": "\\begin{align*} \\tilde { L } _ { \\lambda } ( E ) = L _ { \\lambda } ( E ) - \\ln { 2 } . \\end{align*}"}
+{"id": "3912.png", "formula": "\\begin{align*} h ( x , y ) \\leq \\frac { 1 } { 2 \\pi } \\ln \\frac { 1 } { \\max \\{ | x - y | , d i s t ( x , \\partial \\Omega ) , d i s t ( y , \\partial \\Omega ) \\} } , \\end{align*}"}
+{"id": "3812.png", "formula": "\\begin{align*} | | \\mathcal { A } _ q ^ p | | ^ 2 _ { L ^ 2 ( \\mathbb R , \\mathbb { H } ) } = E _ q \\left ( | p | ^ 2 \\right ) . \\end{align*}"}
+{"id": "5203.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } A _ { t } \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) \\dd x = - \\mu \\int _ { \\mathbb { R } ^ { 3 } } \\nabla \\times B \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) \\dd x . \\end{align*}"}
+{"id": "7866.png", "formula": "\\begin{align*} Z ( v , s ) & = \\Psi ^ q + v + \\psi _ v ( 0 , p _ 0 ) v + s \\psi _ p ( 0 , p _ 0 ) p ' ( 0 ) + \\mathcal O ( \\norm { ( v , s ) } ^ 2 ) \\\\ & = \\Psi ^ q + v + \\mathcal O ( \\norm { ( v , s ) } ^ 2 ) , \\end{align*}"}
+{"id": "1258.png", "formula": "\\begin{align*} \\int _ \\Omega \\left ( \\sum _ { \\Delta \\Subset \\Z ^ d } h _ \\Delta ( \\eta ) \\right ) ^ 2 \\mu ( d \\eta ) = 0 \\end{align*}"}
+{"id": "7462.png", "formula": "\\begin{align*} N _ f ( z ) & = \\exp \\left ( \\textstyle \\sum _ { k = 1 } ^ { + \\infty } \\ , \\frac { ( - 1 ) ^ { k + 1 } } { k } z ^ k - \\frac { ( - 1 ) ^ { k + 1 } } { k } ( m z ) ^ k \\right ) \\\\ & = e ^ { \\ln ( 1 + z ) - \\ln ( 1 + m z ) } \\\\ & = \\dfrac { 1 + z } { 1 + m z } . \\end{align*}"}
+{"id": "4958.png", "formula": "\\begin{align*} & V ( \\xi _ 0 , n , x ) = \\sup _ { \\theta \\in \\Theta _ n } E _ { \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ n Z ^ \\theta _ i \\biggr ) \\biggr ] . \\end{align*}"}
+{"id": "6485.png", "formula": "\\begin{align*} \\psi _ { \\Delta , \\beta } ( p ) = \\psi _ { \\Delta , \\ \\beta } [ \\psi ] ( p ) \\stackrel { d e f } { = } \\frac { p } { p - \\Delta } \\ \\beta ^ p \\ \\psi ( p ) , \\ \\Delta < p \\le b , \\ \\beta > 1 . \\end{align*}"}
+{"id": "7131.png", "formula": "\\begin{align*} g ' ( t _ 0 ) = & \\frac 1 { a ( t _ 0 ) d ( t _ 0 ) - b ( t _ 0 ) c ( t _ 0 ) } ( ( a ( t _ 0 ) c ' ( t _ 0 ) - a ' ( t _ 0 ) c ( t _ 0 ) ) t ^ 2 \\\\ & + ( a ' ( t _ 0 ) d ( t _ 0 ) - b ( t _ 0 ) c ' ( t _ 0 ) - a ( t _ 0 ) d ' ( t _ 0 ) + b ' ( t _ 0 ) c ( t _ 0 ) ) t + ( b ' ( t _ 0 ) d ( t _ 0 ) - d ' ( t _ 0 ) b ( t _ 0 ) ) ) \\frac { \\partial } { \\partial t } \\end{align*}"}
+{"id": "5787.png", "formula": "\\begin{align*} F _ 1 = \\frac { 1 } { \\sqrt { | c _ 1 | } } \\ \\langle \\langle \\nu _ 1 \\cdot \\varphi , \\varphi \\rangle \\rangle . \\end{align*}"}
+{"id": "5871.png", "formula": "\\begin{align*} \\Gamma \\left ( X , Y , Z \\right ) = C \\left ( X , J Y , Z \\right ) + g \\left ( X , \\left ( \\nabla _ { Z } J \\right ) Y \\right ) . \\end{align*}"}
+{"id": "1294.png", "formula": "\\begin{align*} \\forall \\Delta \\Subset \\Z ^ d \\ \\forall \\xi _ { \\Delta } : \\gamma _ { \\Delta } ( \\xi _ { \\Delta } \\lvert \\eta _ { \\Delta ^ c } ) = \\nu ( \\xi _ { \\Delta } \\lvert \\eta _ { \\Delta ^ c } ) \\eta \\in \\Omega , \\end{align*}"}
+{"id": "9296.png", "formula": "\\begin{align*} d _ 1 K _ { m , \\epsilon } = \\frac { \\kappa _ m d _ 1 | q | ^ 2 } { ( | q | ^ 2 + \\epsilon ) ^ { \\kappa _ m + 1 } } , \\end{align*}"}
+{"id": "8466.png", "formula": "\\begin{align*} Q _ { j , - } ( x ; k ) = \\mathcal { P } ^ - \\left ( Q _ { j , - } ( x ; k ) J + D _ j \\right ) ( z ) , z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "2471.png", "formula": "\\begin{align*} f ( t , q ^ { \\prime } ) = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\left ( - \\frac { q ^ { \\prime } } { 1 - q ^ { \\prime } } \\right ) _ { 2 n } ( 1 - q ^ { \\prime } ) ^ { 2 n } \\frac { ( \\alpha t ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! } = \\sinh _ { q ^ { \\prime } } \\left ( \\alpha t \\right ) . \\end{align*}"}
+{"id": "2318.png", "formula": "\\begin{align*} \\left ( \\rho ^ \\star \\right ) ^ { s y m } - \\left ( R i c ^ g \\right ) ^ { J , + } = \\frac { s ^ \\star - s ^ g } { 4 } g . \\end{align*}"}
+{"id": "3761.png", "formula": "\\begin{align*} \\# R ^ 2 _ { n _ 0 ; k _ 1 , k _ 2 , \\cdots , k _ { n _ 0 } } = q ^ { n _ 0 - 1 } \\# R ^ 2 _ { n _ 0 ; k _ 1 - 1 , k _ 2 , \\cdots , k _ { n _ 0 } } \\end{align*}"}
+{"id": "6706.png", "formula": "\\begin{align*} 2 \\hat \\beta ( l _ n , \\xi ) - 2 \\hat \\beta ( l _ m , \\xi ) = h ( l _ n \\xi , l _ n \\eta , l _ n \\omega ) - h ( l _ m \\xi , l _ m \\eta , l _ m \\omega ) . \\end{align*}"}
+{"id": "3037.png", "formula": "\\begin{align*} \\left \\langle Z _ { \\kappa } \\left ( x , \\xi \\right ) , Z _ { \\kappa } \\left ( y , \\eta \\right ) \\right \\rangle = \\tfrac { 1 } { 4 } \\left ( \\left \\langle x , \\eta \\right \\rangle + \\left \\langle y , \\xi \\right \\rangle \\right ) \\end{align*}"}
+{"id": "8814.png", "formula": "\\begin{align*} \\Pi _ { \\mathrm { k e r } A } ( B ( y , z ) + B ( z , y ) ) = 0 \\forall z \\in \\mathrm { k e r } A , y \\in \\mathrm { k e r } A ^ \\perp \\end{align*}"}
+{"id": "4561.png", "formula": "\\begin{align*} \\lambda _ 1 & = q \\sqrt { q } ( z _ 1 + z _ 2 + z _ 3 + z _ 4 ) \\\\ \\lambda _ 2 & = q ^ 2 ( z _ 1 z _ 2 + z _ 1 z _ 3 + z _ 1 z _ 4 + z _ 2 z _ 3 + z _ 2 z _ 4 + z _ 3 z _ 4 ) \\\\ \\lambda _ 3 & = q \\sqrt { q } \\left ( \\frac { 1 } { z _ 1 } + \\frac { 1 } { z _ 2 } + \\frac { 1 } { z _ 3 } + \\frac { 1 } { z _ 4 } \\right ) \\end{align*}"}
+{"id": "303.png", "formula": "\\begin{align*} \\mu \\left ( X \\backslash \\bigcup _ { k = 1 } ^ n \\mathcal { R } _ k \\right ) = 0 . \\end{align*}"}
+{"id": "8248.png", "formula": "\\begin{align*} \\frac { d \\psi _ i } { d t } = \\Psi ^ { n } _ i ( \\psi ) , \\end{align*}"}
+{"id": "4371.png", "formula": "\\begin{align*} \\sup \\limits _ i \\int _ K | \\tilde { F } _ i - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f F ^ { 1 + \\delta } | ^ 2 _ h < + \\infty . \\end{align*}"}
+{"id": "665.png", "formula": "\\begin{align*} Y _ { l _ 1 , m _ 1 } Y _ { l _ 2 , m _ 2 } = \\sum \\limits _ { l = | l _ 1 - l _ 2 | } ^ { l _ 1 + l _ 2 } \\sqrt { \\frac { ( 2 l _ 1 + 1 ) ( 2 l _ 2 + 1 ) } { 4 \\pi ( 2 l + 1 ) } } \\left \\langle l _ 1 , 0 , l _ 2 , 0 \\right | \\left . l , 0 \\right \\rangle \\end{align*}"}
+{"id": "2880.png", "formula": "\\begin{align*} & \\sum _ { y = x - n - 1 } ^ { x + n } \\bar K ^ { ( n ) } _ 1 ( { y , x } ) = \\frac { 1 } { n + 1 } \\sum _ { j = 0 } ^ n \\Phi \\left ( \\frac { j } { n + 1 } , \\frac { j } { n + 1 } \\right ) \\cos \\left ( \\frac { j \\ell } { n + 1 } \\right ) \\\\ & = G _ { \\omega _ 0 } ( \\ell ) + o ( 1 ) , \\end{align*}"}
+{"id": "2559.png", "formula": "\\begin{align*} M = \\sup \\{ N _ { n } : n = 1 , 2 , 3 , \\ldots \\} < \\infty . \\end{align*}"}
+{"id": "4608.png", "formula": "\\begin{align*} I _ R & = ( { x _ 1 } ^ 2 { x _ 2 } ^ 2 , { x _ 2 } ^ 2 { x _ 3 } ^ 2 , \\cdots , { x _ n } ^ 2 { x _ { n + 1 } } ^ 2 , x _ 1 { x _ 2 } ^ 2 + { x _ 2 } ^ 2 x _ 3 , x _ 2 { x _ 3 } ^ 2 + { x _ 3 } ^ 2 x _ 4 , \\cdots , x _ { n - 1 } { x _ n } ^ 2 + { x _ n } ^ 2 x _ { n + 1 } ) \\\\ & = ( x _ 1 { x _ 2 } ^ 2 + { x _ 2 } ^ 2 x _ 3 , x _ 2 { x _ 3 } ^ 2 + { x _ 3 } ^ 2 x _ 4 , \\cdots , x _ { n - 1 } { x _ n } ^ 2 + { x _ n } ^ 2 x _ { n + 1 } ) . \\end{align*}"}
+{"id": "1202.png", "formula": "\\begin{align*} g ( x ) = \\lim _ { n \\rightarrow \\infty } 2 ^ { n } f ( 2 ^ { n } x ) \\end{align*}"}
+{"id": "4327.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t _ 1 \\} } | \\tilde { F } _ 1 - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) F _ { t _ 0 } | ^ 2 _ h e ^ { v _ { t _ 0 , B } ( \\Psi ) - \\Psi } c ( - v _ { t _ 0 , B } ( \\Psi ) ) \\\\ \\le & ( \\int _ { t _ 1 } ^ { t _ 0 + B } c ( s ) e ^ { - s } d s ) \\int _ { M } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | F _ { t _ 0 } | ^ 2 _ h e ^ { - \\Psi } < + \\infty . \\end{align*}"}
+{"id": "8428.png", "formula": "\\begin{align*} w - \\widetilde { w } = \\left ( \\bar { u } _ { x x } + \\frac { 1 } { 2 i } | \\tilde { u } _ x | ^ 2 \\bar { u } _ x \\right ) e ^ { i c _ - ( x ) } - \\left ( \\bar { \\tilde { u } } _ { x x } + \\frac { 1 } { 2 i } | \\tilde { u } _ x | ^ 2 \\bar { \\tilde { u } } _ x \\right ) e ^ { i c _ - ( x ) } ( \\tilde { u } ) . \\end{align*}"}
+{"id": "2017.png", "formula": "\\begin{align*} H ( \\mu ) = \\int _ { \\R ^ m } h ( y ) \\mu ( d y ) , \\end{align*}"}
+{"id": "3654.png", "formula": "\\begin{align*} D _ { y , 2 } & = \\frac { L _ { x y } } { 2 \\mu _ { y } } D _ { x , 2 } + \\frac { 1 } { 2 L _ { x y } } \\left ( \\sqrt { n \\sigma _ { F , r } ^ 2 } + L _ { x } \\sqrt { n } M _ x + \\| \\nabla F ( X ^ * ) \\| \\right ) \\\\ & + \\frac { 1 } { 2 \\mu _ { y } } \\left ( \\sqrt { n \\sigma _ { G , r } ^ 2 } + L _ { y } \\sqrt { n } M _ y + \\| \\nabla G ( Y ^ * ) \\| \\right ) , \\end{align*}"}
+{"id": "5776.png", "formula": "\\begin{align*} \\nabla _ X \\psi = \\frac { 1 } { 2 } X _ 1 \\cdot V \\cdot \\psi - \\frac { 1 } { 2 } S ( X ) \\cdot N \\cdot \\psi . \\end{align*}"}
+{"id": "7196.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } e ^ { - t \\tau _ k } = \\int _ { \\partial \\Omega } \\bigg [ \\frac { 1 } { ( 2 \\pi ) ^ { n - 1 } } \\int _ { \\mathbb { R } ^ { n - 1 } } \\bigg ( \\frac { i } { 2 \\pi } \\int _ { \\mathcal { C } } e ^ { - t \\tau } \\sum _ { j \\leqslant - 1 } \\operatorname { T r } \\phi _ { j } \\ , d \\tau \\bigg ) \\ , d \\xi \\bigg ] \\ , d S . \\end{align*}"}
+{"id": "5999.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\int _ { \\R ^ { 2 } } | u _ { l } | ^ { 4 } d x = \\int _ { \\R ^ { 2 } } | \\nabla u _ { l } | ^ { 2 } d x = \\int _ { \\R ^ { 2 } } \\frac { u _ { l } ^ { 2 } } { | x | ^ { 2 } } \\Big ( \\int _ { 0 } ^ { | x | } \\frac { s } { 2 } u _ { l } ^ { 2 } ( s ) d s \\Big ) ^ { 2 } d x = \\frac { 1 6 \\pi l ^ { 2 } } { 3 } . \\end{align*}"}
+{"id": "7444.png", "formula": "\\begin{align*} \\delta _ { m , n } & = \\frac { m } { n } \\bigg ( 1 + \\sqrt { n - 1 } \\sqrt { \\frac { n - m } { m } } \\bigg ) < \\frac { m } { n } \\sqrt { 1 + n - 1 } \\sqrt { 1 + \\frac { n - m } { m } } = \\sqrt { m } . \\end{align*}"}
+{"id": "8096.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = v & & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta v & = | \\nabla u | ^ 2 & & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "6542.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\pi ( a ) } ( x + p _ i ) = ( a + 1 ) ^ { \\beta ( a + 1 ) } \\prod _ { i = 1 } ^ { \\pi ( a ) } p _ i ^ { \\alpha _ i ' } . \\end{align*}"}
+{"id": "3194.png", "formula": "\\begin{align*} \\mathcal { A } ^ 2 = \\frac { 1 } { 1 6 } \\begin{vmatrix} 0 & 1 & 1 & 1 \\\\ 1 & 0 & L _ { 1 , 2 } ^ 2 & L _ { 1 , 3 } ^ 2 \\\\ 1 & L _ { 1 , 2 } ^ 2 & 0 & L _ { 2 , 3 } ^ 2 \\\\ 1 & L _ { 1 , 3 } ^ 2 & L _ { 2 , 3 } ^ 2 & 0 \\end{vmatrix} . \\end{align*}"}
+{"id": "1399.png", "formula": "\\begin{align*} \\int _ { W ^ d } ( f _ i ( x _ j ) ) _ { i , j = 1 } ^ d ( g _ i ( x _ j ) ) _ { i , j = 1 } ^ d \\prod _ { i = 1 } ^ d \\nu ( d x _ i ) = \\left ( \\int _ { \\mathbb { R } } f _ i ( x ) g _ j ( x ) \\nu ( d x ) \\right ) _ { i , j = 1 } ^ d . \\end{align*}"}
+{"id": "5092.png", "formula": "\\begin{align*} \\phi _ t + u \\cdot \\nabla \\phi = 0 . \\end{align*}"}
+{"id": "7349.png", "formula": "\\begin{align*} ( \\delta _ n ^ k ) ' = \\sum _ { J = \\{ i _ 0 < \\hdots < \\widehat { i _ k } < \\hdots < i _ n \\} \\subset K = \\{ i _ 0 < \\hdots < i _ n \\} } ( \\nu _ { K } ^ J ) _ ! \\end{align*}"}
+{"id": "4844.png", "formula": "\\begin{align*} & ( r + \\nu ) v ( s , i ) - H \\left ( s , i , \\partial _ s v ( s , i ) , \\partial _ i v ( s , i ) \\right ) = 0 , \\\\ - & ( r + \\nu ) v ( s , i ) + H \\left ( s , i , \\partial _ s v ( s , i ) , \\partial _ i v ( s , i ) \\right ) = 0 , \\end{align*}"}
+{"id": "3589.png", "formula": "\\begin{align*} \\hat { p } ^ i _ { j k } = \\frac { \\hat { p } ^ i _ j ( B ) \\hat { p } ^ i _ k ( C ) } { \\hat { p } ^ i ( D ) } . \\end{align*}"}
+{"id": "4485.png", "formula": "\\begin{align*} \\langle v , w \\rangle : = \\sum _ { i \\in Q _ 0 } v _ i w _ i - \\sum _ { h \\in Q _ 1 } v _ { h ' } w _ { h '' } \\ , \\ , v , w \\in \\mathbb { Z } ^ n . \\end{align*}"}
+{"id": "2234.png", "formula": "\\begin{align*} \\frac { 1 } { t } \\int _ 0 ^ t \\bigg | \\int _ { \\mathbb { R } } e ^ { - 2 \\pi i s x } d \\mu _ { \\delta _ 0 } ^ { \\triangle } ( x ) \\bigg | ^ 2 d s = O ( \\log ( t ) / t ) ; \\end{align*}"}
+{"id": "6902.png", "formula": "\\begin{align*} V _ { \\mathrm { o r i g } } = \\sup _ { Q \\in \\P ( \\R ^ n ) } \\left ( \\int _ { \\R ^ n } g \\ , d Q - \\frac { 1 } { n } H ( Q \\ , | \\ , \\gamma _ T ) \\right ) = \\frac { 1 } { n } \\log \\int _ { \\R ^ n } e ^ { n g } \\ , d \\gamma _ T , \\end{align*}"}
+{"id": "1240.png", "formula": "\\begin{align*} \\sup _ { y \\in \\Z ^ d } \\sum _ { \\Delta \\ni y } \\sum _ { z \\neq y } \\sum _ { \\xi _ { \\Delta } } \\sum _ { i = 1 } ^ q \\norm { \\nabla ^ i _ z \\hat { c } _ \\Delta ( \\cdot , \\xi _ \\Delta ) } _ \\infty < \\infty . \\end{align*}"}
+{"id": "8892.png", "formula": "\\begin{align*} S = \\frac { ( 4 \\pi ) ^ { 2 k - 3 } } { \\Gamma ( 2 k - 3 ) } \\left ( 1 + C ( p ) + O ( k ^ { - \\alpha } D ^ { - \\alpha } + p ^ { - 5 / 4 + \\epsilon } k ^ { - 1 3 / 1 2 } D ^ { 7 / 8 + \\epsilon } ) \\right ) . \\end{align*}"}
+{"id": "7809.png", "formula": "\\begin{align*} F ^ q ( v , p ) = 0 \\end{align*}"}
+{"id": "1826.png", "formula": "\\begin{align*} ( \\delta ^ \\nabla \\rho ) ( X _ 1 ) = - \\sum _ i ( \\nabla _ { e _ i } \\rho ) ( e _ i , X _ 1 ) \\ , . \\end{align*}"}
+{"id": "3880.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta h _ { \\hat { x } } ( x , y ) = 0 , \\ \\ & x \\in T _ { \\hat { x } } ( \\Omega ) , \\\\ h _ { \\hat { x } } ( x , y ) = \\frac { 1 } { 2 \\pi } \\ln \\frac { 1 } { | x - y | } , \\ \\ & x \\in \\partial T _ { \\hat { x } } ( \\Omega ) . \\end{cases} \\end{align*}"}
+{"id": "6704.png", "formula": "\\begin{align*} h ( l \\xi , l \\eta , l \\omega ) - h ( \\xi , \\eta , \\omega ) = 2 \\hat \\beta ( l , \\xi ) . \\end{align*}"}
+{"id": "4369.png", "formula": "\\begin{align*} & \\int _ { X } | \\tilde { F } _ i - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f F ^ { 1 + \\delta } | ^ 2 _ h e ^ { v _ { t _ 0 , B } ( \\Psi ) - \\delta \\tilde { M } } c ( - v _ { t _ 0 , B } ( \\Psi ) ) \\\\ \\le & ( \\frac { 1 } { \\delta } c _ i ( T ) e ^ { - T } + \\int _ { T } ^ { t _ 0 + B } c _ i ( s ) e ^ { - s } d s ) \\int _ { X } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f F | ^ 2 _ h , \\end{align*}"}
+{"id": "8999.png", "formula": "\\begin{align*} ( - 1 ) ^ { p ( \\eta , j ) - 1 } i _ { \\eta \\cup j } & = ( - 1 ) ^ { p ( \\eta , j ) - 1 } ( - 1 ) ^ { | \\pi ( \\eta \\cup j ) | } { \\prod _ { r = 1 } ^ { d + 1 } U ^ { \\eta \\cup j } _ { r } } \\\\ & = U { \\prod _ { r = m + 1 } ^ M U ^ { \\eta \\cup j } _ { r } } , \\end{align*}"}
+{"id": "4913.png", "formula": "\\begin{align*} a _ { i } = & \\frac { 1 } { [ i ^ 2 + ( \\alpha - 2 ) i + ( 1 - \\alpha ) ] } \\bigg \\{ ( 1 - \\delta _ { i , 2 } ) \\ , \\sum _ { r = 2 } ^ { i - 1 } \\ , a _ { r } \\ , a _ { i + 1 - r } \\ , [ r ( 1 - \\alpha ) ( i - r ) \\\\ & - r ( r - 1 ) ] + \\sum _ { r = 1 } ^ { i - 1 } \\sum _ { s = 1 } ^ { i - r } \\ , a _ { r } \\ , a _ { s } \\ , a _ { i + 1 - r - s } \\ , [ r ( r - \\alpha ) + s ( \\alpha + \\beta - 2 ) \\\\ & \\times ( i + 1 - r - s ) ] \\bigg \\} , \\end{align*}"}
+{"id": "6686.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } ( \\eta \\gamma ^ n ) ^ + = \\eta \\gamma ^ + . \\end{align*}"}
+{"id": "9069.png", "formula": "\\begin{align*} 0 \\rightarrow E \\rightarrow \\bigoplus _ { j = 1 } ^ n E _ j \\rightarrow \\mathcal { T } _ E \\rightarrow 0 . \\end{align*}"}
+{"id": "1436.png", "formula": "\\begin{align*} \\widetilde { G } _ n ( x , n + z ) & = \\frac { n ^ { n d } } { ( n ! ) ^ d } \\det ( ( 1 + z _ j / n - x _ i / n ) ^ { n - 1 } e ^ { - ( n + z _ j - x _ i ) } ) _ { i , j = 1 } ^ d \\\\ & = \\frac { n ^ { n d } } { ( n ! ) ^ d } \\det ( e ^ { ( n - 1 ) \\log ( 1 + z _ j / n - x _ i / n ) } e ^ { - ( n + z _ j - x _ i ) } ) _ { i , j = 1 } ^ d . \\end{align*}"}
+{"id": "1870.png", "formula": "\\begin{align*} \\frac { 1 } { ( M ) } \\mathcal { Y M } _ e ^ 0 ( { \\nabla } ) = \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } { 2 } ) - 1 = \\exp \\bigg ( \\frac { 1 } { ( M ) } \\mathcal { Y M } ( \\nabla ) \\bigg ) - 1 \\end{align*}"}
+{"id": "1127.png", "formula": "\\begin{align*} P _ { \\min } ^ { \\rm { N } } \\left ( { \\overline S , \\overline \\varepsilon , \\overline R } \\right ) = { p _ K } \\left ( { \\overline S , \\overline \\varepsilon , W } \\right ) + \\frac { { { p _ K } \\left ( { \\overline S , \\overline \\varepsilon , W } \\right ) { { \\left | { { { \\widetilde h } _ b } } \\right | } ^ 2 } + W { N _ 0 } } } { { { { \\left | { { { \\widetilde h } _ b } } \\right | } ^ 2 } } } \\left ( { { 2 ^ { \\frac { { \\overline R } } { W } } } - 1 } \\right ) . \\end{align*}"}
+{"id": "6487.png", "formula": "\\begin{align*} & E ^ 0 _ { k , m } = \\{ n \\in \\mathbb { N } ^ k : 0 \\le n \\le m \\} \\\\ & E ^ 1 _ { k , m } = \\{ n + v _ i : n , n + e _ i \\in E ^ 0 _ { k , m } \\} \\\\ & r ( n + v _ i ) = n , s ( n + v _ i ) = n + e _ i , c ( n + v _ i ) = c _ i \\end{align*}"}
+{"id": "347.png", "formula": "\\begin{align*} { f ^ { i } } : = \\begin{cases} \\large ( 1 , - ( e _ { i } ' + e _ { i + 1 } ' ) , e _ { i } ' \\large ) ^ { ' } & i = 1 , 2 , \\dotsc , n - 2 \\\\ \\large ( 1 , ~ - ( e _ { 1 } ' + e _ { n } ' ) , ~ e _ { n - 1 } ' \\large ) ' & i = n - 1 . \\end{cases} \\end{align*}"}
+{"id": "72.png", "formula": "\\begin{align*} \\ell ( x ) = \\sum _ { \\alpha \\in \\Phi } \\max ( 0 , \\ell ( x , \\alpha ) ) . \\end{align*}"}
+{"id": "205.png", "formula": "\\begin{align*} \\theta _ { i , ( 1 ) } ( v _ { \\alpha , k } ) = - \\sum _ { j = 1 } ^ d a _ { i , j } v _ { \\alpha + 1 _ j , k } , \\end{align*}"}
+{"id": "2644.png", "formula": "\\begin{align*} [ x , y ] = x \\cdot D ( y ) - \\varepsilon ( x , y ) y \\cdot D ( x ) . \\end{align*}"}
+{"id": "2417.png", "formula": "\\begin{align*} f ( t ) = \\lim _ { n \\to \\infty } \\frac { ( - 1 ) ^ { n } } { n ! } \\ , s ^ { n + 1 } F ^ { n } ( s ) | _ { s = n / t } . \\end{align*}"}
+{"id": "9014.png", "formula": "\\begin{align*} i _ { 1 3 5 } = - x _ 1 D _ 2 D _ 4 , i _ { 1 2 5 } = x _ 1 D _ 2 D _ 4 , i _ { 1 2 3 } = - x _ 1 D _ 2 D _ 3 , i _ { 2 3 4 } = - D _ 1 D _ 2 x _ 4 . \\end{align*}"}
+{"id": "5001.png", "formula": "\\begin{align*} f ( A ) = \\begin{cases} 1 , & s < - \\frac d 2 , \\\\ ( \\log A ) ^ \\frac 1 2 , & s = - \\frac d 2 , \\\\ A ^ { \\frac { d } { 2 } + s } , & s > - \\frac d 2 . \\end{cases} \\end{align*}"}
+{"id": "7810.png", "formula": "\\begin{align*} \\Phi ( v , p ) = 0 , \\end{align*}"}
+{"id": "851.png", "formula": "\\begin{align*} \\partial ^ \\square H = \\Delta _ \\Gamma V + V \\big | \\nabla _ \\Gamma \\nu \\big | ^ 2 \\end{align*}"}
+{"id": "272.png", "formula": "\\begin{align*} ( p _ n ^ { - 1 } \\cdot w _ n ) _ h = \\left ( \\frac { 1 } { t } - 1 \\right ) ( x _ { p _ n } , y _ { p _ n } ) , \\end{align*}"}
+{"id": "3679.png", "formula": "\\begin{align*} X _ i = 0 , \\omega _ { i j } V _ i V _ j = \\tfrac { 1 } { 2 } h _ { i j } V _ i V _ j , \\omega _ { i a } V _ i = h _ { i a } V _ i , \\omega _ { a b } = 0 \\end{align*}"}
+{"id": "505.png", "formula": "\\begin{align*} \\sum _ { m _ 1 , \\ldots , m _ n = 0 } ^ { \\infty } \\lambda _ { m _ 1 , \\ldots , m _ n } a _ 1 ^ { m _ 1 } \\cdots a _ n ^ { m _ n } \\end{align*}"}
+{"id": "4195.png", "formula": "\\begin{align*} I _ \\varepsilon ( u _ \\varepsilon ) = c _ \\varepsilon , \\ \\ I ' _ \\varepsilon ( u _ \\varepsilon ) = 0 . \\end{align*}"}
+{"id": "270.png", "formula": "\\begin{align*} d _ H ( \\xi , w _ n ) = | \\xi ^ { - 1 } \\cdot w _ n | _ G \\to 0 , \\end{align*}"}
+{"id": "4726.png", "formula": "\\begin{align*} H _ k : = \\begin{pmatrix} A _ k & b _ k \\\\ b _ k ^ T & c _ k \\end{pmatrix} \\ \\ k = 1 , . . . , m H _ J : = \\begin{pmatrix} A _ J & b _ k \\\\ b _ J ^ T & c _ J \\end{pmatrix} \\end{align*}"}
+{"id": "4989.png", "formula": "\\begin{align*} \\underline { h } _ { \\mu } ( f _ { 1 , \\infty } , x ) : = \\lim _ { \\epsilon \\to 0 } \\liminf _ { n \\to \\infty } \\dfrac { - \\log \\mu ( B _ { n } ( x , \\epsilon ) ) } { n } . \\end{align*}"}
+{"id": "3189.png", "formula": "\\begin{align*} \\lambda ( - 1 + s ^ 2 ) + \\mu { ( - 1 + s ) } ^ 2 = 0 \\frac { \\lambda } { \\mu } = \\frac { 1 - s } { 1 + s } . \\end{align*}"}
+{"id": "3611.png", "formula": "\\begin{align*} \\Gamma _ k = \\{ \\lambda \\in \\mathbb { R } ^ n : \\sigma _ j ( \\lambda ) > 0 , 1 \\leq j \\leq k \\} . \\end{align*}"}
+{"id": "1528.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } g ( 0 ) = e \\\\ \\frac { d g ( t ) } { d t } g ( t ) ^ { - 1 } = v ( t ) \\end{array} \\right . \\end{align*}"}
+{"id": "9243.png", "formula": "\\begin{align*} \\begin{aligned} { \\Gamma } _ m = \\{ \\mathcal { H } _ 1 ( A ) > 0 \\} \\cap \\cdots \\{ \\mathcal { H } _ m ( A ) > 0 \\} . \\end{aligned} \\end{align*}"}
+{"id": "232.png", "formula": "\\begin{align*} k ^ + = k _ { \\mathcal { W } , g } ^ + & = \\min \\{ k \\mid 0 \\le k \\le l _ S ( g ) \\sigma ( k ) = \\sigma ^ + \\} , \\\\ k ^ - = k _ { \\mathcal { W } , g } ^ - & = \\min \\{ k \\mid 0 \\le k \\le l _ S ( g ) \\sigma ( k ) = \\sigma ^ - \\} \\end{align*}"}
+{"id": "2838.png", "formula": "\\begin{align*} ( \\mathbf q , \\mathbf p ) = ( q _ 0 , \\dots , q _ n , p _ 0 , \\dots , p _ n ) \\in \\R ^ { n + 1 } \\times \\R ^ { n + 1 } . \\end{align*}"}
+{"id": "8405.png", "formula": "\\begin{align*} & m = \\int _ { - \\infty } ^ x e ^ { 2 i z ( x - y ) } w ( y ) d y , w ( x ) = - \\partial _ x ( u _ x e ^ { i c _ - ( x ) } ) , \\\\ & n = \\int _ { - \\infty } ^ x e ^ { - 2 i z ( x - y ) } u _ y ( y ) e ^ { i c _ - ( x ) } d y . \\end{align*}"}
+{"id": "6379.png", "formula": "\\begin{align*} \\int ^ \\infty _ 0 f ( x ) ^ \\alpha \\mathrm { d } x = \\left [ \\frac { 2 \\theta } { \\operatorname { B } ( a , b ) } \\right ] ^ \\alpha \\frac { \\Gamma \\left ( \\frac { 3 \\alpha - 1 } { 2 } \\right ) } { 2 \\theta ^ { \\frac { 3 ( \\alpha - 1 ) } { 2 } + 1 } } \\sum ^ \\infty _ { n = 0 } \\frac { ( - 1 ) ^ n } { ( a \\alpha + n ) ^ { \\frac { 3 \\alpha - 1 } { 2 } } } \\frac { \\Gamma ( \\alpha ( b - 1 ) + 1 ) } { \\Gamma ( \\alpha ( b - 1 ) + 1 - n ) n ! } . \\end{align*}"}
+{"id": "6324.png", "formula": "\\begin{align*} z _ 0 x _ 0 ^ { p ^ { h _ i - r } } + z _ 1 x _ 1 ^ { p ^ { h _ i - r } } + \\cdots + z _ n x _ n ^ { p ^ { h _ i - r } } = 0 . \\end{align*}"}
+{"id": "7778.png", "formula": "\\begin{align*} s _ { h , q } : = \\frac { 1 } { q } \\sum _ { g = 0 } ^ { q - 1 } \\zeta _ q ^ { ( h + 1 ) g } ~ \\frac { p ' ( \\zeta _ q ^ g ) } { p ( \\zeta _ q ^ g ) } . \\end{align*}"}
+{"id": "8207.png", "formula": "\\begin{align*} T = \\{ k _ 1 , k _ 2 , \\ldots , k _ m \\} \\mbox { \\ \\ \\ s u c h t h a t } k _ i < k _ { i + 1 } , \\mbox { f o r } 1 \\le i < m . \\end{align*}"}
+{"id": "6068.png", "formula": "\\begin{align*} L [ \\nu _ { n + 1 } ] = - m \\eta x _ { n + 1 } ^ { m - 1 } . \\end{align*}"}
+{"id": "6990.png", "formula": "\\begin{align*} X ( t ) & = x + \\int _ 0 ^ t A ( s , \\overline { X } ( s ) ) d s + \\int _ 0 ^ t B ( s , \\overline { X } ( s ) ) d W ( s ) \\\\ & \\quad \\ + \\int _ 0 ^ t \\int _ { \\{ | z | < 1 \\} } H ( s , \\overline { X } ( s ) , z ) \\widetilde { N } ( d s , d z ) \\\\ & \\quad \\ + \\int _ 0 ^ t \\int _ { \\{ | z | \\ge 1 \\} } J ( s , \\overline { X } ( s ) , z ) N ( d s , d z ) . \\end{align*}"}
+{"id": "2607.png", "formula": "\\begin{align*} R \\cdot R ( x , J x , J x , x ; u , J u ) & = - R \\cdot R ( J x , x , u , J u ; x , J x ) - R \\cdot R ( u , J u , x , J x ; J x , x ) \\\\ & = 2 \\ , R \\cdot R ( u , J u , x , J x ; x , J x ) . \\end{align*}"}
+{"id": "8645.png", "formula": "\\begin{align*} \\ \\begin{aligned} y \\left [ n \\right ] = & \\left ( { \\sum \\limits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ l } } } \\right ) s \\left [ { n - { n _ { \\max } } } \\right ] + \\\\ & \\sum \\limits _ { l = 1 } ^ L { \\sum \\limits _ { l ' \\ne l } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ { l ' } } s \\left [ { n - { n _ { \\max } } + { n _ { l ' } } - { n _ l } } \\right ] } } + z \\left [ n \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "866.png", "formula": "\\begin{align*} x x ' + x y ' + y y ' = 0 \\ , . \\end{align*}"}
+{"id": "144.png", "formula": "\\begin{align*} l _ { x y } ( i ) = c _ { x y } \\lambda _ i \\prod _ { z \\in \\partial x \\setminus \\{ y \\} } \\sum _ { j \\in \\Z } a _ { i j } l _ { z x } ( j ) \\end{align*}"}
+{"id": "2229.png", "formula": "\\begin{align*} R M S E = \\sqrt { \\frac { 1 } { L } \\sum \\limits _ { l = 1 } ^ { L } ( { \\hat \\theta _ { l } - \\theta _ 0 } ) ^ 2 } \\end{align*}"}
+{"id": "7147.png", "formula": "\\begin{align*} S \\textbf { \\textit { u } } : = \\nabla \\textbf { \\textit { u } } + \\nabla \\textbf { \\textit { u } } ^ t , \\textbf { \\textit { u } } \\in ( C ^ 1 ( \\Omega ) ) ^ n , \\end{align*}"}
+{"id": "7133.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathcal P _ X ^ D ( \\alpha _ 1 , \\dots , \\alpha _ r ) _ x & = \\mathcal P _ { X , x } & \\\\ \\mathcal P _ X ^ D ( \\alpha _ 1 , \\dots , \\alpha _ r ) _ x & = \\alpha _ i & \\end{aligned} \\right . \\end{align*}"}
+{"id": "7418.png", "formula": "\\begin{align*} r _ { \\ell - c } = s _ i \\end{align*}"}
+{"id": "5552.png", "formula": "\\begin{align*} A ^ * ( y ) = \\left [ \\hat { A } \\circ \\hat { \\sigma } ^ { - 1 } + W \\circ \\hat { \\sigma } ^ { - 1 } - W \\right ] ( y | x ) , \\end{align*}"}
+{"id": "6619.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c } \\varepsilon _ 1 \\mu & - \\varepsilon _ 1 \\mu \\\\ \\varepsilon _ 3 \\mu & - \\varepsilon _ { 3 } \\mu \\end{array} \\right ) \\end{align*}"}
+{"id": "7007.png", "formula": "\\begin{align*} \\aligned e _ 1 & \\ , : = \\ , ( 1 - y ) \\ , \\partial _ x - z \\ , \\partial _ y + x \\ , \\partial _ u , \\\\ e _ 2 & \\ , : = \\ , ( 1 - y ) \\ , \\partial _ y - 2 z \\ , \\partial _ z + u \\ , \\partial _ u , \\\\ e _ 3 & \\ , : = \\ , u \\ , \\partial _ x - \\tfrac { 4 } { 3 } \\ , x \\ , \\partial _ y + ( 1 - y ) \\ , \\partial _ z , \\\\ e _ 4 & \\ , : = \\ , x \\ , \\partial _ x - z \\ , \\partial _ z + 2 \\ , u \\ , \\partial _ u . \\endaligned \\end{align*}"}
+{"id": "7414.png", "formula": "\\begin{align*} [ \\beta ^ t _ { a , b , n } \\ , , \\ , \\gamma ^ t _ { a , b , m } ] = \\delta _ { n + m , 0 } \\end{align*}"}
+{"id": "2637.png", "formula": "\\begin{align*} a s ^ { l } _ A ( x , y , z ) = ( x \\cdot y ) \\diamond \\alpha ( z ) - \\alpha ( x ) \\diamond ( y \\cdot z ) = 0 . \\end{align*}"}
+{"id": "1802.png", "formula": "\\begin{align*} L ( f , 3 ) & = \\frac { 4 \\pi ^ { 3 } } { 9 \\sqrt { 3 } } \\int _ { 0 } ^ { \\infty } c ^ { 3 } ( e ^ { - 2 \\pi u } ) \\sum _ { k , r = 1 } ^ { \\infty } \\frac { \\chi _ { - 3 } ( k r ) } { k } \\left ( e ^ { - \\frac { 2 \\pi k r } { 9 u } } - e ^ { - \\frac { 2 \\pi k r } { 3 u } } \\right ) u d u . \\end{align*}"}
+{"id": "4610.png", "formula": "\\begin{align*} \\frac { { x _ { n + 1 } } ^ 2 } { g } ( x _ { k } { x _ { k + 1 } } ^ 2 + { x _ { k + 1 } } ^ 2 x _ { k + 2 } ) = \\begin{cases} x _ { n + 1 } & ( k = n - 1 ) \\\\ 0 & ( k : ) , \\end{cases} \\end{align*}"}
+{"id": "4925.png", "formula": "\\begin{align*} J ( q ) = \\int _ { 0 } ^ { I _ { \\Phi \\left ( \\frac { q - \\mu } { \\sigma } \\right ) } ( \\alpha , \\beta ) } \\ , Q ( u ) \\mathrm { d } u = \\sum _ { i = 0 } ^ { \\infty } \\frac { f _ i \\ , \\ , \\ , I _ { \\Phi \\left ( \\frac { q - \\mu } { \\sigma } \\right ) } ( \\alpha , \\beta ) ^ { ( i / \\alpha + 1 ) } } { i / \\alpha + 1 } . \\end{align*}"}
+{"id": "610.png", "formula": "\\begin{align*} \\pi \\circ \\phi = \\mbox { i d } _ Y , \\end{align*}"}
+{"id": "2747.png", "formula": "\\begin{align*} B ( s , 2 ^ { - \\beta } | x | ^ \\beta ) \\cap B ( s ' , 2 ^ { - \\beta } | x | ^ \\beta ) = \\emptyset \\quad \\ s , s ' \\in S \\cap B ( x , r ) . \\end{align*}"}
+{"id": "7620.png", "formula": "\\begin{align*} f _ n ( t , z , x ) : = \\sup _ { \\| q \\| \\le n } \\Big ( q \\cdot z - g ( t , q , x ) \\Big ) . \\end{align*}"}
+{"id": "6951.png", "formula": "\\begin{align*} M _ n ^ { \\psi } ( Q ) / n = T _ { W _ { n J } , \\psi } ( \\mu _ n ( Q ) ) - I ( \\mu _ n ( Q ) ) . \\end{align*}"}
+{"id": "1749.png", "formula": "\\begin{align*} A _ { i j } = \\int _ { Y ^ { \\ast } } \\big ( \\nabla \\chi _ i + e _ i \\big ) \\cdot \\big ( \\nabla \\chi _ j + e _ j \\big ) d y , \\end{align*}"}
+{"id": "1884.png", "formula": "\\begin{align*} \\psi ^ 0 ( t ) + \\sum _ { j \\ge 4 } \\big ( \\psi ( 2 ^ j t ) + \\psi ( - 2 ^ j t ) + \\psi ( 2 ^ j ( t + \\pi ) ) + \\psi ( 2 ^ j ( \\pi - t ) ) \\big ) = 1 \\end{align*}"}
+{"id": "2430.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 } s L _ { q } [ f ( t ) ] = \\lim _ { t \\rightarrow \\infty } \\frac { f ( t ) } { 1 + ( 1 - q ) } . \\end{align*}"}
+{"id": "2478.png", "formula": "\\begin{align*} Z _ { q } ( \\beta ) = \\frac { 1 } { \\left ( \\hbar \\omega \\right ) ^ { D N } } \\left ( \\frac { \\Gamma \\left ( \\frac { 1 } { 1 - q } + 1 \\right ) } { ( 1 - q ) ^ { D N } \\ ; \\Gamma \\left ( \\frac { 1 } { 1 - q } + D N + 1 \\right ) } \\right ) \\frac { 1 } { \\beta ^ { D N } } . \\end{align*}"}
+{"id": "1849.png", "formula": "\\begin{align*} d _ e = \\sup _ { x \\in M } \\frac { \\exp \\big ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ( x ) \\big ) } { \\exp \\big ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ( x ) \\big ) - 1 } \\ , . \\end{align*}"}
+{"id": "7466.png", "formula": "\\begin{align*} N _ f ( z ) = \\dfrac { ( 1 + z ) ( 1 + \\det ( M ) z ) } { ( 1 - z ) ( 1 - \\det ( M ) z ) } . \\end{align*}"}
+{"id": "8969.png", "formula": "\\begin{align*} \\Upsilon ^ { ( i - 1 ) } = \\Delta ^ { ( i - 1 ) } , b _ { i - 1 } ( \\Upsilon ) = b _ { i - 1 } ( \\Delta ) , b _ { i } ( \\Upsilon ) = 0 . \\end{align*}"}
+{"id": "6926.png", "formula": "\\begin{align*} g _ 1 ^ { \\prime + } ( 0 ) - g _ 2 ^ { \\prime + } ( 0 ) & = \\sum _ { i = 1 } ^ n \\int _ \\R \\big ( \\hat { f } ' _ i ( x _ i ) Q _ i ( x _ i ) - Q _ i ' ( x _ i ) \\big ) \\big ( T _ i ( x _ i ) - x _ i \\big ) d x _ i . \\end{align*}"}
+{"id": "4060.png", "formula": "\\begin{align*} e _ 2 ( G ^ { ( \\lambda , C ) } ) = 2 - I ( G ^ { ( \\lambda , C ) } ) = 2 ( 1 - I ( C ) ) = 2 e _ 2 ( C ) . \\end{align*}"}
+{"id": "6696.png", "formula": "\\begin{align*} | \\delta ( \\zeta , \\gamma ^ + ) | = | \\delta ( \\zeta \\theta ^ { - 1 } \\theta , \\gamma ^ + ) | = | \\delta ( \\zeta \\theta ^ { - 1 } , \\theta \\gamma ^ + ) + \\delta ( \\theta , \\gamma ^ + ) | \\leq 4 M . \\end{align*}"}
+{"id": "8485.png", "formula": "\\begin{align*} & \\mathcal { P } ^ + \\left ( f ( z ) e ^ { - 2 i z x } \\right ) = \\int _ { 2 x } ^ { + \\infty } \\widehat { f } ( \\xi ) e ^ { i z ( \\xi - 2 x ) } \\mathrm { d } \\xi . \\end{align*}"}
+{"id": "6493.png", "formula": "\\begin{align*} \\lambda _ x | _ { E _ { 2 , m } } = \\lambda _ y , \\lambda _ x | _ { E _ { 2 , m + e _ i - e _ j } } = \\lambda ^ j \\lambda _ x | _ { E _ { 2 , [ m - e _ j , m + e _ i ] } } = \\phi ^ j \\end{align*}"}
+{"id": "827.png", "formula": "\\begin{align*} g \\big ( c ( t , x ) \\big ) \\begin{cases} \\leq g \\big ( c _ 0 ( x ) \\big ) + \\tilde { \\varepsilon } , \\\\ \\geq g \\big ( c _ 0 ( x ) \\big ) - \\tilde { \\varepsilon } \\end{cases} \\end{align*}"}
+{"id": "4082.png", "formula": "\\begin{align*} \\sigma _ 0 = \\min \\{ \\frac { p _ 0 } { \\nu _ 0 } , q _ 0 \\} . \\end{align*}"}
+{"id": "1902.png", "formula": "\\begin{align*} \\textsf { E } _ i s ^ { Z ( n ) } : = \\sum _ { j \\in { \\mathcal { S } } } { P _ { i j } ( n ) s ^ j } = \\bigl [ { f _ n ( s ) } \\bigr ] ^ i , \\end{align*}"}
+{"id": "9194.png", "formula": "\\begin{align*} \\tau ( x ) = \\dfrac { \\gamma } { 2 } \\int _ { - \\infty } ^ 0 e ^ { \\gamma \\tau } a _ { 0 , \\delta } ( \\hat { x } _ a ( \\tau , x ) ) \\ , d \\tau . \\end{align*}"}
+{"id": "5881.png", "formula": "\\begin{align*} \\tau _ { n , H } ^ M : = \\inf \\{ t \\geq 0 : \\Vert Y _ n ( t ) \\Vert _ H > M \\} \\wedge T . \\end{align*}"}
+{"id": "7054.png", "formula": "\\begin{align*} \\aligned e _ 1 & \\ , : = \\ , \\big ( x - y - \\tfrac { 1 0 } { 9 } \\ , u + 1 \\big ) \\ , \\partial _ x + \\big ( \\tfrac { 1 0 } { 9 } \\ , x - y - \\tfrac { 1 0 } { 9 } \\ , u \\big ) \\ , \\partial _ y + \\big ( x + 2 \\ , u \\big ) \\ , \\partial _ u , \\\\ e _ 2 & \\ , : = \\ , \\big ( u - 2 \\ , x \\big ) \\ , \\partial _ x + \\big ( \\tfrac { 4 } { 3 } \\ , x - y + \\tfrac { 8 } { 9 } \\ , u + 1 \\big ) \\ , \\partial _ y - 3 \\ , u \\ , \\partial _ u , \\endaligned \\end{align*}"}
+{"id": "8507.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ x \\left ( e ^ { 2 i ( c _ - ( x ) + c ) } u ( x ) \\right ) & = I _ 1 ' ( x ) + I _ 2 ' ( x ) . \\end{aligned} \\end{align*}"}
+{"id": "6672.png", "formula": "\\begin{align*} \\begin{aligned} & | h ^ { \\epsilon } ( t ) | \\leq B \\left ( \\rho ^ { 2 } + | Y ^ { \\epsilon } ( t - ) | ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } } , \\\\ & | \\gamma ^ { \\epsilon } ( t ) | \\leq M \\left ( \\rho ^ { 2 } + | Y ^ { \\epsilon } ( t - ) | ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } } . \\end{aligned} \\end{align*}"}
+{"id": "1725.png", "formula": "\\begin{align*} \\widetilde { W } & = ( \\beta _ 1 , \\dots , \\beta _ N ) , \\\\ \\Phi ( t ) ( v ) & = \\sum _ { j = 1 } ^ N g ( y ) \\mu _ j e _ j \\langle v , f _ j \\rangle , \\ v \\in \\mathbb { R } ^ N , \\\\ ( f _ j ) _ { 1 \\le j \\le N } & : \\ \\mathbb { R } ^ N . \\end{align*}"}
+{"id": "1385.png", "formula": "\\begin{align*} \\lambda _ d h ( x ) & = h _ { x _ d } ( x ) , x _ d = x _ { d - 1 } \\\\ \\lambda _ { d - 1 } h ( x ) & = h _ { x _ { d - 1 } } ( x ) , x _ { d - 1 } = x _ { d - 2 } \\\\ & \\ldots \\\\ \\lambda _ 2 h ( x ) & = h _ { x _ { 2 } } ( x ) , x _ { 2 } = x _ 1 . \\end{align*}"}
+{"id": "3046.png", "formula": "\\begin{align*} \\left ( \\nabla _ { x } b \\right ) _ { v } = C _ { x ^ { b } } \\circ b \\left ( v \\right ) \\end{align*}"}
+{"id": "5411.png", "formula": "\\begin{align*} \\partial \\omega = \\partial _ 1 \\omega \\cup \\partial _ 2 \\omega \\cup \\partial _ 3 \\omega , \\end{align*}"}
+{"id": "4150.png", "formula": "\\begin{align*} \\langle \\lambda , \\lambda ' \\rangle : = \\sum _ { \\alpha = 1 } ^ N \\lambda _ { \\alpha } \\lambda ' _ { \\alpha } , \\quad \\ , \\lambda = ( \\lambda _ { \\alpha } ) _ { 1 \\le \\alpha \\le N } , \\ , \\ , \\ , \\lambda ' = ( \\lambda ' _ { \\alpha } ) _ { 1 \\le \\alpha \\le N } \\in \\mathbb { C } ^ N . \\end{align*}"}
+{"id": "444.png", "formula": "\\begin{align*} \\mu ' ( d x ) = e ^ { - \\lambda } \\delta _ \\gamma ( d x ) + ( 1 - e ^ { - \\lambda } ) f ( x - \\gamma ) d x , \\gamma \\in \\R , \\end{align*}"}
+{"id": "7686.png", "formula": "\\begin{align*} | r ^ { N , i } ( t , \\boldsymbol { x } ) | \\leq \\frac { C } { N } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } | x ^ i - x ^ j | \\Big ) . \\end{align*}"}
+{"id": "6971.png", "formula": "\\begin{align*} | t _ n | _ p = 1 . \\end{align*}"}
+{"id": "1081.png", "formula": "\\begin{align*} w s _ \\alpha ( - w ^ { - 1 } \\gamma ) = - ( w s _ \\alpha w ^ { - 1 } ) \\gamma = - s _ { w \\alpha } ( \\gamma ) = - ( \\gamma + w \\alpha ) \\in \\Phi ^ - . \\end{align*}"}
+{"id": "2170.png", "formula": "\\begin{align*} j ^ * : = \\arg \\min _ { j \\in [ 1 , C ] } { \\mathcal { L } ^ { * } ( { \\bf { c } } _ { j } ^ { ( i ) } ) } , \\end{align*}"}
+{"id": "8007.png", "formula": "\\begin{align*} R _ 2 ( g , \\rho ) ( f ) & = \\frac { L } { 8 \\pi _ N ( x ) } \\frac { 1 } { L ^ 2 \\pi _ N ( x ) } \\sum _ { ( p , q ) \\atop { p \\neq q \\leq x \\atop { ( p , N ) = ( q , N ) = 1 } } } \\left [ \\sum _ { l \\geq 0 } U ( l ) a _ f ( p ^ { 2 l } ) \\right ] \\left [ \\sum _ { l ' \\geq 0 } U ( l ' ) a _ f ( q ^ { 2 l ' } ) \\right ] \\\\ & \\left [ 4 G ( 0 ) + \\sum _ { n \\geq 1 } 2 G ( n ) ( a _ f ( p ^ { 2 n } ) - a _ f ( p ^ { 2 n - 2 } ) ) ( a _ f ( q ^ { 2 n } ) - a _ f ( q ^ { 2 n - 2 } ) ) \\right ] \\\\ \\end{align*}"}
+{"id": "487.png", "formula": "\\begin{align*} a = \\int _ { 0 } ^ 1 ( e ^ { \\gamma x } - 1 ) x g ( x ) d x , b = 0 \\nu ( d x ) = { \\bf 1 } _ { \\{ 0 < x \\le c _ 1 \\} } e ^ { \\gamma x } g ( x ) d x . \\end{align*}"}
+{"id": "3400.png", "formula": "\\begin{align*} t \\alpha _ { t } - ( \\alpha _ { 0 } + \\alpha _ { 1 } + \\ldots + \\alpha _ { t - 2 } + \\alpha _ { t - 1 } ) = \\sum _ { k = 1 } ^ { t } k ( \\alpha _ { k } - \\alpha _ { k - 1 } ) = \\sum _ { k = 1 } ^ { t } \\frac { ( - 1 ) ^ { k } } { ( k - 1 ) ! } = - \\alpha _ { t - 1 } . \\end{align*}"}
+{"id": "3744.png", "formula": "\\begin{align*} \\mathbf { Q } _ { 0 , n } f & = \\frac { q t } { q t - 1 } h _ n [ ( 1 - q t ) X / ( q t ) ] \\cdot f \\\\ \\mathbf { Q } _ { 1 , n } f & = \\mathbf { D } _ n f . \\end{align*}"}
+{"id": "6110.png", "formula": "\\begin{align*} N = j ^ { ( 1 ) } + j ^ { ( 2 ) } - j ^ { ( 3 ) } + j ^ { ( 4 ) } + j ^ { ( 0 ) } \\ , . \\end{align*}"}
+{"id": "6382.png", "formula": "\\begin{align*} f _ { i : n } ( x ) = \\sum _ { l = 0 } ^ { n - i } \\sum _ { m _ { 1 } = 0 } ^ { \\infty } \\cdots \\sum _ { m _ { i + l - 1 } = 0 } ^ { \\infty } \\delta _ { i , l } f _ { i , l } ( x ) , \\end{align*}"}
+{"id": "9098.png", "formula": "\\begin{align*} \\sum \\limits _ { i = j + 1 } ^ { n - 1 } - k _ i \\geq \\sum \\limits _ { i = j + 1 } ^ { n - 1 } - r = - ( n - j - 1 ) r . \\end{align*}"}
+{"id": "8618.png", "formula": "\\begin{align*} \\psi _ A ( u _ 1 , \\dots , u _ k ) = \\frac { V ( A [ n - k ] , [ 0 , u _ 1 ] , \\dots , , [ 0 , u _ k ] ) } { | A | } . \\end{align*}"}
+{"id": "7989.png", "formula": "\\begin{align*} l \\sum h _ { k } \\nabla _ { k } \\chi & = l \\sum h _ { k } \\left [ \\frac { - f _ { k } } { f } + \\frac { h _ { k } } { h } + \\frac { \\varphi ^ { ' } } { \\varphi } h _ { k } + \\frac { ( \\nabla G \\cdot e _ { i } ) \\cdot w _ { k i } } { G } \\right ] \\\\ & \\leq C _ { 1 } l + l \\frac { \\nabla G \\cdot e _ { k } } { G } h _ { k } w _ { k k } = C _ { 1 } l + l \\frac { \\nabla G \\cdot e _ { k } } { G } \\rho \\rho _ { k } \\leq C _ { 2 } l , \\end{align*}"}
+{"id": "6455.png", "formula": "\\begin{align*} k < & \\Big ( \\Pi _ { i = j } ^ { n - 1 } \\big ( r _ i + 1 \\big ) \\Big ) h _ j , \\end{align*}"}
+{"id": "8468.png", "formula": "\\begin{align*} Q _ { j , - } - \\mathcal { P } ^ - \\left ( Q _ { j , - } J \\right ) = \\mathcal { P } ^ - ( D _ j ) , Q _ { j , + } - \\mathcal { P } ^ - \\left ( Q _ { j , - } J \\right ) = \\mathcal { P } ^ + ( D _ j ) . \\end{align*}"}
+{"id": "289.png", "formula": "\\begin{align*} u _ \\ast ( \\eta _ 0 ) \\leq \\varphi ( p _ 0 ) = u _ \\ast ( p _ 0 ) , \\end{align*}"}
+{"id": "8095.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = v ^ { p } & & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta v & = f ( | \\nabla u | ) & & \\quad \\mbox { i n } \\Omega . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3388.png", "formula": "\\begin{align*} c _ k = \\frac 1 2 \\sum _ { \\substack { i + j = k \\\\ 0 \\leq i , j \\leq N } } ( i - j ) ^ 2 . \\end{align*}"}
+{"id": "7876.png", "formula": "\\begin{align*} \\abs { a ^ \\textrm { a p p } ( s ( \\tau ) ) - a ( \\tau ) } = \\abs { \\sqrt { \\frac { - \\gamma _ 2 s ( \\tau ) } { \\gamma _ 1 } } - \\tau } = \\mathcal O ( \\tau ^ 3 ) = \\mathcal O ( s ^ \\frac 3 2 ) . \\end{align*}"}
+{"id": "5797.png", "formula": "\\begin{align*} 2 / \\mu \\ | h _ z | ^ 2 = \\mu / 2 \\ | T | ^ 2 = \\mu / 2 \\ ( 1 - \\nu ^ 2 ) = \\mu / 2 - \\tau _ 0 / { 2 \\mu } \\end{align*}"}
+{"id": "8845.png", "formula": "\\begin{align*} d ( X _ { t , 3 } - x _ { t , 3 } ) = - \\bar { x } _ { t , 1 } x _ { t , 2 } d t + x _ { t , 1 } ( x _ { t , 2 } - x _ { 0 , 2 } ) d t + \\bar { x } _ { t , 1 } ( x _ { t , 2 } - x _ { 0 , 2 } ) d t - x _ { t , 4 } x _ { t , 2 } d t + a _ 3 x _ { t , 3 } - \\sigma _ 3 d W _ t ^ { ( 3 ) } , \\end{align*}"}
+{"id": "822.png", "formula": "\\begin{align*} H _ { | ( x , \\varphi ) } = \\frac { 1 } { \\sqrt { 1 + | w ' ( x ) | ^ 2 } } \\left ( \\frac { w '' ( x ) } { 1 + | w ' ( x ) | ^ 2 } - \\frac { 1 } { w ( x ) } \\right ) . \\end{align*}"}
+{"id": "8148.png", "formula": "\\begin{align*} \\frac \\varpi \\pi x ' = \\varphi _ M ^ { - 1 } ( \\frac { \\varpi ^ p } \\pi x ' ) . \\end{align*}"}
+{"id": "1933.png", "formula": "\\begin{align*} f ( m ; z ) = \\sum _ { n = 0 } ^ \\infty b _ 3 ( n ) q ^ { n + \\frac { m ( 3 a + b ) + 2 } { 2 4 } } \\cdot \\prod _ { n = 1 } ^ \\infty ( 1 - q ^ { 3 m n } ) ^ a ( 1 - q ^ { m n } ) ^ b . \\end{align*}"}
+{"id": "6450.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu ( T ^ { m _ n + j } A \\cap B ) = \\lim _ { n \\to \\infty } \\mu ( T ^ { m _ n } A \\cap T ^ { - j } B ) = \\mu ( A ) \\mu ( T ^ { - j } B ) = \\mu ( A ) \\mu ( B ) . \\end{align*}"}
+{"id": "7245.png", "formula": "\\begin{align*} D ( x ) : = \\det \\left [ \\begin{array} { c c c c c } \\alpha _ 1 & \\alpha _ 1 ^ q & \\alpha _ 1 ^ { q ^ 2 } & \\cdots & \\alpha _ 1 ^ { q ^ { n } } \\\\ \\alpha _ 2 & \\alpha _ 2 ^ q & \\alpha _ 2 ^ { q ^ 2 } & \\cdots & \\alpha _ 2 ^ { q ^ { n } } \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha _ n & \\alpha _ n ^ q & \\alpha _ n ^ { q ^ 2 } & \\cdots & \\alpha _ n ^ { q ^ { n } } \\\\ x & x ^ q & x ^ { q ^ 2 } & \\cdots & x ^ { q ^ { n } } \\\\ \\end{array} \\right ] \\end{align*}"}
+{"id": "6404.png", "formula": "\\begin{align*} J \\nabla _ g f = X _ v \\end{align*}"}
+{"id": "2372.png", "formula": "\\begin{align*} \\mathbb { T } \\left ( \\mathcal { H } _ { \\ast } , \\{ \\mathbf { h } _ p \\} _ { p = 0 } ^ { 1 1 } , \\{ 0 \\} _ { p = 0 } ^ { 1 1 } \\right ) = \\prod _ { p = 0 } ^ { 1 1 } \\left [ \\mathbf { h } ' _ p , \\mathbf { h } _ p \\right ] ^ { ( - 1 ) ^ { ( p + 1 ) } } . \\end{align*}"}
+{"id": "5184.png", "formula": "\\begin{align*} & < v _ t , \\xi > + \\int _ { \\mathbb { R } ^ { 3 } } ( ( v + u _ { \\infty } ) \\cdot \\nabla v - ( b + B _ { \\infty } ) \\cdot \\nabla b ) \\cdot \\xi \\dd x + \\nu \\int _ { \\mathbb { R } ^ { 3 } } \\nabla v : \\nabla \\xi \\dd x = 0 , \\\\ & < b _ t , \\zeta > + \\int _ { \\Omega } ( b \\times v + B _ { \\infty } \\times v + b \\times u _ { \\infty } ) \\cdot \\nabla \\times \\zeta \\dd x + \\mu \\int _ { \\Omega } \\nabla \\times b \\cdot \\nabla \\times \\zeta \\dd x = 0 , \\end{align*}"}
+{"id": "8137.png", "formula": "\\begin{align*} M _ { \\hat I } = \\varprojlim _ n M / ( x _ 1 ^ n , \\dots , x _ k ^ n ) . \\end{align*}"}
+{"id": "6310.png", "formula": "\\begin{align*} \\lambda _ t = \\prod _ { i = 1 } ^ { t } ( 1 - \\alpha _ i ) \\leq \\prod _ { i = 1 } ^ { t } \\left ( 1 - \\sqrt { \\mu \\gamma _ i } \\right ) . \\end{align*}"}
+{"id": "6566.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } & \\delta n \\xi ( \\cdot , t ) - \\int _ { 0 } ^ { t } \\int _ { \\Omega } \\delta n \\xi _ { t } + \\int _ { 0 } ^ { t } \\int _ { \\Omega } \\nabla \\delta n \\cdot \\nabla \\xi \\\\ & = \\int _ { 0 } ^ { t } \\int _ { \\Omega } ( \\delta n S ( x , \\tilde { n } _ { 1 } , c _ { 1 } ) \\cdot \\nabla c _ { 1 } + n _ { 2 } Z \\cdot \\nabla c _ { 1 } + n _ { 2 } S ( x , \\tilde { n } _ { 2 } , c _ { 2 } ) \\cdot \\nabla \\delta c ) \\cdot \\nabla \\xi , \\end{aligned} \\end{align*}"}
+{"id": "7960.png", "formula": "\\begin{align*} H _ N ( t ) - \\widetilde H _ N ( t ) = \\int _ { \\Omega _ N } g _ { N , t } \\ , d \\mu _ { N , t } , g _ { N , t } : = \\log \\left ( \\frac { d \\tilde \\nu _ { N , t } } { d \\nu _ { N , t } } \\right ) . \\end{align*}"}
+{"id": "8361.png", "formula": "\\begin{align*} & | a ( k ) | ^ 2 + | b ( k ) | ^ 2 = 1 , k \\in \\mathbb { R } , \\\\ & | a ( k ) | ^ 2 - | b ( k ) | ^ 2 = 1 , k \\in i \\mathbb { R } . \\end{align*}"}
+{"id": "3216.png", "formula": "\\begin{align*} \\sum _ i \\mu _ i \\left ( \\sum _ j \\nu _ j \\gamma _ { i , j } \\right ) = \\sum _ { i , j } ( \\mu _ i \\nu _ { i , j } ) \\gamma _ { i , j } , \\end{align*}"}
+{"id": "8956.png", "formula": "\\begin{align*} \\| { F } ^ { ( \\delta ) } _ { 2 , N } v _ 0 \\| _ p ^ p \\lesssim \\sum _ { | k | \\le N } \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { - p ' } \\bigl | G _ { 2 , k } ^ { ( \\delta ) } \\bigr | ^ { p ' } = : \\bigl [ \\mathbf { G } ^ { ( \\delta ) } _ { 2 , N } ( g ) \\bigr ] ^ { p ' } . \\end{align*}"}
+{"id": "1844.png", "formula": "\\begin{align*} \\nabla X ^ { \\flat } ( Y , Z ) = \\nabla _ Y X ^ { \\flat } Z = \\langle \\nabla _ Y X , Z \\rangle \\ , , \\forall \\ , X , Y \\in \\Gamma ( M ) \\ , . \\end{align*}"}
+{"id": "1668.png", "formula": "\\begin{align*} ( \\pi \\otimes ( \\psi \\circ \\det ) , ( \\psi ( \\varpi _ v ) \\eta _ v ) _ v ) = ( \\pi , ( \\psi ( \\varpi _ v ) \\eta _ v ) _ v ) \\end{align*}"}
+{"id": "8690.png", "formula": "\\begin{align*} s _ \\Sigma ( \\bar x ) & \\le s _ \\Sigma ( x ) + \\frac { 1 } { \\lambda } d ( x , \\bar x ) \\le r _ m + \\frac { \\bar K } \\lambda r _ { m ' } < \\left ( \\frac 1 { \\bar K ^ 2 } + \\frac 1 { \\lambda \\bar K } \\right ) r _ { m ' - 2 } \\le \\frac { r _ { m ' - 2 } } { \\bar K } \\le \\frac { r _ i } { \\bar K } . \\end{align*}"}
+{"id": "7105.png", "formula": "\\begin{align*} \\frac { | \\xi - z | ^ 2 } { | \\xi - z ^ * | ^ 2 | z | ^ 2 } & = 1 - \\frac { | \\xi z - z ^ * z | ^ 2 - | \\xi - z | ^ 2 } { | \\xi - z ^ * | ^ 2 | z | ^ 2 } \\\\ & = 1 - \\frac { ( | \\xi | ^ 2 | z | ^ 2 - 2 { \\rm R e } \\ , ( \\xi \\bar z ) + 1 ) - ( | \\xi | ^ 2 - 2 { \\rm R e } \\ , ( \\xi \\bar z ) + | z | ^ 2 ) } { | \\xi - z ^ * | ^ 2 | z | ^ 2 } \\\\ & = 1 - \\frac { ( 1 - | \\xi | ^ 2 ) ( 1 - | z | ^ 2 ) } { | \\xi - z ^ * | ^ 2 | z | ^ 2 } \\end{align*}"}
+{"id": "6746.png", "formula": "\\begin{align*} \\mathrm { E } ( X ^ n ) & = \\frac { 1 } { \\sigma \\mathrm { B } ( \\alpha , \\beta ) } \\sum _ { j = 0 } ^ { \\infty } ( - 1 ) ^ j \\binom { \\beta - 1 } { j } \\int _ { - \\infty } ^ \\infty x ^ n \\left [ \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] ^ { j + \\alpha - 1 } \\phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\mathrm { d } x . \\end{align*}"}
+{"id": "3128.png", "formula": "\\begin{align*} a \\cdot b = \\varrho ( T ( a ) ) b , \\ \\forall \\ a , \\ b \\in V . \\end{align*}"}
+{"id": "150.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } z _ i = \\lambda _ i \\cdot \\left ( { 1 \\over 1 + \\sum _ { j \\in \\mathbb Z _ 0 } \\widetilde { z } _ j } \\right ) ^ k , \\ \\ i \\in \\mathbb Z _ 0 , \\\\ \\widetilde { z } _ i = \\lambda _ i \\cdot \\left ( { 1 \\over 1 + \\sum _ { j \\in \\mathbb Z _ 0 } z _ j } \\right ) ^ k , \\ \\ i \\in \\mathbb Z _ 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "2045.png", "formula": "\\begin{align*} \\big | \\big ( \\Gamma ( g , h ) , \\ \\omega \\big ) _ { L _ v ^ 2 } \\big | : = \\Big | \\int _ { \\mathbb R ^ 3 } \\Gamma ( g , h ) \\ , \\omega d v \\Big | \\leq C _ 1 \\norm { g } _ { L _ v ^ 2 } \\norm { \\psi ( v , D _ v ) h } _ { L _ v ^ 2 } \\norm { \\psi ( v , D _ v ) \\omega } _ { L _ v ^ 2 } , \\end{align*}"}
+{"id": "3103.png", "formula": "\\begin{align*} \\bar { \\theta } ^ k & = \\Pi _ { \\mathcal { K } , G } \\big \\{ \\bar { \\theta } ^ k - G ^ { - 1 } [ T ( \\tilde { \\theta } ^ k ) + G ( \\bar { \\theta } ^ k - \\tilde { \\theta } ^ k ) ] \\big \\} \\\\ & = \\Pi _ { \\mathcal { K } } \\big \\{ \\bar { \\theta } ^ k - [ T ( \\tilde { \\theta } ^ k ) + G ( \\bar { \\theta } ^ k - \\tilde { \\theta } ^ k ) ] \\big \\} . \\end{align*}"}
+{"id": "7342.png", "formula": "\\begin{align*} \\Pi _ S ( X ) \\rightarrow \\oplus _ { i \\in I } \\Pi _ S ( D _ i , N _ i ) \\rightarrow \\cdots \\rightarrow \\oplus _ { J \\subset I , \\sharp J = n } \\Pi _ S ( D _ J , N _ J ) \\rightarrow \\cdots \\end{align*}"}
+{"id": "4114.png", "formula": "\\begin{align*} \\forall x \\in E , \\forall \\lambda \\in K _ v , \\ \\sigma _ v ( x \\otimes \\lambda ) = \\sigma ( x ) \\otimes \\lambda \\end{align*}"}
+{"id": "3399.png", "formula": "\\begin{align*} \\alpha _ { 0 } + \\alpha _ { 1 } + \\ldots + \\alpha _ { t - 2 } = t \\alpha _ { t } . \\end{align*}"}
+{"id": "1679.png", "formula": "\\begin{align*} \\dot { x } ( t ) = A x ( t ) + A _ d x ( t - h ) , \\forall t \\geq 0 , \\end{align*}"}
+{"id": "1922.png", "formula": "\\begin{align*} \\begin{cases} A ^ \\top \\nabla ^ { ( 1 , 0 ) } \\eta ( A z _ \\star , y _ \\star ) - y _ \\star ^ { ( 1 ) } \\cdot e _ d ^ { ( 1 ) } = \\textbf { 0 } _ d , \\\\ \\nabla ^ { ( 0 , 1 ) } \\eta ( A z _ \\star , y _ \\star ) - \\tfrac { 1 } { p } \\sum _ { i = 2 } ^ { 2 T } ( y _ \\star ^ { ( i ) } ) ^ p \\cdot e _ d ^ { ( i ) } - ( z _ \\star ^ { ( 1 ) } - 2 T + \\tfrac { 1 } { p } ) \\cdot e _ d ^ { ( 1 ) } = \\textbf { 0 } _ d . \\end{cases} \\end{align*}"}
+{"id": "8269.png", "formula": "\\begin{align*} \\int _ { t _ 1 } ^ { t _ 2 } g _ { n + 1 } \\psi _ { n } ( s ) \\sum _ { j = 1 } ^ { n } j V _ { n , j } \\psi _ j ( s ) d s \\to 0 . \\end{align*}"}
+{"id": "4177.png", "formula": "\\begin{align*} x _ 2 \\partial _ { x _ 1 } v _ 3 - x _ 1 \\partial _ { x _ 2 } v _ 3 + k \\partial _ { x _ 3 } v _ 3 = 0 . \\end{align*}"}
+{"id": "1391.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } x ^ { \\alpha } e ^ { - x } L _ n ^ { ( \\alpha ) } ( x ) L _ m ^ { ( \\alpha ) } ( x ) d x = \\frac { \\Gamma ( n + \\alpha + 1 ) } { n ! } \\delta _ { n m } . \\end{align*}"}
+{"id": "81.png", "formula": "\\begin{align*} \\ell ( x , v ( \\beta _ i + \\beta _ \\ell ) ) \\geq 0 \\langle \\mu , v \\beta _ i + v \\beta _ \\ell \\rangle = - 2 . \\end{align*}"}
+{"id": "226.png", "formula": "\\begin{align*} \\prod _ { k = \\kappa ( j _ 1 ) + 1 } ^ { \\kappa ( j _ 2 ) } ( s _ { \\iota ( k ) } ) _ { | \\ , i ( g ) - \\sigma ( k - 1 ) } & = \\prod _ { j = j _ 1 + 1 } ^ { j _ 2 } \\ ; \\prod _ { k = \\kappa ( j - 1 ) + 1 } ^ { \\kappa ( j ) } ( s _ { \\iota ( k ) } ) _ { | \\ , i ( g ) - \\sigma ( k - 1 ) } \\\\ & = \\prod _ { j = j _ 1 + 1 } ^ { j _ 2 } ( s _ { \\iota ( \\kappa ( j ) ) } ) _ { | \\ , i ( g ) - \\sigma ( \\kappa ( j ) - 1 ) } \\ne 1 , \\end{align*}"}
+{"id": "5169.png", "formula": "\\begin{align*} | | b _ n - b _ { 1 , n } - b _ { 2 , n } | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } ^ { 2 } \\leq C \\int _ { D ( 0 , R _ n ) \\backslash D ( 0 , R _ 0 ) } ( | \\nabla \\phi _ n | ^ { 2 } + | G _ n | ^ { 2 } ) \\frac { 1 } { r } \\dd z \\dd r = C | | b _ n | | _ { L ^ { 2 } ( B ( 0 , R _ n ) \\backslash B ( 0 , R _ 0 ) ) } ^ { 2 } . \\end{align*}"}
+{"id": "118.png", "formula": "\\begin{align*} \\tilde { D } _ 1 & : = [ \\gamma / 2 , \\gamma ] \\times [ - \\sqrt { 1 + \\alpha } ~ \\gamma ^ 2 , \\sqrt { 1 + \\alpha } ~ \\gamma ^ 2 ] , \\\\ \\tilde { D } _ 2 & : = [ N , N + \\gamma ] \\times [ \\sqrt { 1 + \\alpha } ~ N ^ { \\frac { \\alpha + 2 } { 2 } } , \\sqrt { 1 + \\alpha } ~ N ^ { \\frac { \\alpha + 2 } { 2 } } + \\gamma ^ 2 ] . \\end{align*}"}
+{"id": "6851.png", "formula": "\\begin{align*} I _ { 1 , n } ( x ) = I _ { 1 , n - 1 } ( x ) + e ^ { \\frac { x } { 2 } } g ( \\frac { i } { 2 } , x ) b _ { n - 1 } ( x ) - \\int _ { - \\infty } ^ { x } \\left ( g ( \\frac { i } { 2 } , t ) e ^ { \\frac { t } { 2 } } \\right ) ^ { \\prime } b _ { n - 1 } ( t ) d t , \\end{align*}"}
+{"id": "3009.png", "formula": "\\begin{align*} \\| x ( t ) - x ^ j ( t ) \\| = \\| w ( t - \\tau ) - w ^ j ( t - \\tau ) \\| , t \\in [ \\tau - h , \\tau ) , \\end{align*}"}
+{"id": "3531.png", "formula": "\\begin{align*} - \\sum _ { j = 0 } ^ { p } B _ j n ^ { p - j } \\binom { p } { j } \\sum _ { K = 0 } ^ { p - j } ( - 1 ) ^ { K + j } \\binom { p - j } { K } . \\end{align*}"}
+{"id": "8905.png", "formula": "\\begin{align*} \\mathbf { f } = & \\log 3 \\mathbf { e } _ 1 + \\log 3 \\mathbf { e } _ 2 + \\log 3 \\mathbf { e } _ 3 + \\log \\frac { 4 } { 3 } \\mathbf { e } _ { 1 2 3 ' } \\\\ = & [ \\log 4 , \\log 4 , \\log 4 , \\log 1 6 , \\log 1 6 , \\log 1 6 , \\log 4 8 ] ^ \\intercal \\end{align*}"}
+{"id": "3368.png", "formula": "\\begin{align*} J _ n & \\leq \\sum _ { k = 1 } ^ { n } \\| f \\| _ \\infty e ^ { k \\| V \\| _ \\infty } \\| \\mathcal { B } _ n ^ V \\textbf { 1 } \\| _ { \\infty } ~ q ^ k \\| u _ 0 - u _ 0 ' \\| \\\\ & + e ^ { n \\| V \\| _ \\infty } \\| \\nabla f \\| _ \\infty q ^ n \\| u _ 0 - u _ 0 ' \\| . \\end{align*}"}
+{"id": "5618.png", "formula": "\\begin{align*} 1 - \\left ( 1 - e ^ { a _ { \\alpha + 1 } + d _ { \\alpha + 1 } - d _ \\alpha } \\right ) \\ . \\left ( e ^ { - c _ { n + 1 } } - 1 \\right ) & = 1 - \\left ( 1 - \\frac { 1 } { 2 } \\right ) \\left ( \\frac { 3 } { 2 } - 1 \\right ) \\\\ & = 1 - \\frac { 1 } { 4 } \\\\ & = \\frac { 3 } { 4 } \\\\ & \\neq \\frac { 2 } { 3 } \\\\ & = e ^ c \\ . \\end{align*}"}
+{"id": "4960.png", "formula": "\\begin{align*} E _ { \\xi _ 1 ^ \\theta ( u ) } \\biggl [ \\varphi \\biggl ( x + u + \\sum _ { i = 2 } ^ { n } Z _ i ^ { \\theta [ u ] } \\biggr ) \\biggr ] = E _ { \\xi _ 1 ^ \\theta ( u ) } \\biggl [ \\varphi \\biggl ( x + u + \\sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \\rho } \\biggr ) \\biggr ] . \\end{align*}"}
+{"id": "1908.png", "formula": "\\begin{align*} w _ n ( s ) = s { \\frac { f ' _ n ( q s ) } { \\beta ^ n } } n \\in { \\mathbb { N } } . \\end{align*}"}
+{"id": "2565.png", "formula": "\\begin{align*} p ^ { n _ { s } ^ { \\prime } - n _ { 1 } ^ { \\prime } } q ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { 2 } } N _ { n _ { 1 } } - p ^ { n _ { 2 } ^ { \\prime } + n _ { s } ^ { \\prime } - 2 n _ { 1 } ^ { \\prime } } a _ { n _ { 1 } } { N _ { n _ { 2 } } } = p ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } q ^ { n _ { s } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { s } } \\frac { N _ { n _ { 2 } } N _ { n _ { 1 } } } { N _ { n _ { s } } } . \\end{align*}"}
+{"id": "3673.png", "formula": "\\begin{align*} X _ { i ; j k } - X _ { i ; k j } & = R _ { k j \\ell i } X ^ \\ell \\\\ X _ { j ; k i } - X _ { j ; i k } & = R _ { i k \\ell j } X ^ \\ell \\\\ X _ { k ; i j } - X _ { k ; j i } & = R _ { j i \\ell k } X ^ \\ell . \\end{align*}"}
+{"id": "6614.png", "formula": "\\begin{align*} g ( w _ 1 , w _ 1 ) = \\varepsilon , \\ , g ( w _ 1 , w _ 2 ) = g ( w _ 1 , w _ 3 ) = g ( w _ 2 , w _ 2 ) = g ( w _ 3 , w _ 3 ) = 0 , \\ , g ( w _ 2 , w _ 3 ) = \\frac \\varepsilon 2 \\end{align*}"}
+{"id": "1852.png", "formula": "\\begin{align*} \\aligned & \\sum _ { i = 1 } ^ { n - 1 } \\langle i _ { e _ i } R ^ { \\nabla } , i _ { e _ i } R ^ { \\nabla } \\rangle + \\langle i _ { \\frac \\partial { \\partial r } } R ^ { \\nabla } , i _ { \\frac \\partial { \\partial r } } R ^ { \\nabla } \\rangle \\\\ & = \\sum _ { 1 \\le j _ 1 \\le n } \\sum _ { i = 1 } ^ { n } \\langle R ^ \\nabla ( e _ i , e _ { j _ 1 } ) , R ^ \\nabla ( e _ i , e _ { j _ 1 } ) \\rangle \\\\ & = 2 | | R ^ \\nabla | | ^ 2 \\ , , \\endaligned \\end{align*}"}
+{"id": "4803.png", "formula": "\\begin{align*} a ( u , v ) : = ( \\triangle u , \\Delta v ) . \\end{align*}"}
+{"id": "1121.png", "formula": "\\begin{align*} { S _ { \\min } ^ { \\rm { N } } } = \\frac { { W I } } { { K L } } \\overline \\varepsilon . \\end{align*}"}
+{"id": "6459.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } \\epsilon _ i < \\infty . \\end{align*}"}
+{"id": "5458.png", "formula": "\\begin{align*} d = 2 n + 1 = 4 l + 3 \\equiv 3 \\mod 4 \\end{align*}"}
+{"id": "2669.png", "formula": "\\begin{align*} \\mathcal { G } ( t ) = \\lambda ( t ) \\mathcal { E } ( t ) + \\mu \\alpha ( t ) \\mathcal { E } ^ r ( t ) \\langle P ( u _ t ) , u \\rangle + \\nu \\alpha ( t ) \\delta ^ { \\frac { 1 } { \\ell } } ( t ) \\mathcal { E } ^ { r + \\frac { 1 } { q } + \\frac { 1 } { \\ell ^ { \\prime } } } ( t ) . \\end{align*}"}
+{"id": "1154.png", "formula": "\\begin{align*} d X _ { t } ^ { i } = \\left ( \\frac { 1 } { n - 1 } \\sum _ { j \\neq i } K \\left ( X _ { t } ^ { i } - X _ { t } ^ { j } \\right ) \\right ) d t + d W _ { t } ^ { i } , i = 1 \\cdots n , \\end{align*}"}
+{"id": "8588.png", "formula": "\\begin{align*} | Z | = \\ ! \\ ! \\ ! \\sum _ { 1 \\le i _ 1 < \\cdots < i _ n \\le m } & | \\det ( u _ { i _ 1 } , \\dots , u _ { i _ n } ) | \\\\ & | P _ { e ^ \\bot } Z | = \\ ! \\ ! \\ ! \\sum _ { 1 \\le i _ 2 < \\cdots < i _ n \\le m } | \\det ( P _ { e ^ \\bot } u _ { i _ 2 } , \\dots , P _ { e ^ \\bot } u _ { i _ n } ) | . \\end{align*}"}
+{"id": "8408.png", "formula": "\\begin{align*} q ( x ; z ) : = [ \\partial _ z \\Psi ^ - _ { 1 1 } ( x ; z ) , \\ \\ \\partial _ z \\Psi ^ - _ { 2 1 } ( x ; z ) - 2 i x \\Psi ^ - _ { 2 1 } ( x ; z ) ] ^ T , \\end{align*}"}
+{"id": "7172.png", "formula": "\\begin{align*} I _ { n + 1 } \\frac { \\partial ^ 2 } { \\partial x _ n ^ 2 } + B \\frac { \\partial } { \\partial x _ n } + C = \\Bigl ( I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } + B - Q \\Bigr ) \\Bigl ( I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } + Q \\Bigr ) . \\end{align*}"}
+{"id": "2606.png", "formula": "\\begin{align*} R \\cdot R ( x , J x , J x , x ; u , J u ) = \\frac { 1 } { 2 } \\ , L ( p , \\bar { \\pi } ^ h ) \\ , Q ^ c ( g , R ) ( x , J x , J x , x ; u , J u ) . \\end{align*}"}
+{"id": "8058.png", "formula": "\\begin{align*} R _ { k } = \\log _ 2 \\left ( 1 + \\gamma _ { k } \\right ) \\end{align*}"}
+{"id": "5285.png", "formula": "\\begin{align*} \\hat { L } _ 0 = \\{ 0 \\} \\sqcup \\bigsqcup _ { m = 1 } ^ \\infty \\bigsqcup _ { \\gamma \\in \\Gamma / ( \\Gamma \\cap N ) } \\{ \\gamma \\cdot ( 0 , 0 , 0 , m ) \\} \\sqcup \\bigsqcup _ { m = 1 } ^ \\infty \\bigsqcup _ { n = 0 } ^ { 3 m - 1 } \\bigsqcup _ { \\gamma \\in \\Gamma } \\{ \\gamma \\cdot ( 0 , 0 , 3 m , n ) \\} . \\end{align*}"}
+{"id": "4300.png", "formula": "\\begin{align*} \\mathcal { H } ^ 2 ( t ; c , f ) : = \\Bigg \\{ \\tilde { f } : \\int _ { \\{ \\Psi < - t \\} } | \\tilde { f } | ^ 2 _ h c ( - \\Psi ) < + \\infty , \\ \\tilde { f } \\in H ^ 0 ( \\{ \\Psi < - t \\} , \\mathcal { O } ( K _ M \\otimes E ) ) \\\\ \\& ( \\tilde { f } - f ) _ { z _ 0 } \\in \\mathcal { O } ( K _ M ) _ { z _ 0 } \\otimes J _ { z _ 0 } , z _ 0 \\in Z _ 0 \\Bigg \\} , \\end{align*}"}
+{"id": "4129.png", "formula": "\\begin{align*} L u : = \\big ( a _ { j k } ^ { \\alpha \\beta } \\partial _ j \\partial _ k u _ \\beta \\big ) _ { 1 \\leq \\alpha \\leq N } , \\end{align*}"}
+{"id": "8960.png", "formula": "\\begin{align*} I _ 2 ^ p : = \\sum _ { | k | \\le N } \\int _ { \\eta _ { k } } ^ { \\eta _ { k + 1 } } v _ 1 ^ { p } ( x ) \\frac { \\Bigl | \\bigl [ V _ 1 ^ - ( x ) \\bigr ] ' \\Bigr | ^ p } { \\bigl [ V _ 1 ^ - ( x ) \\bigr ] ^ { 2 p } } \\biggl ( \\int _ { a ( x ) } ^ { \\eta _ { k } } v _ 1 ^ { - p ' } ( t ) \\biggl ( \\int _ { a ( x ) } ^ t v _ 1 ^ { - p ' } \\biggr ) ^ { 1 - \\delta } [ V _ 1 ( t ) ] ^ { \\delta } \\bigl | G ^ { ( \\delta ) } _ { 2 , k } ( t ) \\bigr | ^ { p ' - 1 } \\ , d t \\biggr ) ^ p \\ , d x \\end{align*}"}
+{"id": "6906.png", "formula": "\\begin{align*} 0 \\le V _ { \\mathrm { o r i g } } - V _ { \\mathrm { d e t } } \\le \\frac { n T ^ 2 } { 2 } \\sum _ { i , j = 1 } ^ n \\int _ { \\R ^ n } | \\partial _ { i j } g | ^ 2 \\ , d \\gamma _ { y ^ * , T } , \\end{align*}"}
+{"id": "7112.png", "formula": "\\begin{align*} \\int _ { t _ 0 } ^ { t _ 1 } \\int _ { \\Omega } \\psi ( x ) ^ T u ( t , x ) \\ , \\omega ( t , x ) d x d t = \\int _ { \\Omega } \\phi ( x ) \\omega ( t _ 1 , x ) d x - \\int _ { \\Omega } \\phi ( x ) \\omega ( t _ 0 , x ) d x \\end{align*}"}
+{"id": "4476.png", "formula": "\\begin{align*} \\frac { 3 ^ { 3 } } { 4 ^ { 3 } } ( a _ { 3 } ^ { * } ) ^ { 3 } a _ { 4 } ^ { * } \\leq a _ { 1 } ^ { * } a _ { 2 } ^ { * } a _ { 3 } ^ { * } a _ { 4 } ^ { * } = \\frac { p ^ { 3 } } { 1 6 } \\end{align*}"}
+{"id": "4183.png", "formula": "\\begin{align*} \\begin{cases} \\nabla w \\cdot \\nabla ^ \\perp \\varphi = 0 , \\\\ w = \\mathcal { L } _ H \\varphi , \\\\ \\varphi | _ { \\partial \\Omega } = 0 . \\end{cases} \\end{align*}"}
+{"id": "6506.png", "formula": "\\begin{align*} \\lambda ( q ( c ( z _ 1 z _ 2 . . . z _ { m - 1 } ) ) , q ( c ( z _ 1 z _ 2 . . . z _ { m - 1 } ) ) l _ m ) = z _ m \\end{align*}"}
+{"id": "1071.png", "formula": "\\begin{align*} W ^ J = \\{ w \\in W \\mid w ( J ) \\subseteq \\Phi ^ + \\} . \\end{align*}"}
+{"id": "410.png", "formula": "\\begin{align*} d ( p , g ) \\leq L d ( p , \\mathcal { F } _ g ) = L d ( p , \\mathcal { F } _ q ) \\leq L d ( p , q ) < L \\frac r L = r , \\end{align*}"}
+{"id": "2154.png", "formula": "\\begin{align*} \\Re \\mathcal { P } [ \\varrho ] = - \\frac { 5 } { 2 } a ^ 3 | w | ^ 4 \\left ( - 2 a | w | ^ 2 + \\sqrt { 5 } | w | ^ 2 - 2 \\Re ( \\varrho _ z ^ 2 ) \\right ) , \\end{align*}"}
+{"id": "73.png", "formula": "\\begin{align*} & \\ell ( x , w ' \\alpha ) + \\ell ( x ' , \\alpha ) \\\\ = & \\ , \\langle \\mu , w ' \\alpha \\rangle + \\langle \\mu ' , \\alpha \\rangle - \\Phi ^ + ( w w ' \\alpha ) + \\Phi ^ + ( w ' \\alpha ) - \\Phi ^ + ( w ' \\alpha ) + \\Phi ^ + ( \\alpha ) \\\\ = & \\ , \\langle ( w ' ) ^ { - 1 } \\mu + \\mu ' , \\alpha \\rangle - \\Phi ^ + ( w w ' \\alpha ) + \\Phi ^ + ( \\alpha ) = \\ell ( x x ' , \\alpha ) . \\end{align*}"}
+{"id": "8793.png", "formula": "\\begin{align*} \\frac { d ^ j } { d t ^ j } \\tilde { X } _ t | _ { t = 0 } \\end{align*}"}
+{"id": "6711.png", "formula": "\\begin{align*} \\begin{array} { l l } \\begin{pmatrix} \\lambda & 0 \\\\ 0 & \\lambda ^ { - 1 } \\end{pmatrix} , & \\lambda > 1 , \\\\ \\begin{pmatrix} a _ 1 & b _ 1 \\\\ c _ 1 & d _ 1 \\end{pmatrix} , & a _ 1 d _ 1 - b _ 1 c _ 1 = 1 , a _ 1 + b _ 1 = c _ 1 + d _ 1 > 0 \\end{array} \\end{align*}"}
+{"id": "2636.png", "formula": "\\begin{align*} ( x \\cdot y ) \\diamond \\alpha ( z ) = \\alpha ( x ) \\diamond ( y \\cdot z ) . \\end{align*}"}
+{"id": "4860.png", "formula": "\\begin{align*} \\Gamma ( x , y ) : = - \\frac { 1 } { e ^ { - { \\psi } } } \\left ( d \\phi ( x ) \\int _ { - \\infty } ^ y e ^ { - { \\psi ( x , v ) } } d v - \\int _ { - \\infty } ^ y e ^ { - { \\psi ( x , v ) } } d _ x { \\psi } ( x , v ) d v \\right ) . \\end{align*}"}
+{"id": "7423.png", "formula": "\\begin{align*} \\psi ( p ) = \\psi _ { p _ 0 } [ \\eta ] ( p ) \\stackrel { d e f } { = } \\inf _ { r \\in R ( p ) } g ( p _ 0 , r , | \\eta | _ r ) , \\ p \\le p _ 0 . \\end{align*}"}
+{"id": "4326.png", "formula": "\\begin{align*} \\frac { H ( r _ 1 ) - H ( r _ 2 ) } { r _ 1 - r _ 2 } \\leq \\liminf \\limits _ { r _ 3 \\to r _ 2 - 0 } \\frac { H ( r _ 3 ) - H ( r _ 2 ) } { r _ 3 - r _ 2 } \\end{align*}"}
+{"id": "7372.png", "formula": "\\begin{gather*} \\int Q \\bigl ( \\oplus _ { i = 1 } ^ n f _ i \\bigr ) \\ , P _ \\mathcal { U } ( d Q ) = \\sum _ { i = 1 } ^ n \\mu _ i ( f _ i ) \\quad f _ i \\in B ( \\Omega _ i , \\mathcal { A } _ i ) i \\in I ; \\end{gather*}"}
+{"id": "628.png", "formula": "\\begin{align*} L ( 1 , \\chi _ K ) = \\frac { 2 \\pi h _ K } { w _ K \\sqrt { \\Delta _ K } } . \\end{align*}"}
+{"id": "8403.png", "formula": "\\begin{align*} ( I - F ) \\Psi ^ - = I , \\end{align*}"}
+{"id": "2334.png", "formula": "\\begin{align*} \\rho ^ \\star ( \\theta ^ \\sharp , X ) & = D ^ g \\theta ( J \\theta ^ \\sharp , J X ) + \\frac { 1 } { 2 } g ( D ^ g _ { J X } \\theta , J \\theta ) + \\frac { 1 } { 2 } g ( D ^ g _ X \\theta , \\theta ) + g ( d \\theta , N _ X ) , \\\\ & = \\frac { 1 } { 2 } D ^ g \\theta ( J \\theta ^ \\sharp , J X ) + \\frac { 1 } { 2 } g ( D ^ g _ X \\theta , \\theta ) + g ( d \\theta , N _ X ) , \\\\ & = - \\frac { 1 } { 2 } ( d \\theta ) ^ { J , - } ( \\theta ^ \\sharp , X ) + g ( d \\theta , N _ X ) . \\end{align*}"}
+{"id": "7120.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\mathsf { g } _ { \\mathbb { D } , \\tilde { \\mathbb { F } } ( \\mathsf { S } ) } ( \\tilde { F } ( L ) ) & = & \\tilde { F } ( \\mathsf { g } _ { \\mathbb { C } , \\mathsf { S } } ( L ) ) \\circ \\Phi _ { S , S } \\\\ & = & \\tilde { F } ( \\mathsf { d } _ { \\mathbb { C } , \\mathsf { S } } ( R ) ) \\circ \\Phi _ { S , S } \\\\ & = & \\mathsf { d } _ { \\mathbb { D } , \\tilde { \\mathbb { F } } ( \\mathsf { S } ) } ( \\tilde { F } ( R ) ) . \\end{array} \\end{align*}"}
+{"id": "5390.png", "formula": "\\begin{align*} ( B \\Psi - \\Phi ) _ n = - B + 2 B K x _ n - \\Phi _ n = 0 , \\end{align*}"}
+{"id": "7479.png", "formula": "\\begin{align*} v : = \\dfrac { u - ( u ) _ { B _ 1 } } { \\| \\nabla u \\| _ { L ^ 2 ( B _ 1 ) } } , \\end{align*}"}
+{"id": "8238.png", "formula": "\\begin{align*} | ( \\mathcal { R } _ \\lambda ( \\sigma ) v _ T , v _ T ) | = \\prod _ { j } \\frac { 1 } { | a _ { i _ j + 1 } - a _ { i _ j } | } , \\end{align*}"}
+{"id": "1727.png", "formula": "\\begin{align*} | | U ( t , 0 ) G ( x ) - G ( x ) | | _ { L ^ { \\infty } ( 0 , \\tau _ 1 ; L ^ 2 ) } = & | | ( U ( t , 0 ) - I ) e ^ { ( \\alpha - 1 ) W } g ( x ) | | _ { L ^ { \\infty } ( 0 , \\tau _ 1 ; L ^ 2 ) } \\\\ \\le & | | U ( t , 0 ) - I | | | | e ^ { ( \\alpha - 1 ) W } | | _ { L ^ { \\infty } ( 0 , \\tau _ 1 ; L ^ { \\infty } ) } | | g ( x ) | | _ { L ^ 2 } \\\\ \\lesssim & | | g ( x ) | | _ { L ^ 2 } \\lesssim | | x | | _ { H ^ 2 } ^ { \\alpha } . \\end{align*}"}
+{"id": "5115.png", "formula": "\\begin{align*} \\nabla _ { z , r } \\eta ( z , r ) = \\int _ { 0 } ^ { \\infty } \\nabla _ { z , r } { \\mathcal { G } } * _ 1 G ( \\cdot , r ' ) \\frac { 1 } { r ' } \\dd r ' . \\end{align*}"}
+{"id": "943.png", "formula": "\\begin{align*} \\partial _ t \\begin{pmatrix} w \\\\ \\overline { w } \\end{pmatrix} & = i t ^ { - 1 } | W _ 1 | ^ 2 P \\begin{pmatrix} i \\mu & 0 \\\\ 0 & - i \\mu \\end{pmatrix} P ^ { - 1 } \\begin{pmatrix} w \\\\ \\overline { w } \\end{pmatrix} + \\begin{pmatrix} S _ 2 + e ^ { 3 i \\lambda _ 1 ( \\Phi _ 1 - \\theta _ 1 ) } R _ 2 \\\\ \\overline { S _ 2 + e ^ { 3 i \\lambda _ 1 ( \\Phi _ 1 - \\theta _ 1 ) } R _ 2 } \\end{pmatrix} . \\end{align*}"}
+{"id": "497.png", "formula": "\\begin{align*} f _ { + a } ( x ) = e ^ { - \\lambda _ s } f _ r ( x ) + ( 1 - e ^ { - \\lambda _ s } ) f _ r \\ast f _ s ( x ) . \\end{align*}"}
+{"id": "7419.png", "formula": "\\begin{align*} Z _ { i , j } = X _ { i , j } \\ , { \\rm c h } _ { \\rm r e f } [ B ( m _ i , m _ j , s _ j - s _ i ) ] . \\end{align*}"}
+{"id": "9064.png", "formula": "\\begin{align*} v _ 0 = \\sum Y _ { \\ 1 } ^ { - } v _ k . \\end{align*}"}
+{"id": "5540.png", "formula": "\\begin{align*} a \\cdot b a & = b , \\\\ a b \\cdot c & = a \\cdot ( b \\cdot c ) , \\\\ ( a \\cdot b ) a & = ( b \\cdot a ) b , \\end{align*}"}
+{"id": "6125.png", "formula": "\\begin{align*} \\lambda ( m ) \\ , r _ n ( m ) = A _ { n } \\ r _ { n + 1 } ( m ) - ( A _ n + C _ n ) \\ , r _ n ( m ) + C _ n \\ r _ { n - 1 } ( m ) \\ , , \\end{align*}"}
+{"id": "960.png", "formula": "\\begin{align*} \\| \\mathcal { R } _ 1 \\| _ { L ^ p } & { } = t ^ { - \\frac 1 2 + \\frac 1 p } \\| ( \\mathcal { F } M ( t ) \\mathcal { F } ^ { - 1 } - 1 ) F _ 1 ( t ) \\| _ { L ^ p } . \\end{align*}"}
+{"id": "8990.png", "formula": "\\begin{align*} D _ { q } = \\sum _ { r = 1 } ^ { q } x _ { r } . \\end{align*}"}
+{"id": "4320.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi < - t \\} } | F _ t + \\alpha q | ^ 2 _ h c ( - \\Psi ) - \\int _ { \\{ \\psi < - t \\} } | F _ t | ^ 2 _ h c ( - \\Psi ) \\geq 0 \\end{align*}"}
+{"id": "6364.png", "formula": "\\begin{align*} \\operatorname { E } ( X ) = \\frac { \\sqrt { \\pi \\theta } } { \\operatorname { B } ( a , b ) } \\sum _ { n = 0 } ^ b \\binom { b - 1 } { n } \\frac { 1 } { \\sqrt { a + n } } . \\end{align*}"}
+{"id": "7600.png", "formula": "\\begin{align*} F ^ 1 ( \\phi _ i x ) = F ^ i x F _ 1 ( \\phi _ i x ) = F _ i x \\end{align*}"}
+{"id": "3976.png", "formula": "\\begin{align*} c _ k = \\sum _ { i = 1 } ^ { t - 1 } c _ i a _ i + c _ { k } \\sum _ { i \\in \\Omega _ t ^ o } a _ i + c _ { k + 1 } \\sum _ { i \\in \\Omega _ t ^ e } a _ i . \\end{align*}"}
+{"id": "344.png", "formula": "\\begin{align*} \\begin{cases} 0 & { i = j } \\\\ d _ { i j } & { \\mbox { e l s e } } . \\\\ \\end{cases} \\end{align*}"}
+{"id": "4127.png", "formula": "\\begin{align*} g \\rightarrow g ^ { \\sigma } : = \\sigma ^ { - 1 } ( g ^ { - 1 } ) ^ { \\vee } \\sigma \\end{align*}"}
+{"id": "4907.png", "formula": "\\begin{align*} \\tau _ { s , r } = & 2 ^ { s / 2 } \\pi ^ { - ( r + 1 / 2 ) } \\sum _ { \\substack { l = 0 \\\\ } } ^ r \\binom { r } { l } 2 ^ { - l } \\pi ^ l \\Gamma \\left ( \\frac { s + r - l + 1 } { 2 } \\right ) \\times \\\\ & F _ A ^ { ( r - l ) } \\left ( \\frac { s + r - l + 1 } { 2 } ; \\frac { 1 } { 2 } , \\ldots , \\frac { 1 } { 2 } ; \\frac { 3 } { 2 } , \\ldots , \\frac { 3 } { 2 } ; - 1 , \\ldots , - 1 \\right ) , \\end{align*}"}
+{"id": "2739.png", "formula": "\\begin{align*} \\Omega = \\{ x \\in \\R ^ 3 \\colon x _ 3 > | x _ 1 | + | x _ 2 | \\} \\end{align*}"}
+{"id": "4717.png", "formula": "\\begin{align*} \\frac { \\partial \\tilde x } { \\partial y } = x \\cdot \\frac { 1 } { 3 } h ^ { - \\frac { 2 } { 3 } } \\cdot h _ y ' , \\end{align*}"}
+{"id": "2205.png", "formula": "\\begin{align*} \\mathbf { a } ( \\theta _ { 0 } ) ^ { T } \\mathbf { v } ^ { 0 } = b _ 0 + b _ 1 e ^ { - j \\phi } + . . . + b _ { N - 1 } e ^ { - j ( N - 1 ) \\phi } \\end{align*}"}
+{"id": "9017.png", "formula": "\\begin{align*} i _ { 1 4 6 } & = - x _ 1 D _ 3 D _ 5 , & i _ { 1 2 6 } & = x _ 1 D _ 2 D _ 5 , & i _ { 1 2 4 } & = - x _ 1 D _ 2 D _ 4 , \\\\ i _ { 2 3 6 } & = - D _ 1 x _ 3 D _ 5 , & i _ { 2 3 4 } & = D _ 1 x _ 3 D _ 4 , & i _ { 2 4 5 } & = - D _ 1 D _ 3 x _ 5 . \\end{align*}"}
+{"id": "695.png", "formula": "\\begin{align*} S : = F \\cup \\{ g \\in G : \\phi _ { g , 1 } \\neq 0 \\} \\cup \\{ g \\in G : \\psi _ { g , 1 } \\neq 0 \\} , \\end{align*}"}
+{"id": "9106.png", "formula": "\\begin{align*} \\mathcal { Q } = \\bigoplus _ { i = 1 } ^ n \\mathcal { Q } _ i \\mathcal { R } = \\bigoplus _ { i = 1 } ^ { n - 1 } J _ { p _ i * } ( J _ { p _ i } ^ * ( \\mathcal { Q } _ { i + 1 } ) ) . \\end{align*}"}
+{"id": "4678.png", "formula": "\\begin{align*} | \\rho ( x ) | = ( x , \\Sigma ) : = \\inf \\{ d ( x , y ) \\mid y \\in \\Sigma \\ , \\} . \\end{align*}"}
+{"id": "4836.png", "formula": "\\begin{align*} \\begin{cases} | \\mathcal { K } ( c ' , x ) | \\leqslant ( c ' + 1 ) C , & | x | < 1 , \\\\ \\mathcal { K } ( c ' , x ) \\leqslant ( - \\alpha + \\frac { \\beta } { | x | } + \\frac { C } { | x | } + c ' C ) , & | x | \\geqslant 1 , \\end{cases} \\end{align*}"}
+{"id": "3547.png", "formula": "\\begin{align*} \\theta _ 4 ( q ) \\theta _ 4 ( q ^ 3 ) + \\dfrac { 1 } { 2 } \\theta _ 2 ( q ) \\theta _ 2 ( q ^ 3 ) & = \\dfrac { 1 } { 2 } \\theta _ 4 ( q ) \\theta _ 4 ( q ^ 3 ) + \\dfrac { 1 } { 2 } \\theta _ 4 ( q ) \\theta _ 4 ( q ^ 3 ) + \\dfrac { 1 } { 2 } \\theta _ 2 ( q ) \\theta _ 2 ( q ^ 3 ) \\\\ & = \\dfrac { 1 } { 2 } \\theta _ 3 ( q ) \\theta _ 3 ( q ^ 3 ) + \\dfrac { 1 } { 2 } \\theta _ 4 ( q ) \\theta _ 4 ( q ^ 3 ) . \\end{align*}"}
+{"id": "5217.png", "formula": "\\begin{align*} & \\int _ 0 ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { j , + } ^ { 2 } ( \\partial _ t \\varphi + ( v _ j + u _ { \\infty } ) \\cdot \\nabla \\varphi ) \\dd x \\dd s + \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } ^ { 2 } \\varphi _ 0 \\dd x \\\\ & = - \\mu _ j \\int _ 0 ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( \\Phi _ { j , + } ^ { 2 } \\Delta \\varphi + \\frac { 2 } { r } \\partial _ r \\Phi _ { j , + } ^ { 2 } \\varphi - 2 | \\nabla \\Phi _ { j , + } | ^ { 2 } \\varphi \\right ) \\dd x \\dd s . \\end{align*}"}
+{"id": "8332.png", "formula": "\\begin{align*} \\psi ^ \\pm ( x , t ; k ) = e ^ { - i c _ - ( x ) \\widehat { \\sigma } _ 3 } \\mu ^ \\pm ( x , t ; k ) e ^ { i c _ + ( x ) \\sigma _ 3 } , \\end{align*}"}
+{"id": "1367.png", "formula": "\\begin{align*} K _ { \\lambda } = \\frac { 1 } { ( 2 \\pi ) ^ { d / 2 } \\prod _ { j = 1 } ^ { d - 1 } j ! } \\int _ { W ^ d } e ^ { - \\sum _ { j = 1 } ^ d ( \\lambda _ j - \\bar { \\lambda } ) z _ j } \\hat { h } ^ { ( \\bar { \\lambda } , \\ldots , \\bar { \\lambda } ) } ( z ) d z _ 1 \\ldots d z _ d . \\end{align*}"}
+{"id": "4795.png", "formula": "\\begin{align*} F _ n ( \\eta ) : = T _ n - \\frac { 1 } { \\eta } I _ n , \\end{align*}"}
+{"id": "4397.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\left | \\frac { \\zeta ' } { \\zeta } ( \\sigma + i t ) \\right | ^ 2 d t = O \\left ( \\frac { T } { ( \\sigma - 1 / 2 ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "3930.png", "formula": "\\begin{align*} z _ { 0 } & = x , \\\\ z _ { \\ell } & = \\sigma \\left ( W _ { \\ell } z _ { \\ell - 1 } + b _ { \\ell } \\right ) , \\ell = 1 , \\ldots , L - 1 , \\\\ \\mathcal { N } ( x ) & = W _ { L } z _ { L - 1 } + b _ { L } . \\end{align*}"}
+{"id": "8157.png", "formula": "\\begin{align*} f ( g ( x ) ) ^ { ( n ) } = \\sum _ { \\pi \\in \\Pi _ n } f ^ { ( | \\pi | ) } ( g ( x ) ) \\prod _ { B \\in \\pi } g ^ { ( | B | ) } ( x ) \\end{align*}"}
+{"id": "6797.png", "formula": "\\begin{align*} B _ { m n } ( x ) : = ( - 1 ) ^ { m + n } \\left ( \\sum _ { k = 1 } ^ { N } \\alpha _ { k } ^ { - } e ^ { 2 \\tau _ { k } x } \\frac { \\left ( \\frac { 1 } { 2 } - \\tau _ { k } \\right ) ^ { m + n } } { \\left ( \\frac { 1 } { 2 } + \\tau _ { k } \\right ) ^ { m + n + 2 } } + \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } s ^ { - } \\left ( \\rho \\right ) e ^ { - 2 i \\rho x } \\frac { \\left ( \\frac { 1 } { 2 } + i \\rho \\right ) ^ { m + n } } { \\left ( \\frac { 1 } { 2 } - i \\rho \\right ) ^ { m + n + 2 } } d \\rho \\right ) , \\end{align*}"}
+{"id": "8425.png", "formula": "\\begin{align*} \\begin{aligned} \\left | e ^ { - i c _ - ( x ) } - e ^ { - i \\tilde { c } _ - ( x ) } \\right | & = \\left | e ^ { \\frac { 1 } { 2 i } \\int _ { - \\infty } ^ x \\left ( | u _ y ( y ) | ^ 2 - | \\tilde { u } _ y ( y ) | ^ 2 \\right ) d y } - 1 \\right | \\leq 2 \\delta c _ 1 \\| u _ y - \\tilde { u } _ y \\| _ { L ^ 2 } . \\end{aligned} \\end{align*}"}
+{"id": "5996.png", "formula": "\\begin{align*} \\| ( u , v ) \\| _ { E } ^ { 2 } = \\| u \\| _ { H _ { r } ^ { 1 } ( \\R ^ { 2 } ) } ^ { 2 } + \\| v \\| _ { H _ { r } ^ { 1 } ( \\R ^ { 2 } ) } ^ { 2 } . \\end{align*}"}
+{"id": "5626.png", "formula": "\\begin{align*} \\frac { \\partial p } { \\partial x } & = h ^ x ( x , 0 ) + \\int _ 0 ^ y \\frac { \\partial h ^ y } { \\partial x } ( x , s ) \\ , d s \\\\ & = h ^ x ( x , 0 ) + \\int _ 0 ^ y g ^ { x y } ( x , s ) \\ , d s \\end{align*}"}
+{"id": "1280.png", "formula": "\\begin{align*} \\norm { \\nabla ^ i _ z c _ { \\Delta } ( \\cdot , \\xi _ { \\Delta } ) } _ \\infty = \\norm { \\nabla ^ i _ { 0 } c _ { \\Delta - z } ( \\cdot , \\xi _ { \\Delta - z } ) } _ \\infty \\norm { \\nabla ^ i _ z \\hat { c } _ { \\Delta } ( \\cdot , \\xi _ { \\Delta } ) } _ \\infty = \\norm { \\nabla ^ i _ { 0 } \\hat { c } _ { \\Delta - z } ( \\cdot , \\xi _ { \\Delta - z } ) } _ \\infty . \\end{align*}"}
+{"id": "5678.png", "formula": "\\begin{align*} & ( p _ { 1 } , \\ldots , p _ { k - 1 } , f _ { k } , \\ldots , f _ { n } ) = \\\\ & \\Big [ p _ { 1 } , \\ldots , p _ { k } , f _ { k + 1 } \\circ ( p _ { 1 } , \\ldots , p _ { k - 1 } , \\widetilde { f } _ { k } , p _ { k + 1 } , \\ldots , p _ { n } ) , \\ldots , f _ { n } \\circ ( p _ { 1 } , \\ldots , p _ { k - 1 } , \\widetilde { f } _ { k } , p _ { k + 1 } , \\ldots , p _ { n } ) \\Big ] \\circ \\\\ & ( p _ { 1 } , \\ldots , p _ { k - 1 } , f _ { k } , p _ { k + 1 } , \\ldots , p _ { n } ) . \\end{align*}"}
+{"id": "4108.png", "formula": "\\begin{align*} L = \\bigoplus _ { i = 1 } ^ \\ell \\mathfrak a _ i b _ i . \\end{align*}"}
+{"id": "7977.png", "formula": "\\begin{align*} \\Theta _ { \\iota } = ( f ^ { - 1 } ) _ { \\iota } \\varphi h G + f ^ { - 1 } G \\varphi h _ { \\iota } \\left ( 1 + \\frac { \\varphi ^ { ' } } { \\varphi } h \\right ) + f ^ { - 1 } \\varphi h G _ { \\iota } . \\end{align*}"}
+{"id": "2071.png", "formula": "\\begin{align*} C _ { \\mu } = \\{ x \\in H : \\mathcal { I } - d _ { - } ( x , H ) < 1 - \\mu \\} . \\end{align*}"}
+{"id": "7624.png", "formula": "\\begin{align*} g _ n ( t , q , x , \\mu ) : = g \\big ( t , \\frac { q } { \\sqrt { n } } , x , \\mu \\big ) \\end{align*}"}
+{"id": "3532.png", "formula": "\\begin{align*} - B _ p n ^ { p - p } \\binom { p } { p } \\sum _ { K = 0 } ^ { 0 } ( - 1 ) ^ { K + p } \\binom { p - p } { K } , \\end{align*}"}
+{"id": "9077.png", "formula": "\\begin{align*} \\tilde { \\sigma } : \\oplus _ { i = 1 } ^ n j _ { i * } ( E _ { i } ) \\rightarrow \\oplus _ { i = 1 } ^ { n - 1 } j _ { p _ i * } ( j ^ * _ { p _ i } ( j _ { ( i + 1 ) * } ( E _ { i + 1 } ) ) ) \\end{align*}"}
+{"id": "2695.png", "formula": "\\begin{align*} m _ k ( x _ k + s ) = \\phi ( x _ k ) + \\langle g _ k , s \\rangle + \\frac { 1 } { 2 } \\langle H _ k s , s \\rangle \\end{align*}"}
+{"id": "709.png", "formula": "\\begin{align*} \\aligned c + 1 + \\frac { 1 } { 2 } \\| u _ n \\| _ { E _ a } \\geq \\Phi _ a ( u _ n ) - \\frac { 1 } { 2 } \\langle \\Phi ' _ { j _ n } ( u _ n ) , u _ n \\rangle = \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 2 p } \\right ) | u _ n | _ { H L } ^ { 2 p } \\endaligned \\end{align*}"}
+{"id": "6265.png", "formula": "\\begin{align*} \\phi _ { i s } ( \\partial _ i ) = - \\frac { \\Gamma _ { i s } ^ s } { \\varphi _ i } , \\quad \\forall s \\neq i , \\end{align*}"}
+{"id": "8682.png", "formula": "\\begin{align*} P _ n P _ { n ' } = \\begin{cases} P _ n , & n ' = n , \\\\ 0 , & n ' \\ne n . \\end{cases} \\end{align*}"}
+{"id": "5573.png", "formula": "\\begin{align*} A ( y _ 1 x ) - A ( y _ 1 x ' ) = A ( y _ 1 x ' ) = A ( 0 1 ^ k 0 . . . ) - A ( 0 ^ \\infty ) = b _ k - a \\end{align*}"}
+{"id": "6127.png", "formula": "\\begin{align*} \\widetilde { \\lambda } ( m ) \\ r _ n ( m ) = E _ { n } \\ \\widetilde { r } _ { n + 1 } ( m ) + F _ n \\ \\widetilde { r } _ { n } ( m ) + G _ n \\ \\widetilde { r } _ { n - 1 } ( m ) \\ , , \\end{align*}"}
+{"id": "8943.png", "formula": "\\begin{align*} a _ { P _ T } ( T ) = \\delta ( T , T ) + O _ { \\epsilon } ( k ^ { - 2 / 3 } \\det ( T ) ^ { 1 + \\epsilon } ) . \\end{align*}"}
+{"id": "5617.png", "formula": "\\begin{align*} \\begin{cases} d _ n = \\log ( 1 - e ^ { a _ { n + 1 } } ) \\\\ b _ n = \\log ( 1 - e ^ { c _ { n + 1 } } ) \\end{cases} \\ \\forall \\ n \\in \\N \\ , \\end{align*}"}
+{"id": "2384.png", "formula": "\\begin{align*} [ \\mathbf { h } ' _ { { 3 i + 2 } } , \\mathbf { h } _ { { 3 i + 2 } } ] = 1 . \\end{align*}"}
+{"id": "5975.png", "formula": "\\begin{align*} \\int | f | ^ 4 \\lesssim \\sum _ { \\substack { R ^ { \\frac { 1 } { 2 } } \\le W \\le R \\\\ W \\ , } } \\sum _ { \\ell ( \\tau ) = \\frac { W } { R } } \\sum _ { U \\| U _ { \\tau , R } } | U | ^ { - 1 } \\Big ( \\int \\sum _ { \\theta \\subset \\tau } | f _ \\theta | ^ 2 W _ U \\Big ) ^ 2 . \\end{align*}"}
+{"id": "343.png", "formula": "\\begin{align*} \\vert S \\cap [ 1 , n ] \\vert \\leq \\sum _ { j = 1 } ^ k \\vert S _ j \\vert \\ll \\sum _ { j = 1 } ^ k \\frac { 2 ^ j \\log c _ j } { j } \\ll \\frac { 2 ^ k \\log c _ k } { k } = o ( \\alpha ( n ) \\cdot n / \\log n ) \\end{align*}"}
+{"id": "8883.png", "formula": "\\begin{align*} \\theta _ { S , \\mu } \\left ( \\gamma ( \\tau , z ) \\right ) ( c \\tau + d ) ^ { - \\frac { g } { 2 } } e \\left ( - c S [ z ] ( c \\tau + d ) ^ { - 1 } \\right ) = \\sum _ { \\eta \\in ( 2 S ) ^ { - 1 } \\mathbb Z ^ g / \\mathbb Z ^ g } \\varepsilon _ { S } ( \\eta , \\mu ; \\gamma ) \\theta _ { S , \\eta } ( \\tau , z ) , \\end{align*}"}
+{"id": "7274.png", "formula": "\\begin{align*} a _ { n - i } ( \\gamma ^ { ( q ^ { n - i } - q ^ n ) } - 1 ) = 0 \\end{align*}"}
+{"id": "5396.png", "formula": "\\begin{align*} F ( r ) = \\sigma _ k ^ { 1 / k } ( \\lambda ( r ) ) \\mbox { w i t h } \\lambda ( r ) \\in \\Gamma _ k \\end{align*}"}
+{"id": "2753.png", "formula": "\\begin{align*} | G ( 0 , 0 , R ) | = | Z ( 0 , 0 , R ) + ( 0 , 0 , R ) | = \\exp R + R \\geq \\exp R \\end{align*}"}
+{"id": "295.png", "formula": "\\begin{align*} u ( \\eta ) - \\frac { \\tilde { d } _ H ( \\xi , \\eta ) ^ \\beta } { \\delta ^ \\beta } \\leq u ^ \\delta _ { \\beta } ( \\xi ) < u ^ \\delta _ \\beta ( p _ 0 ) = u ( q _ { \\delta , \\beta } ) - \\frac { \\tilde { d } _ H ( p _ 0 , q _ { \\delta , \\beta } ) ^ \\beta } { \\delta ^ \\beta } \\end{align*}"}
+{"id": "206.png", "formula": "\\begin{align*} | B _ { G , S } ( n ) \\cap F | = | B _ { F , \\{ x , y \\} } ( n ) | | B _ { G , S } ( n ) | = | B _ { F , \\{ x , y \\} } ( n ) | + 2 \\sum _ { i = 1 } ^ { n } | B _ { F , \\{ x , y \\} } ( n - i ) | . \\end{align*}"}
+{"id": "6155.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 m } \\left ( \\sum _ i p _ i ^ 2 + \\lambda \\left ( \\sum _ i x _ i p _ i \\right ) ^ 2 \\right ) + { \\cal V } ( r ) = \\frac { 1 } { 2 m } \\left ( ( 1 + \\lambda r ^ 2 ) \\sum _ i p _ i ^ 2 - \\lambda \\sum _ { i < j } J _ { i j } ^ 2 \\right ) + { \\cal V } ( r ) , \\end{align*}"}
+{"id": "315.png", "formula": "\\begin{align*} f _ \\alpha = \\frac 1 2 | d ( ( h _ \\alpha ) _ i + ( h _ \\alpha ) _ j ) | , & \\ g _ \\alpha = \\frac 1 2 | d ( ( h _ \\alpha ) _ i - ( h _ \\alpha ) _ j ) | , \\\\ f = \\frac 1 2 | d ( h _ i + h _ j ) | , & g = \\frac 1 2 | d ( h _ i - h _ j ) | . \\end{align*}"}
+{"id": "1769.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\epsilon ^ { \\frac { 1 } { p } } \\| u _ { \\epsilon } \\| _ { L ^ p ( ( 0 , T ) \\times \\Gamma _ { \\epsilon } ) } = \\| u _ 0 \\| _ { L ^ p ( ( 0 , T ) \\times \\Omega \\times \\Gamma ) } . \\end{align*}"}
+{"id": "4676.png", "formula": "\\begin{align*} [ \\kappa ] _ n ^ 2 \\stackrel { d e f } { = } \\kappa _ 1 ^ 2 + \\sum _ { i = 2 } ^ n ( \\kappa _ i - \\kappa _ { i - 1 } ) ^ 2 , \\ n \\ge 2 . \\end{align*}"}
+{"id": "3803.png", "formula": "\\begin{align*} D _ * ^ q ( g ( z ) ) & = D _ * ^ q ( z ^ q f ) ( z ) \\\\ & = \\sum _ { n = 1 } ^ { \\infty } b _ n \\frac { \\Gamma ( q n + 1 ) } { \\Gamma ( q ( n - 1 ) + 1 } z ^ { q ( n - 1 ) } , \\\\ & = \\sum _ { n = 1 } ^ { \\infty } a _ { m - 1 } \\frac { \\Gamma ( q n + 1 ) } { \\Gamma ( q ( n - 1 ) + 1 ) } z ^ { q ( n - 1 ) } . \\end{align*}"}
+{"id": "3964.png", "formula": "\\begin{align*} ( \\mathbf { u } - \\mathbf { v } ) ( \\underbrace { G _ 2 , G _ 3 , \\cdots , G _ { 2 k } } _ { \\bar { G } } ) = ( 0 , 0 , \\cdots , 0 ) . \\end{align*}"}
+{"id": "4499.png", "formula": "\\begin{align*} [ \\omega ] = 2 [ H ] - [ E ] , [ \\alpha ] = 6 [ H ] + [ E ] . \\end{align*}"}
+{"id": "1737.png", "formula": "\\begin{align*} \\abs { \\omega _ n ( t _ 1 ) - \\omega _ n ( t _ 2 ) } = \\omega _ n ( t _ 2 ) - \\omega _ n ( t _ 1 ) \\leq C _ 1 \\omega _ n ( t _ 2 - t _ 1 ) \\leq C _ 1 \\omega ( t _ 2 - t _ 1 ) . \\end{align*}"}
+{"id": "1670.png", "formula": "\\begin{align*} X = \\begin{pmatrix} - 1 & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "4929.png", "formula": "\\begin{align*} T _ \\sigma ( f , g ) ( x ) = \\langle K _ \\sigma ( x , y , z ) , ( f \\otimes g ) ( y , z ) \\rangle , \\end{align*}"}
+{"id": "7088.png", "formula": "\\begin{gather*} \\mathcal { A } _ { j m } ( x , \\xi ) = \\int _ { B _ 1 ( 0 ) } \\phi ( y ) \\mathcal { A } _ j ( x + m y , \\xi ) d y , \\end{gather*}"}
+{"id": "8324.png", "formula": "\\begin{align*} & \\sigma _ 3 = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} , \\ \\ \\ \\ P = \\begin{pmatrix} 0 & u \\\\ - \\bar { u } & 0 \\end{pmatrix} , \\\\ & \\eta = \\sqrt { \\alpha } \\left ( k - \\frac { \\beta } { 2 k } \\right ) , H = \\alpha k U _ x + \\frac { i \\alpha \\beta ^ 2 } { 2 } \\sigma _ 3 \\left ( \\frac { 1 } { k } P - P ^ 2 \\right ) . \\end{align*}"}
+{"id": "5333.png", "formula": "\\begin{align*} u _ n = P _ 1 ( n ) \\alpha _ 1 ^ n + \\cdots + P _ s ( n ) \\alpha _ s ^ n \\end{align*}"}
+{"id": "3287.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\left | \\Delta _ n ^ { - \\frac 1 2 } \\langle f _ 1 , f _ 2 \\rangle \\sum _ { i = 1 } ^ n \\int _ { ( i - 1 ) \\Delta _ n } ^ { i \\Delta _ n } \\langle ( \\mathcal S ( i \\Delta _ n - s ) - I ) f _ 1 , f _ 2 \\rangle d s \\right | = \\infty . \\end{align*}"}
+{"id": "7081.png", "formula": "\\begin{align*} \\int _ { B _ R } \\dfrac { | \\tau _ h D v | ^ p } { | h | ^ { p \\frac { 2 \\gamma } { p } } } = \\int _ { B _ R } \\dfrac { | \\tau _ h D v | ^ p } { | h | ^ { 2 \\gamma } } d x \\le c \\int _ { B _ R } \\dfrac { | \\tau _ h V _ p ( D v ) | ^ 2 } { | h | ^ { 2 \\gamma } } d x h . \\end{align*}"}
+{"id": "8601.png", "formula": "\\begin{align*} P _ Z ( t ) = | Z + t B _ 2 ^ n | = | B _ 2 ^ 2 | \\int _ { t B _ 2 ^ { n - 2 } } \\left ( 1 + \\sqrt { t ^ 2 - | x | ^ 2 } \\right ) ^ { 2 } d x = t ^ { n - 2 } | B _ 2 ^ { 2 } | \\int _ { B _ 2 ^ { n - 2 } } \\left ( 1 + t \\sqrt { 1 - | x | ^ 2 } \\right ) ^ { 2 } d x . \\end{align*}"}
+{"id": "4855.png", "formula": "\\begin{align*} \\Phi : = { \\sup } ^ * \\left \\{ \\zeta : \\begin{array} { l } \\zeta \\overline { \\Omega } \\times \\mathbb R ^ m , \\\\ F \\star \\mathcal P \\Omega \\times \\mathbb R ^ m \\Phi | _ { \\partial \\Omega \\times \\mathbb R ^ m } \\le \\phi \\end{array} \\right \\} . \\end{align*}"}
+{"id": "4730.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { m } \\gamma _ { k } ( f _ { k } ( x ) + C _ { k } ) \\ge 0 \\ \\ \\forall \\ x \\in \\mathbb { R } ^ { n } , \\ \\ C _ k \\in \\mathbb { R } _ + . \\end{align*}"}
+{"id": "8520.png", "formula": "\\begin{align*} \\mathcal { P } ^ { \\pm } ( h ) ( z ) = \\pm \\frac { 1 } { 2 } h ( z ) - \\frac { i } { 2 } \\mathcal { H } ( h ) ( z ) z \\in \\mathbb { R } . \\end{align*}"}
+{"id": "8333.png", "formula": "\\begin{align*} \\mu ^ { \\pm } ( x , t ; k ) = I + \\int _ { \\pm \\infty } ^ { x } \\mathrm { e } ^ { - \\mathrm { i } k ^ 2 \\left ( x - y \\right ) \\widehat { \\sigma } _ 3 } \\left ( P _ 1 \\mu ^ { \\pm } \\right ) \\left ( y , t ; k \\right ) d y , \\end{align*}"}
+{"id": "893.png", "formula": "\\begin{align*} 2 ( r _ 0 ^ 2 + r _ 0 q _ 0 + q _ 0 ^ 2 ) = n \\ , , \\end{align*}"}
+{"id": "9261.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ k \\wedge \\beta _ n ^ { n - m } \\wedge \\omega \\geq 0 . \\end{aligned} \\end{align*}"}
+{"id": "1302.png", "formula": "\\begin{align*} S _ { i } = \\{ X _ { i } , X _ { i + 1 } , X _ { i + 2 } , X _ { i + 3 } \\} , \\overline { S } _ i = \\{ X _ { i - 2 } , X _ { i - 1 } \\} , i \\in \\mathbb { Z } / 6 \\mathbb { Z } . \\end{align*}"}
+{"id": "5804.png", "formula": "\\begin{align*} d g _ 1 '' = ( \\log \\sqrt { \\tau _ 0 } ) _ z d z \\ g '' _ 1 + e ^ { 2 i \\theta } \\frac { 1 } { \\sqrt { \\tau _ 0 } } \\left ( Q _ 0 d z + \\frac { \\tau _ 0 } { 4 } d \\overline { z } \\right ) \\ J \\widehat { g _ 1 '' } I ; \\end{align*}"}
+{"id": "1900.png", "formula": "\\begin{align*} \\sin S _ c ( x , y ) \\partial _ x S _ c ( x , y ) = \\mathrm F ( x , y ) x + \\mathrm G ( x , y ) y , \\end{align*}"}
+{"id": "1028.png", "formula": "\\begin{align*} & \\langle \\mu , \\alpha \\rangle + \\chi _ R ( v ' \\alpha ) + \\chi _ L ( - w ' v ' \\alpha ) \\\\ \\leq & \\langle \\mu , \\alpha \\rangle + 1 - \\chi _ R ( - v ' \\alpha ) + \\chi _ L ( - w ' v ' \\alpha ) \\\\ = & 1 - \\prescript L { } { } \\ell { } ^ R ( x ' , - \\alpha ) \\leq 0 . \\end{align*}"}
+{"id": "764.png", "formula": "\\begin{align*} B _ 2 & = \\int _ { | u - t | > 2 ^ { 2 s } \\ell ( Q ) ^ { 2 s } } \\frac { | P _ s * ( \\phi \\mu ) ( x , u ) - P _ s * ( \\phi \\mu ) ( y , u ) | } { | u - t | ^ { 2 - \\frac 1 { 2 s } } } \\ , d u \\\\ & = \\sum _ k \\int _ { \\mathcal { A } _ k } \\frac { | P _ s * ( \\phi \\mu ) ( x , u ) - P _ s * ( \\phi \\mu ) ( y , u ) | } { | u - t | ^ { 2 - \\frac 1 { 2 s } } } \\ , d u . \\end{align*}"}
+{"id": "3739.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { | I | = 0 , 1 , \\dots , } \\frac { 1 } { | I | ! } \\partial ^ I f ( p ) ( x - p ) ^ I \\mbox { f o r a l l } x \\in U \\end{align*}"}
+{"id": "2916.png", "formula": "\\begin{align*} { \\cal I } : = \\frac { 1 } { 2 \\pi i } \\int _ { C } f ( \\zeta ) \\dd \\zeta \\end{align*}"}
+{"id": "5463.png", "formula": "\\begin{align*} \\beta _ n : = ( n + 1 ) h _ A + n ( \\widetilde b + a ) + a \\end{align*}"}
+{"id": "7266.png", "formula": "\\begin{align*} \\epsilon _ s P ( x _ 1 , \\ldots , x _ n ) = P ( v _ 1 , \\ldots , v _ n ) = 0 . \\end{align*}"}
+{"id": "5488.png", "formula": "\\begin{align*} f '' ( q ) = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { ( 2 - q + k ) ^ 2 } - q ^ 2 - \\frac { 5 } { 6 } . \\end{align*}"}
+{"id": "1648.png", "formula": "\\begin{align*} \\theta _ { r + 1 } ( \\hat { f } ) ( \\gamma _ 1 , \\dots , \\gamma _ r , \\gamma _ { r + 1 } ) = \\theta _ r ( f ) ( \\gamma _ 1 , \\dots , \\gamma _ r \\gamma _ { r + 1 } ) , \\end{align*}"}
+{"id": "7684.png", "formula": "\\begin{align*} \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\Psi } ( t , \\boldsymbol { x } ) ) } ^ { N , i } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } h \\Big ( \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\Psi } ( t , \\boldsymbol { x } ) ) } ^ { N , j } \\Big ) = - R ^ { - 1 } B \\Big [ P _ t \\nu ^ { N , i } _ { \\boldsymbol { x } } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } \\Psi ^ { N , j } ( t , \\boldsymbol { x } ) \\Big ] . \\end{align*}"}
+{"id": "7796.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Theta ( t , x ) = \\int _ \\S \\int _ \\S W ( z , y , x ) \\sin ( \\Theta ( t , z ) + \\Theta ( t , y ) - 2 \\Theta ( t , x ) ) \\ \\d y \\d z , \\end{align*}"}
+{"id": "5928.png", "formula": "\\begin{align*} & \\mathbb { E } \\int _ 0 ^ T \\psi ( t ) \\Big [ \\Vert X ( t \\wedge \\tau _ u ^ M ) \\Vert _ H ^ 2 - \\Vert x \\Vert _ H ^ 2 \\Big ] d t \\\\ = & \\mathbb { E } \\int _ 0 ^ T \\psi ( t ) \\int _ 0 ^ { t \\wedge \\tau _ { u } ^ M } \\Big [ 2 \\langle \\mathcal { A } ( s ) , X ( s ) \\rangle + \\Vert \\mathcal { B } ( s ) \\Vert _ { L _ 2 } ^ 2 \\Big ] d s d t . \\end{align*}"}
+{"id": "1155.png", "formula": "\\begin{align*} d X _ { t } = \\left ( \\langle \\mu _ t , K ( X _ t - \\cdot ) \\rangle \\right ) d t + d W _ t , \\mu _ t = \\operatorname { L a w } ( X _ t ) . \\end{align*}"}
+{"id": "2600.png", "formula": "\\begin{align*} R \\cdot R = f \\ , \\ , Q ^ c ( g , R ) , \\end{align*}"}
+{"id": "1196.png", "formula": "\\begin{align*} M _ { N C } ( n , d ) \\le H _ t ( n , d ) 2 ^ d + \\left ( \\binom { n } { d } - H _ t ( n , d ) \\right ) \\frac { 2 ^ { d + 1 } } { t + 1 } = \\left ( h _ t ( n , d ) + ( 1 - h _ t ( n , d ) ) \\frac { 2 } { t + 1 } \\right ) \\cdot 2 ^ d \\binom { n } { d } \\enspace . \\end{align*}"}
+{"id": "6023.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\R ^ { 2 } } ( | \\nabla u _ { n } | ^ { 2 } + u _ { n } ^ { 2 } ) d x = 0 , \\end{align*}"}
+{"id": "5385.png", "formula": "\\begin{align*} \\nabla _ i u : = \\nabla _ { e _ i } u = e _ i ^ j \\partial _ j u = e _ i ^ j u _ j \\end{align*}"}
+{"id": "2941.png", "formula": "\\begin{align*} c ^ { - 1 } \\left ( \\bar A \\right ) \\cap c ^ { - 1 } \\left ( X \\backslash A \\right ) = c ^ { - 1 } \\left ( \\bar A \\cap ( X \\backslash A ) \\right ) \\neq \\emptyset . \\end{align*}"}
+{"id": "4303.png", "formula": "\\begin{align*} & \\int _ { M } | \\tilde { F } - ( 1 - b _ { t _ 0 , B } ( \\Psi _ 1 ) ) f F ^ { 1 + \\delta } | ^ 2 _ { \\tilde { h } } e ^ { v _ { t _ 0 , B } ( \\Psi _ 1 ) - \\delta \\tilde { M } } c ( - v _ { t _ 0 , B } ( \\Psi _ 1 ) ) \\\\ \\le & \\left ( \\frac { 1 } { \\delta } c ( T _ 1 ) e ^ { - T _ 1 } + \\int _ { T _ 1 } ^ { t _ 0 + B } c ( t ) e ^ { - t } d t \\right ) \\int _ M \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi _ 1 < - t _ 0 \\} } { | f F | } ^ 2 _ { \\tilde { h } } , \\end{align*}"}
+{"id": "8627.png", "formula": "\\begin{align*} \\mathbb { E } _ { n , m } [ \\chi _ { n , m } ( w ' , \\cdot ) | O ( \\chi _ { n , m } ' ( w ' , M , \\cdot ) ) = O ( \\chi _ { n , m } ' ( w , M , \\cdot ) ) ] \\\\ = \\mathbb { E } _ { n , m } [ \\chi _ { n , m } ( w ' , \\cdot ) | w , M ] \\end{align*}"}
+{"id": "5931.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\lim _ { n \\rightarrow \\infty } I \\leq \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ { T } f ( s ) \\Vert X _ n ( s ) - X ( s ) \\Vert _ H ^ 2 d s = 0 . \\end{align*}"}
+{"id": "2295.png", "formula": "\\begin{align*} \\omega _ j = ( e _ 1 + \\dots + e _ j ) | _ { \\C ^ n _ 0 } \\end{align*}"}
+{"id": "4188.png", "formula": "\\begin{align*} - \\varepsilon ^ 2 ( K ( x ) \\nabla u ) + V ( x ) u = u ^ { p } , \\ \\ \\ \\ x \\in \\mathbb { R } ^ n , \\end{align*}"}
+{"id": "6804.png", "formula": "\\begin{align*} & a _ { m } ( x ) + \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ { n + m + 1 } } { 2 \\pi } a _ { n } ( x ) \\int _ { - \\infty } ^ { \\infty } \\frac { \\varphi \\left ( \\tau \\right ) z ^ { n + m + 1 } ( \\tau ) d \\tau } { \\left ( \\frac { 1 } { 2 } - i \\tau \\right ) \\left ( \\frac { 1 } { 2 } + i \\tau \\right ) } \\\\ & = \\frac { ( - 1 ) ^ { m } } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\frac { \\varphi \\left ( \\tau \\right ) z ^ { m + 1 } ( \\tau ) d \\tau } { \\frac { 1 } { 2 } + i \\tau } , m = 0 , 1 , \\ldots . \\end{align*}"}
+{"id": "3777.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\geq p } t e l ( \\mathcal { C } ) : = \\lbrace x \\in t e l ( \\mathcal { C } ) \\mid \\mathcal { A } ( x ) \\geq p \\rbrace . \\end{align*}"}
+{"id": "3958.png", "formula": "\\begin{align*} ( f _ 1 ( \\mathbf { x } ) , f _ 2 ( \\mathbf { x } ) , \\cdots , f _ { n - 1 } ( \\mathbf { x } ) ) = ( f _ 2 ( \\mathbf { y } ) , f _ 3 ( \\mathbf { y } ) , \\cdots , f _ n ( \\mathbf { y } ) ) = ( f _ 2 ( \\mathbf { x } ) , f _ 3 ( \\mathbf { x } ) , \\cdots , f _ n ( \\mathbf { x } ) ) . \\end{align*}"}
+{"id": "2093.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ S g ( x ) p _ { S } ( x ) \\ , d U ( x ) & = \\int _ S \\int _ S g ( x ) 1 _ { B _ y } ( x ) \\ , d U ( y ) \\ , d U ( x ) \\\\ & = \\int _ S \\left ( \\frac { 1 } { U ( B _ y ) } \\int _ { B _ y } g ( x ) \\ , d x \\right ) \\ , d U ( y ) > \\int _ S g ( y ) d U ( y ) = 0 , \\end{aligned} \\end{align*}"}
+{"id": "2957.png", "formula": "\\begin{align*} \\Phi ( t , c ( q ) ) = \\lim _ { n \\to \\infty } \\Phi \\left ( t , c \\left ( q - \\frac { 1 } { n } \\right ) \\right ) \\in \\overline { X \\backslash \\bar A } \\subset X \\backslash A , \\end{align*}"}
+{"id": "7694.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\underset { i = 1 } { \\overset { N } { \\sum } } \\mathbb { E } \\Big [ \\sup _ { 0 \\leq s \\leq t } | x _ s ^ i - x _ s ^ { * , i } | ^ 2 \\Big ] \\leq \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N ^ 2 } \\sum _ { i , j = 1 } ^ N \\mathbb E \\big [ | \\xi ^ i - \\xi ^ j | ^ 2 \\big ] \\Big ) . \\end{align*}"}
+{"id": "6366.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\frac { \\exp [ - ( a + n ) y ] } { y } \\mathrm { d } y = \\operatorname { E } _ 1 ( 0 ) , \\end{align*}"}
+{"id": "8250.png", "formula": "\\begin{align*} \\mu ^ n _ g ( t ) : = \\sum _ { i = 1 } ^ n g _ i \\psi _ i ( t ) , \\end{align*}"}
+{"id": "9312.png", "formula": "\\begin{align*} & b _ 1 z _ 1 ^ 2 - b _ 2 z _ 2 ^ 2 = 4 ^ { m } p , \\\\ & b _ 1 z _ 1 ^ 2 - b _ 1 b _ 2 z _ 3 ^ 2 = - p , \\end{align*}"}
+{"id": "8158.png", "formula": "\\begin{align*} f ( g ( x ) ) ^ { ( n ) } = \\sum _ { k _ 1 + 2 k _ 2 + \\cdots + n k _ n = n } \\frac { n ! } { k _ 1 ! 1 ! ^ { k _ 1 } \\cdots k _ n ! n ! ^ { k _ n } } f ^ { ( k _ 1 + \\cdots + k _ n ) } ( g ( x ) ) \\prod _ { i = 1 } ^ n g ^ { ( i ) } ( x ) ^ { k _ i } , \\end{align*}"}
+{"id": "6597.png", "formula": "\\begin{align*} 0 \\le c _ { r } = r ^ { 1 - d } \\int _ { 0 } ^ { r } \\rho ^ { d - 1 } n c \\ , d \\rho \\le \\frac { \\gamma M _ { 0 } } { \\sigma _ { d } } r \\quad \\mbox { f o r } r \\in ( 0 , R ] . \\end{align*}"}
+{"id": "4066.png", "formula": "\\begin{align*} r _ n ^ { - 1 } n ^ { 2 H - 1 / 2 } \\sum _ { j \\in [ n ] } T ^ { k _ 1 } _ { n , j } T ^ { k _ 2 } _ { n , j } = n \\sum _ { j \\in [ n ] } T ^ { k _ 1 } _ { n , j } T ^ { k _ 2 } _ { n , j } \\end{align*}"}
+{"id": "9231.png", "formula": "\\begin{align*} x _ n ( t ) : = x _ a ( t - n { \\bar { t } } / { \\gamma } , x ( n { \\bar { t } } / { \\gamma } ) ) \\forall \\ , t \\in I _ n . \\end{align*}"}
+{"id": "8508.png", "formula": "\\begin{align*} \\| \\langle x \\rangle I _ 1 ^ { \\prime } ( x ) \\| _ { L ^ 2 ( \\mathbb { R } ^ + ) } = 2 \\pi ^ { - 1 } \\| \\langle x \\rangle \\widehat { \\bar { r } _ 1 ( z ) } ( 2 x ) \\| _ { L ^ 2 ( \\mathbb { R } ) } \\leq c \\| r _ 1 ( z ) \\| _ { H ^ 1 ( \\mathbb { R } ) } , \\end{align*}"}
+{"id": "6130.png", "formula": "\\begin{align*} \\widetilde { \\mu } \\ , ' ( n ) \\ r _ n ( m ) = H ' _ { m } \\ \\widetilde { r } \\ , ' _ { n } ( m + 1 ) + I ' _ m \\ \\widetilde { r } \\ , ' _ { n } ( m ) + J ' _ m \\ \\widetilde { r } \\ , ' _ { n } ( m - 1 ) \\ , , \\end{align*}"}
+{"id": "4953.png", "formula": "\\begin{align*} h ( x , u ) = E _ { \\xi _ 1 ^ \\theta ( u ) } \\left [ \\varphi \\Bigl ( x + u + \\sum _ { i = 2 } ^ { n } Z _ i ^ { \\theta [ u ] } \\Bigr ) \\right ] = E _ { \\xi _ 1 ^ \\theta ( u ) } \\left [ \\varphi \\Bigl ( x + u + \\sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \\rho } \\Bigr ) \\right ] . \\end{align*}"}
+{"id": "419.png", "formula": "\\begin{align*} \\phi ( g _ 2 H ) H = \\pi _ N ( g _ 2 ) \\psi ( \\pi _ N ( g _ 2 ) ) H = g _ 2 H . \\end{align*}"}
+{"id": "1620.png", "formula": "\\begin{align*} \\d : C ^ 0 _ c ( G , V ) ^ G \\to C ^ 1 _ c ( G , V ) ^ G , ( \\d v ) ( x ) = x \\cdot v - v . \\end{align*}"}
+{"id": "521.png", "formula": "\\begin{align*} G ( R ) : = \\sum _ { k = - \\infty } ^ { \\infty } R ^ { d / 4 } 2 ^ { k d / 2 } ( 1 + R ^ { 1 / 2 } 2 ^ { k } ) ^ { - \\alpha - 1 / 2 } \\lesssim 1 . \\end{align*}"}
+{"id": "7565.png", "formula": "\\begin{align*} ( M - X ^ m ( X ^ k - 1 ) I _ p ) \\begin{pmatrix} X _ 1 X _ 2 \\cdots X _ { p - 1 } \\\\ X _ 2 \\cdots X _ { p - 1 } \\\\ \\vdots \\\\ X _ { p - 1 } \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\end{pmatrix} \\end{align*}"}
+{"id": "3364.png", "formula": "\\begin{align*} t \\alpha = \\Lambda ^ 0 < \\dots < \\Lambda ^ L = t \\beta . \\end{align*}"}
+{"id": "57.png", "formula": "\\begin{align*} ( w \\varepsilon ^ \\mu ) ( \\alpha , k ) = ( w \\alpha , k - \\langle \\mu , \\alpha \\rangle ) . \\end{align*}"}
+{"id": "8812.png", "formula": "\\begin{align*} \\begin{aligned} ( \\Pi _ V \\tilde { Y } _ { \\tau / 2 } ) & = \\int _ 0 ^ { \\tau / 2 } \\left \\| \\Pi _ V e ^ { | z | J ^ \\perp ( t - s ) } P ^ { - 1 } \\Pi _ { \\mathrm { k e r } A ^ \\perp } \\sigma \\right \\| _ F ^ 2 d s \\ge c _ 1 \\tau \\end{aligned} \\end{align*}"}
+{"id": "1720.png", "formula": "\\begin{align*} A ( t ) U ( t , 0 ) x = & - i ( \\Delta + b ( t , \\xi ) \\cdot \\nabla + c ( t , \\xi ) ) U ( t , 0 ) x \\\\ = & - i \\Delta U ( t , 0 ) x - i b ( t , \\xi ) \\cdot \\nabla U ( t , 0 ) x - i c ( t , \\xi ) U ( t , 0 ) x \\\\ = & - i \\Delta U ( t , 0 ) x - 2 i ( \\nabla W ) ( t , \\xi ) \\cdot \\nabla U ( t , 0 ) x - i \\sum _ { j = 1 } ^ d ( \\partial _ j W ( t , \\xi ) ) ^ 2 U ( t , 0 ) x \\\\ & - i ( \\Delta W ) ( t , \\xi ) U ( t , 0 ) x - \\mu ( \\xi ) U ( t , 0 ) x - \\widetilde { \\mu } ( \\xi ) U ( t , 0 ) x . \\end{align*}"}
+{"id": "5956.png", "formula": "\\begin{align*} d u ( t ) = \\big [ - \\Delta ^ 2 u + \\Delta \\varphi ( u ) \\big ] d t + B ( t , u ) d W ( t ) , \\end{align*}"}
+{"id": "5587.png", "formula": "\\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) = A ( 1 ^ \\infty ) - A ( 1 ^ n 0 ^ \\infty ) = c - c _ n \\ . \\end{align*}"}
+{"id": "2382.png", "formula": "\\begin{align*} [ \\mathbf { h } ' _ { { 3 i + 1 } } , \\mathbf { h } _ { { 3 i + 1 } } ] = 1 . \\end{align*}"}
+{"id": "3152.png", "formula": "\\begin{align*} & \\sum _ { y \\in \\mathbb { F } _ q } \\chi _ 4 ( ( y - 1 ) ( y - a ) ) = \\sum _ { y ' \\in \\mathbb { F } _ q } \\chi _ 4 ( y ' ( y ' + 1 - a ) ) \\\\ & = \\sum _ { y '' \\in \\mathbb { F } _ q } \\chi _ 4 ( ( 1 - a ) y '' ) \\chi _ 4 ( ( 1 - a ) ( y '' + 1 ) ) = \\varphi ( 1 - a ) \\sum _ { y '' \\in \\mathbb { F } _ q } \\chi _ 4 ( y '' ( y '' + 1 ) ) \\\\ & = \\varphi ( 1 - a ) \\sum _ { y '' \\in \\mathbb { F } _ q } \\chi _ 4 ( - y '' ( - y '' + 1 ) ) \\\\ & = \\varphi ( 1 - a ) J ( \\chi _ 4 , \\chi _ 4 ) , \\end{align*}"}
+{"id": "9202.png", "formula": "\\begin{align*} \\epsilon _ 2 ( \\eta _ a , t ) = \\int _ N ^ t h ( x _ a + \\delta u ( \\tau ) ) - \\dfrac { a _ { 0 , \\delta } ( x _ a ) } { 2 } \\ , d \\tau \\end{align*}"}
+{"id": "140.png", "formula": "\\begin{align*} \\| S _ T ^ { \\infty } ( \\phi _ n ) - S _ T ^ { \\infty } ( \\phi _ m ) \\| _ { C ( [ - 1 , 1 ] ; H ^ { s , 0 } ) } & \\leqslant \\| S _ T ^ { \\infty } ( \\phi _ n ) - S _ T ^ { \\infty } ( \\phi _ n ^ K ) \\| _ { C ( [ - 1 , 1 ] ; H ^ { s , 0 } ) } \\\\ & + \\| S _ T ^ { \\infty } ( \\phi _ m ) - S _ T ^ { \\infty } ( \\phi _ m ^ K ) \\| _ { C ( [ - 1 , 1 ] ; H ^ { s , 0 } ) } \\\\ & + \\| S _ T ^ { \\infty } ( \\phi _ n ^ K ) - S _ T ^ { \\infty } ( \\phi _ m ^ K ) \\| _ { C ( [ - 1 , 1 ] ; H ^ { s , 0 } ) } . \\end{align*}"}
+{"id": "5968.png", "formula": "\\begin{align*} \\Gamma _ k : = \\bigg \\{ f \\in L ^ p ( [ 0 , T ] , H ) : \\int _ 0 ^ { T - \\delta _ k } \\Vert f ( t + \\delta _ k ) - f ( t ) \\Vert _ Y ^ p d t \\leq \\frac { 1 } { k } \\bigg \\} . \\end{align*}"}
+{"id": "3855.png", "formula": "\\begin{align*} \\gamma ( s , \\tau ) = \\left ( r ^ * \\cos \\left ( \\frac { s } { r ^ * } \\right ) , r ^ * \\sin \\left ( \\frac { s } { r ^ * } \\right ) , \\frac { c } { 4 \\pi r ^ * } \\tau \\right ) ^ t , \\end{align*}"}
+{"id": "8080.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ M \\mathbb { E } \\left [ y _ i ^ * y _ i | \\right . \\left . \\hat { \\mathbf { H } } \\right ] = & \\sum _ { j = 1 } ^ { M } a _ j ^ 2 \\left ( \\sum _ { l = 1 } ^ M \\lvert \\hat { \\phi } ^ { \\left ( l , j \\right ) } \\rvert ^ 2 + M \\sigma _ { e _ i } ^ 2 \\lVert \\mathbf { p } _ j \\rVert ^ 2 \\right ) + a _ c ^ 2 \\left ( \\sum _ { i = 1 } ^ M \\lvert \\hat { \\phi } ^ { \\left ( i , c \\right ) } \\rvert ^ 2 + M \\sigma _ { e , i } ^ 2 \\lVert \\mathbf { p } _ c \\rVert ^ 2 \\right ) + M \\sigma _ n ^ 2 . \\end{align*}"}
+{"id": "7283.png", "formula": "\\begin{align*} [ E : F ] = [ F ( \\alpha , \\gamma ) : F ] = [ F ( \\alpha , \\gamma ) : F ( \\gamma ) ] [ F ( \\gamma ) : F ] \\leq m ( q - 1 ) \\end{align*}"}
+{"id": "8700.png", "formula": "\\begin{align*} | { \\bf r ' } - { \\bf q } | & = \\frac { 1 } { | { \\bf r } | } | { \\bf r } - | { \\bf r } | { \\bf q } | \\leq 2 | { \\bf r } - | \\bf { r } | { \\bf q } | \\\\ & \\leq 2 \\left ( | { \\bf r } - { \\bf q } | + | 1 - | \\bf { r } | | \\cdot | { \\bf q } | \\right ) < 2 ( \\epsilon + \\epsilon \\cdot 1 ) = 4 \\epsilon , \\end{align*}"}
+{"id": "4570.png", "formula": "\\begin{align*} & \\frac { \\lambda _ 3 } { q ^ 2 } a _ { 1 , 0 , 0 } = a _ { 0 , 0 , 0 } + ( q ^ 3 + q ^ 2 + q ) a _ { 2 , 1 , 1 } \\\\ & \\frac { \\lambda _ 1 } { q } a _ { 2 , 1 , 0 } = ( q ^ 3 + q ^ 2 ) a _ { 2 , 1 , 1 } + q a _ { 2 , 2 , 0 } + a _ { 3 , 1 , 0 } . \\end{align*}"}
+{"id": "547.png", "formula": "\\begin{align*} w _ 1 v _ 1 & = ( a _ { - 1 } - a _ 1 ) ( 2 a _ 0 - ( a _ { - 1 } + a _ 1 ) - 4 s _ { 1 } ) \\\\ & = ( a _ { - 1 } - a _ 1 ) - ( a _ { - 1 } - a _ 1 ) - 4 \\left ( - \\tfrac { 3 } { 4 } ( a _ { - 1 } - a _ 1 ) + \\tfrac { 3 } { 8 } ( a _ { - 2 } + a _ 0 - a _ 0 - a _ 2 ) \\right ) \\\\ & = - \\tfrac { 3 } { 2 } \\left ( a _ { - 2 } - 2 a _ { - 1 } + 2 a _ 1 - a _ 2 \\right ) , \\end{align*}"}
+{"id": "6274.png", "formula": "\\begin{align*} d \\phi _ { i j } ( \\partial _ j , \\partial _ r ) = \\partial _ j ( \\phi _ { i j } ( \\partial _ r ) ) - \\partial _ r ( \\phi _ { i j } ( \\partial _ j ) ) = - \\partial _ r \\Big ( \\frac { \\Gamma _ { j i } ^ i } { \\varphi _ j } \\Big ) = \\frac { - \\partial _ r \\Gamma _ { j i } ^ i + 2 \\Gamma _ { r j } ^ j \\Gamma _ { j i } ^ i } { \\varphi _ j } , \\end{align*}"}
+{"id": "2729.png", "formula": "\\begin{align*} M ( r , f ) = \\max _ { | x | = r } | f ( x ) | . \\end{align*}"}
+{"id": "5198.png", "formula": "\\begin{align*} A _ t \\cdot r \\nabla \\theta & = \\frac { 1 } { r } \\Phi _ t , \\\\ ( B \\times u ) \\cdot r \\nabla \\theta & = \\frac { 1 } { r } u \\cdot \\nabla \\Phi , \\\\ \\nabla \\times B \\cdot r \\nabla \\theta & = - \\frac { 1 } { r } \\left ( \\Delta - \\frac { 2 } { r } \\partial _ r \\right ) \\Phi . \\end{align*}"}
+{"id": "642.png", "formula": "\\begin{align*} & \\Pr ( \\phi _ { 2 , j } = 1 ) \\\\ & = \\sum _ { j ' = 1 } ^ n \\Pr ( \\{ \\phi _ { 2 , j } = 1 \\} \\cap \\{ \\phi _ { 1 , j ' } = 1 \\} ) \\\\ & = \\sum _ { j ' \\neq j } \\Pr ( \\{ \\phi _ { 2 , j } = 1 \\} | \\{ \\phi _ { 1 , j ' } = 1 \\} ) \\Pr ( \\phi _ { 1 , j ' } = 1 ) \\\\ & = \\sum _ { j ' \\neq j } \\frac { 1 } { n - 1 } \\cdot \\frac { 1 } { n } = \\frac { 1 } { n } \\end{align*}"}
+{"id": "8002.png", "formula": "\\begin{align*} & \\frac { 1 } { | \\mathcal F _ { N , k } | } \\sum _ { f \\in \\mathcal F _ { N , k } } R _ 2 ( g , \\rho ) ( f ) = \\frac { T ( g , \\rho ) } { 4 L } \\\\ & + \\O \\left ( \\frac { 1 } { L } \\right ) + \\O \\left ( \\frac { L ( \\log \\log x ) ^ 2 } { \\pi _ N ( x ) } \\right ) + \\O \\left ( \\frac { x ^ { \\pi _ N ( x ) c } 8 ^ { \\nu ( N ) } } { k N } \\right ) , \\end{align*}"}
+{"id": "4546.png", "formula": "\\begin{align*} & | \\Gamma _ { \\ell , m , 0 } | = ( q - 1 ) ^ 3 ( q + 1 ) q ^ { 3 \\ell + m + 6 } \\\\ & | \\Gamma _ { \\ell , m , m } | = ( q - 1 ) ^ 3 ( q + 1 ) q ^ { 3 \\ell + 6 } \\\\ & | \\Gamma _ { \\ell , \\ell , m } | = ( q - 1 ) ^ 3 ( q + 1 ) q ^ { 4 \\ell - m + 6 } . \\end{align*}"}
+{"id": "4788.png", "formula": "\\begin{align*} a ( u _ n , v _ n ) = ( f _ n , v _ n ) v _ n \\in X _ n , \\end{align*}"}
+{"id": "3135.png", "formula": "\\begin{align*} \\overbrace { [ x + a , y + b ] } = & ( x + a ) \\star ( y + b ) - ( y + b ) \\star ( x + a ) \\\\ = & x \\ast y + \\mathfrak { l } ( x ) b + \\mathfrak { r } ( y ) a - y \\ast x - \\mathfrak { l } ( y ) a - \\mathfrak { r } ( x ) b \\\\ = & [ x , y ] + ( \\mathfrak { l } - \\mathfrak { r } ) ( x ) b - ( \\mathfrak { l } - \\mathfrak { r } ) ( y ) a . \\end{align*}"}
+{"id": "5657.png", "formula": "\\begin{align*} \\begin{array} { l l } s _ { m } ^ { - 1 } E ^ { 2 } _ { p q } & = s _ { m } ^ { - 1 } H _ { p } ( O _ { n - k , n - k } ; H _ { q } ( L _ { k } ) ) \\\\ & \\\\ & \\cong H _ { p } ( O _ { n - k , n - k } ; s _ { m } ^ { - 1 } H _ { q } ( L _ { k } ) ) \\end{array} \\Rightarrow s _ { m } ^ { - 1 } H _ { p + q } ( T _ { k } ) , \\end{align*}"}
+{"id": "7226.png", "formula": "\\begin{align*} \\dim ( K _ i \\cap K _ j ) = 1 \\quad \\quad K _ 1 \\cap K _ 2 \\cap K _ 3 = 0 \\quad . \\end{align*}"}
+{"id": "8043.png", "formula": "\\begin{align*} \\mathbf { x } ^ { \\left ( \\right ) } = & \\mathbf { P } ^ { \\left ( \\right ) } \\mathbf { A } ^ { \\left ( \\right ) } \\mathbf { s } ^ { \\left ( \\right ) } = a _ c s _ c \\mathbf { p } _ c + \\sum _ { m = 1 } ^ { M } a _ m s _ m \\mathbf { p } _ m , \\end{align*}"}
+{"id": "5631.png", "formula": "\\begin{align*} g _ \\lambda ( z ) = \\begin{cases} 0 & z = \\lambda , \\\\ ( z - \\lambda ) ^ { - 1 } & z \\neq \\lambda , \\end{cases} \\end{align*}"}
+{"id": "3665.png", "formula": "\\begin{align*} Z ^ { ( i ) } = \\partial _ i \\mbox { a n d } Z ^ { ( i , j ) } = x _ i \\partial _ j - x _ j \\partial _ i \\mbox { f o r } i , j = 1 , \\dots , n . \\end{align*}"}
+{"id": "6956.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } T _ { W , \\psi } ( \\mu _ n ( Q ^ n ) ) = T _ { W , \\psi } ( \\mu ) , I ( \\mu _ n ( Q ^ n ) ) \\le I ( \\mu ) , \\ \\forall n . \\end{align*}"}
+{"id": "3405.png", "formula": "\\begin{align*} \\beta ( \\alpha ) = \\limsup _ { n \\to \\infty } \\frac { \\ln q _ { n + 1 } } { q _ n } = \\limsup _ { n \\to \\infty } \\frac { - \\ln \\| n \\alpha \\| } { n } , \\end{align*}"}
+{"id": "1201.png", "formula": "\\begin{align*} \\Psi ( x ) = \\lim _ { n \\rightarrow \\infty } \\max \\{ | 2 | ^ { k } \\mu ( 0 , 2 ^ { k + 1 } x ) ; 0 \\leq k < n \\} \\end{align*}"}
+{"id": "3721.png", "formula": "\\begin{align*} p ^ \\intercal & = \\lim _ { t _ j \\to 0 } \\frac { \\big ( h ( t _ j ) - h \\big ) ^ \\intercal } { t _ j } = 0 \\\\ H ' ( p - L _ X h ) & = \\lim _ { t _ j \\to 0 } \\frac { 1 } { t _ j } H ' | _ { g _ { t _ j } } \\big ( h ( t _ j ) - \\psi ^ * _ t ( h ) \\big ) \\\\ & = - \\lim _ { t _ j \\to 0 } \\frac { 1 } { t _ j } \\psi _ t ^ * \\big ( H ' | _ { g } ( h ) \\big ) = - X ( H ' ( h ) ) , \\end{align*}"}
+{"id": "8045.png", "formula": "\\begin{align*} \\mathbf { y } _ k = & a _ c s _ c \\mathbf { H } _ k \\mathbf { p } _ c + \\mathbf { H } _ k \\sum _ { i \\in \\mathcal { M } _ k } a _ i s _ i \\mathbf { p } _ i + \\mathbf { H } _ k \\sum \\limits _ { \\substack { l = 1 \\\\ l \\neq k } } ^ { K } \\sum \\limits _ { j \\in \\mathcal { M } _ l } a _ j s _ j \\mathbf { p } _ j + \\mathbf { n } _ k . \\end{align*}"}
+{"id": "2347.png", "formula": "\\begin{align*} y _ t = \\frac { y _ { x x } } { 1 + e ^ { 2 \\phi } y _ x ^ 2 } + y _ x \\phi ' ( x ) \\left ( 1 + \\frac { 1 } { 1 + e ^ { 2 \\phi } y _ x ^ 2 } \\right ) = \\frac { \\partial } { \\partial x } \\left ( e ^ { - \\phi } \\tan ^ { - 1 } ( y _ x e ^ \\phi ) \\right ) + \\phi ' ( x ) \\left ( y _ x + e ^ { - \\phi } \\tan ^ { - 1 } ( y _ x e ^ \\phi ) \\right ) . \\end{align*}"}
+{"id": "1733.png", "formula": "\\begin{align*} M _ \\omega ( f ) ( x ) = \\begin{cases} f ( x ) \\omega ( x ) & x \\in K \\\\ 0 & x \\notin K . \\end{cases} \\ , \\end{align*}"}
+{"id": "2771.png", "formula": "\\begin{align*} h _ m ( V _ m ( x ) ) = K ( t _ m ) , \\end{align*}"}
+{"id": "7216.png", "formula": "\\begin{align*} d u = \\Delta _ p u \\ , d t + \\kappa \\Delta u \\ , d t + \\sum _ { k \\in \\Z ^ d } \\sum _ { i = 1 } ^ { d - 1 } \\xi _ { k , i } ( x , t ) \\cdot \\nabla u \\ , d B _ t ^ { k , i } , \\end{align*}"}
+{"id": "4086.png", "formula": "\\begin{align*} & \\hphantom { { } = { } } \\phi \\circ ( \\varphi \\circ f ) ( x ) \\\\ * & = \\widehat { \\varphi } ( f ( o ) ) + D \\phi ( p ) ^ i { } _ \\alpha a ^ \\alpha { } _ \\beta x ^ \\beta \\\\ * & \\hphantom { { } = { } } + \\frac 1 2 \\left ( H \\phi ( p ) ^ i { } _ { \\alpha \\beta } a ^ \\alpha { } _ j a ^ \\beta { } _ k + D \\phi ( p ) ^ i { } _ \\alpha b ^ \\alpha { } _ { j k } \\right ) x ^ j x ^ k + ( ) . \\end{align*}"}
+{"id": "9091.png", "formula": "\\begin{align*} \\sum \\limits _ { j = l } ^ { n - 1 } \\chi _ j \\leqslant ( \\sum \\limits _ { j = l } ^ { n } ) \\end{align*}"}
+{"id": "2467.png", "formula": "\\begin{align*} a = \\frac { 1 } { 1 - q ^ { \\prime } } ; b = \\frac { 1 } { 1 - q } + 2 . \\end{align*}"}
+{"id": "8017.png", "formula": "\\begin{align*} a _ f ( p _ 1 ^ { 2 l _ 1 } ) a _ f ( p _ 2 ^ { 2 l _ 2 } ) \\dots a _ f ( p _ r ^ { 2 l _ r } ) a _ f ( q _ 1 ^ { 2 l _ 1 ' } ) a _ f ( q _ 2 ^ { 2 l _ 2 ' } ) \\dots a _ f ( q _ r ^ { 2 l _ r ' } ) = \\prod _ { u = 1 } ^ t \\left ( \\prod _ { i _ u \\in \\mathcal I ( s _ u ) } a _ f ( s _ u ^ { 2 i _ u } ) \\right ) , \\end{align*}"}
+{"id": "1125.png", "formula": "\\begin{align*} P _ { \\min } ^ { \\rm { O } } \\left ( { \\overline S , \\overline \\varepsilon , \\overline R | { W _ s } } \\right ) = \\frac { { \\left ( { W - { W _ s } } \\right ) { N _ 0 } } } { { { { \\left | { { h _ b } } \\right | } ^ 2 } } } \\left ( { { 2 ^ { \\frac { { \\overline R } } { { \\left ( { W - { W _ s } } \\right ) } } } } - 1 } \\right ) + { p _ K } \\left ( { \\overline S , { W _ s } } \\right ) , \\end{align*}"}
+{"id": "4840.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} S ^ \\prime _ t & = - \\beta ( 1 - \\theta L _ t ) S _ t ( 1 - \\theta L _ t ) I _ t , & S _ 0 & = s _ 0 , \\\\ I ^ \\prime _ t & = \\beta ( 1 - \\theta L _ t ) S _ t ( 1 - \\theta L _ t ) I _ t - \\gamma I _ t - I _ t \\phi \\left ( I _ t \\right ) , & I _ 0 & = i _ 0 , \\\\ R ^ \\prime _ t & = \\gamma I _ t , & R _ 0 & = r _ 0 , \\\\ D ^ \\prime _ t & = I _ t \\phi \\left ( I _ t \\right ) , & D _ 0 & = d _ 0 . \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "3132.png", "formula": "\\begin{align*} \\alpha ( e _ 1 ) = e _ 1 , \\alpha ( e _ 2 ) = e _ 2 , \\alpha ( e _ 3 ) = e _ 3 , \\alpha ( e _ 4 ) = \\lambda _ 2 e _ 3 + e _ 4 , \\alpha ( e _ 5 ) = \\frac { a _ 4 } { a _ 5 } \\lambda _ 2 e _ 3 + e _ 5 , \\end{align*}"}
+{"id": "28.png", "formula": "\\begin{align*} f _ \\alpha f _ \\beta \\cdot v _ { \\omega _ i } + \\sum _ { \\mathbf { \\gamma } } c _ { \\mathbf { \\gamma } } f _ { \\gamma _ 1 } f _ { \\gamma _ 2 } \\cdot v _ { \\omega _ i } = 0 . \\end{align*}"}
+{"id": "7978.png", "formula": "\\begin{align*} G _ { \\iota } = ( \\nabla G \\cdot e _ { \\iota } ) w _ { \\iota \\iota } , \\end{align*}"}
+{"id": "4981.png", "formula": "\\begin{align*} \\Delta _ n ( x , \\xi _ 0 ) : = & W ( \\xi _ 0 , n , x , \\mathsf { L } ^ n ) - W ( \\xi _ 0 , n , x , \\mathsf { R } ^ n ) \\\\ = & E _ { \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ { \\mathsf { L } ^ n } \\biggr ) \\biggr ] - E _ { \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ { \\mathsf { R } ^ n } \\biggr ) \\biggr ] . \\end{align*}"}
+{"id": "8648.png", "formula": "\\begin{align*} { \\bf { h } } _ l ^ H { { \\bf { f } } _ { l ' } } = 0 , \\ \\forall l \\ne l ' . \\end{align*}"}
+{"id": "7961.png", "formula": "\\begin{align*} \\phi _ x ^ \\ell : = \\frac 1 \\ell \\sum _ { y = 0 } ^ { \\ell - 1 } \\phi _ { x + y } , \\bar \\phi _ x ^ \\ell : = E _ { \\tilde \\nu _ { N , t } } \\left [ \\phi _ x ^ \\ell ~ \\Big | ~ \\sum _ { y = 0 } ^ { \\ell - 1 } r _ { x + y } \\right ] . \\end{align*}"}
+{"id": "1868.png", "formula": "\\begin{align*} \\begin{aligned} L ^ 2 & \\ge 4 \\pi A ( 1 - \\frac 1 { 4 \\pi } A ) \\\\ & ` ` = \" \\ , \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , . \\end{aligned} \\end{align*}"}
+{"id": "1062.png", "formula": "\\begin{align*} T _ x T _ y = & T _ { x y } \\qquad \\qquad \\qquad \\ell ( x y ) = \\ell ( x ) + \\ell ( y ) , \\\\ T _ s ^ 2 = & ( v - v ^ { - 1 } ) T _ s + 1 ~ ~ s \\in \\widetilde W . \\end{align*}"}
+{"id": "8966.png", "formula": "\\begin{align*} [ \\tau ] = [ v _ { j _ 1 } , v _ { j _ 2 } , \\dots , v _ { j _ { i + 1 } } ] = ( - 1 ) ^ { | \\pi | } [ v _ { j _ { \\pi ( 1 ) } } , v _ { j _ { \\pi ( 2 ) } } , \\dots , v _ { j _ { \\pi ( i + 1 ) } } ] , \\end{align*}"}
+{"id": "6620.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c } \\varepsilon _ 1 \\mu & \\varepsilon _ 1 \\nu \\\\ \\varepsilon _ 3 \\mu & \\varepsilon _ { 3 } \\nu \\end{array} \\right ) \\end{align*}"}
+{"id": "9295.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { H } _ m ( K _ m ) = C _ { m , n } \\delta _ a \\end{aligned} \\end{align*}"}
+{"id": "7551.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ p \\sum _ { s = 1 } ^ q \\alpha _ { r s } ( ( X ^ * X - \\| X \\| ^ 2 ) e _ r \\otimes f _ s ) = 0 . \\end{align*}"}
+{"id": "7437.png", "formula": "\\begin{align*} & a _ k ( x ) \\Big ( \\frac { \\textup { d } } { \\textup { d } x } \\Big ) ^ 2 \\big ( ( 1 - x ) ^ { p _ k ( k - j ) / k } x ^ { p _ k j / k } \\big ) + b _ k ( x ) \\frac { \\textup { d } } { \\textup { d } x } \\big ( ( 1 - x ) ^ { p _ k ( k - j ) / k } x ^ { p _ k j / k } \\big ) \\\\ & + p _ k c _ k ( x ) ( 1 - x ) ^ { p _ k ( k - j ) / k } x ^ { p _ k j / k } \\\\ & = \\frac { p _ k ^ 2 } { k ^ 2 } \\big ( ( 1 - x ) ^ { p _ k / k } - x ^ { p _ k / k } \\big ) ( 1 - x ) ^ { p _ k ( k - j ) / k } x ^ { p _ k j / k } \\big ( j ^ 2 ( 1 - x ) ^ { p _ k / k } - ( k - j ) ^ 2 x ^ { p _ k / k } \\big ) . \\end{align*}"}
+{"id": "7967.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } \\frac { \\partial X ( x , t ) } { \\partial t } = \\frac { 1 } { f ( v ) } \\sigma _ { k } ( x , t ) \\varphi ( X \\cdot v ) ( X \\cdot v ) G ( X ) v - X ( x , t ) ; \\\\ X ( x , 0 ) = X _ { 0 } ( x ) , \\end{array} \\right . \\end{align*}"}
+{"id": "2473.png", "formula": "\\begin{align*} Z _ { q } ( \\beta ) = \\frac { 1 } { h ^ { D N } N ! } \\displaystyle \\int _ { 0 } ^ { \\infty } \\ ; d \\bar { x } \\ ; d \\bar { p } \\ ; \\exp _ { q } \\left ( - \\beta \\displaystyle \\sum _ { i = 1 } ^ { D N } \\frac { p _ { i } ^ { 2 } } { 2 m } \\right ) , \\end{align*}"}
+{"id": "7932.png", "formula": "\\begin{align*} ( A ^ { \\tilde \\Psi } [ \\eta _ 1 , \\dots , \\eta _ n ] ) ( x ) = f ( x ) - g ( x ) . \\end{align*}"}
+{"id": "53.png", "formula": "\\begin{align*} | | u \\psi ^ { 1 / 2 } | | _ { L ^ 2 ( B _ { R _ 2 } ) } \\not = 0 . \\end{align*}"}
+{"id": "1851.png", "formula": "\\begin{align*} \\nabla _ { e _ i } X = r \\nabla _ { e _ i } \\frac \\partial { \\partial r } = r H e s s ( r ) ( e _ i , e _ j ) e _ j \\ , . \\end{align*}"}
+{"id": "692.png", "formula": "\\begin{align*} \\bar \\phi : = ( \\bar \\phi _ { v , v ' } ) _ { V \\times V ' } \\colon X ^ { ( V ' ) } \\to X ^ { ( V ) } \\qquad \\qquad \\bar \\psi : = ( \\bar \\psi _ { v ' , v '' } ) _ { V ' \\times V '' } \\colon X ^ { ( V '' ) } \\to X ^ { ( V ' ) } , \\quad \\end{align*}"}
+{"id": "4104.png", "formula": "\\begin{align*} \\mathcal { P } ^ 1 _ 0 ( T M ) & = \\{ j ^ 1 _ { ( o , o ) } ( \\{ f , g \\} ) \\mid \\{ f , g \\} \\in \\mathcal { T } _ 0 \\} . \\end{align*}"}
+{"id": "1112.png", "formula": "\\begin{align*} { S ^ { \\rm { S - N } } } = \\frac { { { W _ m } I } } { { K L } } { \\widetilde \\varepsilon _ K } \\left ( { { \\gamma ^ { \\rm { S - N } } } } \\right ) , \\end{align*}"}
+{"id": "7754.png", "formula": "\\begin{align*} \\max _ { i > j } | x _ i | < | x _ { j } | < \\min _ { i < j } | x _ i | ~ { \\rm w h e r e } ~ | x _ 0 | = + \\infty | ~ { \\rm a n d } ~ | x _ { d + 1 } | = 0 . \\end{align*}"}
+{"id": "4142.png", "formula": "\\begin{align*} u ( x ' , t ) : = P _ t ^ L * f ( x ' ) , ( x ' , t ) \\in \\R ^ n _ + . \\end{align*}"}
+{"id": "1147.png", "formula": "\\begin{align*} { S } ^ { \\rm P R } ( { \\cal A } ) = \\kappa ( \\rho ) W | _ { \\mathcal D } \\in \\R ^ { N L \\times N L } \\ , , \\end{align*}"}
+{"id": "3005.png", "formula": "\\begin{align*} J ( \\tau , w ( \\cdot ) , u ( \\cdot ) ) = \\sigma ( x ( \\vartheta ) , x _ \\vartheta ( \\cdot ) ) + \\int _ \\tau ^ \\vartheta f ^ 0 ( t , x ( t ) , x ( t - h ) , u ( t ) ) \\mathrm { d } t , \\end{align*}"}
+{"id": "8771.png", "formula": "\\begin{align*} { J _ a ^ b } : = & \\{ \\sum _ { k = 0 } ^ 3 \\psi _ k x _ k \\in \\mathbb H \\ \\mid \\ a _ k < x _ k < b _ k , \\ \\ k = 0 , 1 , 2 , 3 \\} \\\\ = & ( a _ 0 , b _ 0 ) \\times ( a _ 1 , b _ 1 ) \\times ( a _ 2 , b _ 2 ) \\times ( a _ 3 , b _ 3 ) , \\end{align*}"}
+{"id": "5971.png", "formula": "\\begin{align*} 1 \\mapsto \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} , \\ i \\mapsto \\begin{pmatrix} \\sqrt { a } & 0 \\\\ 0 & - \\sqrt { a } \\end{pmatrix} , \\ j \\mapsto \\begin{pmatrix} 0 & \\sqrt { b } \\\\ \\sqrt { b } & 0 \\end{pmatrix} , \\ i j \\mapsto \\begin{pmatrix} 0 & \\sqrt { a b } \\\\ - \\sqrt { a b } & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "5979.png", "formula": "\\begin{align*} \\# \\{ \\tau _ { s _ k } \\in { \\bf { S } } _ { s _ k } : f ^ 1 _ { \\tau _ { s _ k } } \\not \\equiv 0 \\} \\prod _ { i = 1 } ^ { k - 1 } \\max _ { \\tau _ { s _ { i + 1 } } \\in { \\bf { S } } _ { s _ { i + 1 } } } \\# \\{ \\tau _ { s _ i } \\in { \\bf { S } } _ { s _ i } : \\tau _ { s _ i } \\subset \\tau _ { s _ { i + 1 } } , f _ { \\tau _ { s _ i } } ^ 1 \\not \\equiv 0 \\} . \\end{align*}"}
+{"id": "9233.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ( ( n + 1 ) \\bar { t } / \\gamma ) \\| _ { \\mathcal { A } } & \\le \\beta ( \\| x ( n \\bar { t } / \\gamma ) \\| _ { \\mathcal { A } } , \\bar { t } ) + \\tilde { d } & & \\forall \\ , n \\in \\mathbb { N } . \\end{aligned} \\end{align*}"}
+{"id": "7317.png", "formula": "\\begin{align*} T \\in Z ( V ) \\iff \\mu _ T ( x , \\mu _ T ( y , z ) ) = 0 \\forall x , y , z \\in V . \\end{align*}"}
+{"id": "1011.png", "formula": "\\begin{align*} \\prescript L { } { } \\widetilde W { } ^ R \\cap \\left ( \\widetilde W _ L x \\widetilde W _ R \\right ) = & \\left \\{ \\prescript L { } { } x { } ^ R \\right \\} , \\\\ \\prescript { - L } { } { } \\widetilde W { } ^ { - R } \\cap \\left ( \\widetilde W _ L x \\widetilde W _ R \\right ) = & \\left \\{ \\prescript { - L } { } { } x { } ^ { - R } \\right \\} . \\end{align*}"}
+{"id": "2743.png", "formula": "\\begin{align*} \\beta = \\frac { 3 - \\rho } { 2 } \\end{align*}"}
+{"id": "3286.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\mathbb P \\left [ \\sup _ { t \\in [ 0 , T ] } | ( 1 ) _ n | > \\epsilon \\right ] \\\\ \\leq & \\lim _ { n \\to \\infty } \\mathbb P \\left [ \\sup _ { t \\in [ 0 , T ] } | ( 1 . 1 ) _ { n , N } | > \\epsilon \\right ] + \\sup _ { n \\in \\mathbb N } \\mathbb P \\left [ \\sup _ { t \\in [ 0 , T ] } | ( 1 . 2 ) _ { n , N } | > \\epsilon \\right ] \\\\ = & 0 + \\frac 1 { \\epsilon } \\sup _ { n \\in \\mathbb N } \\mathbb E \\left [ \\sup _ { t \\in [ 0 , T ] } | ( 1 . 2 ) _ { n , N } | \\right ] \\leq \\delta . \\end{align*}"}
+{"id": "3898.png", "formula": "\\begin{align*} 0 = & p s _ { \\delta _ N , i } \\left ( \\frac { \\delta _ N } { s _ { \\delta _ N , i } } \\right ) ^ { \\frac { 2 p } { p - 1 } } \\int _ { \\mathbb { R } ^ 2 } \\phi ( T _ { z _ { N , i } } y ) ^ { p - 1 } _ + \\phi ' ( T _ { z _ { N , i } } y ) \\frac { ( T _ { z _ { N , i } } ) _ h ^ t \\cdot T _ { z _ { N , i } } y } { | T _ { z _ { N , i } } y | } \\tilde { u } _ { N , i } ( y ) d y + O \\left ( \\frac { \\varepsilon _ N ^ 2 } { | \\ln \\varepsilon _ N | ^ { p - 1 } } \\right ) . \\end{align*}"}
+{"id": "5280.png", "formula": "\\begin{align*} h g = n _ { u _ 1 } a _ { t _ 1 } n _ { u _ 2 } a _ { t _ 2 } ( n _ { u _ 2 } a _ { t _ 2 } ) ^ { - 1 } k _ { \\theta _ 1 } ( n _ { u _ 2 } a _ { t _ 2 } ) k _ { \\theta _ 2 } . \\end{align*}"}
+{"id": "8518.png", "formula": "\\begin{align*} \\delta _ { \\pm } ( z ) = \\exp [ \\mathcal { P } ^ { \\pm } \\log \\left ( 1 + \\bar { r } _ 1 r _ 2 \\right ) ] , z \\in \\mathbb { R } . \\end{align*}"}
+{"id": "6226.png", "formula": "\\begin{align*} K _ { j } \\nu ( x ) = \\int _ { X } k _ j ( x , y ) d \\nu ( y ) , K ^ { * } \\nu ( x ) = \\sup _ { j } \\vert K _ j \\nu ( x ) \\vert . \\end{align*}"}
+{"id": "5204.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } A _ t \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) \\chi _ R \\rho \\dd x \\dd t = - \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ + \\frac { G } { r ^ { 2 } } \\chi _ R \\dot { \\rho } \\dd x \\dd t - \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } ( \\Phi _ + \\nabla \\theta \\times A _ t ) \\cdot \\nabla \\chi _ R \\rho \\dd x \\dd t . \\end{align*}"}
+{"id": "1539.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } L ( 0 ) = & L _ 0 \\\\ \\frac { d } { d t _ n } L = & \\left [ ( L ^ n ) _ D , L \\right ] \\ ; . \\end{array} \\right . \\end{align*}"}
+{"id": "4117.png", "formula": "\\begin{align*} F ^ { \\perp } : = \\left \\lbrace \\varphi \\in E ^ { \\vee } \\mid \\varphi ( F ) = 0 \\right \\rbrace . \\end{align*}"}
+{"id": "2886.png", "formula": "\\begin{align*} \\Big | \\sum _ { x , y = 0 } ^ n ( \\delta _ { x , y } - M _ { x , y } ( m ) ) f _ y ^ \\star f _ x \\Big | \\ge { \\frak C } \\sum _ { x = 0 } ^ n | f _ x | ^ 2 , ( f _ 0 , \\ldots , f _ n ) \\in \\mathbb C ^ { n + 1 } \\end{align*}"}
+{"id": "6767.png", "formula": "\\begin{align*} e ( - \\rho , x ) = \\overline { e ( \\rho , x ) } , g ( - \\rho , x ) = \\overline { g ( \\rho , x ) } . \\end{align*}"}
+{"id": "4180.png", "formula": "\\begin{align*} K _ H ( x _ 1 , x _ 2 ) = \\frac { 1 } { k ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 } \\begin{pmatrix} k ^ 2 + x _ 2 ^ 2 & - x _ 1 x _ 2 \\\\ - x _ 1 x _ 2 & k ^ 2 + x _ 1 ^ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "2431.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 } s L _ { q } [ f ( t ) ] = \\frac { f ( \\infty ) } { 1 + ( 1 - q ) } . \\end{align*}"}
+{"id": "5141.png", "formula": "\\begin{align*} \\nabla \\times U = \\mu 1 _ { ( 0 , \\infty ) } ( \\phi - \\phi _ { \\infty } ) U . \\end{align*}"}
+{"id": "5673.png", "formula": "\\begin{align*} \\tau _ { 0 } ( v _ { 1 } , \\dots , v _ { p - 2 } ) & : = ( e _ { 1 } , e _ { 2 } , \\bar { v } _ { 1 } , \\dots , \\bar { v } _ { p - 2 } ) \\\\ \\tau _ { 1 } ( v _ { 1 } , \\dots , v _ { p - 2 } ) & : = ( e _ { 1 } , e _ { 2 } - e _ { 1 } , \\bar { v } _ { 1 } , \\dots , \\bar { v } _ { p - 2 } ) \\\\ \\tau _ { 2 } ( v _ { 1 } , \\dots , v _ { p - 2 } ) & : = ( e _ { 2 } , e _ { 2 } - e _ { 1 } , \\bar { v } _ { 1 } , \\dots , \\bar { v } _ { p - 2 } ) , \\end{align*}"}
+{"id": "3206.png", "formula": "\\begin{align*} h _ * ( \\tau ) ( a ) = \\int _ { X _ A } \\hat a \\circ h \\ , d \\mu _ \\tau \\quad a \\in A . \\end{align*}"}
+{"id": "7336.png", "formula": "\\begin{align*} t ^ n | K | \\big ( \\sum _ { m = k } ^ n & { { n } \\choose { m } } V ( K [ m ] , ( U + V ) [ n - m ] ) t ^ m \\big ) \\\\ & \\le c _ n \\big ( \\sum _ { m = i } ^ n { { n } \\choose { m } } V ( K [ m ] , U [ n - m ] ) t ^ m \\big ) \\big ( \\sum _ { m = j } ^ n { { n } \\choose { m } } V ( K [ m ] , V [ n - m ] ) t ^ m \\big ) . \\end{align*}"}
+{"id": "8917.png", "formula": "\\begin{align*} \\frac { ( 2 k ) ! } { k ! } ( 1 + \\rho ) ^ { k } = \\sum _ { j = 0 } ^ { k } \\binom { 2 k } { 2 j } ( 1 + \\rho ) ^ { 2 k - 2 j } ( 1 - \\rho ^ { 2 } ) ^ { j } ( 2 j - 1 ) ! ! ( 2 k - 2 j - 1 ) ! ! . \\end{align*}"}
+{"id": "4718.png", "formula": "\\begin{align*} \\frac { \\partial \\tilde y } { \\partial x } = y \\cdot \\frac { 1 } { 4 } g ^ { - \\frac { 3 } { 4 } } \\cdot \\biggl ( \\frac { 1 } { 6 } \\int \\limits _ { 0 } ^ 1 f _ { x y ^ { 4 } } ^ { ( 5 ) } ( x , s y ) ( 1 - s ) ^ 3 \\ , d s \\biggr ) , \\end{align*}"}
+{"id": "3755.png", "formula": "\\begin{align*} Q ^ i _ { n _ 0 ; l _ 1 , \\ldots , l _ { n _ 0 } } : = \\{ A \\in P _ { n _ 0 ; l _ 1 , l _ 2 , \\ldots , l _ { n _ 0 } } : \\substack { \\alpha ^ { ( l _ i ) } _ i , \\ldots , \\alpha ^ { ( l _ { n _ 0 } ) } _ { n _ 0 } \\\\ \\alpha ^ { ( l _ { i + 1 } ) } _ { i + 1 } , \\ldots , \\alpha ^ { ( l _ { n _ 0 } ) } _ { n _ 0 } } \\} , i = 1 , 2 , \\ldots , n _ 0 \\end{align*}"}
+{"id": "5894.png", "formula": "\\begin{align*} \\overline { X } ( t ) : = x + \\int _ 0 ^ t \\widetilde { \\mathcal { A } } ( s ) d s + \\int _ 0 ^ t \\widetilde { \\mathcal { B } } ( s ) d \\widetilde { W } ( s ) . \\end{align*}"}
+{"id": "8695.png", "formula": "\\begin{align*} \\tilde W \\cap h _ r ( \\tilde W ) = ( S _ 0 ^ - \\cap E _ r ) \\cap h _ r ( S _ 0 ^ - \\cap E _ r ) = ( S _ 0 ^ - \\cap h _ r ( S _ 0 ^ - ) ) \\cap E _ r = \\emptyset \\cap E _ r = \\emptyset , \\end{align*}"}
+{"id": "5480.png", "formula": "\\begin{align*} \\Delta \\Delta \\Psi _ \\delta ( x ) = p ( p + 2 ) \\left ( | x | ^ 2 + \\delta \\right ) ^ { - \\frac { p } { 2 } - 4 } ( A | x | ^ 4 + B | x | ^ 2 + C ) , \\end{align*}"}
+{"id": "3147.png", "formula": "\\begin{align*} \\| f '' \\| _ 1 = \\sum _ { n \\in \\Z } | f ( n + 1 ) + f ( n - 1 ) - 2 f ( n ) | . \\end{align*}"}
+{"id": "3155.png", "formula": "\\begin{align*} \\sum \\limits _ { x , y \\in \\mathbb { F } _ q , x \\neq 1 } \\overline { \\chi _ 4 } ( x ) \\chi _ 4 ( y ) \\chi _ 4 ( 1 - y ) \\overline { \\chi _ 4 } ( x - y ) = 1 - \\rho . \\end{align*}"}
+{"id": "9045.png", "formula": "\\begin{align*} L ( E , \\epsilon ) = \\max \\{ \\log | \\lambda | + | \\epsilon | , 0 \\} , \\forall \\ E \\in \\Sigma _ { 2 \\lambda \\cos , \\alpha } . \\end{align*}"}
+{"id": "368.png", "formula": "\\begin{align*} C _ { k } ( r , s ) = \\cos { ( \\frac { 2 \\pi } { n - 1 } ( r - s ) k ) } ~ ~ \\mbox { a n d } ~ ~ \\widetilde { C _ { k } } ( r , s ) = \\cos { ( \\frac { 2 \\pi } { n - 1 } ( r - s - 1 ) k ) } . \\end{align*}"}
+{"id": "3380.png", "formula": "\\begin{align*} S ( x ; N ) = \\sum _ { n = 1 } ^ N x ^ n , S _ 1 ( x ; N ) = \\sum _ { n = 1 } ^ N n x ^ n \\textrm { a n d } S _ 2 ( x ; N ) = \\sum _ { n = 1 } ^ N n ^ 2 x ^ n . \\end{align*}"}
+{"id": "4.png", "formula": "\\begin{align*} \\beta _ 1 \\oplus \\beta _ 2 = \\begin{cases} \\emptyset & i _ 1 = j _ 1 i _ 2 < j _ 1 - 1 j _ 2 \\leq i _ 2 , \\\\ \\{ ( \\alpha _ { i _ 1 , j _ 2 } , \\alpha _ { j _ 1 , i _ 2 } ) \\} & j _ 1 - 1 \\leq i _ 2 < j _ 2 . \\end{cases} \\end{align*}"}
+{"id": "8731.png", "formula": "\\begin{align*} \\mu _ n = \\frac { 1 } { \\left [ G : K \\right ] } \\sum _ { g \\in G / K } g _ * \\nu _ n \\end{align*}"}
+{"id": "1682.png", "formula": "\\begin{align*} r = \\norm { A } + \\norm { A _ d } , \\end{align*}"}
+{"id": "7470.png", "formula": "\\begin{align*} \\det \\begin{bmatrix} 1 & 1 & 1 \\\\ 2 \\xi _ 1 ^ 1 & 2 \\xi _ 1 ^ 2 & 2 \\xi _ 1 ^ 3 \\\\ 3 ( \\xi _ 1 ^ 1 ) ^ 2 & 3 ( \\xi _ 1 ^ 2 ) ^ 2 & 3 ( \\xi _ 1 ^ 3 ) ^ 2 \\end{bmatrix} & = 6 ( \\xi _ 2 - \\xi _ 1 ) ( \\xi _ 3 - \\xi _ 1 ) ( \\xi _ 3 - \\xi _ 2 ) , \\end{align*}"}
+{"id": "383.png", "formula": "\\begin{align*} B _ { k } : = \\frac { 2 } { ( n - 1 ) ( 2 \\phi _ k + \\frac { 1 } { 2 \\phi _ k } ) ^ { 2 } } \\left [ \\begin{array} { c c c } 0 & \\ 0 & \\ 0 \\\\ \\ 0 & \\frac { C _ k } { 4 \\phi _ { k } ^ { 2 } } & \\frac { C _ k + \\widetilde { C _ { k } } } { 4 \\phi _ k ^ { 2 } } \\\\ \\ 0 & \\frac { C _ k + \\widetilde { C _ { k } } ^ { ' } } { 4 \\phi _ { k } ^ { 2 } } & C _ k \\\\ \\end{array} \\right ] ~ ~ ~ k \\in \\nabla , \\end{align*}"}
+{"id": "398.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\mathrm { d i s t } _ { \\mathbb { U } } ( \\varphi ( t , \\tau - t , \\theta _ { - t } \\omega ) \\mathcal { D } ( \\tau - t , \\theta _ { - t } \\omega ) , \\mathcal { A } ( \\tau , \\omega ) ) = 0 . \\end{align*}"}
+{"id": "5036.png", "formula": "\\begin{align*} \\left | \\dfrac { \\gamma \\alpha \\omega - \\gamma \\omega } { - \\gamma \\alpha \\omega } \\right | = \\dfrac { | \\alpha \\omega - \\omega | } { | - \\gamma \\alpha \\omega | | c \\omega + d | | c \\alpha \\omega + d | } = \\dfrac { 2 ^ { - 1 } } { | a \\alpha \\omega + b | | c \\omega + d | } . \\end{align*}"}
+{"id": "6512.png", "formula": "\\begin{align*} \\lambda _ x | _ { E _ w } = \\lambda _ y | _ { E _ w } , \\lambda _ x ( w , w b ) = f \\lambda _ x ( w b ) = s ( f ) \\end{align*}"}
+{"id": "4704.png", "formula": "\\begin{align*} \\mu _ { i + 1 } = ( I - \\gamma g _ { \\mu _ i } ) \\# \\mu _ i . \\end{align*}"}
+{"id": "3437.png", "formula": "\\begin{align*} \\| \\theta - \\frac { 1 } { 2 } + \\tilde { m } _ n \\alpha \\| = \\inf _ { k \\in [ - q _ n / 2 , q _ n / 2 ) } \\| \\theta - \\frac { 1 } { 2 } + k \\alpha \\| . \\end{align*}"}
+{"id": "4851.png", "formula": "\\begin{align*} & p _ i \\circ H ( X _ 1 , . . . , X _ n , 0 ) \\\\ = & p _ i \\circ \\Delta \\circ F ( c _ 1 , . . . , c _ n ) \\\\ = & p _ i ( F ( c _ 1 , \\dots , c _ n ) , \\dots , F ( c _ 1 , \\dots , X _ n ) ) \\\\ = & F ( c _ 1 , \\dots , c _ n ) \\\\ = & p _ j \\circ \\Delta \\circ F ( c _ 1 , . . . , c _ n ) \\\\ = & p _ j \\circ H ( X _ 1 , . . . , X _ n , 0 ) . \\end{align*}"}
+{"id": "2674.png", "formula": "\\begin{align*} \\Bigl ( \\frac { 1 } { \\lambda ( t ) } \\mathcal { G } ( t ) \\Bigr ) ^ { \\prime } \\leq - \\frac { k _ 3 } { k _ 2 } \\frac { \\alpha ( t ) } { \\lambda ( t ) } \\Bigl [ \\frac { 1 } { \\lambda ( t ) } \\mathcal { G } ( t ) \\Bigr ] . \\end{align*}"}
+{"id": "2186.png", "formula": "\\begin{align*} \\mathbf { v } ^ { 0 } = \\sum \\limits _ { k = 1 } ^ { N } \\alpha _ { k } \\mathbf { u _ { k } } , \\end{align*}"}
+{"id": "7860.png", "formula": "\\begin{align*} \\psi ^ \\dagger _ p ( 0 , p _ 0 ) = 0 . \\end{align*}"}
+{"id": "3696.png", "formula": "\\begin{align*} 0 = S ' ( h , v ) - S ' \\big ( L _ X \\bar g , X ( \\bar u ) \\big ) = S ' ( 0 , v - X ( \\bar u ) ) = 0 \\mbox { i n } M \\setminus \\Omega . \\end{align*}"}
+{"id": "257.png", "formula": "\\begin{align*} X _ j ( X _ i ^ r b _ i ) = \\left \\{ \\begin{array} { l l } b _ 1 & \\\\ 0 & \\end{array} \\right . \\end{align*}"}
+{"id": "51.png", "formula": "\\begin{align*} \\operatorname { d i v } ( \\rho ^ { - Q + 2 } Z ) = 2 \\rho ^ { - Q + 2 } \\end{align*}"}
+{"id": "7585.png", "formula": "\\begin{align*} \\begin{gathered} \\lim _ { j \\to \\infty } ( T \\times T ) ^ { c _ 1 ( j ) } ( x _ { 0 0 } , x _ { 0 1 } ) = ( x _ { 1 0 } , x _ { 1 1 } ) \\\\ \\lim _ { m \\to \\infty } ( T \\times T ) ^ { c _ 2 ( m ) } ( x _ { 0 0 } , x _ { 1 0 } ) = ( x _ { 0 1 } , x _ { 1 1 } ) \\end{gathered} \\end{align*}"}
+{"id": "2297.png", "formula": "\\begin{align*} \\mu _ 0 = ( m _ 1 - m _ 2 ) \\omega _ 1 + \\dots + ( m _ { n - 1 } - m _ n ) \\omega _ { n - 1 } . \\end{align*}"}
+{"id": "2106.png", "formula": "\\begin{align*} h _ c ( K _ { 1 , m } ) - h _ c ( K _ { 1 , m - b } ) & = f _ c ( m ) - f _ c ( m - b ) + b \\Delta _ c \\left ( 1 \\right ) \\\\ & : = R H S ( m , b , c ) . \\end{align*}"}
+{"id": "2435.png", "formula": "\\begin{align*} L _ { q } [ f ( a t ) ] = \\frac { 1 } { a } \\ , \\int _ { 0 } ^ { \\infty } d x \\bigg [ 1 - ( 1 - q ) \\frac { s x } { a } \\bigg ] ^ { \\frac { 1 } { 1 - q } } f ( x ) = \\frac { 1 } { a } F _ { q } ( s / a ) . \\end{align*}"}
+{"id": "8810.png", "formula": "\\begin{align*} \\Pi _ { \\tilde { V } } e ^ { J _ { z / | z | } ^ \\perp t } x = e ^ { \\lambda t } ( \\Pi _ { \\tilde { V } } x + t \\Pi _ { V } x ) . \\end{align*}"}
+{"id": "8564.png", "formula": "\\begin{align*} \\kappa ( t ) = h _ { \\alpha } ( t ) \\cdot \\ , \\kappa _ 1 ( t ) , \\ \\kappa _ 1 ( t ) = \\sum _ { k = 0 } ^ { + \\infty } \\ , a _ k t ^ k , \\ a _ 0 \\not = 0 , \\ 0 < \\alpha < 1 , \\end{align*}"}
+{"id": "149.png", "formula": "\\begin{align*} x _ 0 = 1 - \\frac { x _ 0 \\sum _ { l \\in \\mathbb Z _ 0 } { \\lambda _ l z _ l } } { 1 + \\sum _ { l \\in \\mathbb Z _ 0 } { \\lambda _ l z _ l } } , \\ \\ \\ x _ j = \\frac { x _ 0 \\lambda _ { j } z _ { j } } { 1 + \\sum _ { l \\in \\mathbb Z _ 0 } { \\lambda _ l z _ l } } , \\ \\ \\ j \\in \\mathbb Z _ 0 . \\end{align*}"}
+{"id": "5765.png", "formula": "\\begin{align*} \\mathcal { P } ' \\left ( \\partial _ Y \\Phi ( X ) \\right ) = - \\sum _ { i = 1 , 2 } \\sqrt { c _ i } \\ \\langle Y _ i , X \\rangle \\mathcal { P } ' ( N _ i ) + \\Phi ( \\mathcal { P } ( B ( Y , X _ T ) ) - \\mathcal { P } ( B ^ * ( Y , X _ N ) ) ) ; \\end{align*}"}
+{"id": "4211.png", "formula": "\\begin{align*} \\ln \\frac { \\sigma } { \\tau } \\int _ { \\Omega } q ^ 2 ( K _ H ( x ) \\nabla w ^ { \\sigma , \\tau } _ \\varepsilon | \\nabla w ^ { \\sigma , \\tau } _ \\varepsilon ) d x = \\frac { 1 } { \\varepsilon ^ 2 } \\int _ { \\Omega } \\left ( u _ \\varepsilon - q \\ln \\frac { 1 } { \\varepsilon } \\right ) _ + ^ { p } q d x . \\end{align*}"}
+{"id": "992.png", "formula": "\\begin{align*} L : = \\sum _ { ( P , H ) \\in \\Omega _ 3 } ( | W | - q ^ { d - 1 } ) \\le \\delta q ^ { d + 1 } . \\end{align*}"}
+{"id": "7124.png", "formula": "\\begin{align*} \\begin{array} { l l l } f ^ \\sharp _ L ( L ' \\circ L , R \\circ R ' ) ( f _ L ( s ) ( t ) ) & = & f _ L ( L ' ( L ( s ) ) ) ( t ) \\\\ & = & f ^ \\sharp _ L ( L ' , R ' ) ( f _ L ( L ( s ) ) ( t ) ) \\\\ & = & f ^ \\sharp _ L ( L ' , R ' ) ( f ^ \\sharp _ L ( L , R ) ( f _ L ( s ) ( t ) ) . \\end{array} \\end{align*}"}
+{"id": "1492.png", "formula": "\\begin{align*} \\phi _ 3 ' = - ( c \\phi _ 3 + \\phi _ 2 g ( \\phi _ 1 ) ) , \\end{align*}"}
+{"id": "3496.png", "formula": "\\begin{align*} | \\phi ( z _ { t + 1 } ' ) | \\leq \\begin{cases} | \\phi ( x _ 0 ^ - ) | , z _ { t + 1 } ' \\in I ^ - \\\\ | \\phi ( \\ell q _ n + m _ n ) | , z _ { t + 1 } ' = \\ell q _ n + m _ n \\\\ | \\phi ( x _ 0 ^ + ) , z _ { t + 1 } ' \\in I ^ + \\end{cases} \\end{align*}"}
+{"id": "3583.png", "formula": "\\begin{align*} L ^ { i } = \\{ p \\in \\Delta _ { m - 1 } \\vert \\ , A ^ { i } p = A ^ { i } d \\} \\end{align*}"}
+{"id": "1001.png", "formula": "\\begin{align*} \\ell ( x , \\alpha ) : = \\langle \\mu , \\alpha \\rangle + \\Phi ^ + ( \\alpha ) - \\Phi ^ + ( w \\alpha ) . \\end{align*}"}
+{"id": "8675.png", "formula": "\\begin{align*} { { \\bf { \\bar X } } ^ { \\star } } = { \\left [ { { \\bf { \\tilde A } } } \\right ] _ { : , 1 : { M _ t } } } , \\end{align*}"}
+{"id": "5660.png", "formula": "\\begin{align*} G O _ { n , n } ( R ) : = \\{ A \\in G L _ { 2 n } ( R ) | \\ , \\ , ^ { t } A \\psi _ { 2 n } A = a \\psi _ { 2 n } , \\ , \\ , \\ , \\ , a \\in R ^ { * } \\} . \\end{align*}"}
+{"id": "464.png", "formula": "\\begin{align*} L _ { f _ 1 } ( z ) = ( e ^ { \\lambda \\Lambda _ 1 L _ { g _ 1 } ( z ) } - 1 ) / ( e ^ { \\lambda \\Lambda _ 1 } - 1 ) , \\end{align*}"}
+{"id": "5598.png", "formula": "\\begin{align*} A ( \\tau _ y ( x ) ) + \\sum _ { n = 1 } ^ N \\left [ \\hat { A } \\circ \\hat { \\sigma } ^ { - ( n + 1 ) } ( y | x ) - \\hat { A } \\circ \\hat { \\sigma } ^ { - n } ( \\sigma ( y ) | x ' ) - \\hat { A } \\circ \\hat { \\sigma } ^ { - n } ( y | x ) + \\hat { A } \\circ \\hat { \\sigma } ^ { - n } ( y | x ' ) \\right ] & = \\\\ A ( \\tau _ { y , N + 1 } ( x ) ) + \\sum _ { n = 1 } ^ N \\left [ \\hat { A } \\circ \\hat { \\sigma } ^ { - n } ( y | x ' ) - \\hat { A } \\circ \\hat { \\sigma } ^ { - n } ( \\sigma ( y ) | x ' ) \\right ] \\ . \\end{align*}"}
+{"id": "2070.png", "formula": "\\begin{align*} \\mathcal { I } - d ^ { - } ( p , H ) & = 1 - \\mathcal { I } - d ^ { - } ( p , H ^ { c } ) = 1 - \\mathcal { I } - \\limsup y _ n = 1 + \\mathcal { I } - \\liminf ( - y _ n ) \\\\ & = \\mathcal { I } - \\liminf ( 1 - y _ n ) = \\mathcal { I } - \\liminf x _ n \\\\ & = \\mathcal { I } - d _ { - } ( p , H ) \\end{align*}"}
+{"id": "988.png", "formula": "\\begin{align*} f ( \\delta _ 1 ) & = - 2 q ^ { 2 d + 1 } + \\left ( \\frac { 3 } { 2 } \\alpha ^ 2 + \\frac { 2 1 } { 2 } \\alpha + 1 2 \\right ) q ^ { 2 d } + ( \\alpha ^ 2 + 6 \\alpha + 9 ) q ^ { 2 d - 1 } \\\\ & \\hphantom { \\le { } } \\mathrel { + } ( 4 \\alpha ^ 2 + 2 4 \\alpha + 3 6 ) q ^ { 2 d - 2 } + q ^ { d + 4 } + 4 q ^ { d + 3 } - \\left ( \\frac { \\alpha } { 2 } + 4 \\right ) q ^ { d + 2 } \\\\ & \\hphantom { \\le { } } \\mathrel { - } ( 2 \\alpha + 6 ) q ^ { d + 1 } - ( 8 \\alpha + 2 4 ) q ^ d + q ^ 3 + 4 q ^ 2 \\end{align*}"}
+{"id": "5546.png", "formula": "\\begin{align*} ( x , y ) = ( x , 1 ) ( 1 , y ) = ( 1 \\cdot x , y \\cdot 1 ) ( 1 , y ) = ( x , 1 ) \\wedge ( 1 , y ) \\end{align*}"}
+{"id": "917.png", "formula": "\\begin{align*} W = \\left \\{ \\begin{aligned} & 2 \\lambda _ 1 W _ 1 \\Im \\left [ W _ 1 \\overline { W _ 2 } \\right ] & \\ \\lambda _ 6 = \\lambda _ 1 , \\\\ & - 6 i \\lambda _ 1 W _ 1 \\Re \\left [ W _ 1 \\overline { W _ 2 } \\right ] & \\ \\ \\lambda _ 6 = 3 \\lambda _ 1 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7255.png", "formula": "\\begin{align*} \\epsilon _ r P ( x _ 1 , \\ldots , x _ n ) = P ( v _ 1 , \\ldots , v _ n ) . \\end{align*}"}
+{"id": "4635.png", "formula": "\\begin{align*} \\tau = \\tau ' v , v = \\sum _ { r = 1 } ^ j \\lambda _ r ( e _ { i _ { r , 1 } } + \\dots + e _ { i _ { r , d _ r } } ) \\end{align*}"}
+{"id": "4551.png", "formula": "\\begin{align*} & A _ { w , 1 } f ( v _ { \\ell , 0 , 0 } ) = ( q ^ 3 + q ^ 2 + q ) f ( v _ { \\ell , 1 , 0 } ) + f ( v _ { \\ell + 1 , 0 , 0 } ) \\\\ & A _ { w , 2 } f ( v _ { \\ell , 0 , 0 } ) = ( q ^ 4 + q ^ 3 + q ^ 2 ) f ( v _ { \\ell , 1 , 1 } ) + ( q ^ 2 + q + 1 ) f ( v _ { \\ell + 1 , 1 , 0 } ) \\\\ & A _ { w , 3 } f ( v _ { \\ell , 0 , 0 } ) = q ^ 3 f ( v _ { \\ell - 1 , 0 , 0 } ) + ( q ^ 2 + q + 1 ) f ( v _ { \\ell + 1 , 1 , 1 } ) . \\end{align*}"}
+{"id": "1300.png", "formula": "\\begin{align*} N ( x _ a ; x _ b , x _ c , x _ d ; \\alpha , \\beta ) - D ( x _ a ; x _ b , x _ c , x _ d ; \\alpha , \\beta ) = 0 , \\end{align*}"}
+{"id": "5145.png", "formula": "\\begin{align*} \\phi \\geq 0 , h = 2 \\mu \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi _ { \\infty } ) _ { + } ^ { 2 } \\frac { 1 } { r ^ { 2 } } \\dd x , G = \\mu ( \\phi - \\phi _ { \\infty } ) _ { + } . \\end{align*}"}
+{"id": "4699.png", "formula": "\\begin{align*} R = \\max _ { q \\in D } { \\rm d i s t } ( q , p ) \\end{align*}"}
+{"id": "2549.png", "formula": "\\begin{align*} \\tilde { \\mathbf { U } } = \\mathbf { U } _ { x _ w } - \\omega \\mathbf { U } _ { x _ o } . \\end{align*}"}
+{"id": "5325.png", "formula": "\\begin{align*} \\widetilde { e } _ a ( z , t ) = \\frac { k ! ( n - 1 ) ! } { ( k + n - 1 ) ! } \\ , e _ { k , \\lambda } ^ { n - 1 } ( z , t ) , \\ , \\ , \\chi _ \\tau ( z , t ) = ( n - 1 ) ! 2 ^ { n - 1 } \\frac { J _ { n - 1 } ( \\sqrt { \\tau } | z | ) } { ( \\sqrt { \\tau } | z | ) ^ { n - 1 } } , \\ , \\tau \\geq 0 . \\end{align*}"}
+{"id": "5403.png", "formula": "\\begin{align*} \\begin{aligned} F ^ { i j } ( b _ { n - 1 } ( A _ 1 v - A _ 2 w ) \\pm ( u - \\varphi ) ) _ { i j } \\geq \\ , & 0 \\mbox { i n } \\omega \\\\ b _ { n - 1 } ( A _ 1 v - A _ 2 w ) \\pm ( u - \\varphi ) \\leq \\ , & 0 \\mbox { o n } \\partial \\omega \\end{aligned} \\end{align*}"}
+{"id": "6277.png", "formula": "\\begin{align*} \\omega _ { i j } ^ k = \\left \\{ \\begin{array} { l r } - 2 \\Gamma _ { i j } ^ j & k = j \\neq i , \\\\ 2 \\Gamma _ { i j } ^ j & k = i \\neq j , \\\\ 0 & \\ , . \\end{array} \\right . \\end{align*}"}
+{"id": "1003.png", "formula": "\\begin{align*} \\ell ^ { \\frac \\infty 2 } ( x ) : = \\ell ( w ) + \\langle \\mu , 2 \\rho \\rangle . \\end{align*}"}
+{"id": "4826.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n ( U _ n ) _ { i j } ^ 2 = \\sum _ { i = 1 } ^ n ( U _ n ) _ { j i } ^ 2 = 1 , 1 \\leq j \\leq n , \\end{align*}"}
+{"id": "6750.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty z ^ i \\ , \\phi _ s ( z ) \\ , [ \\Phi _ s ( z ) ] ^ { j + \\alpha - 1 } \\mathrm { d } z = & \\sum _ { k = 0 } ^ \\infty v _ k ( j + \\alpha - 1 ) \\ , \\int _ 0 ^ \\infty z ^ i \\ , \\phi _ s ( z ) \\ , \\Phi _ s ( z ) ^ k \\mathrm { d } z \\\\ & + \\sum _ { k = 0 } ^ \\infty ( - 1 ) ^ { i + k } \\ , \\binom { j + \\alpha - 1 } { k } \\ , \\int _ 0 ^ \\infty z ^ i \\ , \\phi _ s ( z ) \\ , \\Phi _ s ( z ) ^ k \\mathrm { d } z . \\end{align*}"}
+{"id": "1685.png", "formula": "\\begin{align*} \\forall \\tau \\in [ - h , 0 ] , \\varphi ( \\tau ) = \\underbrace { \\ell _ n ^ \\top ( \\tau ) \\Phi _ n } _ { \\varphi _ n ( \\tau ) } + \\tilde { \\varphi } _ { n } ( \\tau ) , \\end{align*}"}
+{"id": "3058.png", "formula": "\\begin{align*} \\rho _ E ( \\lambda , z ) = \\sum _ { j = 1 } ^ l m _ j n _ j z + \\sum _ { i = 0 } ^ n k _ i \\lambda _ i . \\end{align*}"}
+{"id": "5372.png", "formula": "\\begin{align*} H = \\Delta u + \\frac { B } { 2 } | x | ^ 2 , \\end{align*}"}
+{"id": "6680.png", "formula": "\\begin{align*} B _ { \\gamma x , \\gamma y } ( \\gamma ^ + ) = B _ { \\gamma x , \\gamma y } ( \\gamma \\gamma ^ + ) = B _ { x , y } ( \\gamma ^ + ) . \\end{align*}"}
+{"id": "8659.png", "formula": "\\begin{align*} \\begin{aligned} { { \\bf { x } } _ { \\rm t } } \\left [ { m , n } \\right ] & = \\frac { 1 } { { \\sqrt K } } \\sum \\limits _ { k = 0 } ^ { K - 1 } { { { \\bf { x } } _ { \\rm f } } \\left [ { m , k } \\right ] { e ^ { j \\frac { { 2 \\pi } } { K } k n } } } \\\\ & = \\frac { 1 } { { \\sqrt K } } \\sum \\limits _ { k = 0 } ^ { K - 1 } { { { \\bf { u } } _ k } s \\left [ { m , k } \\right ] { e ^ { j \\frac { { 2 \\pi } } { K } k n } } } , \\ n = 0 , \\cdots , K - 1 . \\end{aligned} \\end{align*}"}
+{"id": "3651.png", "formula": "\\begin{align*} \\beta _ y = \\min \\left \\{ \\frac { 1 } { 2 L _ x } , \\frac { 1 } { 2 \\eta _ y L _ { x y } ^ 2 } \\right \\} . \\end{align*}"}
+{"id": "2845.png", "formula": "\\begin{align*} & \\tilde p _ x ( \\ell ) = \\frac { 1 } { \\theta _ n } \\int _ 0 ^ { \\theta _ n } e ^ { - 2 \\pi i \\ell t / \\theta _ n } \\bar p _ x ( t ) \\dd t , \\\\ & \\tilde q _ x ( \\ell ) = \\frac { 1 } { \\theta _ n } \\int _ 0 ^ { \\theta _ n } e ^ { - 2 \\pi i \\ell t / \\theta _ n } \\bar q _ x ( t ) \\dd t , \\ell \\in \\mathbb Z . \\end{align*}"}
+{"id": "2161.png", "formula": "\\begin{align*} \\Re \\mathcal { P } [ \\varrho ] = P ( a , b , X , Y ) + P ( b , a , Y , X ) , \\end{align*}"}
+{"id": "2156.png", "formula": "\\begin{align*} \\varrho _ { Z Z } ( L , L ) = \\overline { \\varrho _ { Z Z } ( L , L ) } = \\varrho _ { Z Z } ( N , N ) = \\overline { \\varrho _ { Z Z } ( N , N ) } = a ( 1 - \\tau ^ 2 ) ( \\gamma ^ 2 + \\alpha ^ 2 ) , \\end{align*}"}
+{"id": "2603.png", "formula": "\\begin{align*} R \\cdot R ( x , J x , J x , x ; u , J u ) = f ( p ) \\ , \\ , Q ^ c ( g , R ) R ( x , J x , J x , x ; u , J u ) , \\end{align*}"}
+{"id": "7109.png", "formula": "\\begin{align*} \\varphi _ { r , h } ( t , x ) : = g _ r ( t ) f _ h ( \\phi ( x ) ) . \\end{align*}"}
+{"id": "8811.png", "formula": "\\begin{align*} \\tilde { Y } _ t = e ^ { | z | J ^ \\perp t } \\tilde { Y } _ 0 + \\int _ 0 ^ t e ^ { | z | J ^ \\perp ( t - s ) } P ^ { - 1 } \\Pi _ { \\mathrm { k e r } A ^ \\perp } \\sigma d W _ s . \\end{align*}"}
+{"id": "9086.png", "formula": "\\begin{align*} \\chi _ i \\leqslant w _ i \\chi + k _ i , ( i = 1 , \\dots , n - 1 ) \\end{align*}"}
+{"id": "3968.png", "formula": "\\begin{align*} G ^ \\prime = \\begin{pmatrix} 1 & 1 & \\cdots & 1 \\\\ g _ { 2 1 } ^ \\prime & g _ { 2 2 } ^ \\prime & \\cdots & g _ { 2 n } ^ \\prime \\\\ \\vdots & \\vdots & & \\vdots \\\\ g _ { k 1 } ^ \\prime & g _ { k 2 } ^ \\prime & \\cdots & g _ { k n } ^ \\prime \\end{pmatrix} = \\begin{pmatrix} G _ 1 ^ \\prime & G _ 2 ^ \\prime & \\cdots & G _ n ^ \\prime \\end{pmatrix} . \\end{align*}"}
+{"id": "8444.png", "formula": "\\begin{align*} & \\| f \\| _ { L ^ p ( \\Omega ) } ^ p = \\int _ { \\Omega } | f | ^ { p } d x = \\int _ { \\Omega } | f | ^ 2 | f | ^ { p - 2 } d x \\leq c \\int _ { \\Omega } | f | ^ 2 d x = c \\| u \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "3211.png", "formula": "\\begin{align*} X _ { p , q } = \\{ f \\in C ( X , M _ p \\otimes M _ q ) \\mid f ( x _ 0 ) \\in M _ p \\otimes 1 _ q , \\ : f ( x _ 1 ) \\in 1 _ p \\otimes M _ q \\} . \\end{align*}"}
+{"id": "4745.png", "formula": "\\begin{align*} & \\| \\alpha _ * u _ * + \\lambda x _ v + ( 1 - \\lambda ) x _ w \\| ^ 2 = \\\\ & = \\alpha _ * ^ 2 \\| u _ * \\| ^ 2 + \\| \\lambda x _ v + ( 1 - \\lambda ) x _ w \\| ^ 2 + 2 \\alpha _ * \\langle u _ * , \\lambda x _ v + ( 1 - \\lambda ) x _ w \\rangle = \\\\ & = \\lambda \\| x _ v \\| ^ 2 + ( 1 - \\lambda ) \\| x _ w \\| ^ 2 \\end{align*}"}
+{"id": "6015.png", "formula": "\\begin{align*} c = \\mathop { \\inf } \\limits _ { \\gamma \\in \\Gamma } \\mathop { \\max } \\limits _ { t \\in [ 0 , 1 ] } I ( \\gamma ( t ) ) , \\end{align*}"}
+{"id": "8438.png", "formula": "\\begin{align*} & k ^ { - 1 } b ( k ) = \\int _ { - \\infty } ^ { \\infty } \\bar { u } _ y ( y ) \\left ( \\psi ^ - _ { 1 1 } - 1 \\right ) e ^ { - 2 i z y } d y + \\int _ { - \\infty } ^ { \\infty } \\bar u _ y ( y ) e ^ { - 2 i z y } d y . \\end{align*}"}
+{"id": "4297.png", "formula": "\\begin{align*} D _ \\mu & = \\partial _ \\mu - i g \\frac { \\lambda ^ a } { 2 } A ^ a _ \\mu \\\\ F ^ a _ { \\mu \\nu } & = \\partial _ \\mu A ^ a _ \\nu - \\partial _ \\nu A ^ a _ \\mu + g f ^ { a b c } A ^ b _ \\mu A ^ a _ \\nu \\end{align*}"}
+{"id": "2138.png", "formula": "\\begin{align*} \\varrho _ { Z Z } ( L , L ) = - a \\varrho _ w ^ 2 - b \\varrho _ z ^ 2 \\end{align*}"}
+{"id": "1717.png", "formula": "\\begin{align*} \\partial _ t U ( t , s ) x & = A ( t ) U ( t , s ) x , \\ t \\ge s , \\\\ \\partial _ s U ( t , s ) x & = - U ( t , s ) A ( s ) x , \\ t \\ge s , \\end{align*}"}
+{"id": "8278.png", "formula": "\\begin{align*} e _ { n + 1 } ^ i = e _ n ^ i + \\int _ { t _ n } ^ { t _ { n + 1 } } ( b ^ i ( X _ t ) - b ^ i ( \\tilde X _ n ) ) \\d t . \\end{align*}"}
+{"id": "3383.png", "formula": "\\begin{align*} \\tilde { p } ( n ) = p ( n + N ) = C p _ 1 ^ { - n } \\ 1 _ { [ - N , 0 ] } ( n ) + C p _ 2 ^ { n } \\ 1 _ { [ 1 , K ] } ( n ) . \\end{align*}"}
+{"id": "3810.png", "formula": "\\begin{align*} \\mathcal { B } _ q ( \\phi ) ( p ) : = \\displaystyle \\int _ { \\mathbb { R } } A _ q ( p , x ) \\phi ( x ) d x . \\end{align*}"}
+{"id": "1421.png", "formula": "\\begin{align*} \\P _ x ( \\rho > n ) = \\int _ { W ^ d } \\det ( f _ { n } ( y _ j - x _ i ) ) _ { i , j = 1 } ^ d d y _ 1 \\ldots d y _ d . \\end{align*}"}
+{"id": "5406.png", "formula": "\\begin{align*} \\begin{aligned} | F ^ { i j } ( \\nabla _ \\alpha u ) _ { i j } | \\leq \\ , & | \\nabla _ \\alpha f ^ { 1 / k } | + C \\sum _ { i , j = 1 } ^ n F ^ { i j } u _ { i j } + C | D u | \\sum _ { i = 1 } ^ n F ^ { i i } \\\\ \\leq \\ , & C f ^ { \\frac { 1 } { k } - \\frac { 1 } { 2 ( k - 1 ) } } + C b _ { n - 1 } \\sum _ { i = 1 } ^ n F ^ { i i } \\\\ \\leq \\ , & C b _ { n - 1 } ^ { 1 / 2 - 1 / k } + C b _ { n - 1 } \\sum _ { i = 1 } ^ n F ^ { i i } \\end{aligned} \\end{align*}"}
+{"id": "5160.png", "formula": "\\begin{align*} I _ { h _ 1 + h _ 2 } = I _ { \\vartheta h _ 2 } < \\vartheta I _ { h _ 2 } = \\left ( \\frac { h _ 1 } { h _ 2 } + 1 \\right ) I _ { h _ 2 } < I _ { h _ 1 } + I _ { h _ 2 } . \\end{align*}"}
+{"id": "7908.png", "formula": "\\begin{align*} H ( q , r ) = \\frac { \\hat W _ r ( 0 ) + \\hat W _ r ( 2 q ) - 2 \\hat W _ r ( q ) } { \\hat W _ r ( q ) } = u ( q r ) , \\end{align*}"}
+{"id": "8669.png", "formula": "\\begin{align*} \\begin{aligned} \\mathop { \\max } \\limits _ { \\left \\{ { { { \\bf { w } } _ k } } \\right \\} _ { k = 0 } ^ { K - 1 } } & \\ \\frac { 1 } { { K } } \\sum \\limits _ { k = 0 } ^ { K - 1 } { { { \\log } _ 2 } \\left ( { 1 + { { \\left | { { \\bf { e } } _ k ^ H { \\bf { V } } _ k ^ H { { \\bf { w } } _ k } } \\right | } ^ 2 } } \\right ) } \\\\ { \\rm { s . t . } } & \\ \\ \\sum \\limits _ { k = 0 } ^ { K - 1 } { { { \\left \\| { { { \\bf { V } } _ k ^ H } { { \\bf { w } } _ k } } \\right \\| } ^ 2 } } \\le K P . \\end{aligned} \\end{align*}"}
+{"id": "7115.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\mu \\circ ( ( f \\otimes \\eta ) \\otimes \\eta ) & = & \\mu \\circ ( i d _ M \\otimes \\eta ) \\circ ( ( f \\circ \\eta ) \\otimes i d _ I ) \\\\ & = & \\rho _ M \\circ ( ( f \\circ \\eta ) \\otimes i d _ I ) \\\\ & = & ( f \\circ \\eta ) \\circ \\rho _ I \\\\ & & \\mbox { ( b y n a t u r a l i t y o f $ \\rho $ ) } \\end{array} \\end{align*}"}
+{"id": "7711.png", "formula": "\\begin{align*} g _ s ( i ) : = \\frac { c ' } { 2 \\sqrt { m _ s } } 1 6 ^ { - i } m _ s n ^ { - \\alpha } , \\end{align*}"}
+{"id": "570.png", "formula": "\\begin{align*} \\frac { e ^ { p ( \\zeta - z ) } } { \\zeta - z } = - \\int _ { p - i e ^ { i \\alpha } [ 0 , + \\infty ) } e ^ { \\omega ( \\zeta - z ) } d \\omega , \\quad \\zeta \\in e ^ { - i \\alpha } [ 0 , + \\infty ) , \\ ; z \\in \\Delta . \\end{align*}"}
+{"id": "2982.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } \\frac { 1 } { \\# ( \\Lambda _ N ^ { \\vec { v } } ( 1 ) ) } \\sum _ { ( m , n ) \\in \\Lambda _ N ^ { \\vec { v } } ( 1 ) } \\max _ { 1 \\leq i < j \\leq k } d ( T ^ { ( m , n ) } x _ i , T ^ { ( m , n ) } x _ j ) = 0 . \\end{align*}"}
+{"id": "8030.png", "formula": "\\begin{align*} & C _ 2 ( t ) = \\sum _ { u : \\ , i _ u \\geq 3 \\atop { 2 \\leq d _ u < i _ u } } d _ u = \\sum _ { i \\geq 3 } \\sum _ { d = 2 } ^ { i - 1 } d x _ { i , d } \\\\ & = \\sum _ { i \\geq 3 } \\sum _ { d = 2 } ^ i d x _ { i , d } - \\sum _ { i \\geq 3 } i x _ { i , i } , \\\\ \\end{align*}"}
+{"id": "865.png", "formula": "\\begin{align*} x y ' - x ' y = n \\ , . \\end{align*}"}
+{"id": "669.png", "formula": "\\begin{align*} b ^ n _ l = - \\frac { 1 } { n } a ^ n _ l \\ , . \\end{align*}"}
+{"id": "2592.png", "formula": "\\begin{align*} 0 = T _ 1 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , v , J v ) = T _ 1 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , J x _ 5 , J x _ 6 ) + T _ 1 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 6 , - x _ 5 ) . \\end{align*}"}
+{"id": "6838.png", "formula": "\\begin{align*} \\left ( \\left ( I + \\mathbf { H } ( \\varphi ) \\right ) \\frac { \\partial y } { \\partial x } \\right ) \\left ( \\rho \\right ) = - \\mathbf { H } ( \\frac { \\partial \\varphi } { \\partial x } ) \\left ( 1 \\right ) \\left ( \\rho \\right ) - \\mathbf { H } ( \\frac { \\partial \\varphi } { \\partial x } ) \\left ( y \\right ) \\left ( \\rho \\right ) . \\end{align*}"}
+{"id": "746.png", "formula": "\\begin{align*} P _ s ( x , t ) = \\frac 1 { | x | ^ N } F _ s \\left ( \\frac { t } { | x | ^ { 2 s } } \\right ) \\end{align*}"}
+{"id": "5006.png", "formula": "\\begin{align*} x _ { m } < y _ { m } , \\ ; \\forall \\ ; m \\ ; \\in \\mathbb { N } \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\lim _ { x \\rightarrow \\infty } y _ { m } = \\infty . \\\\ \\end{align*}"}
+{"id": "813.png", "formula": "\\begin{align*} \\partial ^ \\square H = \\Delta _ \\Gamma V + V \\big | \\nabla _ \\Gamma \\nu \\big | ^ 2 . \\end{align*}"}
+{"id": "7721.png", "formula": "\\begin{align*} | M ^ { ( k ) } _ { k + 1 , k + 1 } | & = M _ { k + 1 , k + 1 } - M _ { k + 1 , [ k ] } M ^ { - 1 } _ { [ k ] , [ k ] } M _ { [ k ] , k + 1 } \\\\ & = | \\langle v _ k ( M ) , M _ { k + 1 , [ k + 1 ] } ^ \\top \\rangle | \\\\ & = \\| v _ k ( M ) \\| _ 2 \\cdot { \\rm d i s t } ( H , M _ { k + 1 , [ k + 1 ] } ^ \\top ) \\ge 1 \\cdot s _ { \\min } ( M _ { [ k + 1 ] , [ k + 1 ] } ) , \\end{align*}"}
+{"id": "4192.png", "formula": "\\begin{align*} I _ \\varepsilon ( t ( u ) u ) = \\max \\{ I _ \\varepsilon ( t u ) , t > 0 \\} . \\end{align*}"}
+{"id": "8971.png", "formula": "\\begin{align*} \\hat { k } _ d ( \\Delta ) = \\sum _ { \\Upsilon \\in \\mathcal { T } _ d ( \\Delta ) } x _ { \\Upsilon } \\ , { \\mathbf { t } _ { d - 1 } ( \\Upsilon ) ^ 2 } . \\end{align*}"}
+{"id": "8571.png", "formula": "\\begin{align*} k ( t ) = t ^ { \\beta - 1 } \\ , E _ { \\alpha , \\beta } ( - t ^ \\alpha ) , \\end{align*}"}
+{"id": "1557.png", "formula": "\\begin{align*} F ( \\mathbf { v } ) = \\begin{bmatrix} D & 0 & \\ell \\sin \\Psi & - \\cos \\Psi \\\\ 0 & D & - \\ell \\cos \\Psi & - \\sin \\Psi \\\\ \\frac { - \\ell \\sin \\Psi } { R ^ 2 } & - \\kappa \\ell & D + \\frac { \\ell \\cos \\Psi } { R } & \\frac { \\sin \\Psi } { R } - \\kappa U \\\\ D ^ 0 _ { - 1 } & 0 & 0 & 0 \\\\ D ^ 0 _ { 1 } & 0 & 0 & 0 \\\\ 0 & 0 & D ^ 0 _ { - 1 } & 0 \\\\ 0 & 0 & D ^ 0 _ { 1 } & 0 \\end{bmatrix} \\mathbf { v } . \\end{align*}"}
+{"id": "5454.png", "formula": "\\begin{align*} \\Omega _ X ^ { [ p ] } : = j _ * \\Omega ^ p _ { X ^ { s m } } = ( \\Omega ^ p _ X ) ^ { \\vee \\vee } \\end{align*}"}
+{"id": "6439.png", "formula": "\\begin{align*} \\begin{aligned} & T ^ { \\star } ( k _ 1 , \\ldots , k _ n ; ( \\alpha , \\beta ) ) - T ^ { \\star } ( k _ n , k _ 1 , \\ldots , k _ { n - 1 } ; ( \\alpha , \\beta ) ) \\\\ = & ( k _ n - 1 ) Z ( n | k - n + 1 ; ( \\alpha , \\beta ) ) + Z ( n + 1 | k - n ; ( \\alpha , \\beta ) ) \\\\ & - \\sum _ { j = 0 } ^ { k _ n - 2 } Z ^ { \\star } _ { I } ( j + 1 , k _ 1 , \\ldots , k _ { n - 1 } , k _ n - j ; ( \\alpha , \\beta ) ) \\end{aligned} \\end{align*}"}
+{"id": "84.png", "formula": "\\begin{align*} \\varepsilon ^ { w s _ \\alpha ( \\mu - \\alpha ^ \\vee ) } = s _ { w \\alpha } \\varepsilon ^ { - w \\alpha ^ \\vee } x < x \\end{align*}"}
+{"id": "8827.png", "formula": "\\begin{align*} \\lambda _ { + , 1 } = 2 \\sqrt { \\delta ( 1 - \\delta ) } , E _ { + , 1 } = \\left \\{ \\begin{pmatrix} - \\bar { a } _ 1 \\sqrt { \\frac { \\delta } { \\delta - 1 } } \\\\ \\bar { b } _ 1 \\sqrt { \\frac { \\delta } { \\delta - 1 } } \\\\ 0 \\\\ 1 \\end{pmatrix} , \\begin{pmatrix} \\bar { b } _ 1 \\sqrt { \\frac { \\delta } { \\delta - 1 } } \\\\ \\bar { a } _ 1 \\sqrt { \\frac { \\delta } { \\delta - 1 } } \\\\ 1 \\\\ 0 \\end{pmatrix} \\right \\} , \\end{align*}"}
+{"id": "6583.png", "formula": "\\begin{align*} \\sup _ { t < T _ { \\rm m a x } } \\int _ { \\Omega } ( n \\log n ( \\cdot , t ) + e ^ { - 1 } ) \\le \\sum _ { i = 1 } ^ { N } \\sup _ { t < T _ { \\rm m a x } } \\int _ { \\Omega \\cap B _ { \\frac { \\delta } { 2 } } ( q _ { i } ) } ( n \\log n ( \\cdot , t ) + e ^ { - 1 } ) \\le C . \\end{align*}"}
+{"id": "6772.png", "formula": "\\begin{align*} B ( x , t ) = \\sum _ { n = 0 } ^ { \\infty } b _ { n } ( x ) L _ { n } ( x - t ) e ^ { - \\frac { x - t } { 2 } } \\end{align*}"}
+{"id": "7477.png", "formula": "\\begin{align*} \\rho ^ { n + 2 } \\int _ { B _ { \\rho / 2 } ( y ) } | \\nabla u | ^ 2 \\ , d x & \\leq C \\delta \\rho ^ { n + 2 } \\int _ { B _ { \\rho } ( y ) } | \\nabla u | ^ 2 \\ , d x + C \\delta ^ { - 2 - \\frac { 3 n } { 2 } } \\left ( \\int _ { B _ { \\rho } ( y ) } | u | \\ , d x \\right ) ^ 2 \\\\ & \\leq C \\delta \\rho ^ { n + 2 } \\int _ { B _ { \\rho } ( y ) } | \\nabla u | ^ 2 \\ , d x + C \\delta ^ { - 2 - \\frac { 3 n } { 2 } } \\left ( \\int _ { B _ { 1 } } | u | \\ , d x \\right ) ^ 2 . \\end{align*}"}
+{"id": "5946.png", "formula": "\\begin{align*} F ( t ) = \\int _ { \\mathcal { O } } f _ 1 ( t , x ) ^ { \\frac { \\alpha } { \\alpha - 1 } } d x . \\end{align*}"}
+{"id": "171.png", "formula": "\\begin{align*} ( f _ i ^ { p ^ m } ) \\cdot T _ i ^ { \\pm \\frac { 1 } { p ^ n } } - T _ i ^ { \\pm \\frac { 1 } { p ^ n } } = ( \\zeta _ { p ^ { n - m } } ^ { \\pm 1 } - 1 ) T _ { i } ^ { \\pm \\frac { 1 } { p ^ n } } , \\end{align*}"}
+{"id": "5838.png", "formula": "\\begin{align*} { f _ { \\bar i } } \\left ( { { { \\bf { x } } _ b } , t + \\Delta t } \\right ) = f _ i ^ * \\left ( { { { \\bf { x } } _ b } , t } \\right ) , \\end{align*}"}
+{"id": "7512.png", "formula": "\\begin{align*} & \\sum _ { j = 0 } ^ { \\infty } r _ { j } ^ { 2 - n } \\int _ { B _ { r _ { j + 1 } } \\setminus B _ { r _ { j + 2 } } } | \\nabla u | ^ 2 \\d x \\\\ & \\leq C \\int _ { B _ { 1 } } r ^ { 2 - n } u _ r ^ 2 \\d x + C \\varepsilon ^ 2 \\int _ { B _ { 1 } } r ^ { 3 - n } | \\nabla u | ^ 2 \\d x \\varepsilon \\leq \\varepsilon _ 0 . \\end{align*}"}
+{"id": "3178.png", "formula": "\\begin{align*} ( l _ 1 , l _ 2 ) = \\left \\{ \\begin{array} { l l l } ( 1 , 1 ) , & \\hbox { i f $ i = 1 $ , } \\\\ ( 1 , - 1 ) , & \\hbox { i f $ i = 2 $ , } \\\\ ( - 1 , 1 ) , & \\hbox { i f $ i = 3 $ , } \\\\ ( - 1 , - 1 ) , & \\hbox { i f $ i = 4 $ . } \\end{array} \\right . \\end{align*}"}
+{"id": "1905.png", "formula": "\\begin{align*} R _ { n + 1 } ( s ) = \\beta R _ n ( s ) - { \\frac { f '' \\bigl ( \\zeta _ n ( s ) \\bigr ) } { 2 } } R _ n ^ 2 ( s ) \\end{align*}"}
+{"id": "2570.png", "formula": "\\begin{align*} \\frac { l _ { k _ { 1 } } } { \\rho ^ { k _ { 1 } } N _ { k _ { 1 } } } - \\frac { p ^ { l _ { k } } b _ { k } } { N _ { k } \\rho ^ { k + r l _ { k } } } = \\frac { p ^ { l _ { t } } b _ { t } } { N _ { t } \\rho ^ { t + r l _ { t } } } . \\end{align*}"}
+{"id": "875.png", "formula": "\\begin{align*} x x ' + x y ' + y y ' = - n \\ , . \\end{align*}"}
+{"id": "9280.png", "formula": "\\begin{align*} C _ m ( K ) = \\int _ { K } ( \\Delta \\omega ^ * ( q , K , \\Omega ) ) ^ m \\wedge \\beta _ n ^ { n - m } . \\end{align*}"}
+{"id": "8341.png", "formula": "\\begin{align*} u ( x , t ) = \\lim _ { k \\rightarrow 0 } ( k ^ { - 1 } \\varphi ( x , t ; k ) ) _ { 1 2 } . \\end{align*}"}
+{"id": "365.png", "formula": "\\begin{align*} D y + \\gamma y = \\frac { ( n + 4 ) ( - 2 n + 7 ) } { 3 ( 2 n - 1 ) } \\ 1 . \\end{align*}"}
+{"id": "5583.png", "formula": "\\begin{align*} A ( y _ 1 x ) - A ( y _ 1 x ' ) = A ( 1 0 ^ k 1 . . . ) - A ( 1 0 ^ \\infty ) = d _ k - d \\ . \\end{align*}"}
+{"id": "8217.png", "formula": "\\begin{align*} x _ { i , 1 , k } & = k - i , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 1 , k } & = 6 m + i - k + 1 , \\\\ x _ { i , 2 , k } & = 2 m + i , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 2 , k } & = 4 m - i + 1 , \\\\ x _ { i , 3 , k } & = m - i + k , \\mbox { a n d } \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 3 , k } & = 5 m + i - k + 1 . \\end{align*}"}
+{"id": "9122.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { j + 1 } \\chi _ i - r ( j + 1 ) < \\chi = \\sum \\limits _ { i = 1 } ^ n \\chi _ i - ( n - 1 ) r \\end{align*}"}
+{"id": "2522.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbf { x } ( \\alpha ) \\triangleq \\mathbf { x } + \\alpha \\Delta \\mathbf { x } , \\mathbf { y } ( \\alpha ) \\triangleq \\mathbf { y } + \\alpha \\Delta \\mathbf { y } , \\mathbf { s } ( \\alpha ) \\triangleq \\mathbf { s } + \\alpha \\Delta \\mathbf { s } . \\end{aligned} \\end{align*}"}
+{"id": "5967.png", "formula": "\\begin{align*} K _ M : = \\bigg \\{ f \\in L ^ p ( [ 0 , T ] , H ) : \\int _ 0 ^ T \\Vert f ( t ) \\Vert _ V ^ p d t \\leq M \\bigg \\} . \\end{align*}"}
+{"id": "8591.png", "formula": "\\begin{align*} \\sum _ { 1 \\le i < j < k \\le M } \\left | \\det \\ ! \\ ! \\begin{pmatrix} x _ i & x _ j & x _ k \\\\ y _ i & y _ j & y _ k \\\\ z _ i & z _ j & z _ k \\end{pmatrix} \\right | \\sum _ { i = 1 } ^ M | z _ i | \\le \\ ! \\ ! \\ ! \\sum _ { 1 \\le i < j \\le M } \\left | \\det \\ ! \\ ! \\begin{pmatrix} y _ i & y _ j \\\\ z _ i & z _ j \\end{pmatrix} \\right | \\sum _ { 1 \\le i < j \\le M } \\left | \\det \\ ! \\ ! \\begin{pmatrix} x _ i & x _ j \\\\ z _ i & z _ j \\end{pmatrix} \\right | \\end{align*}"}
+{"id": "342.png", "formula": "\\begin{align*} \\vert S _ k \\vert \\leq \\sum _ { 2 ^ k / c _ k < p \\leq 2 ^ k } \\frac { 2 ^ k } { p } & = 2 ^ k ( \\log \\log 2 ^ k - \\log \\log ( 2 ^ k / c _ k ) + O ( 1 / \\log ( 2 ^ k / c _ k ) ) ) \\\\ & = 2 ^ k \\log \\frac { k \\log 2 } { k \\log 2 - \\log c _ k } + O ( 2 ^ k / k ) \\\\ & = 2 ^ k \\log \\left ( 1 + \\frac { \\log c _ k } { k \\log 2 - \\log c _ k } \\right ) + O ( 2 ^ k / k ) \\\\ & = \\frac { 2 ^ k \\log c _ k } { k \\log 2 } + o ( ( 2 ^ k \\log c _ k ) / k ) \\end{align*}"}
+{"id": "3003.png", "formula": "\\begin{align*} \\tilde { \\varphi } ( t , x ) = \\varphi ( t , x , \\kappa _ t ( \\cdot ) ) , ( t , x ) \\in \\mathbb [ \\tau , \\vartheta ] \\times \\mathbb R ^ n , \\kappa ( \\cdot ) \\in \\Lambda _ 0 ( \\tau , z , w ( \\cdot ) ) , \\end{align*}"}
+{"id": "1992.png", "formula": "\\begin{align*} \\sigma ( x , T ( x ) ) = f ( x - T ( x ) ) + f ( - x - T ( x ) ) = 0 . \\end{align*}"}
+{"id": "8614.png", "formula": "\\begin{align*} h _ 2 ( t ) = \\frac { | A \\oplus _ 2 ( \\sqrt { t } B ) | ^ 2 } { | P _ { u ^ \\bot } ( A \\oplus _ 2 ( \\sqrt { t } B ) ) | ^ 2 _ { n - 1 } } \\end{align*}"}
+{"id": "4020.png", "formula": "\\begin{align*} \\varphi ^ L _ { \\sigma , s ' } ( D ) = \\varphi ^ L _ { \\sigma , s ' } ( C _ i ) \\varphi ^ L _ { \\sigma , s ' } ( E ) = \\emptyset . \\end{align*}"}
+{"id": "8103.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } \\frac { d s } { \\left ( \\int _ { 0 } ^ { s } F ( t ) d t \\right ) ^ { \\frac { p } { 2 p - q + 1 } } } < \\infty \\quad \\mbox { a n d } \\int _ { 1 } ^ { \\infty } \\frac { s \\ d s } { \\left ( \\int _ { 0 } ^ { s } F ( t ) d t \\right ) ^ { \\frac { p } { 2 p - q + 1 } } } = \\infty . \\end{align*}"}
+{"id": "4002.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t u - \\nabla \\cdot ( A \\nabla u ) = & \\ F , & ( 0 , T ] \\times \\Omega , \\\\ u ( 0 , \\cdot ) = & \\ u _ 0 , \\ , & \\Omega , \\\\ u = & \\ g _ D ^ { } , & ( 0 , T ] \\times \\Gamma _ D ^ { } , \\\\ \\mathbf { n } \\cdot A \\nabla u = & \\ g _ N ^ { } , & ( 0 , T ] \\times \\Gamma _ N ^ { } , \\end{aligned} \\end{align*}"}
+{"id": "6152.png", "formula": "\\begin{align*} X _ { t } ^ 1 & = X _ { t _ k } ^ 1 + \\mu ( t _ k , \\mathcal { X } ^ { 1 } ) ( t - t _ k ) + \\sigma ( t _ k , \\mathcal { X } ^ { 1 } ) ( W _ t - W _ { t _ k } ) \\\\ & = X _ { t _ k } ^ 1 + \\int _ { t _ k } ^ { t } \\mu ( { \\delta _ 1 } ( s ) , \\mathcal { X } ^ { 1 } ) \\ , d s + \\int _ { t _ k } ^ { t } \\sigma ( { \\delta _ 1 } ( s ) , \\mathcal { X } ^ { 1 } ) \\ , d W _ s . \\end{align*}"}
+{"id": "3042.png", "formula": "\\begin{align*} X ^ { \\prime } \\left ( 0 \\right ) = Z \\left ( x , 0 \\right ) k ^ { \\prime } \\left ( 0 \\right ) = Z \\left ( 0 , x ^ { b } \\right ) \\end{align*}"}
+{"id": "2074.png", "formula": "\\begin{align*} \\mathcal { I } - d _ - ( x , A ) + \\mathcal { I } - d _ - ( x , B ) \\leq \\mathcal { I } - \\liminf \\{ a _ n + b _ n \\} & \\leq \\mathcal { I } - \\liminf \\{ 1 + p _ n \\} \\\\ & = 1 + \\mathcal { I } - \\liminf p _ n \\\\ & = 1 + \\mathcal { I } - d _ - ( x , A \\cap B ) \\end{align*}"}
+{"id": "3282.png", "formula": "\\begin{align*} \\mathbb E \\left [ \\langle \\tilde Z _ n ^ N ( i ) , \\tilde Z _ n ^ N ( j ) \\rangle _ { \\mathcal H } \\right ] = & \\mathbb E \\left [ \\mathbb E \\left [ \\langle \\tilde Z _ n ^ N ( i ) , \\tilde Z _ n ^ N ( j ) \\rangle _ { \\mathcal H } | \\mathcal F _ { ( i - 1 ) \\Delta _ n } \\right ] \\right ] \\\\ = & \\mathbb E \\left [ \\langle \\mathbb E \\left [ \\tilde Z _ n ^ N ( i ) | \\mathcal F _ { ( i - 1 ) \\Delta _ n } \\right ] , \\tilde Z _ n ^ N ( j ) \\rangle _ { \\mathcal H } \\right ] = 0 . \\end{align*}"}
+{"id": "6340.png", "formula": "\\begin{align*} f ( x ) = \\frac { 2 \\theta } { \\operatorname { B } ( a , b ) \\ , x ^ 3 } \\exp \\left ( - \\frac { a \\theta } { x ^ 2 } \\right ) \\left [ 1 - \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) \\right ] ^ { b - 1 } . \\end{align*}"}
+{"id": "7045.png", "formula": "\\begin{align*} \\aligned A _ { 1 , 1 } & \\ , : = \\ , \\big ( - F _ { 4 , 1 } + 1 \\big ) \\ , T _ 1 - 2 \\ , T _ 2 , \\\\ B _ 2 & \\ , : = \\ , - \\ , \\tfrac { 1 } { 6 } \\ , F _ { 5 , 0 } \\ , T _ 1 - \\tfrac { 1 } { 6 } \\ , F _ { 4 , 1 } \\ , T _ 2 - \\tfrac { 2 } { 3 } \\ , A _ { 2 , 1 } . \\endaligned \\end{align*}"}
+{"id": "7009.png", "formula": "\\begin{align*} u \\ , = \\ , F ( x , y ) \\ \\ \\ \\ \\ \\ \\ \\ { \\scriptstyle { ( F ( 0 , 0 ) \\ , = \\ , 0 ) } } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ v \\ , = \\ , G ( r , s ) \\ \\ \\ \\ \\ \\ \\ \\ { \\scriptstyle { ( 0 \\ , = \\ , G ( 0 , 0 ) ) } } , \\end{align*}"}
+{"id": "7191.png", "formula": "\\begin{align*} e ^ { - t \\Lambda _ g } \\textbf { \\textit { V } } ( x ^ { \\prime } ) = \\int _ { \\partial \\Omega } \\textbf { \\textit { K } } ( t , x ^ { \\prime } , y ^ { \\prime } ) \\textbf { \\textit { V } } ( x ^ { \\prime } ) \\ , d S , \\textbf { \\textit { V } } \\in ( L ^ { 2 } ( \\partial \\Omega ) ) ^ { n + 1 } . \\end{align*}"}
+{"id": "3485.png", "formula": "\\begin{align*} \\ln \\left \\lbrace \\prod _ { l \\neq j } \\frac { | \\sin \\pi ( \\theta - \\theta _ l ) | } { | \\sin \\pi ( \\theta _ j - \\theta _ l ) | } \\right \\rbrace = \\sum _ { l \\neq j } \\ln | \\sin \\pi ( \\theta - \\theta _ l ) | - \\sum _ { l \\neq j } \\ln | \\sin \\pi ( \\theta _ j - \\theta _ l ) | < ( 2 s q _ { n - n _ 0 } - 1 ) \\epsilon . \\end{align*}"}
+{"id": "6878.png", "formula": "\\begin{align*} \\tilde z = \\Psi _ 2 ^ { - 1 } ( \\hat z ) = \\tilde z _ 3 + \\frac { ( \\tilde z _ 3 - \\tilde z _ 1 ) ( \\tilde z _ 2 - \\tilde z _ 3 ) ( \\hat z _ 2 - \\hat z _ 1 ) ( \\hat z - \\hat z _ 3 ) } { ( \\tilde z _ 2 - \\tilde z _ 1 ) ( \\hat z _ 2 - \\hat z _ 3 ) ( \\hat z - \\hat z _ 1 ) - ( \\tilde z _ 2 - \\tilde z _ 3 ) ( \\hat z _ 2 - \\hat z _ 1 ) ( \\hat z - \\hat z _ 3 ) } \\ , . \\end{align*}"}
+{"id": "2239.png", "formula": "\\begin{align*} \\begin{cases} H _ 0 \\psi \\ , \\ , = - \\triangle \\psi , \\\\ H _ N \\psi = - \\triangle \\psi + \\displaystyle \\sum _ { j = 1 } ^ N v _ j \\langle \\psi , \\delta _ j \\rangle \\delta _ j , N \\geq 1 , \\end{cases} \\end{align*}"}
+{"id": "8463.png", "formula": "\\begin{align*} Q _ { j , + } ( x ; k ) = ( I + J ) Q _ { j , - } ( x ; k ) + D _ j , k \\in \\mathbb { R } \\cup i \\mathbb { R } , \\end{align*}"}
+{"id": "3571.png", "formula": "\\begin{align*} \\varphi ( - t ) ^ 2 G ( - t ) H ( t ) - \\varphi ( t ) ^ 2 G ( t ) H ( t ) = 2 t \\varphi ( t ^ 4 ) ^ 2 H ( t ^ 4 ) H ( t ) \\end{align*}"}
+{"id": "4587.png", "formula": "\\begin{align*} | \\langle B ^ { \\circ } S _ { T - t } ^ { \\circ } x ^ { \\circ } , u \\rangle | = | \\langle x ^ { \\circ } , S _ { T - t } B u \\rangle | \\leq C p _ { \\gamma _ U } ( u ) \\end{align*}"}
+{"id": "5287.png", "formula": "\\begin{align*} \\lim _ { z \\downarrow 0 } \\int _ 0 ^ \\infty w _ z ( \\lambda ) \\lambda ^ { 4 ( 1 - s + w ) } \\frac { d \\lambda } { \\lambda } \\int _ 0 ^ 1 d u \\int _ 0 ^ \\infty \\frac { d t } { t ^ 3 } \\phi ( n _ { u } ^ t a _ { \\frac { 1 } { t } } ) \\sum _ { m = 1 } ^ \\infty \\hat { F } _ { \\pm } ( \\lambda a _ { \\frac { 1 } { t } } \\cdot ( 0 , 0 , 0 , m ) ) \\end{align*}"}
+{"id": "5676.png", "formula": "\\begin{align*} \\ker \\left ( H _ { q } ( O _ { n - 1 , n - 1 } ) \\xrightarrow { i _ { * } } H _ { q } ( O _ { n , n } ) ) \\right ) & = { } _ { m } E ^ { 2 } _ { 1 , q } \\\\ & \\cong { } _ { m } E ^ { \\infty } _ { 1 , q } \\\\ & = 0 \\end{align*}"}
+{"id": "4833.png", "formula": "\\begin{align*} \\| \\mathcal { D } ( a ( \\tau ; \\mu ) ) - \\mu ^ 0 \\| _ L ^ * \\leqslant 2 \\frac { M ' } { M _ 1 ( \\tau ) ^ { 2 m _ 0 ' } } + \\tilde g _ { M _ 1 ( \\tau ) } ( \\tau ) = : { \\hat g } _ { M ' } ( \\tau ) . \\end{align*}"}
+{"id": "5134.png", "formula": "\\begin{align*} h = H [ b ] = 2 \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi _ { \\infty } ) _ { + } \\frac { G } { r ^ { 2 } } \\dd x = 0 . \\end{align*}"}
+{"id": "3602.png", "formula": "\\begin{align*} \\sum _ { \\substack { U \\subset F _ s : \\\\ U \\cap F _ { \\ell + 1 } = T } } \\alpha ^ s _ U . \\end{align*}"}
+{"id": "4405.png", "formula": "\\begin{align*} \\sum _ { n < x ^ 2 } \\frac { \\Lambda _ x ( n ) ^ 2 } { n ^ { 2 \\sigma } } & < \\frac { 1 } { ( 2 \\sigma - 1 ) ^ 2 } + \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { ( 2 \\sigma + 2 n ) ^ 2 } + \\sum _ \\gamma \\frac { 1 } { | 2 \\sigma - 1 / 2 + i \\gamma | ^ 2 } \\\\ & \\leq \\frac { 1 } { 4 ( \\sigma - 1 / 2 ) ^ 2 } + \\frac { \\pi ^ 2 } { 8 } - 1 + \\sum _ \\gamma \\frac { 1 } { \\gamma ^ 2 } , \\end{align*}"}
+{"id": "12.png", "formula": "\\begin{align*} \\beta = \\alpha - \\alpha _ s a _ s = 1 \\end{align*}"}
+{"id": "3652.png", "formula": "\\begin{align*} \\Psi ^ k & = \\frac { 1 } { \\eta _ x } \\| x ^ k - x ^ * \\| ^ 2 + \\frac { 1 } { \\eta _ y } \\| y ^ k - y ^ * \\| ^ 2 + \\frac { 2 } { \\sigma _ x } B _ f ( x _ f ^ k , x ^ * ) + \\frac { 2 } { \\sigma _ y } B _ g ( y _ f ^ k , y ^ * ) \\\\ & + \\frac { 1 } { 4 \\eta _ y } \\| y ^ k - y ^ { k - 1 } \\| ^ 2 - 2 \\langle y ^ k - y ^ { k - 1 } , A ( x ^ k - x ^ * ) \\rangle . \\end{align*}"}
+{"id": "727.png", "formula": "\\begin{align*} p ^ { \\varepsilon } ( x , t ) = \\mathrm { e x p } \\Big ( - \\frac { 1 } { \\varepsilon } y ( t ) h ( x ) \\Big ) q ^ { \\varepsilon } ( x , t ) , \\end{align*}"}
+{"id": "7596.png", "formula": "\\begin{align*} F ^ * x = ( x _ \\epsilon : \\epsilon \\ne \\vec 0 ) \\end{align*}"}
+{"id": "6004.png", "formula": "\\begin{align*} & f _ { 1 } ( t ) = t ^ { 2 } + \\frac { 1 } { 2 } t ^ { 4 } - ( 1 + b ) t ^ { 2 p } , \\\\ & f _ { 1 } ' ( t ) = 2 t + 2 t ^ { 3 } - 2 p ( 1 + b ) t ^ { 2 p - 1 } = 2 t \\Big ( 1 + t ^ { 2 } - 2 p ( 1 + b ) t ^ { 2 p - 2 } \\Big ) , \\end{align*}"}
+{"id": "747.png", "formula": "\\begin{align*} | \\partial _ t ^ { 1 - \\frac 1 { 2 s } } F _ s ( t ) | & \\leq \\int \\frac { | F _ s ( r ) - F _ s ( t ) | } { | r - t | ^ { 2 - \\frac 1 { 2 s } } } d r = \\int _ { | r | \\leq | t | / 2 } \\frac { | F _ s ( r ) - F _ s ( t ) | } { | r - t | ^ { 2 - \\frac 1 { 2 s } } } d r \\\\ & \\quad \\quad \\quad + \\int _ { | t | / 2 < | r | \\leq 2 | t | } \\frac { | F _ s ( r ) - F _ s ( t ) | } { | r - t | ^ { 2 - \\frac 1 { 2 s } } } d r + \\int _ { | r | > 2 | t | } \\frac { | F _ s ( r ) - F _ s ( t ) | } { | r - t | ^ { 2 - \\frac 1 { 2 s } } } d r \\\\ & = I _ 1 + I _ 2 + I _ 3 . \\end{align*}"}
+{"id": "3509.png", "formula": "\\begin{align*} F _ { n - 1 } - { ( n - 1 ) } ^ { p } \\ , \\ , \\ , \\ , \\ , \\ & = \\ \\sum _ { i = 1 } ^ { n - 1 } \\left ( i ^ { p } \\ - \\ 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } i ^ { p - 2 j - 1 } \\right ) \\cdot F _ { n - 1 - i } , \\\\ F _ n - n ^ { p } \\ , \\ , \\ , \\ , \\ , \\ & = \\ \\sum _ { i = 1 } ^ { n } \\left ( i ^ { p } \\ - \\ 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } i ^ { p - 2 j - 1 } \\right ) \\cdot F _ { n - i } . \\end{align*}"}
+{"id": "3932.png", "formula": "\\begin{align*} \\nabla \\mathcal { N } ( x ) = W _ { L } \\cdot \\sigma ' ( N ^ { ( L - 1 ) } ( x ) ) \\cdot W _ { L - 1 } \\cdot \\ldots \\cdot \\sigma ' ( N ^ { ( 1 ) } ( x ) ) \\cdot W _ { 1 } , \\end{align*}"}
+{"id": "8950.png", "formula": "\\begin{gather*} G ^ { ( \\delta ) } ( t ) = G _ { 1 , k } ^ { ( \\delta ) } ( t ) + G _ { 2 , k } ^ { ( \\delta ) } ( t ) , \\\\ G _ { 1 , k } ^ { ( \\delta ) } ( t ) : = { V ^ { \\delta } _ 1 ( t ) } \\int _ t ^ { \\eta _ k } \\frac { g ( x ) } { V _ 1 ( x ) } \\Bigl ( \\int _ { a ( x ) } ^ t v _ 1 ^ { - p ' } \\Bigr ) ^ { 1 - \\delta } d x , \\\\ G _ { 2 , k } ^ { ( \\delta ) } ( t ) : = { V ^ { \\delta } _ 1 ( t ) } \\int _ { \\eta _ k } ^ { a ^ { - 1 } ( t ) } \\frac { g ( x ) } { V _ 1 ( x ) } \\Bigl ( \\int _ { a ( x ) } ^ t v _ 1 ^ { - p ' } \\Bigr ) ^ { 1 - \\delta } d x . \\end{gather*}"}
+{"id": "1220.png", "formula": "\\begin{align*} \\delta _ x ( f ) : = \\sup _ { \\eta , \\xi : \\ \\eta _ { x ^ c } = \\xi _ { x ^ c } } \\abs { f ( \\eta ) - f ( \\xi ) } \\end{align*}"}
+{"id": "9217.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\eta ( t ) - \\eta _ a ( t ) \\| _ 2 \\le & \\ , \\| \\eta ( t ) - z ( t ) \\| _ 1 + \\gamma \\bar { k } _ 3 ( L _ r , \\delta ) \\\\ = & \\ , f ( t ) + \\gamma \\bar { k } _ 3 ( L _ r , \\delta ) . \\end{aligned} \\end{align*}"}
+{"id": "4942.png", "formula": "\\begin{align*} \\sum _ { \\gamma } f _ { \\frac { 1 } { 2 } } ( t - \\gamma ) \\leq \\frac { \\widehat { h } ( 0 ) } { 2 \\pi } \\log t + \\frac { 1 } { \\pi } \\sum _ { n = 2 } ^ \\infty \\frac { \\Lambda ( n ) } { \\sqrt { n } } \\left | \\widehat { h } \\left ( \\frac { \\log n } { 2 \\pi } \\right ) \\right | . \\end{align*}"}
+{"id": "6334.png", "formula": "\\begin{align*} D _ x ^ { ( 1 ) } x _ n + D _ x ^ { ( 1 ) } x _ { n - 1 } x ^ { p ^ { s _ i } } + D _ x ^ { ( 1 ) } x _ { n - 2 } x ^ { 2 p ^ { s _ i } } + \\cdots + x ^ { ( n - 1 ) p ^ { s _ i } } = 0 . \\end{align*}"}
+{"id": "6372.png", "formula": "\\begin{align*} \\delta _ 1 ( X ) = 2 \\mu F ( \\mu ) - 2 J ( \\mu ) \\delta _ 2 ( X ) = \\mu - 2 J ( m ) , \\end{align*}"}
+{"id": "2595.png", "formula": "\\begin{align*} Q ^ c ( g , R ) ( \\pi ; \\bar { \\pi } ) = Q ( g , R ) ( \\pi ; \\bar { \\pi } ) + Q ( g , R ) ( \\pi ; J \\bar { \\pi } ) , \\end{align*}"}
+{"id": "8267.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { t _ 1 } ^ { t _ 2 } \\sum _ { i = m } ^ { n } \\sum _ { j = n + 1 } ^ { \\infty } g _ i V _ { i , j } \\psi _ i ( s ) \\psi _ j ( s ) d s = 0 \\end{align*}"}
+{"id": "2035.png", "formula": "\\begin{align*} a \\mp _ 1 b & = \\phantom { \\pm _ 2 } x \\\\ b \\pm _ 1 a & = \\pm _ 2 1 \\end{align*}"}
+{"id": "864.png", "formula": "\\begin{align*} x y ' - x ' y = 0 \\ , . \\end{align*}"}
+{"id": "1381.png", "formula": "\\begin{align*} g ( x ) = \\prod _ { j = 1 } ^ d \\lambda _ j \\int _ { x _ d } ^ \\infty d b _ d \\int _ { x _ { d - 1 } } ^ { b _ { d } } d b _ { d - 1 } \\cdots \\int _ { x _ 1 } ^ { b _ { 2 } } d b _ 1 g ( b _ 1 , \\ldots , b _ d ) . \\end{align*}"}
+{"id": "8987.png", "formula": "\\begin{align*} \\hat { k } _ d ( \\Delta ) = \\prod _ { q , i } x _ { q , i } ^ { e ( q , i ) } \\prod _ { \\rho \\in \\Gamma _ { d - 1 } } D _ { m ( \\rho ) , k ( \\rho ) } \\end{align*}"}
+{"id": "5046.png", "formula": "\\begin{align*} u = u _ { \\infty } , B ( x , t ) = U ( x - u _ { \\infty } t ) . \\end{align*}"}
+{"id": "6722.png", "formula": "\\begin{align*} \\Phi _ s ( z ) = \\begin{cases} \\frac { \\Gamma ( 1 / s , ( - z ) ^ s ) } { 2 \\Gamma ( 1 / s ) } , & z \\leq 0 , \\\\ 1 - \\frac { \\Gamma ( 1 / s , z ^ s ) } { 2 \\Gamma ( 1 / s ) } , & z > 0 . \\end{cases} \\end{align*}"}
+{"id": "3241.png", "formula": "\\begin{align*} d Y _ t = \\kappa \\partial _ { x x } Y _ t d t + Q ^ { \\frac 1 2 } d W _ t . \\end{align*}"}
+{"id": "9013.png", "formula": "\\begin{align*} \\frac { ( - 1 ) ^ { | \\pi ( \\tau ) | } x ^ \\tau \\langle V , z _ \\tau \\rangle } { x _ { j _ { s + 1 } } U } & = - \\prod _ { r = 1 } ^ { s } D _ { j _ { r } - 1 } ~ - ~ \\sum _ { q = 1 } ^ { s } \\left ( \\prod _ { r = 2 } ^ q D _ { j _ { r - 1 } } \\right ) x _ { j _ q } \\left ( \\prod _ { r = q + 1 } ^ { s } D _ { j _ { r } - 1 } \\right ) ~ + ~ \\prod _ { r = 2 } ^ { s + 1 } D _ { j _ { r - 1 } } \\end{align*}"}
+{"id": "8998.png", "formula": "\\begin{align*} U = ( - 1 ) ^ { p ( \\eta , j ) - 1 } ( - 1 ) ^ { | \\pi ( \\eta \\cup j ) | } \\prod _ { r \\notin [ m + 1 , M ] } { U ^ { \\eta \\cup j } _ { r } } . \\end{align*}"}
+{"id": "5995.png", "formula": "\\begin{align*} \\| ( u , v ) \\| _ { E } ^ { 2 } : = \\int _ { \\R ^ { 2 } } \\big ( | \\nabla u | ^ { 2 } + u ^ { 2 } + | \\nabla v | ^ { 2 } + \\omega v ^ { 2 } \\big ) d x . \\end{align*}"}
+{"id": "6109.png", "formula": "\\begin{align*} a _ { 1 2 } = - j ^ { ( 1 ) } - j ^ { ( 2 ) } - 1 \\ , , a _ { 1 2 3 } = - j ^ { ( 4 ) } - j ^ { ( 0 ) } - 1 \\ , , \\end{align*}"}
+{"id": "2692.png", "formula": "\\begin{align*} s _ d ( n ) = [ x ^ n ] y & \\sim \\sqrt { \\frac { \\phi ( s ) } { 2 \\pi \\phi '' ( s ) } } \\ ; n ^ { - 3 / 2 } \\ , \\phi ' ( s ) ^ n \\\\ & = \\sqrt { \\frac { s / M _ d ( s ) } { 2 \\pi \\big ( - s M _ d '' ( s ) / M _ d ( s ) ^ 2 \\big ) } } \\ ; n ^ { - 3 / 2 } \\left ( \\frac { 1 } { M _ d ( s ) } \\right ) ^ n \\\\ & \\ ; = \\ ; \\frac { 1 } { \\sqrt { - 2 \\pi M _ d '' ( s ) } } \\ , n ^ { - 3 / 2 } \\ , M _ d ( s ) ^ { 1 / 2 - n } . \\end{align*}"}
+{"id": "6202.png", "formula": "\\begin{align*} V _ 2 ( r ) = \\frac { L ' ( L ' + 1 ) } { r ^ 2 } - \\frac { Q ' } { r } f + \\kappa B ' _ 1 \\frac { r } { f } + \\kappa B ' _ 2 \\frac { 1 } { f ^ 2 } + \\kappa B ' _ 3 \\frac { r } { f ^ 3 } + \\kappa B ' _ 4 \\frac { 1 } { f ^ 4 } + R , \\end{align*}"}
+{"id": "1209.png", "formula": "\\begin{align*} \\Phi : \\R \\to \\R , \\qquad \\Phi ( u ) : = \\begin{cases} u - u \\log ( u ) - 1 , & u > 0 , \\\\ \\ - 1 , & u \\leq 0 , \\end{cases} \\end{align*}"}
+{"id": "5609.png", "formula": "\\begin{align*} \\hat { A } ^ * ( y | x ) = A ( y _ 2 y _ 1 x ) + \\left [ A ( y _ 1 x ' ) - A ( y _ 2 x ' ) \\right ] . \\end{align*}"}
+{"id": "3724.png", "formula": "\\begin{align*} L _ X g = h . \\end{align*}"}
+{"id": "8848.png", "formula": "\\begin{align*} V _ K = \\frac { 1 } { \\abs { x _ 1 } } \\chi _ T ( x _ 1 ) + \\sqrt { K } \\left ( 1 - \\frac { K } { 3 2 } \\frac { | x _ 1 | ^ 2 } { R ^ 2 } \\right ) \\chi _ D ( x _ 1 ) , \\end{align*}"}
+{"id": "9332.png", "formula": "\\begin{align*} \\mathbf { Y } = \\mathbf { F X F } ^ \\top , \\end{align*}"}
+{"id": "7455.png", "formula": "\\begin{align*} \\left ( t _ i ^ 2 + c - c \\ , ( U ^ \\top U ) _ { i , i } \\right ) U ^ \\top _ { i , h } = \\mu _ h U ^ \\top _ { i , h } . \\end{align*}"}
+{"id": "8946.png", "formula": "\\begin{align*} \\| v \\| _ { - m , p ' } : = \\sup _ { 0 \\not = u \\in W ^ { p , m } _ 0 } \\frac { | \\langle u , v \\rangle | } { \\| u \\| _ { m , p } } . \\end{align*}"}
+{"id": "6759.png", "formula": "\\begin{align*} \\Phi _ s ( z ) ^ \\alpha = \\sum _ { m = 0 } ^ \\infty \\sum _ { r = 0 } ^ m ( - 1 ) ^ { m + r } \\ , \\binom { \\alpha } { m } \\ , \\binom { m } { r } \\ , \\Phi _ s ( z ) ^ r . \\end{align*}"}
+{"id": "3601.png", "formula": "\\begin{align*} S \\cap [ q ] = ( U \\cap [ q ] ) \\cup ( T \\cap [ q ] ) = ( T \\cap F _ { \\ell + 1 } ) \\cup T = T . \\end{align*}"}
+{"id": "5383.png", "formula": "\\begin{align*} \\begin{aligned} 0 \\geq \\ , & S _ k ^ { i j } ( A \\Psi \\pm T _ \\alpha ( u - \\varphi ) ) _ { i j } \\geq A \\eta _ 0 S _ { k - 1 } - C \\Big ( S _ { k - 1 } + f ^ { 1 - 1 / ( k - 1 ) } \\Big ) \\\\ \\geq \\ , & \\frac { A } { 2 } \\eta _ 0 S _ { k - 1 } + \\frac { A } { 2 } \\eta _ 0 c _ 0 f ^ { 1 - 1 / ( k - 1 ) } - C \\Big ( S _ { k - 1 } + f ^ { 1 - 1 / ( k - 1 ) } \\Big ) > 0 \\end{aligned} \\end{align*}"}
+{"id": "7991.png", "formula": "\\begin{align*} \\partial _ { t } h = P - h > 0 { \\rm f o r \\ a l l } \\ t \\geq 0 . \\end{align*}"}
+{"id": "8730.png", "formula": "\\begin{align*} \\mu = \\frac { 1 } { \\left [ G : K \\right ] } \\sum _ { g \\in G / K } g _ * \\nu = \\frac { 1 } { \\left [ G : H \\right ] } \\sum _ { g \\in G / H } g _ * \\nu . \\end{align*}"}
+{"id": "4756.png", "formula": "\\begin{align*} & ( 1 . ) \\ x ^ { * } ( 1 + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k 2 a _ { k } ) = z - \\sum \\limits _ { k = 1 } ^ m \\gamma _ k b _ { k } \\ \\ \\\\ & ( 2 . ) \\ \\gamma _ k [ a _ { k } \\| x ^ { * } \\| ^ { 2 } + b _ { k } ^ { T } x ^ { * } + c _ { k } ] = 0 \\ \\ k = 1 , . . . , m \\ \\ \\\\ & ( 3 . ) \\ 1 + 2 \\sum \\limits _ { k = 1 } ^ m \\gamma _ k a _ k \\ge 0 , \\ x ^ { * } \\in A \\ . \\end{align*}"}
+{"id": "326.png", "formula": "\\begin{align*} \\lim _ { A \\rightarrow \\infty } \\int _ { | x | > A } | g _ { b , \\gamma } f ( x ) | ^ p w ( x ) d x = 0 , \\end{align*}"}
+{"id": "1910.png", "formula": "\\begin{align*} \\pi _ j : = \\lim _ { n \\to \\infty } { \\mathcal { Q } } _ { i j } ( n ) = j q ^ { j - 1 } \\nu _ { j } i , j \\in { \\mathcal { E } } , \\end{align*}"}
+{"id": "408.png", "formula": "\\begin{align*} \\pi ( B _ { d _ X } ( x , r ) ) = B _ { d _ Y } ( \\pi ( x ) , r ) , \\forall x \\in X , \\forall r > 0 . \\end{align*}"}
+{"id": "4993.png", "formula": "\\begin{align*} u _ N = z + v _ N \\end{align*}"}
+{"id": "5529.png", "formula": "\\begin{align*} p \\ast q : = \\sum _ { n \\geq 1 } p _ n q _ n T ^ n . \\end{align*}"}
+{"id": "5746.png", "formula": "\\begin{align*} h ' ( x ) = \\beta ^ { q + 1 } ( x + v ) ^ { q + 1 } h ( \\mu _ { \\alpha , \\beta , u , v } ( x ) ) , \\end{align*}"}
+{"id": "7906.png", "formula": "\\begin{align*} \\hat W _ r ( k ) = \\frac { 2 } { \\pi k } \\sin ( 2 \\pi k r ) = \\frac { 1 } { k } f ( k r ) , \\end{align*}"}
+{"id": "532.png", "formula": "\\begin{align*} B = & \\ - \\Big [ ( \\sinh 2 t ) ^ { - \\frac { 1 } { 2 } } \\frac { ( x - y ) ^ { 2 } } { ( \\sinh 2 t ) ^ { 2 } } e ^ { - \\varphi ( t , x , y ) } - ( \\sinh 2 t ) ^ { - \\frac { 1 } { 2 } } \\frac { x y } { ( \\cosh t ) ^ { 2 } } e ^ { - \\varphi ( t , x , y ) } \\Big ] \\\\ = : & \\ - [ B _ { 1 } - B _ { 2 } ] . \\end{align*}"}
+{"id": "6768.png", "formula": "\\begin{align*} e ( \\rho , x ) = e ^ { i \\rho x } + \\int _ { x } ^ { \\infty } A ( x , t ) e ^ { i \\rho t } d t \\end{align*}"}
+{"id": "1563.png", "formula": "\\begin{align*} \\mathcal O _ K ^ { \\times , + } : = \\{ a \\in \\mathcal O _ K ^ \\times , \\sigma _ i ( a ) > 0 , 1 \\le i \\le s \\} , \\end{align*}"}
+{"id": "7325.png", "formula": "\\begin{align*} \\rho ^ { + } ( A ) = \\inf _ { G \\supseteq A } \\rho ( G ) , \\end{align*}"}
+{"id": "5695.png", "formula": "\\begin{align*} F ( x ) + F ( x + a ) = b \\end{align*}"}
+{"id": "8213.png", "formula": "\\begin{align*} x _ { i , 1 , k } & = 6 m - 2 i + 1 , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 1 , k } & = 2 i , \\\\ x _ { i , 2 , k } & = 2 i - 1 , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 2 , k } & = 6 m - 2 i + 2 , \\\\ x _ { i , 3 , k } & = 3 m - 2 i + k + 1 , \\mbox { a n d } \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 3 , k } & = 3 m + 2 i - k . \\end{align*}"}
+{"id": "2680.png", "formula": "\\begin{align*} \\mu _ d ( n ) = \\prod _ { i = 1 } ^ k ( - 1 ) ^ { m _ i } \\binom { d } { m _ i } . \\end{align*}"}
+{"id": "939.png", "formula": "\\begin{align*} \\begin{pmatrix} w ( t ) \\\\ \\overline { w ( t ) } \\end{pmatrix} & = P Q ( t ) P ^ { - 1 } \\begin{pmatrix} w ( 1 ) \\\\ \\overline { w ( 1 ) } \\end{pmatrix} \\\\ & + P Q ( t ) \\int ^ t _ 1 Q ( \\tau ) ^ { - 1 } P ^ { - 1 } \\begin{pmatrix} S _ 2 + e ^ { 3 i \\lambda _ 1 ( \\Phi _ 1 - \\theta _ 1 ) } R _ 2 \\\\ \\overline { S _ 2 + e ^ { 3 i \\lambda _ 1 ( \\Phi _ 1 - \\theta _ 1 ) } R _ 2 } \\end{pmatrix} ( \\tau ) \\ d \\tau \\end{align*}"}
+{"id": "1889.png", "formula": "\\begin{align*} { I } _ { l } ^ { k , \\pm } = \\int ( \\psi ^ { k , \\pm } _ l \\psi _ j \\mathfrak a ) ( s ) e ^ { i \\lambda \\mathcal P ( x , y , s ) } d s , \\ \\ { I } _ { \\ast } = \\int ( \\psi _ \\ast \\psi _ j \\mathfrak a ) ( s ) e ^ { i \\lambda \\mathcal P ( x , y , s ) } d s . \\end{align*}"}
+{"id": "2356.png", "formula": "\\begin{align*} F ( 0 , t ) - h _ 0 & = - h _ 0 \\leq 0 = y _ n ( 0 , t ) , \\\\ F ( k + 1 , t ) - h _ 0 & = 4 ( k + 1 ) - h _ 0 \\geq 3 > y _ n ( k + 1 , t ) . \\end{align*}"}
+{"id": "7561.png", "formula": "\\begin{align*} | f ( \\lambda ) | \\leq \\underbrace { \\frac { 1 } { | a _ k | } } _ { = : C _ p } \\sup _ { | z | = 1 } | p ( \\lambda + z \\cdot v ) f ( \\lambda + z \\cdot v ) | \\leq C _ p \\sup _ { \\mu \\in \\C ^ 2 , | \\mu | \\leq 1 } | p ( \\lambda + \\mu ) f ( \\lambda + \\mu ) | , \\end{align*}"}
+{"id": "7157.png", "formula": "\\begin{align*} \\operatorname { g r a d } \\operatorname { d i v } \\textbf { \\textit { u } } = g ^ { i j } \\Bigl ( \\frac { \\partial ^ 2 u ^ k } { \\partial x _ j \\partial x _ k } + \\Gamma ^ l _ { k l } \\frac { \\partial u ^ k } { \\partial x _ j } + \\frac { \\partial \\Gamma ^ l _ { k l } } { \\partial x _ j } u ^ k \\Bigr ) \\frac { \\partial } { \\partial x _ i } . \\end{align*}"}
+{"id": "8449.png", "formula": "\\begin{align*} r _ 1 ( z ) : = - \\frac { b ( k ) } { 2 i k a ( k ) } , r _ 2 ( z ) : = \\frac { 2 i k b ( k ) } { a ( k ) } , z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "2167.png", "formula": "\\begin{align*} \\begin{aligned} \\min _ { \\substack { { \\alpha _ z , \\ss _ { z } , { \\boldsymbol { \\theta } } } } } & \\mathcal { L } , \\end{aligned} \\end{align*}"}
+{"id": "7655.png", "formula": "\\begin{align*} \\partial _ x U ( t , x , \\nu ) = P _ t , \\quad \\partial _ { x x } U ( t , x , \\nu ) = \\partial _ { x \\nu } U ( t , x , \\nu ) = 0 , \\quad \\partial _ { \\nu } U ( t , x , \\nu ) = \\partial _ { \\nu } \\Phi ( t , \\nu ) , \\quad \\partial _ { \\nu \\nu } U ( t , x , \\nu ) = \\partial _ { \\nu \\nu } \\Phi ( t , \\nu ) \\end{align*}"}
+{"id": "3881.png", "formula": "\\begin{align*} G _ { \\hat { x } } ( x , y ) = \\frac { 1 } { 2 \\pi } \\ln \\frac { 1 } { | x - y | } - h _ { \\hat { x } } ( x , y ) , \\ \\ \\forall x , y \\in T _ { \\hat { x } } ( \\Omega ) . \\end{align*}"}
+{"id": "7280.png", "formula": "\\begin{align*} a _ { n - i } ( \\gamma ^ { ( q ^ { n - i } - q ^ n ) } - 1 ) = 0 \\end{align*}"}
+{"id": "3422.png", "formula": "\\begin{align*} | \\phi ( y ) | \\leq \\frac { | \\tilde { P } _ { x _ 2 - y } ( \\theta _ { y + 1 } ) | } { | \\tilde { P } _ { x _ 2 - x _ 1 + 1 } ( \\theta _ { x _ 1 } ) | } \\prod _ { k = x _ 1 } ^ y | \\cos ( \\pi \\theta _ k ) | \\cdot | \\phi ( x _ 1 - 1 ) | + \\frac { | \\tilde { P } _ { y - x _ 1 } ( \\theta _ { x _ 1 } ) | } { | \\tilde { P } _ { x _ 2 - x _ 1 + 1 } ( \\theta _ { x _ 1 } ) | } \\prod _ { k = y } ^ { x _ 2 } | \\cos ( \\pi \\theta _ k ) | \\cdot | \\phi ( x _ 2 + 1 ) | . \\end{align*}"}
+{"id": "221.png", "formula": "\\begin{align*} \\lambda = \\lambda _ S ( G ) > 1 \\end{align*}"}
+{"id": "8824.png", "formula": "\\begin{align*} \\begin{cases} d Y _ t = ( \\Pi _ { V _ j ^ \\perp } L _ { \\Pi _ { V _ j } x _ 0 } \\Pi _ { V _ j ^ \\perp } ) Y _ t d t + \\Pi _ { V _ j ^ \\perp } \\sigma d W _ t \\\\ Y _ 0 = \\Pi _ { V _ j ^ \\perp } x _ 0 . \\end{cases} \\end{align*}"}
+{"id": "3421.png", "formula": "\\begin{align*} F _ k ( \\theta , E ) = \\left ( \\begin{array} { c c } \\tilde { P } _ k ( \\theta , E ) & - \\tilde { P } _ { k - 1 } ( \\theta + \\alpha , E ) \\cos { \\pi \\theta } \\\\ \\tilde { P } _ { k - 1 } ( \\theta , E ) \\cos { \\pi ( \\theta + ( k - 1 ) \\alpha ) } & - \\tilde { P } _ { k - 2 } ( \\theta + \\alpha , E ) \\cos { \\pi \\theta } \\cos { \\pi ( \\theta + ( k - 1 ) \\alpha ) } \\end{array} \\right ) . \\end{align*}"}
+{"id": "421.png", "formula": "\\begin{align*} \\begin{aligned} \\pi ( B ( x ' , r ) ) & = L _ { n } C _ { h ' } C _ { h ^ { - 1 } } L _ { n ^ { - 1 } } ( \\pi ( B ( x , r ) ) ) , \\end{aligned} \\end{align*}"}
+{"id": "8176.png", "formula": "\\begin{align*} \\int _ { R _ { 0 } / 2 } ^ { \\infty } \\nu \\xi ( \\nu , s ) \\ d \\nu = 0 , s \\in ( 0 , t _ { 0 } ) . \\end{align*}"}
+{"id": "1907.png", "formula": "\\begin{align*} w _ n ^ { ( i ) } ( s ) = \\left [ { { \\frac { f _ n ( q s ) } { q } } } \\right ] ^ { i - 1 } w _ n ( s ) , \\end{align*}"}
+{"id": "4254.png", "formula": "\\begin{align*} ( N _ j ( \\varphi ) - P ) & = S ( I - T _ 1 + P ) ( N _ j ( \\varphi ) - P ) \\\\ & = S ( I - C _ \\varphi + P ) ( M _ j ( \\varphi ) - P ) \\\\ & = \\dfrac { 1 } { j } S ( C _ \\varphi - C _ { \\varphi _ { j + 1 } } ) \\rightarrow 0 , \\end{align*}"}
+{"id": "9169.png", "formula": "\\begin{align*} \\dfrac { \\partial b _ { 1 , \\delta } ( x ) } { \\partial x } = \\int _ { 0 } ^ { 1 } \\left . \\dfrac { \\partial h ( s ) } { \\partial s } \\right | _ { x + \\delta \\sin ( 2 \\pi t ) } \\sin ( 2 \\pi t ) d t \\end{align*}"}
+{"id": "7846.png", "formula": "\\begin{align*} \\psi ^ \\dagger _ v ( 0 , p _ 0 ) u _ \\ell & = 0 , \\\\ \\psi ^ \\dagger _ { v v } ( 0 , p _ 0 ) [ u _ \\ell , u _ \\ell ] & = - \\frac { c _ 2 ( q , \\ell , p _ 0 ) } { c _ 1 ( q , 2 \\ell , p _ 0 ) } u _ { 2 \\ell } . \\end{align*}"}
+{"id": "6618.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c } - \\varepsilon _ 1 \\mu & - \\varepsilon _ 1 \\mu \\\\ \\varepsilon _ 3 \\mu & \\varepsilon _ { 3 } \\mu \\end{array} \\right ) \\end{align*}"}
+{"id": "6056.png", "formula": "\\begin{align*} p \\Delta \\left ( H - \\frac { \\mu } { n } \\right ) ^ { p - 1 } + p \\lvert A \\rvert ^ 2 \\left ( H - \\frac { \\mu } { n } \\right ) ^ { p - 1 } - n ^ 2 H \\left ( \\left ( H - \\frac { \\mu } { n } \\right ) ^ p + \\varsigma \\right ) = 0 , \\end{align*}"}
+{"id": "6084.png", "formula": "\\begin{align*} N ( u '' , V ' ) = N ( u , V ' ) . \\end{align*}"}
+{"id": "5716.png", "formula": "\\begin{align*} \\phi _ { f ' } ( x ) = \\alpha \\left ( a ( t x + u ) ^ { q + 1 } + b ( t x + u ) ^ q ( v x + w ) + c ( t x + u ) ( v x + w ) ^ q + d ( v x + w ) ^ { q + 1 } \\right ) , \\end{align*}"}
+{"id": "807.png", "formula": "\\begin{align*} \\lim _ { R \\searrow 0 } F ( R ) & = \\int _ { R _ 0 } ^ 0 \\frac { 1 } { f ( z ) } \\ , \\mathrm { d } z = - \\int _ 0 ^ { R _ 0 } \\frac { 1 } { f ( z ) } \\ , \\mathrm { d } z > 0 , \\\\ \\lim _ { R \\nearrow \\infty } F ( R ) & = \\int _ { R _ 0 } ^ { \\infty } \\frac { 1 } { f ( z ) } \\ , \\mathrm { d } z < 0 \\end{align*}"}
+{"id": "1057.png", "formula": "\\begin{align*} [ b _ x ] = \\max \\{ [ y ] \\mid y \\leq x \\} = \\max \\{ [ y ] \\mid y \\leq x y \\} . \\end{align*}"}
+{"id": "3676.png", "formula": "\\begin{align*} L _ X g ( V , \\cdot ) = h ( V , \\cdot ) \\mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \\Sigma . \\end{align*}"}
+{"id": "3318.png", "formula": "\\begin{align*} | \\sigma | = | \\sigma ' | . \\end{align*}"}
+{"id": "5944.png", "formula": "\\begin{align*} \\langle A ( t , u ) , v \\rangle = - & \\int _ { \\mathcal { O } } \\Big \\{ \\sum _ { i = 1 } ^ { d } a _ i \\big ( t , x , u ( x ) , \\nabla u ( x ) \\big ) \\partial _ i u ( x ) \\\\ & ~ ~ ~ ~ ~ ~ + a _ 0 \\big ( t , x , , u ( x ) , \\nabla u ( x ) \\big ) v ( x ) \\Big \\} d x . \\end{align*}"}
+{"id": "7783.png", "formula": "\\begin{align*} q - h \\ge \\log _ { \\theta } \\Big ( 1 + \\frac { d \\rho } { \\epsilon _ 0 } \\Big ) ~ { \\rm f o r } ~ h = 0 , 1 , \\dots , q - 1 , \\end{align*}"}
+{"id": "5196.png", "formula": "\\begin{align*} a ^ { \\varepsilon } _ { t } + ( B \\times u ) ^ { \\varepsilon } + \\nabla Q ^ { \\varepsilon } = - \\mu \\nabla \\times b ^ { \\varepsilon } \\textrm { o n } \\ L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) . \\end{align*}"}
+{"id": "1762.png", "formula": "\\begin{align*} - \\nabla _ y \\cdot \\big ( D ( \\nabla _ y \\tilde { w } _ i + \\nabla _ y ( S _ 0 ^ { - 1 } ) ^ T e _ i \\big ) & = 0 & \\mbox { i n } & Y ^ { \\ast } ( t , x ) , \\\\ - D ( \\nabla _ y \\tilde { w } _ i + \\nabla _ y ( S _ 0 ^ { - 1 } ) ^ T e _ i ) \\cdot \\nu & = 0 & \\mbox { o n } & \\Gamma ( t , x ) , \\\\ \\int _ { Y ^ { \\ast } ( t , x ) } J _ 0 ^ { - 1 } ( t , x , S _ 0 ^ { - 1 } ( t , x , y ) ) \\tilde { w } _ i ( t , x , y ) d y & = 0 , & \\tilde { w } _ i & \\mbox { i s } Y \\mbox { - p e r i o d i c } . \\end{align*}"}
+{"id": "2859.png", "formula": "\\begin{align*} \\tilde S ^ { ( p ) } _ { j , j ' } = \\tilde F _ { j , j ' } - \\frac { ( \\mu _ { j } - \\mu _ { j ' } ) ^ 2 } { ( 4 \\gamma ) ^ 2 } \\tilde S ^ { ( q ) } _ { j , j ' } . \\end{align*}"}
+{"id": "2974.png", "formula": "\\begin{align*} H ( x , y ) : = ( x , y + h ( x ) ) \\end{align*}"}
+{"id": "7633.png", "formula": "\\begin{align*} \\mathcal { V } ( \\mu , \\nu ; \\alpha ^ { * , \\xi } ) = \\underset { \\alpha \\in \\mathcal { U } _ { a d } ( 0 , T ) } { \\inf } \\mathcal { J ( } \\mu , \\nu ; \\alpha ) . \\end{align*}"}
+{"id": "7652.png", "formula": "\\begin{align*} \\widetilde k ( t , x , y ) : = \\rho ( - R ^ { - 1 } B P _ t \\eta _ t ^ { - 1 } x - R ^ { - 1 } B \\eta _ t y ) \\quad \\end{align*}"}
+{"id": "4976.png", "formula": "\\begin{align*} \\begin{aligned} & W \\Bigl ( \\frac { \\xi _ 0 f _ 1 ( u ) } { \\xi _ 0 f _ 1 ( u ) + ( 1 - \\xi _ 0 ) f _ 2 ( u ) } , k , x + u , \\mathsf { M } ^ { k } \\Bigr ) \\\\ = & W \\Bigl ( \\frac { ( 1 - \\xi _ 0 ) f _ 2 ( u ) } { \\xi _ 0 f _ 1 ( u ) + ( 1 - \\xi _ 0 ) f _ 2 ( u ) } , k , x + u , \\mathsf { M } ^ { k } \\Bigr ) . \\end{aligned} \\end{align*}"}
+{"id": "4428.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ n \\frac { \\mu ( k ) } { k } ( I - S ) h _ k \\to 1 - z \\ H ^ p \\end{align*}"}
+{"id": "2703.png", "formula": "\\begin{align*} \\zeta _ T = \\zeta _ 0 + \\sum _ { k = 0 } ^ { T - 1 } - \\Theta _ k \\Lambda _ k ' \\log \\gamma + ( 1 - \\Theta _ k ) \\Lambda _ k \\log \\gamma \\ge 0 , \\end{align*}"}
+{"id": "5544.png", "formula": "\\begin{align*} x = \\bigwedge \\{ p \\in P ( X ) \\ : | \\ : x \\le p \\} \\end{align*}"}
+{"id": "5539.png", "formula": "\\begin{align*} x \\cdot x = x \\cdot 1 = 1 , 1 \\cdot x = x \\end{align*}"}
+{"id": "5832.png", "formula": "\\begin{align*} { { { \\bf { m } } } ^ { \\left ( { e q } \\right ) } } = { \\bf { M } } { { \\bf { g } } ^ { \\left ( { e q } \\right ) } } = { \\left [ { T , 0 , 0 , 0 , \\overline w T , 0 , 0 } \\right ] ^ { \\rm T } } . \\end{align*}"}
+{"id": "2132.png", "formula": "\\begin{align*} L \\left ( | \\varrho _ { Z Z } ( L , L ) | ^ 2 \\right ) = - 2 \\varrho _ { Z Z } ( L , L ) \\overline { \\varrho _ { Z Z } ( N , L ) } - \\varrho _ { Z Z Z } ( L , L , L ) \\overline { \\varrho _ { Z Z } ( L , L ) } , \\end{align*}"}
+{"id": "5725.png", "formula": "\\begin{align*} A ^ { q - 1 } = \\frac { r } { ( r + 1 ) ^ q } , \\end{align*}"}
+{"id": "9228.png", "formula": "\\begin{align*} \\begin{aligned} \\gamma _ 2 ^ \\star : = & \\ , \\dfrac { d } { 6 M _ \\mathcal { D } T e ^ { L _ \\mathcal { D } \\bar { t } } } \\\\ \\gamma _ 3 ^ \\star : = & \\ , \\dfrac { d } { 9 M _ \\mathcal { D } T \\left ( e ^ { L _ \\mathcal { D } \\bar { t } } - 1 \\right ) } \\\\ \\gamma _ 4 ^ \\star : = & \\ , \\dfrac { d } { 6 M _ \\mathcal { D } T } , \\end{aligned} \\end{align*}"}
+{"id": "603.png", "formula": "\\begin{align*} E \\big [ | Y _ { t _ { 0 } } | _ { V _ { t _ { 0 } } } ^ { p q } \\big ] \\quad E \\bigg [ \\bigg ( \\int _ { t _ { 0 } } ^ { t } | Y _ { s } | _ { V _ { s } } ^ { p - 2 } \\sum _ { k = 1 , \\ , \\alpha _ { k } = 0 } ^ { l } \\null _ { k } \\eta _ { s } ^ { + } \\ , d s \\bigg ) ^ { q } \\bigg ] \\end{align*}"}
+{"id": "8275.png", "formula": "\\begin{align*} \\mathbb E | X _ t ^ i | ^ 2 & = \\mathbb E | X _ 0 ^ i | ^ 2 + 2 \\int _ 0 ^ t X _ s ^ i \\cdot b ^ i ( X _ s ) \\d s + d \\sigma ^ 2 t \\\\ & = \\mathbb E | X _ 0 ^ i | ^ 2 + 2 \\int _ 0 ^ t X _ s ^ i \\cdot \\big ( b ( X _ s ^ i ) + \\gamma ^ i ( X _ s ) \\big ) \\d s + d \\sigma ^ 2 t . \\end{align*}"}
+{"id": "7564.png", "formula": "\\begin{align*} g _ { i , j } = q _ i ( X ^ k - 1 ) \\left ( \\sum _ { n = 0 } ^ { m - 1 } a _ { i , j + n p } X ^ { m - 1 - n } \\right ) + q _ i \\sum _ { n = 0 } ^ { k - 1 } a _ { i , j + ( m + n ) p } X ^ { k - 1 - n } . \\end{align*}"}
+{"id": "9224.png", "formula": "\\begin{align*} \\begin{aligned} \\| f ( z ( \\tau ) , \\tau ) - f ( x _ a ( \\tau ) , \\tau ) \\| \\le & \\ , L _ \\mathcal { D } \\| z ( \\tau ) - x _ a ( \\tau ) \\| \\\\ \\le & \\ , \\gamma 2 L _ \\mathcal { D } M _ \\mathcal { D } T \\\\ \\| f ( x ( \\tau ) , \\tau ) - f ( z ( \\tau ) , \\tau ) \\| \\le & \\ , L _ \\mathcal { D } \\| x ( \\tau ) - z ( \\tau ) \\| . \\end{aligned} \\end{align*}"}
+{"id": "2870.png", "formula": "\\begin{align*} J \\varphi ( 1 ) = \\frac { D } { 4 \\gamma } \\int _ 0 ^ 1 \\varphi '' ( u ) e _ { \\rm t h m } ( u ) d u , \\end{align*}"}
+{"id": "482.png", "formula": "\\begin{align*} \\frac { g _ u ^ { \\ast 2 } \\ast f _ r ( x ) } { g _ u ( x ) } = e ^ { \\Lambda _ u } \\frac { 2 } { \\Lambda _ u ^ 2 } \\frac { f ( x ) } { g _ u ( x ) } - \\frac { 2 } { \\Lambda _ u ^ 2 } \\frac { f _ r ( x ) } { g _ u ( x ) } - \\frac { 2 } { \\Lambda _ u ^ 2 g _ u ( x ) } \\sum _ { n \\neq 2 } ^ \\infty \\frac { \\Lambda _ u ^ n } { n ! } g _ u ^ { \\ast n } \\ast f _ r ( x ) \\end{align*}"}
+{"id": "577.png", "formula": "\\begin{align*} g _ { \\theta _ 1 } ( \\omega ( s ) ) = g _ { \\theta _ 2 } ( \\omega ( s ) ) , s > > 1 . \\end{align*}"}
+{"id": "1946.png", "formula": "\\begin{align*} = \\frac { \\cos ( \\pi \\nu ) } { \\sqrt { \\pi } } ~ e ^ { - z } ~ G _ { { 1 } , { 2 } } ^ { { 2 } , { 1 } } \\left ( 2 z \\bigg { | } \\begin{array} { l l l } \\frac { 1 } { 2 } \\\\ \\nu ~ , ~ - \\nu \\end{array} \\right ) , \\end{align*}"}
+{"id": "101.png", "formula": "\\begin{align*} \\nu ( b ) = \\pi _ { J } ( \\lambda _ G ( b ) ) \\iff J _ 1 \\subseteq J \\subseteq J _ 2 . \\end{align*}"}
+{"id": "6475.png", "formula": "\\begin{align*} K _ p ( \\theta ) = | | \\kappa _ p ( \\xi ) | | _ { \\theta } \\le v ( \\theta , p , r ) \\ | | \\xi | | _ p ^ r , \\ \\end{align*}"}
+{"id": "6714.png", "formula": "\\begin{align*} \\lambda ^ 2 = \\frac { a - 1 } { 1 - d } . \\end{align*}"}
+{"id": "5490.png", "formula": "\\begin{align*} H ( p , s ) = G ( p , s ) - F ( p , s ) . \\end{align*}"}
+{"id": "1520.png", "formula": "\\begin{align*} \\min \\| F _ i ( x ) - \\bar { F } _ i ( x ) \\tilde { H } ( x ) \\| i = 1 , \\dots , n . \\end{align*}"}
+{"id": "3363.png", "formula": "\\begin{align*} N e w t o n ( F ) = C o n v ( \\bigcup \\limits _ { i = 0 } ^ m N e w t o n ( s _ { \\lambda ^ { ( i ) } } ) ) . \\end{align*}"}
+{"id": "7410.png", "formula": "\\begin{align*} M ^ { ( i ) } = M ^ { ( i - 1 ) } _ 2 , M ^ { ( i ) } = M ^ { ( i ) } _ 1 + M ^ { ( i ) } _ 2 , M ^ { ( i ) } _ 1 = N _ i ~ , M ^ { ( i ) } _ 2 = N _ { ( i + 1 ) } + \\dots + N _ L ~ . \\end{align*}"}
+{"id": "4017.png", "formula": "\\begin{align*} E = ( C _ i \\setminus D ) \\cup \\{ c \\} . \\end{align*}"}
+{"id": "8706.png", "formula": "\\begin{align*} \\varphi ^ g ( n ) = \\varphi ( n ^ g ) \\forall g \\in G , \\ , n \\in N . \\end{align*}"}
+{"id": "3764.png", "formula": "\\begin{align*} T _ w : = T _ { s _ 1 } T _ { s _ 2 } \\ldots T _ { s _ { \\ell ( w ) } } , \\end{align*}"}
+{"id": "7705.png", "formula": "\\begin{align*} \\| P _ { i + 1 } ' v \\| _ 2 & \\geq 1 - \\frac { 4 \\tau } { c ' \\tilde \\varepsilon \\sqrt { x } \\ , h } - \\sum _ { j = 1 } ^ i \\frac { 2 ^ { i + 4 - j } \\sqrt { \\ell _ j / x } \\ , \\ , \\tau } { c ' \\tilde \\varepsilon h ^ 2 \\ , g \\big ( \\lfloor ( 1 - ( 1 + \\tilde \\varepsilon ) ^ { - j } ) r \\rfloor \\big ) } \\\\ & \\geq 1 - 4 h ^ 2 - h \\ , \\sum _ { j = 1 } ^ i \\frac { 2 ^ { i + 4 - j } \\sqrt { 2 \\tilde \\varepsilon ( 1 + \\tilde \\varepsilon ) ^ { i - j } } } { 4 ^ { i - j } } > 1 - 1 - 4 h ^ 2 - 6 4 h \\geq 1 / 2 , \\end{align*}"}
+{"id": "4305.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t _ 1 \\} } | F _ { \\delta } - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f F | ^ 2 _ { \\tilde { h } } e ^ { v _ { t _ 0 , B } ( \\Psi ) } c ( - v _ { t _ 0 , B } ( \\Psi ) ) \\\\ \\le & \\bigg ( \\frac { 1 } { \\delta } c ( t _ 1 ) e ^ { - t _ 1 } + \\int _ { t _ 1 } ^ { t _ 0 + B } c ( s ) e ^ { - s } d s \\bigg ) \\int _ { M } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } { | f F | } ^ 2 _ { \\tilde { h } } . \\end{align*}"}
+{"id": "29.png", "formula": "\\begin{align*} f ^ { \\mathbf { s } } \\cdot v _ \\lambda = \\sum _ { \\mathbf { t } < \\mathbf { s } } c _ { \\mathbf { t } } f ^ { \\mathbf { t } } \\cdot v _ \\lambda \\end{align*}"}
+{"id": "1361.png", "formula": "\\begin{align*} k : = \\sup _ { y _ 1 , y _ 2 \\in Y } d ( \\phi ( y _ 1 ) , \\pi ^ { - 1 } ( y _ 2 ) ) , \\end{align*}"}
+{"id": "3944.png", "formula": "\\begin{align*} \\mathcal { J } ' ( \\overline { u } ; h ) & = \\partial _ { y } J ( \\overline { y } , \\overline { u } ) S ' ( \\overline { u } ; h ) + \\partial _ { u } J ( \\overline { y } , \\overline { u } ) h \\\\ & = \\langle \\overline { y } - g , S ' ( \\overline { u } ; h ) \\rangle + \\alpha \\langle \\overline { u } , h \\rangle , \\end{align*}"}
+{"id": "9258.png", "formula": "\\begin{align*} \\Delta \\| q \\| ^ 2 = d _ 0 d _ 1 \\| q \\| ^ 2 = 8 { \\beta } _ n . \\end{align*}"}
+{"id": "876.png", "formula": "\\begin{align*} x x ' + x ' y + y y ' = - 2 n \\ , . \\end{align*}"}
+{"id": "7654.png", "formula": "\\begin{align*} U ( t , x , \\nu ) : = P _ t x + \\Phi ( t , \\nu ) \\end{align*}"}
+{"id": "2056.png", "formula": "\\begin{align*} A _ { j , 2 k - j } ( g , h ) & = \\sum _ { \\ell + 2 p = j , \\ \\ell + 2 q = 2 k - j } c ^ { k , j } _ { \\ell , p , q } ( \\Lambda ^ \\ell M ^ p g ) \\ , \\cdot \\ , ( \\Lambda ^ { \\ell } M ^ q h ) , \\end{align*}"}
+{"id": "8773.png", "formula": "\\begin{align*} \\mu _ { V ^ u _ R ( p _ i ) } ( V ^ u _ { R _ i } ( p _ i ) ) & = \\int _ { V ^ u _ { R _ i } ( x ) } J _ { p _ i , R } ^ { \\mathcal { P } } ( y ) d \\mu _ { V ^ u _ { R } ( x ) } \\\\ & = \\int _ { V ^ u _ { R _ i } ( x ) \\cap E _ i } J _ { p _ i , R } ^ { \\mathcal { P } } ( y ) d \\mu _ { V ^ u _ { R } ( x ) } + \\int _ { V ^ u _ { R _ i } ( x ) \\cap E _ i ^ c } J _ { p _ i , R } ^ { \\mathcal { P } } ( y ) d \\mu _ { V ^ u _ { R } ( x ) } . \\end{align*}"}
+{"id": "8640.png", "formula": "\\begin{align*} y \\left [ n \\right ] = { { \\bf { h } } ^ H } \\left [ n \\right ] * { \\bf { x } } \\left [ n \\right ] + z \\left [ n \\right ] = \\sum \\limits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { \\bf { x } } \\left [ { n - { n _ l } } \\right ] } + z \\left [ n \\right ] , \\end{align*}"}
+{"id": "6964.png", "formula": "\\begin{align*} \\Phi _ n ( x , y ) = \\prod ^ n _ { \\substack { j = 1 \\\\ ( j , n ) = 1 } } ( x - \\zeta ^ j _ n y ) . \\end{align*}"}
+{"id": "5790.png", "formula": "\\begin{align*} [ T ] + \\nu [ N ] = \\frac { 2 } { \\mu } h _ z J + \\nu I , \\end{align*}"}
+{"id": "8311.png", "formula": "\\begin{align*} M = \\left ( \\begin{aligned} & - 1 2 . 4 6 \\\\ & 0 . 5 2 - 1 \\end{aligned} \\right ) , A = \\left ( \\begin{aligned} & 1 . 1 0 \\\\ & \\ , \\ , 0 5 0 \\end{aligned} \\right ) . \\end{align*}"}
+{"id": "7851.png", "formula": "\\begin{align*} \\hat \\Phi ^ \\dagger ( 0 , 0 ) & = 0 , & \\hat \\Phi ^ \\dagger _ a ( 0 , 0 ) & = 0 , \\\\ \\hat \\Phi ^ \\dagger _ { a a } ( 0 , 0 ) & = 0 , & \\hat \\Phi ^ \\dagger _ { a a a } ( 0 , 0 ) & = 6 \\gamma _ 1 , \\end{align*}"}
+{"id": "5603.png", "formula": "\\begin{align*} d _ \\alpha + ( b _ 1 - a _ { \\alpha + 1 } ) + \\sum _ { j = 2 } ^ n ( a _ j - a _ { \\alpha + j } ) = d _ { \\alpha + n } \\ , \\end{align*}"}
+{"id": "3162.png", "formula": "\\begin{align*} { _ { 3 } } F _ { 2 } \\left ( \\begin{array} { c c c } \\chi _ 4 , & \\overline { \\chi _ 4 } , & \\overline { \\chi _ 4 } \\\\ & \\varphi , & \\varepsilon \\end{array} | 1 \\right ) & = \\frac { 1 } { q ^ 2 } J ( \\chi _ 8 , \\chi _ 8 ^ 2 ) \\times 2 R e ( J ( \\chi _ 8 , \\chi _ 8 ) ) \\times \\chi _ 8 ( - 1 ) \\\\ & = \\frac { 1 } { q ^ 2 } [ - 2 u ( - p ) ^ t ] , \\end{align*}"}
+{"id": "5970.png", "formula": "\\begin{align*} \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } \\left ( X _ n \\notin \\Upsilon \\right ) \\leq & \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } \\left ( X _ n \\notin K _ M \\right ) + \\sum _ { k = 1 } ^ { \\infty } \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } \\left ( X _ n \\notin \\Gamma _ k \\right ) \\\\ \\leq & \\frac { \\varepsilon } { 2 } + \\sum _ { k = 1 } ^ { \\infty } \\frac { \\varepsilon } { 2 ^ { k + 1 } } \\leq \\varepsilon . \\end{align*}"}
+{"id": "1878.png", "formula": "\\begin{align*} F _ { { \\Phi } _ x } ( v ) = \\sum _ { i = 1 } ^ m \\big ( 4 \\langle B ( v , e _ i ) , B ( v , e _ i ) \\rangle - \\langle B ( v , v ) , B ( e _ i , e _ i ) \\rangle \\big ) , \\end{align*}"}
+{"id": "8383.png", "formula": "\\begin{align*} \\Psi ^ - _ 1 = e _ 1 + F \\Psi ^ - _ 1 . \\end{align*}"}
+{"id": "2153.png", "formula": "\\begin{align*} | \\varrho _ z | ^ 2 = \\left ( \\frac { a \\sqrt { 5 } } { 2 } - 1 \\right ) | w | ^ 2 . \\end{align*}"}
+{"id": "2326.png", "formula": "\\begin{align*} \\Delta ^ g \\left ( s ^ H - \\| N \\| ^ 2 \\right ) = \\Delta ^ g ( \\| \\theta \\| ^ 2 ) - \\delta ^ g D ^ g _ { \\theta ^ \\sharp } \\theta - \\delta ^ g D ^ g _ { J \\theta ^ \\sharp } J \\theta , \\end{align*}"}
+{"id": "1822.png", "formula": "\\begin{align*} \\frac { d | | R ^ { \\nabla } | | _ { g _ s } ^ 2 } { d s } _ { \\big { | } _ { s = 0 } } = - \\sum _ { i , j } \\langle i _ { e _ i } R ^ { \\nabla } , i _ { e _ j } R ^ { \\nabla } \\rangle \\delta g ( e _ i , e _ j ) \\end{align*}"}
+{"id": "2085.png", "formula": "\\begin{align*} m ( I _ k \\cap ( H \\setminus P ) ) & \\leq \\sum _ { n = n _ k } ^ { \\infty } m ( I _ k \\cap ( H _ n \\setminus K ) \\cap ( \\bigcap _ { m = 1 } ^ { \\infty } ( H _ n \\setminus P _ m ) ) ) \\\\ & \\leq \\sum _ { n = n _ k } ^ { \\infty } m ( H _ n \\setminus P _ n ) \\\\ & < \\sum _ { n = n _ k } ^ { \\infty } \\frac { 1 } { 2 ^ { n + 1 } } = \\frac { 1 } { 2 ^ { n _ k } } \\end{align*}"}
+{"id": "1736.png", "formula": "\\begin{align*} f \\square g ( s ) = \\inf _ { a + b = s } f ( a ) + g ( b ) \\mbox { f o r a l l } s \\geq 0 . \\end{align*}"}
+{"id": "9177.png", "formula": "\\begin{align*} \\begin{aligned} \\left | ( h ( x _ a + \\delta u ( \\tau ) ) \\pm h ( x _ a ) ) u ( \\tau ) - \\dfrac { b _ { 1 , \\delta } ( x _ a ) } { 2 } \\right | \\le \\\\ \\left | ( h ( x _ a + \\delta u ( \\tau ) ) - h ( x _ a ) ) u ( \\tau ) \\right | + \\left | \\dfrac { b _ { 1 , \\delta } ( x _ a ) } { 2 } \\right | \\\\ + | h ( x _ a ) | | u ( \\tau ) | . \\end{aligned} \\end{align*}"}
+{"id": "5882.png", "formula": "\\begin{align*} \\tau _ { n } ^ M : = & T \\wedge \\inf \\Big \\{ t \\geq 0 : \\Vert Y _ n ( t ) \\Vert _ H ^ 2 > M \\Big \\} \\\\ & \\wedge \\inf \\Big \\{ t \\geq 0 : \\int _ 0 ^ t \\Vert Y _ n ( s ) \\Vert _ { V } ^ { \\alpha } d s > M \\Big \\} . \\end{align*}"}
+{"id": "4857.png", "formula": "\\begin{align*} i _ { \\Gamma } ^ * A : = ( I , \\Gamma ) ^ T A ( I , \\Gamma ) = B + C \\Gamma + \\Gamma ^ T C ^ T + \\Gamma D \\Gamma ^ T . \\end{align*}"}
+{"id": "7665.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t c ( t , \\nu ) + \\partial _ { \\nu } c ( t , \\nu ) A \\nu + \\frac { 1 } { 2 } \\partial _ { \\nu \\nu } c ( t , \\nu ) \\sigma _ 0 ^ 2 = 0 , \\\\ & c ( T , \\nu ) = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "1970.png", "formula": "\\begin{align*} \\int _ { F _ \\nu ^ n } f _ \\nu ( x _ \\nu ) f _ \\nu ( \\delta _ \\nu x _ \\nu ) d \\alpha _ \\nu ^ n & = \\alpha _ \\nu ^ n ( A _ \\nu \\cap \\delta _ \\nu ^ { - 1 } A _ \\nu ) \\\\ & \\leq \\min ( \\alpha ^ n _ \\nu ( A _ \\nu ) , \\| \\delta _ \\nu \\| ^ { - n } _ \\nu \\alpha ^ n _ \\nu ( A _ \\nu ) ) . \\end{align*}"}
+{"id": "5031.png", "formula": "\\begin{align*} f ( ( a + 1 ) y + b ) - f ( a y + b ) & = y + y \\int _ { u = a + \\frac b y } ^ { a + 1 + \\frac b y } \\left ( \\log ( u ) + \\log ( y ) \\right ) \\d u \\\\ \\end{align*}"}
+{"id": "6663.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { m \\to \\infty } \\limsup _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\sup _ { t \\in [ 0 , 1 ] } | X ^ { \\epsilon } ( t ) - \\varphi ( t ) | < \\delta , \\tilde { \\chi } < 1 \\Big ) = - \\infty . \\end{align*}"}
+{"id": "4473.png", "formula": "\\begin{align*} \\frac { p ^ { 2 } } { 4 } = \\frac { p ^ { 3 } } { 1 6 ( p / 4 ) } \\leq \\frac { p ^ { 3 } } { 1 6 a _ { 1 } ^ { * } } = a _ { 2 } ^ { * } a _ { 3 } ^ { * } a _ { 4 } ^ { * } \\leq \\left ( \\frac { 4 } { 3 } \\right ) ^ { 3 } ( a _ { 2 } ^ { * } ) ^ { 3 } \\end{align*}"}
+{"id": "1054.png", "formula": "\\begin{align*} \\rho _ { x , \\sigma } ( u ) : = \\rho _ x ( \\prescript { \\sigma ^ { - 1 } } { } ( u ) ) . \\end{align*}"}
+{"id": "6508.png", "formula": "\\begin{align*} q ( c ( z ' ) ) = & q ( c ( z _ 1 . . . z _ { n - 2 } ) c ( z _ { n - 1 } ' z _ n ' z _ { n + 1 } ' ) \\\\ = & q ( c ( z _ 1 . . . z _ { n - 2 } ) c _ a c _ b c _ b ) \\\\ = & q ( c ( z _ 1 . . . z _ { n - 2 } ) ) a b ^ 2 \\\\ = & q ( c ( z _ 1 . . . z _ { n - 2 } ) ) b a \\\\ = & q ( c ( z ) ) \\end{align*}"}
+{"id": "7018.png", "formula": "\\begin{align*} u \\ , = \\ , \\tfrac { x ^ 2 } { 2 } + F _ { 2 , 1 } \\ , \\tfrac { x ^ 2 y } { 2 } + { \\rm O } _ { x , y } ( 4 ) , \\end{align*}"}
+{"id": "5522.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\left ( \\frac { 1 } { ( 1 + n ) ^ 2 } - \\frac { 1 } { \\left ( 1 + n + \\frac { 1 } { 2 } \\right ) ^ 2 } - \\frac { 1 } { \\left ( n + 2 \\right ) ^ 2 } \\right ) = 5 - \\frac { \\pi ^ 2 } { 2 } > 0 , \\end{align*}"}
+{"id": "504.png", "formula": "\\begin{align*} \\bigcup _ { \\rho _ i > r _ i } K \\ < \\frac { x _ 1 } { \\rho _ 1 } , \\ldots , \\frac { x _ 1 } { \\rho _ n } \\ > = \\bigcup _ { \\rho _ i > r _ i } T ( \\rho _ 1 , \\ldots , \\rho _ n ) \\end{align*}"}
+{"id": "3992.png", "formula": "\\begin{align*} \\gamma _ i + \\bar { \\gamma _ j } = 1 \\ , \\ , { \\rm a n d } \\ , \\ , \\gamma _ 3 + \\bar { \\gamma } _ 4 = 1 , \\end{align*}"}
+{"id": "5273.png", "formula": "\\begin{align*} k _ \\theta \\cdot f _ { - } & = \\frac { 1 } { \\sqrt { 2 } } ( s ( \\theta ) , c ( \\theta ) , s ( \\theta ) , c ( \\theta ) ) , \\\\ k _ \\theta \\cdot f _ { + } & = \\frac { 1 } { ( 1 0 8 ) ^ { \\frac { 1 } { 4 } } } ( s ( 3 \\theta ) , 3 c ( 3 \\theta ) , - 3 s ( 3 \\theta ) , - c ( 3 \\theta ) ) , \\end{align*}"}
+{"id": "5510.png", "formula": "\\begin{align*} f ( p ) = 0 . 8 8 ( 8 / 3 ) ^ 2 ( D ( p ) - 1 ) + \\frac { 3 2 - ( p / 2 + 2 ) ( p / 2 + 3 ) } { 3 6 } - b \\frac { \\left ( 1 6 / 3 \\right ) ^ { p / 2 } } { 4 - p } , \\end{align*}"}
+{"id": "9023.png", "formula": "\\begin{align*} \\hat { k } _ d ( 1 \\ast \\Lambda ) = \\prod _ { \\sigma \\in 1 \\ast \\Lambda _ { d - 1 } } \\prod _ { i \\in \\sigma } x _ i = \\prod _ { \\rho \\in \\Lambda _ { d - 1 } } x _ 1 \\prod _ { i \\in \\rho } x _ i = x _ 1 ^ { \\abs { \\Lambda _ { d - 1 } } } \\prod _ i x _ i ^ { \\deg _ \\Lambda ( i ) } . \\end{align*}"}
+{"id": "4812.png", "formula": "\\begin{align*} a _ n ( u _ n , v _ n ) = ( f _ n , v _ n ) \\quad \\ , v \\in X _ n , \\end{align*}"}
+{"id": "4208.png", "formula": "\\begin{align*} \\int _ { A _ \\varepsilon } \\left ( K _ H ( x ) \\nabla \\left ( u _ \\varepsilon - q \\ln \\frac { 1 } { \\varepsilon } \\right ) | \\nabla \\left ( u _ \\varepsilon - q \\ln \\frac { 1 } { \\varepsilon } \\right ) \\right ) d x = \\frac { 1 } { \\varepsilon ^ 2 } \\int _ { A _ \\varepsilon } \\left ( u _ \\varepsilon - q \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ { p + 1 } d x , \\end{align*}"}
+{"id": "4072.png", "formula": "\\begin{align*} e ( 2 H - 5 / 2 , G ^ { N ( 2 , 0 ) } _ { 2 } ) = 2 H - 5 / 2 + ( 2 - 1 - 0 ) + ( 2 - 1 - 1 ) = 2 H - 3 / 2 . \\end{align*}"}
+{"id": "8037.png", "formula": "\\begin{align*} \\sigma _ { p q } ( t ) = \\sigma _ { x y } ( ( 1 - t ) a + t b ) . \\end{align*}"}
+{"id": "6425.png", "formula": "\\begin{align*} \\log \\det ( u _ { , i j } ) = - v _ j x ^ j + u _ { , i } \\xi ^ i + c , \\end{align*}"}
+{"id": "6149.png", "formula": "\\begin{align*} X _ { r } ^ 0 = \\xi _ r \\quad { X } _ t ^ 0 = \\xi _ 0 + \\int _ { 0 } ^ { t } \\mu ( s , X ^ 0 ) \\ , d s + \\int _ { 0 } ^ { t } \\sigma ( s , X ^ 0 ) \\ , d W _ s , \\end{align*}"}
+{"id": "5482.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\int _ { S ^ { d - 1 } } | e _ 1 + \\sqrt { t } \\xi | ^ { - p } \\dd \\xi = \\frac { p ( p - d + 2 ) } { 2 } \\int _ { B _ 2 ^ d } | e _ 1 + \\sqrt { t } x | ^ { - p - 2 } \\dd x . \\end{align*}"}
+{"id": "307.png", "formula": "\\begin{align*} \\mathfrak { J } _ k : = \\{ x \\in E \\ , : \\ , d ( x ) = k \\ , \\ , \\ , \\ , i ( x ) < k \\} \\end{align*}"}
+{"id": "1650.png", "formula": "\\begin{align*} \\theta _ n ^ { u n i v } ( f ) ( \\gamma _ 1 , \\dots , \\gamma _ n ) = f ( \\rho ^ { u n i v } ( \\gamma _ 1 ) , \\dots , \\rho ^ { u n i v } ( \\gamma _ n ) ) . \\end{align*}"}
+{"id": "8547.png", "formula": "\\begin{align*} ( h _ \\alpha \\ , * \\ h _ { 1 - \\alpha } ) ( t ) \\ , = \\ , \\int _ 0 ^ t \\ , h _ \\alpha ( t - \\tau ) \\ , h _ { 1 - \\alpha } ( \\tau ) \\ , d \\tau \\ , = \\ , h _ 1 ( t ) \\ , = \\ , \\{ 1 \\} , \\ t > 0 , \\ 0 < \\alpha < 1 \\end{align*}"}
+{"id": "3477.png", "formula": "\\begin{align*} \\prod _ { j = x _ 1 } ^ k | \\cos ( \\pi \\theta _ j ) | \\leq C ( \\varepsilon ) e ^ { - ( \\ln 2 - \\varepsilon ) | x _ 1 - k | } , \\prod _ { j = k } ^ { x _ 2 } | \\cos ( \\pi \\theta _ j ) | \\leq C ( \\varepsilon ) e ^ { - ( \\ln 2 - \\varepsilon ) | x _ 2 - k | } c _ { n , \\ell } . \\end{align*}"}
+{"id": "1253.png", "formula": "\\begin{align*} \\lim _ { \\Lambda \\uparrow \\Z ^ d } \\frac { 1 } { \\mu ( \\eta _ { \\Lambda } ) } \\int \\mathbf { 1 } _ { \\eta _ { \\Lambda } } ( \\xi ) f ( \\xi ) \\mu ( d \\xi ) = f ( \\eta ) . \\end{align*}"}
+{"id": "125.png", "formula": "\\begin{align*} f _ i : = \\mathcal { F } ( ( \\gamma ( N _ 3 ^ { ( 5 - 2 \\alpha ) + \\varepsilon } t - n ) \\mathbf { 1 } _ { [ 0 , T ] } ( t ) \\tilde { u _ i } ) ) , i = 1 , 2 , 3 . \\end{align*}"}
+{"id": "4442.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } M ^ * _ v ( t ) = \\lim _ { t \\to \\infty } \\big [ a _ 1 M ^ * _ { v ^ { ( 1 ) } } ( t ) + \\ldots + a _ N M ^ * _ { v ^ { ( N ) } } ( t ) \\big ] = a _ 1 W ( v ^ { ( 1 ) } , x ) + a _ 2 W ( v ^ { ( 2 ) } , x ) . \\end{align*}"}
+{"id": "789.png", "formula": "\\begin{align*} V & = g ( c ) H , \\\\ \\partial ^ \\square c & = \\Delta _ \\Gamma \\big ( G ' ( c ) \\big ) + c H V . \\end{align*}"}
+{"id": "4392.png", "formula": "\\begin{align*} & \\int _ { X } | \\tilde { F } - ( 1 - b ( \\Psi ) ) f F ^ { 1 + \\delta } | _ { h } ^ 2 e ^ { v ( \\Psi ) - \\delta \\max \\{ \\psi + T , 2 \\log | F | \\} } c ( - v ( \\Psi ) ) \\\\ \\le & \\bigg ( \\sup _ { X } e ^ { - u ( - v ( \\Psi ) ) } \\bigg ) \\int _ { X } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f F | ^ 2 _ h < + \\infty . \\end{align*}"}
+{"id": "4766.png", "formula": "\\begin{align*} \\inf _ { x \\in E , p _ n x = x _ n } \\| x \\| _ E \\le b \\| x _ n \\| _ { E _ n } ( n \\in \\mathbb N , \\forall x _ n \\in E _ n ) \\end{align*}"}
+{"id": "8264.png", "formula": "\\begin{align*} g _ i ^ A = i \\wedge A , \\end{align*}"}
+{"id": "1480.png", "formula": "\\begin{align*} | J _ { z _ { \\infty } } ( s ) | \\leq \\sum _ { \\{ i \\geq 1 : s _ { \\infty } ( i ) = s \\} } | J _ { \\infty } ( i ) | , \\end{align*}"}
+{"id": "2582.png", "formula": "\\begin{align*} Q ( g , R ) ( X _ 1 , X _ 2 , X _ 3 , X _ 4 ; X , Y ) = - ( ( X \\wedge _ g Y ) \\cdot R ) ( X _ 1 , X _ 2 , X _ 3 , X _ 4 ) \\\\ = R ( ( X \\wedge _ g Y ) X _ 1 , X _ 2 , X _ 3 , X _ 4 ) + R ( X _ 1 , ( X \\wedge _ g Y ) X _ 2 , X _ 3 , X _ 4 ) \\\\ + R ( X _ 1 , X _ 2 , ( X \\wedge _ g Y ) X _ 3 , X _ 4 ) + R ( X _ 1 , X _ 2 , X _ 3 , ( X \\wedge _ g Y ) X _ 4 ) , \\end{align*}"}
+{"id": "3117.png", "formula": "\\begin{align*} \\langle \\cdot , \\cdot \\rangle : V ^ * \\times V \\rightarrow \\mathbb { K } , a ^ * ( b ) = \\langle a ^ * , b \\rangle = \\langle b , a ^ * \\rangle , \\ ; \\ ; \\forall a ^ * \\in V ^ * , b \\in V . \\end{align*}"}
+{"id": "9166.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { 0 } ^ { 1 / 2 } \\left [ m ( x + \\delta \\sin ( 2 \\pi t ) ) - m ( x ) \\right ] \\sin ( 2 \\pi t ) \\ , d t \\\\ \\int _ { 0 } ^ { 1 / 2 } \\left [ m ( x ) - m ( x - \\delta \\sin ( 2 \\pi t ) ) \\right ] \\sin ( 2 \\pi t ) \\ , d t . \\end{aligned} \\end{align*}"}
+{"id": "7874.png", "formula": "\\begin{align*} \\Psi ^ q + v + \\psi ( v , p ( s ) ) = \\Psi ^ q + v + \\frac 1 2 \\psi _ { v v } ( 0 , p _ 0 ) [ v , v ] + O ( \\norm { ( v , s ) } ^ 3 ) \\end{align*}"}
+{"id": "3678.png", "formula": "\\begin{align*} X _ { i ; j } V _ j & = \\omega _ { i j } V _ j \\\\ \\omega _ { i j ; \\ell } V _ \\ell & = - R _ { k \\ell i j } X _ k V _ \\ell + \\tfrac { 1 } { 2 } \\left ( h _ { i j , \\ell } + h _ { i \\ell , j } - h _ { j \\ell , i } \\right ) V _ \\ell \\end{align*}"}
+{"id": "1230.png", "formula": "\\begin{align*} \\nabla _ \\Lambda f ( \\eta ) : = \\sum _ { \\xi _ \\Lambda } [ f ( \\xi _ \\Lambda \\eta _ { \\Lambda ^ c } ) - f ( \\eta ) ] . \\end{align*}"}
+{"id": "7913.png", "formula": "\\begin{align*} h ( \\upsilon ) = \\frac 1 8 \\left ( - f ( \\upsilon ) + 8 \\upsilon - f ( 2 \\upsilon ) + \\frac 1 3 f ( 3 \\upsilon ) \\right ) . \\end{align*}"}
+{"id": "7318.png", "formula": "\\begin{align*} \\mu _ T ( e _ i , \\mu _ T ( e _ j , e _ k ) ) = 0 \\forall i , j , k \\in \\{ 1 , \\ldots , n \\} \\end{align*}"}
+{"id": "2310.png", "formula": "\\begin{align*} e ( \\sum _ { s = 1 } ^ { s _ 0 } 2 ^ { - s } \\sum _ { i + j = k - s } x _ i y _ j ) = \\prod _ { s = 1 } ^ { s _ 0 } e ( 2 ^ { - s } \\sum _ { i + j = k - s } x _ i y _ j ) . \\end{align*}"}
+{"id": "7373.png", "formula": "\\begin{gather*} P ( \\abs { f } ) \\le j + P \\bigl \\{ \\abs { f } \\ , 1 ( \\abs { f } > j ) \\bigr \\} = j + \\sup _ k P \\bigl \\{ \\abs { f } \\wedge k \\ , 1 ( \\abs { f } > j ) \\bigr \\} \\\\ = j + \\sup _ k \\int Q \\bigl \\{ \\abs { f } \\wedge k \\ , 1 ( \\abs { f } > j ) \\bigr \\} \\ , P _ \\mathcal { U } ( d Q ) \\le j + \\sup _ { Q \\in \\mathcal { U } } Q \\bigl \\{ \\abs { f } \\ , 1 ( \\abs { f } > j ) \\bigr \\} . \\end{gather*}"}
+{"id": "2348.png", "formula": "\\begin{align*} \\mathcal { H } ( x ) & : = x _ t - \\nu ( x _ y ) x _ { y y } + \\phi ' ( x ) ( 1 + \\nu ( x _ y ) x _ y ^ 2 ) , \\\\ \\mathcal { V } ( y ) & : = y _ t - \\mu ( y _ x ) y _ { x x } - \\phi ' ( x ) ( 1 + \\mu ( y _ x ) ) y _ x , \\end{align*}"}
+{"id": "2475.png", "formula": "\\begin{align*} g ( E ) = \\frac { V ^ { N } ( 2 \\pi m ) ^ { \\frac { D N } { 2 } } } { h ^ { D N } N ! \\left ( \\frac { D N } { 2 } - 1 \\right ) ! } E ^ { \\frac { D N } { 2 } - 1 } , \\end{align*}"}
+{"id": "4600.png", "formula": "\\begin{align*} 2 \\widetilde y _ n - \\widetilde x _ 0 & = 2 e ^ { r y _ n } - e ^ { r x _ 0 } \\\\ & \\leq e ^ { r ( y _ n + a ) } - e ^ { r x _ 0 } \\\\ & \\leq e ^ { r x _ { n + 1 } } - e ^ { r x _ 0 } \\\\ & = \\widetilde x _ { n + 1 } - e ^ { r x _ 0 } \\\\ & < \\widetilde { x } _ { n + 1 } , \\end{align*}"}
+{"id": "413.png", "formula": "\\begin{align*} \\phi _ { \\hat g } ( g H ) : = { \\hat g } \\phi ( { \\hat g } ^ { - 1 } g H ) , \\forall g H \\in G / H . \\end{align*}"}
+{"id": "3029.png", "formula": "\\begin{align*} \\Pi : \\mathbb { R } ^ { 3 } \\rtimes S ^ { 3 } \\rightarrow \\mathbb { R } ^ { 3 } \\rtimes S O _ { 3 } \\Pi \\left ( x , q \\right ) = \\left ( x , I _ { q } \\right ) \\end{align*}"}
+{"id": "4816.png", "formula": "\\begin{align*} a _ n ( u _ n , v _ n ) : = \\sum _ { K \\in { \\mathcal T } _ n } \\int _ K D ^ 2 u _ n : D ^ 2 v _ n d K , u _ n , v _ n \\in X _ n . \\end{align*}"}
+{"id": "5819.png", "formula": "\\begin{align*} \\psi = \\sqrt { \\frac { { 2 \\Delta { t ^ 2 } \\left ( { { p _ { E O S } } - \\rho \\hat c _ s ^ 2 } \\right ) } } { { G \\Delta { x ^ 2 } } } } . \\end{align*}"}
+{"id": "2399.png", "formula": "\\begin{align*} L = \\frac { 0 . 9 \\log n } { - \\log \\mu [ A , \\infty ) } . \\end{align*}"}
+{"id": "7680.png", "formula": "\\begin{align*} \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) ) } ^ { N , i } : = k _ i ^ N ( t , \\nu ^ { N , 1 } _ { \\boldsymbol { x } } , \\ldots , \\nu ^ { N , N } _ { \\boldsymbol { x } } , \\lambda ^ { N , 1 } _ { \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) } , \\ldots , \\lambda ^ { N , N } _ { \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) } ) , \\quad \\mu ^ i : = k \\big ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } , \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } ) \\big ) , \\end{align*}"}
+{"id": "2281.png", "formula": "\\begin{align*} f ( [ V ] , x ) = a \\circ \\widehat { \\varphi } ( [ I _ n , V ] , x ) = a ( D ( x ) V ) , \\end{align*}"}
+{"id": "8451.png", "formula": "\\begin{align*} \\Psi ^ \\pm _ 2 ( x ; z ) = \\frac { 1 } { 2 i k } \\begin{pmatrix} \\psi ^ \\pm _ { 1 2 } \\\\ - \\bar { u } _ x \\psi ^ \\pm _ { 1 2 } + 2 i k \\psi ^ \\pm _ { 2 2 } \\end{pmatrix} , \\end{align*}"}
+{"id": "6974.png", "formula": "\\begin{align*} \\Phi _ { m , e } ^ { ( 1 ) } ( \\lambda _ 1 , \\lambda _ 2 ) \\Phi _ { m , e } ^ { ( - 1 ) } ( \\lambda _ 1 , \\lambda _ 2 ) = \\Phi _ { m e } ( \\lambda _ 1 , \\lambda _ 2 ) . \\end{align*}"}
+{"id": "3977.png", "formula": "\\begin{align*} c _ { k + 1 } - c _ { k } = ( c _ { k + 1 } - c _ { k } ) \\left ( \\sum _ { i \\in \\Omega _ t ^ o } a _ i - \\sum _ { i \\in \\Omega _ t ^ e } a _ i \\right ) . \\end{align*}"}
+{"id": "1446.png", "formula": "\\begin{align*} \\mathbf { R } _ { X } = \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } \\mathbf { x } \\left [ l \\right ] \\mathbf { x } \\left [ l \\right ] ^ { H } = \\mathbf { P } \\mathbf { P } ^ { H } . \\end{align*}"}
+{"id": "5994.png", "formula": "\\begin{align*} & B ( u ) = \\int _ { \\R ^ { 2 } } \\frac { u ^ { 2 } } { | x | ^ { 2 } } \\Big ( \\int _ { 0 } ^ { | x | } \\frac { s } { 2 } u ^ { 2 } ( s ) d s \\Big ) ^ { 2 } d x , \\\\ & F ( u , v ) = \\int _ { \\R ^ { 2 } } \\big ( u ^ { 2 p } + v ^ { 2 p } + 2 b | u v | ^ { p } \\big ) d x . \\end{align*}"}
+{"id": "1288.png", "formula": "\\begin{align*} \\forall n \\in \\N : \\ c ^ { B _ { n - 1 } ( x ( \\Delta ) ) } _ { \\Delta } ( \\eta _ { \\Lambda _ n } , \\xi _ { \\Delta } ) = 0 . \\end{align*}"}
+{"id": "6642.png", "formula": "\\begin{align*} \\Theta _ { \\epsilon } ( t ) = \\frac { \\epsilon } { 2 } - \\epsilon ^ { 2 } \\int ^ t _ 0 \\theta _ { \\epsilon } ( s - ) \\widetilde { N } ^ { \\epsilon ^ { - 2 } } ( d s ) - \\frac { \\epsilon ^ { 2 } } { 2 } \\theta _ { \\epsilon } ( t ) . \\end{align*}"}
+{"id": "6648.png", "formula": "\\begin{align*} \\mathcal { I } ^ { \\varphi } ( \\varphi ) = \\mathcal { I } ( \\varphi ) . \\end{align*}"}
+{"id": "529.png", "formula": "\\begin{align*} T _ { t } ^ { \\alpha } f ( x ) = \\sum _ { n = 0 } ^ { \\infty } ( 2 n + d ) ^ { - \\alpha } e ^ { ( 2 n + d ) i t } P _ { n } f ( x ) . \\end{align*}"}
+{"id": "5187.png", "formula": "\\begin{align*} A _ t \\cdot B \\dot { g } ( \\Phi ) = \\partial _ t \\left ( g ( \\Phi ) \\frac { G } { r ^ { 2 } } \\right ) + \\nabla \\cdot ( g ( \\Phi ) \\nabla \\theta \\times A _ t ) . \\end{align*}"}
+{"id": "2403.png", "formula": "\\begin{align*} 2 ( r - k ) \\beta \\ge s ' = \\log t - s - 1 \\ge \\log t / 3 . \\end{align*}"}
+{"id": "7001.png", "formula": "\\begin{align*} [ J _ 0 ^ { \\alpha } , J _ \\pm ^ { \\alpha } ] = \\pm \\omega ( \\alpha ) J _ \\pm ^ \\alpha \\end{align*}"}
+{"id": "4353.png", "formula": "\\begin{align*} \\frac { 1 } { r _ 1 ^ 2 } \\int _ { \\{ \\Psi = - \\infty \\} } | f | ^ 2 _ h & = \\lim _ { p \\rightarrow 0 + 0 } \\frac { 1 } { r _ 1 ^ 2 } \\int _ { \\{ p \\Psi < 2 \\log r _ 1 \\} } | f | ^ 2 _ h \\\\ & \\ge G ( 0 ; c \\equiv 1 , \\Psi , h , I _ + ( h , 2 a _ { z _ 0 } ^ f ( \\Psi ; h ) \\Psi ) _ { z _ 0 } , f ) . \\end{align*}"}
+{"id": "6356.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\frac { h ( x ) } { \\exp ( - a \\theta / x ^ 2 ) / x ^ 3 } = & \\frac { 2 \\theta } { \\operatorname { B } ( a , b ) } \\lim _ { x \\to 0 } \\frac { \\left [ 1 - \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) \\right ] ^ { b - 1 } } { \\operatorname { I } _ { 1 - \\exp ( - \\theta / x ^ 2 ) } ( b , a ) } \\\\ = & \\frac { 2 \\theta } { \\operatorname { B } ( a , b ) } . \\end{align*}"}
+{"id": "3701.png", "formula": "\\begin{align*} ( \\hat h , \\hat v ) = \\left \\{ \\begin{array} { l l } ( h , v ) & \\mbox { i n } M \\setminus \\Omega \\\\ ( 0 , 0 ) & \\mbox { i n } \\overline U \\end{array} \\right . . \\end{align*}"}
+{"id": "1514.png", "formula": "\\begin{align*} F _ i ( x ) = f _ { i m } x ^ m + \\dots + f _ { i 0 } x ^ 0 , i = 1 , \\dots , n , f _ { 1 m } \\neq 0 . \\end{align*}"}
+{"id": "6690.png", "formula": "\\begin{align*} \\delta ( \\eta , \\gamma ^ + ) = \\delta ( \\eta \\gamma ^ n , \\gamma ^ + ) = \\delta ( \\eta \\gamma ^ n , \\gamma ^ + ) + \\delta ( \\eta \\gamma ^ n , ( \\eta \\gamma ^ n ) ^ - ) . \\end{align*}"}
+{"id": "1643.png", "formula": "\\begin{align*} H ^ \\ast ( - , - ) = \\bigoplus _ i H ^ i ( - , - ) \\end{align*}"}
+{"id": "5783.png", "formula": "\\begin{align*} ( T _ 1 - f _ 1 ) \\cdot ( T _ 2 + f _ 2 ) \\cdot \\Psi _ 1 = i \\Psi _ 1 \\end{align*}"}
+{"id": "6578.png", "formula": "\\begin{align*} \\nabla ( c \\varphi ) \\cdot \\nu = \\nabla c \\cdot \\nu \\varphi + c \\nabla \\varphi \\cdot \\nu = ( \\gamma - c ) \\varphi + c \\nabla \\varphi \\cdot \\nu \\quad \\mbox { o n } \\ , \\ , \\partial \\Omega , \\end{align*}"}
+{"id": "3908.png", "formula": "\\begin{align*} \\begin{cases} - \\delta ^ 2 ( K _ H ( x ) \\nabla v ) = \\left ( v - \\left ( \\frac { \\alpha | x | ^ 2 } { 2 } + \\beta \\right ) \\right ) ^ { p } _ + , \\ \\ & x \\in B _ { R ^ * } ( 0 ) , \\\\ v = 0 , \\ \\ & x \\in \\partial B _ { R ^ * } ( 0 ) . \\end{cases} \\end{align*}"}
+{"id": "4046.png", "formula": "\\begin{align*} \\sup _ { n \\in \\mathbb { N } } \\sup _ { j \\in [ n ] ^ V } \\sup _ { s _ 1 , . . . , s _ i \\in [ 0 , T ] } \\norm { D ^ i _ { s _ 1 , . . . , s _ i } A _ n ( j ) } _ { L ^ p ( P ) } < \\infty \\end{align*}"}
+{"id": "6094.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\big ( \\mu _ n ^ { ( 1 2 ) } - \\mu _ { n + 1 } ^ { ( 1 2 ) } \\big ) ^ 2 = \\big ( \\mu _ n ^ { ( 1 2 ) } + \\mu _ { n + 1 } ^ { ( 1 2 ) } \\big ) \\ , . \\end{align*}"}
+{"id": "9206.png", "formula": "\\begin{align*} \\begin{aligned} | \\bar { y } - \\bar { y } _ a | = & \\ , | \\bar { y } - \\bar { y } _ a \\pm z _ 2 | \\le | \\bar { y } - z _ 2 | + | z _ 2 - \\bar { y } _ a | \\\\ \\le & \\ , | \\bar { y } - z _ 2 | + \\gamma | \\epsilon _ 2 ( \\eta _ a , t ) | \\\\ \\le & \\ , | \\bar { y } - z _ 2 | + \\gamma L _ r 2 \\delta \\end{aligned} \\end{align*}"}
+{"id": "1560.png", "formula": "\\begin{align*} \\texttt { l e n g t h ( f i n d ( d i f f ( s i g n ( d i f f ( ( S 2 * v ( 2 * n + 1 : 3 * n ) ) ) ) ) ) ) > = 2 } \\end{align*}"}
+{"id": "3232.png", "formula": "\\begin{align*} 0 \\leq \\sum _ { i = 1 } ^ { N - 1 } ( x _ { i + 1 } - x _ i ) ^ 2 = \\sum _ { i = 1 } ^ { N - 1 } x _ { i + 1 } ^ 2 + \\sum _ { i = 1 } ^ { N - 1 } x _ i ^ 2 - 2 \\sum _ { i = 1 } ^ { N - 1 } x _ { i + 1 } x _ i \\leq 2 \\left [ \\sum _ { i = 1 } ^ { N } x _ i ^ 2 - \\sum _ { i = 1 } ^ { N - 1 } x _ { i + 1 } x _ i \\right ] . \\end{align*}"}
+{"id": "6247.png", "formula": "\\begin{align*} \\beta ( C _ S X , Y ) = \\beta ( X , C _ S Y ) , \\quad \\forall S \\in \\Gamma , \\quad \\forall X , Y \\in \\Gamma ^ \\perp . \\end{align*}"}
+{"id": "1398.png", "formula": "\\begin{align*} \\widetilde G _ n ( x , z ) d z & : = \\P _ x ( S _ n \\in d z ; \\rho > n ) \\\\ & = \\prod _ { j = 1 } ^ d \\lambda _ j e ^ { - \\sum _ { j = 1 } ^ d \\lambda _ j ( z _ j - x _ i ) } \\det ( q _ n ( z _ j - x _ i ) ) _ { i , j = 1 } ^ d d z . \\end{align*}"}
+{"id": "809.png", "formula": "\\begin{align*} T = F ( 0 ) = - \\int _ 0 ^ { R _ 0 } \\frac { 1 } { f ( z ) } \\ , \\mathrm { d } z = \\frac { 1 } { d } \\int _ 0 ^ { R _ 0 } \\frac { z } { g \\left ( \\frac { m } { \\alpha _ d } z ^ { - d } \\right ) } \\ , \\mathrm { d } z \\end{align*}"}
+{"id": "6898.png", "formula": "\\begin{align*} y = \\mathbf { X } \\beta + \\varepsilon , \\varepsilon \\sim \\gamma _ { \\sigma ^ 2 } , \\end{align*}"}
+{"id": "1795.png", "formula": "\\begin{align*} L ( f , 2 ) = \\frac { 4 \\pi ^ { 2 } } { 3 } \\int _ { 0 } ^ { \\infty } b ^ { 2 } ( e ^ { - 2 \\pi u } ) c ( e ^ { - 6 \\pi u } ) u d u . \\end{align*}"}
+{"id": "3578.png", "formula": "\\begin{align*} I ( A ) = \\langle p ^ u - p ^ v \\ , | \\ , u , v \\in \\Z _ { \\geq 0 } ^ m \\ , \\ , \\ , \\ , A u = A v \\rangle \\end{align*}"}
+{"id": "1576.png", "formula": "\\begin{align*} h _ { M , K } : = h _ D + e ^ { 2 \\varphi } \\Tilde g _ K \\end{align*}"}
+{"id": "3006.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } t } \\Big ( x ( t ) - g ( t , x ( t - h ) ) \\Big ) = f ( t , x ( t ) , x ( t - h ) , u ( t ) ) t \\in [ \\tau , \\vartheta ] . \\end{align*}"}
+{"id": "8113.png", "formula": "\\begin{align*} e _ { 2 , i k } \\ ! + \\ ! \\gamma _ { i k } \\frac { \\sum _ { j \\neq i } | e ^ { ( n ) } _ { 1 , j i k } | ^ { 2 } } { ( 1 + \\nu _ { i k } \\gamma _ { i k } ) ^ { 2 } } \\ ! - \\ ! 2 \\Re { \\{ e _ { 3 , i k } e ^ { ( n ) } _ { 1 , i i k } \\} } \\ ! - \\ ! 2 \\nu _ { i k } | e _ { 3 , i k } | ^ { 2 } \\ ! = \\ ! 0 . \\end{align*}"}
+{"id": "8230.png", "formula": "\\begin{align*} | \\beta _ k - \\alpha _ k | \\leq C _ 1 \\frac { 2 \\pi } { N } \\left ( s _ k , \\{ z \\in \\mathbb { D } : | b ( z ) | \\leq C \\} \\right ) \\ , k = 1 , \\ldots , N . \\end{align*}"}
+{"id": "7741.png", "formula": "\\begin{align*} f ( x ) = f _ { s ( x ) } ( x ) \\end{align*}"}
+{"id": "197.png", "formula": "\\begin{align*} B _ { \\infty , \\infty } ^ { \\Gamma _ 1 \\times \\Gamma _ 2 - l a } = \\varinjlim _ { M \\to \\infty } B _ { \\infty , \\infty } ^ { p ^ n \\Gamma _ 1 - a n , p ^ n \\Gamma _ 2 } \\langle z _ { k } - z _ { k , M } : \\ ; 1 \\leq k \\leq d \\rangle . \\end{align*}"}
+{"id": "4019.png", "formula": "\\begin{align*} \\varphi ^ L _ { \\sigma , s ' } ( D ) = \\{ d \\} \\varphi ^ L _ { \\sigma , s ' } ( E ) = \\varphi ^ L _ { \\sigma , s ' } ( C _ i ) . \\end{align*}"}
+{"id": "8287.png", "formula": "\\begin{align*} \\d Y _ t ^ i = b ^ i ( Y _ t ) \\d t + \\sigma \\d W _ t ^ i , ~ ~ \\tilde Y _ { n + 1 } ^ i = \\tilde Y _ n ^ i + b ^ i ( \\tilde Y _ n ) \\tau + \\sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , ~ ~ i = 1 , \\cdots , N . \\end{align*}"}
+{"id": "4310.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi _ 1 < - t _ 1 \\} \\cap D _ 4 } | f | ^ 2 _ h e ^ { - \\Psi _ 1 } \\\\ = & \\int _ { \\{ \\Psi _ 1 < - t _ 1 \\} \\cap D _ 4 } | f F ^ 2 | ^ 2 _ h e ^ { - \\varphi _ 1 - \\Psi _ 1 } \\\\ \\le & \\int _ { \\{ \\Psi _ 1 < - t _ 1 \\} \\cap D _ 4 } | \\tilde { F } | ^ 2 _ h e ^ { - \\varphi _ 1 - \\Psi _ 1 } + \\int _ { \\{ \\Psi _ 1 < - t _ 1 \\} \\cap D _ 4 } | \\tilde { F } - f F ^ 2 | ^ 2 _ h e ^ { - \\varphi _ 1 - \\Psi _ 1 } \\\\ < & + \\infty . \\end{align*}"}
+{"id": "4099.png", "formula": "\\begin{align*} & \\hphantom { { } = { } } ( x ^ i , ( a ^ i { } _ j , b ^ i { } _ j ) , ( a ^ i { } _ { j k } , b ^ i { } _ { j k } ) ) . ( g , X ) \\\\ * & = ( x ^ i , ( a ^ i { } _ \\alpha g ^ \\alpha { } _ j , h ^ i { } _ \\alpha b ^ \\alpha { } _ \\beta g ^ \\beta { } _ j + X ^ i { } _ j ) , ( a ^ i { } _ { \\alpha j } g ^ \\alpha { } _ k , h ^ i { } _ \\alpha b ^ \\alpha { } _ { \\beta k } g ^ \\beta { } _ j ) ) , \\end{align*}"}
+{"id": "7623.png", "formula": "\\begin{align*} & f _ n \\big ( t , ( \\partial _ x v _ n \\Sigma ) ( t , x ) , x \\big ) = ( \\partial _ x v _ n \\Sigma \\cdot q ) ( t , x ) - g _ n \\big ( t , q ( t , x ) , x \\big ) . \\end{align*}"}
+{"id": "5885.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } \\Big ( | \\langle Y _ n ( \\zeta _ n + \\delta ) - Y _ n ( \\zeta _ n ) , e \\rangle | > \\varepsilon \\Big ) = 0 , \\end{align*}"}
+{"id": "6693.png", "formula": "\\begin{align*} | \\delta ( \\zeta , \\gamma ^ + ) | = | f ( \\zeta \\gamma ^ + , \\gamma ^ - ) - f ( \\gamma ^ + , \\gamma ^ - ) | \\leq 2 M . \\end{align*}"}
+{"id": "8904.png", "formula": "\\begin{align*} { h _ { \\alpha } } = \\log m _ \\alpha , \\forall \\emptyset \\neq \\alpha \\subseteq [ n ] \\end{align*}"}
+{"id": "6631.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\frac { \\lambda ( \\epsilon ) } { \\epsilon } = \\kappa > 0 , \\end{align*}"}
+{"id": "7737.png", "formula": "\\begin{align*} 1 = \\langle \\varpi \\otimes \\varpi ( \\cdot ) \\xi ' _ i , \\xi ' _ i \\rangle i . \\end{align*}"}
+{"id": "5030.png", "formula": "\\begin{align*} I _ 2 = ( y - 1 ) \\cdot f ( y ) + y \\left ( f \\left ( \\left ( a + 1 + \\frac { b } { y } \\right ) ( y - 1 ) \\right ) - f \\left ( \\left ( a + \\frac { b } { y } \\right ) ( y - 1 ) \\right ) \\right ) . \\\\ \\end{align*}"}
+{"id": "4228.png", "formula": "\\begin{align*} \\dot { s } ( e - \\varepsilon ( y ) ) \\lambda _ i = \\sum _ { j , \\mu } g ^ i _ { j , \\mu } ( e - \\varepsilon ( y ^ { - 1 } x ^ i _ { j , \\mu } ) ) \\lambda _ j . \\end{align*}"}
+{"id": "7005.png", "formula": "\\begin{align*} u \\ , = \\ , F \\big ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , \\dots , x _ n \\big ) , \\end{align*}"}
+{"id": "498.png", "formula": "\\begin{align*} f _ { a r } ( x ) = \\frac { f _ { + a } ( x ) - e ^ { - \\lambda _ s } f _ r ( x ) } { 1 - e ^ { - \\lambda _ s } } = f _ r \\ast f _ s ( x ) , \\end{align*}"}
+{"id": "9142.png", "formula": "\\begin{align*} \\dot x _ a = - { \\gamma \\over 2 } b _ { 1 , \\delta } ( x _ a ) \\ , . \\end{align*}"}
+{"id": "5449.png", "formula": "\\begin{align*} 0 = V _ 0 \\subset V _ 1 \\subset V _ 2 \\subset \\cdots \\subset V _ n \\subseteq V , \\end{align*}"}
+{"id": "5578.png", "formula": "\\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) = A ( 1 ^ \\infty ) - A ( 1 ^ n 0 ^ \\infty ) = \\begin{cases} c - d \\ , \\\\ c - c _ n \\ , \\end{cases} \\ , \\end{align*}"}
+{"id": "2531.png", "formula": "\\begin{align*} \\mathbf { z } ( \\alpha ) = ( 1 - \\alpha ) ( \\mathbf { w } _ { x s } - \\mu \\mathbf { e } ) + \\alpha ( \\mathbf { W } _ { x s } - \\mathbf { R } _ { x s } ) \\widehat { \\Delta \\mathbf { x } } + \\alpha ^ 2 \\widehat { \\Delta \\mathbf { x } } \\circ \\widehat { \\Delta \\mathbf { s } } , \\end{align*}"}
+{"id": "9100.png", "formula": "\\begin{align*} ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + ( j - 1 ) r \\leq \\chi _ j , \\end{align*}"}
+{"id": "7357.png", "formula": "\\begin{gather*} \\int f \\ , d P = \\sup _ k \\int f \\wedge k \\ , d P . \\end{gather*}"}
+{"id": "6494.png", "formula": "\\begin{align*} \\mu | _ { E _ { 2 , m } } = \\lambda _ y = \\lambda _ x | _ { E _ { 2 , m } } . \\end{align*}"}
+{"id": "378.png", "formula": "\\begin{align*} D ^ { \\dagger } = - \\frac { 1 } { 2 } L _ { e v e n } + \\frac { n - 1 } { 2 } ( D ^ \\dag \\ 1 ) ( D ^ \\dag \\ 1 ) ^ { ' } . \\end{align*}"}
+{"id": "6319.png", "formula": "\\begin{align*} S _ 1 : Y ^ { p ^ 2 } + X ^ { 2 p + 1 } + X ^ { 3 p } Y + Z = 0 \\end{align*}"}
+{"id": "1823.png", "formula": "\\begin{align*} \\frac d { d s } \\ , d v _ { g _ s } \\ , _ { \\big { | } _ { s = 0 } } = \\frac 1 2 \\langle g , \\delta g \\rangle d v _ g \\ , . \\end{align*}"}
+{"id": "3273.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sup _ { n \\in \\mathbb N } \\sum _ { k \\geq N } \\mathbb E \\left [ \\langle Y _ n , e _ k \\rangle ^ 2 \\right ] = 0 , \\end{align*}"}
+{"id": "6254.png", "formula": "\\begin{align*} \\nabla ^ \\perp _ T \\eta _ i = 0 , \\quad \\forall T \\in \\Gamma . \\end{align*}"}
+{"id": "8563.png", "formula": "\\begin{align*} C _ { - 1 } ( 0 , + \\infty ) \\ , : = \\ , \\{ f : \\ f ( t ) = t ^ { p } f _ 1 ( t ) , \\ t > 0 , \\ p > - 1 , \\ f _ 1 \\in C [ 0 , + \\infty ) \\} . \\end{align*}"}
+{"id": "2759.png", "formula": "\\begin{align*} K _ m = B \\ ! \\left ( h _ m ( x ) , \\frac 1 4 | h _ m ( x ) | \\right ) \\end{align*}"}
+{"id": "1408.png", "formula": "\\begin{align*} G _ n ( x , z ) d z = \\sum _ { \\pi \\in \\mathcal S _ d } ( \\pi ) \\P _ x ( \\widehat S _ i ( n + d - \\pi ( i ) ) \\in d z _ { \\pi ( i ) } , i = 1 , \\ldots , d ) . \\end{align*}"}
+{"id": "7286.png", "formula": "\\begin{align*} l . \\psi \\varphi = \\psi \\underbrace { c _ { g ' } ( l ) } _ { \\in K } \\varphi = \\psi \\varphi c _ { g } ( c _ { g ' } ( l ) ) = \\psi \\varphi c _ { g ' g } ( l ) \\end{align*}"}
+{"id": "4131.png", "formula": "\\begin{align*} u ( x ' , t ) : = P _ t ^ L * f ( x ' ) , ( x ' , t ) \\in \\R ^ n _ + , \\end{align*}"}
+{"id": "1473.png", "formula": "\\begin{align*} 0 = \\varphi ' ( 0 ) = - \\frac { \\alpha \\sqrt { M } } { 2 \\sqrt { 2 H } } + \\beta M ^ { 2 } \\left ( z _ { 0 } ^ { 2 } - H ^ { 2 } \\right ) . \\end{align*}"}
+{"id": "4356.png", "formula": "\\begin{align*} \\mu _ 2 ( \\{ x \\in ( 0 , s _ 0 ] : g _ 2 ( x ) \\geq \\tilde a ( - \\log s ) s ^ { - 1 } \\} ) = \\mu _ 2 ( \\{ 0 < x \\leq C s \\} ) = C s \\end{align*}"}
+{"id": "792.png", "formula": "\\begin{align*} \\int _ { \\Gamma _ t } ( c + t w ) \\circ \\theta _ t ^ { - 1 } \\ , \\mathrm { d } \\mathcal { H } ^ d _ { \\phantom { x } | t = 0 } = \\int _ \\Sigma c \\ , \\mathrm { d } \\mathcal { H } ^ d = m \\end{align*}"}
+{"id": "5195.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } ( a _ { t } ^ { \\varepsilon } + ( B \\times u ) ^ { \\varepsilon } + \\mu \\nabla \\times b ^ { \\varepsilon } ) \\cdot \\nabla \\times \\zeta \\dd x = 0 . \\end{align*}"}
+{"id": "5708.png", "formula": "\\begin{align*} a x ^ q - b x = 0 \\end{align*}"}
+{"id": "1105.png", "formula": "\\begin{align*} R _ { b \\to s } ^ { \\rm { N } } = W { \\log _ 2 } \\left ( { 1 + \\frac { { { p _ b } { { \\left | { { h _ s } } \\right | } ^ 2 } } } { { { p _ s } { { \\left | { { h _ s } } \\right | } ^ 2 } + W { N _ 0 } } } } \\right ) . \\end{align*}"}
+{"id": "3334.png", "formula": "\\begin{align*} \\chi _ 1 = \\psi _ 1 , \\ldots , \\chi _ { j - 1 } = \\psi _ { j - 1 } , \\chi _ j = \\sigma < _ j \\psi _ j . \\end{align*}"}
+{"id": "1103.png", "formula": "\\begin{align*} { R ^ { \\rm { O } } } = { W _ b } { \\log _ 2 } \\left ( { 1 + \\frac { { { p _ b } { { \\left | { { h _ b } } \\right | } ^ 2 } } } { { { W _ b } { N _ 0 } } } } \\right ) . \\end{align*}"}
+{"id": "1445.png", "formula": "\\begin{align*} \\mathbf { x } \\left [ l \\right ] = \\mathbf { P } \\mathbf { s } \\left [ l \\right ] = \\mathbf { p } _ { \\mathrm { c } } s _ { \\mathrm { c } } \\left [ l \\right ] + \\sum _ { k \\in \\mathcal { K } } \\mathbf { p } _ { \\mathrm { p } , k } s _ { \\mathrm { p } , k } \\left [ l \\right ] , \\end{align*}"}
+{"id": "4075.png", "formula": "\\begin{align*} G _ \\infty = 2 c _ H ^ 2 T ^ { 4 H - 1 } \\int ^ T _ 0 ( V ^ { [ 1 ] } ( X _ { t } ) ) ^ 4 d t . \\end{align*}"}
+{"id": "5164.png", "formula": "\\begin{align*} \\rho _ n = \\left ( | \\nabla \\phi _ n | ^ { 2 } + | G _ n | ^ { 2 } \\right ) \\frac { 2 \\pi } { r } . \\end{align*}"}
+{"id": "5986.png", "formula": "\\begin{align*} U _ { \\tau , R } : = \\{ x \\in \\R ^ 3 : | x \\cdot { \\bf { c } } ( l \\sigma ^ { - 1 } R ^ { - 1 / 2 } ) | \\le \\sigma ^ { - 2 } \\quad & \\quad | x \\cdot { \\bf { n } } ( l \\sigma ^ { - 1 } R ^ { - 1 / 2 } ) | \\le R \\\\ & \\qquad | x \\cdot { \\bf { t } } ( l \\sigma ^ { - 1 } R ^ { - 1 / 2 } ) | \\le \\sigma ^ { - 1 } R ^ { 1 / 2 } \\} . \\end{align*}"}
+{"id": "6396.png", "formula": "\\begin{align*} g = D ^ 2 u . \\end{align*}"}
+{"id": "9174.png", "formula": "\\begin{align*} \\begin{aligned} | x _ a ( t ) - x ^ \\star | & \\leq \\beta ( | x _ a ( 0 ) | _ { { \\cal A } _ \\delta } , 0 ) + \\delta \\leq \\beta ( | x _ a ( 0 ) - x ^ \\star | , 0 ) + \\delta \\\\ & \\leq \\beta ( r _ 0 , 0 ) + \\delta = r - d - \\delta \\end{aligned} \\end{align*}"}
+{"id": "6801.png", "formula": "\\begin{align*} \\left ( \\left ( I + \\mathbf { H } ( \\varphi ) \\right ) y \\right ) \\left ( \\rho \\right ) = - \\mathbf { H } ( \\varphi ) \\left ( 1 \\right ) \\left ( \\rho \\right ) \\end{align*}"}
+{"id": "4097.png", "formula": "\\begin{align*} ( \\Theta ^ { r ( p _ 1 , \\ldots , p _ { r - 1 } ) } ) ^ i { } _ { j _ 1 , \\ldots , j _ { r - 1 } } & = - v ^ i { } _ \\alpha u ^ \\alpha { } _ { j _ 1 , \\ldots , \\underset { \\stackrel { \\frown } { p _ { r - 1 } } } { l } , \\ldots , j _ { r - 1 } , \\beta } v ^ \\beta { } _ \\gamma d u ^ \\gamma \\wedge v ^ l { } _ \\delta d u ^ \\delta \\\\ * & \\hphantom { { } = { } } + ( \\textit { t e r m s w h i c h d o n o t i n v o l v e $ u ^ i { } _ { j _ 1 , \\ldots , j _ r } $ } ) . \\end{align*}"}
+{"id": "9109.png", "formula": "\\begin{align*} ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + r ( j - 1 ) < \\chi _ j < ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + r j , \\end{align*}"}
+{"id": "1692.png", "formula": "\\begin{align*} \\varphi ( \\tau ) = \\left \\{ \\begin{array} { l l } \\ell _ n ^ \\top ( \\tau ) \\Phi _ n , & \\forall \\tau \\in [ - h , 0 ) , \\\\ x , \\quad & \\tau = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "3560.png", "formula": "\\begin{align*} c _ { \\Omega _ 4 } ^ { \\Omega _ 4 } & = q ^ { - 1 / 6 } \\varphi _ 1 ^ { - 4 } ( \\chi _ { 3 , 4 } ^ { 1 , 2 } \\chi _ { 4 , 5 } ^ { 2 , 1 } ) \\\\ & = q ^ { - 1 / 6 } \\varphi _ 1 ^ { - 4 } ( q ^ { 1 / 2 4 } V ( q ) ^ { - 1 } ) ( q ^ { 4 9 / 1 2 0 } H _ 2 V ( q ) ^ { - 1 } ) \\\\ & = q ^ { 1 7 / 6 0 } \\varphi _ 1 ^ { - 4 } ( \\varphi _ 1 ^ { - 2 } \\varphi _ 2 ^ 2 ) H _ 2 \\\\ & = q ^ { 1 7 / 6 0 } \\varphi _ 1 ^ { - 6 } \\varphi _ 2 ^ 2 H _ 2 . \\end{align*}"}
+{"id": "1299.png", "formula": "\\begin{align*} \\frac { N ( x _ a ; x _ b , x _ c , x _ d ; \\alpha , \\beta ) } { D ( x _ a ; x _ b , x _ c , x _ d ; \\alpha , \\beta ) } = 1 , \\end{align*}"}
+{"id": "8294.png", "formula": "\\begin{align*} \\gamma ^ i ( x ) = \\frac 1 { N - 1 } \\sum _ { j \\neq i } K ( x ^ i - x ^ j ) . \\end{align*}"}
+{"id": "3094.png", "formula": "\\begin{align*} \\Pi _ { \\mathcal { K } , G } ( \\theta ) = \\big \\{ \\bar \\theta \\in \\mathcal { K } \\ \\big | \\ \\norm { \\bar \\theta - \\theta } _ { \\mathcal { L } ^ 2 , G } = \\min _ { \\vartheta \\in \\mathcal { K } } \\norm { \\vartheta - \\theta } _ { \\mathcal { L } ^ 2 , G } \\big \\} . \\end{align*}"}
+{"id": "1067.png", "formula": "\\begin{align*} 2 \\rho - s _ \\alpha ( 2 \\rho ) = & \\sum _ { \\beta \\in I } \\left ( \\beta - s _ \\alpha ( \\beta ) \\right ) + \\sum _ { \\beta \\in \\Phi ^ + \\setminus I } \\left ( \\beta - s _ \\alpha ( \\beta ) \\right ) \\\\ = & 2 \\sum _ { \\beta \\in I } \\beta . \\end{align*}"}
+{"id": "5136.png", "formula": "\\begin{align*} \\dot { s } ( 0 ) = - \\frac { \\partial _ { \\tau } j ( 0 , 0 ) } { \\partial _ { s } j ( 0 , 0 ) } . \\end{align*}"}
+{"id": "8650.png", "formula": "\\begin{align*} \\begin{aligned} y \\left [ i \\right ] & = \\sum \\limits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { \\bf { d } } \\left [ { i - { n _ l } } \\right ] } + z \\left [ i \\right ] \\\\ & = \\sum \\limits _ { t = 0 } ^ { { n _ { { \\rm { s p a n } } } } } { { { \\bf { g } } ^ H } \\left [ t \\right ] { \\bf { d } } \\left [ { i - t - { n _ { \\min } } } \\right ] } + z \\left [ i \\right ] , \\end{aligned} \\end{align*}"}
+{"id": "4376.png", "formula": "\\begin{align*} v _ \\epsilon { ' } ( t ) = \\int _ { - \\infty } ^ { t } ( \\frac { 1 } { B - 4 \\epsilon } \\mathbb { I } _ { ( - t _ 0 - B + 2 \\epsilon , - t _ 0 - 2 \\epsilon ) } * \\rho _ { \\frac { 1 } { 4 } \\epsilon } ) ( s ) d s . \\end{align*}"}
+{"id": "5055.png", "formula": "\\begin{align*} \\nabla p = B \\cdot \\nabla B - \\nu ( - \\Delta ) ^ { \\kappa } u . \\end{align*}"}
+{"id": "7998.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } V _ 1 ( u _ n ) = \\frac { 1 } { 4 } V _ 2 ( u _ n ) + \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { p } \\right ) | u _ n | ^ p _ p - c + o ( 1 ) \\leq C | u _ n | ^ 4 _ { \\frac { 8 } { 3 } } + C | u _ n | ^ p _ p - c + o ( 1 ) \\leq C , \\end{align*}"}
+{"id": "535.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\\\ \\frac { 1 } { 2 } & \\frac { 1 } { 2 } & 0 & 1 & 1 & - 1 \\\\ \\frac { 1 } { 2 } & 0 & \\frac { 1 } { 2 } & 1 & 1 & - 1 \\\\ - \\frac { 3 } { 4 } & \\frac { 3 } { 8 } & \\frac { 3 } { 8 } & \\frac { 3 } { 2 } & - 1 & 1 \\\\ 0 & 0 & 0 & 0 & \\frac { 3 } { 2 } & - 1 \\\\ 0 & 0 & 0 & 0 & - 1 & \\frac { 3 } { 2 } \\end{pmatrix} \\end{align*}"}
+{"id": "8733.png", "formula": "\\begin{align*} \\nu _ { y _ 2 } = \\mu * \\delta _ { s ( y _ 2 ) } = \\mu * \\delta _ { g k _ 0 s ( y _ 1 ) } = g \\left ( \\mu * k _ 0 \\delta _ { s ( y _ 1 ) } \\right ) = g \\nu _ { y _ 1 } . \\end{align*}"}
+{"id": "329.png", "formula": "\\begin{align*} L _ { 1 2 } \\leq & C \\Big ( \\int _ { 1 } ^ { \\infty } \\Big ( \\frac { \\sqrt { t } } { \\rho ( x ) } \\Big ) ^ { 2 ( M - n ) } \\Big ( 1 + \\frac { \\sqrt { t } } { \\rho ( x ) } \\Big ) ^ { - 2 N } \\Big ) ^ { 1 / 2 } M _ { V , \\eta } f ( x ) \\\\ & \\leq C M _ { V , \\eta } f ( x ) . \\end{align*}"}
+{"id": "2405.png", "formula": "\\begin{align*} \\displaystyle x ^ { p } = ( I _ { n } + p ^ { m } a ) ^ { p } = I _ { n } + \\sum _ { k = 1 } ^ { p } { p \\choose k } p ^ { m k } a ^ { m } \\equiv I _ { n } \\bmod p ^ { m + 1 } \\ , \\end{align*}"}
+{"id": "6008.png", "formula": "\\begin{align*} E _ { \\omega } : = \\min \\{ J _ { \\omega } ( u ) : u \\in H _ { r } ^ { 1 } ( \\R ^ { 2 } ) \\setminus \\{ 0 \\} , J ' _ { \\omega } ( u ) = 0 \\} , \\end{align*}"}
+{"id": "3047.png", "formula": "\\begin{align*} b \\left ( \\ell \\left ( \\pi \\right ) \\alpha \\left ( t \\right ) \\right ) \\left ( z \\right ) = R _ { \\pi \\alpha \\left ( t \\right ) } \\left ( z \\right ) = \\alpha \\left ( t \\right ) z \\overline { \\alpha \\left ( t \\right ) } \\end{align*}"}
+{"id": "6420.png", "formula": "\\begin{align*} \\Delta _ g \\phi & = g ^ { p q } \\phi _ { , p q } - \\frac { 1 } { 2 } \\xi ^ p \\phi _ { , p } + \\frac { 1 } { 2 } g ^ { p q } v _ p \\phi _ { , q } \\end{align*}"}
+{"id": "5795.png", "formula": "\\begin{align*} a = - \\frac { 1 } { \\mu } \\left ( \\partial _ y \\mu \\ d x - \\partial _ x \\mu \\ d y \\right ) = ( \\frac { 1 } { \\mu } d \\mu - \\frac { 2 } { \\mu } \\mu _ z d z ) I \\end{align*}"}
+{"id": "5625.png", "formula": "\\begin{align*} f \\left ( x , \\frac { 1 } { 2 ^ n } \\right ) = \\frac { h _ n ( x ) } { 2 ^ { n + 1 } } f ( x , 0 ) = 0 . \\end{align*}"}
+{"id": "1562.png", "formula": "\\begin{align*} v ^ k \\gamma v ^ { - k } ( a , x ) = ( R _ v ^ k R _ \\gamma R _ v ^ { - k } a + R _ v ^ k t _ \\gamma , x ) . \\end{align*}"}
+{"id": "6890.png", "formula": "\\begin{align*} \\log \\int _ { \\R ^ n } e ^ { f ( x ) } \\ , d x - \\int _ { \\R ^ n } f \\ , d Q + H ( Q ) = H ( Q \\ , | \\ , P ) \\end{align*}"}
+{"id": "3729.png", "formula": "\\begin{align*} g ( \\omega ( Y ) , Z ) + g ( \\omega ( Z ) , Y ) = h ( Y , Z ) \\end{align*}"}
+{"id": "6019.png", "formula": "\\begin{align*} c _ { 1 } : = \\mathop { \\inf } \\limits _ { ( u , v ) \\in E \\setminus \\{ ( 0 , 0 ) \\} } \\mathop { \\max } \\limits _ { t \\geq 0 } I ( t u , t v ) . \\end{align*}"}
+{"id": "7305.png", "formula": "\\begin{align*} P _ j = \\sum \\limits _ { \\forall v } \\sum \\limits _ { \\forall q } p _ { v , j } ^ q + \\sum \\limits _ { \\forall k , k \\ne 0 } \\sum \\limits _ { \\forall m } p _ { k , j } ^ m + \\sum \\limits _ { \\forall n } p _ { 0 , j } ^ n , \\end{align*}"}
+{"id": "5866.png", "formula": "\\begin{align*} Z G \\left ( X , Y \\right ) = G \\left ( \\nabla _ { Z } X , Y \\right ) + G \\left ( X , \\nabla _ { Z } ^ { \\dag } Y \\right ) , \\end{align*}"}
+{"id": "2453.png", "formula": "\\begin{align*} f ( t ) = \\lim _ { k \\rightarrow \\infty } \\frac { ( - 1 ) ^ { k } } { \\Gamma ( k + 1 ) } F _ { q } ^ { ( k ) } ( s ) ( 2 - q ) s ^ { k + 1 } | _ { s = \\frac { k \\xi _ { m } } { t } } . \\end{align*}"}
+{"id": "4654.png", "formula": "\\begin{align*} T _ h ^ { ( N ) } : L _ { p _ 1 } ( \\mathbb { R } , S _ { q _ 1 } ^ N ) \\times L _ { p _ 2 } ( \\mathbb { R } , S _ { q _ 2 } ^ N ) \\rightarrow L _ { p } ( \\mathbb { R } , S _ { q } ^ N ) \\end{align*}"}
+{"id": "9116.png", "formula": "\\begin{align*} \\frac { \\sum \\limits _ { i = 1 } ^ { j + 1 } \\chi _ i - r ( j + 1 ) } { \\chi } - \\sum \\limits _ { i = 1 } ^ j w _ i > w _ { j + 1 } > \\frac { \\sum \\limits _ { i = 1 } ^ { j + 1 } \\chi _ i - r j } { \\chi } - \\sum \\limits _ { i = 1 } ^ j w _ i . \\end{align*}"}
+{"id": "2151.png", "formula": "\\begin{align*} \\Re \\mathcal { P } [ \\varrho ] = - \\frac { 1 } { 4 } a \\left ( 2 a - \\left ( 5 a ^ 2 - 6 a + 4 \\right ) \\tau + ( 8 - 8 a ) \\tau ^ 2 - \\tau ^ 3 \\right ) < 0 \\end{align*}"}
+{"id": "7247.png", "formula": "\\begin{align*} L ( \\lambda \\alpha ) = \\lambda L ( \\alpha ) = 0 \\end{align*}"}
+{"id": "1323.png", "formula": "\\begin{align*} f _ i ( x ) : = \\begin{cases} & e ^ { - i / x } \\\\ & 0 \\end{cases} \\end{align*}"}
+{"id": "3924.png", "formula": "\\begin{align*} \\phi ^ { \\zeta \\theta } ( e _ { v w } ) : = ( \\phi ^ \\zeta ) ^ \\theta ( e _ { v w } ) & = \\tau ^ \\zeta ( v , e ) \\theta ( v ) + \\phi ^ \\zeta ( e _ { v w } ) + \\tau ^ \\zeta ( w , e ) \\theta ( w ) \\\\ & = \\zeta ( v ) \\tau ( v , e ) \\theta ( v ) + \\phi ( e _ { v w } ) + \\zeta ( w ) \\tau ( w , e ) \\theta ( w ) \\\\ & = \\tau ( v , e ) \\theta ^ \\zeta ( v ) + \\phi ( e _ { v w } ) + \\tau ( w , e ) \\theta ^ \\zeta ( w ) \\\\ & = \\phi ^ { \\theta ^ \\zeta } ( e _ { v w } ) = \\phi ^ { \\theta ^ \\zeta \\zeta } ( e _ { v w } ) \\end{align*}"}
+{"id": "9271.png", "formula": "\\begin{align*} \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ k \\wedge \\beta _ n ^ { n - m } ( \\omega ) = \\int u _ k \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ { k - 1 } \\wedge \\beta _ n ^ { n - m } \\wedge \\Delta \\omega , k = 1 , \\dots , m , \\end{align*}"}
+{"id": "6273.png", "formula": "\\begin{align*} \\alpha ^ h ( \\partial _ j , \\partial _ k ) = 0 \\quad \\forall j \\neq k . \\end{align*}"}
+{"id": "9150.png", "formula": "\\begin{align*} \\dot { { x } } _ a & = - \\gamma \\dfrac { b _ { 1 , \\delta } ( x _ a ) } { 2 } \\\\ \\dot { \\bar { y } } _ a & = - \\gamma \\ , \\bar { y } _ a + \\gamma \\ , \\dfrac { a _ { 0 , \\delta } ( { x } _ a ) } { 2 } \\end{align*}"}
+{"id": "3119.png", "formula": "\\begin{align*} \\varrho ^ \\star ( [ [ x , y ] , \\alpha ( z ) ] ) ( \\beta ^ { - 2 } ) ^ * & = \\varrho ^ { \\star } ( \\alpha ^ { 2 } ( z ) ) \\varrho ^ { \\star } ( \\alpha ( y ) ) \\varrho ^ { \\star } ( x ) - \\varrho ^ { \\star } ( \\alpha ^ { 2 } ( y ) ) \\varrho ^ { \\star } ( \\alpha ( x ) ) \\varrho ^ { \\star } ( z ) \\\\ & \\quad - \\varrho ^ { \\star } ( \\alpha ( [ z , x ] ) ) \\varrho ^ { \\star } ( \\alpha ( y ) ) ( \\beta ^ { - 1 } ) ^ * + \\varrho ^ { \\star } ( \\alpha ^ { 2 } ( x ) ) \\varrho ^ { \\star } ( [ y , z ] ) ( \\beta ^ { - 1 } ) ^ * . \\end{align*}"}
+{"id": "5004.png", "formula": "\\begin{align*} t = \\sum _ { j = 1 } ^ \\infty \\frac { b _ j } { 2 ^ j } \\end{align*}"}
+{"id": "40.png", "formula": "\\begin{align*} B _ { r } : = \\left \\{ g \\in \\mathbb { G } \\ ; | \\ ; \\Gamma ( g , e ) > \\frac { 1 } { r ^ { Q - 2 } } \\right \\} . \\end{align*}"}
+{"id": "6473.png", "formula": "\\begin{align*} | | \\xi | | _ p = [ { \\bf E } | \\xi ^ p | ] ^ { 1 / p } \\le C \\ \\frac { p } { p - 1 } \\ \\beta ^ p \\ | X | _ p , \\ p > 1 . \\end{align*}"}
+{"id": "2741.png", "formula": "\\begin{align*} \\begin{aligned} T ( s ) & = [ 2 s _ 1 - 1 , 2 s _ 1 + 1 ] \\times [ 2 s _ 2 - 1 , 2 s _ 2 + 1 ] \\times \\R \\\\ & = \\{ x \\in \\R ^ 3 \\colon | x _ j - 2 s _ j | \\leq 1 \\ \\ j \\in \\{ 1 , 2 \\} \\} . \\end{aligned} \\end{align*}"}
+{"id": "1150.png", "formula": "\\begin{align*} { S } ^ { \\rm P R \\ , s t r u c t } ( { \\cal A } ) = \\kappa ( \\rho ) W | _ { \\mathcal S } \\in \\R ^ { N L \\times N L } \\ , \\ , , \\end{align*}"}
+{"id": "7969.png", "formula": "\\begin{align*} \\frac { \\partial h } { \\partial t } \\leq \\frac { 1 } { f _ { \\min } } h [ h ^ { - \\varepsilon } \\beta _ { 0 } - f _ { \\min } ] . \\end{align*}"}
+{"id": "4565.png", "formula": "\\begin{align*} a _ { \\ell + 2 , 0 , 0 } - { q } ^ { 1 / 2 } \\sigma _ 1 ( \\textbf { z } ) a _ { \\ell + 1 , 0 , 0 } + q \\sigma _ 2 ( \\textbf { z } ) a _ { \\ell , 0 , 0 } - q ^ { 3 / 2 } \\sigma _ 3 ( \\textbf { z } ) a _ { \\ell - 1 , 0 , 0 } + q ^ 2 \\sigma _ 4 ( \\textbf { z } ) a _ { \\ell - 2 , 0 , 0 } = 0 . \\end{align*}"}
+{"id": "7165.png", "formula": "\\begin{align*} \\Lambda _ { g } = A \\Bigl ( - \\frac { \\partial } { \\partial x _ n } \\Bigr ) - D , \\end{align*}"}
+{"id": "5224.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\dot { \\rho } ( t ) \\left ( \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { + } \\frac { G } { r ^ { 2 } } \\dd x \\right ) \\dd t = - \\rho ( 0 ) \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } \\frac { G _ 0 } { r ^ { 2 } } \\dd x . \\end{align*}"}
+{"id": "7385.png", "formula": "\\begin{align*} v _ \\beta = y ^ { 2 ( r _ 0 + \\cdots + r _ { a ( \\beta ) } - \\beta + 1 ) } . \\end{align*}"}
+{"id": "5286.png", "formula": "\\begin{align*} \\lim _ { z \\downarrow 0 } \\hat { F } _ { \\pm } ( 0 ) \\int _ 0 ^ \\infty w _ z ( \\lambda ) \\lambda ^ { 4 ( 1 - s + w ) } \\frac { d \\lambda } { \\lambda } \\int _ { G ^ 1 / \\Gamma } \\phi ( g ^ { - 1 } ) d g = 0 \\end{align*}"}
+{"id": "3413.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} \\phi ( k + 1 ) \\\\ \\phi ( k ) \\end{matrix} \\right ) = A ( \\theta + k \\alpha , E ) \\left ( \\begin{matrix} \\phi ( k ) \\\\ \\phi ( k - 1 ) \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "5939.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\lim _ { n \\rightarrow \\infty } I I = 0 . \\end{align*}"}
+{"id": "7193.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } \\operatorname { T r } \\textbf { \\textit { K } } ( t , x ^ { \\prime } , x ^ { \\prime } ) \\ , d S = \\sum _ { k = 1 } ^ { \\infty } e ^ { - t \\tau _ k } \\end{align*}"}
+{"id": "6765.png", "formula": "\\begin{align*} - y ^ { \\prime \\prime } + q ( x ) y = \\lambda y , \\quad - \\infty < x < \\infty \\end{align*}"}
+{"id": "5953.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathbb { T } ^ d } \\langle a ( u _ n ( x ) ) \\nabla u _ n ( x ) + b ( u _ n ( x ) ) , \\nabla u _ n ( x ) - \\nabla v ( x ) \\rangle d x \\\\ = & \\int _ { \\mathbb { T } ^ d } \\langle a ( u ( x ) ) \\nabla u ( x ) + b ( u ( x ) ) , \\nabla u ( x ) - \\nabla v ( x ) \\rangle d x . \\end{align*}"}
+{"id": "7055.png", "formula": "\\begin{align*} u \\ , = \\ , F ( x , y , z ) \\ \\ \\ \\ \\ \\ \\ \\ { \\scriptstyle { ( F ( 0 , 0 , 0 ) \\ , = \\ , 0 ) } } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ v \\ , = \\ , G ( r , s , t ) \\ \\ \\ \\ \\ \\ \\ \\ { \\scriptstyle { ( 0 \\ , = \\ , G ( 0 , 0 , 0 ) ) } } , \\end{align*}"}
+{"id": "655.png", "formula": "\\begin{align*} f ( s ) = 2 + e ^ { - 3 s } ( s + 1 ) \\left ( 1 - \\frac { s ^ 2 } { 2 } - \\frac { s ^ 3 } { 3 } - \\frac { s ^ 4 } { 8 } \\right ) + e ^ { - 2 s } \\left ( 2 s + \\frac { 5 s ^ 2 } { 2 } + \\frac { s ^ 3 } { 2 } \\right ) - e ^ { - s } \\left ( 3 + 2 s \\right ) . \\end{align*}"}
+{"id": "4580.png", "formula": "\\begin{align*} x ( t ) = S _ { t } x _ { 0 } + \\int _ { 0 } ^ { t } S _ { t - r } B u ( r ) \\d r ( t \\in [ 0 , T ] ) \\end{align*}"}
+{"id": "7399.png", "formula": "\\begin{align*} \\langle a , b \\rangle : = ( f | [ a , b ] ) . \\end{align*}"}
+{"id": "6922.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\partial _ i f ( x ) \\big ( T _ i ( x _ i ) - x _ i \\big ) Q ( x ) d x & = \\int _ { \\R } \\hat { f } ' _ i ( x _ i ) \\big ( T _ i ( x _ i ) - x _ i \\big ) Q _ i ( x _ i ) d x _ i , \\end{align*}"}
+{"id": "6738.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } \\sigma } \\ell = & - \\frac { 1 } { \\sigma } - \\frac { \\alpha - 1 } { \\sigma ^ 2 } ( x - \\mu ) \\frac { \\phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) } { \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) } + \\frac { \\beta - 1 } { \\sigma ^ 2 } ( x - \\mu ) \\frac { \\phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) } { 1 - \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) } - \\frac { s } { \\sigma } \\left | \\frac { x - \\mu } { \\sigma } \\right | ^ s , \\end{align*}"}
+{"id": "165.png", "formula": "\\begin{align*} C ( G , F ) : = C ( G , K ) \\widehat { \\otimes } _ { K } F . \\end{align*}"}
+{"id": "1282.png", "formula": "\\begin{align*} s _ n ( \\nu | \\mu ) : = \\sum _ { \\eta \\in \\Omega _ { \\Lambda _ n } } \\sum _ { \\Delta \\subset \\Tilde { \\Lambda } _ n } \\sum _ { \\xi _ { \\Delta } \\neq \\eta _ { \\Delta } } f ( \\nu , \\Lambda _ n , \\eta , \\xi _ { \\Delta } ) \\hat { c } ^ { B _ { n - 1 } ( x ( \\Delta ) ) } _ { \\Delta } ( \\xi _ { \\Delta } \\eta _ { \\Delta ^ c } , \\eta _ { \\Delta } ) , \\end{align*}"}
+{"id": "1327.png", "formula": "\\begin{align*} \\omega ( v _ H , - ) = d H \\end{align*}"}
+{"id": "659.png", "formula": "\\begin{align*} \\Box A _ \\nu - \\mu ^ 2 A _ \\nu = 0 ~ , \\nabla ^ \\nu A _ \\nu = 0 \\ , , \\end{align*}"}
+{"id": "9197.png", "formula": "\\begin{align*} \\epsilon ( \\eta _ a , t ) : = \\int _ 0 ^ t F ( \\eta _ a , \\tau ) - F _ a ( \\eta _ a ) \\ , d \\tau \\end{align*}"}
+{"id": "3748.png", "formula": "\\begin{align*} \\psi \\left ( d _ - ^ { \\ell } L ( 0 v , 0 w ) \\right ) & = \\psi \\left ( t ^ { - \\ell } d _ { - } ^ { \\ell + 1 } L ( v 1 , w 1 ) + q t ^ { - \\ell } d _ - ^ { \\ell } L ( v 0 , w 0 ) \\right ) \\\\ & = t ^ { - \\ell } \\psi \\left ( d _ { - } ^ { \\ell + 1 } L ( v 1 , w 1 ) \\right ) + q t ^ { - \\ell } \\left ( d _ - ^ { \\ell } L ( v 0 , w 0 ) \\right ) \\\\ & = t ^ { - \\ell } p ( v 1 , w 1 ) + q t ^ { - \\ell } p ( v 0 , w 0 ) \\\\ & = p ( 0 v , 0 w ) . \\end{align*}"}
+{"id": "6440.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { \\begin{subarray} { l } k _ 1 + \\cdots + k _ n = k \\\\ k _ i \\in \\mathbb { Z } _ { \\ge 1 } ( i = 1 , \\ldots , n - 1 ) , \\\\ k _ n \\in \\mathbb { Z } _ { \\ge 2 } \\end{subarray} } Z ^ { \\star } _ { I } ( \\{ k _ i \\} _ { i = 1 } ^ { n } ; ( \\alpha , \\beta ) ) \\\\ = & \\binom { k - 2 } { n - 1 } Z ( n - 1 | k - n + 1 ; ( \\alpha , \\beta ) ) + \\binom { k - 2 } { n - 2 } Z ( n | k - n ; ( \\alpha , \\beta ) ) \\end{aligned} \\end{align*}"}
+{"id": "1061.png", "formula": "\\begin{align*} \\overline { I x I y I } = \\overline { I ( x \\ast y ) I } \\end{align*}"}
+{"id": "3381.png", "formula": "\\begin{align*} Z ( x , y ; N , K ) = 1 + S ( x , N ) + S ( y , K ) . \\end{align*}"}
+{"id": "8377.png", "formula": "\\begin{align*} \\Phi ( x ; z ) = A ( x ; k ) \\psi ( x ; k ) B ( x , k ) , \\end{align*}"}
+{"id": "2754.png", "formula": "\\begin{align*} \\prod _ { j = M } ^ { m - 1 } | h _ { j } ( x ) | \\leq \\prod _ { j = M } ^ { m - 1 } | h _ { m } ( x ) | ^ { C ^ { j - m } } = | h _ { m } ( x ) | ^ { \\gamma _ m } \\end{align*}"}
+{"id": "2895.png", "formula": "\\begin{align*} { g _ { x , x ' , y } ( s ) : = \\sum _ { \\ell = 0 } ^ { + \\infty } \\Big [ e ^ { - A ( s + \\ell \\theta ) } \\Big ] _ { x + n + 1 , y + n + 1 } \\Big [ e ^ { - A ^ T ( s + \\ell \\theta ) } \\Big ] _ { y + n + 1 , x ' + n + 1 } . } \\end{align*}"}
+{"id": "1197.png", "formula": "\\begin{align*} M _ { N C } ( n , d ) \\le \\left ( 2 \\sqrt { \\frac { 2 } { d + 1 } } - \\frac { 2 } { d + 1 } \\right ) \\cdot 2 ^ d \\binom { n } { d } \\enspace . \\end{align*}"}
+{"id": "6908.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } f \\ , d Q - H ( Q ) = - H ( Q \\ , | \\ , P ) + \\log Z . \\end{align*}"}
+{"id": "4692.png", "formula": "\\begin{align*} { \\tilde \\Sigma } ^ t = \\{ x \\in M \\mid { \\rm d i s t } ( x , \\tilde \\Sigma ) \\leq t \\} . \\end{align*}"}
+{"id": "1419.png", "formula": "\\begin{align*} \\P ( \\tau > n ) & = \\int _ { W ^ d } \\det ( q _ { n + i - j } ( y _ j - x _ i ) ) _ { i , j = 1 } ^ d \\prod _ { i = 1 } ^ d \\lambda _ i ^ n e ^ { - \\sum _ { i = 1 } ^ d \\lambda _ i ( y _ i - x _ i ) } d y _ 1 \\ldots d y _ d , \\\\ \\P ( \\rho > n ) & = \\int _ { W ^ d } \\det ( q _ { n } ( y _ j - x _ i ) ) _ { i , j = 1 } ^ d \\prod _ { i = 1 } ^ d \\lambda _ i ^ n e ^ { - \\sum _ { i = 1 } ^ d \\lambda _ i ( y _ i - x _ i ) } d y _ 1 \\ldots d y _ d . \\end{align*}"}
+{"id": "914.png", "formula": "\\begin{align*} g ( u ) = e \\left ( \\frac { m \\sqrt { X } } { u ^ 2 + u v + v ^ 2 } - \\frac { m u } { v ( u ^ 2 + u v + v ^ 2 ) } \\right ) \\ , , \\end{align*}"}
+{"id": "2517.png", "formula": "\\begin{align*} ( \\mathbf { D } ^ { - T } \\mathbf { d } _ { h x } ) ^ { T } ( \\mathbf { D } \\mathbf { d } _ { h s } ) = \\mathbf { d } _ { h x } ^ { T } \\mathbf { d } _ { h s } = 0 . \\end{align*}"}
+{"id": "2340.png", "formula": "\\begin{align*} \\begin{cases} 2 a ^ 2 + 2 | v | ^ 2 + b \\cdot v = 0 \\\\ 2 a b + A ^ t b + 2 A ^ t v = 0 \\end{cases} \\end{align*}"}
+{"id": "1510.png", "formula": "\\begin{align*} g ( s ) : = G ( s , s ) = \\frac { ( s - a ) ^ { \\alpha - 1 } ( b - s ) ^ { \\alpha - 1 - \\beta } } { ( b - a ) ^ { \\alpha - 1 - \\beta } \\Gamma ( \\alpha ) } . \\end{align*}"}
+{"id": "3642.png", "formula": "\\begin{align*} L _ { x } & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n L _ { x , i } , ~ \\mu _ { x } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mu _ { x , i } , ~ \\sigma _ f ^ 2 = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sigma _ { f , i } ^ 2 , \\\\ L _ { y } & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n L _ { y , i } , ~ \\mu _ { y } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mu _ { y , i } , ~ \\sigma _ g ^ 2 = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sigma _ { g , i } ^ 2 \\end{align*}"}
+{"id": "6093.png", "formula": "\\begin{align*} j _ n ^ { ( 1 2 ) } = n + a _ { 1 2 } \\ , , \\end{align*}"}
+{"id": "6931.png", "formula": "\\begin{align*} \\frac { 1 } { \\binom { n } { k } } \\sum _ { S \\subset [ n ] , \\ , | S | = k } \\mathcal { W } _ 2 ^ 2 ( P _ { S } , Q _ { S } ) = \\frac { 1 } { | \\Pi | } \\sum _ { ( S _ 1 , \\ldots , S _ m ) \\in \\Pi } \\frac { 1 } { m } \\sum _ { i = 1 } ^ m \\mathcal { W } _ 2 ^ 2 ( P _ { S _ i } , Q _ { S _ i } ) . \\end{align*}"}
+{"id": "7091.png", "formula": "\\begin{align*} U ^ * f ^ z _ { i j } U & = e ^ z _ { k j } 1 \\leq k , j \\leq n , \\\\ U ^ * f ^ z _ { ( n + 1 ) k } U & = \\frac { d } { \\norm { d } } e ^ z _ { ( n + 1 ) k } 1 \\leq k \\leq n , \\\\ U ^ * f ^ z _ { k ( n + 1 ) } U & = e ^ z _ { k ( n + 1 ) } \\frac { d ^ * } { \\norm { d } } 1 \\leq k \\leq n , \\\\ U ^ * f ^ z _ { ( n + 1 ) ( n + 1 ) } U & = \\frac { d } { \\norm { d } } e ^ z _ { ( n + 1 ) ( n + 1 ) } \\frac { d ^ * } { \\norm { d } } . \\end{align*}"}
+{"id": "1488.png", "formula": "\\begin{align*} \\sup \\{ \\Re \\lambda : \\lambda \\in \\operatorname { S p } ( \\mathcal { L } _ { \\alpha } ) \\} & = \\sup \\{ \\Re \\lambda : \\lambda \\in \\operatorname { S p } ( \\hat { \\mathcal { L } } ) \\} \\\\ & = \\max ( \\alpha ^ 2 - c \\alpha , \\epsilon \\alpha ^ 2 - c \\alpha - \\kappa e ^ { - \\kappa } ) \\\\ & = \\alpha ^ 2 - c \\alpha . \\end{align*}"}
+{"id": "4315.png", "formula": "\\begin{align*} \\mathcal { H } ^ 2 ( t ; c , f , H ) : = \\Bigg \\{ \\tilde { f } : \\int _ { \\{ \\Psi < - t \\} } | \\tilde { f } | ^ 2 _ h c ( - \\Psi ) < + \\infty , \\ \\tilde { f } \\in H ^ 0 ( \\{ \\Psi < - t \\} , \\mathcal { O } ( K _ M \\otimes E ) ) \\\\ \\& ( \\tilde { f } - f ) _ { z _ 0 } \\in \\mathcal { O } ( K _ M ) _ { z _ 0 } \\otimes ( J _ { z _ 0 } \\cap H _ { z _ 0 } ) , \\ z _ 0 \\in Z _ 0 \\Bigg \\} , \\end{align*}"}
+{"id": "736.png", "formula": "\\begin{align*} \\frac { C } { \\sqrt { R } } - \\frac { C ^ 2 } { 4 R ^ 2 } = \\frac { C } { \\sqrt { R } } \\left [ 1 - \\frac { C } { 4 R \\sqrt { R } } \\right ] \\geq 0 . \\end{align*}"}
+{"id": "5440.png", "formula": "\\begin{align*} V i r = \\mathbb { C } [ \\partial ] L , [ L _ \\lambda L ] = ( \\partial + 2 \\lambda ) L . \\end{align*}"}
+{"id": "5963.png", "formula": "\\begin{align*} I V = - \\Vert \\Delta ( n - \\widetilde { n } ) \\Vert _ { L ^ 2 } ^ 2 = - \\Vert n - \\widetilde { n } \\Vert _ { H ^ 2 } ^ 2 + \\Vert n - \\widetilde { n } \\Vert _ { H ^ 1 } ^ 2 . \\end{align*}"}
+{"id": "8944.png", "formula": "\\begin{align*} w _ 1 ( \\eta ) = \\epsilon + \\begin{cases} \\frac { 3 } { 2 } - \\frac { 3 \\eta } { 2 } & ( 0 \\leq \\eta \\leq \\frac { 1 } { 2 } ) \\\\ 1 - \\frac { \\eta } { 2 } & ( \\frac { 1 } { 2 } \\leq \\eta \\leq 1 ) \\\\ \\frac { 3 } { 2 } - \\eta & ( 1 \\leq \\eta \\leq \\frac { 3 } { 2 } ) \\\\ 0 & ( \\frac { 3 } { 2 } \\leq \\eta \\leq 2 ) \\end{cases} . \\end{align*}"}
+{"id": "501.png", "formula": "\\begin{align*} a = \\int _ { 0 } ^ 1 ( e ^ { \\gamma x } - 1 ) x g ( x ) d x , b = 0 \\nu ( d x ) = { \\bf 1 } _ { \\{ 0 < x \\le c _ 1 \\} } e ^ { \\gamma x } g ( x ) d x . \\end{align*}"}
+{"id": "4664.png", "formula": "\\begin{align*} \\eta \\{ F \\} \\stackrel { d e f } { = } { \\bf E } \\eta / F . \\end{align*}"}
+{"id": "4552.png", "formula": "\\begin{align*} & | \\Gamma _ { \\ell , 0 , 0 } \\cap \\Gamma _ { \\ell , 1 , 0 } | = ( q - 1 ) ^ 3 ( q + 1 ) q ^ { 3 \\ell + 5 } \\\\ & | \\Gamma _ { \\ell , 0 , 0 } \\cap \\Gamma _ { \\ell , 1 , 1 } | = ( q - 1 ) ^ 3 ( q + 1 ) q ^ { 3 \\ell + 4 } \\\\ & | \\Gamma _ { \\ell , 0 , 0 } \\cap \\Gamma _ { \\ell + 1 , 0 , 0 } | = ( q - 1 ) ^ 3 ( q ^ 2 + q + 1 ) ( q + 1 ) q ^ { 3 \\ell + 6 } \\\\ & | \\Gamma _ { \\ell , 0 , 0 } \\cap \\Gamma _ { \\ell + 1 , 1 , 0 } | = | \\Gamma _ { \\ell , 0 , 0 } \\cap \\Gamma _ { \\ell + 1 , 1 , 1 } | = ( q - 1 ) ^ 3 ( q + 1 ) q ^ { 3 \\ell + 6 } . \\end{align*}"}
+{"id": "6568.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta \\delta c + \\tilde { n } _ { 1 } \\delta c = - c _ { 2 } \\delta \\tilde { n } , & x \\in \\Omega , \\\\ \\nabla \\delta c \\cdot \\nu + \\delta c = 0 , & x \\in \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "6436.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { i = 1 } ^ { n } \\sum _ { j = 0 } ^ { k _ i - 2 } Z _ { I } ( j + 1 , k _ { i + 1 } , \\ldots , k _ n , k _ 1 , \\ldots , k _ { i - 1 } , k _ i - j ; ( \\alpha , \\beta ) ) \\\\ = & \\sum _ { i = 1 } ^ { n } Z _ { I I } ( k _ { i + 1 } , \\ldots , k _ n , k _ 1 , \\ldots , k _ { i - 1 } , k _ i + 1 ; ( \\alpha , \\beta ) ) , \\end{aligned} \\end{align*}"}
+{"id": "6240.png", "formula": "\\begin{align*} \\tau = \\frac { \\partial _ v \\Gamma _ { u v } ^ v - 2 \\Gamma _ { u v } ^ u \\Gamma _ { v u } ^ v } { \\partial _ u \\Gamma _ { v u } ^ u - 2 \\Gamma _ { u v } ^ u \\Gamma _ { v u } ^ v } \\neq 1 \\end{align*}"}
+{"id": "3675.png", "formula": "\\begin{align*} & 2 X _ { i ; j k } - ( X _ { i ; j k } + X _ { j ; i k } ) - ( X _ { i ; k j } + X _ { k ; i j } ) + ( X _ { j ; k i } + X _ { k ; j i } ) \\\\ & = 2 X _ { i ; j k } - h _ { i j ; k } - h _ { i k ; j } + h _ { j k ; i } . \\end{align*}"}
+{"id": "673.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | e ^ { i k \\phi } \\left | l _ 2 , m _ 2 \\right \\rangle = \\sum \\limits _ { l = | l _ 1 - l _ 2 | } ^ { l _ 1 + l _ 2 } \\frac { ( - 1 ) ^ { m _ 1 } } { 2 } \\sqrt { \\frac { ( 2 l _ 1 + 1 ) ( 2 l _ 2 + 1 ) ( l + k ) ! } { ( l - k ) ! } } \\end{align*}"}
+{"id": "5758.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = \\frac { 1 } { 2 } ( \\sqrt { c _ 1 } \\ X _ 1 \\cdot \\nu _ 1 + \\sqrt { c _ 2 } \\ X _ 2 \\cdot \\nu _ 2 ) \\cdot \\varphi - \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi . \\end{align*}"}
+{"id": "2237.png", "formula": "\\begin{align*} d \\mu _ { \\beta , f , g } ( x ) = K _ { \\beta , f } ( x ) g ( x ) \\ , d x , \\end{align*}"}
+{"id": "2082.png", "formula": "\\begin{align*} h _ k = \\frac { m ( H \\cap I _ { k } ) } { m ( I _ { k } ) } = \\frac { m ( ( A \\cup ( H \\setminus A ) ) \\cap I _ k ) } { m ( I _ k ) } & = \\frac { m ( A \\cap I _ { k } ) } { m ( I _ { k } ) } + \\frac { m ( ( H \\setminus A ) \\cap I _ { k } ) } { m ( I _ { k } ) } \\\\ & = \\frac { m ( A \\cap I _ { k } ) } { m ( I _ { k } ) } = a _ k \\end{align*}"}
+{"id": "7502.png", "formula": "\\begin{align*} \\| \\varphi ^ { + } \\| _ { L ^ { m _ { k } } ( \\partial B _ 1 ) } & \\leq ( C m _ 0 ) ^ { \\frac { 1 } { m _ { 0 } } \\sum _ { j = 0 } ^ { k - 1 } \\left ( \\frac { 2 } { 2 ^ { \\star } } \\right ) ^ { j } } \\left ( \\frac { 2 ^ { \\star } } { 2 } \\right ) ^ { \\frac { 1 } { m _ { 0 } } \\sum _ { j = 0 } ^ { k - 1 } j \\left ( \\frac { 2 } { 2 ^ { \\star } } \\right ) ^ { j } } \\| \\varphi ^ { + } \\| _ { L ^ { m _ { 0 } } ( \\partial B _ 1 ) } . \\end{align*}"}
+{"id": "4217.png", "formula": "\\begin{align*} & d _ { \\mathrm { s t } } = \\int d z \\sum _ { \\alpha \\in \\Delta _ + } e _ { \\alpha } ( z ) \\otimes \\psi _ { \\alpha _ i } ^ * ( z ) - \\frac { 1 } { 2 } \\sum _ { \\alpha , \\beta , \\gamma \\in \\Delta _ + } c _ { \\alpha , \\beta } ^ \\gamma : \\psi _ \\gamma ( z ) \\psi _ \\alpha ^ * ( z ) \\psi _ \\beta ^ * ( z ) : , \\\\ & d _ { \\chi } = \\int d z \\sum _ { \\alpha \\in \\Pi } \\psi ^ * _ \\alpha ( z ) = \\sum _ { \\alpha \\in \\Pi } \\psi ^ * _ { \\alpha , 1 } . \\end{align*}"}
+{"id": "3512.png", "formula": "\\begin{align*} A _ { p } \\ = \\ ( - 1 ) ^ p + 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } A _ { p - 2 j - 1 } . \\end{align*}"}
+{"id": "7257.png", "formula": "\\begin{align*} ( i _ 1 \\alpha _ 1 + i _ 2 \\alpha _ 2 + \\cdots + i _ n \\alpha _ n ) ^ { p - 1 } = \\sum _ { k _ 1 + \\cdots + k _ n = p - 1 } \\binom { p - 1 } { k _ 1 , \\ldots , k _ n } i _ 1 ^ { k _ 1 } \\cdots i _ n ^ { k _ n } \\alpha _ 1 ^ { k _ 1 } \\cdots \\alpha _ n ^ { k _ n } \\end{align*}"}
+{"id": "5945.png", "formula": "\\begin{align*} \\delta \\in [ 0 , 1 ] \\frac { 1 } { p } = \\big ( \\frac { 1 } { \\alpha } - \\frac { 1 } { d } \\big ) \\delta + \\frac { 1 - \\delta } { 2 } . \\end{align*}"}
+{"id": "3395.png", "formula": "\\begin{align*} \\mu ( n ) = \\frac { 1 } { f ( x _ 0 ) } \\cdot \\frac { x _ 0 ^ n } { n ! } \\ 1 _ { [ k , l ] } ( n ) , \\textrm { w h e r e } f ( x ) = \\sum _ { i = k } ^ l \\frac { x ^ i } { i ! } . \\end{align*}"}
+{"id": "1341.png", "formula": "\\begin{align*} \\frac { d ( p _ { n + 1 } , q _ { n + 1 } ) } { 1 + d ( p _ { n + 1 } , q _ { n + 1 } ) } \\min \\bigl \\{ 1 , \\dfrac { d ( a _ j , b _ j ) } { C _ n \\alpha _ j ' } \\bigr \\} & \\leq \\frac { d ( c _ j , d _ j ) } { \\alpha _ { m + j } + \\beta _ k } \\\\ & = \\dfrac { 1 } { \\frac { \\alpha _ j ' } { d ( a _ j , b _ j ) } + \\frac { \\lambda _ { n + 1 } } { d ( p _ { n + 1 } , q _ { n + 1 } ) } } . \\end{align*}"}
+{"id": "7381.png", "formula": "\\begin{align*} U _ { \\bf n } = \\rm { d i a g } ( \\ \\underbrace { 1 , \\ldots 1 } _ { n _ 0 } , \\underbrace { \\omega , \\ldots , \\omega } _ { n _ 1 } , \\ldots , \\underbrace { \\omega ^ { \\ell - 1 } , \\ldots , \\omega ^ { \\ell - 1 } } _ { n _ { \\ell - 1 } } \\ ) . \\end{align*}"}
+{"id": "8353.png", "formula": "\\begin{align*} \\mathcal { G } = \\left \\lbrace u _ 0 ( x ) : \\ u _ 0 ( x ) \\in H ^ { 3 } ( \\mathbb { R } ) \\cap H ^ { 2 , 1 } ( \\mathbb { R } ) , \\ a ( k ) \\neq 0 \\mathbb { C } ^ - \\right \\rbrace , \\end{align*}"}
+{"id": "2796.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { K _ { N } ^ { 2 } } \\mathbb { E } \\left ( | \\hat { \\Gamma } _ { N } ^ { D , M } ( 0 ) \\cap A _ { N } | - e ^ { \\pi \\lambda b } | \\hat { \\Gamma } _ { N } ^ { D , M } ( b ) \\cap A _ { N } | \\right ) ^ { 2 } = 0 . \\end{align*}"}
+{"id": "493.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } e ^ { \\gamma x } ( f _ a ( x ) + f _ r ( x ) ) = 0 . \\end{align*}"}
+{"id": "2564.png", "formula": "\\begin{align*} ( \\frac { q } { p } ) ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } \\frac { a _ { n _ { 2 } } } { N _ { n _ { 2 } } } - \\frac { a _ { n _ { 1 } } } { N _ { n _ { 1 } } } = ( \\frac { q } { p } ) ^ { n _ { s } ^ { \\prime } - n _ { 1 } ^ { \\prime } } \\frac { a _ { n _ { s } } } { N _ { n _ { s } } } . \\end{align*}"}
+{"id": "3565.png", "formula": "\\begin{align*} c _ { \\Omega _ 4 } ^ { \\Omega _ 0 } & = q ^ { - 1 / 6 } \\varphi ( q ) ^ { - 4 } \\dfrac { 1 } { 2 } q ^ { 1 / 2 0 } ( \\varphi ( - q ^ { 1 / 2 } ) ^ 2 \\varphi ( q ) ^ { - 2 } H ( - q ^ { 1 / 2 } ) - \\varphi ( q ^ { 1 / 2 } ) ^ 2 \\varphi ( q ) ^ { - 2 } H ( q ^ { 1 / 2 } ) ) \\\\ & = q ^ { - 7 / 6 0 } \\dfrac { 1 } { 2 } \\varphi ( q ) ^ { - 6 } ( \\varphi ( - q ^ { 1 / 2 } ) ^ 2 H ( - q ^ { 1 / 2 } ) - \\varphi ( q ^ { 1 / 2 } ) ^ 2 H ( q ^ { 1 / 2 } ) ) . \\end{align*}"}
+{"id": "2108.png", "formula": "\\begin{align*} R L ( b , c ) & = f _ c \\left ( \\binom { b } { 2 } + b \\right ) - f _ c \\left ( \\binom { b } { 2 } \\right ) + b \\Delta _ c ( 1 ) \\\\ & = \\sum _ { i = 1 } ^ b \\left [ f _ c \\left ( \\binom { b } { 2 } + i \\right ) - f _ c \\left ( \\binom { b } { 2 } + i - 1 \\right ) \\right ] + b \\Delta _ c ( 1 ) \\\\ & = \\sum _ { i = 1 } ^ b \\Delta _ c \\left ( \\binom { b } { 2 } + i \\right ) + b \\Delta _ c ( 1 ) , \\end{align*}"}
+{"id": "8955.png", "formula": "\\begin{align*} V _ 1 ( t ) = 2 V _ 1 ^ + ( t ) \\le 2 \\int _ { a ( x ) } ^ { b ( \\eta _ k ) } v _ 1 ^ { - p ' } \\le 2 V _ 1 ( x ) = 4 V _ 1 ^ - ( x ) , \\eta _ k \\le x \\le \\eta _ { k + 1 } , \\end{align*}"}
+{"id": "8644.png", "formula": "\\begin{align*} \\begin{aligned} y \\left [ n \\right ] = & \\sum \\limits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ l } s \\left [ { n - { \\kappa _ l } - { n _ l } } \\right ] } + \\\\ & \\sum \\limits _ { l = 1 } ^ L { \\sum \\limits _ { l ' \\ne l } ^ L { { \\bf { h } } _ l ^ H { { \\bf { f } } _ { l ' } } s \\left [ { n - { \\kappa _ { l ' } } - { n _ l } } \\right ] } } + z \\left [ n \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "833.png", "formula": "\\begin{align*} \\Gamma _ 0 ^ \\varepsilon = \\big \\{ F ( z ) + \\varepsilon \\rho _ 0 \\nu _ \\Sigma ( z ) \\ , \\big | \\ , z \\in M \\big \\} \\end{align*}"}
+{"id": "2203.png", "formula": "\\begin{align*} \\phi = 2 \\pi \\frac { d } { \\lambda } { s i n } \\theta _ 0 \\end{align*}"}
+{"id": "4278.png", "formula": "\\begin{align*} n ^ { 3 } + \\left ( n + 1 \\right ) ^ { 3 } - \\left ( \\beta n + \\beta ^ { 2 } + \\beta + 1 \\right ) ^ { 3 } + \\left ( \\beta n + \\beta ^ { 2 } + 1 \\right ) ^ { 3 } = 0 \\end{align*}"}
+{"id": "5124.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 } _ { + } } ( \\phi - \\phi _ { \\infty } ) _ { + } ^ { q } \\frac { 1 } { r } \\dd z \\dd r \\lesssim \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\phi ^ { l + q } \\frac { 1 } { r ^ { 2 l + 1 } } \\dd z \\dd r = \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\phi ^ { p } \\frac { 1 } { r ^ { p / 2 + 2 } } \\dd z \\dd r \\lesssim | | \\nabla \\phi | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } _ { + } ; r ^ { - 1 } ) } ^ { p } . \\end{align*}"}
+{"id": "3180.png", "formula": "\\begin{align*} k _ 3 ( \\langle H \\rangle ) = \\frac { q - 1 } { 7 6 8 } & \\left [ 2 ( q ^ 2 - 2 0 q + 8 1 ) + 2 u ( - p ) ^ t + 3 q ^ 2 \\cdot { _ { 3 } } F _ { 2 } \\left ( \\begin{array} { c c c } \\chi _ 4 , & \\chi _ 4 , & \\overline { \\chi _ 4 } \\\\ & \\varepsilon , & \\varepsilon \\end{array} | 1 \\right ) \\right ] . \\end{align*}"}
+{"id": "7751.png", "formula": "\\begin{align*} p _ 0 ( x ) : = p ( x ) , ~ p _ { h + 1 } ( x ) : = ( - 1 ) ^ d p _ h ( \\sqrt x ~ ) p _ h ( \\sqrt { - x } ~ ) , ~ h = 0 , 1 , \\dots , \\end{align*}"}
+{"id": "6828.png", "formula": "\\begin{align*} F _ { - } ( s ) : = \\beta ( s ) - d ( t ) \\beta ^ { - \\frac { 1 } { 2 } } ( s ) \\leq \\rho \\leq \\beta ( s ) + d ( t ) \\beta ^ { - \\frac { 1 } { 2 } } ( s ) = : F _ { + } ( s ) . \\end{align*}"}
+{"id": "5710.png", "formula": "\\begin{align*} \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} \\begin{pmatrix} x \\\\ F ( x ) \\end{pmatrix} + \\begin{pmatrix} u \\\\ v \\end{pmatrix} = \\begin{pmatrix} \\pi ( x ) \\\\ F ' ( \\pi ( x ) ) \\end{pmatrix} . \\end{align*}"}
+{"id": "853.png", "formula": "\\begin{align*} K ( r , h ) = \\sum \\limits _ { \\alpha \\leq x \\leq \\beta \\atop { ( x , r ) = 1 } } e \\left ( \\frac { h \\overline { x } _ { | r | } } { r } \\right ) \\ , , \\end{align*}"}
+{"id": "8302.png", "formula": "\\begin{align*} \\begin{aligned} & \\dot y _ { b e n } = a y '' _ { b e n } + M y _ { b e n } , x \\in ( 0 , R ) \\\\ & \\dot y _ { n b } = b y '' _ { n b } + ( M - \\mu \\ 1 ) y _ { n b } , x \\in ( R , R + r ) , \\end{aligned} \\end{align*}"}
+{"id": "6099.png", "formula": "\\begin{align*} { \\widehat { \\mathcal { Q } } _ n } ( z ) = \\frac { ( z - j ^ { ( 0 ) } + j ^ { ( 4 ) } ) ( z + j ^ { ( 0 ) } + j ^ { ( 4 ) } + 1 ) ( z - j ^ { ( 3 ) } + j _ n ^ { ( 1 2 ) } ) ( z - j ^ { ( 3 ) } - j _ n ^ { ( 1 2 ) } - 1 ) } { 2 z \\ , ( 2 z - 1 ) } \\ , . \\end{align*}"}
+{"id": "8506.png", "formula": "\\begin{align*} e ^ { i c _ + ( x ) } \\partial _ { x } \\left ( \\bar { u } _ x ( x ) e ^ { i c _ + ( x ) } \\right ) & = - \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } r _ 2 ( z ) e ^ { 2 i s x } \\mathrm { d } z - \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } r _ 2 ( z ) e ^ { 2 i z x } \\left [ M _ { + , 2 2 } - 1 \\right ] d z \\\\ & : = I _ 3 ( x ) + I _ 4 ( x ) . \\end{align*}"}
+{"id": "4665.png", "formula": "\\begin{align*} \\ T _ { \\tau } ( t ) \\stackrel { d e f } { = } { \\bf P } ( | \\tau | \\ge t ) , \\ t \\ge 0 . \\end{align*}"}
+{"id": "7239.png", "formula": "\\begin{align*} L ( x ) = x ^ { 2 ^ r } + x ^ 2 + t x \\end{align*}"}
+{"id": "7927.png", "formula": "\\begin{align*} G ( \\Psi ) ( x ) = \\int _ \\S W _ r ( x - y ) \\sin ( \\Psi ( y ) - \\Psi ( x ) ) \\ \\d y - \\int _ \\S W _ r ( y ) \\sin ( \\Psi ( y ) ) \\ \\d y . \\end{align*}"}
+{"id": "8760.png", "formula": "\\begin{align*} E _ { \\psi } ( w ' - z ' ) = K ( T ( w ' ) - T ( z ' ) ) = K \\circ T ( w ' - z ' ) = \\frac { 1 } { 2 \\pi i } \\sum _ { n = 0 } ^ { \\infty } U _ n ( z ' ) V _ n ( w ' ) , \\end{align*}"}
+{"id": "6201.png", "formula": "\\begin{align*} E _ 0 = \\kappa \\left ( \\frac { B _ 3 } { 2 \\sqrt { \\kappa B _ 4 } } + L + 2 \\right ) ^ 2 - \\left ( \\frac { Q } { 2 ( L + 1 ) } - \\sqrt { \\kappa B _ 4 } \\right ) ^ 2 , \\end{align*}"}
+{"id": "3066.png", "formula": "\\begin{align*} \\mathfrak X ( Z ) : = \\overline { \\mathfrak X ^ 0 ( Z ) } \\subseteq \\P ^ { n + 1 } \\times \\P ^ n . \\end{align*}"}
+{"id": "3492.png", "formula": "\\begin{align*} \\phi ( y ) = \\sum _ { \\substack { z _ 1 , , , z _ t , z _ { t + 1 } \\\\ z _ { i + 1 } \\in I ( z _ i ' ) } } G _ { I ( y ) } ( y , z _ 1 ) G _ { I ( z _ 1 ' ) } ( z _ 1 ' , z _ 2 ) \\cdots G _ { I ( z _ t ' ) } ( z _ t ' , z _ { t + 1 } ) \\phi ( { z _ { t + 1 } ' } ) , \\end{align*}"}
+{"id": "7506.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } \\div ( A ( x ) \\nabla \\varphi _ k ) = 0 & B _ 1 \\\\ N \\varphi _ k = g _ k & \\partial B _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "4345.png", "formula": "\\begin{align*} & \\int _ { \\{ - t ' _ 3 \\le \\Psi < - t ' _ 4 \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 e ^ { - \\varphi } \\\\ \\le & \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\} } | \\tilde F | ^ 2 _ h + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\backslash N \\} } | F _ { t _ 0 , \\tilde { c } } - \\tilde F | ^ 2 _ h \\\\ & + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\cap N \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h . \\end{align*}"}
+{"id": "5910.png", "formula": "\\begin{align*} \\varphi ( t ) : = \\exp \\left ( - \\int _ 0 ^ t \\big [ f ( r ) + \\rho ( X ( r ) ) + \\eta ( X ^ { \\prime } ( r ) ) \\big ] d r \\right ) . \\end{align*}"}
+{"id": "7100.png", "formula": "\\begin{align*} \\partial _ t \\omega + u \\cdot \\nabla \\omega = 0 \\end{align*}"}
+{"id": "2003.png", "formula": "\\begin{align*} 0 = F ^ { ( n ) } ( x ) + F ^ { ( n ) } ( x + F ( x ) ) ( 1 + F ^ { \\prime } ( x ) ) ^ { n } + \\sum _ { j = 1 } ^ { n - 1 } F ^ { ( j ) } ( x + F ( x ) ) G _ { j } ^ { n } ( x ) , \\end{align*}"}
+{"id": "9125.png", "formula": "\\begin{align*} E ^ { \\pm } ( \\Q _ { p ^ g , \\infty } ) = \\bigcup _ { n \\geq - 1 } E ^ { \\pm } ( \\Q _ { p ^ g , n } ) . \\end{align*}"}
+{"id": "2194.png", "formula": "\\begin{align*} S ( \\theta ) = \\frac { 1 } { \\| \\mathbf { a } ^ H ( \\theta ) \\mathbf { U } _ V \\| ^ 2 } \\end{align*}"}
+{"id": "5502.png", "formula": "\\begin{align*} \\tilde H ( p , s ) = s ^ { - p / 2 } 2 ^ { p / 2 } F ( p , 2 ) - F ( p , s ) , 2 < p < 3 , \\ s > 1 . \\end{align*}"}
+{"id": "7480.png", "formula": "\\begin{align*} \\left ( \\int _ { B _ 1 } | v | ^ p \\d x \\right ) ^ { \\frac { 1 } { p } } \\leq C \\left ( \\int _ { B _ 1 } | \\nabla v | ^ 2 \\d x \\right ) ^ { \\frac { 1 } { 2 } } = C . \\end{align*}"}
+{"id": "3261.png", "formula": "\\begin{align*} \\mathbb P \\left [ \\int _ 0 ^ T \\| \\alpha _ s \\| ^ { \\frac m 2 } + \\| \\sigma _ s \\| _ { L _ { } ( U , H ) } ^ m d s < \\infty \\right ] = 1 . \\end{align*}"}
+{"id": "558.png", "formula": "\\begin{align*} \\Omega _ { \\theta } : = \\lbrace \\omega \\colon R e \\left ( \\omega e ^ { i \\theta } \\right ) < - I _ f ( \\theta ) \\rbrace , \\end{align*}"}
+{"id": "8360.png", "formula": "\\begin{align*} \\lim _ { | z | \\rightarrow \\infty } z \\mathcal { C } ( h ) ( z ) = - \\frac { 1 } { 2 \\pi i } \\int _ { \\mathbb { R } } h ( s ) d s . \\end{align*}"}
+{"id": "680.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | \\cos ( n \\theta ) \\left | l _ 2 , m _ 2 \\right \\rangle = \\sum \\limits _ { l = | l _ 1 - l _ 2 | } ^ { l _ 1 + l _ 2 } \\sqrt { \\left ( l _ 1 + \\frac { 1 } { 2 } \\right ) \\left ( l _ 2 + \\frac { 1 } { 2 } \\right ) } ( - 1 ) ^ { m _ 1 } \\end{align*}"}
+{"id": "3538.png", "formula": "\\begin{align*} A _ k \\ & = \\ \\sum _ { j = 0 } ^ { k } \\ , j ! \\ , F _ { j + 1 } \\begin{Bmatrix} k \\\\ j \\end{Bmatrix} , \\\\ B _ k \\ & = \\ \\sum _ { j = 0 } ^ { k } \\ , j ! \\ , F _ { j + 2 } \\begin{Bmatrix} k \\\\ j \\end{Bmatrix} . \\end{align*}"}
+{"id": "9027.png", "formula": "\\begin{align*} ( H _ { \\lambda \\exp , \\alpha } \\psi ) _ { n } = \\psi _ { n + 1 } + \\psi _ { n - 1 } + \\lambda { \\rm e } ^ { { \\rm i } ( x + n \\alpha ) } \\psi _ { n } , \\end{align*}"}
+{"id": "4337.png", "formula": "\\begin{align*} \\int _ { \\{ - t ' _ 1 \\le \\Psi < - t ' _ 2 \\} } | \\tilde { F } | ^ 2 _ h \\ge \\frac { G ( T _ 1 ) } { \\int _ { T _ 1 } ^ { + \\infty } c ( s ) e ^ { - s } d s } \\int _ { t ' _ 2 } ^ { t ' _ 1 } e ^ { - s } d s . \\end{align*}"}
+{"id": "3170.png", "formula": "\\begin{align*} & k _ 3 ( \\langle H \\rangle , a ) \\\\ & = \\frac { 1 } { 2 \\times 4 ^ 5 } \\sum _ { x \\neq 0 , a } \\sum _ { y \\neq 0 , a , x } [ ( 2 + h \\chi _ 4 ( a - x ) + \\overline { h } \\overline { \\chi _ 4 } ( a - x ) ) \\\\ & \\times ( 2 + h \\chi _ 4 ( a - y ) + \\overline { h } \\overline { \\chi _ 4 } ( a - y ) ) ( 2 + h \\chi _ 4 ( x - y ) + \\overline { h } \\overline { \\chi _ 4 } ( x - y ) ) \\\\ & \\times ( 2 + h \\chi _ 4 ( x ) + \\overline { h } \\overline { \\chi _ 4 } ( x ) ) ( 2 + h \\chi _ 4 ( y ) + \\overline { h } \\overline { \\chi _ 4 } ( y ) ) ] . \\end{align*}"}
+{"id": "6528.png", "formula": "\\begin{align*} q _ 1 q _ 2 q _ 3 q _ 4 = p _ 1 ^ { \\alpha _ 1 } p _ 2 ^ { \\alpha _ 2 } p _ 3 ^ { \\alpha _ 3 } p _ 4 ^ { \\alpha _ 4 } p _ 5 . \\end{align*}"}
+{"id": "8738.png", "formula": "\\begin{align*} \\pi = \\pi _ 1 \\oplus \\pi _ 2 \\oplus \\pi _ 3 \\tau = t s _ 1 \\tau _ 1 \\oplus ( 1 - t ) s _ 2 \\tau _ 2 \\oplus ( t ( 1 - s _ 1 ) + ( 1 - t ) ( 1 - s _ 2 ) ) \\tau _ 3 . \\end{align*}"}
+{"id": "2027.png", "formula": "\\begin{align*} & f ( t , y , z , u , \\mu ) = \\hat f \\big ( t , y , z , u , \\int \\varphi d \\mu \\big ) , \\ , \\ , \\ , \\ , g ( t , y , z , u , \\mu ) = \\hat g \\big ( t , y , z , u , \\int \\phi d \\mu \\big ) , \\\\ & h ( t , y , z , u , \\mu ) = \\hat h \\big ( t , y , z , u , \\int \\psi d \\mu \\big ) , \\ , \\ , \\ , \\ , \\Phi ( y , \\mu _ y ) = \\hat \\Phi \\big ( y , \\int \\gamma d \\mu _ y \\big ) , \\end{align*}"}
+{"id": "4769.png", "formula": "\\begin{align*} \\sigma ( F ) = \\Omega \\setminus \\rho ( F ) . \\end{align*}"}
+{"id": "1674.png", "formula": "\\begin{align*} V _ { p , \\alpha _ z } = \\bigoplus _ { v \\in S _ { i n } } V _ { v , \\alpha _ { z , v } } \\oplus \\bigoplus _ { v \\in S _ { s p } } V _ v ^ + . \\end{align*}"}
+{"id": "9323.png", "formula": "\\begin{align*} F ( a , b , c ; z ) & = \\sum _ { i = 0 } ^ \\infty \\frac { ( a ) _ i ( b ) _ i } { ( c ) _ i } \\frac { z ^ i } { i ! } \\\\ & = 1 + \\frac { a b } { c } \\frac { z } { 1 ! } + \\frac { a ( a + 1 ) b ( b + 1 ) } { c ( c + 1 ) } \\frac { z ^ 2 } { 2 ! } + \\cdots . \\end{align*}"}
+{"id": "7933.png", "formula": "\\begin{align*} \\norm { f } _ { L ^ 2 } ^ 2 & = \\int _ \\S \\left ( \\int _ \\S W _ r ( x - y ) \\prod _ { i = 1 } ^ n ( \\eta _ i ( y ) - \\eta _ i ( x ) ) \\sin ^ { [ n ] } ( \\tilde \\Psi ( y ) - \\tilde \\Psi ( x ) ) \\ \\d y \\right ) ^ 2 \\d x \\\\ & \\le \\int _ \\S \\int _ \\S \\prod _ { i = 1 } ^ n ( \\eta _ i ( y ) - \\eta _ i ( x ) ) ^ 2 \\ \\d y \\d x \\\\ & = \\prod _ { i = 1 } ^ n \\norm { \\partial \\eta _ i } _ { L ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "1923.png", "formula": "\\begin{align*} \\nabla ^ { ( 1 , 0 ) } \\eta ( A z _ \\star , y _ \\star ) & = \\sum _ { i = 1 } ^ d ( ( A z _ \\star ) ^ { ( i ) } ) ^ { p - 1 } y _ \\star ^ { ( i ) } e _ d ^ { ( i ) } , \\\\ \\nabla ^ { ( 0 , 1 ) } \\eta ( A z _ \\star , y _ \\star ) & = \\tfrac { 1 } { p } \\left ( \\sum _ { i = 1 } ^ d ( ( A z _ \\star ^ { ( i ) } ) ) ^ p e _ d ^ { ( i ) } \\right ) . \\end{align*}"}
+{"id": "5734.png", "formula": "\\begin{align*} \\frac { x ^ { q + 1 } + x } { y ^ { q + 1 } + y } = d , \\end{align*}"}
+{"id": "5455.png", "formula": "\\begin{align*} v _ 1 \\cdot v _ 2 = - \\int _ X v _ 1 \\wedge v _ 2 ^ \\vee . \\end{align*}"}
+{"id": "3179.png", "formula": "\\begin{align*} k _ 3 ( \\langle H \\rangle , 1 ) = \\frac { 1 } { 1 2 8 } \\left [ 2 ( q ^ 2 - 2 0 q + 8 1 ) + 2 u ( - p ) ^ t + 3 q ^ 2 \\cdot { _ { 3 } } F _ { 2 } \\left ( \\begin{array} { c c c } \\chi _ 4 , & \\chi _ 4 , & \\overline { \\chi _ 4 } \\\\ & \\varepsilon , & \\varepsilon \\end{array} | 1 \\right ) \\right ] . \\end{align*}"}
+{"id": "3276.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sup _ { n \\in \\mathbb N } \\sum _ { k \\geq N } \\langle \\mathbb E [ X _ n ] e _ k , e _ k \\rangle = 0 . \\end{align*}"}
+{"id": "3537.png", "formula": "\\begin{align*} 0 ^ p \\cdot F _ 2 + 1 ^ p \\cdot F _ 1 + 2 ^ p \\cdot F _ 0 \\ & = \\ A _ p F _ 2 + B _ p F _ 3 - \\sum _ { k = 0 } ^ p \\binom { p } { k } B _ k 2 ^ { p - k } , \\intertext { a n d i f w e s i m p l i f y t h i s a n d p u l l o u t t h e $ k = p $ t e r m f r o m t h e s u m , w e g e t } 1 \\ & = \\ A _ p + 2 B _ p - B _ p - \\sum _ { k = 0 } ^ { p - 1 } \\binom { p } { k } B _ k 2 ^ { p - k } . \\end{align*}"}
+{"id": "6526.png", "formula": "\\begin{align*} q _ 1 q _ 2 q _ 3 q _ 4 = p _ 1 ^ { \\alpha _ 1 } p _ 2 ^ { \\alpha _ 2 } p _ 3 ^ { \\alpha _ 3 } p _ 4 ^ { \\alpha _ 4 } . \\end{align*}"}
+{"id": "8890.png", "formula": "\\begin{align*} \\Delta _ 1 ( m , n ) = \\delta ( m , n ) + O ( ( m n ) ^ { 3 / 8 + \\epsilon } k ^ { - 1 3 / 1 2 } ) . \\end{align*}"}
+{"id": "2053.png", "formula": "\\begin{align*} \\forall \\ t \\in [ t _ 0 , 1 ] , \\partial _ { \\eta _ 1 } ^ { 2 k } \\rho _ k = ( 2 k ) ! ( t - t _ 0 ) ^ k \\leq ( 2 k ) ! \\rho _ { 0 } . \\end{align*}"}
+{"id": "4698.png", "formula": "\\begin{align*} k ( M ) = \\inf \\frac { { \\rm V o l } _ { n - 1 } ( \\Sigma ) } { { \\rm V o l } _ n ( A ) | \\log ( { \\rm V o l } _ n ( A ) ) | } , \\end{align*}"}
+{"id": "3542.png", "formula": "\\begin{align*} \\tau ( \\alpha ' _ 1 ) & = \\alpha ' _ 6 , & \\tau ( \\alpha ' _ 2 ) & = \\alpha ' _ 5 , & \\tau ( \\alpha ' _ 3 ) & = \\alpha ' _ 3 , \\\\ \\tau ( \\alpha ' _ 4 ) & = \\alpha ' _ 4 , & \\tau ( \\alpha ' _ 5 ) & = \\alpha ' _ 2 , & \\tau ( \\alpha ' _ 6 ) & = \\alpha ' _ 1 . \\end{align*}"}
+{"id": "5600.png", "formula": "\\begin{align*} A ^ * ( y ) \\ , = \\ , \\hat { A } ^ * ( y | x ) : = A ( \\tau _ y ( x ) ) + W \\circ \\hat { \\sigma } ^ { - 1 } ( y | x ) - W ( y | x ) \\ , \\end{align*}"}
+{"id": "8170.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { R _ { 0 } } \\mu \\xi ( \\mu , t _ { 0 } ) \\ d \\mu = \\int _ { 0 } ^ { R _ { 0 } } \\mu \\xi ^ { \\mathrm { i n } } ( \\mu ) \\ d \\mu . \\end{align*}"}
+{"id": "6593.png", "formula": "\\begin{align*} c _ { r } = r ^ { 1 - d } \\int _ { 0 } ^ { r } \\rho ^ { d - 1 } n c \\ , d \\rho < \\frac { M _ { 0 } m _ { 0 } } { \\sigma _ { d } } r \\quad \\mbox { f o r a l l } r \\in ( 0 , R ] . \\end{align*}"}
+{"id": "5788.png", "formula": "\\begin{align*} S = \\left ( \\begin{array} { c c } 1 / 2 + \\alpha & \\beta \\\\ \\beta & 1 / 2 - \\alpha \\end{array} \\right ) \\end{align*}"}
+{"id": "8803.png", "formula": "\\begin{align*} ( \\Pi _ V \\tilde { Y } _ { t _ * } ) = \\int _ 0 ^ { t _ * } \\left \\| \\Pi _ V e ^ { | z | J ^ \\perp ( t _ * - s ) } P ^ { - 1 } \\Pi _ { \\mathrm { k e r } A ^ \\perp } \\sigma \\right \\| _ F ^ 2 d s , \\end{align*}"}
+{"id": "5438.png", "formula": "\\begin{align*} \\sigma _ { k - 1 } ( b ) u _ { n n } ( 0 ) - \\sum _ { \\alpha \\leq n - 1 } u _ { n \\alpha } ^ 2 ( 0 ) \\sigma _ { k - 2 ; \\alpha } ( b ) + \\sigma _ k ( b ) = f ( 0 ) . \\end{align*}"}
+{"id": "2856.png", "formula": "\\begin{align*} D _ 2 ( { \\frak y } ) = 4 \\gamma \\begin{bmatrix} T _ - & 0 & 0 & \\dots & 0 \\\\ 0 & y _ 1 & 0 & \\dots & 0 \\\\ 0 & 0 & y _ 2 & \\dots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\dots & y _ n \\end{bmatrix} . \\end{align*}"}
+{"id": "4122.png", "formula": "\\begin{align*} H _ { m i n } \\left ( ( E _ 1 [ t ] \\times E _ 1 [ t ] ^ { \\vee } ) \\otimes F \\right ) & = H _ { m i n } \\left ( E _ 1 [ t ] \\otimes F \\times E _ 1 [ t ] ^ { \\vee } \\otimes F \\right ) \\\\ & = \\min \\left ( H _ { m i n } ( E _ 1 [ t ] \\otimes F ) , H _ { m i n } ( E _ 1 [ t ] ^ { \\vee } \\otimes F ) \\right ) \\\\ & \\leq H _ { m i n } ( E _ 1 [ t ] \\otimes F ) = t H _ { m i n } ( E _ 1 \\otimes F ) \\leq t H _ { m i n } ( E _ 1 ) H _ { m i n } ( F ) \\end{align*}"}
+{"id": "3952.png", "formula": "\\begin{align*} - \\Delta ( v \\psi _ { m } ) ( x ) + F ' ( \\overline { y } ; v \\psi _ { m } ) ( x ) = 0 , x \\in \\mathcal { A } , \\end{align*}"}
+{"id": "7078.png", "formula": "\\begin{align*} \\frac { \\tau ^ 2 _ h ( \\varphi v ) ( x ) } { | h | ^ \\alpha } & = \\frac { \\tau _ h ( \\varphi ( x + h ) \\tau _ h v ( x ) + v ( x ) \\tau _ h \\varphi ( x ) ) } { | h | ^ \\alpha } \\\\ & = \\varphi ( x + 2 h ) \\frac { \\tau ^ 2 _ h v ( x ) } { | h | ^ \\alpha } + 2 \\frac { \\tau _ h v ( x ) \\tau _ h \\varphi ( x + h ) } { | h | ^ \\alpha } + v ( x ) \\frac { \\tau ^ 2 _ { h } \\varphi ( x ) } { | h | ^ \\alpha } \\end{align*}"}
+{"id": "3631.png", "formula": "\\begin{align*} ( \\tilde { \\kappa } _ 1 ) _ { i i } = \\tilde { h } _ { 1 1 i i } + 2 \\sum _ { p \\neq 1 } \\frac { \\tilde { h } _ { 1 p i } ^ 2 } { \\tilde { \\kappa } _ 1 - \\tilde { \\kappa } _ p } = h _ { 1 1 i i } + 2 \\sum _ { p \\neq 1 } \\frac { h _ { 1 p i } ^ 2 } { \\kappa _ 1 - \\tilde { \\kappa } _ p } . \\end{align*}"}
+{"id": "8123.png", "formula": "\\begin{align*} U _ { i j } \\subseteq U _ { i } \\subseteq U U \\backslash U _ { i } = B _ { i } \\backslash U _ { i } U _ { j } \\backslash U _ { i j } = B _ { i } \\backslash U _ { i j } . \\end{align*}"}
+{"id": "6067.png", "formula": "\\begin{align*} 0 = \\int _ \\Sigma \\Delta x _ { n + 1 } \\ , d A = \\int _ \\Sigma \\left ( \\eta x _ { n + 1 } ^ m + \\lambda \\right ) \\nu _ { n + 1 } \\ , d A = \\int _ \\Sigma \\eta x _ { n + 1 } ^ m \\nu _ { n + 1 } \\ , d A , \\end{align*}"}
+{"id": "4909.png", "formula": "\\begin{align*} F ( x ) = I _ { \\Phi ( x ) } ( \\alpha , \\beta ) = \\frac { 1 } { \\mathrm { B } ( \\alpha , \\beta ) } \\int _ 0 ^ { \\Phi ( x ) } \\omega ^ { \\alpha - 1 } \\ , ( 1 - \\omega ) ^ { \\beta - 1 } \\mathrm { d } \\omega . \\end{align*}"}
+{"id": "9002.png", "formula": "\\begin{align*} \\sum _ { \\ell _ { q - 1 } < j < \\ell _ q } ( - 1 ) ^ { p ( \\eta , j ) - 1 } i _ { \\eta \\cup j } & = U \\left ( \\prod _ { r = m + 1 } ^ { q - 1 } { D _ { \\ell _ { r } - 1 } } \\prod _ { r = q + 1 } ^ { M } { D _ { \\ell _ { r - 1 } } } \\right ) \\left ( - \\sum _ { \\ell _ { q - 1 } < j < \\ell _ q } { x _ { j } } \\right ) \\\\ & = U \\left ( \\prod _ { r = m + 1 } ^ { q - 1 } { D _ { \\ell _ { r } - 1 } } \\prod _ { r = q + 1 } ^ { M } { D _ { \\ell _ { r - 1 } } } \\right ) \\left ( D _ { \\ell _ { q - 1 } } - D _ { \\ell _ q - 1 } \\right ) \\\\ & = U ( B ( q - 1 ) - B ( q ) ) . \\end{align*}"}
+{"id": "1276.png", "formula": "\\begin{align*} ( r _ n \\omega ) _ y = \\begin{cases} \\omega _ y , & y \\in \\Lambda _ n , \\\\ \\ 1 , & . \\end{cases} \\end{align*}"}
+{"id": "4495.png", "formula": "\\begin{align*} Z _ { V , [ \\alpha ] } ( t ) = - \\int _ { V } \\alpha + \\sqrt { - 1 } t \\int _ { V } \\omega . \\end{align*}"}
+{"id": "8993.png", "formula": "\\begin{align*} \\sum _ { q = 1 } ^ { s } \\sum _ { \\ell _ { q - 1 } < j < \\ell _ q } ( - 1 ) ^ { p ( \\eta , j ) - 1 } i _ { \\eta \\cup j } & = \\sum _ { q = 1 } ^ { s } \\left ( \\prod _ { r = s + 1 } ^ { d + 1 } { D _ { \\ell _ { r } - 1 } } \\right ) \\left ( A ( q - 1 ) - A ( q ) \\right ) \\\\ & = \\left ( \\prod _ { r = s + 1 } ^ { d + 1 } { D _ { \\ell _ { r } - 1 } } \\right ) \\sum _ { q = 1 } ^ s \\left ( A ( q - 1 ) - A ( q ) \\right ) \\\\ & = - A ( s ) \\prod _ { r = s + 1 } ^ { d + 1 } { D _ { \\ell _ { r } - 1 } } = - \\prod _ { r = 1 } ^ { d + 1 } { D _ { \\ell _ { r } - 1 } } . \\end{align*}"}
+{"id": "897.png", "formula": "\\begin{align*} \\Omega ( X , q , n ) = \\frac { X } { q } + \\mathcal { O } ( 1 ) \\ , . \\end{align*}"}
+{"id": "5457.png", "formula": "\\begin{align*} \\int _ X \\alpha ^ { 2 n } = c _ X \\cdot q _ X ( \\alpha ) ^ n . \\end{align*}"}
+{"id": "8225.png", "formula": "\\begin{align*} \\sigma _ b \\left ( Q ( z ) \\right ) = 0 \\quad | b ( z ) | \\geq C _ 1 \\ . \\end{align*}"}
+{"id": "6466.png", "formula": "\\begin{align*} \\lim _ { m _ i \\in M _ 2 , m _ i \\to \\infty } \\mu ( T ^ { m _ i } A \\cap B ) = \\mu ( A ) \\mu ( B ) . \\end{align*}"}
+{"id": "5133.png", "formula": "\\begin{align*} \\frac { \\partial j } { \\partial s } ( 0 , 0 ) = 2 \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi _ 0 1 _ { ( 0 , \\infty ) } ( \\phi - \\phi _ { \\infty } ) G + ( \\phi - \\phi _ { \\infty } ) _ { + } G _ 0 ) \\frac { 1 } { r ^ { 2 } } \\dd x . \\end{align*}"}
+{"id": "5269.png", "formula": "\\begin{align*} I ( \\lambda ) = \\int _ a ^ b e ^ { i \\lambda \\phi ( x ) } \\psi ( x ) d x \\end{align*}"}
+{"id": "1460.png", "formula": "\\begin{align*} c _ { 1 } : = \\frac { 5 } { 4 } \\left ( \\frac { \\alpha ^ { 4 } \\beta } { 3 } \\right ) ^ { 1 / 5 } , c _ { 2 } : = 2 0 \\left ( \\frac { 2 \\alpha ^ { 6 } } { 3 \\beta } \\right ) ^ { 1 / 5 } , c _ { 3 } : = \\frac { 5 } { 4 } \\left ( \\frac { 3 \\alpha ^ { 6 } } { \\beta } \\right ) ^ { 1 / 5 } . \\end{align*}"}
+{"id": "6034.png", "formula": "\\begin{align*} o _ { k } ( 1 ) & = \\frac { 1 } { 2 } \\int _ { \\R ^ { 2 } } ( | \\nabla u _ { k } | ^ { 2 } + u _ { k } ^ { 2 } + | \\nabla v _ { k } | ^ { 2 } + \\omega v _ { k } ^ { 2 } ) d x + \\frac { 1 } { 2 } ( B ( u _ { k } ) + B ( v _ { k } ) ) - \\frac { 1 } { 2 p } F ( u _ { k } , v _ { k } ) \\\\ & \\geq \\Big ( \\frac { 1 } { 2 } - \\frac { \\alpha } { 2 ( p \\alpha - 1 ) } \\Big ) \\| ( u _ { k } , v _ { k } ) \\| _ { E } ^ { 2 } . \\end{align*}"}
+{"id": "3063.png", "formula": "\\begin{align*} \\Lambda _ E = \\left \\{ ( \\lambda , z ) \\in { \\Lambda } \\mid \\rho _ E ( \\lambda , z ) = 0 \\right \\} \\end{align*}"}
+{"id": "1740.png", "formula": "\\begin{align*} \\int _ { f ( [ 0 , 1 ] ) } \\abs { \\varphi \\circ f ^ { - 1 } ( t ) } d t = \\int _ { [ 0 , 1 ] } \\abs { \\varphi \\circ f ^ { - 1 } ( f ( t ) ) } \\ ; | f ' ( t ) | d t \\leq \\norm { \\varphi } _ 1 \\norm { f } _ L . \\end{align*}"}
+{"id": "9099.png", "formula": "\\begin{align*} \\sum \\limits _ { i = j + 1 } ^ { n - 1 } k _ i + ( n - 2 ) r \\geq \\sum \\limits _ { i = j + 1 } ^ { n - 1 } r = - ( n - j - 1 ) + ( n - 2 ) r = ( j - 1 ) r \\end{align*}"}
+{"id": "6685.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } ( \\eta \\gamma ^ n ) ^ k x = ( \\eta \\gamma ^ n ) ^ + . \\end{align*}"}
+{"id": "9020.png", "formula": "\\begin{align*} \\dfrac { \\hat { k } _ 2 { ( \\Delta ) } } { \\hat { k } _ 2 ( \\Psi ) } = \\dfrac { D _ { 2 } D _ { 4 } D _ { 6 } } { D _ { 1 } D _ { 3 } D _ { 5 } } . \\end{align*}"}
+{"id": "6943.png", "formula": "\\begin{align*} H ( \\overline { Q } ) \\le \\frac 1 n \\sum _ { i = 1 } ^ n H ( Q _ i ) . \\end{align*}"}
+{"id": "8300.png", "formula": "\\begin{align*} M _ { j j } = - m _ j , \\ , j = 1 , \\cdots , n , M _ { j + 1 , j } = b _ j , \\ , j = 1 , \\cdots , n - 1 , M _ { 1 n } = b _ n , \\end{align*}"}
+{"id": "7340.png", "formula": "\\begin{align*} V ( A , - C ) = V ( A , C ) + \\sqrt { | A | \\ , | C | } . \\end{align*}"}
+{"id": "523.png", "formula": "\\begin{align*} H ^ { t } f ( x ) = \\sum _ { n = 0 } ^ { \\infty } e ^ { - N t } P _ { n } f ( x ) \\end{align*}"}
+{"id": "1040.png", "formula": "\\begin{align*} \\rho _ x & = \\rho _ { \\omega \\ast r _ { a _ 1 } \\ast \\cdots \\ast r _ { a _ n } } = \\rho _ { r _ { a _ n } } \\circ \\cdots \\circ \\rho _ { r _ { a _ 1 } } \\circ \\rho _ \\omega . \\end{align*}"}
+{"id": "4970.png", "formula": "\\begin{align*} W ( \\xi _ 0 , n , x , \\mathsf { M } ^ n ) = W ( 1 - \\xi _ 0 , n , x , \\mathsf { M } ^ n ) , ~ \\forall \\xi _ 0 \\in [ 0 , 1 ] , \\forall x \\in \\mathbb { R } . \\end{align*}"}
+{"id": "7093.png", "formula": "\\begin{align*} ( X _ 4 \\cup \\{ c _ 4 \\} ) \\cap { \\cal N } = ( X _ 1 \\cup \\{ c _ 1 \\} ) \\cap { \\cal N } = \\{ c _ 2 , c _ 3 \\} \\cap { \\cal N } = \\emptyset . \\end{align*}"}
+{"id": "8328.png", "formula": "\\begin{align*} D _ + = \\{ k : { \\rm I m } k ^ 2 > 0 \\} , \\ \\ D _ - = \\{ k : { \\rm I m } k ^ 2 < 0 \\} ; \\end{align*}"}
+{"id": "5554.png", "formula": "\\begin{align*} \\int e ^ { W ( y | x ) } d \\nu _ { A ^ { * } } ( y ) = \\varphi _ A ( x ) \\end{align*}"}
+{"id": "1597.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { \\Phi } _ { \\mathrm { r } } = \\mathsf { b l k d i a g } ( \\mathbf { \\Phi } _ { \\mathrm { r } , 1 } , \\ldots , \\mathbf { \\Phi } _ { \\mathrm { r } , G } ) , \\\\ \\mathbf { \\Phi } _ { \\mathrm { t } } = \\mathsf { b l k d i a g } ( \\mathbf { \\Phi } _ { \\mathrm { t } , 1 } , \\ldots , \\mathbf { \\Phi } _ { \\mathrm { t } , G } ) , \\end{aligned} \\end{align*}"}
+{"id": "5363.png", "formula": "\\begin{align*} \\sigma _ { k } ( \\lambda ) = \\sum _ { i _ { 1 } < \\ldots < i _ { k } } \\lambda _ { i _ { 1 } } \\ldots \\lambda _ { i _ { k } } , \\ \\ k = 1 , \\ldots , n , \\end{align*}"}
+{"id": "4477.png", "formula": "\\begin{align*} \\frac { p } { p + 1 } \\sum _ { k = 1 } ^ { \\infty } \\sum _ { n = k ^ { 2 } p } ^ { ( k + 1 ) ^ { 2 } p } \\frac { 2 \\cdot 2 ^ { \\omega ( \\gcd ( n , p ) ) } r _ { Q ^ { * } } ( n ) ^ { 2 } } { n } \\sum _ { d = 1 } ^ { \\infty } \\psi \\left ( d \\sqrt { \\frac { n } { p } } \\right ) . \\end{align*}"}
+{"id": "4241.png", "formula": "\\begin{align*} V ^ { - 1 } ( \\rho \\circ \\varphi ) V = V ^ { - 1 } \\circ \\rho \\circ V ( V ^ { - 1 } \\circ \\varphi \\circ V ) = 0 _ s \\oplus \\begin{bmatrix} ( V ^ { - 1 } \\varphi V ) ^ { s + 1 } ( z ) \\\\ \\vdots \\\\ ( V ^ { - 1 } \\varphi V ) ^ n ( z ) \\\\ \\end{bmatrix} . \\end{align*}"}
+{"id": "5763.png", "formula": "\\begin{align*} \\mathcal { D } + \\mathcal { E } = 0 . \\end{align*}"}
+{"id": "8119.png", "formula": "\\begin{align*} 1 - n + m - t = 0 , \\end{align*}"}
+{"id": "3133.png", "formula": "\\begin{align*} & \\omega ( \\alpha ( x ) , \\alpha ( y ) ) = \\omega ( x , y ) , \\displaystyle \\circlearrowleft _ { x , y , z } \\omega ( [ x , y ] , \\alpha ( z ) ) = 0 . \\end{align*}"}
+{"id": "1982.png", "formula": "\\begin{align*} z \\in g ( [ - a , a ] ) = [ - a - T ( - a ) , a - T ( a ) ] = [ - a - T ( a ) , a - T ( a ) ] . \\end{align*}"}
+{"id": "6725.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\phi _ s ( z ) = \\lim _ { s \\to \\infty } \\frac { s } { 2 \\Gamma ( 1 / s ) } \\exp ( - | z | ^ s ) = \\begin{cases} \\frac { 1 } { 2 } , & | z | < 1 , \\\\ 0 , & \\end{cases} \\end{align*}"}
+{"id": "1496.png", "formula": "\\begin{align*} \\phi _ 3 = - c \\phi _ 1 - \\frac { \\epsilon } { \\kappa } \\phi _ 4 - \\frac { c } { \\kappa } \\phi _ 2 + \\frac { c } { \\kappa } \\rightarrow 0 , \\ \\phi _ 1 \\rightarrow \\phi _ 1 ^ * , \\ \\phi _ 2 \\rightarrow 0 \\ \\ \\phi _ 4 \\rightarrow 0 \\ \\ z \\rightarrow - \\infty \\end{align*}"}
+{"id": "4511.png", "formula": "\\begin{align*} \\phi ( k - 1 ) & = k - 1 + t ' - \\frac { ( s _ 1 - 1 ) t ' + s _ 1 d ' } { k - 1 } \\\\ & < k - 1 + t ' - \\frac { s } { k } ( t ' + d ' ) \\le k - 1 + t ' - s + d = k + t ' - s _ 1 + d \\end{align*}"}
+{"id": "2988.png", "formula": "\\begin{align*} \\widetilde \\mu _ { ( x , 0 ) } = \\widehat \\mu _ { x } \\times \\delta _ 0 \\end{align*}"}
+{"id": "1846.png", "formula": "\\begin{align*} ( i _ X S _ { e , \\mathcal { Y M } ^ 0 } ) = \\langle S _ { e , \\mathcal { Y M } ^ 0 } , \\nabla X ^ { \\flat } \\rangle + ( S _ { e , \\mathcal { Y M } ^ 0 } ) ( X ) \\ , , \\forall \\ , X \\in \\Gamma ( M ) \\ , . \\end{align*}"}
+{"id": "5911.png", "formula": "\\begin{align*} \\sigma _ { n } ^ M : = & T \\wedge \\inf \\Big \\{ t \\geq 0 : \\Vert X ( t , x _ n ) \\Vert _ H > M \\Big \\} \\\\ & \\wedge \\inf \\Big \\{ t \\geq 0 : \\int _ 0 ^ t \\Vert X ( s , x _ n ) \\Vert _ { V } ^ { \\alpha } d s > M \\Big \\} \\\\ & \\wedge \\inf \\Big \\{ t \\geq 0 : \\Vert X ( t , x ) \\Vert _ H > M \\Big \\} \\\\ & \\wedge \\inf \\Big \\{ t \\geq 0 : \\int _ 0 ^ t \\Vert X ( s , x ) \\Vert _ { V } ^ { \\alpha } d s > M \\Big \\} . \\end{align*}"}
+{"id": "2786.png", "formula": "\\begin{align*} \\mathbb { P } ( h ^ { D _ { N } } ( x ) \\geq a _ { N } + b ) & = ( 1 + o ( 1 ) ) \\frac { 1 } { \\sqrt { 2 \\pi } } \\frac { \\sqrt { \\gamma \\log N + s _ { D } ( x ) + o ( 1 ) } } { a _ { N } + b } e ^ { - \\frac { ( a _ { N } + b ) ^ { 2 } } { 2 ( \\gamma \\log N + s _ { D } ( x ) + o ( 1 ) ) ^ { 2 } } } \\\\ & = ( 1 + o ( 1 ) ) \\frac { 1 } { 4 \\lambda \\sqrt { \\pi } \\sqrt { \\log N } } e ^ { - \\frac { a _ { N } ^ { 2 } } { 2 \\gamma \\log N } } e ^ { - \\pi \\lambda b } e ^ { \\frac { 4 \\lambda ^ { 2 } s _ { D } ( x ) } { \\gamma } } . \\end{align*}"}
+{"id": "4864.png", "formula": "\\begin{align*} \\partial _ y T ( x _ 0 , y ) = 1 . \\end{align*}"}
+{"id": "3225.png", "formula": "\\begin{align*} \\mathcal S ( t ) h ( x ) : = \\begin{cases} h ( x + t ) , & x + t \\leq 1 , \\\\ 0 , & x + t > 1 . \\end{cases} \\end{align*}"}
+{"id": "7636.png", "formula": "\\begin{align*} \\alpha _ t ^ { \\xi } = - R ^ { - 1 } B y _ t ^ { \\xi } - h ( \\mu _ t ) . \\end{align*}"}
+{"id": "5925.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\mathbb { P } ( \\tau _ u ^ M < T ) = 0 . \\end{align*}"}
+{"id": "9274.png", "formula": "\\begin{align*} & \\sum _ { A = 0 } ^ { 2 n - 1 } \\left [ d _ 1 \\left ( u - v \\right ) \\wedge ( \\triangle v ) ^ { p - 1 } \\wedge ( \\triangle u ) ^ { m - p } \\wedge \\beta _ n ^ { n - m } \\right ] _ A \\cdot \\nabla _ { A 0 ' } \\varrho ( q ) \\ , \\Omega _ { 2 n } \\\\ = & d _ 0 \\varrho ( q ) \\wedge d _ 1 \\left ( u - v \\right ) \\wedge ( \\triangle v ) ^ { p - 1 } \\wedge ( \\triangle u ) ^ { m - p } \\wedge \\beta _ n ^ { n - m } , \\end{align*}"}
+{"id": "1417.png", "formula": "\\begin{align*} \\P _ x ( \\rho > n ) & = \\int _ { W ^ d } \\P _ x ( \\rho > [ n / 2 ] , S _ { [ n / 2 ] } \\in d y ) \\P _ y ( \\rho > n - [ n / 2 ] ) \\\\ & \\le \\int _ { W ^ d \\cap \\{ \\max _ j ( y _ j - y _ { j - 1 } ) \\le \\sqrt n \\} } \\P _ x ( \\rho > [ n / 2 ] , S _ { [ n / 2 ] } \\in d y ) \\P _ y ( \\rho > n - [ n / 2 ] ) \\\\ & + \\sum _ { j = 2 } ^ d \\int _ { W ^ d \\cap \\{ ( y _ j - y _ { j - 1 } ) > \\sqrt n \\} } \\P _ x ( \\rho > [ n / 2 ] , S _ { [ n / 2 ] } \\in d y ) \\P _ y ( \\rho > n - [ n / 2 ] ) \\\\ & = : P _ 1 + \\sum _ { j = 2 } ^ d P _ j . \\end{align*}"}
+{"id": "814.png", "formula": "\\begin{align*} \\mathcal { L } w = - \\mathcal { L } H = \\partial ^ \\square H - \\big ( \\Delta _ \\Gamma V + V \\big | \\nabla _ \\Gamma \\nu \\big | ^ 2 \\big ) = 0 \\ , [ 0 , T ] \\times M \\end{align*}"}
+{"id": "6850.png", "formula": "\\begin{align*} J _ { 2 , n } ( x ) = J _ { 2 , n - 1 } ( x ) - e ^ { - \\frac { x } { 2 } } \\eta ( x ) a _ { n - 1 } ( x ) - \\int _ { x } ^ { \\infty } \\left ( \\eta ( t ) e ^ { - \\frac { t } { 2 } } \\right ) ^ { \\prime } a _ { n - 1 } ( t ) d t \\end{align*}"}
+{"id": "3833.png", "formula": "\\begin{align*} A \\cdot B = \\sum _ { k = 1 } ^ 3 \\frac { 1 } { | V _ k | ^ 2 } ( A \\cdot V _ k ) ( B \\cdot V _ k ) + \\sum _ { k = 1 } ^ 6 \\frac { 1 } { | W _ k | ^ 2 } ( A \\cdot W _ k ) ( B \\cdot W _ k ) . \\end{align*}"}
+{"id": "9015.png", "formula": "\\begin{align*} i _ { 1 2 4 } = - x _ 1 D _ 2 x _ 4 , i _ { 1 3 4 } = x _ 1 D _ 2 x _ 4 . \\end{align*}"}
+{"id": "7287.png", "formula": "\\begin{align*} ( \\bigoplus _ { i \\in I _ j } V _ i \\to W _ j , \\bigoplus _ { j \\in J } W _ j \\to U ) \\mapsto ( \\bigoplus _ { i \\in I } V _ i = \\bigoplus _ { j \\in J } \\bigoplus _ { i \\in I _ j } V _ i \\to \\bigoplus _ { j \\in J } W _ j \\to U ) \\end{align*}"}
+{"id": "449.png", "formula": "\\begin{align*} \\mu ( d x ) = e ^ { - \\lambda } \\delta _ 0 ( d x ) + ( 1 - e ^ { - \\lambda } ) f ( x ) d x , \\end{align*}"}
+{"id": "8902.png", "formula": "\\begin{align*} \\mathbf { h } = \\lambda _ { 1 } \\mathbf { e } _ 1 + \\lambda _ { 2 } \\mathbf { e } _ 2 + \\lambda _ { 3 } \\mathbf { e } _ 3 + \\lambda _ { 1 2 3 ' } \\mathbf { e } _ { 1 2 3 ' } \\end{align*}"}
+{"id": "982.png", "formula": "\\begin{align*} \\left | \\bigcup _ { { F \\in C _ 2 } } F \\cap M \\right | & \\le \\left ( c _ 2 - q ^ { d - 1 } \\right ) ( \\Delta + z q ^ { d ^ 2 - d } \\theta _ { d - 1 } ) + q ^ { d - 1 } ( \\Delta + z ' q ^ { d ^ 2 - d } \\theta _ { d - 1 } ) \\\\ & = c _ 2 ( \\Delta + z q ^ { d ^ 2 - d } \\theta _ { d - 1 } ) + \\underbrace { ( z ' - z ) } _ { \\le \\theta _ d } q ^ { d ^ 2 - 1 } \\theta _ { d - 1 } . \\end{align*}"}
+{"id": "5460.png", "formula": "\\begin{align*} d = 2 n + 1 = 4 l + 3 \\equiv 3 \\mod 4 \\end{align*}"}
+{"id": "13.png", "formula": "\\begin{align*} \\beta = \\alpha + \\alpha _ j j \\geq s . \\end{align*}"}
+{"id": "3015.png", "formula": "\\begin{align*} \\delta \\leq \\min \\{ \\delta _ * , \\delta _ \\circ \\} , \\delta < \\epsilon _ \\circ / ( \\lambda _ \\varphi ( 1 + \\lambda _ L ) ) , \\lambda _ L = \\max \\{ \\| l \\| \\ , | \\ , l \\in L \\} . \\end{align*}"}
+{"id": "5800.png", "formula": "\\begin{align*} g ' _ 2 = - \\frac { 1 } { \\nu } I \\widehat { g _ 1 ' } I + \\frac { 2 i } { \\mu \\nu } h _ z J g _ 1 ' \\end{align*}"}
+{"id": "7041.png", "formula": "\\begin{align*} F _ { 7 , 0 } \\ , = : \\ , \\theta \\ , \\in \\ , \\R , \\end{align*}"}
+{"id": "5531.png", "formula": "\\begin{align*} h \\cdot ( g \\cdot \\gamma ) & = \\theta _ n \\circ ( h \\times g \\cdot \\gamma ) \\circ i , \\\\ ( h g ) \\cdot \\gamma & = \\theta _ n \\circ ( h g \\times \\gamma ) \\circ i . \\end{align*}"}
+{"id": "1941.png", "formula": "\\begin{align*} F ( m ; z ) = \\sum _ { n = 0 } ^ \\infty b _ 5 \\left ( \\frac { m n - 1 } { 6 } \\right ) q ^ n \\in S _ { 2 m - 2 } ( \\Gamma _ 0 ( 1 8 0 ) , \\chi _ 5 ) _ m . \\end{align*}"}
+{"id": "473.png", "formula": "\\begin{align*} \\frac { f ( x + y ) } { f ( x ) } & = \\frac { \\sup _ { t \\ge x + y } f ( t ) } { \\inf _ { s \\in [ x _ 0 , x ] } f ( s ) } \\frac { f ( x + y ) } { \\sup _ { t \\ge x + y } f ( t ) } \\frac { \\inf _ { s \\in [ x _ 0 , x ] } f ( s ) } { f ( x ) } \\\\ & \\ge \\frac { F ( x + y + \\Delta ) } { F ( x - c + \\Delta ) } \\frac { f ( x + y ) } { \\sup _ { t \\ge x + y } f ( t ) } \\frac { \\inf _ { s \\in [ x _ 0 , x ] } f ( s ) } { f ( x ) } \\end{align*}"}
+{"id": "6173.png", "formula": "\\begin{align*} V _ 2 ( r ; l , Q ) = \\frac { ( L + 1 ) ( L + 2 ) } { r ^ 2 } - \\frac { Q } { r } f ( r ) - E _ 0 = V _ 1 ( r ; l + 1 , Q ) . \\end{align*}"}
+{"id": "6398.png", "formula": "\\begin{align*} \\omega = \\sum _ { i = 1 } ^ n d x ^ i \\wedge d \\theta _ i . \\end{align*}"}
+{"id": "3962.png", "formula": "\\begin{align*} \\mathbf { c } ^ { ( n - 2 k + 1 ) } = ( \\diamond , \\diamond , \\cdots , \\diamond , \\underbrace { 1 , 1 , \\cdots , 1 } _ { 2 k } ) . \\end{align*}"}
+{"id": "1787.png", "formula": "\\begin{align*} a ( q ) & : = \\sum _ { m , n \\in \\mathbb { Z } } q ^ { m ^ { 2 } + m n + n ^ { 2 } } , \\\\ b ( q ) & : = \\sum _ { m , n \\in \\mathbb { Z } } \\omega ^ { m - n } q ^ { m ^ { 2 } + m n + n ^ { 2 } } , \\\\ c ( q ) & : = \\sum _ { m , n \\in \\mathbb { Z } } q ^ { \\left ( m + \\frac { 1 } { 3 } \\right ) ^ { 2 } + \\left ( m + \\frac { 1 } { 3 } \\right ) \\left ( n + \\frac { 1 } { 3 } \\right ) + \\left ( n + \\frac { 1 } { 3 } \\right ) ^ { 2 } } , \\end{align*}"}
+{"id": "967.png", "formula": "\\begin{align*} a ' + n a = c _ 0 + c _ 1 p ^ { i _ { 1 , n } } + \\cdots + c _ { t - 1 } p ^ { i _ { t - 1 , n } } \\end{align*}"}
+{"id": "7358.png", "formula": "\\begin{gather*} \\lim _ k \\sup _ { P \\in M } P \\bigl \\{ \\abs { f } \\ , 1 ( \\abs { f } > k ) \\bigr \\} = 0 \\quad \\quad f \\in F . \\end{gather*}"}
+{"id": "7898.png", "formula": "\\begin{align*} X ( q , r ) = \\frac 1 q \\iota ( q r ) \\end{align*}"}
+{"id": "4962.png", "formula": "\\begin{align*} \\begin{aligned} \\theta ^ { [ 1 ] } _ 1 = & \\pi ^ { [ 1 ] } _ 1 ( \\xi _ 0 , x ) \\\\ : = & \\begin{cases} 1 , \\\\ 0 , \\end{cases} \\end{aligned} \\end{align*}"}
+{"id": "2189.png", "formula": "\\begin{align*} \\lambda _ { 1 } = R ( \\mathbf R , \\mathbf { v } _ { \\lambda _ 1 } ) = \\frac { { \\mathbf { v } _ { \\lambda _ 1 } } ^ T \\mathbf R \\mathbf { v } _ { \\lambda _ 1 } } { { \\mathbf { v } _ { \\lambda _ 1 } } ^ T \\mathbf { v } _ { \\lambda _ 1 } } \\end{align*}"}
+{"id": "1049.png", "formula": "\\begin{align*} d ( v _ 1 \\Rightarrow w _ 2 v _ 2 ) = & d ( v _ 1 \\Rightarrow w _ 2 v _ 2 ' ) + d ( w _ 2 v _ 2 ' \\Rightarrow w _ 2 v _ 2 ) . \\end{align*}"}
+{"id": "2627.png", "formula": "\\begin{align*} \\varepsilon ( h , y ) \\varepsilon ( x , y + z ) [ [ \\alpha ( y ) , h \\cdot z ] , \\alpha ^ 2 ( x ) ] = & 2 \\varepsilon ( x , y + z ) [ \\alpha ( h ) \\cdot [ y , z ] , \\alpha ^ 2 ( x ) ] \\\\ & - \\varepsilon ( x , y + z ) [ [ h \\cdot y , \\alpha ( z ) ] , \\alpha ^ 2 ( x ) ] = 0 . \\end{align*}"}
+{"id": "5889.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 + } \\sup _ { n \\in \\mathbb { N } } \\mathbb { E } \\int _ 0 ^ { T - \\delta } \\Vert Y _ n ^ M ( t + \\delta ) - Y _ n ^ M ( t ) \\Vert _ { H } ^ { \\alpha } d t = 0 , \\end{align*}"}
+{"id": "207.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sqrt [ n ] { | B _ { F , \\{ x , y \\} } ( n ) | } = \\lim _ { n \\to \\infty } \\sqrt [ n ] { | B _ { G , S } ( G ) | } = 3 . \\end{align*}"}
+{"id": "5097.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 } } | b | ^ { 2 } \\dd x ' = \\int _ { \\mathbb { R } ^ { 2 } } \\left ( | \\nabla \\phi | ^ { 2 } + | G | ^ { 2 } \\right ) \\dd x ' . \\end{align*}"}
+{"id": "2402.png", "formula": "\\begin{align*} M ^ \\sharp ( T ) = \\left ( \\frac { 1 } { D + 1 } + \\lambda \\right ) D \\theta M ^ \\otimes ( T ) . \\end{align*}"}
+{"id": "718.png", "formula": "\\begin{align*} \\left ( \\frac { N - 2 } { 2 } - \\frac { N + \\alpha } { 2 p } \\right ) \\int | \\nabla u | ^ 2 d x = \\lambda \\left ( \\frac { N } { 2 } - \\frac { N + \\alpha } { 2 p } \\right ) \\int | u | ^ 2 d x . \\end{align*}"}
+{"id": "3805.png", "formula": "\\begin{align*} [ D _ * ^ q , M _ { z ^ q } ] ( f ) & = D ^ q _ * ( z ^ q f ) ( z ) - z ^ q D _ * ^ q ( f ) ( z ) \\\\ & = a _ 0 \\Gamma ( q + 1 ) + \\sum _ { n = 1 } ^ { \\infty } z ^ { q n } a _ n \\left [ \\underbrace { \\frac { \\Gamma ( q ( n + 1 ) + 1 ) } { \\Gamma ( q n + 1 ) } - \\frac { \\Gamma ( q n + 1 ) } { \\Gamma ( q ( n - 1 ) + 1 ) } } _ { : = \\beta _ { m , q } } \\right ] \\\\ & = a _ 0 \\Gamma ( q + 1 ) + \\sum _ { n = 1 } ^ { \\infty } a _ n \\beta _ { m , q } \\ ; z ^ { q n } . \\end{align*}"}
+{"id": "8327.png", "formula": "\\begin{align*} \\psi ^ { \\pm } ( x , t ; k ) = I + k \\int _ { \\pm \\infty } ^ { x } e ^ { - 2 i k ^ 2 ( x - y ) \\widehat { \\sigma } _ 3 } P _ y ( y ) \\psi ^ { \\pm } ( y , t ; k ) d y . \\end{align*}"}
+{"id": "3588.png", "formula": "\\begin{align*} \\mathrm { L i f t } ( x ^ 1 _ 1 - x ^ 1 _ 2 ) & = & & \\{ z ^ 1 _ { 1 1 } - z ^ 1 _ { 2 1 } , z ^ 1 _ { 1 2 } - z ^ 1 _ { 2 2 } \\} \\\\ \\mathrm { L i f t } ( y ^ 1 _ 1 y ^ 2 _ 2 - y ^ 1 _ 2 y ^ 2 _ 1 ) & = & & \\{ z ^ 1 _ { 1 1 } z ^ 2 _ { 1 2 } - z ^ 1 _ { 1 2 } z ^ 2 _ { 1 1 } , z ^ 1 _ { 1 1 } z ^ 2 _ { 2 2 } - z ^ 1 _ { 1 2 } z ^ 2 _ { 2 1 } , z ^ 1 _ { 2 1 } z ^ 2 _ { 1 2 } - z ^ 1 _ { 2 2 } z ^ 2 _ { 1 1 } , z ^ 1 _ { 2 1 } z ^ 2 _ { 2 2 } - z ^ 1 _ { 2 2 } z ^ 2 _ { 2 1 } \\} \\end{align*}"}
+{"id": "7804.png", "formula": "\\begin{align*} F ^ q _ v ( 0 , p ) [ u _ k ] & = c _ 1 ( q , k , p ) u _ k , \\\\ F ^ q _ v ( 0 , p ) [ w _ k ] & = c _ 1 ( q , k , p ) w _ k , \\end{align*}"}
+{"id": "6479.png", "formula": "\\begin{align*} | | \\tau | | G \\psi \\stackrel { d e f } { = } \\sup _ { p \\in ( a , b ) } \\left \\{ \\ \\frac { | | \\tau | | L _ p ( \\Omega ) } { \\psi ( p ) } \\ \\right \\} = \\sup _ { p \\in ( a , b ) } \\left \\{ \\ \\frac { | | \\tau | | _ p } { \\psi ( p ) } \\ \\right \\} . \\end{align*}"}
+{"id": "8142.png", "formula": "\\begin{align*} H ^ n ( G , M ) = H ^ n \\mathcal C ^ \\bullet , . \\end{align*}"}
+{"id": "3419.png", "formula": "\\begin{align*} \\prod _ { \\ell = \\ell _ 1 } ^ { \\ell _ 2 } | \\cos ( \\pi ( \\theta + \\ell \\alpha ) ) | \\leq C ( \\varepsilon ) e ^ { ( \\ell _ 2 - \\ell _ 1 ) ( - \\ln { 2 } + \\varepsilon ) } \\inf _ { j = \\ell _ 1 } ^ { \\ell _ 2 } | \\cos ( \\pi ( \\theta + j \\alpha ) ) | , \\end{align*}"}
+{"id": "6608.png", "formula": "\\begin{align*} R i c ^ g + \\nabla \\tau = 0 , \\end{align*}"}
+{"id": "6258.png", "formula": "\\begin{align*} \\nabla _ X ^ \\perp \\eta _ i = \\sum _ { j } \\phi _ { i j } ( X ) \\varphi _ j \\eta _ j . \\end{align*}"}
+{"id": "8654.png", "formula": "\\begin{align*} { { \\bf { h } } _ l ^ H } { { \\bf { F } } _ { l ' } } = { { \\bf { 0 } } _ { 1 \\times M _ t } } , \\ \\forall l \\ne l ' . \\end{align*}"}
+{"id": "7211.png", "formula": "\\begin{align*} \\xi _ { k , i } ( x ) = \\left \\{ \\begin{array} { l l } 2 \\sqrt { C _ { d } \\kappa } \\ , \\theta _ { k } { \\rm R e } ( \\sigma _ { k , i } ( x ) ) , & k \\in \\Z _ { + } ^ d , \\\\ 2 \\sqrt { C _ { d } \\kappa } \\ , \\theta _ { k } { \\rm I m } ( \\sigma _ { k , i } ( x ) ) , & k \\in \\Z _ { - } ^ d , \\end{array} \\right . \\end{align*}"}
+{"id": "6491.png", "formula": "\\begin{align*} \\lambda _ x | _ { E _ { 2 , m } } = \\lambda _ y , \\lambda _ x ( m + v _ i ) = f \\lambda _ x ( m + e _ i ) = s ( f ) \\end{align*}"}
+{"id": "3672.png", "formula": "\\begin{align*} X _ { i ; j k } = - R _ { \\ell k j i } X ^ \\ell + \\tfrac { 1 } { 2 } ( h _ { i j ; k } + h _ { i k ; j } - h _ { j k ; i } ) . \\end{align*}"}
+{"id": "1315.png", "formula": "\\begin{align*} A ' ( t ) & = 2 | | u _ t | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } + 2 { \\rm R e } \\langle u _ { t t } , u \\rangle + 2 b { \\rm R e } \\int _ { \\mathbb { H } ^ n } \\overline { u } u _ t d x \\\\ & = 2 | | u _ t | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } - 2 I ( u ) , \\ , \\ , \\ , 0 \\leq t < T _ 1 . \\end{align*}"}
+{"id": "2689.png", "formula": "\\begin{align*} d a _ d ( n - 2 ) & \\leq a _ d ( n - 1 ) , \\\\ - \\binom { d } { 2 } a _ d ( n - 3 ) & \\leq - d ^ 3 \\binom { d } { 2 } a _ d ( n - 6 ) , \\\\ d a _ d ( n - 4 ) & \\leq d ( d + 1 ) ^ 2 a _ d ( n - 6 ) , \\\\ - d ^ 2 a _ d ( n - 5 ) & \\leq - d ^ 3 a _ d ( n - 6 ) . \\end{align*}"}
+{"id": "3148.png", "formula": "\\begin{align*} \\begin{aligned} \\| f '' \\| _ 1 & \\leq \\limsup _ { n \\rightarrow \\infty } | f ( n + 1 ) - f ( n ) | + \\limsup _ { n \\rightarrow \\infty } | f ( - n - 1 ) - f ( - n ) | \\\\ & + 2 \\sum _ { n \\in \\partial _ l S _ - f } f ( n ) - f ( n - 1 ) + 2 \\sum _ { n \\in \\partial _ r S _ - f } f ( n ) - f ( n + 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "3257.png", "formula": "\\begin{align*} N _ { k , l } = \\frac 1 { \\sqrt 2 } \\sum _ { c , b = 1 } ^ d \\int _ 0 ^ t \\hat { \\sigma } _ { k l , b c } ( s ) + \\hat { \\sigma } _ { l k , b c } ( s ) d B ^ { c b } _ s . \\end{align*}"}
+{"id": "7299.png", "formula": "\\begin{align*} L \\left ( d _ { 0 , j } \\right ) = 1 2 8 . 1 + 3 7 . 6 { \\log _ { 1 0 } } \\left ( { { d _ { { { 0 , j } } } } } \\right ) , \\end{align*}"}
+{"id": "6587.png", "formula": "\\begin{align*} Q ( r , t ) : = \\int _ { B _ { r } ( 0 ) } n ( x , t ) \\ , d x = \\sigma _ { d } \\int _ { 0 } ^ { r } \\rho ^ { d - 1 } n ( \\rho , t ) \\ , d \\rho \\end{align*}"}
+{"id": "2217.png", "formula": "\\begin{align*} \\mathbf { a } ( \\theta _ { 0 } ) ^ { T } \\mathbf { v } ^ { 0 } & = e ^ { - j A } + e ^ { - j 2 A } e ^ { - j \\phi } + . . . + e ^ { - j N A } e ^ { - j ( N - 1 ) \\phi } \\\\ & = e ^ { - j A } + e ^ { - j ( 2 A + \\phi ) } + . . . + e ^ { - j ( N A + ( N - 1 ) \\phi ) } \\\\ & = \\frac { e ^ { - j A } ( 1 - ( e ^ { - j A } e ^ { - j \\phi } ) ^ N ) } { 1 - ( e ^ { - j A } e ^ { - j \\phi } ) } \\end{align*}"}
+{"id": "3338.png", "formula": "\\begin{align*} \\phi _ j \\cap \\psi _ j = ( \\alpha _ j \\setminus \\{ x \\} ) \\cap \\alpha _ j = \\alpha _ j \\setminus \\{ x \\} = \\psi _ j \\setminus \\{ x \\} . \\end{align*}"}
+{"id": "5095.png", "formula": "\\begin{align*} 0 = \\lim _ { n \\to \\infty } \\left \\{ | | u _ n | | _ { L ^ { 2 } } + | | B _ n - B | | _ { L ^ { 2 } } \\right \\} \\geq \\liminf _ { n \\to \\infty } \\left \\{ | | u _ n | | _ { L ^ { 2 } } + \\inf _ { U \\in { \\mathcal { S } } _ h } | | B _ n - U | | _ { L ^ { 2 } } \\right \\} \\geq \\frac { \\varepsilon _ 0 } { 2 } > 0 . \\end{align*}"}
+{"id": "5512.png", "formula": "\\begin{align*} R ( p ) = 0 . 8 8 ( 8 / 3 ) ^ 2 ( D ( p ) - 1 ) + \\frac { 8 } { 9 } \\end{align*}"}
+{"id": "5244.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi _ t & = { \\mathcal { L } } \\varphi + \\mathbb { P } \\nabla \\cdot F \\textrm { o n } \\ L ^ { 2 } _ { \\sigma } ( \\mathbb { R } ^ { 3 } ) . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "333.png", "formula": "\\begin{align*} s _ \\lambda = \\det ( h _ { \\lambda _ j - j + i } ) _ { i , j = 1 } ^ \\ell = \\sum _ { \\pi \\in S _ \\ell } \\operatorname { s g n } ( \\pi ) h _ { \\pi ( \\lambda ) } \\end{align*}"}
+{"id": "216.png", "formula": "\\begin{align*} \\sigma _ { S , W } \\colon [ 0 , l ] _ \\mathbb { Z } \\to \\mathbb { Z } , k \\mapsto - \\sum _ { j = 1 } ^ k \\rho \\big ( s _ { \\iota ( j ) } \\big ) . \\end{align*}"}
+{"id": "7355.png", "formula": "\\begin{gather*} \\Gamma _ 0 = \\bigl \\{ P \\in \\Gamma : ( \\pi _ 1 , \\ldots , \\pi _ n ) P \\bigr \\} . \\end{gather*}"}
+{"id": "2031.png", "formula": "\\begin{align*} u _ t = - \\tfrac { 1 } { h _ 3 } ( f _ 3 p _ t + g _ 3 q _ t ) , \\end{align*}"}
+{"id": "7404.png", "formula": "\\begin{align*} ( \\mu ) _ \\infty = \\prod _ { n = 1 } ^ \\infty ( 1 - z ^ { n - 1 } \\mu ) . \\end{align*}"}
+{"id": "902.png", "formula": "\\begin{align*} \\sigma = \\sum \\limits _ { d = 1 } ^ \\infty \\frac { \\mu ( d ) \\lambda ( d ^ 2 ) } { d ^ 2 } \\ , . \\end{align*}"}
+{"id": "7142.png", "formula": "\\begin{align*} \\begin{aligned} 0 & \\to \\left ( T _ { \\mu _ g ( \\kappa ) ^ { - 1 } ( X , D , \\alpha , m ) } \\right ) _ { ( X , D , \\alpha , m ) } \\to \\left ( T _ { \\mathcal B _ g ( \\kappa ) } \\right ) _ { ( X , D , \\alpha , p , m ) } \\to \\left ( T _ { \\mathcal A _ g ( \\kappa ) } \\right ) _ { ( X , D , \\alpha , m ) } \\\\ & \\xrightarrow { \\left ( d \\mathfrak c _ g ( \\kappa ) \\right ) _ { ( X , D , \\alpha , m ) } } H ^ 1 \\left ( X , K _ X ^ { \\otimes 2 } ( - D ) \\right ) \\to 0 \\end{aligned} \\end{align*}"}
+{"id": "1565.png", "formula": "\\begin{align*} \\omega _ { 2 2 } = 1 \\omega _ { 1 2 } = 0 . \\end{align*}"}
+{"id": "5901.png", "formula": "\\begin{align*} g _ n ( t , \\omega ) : = & \\langle A ( t , X _ n ( t , \\omega ) ) , X _ n ( t , \\omega ) - X ( t , \\omega ) \\rangle , \\\\ F _ n ( t , \\omega ) : = & f ( t ) ( 2 + \\Vert X _ n ( t , \\omega ) \\Vert _ H ^ 2 ) + C \\Vert X ( t , \\omega ) \\Vert _ V ^ { \\alpha } \\\\ & + C \\Vert X _ n ( t , \\omega ) \\Vert _ H ^ { \\beta ( \\alpha - 1 ) } \\Vert X ( t , \\omega ) \\Vert _ V ^ { \\alpha } . \\end{align*}"}
+{"id": "7362.png", "formula": "\\begin{gather*} H = \\{ \\pi _ 1 \\} \\cup \\bigl \\{ \\bigl ( \\pi _ { i + 1 } - \\pi _ i \\bigr ) \\ , g ( \\pi _ 1 , \\ldots , \\pi _ i ) : 1 \\le i < n , \\ , \\ , g \\in B ( \\mathbb { R } ^ i , \\mathcal { B } ( \\mathbb { R } ^ i ) ) \\bigr \\} ; \\end{gather*}"}
+{"id": "7025.png", "formula": "\\begin{align*} \\aligned B _ 1 & \\ , : = \\ , - \\ , \\tfrac { 1 } { 3 } \\ , F _ { 3 , 1 } \\ , T _ 2 - A _ { 2 , 1 } , \\\\ A _ { 2 , 2 } & \\ , = \\ , - \\ , F _ { 3 , 1 } \\ , T _ 1 - T _ 2 . \\endaligned \\end{align*}"}
+{"id": "7155.png", "formula": "\\begin{align*} \\operatorname { g r a d } f = \\nabla ^ i f \\frac { \\partial } { \\partial x _ i } = g ^ { i j } \\frac { \\partial f } { \\partial x _ i } \\frac { \\partial } { \\partial x _ j } , f \\in C ^ { \\infty } ( \\Omega ) , \\end{align*}"}
+{"id": "4527.png", "formula": "\\begin{align*} v _ i ( x , t ) = S _ { d , i } ( t , t _ { i , e x } ( x , t ) ) S _ { c , i } ( t _ { i , e x } ( x , t ) , t _ { i , e n } ( x , t ) ) S _ { d , i } ( t _ { i , e n } ( x , t ) , 0 ) v _ { i , 0 } ( x ) . \\end{align*}"}
+{"id": "4246.png", "formula": "\\begin{align*} | z - w _ R | ^ 2 & = | P _ w ( z ) - w _ R | ^ 2 + | Q _ w ( z ) | ^ 2 \\\\ & = R ^ 2 s \\Big ( \\dfrac { | P _ w ( z ) - w _ R | ^ 2 } { R ^ 2 s } + \\dfrac { | Q _ w ( z ) | ^ 2 } { R ^ 2 s } \\Big ) \\\\ & < R ^ 2 s \\Big ( \\dfrac { | P _ w ( z ) - w _ R | ^ 2 } { R ^ 2 s ^ 2 } + \\dfrac { | Q _ w ( z ) | ^ 2 } { R ^ 2 s } \\Big ) = R ^ 2 s . \\end{align*}"}
+{"id": "6500.png", "formula": "\\begin{align*} E _ w ^ 0 & = \\{ z \\in B S ( 2 , 1 ) ^ + : z \\le w \\} , & E _ w ^ 1 & = \\{ ( z , z a ) : z , z a \\in E _ w ^ 0 \\} \\bigcup \\{ ( z , z b ) : z , z b \\in E _ w ^ 0 \\} \\\\ & & & = \\{ ( z , z l ) : z , z l \\in E _ w ^ 0 l = a , b \\} \\end{align*}"}
+{"id": "9235.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ( 2 \\bar { t } / \\gamma ) \\| _ { \\mathcal { A } } & \\le \\beta ( \\| x ( \\bar { t } / \\gamma ) \\| _ { \\mathcal { A } } , \\bar { t } ) + \\tilde { d } \\\\ & \\le \\beta ( \\underline { \\rho } , \\bar { t } ) + \\tilde { d } \\\\ & \\le \\beta ( \\rho , \\bar { t } ) + \\tilde { d } \\\\ & \\le \\underline { \\rho } \\end{aligned} \\end{align*}"}
+{"id": "5714.png", "formula": "\\begin{align*} f ' ( x , y ) & = \\alpha ( f \\circ L ( x , y ) ) , \\\\ & = \\alpha \\left ( a ( t x + u y ) ^ { q + 1 } + b ( t x + u y ) ^ q ( v x + w y ) + c ( t x + u y ) ( v x + w y ) ^ q + d ( v x + w y ) ^ { q + 1 } \\right ) , \\end{align*}"}
+{"id": "5550.png", "formula": "\\begin{align*} x \\cdot ( x \\wedge y ) = x \\cdot y \\in I . \\end{align*}"}
+{"id": "6390.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\widetilde { \\mathcal { R } } q ) : = \\pi _ { \\widehat { Q } _ h ^ \\perp \\cap \\widetilde { P } _ { k - 1 } ^ { \\mathrm { d i s c } } ( \\mathcal { T } ) } q = \\begin{cases} \\pi _ { \\widehat { Q } _ h ^ \\perp } q & k \\geq d , \\\\ \\pi _ { \\widetilde { P } _ { k - 1 } ^ { \\mathrm { d i s c } } ( \\mathcal { T } ) } q & k < d , \\end{cases} \\quad q \\in Q , \\end{align*}"}
+{"id": "6896.png", "formula": "\\begin{align*} \\delta _ \\square ( W _ 1 , W _ 2 ) : = \\inf _ { \\varphi } d _ \\square ( W _ 1 , W _ 2 ^ \\varphi ) , \\end{align*}"}
+{"id": "5158.png", "formula": "\\begin{align*} H [ t b ] = 2 t ^ { 2 } \\mu \\int _ { \\mathbb { R } ^ { 3 } } \\left ( \\phi - \\frac { \\phi _ { \\infty } } { t } \\right ) _ { + } ^ { 2 } \\frac { 1 } { r ^ { 2 } } \\dd x > 2 t ^ { 2 } \\mu \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi _ { \\infty } ) _ { + } ^ { 2 } \\frac { 1 } { r ^ { 2 } } \\dd x = t ^ { 2 } H [ b ] = t ^ { 2 } h . \\end{align*}"}
+{"id": "1365.png", "formula": "\\begin{align*} \\mathfrak { X } = \\frac { \\prod _ { j = 1 } ^ d \\Gamma ( j / 2 ) } { \\pi ^ { d / 2 } \\prod _ { j = 1 } ^ { d - 1 } j ! } . \\end{align*}"}
+{"id": "8894.png", "formula": "\\begin{align*} \\Gamma ( k - 3 / 2 ) S ( D _ 1 , D _ 2 ) = 2 \\pi ^ { k - \\frac { 3 } { 2 } } D _ 2 ^ { k - \\frac { 3 } { 2 } } p _ { D _ 2 } ( D _ 1 ) . \\end{align*}"}
+{"id": "5043.png", "formula": "\\begin{align*} U \\cdot \\nabla f = 0 , \\end{align*}"}
+{"id": "2155.png", "formula": "\\begin{align*} \\Im ( \\varrho _ z ^ 2 + \\varrho _ w ^ 2 ) = \\Im ( \\varrho _ { Z Z } ( L , L ) ) / a = 0 . \\end{align*}"}
+{"id": "5120.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\frac { r ' ( r r ' ) ^ { 2 \\tilde { \\tau } + 1 } } { | r - r ' | ^ { 4 \\tilde { \\tau } - 2 / q } r ' } \\dd r ' = C r ^ { 2 / q } . \\end{align*}"}
+{"id": "500.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } f _ { + a } ( x ) / f _ s ( x ) = 1 - e ^ { - \\lambda _ s } . \\end{align*}"}
+{"id": "2646.png", "formula": "\\begin{align*} ( q ; q ) _ { \\infty } ^ { 3 } = \\sum _ { j = 0 } ^ { \\infty } ( - 1 ) ^ { j } ( 2 j + 1 ) q ^ { \\frac { j ( j + 1 ) } { 2 } } . \\end{align*}"}
+{"id": "8367.png", "formula": "\\begin{align*} & N ( x ; k ) = I + \\frac { 1 } { 2 \\pi i } \\int _ { \\mathbb { R } \\cup i \\mathbb { R } } \\frac { \\varrho ( x ; s ) J } { s - k } d s = I + \\mathcal { C } ( \\varrho J ) ( z ) , \\end{align*}"}
+{"id": "3524.png", "formula": "\\begin{align*} - n ^ p - \\sum _ { K = 1 } ^ { p } \\binom { p } { K } \\ , B _ K \\ , n ^ { p - K } . \\end{align*}"}
+{"id": "4478.png", "formula": "\\begin{align*} \\frac { ( a _ { 4 } ^ { * } ) ^ { 1 / 3 } } { \\sqrt { p } } \\sum _ { k = 1 } ^ { \\infty } k ^ { 3 / 2 } ( 1 3 1 9 5 8 . 3 6 ( 4 k ^ { 3 } + 6 k ^ { 2 } + 4 k + 1 ) + 2 6 3 9 1 9 . 6 ( k + 1 ) ^ { 4 } ) M ( 2 5 . 0 9 ( k + 1 ) ^ { 2 } p ^ { 3 5 / 6 } ) e ^ { - 4 \\pi k } . \\end{align*}"}
+{"id": "1351.png", "formula": "\\begin{align*} t _ { k + 1 } = \\max \\bigl \\{ t \\colon d ( \\gamma _ { a _ 1 , b _ 1 } ( t ) , u _ { k } ) = s \\bigr \\} \\end{align*}"}
+{"id": "6522.png", "formula": "\\begin{align*} B ( L X ) + D ( L X ) + ( n - 1 ) [ \\beta ( L X ) + \\gamma ( L X ) ] = 0 . \\end{align*}"}
+{"id": "8418.png", "formula": "\\begin{align*} \\Psi ^ - _ { 1 1 } ( x ; z ) = e ^ { - i c _ - ( x ) } + \\frac { 1 } { 2 i } e ^ { - i c _ - ( x ) } \\int _ { - \\infty } ^ x u _ y ( y ) e ^ { i c _ - ( y ) } \\Psi ^ - _ { 2 1 } ( x ; z ) d y , \\end{align*}"}
+{"id": "8343.png", "formula": "\\begin{align*} C ^ - ( k ) = e ^ { - i c \\sigma _ 3 } , C ^ + ( k ) = { I } . \\end{align*}"}
+{"id": "4991.png", "formula": "\\begin{align*} \\underline { h } _ { \\mu } ( f _ { 1 , \\infty } ) : = \\int \\underline { h } _ { \\mu } ( f _ { 1 , \\infty } , x ) \\ d \\mu ( x ) . \\end{align*}"}
+{"id": "6950.png", "formula": "\\begin{align*} T _ { W _ { n J } , \\psi } ( \\mu _ n ( Q ) ) & = \\frac { 1 } { n } \\sum _ { i , j = 1 } ^ n J _ { i j } \\int _ \\R \\int _ \\R \\psi ( x , y ) Q _ i ( d x ) Q _ j ( d y ) , \\\\ I ( \\mu _ n ( Q ) ) & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n H ( Q _ i | \\rho ) , \\end{align*}"}
+{"id": "7795.png", "formula": "\\begin{align*} \\Theta ^ q ( x ) = 2 \\pi q x + \\alpha . \\end{align*}"}
+{"id": "7622.png", "formula": "\\begin{align*} \\ \\begin{cases} \\partial _ t v _ n + \\partial _ x v _ n \\cdot B + \\frac { 1 } { 2 } \\mathrm { T r } \\big [ \\partial _ { x x } v _ n \\Sigma \\Sigma ^ \\top \\big ] + f _ n ( t , \\partial _ { x } v _ n \\Sigma , x ) [ 0 , 1 ) \\times \\R ^ \\ell = 0 , \\\\ v _ n ( 1 , x ) = F ( x ) x \\in \\R ^ \\ell \\end{cases} \\end{align*}"}
+{"id": "5059.png", "formula": "\\begin{align*} \\partial _ t \\Phi _ { j } + u _ j \\cdot \\nabla \\Phi _ { j } = \\mu _ j \\left ( \\Delta - \\frac { 2 } { r } \\partial _ r \\right ) \\Phi _ j . \\end{align*}"}
+{"id": "597.png", "formula": "\\begin{align*} | x | _ { A } : = | A x | \\end{align*}"}
+{"id": "5942.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d } [ a _ i ( t , x , u , z ) - a _ i ( t , x , u , \\tilde { z } ) ] ( z _ i - \\tilde { z } _ i ) > 0 . \\end{align*}"}
+{"id": "7557.png", "formula": "\\begin{align*} \\prescript { } { l } { v } _ j ( \\lambda ) : = \\prescript { } { l } { \\tilde { h } } _ j ( \\lambda ) q _ { l , n _ j } ( \\lambda ) = \\prescript { } { l } { \\beta } ( \\lambda ) \\prescript { } { l } { R } _ j ( \\lambda ) q _ { l , n _ j } ( \\lambda ) . \\end{align*}"}
+{"id": "837.png", "formula": "\\begin{align*} H ( \\rho ^ \\varepsilon ) ( 0 , z _ l ) = \\frac { - 1 } { R + \\varepsilon \\rho _ 0 } \\phantom { x x x } \\phantom { x x x } H ( \\rho ^ \\varepsilon ) ( 0 , z _ r ) = \\frac { 1 } { R - \\varepsilon \\rho _ 0 } \\end{align*}"}
+{"id": "8489.png", "formula": "\\begin{align*} & \\sup _ { x \\in \\left ( x _ { 0 } , \\infty \\right ) } \\left \\| \\langle x \\rangle \\mathcal { P } ^ { + } \\left ( z ^ { - i } \\bar { r } _ 1 ( z ) e ^ { - 2 i z x } \\right ) \\right \\| _ { L _ z ^ 2 } \\leq \\left \\| z ^ { - i } \\bar r _ 1 ( z ) \\right \\| _ { H ^ 1 _ z } , i = 0 , 1 , \\\\ & \\sup _ { x \\in \\left ( x _ 0 , \\infty \\right ) } \\left \\| \\langle x \\rangle \\mathcal { P } ^ - \\left ( r _ 2 ( z ) e ^ { 2 i z x } \\right ) \\right \\| _ { L _ { z } ^ 2 } \\leq \\left \\| r _ 2 ( z ) \\right \\| _ { H ^ 1 _ z } . \\end{align*}"}
+{"id": "6016.png", "formula": "\\begin{align*} g ( t ) : = I ( t u , t v ) = \\frac { t ^ { 2 } } { 2 } \\| ( u , v ) \\| _ { E } ^ { 2 } + \\frac { t ^ { 6 } } { 2 } \\Big ( B ( u ) + B ( v ) \\Big ) - \\frac { t ^ { 2 p } } { 2 p } F ( u , v ) . \\end{align*}"}
+{"id": "7669.png", "formula": "\\begin{align*} \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } ^ * _ t } + \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } h ( \\mu ^ { N , j } _ { \\boldsymbol { \\alpha } ^ * _ t } ) = \\Delta _ { * , t } ^ { N , i } : = - R ^ { - 1 } B \\lambda ^ { N , i } _ { \\boldsymbol { y } _ t ^ { * } } , \\end{align*}"}
+{"id": "9044.png", "formula": "\\begin{align*} L ( E , \\epsilon ) = \\max \\{ \\log | \\lambda | + | \\epsilon | , L ( E , 0 ) \\} . \\end{align*}"}
+{"id": "4849.png", "formula": "\\begin{align*} & G ( c _ 1 , \\dots , c _ n , 0 ) \\\\ & = ( H _ 1 ( c _ 1 , \\dots , c _ n , 0 ) , \\dots , H _ n ( c _ 1 , \\dots , c _ n , 0 ) ) \\\\ & = ( p _ 1 ( c _ 1 , \\dots , c _ n ) , \\dots , p _ 1 ( c _ 1 , \\dots , c _ n ) ) \\\\ & = ( c _ 1 , \\dots , c _ 1 ) \\\\ & = \\Delta _ n p _ 1 ( c _ 1 , \\dots , c _ n ) , \\end{align*}"}
+{"id": "175.png", "formula": "\\begin{align*} ( \\prod _ { I } \\widehat { \\bigoplus } _ { I } A ^ + ) [ \\frac { 1 } { p } ] = ( \\prod _ { I } \\widehat { \\bigoplus } _ { I } A ^ { H ' , + } ) [ \\frac { 1 } { p } ] \\oplus ( \\prod _ { I } \\widehat { \\bigoplus } _ { I } X ^ + _ { H ' , n } ) [ \\frac { 1 } { p } ] . \\end{align*}"}
+{"id": "5107.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\nabla \\phi \\cdot \\nabla \\tilde { \\phi } \\frac { 2 \\pi ^ { 2 } } { r } \\dd z \\dd r = \\int _ { \\mathbb { R } ^ { 5 } } \\nabla _ y \\varphi \\cdot \\nabla _ y \\tilde { \\varphi } \\dd y , \\varphi = \\frac { \\phi } { r ^ { 2 } } , \\tilde { \\varphi } = \\frac { \\tilde { \\phi } } { r ^ { 2 } } . \\end{aligned} \\end{align*}"}
+{"id": "7925.png", "formula": "\\begin{align*} q _ 1 + q _ 2 & = 2 ( \\hat W _ { r _ 0 } ( q - 1 ) - \\hat W _ { r _ 0 } ( q ) - \\hat W _ { r _ 0 } ( q + 1 ) + \\hat W _ { r _ 0 } ( q + 2 ) ) \\\\ & = 2 ( \\hat W _ { r _ 0 } ( q ) - 2 \\hat W _ { r _ 0 } ( q + 1 ) + \\hat W _ { r _ 0 } ( q + 2 ) ) \\\\ & > 0 . \\end{align*}"}
+{"id": "7707.png", "formula": "\\begin{align*} i _ { k + 1 } ( B _ { I , [ m ] } ) & = { \\rm a r g m a x } _ { i \\in I \\backslash I _ k ( B _ { I , [ m ] } ) } | \\langle v _ { k + 1 } ( B _ { I , [ m ] } ) , B _ { i , [ m ] } ^ \\top \\rangle | \\\\ & = { \\rm a r g m a x } _ { i \\in [ n ] \\backslash I _ k ( B ) } | \\langle v _ { k + 1 } ( B ) , B _ { i , [ m ] } ^ \\top \\rangle | = i _ { k + 1 } ( B ) . \\end{align*}"}
+{"id": "7676.png", "formula": "\\begin{align*} \\left . \\begin{aligned} & \\boldsymbol { y } _ t ^ { * } : = ( y _ t ^ { * , i } ) _ { 1 \\leq i \\leq N } : = ( P _ t x _ t ^ { * , i } + \\varphi _ t ^ { * , i } ) _ { 1 \\leq i \\leq N } , \\\\ & \\boldsymbol { z } _ t ^ { * } : = ( z _ t ^ { * , i , j } ) _ { 1 \\leq i , j \\leq N } : = ( P _ t \\sigma + \\Lambda _ t ^ { * , i , j } ) _ { 1 \\leq i , j \\leq N } , \\boldsymbol { z } _ t ^ { 0 , * } : = ( z _ t ^ { 0 , * , i } ) _ { 1 \\leq i \\leq N } : = ( P _ t \\sigma _ 0 + \\Lambda _ t ^ { 0 , * , i } ) _ { 1 \\leq i \\leq N } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "8539.png", "formula": "\\begin{align*} \\psi _ 1 ^ - ( x , t ; k ) = a ( k ) \\psi _ 1 ^ + ( x , t ; k ) + b ( k ) e ^ { 2 i k ^ 2 x + 2 i \\eta ^ 2 t } \\psi _ 2 ^ + ( x , t ; k ) , k \\in \\mathbb { R } \\cup i \\mathbb { R } , \\end{align*}"}
+{"id": "4051.png", "formula": "\\begin{align*} \\max _ { v \\in \\Lambda } e ( \\alpha ^ { ( v ) } , G ^ { ( v ) } ) = e ( 0 , G ) . \\end{align*}"}
+{"id": "7080.png", "formula": "\\begin{align*} \\displaystyle \\int _ { B _ \\rho } \\dfrac { | \\tau _ h ( \\tau _ h v ( x ) ) | ^ { p } } { | h | ^ { ( 1 + \\gamma ) p } } d x \\leq & c | h | ^ p \\displaystyle \\int _ { B _ R } \\dfrac { | \\tau _ h D v ( x ) | ^ { p } } { | h | ^ { ( 1 + \\gamma ) p } } d x \\\\ = & c \\displaystyle \\int _ { B _ R } \\dfrac { | \\tau _ h D v ( x ) | ^ { p } } { | h | ^ { \\gamma p } } d x \\le c [ D v ] _ { B ^ { \\gamma } _ { p , \\infty } ( B _ R ) } ^ p , \\end{align*}"}
+{"id": "8189.png", "formula": "\\begin{align*} S & = x _ { i , 1 } + x _ { i + 1 , 1 } + x _ { i , n } + x _ { i + 1 , n } \\mbox { a n d } \\\\ S & = x _ { i + 1 , 1 } + x _ { i + 2 , 1 } + x _ { i + 1 , n } + x _ { i + 2 , n } , \\mbox { w h e r e } i = 1 , 2 , \\ldots , m . \\end{align*}"}
+{"id": "6393.png", "formula": "\\begin{align*} \\phi = \\frac { 1 } { 2 } \\left ( u _ { , p } \\xi ^ p + v _ q x ^ q \\right ) . \\end{align*}"}
+{"id": "5624.png", "formula": "\\begin{align*} \\| f | \\sigma _ 1 \\| _ { B V ( \\sigma _ 1 ) } = \\lim _ { n \\to \\infty } \\| p _ n | \\sigma _ 1 \\| _ { B V ( \\sigma _ 1 ) } \\leq \\lim _ { n \\to \\infty } \\| p _ n \\| _ { B V ( \\sigma ) } = \\| f \\| _ { B V ( \\sigma ) } . \\end{align*}"}
+{"id": "8458.png", "formula": "\\begin{align*} T - \\mathcal { P } ^ + \\left ( T R _ + \\right ) - \\mathcal { P } ^ - \\left ( T R _ - \\right ) = F , \\end{align*}"}
+{"id": "6142.png", "formula": "\\begin{align*} f ( x , x , z ) - f ( x , z , z ) = & d _ 1 ( x , z ) h _ { 0 0 } f ( x , y , z ) + h _ { 1 0 } ( d _ 0 ( x , y ) f ( x , y , z ) ) \\\\ & + h _ { 0 1 } ( d _ 0 ( y , z ) f ( x , y , z ) ) , \\end{align*}"}
+{"id": "1694.png", "formula": "\\begin{align*} V _ n ( \\varphi ) = \\begin{bsmallmatrix} x \\\\ \\Phi _ n \\end{bsmallmatrix} ^ { \\top } \\mathbf { P } _ n \\begin{bsmallmatrix} x \\\\ \\Phi _ n \\end{bsmallmatrix} \\leq 0 , \\ ; \\forall n \\geq \\mathcal { N } ( \\mathcal { E } ( \\eta _ 0 ) ) , \\end{align*}"}
+{"id": "6035.png", "formula": "\\begin{align*} ( S _ { \\omega } ) ^ { \\delta } : = \\{ u \\in H _ { r } ^ { 1 } ( \\R ^ { 2 } ) | u = \\widetilde { u } + \\overline { u } , \\widetilde { u } \\in S _ { \\omega } , \\| \\overline { u } \\| _ { H _ { r } ^ { 1 } ( \\R ^ { 2 } ) } \\leq \\delta \\} , \\end{align*}"}
+{"id": "4908.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { \\mathrm { B } ( \\alpha , \\beta ) } \\ , \\phi ( x ) \\ , \\Phi ( x ) ^ { \\alpha - 1 } \\ , [ 1 - \\Phi ( x ) ] ^ { \\beta - 1 } , \\end{align*}"}
+{"id": "2230.png", "formula": "\\begin{align*} x ^ { ( n + 1 ) } = T \\left ( x ^ { ( n ) } \\right ) ( \\forall n \\in \\mathbb { N } ) \\end{align*}"}
+{"id": "3404.png", "formula": "\\begin{align*} ( H _ { \\lambda , \\alpha , \\theta } u ) _ n = u _ { n + 1 } + u _ { n - 1 } + \\lambda \\tan ( \\pi ( \\theta + n \\alpha ) u _ n ) , \\end{align*}"}
+{"id": "6788.png", "formula": "\\begin{align*} e ( z ) = 1 + \\left ( z + 1 \\right ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } z ^ { n } a _ { n } ( 0 ) , \\end{align*}"}
+{"id": "7881.png", "formula": "\\begin{align*} \\kappa ( s ) : = \\sup _ { \\stackrel { k \\in \\N } { k \\neq \\ell } } c _ 1 ( q , k , p ( s ) ) \\end{align*}"}
+{"id": "4518.png", "formula": "\\begin{align*} \\tau ^ * = \\bar { \\tau } \\Leftrightarrow \\dfrac { | \\lambda _ i | } { | \\lambda _ { i + 1 } | } = \\dfrac { | \\lambda _ { i + 1 } | } { | \\lambda _ { i + 2 } | } \\quad \\forall i \\in [ 1 , p - 2 ] \\quad \\forall i \\in [ p + 1 , n - 2 ] ) , \\end{align*}"}
+{"id": "360.png", "formula": "\\begin{align*} \\begin{aligned} w _ k & = ( \\frac { 1 } { 4 \\cos ^ { 2 } ( \\frac { \\pi k } { n - 1 } ) } + 1 ) ( - 8 \\cos ^ { 2 } ( \\frac { \\pi k } { n - 1 } ) ) u _ k \\\\ & = - 2 ( 1 + 4 \\cos ^ { 2 } ( \\frac { \\pi k } { n - 1 } ) ) u _ k . \\end{aligned} \\end{align*}"}
+{"id": "9090.png", "formula": "\\begin{align*} \\mu _ w ( \\bigoplus _ { j = l } ^ { n - 1 } K _ j ' ) = \\frac { ( \\bigoplus _ { j = l } ^ { n - 1 } K _ j ' ) } { ( \\sum \\limits _ { j = l } ^ { n - 1 } w _ j ) r } \\geq \\frac { \\chi ( E ) } { r } \\end{align*}"}
+{"id": "3248.png", "formula": "\\begin{align*} \\rho _ { \\Sigma } ( m ) = \\mathbb E [ U ^ { \\otimes m } ] . \\end{align*}"}
+{"id": "643.png", "formula": "\\begin{align*} [ a ] \\ast [ b ] & = \\{ [ d ] \\in G / H : \\mbox { t h e r e e x i s t } a ' , b ' , d ' \\in G \\\\ & \\mbox { w i t h } [ d ' ] = [ d ] , \\ , [ a ' ] = [ a ] , \\ , [ b ' ] = [ b ] \\mbox { a n d } d ' \\in a ' \\ast b ' \\} . \\end{align*}"}
+{"id": "4349.png", "formula": "\\begin{align*} & \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in V \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h \\\\ = & \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in V \\} } | \\tilde F | ^ 2 _ h + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in V \\backslash N \\} } | F _ { t _ 0 , \\tilde { c } } - \\tilde F | ^ 2 _ h \\\\ & + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in V \\cap N \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h \\end{align*}"}
+{"id": "1070.png", "formula": "\\begin{align*} \\langle \\alpha ^ \\vee , 2 \\rho \\rangle = 2 + \\sum _ { \\substack { \\alpha \\neq \\beta \\in \\Phi ^ + \\\\ s _ \\alpha ( \\beta ) \\in \\Phi ^ - } } \\langle \\alpha ^ \\vee , \\beta \\rangle \\geq 2 + \\# \\{ \\beta \\in \\Phi ^ + \\setminus \\{ \\alpha \\} \\mid s _ \\alpha ( \\beta ) \\in \\Phi ^ - \\} = 1 + \\ell ( s _ \\alpha ) . \\end{align*}"}
+{"id": "5558.png", "formula": "\\begin{align*} W ( y | x ) = \\sum _ { n \\geq 1 } A ( y _ n , . . . , y _ 1 , x _ 1 , x _ 2 , . . . ) - A ( y _ n , . . . , y _ 1 , x _ 1 ' , x _ 2 ' , . . . ) , \\end{align*}"}
+{"id": "7549.png", "formula": "\\begin{align*} \\langle x | A _ j x \\rangle & = \\langle U x | U A _ j U ^ * ( U x ) \\rangle \\\\ & = \\left \\langle \\left ( \\sqrt { s _ 1 } y _ 1 , \\dots , \\sqrt { s _ m } y _ m , 0 , \\dots , 0 \\right ) | \\left ( \\sqrt { s _ 1 } A _ { j , 1 } y _ 1 , \\dots , \\sqrt { s _ m } A _ { j , m } y _ m , 0 , \\dots , 0 \\right ) \\right \\rangle \\\\ & = \\sum \\limits _ { k = 1 } ^ m s _ k \\langle y _ k | A _ { j , k } y _ k \\rangle . \\end{align*}"}
+{"id": "3817.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla ^ 2 _ U \\eta ( U , x , t ) - ( \\nabla _ V \\tilde { \\eta } ) & ( A ( U , x , t ) , x , t ) \\cdot \\nabla ^ 2 _ U A ( U , x , t ) = \\\\ & = ( \\nabla _ V ^ 2 \\tilde { \\eta } ) ( A ( U , x , t ) , x , t ) : ( \\nabla _ U A ( U , x , t ) , \\nabla _ U A ( U , x , t ) ) \\ ; . \\end{aligned} \\end{align*}"}
+{"id": "8567.png", "formula": "\\begin{align*} \\kappa ( t ) = h _ { \\alpha , \\rho } ( t ) , \\ 0 < \\alpha < 1 , \\ \\rho \\ge 0 , \\end{align*}"}
+{"id": "4430.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { \\mu ( k ) } { k } = 0 \\ \\ \\mathrm { a n d } \\ \\ \\sum _ { k = 1 } ^ \\infty \\frac { \\mu ( k ) \\log k } { k } = - 1 \\end{align*}"}
+{"id": "6133.png", "formula": "\\begin{align*} P _ { m ; l } \\otimes P _ { m ' ; l ' } \\cong \\bigoplus \\limits _ { \\nu = \\vert l - l ' \\vert \\mathrm { s t e p } 2 } ^ { \\mathrm { m i n \\left ( l + l ' , 2 d - 4 - l - l ' \\right ) } } P _ { m + m ' + \\frac { 1 } { 2 } \\left ( l + l ' - \\nu \\right ) ; \\nu } \\end{align*}"}
+{"id": "5874.png", "formula": "\\begin{gather*} \\begin{aligned} & 2 \\langle A ( t , u ) - A ( t , v ) , u - v \\rangle + \\Vert B ( t , u ) - B ( t , v ) \\Vert _ { L _ 2 } ^ 2 \\\\ \\leq & [ f ( t ) + \\rho ( u ) + \\eta ( v ) ] \\Vert u - v \\Vert _ { H } ^ 2 , \\\\ \\end{aligned} \\\\ | \\rho ( u ) | + | \\eta ( u ) | \\leq C ( 1 + \\Vert u \\Vert _ V ^ { \\alpha } ) ( 1 + \\Vert u \\Vert _ H ^ { \\gamma } ) , \\end{gather*}"}
+{"id": "6447.png", "formula": "\\begin{align*} Z _ { I } ^ { \\star } ( 1 , \\{ 2 \\} ^ { s - 1 } ; ( \\alpha , \\beta , \\gamma ) ) = Z ( s - 1 | s - 1 | 1 ; ( \\alpha , \\beta , \\gamma ) ) + Z ( s - 1 | s - 1 | 1 ; ( \\alpha , \\beta , \\alpha + \\beta - \\gamma ) ) \\end{align*}"}
+{"id": "2885.png", "formula": "\\begin{align*} & \\sum _ { x , x ' = 0 } ^ n \\Big ( \\delta _ { x , x ' } - M _ { x , x ' } ( m ) \\Big ) \\tilde V _ { x ' } ( m ) \\tilde V ^ { \\star } _ { x } ( m ) \\\\ & = { - \\tilde V _ { 0 } ( m ) \\sum _ { x = 0 } ^ n M _ { x , 0 } ( m ) \\tilde V ^ { \\star } _ { x } ( m ) } + \\sum _ { x = 0 } ^ n \\tilde v _ x ( m ) \\tilde V ^ { \\star } _ { x } ( m ) . \\end{align*}"}
+{"id": "3884.png", "formula": "\\begin{align*} \\frac { 1 } { \\ln \\frac { R } { s _ { \\delta , j } } } = \\frac { 1 } { \\ln \\frac { R } { \\varepsilon } } + O \\left ( \\frac { \\ln | \\ln \\varepsilon | } { | \\ln \\varepsilon | ^ 2 } \\right ) . \\end{align*}"}
+{"id": "2683.png", "formula": "\\begin{align*} y ^ { \\left ( \\prod _ { i = 1 } ^ d r _ i \\right ) n } = \\sum _ { a _ 1 , \\ldots , a _ d \\geq 1 } S _ { d , ( a _ 1 r _ 1 , \\ldots , a _ d r _ d n ) } ( x ) . \\end{align*}"}
+{"id": "4972.png", "formula": "\\begin{align*} W ( \\xi _ 0 , k , x , \\mathsf { L } ^ k ) = W ( 1 - \\xi _ 0 , k , x , \\mathsf { R } ^ k ) , ~ \\forall \\xi _ 0 \\in [ 0 , 1 ] , \\forall x \\in \\mathbb { R } , \\end{align*}"}
+{"id": "5421.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\sum _ { \\beta = 1 } ^ { \\alpha } b _ \\beta \\bar { x } _ \\beta ^ 2 \\leq C b _ \\alpha \\bar { x } _ n + b _ \\alpha O ( | \\bar { x } ' | ^ 3 ) \\leq C b _ \\alpha \\bar { x } _ n + C \\delta ^ 3 b _ \\alpha ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "6800.png", "formula": "\\begin{align*} b _ { m } ( x ) + \\sum _ { n = 0 } ^ { \\infty } b _ { n } ( x ) B _ { m n } ( x ) = s _ { m } ( x ) , m = 0 , 1 , \\ldots . \\end{align*}"}
+{"id": "9201.png", "formula": "\\begin{align*} \\epsilon _ 1 ( \\eta _ a , t ) = \\int _ { N } ^ { t } \\varepsilon _ \\delta ( { x } _ a , \\bar { y } _ a , \\tau ) u ( \\tau ) - b _ { 1 , \\delta } ( x _ a ) / 2 \\ , d \\tau \\end{align*}"}
+{"id": "4422.png", "formula": "\\begin{align*} z ^ k & \\mapsto f _ k ( s ) = - \\frac { 1 } { s } \\left ( ( k + 1 ) ^ { 1 - s } - k ^ { 1 - s } \\right ) & & ( k \\in \\mathbb { N ^ * } ) \\\\ 1 & \\mapsto - \\frac { 1 } { s } \\end{align*}"}
+{"id": "2975.png", "formula": "\\begin{align*} h ( x ) = \\begin{cases} - P ^ { - 1 } ( x ) & ( x \\geq 0 ) \\\\ x & ( x < 0 ) , \\end{cases} \\end{align*}"}
+{"id": "815.png", "formula": "\\begin{align*} H ( t ) = - w ( t ) = 0 & M t \\in ( 0 , t _ 0 ] \\\\ H ( t ) = - w ( t ) > 0 & M t \\in ( t _ 0 , T ] . \\end{align*}"}
+{"id": "5221.png", "formula": "\\begin{align*} | \\Phi _ { j , + } - \\Phi _ { + } | & = | \\Phi _ { j , + } - \\Phi _ { + } | ( 1 _ { ( 0 , \\infty ) } ( \\Phi _ j ) + 1 _ { ( 0 , \\infty ) } ( \\Phi ) - 1 _ { ( 0 , \\infty ) } ( \\Phi _ j ) 1 _ { ( 0 , \\infty ) } ( \\Phi ) ) \\\\ & \\leq | \\Phi _ { j , + } - \\Phi _ { + } | ( 1 _ { ( 0 , \\infty ) } ( \\Phi _ j ) + 1 _ { ( 0 , \\infty ) } ( \\Phi ) ) . \\end{align*}"}
+{"id": "6785.png", "formula": "\\begin{align*} \\alpha _ { k } ^ { \\pm } = \\frac { \\left ( d _ { k } \\right ) ^ { \\pm 1 } } { i a ^ { \\prime } ( \\rho _ { k } ) } \\end{align*}"}
+{"id": "8131.png", "formula": "\\begin{align*} f : = \\varinjlim _ i f _ i \\colon \\varinjlim _ i \\mathcal A _ i \\to \\varinjlim _ i \\mathcal B _ i \\end{align*}"}
+{"id": "1608.png", "formula": "\\begin{align*} \\mathbf { \\Xi } _ g ^ v = - \\triangledown _ { \\mathcal { M } _ g } \\tilde { f } _ g ( \\mathbf { \\Phi } _ { g } ^ v ) + \\mu _ g ^ v \\mathsf { P r } _ { \\mathbf { \\Phi } _ { g } ^ v } ( \\mathbf { \\Xi } _ g ^ { v - 1 } ) , \\forall g \\in \\mathcal { G } , \\end{align*}"}
+{"id": "351.png", "formula": "\\begin{align*} \\theta _ { k } : = - 8 \\cos ^ { 2 } ( \\frac { \\pi k } { n - 1 } ) - 2 ~ ~ ~ k = 1 , \\dotsc , n - 2 . \\end{align*}"}
+{"id": "4694.png", "formula": "\\begin{align*} { \\rm V o l } ( { \\tilde \\Sigma } ^ { 3 r } ) \\leq C ( n ) \\sum _ { i = 1 } ^ s { \\rm V o l } ( A \\cap B _ { p _ i } ( r ) ) \\leq C ( n ) \\ , r \\sum _ { i = 1 } ^ s { \\rm V o l } ( \\Sigma \\cap B _ { p _ i } ( 3 r ) ) . \\end{align*}"}
+{"id": "8422.png", "formula": "\\begin{align*} \\begin{aligned} \\widetilde { m } ( x ; z ) & = z \\int _ { - \\infty } ^ { x } e ^ { 2 i z ( x - y ) } w ( y ) d y + \\frac { 1 } { 2 i } w ( x ) \\\\ & - \\frac { 1 } { 2 i } \\int _ { - \\infty } ^ x e ^ { 2 i z ( x - y ) } \\left [ \\left ( 2 i \\bar { u } _ { y y } ( y ) + \\bar { u } _ y ( y ) | u _ y ( y ) | ^ { 2 } \\right ) \\widehat { \\Psi } ^ - _ { 1 1 } ( y ) + | u _ y ( y ) | ^ 2 \\widehat { \\Psi } ^ - _ { 2 1 } ( y ) \\right ] d y , \\end{aligned} \\end{align*}"}
+{"id": "7790.png", "formula": "\\begin{align*} \\Theta ^ M ( t , x ) = \\phi _ i ( t ) x \\in \\left [ \\frac { i - 1 } M , \\frac i M \\right ) . \\end{align*}"}
+{"id": "6780.png", "formula": "\\begin{align*} e ^ { \\prime } ( \\rho , x ) = e ^ { i \\rho x } \\left ( \\frac { z - 1 } { 2 ( z + 1 ) } - \\frac { 1 } { 2 } \\int _ { x } ^ { \\infty } q ( t ) d t + ( z + 1 ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } z ^ { n } d _ { n } ( x ) \\right ) \\end{align*}"}
+{"id": "2780.png", "formula": "\\begin{align*} G ^ { D _ { N } } ( [ x N ] , [ y N ] ) = \\tilde { G } ^ { D } ( x , y ) + o ( 1 ) \\end{align*}"}
+{"id": "3741.png", "formula": "\\begin{align*} \\Delta x _ k + \\bar u ^ { - 1 } \\nabla \\bar u \\cdot \\nabla x _ k = - \\Gamma ^ k + \\bar u ^ { - 1 } \\bar g ^ { i k } \\frac { \\partial \\bar u } { \\partial x _ i } . \\end{align*}"}
+{"id": "8514.png", "formula": "\\begin{align*} \\begin{aligned} & \\| u - \\tilde { u } \\| _ { H ^ 3 \\left ( \\mathbb { R } ^ { + } \\right ) \\cap H ^ { 2 , 1 } \\left ( \\mathbb { R } ^ + \\right ) } \\\\ \\leq & c \\left ( \\left \\| z ^ { - 1 } ( r _ 1 - \\tilde { r } _ 1 ) \\right \\| _ { H ^ 1 \\cap L ^ { 2 , 1 } } + \\left \\| r _ 1 - \\tilde { r } _ 1 \\right \\| _ { H ^ 1 \\cap L ^ { 2 , 1 } } + \\left \\| r _ 2 - \\tilde { r } _ 2 \\right \\| _ { H ^ 1 \\cap L ^ { 2 , 1 } } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "606.png", "formula": "\\begin{align*} \\phi _ q ( n ) : = ( \\pi _ H ( q ^ { - 1 } n ) ) ^ { - 1 } \\phi ( \\pi _ N ( q ^ { - 1 } n ) ) , \\mbox { f o r a l l } n \\in E _ q , \\end{align*}"}
+{"id": "5225.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ { 3 } } \\left ( | v | ^ { 2 } + | b | ^ { 2 } \\right ) \\dd x \\leq \\int _ { \\mathbb { R } ^ { 3 } } \\left ( | v _ 0 | ^ { 2 } + | b _ 0 | ^ { 2 } \\right ) \\dd x , \\\\ & \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi _ { \\infty } ) _ { + } \\frac { G } { r ^ { 2 } } \\dd x = \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi _ 0 - \\phi _ { \\infty } ) _ { + } \\frac { G _ 0 } { r ^ { 2 } } \\dd x , \\end{align*}"}
+{"id": "4004.png", "formula": "\\begin{align*} \\theta = \\big \\{ W ^ 1 , b ^ 1 , \\big ( U ^ { z , l } , W ^ { z , l } , b ^ { z , l } \\big ) ^ L _ { l = 1 } , \\big ( U ^ { g , l } , W ^ { g , l } , b ^ { g , l } \\big ) ^ L _ { l = 1 } , \\big ( U ^ { r , l } , W ^ { r , l } , b ^ { r , l } \\big ) ^ L _ { l = 1 } , \\big ( U ^ { h , l } , W ^ { h , l } , b ^ { h , l } \\big ) ^ L _ { l = 1 } , W , b \\big \\} \\end{align*}"}
+{"id": "1799.png", "formula": "\\begin{align*} c ( q ) & = 3 \\sum _ { r , s = 1 } ^ { \\infty } \\chi _ { - 3 } ( r ) \\left ( q ^ { \\frac { r s } { 3 } } - q ^ { r s } \\right ) , \\\\ b ( q ) c ( q ^ { 3 } ) & = 3 \\sum _ { n , k = 1 } ^ { \\infty } \\chi _ { - 3 } ( n k ) k q ^ { n k } , \\end{align*}"}
+{"id": "9252.png", "formula": "\\begin{align*} \\begin{aligned} d _ \\alpha F = \\sum _ { I } \\sum _ { A = 0 } ^ { 2 n - 1 } \\nabla _ { A \\alpha } f _ I \\omega ^ A \\wedge \\omega ^ I , \\end{aligned} \\end{align*}"}
+{"id": "8226.png", "formula": "\\begin{align*} \\sup _ { z \\in \\mathbb { D } } \\bigg | \\frac { 1 - \\overline { b ( a ) } b ( z ) } { 1 - \\overline { a } z } \\bigg | = o \\left ( \\frac { 1 } { 1 - | a | ^ 2 } \\right ) \\ , | b ( a ) | \\to 1 , \\end{align*}"}
+{"id": "8255.png", "formula": "\\begin{align*} \\frac { d \\nu ^ { n } _ m ( t ) } { d t } = \\Big ( \\sum _ { i = m } ^ { n - 1 } \\sum _ { j = 1 } ^ { i } j V _ { i , j } \\psi ^ { n } _ i \\psi ^ { n } _ j + m \\psi ^ { n } _ { m - 1 } \\sum _ { j = 1 } ^ { m - 1 } j V _ { m - 1 , j } \\psi ^ { n } _ j - \\sum _ { i = m } ^ { n - 1 } \\sum _ { j = i } ^ { n - 1 } i V _ { i , j } \\psi ^ { n } _ i \\psi ^ { n } _ j \\Big ) ( t ) . \\end{align*}"}
+{"id": "5362.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\sigma _ k \\big ( \\lambda ( D ^ { 2 } u ) \\big ) & = f \\ ; \\ ; \\mbox { i n } ~ \\Omega , \\\\ u & = \\varphi \\ ; \\ ; \\mbox { o n } ~ \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7753.png", "formula": "\\begin{align*} - \\frac { p _ { d - j + 1 } ^ { ( h ) } } { p _ { d - j } ^ { ( h ) } } \\approx x ^ { ( h ) } _ { j } = x _ { j } ^ { 2 ^ h } \\end{align*}"}
+{"id": "9137.png", "formula": "\\begin{align*} \\dot x = \\varphi _ { \\epsilon , r _ 0 } ( x , h ( x ) , t ) x ( 0 ) = x _ 0 , \\end{align*}"}
+{"id": "4590.png", "formula": "\\begin{align*} \\frac { g ( t ) - g ( r ) } { t - r } = \\lambda \\frac { g ( t ) - g ( s ) } { t - s } + ( 1 - \\lambda ) \\frac { g ( s ) - g ( r ) } { s - r } . \\end{align*}"}
+{"id": "2675.png", "formula": "\\begin{align*} \\bigl ( | u _ t | ^ { \\ell - 2 } u _ t \\bigr ) _ t - a \\nabla \\cdot \\bigl ( | \\nabla u | ^ { q - 2 } \\nabla u \\bigr ) + b | u _ t | ^ { m - 2 } u _ t = 0 , \\end{align*}"}
+{"id": "3150.png", "formula": "\\begin{align*} { _ { n + 1 } } F _ n \\left ( \\begin{array} { c c c c } A _ 0 , & A _ 1 , & \\ldots , & A _ n \\\\ & B _ 1 , & \\ldots , & B _ n \\end{array} \\mid x \\right ) : = \\frac { q } { q - 1 } \\sum _ { \\chi \\in \\widehat { \\mathbb { F } _ q ^ \\times } } { A _ 0 \\chi \\choose \\chi } { A _ 1 \\chi \\choose B _ 1 \\chi } \\cdots { A _ n \\chi \\choose B _ n \\chi } \\chi ( x ) . \\end{align*}"}
+{"id": "4141.png", "formula": "\\begin{align*} { \\rm I } & : = \\bigg ( \\frac { 1 } { | Q | } \\iint _ { T _ Q } | \\Theta ( ( f - f _ Q ) \\mathbf { 1 } _ { 4 Q } ) ( x ' , t ) | ^ 2 \\frac { d x ' d t } { t } \\bigg ) ^ { \\frac 1 2 } , \\\\ { \\rm I I } & : = \\bigg ( \\frac { 1 } { | Q | } \\iint _ { T _ Q } | \\Theta ( ( f - f _ Q ) \\mathbf { 1 } _ { \\R ^ { n - 1 } \\setminus 4 Q } ) ( x ' , t ) | ^ 2 \\frac { d x ' d t } { t } \\bigg ) ^ { \\frac 1 2 } . \\end{align*}"}
+{"id": "311.png", "formula": "\\begin{align*} \\lim _ { \\ell \\to + \\infty } \\lambda ^ \\ell ( p ) = \\lambda ^ \\infty ( p ) \\end{align*}"}
+{"id": "8218.png", "formula": "\\begin{align*} x _ { i , 1 , k } & = 2 m - i + k , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 1 , k } & = 4 m + i - k + 1 , \\\\ x _ { i , 2 , k } & = i , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 2 , k } & = 6 m - i + 1 , \\\\ x _ { i , 3 , k } & = 3 m - i + k , \\mbox { a n d } \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 3 , k } & = 3 m + i - k + 1 . \\end{align*}"}
+{"id": "7402.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - y } = \\sum _ { n = 0 } ^ \\infty y ^ n . \\end{align*}"}
+{"id": "3329.png", "formula": "\\begin{align*} \\chi _ i = \\left \\{ \\begin{aligned} & \\psi _ i & & ( i \\neq j ) \\\\ & \\sigma & & ( i = j ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3938.png", "formula": "\\begin{align*} - \\Delta z _ { n } + \\frac { F ( y _ { n } ) - F ( y ) } { t _ { n } } = h _ { n } . \\end{align*}"}
+{"id": "7330.png", "formula": "\\begin{align*} v ( x ) = \\left | \\sum _ { i = 1 } ^ k x _ i B _ i \\right | \\end{align*}"}
+{"id": "9232.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ( t ) \\| _ { \\mathcal { A } } & \\le \\| x _ { n } ( t ) \\| _ { \\mathcal { A } } + \\| x ( t ) - x _ { n } ( t ) \\| \\\\ & \\le \\beta ( \\| x ( n \\bar { t } / \\gamma ) \\| _ { \\mathcal { A } } , \\gamma t - n \\bar { t } ) + \\tilde { d } \\\\ & \\le \\beta ( \\| x ( n \\bar { t } / \\gamma ) \\| _ { \\mathcal { A } } , 0 ) + \\tilde { d } & & \\forall \\ , t \\in I _ n , \\ , \\forall \\ , n \\in \\mathbb { N } \\end{aligned} \\end{align*}"}
+{"id": "7803.png", "formula": "\\begin{align*} \\hat W _ r ( k ) = \\begin{cases} \\frac { 2 \\sin ( 2 \\pi k r ) } { \\pi k } & k \\neq 0 \\\\ 4 r & \\quad k = 0 \\end{cases} . \\end{align*}"}
+{"id": "1046.png", "formula": "\\begin{align*} \\ell ( x _ \\ast ) = \\ell ( x _ 1 ' ) + \\ell ( x _ 2 ) = \\ell ( x _ 1 ) + \\ell ( x _ 2 ) - d ( v _ 1 \\Rightarrow w _ 2 v _ 2 ) . \\end{align*}"}
+{"id": "2960.png", "formula": "\\begin{align*} \\Psi ( t , x , y ) = ( x e ^ { - t } , y e ^ { - t } ) . \\end{align*}"}
+{"id": "358.png", "formula": "\\begin{align*} \\nu _ k = \\frac { 2 ( 1 + \\omega ^ { - k } ) } { 8 \\cos ^ { 2 } ( \\frac { \\pi k } { n - 1 } ) } \\ 1 _ { n - 1 } ' v _ { k } + 2 \\ 1 _ { n - 1 } ' v _ { k } . \\end{align*}"}
+{"id": "7763.png", "formula": "\\begin{align*} p ( x ) \\Longleftrightarrow t _ { c , \\rho } ( y ) = p \\Big ( \\frac { x - c } { \\rho } \\Big ) , \\end{align*}"}
+{"id": "6158.png", "formula": "\\begin{align*} \\left ( - ( 1 - \\kappa r ^ 2 ) \\frac { d ^ 2 } { d r ^ 2 } - ( d - 1 - d \\kappa r ^ 2 ) \\frac { 1 } { r } \\frac { d } { d r } + \\frac { l ( l + d - 2 ) } { r ^ 2 } - \\frac { Q } { r } \\sqrt { 1 - \\kappa r ^ 2 } - { \\cal E } \\right ) R ( r ) = 0 , \\end{align*}"}
+{"id": "8833.png", "formula": "\\begin{align*} \\begin{cases} d E _ t = X _ { t , 1 } X _ { 0 , n - 1 } d t + d W _ t \\\\ d X _ { t , 1 } = X _ { 0 , 2 } ( X _ { 0 , n } + E _ t ) d t \\\\ d X _ { t , 3 } = - X _ { t , 1 } X _ { 0 , 2 } d t . \\end{cases} \\end{align*}"}
+{"id": "96.png", "formula": "\\begin{align*} \\nu ( b _ x ) = \\max _ J \\pi _ J ( \\lambda ( b _ x ) ) , \\end{align*}"}
+{"id": "3093.png", "formula": "\\begin{align*} \\langle \\theta , \\vartheta \\rangle _ { \\mathcal { L } ^ 2 , { G } } = \\langle \\theta , G \\vartheta \\rangle _ { \\mathcal { L } ^ 2 } = \\beta r \\langle x , u \\rangle _ { \\mathcal { L } ^ 2 } + \\beta \\langle y , v \\rangle _ { \\mathcal { L } ^ 2 } + \\frac { 1 } { \\beta } \\langle \\lambda , w \\rangle _ { \\mathcal { L } ^ 2 } , \\end{align*}"}
+{"id": "4470.png", "formula": "\\begin{align*} a & = a _ { 3 } ^ { * } + a _ { 4 } ^ { * } n _ { 4 3 } ^ { 2 } \\leq \\frac { 4 p } { 9 } + \\frac { p ^ { 3 / 4 } } { 3 \\sqrt { 3 } } \\leq 0 . 5 4 p | b | = 2 a _ { 4 } ^ { * } | n _ { 4 3 } | \\leq \\frac { 4 p ^ { 3 / 4 } } { 3 \\sqrt { 3 } } c = a _ { 4 } ^ { * } \\leq \\frac { 4 p ^ { 3 / 4 } } { 3 \\sqrt { 3 } } \\\\ | b ^ { 2 } - 4 a c | & = | - 4 a _ { 3 } ^ { * } a _ { 4 } ^ { * } | \\leq \\frac { 6 4 p ^ { 7 / 4 } } { 2 7 \\sqrt { 3 } } \\\\ \\end{align*}"}
+{"id": "8628.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { m = 1 } ^ n \\mathbb { E } _ { n , m } ' [ \\| \\chi _ { n , m } ' \\| _ { L ^ 2 ( \\mu ) } ^ 2 : \\| \\chi _ { n , m } ' \\| _ { L ^ 2 ( \\mu ) } > \\varepsilon ] = 0 \\end{align*}"}
+{"id": "9337.png", "formula": "\\begin{align*} \\min _ { \\xi , \\eta } p \\left ( \\xi , \\eta \\right ) = \\min \\left \\{ \\min _ { \\xi } p \\left ( \\xi , 1 \\right ) , \\min _ { \\xi } p \\left ( \\xi , \\frac { 9 9 } { 1 0 0 } \\right ) \\right \\} . \\end{align*}"}
+{"id": "9288.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ B u _ { p + N } \\Delta u _ p \\wedge \\alpha & = B _ { p , N } + \\int _ B u _ p \\Delta u _ { p + N } \\wedge \\alpha \\geq B _ { p , N } + \\int _ B u _ { p + N } \\Delta u _ { p + N } \\wedge \\alpha , \\end{aligned} \\end{align*}"}
+{"id": "6659.png", "formula": "\\begin{align*} & \\lim _ { m \\to \\infty } \\lim _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\sup _ { t \\in [ 0 , 1 ] } | \\mathcal { R } _ { 4 , \\epsilon } ( t ) | \\geq \\sup _ { u \\in [ 0 , 1 ] } | \\varphi ( u ) - \\varphi _ m ( u ) | ^ { 1 / 2 } \\Big ) \\\\ & \\leq - \\lim _ { m \\to \\infty } \\frac { c \\eta } { L ^ { 2 } \\sup _ { t \\in [ 0 , 1 ] } | \\varphi ( t ) - \\varphi _ m ( t ) | } = - \\infty . \\end{align*}"}
+{"id": "8063.png", "formula": "\\begin{align*} e _ { a _ j } \\left [ t \\right ] = a _ j \\left [ t \\right ] - a _ j ^ { \\left ( o \\right ) } , \\end{align*}"}
+{"id": "3233.png", "formula": "\\begin{align*} 0 = \\lim _ { n \\to \\infty } \\ \\sum _ { i = 1 } ^ { n } \\mathbb E \\left [ \\| ( \\mathcal S ( \\Delta _ n ) - I ) Y _ { ( i - 1 ) \\Delta _ n } ) \\| ^ 4 \\right ] \\geq & \\lim _ { n \\to \\infty } \\ \\sum _ { i = 1 } ^ { n } \\mathbb E \\left [ \\| ( \\mathcal S ( \\Delta _ n ) - I ) Y _ { ( i - 1 ) \\Delta _ n } ) \\| ^ 2 \\right ] ^ 2 \\\\ = & \\lim _ { n \\to \\infty } \\Delta _ n ^ { 2 + 4 \\mathfrak H } \\sum _ { i = 1 } ^ n ( i - 1 ) ^ 2 > 0 , \\end{align*}"}
+{"id": "6502.png", "formula": "\\begin{align*} \\lambda | ^ * _ { [ w _ 1 , w _ 2 ] } ( a ) = \\lambda ( w _ 1 a ) \\end{align*}"}
+{"id": "1788.png", "formula": "\\begin{align*} a ^ { 3 } ( q ) = b ^ { 3 } ( q ) + c ^ { 3 } ( q ) . \\end{align*}"}
+{"id": "8895.png", "formula": "\\begin{align*} \\frac { D } { a ^ 2 } = \\frac { f ^ 2 D _ 0 } { a ^ 2 } = \\frac { ( f _ o / a _ o ) ^ 2 f _ 2 ^ 2 D _ 0 } { a _ 2 ^ 2 } = \\frac { ( f _ o / a _ o ) ^ 2 ( f _ 2 / 2 ^ { \\beta - 1 } ) ^ 2 D _ 0 } { 4 } = A ^ 2 \\cdot m , \\end{align*}"}
+{"id": "8708.png", "formula": "\\begin{align*} \\varphi _ { | N } = \\frac { 1 } { n } ( \\psi _ 1 + \\cdots + \\psi _ n ) \\end{align*}"}
+{"id": "583.png", "formula": "\\begin{align*} A _ k = \\begin{pmatrix} M _ 1 ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) \\\\ M _ 2 ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) \\\\ \\vdots \\\\ M _ { k - 1 } ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) \\\\ M _ { k + 1 } ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) \\\\ \\vdots \\\\ M _ { 2 d - 1 } ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) \\\\ M _ { 2 d } ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "5938.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\lim _ { n \\rightarrow 0 } I I _ 3 = 0 . \\end{align*}"}
+{"id": "7194.png", "formula": "\\begin{align*} e ^ { - t \\Lambda _ g } = \\frac { i } { 2 \\pi } \\int _ { \\mathcal { C } } e ^ { - t \\tau } ( \\Lambda _ g - \\tau I ) ^ { - 1 } \\ , d \\tau , \\end{align*}"}
+{"id": "2737.png", "formula": "\\begin{align*} \\varphi ( x _ 1 , x _ 2 , x _ 3 ) = ( x _ 1 , x _ 2 , x _ 3 - | x _ 1 | - | x _ 2 | ) \\end{align*}"}
+{"id": "8561.png", "formula": "\\begin{align*} ( D ^ { \\alpha , 0 } _ { 0 + } \\ , f ) ( x ) = ( I _ { 0 + } ^ { 0 } \\ , \\frac { d } { d t } \\ , I _ { 0 + } ^ { 1 - \\alpha } \\ , f ) ( t ) \\ , = \\ , \\frac { d } { d t } \\ , ( I _ { 0 + } ^ { 1 - \\alpha } \\ , f ) ( t ) \\ , = \\ , ( D ^ \\alpha _ { 0 + } \\ , f ) ( t ) , \\ 0 \\le \\alpha < 1 , \\end{align*}"}
+{"id": "2271.png", "formula": "\\begin{align*} a ( Z ) = a \\big ( ( Z ^ * Z ) ^ \\frac { 1 } { 2 } \\big ) \\end{align*}"}
+{"id": "1499.png", "formula": "\\begin{align*} u '' + q ( t ) u = 0 , t \\in ( a , b ) \\end{align*}"}
+{"id": "9088.png", "formula": "\\begin{align*} \\chi _ j = \\chi - \\sum \\limits _ { i \\neq j } \\chi _ i - ( n - 1 ) r \\end{align*}"}
+{"id": "6353.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { h ( x ) } { 1 / x } = & \\lim _ { x \\to \\infty } \\frac { 2 \\theta } { \\operatorname { B } ( a , b ) } \\frac { 1 } { x ^ 3 } \\frac { \\left [ 1 - \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) \\right ] ^ { b - 1 } \\exp \\left ( - \\frac { a \\theta } { x ^ 2 } \\right ) } { \\operatorname { I } _ { 1 - \\exp ( - \\theta / x ^ 2 ) } ( b , a ) } x . \\end{align*}"}
+{"id": "3330.png", "formula": "\\begin{align*} \\chi _ 1 = \\psi _ 1 , \\ldots , \\chi _ { j - 1 } = \\psi _ { j - 1 } , \\chi _ j = \\sigma < _ K \\psi _ j . \\end{align*}"}
+{"id": "2152.png", "formula": "\\begin{align*} \\tau = \\frac { \\varrho ^ 2 _ z } { | w | ^ 2 } < 0 . \\end{align*}"}
+{"id": "5290.png", "formula": "\\begin{align*} \\Sigma ^ { \\pm } ( s - w , \\phi , F _ { \\pm } ) = & \\lim _ { z \\downarrow 0 } \\sum _ { \\varepsilon = \\pm } \\int _ 0 ^ \\infty w _ z ( \\lambda ) \\lambda ^ { 4 ( 1 - s + w ) - 1 } \\int _ 0 ^ \\infty \\\\ & \\times \\sum _ { \\ell , m = 1 } ^ \\infty \\rho _ \\phi ( 3 \\ell m ) K _ \\nu ( 6 \\pi \\ell m t ^ 2 ) \\hat { F } _ 4 \\left ( 0 , 0 , \\frac { 3 m \\lambda } { t } , \\frac { \\varepsilon 3 \\ell m t ^ 3 } { \\lambda } \\right ) \\frac { d \\lambda } { \\lambda } t d t . \\end{align*}"}
+{"id": "2828.png", "formula": "\\begin{align*} \\mathbf { H } = | \\mathbf { H } | \\odot e ^ { j \\angle \\mathbf { H } } , \\end{align*}"}
+{"id": "1918.png", "formula": "\\begin{align*} \\| x _ { k + 1 } - x ^ \\star \\| ^ 2 \\leq \\tfrac { 1 } { 2 } \\| x _ k - x ^ \\star \\| ^ 2 , \\textnormal { f o r a l l } k = 0 , 1 , 2 , \\ldots , T . \\end{align*}"}
+{"id": "7057.png", "formula": "\\begin{align*} G _ { 4 , 1 , 0 } \\ , = \\ , \\pm \\ , 1 , \\end{align*}"}
+{"id": "6448.png", "formula": "\\begin{align*} K = \\bigcup _ { n = 1 } ^ { \\infty } [ 2 ^ { 2 n } , 2 ^ { 2 n + 1 } - 1 ] \\cap \\N . \\end{align*}"}
+{"id": "6679.png", "formula": "\\begin{align*} B _ { x , \\gamma x } ( \\gamma ^ + ) = B _ { y , \\gamma y } ( \\gamma ^ + ) + B _ { x , y } ( \\gamma ^ + ) - B _ { \\gamma x , \\gamma y } ( \\gamma ^ + ) . \\end{align*}"}
+{"id": "9198.png", "formula": "\\begin{align*} \\begin{aligned} \\epsilon ( \\eta _ a , t ) = & \\sum _ { n = 1 } ^ N \\int _ { n - 1 } ^ { n } F ( \\eta _ a , \\tau ) - F _ a ( \\eta _ a ) \\ , d \\tau \\\\ & + \\int _ { N } ^ { t } F ( \\eta _ a , \\tau ) - F _ a ( \\eta _ a ) \\ , d \\tau \\\\ = & \\int _ { N } ^ { t } F ( \\eta _ a , \\tau ) - F _ a ( \\eta _ a ) \\ , d \\tau . \\end{aligned} \\end{align*}"}
+{"id": "5982.png", "formula": "\\begin{align*} \\Big \\{ ( \\nu _ 1 , \\nu _ 2 , \\nu _ 3 ) : \\nu _ 1 = \\frac { 1 } { 2 } \\nu _ 2 ^ 2 , 1 - \\frac { 1 } { K } \\le \\nu _ 3 \\le 1 , \\Big | \\frac { \\nu _ 2 } { \\nu _ 3 } \\Big | \\le 1 \\Big \\} \\end{align*}"}
+{"id": "7277.png", "formula": "\\begin{align*} [ F ( \\gamma , \\beta ) : F ] = [ F ( \\gamma , \\beta ) : F ( \\beta ) ] [ F ( \\beta ) : F ] = [ F ( \\gamma , \\beta ) : F ( \\beta ) ] ( q ^ n - 1 ) . \\ \\end{align*}"}
+{"id": "4629.png", "formula": "\\begin{align*} \\phi ( \\sigma , \\tau _ 1 \\tau _ 2 ) = \\widetilde \\sigma \\widetilde { \\tau _ 1 } \\widetilde { \\tau _ 2 } { \\widetilde { \\sigma } } ^ { - 1 } { \\widetilde { \\tau _ 2 } } ^ { - 1 } { \\widetilde { \\tau _ 1 } } ^ { - 1 } , \\phi ( \\sigma , \\tau _ 1 ) \\phi ( \\sigma , \\tau _ 2 ) = \\widetilde \\sigma \\widetilde { \\tau _ 1 } { \\widetilde \\sigma } ^ { - 1 } { \\widetilde { \\tau _ 1 } } ^ { - 1 } [ \\widetilde \\sigma , \\widetilde { \\tau _ 2 } ] \\end{align*}"}
+{"id": "7708.png", "formula": "\\begin{align*} { \\rm a r g m a x } _ { j \\in [ n ] \\backslash I } | \\langle v _ { k + 1 } ( B ) , ( B _ { j , [ m ] } ) ^ \\top \\rangle | & \\le | \\langle v _ { k + 1 } ( B _ { I , [ m ] } ) , ( B _ { i _ { k + 1 } ( B _ { I , [ m ] } ) , [ m ] } ) ^ \\top \\rangle | \\\\ & = { \\rm a r g m a x } _ { i \\in I \\backslash I _ { k } ( B ) } | \\langle e _ 1 , ( B _ { i , [ m ] } ) ^ \\top \\rangle | , \\end{align*}"}
+{"id": "6703.png", "formula": "\\begin{align*} h ( s \\xi , s \\eta , s \\omega ) - h ( \\xi , \\eta , \\omega ) = 2 \\hat \\beta ( s , \\xi ) . \\end{align*}"}
+{"id": "1443.png", "formula": "\\begin{align*} & \\P _ x ( Z _ d ( n _ 1 ) \\leq \\xi _ 1 , \\ldots , Z _ d ( n _ m ) \\leq \\xi _ m ) \\\\ & = \\frac { 1 } { h ( x ^ 0 ) } \\int _ { I ^ { \\xi _ 1 } } \\ldots \\int _ { I ^ { \\xi _ m } } \\det ( f _ { n _ 1 - d + i } ( x _ j ^ 1 - x _ i ^ 0 ) ) \\det ( f _ { n _ 2 - n _ 1 } ( x _ j ^ 2 - x _ i ^ 1 ) ) \\\\ & \\ldots \\det ( f _ { n _ m - n _ { m - 1 } } ( x _ j ^ m - x _ i ^ { m - 1 } ) ) \\Delta ( x ^ m ) \\prod _ { k = 1 } ^ m \\prod _ { j = 1 } ^ d d x _ j ^ k . \\end{align*}"}
+{"id": "7769.png", "formula": "\\begin{align*} - \\frac { p _ { d - 1 } ^ { ( h ) } } { p _ { d } ^ { ( h ) } } = - \\frac { p ' _ h ( 0 ) } { p _ h ( 0 ) } = \\sum _ { j = 1 } ^ d x _ j ^ { - k } , ~ - \\frac { p _ { 1 } ^ { ( h ) } } { p _ { 0 } ^ { ( h ) } } = - \\frac { p ' _ { h , { \\rm r e v } } ( 0 ) } { p _ { h , { \\rm r e v } } ( 0 ) } = \\sum _ { j = 1 } ^ d x _ j ^ { k } ~ { \\rm f o r } ~ k = 2 ^ h . \\end{align*}"}
+{"id": "6043.png", "formula": "\\begin{align*} I ( u _ { n } , v _ { n } ) = \\Big ( \\frac { 1 } { 2 } - \\frac { \\alpha } { 2 ( p \\alpha - 1 ) } \\Big ) a ( u _ { n } , v _ { n } ) + \\Big ( \\frac { 1 } { 2 } - \\frac { \\alpha - 1 } { 2 ( p \\alpha - 1 ) } \\Big ) b ( u _ { n } , v _ { n } ) + \\Big ( \\frac { 1 } { 2 } - \\frac { 3 \\alpha - 2 } { 2 ( p \\alpha - 1 ) } \\Big ) c ( u _ { n } , v _ { n } ) . \\end{align*}"}
+{"id": "2101.png", "formula": "\\begin{align*} T ^ { - 1 } ( B ) = \\{ \\theta \\in \\Theta : T ( \\theta ) \\in B \\} = \\textstyle \\bigcup _ { \\sigma \\in S _ k } B [ \\sigma ] . \\end{align*}"}
+{"id": "122.png", "formula": "\\begin{align*} u = S _ T ^ { \\infty } ( u _ 0 ) \\in C ( [ - T , T ] ; H ^ { \\infty , 0 } ( \\R ^ 2 ) ) \\end{align*}"}
+{"id": "7217.png", "formula": "\\begin{align*} \\xi _ { k , i } ( x , t ) = \\left \\{ \\begin{array} { l l } 2 \\sqrt { C _ { d } \\kappa } \\ , \\theta _ { k } ( t ) { \\rm R e } ( \\sigma _ { k , i } ( x ) ) , & k \\in \\Z _ { + } ^ d , \\\\ 2 \\sqrt { C _ { d } \\kappa } \\ , \\theta _ { k } ( t ) { \\rm I m } ( \\sigma _ { k , i } ( x ) ) , & k \\in \\Z _ { - } ^ d , \\end{array} \\right . \\end{align*}"}
+{"id": "7454.png", "formula": "\\begin{align*} ( U ^ \\top U ) _ { i , i } - \\frac { m } { n } = \\frac { \\sqrt { m - m ^ 2 / n } } { \\sqrt { 1 - 1 / n } } \\bigg ( t _ i ^ 2 - \\frac { 1 } { n } \\bigg ) \\mbox { f o r a l l } i \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
+{"id": "5961.png", "formula": "\\begin{align*} I & = - \\Vert u - \\widetilde { u } \\Vert _ { V } ^ 2 , \\\\ I I & \\leq \\varepsilon \\Vert u - \\widetilde { u } \\Vert _ { V } ^ 2 + C _ { \\varepsilon } \\Vert u \\Vert _ { V } ^ 2 \\Vert u - \\widetilde { u } \\Vert _ { H } ^ 2 . \\end{align*}"}
+{"id": "1153.png", "formula": "\\begin{align*} H \\left ( \\nu \\mid \\mu \\right ) : = \\int \\frac { d \\nu } { d \\mu } \\log \\frac { d \\nu } { d \\mu } d \\mu , \\end{align*}"}
+{"id": "5816.png", "formula": "\\begin{align*} { { \\bf { F } } _ i } \\left ( { { \\bf { x } } , t } \\right ) = f _ i ^ { \\left ( { e q } \\right ) } \\left ( { { { \\bf { u } } ^ { e q } } + \\Delta { \\bf { u } } } \\right ) - f _ i ^ { \\left ( { e q } \\right ) } \\left ( { { { \\bf { u } } ^ { e q } } } \\right ) , \\end{align*}"}
+{"id": "1442.png", "formula": "\\begin{align*} C _ { \\lambda } & = \\chi \\int _ { W ^ d } e ^ { - \\sum _ { j = 1 } ^ d ( \\lambda _ j - \\bar { \\lambda } ) z _ j } \\Delta ( z ) d z _ 1 \\ldots d z _ d . \\end{align*}"}
+{"id": "5630.png", "formula": "\\begin{align*} \\norm { \\tfrac { 1 } { p _ n } - \\tfrac { 1 } { f } } _ \\infty = \\norm { \\tfrac { p _ n - f } { p _ n f } } _ \\infty \\leq \\tfrac { 2 } { \\delta ^ 2 } \\| p _ n - f \\| _ \\infty \\to 0 . \\end{align*}"}
+{"id": "8182.png", "formula": "\\begin{align*} \\mathcal { M } ^ { r } ( t ) \\geq \\mathcal { M } ^ { r } ( \\delta ) + ( r - 1 ) \\theta _ { 1 } \\varrho _ { 0 } ^ { 1 - \\sigma } \\int _ { \\delta } ^ { t } ( \\mathcal { M } ^ { r } ( s ) ) ^ { \\sigma + 1 } d s , 0 < \\delta < t < t _ { 0 } < T _ { \\mathrm { g e l } } . \\end{align*}"}
+{"id": "2033.png", "formula": "\\begin{align*} \\theta _ 1 & = \\frac { 1 } { 2 } ( \\arg \\alpha _ 1 - \\arg \\alpha _ 2 + \\arg \\beta _ 1 - \\arg \\beta _ 2 ) , & \\delta _ 1 & = u ( \\theta _ 1 ) \\\\ \\theta _ 2 & = \\frac { 1 } { 2 } ( \\arg \\alpha _ 1 - \\arg \\alpha _ 2 - \\arg \\beta _ 1 + \\arg \\beta _ 2 ) , & \\delta _ 2 & = u ( \\theta _ 2 ) \\end{align*}"}
+{"id": "2667.png", "formula": "\\begin{align*} \\mathcal { H } ( t ) = \\lambda ( t ) \\mathcal { E } ( t ) + \\mu \\alpha ( t ) \\mathcal { E } ^ r ( t ) \\langle P ( u _ t ) , u \\rangle \\end{align*}"}
+{"id": "1916.png", "formula": "\\begin{align*} \\| \\tilde { x } _ { T _ { \\textnormal { i n n e r } } } - x ^ \\star \\| ^ 2 \\leq \\tfrac { 1 } { \\mu ( \\sum _ { k = 1 } ^ { T _ { \\textnormal { i n n e r } } } \\lambda _ k ) } \\left ( \\sum _ { k = 1 } ^ { T _ { \\textnormal { i n n e r } } } \\lambda _ k \\langle F ( x _ k ) , x _ k - x ^ \\star \\rangle \\right ) . \\end{align*}"}
+{"id": "94.png", "formula": "\\begin{align*} B ( G ) _ x : = \\{ [ b ] \\in B ( G ) \\mid I x I \\cap [ b ] \\neq \\emptyset \\} . \\end{align*}"}
+{"id": "3143.png", "formula": "\\begin{align*} \\alpha \\circ \\mathcal { T } ( a ) & = \\alpha ( T \\varrho ( x ) ( a ) + [ T ( a ) , x ] ) = T ( \\beta ( \\varrho ( x ) ( a ) ) + \\alpha ( [ T ( a ) , x ] ) ) \\\\ & = T ( \\varrho ( \\alpha ( x ) ) ( \\beta ( a ) ) ) + [ \\alpha ( T ( a ) ) , \\alpha ( x ) ] ) ) = T ( \\varrho ( x ) ( \\beta ( a ) ) ) + [ T ( \\beta ( a ) ) , x ] ) ) \\\\ & = \\mathcal { T } ( \\beta ( a ) ) . \\end{align*}"}
+{"id": "7822.png", "formula": "\\begin{align*} 0 & = B _ \\phi 0 \\\\ & = B _ \\phi ( I - Q ) F ^ q ( u + \\psi ( u , p ) , p ) \\\\ & = ( I - Q ) B _ \\phi F ^ q ( u + \\psi ( u , p ) , p ) \\\\ & = ( I - Q ) F ^ q ( B _ \\phi [ u + \\psi ( u , p ) ] , p ) \\\\ & = ( I - Q ) F ^ q ( B _ \\phi u + B _ \\phi \\psi ( u , p ) , p ) , \\end{align*}"}
+{"id": "4841.png", "formula": "\\begin{align*} & S _ t = I _ t = 0 , & & R _ t = R _ { \\tau ^ - } + S _ { \\tau ^ - } + I _ { \\tau ^ - } , & & D _ t = D _ { \\tau ^ - } . \\end{align*}"}
+{"id": "6355.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { h ( x ) } { 1 / x } = & 2 \\theta ( b - 1 ) \\lim _ { x \\to \\infty } \\left [ \\frac { 1 / x ^ 2 } { 1 - \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) } \\right ] - 2 . \\end{align*}"}
+{"id": "2718.png", "formula": "\\begin{align*} f _ \\mathcal { B } ( x ) = \\frac { 1 } { | \\mathcal { B } | } \\sum _ { d \\in \\mathcal { B } } l ( x , d ) \\ \\ g _ \\mathcal { B } ( x ) = \\frac { 1 } { | \\mathcal { B } | } \\sum _ { d \\in \\mathcal { B } } \\nabla _ x l ( x , d ) . \\end{align*}"}
+{"id": "3886.png", "formula": "\\begin{align*} \\frac { \\partial \\hat { q } _ { \\delta , i } } { \\partial z _ { i , h } } = O ( 1 ) , \\end{align*}"}
+{"id": "1107.png", "formula": "\\begin{align*} R _ { b \\to b } ^ { \\rm { N } } = W { \\log _ 2 } \\left ( { 1 + \\frac { { { p _ b } { { \\left | { { h _ b } } \\right | } ^ 2 } } } { { { p _ s } { { \\left | { { h _ b } } \\right | } ^ 2 } + W { N _ 0 } } } } \\right ) . \\end{align*}"}
+{"id": "3494.png", "formula": "\\begin{align*} \\prod _ { \\ell = z _ { j + 1 } } ^ { z _ j ' } | \\cos ( \\pi ( \\theta + \\ell \\alpha ) ) | \\leq e ^ { | z _ j ' - z _ { j + 1 } | ( - \\ln 2 + \\varepsilon ) } . \\end{align*}"}
+{"id": "556.png", "formula": "\\begin{align*} & \\gamma \\colon \\mathbb { R } \\rightarrow \\mathbb { C } , \\\\ & \\gamma ( t ) = p - i e ^ { i \\alpha } | t | , \\quad t \\in ( - \\infty , 0 ] , \\\\ & \\gamma ( t ) = p + i e ^ { - i \\alpha } | t | , \\quad t \\in ( 0 , + \\infty ] , \\end{align*}"}
+{"id": "2645.png", "formula": "\\begin{align*} ( q ; q ) _ { \\infty } = \\sum _ { j = - \\infty } ^ { \\infty } ( - 1 ) ^ { j } q ^ { \\frac { j ( 3 j - 1 ) } { 2 } } = 1 + \\sum _ { j = 1 } ^ { \\infty } ( - 1 ) ^ { j } q ^ { \\frac { j ( 3 j \\pm 1 ) } { 2 } } , \\end{align*}"}
+{"id": "8559.png", "formula": "\\begin{align*} ( _ * D ^ \\alpha _ { 0 + } \\ , f ) ( t ) : = ( I ^ { 1 - \\alpha } _ { 0 + } \\ , f ^ \\prime ) ( t ) \\ , = \\ , ( D ^ \\alpha _ { 0 + } \\ , f ) ( t ) - f ( 0 ) h _ { 1 - \\alpha } ( t ) , \\ t > 0 . \\end{align*}"}
+{"id": "1403.png", "formula": "\\begin{align*} & \\int _ { \\{ z ^ i \\in W ^ d , \\forall i = 1 , \\ldots , n - 1 \\} } \\prod _ { j = 1 } ^ d \\lambda _ j ^ n e ^ { - \\sum _ { j = 1 } ^ d \\lambda _ j ( z _ j ^ i - z _ j ^ { i - 1 } ) } 1 _ { \\{ z _ j ^ i > z _ j ^ { i - 1 } \\} } \\\\ & = \\prod _ { j = 1 } ^ d \\lambda _ j ^ n e ^ { - \\sum _ { j = 1 } ^ d \\lambda _ j ( z _ j - x _ j ) } \\int _ { \\{ z ^ i \\in W ^ d , \\forall i = 1 , \\ldots , n - 1 \\} } 1 _ { \\{ z _ j ^ i > z _ j ^ { i - 1 } \\} } . \\end{align*}"}
+{"id": "2574.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( \\hat { \\mu } \\right ) = \\bigcup _ { i = 1 } ^ { r } \\mathcal { Z } \\left ( \\hat { \\nu } _ { i } \\right ) , \\ \\mathcal { Z } \\left ( \\hat { \\nu } _ { i } \\right ) = \\rho ^ { - i } \\bigcup _ { j = 0 } ^ { \\infty } \\left ( \\frac { q } { p } \\right ) ^ { j } \\frac { a _ { j } } { N _ { j } } , \\ a _ { j } \\in \\mathbb { Z } \\setminus N _ { j } \\mathbb { Z } . \\end{align*}"}
+{"id": "1277.png", "formula": "\\begin{align*} \\frac { \\mu ( \\eta _ { \\Lambda _ n } ) } { \\mu ( \\xi _ \\Delta \\eta _ { \\Lambda _ n \\setminus \\Delta } ) } = \\frac { \\mu ( \\eta _ \\Delta | \\eta _ { \\Lambda _ n \\setminus \\Delta } ) } { \\mu ( \\xi _ \\Delta | \\eta _ { \\Lambda _ n \\setminus \\Delta } ) } . \\end{align*}"}
+{"id": "7030.png", "formula": "\\begin{align*} F _ { 2 , 5 } \\ , = \\ , 5 ! \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0 \\ , = \\ , F _ { 0 , 7 } \\ , = \\ , F _ { 1 , 6 } \\ , = \\ , F _ { 3 , 4 } \\ , = \\ , F _ { 4 , 3 } \\ , = \\ , F _ { 5 , 2 } \\ , = \\ , F _ { 6 , 1 } \\ , = \\ , F _ { 7 , 0 } . \\end{align*}"}
+{"id": "5194.png", "formula": "\\begin{align*} \\nabla \\times ( a _ { t } ^ { \\varepsilon } + ( B \\times u ) ^ { \\varepsilon } + \\mu \\nabla \\times b ^ { \\varepsilon } ) = 0 \\textrm { o n } \\ ( L ^ { 2 } _ { \\sigma } \\cap H ^ { 1 } ) ( \\mathbb { R } ^ { 3 } ) ^ { * } , \\end{align*}"}
+{"id": "5998.png", "formula": "\\begin{align*} \\Big \\{ u _ { l } = \\frac { \\sqrt { 8 } l } { 1 + | l x | ^ { 2 } } \\in H _ { r } ^ { 1 } ( \\R ^ { 2 } ) \\Big | \\ , \\ , l \\in ( 0 , \\infty ) \\Big \\} , \\end{align*}"}
+{"id": "2614.png", "formula": "\\begin{align*} & \\varepsilon ( i , j ) = ( - 1 ) ^ { i j } , \\\\ & e _ 1 \\cdot e _ 2 = e _ 2 \\cdot e _ 1 = - 2 e _ 3 , \\\\ & \\begin{array} [ t ] { l } \\alpha ( e _ 1 ) = \\sqrt { 2 } e _ 1 , \\alpha ( e _ 2 ) = e _ 3 - e _ 2 , \\\\ \\alpha ( e _ 3 ) = e _ 3 . \\end{array} \\end{align*}"}
+{"id": "6632.png", "formula": "\\begin{align*} \\begin{aligned} \\epsilon \\Theta _ { \\epsilon } ( t ) = \\epsilon ^ { 2 } \\int ^ { \\epsilon ^ { - 2 } t } _ { 0 } \\xi ( r ) d r , \\end{aligned} \\end{align*}"}
+{"id": "4706.png", "formula": "\\begin{align*} \\| g ( x ) \\| _ 2 ^ { 2 } = \\sum _ { i = 1 } ^ { d } \\left \\langle k ( x , . ) , g _ { i } \\right \\rangle _ { \\mathcal { H } _ { 0 } } ^ { 2 } \\leq \\| k ( x , . ) \\| _ { \\mathcal { H } _ { 0 } } ^ { 2 } \\| g \\| _ { \\mathcal { H } } ^ { 2 } \\leq B ^ { 2 } \\| g \\| _ { \\mathcal { H } } ^ { 2 } . \\end{align*}"}
+{"id": "2757.png", "formula": "\\begin{align*} D \\Lambda ( x ) = D Z ( \\Lambda ( x ) ) ^ { - 1 } \\end{align*}"}
+{"id": "2492.png", "formula": "\\begin{align*} \\mathcal { G } \\triangleq \\left \\{ \\Theta \\mathbf { G } : \\mathbf { G } = \\left ( { { \\mathbf { G } } ^ { 1 } } , \\cdots , { { \\mathbf { G } } ^ { k } } \\right ) , \\mathbf { G } ^ i \\in \\mathbf { R } ^ { n _ i \\times n _ i } , ( \\mathbf { G } ^ i ) ^ T \\mathbf { Q } ^ i \\mathbf { G } ^ i = \\mathbf { Q } ^ i \\right \\} \\end{align*}"}
+{"id": "3507.png", "formula": "\\begin{align*} F _ n - n ^ { p } \\ = \\ \\sum _ { i = 1 } ^ { n } \\left ( i ^ { p } - 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } i ^ { p - 2 j - 1 } \\right ) \\cdot F _ { n - i } . \\end{align*}"}
+{"id": "2140.png", "formula": "\\begin{align*} \\tilde { u } ^ 2 - \\tilde { v } ^ 2 & = \\tilde { y } ^ 2 - \\tilde { x } ^ 2 , \\\\ \\tilde { u } \\tilde { v } & = - \\tilde { x } \\tilde { y } . \\end{align*}"}
+{"id": "8526.png", "formula": "\\begin{align*} M _ { \\delta , \\pm } ( x ; z ) : = M _ { \\pm } ( x ; z ) \\delta _ { \\pm } ( z ) ^ { - \\delta _ 3 } , \\end{align*}"}
+{"id": "5077.png", "formula": "\\begin{align*} j ( \\tau , s ) = \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } ( U + \\tau \\tilde { U } + s U _ 0 ) \\cdot ( U + \\tau \\tilde { U } + s U _ 0 ) \\dd x . \\end{align*}"}
+{"id": "8066.png", "formula": "\\begin{align*} a _ c \\left [ t + 1 \\right ] = & a _ c \\left [ t \\right ] - 4 \\mu a _ c \\left [ t \\right ] \\sum _ { j = 1 } ^ { M } \\lvert \\phi ^ { \\left ( j , c \\right ) } \\rvert ^ 2 - 2 \\mu \\sum _ { l = 1 } ^ { M } \\Re \\left \\{ \\phi ^ { \\left ( l , c \\right ) } \\right \\} + 2 \\mu a _ c \\left [ t \\right ] f ^ { \\left ( c \\right ) } , \\end{align*}"}
+{"id": "8713.png", "formula": "\\begin{align*} f ( x ) _ { i j } = p _ { 1 } \\pi ( t _ { i } ^ { - 1 } ) x \\pi ( t _ { j } ) p _ { 1 } \\forall x \\in M , \\ ; \\forall i , j \\in \\{ 1 , \\ldots , n \\} \\end{align*}"}
+{"id": "234.png", "formula": "\\begin{align*} \\rho ( \\dot x ( J ) ) = \\rho ( x ) = - \\sigma ^ + , \\rho ( \\dot y ( J ) ) = \\rho ( y ) = - \\sigma ^ - + \\sigma ^ + , \\rho ( \\dot z ( J ) ) = \\rho ( z ) = \\sigma ^ { - } \\end{align*}"}
+{"id": "2884.png", "formula": "\\begin{align*} \\sum _ { x = 0 } ^ n | \\tilde V _ x ( m ) | ^ 2 = \\sum _ { x = 0 } ^ n \\sum _ { x ' = 1 } ^ n M _ { x , x ' } ( m ) \\tilde V _ { x ' } ( m ) \\tilde V ^ { \\star } _ { x } ( m ) + \\sum _ { x = 0 } ^ n \\tilde v _ x ( m ) \\tilde V ^ { \\star } _ { x } ( m ) . \\end{align*}"}
+{"id": "5905.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ T g _ n ( t ) d t = 0 . \\end{align*}"}
+{"id": "1773.png", "formula": "\\begin{align*} G = F Q + R , \\end{align*}"}
+{"id": "1128.png", "formula": "\\begin{align*} { \\widetilde p _ { b , m } ^ * } = { \\widetilde \\lambda ^ * } { W _ m } - \\frac { 1 } { { { { \\widetilde h } _ m } } } , { \\widetilde p _ { b , o } ^ * } = { \\widetilde \\lambda ^ * } \\left ( { W - { W _ m } } \\right ) - \\frac { 1 } { { { { \\widetilde h } _ b } } } , \\end{align*}"}
+{"id": "8151.png", "formula": "\\begin{align*} \\mathfrak X _ \\pi : = V ( \\pi ) \\subset \\mathfrak X . \\end{align*}"}
+{"id": "2874.png", "formula": "\\begin{align*} \\bar K _ 1 ^ { ( n ) } ( x , y ) : = \\frac { 1 } { 4 ( n + 1 ) ^ 2 } \\sum _ { j , j ' = - n - 1 } ^ n \\Phi \\left ( \\frac { j } { n + 1 } , \\frac { j ' } { n + 1 } \\right ) \\cos \\left ( \\frac { \\pi j ( y - x ) } { n + 1 } \\right ) \\cos \\left ( \\frac { \\pi j ' ( y - x ) } { n + 1 } \\right ) , \\end{align*}"}
+{"id": "5681.png", "formula": "\\begin{align*} p ' _ { 1 } \\circ ( p _ { 1 2 } \\circ f \\circ \\iota _ { 1 2 } ) = p _ { 1 } \\circ f \\circ \\iota _ { 1 2 } = p _ { 1 } \\circ f \\circ l _ { 1 } \\circ \\iota _ { 1 2 } = p _ { 1 } \\circ f \\circ \\iota _ { 1 2 } \\circ l ' _ { 1 } \\\\ = p ' _ { 1 } \\circ ( p _ { 1 2 } \\circ f \\circ \\iota _ { 1 2 } ) \\circ l ' _ { 1 } , \\end{align*}"}
+{"id": "7260.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { q - 1 } \\binom { n + i - 1 } { i } = \\frac { q ( q + 1 ) \\cdots ( q + n - 1 ) } { n ! } . \\end{align*}"}
+{"id": "8209.png", "formula": "\\begin{align*} x _ { 2 i - 1 , n + 1 - j } = \\tfrac { 1 } { 2 } S - x _ { 2 i - 1 , j } . \\end{align*}"}
+{"id": "2362.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { Q } ( u ) = 0 & \\ \\Omega _ { T } \\\\ u = u _ 0 & \\ \\overline { \\Omega } \\times \\{ 0 \\} \\\\ u = \\psi & \\ \\{ a , b \\} \\times [ 0 , T ] \\end{cases} \\end{align*}"}
+{"id": "1485.png", "formula": "\\begin{align*} M ( \\xi ) = - ( \\xi _ 1 ^ 2 + \\xi _ 2 ^ 2 + \\cdots + \\xi _ d ^ 2 ) \\begin{pmatrix} 1 & 0 \\\\ 0 & \\epsilon \\end{pmatrix} + i \\xi _ 1 c \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} 0 & e ^ { - \\kappa } \\\\ 0 & - \\kappa e ^ { - \\kappa } \\end{pmatrix} , \\end{align*}"}
+{"id": "1243.png", "formula": "\\begin{align*} \\forall z \\in \\Z ^ d : \\sum _ { \\Delta \\Subset \\Z ^ d } \\sum _ { i = 1 } ^ q \\norm { \\nabla ^ i _ z \\gamma _ \\Delta ( \\cdot | \\cdot ) } _ \\infty < \\infty , \\end{align*}"}
+{"id": "3237.png", "formula": "\\begin{align*} \\mathbb E \\left [ \\left \\| \\sum _ { i = 1 } ^ { n } \\int _ { 0 } ^ { ( i - 1 ) \\Delta _ n } ( \\mathcal S ( \\Delta _ n ) - I ) \\Sigma _ s ^ { \\mathcal S _ n } ( \\mathcal S ( \\Delta _ n ) - I ) ^ * d s \\right \\| _ { \\mathcal H } ^ 2 \\right ] \\geq C n ^ { 1 - 4 H } . \\end{align*}"}
+{"id": "8741.png", "formula": "\\begin{align*} \\overline { M } = \\overline { C } \\cap ( N K \\cap H ) / K . \\end{align*}"}
+{"id": "4837.png", "formula": "\\begin{align*} N _ j ( t ) - N _ j ( 0 ) = \\sum _ { k = 1 } ^ d \\int _ 0 ^ t \\Psi _ { j k } ( s ) d \\beta ^ c _ k ( s ) \\forall \\ , 0 \\le t \\le T , \\ ; j = 1 , \\dots , d . \\end{align*}"}
+{"id": "2846.png", "formula": "\\begin{align*} b - a = \\frac { 1 } { 2 } , a \\le 0 \\quad \\mbox { a n d } b \\ge 0 . \\end{align*}"}
+{"id": "443.png", "formula": "\\begin{align*} f \\ast g ( x ) = \\int _ { - \\infty } ^ \\infty f ( x - y ) g ( y ) d y f ^ { \\ast n } ( x ) . \\end{align*}"}
+{"id": "1894.png", "formula": "\\begin{align*} \\partial _ x ^ \\alpha \\partial _ y ^ \\beta \\Phi _ s ( x , y ) = O ( \\mu ^ { \\frac 3 2 - | \\alpha | - | \\beta | } ) , \\ | \\alpha | , | \\beta | \\ge 1 . \\end{align*}"}
+{"id": "69.png", "formula": "\\begin{align*} \\ell ( x , v \\alpha ) = \\langle v ^ { - 1 } \\mu , \\alpha \\rangle - \\Phi ^ + ( - v \\alpha ) + \\Phi ^ + ( - w v \\alpha ) \\geq 0 . \\end{align*}"}
+{"id": "6615.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c } \\varepsilon _ 1 \\lambda & - \\varepsilon _ 1 \\mu \\\\ \\varepsilon _ 3 \\mu & - \\varepsilon _ { 3 } \\lambda \\end{array} \\right ) \\end{align*}"}
+{"id": "9133.png", "formula": "\\begin{align*} \\mbox { $ b _ { m a x p r i m } ( G ) : = \\max \\{ b ( G ^ \\Omega ) \\mid G \\Omega \\} $ . } \\end{align*}"}
+{"id": "6113.png", "formula": "\\begin{align*} \\psi ^ { + 0 } _ { n , p } = \\psi ^ { - 0 } _ { n , p } = : \\psi _ { n , p } \\ , , \\rho ^ { 0 + } _ { n , p } = \\rho ^ { 0 - } _ { n , p } = : \\rho _ { n , p } \\ , , \\end{align*}"}
+{"id": "2223.png", "formula": "\\begin{align*} & \\mathbf { v } ^ { 0 } = [ \\cdots , 1 , 0 , \\cdots , 1 , \\cdots , 1 , \\cdots , 1 , \\cdots ] ^ { T } , \\\\ & m , n , k , l = 1 , \\cdots , N . \\end{align*}"}
+{"id": "1637.png", "formula": "\\begin{align*} \\beta \\left ( R ( r + s ) \\right ) & = \\int ^ { T _ 2 } _ { T _ 1 } G ( T _ 2 , \\tau ) f _ \\uparrow \\left ( r ( \\tau ) + s ( \\tau ) \\right ) d \\tau \\\\ & \\le f _ \\uparrow \\left ( b + 4 ^ k c \\right ) \\int ^ { T _ 2 } _ { T _ 1 } G ( T _ 2 , \\tau ) \\ ; d \\tau \\\\ & < b . \\end{align*}"}
+{"id": "6337.png", "formula": "\\begin{align*} G ( x ; \\theta ) = \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) , x > 0 , \\theta > 0 . \\end{align*}"}
+{"id": "8800.png", "formula": "\\begin{align*} \\begin{cases} d \\tilde { Y } _ t = | z | J ^ \\perp \\tilde { Y } _ t d t + P ^ { - 1 } \\Pi _ { \\mathrm { k e r } A ^ \\perp } \\sigma d W _ s \\\\ \\tilde { Y } _ 0 = P ^ { - 1 } Y _ 0 \\in \\C ^ n . \\end{cases} \\end{align*}"}
+{"id": "4322.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi < - t \\} } | F _ t + q | ^ 2 _ h c ( - \\Psi ) = \\int _ { \\{ \\Psi < - t \\} } ( | F _ t | ^ 2 _ h + | q | ^ 2 _ h ) c ( - \\Psi ) . \\end{align*}"}
+{"id": "5407.png", "formula": "\\begin{align*} | F ^ { i j } ( \\nabla _ \\alpha \\varphi ) _ { i j } | \\leq C \\sqrt { b _ { n - 1 } } \\sum _ { i = 1 } ^ n F ^ { i i } \\mbox { i n } \\omega _ 1 . \\end{align*}"}
+{"id": "8413.png", "formula": "\\begin{align*} \\nu ( x ; z ) : = - \\frac { 1 } { 2 i } \\left [ \\left ( 2 i \\bar { u } _ { x x } ( x ) + | u _ x ( x ) | ^ 2 \\bar { u } _ x ( x ) \\right ) \\Psi ^ - _ { 1 1 } ( x ; z ) + | u _ x ( x ) | ^ 2 \\Psi ^ - _ { 2 1 } ( x ; z ) \\right ] . \\end{align*}"}
+{"id": "2158.png", "formula": "\\begin{align*} \\alpha ^ 2 + \\gamma ^ 2 = \\frac { 1 - a ^ 2 } { a \\tau ^ 2 - a + \\tau ^ 2 + 1 } . \\end{align*}"}
+{"id": "3819.png", "formula": "\\begin{align*} G \\cdot R - \\bar G \\cdot \\bar R - \\bar R \\cdot \\nabla \\bar G ( \\nabla & \\bar A ) ^ { - 1 } ( A - \\bar A ) - \\bar G \\cdot ( P - \\bar P ) = ( G - \\bar G ) \\cdot ( R - \\bar R ) \\\\ & + \\bar R \\cdot G ( U | \\bar U ; x , t ) + \\bar G \\cdot \\left [ A _ t - \\bar A _ t + f _ { \\alpha , x _ \\alpha - } \\bar f _ { \\alpha , x _ \\alpha } \\right ] \\ , . \\end{align*}"}
+{"id": "6435.png", "formula": "\\begin{align*} \\sum _ { 0 \\le m _ 1 { \\le } \\cdots { \\le } m _ { n } < \\infty } \\frac { ( \\alpha ) _ { m _ 1 } } { { m _ 1 } ! } \\frac { { m _ n } ! } { ( \\alpha ) _ { m _ n } } \\left \\{ \\prod _ { i = 1 } ^ { n } \\frac { 1 } { ( m _ i + \\alpha ) ^ { a _ i } ( m _ i + \\beta ) ^ { b _ i } } \\right \\} , \\end{align*}"}
+{"id": "7775.png", "formula": "\\begin{align*} \\gamma _ { 1 , k , m } \\le \\epsilon = 1 / 2 ^ b ~ { \\rm f o r } ~ k \\ge ( b + 2 ) \\log _ { \\Delta _ { 1 , m } } ( 2 ) ~ { \\rm i f } ~ 2 \\Delta _ { 1 , m } ^ k \\le 1 . \\end{align*}"}
+{"id": "7976.png", "formula": "\\begin{align*} \\nabla _ { 1 1 } \\Theta - \\frac { k } { k + 1 } \\frac { ( \\nabla _ { 1 } \\Theta ) ^ { 2 } } { \\Theta } + ( k + 1 ) \\Theta = ( k + 1 ) \\Theta ^ { \\frac { k } { k + 1 } } \\left ( \\Theta ^ { \\frac { 1 } { k + 1 } } + ( \\Theta ^ { \\frac { 1 } { k + 1 } } ) _ { \\iota \\iota } \\right ) = ( k + 1 ) \\Theta ( 1 + \\Theta ^ { - \\frac { 1 } { k + 1 } } \\Theta ^ { \\frac { 1 } { k + 1 } } _ { \\iota \\iota } ) . \\end{align*}"}
+{"id": "4289.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } u ^ h ( t , x ) = \\frac { \\partial } { \\partial t } \\left ( \\sum _ i w _ i ( t ) \\sigma _ i ( x ) \\right ) = \\sum _ i \\sigma _ i ( x ) \\frac { d } { d t } w _ i ( t ) , \\end{align*}"}
+{"id": "1434.png", "formula": "\\begin{align*} \\P _ { ( x _ 1 , x _ 2 ) } ( \\rho \\leq n ) & = 2 \\P _ { ( x _ 1 , x _ 2 ) } ( S _ 2 ( n ) < S _ 1 ( n ) ) . \\end{align*}"}
+{"id": "7162.png", "formula": "\\begin{align*} & g _ { j k } ( x _ 0 ) = g ^ { j k } ( x _ 0 ) = \\delta _ { j k } , 1 \\leqslant j , k \\leqslant n , \\\\ & \\frac { \\partial g _ { j k } } { \\partial x _ \\alpha } ( x _ 0 ) = \\frac { \\partial g ^ { j k } } { \\partial x _ \\alpha } ( x _ 0 ) = 0 , 1 \\leqslant \\alpha \\leqslant n - 1 , \\end{align*}"}
+{"id": "3971.png", "formula": "\\begin{align*} ( \\mathbf { x } , - \\mathbf { y } ) M _ { I J } = 0 . \\end{align*}"}
+{"id": "1394.png", "formula": "\\begin{align*} \\lim _ { \\lambda _ 1 , \\ldots , \\lambda _ d \\rightarrow 1 } \\frac { h ^ { ( \\lambda _ 1 , \\ldots , \\lambda _ d ) } ( x ) } { \\Delta ( \\lambda ) } = \\frac { h ( x ) } { \\prod _ { j = 1 } ^ { d - 1 } j ! } . \\end{align*}"}
+{"id": "4368.png", "formula": "\\begin{align*} | c _ i ( t ) e ^ { - t } - \\tilde { h } ^ { - } ( t ) | & \\le \\int _ { - \\frac { 1 } { i } } ^ 0 | \\tilde { h } ( t + y ) - h ^ { - } ( t ) | g _ i ( y ) d y \\\\ & + \\int _ { 0 } ^ { \\frac { 1 } { i } } \\tilde { h } ( t + y ) g _ i ( y ) d y . \\end{align*}"}
+{"id": "8872.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } & \\prod _ { k = 1 } ^ { j - 1 } f _ k ( T _ { 1 } ^ { p _ { 1 , k } ( n ) } \\cdots T _ { d } ^ { p _ { d , k } ( n ) } x ) ( f _ j - \\mathbb { E } ( f _ j | \\mathcal { A } _ { g _ { j _ 0 } } ) ) ( T _ { 1 } ^ { p _ { 1 , j } ( n ) } \\cdots T _ { d } ^ { p _ { d , j } ( n ) } x ) \\cdot \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\prod _ { l = j + 1 } ^ { m } \\mathbb { E } ( f _ { l } | \\mathcal { A } ) ( T _ { 1 } ^ { p _ { 1 , l } ( n ) } \\cdots T _ { d } ^ { p _ { d , l } ( n ) } x ) \\rightarrow 0 . \\end{align*}"}
+{"id": "1715.png", "formula": "\\begin{align*} y = U ( t , 0 ) x - \\lambda i \\int _ 0 ^ t U ( t , s ) e ^ { ( \\alpha - 1 ) W ( s ) } g ( y ( s ) ) d s . \\end{align*}"}
+{"id": "4439.png", "formula": "\\begin{align*} E [ M ^ * _ v ( t ) ^ 2 ] \\le e ^ { - 2 ( \\lambda ^ * - \\lambda ) t } E [ M _ v ( 0 ) ^ 2 ] + \\left \\{ \\begin{aligned} & 4 | v | ^ 2 \\| x \\| t e ^ { - 2 ( \\lambda ^ * - \\lambda ) t } & & \\lambda \\ge 1 , \\\\ & 4 | v | ^ 2 \\| x \\| \\frac { 1 } { 1 - \\lambda } e ^ { - 2 ( \\lambda ^ * - 1 ) t } & & \\lambda < 1 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3466.png", "formula": "\\begin{align*} r _ { \\ell } \\leq e ^ { 3 9 \\varepsilon q _ n } \\frac { e ^ { \\beta _ n q _ n } } { \\max ( | \\ell | , 1 ) } e ^ { - q _ n L } \\max \\begin{cases} c _ { n , \\ell - 1 } r _ { \\ell - 1 } \\\\ \\max ( | \\ell | , e ^ { \\delta _ n q _ n } , 1 ) e ^ { - \\beta _ n q _ n } r _ { \\ell } \\\\ c _ { n , \\ell } r _ { \\ell + 1 } \\end{cases} . \\end{align*}"}
+{"id": "2178.png", "formula": "\\begin{align*} \\tau _ { n } = \\tau _ { 0 } - ( d _ { n } / c ) { \\sin } \\theta _ 0 , ~ ~ n = 1 , 2 , . . . , N , \\end{align*}"}
+{"id": "486.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { f ( x ) } { g _ 1 ( x ) } = \\lim _ { x \\to \\infty } \\frac { f ( x ) } { f _ u ( x ) } \\frac { f _ u ( x ) } { g _ u ( x ) } \\frac { g _ u ( x ) } { g _ 1 ( x ) } = \\nu ( ( 1 , \\infty ) ) . \\end{align*}"}
+{"id": "725.png", "formula": "\\begin{align*} \\ell ( \\omega , x , s ) : = \\frac { 1 } { 2 } \\| \\omega \\| ^ 2 + \\frac { 1 } { 2 } \\| \\dot y ( s ) - h ( x ) \\| ^ 2 . \\end{align*}"}
+{"id": "4262.png", "formula": "\\begin{align*} u _ { g , n } = \\sum _ { i = 0 } ^ n ( - 2 ) ^ i w _ { g , n - i } , \\end{align*}"}
+{"id": "2863.png", "formula": "\\begin{align*} M _ { x , y } : = \\sum _ { j , j ' = 0 } ^ n \\Theta ( \\mu _ j , \\mu _ { j ' } ) \\psi _ j ( x ) \\psi _ { j ' } ( x ) \\psi _ j ( y ) \\psi _ { j ' } ( y ) . \\end{align*}"}
+{"id": "1116.png", "formula": "\\begin{align*} { R ^ { \\rm { S - N } } } = R _ { b , m } ^ { \\rm { S - N } } + R _ { b , o } ^ { \\rm { S - N } } . \\end{align*}"}
+{"id": "8124.png", "formula": "\\begin{align*} \\bigcap _ { i = 1 } ^ { n } \\left ( B \\left ( X _ { i } , r \\right ) ^ { c } \\cap C _ { j } \\right ) \\neq \\emptyset . \\end{align*}"}
+{"id": "1000.png", "formula": "\\begin{align*} r _ a = s _ \\alpha \\varepsilon ^ { k \\alpha ^ \\vee } \\in \\widetilde W . \\end{align*}"}
+{"id": "9021.png", "formula": "\\begin{align*} \\dfrac { \\hat { k } _ 1 { ( G ) } } { \\hat { k } _ 1 ( G \\setminus \\sigma ) } = \\dfrac { D _ { \\ell _ 1 } D _ { \\ell _ 2 } } { D _ { \\ell _ 1 - 1 } D _ { \\ell _ 2 - 1 } } = \\dfrac { D _ { \\ell _ 2 } } { D _ { \\ell _ 1 - 1 } } . \\end{align*}"}
+{"id": "2198.png", "formula": "\\begin{align*} ( \\mathbf { v } ^ { 0 } ) ^ { H } = \\frac { \\sum \\limits _ { j = 1 } ^ { N } \\mathbf { R } _ { . j } } { \\mathbf { S ( R ) } } \\end{align*}"}
+{"id": "8283.png", "formula": "\\begin{align*} \\d Y _ t ^ i = \\bigg ( b ( Y _ t ^ i ) + \\frac 1 { p - 1 } \\sum _ { j \\neq i , j \\in \\C } K ( Y _ t ^ i - Y _ t ^ j ) \\bigg ) \\d t + \\sigma \\d W _ t ^ i , ~ ~ i \\in \\C , ~ ~ t \\in [ t _ n , t _ { n + 1 } ) . \\end{align*}"}
+{"id": "7320.png", "formula": "\\begin{align*} & 1 * x : = x x \\in \\widetilde { A } , \\\\ & a * a ' : = a a ' + ( a | a ' ) y a , a ' \\in A , \\\\ & a * y : = 0 , y * a = 0 a \\in A , \\\\ & y * y : = 0 . \\end{align*}"}
+{"id": "7344.png", "formula": "\\begin{align*} \\varphi _ { n } = \\varphi _ { 1 } \\circ \\psi _ { 2 } \\cdots \\circ \\psi _ { n } \\colon D _ { n } \\to D _ { n - 1 } \\to \\cdots D _ { 2 } \\to D _ { 1 } \\to { \\mathbf { A } } _ { k } ^ { 2 } \\end{align*}"}
+{"id": "6499.png", "formula": "\\begin{align*} B S ( m , n ) = \\langle a , b \\mid a b ^ m = b ^ n a \\rangle \\end{align*}"}
+{"id": "3472.png", "formula": "\\begin{align*} r _ { \\ell } ^ + \\leq & e ^ { 5 0 \\varepsilon q _ n } \\frac { e ^ { - q _ n L } } { \\max ( | \\ell | , 1 ) } \\max ( c _ { n , \\ell - 1 } r _ { \\ell - 1 } ^ - , r _ { \\ell - 1 } ^ + , r _ { \\ell } ^ + , r _ { \\ell + 1 } ^ - , r _ { \\ell + 1 } ^ + ) \\times \\begin{cases} \\max ( | \\ell | , e ^ { \\delta _ n q _ n } , 1 ) , & \\beta _ n \\geq \\delta _ n + 2 0 0 \\varepsilon \\\\ e ^ { \\beta _ n q _ n } , & \\beta _ n < \\delta _ n + 2 0 0 \\varepsilon \\end{cases} . \\end{align*}"}
+{"id": "7812.png", "formula": "\\begin{align*} ( I - Q ) F ^ q ( v + \\psi ( v , p ) , p ) = 0 \\end{align*}"}
+{"id": "861.png", "formula": "\\begin{align*} ( x , y ) = ( x ' , y ' ) = 1 \\end{align*}"}
+{"id": "5428.png", "formula": "\\begin{align*} u ( x ) \\geq l ( y ' ) + \\frac { 1 } { 4 } \\sum _ { \\beta = 1 } ^ { n - 1 } b _ \\beta y _ \\beta ^ 2 - C b _ \\alpha | \\hat { y } | ^ 2 \\end{align*}"}
+{"id": "1043.png", "formula": "\\begin{align*} \\ell ( x ' , s _ \\alpha u \\beta ) = \\ell ( x , u \\beta ) + \\ell ( s _ \\alpha , - u \\beta ) = \\begin{cases} \\ell ( x , u \\beta ) , & u \\beta \\neq \\pm \\alpha , \\\\ - \\ell ( x , \\alpha ) + 1 > 0 , & u \\beta = - \\alpha , \\\\ \\ell ( x , \\alpha ) - 1 < 0 , & u \\beta = \\alpha . \\end{cases} \\end{align*}"}
+{"id": "5293.png", "formula": "\\begin{align*} p = 1 \\oplus 0 \\oplus 0 \\oplus \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} , q = 0 \\oplus 1 \\oplus 0 \\oplus \\begin{pmatrix} a ^ 2 & a b \\\\ a b & b ^ 2 \\end{pmatrix} , \\end{align*}"}
+{"id": "4243.png", "formula": "\\begin{align*} & \\lim _ { j \\rightarrow \\infty } \\sup _ { z \\in V ^ { - 1 } \\mathbb { B } _ n } \\Big { | } ( ( V ^ { - 1 } \\varphi _ { k j } V ) ^ 1 , . . . , ( V ^ { - 1 } \\varphi _ { k j } V ) ^ s ) ( z ) \\Big { | } \\\\ & \\qquad = \\lim _ { j \\rightarrow \\infty } \\| V ^ { - 1 } ( \\varphi _ { k j } - \\rho ) \\| _ \\infty \\\\ & \\leq C \\lim _ { j \\rightarrow \\infty } \\| \\varphi _ { k j } - \\rho \\| _ \\infty = 0 . \\end{align*}"}
+{"id": "4919.png", "formula": "\\begin{align*} \\Phi ( x ) ^ { \\delta } = \\sum _ { r = 0 } ^ \\infty \\ , s _ r ( \\delta ) \\ , \\Phi ( x ) ^ r , \\end{align*}"}
+{"id": "4362.png", "formula": "\\begin{align*} v = D '' u _ \\epsilon + P _ { \\omega _ { \\epsilon } , h } ( \\sqrt { \\lambda } \\tau _ \\epsilon ) \\end{align*}"}
+{"id": "5181.png", "formula": "\\begin{align*} I _ h \\leq E [ b ] \\leq \\liminf _ { n \\to \\infty } E [ b _ n ] = I _ h . \\end{align*}"}
+{"id": "4956.png", "formula": "\\begin{align*} E _ { \\xi _ 1 ^ \\theta ( u ) } \\left [ \\varphi \\Bigl ( x + u + \\sum _ { i = 2 } ^ { n } Z _ i ^ { \\theta [ u ] } \\Bigr ) \\right ] = E _ { \\xi _ 1 ^ \\theta ( u ) } \\left [ \\varphi \\Bigl ( x + u + \\sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \\rho } \\Bigr ) \\right ] . \\end{align*}"}
+{"id": "6123.png", "formula": "\\begin{align*} \\rho _ { n , p } & = \\Big [ { \\frac { p ( 2 j ^ { ( 4 ) } + 1 - p ) ( 2 j ^ { ( 0 ) } + 1 - p ) ( 2 j ^ { ( 4 ) } + 2 j ^ { ( 0 ) } + 2 - p ) ( N - n - p + 2 j ^ { ( 3 ) } + 2 ) ( N - n - p + 1 ) } { ( 2 j ^ { ( 0 ) } + 2 j ^ { ( 4 ) } + 2 - 2 p ) ^ 2 ( 2 j ^ { ( 0 ) } + 2 j ^ { ( 4 ) } + 1 - 2 p ) ( 2 j ^ { ( 0 ) } + 2 j ^ { ( 4 ) } + 3 - 2 p ) } } \\\\ & \\qquad \\times { ( p - n - N + 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } ) ( n - p - N + 2 j ^ { ( 0 ) } + 2 j ^ { ( 4 ) } + 1 ) } \\Big ] ^ { \\frac 1 2 } \\ , , \\end{align*}"}
+{"id": "480.png", "formula": "\\begin{align*} f _ u ( x ) = ( e ^ { \\Lambda _ u } - 1 ) ^ { - 1 } \\sum _ { n = 1 } ^ \\infty ( \\Lambda _ u ^ n / n ! ) g _ u ^ { \\ast n } ( x ) . \\end{align*}"}
+{"id": "5149.png", "formula": "\\begin{align*} E [ \\tilde { b } ] = \\pi \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\left ( | \\nabla \\phi ^ { * } | ^ { 2 } + \\mu ^ { 2 } ( \\phi ^ { * } - \\phi ) _ { + } ^ { 2 } \\right ) \\frac { 1 } { r } \\dd z \\dd r \\leq \\pi \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\left ( | \\nabla \\phi | ^ { 2 } + \\mu ^ { 2 } ( \\phi - \\phi ) _ { + } ^ { 2 } \\right ) \\frac { 1 } { r } \\dd z \\dd r = E [ b ] . \\end{align*}"}
+{"id": "277.png", "formula": "\\begin{align*} T ^ 2 [ f ] ( p ) = \\begin{cases} z ^ 2 - 1 & \\ | z | \\ge 1 , \\\\ 0 & \\ | z | < 1 . \\end{cases} \\end{align*}"}
+{"id": "8619.png", "formula": "\\begin{align*} \\lim \\limits _ { \\varepsilon \\to 0 } \\frac { | A \\oplus _ p ( \\varepsilon ^ { \\frac { 1 } { p } } \\ , B ) | - | A | } { \\varepsilon } = \\frac { 1 } { p } \\int \\limits _ { S ^ { n - 1 } } h ^ p _ B ( u ) h _ A ^ { 1 - p } d S _ A ( u ) , \\end{align*}"}
+{"id": "1241.png", "formula": "\\begin{align*} \\sum _ { \\Delta \\Subset \\Z ^ d } \\sum _ { \\xi _ \\Delta } \\sum _ { i = 1 } ^ q \\norm { \\nabla ^ i _ z \\hat { c } _ \\Delta ( \\cdot , \\xi _ \\Delta ) } _ \\infty < \\infty . \\end{align*}"}
+{"id": "2169.png", "formula": "\\begin{align*} { \\bf { c } } _ j = [ x _ j , y _ j ] ^ { } \\in \\mathbb { R } ^ 2 , j \\in [ 1 , C ] , \\end{align*}"}
+{"id": "1879.png", "formula": "\\begin{align*} { F _ { \\Phi _ { ( 3 ) } } } _ x ( v ) = \\sum _ { i = 1 } ^ { m } \\big ( 6 \\langle B ( v , e _ { i } ) , B ( v , e _ { i } ) \\rangle _ { \\mathbb R ^ q } - \\langle B ( v , v ) , B ( e _ { i } , e _ { i } ) \\rangle _ { \\mathbb R ^ q } \\big ) , \\end{align*}"}
+{"id": "6280.png", "formula": "\\begin{align*} \\mathcal { U } = \\{ u \\in \\hat { \\mathcal { N } } _ p ( q ) : u \\} \\subseteq \\hat { \\mathcal { N } } _ p ( q ) \\cap \\{ u = ( u _ i ) _ i : 1 + \\sum _ i u _ i = 0 \\} \\cong \\mathbb { R } ^ { p + 1 - k } . \\end{align*}"}
+{"id": "9072.png", "formula": "\\begin{align*} G r _ w ( E ) = \\bigoplus _ { j = 1 } ^ { k } E ^ j / E ^ { j - 1 } , \\end{align*}"}
+{"id": "6869.png", "formula": "\\begin{align*} \\tilde z = \\Psi _ 1 ( w ) \\end{align*}"}
+{"id": "4918.png", "formula": "\\begin{align*} \\mathrm { E } ( Z ^ s ) = \\int _ { 0 } ^ { 1 } \\ , \\left ( \\sum _ { i = 0 } ^ { \\infty } f _ i \\ , \\ , u ^ { i / \\alpha } \\right ) ^ s \\mathrm { d } u = \\sum _ { i = 0 } ^ { \\infty } \\ , \\frac { g _ { s , i } } { i / \\alpha + 1 } , \\end{align*}"}
+{"id": "1156.png", "formula": "\\begin{align*} K ( x ) = \\frac { 1 } { 2 \\pi } \\frac { x ^ \\perp } { | x | ^ 2 } + \\frac { 1 } { 2 \\pi } \\sum _ { k \\in \\mathbb { Z } ^ 2 , k \\neq 0 } \\frac { ( x - k ) ^ \\perp } { | x - k | ^ 2 } , \\end{align*}"}
+{"id": "4199.png", "formula": "\\begin{align*} \\left ( K _ H ( x ) \\nabla q | \\nabla \\left ( \\frac { u ^ 2 } { q } \\right ) \\right ) = \\left ( K _ H ( x ) \\nabla u | \\nabla u \\right ) - q ^ 2 \\left ( K _ H ( x ) \\nabla \\left ( \\frac { u } { q } \\right ) | \\nabla \\left ( \\frac { u } { q } \\right ) \\right ) . \\end{align*}"}
+{"id": "708.png", "formula": "\\begin{align*} Q _ \\lambda ( \\zeta _ \\lambda ) = O ( b - \\lambda ) ~ ~ ~ ~ \\lambda \\to b ^ - \\end{align*}"}
+{"id": "8504.png", "formula": "\\begin{align*} & \\| I _ 1 ( x ) \\| _ { L ^ { 2 , 1 } } = \\frac { 1 } { \\pi } \\| \\widehat { ( z ^ { - 1 } \\bar { r } _ 1 ) } ( 2 x ) \\| _ { L ^ { 2 , 1 } _ x } = \\frac { 1 } { \\pi } \\| z ^ { - 1 } \\bar { r } _ 1 ( z ) \\| _ { H ^ 1 _ z } . \\end{align*}"}
+{"id": "2185.png", "formula": "\\begin{align*} \\mathbf { R } = \\frac { 1 } { K } \\sum \\limits _ { n = 1 } ^ { K } \\overline { \\mathbf { X } } ( n ) \\overline { \\mathbf { X } } ^ { H } ( n ) , \\end{align*}"}
+{"id": "7691.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } ^ { \\boldsymbol { \\xi } } [ | r ^ { N , i } ( t , \\boldsymbol { x } ^ * _ t ) | ^ 2 ] \\leq & \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } \\mathbb { E } ^ { \\boldsymbol { \\xi } } [ | x ^ { * , i } _ t - x ^ { * , j } _ t | ^ 2 ] \\Big ) \\\\ \\leq & \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 ] \\Big ) . \\end{aligned} \\end{align*}"}
+{"id": "2840.png", "formula": "\\begin{align*} S _ - = T _ - \\partial _ { p _ 0 } ^ 2 - p _ 0 \\partial _ { p _ 0 } . \\end{align*}"}
+{"id": "6346.png", "formula": "\\begin{align*} F ( x ) = \\frac { 1 } { \\operatorname { B } ( a , b ) } \\sum ^ { \\infty } _ { n = 0 } \\frac { ( - 1 ) ^ n \\ , \\Gamma ( b ) } { ( a + n ) \\ , \\Gamma ( b - n ) \\ , n ! } \\exp \\left \\{ - \\frac { ( a + n ) \\theta } { x ^ 2 } \\right \\} . \\end{align*}"}
+{"id": "5775.png", "formula": "\\begin{align*} V \\cdot \\psi _ 1 = \\psi _ 2 . \\end{align*}"}
+{"id": "4573.png", "formula": "\\begin{align*} C _ { i j k } + C _ { i , j , l } & = B _ { i , j , 0 } \\\\ z _ k C _ { i j k } + s _ l C _ { i , j , l } & = q ^ { 3 / 2 } B _ { i , j , 1 } , \\end{align*}"}
+{"id": "7067.png", "formula": "\\begin{align*} A _ { 1 , 1 } \\ , : = \\ , \\Big ( - \\tfrac { 3 } { 2 } \\ , F _ { 5 , 1 , 0 , 0 } - \\tfrac { 1 } { 2 } \\ , F _ { 7 , 0 , 0 , 1 } \\ , T _ 1 \\Big ) \\ , T _ 1 - T _ 2 + \\tfrac { 5 } { 4 } \\ , T _ 4 , \\end{align*}"}
+{"id": "4512.png", "formula": "\\begin{align*} \\frac { ( k - ( t ' + d ' ) ) ( t ' - d ' ) } { t ' + d ' } & = \\frac { ( k - ( t ' + d ' ) ) ( ( t ' + d ' ) - 2 d ' ) } { t ' + d ' } \\\\ & = k + 2 d ' - \\frac { 2 k d ' } { t ' + d ' } - ( t ' + d ' ) \\le k + 2 d ' - 2 \\sqrt { 2 k d ' } = ( \\sqrt { k } - \\sqrt { 2 d ' } ) ^ 2 \\end{align*}"}
+{"id": "7132.png", "formula": "\\begin{align*} D = \\sum _ { i = 1 } ^ r n _ i \\cdot y _ i \\end{align*}"}
+{"id": "7441.png", "formula": "\\begin{align*} \\mathfrak { W } _ 2 ^ { [ k , k ] } ( 1 + \\varepsilon , 1 - \\varepsilon ) & = 1 - \\frac { k - 1 } { 2 ( 2 k - 1 ) } \\varepsilon ^ 2 + O ( \\varepsilon ^ 3 ) , \\\\ \\mathfrak { M } _ 2 ^ { [ r ] } ( 1 + \\varepsilon , 1 - \\varepsilon ) & = 1 - \\frac { 1 - r } { 2 } \\varepsilon ^ 2 + O ( \\varepsilon ^ 3 ) \\end{align*}"}
+{"id": "6701.png", "formula": "\\begin{align*} \\hat \\beta ( s , \\xi ) = \\varphi _ s ( \\xi ) + \\frac { f ( s \\xi , s \\eta ) - f ( \\xi , \\eta ) - \\varphi _ s ( \\xi ) - \\varphi _ s ( \\eta ) } { 2 } . \\end{align*}"}
+{"id": "9118.png", "formula": "\\begin{align*} ( n - 2 ) r < \\sum \\limits _ { i = 2 } ^ n \\chi _ i , \\end{align*}"}
+{"id": "8794.png", "formula": "\\begin{align*} \\left | \\Pi _ { \\mathrm { k e r } A ^ \\perp } \\frac { d ^ j } { d t ^ j } X _ t | _ { t = 0 } \\right | \\geq \\frac { 1 } { C _ J } K ^ { j + 1 } . \\end{align*}"}
+{"id": "7387.png", "formula": "\\begin{align*} V : = U ( \\mathfrak g ) \\otimes _ { U ( \\mathfrak g _ { \\geq 0 } ) } \\mathbb C _ x . \\end{align*}"}
+{"id": "4544.png", "formula": "\\begin{align*} A _ i f ( x ) : = \\sum _ { \\substack { y \\sim x \\\\ \\tau ( y ) = \\tau ( x ) + i } } f ( y ) , \\end{align*}"}
+{"id": "4295.png", "formula": "\\begin{align*} \\Delta t b _ k \\mathcal { L } ( \\tilde { u } ^ { [ n + k ] } ( x ) ) + a _ k \\tilde { u } ^ { [ n + k ] } ( x ) = - \\sum _ { j = 0 } ^ { k - 1 } a _ j \\tilde { u } ^ { [ n + j ] } ( x ) + \\Delta t b _ k f ( t _ { n + k } , x ) , \\end{align*}"}
+{"id": "4639.png", "formula": "\\begin{align*} \\tau _ y = \\tau ' _ y v _ y , v _ y = \\sum _ { \\substack { 1 \\leq r \\leq j \\\\ C _ r \\in O _ y } } \\lambda _ r ( e _ { i _ { r , 1 } } + \\dots + e _ { i _ { r , d _ r } } ) \\end{align*}"}
+{"id": "113.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } \\partial _ t u - D ^ { 2 } _ x \\partial _ { x } u - \\partial _ { x } ^ { - 1 } \\partial _ y ^ 2 u & = u \\partial _ x u , ( t , x , y ) \\in \\R \\times \\R \\times \\R , \\\\ u ( 0 ) & = u _ 0 \\in H ^ { s _ 1 , s _ 2 } ( \\R ^ 2 ) , \\end{array} \\right . \\end{align*}"}
+{"id": "2898.png", "formula": "\\begin{align*} { \\bf c } _ { 2 } ( t ) = { \\cal L } { \\cal T } ( { \\bf c } _ { 2 } ) ( t ) + \\bar { \\bf p } ^ 2 ( t ) , \\end{align*}"}
+{"id": "1888.png", "formula": "\\begin{align*} A , { \\tilde A } \\subset \\mathbb A _ 0 : = \\big \\{ x : | x | ^ { - 1 } x \\in \\mathbb S \\big \\} . \\end{align*}"}
+{"id": "4707.png", "formula": "\\begin{align*} \\begin{cases} \\gamma & \\leq { ( \\alpha - 1 ) } \\min \\{ 1 , { 1 } / { L _ 1 } \\} \\left [ { \\alpha B ^ 2 \\left ( C _ 0 + 1 \\right ) } \\right ] ^ { - 1 } ; \\\\ \\gamma & \\leq B ^ { - 2 } \\Big [ \\alpha ^ 2 + ( e - 1 ) \\Big ( L _ { 0 } + L _ 1 C _ 0 \\Big ) \\Big ] ^ { - 1 } . \\end{cases} \\end{align*}"}
+{"id": "8939.png", "formula": "\\begin{gather*} E ( X - Y ) ^ { 2 k } \\allowbreak = \\allowbreak E ( X + \\rho Y - ( 1 + \\rho ) Y ) ^ { 2 k } \\allowbreak = \\\\ \\allowbreak \\sum _ { j = 0 } ^ { 2 k } \\binom { 2 k } { j } ( - 1 ) ^ { j } ( 1 + \\rho ) ^ { 2 k - j } E ( X + \\rho Y ) ^ { j } Y ^ { 2 k - j } \\allowbreak = \\\\ \\allowbreak \\sum _ { j = 0 } ^ { 2 k } \\binom { 2 k } { j } ( - 1 ) ^ { j } ( 1 + \\rho ) ^ { 2 k - j } E ( E ( X + \\rho Y ) ^ { j } | Y ) Y ^ { 2 k - j } \\allowbreak . \\end{gather*}"}
+{"id": "2117.png", "formula": "\\begin{align*} \\psi _ * \\mu \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) = \\mu ( \\psi ^ { - 1 } \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) ) = \\mu ( \\pi \\big ( B ( \\psi ( y ) , r ) \\cap \\psi ( Y ) \\big ) ) , \\end{align*}"}
+{"id": "8819.png", "formula": "\\begin{align*} \\Pi _ { \\mathrm { s p a n } \\{ P _ x ^ { - 1 } v \\} } J _ x ^ \\perp y = \\lambda J _ x \\Pi _ { \\mathrm { s p a n } \\{ P _ x ^ { - 1 } v \\} } y \\forall y \\in \\C ^ n . \\end{align*}"}
+{"id": "5656.png", "formula": "\\begin{align*} s _ { m } ^ { - 1 } H _ { n } ( G ) = 0 . \\end{align*}"}
+{"id": "8766.png", "formula": "\\begin{align*} ( { } _ a D _ g ^ { \\alpha } f ) ( x ) = & \\left ( \\frac { 1 } { g ' ( y ) } \\frac { d } { d y } \\right ) ( { } _ { a } I _ g ^ { 1 - \\alpha } f ) ( y ) \\\\ = & \\frac { \\left ( \\displaystyle \\frac { 1 } { g ' ( y ) } \\displaystyle \\frac { d } { d y } \\right ) } { \\Gamma ( 1 - \\alpha ) } \\int _ a ^ x \\frac { f ( y ) g ' ( y ) } { ( g ( x ) - g ( y ) ) ^ { \\alpha } } d y . \\end{align*}"}
+{"id": "4637.png", "formula": "\\begin{align*} \\tau ' _ y ( i ) = \\begin{cases} \\tau ' ( i ) & i \\in I _ y \\\\ i & i \\notin I _ y . \\end{cases} \\end{align*}"}
+{"id": "1967.png", "formula": "\\begin{align*} & \\left ( 1 + \\frac { q ^ { m - 1 } - q } { q - 1 } \\right ) Q ^ { ( m ) } _ \\nu ( k ) = \\left ( \\frac { q ^ { k + 1 } - 1 } { q - 1 } + \\frac { q ^ k ( q ^ { m - 1 } - q ) } { q - 1 } \\right ) Q ^ { ( m - 1 ) } _ \\nu ( k ) \\\\ & \\Rightarrow \\frac { q ^ { m - 1 } - 1 } { q - 1 } Q ^ { ( m ) } _ \\nu ( k ) = \\frac { q ^ { m + k - 1 } - 1 } { q - 1 } Q ^ { ( m - 1 ) } _ \\nu ( k ) \\\\ & \\Rightarrow Q ^ { ( m ) } _ \\nu ( k ) = \\frac { q ^ { m + k - 1 } - 1 } { q ^ { m - 1 } - 1 } Q ^ { ( m - 1 ) } _ \\nu ( k ) . \\end{align*}"}
+{"id": "6533.png", "formula": "\\begin{align*} x ^ { \\pi ( a ) } + c _ { \\pi ( a ) - 1 } x ^ { \\pi ( a ) - 1 } + c _ { \\pi ( a ) - 2 } x ^ { \\pi ( a ) - 2 } + \\cdots c _ 1 x + c _ 0 = ( x - r _ 1 ) ( x - r _ 2 ) \\cdots ( x - r _ { \\pi ( a ) } ) \\end{align*}"}
+{"id": "3105.png", "formula": "\\begin{align*} E r r ( x ^ { k } ) : = \\max _ { i } | x ^ { k } ( \\hat \\omega _ { i } ) \\ ! - \\ ! \\Pi _ { C ( \\hat \\o _ { i } ) } ( x ^ { k } ( \\hat \\omega _ { i } ) ) \\ ! - \\ ! \\hat F ( x ^ { k } ) ( \\hat \\omega _ { i } ) \\ ! + \\ ! \\lambda ^ { k } ( \\hat \\omega _ { i } ) ) | \\ ! + \\ ! \\sum _ { i = 1 } ^ { m } | x ^ { k } ( \\hat \\omega _ { i } ) \\ ! - \\ ! y ^ { k } ( \\hat \\omega _ { i } ) | ^ 2 p _ { i } < \\varepsilon \\end{align*}"}
+{"id": "9138.png", "formula": "\\begin{align*} \\dot { x } = - \\gamma \\ , y _ \\delta ( x , t ) \\ , u ( t ) x ( 0 ) = x _ 0 \\end{align*}"}
+{"id": "8261.png", "formula": "\\begin{align*} \\frac { d \\xi ^ { n } _ m ( t ) } { d t } = e ^ { - t } \\left [ \\frac { d \\mu ^ { n } _ 1 ( t ) } { d t } - \\sum _ { i = 1 } ^ { m - 1 } i \\frac { d \\psi ^ { n } _ i ( t ) } { d t } \\right ] - e ^ { - t } \\left [ \\mu ^ { n } _ 1 ( t ) - \\sum _ { i = 1 } ^ { m - 1 } i \\psi ^ { n } _ i ( t ) + ( 2 m + 2 ) \\mu ^ { n } _ 1 ( 0 ) ^ 2 \\right ] \\end{align*}"}
+{"id": "6431.png", "formula": "\\begin{align*} C _ k ( t ) ^ { 2 \\beta - m + 1 } \\sum _ { i \\geq 0 } ( - 1 ) ^ i \\binom { \\alpha - m - ( k - 1 ) i } { i } t ^ i - C _ k ( t ) ^ { \\beta - m } \\sum _ { i \\geq 0 } ( - 1 ) ^ i \\binom { \\alpha - m - 1 - ( k - 1 ) i } { i } t ^ i . \\end{align*}"}
+{"id": "3122.png", "formula": "\\begin{align*} [ T ( a ) , T ( b ) ] _ { \\alpha } & = \\alpha [ T ( a ) , T ( b ) ] = \\alpha \\big ( T ( \\varrho ( T ( a ) ) b - \\varrho ( T ( b ) ) a \\big ) , \\\\ T \\big ( \\tilde { \\varrho } ( T ( a ) ) b - \\tilde { \\varrho } ( T ( b ) ) a \\big ) & = T \\big ( ( \\varrho ( \\alpha ( T ( a ) ) ) \\beta ( b ) - \\varrho ( \\alpha ( T ( b ) ) ) \\beta ( a ) \\big ) . \\end{align*}"}
+{"id": "1108.png", "formula": "\\begin{align*} { R ^ { \\rm { N } } } = \\min \\left \\{ { R _ { b \\to s } ^ { \\rm { N } } , R _ { b \\to b } ^ { \\rm { N } } } \\right \\} \\triangleq W { \\log _ 2 } \\left ( { 1 + \\frac { { { p _ b } { { \\left | { { { \\widetilde h } _ b } } \\right | } ^ 2 } } } { { { p _ s } { { \\left | { { { \\widetilde h } _ b } } \\right | } ^ 2 } + W { N _ 0 } } } } \\right ) , \\end{align*}"}
+{"id": "1540.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c l } \\frac { d } { d t _ n } L & = & \\left [ ( L ^ n ) _ D ( t ) , L ( t ) \\right ] = - \\left [ ( L ^ n ) _ S ( t ) , L ( t ) \\right ] \\\\ L ( 0 ) & = & L _ 0 \\\\ \\end{array} \\right . \\ ; , \\end{align*}"}
+{"id": "7602.png", "formula": "\\begin{align*} B _ 1 + \\cdots + B _ k \\subset \\{ n \\in \\N : T ^ n x _ { \\vec 0 } \\in E \\} = A \\end{align*}"}
+{"id": "1228.png", "formula": "\\begin{align*} \\hat { c } _ { \\Delta } ( \\eta , \\xi _ { \\Delta } ) : = c _ { \\Delta } ( \\xi _ { \\Delta } \\eta _ { \\Delta ^ c } , \\eta _ { \\Delta } ) \\frac { \\gamma _ { \\Delta } ( \\xi _ { \\Delta } | \\eta _ { \\Delta ^ c } ) } { \\gamma _ { \\Delta } ( \\eta _ { \\Delta } | \\eta _ { \\Delta ^ c } ) } , \\end{align*}"}
+{"id": "7024.png", "formula": "\\begin{align*} \\aligned u & \\ , = \\ , \\tfrac { x ^ 2 } { 2 } + \\tfrac { x ^ 2 y } { 2 } + F _ { 4 , 0 } \\ , \\tfrac { x ^ 4 } { 2 4 } + F _ { 3 , 1 } \\ , \\tfrac { x ^ 3 y } { 6 } + \\underline { \\tfrac { x ^ 2 y ^ 2 } { 2 } + 0 + 0 } + { \\rm O } _ { x , y } ( 5 ) , \\\\ v & \\ , = \\ , \\tfrac { r ^ 2 } { 2 } + \\tfrac { r ^ 2 s } { 2 } + G _ { 4 , 0 } \\ , \\tfrac { r ^ 4 } { 2 4 } + G _ { 3 , 1 } \\ , \\tfrac { r ^ 3 s } { 6 } + \\underline { \\tfrac { r ^ 2 s ^ 2 } { 2 } + 0 + 0 } + { \\rm O } _ { r , s } ( 5 ) . \\endaligned \\end{align*}"}
+{"id": "3048.png", "formula": "\\begin{align*} \\left \\langle \\left ( x , y \\right ) , \\left ( x ^ { \\prime } , y ^ { \\prime } \\right ) \\right \\rangle = \\tfrac { 1 } { 2 } \\left ( \\left \\langle x , x ^ { \\prime } \\right \\rangle - \\left \\langle y , y ^ { \\prime } \\right \\rangle \\right ) \\end{align*}"}
+{"id": "5264.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( a , b , c ; z ) = \\frac { \\Gamma ( c ) } { \\Gamma ( b ) \\Gamma ( c - b ) } \\int _ 0 ^ 1 t ^ { b - 1 } ( 1 - t ) ^ { c - b - 1 } ( 1 - t z ) ^ { - a } d t , | z | < 1 . \\end{align*}"}
+{"id": "3084.png", "formula": "\\begin{align*} \\Pi _ { \\mathcal { S } } ( x ) : = \\Big \\{ \\bar x \\in \\mathcal { S } \\ \\Big | \\ \\| x - \\bar x \\| _ { \\mathcal { H } } = \\inf _ { y \\in \\mathcal { S } } \\| x - y \\| _ { \\mathcal { H } } \\Big \\} . \\end{align*}"}
+{"id": "1090.png", "formula": "\\begin{align*} r _ { b } w \\varepsilon ^ \\lambda = s _ \\beta w \\varepsilon ^ { \\lambda + \\Phi ^ + ( - \\beta ) w ^ { - 1 } \\beta ^ \\vee } \\in \\widetilde W _ J \\cdot w \\varepsilon ^ \\lambda \\end{align*}"}
+{"id": "9140.png", "formula": "\\begin{align*} \\begin{aligned} & y _ \\delta ( x _ a , t ) = \\dfrac { a _ { 0 , \\delta } ( x _ a ) } { 2 } + \\\\ & \\sum _ { k = 1 } ^ { \\infty } a _ { k , \\delta } ( x _ a ) \\cos \\left ( k 2 \\pi t \\right ) + b _ { k , \\delta } ( x _ a ) \\sin \\left ( k 2 \\pi t \\right ) \\end{aligned} \\end{align*}"}
+{"id": "5242.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( | v | ^ { 2 } + | b | ^ { 2 } \\right ) \\dd x + \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( \\nu | \\nabla v | ^ { 2 } + \\mu | \\nabla b | ^ { 2 } \\right ) \\dd x \\dd s = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( | v _ 0 | ^ { 2 } + | b _ 0 | ^ { 2 } \\right ) \\dd x , \\end{align*}"}
+{"id": "1549.png", "formula": "\\begin{align*} v + \\dfrac { 1 } { 2 } \\varphi _ \\alpha \\circ ( u - v ) & { } = \\dfrac { u + h ( u , v ) } { 2 } , & u - \\dfrac { 1 } { 2 } \\varphi _ \\alpha \\circ ( u - v ) & { } = \\dfrac { v + k ( u , v ) } { 2 } . \\end{align*}"}
+{"id": "8139.png", "formula": "\\begin{align*} X ^ \\sigma : = \\lim X \\in \\mathcal C \\end{align*}"}
+{"id": "7850.png", "formula": "\\begin{align*} F ^ { q , \\dagger } _ v ( 0 , p _ 0 ) \\psi ^ \\dagger _ { v v } ( 0 , p _ 0 ) [ u _ \\ell , u _ \\ell ] = - c _ 2 ( q , \\ell , p _ 0 ) u _ { 2 \\ell } . \\end{align*}"}
+{"id": "1069.png", "formula": "\\begin{align*} \\langle \\alpha ^ \\vee , 2 \\rho \\rangle = 2 + \\sum _ { \\substack { \\alpha \\neq \\beta \\in \\Phi ^ + \\\\ s _ \\alpha ( \\beta ) \\in \\Phi ^ - } } \\langle \\alpha ^ \\vee , \\beta \\rangle . \\end{align*}"}
+{"id": "2411.png", "formula": "\\begin{align*} U = \\left \\{ \\begin{pmatrix} 1 & u \\\\ 0 & 1 \\end{pmatrix} \\colon u \\in \\Z _ { p } \\right \\} \\ , \\end{align*}"}
+{"id": "2745.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} G ( H ( x ) ) & \\ x \\in T , \\\\ H ( x ) & \\ x \\notin T . \\end{cases} \\end{align*}"}
+{"id": "6719.png", "formula": "\\begin{align*} A ( y ) : = A i ( \\gamma ( y - y _ c ) ) , \\end{align*}"}
+{"id": "3841.png", "formula": "\\begin{align*} X _ { n + 1 } = L _ n ( X _ n , Y _ { n + 1 } ) + F _ n ( X _ n , Y _ { n + 1 } ) + G _ n ( X _ n , Y _ { n + 1 } , \\xi _ { n + 1 } ) \\end{align*}"}
+{"id": "8205.png", "formula": "\\begin{align*} x _ { m , j } + x _ { m , j + 1 } + x _ { 1 , n + 1 - j } + x _ { 1 , n - j } & = x _ { m , j } + x _ { m , j + 1 } \\\\ & + ( \\tfrac { 1 } { 2 } S - x _ { 1 , j } ) + ( \\tfrac { 1 } { 2 } S - x _ { 1 , j + 1 } ) \\\\ & = a _ j + S - a _ j = S . \\end{align*}"}
+{"id": "2666.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) = \\mathcal { E } ( 0 ) - \\int ^ t _ 0 \\langle B ( s , x , u _ t ( s ) ) , u _ t ( s ) \\rangle d s t \\in J . \\end{align*}"}
+{"id": "7163.png", "formula": "\\begin{align*} H = \\sum _ \\alpha \\kappa _ \\alpha = - \\Gamma ^ { \\alpha } _ { n \\alpha } = - \\frac { 1 } { 2 } \\frac { \\partial g _ { \\alpha \\alpha } } { \\partial x _ n } , \\end{align*}"}
+{"id": "3238.png", "formula": "\\begin{align*} Y _ { i \\Delta _ n } ( j \\Delta _ n ) : = Y _ { t _ i } ( x _ j ) , i , j = 1 , . . . , n , \\end{align*}"}
+{"id": "935.png", "formula": "\\begin{align*} \\begin{aligned} J \\mathcal { N } _ 2 ( u _ 1 , u _ 2 ) & = 2 | u _ 1 | ^ 2 J u _ 2 - u ^ 2 _ 1 \\overline { J u _ 2 } \\\\ & + 2 ( \\overline { u } _ 1 u _ 2 J u _ 1 - u _ 1 u _ 2 \\overline { J u _ 1 } ) + 2 u _ 1 \\overline { u } _ 2 J u _ 1 . \\end{aligned} \\end{align*}"}
+{"id": "6731.png", "formula": "\\begin{align*} \\ell ( \\mathbf { \\theta } ) = & - \\log \\mathrm { B } ( \\alpha , \\beta ) - \\log \\sigma + ( \\alpha - 1 ) \\log \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\\\ & + ( \\beta - 1 ) \\log \\left [ 1 - \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] + \\log \\phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) . \\end{align*}"}
+{"id": "1320.png", "formula": "\\begin{align*} ( M ' ( t ) ) ^ 2 & = 4 \\left ( | { \\rm R e } ( u , u _ t ) | ^ 2 + 2 b { \\rm R e } ( u , u _ t ) \\int _ 0 ^ t { \\rm R e } ( u ( s ) , u _ s ( s ) ) d s + b ^ 2 \\left ( \\int _ 0 ^ t { \\rm R e } ( u ( s ) , u _ s ( s ) ) d s \\right ) ^ 2 \\right ) \\\\ & \\leq 4 \\left ( | | u | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } + b \\int _ { 0 } ^ t | | u ( s ) | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } d s \\right ) \\left ( | | u _ t | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } + \\int _ { 0 } ^ t | | u _ s ( s ) | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } d s \\right ) , \\end{align*}"}
+{"id": "2619.png", "formula": "\\begin{align*} [ x , y ] = x \\cdot y - \\varepsilon ( x , y ) y \\cdot x . \\end{align*}"}
+{"id": "1534.png", "formula": "\\begin{align*} \\widehat { \\mathcal D } _ { \\hat { A } } = \\left \\{ P = \\sum _ { \\alpha \\in \\mathbb { Z } } a _ { \\alpha } \\ , D ^ { \\alpha } \\ ; | \\ ; P \\in \\widehat { \\Psi } ( \\hat { A } ) \\mbox { a n d } a _ \\alpha = 0 \\mbox { f o r } \\alpha < 0 \\right \\} \\ ; . \\end{align*}"}
+{"id": "5334.png", "formula": "\\begin{align*} w _ n = \\sum _ { i = 0 } ^ { r + \\tilde { r } - 1 } c _ i w _ { n + i } \\end{align*}"}
+{"id": "8404.png", "formula": "\\begin{align*} ( I - F ) \\left ( \\Psi _ - - e ^ { - i c _ - ( x ) \\sigma _ 3 } \\right ) = \\begin{pmatrix} 0 & n \\\\ m & 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "8537.png", "formula": "\\begin{align*} & \\sup _ { x \\in \\mathbb { R } } \\left \\| \\mathcal { P } ^ + \\left ( z ^ { - i } \\bar { r } _ 1 ( z ) e ^ { 2 i z x } \\right ) \\right \\| _ { L _ z ^ \\infty } \\leq \\frac { 1 } { \\sqrt { 2 } } \\left \\| z ^ { - i } r _ 1 ( z ) \\right \\| _ { H ^ 1 } , i = 0 , 1 , \\\\ & \\sup _ { x \\in \\mathbb { R } } \\left \\| \\mathcal { P } ^ - \\left ( r _ 2 ( z ) \\mathrm { e } ^ { - 2 i z x } \\right ) \\right \\| _ { L _ { z } ^ { \\infty } } \\leq \\frac { 1 } { \\sqrt { 2 } } \\left \\| r _ 2 ( z ) \\right \\| _ { H ^ { 1 } } . \\end{align*}"}
+{"id": "857.png", "formula": "\\begin{align*} \\Gamma ( X ) = \\sigma X + \\mathcal { O } \\left ( X ^ { \\frac { 4 } { 5 } + \\varepsilon } \\right ) \\end{align*}"}
+{"id": "8111.png", "formula": "\\begin{align*} y _ { i k } = { \\bf { w } } ^ { H } _ { i } { \\bf { h } } _ { i k } s _ { i } + \\sum _ { j \\neq i } { \\bf { w } } ^ { H } _ { j } { \\bf { h } } _ { i k } s _ { j } + n _ { i k } \\end{align*}"}
+{"id": "7262.png", "formula": "\\begin{align*} P ( x _ 1 , x _ 2 ) = x _ 2 ^ { q + 1 } + a x _ 1 ^ q x _ 2 + b x _ 1 ^ { q + 1 } \\end{align*}"}
+{"id": "7697.png", "formula": "\\begin{align*} { \\bf g _ { \\rm G E P P } } ( A ) : = \\frac { \\max _ { k , i , j \\in [ n ] } | \\hat A ^ { ( k - 1 ) } _ { i , j } | } { \\max _ { i , j \\in [ n ] } | A _ { i , j } | } , \\end{align*}"}
+{"id": "3677.png", "formula": "\\begin{align*} \\nabla _ V X & = \\omega ( V ) \\\\ \\nabla _ V \\omega & = - R ( X , V ) + T _ h ( V , \\cdot ) . \\end{align*}"}
+{"id": "782.png", "formula": "\\begin{align*} V = H \\end{align*}"}
+{"id": "2128.png", "formula": "\\begin{align*} Z _ 1 = L : = \\varrho _ { w } \\partial _ z - \\varrho _ z \\partial _ w , \\end{align*}"}
+{"id": "638.png", "formula": "\\begin{align*} \\rho ^ * ( d x \\wedge d z ) & = ( ( \\cos \\theta + 1 ) d t - t \\sin \\theta \\ d \\theta ) \\wedge ( \\sin \\theta \\ d t + t \\cos \\theta \\ d \\theta ) \\\\ & = t \\cos \\theta ( \\cos \\theta + 1 ) d t \\wedge d \\theta - t \\sin ^ 2 \\theta d \\theta \\wedge d t \\\\ & = ( t \\cos ^ 2 \\theta + t \\cos \\theta + t \\sin ^ 2 \\theta ) d t \\wedge d \\theta \\\\ & = t ( 1 + \\cos \\theta ) d t \\wedge d \\theta \\end{align*}"}
+{"id": "2079.png", "formula": "\\begin{align*} \\bigg | \\frac { 1 } { k } \\int _ { r } ^ { r + k } g ( t ) d t - g ( r ) \\bigg | & = \\bigg | \\frac { 1 } { k } \\int _ { r } ^ { r + k } ( g ( t ) - g ( r ) ) d t \\bigg | \\\\ & \\leq \\frac { 1 } { k } \\int _ { r } ^ { r + k } | g ( t ) - g ( r ) | d t \\\\ & = \\frac { 1 } { k } \\int _ { [ r , r + k ] \\cap B _ r } | g ( t ) - g ( r ) | d t + \\frac { 1 } { k } \\int _ { [ r , r + k ] \\setminus B _ r } | g ( t ) - g ( r ) | d t \\end{align*}"}
+{"id": "5140.png", "formula": "\\begin{align*} U & = b + B _ { \\infty } = \\nabla \\times ( ( \\phi - \\phi _ { \\infty } ) \\nabla \\theta ) + G \\nabla \\theta , \\\\ \\nabla \\times U & = \\nabla \\times ( G \\nabla \\theta ) - L \\phi \\nabla \\theta . \\end{align*}"}
+{"id": "2147.png", "formula": "\\begin{align*} \\Re \\mathcal { P } [ \\varrho ] = \\frac { a } { 4 } | w | ^ 6 \\left ( 4 \\tau ^ 6 + ( 8 a + 8 ) \\tau ^ 4 + ( 4 + 6 a + 5 a ^ 2 ) \\tau ^ 2 - 2 a \\right ) , \\end{align*}"}
+{"id": "8718.png", "formula": "\\begin{align*} S ( \\zeta ) = \\mathrm { s t a b } _ N ( \\psi _ \\zeta ) \\end{align*}"}
+{"id": "5958.png", "formula": "\\begin{align*} \\mathbb { H } : = H \\times [ H ^ 1 ( \\mathcal { O } ) ^ 3 ] , \\mathbb { V } : = V \\times \\Big \\{ n \\in H ^ 2 ( \\mathcal { O } ) ^ 3 : \\frac { \\partial n } { \\partial \\nu } = 0 \\Big \\} , \\end{align*}"}
+{"id": "1353.png", "formula": "\\begin{align*} y _ 1 & = \\sum _ { k = 1 } ^ { K } \\bigl ( \\frac { d ( u _ { 2 k - 2 } , u _ { 2 k - 1 } ) } { d ( a _ 1 , b _ 1 ) } \\ , m _ { u _ { 2 k - 2 } , u _ { 2 k - 1 } } - \\frac { d ( u _ { 2 k - 1 } , u _ { 2 k } ) } { d ( a _ 1 , b _ 1 ) } \\ , m _ { u _ { 2 k - 1 } , u _ { 2 k } } \\bigr ) . \\end{align*}"}
+{"id": "1914.png", "formula": "\\begin{align*} { \\frac { \\partial { R _ n ( s ) } } { \\partial { s } } } = - { \\frac { \\mathcal { A } ^ { 2 } _ { \\gamma } ( s ) } { ( q - s ) ^ { 2 } } } \\ , \\beta ^ n \\cdot \\bigl ( { 1 + o ( 1 ) } \\bigr ) n \\to \\infty , \\end{align*}"}
+{"id": "3312.png", "formula": "\\begin{align*} \\langle F _ 1 , \\ldots , F _ t \\rangle = \\{ \\sigma \\subset V \\ | \\ \\sigma \\subset F _ i i \\in [ t ] \\} . \\end{align*}"}
+{"id": "5652.png", "formula": "\\begin{align*} E ^ { 2 } _ { p , q } = H _ { p } ( O _ { n - k , n - k } ; H _ { q } ( L _ { k } ) ) \\Rightarrow H _ { p + q } ( T _ { k } ) . \\end{align*}"}
+{"id": "8091.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ M \\sum \\limits _ { \\substack { j = 1 \\\\ j \\neq l } } ^ M \\mathbb { E } \\left [ y _ l ^ * y _ j \\right ] = & 2 \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { q = i + 1 } ^ { M } \\sum _ { r = 1 } ^ M a _ r ^ 2 \\Re \\left [ \\phi ^ { \\left ( i , r \\right ) ^ * } \\phi ^ { \\left ( q , r \\right ) } \\right ] + \\sum _ { l = 1 } ^ M \\sum \\limits _ { \\substack { j = 1 \\\\ j \\neq l } } ^ M a _ c ^ 2 \\phi ^ { \\left ( l , c \\right ) ^ * } \\phi ^ { \\left ( j , c \\right ) } . \\end{align*}"}
+{"id": "3850.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\mathbf { v } + ( \\mathbf { v } \\cdot \\nabla ) \\mathbf { v } = - \\nabla P , \\ \\ & D \\times ( 0 , T ) , \\\\ \\nabla \\cdot \\mathbf { v } = 0 , \\ \\ & D \\times ( 0 , T ) , \\\\ \\mathbf { v } ( \\cdot , 0 ) = \\mathbf { v } _ 0 ( \\cdot ) , \\ \\ & D , \\end{cases} \\end{align*}"}
+{"id": "5013.png", "formula": "\\begin{align*} S t _ { D N M } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( 1 , z ) - 1 | | _ { \\infty } = 0 , \\end{align*}"}
+{"id": "3750.png", "formula": "\\begin{align*} p ( 1 v , 0 w ) = p ( v 1 , w \\bullet ) = \\psi \\left ( d _ { - } ^ { \\ell + 1 } L ( v 1 , w \\bullet ) \\right ) . \\end{align*}"}
+{"id": "2418.png", "formula": "\\begin{align*} L _ { q } [ f ( t ) ] ( s ) = F _ { q } ( s ) = \\int _ { 0 } ^ { \\infty } f ( t ) \\exp _ { q } ( - s t ) d t . \\end{align*}"}
+{"id": "1876.png", "formula": "\\begin{align*} \\begin{aligned} \\pi _ 1 ( M ) & = \\pi _ 2 ( M ) = 0 \\\\ \\big ( \\operatorname { r e s p . } \\ , \\pi _ 1 ( M ) & = \\cdots = \\pi _ { [ p ] } = 0 \\ , \\big ) . \\end{aligned} \\end{align*}"}
+{"id": "8325.png", "formula": "\\begin{align*} \\psi ( x , t ; k ) = \\phi ( x , t ; k ) e ^ { i ( k ^ 2 x + \\eta ^ 2 t ) \\sigma _ 3 } , \\end{align*}"}
+{"id": "7616.png", "formula": "\\begin{align*} 0 & = e _ y ( y , v ) = f ^ 2 ( y , v ) = f ( y , y ) f ( y , v ) + \\sum _ { y < u < v } f ( y , u ) f ( u , v ) \\\\ & = \\pm f ( y , v ) + \\sum _ { y < u < v } f ( y , u ) f ( u , v ) , \\end{align*}"}
+{"id": "5098.png", "formula": "\\begin{align*} a ( x _ 1 , x _ 2 ) = \\nabla \\times ( \\eta ( x _ 1 , x _ 2 ) \\nabla z ) + \\phi ( x _ 1 , x _ 2 ) \\nabla z , \\end{align*}"}
+{"id": "6433.png", "formula": "\\begin{align*} \\lfloor x \\rfloor + \\Big \\lfloor x + \\frac 1 n \\Big \\rfloor + \\ldots + \\Big \\lfloor x + \\frac { n - 1 } n \\Big \\rfloor = \\lfloor n x \\rfloor . \\end{align*}"}
+{"id": "8277.png", "formula": "\\begin{align*} X _ { t _ { n + 1 } } ^ i = X _ { t _ n } ^ i + \\int _ { t _ n } ^ { t _ { n + 1 } } b ^ i ( X _ t ) \\d t + \\sigma W _ \\tau ^ i , ~ ~ ~ ~ \\tilde X _ { n + 1 } ^ i = \\tilde X _ n + \\int _ { t _ n } ^ { t _ { n + 1 } } b ^ i ( \\tilde X _ n ) \\d t + \\sigma W _ \\tau ^ i , \\end{align*}"}
+{"id": "2043.png", "formula": "\\begin{align*} M = - \\int ^ t _ { t _ 0 } | \\partial _ { v _ 1 } + ( r - t _ 0 ) \\partial _ { x _ 1 } | ^ 2 d r = - ( t - t _ 0 ) \\partial _ { v _ 1 } ^ 2 - ( t - t _ 0 ) ^ 2 \\partial _ { x _ 1 } \\partial _ { v _ 1 } - \\frac { ( t - t _ 0 ) ^ 3 } { 3 } \\partial _ { x _ 1 } ^ 2 , \\end{align*}"}
+{"id": "3271.png", "formula": "\\begin{align*} A _ k ^ { \\epsilon } : = \\left \\lbrace h \\in H : \\sum _ { l \\geq N _ k ^ { \\epsilon } } \\langle h , e _ l \\rangle ^ 2 \\leq \\frac 1 { l _ k } \\right \\rbrace . \\end{align*}"}
+{"id": "625.png", "formula": "\\begin{align*} O _ { \\gamma } ( f _ 0 ) = 2 \\frac { q ^ { d _ \\gamma + 1 } - 1 } { q - 1 } . \\end{align*}"}
+{"id": "8726.png", "formula": "\\begin{align*} \\overline { \\pi } _ 1 ( q _ 1 ) \\overline { \\pi } _ 1 ( q _ 2 ) = c _ { \\chi _ 1 } ( q _ 1 , q _ 2 ) \\overline { \\pi } _ 1 ( q _ 1 q _ 2 ) \\forall q _ 1 , q _ 2 \\in Q \\end{align*}"}
+{"id": "2899.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\theta } { \\frak G } _ { x , y } ( s ) \\dd s = M _ { x , y } \\end{align*}"}
+{"id": "7950.png", "formula": "\\begin{align*} \\prod _ { p \\leq X } \\left ( 1 - \\frac { 1 } { p } \\right ) \\left ( 1 + \\frac { \\Delta _ K ( p ) \\iota _ { K , f } ( p ) } { p } \\right ) \\asymp \\exp \\left ( \\sum _ { p \\leq X } \\frac { \\Delta _ K ( p ) \\iota _ { K , f } ( p ) - 1 } { p } \\right ) & = \\exp \\left ( \\sum _ { p \\leq X } \\frac { \\Delta _ K ( p ) - 1 } { p } + O _ { K , f } ( 1 ) \\right ) \\\\ & \\asymp _ { K , f } \\prod _ { p \\leq X } \\left ( 1 + \\frac { \\Delta _ K ( p ) - 1 } { p } \\right ) , \\end{align*}"}
+{"id": "1404.png", "formula": "\\begin{align*} & \\int _ { W ^ d } \\det ( q _ { n + i - j } ( y _ j - x _ i ) ) _ { i , j = 1 } ^ d \\det ( q _ { m + i - j } ( z _ j - y _ i ) ) _ { i , j = 1 } ^ d d y _ 1 \\ldots d y _ d \\\\ & = \\det ( q _ { n + m + i - j } ( z _ j - x _ i ) ) _ { i , j = 1 } ^ d . \\end{align*}"}
+{"id": "4312.png", "formula": "\\begin{align*} | f _ x | ^ 2 _ h e ^ { - \\varphi _ 1 } c ( - \\Psi _ 1 ) \\ge \\min \\{ \\tilde { C } _ K , C _ 1 C _ 2 \\} | f _ x | ^ 2 _ { \\hat { h } } , \\end{align*}"}
+{"id": "5533.png", "formula": "\\begin{align*} \\int _ { \\mathfrak U } | \\omega | = { \\phi _ s } _ ! \\int _ { \\mathfrak Y } | \\omega | , \\end{align*}"}
+{"id": "9000.png", "formula": "\\begin{align*} B ( q ) = B ( q , m , M ) = \\left ( \\prod _ { r = m + 1 } ^ { q } { D _ { \\ell _ { r } - 1 } } \\prod _ { r = q + 1 } ^ { M } { D _ { \\ell _ { r - 1 } } } \\right ) . \\end{align*}"}
+{"id": "3369.png", "formula": "\\begin{align*} & B _ i = \\{ \\omega : \\| u _ i - u _ i ' \\| \\leq q \\| u _ { i - 1 } - u _ { i - 1 } ' \\| \\} \\\\ & \\boldsymbol { A } _ { 0 } = B _ { m } ^ c \\\\ & \\boldsymbol { A } _ k = \\big ( \\cap _ { i = m } ^ { k + m - 1 } B _ { i } \\big ) \\cap B _ { k + m } ^ c , \\forall 1 \\leq k \\leq n - 1 , \\\\ & \\boldsymbol { A } _ { n } = \\cap _ { i = m } ^ { n + m - 1 } B _ i . \\end{align*}"}
+{"id": "8265.png", "formula": "\\begin{align*} \\int _ { t _ 1 } ^ { t _ 2 } \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j = 1 } ^ { i } j ( g _ { i + 1 } - g _ i ) V _ { i , j } \\psi _ j ( s ) \\psi _ i ( s ) d s < \\infty \\end{align*}"}
+{"id": "3010.png", "formula": "\\begin{align*} \\lambda _ j = 2 \\theta ^ { \\alpha _ X } _ { \\gamma } ( 0 ) \\lambda _ H ( 1 + 2 \\lambda _ g ) + \\lambda _ { j + 1 } ( 2 + \\lambda _ g ) . \\end{align*}"}
+{"id": "145.png", "formula": "\\begin{align*} l _ { x y } ( i ) = { \\lambda _ i \\over \\lambda _ 0 } \\prod _ { z \\in \\partial x \\setminus \\{ y \\} } { a _ { i 0 } + \\sum _ { j \\in \\Z _ 0 } a _ { i j } l _ { z x } ( j ) \\over a _ { 0 0 } + \\sum _ { j \\in \\Z _ 0 } a _ { 0 j } l _ { z x } ( j ) } . \\end{align*}"}
+{"id": "6329.png", "formula": "\\begin{align*} y ^ { q ^ 2 } - y + ( x ^ { q ^ 2 } - x ) ( x ^ { q ^ 3 } + x ^ q ) = 0 , \\end{align*}"}
+{"id": "4520.png", "formula": "\\begin{align*} \\begin{cases} t _ { i , e n } ( x _ 0 , t _ 0 ) = \\max \\left ( 0 , t _ 0 - \\dfrac { x _ 0 + R } { \\lambda _ i } \\right ) \\forall \\ : i \\in [ 1 , p ] , \\\\ t _ { i , e n } ( x _ 0 , t _ 0 ) = \\max \\left ( 0 , t _ 0 - \\dfrac { x _ 0 - R } { \\lambda _ i } \\right ) \\forall \\ : i \\in [ p + 1 , n ] . \\end{cases} \\end{align*}"}
+{"id": "7666.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x ^ { * , i } _ t = & ~ [ A x ^ { * , i } _ t + B \\alpha ^ { * , i } _ t ] d t + \\sigma d W ^ i _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x ^ { * , i } _ 0 = & ~ \\xi ^ i . \\end{aligned} \\right . \\end{align*}"}
+{"id": "4173.png", "formula": "\\begin{align*} \\mathbf { h } ( H _ { \\rho } ( x ) ) = R _ \\rho \\mathbf { h } ( x ) \\end{align*}"}
+{"id": "7953.png", "formula": "\\begin{align*} \\varrho _ N ^ { \\rm i n i } ( u ) : = \\varrho _ * + N ^ { - \\alpha } \\rho ^ { \\rm i n i } ( u ) , \\end{align*}"}
+{"id": "8580.png", "formula": "\\begin{align*} k _ 1 ( t ) = h _ { \\gamma } ( t ) \\cdot \\ , f _ 2 ( t ) , \\ f _ 2 ( t ) = \\sum _ { k = 0 } ^ { + \\infty } \\ , b _ k t ^ k , \\ b _ 0 \\not = 0 , \\ 0 < \\gamma < 1 - \\alpha . \\end{align*}"}
+{"id": "6515.png", "formula": "\\begin{align*} & s ( \\mu \\nu ) = \\mu \\nu ( w _ 2 w _ 3 ) = \\mu \\nu | ^ * _ { E _ [ w _ 2 , w _ 2 w _ 3 ] } ( w _ 3 ) = \\nu ( w _ 3 ) = s ( \\nu ) , \\\\ & r ( \\mu \\nu ) = \\mu \\nu ( e ) = \\mu \\nu | _ { E _ { w _ 2 } } ( e ) = \\mu ( e ) = r ( \\mu ) \\end{align*}"}
+{"id": "3823.png", "formula": "\\begin{align*} | f _ \\alpha ( U | \\bar U ; x , t ) | & \\le | f _ \\alpha - \\bar f _ \\alpha - \\nabla \\bar f _ \\alpha ( U - \\bar U ) ) | \\\\ & + \\left | \\nabla \\bar f _ \\alpha \\nabla \\bar A ^ { - 1 } \\big ( A ( U , x , t ) - \\bar A - \\nabla \\bar A ( U - \\bar U ) \\big ) \\right | \\\\ & \\le k _ 4 | A ( U , x , t ) - \\bar A | ^ 2 \\ , , \\end{align*}"}
+{"id": "1052.png", "formula": "\\begin{align*} x ^ { \\ast , \\sigma , n } = x ^ { \\ast , \\sigma , n - 1 } \\ast \\left ( \\prescript { \\sigma ^ { n - 1 } } { } x \\right ) = x ^ { \\ast , \\sigma , n - 1 } \\cdot \\left ( \\prescript { \\sigma ^ { n - 1 } } { } x _ n \\right ) . \\end{align*}"}
+{"id": "6708.png", "formula": "\\begin{align*} \\lim _ { n , m \\rightarrow \\infty } \\Omega ( m , n ) = 0 \\end{align*}"}
+{"id": "7882.png", "formula": "\\begin{align*} F ^ { q , \\dagger } _ v ( v ( \\tau ) , p ( s ( \\tau ) ) ) ( u _ \\ell + \\omega ( \\tau ) ) = \\nu ( \\tau ) ( u _ \\ell + \\omega ( \\tau ) ) , \\end{align*}"}
+{"id": "2577.png", "formula": "\\begin{align*} q ^ { k s } \\pmod { N _ { n _ { 1 } } } = 1 \\end{align*}"}
+{"id": "3567.png", "formula": "\\begin{align*} \\varphi _ 2 ^ 2 G _ 2 - \\varphi _ \\frac { 1 } { 2 } ^ 2 H _ \\frac { 1 } { 2 } = \\dfrac { 1 } { 2 } \\varphi ( q ) ^ { - 6 } ( \\varphi ( - q ^ { 1 / 2 } ) ^ 2 H ( - q ^ { 1 / 2 } ) - \\varphi ( q ^ { 1 / 2 } ) ^ 2 H ( q ^ { 1 / 2 } ) ) \\end{align*}"}
+{"id": "3379.png", "formula": "\\begin{align*} M ( X ) = \\max _ z p ( z ) = p ( 0 ) \\leq q ( 0 ) = \\max _ z q ( z ) = M ( Y ) . \\end{align*}"}
+{"id": "7959.png", "formula": "\\begin{align*} H _ N ( t ) = H ( f _ { N , t } ; \\nu _ { N , t } ) : = \\int _ { \\Omega _ N } f _ { N , t } \\log f _ { N , t } d \\nu _ { N , t } . \\end{align*}"}
+{"id": "1008.png", "formula": "\\begin{align*} x '' : = w '' \\varepsilon ^ { \\mu '' } : = x ' r _ { a } = w ' s _ \\alpha \\varepsilon ^ { \\mu ' - \\langle \\mu ' , \\alpha \\rangle \\alpha ^ \\vee } < x ' . \\end{align*}"}
+{"id": "1465.png", "formula": "\\begin{align*} \\lim _ { a \\to 0 ^ { + } } \\Gamma _ { a } = \\lim _ { a \\to 0 ^ { + } } \\Gamma _ { a } ^ { + } = \\lim _ { a \\to 0 ^ { + } } \\Gamma _ { a } ^ { - } = 0 . \\end{align*}"}
+{"id": "5811.png", "formula": "\\begin{align*} \\underline { u } = \\frac { 1 } { 2 } \\sum _ { 1 \\leq j < k \\leq N } \\langle u ( e _ j ) , e _ k \\rangle \\ e _ j \\cdot e _ k . \\end{align*}"}
+{"id": "3653.png", "formula": "\\begin{align*} D = \\max \\left \\{ D _ { x , 1 } \\sqrt { \\delta ' } + D _ { x , 2 } , D _ { y , 1 } \\sqrt { \\delta ' } + D _ { y , 2 } \\right \\} , \\end{align*}"}
+{"id": "3533.png", "formula": "\\begin{align*} ( - 1 ) ^ k A _ k \\ & = \\ \\sum _ { j = 0 } ^ k \\binom { k } { j } B _ j ( - 1 ) ^ j , \\\\ B _ k \\ & = \\ \\sum _ { j = 0 } ^ k \\binom { k } { j } A _ j . \\end{align*}"}
+{"id": "1639.png", "formula": "\\begin{align*} \\psi ( S ( r + s ) ) & = \\int ^ { T _ 2 } _ { T _ 1 } G ( T _ 2 , \\tau ) f _ \\downarrow \\left ( r ( \\tau ) + s ( \\tau ) \\right ) d \\tau \\\\ & \\ge f _ \\downarrow \\left ( b + d \\right ) \\int ^ { T _ 2 } _ { T _ 1 } G ( T _ 2 , \\tau ) \\ ; d \\tau \\\\ & > d . \\end{align*}"}
+{"id": "4022.png", "formula": "\\begin{align*} \\beta ' = \\beta \\pi _ { 2 3 } , \\delta ' = \\delta \\pi _ { 2 3 } , \\gamma ' = \\gamma \\pi _ { 1 2 } , \\gamma '' = \\pi _ { 2 3 } \\gamma \\pi _ { 2 3 } . \\end{align*}"}
+{"id": "8034.png", "formula": "\\begin{align*} \\sum _ { t = 2 } ^ { 2 r } \\frac { L ^ { 3 r - 2 t } } { \\pi _ N ( x ) ^ { 2 r - t } } = \\frac { L ^ 2 } { \\pi _ N ( x ) ^ 2 } + \\frac { 1 } { \\pi _ N ( x ) } + \\frac { 1 } { L ^ 2 } . \\end{align*}"}
+{"id": "8978.png", "formula": "\\begin{align*} R _ { \\sigma } = \\dfrac { 1 } { x _ { 2 2 2 } } \\Big ( \\dfrac { x _ { 2 2 2 } ^ * - x _ { 1 1 1 } } { x _ { 2 2 2 } ^ * } \\Big ) . \\end{align*}"}
+{"id": "6290.png", "formula": "\\begin{align*} A _ k ( a _ j ) = A _ k ( b _ j ) = 0 . \\end{align*}"}
+{"id": "2825.png", "formula": "\\begin{align*} \\mathcal { F } ^ { ( k ) } \\left ( \\mathbf { x } ^ { ( k ) } \\right ) = s o f t \\left ( \\mathcal { F } ^ { ( k ) } \\left ( \\mathbf { r } ^ { ( k ) } \\right ) , \\theta ^ { ( k ) } \\right ) , \\end{align*}"}
+{"id": "5227.png", "formula": "\\begin{align*} & H _ { \\gamma } [ b _ n ] = H _ { \\gamma } [ b _ { 0 , n } ] = h _ n , \\\\ & I _ { h _ n , \\gamma } \\leq { \\mathcal { E } } [ v _ n , b _ n ] \\leq { \\mathcal { E } } [ v _ { 0 , n } , b _ { 0 , n } ] . \\end{align*}"}
+{"id": "2451.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } ( 1 - ( 1 - q ) k y ) ^ { \\frac { 1 } { 1 - q } - k } y ^ { k } d y = \\frac { \\Gamma ( k + 1 ) \\Gamma \\left ( \\frac { 2 - q } { 1 - q } - k \\right ) } { \\Gamma \\left ( \\frac { 3 - 2 q } { 1 - q } \\right ) } . \\end{align*}"}
+{"id": "7869.png", "formula": "\\begin{align*} H = \\begin{pmatrix} \\psi _ { v v } ( 0 , p _ 0 ) [ v , v ] & \\psi _ { v p } ( 0 , p _ 0 ) [ v , p ' ( 0 ) ] \\\\ \\psi _ { v p } ( 0 , p _ 0 ) [ v , p ' ( 0 ) ] & \\psi _ { p p } ( 0 , p _ 0 ) [ p ' ( 0 ) , p ' ( 0 ) ] + \\psi _ p ( 0 , p _ 0 ) p '' ( 0 ) \\end{pmatrix} \\end{align*}"}
+{"id": "3163.png", "formula": "\\begin{align*} k _ 3 ( P ^ \\ast ( q ) ) & = \\frac { 1 } { 3 } \\times q \\times k _ 3 ( P ^ \\ast ( q ) , 0 ) \\\\ & = \\frac { q } { 3 } \\times \\langle H \\rangle . \\end{align*}"}
+{"id": "3176.png", "formula": "\\begin{align*} l = \\left \\{ \\begin{array} { l l l } 1 , & \\hbox { i f $ i $ i s o d d , } \\\\ - 1 , & \\hbox { ; } \\end{array} \\right . \\end{align*}"}
+{"id": "6367.png", "formula": "\\begin{align*} Q ( u ) = F ^ { - 1 } ( u ) = \\sqrt { - \\frac { \\theta } { \\log ( \\operatorname { I } ^ { - 1 } _ { u } ( a , b ) ) } } , 0 < u < 1 , \\end{align*}"}
+{"id": "6840.png", "formula": "\\begin{align*} a _ { m } ( x ) + \\sum _ { n = 0 } ^ { N _ { s } } a _ { n } ( x ) A _ { m n } ( x ) = r _ { m } ( x ) , m = 0 , \\ldots , N _ { s } \\end{align*}"}
+{"id": "6945.png", "formula": "\\begin{align*} \\phi ( x , y ) = \\sum _ { i = 1 } ^ L c _ i a _ i ( x ) b _ i ( y ) , \\end{align*}"}
+{"id": "707.png", "formula": "\\begin{align*} J ( P _ 0 \\Psi _ { R ( \\lambda ) } ) = O ( | b - \\lambda | ^ { 2 p } ) ~ ~ \\lambda \\to b . \\end{align*}"}
+{"id": "4123.png", "formula": "\\begin{align*} H _ { m i n } ( E \\otimes F _ { \\ell - 1 } ) = H _ { m i n } ( E \\otimes F _ 1 ) . \\end{align*}"}
+{"id": "3650.png", "formula": "\\begin{align*} \\alpha _ y = \\mu _ y , \\end{align*}"}
+{"id": "1083.png", "formula": "\\begin{align*} \\Phi ^ + _ J ( \\alpha ) : = \\begin{cases} 1 , & \\alpha \\in \\Phi ^ + _ J , \\\\ 0 , & \\alpha \\in \\Phi \\setminus \\Phi ^ + _ J . \\end{cases} . \\end{align*}"}
+{"id": "8448.png", "formula": "\\begin{align*} R ( x ; z ) = \\begin{pmatrix} \\bar { r } _ 1 ( z ) r _ 2 ( z ) & \\bar { r } _ 1 ( z ) \\mathrm { e } ^ { - 2 i z x } \\\\ r _ 2 ( z ) \\mathrm { e } ^ { 2 i z x } & 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "6541.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\pi ( a ) } q _ i ' = ( a + 1 ) ^ { \\beta ( a + 1 ) } \\prod _ { i = 1 } ^ { \\pi ( a ) } p _ i ^ { \\alpha _ i ' } \\end{align*}"}
+{"id": "615.png", "formula": "\\begin{align*} \\frac { d ( \\phi ( y _ 1 ) , \\phi ( y _ 2 ) ) } { d ( \\phi ( y _ 1 ) , \\phi ( y _ 3 ) ) } \\leq \\limsup _ { \\substack { x , x ' \\in \\phi ( Y ) , \\ , \\pi ( x ) = \\pi ( x ' ) \\\\ x \\to x ' } } \\ , \\eta \\left ( \\frac { d ( x , \\pi ^ { - 1 } ( y _ 1 ) ) } { d ( x ' , \\pi ^ { - 1 } ( y _ 1 ) ) } \\right ) < H . \\end{align*}"}
+{"id": "6699.png", "formula": "\\begin{align*} ( \\langle \\gamma _ 1 \\rangle , \\langle \\gamma _ 2 \\rangle ) \\in R _ \\alpha \\qquad \\Longleftrightarrow \\ell _ \\alpha ( \\langle \\gamma _ 1 \\rangle ) = \\ell _ \\alpha ( \\langle \\gamma _ 2 \\rangle ) . \\end{align*}"}
+{"id": "7343.png", "formula": "\\begin{align*} \\mathcal { C } _ { i j } & \\simeq \\Omega _ { X } | _ { D _ { i j } } & \\mathcal { C } _ { i j } ^ l & \\simeq \\omega _ l | _ { D _ { i j } } , l = i , j \\\\ \\omega _ X | _ { D _ i } & \\simeq \\mathcal { C } _ i \\ \\otimes \\omega _ i , & \\det ( \\mathcal { C } _ { i j } ) & \\simeq \\mathcal { C } _ { i j } ^ l \\otimes \\mathcal { C } _ l | _ { D _ { i j } } l = i , j \\end{align*}"}
+{"id": "691.png", "formula": "\\begin{align*} S : = \\{ 1 \\} \\cup \\{ g \\in G : \\phi _ { g , 1 } \\neq 0 \\} \\cup \\{ g \\in G : \\psi _ { g , 1 } \\neq 0 \\} . \\end{align*}"}
+{"id": "9171.png", "formula": "\\begin{align*} \\begin{aligned} \\left . \\dfrac { \\partial h ( s ) } { \\partial s } \\right | _ { s = x + \\delta \\sin ( 2 \\pi t ) } = & \\ , \\left . \\dfrac { \\partial h ( s ) } { \\partial s } \\right | _ x \\\\ & + \\left . \\dfrac { \\partial ^ 2 h ( s ) } { \\partial s ^ 2 } \\right | _ { x } \\delta \\sin ( 2 \\pi t ) + O ( \\delta ^ 2 ) \\\\ \\end{aligned} \\end{align*}"}
+{"id": "1236.png", "formula": "\\begin{align*} \\sum _ { \\xi _ { \\Delta } } \\int _ { \\Omega } c _ { \\Delta } ( \\omega , \\xi _ { \\Delta } ) f ( \\omega ) g ( \\xi _ { \\Delta } \\omega _ { \\Delta ^ c } ) \\mu ( d \\omega ) = \\sum _ { \\zeta _ { \\Delta } } \\int _ { \\Omega } \\hat { c } _ { \\Delta } ( \\omega , \\zeta _ { \\Delta } ) f ( \\zeta _ { \\Delta } \\omega _ { \\Delta ^ c } ) g ( \\omega ) \\mu ( d \\omega ) , \\end{align*}"}
+{"id": "3802.png", "formula": "\\begin{align*} g ( z ) = \\sum _ { m = 0 } ^ { \\infty } a _ m z ^ { q ( m + 1 ) } = \\sum _ { n = 1 } ^ { \\infty } a _ { n - 1 } z ^ { q n } = \\sum _ { n = 0 } ^ { \\infty } b _ n z ^ { q n } , b _ n = \\begin{cases} a _ { n - 1 } & , n \\geq 1 , \\\\ 0 & , . \\end{cases} \\end{align*}"}
+{"id": "7706.png", "formula": "\\begin{align*} K _ r ( B ) = K _ r ( B _ { [ n ] , [ r ] } ) \\times \\R ^ { m - r } \\mbox { a n d } \\sigma _ m ( K _ r ( B ) ) = \\sigma _ r ( K _ r ( B _ { [ n ] , [ r ] } ) ) , r \\in [ m - 1 ] , \\end{align*}"}
+{"id": "3432.png", "formula": "\\begin{align*} j _ 1 \\alpha - \\frac { p } { q _ n } = j _ 1 \\frac { p _ n } { q _ n } - \\frac { p } { q _ n } + j _ 1 ( \\alpha - \\frac { p _ n } { q _ n } ) = k + j _ 1 ( \\alpha - \\frac { p _ n } { q _ n } ) , \\end{align*}"}
+{"id": "8321.png", "formula": "\\begin{align*} & u _ { x t } + \\alpha \\beta ^ 2 u - 2 i \\alpha \\beta u _ x - \\alpha u _ { x x } - i \\alpha \\beta ^ 2 | u | ^ 2 u _ x = 0 , \\\\ & u ( x , t = 0 ) = u _ 0 ( x ) , \\end{align*}"}
+{"id": "3195.png", "formula": "\\begin{align*} V ( x ) = & v o l ( B _ 1 ( x ) \\cap \\tau _ 1 ) + v o l ( B _ 2 ( x ) \\cap \\tau _ 2 ) \\\\ = & V ( 0 ) \\frac { e ^ { 2 x } + e ^ { - 2 x } } { 2 } = V ( 0 ) \\cosh \\left ( 2 x \\right ) \\end{align*}"}
+{"id": "7942.png", "formula": "\\begin{align*} A ^ { \\tilde \\Psi + \\eta _ n } [ \\eta _ 1 , \\dots , \\eta _ { n - 1 } ] - A ^ { \\tilde \\Psi } [ \\eta _ 1 , \\dots , \\eta _ { n - 1 } ] - A ^ { \\tilde \\Psi } [ \\eta _ 1 , \\dots , \\eta _ n ] = g _ 1 ( x ) + g _ 2 ( x ) . \\end{align*}"}
+{"id": "7450.png", "formula": "\\begin{align*} \\frac { k l } { m } , k \\ , \\sqrt { \\frac { l - m } { m ( l - 1 ) } } , l \\ , \\sqrt { \\frac { k - m } { m ( k - 1 ) } } . \\end{align*}"}
+{"id": "2516.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbf { A } \\mathbf { d } _ { h x } = 0 , \\\\ & { { \\mathbf { A } } ^ { T } } \\mathbf { d } _ { h y } + \\mathbf { C } d _ { h x } + \\mathbf { d } _ { h s } = 0 , \\\\ & \\bar { \\mathbf { X } } \\mathbf { D } \\mathbf { d } _ { h s } + \\bar { \\mathbf { S } } \\mathbf { D } ^ { - T } \\mathbf { d } _ { h x } = 0 , \\end{aligned} \\end{align*}"}
+{"id": "4571.png", "formula": "\\begin{align*} q z _ i ^ 2 A _ { i , 2 , 0 } = \\frac { \\lambda _ 1 } { q } z _ i ^ 2 A _ { i , 1 , 0 } - \\frac { \\lambda _ 3 ( q + 1 ) } { q \\sqrt { q } ( q ^ 2 + q + 1 ) } z _ i A _ { i , 0 , 0 } + \\frac { q + 1 } { q ^ 2 + q + 1 } A _ { i , 0 , 0 } - \\sqrt { q } z _ i ^ 3 A _ { i , 1 , 0 } . \\end{align*}"}
+{"id": "3083.png", "formula": "\\begin{align*} \\textstyle 1 , \\binom { n } { 1 } , \\binom { n } { 2 } , \\dotsc , \\binom { n } { n - 1 } , 1 . \\end{align*}"}
+{"id": "6115.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathfrak { s } \\ , ( z _ 1 , z _ 2 , z _ 3 , z _ 4 , z _ 0 ) = ( z _ 0 , z _ 4 , z _ 3 , z _ 2 , z _ 1 ) \\ , , \\\\ & \\mathfrak { t } \\ , ( z _ 1 , z _ 2 , z _ 3 , z _ 4 , z _ 0 ) = ( z _ 2 , - z _ 3 - 1 , - z _ 1 - 1 , z _ 4 , z _ 0 ) \\ , , \\\\ & \\mathfrak { i } \\ , ( z _ 1 , z _ 2 , z _ 3 , z _ 4 , z _ 0 ) = ( z _ 1 , z _ 2 , z _ 3 , z _ 0 , z _ 4 ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "4616.png", "formula": "\\begin{align*} & \\frac { d } { d r } \\log _ { m + 1 } \\frac { e } { r } = \\frac { 1 } { \\prod _ { k = 1 } ^ m \\log _ k \\frac { e } { r } } \\left ( - \\frac { 1 } { r } \\right ) , \\\\ & \\frac { d } { d r } \\log _ { m + 1 } \\frac { 1 } { C r ^ \\alpha } = \\frac { 1 } { \\prod _ { k = 1 } ^ m \\log _ k \\frac { 1 } { C r ^ \\alpha } } \\left ( - \\frac { \\alpha } { r } \\right ) . \\end{align*}"}
+{"id": "601.png", "formula": "\\begin{align*} \\delta _ { p , c } : = \\sum _ { k = 1 } ^ { l } ( 2 - \\alpha _ { k } - \\beta _ { k } ) c _ { k } ^ { \\beta _ { k } } \\zeta _ { k } ^ { \\frac { p } { 2 - \\alpha _ { k } - \\beta _ { k } } } \\big [ \\null _ { k } \\eta \\big ] _ { \\frac { p } { 2 - \\alpha _ { k } } } , \\end{align*}"}
+{"id": "2817.png", "formula": "\\begin{align*} \\mathbf { H } = \\left [ \\mathbf { J } _ { 1 } , \\ \\mathbf { J } _ { 2 } \\right ] \\begin{bmatrix} \\mathbf { \\Sigma } _ { 1 } & \\ \\\\ \\ & \\mathbf { \\Sigma } _ { 2 } \\end{bmatrix} \\left [ \\mathbf { J } _ { 1 } , \\ \\mathbf { J } _ { 2 } \\right ] ^ { \\mathbf { H } } , \\end{align*}"}
+{"id": "3717.png", "formula": "\\begin{align*} \\big ( h ( t _ j ) , v ( t _ j ) \\big ) & = \\big ( \\hat h ( t _ j ) , \\hat v ( t _ j ) \\big ) - \\big ( L _ V g _ { t _ j } , V ( u _ { t _ j } ) \\big ) \\\\ ( p , z ) & = ( \\hat p , \\hat z ) - \\Big ( L _ V L _ X g _ a , V \\big ( X ( u _ a ) \\big ) \\Big ) \\end{align*}"}
+{"id": "1390.png", "formula": "\\begin{align*} e ^ { \\sum _ { j = 1 } ^ d x _ j } \\det ( ( - 1 ) ^ { j - 1 } \\phi ^ { ( 1 - j ) } _ i ( x _ j ) ) _ { i , j = 1 } ^ d = e ^ { \\sum _ { j = 1 } ^ d x _ j } \\det ( ( - 1 ) ^ { d - j } \\phi ^ { ( d - j ) } _ i ( x _ j ) ) _ { i , j = 1 } ^ d . \\end{align*}"}
+{"id": "2613.png", "formula": "\\begin{align*} x \\cdot y = \\varepsilon ( x , y ) y \\cdot x , \\end{align*}"}
+{"id": "1873.png", "formula": "\\begin{align*} F _ { p , x } ( v ) = ( p - 2 ) \\langle \\mathsf B ( v , v ) , B ( v , v ) \\rangle _ { \\mathbb R ^ q } + \\langle Q ^ M _ x ( v ) , v \\rangle _ M , \\end{align*}"}
+{"id": "8821.png", "formula": "\\begin{align*} \\Pi _ { \\mathrm { k e r } A } B ( v _ 1 , v _ 2 ) + \\Pi _ { \\mathrm { k e r } A } B ( v _ 2 , v _ 1 ) = 0 \\end{align*}"}
+{"id": "6474.png", "formula": "\\begin{align*} \\exists \\ \\theta > 1 \\ \\Rightarrow \\ K _ p ( \\theta ) : = | | \\kappa _ p ( \\xi ) | | _ { \\theta } < \\infty , \\end{align*}"}
+{"id": "1509.png", "formula": "\\begin{align*} g _ 1 ( t , s ) & = \\frac { ( t - a ) ^ { \\alpha - 1 } ( b - s ) ^ { \\alpha - 1 - \\beta } } { ( b - a ) ^ { \\alpha - 1 - \\beta } } - ( t - s ) ^ { \\alpha - 1 } \\\\ & = ( t - a ) ^ { \\alpha - 1 } \\left ( \\frac { b - s } { b - a } \\right ) ^ { \\alpha - 1 - \\beta } - ( t - a ) ^ { \\alpha - 1 } \\left ( \\frac { t - s } { t - a } \\right ) ^ { \\alpha - 1 } \\\\ & = ( t - a ) ^ { \\alpha - 1 } \\left [ \\left ( \\frac { b - s } { b - a } \\right ) ^ { \\alpha - 1 - \\beta } - \\left ( \\frac { t - s } { t - a } \\right ) ^ { \\alpha - 1 } \\right ] . \\end{align*}"}
+{"id": "4215.png", "formula": "\\begin{align*} \\mathcal { D } _ { G , \\kappa } ^ { \\mathrm { c h } } \\simeq \\bigoplus _ { \\lambda \\in P ^ + } \\mathbb { V } ^ { \\kappa } _ \\lambda \\otimes \\mathbb { V } ^ { \\kappa ^ * } _ { \\lambda ^ * } , \\frac { 1 } { \\kappa + h ^ \\vee } + \\frac { 1 } { \\kappa ^ * + h ^ \\vee } = 0 . \\end{align*}"}
+{"id": "3857.png", "formula": "\\begin{align*} \\gamma _ i ( s , \\tau ) = \\bar { Q } _ { \\frac { 2 \\pi ( i - 1 ) } { m } } \\gamma ( s , \\tau ) . \\end{align*}"}
+{"id": "1412.png", "formula": "\\begin{align*} \\left | \\mathrm { d e t } { \\left ( e ^ { - i \\theta _ i x _ j } \\right ) } _ { i , j = 1 } ^ { d } \\right | \\le C _ d \\Delta ( \\theta ) \\Delta ( x ) . \\end{align*}"}
+{"id": "3628.png", "formula": "\\begin{align*} \\tilde { Q } = \\ln \\tilde { \\kappa } _ 1 - N \\ln \\nu ^ { n + 1 } . \\end{align*}"}
+{"id": "1981.png", "formula": "\\begin{align*} f ( z ) & = \\frac 1 2 h ( g ^ { - 1 } ( z ) ) . \\end{align*}"}
+{"id": "657.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | e ^ { i k \\phi } \\cos ( n \\theta ) \\left | l _ 2 , m _ 2 \\right \\rangle = \\end{align*}"}
+{"id": "179.png", "formula": "\\begin{align*} ( \\gamma - 1 ) ^ { - 1 } = - \\sum _ { i = 0 } ^ \\infty \\gamma ^ { - 1 } ( \\gamma ^ { - 1 } - 1 ) ^ { - ( i + 1 ) } \\otimes ( \\gamma - 1 ) ^ { i } \\end{align*}"}
+{"id": "1374.png", "formula": "\\begin{align*} ( S _ j ( n ) : 1 \\leq j \\leq d , n \\geq 0 ) = ( T _ j ( n + d - j ) : 1 \\leq j \\leq d , n \\geq 0 ) . \\end{align*}"}
+{"id": "2404.png", "formula": "\\begin{align*} \\mathcal { L } ( B _ v ) : = \\begin{cases} \\{ u \\in B _ v : \\textnormal { d i s t } ( u , v ) = r \\} & \\textnormal { i f } B _ v \\textnormal { i s a t r e e } , \\\\ \\{ u \\in B _ v : \\textnormal { d i s t } ( u , v ) \\geq r \\textnormal { a n d } \\textnormal { d i s t } ( u , C _ v ) \\geq r - h \\} & \\textnormal { i f } C _ v \\textnormal { e x i s t s } , \\end{cases} \\end{align*}"}
+{"id": "2965.png", "formula": "\\begin{align*} \\tilde F _ E ( t ) = \\begin{cases} F _ E ( t + \\alpha ) - \\alpha & ( t \\in [ 0 , 1 - \\alpha ) ) \\\\ F _ E ( t + \\alpha - 1 ) - \\alpha & ( t \\in [ 1 - \\alpha , 1 ) ) . \\end{cases} \\end{align*}"}
+{"id": "5377.png", "formula": "\\begin{align*} T _ \\alpha = \\partial _ { \\alpha } + \\sum _ { \\beta < n } B _ { \\alpha \\beta } ( x _ { \\beta } \\partial _ { n } - x _ { n } \\partial _ { \\beta } ) , \\ ; \\ ; \\mbox { f o r } \\alpha < n , \\end{align*}"}
+{"id": "662.png", "formula": "\\begin{align*} [ \\Box - \\mu _ { \\rm e f f } ^ 2 ( r , \\theta ) ] \\varphi = 0 \\ , . \\end{align*}"}
+{"id": "5167.png", "formula": "\\begin{align*} \\phi _ { 2 , n } = \\phi _ { n } ( 1 - \\chi _ { R _ n / 2 } ) , G _ { 2 , n } = G _ { n } ( 1 - \\chi _ { R _ n / 2 } ) , \\end{align*}"}
+{"id": "4754.png", "formula": "\\begin{align*} M i n i m i z e _ { x \\in \\mathbb { R } ^ n } \\ \\ J ( x ) : = 1 / 2 \\| x - z \\| ^ 2 \\ \\ s . t . \\ x \\in A \\end{align*}"}
+{"id": "3309.png", "formula": "\\begin{align*} \\mathcal { Z } _ K ( D ^ 2 , S ^ 1 ) = \\mathcal { Z } _ { \\mathcal { Z } ^ * _ K ( \\Delta ^ 1 , \\partial \\Delta ^ 1 ) } ( D ^ 1 , S ^ 0 ) . \\end{align*}"}
+{"id": "1755.png", "formula": "\\begin{align*} \\begin{aligned} - \\nabla _ y \\cdot \\left ( D _ 0 \\big ( \\nabla _ y w _ i + e _ i \\big ) \\right ) & = 0 \\mbox { i n } ( 0 , T ) \\times \\Omega \\times Y ^ { \\ast } , \\\\ - D _ 0 \\big ( \\nabla _ y w _ i + e _ i \\big ) \\cdot \\nu & = 0 \\mbox { o n } ( 0 , T ) \\times \\Omega \\times \\Gamma , \\\\ w _ i \\mbox { i s } Y \\mbox { - p e r i o d i c , } \\int _ { Y ^ { \\ast } } & w _ i ( t , x , y ) d y = 0 \\mbox { f . a . e . } ( t , x ) \\in ( 0 , T ) \\times \\Omega . \\end{aligned} \\end{align*}"}
+{"id": "4181.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t w + \\nabla w \\cdot \\nabla ^ \\perp \\varphi = 0 , \\\\ w = \\mathcal { L } _ H \\varphi , \\\\ \\nabla ^ \\perp \\varphi \\cdot \\nu | _ { \\partial \\Omega } = v _ n | _ { \\partial \\Omega } . \\end{cases} \\end{align*}"}
+{"id": "8578.png", "formula": "\\begin{align*} ( h _ \\alpha \\ , * \\ , h _ \\beta ) ( t ) = h _ { \\alpha + \\beta } ( t ) , \\ t > 0 , \\ \\alpha > 0 , \\ \\beta > 0 \\end{align*}"}
+{"id": "8831.png", "formula": "\\begin{align*} \\begin{cases} d X _ { t , n } = X _ { t , 1 } X _ { t , n - 1 } d t + \\sigma _ n d W _ t ^ { ( n ) } \\\\ d X _ { t , 1 } = X _ { t , 2 } X _ { t , n } d t \\\\ d X _ { t , 3 } = - X _ { t , 1 } X _ { t , 2 } d t \\\\ d X _ { t , j } = 0 & j \\not \\in \\{ n , 1 , 3 \\} . \\end{cases} \\end{align*}"}
+{"id": "3438.png", "formula": "\\begin{align*} c _ { n , \\ell } : = | \\cos ( \\pi \\theta _ { m _ n + \\ell q _ n } ) | . \\end{align*}"}
+{"id": "2077.png", "formula": "\\begin{align*} g ( x ) = \\lim _ { n \\rightarrow { \\infty } } g _ n ( x ) \\end{align*}"}
+{"id": "1837.png", "formula": "\\begin{align*} \\delta ^ \\nabla \\big ( F ^ { \\prime } ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) R ^ \\nabla \\big ) = 0 \\ , . \\end{align*}"}
+{"id": "3865.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t w + \\nabla ^ \\perp \\varphi \\cdot \\nabla w = 0 , \\ \\ & \\ B _ { R ^ * } ( 0 ) , \\\\ w = \\mathcal { L } _ H \\varphi , \\ \\ & \\ B _ { R ^ * } ( 0 ) , \\\\ \\varphi = 0 , \\ \\ & \\ \\partial B _ { R ^ * } ( 0 ) . \\end{cases} \\end{align*}"}
+{"id": "8534.png", "formula": "\\begin{align*} F _ { \\delta } ( x ; z ) : = \\left ( \\mathcal { P } ^ + \\left ( r _ { \\delta , 2 } ( z ) e ^ { 2 i z x } \\right ) e _ 2 , \\mathcal { P } ^ - \\left ( \\bar { r } _ { \\delta , 1 } ( z ) e ^ { - 2 i z x } \\right ) e _ 1 \\right ) . \\end{align*}"}
+{"id": "1059.png", "formula": "\\begin{align*} G ( \\breve F ) = \\bigsqcup _ { x \\in \\widetilde W } I x I . \\end{align*}"}
+{"id": "1159.png", "formula": "\\begin{align*} d X _ t = \\left ( b _ 0 ( t , X ) + \\langle b ( t , X , \\cdot ) , \\mu \\rangle \\right ) d t + d W _ t , \\quad \\operatorname { L a w } ( X ) = \\mu , \\quad \\operatorname { L a w } ( X _ 0 ) = \\mu _ 0 . \\end{align*}"}
+{"id": "5420.png", "formula": "\\begin{align*} \\begin{aligned} u ( \\bar { x } ) & = \\frac { 1 } { 2 } \\sum _ { \\beta = 1 } ^ { n - 1 } b _ \\beta \\bar { x } _ \\beta ^ 2 + + \\frac { 1 } { 6 } \\sum _ { \\xi , \\beta , \\gamma } \\varphi _ { \\xi \\beta \\gamma } ( 0 ) \\bar { x } _ \\xi \\bar { x } _ \\beta \\bar { x } _ \\gamma + O ( | \\bar { x } ' | ^ 4 ) \\\\ & \\leq u ( x _ 0 ) + \\frac { 1 } { 2 } \\sum _ { \\beta = 1 } ^ { \\alpha } b _ \\beta \\bar { x } _ \\beta ^ 2 + C \\delta ^ 3 b _ \\alpha ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "367.png", "formula": "\\begin{align*} y y ' = \\left [ \\begin{array} { c c c c } 1 & \\frac { ( n - 2 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } ' & \\frac { - ( n + 1 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } ' \\\\ \\frac { ( n - 2 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } & \\frac { ( n - 2 ) ^ 2 } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } & \\frac { - ( n - 2 ) ( n + 1 ) } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } \\\\ \\frac { - ( n + 1 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } & \\frac { - ( n - 2 ) ( n + 1 ) } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } & \\frac { ( n + 1 ) ^ 2 } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } \\end{array} \\right ] . \\end{align*}"}
+{"id": "7294.png", "formula": "\\begin{align*} \\lambda _ { f \\times i d _ { X } } [ ( f \\times i d _ { X } ) ( r , x ) , \\alpha ] ( 0 ) = ( f ( r ) , x ) = ( f \\times i d _ { X } ) ( r , x ) \\end{align*}"}
+{"id": "9289.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { B } ( u _ p - u _ { p + N } ) ( \\Delta u _ p ) ^ k \\wedge ( \\Delta u _ { p + N } ) ^ { m - k } \\wedge \\beta _ n ^ m \\\\ \\leq & \\sigma ( v , p , N ) + \\int _ B u _ p ( \\Delta u _ p ) ^ m \\wedge \\beta _ n ^ { n - m } - \\int _ B u _ { p + N } ( \\Delta u _ { p + N } ) ^ m \\wedge \\beta _ n ^ { n - m } \\end{aligned} \\end{align*}"}
+{"id": "3328.png", "formula": "\\begin{align*} \\phi _ 1 = \\psi _ 1 , \\ldots , \\phi _ { j - 1 } = \\psi _ { j - 1 } , \\phi _ j < _ K \\psi _ j \\end{align*}"}
+{"id": "2344.png", "formula": "\\begin{align*} \\frac { \\partial R } { \\partial t } ( t ) = - \\frac { \\partial \\phi } { \\partial r } ( R ( t ) ) . \\end{align*}"}
+{"id": "4675.png", "formula": "\\begin{align*} { \\bf P } ( \\lim _ { n \\to \\infty } \\kappa _ n = \\kappa ) = 1 . \\end{align*}"}
+{"id": "2529.png", "formula": "\\begin{align*} \\begin{aligned} & \\hat { \\mathbf { r } } _ p ( \\alpha ) = ( 1 - ( 1 - \\nu ) \\alpha ) \\hat { \\mathbf { r } } _ p \\\\ & \\hat { \\mathbf { r } } _ d ( \\alpha ) = ( 1 - ( 1 - \\nu ) \\alpha ) \\hat { \\mathbf { r } } _ d \\\\ & \\mu ( \\alpha ) = ( 1 - ( 1 - \\nu ) \\alpha ) \\mu , \\end{aligned} \\end{align*}"}
+{"id": "8922.png", "formula": "\\begin{gather*} \\sum _ { m = 0 } ^ { n } ( - 1 ) ^ { n - m } \\binom { n } { m } 2 ^ { - m } \\sum _ { j = 0 } ^ { m } \\binom { m } { j } \\left ( \\beta \\right ) ^ { \\left ( n - j \\right ) } \\left ( \\beta + m - j \\right ) ^ { \\left ( j \\right ) } \\\\ = \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ \\frac { n ! } { 2 ^ { n / 2 } ( n / 2 ) ! } \\left ( \\beta \\right ) ^ { \\left ( n / 2 \\right ) } & & n \\end{array} \\right . . \\end{gather*}"}
+{"id": "3442.png", "formula": "\\begin{align*} | m _ n + \\ell _ n q _ n - \\tilde { m } _ n | = q _ { n + 1 } , \\end{align*}"}
+{"id": "4748.png", "formula": "\\begin{align*} & \\Delta : = ( \\langle u _ * , ( \\lambda x _ v + ( 1 - \\lambda ) x _ w ) \\rangle ) ^ 2 + \\lambda ( 1 - \\lambda ) \\| u _ * \\| ^ 2 \\| x _ v - x _ w \\| ^ 2 \\ge 0 , \\\\ \\end{align*}"}
+{"id": "8090.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M } \\mathbb { E } \\left [ y _ i ^ * y _ i \\right ] = \\sum _ { l = 1 } ^ M a _ c ^ 2 \\lvert \\mathbf { h } _ { l , * } \\mathbf { p } _ c \\rvert ^ 2 + \\sum _ { i = 1 } ^ M \\sum _ { j = 1 } ^ M a _ j ^ 2 \\lvert \\mathbf { h } _ { i , * } \\mathbf { p } _ j \\rvert ^ 2 + M \\sigma _ n ^ 2 , \\end{align*}"}
+{"id": "2716.png", "formula": "\\begin{align*} \\begin{aligned} & ~ & & y _ 2 ^ 2 - 2 L _ 1 y _ 1 y _ 2 + ( L _ 1 \\| x _ k \\| ) ^ 2 \\le ( \\kappa _ { \\rm e g } \\delta _ k + \\epsilon _ g ) ^ 2 \\ ; \\ ; & & \\\\ \\end{aligned} \\end{align*}"}
+{"id": "855.png", "formula": "\\begin{align*} \\Gamma ( X ) = \\sum \\limits _ { 1 \\leq n \\leq X } \\mu ^ 2 ( n ^ 2 + n + 1 ) \\ , . \\end{align*}"}
+{"id": "7213.png", "formula": "\\begin{align*} d u = \\Delta _ p u \\ , d t + \\kappa \\Delta u \\ , d t + \\sum _ { k \\in \\Z ^ d } \\sum _ { i = 1 } ^ { d - 1 } \\xi _ { k , i } ( x ) \\cdot \\nabla u \\ , d B _ t ^ { k , i } , \\end{align*}"}
+{"id": "8197.png", "formula": "\\begin{align*} x _ { 2 i - 1 , j } & + x _ { 2 i - 1 , j + 1 } + x _ { 2 i , j } + x _ { 2 i , j + 1 } = S \\\\ & = x _ { 2 i , j } + x _ { 2 i , j + 1 } + x _ { 2 i + 1 , j } + x _ { 2 i + 1 , j + 1 } , \\end{align*}"}
+{"id": "5643.png", "formula": "\\begin{align*} d _ { i } ( v _ { 1 } , \\dots , v _ { k } ) : = ( v _ { 1 } , \\dots , \\hat { v } _ { i } , \\dots , v _ { k } ) . \\end{align*}"}
+{"id": "733.png", "formula": "\\begin{align*} b ^ { \\varepsilon } = g + \\frac { 1 } { 2 } \\left ( \\partial _ x v + \\partial _ x \\tilde w ^ \\varepsilon \\right ) . \\end{align*}"}
+{"id": "3592.png", "formula": "\\begin{align*} Z ( \\theta ) = \\sum \\limits _ { i \\in R } \\prod \\limits _ { F \\in \\mathrm { f a c e t } ( \\Gamma ) } \\theta ^ { F } _ { i _ { F } } . \\end{align*}"}
+{"id": "1551.png", "formula": "\\begin{align*} u _ t - v _ s & { } = ( 1 - t ) u + t ( v - \\alpha ) - ( 1 - s ) v - s ( u + \\alpha ) \\\\ & { } = ( 1 - t - s ) ( u - v ) - ( t + s ) \\alpha < - \\alpha . \\end{align*}"}
+{"id": "5711.png", "formula": "\\begin{align*} A _ 1 \\circ F \\circ A _ 2 ( x ) + A _ 3 ( x ) = F ' ( x ) , \\end{align*}"}
+{"id": "4739.png", "formula": "\\begin{align*} \\Tilde { x } \\in \\mathcal { S } ^ n : = \\{ x \\in \\mathbb { R } ^ n \\ | \\ \\| x \\| ^ 2 = \\lambda \\| x _ v \\| ^ 2 + ( 1 - \\lambda ) \\| x _ w \\| ^ 2 \\} . \\end{align*}"}
+{"id": "3800.png", "formula": "\\begin{align*} \\begin{aligned} [ D _ * ^ q , M _ { z ^ q } ] & = D ^ q _ * M _ { z ^ q } ( f ) ( z ) - M _ { z ^ q } D _ * ^ q ( f ) ( z ) \\\\ & = D ^ q _ * ( z ^ q f ) ( z ) - z ^ q D _ * ^ q ( f ) ( z ) . \\end{aligned} \\end{align*}"}
+{"id": "3731.png", "formula": "\\begin{align*} \\nabla _ V \\big ( \\nabla _ Y \\hat X - \\omega ( Y ) \\big ) & = \\big ( \\nabla _ Y \\omega - R ( Y , \\hat X ) \\big ) V - T _ h ( V , Y ) \\\\ \\nabla _ V \\left ( \\nabla _ Y \\omega - R ( Y , \\hat X ) \\right ) & = - R \\big ( \\nabla _ Y \\hat X - \\omega ( Y ) , V \\big ) \\\\ & - R ( \\omega ( Y ) , V ) + \\nabla _ Y ( T _ h ( V ) ) + R ( V , Y ) \\omega \\\\ & + ( \\nabla _ { \\hat X } R ) ( V , Y ) - R ( Y , \\omega ( V ) ) . \\end{align*}"}
+{"id": "9242.png", "formula": "\\begin{align*} \\begin{aligned} \\det ( s I + A ) & = \\prod \\limits _ { k = 1 } ^ n ( s + \\lambda _ k ( A ) ) = \\sum \\limits _ { m = 0 } ^ n \\sum \\limits _ { 1 \\leq j _ 1 < \\dots < j _ m \\leq n } \\lambda _ { j _ 1 } ( A ) \\cdots \\lambda _ { j _ m } ( A ) s ^ { n - m } = \\sum \\limits _ { m = 0 } ^ n \\mathcal { H } _ m ( A ) s ^ { n - m } \\end{aligned} \\end{align*}"}
+{"id": "1203.png", "formula": "\\begin{align*} \\lim _ { j \\rightarrow \\infty } \\lim _ { n \\rightarrow \\infty } \\max \\{ | 2 | ^ { k } \\mu ( 0 , \\ 2 ^ { k + 1 } x ) ; j \\leq k < n + j \\} = 0 \\end{align*}"}
+{"id": "2210.png", "formula": "\\begin{align*} \\mathbf { v } ^ { 0 } = [ 1 , - 1 , 1 , - 1 , \\cdots ] ^ { T } . \\end{align*}"}
+{"id": "2731.png", "formula": "\\begin{align*} M ( \\Gamma ) = \\inf _ { \\rho \\in \\mathcal { F } ( \\Gamma ) } \\int _ { \\mathbb { R } ^ d } \\rho ^ d d m , \\end{align*}"}
+{"id": "1675.png", "formula": "\\begin{align*} G _ B ( R ) = ( B \\otimes _ F R ) ^ \\times \\end{align*}"}
+{"id": "1438.png", "formula": "\\begin{align*} & \\widetilde { G } _ n ( x , n + z ) = \\frac { n ^ { n d } e ^ { - n d } } { ( n ! ) ^ d } \\det ( e ^ { ( n - 1 ) L _ M ( 1 + z _ j / n - x _ i / n ) - ( z _ j - x _ i ) } ) ) _ { i , j = 1 } ^ d + O ( n ^ { - \\alpha } ) \\\\ & \\sim ( 2 \\pi n ) ^ { - d / 2 } \\det \\left ( \\sum _ { p = 0 } ^ M \\frac { 1 } { p ! } \\left [ ( n - 1 ) \\sum _ { k = 2 } ^ M \\frac { ( x _ i - z _ j ) ^ k } { k n ^ k } + \\frac { x _ i - z _ j } { n ^ 2 } \\right ] ^ p \\right ) _ { i , j = 1 } ^ d + O ( n ^ { - \\alpha } ) . \\end{align*}"}
+{"id": "2187.png", "formula": "\\begin{align*} \\underset { n \\rightarrow \\infty } { \\lim } \\mathbf { v } ^ { n } = \\underset { n \\rightarrow \\infty } { \\lim } { \\lambda } ^ { n } _ { 1 } \\alpha _ { 1 } \\mathbf { u _ { 1 } } , \\end{align*}"}
+{"id": "1324.png", "formula": "\\begin{align*} \\widehat { H } : = \\tau H ( x , \\xi / \\tau ) . \\end{align*}"}
+{"id": "7421.png", "formula": "\\begin{align*} | { \\sf N } _ 1 ^ { > 0 } ( i ; c ) | = \\left \\{ \\begin{array} { l l } | { \\sf N } _ 2 ^ { \\geq 0 } ( i ; c ) | & { \\rm f o r } \\ - \\ell + 1 \\leq c \\leq \\ell - 1 , \\ c \\neq 0 \\\\ & \\\\ | { \\sf N } _ 2 ^ { \\geq 0 } ( i ; c ) | - 1 & { \\rm f o r } \\ c = 0 . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "7955.png", "formula": "\\begin{align*} H ( f ; \\mu ) : = \\int _ { \\Omega _ N ^ d } f \\log f d \\mu = \\sum _ { \\eta \\in \\Omega _ N ^ d } f ( \\eta ) \\log f ( \\eta ) \\mu ( \\eta ) . \\end{align*}"}
+{"id": "7486.png", "formula": "\\begin{align*} \\eta : = | x | _ { A ^ { - 1 } ( 0 ) } ^ { - a / 2 } \\zeta , \\end{align*}"}
+{"id": "5709.png", "formula": "\\begin{align*} D / D = \\{ 1 \\} ^ { [ k ] } \\cup ( G \\setminus \\{ 1 \\} ) ^ { [ \\lambda ] } . \\end{align*}"}
+{"id": "9204.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\eta ( t ) & - \\eta _ a ( t ) \\| = \\| \\eta ( t ) - \\eta _ a ( t ) \\pm z ( t ) \\| \\\\ \\le & \\| \\eta ( t ) - z ( t ) \\| + \\| z ( t ) - \\eta _ a ( t ) \\| \\\\ \\le & \\| \\eta ( t ) - z ( t ) \\| + \\gamma \\| \\epsilon ( \\eta _ a ( t ) , t ) \\| \\\\ \\le & \\| \\eta ( t ) - z ( t ) \\| + \\gamma \\bar { k } _ 3 ( L _ r , \\delta ) . \\end{aligned} \\end{align*}"}
+{"id": "3160.png", "formula": "\\begin{align*} & J ( \\chi _ 8 , \\chi _ 8 ^ 6 ) = \\chi _ 8 ( - 1 ) J ( \\chi _ 8 , \\chi _ 8 ) , \\\\ & J ( \\chi _ 8 , \\chi _ 8 ^ 5 ) = \\chi _ 8 ( - 1 ) J ( \\chi _ 8 , \\chi _ 8 ^ 2 ) , \\\\ & J ( \\chi _ 8 ^ 5 , \\chi _ 8 ^ 6 ) = \\chi _ 8 ( - 1 ) \\overline { J ( \\chi _ 8 , \\chi _ 8 ) } . \\end{align*}"}
+{"id": "5166.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 } _ { + } } | \\nabla \\phi _ { 1 , n } | ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r - \\int _ { D ( 0 , R _ 0 ) } | \\nabla \\phi _ { n } | ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r & = \\int _ { D ( 0 , 2 R _ 0 ) \\backslash D ( 0 , R _ 0 ) } | \\nabla \\phi _ { n } \\chi _ { R _ 0 } + \\phi _ { n } \\nabla \\chi _ { R _ 0 } | ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r \\\\ & \\leq C \\int _ { D ( 0 , 2 R _ 0 ) \\backslash D ( 0 , R _ 0 ) } | \\nabla \\phi _ { n } | ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r . \\end{align*}"}
+{"id": "3536.png", "formula": "\\begin{align*} A _ k \\ & = \\ \\sum _ { j = 0 } ^ { k - 1 } ( 2 ^ { k - j } - 1 ) \\binom { k } { j } A _ j , \\\\ B _ k \\ & = \\ \\sum _ { j = 0 } ^ { k - 1 } ( 2 ^ { k - j } - 1 ) \\binom { k } { j } B _ j \\ \\ \\ + 1 . \\end{align*}"}
+{"id": "3488.png", "formula": "\\begin{align*} | \\phi ( 0 ) | \\leq & | G _ I ( x _ 1 , 0 ) | \\cdot | \\phi ( x _ 1 - 1 ) | + | G _ I ( x _ 2 , 0 ) | \\cdot | \\phi ( x _ 2 + 1 ) | \\\\ = & \\frac { | \\tilde { P } _ { x _ 2 } ( \\theta _ 1 ) | } { | \\tilde { P } _ I ( \\theta _ { x _ 1 } ) | } \\prod _ { j = x _ 1 } ^ 0 | \\cos ( \\pi ( \\theta _ j ) ) | \\cdot | \\phi ( x _ 1 - 1 ) | + \\frac { | \\tilde { P } _ { - x _ 1 } ( \\theta _ { x _ 1 } ) | } { | \\tilde { P } _ I ( \\theta _ { x _ 1 } ) | } \\prod _ { j = 0 } ^ { x _ 2 } | \\cos ( \\pi ( \\theta _ j ) ) | \\cdot | \\phi ( x _ 2 + 1 ) | . \\end{align*}"}
+{"id": "6029.png", "formula": "\\begin{align*} \\widetilde { E } _ { \\omega } : = \\inf _ { u \\in \\mathcal { M } _ { \\omega } } J _ { \\omega } ( u ) \\end{align*}"}
+{"id": "3142.png", "formula": "\\begin{align*} \\varrho ( [ x , y ] ) \\varrho ( x ) = 0 , \\quad \\forall \\ y \\in A . \\end{align*}"}
+{"id": "7847.png", "formula": "\\begin{align*} ( I - Q ^ \\dagger ) F ^ { q , \\dagger } _ v ( v + \\psi ^ \\dagger ( v , p ) , p ) [ v _ 1 + \\psi ^ \\dagger _ v ( v , p ) v _ 1 ] = 0 \\end{align*}"}
+{"id": "5262.png", "formula": "\\begin{align*} \\rho _ \\phi ( m ) \\rho _ \\phi ( n ) = \\sum _ { d | ( m , n ) } \\rho _ \\phi \\left ( \\frac { m n } { d ^ 2 } \\right ) . \\end{align*}"}
+{"id": "3372.png", "formula": "\\begin{align*} \\tilde { M } _ \\tau ^ \\dagger \\ , \\tilde { M } _ \\tau = \\begin{pmatrix} X ^ 2 & 0 \\\\ 0 & 0 \\end{pmatrix} + \\tau \\begin{pmatrix} 0 & e ^ { i \\varphi } X D \\\\ e ^ { - i \\varphi } D ^ \\dagger X & 0 \\end{pmatrix} + \\tau ^ 2 \\ : \\begin{pmatrix} D D ^ \\dagger + E & 0 \\\\ 0 & D ^ \\dagger D \\end{pmatrix} \\ : . \\end{align*}"}
+{"id": "2791.png", "formula": "\\begin{align*} s _ { r D } ( r x ) = s _ { D } ( x ) + \\gamma \\log r \\end{align*}"}
+{"id": "1904.png", "formula": "\\begin{align*} R _ { n + 1 } ( s ) = f ' \\bigl ( { \\xi _ n ( s ) } \\bigr ) R _ n ( s ) , \\end{align*}"}
+{"id": "3490.png", "formula": "\\begin{align*} \\sum _ { j \\neq k } \\ln | \\sin \\pi ( \\theta _ k - \\theta _ j ) | = \\sum _ { j \\in I _ 0 , j \\neq k } \\ln { | \\sin \\pi ( k - j ) \\alpha | } + \\sum _ { j \\in I _ { \\ell } } \\ln { | \\sin \\pi ( k - j ) \\alpha | } = : \\sum _ { 1 } + \\sum _ { 2 } . \\end{align*}"}
+{"id": "1689.png", "formula": "\\begin{align*} \\Delta _ n ( \\tau _ 1 , \\tau _ 2 ) \\ ! = \\ ! \\begin{bmatrix} \\frac { 2 } { h } U ^ \\top ( h + \\tau _ 2 ) A _ d \\\\ 2 A _ d ^ \\top U ( \\tau _ 1 - \\tau _ 2 ) A _ d \\\\ A _ d ^ { \\top } U ( \\tau _ 1 - \\tau _ 2 ) A _ d \\\\ \\frac { 1 } { h } I _ m \\end{bmatrix} . \\end{align*}"}
+{"id": "2820.png", "formula": "\\begin{align*} y _ { m } = \\mathbf { h } _ { m } ^ H \\mathbf { w } _ { m } x _ { m } + n _ { m } , \\end{align*}"}
+{"id": "6207.png", "formula": "\\begin{align*} B _ 3 = \\frac { 2 \\kappa ( 2 L + 3 ) \\Q [ ( 2 L + 1 ) \\Q ^ 2 - 6 \\kappa ] } { ( \\Q ^ 2 + 3 \\kappa ) ^ 2 } , B _ 4 = \\frac { \\kappa ( 2 L + 3 ) ^ 2 \\Q ^ 2 } { ( \\Q ^ 2 + 3 \\kappa ) ^ 2 } . \\end{align*}"}
+{"id": "4620.png", "formula": "\\begin{align*} B _ n = ( \\mathbb Z / 2 \\mathbb Z ) ^ n \\rtimes S _ n \\end{align*}"}
+{"id": "5026.png", "formula": "\\begin{align*} h ( T ) & = y q \\log ( y q ) + \\sum _ { k = 1 } ^ { r } \\left ( \\Delta ( i _ k ) + \\Delta ( j _ k ) \\right ) \\\\ & \\le y q \\log ( y q ) + \\sum _ { k = 1 } ^ { r } \\left ( \\Delta ( q + 1 ) + \\Delta ( k ) \\right ) \\\\ & = h ( T ' ) , \\end{align*}"}
+{"id": "1925.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty b _ k ( n ) q ^ n = \\prod _ { n = 1 } ^ \\infty \\frac { 1 - q ^ { k n } } { 1 - q ^ n } . \\end{align*}"}
+{"id": "6032.png", "formula": "\\begin{align*} \\widetilde { E } _ { \\omega } \\leq J _ { \\omega } ( u ) \\leq \\liminf _ { n \\rightarrow \\infty } J _ { \\omega } ( u _ { n } ) = \\widetilde { E } _ { \\omega } . \\end{align*}"}
+{"id": "6134.png", "formula": "\\begin{align*} ( \\lambda _ { S } ) _ { i j } : = \\delta _ { i , 0 } L _ { ( P _ S ) _ i } \\mbox { a n d } ( \\rho _ { S } ) _ { i j } : = \\delta _ { j , 0 } R _ { ( P _ S ) _ j } , \\end{align*}"}
+{"id": "524.png", "formula": "\\begin{align*} k _ { t } ( x , y ) = ( \\sinh 2 t ) ^ { - d / 2 } e ^ { \\Psi _ t ( x , y ) } , \\end{align*}"}
+{"id": "1245.png", "formula": "\\begin{align*} a c - b d = \\frac { 1 } { 2 } [ ( a - b ) ( c + d ) + ( a + b ) ( c - d ) ] , \\end{align*}"}
+{"id": "8704.png", "formula": "\\begin{align*} ( T _ { B _ 1 , 2 } , \\dots , T _ { B _ 1 , 1 6 } ) = ( 0 , - 1 0 , { - 3 } / { - 1 2 } / { - 2 1 } , - 1 0 ) . \\end{align*}"}
+{"id": "6782.png", "formula": "\\begin{align*} q = \\frac { a _ { 0 } ^ { \\prime \\prime } - a _ { 0 } ^ { \\prime } } { a _ { 0 } + 1 } \\end{align*}"}
+{"id": "5152.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 } _ { + } \\backslash Q } ( \\phi - \\phi _ { \\infty } ) _ { + } ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r = \\int _ { \\{ r \\geq R \\} } ( \\phi - \\phi _ { \\infty } ) _ { + } ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r + 2 \\int _ { \\{ z \\geq Z , r < R \\} } ( \\phi - \\phi _ { \\infty } ) _ { + } ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r . \\end{align*}"}
+{"id": "6969.png", "formula": "\\begin{align*} \\Psi _ { n , e } ^ { ( i ) } ( x , y ) = \\prod ^ { n e } _ { \\substack { j = 1 \\\\ ( j , n e ) > 1 \\\\ j \\equiv i \\bmod e } } ( x - \\zeta ^ j _ { n e } y ) . \\end{align*}"}
+{"id": "3061.png", "formula": "\\begin{align*} \\pi _ 1 ^ * ( d \\log \\varphi _ { \\lambda , z } ) = \\left ( \\sum _ { Z \\in \\mathcal I } m _ Z \\rho _ { Z } \\right ) \\frac { d g } { g } + \\eta \\end{align*}"}
+{"id": "4467.png", "formula": "\\begin{align*} & a = a _ { 3 } ^ { * } + a _ { 4 } ^ { * } n _ { 4 3 } ^ { 2 } b = 2 a _ { 4 } ^ { * } n _ { 4 3 } c = a _ { 4 } ^ { * } d = 2 a _ { 3 } ^ { * } n _ { 3 1 } x _ { 1 } + 2 a _ { 3 } ^ { * } n _ { 3 2 } x _ { 2 } + 2 a _ { 4 } ^ { * } n _ { 4 1 } n _ { 4 3 } x _ { 1 } + 2 a _ { 4 } ^ { * } n _ { 4 2 } n _ { 4 3 } x _ { 2 } \\\\ & e = 2 a _ { 4 } ^ { * } n _ { 4 3 } ( n _ { 4 1 } x _ { 1 } + n _ { 4 2 } x _ { 2 } ) f = Q ^ { * } ( x _ { 1 } , x _ { 2 } , 0 , 0 ) - n . \\end{align*}"}
+{"id": "1806.png", "formula": "\\begin{align*} F : [ 0 , \\infty ) \\to [ 0 , \\infty ) \\ , \\ , \\ , \\ , \\ , C ^ 2 \\ , \\ , \\ , F ( 0 ) = 0 . \\end{align*}"}
+{"id": "2502.png", "formula": "\\begin{align*} \\hat { \\mathbf { D } } = \\begin{pmatrix} \\begin{aligned} & \\mathbf { D } & 0 \\\\ & 0 & 1 \\end{aligned} \\end{pmatrix} . \\end{align*}"}
+{"id": "4750.png", "formula": "\\begin{align*} & M i n i m i z e _ { x \\in \\mathbb { R } ^ n } \\ , T r ( A _ J , X ) + 2 b _ J ^ T x \\\\ & s . t . \\ \\ T r ( A _ k , X ) + 2 b _ k ^ T x + c _ k \\le 0 \\ , k = 1 , . . . , m \\\\ & \\begin{pmatrix} 1 & x ^ T \\\\ x & X \\end{pmatrix} \\succeq 0 \\end{align*}"}
+{"id": "6116.png", "formula": "\\begin{align*} \\mu _ m ^ { ( 1 2 ) } \\big ( \\mathfrak { s } \\boldsymbol { J } \\big ) = \\mu _ m ^ { ( 1 2 3 ) } \\big ( \\boldsymbol { J } \\big ) \\mu _ m ^ { ( 1 2 3 ) } \\big ( \\mathfrak { s } \\boldsymbol { J } \\big ) = \\mu _ m ^ { ( 1 2 ) } \\big ( \\boldsymbol { J } \\big ) . \\end{align*}"}
+{"id": "2162.png", "formula": "\\begin{align*} P ( a , b , X , Y ) = \\frac { 1 } { 2 } a ( b ^ 2 + 4 ) | X | ^ 2 \\Im ( X ) + \\frac { 5 } { 2 } a ^ 2 b \\Im ( X ^ 2 \\overline { Y } ) - b ^ 2 | X | ^ 3 \\\\ + 4 a ( 1 - b ^ 2 ) | X Y | \\Im ( X ) + ( 4 a ^ 2 + 3 b ^ 2 ) | X ^ 2 Y | \\\\ - 1 0 a b | X | \\Re ( X \\overline { Y } ) + \\frac { 1 } { 2 } b ( 4 + a ^ 2 + 5 b ^ 2 ) | X | ^ 2 \\Im ( Y ) , \\end{align*}"}
+{"id": "5632.png", "formula": "\\begin{align*} g _ \\xi ( z ) = \\begin{cases} 0 & z \\in U ^ c , \\\\ \\frac { g ( z ) } { \\xi - z } & z \\in U \\end{cases} \\end{align*}"}
+{"id": "2704.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { T - 1 } - \\Theta _ k \\Lambda _ k ' + ( 1 - \\Theta _ k ) \\Lambda _ k \\le - \\frac { \\zeta _ 0 } { \\log \\gamma } = \\max \\left \\{ \\log _ \\gamma \\frac { \\bar \\Delta ' } { \\delta _ 0 } , 0 \\right \\} . \\end{align*}"}
+{"id": "6940.png", "formula": "\\begin{align*} M _ n ( Q ) & : = \\int _ { \\R ^ n } f \\ , d Q - H ( Q ) \\\\ & = \\sum _ { i = 1 } ^ n \\int _ { \\R } V ( x ) \\ , Q _ i ( d x ) + \\frac 1 2 \\sum _ { i , j = 1 } ^ n J _ { i j } \\int _ { \\R } \\int _ { \\R } K ( x - y ) \\ , Q _ i ( d x ) Q _ j ( d y ) - \\sum _ { i = 1 } ^ n H ( Q _ i ) , \\end{align*}"}
+{"id": "3365.png", "formula": "\\begin{align*} N e w t o n ( F _ t ) & = C o n v ( \\bigcup \\limits _ { i = 0 } ^ m N e w t o n ( s _ { \\Lambda ^ { ( i ) } } ) ) \\\\ & = t C o n v ( \\bigcup \\limits _ { i = 0 } ^ m N e w t o n ( s _ { \\lambda ^ { ( i ) } } ) ) \\\\ & = t N e w t o n ( F ) . \\end{align*}"}
+{"id": "8543.png", "formula": "\\begin{align*} \\begin{aligned} & \\| u ( x , t _ 2 ) - u ( x , t _ 1 ) \\| _ { H ^ 3 \\cap H ^ { 2 , 1 } } \\leq c \\| r _ { 1 , 2 } ( z ) ( e ^ { 2 i \\eta ^ 2 t _ 2 } - e ^ { 2 i \\eta ^ 2 t _ 1 } ) \\| _ { H ^ 1 \\cap H ^ { 2 , 1 } } \\\\ & \\leq c | t _ 2 - t _ 1 | \\| r _ { 1 , 2 } ( z ) \\| _ { H ^ 1 \\cap L ^ { 2 , 1 } } , \\end{aligned} \\end{align*}"}
+{"id": "7526.png", "formula": "\\begin{align*} H = \\left \\{ S \\in V ^ { ( k - 1 ) } : \\ : \\deg _ { G } ( S ) \\geq \\sqrt { d } \\right \\} , \\end{align*}"}
+{"id": "5132.png", "formula": "\\begin{align*} j ( \\tau , s ) = H [ b + \\tau \\tilde { b } + s b _ 0 ] , \\tau , s \\in \\mathbb { R } . \\end{align*}"}
+{"id": "477.png", "formula": "\\begin{align*} \\mu _ { r _ 1 } \\ast \\mu _ { r _ 3 } ( d x ) : = e ^ { - c _ { 1 3 } } \\delta _ 0 ( d x ) + ( 1 - e ^ { - c _ { 1 3 } } ) f _ { 1 3 } ( x ) d x , \\end{align*}"}
+{"id": "430.png", "formula": "\\begin{align*} a _ { i } = Q _ { i } \\kappa . \\end{align*}"}
+{"id": "1864.png", "formula": "\\begin{align*} 1 6 \\ , \\mathcal A = 1 6 \\ , a \\cdot b \\le ( 2 a + 2 b ) ^ 2 \\ = { \\mathcal L } ^ 2 \\end{align*}"}
+{"id": "7944.png", "formula": "\\begin{align*} c = \\prod _ { i = 1 } ^ { n - 1 } \\norm { \\eta _ i } _ { H ^ 1 } ^ 2 \\left ( 1 + ( n + 3 ) \\left ( \\frac { n + 6 } { 4 } + \\frac 1 4 \\norm { \\tilde \\Psi } _ { H ^ 1 } \\right ) \\right ) \\end{align*}"}
+{"id": "4592.png", "formula": "\\begin{align*} g ( x - k \\gamma ) & = g ( x ) - \\gamma \\sum _ { j = 1 } ^ k \\frac { g ( x - ( j - 1 ) \\gamma ) - g ( x - j \\gamma ) } { \\gamma } \\\\ & < g ( x ) - \\gamma \\sum _ { j = 1 } ^ k ( \\Lambda + \\varepsilon ) ^ { - j } \\frac { \\Lambda - 1 + \\delta } { \\gamma } g ( x ) \\\\ & = g ( x ) \\left ( 1 - ( \\Lambda - 1 + \\delta ) \\sum _ { j = 1 } ^ k ( \\Lambda + \\varepsilon ) ^ { - j } \\right ) \\\\ & < 0 , \\end{align*}"}
+{"id": "8979.png", "formula": "\\begin{align*} \\dfrac { \\hat { k } _ 2 ( \\Delta ) } { \\hat { k } _ 2 ( \\Psi ) } = \\dfrac { 1 } { 1 - x _ { \\sigma } R _ { \\sigma } } = \\dfrac { x _ { 2 2 2 } ^ * } { x _ { 1 1 1 } } = \\dfrac { ( x _ { 1 , 1 } + x _ { 1 , 2 } ) ( x _ { 2 , 1 } + x _ { 2 , 2 } ) ( x _ { 3 , 1 } + x _ { 3 , 2 } ) } { x _ { 1 , 1 } x _ { 2 , 1 } x _ { 3 , 1 } } , \\end{align*}"}
+{"id": "269.png", "formula": "\\begin{align*} \\varphi _ n ( p ) = \\frac { 1 } { n } \\langle p , \\xi \\rangle + n \\left ( ( x _ p y _ \\xi - y _ p x _ \\xi ) ^ 2 + z _ p ^ 2 \\right ) , p \\in \\Omega . \\end{align*}"}
+{"id": "9107.png", "formula": "\\begin{align*} ( \\sigma _ 1 , \\dots , \\sigma _ { n - 1 } ) \\subset \\bigoplus _ { i = 1 } ^ { n - 1 } ( E _ { i , q _ i } , E _ { i + 1 , q _ i ' } ) . \\end{align*}"}
+{"id": "5730.png", "formula": "\\begin{align*} b = r ^ { q + 1 } + r = \\left ( \\frac { h } { h + h ^ q } \\right ) ^ { q + 1 } + \\frac { h } { h + h ^ q } = \\frac { h ^ { q + 1 } + h ( h ^ q + h ) ^ q } { ( h ^ q + h ) ^ { q + 1 } } = \\frac { h ^ { q ^ 2 + 1 } } { ( h ^ q + h ) ^ { q + 1 } } . \\end{align*}"}
+{"id": "1476.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\int _ { a } ^ { b } | u _ { n } ' ( x ) | \\ , d x - | D z _ { n } | ( ( a , b ) ) = 0 . \\end{align*}"}
+{"id": "3564.png", "formula": "\\begin{align*} \\varphi ( - q ^ { 1 / 2 } ) ^ 2 G ( - q ^ { 1 / 2 } ) - \\varphi ( q ^ { 1 / 2 } ) ^ 2 G ( q ^ { 1 / 2 } ) = 2 q ^ { 1 / 2 } \\varphi ( q ^ 2 ) ^ 2 H ( q ^ 2 ) . \\end{align*}"}
+{"id": "4203.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( K _ H ( x ) \\nabla v ^ \\tau _ \\varepsilon | \\nabla v ^ \\tau _ \\varepsilon ) d x = & 2 \\pi q ^ 2 \\sqrt { ( K _ { H } ) _ { 1 1 } ( K _ { H } ) _ { 2 2 } } ( \\bar { x } ) \\ln \\frac { 1 } { \\varepsilon } + O ( 1 ) + C | \\ln \\tau | \\\\ = & 2 \\pi q ^ 2 \\sqrt { d e t ( K _ { H } ) } ( \\bar { x } ) \\ln \\frac { 1 } { \\varepsilon } + O ( 1 ) + C | \\ln \\tau | , \\end{align*}"}
+{"id": "8752.png", "formula": "\\begin{align*} \\mathcal D _ { \\psi , z ' } ( g ) = \\left ( \\frac { d } { d \\bar z } g \\circ T ^ { - 1 } \\right ) \\circ T . \\end{align*}"}
+{"id": "1478.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } | J _ { n } ( i ) | \\leq \\left ( \\sum _ { i = 1 } ^ { \\infty } | J _ { n } ( i ) | ^ { 1 / 2 } \\right ) ^ { 2 } \\leq M ^ { 2 } \\forall n \\geq 1 . \\end{align*}"}
+{"id": "4032.png", "formula": "\\begin{align*} M _ I : = M _ { \\varrho ( I ) } \\textrm { a n d } R _ I : = R _ { \\varrho ( I ) } . \\end{align*}"}
+{"id": "4410.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\left | \\frac { \\theta _ 0 ( e ^ { \\kappa + \\tau } ) - \\theta _ 0 ( e ^ \\tau ) - \\delta e ^ \\tau } { e ^ { \\sigma \\tau } } \\right | ^ 2 d \\tau = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ \\infty \\left | \\frac { e ^ { \\kappa s } - 1 } { s } \\right | ^ 2 \\left | \\frac { \\zeta ' } { \\zeta } ( s ) + g ( s ) \\right | ^ 2 d t . \\end{align*}"}
+{"id": "7171.png", "formula": "\\begin{align*} A ^ { - 1 } \\mathcal { L } _ g = \\Bigl ( I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } + B - Q \\Bigr ) \\Bigl ( I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } + Q \\Bigr ) \\end{align*}"}
+{"id": "3258.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ] } | ( 1 . 1 ) _ { n , N } | \\leq \\Delta _ n ^ { \\frac 1 2 } \\sum _ { j = 1 } ^ { N - 1 } \\int _ 0 ^ T \\| \\Sigma _ s \\| _ { } d s \\| h \\| \\| g \\| \\sup _ { t \\in [ 0 , T ] } \\| \\mathcal S ( t ) \\| _ { } ^ 2 , \\end{align*}"}
+{"id": "3078.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ n a _ { i j } = k , \\forall \\ , i = 0 , \\dotsc , n . \\end{align*}"}
+{"id": "6200.png", "formula": "\\begin{align*} \\xi = - L - 1 , \\eta = \\frac { Q } { 2 ( L + 1 ) } - \\sqrt { \\kappa B _ 4 } , \\zeta = \\kappa \\left ( \\frac { B _ 3 } { 2 \\sqrt { \\kappa B _ 4 } } + 1 \\right ) , \\sigma = \\sqrt { \\kappa B _ 4 } , \\end{align*}"}
+{"id": "2050.png", "formula": "\\begin{align*} \\bar a _ { i , j } \\big ( \\sqrt { \\mu } \\partial _ { v _ j } g \\big ) = a _ { i , j } * \\Big ( \\partial _ { v _ j } ( \\sqrt { \\mu } g ) - g \\partial _ { v _ j } \\sqrt { \\mu } \\Big ) = \\big ( \\partial _ { v _ j } a _ { i , j } \\big ) * \\big ( \\sqrt { \\mu } g \\big ) + \\frac 1 2 a _ { i , j } * \\big ( v _ j \\sqrt { \\mu } g \\big ) , \\end{align*}"}
+{"id": "33.png", "formula": "\\begin{align*} \\nabla _ { H } u = \\sum _ { i = 1 } ^ m X _ i u X _ i \\end{align*}"}
+{"id": "9247.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial u } { \\partial \\overline { q _ l } } = \\partial _ { x _ { 4 l } } + \\mathbf { i } \\partial _ { x _ { 4 l + 1 } } + \\mathbf { j } \\partial _ { x _ { 4 l + 2 } } + \\mathbf { k } \\partial _ { x _ { 4 l + 3 } } , \\end{aligned} \\end{align*}"}
+{"id": "1598.png", "formula": "\\begin{align*} \\mathbf { \\Phi } _ { \\mathrm { r } , g } ^ { H } \\mathbf { \\Phi } _ { \\mathrm { r } , g } + \\mathbf { \\Phi } _ { \\mathrm { t } , g } ^ { H } \\mathbf { \\Phi } _ { \\mathrm { t } , g } = \\mathbf { I } _ { \\bar { M } } , \\forall g \\in \\mathcal { G } . \\end{align*}"}
+{"id": "7098.png", "formula": "\\begin{align*} \\partial _ t u + ( u \\cdot \\nabla ) \\ , u & = - \\nabla p , \\\\ \\nabla \\cdot u & = 0 \\end{align*}"}
+{"id": "4880.png", "formula": "\\begin{align*} \\begin{cases} \\dim H ^ 1 ( S , T _ S ) = 6 8 ; \\\\ \\dim H ^ 2 ( S , T _ S ) = 6 . \\end{cases} \\end{align*}"}
+{"id": "5685.png", "formula": "\\begin{align*} \\varphi ( f \\circ g ) = ( ( f \\circ g ) _ { 1 1 } , \\ldots , ( f \\circ g ) _ { n n } ) = ( f _ { 1 1 } \\circ g _ { 1 1 } , \\ldots , f _ { n n } \\circ g _ { n n } ) \\\\ = ( f _ { 1 1 } , \\ldots , f _ { n n } ) \\circ ( g _ { 1 1 } , \\ldots , g _ { n n } ) = \\varphi ( f ) \\circ \\varphi ( g ) . \\end{align*}"}
+{"id": "6716.png", "formula": "\\begin{align*} \\pi _ { \\lambda , c _ 1 } ' ( \\prod _ { i = 1 } ^ g [ \\alpha _ i , \\beta _ i ] \\prod _ { j = 1 } ^ n \\gamma _ j ) = \\pm \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "5391.png", "formula": "\\begin{align*} u _ { n n } \\sigma _ { k - 1 } ( b ) - \\sum _ { \\alpha = 1 } ^ { n - 1 } u _ { \\alpha n } ^ 2 \\sigma _ { k - 2 ; \\alpha } ( b ) + \\sigma _ k ( b ) = f . \\end{align*}"}
+{"id": "1002.png", "formula": "\\begin{align*} \\ell ( x ) + 1 = \\ell ( x ' ) \\geq \\ell ( x ) + d ( v ' \\Rightarrow v ) + d ( w v \\Rightarrow w ' v ' ) \\geq \\ell ( x ) + 1 . \\end{align*}"}
+{"id": "7339.png", "formula": "\\begin{align*} 1 _ { B - A } * 1 _ { A - C } ( b - c ) & = \\int _ { \\R ^ n } 1 _ { B - A } ( z ) 1 _ { A - C } ( b - c - z ) d z = \\int _ { \\R ^ n } 1 _ { B - A } ( b - y ) 1 _ { A - C } ( y - c ) d y \\\\ & \\ge \\int _ { A } 1 _ { B - A } ( b - y ) 1 _ { A - C } ( y - c ) d y = | A | . \\end{align*}"}
+{"id": "6538.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\pi ( a ) - 1 } q _ i \\neq \\prod _ { i = 1 } ^ { \\pi ( a ) - 1 } p _ i ^ { \\alpha _ i } . \\end{align*}"}
+{"id": "8787.png", "formula": "\\begin{align*} D ( x ) : = x \\cdot A x . \\end{align*}"}
+{"id": "5392.png", "formula": "\\begin{align*} \\{ u _ { \\alpha \\beta } ( 0 ) \\} _ { 1 \\leq \\alpha , \\beta \\leq n - 1 } = \\mathrm { d i a g } \\{ b _ 1 , \\ldots , b _ { n - 1 } \\} . \\end{align*}"}
+{"id": "5329.png", "formula": "\\begin{align*} f ( z , t ) = \\sum _ { ( p , q ) \\in \\mathbb { N } ^ 2 } \\sum _ { j = 1 } ^ { d ( p , q ) } g _ { ( p , q ) , j } ( r , t ) \\ , P _ { p , q } ^ j ( z ) , \\ , \\ , \\ , z = r \\omega \\end{align*}"}
+{"id": "5412.png", "formula": "\\begin{align*} u ( x ) \\geq \\left ( \\frac { 1 } { 2 } - C \\delta \\right ) \\sum _ { \\beta = \\alpha + 1 } ^ { n - 1 } b _ \\beta x _ \\beta ^ 2 - C b _ \\alpha | \\hat { x } | ^ 2 \\geq \\frac { 1 } { 4 } \\sum _ { \\beta = \\alpha + 1 } ^ { n - 1 } b _ \\beta x _ \\beta ^ 2 - C b _ \\alpha | \\hat { x } | ^ 2 \\end{align*}"}
+{"id": "2177.png", "formula": "\\begin{align*} v ( x ) & = \\underbrace { v ( x _ 0 ) } _ { = 0 } + \\nabla v ( x _ 0 ) \\cdot ( x - x _ 0 ) + o \\left ( \\abs { x - x _ 0 } \\right ) \\\\ [ 7 p t ] & = - \\frac { \\partial v } { \\partial \\nu } ( x _ 0 ) \\abs { x - x _ 0 } + o \\left ( \\abs { x - x _ 0 } \\right ) \\\\ [ 5 p t ] & \\ge \\tau \\abs { x - x _ 0 } + o \\left ( \\abs { x - x _ 0 } \\right ) \\\\ [ 5 p t ] & > \\frac { \\tau } { 2 } \\abs { x - x _ 0 } = \\frac { \\tau } { 2 } d ( x , \\partial \\Omega ) \\end{align*}"}
+{"id": "5479.png", "formula": "\\begin{align*} \\kappa _ { p , d } = 2 ^ { 1 - p } \\frac { \\Gamma \\left ( \\frac { d - p } { 2 } \\right ) } { \\Gamma \\left ( \\frac { d } { 2 } \\right ) \\Gamma \\left ( \\frac { p } { 2 } \\right ) } . \\end{align*}"}
+{"id": "780.png", "formula": "\\begin{align*} V & = \\big ( G ( c ) - G ' ( c ) c \\big ) H , \\\\ \\partial ^ \\square c & = \\Delta _ \\Gamma \\big ( G ' ( c ) \\big ) + c H V . \\end{align*}"}
+{"id": "1141.png", "formula": "\\begin{align*} x _ { N ( \\ell - 1 ) + j } = \\max _ { 1 \\leq q \\leq N L } x _ q , y _ { N ( k - 1 ) + i } = \\max _ { 1 \\leq q \\leq N L } y _ q . \\end{align*}"}
+{"id": "2294.png", "formula": "\\begin{align*} \\mu _ 1 \\prec \\mu _ 2 \\Longleftrightarrow \\mu _ 2 - \\mu _ 1 = \\alpha _ 1 + \\dots + \\alpha _ m \\alpha _ 1 , \\dots , \\alpha _ m \\in \\Phi _ + . \\end{align*}"}
+{"id": "4741.png", "formula": "\\begin{align*} & b _ k ^ T \\Tilde { x } \\le b _ k ^ T ( \\lambda x _ v + ( 1 - \\lambda ) x _ w ) , \\\\ & b _ k ^ T ( \\Tilde { x } - ( \\lambda x _ v + ( 1 - \\lambda ) x _ w ) ) \\le 0 . \\end{align*}"}
+{"id": "2891.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n \\sum _ { x = 0 } ^ n \\int _ { 0 } ^ { + \\infty } \\varphi \\left ( t , \\frac x n \\right ) \\mathcal E _ x ( n ^ 2 t ) \\dd t = \\int _ { 0 } ^ { + \\infty } \\int _ 0 ^ 1 \\varphi ( t , u ) T ( t , u ) \\dd t \\dd u , \\quad \\mbox { i n p r o b a b i l i t y , } \\end{align*}"}
+{"id": "6369.png", "formula": "\\begin{align*} q _ { i } = & \\frac { 1 } { i ^ { 2 } + ( a - 2 ) i + ( 1 - a ) } \\left \\{ ( 1 - \\delta _ { i , 2 } ) \\sum _ { r = 2 } ^ { i - 1 } q _ { r } q _ { i + 1 - r } [ r ( 1 - a ) ( i - r ) - r ( r - 1 ) ] + \\right . \\\\ & \\left . \\sum _ { r = 1 } ^ { i - 1 } \\sum _ { s = 1 } ^ { i - r } q _ { r } q _ { s } q _ { i + 1 - r - s } [ r ( r - a ) + s ( a + b - 2 ) ( i + 1 - r - s ) ] \\right \\} , \\end{align*}"}
+{"id": "2228.png", "formula": "\\begin{align*} C _ { R P I - P R } = O \\{ ( \\beta + 8 ) N ^ 2 + N K ( 2 N + 3 ) - 1 1 N + 4 \\} \\end{align*}"}
+{"id": "7049.png", "formula": "\\begin{align*} \\aligned 0 & \\ , = \\ , - \\ , \\tfrac { 1 } { 1 0 } \\ , F _ { 5 , 0 } + \\tfrac { 1 } { 1 2 0 } \\ , F _ { 6 , 0 } + \\tfrac { 1 } { 3 6 } \\ , F _ { 5 , 1 } , \\\\ 0 & \\ , = \\ , - \\tfrac { 7 } { 1 2 0 } \\ , F _ { 5 , 0 } + \\tfrac { 1 } { 1 2 0 } \\ , F _ { 5 , 1 } + \\tfrac { 2 } { 2 7 } , \\endaligned \\end{align*}"}
+{"id": "4529.png", "formula": "\\begin{align*} \\begin{cases} v _ i ( x , t ) = S _ { d , i } ( t , 0 ) v _ { i , 0 } ( x ) & \\forall \\ : i \\in [ 1 , p ] , \\\\ v _ i ( x , t ) = S _ { d , i } ( t , t _ { i , e x } ( x , t ) ) S _ { c , i } ( t _ { i , e x } , t _ { i , e n } ( x , t ) ) S _ { d , i } ( t _ { i , e n } ( x , t ) , 0 ) v _ { i , 0 } ( x ) & \\forall \\ : i \\in [ p + 1 , n ] ; \\end{cases} \\end{align*}"}
+{"id": "7615.png", "formula": "\\begin{align*} f ( u , y ) = \\mp \\sum _ { u < v < y } f ( u , v ) f ( v , y ) . \\end{align*}"}
+{"id": "6169.png", "formula": "\\begin{align*} \\hat { H } ' _ { 1 , 2 } = \\hat { \\pi } _ r ^ 2 + V ' _ { 1 , 2 } ( r ) + E ' _ 0 , V ' _ { 1 , 2 } ( r ) = W ^ { \\prime 2 } ( r ) \\mp f ( r ) \\frac { d W ' } { d r } , \\end{align*}"}
+{"id": "4057.png", "formula": "\\begin{align*} u _ n ( A ' ) = n ^ { 2 H - 1 / 2 } \\sum _ { j \\in [ n ] } A ' _ n ( j ) I _ 1 ( 1 _ j ) 1 _ j . \\end{align*}"}
+{"id": "4016.png", "formula": "\\begin{align*} D = \\{ c \\} \\cup \\{ d \\in C _ i \\mid \\varphi _ { \\sigma , s } ^ L ( d ) = a \\} \\end{align*}"}
+{"id": "8610.png", "formula": "\\begin{align*} d ( z _ n , H _ n ) ^ 2 = | z _ n - h _ n | ^ 2 = \\| P _ E z _ n - P _ E h _ n \\| ^ 2 + \\| P _ { E ^ \\bot } z _ n - P _ { E ^ \\bot } h _ n \\| ^ 2 . \\end{align*}"}
+{"id": "1198.png", "formula": "\\begin{align*} f ( x + y ) = \\frac { f ( x ) f ( y ) } { f ( x ) + f ( y ) } \\ \\ \\ ( x , y \\in \\mathbb { R } - \\{ 0 \\} ) . \\end{align*}"}
+{"id": "1607.png", "formula": "\\begin{align*} \\triangledown \\tilde { f } _ g ( \\mathbf { \\Phi } _ { g } ) = 2 \\mathbf { Z } _ { g } \\mathbf { \\Phi } _ { g } \\mathbf { Y } _ { g } - 2 \\mathbf { X } _ { g } ^ H , \\forall g \\in \\mathcal { G } . \\end{align*}"}
+{"id": "5696.png", "formula": "\\begin{align*} F ( X ) = \\sum _ { i = 0 } ^ { 2 ^ n - 1 } A _ i X ^ i . \\end{align*}"}
+{"id": "5074.png", "formula": "\\begin{align*} \\mathcal { I } _ { h } = \\begin{cases} \\ h \\mathcal { I } _ { 1 } & h > 0 , \\\\ \\ - h \\mathcal { I } _ { - 1 } & h < 0 , \\\\ \\ 0 & h = 0 , \\end{cases} S _ { h } = \\begin{cases} \\ h ^ { 1 / 2 } S _ { 1 } & h > 0 , \\\\ \\ ( - h ) ^ { 1 / 2 } S _ { - 1 } & h < 0 , \\\\ \\ \\emptyset & h = 0 . \\end{cases} \\end{align*}"}
+{"id": "2658.png", "formula": "\\begin{align*} \\sum \\limits _ { n = 0 } ^ { \\infty } g _ { o } ( n , \\alpha , k , p ) - g _ { e } ( n , \\alpha , k , p ) q ^ { n } = \\frac { ( q ^ { k } ; q ^ { k } ) _ { \\infty } } { ( q ; q ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { p n + \\alpha } } { 1 + q ^ { p n + \\alpha } } . \\end{align*}"}
+{"id": "3451.png", "formula": "\\begin{align*} \\phi ( m _ n + 1 ) + \\phi ( m _ n - 1 ) + \\lambda \\tan ( \\pi \\theta _ { m _ n } ) \\phi ( m _ n ) = E \\phi ( m _ n ) \\end{align*}"}
+{"id": "7195.png", "formula": "\\begin{align*} \\operatorname { T r } \\textbf { \\textit { K } } ( t , x ^ { \\prime } , x ^ { \\prime } ) = \\frac { 1 } { ( 2 \\pi ) ^ { n - 1 } } \\int _ { \\mathbb { R } ^ { n - 1 } } \\bigg ( \\frac { i } { 2 \\pi } \\int _ { \\mathcal { C } } e ^ { - t \\tau } \\sum _ { j \\leqslant - 1 } \\operatorname { T r } \\phi _ { j } \\ , d \\tau \\bigg ) \\ , d \\xi . \\end{align*}"}
+{"id": "1606.png", "formula": "\\begin{align*} \\mathsf { T } _ { \\mathbf { \\Phi } _ { g } } \\mathcal { M } _ g = \\{ \\mathbf { T } _ { g } \\in \\mathbb { C } ^ { 2 \\bar { M } \\times \\bar { M } } : \\Re \\{ \\mathbf { \\Phi } _ { g } ^ H \\mathbf { T } _ { g } \\} = \\mathbf { 0 } _ { \\bar { M } } \\} , \\forall g \\in \\mathcal { G } . \\end{align*}"}
+{"id": "7189.png", "formula": "\\begin{align*} p _ 1 & = A q _ 1 - d _ 1 , \\\\ p _ 0 & = A q _ 0 - d _ 0 , \\\\ p _ { - m } & = A q _ { - m } , m \\geqslant 1 , \\end{align*}"}
+{"id": "1018.png", "formula": "\\begin{align*} & \\prescript L { } { } \\ell { } ^ R ( x , \\cdot ) : \\Phi \\rightarrow \\mathbb Z , \\alpha \\mapsto \\prescript L { } { } \\ell { } ^ R ( x , \\alpha ) , \\\\ & \\prescript L { } { } \\ell { } ^ R ( x , \\alpha ) : = \\langle \\mu , \\alpha \\rangle + \\chi _ R ( \\alpha ) - \\chi _ L ( w \\alpha ) . \\end{align*}"}
+{"id": "8350.png", "formula": "\\begin{align*} a ( - k ) = a ( k ) , k \\in \\overline { D } _ + , b ( - k ) = - b ( k ) , \\operatorname { I m } k ^ { 2 } = 0 . \\end{align*}"}
+{"id": "3706.png", "formula": "\\begin{align*} h _ { i j } & = \\tfrac { 2 } { n - 2 } c _ { i j } | x | ^ { 2 - n } + O ^ { 2 , \\alpha } ( | x | ^ { \\max \\{ - 2 q , 1 - n \\} } ) \\\\ v & = - c | x | ^ { 2 - n } + O ^ { 2 , \\alpha } ( | x | ^ { \\max \\{ - 2 - q , 1 - n \\} } ) . \\end{align*}"}
+{"id": "4568.png", "formula": "\\begin{align*} a _ { \\ell + 2 , m + 1 , 0 } + q a _ { \\ell + 1 , m + 2 , 0 } - \\frac { \\lambda _ 1 } { q } a _ { \\ell + 1 , m + 1 , 0 } + \\frac { \\lambda _ 3 } { q } a _ { \\ell , m , 0 } - q a _ { \\ell - 1 , m , 0 } - a _ { \\ell , m - 1 , 0 } = 0 . \\end{align*}"}
+{"id": "6827.png", "formula": "\\begin{align*} d = \\left ( \\Lambda ^ { - \\frac { 1 } { 2 } } ( s ) \\beta ( s ) \\right ) \\beta ^ { \\frac { 1 } { 2 } } ( s ) = \\left ( \\frac { \\beta ^ { 3 } ( s ) } { \\Lambda ( s ) } \\right ) ^ { \\frac { 1 } { 2 } } = \\frac { 1 } { 4 \\sqrt [ 4 ] { t / 2 } } . \\end{align*}"}
+{"id": "8709.png", "formula": "\\begin{align*} \\varphi _ { | N } = \\frac { 1 } { n } ( \\psi _ 1 + \\cdots + \\psi _ n ) \\end{align*}"}
+{"id": "8672.png", "formula": "\\begin{align*} { { \\bf { \\bar X } } ^ { \\star } } = \\left [ { { { \\bf { W } } ^ { \\star } } , { { \\bf { 0 } } _ { \\left ( { \\sum { { { \\bar r } _ { l ' } } } } \\right ) \\times \\left ( { { M _ t } - K } \\right ) } } } \\right ] , \\end{align*}"}
+{"id": "7467.png", "formula": "\\begin{align*} \\int | h _ X | ^ 2 w _ U & = \\int h _ X \\overline { h _ X w _ U } = \\int \\widehat { h _ X } \\overline { \\widehat { h _ X } * \\widehat { w _ U } } \\end{align*}"}
+{"id": "5548.png", "formula": "\\begin{align*} ( a \\wedge b ) \\cdot c & = ( a \\cdot b ) \\ : \\cdot \\ : ( a \\cdot c ) \\\\ a \\cdot ( b \\wedge c ) & = ( a \\cdot b ) \\wedge ( a \\cdot c ) . \\end{align*}"}
+{"id": "180.png", "formula": "\\begin{align*} V ^ { H ' } = \\varinjlim _ { n } X _ { H ' , n } \\widehat { \\otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V ) \\oplus \\varinjlim _ { n } A _ { H ' , n } \\widehat { \\otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V ) . \\end{align*}"}
+{"id": "2321.png", "formula": "\\begin{align*} R ^ \\nabla _ { X , Y } Z = R ^ W _ { X , Y } Z & - \\frac { 1 } { 2 } ( d J \\theta ) ( X , Y ) J Z - \\frac { 1 } { 2 } ( d \\theta ) ( X , Y ) Z \\\\ & - \\frac { 1 } { 2 } \\left ( D ^ W _ X \\left ( D ^ W _ Y J \\right ) - D ^ W _ Y \\left ( D ^ W _ X J \\right ) - D ^ W _ { [ X , Y ] } J \\right ) J Z \\\\ & + \\frac { 1 } { 4 } \\left ( \\left ( D ^ W _ X J \\right ) \\left ( D ^ W _ Y J \\right ) - \\left ( D ^ W _ Y J \\right ) \\left ( D ^ W _ X J \\right ) \\right ) Z . \\end{align*}"}
+{"id": "993.png", "formula": "\\begin{align*} L & \\le ( \\theta _ { d + 1 } - | W | ) ( | W | - q ^ { d - 1 } ) \\\\ & = ( | W | - q ) q ^ { d + 1 } - ( | W | - y ) ^ 2 + y ^ 2 - \\theta _ { d + 1 } q ^ { d - 1 } + q ^ { d + 2 } \\end{align*}"}
+{"id": "7996.png", "formula": "\\begin{align*} V ' _ i ( u ) [ v ] = 4 B _ i ( u ^ 2 , u v ) , \\ \\forall u , v \\in X , \\ i = 0 , 1 , 2 . \\end{align*}"}
+{"id": "3637.png", "formula": "\\begin{align*} 2 \\frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \\kappa _ 1 ( \\kappa _ 1 - \\tilde { \\kappa } _ i ) } - \\frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \\kappa _ 1 ^ 2 } = \\frac { F ^ { i i } ( \\kappa _ 1 + \\tilde { \\kappa } _ i ) } { \\kappa _ 1 ^ 2 ( \\kappa _ 1 - \\tilde { \\kappa } _ i ) } h _ { 1 1 i } ^ 2 \\geq c ( n ) \\frac { F ^ { i i } h _ { 1 1 i } ^ 2 } { \\kappa _ 1 ^ 2 } , \\end{align*}"}
+{"id": "6778.png", "formula": "\\begin{align*} e ( \\rho , x ) = e ^ { i \\rho x } \\left ( 1 + \\left ( z + 1 \\right ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } z ^ { n } a _ { n } ( x ) \\right ) \\end{align*}"}
+{"id": "5829.png", "formula": "\\begin{align*} { \\rm O } ( \\epsilon ^ 1 ) : \\rho c _ v \\partial _ { t _ 1 } g _ i ^ { ( 0 ) } + d _ { 1 i } g _ i ^ { ( 0 ) } = - \\frac { 1 } { \\Delta t } \\left ( { { { \\bf { M } } } } ^ { - 1 } { \\Lambda } { { { \\bf { M } } } } \\right ) _ { i j } g _ j ^ { ( 1 ) } + { { w } _ i \\bar F ^ { ( 1 ) } } , \\end{align*}"}
+{"id": "7153.png", "formula": "\\begin{align*} Q ^ 2 - B Q - \\Bigl [ I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } , Q \\Bigr ] + C = 0 , \\end{align*}"}
+{"id": "3359.png", "formula": "\\begin{align*} C o n v ( \\bigcup \\limits _ { \\mu } S u p p ( s _ \\mu ) ) = C o n v ( \\bigcup \\limits _ { i = 0 } ^ l N e w t o n ( s _ { \\lambda ^ i } ) ) . \\end{align*}"}
+{"id": "6848.png", "formula": "\\begin{align*} a _ { 0 } ( x ) & = e ( \\frac { i } { 2 } , x ) e ^ { \\frac { x } { 2 } } - 1 \\quad b _ { 0 } ( x ) = g ( \\frac { i } { 2 } , x ) e ^ { - \\frac { x } { 2 } } - 1 , \\\\ d _ { 0 } ( x ) & = a _ { 0 } ^ { \\prime } ( x ) - \\frac { a _ { 0 } ( x ) } { 2 } + \\frac { 1 } { 2 } \\int _ { x } ^ { \\infty } q ( t ) d t , \\\\ c _ { 0 } ( x ) & = b _ { 0 } ^ { \\prime } ( x ) + \\frac { b _ { 0 } ( x ) } { 2 } - \\frac { 1 } { 2 } \\int _ { - \\infty } ^ { x } q ( t ) d t \\end{align*}"}
+{"id": "4280.png", "formula": "\\begin{align*} n = \\frac { \\beta \\left ( 3 \\beta ^ { 2 } + 4 \\right ) + 1 } { 2 } \\end{align*}"}
+{"id": "850.png", "formula": "\\begin{align*} \\nabla _ \\Sigma f ( p ) = 0 \\phantom { x x } \\phantom { x x } D _ \\Sigma ^ 2 f ( p ) \\leq 0 . \\end{align*}"}
+{"id": "636.png", "formula": "\\begin{align*} g _ 0 = \\begin{bmatrix} \\sqrt { x _ 0 } & 0 \\\\ \\sqrt { - y _ 0 } & \\frac { 1 } { \\sqrt { x _ 0 } } \\end{bmatrix} = \\begin{bmatrix} \\sqrt { x _ 0 } & 0 \\\\ 0 & \\frac { 1 } { \\sqrt { x _ 0 } } \\\\ \\end{bmatrix} \\cdot \\begin{bmatrix} 1 & 0 \\\\ \\sqrt { - { x _ 0 } { y _ 0 } } & 1 \\\\ \\end{bmatrix} . \\end{align*}"}
+{"id": "1292.png", "formula": "\\begin{align*} \\forall \\psi \\in \\Omega _ { \\Tilde { \\Lambda } _ m \\setminus \\Lambda _ n } \\ \\forall \\Delta \\subset \\Lambda _ n \\ \\forall \\xi _ { \\Delta } : \\nu ( \\xi _ { \\Delta } \\eta _ { \\Lambda _ n \\setminus \\Delta } \\psi _ { \\Tilde { \\Lambda } _ m \\setminus \\Lambda _ n } ) = 0 . \\end{align*}"}
+{"id": "1375.png", "formula": "\\begin{align*} \\lim _ { \\lambda _ 1 , \\ldots , \\lambda _ d \\rightarrow 1 } \\frac { h ^ { ( \\lambda _ 1 , \\ldots , \\lambda _ d ) } ( x ) } { \\Delta ( \\lambda ) } = \\frac { h ( x ) } { \\prod _ { j = 1 } ^ { d - 1 } j ! } , \\lim _ { \\lambda _ 1 , \\ldots , \\lambda _ d \\rightarrow 1 } \\frac { \\mathfrak { h } ^ { ( \\lambda _ 1 , \\ldots , \\lambda _ d ) } ( x ) } { \\Delta ( \\lambda ) } = \\frac { \\mathfrak { h } ( x ) } { \\prod _ { j = 1 } ^ { d - 1 } j ! } . \\end{align*}"}
+{"id": "3559.png", "formula": "\\begin{align*} c _ { \\Omega _ 4 } ^ { \\Omega _ 4 } & = q ^ { - 1 / 6 } \\varphi _ 1 ^ { - 4 } ( \\chi _ { 3 , 4 } ^ { 1 , 2 } \\chi _ { 4 , 5 } ^ { 2 , 2 } ) \\\\ & = q ^ { - 1 / 6 } \\varphi _ 1 ^ { - 4 } ( q ^ { 1 / 2 4 } V ( q ) ^ { - 1 } ) ( q ^ { 1 / 1 2 0 } G _ 2 V ( q ) ^ { - 1 } ) \\\\ & = q ^ { - 7 / 6 0 } \\varphi _ 1 ^ { - 4 } ( \\varphi _ 1 ^ { - 2 } \\varphi _ 2 ^ 2 ) G _ 2 \\\\ & = q ^ { - 7 / 6 0 } \\varphi _ 1 ^ { - 6 } \\varphi _ 2 ^ 2 G _ 2 . \\end{align*}"}
+{"id": "9209.png", "formula": "\\begin{align*} \\left | \\left . \\dfrac { \\partial \\varepsilon _ \\delta ( x , \\bar { y } _ a , t ) - b _ { 1 , \\delta } ( x ) / 2 } { \\partial x } \\right | _ { x = x _ a } \\right | \\le 2 L _ r . \\end{align*}"}
+{"id": "41.png", "formula": "\\begin{align*} \\rho ( g ) = \\tilde { \\Gamma } ( g ) ^ { \\frac { - 1 } { Q - 2 } } \\end{align*}"}
+{"id": "3482.png", "formula": "\\begin{align*} e ^ { ( \\delta _ n - L + 5 0 \\varepsilon ) t q _ n } r _ { \\ell _ t } \\leq e ^ { ( \\delta _ n - L + 5 0 \\varepsilon ) t _ 0 q _ n } \\max _ { 1 \\leq j \\leq y _ n - 1 } r _ j \\leq & C _ 0 y _ n q _ n e ^ { ( \\delta _ n - L + 5 0 \\varepsilon ) t _ 0 q _ n } \\\\ \\leq & e ^ { \\varepsilon y _ n q _ n } e ^ { ( \\delta _ n - L + 5 0 \\varepsilon ) \\frac { y _ n } { 2 } q _ n } \\\\ = & e ^ { ( \\delta _ n - L + 5 2 \\varepsilon ) \\frac { y _ n } { 2 } q _ n } \\\\ \\leq & e ^ { ( \\delta _ n - L + 5 2 \\varepsilon ) \\ell _ 0 q _ n } . \\end{align*}"}
+{"id": "4089.png", "formula": "\\begin{align*} \\sigma ( ( u ^ i , u ^ i { } _ j ) . a ) & = \\sigma ( u ^ i , u ^ i { } _ \\alpha a ^ \\alpha { } _ j ) \\\\ * & = ( u ^ i , u ^ i { } _ \\alpha a ^ \\alpha { } _ j , - \\Gamma ^ i { } _ { \\alpha \\beta } u ^ \\alpha { } _ \\gamma u ^ \\beta { } _ \\delta a ^ \\gamma { } _ j a ^ \\delta { } _ k ) \\\\ * & = \\sigma ( u ^ i , u ^ i { } _ j ) . a \\end{align*}"}
+{"id": "1027.png", "formula": "\\begin{align*} & \\langle \\mu , \\alpha \\rangle + \\chi _ R ( v ' \\alpha ) + \\chi _ L ( - w ' v ' \\alpha ) \\\\ \\leq & \\langle \\mu , \\alpha \\rangle + \\chi _ R ( v ' \\alpha ) + 1 - \\chi _ L ( w ' v ' \\alpha ) \\\\ = & \\prescript L { } { } \\ell { } ^ R ( x ' , \\alpha ) + 1 \\leq 0 . \\end{align*}"}
+{"id": "8167.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t _ { 0 } } \\bigg ( \\int _ { R _ { 0 } } ^ { \\infty } \\nu \\xi ( \\nu , s ) d \\nu \\bigg ) ^ { 2 } \\ d s = 0 \\end{align*} % \\end{align*}"}
+{"id": "2363.png", "formula": "\\begin{align*} \\mathcal { Q } ( u ) : = u _ t - \\frac { \\partial } { \\partial x } \\left ( a ( x , u , u _ x ) \\right ) + b ( x , u , u _ x ) , \\end{align*}"}
+{"id": "7481.png", "formula": "\\begin{align*} & \\int _ { B _ { 1 / 2 } } | \\nabla v | ^ { 3 - 3 \\theta } \\d x \\leq C \\end{align*}"}
+{"id": "2538.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { \\Gamma } \\triangleq \\frac { 4 \\left ( \\gamma ^ 2 + \\delta ^ 2 \\right ) } { \\left ( 1 - 3 \\gamma \\right ) ^ 2 } \\left ( 1 - \\frac { \\delta } { \\sqrt { 2 k } } \\right ) ^ { - 1 } < \\gamma , \\\\ & \\nu = 1 - \\delta / \\sqrt { 2 k } . \\end{aligned} \\end{align*}"}
+{"id": "3223.png", "formula": "\\begin{align*} \\Sigma _ t : = \\sigma _ t \\sigma _ t ^ * \\forall t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "4063.png", "formula": "\\begin{align*} \\bar \\beta _ n ( G ) = O ( n ^ { e ( G ) } ) , \\end{align*}"}
+{"id": "4758.png", "formula": "\\begin{align*} & ( 1 . ) \\ w : = 1 + 2 \\gamma _ 1 a _ 1 = 0 \\Rightarrow \\gamma _ 1 = - \\frac { 1 } { 2 a _ 1 } > 0 , \\ \\ \\\\ & ( 2 . ) \\ b _ 2 ^ T x ^ * \\le a _ { 2 } \\frac { c _ { 1 } } { a _ { 1 } } - c _ { 2 } \\ \\\\ & ( 3 . ) \\ \\| x ^ { * } \\| ^ { 2 } = - \\frac { c _ { 1 } } { a _ { 1 } } \\ \\end{align*}"}
+{"id": "2048.png", "formula": "\\begin{align*} a _ g = \\abs v ^ \\gamma * ( \\sqrt \\mu g ) , \\ \\boldsymbol { A } _ g = ( a _ { 1 , g } , a _ { 2 , g } , a _ { 3 , g } ) = \\abs v ^ \\gamma * ( \\sqrt \\mu \\partial _ v g ) \\textrm { a n d } \\boldsymbol { B } _ { g } = ( b _ { 1 , g } , b _ { 2 , g } , b _ { 3 , g } ) = \\abs v ^ \\gamma * ( v \\sqrt \\mu g ) , \\end{align*}"}
+{"id": "3643.png", "formula": "\\begin{align*} & \\mathbb { E } \\widetilde { g } _ { \\delta , L , \\mu } ( x ) = g _ { \\delta , L , \\mu } ( x ) , \\\\ & \\mathbb { E } \\| \\widetilde { g } _ { \\delta , L , \\mu } ( x ) - g _ { \\delta , L , \\mu } ( x ) \\| ^ 2 \\leq \\frac { \\sum _ { i = 1 } ^ n \\sigma _ { f , i } ^ 2 } { n ^ 2 r } = \\frac { \\sigma _ f ^ 2 } { n r } . \\end{align*}"}
+{"id": "4168.png", "formula": "\\begin{align*} \\frac { \\alpha } { 2 \\beta } \\min \\{ \\kappa - r , \\lambda \\} \\leq \\frac { \\alpha } { 2 \\beta } \\min \\{ \\kappa - r , r \\} \\leq \\frac { 1 } { 2 } \\frac { \\alpha \\kappa } { 2 \\beta } = \\frac { 1 } { 2 } \\min \\{ \\alpha + \\gamma - \\frac { 1 } { 2 } , \\alpha \\} \\leq \\frac { \\alpha } { 2 } \\leq \\min \\{ \\alpha - \\varepsilon , \\frac { 1 } { 2 } \\} . \\end{align*}"}
+{"id": "3251.png", "formula": "\\begin{align*} \\mathbb P \\left [ \\int _ 0 ^ T \\| \\alpha _ s \\| ^ { \\frac { 2 m } { 2 + m } } d s + \\int _ 0 ^ T \\| \\sigma _ s \\| _ { L _ { } ( U , H ) } ^ m d s < \\infty \\right ] = 1 . \\end{align*}"}
+{"id": "7140.png", "formula": "\\begin{align*} \\chi _ { \\alpha } ( V ) = \\left ( ( n + 2 ) a _ 2 + 2 ( n + 3 ) a _ 3 z + 3 ( n + 4 ) a _ 4 z ^ 2 + \\cdots + n ( 2 n + 1 ) a _ { n + 1 } z ^ { n - 1 } \\right ) d z \\end{align*}"}
+{"id": "7152.png", "formula": "\\begin{align*} A ^ { - 1 } \\mathcal { L } _ g = I _ { n + 1 } \\frac { \\partial ^ 2 } { \\partial x _ n ^ 2 } + B \\frac { \\partial } { \\partial x _ n } + C = \\Bigl ( I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } + B - Q \\Bigr ) \\Bigl ( I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } + Q \\Bigr ) , \\end{align*}"}
+{"id": "8204.png", "formula": "\\begin{align*} x _ { i , 1 } + x _ { i + 1 , 1 } + x _ { i , n } + x _ { i + 1 , n } = ( \\tfrac { 1 } { 2 } S - x _ { i , n } ) + ( \\tfrac { 1 } { 2 } S - x _ { i + 1 , n } ) + x _ { i , n } + x _ { i + 1 , n } = S . \\end{align*}"}
+{"id": "4102.png", "formula": "\\begin{align*} & \\hphantom { { } = { } } D _ { ( o , o ) } ( \\{ f , g \\} . \\{ f ' , g ' \\} ) \\\\ * & = \\begin{pmatrix} D f ( f ' ( o ) ) D f ' ( o ) \\\\ D g ( f ' ( o ) ) D f ' ( o ) + H f ( f ' ( o ) ) D f ' ( o ) g ' ( o ) + D f ( f ' ( o ) ) D g ' ( o ) & D f ( f ' ( o ) ) D f ' ( o ) \\end{pmatrix} \\end{align*}"}
+{"id": "2990.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\widetilde { \\mu } } = \\mathcal { D } \\times \\mathcal { C } . \\end{align*}"}
+{"id": "2020.png", "formula": "\\begin{align*} \\hat y _ t ^ \\varepsilon = \\frac { y ^ \\varepsilon _ t - y _ t } { \\varepsilon } - K _ t ~ ~ \\hat z _ t ^ \\varepsilon = \\frac { z ^ \\varepsilon _ t - z _ t } { \\varepsilon } - L _ t . \\end{align*}"}
+{"id": "4648.png", "formula": "\\begin{align*} A ( G ) : = L _ 2 ( G ) \\ast L _ 2 ( G ) = \\{ s \\mapsto \\langle \\l _ s \\xi , \\eta \\rangle \\mid \\xi , \\eta \\in L _ 2 ( G ) \\} . \\end{align*}"}
+{"id": "8069.png", "formula": "\\begin{align*} \\sigma _ { e _ c } ^ 2 \\left [ t + 1 \\right ] = & \\left \\{ 1 - 2 \\mu \\left ( 2 \\sum _ { j = 1 } ^ { M } \\lvert \\phi ^ { \\left ( j , c \\right ) } \\rvert ^ 2 - f ^ { \\left ( c \\right ) } \\right ) \\right \\} \\sigma _ { e _ c } ^ 2 \\left [ t \\right ] \\\\ & - 4 \\left ( \\frac { _ { } \\left ( a _ c ^ { \\left ( o \\right ) } \\right ) } { a _ c ^ { \\left ( o \\right ) } } + a _ c ^ { \\left ( o \\right ) } \\sum _ { j = 1 } ^ { M } \\Re \\left \\{ \\phi ^ { \\left ( j , c \\right ) } \\right \\} \\right ) \\end{align*}"}
+{"id": "8093.png", "formula": "\\begin{align*} \\nu _ { 1 5 , Z ( \\mathbb { Z } _ { 1 5 } ) } = \\big \\{ & \\{ 3 , 3 \\} , \\{ 3 , 5 \\} , \\{ 3 , 6 \\} , \\{ 3 , 9 \\} , \\{ 3 , 1 0 \\} , \\{ 3 , 1 2 \\} , \\{ 5 , 5 \\} , \\{ 5 , 6 \\} , \\\\ & \\{ 5 , 9 \\} , \\{ 5 , 1 0 \\} , \\{ 5 , 1 2 \\} , \\{ 6 , 9 \\} , \\{ 9 , 1 0 \\} , \\{ 9 , 1 2 \\} \\big \\} . \\end{align*}"}
+{"id": "3519.png", "formula": "\\begin{align*} \\mathcal { C } _ n ^ { ( p ) } \\ = \\ ( - 1 ) ^ p F _ n - n ^ { p } + 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } \\mathcal { C } _ n ^ { ( p - 2 j - 1 ) } . \\end{align*}"}
+{"id": "595.png", "formula": "\\begin{align*} \\gamma _ { p } & : = p \\eta _ { 1 } + ( p - 1 ) \\eta _ { 2 } + c _ { p } \\big ( p \\hat { \\eta } _ { 1 } ^ { 2 } + 2 ( p - 1 ) \\hat { \\eta } _ { 1 } \\hat { \\eta } _ { 2 } + ( p - 2 ) \\hat { \\eta } _ { 2 } ^ { 2 } \\big ) \\\\ \\delta & : = \\eta _ { 2 } + 2 c _ { p } \\big ( \\hat { \\eta } _ { 1 } \\hat { \\eta } _ { 2 } + \\hat { \\eta } _ { 2 } ^ { 2 } \\big ) . \\end{align*}"}
+{"id": "7364.png", "formula": "\\begin{gather*} f \\in L f \\le c P ( c > f + \\epsilon ) = 0 \\epsilon > 0 . \\end{gather*}"}
+{"id": "7527.png", "formula": "\\begin{align*} \\deg _ G ( v ) = \\frac { \\sum _ { v \\in S \\in V ^ { ( k - 1 ) } } \\deg _ G ( S ) } { k - 1 } \\geq \\frac { \\deg _ H ( v ) \\sqrt { d } } { k - 1 } > d , \\end{align*}"}
+{"id": "8852.png", "formula": "\\begin{align*} | Z _ t | ^ 2 = | Z _ 0 | ^ 2 \\end{align*}"}
+{"id": "4361.png", "formula": "\\begin{align*} C ^ { \\alpha } _ { i } G _ { \\alpha \\beta } ^ T H ^ { \\beta } _ 1 \\overline { G _ { \\alpha \\beta } } = G _ { \\alpha \\beta } ^ T H ^ { \\beta } _ 1 \\overline { G _ { \\alpha \\beta } } ( \\overline { C ^ { \\alpha } _ { i } } ^ { T } ) \\end{align*}"}
+{"id": "435.png", "formula": "\\begin{align*} \\kappa = \\sum _ { j = 1 } ^ { n } P _ { j } a _ { j } , \\end{align*}"}
+{"id": "2939.png", "formula": "\\begin{align*} \\left \\langle \\overset { \\cdot } { K } _ { \\alpha } \\left ( \\omega \\right ) , \\varphi \\right \\rangle : = - \\left \\langle K _ { \\alpha } \\left ( \\omega \\right ) , \\varphi ^ { \\prime } \\right \\rangle : = \\int _ { \\mathbb { R } _ { + } } K _ { \\alpha } \\left ( t , \\omega \\right ) \\varphi ^ { \\prime } \\left ( t \\right ) d t \\end{align*}"}
+{"id": "1251.png", "formula": "\\begin{align*} - \\infty < \\inf _ { \\xi : \\xi _ { \\Lambda } = \\eta _ { \\Lambda } } f ( \\xi ) \\leq f ( \\eta ) \\leq \\sup _ { \\xi : \\xi _ { \\Lambda } = \\eta _ { \\Lambda } } f ( \\xi ) < \\infty . \\end{align*}"}
+{"id": "5528.png", "formula": "\\begin{align*} [ X \\times _ k \\mathbb A _ k ^ n \\to S , \\sigma ] = [ X \\times _ k \\mathbb A _ k ^ n \\to S , \\sigma ' ] \\end{align*}"}
+{"id": "4697.png", "formula": "\\begin{align*} \\left | \\int _ M f ( x ) \\right | = \\left | \\int _ M ( f ( x ) - \\tilde f ( x ) ) \\right | \\leq { \\rm V o l } ( \\tilde \\Sigma ^ r ) \\cdot { \\rm V o l } ( M ) . \\end{align*}"}
+{"id": "4759.png", "formula": "\\begin{align*} f _ { 1 } ( x ^ { * } ) = a _ { 1 } \\frac { \\gamma _ { 1 } ^ { 2 } } { w ^ { 2 } } b ^ { T } _ { 1 } b _ { 1 } - \\frac { 1 } { w } \\gamma _ { 1 } b _ { 1 } ^ { T } b _ { 1 } + c _ { 1 } = 0 \\end{align*}"}
+{"id": "1771.png", "formula": "\\begin{align*} g ( u , c ) = \\frac { \\vartheta } { 1 - \\frac { 1 } { \\psi } } ( 1 - r ) u \\bigg [ \\Big ( \\frac { c } { \\left ( ( 1 - r ) u \\right ) ^ { \\frac { 1 } { 1 - r } } } \\Big ) ^ { 1 - \\frac { 1 } { \\psi } } - 1 \\bigg ] , \\end{align*}"}
+{"id": "5081.png", "formula": "\\begin{align*} f _ { 1 } ^ { + } = 2 \\mathcal { I } _ 1 = \\int _ { \\Omega } | U | ^ { 2 } \\dd x = \\int _ { \\Omega } U \\cdot \\nabla \\times ( \\textrm { c u r l } ^ { - 1 } U ) \\dd x = f \\int _ { \\Omega } U \\cdot \\textrm { c u r l } ^ { - 1 } U \\dd x = f . \\end{align*}"}
+{"id": "2129.png", "formula": "\\begin{align*} \\partial \\varrho ( \\xi ) = 1 , \\bar { \\partial } \\partial \\varrho \\rfloor \\xi = 0 . \\end{align*}"}
+{"id": "5394.png", "formula": "\\begin{align*} \\varphi ( x ' , \\rho ( x ' ) ) = \\frac { 1 } { 2 } \\sum _ { \\alpha \\leq n - 1 } b _ \\alpha x _ \\alpha ^ 2 + \\frac { 1 } { 6 } \\sum _ { \\alpha , \\beta , \\gamma \\leq n - 1 } \\varphi _ { \\alpha \\beta \\gamma } ( 0 ) x _ \\alpha x _ \\beta x _ \\gamma + O ( | x ' | ^ 4 ) , \\end{align*}"}
+{"id": "4231.png", "formula": "\\begin{align*} & \\ \\sum _ { j , \\mu } g ^ i _ { j , \\mu } \\sum _ { k , \\nu } g ^ j _ { k , \\nu } ( \\varepsilon ( z ^ { - 1 } x ^ j _ { k , \\nu } ) - \\varepsilon ( ( x ^ i _ { j , \\mu } + z ) ^ { - 1 } x ^ j _ { k , \\nu } ) ) \\lambda _ k \\\\ & \\ = \\sum _ { k , \\nu } g ^ i _ { k , \\nu } ( \\varepsilon ( - z ^ { - 1 } ( 1 + x ^ i _ { k , \\nu } ) ) - \\varepsilon ( z ^ { - 1 } ( z - 1 ) ^ { - 1 } ( x ^ i _ { k , \\nu } - z + 1 ) ) ) \\lambda _ k \\end{align*}"}
+{"id": "354.png", "formula": "\\begin{align*} L _ { e v e n } : = A + \\sum _ { k = 1 } ^ { \\frac { n - 2 } { 2 } } B _ k . \\end{align*}"}
+{"id": "5023.png", "formula": "\\begin{align*} h ( G ) \\le ( m - y ) \\log ( m - y ) + y \\log ( y ) + m \\left ( \\log ( m ) - \\log ( y ) \\right ) - ( m - y ) \\left ( \\log ( m - y ) - \\log ( y ) \\right ) = m \\log m . \\end{align*}"}
+{"id": "7053.png", "formula": "\\begin{align*} F _ { 6 , 1 } \\ , : = \\ , \\tfrac { 1 4 0 } { 3 } , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ F _ { 7 , 0 } \\ , : = \\ , - \\ , \\tfrac { 2 8 0 } { 2 7 } . \\end{align*}"}
+{"id": "5570.png", "formula": "\\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) = A ( 0 ^ n 1 ^ \\infty ) - A ( 0 ^ \\infty ) = a _ n - a \\ . \\end{align*}"}
+{"id": "9031.png", "formula": "\\begin{align*} L ( E ) = \\max \\bigg \\{ \\log \\Big | \\frac { E } { 2 } + \\frac { \\sqrt { E ^ { 2 } - 4 } } { 2 } \\Big | , \\log | \\lambda | \\bigg \\} , \\forall \\ E \\in \\mathbb { C } . \\end{align*}"}
+{"id": "8635.png", "formula": "\\begin{align*} \\mathbb { E } _ { n , m } ' [ \\beta \\gamma ] \\\\ = \\mathbb { E } _ { \\mathcal { M } } [ \\mathbb { 1 } _ { X _ 0 ( M ) } ( x _ 1 ) \\mathbb { 1 } _ { X _ 1 ( M ) } ( x _ 2 ) \\mathbb { E } _ { n , m } [ \\chi _ { n , m } ( w , x _ 1 ) \\chi _ { n , m } ( w , x _ 2 ) | M ] \\end{align*}"}
+{"id": "8743.png", "formula": "\\begin{align*} ( D _ { a ^ + } ^ { \\alpha } f ) ( x ) : = \\frac { d } { d x } \\left [ ( { \\bf I } _ { a ^ + } ^ { 1 - \\alpha } f ) ( x ) \\right ] \\end{align*}"}
+{"id": "849.png", "formula": "\\begin{align*} \\partial _ t c _ { \\min \\ , | t } & = \\partial ^ \\square c _ { | ( t , p ) } = \\Delta _ \\Gamma G ' ( c ) + c V H _ { \\ , | ( t , p ) } \\\\ & = G '' ( c ) \\Delta _ \\Gamma c + G ''' ( c ) \\big | \\nabla _ \\Gamma c \\big | ^ 2 + g ( c ) H ^ 2 c _ { \\ , | ( t , p ) } \\geq g ( c ) H ^ 2 c _ { \\ , | ( t , p ) } . \\end{align*}"}
+{"id": "9119.png", "formula": "\\begin{align*} \\frac { \\sum \\limits _ { i = 1 } ^ j \\chi _ i - r j } { \\chi } < \\sum \\limits _ { i = 1 } ^ { j } w _ i < \\frac { \\sum \\limits _ { i = 1 } ^ j \\chi _ i - r ( j - 1 ) } { \\chi } . \\end{align*}"}
+{"id": "3830.png", "formula": "\\begin{align*} \\begin{aligned} \\Big [ | \\langle \\boldsymbol { \\nu } , A \\rangle - \\bar A | + | \\langle \\boldsymbol { \\nu } , f _ \\alpha \\rangle - \\bar f _ \\alpha | \\Big ] & \\le c _ 3 \\langle \\boldsymbol { \\nu } , \\eta ( \\lambda | \\bar U ; x , t ) \\rangle = c _ 3 \\mathcal { H } ( \\boldsymbol { \\nu } , U , \\bar { U } ; x , t ) \\ ; . \\end{aligned} \\end{align*}"}
+{"id": "9192.png", "formula": "\\begin{align*} \\begin{aligned} | x ( 2 \\bar { t } / \\gamma ) | _ { \\mathcal { A } _ \\delta } & \\le \\kappa \\bar c _ 0 e ^ { - \\gamma \\lambda \\bar { t } / \\gamma } + \\bar c \\\\ & \\le \\kappa \\bar c _ 0 e ^ { - \\lambda \\bar { t } _ 2 } + \\bar c < \\bar c _ 0 . \\end{aligned} \\end{align*}"}
+{"id": "283.png", "formula": "\\begin{align*} \\eta ^ \\lambda = ( 1 - \\lambda ) p _ 0 + \\lambda \\eta _ 0 \\in \\Omega \\end{align*}"}
+{"id": "7059.png", "formula": "\\begin{align*} F _ { 6 , 1 , 0 } \\ , = \\ , 5 \\ , F _ { 6 , 0 , 0 } . \\end{align*}"}
+{"id": "6196.png", "formula": "\\begin{align*} & B _ 1 = 2 ( L + 1 ) ( L + 2 ) \\Q , B _ 2 = L ( L + 1 ) , \\\\ & W ( r ) = - \\frac { L + 1 } { r } f + ( L + 2 ) \\Q + \\kappa ( L + 1 ) \\frac { r } { f } , \\\\ & W ' ( r ) = - \\frac { L + 2 } { r } f + ( L + 1 ) \\Q + \\kappa ( L + 2 ) \\frac { r } { f } , \\\\ & E _ 0 = 4 \\kappa ( L + 1 ) ^ 2 - ( L + 2 ) ^ 2 \\Q ^ 2 , E _ 1 = 4 \\kappa ( L + 2 ) ^ 2 - ( L + 1 ) ^ 2 \\Q ^ 2 . \\end{align*}"}
+{"id": "3966.png", "formula": "\\begin{align*} G = \\begin{pmatrix} \\mathbf { 1 } _ { n - 1 } & 1 \\\\ G _ { n - 1 } & \\mathbf { 0 } _ { n - 1 } ^ T \\end{pmatrix} , \\end{align*}"}
+{"id": "4554.png", "formula": "\\begin{align*} & A _ { w , 1 } f ( v _ { \\ell , \\ell , \\ell } ) = q ^ 3 f ( v _ { \\ell - 1 , \\ell - 1 , \\ell - 1 } ) + ( q ^ 2 + q + 1 ) f ( v _ { \\ell + 1 , \\ell , \\ell } ) \\\\ & A _ { w , 2 } f ( v _ { \\ell , \\ell , \\ell } ) = ( q ^ 4 + q ^ 3 + q ^ 2 ) f ( v _ { \\ell , \\ell - 1 , \\ell - 1 } ) + ( q ^ 2 + q + 1 ) f ( v _ { \\ell + 1 , \\ell + 1 , \\ell } ) \\\\ & A _ { w , 3 } f ( v _ { \\ell , \\ell , \\ell } ) = ( q ^ 3 + q ^ 2 + q ) f ( v _ { \\ell , \\ell , \\ell - 1 } ) + f ( v _ { \\ell + 1 , \\ell + 1 , \\ell + 1 } ) . \\end{align*}"}
+{"id": "1973.png", "formula": "\\begin{align*} F ( x + F ( x ) ) = - F ( x ) , \\end{align*}"}
+{"id": "515.png", "formula": "\\begin{align*} S _ { R } ^ { \\alpha } ( x , y ) = \\sum _ { n = 0 } ^ { \\infty } \\Big ( 1 - \\frac { 2 n + d } { R } \\Big ) _ { + } ^ { \\alpha } \\Phi _ { n } ( x , y ) , \\end{align*}"}
+{"id": "2041.png", "formula": "\\begin{align*} [ T _ 1 , T _ 2 ] = T _ 1 T _ 2 - T _ 2 T _ 1 . \\end{align*}"}
+{"id": "2091.png", "formula": "\\begin{align*} \\| g \\| _ { 1 } \\geq \\int _ { \\triangle \\setminus \\triangle _ { \\delta } } g d U ( x ) = \\int _ { \\triangle \\setminus \\triangle _ { \\delta } } g ^ { + } d U ( x ) + \\int _ { \\triangle \\setminus \\triangle _ { \\delta } } g ^ { - } d U ( x ) \\geq ( 3 / 2 0 ) \\cdot \\| g \\| _ { 1 } . \\end{align*}"}
+{"id": "7200.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n - 1 } } e ^ { - C | \\xi | } \\ , d \\xi = \\frac { \\Gamma ( n - 1 ) \\operatorname { v o l } ( \\mathbb { S } ^ { n - 2 } ) } { C ^ { n - 1 } } , C > 0 , \\ n \\geqslant 2 , \\end{align*}"}
+{"id": "8837.png", "formula": "\\begin{align*} d ( X _ { t , n } - x _ { t , n } ) = x _ { 0 , 1 } ( X _ { t , n - 1 } - x _ { t , n - 1 } ) d t + x _ { t , n - 1 } ( x _ { 0 , 1 } - x _ { t , 1 } ) d t + a _ n x _ { t , n } d t + x _ { t , n - 2 } x _ { t , n - 1 } d t \\end{align*}"}
+{"id": "6836.png", "formula": "\\begin{align*} L _ { k } ^ { \\left ( 2 \\right ) } \\left ( s , \\rho \\right ) = \\int _ { - \\infty } ^ { \\infty } \\frac { e ^ { i S ( \\tau , x - s ) } \\tau ^ { k - 2 } } { \\tau - \\rho } d \\tau . \\end{align*}"}
+{"id": "8656.png", "formula": "\\begin{align*} { \\bf { \\bar H } } _ { l ' } ^ H { { { \\bf { F } } } _ { l ' } } = { { \\bf { 0 } } _ { \\left | { { \\cal L } _ { l ' } } \\right | \\times { M _ t } } } . \\end{align*}"}
+{"id": "4374.png", "formula": "\\begin{align*} v _ \\epsilon ( t ) : = & \\int _ { - \\infty } ^ { t } ( \\int _ { - \\infty } ^ { t _ 1 } ( \\frac { 1 } { B - 4 \\epsilon } \\mathbb { I } _ { ( - t _ 0 - B + 2 \\epsilon , - t _ 0 - 2 \\epsilon ) } * \\rho _ { \\frac { 1 } { 4 } \\epsilon } ) ( s ) d s ) d t _ 1 \\\\ & - \\int _ { - \\infty } ^ { - t _ 0 } ( \\int _ { - \\infty } ^ { t _ 1 } ( \\frac { 1 } { B - 4 \\epsilon } \\mathbb { I } _ { ( - t _ 0 - B + 2 \\epsilon , - t _ 0 - 2 \\epsilon ) } * \\rho _ { \\frac { 1 } { 4 } \\epsilon } ) ( s ) d s ) d t _ 1 - t _ 0 , \\end{align*}"}
+{"id": "100.png", "formula": "\\begin{align*} \\nu ( x ) = \\pi _ J ( v ^ { - 1 } \\mu ) . \\end{align*}"}
+{"id": "8452.png", "formula": "\\begin{align*} \\lim _ { | z | \\rightarrow \\infty } \\Psi ^ \\pm _ 2 ( x ; z ) = e ^ { - i c _ \\pm } e _ 2 , \\end{align*}"}
+{"id": "5218.png", "formula": "\\begin{align*} \\int _ 0 ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { + } ^ { 2 } ( \\partial _ t \\varphi + ( v + u _ { \\infty } ) \\cdot \\nabla \\varphi ) \\dd x \\dd s = - \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } ^ { 2 } \\varphi _ 0 \\dd x . \\end{align*}"}
+{"id": "7377.png", "formula": "\\begin{gather*} f = \\bigl ( \\pi _ { i + 1 } - \\pi _ i \\bigr ) \\ , 1 ( \\pi _ 1 \\in A _ 1 ) \\ , \\ldots \\ , 1 ( \\pi _ i \\in A _ i ) , \\end{gather*}"}
+{"id": "3498.png", "formula": "\\begin{align*} \\phi ( y ) = \\sum _ { s ; z _ { i + 1 } \\in I ( z _ i ' ) } G _ { I ( y ) } ( y , z _ 1 ) G _ { I ( z _ 1 ' ) } ( z _ 1 ' , z _ 2 ) \\cdots G _ { I ( z _ t ' ) } ( z _ t ' , z _ { t + 1 } ) \\phi ( { z _ { t + 1 } ' } ) , \\end{align*}"}
+{"id": "7356.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ n \\mu _ i ( f _ i ) \\ge \\inf _ { Q \\in \\mathcal { U } } Q \\Bigl ( \\ , \\sum _ { i = 1 } ^ n f _ i \\circ \\pi _ i \\Bigr ) \\end{gather*}"}
+{"id": "8576.png", "formula": "\\begin{align*} ( \\kappa \\ , * \\ , k _ 1 \\ , * \\ , k _ 2 ) ( t ) \\ , = \\ , h _ 1 ( t ) \\ , = \\ , \\{ 1 \\} , \\ , t > 0 \\end{align*}"}
+{"id": "4186.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { L } _ H q = 0 , \\\\ \\nabla ^ { \\perp } q \\cdot \\nu | _ { \\partial \\Omega } = - v _ n | _ { \\partial \\Omega } . \\\\ \\end{cases} \\end{align*}"}
+{"id": "8120.png", "formula": "\\begin{align*} C ( G , x ) = ( 1 + x ) ( 1 + ( n - 1 ) x + ( m - n + 1 ) x ^ 2 ) . \\end{align*}"}
+{"id": "6532.png", "formula": "\\begin{align*} P ( x ) = x ^ { \\pi ( a ) } + c _ { \\pi ( a ) - 1 } x ^ { \\pi ( a ) - 1 } + c _ { \\pi ( a ) - 2 } x ^ { \\pi ( a ) - 2 } + \\cdots c _ 1 x + c _ 0 . \\end{align*}"}
+{"id": "3967.png", "formula": "\\begin{align*} M _ { I J } = \\begin{pmatrix} G _ { I _ 1 } & G _ { I _ 2 } & \\cdots & G _ { I _ { 2 k } } \\\\ G _ { J _ 1 } & G _ { J _ 2 } & \\cdots & G _ { J _ { 2 k } } \\end{pmatrix} , \\end{align*}"}
+{"id": "6671.png", "formula": "\\begin{align*} \\begin{aligned} Y ^ { \\epsilon } ( t ) & = y ^ { \\epsilon } _ { 0 } + \\int ^ { t } _ { 0 } h ^ { \\epsilon } ( s ) d s + \\lambda ( \\epsilon ) \\epsilon \\int ^ { t } _ { 0 } \\gamma ^ { \\epsilon } ( s ) \\tilde { N } ^ { \\epsilon ^ { - 2 } } ( d s ) . \\end{aligned} \\end{align*}"}
+{"id": "6394.png", "formula": "\\begin{align*} \\log \\det ( u _ { , i j } ) = - v _ j x ^ j + u _ { , i } \\xi ^ i + c , \\end{align*}"}
+{"id": "4737.png", "formula": "\\begin{align*} J ( x ) + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k f _ k ( x ) \\ge J ( x ^ * ) + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k f _ k ( x ^ * ) = J ( x ^ * ) \\ \\ \\forall \\ , x \\in \\mathbb { R } ^ n . \\end{align*}"}
+{"id": "5191.png", "formula": "\\begin{align*} A _ t & = a _ t , \\\\ B \\times u & = b \\times v + b \\times u _ { \\infty } + B _ { \\infty } \\times v , \\\\ \\nabla \\times B & = \\nabla \\times b . \\end{align*}"}
+{"id": "2889.png", "formula": "\\begin{align*} & { \\frak C } ^ 2 \\sum _ { x = 0 } ^ n | \\tilde V _ x ( m ) | ^ 2 \\le \\left [ { \\frak M } | \\tilde V _ 0 ( m ) | + \\left \\{ \\sum _ { x = 0 } ^ n | \\tilde v _ { x } ( m ) | ^ 2 \\right \\} ^ { 1 / 2 } \\right ] ^ 2 \\le 2 { \\frak M } ^ 2 | \\tilde V _ 0 ( m ) | ^ 2 + 2 \\sum _ { x = 0 } ^ n | \\tilde v _ { x } ( m ) | ^ 2 . \\end{align*}"}
+{"id": "4239.png", "formula": "\\begin{align*} \\beta ( z , w ) = \\dfrac { 1 } { 2 } \\log \\dfrac { 1 + | \\varphi _ z ( w ) | } { 1 - | \\varphi _ z ( w ) | } , z , w \\in \\mathbb { B } _ n . \\end{align*}"}
+{"id": "460.png", "formula": "\\begin{align*} f _ e ( x ) = ( e ^ { \\lambda \\Lambda _ e } - 1 ) ^ { - 1 } \\sum _ { n = 1 } ^ \\infty ( ( \\lambda \\Lambda _ { e } ) ^ n / n ! ) \\ , g _ { e } ^ { \\ast n } ( x ) \\end{align*}"}
+{"id": "7284.png", "formula": "\\begin{align*} \\pi ^ G _ k ( X ) = \\begin{cases} \\displaystyle \\operatorname * { c o l i m } _ { V \\in s ( \\mathcal { U } _ G ) } \\ ; \\ ; [ S ^ { k + V } , X ( V ) ] _ * ^ G & \\mathrm { f o r } \\ ; \\ ; k \\geq 0 \\\\ \\displaystyle \\operatorname * { c o l i m } _ { { V \\in s ( \\mathcal { U } _ G ) } } \\ ; \\ ; [ S ^ { k + V } , X ( \\mathbb { R } ^ { - k } \\oplus V ) ] _ * ^ G & \\mathrm { f o r } \\ ; \\ ; k \\leq 0 \\end{cases} \\end{align*}"}
+{"id": "8012.png", "formula": "\\begin{align*} \\frac { 1 } { | \\mathcal F _ { N , k } | } \\sum _ { ( p _ 1 , p _ 2 , \\dots , p _ t ) } ^ { ( 1 ) } \\sum _ { f \\in \\mathcal F _ { N , k } } a _ f ( p _ 1 ^ { 2 m _ 1 } p _ 2 ^ { 2 m _ 2 } \\dots p _ t ^ { 2 m _ t } ) = \\pi _ N ( x ) ( \\pi _ N ( x ) - 1 ) ( \\pi _ N ( x ) - 2 ) \\dots ( \\pi _ N ( x ) - ( t - 1 ) ) . \\end{align*}"}
+{"id": "404.png", "formula": "\\begin{align*} \\pi \\circ \\phi = \\mbox { i d } _ Y , \\end{align*}"}
+{"id": "2557.png", "formula": "\\begin{align*} ( q _ { k + 1 } + \\dots + q _ n ) ^ 2 & = ( 2 m - S ^ + _ k ) ^ 2 \\leq ( n - k ) ( q ^ 2 _ { k + 1 } + \\dots + q ^ 2 _ n ) \\\\ & = ( n - k ) \\Big ( 2 m + \\displaystyle \\sum _ { v _ i \\in V ( G ) } d ^ 2 _ { v _ i } - ( q ^ 2 _ 1 + \\dots + q ^ 2 _ k ) \\Big ) \\\\ & \\leq ( n - k ) \\Big ( 2 m + \\displaystyle \\sum _ { v _ i \\in V ( G ) } d ^ 2 _ { v _ i } - \\frac { { S ^ + _ k } ^ 2 } { k } \\Big ) . \\end{align*}"}
+{"id": "2073.png", "formula": "\\begin{align*} | b - x _ { k _ 0 } | \\leq | b - x | + | x - x _ { k _ 0 } | \\leq \\frac { m ( J ) } { 2 } + \\frac { m ( J _ { k _ 0 } ) } { 2 } < m ( J _ { k _ 0 } ) + \\frac { m ( J _ { k _ 0 } ) } { 2 } = \\frac { 3 } { 2 } m ( J _ { k _ 0 } ) < 2 m ( J _ { k _ 0 } ) \\end{align*}"}
+{"id": "4061.png", "formula": "\\begin{align*} a _ { n , k } = \\frac 1 n \\sum _ { j \\in [ n ] ^ k } \\rho _ H ( j _ 1 - j _ 2 ) \\rho _ H ( j _ 2 - j _ 3 ) \\cdots \\rho _ H ( j _ { k - 1 } - j _ k ) \\rho _ H ( j _ k - j _ 1 ) . \\end{align*}"}
+{"id": "7968.png", "formula": "\\begin{align*} & \\partial _ { t } \\left ( \\frac { \\rho ( u , t ) ^ { 2 } } { 2 } \\right ) - \\Theta \\sigma ^ { i j } _ { k } \\nabla _ { i j } \\left ( \\frac { \\rho ( u , t ) ^ { 2 } } { 2 } \\right ) \\\\ & = ( k + 1 ) h \\Theta \\sigma _ { k } - \\rho ^ { 2 } + \\sigma _ { k } \\nabla _ { i } h \\nabla _ { j } \\Theta - \\Theta \\sigma ^ { i j } _ { k } w _ { m i } w _ { m j } . \\end{align*}"}
+{"id": "1182.png", "formula": "\\begin{align*} Z _ t ^ n = 1 + \\sum _ { i = 1 } ^ n \\int _ 0 ^ t Z _ s ^ n \\triangle K _ s ^ { i , n } \\cdot d W _ s ^ i . \\end{align*}"}
+{"id": "7051.png", "formula": "\\begin{align*} F _ { 0 , 7 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 1 , 6 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 2 , 5 } \\ , : = \\ , 1 2 0 , \\ \\ \\ \\ \\ F _ { 3 , 4 } \\ , : = \\ , 2 4 0 , \\ \\ \\ \\ \\ F _ { 4 , 3 } \\ , : = \\ , 9 0 , \\ \\ \\ \\ \\ F _ { 5 , 2 } \\ , : = \\ , 5 0 \\ , F _ { 5 , 0 } - \\tfrac { 8 0 0 } { 9 } , \\end{align*}"}
+{"id": "5108.png", "formula": "\\begin{align*} \\begin{aligned} | | \\phi | | _ { L ^ { 1 } ( \\mathbb { R } ^ { 3 } ) } & = \\frac { 1 } { \\pi } | | \\varphi | | _ { L ^ { 1 } ( \\mathbb { R } ^ { 5 } ) } , \\\\ | | \\phi | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } _ { + } ; r ^ { - 1 } ) } & = \\frac { 1 } { \\sqrt { 2 } \\pi } | | \\varphi | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 5 } ) } . \\end{aligned} \\end{align*}"}
+{"id": "9062.png", "formula": "\\begin{align*} X _ { - \\beta _ m } X _ { - \\beta _ { m - 1 } } \\cdots X _ { - \\beta _ { m - k + 1 } } v _ k = \\prod _ { l = 1 } ^ { k } ( \\lambda _ { m - l + k } + \\mu _ 1 - l + 1 ) v _ 0 . \\end{align*}"}
+{"id": "1686.png", "formula": "\\begin{align*} V _ n ( \\varphi ) = \\begin{bsmallmatrix} \\varphi ( 0 ) \\\\ \\Phi _ n \\end{bsmallmatrix} ^ { \\top } \\mathbf { P } _ n \\begin{bsmallmatrix} \\varphi ( 0 ) \\\\ \\Phi _ n \\end{bsmallmatrix} , \\end{align*}"}
+{"id": "5012.png", "formula": "\\begin{align*} S t _ { D N M } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( f , z ) - f ( z ) | | _ { \\infty } = 0 \\end{align*}"}
+{"id": "5669.png", "formula": "\\begin{align*} { } _ { m } E ^ { 1 } _ { p , q } = s _ { m } ^ { - 1 } H _ { q } ( O _ { n , n } , C _ { p } ( n ) ) \\cong H _ { q } ( O _ { n - p , n - p } ) \\end{align*}"}
+{"id": "4147.png", "formula": "\\begin{align*} \\Phi ( x ' ) : = \\partial _ n K ^ L ( x ' , 1 ) , x ' \\in \\R ^ { n - 1 } . \\end{align*}"}
+{"id": "7031.png", "formula": "\\begin{align*} u \\ , = \\ , \\frac { 1 } { 2 } \\ , \\frac { x ^ 2 } { 1 - y } , \\end{align*}"}
+{"id": "3627.png", "formula": "\\begin{align*} \\tilde { \\kappa } _ i = \\begin{cases} \\kappa _ 1 , & i = 1 , \\\\ \\kappa _ i - 1 , & i > 1 . \\end{cases} \\end{align*}"}
+{"id": "2670.png", "formula": "\\begin{align*} | \\frac { 1 } { \\lambda ( t ) } \\mathcal { G } ( t ) - \\mathcal { E } ( t ) | & \\leq ( \\mu c _ 1 + \\nu ) \\frac { \\alpha ( t ) \\delta ^ { \\frac { 1 } { \\ell } } ( t ) } { \\lambda ( t ) } \\mathcal { E } ( t ) \\mathcal { E } ^ { ( r + \\frac { 1 } { \\ell ^ { \\prime } } + \\frac { 1 } { q } - 1 ) } ( t ) \\\\ & \\leq ( \\mu c _ 1 + \\nu ) c ^ * _ { _ E } \\mathcal { E } ( t ) , \\end{align*}"}
+{"id": "2247.png", "formula": "\\begin{align*} ( A , B ) \\cdot Z = A Z B ^ { - 1 } \\end{align*}"}
+{"id": "7815.png", "formula": "\\begin{align*} \\langle z _ 1 ^ * , v \\rangle = \\hat z _ 1 \\cdot v , \\langle z _ 2 ^ * , v \\rangle = \\hat z _ 2 \\cdot v , \\end{align*}"}
+{"id": "2869.png", "formula": "\\begin{align*} e _ { \\rm t h m } ( u ) = T ( u ) , u \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "5538.png", "formula": "\\begin{align*} \\int _ { \\mathfrak Y ( n ) } | \\mathfrak h ( n ) _ { \\eta } ^ * \\omega ( n ) | = ( \\phi _ 0 ) _ ! \\int _ { S m ( \\widetilde { \\mathfrak Y ( n ) } ) } | \\phi _ { \\eta } ^ * \\mathfrak h ( n ) _ { \\eta } ^ * \\omega ( n ) | . \\end{align*}"}
+{"id": "4024.png", "formula": "\\begin{align*} \\varphi ^ L _ { \\sigma , s } ( x ) = a < b = \\varphi ^ L _ { \\sigma , s } ( y ) . \\end{align*}"}
+{"id": "5846.png", "formula": "\\begin{align*} { f _ 6 } = { f _ 5 } - \\frac { 1 } { 3 } \\rho { u _ z } . \\end{align*}"}
+{"id": "7587.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } T ^ { c _ 1 ( n ) } x _ { 0 0 } = x _ { 1 0 } , \\textup { a n d } \\lim _ { m \\to \\infty } T ^ { c _ 2 ( m ) } x _ { 1 0 } = x _ { 1 1 } . \\end{align*}"}
+{"id": "4483.png", "formula": "\\begin{align*} \\mathcal { S } ^ { k - 1 } : = \\{ \\underline { x } \\in [ 0 , 1 ] ^ { k } : | | \\underline { x } | | = 1 \\} , ~ \\mathcal { D } ^ { k } ( A ) : = \\{ \\rho ( \\underline { a } ) : \\underline { a } \\in A ^ { k } \\} ~ \\mbox { a n d } ~ \\mathcal { D } ^ { \\underline { k } } ( A ) : = \\{ \\rho ( \\underline { a } ) : \\underline { a } \\in A ^ { \\underline { k } } \\} , \\end{align*}"}
+{"id": "7692.png", "formula": "\\begin{align*} \\underset { 0 \\leq t \\leq T } { \\sup } \\Bigg [ \\frac { 1 } { N } \\underset { i = 1 } { \\overset { N } { \\sum } } \\mathbb { E } ^ { \\boldsymbol { \\xi } } [ | \\phi ^ { N , i } ( t , \\boldsymbol { x } ^ * _ t ) - \\Psi ^ { N , i } ( t , \\boldsymbol { x } ^ * _ t ) | ^ 2 ] \\Bigg ] \\leq \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N ^ 2 } \\underset { i , j = 1 } { \\overset { N } { \\sum } } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 \\Big ) \\end{align*}"}
+{"id": "2352.png", "formula": "\\begin{align*} c ( 0 ) ( 1 - y ) + c y = c , c ( t ) ( 1 - 0 ) + c \\cdot 0 = c ( t ) , c ( t ) ( 1 - 1 ) + c \\cdot 1 = c . \\end{align*}"}
+{"id": "3587.png", "formula": "\\begin{align*} \\mathrm { L i f t } ( f ) = \\left \\lbrace \\prod _ { \\alpha = 1 } ^ d z ^ { i _ { \\alpha } } _ { j _ { \\alpha } k _ { \\alpha } } - \\prod _ { \\alpha = 1 } ^ d z ^ { i _ { \\alpha } } _ { j ' _ { \\alpha } k _ { \\alpha } } \\mid k _ { \\alpha } \\in [ t _ { i _ { \\alpha } } ] \\alpha \\in [ d ] \\right \\rbrace . \\end{align*}"}
+{"id": "3151.png", "formula": "\\begin{align*} \\frac { 2 + h \\chi _ 4 ( x ) + \\overline { h } \\overline { \\chi _ 4 } ( x ) } { 4 } = \\left \\{ \\begin{array} { l l l } 1 , & \\hbox { i f $ \\chi _ 4 ( x ) \\in \\{ 1 , \\chi _ 4 ( g ) \\} $ ; } \\\\ 0 , & \\hbox { } \\end{array} \\right . \\end{align*}"}
+{"id": "8102.png", "formula": "\\begin{align*} \\int _ { 1 } ^ \\infty \\frac { d s } { \\Big ( \\displaystyle \\int _ 0 ^ s F ( t ) d t \\Big ) ^ { p / ( 2 p - q + 1 ) } } = \\infty . \\end{align*}"}
+{"id": "5572.png", "formula": "\\begin{align*} W ( [ 0 ^ l 1 ] | 1 ^ \\infty ) = ( b - a ) + \\sum _ { n = 2 } ^ l ( a _ n - a ) + d _ l - d \\ . \\end{align*}"}
+{"id": "8036.png", "formula": "\\begin{align*} \\begin{aligned} f ( t , s ) = F _ { \\boldsymbol { \\omega } \\boldsymbol { \\eta } } \\big ( L ^ { - 1 } _ { \\boldsymbol { \\omega } } ( t ) , K ^ { - 1 } _ { \\boldsymbol { \\eta } } ( s ) , f \\big ( L ^ { - 1 } _ { \\boldsymbol { \\omega } } ( t ) , K ^ { - 1 } _ { \\boldsymbol { \\eta } } ( s ) \\big ) \\big ) , ~ ~ ~ ~ ~ \\forall ~ ( t , s ) \\in L _ { \\boldsymbol { \\omega } } ( S G ' ) \\times K _ { \\boldsymbol { \\eta } } ( S G '' ) , \\end{aligned} \\end{align*}"}
+{"id": "964.png", "formula": "\\begin{align*} N : = \\left \\{ n \\in \\N _ 0 \\colon f ( \\Phi ^ n ( \\alpha ) ) \\in G \\right \\} \\end{align*}"}
+{"id": "4155.png", "formula": "\\begin{align*} H = \\Theta : L ^ 2 ( \\R ^ { n - 1 } , \\mathbb { C } ^ N ) \\to L ^ 2 ( \\R ^ n _ + , d x ' d t / t ) ^ N . \\end{align*}"}
+{"id": "3347.png", "formula": "\\begin{align*} \\chi _ i = \\left \\{ \\begin{aligned} & \\phi ^ s _ i & & ( i \\neq j ) \\\\ & \\mu & & ( i = j ) \\end{aligned} \\right . \\end{align*}"}
+{"id": "4779.png", "formula": "\\begin{align*} - \\Delta u = \\lambda u & D , \\\\ [ 1 m m ] u = 0 & \\partial D . \\end{align*}"}
+{"id": "1186.png", "formula": "\\begin{align*} C ( \\alpha ) : = \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { 1 + 0 . 1 j } \\frac { 1 } { j ^ \\alpha } < \\infty . \\end{align*}"}
+{"id": "4487.png", "formula": "\\begin{align*} X ^ v \\circ X ^ w = t ^ { - \\langle v , w \\rangle } X ^ { v + w } \\ , v , w \\in \\mathbb { N } ^ n . \\end{align*}"}
+{"id": "564.png", "formula": "\\begin{align*} k ( \\theta ) = I _ f ( \\theta ) \\end{align*}"}
+{"id": "1556.png", "formula": "\\begin{align*} R ( \\tau ) \\Psi ^ \\prime ( \\tau ) + \\ell \\sin \\Psi ( \\tau ) - \\kappa \\ell R ( \\tau ) U ( \\tau ) = 0 , \\end{align*}"}
+{"id": "6983.png", "formula": "\\begin{align*} ( \\theta _ 1 , \\theta _ 2 ) = ( b ^ { - x } \\alpha ^ y , a ^ { - x } \\beta ^ y ) . \\end{align*}"}
+{"id": "2088.png", "formula": "\\begin{align*} g ( x ) = \\left \\{ \\begin{array} { r c l } \\frac { 1 } { \\inf \\{ \\beta : x \\in Q _ { \\beta } \\} } & \\mbox { i f } & x \\in H \\\\ 0 & \\mbox { i f } & x \\notin H \\end{array} \\right . \\end{align*}"}
+{"id": "6046.png", "formula": "\\begin{align*} 2 \\alpha ^ { 2 } a ( u , v ) + 2 ( \\alpha - 1 ) ^ { 2 } b ( u , v ) + 2 ( 3 \\alpha - 2 ) ^ { 2 } c ( u , v ) - \\frac { 2 ( p \\alpha - 1 ) ^ { 2 } } { p } F ( u , v ) = 0 . \\end{align*}"}
+{"id": "2821.png", "formula": "\\begin{align*} \\mathbf { F } _ d ^ H \\mathbf { H } _ { s f } \\mathbf { F } _ a = \\mathbf { H } _ { a d } , \\end{align*}"}
+{"id": "4355.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < 0 \\} } | f | ^ 2 _ h e ^ { - \\Psi } \\\\ = & \\int _ { \\{ \\Psi < 0 \\} } | f | ^ 2 _ h e ^ { - \\Psi + a \\Psi } + \\int _ { 0 } ^ { + \\infty } \\left ( \\int _ { \\{ \\Psi < - \\frac { l } { a } \\} } | f | ^ 2 _ h e ^ { - \\Psi + a \\Psi } \\right ) e ^ { l } d l \\\\ \\ge & \\left ( 1 + \\int _ 0 ^ { + \\infty } e ^ { - \\frac { - 1 + q ' } { a } l } d l \\right ) \\frac { 1 } { K _ { \\Psi , f , h , a } ( z _ 0 ) } \\\\ = & \\frac { a + q ' - 1 } { q ' - 1 } \\cdot \\frac { 1 } { K _ { \\Psi , f , h , a } ( z _ 0 ) } \\end{align*}"}
+{"id": "7683.png", "formula": "\\begin{align*} \\mu _ { ( \\boldsymbol { x } , \\boldsymbol { \\Psi } ( t , \\boldsymbol { x } ) ) } ^ { N , i } : = k _ i ^ N ( t , \\nu ^ { N , 1 } _ { \\boldsymbol { x } } , \\ldots , \\nu ^ { N , N } _ { \\boldsymbol { x } } , \\lambda ^ { N , 1 } _ { \\boldsymbol { \\Psi } ( t , \\boldsymbol { x } ) } , \\ldots , \\lambda ^ { N , N } _ { \\boldsymbol { \\Psi } ( t , \\boldsymbol { x } ) } ) , \\end{align*}"}
+{"id": "8266.png", "formula": "\\begin{align*} \\int _ { t _ 1 } ^ { t _ 2 } \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j = 1 } ^ { i } g _ j V _ { i , j } \\psi _ i ( s ) \\psi _ j ( s ) d s < \\infty \\end{align*}"}
+{"id": "1821.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d s } \\mathcal { Y M } _ { e , g _ s } ( { \\nabla } ) _ { \\big { | } _ { s = 0 } } & = \\frac 1 2 \\int _ M \\langle S _ { e , \\mathcal { Y M } } , \\delta g \\rangle d v _ g \\end{aligned} \\end{align*}"}
+{"id": "4617.png", "formula": "\\begin{align*} ( \\mathbf { I } - \\mathbf { P } ) ( v _ i v _ j \\sqrt { \\mu } ) = \\Big \\{ v _ i v _ j - \\frac { | v | ^ 2 } { 3 } \\delta _ { i j } \\Big \\} \\sqrt { \\mu } . \\end{align*}"}
+{"id": "6732.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } s } \\Gamma ( 1 / s , x ^ s ) & = - \\frac { 1 } { s ^ 2 } \\Big [ \\log ( x ^ s ) \\Gamma ( 1 / s , x ^ s ) + x ^ s T ( 3 , 1 / s , x ^ s ) \\Big ] - x \\exp ( - x ^ s ) \\log ( x ) \\\\ & \\triangleq \\tilde \\psi ( s , x ) , \\end{align*}"}
+{"id": "3131.png", "formula": "\\begin{align*} \\mathcal { R } ( e _ 1 ) = e _ 1 + a _ { 4 } e _ 4 + a _ { 5 } e _ { 5 } , \\ \\mathcal { R } ( e _ 2 ) = b e _ 3 , \\ \\mathcal { R } ( e _ 3 ) = \\mathcal { R } ( e _ 5 ) = 0 , \\mathcal { R } ( e _ 4 ) = \\frac { - b } { a _ { 5 } } e _ { 2 } , \\end{align*}"}
+{"id": "2620.png", "formula": "\\begin{align*} & e _ 1 \\cdot e _ 2 = e _ 3 , e _ 2 \\cdot e _ 1 = - e _ 3 , \\\\ & \\varepsilon ( i , j ) = ( - 1 ) ^ { i j } , \\\\ & \\begin{array} [ t ] { l } \\alpha ( e _ 1 ) = - 2 e _ 1 , \\alpha ( e _ 2 ) = e _ 3 , \\\\ \\alpha ( e _ 3 ) = e _ 2 - e _ 3 . \\end{array} \\end{align*}"}
+{"id": "258.png", "formula": "\\begin{align*} X _ j ( X _ 1 ^ r b _ 1 ) = \\left \\{ \\begin{array} { l l } \\sum _ { i = 2 } ^ { k } X _ i ^ { p - 1 } b _ i & \\\\ 0 & \\end{array} \\right . \\end{align*}"}
+{"id": "7434.png", "formula": "\\begin{align*} \\frac { 3 ^ { 9 ( 2 k - p _ k ) / 5 + 1 } } { 1 0 ^ { 2 k - p _ k + 1 } } & > \\frac { 3 ^ { ( 9 / 1 0 ) \\log _ 2 ( \\pi k ) + 1 } } { 1 0 ^ { ( 1 / 2 ) \\log _ 2 ( \\pi k ) + 4 1 / 4 0 } } = \\frac { 3 } { 1 0 ^ { 4 1 / 4 0 } } ( \\pi k ) ^ { ( 9 / 1 0 ) \\log _ 2 3 - ( 1 / 2 ) \\log _ 2 1 0 } \\\\ & > \\frac { 1 } { 4 } ( \\pi k ) ^ { - 1 / 4 } > ( 2 \\pi k ) ^ { - 1 / 2 } , \\end{align*}"}
+{"id": "2489.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbf { A x } - \\mathbf { b } \\tau - { { \\mathbf { r } } _ { p } } \\nu = 0 , \\\\ & - { { \\mathbf { A } } ^ { T } } \\mathbf { y } + \\mathbf { c } \\tau - \\mathbf { s } - { { \\mathbf { r } } _ { d } } \\nu = 0 , \\\\ & { { \\mathbf { b } } ^ { T } } \\mathbf { y } - { { \\mathbf { c } } ^ { T } } \\mathbf { x } - \\kappa - { { r } _ { g } } \\nu = 0 , \\\\ & \\mathbf { x } \\circ \\mathbf { s } = \\mu \\mathbf { e } , \\\\ & \\kappa \\tau = \\mu \\end{aligned} \\end{align*}"}
+{"id": "5563.png", "formula": "\\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . 1 x ' ) = A ( 0 ^ { k + n } 1 . . . ) - A ( 0 ^ \\infty ) = a _ { k + n } - a \\end{align*}"}
+{"id": "600.png", "formula": "\\begin{align*} \\gamma _ { p , c } : = \\sum _ { k = 1 } ^ { l } ( p - 2 + \\alpha _ { k } + \\beta _ { k } ) c _ { k } ^ { \\beta _ { k } } \\big [ \\null _ { k } \\eta \\big ] _ { \\frac { p } { 2 - \\alpha _ { k } } } \\end{align*}"}
+{"id": "6315.png", "formula": "\\begin{align*} \\frac { 1 } { \\rho _ K } \\| ( x ^ { K + 1 } , \\lambda ^ { K + 1 } ) - ( x ^ K , \\lambda ^ K ) \\| \\leq \\frac { r _ 0 + \\theta } { \\rho _ 0 \\zeta ^ K } \\overset { \\eqref { K } } { \\leq } \\frac { \\varepsilon } { 2 } , \\eta _ K = \\eta _ 0 \\sigma ^ K \\overset { \\eqref { K } } { \\leq } \\frac { \\varepsilon } { 2 } , \\end{align*}"}
+{"id": "3774.png", "formula": "\\begin{align*} \\mathcal { A } _ { H _ { t } } ( [ \\gamma , u ] ) = \\int _ { \\gamma } H _ { t } + \\int u ^ { * } \\omega . \\end{align*}"}
+{"id": "1217.png", "formula": "\\begin{align*} \\mu ( \\gamma _ \\Lambda ( \\eta _ \\Lambda | \\cdot ) ) = \\mu ( \\eta _ \\Lambda ) . \\end{align*}"}
+{"id": "8105.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { v ' ( r ) ^ { 2 } } { 2 } & \\leq \\int _ { 0 } ^ { r } t ^ { b } v ^ { q } ( t ) v ' ( t ) f ( w ( t ) ) d t \\\\ & \\leq r ^ { b } f ( w ( r ) ) \\int _ { 0 } ^ { r } \\left ( \\frac { v ^ { q + 1 } ( t ) } { q + 1 } \\right ) ' d t \\\\ & \\leq C f ( w ( r ) ) v ^ { q + 1 } ( r ) \\quad \\mbox { f o r a l l } \\ 0 < r < R , \\end{aligned} \\end{align*}"}
+{"id": "4460.png", "formula": "\\begin{align*} a _ { E ^ \\ast } ( n ) = \\frac { 2 } { L ( - 1 , \\chi _ { p } ) } \\sum _ { d | n } d ( \\chi _ { p } ( d ) - \\chi _ { p } ( n / d ) ) \\end{align*}"}
+{"id": "7103.png", "formula": "\\begin{align*} \\int _ \\Omega u ( t , x ) \\cdot \\nabla h ( x ) \\ , d x = 0 \\forall h \\in H ^ 1 _ { \\rm l o c } ( \\Omega ) \\end{align*}"}
+{"id": "4683.png", "formula": "\\begin{align*} \\Delta \\rho _ 1 ( x ) \\leq \\psi _ { K , - H } ( \\rho _ 1 ( x ) ) = - \\psi _ { K , H } ( \\rho ( x ) ) . \\end{align*}"}
+{"id": "3228.png", "formula": "\\begin{align*} F _ i : = Y _ { \\frac i 4 } = & \\mathcal S \\left ( \\frac 1 4 \\right ) Y _ { \\frac { i - 1 } 4 } + \\int _ { \\frac { i - 1 } 4 } ^ { \\frac i 4 } \\mathcal S \\left ( \\frac i 4 - s \\right ) \\alpha _ s d s + \\int _ { \\frac { i - 1 } 4 } ^ { \\frac i 4 } \\mathcal S \\left ( \\frac i 4 - s \\right ) \\sigma _ s d W _ s \\\\ = & \\mathcal S \\left ( \\frac 1 4 \\right ) Y _ { \\frac { i - 1 } 4 } + \\mu _ i + \\epsilon _ i , \\end{align*}"}
+{"id": "2717.png", "formula": "\\begin{align*} \\begin{aligned} & \\min _ { y _ 1 , y _ 2 } & & y _ 1 \\ ; \\ ; & & \\\\ & & & y _ 2 ^ 2 - 2 L _ 1 y _ 1 y _ 2 + ( L _ 1 \\| x _ k \\| ) ^ 2 \\le ( \\kappa _ { \\rm e g } \\delta _ k + \\epsilon _ g ) ^ 2 \\ ; \\ ; & & \\\\ & ~ & & | y _ 1 | \\le \\| x _ k \\| y _ 2 \\ge \\min \\{ 1 0 ^ { - 6 } , 1 0 ^ { - 2 } \\| L _ 1 x _ k \\| \\} . & & \\end{aligned} \\end{align*}"}
+{"id": "607.png", "formula": "\\begin{align*} C ^ r _ { N , H } ( \\alpha ) : = \\{ p \\in G = H \\ltimes N \\ , : \\ , d ( 1 , \\tilde \\pi _ N ( p ) ) \\leq \\alpha d ( 1 , \\tilde \\pi _ H ( p ) ) \\} , \\end{align*}"}
+{"id": "5188.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { + } \\frac { G } { r ^ { 2 } } \\dd x + \\mu \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { 3 } } \\nabla \\times B \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) \\dd x \\dd s & = \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } \\frac { G _ 0 } { r ^ { 2 } } \\dd x , \\\\ \\int _ { \\mathbb { R } ^ { 3 } } \\Phi ^ { 2 } _ { + } \\dd x + 2 \\mu \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { 3 } } | \\nabla \\Phi _ { + } | ^ { 2 } \\dd x \\dd s & = \\int _ { \\mathbb { R } ^ { 3 } } \\Phi ^ { 2 } _ { 0 , + } \\dd x . \\end{align*}"}
+{"id": "5029.png", "formula": "\\begin{align*} I _ 1 = y \\cdot f ( y - 1 ) + ( y - 1 ) \\left ( f ( ( a + 1 ) y + b ) - f ( a y + b ) \\right ) \\end{align*}"}
+{"id": "9265.png", "formula": "\\begin{align*} \\phi = \\left \\{ \\begin{array} { l l } u , & { \\rm o n } \\Omega \\setminus D , \\\\ \\max \\{ u , v \\} , & { \\rm o n } D , \\end{array} \\right . \\end{align*}"}
+{"id": "1839.png", "formula": "\\begin{align*} S _ { e , \\mathcal { Y M } ^ 0 } = S _ { e , \\mathcal { Y M } } = 0 \\ , . \\end{align*}"}
+{"id": "7457.png", "formula": "\\begin{align*} \\nu = [ h , \\eta ] = h \\eta h ^ { - 1 } \\eta ^ { - 1 } = \\rho \\eta ^ { g } \\rho ^ { - 1 } \\eta ^ { - 1 } \\in \\Gamma ^ G . \\end{align*}"}
+{"id": "7033.png", "formula": "\\begin{align*} F _ { 5 , 1 } \\ , = \\ , 4 , \\ \\ \\ \\ \\ F _ { 2 , 4 } \\ , = \\ , 4 ! \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0 \\ , = \\ , F _ { 0 , 6 } \\ , = \\ , F _ { 1 , 5 } \\ , = \\ , F _ { 3 , 3 } \\ , = \\ , F _ { 4 , 2 } , \\end{align*}"}
+{"id": "6162.png", "formula": "\\begin{align*} E _ { n _ r } ( l , Q ) = - \\frac { Q ^ 2 } { 4 n ^ 2 } + \\kappa n ^ 2 , n = n _ r + l + \\frac { d - 1 } { 2 } , n _ r = 0 , 1 , 2 , \\ldots , \\end{align*}"}
+{"id": "433.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { n } k _ { r s } \\theta ^ { s - 1 } & \\overset { \\eqref { e q : a _ i - k _ i j - t h e t a } } { { = } } a _ { r } = \\sum _ { j = 1 } ^ { n } \\Big ( \\sum _ { h = 1 } ^ { n } p _ { r j h } \\theta ^ { h - 1 } \\Big ) a _ { j } = \\sum _ { j = 1 } ^ { n } \\Big ( \\sum _ { h = 1 } ^ { n } p _ { r j h } \\theta ^ { h - 1 } \\Big ) \\Big ( \\sum _ { l = 1 } ^ { n } k _ { j l } \\theta ^ { l - 1 } \\Big ) \\\\ & \\overset { \\eqref { e q : t h e t a - i - t h e t a - j - c _ i j k } } { = } \\sum _ { j , h , l = 1 } ^ { n } \\sum _ { s = 1 } ^ { n } p _ { r j h } k _ { j l } c _ { h l s } \\theta ^ { s - 1 } . \\end{align*}"}
+{"id": "8492.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { P } ^ { + } \\left ( z ^ { - 1 } \\bar { r } _ 1 ( z ) e ^ { - 2 i z x } \\right ) & = \\int _ { 2 x } ^ { \\infty } \\widehat { ( z ^ { - 1 } \\bar { r } _ 1 ( z ) ) } ( \\xi ) e ^ { i z ( y - 2 x ) } d \\xi . \\end{aligned} \\end{align*}"}
+{"id": "6545.png", "formula": "\\begin{align*} 2 a = ( a + b ) + ( a - b ) \\end{align*}"}
+{"id": "7739.png", "formula": "\\begin{align*} v [ j ] \\mathrel { : = } v [ j ] + v [ t ] w _ { t } [ j ] \\end{align*}"}
+{"id": "5762.png", "formula": "\\begin{align*} \\mathcal { A } = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ p e _ j \\cdot \\big ( ( \\widetilde { \\nabla } _ Y B ) ( X , e _ j ) - ( \\widetilde { \\nabla } _ X B ) ( Y , e _ j ) \\big ) , \\end{align*}"}
+{"id": "1672.png", "formula": "\\begin{align*} S _ p = S _ { i n } \\cup S _ { s p } , \\end{align*}"}
+{"id": "546.png", "formula": "\\begin{align*} a _ 0 y _ k = \\tfrac { 3 } { 8 } ( { x _ { - k } } ^ { \\tau _ 0 } + x _ { - k } ) + \\tfrac { 3 } { 2 } y _ k - \\left ( p _ k ( \\ 1 ) - p _ k ( \\ 2 ) \\right ) \\end{align*}"}
+{"id": "2272.png", "formula": "\\begin{align*} a ( Z ) = a \\big ( ( Z Z ^ * ) ^ \\frac { 1 } { 2 } \\big ) \\end{align*}"}
+{"id": "3692.png", "formula": "\\begin{align*} X - W = c _ i W ^ { ( i ) } + O ^ { 3 , \\alpha } ( | x | ^ { - q } ) . \\end{align*}"}
+{"id": "4406.png", "formula": "\\begin{align*} \\sum _ { \\substack { m , n < x ^ 2 \\\\ m \\neq n } } \\frac { 1 } { | m - n | } = 2 \\sum _ { m < x ^ 2 } \\sum _ { n < m } \\frac { 1 } { m - n } & < 2 \\sum _ { m < x ^ 2 } \\sum _ { k < x ^ 2 } \\frac { 1 } { k } < 4 x ^ 2 \\log x + 2 \\gamma x ^ 2 + 1 , \\end{align*}"}
+{"id": "1990.png", "formula": "\\begin{align*} h ( g ^ { - 1 } ( z ) ) & = h ( - g ^ { - 1 } ( z - 2 g ^ { - 1 } ( z ) ) ) = - h ( g ^ { - 1 } ( z - 2 g ^ { - 1 } ( z ) ) ) , \\\\ f ( z ) & = - f ( z - 2 g ^ { - 1 } ( z ) ) . \\end{align*}"}
+{"id": "3813.png", "formula": "\\begin{align*} \\mathcal { B } _ q ( \\psi _ m ) ( p ) = \\displaystyle \\frac { p ^ m } { \\sqrt { \\Gamma ( q m + 1 ) } } , \\end{align*}"}
+{"id": "2953.png", "formula": "\\begin{align*} h ( x , y ) : = \\begin{cases} ( x , y ) & ( x < 0 ) \\\\ ( x , x + y ) & ( x \\geq 0 ) . \\end{cases} \\end{align*}"}
+{"id": "4488.png", "formula": "\\begin{align*} \\sigma _ i : V _ i & \\ , \\ , \\to \\ , \\ , \\bigoplus _ { h ' = i } V _ { h '' } , & { } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\tau _ i : \\bigoplus _ { h '' = i } V _ { h ' } & \\ , \\ , \\to \\ , \\ , V _ i , \\\\ v _ i & \\ , \\ , \\mapsto \\ , \\ , ( f _ h ( v _ i ) ) _ { h ' = i } , & { } ( v _ { h ' } ) & \\ , \\ , \\mapsto \\ , \\ , \\sum _ { h '' = i } f _ h ( v _ { h ' } ) . \\end{align*}"}
+{"id": "1379.png", "formula": "\\begin{align*} h ( x ) & = \\prod _ { j = 1 } ^ d \\lambda _ j \\int _ 0 ^ \\infty d a _ d \\int _ 0 ^ { x _ d - x _ { d - 1 } + a _ d } d a _ { d - 1 } \\ldots \\int _ 0 ^ { x _ { 2 } - x _ 1 + a _ { 2 } } d a _ 1 \\\\ & \\times \\left ( e ^ { - \\sum _ { i = 1 } ^ d \\lambda _ i a _ i } h ( x _ 1 + a _ 1 , \\ldots , x _ d + a _ d ) \\right ) . \\end{align*}"}
+{"id": "1779.png", "formula": "\\begin{align*} f = \\prod _ { g \\in \\mathcal G } g . \\end{align*}"}
+{"id": "1915.png", "formula": "\\begin{align*} \\textbf { I } = \\sum _ { k = 1 } ^ T \\langle \\lambda _ k F ( x _ k ) , x _ 0 - x \\rangle = \\sum _ { k = 1 } ^ T \\langle s _ { k - 1 } - s _ k , x _ 0 - x \\rangle = \\langle s _ 0 - s _ T , x _ 0 - x \\rangle = \\langle s _ T , x - x _ 0 \\rangle . \\end{align*}"}
+{"id": "1242.png", "formula": "\\begin{align*} \\forall z \\in \\Z ^ d : \\sum _ { \\Delta \\Subset \\Z ^ d } \\sum _ { \\xi _ \\Delta } \\sum _ { i = 1 } ^ q \\norm { \\nabla ^ i _ z c _ \\Delta ( \\cdot , \\xi _ \\Delta ) } _ \\infty < \\infty , \\end{align*}"}
+{"id": "945.png", "formula": "\\begin{align*} \\| R _ { 2 1 } \\| _ { L ^ \\infty } \\lesssim { } & t ^ { - \\frac 5 4 } \\| \\mathcal { N } _ 2 ( U ( - 1 / t ) v _ 1 , U ( - 1 / t ) v _ 2 ) \\| _ { H ^ 1 } \\\\ \\lesssim { } & t ^ { - \\frac 5 4 } \\| U ( - 1 / t ) v _ 1 \\| _ { H ^ 1 } ^ 2 \\| U ( - 1 / t ) v _ 2 \\| _ { H ^ 1 } = t ^ { - \\frac 5 4 } \\| v _ 1 \\| _ { H ^ 1 } ^ 2 \\| v _ 2 \\| _ { H ^ 1 } \\end{align*}"}
+{"id": "3167.png", "formula": "\\begin{align*} & \\langle H \\rangle \\\\ = & \\frac { 1 } { 2 \\times 4 ^ 3 } \\sum \\limits _ { x \\neq 0 } [ ( 2 + h \\chi _ 4 ( x ) + \\overline { h } \\overline { \\chi _ 4 } ( x ) ) ( 4 ( q - 3 ) - 4 h \\chi _ 4 ( x ) - 4 \\overline { h } \\overline { \\chi _ 4 } ( x ) ) ] \\\\ = & \\frac { 1 } { 2 \\times 4 ^ 3 } \\sum \\limits _ { x \\neq 0 } [ 8 ( q - 5 ) + ( 4 h ( q - 3 ) - 8 h ) \\chi _ 4 ( x ) + ( 4 \\overline { h } ( q - 3 ) - 8 \\overline { h } ) \\overline { \\chi _ 4 } ( x ) ] \\\\ = & \\frac { ( q - 1 ) ( q - 5 ) } { 1 6 } . \\end{align*}"}
+{"id": "8884.png", "formula": "\\begin{align*} \\mathbb B _ { k , S , g } ( \\tau , z ) = v ^ { k } e ^ { - 4 \\pi S [ y ^ t ] / v } B _ { k , S , g } ( \\tau , z ; \\tau , z ) . \\end{align*}"}
+{"id": "2125.png", "formula": "\\begin{align*} z & = \\frac { 1 } { \\sqrt { 1 - a + \\tau ^ 2 ( 1 + a ) } } \\left ( \\frac { \\sqrt { 1 - a } \\ , \\cos ( t ) } { \\sqrt { 1 + a } } + \\frac { i \\ , \\tau \\ , \\sqrt { 1 + a } \\ , \\sin ( t ) } { \\sqrt { 1 - a } } \\right ) , \\\\ w & = \\frac { 1 } { \\sqrt { 1 - a + \\tau ^ 2 ( 1 + a ) } } \\left ( \\frac { \\sqrt { 1 - a } \\ , \\sin ( t ) } { \\sqrt { 1 + a } } - \\frac { i \\ , \\tau \\ , \\sqrt { 1 + a } \\ , \\cos ( t ) } { \\sqrt { 1 - a } } \\right ) , \\end{align*}"}
+{"id": "878.png", "formula": "\\begin{align*} x y ' - x ' y = - n \\ , . \\end{align*}"}
+{"id": "6691.png", "formula": "\\begin{align*} \\delta ( \\eta \\gamma ^ n , \\gamma ^ + ) + \\delta ( \\eta \\gamma ^ n , ( \\eta \\gamma ^ n ) ^ - ) = f ( \\eta \\gamma ^ n \\gamma ^ + , \\eta \\gamma ^ n ( \\eta \\gamma ^ n ) ^ - ) ) - f ( \\gamma ^ + , ( \\eta \\gamma ^ n ) ^ - ) ) . \\end{align*}"}
+{"id": "5256.png", "formula": "\\begin{align*} D ( M _ { q } ) = \\{ f \\in \\ell ^ { \\infty } \\ ; | \\ ; q f \\in \\ell ^ { \\infty } \\} \\end{align*}"}
+{"id": "6841.png", "formula": "\\begin{align*} b _ { m } ( x ) + \\sum _ { n = 0 } ^ { N _ { s } } b _ { n } ( x ) B _ { m n } ( x ) = s _ { m } ( x ) , m = 0 , \\ldots , N _ { s } \\end{align*}"}
+{"id": "9025.png", "formula": "\\begin{align*} H _ { V } = \\Delta + V ( \\cdot ) , \\end{align*}"}
+{"id": "499.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { f _ s ( x ) } { f ( x ) } = \\lim _ { x \\to \\infty } \\frac { f _ { + a } ( x ) } { f ( x ) } \\cdot \\frac { f _ { a r } ( x ) } { f _ { + a } ( x ) } \\cdot \\frac { f _ s ( x ) } { f _ { a r } ( x ) } = ( 1 - e ^ { - \\lambda _ s } ) ^ { - 1 } . \\end{align*}"}
+{"id": "7984.png", "formula": "\\begin{align*} \\partial _ { t } E = - w ^ { 1 1 } \\partial _ { t } w _ { 1 1 } - d \\frac { h _ { t } } { h } + l \\rho \\rho _ { t } = - w ^ { 1 1 } ( h _ { 1 1 t } + h _ { t } ) - d \\frac { h _ { t } } { h } + l \\rho \\rho _ { t } . \\end{align*}"}
+{"id": "8678.png", "formula": "\\begin{align*} { P _ { e , { \\rm { O F D M } } } } = \\frac { 1 } { K ' } \\sum \\limits _ { k = 1 } ^ { K ' } { { P _ e } \\left ( { \\frac { { { K ' } { p _ k ^ { \\star } } { { \\left \\| { { \\bf { h } } \\left [ k \\right ] } \\right \\| } ^ 2 } } } { { \\left ( { { K ' } + { { \\tilde n } _ { { \\rm { s p a n } } } } } \\right ) { \\sigma ^ 2 } / { K ' } } } } \\right ) } , \\end{align*}"}
+{"id": "6776.png", "formula": "\\begin{align*} L b _ { 0 } + b _ { 0 } ^ { \\prime } = q , \\end{align*}"}
+{"id": "8014.png", "formula": "\\begin{align*} & T _ { 1 , r } ( \\underline { p , q } ) = ( 4 ( G ( 0 ) ) ^ r \\sum _ { l _ 1 , l _ 2 , \\dots , l _ r \\atop { 0 \\leq l _ i \\leq L } } U ( l _ 1 ) U ( l _ 2 ) \\dots U ( l _ r ) \\sum _ { l _ 1 ' , l _ 2 ' , \\dots , l _ r ' \\atop { 0 \\leq l _ i ' \\leq L } } U ( l _ 1 ' ) U ( l _ 2 ' ) \\dots U ( l _ r ' ) \\\\ & a _ f ( p _ 1 ^ { 2 l _ 1 } ) a _ f ( p _ 2 ^ { 2 l _ 2 } ) \\dots a _ f ( p _ r ^ { 2 l _ r } ) a _ f ( q _ 1 ^ { 2 l _ 1 ' } ) a _ f ( q _ 2 ^ { 2 l _ 2 ' } ) \\dots a _ f ( q _ r ^ { 2 l _ r ' } ) , \\\\ \\end{align*}"}
+{"id": "2339.png", "formula": "\\begin{align*} \\left ( \\mathcal { L } _ { J \\theta ^ \\sharp } J \\right ) ^ { a n t i - s y m } = J \\left ( \\mathcal { L } _ { \\theta ^ \\sharp } J \\right ) ^ { a n t i - s y m } . \\end{align*}"}
+{"id": "3837.png", "formula": "\\begin{align*} T _ u v = \\sum _ j S _ { j - 1 } u \\Delta _ j v , R ( u , v ) = \\sum _ { | j - j ' | \\leq 1 } \\Delta _ j u \\Delta _ j ' v . \\end{align*}"}
+{"id": "485.png", "formula": "\\begin{align*} 2 & = \\limsup _ { x \\to \\infty } \\frac { g _ u ^ { \\ast 2 } \\ast f _ r ( x ) } { g _ u ( x ) } \\\\ & \\ge \\limsup _ { x \\to \\infty } \\frac { g _ u ^ { \\ast 2 } ( x ) } { g _ u ( x ) } \\liminf _ { x \\to \\infty } \\frac { g _ u ^ { \\ast 2 } \\ast f _ r ( x ) } { g _ u ^ { \\ast 2 } ( x ) } \\\\ & \\ge \\limsup _ { x \\to \\infty } \\frac { g _ u ^ { \\ast 2 } ( x ) } { g _ u ( x ) } , \\end{align*}"}
+{"id": "7003.png", "formula": "\\begin{align*} ( g _ { j k } ) = 2 \\left ( \\begin{array} { c c c } \\omega ^ 2 & 0 & 0 \\\\ 0 & 0 & 2 \\omega ^ { - 2 p + 2 } \\\\ 0 & 2 \\omega ^ { - 2 p + 2 } & 0 \\end{array} \\right ) . \\end{align*}"}
+{"id": "2259.png", "formula": "\\begin{align*} ( x _ j ^ 2 - x _ k ^ 2 ) a _ { j k } b _ { j k } = 0 . \\end{align*}"}
+{"id": "955.png", "formula": "\\begin{align*} \\mathcal { E } _ { j } : = - { { \\mathcal L } } \\widetilde { u } _ { \\mathrm { a p } , j } + \\tilde { { \\mathcal N } } _ { j } ( u _ { \\mathrm { a p } , 1 } , u _ { \\mathrm { a p } , 2 } ) . \\end{align*}"}
+{"id": "1518.png", "formula": "\\begin{align*} \\gcd ( F _ 1 ( x ) , \\dots , F _ n ( x ) ) = x ^ d + c _ { d , 1 } x ^ { d - 1 } + \\dots + c _ { 1 , 1 } x ^ 0 . \\end{align*}"}
+{"id": "1746.png", "formula": "\\begin{align*} \\partial _ t R _ 0 = g ( u _ 0 , R _ 0 ) . \\end{align*}"}
+{"id": "7482.png", "formula": "\\begin{align*} - x \\cdot \\nabla ( L u ) = f ' ( u ) \\left ( x \\cdot \\nabla u \\right ) , \\end{align*}"}
+{"id": "6842.png", "formula": "\\begin{align*} I ( x ) = \\int _ { - \\pi } ^ { \\pi } s \\left ( \\frac { i ( 1 - e ^ { i \\theta } ) } { 2 ( 1 + e ^ { i \\theta } ) } \\right ) \\exp \\left ( \\frac { e ^ { i \\theta } - 1 } { e ^ { i \\theta } + 1 } x \\right ) e ^ { i \\theta ( m + 1 ) } \\left ( e ^ { i \\theta } + 1 \\right ) ^ { n - m - 2 } d \\theta . \\end{align*}"}
+{"id": "2634.png", "formula": "\\begin{align*} [ x \\cdot y , \\alpha ( z ) ] = \\varepsilon ( y , z ) [ x , z ] \\cdot \\alpha ( y ) + \\alpha ( x ) \\cdot [ y , z ] . \\end{align*}"}
+{"id": "469.png", "formula": "\\begin{align*} ( e ^ { \\lambda \\Lambda _ 1 } - 1 ) f _ 1 ( x ) / \\breve g _ 1 ( x ) - \\lambda \\Lambda _ 1 g _ 1 ( x ) / \\breve g _ 1 ( x ) & = \\sum _ { n = 2 } ^ \\infty ( ( \\lambda \\Lambda _ 1 ) ^ n / n ! ) \\ , g ^ { \\ast n } _ 1 ( x ) / \\breve g _ 1 ( x ) \\\\ & = \\sum _ { n = 2 } ^ \\infty ( ( \\lambda \\Lambda _ 1 ) ^ n / n ! ) \\ , \\breve g ^ { \\ast n } _ 1 ( x ) / \\breve g _ 1 ( x ) \\end{align*}"}
+{"id": "2459.png", "formula": "\\begin{align*} f ( t ) = \\exp _ { q ^ { \\prime } } \\left ( \\pm \\sigma _ { \\epsilon } \\ ; t \\right ) = { } _ { 1 } F _ { 0 } \\left ( - \\frac { 1 } { 1 - q ^ { \\prime } } ; \\ ; ; \\mp ( 1 - q ^ { \\prime } ) \\sigma _ { \\epsilon } \\ ; t \\right ) . \\end{align*}"}
+{"id": "1435.png", "formula": "\\begin{align*} \\P _ { ( x _ 1 , x _ 2 ) } ( \\rho > n ) = 2 \\int _ 0 ^ { x _ 2 - x _ 1 } \\frac { 1 } { 2 ^ { n - 1 / 2 } \\sqrt { \\pi } \\Gamma ( n ) } z ^ { n - 1 / 2 } K _ { n - 1 / 2 } ( z ) d z \\end{align*}"}
+{"id": "7192.png", "formula": "\\begin{align*} \\textbf { \\textit { K } } ( t , x ^ { \\prime } , y ^ { \\prime } ) = e ^ { - t \\Lambda _ g } \\delta ( x ^ { \\prime } - y ^ { \\prime } ) = \\sum _ { k = 1 } ^ { \\infty } e ^ { - t \\tau _ k } \\textbf { \\textit { U } } _ k ( x ^ { \\prime } ) \\otimes \\textbf { \\textit { U } } _ k ( y ^ { \\prime } ) . \\end{align*}"}
+{"id": "4453.png", "formula": "\\begin{align*} | a _ { C } ( n ) | \\leq \\left ( \\sum _ { i = 1 } ^ s | c _ { i } | \\right ) \\tau ( n ) \\sqrt { n } \\end{align*}"}
+{"id": "2575.png", "formula": "\\begin{align*} \\mathcal { Z } \\left ( \\hat { \\mu } \\right ) \\subset \\bigcup _ { i = 1 } ^ { r } \\rho ^ { - i } \\bigcup _ { j = 0 } ^ { \\infty } \\frac { b _ { j } } { p ^ { j } N _ { j } } , \\ 1 \\leq i \\leq r , \\ b _ { j } \\in \\mathbb { Z } \\setminus N _ { j } \\mathbb { Z } . \\end{align*}"}
+{"id": "3022.png", "formula": "\\begin{align*} C : \\mathbb { R } ^ { 3 } \\rightarrow \\mathfrak { s o } _ { 3 } \\xi \\mapsto C _ { \\xi } C _ { \\xi } \\left ( y \\right ) = \\xi \\times y \\end{align*}"}
+{"id": "5960.png", "formula": "\\begin{align*} A ( X ) : = \\left ( \\begin{array} { c } P _ H [ \\Delta u - ( u \\cdot \\nabla ) u - \\nabla n \\cdot \\Delta n ] \\\\ \\Delta n - ( u \\cdot \\nabla ) n - \\varphi ( n ) \\end{array} \\right ) . \\end{align*}"}
+{"id": "1950.png", "formula": "\\begin{align*} B _ { p , \\nu } ( x , y ) = \\sqrt { \\frac { 2 p } { \\pi } } \\displaystyle { \\int _ { 0 } ^ { 1 } t ^ { x - \\frac { 3 } { 2 } } ( 1 - t ) ^ { y - \\frac { 3 } { 2 } } } K _ { \\nu + \\frac { 1 } { 2 } } \\left ( \\frac { p } { t ( 1 - t ) } \\right ) d t , \\end{align*}"}
+{"id": "8313.png", "formula": "\\begin{align*} \\sqrt { \\mu _ 1 ( 0 ) } \\tanh ( \\sqrt { \\mu _ 1 ( 0 ) } / 2 ) > 0 . 0 6 7 \\tan ( 1 . 3 4 ) = 0 . 2 9 . \\end{align*}"}
+{"id": "8056.png", "formula": "\\begin{align*} \\gamma _ k = \\frac { a _ k ^ 2 \\lvert \\mathbf { \\hat { h } } _ k ^ { \\textrm { T } } \\mathbf { p } _ k \\rvert ^ 2 } { \\sum \\limits _ { \\substack { i = 1 \\\\ i \\neq k } } ^ K a _ i ^ 2 \\lvert \\mathbf { h } _ k \\mathbf { p } _ i \\rvert ^ 2 + \\sigma _ w ^ 2 } . \\end{align*}"}
+{"id": "3495.png", "formula": "\\begin{align*} | z _ j ' - z _ { j + 1 } | \\geq \\frac { 1 } { 4 } I ( z _ j ' ) \\geq \\frac { 1 } { 8 } \\min ( ( z _ j ' - \\ell q _ n , ( \\ell + 1 ) q _ n - z _ j ' ) ) \\geq \\frac { b _ n } { 8 } = \\frac { \\tau _ n } { 8 } q _ n . \\end{align*}"}
+{"id": "2456.png", "formula": "\\begin{align*} F _ { q } ^ { k } ( s ) = \\frac { 1 } { Q _ { m } ( 2 - q ) } ( - 1 ) ^ { k } \\frac { \\Gamma ( m + k ) } { s ^ { m + k } } . \\end{align*}"}
+{"id": "4053.png", "formula": "\\begin{align*} \\phi _ H ( I ) = - 1 / 2 + ( ( 1 / 2 - H ) I ) \\vee ( - 1 / 2 ) \\quad \\ ; I \\geq 2 . \\end{align*}"}
+{"id": "9239.png", "formula": "\\begin{align*} \\begin{aligned} P ( s I + x ) = \\det ( s I + \\operatorname { d i a g } ( \\lambda _ 1 , \\dots , \\lambda _ n ) ) = \\prod \\limits _ 1 ^ n ( s + \\lambda _ k ) . \\end{aligned} \\end{align*}"}
+{"id": "7096.png", "formula": "\\begin{align*} \\mbox { $ ( ( X _ { 2 , 6 } \\setminus N ( c _ 2 ) ) \\cup \\{ c _ 6 \\} ) \\cap { \\cal N } = \\emptyset $ . } \\end{align*}"}
+{"id": "510.png", "formula": "\\begin{align*} \\phi _ { n } ( t ) = ( 2 ^ { n } \\sqrt { \\pi } n ! ) ^ { - 1 / 2 } \\exp ( - t ^ { 2 } / 2 ) H _ { n } ( t ) , \\ n = 0 , 1 , 2 , . . . , \\end{align*}"}
+{"id": "3159.png", "formula": "\\begin{align*} { _ { 3 } } F _ { 2 } \\left ( \\begin{array} { c c c } \\chi _ 4 , & \\chi _ 4 , & \\chi _ 4 \\\\ & \\varepsilon , & \\varepsilon \\end{array} \\mid 1 \\right ) & = \\binom { \\chi _ 8 } { \\chi _ 8 ^ 2 } \\binom { \\chi _ 8 } { \\chi _ 8 ^ 3 } + \\binom { \\chi _ 8 ^ 5 } { \\chi _ 8 ^ 2 } \\binom { \\chi _ 8 ^ 5 } { \\overline { \\chi _ 8 } } \\\\ & = \\frac { \\chi _ 8 ( - 1 ) } { q ^ 2 } [ J ( \\chi _ 8 , \\chi _ 8 ^ 6 ) J ( \\chi _ 8 , \\chi _ 8 ^ 5 ) + J ( \\chi _ 8 ^ 5 , \\chi _ 8 ^ 6 ) J ( \\chi _ 8 ^ 5 , \\chi _ 8 ) ] . \\end{align*}"}
+{"id": "2866.png", "formula": "\\begin{align*} \\sup _ { x } | G _ x ^ { ( n ) } | \\le \\frac { C } { n } , n = 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "698.png", "formula": "\\begin{align*} \\aligned \\Phi _ \\lambda ( u ) = \\frac { 1 } { 2 } Q _ \\lambda ( u ) - \\frac { 1 } { 2 p } J ( u ) , \\endaligned \\end{align*}"}
+{"id": "3462.png", "formula": "\\begin{align*} & \\prod _ { j = k } ^ { x _ 2 } | \\cos ( \\pi \\theta _ j ) ) | \\leq C ( \\varepsilon ) c _ { n , \\ell } c _ { n , \\ell + 1 } e ^ { ( - \\ln { 2 } + \\varepsilon ) | x _ 2 - k | } , \\ \\ \\\\ & \\prod _ { j = x _ 1 } ^ k | \\cos ( \\pi \\theta _ j ) ) | \\leq C ( \\varepsilon ) e ^ { ( - \\ln 2 + \\varepsilon ) | k - x _ 1 | } . \\end{align*}"}
+{"id": "6119.png", "formula": "\\begin{align*} g _ I = \\big ( g _ { \\{ i \\} } \\cup g _ { I \\setminus \\{ i \\} } \\big ) \\setminus \\big ( g _ { \\{ i \\} } \\cap g _ { I \\setminus \\{ i \\} } \\big ) \\ , , \\forall i \\in I . \\end{align*}"}
+{"id": "4357.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\int _ { \\{ \\Psi < - l \\} } | f | _ h ^ 2 c _ t ^ n ( - \\Psi ) = \\int _ { \\{ - t \\leq \\Psi < - l \\} } | f | _ h ^ 2 . \\end{align*}"}
+{"id": "193.png", "formula": "\\begin{gather*} \\theta _ { j , ( 1 ) } ( Y _ { k , ( 1 ) } ) = \\delta _ { j , k } = \\theta _ { j , ( 2 ) } ( Y _ { k , ( 2 ) } ) . \\end{gather*}"}
+{"id": "5405.png", "formula": "\\begin{align*} \\epsilon _ 1 b _ { n - 1 } u _ n ( x ) \\leq u ( y ) - u ( x ) + \\sum _ { \\alpha = 1 } ^ { n - 1 } ( y _ \\alpha - x _ \\alpha ) u _ \\alpha ( x ) \\leq C b _ { n - 1 } ^ 2 \\end{align*}"}
+{"id": "9153.png", "formula": "\\begin{align*} \\begin{aligned} h ( x ) = & \\ , h _ 0 + A \\sin ( 1 0 x ) \\\\ & \\ , + \\left \\{ \\begin{array} { c c } ( x - \\pi ) ^ 2 - 1 & x < \\pi \\\\ \\cos ( x - \\pi ) - 2 & x \\in [ \\pi , \\ , 2 \\pi ) \\\\ ( x - 2 \\pi ) ^ 2 - 3 & x \\ge 2 \\pi \\end{array} \\right . \\end{aligned} \\end{align*}"}
+{"id": "6250.png", "formula": "\\begin{align*} \\eta _ i : = \\frac { \\alpha ^ g ( X _ i , X _ i ) } { \\langle A X _ i , X _ i \\rangle } \\in \\Gamma ( T ^ \\perp _ g M \\otimes \\mathbb { C } ) . \\end{align*}"}
+{"id": "5330.png", "formula": "\\begin{align*} f ( z , t ) = \\sum _ { \\delta \\in \\widehat { K } _ 0 } f _ \\delta ( z , t ) , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , f _ \\delta ( z , t ) = \\sum _ { j = 1 } ^ { d ( p , q ) } g _ { ( p , q ) , j } ( r , t ) \\ , P _ { p , q } ^ j ( z ) , \\ , \\ , \\ , \\delta = ( p , q ) . \\end{align*}"}
+{"id": "4143.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n - 1 } } \\nabla ^ k [ P _ t ^ L ( x ' - y ' ) ] d y ' = 0 , \\quad \\forall t > 0 , \\ , k \\in \\N . \\end{align*}"}
+{"id": "187.png", "formula": "\\begin{align*} \\widetilde { X } = \\varprojlim _ { G ' \\subset G } X _ { G ' } . \\end{align*}"}
+{"id": "7085.png", "formula": "\\begin{align*} \\varphi _ 2 ( x ) = u ( x ) + t v _ 2 ( x ) \\end{align*}"}
+{"id": "4773.png", "formula": "\\begin{align*} T x = \\frac { 1 } { \\lambda } x . \\end{align*}"}
+{"id": "3201.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\mu ( h ^ { - k } ( A ) \\cap B ) = \\mu ( A ) \\mu ( B ) ; \\end{align*}"}
+{"id": "3429.png", "formula": "\\begin{align*} \\delta _ n : = \\frac { \\ln \\| q _ n ( \\theta - \\frac { 1 } { 2 } ) \\| - \\ln \\| q _ n \\alpha \\| } { q _ n } . \\end{align*}"}
+{"id": "7642.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { * , \\xi } = & ~ [ ( A - B ^ 2 R ^ { - 1 } P _ t ) x _ t ^ { * , \\xi } - B ^ 2 R ^ { - 1 } \\varphi _ t ^ { * , \\xi } - B h ( \\mu _ t ^ { * , \\xi } ) \\\\ & + f ( \\nu _ t ^ { * , \\xi } ) + b ( \\mu _ t ^ { * , \\xi } ) ] d t + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ 0 ^ { * , \\xi } = & ~ \\xi , \\end{aligned} \\right . \\end{align*}"}
+{"id": "2049.png", "formula": "\\begin{align*} a _ { i , i } ( z ) = \\abs { z } ^ \\gamma \\sum _ { \\stackrel { 1 \\leq j \\leq 3 } { j \\neq i } } z _ j ^ 2 , a _ { i , j } ( z ) = - \\abs { z } ^ \\gamma z _ i z _ j \\ \\ { \\rm f o r } \\ \\ i \\neq j , \\end{align*}"}
+{"id": "5189.png", "formula": "\\begin{align*} A _ t + \\mathbb { P } ( B \\times u ) = - \\mu \\nabla \\times B \\textrm { o n } \\ L ^ { 2 } _ { \\sigma } ( \\mathbb { R } ^ { 3 } ) . \\end{align*}"}
+{"id": "299.png", "formula": "\\begin{align*} E _ 1 & = \\textstyle \\bigcup _ { \\lambda \\in [ 0 , 1 ] } \\delta _ { \\lambda } ( ( 1 , 1 , 1 ) ) \\cup \\delta _ { \\lambda } ( ( - 1 , - 1 , 1 ) ) , \\\\ E _ 2 & = ( \\pi ( 0 , 1 ) \\times \\{ 0 \\} ) \\cup ( \\pi ( 0 , 1 ) \\times \\{ t \\} ) \\cup ( \\{ ( 0 , 0 ) \\} \\times ( 0 , 1 ) ) , \\\\ E _ 3 & = \\{ ( x , 0 , 0 ) : x \\in ( - 1 , 1 ) \\} , \\end{align*}"}
+{"id": "6624.png", "formula": "\\begin{align*} h = \\nu & = 0 \\\\ \\lambda \\varepsilon _ 1 \\delta _ 1 = \\lambda \\varepsilon _ 1 \\delta _ 4 & = 0 \\\\ \\varepsilon _ 2 \\delta _ 2 & = \\varepsilon _ 3 \\delta _ 3 \\\\ \\varepsilon _ 2 \\delta _ 5 & = \\varepsilon _ 3 \\delta _ 6 \\\\ \\lambda ^ 2 + \\rho ^ 2 - \\varepsilon _ 1 \\mu \\delta _ 1 + \\varepsilon _ 2 \\rho \\delta _ 2 & = 0 \\\\ \\lambda ^ 2 + \\rho ^ 2 - \\varepsilon _ 1 \\mu \\delta _ 4 + \\varepsilon _ 2 \\rho \\delta _ 5 & = 0 . \\end{align*}"}
+{"id": "450.png", "formula": "\\begin{align*} f ( x ) = ( e ^ \\lambda - 1 ) ^ { - 1 } \\sum _ { n = 1 } ^ \\infty ( \\lambda ^ n / n ! ) \\ , g ^ { \\ast n } ( x ) . \\end{align*}"}
+{"id": "3905.png", "formula": "\\begin{align*} I _ \\delta ( u ) = \\frac { \\delta ^ 2 } { 2 } \\int _ { \\Omega } \\left ( K ( x ) \\nabla u | \\nabla u \\right ) - \\frac { 1 } { p + 1 } \\int _ { \\Omega } ( u - q ) ^ { p + 1 } _ + . \\end{align*}"}
+{"id": "9066.png", "formula": "\\begin{align*} \\lambda _ { m - l + 1 } + \\mu _ { 1 } = l - 1 . \\end{align*}"}
+{"id": "9009.png", "formula": "\\begin{align*} \\langle V , z _ \\tau \\rangle = v _ \\tau + \\sum _ { q = 1 } ^ { s + 1 } ( - 1 ) ^ q v _ { \\tau _ q } , \\end{align*}"}
+{"id": "5692.png", "formula": "\\begin{align*} 0 \\in ( F _ k + B ) ( z ^ k _ * ) = ( F + B ) ( z ^ k _ * ) + \\frac { 1 } { \\rho _ k } ( z ^ k _ * - z ^ k ) . \\end{align*}"}
+{"id": "3349.png", "formula": "\\begin{align*} K _ i = \\langle \\{ a _ i , b _ i , c _ i \\} , \\{ a _ i , c _ i , d _ i \\} , \\{ b _ i , d _ i \\} \\rangle , \\ L _ i = \\langle \\{ a _ i , b _ i , c _ i \\} , \\{ b _ i , d _ i \\} \\rangle . \\end{align*}"}
+{"id": "7023.png", "formula": "\\begin{align*} 0 \\ , \\neq \\ , \\left \\vert \\ ! \\begin{array} { c c c } a _ { 1 , 1 } & 0 & b _ 1 \\\\ a _ { 2 , 1 } & a _ { 2 , 2 } & b _ 2 \\\\ 0 & 0 & a _ { 1 , 1 } ^ 2 \\end{array} \\ ! \\right \\vert \\ , = \\ , a _ { 1 , 1 } \\ , a _ { 2 , 2 } \\ , a _ { 1 , 1 } ^ 2 . \\end{align*}"}
+{"id": "9257.png", "formula": "\\begin{align*} 8 ^ { n - m } ( \\Delta u ) ^ m \\wedge \\beta _ n ^ { n - m } = n ! \\det ( A , \\dots , A , 8 I , \\dots , 8 I ) \\Omega _ { 2 n } , \\end{align*}"}
+{"id": "1997.png", "formula": "\\begin{align*} f ( x ) & = \\log ( x + 2 ) + 2 , \\\\ \\sigma ( x , t ) & = \\log ( x - t + 2 ) + \\log ( - x - t + 2 ) + 4 , \\\\ \\mu ( x , t ) & = \\log ( x - t + 2 ) - \\log ( - x - t + 2 ) , \\\\ \\sigma ( x , 0 ) & = \\log ( 2 - x ) + \\log ( x + 2 ) + 4 , \\\\ \\mu ( x , 0 ) & = \\log ( x + 2 ) - \\log ( 2 - x ) . \\end{align*}"}
+{"id": "7940.png", "formula": "\\begin{align*} c = 4 + ( n + 3 ) \\left ( n + 2 + \\norm { \\tilde \\Psi } _ { H ^ 1 } ^ 2 \\right ) . \\end{align*}"}
+{"id": "5303.png", "formula": "\\begin{align*} \\hat { f } ( \\pi ) = \\int _ { G } f ( g ) \\pi ( g ) d g \\end{align*}"}
+{"id": "5718.png", "formula": "\\begin{align*} f ' ( x ) = f ( x + r ) = a x ^ { q + 1 } + b ' x ^ q + c ' x . \\end{align*}"}
+{"id": "842.png", "formula": "\\begin{align*} - H ( \\rho ^ \\varepsilon _ 0 ) ( z _ l ) \\geq \\frac { 1 } { 2 R } \\phantom { x x } \\phantom { x x } H ( \\rho ^ \\varepsilon _ 0 ) ( z _ l ) + H ( \\rho ^ \\varepsilon _ 0 ) ( z _ r ) = \\frac { 2 \\varepsilon \\rho _ 0 } { R ^ 2 - \\varepsilon ^ 2 \\rho _ 0 ^ 2 } \\leq \\frac { 4 \\varepsilon \\rho _ 0 } { R ^ 2 } \\end{align*}"}
+{"id": "6363.png", "formula": "\\begin{align*} \\operatorname { E } ( X ^ r ) = \\frac { \\theta ^ { r / 2 } } { \\operatorname { B } ( a , b ) } \\ , S _ r ( a , b ) , r < 2 . \\end{align*}"}
+{"id": "8268.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | g _ { n + 1 } | \\sum _ { i = n + 1 } ^ { \\infty } \\psi _ i ( t _ p ) = 0 . \\end{align*}"}
+{"id": "6682.png", "formula": "\\begin{align*} d \\mu ( \\xi , \\eta ) = d _ \\epsilon ( \\xi , \\eta ) ^ { - 2 h ( g ) / \\epsilon } \\cdot d v _ p ( \\xi ) d v _ p ( \\eta ) \\end{align*}"}
+{"id": "5663.png", "formula": "\\begin{align*} & P \\otimes _ { G } Q \\rightarrow P \\otimes _ { G } Q \\\\ & p \\otimes q \\mapsto p \\kappa ^ { - 1 } \\otimes \\kappa q = p \\otimes q \\end{align*}"}
+{"id": "4969.png", "formula": "\\begin{align*} \\begin{aligned} \\mathsf { M } ^ n = & \\{ \\mathsf { m } _ 1 , \\cdots , \\mathsf { m } _ n \\} \\\\ = & \\left \\{ g \\bigl ( \\xi _ 0 \\bigr ) , g \\bigl ( \\xi _ 1 ^ { \\mathsf { M } } \\bigr ) , \\cdots , g \\bigl ( \\xi _ { n - 1 } ^ { \\mathsf { M } } \\bigr ) \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "2094.png", "formula": "\\begin{align*} p _ { S } ( x ) \\le \\sum _ { i = 1 } ^ { d + 1 } \\frac { d y } { \\Omega _ d \\cdot d ( y , H _ i ) ^ d } \\le \\frac { d + 1 } { \\Omega _ d } \\cdot \\sup _ { x \\in \\mathbb R ^ n } \\int _ { P _ { 1 , x } } \\frac { d y } { d ( y , H _ 1 ) ^ d } . \\end{align*}"}
+{"id": "4856.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c } B & C \\\\ C ^ T & D \\end{array} \\right ) \\end{align*}"}
+{"id": "1310.png", "formula": "\\begin{align*} I ( u ) = m | | u | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } + | | \\nabla _ { H } u | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } - { \\rm R e } \\int _ { \\mathbb { H } ^ n } f ( u ) \\overline { u } d x . \\end{align*}"}
+{"id": "6551.png", "formula": "\\begin{align*} n _ t = \\nabla \\cdot ( \\nabla n - \\chi ( c ) n \\nabla c ) , \\quad c _ { t } = \\Delta c - n c , \\qquad x \\in \\Omega , \\ , t > 0 , \\end{align*}"}
+{"id": "7992.png", "formula": "\\begin{align*} C \\geq h ( x , t ) - h ( x , 0 ) = \\int _ { 0 } ^ { t } ( P - h ) ( x , t ) d t . \\end{align*}"}
+{"id": "252.png", "formula": "\\begin{align*} u _ \\alpha - 1 & = \\sum _ { j = 1 } ^ s a _ j ( \\sum _ { i = 1 } ^ k w _ { j i } X _ i ) + \\sum _ { j = 1 } ^ t b _ j ( \\sum _ { i = 1 } ^ k v _ { j i } X _ i ) \\\\ & \\equiv \\sum _ { j = 1 } ^ s a _ j ( \\prod _ { i = 1 } ^ k g _ i ^ { w _ { j i } } - 1 ) + \\sum _ { j = 1 } ^ t b _ j ( \\prod _ { i = 1 } ^ k g _ i ^ { v _ { j i } } - 1 ) \\mod J ^ 2 \\\\ & = \\sum _ { j = 1 } ^ s a _ j ( f _ j - 1 ) + \\sum _ { j = 1 } ^ t b _ j ( h _ j - 1 ) , \\end{align*}"}
+{"id": "6276.png", "formula": "\\begin{align*} \\sum _ { k } \\varphi _ k ( d \\phi _ { i k } + \\Omega _ { i j } ) & = \\sum _ k \\varphi _ k d \\phi _ { i k } + \\sum _ l \\varphi _ l \\phi _ { i l } \\wedge \\Big ( \\sum _ k \\varphi _ k \\phi _ { k l } \\Big ) = \\sum _ k \\varphi _ k d \\phi _ { i k } + \\sum _ l \\varphi _ l \\phi _ { i l } \\wedge ( 2 \\varphi _ l \\phi _ { l l } ) \\\\ & = \\sum _ k \\varphi _ k d \\phi _ { i k } - \\sum _ { l } \\phi _ { i l } \\wedge d \\varphi _ l = d \\Big ( \\sum _ k \\varphi _ k \\phi _ { i k } \\Big ) = 0 . \\end{align*}"}
+{"id": "8487.png", "formula": "\\begin{align*} & \\overline { \\mathcal { P } ^ - ( f ( z ) e ^ { 2 i z x } ) } = - \\lim _ { \\varepsilon \\rightarrow 0 } \\frac { 1 } { 2 \\pi i } \\int _ { R } \\frac { \\bar { f } ( s ) e ^ { - 2 i s x } } { s - ( z + i \\varepsilon ) } d s = - \\mathcal { P } ^ + ( \\bar f ( z ) e ^ { - 2 i z x } ) \\\\ & = - \\int _ { 2 x } ^ { + \\infty } \\widehat { \\bar { f } \\ } ( \\xi ) e ^ { i z ( \\xi - 2 x ) } \\mathrm { d } \\xi . \\end{align*}"}
+{"id": "7867.png", "formula": "\\begin{align*} Z ^ 1 ( v , s ) = \\Psi ^ q + v \\end{align*}"}
+{"id": "2678.png", "formula": "\\begin{align*} u _ { t t } - \\Delta u + b ( t , x ) | u _ t | ^ { m - 1 } u _ t + | u | ^ { p - 1 } u = 0 , \\end{align*}"}
+{"id": "2691.png", "formula": "\\begin{align*} \\phi ' ( z ) = \\frac { M _ d ( z ) - z M _ d ' ( z ) } { M _ d ( z ) ^ 2 } , \\phi '' ( z ) = \\frac { - z M _ d ( z ) M _ d '' ( z ) - 2 M _ d ( z ) M _ d ' ( z ) + 2 z ( M _ d ' ( z ) ) ^ 2 } { M _ d ( z ) ^ 3 } . \\end{align*}"}
+{"id": "4044.png", "formula": "\\begin{align*} G ^ { ( 1 ) } \\vee . . . \\vee G ^ { ( k ) } = \\vee _ { \\gamma \\in \\Gamma } G ^ { ( \\gamma ) } . \\end{align*}"}
+{"id": "6783.png", "formula": "\\begin{align*} q = \\frac { b _ { 0 } ^ { \\prime \\prime } + b _ { 0 } ^ { \\prime } } { b _ { 0 } + 1 } . \\end{align*}"}
+{"id": "7704.png", "formula": "\\begin{align*} \\sum _ { q = r + 1 - \\sum _ { d = j } ^ { i + 1 } \\ell _ { d } } ^ { r - \\sum _ { d = j + 1 } ^ { i + 1 } \\ell _ { d } } a _ q ^ 2 \\leq \\frac { \\beta ^ 2 } { g \\big ( \\lfloor ( 1 - ( 1 + \\tilde \\varepsilon ) ^ { - j } ) r \\rfloor \\big ) ^ 2 } \\leq \\frac { h ^ 2 } { 1 6 ^ 2 } \\cdot 1 6 ^ { j - i } . \\end{align*}"}
+{"id": "2892.png", "formula": "\\begin{align*} { \\bf c } _ { 2 , \\mu } ( t ) = \\left ( \\begin{array} { c } C ^ { ( p ) } _ { 0 , 0 , \\mu } ( t ) \\\\ \\vdots \\\\ C ^ { ( p ) } _ { n , n , \\mu } ( t ) \\end{array} \\right ) \\end{align*}"}
+{"id": "6451.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { r _ n ^ 2 } { h _ n } = 0 . \\end{align*}"}
+{"id": "6856.png", "formula": "\\begin{align*} 0 < | b _ T ( x _ 0 , b _ 0 , \\xi _ 0 ) | = | b ( T , x _ T ) | . \\end{align*}"}
+{"id": "6993.png", "formula": "\\begin{align*} P ^ X & ( s , x ; t , { \\cal X } ) \\\\ & = e ^ { - \\nu ( \\{ | z | \\ge 1 \\} ) ( t - s ) } P ^ Z ( s , x ; t , { \\cal X } ) \\\\ & \\ \\ \\ + \\int _ { s } ^ t \\int _ { \\{ | x _ 2 | \\ge 1 \\} } e ^ { - \\nu ( \\{ | z | \\ge 1 \\} ) ( t _ 1 - s ) } P ^ X ( t _ 1 , x _ 1 + J ( t _ 1 , x _ 1 , x _ 2 ) ; t , { \\cal X } ) P ^ Z ( s , x ; t _ 1 , d x _ 1 ) \\nu ( d x _ 2 ) d t _ 1 . \\end{align*}"}
+{"id": "7004.png", "formula": "\\begin{align*} u \\ , = \\ , \\frac { 1 } { 3 \\ , z ^ 2 } \\Big \\{ \\big ( 1 - 2 \\ , y + y ^ 2 - 2 \\ , x z \\big ) ^ { 3 / 2 } - ( 1 - y ) \\ , \\big ( 1 - 2 \\ , y + y ^ 2 - 3 \\ , x z \\big ) \\Big \\} , \\end{align*}"}
+{"id": "5316.png", "formula": "\\begin{align*} \\rho ^ { \\lambda } _ { k } ( f ) e ^ { n - 1 } _ { k , \\lambda } ( z , t ) = e ^ { i \\lambda t } f ^ { - \\lambda } \\ast _ { - \\lambda } \\varphi _ { k , \\lambda } ^ { n - 1 } ( z ) . \\end{align*}"}
+{"id": "520.png", "formula": "\\begin{align*} S _ { R } ^ { \\alpha } f ( x ) = \\sum _ { k = - \\infty } ^ { \\infty } S _ { R } ^ { \\alpha } f _ { k } ( x ) . \\end{align*}"}
+{"id": "7726.png", "formula": "\\begin{align*} \\mathbb { E } { \\bf g } _ { \\rm G E } ( G ) = + \\infty . \\end{align*}"}
+{"id": "7958.png", "formula": "\\begin{align*} g _ d ( \\ell ) = \\begin{cases} \\ell , & d = 1 , \\\\ \\log \\ell , & d = 2 , \\\\ 1 , & d \\geq 3 , \\end{cases} \\end{align*}"}
+{"id": "7729.png", "formula": "\\begin{align*} \\Gamma = \\bigoplus _ { n = 1 } ^ \\infty \\Gamma _ n , F = \\prod _ { n = 1 } ^ \\infty F _ n \\quad G = \\Gamma \\rtimes F \\end{align*}"}
+{"id": "7415.png", "formula": "\\begin{align*} _ { \\rm r e f } [ V _ \\lambda ] = q ^ { - \\frac { c } { 2 4 } } q ^ { \\frac { ( \\lambda | \\lambda + 2 \\rho ) } { 2 ( k + h ^ \\vee ) } - \\lambda ( x + y ) } \\ \\prod _ { i = 1 } ^ L X _ i \\ \\prod _ { 1 \\leq i < j \\leq L } X _ { i , j } \\end{align*}"}
+{"id": "8441.png", "formula": "\\begin{align*} K f = \\int _ { - \\infty } ^ x u _ y ( y ) e ^ { 2 i z y } d y \\int _ { - \\infty } ^ { y } \\bar { u } _ s ( s ) e ^ { - 2 i z s } f d s , \\end{align*}"}
+{"id": "8756.png", "formula": "\\begin{align*} f ( z ) - f ( 0 ) & = \\int _ { 0 } ^ 1 \\left ( \\frac { \\partial f } { \\partial x } ( t z ) x - \\frac { \\psi _ 0 } { \\psi _ 1 } y \\frac { \\partial f } { \\partial x } ( t z ) \\right ) d t \\\\ & = \\int _ { 0 } ^ 1 \\left ( x - \\frac { \\psi _ 0 } { \\psi _ 1 } y \\right ) \\frac { \\partial f } { \\partial x } ( t z ) d t = \\left ( x - \\frac { \\psi _ 0 } { \\psi _ 1 } y \\right ) h _ 1 ( z ) , \\\\ \\end{align*}"}
+{"id": "2671.png", "formula": "\\begin{align*} \\mathcal { H } ^ { \\prime } ( t ) = & \\lambda ( t ) \\mathcal { E } ^ { \\prime } ( t ) + \\lambda ^ { \\prime } ( t ) \\mathcal { E } ( t ) + \\mu \\alpha ^ { \\prime } ( t ) \\mathcal { E } ^ r ( t ) \\langle P ( u _ t ) , u \\rangle + \\mu \\alpha ( t ) \\mathcal { E } ^ r ( t ) \\frac { d } { d t } \\langle P ( u _ t ) , u \\rangle \\\\ & + \\mu r \\alpha ( t ) \\mathcal { E } ^ { r - 1 } ( t ) \\mathcal { E } ^ { \\prime } ( t ) \\langle P ( u _ t ) , u \\rangle . \\end{align*}"}
+{"id": "3335.png", "formula": "\\begin{align*} \\chi _ i \\cap \\psi _ i & = \\left \\{ \\begin{aligned} & \\psi _ i & & ( i \\neq j ) \\\\ & \\sigma \\cap \\psi _ j & & ( i = j ) \\end{aligned} \\right . \\\\ & = \\left \\{ \\begin{aligned} & \\psi _ i & & ( i \\neq j ) \\\\ & \\psi _ j \\setminus \\{ x \\} & & ( i = j ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1129.png", "formula": "\\begin{align*} P _ { \\min } ^ { \\rm { S - N } } \\left ( { \\overline S , \\overline \\varepsilon , \\overline R } \\right ) = \\mathop { \\arg \\min } \\limits _ { { W _ m } \\in \\left [ { 0 , W } \\right ] } \\ ; \\left ( { { p _ K } \\left ( { \\bar S , \\bar \\varepsilon , { W _ m } } \\right ) + p _ { b , m } ^ * + p _ { b , o } ^ * } \\right ) . \\end{align*}"}
+{"id": "8920.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { n } \\left ( - 1 \\right ) ^ { m } \\sum _ { k = 0 } ^ { m } \\binom { m } { k } \\binom { n - m } { k } \\rho ^ { k } = \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ ( 1 - \\rho ) ^ { n / 2 } & & n \\end{array} \\right . . \\end{align*}"}
+{"id": "8305.png", "formula": "\\begin{align*} \\begin{aligned} & A _ { b e n } y '' _ { b e n } + M _ { b e n } y _ { b e n } = E y _ { b e n } \\\\ & A _ { n b } y '' _ { n b } + M _ { n b } y _ { n b } = E y _ { n b } . \\end{aligned} \\end{align*}"}
+{"id": "5748.png", "formula": "\\begin{align*} \\gamma ( u ^ q + u ) + ( \\gamma + 1 ) ( v ^ q + v ) & = c , \\\\ \\gamma u ^ { q + 1 } + ( \\gamma + 1 ) v ^ { q + 1 } & = d , \\end{align*}"}
+{"id": "1521.png", "formula": "\\begin{align*} F _ i ( x ) = \\tilde { F } _ i ( x ) H ( x ) + \\frac { e } { \\| \\hat { F } _ i ( x ) \\| } \\hat { F } _ i ( x ) . \\end{align*}"}
+{"id": "7938.png", "formula": "\\begin{align*} \\norm { h _ 1 } _ { L ^ 2 } ^ 2 & = \\int _ \\S \\left [ \\prod _ { i = 1 } ^ n ( \\eta _ i ( x + r ) - \\eta _ i ( x ) ) \\sin ^ { [ n ] } ( \\tilde \\Psi ( x + r ) - \\tilde \\Psi ( x ) ) \\right ] ^ 2 \\ \\d x \\\\ & \\le \\int _ \\S \\prod _ { i = 1 } ^ n ( \\eta _ i ( x + r ) - \\eta _ i ( x ) ) ^ 2 \\ \\d x \\\\ & \\le \\prod _ { i = 1 } ^ n \\norm { \\partial \\eta _ i } _ { L ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "3763.png", "formula": "\\begin{align*} T _ s ^ 2 = & ( q - 1 ) T _ s + q & & \\\\ T _ s T _ { s ' } = & T _ { s ' } T _ s & & \\\\ T _ s T _ { s ' } T _ s = & T _ { s ' } T _ s T _ { s ' } & & \\end{align*}"}
+{"id": "2100.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Pr ( K = k ( \\theta _ 0 ) \\mid X _ 1 , \\ldots , X _ n ) = 1 ~ ~ ~ \\mathrm { a . s . } [ P _ { \\theta _ 0 } ] . \\end{align*}"}
+{"id": "2212.png", "formula": "\\begin{align*} \\phi = \\frac { ( 2 n + 1 ) \\pi } { N } , \\end{align*}"}
+{"id": "4367.png", "formula": "\\begin{align*} ( \\sqrt { - 1 } & \\Theta _ { T M } + \\varpi \\otimes I d _ { T M } ) ( \\kappa _ 1 \\otimes \\kappa _ 2 , \\kappa _ 1 \\otimes \\kappa _ 2 ) \\ge 0 \\\\ & \\forall \\kappa _ 1 , \\kappa _ 2 \\in T M w i t h \\langle \\kappa _ 1 , \\kappa _ 2 \\rangle = 0 \\end{align*}"}
+{"id": "7610.png", "formula": "\\begin{align*} \\tilde { \\xi } ( g \\gamma ) = \\tilde { \\xi } ( g ) , \\qquad g \\in G ^ \\circ , \\gamma \\in \\Gamma ^ \\circ , \\end{align*}"}
+{"id": "1877.png", "formula": "\\begin{align*} \\begin{aligned} & { \\rm ( a ) } \\ \\pi _ 1 ( M ) = \\pi _ 2 ( M ) = 0 \\ , . \\\\ & { \\rm ( b ) } \\ \\ , M \\ , . \\\\ & { \\rm ( c ) } \\ \\ , M \\ , . \\end{aligned} \\end{align*}"}
+{"id": "4034.png", "formula": "\\begin{align*} \\kappa _ { V , I } : = \\sum _ { J \\subset I } ( - 1 ) ^ { | J | - | I | } K _ { V , J } . \\end{align*}"}
+{"id": "255.png", "formula": "\\begin{align*} u _ \\alpha - 1 & = \\sum _ { i = 1 } ^ s a _ i \\left ( \\sum _ { j = 1 } ^ k w _ { i j } X _ j \\right ) + \\sum _ { i = 1 } ^ t b _ i X _ { j _ i } \\\\ & \\equiv \\sum _ { i = 1 } ^ s a _ i \\left ( \\prod _ { j = 1 } ^ k g _ j ^ { w _ { i j } } - 1 \\right ) + \\sum _ { i = 1 } ^ t b _ i ( g _ { j _ i } - 1 ) ) ( \\mod J ^ 2 ) \\\\ & = \\sum _ { i = 1 } ^ s a _ i ( f _ i - 1 ) + \\sum _ { i = 1 } ^ t b _ i ( g _ { j _ i } - 1 ) , \\end{align*}"}
+{"id": "327.png", "formula": "\\begin{align*} L _ 2 ( x ) = C \\displaystyle | b ( x + h ) - b ( x ) | g _ \\gamma ( f ) ( x ) \\leq C \\displaystyle | h | g ( f ) ( x ) . \\end{align*}"}
+{"id": "560.png", "formula": "\\begin{align*} \\Omega : = \\bigcup _ { \\theta \\colon e ^ { i \\theta } [ 0 , + \\infty ) \\subset \\bar { \\Delta } } \\Omega _ { \\theta } , \\end{align*}"}
+{"id": "6866.png", "formula": "\\begin{align*} \\hat z _ k = { \\tt h e t ( ( k - 1 ) n / 4 + 1 ) } , k = 1 , 2 , 3 , 4 . \\end{align*}"}
+{"id": "2095.png", "formula": "\\begin{align*} H _ { x , u } ^ + & = \\{ y \\in \\mathbb R ^ n : \\langle y , u \\rangle \\ge \\langle x , u \\rangle \\} \\\\ H _ { x , u } & = \\partial H _ { x , u } ^ + = \\{ y \\in \\mathbb R ^ n : \\langle y , u \\rangle = \\langle x , u \\rangle \\} , \\end{align*}"}
+{"id": "774.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } \\alpha _ i f _ i ( x , t , u ) \\le k _ 0 + k _ 1 \\sum _ { i = 1 } ^ m u _ i , \\forall ( x , t ) \\in \\Omega \\times \\R _ + , \\end{align*}"}
+{"id": "304.png", "formula": "\\begin{align*} d ( x ) & : = \\max \\{ 1 \\le l \\le n \\ , : \\ , \\ , \\\\ & \\qquad \\} . \\end{align*}"}
+{"id": "2361.png", "formula": "\\begin{align*} \\mathcal { Q } ( u ) : = u _ t - A ( x , u , u _ x ) u _ { x x } + B ( x , u , u _ x ) , \\end{align*}"}
+{"id": "4508.png", "formula": "\\begin{align*} \\begin{aligned} \\abs { c _ 2 ( S ^ 2 \\setminus S _ i ) } \\le \\abs { c _ 1 ( S ^ 1 \\setminus S _ i ) } + \\abs { c _ 2 ( P ^ 2 ) } & \\le \\abs { S } - \\abs { S \\cap S _ i } + q - q _ i + s _ 2 \\\\ & = ( 2 k - t ' - p ) - \\abs { S \\cap S _ i } + q - q _ i + ( p - q - s _ 1 ) \\\\ & = 2 k - t ' - s _ 1 - \\abs { S \\cap S _ i } - q _ i . \\end{aligned} \\end{align*}"}
+{"id": "4563.png", "formula": "\\begin{align*} & \\lambda _ 1 f ( v _ { \\ell + 1 , 0 , 0 } ) = ( q ^ 3 + q ^ 2 + q ) f ( v _ { \\ell + 1 , 1 , 0 } ) + f ( v _ { \\ell + 2 , 0 , 0 } ) \\\\ & \\lambda _ 2 f ( v _ { \\ell , 0 , 0 } ) = ( q ^ 4 + q ^ 3 + q ^ 2 ) f ( v _ { \\ell , 1 , 1 } ) + ( q ^ 2 + q + 1 ) f ( v _ { \\ell + 1 , 1 , 0 } ) \\\\ & \\lambda _ 3 f ( v _ { \\ell - 1 , 0 , 0 } ) = q ^ 3 f ( v _ { \\ell - 2 , 0 , 0 } ) + ( q ^ 2 + q + 1 ) f ( v _ { \\ell , 1 , 1 } ) . \\end{align*}"}
+{"id": "5627.png", "formula": "\\begin{align*} | F _ y ( x , y ) - p _ y ( x , y ) | = | F _ y ( x , y ) - h ^ y ( x , y ) | < 2 \\varepsilon . \\end{align*}"}
+{"id": "4915.png", "formula": "\\begin{align*} Q _ { \\mathrm { S N } } ( u ) = \\sum _ { i = 0 } ^ { \\infty } c _ k \\ , \\ , ( u - 1 / 2 ) ^ { k } , \\end{align*}"}
+{"id": "7821.png", "formula": "\\begin{align*} ( I - Q ) F ^ q ( B _ \\phi u + \\psi ( B _ \\phi u , p ) , p ) = 0 . \\end{align*}"}
+{"id": "247.png", "formula": "\\begin{align*} p _ k : = \\sum _ { i = 1 } ^ { k } \\left ( x _ 1 \\cdots \\widehat { x _ i } \\cdots x _ k \\right ) ^ { p - 1 } \\end{align*}"}
+{"id": "6037.png", "formula": "\\begin{align*} o _ { k } ( 1 ) = \\int _ { \\R ^ { 2 } } | \\nabla \\widetilde { u } _ { k } | ^ { 2 } + \\widetilde { u } _ { k } ^ { 2 } + 3 B ( \\widetilde { u } _ { k } ) - \\int _ { \\R ^ { 2 } } \\widetilde { u } _ { k } ^ { 2 p } , \\end{align*}"}
+{"id": "4807.png", "formula": "\\begin{align*} F ( \\eta ) : = T - \\frac { 1 } { \\eta } I , \\eta \\in \\Omega . \\end{align*}"}
+{"id": "3402.png", "formula": "\\begin{align*} ( n + 1 ) b _ { n + 1 } = ( n + e ^ t ) b _ n + ( 1 - e ^ t ) b _ { n - 1 } , b _ 0 = 1 , \\ b _ 1 = e ^ t . \\end{align*}"}
+{"id": "8228.png", "formula": "\\begin{align*} \\int _ { \\{ | t - \\xi | > 1 - r \\} } \\frac { 1 - r } { | 1 - \\bar { t } r \\xi | | 1 - \\bar { t } \\xi | } d \\mu ( t ) \\leq & \\sum _ { n = 0 } ^ \\infty \\int _ { I ( n + 1 ) \\setminus I ( n ) } \\frac { 1 - r } { 2 ^ { 2 n } ( 1 - r ) ^ 2 } d \\mu ( t ) \\\\ = & \\sum _ { n = 0 } ^ { \\infty } \\frac { \\mu ( I ( n + 1 ) \\setminus I ( n ) ) } { 2 ^ { 2 n } ( 1 - r ) } \\leq C _ 2 \\ , \\end{align*}"}
+{"id": "2179.png", "formula": "\\begin{align*} \\mathbf { y } ( t ) = \\mathbf { a } ( \\theta _ { 0 } ) s ( t ) + \\mathbf { v } ( t ) , \\end{align*}"}
+{"id": "2395.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\subset \\mu _ \\ell } c _ { \\ell } x _ { \\ell , \\omega ( \\mu _ \\ell ) - \\omega ( \\kappa ) } + c _ { \\kappa } & = 0 \\sigma \\subset \\kappa \\\\ \\sum _ { \\sigma \\subset \\mu _ \\ell } c _ { \\ell } x _ { \\ell , \\omega ( \\mu _ \\ell ) - \\omega ( \\kappa ) } & = 0 \\sigma \\not \\subset \\kappa . \\end{align*}"}
+{"id": "4591.png", "formula": "\\begin{align*} ( \\Lambda - 1 + \\delta ) \\sum _ { j = 1 } ^ k ( \\Lambda + \\varepsilon ) ^ { - j } > 1 , \\end{align*}"}
+{"id": "9291.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ k & \\wedge \\beta _ n ^ { n - m } ( \\omega ) = \\int u _ k \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ { k - 1 } \\wedge \\beta _ n ^ { n - m } \\wedge \\Delta \\omega \\end{aligned} \\end{align*}"}
+{"id": "5.png", "formula": "\\begin{align*} \\beta _ 1 \\oplus \\beta _ 2 = \\begin{cases} \\emptyset & i _ 2 < j _ 1 , \\\\ \\{ ( \\alpha _ { i _ 1 , \\overline { j } _ 1 } , \\alpha _ { j _ 2 , \\overline { i } _ 2 } ) , ( \\alpha _ { i _ 1 , \\overline { j } _ 2 } , \\alpha _ { j _ 1 , \\overline { i } _ 2 } ) \\} & j _ 1 \\leq j _ 2 < i _ 2 , \\\\ \\{ ( \\alpha _ { i _ 1 , \\overline { j } _ 1 } , \\alpha _ { i _ 2 , \\overline { j } _ 2 } ) \\} & j _ 1 \\leq i _ 2 \\leq j _ 2 . \\\\ \\end{cases} \\end{align*}"}
+{"id": "4386.png", "formula": "\\begin{align*} \\sup _ { \\epsilon } \\int _ { X _ j } | ( 1 - v ' _ { \\epsilon } ( \\Psi ) ) f F ^ { 1 + \\delta } | _ { h _ i } ^ 2 \\le c _ i ( \\sup _ { X _ j } | F | ^ { 2 + 2 \\delta } ) \\int _ { X _ j } \\mathbb { I } _ { \\{ \\Psi < - t _ 0 \\} } | f | _ { \\hat { h } } ^ 2 < + \\infty , \\end{align*}"}
+{"id": "3727.png", "formula": "\\begin{align*} \\nabla _ { V } \\hat X & = \\omega ( V ) \\\\ \\nabla _ V \\omega & = - R ( \\hat X , V ) + T _ h ( V ) . \\end{align*}"}
+{"id": "2202.png", "formula": "\\begin{align*} \\mathbf { a } ( \\theta _ { 0 } ) & = [ 1 , e ^ { - j 2 \\pi \\frac { d } { \\lambda } { s i n } \\theta _ 0 } , \\cdots , e ^ { - j 2 \\pi ( N - 1 ) \\frac { d } { \\lambda } { s i n } \\theta _ 0 } ] ^ { T } \\\\ & = [ 1 , e ^ { - j \\phi } , e ^ { - j 2 \\phi } , \\cdots , e ^ { - j ( N - 1 ) \\phi } ] ^ { T } , \\end{align*}"}
+{"id": "3198.png", "formula": "\\begin{align*} \\delta _ { o p t } ( \\Gamma ) = \\frac { \\frac { 1 } { 1 6 \\sqrt { 3 } } + \\frac { 1 } { 2 4 } \\left ( 2 \\sqrt { 3 } + 3 \\right ) } { v o l ( \\widehat { A V } _ 3 ) } \\approx 0 . 8 3 8 8 2 5 . \\end{align*}"}
+{"id": "2015.png", "formula": "\\begin{align*} \\widetilde { H } ( x _ 0 + \\Delta x , Y _ 0 + \\Delta Y ) - \\widetilde { H } ( x _ 0 , Y _ 0 ) = [ D \\widetilde { H } ] ( x _ 0 , Y _ 0 ) ( \\Delta x , \\Delta Y ) + o ( | \\Delta x | + \\| \\Delta Y \\| _ { L ^ 2 } ) . \\end{align*}"}
+{"id": "9336.png", "formula": "\\begin{align*} \\min _ { \\xi , \\eta } p ( \\xi , \\eta ) = \\min \\left \\{ \\min _ { \\xi } p \\left ( \\xi , \\frac 1 2 \\right ) , \\min _ { \\xi } p \\left ( \\xi , \\frac { 9 9 } { 1 0 0 } \\right ) \\right \\} . \\end{align*}"}
+{"id": "4513.png", "formula": "\\begin{align*} \\omega : = \\R \\setminus B _ R ( 0 ) = \\{ x \\in \\R : \\Vert x \\Vert \\ge R \\} \\end{align*}"}
+{"id": "5332.png", "formula": "\\begin{align*} \\alpha _ { k + 1 } & = P ( k + 1 ) - \\sum _ { i = 0 } ^ k \\alpha _ i \\binom { k + 1 } { i } \\\\ \\vert \\alpha _ { k + 1 } \\vert & \\leq 1 + \\sum _ { i = 0 } ^ k 2 ^ i i ! \\binom { k + 1 } { i } \\\\ & = 2 ^ { k + 1 } ( k + 1 ) ! \\left ( \\frac { 1 } { 2 ^ { k + 1 } ( k + 1 ) ! } + \\sum _ { i = 0 } ^ k \\frac { 1 } { 2 ^ { k + 1 - i } ( k + 1 - i ) ! } \\right ) \\\\ & < 2 ^ { k + 1 } ( k + 1 ) ! \\left ( \\frac { 1 } { 8 } + e ^ { 1 / 2 } - 1 \\right ) < 2 ^ { k + 1 } ( k + 1 ) ! \\end{align*}"}
+{"id": "8079.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ y _ l ^ * y _ j | \\hat { \\mathbf { H } } \\right ] = & a _ c ^ 2 \\hat { \\phi } ^ { \\left ( l , c \\right ) ^ * } \\hat { \\phi } ^ { \\left ( j , c \\right ) } + \\sum _ { q = 1 } ^ M a _ q ^ 2 \\hat { \\phi } ^ { \\left ( l , q \\right ) ^ * } \\hat { \\phi } ^ { \\left ( j , q \\right ) } . \\end{align*}"}
+{"id": "1578.png", "formula": "\\begin{align*} \\vert \\{ g \\in G \\ \\vert \\ ( g K ) \\cap K \\neq \\emptyset \\} \\vert = \\sum \\limits _ { j \\in J } \\vert I _ j \\vert < + \\infty , \\end{align*}"}
+{"id": "5664.png", "formula": "\\begin{align*} \\kappa : = D _ { a , n } = \\begin{pmatrix} a \\\\ & a ^ { - 1 } \\\\ & & \\ddots \\\\ & & & a \\\\ & & & & a ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
+{"id": "5346.png", "formula": "\\begin{align*} c _ k : = \\frac { N _ k ( A ) } { r ( A ) ^ k } \\ \\end{align*}"}
+{"id": "996.png", "formula": "\\begin{align*} S = \\{ s _ \\alpha \\mid \\alpha \\in \\Delta \\} \\subseteq W . \\end{align*}"}
+{"id": "8620.png", "formula": "\\begin{align*} V _ p ( A [ n - 1 ] , B ) = \\frac { 1 } { n } \\int \\limits _ { S ^ { n - 1 } } h ^ p _ B ( u ) h _ A ^ { 1 - p } d S _ A ( u ) . \\end{align*}"}
+{"id": "3497.png", "formula": "\\begin{align*} \\phi ( y ) = \\sum _ { z \\in \\partial I ( y ) } G _ { I ( y ) } ( z , y ) \\phi ( z ' ) . \\end{align*}"}
+{"id": "2900.png", "formula": "\\begin{align*} 1 = \\sum _ { y = 0 } ^ n \\int _ 0 ^ { \\theta } { \\frak G } _ { x , y } ( s ) \\dd s , x = 0 , 1 , \\ldots , n \\end{align*}"}
+{"id": "6576.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } \\nabla c \\cdot \\nu c \\psi ^ { 2 } = \\int _ { \\partial \\Omega \\cap B _ { \\eta } } \\nabla c \\cdot \\nu c \\psi ^ { 2 } . \\end{align*}"}
+{"id": "6937.png", "formula": "\\begin{align*} H \\bigg ( \\mathrm { L a w } \\bigg ( \\sum _ { i = 1 } ^ n t _ i X _ i \\bigg ) \\bigg ) \\le \\sum _ { i = 1 } ^ n t _ i H ( Q _ i ) . \\end{align*}"}
+{"id": "1654.png", "formula": "\\begin{align*} \\chi \\cdot ( [ K \\delta K ] ( \\phi ) ) ( \\sigma , g ) & = \\chi ( \\det g ) ( [ K \\delta K ] ( \\phi ) ) ( \\sigma , g ) \\\\ & = \\chi ( \\det g ) \\sum _ i \\delta _ i \\cdot \\phi ( \\sigma , g \\delta _ i ) . \\end{align*}"}
+{"id": "2543.png", "formula": "\\begin{align*} \\mathbf { q } _ c = ( \\mathbf { x } _ c , \\mathbf { y } _ c , \\mathbf { s } _ c , \\kappa _ c , \\tau _ c ) = ( \\mathbf { e } , 0 , \\mathbf { e } , 1 , 1 ) . \\end{align*}"}
+{"id": "8781.png", "formula": "\\begin{align*} L _ x v = B ( x , v ) + B ( v , x ) \\end{align*}"}
+{"id": "5527.png", "formula": "\\begin{align*} \\tilde h _ d ( \\alpha _ d ) & \\leq O ( 1 ) + \\log \\frac { d \\cdot \\Gamma ( \\frac { d } { 2 } ) } { \\alpha _ d d ^ { \\frac { d - 1 } { 2 } } } + d \\log 2 \\\\ & = O ( 1 ) + \\log \\frac { \\Gamma ( \\frac { d } { 2 } ) } { C e ^ { - c d } d ^ { \\frac { d - 1 } { 2 } } } + d \\log 2 . \\end{align*}"}
+{"id": "4411.png", "formula": "\\begin{align*} \\left | \\frac { e ^ { \\kappa s } - 1 } { s } \\right | & = \\left | \\sum _ { n = 0 } ^ \\infty \\frac { \\kappa ^ { n + 1 } s ^ n } { ( n + 1 ) ! } \\right | \\leq e ^ { \\frac { 3 } { 4 } \\kappa } \\kappa , \\end{align*}"}
+{"id": "653.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { k _ 2 } \\left ( 1 - \\frac { i } { n - X ( t ) } \\right ) = 1 + O ( 1 / n ) . \\end{align*}"}
+{"id": "2534.png", "formula": "\\begin{align*} \\mathbf { w } _ { x s } ^ T \\mathbf { e } = \\mathbf { s } ^ T \\mathbf { T } _ x \\mathbf { e } = \\mathbf { s } ^ T \\mathbf { x } = k \\mu , \\end{align*}"}
+{"id": "8198.png", "formula": "\\begin{align*} x _ { 2 i , n - j } & + x _ { 2 i , n - j + 1 } + x _ { 2 i + 1 , n - j } + x _ { 2 i + 1 , n - j + 1 } = S \\\\ & = x _ { 2 i + 1 , n - j } + x _ { 2 i + 1 , n - j + 1 } + x _ { 2 i + 2 , n - j } + x _ { 2 i + 2 , n - j + 1 } , \\end{align*}"}
+{"id": "142.png", "formula": "\\begin{align*} H ^ { \\lambda } _ { G } ( \\sigma ) = \\left \\{ \\begin{array} { l l } J \\sum \\limits _ { x \\in { V } } { \\ln \\lambda _ { \\sigma ( x ) } , } \\ \\ \\ $ i f $ \\sigma \\in \\Omega ^ G $ , $ \\\\ + \\infty , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $ \\ i f $ \\sigma \\ \\notin \\Omega ^ G $ , $ \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "4391.png", "formula": "\\begin{align*} & \\int _ { X _ j } | \\tilde { F } - ( 1 - b ( \\Psi ) ) f F ^ { 1 + \\delta } | _ { h } ^ 2 e ^ { v ( \\Psi ) - \\delta \\max \\{ \\psi + T , 2 \\log | F | \\} } c ( - v ( \\Psi ) ) \\\\ \\le & \\bigg ( \\sup _ { X } e ^ { - u ( - v ( \\Psi ) ) } \\bigg ) \\int _ { X } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f F | ^ 2 _ h < + \\infty . \\end{align*}"}
+{"id": "2963.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha & : = \\sup \\{ t \\in [ 0 , 1 ) \\mid F _ E ( t ) > t \\} , \\\\ \\beta & : = \\inf \\{ t \\in [ 0 , 1 ) \\mid F _ E ( t ) < t \\} , \\end{aligned} \\end{align*}"}
+{"id": "6449.png", "formula": "\\begin{align*} \\mu ( T ^ { 2 r _ n } A \\cap A ) & \\geq \\mu ( A \\cap T ^ { - r _ n } A \\cap T ^ { r _ n } A ) \\\\ & > 2 \\big ( \\beta - \\frac { 1 } { 2 } \\big ) \\mu ( A ) \\\\ & = \\big ( 2 \\beta - 1 \\big ) \\mu ( A ) > \\mu ( A ) ^ 2 . \\end{align*}"}
+{"id": "2983.png", "formula": "\\begin{align*} & \\limsup _ { N \\to \\infty } \\frac { 1 } { \\# ( \\Lambda _ N ^ { \\vec { v } } ( 1 ) ) } \\sum _ { ( m , n ) \\in \\Lambda _ N ^ { \\vec { v } } ( 1 ) } \\min _ { 1 \\leq i < j \\leq k } d ( T ^ { ( m , n ) } x _ i , T ^ { ( m , n ) } x _ j ) \\\\ \\geq & \\limsup _ { N \\to \\infty } \\frac { 1 } { 2 N } \\sum _ { n = 0 } ^ { N - 1 } \\min _ { 1 \\leq i < j \\leq k } { d } ( T ^ { ( n , [ n \\beta + t ] ) } x _ i , T ^ { ( n , [ n \\beta + t ] ) } x _ j ) \\geq \\eta / 2 > 0 . \\end{align*}"}
+{"id": "5741.png", "formula": "\\begin{align*} d I _ 3 \\cap ( I _ 1 \\cup I _ 2 \\cup I _ 3 ) = \\emptyset , \\end{align*}"}
+{"id": "3887.png", "formula": "\\begin{align*} \\frac { \\partial s _ { \\delta , i } } { \\partial z _ { i , h } } = O ( \\delta | \\ln \\delta | ^ { \\frac { p - 1 } { 2 } } ) . \\end{align*}"}
+{"id": "438.png", "formula": "\\begin{align*} \\alpha v = \\theta v , \\end{align*}"}
+{"id": "2950.png", "formula": "\\begin{align*} \\begin{aligned} h \\left ( E ^ \\Phi _ A ( x ) \\right ) & = E ^ \\Psi _ B \\left ( h ( x ) \\right ) \\\\ h \\left ( P ^ \\Phi _ { \\bar A } ( x ) \\right ) & = P ^ \\Psi _ { \\bar B } \\left ( h ( x ) \\right ) . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "4792.png", "formula": "\\begin{align*} X _ n : = \\{ v \\in L ^ 2 ( D ) : v | _ { K } \\in P ^ \\ell ( K ) , K \\in { \\mathcal T } _ n \\} , \\end{align*}"}
+{"id": "7798.png", "formula": "\\begin{align*} F ^ q ( v , p ) : = F ( \\Psi ^ q + v , p ) , F ^ q \\colon X \\times \\mathcal P \\to X . \\end{align*}"}
+{"id": "693.png", "formula": "\\begin{align*} \\ell \\left ( \\bar \\phi \\left ( X ^ { ( N _ { r _ 0 } ( v ) ) } \\right ) \\right ) \\leq \\ell \\left ( X ^ { ( N _ { r _ 0 } ( v ) ) } \\right ) - 1 = | N _ { r _ 0 } \\left ( B \\right ) | \\ell ( X ) - 1 . \\end{align*}"}
+{"id": "2175.png", "formula": "\\begin{gather*} \\gamma ^ + ( u ) ( x ) \\ne \\gamma ^ - ( u ) ( x ) , \\\\ [ 5 p t ] \\gamma ^ + ( u ) + s \\gamma ^ + ( v ) ( x ) = \\gamma ^ - ( u ) + s \\gamma ^ - ( v ) ( x ) , \\end{gather*}"}
+{"id": "3634.png", "formula": "\\begin{align*} \\sum _ i F ^ { i i } h _ { i i 1 1 } = - \\sum _ { p , q , r , s } F ^ { p q , r s } h _ { p q 1 } h _ { r s 1 } = - \\sum _ { p \\neq q } F ^ { p p , q q } h _ { p p 1 } h _ { q q 1 } + \\sum _ { p \\neq q } F ^ { p p , q q } h _ { p q 1 } ^ 2 . \\end{align*}"}
+{"id": "2477.png", "formula": "\\begin{align*} Z _ { q } ( \\beta ) = \\frac { 1 } { h ^ { D N } } \\displaystyle \\int _ { 0 } ^ { \\infty } \\ ; d \\bar { x } \\ ; d \\bar { p } \\ ; \\exp _ { q } \\left ( - \\beta \\displaystyle \\sum _ { i = 1 } ^ { D N } \\left ( \\frac { p _ { i } ^ { 2 } } { 2 m } + \\frac { 1 } { 2 } m \\omega ^ { 2 } x _ { i } ^ { 2 } \\right ) \\right ) , \\end{align*}"}
+{"id": "1424.png", "formula": "\\begin{align*} ( - 1 ) ^ n { k - 1 / 2 \\choose n } = \\sum _ { j = 0 } ^ { N - 1 } a _ { j } ^ { ( k ) } ( n + 1 ) ^ { - k - 1 / 2 - j } + O ( ( n + 1 ) ^ { - N - 1 / 2 } ) . \\end{align*}"}
+{"id": "6895.png", "formula": "\\begin{align*} d _ \\square ( W _ 1 , W _ 2 ) : = \\sup _ { S , T \\subset [ 0 , 1 ] } \\bigg | \\int _ { S \\times T } \\big ( W _ 1 ( u , v ) - W _ 2 ( u , v ) \\big ) d u d v \\bigg | , \\end{align*}"}
+{"id": "1179.png", "formula": "\\begin{align*} \\sup _ { n } \\mathbb { E } _ { \\mathbb { P } } \\left [ \\exp \\left \\{ \\kappa \\sum _ { i = 1 } ^ { n } \\int _ { T _ { 0 } } ^ { T _ { 0 } + \\delta } \\triangle K _ { t } ^ { i , n } \\cdot d W _ { t } ^ { i } - \\frac { \\kappa } { 2 } \\sum _ { i = 1 } ^ n \\int _ { T _ { 0 } } ^ { T _ { 0 } + \\delta } \\left | \\triangle K _ { t } ^ { i , n } \\right | ^ { 2 } d t \\right \\} \\right ] \\end{align*}"}
+{"id": "9173.png", "formula": "\\begin{align*} \\begin{aligned} b _ { 1 , \\delta } ( x ^ \\star ) = & \\ , 2 \\int _ { 0 } ^ { 1 / 2 } h ( x ^ \\star + \\delta \\sin ( 2 \\pi t ) ) \\sin ( 2 \\pi t ) \\ , d t \\\\ & - 2 \\int _ { 0 } ^ { 1 / 2 } h ( x ^ \\star + \\delta \\sin ( 2 \\pi t ) ) \\sin ( 2 \\pi t ) \\ , d t = 0 . \\end{aligned} \\end{align*}"}
+{"id": "7225.png", "formula": "\\begin{align*} - K _ F = ( n - 1 ) A _ X \\vert _ F , \\end{align*}"}
+{"id": "2893.png", "formula": "\\begin{align*} \\Big ( ( \\bar { \\bf q } ( 0 ) , \\bar { \\bf p } ( 0 ) ) , C ( 0 ) \\Big ) = \\Big ( ( \\bar { \\bf q } ( \\theta ) , \\bar { \\bf p } ( \\theta ) ) , C ( \\theta ) \\Big ) . \\end{align*}"}
+{"id": "5452.png", "formula": "\\begin{align*} \\phi _ { \\lambda } ( [ a _ { \\mu } b ] ) = [ \\phi _ { \\lambda } ( a ) _ { \\lambda + \\mu } b ] + [ a _ { \\mu } \\phi _ { \\lambda } ( b ) ] , \\forall \\ a , b \\in \\mathcal { A } . \\end{align*}"}
+{"id": "4052.png", "formula": "\\begin{align*} e ( G ^ { ( \\lambda , C ) } ) = 2 - I ( G ^ { ( \\lambda , C ) } ) = 2 ( 1 - I ( C ) ) = 2 e ( C ) . \\end{align*}"}
+{"id": "3030.png", "formula": "\\begin{align*} \\left ( q , \\left ( x , p \\right ) \\right ) \\mapsto L _ { \\left ( 0 , q \\right ) } R _ { \\left ( 0 , \\bar { q } \\right ) } \\left ( x , p \\right ) = \\left ( q x \\bar { q } , q p \\bar { q } \\right ) = \\left ( I _ { q } \\left ( x \\right ) , q p \\bar { q } \\right ) \\end{align*}"}
+{"id": "2406.png", "formula": "\\begin{align*} ( I _ { n } + p ^ { m - 1 } x _ { r } ) ^ { p } = I _ { n } + p ^ { m } a + p ^ { m + r } c \\ . \\end{align*}"}
+{"id": "8317.png", "formula": "\\begin{align*} \\begin{aligned} & z '' _ { b e n } + ( M _ { b e n } - E ) A _ { b e n } ^ { - 1 } z _ { b e n } = 0 \\\\ & z '' _ { n b } + ( M _ { n b } - E ) A _ { n b } ^ { - 1 } z _ { n b } = 0 , \\end{aligned} \\end{align*}"}
+{"id": "1545.png", "formula": "\\begin{align*} \\varphi _ \\alpha ( z ) = ( ( z + \\alpha ) \\vee 0 ) + ( ( z - \\alpha ) \\wedge 0 ) . \\end{align*}"}
+{"id": "3479.png", "formula": "\\begin{align*} \\prod _ { j = x _ 1 } ^ k | \\cos ( \\pi \\theta _ j ) | \\leq e ^ { - ( \\ln 2 - \\varepsilon ) | x _ 1 - k | } c _ { n , \\ell - 1 } , \\prod _ { j = k } ^ { x _ 2 } | \\cos ( \\pi \\theta _ j ) | \\leq e ^ { - ( \\ln 2 - \\varepsilon ) | x _ 2 - k | } c _ { n , \\ell } . \\end{align*}"}
+{"id": "9151.png", "formula": "\\begin{align*} \\mbox { \\rm g r a p h } \\left . \\tau \\right | _ { { \\cal A } _ \\delta } = \\{ ( x _ a , y _ a ) \\in { \\cal A } _ \\delta \\times \\mathbb { R } \\ ; : \\ ; y _ a = \\tau ( x _ a ) \\} \\end{align*}"}
+{"id": "1752.png", "formula": "\\begin{align*} \\nabla \\partial _ t R _ 0 = \\partial _ 1 g ( u _ 0 , R _ 0 ) \\nabla u _ 0 + \\partial _ 2 g ( u _ 0 , R _ 0 ) \\nabla R _ 0 \\in L ^ 2 ( ( 0 , T ) \\times \\Omega ) \\end{align*}"}
+{"id": "7781.png", "formula": "\\begin{align*} q - h \\ge \\log _ { \\theta } \\Big ( 1 + \\frac { d } { \\epsilon _ 0 } \\Big ) = \\log _ { \\theta } ( 1 + b _ 0 d ) ~ { \\rm f o r } ~ h = 0 , 1 , \\dots , q - 1 . \\end{align*}"}
+{"id": "1511.png", "formula": "\\begin{align*} g ' ( s ) & = \\frac { ( \\alpha - 1 ) ( s - a ) ^ { \\alpha - 2 } ( b - s ) ^ { \\alpha - 1 - \\beta } - ( \\alpha - 1 - \\beta ) ( s - a ) ^ { \\alpha - 1 } ( b - s ) ^ { \\alpha - 2 } } { ( b - a ) ^ { \\alpha - 1 - \\beta } \\Gamma ( \\alpha ) } \\\\ & = \\frac { ( s - a ) ^ { \\alpha - 2 } ( b - s ) ^ { \\alpha - 2 - \\beta } \\left [ ( \\alpha - 1 ) ( b - s ) - ( \\alpha - 1 - \\beta ) ( s - a ) \\right ] } { ( b - a ) ^ { \\alpha - 1 - \\beta } \\Gamma ( \\alpha ) } . \\end{align*}"}
+{"id": "3656.png", "formula": "\\begin{align*} D _ { x , 2 } & = \\frac { 1 } { 2 L _ { x y } } \\left ( \\sqrt { n \\sigma _ { G , r } ^ 2 } + L _ { y } \\sqrt { n } M _ y + \\| \\nabla G ( Y ^ * ) \\| \\right ) \\\\ & + \\frac { 1 } { 2 \\mu _ { x } } \\left ( \\sqrt { n \\sigma _ { F , r } ^ 2 } + L _ { x } \\sqrt { n } M _ x + \\| \\nabla F ( X ^ * ) \\| \\right ) , \\end{align*}"}
+{"id": "4119.png", "formula": "\\begin{align*} L ^ { * } : = \\left \\lbrace y \\in E \\mid h \\left ( L , y \\right ) \\subset { \\mathcal O } _ K \\right \\rbrace . \\end{align*}"}
+{"id": "2647.png", "formula": "\\begin{align*} F ( z , q ) & = \\sum \\limits _ { n = 1 } ^ { \\infty } z ^ { n } ( q ^ { ( p - r ) n } + q ^ { ( p - r + p ) n } + q ^ { ( p - r + 2 p ) n } + \\cdots + q ^ { ( p - r + 1 ) n } + q ^ { ( p - r + 1 + p ) n } \\\\ \\\\ & + q ^ { ( p - r + 1 + 2 p ) n } + \\cdots ) \\prod \\limits _ { j \\neq n , j = 1 } ^ { \\infty } ( 1 + q ^ { j } ) \\end{align*}"}
+{"id": "1707.png", "formula": "\\begin{align*} \\frac { d y } { d t } & = A ( t ) y , \\\\ y ( s ) & = x , s \\le t < \\infty , \\end{align*}"}
+{"id": "5322.png", "formula": "\\begin{align*} f ( z , t ) = \\int _ \\Omega \\int _ { \\C ^ n } F ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \\ , d \\nu ( a ) . \\end{align*}"}
+{"id": "298.png", "formula": "\\begin{align*} E = ( \\pi ( 0 , r ) \\times ( - \\delta , 0 ) ) \\cup ( \\pi ( 0 , r ) \\times ( t , t + \\delta ) ) \\end{align*}"}
+{"id": "4634.png", "formula": "\\begin{align*} C _ 1 = ( i _ { 1 , 1 } \\dots i _ { 1 , d _ 1 } ) , \\dots , C _ j = ( i _ { j , 1 } \\dots i _ { j , d _ j } ) , \\sum _ { r = 1 } ^ j d _ r = n , \\end{align*}"}
+{"id": "7861.png", "formula": "\\begin{align*} \\Phi ^ \\dagger _ { v p } ( v , p ) [ v _ 1 , p _ 1 ] & = Q ^ \\dagger F ^ { q , \\dagger } _ { v v } ( v + \\psi ^ \\dagger ( v , p ) , p ) [ v _ 1 + \\psi _ v ^ \\dagger ( v , p ) v _ 1 ] \\psi ^ \\dagger _ p ( v , p ) p _ 1 \\\\ & + Q ^ \\dagger F ^ { q , \\dagger } _ { v p } ( v + \\psi ^ \\dagger ( v , p ) , p ) [ v _ 1 + \\psi _ v ^ \\dagger ( v , p ) v _ 1 , p _ 1 ] \\\\ & + Q ^ \\dagger F ^ { q , \\dagger } _ { v } ( v + \\psi ^ \\dagger ( v , p ) , p ) \\psi ^ \\dagger _ { v p } ( v , p ) [ v _ 1 , p _ 1 ] \\end{align*}"}
+{"id": "4994.png", "formula": "\\begin{align*} u = z + v \\end{align*}"}
+{"id": "801.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } t } \\int _ { \\Gamma } 1 \\ , \\mathrm { d } \\mathcal { H } ^ d = - \\int _ { \\Gamma } H V \\ , \\mathrm { d } \\mathcal { H } ^ d = - \\int _ { \\Gamma } g ( c ) H ^ 2 \\ , \\mathrm { d } \\mathcal { H } ^ d . \\end{align*}"}
+{"id": "4452.png", "formula": "\\begin{align*} \\beta _ { \\infty } ( n ) = \\frac { 4 \\pi ^ { 2 } n } { \\sqrt { \\det ( A ) } } . \\end{align*}"}
+{"id": "3371.png", "formula": "\\begin{align*} P ( x , y ) & : = \\pi _ x y \\ : : \\ : S _ y \\rightarrow S _ x \\\\ & \\ , \\ , = \\pi _ x y \\ , U _ { \\xi } \\ : : \\ : S _ x \\rightarrow S _ x \\ : . \\end{align*}"}
+{"id": "5435.png", "formula": "\\begin{align*} m ^ { - 1 / 2 ( k - 1 ) } \\leq ( b _ \\alpha ^ { k - 1 } ) ^ { - 1 / 2 ( k - 1 ) } = b _ \\alpha ^ { - 1 / 2 } . \\end{align*}"}
+{"id": "2906.png", "formula": "\\begin{align*} & = \\frac { 1 } { ( n + 1 ) ^ 2 [ \\exp \\left \\{ \\frac { i \\pi x } { n + 1 } \\right \\} - 1 ] } \\sum _ { j , j ' = 0 } ^ n \\Big [ \\Phi \\left ( \\frac { j - 1 } { n + 1 } , \\frac { j ' } { n + 1 } \\right ) - \\Phi \\left ( \\frac { j } { n + 1 } , \\frac { j ' } { n + 1 } \\right ) \\Big ] \\\\ & \\times \\exp \\left \\{ \\frac { 2 i \\pi j x } { n + 1 } \\right \\} \\exp \\left \\{ \\frac { 2 i \\pi j ' x } { n + 1 } \\right \\} \\end{align*}"}
+{"id": "6212.png", "formula": "\\begin{align*} f ( r ) \\frac { d W _ + ( r ) } { d r } = W _ + ( r ) W _ - ( r ) + E _ 1 - E _ 0 . \\end{align*}"}
+{"id": "1636.png", "formula": "\\begin{align*} \\alpha \\left ( R ( r + s ) \\right ) & = \\int ^ { T _ 2 } _ { T _ 1 } G \\left ( \\frac { 3 T _ 1 + T _ 2 } { 4 } , \\tau \\right ) f _ \\uparrow \\left ( r ( \\tau ) + s ( \\tau ) \\right ) d \\tau \\\\ & \\ge \\int ^ { T _ 2 } _ { \\frac { 3 T _ 1 + T _ 2 } { 4 } } G \\left ( \\frac { 3 T _ 1 + T _ 2 } { 4 } , \\tau \\right ) f _ \\uparrow \\left ( r ( \\tau ) + s ( \\tau ) \\right ) d \\tau \\\\ & \\ge f _ \\uparrow \\left ( a + 4 ^ { - k } d \\right ) \\int ^ { T _ 2 } _ { \\frac { 3 T _ 1 + T _ 2 } { 4 } } G \\left ( \\frac { 3 T _ 1 + T _ 2 } { 4 } , \\tau \\right ) d \\tau \\\\ & > a . \\end{align*}"}
+{"id": "1890.png", "formula": "\\begin{align*} \\ell _ { \\lambda ' } ( \\rho ) = \\begin{cases} \\ \\ \\ ( \\lambda / \\lambda ' ) ^ { \\frac 2 3 } \\rho , & \\lambda \\ge \\lambda ' , \\\\ [ 2 p t ] \\ , ( \\lambda ' - \\lambda ) / \\lambda ' + \\rho , & \\lambda < \\lambda ' . \\end{cases} \\end{align*}"}
+{"id": "5381.png", "formula": "\\begin{align*} W : = \\max _ { \\overline { \\omega } _ \\delta } ( A \\Psi \\pm T _ \\alpha ( u - \\varphi ) ) \\end{align*}"}
+{"id": "5401.png", "formula": "\\begin{align*} L ( x ) : = \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( y ' ) ( x _ \\beta - y _ \\beta ) . \\end{align*}"}
+{"id": "9057.png", "formula": "\\begin{align*} \\mu _ t = j _ t - \\frac { L } { 2 } \\medspace ( 1 \\leq t \\leq n ) , \\end{align*}"}
+{"id": "8032.png", "formula": "\\begin{align*} E _ a ( x ) \\ll \\begin{cases} \\pi _ N ( x ) ^ { j - a } L ^ { 2 r } & a \\leq j \\\\ L ^ { 2 r - ( a - j ) } & a > j . \\\\ \\end{cases} \\end{align*}"}
+{"id": "6802.png", "formula": "\\begin{gather*} y \\left ( \\rho \\right ) = e ^ { - i \\rho x } e \\left ( \\rho , x \\right ) - 1 , \\\\ \\left ( \\mathbf { H } ( \\varphi ) y \\right ) \\left ( \\rho \\right ) = \\frac { 1 } { 2 \\pi i } \\int _ { - \\infty } ^ { \\infty } \\frac { \\varphi \\left ( \\tau \\right ) y \\left ( \\tau \\right ) } { \\tau + \\rho } d \\tau , \\quad \\rho \\in \\mathbb { R } \\end{gather*}"}
+{"id": "3208.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ X ( f \\circ h ^ n ) g \\ ; d \\mu = \\int _ X f \\ , d \\mu \\cdot \\int _ X g \\ , d \\mu \\end{align*}"}
+{"id": "2910.png", "formula": "\\begin{align*} A S ( m ) + S ( m ) A ^ T + \\frac { 2 \\pi i m } { \\theta } S ( m ) = \\Sigma _ 2 ( g ) . \\end{align*}"}
+{"id": "7341.png", "formula": "\\begin{align*} V ( A , - C ) - \\sqrt { | A | \\ , | C | } = | A | . \\end{align*}"}
+{"id": "5799.png", "formula": "\\begin{align*} \\Im m ( a _ 0 \\overline { a _ 1 } ) = \\Im m ( a _ 2 \\overline { a _ 3 } ) ; \\end{align*}"}
+{"id": "4932.png", "formula": "\\begin{align*} b _ j = 0 \\end{align*}"}
+{"id": "8062.png", "formula": "\\begin{align*} a _ j \\left [ t + 1 \\right ] = & a _ j \\left [ t \\right ] - 4 \\mu a _ j \\left [ t \\right ] \\sum _ { l = 1 } ^ { M } \\lvert \\phi ^ { \\left ( l , j \\right ) } \\rvert ^ 2 + 2 \\mu \\Re \\left \\{ \\phi ^ { \\left ( j , j \\right ) } \\right \\} - 4 \\mu a _ j \\left [ t \\right ] \\sum _ { q = 1 } ^ { M - 1 } \\sum _ { r = q + 1 } ^ { M } \\Re \\left \\{ \\phi ^ { \\left ( q , j \\right ) ^ * } \\phi ^ { \\left ( r , j \\right ) } \\right \\} . \\end{align*}"}
+{"id": "2542.png", "formula": "\\begin{align*} \\Delta \\mathbf { A } = \\mathbf { A } - \\mathbf { A } _ o , \\ \\Delta \\mathbf { b } = \\mathbf { b } - \\mathbf { b } _ o , \\ \\Delta \\mathbf { c } = \\mathbf { c } - \\mathbf { c } _ o . \\end{align*}"}
+{"id": "1683.png", "formula": "\\begin{align*} \\begin{array} { c } \\forall k \\in \\mathbb { N } , \\ ; l _ k ( \\tau ) = ( - 1 ) ^ k \\overset { k } { \\underset { j = 0 } { \\sum } } ( - 1 ) ^ j ( \\begin{smallmatrix} k \\\\ j \\end{smallmatrix} ) ( \\begin{smallmatrix} k + j \\\\ j \\end{smallmatrix} ) \\left ( \\frac { \\tau + h } { h } \\right ) ^ j , \\end{array} \\end{align*}"}
+{"id": "4669.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\frac { \\psi ( p ) } { p } = 0 . \\end{align*}"}
+{"id": "5828.png", "formula": "\\begin{align*} { \\rm O } ( \\epsilon ^ 0 ) : g _ i ^ { ( 0 ) } = g _ i ^ { ( e q ) } , \\end{align*}"}
+{"id": "290.png", "formula": "\\begin{align*} U ^ \\ast = ( U ^ \\ast ) _ \\ast = K \\end{align*}"}
+{"id": "8879.png", "formula": "\\begin{align*} \\lambda _ { k , S ; ( n , g ) } ( l , r ) : = \\frac { \\Gamma _ n ( k - \\frac { n + g + 1 } { 2 } ) ( 4 l - S ^ { - 1 } [ r ] ) ^ { - k + ( n + g + 1 ) / 2 } } { ( \\pi ) ^ { n k - n ( n + g + 1 ) / 2 } ( \\det 2 S ) ^ { n / 2 } } . \\end{align*}"}
+{"id": "1176.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\mathbb { P } } \\left [ \\left | \\frac { 1 } { t ^ { \\frac { 1 } { 2 } - H } } \\sum _ { j = 2 } ^ n \\triangle k _ t ^ { 1 , j , n } \\right | ^ { 2 p } \\right ] \\leq p ! \\left ( ( n - 1 ) \\beta \\right ) ^ { p } . \\end{align*}"}
+{"id": "2385.png", "formula": "\\begin{align*} \\mathbb { T } ( \\mathcal { H } _ { \\ast } , \\{ \\mathbf { h } _ p \\} _ { p = 0 } ^ { 1 1 } , \\{ 0 \\} _ { p = 0 } ^ { 1 1 } ) = \\prod _ { p = 0 } ^ { 1 1 } \\left [ \\mathbf { h } ' _ p , \\mathbf { h } _ p \\right ] ^ { ( - 1 ) ^ { ( p + 1 ) } } = 1 . \\end{align*}"}
+{"id": "36.png", "formula": "\\begin{align*} \\Delta _ { \\lambda } \\xi = \\lambda { \\xi } _ 1 + . . . . . \\lambda ^ { h } { \\xi } _ h , \\end{align*}"}
+{"id": "1388.png", "formula": "\\begin{align*} g _ { x _ j } ( x ) = 0 , x _ j = x _ { j - 1 } \\end{align*}"}
+{"id": "6292.png", "formula": "\\begin{align*} \\begin{array} { r l } \\min \\limits _ { x \\in X } \\{ f ( x ) : g _ i ( x ) \\leq 0 , \\ i = 1 , \\ldots , m _ 1 ; h _ j ( x ) = 0 , \\ j = 1 , \\ldots , m _ 2 \\} . \\end{array} \\end{align*}"}
+{"id": "7975.png", "formula": "\\begin{align*} w ^ { 1 1 } w _ { 1 1 i } = - \\frac { h _ { i } } { h } . \\end{align*}"}
+{"id": "502.png", "formula": "\\begin{align*} e ^ { \\gamma x } f _ r ( x ) = \\int _ { - \\infty } ^ \\infty e ^ { \\gamma ( x - y ) } f _ { r + } ( x - y ) e ^ { \\gamma y } \\mu _ - ( d y ) \\le c \\int _ { - \\infty } ^ \\infty e ^ { \\gamma y } \\mu _ - ( d x ) < \\infty . \\end{align*}"}
+{"id": "2952.png", "formula": "\\begin{align*} \\Phi ( t , x , y ) : = \\begin{cases} ( x + t , y - t ) & ( x \\geq 0 x + t \\geq 0 ) \\\\ ( x + t , y - x - t ) & ( x < 0 x + t \\geq 0 ) \\\\ ( x + t , x + y ) & ( x \\geq 0 x + t < 0 ) \\\\ ( x + t , y ) & ( x < 0 x + t < 0 ) \\ \\end{cases} \\end{align*}"}
+{"id": "1832.png", "formula": "\\begin{align*} \\aligned \\exp \\ , ( \\frac 1 2 | | R ^ { \\nabla ^ t } | | ^ 2 ) = \\exp \\ , ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 + t \\langle R ^ \\nabla , d ^ \\nabla A \\rangle + \\varepsilon ( t ^ 2 ) ) \\ , , \\endaligned \\end{align*}"}
+{"id": "5865.png", "formula": "\\begin{align*} Z g \\left ( X , Y \\right ) = g \\left ( \\nabla _ { Z } X , Y \\right ) + g \\left ( X , \\nabla _ { Z } ^ { \\ast } Y \\right ) \\end{align*}"}
+{"id": "5936.png", "formula": "\\begin{align*} \\begin{cases} 2 + \\gamma + \\frac { 2 \\theta } { \\alpha } < p , & \\alpha \\leq 2 , \\\\ 4 + \\gamma + \\theta - \\alpha < p , & \\alpha > 2 . \\end{cases} \\end{align*}"}
+{"id": "2368.png", "formula": "\\begin{align*} \\begin{array} { c c c } C _ { \\ast } : = C _ { \\ast } ( K ; \\mathrm { A d } _ { \\rho } ) = ( 0 \\stackrel { \\partial _ 1 } { \\longrightarrow } C _ { 0 } ( K ; { \\mathrm { A d } _ \\rho } ) \\stackrel { \\partial _ 0 } { \\longrightarrow } 0 ) . \\end{array} \\end{align*}"}
+{"id": "6518.png", "formula": "\\begin{align*} & s ( w _ 1 w _ 2 ) = \\star = s ( w _ 2 ) \\\\ & r ( w _ 1 w _ 2 ) = \\star = r ( w _ 1 ) \\end{align*}"}
+{"id": "9111.png", "formula": "\\begin{align*} - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + ( j - 1 ) r < \\chi _ j < - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + j r , \\end{align*}"}
+{"id": "2697.png", "formula": "\\begin{align*} \\mathcal { F } _ { k - 1 } = \\sigma \\left ( \\left ( \\xi ^ { ( 0 ) } _ 0 , { \\xi ^ { ( 0 + ) } _ { 0 } } , \\xi ^ { ( 1 ) } _ 0 , \\xi ^ { ( 2 ) } _ 0 \\right ) , \\dots , \\left ( \\xi ^ { ( 0 ) } _ { k - 1 } , { \\xi ^ { ( 0 + ) } _ { k - 1 } } , \\xi ^ { ( 1 ) } _ { k - 1 } , \\xi ^ { ( 2 ) } _ { k - 1 } \\right ) \\right ) . \\end{align*}"}
+{"id": "5137.png", "formula": "\\begin{align*} \\mu = \\frac { 2 } { \\partial _ { s } j ( 0 , 0 ) } \\left ( \\int _ { \\mathbb { R } ^ { 3 } } b \\cdot b _ 0 \\dd x \\right ) . \\end{align*}"}
+{"id": "922.png", "formula": "\\begin{align*} & u _ 1 ( t ) = M ( t ) D ( t ) F _ 1 ( t ) + O ( t ^ { - \\frac { 3 } { 4 } + C \\varepsilon _ 1 ^ 2 } ) , \\\\ & u _ 2 ( t ) = M ( t ) D ( t ) F _ 2 ( t ) + O ( t ^ { - \\frac { 3 } { 4 } + C \\varepsilon _ 1 ^ 2 } ) \\end{align*}"}
+{"id": "3352.png", "formula": "\\begin{align*} V ( C _ 4 ) = \\{ 1 , 2 , 3 , 4 \\} , & & E ( C _ 4 ) = \\{ 1 2 , 2 3 , 3 4 , 4 1 \\} . \\end{align*}"}
+{"id": "6469.png", "formula": "\\begin{align*} \\mu \\Big ( \\bigcap _ { j = 0 } ^ { t } T ^ { j h ( C ) } J \\Big ) \\geq ( 1 - \\epsilon ) \\mu ( J ) . \\end{align*}"}
+{"id": "1524.png", "formula": "\\begin{align*} ( H f ) ( z _ 1 , \\ldots , z _ N ) & = \\frac { d } { d s } ( e ^ { s H } f ) ( z _ 1 , \\ldots , z _ N ) | _ { s = 0 } \\\\ & = \\frac { d } { d s } \\big [ ( w \\chi ) ( e ^ { s H } ) \\cdot f \\big ( ( w \\alpha _ 1 ) ( e ^ { - s H } ) z _ 1 , \\ldots , ( w \\alpha _ N ) ( e ^ { - s H } ) z _ N \\big ) \\big ] | _ { s = 0 } \\\\ & = d ( w \\chi ) ( H ) f ( z _ 1 , \\ldots , z _ N ) - \\sum _ { r = 1 } ^ N \\sum _ { s = 1 } ^ { \\ell } d ( w \\alpha _ r ) ( H ) z _ { r s } \\frac { \\partial f } { \\partial z _ { r s } } ( z _ 1 , \\ldots , z _ N ) , \\end{align*}"}
+{"id": "4887.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { \\sigma \\mathrm { B } ( \\alpha , \\beta ) } \\ , \\left [ \\Phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] ^ { \\alpha - 1 } \\ , \\left [ 1 - \\Phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] ^ { \\beta - 1 } \\ , \\phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) , x \\in \\mathbb { R } , \\end{align*}"}
+{"id": "5231.png", "formula": "\\begin{align*} - \\left ( \\partial _ { \\rho } ^ { 2 } + \\frac { 4 } { \\rho } \\partial _ { \\rho } \\right ) \\varphi = \\mu ^ { 2 } \\left ( \\varphi - \\frac { W } { 2 } \\right ) _ { + } , \\rho > 0 . \\end{align*}"}
+{"id": "9260.png", "formula": "\\begin{align*} \\begin{aligned} ( \\Delta u ) ^ { k } \\wedge \\beta _ n ^ { n - k } \\geq 0 , k = 1 , 2 , \\dots , m . \\end{aligned} \\end{align*}"}
+{"id": "7534.png", "formula": "\\begin{align*} \\deg _ H ( S ) = \\deg _ F ( S ) \\leq 1 , \\end{align*}"}
+{"id": "1558.png", "formula": "\\begin{align*} \\begin{bmatrix} D \\Psi - \\kappa \\ell U & - \\kappa \\ell R & R D + \\ell \\cos \\Psi & \\sin \\Psi - \\kappa U R \\end{bmatrix} . \\end{align*}"}
+{"id": "2752.png", "formula": "\\begin{align*} | H ( 0 , 0 , R ) | = | Z ( 0 , 0 , R ) | = \\exp R \\end{align*}"}
+{"id": "2552.png", "formula": "\\begin{align*} \\psi _ o = \\frac { { { \\mathbf { e } } ^ { T } } } { k } \\left ( { { \\mathbf { x } } _ { o } } + { { \\mathbf { s } } _ { o } } \\right ) . \\end{align*}"}
+{"id": "3384.png", "formula": "\\begin{align*} P ( x , y ) = Z ^ 4 - Z ^ 3 - Z ( S _ 2 ( x ; N ) + S _ 2 ( y ; K ) ) + ( S _ 1 ( x ; N ) - S _ 1 ( y ; K ) ) ^ 2 , Z = Z ( x , y ; N , K ) . \\end{align*}"}
+{"id": "8263.png", "formula": "\\begin{align*} \\frac { d \\xi ^ { n } _ m ( t ) } { d t } \\leq & e ^ { - t } \\left [ \\sum _ { i = 1 } ^ { m - 1 } \\sum _ { j = i } ^ { n - 1 } j ( i + j ) \\psi ^ { n } _ i \\psi ^ { n } _ j + ( m - 1 ) \\psi ^ { n } _ { m - 1 } \\sum _ { j = 1 } ^ { m - 1 } j ( m - 1 + j ) \\psi ^ { n } _ j - 2 \\mu ^ { n } _ 1 ( 0 ) ^ 2 - 2 m \\mu ^ { n } _ 1 ( 0 ) ^ 2 \\right ] . \\end{align*}"}
+{"id": "9183.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } ( t ) & - \\dot { z } ( t ) = \\\\ & - \\gamma ( h ( x ( t ) + \\delta u ( t ) ) - h ( z ( t ) + \\delta u ( t ) ) ) u ( t ) \\\\ & - \\gamma ( h ( z ( t ) + \\delta u ( t ) ) - h ( x _ a ( t ) + \\delta u ( t ) ) ) u ( t ) \\\\ & - \\left . \\gamma ^ 2 \\dfrac { \\partial \\epsilon ( x , t ) } { \\partial x } \\right | _ { x = x _ a ( t ) } \\dfrac { b _ { 1 , \\delta } ( x _ a ( t ) ) } { 2 } . \\end{aligned} \\end{align*}"}
+{"id": "9094.png", "formula": "\\begin{align*} \\chi _ j \\leq \\chi ( \\sum \\limits _ { i = 1 } ^ { j } w _ i ) - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + \\sum \\limits _ { i = 1 } ^ { j } k _ i + ( j - 1 ) r . \\end{align*}"}
+{"id": "2648.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k x ^ { k } = \\frac { x ( 1 - x ^ { n + 1 } ) - ( n + 1 ) ( 1 - x ) x ^ { n + 1 } } { ( 1 - x ) ^ 2 } . \\end{align*}"}
+{"id": "9058.png", "formula": "\\begin{align*} X _ { \\alpha } v _ k = \\sum _ { i = 1 } ^ { k } X _ { m - k + 1 } \\cdots X _ { m - k + i - 1 } [ X _ { \\alpha } , X _ { m - k + i } ] X _ { m - k + i + 1 } \\cdots X _ m v _ 0 \\end{align*}"}
+{"id": "8329.png", "formula": "\\begin{align*} & \\psi ^ { \\pm } ( x , t ; k ) = \\sigma _ 2 \\overline { \\psi ^ { \\pm } ( x , t ; \\bar { k } ) } \\sigma _ 2 , \\ \\ \\ \\psi ^ { \\pm } ( x , t ; k ) = \\sigma _ 3 \\psi ^ { \\pm } ( x , t ; - k ) \\sigma _ 3 , \\end{align*}"}
+{"id": "6268.png", "formula": "\\begin{align*} \\partial _ j \\varphi _ i = 2 \\Gamma ^ i _ { j i } \\varphi _ i \\quad \\forall i \\neq j . \\end{align*}"}
+{"id": "705.png", "formula": "\\begin{align*} \\lim \\limits _ { R \\to \\infty } R ^ { N p - N - \\alpha } J ( \\Psi _ { R } ) = [ M ( | \\Psi | ^ p ) ] ^ 2 J ( \\eta ) . \\end{align*}"}
+{"id": "4593.png", "formula": "\\begin{align*} \\frac { g ' ( x + \\gamma ) } { g ' ( x ) } = e ^ { a \\gamma } > 1 , \\end{align*}"}
+{"id": "6591.png", "formula": "\\begin{align*} F ( r , t ) : = ( Q ( r , t ) - W ( r , t ) ) \\exp ( - t ) \\end{align*}"}
+{"id": "4782.png", "formula": "\\begin{align*} a ( u , v ) = \\int _ D \\nabla u \\cdot \\nabla \\bar { v } ~ d x , ( f , v ) = \\int _ D f \\bar { v } ~ d x . \\end{align*}"}
+{"id": "235.png", "formula": "\\begin{align*} g = \\dot x ( J ) \\ , \\dot y ( J ) \\ , \\dot z ( J ) . \\end{align*}"}
+{"id": "1064.png", "formula": "\\begin{align*} \\langle \\omega _ a , \\beta \\rangle = \\begin{cases} 1 , & \\alpha = \\beta , \\\\ 0 , & \\alpha \\neq \\beta . \\end{cases} \\end{align*}"}
+{"id": "52.png", "formula": "\\begin{align*} [ X _ i , \\rho ^ { - Q + 2 } Z ] v = \\rho ^ { - Q + 2 } [ X _ i , Z ] v + X _ i ( \\rho ^ { - Q + 2 } ) Z v = \\rho ^ { - Q + 2 } X _ i v + ( 2 - Q ) \\rho ^ { - Q + 1 } X _ i \\rho \\ ; Z v . \\end{align*}"}
+{"id": "489.png", "formula": "\\begin{align*} \\lim _ { z \\to - \\infty } \\lim _ { x \\to \\infty } I _ k ^ x ( z ) = \\lim _ { z \\to - \\infty } \\lim _ { x \\to \\infty } J _ k ^ x ( z ) = 1 , \\end{align*}"}
+{"id": "997.png", "formula": "\\begin{align*} s _ \\alpha ( \\mu ) = \\mu - \\langle \\mu , \\alpha \\rangle \\alpha ^ \\vee , \\alpha \\in \\Phi , ~ \\mu \\in X _ \\ast . \\end{align*}"}
+{"id": "7917.png", "formula": "\\begin{align*} \\hat W _ r ( m _ 2 ) = \\frac { m _ 1 } { m _ 2 } \\hat W _ { \\frac { r m _ 2 } { m _ 1 } } ( m _ 1 ) . \\end{align*}"}
+{"id": "7974.png", "formula": "\\begin{align*} \\nabla _ { j } \\frac { P } { h } = - \\frac { P } { h } \\frac { B \\nabla _ { j } \\frac { \\rho ^ { 2 } } { 2 } } { 1 - \\frac { B \\rho ^ { 2 } } { 2 } } = - \\frac { P } { h } \\frac { B w _ { j m } h _ { m } } { 1 - \\frac { B \\rho ^ { 2 } } { 2 } } . \\end{align*}"}
+{"id": "8194.png", "formula": "\\begin{align*} x _ { 1 , 2 j } & + x _ { 1 , 2 j + 1 } + x _ { m , ( n + 1 ) - 2 j } + x _ { m , ( n + 1 ) - ( 2 j + 1 ) } \\\\ & = x _ { 1 , 2 j - 1 } + x _ { 1 , 2 j } + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { m , 2 j } \\bigr ) + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { m , 2 j + 1 } \\bigr ) \\\\ & = \\bigl ( ( n _ 1 + 1 ) m + 1 \\bigr ) + S - \\bigl ( ( n _ 1 + 1 ) m + 1 \\bigr ) = S . \\end{align*}"}
+{"id": "6861.png", "formula": "\\begin{align*} f ( z _ 1 ) = 0 , \\ f ( z _ 2 ) = 1 , \\ f ( z _ 3 ) = 1 + \\i h , \\ f ( z _ 4 ) = \\i h . \\end{align*}"}
+{"id": "4196.png", "formula": "\\begin{align*} c _ \\varepsilon = I _ \\varepsilon ( u _ \\varepsilon ) \\geq \\inf \\limits _ { w \\in \\mathcal { N } _ \\varepsilon } I _ \\varepsilon ( w ) . \\end{align*}"}
+{"id": "8721.png", "formula": "\\begin{align*} s ( q _ 1 ) s ( q _ 2 ) = c ( q _ 1 , q _ 2 ) s ( q _ 1 q _ 2 ) \\forall q _ 1 , q _ 2 \\in Q . \\end{align*}"}
+{"id": "7071.png", "formula": "\\begin{align*} \\aligned F _ { 7 , 0 , 0 , 0 } & \\ , = \\ , 0 , & \\ \\ \\ \\ \\ \\ \\ \\ \\ F _ { 7 , 1 , 0 , 0 } & \\ , = \\ , 0 , & \\ \\ \\ \\ \\ \\ \\ \\ \\ F _ { 8 , 0 , 0 , 0 } & \\ , = \\ , - \\ , \\tfrac { 5 6 } { 7 5 } , \\\\ F _ { 7 , 1 , 0 , 0 } & \\ , = \\ , 0 , & \\ \\ \\ \\ \\ \\ \\ \\ \\ F _ { 8 , 0 , 1 , 0 } & \\ , = \\ , 0 , & \\ \\ \\ \\ \\ \\ \\ \\ \\ F _ { 8 , 1 , 0 , 0 } & \\ , = \\ , - \\ , \\tfrac { 3 9 2 } { 7 5 } , \\ \\ \\ \\ \\ \\ \\ \\ \\ F _ { 9 , 0 , 0 , 0 } \\ , = \\ , 0 . \\endaligned \\end{align*}"}
+{"id": "2112.png", "formula": "\\begin{align*} \\frac { d } { d b } \\left ( R L ( b , c ) - L L ( b , c ) \\right ) = J _ 1 ( b , c ) + J _ 2 ( b , c ) + J _ 3 ( b , c ) , \\end{align*}"}
+{"id": "5082.png", "formula": "\\begin{align*} { \\mathcal { S } } _ h = \\begin{cases} \\ \\{ U _ j ^ { + } \\} _ { j = 1 } ^ { N ^ { + } } & h > 0 , \\\\ \\ \\{ U _ j ^ { - } \\} _ { j = 1 } ^ { N ^ { - } } & h < 0 , \\\\ \\ \\emptyset & h = 0 . \\end{cases} \\end{align*}"}
+{"id": "6914.png", "formula": "\\begin{align*} Q _ i ( d x _ i ) & = Z _ i ^ { - 1 } e ^ { \\hat { f } _ i ( x _ i ) } \\ , d x _ i , \\hat { f } _ i : \\R \\to \\R \\cup \\{ - \\infty \\} \\\\ \\hat { f } _ i ( x _ i ) & : = \\int _ { \\R ^ { n - 1 } } f ( x _ 1 , \\ldots , x _ n ) \\ , \\prod _ { j \\neq i } Q ^ * _ j ( d x _ j ) , i \\in [ n ] . \\end{align*}"}
+{"id": "8251.png", "formula": "\\begin{align*} \\frac { d \\mu _ g ^ n } { d t } = \\sum _ { i = 1 } ^ { n - 1 } \\sum _ { j = i } ^ { n - 1 } ( i g _ { j + 1 } - i g _ j - g _ i ) V _ { i , j } \\psi _ i \\psi _ j \\end{align*}"}
+{"id": "8004.png", "formula": "\\begin{align*} & \\frac { 1 } { | \\mathcal F _ { N , k } | } \\sum _ { f \\in \\mathcal F _ { N , k } } \\left ( R _ 2 ( g , \\rho ) ( f ) - \\frac { T ( g , \\rho ) } { 4 L } \\right ) ^ 2 \\\\ & \\ll \\frac { 1 } { L } + \\frac { L ^ 2 ( \\log \\log x ) ^ 2 } { \\pi _ N ( x ) ^ 2 } + \\frac { L ( \\log \\log x ) ^ 2 } { \\pi _ N ( x ) } + \\frac { L ^ { 1 / 2 } \\log \\log x } { \\pi _ N ( x ) ^ { 1 / 2 } } \\\\ & + \\frac { x ^ { C \\pi _ N ( x ) } 8 ^ { \\nu ( N ) } } { k { N } } , \\end{align*}"}
+{"id": "471.png", "formula": "\\begin{align*} g ^ { \\ast 2 } ( x ) = ( 2 / \\lambda ^ 2 e ^ { \\lambda } ) \\big \\{ ( 1 - e ^ { - \\lambda } ) f ( x ) - e ^ { - \\lambda } \\sum _ { n \\neq 2 } ^ \\infty ( \\lambda ^ n / n ! ) \\ , g ^ { \\ast n } ( x ) \\big \\} . \\end{align*}"}
+{"id": "9245.png", "formula": "\\begin{align*} \\begin{aligned} \\binom { n } { m } \\det ( A _ 1 , \\dots , A _ m , I , \\dots , I ) \\geq \\mathcal { H } _ m ( A _ 1 ) ^ { \\frac { 1 } { m } } \\dots \\mathcal { H } _ m ( A _ m ) ^ { \\frac { 1 } { m } } . \\end{aligned} \\end{align*}"}
+{"id": "5465.png", "formula": "\\begin{align*} \\lambda _ { \\phi _ { \\psi } } ( w ) = \\bigoplus _ { i = 0 } ^ { a } \\bigoplus _ { j = 0 } ^ { d } | w | ^ { \\frac { a + d } { 2 } - ( i + j ) } \\end{align*}"}
+{"id": "4230.png", "formula": "\\begin{align*} & \\ \\dot { s } \\epsilon ( z ) \\dot { s } ( e - \\varepsilon ( 1 ) ) \\lambda _ i = \\varepsilon ( - z ^ { - 1 } ) \\dot { s } ( \\varepsilon ( - z ) - \\varepsilon ( z ^ 2 - z ) ) \\lambda _ i \\\\ & \\ = \\sum _ { k , \\nu } g ^ i _ { k , \\nu } ( \\varepsilon ( - z ^ { - 1 } ( 1 + x ^ i _ { k , \\nu } ) ) - \\varepsilon ( z ^ { - 1 } ( z - 1 ) ^ { - 1 } ( x ^ i _ { k , \\nu } - z + 1 ) ) ) \\lambda _ k . \\end{align*}"}
+{"id": "2503.png", "formula": "\\begin{align*} \\mu = ( \\mathbf { x } ^ T \\mathbf { s } + \\kappa \\tau ) / ( k + 1 ) = \\hat { \\mathbf { x } } ^ T \\hat { \\mathbf { s } } / ( k + 1 ) \\end{align*}"}
+{"id": "3929.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta y + \\mathcal { N } ( \\cdot , y ) & = u , \\ ; \\ ; \\Omega , \\\\ y & = 0 , \\ ; \\ ; \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3504.png", "formula": "\\begin{align*} { c _ { i } } | W _ i \\cap C ^ { ( i ) } | = { \\beta _ { j + i - 1 } } | W _ { i - 1 } \\cap C ^ { ( i - 1 ) } | . \\end{align*}"}
+{"id": "3723.png", "formula": "\\begin{align*} r _ m = ( 2 m ) ^ { \\frac { 1 } { n - 2 } } , g _ m = \\left ( 1 - \\tfrac { 2 m } { r ^ { n - 2 } } \\right ) ^ { - 1 } d r ^ 2 + r ^ 2 g _ { S ^ { n - 1 } } , u _ m = \\sqrt { 1 - \\tfrac { 2 m } { r ^ { n - 2 } } } . \\end{align*}"}
+{"id": "3766.png", "formula": "\\begin{align*} r _ { i , j } ( w ) . : = | \\{ k ; k \\leq i , w ( k ) \\leq j \\} | . \\end{align*}"}
+{"id": "2597.png", "formula": "\\begin{align*} Q ^ c ( g , R ) ( \\pi ; \\bar { \\pi } ^ h ) = 2 Q ( g , R ) ( \\pi ; \\bar { \\pi } ^ h ) , \\end{align*}"}
+{"id": "8900.png", "formula": "\\begin{align*} \\lambda _ { 1 2 3 ' } = \\log m , m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "3758.png", "formula": "\\begin{align*} \\# R ^ i _ { n _ 0 ; l _ 1 , \\cdots , l _ { n _ 0 } } = \\underset { \\tilde { l } = ( \\tilde { l } _ { i } , \\tilde { l } _ { i + 1 } , \\cdots , \\tilde { l } _ { n _ 0 } ) \\in L } { \\sum } c _ { \\tilde { l } } \\# P _ { l _ 1 , l _ 2 , \\cdots , l _ { i - 1 } , \\tilde { l } _ i , \\tilde { l } _ { i + 1 } , \\cdots , \\tilde { l } _ { n _ 0 } } \\end{align*}"}
+{"id": "2118.png", "formula": "\\begin{align*} A x ^ 2 + B y ^ 2 + C u ^ 2 + D v ^ 2 = 1 , A , B , C , D \\in \\mathbb { R } _ { > 0 } . \\end{align*}"}
+{"id": "8901.png", "formula": "\\begin{align*} \\lambda _ { 1 2 3 ' } = \\log m , m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "7275.png", "formula": "\\begin{align*} a _ { n - i } ( \\gamma ^ { ( q ^ { n - i } - q ^ n ) } - 1 ) = 0 \\end{align*}"}
+{"id": "198.png", "formula": "\\begin{align*} D _ { \\theta _ { ( 2 ) } } ( f ) : = D ^ d _ { \\theta _ { ( 2 ) } } \\circ \\cdots \\circ D ^ 2 _ { \\theta _ { ( 2 ) } } \\circ D ^ 1 _ { \\theta _ { ( 2 ) } } ( f ) . \\end{align*}"}
+{"id": "8795.png", "formula": "\\begin{align*} \\left | \\Pi _ { \\mathrm { k e r } A ^ \\perp } \\frac { d ^ \\ell } { d t ^ \\ell } X _ t | _ { t = 0 } \\right | \\ge \\frac { \\gamma } { C _ J } K ^ { \\ell + 1 } , \\end{align*}"}
+{"id": "9024.png", "formula": "\\begin{align*} x = \\begin{pmatrix} - \\sigma & - \\tau & - 1 \\\\ - \\tau & - 1 & - \\sigma \\\\ 1 & \\sigma & \\tau \\end{pmatrix} , \\ y = \\begin{pmatrix} 1 & \\sigma & \\tau \\\\ \\sigma & \\tau & 1 \\\\ \\tau & 1 & \\sigma \\end{pmatrix} , \\end{align*}"}
+{"id": "1416.png", "formula": "\\begin{align*} \\mathrm { e } ^ { \\lambda \\sum _ { i = 1 } ^ d ( y _ i - x _ i ) - d n \\ln ( 1 + \\lambda ) } \\le \\mathrm { e } ^ { \\lambda d ( y _ 1 - x _ 1 ) + d ^ 2 - n d \\lambda + n d \\lambda ^ 2 } \\le C _ d \\mathrm { e } ^ { - d \\frac { ( y _ 1 - x _ 1 - n ) } { \\sqrt n } } . \\end{align*}"}
+{"id": "8412.png", "formula": "\\begin{align*} \\begin{aligned} \\Psi ^ - _ { 2 1 } ( x ; z ) & = \\int _ { - \\infty } ^ { x - \\delta } e ^ { 2 i z ( x - y ) } \\nu ( y ; z ) d y + \\nu ( x ; z ) \\int _ { x - \\delta } ^ { x } e ^ { 2 i z ( x - y ) } \\mathrm { d } y \\\\ & + \\int _ { x - \\delta } ^ { x } e ^ { 2 i z ( x - y ) } ( \\nu ( y ; z ) - \\nu ( x ; z ) ) \\mathrm { d } y \\equiv I _ 1 + I _ 2 + I _ 3 , \\end{aligned} \\end{align*}"}
+{"id": "4923.png", "formula": "\\begin{align*} M ( - t ) = \\frac { 1 } { \\sqrt 2 \\pi } \\sum _ { r , j = 0 } ^ \\infty \\ , \\pi _ r \\ , \\ , c _ { r , j } \\ , \\ , J ( t , j ) . \\end{align*}"}
+{"id": "2135.png", "formula": "\\begin{align*} 0 = \\varrho _ { Z Z } ( L , L ) , \\end{align*}"}
+{"id": "7335.png", "formula": "\\begin{align*} | A | \\ , | B _ 1 + \\dots + B _ m | \\le \\prod \\limits _ { i = 1 } ^ m | A + B _ i | , \\end{align*}"}
+{"id": "2530.png", "formula": "\\begin{align*} \\mathbf { D } = \\mathbf { I } . \\end{align*}"}
+{"id": "4134.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n - 1 } } P ^ L ( x ' ) \\ , d x ' = I _ { N \\times N } , \\end{align*}"}
+{"id": "5431.png", "formula": "\\begin{align*} - \\epsilon ^ 2 \\delta ^ 2 b _ { \\alpha } u _ n ( x ) \\leq u ( 0 ) - u ( x ) + \\sum _ { \\beta = 1 } ^ { n - 1 } x _ \\beta u _ \\beta ( x ) \\leq C b _ \\alpha ^ 2 \\end{align*}"}
+{"id": "7011.png", "formula": "\\begin{align*} u \\ , = \\ , \\tfrac { x ^ 2 } { 2 } + { \\rm O } _ { x , y } ( 3 ) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ v \\ , = \\ , \\tfrac { r ^ 2 } { 2 } + { \\rm O } _ { r , s } ( 3 ) . \\end{align*}"}
+{"id": "9290.png", "formula": "\\begin{align*} A _ { p , N } & \\rightarrow \\sum _ { A = 0 } ^ { 2 n - 1 } \\int _ S \\left [ v _ p ( d _ 1 v _ { p } \\wedge \\alpha ) _ A \\tau ( \\mathbf { { n } } ) _ { A 0 } + v _ { p } ( d _ 0 v _ p \\wedge \\alpha ) _ A \\tau ( \\mathbf { { n } } ) _ { A 1 } \\right ] d S \\\\ & = \\int _ B ( d _ 0 v _ p \\wedge d _ 1 v _ { p } + v _ p d _ 0 d _ 1 v _ { p } ) \\wedge \\alpha + \\int _ B ( d _ 1 v _ { p } \\wedge d _ 0 v _ p + v _ p d _ 1 d _ 0 v _ { p } ) \\wedge \\alpha = 0 , \\end{align*}"}
+{"id": "6235.png", "formula": "\\begin{align*} D _ n ( x - ( I - R ^ { - 1 } ) ^ { - 1 } b ) & = \\prod _ { k = 0 } ^ { n } m _ \\tau ( R ^ k ( x - ( I - R ^ { - 1 } ) ^ { - 1 } b ) ) \\\\ & = \\prod _ { k = 0 } ^ { n } m _ \\tau ( \\mathcal { R } ^ k x - \\mathcal { R } ^ k ( ( I - R ^ { - 1 } ) ^ { - 1 } b ) ) \\\\ & = \\prod _ { k = 0 } ^ { n } m _ \\tau ( \\mathcal { R } ^ k x - \\mathcal { R } ^ k ( \\sum \\limits _ { i = 0 } ^ { \\infty } R ^ { - i } b ) ) \\\\ & = \\prod _ { k = 0 } ^ { n } m _ \\tau ( \\mathcal { R } ^ k x - ( I - R ^ { - 1 } ) ^ { - 1 } b ) . \\end{align*}"}
+{"id": "5123.png", "formula": "\\begin{align*} | \\{ x \\in \\mathbb { R } ^ { 3 } \\ | \\ \\phi ( x ) > \\phi _ { \\infty } \\} | = \\int _ { \\{ \\phi > \\phi _ { \\infty } \\} } \\dd x = 2 \\pi \\int _ { \\{ \\phi > \\phi _ { \\infty } \\} } \\frac { 1 } { r } \\dd z \\dd r \\leq 2 \\pi \\int _ { \\{ \\phi > r ^ { 2 } \\} } \\phi ^ { 2 } \\frac { 1 } { r ^ { 3 } } \\dd z \\dd r \\lesssim | | \\nabla \\phi | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } _ { + } ; r ^ { - 1 } ) } ^ { 2 } . \\end{align*}"}
+{"id": "7872.png", "formula": "\\begin{align*} 0 & = ( I - Q ) F _ { \\Psi \\Psi } ( \\Psi ^ q , p _ 0 ) [ v _ 1 , \\psi _ p ( 0 , p _ 0 ) p ' ( 0 ) ] \\\\ & + ( I - Q ) F _ { \\Psi p } ( \\Psi ^ q , p _ 0 ) [ v _ 1 , p ' ( 0 ) ] \\\\ & + ( I - Q ) F _ \\Psi ( \\Psi ^ q , p _ 0 ) \\psi _ { v p } ( 0 , p _ 0 ) [ v _ 1 , p ' ( 0 ) ] , \\end{align*}"}
+{"id": "6327.png", "formula": "\\begin{align*} z _ 0 x _ 0 ^ { p ^ { h _ i - r } } + z _ 1 x _ 1 ^ { p ^ { h _ i - r } } + \\cdots + z _ n x _ n ^ { p ^ { h _ i - r } } = 0 . \\end{align*}"}
+{"id": "2701.png", "formula": "\\begin{align*} h ( \\gamma \\bar \\Delta ) \\sum _ { k = 0 } ^ { T - 1 } \\Theta _ k \\Lambda _ k ' < \\phi ( x _ 0 ) - \\hat \\phi + \\sum _ { k = 0 } ^ { T - 1 } \\Theta _ k ' \\left ( \\mathcal { E } _ k + \\mathcal { E } _ k ^ + + r \\right ) . \\end{align*}"}
+{"id": "5650.png", "formula": "\\begin{align*} \\langle e _ { i } , A e _ { j } \\rangle = \\langle A e _ { i } , A e _ { j } \\rangle = \\langle e _ { i } , e _ { j } \\rangle = 0 \\end{align*}"}
+{"id": "5388.png", "formula": "\\begin{align*} \\begin{aligned} G ^ { \\alpha \\beta } _ 0 \\sigma _ { \\alpha \\beta } \\geq \\ , & G ( \\sigma _ { \\alpha \\beta } - \\theta \\delta _ { \\alpha \\beta } ) + \\theta \\sum _ { \\alpha = 1 } ^ { n - 1 } G ^ { \\alpha \\alpha } _ 0 \\\\ \\geq \\ , & \\gamma + \\theta \\sum _ { \\alpha = 1 } ^ { n - 1 } G ^ { \\alpha \\alpha } _ 0 \\end{aligned} \\end{align*}"}
+{"id": "3839.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & k _ 1 = K _ 1 - \\gamma _ 1 , k _ 2 = K _ 2 - \\gamma _ 2 , k _ 3 = K _ 3 - \\gamma _ 3 , \\\\ & k _ { 1 1 } = K _ 4 - \\gamma _ 1 , k _ { 2 2 } = K _ 5 - \\gamma _ 2 , k _ { 3 3 } = K _ 6 - \\gamma _ 3 , \\\\ & k _ { 3 1 } = K _ 7 - \\gamma _ 1 , k _ { 1 2 } = K _ 8 - \\gamma _ 2 , k _ { 2 3 } = K _ 9 - \\gamma _ 3 , \\\\ & k _ { 2 1 } = K _ { 1 0 } - \\gamma _ 1 , k _ { 3 2 } = K _ { 1 1 } - \\gamma _ 2 , k _ { 1 3 } = K _ { 1 2 } - \\gamma _ 3 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6233.png", "formula": "\\begin{align*} D _ { n + 1 } ( x ) = m _ \\tau ( x ) D _ n ( R ^ { t } x ) . \\end{align*}"}
+{"id": "2815.png", "formula": "\\begin{align*} \\mu _ k ^ { \\mathrm { o p t } } = \\frac { \\sum _ { l \\in \\mathcal { M } _ { k } } \\overline { \\mathbf { h } } _ { k l } ^ { \\mathbf { H } } \\mathbf { z } _ { k l } } { \\sum _ { i \\in \\mathcal { K } _ { l } } \\sum _ { l \\in \\mathcal { M } _ { i } } \\overline { \\mathbf { h } } _ { k l } ^ { \\mathbf { H } } \\mathbf { z } _ { i l } \\mathbf { z } _ { i l } ^ { \\mathbf { H } } \\overline { \\mathbf { h } } _ { k l } + \\delta _ { } ^ 2 } . \\end{align*}"}
+{"id": "1760.png", "formula": "\\begin{align*} \\partial _ t u _ 0 = \\bar { J } _ 0 ^ { - 1 } \\partial _ t ( \\bar { J } _ 0 u _ 0 ) - \\bar { J } _ 0 ^ { - 1 } u _ 0 \\partial _ t \\bar { J } _ 0 \\in L ^ 2 ( ( 0 , T ) , W ^ { 1 , q } ( \\Omega ) ' ) . \\end{align*}"}
+{"id": "7307.png", "formula": "\\begin{align*} { \\rm E E } _ { v , j } = \\frac { { \\overline { \\rho } _ { v , j } B _ { \\rm V L C } \\log _ 2 \\left ( { 1 + { \\frac { \\exp ( 1 ) } { 2 \\pi } } \\frac { { \\overline P } _ v { \\left ( { { R _ { \\rm { P D } } } { \\overline { G } _ { v , j } } } \\right ) } ^ 2 } { { \\sum \\limits _ { v ' \\ne v } { \\overline P } _ { v ' } { \\left ( { { R _ { \\rm { P D } } } { \\overline { G } _ { v ' , j } } } \\right ) } ^ 2 } + N _ { \\rm V L C } B _ { \\rm V L C } } } \\right ) } } { P _ { \\rm V L C } + P _ v } . \\end{align*}"}
+{"id": "2483.png", "formula": "\\begin{align*} k = l + m , \\end{align*}"}
+{"id": "224.png", "formula": "\\begin{align*} g _ { | i ( g ) } = \\prod _ { k = 1 } ^ { l _ S ( g ) } ( s _ { \\iota ( k ) } ) _ { | \\ , i ( g ) - \\sigma ( k - 1 ) } . \\end{align*}"}
+{"id": "2241.png", "formula": "\\begin{align*} \\frac { 1 } { t } \\int _ 0 ^ t \\vert \\langle \\psi _ { \\beta , f _ \\delta } , e ^ { - i s H _ N } \\psi _ { \\beta , f _ \\delta } \\rangle \\vert ^ 2 \\ , d s = O ( t ^ { - 2 ( 1 - \\beta ) } ) \\end{align*}"}
+{"id": "7868.png", "formula": "\\begin{align*} \\Psi ^ q + v + \\psi ( v , p ( s ) ) = \\Psi ^ q + v + \\frac 1 2 ( v , p ( s ) - p ( 0 ) ) H \\begin{pmatrix} v \\\\ p ( s ) - p ( 0 ) \\end{pmatrix} + O ( \\norm { ( v , s ) } ^ 3 ) , \\end{align*}"}
+{"id": "8177.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\lambda } \\mu [ \\xi ( \\mu , t ) - \\xi ( \\mu , \\tau ) ] \\ d \\mu = - \\int _ { \\tau } ^ { t } \\int _ { \\lambda } ^ { \\infty } \\int _ { 0 } ^ { \\lambda } \\nu \\Lambda ( \\mu , \\nu ) \\xi ( \\mu , s ) \\xi ( \\nu , s ) \\ d \\nu d \\mu d s . \\end{align*} % \\end{align*}"}
+{"id": "4723.png", "formula": "\\begin{align*} f _ k ( x ) : = x ^ T A _ k x + 2 b _ k ^ T x + c _ k \\end{align*}"}
+{"id": "1622.png", "formula": "\\begin{align*} C _ H ( C , M ^ \\vee ) : = \\bigoplus _ { n \\geq 0 } \\ , C _ { n , H } ( C , M ^ \\vee ) \\end{align*}"}
+{"id": "568.png", "formula": "\\begin{align*} \\frac { 1 } { z } = - \\int _ 0 ^ { + \\infty } e ^ { \\omega z } d \\omega , \\quad R e ( z ) < 0 . \\end{align*}"}
+{"id": "4869.png", "formula": "\\begin{align*} \\Gamma ( x _ 0 , y ) = d _ x T ( x _ 0 , y ) . \\end{align*}"}
+{"id": "9227.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ( t ) - x _ a ( t ) \\| \\le & \\ , \\| x ( 0 ) - z ( 0 ) \\| e ^ { \\gamma L _ \\mathcal { D } t } \\\\ & \\ , + \\gamma 3 M _ \\mathcal { D } T \\left ( e ^ { \\gamma L _ \\mathcal { D } t } - 1 \\right ) \\\\ & \\ , + \\gamma 2 M _ \\mathcal { D } T . \\end{aligned} \\end{align*}"}
+{"id": "2853.png", "formula": "\\begin{align*} { \\bf X } ' ( t ) = e ^ { - A t } { \\bf X } ' ( 0 ) + \\int _ { 0 } ^ t e ^ { - A ( t - s ) } \\Sigma \\Big ( { \\bf p } ( s - ) \\Big ) \\dd M ( s ) , t \\ge 0 . \\end{align*}"}
+{"id": "3877.png", "formula": "\\begin{align*} \\begin{cases} - \\delta ^ 2 ( K ( \\hat { x } ) \\nabla v ) = ( v - \\hat { q } ) ^ { p } _ + , \\ \\ & \\ T _ { \\hat { x } } ^ { - 1 } ( B _ R ( 0 ) ) , \\\\ v = 0 , \\ \\ & \\ \\partial T _ { \\hat { x } } ^ { - 1 } ( B _ R ( 0 ) ) . \\end{cases} \\end{align*}"}
+{"id": "3242.png", "formula": "\\begin{align*} \\mathcal S ( t ) f = \\sum _ { j = 1 } ^ { \\infty } e ^ { t \\lambda _ j } \\langle e _ j , f \\rangle e _ j , \\end{align*}"}
+{"id": "2258.png", "formula": "\\begin{align*} a _ { j k } x _ k = - x _ j b _ { j k } , x _ j a _ { j k } = - b _ { j k } x _ k , \\end{align*}"}
+{"id": "3917.png", "formula": "\\begin{align*} | G _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) | + | \\nabla G _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) | \\le C , \\ \\ 1 \\leq i \\neq j \\leq m , \\end{align*}"}
+{"id": "305.png", "formula": "\\begin{align*} i ( x ) & : = \\min \\{ 1 \\le l \\le n \\ , : \\ , \\\\ & \\} , \\end{align*}"}
+{"id": "2755.png", "formula": "\\begin{align*} \\gamma _ m = \\sum _ { j = M } ^ { m - 1 } C ^ { j - m } = \\sum _ { k = 1 } ^ { m - M } C ^ { - k } \\leq \\frac { 1 } { C - 1 } . \\end{align*}"}
+{"id": "1235.png", "formula": "\\begin{align*} \\hat { c } _ { \\Delta } ( \\eta , \\xi _ { \\Delta } ) : = c _ { \\Delta } ( \\xi _ { \\Delta } \\eta _ { \\Delta ^ c } , \\eta _ { \\Delta } ) \\frac { \\gamma _ { \\Delta } ( \\xi _ { \\Delta } | \\eta _ { \\Delta ^ c } ) } { \\gamma _ { \\Delta } ( \\eta _ { \\Delta } | \\eta _ { \\Delta ^ c } ) } . \\end{align*}"}
+{"id": "214.png", "formula": "\\begin{align*} g = W ( s _ 1 , \\ldots , s _ d ) = \\prod _ { k = 1 } ^ l s _ { \\iota ( k ) } ; \\end{align*}"}
+{"id": "370.png", "formula": "\\begin{align*} F _ k ( r , s ) = \\cos { ( \\frac { 2 \\pi } { n - 1 } ( r - s + 1 ) k ) } . \\end{align*}"}
+{"id": "6968.png", "formula": "\\begin{align*} \\Phi _ { n , e } ^ { ( i ) } ( x , y ) = \\prod _ { \\substack { m | n \\\\ ( m , e ) = 1 \\\\ \\overline m m \\equiv i \\bmod e } } ( x ^ { n / m } - \\zeta ^ { \\overline m } _ { e } y ^ { n / m } ) ^ { \\mu ( m ) } . \\end{align*}"}
+{"id": "7994.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta u + V ( x ) u + \\gamma \\phi ( x ) u = b \\ , | u | ^ { p - 2 } u & \\mathbb { R } ^ 2 \\\\ \\Delta \\phi = u ^ 2 & \\mathbb { R } ^ 2 , \\end{array} \\right . \\end{align*}"}
+{"id": "898.png", "formula": "\\begin{align*} \\Sigma \\big ( X , q \\big ) = \\sum \\limits _ { 1 \\leq n \\leq q \\atop { n ^ 2 + n + 1 \\equiv 0 \\ , ( q ) } } \\Omega ( X , q , n ) \\ , . \\end{align*}"}
+{"id": "6238.png", "formula": "\\begin{align*} ( \\partial _ v \\Gamma _ { u v } ^ v - 2 \\Gamma _ { u v } ^ v \\Gamma _ { v u } ^ v ) \\tau = \\partial _ u \\Gamma _ { v u } ^ u - 2 \\Gamma _ { u v } ^ u \\Gamma _ { v u } ^ v . \\end{align*}"}
+{"id": "5083.png", "formula": "\\begin{align*} { \\mathcal { S } } _ 1 = \\left \\{ U \\in L ^ { 2 } _ { \\sigma } ( \\Omega ) \\ \\middle | \\ \\nabla \\times U = f _ 1 ^ { + } U , \\ \\ \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } U \\cdot U \\dd x = 1 \\right \\} . \\end{align*}"}
+{"id": "1270.png", "formula": "\\begin{align*} \\norm { \\varphi _ n ( r _ m \\cdot ) - \\varphi _ n ( r _ { m + 1 } \\cdot ) } _ \\infty \\leq \\sum _ { \\abs { z } = m } \\sum _ { i = 1 } ^ q \\norm { \\nabla ^ i _ z \\varphi _ n } _ \\infty . \\end{align*}"}
+{"id": "2329.png", "formula": "\\begin{align*} ( R i c ^ g ) ^ { J , - } & = - \\frac { 1 } { 2 } ( \\theta \\otimes \\theta ) ^ { J , - } . \\end{align*}"}
+{"id": "6338.png", "formula": "\\begin{align*} F ( x ) = \\operatorname { I } _ { \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) } ( a , b ) = \\frac { 1 } { \\operatorname { B } ( a , b ) } \\int _ 0 ^ { \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) } \\omega ^ { a - 1 } ( 1 - \\omega ) ^ { b - 1 } \\mathrm { d } \\omega , \\end{align*}"}
+{"id": "5367.png", "formula": "\\begin{align*} \\sum _ i S ^ { i i } _ k = ( n - k + 1 ) S _ { k - 1 } \\geq c _ 0 S _ k ^ { 1 - 1 / ( k - 1 ) } S _ 1 ^ { 1 / ( k - 1 ) } \\end{align*}"}
+{"id": "1630.png", "formula": "\\begin{align*} \\left < K _ i , K _ j \\right > = q ^ { - a _ { i j } / 2 } , \\left < E _ i , K _ j \\right > = \\left < K _ i , F _ j \\right > = 0 , \\left < E _ i , F _ j \\right > = \\frac { \\delta _ { i j } } { q _ j ^ { - 1 } - q _ j } \\end{align*}"}
+{"id": "2908.png", "formula": "\\begin{align*} \\tilde H _ y ^ { ( n ) } = \\frac { 1 } { 2 ( n + 1 ) } \\sum _ { j = - n - 1 } ^ n \\cos \\left ( \\frac { \\pi j ( 2 y + 1 ) } { n + 1 } \\right ) \\Xi \\left ( \\frac { j } { n + 1 } \\right ) . \\end{align*}"}
+{"id": "925.png", "formula": "\\begin{align*} u _ { \\mathrm { a p } , j } = M ( t ) D ( t ) F _ { j } ( t , \\cdot ) = t ^ { - \\frac 1 2 } F _ { j } ( t , \\tfrac { x } { t } ) e ^ { \\frac { i x ^ { 2 } } { 2 t } - i \\frac { \\pi } { 4 } } , j = 1 , 2 , \\end{align*}"}
+{"id": "7722.png", "formula": "\\begin{align*} x \\ , h _ s ( i ) ^ { \\tilde \\varepsilon x / 6 4 } \\le x \\exp \\bigg ( - \\frac { 1 2 8 \\ , \\log n } { \\tilde \\varepsilon ^ 2 ( 1 + \\tilde \\varepsilon ) ^ { - i } m _ s } \\frac { \\tilde \\varepsilon x } { 6 4 } \\bigg ) \\le x \\exp ( - 2 \\log n ) = n ^ { - 1 } , \\end{align*}"}
+{"id": "3177.png", "formula": "\\begin{align*} m = \\left \\{ \\begin{array} { l l l } x , & \\hbox { i f $ i \\in \\{ 1 , 2 \\} $ , } \\\\ 1 - x , & \\hbox { } \\end{array} \\right . \\end{align*}"}
+{"id": "7164.png", "formula": "\\begin{align*} ( a ^ \\alpha _ \\beta ) : = \\begin{pmatrix} a ^ 1 _ 1 & \\dots & a ^ 1 _ { n - 1 } \\\\ \\vdots & \\ddots & \\vdots \\\\ a ^ { n - 1 } _ 1 & \\dots & a ^ { n - 1 } _ { n - 1 } \\\\ \\end{pmatrix} , \\end{align*}"}
+{"id": "6712.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\begin{pmatrix} a _ i & b _ i \\\\ c _ i & d _ i \\end{pmatrix} , & a _ i d _ i - b _ i c _ i = 1 , & c _ i > 0 , \\\\ \\begin{pmatrix} a ' _ i & b ' _ i \\\\ c ' _ i & d ' _ i \\end{pmatrix} , & a ' _ i d ' _ i - b ' _ i c ' _ i = 1 , & c ' _ i > 0 , \\end{array} \\end{align*}"}
+{"id": "7817.png", "formula": "\\begin{align*} A _ \\phi = \\begin{pmatrix} \\cos ( 2 \\pi \\ell \\phi ) & \\sin ( 2 \\pi \\ell \\phi ) \\\\ - \\sin ( 2 \\pi \\ell \\phi ) & \\cos ( 2 \\pi \\ell \\phi ) \\end{pmatrix} \\end{align*}"}
+{"id": "4783.png", "formula": "\\begin{align*} a ( u , v ) = \\lambda ( u , v ) ~ v \\in H ^ 1 _ 0 ( D ) , \\end{align*}"}
+{"id": "3983.png", "formula": "\\begin{align*} \\bar { \\mathbf { a } } + \\bar { \\mathbf { b } } = ( \\bar { \\alpha } _ 1 + \\bar { \\beta } _ 1 , a _ 1 + \\bar { \\beta } _ 2 , a _ 2 + b _ 1 , a _ 3 + b _ 2 , \\cdots , a _ { 2 k - 2 } + b _ { 2 k - 3 } , a _ { 2 k - 1 } + b _ { 2 k - 2 } , \\bar { \\alpha } _ 2 + b _ { 2 k - 1 } , \\bar { \\alpha } _ 3 + \\bar { \\beta } _ 3 , \\bar { \\alpha } _ 4 + \\bar { \\beta } _ 4 ) . \\end{align*}"}
+{"id": "65.png", "formula": "\\begin{align*} v = v _ 1 , \\dotsc , v _ k \\in W \\end{align*}"}
+{"id": "6645.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\sup _ { 0 \\leq t - s \\leq \\delta , t , s \\in [ 0 , 1 ] } \\big | X ^ { \\epsilon } ( t ) - X ^ { \\epsilon } ( s ) \\big | > \\delta _ 0 \\Big ) = - \\infty . \\end{align*}"}
+{"id": "906.png", "formula": "\\begin{align*} & \\mathcal { N } ( d ) = \\{ n \\in \\mathbb { N } \\ ; : \\ ; 1 \\leq n \\leq d , \\ ; \\ ; n ^ 2 + n + 1 \\equiv 0 \\ , ( d ) \\} \\ , , \\\\ & \\mathcal { N } ' ( d ) = \\{ n \\in \\mathbb { N } \\ ; : \\ ; 1 \\leq n \\leq d ^ 2 , \\ ; \\ ; n ^ 2 + n + 1 \\equiv 0 \\ , ( d ^ 2 ) \\} \\ , . \\end{align*}"}
+{"id": "8734.png", "formula": "\\begin{align*} x _ i ( g ) = x ( f _ i ( g ) ) \\forall g \\in G . \\end{align*}"}
+{"id": "4257.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\inf _ { k \\in { \\mathbb N } } \\left ( \\frac { \\mu ( f ^ { k + n } ( W ) ) } { \\mu ( f ^ { k } ( W ) ) } \\right ) = \\infty \\ \\ \\& \\ \\ \\lim _ { n \\rightarrow \\infty } \\inf _ { k \\in - { \\mathbb N } _ 0 } \\left ( \\frac { \\mu ( f ^ { k - n } ( W ) ) } { \\mu ( f ^ { k } ( W ) ) } \\right ) = \\infty \\tag * { $ \\mathcal { U E } 3 $ } \\end{align*}"}
+{"id": "4281.png", "formula": "\\begin{align*} a = \\frac { \\left ( \\beta - 3 \\right ) n + 2 \\beta } { 3 } \\end{align*}"}
+{"id": "479.png", "formula": "\\begin{align*} e ^ { \\gamma x } f _ r \\ast f _ G ( x ) = \\int _ { - \\infty } ^ { x / 2 } \\big \\{ e ^ { \\gamma ( x - y ) } f _ r ( x - y ) e ^ { \\gamma y } f _ G ( y ) + e ^ { \\gamma ( x - y ) } f _ G ( x - y ) e ^ { \\gamma y } f _ r ( y ) \\big \\} d y < \\infty \\end{align*}"}
+{"id": "6770.png", "formula": "\\begin{align*} A ( x , x ) = \\frac { 1 } { 2 } \\int _ { x } ^ { \\infty } q ( t ) d t , \\end{align*}"}
+{"id": "3207.png", "formula": "\\begin{align*} d _ A ( f , g ) = \\begin{cases} \\left | e ^ { 2 \\pi i d ' _ A ( f , g ) } - 1 \\right | & d ' _ A ( f , g ) < \\frac { 1 } { 2 } \\\\ 2 & d ' _ A ( f , g ) \\ge \\frac { 1 } { 2 } . \\end{cases} \\end{align*}"}
+{"id": "4525.png", "formula": "\\begin{align*} v _ 1 ( x , t ) = S _ { d , 1 } ( t , t _ { 1 , e x } ( x , t ) ) S _ { c , 1 } ( t _ { 1 , e x } ( x , t ) , t _ { 1 , e n } ( x , t ) ) S _ { d , 1 } ( t _ { 1 , e n } ( x , t ) , 0 ) v _ { 1 , 0 } ( x ) . \\end{align*}"}
+{"id": "3191.png", "formula": "\\begin{align*} L ( x ) = 2 \\sinh { \\left ( \\tfrac { x } { 2 } \\right ) } . \\end{align*}"}
+{"id": "516.png", "formula": "\\begin{align*} S _ { R } ^ { \\alpha } = C _ { \\beta , \\delta } \\int _ { 0 } ^ { 1 } ( 1 - t ) ^ { \\beta - 1 } t ^ \\delta S _ { R t } ^ { \\delta } d t , \\end{align*}"}
+{"id": "4449.png", "formula": "\\begin{align*} \\Theta _ { Q } ( z ) = E ( z ) + C ( z ) = \\sum _ { n = 0 } ^ { \\infty } a _ { E } ( n ) q ^ { n } + \\sum _ { n = 1 } ^ { \\infty } a _ { C } ( n ) q ^ { n } . \\end{align*}"}
+{"id": "3577.png", "formula": "\\begin{align*} \\phi _ A ( t _ 1 , \\hdots , t _ n ) = \\left ( \\prod _ { i = 1 } ^ n t _ i ^ { a _ { i 1 } } , \\prod _ { i = 1 } ^ n t _ i ^ { a _ { i 2 } } , \\hdots , \\prod _ { i = 1 } ^ n t _ i ^ { a _ { i m } } \\right ) . \\end{align*}"}
+{"id": "6017.png", "formula": "\\begin{align*} g ( \\overline { t } ) = \\mathop { \\max } \\limits _ { t > 0 } g ( t ) . \\end{align*}"}
+{"id": "873.png", "formula": "\\begin{align*} x x ' + x y ' + y y ' = 2 n \\ , . \\end{align*}"}
+{"id": "8099.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u & = f ( u ) & & \\quad \\mbox { i n } \\Omega , \\\\ u & = \\infty & & \\quad \\mbox { o n } \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4090.png", "formula": "\\begin{align*} d ^ \\nu { } _ { j ^ 1 _ p ( s ) } X & = \\pi ^ 1 { } _ * X - s _ * \\overline { \\pi } ^ 1 _ * X , \\\\ * \\theta _ { j ^ 1 _ p ( s ) } ( X ) & = s ( p ) ^ { - 1 } ( d ^ \\nu { } _ { j ^ 1 _ p ( s ) } X ) . \\end{align*}"}
+{"id": "2774.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { a _ { N } } { \\log N } = 2 \\lambda \\sqrt { 2 \\gamma } , \\end{align*}"}
+{"id": "5687.png", "formula": "\\begin{align*} \\gamma _ t = \\gamma _ 0 \\delta ^ { n _ t } , \\ \\beta _ t = \\frac { \\gamma _ { t - 1 } ( 1 - \\kappa _ t ) } { \\gamma _ t ( 1 - \\kappa _ { t - 1 } ) } \\left ( 1 + \\frac { 2 \\mu \\gamma _ { t - 1 } } { 1 - \\kappa _ { t - 1 } } \\right ) ^ { - 1 } , \\ \\alpha _ t = \\frac { \\kappa _ { t - 1 } \\gamma _ { t } \\beta _ t } { \\gamma _ { t - 1 } } , \\end{align*}"}
+{"id": "8871.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } & \\prod _ { k = 1 } ^ { j - 1 } f _ k ( T _ { 1 } ^ { p _ { 1 , k } ( n ) } \\cdots T _ { d } ^ { p _ { d , k } ( n ) } x ) ( f _ j - \\mathbb { E } ( f _ j | \\mathcal { A } ) ) ( T _ { 1 } ^ { p _ { 1 , j } ( n ) } \\cdots T _ { d } ^ { p _ { d , j } ( n ) } x ) \\cdot \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\prod _ { l = j + 1 } ^ { m } \\mathbb { E } ( f _ { l } | \\mathcal { A } ) ( T _ { 1 } ^ { p _ { 1 , l } ( n ) } \\cdots T _ { d } ^ { p _ { d , l } ( n ) } x ) \\rightarrow 0 \\end{align*}"}
+{"id": "2051.png", "formula": "\\begin{align*} \\forall \\ \\sigma \\geq 0 , \\widehat { M ^ \\sigma g } ( m , \\eta ) = \\rho _ \\sigma \\hat g ( m , \\eta ) \\end{align*}"}
+{"id": "980.png", "formula": "\\begin{align*} | F \\cap M | \\le \\begin{cases} \\Delta & U \\le H , \\\\ \\Delta + | H \\cap W | q ^ { d ^ 2 - d } \\theta _ { d - 1 } & . \\end{cases} \\end{align*}"}
+{"id": "5129.png", "formula": "\\begin{align*} - L \\phi = \\mu ^ { 2 } ( \\phi - \\phi _ \\infty ) _ { + } \\textrm { i n } \\ \\mathbb { R } ^ { 2 } _ { + } , \\phi = 0 \\textrm { o n } \\ \\partial \\mathbb { R } ^ { 2 } _ { + } , \\end{align*}"}
+{"id": "7010.png", "formula": "\\begin{align*} F \\ , = \\ , \\sum _ { i + j \\geqslant 1 } \\ , F _ { j , k } \\ , \\frac { x ^ i } { i ! } \\ , \\frac { y ^ j } { j ! } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ G \\ , = \\ , \\sum _ { k + l \\geqslant 1 } \\ , G _ { k , l } \\ , \\frac { r ^ k } { k ! } \\ , \\frac { s ^ l } { l ! } . \\end{align*}"}
+{"id": "7786.png", "formula": "\\begin{align*} \\frac { 1 } { | x _ d | } \\le \\frac { 1 } { d } \\Big | \\frac { p ' ( c ) } { p ( c ) } \\Big | = \\frac { 1 } { d } \\Big | \\sum _ { j = 1 } ^ d \\frac { 1 } { c - x _ j } \\Big | \\end{align*}"}
+{"id": "8809.png", "formula": "\\begin{align*} \\Pi _ V J _ { z / | z | } ^ \\perp x = \\lambda J _ { z / | z | } ^ \\perp \\Pi _ V x \\forall x \\in \\C ^ n \\end{align*}"}
+{"id": "8460.png", "formula": "\\begin{align*} \\partial _ x T - \\mathcal { P } ^ + \\left ( \\partial _ x T \\right ) R _ + - \\mathcal { P } ^ - \\left ( \\partial _ x T \\right ) R _ - = \\widetilde { F } , \\end{align*}"}
+{"id": "3146.png", "formula": "\\begin{align*} [ N ( x ) , N ( y ) ] & = N ( x ) \\cdot N ( y ) - N ( y ) \\cdot N ( x ) \\\\ & = N \\big ( N ( x ) \\cdot y + x \\cdot N ( y ) - N ( x \\cdot y ) - N ( y ) \\cdot x - y \\cdot N ( x ) + N ( y \\cdot x ) \\big ) \\\\ & = N \\big ( [ N ( x ) , y ] + [ x , N ( y ) ] - N [ x , y ] \\big ) . \\end{align*}"}
+{"id": "1328.png", "formula": "\\begin{align*} \\omega ( v _ \\lambda , - ) = \\lambda , \\end{align*}"}
+{"id": "3057.png", "formula": "\\begin{align*} \\rho _ E ( \\lambda , z ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\gamma } \\omega _ { \\lambda , z } \\end{align*}"}
+{"id": "3769.png", "formula": "\\begin{align*} C ' _ w C ' _ s = & C ' _ { w s } + \\sum _ { \\substack { z \\leq w \\\\ z s < z } } \\mu ( z , w ) C ' _ z , \\\\ C ' _ s C ' _ w = & ( q ^ { - \\frac { 1 } { 2 } } + q ^ { \\frac { 1 } { 2 } } ) C ' _ w , \\end{align*}"}
+{"id": "6243.png", "formula": "\\begin{align*} Q _ { i j } ( h ) = \\partial ^ 2 _ { i j } h - \\Gamma _ { j i } ^ i \\partial _ i h - \\Gamma _ { i j } ^ j \\partial _ j h + g _ { i j } h = 0 , \\forall i < j , \\end{align*}"}
+{"id": "3336.png", "formula": "\\begin{align*} \\chi \\cap \\psi = \\psi \\setminus \\{ x \\} . \\end{align*}"}
+{"id": "7168.png", "formula": "\\begin{align*} ( \\nu S \\textbf { \\textit { u } } ) ^ i = - ( \\nabla ^ i u _ n + \\nabla _ n u ^ i ) . \\end{align*}"}
+{"id": "1936.png", "formula": "\\begin{align*} f ( m ; z ) : = \\frac { \\eta ( 5 z ) } { \\eta ( z ) } \\eta ^ a ( 5 m z ) \\eta ^ b ( m z ) , \\end{align*}"}
+{"id": "6211.png", "formula": "\\begin{align*} W _ + ( r ) = W ' ( r ) + W ( r ) , W _ - ( r ) = W ' ( r ) - W ( r ) , \\end{align*}"}
+{"id": "1426.png", "formula": "\\begin{align*} ( - 1 ) ^ n { k - 1 / 2 \\choose n } = \\frac { ( - 1 ) ^ k \\Gamma ( k + 1 / 2 ) \\Gamma ( n + 1 / 2 - k ) } { \\pi \\Gamma ( n + 1 ) } . \\end{align*}"}
+{"id": "1830.png", "formula": "\\begin{align*} R ^ { \\nabla ^ t } = R ^ \\nabla + t ( d ^ \\nabla A ) + t ^ 2 [ A , A ] \\ , , \\end{align*}"}
+{"id": "4821.png", "formula": "\\begin{align*} s _ { \\min } ( A ) = s _ n ( A ) = \\min \\limits _ { x : \\ , \\lVert x \\rVert _ 2 = 1 } \\lVert A x \\rVert _ 2 , s _ { \\max } ( A ) = s _ 1 ( A ) = \\max \\limits _ { x : \\ , \\lVert x \\rVert _ 2 = 1 } \\lVert A x \\rVert _ 2 . \\end{align*}"}
+{"id": "5000.png", "formula": "\\begin{align*} u _ n & = \\sum _ { j = 0 } ^ \\infty \\Xi _ j ( \\vec u _ { 0 , n } ) = \\sum _ { j = 0 } ^ \\infty \\Xi _ j ( \\vec u _ { 0 } + \\vec \\phi _ n ) , \\end{align*}"}
+{"id": "5705.png", "formula": "\\begin{align*} f = ( a _ 0 , b _ 0 , c _ 0 , d _ 0 ) _ q , \\end{align*}"}
+{"id": "7745.png", "formula": "\\begin{align*} F ( x ) = \\max _ { i = 1 , \\ldots , 2 ^ k } x _ i . \\end{align*}"}
+{"id": "7597.png", "formula": "\\begin{align*} F _ 1 x & = ( x _ { 0 0 0 } , x _ { 0 0 1 } , x _ { 0 1 0 } , x _ { 0 1 1 } ) \\\\ F _ 2 x & = ( x _ { 0 0 0 } , x _ { 0 0 1 } , x _ { 1 0 0 } , x _ { 1 0 1 } ) \\\\ F _ 3 x & = ( x _ { 0 0 0 } , x _ { 0 1 0 } , x _ { 1 0 0 } , x _ { 1 1 0 } ) \\end{align*}"}
+{"id": "15.png", "formula": "\\begin{align*} \\overline { \\mathbf { p } } ^ { \\oplus } = \\{ \\alpha _ { i _ k , j _ \\ell } \\mid 1 \\leq k , \\ell \\leq n \\} . \\end{align*}"}
+{"id": "3424.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\int _ { 0 } ^ { 1 } \\ln | \\tilde { P } _ k ( \\theta ) | \\mathrm { d } \\theta = \\frac { 1 } { k } \\int _ { 0 } ^ { 1 } \\ln | \\tilde { P } _ k ( 2 \\theta ) | \\mathrm { d } \\theta \\geq L - \\ln 2 = \\tilde { L } \\end{align*}"}
+{"id": "3783.png", "formula": "\\begin{align*} K _ q ( z , w ) = E _ q ( z \\overline { w } ) : = \\displaystyle \\sum _ { n = 0 } ^ \\infty \\frac { ( z \\overline { w } ) ^ n } { \\Gamma ( q n + 1 ) } , \\end{align*}"}
+{"id": "452.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\frac { g _ + ( b ^ n ( x _ 0 + v ) ) } { n g _ + ( b ^ n x _ 0 ) } = \\liminf _ { n \\to \\infty } \\frac { \\nu \\big ( ( b ^ n ( x _ 0 + v ) , b ^ n ( x _ 0 + v ) + 1 ] \\big ) } { n \\nu \\big ( ( b ^ n x _ 0 , b ^ n x _ 0 + 1 ] \\big ) } \\ge c _ 0 , \\end{align*}"}
+{"id": "5492.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\left ( 1 - \\frac { u ^ 2 } { 6 s } + \\frac { u ^ 4 } { 7 2 s ^ 2 } \\right ) e ^ { - u } u ^ { p / 2 - 1 } \\dd u = \\Gamma \\left ( p / 2 \\right ) - \\frac { \\Gamma ( p / 2 + 2 ) } { 6 s } + \\frac { \\Gamma ( p / 2 + 4 ) } { 7 2 s ^ 2 } \\end{align*}"}
+{"id": "1932.png", "formula": "\\begin{align*} f ( m ; z ) : = \\frac { \\eta ( 3 z ) } { \\eta ( z ) } \\eta ^ a ( 3 m z ) \\eta ^ b ( m z ) , \\end{align*}"}
+{"id": "1594.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { \\Phi } _ { \\mathrm { r } } & = \\mathsf { d i a g } ( \\phi _ { \\mathrm { r } , 1 } , \\ldots , \\phi _ { \\mathrm { r } , M } ) , ~ ~ \\mathbf { \\Phi } _ { \\mathrm { t } } & = \\mathsf { d i a g } ( \\phi _ { \\mathrm { t } , 1 } , \\ldots , \\phi _ { \\mathrm { t } , M } ) , \\end{aligned} \\end{align*}"}
+{"id": "3157.png", "formula": "\\begin{align*} \\sum _ x \\overline { \\chi _ 4 } ( x ) \\overline { \\chi _ 4 } ( x - y ) & = \\sum _ { x ' } \\overline { \\chi _ 4 } ( x ' y ) \\overline { \\chi _ 4 } ( x ' y - y ) \\\\ & = \\varphi ( y ) \\rho . \\end{align*}"}
+{"id": "716.png", "formula": "\\begin{align*} \\int | \\nabla u | ^ 2 d x - \\lambda \\int | u | ^ 2 d x - \\int ( I _ \\alpha \\ast | u | ^ { p } ) | u | ^ p d x = 0 . \\end{align*}"}
+{"id": "8826.png", "formula": "\\begin{align*} \\begin{aligned} d u _ m & = i 2 ^ m \\left ( \\overline { u _ { m + 1 } } u _ { m + 2 } - \\frac { \\delta } { 2 } \\overline { u _ { m - 1 } } u _ { m + 1 } - \\frac { \\delta - 1 } { 4 } u _ { m - 2 } u _ { m - 1 } \\right ) \\\\ & - \\delta 2 ^ { 2 m } u _ m + q _ m d W _ t ^ { ( m ; R ) } + i p _ m d W _ t ^ { ( m ; I ) } , \\end{aligned} \\end{align*}"}
+{"id": "1548.png", "formula": "\\begin{align*} P _ C ( u , v ) = ( u _ { 1 / 2 } , v _ { 1 / 2 } ) . \\end{align*}"}
+{"id": "6333.png", "formula": "\\begin{align*} ( z - z ^ q ) + u ^ { p ^ { h - r } } ( y - y ^ q ) + u ^ { 2 p ^ { h - r } } ( x - x ^ q ) = 0 . \\end{align*}"}
+{"id": "8359.png", "formula": "\\begin{align*} \\mathcal { P } ^ { \\pm } ( h ) ( z ) : = \\lim _ { \\varepsilon \\downarrow 0 } \\frac { 1 } { 2 \\pi i } \\int _ { \\mathbb { R } } \\frac { h ( s ) } { s - ( z \\pm i \\varepsilon ) } d s z \\in \\mathbb { R } . \\end{align*}"}
+{"id": "567.png", "formula": "\\begin{align*} 2 \\pi i \\cdot f ( z ) = \\int _ { e ^ { - i \\alpha } [ 0 , + \\infty ) } \\frac { f ( \\zeta ) e ^ { p ( \\zeta - z ) } } { \\zeta - z } d \\zeta - \\int _ { e ^ { i \\alpha } [ 0 , + \\infty ) } \\frac { f ( \\zeta ) e ^ { p ( \\zeta - z ) } } { \\zeta - z } d \\zeta , z \\in \\Delta . \\end{align*}"}
+{"id": "4872.png", "formula": "\\begin{align*} \\Phi : = \\sup \\{ \\zeta \\in U S C ( \\overline { \\Omega } \\times \\mathbb R ^ m ) : \\zeta | _ { \\Omega \\times \\mathbb R ^ m } F \\star \\mathcal P \\zeta | _ { \\partial \\Omega \\times \\mathbb R ^ m } \\le \\phi \\} \\end{align*}"}
+{"id": "4156.png", "formula": "\\begin{align*} | H ( x ' , t ) | & = \\bigg | \\int _ { Q } ( \\theta ( x ' , t ; y ' ) - \\theta ( x ' , t ; x ' _ Q ) ) h ( y ' ) \\ , d y ' \\bigg | \\\\ & = \\sup _ { y ' \\in Q } | \\nabla _ { y ' } \\theta ( x ' , t ; y ' ) | \\ , \\| h \\| _ { L ^ 1 ( Q ) } \\\\ & \\lesssim \\sup _ { y ' \\in Q } \\frac { t } { | ( x ' - y ' , t ) | ^ { n + 1 } } \\ , \\| h \\| _ { L ^ 1 ( Q ) } \\\\ & \\lesssim \\frac { t \\ , \\ell ( Q ) } { ( 2 ^ k \\ell ( Q ) ) ^ { n + 1 } } \\| h \\| _ { L ^ 1 ( Q ) } , \\end{align*}"}
+{"id": "6239.png", "formula": "\\begin{align*} \\partial _ u \\Gamma _ { v u } ^ u = 2 \\Gamma _ { u v } ^ u \\Gamma _ { v u } ^ v = \\partial _ v \\Gamma _ { u v } ^ v . \\end{align*}"}
+{"id": "2491.png", "formula": "\\begin{align*} \\mathbf { Q } ^ i = \\begin{pmatrix} 1 & 0 \\\\ 0 & - \\mathbf { I } _ { n _ i - 1 } \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "6580.png", "formula": "\\begin{align*} \\begin{aligned} \\| ( \\gamma - c ) \\varphi \\| _ { W ^ { \\frac { 1 } { 3 } , \\frac { 3 } { 2 } } ( \\partial \\Omega ) } + & \\| c \\nabla \\varphi \\cdot \\nu \\| _ { W ^ { \\frac { 1 } { 3 } , \\frac { 3 } { 2 } } ( \\partial \\Omega ) } \\\\ & \\le C ( \\gamma \\| \\varphi \\| _ { W ^ { 2 , \\frac { 3 } { 2 } } ( \\Omega ) } + \\| \\nabla c \\| _ { L ^ { 2 } ( \\Omega ) } \\| \\varphi \\| _ { W ^ { 1 , 6 } ( \\Omega ) } ) . \\end{aligned} \\end{align*}"}
+{"id": "2528.png", "formula": "\\begin{align*} \\mathbf { x } { { \\left ( \\alpha \\right ) } ^ { T } } \\mathbf { s } \\left ( \\alpha \\right ) = \\left ( 1 - \\left ( 1 - \\nu \\right ) \\alpha \\right ) { \\mathbf { x } } ^ { T } \\mathbf { s } . \\end{align*}"}
+{"id": "9229.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ( t ) \\| _ { \\mathcal { A } } & \\le \\| x ( t ) - x _ a ( t ) \\| + \\| x _ a ( t ) \\| _ \\mathcal { A } \\\\ & \\le \\tilde { d } + \\beta ( \\| x _ 0 \\| _ { \\mathcal { A } } , \\gamma t ) & & \\forall \\ , t \\in [ 0 , \\bar { t } / \\gamma ] \\end{aligned} \\end{align*}"}
+{"id": "8385.png", "formula": "\\begin{align*} F f ( x ; z ) = ( h _ { 1 } , h _ { 2 } ) ^ T , \\end{align*}"}
+{"id": "7956.png", "formula": "\\begin{align*} H _ { N } ( t ) : = H ( f _ { N , t } ; \\nu _ { N , t } ) , \\forall \\ , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "4885.png", "formula": "\\begin{align*} \\mathcal { M } _ { ( i , j ) } [ m , n ] : = \\mathcal { M } _ { ( i + m , j + n ) } \\ , , \\end{align*}"}
+{"id": "4957.png", "formula": "\\begin{align*} \\begin{aligned} \\theta ^ { [ n ] } _ 1 & = \\pi ^ { [ n ] } _ 1 ( \\xi _ 0 , x ) , \\\\ \\theta ^ { [ n ] } _ i & = \\pi ^ { [ n ] } _ i ( \\xi _ 0 , x , X _ 1 , Y _ 1 , \\dots , X _ { i - 1 } , Y _ { i - 1 } ) , ~ i \\geq 2 . \\end{aligned} \\end{align*}"}
+{"id": "4734.png", "formula": "\\begin{align*} \\Tilde { L } ( x ) : = \\gamma _ 0 f _ 0 ( x ) + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k f _ k ( x ) \\ge 0 \\ \\ x \\in \\mathbb { R } ^ n . \\end{align*}"}
+{"id": "901.png", "formula": "\\begin{align*} \\Gamma _ 1 ( X ) & = X \\sum \\limits _ { 1 \\leq d \\leq z } \\frac { \\mu ( d ) \\lambda ( d ^ 2 ) } { d ^ 2 } + \\mathcal { O } \\big ( z X ^ \\varepsilon \\big ) \\\\ & = \\sigma X - X \\sum \\limits _ { d > z } \\frac { \\mu ( d ) \\lambda ( d ^ 2 ) } { d ^ 2 } + \\mathcal { O } \\big ( z X ^ \\varepsilon \\big ) \\ , , \\end{align*}"}
+{"id": "4998.png", "formula": "\\begin{align*} [ - T , T ] = \\bigcup _ { j = - [ T / \\tau ] - 1 } ^ { [ T / \\tau ] } I _ j , I _ j = [ j \\tau , ( j + 1 ) \\tau ] \\cap [ - T , T ] , \\end{align*}"}
+{"id": "8024.png", "formula": "\\begin{align*} a _ { i , d } ( \\underline { b } ) + a ' _ { i , d } ( \\underline { b } ) = x _ { i , d } i , d , \\end{align*}"}
+{"id": "9221.png", "formula": "\\begin{align*} \\epsilon ( x , t ) : = \\int _ 0 ^ t f ( x , \\tau ) - f _ a ( x ) \\ , d \\tau , \\end{align*}"}
+{"id": "6561.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla v _ { - } | ^ { 2 } + \\int _ { \\Omega } u | v _ { - } | ^ { 2 } + \\int _ { \\partial \\Omega } | v _ { - } | ^ { 2 } = - \\eta \\int _ { \\partial \\Omega } v _ { - } . \\end{align*}"}
+{"id": "6418.png", "formula": "\\begin{align*} g ^ { q p } u _ { , p q i } = - v _ i + u _ { , i q } \\xi ^ q \\end{align*}"}
+{"id": "3599.png", "formula": "\\begin{align*} d ^ D _ t = \\sum _ { u , v : t ( u , v ) = t } \\sum _ { s = 0 } ^ { \\omega _ { u v } - 1 } d _ { ( u , v , s ) } . \\end{align*}"}
+{"id": "8951.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty g ( x ) f ( x ) \\ , d x = \\sum _ { | k | \\le N } \\biggl \\{ \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { - p ' } ( t ) \\bigl | G _ { 1 , k } ^ { ( \\delta ) } ( t ) \\bigr | ^ { p ' } \\ , d t + \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { - p ' } ( t ) \\bigl | G _ { 2 , k } ^ { ( \\delta ) } ( t ) \\bigr | ^ { p ' } \\ , d t \\biggr \\} . \\end{align*}"}
+{"id": "7629.png", "formula": "\\begin{align*} \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } } : = \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } \\alpha ^ j , \\nu ^ { N , i } _ { \\boldsymbol { x } } : = \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } x ^ j . \\end{align*}"}
+{"id": "5491.png", "formula": "\\begin{align*} U ( p , s ) & = \\frac { 4 ^ p ( 2 \\pi \\cdot 1 5 ^ { 1 / 2 } ) ^ { - s / 2 } } { 3 s / 2 - p } \\\\ & \\qquad + 2 ^ { 3 p / 2 - 1 } s ^ { - p / 2 } \\left ( \\Gamma \\left ( p / 2 \\right ) - \\frac { \\Gamma ( p / 2 + 2 ) } { 6 s } + \\frac { \\Gamma ( p / 2 + 4 ) } { 7 2 s ^ 2 } \\right ) \\end{align*}"}
+{"id": "4747.png", "formula": "\\begin{align*} \\alpha _ * ^ 2 \\| u _ * \\| ^ 2 + 2 \\alpha _ * \\langle u _ * , \\lambda x _ v + ( 1 - \\lambda ) x _ w \\rangle - \\lambda ( 1 - \\lambda ) \\| x _ v - x _ w \\| ^ 2 = 0 \\end{align*}"}
+{"id": "9076.png", "formula": "\\begin{align*} \\tilde { \\sigma } _ i ( u _ i , u _ { i + 1 } ) = { \\rho _ { p _ i } ' } ^ { - 1 } ( \\sigma _ i \\circ \\rho _ i ( u _ i ) - \\rho _ { i + 1 } ' ( u _ { i + 1 } ) ) \\end{align*}"}
+{"id": "1805.png", "formula": "\\begin{align*} b ( q ^ { \\frac { 1 } { 3 } } ) = a ( q ) - c ( q ) , \\end{align*}"}
+{"id": "7986.png", "formula": "\\begin{align*} \\frac { h _ { j t } + h _ { j } } { h _ { t } + h } & = w ^ { i k } \\nabla _ { j } w _ { i k } + \\nabla _ { j } \\chi \\\\ & = w ^ { i i } ( h _ { j i i } + h _ { i } \\delta _ { i j } ) + \\nabla _ { j } \\chi , \\end{align*}"}
+{"id": "2300.png", "formula": "\\begin{align*} U \\cdot ( w ^ * \\otimes w ) = ( U \\cdot w ^ * ) \\otimes w , \\end{align*}"}
+{"id": "8464.png", "formula": "\\begin{align*} Q _ { j , \\pm } ( x ; k ) : = M _ { \\pm } ( x ; z ) V _ j ( k ) - V _ j ( k ) , D _ j = V _ j ( k ) J , j = 1 , 2 . \\end{align*}"}
+{"id": "2813.png", "formula": "\\begin{align*} \\mathbf { U } _ { k l } \\mathbf { h } _ { k l } = \\left [ \\mathbf { a } ( \\overline { \\theta } _ { 1 } ) , \\mathbf { a } ( \\overline { \\theta } _ { 2 } ) , . . . , \\mathbf { a } ( \\overline { \\theta } _ { N } ) \\right ] ^ { \\mathbf { H } } \\mathbf { h } _ { k l } , \\end{align*}"}
+{"id": "8053.png", "formula": "\\begin{align*} a _ c \\left [ t + 1 \\right ] & = a _ c \\left [ t \\right ] - \\mu \\frac { \\partial \\mathbb { E } \\left [ \\varepsilon \\right ] } { \\partial a _ c } \\\\ a _ i \\left [ t + 1 \\right ] & = a _ i \\left [ t \\right ] - \\mu \\frac { \\partial \\mathbb { E } \\left [ \\varepsilon \\right ] } { \\partial a _ i } . \\end{align*}"}
+{"id": "3049.png", "formula": "\\begin{align*} \\ell _ { v } = Z _ { 1 } \\left ( v , v \\right ) \\rho _ { v } = Z _ { 1 } \\left ( v , - v \\right ) . \\end{align*}"}
+{"id": "6095.png", "formula": "\\begin{align*} \\mu _ { n + 2 } ^ { ( 1 2 ) } - 2 \\mu _ { n + 1 } ^ { ( 1 2 ) } + \\mu _ n ^ { ( 1 2 ) } = 2 \\ , , \\end{align*}"}
+{"id": "1583.png", "formula": "\\begin{align*} X _ 1 : = ( \\ln \\vert \\alpha \\vert , \\ln \\vert \\sigma ( \\alpha ) \\vert , \\ln \\vert \\sigma ^ 2 ( \\alpha ) \\vert ) , & & X _ 2 : = ( \\ln \\vert \\sigma ( \\alpha ) \\vert , \\ln \\vert \\sigma ^ 2 ( \\alpha ) \\vert , \\ln \\vert \\alpha \\vert ) \\end{align*}"}
+{"id": "8569.png", "formula": "\\begin{align*} h _ { \\alpha , \\rho } ( t ) = \\frac { t ^ { \\alpha - 1 } } { \\Gamma ( \\alpha ) } \\ , e ^ { - \\rho t } , \\ \\ \\alpha > 0 , \\ \\rho \\in \\R , \\ t > 0 . \\end{align*}"}
+{"id": "2265.png", "formula": "\\begin{align*} a ( U _ 1 ^ { - 1 } Z V _ 1 ) = a ( U _ 1 ^ { - 1 } U D ( x ) V V _ 1 ) = a ( D ( x ) ) = a ( U D ( x ) V ) = a ( Z ) , \\end{align*}"}
+{"id": "6210.png", "formula": "\\begin{align*} W ^ 2 ( r ) + f ( r ) \\frac { d W ( r ) } { d r } = W ^ { \\prime 2 } ( r ) - f ( r ) \\frac { d W ' ( r ) } { d r } + E _ 1 - E _ 0 , \\end{align*}"}
+{"id": "6097.png", "formula": "\\begin{align*} \\xi _ { 1 2 } & = ( \\mu ^ { ( 3 ) } - \\mu ^ { ( 2 ) } ) ( \\mu ^ { ( 1 ) } - \\mu _ p ^ { ( 1 2 3 ) } ) \\ , , \\\\ \\xi _ { 2 3 } & = ( \\mu ^ { ( 1 ) } - \\mu ^ { ( 2 ) } ) ( \\mu ^ { ( 3 ) } - \\mu _ p ^ { ( 1 2 3 ) } ) \\ , , \\\\ \\xi _ 0 & = ( \\mu ^ { ( 1 ) } \\mu ^ { ( 3 ) } - \\mu ^ { ( 2 ) } \\mu _ p ^ { ( 1 2 3 ) } ) ( \\mu _ p ^ { ( 1 2 3 ) } - \\mu ^ { ( 1 ) } + \\mu ^ { ( 2 ) } - \\mu ^ { ( 3 ) } ) + ( \\mu ^ { ( 2 ) } + \\mu ^ { ( 3 ) } ) ( \\mu ^ { ( 1 ) } + \\mu ^ { ( 1 2 3 ) } ) \\ , . \\end{align*}"}
+{"id": "5915.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ T \\Vert X ( t , x _ n ) - X ( t , x ) \\Vert _ H ^ 2 d t = 0 . \\end{align*}"}
+{"id": "6120.png", "formula": "\\begin{align*} & C _ { 1 3 } = - C _ { 2 3 } + C _ { 1 2 3 } - C _ { 1 2 } + C _ 1 + C _ 2 + C _ 3 \\\\ & C _ { 2 4 } = C _ { 2 3 4 } - C _ { 2 3 } - C _ { 3 4 } + C _ 2 + C _ 3 + C _ 4 \\ , , \\\\ & C _ { 1 4 } = C _ { 1 2 3 4 } - C _ { 1 2 3 } - C _ { 2 3 4 } + C _ { 2 3 } + C _ 1 + C _ 4 \\ , , \\\\ & C _ { 1 2 4 } = C _ { 1 2 3 4 } - C _ { 1 2 3 } - C _ { 3 4 } + C _ { 1 2 } + C _ { 3 } + C _ { 4 } \\ , , \\\\ & C _ { 1 3 4 } = C _ { 1 2 3 4 } - C _ { 2 3 4 } - C _ { 1 2 } + C _ { 3 4 } + C _ 1 + C _ 2 \\ , . \\end{align*}"}
+{"id": "2419.png", "formula": "\\begin{align*} L _ { q } ^ { - 1 } [ F _ { q } ( s ) ] ( t ) = f ( t ) = \\frac { 2 - q } { 2 \\pi i } \\int _ { c - i \\infty } ^ { c + i \\infty } F _ { q } ( s ) [ \\exp _ { q } ( - s t ) ] ^ { 2 q - 3 } d s , \\end{align*}"}
+{"id": "4761.png", "formula": "\\begin{align*} & ( 1 . ) \\ w : = 1 + 2 \\gamma _ 1 a _ 1 + 2 \\gamma _ 2 a _ 2 \\ge 0 , \\ \\ \\\\ & ( 2 . ) \\ f _ { i } ( x ^ { * } ) = a _ { i } \\| x ^ { * } \\| ^ { 2 } + b _ { i } ^ { T } x ^ { * } + c _ { i } = 0 \\ \\ i + 1 , 2 \\ \\\\ & ( 3 . ) \\ \\gamma _ 1 b _ { 1 } + \\gamma _ 2 b _ 2 = - x ^ { * } w \\ \\end{align*}"}
+{"id": "1425.png", "formula": "\\begin{align*} a _ 0 ^ { ( k ) } = \\frac { ( - 1 ) ^ k \\Gamma ( k + 1 / 2 ) } { \\pi } . \\end{align*}"}
+{"id": "3210.png", "formula": "\\begin{align*} h ( t ) = \\begin{cases} t + 2 ^ \\alpha t ^ { 1 + \\alpha } & \\ 0 \\le t < \\frac { 1 } { 2 } \\\\ 2 t - 1 & \\ \\frac { 1 } { 2 } \\le t \\le 1 . \\end{cases} \\end{align*}"}
+{"id": "7251.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ m b _ i ( x ^ { q ^ i } \\ \\ f ) = 0 . \\end{align*}"}
+{"id": "4624.png", "formula": "\\begin{align*} \\sigma ( x _ i ) = x _ { \\sigma ( i ) } , \\ \\sigma ( \\epsilon ) = \\epsilon , \\sigma \\in S _ n , \\ 1 \\leq i \\leq n . \\end{align*}"}
+{"id": "220.png", "formula": "\\begin{align*} D = D ( S , u ) = l _ S ( u ) + 2 \\max \\left \\{ l _ S \\big ( t ^ j \\big ) \\mid - r _ S \\le j \\le r _ S \\right \\} \\in \\mathbb { N } . \\end{align*}"}
+{"id": "3849.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ { k - 1 } \\beta ^ k _ l = \\sum _ { l = 0 } ^ { k - 1 } \\sum _ { j = 0 } ^ l ( k - j ) ^ { r - 1 - p / 2 } = \\sum _ { j = 0 } ^ { k - 1 } ( k - j ) ^ { r - p / 2 } \\lq \\zeta ( p / 2 - r ) . \\end{align*}"}
+{"id": "103.png", "formula": "\\begin{align*} ( X _ \\ast ( T ) _ { \\Gamma _ 0 } \\otimes \\mathbb Q ) ^ { \\sigma w } = \\{ \\mu \\in X _ \\ast ( T ) _ { \\Gamma _ 0 } \\otimes \\mathbb Q \\mid \\sigma ( \\mu ) = \\mu \\langle \\mu , \\Phi \\rangle = \\{ 0 \\} \\} . \\end{align*}"}
+{"id": "2274.png", "formula": "\\begin{align*} a ( U Z ) = a \\big ( ( ( U Z ) ^ * ( U Z ) ) ^ \\frac { 1 } { 2 } \\big ) = a \\big ( ( Z ^ * U ^ * U Z ) ^ \\frac { 1 } { 2 } \\big ) = a \\big ( ( Z Z ^ * ) ^ \\frac { 1 } { 2 } \\big ) = a ( Z ) \\end{align*}"}
+{"id": "6504.png", "formula": "\\begin{align*} d ^ * ( \\lfloor x \\rfloor ) & = q ( c ( \\lfloor x \\rfloor ) ) \\\\ & = q ( c _ b c _ b c _ a c _ a ) \\\\ & = b ^ 2 a ^ 2 \\end{align*}"}
+{"id": "2578.png", "formula": "\\begin{align*} R ( X , Y ) Z = \\nabla _ X \\nabla _ Y Z - \\nabla _ Y \\nabla _ X Z - \\nabla _ { [ X , Y ] } Z , \\end{align*}"}
+{"id": "5785.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = - \\frac { 1 } { 2 } ( \\sqrt { | c _ 1 | } \\ X _ 1 \\cdot \\nu _ 1 + \\sqrt { | c _ 2 | } \\ X _ 2 \\cdot \\nu _ 2 ) \\cdot \\varphi - \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi \\end{align*}"}
+{"id": "4059.png", "formula": "\\begin{align*} D ^ { k } B ^ C _ n ( j _ { V _ C } ) = D ^ { k } \\check { B } ^ C _ n ( j _ { V _ C } ) \\end{align*}"}
+{"id": "7448.png", "formula": "\\begin{align*} \\cos ^ 2 \\theta & = \\frac { 1 + \\cos ( 2 \\theta ) } { 2 } = \\frac { \\sqrt { m } - \\delta _ { m , l } } { 2 \\sqrt { m } - \\delta _ { m , k } - \\delta _ { m , l } } , \\\\ \\sin ^ 2 \\theta & = \\frac { 1 - \\cos ( 2 \\theta ) } { 2 } = \\frac { \\sqrt { m } - \\delta _ { m , k } } { 2 \\sqrt { m } - \\delta _ { m , k } - \\delta _ { m , l } } , \\end{align*}"}
+{"id": "4427.png", "formula": "\\begin{align*} \\liminf _ { i \\to \\infty } \\frac { w _ { n _ i } } { n _ i ^ { r - \\frac { 1 } { 2 } } } = 0 \\limsup _ { i \\to \\infty } \\frac { w _ { n _ i } } { n _ i ^ { r - \\frac { 1 } { 2 } } } = \\infty \\end{align*}"}
+{"id": "4677.png", "formula": "\\begin{align*} h ( M ) = \\inf \\frac { _ { n - 1 } ( \\Sigma ) } { \\min \\{ _ n ( A ) , _ n ( B ) \\} } , \\end{align*}"}
+{"id": "349.png", "formula": "\\begin{align*} \\lambda _ { j } : = 3 n - 8 - ( - 1 ) ^ { j } \\sqrt { 5 } \\sqrt { n ( 2 n - 9 ) + 1 2 } ~ ~ ~ ~ j = 1 , 2 \\end{align*}"}
+{"id": "6430.png", "formula": "\\begin{align*} [ q ^ \\alpha r ^ \\beta ] C _ k ( q , r , t ) = C _ k ( t ) ^ { \\beta + 1 } \\sum _ { i \\geq 0 } ( - 1 ) ^ i \\binom { \\alpha - ( k - 1 ) i } { i } t ^ i - \\sum _ { i \\geq 0 } ( - 1 ) ^ i \\binom { \\alpha - \\beta - 1 - ( k - 1 ) i } { i } t ^ i . \\end{align*}"}
+{"id": "2176.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u _ 0 = f & \\Omega , \\\\ u _ 0 = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "1099.png", "formula": "\\begin{align*} S = \\frac { { W I } } { { K L } } \\varepsilon \\left ( { K , \\gamma } \\right ) . \\end{align*}"}
+{"id": "3077.png", "formula": "\\begin{align*} q _ n = x _ 1 ^ 2 + x _ 0 x _ 1 + x _ 0 x _ 2 + \\dotsb + x _ 0 x _ n . \\end{align*}"}
+{"id": "6086.png", "formula": "\\begin{align*} 1 = N _ 0 \\rhd N _ 1 \\rhd \\cdots \\rhd N _ r = G , \\end{align*}"}
+{"id": "2699.png", "formula": "\\begin{align*} h ( \\delta ) = C _ 3 \\delta ^ 2 \\ ; \\ ; \\ ; \\ ; C _ 3 \\stackrel { \\rm d e f } { = } \\frac { 1 } { 2 } \\eta _ 1 \\eta _ 2 \\kappa _ { \\rm f c d } \\min \\left \\{ \\frac { \\eta _ 2 } { \\kappa _ { \\rm b h m } } , 1 \\right \\} . \\end{align*}"}
+{"id": "6809.png", "formula": "\\begin{align*} G _ { - } ( \\rho ) = \\frac { 1 } { 2 i \\rho } \\int _ { 0 } ^ { \\infty } q ( s ) e ^ { - 2 i s \\rho } d s + \\frac { 1 } { \\left ( 2 i \\rho \\right ) ^ { 2 } } \\int _ { 0 } ^ { \\infty } Q ^ { \\prime } ( s ) e ^ { - 2 i s \\rho } d s \\end{align*}"}
+{"id": "6085.png", "formula": "\\begin{align*} z ^ j \\cdot \\left ( \\prod _ { i = 1 } ^ n x _ i ^ { \\alpha _ i } . y _ i ^ { \\beta _ i } \\right ) ^ k , 0 \\le j \\le p - 1 , 1 \\le k \\le p - 1 . \\end{align*}"}
+{"id": "3059.png", "formula": "\\begin{align*} \\rho _ { Z } = \\begin{cases} n _ j z , & Z = V ( f _ j ) \\\\ \\lambda _ j , & Z = V ( x _ j ) . \\end{cases} \\end{align*}"}
+{"id": "4680.png", "formula": "\\begin{align*} \\psi _ { 0 , H } ( t ) = \\frac { ( n - 1 ) H } { ( n - 1 ) + t H } . \\end{align*}"}
+{"id": "1591.png", "formula": "\\begin{align*} \\mathbf { H } _ { \\mathrm { a l l } , k } = \\left \\{ \\begin{array} { c c } \\mathbf { H } _ { \\mathrm { d } , k } + \\overline { \\mathbf { H } } _ { k , 1 } \\mathbf { \\Phi } _ { 1 , 1 } \\overline { \\mathbf { G } } _ { 1 } , & k \\in \\mathcal { K } _ { 1 } , \\\\ \\mathbf { H } _ { \\mathrm { d } , k } + \\overline { \\mathbf { H } } _ { k , 2 } \\mathbf { \\Phi } _ { 2 , 1 } \\overline { \\mathbf { G } } _ { 1 } , & k \\in \\mathcal { K } _ { 2 } . \\end{array} \\right . \\end{align*}"}
+{"id": "6414.png", "formula": "\\begin{align*} g ^ { q p } u _ { , p q i } = - v _ { i } + u _ { , i q } \\xi ^ q \\end{align*}"}
+{"id": "5601.png", "formula": "\\begin{align*} b _ n = a _ { \\alpha + 1 } + ( d _ { \\alpha + 1 } - d _ \\alpha ) = b \\ , \\end{align*}"}
+{"id": "6230.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\mathcal { R } } & = \\{ f = \\frac { \\sum _ { h = 1 } ^ { H } c _ h \\chi _ { Q _ h } } { \\sum _ { i = 1 } ^ { H } c _ h \\nu ( Q _ h ) } : \\ ; c _ h \\in \\mathbb { R } ^ + , \\ ; Q _ i \\cap Q _ j = \\emptyset \\} , \\end{align*}"}
+{"id": "1599.png", "formula": "\\begin{align*} \\begin{aligned} \\gamma _ { k } = & \\frac { | \\tilde { \\mathbf { h } } _ { k } ^ H \\mathbf { w } _ { k } | ^ 2 } { \\sum _ { p \\in \\mathcal { K } , p \\ne k } | \\tilde { \\mathbf { h } } _ { k } ^ H \\mathbf { w } _ { p } | ^ 2 + \\sigma _ { k } ^ 2 } , \\forall i \\in \\{ \\mathrm { t , r } \\} , \\forall k \\in \\mathcal { K } _ i . \\end{aligned} \\end{align*}"}
+{"id": "2535.png", "formula": "\\begin{align*} ( \\mathbf { w } _ { x s } - \\mu \\mathbf { e } ) ^ T \\mathbf { e } = 0 . \\end{align*}"}
+{"id": "8772.png", "formula": "\\begin{align*} D _ { a _ j ^ + , g _ j } ^ { 1 - \\alpha _ j } [ 1 ] = & \\frac { 1 } { \\Gamma ( 1 - \\alpha _ j ) } \\frac { 1 } { g ' _ j ( x _ j ) } \\frac { \\partial } { \\partial x _ j } \\int _ { a _ j } ^ { x _ j } \\frac { g ' _ j ( y _ j ) } { ( g _ j ( x _ j ) - g _ j ( y _ j ) ) ^ { 1 - \\alpha _ j } } d y _ j \\\\ = & \\frac { 1 } { \\Gamma ( 1 - \\alpha _ j ) ( g _ j ( x _ j ) - g _ j ( a ) ) ^ { 1 - \\alpha _ j } } , \\end{align*}"}
+{"id": "4711.png", "formula": "\\begin{align*} \\tilde U ^ { i j } D _ { i j } \\tilde w = - \\gamma ^ { - 1 } f , \\end{align*}"}
+{"id": "2725.png", "formula": "\\begin{align*} d = R \\sqrt { 2 } \\sqrt { \\frac { n + 1 } { n } } \\end{align*}"}
+{"id": "1016.png", "formula": "\\begin{align*} \\{ \\tilde x \\in \\widetilde W \\mid \\exists x _ L \\in \\widetilde W _ L , x _ R \\in \\widetilde W _ R : ~ x = x _ L \\tilde x x _ R \\ell ( x ) = \\ell ( x _ L ) + \\ell ( \\tilde x ) + \\ell ( x _ R ) \\} , \\end{align*}"}
+{"id": "7000.png", "formula": "\\begin{align*} Q _ { n + 1 } ( \\lambda ) = \\frac { 1 } { d _ n } [ ( \\lambda - b _ n ) Q _ n ( x ) - c _ { n - 1 } Q _ { n - 1 } ( \\lambda ) ] . \\end{align*}"}
+{"id": "2434.png", "formula": "\\begin{align*} L _ { q } [ f ( a t ) ] = \\int _ { 0 } ^ { \\infty } d t [ 1 - ( 1 - q ) s t ] ^ { \\frac { 1 } { 1 - q } } f ( a t ) . \\end{align*}"}
+{"id": "3461.png", "formula": "\\begin{align*} | \\phi ( k ) | \\leq & \\frac { | \\tilde { P } _ { x _ 2 - k } ( \\theta _ { k + 1 } ) | } { | \\tilde { P } _ { 2 q _ n - 1 } ( \\theta _ { x _ 1 } ) | } \\prod _ { j = x _ 1 } ^ k | \\cos ( \\pi \\theta _ j ) | \\ , | \\phi ( x _ 1 - 1 ) | + \\frac { | \\tilde { P } _ { k - x _ 1 } ( \\theta _ { x _ 1 } ) | } { | \\tilde { P } _ { 2 q _ n - 1 } ( \\theta _ { x _ 1 } ) | } \\prod _ { j = k } ^ { x _ 2 } | \\cos ( \\pi \\theta _ j ) | \\ , | \\phi ( x _ 2 + 1 ) | \\end{align*}"}
+{"id": "6072.png", "formula": "\\begin{align*} x ^ G = \\{ g ^ { - 1 } x g \\mid g \\in G \\} , \\end{align*}"}
+{"id": "7334.png", "formula": "\\begin{align*} \\Omega & = \\{ ( x , y ) \\in \\R _ + ^ 2 : x + y \\le 1 , | 1 / 2 - x | \\le | 1 / 2 - y | \\} \\\\ & = \\{ ( x , 1 - x ) ; 0 \\le x \\le 1 / 2 \\} \\cup \\{ ( x , y ) \\in \\R _ + ^ 2 ; y \\le \\min ( x , 1 - x ) \\} . \\end{align*}"}
+{"id": "5057.png", "formula": "\\begin{align*} \\begin{aligned} & I _ { h , W , \\gamma } = \\\\ & \\inf \\left \\{ \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } | b | ^ { 2 } \\dd x \\ \\middle | \\ b = \\nabla \\times ( \\phi \\nabla \\theta ) + G \\nabla \\theta \\in L ^ { 2 } _ { \\sigma , \\textrm { a x i } } ( \\mathbb { R } ^ { 3 } ) , 2 \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi _ { \\infty } ) _ { + } \\frac { G } { r ^ { 2 } } \\dd x = h \\ \\right \\} , \\end{aligned} \\end{align*}"}
+{"id": "3243.png", "formula": "\\begin{align*} \\dot H ^ r : = D \\left ( \\partial _ { x x } ^ { \\frac r 2 } \\right ) = \\left \\{ h \\in H : \\| h \\| _ { \\dot H ^ r } ^ 2 : = \\sum _ { j = 1 } ^ { \\infty } \\lambda _ j ^ { r } \\langle e _ j , h \\rangle ^ 2 < \\infty \\right \\} . \\end{align*}"}
+{"id": "6912.png", "formula": "\\begin{align*} f ( x ) \\le a - b | x | ^ 2 , x \\in \\R ^ n , \\ a : = f ( 0 ) + \\kappa ^ { - 1 } | \\nabla f ( 0 ) | ^ 2 , \\ b : = \\kappa / 4 . \\end{align*}"}
+{"id": "9022.png", "formula": "\\begin{align*} \\dfrac { \\hat { k } _ 1 { ( G ) } } { \\hat { k } _ 1 ( G \\setminus \\sigma ) } = \\dfrac { D _ { \\ell _ 1 } D _ { \\ell _ 2 } } { D _ { \\ell _ 1 - 1 } D _ { \\ell _ 2 - 1 } } . \\end{align*}"}
+{"id": "5348.png", "formula": "\\begin{align*} p ^ { C _ 5 } = e . \\end{align*}"}
+{"id": "6905.png", "formula": "\\begin{align*} V _ { \\mathrm { d e t } } = \\sup _ { y \\in \\R ^ n } \\left ( \\int _ { \\R ^ n } g \\ , d \\gamma _ { y , T } - \\frac { 1 } { n } H ( \\gamma _ { y , T } \\ , | \\ , \\gamma _ T ) \\right ) = \\sup _ { y \\in \\R ^ n } \\left ( \\int _ { \\R ^ n } g \\ , d \\gamma _ { y , T } - \\frac { | y | ^ 2 } { 2 n T } \\right ) , \\end{align*}"}
+{"id": "4579.png", "formula": "\\begin{align*} \\dot { x } ( t ) & = - A x ( t ) + B u ( t ) \\quad ( t > 0 ) , \\\\ x ( 0 ) & = x _ 0 \\in X , \\end{align*}"}
+{"id": "1567.png", "formula": "\\begin{align*} u \\cdot ( v _ 1 , w _ 1 , x _ 2 + i y _ 2 ) & = ( \\sigma _ 1 ( u ) v _ 1 , \\sigma _ 1 ( u ) ^ { - \\frac { 1 } { 2 } } w _ 1 , \\sigma _ 2 ( u ) ( x _ 2 + i y _ 2 ) ) \\\\ & = ( \\sigma _ 1 ( u ) v _ 1 , \\vert \\sigma _ 2 ( u ) \\vert w _ 1 , \\sigma _ 2 ( u ) ( x _ 2 + i y _ 2 ) ) . \\end{align*}"}
+{"id": "1178.png", "formula": "\\begin{align*} H \\left ( P ^ { ( n , n ) } [ t ] \\mid \\mu ^ { \\otimes n } [ t ] \\right ) = \\mathbb { E } _ \\mathbb { P } [ Z _ t ^ n \\log Z _ t ^ n ] . \\end{align*}"}
+{"id": "5099.png", "formula": "\\begin{align*} b ( x ) = \\nabla \\times ( \\phi ( z , r ) \\nabla \\theta ) + G ( z , r ) \\nabla \\theta . \\end{align*}"}
+{"id": "8914.png", "formula": "\\begin{align*} h _ { 1 } + h _ { 2 } = h _ { 1 2 } . \\end{align*}"}
+{"id": "3848.png", "formula": "\\begin{align*} x _ { n + 1 } = \\begin{cases} x _ n - 1 , & x _ n > 1 \\\\ n + 1 , & x _ n = 1 . \\end{cases} \\end{align*}"}
+{"id": "5125.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi _ { \\infty } ) _ { + } ^ { q } \\dd x \\lesssim \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\phi ^ { q + l } \\frac { 1 } { r ^ { 2 l - 1 } } \\dd z \\dd r = \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\phi ^ { p } \\frac { 1 } { r ^ { p / 2 + 2 } } \\dd z \\dd r \\lesssim | | \\nabla \\phi | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } _ { + } ; r ^ { - 1 } ) } ^ { p } . \\end{align*}"}
+{"id": "1158.png", "formula": "\\begin{align*} d X _ t ^ i = \\left ( b _ 0 ( t , X ^ i ) + \\frac { 1 } { n - 1 } \\sum _ { j = 1 , j \\neq i } ^ n b \\left ( t , X ^ i , X ^ j \\right ) \\right ) d t + d W ^ i _ t , i = 1 , \\cdots , n , \\end{align*}"}
+{"id": "4466.png", "formula": "\\begin{align*} P ( x _ { 3 } , y _ { 3 } ) : = a x _ { 3 } ^ { 2 } + b x _ { 3 } x _ { 4 } + c x _ { 4 } ^ { 2 } + d x _ { 3 } + e x _ { 4 } + f = 0 , \\end{align*}"}
+{"id": "1283.png", "formula": "\\begin{align*} s ( \\nu | \\mu ) : = \\lim _ { n \\to \\infty } \\frac { 1 } { \\abs { \\Lambda _ n } } s _ n ( \\nu | \\mu ) . \\end{align*}"}
+{"id": "2487.png", "formula": "\\begin{align*} \\mathbf { x } \\circ \\mathbf { s } = ( \\mathbf { x } ) ( \\mathbf { s } ) \\mathbf { e } , \\end{align*}"}
+{"id": "2021.png", "formula": "\\begin{align*} H ( t , y , z , u , \\mu , p , q ) : = \\big < f ( t , y , z , u , \\mu ) , p \\big > + \\big < g ( t , y , z , u , \\mu ) , q \\big > + h ( t , y , z , u , \\mu ) . \\end{align*}"}
+{"id": "3141.png", "formula": "\\begin{align*} [ [ x , y ] , [ x , z ] ] = 0 , \\forall \\ y , z \\in A . \\end{align*}"}
+{"id": "58.png", "formula": "\\begin{align*} \\Phi ^ + : \\Phi \\rightarrow \\{ 0 , 1 \\} , \\alpha \\mapsto \\begin{cases} 1 , & \\alpha \\in \\Phi ^ + , \\\\ 0 , & \\alpha \\in \\Phi ^ - . \\end{cases} \\end{align*}"}
+{"id": "2325.png", "formula": "\\begin{align*} ( R i c ^ g ) ^ { J , + } & = \\frac { s ^ H } { 4 } g - \\frac { 1 } { 4 } \\| N \\| ^ 2 g - \\frac { 1 } { 2 } ( \\theta \\otimes \\theta ) ^ { J , + } . \\end{align*}"}
+{"id": "1340.png", "formula": "\\begin{align*} \\delta _ j ^ 1 & = \\min \\bigl \\{ 1 , \\frac { d ( a _ j , c _ j ) } { C _ { n + 1 } \\alpha _ { j } } \\bigr \\} , \\\\ \\delta _ j ^ 2 & = \\min \\bigl \\{ 1 , \\frac { d ( c _ j , d _ j ) } { C _ { n + 1 } ( \\alpha _ { m + j } + \\beta _ k ) } \\bigr \\} , \\\\ \\delta _ j ^ 3 & = \\min \\bigl \\{ 1 , \\frac { d ( d _ j , b _ j ) } { C _ { n + 1 } \\alpha _ { 2 m + j } } \\bigr \\} . \\end{align*}"}
+{"id": "8763.png", "formula": "\\begin{align*} p _ { _ { l , 2 } } \\varphi _ l = - { \\bf i } q _ { _ { l , 1 } } \\vartheta _ l \\ \\textrm { o n } \\ \\C \\ \\textrm { f o r } \\ l = 1 , 2 . \\end{align*}"}
+{"id": "4612.png", "formula": "\\begin{align*} & \\o ^ \\beta ( t , x ) = \\o _ 0 ^ \\beta ( X ^ \\beta ( 0 ; t , x ) ) , \\\\ & \\frac { d } { d s } X ^ \\beta ( s ; t , x ) = u ^ \\beta ( s , X ^ \\beta ( s ; t , x ) ) , \\ \\ X ^ \\beta ( s ; t , x ) | _ { s = t } = x . \\end{align*}"}
+{"id": "7843.png", "formula": "\\begin{align*} F ^ { q , \\dagger } _ { v v } ( 0 , p ) [ u _ k , u _ k ] & = c _ 2 ( q , k , p ) u _ { 2 k } \\end{align*}"}
+{"id": "3858.png", "formula": "\\begin{align*} \\mathbf { v } \\cdot \\overrightarrow { \\zeta } = 0 , \\end{align*}"}
+{"id": "8962.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { | k | \\ge n } \\biggl \\{ \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { - p ' } ( t ) \\bigl | G _ { 1 , k } ^ { ( 0 ) } ( t ) \\bigr | ^ { p ' } \\ , d t + \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { - p ' } ( t ) \\bigl | G _ { 2 , k } ^ { ( 0 ) } ( t ) \\bigr | ^ { p ' } \\ , d t \\\\ + \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { - p ' } ( t ) \\bigl | G _ { 1 , k } ^ { ( 1 ) } ( t ) \\bigr | ^ { p ' } \\ , d t + \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { - p ' } ( t ) \\bigl | G _ { 2 , k } ^ { ( 1 ) } ( t ) \\bigr | ^ { p ' } \\ , d t \\biggr \\} = 0 . \\end{align*}"}
+{"id": "9038.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { { \\bf k } \\in \\Gamma _ { r } \\backslash \\Lambda _ { 1 } } \\begin{pmatrix} a _ { \\bf k } & 0 \\\\ 0 & - a _ { \\bf k } \\end{pmatrix} { \\rm e } ^ { { \\rm i } \\langle { \\bf k } , x \\rangle } + \\sum _ { { \\bf k } \\in \\Gamma _ { r } \\backslash \\Lambda _ { 2 } } \\begin{pmatrix} 0 & b _ { \\bf k } \\\\ c _ { \\bf k } & 0 \\end{pmatrix} { \\rm e } ^ { { \\rm i } \\langle { \\bf k } , x \\rangle } . \\end{align*}"}
+{"id": "5338.png", "formula": "\\begin{align*} \\vert \\Delta _ { n , k , m } \\vert & \\leq ( m + 1 ) ! C _ 1 ^ { m + 1 } \\left ( ( d _ n + k + 1 ) C _ 3 ^ { 2 n } C _ 2 ^ { d _ n + k } \\right ) ^ { m + 1 } r ^ { - m ( m + 1 ) } \\\\ & < ( C _ 3 ^ 3 r ^ { - 1 } ) ^ { m ( m + 1 ) } \\end{align*}"}
+{"id": "8234.png", "formula": "\\begin{align*} D _ { \\xi _ 0 } ( k ) : = \\left \\lbrace z \\in \\mathbb { C } : \\ | z - \\xi _ 0 | < k \\mu ( \\xi _ 0 ) \\right \\rbrace \\ . \\end{align*}"}
+{"id": "2490.png", "formula": "\\begin{align*} \\mathcal { G } ^ i \\triangleq \\left \\{ \\theta ^ i \\mathbf { G } ^ i : \\theta ^ i > 0 , \\mathbf { G } ^ i \\in \\mathbf { R } ^ { n _ i \\times n _ i } , ( \\mathbf { G } ^ i ) ^ T \\mathbf { Q } ^ i \\mathbf { G } ^ i = \\mathbf { Q } ^ i \\right \\} , \\end{align*}"}
+{"id": "2263.png", "formula": "\\begin{align*} 0 = D ( i y ) D ( x ) - D ( x ) D ( i y ) = D ( v ) , \\end{align*}"}
+{"id": "1420.png", "formula": "\\begin{align*} p _ { x _ 1 , x _ 2 } ( n ) = ( - 1 ) ^ n \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ { k + 1 } { k / 2 - 1 \\choose n } \\frac { ( x _ 2 - x _ 1 ) ^ k } { k ! } . \\end{align*}"}
+{"id": "1883.png", "formula": "\\begin{align*} \\Pi _ \\lambda f = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } e ^ { i \\frac t 2 ( \\lambda - \\mathcal H ) } f d t , f \\in \\mathcal S ( \\mathbb R ^ d ) . \\end{align*}"}
+{"id": "1513.png", "formula": "\\begin{align*} f ' ( t ) = \\frac { ( t - a ) ^ { \\alpha - 2 } } { \\Gamma ( \\alpha ) } \\left [ \\frac { \\alpha - 1 } { \\alpha - \\beta } ( b - a ) - ( t - a ) \\right ] . \\end{align*}"}
+{"id": "1278.png", "formula": "\\begin{align*} a d - b c = \\frac { 1 } { 2 } [ ( a - b ) ( c + d ) - ( a + b ) ( c - d ) ] \\end{align*}"}
+{"id": "1311.png", "formula": "\\begin{align*} { \\rm R e } \\langle u _ { t t } , v \\rangle + { \\rm R e } \\int _ { \\mathbb { H } ^ n } \\nabla _ { H } u \\cdot \\nabla _ { H } v d x + m { \\rm R e } \\int _ { \\mathbb { H } ^ n } u v d x + b { \\rm R e } \\int _ { \\mathbb { H } ^ n } u _ t v d x \\ = { \\rm R e } \\int _ { \\mathbb { H } ^ n } f ( u ) v d x , \\end{align*}"}
+{"id": "6839.png", "formula": "\\begin{align*} - y ^ { \\prime \\prime } + q ( x ) y + \\frac { 1 } { 4 } y = 0 . \\end{align*}"}
+{"id": "3305.png", "formula": "\\begin{align*} \\abs { F ( k p _ i ) - \\left ( 1 - \\frac { 1 } { p _ i } \\right ) F ( k ) } \\le \\sum _ { d | k } \\mu ^ 2 ( d ) \\left ( 1 + \\frac 1 p _ i \\right ) \\sqrt p ( \\log p + 1 ) = ( 1 + 1 / p _ i ) W ( k ) \\sqrt { p } ( \\log p + 1 ) \\end{align*}"}
+{"id": "6986.png", "formula": "\\begin{align*} | u _ { m ( p ) - 1 } \\dots u _ n | _ p \\geq \\prod ^ { m ( p ) - n } _ { r = 1 } ( | t _ r | _ p | a ' b ' | _ p ) . \\end{align*}"}
+{"id": "2481.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathcal { K } _ { L } ^ { n } = \\{ \\mathbf { v } \\in { { \\mathbf { R } } ^ { n } } : \\mathbf { v } \\ge 0 \\} , n \\ge 1 , \\\\ & \\mathcal { K } _ { S } ^ { n } = \\{ \\mathbf { v } \\in { { \\mathbf { R } } ^ { n } } : { { \\mathbf { v } } _ { 1 } } \\ge \\left \\| { { \\mathbf { v } } _ { 2 : n } } \\right \\| \\} , n \\ge 2 , \\end{aligned} \\end{align*}"}
+{"id": "5301.png", "formula": "\\begin{align*} l ( \\hat { y } p ) = l ( y _ { p _ 1 } p ) = l ( \\Psi ( v _ 1 ) v _ 1 ^ * p ) = l ( \\Psi ( v _ 1 ) v _ 1 ^ * p v _ 1 ) = l ( \\Psi ( v _ 1 v _ 1 ^ * p v _ 1 ) ) = l ( \\Psi ( p v _ 1 ) ) = \\Phi ( p ) . \\end{align*}"}
+{"id": "6020.png", "formula": "\\begin{align*} & b _ { 1 } = \\Bigg ( \\frac { ( p - 1 ) 3 ^ { \\frac { p } { p - 1 } } \\| u _ { 1 } \\| _ { 2 p } ^ { 2 p } } { p E _ { \\omega } } \\Bigg ) ^ { p - 1 } , \\ \\ \\ b _ { 2 } = \\Bigg ( \\frac { ( p - 1 ) 3 ^ { \\frac { p } { p - 1 } } \\| u _ { \\omega } \\| _ { 2 p } ^ { 2 p } } { p E _ { 1 } } \\Bigg ) ^ { p - 1 } , \\\\ & b _ { 3 } = \\Bigg ( \\frac { ( p - 1 ) 3 ^ { \\frac { 3 } { p - 3 } } \\| u _ { 1 } \\| _ { 2 p } ^ { 2 p } } { p E _ { \\omega } } \\Bigg ) ^ { p - 1 } , \\ \\ \\ b _ { 4 } = \\Bigg ( \\frac { ( p - 1 ) 3 ^ { \\frac { 3 } { p - 3 } } \\| u _ { \\omega } \\| _ { 2 p } ^ { 2 p } } { p E _ { 1 } } \\Bigg ) ^ { p - 1 } . \\end{align*}"}
+{"id": "5274.png", "formula": "\\begin{align*} a _ { \\frac { 1 } { t } } n _ { - u } \\cdot f _ { - } & = \\frac { 1 } { \\sqrt { 2 } } \\left ( 0 , t , \\frac { - 2 u } { t } , \\frac { 1 + u ^ 2 } { t ^ 3 } \\right ) , \\\\ a _ { \\frac { 1 } { t } } n _ { - u } \\cdot f _ { + } & = \\frac { 1 } { ( 1 0 8 ) ^ { \\frac { 1 } { 4 } } } \\left ( 0 , 3 t , \\frac { - 6 u } { t } , \\frac { - 1 + 3 u ^ 2 } { t ^ 3 } \\right ) . \\end{align*}"}
+{"id": "437.png", "formula": "\\begin{align*} \\kappa \\begin{pmatrix} 1 \\\\ \\vdots \\\\ \\theta ^ { n - 1 } \\end{pmatrix} = \\omega ^ { - 1 } \\begin{pmatrix} a _ { 1 } \\\\ \\vdots \\\\ a _ { n } \\end{pmatrix} , \\end{align*}"}
+{"id": "7648.png", "formula": "\\begin{align*} \\mu _ t ^ { * , \\xi } = k ( t , \\nu _ t ^ { * , \\xi } , \\varphi _ t ^ { * , \\xi } ) . \\end{align*}"}
+{"id": "7813.png", "formula": "\\begin{align*} \\langle z _ 1 ^ * , v \\rangle & = 2 \\int _ \\S \\sin ( 2 \\pi \\ell x ) v ( x ) \\ \\d x , \\\\ \\langle z _ 2 ^ * , v \\rangle & = - 2 \\int _ \\S \\cos ( 2 \\pi \\ell x ) v ( x ) \\ \\d x , \\end{align*}"}
+{"id": "8395.png", "formula": "\\begin{align*} \\psi ^ - _ { 2 1 } ( x ; k ) = \\frac { 1 } { 2 i k } \\left ( \\bar { u } _ x \\Psi ^ - _ { 1 1 } ( x ; z ) + \\Psi ^ - _ { 2 1 } ( x ; z ) \\right ) . \\end{align*}"}
+{"id": "4716.png", "formula": "\\begin{align*} \\frac { \\partial \\tilde x } { \\partial x } = h ^ { \\frac { 1 } { 3 } } + x \\cdot \\frac { 1 } { 3 } h ^ { - \\frac { 2 } { 3 } } \\cdot h _ x ' , \\end{align*}"}
+{"id": "4441.png", "formula": "\\begin{align*} \\big | M ^ * _ v ( t ) - M ^ * _ v ( \\delta n ) \\big | & \\le e ^ { - ( \\lambda ^ * - \\lambda ) t } \\big | M _ v ( t ) - M _ v ( \\delta n ) \\big | + \\big ( e ^ { - ( \\lambda ^ * - \\lambda ) \\delta n } - e ^ { - ( \\lambda ^ * - \\lambda ) t } \\big ) | M _ v ( \\delta n ) | \\\\ & \\le \\delta e ^ { - ( \\lambda ^ * - \\lambda ) ( t - \\delta n ) } + \\delta \\big ( 1 - e ^ { - ( \\lambda ^ * - \\lambda ) ( t - \\delta n ) } \\big ) = \\delta , \\end{align*}"}
+{"id": "2964.png", "formula": "\\begin{align*} \\tilde c ( t ) : = c ( t + \\alpha ) . \\end{align*}"}
+{"id": "7759.png", "formula": "\\begin{align*} \\frac { p ' _ { h + 1 } ( x ) } { p _ { h + 1 } ( x ) } = \\frac { 1 } { 2 \\sqrt x } \\Big ( \\frac { p _ h ( \\sqrt x ) } { p _ h ( \\sqrt { - x } ) } - \\frac { p _ h ( \\sqrt { - x } ) } { p ' _ h ( \\sqrt x ) } \\Big ) , ~ h = 0 , 1 , \\dots , \\end{align*}"}
+{"id": "6104.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { 2 } \\Big ( \\psi ^ { - 0 } _ { n - 1 , p } \\psi ^ { - 0 } _ { n , p } \\rho ^ { 0 + } _ { n , p } - 2 \\psi ^ { - 0 } _ { n - 1 , p } \\rho ^ { 0 + } _ { n - 1 , p } \\psi ^ { - 0 } _ { n , p - 1 } + \\rho ^ { 0 + } _ { n - 2 , p } \\psi ^ { - 0 } _ { n - 1 , p - 1 } \\psi ^ { - 0 } _ { n , p - 1 } \\Big ) & = & - \\varphi ^ { - + } _ { n - 1 , p } \\psi ^ { - 0 } _ { n , p - 1 } \\\\ & & = & - \\psi ^ { - 0 } _ { n - 1 , p } \\varphi ^ { - + } _ { n , p } \\ , . \\end{aligned} \\end{align*}"}
+{"id": "1748.png", "formula": "\\begin{align*} A ^ { - 1 } - B ^ { - 1 } = A ^ { - 1 } ( B - A ) B ^ { - 1 } , \\end{align*}"}
+{"id": "6871.png", "formula": "\\begin{align*} { \\tt P s i 1 = p l g c i r m a p ( v e r s , h a l p h a ) } \\end{align*}"}
+{"id": "5519.png", "formula": "\\begin{align*} h _ d ( q ) = \\log ( c _ { d , 2 } ( q ) ^ q ) - \\log ( c _ { d , \\infty } ( q ) ^ q ) . \\end{align*}"}
+{"id": "666.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | e ^ { i k \\phi } \\cos ( n \\theta ) \\left | l _ 2 , m _ 2 \\right \\rangle = \\end{align*}"}
+{"id": "1223.png", "formula": "\\begin{align*} \\sup _ { y \\in \\Z ^ d } \\sum _ { x \\neq y } \\sum _ { \\Delta \\ni y } \\sum _ { \\xi _ \\Delta } \\sum _ { i = 1 } ^ q \\norm { \\nabla ^ i _ x c _ \\Delta ( \\cdot , \\xi _ \\Delta ) } _ \\infty < \\infty . \\end{align*}"}
+{"id": "2707.png", "formula": "\\begin{align*} \\frac { 2 \\epsilon _ f + 4 / a + r } { h ( \\gamma \\bar \\Delta ) } T + \\left [ \\sum _ { k = 0 } ^ { T - 1 } \\Theta _ k ( 1 - \\Lambda _ k ' ) + ( 1 - \\Theta _ k ) \\Lambda _ k \\right ] < \\left ( \\frac { 2 \\epsilon _ f + 4 / a + r } { h ( \\gamma \\bar \\Delta ) } + \\frac { 1 } { 2 } \\right ) T + \\frac { 1 } { 2 } \\left | \\log _ \\gamma \\frac { \\bar \\Delta } { \\delta _ 0 } \\right | + \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "2779.png", "formula": "\\begin{align*} G ^ { D _ { N } } ( [ x N ] , [ x N ] ) = \\gamma \\log N + s _ { D } ( x ) + o ( 1 ) \\end{align*}"}
+{"id": "6760.png", "formula": "\\begin{align*} \\left [ \\sum _ { m = 0 } ^ \\infty \\frac { ( - 1 ) ^ m } { ( 1 / s + m ) m ! } y ^ m \\right ] ^ j = \\sum _ { m = 0 } ^ \\infty c _ { m , j } y ^ { m } , \\end{align*}"}
+{"id": "3476.png", "formula": "\\begin{align*} \\begin{cases} | \\phi ( x _ 1 - 1 ) | \\leq e ^ { 3 0 \\varepsilon q _ n } \\max \\{ e ^ { - ( x _ 1 - ( \\ell - 1 ) q _ n ) L } r _ { \\ell - 1 } ^ + , e ^ { - ( \\ell q _ n - x _ 1 ) L } r _ { \\ell } ^ - \\} \\\\ | \\phi ( x _ 2 + 1 ) | \\leq e ^ { 3 0 \\varepsilon q _ n } \\max \\{ e ^ { - ( x _ 2 - ( \\ell + 1 ) q _ n ) L } r _ { \\ell + 1 } ^ + , e ^ { - ( ( \\ell + 2 ) q _ n - x _ 2 ) L } r _ { \\ell + 2 } ^ - \\} \\end{cases} \\end{align*}"}
+{"id": "8446.png", "formula": "\\begin{align*} M ( x ; z ) : = \\begin{cases} e ^ { i c _ + ( x ) \\sigma _ 3 } \\left ( \\frac { \\Psi _ 1 ^ - ( x ; z ) } { a ( z ) } , \\Psi ^ + _ 2 ( x ; z ) \\right ) , \\ ; z \\in \\mathbb { C } ^ + , \\\\ [ 5 p t ] e ^ { i c _ + ( x ) \\sigma _ 3 } \\left ( \\Psi _ 1 ^ + ( x ; z ) , \\frac { \\Psi ^ - _ 2 ( x ; z ) } { \\bar { a } ( z ) } \\right ) , \\ ; z \\in \\mathbb { C } ^ - , \\end{cases} \\end{align*}"}
+{"id": "3310.png", "formula": "\\begin{align*} \\mathcal { Z } ^ * _ K ( \\underline { \\mathrm { p t } ^ { J } } , \\{ \\emptyset \\} ) ( \\underline { \\mathrm { p t } ^ J } , \\{ \\emptyset \\} ) & = \\left \\{ \\left ( { \\bigsqcup } _ { j _ i } \\mathrm { p t } , \\{ \\emptyset \\} \\right ) \\right \\} _ { i \\in \\{ 1 , \\ldots , m \\} } . \\end{align*}"}
+{"id": "7902.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Theta ( t , x ) = \\int _ \\S \\int _ \\S W _ r ( z - x ) W _ r ( y - x ) \\sin ( \\Theta ( t , z ) + \\Theta ( t , y ) - 2 \\Theta ( t , x ) ) \\ \\d y \\d z \\end{align*}"}
+{"id": "1813.png", "formula": "\\begin{align*} | | R ^ { \\nabla } | | ^ 2 _ x = \\sum _ { i < j } | | R ^ { \\nabla } _ { e _ i , e _ j } | | ^ 2 _ x \\ , \\end{align*}"}
+{"id": "3219.png", "formula": "\\begin{align*} ( x , . . . , x ) \\in X ^ { \\sum _ i M _ i } = p _ X ^ { \\otimes \\sum _ i M _ i } [ \\gamma _ { 1 , 1 } , . . . , \\gamma _ { N , M _ N } ] , \\end{align*}"}
+{"id": "4776.png", "formula": "\\begin{align*} ( p _ n f , x _ n ) = ( f , x _ n ) x _ n \\in X _ n . \\end{align*}"}
+{"id": "2441.png", "formula": "\\begin{align*} I = L _ { q } [ f ( t ) ] [ \\exp _ { q } ( - s t _ { 0 } ) ] ^ { 2 - q } . \\end{align*}"}
+{"id": "2794.png", "formula": "\\begin{align*} \\sum _ { \\substack { x , y \\in D _ { N } ^ { \\varepsilon } \\\\ \\| x - y \\| > K _ { N } ^ { 1 / 8 } } } \\mathbb { P } ( x , y \\in \\hat { \\Gamma } _ { N } ^ { D , M } ( b , b ' ) ) \\leq c ' \\frac { K _ { N } } { N ^ { 4 } } \\sum _ { k = k _ { N } } ^ { n } \\frac { \\sqrt { n } } { k \\sqrt { n - k + 1 } } N ^ { 4 } e ^ { 4 k - \\frac { k a _ { N } ^ { 2 } } { 2 \\gamma n ^ { 2 } } + \\tilde { c } M ( n - k ) ^ { 3 / 4 } } . \\end{align*}"}
+{"id": "3304.png", "formula": "\\begin{align*} F ( k p _ i ) = \\sum _ { d | k p _ i } \\mu ( d ) N ( d ) = \\sum _ { d | k } \\mu ( d ) ( N ( d ) - N ( d p _ i ) ) = \\sum _ { d | k } \\frac { \\mu ( d ) ( p + 1 ) } { 2 d } \\left ( 1 - \\frac 1 { p _ i } \\right ) + \\sum _ { d | k } \\mu ( d ) ( \\xi ( d ) - \\xi ( d p _ i ) ) . \\end{align*}"}
+{"id": "2812.png", "formula": "\\begin{align*} \\mathbf { a } ( \\theta ) = \\frac { 1 } { \\sqrt { N } } { [ e ^ { - j 2 \\pi \\theta v } ] } _ { v \\in \\mathcal { V } } , \\end{align*}"}
+{"id": "4094.png", "formula": "\\begin{align*} ( a b ) ^ i { } _ { j k l } & = a ^ i { } _ { \\alpha \\beta \\gamma } b ^ \\alpha { } _ j b ^ \\beta { } _ k b ^ \\gamma { } _ l + a ^ i { } _ { \\alpha \\beta } b ^ \\alpha { } _ { j l } b ^ \\gamma { } _ k + a ^ i { } _ { \\alpha \\beta } b ^ \\alpha { } _ j b ^ \\gamma { } _ { k l } + a ^ i { } _ { \\alpha \\beta } b ^ \\beta { } _ l b ^ \\gamma { } _ { j k } + a ^ i { } _ \\alpha b ^ \\alpha { } _ { j k l } \\end{align*}"}
+{"id": "8961.png", "formula": "\\begin{align*} \\sum _ { | k | \\le N } \\int _ { \\eta _ { k } } ^ { \\eta _ { k + 1 } } \\frac { v _ 1 ^ { - p ' } ( x ) } { \\bigl [ V _ 1 ^ - ( x ) \\bigr ] ^ { p } } \\biggl ( \\int _ { a ( x ) } ^ { \\eta _ { k } } v _ 1 ^ { - p ' } ( t ) \\bigl | G ^ { ( \\delta ) } _ { 2 , k } ( t ) \\bigr | ^ { p ' - 1 } \\ , d t \\biggr ) ^ p \\ , d x \\lesssim \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { - p ' } \\bigl | G _ { 2 , k } ^ { ( \\delta ) } \\bigr | ^ { p ' } = \\bigl [ \\mathbf { G } ^ { ( \\delta ) } _ { 2 , N } ( g ) \\bigr ] ^ { p ' - 1 } , \\end{align*}"}
+{"id": "9317.png", "formula": "\\begin{align*} p ^ { 2 } \\Tilde { z } _ { 1 } ^ { 2 } - p z _ { 2 } ^ { 2 } = 4 ^ { m } p \\implies - z _ { 2 } ^ { 2 } \\equiv 4 ^ { m } \\pmod p \\end{align*}"}
+{"id": "7653.png", "formula": "\\begin{align*} \\widetilde h ( t , x , y ) = h ( \\widetilde k ( t , x , y ) ) \\quad \\widetilde b ( t , x , y ) = b ( \\widetilde k ( t , x , y ) ) . \\end{align*}"}
+{"id": "8118.png", "formula": "\\begin{align*} f = t + 1 , \\end{align*}"}
+{"id": "3888.png", "formula": "\\begin{align*} - \\Delta w = w ^ p _ + , \\ \\ \\ \\mathbb { R } ^ 2 . \\end{align*}"}
+{"id": "3332.png", "formula": "\\begin{align*} \\chi \\cap \\psi = \\psi \\setminus \\{ x \\} . \\end{align*}"}
+{"id": "7301.png", "formula": "\\begin{align*} R _ { 0 , j } ^ n = B _ { \\rm R F } \\log _ 2 \\left ( 1 + \\frac { p _ { 0 , j } ^ n \\left | G _ { 0 , j } ^ m \\right | ^ 2 } { N _ { \\rm R F } B _ { \\rm R F } } \\right ) , \\end{align*}"}
+{"id": "2568.png", "formula": "\\begin{align*} q ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { 2 } } - p ^ { n _ { 2 } ^ { \\prime } - n _ { s } ^ { \\prime } } a _ { n _ { s } } N _ { n _ { 2 } } \\frac { q ^ { n _ { s } ^ { \\prime } - n _ { 1 } ^ { \\prime } } } { N _ { n _ { s } } } = p ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { 1 } } \\frac { N _ { n _ { 2 } } } { N _ { n _ { 1 } } } . \\end{align*}"}
+{"id": "1305.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } ( t ) - \\mathcal { L } u ( t ) + b u _ t ( t ) + m u ( t ) = f ( u ) , & t > 0 , \\\\ u ( x , 0 ) = u _ 0 ( x ) , \\ , \\ , \\ , & u _ 0 \\in H ^ 1 _ { \\mathcal { L } } ( \\mathbb { H } ^ n ) , \\\\ u _ t ( x , 0 ) = u _ 1 ( x ) , \\ , \\ , \\ , & u _ 1 \\in L ^ 2 ( \\mathbb { H } ^ n ) , \\end{cases} \\end{align*}"}
+{"id": "8768.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ 3 x _ k \\psi _ k \\rightarrow ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) . \\end{align*}"}
+{"id": "8459.png", "formula": "\\begin{align*} & R _ + ( x ; z ) = \\begin{pmatrix} 0 & \\bar { r } _ 1 ( z ) \\mathrm { e } ^ { - 2 i z x } \\\\ 0 & 0 \\end{pmatrix} , R _ - ( x ; z ) = \\begin{pmatrix} 0 & 0 \\\\ r _ 2 ( z ) \\mathrm { e } ^ { 2 i z x } & 0 \\end{pmatrix} , \\\\ & F = \\left ( \\mathcal { P } ^ { - } ( r _ 2 ( z ) \\mathrm { e } ^ { 2 i z x } ) e _ 2 , \\mathcal { P } ^ + ( \\bar { r } _ 1 ( z ) e ^ { - 2 i z x } ) e _ 1 \\right ) , \\end{align*}"}
+{"id": "1139.png", "formula": "\\begin{align*} S ^ { \\rm P R } ( B ) = \\kappa ( \\rho ) W \\in \\R ^ { N L \\times N L } , \\end{align*}"}
+{"id": "5878.png", "formula": "\\begin{align*} W _ n ( t ) = Q _ n W ( t ) : = \\sum _ { i = 1 } ^ n \\langle W ( t ) , h _ i \\rangle h _ i , \\end{align*}"}
+{"id": "3950.png", "formula": "\\begin{align*} \\langle \\overline { y } - g , S ' ( \\overline { u } ; h ) \\rangle = \\langle \\overline { y } - g , z \\rangle = \\langle \\zeta z , p \\rangle + \\langle - \\Delta z , p \\rangle = \\langle \\zeta z , p \\rangle - \\langle F ' ( \\overline { y } ; z ) , p \\rangle + \\langle h , p \\rangle \\ge - \\alpha \\langle \\overline { u } , h \\rangle . \\end{align*}"}
+{"id": "6843.png", "formula": "\\begin{align*} q ( x ) = x e ^ { - x ^ { 2 } } . \\end{align*}"}
+{"id": "7875.png", "formula": "\\begin{align*} Z ^ 2 ( v , s ) = \\Psi ^ q + v + \\frac 1 2 \\psi _ { v v } ( 0 , p _ 0 ) [ v , v ] \\end{align*}"}
+{"id": "6484.png", "formula": "\\begin{align*} \\psi _ { \\tau } ( p ) = | | \\tau | | _ p \\le C _ 1 ( \\beta , \\gamma , L ) \\ ( \\beta - p ) ^ { - ( \\gamma + 1 ) / \\beta } \\ L ^ { 1 / \\beta } ( 1 / ( \\beta - p ) ) , \\end{align*}"}
+{"id": "3747.png", "formula": "\\begin{align*} \\psi \\left ( L ( 0 ^ M , 0 ^ N ) \\right ) & = \\psi \\left ( ( 1 - q ) ^ { - 1 } d _ - L ( 0 ^ { M - 1 } 1 , 0 ^ { n - 1 } 1 \\right ) \\\\ & = ( 1 - q ) ^ { - 1 } \\psi \\left ( d _ - L ( 0 ^ { M - 1 } 1 , 0 ^ { n - 1 } 1 \\right ) \\\\ & = ( 1 - q ) ^ { - 1 } p ( 0 ^ { M - 1 } 1 , 0 ^ { N - 1 } ) \\\\ & = p ( 0 ^ M , 0 ^ N ) \\end{align*}"}
+{"id": "4533.png", "formula": "\\begin{align*} \\mathcal { I } ( x , t ) = \\bigcup _ { i = 1 } ^ p [ t _ { i , e n } ( x , t ) , t _ { i , e x } ( x , t ) ] \\end{align*}"}
+{"id": "1767.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\| u _ { \\epsilon } \\| _ { L ^ p ( ( 0 , T ) \\times \\Omega ) } = \\| u _ 0 \\| _ { L ^ p ( ( 0 , T ) \\times \\Omega \\times Y ) } . \\end{align*}"}
+{"id": "1529.png", "formula": "\\begin{align*} A = B \\circ g \\end{align*}"}
+{"id": "7635.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { \\xi } = & ~ [ A x _ t ^ { \\xi } - B ^ 2 R ^ { - 1 } y _ t ^ { \\xi } - B h ( \\mu _ t ) + f ( \\nu _ t ) + b ( \\mu _ t ) ] d t \\\\ & + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ d y _ t ^ { \\xi } = & - [ A y _ t ^ { \\xi } + Q x _ t ^ { \\xi } + Q l ( \\nu _ t ) ] d t + z _ t ^ { \\xi } d W _ t + z ^ { 0 , \\xi } _ t d W ^ 0 _ t , \\\\ x _ 0 ^ { \\xi } = & ~ \\xi , ~ y _ T ^ { \\xi } = G ( x _ T ^ \\xi + g ( \\nu _ T ) ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "6928.png", "formula": "\\begin{align*} \\log \\int _ { \\R ^ n } e ^ { f } \\ , d \\gamma _ t - \\sup _ { y \\in \\R ^ n } \\left ( \\int _ { \\R ^ n } f \\ , d \\gamma _ { y , t } - H ( \\gamma _ { y , t } \\ , | \\ , \\gamma _ t ) \\right ) = \\inf _ { y \\in \\R ^ n } H ( \\gamma _ { y , t } \\ , | \\ , P ) . \\end{align*}"}
+{"id": "2102.png", "formula": "\\begin{align*} \\int _ B \\tilde { p } ( X _ { 1 : n } | \\theta ) d \\tilde { Q } _ k ( \\theta ) = \\sum _ { \\sigma \\in S _ k } \\int _ B p ( X _ { 1 : n } | \\theta ) d Q _ k ( \\theta [ \\sigma ] ) = \\int _ { \\bigcup _ { \\sigma \\in S _ k } B [ \\sigma ] } p ( X _ { 1 : n } | \\theta ) d Q _ k ( \\theta ) \\end{align*}"}
+{"id": "4649.png", "formula": "\\begin{align*} T ^ { ( N ) } : M _ N ( E _ 1 ) \\times \\ldots \\times M _ N ( E _ n ) \\to M _ N ( E ) \\end{align*}"}
+{"id": "2934.png", "formula": "\\begin{align*} \\lim \\sup _ { t \\rightarrow \\infty } t ^ { - 1 / \\eta } L ^ { \\ast } \\left ( t \\right ) = \\infty \\end{align*}"}
+{"id": "7267.png", "formula": "\\begin{align*} L = a _ n x ^ { q ^ n } + a _ { n - 1 } x ^ { q ^ { n - 1 } } + \\cdots + a _ 0 x , \\end{align*}"}
+{"id": "2686.png", "formula": "\\begin{align*} a _ d ( n ) = \\sum _ { k = 2 } ^ { n + 1 } - \\mu _ d ( k ) a _ d ( n + 1 - k ) \\end{align*}"}
+{"id": "9108.png", "formula": "\\begin{align*} 0 \\rightarrow E \\rightarrow \\bigoplus _ { j = 1 } ^ n E _ j \\rightarrow \\mathcal { T } _ E \\rightarrow 0 , \\end{align*}"}
+{"id": "5810.png", "formula": "\\begin{align*} g _ 1 ' : = \\frac { 1 } { 2 } ( 1 + i I ) v ' \\end{align*}"}
+{"id": "1500.png", "formula": "\\begin{align*} ( r ( t ) u ' ) ' + q ( t ) u = 0 , \\end{align*}"}
+{"id": "4080.png", "formula": "\\begin{align*} d \\big ( ( t , x ) , ( s , y ) \\big ) = | t - s | ^ { \\frac { 1 } { 2 } } + | x - y | , \\end{align*}"}
+{"id": "5616.png", "formula": "\\begin{align*} \\frac { 1 - e ^ { a _ { \\alpha + n + 1 } + a _ { \\alpha + n + 2 } } } { \\left ( e ^ { - a _ { \\alpha + 1 } } - 1 \\right ) \\left ( 1 - e ^ { a _ { n + 1 } } \\right ) \\exp \\left [ \\sum _ { j = 2 } ^ n ( a _ j - a _ { \\alpha + j } ) \\right ] } \\end{align*}"}
+{"id": "5481.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( a , b ; c ; z ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( a ) _ k ( b ) _ k } { ( c ) _ k } \\frac { z ^ k } { k ! } . \\end{align*}"}
+{"id": "3955.png", "formula": "\\begin{align*} M _ { I J } = \\begin{pmatrix} G _ { I _ 1 } & G _ { I _ 2 } & \\cdots & G _ { I _ s } \\\\ G _ { J _ 1 } & G _ { J _ 2 } & \\cdots & G _ { J _ s } \\end{pmatrix} . \\end{align*}"}
+{"id": "2377.png", "formula": "\\begin{align*} C _ 2 ( \\mathcal { H } _ { \\ast } ) = \\mathrm { I m } \\ , \\varphi _ 1 \\oplus s _ { _ { 2 } } ( \\mathrm { I m } \\ , \\partial ' _ 0 ) = s _ { _ { 2 } } ( \\mathrm { I m } \\ , \\partial ' _ 0 ) . \\end{align*}"}
+{"id": "6159.png", "formula": "\\begin{align*} \\left ( - \\sqrt { f ( r ) } \\frac { d } { d r } f ( r ) \\frac { d } { d r } \\sqrt { f ( r ) } + V ( r ) - E \\right ) \\psi ( r ) = 0 , \\end{align*}"}
+{"id": "4558.png", "formula": "\\begin{align*} & \\lambda _ 1 f ( v _ { \\ell + 1 , 0 , 0 } ) = ( q ^ 3 + q ^ 2 + q ) f ( v _ { \\ell + 1 , 1 , 0 } ) + f ( v _ { \\ell + 2 , 0 , 0 } ) \\\\ & \\lambda _ 2 f ( v _ { \\ell , 0 , 0 } ) = ( q ^ 4 + q ^ 3 + q ^ 2 ) f ( v _ { \\ell , 1 , 1 } ) + ( q ^ 2 + q + 1 ) f ( v _ { \\ell + 1 , 1 , 0 } ) \\\\ & \\lambda _ 3 f ( v _ { \\ell - 1 , 0 , 0 } ) = q ^ 3 f ( v _ { \\ell - 2 , 0 , 0 } ) + ( q ^ 2 + q + 1 ) f ( v _ { \\ell , 1 , 1 } ) . \\end{align*}"}
+{"id": "7048.png", "formula": "\\begin{align*} A _ { 2 , 1 } \\ , : = \\ , \\big ( - \\tfrac { 1 } { 2 } \\ , F _ { 5 , 1 } + 2 \\ , F _ { 5 , 0 } \\big ) \\ , T _ 1 - \\tfrac { 4 } { 3 } \\ , T _ 2 . \\end{align*}"}
+{"id": "8605.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ n [ 0 , z _ i ] \\right | _ n ^ 2 = \\sum _ { | I | = n } ( \\det \\ U _ I ) ^ 2 . \\end{align*}"}
+{"id": "1474.png", "formula": "\\begin{gather*} L \\leq L _ { n , k + 1 } < L _ { n , k } < L + 1 , \\\\ L _ { n , k } - L _ { n , k + 1 } = \\frac { 1 } { 2 ^ { n - k } } , \\end{gather*}"}
+{"id": "7859.png", "formula": "\\begin{align*} ( I - Q ^ \\dagger ) F ^ { q , \\dagger } _ v ( 0 , p _ 0 ) \\psi ^ \\dagger _ p ( 0 , p _ 0 ) + ( I - Q ^ \\dagger ) F ^ { q , \\dagger } _ p ( 0 , p _ 0 ) = 0 . \\end{align*}"}
+{"id": "5759.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = \\frac { 1 } { 2 } ( \\sqrt { c _ 1 } \\ X _ 1 \\cdot \\nu _ 1 + \\sqrt { c _ 2 } \\ X _ 2 \\cdot \\nu _ 2 ) \\cdot \\varphi - \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi \\end{align*}"}
+{"id": "8586.png", "formula": "\\begin{align*} \\left | P _ { [ u _ 1 , \\dots , u _ r ] ^ \\bot } K \\right | _ { n - r } = \\frac { n ! } { ( n - r ) ! } V ( K [ n - r ] , [ 0 , u _ 1 ] , \\dots , [ 0 , u _ r ] ) . \\end{align*}"}
+{"id": "8334.png", "formula": "\\begin{align*} & P _ 1 = e ^ { \\frac { i } { 2 } \\int _ { - \\infty } ^ x | u _ y ( y , t ) | ^ 2 d y \\widehat { \\sigma } _ { 3 } } ( k P _ x + \\frac { i } { 2 } | u _ x | ^ 2 \\sigma _ { 3 } ) . \\end{align*}"}
+{"id": "4820.png", "formula": "\\begin{align*} s _ i ( A ) = \\min \\limits _ { E : \\ ; \\dim ( E ) = n - i + 1 } \\max \\limits _ { x \\in E , \\ , \\lVert x \\rVert _ 2 = 1 } \\lVert A x \\rVert _ 2 , 1 \\leq i \\leq n , \\end{align*}"}
+{"id": "870.png", "formula": "\\begin{align*} x ' = - y \\end{align*}"}
+{"id": "3039.png", "formula": "\\begin{align*} \\mathcal { I } : G _ { \\kappa } \\rightarrow S O \\left ( M _ { k } \\right ) \\mathcal { I } \\left ( g \\right ) = \\left ( d g \\right ) _ { o } \\end{align*}"}
+{"id": "3136.png", "formula": "\\begin{align*} a \\succ b = \\mathfrak { l } ( T ( a ) ) b \\ \\ \\ \\ a \\prec b = \\mathfrak { r } ( T ( b ) ) a . \\end{align*}"}
+{"id": "4121.png", "formula": "\\begin{align*} H _ { m i n } \\left ( ( E _ 1 [ t ] \\times E _ 1 [ t ] ^ { \\vee } ) \\otimes F \\right ) & = H _ { m i n } ( E _ 1 [ t ] \\times E _ 1 [ t ] ^ { \\vee } ) H _ { m i n } ( F ) \\\\ & = H _ { m i n } ( E _ 1 [ t ] ) H _ { m i n } ( F ) \\\\ & = t H _ { m i n } ( E _ 1 ) H _ { m i n } ( F ) . \\end{align*}"}
+{"id": "6316.png", "formula": "\\begin{align*} z _ 0 ^ { p ^ r } x _ 0 + \\cdots + z _ n ^ { p ^ r } x _ n = 0 . \\end{align*}"}
+{"id": "8025.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ j x _ { i , d } = x _ i , \\ , \\sum _ { i = 1 } ^ r x _ i = t . \\end{align*}"}
+{"id": "8211.png", "formula": "\\begin{align*} x _ { 2 i - 1 , j } = ( - 1 ) ^ { j + 1 } k _ { \\alpha ( i ) , \\rho ( i ) } + \\sum _ { \\ell = 1 } ^ { j - 1 } ( - 1 ) ^ { \\ell + 1 } a _ { j - \\ell } . \\end{align*}"}
+{"id": "2359.png", "formula": "\\begin{align*} \\mathcal { L } ( u ) : = u _ t - A ( x , t ) u _ { x x } + B ( x , t ) u _ x + C ( x , t ) u , \\end{align*}"}
+{"id": "284.png", "formula": "\\begin{align*} v ( p ) = \\sup \\{ u ( p ) : \\} , p \\in \\Omega . \\end{align*}"}
+{"id": "7460.png", "formula": "\\begin{align*} N ( f ) & = \\frac { 1 } { d } \\sum _ { i = 0 } ^ { d - 1 } \\lvert \\det ( I - A ^ i M ) \\rvert \\\\ & = \\frac { 1 } { d } \\sum _ { i = 0 } ^ { d - 1 } \\lvert \\det ( A ^ { - i } ) \\det ( I - A ^ i M ) \\det ( A ^ i ) \\rvert \\\\ & = \\frac { 1 } { d } \\sum _ { i = 0 } ^ { d - 1 } \\lvert \\det ( I - M A ^ i ) \\rvert \\\\ & = \\frac { 1 } { d } \\sum _ { i = 0 } ^ { d - 1 } \\lvert \\det ( I - M ) \\rvert = N ( \\tilde f ) . \\end{align*}"}
+{"id": "5645.png", "formula": "\\begin{align*} E ^ { 2 } _ { p , q } ( n ) & = H _ { p } ( O _ { n , n } , H _ { q } ( C _ { * } ( n ) ) ) \\Rightarrow H _ { p + q } ( O _ { n , n } , C _ { * } ( n ) ) \\\\ E ^ { 1 } _ { p , q } ( n ) & = H _ { q } ( O _ { n , n } , C _ { p } ( n ) ) \\Rightarrow H _ { p + q } ( O _ { n , n } , C _ { * } ( n ) ) . \\end{align*}"}
+{"id": "3775.png", "formula": "\\begin{align*} \\mu ( [ \\gamma , u ] ) = n + C Z ( [ \\gamma , u ] ) \\end{align*}"}
+{"id": "5429.png", "formula": "\\begin{align*} y _ n = \\rho ( x ) - \\rho ( x _ 0 ) \\geq \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( x ' _ 0 ) y _ \\beta + \\kappa | y ' | ^ 2 , \\end{align*}"}
+{"id": "1229.png", "formula": "\\begin{align*} \\sum _ { \\Delta \\Subset \\Z ^ d } \\sum _ { \\xi _ { \\Delta } } \\nabla _ { \\Lambda } \\left ( c _ { \\Delta } ( \\cdot , \\xi _ { \\Delta } ) - \\hat { c } ( \\cdot , \\xi _ { \\Delta } ) \\right ) ( \\eta ) = 0 , \\end{align*}"}
+{"id": "1207.png", "formula": "\\begin{align*} \\lambda _ { 2 } \\left ( A _ { 3 } \\right ) \\leq \\lambda _ { 2 } \\left ( M _ { 3 } \\right ) + \\lambda _ { 3 } \\left ( D _ { 3 } \\right ) = \\lambda _ { 2 } \\left ( M _ { 3 } \\right ) + a _ { 3 } . \\end{align*}"}
+{"id": "5352.png", "formula": "\\begin{align*} \\epsilon ( \\pi ^ * D , y ) = \\epsilon ( D , \\pi ( y ) ) . \\end{align*}"}
+{"id": "1257.png", "formula": "\\begin{align*} h _ { \\Delta } ( \\eta ) : = \\sum _ { \\zeta _ { \\Delta } } \\nabla _ { \\Lambda } ( c _ { \\Delta } ( \\cdot , \\zeta _ { \\Delta } ) - \\hat { c } _ { \\Delta } ( \\cdot , \\zeta _ { \\Delta } ) ) ( \\eta ) , \\eta \\in \\Omega . \\end{align*}"}
+{"id": "8047.png", "formula": "\\begin{align*} \\mathbf { T } = \\begin{bmatrix} 1 & 1 & \\cdots & 1 \\\\ 1 & 0 & \\cdots & 0 \\\\ 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 \\end{bmatrix} . \\end{align*}"}
+{"id": "8476.png", "formula": "\\begin{align*} \\left ( Q _ - J + D _ j \\right ) _ { i 1 } = \\left ( M _ - V _ j J \\right ) _ { i 1 } = \\left ( M _ - R V _ j \\right ) _ { i 1 } = \\left ( M _ - R \\right ) _ { i 1 } , i = 1 , 2 , \\end{align*}"}
+{"id": "9282.png", "formula": "\\begin{align*} \\begin{aligned} C _ m ( U ) & = \\sup \\left \\{ \\int _ { U } ( \\Delta u ) ^ m \\wedge \\beta _ n ^ { n - m } : u \\in Q S H _ m ( \\Omega ) \\cap C ( \\Omega ) , - 1 \\leq u < 0 \\right \\} \\\\ & = \\sup \\left \\{ \\int _ { U } ( \\Delta u ) ^ m \\wedge \\beta _ n ^ { n - m } : u \\in Q S H _ m ( \\Omega ) \\cap C ^ { \\infty } ( \\Omega ) , - 1 \\leq u < 0 \\right \\} \\end{aligned} \\end{align*}"}
+{"id": "9004.png", "formula": "\\begin{align*} \\sum _ { \\ell _ { m } \\leq j \\leq \\ell _ { M } } { ( - 1 ) ^ { p ( \\eta , j ) - 1 } i _ { \\eta \\cup j } } & = U \\left ( - B ( m ) + \\sum _ { q = m + 1 } ^ { M - 1 } \\left ( B ( q - 1 ) - B ( q ) \\right ) + B ( M - 1 ) \\right ) \\\\ & = 0 \\end{align*}"}
+{"id": "1133.png", "formula": "\\begin{align*} \\mathbf { c } _ Y ^ T \\mathbf { c } _ X = \\| \\mathbf { c } _ X \\| _ 2 \\| \\mathbf { c } _ Y \\| _ 2 \\cos \\phi , \\end{align*}"}
+{"id": "1872.png", "formula": "\\begin{align*} \\langle Q ^ M _ x ( v ) , v \\rangle _ M = \\sum ^ m _ { i = 1 } \\bigg ( 2 \\langle B ( v , e _ i ) , B ( v , e _ i ) \\rangle _ { \\mathbb R ^ q } - \\langle B ( v , v ) , B ( e _ i , e _ i ) \\rangle _ { \\mathbb R ^ q } \\bigg ) \\end{align*}"}
+{"id": "4319.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi < - t \\} } | \\frac { f _ 1 + f _ 2 } { 2 } | ^ 2 _ h c ( - \\Psi ) < G ( t ) \\end{align*}"}
+{"id": "1797.png", "formula": "\\begin{align*} L ( f , 2 ) = \\frac { 4 \\pi ^ { 2 } } { 2 7 \\sqrt { 3 } } \\int _ { 0 } ^ { \\infty } c ^ { 2 } ( e ^ { - \\frac { 2 \\pi } { 3 u } } ) b ( e ^ { - \\frac { 2 \\pi } { 9 u } } ) \\frac { d u } { u ^ { 2 } } . \\end{align*}"}
+{"id": "2014.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d y _ t = & f ( t , y _ t , z _ t , u _ t , \\mathcal L ( y _ t , z _ t , u _ t ) ) d t + g ( t , y _ t , z _ t , u _ t , \\mathcal L ( y _ t , z _ t , u _ t ) ) d \\overleftarrow B _ t \\\\ & - z _ t d W _ t , \\ , \\ , \\ , \\ , t \\in [ 0 , T ] , \\\\ y _ T = & \\xi . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3746.png", "formula": "\\begin{align*} L ( 0 0 0 1 1 0 , 0 1 1 0 ) & = t ^ { - 4 } d _ - d _ = d _ + d _ + d _ + ( 1 ) + t ^ { - 4 } q d _ = d _ = d _ + d _ + ( 1 ) \\\\ & = - t ^ { - 2 } y _ 1 s _ 1 + t ^ { - 3 } q y _ 1 y _ 2 \\in V _ 2 . \\end{align*}"}
+{"id": "4494.png", "formula": "\\begin{align*} 1 + \\sum _ { v ^ * } \\ ! \\Bigg ( \\prod _ { s \\ge 1 } \\frac { q ^ { \\langle v ^ s \\ ! , \\ , v ^ s \\rangle } } { q ^ { \\langle \\beta ^ s \\ ! , \\ , \\beta ^ s \\rangle } } \\frac { c ( v ^ s , q ) } { \\mathit { g l } ( v ^ s , q ) } X ^ { s v ^ s } \\Bigg ) = \\ , \\mathrm { E x p } \\Bigg ( \\sum _ { v \\in \\mathbb { N } ^ n \\backslash \\{ 0 \\} } \\ ! \\frac { a ( v , q ) } { q - 1 } X ^ v \\Bigg ) , \\end{align*}"}
+{"id": "1231.png", "formula": "\\begin{align*} \\sum _ { \\Delta \\Subset \\Z ^ d } \\sum _ { \\xi _ { \\Delta } } \\nabla _ { \\Lambda } \\left ( c _ { \\Delta } ( \\cdot , \\xi _ { \\Delta } ) - \\hat { c } ( \\cdot , \\xi _ { \\Delta } ) \\right ) ( \\eta ) = 0 . \\end{align*}"}
+{"id": "6075.png", "formula": "\\begin{align*} 2 m + n = \\sum _ { x \\in G } { \\frac { | G | } { | x ^ G | } } = | G | \\cdot \\sum _ { x \\in G } { \\frac { 1 } { | x ^ G | } } = n \\cdot k . \\end{align*}"}
+{"id": "4566.png", "formula": "\\begin{align*} & A _ { 1 , 0 , 0 } + A _ { 2 , 0 , 0 } + A _ { 3 , 0 , 0 } + A _ { 4 , 0 , 0 } = 1 \\\\ & z _ 1 A _ { 1 , 0 , 0 } + z _ 2 A _ { 2 , 0 , 0 } + z _ 3 A _ { 3 , 0 , 0 } + z _ 4 A _ { 4 , 0 , 0 } = a _ { 1 , 0 , 0 } / \\sqrt { q } \\\\ & z _ 1 ^ 2 A _ { 1 , 0 , 0 } + z _ 2 ^ 2 A _ { 2 , 0 , 0 } + z _ 3 ^ 2 A _ { 3 , 0 , 0 } + z _ 4 ^ 2 A _ { 4 , 0 , 0 } = a _ { 2 , 0 , 0 } / { q } \\\\ & z _ 1 ^ 3 A _ { 1 , 0 , 0 } + z _ 2 ^ 3 A _ { 2 , 0 , 0 } + z _ 3 ^ 3 A _ { 3 , 0 , 0 } + z _ 4 ^ 3 A _ { 4 , 0 , 0 } = a _ { 3 , 0 , 0 } / q \\sqrt { q } , \\end{align*}"}
+{"id": "671.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | e ^ { i k \\phi } \\cos ( n \\theta ) \\left | l _ 2 , m _ 2 \\right \\rangle = ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
+{"id": "7416.png", "formula": "\\begin{align*} _ { \\rm r e f } [ V _ \\lambda ] = q ^ { - \\frac { c } { 2 4 } } q ^ { \\frac { ( \\lambda | \\lambda + 2 \\rho ) } { 2 ( k + h ^ \\vee ) } - \\lambda ( x + y ) } \\ \\prod _ { i = 1 } ^ L X _ i \\ \\prod _ { 1 \\leq i < j \\leq L } X _ { i , j } \\end{align*}"}
+{"id": "5571.png", "formula": "\\begin{align*} A ( y _ { l + 1 } . . . y _ 1 x ) - A ( y _ { l + 1 } . . . y _ 1 x ' ) = A ( 1 0 ^ l 1 ^ \\infty ) - A ( 1 0 ^ \\infty ) = d _ l - d \\ , \\end{align*}"}
+{"id": "7659.png", "formula": "\\begin{align*} \\begin{aligned} & V ( t _ 0 , x _ 0 , \\nu _ 0 ) \\\\ : = & \\frac { 1 } { 2 } \\mathbb { E } \\Big \\{ \\int _ { t _ 0 } ^ T \\Big [ Q \\left ( x _ t ^ { * , t _ 0 , x _ 0 , \\xi } + l ( \\nu _ t ^ { * , t _ 0 , \\xi } ) \\right ) ^ 2 + R \\left ( \\alpha _ t ^ { * , t _ 0 , x _ 0 , \\xi } + h ( \\mu _ t ^ { * , t _ 0 , \\xi } ) \\right ) ^ 2 \\Big ] d t + G \\left ( x _ T ^ { * , t _ 0 , x _ 0 , \\xi } + g ( \\mu _ T ^ { * , t _ 0 , \\xi } ) \\right ) ^ 2 \\Big \\} \\end{aligned} \\end{align*}"}
+{"id": "9200.png", "formula": "\\begin{align*} \\left | \\dfrac { \\partial a _ { 0 , \\delta } ( x ) } { \\partial x } \\right | _ { x = x _ a } \\le 2 \\int _ 0 ^ 1 \\left | \\left . \\dfrac { \\partial h ( x ) } { \\partial x } \\right | _ { x = x _ a + \\delta u ( t ) } \\right | \\ , d t \\le 2 L _ r . \\end{align*}"}
+{"id": "3275.png", "formula": "\\begin{align*} \\sum _ { k \\geq N } \\langle X _ n e _ k , e _ k \\rangle = \\sum _ { k \\geq N } \\| X _ n ^ { \\frac 1 2 } e _ k \\| ^ 2 \\geq \\sum _ { k , l \\geq N } \\langle X _ n ^ { \\frac 1 2 } e _ k , e _ l \\rangle ^ 2 = \\sum _ { k , l \\geq N } \\langle X _ n ^ { \\frac 1 2 } , e _ k \\otimes e _ l \\rangle _ { \\mathcal H } ^ 2 . \\end{align*}"}
+{"id": "3931.png", "formula": "\\begin{align*} \\sigma ( t ) = \\max ( t , 0 ) , t \\in \\mathbb { R } . \\end{align*}"}
+{"id": "8068.png", "formula": "\\begin{align*} e _ c \\left [ t + 1 \\right ] = & \\left \\{ 1 - 4 \\mu \\sum _ { j = 1 } ^ { M } \\lvert \\phi ^ { \\left ( j , c \\right ) } \\rvert ^ 2 - 2 \\mu f ^ { \\left ( c \\right ) } \\vphantom { \\sum _ { j = 1 } ^ { M } } \\right \\} e _ c \\left [ t \\right ] - 2 \\mu \\left ( 2 \\sum _ { j = 1 } ^ { M } \\lvert \\phi ^ { \\left ( j , c \\right ) } \\rvert ^ 2 + \\mu f ^ { \\left ( c \\right ) } \\right ) a _ c ^ { \\left ( o \\right ) } \\\\ & - 2 \\mu \\sum _ { l } ^ { M } \\Re \\left \\{ \\phi ^ { \\left ( l , c \\right ) } \\right \\} . \\end{align*}"}
+{"id": "7407.png", "formula": "\\begin{align*} [ V ^ W ] = [ V \\otimes F ' ] \\end{align*}"}
+{"id": "215.png", "formula": "\\begin{align*} g = \\widetilde { g } \\ , t ^ { - \\sigma ( l ) } \\widetilde { g } = \\prod _ { k = 1 } ^ l \\widetilde { s _ { \\iota ( k ) } } ^ { \\ , t ^ { \\sigma ( k - 1 ) } } , \\end{align*}"}
+{"id": "5794.png", "formula": "\\begin{align*} [ T ] = \\frac { 1 } { \\mu } J ( \\partial _ x h + I \\partial _ y h ) = \\frac { 2 } { \\mu } h _ z J \\end{align*}"}
+{"id": "7862.png", "formula": "\\begin{align*} \\Phi ^ \\dagger _ { v p } ( 0 , p _ 0 ) [ u _ \\ell , p _ 1 ] & = Q ^ \\dagger F ^ { q , \\dagger } _ { v p } ( 0 , p _ 0 ) [ u _ \\ell , p _ 1 ] \\\\ & = D _ p c _ 1 ( q , k , p ) \\Big | _ { p = p _ 0 } p _ 1 u _ \\ell . \\end{align*}"}
+{"id": "14.png", "formula": "\\begin{align*} \\gamma _ 1 + \\gamma _ 2 = \\beta _ 1 + \\beta _ 2 \\gamma _ 1 \\leq \\beta _ 1 , \\beta _ 2 \\leq \\gamma _ 2 \\end{align*}"}
+{"id": "724.png", "formula": "\\begin{align*} R _ { K } = \\log _ { 2 } \\bigg ( 1 + \\frac { P _ { K } { \\lvert \\mathbf { d } _ { K } ^ { H } \\mathbf { h } _ { K } \\rvert } ^ { 2 } } { \\sum _ { \\pi _ { k ' , i ' } > \\pi _ { K } } P _ { k ' , i ' } { \\lvert \\mathbf { d } _ { k ' , i ' } ^ { H } \\mathbf { h } _ { k ' } \\rvert } ^ { 2 } + \\sigma _ { \\textrm { u l } } ^ { 2 } } \\bigg ) . \\end{align*}"}
+{"id": "572.png", "formula": "\\begin{align*} \\omega ^ { \\prime } = p - \\omega . \\end{align*}"}
+{"id": "5773.png", "formula": "\\begin{align*} \\nabla _ X \\psi _ 1 = \\frac { 1 } { 2 } X _ 1 \\cdot \\psi _ 1 - \\frac { 1 } { 2 } S ( X ) \\cdot N \\cdot \\psi _ 1 \\end{align*}"}
+{"id": "273.png", "formula": "\\begin{align*} T _ { e u c l } ( w ) = \\inf \\left \\{ \\max \\{ f ( p ) , f ( q ) \\} : w \\in [ p , q ] , p \\in \\Omega , q \\in \\Omega \\right \\} , w \\in \\R ^ 3 . \\end{align*}"}
+{"id": "5830.png", "formula": "\\begin{align*} { \\rm O } ( \\epsilon ^ 0 ) : { { { \\bf { m } } } } ^ { ( 0 ) } = { { { \\bf { m } } } } ^ { ( e q ) } , \\end{align*}"}
+{"id": "7384.png", "formula": "\\begin{align*} v ^ { { \\rm c o l } ( \\mu ) } \\prod _ { ( i , j ) \\in \\mu } X _ { a - j + 1 } = \\prod _ { i = 1 } ^ { { \\rm c o l } ( \\mu ) } \\bigg ( v \\prod _ { j = 1 } ^ { \\mu ' _ i } X _ { a - j + 1 } \\bigg ) , \\end{align*}"}
+{"id": "9250.png", "formula": "\\begin{align*} \\beta _ n = \\sum \\limits _ { l = 0 } ^ { n - 1 } \\omega ^ l \\wedge \\omega ^ { n + l } , \\beta _ n ^ n = \\wedge ^ n \\beta _ n = n ! ~ \\Omega _ { 2 n } , \\end{align*}"}
+{"id": "6401.png", "formula": "\\begin{align*} ( ( \\C ^ * ) ^ n , \\omega _ { \\mathrm { l o g } } ) , \\omega _ { \\mathrm { l o g } } = \\i \\sum \\frac { d z ^ i } { z _ i } \\wedge \\frac { d \\bar { z } ^ i } { \\bar { z _ i } } , \\end{align*}"}
+{"id": "4574.png", "formula": "\\begin{align*} f ( v _ { \\ell , m , n } ) = \\sqrt { q } ^ { 3 \\ell + m - n } \\sum _ { \\substack { ( i , j ) \\in \\{ 3 , 4 \\} ^ 2 \\\\ i \\neq j } } & \\{ D _ { 1 1 i } z _ 1 ^ { \\ell + m } z _ i ^ n + D _ { 1 i 1 } z _ 1 ^ { \\ell + n } z _ i ^ m + D _ { 1 i j } z _ 1 ^ \\ell z _ i ^ m z _ j ^ n + D _ { i 1 1 } z _ i ^ \\ell z _ 1 ^ { m + n } \\\\ & + D _ { i 1 j } z _ i ^ \\ell z _ 1 ^ m z _ j ^ n + D _ { i j 1 } z _ i ^ \\ell z _ j ^ m z _ 1 ^ n \\} , \\\\ \\end{align*}"}
+{"id": "6675.png", "formula": "\\begin{align*} \\begin{aligned} & P \\Big ( \\sup _ { t \\in [ \\iota _ 1 , \\tau ] } \\lambda ( \\epsilon ) \\epsilon \\int ^ { t } _ { \\iota _ 1 } \\gamma ( s ) \\tilde { N } ^ { \\epsilon ^ { - 2 } } ( d s ) > a \\Big ) \\\\ & \\leq \\exp \\left \\{ - \\lambda ^ { - 2 } ( \\epsilon ) \\Big ( \\eta a - \\frac { 1 } { 2 } \\eta ^ 2 \\rho ^ 2 ( \\iota _ 2 - \\iota _ 1 ) e ^ { \\epsilon \\lambda ^ { - 1 } ( \\epsilon ) \\eta \\rho } \\Big ) \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "5041.png", "formula": "\\begin{align*} B \\cdot n = 0 . \\end{align*}"}
+{"id": "8088.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ y _ i ^ * y _ i \\right ] = a _ c ^ 2 \\lvert \\mathbf { h } _ { i , * } \\mathbf { p } _ c \\rvert ^ 2 + \\sum _ { l = 1 } ^ M a _ l ^ 2 \\lvert \\mathbf { h } _ { i , * } \\mathbf { p } _ l \\rvert ^ 2 + \\sigma _ n ^ 2 . \\end{align*}"}
+{"id": "7406.png", "formula": "\\begin{align*} [ V _ \\lambda ] = q ^ { - \\frac { c } { 2 4 } } q ^ { \\frac { ( \\lambda | \\lambda + 2 \\rho ) } { 2 ( k + h ^ \\vee ) } - \\lambda ( x + y ) } \\ \\prod _ { i = 1 } ^ L X _ i \\ \\prod _ { 1 \\leq i < j \\leq L } X _ { i , j } \\ \\Big \\vert _ { y = 1 } \\end{align*}"}
+{"id": "7858.png", "formula": "\\begin{align*} ( I - Q ^ \\dagger ) F ^ { q , \\dagger } _ v ( v + \\psi ^ \\dagger ( v , p ) , p ) \\psi _ p ( v , p ) + ( I - Q ^ \\dagger ) F ^ { q , \\dagger } _ p ( v + \\psi ^ \\dagger ( v , p ) , p ) = 0 . \\end{align*}"}
+{"id": "8104.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\triangle u & = | x | ^ a v ^ p & & \\quad \\mbox { i n } \\ B _ { R } , \\\\ \\triangle v & = | x | ^ { b } v ^ q | \\nabla u | ^ { s } & & \\quad \\mbox { i n } \\ B _ { R } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2105.png", "formula": "\\begin{align*} h _ c ( G ) - h _ c ( G \\backslash v ) & = f _ c ( b ) - f _ c ( 0 ) + \\sum _ { i = 1 } ^ b \\Delta _ c ( d _ i ) \\\\ & \\le f _ c ( b ) - f _ c ( 0 ) + b \\cdot \\Delta _ c \\left ( \\frac { m + \\binom { b } { 2 } } { b } \\right ) \\\\ & : = L H S ( m , b , c ) . \\end{align*}"}
+{"id": "8421.png", "formula": "\\begin{align*} z m ( x ; z ) e _ 2 - ( I - F ) \\left ( \\widehat { \\Psi } ^ - _ { 1 1 } e _ 1 + \\widehat { \\Psi } ^ - _ { 2 1 } e _ 2 \\right ) = \\widetilde { m } ( x ; z ) e _ 2 , \\end{align*}"}
+{"id": "4330.png", "formula": "\\begin{align*} & \\epsilon \\int _ { \\{ \\Psi < - t _ 2 \\} } | \\tilde { F } _ { t _ 1 } - F _ { t _ 0 } | ^ 2 _ h e ^ { - \\Psi } \\\\ \\le & \\int _ { \\{ \\Psi < - t _ 2 \\} } | \\tilde { F } _ { t _ 1 } - F _ { t _ 0 } | ^ 2 _ h ( e ^ { - \\Psi - t _ 0 } c ( t _ 0 ) - c ( - \\Psi ) ) \\\\ \\le & \\int _ { \\{ \\Psi < - t _ 0 \\} } | \\tilde { F } _ { t _ 1 } - F _ { t _ 0 } | ^ 2 _ h ( e ^ { - \\Psi - t _ 0 } c ( t _ 0 ) - c ( - \\Psi ) ) \\\\ = & 0 . \\end{align*}"}
+{"id": "987.png", "formula": "\\begin{align*} | H \\cap W ' | q ^ { d ^ 2 - d } \\theta _ { d - 1 } \\le \\begin{cases} | W ' \\cap \\tau _ i | q ^ { d ^ 2 - d } \\theta _ { d - 1 } & H \\cap U = \\tau _ i i \\in \\{ 1 , 2 \\} , \\\\ ( q ^ { d - 1 } + \\theta _ { d - 1 } ) q ^ { d ^ 2 - d } \\theta _ { d - 1 } & \\end{cases} \\end{align*}"}
+{"id": "8607.png", "formula": "\\begin{align*} \\frac { \\sum _ { | I | = n } ( \\det \\ U _ I ) ^ 2 } { \\sum _ { | I | = n - 1 } ( \\det \\ U ' _ I ) ^ 2 } \\ge \\frac { \\sum _ { | I | = n } ( \\det \\ V _ I ) ^ 2 } { \\sum _ { | I | = n - 1 } ( \\det \\ V ' _ I ) ^ 2 } + \\frac { \\sum _ { | I | = n } ( \\det \\ W _ I ) ^ 2 } { \\sum _ { | I | = n - 1 } ( \\det \\ W ' _ I ) ^ 2 } . \\end{align*}"}
+{"id": "1903.png", "formula": "\\begin{align*} Q ( n ) = \\mathcal { K } m ^ { n } \\bigl ( { 1 + o ( 1 ) } \\bigr ) n \\to \\infty , \\end{align*}"}
+{"id": "4987.png", "formula": "\\begin{align*} \\mathcal { W } _ { f _ { 1 , \\infty } } ( n , \\alpha , \\epsilon , g , \\psi ) : = \\inf \\bigg \\{ \\sum _ { i } c _ { i } ^ { - \\alpha n _ { i } + S _ { 1 , n _ { i } } \\psi ( x _ { i } , \\epsilon ) } \\bigg \\} , \\end{align*}"}
+{"id": "9281.png", "formula": "\\begin{align*} C _ m ( K ) = \\sup \\left \\{ \\int _ { K } \\Delta u _ 1 \\wedge \\cdots \\wedge \\Delta u _ m \\wedge \\beta _ n ^ { n - m } ; u _ j \\in Q S H _ m ( \\Omega ) \\cap C ( \\Omega ) , - 1 \\leq u _ j < 0 \\right \\} . \\end{align*}"}
+{"id": "6698.png", "formula": "\\begin{align*} \\beta _ i ( \\gamma , x ) + \\beta _ i ( \\gamma , y ) = f _ i ( \\gamma x , \\gamma y ) - f _ i ( x , y ) ( \\gamma \\in \\Gamma , \\ x \\ne y \\in X , \\ i = 1 , 2 ) . \\end{align*}"}
+{"id": "6480.png", "formula": "\\begin{align*} \\psi _ { \\tau } ( p ) \\stackrel { d e f } { = } \\left [ \\ p \\int _ 0 ^ { \\infty } \\ t ^ { p - 1 } \\ T _ { \\tau } ( t ) \\ d t \\ \\right ] ^ { 1 / p } = | | \\tau | | L _ p ( \\Omega ) , \\end{align*}"}
+{"id": "4384.png", "formula": "\\begin{align*} & \\int _ { X _ j } | F _ { \\epsilon , j } - ( 1 - v ' _ { \\epsilon } ( \\Psi ) ) f F ^ { 1 + \\delta } | ^ 2 _ h e ^ { v _ \\epsilon ( \\Psi ) - \\delta \\max \\{ \\psi + T , 2 \\log | F | \\} } c ( - v _ \\epsilon ( \\Psi ) ) \\\\ \\le & \\int _ { X _ j } v '' _ { \\epsilon } ( \\Psi ) | f F | ^ 2 _ h e ^ { - u ( - v _ { \\epsilon } ( \\Psi ) ) } < + \\infty . \\end{align*}"}
+{"id": "1157.png", "formula": "\\begin{align*} \\left ( \\int \\varphi d | \\mu - \\nu | \\right ) ^ 2 \\leq 4 C H \\left ( \\mu \\mid \\nu \\right ) , C = \\frac { 1 } { 6 } \\int \\varphi ^ 2 d \\mu + \\frac { 1 } { 3 } \\int \\varphi ^ 2 d \\nu . \\end{align*}"}
+{"id": "8154.png", "formula": "\\begin{align*} \\varprojlim _ { n \\in \\Delta } \\mathcal F ( Y _ n ) = \\varprojlim _ { n \\in \\Delta } \\varprojlim _ { k \\in \\Delta } \\mathcal F ( \\tilde Y _ { n , k } ) = \\varprojlim _ { k \\in \\Delta } \\left ( \\varprojlim _ { n \\in \\Delta } \\mathcal F ( \\tilde Y _ { n , k } ) \\right ) . \\end{align*}"}
+{"id": "3127.png", "formula": "\\begin{align*} [ x , y ] = x \\cdot y - y \\cdot x , \\end{align*}"}
+{"id": "7555.png", "formula": "\\begin{align*} \\overline { p ( z ) } = \\sum _ { n = 0 } ^ k \\overline { a } _ n \\overline { z } ^ n = \\sum _ { n = 0 } ^ k \\overline { a } _ n z ^ { - n } = z ^ { - k } \\sum _ { n = 0 } ^ \\infty \\overline { a } _ n z ^ { k - n } = z ^ { - k } q ( z ) . \\end{align*}"}
+{"id": "7089.png", "formula": "\\begin{align*} \\int _ { B _ R } F ( x , D \\tilde { u } ) d x = \\lim _ { j _ 0 \\to + \\infty } \\int _ { B _ R } F _ { j _ 0 } ( x , D \\tilde { u } ) d x \\leq \\int _ { B _ R } F ( x , D u ) d x , \\end{align*}"}
+{"id": "4035.png", "formula": "\\begin{align*} K _ { V , I } = \\sum _ { J \\subset I } \\kappa _ { V , J } . \\end{align*}"}
+{"id": "863.png", "formula": "\\begin{align*} x = x ' \\ , , y = y ' \\ , . \\end{align*}"}
+{"id": "4040.png", "formula": "\\begin{align*} X _ t = X _ 0 + \\int _ 0 ^ t V ^ { [ 2 ] } ( X _ s ) d s + \\int _ 0 ^ t V ^ { [ 1 ] } ( X _ s ) d B _ s \\end{align*}"}
+{"id": "3893.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m \\sum _ { h = 1 } ^ 2 C _ { j , h } \\int _ { \\Omega } \\frac { \\partial } { \\partial z _ { j , h } } ( - \\delta ^ 2 ( K ( z _ j ) \\nabla V _ { \\delta , Z , j } ) ) \\frac { \\partial V _ { \\delta , Z , i } } { \\partial z _ { i , \\hbar } } = \\int _ { \\Omega } u \\frac { \\partial V _ { \\delta , Z , i } } { \\partial z _ { i , \\hbar } } , \\ \\ \\forall i = 1 , \\cdots , m , \\ \\hbar = 1 , 2 . \\end{align*}"}
+{"id": "6790.png", "formula": "\\begin{align*} E ( z ) = \\frac { z - 1 } { 2 ( z + 1 ) } - \\frac { 1 } { 2 } \\int _ { 0 } ^ { \\infty } q ( t ) d t + ( z + 1 ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } z ^ { n } d _ { n } ( 0 ) \\end{align*}"}
+{"id": "3738.png", "formula": "\\begin{align*} { \\bf h } | _ { \\partial \\bf N } & = \\pm 2 \\bar u v d t ^ 2 + h ^ \\intercal \\\\ A ' | _ { \\bar { \\bf g } } ( \\bf h ) & = \\pm \\big ( v \\nu ( \\bar u ) + \\bar u ( \\nu ( u ) ) ' \\big ) d t ^ 2 + A ' | _ g ( h ) . \\end{align*}"}
+{"id": "6873.png", "formula": "\\begin{align*} \\hat z = \\Psi _ 2 ( \\tilde z ) = \\hat z _ 3 + \\frac { ( \\hat z _ 3 - \\hat z _ 1 ) ( \\hat z _ 2 - \\hat z _ 3 ) ( \\tilde z _ 2 - \\tilde z _ 1 ) ( \\tilde z - \\tilde z _ 3 ) } { ( \\hat z _ 2 - \\hat z _ 1 ) ( \\tilde z _ 2 - \\tilde z _ 3 ) ( \\tilde z - \\tilde z _ 1 ) - ( \\hat z _ 2 - \\hat z _ 3 ) ( \\tilde z _ 2 - \\tilde z _ 1 ) ( \\tilde z - \\tilde z _ 3 ) } \\ , , \\end{align*}"}
+{"id": "7008.png", "formula": "\\begin{align*} \\aligned r & \\ , : = \\ , a _ { 1 , 1 } \\ , x + a _ { 1 , 2 } \\ , y + b _ 1 \\ , u , \\\\ s & \\ , : = \\ , a _ { 2 , 1 } \\ , x + a _ { 2 , 2 } \\ , y + b _ 2 \\ , u , \\\\ v & \\ , : = \\ , c _ 1 \\ , x + c _ 2 \\ , y + d \\ , u , \\endaligned \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0 \\ , \\neq \\ , \\left \\vert \\ ! \\begin{array} { c c c } a _ { 1 , 1 } & a _ { 1 , 2 } & b _ 1 \\\\ a _ { 2 , 1 } & b _ { 2 , 2 } & b _ 2 \\\\ c _ 1 & c _ 2 & d \\end{array} \\ ! \\right \\vert . \\end{align*}"}
+{"id": "6639.png", "formula": "\\begin{align*} \\widetilde { I } _ { t _ { 1 } , \\cdots , t _ { k } } ( x _ { 1 } , \\cdots , x _ { k } ) = \\sup _ { \\delta > 0 } \\liminf _ { m \\to \\infty } \\inf _ { | y - x | \\le \\delta } I _ { m , t _ { 1 } , \\cdots , t _ { k } } ( y _ { 1 } , \\cdots , y _ { k } ) , ~ ~ x = ( x _ { 1 } , \\cdots , x _ { k } ) . \\end{align*}"}
+{"id": "5507.png", "formula": "\\begin{align*} f ( x ) = \\log \\Gamma ( x ) - \\log \\Gamma \\left ( x + \\frac 1 2 \\right ) - \\log \\Gamma \\left ( x + \\frac 3 2 \\right ) \\end{align*}"}
+{"id": "3742.png", "formula": "\\begin{align*} - \\Gamma ^ k + \\bar u ^ { - 1 } \\bar g ^ { i k } \\frac { \\partial \\bar u } { \\partial x _ i } = 0 . \\end{align*}"}
+{"id": "6103.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { 2 } \\Big ( \\psi ^ { + 0 } _ { n , p } \\psi ^ { + 0 } _ { n - 1 , p } \\rho ^ { 0 + } _ { n - 2 , p } - 2 \\psi ^ { + 0 } _ { n , p } \\rho ^ { 0 + } _ { n - 1 , p } \\psi ^ { + 0 } _ { n - 1 , p - 1 } + \\rho ^ { 0 + } _ { n , p } \\psi ^ { + 0 } _ { n , p - 1 } \\psi ^ { + 0 } _ { n - 1 , p - 1 } \\Big ) & = & - \\varphi ^ { + + } _ { n , p } \\psi ^ { + 0 } _ { n - 1 , p - 1 } \\\\ & & = & - \\psi ^ { + 0 } _ { n , p } \\varphi ^ { + + } _ { n - 1 , p } \\ , . \\end{aligned} \\end{align*}"}
+{"id": "2629.png", "formula": "\\begin{align*} & \\circlearrowleft _ { x , y , z } \\varepsilon ( z , x ) \\Big ( \\varepsilon ( h , x + y ) [ [ \\alpha ( x ) , \\alpha ( y ) ] , \\alpha ( h \\cdot z ) ] + \\varepsilon ( x + y , z ) [ [ h \\cdot z , \\alpha ( x ) ] , \\alpha ^ 2 ( y ) ] \\\\ & + \\varepsilon ( x , y + z ) \\big ( 2 [ \\alpha ( h ) \\cdot [ y , z ] , \\alpha ^ 2 ( x ) ] - \\varepsilon ( x , y + z ) [ [ h \\cdot y , \\alpha ( z ) ] , \\alpha ^ 2 ( x ) ] \\big ) \\Big ) = 0 . \\end{align*}"}
+{"id": "3926.png", "formula": "\\begin{align*} W _ { 1 } = \\left \\langle \\left [ \\theta _ { 1 } \\right ] , \\left [ \\theta _ { 2 } \\right ] , \\dots , \\left [ \\theta _ { s } \\right ] \\right \\rangle , W _ { 2 } = \\left \\langle \\left [ \\vartheta _ { 1 } \\right ] , \\left [ \\vartheta _ { 2 } \\right ] , \\dots , \\left [ \\vartheta _ { s } \\right ] \\right \\rangle \\in G _ { s } \\left ( { \\rm H ^ { 2 } } \\left ( { \\bf A } , \\mathbb C \\right ) \\right ) , \\end{align*}"}
+{"id": "6344.png", "formula": "\\begin{align*} ( 1 - z ) ^ { b - 1 } = \\sum ^ { \\infty } _ { n = 0 } \\frac { ( - 1 ) ^ { n } \\ , \\Gamma ( b ) } { \\Gamma ( b - n ) \\ , n ! } z ^ n , \\end{align*}"}
+{"id": "2044.png", "formula": "\\begin{align*} [ M , \\ , \\ , \\partial _ t + v \\ , \\cdot \\ , \\partial _ x ] = \\partial _ { v _ 1 } ^ 2 , \\end{align*}"}
+{"id": "3270.png", "formula": "\\begin{align*} \\sup _ { n \\in \\mathbb N } \\mathbb P \\left [ \\sum _ { l \\geq N _ k ^ { \\epsilon } } \\langle Y _ n , e _ l \\rangle ^ 2 > \\frac 1 { l _ k } \\right ] \\leq \\epsilon \\frac 1 { l _ k ^ 2 \\sum _ { j = 1 } ^ { \\infty } \\frac 1 { l _ j ^ 2 } } . \\end{align*}"}
+{"id": "8023.png", "formula": "\\begin{align*} v ( u ) \\leq d _ u ( \\pi _ N ( x ) + L ) + ( i _ u - d _ u ) L = d _ u ( \\pi _ N ( x ) ) + i _ u L < 2 i _ u \\pi _ N ( x ) , L < \\pi _ N ( x ) . \\end{align*}"}
+{"id": "1776.png", "formula": "\\begin{align*} \\widetilde { v } _ 2 = \\frac { a _ { 2 , 1 } + a _ { 2 , 2 } v _ 2 + v _ 1 \\Phi _ 2 } { a _ { 1 , 1 } + a _ { 1 , 2 } v _ 2 + v _ 1 \\Phi _ 1 } \\end{align*}"}
+{"id": "8802.png", "formula": "\\begin{align*} \\tilde { Y } _ t = e ^ { | z | J ^ \\perp t } \\tilde { Y } _ 0 + \\int _ 0 ^ t e ^ { | z | J ^ \\perp ( t - s ) } P ^ { - 1 } \\Pi _ { \\mathrm { k e r } A ^ \\perp } \\sigma d W _ s . \\end{align*}"}
+{"id": "6715.png", "formula": "\\begin{align*} c ^ 2 _ 1 [ \\frac { b c ( a + b - 1 ) } { ( 1 - a ) ^ 2 ( c + d - 1 ) } - \\frac { c + d - 1 } { 1 - d } ] = 1 . \\end{align*}"}
+{"id": "6516.png", "formula": "\\begin{align*} x _ { \\lambda } x _ { \\mu } x _ { \\nu } = ( x _ { \\lambda } x _ { \\mu } ) x _ { \\nu } ( \\lambda \\mu ) \\nu \\end{align*}"}
+{"id": "7489.png", "formula": "\\begin{align*} & | \\nabla u | _ { A ( 0 ) } ^ 2 \\Big ( ( n - 2 ) \\eta ^ 2 + x \\cdot \\nabla ( \\eta ^ 2 ) \\Big ) \\\\ & = ( n - 2 - a ) | x | _ { A ^ { - 1 } ( 0 ) } ^ { - a } | \\nabla u | _ { A ( 0 ) } ^ 2 \\zeta ^ 2 + | x | _ { A ^ { - 1 } ( 0 ) } ^ { - a } | \\nabla u | _ { A ( 0 ) } ^ 2 ( x \\cdot \\nabla ( \\zeta ^ 2 ) ) . \\end{align*}"}
+{"id": "496.png", "formula": "\\begin{align*} e ^ { \\gamma x } f _ r \\ast f _ G ( x ) = \\int _ { - \\infty } ^ { x / 2 } \\big \\{ e ^ { \\gamma ( x - y ) } f _ r ( x - y ) e ^ { \\gamma y } f _ G ( y ) + e ^ { \\gamma ( x - y ) } f _ G ( x - y ) e ^ { \\gamma y } f _ r ( y ) \\big \\} d y < \\infty \\end{align*}"}
+{"id": "7393.png", "formula": "\\begin{align*} [ V _ f ] = \\prod _ { i = 0 } ^ r e ^ { f ( x _ i ) \\omega _ i } \\frac { 1 } { D } . \\end{align*}"}
+{"id": "2171.png", "formula": "\\begin{align*} \\gamma ( { \\bf { v } } _ n , { \\bf { u } } _ k , \\psi _ n ) = 1 5 \\left ( | c o s ( \\rho _ { n , k } ) | + 2 | s i n ( \\varphi _ { n , k } + \\psi _ n ) | \\right ) , \\end{align*}"}
+{"id": "6730.png", "formula": "\\begin{align*} J _ { i , k } ^ { ( s ) } = \\int _ 0 ^ \\infty z ^ i \\ , \\phi _ s ( z ) \\ , \\Phi _ s ( z ) ^ k \\mathrm { d } z . \\end{align*}"}
+{"id": "7177.png", "formula": "\\begin{align*} q _ 1 = \\begin{pmatrix} | \\xi | I _ { n - 1 } + \\frac { s _ { 1 } } { | \\xi | } ( g ^ { \\alpha \\beta } \\xi _ \\alpha \\xi _ \\gamma ) & i s _ { 1 } ( g ^ { \\alpha \\beta } \\xi _ \\beta ) & 0 \\\\ [ 1 m m ] i s _ { 1 } ( \\xi _ \\beta ) & ( 1 - s _ { 1 } ) | \\xi | & 0 \\\\ [ 1 m m ] 0 & 0 & | \\xi | \\end{pmatrix} , \\end{align*}"}
+{"id": "4321.png", "formula": "\\begin{align*} \\mathfrak { R } \\int _ { \\{ \\Psi < - t \\} } \\langle F _ t , \\bar { q } \\rangle _ h c ( - \\Psi ) = 0 , \\end{align*}"}
+{"id": "6764.png", "formula": "\\begin{align*} q = \\frac { b _ { 0 } ^ { \\prime \\prime } + b _ { 0 } ^ { \\prime } } { b _ { 0 } + 1 } . \\end{align*}"}
+{"id": "879.png", "formula": "\\begin{align*} x ' y - x y ' = n \\ , . \\end{align*}"}
+{"id": "2182.png", "formula": "\\begin{align*} \\mathbf { R } _ y = \\sum \\limits _ { i = 1 } ^ { N } \\lambda _ { i } \\mathbf { e } _ { i } \\mathbf { e } _ { i } ^ { H } = \\mathbf { U _ S \\Sigma _ S U _ S ^ { H } + U _ V \\Sigma _ V U _ V ^ { H } } . \\end{align*}"}
+{"id": "8839.png", "formula": "\\begin{align*} \\begin{cases} d X _ { t , 3 } = - X _ { t , 1 } X _ { t , 2 } d t \\\\ d X _ { t , j } = 0 & j \\neq 3 \\end{cases} \\end{align*}"}
+{"id": "2145.png", "formula": "\\begin{align*} \\mathcal { P } [ \\varrho ] = 0 . \\end{align*}"}
+{"id": "3798.png", "formula": "\\begin{align*} S _ q = B _ q \\circ \\mathcal F \\circ B _ q ^ { - 1 } . \\end{align*}"}
+{"id": "422.png", "formula": "\\begin{align*} f ( x ) = x ^ { n } + k _ { n - 1 } x ^ { n - 1 } + \\dots + k _ { 1 } x + k _ { 0 } \\in A [ x ] \\end{align*}"}
+{"id": "7202.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n - 1 } } e ^ { - C | \\xi | } \\frac { \\xi _ { \\alpha } ^ 2 } { | \\xi | ^ 2 } \\ , d \\xi = \\frac { \\Gamma ( n - 1 ) \\operatorname { v o l } ( \\mathbb { S } ^ { n - 2 } ) } { ( n - 1 ) C ^ { n - 1 } } , C > 0 , \\ n \\geqslant 2 , \\end{align*}"}
+{"id": "8185.png", "formula": "\\begin{align*} \\Lambda ( \\mu , \\nu ) = \\varphi ( \\mu ) + \\varphi ( \\nu ) + \\Psi ( \\mu , \\nu ) , ( \\mu , \\nu ) \\in ( 0 , \\infty ) ^ { 2 } , \\end{align*}"}
+{"id": "7649.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d \\nu _ t ^ { * , \\xi } = & [ ( A - B ^ 2 R ^ { - 1 } P _ t ) \\nu _ t ^ { * , \\xi } - B ^ 2 R ^ { - 1 } \\varphi _ t ^ { * , \\xi } - B h ( \\mu _ t ^ { * , \\xi } ) + f ( \\nu _ t ^ { * , \\xi } ) + b ( \\mu _ t ^ { * , \\xi } ) ] d t + \\sigma _ 0 d W ^ 0 _ t , \\\\ \\nu _ 0 ^ { * , \\xi } = & ~ \\mathbb { E } [ \\xi ] . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6876.png", "formula": "\\begin{align*} w = f ( z ) = ( \\Psi \\circ \\Phi ) ( z ) = ( \\Psi _ 1 ^ { - 1 } \\circ \\Psi _ 2 ^ { - 1 } \\circ \\Phi ) ( z ) \\end{align*}"}
+{"id": "3518.png", "formula": "\\begin{align*} - F _ n - n ^ p \\ \\ = \\ \\ \\mathcal { C } _ n ^ { ( p ) } \\ - \\ 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } \\mathcal { C } _ n ^ { ( p - 2 j - 1 ) } \\mbox { f o r $ p $ o d d } . \\end{align*}"}
+{"id": "5058.png", "formula": "\\begin{align*} - L \\phi = \\mu ^ { 2 } ( \\phi - \\phi _ { \\infty } ) _ { + } \\textrm { i n } \\ \\mathbb { R } ^ { 2 } _ { + } , \\phi = 0 \\textrm { o n } \\ \\partial \\mathbb { R } ^ { 2 } _ { + } , \\end{align*}"}
+{"id": "9110.png", "formula": "\\begin{align*} ( \\sum \\limits _ { i = 1 } ^ { j } w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + ( j - 1 ) r < \\chi _ j < ( \\sum \\limits _ { i = 1 } ^ { j } w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + j r , \\end{align*}"}
+{"id": "5128.png", "formula": "\\begin{align*} I _ { h , \\gamma } & = \\inf \\left \\{ E [ b ] \\ \\Bigg | \\ b \\in L ^ { 2 } _ { \\sigma , \\textrm { a x i } } ( \\mathbb { R } ^ { 3 } ) , \\ H [ b ] = h \\right \\} , \\\\ S _ { h , \\gamma } & = \\left \\{ b \\in L ^ { 2 } _ { \\sigma , \\textrm { a x i } } ( \\mathbb { R } ^ { 3 } ) \\ \\middle | \\ E [ b ] = I _ { h , \\gamma } , \\ H [ b ] = h \\right \\} . \\end{align*}"}
+{"id": "8382.png", "formula": "\\begin{align*} ( F f ) ( x ; z ) : = \\int _ { - \\infty } ^ x \\left ( 1 , e ^ { 2 i z ( x - y ) } \\right ) \\widetilde { Q } f ( y ) \\mathrm { d } y , \\end{align*}"}
+{"id": "2349.png", "formula": "\\begin{align*} V _ n ( s ) : = \\begin{cases} \\mathcal { V } ( y ) = 0 & \\ \\Omega _ s \\\\ y = Y _ { n } & \\ \\Gamma _ s \\\\ \\end{cases} \\end{align*}"}
+{"id": "212.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sqrt [ n ] { \\binom { n + \\lceil \\alpha n \\rceil } { \\lceil \\alpha n \\rceil } } \\le \\frac { ( 1 + \\alpha ) ^ { 1 + \\alpha } } { \\alpha ^ { \\alpha } } . \\end{align*}"}
+{"id": "1212.png", "formula": "\\begin{align*} \\frac { \\mu ( x ) } { \\nu ( x ) } \\frac { \\nu ( y ) } { \\mu ( y ) } = 1 \\end{align*}"}
+{"id": "2829.png", "formula": "\\begin{align*} \\angle \\mathbf { H } ^ { } _ { m , n } = \\angle e ^ { - j \\theta _ { m , n } } , \\forall m \\in \\left [ R _ { d } \\right ] , n \\in \\left [ N _ { b } \\right ] , \\end{align*}"}
+{"id": "6167.png", "formula": "\\begin{align*} \\hat { A } ^ { \\pm } = \\mp \\sqrt { f ( r ) } \\frac { d } { d r } \\sqrt { f ( r ) } + W ( r ) . \\end{align*}"}
+{"id": "990.png", "formula": "\\begin{align*} | W | q ^ { d ^ 2 - 1 } \\theta _ d = | M | & \\le \\delta ( \\alpha + 4 ) q ^ { d ^ 2 + d - 2 } + \\sum _ { l \\in \\Omega _ 1 } | l \\cap W | q ^ { d ^ 2 - 1 } \\\\ & \\hphantom { \\le { } } \\mathrel { + } \\omega _ 2 | W | q ^ { d ^ 2 - d - 1 } + \\sum _ { \\mathclap { ( P , H ) \\in \\Omega _ 3 } } | W | q ^ { d ^ 2 - d } + | \\widehat { \\Omega } _ 4 | . \\end{align*}"}
+{"id": "3847.png", "formula": "\\begin{align*} X _ { n + 1 } = A X _ { n } + B u _ n + \\xi _ { n + 1 } \\end{align*}"}
+{"id": "7182.png", "formula": "\\begin{align*} E _ { - m } = b _ 0 q _ { - m } + \\frac { \\partial q _ { - m } } { \\partial x _ n } - i \\sum _ \\alpha \\frac { \\partial b _ 1 } { \\partial \\xi _ \\alpha } \\frac { \\partial q _ { - m } } { \\partial x _ \\alpha } - \\sum _ { \\substack { - m \\leqslant j , k \\leqslant 1 \\\\ | J | = j + k + m } } \\frac { ( - i ) ^ { | J | } } { J ! } \\partial _ { \\xi } ^ { J } q _ j \\ , \\partial _ { x ^ \\prime } ^ { J } q _ k , m \\geqslant 0 . \\end{align*}"}
+{"id": "7905.png", "formula": "\\begin{align*} \\lambda _ k = \\frac 1 8 \\sum _ { \\ell \\in \\Z } \\Big [ & \\hat W _ r ( - k + \\ell - q ) \\hat W _ r ( \\ell + q ) + \\hat W _ r ( - k + \\ell + q ) \\hat W _ r ( \\ell - q ) \\\\ & + \\hat W _ r ( \\ell - q ) \\hat W _ r ( k + q + \\ell ) + \\hat W _ r ( \\ell + q ) \\hat W _ r ( k - q + \\ell ) \\\\ & - 4 \\hat W _ r ( \\ell - q ) \\hat W _ r ( \\ell + q ) \\Big ] \\end{align*}"}
+{"id": "8932.png", "formula": "\\begin{align*} E X ^ { n } = \\int _ { 0 } ^ { \\infty } x ^ { n } f _ { g } ( x | \\beta ) d x = \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } . \\end{align*}"}
+{"id": "3900.png", "formula": "\\begin{align*} | | u _ N | | _ { L ^ \\infty ( B _ { L s _ { \\delta _ N , i } } ( z _ { N , i } ) ) } = o ( 1 ) . \\end{align*}"}
+{"id": "7888.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Psi ( t , x ) & = - \\int _ \\S W _ r ( x - y ) \\sin ( \\Psi ( t , y ) - \\Psi ( t , x ) ) \\ \\d y + \\int _ \\S W _ r ( y ) \\sin ( \\Psi ( y ) ) \\ \\d y . \\end{align*}"}
+{"id": "2238.png", "formula": "\\begin{align*} d \\mu _ { \\beta , f , g } ( x ) = K _ { \\beta , f } ( x ) g ( x ) \\ , d x . \\end{align*}"}
+{"id": "4638.png", "formula": "\\begin{align*} \\bigcup _ { y = 1 } ^ z I _ y = \\{ 1 , \\dots , n \\} \\end{align*}"}
+{"id": "8026.png", "formula": "\\begin{align*} \\sum _ { u = 1 } ^ t i _ u = 2 r , \\end{align*}"}
+{"id": "5820.png", "formula": "\\begin{align*} { f _ i } \\left ( { { \\bf { x } } + { { \\bf { c } } _ i } \\Delta t , t + \\Delta t } \\right ) = { f _ i } ^ * \\left ( { { \\bf { x } } , t } \\right ) . \\end{align*}"}
+{"id": "884.png", "formula": "\\begin{align*} r ^ 2 + r q + q ^ 2 = z ^ 2 q ^ 2 - 2 z q a n + a ^ 2 n ^ 2 + z q ^ 2 - a n q + q ^ 2 \\equiv ( z ^ 2 + z + 1 ) q ^ 2 \\ , ( n ) \\ , . \\end{align*}"}
+{"id": "133.png", "formula": "\\begin{align*} \\sup _ { t _ N \\in [ - T , T ] } \\| P _ { N } v ( t _ N ) \\| _ { L ^ 2 } ^ 2 & \\lesssim \\| P _ N \\phi \\| _ { L ^ 2 } ^ 2 \\\\ & + \\sup _ { t _ N \\in [ - T , T ] } \\Big | \\int _ { [ 0 , t _ N ] \\times \\R ^ 2 } P _ N v P _ N ( \\partial _ x ( v ( u _ 1 + u _ 2 ) ) ) ~ d x d y d t \\Big | . \\end{align*}"}
+{"id": "4997.png", "formula": "\\begin{align*} \\Xi = ( v _ 0 , v _ 1 , f _ 1 , f _ 2 , f _ 3 ) \\end{align*}"}
+{"id": "4283.png", "formula": "\\begin{align*} n & = \\frac { 9 \\left ( 2 i - 1 \\right ) ^ { 3 } - 1 } { 2 } \\\\ N & = \\frac { 2 7 \\left ( 2 i - 1 \\right ) ^ { 3 } \\left ( 2 7 \\left ( 2 i - 1 \\right ) ^ { 6 } + 1 \\right ) } { 4 } \\\\ n + a & = \\frac { 3 \\left ( 2 i - 1 \\right ) \\left ( 3 \\left ( 2 i - 1 \\right ) ^ { 3 } + 1 \\right ) } { 2 } \\\\ n + b & = - \\frac { 3 \\left ( 2 i - 1 \\right ) \\left ( 3 \\left ( 2 i - 1 \\right ) ^ { 3 } - 1 \\right ) } { 2 } \\end{align*}"}
+{"id": "1238.png", "formula": "\\begin{align*} \\rho ( \\xi _ { \\Delta } \\lvert \\eta _ { \\Lambda \\setminus \\Delta } ) = \\prod _ { j = 1 } ^ { k - 1 } \\rho ( \\eta _ { i _ j } | \\eta _ { [ i _ { j + 1 } , i _ k ] } \\eta _ { \\Lambda \\setminus \\Delta } ) . \\end{align*}"}
+{"id": "6815.png", "formula": "\\begin{align*} \\varphi _ { 0 } \\left ( \\rho \\right ) = G _ { - } \\left ( \\rho \\right ) e ^ { i S ( \\rho , x ) } \\end{align*}"}
+{"id": "6217.png", "formula": "\\begin{align*} E _ 1 - E _ 0 & = ( 2 L + 3 ) ( \\Q ^ 2 + 4 \\kappa ) , \\\\ E _ 1 - E _ 0 & = ( 2 L + 3 ) \\frac { ( \\Q ^ 2 + \\kappa ) ( \\Q ^ 2 + 9 \\kappa ) } { \\Q ^ 2 + 3 \\kappa } , \\\\ E _ 1 - E _ 0 & = ( 2 L + 3 ) \\frac { ( \\Q ^ 2 + 4 \\kappa ) ( \\Q ^ 2 + 1 6 \\kappa ) } { \\Q ^ 2 + 1 0 \\kappa } , \\end{align*}"}
+{"id": "5782.png", "formula": "\\begin{align*} | \\Psi _ 1 | ^ 2 + | \\Psi _ 2 | ^ 2 = 1 . \\end{align*}"}
+{"id": "7801.png", "formula": "\\begin{align*} \\eta ( x ) = \\sum _ { k = 1 } ^ \\infty a _ k \\sin ( 2 \\pi k x ) + b _ k ( 1 - \\cos ( 2 \\pi k x ) ) , \\end{align*}"}
+{"id": "3793.png", "formula": "\\begin{align*} | | B _ q ( \\psi ) | | _ { M L _ q ( \\mathbb { C } ) } = | | \\psi | | _ { L ^ 2 ( \\mathbb { R } ) } . \\end{align*}"}
+{"id": "2111.png", "formula": "\\begin{align*} \\frac { d } { d b } R L ( b , c ) = & \\frac { 2 b + 1 } { 2 } \\left ( \\log \\left ( \\binom { b + 1 } { 2 } + c \\right ) + 1 \\right ) \\\\ & - \\frac { 2 b - 1 } { 2 } \\left ( \\log \\left ( \\binom { b } { 2 } + c \\right ) + 1 \\right ) + f _ c ( 1 ) - f _ c ( 0 ) \\\\ \\frac { d } { d b } L L ( b , c ) = & ( b + 1 ) \\left ( \\log \\left ( b + c \\right ) + 1 \\right ) \\\\ & - b \\left ( \\log \\left ( b + c - 1 \\right ) + 1 \\right ) + f _ c ( b ) - f _ c ( b - 1 ) \\end{align*}"}
+{"id": "2554.png", "formula": "\\begin{align*} \\begin{aligned} & \\omega \\in [ 0 , 1 ] { { \\xi } _ { o } } \\le 0 \\\\ & \\omega \\in \\left [ \\frac { { { \\xi } _ { o } } } { { { \\xi } _ { o } } + \\left ( \\gamma - { { \\gamma } _ { o } } \\right ) \\mu ( { { \\mathbf { q } } _ { o } } ) } , 1 \\right ] , { { \\xi } _ { o } } > 0 . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "9225.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\| \\dfrac { \\partial \\epsilon ( s , t ) } { \\partial s } \\right \\| \\le \\int _ { N T } ^ t \\left \\| \\dfrac { \\partial f ( s , \\tau ) } { \\partial s } \\right \\| + \\left \\| \\dfrac { \\partial f _ a ( s ) } { \\partial s } \\right \\| \\ , d \\tau \\le 2 L _ \\mathcal { D } T , \\end{aligned} \\end{align*}"}
+{"id": "2769.png", "formula": "\\begin{align*} 4 N _ 1 \\leq \\begin{cases} C _ 6 & \\ \\tau \\leq t ^ { \\beta } , \\\\ \\displaystyle C _ 6 t ^ { - 2 \\beta } \\tau ^ 2 & \\ t ^ { \\beta } \\leq \\tau . \\\\ \\end{cases} \\end{align*}"}
+{"id": "3294.png", "formula": "\\begin{align*} \\int _ 0 ^ { 2 - i \\Delta _ n + s } | x - y | ^ { 2 \\mathfrak H - 1 } d y \\leq \\int _ 0 ^ 2 | x - y | ^ { 2 \\mathfrak H - 1 } d y = & \\int _ 0 ^ x ( x - y ) ^ { 2 \\mathfrak H - 1 } d y + \\int _ { x } ^ { 2 } ( y - x ) ^ { 2 \\mathfrak H - 1 } d y \\\\ = & \\frac { x ^ { 2 \\mathfrak H } + ( 2 - x ) ^ { 2 \\mathfrak H } } { 2 \\mathfrak H } \\\\ \\leq & \\frac { 2 ^ { 2 \\mathfrak H } } { \\mathfrak H } , \\end{align*}"}
+{"id": "3978.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k a _ i \\neq 1 \\ , \\ , { \\rm a n d } \\ , \\ , \\sum _ { i \\in \\Omega _ t ^ o } a _ i - \\sum _ { i \\in \\Omega _ t ^ e } a _ i \\neq 1 . \\end{align*}"}
+{"id": "465.png", "formula": "\\begin{align*} \\lambda \\Lambda _ 1 L _ { g _ 1 } ( z ) = \\log \\big ( 1 - ( 1 - e ^ { \\lambda \\Lambda _ 1 } ) L _ { f _ 1 } ( z ) \\big ) . \\end{align*}"}
+{"id": "2705.png", "formula": "\\begin{align*} \\frac { 2 \\epsilon _ f + r } { h ( \\gamma \\bar \\Delta ) } \\left [ \\sum _ { k = 0 } ^ { T - 1 } 1 \\right ] - \\left [ \\sum _ { k = 0 } ^ { T - 1 } ( 1 - \\Theta _ k ) ( 1 - \\Lambda _ k ) + \\Theta _ k \\Lambda _ k ' \\right ] < \\left ( \\frac { 2 \\epsilon _ f + r } { h ( \\gamma \\bar \\Delta ) } - \\frac { 1 } { 2 } \\right ) T + \\frac { 1 } { 2 } \\left | \\log _ \\gamma \\frac { \\bar \\Delta } { \\delta _ 0 } \\right | + \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "4103.png", "formula": "\\begin{align*} & \\hphantom { { } = { } } D _ { ( o , o ) } ( \\{ f , g \\} . \\{ f ' , g ' \\} ) \\\\ * & = \\begin{pmatrix} D f ( o ) D f ' ( o ) \\\\ D g ( o ) D f ' ( o ) + D f ( o ) D g ' ( o ) & D f ( o ) D f ' ( o ) \\end{pmatrix} \\\\ * & = \\begin{pmatrix} D f ( o ) \\\\ D g ( o ) & D f ( o ) \\end{pmatrix} \\begin{pmatrix} D f ' ( o ) \\\\ D g ' ( o ) & D f ' ( o ) \\end{pmatrix} . \\end{align*}"}
+{"id": "7201.png", "formula": "\\begin{align*} \\frac { i } { ( 2 \\pi ) ^ { n } } \\int _ { \\mathbb { R } ^ { n - 1 } } \\int _ { \\mathcal { C } } e ^ { - t \\tau } \\operatorname { T r } \\phi _ { - 1 } \\ , d \\tau \\ , d \\xi = a _ 0 ( x ) t ^ { 1 - n } , \\end{align*}"}
+{"id": "7232.png", "formula": "\\begin{align*} A : = \\left [ \\begin{array} { c c c c c } \\alpha _ 1 & \\alpha _ 1 ^ q & \\alpha _ 1 ^ { q ^ 2 } & \\cdots & \\alpha _ 1 ^ { q ^ { n } } \\\\ \\alpha _ 2 & \\alpha _ 2 ^ q & \\alpha _ 2 ^ { q ^ 2 } & \\cdots & \\alpha _ 2 ^ { q ^ { n } } \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha _ n & \\alpha _ n ^ q & \\alpha _ n ^ { q ^ 2 } & \\cdots & \\alpha _ n ^ { q ^ { n } } \\\\ x & x ^ q & x ^ { q ^ 2 } & \\cdots & x ^ { q ^ { n } } \\\\ \\end{array} \\right ] . \\end{align*}"}
+{"id": "6781.png", "formula": "\\begin{align*} g ^ { \\prime } ( \\rho , x ) = e ^ { - i \\rho x } \\left ( - \\frac { z - 1 } { 2 ( z + 1 ) } + \\frac { 1 } { 2 } \\int _ { - \\infty } ^ { x } q ( t ) d t + ( z + 1 ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } z ^ { n } c _ { n } ( x ) \\right ) \\end{align*}"}
+{"id": "916.png", "formula": "\\begin{align*} \\mathcal { L } u = \\lambda | u | ^ 2 u , \\lambda \\in \\R \\end{align*}"}
+{"id": "1978.png", "formula": "\\begin{align*} G ( x - 2 G ( x ) ) = - G ( x ) \\end{align*}"}
+{"id": "68.png", "formula": "\\begin{align*} & \\abs { \\ell ( x , \\alpha + \\beta ) - \\ell ( x , \\alpha ) - \\ell ( x , \\beta ) } \\\\ = & \\lvert \\underbrace { \\Phi ^ + ( \\alpha + \\beta ) - \\Phi ^ + ( \\alpha ) - \\Phi ^ + ( \\beta ) } _ { \\in \\{ - 1 , 0 \\} } \\underbrace { - \\Phi ^ + ( w ( \\alpha + \\beta ) ) + \\Phi ^ + ( w \\alpha ) + \\Phi ^ + ( w \\beta ) } _ { \\in \\{ 0 , 1 \\} } \\rvert \\leq 1 . \\end{align*}"}
+{"id": "275.png", "formula": "\\begin{align*} T [ f ] ( p ) = \\begin{cases} z ^ 2 - 1 & \\ | z | \\ge 1 , \\\\ 0 & \\ | z | < 1 \\ \\ ( x , y ) \\neq ( 0 , 0 ) , \\\\ 1 - z ^ 2 & \\ | z | < 1 \\ \\ ( x , y ) = ( 0 , 0 ) , \\\\ \\end{cases} \\end{align*}"}
+{"id": "2428.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } s L _ { q } [ f ( t ) ] = \\lim _ { s \\rightarrow \\infty } \\frac { f ( t ) } { 1 + ( 1 - q ) } . \\end{align*}"}
+{"id": "309.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\downarrow 0 } \\frac { \\mu ( B _ r ( x ) \\cap \\mathfrak { J } _ k ) } { \\mu ( B _ r ( x ) ) } = 1 . \\end{align*}"}
+{"id": "2163.png", "formula": "\\begin{align*} g _ { z } = \\phi _ { z } ( { \\bf { v } } , { \\bf { u } } ) + \\gamma ( { \\bf { v } } , { \\bf { u } } , \\psi ) + \\eta _ { z } , \\end{align*}"}
+{"id": "2664.png", "formula": "\\begin{align*} \\langle P ( u _ t ( s ) ) , \\phi ( s ) \\rangle _ V \\Bigr | ^ t _ 0 = \\int ^ t _ 0 \\Bigl [ \\langle P ( u _ t ( s ) ) , \\phi _ t ( s ) \\rangle _ V - \\langle A ( u ( s ) ) , \\phi ( s ) \\rangle _ X - \\langle B ( s , x , u _ t ( s ) ) , \\phi ( s ) \\rangle _ W \\Bigr ] d s \\end{align*}"}
+{"id": "2207.png", "formula": "\\begin{align*} \\mathbf { a } ( \\theta _ { 0 } ) ^ { T } \\mathbf { v } ^ { 0 } & = 1 + e ^ { - j \\phi } + e ^ { - j 2 \\phi } + . . . + e ^ { - j ( N - 1 ) \\phi } \\\\ & = \\frac { 1 - ( e ^ { - j \\phi } ) ^ N } { 1 - e ^ { - j \\phi } } \\end{align*}"}
+{"id": "288.png", "formula": "\\begin{align*} U _ { r _ n } ( q _ n ) = \\varphi ( q _ n ) + \\delta _ n , \\end{align*}"}
+{"id": "1535.png", "formula": "\\begin{align*} \\widehat { \\Psi } ( \\hat { A } ) ^ { \\times } = \\{ P \\in \\widehat { \\Psi } ( \\mathcal { O } _ { \\hat { A } } ) \\ ; | \\ ; \\pi ( P ) \\in G _ { \\hat { A } } \\} \\end{align*}"}
+{"id": "8060.png", "formula": "\\begin{align*} S _ r = \\min _ { k } \\mathbb { E } \\left [ \\bar { R _ { c , k } } \\right ] + \\sum _ { k = 1 } ^ K \\mathbb { E } \\left [ \\bar { R } _ k \\right ] , \\end{align*}"}
+{"id": "4980.png", "formula": "\\begin{align*} \\xi _ 2 ^ { \\mathsf { U } ^ n } ( u , s ) = & \\frac { \\xi _ 0 f _ 1 ( u ) f _ 2 ( s ) } { \\xi _ 0 f _ 1 ( u ) f _ 2 ( s ) + ( 1 - \\xi _ 0 ) f _ 2 ( u ) f _ 1 ( s ) } = \\xi _ 2 ^ { \\mathsf { V } ^ n } ( s , u ) . \\end{align*}"}
+{"id": "4928.png", "formula": "\\begin{align*} \\mathrm { H } ( X ) = & \\ln \\left \\{ \\sqrt { 2 \\pi } \\sigma \\mathrm { B } ( \\alpha , \\beta ) \\right \\} + \\frac { 1 } { 2 \\sigma ^ 2 } \\left [ \\mathrm { E } ( X ^ 2 ) - 2 \\mu \\mathrm { E } ( X ) + \\mu ^ 2 \\right ] \\\\ & + \\frac { 1 } { \\mathrm { B } ( \\alpha , \\beta ) } \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n } \\left [ ( \\alpha - 1 ) \\mathrm { B } ( \\alpha , n + \\beta ) + ( \\beta - 1 ) \\mathrm { B } ( n + \\alpha , \\beta ) \\right ] . \\end{align*}"}
+{"id": "7929.png", "formula": "\\begin{align*} \\abs { \\eta ( y ) - \\eta ( x ) } ^ 2 & = \\abs { \\int _ x ^ y \\partial \\eta ( z ) \\ \\d z } ^ 2 \\\\ & \\le \\int _ \\S \\partial \\eta ( z ) ^ 2 \\ \\d z \\\\ & = \\norm { \\partial \\eta } _ { L ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "4926.png", "formula": "\\begin{align*} & \\ln \\left \\{ \\Phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right \\} = \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ n } { n } \\ , \\left [ 1 - \\Phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] ^ n \\end{align*}"}
+{"id": "4541.png", "formula": "\\begin{align*} \\omega ^ * : = \\R \\setminus [ - R , 0 ] = ( - \\infty , - R ) . \\end{align*}"}
+{"id": "3209.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu \\left ( \\left \\{ x \\in X \\mid \\frac { S _ n f ( x ) } { \\sqrt { n } } \\le z \\right \\} \\right ) = \\frac { 1 } { \\sigma _ f \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ z \\exp \\left ( - \\frac { t ^ 2 } { 2 \\sigma _ f ^ 2 } \\right ) d t . \\end{align*}"}
+{"id": "467.png", "formula": "\\begin{align*} \\lambda _ 1 \\Lambda _ 1 g _ 1 ( x ) \\stackrel { a . e . } { = } - \\sum _ { n = 1 } ^ \\infty n ^ { - 1 } ( 1 - e ^ { \\lambda \\Lambda _ 1 } ) ^ n f _ 1 ^ { \\ast n } ( x ) = : \\lambda _ 1 \\Lambda _ 1 \\breve g _ 1 ( x ) , \\end{align*}"}
+{"id": "4387.png", "formula": "\\begin{align*} & \\int _ K | F _ { j } - ( 1 - b ( \\Psi ) ) f F ^ { 1 + \\delta } | _ { h } ^ 2 e ^ { v ( \\Psi ) - \\delta \\max \\{ \\psi + T , 2 \\log | F | \\} } c ( - v ( \\Psi ) ) \\\\ \\le & \\bigg ( \\sup _ { X _ j } e ^ { - u ( - v ( \\Psi ) ) } \\bigg ) \\int _ { X _ j } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f F | ^ 2 _ h . \\end{align*}"}
+{"id": "4110.png", "formula": "\\begin{align*} H _ r ( E [ t ] ) = t H _ r ( E ) . \\end{align*}"}
+{"id": "2350.png", "formula": "\\begin{align*} H _ { c } ( s ) : = \\begin{cases} \\mathcal { H } ( x ) = 0 & \\ ( 0 , 1 ) \\times ( 0 , s ) \\\\ x = c & \\ [ 0 , 1 ] \\times \\{ 0 \\} \\\\ x ( 0 , t ) = c ( t ) , \\ x ( 1 , t ) = c & \\forall t \\in ( 0 , s ) . \\end{cases} \\end{align*}"}
+{"id": "8181.png", "formula": "\\begin{align*} \\mathcal { M } ^ { r } ( t ) - \\mathcal { M } ^ { r } ( \\delta ) = \\int _ { \\delta } ^ { t } \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\mu } \\omega ( \\mu , \\nu ) \\xi ( \\mu , s ) \\xi ( \\nu , s ) \\Lambda ( \\mu , \\nu ) \\ d \\nu d \\mu d s , \\end{align*}"}
+{"id": "6452.png", "formula": "\\begin{align*} \\vec { r } = ( r _ 1 , r _ 2 , \\ldots ) . \\end{align*}"}
+{"id": "3045.png", "formula": "\\begin{align*} R _ { x } = I \\left ( \\cos \\left ( \\left \\vert x \\right \\vert / 2 \\right ) + \\sin \\left ( \\left \\vert x \\right \\vert / 2 \\right ) x / \\left \\vert x \\right \\vert \\right ) \\end{align*}"}
+{"id": "5200.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ + \\frac { G } { r ^ { 2 } } \\dd x = - \\mu \\int _ { \\mathbb { R } ^ { 3 } } \\nabla \\times B \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) \\dd x , \\end{align*}"}
+{"id": "5400.png", "formula": "\\begin{align*} \\begin{aligned} x _ n - y _ n = \\ , & \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( y ' ) ( x _ \\beta - y _ \\beta ) + O ( | x ' - y ' | ^ 2 ) \\\\ \\geq \\ , & \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( y ' ) ( x _ \\beta - y _ \\beta ) - \\kappa | x ' - y ' | ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "6221.png", "formula": "\\begin{align*} - E _ 0 & = \\frac { 1 } { 4 } \\biggl \\{ - ( 2 L + 2 ) ^ 2 \\kappa + \\Q ^ 2 [ ( 2 L + 3 ) a _ 0 + 1 ] ^ 2 - [ ( 2 L + 3 ) a _ 1 - m ^ 2 ] \\kappa \\\\ & \\quad { } - 2 ( 2 L + 2 ) [ ( 2 L + 3 ) a _ 1 - m ] \\kappa \\biggr \\} . \\end{align*}"}
+{"id": "8679.png", "formula": "\\begin{align*} \\begin{aligned} \\mathop { \\max } \\limits _ { \\{ { \\bf { \\bar c } } _ k \\} _ { k = 0 } ^ { K - 1 } } & \\ \\frac { 1 } { K } \\sum \\limits _ { k = 0 } ^ { K - 1 } { \\log _ 2 } \\left ( { 1 + { { \\left | { \\bf { \\bar e } } _ k ^ H { { \\bf { \\bar c } } _ k } \\right | } ^ 2 } } \\right ) \\\\ { \\rm { s . t . } } & \\ \\ \\sum \\limits _ { k = 0 } ^ { K - 1 } { { { \\left \\| { { \\bf { \\bar c } } _ k } \\right \\| } ^ 2 } } \\le K P . \\end{aligned} \\end{align*}"}
+{"id": "762.png", "formula": "\\begin{align*} \\frac 1 { | y | ^ { N - 1 } \\ , | \\bar y | _ p ^ { 2 s } } * ( \\chi _ { Q } m ) ( \\bar x ) & \\lesssim \\int _ { y \\in Q _ 1 } \\frac 1 { | x - y | ^ { N - 1 + s / 2 } } d y \\ , \\int _ { u \\in I _ Q } \\frac 1 { | t - u | ^ { 3 / 4 } } \\ , d u \\\\ & \\lesssim \\ell ( Q ) ^ { 1 - s / 2 } \\ , ( \\ell ( Q ) ^ { 2 s } ) ^ { 1 / 4 } = \\ell ( Q ) . \\end{align*}"}
+{"id": "7952.png", "formula": "\\begin{align*} \\partial _ t \\varrho + m \\cdot \\nabla J ( \\varrho ) = 0 , \\varrho ( 0 , \\cdot ) = \\varrho ^ { \\rm i n i } , \\end{align*}"}
+{"id": "5015.png", "formula": "\\begin{align*} S t _ { D N M } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( z ^ { 2 } , z ) - z ^ { 2 } | | _ { \\infty } = 0 . \\\\ \\end{align*}"}
+{"id": "528.png", "formula": "\\begin{align*} \\mu ( n ) = ( 2 n + d ) ^ { - \\alpha } e ^ { ( 2 n + d ) i t } . \\end{align*}"}
+{"id": "6014.png", "formula": "\\begin{align*} I ( t u , t v ) = \\frac { t ^ { 2 } } { 2 } \\| ( u , v ) \\| _ { E } ^ { 2 } + \\frac { t ^ { 6 } } { 2 } \\Big ( B ( u ) + B ( v ) \\Big ) - \\frac { t ^ { 2 p } } { 2 p } F ( u , v ) , \\end{align*}"}
+{"id": "2246.png", "formula": "\\begin{align*} a ( Z ) = a \\big ( ( Z Z ^ * ) ^ \\frac { 1 } { 2 } \\big ) \\end{align*}"}
+{"id": "7380.png", "formula": "\\begin{align*} ( r _ 0 , \\ldots , r _ { \\ell - 1 } ) = \\big ( \\ , \\underbrace { s _ L , \\ldots , s _ L } _ { m _ L } , \\ldots , \\underbrace { s _ 1 , \\ldots , s _ 1 } _ { m _ 1 } \\ , \\big ) , 0 < s _ 1 < \\cdots s _ L , m _ i > 0 , \\ 1 \\leq i \\leq L . \\end{align*}"}
+{"id": "6688.png", "formula": "\\begin{align*} c ( \\gamma , \\gamma ^ - ) = - c ( \\gamma , \\gamma ^ + ) . \\end{align*}"}
+{"id": "4772.png", "formula": "\\begin{align*} \\epsilon _ n = \\max _ { | \\eta - \\lambda | \\le \\delta } \\max _ { v \\in G ( \\lambda ) } \\| F _ n ( \\eta ) p _ n v - q _ n F ( \\eta ) v \\| _ { X _ n } \\end{align*}"}
+{"id": "6649.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { m \\to \\infty } \\limsup _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\sup _ { t \\in [ 0 , \\chi \\wedge 1 ] } | \\hat { X } ^ { \\epsilon } ( t ) - X ^ { \\epsilon } ( t ) | \\geq \\delta ^ { 1 / 4 } \\Big ) = - \\infty , \\end{align*}"}
+{"id": "7271.png", "formula": "\\begin{align*} \\beta ^ { q ^ n } + a _ { n - k } \\beta ^ { q ^ { n - k } } + \\cdots + a _ 0 \\beta = 0 \\end{align*}"}
+{"id": "8497.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\| M _ { + , 1 2 } ( x ; z ) \\right \\| _ { L _ { z } ^ { 2 } } \\leq c \\left \\| \\mathcal { P } ^ + \\left ( \\bar { r } _ 1 ( z ) \\mathrm { e } ^ { - 2 i z x } \\right ) \\right \\| _ { L _ { z } ^ 2 } , \\end{aligned} \\end{align*}"}
+{"id": "4296.png", "formula": "\\begin{align*} \\| \\tilde { X } ( k ) \\| ^ 2 \\leq \\frac { \\sum \\limits _ { i = 1 } ^ { p } \\left \\| \\tilde { Y } _ i ( k ) \\right \\| ^ 2 + \\sum \\limits _ { j = 1 } ^ { q } \\left \\| \\tilde { Z } _ j ( k ) \\right \\| ^ 2 } { p + q } . \\end{align*}"}
+{"id": "1546.png", "formula": "\\begin{align*} h ( u _ t , v _ s ) & { } = u _ { 1 - s } , & k ( u _ t , v _ s ) & { } = v _ { 1 - t } , \\end{align*}"}
+{"id": "8223.png", "formula": "\\begin{align*} M P A P P ( m , n ) = & \\biggl \\{ A \\in P A P P ( m , n ) : A \\mbox { i s t h e m i n i m a l e l e m e n t i n t h e s e t } \\\\ & \\{ A , \\sigma ( A ) \\} \\mbox { u n d e r t h e l e x i c o g r a p h i c o r d e r } \\biggr \\} \\end{align*}"}
+{"id": "794.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } t } _ { | t = 0 } \\int _ { \\Gamma _ t } ( c + t w ) \\circ \\theta _ t ^ { - 1 } \\ , \\mathrm { d } \\mathcal { H } ^ d & = \\int _ \\Sigma \\partial ^ \\square \\big ( ( c + t w ) \\circ \\theta _ t ^ { - 1 } \\big ) - \\big ( ( c + t w ) \\circ \\theta _ t ^ { - 1 } \\big ) _ { | t = 0 } H V \\ , \\mathrm { d } \\mathcal { H } ^ d \\\\ & = \\int _ \\Sigma w - c H V \\ , \\mathrm { d } \\mathcal { H } ^ d = 0 \\end{align*}"}
+{"id": "1319.png", "formula": "\\begin{align*} \\int _ { 0 } ^ t \\frac { d } { d s } | | u ( s ) | | _ { L ^ 2 ( \\mathbb { H } ^ n ) } ^ 2 d s = | | u ( t ) | | _ { L ^ 2 ( \\mathbb { H } ^ n ) } ^ 2 - | | u ( 0 ) | | _ { L ^ 2 ( \\mathbb { H } ^ n ) } ^ 2 . \\end{align*}"}
+{"id": "7934.png", "formula": "\\begin{align*} \\norm { g } _ { L ^ 2 } ^ 2 \\le \\prod _ { i = 1 } ^ n \\norm { \\eta _ i } _ { L ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "7566.png", "formula": "\\begin{align*} M = \\begin{pmatrix} g _ { 1 , p - 1 } & X g _ { 1 , 0 } & \\cdots & X g _ { 1 , p - 3 } & X g _ { 1 , p - 2 } \\\\ g _ { 2 , p - 2 } & g _ { 2 , p - 1 } & \\cdots & X g _ { 2 , p - 4 } & X g _ { 2 , p - 3 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ g _ { p - 1 , 1 } & g _ { p - 1 , 2 } & \\cdots & g _ { p - 1 , p - 1 } & X g _ { p - 1 , 0 } \\\\ g _ { 0 , 0 } & g _ { 0 , 1 } & \\cdots & g _ { 0 , p - 2 } & g _ { 0 , p - 1 } \\end{pmatrix} \\end{align*}"}
+{"id": "1088.png", "formula": "\\begin{align*} w \\varepsilon ^ \\lambda = y \\cdot w ' \\varepsilon ^ { \\lambda ' } . \\end{align*}"}
+{"id": "4884.png", "formula": "\\begin{align*} \\mathcal { C } \\big ( \\mathcal { U } _ { ( i , j ) } \\big ) \\cong \\begin{cases} C ^ \\infty \\big ( \\mathcal { U } _ { ( 0 , 0 ) } \\big ) & \\\\ \\mathcal { S } \\big ( \\mathcal { U } _ { ( i , j ) } \\big ) & \\\\ \\mathcal { A } \\big ( \\mathcal { U } _ { ( i , j ) } \\big ) & \\end{cases} \\ , , \\end{align*}"}
+{"id": "8727.png", "formula": "\\begin{align*} \\eta ( q _ 1 ) \\eta ( q _ 2 ) \\overline { \\pi } ( q _ 1 ) \\overline { \\pi } ( q _ 2 ) = \\eta ( q _ 1 q _ 2 ) c _ { \\chi _ 1 } ( q _ 1 , q _ 2 ) \\overline { \\pi } ( q _ 1 q _ 2 ) . \\end{align*}"}
+{"id": "3818.png", "formula": "\\begin{align*} \\sum _ { \\alpha , \\beta = 1 } ^ d \\xi _ \\alpha \\cdot \\left ( \\nabla G ( U , x , t ) ^ T B _ { \\alpha \\beta } ( U , x , t ) \\ , \\xi _ \\beta \\right ) = \\sum _ { \\alpha , \\beta = 1 } ^ d \\sum _ { i , j = 1 } ^ n \\xi _ \\alpha ^ i D _ { \\alpha \\beta } ^ { i j } \\xi _ \\beta ^ j > 0 \\end{align*}"}
+{"id": "3455.png", "formula": "\\begin{align*} F ( \\theta _ { ( \\ell + 1 ) q _ n + m _ n } ) F _ { q _ n - 1 } ( \\theta _ { \\ell q _ n + m _ n + 1 } ) = : \\left ( \\begin{matrix} b _ 1 & b _ 2 \\\\ b _ 3 & b _ 4 \\end{matrix} \\right ) , \\end{align*}"}
+{"id": "7459.png", "formula": "\\begin{align*} G : = { \\rm G a l } ( M ' / K ' ) = { \\rm G a l } ( M ' / M _ 1 ' ) \\times { \\rm G a l } ( M ' / M _ 2 ' ) \\cong C _ 2 \\times C _ 2 \\end{align*}"}
+{"id": "7793.png", "formula": "\\begin{align*} W _ r ( x ) : = \\begin{cases} 1 & \\min ( x , 1 - x ) \\le r \\\\ 0 & \\end{cases} . \\end{align*}"}
+{"id": "5772.png", "formula": "\\begin{align*} F _ 1 = \\frac { 1 } { \\sqrt { c _ 1 } } \\langle \\langle \\nu _ 1 \\cdot \\varphi , \\varphi \\rangle \\rangle . \\end{align*}"}
+{"id": "1092.png", "formula": "\\begin{align*} \\varepsilon ^ { w ( \\lambda - \\lambda ' ) } = w \\varepsilon ^ \\lambda \\varepsilon ^ { - \\lambda ' } w ^ { - 1 } = y w ' \\varepsilon ^ { \\lambda ' } \\varepsilon ^ { - \\lambda ' } w ^ { - 1 } = y \\underbrace { w ' w ^ { - 1 } } _ { \\in W _ J } \\in \\widetilde W _ J . \\end{align*}"}
+{"id": "1393.png", "formula": "\\begin{align*} x _ s + \\Psi _ s & \\stackrel { d } { = } x _ { s + 1 } + V _ { s + 1 } ^ { r } + \\zeta ^ { ( 1 ) } \\\\ x _ { s + 1 } + \\Psi _ { s + 1 } & \\stackrel { d } { = } x _ { s + 1 } + V _ { s + 1 } ^ { r } + \\zeta ^ { ( 2 ) } \\end{align*}"}
+{"id": "3339.png", "formula": "\\begin{align*} \\chi _ i = \\left \\{ \\begin{aligned} & \\psi _ i & & ( i \\neq j ) \\\\ & \\sigma & & ( i = j ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6177.png", "formula": "\\begin{align*} & \\xi ( \\xi + 1 ) = L ( L + 1 ) , \\\\ & - \\kappa \\xi ^ 2 + 2 \\xi \\zeta + \\eta ^ 2 - \\frac { \\zeta ^ 2 } { \\kappa } = - E _ 0 , \\\\ & 2 \\xi \\eta = - Q , \\\\ & 2 \\eta \\zeta = \\kappa B _ 1 , \\\\ & \\frac { \\zeta } { \\kappa } \\left ( \\frac { \\zeta } { \\kappa } - 1 \\right ) = B _ 2 . \\end{align*}"}
+{"id": "6636.png", "formula": "\\begin{align*} & \\limsup _ { m \\to \\infty } \\limsup _ { \\epsilon \\to 0 } \\epsilon ^ { 2 } \\log \\mathbb P \\Big ( \\epsilon \\left | G _ { m , \\epsilon } ( t _ 1 , \\cdots , t _ k ) - G _ { \\epsilon } ( t _ 1 , \\cdots , t _ k ) \\right | > \\delta \\Big ) = - \\infty , \\end{align*}"}
+{"id": "5112.png", "formula": "\\begin{align*} \\int _ { D ( 0 , 2 ) \\backslash D ( 0 , 1 ) } | \\phi | ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r = 1 . \\end{align*}"}
+{"id": "5347.png", "formula": "\\begin{align*} c _ k = 0 \\ \\ c _ k = \\left ( \\frac { 1 } { p ^ { d / \\tau } } \\right ) ^ { p ^ { v _ p ( k ) } } \\ \\end{align*}"}
+{"id": "2826.png", "formula": "\\begin{align*} \\mathbf { x } ^ { ( k ) } = \\mathbf { r } ^ { ( k ) } + \\mathcal { B } ^ { ( k ) } \\left [ \\widetilde { \\mathcal { H } } ^ { ( k ) } \\left [ \\mbox { s o f t } \\left [ \\mathcal { H } ^ { ( k ) } \\left ( \\mathcal { M } ^ { ( k ) } \\left ( \\mathbf { r } ^ { ( k ) } \\right ) \\right ) , \\theta ^ { ( k ) } \\right ] \\right ] \\right ] \\end{align*}"}
+{"id": "9328.png", "formula": "\\begin{align*} \\P \\left ( \\frac { 1 } { m } \\sum _ { i = 1 } ^ m U _ i \\geq \\left ( \\frac { \\pi } { 4 } + \\frac { 1 9 9 \\pi } { 3 2 0 0 k } \\right ) \\| \\mathbf { x } \\| \\right ) \\leq 2 \\exp \\left ( - \\frac { c _ 8 m } { k } \\right ) . \\end{align*}"}
+{"id": "6560.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla ( v - \\tilde { v } ) | ^ { 2 } + \\int _ { \\partial \\Omega } | v - \\tilde { v } | ^ { 2 } + \\int _ { \\Omega } u | v - \\tilde { v } | ^ { 2 } = 0 , \\end{align*}"}
+{"id": "8622.png", "formula": "\\begin{align*} | K \\oplus _ p L | = \\frac { \\Gamma ( \\frac { n _ 1 } { q } + 1 ) \\Gamma ( \\frac { n _ 2 } { q } + 1 ) } { \\Gamma ( \\frac { n _ 1 + n _ 1 } { q } + 1 ) } | K | | L | = c _ { n _ 1 , n _ 2 , q } | K | | L | , \\end{align*}"}
+{"id": "9180.png", "formula": "\\begin{align*} \\begin{aligned} | x ( t ) - x _ a ( t ) | & = | x ( t ) - z ( t ) - x _ a ( t ) + z ( t ) | \\\\ & \\leq | x ( t ) - z ( t ) | + \\gamma \\bar \\epsilon ( L _ r , M _ r , \\delta ) . \\end{aligned} \\end{align*}"}
+{"id": "8896.png", "formula": "\\begin{align*} W ( F ) ( \\tau , \\tau ' ) : = F _ { z = 0 } . \\end{align*}"}
+{"id": "8817.png", "formula": "\\begin{align*} \\P \\left ( \\Omega _ 0 = \\left \\{ \\omega \\in \\Omega : \\int _ 0 ^ \\eta | y _ t | ^ 2 d t \\le \\sqrt { \\delta _ 1 } \\eta K ^ { 2 r } , \\sup _ { 0 \\le t \\le \\eta } | z _ t - z _ 0 | \\le R K ^ { \\frac { r - 1 } { 3 } } \\right \\} \\right ) \\ge 1 - \\frac { \\beta } { 2 } . \\end{align*}"}
+{"id": "7505.png", "formula": "\\begin{align*} g _ k : = { \\rm s g n } ( v ) \\star \\eta _ k , \\end{align*}"}
+{"id": "8670.png", "formula": "\\begin{align*} { { \\bf { w } } _ k ^ { \\star } } = \\sqrt { { \\mu _ k ^ { \\star } } } { { \\bf { B } } _ k } { \\bf { \\Sigma } } _ k ^ { - 1 } \\frac { { { \\bf { A } } _ k ^ H { { \\bf { e } } _ k } } } { { \\left \\| { { \\bf { A } } _ k ^ H { { \\bf { e } } _ k } } \\right \\| } } , \\ \\forall k , \\end{align*}"}
+{"id": "8697.png", "formula": "\\begin{align*} q ( A ) = \\sum _ i q ( E _ i ) p ( A | E _ i ) , A \\in \\bf { A } . \\end{align*}"}
+{"id": "5499.png", "formula": "\\begin{align*} L ( p ) = \\frac { 6 7 } { 8 0 } + \\frac { 1 3 } { 4 0 ( p + 2 ) } + \\frac { 3 7 7 } { 1 5 3 6 0 0 } p + c _ 1 \\cdot p 1 0 ^ p + c _ 2 \\cdot p 5 ^ p + \\sum _ i \\lambda _ i p a _ i ^ p \\end{align*}"}
+{"id": "8424.png", "formula": "\\begin{align*} \\begin{aligned} k ^ { - 1 } \\psi ^ { \\pm } _ { 2 1 } ( x ; k ) & = - \\int _ { \\pm \\infty } ^ { x } e ^ { 2 i z ( x - y ) } \\bar { u } _ y e ^ { - i c _ \\pm } d y \\\\ & - \\int _ { \\pm \\infty } ^ { x } e ^ { 2 i z ( x - y ) } \\bar { u } _ y \\left ( \\psi ^ { \\pm } _ { 1 1 } ( y ; k ) - e ^ { - i c _ \\pm } \\right ) d y . \\end{aligned} \\end{align*}"}
+{"id": "701.png", "formula": "\\begin{align*} \\aligned \\int ( I _ \\alpha \\ast f ) f d x = \\int \\left ( I _ { \\frac { \\alpha } { 2 } } \\ast f \\right ) ^ 2 d x \\geq 0 \\endaligned \\end{align*}"}
+{"id": "4775.png", "formula": "\\begin{align*} F ( \\lambda ) x = 0 . \\end{align*}"}
+{"id": "2345.png", "formula": "\\begin{align*} & \\kappa = \\frac { \\phi ' e ^ { \\phi } ( e ^ { 2 \\phi } + 2 x _ y ^ 2 ) - e ^ { \\phi } x _ { y y } } { ( e ^ { 2 \\phi } + x _ y ^ 2 ) ^ { \\frac { 3 } { 2 } } } , \\ x ( y , t ) , \\\\ & \\kappa = \\frac { \\phi ' e ^ { \\phi } y _ x ( y _ x ^ 2 e ^ { 2 \\phi } + 2 ) + e ^ { \\phi } y _ { x x } } { ( 1 + e ^ { 2 \\phi } y _ x ^ 2 ) ^ { \\frac { 3 } { 2 } } } , \\ y ( x , t ) . \\end{align*}"}
+{"id": "8842.png", "formula": "\\begin{align*} S ( t ) : = \\exp \\left ( t \\begin{pmatrix} 0 & x _ { 0 , n - 1 } \\\\ x _ { 0 , 2 } & 0 \\end{pmatrix} \\right ) . \\end{align*}"}
+{"id": "5079.png", "formula": "\\begin{align*} 0 = \\frac { \\dd } { \\dd \\tau } \\int _ { \\Omega } | U + \\tau \\tilde { U } + s ( \\tau ) U _ 0 | ^ { 2 } \\dd x \\Bigg | _ { \\tau = 0 } = 2 \\int _ { \\Omega } U \\cdot ( \\tilde { U } + \\dot { s } ( 0 ) U _ 0 ) \\dd x . \\end{align*}"}
+{"id": "6710.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ g [ \\alpha _ i , \\beta _ i ] \\prod _ { j = 1 } ^ n \\gamma _ j = e . \\end{align*}"}
+{"id": "5008.png", "formula": "\\begin{align*} S t _ { D N P } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( f , z ) - f ( z ) | | _ { \\infty } = 0 \\end{align*}"}
+{"id": "3285.png", "formula": "\\begin{align*} \\mathcal S ( t - s ) e _ i - e _ i = \\int _ s ^ t \\mathcal S ( u - s ) \\mathcal A e _ i d u , \\end{align*}"}
+{"id": "527.png", "formula": "\\begin{align*} \\partial _ { t } ^ { k + 1 } U ( x , t ) = \\int _ { \\R ^ { d } } \\partial _ { t } ^ { k } M ( t / 2 , x , y ) \\partial _ { t } u ( y , t / 2 ) d y . \\end{align*}"}
+{"id": "6220.png", "formula": "\\begin{align*} & \\frac { 1 } { 4 } \\biggl \\{ - 2 ( 2 L + 2 ) \\Q [ ( 2 L + 3 ) a _ 0 + 1 ] - 2 ( 2 L + 2 ) ( 2 L + 3 ) \\sum _ { k = 1 } ^ { [ m / 2 ] } a _ { 2 k } \\biggr \\} \\frac { f } { r } \\\\ & = - 2 ( L + 1 ) ( L + 2 ) \\Q \\frac { f } { r } = - Q \\frac { f } { r } , \\end{align*}"}
+{"id": "3440.png", "formula": "\\begin{align*} \\delta _ n < \\beta _ n = \\frac { \\ln q _ { n + 1 } } { q _ n } < \\frac { \\ln ( 4 q _ n ) } { q _ n } , \\end{align*}"}
+{"id": "5771.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = \\frac { 1 } { 2 } \\sqrt { c _ 1 } \\ X _ 1 \\cdot \\nu _ 1 \\cdot \\varphi - \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi \\end{align*}"}
+{"id": "83.png", "formula": "\\begin{align*} p : w = w _ 1 \\rightarrow w _ 2 \\rightarrow \\cdots \\rightarrow w _ { \\ell ( p ) + 1 } = w ' . \\end{align*}"}
+{"id": "5495.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial s } f ( p , s ) = \\log a - \\frac { p / 2 + 2 } { s } \\geq \\log a - \\frac { p _ k / 2 + 2 } { s _ k } \\end{align*}"}
+{"id": "9139.png", "formula": "\\begin{align*} \\dot { { x } } _ a = - \\gamma \\int _ 0 ^ 1 y _ \\delta ( { x } _ a , t ) u ( t ) d t \\ , . \\end{align*}"}
+{"id": "7518.png", "formula": "\\begin{align*} \\int _ { B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x & \\leq \\theta \\int _ { B _ { 8 \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x \\rho \\leq 1 / 8 \\varepsilon \\leq \\varepsilon _ 0 , \\end{align*}"}
+{"id": "4288.png", "formula": "\\begin{align*} u ( t , x ) \\approx u ^ h ( t , x ) = \\sum _ i w _ i ( t ) \\sigma _ i ( x ) . \\end{align*}"}
+{"id": "2495.png", "formula": "\\begin{align*} \\mathbf { D } = ( \\mathbf { \\Theta } \\mathbf { G } ) ^ { - 1 } . \\end{align*}"}
+{"id": "6175.png", "formula": "\\begin{align*} V ( r ) = \\frac { L ( L + 1 ) } { r ^ 2 } - \\frac { Q } { r } f + \\kappa B _ 1 \\frac { r } { f } + \\kappa B _ 2 \\frac { 1 } { f ^ 2 } , B _ 2 > 0 , \\end{align*}"}
+{"id": "4219.png", "formula": "\\begin{align*} \\mathbb { M } ( \\theta ) _ J = \\sum _ { w \\in X _ J } \\Bbbk { \\bf U } \\dot { w } \\eta ( \\theta ) _ J = \\sum _ { w \\in X _ J } \\Bbbk { \\bf U } _ { w _ J w ^ { - 1 } } \\dot { w } \\eta ( \\theta ) _ J . \\end{align*}"}
+{"id": "7628.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x ^ i _ t = & ~ [ A x ^ i _ t + B \\alpha ^ i _ t + f ( \\nu ^ { N , i } _ { \\boldsymbol { x } _ t } ) + b ( \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } _ t } ) ] d t + \\sigma d W ^ i _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x ^ i _ 0 = & ~ \\xi ^ i , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1926.png", "formula": "\\begin{align*} \\begin{aligned} p ( 5 n + 4 ) & \\equiv 0 \\ ( \\mathrm { m o d } \\ 5 ) , \\\\ p ( 7 n + 5 ) & \\equiv 0 \\ ( \\mathrm { m o d } \\ 7 ) , \\\\ p ( 1 1 n + 6 ) & \\equiv 0 \\ ( \\mathrm { m o d } \\ 1 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "2537.png", "formula": "\\begin{align*} d _ { 2 } ( \\mathbf { x } , \\mathbf { s } ) = \\sqrt { 2 } \\| \\mathbf { w } _ { x s } - \\mu \\mathbf { e } \\| = \\min _ { \\mathbf { D } \\in \\mathcal { G } } \\| ( \\mathbf { D } ^ { - T } \\mathbf { x } ) \\circ ( \\mathbf { D } \\mathbf { s } ) - \\mu \\mathbf { e } \\| \\end{align*}"}
+{"id": "3851.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\mathbf { w } + ( \\mathbf { v } \\cdot \\nabla ) \\mathbf { w } = ( \\mathbf { w } \\cdot \\nabla ) \\mathbf { v } , \\ \\ & D \\times ( 0 , T ) , \\\\ \\mathbf { w } ( \\cdot , 0 ) = \\nabla \\times \\mathbf { v } _ 0 ( \\cdot ) , \\ \\ & D . \\end{cases} \\end{align*}"}
+{"id": "3397.png", "formula": "\\begin{align*} h ' ( x ) = - \\frac { \\log { \\frac { f ' ( x ) } { f ( x ) } } } { f ^ 2 ( x ) } \\cdot ( - x ( f ' ( x ) ) ^ 2 + x f ( x ) f '' ( x ) + f ( x ) f ' ( x ) ) . \\end{align*}"}
+{"id": "7532.png", "formula": "\\begin{align*} \\deg _ H ( x ) = \\deg _ { H _ i } ( x ) + \\deg _ F ( x ) \\leq \\binom { p - 1 } { k - 1 } + \\frac { d } { 2 } \\leq \\frac { d } { 2 } + \\frac { d } { 2 } = d . \\end{align*}"}
+{"id": "6121.png", "formula": "\\begin{align*} [ C _ { 1 2 } \\ , , \\ , C _ { 3 4 } ] = 0 \\ , , [ C _ { 1 2 } \\ , , \\ , C _ { 1 2 3 } ] = 0 \\ , , [ C _ { 2 3 } \\ , , \\ , C _ { 1 2 3 } ] = 0 \\ , , [ C _ { 2 3 } \\ , , \\ , C _ { 2 3 4 } ] = 0 \\ , , [ C _ { 3 4 } \\ , , \\ , C _ { 2 3 4 } ] = 0 \\ , . \\end{align*}"}
+{"id": "1264.png", "formula": "\\begin{align*} \\varphi _ n ( \\eta ) : = \\sum _ { \\Delta \\cap \\Delta _ n \\neq \\emptyset } \\sum _ { \\xi _ { \\Delta } } \\left ( c _ \\Delta ( \\eta , \\xi _ \\Delta ) - \\hat { c } _ \\Delta ( \\eta , \\xi _ \\Delta ) \\right ) , \\end{align*}"}
+{"id": "7040.png", "formula": "\\begin{align*} \\aligned F _ { 0 , 8 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 1 , 7 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 2 , 6 } \\ , : = \\ , 7 2 0 , \\ \\ \\ \\ \\ F _ { 3 , 5 } & \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 4 , 4 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 5 , 3 } \\ , : = \\ , 1 2 0 , \\\\ F _ { 6 , 2 } & \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 7 , 1 } \\ , : = \\ , 6 \\ , F _ { 7 , 0 } , \\ \\ \\ \\ \\ F _ { 8 , 0 } \\ , : = \\ , \\tfrac { 3 5 } { 2 } , \\endaligned \\end{align*}"}
+{"id": "8021.png", "formula": "\\begin{align*} & m _ i = l _ i + n _ i l _ i - n _ i \\ , ( l _ i \\geq n _ i ) n _ i - l _ i - 1 \\ , ( l _ i < n _ i ) , \\\\ & m ' _ i = l ' _ i + n _ i l ' _ i - n _ i \\ , ( l ' _ i \\geq n _ i ) n _ i - l ' _ i - 1 \\ , ( l ' _ i < n _ i ) . \\end{align*}"}
+{"id": "686.png", "formula": "\\begin{align*} N _ n ( B ) = \\left \\{ \\sum _ { i = 1 } ^ k \\lambda _ i e _ i : \\lambda _ i \\in \\{ - n , \\dots , 0 , \\dots , n \\} \\sum _ { i = 1 } ^ k | \\lambda _ i | \\leq n \\right \\} \\ , . \\end{align*}"}
+{"id": "4578.png", "formula": "\\begin{align*} p ( f ( t _ { n } ) - f ( t ) ) & = p ( S _ { T - t _ { n } } v ( t _ { n } ) - S _ { T - t } v ( t ) ) \\\\ & \\leq p ( S _ { T - t _ { n } } v ( t _ { n } ) - S _ { { T } - t _ { n } } v ( t ) ) + p ( S _ { T - t _ { n } } v ( t ) - S _ { T - t } v ( t ) ) \\\\ & \\leq p ( S _ { T - t _ { n } } x _ { n } ) + p ( ( S _ { T - t _ { n } } - S _ { T - t } ) v ( t ) ) \\\\ & \\leq \\sup _ { s \\in [ 0 , T ] } p ( S _ { s } x _ { n } ) + p ( ( S _ { T - t _ { n } } - S _ { T - t } ) v ( t ) ) . \\end{align*}"}
+{"id": "984.png", "formula": "\\begin{align*} | T \\cap M ' | & = | H \\cap W ' | q ^ { d ^ 2 - d } \\\\ & \\le \\begin{cases} | W ' \\cap \\tau _ i | q ^ { d ^ 2 - d } & H \\cap U = \\tau _ i i \\in \\{ 1 , 2 \\} , \\\\ ( q ^ { d - 1 } + \\theta _ { d - 1 } ) q ^ { d ^ 2 - d } & \\end{cases} \\end{align*}"}
+{"id": "2218.png", "formula": "\\begin{align*} & \\phi = \\frac { 2 \\pi Z } { N } - A = 2 \\pi \\frac { d } { \\lambda } { s i n } \\theta _ 0 \\\\ & { s i n } \\theta _ 0 = \\frac { \\lambda ( Z \\sqrt { N } - 1 ) } { N \\sqrt { N } d } . \\end{align*}"}
+{"id": "9018.png", "formula": "\\begin{align*} i _ { 1 3 6 } & = x _ 1 x _ 3 D _ 5 , & i _ { 2 3 5 } & = D _ 1 x _ 3 x _ 5 , & i _ { 1 3 4 } & = - x _ 1 x _ 3 D _ 4 , \\\\ i _ { 1 4 5 } & = x _ 1 D _ 3 x _ 5 , & i _ { 1 2 5 } & = - x _ 1 D _ 2 x _ 5 , & i _ { 1 2 3 } & = 0 . \\end{align*}"}
+{"id": "301.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } f ( t ) = 0 f ( T ) \\leq \\frac { 1 } { 1 6 n } \\cdot \\end{align*}"}
+{"id": "3610.png", "formula": "\\begin{align*} \\mathbb { H } ^ { n + 1 } = \\{ ( x , x _ { n + 1 } ) \\in \\mathbb { R } ^ { n + 1 } : x _ { n + 1 } > 0 \\} , d s ^ 2 = \\frac { 1 } { x _ { n + 1 } ^ 2 } \\sum _ { i = 1 } ^ { n + 1 } d x _ i ^ 2 , \\end{align*}"}
+{"id": "6817.png", "formula": "\\begin{align*} h _ { p } ^ { \\left ( 0 \\right ) } = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\varphi _ { 0 } ^ { - } \\left ( \\tau \\right ) \\frac { z ^ { p } ( \\tau ) } { \\left ( \\frac { 1 } { 2 } - i \\tau \\right ) ^ { 2 } } d \\tau , p = 1 , 2 , \\ldots \\end{align*}"}
+{"id": "75.png", "formula": "\\begin{align*} k ( w t ^ \\mu , \\alpha ) = \\langle \\mu , \\alpha ^ \\vee \\rangle + \\Phi ^ + ( ( \\alpha ) ( w ^ { - 1 } ) ) - \\Phi ^ + ( \\alpha ) . \\end{align*}"}
+{"id": "2586.png", "formula": "\\begin{align*} & R \\cdot R ( J X _ 1 , J X _ 2 , X _ 3 , X _ 4 ; X , Y ) = R \\cdot R ( X _ 1 , X _ 2 , J X _ 3 , J X _ 4 ; X , Y ) \\\\ & = R \\cdot R ( X _ 1 , X _ 2 , X _ 3 , X _ 4 ; J X , J Y ) = R \\cdot R ( X _ 1 , X _ 2 , X _ 3 , X _ 4 ; X , Y ) , \\end{align*}"}
+{"id": "5774.png", "formula": "\\begin{align*} \\nabla _ X \\psi _ 2 = - \\frac { 1 } { 2 } X _ 1 \\cdot \\psi _ 2 - \\frac { 1 } { 2 } S ( X ) \\cdot N \\cdot \\psi _ 2 \\end{align*}"}
+{"id": "5307.png", "formula": "\\begin{align*} f ( x ) = ( 2 \\pi ) ^ { - n / 2 } \\int _ { - \\infty } ^ \\infty \\int _ { S ^ { n - 1 } } \\widehat { f } ( \\lambda , \\omega ) e _ { \\lambda , \\omega } ( x ) | \\lambda | ^ { n - 1 } d \\omega \\ , d \\lambda \\end{align*}"}
+{"id": "6602.png", "formula": "\\begin{align*} \\mathcal { G } ^ { \\mathrm { e n d } } = \\left ( \\begin{array} { c c } A & g ^ { - 1 } \\\\ h & A ^ * \\end{array} \\right ) \\end{align*}"}
+{"id": "3106.png", "formula": "\\begin{align*} A = \\{ \\omega \\in \\Omega \\mid \\exists ~ x \\in C ( \\omega ) ~ s . t . ~ \\langle \\mathcal { F } ( x ^ * ) ( \\omega ) , x - x ^ * ( \\omega ) \\rangle < 0 \\} . \\end{align*}"}
+{"id": "6563.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 0 = \\Delta c - \\tilde { n } c , & x \\in \\Omega , \\\\ \\nabla c \\cdot \\nu = \\gamma - c , & x \\in \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "9303.png", "formula": "\\begin{align*} D _ j ( U ^ { i j } D _ i w ) = f \\quad \\ , \\ , \\ , \\ , B _ 1 \\end{align*}"}
+{"id": "5422.png", "formula": "\\begin{align*} \\begin{aligned} u ( \\bar { x } ) \\leq \\ , & u ( x ^ * ) + C \\epsilon _ 0 \\delta ^ 2 b _ { \\alpha } ^ 2 + C b _ { \\alpha } \\bar { x } _ n + C \\delta ^ 3 b _ \\alpha ^ 2 \\\\ \\leq \\ , & u ( x ^ * ) + C \\left ( \\epsilon _ 0 + \\delta \\right ) \\delta ^ 2 b _ \\alpha ^ 2 \\leq \\left ( 1 + \\frac { c _ 0 } { 3 } \\right ) u ( x ^ * ) \\end{aligned} \\end{align*}"}
+{"id": "845.png", "formula": "\\begin{align*} \\mathcal { L } w = - \\mathcal { L } c = \\partial ^ \\square c - \\big ( \\Delta _ \\Gamma G ' ( c ) + c V H \\big ) = 0 [ 0 , T ] \\times M \\end{align*}"}
+{"id": "891.png", "formula": "\\begin{align*} r ^ 2 + r q + q ^ 2 = 2 n \\ , . \\end{align*}"}
+{"id": "1930.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty b _ 3 \\left ( \\frac { m n - 1 } { 1 2 } \\right ) q ^ n \\in S _ { 3 m - 3 } ( \\Gamma _ 0 ( 4 3 2 ) , \\chi _ { 1 2 } ) _ m , \\end{align*}"}
+{"id": "8401.png", "formula": "\\begin{align*} \\lim _ { | z | \\rightarrow \\infty } \\Psi ^ - _ 1 ( x ; z ) = e ^ { - i c _ - ( x ) } e _ 1 . \\end{align*}"}
+{"id": "4589.png", "formula": "\\begin{align*} F = \\bigcup _ { n = 0 } ^ \\infty K _ n , \\end{align*}"}
+{"id": "8933.png", "formula": "\\begin{align*} H _ { j , n } = \\frac { \\sqrt { n ! } } { \\Gamma ( \\beta ) \\sqrt { \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } } } \\int _ { 0 } ^ { \\infty } x ^ { j } L _ { n } ^ { ( \\beta ) } ( x ) x ^ { \\beta - 1 } \\exp ( - x ) d x . \\end{align*}"}
+{"id": "362.png", "formula": "\\begin{align*} S ' u _ k + T v _ k = ( \\frac { - 1 } { 8 \\cos ^ { 2 } ( \\frac { \\pi k } { n - 1 } ) } ) S ' S v _ k + T v _ k . \\end{align*}"}
+{"id": "6758.png", "formula": "\\begin{align*} \\Phi _ s ( z ) ^ \\alpha = \\{ 1 - [ 1 - \\Phi _ s ( z ) ] \\} ^ \\alpha = \\sum _ { j = 0 } ^ \\infty ( - 1 ) ^ j \\ , \\binom { \\alpha } { j } \\ , [ 1 - \\Phi _ s ( z ) ] ^ j . \\end{align*}"}
+{"id": "7970.png", "formula": "\\begin{align*} \\frac { \\partial h } { \\partial t } \\geq \\frac { 1 } { f _ { \\max } } h [ h ^ { - \\varepsilon } \\beta _ { 1 } - f _ { \\max } ] . \\end{align*}"}
+{"id": "5900.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\langle A _ 0 ( u _ n ) , u _ n - u \\rangle = 0 . \\end{align*}"}
+{"id": "5875.png", "formula": "\\begin{align*} X ( t ) = X _ 0 + \\int _ 0 ^ t A ( s , \\bar { X } ( s ) ) d s + \\int _ 0 ^ t B ( s , \\bar { X } ( s ) ) d W ( s ) , \\quad \\forall \\ t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "2625.png", "formula": "\\begin{align*} \\begin{array} { l } \\circlearrowleft _ { x , y , z } \\varepsilon ( z , x ) \\Big ( [ \\alpha ( h ) , [ x , y ] , \\alpha ^ 2 ( z ) ] \\Big ) \\\\ + \\circlearrowleft _ { x , y , z } \\varepsilon ( z , x ) \\Big ( \\varepsilon ( h , x + y ) [ \\alpha ( [ x , y ] ) , \\alpha ( h \\cdot z ) ] \\Big ) = 0 . \\end{array} \\end{align*}"}
+{"id": "1491.png", "formula": "\\begin{align*} \\phi _ 2 ' = \\phi _ 4 , \\end{align*}"}
+{"id": "7496.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } \\div ( A ( x ) \\nabla \\varphi ) = 0 & B _ 1 \\\\ A ( x ) \\nabla \\varphi \\cdot \\nu = g & \\partial B _ 1 , \\end{array} \\right . \\end{align*}"}
+{"id": "8429.png", "formula": "\\begin{align*} a ( z ) = 1 + k \\int _ { \\mathbb { R } } u _ y ( y ) \\psi ^ - _ { 2 1 } ( x ; k ) \\mathrm { d } y . \\end{align*}"}
+{"id": "2785.png", "formula": "\\begin{align*} \\mathbb { P } ( h ^ { U } ( x ) \\geq a ) & \\leq \\mathbb { P } ( h ^ { U } ( x ) \\geq a , \\varphi ^ { V , U } ( x ) \\geq 0 ) + \\mathbb { P } ( h ^ { U } ( x ) \\geq a , \\varphi ^ { V , U } ( x ) \\leq 0 ) \\\\ & = 2 \\mathbb { P } ( h ^ { U } ( x ) \\geq a , \\varphi ^ { V , U } ( x ) \\geq 0 ) \\leq 2 \\mathbb { P } ( h ^ { V } ( x ) \\geq a ) . \\end{align*}"}
+{"id": "7027.png", "formula": "\\begin{align*} B _ 2 \\ , : = \\ , - \\ , \\tfrac { 1 } { 6 } \\ , F _ { 5 , 0 } \\ , T _ 1 - 0 . \\end{align*}"}
+{"id": "7583.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\lim _ { m \\to \\infty } T ^ { c _ 1 ( j ) + c _ 2 ( m ) } a = \\lim _ { m \\to \\infty } \\lim _ { j \\to \\infty } T ^ { c _ 1 ( j ) + c _ 2 ( m ) } a . \\end{align*}"}
+{"id": "3260.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb E \\left [ Z g ( U _ n ( Y ) ) \\right ] = \\tilde { \\mathbb E } \\left [ Z g ( U ( Y ) ) \\right ] , \\end{align*}"}
+{"id": "2714.png", "formula": "\\begin{align*} \\begin{aligned} & ~ & & \\| g _ k - \\nabla \\phi ( x _ k ) \\| = \\| g _ k - L _ 1 x _ k \\| \\le \\kappa _ { \\rm e g } \\delta _ k + \\epsilon _ g \\ ; \\ ; & & \\end{aligned} \\end{align*}"}
+{"id": "1527.png", "formula": "\\begin{align*} \\frac { d g ( t ) } { d t } g ( t ) ^ { - 1 } = v ( t ) . \\end{align*}"}
+{"id": "6713.png", "formula": "\\begin{align*} ( \\pi ( \\prod _ { i = 2 } ^ g [ \\alpha _ i . \\beta _ i ] \\prod _ { 1 = 1 } ^ n \\gamma _ j ) ) ^ { - 1 } = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} . \\end{align*}"}
+{"id": "6579.png", "formula": "\\begin{align*} \\begin{aligned} & \\| c \\varphi \\| _ { W ^ { 1 , 6 } ( \\Omega ) } \\\\ & \\le C ( \\| \\Delta ( c \\varphi ) \\| _ { L ^ { \\frac { 3 } { 2 } } ( \\Omega ) } + \\| c \\varphi \\| _ { L ^ { \\frac { 3 } { 2 } } ( \\Omega ) } + \\| ( \\gamma - c ) \\varphi \\| _ { W ^ { \\frac { 1 } { 3 } , \\frac { 3 } { 2 } } ( \\partial \\Omega ) } + \\| c \\nabla \\varphi \\cdot \\nu \\| _ { W ^ { \\frac { 1 } { 3 } , \\frac { 3 } { 2 } } ( \\partial \\Omega ) } ) . \\end{aligned} \\end{align*}"}
+{"id": "418.png", "formula": "\\begin{align*} \\begin{aligned} d ( x , x ' ) & \\leq d ( x , x H _ 2 \\cap x ' H _ 1 ) + d ( x H _ 2 \\cap x ' H _ 1 , x ' ) \\leq 2 d ( x ' , x H _ 2 \\cap x ' H _ 1 ) \\\\ & \\leq 2 d ( x ' , x ' H _ 1 ) \\leq 2 d ( 1 , \\pi _ { H _ 1 } ( x ^ { - 1 } x ' ) ) . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "3340.png", "formula": "\\begin{align*} \\chi _ i = \\left \\{ \\begin{aligned} & \\psi _ i & & ( i \\neq j ) \\\\ & \\phi _ j & & ( i = j ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6058.png", "formula": "\\begin{align*} n H = \\eta x _ { n + 1 } ^ m + \\lambda . \\end{align*}"}
+{"id": "8996.png", "formula": "\\begin{align*} ( - 1 ) ^ { p ( \\eta , \\ell _ { m } ) - 1 } i _ { \\eta \\cup \\ell _ { m } } & = - ( - 1 ) ^ { p ( \\eta , \\ell _ { m } ) } ( - 1 ) ^ { | \\pi ( \\eta \\cup \\ell _ { m } ) | } { \\prod _ { r = 1 } ^ { d + 1 } U ^ { \\eta \\cup \\ell _ { m } } _ { r } } \\\\ & = - U { \\prod _ { r = m + 1 } ^ M U ^ { \\eta \\cup \\ell _ { m } } _ { r } } , \\end{align*}"}
+{"id": "264.png", "formula": "\\begin{align*} d ( D , E ) = \\max \\left \\{ \\sup _ { p \\in D } d ( p , E ) , \\ \\sup _ { p \\in E } d ( p , D ) \\right \\} . \\end{align*}"}
+{"id": "2513.png", "formula": "\\begin{align*} \\mathbf { R } _ { x s } ^ i \\triangleq \\begin{pmatrix} & \\dot { \\mathbf { w } } _ 1 & \\dot { \\mathbf { w } } _ { 2 : n _ i } ^ T \\\\ & \\dot { \\mathbf { w } } _ { 2 : n _ i } & \\tilde { \\mathbf { R } } _ { x s } ^ i \\end{pmatrix} \\end{align*}"}
+{"id": "4093.png", "formula": "\\begin{align*} ( F \\circ G ) ^ 2 ( x ) & = ( f ^ i ( g ^ m ( x ) ) , f ^ i { } _ \\alpha ( g ^ m ( x ) ) g ^ \\alpha { } _ j ( x ) , \\\\ * & \\hphantom { { } = { } } f ^ i { } _ { \\alpha \\beta } ( g ^ m ( x ) ) g ^ \\alpha { } _ j ( x ) g ^ \\beta { } _ k ( x ) + f ^ i { } _ \\alpha ( g ^ m ( x ) ) g ^ \\alpha { } _ { j k } ( x ) ) . \\end{align*}"}
+{"id": "4921.png", "formula": "\\begin{align*} M ( - t ) = \\frac { 1 } { \\sqrt 2 \\pi } \\sum _ { r , j = 0 } ^ \\infty \\ , \\pi _ r \\ , \\ , c _ { r , j } \\ , \\int _ { - \\infty } ^ { \\infty } \\ , x ^ { j } \\ , \\exp \\left ( - t x - \\frac { x ^ 2 } { 2 } \\right ) \\mathrm { d } x . \\end{align*}"}
+{"id": "6000.png", "formula": "\\begin{align*} \\| u \\| _ { 2 } ^ { 2 } + \\omega \\| v \\| _ { 2 } ^ { 2 } + 2 \\int _ { \\R ^ { 2 } } \\bigg ( \\frac { h ^ { 2 } ( | x | ) } { | x | ^ { 2 } } u ^ { 2 } + \\frac { g ^ { 2 } ( | x | ) } { | x | ^ { 2 } } v ^ { 2 } \\bigg ) d x = \\frac { 1 } { p } \\Big ( \\| ( u , v ) \\| _ { 2 p } ^ { 2 p } + 2 b \\| u v \\| _ { p } ^ { p } \\Big ) . \\end{align*}"}
+{"id": "6300.png", "formula": "\\begin{align*} R _ t = \\frac { 1 } { 2 } \\mu \\gamma _ t ( \\alpha _ t ^ { - 1 } - 1 ) \\| x ^ t - y ^ t \\| ^ 2 - \\frac { 1 } { 2 \\alpha _ t } \\| x ^ { t + 1 } - y ^ t \\| ^ 2 . \\end{align*}"}
+{"id": "7199.png", "formula": "\\begin{align*} \\frac { i } { 2 \\pi } \\int _ { \\mathcal { C } } e ^ { - t \\tau } \\operatorname { T r } \\phi _ { - 1 } \\ , d \\tau = e ^ { - t \\alpha | \\xi | } + ( n - 2 ) e ^ { - t \\mu | \\xi | } + e ^ { - 2 t \\mu | \\xi | } + e ^ { - 2 t \\mu s _ 1 | \\xi | } . \\end{align*}"}
+{"id": "4242.png", "formula": "\\begin{align*} V ^ { - 1 } \\rho \\circ \\varphi V = V ^ { - 1 } \\varphi \\circ \\rho V = V ^ { - 1 } \\rho V . \\end{align*}"}
+{"id": "7900.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Theta ( t , x ) = \\int _ \\S \\int _ \\S W _ r ( m _ 1 z + m _ 2 y + m _ 3 x ) ~ \\sin ( n _ 1 \\Theta ( t , z ) + n _ 2 \\Theta ( t , y ) + n _ 3 \\Theta ( t , x ) ) \\ \\d y \\d z , \\end{align*}"}
+{"id": "7671.png", "formula": "\\begin{align*} \\{ \\lambda _ { t } ^ { N , j } \\} _ { 1 \\leq j \\leq N } \\subset \\Big \\{ \\lambda \\in \\mathbb C \\ , : \\ , \\left | \\lambda - 1 \\right | \\leq \\frac { 1 } { N - 1 } \\sum _ { j \\not = i } h ' ( \\mu ^ { N , j } _ { \\boldsymbol { \\alpha } ^ * _ t } ) \\Big \\} . \\end{align*}"}
+{"id": "2084.png", "formula": "\\begin{align*} I _ k \\cap ( H \\setminus P ) = \\bigcup _ { n = n _ k } ^ { \\infty } ( I _ k \\cap ( H _ n \\setminus K ) \\cap ( \\bigcap _ { m = 1 } ^ { \\infty } ( H _ n \\setminus P _ m ) ) ) \\end{align*}"}
+{"id": "2525.png", "formula": "\\begin{align*} { { \\left ( \\mathbf { x } + \\Delta \\mathbf { x } \\right ) } ^ { T } } \\left ( \\mathbf { s } + \\Delta \\mathbf { s } \\right ) = \\nu { { \\mathbf { x } } ^ { T } } \\mathbf { s } + \\Delta \\mathbf { x } ^ T \\Delta \\mathbf { s } . \\end{align*}"}
+{"id": "6658.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\chi _ 3 < 1 \\Big ) \\leq \\lim _ { K \\to + \\infty } \\lim _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\sup _ { t \\in [ 0 , 1 ] } | X ^ { \\epsilon } ( t ) | \\geq K \\Big ) = - \\infty . \\end{aligned} \\end{align*}"}
+{"id": "667.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | e ^ { i k \\phi } \\sin ( n \\theta ) \\left | l _ 2 , m _ 2 \\right \\rangle = \\end{align*}"}
+{"id": "6769.png", "formula": "\\begin{align*} g ( \\rho , x ) = e ^ { - i \\rho x } + \\int _ { - \\infty } ^ { x } B ( x , t ) e ^ { - i \\rho t } d t \\end{align*}"}
+{"id": "4583.png", "formula": "\\begin{align*} - F v = \\norm { v } _ { V } F \\left ( \\frac { - v } { \\norm { v } _ { V } } \\right ) = \\norm { v } _ { V } ( G \\widetilde { w } + z ) = G ( \\norm { v } _ { V } \\widetilde { w } ) + \\norm { v } _ { V } z , \\end{align*}"}
+{"id": "8948.png", "formula": "\\begin{align*} \\int _ { a ( t ) } ^ t v _ 1 ^ { - p ' } = \\int _ t ^ { b ( t ) } v _ 1 ^ { - p ' } , t > 0 , \\end{align*}"}
+{"id": "8165.png", "formula": "\\begin{align*} \\begin{array} { r c l l } \\xi ^ { \\mathrm { i n } } & = & 0 & a . e . ~ \\mbox { i n } ( R _ { 0 } , \\infty ) , \\\\ \\mathcal { I } _ { 1 } & = & 0 & a . e . ~ \\mbox { i n } ( 0 , t _ { 0 } ) \\times ( R _ { 0 } , \\infty ) , \\\\ \\mathcal { I } _ { 2 } & = & 0 & a . e . ~ \\mbox { i n } ~ ( 0 , t _ { 0 } ) \\times ( R _ { 0 } , \\infty ) . \\end{array} \\end{align*}"}
+{"id": "1247.png", "formula": "\\begin{align*} \\int _ \\Omega \\left ( P ( t ) f ( \\eta ) \\right ) g ( \\eta ) \\mu ( d \\eta ) = \\int _ \\Omega f ( \\eta ) ( \\hat { P } ( t ) g ( \\eta ) ) \\mu ( d \\eta ) , t \\geq 0 . \\end{align*}"}
+{"id": "4482.png", "formula": "\\begin{align*} \\sum _ { r _ { Q } ( n ) = 0 } n \\ll \\max \\left \\{ \\frac { p ^ { 3 + \\epsilon } } { ( \\min Q ^ { * } ) ^ { 2 } } , p ^ { 5 / 2 + \\epsilon } \\right \\} . \\end{align*}"}
+{"id": "249.png", "formula": "\\begin{align*} g & = p _ { k - 1 } s + t , \\\\ h & = p _ { k - 1 } s ' + t ' , \\end{align*}"}
+{"id": "5751.png", "formula": "\\begin{align*} \\varphi _ { c , d } ( x ) + \\varphi _ { c , d } ( x + 1 ) = \\frac { x ^ { q + 1 } + d + ( x + 1 ) ^ { q + 1 } + d } { x ^ q + x + c } = \\frac { x ^ q + x + 1 } { x ^ q + x + c } \\ne 0 , \\end{align*}"}
+{"id": "6190.png", "formula": "\\begin{align*} B _ 2 = L ( L + 1 ) , \\zeta = \\kappa ( L + 1 ) , \\end{align*}"}
+{"id": "7458.png", "formula": "\\begin{align*} { \\rm G a l } ( L ' / K ' ) = { \\rm G a l } ( L ' / N _ 1 ) \\times { \\rm G a l } ( L ' / N _ 2 ) . \\end{align*}"}
+{"id": "3909.png", "formula": "\\begin{align*} P _ \\delta ( z ) = \\frac { \\pi \\delta ^ 2 } { \\ln \\frac { R } { \\varepsilon } } q ^ 2 \\sqrt { d e t ( K _ H ) } ( z ) + N _ \\delta ( z ) , \\end{align*}"}
+{"id": "334.png", "formula": "\\begin{align*} \\pi ( \\lambda ) _ { r + 1 } - \\pi ( \\lambda ) _ r - 1 = ( \\pi s _ r ) ( \\lambda ) _ r - \\pi ( \\lambda ) _ r \\end{align*}"}
+{"id": "1273.png", "formula": "\\begin{align*} \\hat { \\Lambda } _ n = [ - 2 ^ n + n + L + 1 , 2 ^ n - n - L - 1 ] ^ d , L : = \\max _ { \\Delta \\ni 0 : c _ { \\Delta } > 0 } ( \\Delta ) + 1 . \\end{align*}"}
+{"id": "1713.png", "formula": "\\begin{align*} \\Delta u ( t ) = U ( t , 0 ) \\Delta u _ 0 & + \\int _ 0 ^ t U ( t , s ) [ 2 i ( D _ j \\nabla \\tilde { b } ^ j ( s ) + \\nabla \\tilde { b } ^ j ( s ) D _ j + \\nabla \\tilde { c } ( s ) ) \\nabla u \\\\ & + i ( D _ j \\Delta \\tilde { b } ^ j ( s ) + \\Delta \\tilde { b } ^ j ( s ) D _ j + \\Delta \\tilde { c } ( s ) ) u + \\Delta f ( s ) ] d s . \\end{align*}"}
+{"id": "7495.png", "formula": "\\begin{align*} \\int _ { B _ { 2 \\rho } } r ^ { 4 - n } | D ^ 2 u | | \\nabla u | \\d x & \\leq C \\left ( \\int _ { B _ { 4 \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x \\right ) ^ { 1 / 2 } \\left ( \\int _ { B _ { 4 \\rho } } r ^ { 4 - n } | \\nabla u | ^ 2 \\d x \\right ) ^ { 1 / 2 } \\\\ & + C \\int _ { B _ { 4 \\rho } } r ^ { 3 - n } | \\nabla u | ^ 2 \\d x , \\end{align*}"}
+{"id": "7922.png", "formula": "\\begin{align*} 1 6 c _ 5 ( q , 1 , p _ 0 ) & = \\hat W _ { r _ 0 } ( q - 2 ) - 4 \\hat W _ { r _ 0 } ( q - 1 ) + 6 \\hat W _ { r _ 0 } ( q ) - 4 \\hat W _ { r _ 0 } ( q + 1 ) + \\hat W _ { r _ 0 } ( q + 2 ) \\\\ & = \\hat W _ { r _ 0 } ( q - 2 ) - 2 \\hat W _ { r _ 0 } ( q ) + \\hat W _ { r _ 0 } ( q + 2 ) = : q _ 1 . \\end{align*}"}
+{"id": "5869.png", "formula": "\\begin{align*} C ^ { \\ast } \\left ( X , Y , Z \\right ) \\equiv \\left ( \\nabla _ { Z } ^ { \\ast } g \\right ) \\left ( X , Y \\right ) = - C \\left ( X , Y , Z \\right ) \\end{align*}"}
+{"id": "7453.png", "formula": "\\begin{align*} \\max \\bigg \\lbrace \\sum _ { i , j = 1 } ^ n & t _ i t _ j | U ^ \\top U | _ { i j } : t \\in \\mathbb { R } ^ n , \\ ; \\| t \\| _ 2 = 1 , U \\in \\mathbb { R } ^ { m \\times n } , \\ ; U U ^ \\top = { \\rm I } _ m \\bigg \\rbrace \\\\ & \\le \\frac { m } { n } \\left ( 1 + \\sqrt { \\frac { ( n - 1 ) ( n - m ) } { m } } \\right ) , \\end{align*}"}
+{"id": "5858.png", "formula": "\\begin{align*} { p _ { E O S } } = \\frac { { \\rho R T } } { { 1 - b \\rho } } - \\frac { { a \\xi \\left ( T \\right ) { \\rho ^ 2 } } } { { 1 + 2 b \\rho - { b ^ 2 } { \\rho ^ 2 } } } , \\end{align*}"}
+{"id": "1645.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\cdot P \\right ) ( X ) = \\det \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} ^ { \\frac { w - n } { 2 } } ( a + c X ) ^ n P \\left ( \\frac { b + d X } { a + c X } \\right ) . \\end{align*}"}
+{"id": "8737.png", "formula": "\\begin{align*} d _ \\textrm { H S } ( A , B ) = d _ \\textrm { H S } ( C A D , C B D ) \\forall A , B , C , D \\in \\mathrm { U } ( n ) . \\end{align*}"}
+{"id": "5569.png", "formula": "\\begin{align*} W ( [ 0 ^ l 1 ] | [ 0 ^ k 1 ] ) = \\sum _ { n = 1 } ^ l ( a _ { n + k } - a ) + d _ { l + k } - d \\ . \\end{align*}"}
+{"id": "5973.png", "formula": "\\begin{align*} \\sigma ( \\sqrt { a } ) = - \\sqrt { a } , \\ & \\sigma ( \\sqrt { d } ) = \\sqrt { d } \\\\ \\tau ( \\sqrt { a } ) = \\sqrt { a } , \\ & \\tau ( \\sqrt { d } ) = - \\sqrt { d } . \\end{align*}"}
+{"id": "8770.png", "formula": "\\begin{align*} { } ^ { \\psi } \\mathcal D \\circ { } ^ { \\psi } \\mathcal T [ \\mathfrak { f } ] = \\mathfrak { f } , \\ \\ \\forall \\mathfrak { f } \\in L _ 2 ( \\Omega , \\mathbb H ) \\cup C ( \\Omega , \\mathbb H ) . \\end{align*}"}
+{"id": "1386.png", "formula": "\\begin{align*} g _ { x _ 1 x _ 2 \\ldots x _ d } = ( - 1 ) ^ d \\prod _ { j = 1 } ^ d \\lambda _ j g . \\end{align*}"}
+{"id": "6666.png", "formula": "\\begin{align*} \\nu ( 1 ) + \\nu ( - 1 ) = 1 , \\ \\ \\ \\ 0 \\leq \\nu ( 1 ) \\leq 1 . \\end{align*}"}
+{"id": "6249.png", "formula": "\\begin{align*} \\varphi _ i : = \\frac { \\langle \\alpha ^ f ( X _ i , X _ i ) , \\alpha ^ f ( X _ i , X _ i ) \\rangle } { \\langle \\beta ( X _ i , X _ i ) , \\beta ( X _ i , X _ i ) \\rangle } \\neq 0 , \\end{align*}"}
+{"id": "5461.png", "formula": "\\begin{align*} d = 4 + 4 n = 8 + 8 l \\equiv 0 \\mod 8 \\end{align*}"}
+{"id": "8918.png", "formula": "\\begin{align*} ( 1 + \\rho ) ^ { k } = \\frac { 1 } { 2 ^ { k } } \\sum _ { j = 0 } ^ { 2 k } \\sum _ { m = 0 } ^ { \\left \\lfloor j / 2 \\right \\rfloor } ( 2 \\rho ) ^ { j - 2 m } \\binom { k } { j - 2 m } \\binom { k - j + 2 m } { m } . \\end{align*}"}
+{"id": "9189.png", "formula": "\\begin{align*} I _ n : = \\left [ n \\dfrac { \\bar { t } } { \\gamma } , \\ , ( n + 1 ) \\dfrac { \\bar { t } } { \\gamma } \\right ) n \\in { \\mathbb { N } } \\end{align*}"}
+{"id": "5243.png", "formula": "\\begin{align*} \\varphi = \\begin{bmatrix} v \\\\ b \\\\ \\end{bmatrix} , F = \\begin{bmatrix} ( d + B _ { \\infty } ) \\otimes b - ( w + u _ { \\infty } ) \\otimes v \\\\ ( d + B _ { \\infty } ) \\otimes v - ( w + u _ { \\infty } ) \\otimes b \\\\ \\end{bmatrix} , { \\mathcal { L } } = \\begin{bmatrix} \\nu & 0 \\\\ 0 & \\mu \\\\ \\end{bmatrix} \\Delta . \\end{align*}"}
+{"id": "4500.png", "formula": "\\begin{align*} \\int _ { X } ( \\alpha + \\sqrt { - 1 } \\omega ) ^ { 2 } = \\int _ { X } ( \\alpha ^ { 2 } - \\omega ^ { 2 } ) + 2 \\sqrt { - 1 } \\int _ { X } \\alpha \\wedge \\omega = 3 2 + 2 6 \\sqrt { - 1 } . \\end{align*}"}
+{"id": "3014.png", "formula": "\\begin{align*} | \\tilde { \\varphi } ( t , x ) - \\tilde { \\varphi } ( t ' , x ' ) | \\leq \\varepsilon , ( t ' , x ' ) \\in O _ * : = O _ \\nu ( t , x ) \\cap ( [ \\tau , \\tau + \\delta _ * ] \\times \\mathbb R ^ n ) . \\end{align*}"}
+{"id": "1621.png", "formula": "\\begin{align*} C ( H , K , M ) : = \\bigoplus _ { n \\geq 0 } C ^ n ( H , K , M ) \\end{align*}"}
+{"id": "2948.png", "formula": "\\begin{align*} T : = T ^ e _ A ( x ) . \\end{align*}"}
+{"id": "8241.png", "formula": "\\begin{align*} ( \\mathcal { R } _ \\lambda ( \\sigma s _ i ) v _ T , v _ T ) = \\pm \\frac { 1 } { a _ { i + 1 } - a _ { i } } ( \\mathcal { R } _ \\lambda ( \\sigma ) v _ T , v _ T ) . \\end{align*}"}
+{"id": "4248.png", "formula": "\\begin{align*} \\lim _ { j \\rightarrow \\infty } M _ j ( C _ \\varphi ) = \\frac { 1 } { k } \\sum _ { i = 0 } ^ { k - 1 } C _ { \\rho \\circ \\varphi _ i } . \\end{align*}"}
+{"id": "3283.png", "formula": "\\begin{align*} N _ { k , l } = \\frac 1 { \\sqrt 2 } \\sum _ { c , b = 1 } ^ d \\int _ 0 ^ t \\left ( \\hat { \\sigma } _ { k l , b c } ( s ) + \\hat { \\sigma } _ { l k , b c } ( s ) \\right ) d B ^ { c b } _ s . \\end{align*}"}
+{"id": "1651.png", "formula": "\\begin{align*} r = \\mathrm { a d } ^ 0 \\rho _ \\pi : G _ { F , S } \\to H ( E ) . \\end{align*}"}
+{"id": "1380.png", "formula": "\\begin{align*} h ( x ) = \\prod _ { j = 1 } ^ d \\lambda _ j \\int _ { x _ d } ^ \\infty d b _ d \\int _ { x _ { d - 1 } } ^ { b _ { d } } d b _ { d - 1 } \\cdots \\int _ { x _ 1 } ^ { b _ { 2 } } d b _ 1 e ^ { \\sum _ { i = 1 } ^ d \\lambda _ i ( x _ i - b _ i ) } h ( b _ 1 , \\ldots , b _ d ) . \\end{align*}"}
+{"id": "7178.png", "formula": "\\begin{align*} ( q _ 1 - b _ 1 ) q _ 0 + q _ 0 q _ 1 = E _ 1 , \\end{align*}"}
+{"id": "5592.png", "formula": "\\begin{align*} A ( y _ { l + 1 } . . . y _ 1 x ) - A ( y _ { l + 1 } . . . y _ 1 x ' ) = A ( 0 1 ^ { l + k } 0 . . . ) - A ( 0 1 ^ l 0 ^ \\infty ) = b _ { l + k } - b _ l \\ . \\end{align*}"}
+{"id": "713.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\aligned & - \\Delta u + V ( x ) u = \\left ( I _ \\alpha \\ast | u | ^ { p } \\right ) | u | ^ { p - 2 } u + \\lambda u ~ \\R ^ N , \\\\ & u \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { N } \\right ) \\endaligned \\end{array} \\right . \\end{align*}"}
+{"id": "9340.png", "formula": "\\begin{align*} \\limsup _ { r \\to 0 ^ + } r ^ { 1 - \\frac { 2 } { p } - \\frac { 3 } { q } } \\| v _ 3 \\| _ { L ^ p _ t L ^ q _ x ( Q _ r ( x _ 0 , r ^ 2 ) ) } = 0 1 \\leq p , q \\leq \\infty \\end{align*}"}
+{"id": "2728.png", "formula": "\\begin{align*} \\psi ( t ) : = \\frac { 2 \\sqrt { n } } { n + 1 } \\Bigl ( t ( n + 1 - t ) \\Bigr ) ^ { 1 / 2 } + \\left | 1 - \\frac { 2 t } { n + 1 } \\right | . \\end{align*}"}
+{"id": "1314.png", "formula": "\\begin{align*} A ( t ) = 2 { \\rm R e } ( u , u _ t ) + b | | u | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } . \\end{align*}"}
+{"id": "2416.png", "formula": "\\begin{align*} F ^ { ( n ) } \\left ( \\frac { n } { x } \\right ) = ( - 1 ) ^ { n } x ^ { n + 1 } \\int _ { 0 } ^ { \\infty } ( y e ^ { - y } ) ^ { n } f ( x y ) d y . \\end{align*}"}
+{"id": "1705.png", "formula": "\\begin{align*} b ( t , \\xi ) = 2 \\nabla W ( t , \\xi ) , \\end{align*}"}
+{"id": "231.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\left | R _ q ( n ) \\right | } { | B _ S ( n ) | } = 0 \\end{align*}"}
+{"id": "6880.png", "formula": "\\begin{align*} u ( z ) = \\Re f ( z ) , z \\in D . \\end{align*}"}
+{"id": "9240.png", "formula": "\\begin{align*} \\mathcal { H } _ m ( A ) : = S _ m ( \\lambda ( A ) ) , \\end{align*}"}
+{"id": "7776.png", "formula": "\\begin{align*} s _ i ' + \\sum _ { j = 1 } ^ { i - 1 } p ' _ { d - j } s ' _ { i - j } = - i p ' _ { d - i } , ~ i = 1 , \\dots , k + 1 . \\end{align*}"}
+{"id": "5918.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { t \\leq T } \\Vert X ( t , x _ n ) - X ( t , x ) \\Vert _ H = 0 , \\ \\mathbb { P } . \\end{align*}"}
+{"id": "1615.png", "formula": "\\begin{align*} \\tilde { \\mathbf { W } } = ( \\mathbf { G } ^ H \\mathbf { \\Phi } ^ H \\mathbf { H } \\mathbf { H } ^ H \\mathbf { \\Phi } \\mathbf { G } + \\sigma ^ 2 \\mathbf { I } _ N ) ^ { - 1 } \\mathbf { G } ^ H \\mathbf { \\Phi } ^ H \\mathbf { H } , \\end{align*}"}
+{"id": "6208.png", "formula": "\\begin{align*} & B _ 1 = 2 \\Q \\frac { 2 ( L + 1 ) ( L + 2 ) \\Q ^ 4 - 3 \\kappa \\Q ^ 2 - 6 \\kappa ^ 2 ( L ^ 2 + 3 L + 3 ) } { ( \\Q ^ 2 + 3 \\kappa ) ^ 2 } , \\\\ & B _ 2 = \\frac { 2 ( 4 L ^ 2 + 1 0 L + 7 ) \\Q ^ 4 + \\kappa ( 4 L ^ 2 + 3 ) \\Q ^ 2 + 1 8 \\kappa ^ 2 } { ( \\Q ^ 2 + 3 \\kappa ) ^ 2 } . \\end{align*}"}
+{"id": "3088.png", "formula": "\\begin{align*} \\tilde { y } ^ { k } = \\Pi _ { \\mathcal { N } } \\{ y ^ k - \\frac { 1 } { \\beta } [ \\lambda ^ k - \\beta ( \\tilde { x } ^ { k } - y ^ k ) ] \\} . \\end{align*}"}
+{"id": "1741.png", "formula": "\\begin{align*} \\overline { \\Omega } = \\bigcup _ { { \\bf k } \\in K _ { \\epsilon } } \\epsilon \\big ( \\overline { Y } + { \\bf k } \\big ) . \\end{align*}"}
+{"id": "8621.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } | x _ 1 | ^ p d S _ { B _ 2 ^ n } ( x ) = ( n + p ) \\int _ { B _ 2 ^ { n - 1 } } | z _ 1 | ^ p d z = \\frac { 2 \\pi ^ \\frac { n - 1 } { 2 } \\Gamma \\left ( \\frac { p + 1 } { 2 } \\right ) } { \\Gamma \\left ( \\frac { p + n } { 2 } \\right ) } . \\end{align*}"}
+{"id": "6987.png", "formula": "\\begin{align*} \\prod ^ { m ( p ) - n } _ { r = 1 } | t _ r | _ p = | t _ l | ^ { s _ 1 } _ p | s _ 1 ! | _ p , \\end{align*}"}
+{"id": "3624.png", "formula": "\\begin{align*} D _ { e _ i } e _ j = 0 , h _ { i j } = \\delta _ { i j } \\kappa _ i , \\kappa _ 1 \\geq \\cdots \\geq \\kappa _ n . \\end{align*}"}
+{"id": "3740.png", "formula": "\\begin{align*} \\Delta x _ k + \\bar u ^ { - 1 } \\nabla \\bar u \\cdot \\nabla x _ k = 0 \\mbox { f o r a l l } k = 1 , \\dots , n . \\end{align*}"}
+{"id": "1929.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty b _ k ( h _ m ( n ) ) q ^ n \\end{align*}"}
+{"id": "7210.png", "formula": "\\begin{align*} W ( t , x ) & = \\sqrt { C _ { d } \\kappa } \\ , \\sum _ { k \\in \\Z ^ d _ 0 } \\sum _ { i = 1 } ^ { d - 1 } \\theta _ { k } \\left \\{ { \\rm R e } ( \\sigma _ { k , i } ( x ) ) { \\rm R e } ( W ^ { k , i } _ { t } ) - { \\rm I m } ( \\sigma _ { k , i } ( x ) ) { \\rm I m } ( W ^ { k , i } _ { t } ) \\right \\} \\\\ & = \\sum _ { k \\in \\Z ^ d _ 0 } \\sum _ { i = 1 } ^ { d - 1 } \\xi _ { k , i } ( x ) B _ t ^ { k , i } . \\end{align*}"}
+{"id": "6170.png", "formula": "\\begin{align*} V ' _ 1 ( r ) + E ' _ 0 = V _ 2 ( r ) + E _ 0 . \\end{align*}"}
+{"id": "5605.png", "formula": "\\begin{align*} \\begin{cases} a _ { \\alpha + n + 1 } - a _ { n + 1 } = d _ { \\alpha + n } - d _ { \\alpha + n + 1 } \\\\ b _ n = a _ { \\alpha + 1 } + ( d _ { \\alpha + 1 } - d _ \\alpha ) = b \\\\ d _ { \\alpha + n } = d _ \\alpha + ( b _ 1 - a _ { \\alpha + 1 } ) + \\sum _ { j = 2 } ^ n ( a _ j - a _ { \\alpha + j } ) \\\\ d = d _ \\alpha + b - a _ { \\alpha + 1 } + \\sum _ { j = 2 } ^ \\infty ( a _ j - a _ { \\alpha + j } ) \\end{cases} \\ , \\end{align*}"}
+{"id": "4198.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( K _ H ( x ) \\nabla u | \\nabla u ) d x = \\int _ { \\Omega } q ^ 2 \\left ( K _ H ( x ) \\nabla \\left ( \\frac { u } { q } \\right ) | \\nabla \\left ( \\frac { u } { q } \\right ) \\right ) d x . \\end{align*}"}
+{"id": "4585.png", "formula": "\\begin{align*} \\dot { x } ( t ) & = - A ^ { \\circ } x ( t ) \\quad ( t > 0 ) , \\\\ y ( t ) & = B ^ { \\circ } x ( t ) \\quad ( t \\geq 0 ) , \\\\ x ( 0 ) & = x _ 0 \\in X ^ \\circ . \\end{align*}"}
+{"id": "1573.png", "formula": "\\begin{align*} e ^ { 2 w ( x ) } : = \\left ( \\int _ { \\Bar { \\mathcal F } _ x } \\dd \\mu _ x \\right ) ^ { - 1 } \\left ( \\int _ { \\Bar { \\mathcal F } _ x } e ^ { 2 { \\bar f } _ x } \\dd \\mu _ x \\right ) , \\end{align*}"}
+{"id": "181.png", "formula": "\\begin{align*} ( \\varinjlim _ { n } A _ { H ' , n } \\widehat { \\otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V ) ) ^ { R G - l a } = \\varinjlim _ { n } A _ { H ' , n } \\widehat { \\otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V ) . \\end{align*}"}
+{"id": "2949.png", "formula": "\\begin{align*} \\begin{aligned} h \\left ( \\mathrm { d o m } \\ , E ^ \\Phi _ A \\right ) & = \\mathrm { d o m } \\ , E ^ \\Psi _ B \\\\ h \\left ( \\mathrm { d o m } \\ , P ^ \\Phi _ { \\bar A } \\right ) & = \\mathrm { d o m } \\ , P ^ \\Psi _ { \\bar B } \\\\ \\end{aligned} \\end{align*}"}
+{"id": "975.png", "formula": "\\begin{align*} I : = \\{ i \\in S \\mid \\} . \\end{align*}"}
+{"id": "6818.png", "formula": "\\begin{align*} h _ { p } ^ { \\left ( 0 \\right ) } = \\frac { 1 } { 2 \\pi i } \\int _ { \\mathbb { T } } \\widetilde { \\varphi } _ { 0 } ^ { - } \\left ( u \\right ) u ^ { p } d u , \\end{align*}"}
+{"id": "5295.png", "formula": "\\begin{align*} l ( \\rho \\circ \\Psi \\circ \\pi ^ { - 1 } ( 1 \\otimes e _ 0 ) ) = l ( \\rho \\circ \\Psi ( p r ) ) = \\rho ( l ( \\Psi ( p r ) ) ) = \\rho ( \\Phi ( p r ) ) . \\end{align*}"}
+{"id": "3550.png", "formula": "\\begin{align*} y ( q ) = \\psi ( q ^ 3 ) \\left ( \\frac { \\varphi _ 6 \\varphi _ 9 ^ 2 } { \\varphi _ 3 \\varphi _ { 1 8 } } - 2 q \\psi ( q ^ 9 ) \\right ) = \\frac { \\varphi _ 6 ^ 2 } { \\varphi _ 3 } \\frac { \\varphi _ 6 \\varphi _ 9 ^ 2 } { \\varphi _ 3 \\varphi _ { 1 8 } } - 2 q \\psi ( q ^ 3 ) \\psi ( q ^ 9 ) = z ( q ^ 3 ) - 2 q \\psi ( q ^ 3 ) \\psi ( q ^ 9 ) \\end{align*}"}
+{"id": "5805.png", "formula": "\\begin{align*} d \\sigma - ( \\log \\sqrt { \\tau _ 0 } ) _ z d z \\sigma = e ^ { 2 i \\theta } \\frac { 1 } { \\sqrt { \\tau _ 0 } } \\left ( Q _ 0 d z + \\frac { \\tau _ 0 } { 4 } d \\overline { z } \\right ) \\ J \\widehat { \\sigma } I \\end{align*}"}
+{"id": "1009.png", "formula": "\\begin{align*} x '' : = w '' \\varepsilon ^ { \\mu '' } : = x ' r _ { \\mathbf a } = w ' s _ \\alpha \\varepsilon ^ { \\mu ' - ( 1 + \\langle \\mu ' , \\alpha \\rangle ) \\alpha ^ \\vee } < x ' . \\end{align*}"}
+{"id": "4974.png", "formula": "\\begin{align*} \\begin{aligned} E _ { \\xi _ 1 ^ { \\mathsf { L } ^ { k + 1 } } ( u ) } \\biggl [ \\varphi \\biggl ( x + u + \\sum _ { i = 2 } ^ { k + 1 } Z _ i ^ { \\mathsf { L } ^ { k + 1 } [ u ] } \\biggr ) \\biggr ] & = E _ { \\xi _ 1 ^ { \\mathsf { L } ^ { k + 1 } } ( u ) } \\biggl [ \\varphi \\biggl ( x + u + \\sum _ { i = 1 } ^ { k } Z _ i ^ { \\mathsf { M } ^ { k } } \\biggr ) \\biggr ] \\\\ & = W ( \\xi _ 1 ^ { \\mathsf { L } ^ { k + 1 } } ( u ) , k , x + u , \\mathsf { M } ^ { k } ) . \\end{aligned} \\end{align*}"}
+{"id": "4534.png", "formula": "\\begin{align*} \\mathcal { Q } = \\int _ { 0 } ^ t v _ 1 ( s , x - \\lambda t + \\lambda s ) \\dd s - \\int _ { t _ { 1 , e n } ( x , t ) } ^ { t _ { 1 , e x } ( x , t ) } v _ 1 ( s , x - \\lambda t + \\lambda s ) \\dd s \\end{align*}"}
+{"id": "5981.png", "formula": "\\begin{align*} \\Gamma _ { \\frac { 1 } { K } } & : = \\Big \\{ ( \\nu _ 1 , \\nu _ 2 , \\nu _ 3 ) : 2 \\nu _ 1 \\nu _ 3 = \\nu _ 2 ^ 2 , 1 - \\frac { 1 } { K } \\le \\nu _ 3 \\le 1 , \\quad \\Big | \\frac { \\nu _ 2 } { \\nu _ 3 } \\Big | \\le 1 \\Big \\} . \\end{align*}"}
+{"id": "604.png", "formula": "\\begin{align*} \\Phi ( n ) : = n \\cdot \\phi ( n ) , \\forall n \\in E . \\end{align*}"}
+{"id": "6223.png", "formula": "\\begin{align*} E _ 1 = \\frac { 1 } { 4 } \\left \\{ - \\Q ^ 2 [ ( 2 L + 3 ) a _ 0 - 1 ] ^ 2 + ( m + 1 ) ^ 2 \\kappa [ ( 2 L + 3 ) a _ 0 + 1 ] ^ 2 \\right \\} . \\end{align*}"}
+{"id": "513.png", "formula": "\\begin{align*} P _ { n } f ( x ) = \\sum _ { | \\nu | = n } \\widehat { f } ( \\nu ) \\Phi _ { \\nu } ( x ) . \\end{align*}"}
+{"id": "3115.png", "formula": "\\begin{align*} J _ { \\alpha } ( \\alpha ( x ) , \\alpha ( y ) , [ x , z ] ) = [ J _ { \\alpha } ( x , y , z ) , \\alpha ^ { 2 } ( x ) ] , \\end{align*}"}
+{"id": "2650.png", "formula": "\\begin{align*} ( q ; q ) _ { \\infty } \\sum \\limits _ { n = 0 } ^ { \\infty } d _ { e } ( n ) q ^ { n } = ( q ^ { 4 } ; q ^ { 4 } ) _ { \\infty } \\sum \\limits _ { n = 1 } ^ { \\infty } \\frac { q ^ { 4 n } } { 1 - q ^ { 4 n } } . \\end{align*}"}
+{"id": "5066.png", "formula": "\\begin{align*} d ( \\phi \\wedge \\beta ) = d \\phi \\wedge \\beta - \\phi \\wedge d \\beta = d \\phi \\wedge \\beta = \\alpha \\wedge \\beta . \\end{align*}"}
+{"id": "3561.png", "formula": "\\begin{align*} c _ { \\Omega _ 0 } ^ { \\Omega _ 0 } & = q ^ { - 1 / 6 } \\varphi ( q ) ^ { - 4 } \\dfrac { 1 } { 2 } q ^ { - 1 / 2 0 } ( \\varphi ( - q ^ { 1 / 2 } ) ^ 2 \\varphi ( q ) ^ { - 2 } G ( - q ^ { 1 / 2 } ) + \\varphi ( q ^ { 1 / 2 } ) ^ 2 \\varphi ( q ) ^ { - 2 } G ( q ^ { 1 / 2 } ) ) \\\\ & = q ^ { - 1 3 / 6 0 } \\dfrac { 1 } { 2 } \\varphi ( q ) ^ { - 6 } ( \\varphi ( - q ^ { 1 / 2 } ) ^ 2 G ( - q ^ { 1 / 2 } ) + \\varphi ( q ^ { 1 / 2 } ) ^ 2 G ( q ^ { 1 / 2 } ) ) . \\end{align*}"}
+{"id": "375.png", "formula": "\\begin{align*} ( 2 \\phi _ { n - 1 - k } + \\frac { 1 } { 2 \\phi _ { n - 1 - k } } ) ^ 2 & = 4 \\phi _ { n - 1 - k } ^ 2 + \\frac { 1 } { 4 \\phi _ { n - 1 - k } ^ 2 } + 2 \\\\ & = 4 \\phi _ { k } ^ 2 + \\frac { 1 } { 4 \\phi _ { k } ^ 2 } + 2 \\\\ & = ( 2 \\phi _ { k } + \\frac { 1 } { 2 \\phi _ { k } } ) ^ 2 . \\end{align*}"}
+{"id": "9104.png", "formula": "\\begin{align*} \\mathcal { E } _ { | _ { x \\times C } } = \\varphi ( x ) , \\end{align*}"}
+{"id": "4007.png", "formula": "\\begin{align*} \\gamma ^ k = \\max _ { x _ n ^ D \\in \\mathcal { N } _ D } 5 0 0 \\cdot \\frac { | \\Gamma _ D | \\mathrm { N } _ I \\lambda _ d ^ 2 } { | \\Omega | \\mathrm { N } _ D \\lambda _ 1 } \\frac { | \\nabla w ( x _ n ^ D ) | ^ 2 } { | \\nabla w ( y _ n ) | ^ 2 } . \\end{align*}"}
+{"id": "4286.png", "formula": "\\begin{align*} n = \\frac { \\beta ^ { 3 } - 3 } { 6 } \\end{align*}"}
+{"id": "6308.png", "formula": "\\begin{align*} \\lambda _ 0 = 1 , \\lambda _ t = \\prod _ { i = 1 } ^ { t } ( 1 - \\alpha _ i ) . \\end{align*}"}
+{"id": "3816.png", "formula": "\\begin{align*} G ( U , x , t ) = ( \\nabla _ V \\tilde { \\eta } ) ( A ( U , x , t ) , x , t ) \\end{align*}"}
+{"id": "9320.png", "formula": "\\begin{align*} 2 z _ { 1 } ^ { 2 } - 1 0 z _ { 2 } ^ { 2 } = 4 p \\implies 2 z _ { 1 } ^ { 2 } \\equiv 4 p \\pmod 5 . \\end{align*}"}
+{"id": "1854.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac 1 { \\rho _ 1 ^ { n - 2 d _ e \\| R ^ \\nabla \\| ^ 2 _ \\infty } } \\int _ { B _ { \\rho _ 1 } ( x _ 0 ) } \\big ( \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) - 1 \\big ) \\ , d v \\\\ & \\leq \\frac 1 { \\rho _ 2 ^ { n - 2 d _ e \\| R ^ \\nabla \\| ^ 2 _ \\infty } } \\int _ { B _ { \\rho _ 2 } ( x _ 0 ) } \\big ( \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) - 1 \\big ) \\ , d v \\ , , \\end{aligned} \\end{align*}"}
+{"id": "2124.png", "formula": "\\begin{align*} 4 s ^ 3 + 8 ( 1 + a ) s ^ 2 + ( 4 + 6 a + 5 a ^ 2 ) s - 2 a = 0 ; \\end{align*}"}
+{"id": "4954.png", "formula": "\\begin{align*} \\theta ' _ i = \\pi _ i ( X _ 2 , Y _ 2 , \\dots , X _ { i - 1 } , Y _ { i - 1 } ) , i \\ge 2 . \\end{align*}"}
+{"id": "5366.png", "formula": "\\begin{align*} S ^ { i j } _ k = \\frac { \\partial S _ k [ D ^ 2 u ] } { \\partial r _ { i j } } . \\end{align*}"}
+{"id": "3222.png", "formula": "\\begin{align*} d Y _ t = ( \\mathcal A Y _ t + \\alpha _ t ) d t + \\sigma _ t d W _ t , t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "544.png", "formula": "\\begin{align*} s _ 1 w _ i & = s _ 1 ( a _ { - i } - a _ i ) \\\\ & = - \\tfrac { 3 } { 4 } w _ i + \\tfrac { 3 } { 8 } ( a _ { - i - 1 } + a _ { - i + 1 } - a _ { i - 1 } - a _ { i + 1 } ) \\end{align*}"}
+{"id": "3205.png", "formula": "\\begin{align*} \\tau ( a ) = \\int _ { X _ A } \\hat a \\ , d \\mu \\quad a \\in A _ { s a } . \\end{align*}"}
+{"id": "2047.png", "formula": "\\begin{align*} \\bar a _ { i , j } ( f ) = a _ { i , j } * f = \\int _ { \\mathbb R ^ 3 } a _ { i , j } ( v - v _ * ) f ( v _ * ) d v _ * . \\end{align*}"}
+{"id": "5369.png", "formula": "\\begin{align*} \\begin{aligned} 0 \\geq & S ^ { i j } _ k ( u _ 1 + B | x | ^ 2 ) _ { i j } = f _ 1 + 2 B \\sum _ i S ^ { i i } _ k \\\\ \\geq & - A f ^ { 1 - 1 / ( k - 1 ) } + 2 c _ 0 B S _ 1 ^ { 1 / ( k - 1 ) } S _ k ^ { 1 - 1 / ( k - 1 ) } \\\\ = & - A f ^ { 1 - 1 / ( k - 1 ) } + 2 c _ 0 B ( \\Delta u ) ^ { 1 / ( k - 1 ) } f ^ { 1 - 1 / ( k - 1 ) } . \\end{aligned} \\end{align*}"}
+{"id": "5351.png", "formula": "\\begin{align*} \\prod _ { 1 \\leq j \\leq N , n _ j \\mid n _ 1 } \\vert \\eta _ { j , ( 1 ) } \\vert _ F = \\frac { 1 } { p ^ { D } } \\end{align*}"}
+{"id": "6810.png", "formula": "\\begin{align*} S \\left ( \\rho , x \\right ) = 8 t \\rho ^ { 3 } + 2 x \\rho . \\end{align*}"}
+{"id": "4700.png", "formula": "\\begin{align*} \\frac D 2 = \\{ q _ 1 \\mid q _ 1 \\overline { q p } q \\in D \\} . \\end{align*}"}
+{"id": "6059.png", "formula": "\\begin{align*} \\phi ( s , t ) = \\gamma ( s ) + \\sum _ { i = 1 } ^ { n - 1 } t _ i w _ i , t = ( t _ 1 , \\ldots , t _ { n - 1 } ) , \\end{align*}"}
+{"id": "1448.png", "formula": "\\begin{align*} \\left [ \\mathbf { F } \\right ] _ { i , j } = \\frac { 2 } { \\sigma _ { m } ^ { 2 } } \\mathrm { R e } \\Big \\{ \\sum _ { l = 1 } ^ { L } \\frac { \\partial \\mbox { \\boldmath $ \\mu $ } \\left [ l \\right ] ^ { H } } { \\partial \\xi _ { i } } \\frac { \\partial \\mbox { \\boldmath $ \\mu $ } \\left [ l \\right ] } { \\partial \\xi _ { j } } \\Big \\} , \\ i , j = 1 , \\cdots , 4 , \\end{align*}"}
+{"id": "7755.png", "formula": "\\begin{align*} q _ 0 ( x ) : = q ( x ) , ~ 2 \\sqrt x q _ { h + 1 } ( x ) : = q _ h ( \\sqrt x ~ ) p _ h ( \\sqrt { - x } ~ ) - q _ h ( \\sqrt { - x } ~ ) p _ h ( \\sqrt x ~ ) , \\end{align*}"}
+{"id": "2141.png", "formula": "\\begin{align*} \\frac { \\tilde { x } ^ 2 } { ( 1 + a ) ( 1 - b ) } + \\frac { \\tilde { y } ^ 2 } { ( 1 - a ) ( 1 + b ) } = \\frac { a b } { a + b } . \\end{align*}"}
+{"id": "3511.png", "formula": "\\begin{align*} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\sum _ { i = 1 } ^ { n } \\left ( i ^ { p } \\ - \\ 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } i ^ { p - 2 j - 1 } \\right ) \\cdot F _ { n + 1 - i } \\end{align*}"}
+{"id": "4555.png", "formula": "\\begin{align*} A _ { w , 1 } f ( v _ { \\ell , m , 0 } ) = & ( q ^ 3 + q ^ 2 ) f ( v _ { \\ell , m , 1 } ) + q f ( v _ { \\ell , m + 1 , 0 } ) + f ( v _ { \\ell + 1 , m , 0 } ) \\\\ A _ { w , 2 } f ( v _ { \\ell , m , 0 } ) = & q ^ 4 f ( v _ { \\ell - 1 , m - 1 , 0 } ) + ( q ^ 3 + q ^ 2 ) f ( v _ { \\ell , m + 1 , 1 } ) \\\\ & + ( q ^ 2 + q ) f ( v _ { \\ell + 1 , m , 1 } ) + f ( v _ { \\ell + 1 , m + 1 , 0 } ) \\\\ A _ { w , 3 } f ( v _ { \\ell , m , 0 } ) = & q ^ 3 f ( v _ { \\ell - 1 , m , 0 } ) + q ^ 2 f ( v _ { \\ell , m - 1 , 0 } ) + ( q + 1 ) f ( v _ { \\ell + 1 , m + 1 , 1 } ) . \\end{align*}"}
+{"id": "4800.png", "formula": "\\begin{align*} \\Delta ^ 2 u = \\lambda u & D , \\\\ [ 1 m m ] u = \\frac { \\partial u } { \\partial \\nu } = 0 & \\partial D . \\end{align*}"}
+{"id": "981.png", "formula": "\\begin{align*} | W | q ^ { d ^ 2 - 1 } \\theta _ d \\le \\sum _ { i = 0 } ^ 3 \\left | \\bigcup _ { F \\in C _ i } F \\cap M \\right | . \\end{align*}"}
+{"id": "7883.png", "formula": "\\begin{align*} \\gamma _ 2 \\dot s ( 0 ) = - \\dot { \\nu } ( 0 ) , \\end{align*}"}
+{"id": "872.png", "formula": "\\begin{align*} ( x - y ) ^ 2 + ( x - y ) ( y + y ' ) + ( y + y ' ) ^ 2 = 3 n \\ , . \\end{align*}"}
+{"id": "5916.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ T \\Vert B ( t , X ( t , x _ n ) ) - B ( t , X ( t , x ) ) \\Vert _ { L _ 2 } ^ 2 d t = 0 . \\end{align*}"}
+{"id": "2351.png", "formula": "\\begin{align*} 1 + 4 ( x - n ) \\leq y _ n ( x , 0 ) , & \\forall x \\in [ 0 , n ] , \\\\ 1 + 4 ( 0 - n ) \\leq 0 = y _ n ( 0 , \\cdot ) , & 1 + 4 ( n - n ) = 1 = y _ n ( n , \\cdot ) . \\end{align*}"}
+{"id": "3375.png", "formula": "\\begin{align*} u = & B _ 1 ( u , u _ 1 ) + B _ 1 ( u _ 2 , u ) + B _ 2 ( b , b _ 1 ) + B _ 2 ( b _ 2 , b ) \\\\ b = & B _ 3 ( u , b _ 1 ) + B _ 3 ( u _ 2 , b ) \\ , . \\end{align*}"}
+{"id": "854.png", "formula": "\\begin{align*} \\lambda ( q ) = \\sum \\limits _ { 1 \\leq n \\leq q \\atop { n ^ 2 + n + 1 \\equiv 0 \\ , ( q ) } } 1 \\ , , \\end{align*}"}
+{"id": "2025.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - d y _ t = & [ - k _ 3 p _ t + f _ 0 ( t ) ] d t + [ - k _ 3 q _ t + g _ 0 ( t ) ] d \\overleftarrow B _ t - z _ t d W _ t , \\\\ d p _ t = & [ k _ 2 y _ t + F _ 0 ( t ) ] d t + [ k _ 2 z _ t + G _ 0 ( t ) ] d W _ t - q _ t d \\overleftarrow B _ t , \\\\ y _ T = & \\xi , \\ , \\ , p _ 0 = y _ 0 + \\Psi _ 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2123.png", "formula": "\\begin{align*} z = \\pm \\sqrt { \\frac { s _ 0 } { ( 1 + a ) ( 1 + a + s _ 0 ) } } , \\ w = \\sqrt { \\frac { 1 + a } { 1 + a + s _ 0 } } \\left ( \\cos ( t ) + i \\sin ( t ) \\right ) , \\end{align*}"}
+{"id": "748.png", "formula": "\\begin{align*} I _ { 1 2 } & = \\frac { | F _ s ( t ) | } { | t | ^ { 1 - \\frac 1 { 2 s } } } \\approx \\frac { t ^ 2 } { t ^ { 2 - \\frac 1 { 2 s } } ( 1 + t ^ { \\frac 1 s } ) ^ { \\frac { N + 2 s } 2 } } = \\frac { t ^ { \\frac 1 { 2 s } } } { ( 1 + t ^ { \\frac 1 s } ) ^ { \\frac { N + 2 s } 2 } } \\lesssim \\min \\left ( 1 , \\frac 1 { t } \\right ) . \\end{align*}"}
+{"id": "6501.png", "formula": "\\begin{align*} E _ { [ w _ 1 , w _ 2 ] } ^ 0 = & \\{ w \\in B S ( 2 , 1 ) ^ + : w _ 1 w , w w _ 2 \\} \\\\ E _ { [ w _ 1 , w _ 2 ] } ^ * = & \\{ x \\in E ^ 1 _ { w _ 2 } : s ( x ) , r ( x ) \\in E _ { [ w _ 1 , w _ 2 ] } ^ 0 \\} \\end{align*}"}
+{"id": "6341.png", "formula": "\\begin{align*} X = \\sqrt { - \\frac { \\theta } { \\log ( Y ) } } \\end{align*}"}
+{"id": "6833.png", "formula": "\\begin{align*} y \\left ( \\rho \\right ) : = y \\left ( \\rho , x \\right ) = \\frac { 1 } { \\frac { 1 } { 2 } - i \\rho } \\sum _ { j = 0 } ^ { \\infty } a _ { j } ( x ) z ^ { n } \\left ( \\rho \\right ) , \\quad \\sum _ { j = 0 } ^ { \\infty } \\left \\vert a _ { j } ( x ) \\right \\vert ^ { 2 } < \\infty . \\end{align*}"}
+{"id": "2370.png", "formula": "\\begin{align*} \\mathbb { T } _ { \\rho } ( \\mathbb { D } ^ 2 , \\{ \\mathbf { h } _ 0 ^ { \\mathbb { D } ^ 2 } \\} ) = \\left [ \\ell _ 0 ( \\mathbf { h } _ 0 ^ { \\mathbb { D } ^ 2 } ) , \\mathbf { c } _ 0 \\right ] . \\end{align*}"}
+{"id": "5483.png", "formula": "\\begin{align*} \\min _ { x > 0 } \\Gamma ( x ) = 0 . 8 8 5 6 0 3 1 9 4 4 1 0 8 8 8 6 8 8 7 . . , \\end{align*}"}
+{"id": "5252.png", "formula": "\\begin{align*} \\gamma ( \\| \\cdot \\| _ { \\infty } , \\tau _ { \\operatorname { c o } } ) = \\beta _ { 0 } = \\mu ( \\mathrm { C } _ { \\operatorname { b } } ( \\Omega ) , \\mathrm { M } _ { \\operatorname { t } } ( \\Omega ) ) \\end{align*}"}
+{"id": "2540.png", "formula": "\\begin{align*} d _ 2 ( \\mathbf { x } ( 1 ) , \\mathbf { s } ( 1 ) ) \\le \\mathbf { z } ( 1 ) \\le \\nu \\tilde { \\Gamma } \\mu = \\tilde { \\Gamma } \\mu ( 1 ) . \\end{align*}"}
+{"id": "5880.png", "formula": "\\begin{align*} \\Vert Y _ n ( t ) \\Vert _ H ^ 2 = & \\Vert P _ n x \\Vert _ H ^ 2 + \\int _ 0 ^ t \\Big [ 2 \\langle A ( s , Y _ n ( s ) ) , Y _ n ( s ) \\rangle + \\Vert P _ n B ( s , Y _ n ( s ) ) Q _ n \\Vert _ { L _ 2 } ^ 2 \\Big ] d s \\\\ & + 2 \\int _ 0 ^ t \\Big ( Y _ n ( s ) , B ( s , Y _ n ( s ) ) Q _ n d W ( s ) \\Big ) . \\end{align*}"}
+{"id": "5253.png", "formula": "\\begin{align*} R ( \\lambda , M _ { q } ) f = \\frac { 1 } { \\lambda - q } f , f \\in \\mathrm { C } _ { \\operatorname { b } } ( \\Omega ) , \\end{align*}"}
+{"id": "2566.png", "formula": "\\begin{align*} p ^ { n _ { s } ^ { \\prime } - n _ { 1 } ^ { \\prime } } q ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { 2 } } - p ^ { n _ { 2 } ^ { \\prime } + n _ { s } ^ { \\prime } - 2 n _ { 1 } ^ { \\prime } } a _ { n _ { 1 } } = p ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } q ^ { n _ { s } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { s } } \\frac { N _ { n _ { 2 } } } { N _ { n _ { s } } } . \\end{align*}"}
+{"id": "4074.png", "formula": "\\begin{align*} \\max _ { \\lambda \\in \\Lambda ^ { M ( 4 , 0 ) } _ { 1 } } e ( \\alpha ^ \\lambda , G ^ \\lambda ) = e ( 6 H - 5 / 2 , G ^ { M ( 3 , 0 ) } _ { 1 } ) + 2 H - 3 / 2 = 4 H - 3 . \\end{align*}"}
+{"id": "940.png", "formula": "\\begin{align*} R _ 2 ( t ) & = - i \\lambda _ 6 t ^ { - 1 } \\mathcal { F } M ( t ) ^ { - 1 } \\mathcal { F } ^ { - 1 } \\mathcal { N } _ 2 ( \\mathcal { F } M { { \\mathcal F } } ^ { - 1 } v _ 1 , \\mathcal { F } M { { \\mathcal F } } ^ { - 1 } v _ 2 ) ( t ) \\\\ & + i \\lambda _ 6 t ^ { - 1 } \\mathcal { N } _ 2 ( v _ 1 , v _ 2 ) ( t ) , \\end{align*}"}
+{"id": "1172.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\mathbb { P } } \\left [ \\left ( \\int _ { T _ { 0 } } ^ { \\left ( T _ { 0 } + \\delta \\right ) \\wedge T } \\left | \\triangle b _ { t } ^ { i , n , \\infty } \\right | ^ { 2 } d t \\right ) ^ { p } \\right ] \\leq \\frac { p ! \\beta ^ { p } \\delta ^ { p } } { n ^ { p } } , i = 1 , \\cdots , n . \\end{align*}"}
+{"id": "4220.png", "formula": "\\begin{align*} E ( \\theta ) _ J = \\sum _ { w \\in Z _ J } \\Bbbk { \\bf U } _ { w _ J w ^ { - 1 } } \\dot { w } C ( \\theta ) _ J . \\end{align*}"}
+{"id": "2002.png", "formula": "\\begin{align*} g ^ { - 1 } ( z ) = \\begin{cases} \\frac 2 3 ( z + 1 ) , & z \\geq 1 / 2 , \\\\ z + 1 / 2 , & - 3 / 2 < z < 1 / 2 , \\\\ 2 ( z + 1 ) , & z \\leq - 3 / 2 . \\end{cases} \\end{align*}"}
+{"id": "2023.png", "formula": "\\begin{align*} \\bar u _ s = \\left \\{ \\begin{array} { l } \\alpha _ s , \\ \\ \\ s \\in [ t , t + \\varepsilon ] , \\\\ u _ s , \\ \\ \\ , \\end{array} \\right . \\end{align*}"}
+{"id": "9147.png", "formula": "\\begin{align*} \\dot x = & \\ , - \\gamma \\left ( y _ \\delta ( { x } , t ) - \\bar { y } \\right ) \\ , u ( t ) & & { x } ( 0 ) = { x } _ 0 \\\\ \\dot { \\bar { y } } = & \\ , \\gamma \\left ( y _ \\delta ( { x } , t ) - \\bar { y } \\right ) & & \\bar { y } ( 0 ) = \\bar { y } _ 0 \\end{align*}"}
+{"id": "2920.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle \\mathcal { A } ^ * \\varphi + \\frac { \\partial f } { \\partial y } ( x , y _ u ) \\varphi = \\frac { \\partial L } { \\partial y } ( x , y _ u , u ) \\Omega , \\\\ \\varphi = 0 \\Gamma . \\end{array} \\right . \\end{align*}"}
+{"id": "8600.png", "formula": "\\begin{align*} \\frac { 2 } { m ( m - 1 ) } \\sum _ { 1 \\le i < j \\le m } | P _ { u _ i ^ \\bot } Z | | P _ { u _ j ^ \\bot } Z | \\le \\left ( \\frac { 1 } { m } \\sum _ { i = 1 } ^ m | P _ { u _ i ^ \\bot } Z | \\right ) ^ 2 . \\end{align*}"}
+{"id": "2086.png", "formula": "\\begin{align*} \\frac { m ( I _ k \\cap H ) } { m ( I _ k ) } & = \\frac { m ( I _ k \\cap P ) } { m ( I _ k ) } + \\frac { m ( I _ k \\cap ( H \\setminus P ) ) } { m ( I _ k ) } \\\\ & < \\frac { m ( I _ k \\cap P ) } { m ( I _ k ) } + \\frac { n _ k + 1 } { 2 ^ { n _ k } } \\end{align*}"}
+{"id": "5941.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d } a _ i ( t , x , u , z ) z _ i + a _ 0 ( t , x , u , z ) u \\geq c _ 3 | z | ^ { \\alpha } - c _ 4 | u | ^ 2 - f _ 2 ( t , x ) . \\end{align*}"}
+{"id": "8641.png", "formula": "\\begin{align*} { { \\bf { h } } _ l } = { \\alpha _ l } { \\bf { a } } \\left ( { { \\theta _ l } } \\right ) = { \\alpha _ l } { \\left [ { 1 , { e ^ { - j \\pi \\cos { \\theta _ l } } } , \\cdots , { e ^ { - j \\pi \\left ( { { M _ t } - 1 } \\right ) \\cos { \\theta _ l } } } } \\right ] ^ T } . \\end{align*}"}
+{"id": "3086.png", "formula": "\\begin{align*} d ( \\theta ^ k , \\tilde { \\theta } ^ { k } , \\zeta ^ k ) : = \\theta ^ k - \\tilde { \\theta } ^ k - G ^ { - 1 } \\zeta ^ k . \\end{align*}"}
+{"id": "5186.png", "formula": "\\begin{align*} \\Phi _ { t } + u \\cdot \\nabla \\Phi & = \\mu \\left ( \\Delta - \\frac { 2 } { r } \\partial _ r \\right ) \\Phi . \\end{align*}"}
+{"id": "1356.png", "formula": "\\begin{align*} J = \\{ j \\in \\{ 1 , \\dotsc , m \\} \\colon u _ j \\in B _ i v _ j \\in B _ i \\} \\end{align*}"}
+{"id": "6100.png", "formula": "\\begin{align*} { \\rho } ^ { 0 0 } _ { n , p } = - \\widehat { \\mathcal { Q } } _ n ( j _ { p + 1 } ^ { ( 1 2 3 ) } ) - \\widehat { \\mathcal { Q } } _ n ( - j _ p ^ { ( 1 2 3 ) } ) + ( j ^ { ( 3 ) } - j ^ { ( 4 ) } - 1 ) ( j ^ { ( 3 ) } - j ^ { ( 4 ) } ) \\ , . \\end{align*}"}
+{"id": "4347.png", "formula": "\\begin{align*} & \\int _ { \\{ - t ' _ 3 \\le \\Psi < - t ' _ 4 \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h \\\\ = & \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\} } | \\tilde F | ^ 2 _ h + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\backslash N \\} } | F _ { t _ 0 , \\tilde { c } } - \\tilde F | ^ 2 _ h \\\\ & + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\cap N \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h . \\end{align*}"}
+{"id": "8246.png", "formula": "\\begin{align*} X = \\{ z = ( z _ k ) : | | z | | < \\infty \\} , \\end{align*}"}
+{"id": "1005.png", "formula": "\\begin{align*} \\widetilde \\omega _ a = \\begin{cases} 0 , & k \\neq 0 , \\\\ \\frac { 1 } { \\langle \\omega _ a , \\theta \\rangle } \\omega _ a , & k = 0 . \\end{cases} \\end{align*}"}
+{"id": "1860.png", "formula": "\\begin{align*} \\operatorname { V o l u m e } \\big ( B _ { \\rho } ( x _ 0 ) \\big ) = o ( { \\rho } ^ { \\lambda } ) \\rho \\rightarrow \\infty \\ , , \\end{align*}"}
+{"id": "2931.png", "formula": "\\begin{align*} \\overline { h } \\left ( r \\right ) = \\int _ { \\left \\vert x \\right \\vert > r } \\nu _ { \\alpha } ^ { \\prime } \\left ( d x \\right ) + r ^ { - 2 } \\int _ { \\left \\vert x \\right \\vert \\leq r } \\left \\vert x \\right \\vert ^ { 2 } \\nu _ { \\alpha } ^ { \\prime } \\left ( d x \\right ) + r ^ { - 1 } \\int _ { 1 < \\left \\vert x \\right \\vert \\leq r } x \\nu _ { \\alpha } ^ { \\prime } \\left ( d x \\right ) \\end{align*}"}
+{"id": "507.png", "formula": "\\begin{align*} \\Omega ^ 1 _ A \\cong \\bigoplus _ { i = 1 } ^ n A . d x _ i , \\end{align*}"}
+{"id": "2839.png", "formula": "\\begin{align*} \\mathcal G _ t = \\mathcal A _ t + \\gamma S _ { } + 2 \\gamma S _ - , \\end{align*}"}
+{"id": "3108.png", "formula": "\\begin{align*} \\Phi ( \\omega ) = \\{ x \\in C ( \\omega ) \\mid \\langle \\mathcal { F } ( x ^ * ) ( \\omega ) , x - x ^ * ( \\omega ) \\rangle < 0 \\} \\end{align*}"}
+{"id": "8935.png", "formula": "\\begin{align*} H _ { j , n } = ( - 1 ) ^ { n } \\binom { j } { n } ( \\beta ) ^ { ( j ) } \\sqrt { \\frac { n ! } { \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } } } . \\end{align*}"}
+{"id": "7588.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } T ^ { c _ 2 ( m ) } x _ { 0 0 } = x _ { 0 1 } \\quad \\textup { a n d } \\lim _ { n \\to \\infty } T ^ { c _ 1 ( n ) } x _ { 0 1 } = x _ { 1 1 } , \\end{align*}"}
+{"id": "3212.png", "formula": "\\begin{align*} \\mu \\left ( \\left \\{ \\tau \\in \\partial _ e ( T ( A ) ) \\mid \\left | \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } \\tau ( \\alpha ^ i ( a ) ) - \\tau _ \\mu ( a ) \\right | > \\varepsilon \\right \\} \\right ) \\le c _ 1 e ^ { - c _ 2 n \\varepsilon ^ 2 } . \\end{align*}"}
+{"id": "2040.png", "formula": "\\begin{align*} \\norm { u } _ { L ^ 1 _ m L ^ 2 _ v } : = \\sum _ { m \\in \\mathbb Z ^ 3 } \\norm { \\mathcal F _ x u ( m , \\cdot ) } _ { L ^ 2 _ v } , \\end{align*}"}
+{"id": "5478.png", "formula": "\\begin{align*} J _ \\nu ( t ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k } { k ! \\Gamma ( k + \\nu + 1 ) } \\left ( \\frac { t } { 2 } \\right ) ^ { 2 k + \\nu } \\end{align*}"}
+{"id": "1469.png", "formula": "\\begin{align*} w _ { \\tau } ( x ) : = \\begin{cases} \\dfrac { M ( a + b ) } { 2 } & a + \\tau < x < b - \\tau , \\\\ [ 1 e x ] w _ { 0 } ( x ) & x \\in ( a , b ) \\setminus ( a + \\tau , b - \\tau ) . \\end{cases} \\end{align*}"}
+{"id": "6638.png", "formula": "\\begin{align*} G _ { m , \\epsilon } ( t ) & = \\int _ 0 ^ 1 K _ m ( t , r ) d \\Theta _ { \\epsilon } ( r ) \\\\ & = \\Theta _ { \\epsilon } ( 1 ) K _ m ( t , 1 ) - \\int ^ { 1 } _ { 0 } \\Theta _ { \\epsilon } ( r ) \\frac { \\partial K _ m ( t , r ) } { \\partial r } d r , \\end{align*}"}
+{"id": "6323.png", "formula": "\\begin{align*} z _ 0 ^ { p ^ r } x _ 0 ^ { q ^ { m _ i } } + z _ 1 ^ { p ^ r } x _ 1 ^ { q ^ { m _ i } } + \\cdots + z _ { n - 1 } ^ { p ^ r } x _ { n - 1 } ^ { q ^ { m _ i } } = 0 \\end{align*}"}
+{"id": "9046.png", "formula": "\\begin{align*} b ( v , w ) = - ( - ) ^ { \\deg v \\deg w } b ( w , v ) , \\end{align*}"}
+{"id": "7390.png", "formula": "\\begin{align*} [ V ] & = \\frac { [ \\mathbb C _ x ] } { D } \\\\ D & = \\prod _ { n = 1 } ^ \\infty ( 1 - e ^ { - n \\delta } ) ^ { r } \\prod _ { n = 1 } ^ \\infty \\prod _ { \\alpha \\in \\Delta _ + ^ 0 } ( 1 - e ^ { - \\alpha } e ^ { ( 1 - n ) \\delta } ) ( 1 - e ^ { \\alpha } e ^ { - n \\delta } ) . \\end{align*}"}
+{"id": "5327.png", "formula": "\\begin{align*} \\widehat { f } ( \\lambda , \\omega ) = c _ { n , m } \\ , P _ m ( \\lambda \\omega ) \\ , \\int _ 0 ^ \\infty g ( r ) \\ , \\frac { J _ { n / 2 + m - 1 } ( \\lambda r ) } { ( \\lambda r ) ^ { n / 2 + m - 1 } } r ^ { n + 2 m - 1 } d r . \\end{align*}"}
+{"id": "8010.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leq x \\\\ ( p , N ) = 1 } } a _ f \\left ( p ^ { 2 l } \\right ) \\ll ( l \\log l ) \\sqrt { x } \\log x \\log ( N ( k - 1 ) ) . \\end{align*}"}
+{"id": "5241.png", "formula": "\\begin{align*} \\begin{aligned} v _ t + ( w + u _ { \\infty } ) \\cdot \\nabla v + \\nabla p & = ( d + B _ { \\infty } ) \\cdot \\nabla b + \\nu \\Delta v , \\\\ b _ t + ( w + u _ { \\infty } ) \\cdot \\nabla b + \\nabla q & = ( d + B _ { \\infty } ) \\cdot \\nabla v + \\mu \\Delta b , \\\\ \\nabla \\cdot v = \\nabla \\cdot b & = 0 , \\\\ w = \\chi _ { \\varepsilon } * v , \\ d & = \\chi _ { \\varepsilon } * b . \\end{aligned} \\end{align*}"}
+{"id": "3362.png", "formula": "\\begin{align*} N e w t o n ( F ) \\cap \\mathbb { Z } ^ m = S u p p ( F ) = \\bigcup \\limits _ { i = 0 } ^ l S u p p ( s _ { \\lambda ^ i } ) = \\bigcup \\limits _ { i = 0 } ^ l N e w t o n ( s _ { \\lambda ^ i } ) \\cap \\mathbb { Z } ^ m . \\end{align*}"}
+{"id": "4988.png", "formula": "\\begin{align*} P _ { \\mu , f _ { 1 , \\infty } } ( x , \\psi ) : = \\lim _ { \\epsilon \\to 0 } \\liminf _ { n \\to \\infty } \\dfrac { - \\log \\mu ( B _ { n } ( x , \\epsilon ) ) + S _ { 1 , n } \\psi ( x ) } { n } \\end{align*}"}
+{"id": "4272.png", "formula": "\\begin{align*} D = 3 \\left ( \\left ( a - b \\right ) ^ { 4 } - \\left ( a ^ { 4 } + b ^ { 4 } + \\left ( a - 1 \\right ) ^ { 4 } + \\left ( b - 1 \\right ) ^ { 4 } \\right ) + 1 \\right ) \\end{align*}"}
+{"id": "2818.png", "formula": "\\begin{align*} \\tilde { \\mathbf { H } } = \\mathbf { J } _ { 1 } \\mathbf { \\Sigma } _ { 1 } \\mathbf { J } _ { 1 } ^ { \\mathbf { H } } . \\end{align*}"}
+{"id": "2148.png", "formula": "\\begin{align*} \\tau = \\frac { \\varrho _ z } { | w | } . \\end{align*}"}
+{"id": "7577.png", "formula": "\\begin{align*} A = \\{ n \\in \\N : T ^ n a \\in E \\} . \\end{align*}"}
+{"id": "1402.png", "formula": "\\begin{align*} G _ n ( x , z ) = \\prod _ { j = 1 } ^ d \\lambda _ j ^ n e ^ { - \\sum _ { j = 1 } ^ d \\lambda _ j ( z _ j - x _ j ) } \\det ( q _ { n + i - j } ( z _ j - x _ i ) ) _ { i , j = 1 } ^ d . \\end{align*}"}
+{"id": "7871.png", "formula": "\\begin{align*} 0 = ( I - Q ) F _ \\Psi ( \\Psi ^ q , p _ 0 ) \\psi _ { p p } ( 0 , p _ 0 ) [ p ' ( 0 ) , p ' ( 0 ) ] . \\end{align*}"}
+{"id": "8108.png", "formula": "\\begin{align*} \\lim \\limits _ { | x | \\nearrow R } u ( x ) = \\lim \\limits _ { | x | \\nearrow R } v ( x ) = \\infty \\end{align*}"}
+{"id": "1453.png", "formula": "\\begin{align*} \\phi ( p ) = ( 1 + | p | ^ { \\gamma } ) ^ { \\alpha } \\quad \\gamma > 1 \\alpha \\in ( 0 , 1 / \\gamma ) . \\end{align*}"}
+{"id": "6728.png", "formula": "\\begin{align*} \\mathrm { E } ( X ^ n ) = & \\frac { \\mu ^ n } { \\mathrm { B } ( \\alpha , \\beta ) } \\sum _ { j = 0 } ^ { \\infty } ( - 1 ) ^ j \\binom { \\beta - 1 } { j } \\sum _ { i = 0 } ^ n \\binom { n } { i } \\left ( \\frac { \\sigma } { \\mu } \\right ) ^ i \\\\ & \\times \\sum _ { k = 0 } ^ \\infty \\left [ v _ k ( j + \\alpha - 1 ) + ( - 1 ) ^ { i + k } \\binom { j + \\alpha - 1 } { k } \\right ] J _ { i , k } ^ { ( s ) } , \\end{align*}"}
+{"id": "533.png", "formula": "\\begin{align*} | B _ { 2 } | \\lesssim t ^ { - \\frac { 1 } { 2 } } t ^ { - 1 } e ^ { - \\frac { 1 } { 1 6 } \\frac { ( x - y ) ^ { 2 } } { t } } = t ^ { - \\frac { 3 } { 2 } } e ^ { - \\frac { 1 } { 1 6 } \\frac { ( x - y ) ^ { 2 } } { t } } . \\end{align*}"}
+{"id": "7785.png", "formula": "\\begin{align*} \\frac { f ' ( x ) } { f ( x ) } ~ { \\rm f o r } ~ f ( x ) = \\frac { p ( x ) } { g ( x ) } , ~ g ( x ) = \\prod _ { j = 1 } ^ k ( x - z _ j ) , \\end{align*}"}
+{"id": "8469.png", "formula": "\\begin{align*} q _ 1 ( z ) : = \\mathcal { C } \\left ( Q _ { j , - } J \\right ) ( z ) q _ 2 ( z ) : = \\mathcal { C } \\left ( Q _ { j , - } J + D _ j \\right ) ^ H ( z ) \\end{align*}"}
+{"id": "8216.png", "formula": "\\begin{align*} x _ { i , 1 , k } & = 2 m - 2 i + 1 , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 1 , k } & = 4 m + 2 i , \\\\ x _ { i , 2 , k } & = 4 m + 2 i - 1 , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 2 , k } & = 2 m - 2 i + 2 , \\\\ x _ { i , 3 , k } & = 4 m - 2 i + k + 1 , \\mbox { a n d } \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 3 , k } & = 2 m + 2 i - k . \\end{align*}"}
+{"id": "3745.png", "formula": "\\begin{align*} d _ + f & = T _ 1 T _ 2 \\ldots T _ { \\ell } f [ X + ( t - 1 ) y _ { \\ell + 1 } ] \\\\ d _ - f & = \\left . - y _ { \\ell } f [ X - ( t - 1 ) y _ { \\ell } ] \\sum _ { i \\geq 0 } h _ i [ - X / y _ { \\ell } ] \\right | _ { y _ { \\ell } ^ 0 } \\end{align*}"}
+{"id": "8674.png", "formula": "\\begin{align*} { { \\bf { \\bar X } } ^ { \\star } } = \\left [ { { \\bf { \\tilde A } } , { { \\bf { 0 } } _ { ( \\sum { { { \\bar r } _ { l ' } } } ) \\times ( { M _ t } - ( \\sum { { { \\bar r } _ { l ' } } } ) ) } } } \\right ] , \\end{align*}"}
+{"id": "4626.png", "formula": "\\begin{align*} \\phi ( \\sigma , \\tau ) = [ \\widetilde \\sigma , \\widetilde \\tau ] = \\widetilde \\sigma \\widetilde \\tau { \\widetilde \\sigma } ^ { - 1 } { \\widetilde \\tau } ^ { - 1 } \\in \\langle \\epsilon \\rangle \\cong \\mathbb Z / 2 \\mathbb Z . \\end{align*}"}
+{"id": "2119.png", "formula": "\\begin{align*} \\alpha | z | ^ 2 + \\beta | w | ^ 2 + \\Re \\left ( a z ^ 2 + b w ^ 2 \\right ) - 1 = 0 , \\end{align*}"}
+{"id": "2693.png", "formula": "\\begin{align*} c _ 1 ( d ) = { } & 2 2 9 3 7 6 d ^ { 1 0 } + 1 9 6 6 0 8 0 d ^ 9 + 7 2 9 4 9 7 6 d ^ 8 + 1 5 3 2 3 1 3 6 d ^ 7 + 2 0 1 2 4 2 8 8 d ^ 6 + 1 7 5 5 5 0 7 2 d ^ 5 \\\\ & + 1 1 1 0 2 4 9 6 d ^ 4 + 5 9 1 7 0 3 2 d ^ 3 + 2 8 0 3 8 4 7 d ^ 2 + 9 3 3 6 3 9 d + 1 3 9 9 6 8 . \\end{align*}"}
+{"id": "946.png", "formula": "\\begin{align*} \\begin{aligned} { \\| J u _ 2 ( t ) \\| _ { L ^ 2 } } \\le { } & \\| J u _ 2 ( 1 ) \\| _ { L ^ 2 } \\\\ & { } + C \\int ^ t _ 1 ( \\| u _ 1 \\| ^ 2 _ { L ^ { \\infty } } \\| J u _ 2 \\| _ { L ^ 2 } + \\| u _ 1 \\| _ { L ^ { \\infty } } \\| u _ 2 \\| _ { L ^ { \\infty } } \\| J u _ 1 \\| _ { L ^ 2 } ) ( \\tau ) \\ d \\tau . \\end{aligned} \\end{align*}"}
+{"id": "6047.png", "formula": "\\begin{align*} 0 = & ( \\alpha ^ { 2 } - \\alpha ( p \\alpha - 1 ) ) a ( u , v ) + ( ( \\alpha - 1 ) ^ { 2 } - ( \\alpha - 1 ) ( p \\alpha - 1 ) ) b ( u , v ) \\\\ & + ( ( 3 \\alpha - 2 ) ^ { 2 } - ( 3 \\alpha - 2 ) ( p \\alpha - 1 ) ) c ( u , v ) \\end{align*}"}
+{"id": "429.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } k _ { i j } c _ { r j k } = \\sum _ { j = 1 } ^ { n } q _ { r i j } k _ { j k } . \\end{align*}"}
+{"id": "6607.png", "formula": "\\begin{align*} R _ { D } ^ \\pm = R _ { D ^ 0 } ^ \\pm + d ^ { D ^ 0 } S | _ { E _ \\pm \\otimes E _ { \\mp } \\otimes E _ \\pm } , \\end{align*}"}
+{"id": "7901.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Theta ( t , x ) = \\int _ { \\S ^ d } W _ r \\left ( \\sum _ { i = 1 } ^ d m _ i y _ i + m _ { d + 1 } x \\right ) ~ \\sin \\left ( \\sum _ { i = 1 } ^ d n _ i \\Theta ( t , y _ i ) + n _ { d + 1 } \\Theta ( t , x ) \\right ) \\ \\d y \\end{align*}"}
+{"id": "7698.png", "formula": "\\begin{align*} \\frac { \\| \\hat x - x \\| _ 2 } { \\| \\hat x \\| _ 2 } = O \\big ( n ^ 4 \\ , { \\bf u } \\ , \\kappa ( A ) \\ , { \\bf g _ { \\rm G E P P } } ( A ) \\big ) , \\end{align*}"}
+{"id": "6656.png", "formula": "\\begin{align*} \\begin{aligned} | \\mathcal { R } _ { 3 , \\epsilon } ( t ) | & \\leq \\vartheta \\lambda ^ { 2 } ( \\epsilon ) \\int _ 0 ^ t 1 + | \\varphi ( s ) | + | { X } ^ { \\epsilon } ( s ) | d s , \\end{aligned} \\end{align*}"}
+{"id": "3481.png", "formula": "\\begin{align*} r _ { \\ell _ 0 } \\leq e ^ { ( \\delta _ n - L + 5 0 \\varepsilon ) q _ n } \\max _ { \\ell _ 1 = \\ell _ 0 \\pm 1 } r _ { \\ell _ 1 } . \\end{align*}"}
+{"id": "2518.png", "formula": "\\begin{align*} \\max _ { i , j } \\left \\| \\lambda _ i ^ j ( \\mathbf { w } _ { \\bar { x } \\bar { s } } ) - \\nu \\right \\| = \\max _ { i , j } \\left \\| \\lambda _ i ^ j ( \\mathbf { w } _ { x s } ) - \\nu \\right \\| \\le \\gamma . \\end{align*}"}
+{"id": "1352.png", "formula": "\\begin{align*} d \\bigl ( u _ { 2 K - 3 } , \\gamma _ { a _ 1 , b _ 1 } ( t _ { 2 K - 2 } ) \\bigr ) = d \\bigl ( u _ { 2 K - 1 } , \\gamma _ { a _ 1 , b _ 1 } ( t _ { 2 K - 2 } ) \\bigr ) , \\end{align*}"}
+{"id": "5027.png", "formula": "\\begin{align*} h ( T ' ) & = y q \\log ( y q ) + r \\Delta ( q + 1 ) + f ( r ) \\\\ & = q y \\log ( q ) + q y \\log ( y ) + r \\log ( q ) + r + f ( r ) + o ( 1 ) \\\\ & = m \\log ( m - r ) - r \\log ( y ) + r + r \\log ( r ) + o ( 1 ) \\\\ & = m \\log ( m ) - r \\log \\left ( \\frac y r \\right ) + o ( 1 ) , \\end{align*}"}
+{"id": "5261.png", "formula": "\\begin{align*} \\phi \\left ( \\begin{pmatrix} 1 & u \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} t & 0 \\\\ 0 & \\frac { 1 } { t } \\end{pmatrix} \\right ) = 2 t \\sum _ { n = 1 } ^ \\infty \\rho _ \\phi ( n ) K _ { \\nu } ( 2 \\pi n t ^ 2 ) \\cos ( 2 \\pi n u ) . \\end{align*}"}
+{"id": "213.png", "formula": "\\begin{align*} G = H \\wr \\langle t \\rangle = A \\rtimes \\langle t \\rangle A = \\bigoplus \\nolimits _ { i \\in \\mathbb { Z } } H ^ { t ^ i } \\end{align*}"}
+{"id": "6228.png", "formula": "\\begin{align*} \\lim \\limits _ { \\alpha \\to \\infty } \\sup _ { \\Vert f \\Vert _ { L ^ 1 ( \\nu ) } \\leq 1 } \\psi _ \\alpha ( K ^ { * } f ) = 0 \\iff \\lim \\limits _ { \\alpha \\to \\infty } \\sup _ { \\omega = \\frac { 1 } { H } \\sum _ { h = 1 } ^ { H } \\delta _ { a _ h } } \\psi _ { \\alpha } ( K ^ * \\omega ) = 0 \\end{align*}"}
+{"id": "2772.png", "formula": "\\begin{align*} \\mu ( B ( x , r ) ) = \\mu ( B ( x , r ) \\cap V _ m ( x ) ) . \\end{align*}"}
+{"id": "8295.png", "formula": "\\begin{align*} \\d X _ t ^ i = \\bigg ( b ( X _ t ^ i ) + \\frac 1 { N - 1 } \\sum _ { j \\neq i } K ( X _ t ^ i - X _ t ^ j ) \\bigg ) \\d t + \\sigma \\d W _ t ^ i . \\end{align*}"}
+{"id": "8788.png", "formula": "\\begin{align*} \\tilde { B } = \\{ z \\in \\R ^ n : | \\Pi _ { \\mathrm { k e r } A ^ \\perp } z | ^ 2 \\le \\delta | \\Pi _ { \\mathrm { k e r } A } z | ^ { 2 r } \\} , \\end{align*}"}
+{"id": "6231.png", "formula": "\\begin{align*} \\Lambda = \\bigcup _ { n = 0 } ^ { \\infty } \\Lambda _ n = \\sum _ { k = 0 } ^ { \\infty } ( R ^ { t } ) ^ { k } L , \\end{align*}"}
+{"id": "6909.png", "formula": "\\begin{align*} H ( Q _ k \\ , | \\ , P ) = - \\log P ( B _ k ) \\to 0 . \\end{align*}"}
+{"id": "186.png", "formula": "\\begin{align*} S _ { H ' } ( C ^ \\bullet ) \\widehat { \\otimes } _ { A _ { H ' , \\infty } } A : = \\varinjlim _ n S _ { H ' , n } ( C ^ \\bullet ) \\widehat { \\otimes } _ { A _ { H ' , n } } A = C ^ \\bullet . \\end{align*}"}
+{"id": "6314.png", "formula": "\\begin{align*} \\rho _ k ^ { - 1 } L _ k \\overset { \\eqref { c o n i c - L } } { = } C + \\rho _ k ^ { - 1 } B + \\rho _ k ^ { - 1 } L _ { \\nabla g } \\sum _ { i = 0 } ^ { k - 1 } \\rho _ i \\eta _ i + \\rho _ k ^ { - 2 } \\overset { \\eqref { d e f 1 } } { \\leq } C + \\rho _ 0 ^ { - 1 } B + \\rho _ 0 ^ { - 1 } L _ { \\nabla g } \\theta + \\rho _ 0 ^ { - 2 } \\overset { \\eqref { d e f 2 } } { = } L . \\end{align*}"}
+{"id": "3056.png", "formula": "\\begin{align*} \\omega _ { \\lambda , z } = \\pi ^ * ( d \\log \\varphi _ { \\lambda , z } ) \\end{align*}"}
+{"id": "704.png", "formula": "\\begin{align*} \\aligned c _ \\lambda + 1 + \\frac { 1 } { 2 } \\| u _ n \\| _ { E _ \\lambda } \\geq \\Phi _ \\lambda ( u _ n ) - \\frac { 1 } { 2 } \\langle \\Phi ' _ { \\lambda } ( u _ n ) , u _ n \\rangle = \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 2 p } \\right ) J ( u _ n ) . \\endaligned \\end{align*}"}
+{"id": "5324.png", "formula": "\\begin{align*} f ^ \\lambda ( z ) = ( 2 \\pi ) ^ { - n } \\ , | \\lambda | ^ n \\ , \\sum _ { k = 0 } ^ \\infty R _ k ^ { n - 1 } ( \\lambda , f ) \\ , \\varphi _ { k , \\lambda } ^ { n - 1 } ( z ) . \\end{align*}"}
+{"id": "131.png", "formula": "\\begin{align*} & ~ N N _ 1 ^ { 5 - 2 \\alpha + \\varepsilon } N _ 1 ^ { - \\frac { 3 \\alpha } { 4 } + \\frac { 1 } { 2 } } N ^ { - \\frac { 1 } { 2 } } \\prod _ { i = 1 } ^ 3 \\sum _ { L _ i \\geqslant N _ 1 ^ { ( 5 - 2 \\alpha ) + } } L _ i ^ { \\frac { 1 } { 2 } } \\| f _ { i , L _ i } \\| _ { L ^ 2 } \\\\ & \\lesssim N ^ { \\frac { 1 } { 2 } } N _ 1 ^ { \\frac { 1 1 } { 2 } - \\frac { 1 1 \\alpha } { 4 } + \\varepsilon } \\| u _ { N _ 1 } \\| _ { F _ { N _ 1 } ( T ) } \\| u _ { N _ 2 } \\| _ { F _ { N _ 2 } ( T ) } \\| u _ N \\| _ { F _ N ( T ) } . \\end{align*}"}
+{"id": "6835.png", "formula": "\\begin{align*} L _ { k } ^ { \\left ( 1 \\right ) } \\left ( s , \\rho \\right ) = \\int _ { - \\infty } ^ { \\infty } \\frac { e ^ { i S ( \\tau , x - s ) } \\tau ^ { k - 1 } } { \\tau - \\rho } d \\tau \\end{align*}"}
+{"id": "5475.png", "formula": "\\begin{align*} c _ { 4 } ( q ) = \\begin{cases} c _ { 4 , 2 } ( q ) , & - 3 < q \\leq - 2 , \\\\ c _ { 4 , \\infty } ( q ) , & - 2 \\leq q < 0 . \\end{cases} \\end{align*}"}
+{"id": "652.png", "formula": "\\begin{align*} M ( t ) = \\sum _ { \\substack { j , h : \\\\ j + h \\in \\{ q - 1 , q \\} } } h \\cdot C _ { j , h } ( t ) . \\end{align*}"}
+{"id": "286.png", "formula": "\\begin{align*} U _ r ( p ) = \\begin{cases} \\max \\{ \\varphi ( p ) + \\delta , u ( p ) \\} & \\\\ u ( p ) & \\end{cases} \\end{align*}"}
+{"id": "6102.png", "formula": "\\begin{align*} \\varphi ^ { 0 0 } _ { n , p } = \\frac { \\mu _ n ^ { ( 1 2 ) } + \\mu ^ { ( 1 ) } - \\mu ^ { ( 2 ) } } { 2 \\mu _ n ^ { ( 1 2 ) } } \\ , \\rho ^ { 0 0 } _ { n , p } - \\frac { 1 } { 2 \\mu _ n ^ { ( 1 2 ) } } \\big ( ( \\mu _ n ^ { ( 1 2 ) } - \\mu ^ { ( 1 ) } - \\mu ^ { ( 2 ) } - \\mu ^ { ( 0 ) } ) \\mu _ n ^ { ( 1 2 ) } + \\mu ^ { ( 0 ) } ( \\mu ^ { ( 1 ) } - \\mu ^ { ( 2 ) } ) \\big ) \\end{align*}"}
+{"id": "8636.png", "formula": "\\begin{align*} \\mathbb { E } _ { n , m } ' [ ( \\alpha + \\beta ) ( \\gamma + \\delta ) ] \\\\ = \\mathbb { E } _ { n , m } [ \\chi _ { n , m } ( w , x _ 1 ) \\chi _ { n , m } ( w , x _ 2 ) ] \\\\ \\\\ = \\mathcal { K } _ { n , m } ( x _ 1 , x _ 2 ) \\end{align*}"}
+{"id": "810.png", "formula": "\\begin{align*} G '' ( c ) = - s c ^ { s - 2 } \\phantom { x x } \\phantom { x x } g ( c ) = G ( c ) - G ' ( c ) c = c ^ s \\end{align*}"}
+{"id": "1572.png", "formula": "\\begin{align*} \\gamma ^ { - 1 } S _ { \\gamma _ N x } \\gamma & = \\lbrace \\gamma ^ { - 1 } \\gamma ' \\gamma , \\gamma ' \\in S _ { \\gamma _ N x } \\rbrace \\\\ & = \\lbrace \\gamma ^ { - 1 } \\gamma ' \\gamma , \\gamma ' _ N \\gamma _ N \\cdot x \\in \\Bar { P } ^ 0 \\gamma _ N x \\rbrace \\\\ & = \\lbrace \\gamma ^ { - 1 } \\gamma ' \\gamma , \\gamma _ N ^ { - 1 } \\gamma ' _ N \\gamma _ N \\cdot x \\in \\Bar { P } ^ 0 x \\rbrace \\\\ & \\subset S _ x , \\end{align*}"}
+{"id": "7223.png", "formula": "\\begin{align*} - K _ X = ( n - 1 ) A _ X , \\end{align*}"}
+{"id": "812.png", "formula": "\\begin{align*} \\Delta _ \\Gamma V + V \\big | \\nabla _ \\Gamma \\nu \\big | ^ 2 = A \\colon D _ \\Gamma ^ 2 H + B \\cdot \\nabla _ \\Gamma H + C H [ 0 , T ] \\times M \\end{align*}"}
+{"id": "6854.png", "formula": "\\begin{align*} J _ { 2 , n } ^ { \\prime } ( x ) = J _ { 2 , n - 1 } ^ { \\prime } ( x ) - e ^ { - \\frac { x } { 2 } } \\eta ( x ) a _ { n - 1 } ^ { \\prime } ( x ) , \\end{align*}"}
+{"id": "8143.png", "formula": "\\begin{align*} \\Gamma ( G , M ( S ) ) = [ H ^ 0 ( G , r ( M ) ( S ) ) \\to H ^ 0 ( G , r ^ 2 ( M ) ( S ) ) \\to \\dots ] , \\end{align*}"}
+{"id": "8253.png", "formula": "\\begin{align*} \\frac { d \\mu _ 1 ^ { n } ( t ) } { d t } = 0 , \\end{align*}"}
+{"id": "1550.png", "formula": "\\begin{align*} u _ t & { } = ( 1 - t ) u + t ( v - \\alpha ) , & v _ s & { } = ( 1 - s ) v + s ( u + \\alpha ) . \\end{align*}"}
+{"id": "4682.png", "formula": "\\begin{align*} T = \\frac { 1 } { \\sqrt K } \\tanh ^ { - 1 } \\frac { ( n - 1 ) \\sqrt K } { - H } \\ \\ \\ \\ K > 0 \\ \\ \\ \\ T = \\frac { ( n - 1 ) } { - H } \\ \\ \\ \\ K = 0 . \\end{align*}"}
+{"id": "2632.png", "formula": "\\begin{align*} [ x , y ] = x \\diamond y - \\varepsilon ( x , y ) y \\diamond x , \\ ; \\ ; \\forall x , y \\in \\mathcal { H } ( A ) . \\end{align*}"}
+{"id": "6157.png", "formula": "\\begin{align*} { \\hat H } & = - \\left ( ( 1 + \\lambda r ^ 2 ) \\hat { \\Delta } + \\lambda r \\frac { \\partial } { \\partial r } + \\lambda { \\hat J } ^ 2 \\right ) + { \\cal V } ( r ) \\\\ & = - \\left ( ( 1 + \\lambda r ^ 2 ) \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + ( d - 1 + d \\lambda r ^ 2 ) \\frac { \\partial } { \\partial r } - \\frac { { \\hat J } ^ 2 } { r ^ 2 } \\right ) + { \\cal V } ( r ) , . \\end{align*}"}
+{"id": "4390.png", "formula": "\\begin{align*} \\int _ { X _ j } | ( 1 - b ( \\Psi ) ) f F ^ { 1 + \\delta } | _ { h _ i } ^ 2 \\le c _ i ( \\sup _ { X _ j } | F | ^ { 2 + 2 \\delta } ) \\int _ { X _ j } \\mathbb { I } _ { \\{ \\Psi < - t _ 0 \\} } | f | _ { \\hat { h } } ^ 2 < + \\infty , \\end{align*}"}
+{"id": "6362.png", "formula": "\\begin{align*} S _ r ( a , b ) = \\Gamma \\left ( 1 - \\frac { r } { 2 } \\right ) \\sum _ { n = 0 } ^ b ( - 1 ) ^ n \\binom { b - 1 } { n } \\frac { 1 } { ( a + n ) ^ { 1 - r / 2 } } , r < 2 . \\end{align*}"}
+{"id": "6267.png", "formula": "\\begin{align*} \\Gamma ^ k _ { i j } = 0 \\quad \\forall i \\neq j \\neq k \\neq i , \\end{align*}"}
+{"id": "4924.png", "formula": "\\begin{align*} \\delta _ 1 = 2 \\big [ \\nu F ( \\nu ) - J ( q ) \\big ] \\quad \\ , \\ , \\ , \\quad \\ , \\ , \\ , \\delta _ 2 = \\nu - 2 \\ , J ( m ) , \\end{align*}"}
+{"id": "512.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { \\nu } \\widehat { f } ( \\nu ) \\Phi _ { \\nu } ( x ) = \\sum _ { n = 0 } ^ { \\infty } P _ { n } f ( x ) , \\end{align*}"}
+{"id": "5436.png", "formula": "\\begin{align*} \\begin{aligned} | F ^ { i j } ( \\nabla _ \\beta u ) _ { i j } | \\leq \\ , & | \\nabla _ \\beta f ^ { 1 / k } | + C \\left ( f ^ { 1 / k } + b _ \\alpha \\sum _ { i = 1 } ^ n F ^ { i i } \\right ) \\\\ \\leq \\ , & C m ^ { 1 / k } m ^ { - 1 / 2 ( k - 1 ) } + C b _ \\alpha \\sum _ { i = 1 } ^ n F ^ { i i } \\\\ \\leq \\ , & C b _ \\alpha ^ { - 1 / 2 } m ^ { 1 / k } + C b _ \\alpha \\sum _ { i = 1 } ^ n F ^ { i i } \\end{aligned} \\end{align*}"}
+{"id": "1406.png", "formula": "\\begin{align*} G _ n ( x , z ) = \\mathrm { d e t } \\left ( f _ { n + i - j } ( z _ j - x _ i ) \\right ) _ { i , j = 1 } ^ d , \\end{align*}"}
+{"id": "1181.png", "formula": "\\begin{align*} \\sup _ { n } \\mathbb { E } _ { \\mathbb { P } } \\left [ \\exp \\left \\{ \\kappa \\sum _ { i = 1 } ^ { n } \\int _ { T _ { 0 } } ^ { T _ { 0 } + \\delta } \\Delta K _ { t } ^ { i , n } \\cdot d W _ { t } ^ { i } - \\frac { \\kappa } { 2 } \\sum _ { i = 1 } ^ n \\int _ { T _ { 0 } } ^ { T _ { 0 } + \\delta } \\left | \\Delta K _ { t } ^ { i , n } \\right | ^ { 2 } d t \\right \\} \\right ] \\end{align*}"}
+{"id": "5415.png", "formula": "\\begin{align*} \\begin{aligned} u ( x _ 0 ) & = \\varphi ( x _ 0 ) = \\frac { 1 } { 2 } \\sum _ { \\beta = \\alpha + 1 } ^ { n - 1 } b _ \\beta x _ \\beta ^ 2 + \\frac { 1 } { 6 } \\sum _ { \\xi , \\beta , \\gamma \\geq \\alpha + 1 } \\varphi _ { \\xi \\beta \\gamma } ( 0 ) x _ \\xi x _ \\beta x _ \\gamma + O ( | \\tilde { x } | ^ 4 ) \\\\ & \\geq \\frac { 1 } { 2 } \\delta ^ 2 b _ \\alpha ^ 2 - C \\delta ^ 3 b _ \\alpha ^ 2 \\geq \\frac { 7 } { 1 6 } \\delta ^ 2 b _ \\alpha ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "689.png", "formula": "\\begin{align*} ( \\beta ^ g \\phi ) _ { g _ 2 , g _ 1 } = \\sum _ { h \\in G } \\beta ^ g _ { g _ 2 , h } \\phi _ { h , g _ 1 } = \\phi _ { g ^ { - 1 } g _ 2 , g _ 1 } \\qquad \\\\ ( \\phi \\beta ^ g ) _ { g _ 2 , g _ 1 } = \\sum _ { h \\in G } \\phi _ { g _ 2 , h } \\beta ^ { g } _ { h , g _ 1 } = \\phi _ { g _ 2 , g g _ 1 } . \\end{align*}"}
+{"id": "8163.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\omega ( \\mu ) \\big [ \\xi ( \\mu , t ) - \\xi ( \\mu , 0 ) \\big ] \\ d \\mu = \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\infty } \\int _ { 0 } ^ { \\mu } \\varpi _ 1 ( \\mu , \\nu ) \\xi ( \\mu , s ) \\xi ( \\nu , s ) \\Lambda ( \\mu , \\nu ) \\ d \\nu d \\mu d s , \\end{align*}"}
+{"id": "2288.png", "formula": "\\begin{align*} \\langle T _ j f , g \\rangle _ \\lambda = \\langle T ^ { ( \\lambda ) } _ { a _ j } f , g \\rangle _ \\lambda \\end{align*}"}
+{"id": "4414.png", "formula": "\\begin{align*} \\int _ { a x } ^ { x } | \\theta ( t + \\delta t ) - \\theta ( t ) - \\delta t | ^ 2 d t & > \\int _ I | \\theta ( t + \\delta t ) - \\theta ( t ) - \\delta t | ^ 2 d t = \\delta ^ 2 \\int _ I t ^ 2 d t \\\\ & > \\delta ^ 2 \\int _ { a x } ^ { x / 2 } t ^ 2 d t = \\frac { \\delta ^ 2 x ^ 3 } { 3 } \\left ( \\frac { 1 } { 8 } - a ^ 3 \\right ) . \\end{align*}"}
+{"id": "2252.png", "formula": "\\begin{align*} ( A , B ) \\cdot a = a \\end{align*}"}
+{"id": "7792.png", "formula": "\\begin{align*} W _ r ^ M ( x ) = a _ i i \\in \\left [ \\frac { i - 1 } M , \\frac i M \\right ) , \\end{align*}"}
+{"id": "5207.png", "formula": "\\begin{align*} \\left | \\int _ { \\mathbb { R } ^ { 3 } } \\nabla \\times B \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) \\dd x \\right | = \\left | \\int _ { \\mathbb { R } ^ { 3 } } \\nabla \\times b \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) \\dd x \\right | \\lesssim | | \\nabla \\times b | | _ { L ^ { 2 } } | | b | | _ { L ^ { 2 } } . \\end{align*}"}
+{"id": "8937.png", "formula": "\\begin{gather*} g ( t ) \\allowbreak = \\allowbreak \\frac { 1 } { ( 1 - t ^ { 2 } ) ^ { \\beta } } \\left ( 1 + \\frac { \\rho t ^ { 2 } } { 1 - t ^ { 2 } } \\right ) ^ { - \\beta } \\allowbreak = \\\\ = \\allowbreak \\frac { 1 } { ( 1 - t ^ { 2 } ( 1 - \\rho ) ) ^ { \\beta } } = \\sum _ { n \\geq 0 } t ^ { 2 n } ( 1 - \\rho ) ^ { n } \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } / n ! . \\end{gather*}"}
+{"id": "6358.png", "formula": "\\begin{align*} \\operatorname { E } ( X ^ r ) & = \\frac { \\theta ^ { r / 2 } } { \\operatorname { B } ( a , b ) } \\int _ 0 ^ \\infty y ^ { - r / 2 } \\exp ( - a y ) \\left \\{ 1 - \\exp ( - y ) \\right \\} ^ { b - 1 } \\mathrm { d } y . \\end{align*}"}
+{"id": "8778.png", "formula": "\\begin{align*} \\mathcal { U } = \\mathrm { k e r } A \\cap \\S ^ { n - 1 } . \\end{align*}"}
+{"id": "2336.png", "formula": "\\begin{align*} \\left ( d J \\theta \\right ) ^ { ( 2 , 0 ) + ( 0 , 2 ) } = - \\left ( d \\theta \\right ) ^ { ( 2 , 0 ) + ( 0 , 2 ) } _ { J \\cdot , \\cdot } . \\end{align*}"}
+{"id": "759.png", "formula": "\\begin{align*} \\gamma _ { \\Theta } ^ s ( E ) = \\sup \\{ | \\langle \\nu , 1 \\rangle | \\} , \\end{align*}"}
+{"id": "620.png", "formula": "\\begin{align*} x = \\epsilon t ^ 2 + 1 , y = 2 t , z = 1 - \\epsilon t ^ 2 , t \\in F . \\end{align*}"}
+{"id": "1626.png", "formula": "\\begin{align*} \\cdot : G \\times G / K \\to G / K , x \\cdot y K : = x y K \\end{align*}"}
+{"id": "5213.png", "formula": "\\begin{align*} \\partial _ t \\Phi ^ { 2 } _ { + } + u \\cdot \\nabla \\Phi ^ { 2 } _ { + } & = 0 \\textrm { i n } \\ \\mathbb { R } ^ { 3 } \\times ( 0 , T ) , \\end{align*}"}
+{"id": "5239.png", "formula": "\\begin{align*} U _ C = \\nabla \\times ( \\Phi _ C \\nabla \\theta ) + G _ C \\nabla \\theta = b ( \\cdot - z _ 0 e _ z ) + B _ { \\infty } . \\end{align*}"}
+{"id": "5170.png", "formula": "\\begin{align*} \\left | H [ b _ 1 ] - H [ b _ 2 ] \\right | & \\lesssim \\left ( \\max _ { i = 1 , 2 } | | b _ i | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } ^ { 5 / 3 } \\right ) | | b _ 1 - b _ 2 | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } , \\textrm { f o r } \\ b _ 1 , b _ 2 \\in L ^ { 2 } _ { \\sigma , \\textrm { a x i } } ( \\mathbb { R } ^ { 3 } ) , \\end{align*}"}
+{"id": "681.png", "formula": "\\begin{align*} \\sin \\theta \\sin ( n \\theta ) = \\frac { 1 } { 2 } \\left [ \\cos ( ( n - 1 ) \\theta ) - \\cos ( ( n + 1 ) \\theta ) \\right ] = \\end{align*}"}
+{"id": "2496.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbf { T } _ { u } \\triangleq \\begin{pmatrix} & \\mathbf { u } _ { 1 } & \\mathbf { u } _ { 2 : n } ^ T \\\\ & \\mathbf { u } _ { 2 : n } & \\beta _ { u } \\mathbf { I } + \\frac { \\mathbf { u } _ { 2 : n } \\mathbf { u } _ { 2 : n } ^ T } { \\beta _ { u } + \\mathbf { u } _ { 1 } } \\end{pmatrix} , \\beta _ { u } = \\sqrt { \\mathbf { u } _ { 1 } ^ 2 - \\| \\mathbf { u } _ { 2 : n } \\| ^ 2 } \\end{aligned} \\end{align*}"}
+{"id": "50.png", "formula": "\\begin{align*} 4 \\beta \\int \\rho ^ { - Q + 2 } Z v \\ ; \\Delta _ { H } v = 2 \\beta \\int \\operatorname { d i v } ( \\rho ^ { - Q + 2 } Z ) | \\nabla _ { H } v | ^ 2 - 4 \\beta \\sum _ { i = 1 } ^ { m } \\int X _ i v [ X _ i , \\rho ^ { - Q + 2 } Z ] v . \\end{align*}"}
+{"id": "2400.png", "formula": "\\begin{align*} \\sum _ { v \\in G _ n } \\deg _ { G _ n } ( v ) = \\sum _ { v \\in G _ n } ( X _ v + \\bar X _ { v } ) \\in ( ( 1 - \\delta ) n d , ( 1 + 2 \\delta ) n d ) \\subset ( n d / 2 , 2 n d ) . \\end{align*}"}
+{"id": "2560.png", "formula": "\\begin{align*} \\mu _ { \\rho , \\{ \\mathcal { D } _ { n } \\} } = \\delta _ { \\rho \\mathcal { D } _ { 1 } } * \\delta _ { \\rho ^ { 2 } \\mathcal { D } _ { 2 } } * \\delta _ { \\rho ^ { 3 } \\mathcal { D } _ { 3 } } * \\cdots . \\end{align*}"}
+{"id": "6136.png", "formula": "\\begin{align*} s _ { a + b , - a - b } = s _ { b , - b } s _ { a , - a } = s _ { a , - a } s _ { b , - b } . \\end{align*}"}
+{"id": "5534.png", "formula": "\\begin{align*} \\int _ { \\mathfrak X } | \\omega | & = { \\pi _ s } _ ! \\int _ { \\mathfrak U } | \\omega | = { \\pi _ s } _ ! \\Big ( { \\psi _ s } _ ! \\int _ { \\mathfrak Y } | \\omega | \\Big ) = { \\phi _ s } _ ! \\int _ { \\mathfrak Y } | \\omega | , \\end{align*}"}
+{"id": "7931.png", "formula": "\\begin{align*} g ( x ) = \\int _ \\S W _ r ( y ) \\prod _ { i = 1 } ^ n \\eta _ i ( y ) \\sin ^ { [ n ] } ( \\tilde \\Psi ( y ) ) \\ \\d y . \\end{align*}"}
+{"id": "2409.png", "formula": "\\begin{align*} \\displaystyle G ^ { a b } \\simeq \\Z _ { p } ^ { r } \\times \\prod _ { s = 1 } ^ { n } \\Z / p ^ { i _ { s } } \\Z \\end{align*}"}
+{"id": "2765.png", "formula": "\\begin{align*} q = k \\eta ( k ) + \\exp ( k \\eta ( k ) ) \\geq 2 0 ( | s _ 1 | + | s _ 2 | ) \\end{align*}"}
+{"id": "1934.png", "formula": "\\begin{align*} \\begin{aligned} & \\ \\ \\ \\ \\eta ^ { a m + 1 } ( 3 z ) \\eta ^ { b m - 1 } ( z ) \\ | \\ U ( m ) \\\\ & \\equiv _ m \\left ( \\sum _ { n = 0 } ^ \\infty b _ 3 ( n ) q ^ { n + \\frac { m ( 3 a + b ) + 2 } { 2 4 } } \\ | \\ U ( m ) \\right ) \\cdot \\prod _ { n = 1 } ^ \\infty ( 1 - q ^ { 3 n } ) ^ a ( 1 - q ^ { n } ) ^ b . \\end{aligned} \\end{align*}"}
+{"id": "2850.png", "formula": "\\begin{align*} \\sum _ x | \\tilde p _ x ( \\ell ) | ^ 2 = \\left ( \\frac { 2 \\pi \\ell } { \\theta _ n } \\right ) ^ 2 \\sum _ x | \\tilde q _ x ( \\ell ) | ^ 2 . \\end{align*}"}
+{"id": "1526.png", "formula": "\\begin{align*} F ( u , q ) = 0 \\end{align*}"}
+{"id": "4193.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } j ( t ) = \\int _ { \\Omega } ( K _ H ( x ) \\nabla u | \\nabla u ) d x > 0 . \\end{align*}"}
+{"id": "494.png", "formula": "\\begin{align*} \\mu _ { r _ 1 } \\ast \\mu _ { r _ 3 } ( d x ) : = e ^ { - c _ { 1 3 } } \\delta _ 0 ( d x ) + ( 1 - e ^ { - c _ { 1 3 } } ) f _ { 1 3 } ( x ) d x , \\end{align*}"}
+{"id": "4161.png", "formula": "\\begin{align*} u _ { \\varepsilon } ( x ' , t ) : = u ( x ' , t + \\varepsilon ) \\quad f _ { \\varepsilon } ( x ' ) : = u ( x ' , \\varepsilon ) , x ' \\in \\R ^ n , \\ , t > 0 . \\end{align*}"}
+{"id": "2977.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\max _ { 1 \\leq i < j \\leq k } \\widetilde { d } ( T _ { \\vec { v } } ^ n ( x _ i , t _ i ) , T _ { \\vec { v } } ^ n ( x _ j , t _ j ) ) = 0 \\end{align*}"}
+{"id": "2862.png", "formula": "\\begin{align*} \\tilde S ^ { ( q ) } _ { j , j ' } = \\frac { 2 \\Theta ( \\mu _ j , \\mu _ { j ' } ) } { \\mu _ { j } + \\mu _ { j ' } } \\tilde F _ { j , j ' } . \\end{align*}"}
+{"id": "6423.png", "formula": "\\begin{align*} \\sigma : = | u _ { , i j k } | ^ 2 \\end{align*}"}
+{"id": "8930.png", "formula": "\\begin{align*} H _ { j , n } = E X ^ { j } h _ { n } ( X ) = \\frac { 1 } { \\sqrt { n ! } } E X ^ { j } H _ { n } ( X ) = \\left \\{ \\begin{array} { c c c } 0 & & n > j j - n \\\\ \\frac { j ! } { 2 ^ { ( j - n ) / 2 } ( ( j - n ) / 2 ) ! \\sqrt { n ! } } & & j - n \\end{array} \\right . \\end{align*}"}
+{"id": "4615.png", "formula": "\\begin{align*} & | X ^ \\beta ( 0 ; t , x ) - X ^ \\beta ( 0 ; t , y ) | \\le | x - y | + \\int _ 0 ^ t \\left | \\frac { d } { d s } X ^ \\beta ( s ; t , x ) - \\frac { d } { d s } X ^ \\beta ( s ; t , y ) \\right | \\dd s \\\\ & = | x - y | + \\int _ 0 ^ t | u ( X ^ \\beta ( s ; t , x ) , s ) - u ( X ^ \\beta ( s ; t , y ) , s ) | \\dd s \\\\ & \\le | x - y | + \\int _ 0 ^ t \\varphi _ \\Theta ( | X ^ \\beta ( s ; t , x ) - X ^ \\beta ( s ; t , y ) | ) B \\dd s . \\end{align*}"}
+{"id": "3090.png", "formula": "\\begin{align*} \\theta ^ { k + 1 } = \\theta ^ { k } - \\alpha d ( \\theta ^ { k } , \\tilde { \\theta } ^ { k } , \\zeta ^ k ) , \\end{align*}"}
+{"id": "8375.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\mathbb { R } } s _ 1 ( z ) s _ 2 ( z ) d z = \\int _ { \\mathbb { R } } \\left [ \\mathcal { P } ^ { - } ( s J ) + s J \\right ] \\left [ \\mathcal { P } ^ { - } ( s J ) \\right ] ^ { H } \\mathrm { d } z = 0 . \\end{aligned} \\end{align*}"}
+{"id": "491.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } f _ s ( x ) / f ( x ) = ( 1 - e ^ { - \\lambda _ s } ) ^ { - 1 } . \\end{align*}"}
+{"id": "2245.png", "formula": "\\begin{align*} a ( Z ) = a \\big ( ( Z ^ * Z ) ^ \\frac { 1 } { 2 } \\big ) \\end{align*}"}
+{"id": "5157.png", "formula": "\\begin{align*} H [ b ] = 2 \\mu \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi _ { \\infty } ) _ { + } ^ { 2 } \\frac { 1 } { r ^ { 2 } } \\dd x . \\end{align*}"}
+{"id": "2193.png", "formula": "\\begin{align*} \\mathbf { U } _ V = \\mathbf { I } - \\mathbf { u } _ { \\lambda _ 1 } ( \\mathbf { u } _ { \\lambda _ 1 } ^ { H } \\mathbf { u } _ { \\lambda _ 1 } ) ^ { - 1 } \\mathbf { u } _ { \\lambda _ 1 } ^ { H } , \\end{align*}"}
+{"id": "2302.png", "formula": "\\begin{align*} \\langle T f , g \\rangle _ \\lambda = \\langle T ^ { ( \\lambda ) } _ a f , g \\rangle _ \\lambda \\end{align*}"}
+{"id": "2795.png", "formula": "\\begin{align*} \\mathbb { P } ( \\hat { \\eta } ^ { D , M } ( D \\times [ b , \\infty ) ) < \\infty ) = 1 , \\end{align*}"}
+{"id": "2308.png", "formula": "\\begin{align*} \\rho ( \\xi , Q _ n ) _ p = \\max _ { z \\in Q _ n ^ \\perp } \\frac { \\langle \\xi , z \\rangle } { \\| z \\| _ { p ' } } \\ge \\frac { \\langle \\xi , P \\xi \\rangle } { \\| P \\xi \\| _ { p ' } } . \\end{align*}"}
+{"id": "3169.png", "formula": "\\begin{align*} k _ 3 ( \\langle H \\rangle , a ) = \\frac { 1 } { 2 } \\times \\mathop { \\sum \\sum } _ { \\chi _ 4 ( x - y ) \\in \\{ 1 , \\chi _ 4 ( g ) \\} } 1 , \\end{align*}"}
+{"id": "7603.png", "formula": "\\begin{align*} Z = \\prod _ { k \\in \\N } Z _ k , \\end{align*}"}
+{"id": "9339.png", "formula": "\\begin{align*} \\min _ { \\xi , \\eta } p ( \\xi , \\eta ) & = \\min \\left \\{ p \\left ( \\frac { 4 7 } { 5 0 } , 1 \\right ) , p \\left ( \\frac { 4 7 } { 5 0 } , \\frac { 9 9 } { 1 0 0 } \\right ) \\right \\} \\\\ & \\ge 0 . 0 0 4 6 . \\end{align*}"}
+{"id": "1490.png", "formula": "\\begin{align*} \\phi _ 1 ' = \\phi _ 3 , \\end{align*}"}
+{"id": "3780.png", "formula": "\\begin{align*} C F ( F ) = \\widehat { t e l } ( C F ( F ) , i d ) \\rightarrow C ^ { L , k } = \\varinjlim _ { p } \\varprojlim _ { q } \\varinjlim _ { n _ { i } } C F ^ { L , k } _ { [ p , q ] } ( H _ { n _ { i } , t } ) . \\end{align*}"}
+{"id": "5358.png", "formula": "\\begin{align*} \\widehat { \\iota } ( D ) = \\frac { \\beta - e } { d } \\end{align*}"}
+{"id": "202.png", "formula": "\\begin{align*} \\theta _ { i , ( 1 ) } ( f ) = \\sum _ { j } - a _ { i , j } \\theta _ { j , ( 2 ) } ( f ) . \\end{align*}"}
+{"id": "5497.png", "formula": "\\begin{align*} A _ 4 \\leq \\int _ { 1 0 } ^ \\infty \\Big ( ( 8 / \\pi ) ^ { 1 / 2 } ( 1 0 0 / 9 9 ) ^ { 1 / 4 } t ^ { - 3 / 2 } \\Big ) ^ { 1 . 3 } t ^ { p - 1 } \\dd t & = \\frac { 2 ^ { 5 3 / 2 0 } } { 1 1 ^ { 1 3 / 4 0 } \\cdot 5 ^ { 3 / 1 0 } ( 3 \\pi ) ^ { 1 3 / 2 0 } } \\frac { 1 0 ^ p } { 3 9 - 2 0 p } \\\\ & \\leq \\frac { 2 ^ { 5 3 / 2 0 } } { 1 1 ^ { 1 3 / 4 0 } \\cdot 5 ^ { 3 / 1 0 } ( 3 \\pi ) ^ { 1 3 / 2 0 } } \\frac { 1 0 ^ p } { 3 4 } . \\end{align*}"}
+{"id": "5326.png", "formula": "\\begin{align*} \\lambda ^ { - m } \\ , \\int _ { S ^ { n - 1 } } \\widehat { f } ( \\lambda , \\omega ) \\ , Y _ m ( \\omega ) \\ , d \\omega = c _ { n , m } \\int _ 0 ^ \\infty r ^ { - m } f _ m ( r ) \\ , \\frac { J _ { n / 2 + m - 1 } ( \\lambda r ) } { ( \\lambda r ) ^ { n / 2 + m - 1 } } r ^ { n + 2 m - 1 } d r \\end{align*}"}
+{"id": "6749.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty z ^ i \\ , \\phi _ s ( z ) \\ , [ \\Phi _ s ( z ) ] ^ { j + \\alpha - 1 } \\mathrm { d } z = & \\int _ 0 ^ \\infty z ^ i \\ , \\phi _ s ( z ) \\ , [ \\Phi _ s ( z ) ] ^ { j + \\alpha - 1 } \\mathrm { d } z \\\\ & + \\int _ 0 ^ \\infty ( - 1 ) ^ i \\ , z ^ i \\ , \\phi _ s ( z ) [ 1 - \\Phi _ s ( z ) ] ^ { j + \\alpha - 1 } \\mathrm { d } z . \\end{align*}"}
+{"id": "8881.png", "formula": "\\begin{align*} \\lambda _ { k , S , D } = \\Gamma ( \\ell ) ( \\det 2 S ) ^ { \\ell - 1 / 2 } 2 ^ { - g / 2 } ( 2 \\pi D ) ^ { - \\ell } \\ , \\ell = k - g / 2 - 1 . \\end{align*}"}
+{"id": "8049.png", "formula": "\\begin{align*} y _ i = a _ c s _ c \\left ( \\hat { \\mathbf { h } } _ { i , * } + \\tilde { \\mathbf { h } } _ { i , * } \\right ) \\mathbf { p } _ c + \\sum _ { j = 1 } ^ { M } a _ j s _ j \\left ( \\hat { \\mathbf { h } } _ { i , * } + \\tilde { \\mathbf { h } } _ { i , * } \\right ) \\mathbf { p } _ j + n _ i . \\end{align*}"}
+{"id": "1010.png", "formula": "\\begin{align*} \\ell ( x s _ \\alpha , s _ \\alpha v \\beta ) = & \\ell ( x , v \\beta ) + \\ell ( s _ \\alpha , s _ \\alpha v \\beta ) \\\\ = ~ \\ , & \\ell ( x , v \\beta ) + \\begin{cases} 1 , & v \\beta = - \\alpha , \\\\ - 1 , & v \\beta = \\alpha , \\\\ 0 , & v \\beta \\neq \\pm \\alpha . \\end{cases} \\end{align*}"}
+{"id": "757.png", "formula": "\\begin{align*} E = \\bigcap _ { k = 0 } ^ \\infty E _ k . \\end{align*}"}
+{"id": "5393.png", "formula": "\\begin{align*} \\tilde { f } ( x ) \\leq \\tilde { f } ( 0 ) + \\sum _ { i = 1 } ^ n \\tilde { f } _ i ( 0 ) x _ i + C | x | ^ 2 \\leq C \\left ( \\sigma _ { k - 1 } ^ { 1 / ( k - 1 ) } ( b ) + \\sigma _ { k - 1 } ^ { 1 / 2 ( k - 1 ) } ( b ) | x ' | + x _ n + | x | ^ 2 \\right ) \\end{align*}"}
+{"id": "2307.png", "formula": "\\begin{align*} d _ n ( K , \\ell _ \\infty ^ { N ' } ) \\le \\delta , K : = \\{ \\xi ' ( \\omega ) \\colon \\Lambda ( \\omega ) = \\Lambda ^ \\circ , \\ ; \\Gamma ( \\omega ) = \\Gamma ^ \\circ \\} . \\end{align*}"}
+{"id": "8945.png", "formula": "\\begin{align*} \\det ( Y ) ^ { k / 2 } | F ( Z ) | \\ll _ { n , \\epsilon } \\begin{cases} k ^ { \\frac { 3 n ( n + 1 ) } { 8 } } \\det ( Y ) ^ { \\frac { n + 1 } { 4 } + \\epsilon } \\\\ k ^ { \\frac { n ( n + 1 ) } { 4 } } \\det ( Y ) ^ { \\frac { 3 ( n + 1 ) } { 8 } + \\epsilon } . \\end{cases} \\end{align*}"}
+{"id": "4623.png", "formula": "\\begin{align*} \\epsilon x _ i = x _ i \\epsilon , 1 \\leq i \\leq n . \\end{align*}"}
+{"id": "2309.png", "formula": "\\begin{align*} | e ( x y / 2 ^ k ) - e ( \\sum _ { s = 1 } ^ { s _ 0 } 2 ^ { - s } \\sum _ { i + j = k - s } x _ i y _ j ) | \\le 2 \\pi k 2 ^ { - s _ 0 } . \\end{align*}"}
+{"id": "4858.png", "formula": "\\begin{align*} 0 = D f | _ { \\Gamma _ 0 } ( \\Delta ) = \\phi ( C \\Delta + \\Delta ^ T C ^ T + \\Delta ^ T D \\Gamma _ 0 + \\Gamma _ 0 ^ T D \\Delta ) . \\end{align*}"}
+{"id": "6863.png", "formula": "\\begin{align*} { \\tt P h i 1 = p l g c i r m a p ( v e r , i n f ) } \\end{align*}"}
+{"id": "4132.png", "formula": "\\begin{align*} u ( x ' , t ) : = P _ t ^ L * f ( x ' ) , ( x ' , t ) \\in \\R ^ n _ + , \\end{align*}"}
+{"id": "3632.png", "formula": "\\begin{align*} h _ { 1 1 i i } = h _ { i i 1 1 } + \\kappa _ 1 ^ 2 \\kappa _ i - \\kappa _ 1 \\kappa _ i ^ 2 - \\kappa _ 1 + \\kappa _ i . \\end{align*}"}
+{"id": "4501.png", "formula": "\\begin{align*} Z _ { V } ( [ \\alpha ] ) = - \\int _ { V } \\alpha + \\sqrt { - 1 } \\int _ { V } \\omega , \\end{align*}"}
+{"id": "4789.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| T _ n p _ n f - p _ n T f \\| = 0 f \\in X . \\end{align*}"}
+{"id": "2172.png", "formula": "\\begin{align*} { \\bf { x } } _ { n , k } = [ \\frac { x [ n ] - x _ k } { \\| { \\bf { v } } _ n - { \\bf { u } } _ k \\| _ 2 } , \\frac { y [ n ] - y _ k } { \\| { \\bf { v } } _ n - { \\bf { u } } _ k \\| _ 2 } , \\frac { z [ n ] } { \\| { \\bf { v } } _ n - { \\bf { u } } _ k \\| _ 2 } , \\psi _ n ] ^ { } . \\end{align*}"}
+{"id": "730.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} g ( x ) & : = { \\rm s g n } ( x ) f ( | x | ) , \\\\ \\tilde V _ w ^ { \\varepsilon } ( x , t ) & : = \\frac { 1 } { 2 } ( h ( | x | ) ) ^ 2 - \\dot y ( t ) h ( | x | ) + \\varepsilon f ' ( | x | ) - \\frac { \\varepsilon } { 2 } y ( t ) h '' ( | x | ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3079.png", "formula": "\\begin{align*} \\varphi _ A ( x _ 1 , \\dots , x _ n ) = ( x _ { 1 } ^ { b _ { 1 1 } } \\cdots x _ { n } ^ { b _ { 1 n } } , \\dotsc , x _ { 1 } ^ { b _ { n 1 } } \\cdots x _ { n } ^ { b _ { n n } } ) \\end{align*}"}
+{"id": "9136.png", "formula": "\\begin{align*} M ^ d \\times ( - \\infty , 0 ) \\ni ( x , z ) \\mapsto \\mathcal { S } ( x , z ) = ( x , \\varrho ( x , z ) ) \\in \\Omega , \\end{align*}"}
+{"id": "5350.png", "formula": "\\begin{align*} \\vert S \\cap [ 1 , n ] \\vert \\leq \\sum _ { \\ell = 1 } ^ U \\vert S _ { \\ell } \\vert = O ( U ^ 2 ) = O ( ( \\log n ) ^ 2 ) = o ( n / \\log n ) \\end{align*}"}
+{"id": "156.png", "formula": "\\begin{align*} e _ { r , t } \\ : = \\ : \\left ( \\Gamma ( { \\textstyle \\frac { t } { t - 1 } } ) + o ( 1 ) \\right ) r ^ { - ( 1 + \\frac { 1 } { t - 1 } ) } \\ ; \\ ; \\ ; \\ ; { \\rm a s } \\ ; r \\to \\infty \\ , . \\end{align*}"}
+{"id": "1931.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty b _ 5 \\left ( \\frac { m n - 1 } { 6 } \\right ) q ^ n \\in S _ { 2 m - 2 } ( \\Gamma _ 0 ( 1 8 0 ) , \\chi _ 5 ) _ m , \\end{align*}"}
+{"id": "6530.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\pi ( a ) } q _ i = \\prod _ { i = 1 } ^ { \\pi ( a ) } p _ i ^ { \\alpha _ i } . \\end{align*}"}
+{"id": "3866.png", "formula": "\\begin{align*} w ( x ' , t ) = W ( \\bar { R } _ { - \\alpha | \\ln \\varepsilon | t } ( x ' ) ) , \\ \\ \\varphi ( x ' , t ) = \\varPhi ( \\bar { R } _ { - \\alpha | \\ln \\varepsilon | t } ( x ' ) ) , \\end{align*}"}
+{"id": "8572.png", "formula": "\\begin{align*} E _ { \\alpha , \\beta } ( z ) = \\sum _ { k = 0 } ^ { + \\infty } \\frac { z ^ k } { \\Gamma ( \\alpha \\ , k + \\beta ) } , \\ \\alpha > 0 , \\ \\beta , z \\in \\C . \\end{align*}"}
+{"id": "1580.png", "formula": "\\begin{align*} h : = b \\vert _ { E _ 1 } + \\sum \\limits _ { k = 2 } ^ \\delta e ^ { 2 f _ k } b \\vert _ { E _ k } + g , \\end{align*}"}
+{"id": "6422.png", "formula": "\\begin{align*} \\phi = \\frac { 1 } { 2 } ( u _ { , p } \\xi ^ p + v _ q x ^ q ) . \\end{align*}"}
+{"id": "3441.png", "formula": "\\begin{align*} \\delta _ n ' = \\frac { \\ln q _ { n + 1 } + \\ln | \\sin ( \\pi ( \\theta - \\frac { 1 } { 2 } + m _ n \\alpha ) ) | } { q _ n } \\leq \\frac { \\ln q _ { n + 1 } + \\ln ( \\pi \\| q _ n \\alpha \\| ) } { q _ n } < \\frac { \\ln \\pi } { q _ n } . \\end{align*}"}
+{"id": "6816.png", "formula": "\\begin{align*} \\varphi _ { 0 } ^ { - } \\left ( \\rho \\right ) : = \\frac { 1 } { 2 \\pi i } \\int _ { - \\infty } ^ { \\infty } \\frac { \\varphi _ { 0 } \\left ( \\tau \\right ) } { \\tau - \\rho } d \\tau , \\quad \\rho \\in \\mathbb { R } , \\end{align*}"}
+{"id": "4862.png", "formula": "\\begin{align*} \\Gamma ( x , y ) & = - \\frac { 1 } { e ^ { - { \\psi } } } \\left ( d \\phi ( x ) \\int _ { - \\infty } ^ y e ^ { - { \\psi ( x , v ) } } d v - \\int _ { - \\infty } ^ y e ^ { - { \\psi ( x , v ) } } d _ x { \\psi } ( x , v ) d v \\right ) \\\\ & = \\frac { 1 } { e ^ { - \\psi } } \\left ( d \\phi ( x ) \\int _ { y } ^ \\infty e ^ { - { \\psi ( x , v ) } } d v - \\int _ { y } ^ \\infty e ^ { - { \\psi ( x , v ) } } d _ x { \\psi } ( x , v ) d v \\right ) . \\end{align*}"}
+{"id": "7647.png", "formula": "\\begin{align*} k ( t , x , y ) : = \\rho ( - R ^ { - 1 } B P _ t x - R ^ { - 1 } B y ) \\quad \\end{align*}"}
+{"id": "8153.png", "formula": "\\begin{align*} f _ * \\mathcal L ^ \\vee ( d ) [ 2 d ] = ( f _ * \\mathcal L ) ^ \\vee \\end{align*}"}
+{"id": "4904.png", "formula": "\\begin{align*} \\lim _ { x \\to - \\infty } \\frac { h _ { } ( x ) } { \\phi ( \\frac { x - \\mu } { \\sigma } ) } = \\frac { 1 } { \\sigma } \\lim _ { x \\to - \\infty } \\frac { \\phi ( \\frac { x - \\mu } { \\sigma } ) } { 1 - \\Phi ( \\frac { x - \\mu } { \\sigma } ) } \\frac { 1 } { \\phi ( \\frac { x - \\mu } { \\sigma } ) } = \\frac { 1 } { \\sigma } \\lim _ { x \\to - \\infty } \\frac { 1 } { 1 - \\Phi ( \\frac { x - \\mu } { \\sigma } ) } = \\frac { 1 } { \\sigma } . \\end{align*}"}
+{"id": "1110.png", "formula": "\\begin{align*} x = \\sqrt { { p _ s } } { x _ s } + \\sqrt { { p _ { b , m } } } { x _ { b , m } } + \\sqrt { { p _ { b , o } } } { x _ { b , o } } , \\end{align*}"}
+{"id": "5648.png", "formula": "\\begin{align*} & u _ { i } + \\ , ^ { t } B \\psi x _ { i } = 0 , \\\\ & c ^ { i } _ { j } + c ^ { j } _ { i } + \\langle x _ { i } , x _ { j } \\rangle = 0 , \\\\ & B \\in O _ { n - k , n - k } . \\end{align*}"}
+{"id": "3526.png", "formula": "\\begin{align*} \\mathcal { S } _ n ^ { ( p ) } \\ = \\ \\left ( \\sum _ { k = 0 } ^ p \\binom { p } { k } ( - 1 ) ^ k A _ k n ^ { p - k } \\right ) F _ n \\ + \\ \\left ( \\sum _ { k = 0 } ^ p \\binom { p } { k } ( - 1 ) ^ k B _ k n ^ { p - k } \\right ) F _ { n + 1 } \\ - \\ ( - 1 ) ^ p B _ p . \\end{align*}"}
+{"id": "3981.png", "formula": "\\begin{align*} \\beta _ 1 + \\bar { \\beta _ j } = 1 \\ , \\ , { \\rm a n d } \\ , \\ , \\beta _ u + \\bar { \\beta } _ v = 1 . \\end{align*}"}
+{"id": "735.png", "formula": "\\begin{align*} R \\sqrt { R } \\geq 8 R \\geq 8 \\max ( 4 , 8 \\| g \\| ) > C = 4 \\max ( 1 , \\sqrt { \\| g \\| } ) , \\end{align*}"}
+{"id": "2436.png", "formula": "\\begin{align*} F _ { q } ( s - s _ { 0 } ) = L _ { q } \\left [ f ( t ) \\exp _ { q } \\left [ \\frac { s _ { 0 } t } { 1 - ( 1 - q ) s t } \\right ] \\right ] . \\end{align*}"}
+{"id": "478.png", "formula": "\\begin{align*} e ^ { \\gamma x } f _ r ( x ) = e ^ { \\gamma x } e ^ { - c _ { 1 3 } } f _ 0 ( x ) & + ( 1 - e ^ { - c _ { 1 3 } } ) \\int _ { - \\infty } ^ { x / 2 } e ^ { \\gamma ( x - y ) } f _ 0 ( x - y ) e ^ { \\gamma y } f _ { 1 3 } ( y ) d y \\\\ & + ( 1 - e ^ { - c _ { 1 3 } } ) \\int _ { - \\infty } ^ { x / 2 } e ^ { \\gamma ( x - y ) } f _ { 1 3 } ( x - y ) e ^ { \\gamma y } f _ 0 ( y ) d y . \\end{align*}"}
+{"id": "8544.png", "formula": "\\begin{align*} \\tilde { u } ( x , t ) = \\begin{cases} u ( x , t ) , & t \\in [ 0 , T _ ] \\\\ u _ 1 ( x , t ) , & t \\in [ T _ , T _ + T _ 1 ] , \\end{cases} \\end{align*}"}
+{"id": "2512.png", "formula": "\\begin{align*} \\mathbf { R } _ { x s } = \\left ( \\mathbf { R } _ { x s } ^ i : i = 1 , \\cdots , k \\right ) , \\end{align*}"}
+{"id": "2904.png", "formula": "\\begin{align*} H _ { x , x ' } ^ { ( n ) } = \\frac { 1 } { ( n + 1 ) ^ 2 } \\sum _ { j , j ' = 0 } ^ n \\Phi \\left ( \\frac { \\theta j } { n + 1 } , \\frac { \\theta ' j ' } { n + 1 } \\right ) \\exp \\left \\{ \\frac { 2 i \\pi j x } { n + 1 } \\right \\} \\exp \\left \\{ \\frac { 2 i \\pi j ' x ' } { n + 1 } \\right \\} \\end{align*}"}
+{"id": "7185.png", "formula": "\\begin{align*} \\frac { \\partial \\textbf { \\textit { W } } } { \\partial t } - ( B - Q ) \\textbf { \\textit { W } } = - \\textbf { \\textit { Y } } . \\end{align*}"}
+{"id": "5957.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u = \\Delta u - ( u \\cdot \\nabla ) u - \\nabla p - \\nabla \\cdot ( \\nabla n \\otimes \\nabla n ) , \\\\ \\nabla \\cdot u = 0 , \\\\ \\partial _ t n = \\Delta n - ( u \\cdot \\nabla ) n - \\Phi ( n ) , \\\\ u = 0 \\quad \\frac { \\partial n } { \\partial \\nu } = 0 \\quad \\partial \\mathcal { O } , \\\\ u ( 0 ) = u _ 0 , n ( 0 ) = n _ 0 , \\end{cases} \\end{align*}"}
+{"id": "263.png", "formula": "\\begin{align*} | p | _ G = \\left ( ( x ^ 2 + y ^ 2 ) ^ 2 + 1 6 z ^ 2 \\right ) ^ { \\frac { 1 } { 4 } } . \\end{align*}"}
+{"id": "5893.png", "formula": "\\begin{align*} \\widetilde { X } _ n ( t ) = & P _ n x + \\int _ 0 ^ t P _ n A ( s , \\widetilde { X } _ n ( s ) ) d s + \\int _ 0 ^ t P _ n B ( s , \\widetilde { X } _ n ( s ) ) Q _ n d \\widetilde { W } ( s ) , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "6627.png", "formula": "\\begin{align*} I ( f ) = \\inf \\left \\{ \\int ^ { 1 } _ { 0 } \\Lambda ^ { * } ( \\phi ( r ) ) d r ; ~ f ( t ) = \\int ^ { 1 } _ { 0 } K ( t , r ) \\phi ( r ) d r , ~ \\phi \\in L ^ 2 ( [ 0 , 1 ] ) , ~ t \\in [ 0 , 1 ] \\right \\} , \\end{align*}"}
+{"id": "8409.png", "formula": "\\begin{align*} ( I - F ) q ( x ; z ) = m _ 1 e _ 1 + m _ 2 e _ 2 + m _ 3 e _ 2 , \\end{align*}"}
+{"id": "1087.png", "formula": "\\begin{align*} \\prescript J { } \\pi ( s _ \\beta w ) = ( w ' , \\mu + \\Phi ^ + ( - \\beta ) w ^ { - 1 } \\beta ^ \\vee ) \\end{align*}"}
+{"id": "537.png", "formula": "\\begin{align*} u _ i & = 3 c _ i + 4 s _ { i } + 4 z _ i \\\\ v _ i & = c _ i - 4 s _ { i } - 4 z _ i . \\end{align*}"}
+{"id": "6555.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta v + u v = f , & x \\in \\Omega , \\\\ \\nabla v \\cdot \\nu + v = \\eta , & x \\in \\partial \\Omega \\end{array} \\right . \\end{align*}"}
+{"id": "8592.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { 1 \\le i < j \\le M } \\left | \\det \\begin{pmatrix} x _ i & x _ j \\\\ z _ i & z _ j \\end{pmatrix} \\right | \\quad \\hbox { a n d } g ( x ) = \\sum _ { 1 \\le i < j < k \\le M } \\left | \\det \\begin{pmatrix} x _ i & x _ j & x _ k \\\\ y _ i & y _ j & y _ k \\\\ z _ i & z _ j & z _ k \\end{pmatrix} \\right | . \\end{align*}"}
+{"id": "2681.png", "formula": "\\begin{align*} x = \\sum _ { n \\geq 1 } \\mu _ d ( n ) y ^ n . \\end{align*}"}
+{"id": "4377.png", "formula": "\\begin{align*} D '' u _ { m , m ' , \\epsilon , j } + P _ { m , m ' } \\big ( \\sqrt { 2 \\pi b N _ 1 \\delta \\tilde \\delta _ m + N _ 1 \\tilde { \\lambda } _ { m ' } } h _ { m , m ' , \\epsilon , j } \\big ) = \\lambda \\end{align*}"}
+{"id": "4184.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { L } _ H \\varphi ( x ) = - \\cdot ( K _ H ( x ) \\nabla \\varphi ( x ) ) = \\frac { 1 } { \\varepsilon ^ 2 } f ( \\varphi ( x ) - \\mu ) , \\ & x \\in \\Omega , \\\\ \\varphi ( x ) = 0 , \\ & x \\in \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "3719.png", "formula": "\\begin{align*} S ' | _ { g _ { t _ j } } \\big ( h ( t _ j ) , v ( t _ j ) \\big ) = 0 \\mbox { a n d } S ' | _ { g _ { t _ j } } \\big ( \\psi _ { t _ j } ^ * ( h , v ) \\big ) = 0 \\mbox { i n } M \\setminus \\Omega . \\end{align*}"}
+{"id": "8755.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial y } ( t z ) = - \\frac { \\psi _ 0 } { \\psi _ 1 } \\frac { \\partial f } { \\partial x } ( t z ) . \\end{align*}"}
+{"id": "1044.png", "formula": "\\begin{align*} p : v = v _ 1 \\xrightarrow { \\alpha _ 1 } v _ 2 \\xrightarrow { \\alpha _ 2 } \\cdots \\xrightarrow { \\alpha _ { n - 1 } } v _ n = v ' \\end{align*}"}
+{"id": "7218.png", "formula": "\\begin{align*} C _ 1 ( \\kappa , 1 , R ) & \\leq \\min \\left \\{ \\frac { 1 } { 1 + \\mu _ 0 ( p - 2 ) R ^ { p - 2 } } , \\frac { 1 } { 3 } \\right \\} , \\\\ \\sup _ { n \\in \\N } C _ 2 ( \\theta ^ { ( n ) } , \\mu _ n ) & \\leq \\min \\left \\{ \\frac { 1 } { 1 + \\mu _ 0 ( p - 2 ) R ^ { p - 2 } } , \\frac { 1 } { 3 } \\right \\} , \\end{align*}"}
+{"id": "1355.png", "formula": "\\begin{align*} A _ i = \\{ t \\in [ 0 , 1 ] \\colon d ( \\gamma ( t ) , p _ i ) = d ( p , q ) / n \\} . \\end{align*}"}
+{"id": "5199.png", "formula": "\\begin{align*} \\frac { 1 } { r } \\left ( \\Phi _ t + u \\cdot \\nabla \\Phi - \\mu \\left ( \\Delta - \\frac { 2 } { r } \\partial _ r \\right ) \\Phi \\right ) = 0 \\textrm { o n } \\ L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) . \\end{align*}"}
+{"id": "7253.png", "formula": "\\begin{align*} g x _ i = a _ { i 1 } x _ 1 + \\cdots + a _ { i n } x _ n \\end{align*}"}
+{"id": "7780.png", "formula": "\\begin{align*} | s _ { h , q } - s _ h | \\le \\epsilon _ 0 = 1 / 2 ^ { b _ 0 } \\end{align*}"}
+{"id": "117.png", "formula": "\\begin{align*} \\hat { \\phi } _ 1 ( \\xi _ 1 , \\eta _ 1 ) & = \\gamma ^ { - \\frac { 3 } { 2 } } \\mathbf { 1 } _ { D _ 1 } ( \\xi _ 1 , \\eta _ 1 ) , \\\\ \\hat { \\phi } _ 2 ( \\xi _ 2 , \\eta _ 2 ) & = \\gamma ^ { - \\frac { 3 } { 2 } } N ^ { - s _ 1 - ( 1 + \\frac { \\alpha } { 2 } ) s _ 2 } \\mathbf { 1 } _ { D _ 2 } ( \\xi _ 2 , \\eta _ 2 ) , \\end{align*}"}
+{"id": "722.png", "formula": "\\begin{align*} \\mathbf { y } = \\sum _ { k \\in \\mathcal { K } } \\mathbf { h } _ { k } x _ { k } + \\mathbf { n } _ { \\textrm { u l } } , \\end{align*}"}
+{"id": "8924.png", "formula": "\\begin{align*} f _ { N } ( x , y | \\rho ) = f _ { N } ( x ) f _ { N } ( x ) \\sum _ { n \\geq 0 } \\rho ^ { n } h _ { n } ( x ) h _ { n } ( y ) , \\end{align*}"}
+{"id": "1387.png", "formula": "\\begin{align*} L \\Delta ( x ) = \\begin{vmatrix} 1 & x _ 1 - 1 & \\cdots & x _ 1 ^ { d - 2 } - ( d - 2 ) x _ 1 ^ { d - 3 } & x _ 1 ^ { d - 1 } - ( d - 1 ) x _ 1 ^ { d - 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 1 & x _ d - 1 & \\cdots & x _ { d } ^ { d - 2 } - ( d - 2 ) x _ { d } ^ { d - 3 } & x _ d ^ { d - 1 } - ( d - 1 ) x _ d ^ { d - 2 } \\end{vmatrix} \\end{align*}"}
+{"id": "3927.png", "formula": "\\begin{align*} { \\bf T } _ { s } ( { \\bf A } ) = \\left \\{ W = \\left \\langle \\left [ \\theta _ { 1 } \\right ] , \\left [ \\theta _ { 2 } \\right ] , \\dots , \\left [ \\theta _ { s } \\right ] \\right \\rangle \\in G _ { s } \\left ( { \\rm H ^ { 2 } } \\left ( { \\bf A } , \\mathbb C \\right ) \\right ) : \\bigcap \\limits _ { i = 1 } ^ { s } \\operatorname { A n n } ( \\theta _ { i } ) \\cap \\operatorname { A n n } ( { \\bf A } ) = 0 \\right \\} , \\end{align*}"}
+{"id": "4383.png", "formula": "\\begin{align*} D '' u _ { \\epsilon , j } = D '' \\left ( ( 1 - v ' _ { \\epsilon } ( \\Psi ) ) f F ^ { 1 + \\delta } \\right ) . \\end{align*}"}
+{"id": "4621.png", "formula": "\\begin{align*} x _ i ^ 2 = \\epsilon ^ 2 = 1 , 1 \\leq i \\leq n , \\end{align*}"}
+{"id": "8566.png", "formula": "\\begin{align*} a _ 0 b _ 0 = 1 , \\ \\sum _ { k = 0 } ^ n \\Gamma ( k + 1 - \\alpha ) \\Gamma ( \\alpha + n - k ) a _ { n - k } b _ k = 0 , \\ n \\ge 1 . \\end{align*}"}
+{"id": "9029.png", "formula": "\\begin{align*} \\Gamma _ { r } = \\mathbb { Z } ^ { d } _ { * } \\cap \\bigg \\{ \\mathbf { k } : \\sum _ { j } \\langle j \\rangle ^ { \\eta } { \\bf k } _ { j } w _ { j } \\geq r | { \\bf k } | _ { \\eta } , \\sum _ { j } w _ { j } = 1 , w _ j > 0 \\bigg \\} , \\end{align*}"}
+{"id": "8751.png", "formula": "\\begin{align*} c _ { \\psi } = \\sin ^ 2 ( \\alpha ) \\int _ 0 ^ { 2 \\pi } \\frac { \\psi _ 0 \\cos \\theta + \\psi _ 1 \\sin \\theta } { \\left ( \\dfrac { \\cos \\theta } { r _ 0 ^ 2 } - \\dfrac { \\sin \\theta \\cos \\alpha } { r _ 0 r _ 1 } \\right ) \\psi _ 0 + \\left ( \\dfrac { \\sin \\theta } { r ^ 2 _ 1 } - \\dfrac { \\cos \\theta \\cos \\alpha } { r _ 0 r _ 1 } \\right ) \\psi _ 1 } d \\theta \\end{align*}"}
+{"id": "5201.png", "formula": "\\begin{align*} | | B 1 _ { ( 0 , \\infty ) } ( \\Phi ) | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } = | | ( b + B _ \\infty ) 1 _ { ( 0 , \\infty ) } ( \\Phi ) | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } \\lesssim | | b | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } . \\end{align*}"}
+{"id": "1655.png", "formula": "\\begin{align*} ( [ K \\delta K ] ( \\chi \\cdot \\phi ) ) ( \\sigma , g ) & = \\sum _ i \\delta _ i \\cdot ( \\chi ( \\phi ) ) ( \\sigma , g \\delta _ i ) \\\\ & = \\sum _ i \\chi ( \\det ( g \\delta _ i ) ) \\delta _ i \\cdot \\phi ( \\sigma , g \\delta _ i ) \\\\ & = \\chi ( \\det \\delta ) \\chi ( \\det g ) \\sum _ i \\delta _ i \\cdot \\phi ( \\sigma , g \\delta _ i ) . \\end{align*}"}
+{"id": "6991.png", "formula": "\\begin{align*} X ^ { ( i ) } ( t ) & = x + \\int _ 0 ^ t A ( s , \\overline { X ^ { ( i ) } } ( s ) , i ) d s + \\int _ 0 ^ t B ( s , \\overline { X ^ { ( i ) } } ( s ) , i ) d W ( s ) \\\\ & \\quad \\ + \\int _ 0 ^ t \\int _ { \\{ | z | < 1 \\} } H ( s , \\overline { X ^ { ( i ) } } ( s ) , i , z ) \\widetilde { N } ( d s , d z ) \\\\ & \\quad \\ + \\int _ 0 ^ t \\int _ { \\{ | z | \\ge 1 \\} } J ( s , \\overline { X ^ { ( i ) } } ( s ) , i , z ) N ( d s , d z ) , \\end{align*}"}
+{"id": "8560.png", "formula": "\\begin{align*} ( D ^ { \\alpha , \\beta } _ { 0 + } \\ , f ) ( t ) = ( I _ { 0 + } ^ { \\beta ( 1 - \\alpha ) } \\ , \\frac { d } { d t } \\ , I _ { 0 + } ^ { ( 1 - \\alpha ) ( 1 - \\beta ) } \\ , f ) ( t ) , \\ \\ 0 \\le \\alpha < 1 , \\ 0 \\le \\beta \\le 1 . \\end{align*}"}
+{"id": "1530.png", "formula": "\\begin{align*} \\mathcal { F } F C l _ { D i f f ( S ^ 1 ) } ^ { * } ( S ^ 1 , V ) = \\mathcal { F } C l ^ { * } ( S ^ 1 , V ) \\rtimes D i f f ( S ^ 1 ) . \\end{align*}"}
+{"id": "7328.png", "formula": "\\begin{align*} | A | ^ { m - 1 } | B _ 1 + \\ldots + B _ m | \\le \\prod _ { i = 1 } ^ m | A + B _ i | . \\end{align*}"}
+{"id": "5559.png", "formula": "\\begin{align*} \\left | \\sum _ { j = 0 } ^ { n - 1 } A ( \\sigma ^ j ( x ) ) - \\sum _ { j = 0 } ^ { n - 1 } A ( \\sigma ^ j ( y ) ) \\right | < \\epsilon \\ , \\end{align*}"}
+{"id": "5034.png", "formula": "\\begin{align*} & u _ \\alpha ( 0 ) = \\pi + \\upsilon , & & \\mbox { w i t h } | \\upsilon | < 2 ^ { - 5 } , \\\\ & u _ \\alpha \\left ( \\frac { 1 } { T } \\right ) = \\pi ^ { - 1 } + \\delta _ { 1 / T } , & & \\mbox { w i t h } | \\delta _ { 1 / T } | < 2 ^ { - 3 } , \\\\ & u _ \\alpha \\left ( \\frac { 1 } { T ^ 2 } \\right ) = \\pi ^ 2 + \\delta _ { 1 / T ^ 2 } , & & \\mbox { w i t h } | \\delta _ { 1 / T ^ 2 } | < 2 ^ { - 6 } . \\end{align*}"}
+{"id": "3728.png", "formula": "\\begin{align*} \\hat X ( p ) = X ( p ) \\mbox { a n d } \\omega ( p ) = ( \\nabla X ) ( p ) . \\end{align*}"}
+{"id": "6259.png", "formula": "\\begin{align*} \\langle \\nabla ^ E _ X [ e _ i ] , [ e _ j ] \\rangle = \\sum _ k \\varphi _ k \\phi _ { i k } ( X ) d _ { k j } = \\phi _ { i j } ( X ) + \\sum _ k \\varphi _ k \\phi _ { i k } ( X ) = \\phi _ { i j } ( X ) , \\end{align*}"}
+{"id": "1331.png", "formula": "\\begin{align*} \\varepsilon ' = \\frac { \\varepsilon d ( p _ { n + 1 } , q _ { n + 1 } ) } { C _ { n + 1 } ( 1 + d ( p _ { n + 1 } , q _ { n + 1 } ) ) } \\end{align*}"}
+{"id": "2526.png", "formula": "\\begin{align*} \\Delta { { \\mathbf { y } } ^ { T } } \\mathbf { A x } - \\Delta { { \\mathbf { x } } ^ { T } } \\left ( { { \\mathbf { A } } ^ { T } } \\mathbf { y } + \\mathbf { C } \\mathbf { x } + \\mathbf { s } \\right ) = 0 . \\end{align*}"}
+{"id": "1214.png", "formula": "\\begin{align*} h ( \\nu | \\mu ) : = \\limsup _ { n \\to \\infty } \\frac { 1 } { \\abs { \\Lambda _ n } } h _ { \\Lambda _ n } ( \\nu | \\mu ) , \\end{align*}"}
+{"id": "5297.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i , j \\leq n } \\Psi ( v _ i v _ i ^ * v ' _ j ) { v ' _ j } ^ * = \\sum _ { j = 1 } ^ n \\Psi \\left ( \\sum _ { i = 1 } ^ n v _ i v _ i ^ * v ' _ j \\right ) { v ' _ j } ^ * = \\sum _ { j = 1 } ^ n \\Psi \\left ( p v ' _ j \\right ) { v ' _ j } ^ * = \\sum _ { j = 1 } ^ n \\Psi ( v ' _ j ) { v ' _ j } ^ * . \\end{align*}"}
+{"id": "1256.png", "formula": "\\begin{align*} \\forall z \\in \\Z ^ d \\ \\forall i = 1 , \\dots , q \\ \\forall \\eta \\in \\Omega : \\sum _ { \\Delta \\Subset \\Z ^ d } \\sum _ { \\xi _ { \\Delta } } \\nabla _ z ^ i \\left ( c _ { \\Delta } ( \\cdot , \\xi _ { \\Delta } ) - \\hat { c } ( \\cdot , \\xi _ { \\Delta } ) \\right ) ( \\eta ) = 0 . \\end{align*}"}
+{"id": "3822.png", "formula": "\\begin{align*} \\begin{aligned} \\eta ( U | \\bar U ; x , t ) & \\doteq \\eta ( U , x , t ) - \\eta ( \\bar U , x , t ) - G ( \\bar U , x , t ) \\cdot ( A ( U , x , t ) - A ( \\bar U , x , t ) ) \\\\ & = \\tilde { \\eta } ( V , x , t ) - \\tilde { \\eta } ( \\bar V , x , t ) - ( \\nabla _ V \\tilde { \\eta } ) ( \\bar V , x , t ) \\ ; ( A ( U , x , t ) - A ( \\bar U , x , t ) ) \\ , \\\\ & \\doteq \\tilde { \\eta } \\big ( V | \\bar V ; x , t \\big ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "3641.png", "formula": "\\begin{align*} F ( X ) = \\sum _ { i = 1 } ^ n f _ i ( x _ i ) , ~ G ( Y ) = \\sum _ { i = 1 } ^ n g _ i ( y _ i ) . \\end{align*}"}
+{"id": "2233.png", "formula": "\\begin{align*} x ^ { ( k + 1 ) } = T \\left ( x ^ { ( k ) } \\right ) \\coloneqq x ^ { ( k ) } - \\frac { f ( x ^ { ( k ) } ) } { f ' ( x ^ { ( k ) } ) } , \\end{align*}"}
+{"id": "2335.png", "formula": "\\begin{align*} \\left ( s ^ H + \\| N \\| ^ 2 - 2 \\| \\theta \\| ^ 2 \\right ) \\| \\theta \\| ^ 2 = 0 . \\end{align*}"}
+{"id": "4028.png", "formula": "\\begin{align*} K _ { \\lambda , \\varrho ( I ) } = \\sum _ { J \\subset I } \\kappa _ { \\lambda , J } , \\end{align*}"}
+{"id": "6187.png", "formula": "\\begin{align*} E ' _ 0 = \\kappa ( L + 2 ) ^ 2 - \\frac { Q ^ 2 } { 4 ( L + 2 ) ^ 2 } + \\frac { \\kappa ( L + 2 ) B _ 1 } { Q } \\left ( \\frac { ( L + 2 ) B _ 1 } { Q } + 2 L + 4 \\right ) . \\end{align*}"}
+{"id": "5778.png", "formula": "\\begin{align*} \\nabla _ X \\psi _ 2 = - \\frac { 1 } { 2 } X _ 1 \\cdot \\psi _ 2 - \\frac { 1 } { 2 } B ( X ) \\cdot \\psi _ 2 . \\end{align*}"}
+{"id": "2387.png", "formula": "\\begin{align*} \\mathbb { T } _ { \\varrho } ( M , \\{ \\mathbf { h } _ p ^ { M } \\} _ { p = 0 } ^ 3 ) = \\mathbb { T } _ { { \\psi _ { _ 1 } } } ( M _ 1 , \\{ \\mathbf { h } _ p ^ { M _ 1 } \\} _ { p = 0 } ^ { 3 } ) \\ ; \\mathbb { T } _ { { \\psi _ { _ 2 } } } ( M _ 2 , \\{ \\mathbf { h } _ p ^ { M _ 2 } \\} _ { p = 0 } ^ { 3 } ) . \\end{align*}"}
+{"id": "62.png", "formula": "\\begin{align*} \\# \\{ \\beta \\in o \\mid \\varphi ( v ' \\beta ) < 0 \\} = & \\# \\{ \\beta \\in - s _ { \\alpha } ( o ) \\mid \\varphi ( v ' \\beta ) < 0 \\} \\\\ = & \\# \\{ \\beta \\in o \\mid \\varphi ( - v \\beta ) < 0 \\} \\\\ \\leq & \\# \\{ \\beta \\in o \\mid \\varphi ( v \\beta ) \\geq 0 \\} \\\\ = & 2 - \\# \\{ \\beta \\in o \\mid \\varphi ( v \\beta ) < 0 \\} . \\end{align*}"}
+{"id": "6437.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { i = 1 } ^ { n } \\sum _ { j = 0 } ^ { k _ i - 2 } Z ^ { \\star } _ { I } ( j + 1 , k _ { i + 1 } , \\ldots , k _ n , k _ 1 , \\ldots , k _ { i - 1 } , k _ i - j ; ( \\alpha , \\beta ) ) \\\\ = & ( k - n ) Z ( n | k - n + 1 ; ( \\alpha , \\beta ) ) + n Z ( n + 1 | k - n ; ( \\alpha , \\beta ) ) \\end{aligned} \\end{align*}"}
+{"id": "6272.png", "formula": "\\begin{align*} \\sum _ s \\phi _ { i s } ( \\partial _ i ) \\varphi _ s & = \\frac { 1 } { 2 } \\partial _ i ( \\varphi ^ { - 1 } _ i ) \\varphi _ i - \\sum _ { s \\neq i } \\frac { \\Gamma _ { i s } ^ s \\varphi _ s } { \\varphi _ i } = \\frac { 1 } { 2 \\varphi _ i } \\partial _ i \\Big ( \\sum _ s \\varphi _ s \\Big ) = 0 . \\end{align*}"}
+{"id": "1840.png", "formula": "\\begin{align*} d ^ \\nabla R ^ \\nabla = 0 \\ , . \\end{align*}"}
+{"id": "6069.png", "formula": "\\begin{align*} Y = \\left ( \\frac { \\eta } { m + 1 } x _ { n + 1 } ^ { m + 1 } + \\lambda x _ { n + 1 } \\right ) e _ { n + 1 } . \\end{align*}"}
+{"id": "4601.png", "formula": "\\begin{align*} 2 \\widetilde y _ n & = 2 e ^ { r n ( A + a ) + r A } \\\\ & > ( e ^ { - r A } + e ^ { r a } ) e ^ { r n ( A + a ) + r A } \\\\ & = e ^ { r n ( A + a ) } + e ^ { r ( n + 1 ) ( A + a ) } \\\\ & = \\widetilde x _ n + \\widetilde x _ { n + 1 } . \\end{align*}"}
+{"id": "1601.png", "formula": "\\begin{align*} \\tau _ { k } ^ \\star = \\frac { \\sqrt { 1 + \\iota _ { k } } \\tilde { \\mathbf { h } } _ { k } ^ H \\mathbf { w } _ { k } } { \\sum _ { p \\in \\mathcal { K } } | \\tilde { \\mathbf { h } } _ { k } ^ H \\mathbf { w } _ { p } | ^ 2 + \\sigma _ { k } ^ 2 } , \\forall k \\in \\mathcal { K } . \\end{align*}"}
+{"id": "4490.png", "formula": "\\begin{align*} H ( v , w , t ^ { - 1 } ) = t ^ { ( _ { \\bullet } v - w ) \\cdot ( _ { \\bullet } v - w ) - _ { \\bullet } v \\cdot _ { \\bullet } v } \\frac { \\mathit { g l } ( _ { \\bullet } v , t ) } { \\mathit { g l } ( _ { \\bullet } v - w , t ) } . \\end{align*}"}
+{"id": "5520.png", "formula": "\\begin{align*} \\tilde { h } _ d ( x ) = x \\log d + \\log \\left ( \\frac { \\Gamma ( x ) } { \\Gamma \\left ( x + \\frac { 1 } { 2 } \\right ) \\Gamma \\left ( x + \\frac { d - 1 } { 2 } \\right ) } \\right ) + \\log \\left ( \\frac { 2 ^ { d - 2 } \\Gamma \\left ( \\frac { d } { 2 } \\right ) ^ 2 } { \\sqrt { \\pi } d ^ { \\frac { d - 1 } { 2 } } } \\right ) . \\end{align*}"}
+{"id": "6387.png", "formula": "\\begin{align*} Q _ h : = { P } _ { k - 1 } ^ { \\mathrm { d i s c } } ( \\mathcal { T } ) \\cap Q . \\end{align*}"}
+{"id": "7728.png", "formula": "\\begin{align*} q _ \\pi ( G _ 0 ) = q _ \\pi ( V \\times K ) = q _ \\pi ( V ) q _ \\pi ( K ) . \\end{align*}"}
+{"id": "1117.png", "formula": "\\begin{align*} P _ { \\min } ^ { { \\rm { S - N } } } \\left ( { \\overline S , \\overline \\varepsilon , \\overline R } \\right ) = \\bigcap { \\left \\{ { P \\in { { \\mathbb { R } } ^ + } : \\left ( { \\overline S , \\overline R } \\right ) \\in { \\mathcal { R } } _ { { \\rm { S v B } } } ^ { { \\rm { S - N } } } \\left ( { W , P , K , \\overline \\varepsilon } \\right ) } \\right \\} } . \\end{align*}"}
+{"id": "8664.png", "formula": "\\begin{align*} \\begin{aligned} { \\bf { d } } \\left [ { i - { \\kappa _ l } } \\right ] & = { { { \\bf { \\bar x } } } _ { \\rm { t } } } \\Big [ { \\left \\lfloor { \\frac { { i - { \\kappa _ l } + { N _ { \\rm { C P } } } } } { { K + { N _ { \\rm { C P } } } } } } \\right \\rfloor , } \\Big . \\\\ & \\ \\ \\ \\ \\Big . { { \\rm { m o d } } \\left ( { i - { \\kappa _ l } + { N _ { \\rm { C P } } } , K + { N _ { \\rm { C P } } } } \\right ) - { N _ { \\rm { C P } } } } \\Big ] . \\end{aligned} \\end{align*}"}
+{"id": "3635.png", "formula": "\\begin{align*} \\sum _ { p \\neq q } \\frac { F ^ { p p , q q } h _ { p q 1 } ^ 2 } { \\kappa _ 1 } & \\geq 2 \\sum _ { i > m } \\frac { F ^ { 1 1 , i i } h _ { 1 1 i } ^ 2 } { \\kappa _ 1 } = 2 \\sum _ { i > m } \\frac { ( F ^ { i i } - F ^ { 1 1 } ) h _ { 1 1 i } ^ 2 } { \\kappa _ 1 ( \\kappa _ 1 - \\kappa _ i ) } \\geq 2 \\sum _ { i > m } \\frac { ( F ^ { i i } - F ^ { 1 1 } ) h _ { 1 1 i } ^ 2 } { \\kappa _ 1 ( \\kappa _ 1 - \\tilde { \\kappa } _ i ) } , \\end{align*}"}
+{"id": "9170.png", "formula": "\\begin{align*} \\begin{aligned} \\dfrac { \\partial h ( s ) } { \\partial s } & = \\left . \\dfrac { \\partial h ( s ) } { \\partial s } \\right | _ x + \\left . \\dfrac { \\partial ^ 2 h ( s ) } { \\partial s ^ 2 } \\right | _ x ( s - x ) + O ( ( s - x ) ^ 2 ) . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "5726.png", "formula": "\\begin{align*} \\frac { 1 } { A r } = h ^ q , \\end{align*}"}
+{"id": "6643.png", "formula": "\\begin{align*} \\begin{aligned} X ^ { \\epsilon } ( t ) & = x _ { 0 } + \\int ^ { t } _ { 0 } b ( X ^ { \\epsilon } ( s ) ) d s - \\lambda ( \\epsilon ) \\epsilon ^ { 2 } \\int ^ { t } _ { 0 } \\sigma ( X ^ { \\epsilon } ( s - ) ) \\theta _ { \\epsilon } ( s - ) \\tilde { N } ^ { \\epsilon ^ { - 2 } } ( d s ) \\\\ & \\ \\ \\ \\ - \\frac { \\lambda ( \\epsilon ) \\epsilon ^ { 2 } } { 2 } \\int _ 0 ^ t \\sigma ( X ^ { \\epsilon } ( s - ) ) d \\theta _ { \\epsilon } ( s ) . \\end{aligned} \\end{align*}"}
+{"id": "8606.png", "formula": "\\begin{align*} | [ - u _ 1 , u _ 1 ] \\oplus _ 2 \\cdots \\oplus _ 2 [ - u _ m , u _ m ] | _ n = | B _ 2 ^ n | \\left | \\sum _ { i = 1 } ^ n [ 0 , z _ i ] \\right | _ n . \\end{align*}"}
+{"id": "6383.png", "formula": "\\begin{align*} \\ell ( a , b , \\theta ) = & n [ \\log ( 2 \\theta ) - \\log \\{ \\operatorname { B } ( a , b ) \\} ] - 3 \\sum ^ n _ { i = 1 } \\log ( x _ { i } ) - a \\theta \\sum ^ n _ { i = 1 } \\frac { 1 } { x ^ 2 _ i } \\\\ & + ( b - 1 ) \\sum ^ { n } _ { i = 1 } \\log \\left \\{ 1 - \\exp \\left ( - \\frac { \\theta } { x _ i ^ 2 } \\right ) \\right \\} . \\end{align*}"}
+{"id": "403.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } \\int _ { \\tau } ^ { \\tau + T } \\widehat { Q } _ 1 ^ k ( t ) \\d t = 0 . \\end{align*}"}
+{"id": "3544.png", "formula": "\\begin{align*} \\psi _ 1 & = \\frac { 1 } { 2 } ( 2 , 2 , 1 , 1 ) , & \\psi _ 2 & = ( 1 , 2 , 1 , 1 ) , & \\psi _ 3 & = \\frac { 1 } { 2 } ( 1 , 2 , 2 , 1 ) , & \\psi _ 4 & = \\frac { 1 } { 2 } ( 1 , 2 , 1 , 2 ) . \\end{align*}"}
+{"id": "8779.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } X _ t = B ( X _ t , X _ t ) \\\\ X _ 0 = x , \\end{cases} \\end{align*}"}
+{"id": "161.png", "formula": "\\begin{align*} { \\rm e x } ( t n , H _ k ^ r ) \\ : \\geq \\ : t ^ r { \\rm e x } ( n , H _ k ^ r ) + \\ , \\frac { 1 } { n } \\binom { t n } { r } - \\ , \\frac { 1 } { n } \\ , t ^ r \\binom { n } { r } - \\frac { 1 } { 2 } \\ , t ^ { r - 1 } ( t - 1 ) \\binom { n - 1 } { r - 2 } . \\end{align*}"}
+{"id": "1095.png", "formula": "\\begin{align*} \\prescript J { } \\pi ( w _ 1 ) = ( w _ 1 ' , \\mu _ 1 ) , \\prescript J { } \\pi ( w _ 2 ) = ( w _ 2 ' , \\mu _ 2 ) . \\end{align*}"}
+{"id": "7531.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ k | V _ i \\setminus X _ i | - \\frac { m ^ k } { 4 } \\geq m ^ k \\left ( \\left ( 1 - \\frac { 1 } { 2 k ^ 2 } \\right ) ^ k - \\frac { 1 } { 4 } \\right ) \\geq m ^ k \\left ( 1 - \\frac { 1 } { 2 k } - \\frac { 1 } { 4 } \\right ) \\geq \\frac { m ^ k } { 2 } \\end{align*}"}
+{"id": "8977.png", "formula": "\\begin{align*} x _ \\Upsilon = \\prod _ { \\tau \\in \\Upsilon _ d } x _ \\tau . \\end{align*}"}
+{"id": "5661.png", "formula": "\\begin{align*} B _ { a } : = \\begin{pmatrix} 1 \\\\ & a \\\\ & & \\ddots \\\\ & & & 1 \\\\ & & & & a \\end{pmatrix} \\end{align*}"}
+{"id": "4425.png", "formula": "\\begin{align*} X : = \\{ \\sum _ { n = 0 } ^ \\infty a _ n z ^ n \\mid ( a _ n n ^ { - \\alpha } ) _ { n \\in \\mathbb { N ^ * } } \\in \\ell ^ 2 \\} \\end{align*}"}
+{"id": "859.png", "formula": "\\begin{align*} x ^ 2 + x y + y ^ 2 = n \\ , , ( x , y ) = 1 \\ , , x \\in \\mathbb { N } \\ , , y \\in \\mathbb { Z } \\setminus \\{ 0 \\} \\end{align*}"}
+{"id": "233.png", "formula": "\\begin{align*} \\dot x ( J ) = \\dot x _ { \\mathcal { W } , g } ( J , u ) , \\dot y ( J ) = \\dot y _ { \\mathcal { W } , g } ( J , u ) , \\dot z ( J ) = \\dot z _ { \\mathcal { W } , g } ( J , u ) \\end{align*}"}
+{"id": "8214.png", "formula": "\\begin{align*} x _ { i , 1 , k } & = 6 m - 3 i + k + 1 , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 1 , k } & = 3 i - k , \\\\ x _ { i , 2 , k } & = 3 i - k - 1 , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 2 , k } & = 6 m - 3 i + k + 2 , \\\\ x _ { i , 3 , k } & = 3 m - 3 i + 2 k + 1 , \\mbox { a n d } \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 3 , k } & = 3 m + 3 i - 2 k . \\end{align*}"}
+{"id": "2227.png", "formula": "\\begin{align*} C _ { R P I - R I } = O \\{ \\beta ( 2 N - 2 ) ^ 2 + 2 N - 3 \\} \\end{align*}"}
+{"id": "5162.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sup _ { z _ 0 \\in \\mathbb { R } } \\int _ { B ( z _ 0 e _ z , R ) } | b _ { n _ { k } } | ^ { 2 } \\dd x = 0 . \\end{align*}"}
+{"id": "640.png", "formula": "\\begin{align*} \\Phi _ { X _ 0 } ( X ) = \\frac { d } { d t } \\big \\vert _ { t = 0 } ( \\varphi _ { X _ 0 } \\circ \\exp ) ( t X ) . \\end{align*}"}
+{"id": "7238.png", "formula": "\\begin{align*} L ( x ) = x ^ { 2 ^ r } + t x . \\end{align*}"}
+{"id": "8245.png", "formula": "\\begin{align*} \\mu _ r ( \\psi ( \\cdot ) ) = \\mu _ r ( \\cdot ) : = \\sum _ { i = 1 } ^ { \\infty } i ^ r \\psi _ i ( \\cdot ) . \\end{align*}"}
+{"id": "3612.png", "formula": "\\begin{align*} \\sigma _ { n - 1 } ( \\kappa ) = \\sigma , \\partial \\Sigma = \\Gamma , \\end{align*}"}
+{"id": "237.png", "formula": "\\begin{align*} \\dot g ( J ) \\in A t ^ { \\rho ( \\dot g ( J ) ) } , \\end{align*}"}
+{"id": "2256.png", "formula": "\\begin{align*} t \\cdot ( U , V ) = ( U D ( \\overline { t } ) , D ( t ) V ) , \\end{align*}"}
+{"id": "6605.png", "formula": "\\begin{align*} \\lambda _ S = \\langle \\alpha , \\beta \\rangle \\alpha - \\langle \\alpha , \\alpha \\rangle \\beta . \\end{align*}"}
+{"id": "4641.png", "formula": "\\begin{align*} \\sum _ { K \\ \\ L } \\ \\sum _ { \\widetilde { K } \\ \\ \\pi ^ { - 1 } ( K ) } ( e _ { \\widetilde { K } } - 1 ) \\langle [ \\widetilde { K } ] , \\alpha \\rangle = 0 . \\end{align*}"}
+{"id": "2039.png", "formula": "\\begin{align*} \\mathcal { L } f = - \\Gamma ( \\sqrt \\mu , f ) - \\Gamma ( f , \\sqrt \\mu ) , \\Gamma ( g , h ) = \\mu ^ { - { 1 \\over 2 } } Q _ L ( \\sqrt { \\mu } g , \\sqrt { \\mu } h ) . \\end{align*}"}
+{"id": "6021.png", "formula": "\\begin{align*} & a ( u , v ) : = \\| \\nabla u \\| _ { 2 } ^ { 2 } + \\| \\nabla v \\| _ { 2 } ^ { 2 } , \\\\ & b ( u , v ) : = \\| u \\| _ { 2 } ^ { 2 } + \\omega \\| v \\| _ { 2 } ^ { 2 } , \\\\ & c ( u , v ) : = B ( u ) + B ( v ) . \\end{align*}"}
+{"id": "6546.png", "formula": "\\begin{align*} P ( 2 a ) = ( 2 a ) ^ { \\pi ( a ) } + d _ { \\pi ( a ) - 1 } ( 2 a ) ^ { \\pi ( a ) - 1 } + d _ { \\pi ( a ) - 2 } ( 2 a ) ^ { \\pi ( a ) - 2 } + \\cdots d _ 1 ( 2 a ) + d _ 0 . \\end{align*}"}
+{"id": "7963.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } \\frac { \\partial X ( x , t ) } { \\partial t } = \\frac { 1 } { f ( v ) } \\sigma _ { k } ( x , t ) \\varphi ( X \\cdot v ) ( X \\cdot v ) G ( X ) v - X ; \\\\ X ( x , 0 ) = X _ { 0 } ( x ) , \\end{array} \\right . \\end{align*}"}
+{"id": "4343.png", "formula": "\\begin{align*} & \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in S _ { i , n } \\backslash U _ k \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h c ( - \\Psi ) \\\\ = & \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in S _ { i , n } \\backslash U _ k \\} } | \\tilde F | ^ 2 _ h c ( - \\Psi ) + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in S _ { i , n } \\backslash U _ k \\} } | F _ { t _ 0 , \\tilde { c } } - \\tilde F | ^ 2 _ h c ( - \\Psi ) . \\end{align*}"}
+{"id": "6975.png", "formula": "\\begin{align*} ( a / b ) ^ k = ( \\alpha / \\beta ) ^ l . \\end{align*}"}
+{"id": "9085.png", "formula": "\\begin{align*} - \\sum \\limits _ { j = 3 } ^ { n - 1 } + ( n - 2 ) r \\geq - ( n - 3 ) r + ( n - 2 ) r = r \\end{align*}"}
+{"id": "5094.png", "formula": "\\begin{align*} & \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B _ { n } \\cdot B _ { n } \\dd x = \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B _ { 0 , n } \\cdot B _ { 0 , n } \\dd x = h _ n , \\\\ & \\mathcal { I } _ { h _ n } \\leq \\frac { 1 } { 2 } \\int _ { \\Omega } \\left ( | u _ { n } | ^ { 2 } + | B _ { n } | ^ { 2 } \\right ) \\dd x \\leq \\frac { 1 } { 2 } \\int _ { \\Omega } \\left ( | u _ { 0 , n } | ^ { 2 } + | B _ { 0 , n } | ^ { 2 } \\right ) \\dd x , \\end{align*}"}
+{"id": "7019.png", "formula": "\\begin{align*} \\aligned A _ { 1 , 2 } & \\ , : = \\ , - \\ , F _ { 2 , 1 } \\ , T _ 1 , \\\\ D & \\ , : = \\ , F _ { 2 , 1 } \\ , T _ 2 + 2 \\ , A _ { 1 , 1 } . \\endaligned \\end{align*}"}
+{"id": "8306.png", "formula": "\\begin{align*} \\begin{aligned} & \\dot y _ { b e n } = A y '' _ { b e n } + M _ { b e n } y _ { b e n } \\\\ & \\dot y _ { n b } = a A y '' _ { n b } + M _ { n b } y _ { n b } \\end{aligned} \\end{align*}"}
+{"id": "55.png", "formula": "\\begin{align*} \\sigma \\left ( \\sum a _ n \\varepsilon ^ n \\right ) = \\sum a _ n ^ q \\varepsilon ^ n . \\end{align*}"}
+{"id": "3576.png", "formula": "\\begin{align*} B ( t ) & = \\frac { 1 } { 2 } ( ( A ( t ) + B ( t ) ) + ( B ( t ) - A ( t ) ) ) \\\\ & = \\varphi ( t ^ 4 ) ^ 2 ( 2 t G ( t ) G ( t ^ 4 ) ) \\\\ & = 2 t \\varphi ( t ^ 4 ) ^ 2 G ( t ^ 4 ) G ( t ) \\end{align*}"}
+{"id": "6525.png", "formula": "\\begin{align*} 2 0 = 1 8 + 2 = 1 7 + 3 = 1 5 + 5 = 1 3 + 7 \\end{align*}"}
+{"id": "5717.png", "formula": "\\begin{align*} f ( x ) = a x ^ { q + 1 } + b x ^ q + c x + d \\end{align*}"}
+{"id": "8443.png", "formula": "\\begin{align*} z ^ { - 1 } k ^ { - 1 } b ( k ) = - \\frac { 1 } { 2 } \\widehat { \\bar { u } _ x K \\psi ^ - _ { 1 1 } } ( z ) + i \\widehat { \\bar { u } } ( z ) . \\end{align*}"}
+{"id": "8887.png", "formula": "\\begin{align*} \\epsilon _ { f \\otimes \\chi _ D } = \\chi _ D ( - r ) \\mu ( q ' ) q '^ { 1 / 2 } \\lambda _ f ( q ' ) \\epsilon _ f . \\end{align*}"}
+{"id": "6325.png", "formula": "\\begin{align*} z _ 0 x _ 0 ^ { p ^ { h _ { n - 1 } - r } } + z _ 1 x _ 1 ^ { p ^ { h _ { n - 1 } - r } } + \\cdots + z _ n x _ n ^ { p ^ { h _ { n - 1 } - r } } = 0 . \\end{align*}"}
+{"id": "5879.png", "formula": "\\begin{align*} Y _ n ( t ) = P _ n x + \\int _ 0 ^ t P _ n A ( s , Y _ n ( s ) ) d s + \\int _ 0 ^ t P _ n B ( s , Y _ n ( s ) ) Q _ n d W ( s ) , \\end{align*}"}
+{"id": "1913.png", "formula": "\\begin{align*} { \\frac { 1 } { \\mathcal { A } _ { \\gamma } ( s ) } } = { \\frac { 1 } { q - s } } + { \\gamma } , \\end{align*}"}
+{"id": "6171.png", "formula": "\\begin{align*} W ( r ) = W ( r ; l , Q ) = - \\frac { L + 1 } { r } f ( r ) + \\frac { Q } { 2 L + 2 } \\end{align*}"}
+{"id": "4905.png", "formula": "\\begin{align*} \\mathrm { E } ( Z ^ s ) = \\sum _ { r = 0 } ^ \\infty w _ { r } ( \\alpha , \\beta ) \\ , \\tau _ { s , r + a - 1 } \\mathrm { E } ( Z ^ s ) = \\sum _ { i , j = 0 } ^ \\infty \\sum _ { r = 0 } ^ j w _ { i , j , r } ( \\alpha , \\beta ) \\ , \\tau _ { s , r } , \\end{align*}"}
+{"id": "4455.png", "formula": "\\begin{align*} \\langle C , C \\rangle = \\sum _ { i = 1 } ^ s | c _ { i } | ^ { 2 } \\langle g _ { i } , g _ { i } \\rangle . \\end{align*}"}
+{"id": "3715.png", "formula": "\\begin{align*} ( g _ t , u _ t ) = \\psi _ t ^ * \\big ( \\left . \\big ( \\bar g , \\bar u \\big ) \\right | _ { M \\setminus \\Omega _ t } \\big ) . \\end{align*}"}
+{"id": "4030.png", "formula": "\\begin{align*} V _ { \\lambda / \\mu } : = L _ T , \\end{align*}"}
+{"id": "2923.png", "formula": "\\begin{align*} & \\| y _ { \\bar u + \\theta ( u - \\bar u ) } - \\bar y \\| _ { C ( \\bar \\Omega ) } = \\| \\phi + \\theta ( y _ u - \\bar y ) \\| _ { C ( \\bar \\Omega ) } \\\\ & \\le ( C _ 2 C _ { f , K _ U } \\sqrt { | \\Omega | } \\| y _ u - \\bar y \\| _ { C ( \\bar \\Omega ) } + 1 ) \\| y _ u - \\bar y \\| _ { C ( \\bar \\Omega ) } . \\end{align*}"}
+{"id": "7394.png", "formula": "\\begin{align*} \\Delta _ + ^ 0 = \\{ \\alpha _ i - \\alpha _ j | 1 \\leq 1 < j \\leq r \\} \\end{align*}"}
+{"id": "8240.png", "formula": "\\begin{align*} \\mathcal { R } _ \\lambda ( s _ i ) v _ T = \\pm \\frac { 1 } { a _ { i + 1 } - a _ { i } } v _ { T } + \\sqrt { 1 - \\frac { 1 } { ( a _ { i + 1 } - a _ { i } ) ^ 2 } } v _ { T ' } , \\end{align*}"}
+{"id": "7626.png", "formula": "\\begin{align*} \\partial _ t \\mu ^ \\phi = - \\partial _ x \\Big \\{ \\Big [ b ( t , \\cdot ) + \\sigma \\phi ( t , \\cdot ) \\Big ] \\mu ^ \\phi \\Big \\} . \\end{align*}"}
+{"id": "3145.png", "formula": "\\begin{align*} N ( x ) \\cdot N ( y ) = N \\big ( N ( x ) \\cdot y + x \\cdot N ( y ) - N ( x \\cdot y ) \\big ) , \\quad \\forall \\ x , y \\in A . \\end{align*}"}
+{"id": "6572.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\psi _ { \\eta } \\| _ { L ^ { 2 } ( \\R ^ { d } ) } ^ { 2 } & = \\sigma _ { d } \\int _ { 0 } ^ { \\eta } | \\ln ( - \\ln r ) - \\ln ( - \\ln \\eta ) | ^ { 2 } r ^ { d - 1 } \\ , d r \\\\ & \\le \\sigma _ { d } \\int _ { 0 } ^ { \\eta } | \\ln ( - \\ln r ) | ^ { 2 } r ^ { d - 1 } d r \\\\ & = \\sigma _ { d } \\int _ { \\ln \\frac { 1 } { \\eta } } ^ { \\infty } | \\ln \\rho | ^ { 2 } e ^ { - d \\rho } d \\rho \\end{aligned} \\end{align*}"}
+{"id": "1397.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ d q _ j ^ { - x _ j } \\left ( \\left ( \\frac { q _ i } { 1 - q _ i } \\right ) ^ { j - 1 } q _ i ^ { x _ j } \\right ) _ { i , j = 1 } ^ d . \\end{align*}"}
+{"id": "8652.png", "formula": "\\begin{align*} \\begin{aligned} y \\left [ i \\right ] & = \\sum \\limits _ { l = 1 } ^ L { { \\bf { h } } _ l ^ H { \\bf { \\bar q } } \\left [ { i - { n _ l } } \\right ] } + z \\left [ i \\right ] \\\\ & = \\sum \\limits _ { l = 1 } ^ L { \\sum \\limits _ { { l ' } = 1 } ^ { L ' } { { \\bf { h } } _ l ^ H { { \\bf { { F } } } _ { l ' } } { \\bf { d } } \\left [ { i - { n _ l } - { \\kappa _ { l ' } } } \\right ] } + z \\left [ i \\right ] } . \\end{aligned} \\end{align*}"}
+{"id": "8435.png", "formula": "\\begin{align*} k ^ { - 1 } b ( k ) = \\Psi ^ + _ { 1 1 } ( 0 ; z ) k ^ { - 1 } \\psi ^ - _ { 2 1 } ( 0 ; k ) - \\Psi ^ - _ { 1 1 } ( 0 ; z ) k ^ { - 1 } \\psi ^ + _ { 2 1 } ( 0 ; k ) \\end{align*}"}
+{"id": "4877.png", "formula": "\\begin{align*} \\psi _ \\epsilon = \\sup _ { \\| x - x ' \\| < \\delta , \\| y - y ' \\| \\le \\delta } u ( x ' , y ' ) \\end{align*}"}
+{"id": "3703.png", "formula": "\\begin{align*} ( \\tilde h , \\tilde v ) : = ( \\hat h , \\hat v ) + \\big ( L _ Y \\bar g , Y ( \\bar u ) \\big ) \\end{align*}"}
+{"id": "8400.png", "formula": "\\begin{align*} \\lim _ { | z | \\rightarrow \\infty } \\Psi ^ - _ { 1 1 } ( x ; z ) = e ^ { - i c _ - ( x ) } . \\end{align*}"}
+{"id": "6664.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\sup _ { t \\in [ 0 , 1 ] } | X ^ { \\epsilon } ( t ) - \\varphi ( t ) | < \\delta , \\tilde { \\chi } _ 1 < 1 \\Big ) = - \\infty . \\end{align*}"}
+{"id": "1165.png", "formula": "\\begin{align*} d X _ t = \\left ( b _ 0 ( t , X _ t ) + \\langle b ( t , X _ t , \\cdot ) , \\mu \\rangle \\right ) d t + d B ^ H _ t , \\quad \\operatorname { L a w } ( X ) = \\mu , \\quad \\operatorname { L a w } ( X _ 0 ) = \\mu _ 0 . \\end{align*}"}
+{"id": "3661.png", "formula": "\\begin{align*} & ( r - 1 ) ^ { n - 1 } \\tfrac { ( r - 1 ) ! } { ( 2 r - 2 ) ! [ ( r - 1 ) ! ] ^ { r - 1 } } \\tfrac { ( r ^ { 2 } - 1 ) ( 2 r - 3 ) ! [ ( r - 2 ) ! ] ^ { r - 1 } r ^ { r - 2 } } { ( r - 1 ) ! } \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ { r } \\tfrac { d _ { i } } { [ ( r - 1 ) ! ] ^ { r } } 2 d _ { i } [ ( r - 1 ) ! ] ^ { r } \\\\ & = ( - 1 ) ^ { r } ( r - 1 ) ^ { n - r } ( r + 1 ) r ^ { r - 2 } \\sum _ { i = 1 } ^ { n } d _ { i } ^ { 2 } . \\end{align*}"}
+{"id": "5926.png", "formula": "\\begin{align*} \\lim _ { \\substack { n \\rightarrow \\infty \\\\ M \\rightarrow \\infty } } \\Vert X _ n ( t \\wedge \\tau _ u ^ M ) - X ( t \\wedge \\tau _ u ^ M ) \\Vert _ H = 0 , \\mathbb { P } \\otimes d t . \\end{align*}"}
+{"id": "6938.png", "formula": "\\begin{align*} H ( \\widetilde { Q } ) \\le \\sum _ { i = 1 } ^ { n } \\tilde { t } _ i H ( Q _ i ) , \\end{align*}"}
+{"id": "1232.png", "formula": "\\begin{align*} \\forall \\eta _ \\Lambda \\in \\Omega _ \\Lambda : \\sum _ { \\Delta \\subset \\Lambda } \\sum _ { \\xi _ \\Delta } \\mu ( \\eta _ \\Lambda ) c _ \\Delta ( \\eta _ \\Lambda , \\xi _ \\Delta ) = \\sum _ { \\Delta \\subset \\Lambda } \\sum _ { \\xi _ \\Delta } \\mu ( \\xi _ \\Delta \\eta _ { \\Lambda \\setminus \\Delta } ) c _ \\Delta ( \\xi _ \\Delta \\eta _ { \\Lambda \\setminus \\Delta } , \\eta _ \\Delta ) , \\end{align*}"}
+{"id": "2069.png", "formula": "\\begin{align*} \\mathcal { I } - d ^ { - } ( p , H ) & = \\mathcal { I } - d _ { - } ( p , H ) = \\mathcal { I } - \\liminf x _ n = \\mathcal { I } - \\liminf ( 1 - y _ n ) \\\\ & = 1 - \\mathcal { I } - \\limsup y _ n = 1 - \\mathcal { I } - d ^ { - } ( p , H ^ { c } ) \\end{align*}"}
+{"id": "2581.png", "formula": "\\begin{align*} R \\cdot R = L \\ , \\ , Q ( g , R ) , \\end{align*}"}
+{"id": "4341.png", "formula": "\\begin{align*} \\int _ { \\{ - t _ 3 \\le \\Psi < - t _ 4 \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h c ( - \\Psi ) = & \\int _ { \\{ - t _ 3 \\le \\Psi < - t _ 4 \\} } | \\tilde { F } | ^ 2 _ h c ( - \\Psi ) \\\\ + & \\int _ { \\{ - t _ 3 \\le \\Psi < - t _ 4 \\} } | F _ { t _ 0 , \\tilde { c } } - \\tilde { F } | ^ 2 _ h c ( - \\Psi ) \\end{align*}"}
+{"id": "8790.png", "formula": "\\begin{align*} \\int _ { A _ { n , j } \\cap A _ n } & \\int _ { \\eta _ j ( K ) } ^ { \\tau _ { n + 1 } ( \\omega ) - \\tau _ n ( \\omega ) } D ^ p ( x _ { \\tau _ n + t } ( \\omega ) ) d t d \\P \\\\ & \\ge c \\delta ^ p K ^ { 2 r p } \\int _ { A _ { n , j } \\cap A _ n \\cap \\{ \\bar { \\tau } \\ge T \\} } ( \\tau _ { n + 1 } ( \\omega ) - \\tau _ n ( \\omega ) ) d \\P - c \\eta _ j ( K ) \\delta ^ p K ^ { 2 r p } \\P ( A _ { n , j } \\cap A _ n ) . \\end{align*}"}
+{"id": "7660.png", "formula": "\\begin{align*} \\alpha ^ { * , t _ 0 , x _ 0 , \\xi } _ t : = - B R ^ { - 1 } U ( t , x _ t ^ { * , t _ 0 , x _ 0 , \\xi } , \\nu _ t ^ { * , t _ 0 , \\xi } ) - h ( \\mu _ t ^ { * , t _ 0 , \\xi } ) . \\end{align*}"}
+{"id": "1898.png", "formula": "\\begin{align*} \\mathbf M = \\frac 1 { 2 | x - y | ^ 2 } \\begin{pmatrix} 2 h ^ 2 & \\ ( 1 + \\cos S _ c ) \\rho h - 2 r h \\\\ 2 \\rho h - ( 1 + \\cos S _ c ) r h & \\ 2 ( r - \\rho ) ^ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "7063.png", "formula": "\\begin{align*} u \\ , = \\ , \\frac { 1 } { 3 \\ , z ^ 2 } \\Big \\{ \\big ( 1 - 2 \\ , y + y ^ 2 - 2 \\ , x z \\big ) ^ { 3 / 2 } - ( 1 - y ) \\ , \\big ( 1 - 2 \\ , y + y ^ 2 - 3 \\ , x z \\big ) \\Big \\} . \\end{align*}"}
+{"id": "2877.png", "formula": "\\begin{align*} & \\sum _ { y = x - n - 1 } ^ { x + n } \\bar K ^ { ( n ) } _ 1 ( { y , x } ) = \\frac { 1 } { n + 1 } \\sum _ { j = 0 } ^ n \\Phi \\left ( \\frac { j } { n + 1 } , \\frac { j } { n + 1 } \\right ) \\\\ & = \\frac { 1 } { n + 1 } \\sum _ { j = 0 } ^ n \\frac 1 { \\mu _ j } = G _ { \\omega _ 0 } ( 0 ) + o ( 1 ) , \\end{align*}"}
+{"id": "3613.png", "formula": "\\begin{align*} \\sigma _ { n - 1 } ( \\kappa ) = \\sigma , \\partial \\Sigma = \\Gamma . \\end{align*}"}
+{"id": "4757.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\| x ^ * \\| ^ 2 = - \\frac { c _ { 1 } } { a _ { 1 } } \\\\ & b _ 2 ^ T x ^ * \\le a _ { 2 } \\frac { c _ { 1 } } { a _ { 1 } } - c _ { 2 } \\end{aligned} \\right . \\end{align*}"}
+{"id": "5674.png", "formula": "\\begin{align*} \\bar { v } _ { i } : = \\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ v _ { i } \\end{pmatrix} \\in R ^ { 2 n } . \\end{align*}"}
+{"id": "7937.png", "formula": "\\begin{align*} u _ j ( x ) & = \\int _ { x - r } ^ { x + r } \\prod _ { \\stackrel { i = 1 } { i \\neq j } } ^ n ( \\eta _ i ( y ) - \\eta _ i ( x ) ) ( - \\partial \\eta _ j ( x ) ) \\sin ^ { [ n ] } ( \\tilde \\Psi ( y ) - \\tilde \\Psi ( x ) ) \\ \\d y , \\\\ q ( x ) & = \\int _ { x - r } ^ { x + r } \\prod _ { i = 1 } ^ n ( \\eta _ i ( y ) - \\eta _ i ( x ) ) \\sin ^ { [ n + 1 ] } ( \\tilde \\Psi ( y ) - \\tilde \\Psi ( x ) ) ( - \\partial \\tilde \\Psi ( x ) ) \\ \\d y \\end{align*}"}
+{"id": "1384.png", "formula": "\\begin{align*} \\left ( \\lambda _ 1 I - \\frac { \\partial } { \\partial x _ 1 } \\right ) \\cdots \\left ( \\lambda _ d I - \\frac { \\partial } { \\partial x _ d } \\right ) h ( x ) = \\prod _ { j = 1 } ^ d \\lambda _ j h ( x ) \\end{align*}"}
+{"id": "4729.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { m } \\gamma _ { k } y _ { k } \\ge 0 \\ \\ \\forall \\ y \\in \\Omega _ { 0 } \\end{align*}"}
+{"id": "8013.png", "formula": "\\begin{align*} & \\frac { 1 } { | \\mathcal F _ { N , k } | } \\sum _ { ( p _ 1 , p _ 2 , \\dots , p _ t ) } ^ { ( 1 ) } \\sum _ { ( m _ 1 , m _ 2 , m _ 3 , \\dots , m _ t ) } ^ { ( a ) } \\sum _ { f \\in \\mathcal F _ { N , k } } a _ f ( p _ 1 ^ { 2 m _ 1 } p _ 2 ^ { 2 m _ 2 } \\dots p _ t ^ { 2 m _ t } ) \\\\ & = \\O \\left ( \\pi _ N ( x ) ^ { a } ( \\log \\log x ) ^ { t - a } \\right ) + \\O \\left ( \\frac { \\pi _ N ( x ) ^ t x ^ { ( 2 M _ 1 + 2 M _ 2 + \\dots + 2 M _ t ) c ' } 8 ^ { \\nu ( N ) } } { k { N } } \\right ) . \\end{align*}"}
+{"id": "4335.png", "formula": "\\begin{align*} & \\limsup _ { n \\to + \\infty } \\sum _ { i = 0 } ^ { n - 1 } \\frac { 1 } { \\inf _ { I _ { i , n } \\backslash U _ k } c ( t ) } \\int _ { I _ { i , n } \\backslash U _ k } c ( s ) e ^ { - s } d s \\\\ \\le & \\limsup _ { n \\to + \\infty } \\sum _ { i = 0 } ^ { n - 1 } \\frac { \\sup _ { I _ { i , n } \\backslash U _ k } c ( t ) } { \\inf _ { I _ { i , n } \\backslash U _ k } c ( t ) } \\int _ { I _ { i , n } \\backslash U _ k } e ^ { - s } d s \\\\ = & \\int _ { ( t ' _ 2 , t ' _ 1 ] \\backslash U _ k } e ^ { - s } d s . \\end{align*}"}
+{"id": "3008.png", "formula": "\\begin{align*} \\hat { \\varphi } ( \\tau , w ( \\cdot ) ) = \\inf \\limits _ { u ( \\cdot ) } J ( \\tau , w ( \\cdot ) , u ( \\cdot ) ) . \\end{align*}"}
+{"id": "702.png", "formula": "\\begin{align*} \\| u \\| _ { \\mathcal Q ^ { \\alpha , p } } : = \\left ( \\int _ { \\mathbb R ^ N } \\left | I _ \\frac \\alpha 2 \\ast | u | ^ { p } \\right | ^ 2 d x \\right ) ^ { \\frac 1 { 2 p } } \\end{align*}"}
+{"id": "4948.png", "formula": "\\begin{align*} W ( \\xi _ 0 , n , x , \\theta ) = E _ { \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\biggr ] , \\end{align*}"}
+{"id": "6224.png", "formula": "\\begin{align*} V ( r ) = \\frac { L ( L + 1 ) } { r ^ 2 } - \\frac { Q } { r } f + \\kappa B _ 1 \\frac { r } { f } + \\kappa B _ 2 \\frac { 1 } { f ^ 2 } + \\kappa B _ 3 \\frac { r } { f ^ 3 } + \\kappa B _ 4 \\frac { 1 } { f ^ 4 } + \\kappa B _ 5 \\frac { r } { f ^ 5 } + \\kappa B _ 6 \\frac { 1 } { f ^ 6 } . \\end{align*}"}
+{"id": "3646.png", "formula": "\\begin{align*} \\beta _ x = \\min \\left \\{ \\frac { 1 } { 2 L _ y } , \\frac { 1 } { 2 \\eta _ x L _ { x y } ^ 2 } \\right \\} . \\end{align*}"}
+{"id": "335.png", "formula": "\\begin{align*} \\iota ( A ) = e _ { r } ^ { ( \\pi s _ r ) ( \\lambda ) _ r - \\pi ( \\lambda ) _ r } ( A ) . \\end{align*}"}
+{"id": "3087.png", "formula": "\\begin{align*} \\tilde { x } ^ { k } = \\Pi _ { \\mathcal { C } } \\{ x ^ k - \\frac { 1 } { \\beta r } [ F ( x ^ { k } ) - \\lambda ^ k + \\beta ( x ^ k - y ^ k ) ] \\} . \\end{align*}"}
+{"id": "4393.png", "formula": "\\begin{align*} & 1 ) ( s + \\frac { s '^ 2 } { u '' s - s '' } ) e ^ { u - t } = \\frac { 1 } { c ( t ) } , \\\\ & 2 ) s ' - s u ' = 1 , \\end{align*}"}
+{"id": "3134.png", "formula": "\\begin{align*} \\alpha \\circ T = T \\circ \\beta ~ ~ ~ ~ T ( a ) \\ast T ( b ) = T \\big ( \\mathfrak { l } ( T ( a ) ) b + \\mathfrak { r } ( T ( b ) ) a \\big ) . \\end{align*}"}
+{"id": "8342.png", "formula": "\\begin{align*} & \\psi ^ \\pm ( x , t ; k ) = e ^ { - i c _ - \\sigma _ { 3 } } \\mu ^ \\pm ( x , t ; k ) e ^ { i c \\sigma _ 3 } , \\\\ & \\mu ^ \\pm ( x , t ; k ) = e ^ { - i c _ - \\sigma _ { 3 } } \\varphi ^ \\pm ( x , t ; k ) e ^ { - i \\left ( k ^ 2 x + \\eta ^ 2 t \\right ) \\sigma _ 3 } C ^ \\pm ( k ) e ^ { i \\left ( k ^ 2 x + \\eta ^ 2 t \\right ) \\sigma _ 3 } e ^ { i c \\sigma _ 3 } . \\end{align*}"}
+{"id": "8135.png", "formula": "\\begin{align*} H ^ { d + 1 } _ c ( A / V ) : = H ^ { d + 1 } ( f _ ! A ) \\ ( = \\pi _ { - d - 1 } ( f _ ! A ) ) \\end{align*}"}
+{"id": "3941.png", "formula": "\\begin{align*} \\nabla \\mathcal { N } _ { \\epsilon } ( x ) = W _ { L } \\cdot \\sigma _ { \\epsilon } ' ( N _ { \\epsilon } ^ { ( L - 1 ) } ( x ) ) \\cdot W _ { L - 1 } \\cdot \\ldots \\cdot \\sigma _ { \\epsilon } ' ( N _ { \\epsilon } ^ { ( 1 ) } ( x ) ) \\cdot W _ { 1 } . \\end{align*}"}
+{"id": "7805.png", "formula": "\\begin{align*} c _ 1 ( q , k , p ) : = \\frac { 1 } { 4 } \\Big ( \\hat W _ r ( q - k ) + \\hat W _ r ( q + k ) - 2 \\hat W _ r ( q ) - ( 4 \\lambda + 2 \\mu ) \\hat W _ r ( q ) \\Big ) , \\end{align*}"}
+{"id": "6787.png", "formula": "\\begin{align*} u ( x , 0 ) = q ( x ) , x \\in \\left ( - \\infty , \\infty \\right ) , \\end{align*}"}
+{"id": "9302.png", "formula": "\\begin{align*} v _ a ( u ) = \\lim _ { r \\to 0 } { \\frac { \\sigma ( a , r ) } { { r } ^ { \\frac { 4 n ( m - 1 ) } { m } } } } \\end{align*}"}
+{"id": "8880.png", "formula": "\\begin{align*} \\lim \\nolimits _ { k \\to \\infty } C ( l ' , r ' ; P _ { k , S } ^ { l , r } ) = \\delta _ S ( l , r ; \\ , l ' , r ' ) , \\end{align*}"}
+{"id": "5076.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\Omega } | U | ^ { 2 } \\dd x = \\frac { 1 } { 2 } \\int _ { \\Omega } U \\cdot \\nabla \\times ( \\textrm { c u r l } ^ { - 1 } U ) \\dd x & = \\frac { 1 } { 2 } \\int _ { \\Omega } \\nabla \\times U \\cdot \\textrm { c u r l } ^ { - 1 } U \\dd x \\\\ & = \\frac { f _ { 1 } ^ { + } } { 2 } \\int _ { \\Omega } U \\cdot \\textrm { c u r l } ^ { - 1 } U \\dd x = { \\mathcal { I } } _ 1 . \\end{align*}"}
+{"id": "2190.png", "formula": "\\begin{align*} \\mathbf { u } _ { \\lambda _ { 1 } } = \\left [ \\begin{array} { l } \\mathbf { u } _ { \\lambda _ { 1 1 } } \\\\ \\mathbf { u } _ { \\lambda _ { 1 2 } } \\end{array} \\right ] \\end{align*}"}
+{"id": "599.png", "formula": "\\begin{align*} \\sup _ { s \\in [ t _ { 0 } , t ] } u ( s \\wedge \\tau _ { k } ) | Y _ { s } ^ { \\tau _ { k } } | _ { V _ { s } ^ { \\tau _ { k } } } ^ { p } & \\leq u ( t _ { 0 } ) | Y _ { t _ { 0 } } | _ { V _ { t _ { 0 } } } ^ { p } + \\int _ { t _ { 0 } } ^ { t \\wedge \\tau _ { k } } | Y _ { s } | _ { V _ { s } } ^ { p - 2 } \\big ( \\dot { u } ( s ) | Y _ { s } | _ { V _ { s } } ^ { 2 } + u ( s ) p \\hat { Z } _ { s } \\big ) ^ { + } \\ , d s \\\\ & + \\sup _ { s \\in [ t _ { 0 } , t ] } I _ { s } ^ { \\tau _ { k } } \\quad \\end{align*}"}
+{"id": "2635.png", "formula": "\\begin{align*} [ x , y ] = x \\diamond y - \\varepsilon ( x , y ) y \\diamond x . \\end{align*}"}
+{"id": "9011.png", "formula": "\\begin{align*} ( - 1 ) ^ { | \\pi ( \\tau ) | } v _ { \\tau } = \\dfrac { \\left ( \\prod _ { r = 1 } ^ { s } D _ { j _ { r } - 1 } \\right ) \\left ( - x _ { j _ { s + 1 } } \\right ) U } { x _ { \\tau } } . \\end{align*}"}
+{"id": "7548.png", "formula": "\\begin{align*} \\boldsymbol { A ^ * A } x = \\| \\boldsymbol { A } \\| ^ 2 x . \\end{align*}"}
+{"id": "3609.png", "formula": "\\begin{align*} f ( \\kappa ) = \\sigma , \\partial \\Sigma = \\Gamma , \\end{align*}"}
+{"id": "172.png", "formula": "\\begin{align*} S _ { H ' } ( F ) : = ( F ^ { H ' } ) ^ { \\Gamma _ { H ' } - l a } . \\end{align*}"}
+{"id": "7679.png", "formula": "\\begin{align*} | k _ i ^ N ( t , \\nu ^ { N , 1 } _ { \\boldsymbol { x } } , \\ldots , \\nu ^ { N , N } _ { \\boldsymbol { x } } , \\lambda ^ { N , 1 } _ { \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) } , \\ldots , \\lambda ^ { N , N } _ { \\boldsymbol { \\phi } ( t , \\boldsymbol { x } ) } ) - k \\big ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } , \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } ) \\big ) | = O \\Big ( \\frac { 1 } { N } \\Big ( 1 + \\frac { 1 } { N } \\underset { j \\neq i } \\sum | x ^ i - x ^ j | \\Big ) \\Big ) ; \\end{align*}"}
+{"id": "7248.png", "formula": "\\begin{align*} \\sigma ( \\alpha + \\beta ) = \\sigma ( \\alpha ) + \\sigma ( \\beta ) \\end{align*}"}
+{"id": "7432.png", "formula": "\\begin{align*} \\phi _ k ' ( t ) & = - ( 2 k - p _ k ) \\log \\frac { 1 - t } { t } + \\frac { 1 - 2 t } { 2 ( 1 - t ) t } , \\\\ \\phi _ k '' ( t ) & = \\frac { 2 k - p _ k } { ( 1 - t ) t } - \\frac { 2 t ^ 2 - 2 t + 1 } { 2 ( 1 - t ) ^ 2 t ^ 2 } \\end{align*}"}
+{"id": "3914.png", "formula": "\\begin{align*} \\partial \\Omega _ \\varepsilon = \\{ x + \\varepsilon \\rho ( x ) \\mathbf { n } _ x \\mid x \\in \\partial \\Omega \\} , \\end{align*}"}
+{"id": "1313.png", "formula": "\\begin{align*} I ( u ) = m | | u | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } + | | \\nabla _ { H } u | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } - { \\rm R e } \\int _ { \\mathbb { H } ^ n } \\overline { u } f ( u ) d x . \\end{align*}"}
+{"id": "1370.png", "formula": "\\begin{align*} K ( n _ i , y ; n _ j , z ) & = - f _ { n _ j - n _ i } ( z - y ) 1 _ { \\{ i < j \\} } \\\\ & + \\sum _ { k , l = 1 } ^ d \\int _ { y } ^ { \\infty } f _ { n _ m - n _ i } ( u - y ) u ^ { k - 1 } d u ( A ^ { - 1 } ) _ { l k } f _ { n _ j - d + l } ( z - x _ l ) \\end{align*}"}
+{"id": "8050.png", "formula": "\\begin{align*} \\frac { \\partial \\mathbb { E } \\left [ \\varepsilon | \\hat { \\mathbf { H } } \\right ] } { \\partial a _ i } = & 4 a _ i \\sum _ { l = 1 } ^ { M - 1 } \\sum _ { q = l + 1 } ^ { M } \\Re \\left [ \\hat { \\phi } ^ { \\left ( l , i \\right ) ^ * } \\hat { \\phi } ^ { \\left ( q , i \\right ) } \\right ] - 2 \\Re { \\left \\{ \\hat { \\phi } ^ { \\left ( i , i \\right ) } \\right \\} } + 4 a _ i \\left ( \\sum _ { l = 1 } ^ M \\lvert \\hat { \\phi } ^ { \\left ( l , i \\right ) } \\rvert ^ 2 + M \\sigma _ { e , i } ^ 2 \\lVert \\mathbf { p } _ i \\rVert ^ 2 \\right ) . \\end{align*}"}
+{"id": "1123.png", "formula": "\\begin{align*} p _ { b , m } ^ * = \\left [ { \\frac { { { W _ m } } } { { { \\lambda ^ * } } } - \\frac { 1 } { { { h _ m } } } } \\right ] _ 0 ^ { \\widetilde P } , p _ { b , o } ^ * = \\left [ { \\frac { { { \\left ( { W - { W _ m } } \\right ) } } } { { { \\lambda ^ * } } } - \\frac { 1 } { { { h _ b } } } } \\right ] _ 0 ^ { \\widetilde P } , \\end{align*}"}
+{"id": "6935.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n \\big ( z _ i ^ 2 - z _ i z _ j \\big ) J _ { i j } K '' ( x _ i - x _ j ) & = \\frac 1 2 \\sum _ { i , j = 1 } ^ n ( z _ i - z _ j ) ^ 2 J _ { i j } K '' ( x _ i - x _ j ) . \\end{align*}"}
+{"id": "4643.png", "formula": "\\begin{align*} \\int _ T \\gamma = \\sum _ { \\widetilde { K } \\ \\ \\pi ^ { - 1 } ( K ) } ( e _ { \\widetilde { K } } - 1 ) \\langle [ \\widetilde { K } ] , \\alpha \\rangle . \\end{align*}"}
+{"id": "2396.png", "formula": "\\begin{align*} T _ n ^ v \\cong \\bigoplus _ { s = q + 1 } ^ { p } R / ( v ( \\kappa ^ n _ s ) / v ( \\mu ^ { n + 1 } _ { \\alpha ( s ) } ) ) , \\end{align*}"}
+{"id": "6598.png", "formula": "\\begin{align*} [ X + \\xi , Y + \\eta ] _ H = \\mathcal { L } _ X ( Y + \\eta ) - \\iota _ Y d \\xi + H ( X , Y , \\cdot ) , \\end{align*}"}
+{"id": "8117.png", "formula": "\\begin{align*} n - m + f = 2 , \\end{align*}"}
+{"id": "7494.png", "formula": "\\begin{align*} & \\frac { ( n - 2 ) ( 1 0 - n ) } { 4 } \\int _ { B _ { \\rho } } r ^ { 2 - n } u _ r ^ 2 \\d x \\\\ & \\leq C \\int _ { B _ { 2 \\rho } \\setminus B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x + C \\varepsilon \\int _ { B _ { 4 \\rho } } r ^ { 3 - n } | \\nabla u | ^ 2 \\d x , \\rho \\leq 1 / 4 \\varepsilon \\leq \\varepsilon _ 0 , \\end{align*}"}
+{"id": "5276.png", "formula": "\\begin{align*} \\partial _ \\theta \\Phi = \\partial _ \\lambda \\Phi = \\partial _ t \\Phi = 0 \\end{align*}"}
+{"id": "3344.png", "formula": "\\begin{align*} \\psi _ i = \\left \\{ \\begin{aligned} & \\sigma ^ 0 & & ( i \\in S ) \\\\ & \\tau & & ( i \\notin S ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5836.png", "formula": "\\begin{align*} c _ s ^ 2 \\partial _ { z 1 } T = - \\frac { 1 } { \\Delta t } { \\varsigma } _ 3 { m } _ 3 ^ { ( 1 ) } , \\end{align*}"}
+{"id": "1058.png", "formula": "\\begin{align*} v ^ { - 1 } \\sum _ { k = 1 } ^ N ( \\sigma \\circ w _ \\infty ) ^ k \\mu _ \\infty \\in X _ \\ast \\otimes \\mathbb Q \\end{align*}"}
+{"id": "5144.png", "formula": "\\begin{align*} \\varphi ( y ) \\geq \\inf \\left \\{ \\varphi ( y ) \\ | \\ | y | = R \\right \\} , | y | \\leq R . \\end{align*}"}
+{"id": "5747.png", "formula": "\\begin{align*} \\alpha ^ { q + 1 } + \\beta ^ { q + 1 } & = 1 , \\\\ \\alpha ^ { q + 1 } u + \\beta ^ { q + 1 } v & = r , \\\\ \\alpha ^ { q + 1 } u ^ q + \\beta ^ { q + 1 } v ^ q & = r + c , \\\\ \\alpha ^ { q + 1 } u ^ { q + 1 } + \\beta ^ { q + 1 } v ^ { q + 1 } & = d . \\end{align*}"}
+{"id": "3708.png", "formula": "\\begin{align*} & S ' ( h , v ) = 0 \\mbox { i n a c o l l a r n e i g h b o r h o o d $ U $ o f $ \\hat \\Sigma $ i n } M \\setminus \\Omega \\\\ & \\left \\{ \\begin{array} { l } h ^ \\intercal = 0 \\\\ H ' ( h ) = 0 \\end{array} \\right . \\mbox { o n } \\hat \\Sigma . \\end{align*}"}
+{"id": "9306.png", "formula": "\\begin{align*} \\sum _ { \\ell \\in L _ { i } } c _ { \\ell } ( 0 ) \\cdot u _ { i } ^ { \\gamma _ { \\ell } } = 0 \\end{align*}"}
+{"id": "4178.png", "formula": "\\begin{align*} x _ 2 v _ 1 - x _ 1 v _ 2 + k v _ 3 = 0 . \\end{align*}"}
+{"id": "5740.png", "formula": "\\begin{align*} u ^ { q + 1 } h ^ q + u ^ q ( h ^ q + h ) + d ^ q h & = 0 , \\\\ h v ^ { q + 1 } + v ( h ^ q + h ) + d h ^ q & = 0 , \\end{align*}"}
+{"id": "6932.png", "formula": "\\begin{align*} A _ 2 ^ 2 \\le \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mathcal { W } _ 1 ^ 2 ( P _ i , Q ^ * _ i ) \\le \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mathcal { W } _ 2 ^ 2 ( P _ i , Q ^ * _ i ) \\le \\frac { 2 R _ f } { \\kappa n } . \\end{align*}"}
+{"id": "888.png", "formula": "\\begin{align*} ( r , q ) = 1 \\ , . \\end{align*}"}
+{"id": "20.png", "formula": "\\begin{align*} \\alpha ^ { \\tau } \\oplus \\beta ^ { \\tau } = \\{ \\gamma _ 1 \\oplus \\gamma _ 2 \\mid \\gamma _ 1 \\in \\alpha ^ { \\tau } , \\gamma _ 2 \\in \\beta ^ { \\tau } \\} ^ { \\tau } . \\end{align*}"}
+{"id": "6412.png", "formula": "\\begin{align*} g = D ^ 2 u , u \\in C ^ \\infty ( N , \\R ) . \\end{align*}"}
+{"id": "5364.png", "formula": "\\begin{align*} \\Gamma _ { k } = \\{ \\lambda \\in \\mathbb { R } ^ { n } : \\sigma _ { j } ( \\lambda ) > 0 , j = 1 , \\ldots , k \\} . \\end{align*}"}
+{"id": "8642.png", "formula": "\\begin{align*} { \\bf { x } } \\left [ n \\right ] = \\sum \\limits _ { l = 1 } ^ L { { { \\bf { f } } _ l } s \\left [ { n - { \\kappa _ l } } \\right ] } , \\end{align*}"}
+{"id": "8009.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leq x \\\\ ( p , N ) = 1 } } a _ f \\left ( p ^ { 2 l } \\right ) \\ll l ^ { 2 } \\pi _ N ( x ) \\left ( x ^ { - \\frac { 1 } { 2 c _ { 9 } l } } + e ^ { - \\frac { c _ { 1 0 } \\log x } { 4 l ^ { 2 } \\log ( 2 k N l ) } } + e ^ { - \\frac { c _ { 1 0 } \\sqrt { \\log x } } { \\sqrt { 2 l } } } \\right ) . \\end{align*}"}
+{"id": "2282.png", "formula": "\\begin{align*} a ( D ( x ) V ) = a ( D ( x ) D ( t ) V _ 1 ) = a ( D ( t ) D ( x ) V _ 1 ) = a ( D ( x ) V _ 1 ) , \\end{align*}"}
+{"id": "5962.png", "formula": "\\begin{align*} J _ 1 + J _ 3 = 0 . \\end{align*}"}
+{"id": "6883.png", "formula": "\\begin{align*} f ( z ) = C \\int _ 0 ^ z \\frac { \\zeta ^ \\alpha ( \\zeta - 1 ) ^ \\beta ( \\zeta - t ) ^ \\gamma } { ( \\zeta - z _ 0 ) ^ 2 ( \\zeta - \\overline { z } _ 0 ) ^ 2 } \\ , d \\zeta + z _ 1 , \\end{align*}"}
+{"id": "4068.png", "formula": "\\begin{align*} n \\sum _ { j \\in [ n ] } T ^ { 1 } _ { n , j } T ^ { 3 } _ { n , j } = O _ M ( n ^ { 1 - 2 H } ) . \\end{align*}"}
+{"id": "2445.png", "formula": "\\begin{align*} D _ { q } ^ { n } ( s ) \\{ L _ { q } [ f ( t ) ] \\} ( s ) = L _ { q } [ ( - t ) ^ { n } f ( t ) ] . \\end{align*}"}
+{"id": "5853.png", "formula": "\\begin{align*} \\delta z = - \\frac { 1 } { 8 } { F _ z } . \\end{align*}"}
+{"id": "6634.png", "formula": "\\begin{align*} \\begin{aligned} G _ { \\epsilon } ( t _ 1 , \\cdots , t _ k ) & = \\Big ( \\int ^ { 1 } _ { 0 } K ( t _ 1 , r ) \\theta _ { \\epsilon } ( r ) d r , \\cdots , \\int ^ { 1 } _ { 0 } K ( t _ k , r ) \\theta _ { \\epsilon } ( r ) d r \\Big ) ^ { \\tau } \\\\ & = \\Big ( G _ { \\epsilon } ( t _ 1 ) , \\cdots , G _ { \\epsilon } ( t _ k ) \\Big ) \\end{aligned} \\end{align*}"}
+{"id": "4285.png", "formula": "\\begin{align*} \\frac { \\left ( 6 n - \\beta ^ { 3 } + 3 \\right ) \\left ( n ^ { 2 } + n + 1 \\right ) } { 3 } = 0 \\end{align*}"}
+{"id": "7943.png", "formula": "\\begin{align*} f ( x _ 0 + a ) = f ( x _ 0 ) + a f ' ( x _ 0 ) + \\frac { f '' ( \\xi ) } { 2 } a ^ 2 \\end{align*}"}
+{"id": "1489.png", "formula": "\\begin{align*} \\operatorname { S p } ( \\mathcal { L } _ { 1 , \\alpha } \\otimes I _ { H ^ k ( \\mathbb { R } ^ { d - 1 } ) } + I _ { H ^ k _ { \\alpha } ( \\mathbb { R } ) } \\otimes \\Delta _ y ) = \\operatorname { S p } ( \\mathcal { L } _ { 1 , \\alpha } ) + \\operatorname { S p } ( \\Delta _ y ) . \\end{align*}"}
+{"id": "2412.png", "formula": "\\begin{align*} F ( u ) = \\int _ { a } ^ { b } f ( x ) \\ , K ( u , x ) d x . \\end{align*}"}
+{"id": "2679.png", "formula": "\\begin{align*} \\mu _ d ( n ) = \\sum _ { \\substack { n _ 1 , \\ldots , n _ d \\geq 1 \\\\ \\prod _ { i = 1 } ^ d n _ i = n } } \\mu ( n _ i ) \\end{align*}"}
+{"id": "8552.png", "formula": "\\begin{align*} \\phi ( t ) = \\frac { d } { d t } ( k \\ , * \\ , f ) ( t ) \\ , = \\ , \\frac { d } { d t } \\ , \\int _ 0 ^ t k ( t - \\tau ) f ( \\tau ) \\ , d \\tau , \\ t > 0 . \\end{align*}"}
+{"id": "6793.png", "formula": "\\begin{align*} F ^ { - } ( x + y ) + B ( x , y ) + \\int _ { - \\infty } ^ { x } B ( x , t ) F ^ { - } ( t + y ) d t = 0 , y < x \\end{align*}"}
+{"id": "2541.png", "formula": "\\begin{align*} \\bar { \\mathbf { A } } = \\mathbf { A } \\mathbf { D } ^ T , \\bar { \\mathbf { C } } = \\mathbf { D } \\mathbf { C } \\mathbf { D } ^ T . \\end{align*}"}
+{"id": "6151.png", "formula": "\\begin{align*} X _ { r } ^ 1 = \\mathcal { X } _ { r } ^ 1 = \\xi _ r , \\quad { X } ^ { 1 } _ t = X _ { t _ k } ^ 1 + \\mu ( t _ k , \\mathcal { X } ^ { 1 } ) ( t - t _ k ) + \\sigma ( t _ k , \\mathcal { X } ^ { 1 } ) ( W _ t - W _ { t _ k } ) . \\end{align*}"}
+{"id": "6289.png", "formula": "\\begin{align*} d \\hat { \\sigma } _ i ( T , X _ j ) = - T ( r ^ j _ i ) + X _ j ( \\lambda _ i ( T ) ) - \\lambda _ i ( \\nabla ^ v _ { X _ j } T ) + \\lambda _ j ( T ) r ^ j _ i = 0 , \\quad \\forall j \\neq i . \\end{align*}"}
+{"id": "4708.png", "formula": "\\begin{align*} \\det D ^ { 2 } u = \\mu , \\end{align*}"}
+{"id": "4910.png", "formula": "\\begin{align*} B = \\frac { Q ( 3 / 4 ) + Q ( 1 / 4 ) - 2 Q ( 1 / 2 ) } { Q ( 3 / 4 ) - Q ( 1 / 4 ) } . \\end{align*}"}
+{"id": "8876.png", "formula": "\\begin{align*} [ ( \\lambda _ 1 , \\mu _ 1 ) , \\kappa _ 1 ] \\cdot [ ( \\lambda _ 2 , \\mu _ 2 ) , \\kappa _ 2 ] : = [ ( \\lambda _ 1 + \\lambda _ 2 , \\mu _ 1 + \\mu _ 2 ) , \\kappa _ 1 + \\kappa _ 2 + \\lambda _ 1 \\mu _ 2 ^ t - \\mu _ 1 \\lambda _ 2 ^ t ] . \\end{align*}"}
+{"id": "8807.png", "formula": "\\begin{align*} \\frac { d } { d t } ( Y _ t - y _ t ) & = \\Pi _ { \\mathrm { k e r } A ^ \\perp } ( B ( Y _ t , z _ 0 ) + B ( z _ 0 , Y _ t ) - B ( y _ t , z _ t ) - B ( z _ t , y _ t ) - B ( y _ t , y _ t ) + A y _ t ) \\\\ & = L _ { z _ 0 } ^ { \\perp } ( Y _ t - y _ t ) + \\Pi _ { \\mathrm { k e r } A ^ \\perp } ( B ( y _ t , z _ 0 - z _ t ) + B ( z _ 0 - z _ t , y _ t ) - B ( y _ t , y _ t ) + A y _ t ) . \\end{align*}"}
+{"id": "5292.png", "formula": "\\begin{align*} p = 1 \\oplus 0 \\oplus 1 \\oplus 0 \\oplus \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} , q = 0 \\oplus 1 \\oplus 1 \\oplus 0 \\oplus \\begin{pmatrix} a ^ 2 & a b \\\\ a b & b ^ 2 \\end{pmatrix} \\end{align*}"}
+{"id": "2137.png", "formula": "\\begin{align*} z & = \\sqrt { \\frac { a } { a + b } } \\left ( \\sqrt { \\frac { 1 - b } { 1 + a } } \\cos ( t ) + i \\sqrt { \\frac { 1 + b } { 1 - a } } \\sin ( t ) \\right ) , \\\\ w & = \\pm \\sqrt { \\frac { b } { a + b } } \\left ( \\sqrt { \\frac { 1 - a } { 1 + b } } \\sin ( t ) - i \\sqrt { \\frac { 1 + a } { 1 - b } } \\cos ( t ) \\right ) . \\end{align*}"}
+{"id": "8233.png", "formula": "\\begin{align*} \\frac { 1 - | b ( r \\xi ) | ^ 2 } { | 1 - b ( r \\xi ) | ^ 2 } = \\int _ { \\partial \\mathbb { D } } \\frac { 1 - r ^ 2 } { | t - r \\xi | ^ 2 } d \\mu ( t ) \\geq \\frac { \\mu \\left ( \\{ t : \\ | t - \\xi | < 1 - r \\} \\right ) } { 4 ( 1 - r ) } \\geq C _ 1 \\ , \\end{align*}"}
+{"id": "2802.png", "formula": "\\begin{align*} \\mathbb { E } e ^ { - \\langle Z _ { \\lambda } ^ { S _ { n } } , f \\rangle } = \\mathbb { E } e ^ { - \\langle Y _ { \\infty } ^ { S _ { n } } , f \\rangle } , \\end{align*}"}
+{"id": "7198.png", "formula": "\\begin{align*} s _ 3 = \\frac { ( \\lambda + \\mu ) ( \\lambda + 2 \\mu ) } { \\lambda + 3 \\mu } , s _ 4 = \\frac { 2 \\mu ( \\lambda + 2 \\mu ) } { \\lambda + 3 \\mu } , t _ 1 = \\frac { \\mu } { 2 | \\xi | } \\biggl ( \\frac { 1 - 2 s _ 1 } { \\tau - 2 \\mu s _ 1 | \\xi | } - \\frac { 1 } { \\tau - 2 \\mu | \\xi | } \\biggr ) . \\end{align*}"}
+{"id": "3608.png", "formula": "\\begin{align*} Y = m ( X ) + \\eta \\end{align*}"}
+{"id": "7904.png", "formula": "\\begin{align*} \\lambda _ k = \\frac 1 4 \\hat W _ r ( q ) \\Big [ \\hat W _ r ( q + k ) + \\hat W _ r ( q - k ) - 2 \\hat W _ r ( q ) \\Big ] \\end{align*}"}
+{"id": "2097.png", "formula": "\\begin{align*} ~ ~ ~ ~ & K \\sim \\pi \\\\ ~ ~ ~ ~ & W \\mid K = k \\ , \\ , \\sim D _ k \\\\ ~ ~ ~ ~ & V \\mid K = k \\ , \\ , \\sim G _ k \\\\ ~ ~ ~ ~ & X _ 1 , \\ldots , X _ n \\mid W , V \\ , \\ , \\sim P _ { W , V } \\end{align*}"}
+{"id": "1144.png", "formula": "\\begin{align*} { \\cal P } = [ O _ N , \\dots , O _ N , P _ { i j } ^ { ( \\ell ) } , O _ N , \\dots , O _ N ] \\in \\R ^ { N \\times N L } \\quad \\mathrm { w i t h } P _ { i j } ^ { ( \\ell ) } = \\mathbf { e } _ i \\mathbf { e } _ j ^ T \\in \\R ^ { N \\times N } . \\end{align*}"}
+{"id": "1716.png", "formula": "\\begin{align*} & \\partial _ t \\left ( \\int _ 0 ^ t U ( t , s ) e ^ { ( \\alpha - 1 ) W ( s ) } g ( y ( s ) ) d s \\right ) \\\\ = & \\int _ 0 ^ t \\partial _ t U ( t , s ) e ^ { ( \\alpha - 1 ) W ( s ) } g ( y ( s ) ) d s + U ( t , t ) e ^ { ( \\alpha - 1 ) W ( t ) } g ( y ( t ) ) \\\\ = & \\int _ 0 ^ t A ( t ) U ( t , s ) e ^ { ( \\alpha - 1 ) W ( s ) } g ( y ( s ) ) d s + e ^ { ( \\alpha - 1 ) W ( t ) } g ( y ( t ) ) . \\end{align*}"}
+{"id": "5177.png", "formula": "\\begin{align*} \\int _ { D ( z _ 0 , R ' ) } | \\phi _ n | ^ { 1 0 / 3 } \\frac { 1 } { r ^ { 1 1 / 3 } } \\dd z \\dd r \\leq C \\left ( \\int _ { D ( z _ 0 , R ' ) } | \\nabla \\phi _ n | ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r \\right ) ^ { 5 / 3 } = C | | \\nabla \\phi _ n | | _ { L ^ { 2 } ( D ( z _ 0 , R ' ) ; r ^ { - 1 } ) } ^ { 1 0 / 3 } . \\end{align*}"}
+{"id": "5320.png", "formula": "\\begin{align*} ( 2 \\pi ) ^ { - n } | \\lambda | ^ n \\ , \\int _ { \\C ^ n } \\widehat { f } ( a , w ) \\varphi _ { k , \\lambda } ^ { n - 1 } ( z - w ) e ^ { \\frac { i } { 2 } \\lambda \\Im ( z \\cdot \\bar { w } ) } d w = f ^ { - \\lambda } \\ast _ { - \\lambda } \\varphi _ { k , \\lambda } ^ { n - 1 } ( z ) = \\widehat { f } ( a , z ) . \\end{align*}"}
+{"id": "8431.png", "formula": "\\begin{align*} \\begin{aligned} \\| a ( z ) - \\hat { a } \\| _ { H _ z ^ 1 ( \\mathbb { R } ) } & \\leq \\sup _ { x \\in \\mathbb { R } } \\| u _ x \\| _ { L ^ 2 _ x } ^ 2 \\sup _ { x \\in \\mathbb { R } } \\| \\Psi ^ - _ { 1 1 } - e ^ { - i c _ - } \\| _ { H _ z ^ 1 } + \\| \\bar { u } _ x \\| _ { L ^ 1 _ x } \\sup _ { x \\in \\mathbb { R } } \\| \\Psi ^ - _ { 2 1 } \\| _ { H _ z ^ 1 } , \\end{aligned} \\end{align*}"}
+{"id": "5803.png", "formula": "\\begin{align*} \\omega : = - p _ 1 ^ * ( \\log \\sqrt { \\tau _ 0 } ) _ z d z + p _ 2 ^ * \\omega _ 0 . \\end{align*}"}
+{"id": "1065.png", "formula": "\\begin{align*} \\langle \\alpha ^ \\vee , 2 \\rho \\rangle = \\frac 1 2 \\left ( \\langle \\alpha ^ \\vee , 2 \\rho \\rangle + \\langle s _ \\alpha ( \\alpha ^ \\vee ) , s _ \\alpha ( 2 \\rho ) \\rangle \\right ) = \\frac 1 2 \\langle \\alpha ^ \\vee , 2 \\rho - s _ \\alpha ( 2 \\rho ) \\rangle . \\end{align*}"}
+{"id": "4760.png", "formula": "\\begin{align*} c _ { 1 } = \\gamma _ { 1 } b _ { 1 } ^ { T } b _ { 1 } , \\ \\ \\gamma _ { 1 } = \\frac { b _ { 1 } ^ { T } b _ { 1 } } { c _ { 1 } } . \\end{align*}"}
+{"id": "677.png", "formula": "\\begin{align*} b ^ n _ l = \\left ( l + \\frac { 1 } { 2 } \\right ) I _ { n , l } ~ , ~ a ^ n _ l = - \\frac { 1 } { n } \\left ( l + \\frac { 1 } { 2 } \\right ) I _ { n , l } \\end{align*}"}
+{"id": "2222.png", "formula": "\\begin{align*} \\mathbf { a } ( \\theta _ { 0 } ) ^ { T } \\mathbf { v } ^ { 0 } & = e ^ { - j m \\phi } + e ^ { - j n \\phi } + e ^ { - j k \\phi } \\\\ & = e ^ { - j m \\phi } ( 1 + e ^ { - j ( n - m ) \\phi } ( 1 + e ^ { - j ( k - n ) \\phi } ) ) \\end{align*}"}
+{"id": "8633.png", "formula": "\\begin{align*} \\| \\chi \\| _ { L ^ 2 ( \\mu ) } ^ 2 = \\frac { \\| \\chi \\| _ { L ^ 2 ( \\mu ) } ^ { 2 + \\delta } } { \\| \\chi \\| _ { L ^ 2 ( \\mu ) } ^ { \\delta } } \\leq \\frac { \\| \\chi \\| _ { L ^ 2 ( \\mu ) } ^ { 2 + \\delta } } { \\varepsilon ^ \\delta } \\end{align*}"}
+{"id": "1939.png", "formula": "\\begin{align*} F ( m ; z ) = \\sum _ { n = 0 } ^ \\infty b _ 3 \\left ( \\frac { m n - 1 } { 1 2 } \\right ) q ^ n , \\end{align*}"}
+{"id": "7309.png", "formula": "\\begin{align*} \\begin{array} { l } R _ { \\max } = \\mathop { \\max } \\limits _ { \\bf p } { R _ T } \\\\ { \\rm { s } } { \\rm { . t } } { \\rm { . } } \\\\ C 2 , \\ , C 3 , \\ , C 8 , \\ , { \\rm a n d } \\ , C 9 . \\end{array} \\end{align*}"}
+{"id": "4952.png", "formula": "\\begin{align*} \\theta _ 2 = \\pi _ 2 ( X _ 1 ) , \\ \\ \\theta _ i = \\pi _ i ( X _ 1 , X _ 2 , Y _ 2 , \\cdots , X _ { i - 1 } , Y _ { i - 1 } ) , ~ i \\geq 3 . \\end{align*}"}
+{"id": "3829.png", "formula": "\\begin{align*} \\begin{aligned} \\Big | \\langle \\boldsymbol { \\nu } , A _ t \\rangle - \\bar A _ t + \\langle \\boldsymbol { \\nu } , f _ { \\alpha , x _ \\alpha } \\rangle - \\bar f _ { \\alpha , x _ \\alpha } ) \\Big | & \\le c _ 3 \\langle \\boldsymbol { \\nu } , \\eta ( \\lambda | \\bar U ; x , t ) \\rangle = c _ 3 \\mathcal { H } ( \\boldsymbol { \\nu } , U , \\bar { U } ; x , t ) \\ ; . \\end{aligned} \\end{align*}"}
+{"id": "895.png", "formula": "\\begin{align*} & \\Gamma _ 1 ( X ) = \\sum \\limits _ { 1 \\leq d \\leq z } \\mu ( d ) \\Sigma \\big ( X , d ^ 2 \\big ) \\ , , \\\\ & \\Gamma _ 2 ( X ) = \\sum \\limits _ { d > z } \\mu ( d ) \\Sigma \\big ( X , d ^ 2 \\big ) \\ , , \\\\ & \\Sigma \\big ( X , d ^ 2 \\big ) = \\sum \\limits _ { 1 \\leq n \\leq X \\atop { n ^ 2 + n + 1 \\equiv 0 \\ , ( d ^ 2 ) } } 1 \\ , , \\\\ & \\sqrt { X } \\leq z < X \\ , , \\end{align*}"}
+{"id": "6481.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\frac { \\psi ( p ) } { p } = 0 . \\end{align*}"}
+{"id": "5823.png", "formula": "\\begin{align*} g _ i ^ { ( e q ) } = { { w } _ i } T , \\end{align*}"}
+{"id": "5235.png", "formula": "\\begin{align*} C _ 1 = \\frac { 3 } { 2 } W \\frac { c _ { 3 / 2 } ^ { 1 / 2 } } { | \\mu | ^ { 3 } J _ { 5 / 2 } ( c _ { 3 / 2 } ) } . \\end{align*}"}
+{"id": "7174.png", "formula": "\\begin{align*} \\Bigl [ I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } , Q \\Bigr ] f = I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } ( Q f ) - Q \\Bigl ( I _ { n + 1 } \\frac { \\partial } { \\partial x _ n } \\Bigr ) f = \\frac { \\partial Q } { \\partial x _ n } f , f \\in C ^ { \\infty } ( \\Omega ) . \\end{align*}"}
+{"id": "7128.png", "formula": "\\begin{align*} \\mathcal S ( f ) = \\left ( \\frac { f '' } { f ' } \\right ) - \\frac 1 2 \\left ( \\frac { f '' } { f ' } \\right ) ^ 2 \\end{align*}"}
+{"id": "776.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m f _ i ( u ) = 0 , \\end{align*}"}
+{"id": "9182.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } ( t ) & - \\dot { z } ( t ) = \\\\ & - \\gamma \\left ( h ( x ( t ) + \\delta u ( t ) ) u ( t ) - \\dfrac { b _ { 1 , \\delta } ( x _ a ( t ) ) } { 2 } \\right ) \\\\ & - \\left . \\gamma ^ 2 \\dfrac { \\partial \\epsilon ( x , t ) } { \\partial x } \\right | _ { x = x _ a ( t ) } \\dfrac { b _ { 1 , \\delta } ( x _ a ( t ) ) } { 2 } \\\\ & + \\gamma \\left . \\dfrac { \\partial \\epsilon ( x , t ) } { \\partial t } \\right | _ { x = x _ a ( t ) } . \\end{aligned} \\end{align*}"}
+{"id": "3190.png", "formula": "\\begin{align*} ( s - 1 ) \\left ( - x ^ 0 x ^ 0 + \\sum _ { i = 1 } ^ 3 ( x ^ i ) ^ 2 \\right ) - ( 1 + s ) { ( x ^ 0 - x ^ 3 ) } ^ 2 & = 0 . \\end{align*}"}
+{"id": "7601.png", "formula": "\\begin{align*} H _ \\epsilon = \\bigcap _ { b \\in B _ 1 \\times \\cdots \\times B _ k } T ^ { - \\overline { \\eta ( \\epsilon ) } \\cdot b } E \\end{align*}"}
+{"id": "2808.png", "formula": "\\begin{align*} \\frac { 2 d + 1 } { 2 d } \\mathbb { E } [ h ^ { V } | \\sigma ( h _ { z } ^ { V } : z \\in V \\backslash U ) ] & = \\mathbb { E } [ \\mathbb { E } [ h _ { x } ^ { V } | \\sigma ( h _ { z } ^ { V } : z \\in V \\backslash \\{ x \\} ) ] | \\sigma ( h _ { z } ^ { V } : z \\in V \\backslash U ) ] \\\\ & = \\mathbb { E } \\Big [ \\frac { 1 } { 2 d } \\sum _ { i } h _ { x \\pm e _ { i } } - \\frac { 1 } { 2 d ^ { 2 } } \\sum _ { i < j } h _ { x \\pm e _ { i } \\pm e _ { j } } - \\frac { 1 } { 4 d ^ { 2 } } \\sum _ { i } h _ { x \\pm 2 e _ { i } } \\Big | \\sigma ( h _ { z } ^ { V } : z \\in V \\backslash U ) \\Big ] . \\end{align*}"}
+{"id": "4447.png", "formula": "\\begin{align*} C ( z ) = \\sum _ { i } c _ { i } g _ { i } ( z ) \\end{align*}"}
+{"id": "6724.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { \\sigma \\mathrm { B } ( \\alpha , \\beta ) } \\ , \\left [ \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] ^ { \\alpha - 1 } \\left [ 1 - \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] ^ { \\beta - 1 } \\phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) . \\end{align*}"}
+{"id": "4256.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\inf _ { k \\in { \\mathbb Z } } \\left ( \\frac { \\mu ( f ^ { k - n } ( W ) ) } { \\mu ( f ^ { k } ( W ) ) } \\right ) = \\infty \\tag * { $ \\mathcal { U E } 2 $ } \\end{align*}"}
+{"id": "8844.png", "formula": "\\begin{align*} \\sup _ { 0 \\le t \\le \\tau _ 3 } \\left | \\int _ 0 ^ t S ( t - s ) \\begin{pmatrix} 0 \\\\ \\sigma _ 1 d W _ s ^ { ( 1 ) } \\\\ \\end{pmatrix} \\right | & \\le R \\sqrt { \\tau _ 3 } , \\\\ \\int _ 0 ^ { \\tau _ 3 } \\sum _ { j = 3 } ^ n | x _ { t , j } | ^ 2 d t & \\le \\sqrt { \\delta } \\tau _ 3 K ^ { 2 r } , \\\\ \\sup _ { 0 \\le t \\le \\tau _ 3 } | W _ t | & \\le R \\sqrt { \\tau _ 3 } , \\\\ \\sup _ { 0 \\le t \\le \\tau _ 3 } | x _ { t , 1 } | & \\le ( \\delta _ 1 ^ { 1 / 4 } + \\delta ^ { 1 / 4 } ) K ^ { 4 r } + 1 . \\end{align*}"}
+{"id": "4768.png", "formula": "\\begin{align*} \\rho ( F ) = \\{ \\eta \\in \\Omega : F ( \\eta ) ^ { - 1 } \\} \\end{align*}"}
+{"id": "5344.png", "formula": "\\begin{align*} \\tilde { A } _ n ( \\ell ) a _ { \\ell } = \\tilde { Q } _ { n , 1 } ( \\ell ) \\tilde { \\beta } _ { n , 1 } ^ { \\ell } + \\cdots + \\tilde { Q } _ { n , \\tilde { s } ( n ) } ( \\ell ) \\tilde { \\beta } _ { n , \\tilde { s } ( n ) } ^ { \\ell } \\end{align*}"}
+{"id": "5155.png", "formula": "\\begin{align*} \\tilde { b } = \\nabla \\times ( \\phi \\nabla \\theta ) + \\frac { 1 } { \\tau } G \\nabla \\theta , \\end{align*}"}
+{"id": "1218.png", "formula": "\\begin{align*} \\lim _ { \\Delta \\uparrow \\Z ^ d } \\sup _ { \\eta , \\xi \\in \\Omega } \\abs { \\gamma _ { \\Lambda } ( \\eta _ \\Lambda | \\eta _ { \\Delta \\setminus \\Lambda } \\xi _ { \\Delta ^ c } ) - \\gamma _ \\Lambda ( \\eta _ { \\Lambda } | \\eta _ { \\Lambda ^ c } ) } = 0 . \\end{align*}"}
+{"id": "7403.png", "formula": "\\begin{align*} [ V _ \\lambda ] = q ^ { - \\frac { c } { 2 4 } } q ^ { \\frac { ( \\lambda | \\lambda + 2 \\rho ) } { 2 ( k + h ^ \\vee ) } - \\lambda ( x ) } \\ \\prod _ { n = 1 } ^ \\infty \\prod _ { \\alpha \\in \\Delta _ { } ^ { 0 } } ( 1 - q ^ { n - \\epsilon ( \\alpha ) } ) ^ { - 1 } \\prod _ { \\alpha \\in \\Delta _ { } ^ { + , \\sharp } } ( 1 - e ^ { \\alpha ( h ) } q ^ { n - \\epsilon ( \\alpha ) } ) ^ { - 1 } ( 1 - e ^ { - \\alpha ( h ) } q ^ { n + \\epsilon ( \\alpha ) - 1 } ) ^ { - 1 } , \\end{align*}"}
+{"id": "1167.png", "formula": "\\begin{align*} \\partial _ t \\rho _ n = - \\sum _ { i = 1 } ^ N \\nabla _ { x _ i } \\cdot \\left ( \\left ( \\frac { 1 } { n - 1 } \\sum _ { j = 1 } ^ n K \\left ( x _ i - x _ j \\right ) \\right ) \\rho _ n \\right ) + \\sum _ { i = 1 } ^ n \\Delta _ { x _ i } \\rho _ n . \\end{align*}"}
+{"id": "8815.png", "formula": "\\begin{align*} \\eta ( K ) = 4 K ^ { \\frac { 2 r - 2 } { 3 } } . \\end{align*}"}
+{"id": "8348.png", "formula": "\\begin{align*} & S ( k ) = I + \\mathcal { O } ( k ^ { - 1 } ) , k \\rightarrow \\infty . \\end{align*}"}
+{"id": "3425.png", "formula": "\\begin{align*} \\max _ { \\theta \\in [ 0 , 1 ] } \\ \\max _ { j = 1 , . . . , k + 1 } \\prod _ { \\ell \\neq j } \\ \\frac { | \\sin \\pi ( \\theta - \\theta _ { \\ell } ) | } { | \\sin \\pi ( \\theta _ j - \\theta _ { \\ell } ) | } < e ^ { \\gamma k } . \\end{align*}"}
+{"id": "2136.png", "formula": "\\begin{align*} \\varrho = 0 , \\varrho _ { Z Z } ( L , L ) \\end{align*}"}
+{"id": "4896.png", "formula": "\\begin{align*} \\alpha ^ * & = 1 + \\lim _ { z \\rightarrow 0 } z \\frac { \\Phi ( z ) } { \\phi ( z ) } \\frac { 1 - \\Phi ( z ) } { 1 - 2 \\Phi ( z ) } = 1 + \\frac { \\sqrt { 2 \\pi } } { 4 } \\lim _ { z \\rightarrow 0 } \\frac { z } { 1 - 2 \\Phi ( z ) } \\\\ & = 1 + \\frac { \\sqrt { 2 \\pi } } { 4 } \\lim _ { z \\rightarrow 0 } \\frac { 1 } { - 2 \\phi ( z ) } = 1 - \\frac { \\pi } { 4 } \\approx 0 . 2 1 4 6 . \\end{align*}"}
+{"id": "178.png", "formula": "\\begin{align*} A \\widehat { \\otimes } _ { A _ { H ' , n } } S _ { H ' , n } ( V ) = V . \\end{align*}"}
+{"id": "3974.png", "formula": "\\begin{align*} c _ { k + 1 } = \\sum _ { i = 1 } ^ k a _ i c _ i , \\ , \\ , c _ j = c _ { n + 1 - j } , \\ , \\ , j = k + 2 , k + 3 , \\cdots , n . \\end{align*}"}
+{"id": "6260.png", "formula": "\\begin{align*} \\gamma ( X _ i , X _ i ) = \\langle A X _ i , X _ i \\rangle [ e _ i ] \\quad \\forall i , \\end{align*}"}
+{"id": "5567.png", "formula": "\\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) = A ( 0 ^ { n + k } 1 . . . ) - A ( 0 ^ \\infty ) = a _ { n + k } - a \\ . \\end{align*}"}
+{"id": "6064.png", "formula": "\\begin{align*} \\kappa ( s ) = \\eta x _ { n + 1 } ^ m ( s ) + \\lambda . \\end{align*}"}
+{"id": "1161.png", "formula": "\\begin{align*} d X _ t ^ i = \\left ( b _ 0 ( t , X ^ i ) + \\frac { 1 } { n - 1 } \\sum _ { j = 1 , j \\neq i } ^ n b \\left ( t , X ^ i , X ^ j \\right ) \\right ) d t + d B ^ { H , i } _ t , i = 1 , \\cdots , n , \\end{align*}"}
+{"id": "4128.png", "formula": "\\begin{align*} \\rho ^ { \\sigma } ( g ) : = \\rho ( g ^ { \\sigma } ) , \\ \\ g \\in G . \\end{align*}"}
+{"id": "8634.png", "formula": "\\begin{align*} \\mathbb { E } _ { n , m } ' [ \\alpha \\gamma ] \\\\ = \\mathbb { E } _ { n , m } ' [ \\chi _ { n , m } ( w , x _ 1 ) \\mathbb { 1 } _ { X _ 1 ( M ) } ( x _ 1 ) \\chi _ { n , m } ( w , x _ 2 ) \\mathbb { 1 } _ { X _ 1 ( M ) } ( x _ 2 ) ] \\\\ = \\mathbb { E } _ { \\mathcal { M } } [ \\mathbb { 1 } _ { X _ 1 ( M ) } ( x _ 1 ) \\mathbb { 1 } _ { X _ 1 ( M ) } ( x _ 2 ) \\mathbb { E } _ { n , m } [ \\chi _ { n , m } ( w , x _ 1 ) \\chi _ { n , m } ( w , x _ 2 ) | M ] ] \\end{align*}"}
+{"id": "739.png", "formula": "\\begin{align*} \\begin{cases} \\forall s \\mbox { a . e . } \\in [ 0 , t ] , \\ : \\forall q \\geq 0 , \\ : ( \\dot z _ \\omega ^ { x , t } ( s ) - \\omega ( s ) ) ( q - z _ \\omega ^ { x , t } ( s ) ) \\leq 0 \\\\ z _ { \\omega } ^ { x , t } ( t ) = x , \\end{cases} \\end{align*}"}
+{"id": "6471.png", "formula": "\\begin{align*} \\mu \\Big ( \\bigcap _ { j = 0 } ^ { t _ n } T ^ { s _ { j , i } } A \\Big ) \\geq ( 1 - \\epsilon _ n ) \\mu ( A ) . \\end{align*}"}
+{"id": "622.png", "formula": "\\begin{align*} \\begin{aligned} ( T \\backslash G ) \\times G & \\to G \\\\ ( g , \\gamma ) & \\mapsto g ^ { - 1 } \\gamma g , \\end{aligned} \\end{align*}"}
+{"id": "8729.png", "formula": "\\begin{align*} \\widehat { A } ^ \\Gamma = \\{ \\chi \\in \\widehat { A } \\ : : \\ : \\gamma \\chi = \\chi \\ , \\forall \\gamma \\in \\Gamma \\} \\end{align*}"}
+{"id": "108.png", "formula": "\\begin{align*} \\langle \\nu ( b ) , 2 \\rho \\rangle \\leq \\ell ( y ) \\leq \\ell ( y ' ) = \\langle \\nu ( b ) , 2 \\rho \\rangle . \\end{align*}"}
+{"id": "7971.png", "formula": "\\begin{align*} 0 = \\nabla _ { i } Q = \\frac { \\nabla _ { i } P } { P } - A \\nabla _ { i } \\left ( \\frac { \\rho ^ { 2 } } { 2 } \\right ) , \\end{align*}"}
+{"id": "4266.png", "formula": "\\begin{align*} 1 7 2 9 = 9 ^ { 3 } + 1 0 ^ { 3 } = 1 2 ^ { 3 } + 1 ^ { 3 } \\end{align*}"}
+{"id": "3630.png", "formula": "\\begin{align*} ( \\tilde { \\kappa } _ 1 ) _ i = \\tilde { h } _ { 1 1 i } = h _ { 1 1 i } , \\end{align*}"}
+{"id": "7244.png", "formula": "\\begin{align*} \\sigma ( \\lambda \\alpha ) = \\sigma ( \\lambda ) \\sigma ( \\alpha ) = \\lambda \\sigma ( \\alpha ) . \\end{align*}"}
+{"id": "1893.png", "formula": "\\begin{align*} \\mathfrak P _ l f ( x ) = 2 ^ { \\frac d 2 j } 2 ^ { - l } \\int \\int e ^ { i \\lambda \\Phi _ s ( x , y ) } \\widetilde A _ s ( x , y ) d s \\ , f ( y ) d y , \\end{align*}"}
+{"id": "4385.png", "formula": "\\begin{align*} & \\lim _ { \\epsilon \\to 0 } \\int _ { X _ j } v '' _ { \\epsilon } ( \\Psi ) | f F | ^ 2 _ { h } e ^ { - u ( - v _ { \\epsilon } ( \\Psi ) ) } \\\\ = & \\int _ { X _ j } v '' ( \\Psi ) | f F | ^ 2 _ h e ^ { - u ( - v ( \\Psi ) ) } \\\\ \\le & \\bigg ( \\sup _ { X _ j } e ^ { - u ( - v ( \\Psi ) ) } \\bigg ) \\int _ { X _ j } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f F | ^ 2 _ h . \\end{align*}"}
+{"id": "877.png", "formula": "\\begin{align*} x x ' + x y ' + y y ' = - 2 n \\ , . \\end{align*}"}
+{"id": "7456.png", "formula": "\\begin{align*} \\eta ( 1 ) = \\gamma ^ { - 1 } \\mbox { a n d } \\eta ( g ) = 1 . \\end{align*}"}
+{"id": "6083.png", "formula": "\\begin{align*} V [ e _ 1 ] = \\{ v \\in V \\setminus V ' \\mid ( v , e _ 1 ) \\in E ( v , f _ 1 ) \\notin E \\} \\end{align*}"}
+{"id": "2842.png", "formula": "\\begin{align*} n J _ n ^ { a , b } = J ^ { a , b } + o ( 1 ) , \\quad \\mbox { a s $ n \\to + \\infty $ } , \\end{align*}"}
+{"id": "4912.png", "formula": "\\begin{align*} x = Q _ { \\mathrm { B } } ( u ) = I _ u ^ { - 1 } ( \\alpha , \\beta ) = \\sum _ { i = 1 } ^ { \\infty } d _ i \\ , u ^ { i / \\alpha } . \\end{align*}"}
+{"id": "6831.png", "formula": "\\begin{align*} \\mathbf { H } ( \\varphi ) = \\mathbf { H } ( \\varphi _ { 0 } ) T ( T _ { + } ) , \\end{align*}"}
+{"id": "6829.png", "formula": "\\begin{align*} \\left \\vert \\widetilde { \\varphi } _ { 0 } ^ { - } \\left ( z ( \\rho ) \\right ) \\right \\vert = c _ { l } \\left \\vert \\frac { 1 } { 2 } - i \\rho \\right \\vert ^ { l + 1 } \\left \\vert J _ { l } \\left ( \\rho \\right ) \\right \\vert \\leq \\operatorname * { C o n s t } \\left ( \\left ( 1 + \\left \\vert \\rho \\right \\vert \\right ) ^ { \\frac { 5 } { 2 } l - \\frac { 1 } { 2 } - 2 \\alpha } \\mathbf { L } ( \\rho ) + 1 \\right ) \\end{align*}"}
+{"id": "3067.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } w _ i \\frac { d y _ i } { y _ i } + w _ { n + 1 } \\frac { d \\varphi } { \\varphi } = 0 \\end{align*}"}
+{"id": "1093.png", "formula": "\\begin{align*} \\prescript J { } \\pi ( w _ 1 ) = ( w _ 1 ' , \\mu _ 1 ) , \\prescript J { } \\pi ( w _ 2 ) = ( w _ 2 ' , \\mu _ 2 ) . \\end{align*}"}
+{"id": "7445.png", "formula": "\\begin{align*} t _ \\theta : = \\begin{bmatrix} \\cos \\theta \\dfrac { 1 } { \\sqrt { k } } \\mathbb { 1 } _ k \\\\ \\hline \\sin \\theta \\dfrac { 1 } { \\sqrt { l } } \\mathbb { 1 } _ l \\end{bmatrix} \\mbox { a n d } U _ \\theta : = \\begin{bmatrix} \\ ; \\cos \\theta \\sqrt { \\dfrac { m } { k } } V & \\vline & \\sin \\theta \\sqrt { \\dfrac { m } { l } } W \\ ; \\end{bmatrix} , \\end{align*}"}
+{"id": "4224.png", "formula": "\\begin{align*} \\overline { \\mathfrak { D } _ { q ^ n } } \\ \\overline { U _ { q ^ n } } \\dot { s } \\varepsilon ( v ) \\zeta = ( A + B + C ) \\overline { U _ { q ^ n } } \\dot { s } \\overline { U ^ * _ { q ^ n } } { \\bf 1 } _ { - } \\in \\Bbbk { \\bf G } \\zeta . \\end{align*}"}
+{"id": "6667.png", "formula": "\\begin{align*} \\Lambda ( \\alpha ) = \\lim _ { \\epsilon \\to 0 } \\Lambda _ { \\epsilon ^ 2 } ( \\alpha ) . \\end{align*}"}
+{"id": "4251.png", "formula": "\\begin{align*} & \\sup \\{ | \\varphi _ { 2 k j _ 0 } ( z ) - \\rho ( z ) | ; \\ z \\in \\mathbb { B } _ n , \\ | \\varphi _ { k j _ 0 } ( z ) | \\leq r \\} \\\\ & = \\sup \\{ | \\varphi _ { k j _ 0 } \\circ \\varphi _ { k j _ 0 } ( z ) - \\rho \\circ \\varphi _ { k j _ 0 } ( z ) | ; \\ z \\in \\mathbb { B } _ n , \\ | \\varphi _ { k j } ( z ) | \\leq r \\} \\\\ & \\leq \\sup \\{ | \\varphi _ { k j _ 0 } ( z ) - \\rho ( z ) | ; \\ z \\in \\mathbb { B } _ n , \\ | z | \\leq r \\} \\leq \\varepsilon \\end{align*}"}
+{"id": "6107.png", "formula": "\\begin{align*} Z ^ { + + } _ { n , p } = \\frac { ( n - p + a _ { 1 2 } - a _ { 1 2 3 } + j ^ { ( 3 ) } + 1 ) ( n - p + a _ { 1 2 } - a _ { 1 2 3 } - j ^ { ( 3 ) } ) } { ( n - p + a _ { 1 2 } - a _ { 1 2 3 } + j ^ { ( 3 ) } ) ( n - p + a _ { 1 2 } - a _ { 1 2 3 } - j ^ { ( 3 ) } - 1 ) } \\ , . \\end{align*}"}
+{"id": "6847.png", "formula": "\\begin{align*} \\xi ( x ) : = g ( \\frac { i } { 2 } , x ) \\int _ { x } ^ { 0 } \\frac { d t } { g ^ { 2 } ( \\frac { i } { 2 } , t ) } . \\end{align*}"}
+{"id": "316.png", "formula": "\\begin{align*} \\rho ( x ) = \\sup _ { r > 0 } \\Big \\{ r : \\frac { 1 } { r ^ { n - 2 } } \\int _ { B ( x , r ) } V ( y ) d y \\leq 1 \\Big \\} . \\end{align*}"}
+{"id": "95.png", "formula": "\\begin{align*} X _ x ( b ) ( \\overline { \\mathbb F _ q } ) = \\{ g \\in G ( \\breve F ) / I \\mid g ^ { - 1 } b \\sigma ( g ) \\in I x I \\} . \\end{align*}"}
+{"id": "8556.png", "formula": "\\begin{align*} ( I ^ 0 _ { 0 + } \\ , f ) ( t ) : = f ( t ) , \\ t > 0 . \\end{align*}"}
+{"id": "8728.png", "formula": "\\begin{align*} \\left [ c _ { \\chi _ 1 \\chi ^ { - 1 } } \\right ] = \\left [ \\beta _ \\eta \\right ] = 0 \\in \\mathrm { H } ^ 2 ( Q , S ^ 1 ) . \\end{align*}"}
+{"id": "8371.png", "formula": "\\begin{align*} G _ - ( x ; k ) = \\mathcal { P } ^ { - } \\left ( G _ { - } J \\right ) ( z ) + \\mathcal { P } ^ { - } \\left ( J \\right ) , \\end{align*}"}
+{"id": "7431.png", "formula": "\\begin{align*} f _ k ( x ) : = \\sum _ { j = 0 } ^ { k } \\dbinom { k } { j } ^ 2 ( 1 - x ) ^ { p _ k ( k - j ) / k } x ^ { p _ k j / k } . \\end{align*}"}
+{"id": "8387.png", "formula": "\\begin{align*} & \\| h _ { 1 } \\| _ { L ^ \\infty } = \\sup _ { x \\in ( - \\infty , x _ 0 ) } \\bigg | \\frac { 1 } { 2 i } \\int _ { - \\infty } ^ x ( | u _ y | ^ 2 f _ 1 + u _ y f _ 2 ) d y \\bigg | \\\\ & \\leq \\frac { 1 } { 2 } ( \\| f _ 1 \\| _ { L ^ \\infty } \\| u _ x \\| _ { L ^ 2 } ^ 2 + \\| f _ 2 \\| _ { L ^ \\infty } \\| u _ x \\| _ { L ^ 1 } ) d y \\\\ & \\leq \\frac { 1 } { 2 } ( \\| u _ x \\| ^ 2 _ { L ^ 2 } + \\| u _ x \\| _ { L ^ 1 } ) \\| f \\| _ { L ^ { \\infty } } . \\end{align*}"}
+{"id": "9006.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ n a _ k + \\sum _ { j = 1 } ^ n \\left ( \\prod _ { k = 1 } ^ { j - 1 } ( a _ k + b _ k ) \\right ) b _ j \\left ( \\prod _ { k = j + 1 } ^ n a _ k \\right ) = \\prod _ { i = 1 } ^ n ( a _ i + b _ i ) , \\end{align*}"}
+{"id": "461.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } e ^ { \\gamma x } f _ 2 ( x ) = \\lim _ { x \\to \\infty } c f _ e ( x ) = c \\lim _ { x \\to \\infty } \\sum _ { n c _ 1 \\ge x } ^ \\infty ( ( \\lambda \\Lambda _ e ) ^ n / n ! ) \\ , g _ { e } ^ { \\ast n } ( x ) = 0 . \\end{align*}"}
+{"id": "5300.png", "formula": "\\begin{align*} \\Phi \\left ( \\bigvee _ { l \\geq j + 1 } p _ l \\right ) ^ { \\perp } \\check { y } h _ n & = \\Phi \\left ( \\bigvee _ { l \\geq j + 1 } p _ l \\right ) ^ { \\perp } \\check { y } ( h _ { n , 1 } + \\cdots + h _ { n , j } ) \\\\ & = \\Phi \\left ( \\bigvee _ { l \\geq j + 1 } p _ l \\right ) ^ { \\perp } \\check { y } ( p _ 1 + \\cdots + p _ j ) h _ n . \\end{align*}"}
+{"id": "1250.png", "formula": "\\begin{align*} \\lim _ { \\Lambda \\uparrow \\Z ^ d } \\frac { 1 } { \\mu ( \\eta _ { \\Lambda } ) } \\int \\mathbf { 1 } _ { \\eta _ { \\Lambda } } ( \\xi ) f ( \\xi ) \\mu ( d \\xi ) = f ( \\eta ) . \\end{align*}"}
+{"id": "106.png", "formula": "\\begin{align*} \\langle \\nu ( x ) , 2 \\rho \\rangle = \\sum _ { \\alpha \\in \\Phi ^ + } \\frac 1 N \\sum _ { k = 1 } ^ N \\ell ( x , ( \\sigma \\circ w ) ^ k v \\alpha ) \\geq \\ell ( x ) . \\end{align*}"}
+{"id": "751.png", "formula": "\\begin{align*} I _ 3 \\leq \\int _ { | r | > 2 | t | } \\frac { | F _ s ( r ) | } { | r - t | ^ { 2 - \\frac 1 { 2 s } } } d r + \\int _ { | r | > 2 | t | } \\frac { | F _ s ( t ) | } { | r - t | ^ { 2 - \\frac 1 { 2 s } } } d r = I _ { 3 1 } + I _ { 3 2 } . \\end{align*}"}
+{"id": "6601.png", "formula": "\\begin{align*} \\langle \\mathcal { G } ^ { \\mathrm { e n d } } \\cdot , \\cdot \\rangle = \\mathcal { G } \\end{align*}"}
+{"id": "7744.png", "formula": "\\begin{align*} R = \\left ( \\prod _ { m = i + 1 } ^ { j - 1 } M _ m ^ T \\bar { Q } _ m \\right ) \\times M _ j ^ T , \\end{align*}"}
+{"id": "4894.png", "formula": "\\begin{align*} \\frac { \\partial f ( 0 ) } { \\partial z } = \\frac { \\partial ^ 2 f ( 0 ) } { \\partial z ^ 2 } = \\frac { \\partial ^ 3 f ( 0 ) } { \\partial z ^ 3 } = 0 \\frac { \\partial ^ 4 f ( 0 ) } { \\partial z ^ 4 } & = \\frac { 4 \\sqrt { 2 } } { \\pi } \\frac { \\Gamma ( 3 / 2 - \\pi / 4 ) } { \\Gamma ( 1 - \\pi / 4 ) } \\left ( \\frac { 3 } { \\pi } - 1 \\right ) < 0 . \\end{align*}"}
+{"id": "1298.png", "formula": "\\begin{align*} a ( x _ a ; x _ c ; \\beta ) - a ( x _ a ; x _ b ; \\alpha ) - a ( x _ a ; x _ d ; { \\beta - \\alpha } ) = 0 , \\\\ c ( x _ a ; x _ c ; \\beta ) - c ( x _ a ; x _ b ; \\alpha ) - a ( x _ a ; x _ d ; { \\beta - \\alpha } ) = 0 , \\end{align*}"}
+{"id": "6335.png", "formula": "\\begin{align*} D _ x ^ { ( 1 ) } x _ { n - 1 } + 2 D _ x ^ { ( 1 ) } x _ { n - 2 } x ^ { p ^ { s _ j } } + \\cdots + ( n - 1 ) x ^ { ( n - 2 ) p ^ { s _ j } } = 0 . \\end{align*}"}
+{"id": "1944.png", "formula": "\\begin{align*} { } _ { p } F _ { q } \\left ( \\ \\begin{array} { l l l } \\alpha _ { 1 } , . . . , \\alpha _ { p } ~ ; ~ \\\\ \\beta _ { 1 } , . . . , \\beta _ { q } ~ ; ~ \\end{array} z \\right ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\alpha _ { 1 } ) _ { n } . . . ( \\alpha _ { p } ) _ { n } } { ( \\beta _ { 1 } ) _ { n } . . . ( \\beta _ { q } ) _ { n } } \\frac { z ^ { n } } { n ! } ~ , \\end{align*}"}
+{"id": "2738.png", "formula": "\\begin{align*} H = Z \\circ \\varphi . \\end{align*}"}
+{"id": "7056.png", "formula": "\\begin{align*} u \\ , = \\ , \\tfrac { x ^ 2 } { 2 } + { \\rm O } _ { x , y , z } ( 3 ) , \\end{align*}"}
+{"id": "463.png", "formula": "\\begin{align*} f _ 1 ( x ) = ( e ^ { \\lambda \\Lambda _ 1 } - 1 ) ^ { - 1 } \\sum _ { n = 1 } ^ { \\infty } ( ( \\lambda \\Lambda _ 1 ) ^ n / n ! ) g _ 1 ^ { \\ast n } ( x ) \\end{align*}"}
+{"id": "5844.png", "formula": "\\begin{align*} \\rho { u _ z } = \\sum \\limits _ { i = 0 } ^ { 1 8 } { { c _ { i z } } { f _ i } + \\frac { { \\Delta t } } { 2 } { F _ z } } . \\end{align*}"}
+{"id": "2009.png", "formula": "\\begin{align*} ( ( G _ \\alpha ( \\epsilon _ N ) \\circ G _ \\alpha T _ \\alpha ( \\iota ) \\circ \\psi _ L ) ( l ) ) ( s ) & = \\epsilon _ N ( ( G _ \\alpha T _ \\alpha ( \\iota ) \\circ \\psi _ L ) ( l ) ( s ) ) \\\\ & = \\epsilon _ N ( T _ \\alpha ( \\iota ) \\circ \\psi _ L ( l ) ( s ) ) \\\\ & = \\epsilon _ N ( T _ \\alpha ( \\iota ) ( s \\otimes l ) ) \\\\ & = \\epsilon _ N ( s \\otimes \\iota ( l ) ) \\\\ & = \\iota ( l ) ( s ) \\end{align*}"}
+{"id": "3420.png", "formula": "\\begin{align*} A _ k ( \\theta , E ) = \\left ( \\begin{array} { c c } P _ k ( \\theta , E ) & - P _ { k - 1 } ( \\theta + \\alpha , E ) \\\\ P _ { k - 1 } ( \\theta , E ) & - P _ { k - 2 } ( \\theta + \\alpha , E ) \\end{array} \\right ) , \\end{align*}"}
+{"id": "8272.png", "formula": "\\begin{align*} u ( t ) & \\ , \\ , \\ , \\leq \\int _ { 0 } ^ { t } \\bigg ( 2 \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j = 1 } ^ { i } j V _ { i , j } | u _ j | \\ , \\psi _ i + \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j = i } ^ { \\infty } V _ { i , j } | u _ j | \\ , \\psi _ i \\bigg ) ( h ) d h . \\end{align*}"}
+{"id": "4917.png", "formula": "\\begin{align*} Q ( u ) = \\sum _ { i = 0 } ^ { \\infty } f _ i \\ , \\ , u ^ { i / \\alpha } , \\end{align*}"}
+{"id": "6557.png", "formula": "\\begin{align*} \\| u _ { l } v _ { l } \\| _ { L ^ { p } ( \\Omega ) } \\le C \\| u _ { l } \\| _ { L ^ { p } ( \\Omega ) } \\| v _ { l } \\| _ { L ^ { 2 } ( \\Omega ) } ^ { \\theta _ { 1 } } \\| v _ { l } \\| _ { W ^ { 2 , p } ( \\Omega ) } ^ { 1 - \\theta _ { 1 } } , \\qquad \\theta _ { 1 } = \\frac { \\frac { 2 } { d } - \\frac { 1 } { p } } { \\frac { 2 } { d } - \\frac { 1 } { p } + \\frac { 1 } { 2 } } \\in ( 0 , 1 ) , \\end{align*}"}
+{"id": "8254.png", "formula": "\\begin{align*} \\nu ^ { n } _ m ( t ) : = \\sum _ { i = m } ^ { n } i \\psi ^ { n } _ i ( t ) , \\end{align*}"}
+{"id": "7740.png", "formula": "\\begin{align*} f ( x ) = \\left \\{ \\begin{array} { l l l } x & & x \\geq 0 \\\\ a ( e ^ { x } - 1 ) & & x < 0 \\end{array} \\right . a > 0 . \\end{align*}"}
+{"id": "4282.png", "formula": "\\begin{align*} \\beta = 3 \\left ( 2 i - 1 \\right ) \\end{align*}"}
+{"id": "7354.png", "formula": "\\begin{gather*} \\Gamma _ 0 = \\bigl \\{ P \\in \\Gamma : P ( \\abs { f } ) < \\infty P ( f ) = 0 f \\in F \\bigr \\} \\end{gather*}"}
+{"id": "5486.png", "formula": "\\begin{align*} f ( q ) \\leq f ( 0 . 3 5 ) < - 0 . 1 2 4 < \\log ( 0 . 8 8 5 ) < \\log \\Gamma ( 2 - q ) = g ( q ) . \\end{align*}"}
+{"id": "8921.png", "formula": "\\begin{gather*} \\sum _ { m = 0 } ^ { n } \\left ( - 1 \\right ) ^ { m } ( 2 m - 1 ) ! ! ( 2 n - 2 m - 1 ) ! ! \\sum _ { k = 0 } ^ { m } \\frac { 1 } { k ! ( m - k ) ! ( n - m - k ) ! } \\frac { x ^ { k } } { ( 2 k - 1 ) ! ! } \\\\ = \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ \\frac { ( n - 1 ) ! ! } { ( n / 2 ) ! } ( 2 - x ) ^ { n / 2 } & & n \\end{array} \\right . . \\end{gather*}"}
+{"id": "928.png", "formula": "\\begin{align*} u _ 1 ( t ) & = M ( t ) D ( t ) \\mathcal { F } M ( t ) \\mathcal { F } ^ { - 1 } v _ 1 ( t ) \\\\ & = M ( t ) D ( t ) v _ 1 ( t ) + O ( t ^ { - 3 / 4 + C _ 1 \\varepsilon ^ 2 _ 1 } ) \\\\ & = M ( t ) D ( t ) e ^ { - 3 i \\lambda _ 1 \\Phi _ 1 ( t ) } \\alpha + O ( t ^ { - 3 / 4 + 2 C _ 1 \\varepsilon ^ 2 _ 1 } ) \\end{align*}"}
+{"id": "7371.png", "formula": "\\begin{gather*} \\lim _ k \\sup _ { Q \\in \\mathcal { U } } Q \\bigl \\{ \\abs { f } \\ , 1 ( \\abs { f } > k ) \\bigr \\} = 0 \\quad f \\in F . \\end{gather*}"}
+{"id": "3443.png", "formula": "\\begin{align*} \\tilde { m } _ n = \\begin{cases} m _ n - q _ { n - 1 } , \\ell _ n = a _ { n + 1 } \\\\ m _ n + q _ { n - 1 } , \\ell _ n = - a _ { n + 1 } \\end{cases} . \\end{align*}"}
+{"id": "8712.png", "formula": "\\begin{align*} \\widehat { \\tau } : x \\mapsto \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\tau _ { 1 } ( x _ { i i } ) \\forall x = ( x _ { i j } ) \\in \\mathcal { M } _ { n } ( M _ { 1 } ) . \\end{align*}"}
+{"id": "4120.png", "formula": "\\begin{align*} b _ { \\sigma \\otimes \\tau } = b _ { \\sigma } \\otimes b _ { \\tau } . \\end{align*}"}
+{"id": "2778.png", "formula": "\\begin{align*} s _ { D } ( x ) : = a ^ { D } ( x , x ) . \\end{align*}"}
+{"id": "3906.png", "formula": "\\begin{align*} \\delta ^ 2 \\int _ { \\Omega } \\left ( ( K ( x ) - K ( z _ j ) ) \\nabla V _ { \\delta , Z , j } | \\nabla V _ { \\delta , Z , i } \\right ) = O \\left ( \\frac { \\delta ^ 2 } { | \\ln \\varepsilon | ^ 2 } \\right ) . \\end{align*}"}
+{"id": "2407.png", "formula": "\\begin{align*} z : = ( I _ { n } + p ^ { m - 1 } x _ { r } ) ^ { - ( p - 1 ) } c x _ { r + 1 } : = x _ { r } - p ^ { r } z \\ . \\end{align*}"}
+{"id": "3674.png", "formula": "\\begin{align*} & X _ { i ; j k } - X _ { i ; k j } + X _ { j ; k i } - X _ { j ; i k } - X _ { k ; i j } + X _ { k ; j i } \\\\ & = ( R _ { k j \\ell i } + R _ { i k \\ell j } - R _ { j i \\ell k } ) X ^ \\ell \\\\ & = 2 R _ { i j \\ell k } X ^ \\ell = - 2 R _ { \\ell k j i } X ^ \\ell \\end{align*}"}
+{"id": "475.png", "formula": "\\begin{align*} 1 = \\lim _ { x \\to \\infty } \\frac { F ( x + y + \\Delta ) } { F ( x - c + \\Delta ) } \\le \\lim _ { x \\to \\infty } \\frac { f ( x + y ) } { f ( x ) } \\le \\lim _ { x \\to \\infty } \\frac { F ( x + y - c + \\Delta ) } { F ( x + \\Delta ) } = 1 \\end{align*}"}
+{"id": "3781.png", "formula": "\\begin{align*} \\mathcal { F } = \\left \\{ f \\in \\mathcal { H } ( \\mathbb { C } ) : \\frac { 1 } { \\pi } \\iint _ { \\mathbb C } | f ( z ) | ^ 2 e ^ { - | z | ^ 2 } d A ( z ) < \\infty \\right \\} , \\end{align*}"}
+{"id": "2548.png", "formula": "\\begin{align*} \\mathbf { T } _ { x } = ( \\mathbf { x } ) - \\mathbf { U } _ { x } , \\mathbf { U } _ { x } \\mathbf { e } = 0 . \\end{align*}"}
+{"id": "7544.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = 1 } ^ d A _ { j } ^ * A _ { j } \\right \\| = \\sum \\limits _ { j = 1 } ^ d \\left \\| X _ { j } ^ * X _ { j } \\right \\| . \\end{align*}"}
+{"id": "3915.png", "formula": "\\begin{align*} \\delta _ \\rho G ( y , z ) = - \\delta _ \\rho h ( y , z ) = \\int _ { \\partial \\Omega } \\frac { \\partial G ( y , x ) } { \\partial \\mathbf { n } _ x } \\frac { \\partial G ( z , x ) } { \\partial \\mathbf { n } _ x } \\rho ( x ) d \\sigma _ x , \\end{align*}"}
+{"id": "6567.png", "formula": "\\begin{align*} Z = S ( x , \\tilde { n } _ { 1 } , c _ { 1 } ) - S ( x , \\tilde { n } _ { 2 } , c _ { 1 } ) + S ( x , \\tilde { n } _ { 2 } , c _ { 1 } ) - S ( x , \\tilde { n } _ { 2 } , c _ { 2 } ) . \\end{align*}"}
+{"id": "6553.png", "formula": "\\begin{align*} \\displaystyle \\int _ { \\Omega } n ( \\cdot , t ) = \\int _ { \\Omega } n _ { 0 } , n ( x , t ) \\ge 0 , 0 < c ( x , t ) < \\gamma \\mbox { f o r } x \\in \\Omega . \\end{align*}"}
+{"id": "5343.png", "formula": "\\begin{align*} \\tilde { H } _ { n , k , m } = \\begin{pmatrix} \\tilde { P } _ { n , k } ( 0 ) a _ 0 & \\tilde { P } _ { n , k } ( 1 ) a _ { 1 } & \\ldots & \\tilde { P } _ { n , k } ( m ) a _ { m } \\\\ \\tilde { P } _ { n , k } ( 1 ) a _ { 1 } & \\tilde { P } _ { n , k } ( 2 ) a _ { 2 } & \\ldots & \\tilde { P } _ { n , k } ( m + 1 ) a _ { m + 1 } \\\\ \\ldots \\\\ \\tilde { P } _ { n , k } ( m ) a _ { m } & \\tilde { P } _ { n , k } ( m + 1 ) a _ { m + 1 } & \\ldots & \\tilde { P } _ { n , k } ( 2 m ) a _ { 2 m } \\end{pmatrix} \\end{align*}"}
+{"id": "4491.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { n ^ 2 } X ^ n } { \\prod _ { i = 0 } ^ { n - 1 } ( q ^ n - q ^ i ) } = \\left ( \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { n ^ 2 - n } X ^ n } { \\prod _ { i = 0 } ^ { n - 1 } ( q ^ n - q ^ i ) } \\right ) \\cdot \\left ( \\sum _ { n = 0 } ^ \\infty X ^ n \\right ) . \\end{align*}"}
+{"id": "2922.png", "formula": "\\begin{align*} & f ( x , y _ { \\bar u + \\theta ( u - \\bar u ) } ) - f ( x , \\bar y ) = \\frac { \\partial f } { \\partial y } ( x , y _ 1 ) ( y _ { \\bar u + \\theta ( u - \\bar u ) } - \\bar y ) \\ , \\ , y _ 1 = \\bar y + \\theta _ 1 ( y _ { \\bar u + \\theta ( u - \\bar u ) } - \\bar y ) , \\\\ & f ( x , y _ u ) - f ( x , \\bar y ) = \\frac { \\partial f } { \\partial y } ( x , y _ 2 ) ( y _ u - \\bar y ) y _ 2 = \\bar y + \\theta _ 2 ( y _ u - \\bar y ) . \\end{align*}"}
+{"id": "7514.png", "formula": "\\begin{align*} \\int _ { B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x & \\leq C \\int _ { B _ { 2 \\rho } } r ^ { 2 - n } u _ r ^ 2 \\d x + C \\varepsilon ^ 2 \\rho \\int _ { B _ { 2 \\rho } } r ^ { 3 - n } | \\nabla u | ^ 2 \\d x \\\\ & \\rho \\leq 1 / 2 \\varepsilon \\leq \\varepsilon _ 0 . \\end{align*}"}
+{"id": "1960.png", "formula": "\\begin{align*} \\int _ { X _ n } \\sum _ { x _ 1 , \\ldots , x _ k \\in \\Z ^ n \\atop \\mathrm { i n d e p . } } f ( x _ 1 g , \\ldots , x _ k g ) d \\mu ( g ) = \\int _ { ( \\R ^ n ) ^ k } f ( x _ 1 , \\ldots , x _ k ) d x _ 1 \\ldots d x _ k . \\end{align*}"}
+{"id": "4633.png", "formula": "\\begin{align*} \\phi ( \\alpha , \\tau ) = \\widetilde \\alpha \\widetilde \\tau { \\widetilde \\alpha } ^ { - 1 } \\cdot { \\widetilde \\tau } ^ { - 1 } = \\alpha ( x _ { i _ 1 } \\cdots x _ { i _ k } ) \\cdot ( x _ { i _ 1 } \\cdots x _ { i _ k } ) ^ { - 1 } = x _ { \\alpha ( i _ 1 ) } \\cdots x _ { \\alpha ( i _ k ) } \\cdot ( x _ { i _ 1 } \\cdots x _ { i _ k } ) ^ { - 1 } = 1 . \\end{align*}"}
+{"id": "2832.png", "formula": "\\begin{align*} \\delta = \\frac { 6 ( \\alpha - 1 ) } { 3 \\alpha - 5 } > 2 \\ , \\ , \\mbox { a n d } \\ , \\ , \\gamma = \\frac { 6 ( \\beta - 1 ) } { 3 \\beta - 5 } > 2 . \\end{align*}"}
+{"id": "592.png", "formula": "\\begin{align*} \\Theta ( \\cdot , \\mu , \\tilde { \\mu } ) : = \\sum _ { \\substack { k = 1 , \\\\ \\beta _ { k } > 0 } } ^ { l } \\zeta _ { k } \\big [ \\null _ { k } \\eta \\big ] _ { \\frac { p } { 1 - \\alpha _ { k } } } \\vartheta ( \\mu , \\tilde { \\mu } ) ^ { \\beta _ { k } } + \\hat { \\zeta } _ { k } ^ { 2 } \\big [ \\null _ { k } \\hat { \\eta } \\big ] _ { \\frac { p } { 1 - \\alpha _ { k } } } ^ { 2 } \\vartheta ( \\mu , \\tilde { \\mu } ) ^ { 2 \\beta _ { k } } . \\end{align*}"}
+{"id": "749.png", "formula": "\\begin{align*} I _ { 2 , N } & : = \\int _ { r \\in N , | t | / 2 \\leq | r | \\leq 2 | t | } \\frac { | F _ s ( r ) - F _ s ( t ) | } { | r - t | ^ { 2 - \\frac 1 { 2 s } } } \\ , d r \\lesssim \\frac 1 { | t | ^ { 2 - \\frac 1 { 2 s } } } \\int _ 0 ^ { 2 | t | } \\frac { r } { ( 1 + r ^ { \\frac 1 s } ) ^ { \\frac { N + 2 s } 2 } } \\ , d r + \\frac { | F _ s ( t ) | } { | t | ^ { 1 - \\frac 1 { 2 s } } } . \\end{align*}"}
+{"id": "5543.png", "formula": "\\begin{align*} a \\cdot b : = \\bigvee \\{ c \\in A \\ : | \\ : c \\wedge a \\le b \\} . \\end{align*}"}
+{"id": "274.png", "formula": "\\begin{align*} Q _ { e u c l } ( f ) = T _ { e u c l } [ f ] \\end{align*}"}
+{"id": "6992.png", "formula": "\\begin{align*} V _ n ' ( | X ( t ) - Y ( t ) | _ H ^ 2 ) & = ( 1 - r ) | X ( t ) - Y ( t ) | _ H ^ { - 2 r } , \\\\ V _ n '' ( | X ( t ) - Y ( t ) | _ H ^ 2 ) & = - r ( 1 - r ) | X ( t ) - Y ( t ) | _ H ^ { - 2 r - 2 } . \\end{align*}"}
+{"id": "6044.png", "formula": "\\begin{align*} I ( t _ { 0 } ^ { \\alpha } u ( t _ { 0 } \\cdot ) , t _ { 0 } ^ { \\alpha } v ( t _ { 0 } \\cdot ) ) & < \\mathop { \\lim } \\limits _ { n \\rightarrow \\infty } \\Big ( \\frac { 1 } { 2 } a ( u _ { n } , v _ { n } ) + \\frac { 1 } { 2 } b ( u _ { n } , v _ { n } ) + \\frac { 1 } { 2 } c ( u _ { n } , v _ { n } ) - \\frac { 1 } { 2 p } F ( u _ { n } , v _ { n } ) \\Big ) \\\\ & = c _ { b } , \\end{align*}"}
+{"id": "3342.png", "formula": "\\begin{align*} ( \\underline { \\Delta ^ { J - 1 } } , \\underline { \\partial \\Delta ^ { J - 1 } } ) & = \\{ ( \\Delta ^ { j _ i - 1 } , \\partial \\Delta ^ { j _ i - 1 } ) \\} _ { i \\in [ m ] } , \\\\ ( \\underline { \\mathrm { p t } ^ J } , \\{ \\emptyset \\} ) & = \\left \\{ \\left ( { \\bigsqcup } _ { j _ i } \\mathrm { p t } , \\{ \\emptyset \\} \\right ) \\right \\} _ { i \\in [ m ] } . \\end{align*}"}
+{"id": "3515.png", "formula": "\\begin{align*} F _ n - n ^ p \\ = \\ \\sum _ { i = 1 } ^ { n } i ^ { p } F _ { n - i } \\ - \\ 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } \\sum _ { i = 1 } ^ n i ^ { p - 2 j - 1 } F _ { n - i } . \\end{align*}"}
+{"id": "3521.png", "formula": "\\begin{align*} - n ^ p - 2 \\sum _ { K = 1 } ^ { p } \\sum _ { j = 0 } ^ { K / 2 } \\binom { p } { 2 j + 1 } \\binom { { p - 2 j - 1 } } { K - 2 j - 1 } B _ { K - 2 j - 1 } \\ , n ^ { p - K } . \\end{align*}"}
+{"id": "399.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to - \\infty } \\| \\varphi ( t , \\tau , \\omega ) x _ { \\tau } - \\varphi _ { \\infty } ( t , \\omega ) x _ 0 \\| _ { \\mathbb { U } } = 0 , t \\geq 0 \\omega \\in \\Omega , \\end{align*}"}
+{"id": "6960.png", "formula": "\\begin{align*} u _ n = r u _ { n - 1 } + s u _ { n - 2 } , \\end{align*}"}
+{"id": "883.png", "formula": "\\begin{align*} r = z q - a n . \\end{align*}"}
+{"id": "596.png", "formula": "\\begin{align*} \\Theta ( \\cdot , \\mu , \\tilde { \\mu } ) : = \\eta _ { 2 } \\vartheta ( \\mu , \\tilde { \\mu } ) + \\hat { \\eta } _ { 2 } ^ { 2 } \\vartheta ( \\mu , \\tilde { \\mu } ) ^ { 2 } . \\end{align*}"}
+{"id": "9001.png", "formula": "\\begin{align*} ( - 1 ) ^ { p ( \\eta , \\ell _ m ) - 1 } i _ { \\eta \\cup \\ell _ m } = - U \\prod _ { r = m + 1 } ^ { M } U ^ { \\eta \\cup \\ell _ m } _ { r } = - U \\prod _ { r = m + 1 } ^ { M } D _ { \\ell _ { r - 1 } } = - U \\cdot B ( m ) . \\end{align*}"}
+{"id": "1838.png", "formula": "\\begin{align*} \\delta ^ \\nabla ( | | R ^ \\nabla | | ^ { p - 2 } R ^ \\nabla ) = 0 \\end{align*}"}
+{"id": "1574.png", "formula": "\\begin{align*} e ^ { 2 \\varphi ( x ) } : = \\int _ { T _ x N } e ^ { 2 w } \\circ \\exp _ x ( v ) \\chi ( \\Vert v \\Vert ) \\dd V _ x , \\end{align*}"}
+{"id": "355.png", "formula": "\\begin{align*} D q _ k = \\left [ \\begin{array} { c c c c } \\ 1 _ { n - 1 } ' u _ k + 2 \\ 1 _ { n - 1 } ' v _ k \\\\ 2 J _ { n - 1 } u _ k - 2 I _ { n - 1 } u _ k + S v _ k \\\\ S ' u _ k + T v _ k \\end{array} \\right ] . \\end{align*}"}
+{"id": "1482.png", "formula": "\\begin{align*} v _ { 1 } ( x ) = \\frac { L _ { 1 } } { L _ { 2 } } v _ { 2 } \\left ( \\frac { L _ { 2 } x } { L _ { 1 } } \\right ) \\forall x \\in ( 0 , L _ { 1 } ) , \\end{align*}"}
+{"id": "4955.png", "formula": "\\begin{align*} \\rho _ i = \\pi _ { i + 1 } ( X _ 1 , Y _ 1 , \\dots , X _ { i - 1 } , Y _ { i - 1 } ) , 1 \\le i \\le n - 1 . \\end{align*}"}
+{"id": "5498.png", "formula": "\\begin{align*} L ( p ) = p \\cdot ( B _ 1 + \\dots + B _ 4 ) , R ( p ) = ( e ^ { 2 / 1 7 } 2 ^ { 3 / 2 } 1 . 7 ^ { - 1 / 2 } ) ^ p \\Gamma \\left ( \\frac { p } { 2 } + 1 \\right ) . \\end{align*}"}
+{"id": "9047.png", "formula": "\\begin{align*} u = \\left ( \\begin{array} { c c } 0 & 1 \\\\ - 1 & 0 \\end{array} \\right ) , \\end{align*}"}
+{"id": "8797.png", "formula": "\\begin{align*} \\frac { d } { d t } S _ X ( t , s ) v & = L _ { X _ t } S _ X ( t , s ) v \\\\ S _ X ( s , s ) v & = v . \\end{align*}"}
+{"id": "8465.png", "formula": "\\begin{align*} Q _ { j , \\pm } ( x ; k ) = \\mathcal { C } \\left ( Q _ { j , - } ( x ; k ) J + D _ j \\right ) ( z ) , z \\in \\mathbb { C } ^ { \\pm } . \\end{align*}"}
+{"id": "7993.png", "formula": "\\begin{align*} \\frac { 1 } { f ( x ) } \\sigma _ { k } ( x ) \\varphi ( h ) G ( \\nabla h ) = 1 \\end{align*}"}
+{"id": "1680.png", "formula": "\\begin{align*} \\Pi ( \\tau _ 1 , \\tau _ 2 ) \\ ! = \\ ! \\begin{bmatrix} \\frac { U ( 0 ) } { h ^ 2 } & \\frac { 1 } { 2 h } U ^ \\top \\ ! ( h + \\tau _ 1 ) A _ d & \\frac { 1 } { 2 h } U ^ \\top \\ ! ( h + \\tau _ 2 ) A _ d \\\\ \\ast & \\frac { 1 } { 2 h } I _ m & \\frac { 1 } { 2 } A _ d ^ { \\top } U ( \\tau _ 1 - \\tau _ 2 ) A _ d \\\\ \\ast & \\ast & \\frac { 1 } { 2 h } I _ m \\end{bmatrix} \\ ! , \\end{align*}"}
+{"id": "6855.png", "formula": "\\begin{align*} d _ { n + 1 } ( x ) = d _ { n } ( x ) + a _ { n + 1 } ^ { \\prime } ( x ) - a _ { n } ^ { \\prime } ( x ) - \\frac { 1 } { 2 } \\left ( a _ { n + 1 } ( x ) + a _ { n } ( x ) \\right ) , n = 0 , 1 , \\ldots \\end{align*}"}
+{"id": "2028.png", "formula": "\\begin{align*} d X _ t ^ i = \\frac { 1 } { N } \\sum \\limits _ { j = 1 } ^ N \\hat b ( t , X _ t ^ i , X _ t ^ j , u _ t ^ i ) d t + \\sigma d W _ t , \\ , \\ , i = 1 , \\cdots , N , \\end{align*}"}
+{"id": "8000.png", "formula": "\\begin{align*} \\rho _ L ( \\theta ) : = \\sum _ { n \\in \\Z } \\rho ( L ( \\theta + n ) ) L = L ( x ) \\geq 1 , \\end{align*}"}
+{"id": "3994.png", "formula": "\\begin{align*} \\bar { \\mathbf { a } } + \\bar { \\mathbf { c } } = ( \\bar { \\alpha } _ 1 + \\bar { \\gamma } _ 1 , \\bar { \\alpha } _ 2 + \\bar { \\gamma } _ 2 , \\bar { \\alpha } _ 3 + \\bar { \\gamma } _ 3 , \\bar { \\alpha } _ 4 + c _ 1 , a _ 1 + c _ 2 , \\cdots , a _ { 2 k - 4 } + c _ { 2 k - 3 } , a _ { 2 k - 3 } + c _ { 2 k - 2 } , a _ { 2 k - 2 } + \\bar { \\gamma } _ 4 , a _ { 2 k - 1 } + c _ { 2 k - 1 } ) . \\end{align*}"}
+{"id": "1974.png", "formula": "\\begin{align*} F ( x + y F ( x ) ) = F ( x ) F ( y ) . \\end{align*}"}
+{"id": "8374.png", "formula": "\\begin{align*} \\oint s _ { 1 } ( z ) s _ 2 ( z ) d z = 0 . \\end{align*}"}
+{"id": "1459.png", "formula": "\\begin{align*} \\left ( v ( a ) , v ' ( a ) , v ( b ) , v ' ( b ) \\strut \\right ) = \\left ( u ( a ) , u ' ( a ) , u ( b ) , u ' ( b ) \\strut \\right ) . \\end{align*}"}
+{"id": "8912.png", "formula": "\\begin{align*} h _ 1 & = \\lambda _ 1 + \\lambda _ { 1 2 } + \\lambda _ { 1 2 3 ' } , \\\\ h _ 2 & = \\lambda _ 2 + \\lambda _ { 1 2 } + \\lambda _ { 1 2 3 ' } , \\\\ h _ 3 & = \\lambda _ 3 + \\lambda _ { 1 2 3 ' } , \\\\ h _ { 1 2 } & = \\lambda _ 1 + \\lambda _ 2 + \\lambda _ { 1 2 } + 2 \\lambda _ { 1 2 3 ' } , \\\\ h _ { 1 3 } & = \\lambda _ 1 + \\lambda _ 3 + \\lambda _ { 1 2 } + 2 \\lambda _ { 1 2 3 ' } , \\\\ h _ { 2 3 } & = \\lambda _ 2 + \\lambda _ 3 + \\lambda _ { 1 2 } + 2 \\lambda _ { 1 2 3 ' } , \\\\ h _ { 1 2 3 } & = \\lambda _ 1 + \\lambda _ 2 + \\lambda _ 3 + \\lambda _ { 1 2 } + 2 \\lambda _ { 1 2 3 ' } . \\end{align*}"}
+{"id": "8825.png", "formula": "\\begin{align*} B ( v _ 1 , v _ 2 ) + B ( v _ 2 , v _ 1 ) & = \\frac { k } { \\ell } ( - \\alpha \\sin \\ell x _ 1 + \\beta \\cos \\ell x _ 1 ) \\cos k x _ 2 - \\frac { \\ell } { k } \\cos k x _ 2 ( - \\alpha \\sin \\ell x _ 1 + \\beta \\cos \\ell x _ 1 ) \\\\ & = \\left ( \\frac { k } { \\ell } - \\frac { \\ell } { k } \\right ) ( - \\alpha \\sin \\ell x _ 1 + \\beta \\cos \\ell x _ 1 ) \\cos k x _ 2 . \\end{align*}"}
+{"id": "3563.png", "formula": "\\begin{align*} \\varphi ( q ^ { 1 / 2 } ) ^ 2 G ( q ^ { 1 / 2 } ) + q ^ { 1 / 2 } \\varphi ( q ^ 2 ) ^ 2 H ( q ^ 2 ) = \\dfrac { 1 } { 2 } ( \\varphi ( - q ^ { 1 / 2 } ) ^ 2 G ( - q ^ { 1 / 2 } ) + \\varphi ( q ^ { 1 / 2 } ) ^ 2 G ( q ^ { 1 / 2 } ) ) \\end{align*}"}
+{"id": "6442.png", "formula": "\\begin{align*} \\begin{aligned} Z ^ { \\star } _ { I } ( 1 , \\{ \\{ 1 \\} ^ { m - 1 } , 2 \\} ^ { n } ; ( \\alpha , \\beta ) ) = Z ( m n | n + 1 ; ( \\alpha , \\beta ) ) + m Z ( m n + 1 | n ; ( \\alpha , \\beta ) ) \\end{aligned} \\end{align*}"}
+{"id": "2992.png", "formula": "\\begin{align*} \\bigvee _ { n = 0 } ^ { N - 1 } \\widetilde \\Gamma _ { p } ^ n \\bigvee _ { ( i , j ) \\in \\Lambda _ N ^ { \\vec { v } } ( 1 ) } T ^ { - ( i , j ) } \\{ A , A ^ c \\} \\times [ 0 , 1 ) . \\end{align*}"}
+{"id": "5902.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Vert X _ n ( t , \\omega ) - X ( t , \\omega ) \\Vert _ H = 0 , \\forall \\ , ( t , \\omega ) \\in \\Gamma , \\end{align*}"}
+{"id": "1467.png", "formula": "\\begin{align*} \\left ( w ( a ) , w ' ( a ) , w ( b ) , w ' ( b ) \\strut \\right ) = ( A _ { 0 } , A _ { 1 } , B _ { 0 } , B _ { 1 } ) , \\end{align*}"}
+{"id": "3940.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\epsilon } ( x , y ) : = \\int _ { \\mathbb { R } } \\rho _ { \\epsilon } ( y - z ) \\mathcal { N } ( x , z ) \\ , d z , \\end{align*}"}
+{"id": "7298.png", "formula": "\\begin{align*} { G _ { 0 , j } ^ n } = { 1 0 ^ { - \\frac { { { { L \\left ( d _ { 0 , j } \\right ) + \\Psi + \\Gamma + { X _ { \\sigma } } } } \\left [ { { \\rm { d B } } } \\right ] } } { { 1 0 } } } } , \\end{align*}"}
+{"id": "3267.png", "formula": "\\begin{align*} \\tilde { \\Delta } _ i ^ n A : = \\int _ { ( i - 1 ) \\Delta _ n } ^ { i \\Delta _ n } \\mathcal S ( i \\Delta _ n - s ) \\alpha _ s d s , \\\\ \\tilde { \\Delta } _ i ^ n M : = \\int _ { ( i - 1 ) \\Delta _ n } ^ { i \\Delta _ n } \\mathcal S ( i \\Delta _ n - s ) \\sigma _ s d W _ s . \\end{align*}"}
+{"id": "8478.png", "formula": "\\begin{align*} & M ^ { \\pm } _ 2 ( x ; z ) - e _ { 2 } = \\mathcal { P } ^ { \\pm } \\left ( M _ { - } ( x ; \\cdot ) R ( x ; \\cdot ) \\right ) _ { 2 1 } , z \\in \\mathbb { R } , \\\\ & 2 i k ( M ^ { \\pm } _ 2 ( x ; z ) - e _ { 2 } ) = \\mathcal { P } ^ { \\pm } \\left ( 2 i k M _ - ( x ; \\cdot ) R ( x ; \\cdot ) \\right ) _ { 2 1 } , z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "844.png", "formula": "\\begin{align*} \\Delta _ \\Gamma G ' ( c ) + c V H & = A : D _ \\Gamma ^ 2 c + B \\cdot \\nabla _ \\Gamma c + C c \\end{align*}"}
+{"id": "7946.png", "formula": "\\begin{align*} & 0 \\leq ( 1 - ( z _ 1 ) ) ( 1 - ( z _ 2 ) ) = 1 - ( z _ 1 ) - ( z _ 2 ) + ( z _ 1 ) ( z _ 2 ) , \\end{align*}"}
+{"id": "6073.png", "formula": "\\begin{align*} \\bar { N } ( x ) = \\{ u \\in G \\mid u = v ( u , v ) \\in E \\} . \\end{align*}"}
+{"id": "5143.png", "formula": "\\begin{align*} - \\Delta _ y \\varphi = \\mu ^ { 2 } \\left ( \\varphi - 1 - \\frac { \\gamma } { r ^ { 2 } } \\right ) _ { + } \\textrm { i n } \\ \\mathbb { R } ^ { 5 } . \\end{align*}"}
+{"id": "4755.png", "formula": "\\begin{align*} & x ^ { * } + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k ( 2 a _ { k } x ^ { * } + b _ { k } ) = z \\ \\ \\\\ & \\gamma _ k f _ k ( x ^ * ) = 0 \\ \\ k = 1 , . . . , m \\ \\ \\\\ & 1 + 2 \\sum \\limits _ { k = 1 } ^ m \\gamma _ k a _ k \\ge 0 , \\ x ^ { * } \\in A \\ \\ \\end{align*}"}
+{"id": "22.png", "formula": "\\begin{align*} s _ i ^ 2 = 1 , ( s _ i s _ j ) ^ 2 = 1 | i - j | > 1 , ( s _ i s _ { i + 1 } ) ^ 3 = 1 i < n - 1 , ( s _ { n - 1 } s _ n ) ^ 4 = 1 . \\end{align*}"}
+{"id": "8210.png", "formula": "\\begin{align*} \\widetilde { T } = \\{ \\{ k _ { i , 1 } , k _ { i , 2 } \\} : 1 \\le i \\le m \\} & \\mbox { s u c h t h a t } k _ { i , 1 } < k _ { i , 2 } , \\mbox { f o r a l l } 1 \\le i \\le m , \\\\ & \\mbox { a n d } k _ { i , 1 } < k _ { i + 1 , 1 } , \\mbox { f o r a l l } 1 \\le i < m . \\end{align*}"}
+{"id": "1850.png", "formula": "\\begin{align*} \\nabla _ { \\frac \\partial { \\partial r } } X = \\frac \\partial { \\partial r } \\ , , \\end{align*}"}
+{"id": "2720.png", "formula": "\\begin{align*} H ( x ) = \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n \\frac { f ( x + \\sigma u _ i + \\sigma u _ j ) - f ( x + \\sigma u _ i ) - f ( x + \\sigma u _ j ) + f ( x ) } { \\sigma ^ 2 } u _ i u _ j ^ \\intercal , \\end{align*}"}
+{"id": "3686.png", "formula": "\\begin{align*} d ( X ( \\bar u ) ) - L _ X g ( \\nabla \\bar u , \\cdot ) & = \\nabla ^ 2 \\bar u ( X , \\cdot ) - \\nabla X ( \\nabla \\bar u , \\cdot ) . \\end{align*}"}
+{"id": "3903.png", "formula": "\\begin{align*} L _ \\delta \\omega _ \\delta = l _ { 1 , \\delta } + l _ { 2 , \\delta } + R _ \\delta ( \\omega _ \\delta ) , \\end{align*}"}
+{"id": "6161.png", "formula": "\\begin{align*} \\left ( - m ^ { - 1 / 4 } ( r ) \\frac { d } { d r } m ^ { - 1 / 2 } ( r ) \\frac { d } { d r } m ^ { - 1 / 4 } ( r ) + V ( r ) - E \\right ) \\psi ( r ) = 0 \\end{align*}"}
+{"id": "1118.png", "formula": "\\begin{align*} { S _ { \\max } } = \\frac { { W I } } { { K L } } { \\widetilde \\varepsilon _ K } \\left ( { \\frac { { P { { \\left | { { h _ s } } \\right | } ^ 2 } } } { { W { N _ 0 } } } } \\right ) , \\end{align*}"}
+{"id": "966.png", "formula": "\\begin{align*} X _ G : = X _ G ( g _ 0 , \\ldots , g _ { N - 1 } ) : = \\left \\{ x \\in X ( K ) \\colon g _ 0 ( x ) , g _ 1 ( x ) , \\dots , g _ { N - 1 } ( x ) \\in G \\right \\} \\end{align*}"}
+{"id": "5069.png", "formula": "\\begin{align*} B = \\sum _ { j } c _ j \\bold { B } _ j , c _ j = \\int _ { \\Omega } B \\cdot \\bold { B } _ { j } \\dd x , A = \\sum _ { j } c _ j \\bold { A } _ { j } . \\end{align*}"}
+{"id": "447.png", "formula": "\\begin{align*} I _ 2 ( x ) = p + q \\int _ { - \\alpha ( x ) } ^ { \\alpha ( x ) } f ( x - y ) f ( y ) / f ( x ) d y \\to 1 , \\end{align*}"}
+{"id": "271.png", "formula": "\\begin{align*} \\nabla _ H \\varphi _ n ( p _ n ) = \\Bigg ( \\frac { x _ \\xi } { n } + 2 n y _ \\xi ( x _ { p _ n } y _ \\xi - y _ { p _ n } x _ \\xi ) - n y _ { p _ n } z _ { p _ n } , \\frac { y _ \\xi } { n } - 2 n x _ \\xi ( x _ { p _ n } y _ \\xi - y _ { p _ n } x _ \\xi ) + n x _ { p _ n } z _ { p _ n } \\Bigg ) . \\end{align*}"}
+{"id": "4498.png", "formula": "\\begin{align*} H ^ { 2 } = 1 , E ^ { 2 } = - 1 , H \\cdot E = 0 , \\end{align*}"}
+{"id": "4564.png", "formula": "\\begin{align*} & \\sqrt { q } \\sigma _ 1 ( \\textbf { z } ) a _ { \\ell + 1 , 0 , 0 } = ( q ^ 3 + q ^ 2 + q ) a _ { \\ell + 1 , 1 , 0 } + a _ { \\ell + 2 , 0 , 0 } \\\\ & { q } \\sigma _ 2 ( \\textbf { z } ) a _ { \\ell , 0 , 0 } = ( q ^ 5 + q ^ 4 + q ^ 3 ) a _ { \\ell , 1 , 1 } + ( q ^ 3 + q ^ 2 + q ) a _ { \\ell + 1 , 1 , 0 } \\\\ & q ^ { 3 / 2 } \\sigma _ 3 ( \\textbf { z } ) a _ { \\ell - 1 , 0 , 0 } = q ^ 2 a _ { \\ell - 2 , 0 , 0 } + ( q ^ 5 + q ^ 4 + q ^ 3 ) a _ { \\ell , 1 , 1 } . \\end{align*}"}
+{"id": "4679.png", "formula": "\\begin{align*} \\psi _ { K , H } ' + \\frac { \\psi _ { K , H } ^ 2 } { n - 1 } - ( n - 1 ) K = 0 , \\psi _ { K , H } ( 0 ) = H . \\end{align*}"}
+{"id": "4475.png", "formula": "\\begin{align*} \\frac { p ^ { 3 } } { 1 6 ( p / 4 ) a _ { 4 } ^ { * } } \\leq \\frac { p ^ { 3 } } { 1 6 a _ { 1 } ^ { * } a _ { 4 } ^ { * } } = a _ { 2 } ^ { * } a _ { 3 } ^ { * } \\leq \\frac { 3 } { 4 } ( a _ { 2 } ^ { * } ) ^ { 2 } . \\end{align*}"}
+{"id": "6039.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } J _ { 1 } ( \\overline { u } _ { k } ) = \\widetilde { E } _ { 1 } , \\ \\ \\ \\lim _ { k \\rightarrow \\infty } J _ { \\omega } \\overline { v } _ { k } = \\widetilde { E } _ { \\omega } . \\end{align*}"}
+{"id": "7145.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { L } _ g \\textbf { \\textit { U } } = 0 & \\ \\Omega , \\\\ \\Lambda _ { g } ( \\textbf { \\textit { U } } ) = \\tau \\textbf { \\textit { U } } & \\ \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "2160.png", "formula": "\\begin{align*} \\Im \\mathcal { P } [ \\varrho ] = \\frac { 1 } { 2 } a ( b ^ 2 - 4 ) | X | ^ 2 \\Re ( X ) + \\frac { 1 } { 2 } b ( a ^ 2 - 4 ) | Y | ^ 2 \\Re ( Y ) \\\\ + \\frac { 5 } { 2 } a ^ 2 b \\Re ( X ^ 2 \\overline { Y } ) + \\frac { 5 } { 2 } a b ^ 2 \\Re ( Y ^ 2 \\overline { X } ) \\\\ - 4 a ( b ^ 2 + 1 ) | X Y | \\Re ( X ) - 4 b ( a ^ 2 + 1 ) | X Y | \\Re ( Y ) \\\\ + \\frac { 1 } { 2 } b ( a ^ 2 + 5 b ^ 2 - 4 ) | X | ^ 2 \\Re ( Y ) + \\frac { 1 } { 2 } a ( b ^ 2 + 5 a ^ 2 - 4 ) | Y | ^ 2 \\Re ( X ) . \\end{align*}"}
+{"id": "1031.png", "formula": "\\begin{align*} \\prescript L { } { } \\ell { } ^ R ( x , - w ^ { - 1 } \\alpha ) = \\prescript L { } { } \\ell { } ^ R ( x , v ( - ( w v ) ^ { - 1 } \\alpha ) ) \\geq 0 \\end{align*}"}
+{"id": "7520.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - L u = f ( u ) & B _ 1 \\\\ u = 0 & \\partial B _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "771.png", "formula": "\\begin{align*} | g ( x , t ) | & = \\left | \\int _ { \\R } \\frac { \\partial _ u ^ M \\varphi ( x , u ) - \\partial _ u ^ M \\varphi ( x , t ) } { | u - t | ^ { \\frac 1 { 2 s } - M + 1 } } d u \\right | = \\left | \\int _ { I _ Q } \\frac { \\partial _ u ^ M \\varphi ( x , u ) } { | u - t | ^ { \\frac 1 { 2 s } - M + 1 } } d u \\right | \\approx \\left | \\int _ { I _ Q } \\frac { \\partial _ u \\varphi ( x , u ) } { | u - t | ^ { \\frac 1 { 2 s } } } d u \\right | . \\end{align*}"}
+{"id": "4160.png", "formula": "\\begin{align*} \\beta _ 1 ( u ) = \\beta _ 2 ( u ) = \\beta _ 3 ( u ) = 0 \\Longrightarrow \\beta _ 3 ( u _ { \\varepsilon } ) = 0 . \\end{align*}"}
+{"id": "6191.png", "formula": "\\begin{align*} V ( r ) = \\frac { L ( L + 1 ) } { r ^ 2 } - \\frac { Q } { r } f + \\kappa Q \\frac { r } { f } + \\kappa \\frac { L ( L + 1 ) } { f ^ 2 } , \\end{align*}"}
+{"id": "9179.png", "formula": "\\begin{align*} & | \\epsilon ( x _ a , t ) | \\le \\bar \\epsilon ( L _ r , M _ r , \\delta ) : = 2 L _ r \\delta + M _ r \\ , . \\end{align*}"}
+{"id": "1836.png", "formula": "\\begin{align*} F ^ { \\prime } ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) \\delta ^ \\nabla R ^ \\nabla - i _ { \\big ( F ^ { \\prime } ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) \\big ) } R ^ \\nabla = 0 \\end{align*}"}
+{"id": "867.png", "formula": "\\begin{align*} x = - y ' \\end{align*}"}
+{"id": "3172.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\frac { q - 1 } { 4 } } k _ 3 ( \\langle H \\rangle , x _ i ) + \\sum _ { i = 1 } ^ { \\frac { q - 1 } { 4 } } k _ 3 ( \\langle H \\rangle , y _ i ) = 3 \\times k _ 3 ( \\langle H \\rangle ) . \\end{align*}"}
+{"id": "9330.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m U _ i < \\sum _ { i = 1 } ^ m V _ i , \\end{align*}"}
+{"id": "1662.png", "formula": "\\begin{align*} \\phi _ x ( t _ v ) = \\phi _ z ( t _ v ) \\end{align*}"}
+{"id": "6126.png", "formula": "\\begin{align*} n ( n + \\alpha + \\beta + 1 ) \\ , r _ n ( m ) = B _ m \\ r _ { n } ( m + 1 ) - ( B _ m + D _ m ) \\ , r _ n ( m ) + D _ m \\ r _ n ( m - 1 ) \\ , , \\end{align*}"}
+{"id": "2751.png", "formula": "\\begin{align*} h _ { 2 n } = f ^ n h _ { 2 n + 1 } = H \\circ f ^ n . \\end{align*}"}
+{"id": "9120.png", "formula": "\\begin{align*} \\frac { \\sum \\limits _ { i = 1 } ^ { j + 1 } \\chi _ i - r ( j + 1 ) } { \\chi } < \\sum \\limits _ { i = 1 } ^ { j + 1 } w _ i < \\frac { \\sum \\limits _ { i = 1 } ^ { j + 1 } \\chi _ i - r j } { \\chi } . \\end{align*}"}
+{"id": "4013.png", "formula": "\\begin{align*} E ^ L _ Y ( u ) \\subset E ^ L _ Y ( s \\sigma ( v ) w ) = \\varphi ^ L _ { \\sigma , s } ( E ^ L _ X ( v ) ) \\end{align*}"}
+{"id": "2012.png", "formula": "\\begin{align*} & k _ \\lambda ( M ) = k _ \\lambda ( M / K ) + k _ \\lambda ( K ' ) \\mbox { i f } \\nu \\neq \\lambda \\\\ & k _ \\lambda ( M ) = k _ \\lambda ( M / K ) + k _ \\lambda ( K ' ) + 1 \\mbox { i f } \\nu = \\lambda \\end{align*}"}
+{"id": "8085.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\mathbf { s } ^ { \\left ( \\right ) ^ H } \\mathbf { y } ' \\right ] = a _ c \\sum _ { j = 1 } ^ { M } \\mathbf { h } _ { j , * } \\mathbf { p } _ c + \\sum _ { l = 1 } ^ M a _ l \\mathbf { h } _ { l , * } \\mathbf { p } _ l . \\end{align*}"}
+{"id": "6234.png", "formula": "\\begin{align*} D _ { n + 1 } ( x ) & = \\sum _ { \\lambda _ { n } \\in \\Lambda _ { n } } \\sum _ { l \\in L } e ^ { 2 \\pi i \\tau ( R ^ t \\lambda _ { n } + l ) \\cdot x } \\\\ & = \\sum _ { l \\in L } e ^ { 2 \\pi i ( \\tau l ) \\cdot x } \\sum _ { \\lambda _ { n } \\in \\Lambda _ { n } } e ^ { 2 \\pi i \\tau ( R ^ t \\lambda _ { n } ) \\cdot x } \\\\ & = m _ \\tau ( x ) D _ n ( R ^ t x ) . \\end{align*}"}
+{"id": "2476.png", "formula": "\\begin{align*} H = \\displaystyle \\sum _ { i = 1 } ^ { D N } \\left ( \\frac { p _ { i } ^ { 2 } } { 2 m } + \\frac { 1 } { 2 } m \\omega ^ { 2 } x _ { i } ^ { 2 } \\right ) , \\end{align*}"}
+{"id": "9.png", "formula": "\\begin{align*} \\alpha \\wedge \\beta : = \\{ \\gamma \\in R ^ + \\mid \\gamma = \\alpha + \\beta - \\delta \\delta \\in \\alpha \\vee \\beta \\} \\end{align*}"}
+{"id": "8252.png", "formula": "\\begin{align*} \\frac { d \\mu _ g ^ n } { d t } & = \\sum _ { i = 1 } ^ { n - 1 } \\sum _ { j = 1 } ^ { i } ( g _ { i + 1 } - g _ i ) j V _ { i , j } \\psi _ i \\psi _ j - \\sum _ { i = 1 } ^ { n - 1 } \\sum _ { j = i } ^ { n - 1 } g _ { i } V _ { i , j } \\psi _ i \\psi _ j , \\end{align*}"}
+{"id": "1337.png", "formula": "\\begin{align*} t _ { k } = \\max \\bigl \\{ t \\in \\bigcup _ { j \\in J } T _ j \\colon t > t _ { k - 1 } \\bigr \\} . \\end{align*}"}
+{"id": "7254.png", "formula": "\\begin{align*} P ^ g ( x _ 1 , \\ldots , x _ n ) = P ( g x _ 1 , \\ldots , g x _ n ) . \\end{align*}"}
+{"id": "5249.png", "formula": "\\begin{align*} \\| x \\| = \\sup _ { p \\in \\Gamma _ { \\tau } } p ( x ) , x \\in X ; \\end{align*}"}
+{"id": "6385.png", "formula": "\\begin{align*} { { P } } _ k ^ { \\mathrm { d i s c } } ( \\mathcal { T } ) : = \\left \\{ q _ h \\in L ^ 2 ( \\Omega ) : q _ h | _ T \\in P _ k ( T ) T \\in \\mathcal { T } \\right \\} . \\end{align*}"}
+{"id": "261.png", "formula": "\\begin{align*} ( x _ p , y _ p , z _ p ) \\cdot ( x _ q , y _ q , z _ q ) = \\left ( x _ p + x _ q , y _ p + y _ q , z _ p + z _ q + \\frac { 1 } { 2 } ( x _ p y _ q - x _ q y _ p ) \\right ) , \\end{align*}"}
+{"id": "4605.png", "formula": "\\begin{align*} f _ 1 ( x _ 1 , x _ 2 ) = x _ 2 g _ 1 ( x _ 1 , x _ 2 ) . \\end{align*}"}
+{"id": "6076.png", "formula": "\\begin{align*} \\Gamma ( G \\times H ) = \\Gamma ( G ) \\boxtimes \\Gamma ( H ) . \\end{align*}"}
+{"id": "3605.png", "formula": "\\begin{align*} \\sigma _ W ^ 2 = 3 2 / \\alpha ^ 2 \\sigma _ Z ^ 2 = 3 2 T ^ 2 / \\alpha ^ 2 \\end{align*}"}
+{"id": "2408.png", "formula": "\\begin{align*} ( I _ { n } + p ^ { i + \\varepsilon } a ) ^ { - 1 } = I _ { n } + \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ { k } p ^ { ( i + \\varepsilon ) k } a ^ { k } \\equiv I _ { n } - p ^ { i + \\varepsilon } a \\bmod p ^ { i + 1 + \\varepsilon } . \\end{align*}"}
+{"id": "7285.png", "formula": "\\begin{align*} [ g ' k g h , \\psi k \\varphi h ] = [ g ' g \\underbrace { c _ g ( k ) h } _ { \\in H } , \\psi k \\varphi h ] = [ g ' g , \\psi k \\varphi h ( c _ { g } ( k ) h ) ^ { - 1 } ] = [ g ' g , \\psi k \\varphi c _ { g ^ { } } ( k ^ { - 1 } ) ] = [ g ' g , \\psi \\varphi ] \\end{align*}"}
+{"id": "1880.png", "formula": "\\begin{align*} \\begin{aligned} { F _ { \\Phi _ { ( 3 ) } } } _ x ( v ) = \\sum _ { i = 1 } ^ { m } \\big ( 6 \\langle B ( v , e _ { i } ) , B ( v , e _ { i } ) \\rangle _ { \\mathbb R ^ q } - \\langle B ( v , v ) , B ( e _ { i } , e _ { i } ) \\rangle _ { \\mathbb R ^ q } \\big ) < 0 \\end{aligned} \\end{align*}"}
+{"id": "6312.png", "formula": "\\begin{align*} F ' _ { k - 1 } ( x ^ k ) = \\nabla f ( x ^ k ) + \\rho _ { k - 1 } ^ { - 1 } ( x ^ k - x ^ { k - 1 } ) + P ' ( x ^ k ) \\in \\partial F _ { k - 1 } ( x ^ k ) , \\| F ' _ { k - 1 } ( x ^ k ) \\| \\leq \\eta _ { k - 1 } . \\end{align*}"}
+{"id": "6633.png", "formula": "\\begin{align*} J ( \\varphi ) = \\inf \\left \\{ \\int ^ { 1 } _ { 0 } \\Lambda ^ { * } ( \\phi ( r ) ) d r ; ~ \\varphi ( t ) = \\int ^ { t } _ { 0 } \\phi ( r ) d r \\in \\mathcal { A C } , ~ t \\in [ 0 , 1 ] \\right \\} , \\end{align*}"}
+{"id": "2061.png", "formula": "\\begin{align*} \\forall \\ N \\leq k , M ^ N ( g h ) = \\sum ^ { 2 N } _ { j = 0 } A _ { j , 2 N - j } ( g , h ) , \\end{align*}"}
+{"id": "152.png", "formula": "\\begin{align*} A ( 1 + B ) ^ k - B ( 1 + A ) ^ k = 0 \\end{align*}"}
+{"id": "6203.png", "formula": "\\begin{align*} & L ' = L + 1 , Q ' = Q , B ' _ 1 = B _ 1 , B ' _ 2 = B _ 2 + \\frac { B _ 3 } { \\sqrt { \\kappa B _ 4 } } + 2 , \\\\ & B ' _ 3 = B _ 3 + 4 \\sqrt { \\kappa B _ 4 } , B ' _ 4 = B _ 4 , R = - E _ 0 . \\end{align*}"}
+{"id": "5161.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sup _ { z _ 0 \\in \\mathbb { R } } \\int _ { D ( z _ 0 , R ) } \\rho _ { n _ { k } } \\dd z \\dd r = 0 . \\end{align*}"}
+{"id": "4124.png", "formula": "\\begin{align*} H _ { m i n } ( E \\otimes ( F _ 1 [ t ] \\times F _ 1 [ t ] ^ { \\vee } ) ) & = H _ { m i n } ( E ) H _ { m i n } ( F _ 1 [ t ] \\times F _ 1 [ t ] ^ { \\vee } ) \\\\ & = t H _ { m i n } ( E ) H _ { m i n } ( F _ 1 ) = t ^ { - 1 } H _ { m i n } ( E ) H _ { m i n } ( F _ 1 ^ { \\vee } ) . \\end{align*}"}
+{"id": "4144.png", "formula": "\\begin{align*} \\theta ^ j _ { \\alpha \\beta } ( x ' , t ; y ' ) : = t \\ , \\partial _ j K ^ L _ { \\alpha \\beta } ( x ' - y ' , t ) , x ' , y ' \\in \\R ^ { n - 1 } , \\ , t > 0 . \\end{align*}"}
+{"id": "2240.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\psi = - i H _ N \\psi , t \\in { \\mathbb { R } } , \\\\ \\psi ( 0 ) = \\psi \\in \\ell ^ 2 ( \\mathbb { N } ) . \\end{cases} \\end{align*}"}
+{"id": "4611.png", "formula": "\\begin{align*} \\frac { \\sum _ { j = 1 } ^ n x _ j x _ { j + 1 } } { g } ( x _ { k } { x _ { k + 1 } } ^ 2 + { x _ { k + 1 } } ^ 2 x _ { k + 2 } ) = \\frac { { x _ { k } } ^ 2 { x _ { k + 1 } } ^ 3 + { x _ { k + 1 } } ^ 3 { x _ { k + 2 } } ^ 2 } { g } = \\frac { x _ { k + 1 } } { 1 } . \\end{align*}"}
+{"id": "1671.png", "formula": "\\begin{align*} \\rho _ v = \\begin{pmatrix} \\chi _ 1 & 0 \\\\ 0 & \\chi _ 2 \\end{pmatrix} \\end{align*}"}
+{"id": "5863.png", "formula": "\\begin{align*} \\centering \\mathbf { M } = \\left [ \\begin{array} { c c c c c c c } 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 0 & 1 & - 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & - 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & - 1 \\\\ 0 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 0 & 1 & 1 & - 1 & - 1 & 0 & 0 \\\\ 0 & 1 & 1 & 0 & 0 & - 1 & - 1 \\end{array} \\right ] . \\end{align*}"}
+{"id": "3265.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { m } \\langle \\tilde { \\Delta } _ { i _ k } ^ n Y , e _ { j _ k } \\rangle = & \\prod _ { k = 1 } ^ { m } \\left ( \\langle \\Delta _ { i _ k } ^ n S , e _ { j _ k } \\rangle + \\Delta _ n ^ { ( i _ k - 1 ) \\Delta _ n } a ^ { j _ k } \\right ) \\\\ = & \\prod _ { k = 1 } ^ { m } \\langle \\Delta _ { i _ k } ^ n S , e _ { j _ k } \\rangle + \\sum _ { y = 1 , x = 2 } ^ m b _ { x y } ( i _ 1 , . . . , i _ m , j _ 1 , . . . , j _ m ) , \\end{align*}"}
+{"id": "5568.png", "formula": "\\begin{align*} A ( y _ { l + 1 } . . . y _ x ) - A ( y _ { l + 1 } . . . y _ 1 x ' ) = A ( 1 0 ^ { l + k } 1 . . . ) - A ( 1 0 ^ \\infty ) = d _ { l + k } - d \\ . \\end{align*}"}
+{"id": "184.png", "formula": "\\begin{align*} R S _ { H ' } ( F ) = S _ { H ' } ( F ) = \\varinjlim _ n \\varprojlim _ k A _ { H ' , n } \\widehat { \\otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V _ k ) . \\end{align*}"}
+{"id": "2493.png", "formula": "\\begin{align*} \\Theta = \\left ( { \\theta } ^ { 1 } \\mathbf { I } _ { n _ 1 } , \\cdots , { \\theta } ^ { k } \\mathbf { I } _ { n _ k } \\right ) . \\end{align*}"}
+{"id": "412.png", "formula": "\\begin{align*} \\begin{aligned} d ( x _ 1 , x _ 2 ) & \\leq d ( x _ 1 , \\bar x _ 2 ) + d ( \\bar x _ 2 , x _ 2 ) \\\\ & \\leq d ( x _ 1 , \\bar x _ 2 ) + L d ( f ( \\bar x _ 2 ) , f ( x _ 2 ) ) \\\\ & = d ( x _ 1 , \\bar x _ 2 ) + L d ( f ( \\bar x _ 2 ) , f ( x _ 1 ) ) \\\\ & \\leq ( 1 + L ^ 2 ) d ( x _ 1 , \\bar x _ 2 ) , \\end{aligned} \\end{align*}"}
+{"id": "7664.png", "formula": "\\begin{align*} \\begin{aligned} & \\bar y _ t ^ { * , t _ 0 , \\xi } : = \\bar U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) , \\bar z _ t ^ { * , t _ 0 , \\xi } : = \\sigma \\partial _ x \\bar U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) , \\\\ & \\bar z _ t ^ { 0 , * , t _ 0 , \\xi } : = \\sigma _ 0 [ \\partial _ x \\bar U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) + \\partial _ \\nu U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\bar \\nu _ t ^ { * , t _ 0 , \\xi } ) ] . \\end{aligned} \\end{align*}"}
+{"id": "4457.png", "formula": "\\begin{align*} \\psi ( x ) = - \\frac { 6 } { \\pi } x K _ { 1 } ( 4 \\pi x ) + 2 4 x ^ { 2 } K _ { 0 } ( 4 \\pi x ) , \\end{align*}"}
+{"id": "5309.png", "formula": "\\begin{align*} f ( x ) = c \\ , \\int _ { - \\infty } ^ \\infty \\int _ { K / M } \\widetilde { f } ( \\lambda , b ) e ^ { ( i \\lambda + \\rho ) A ( x , b ) } | c ( \\lambda ) | ^ { - 2 } d b \\ , d \\lambda \\end{align*}"}
+{"id": "2807.png", "formula": "\\begin{align*} \\tilde { h } _ { x } ^ { V } : = \\sum _ { n = 1 } ^ { | V | } \\varphi _ { n } ( x ) Z _ { n } , x \\in \\mathbb { Z } ^ { d } \\end{align*}"}
+{"id": "5282.png", "formula": "\\begin{align*} \\phi ( h g ) - \\phi ( g ) = \\phi ( n _ { u _ 1 + t _ 1 ^ 2 u _ 2 + t _ 1 ^ 2 t _ 2 ^ 2 u _ 1 ' } a _ { t _ 1 t _ 2 t _ 1 ' } ) - \\phi ( n _ { u _ 2 } a _ { t _ 2 } ) = O \\left ( \\delta t _ 2 ^ { O ( 1 ) } \\right ) . \\end{align*}"}
+{"id": "862.png", "formula": "\\begin{align*} \\beta ( x , y ) = \\beta ( x ' , y ' ) \\ , . \\end{align*}"}
+{"id": "5311.png", "formula": "\\begin{align*} \\widehat { f } ( a , z ) = \\rho _ k ^ \\lambda ( f ) e _ a ( z , 0 ) , \\ , \\ , \\ , \\rho _ k ^ \\lambda ( f ) = \\int _ { G _ n } f ( g ) \\rho _ k ^ \\lambda ( g ) d g . \\end{align*}"}
+{"id": "593.png", "formula": "\\begin{align*} \\Theta ( \\cdot , \\mu , \\tilde { \\mu } ) = [ \\null _ { 2 } \\eta ] _ { p } \\vartheta ( \\mu , \\tilde { \\mu } ) + [ \\null _ { 2 } \\hat { \\eta } ] _ { p } ^ { 2 } \\vartheta ( \\mu , \\tilde { \\mu } ) ^ { 2 } \\end{align*}"}
+{"id": "5296.png", "formula": "\\begin{align*} \\Psi ( v _ i ) v _ i ^ * = \\Psi ( v _ i ) v _ i ^ * p & = \\Psi ( v _ i ) v _ i ^ * ( v ' _ 1 { v ' _ 1 } ^ * + \\cdots + v ' _ n { v ' _ n } ^ * ) \\\\ & = \\Psi ( v _ i ) ( v _ i ^ * v ' _ 1 ) { v ' _ 1 } ^ * + \\cdots + \\Psi ( v _ i ) ( v _ i ^ * v ' _ n ) { v ' _ n } ^ * \\\\ & = \\Psi ( v _ i v _ i ^ * v ' _ 1 ) { v ' _ 1 } ^ * + \\cdots + \\Psi ( v _ i v _ i ^ * v ' _ n ) { v ' _ n } ^ * , \\end{align*}"}
+{"id": "5691.png", "formula": "\\begin{align*} 0 \\in ( F _ k + B ) ( x ) = ( F + B ) ( x ) + \\frac { 1 } { \\rho _ k } ( x - z ^ k ) . \\end{align*}"}
+{"id": "5417.png", "formula": "\\begin{align*} \\xi _ 2 = \\kappa _ 0 \\xi _ 1 ^ 2 + O ( | \\xi _ 1 | ^ 3 ) \\mbox { a s } \\xi _ 1 \\rightarrow 0 \\end{align*}"}
+{"id": "5229.png", "formula": "\\begin{align*} - \\Delta _ { y } \\varphi = \\mu ^ { 2 } \\left ( \\varphi - \\frac { W } { 2 } \\right ) _ { + } \\textrm { i n } \\ \\mathbb { R } ^ { 5 } . \\end{align*}"}
+{"id": "1431.png", "formula": "\\begin{align*} I ( m , a ) = \\frac { 1 } { m ! } \\int _ 0 ^ { \\infty } \\ldots \\int _ 0 ^ { \\infty } e ^ { - t _ 1 - \\ldots - t _ { m } } \\prod _ { j = 1 } ^ { m } t _ j ^ { a } \\Delta ( t _ 1 , \\ldots , t _ { m } ) ^ 2 d t _ 1 \\ldots d t _ { m } . \\end{align*}"}
+{"id": "1036.png", "formula": "\\begin{align*} y _ 1 ' = ( x _ 2 ' ) ^ { - 1 } , y _ 2 ' = ( x _ 1 ' ) ^ { - 1 } , y _ \\ast = x _ \\ast ^ { - 1 } . \\end{align*}"}
+{"id": "1790.png", "formula": "\\begin{align*} f ( q ) : = \\frac { 1 } { 3 } b ^ { 2 } ( q ) c ( q ^ { 3 } ) , \\end{align*}"}
+{"id": "4298.png", "formula": "\\begin{align*} Y _ \\infty = \\left ( \\frac { m } { \\textrm { G e V } } \\right ) ^ { - 3 } \\left [ 1 + \\frac { 3 \\ln ( m / \\textrm { G e V } ) } { 1 5 } + \\frac { \\ln ( c _ 2 / 5 ) } { 1 5 } \\right ] \\end{align*}"}
+{"id": "2756.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { m - 1 } | h _ { j } ( x ) | \\leq | h _ { m } ( x ) | ^ \\delta \\end{align*}"}
+{"id": "660.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | X ( r , \\theta , e ^ { i \\phi } ; a , M ) \\left | l _ 2 , m _ 2 \\right \\rangle = \\end{align*}"}
+{"id": "5219.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\dot { \\rho } ( t ) \\left ( \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { + } ^ { 2 } \\dd x \\right ) \\dd t & = - \\rho ( 0 ) \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } ^ { 2 } \\dd x . \\end{align*}"}
+{"id": "3854.png", "formula": "\\begin{align*} \\partial _ \\tau \\gamma = \\frac { c \\bar { K } } { 4 \\pi } \\mathbf { b } _ { \\gamma ( \\tau ) } . \\end{align*}"}
+{"id": "8411.png", "formula": "\\begin{align*} & \\widehat { \\Psi } ^ \\pm _ { 1 1 } ( x ) : = - \\frac { 1 } { 4 } e ^ { - i c _ \\pm } \\int _ { \\pm \\infty } ^ { x } \\left [ u _ y ( y ) \\bar { u } _ { y y } ( y ) + \\frac { 1 } { 2 i } | u _ y ( y ) | ^ 4 \\right ] d y , \\\\ & \\widehat { \\Psi } ^ \\pm _ { 2 1 } ( x ) = \\frac { 1 } { 2 i } \\partial _ { x } ( \\bar { u } _ { x } ( x ) e ^ { i c _ \\pm ( x ) } ) . \\end{align*}"}
+{"id": "5733.png", "formula": "\\begin{align*} J = \\frac { I _ 1 \\cup I _ 3 ^ { [ 3 ] } } { I _ 1 \\cup I _ 3 ^ { [ 3 ] } } = M \\cup \\frac { I _ 1 } { I _ 1 } \\cup \\frac { I _ 3 ^ { [ 3 ] } } { I _ 1 } . \\end{align*}"}
+{"id": "2609.png", "formula": "\\begin{align*} R \\cdot R ( u , J u , x , J x ; x , J x ) = \\frac { 1 } { 2 } \\ , L ( p , \\bar { \\pi } ^ h ) \\ , Q ^ c ( g , R ) ( u , J u , x , J x ; x , J x ) . \\end{align*}"}
+{"id": "2639.png", "formula": "\\begin{align*} [ x , y ] = x \\cdot y - \\varepsilon ( x , y ) y \\cdot x . \\end{align*}"}
+{"id": "4703.png", "formula": "\\begin{align*} g _ \\mu : = \\arg \\max _ { \\psi \\in \\mathcal { H } } \\left \\{ - \\frac { \\mathrm { d } } { \\mathrm { d } \\gamma } \\mathrm { K L } \\left ( T _ { \\gamma } { \\# } \\mu \\| \\pi \\right ) \\Big | _ { \\gamma = 0 } \\quad \\| \\psi \\| _ { \\mathcal { H } } \\leq 1 \\right \\} . \\end{align*}"}
+{"id": "8122.png", "formula": "\\begin{align*} W _ { 1 } \\left ( \\mu , \\nu \\right ) = \\sup _ { \\left \\Vert f \\right \\Vert _ { L i p } = 1 } \\int f \\left ( d \\mu - d \\nu \\right ) , \\end{align*}"}
+{"id": "682.png", "formula": "\\begin{align*} \\int _ 0 ^ \\pi d \\theta \\sin \\theta \\sin ( n \\theta ) P _ l ( \\cos \\theta ) = \\end{align*}"}
+{"id": "2569.png", "formula": "\\begin{align*} q ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { 2 } } - p ^ { n _ { 2 } ^ { \\prime } - n _ { s } ^ { \\prime } } a _ { n _ { s } } q ^ { n _ { s } ^ { \\prime } - n _ { 1 } ^ { \\prime } } = p ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { 1 } } \\frac { N _ { n _ { 2 } } } { N _ { n _ { 1 } } } , \\end{align*}"}
+{"id": "4834.png", "formula": "\\begin{align*} g _ { M ' } ^ 3 ( \\tau , r ) & \\leqslant 4 ( g _ 0 ( r ^ { - 1 } ) ) ^ { - 2 m ' _ 0 } ( M ' \\vee C _ { m ' _ 0 } ) + 2 r \\\\ & + 2 ( C _ { m ' _ 0 } ( g ( r ^ { - 1 } ) ) + 1 ) \\big ( \\ln \\ln ( r ^ { - 1 } \\vee 3 ) \\big ) ^ { - 1 } + r \\exp \\big ( C \\tau \\ln \\ln ( r ^ { - 1 } \\vee 3 ) \\big ) . \\end{align*}"}
+{"id": "8262.png", "formula": "\\begin{align*} \\frac { d } { d t } \\sum _ { i = 1 } ^ { m - 1 } i \\psi ^ { n } _ i ( t ) = \\sum _ { i = 1 } ^ { m - 2 } \\sum _ { j = 1 } ^ { i } j V _ { i , j } \\psi ^ { n } _ { i } \\psi ^ { n } _ { j } - \\sum _ { i = 1 } ^ { m - 1 } \\sum _ { j = i } ^ { n - 1 } i V _ { i , j } \\psi ^ { n } _ i \\psi ^ { n } _ j - ( m - 1 ) \\psi ^ { n } _ { m - 1 } \\sum _ { j = 1 } ^ { m - 1 } j V _ { m - 1 , j } \\psi ^ { n } _ j . \\end{align*}"}
+{"id": "2298.png", "formula": "\\begin{align*} | \\mu | = m _ 1 + \\dots + m _ n , \\end{align*}"}
+{"id": "7229.png", "formula": "\\begin{align*} ( \\alpha + \\beta ) ^ { q ^ i } = \\alpha ^ { q ^ i } + \\beta ^ { q ^ i } \\end{align*}"}
+{"id": "8854.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( \\sup _ { 0 \\le s \\le \\tau } \\int _ 0 ^ \\tau \\mathbf { 1 } _ { s \\le t } \\| S _ { Z } ( t , s ) \\| ^ 2 d t \\right ) ^ { 1 / 2 } \\le e ^ { C _ 0 } \\left ( \\int _ 0 ^ { \\tau } e ^ { ( t - \\tau _ 1 ) \\epsilon | x _ 0 | } d t \\right ) ^ { 1 / 2 } \\le \\frac { e ^ { C _ 0 } } { \\sqrt { \\epsilon } } | x _ 0 | ^ { \\epsilon - 1 / 2 } \\end{aligned} \\end{align*}"}
+{"id": "7634.png", "formula": "\\begin{align*} \\mu _ t ^ { * , \\xi } = \\mathbb E [ \\alpha ^ { * , \\xi } _ t | \\mathcal { F } _ t ^ { W ^ 0 } ] \\quad \\nu _ t ^ { * , \\xi } = \\mathbb E [ x _ t ^ { \\xi , \\alpha ^ { * , \\xi } } | \\mathcal { F } _ t ^ { W ^ 0 } ] . \\end{align*}"}
+{"id": "7159.png", "formula": "\\begin{align*} R _ { i j } = g ^ { k l } R _ { i k l j } , \\end{align*}"}
+{"id": "8399.png", "formula": "\\begin{align*} \\lim _ { | z | \\rightarrow \\infty } \\Psi ^ - _ { 1 1 } ( x ; z ) = 1 + \\frac { 1 } { 2 i } \\int _ { - \\infty } ^ { x } | u _ y ( y ) | ^ 2 \\lim _ { | z | \\rightarrow \\infty } \\Psi ^ - _ { 1 1 } ( y ; z ) \\mathbf { d } y , \\end{align*}"}
+{"id": "4412.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\left | \\frac { G ( e ^ { \\tau } , \\delta ) } { e ^ { \\sigma \\tau } } \\right | ^ 2 d \\tau = \\int _ 1 ^ \\infty \\left | \\frac { G ( y , \\delta ) } { y ^ { \\sigma + \\frac { 1 } { 2 } } } \\right | ^ 2 d y & = \\int _ 1 ^ \\infty \\left | \\frac { G ( y , \\delta ) } { y ^ { 1 + \\frac { \\alpha } { \\log ( 1 / \\delta ) } } } \\right | ^ 2 d y > \\frac { 1 } { e ^ { \\alpha b } } \\int _ 1 ^ { 1 / \\delta ^ b } \\left | \\frac { G ( y , \\delta ) } { y } \\right | ^ 2 d y \\end{align*}"}
+{"id": "8608.png", "formula": "\\begin{align*} \\left ( \\frac { \\left | \\sum _ { i = 1 } ^ n [ 0 , z _ i ] \\right | } { \\left | \\sum _ { i = 1 } ^ { n - 1 } [ 0 , z _ i ] \\right | } \\right ) ^ 2 \\ge \\left ( \\frac { \\left | \\sum _ { i = 1 } ^ n [ 0 , P _ E z _ i ] \\right | } { \\left | \\sum _ { i = 1 } ^ { n - 1 } [ 0 , P _ E z _ i ] \\right | } \\right ) ^ 2 + \\left ( \\frac { \\left | \\sum _ { i = 1 } ^ n [ 0 , P _ { E ^ \\bot } z _ i ] \\right | } { \\left | \\sum _ { i = 1 } ^ { n - 1 } [ 0 , P _ { E ^ \\bot } z _ i ] \\right | } \\right ) ^ 2 . \\end{align*}"}
+{"id": "3621.png", "formula": "\\begin{align*} \\sum _ i \\sigma _ { n - 1 } ^ { i i } ( \\nu ^ { n + 1 } ) _ { i i } = & \\ 2 \\sum _ i \\sigma _ { n - 1 } ^ { i i } \\frac { u _ i } { u } ( \\nu ^ { n + 1 } ) _ i + ( n - 1 ) \\sigma ( 1 + ( \\nu ^ { n + 1 } ) ^ 2 ) \\\\ & - \\nu ^ { n + 1 } \\left ( \\sum _ i \\sigma _ { n - 1 } ^ { i i } + \\sum _ i \\sigma _ { n - 1 } ^ { i i } \\kappa _ i ^ 2 \\right ) . \\end{align*}"}
+{"id": "761.png", "formula": "\\begin{align*} 0 & \\leq P _ s * ( \\chi _ { B _ 0 } \\mu ) ( \\bar x ) \\lesssim \\int _ { \\bar y \\in B _ 0 } \\frac 1 { | \\bar x - \\bar y | _ p ^ N } \\ , d \\mu ( \\bar y ) \\\\ & \\leq \\int _ { | \\bar x - \\bar y | _ p \\leq 4 d } \\frac 1 { | \\bar x - \\bar y | _ p ^ N } \\ , d \\mu ( \\bar y ) \\lesssim d ^ { 2 s - 1 } = | t - u | ^ { 1 - \\frac 1 { 2 s } } . \\end{align*}"}
+{"id": "2865.png", "formula": "\\begin{align*} & \\sum _ { x , y = 0 } ^ n ( \\delta _ { x , y } - M _ { x , y } ) f _ y f _ x \\ge 2 C _ * \\sum _ { x = 0 } ^ { n - 1 } ( \\nabla f _ x ) ^ 2 \\end{align*}"}
+{"id": "1233.png", "formula": "\\begin{align*} \\forall \\eta _ \\Lambda \\in \\Omega _ \\Lambda : \\sum _ { \\Delta \\subset \\Lambda } \\sum _ { \\xi _ \\Delta } \\left ( c _ \\Delta ( \\eta _ \\Lambda , \\xi _ \\Delta ) - \\frac { \\mu ( \\xi _ \\Delta \\eta _ { \\Lambda \\setminus \\Delta } ) } { \\mu ( \\eta _ \\Lambda ) } c _ \\Delta ( \\xi _ \\Delta \\eta _ { \\Lambda \\setminus \\Delta } , \\eta _ \\Delta ) \\right ) = 0 . \\end{align*}"}
+{"id": "618.png", "formula": "\\begin{align*} \\omega _ { S ^ 1 } = ( x - i y ) ( d x + i d y ) = ( x - i y ) d x + ( y + i x ) d y . \\end{align*}"}
+{"id": "4145.png", "formula": "\\begin{align*} | \\theta ^ j _ { \\alpha \\beta } ( x ' , t ; y ' ) | & = t | \\partial _ j K ^ L _ { \\alpha \\beta } ( x ' - y ' , t ) | \\lesssim \\frac { t } { | ( x ' - y ' , t ) | ^ n } , \\\\ | \\nabla _ { y ' } \\theta ^ j _ { \\alpha \\beta } ( x ' , t ; y ' ) | & \\le t | \\nabla ^ 2 K ^ L _ { \\alpha \\beta } ( x ' - y ' , t ) | \\lesssim \\frac { t } { | ( x ' - y ' , t ) | ^ { n + 1 } } . \\end{align*}"}
+{"id": "4023.png", "formula": "\\begin{align*} \\varphi ^ L _ { \\sigma , s } ( x ) = a = \\varphi ^ L _ { \\sigma , s } ( y ) \\end{align*}"}
+{"id": "3031.png", "formula": "\\begin{align*} d \\varphi _ { r i } \\left ( i \\right ) = \\left . \\frac { d } { d t } \\right \\vert _ { r } \\left ( l \\left ( t \\right ) i , e ^ { \\theta \\left ( t \\right ) i } \\right ) = \\left ( l ^ { \\prime } \\left ( r \\right ) i , \\theta ^ { \\prime } \\left ( r \\right ) i e ^ { \\theta \\left ( r \\right ) i } \\right ) \\end{align*}"}
+{"id": "3130.png", "formula": "\\begin{align*} \\alpha ( e _ 1 ) = e _ 1 + a _ 4 e _ 4 , \\alpha ( e _ 2 ) = - e _ 2 + b _ 3 e _ 3 , \\alpha ( e _ 3 ) = - e _ 3 , \\alpha ( e _ 4 ) = - e _ 4 , \\end{align*}"}
+{"id": "9195.png", "formula": "\\begin{align*} \\begin{aligned} F ( \\eta , t ) & : = \\left ( \\begin{array} { c } - \\varepsilon _ \\delta ( { x } , \\bar { y } , t ) u ( t ) \\\\ \\varepsilon _ \\delta ( { x } , \\bar { y } , t ) \\end{array} \\right ) \\end{aligned} \\end{align*}"}
+{"id": "8557.png", "formula": "\\begin{align*} ( D ^ 0 _ { 0 + } \\ , f ) ( t ) \\ , = \\ , \\frac { d } { d t } \\ , ( I ^ { 1 } _ { 0 + } \\ , f ) ( t ) \\ , = \\ , f ( t ) , \\ t > 0 \\end{align*}"}
+{"id": "4293.png", "formula": "\\begin{align*} \\frac { \\tilde { u } ^ { [ n ] } ( x ) - \\tilde { u } ^ { [ n - 1 ] } ( x ) } { \\Delta t } = \\theta \\left [ \\mathcal { L } ( \\tilde { u } ^ { [ n ] } ) + f ( t _ n , x ) \\right ] + ( 1 - \\theta ) \\left [ \\mathcal { L } ( \\tilde { u } ^ { [ n - 1 ] } ) + f ( t _ { n - 1 } , x ) \\right ] , \\end{align*}"}
+{"id": "7558.png", "formula": "\\begin{align*} \\prescript { } { l } { u } - \\prescript { } { l } { v } _ 1 = Q _ 1 \\prescript { } { l } { f } , \\end{align*}"}
+{"id": "6821.png", "formula": "\\begin{align*} J _ { l , 0 } \\left ( \\rho \\right ) : = \\int _ { \\mathbb { R } } \\frac { \\varphi _ { 0 } \\left ( \\tau \\right ) \\tau ^ { l - 1 } } { \\left ( \\tau - \\rho \\right ) ^ { l + 1 } } d \\tau . \\end{align*}"}
+{"id": "5721.png", "formula": "\\begin{align*} w / v = \\mu ^ { - 1 } ( \\infty ) \\textrm { i s a z e r o o f $ f ' $ } \\iff \\infty \\textrm { i s a z e r o o f } f \\end{align*}"}
+{"id": "4751.png", "formula": "\\begin{align*} & M i n i m i z e _ { x , y \\in \\mathbb { R } ^ { n } } \\ \\ \\alpha _ J ^ T y + 2 b _ J ^ T x \\\\ & s . t . \\ \\ \\alpha _ k ^ T y + 2 b _ k ^ T x + c _ j \\le 0 \\ \\ ( \\forall \\ , k \\in \\{ 1 , . . . , m \\} ) \\\\ & s . t . \\ \\ x _ i ^ 2 - y _ i = 0 , \\ \\ \\forall \\ , i \\end{align*}"}
+{"id": "2006.png", "formula": "\\begin{align*} \\phi _ \\alpha ( e _ \\alpha ) & = f _ \\alpha \\\\ \\phi _ \\alpha ( f _ \\alpha ) & = e _ \\alpha . \\end{align*}"}
+{"id": "869.png", "formula": "\\begin{align*} x x ' + x y ' + y y ' = n \\ , . \\end{align*}"}
+{"id": "7550.png", "formula": "\\begin{align*} \\lambda = \\sum _ { s = 1 } ^ q \\langle \\sum _ { r = 1 } ^ p \\alpha _ { r s } e _ r , \\sum _ { \\mu = 1 } ^ p \\alpha _ { \\mu s } X e _ { \\mu } \\rangle , \\end{align*}"}
+{"id": "3415.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{matrix} A _ k ( \\theta , E ) = A ( \\theta + ( k - 1 ) \\alpha , E ) \\cdots A ( \\theta + \\alpha , E ) A ( \\theta , E ) \\ \\mathrm { f o r } \\ k \\geq 1 , \\\\ A _ 0 ( \\theta , E ) = \\mathrm { I d } , \\\\ A _ k ( \\theta , E ) = ( A _ { - k } ( \\theta + k \\alpha , E ) ) ^ { - 1 } \\ \\mathrm { f o r } \\ k \\leq - 1 . \\end{matrix} \\right . \\end{align*}"}
+{"id": "8155.png", "formula": "\\begin{align*} \\varprojlim _ { n \\in \\Delta } \\mathcal F ( Y _ n ) = \\varprojlim _ { k \\in \\Delta } \\mathcal C ^ k . \\end{align*}"}
+{"id": "2458.png", "formula": "\\begin{align*} F _ { q } ^ { ( k ) } ( s ) = \\frac { 1 } { Q _ { 1 } ( 2 - q ) } \\left [ \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { \\left ( \\frac { 3 - 2 q } { 1 - q } \\right ) _ { n } } \\frac { \\left ( \\pm \\frac { \\sigma _ { \\epsilon } } { 1 - q } \\right ) ^ { n } } { n ! } ( - 1 ) ^ { k } \\frac { \\Gamma ( n + k + 1 ) } { s ^ { n + k + 1 } } \\right ] . \\end{align*}"}
+{"id": "4787.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| T _ n p _ n f - p _ n T f \\| = 0 f \\in X . \\end{align*}"}
+{"id": "7181.png", "formula": "\\begin{align*} ( q _ 1 - b _ 1 ) q _ { - m - 1 } + q _ { - m - 1 } q _ 1 = E _ { - m } , \\end{align*}"}
+{"id": "4874.png", "formula": "\\begin{align*} \\Phi _ k : = \\sup \\{ \\zeta \\in U S C ( \\overline { \\Omega } \\times \\mathbb R ^ m ) : \\zeta | _ { \\Omega \\times \\mathbb R ^ m } F \\star \\mathcal P \\zeta | _ { \\partial \\Omega \\times \\mathbb R ^ m } \\le \\phi _ { \\tau , k } \\} \\end{align*}"}
+{"id": "3684.png", "formula": "\\begin{align*} L _ X g ( \\nu , \\cdot ) = h ( \\nu , \\cdot ) \\mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \\Sigma . \\end{align*}"}
+{"id": "110.png", "formula": "\\begin{align*} M ( u ) ( t ) & = \\int _ { \\R ^ 2 } u ( x , y ) ^ 2 d x d y , \\\\ E _ { \\alpha } ( u ) ( t ) & = \\int _ { \\R ^ 2 } \\Big ( \\frac { 1 } { 2 } | D _ x ^ { \\frac { \\alpha } { 2 } } u | ^ 2 + \\frac { 1 } { 2 } | \\partial _ x ^ { - 1 } \\partial _ y u | ^ 2 + \\frac { 1 } { 6 } u ^ 3 \\Big ) d x d y . \\end{align*}"}
+{"id": "6192.png", "formula": "\\begin{align*} & W ( r ) = - \\frac { L + 1 } { r } f + \\frac { Q } { 2 ( L + 1 ) } + \\kappa ( L + 1 ) \\frac { r } { f } , \\\\ & W ' ( r ) = - \\frac { L + 2 } { r } f + \\frac { Q } { 2 ( L + 2 ) } + \\kappa ( L + 2 ) \\frac { r } { f } , . \\end{align*}"}
+{"id": "2289.png", "formula": "\\begin{align*} T _ 1 | _ W = T ^ { ( \\lambda ) } _ { a _ 1 } | _ W , T _ 2 | _ W = T ^ { ( \\lambda ) } _ { a _ 2 } | _ W , \\end{align*}"}
+{"id": "6370.png", "formula": "\\begin{align*} B = & \\frac { Q ( 3 / 4 ) - 2 Q ( 1 / 2 ) + Q ( 1 / 4 ) } { Q ( 3 / 4 ) - Q ( 1 / 4 ) } , \\\\ M = & \\frac { Q ( 7 / 8 ) - Q ( 5 / 8 ) - Q ( 3 / 8 ) + Q ( 1 / 8 ) } { Q ( 6 / 8 ) - Q ( 2 / 8 ) } . \\end{align*}"}
+{"id": "7180.png", "formula": "\\begin{align*} ( q _ 1 - b _ 1 ) q _ { - 1 } + q _ { - 1 } q _ 1 = E _ 0 , \\end{align*}"}
+{"id": "9262.png", "formula": "\\begin{align*} \\Delta u \\wedge \\Delta v _ 1 \\wedge \\dots \\wedge \\Delta v _ { m - 1 } \\wedge \\beta _ n ^ { n - m } ( \\omega ) = \\int u \\Delta v _ 1 \\wedge \\dots \\wedge \\Delta v _ { m - 1 } \\wedge \\beta _ n ^ { n - m } \\wedge \\Delta \\omega , \\omega \\in C _ 0 ^ { \\infty } ( \\Omega ) , \\end{align*}"}
+{"id": "721.png", "formula": "\\begin{align*} x _ k = \\begin{cases} \\sqrt { P _ { k , 1 } } s _ { k , 1 } + \\sqrt { P _ { k , 2 } } s _ { k , 2 } , & \\textrm { i f } \\ ; k \\in \\mathcal { K } \\setminus \\{ K \\} , \\\\ \\sqrt { P _ { K } } s _ { K } , & \\textrm { i f } \\ ; k = K . \\end{cases} \\end{align*}"}
+{"id": "551.png", "formula": "\\begin{align*} h g ( h g e _ n h g ) ( h g ^ 2 e _ n h g ^ 2 ) \\cdots ( h g ^ { n - 1 } e _ n h g ^ { n - 1 } ) = ( h g e _ n h g ) ( h g ^ 2 e _ n h g ^ 2 ) \\cdots ( h g ^ { n - 1 } e _ n h g ^ { n - 1 } ) . \\end{align*}"}
+{"id": "1357.png", "formula": "\\begin{align*} | g ( a ) - g ( p _ i ) | & = | f ( a ) - f ( q _ i ) + d ( p _ i , q _ i ) | \\leq d ( a , q _ i ) + d ( p _ i , q _ i ) \\\\ & \\leq d ( a , p _ i ) + 2 d ( p _ i , q _ i ) \\leq d ( a , p _ i ) + 2 \\theta r \\leq ( 1 + 2 \\theta ) d ( a , p _ i ) . \\end{align*}"}
+{"id": "3662.png", "formula": "\\begin{align*} A _ 2 = \\sum _ { i = 1 } ^ { s } \\left ( ( a _ { i } - \\tfrac { r m } { n } ) ^ { 2 } + b _ { i } ^ { 2 } \\right ) = \\sum _ { i = 1 } ^ { s } ( a _ { i } ^ { 2 } + b _ { i } ^ { 2 } ) - ( r - 1 ) ^ { n - 1 } \\tfrac { r ^ { 2 } m ^ { 2 } } { n } \\leq s \\rho ( \\mathcal { L } _ { H } ) ^ { 2 } - ( r - 1 ) ^ { n - 1 } \\tfrac { r ^ { 2 } m ^ { 2 } } { n } . \\end{align*}"}
+{"id": "7990.png", "formula": "\\begin{align*} & l \\sum h _ { k } \\nabla _ { k } \\chi - w ^ { 1 1 } \\nabla _ { 1 1 } \\chi \\\\ & \\leq \\widetilde { C } _ { 1 } l + \\widetilde { C } _ { 2 } d + \\widetilde { C } _ { 3 } w ^ { 1 1 } + \\widetilde { C } _ { 4 } w _ { 1 1 } + \\widetilde { C } _ { 5 } . \\end{align*}"}
+{"id": "3181.png", "formula": "\\begin{align*} \\mathcal { J } & : = \\sum \\limits _ { a _ m \\neq a _ i } \\prod _ { i = 1 } ^ { m - 1 } \\{ 2 + h \\chi _ 4 ( a _ m - a _ i ) + \\overline { h } \\chi _ 4 ^ 3 ( a _ m - a _ i ) \\} \\\\ & = 2 ^ { m - 1 } ( q - m + 1 ) \\\\ & + \\sum \\limits _ { a _ m \\neq a _ i } [ ( 3 ^ { m - 1 } - 1 ) \\chi _ 4 ] \\end{align*}"}
+{"id": "6865.png", "formula": "\\begin{align*} \\mbox { \\tt h e t = P h i 1 . z e t } . \\end{align*}"}
+{"id": "8475.png", "formula": "\\begin{align*} D _ j ( x ; k ) : = V _ j ( k ) J = R ( x ; z ) V _ j ( k ) , \\ j = 1 , 2 . \\end{align*}"}
+{"id": "5897.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Vert X _ n ( t , \\omega ) - X ( t , \\omega ) \\Vert _ H = 0 , \\ a . e . \\ ( t , \\omega ) . \\end{align*}"}
+{"id": "9113.png", "formula": "\\begin{align*} \\begin{cases} \\chi _ 1 \\leq w _ 1 \\chi + r \\\\ \\chi _ 1 \\geq w _ 1 \\chi . \\end{cases} \\end{align*}"}
+{"id": "56.png", "formula": "\\begin{align*} \\widetilde W = N _ G ( T ) ( L ) / ( T ( L ) \\cap I ) \\end{align*}"}
+{"id": "5737.png", "formula": "\\begin{align*} b _ 1 x ^ q + c _ 1 x & = d _ 1 ( y ^ q + y ) , \\\\ b _ 0 x ^ q + c _ 0 x & = d _ 0 ( y ^ q + y ) , \\end{align*}"}
+{"id": "7887.png", "formula": "\\begin{align*} b _ k : = \\begin{cases} 1 & \\min ( \\abs { k } , M - \\abs { k } ) \\le k _ 0 \\\\ r M - k _ 0 & \\min ( \\abs { k } , M - \\abs { k } ) = k _ 0 + 1 \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "1101.png", "formula": "\\begin{align*} { \\gamma ^ { \\rm { O } } } = \\frac { { { p _ s } { { \\left | { { h _ s } } \\right | } ^ 2 } } } { { { W _ s } { N _ 0 } } } , \\end{align*}"}
+{"id": "1063.png", "formula": "\\begin{align*} T _ x T _ y = \\sum _ { z \\in \\widetilde W } f _ { x , y , z } ( v - v ^ { - 1 } ) T _ z \\end{align*}"}
+{"id": "8710.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n p _ { i } = 1 _ { Q } = 1 _ { M } p _ i p _ j = 0 \\forall i \\neq j \\end{align*}"}
+{"id": "8758.png", "formula": "\\begin{align*} f ( z ) = & \\sum _ { n = 0 } ^ { \\infty } \\left ( x - \\frac { \\psi _ 0 } { \\psi _ 1 } y \\right ) ^ n a _ n = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { z + \\bar z } { 2 } - \\frac { \\psi _ 0 } { \\psi _ 1 } \\frac { z - \\bar z } { 2 } \\right ) ^ n a _ n \\\\ = & \\sum _ { n = 0 } ^ { \\infty } \\left ( ( 1 - \\frac { \\psi _ 0 } { \\psi _ 1 } ) z + ( 1 + \\frac { \\psi _ 0 } { \\psi _ 1 } ) \\bar z \\right ) ^ n \\frac { a _ n } { 2 ^ n } , \\end{align*}"}
+{"id": "7490.png", "formula": "\\begin{align*} | \\nabla \\eta | ^ 2 _ { A ( 0 ) } = \\frac { a ^ 2 } { 4 } | x | _ { A ^ { - 1 } ( 0 ) } ^ { - ( a + 2 ) } \\zeta ^ 2 - \\frac { a } { 2 } | x | _ { A ^ { - 1 } ( 0 ) } ^ { - ( a + 2 ) } \\big ( x \\cdot \\nabla ( \\zeta ^ 2 ) \\big ) + | x | _ { A ^ { - 1 } ( 0 ) } ^ { - a } | \\nabla \\zeta | ^ 2 _ { A ( 0 ) } , \\end{align*}"}
+{"id": "941.png", "formula": "\\begin{align*} S _ 2 ( t ) & = i \\eta t ^ { - 1 } ( | e ^ { 3 i \\lambda _ 1 ( \\Phi _ 1 ( t ) - \\theta _ 1 ) } v _ 1 ( t ) | ^ 2 - | W _ 1 | ^ 2 ) w ( t ) \\\\ & - i \\lambda _ 6 t ^ { - 1 } \\{ ( e ^ { 3 i \\lambda _ 1 ( \\Phi _ 1 ( t ) - \\theta _ 1 ) } v _ 1 ( t ) ) ^ 2 - W _ 1 ^ 2 \\} \\overline { w ( t ) } . \\end{align*}"}
+{"id": "1569.png", "formula": "\\begin{align*} \\alpha _ 1 : = \\underset { x \\in K _ 1 } { \\inf } d ^ \\mathcal Z ( x , \\partial \\mathcal Z ) \\beta _ 1 : = \\underset { x \\in K _ 1 } { \\sup } \\ ; d ^ \\mathcal Z ( x , \\partial \\mathcal Z ) \\end{align*}"}
+{"id": "4255.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\inf _ { k \\in { \\mathbb Z } } \\left ( \\frac { \\mu ( f ^ { k + n } ( W ) ) } { \\mu ( f ^ { k } ( W ) ) } \\right ) = \\infty \\tag * { $ \\mathcal { U E } 1 $ } \\end{align*}"}
+{"id": "2649.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } o _ { p , e } ( n ) q ^ { n } & = \\frac { 1 } { 2 } \\left ( H ( 1 , q ) + H ( - 1 , q ) \\right ) \\\\ & = \\frac { ( q ^ { p } ; q ^ { p } ) _ { \\infty } } { ( q ; q ) _ { \\infty } } \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { 2 p n } } { 1 - q ^ { 2 p n } } . \\end{align*}"}
+{"id": "4882.png", "formula": "\\begin{align*} f _ s ( Z ^ n _ s ) = C ^ { ( n ) } _ s \\end{align*}"}
+{"id": "7720.png", "formula": "\\begin{align*} M ^ { ( k ) } _ { j , [ k + 1 , n ] } = M _ { j , [ k + 1 , n ] } - M _ { j , [ k ] } \\big ( M _ { [ k ] , [ k ] } \\big ) ^ { - 1 } M _ { [ k ] , [ k + 1 , n ] } . \\end{align*}"}
+{"id": "6409.png", "formula": "\\begin{align*} \\Phi ( x ) : = \\eta ( r ) s ( x ) \\end{align*}"}
+{"id": "6386.png", "formula": "\\begin{align*} \\widetilde { P } _ { k } ^ { \\mathrm { d i s c } } ( \\mathcal { T } ) : = \\left \\{ q _ h \\in { P } _ { k } ^ { \\mathrm { d i s c } } ( \\mathcal { T } ) : ( q _ h , 1 ) _ T = 0 T \\in \\mathcal { T } \\right \\} . \\end{align*}"}
+{"id": "9264.png", "formula": "\\begin{align*} \\mathcal { H } _ m ( u ) ( q _ 0 ) = \\lambda _ { n } ( q _ 0 ) \\sum \\limits _ { 1 \\leq j _ 2 < \\dots < j _ m \\leq n - 1 } \\lambda _ { j _ 1 } ( q _ 0 ) \\cdots \\lambda _ { j _ { m - 1 } } ( q _ 0 ) + \\sum \\limits _ { 1 \\leq j _ 1 < \\dots < j _ m \\leq n - 1 } \\lambda _ { j _ 1 } ( q _ 0 ) \\cdots \\lambda _ { j _ m } ( q _ 0 ) . \\end{align*}"}
+{"id": "656.png", "formula": "\\begin{align*} y _ { k } ' ( s ) = F _ { k } ( s , y _ { 1 } ( s ) , \\ldots , y _ { a } ( s ) ) \\mbox { w i t h $ y _ { k } ( 0 ) = \\hat { y } _ k $ f o r $ 1 \\le k \\le a $ , } \\end{align*}"}
+{"id": "6879.png", "formula": "\\begin{align*} { \\tt w = e v a l u ( P s i 1 , \\tilde z , { ' v ' } ) } \\end{align*}"}
+{"id": "2283.png", "formula": "\\begin{align*} a ( U D ( x ) V ) = a \\circ \\widehat { \\varphi } ( [ U , V ] , x ) = f ( [ V ] , x ) = a \\circ \\widehat { \\varphi } ( [ I _ n , V ] , x ) = a ( D ( x ) V ) \\end{align*}"}
+{"id": "3071.png", "formula": "\\begin{align*} d _ r = \\sum _ { j = 0 } ^ r \\binom { r } { j } g _ j \\ \\geq \\ \\sum _ { j = 0 } ^ r \\binom { r } { j } = 2 ^ r \\end{align*}"}
+{"id": "6003.png", "formula": "\\begin{align*} 0 & = \\int _ { \\R ^ { 2 } } ( | \\nabla u | ^ { 2 } + u ^ { 2 } + | \\nabla v | ^ { 2 } + \\omega v ^ { 2 } ) d x + 3 \\Big ( B ( u ) + B ( v ) \\Big ) - \\int _ { \\R ^ { 2 } } ( u ^ { 2 p } + v ^ { 2 p } + 2 b | u v | ^ { p } ) d x \\\\ & \\geq \\int _ { \\R ^ { 2 } } ( u ^ { 2 } + \\frac { 1 } { 2 } u ^ { 4 } - ( 1 + b ) u ^ { 2 p } ) d x + \\int _ { \\R ^ { 2 } } ( \\omega v ^ { 2 } + \\frac { 1 } { 2 } v ^ { 4 } - ( 1 + b ) v ^ { 2 p } ) d x . \\end{align*}"}
+{"id": "6026.png", "formula": "\\begin{align*} \\gamma _ { u , v } ( t ) : = ( t ^ { \\alpha } u ( t \\cdot ) , t ^ { \\alpha } v ( t \\cdot ) ) , \\ \\ t \\geq 0 , \\end{align*}"}
+{"id": "4444.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\lambda ^ * t } Z ( t ) = \\frac { W ( v ^ { ( 1 ) } , x ) } { | v ^ { ( 1 ) } | ^ 2 } v ^ { ( 1 ) } + \\frac { W ( v ^ { ( 2 ) } , x ) } { | v ^ { ( 2 ) } | ^ 2 } v ^ { ( 2 ) } . \\end{align*}"}
+{"id": "8926.png", "formula": "\\begin{align*} f _ { G } ( x , y | \\rho ) = f _ { g } ( x | \\beta ) f _ { g } ( y | \\beta ) \\sum _ { j \\geq 0 } \\rho ^ { n } l _ { n } ( x | \\beta ) l _ { n } ( y | \\beta ) , \\end{align*}"}
+{"id": "1050.png", "formula": "\\begin{align*} d ( v _ 1 \\Rightarrow w _ 2 v _ 2 ' ) = & d ( v _ 1 \\Rightarrow w _ 2 v _ 2 ) + d ( w _ 2 v _ 2 \\Rightarrow w _ 2 v _ 2 ' ) . \\end{align*}"}
+{"id": "2397.png", "formula": "\\begin{align*} \\hat \\beta _ { A C } & = A C + \\pi C D + \\pi ^ 2 D A \\\\ \\hat \\beta _ { C B } & = C B + \\pi ^ 4 B D + \\pi ^ 2 D C \\\\ \\hat \\beta _ { B A } & = B A + \\pi ^ 4 A D + \\pi ^ 5 D B . \\end{align*}"}
+{"id": "9168.png", "formula": "\\begin{align*} \\begin{aligned} \\dfrac { b _ { 1 , \\delta } ( x ) } { 2 } & \\geq \\bar b & \\forall \\ , x \\geq x ^ \\star + \\delta \\\\ \\dfrac { b _ { 1 , \\delta } ( x ) } { 2 } & \\leq - \\bar b & \\forall \\ , x \\leq x ^ \\star - \\delta \\ , . \\end{aligned} \\end{align*}"}
+{"id": "346.png", "formula": "\\begin{align*} D ( G _ n ) = \\left [ \\begin{array} { c c c c c c c c c } 0 & \\ 1 _ { n - 1 } ' & 2 \\ 1 _ { n - 1 } ' \\\\ \\ 1 _ { n - 1 } & 2 ( J _ { n - 1 } - I _ { n - 1 } ) & S \\\\ 2 \\ 1 _ { n - 1 } & S ' & T \\end{array} \\right ] , \\end{align*}"}
+{"id": "3514.png", "formula": "\\begin{align*} \\mathcal { C } _ n ^ { ( p ) } \\ = \\ A _ p F _ n + B _ p F _ { n + 1 } - \\sum _ { k = 0 } ^ p \\binom { p } { k } B _ k n ^ { p - k } . \\end{align*}"}
+{"id": "1957.png", "formula": "\\begin{align*} \\varphi _ { - } ( F ( z ) ) = T _ 1 ( \\varphi _ { - } ( z ) ) \\end{align*}"}
+{"id": "6.png", "formula": "\\begin{align*} \\beta = \\alpha + \\alpha _ s a _ s = 1 \\end{align*}"}
+{"id": "6952.png", "formula": "\\begin{align*} C \\le M _ n ^ { \\psi } ( Q ^ n ) / n = T _ { W _ { n J } , \\psi } ( \\mu _ n ( Q ^ n ) ) - I ( \\mu _ n ( Q ^ n ) ) \\le - I ( \\mu _ n ( Q ^ n ) ) . \\end{align*}"}
+{"id": "956.png", "formula": "\\begin{align*} \\begin{aligned} v _ { j } ( t ) & = i \\int _ { t } ^ { \\infty } U ( t - \\tau ) \\left \\{ \\tilde { { \\mathcal N } } _ { j } ( v _ { 1 } + u _ { \\mathrm { a p } , 1 } , v _ { 2 } + u _ { \\mathrm { a p } , 2 } ) - \\tilde { { \\mathcal N } } _ { j } ( u _ { \\mathrm { a p } , 1 } , u _ { \\mathrm { a p } , 2 } ) \\right \\} ( \\tau ) d \\tau \\\\ & + i \\int _ { t } ^ { \\infty } U ( t - \\tau ) \\mathcal { E } _ { j } ( \\tau ) d \\tau + \\mathcal { R } _ j . \\end{aligned} \\end{align*}"}
+{"id": "6550.png", "formula": "\\begin{align*} \\nabla c \\cdot \\nu = ( \\gamma - c ) g \\quad \\mbox { o n } \\partial \\Omega . \\end{align*}"}
+{"id": "1366.png", "formula": "\\begin{align*} \\P _ x ( \\tau > n ) \\sim K _ { \\lambda } n ^ { - d ( d - 1 ) / 2 - d / 2 } e ^ { - \\gamma n } e ^ { \\sum _ { i = 1 } ^ d ( \\lambda _ i - \\bar { \\lambda } ) x _ i } h ^ { ( \\bar { \\lambda } , \\ldots , \\bar { \\lambda } ) } ( x ) , n \\rightarrow \\infty . \\end{align*}"}
+{"id": "1885.png", "formula": "\\begin{align*} \\Pi _ \\lambda [ \\psi _ j ^ { - } ] ( x , y ) & = C _ d \\ , \\overline { \\Pi _ { \\lambda } [ \\psi _ j ^ { + } ] } ( x , y ) , \\\\ [ 4 p t ] \\Pi _ \\lambda [ \\psi _ j ^ { \\pm \\pi } ] ( x , y ) & = C _ d ' \\ , \\Pi _ { \\lambda } [ \\psi _ j ^ { \\mp } ] ( x , - y ) , \\end{align*}"}
+{"id": "8243.png", "formula": "\\begin{align*} \\chi ( g ) = \\lim _ { l \\to \\infty } \\chi _ { _ { ^ { k ( l ) } \\ ! { \\scriptstyle \\lambda } } } ( g ) ~ ~ g \\in \\mathfrak { S } _ { \\widehat { \\mathbf { n } } } . \\end{align*}"}
+{"id": "3527.png", "formula": "\\begin{align*} \\mathcal { S } _ n ^ { ( p ) } \\ & = \\ \\sum _ { i = 0 } ^ { n } \\ , i ^ p \\cdot F _ { i } , \\\\ \\intertext { a n d r e p l a c i n g $ i $ w i t h $ n - i $ w e h a v e } \\mathcal { S } _ n ^ { ( p ) } \\ & = \\ \\sum _ { i = 0 } ^ { n } \\ , ( n - i ) ^ p \\cdot F _ { n - i } \\ = \\ \\sum _ { i = 0 } ^ { n } \\left ( \\sum _ { k = 0 } ^ p \\binom { p } { k } n ^ { p - k } ( - 1 ) ^ k i ^ k \\right ) \\cdot F _ { n - i } . \\end{align*}"}
+{"id": "7127.png", "formula": "\\begin{align*} \\begin{array} { l l l } f _ L ^ M ( ( L ' , R ' ) \\star ( L , R ) ) \\circ u \\\\ & = & f _ L ^ M ( L ' \\circ L , R \\circ R ' ) \\circ f _ L ( s ) \\circ t \\\\ & = & f _ L ( L ' \\circ L \\circ s ) \\circ t \\\\ & = & f _ L ^ M ( L ' , R ' ) \\circ f _ L ( L \\circ s ) \\circ t \\\\ & = & f _ L ^ M ( L ' , R ' ) \\circ f _ L ^ M ( L , R ) \\circ f _ L ( s ) \\circ t \\\\ & = & f _ L ^ M ( L ' , R ' ) \\circ f _ L ^ M ( L , R ) \\circ u . \\end{array} \\end{align*}"}
+{"id": "1506.png", "formula": "\\begin{align*} G ( t , s ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\left \\{ \\begin{array} { l r } \\frac { ( t - a ) ^ { \\alpha - 1 } ( b - s ) ^ { \\alpha - 1 - \\beta } } { ( b - a ) ^ { \\alpha - 1 - \\beta } } - ( t - s ) ^ { \\alpha - 1 } , & a \\le s \\le t \\le b , \\\\ \\\\ \\frac { ( t - a ) ^ { \\alpha - 1 } ( b - s ) ^ { \\alpha - 1 - \\beta } } { ( b - a ) ^ { \\alpha - 1 - \\beta } } , & a \\le t \\le s \\le b . \\end{array} \\right . \\end{align*}"}
+{"id": "6141.png", "formula": "\\begin{align*} d ^ { a } _ 1 ( \\eta ^ a x , y ) = \\eta ^ a d _ 1 ( x , y ) & \\quad & d ^ { a } _ 0 ( \\eta ^ a x , y ) = \\eta ^ { - a } d _ 1 ( x , y ) \\\\ d ^ { b } _ 1 ( y , \\eta ^ { - b } z ) = d _ { 1 } ( y , z ) & \\quad & d ^ { b } _ 0 ( y , \\eta ^ { - b } z ) = d _ { 0 } ( y , z ) \\\\ d _ { 1 } ^ { a + b } ( \\eta ^ a x , \\eta ^ { - b } z ) = \\eta ^ a d _ 1 ( x , z ) & \\quad & d _ { 0 } ^ { a + b } ( \\eta ^ a x , \\eta ^ { - b } z ) = \\eta ^ { - a } d _ 0 ( x , y ) . \\end{align*}"}
+{"id": "2038.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t F + v \\cdot \\partial _ x F = Q _ L ( F , \\ F ) , \\\\ F | _ { t = 0 } = F _ 0 , \\end{cases} \\end{align*}"}
+{"id": "5408.png", "formula": "\\begin{align*} \\begin{aligned} F ^ { i j } \\left ( \\sqrt { b _ { n - 1 } } ( A _ 1 v - A _ 2 w ) \\pm \\nabla _ \\alpha ( u - \\varphi ) \\right ) _ { i j } \\geq \\ , & 0 \\mbox { i n } \\omega _ 1 \\\\ A \\sqrt { b _ { n - 1 } } ( A _ 1 v - A _ 2 w ) \\pm \\nabla _ \\alpha ( u - \\varphi ) \\leq \\ , & 0 \\mbox { o n } \\partial \\omega _ 1 \\end{aligned} \\end{align*}"}
+{"id": "6556.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta v _ { l } + u _ { l } v _ { l } = f , & x \\in \\Omega , \\\\ \\nabla v _ { l } \\cdot \\nu + v _ { l } = \\eta , & x \\in \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "8417.png", "formula": "\\begin{align*} \\partial _ x \\left ( e ^ { i c _ - } \\Psi ^ - _ { 1 1 } ( x ; z ) \\right ) = \\frac { 1 } { 2 i } u _ x ( x ) e ^ { i c _ - ( x ) } \\Psi ^ - _ { 2 1 } ( x ; z ) . \\end{align*}"}
+{"id": "391.png", "formula": "\\begin{align*} { { q _ \\delta { q _ \\delta } } ^ { \\ast } } = \\left [ \\begin{array} { c c c c } 0 & \\ 0 & \\ 0 \\\\ \\ 0 & u _ { \\delta } u _ { \\delta } ' & O \\\\ \\ 0 & O & O \\end{array} \\right ] = \\left [ \\begin{array} { c c c c } 0 & \\ 0 & \\ 0 \\\\ \\ 0 & T & O \\\\ \\ 0 & O & O \\end{array} \\right ] . \\end{align*}"}
+{"id": "6826.png", "formula": "\\begin{align*} M ^ { \\pm } = \\left \\{ s \\in \\left ( 0 , \\infty \\right ) \\mid \\left \\vert \\rho \\mp \\beta ( s ) \\right \\vert \\leq \\Lambda ^ { - \\frac { 1 } { 2 } } \\left ( s \\right ) \\beta ( s ) \\right \\} . \\end{align*}"}
+{"id": "6218.png", "formula": "\\begin{align*} & W _ + ( r ) = ( 2 L + 3 ) \\left ( - \\frac { f } { r } + \\Q a _ 0 + \\kappa \\sum _ { k = 0 } ^ { [ ( m - 1 ) / 2 ] } a _ { 2 k + 1 } \\frac { r } { f ^ { 2 k + 1 } } + \\sum _ { k = 1 } ^ { [ m / 2 ] } a _ { 2 k } \\frac { 1 } { f _ { 2 k } } \\right ) , \\\\ & W _ - ( r ) = - \\frac { f } { r } - \\Q + m \\kappa \\frac { r } { f } , \\\\ & E _ 1 - E _ 0 = ( 2 L + 3 ) a _ 0 [ \\Q ^ 2 + ( m + 1 ) ^ 2 \\kappa ] , . \\end{align*}"}
+{"id": "7668.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d y ^ { * , i } _ t = & - [ A y ^ { * , i } _ t + Q x ^ { * , i } _ t + Q l ( \\nu ^ { N , i } _ { \\boldsymbol { x } ^ * _ t } ) ] d t + \\sum _ { j = 1 } ^ N z _ t ^ { * , i , j } d W ^ j _ t + z ^ { 0 , * , i } d W ^ 0 _ t , \\\\ y ^ { * , i } _ T = & G ( x _ T ^ { * , i } + g ( \\nu ^ { N , i } _ { \\boldsymbol { x } ^ * _ T } ) ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7578.png", "formula": "\\begin{align*} \\delta = \\lim _ { N \\to \\infty } \\dfrac { | A \\cap \\Phi _ N | } { | \\Phi _ N | } \\end{align*}"}
+{"id": "1098.png", "formula": "\\begin{align*} \\eta \\left ( { { \\mathbf { s } } , \\widehat { \\mathbf { s } } } \\right ) = \\frac { { { \\mathbf { B } } \\left ( { \\mathbf { s } } \\right ) { \\mathbf { B } } { { \\left ( { \\widehat { \\mathbf { s } } } \\right ) } ^ T } } } { { \\left \\| { { \\mathbf { B } } \\left ( { \\mathbf { s } } \\right ) } \\right \\| \\left \\| { { \\mathbf { B } } { { \\left ( { \\widehat { \\mathbf { s } } } \\right ) } } } \\right \\| } } , \\end{align*}"}
+{"id": "6588.png", "formula": "\\begin{align*} Q ( R , t ) = \\| n _ { 0 } \\| _ { L ^ { 1 } ( \\Omega ) } , \\end{align*}"}
+{"id": "281.png", "formula": "\\begin{align*} s w _ n = ( s x _ { w _ n } , s y _ { w _ n } , 0 ) \\quad \\ w _ n = ( x _ { w _ n } , y _ { w _ n } , 0 ) , \\end{align*}"}
+{"id": "2231.png", "formula": "\\begin{align*} x ^ { ( k ) } _ { i } = x ^ { ( n ) } _ { i } ( \\forall i \\in \\{ 1 , \\dots , D \\} ) ( \\forall k > n ) . \\end{align*}"}
+{"id": "2269.png", "formula": "\\begin{align*} Z = U P = Q V . \\end{align*}"}
+{"id": "1463.png", "formula": "\\begin{align*} v _ { n } ( y ) = w _ { n } ( y ) - w _ { n } ( 0 ) \\forall y \\in ( A _ { n } , B _ { n } ) , \\end{align*}"}
+{"id": "5275.png", "formula": "\\begin{align*} E ( g _ 1 , w ) = \\int _ { - \\infty } ^ \\infty \\int _ 0 ^ 1 \\int _ 0 ^ \\infty \\int _ 0 ^ \\infty e ^ { i \\Phi \\left ( \\lambda , \\theta , t , u + \\left ( \\frac { t _ 1 } { t } \\right ) ^ 2 u _ 1 \\right ) } W ( \\lambda , t , u ) d \\lambda d \\theta d t d u . \\end{align*}"}
+{"id": "7893.png", "formula": "\\begin{align*} 4 \\lambda + 2 \\mu = \\frac { \\hat W _ r ( 0 ) + \\hat W _ r ( 2 q ) - 2 \\hat W _ r ( q ) } { \\hat W _ r ( q ) } = : H ( q , r ) . \\end{align*}"}
+{"id": "8615.png", "formula": "\\begin{align*} A \\oplus _ 2 \\left ( \\sqrt { ( 1 - \\lambda ) s + \\lambda t } B \\right ) = \\left ( \\sqrt { 1 - \\lambda } ( A \\oplus _ 2 \\sqrt { s } B ) \\right ) \\oplus _ 2 \\left ( \\sqrt { \\lambda } ( A \\oplus _ 2 \\sqrt { t } B ) \\right ) . \\end{align*}"}
+{"id": "7060.png", "formula": "\\begin{align*} G _ { 6 , 0 , 0 } \\ , = \\ , \\pm 1 \\ , = \\ , F _ { 6 , 0 , 0 } , \\end{align*}"}
+{"id": "7241.png", "formula": "\\begin{align*} L = a _ 0 x + a _ 1 x ^ q + a _ 2 x ^ { q ^ 2 } + \\cdots + a _ n x ^ { q ^ n } . \\end{align*}"}
+{"id": "5053.png", "formula": "\\begin{align*} u _ E = U _ C ( x + B _ { \\infty } t ) - B _ { \\infty } , \\end{align*}"}
+{"id": "2089.png", "formula": "\\begin{align*} Y = f ^ { * } ( X ) + \\xi \\end{align*}"}
+{"id": "208.png", "formula": "\\begin{align*} \\lambda _ S ( G ) = \\lim _ { n \\rightarrow \\infty } \\sqrt [ n ] { | B _ S ( n ) | } = \\inf \\big \\{ \\sqrt [ n ] { | B _ S ( n ) | } \\mid n \\in \\mathbb { N } _ 0 \\big \\} \\end{align*}"}
+{"id": "1330.png", "formula": "\\begin{align*} L ( \\gamma ) = \\sup \\bigl \\{ \\sum _ { i = 0 } ^ { n - 1 } d ( \\gamma ( t _ i ) , \\gamma ( t _ { i + 1 } ) ) \\colon n \\in \\mathbb { N } , \\ , \\alpha = t _ 0 \\leq \\dotsb \\leq t _ n = \\beta \\bigr \\} . \\end{align*}"}
+{"id": "664.png", "formula": "\\begin{align*} Y _ { l , m } ( \\theta , \\phi ) = \\sqrt { \\frac { ( 2 l + 1 ) ( l - m ) ! } { 4 \\pi ( l + m ) ! } } P _ l ^ m ( \\cos \\theta ) e ^ { i m \\phi } \\ , , \\end{align*}"}
+{"id": "6012.png", "formula": "\\begin{align*} I ( u , v ) & = \\frac { 1 } { 2 } \\| ( u , v ) \\| _ { E } ^ { 2 } + \\frac { 1 } { 2 } \\Big ( B ( u ) + B ( v ) \\Big ) - F ( u , v ) ; \\\\ & \\geq \\frac { 1 } { 2 } \\| ( u , v ) \\| _ { E } ^ { 2 } - C \\| ( u , v ) \\| _ { E } ^ { 2 p } . \\end{align*}"}
+{"id": "2971.png", "formula": "\\begin{align*} \\begin{aligned} & H ( P ( x ) , 0 ) = ( Q , 0 ) \\left ( H ( x , 0 ) \\right ) \\\\ & \\Psi \\left ( t , H ( x , y ) \\right ) = H \\left ( \\Phi ( t , x , y ) \\right ) \\\\ & \\Psi _ s \\left ( t , H ( x , 0 ) \\right ) = H \\left ( \\Phi _ s ( t , x , 0 ) \\right ) \\end{aligned} \\end{align*}"}
+{"id": "4813.png", "formula": "\\begin{align*} a _ n ( u _ n , v _ n ) = \\lambda _ n ( u _ n , v _ n ) \\quad \\ , v _ n \\in X _ n . \\end{align*}"}
+{"id": "998.png", "formula": "\\begin{align*} \\Phi ^ + ( \\alpha ) : = \\begin{cases} 1 , & \\alpha \\in \\Phi ^ + , \\\\ 0 , & \\alpha \\in \\Phi ^ - . \\end{cases} \\end{align*}"}
+{"id": "3826.png", "formula": "\\begin{align*} \\begin{aligned} | \\langle \\boldsymbol { \\nu } , f _ \\alpha ( \\lambda | \\bar U ; x , t ) \\rangle | & \\le c _ 3 \\langle \\boldsymbol { \\nu } , \\eta ( \\lambda | \\bar U ; x , t ) \\rangle = c _ 3 \\mathcal { H } ( \\boldsymbol { \\nu } , U , \\bar { U } ; x , t ) \\ ; , \\end{aligned} \\end{align*}"}
+{"id": "4047.png", "formula": "\\begin{align*} e ( C ) = 2 - I ( C ) - s ( C ) ; \\end{align*}"}
+{"id": "4025.png", "formula": "\\begin{align*} x ' _ 1 = \\varphi _ { \\sigma , s } ^ L ( x _ 1 ) , x ' _ 2 = \\varphi _ { \\sigma , s } ^ L ( x _ 2 ) , y ' _ 1 = \\varphi _ { \\sigma , p } ^ R ( y _ 1 ) y ' _ 2 = \\varphi _ { \\sigma , p } ^ R ( y _ 2 ) . \\end{align*}"}
+{"id": "1164.png", "formula": "\\begin{align*} d X _ t ^ i = \\left ( b _ 0 ( t , X _ t ^ i ) + \\frac { 1 } { n - 1 } \\sum _ { j = 1 , j \\neq i } ^ n b \\left ( t , X _ t ^ i , X _ t ^ j \\right ) \\right ) d t + d B ^ { H , i } _ t , i = 1 , \\cdots , n , \\end{align*}"}
+{"id": "4069.png", "formula": "\\begin{align*} e _ 2 ( 6 H - 3 / 2 , G ^ { M ( 3 , 0 ) } _ { 2 } , \\emptyset ) & = 6 H - 3 / 2 + ( 1 - 6 H ) \\vee ( - 3 ) = ( - 1 / 2 ) \\vee ( 6 H - 9 / 2 ) . \\end{align*}"}
+{"id": "242.png", "formula": "\\begin{align*} | B _ S ( n + D ' ) | > \\frac { q ( n ) } { 2 } \\ , \\sum _ { \\ell > q ( n ) } \\left | R _ { q , \\ell } ^ \\flat ( n ) \\right | = \\frac { q ( n ) } { 2 } \\ , \\left | R _ { q } ^ \\flat ( n ) \\right | , \\end{align*}"}
+{"id": "5752.png", "formula": "\\begin{align*} \\xi ^ { 2 ^ k } = \\left \\{ \\begin{array} { l l } \\xi + 1 & \\textrm { i f $ k $ i s o d d , } \\\\ \\xi & \\textrm { i f $ k $ i s e v e n } . \\end{array} \\right . \\end{align*}"}
+{"id": "7749.png", "formula": "\\begin{align*} D ^ a _ F ( 0 ) = \\{ M _ 1 ^ T D _ 1 M _ 2 ^ T D _ 2 \\ldots M _ { L - 1 } ^ T D _ { L - 1 } M _ L ^ T \\} \\end{align*}"}
+{"id": "661.png", "formula": "\\begin{align*} S = \\int d ^ 4 x \\sqrt { - g } \\left [ \\frac { R } { 1 6 \\pi } - \\frac { 1 } { 2 } g _ { \\mu \\nu } \\partial ^ \\mu \\Phi \\partial ^ \\nu \\Phi - \\frac { V ( \\Phi ) } { 1 6 \\pi } \\right ] + S ( \\psi _ m , \\mathcal { A } ( \\Phi ) ^ 2 g _ { \\mu \\nu } ) \\ , . \\end{align*}"}
+{"id": "3970.png", "formula": "\\begin{align*} N = \\begin{pmatrix} Q & 0 _ { k \\times k } \\\\ 0 _ { k \\times k } & Q \\end{pmatrix} . \\end{align*}"}
+{"id": "931.png", "formula": "\\begin{align*} u _ 1 ( t ) & = M ( t ) D ( t ) e ^ { - 3 i \\lambda _ 1 | \\alpha | ^ 2 \\log t } W _ 1 + O ( t ^ { - 3 / 4 + C _ 1 \\varepsilon ^ 2 _ 1 } ) \\\\ & = M ( t ) D ( t ) e ^ { - 3 i \\lambda _ 1 | W _ 1 | ^ 2 \\log t } W _ 1 + O ( t ^ { - 3 / 4 + C _ 1 \\varepsilon ^ 2 _ 1 } ) , \\end{align*}"}
+{"id": "4417.png", "formula": "\\begin{align*} \\overline { \\mathrm { s p a n } } _ { H ^ 2 ( \\mathbb { D } ) } \\{ ( I - S ) h _ k \\mid k \\geq 2 \\} = H ^ 2 ( \\mathbb { D } ) . \\end{align*}"}
+{"id": "1104.png", "formula": "\\begin{align*} P _ { \\min } ^ { { \\rm { O } } } \\left ( { \\overline S , \\overline \\varepsilon , \\overline R } \\right ) = \\bigcap { \\left \\{ { P \\in { { \\mathbb { R } } ^ + } : \\left ( { \\overline S , \\overline R } \\right ) \\in { \\mathcal { R } } _ { { \\rm { S v B } } } ^ { { \\rm { O } } } \\left ( { W , P , K , \\overline \\varepsilon } \\right ) } \\right \\} } , \\end{align*}"}
+{"id": "8380.png", "formula": "\\begin{align*} \\Psi ^ { \\pm } ( x ; z ) = I + \\int _ { \\pm \\infty } ^ x e ^ { - i z ( x - y ) \\widehat { \\sigma } _ 3 } \\widetilde { Q } \\Psi ^ { \\pm } ( y ; z ) d y . \\end{align*}"}
+{"id": "6586.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } \\frac { d } { d t } \\int _ { \\Omega } n ^ { 3 } + \\frac { 8 } { 9 } \\int _ { \\Omega } | \\nabla n ^ { \\frac { 3 } { 2 } } | ^ { 2 } = \\frac { 4 } { 3 } \\int _ { \\Omega } n ^ { \\frac { 3 } { 2 } } \\nabla n ^ { \\frac { 3 } { 2 } } \\cdot ( S ( x , n , c ) \\cdot \\nabla c ) . \\end{align*}"}
+{"id": "9185.png", "formula": "\\begin{align*} \\gamma _ 1 ^ \\star : = \\dfrac { \\bar c } { \\bar \\epsilon ( L _ r , M _ r , \\delta ) } \\end{align*}"}
+{"id": "2533.png", "formula": "\\begin{align*} \\begin{aligned} & \\Gamma _ p = 2 \\left ( \\gamma ^ 2 / 2 + ( 1 - \\nu ) ^ 2 k \\right ) ^ { 1 / 2 } / ( 1 - 3 \\gamma ) . \\end{aligned} \\end{align*}"}
+{"id": "4027.png", "formula": "\\begin{align*} \\beta = ( \\beta _ 1 , \\ldots , \\beta _ k ) \\textrm { w i t h } \\sum _ i \\beta _ i = m . \\end{align*}"}
+{"id": "2414.png", "formula": "\\begin{align*} f ( t ) = \\frac { 1 } { 2 \\pi i } \\lim _ { \\tau \\rightarrow \\infty } \\int _ { \\sigma - i \\tau } ^ { \\sigma + i \\tau } \\exp ( s t ) F ( s ) d s . \\end{align*}"}
+{"id": "1681.png", "formula": "\\begin{align*} \\dot { V } ( x _ t ) = - \\norm { x ( t - h ) } ^ 2 , \\end{align*}"}
+{"id": "2290.png", "formula": "\\begin{align*} T \\circ \\pi _ 1 ( h ) = \\pi _ 2 ( h ) \\circ T , \\end{align*}"}
+{"id": "965.png", "formula": "\\begin{align*} n ^ 2 = p ^ { m _ 1 } + p ^ { m _ 2 } + \\cdots + p ^ { m _ r } . \\end{align*}"}
+{"id": "5365.png", "formula": "\\begin{align*} S _ k [ r ] = \\sigma _ k ( \\lambda ( r ) ) \\end{align*}"}
+{"id": "1149.png", "formula": "\\begin{align*} { \\cal W } : = [ W ^ { ( 1 ) } , W ^ { ( 2 ) } , \\dots , W ^ { ( L ) } ] \\in \\R ^ { N \\times N L } . \\end{align*}"}
+{"id": "4370.png", "formula": "\\begin{align*} \\lim _ { i \\to + \\infty } \\int _ T ^ { t _ 0 + B } c _ i ( t _ 1 ) e ^ { - t _ 1 } d t _ 1 = \\int _ { T } ^ { t _ 0 + B } c ( t _ 1 ) e ^ { - t _ 1 } d t _ 1 . \\end{align*}"}
+{"id": "9308.png", "formula": "\\begin{align*} E : y ^ { 2 } = x ( x - 4 ^ { m } p ) ( x + p ) \\end{align*}"}
+{"id": "1295.png", "formula": "\\begin{align*} h _ { \\Lambda } ( \\nu \\lvert \\mu ) = \\sum _ { \\eta _ { \\Lambda } } \\nu ( \\eta _ { \\Lambda } ) \\log \\left ( \\nu ( \\eta _ { \\Lambda } ) \\right ) - \\sum _ { \\eta _ { \\Lambda } } \\nu ( \\eta _ { \\Lambda } ) \\log \\left ( \\mu ( \\eta _ { \\Lambda } ) \\right ) . \\end{align*}"}
+{"id": "5071.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } A \\cdot B \\dd x & \\leq \\frac { 1 } { f _ 1 ^ { + } } \\int _ { \\Omega } | B | ^ { 2 } \\dd x , \\\\ - \\int _ { \\Omega } A \\cdot B \\dd x & \\leq \\frac { 1 } { - f _ 1 ^ { - } } \\int _ { \\Omega } | B | ^ { 2 } \\dd x , \\end{aligned} \\end{align*}"}
+{"id": "6305.png", "formula": "\\begin{align*} F ( x ^ { t + 1 } ) - F ( x ) + \\frac { \\alpha _ t ^ 2 } { 2 \\gamma _ t } \\| x - z ^ { t + 1 } \\| ^ 2 \\leq \\prod _ { i = 1 } ^ t ( 1 - \\alpha _ i ) \\left ( F ( x ^ 1 ) - F ( x ) + \\frac { \\alpha ^ 2 _ { 0 } } { 2 \\gamma _ { 0 } } \\| x - x ^ 1 \\| ^ 2 \\right ) . \\end{align*}"}
+{"id": "8346.png", "formula": "\\begin{align*} S ( k ) = \\begin{pmatrix} a ( k ) & b ( k ) \\\\ [ 3 p t ] - \\overline { b ( \\bar { k } ) } & \\overline { a ( \\bar { k } ) } \\end{pmatrix} , \\ \\ \\det S ( k ) = 1 . \\end{align*}"}
+{"id": "7237.png", "formula": "\\begin{align*} L ( x ) = \\sum _ { i = 0 } ^ r a _ i x ^ { 2 ^ i } \\end{align*}"}
+{"id": "4614.png", "formula": "\\begin{align*} & | \\o ( t , x ) - \\o ^ \\beta ( t , x ) | = | \\o _ 0 ( X ( 0 ; t , x ) ) - \\o _ 0 ^ \\beta ( X ^ \\beta ( 0 ; t , x ) ) | \\\\ & \\le | \\o _ 0 ( X ( 0 ; t , x ) ) - \\o _ 0 ^ \\ell ( X ( 0 ; t , x ) ) | + | \\o _ 0 ^ \\ell ( X ^ \\beta ( 0 ; t , x ) ) - \\o _ 0 ^ \\beta ( X ^ \\beta ( 0 ; t , x ) ) | \\\\ & + | \\o _ 0 ^ \\ell ( X ( 0 ; t , x ) ) - \\o _ 0 ^ \\ell ( X ^ \\beta ( 0 ; t , x ) ) | , \\end{align*}"}
+{"id": "1586.png", "formula": "\\begin{align*} x _ 1 = \\left ( \\begin{matrix} 1 \\\\ \\alpha + \\alpha ^ 2 \\\\ \\alpha \\end{matrix} \\right ) , x _ 2 = \\left ( \\begin{matrix} 1 \\\\ \\sigma ( \\alpha ) + \\sigma ( \\alpha ) ^ 2 \\\\ \\sigma ( \\alpha ) \\end{matrix} \\right ) , x _ 3 = \\left ( \\begin{matrix} 1 \\\\ \\sigma ^ 2 ( \\alpha ) + \\sigma ^ 2 ( \\alpha ) ^ 2 \\\\ \\sigma ^ 2 ( \\alpha ) \\end{matrix} \\right ) , \\end{align*}"}
+{"id": "9068.png", "formula": "\\begin{align*} 0 \\rightarrow \\mathcal { O } _ C \\rightarrow \\bigoplus _ { j = 1 } ^ n \\mathcal { O } _ { C _ j } \\rightarrow \\mathcal { T } \\rightarrow 0 . \\end{align*}"}
+{"id": "2672.png", "formula": "\\begin{align*} \\lambda ^ { \\prime } ( t ) \\mathcal { E } ( t ) + \\mu \\alpha ^ { \\prime } ( t ) \\mathcal { E } ^ r ( t ) \\langle P ( u _ t ) , u \\rangle = \\frac { \\lambda ^ { \\prime } ( t ) } { \\lambda ( t ) } \\mathcal { H } ( t ) + \\mu \\bigl ( \\frac { \\alpha ( t ) } { \\lambda ( t ) } \\bigr ) ^ { \\prime } \\lambda ( t ) \\mathcal { E } ^ r ( t ) \\langle P ( u _ t ) , u \\rangle . \\end{align*}"}
+{"id": "949.png", "formula": "\\begin{align*} Q ( t ) ^ { - 1 } P ^ { - 1 } \\begin{pmatrix} w ( \\xi _ 0 ) \\\\ \\overline { w ( \\xi _ 0 ) } \\end{pmatrix} = \\begin{pmatrix} \\beta _ 1 ( \\xi _ 0 ) \\\\ \\beta _ 2 ( \\xi _ 0 ) \\end{pmatrix} + O ( \\varepsilon _ 1 ^ 2 \\varepsilon _ 2 t ^ { - 1 / 4 + C _ 1 \\varepsilon ^ 2 _ 1 + 2 \\mu C _ 2 \\varepsilon ^ 2 _ 1 + C _ 4 \\varepsilon _ 1 ^ 2 } ) \\end{align*}"}
+{"id": "4797.png", "formula": "\\begin{align*} a _ n ( u _ n , v _ n ) = ( f _ n , v _ n ) v _ n \\in X _ n , \\end{align*}"}
+{"id": "4604.png", "formula": "\\begin{align*} \\frac { x _ i } { g } ( x _ 1 - x _ k ) = \\begin{cases} x _ i & ( i = k ) \\\\ 0 & ( i \\neq k ) . \\end{cases} \\end{align*}"}
+{"id": "3991.png", "formula": "\\begin{align*} \\bar { \\mathbf { a } } + \\bar { \\mathbf { b } } = ( \\bar { \\alpha } _ 1 + \\bar { \\beta } _ 1 , \\bar { \\alpha } _ 2 + \\bar { \\beta } _ 2 , \\bar { \\alpha } _ 3 + \\bar { \\beta } _ 3 , \\bar { \\alpha } _ 4 + b _ 1 , a _ 1 + b _ 2 , \\cdots , a _ { 2 k - 3 } + b _ { 2 k - 2 } , a _ { 2 k - 2 } + b _ { 2 k - 1 } , a _ { 2 k - 1 } + \\bar { \\beta } _ 4 ) . \\end{align*}"}
+{"id": "8381.png", "formula": "\\begin{align*} & \\Psi ^ - _ 1 ( x ; z ) : = e _ 1 + \\int _ { - \\infty } ^ x \\left ( 1 , e ^ { 2 i z ( x - y ) } \\right ) \\widetilde { Q } \\Psi ^ - _ 1 ( y ; z ) d y , \\\\ & \\Psi ^ - _ 2 ( x ; z ) : = e _ 2 + \\int _ { - \\infty } ^ x \\left ( e ^ { - 2 i z ( x - y ) } , 1 \\right ) \\widetilde { Q } \\Psi ^ - _ 2 ( y ; z ) d y . \\end{align*}"}
+{"id": "3594.png", "formula": "\\begin{align*} \\tilde { d } _ { ( u , s ' ) } = \\sum _ { n = 0 } ^ { Y _ u - 1 } d _ { ( u , s ' Y _ u + n ) } \\end{align*}"}
+{"id": "1358.png", "formula": "\\begin{align*} ( f - g ) ( y ) & = ( f - g ) \\bigl ( \\sum _ { j \\in J } \\frac { 1 } { m } \\ , m _ { u _ j , v _ j } \\bigr ) \\leq ( \\| f \\| + \\| g \\| ) \\sum _ { j \\in J } \\frac { 1 } { m } \\| m _ { u _ j , v _ j } \\| \\\\ & \\leq ( 2 + 2 \\theta ) \\frac { 2 m } { m n } = \\frac { 4 ( 1 + \\theta ) } { n } . \\end{align*}"}
+{"id": "1891.png", "formula": "\\begin{align*} \\cos S _ c ( x , y ) = \\tau ^ - ( x , y ) . \\end{align*}"}
+{"id": "6399.png", "formula": "\\begin{align*} \\omega ^ { - 1 } ( d x ^ i , d x ^ j ) = \\omega ^ { - 1 } ( d \\theta _ i , d \\theta _ j ) = 0 , \\omega ^ { - 1 } ( d \\theta _ i , d x ^ j ) = \\delta _ i ^ j , \\end{align*}"}
+{"id": "348.png", "formula": "\\begin{align*} \\langle \\ 1 , f ^ { i } \\rangle = 0 ~ ~ ~ i = 1 , \\dotsc , n - 1 \\end{align*}"}
+{"id": "2653.png", "formula": "\\begin{align*} d _ { e } ( 8 ) & = 5 \\gamma ( 4 ) + \\gamma ( 8 ) \\\\ & = 5 [ \\sigma _ { 0 } ( 1 ) - \\sigma _ { 0 } ( 0 ) ] + [ \\sigma _ { 0 } ( 2 ) + \\sigma _ { 0 } ( 0 ) - \\sigma _ { 0 } ( 1 ) \\\\ & = \\sigma _ { 0 } ( 2 ) + 4 \\sigma _ { 0 } ( 1 ) - 5 \\sigma _ { 0 } ( 0 ) \\\\ & = 2 + 4 ( 1 ) - 5 ( 0 ) \\\\ & = 6 . \\end{align*}"}
+{"id": "8651.png", "formula": "\\begin{align*} { \\bf { \\bar q } } \\left [ i \\right ] = \\sum \\limits _ { l = 1 } ^ { L ' } { { { { \\bf { F } } } _ l } { \\bf { d } } \\left [ { i - { \\kappa _ l } } \\right ] } , \\end{align*}"}
+{"id": "9223.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ( t ) - x _ a ( t ) \\| & \\ , \\le \\| x ( t ) - z ( t ) \\| + \\| z ( t ) - x _ a ( t ) \\| \\\\ & \\ , \\le \\| x ( t ) - z ( t ) \\| + \\gamma \\| \\epsilon ( x _ a ( t ) , t ) \\| \\\\ & \\ , \\le \\| x ( t ) - z ( t ) \\| + \\gamma 2 M _ \\mathcal { D } T . \\end{aligned} \\end{align*}"}
+{"id": "2131.png", "formula": "\\begin{align*} A _ { 1 1 , 0 } = T A _ { 1 1 } - 2 \\Gamma _ { 0 1 } ^ 1 A _ { 1 1 } , \\end{align*}"}
+{"id": "2775.png", "formula": "\\begin{align*} K _ { N } : = \\frac { N ^ { 4 } } { \\sqrt { \\log N } } e ^ { - \\frac { a _ { N } ^ { 2 } } { 2 \\gamma \\log N } } , \\end{align*}"}
+{"id": "1927.png", "formula": "\\begin{align*} \\left ( \\sum _ { n = 0 } ^ \\infty a ( n ) q ^ n \\right ) \\ | \\ U ( j ) : = \\sum _ { n = 0 } ^ \\infty a ( j n ) q ^ n . \\end{align*}"}
+{"id": "6472.png", "formula": "\\begin{align*} \\mu ( X ( s _ 1 ) \\in B _ 1 , \\cdots , X ( s _ r ) \\in B _ r ) = \\frac { 1 } { ( 2 \\pi ) ^ { \\frac { r } { 2 } } . | C | } \\int _ { B _ 1 \\times \\cdots \\times B _ r } \\exp \\big ( - \\frac { 1 } { 2 } . { } ^ { t } { x } . C ^ { - 1 } . x \\big ) d x , \\end{align*}"}
+{"id": "336.png", "formula": "\\begin{align*} c _ \\lambda = \\sum _ { T } \\operatorname { s g n } ( T ) \\end{align*}"}
+{"id": "1038.png", "formula": "\\begin{align*} \\ell ( x _ 1 \\ast x _ 2 ) = \\ell ( y ) + \\ell ( x _ 2 ) \\leq \\ell ( x _ 1 ) + \\ell ( x _ 2 ) - d ( v _ 1 \\Rightarrow w _ 2 v _ 2 ) - d ( w ' w _ 2 v _ 2 \\Rightarrow w _ 1 v _ 1 ) . \\end{align*}"}
+{"id": "7852.png", "formula": "\\begin{align*} \\gamma _ 1 : = \\frac { 1 } { 2 } \\left ( c _ 5 ( q , \\ell , p _ 0 ) - \\frac { c _ 2 ( q , \\ell , p _ 0 ) c _ 3 ( q , 2 \\ell , \\ell , p _ 0 ) } { c _ 1 ( q , 2 \\ell , p _ 0 ) } \\right ) . \\end{align*}"}
+{"id": "3467.png", "formula": "\\begin{align*} R _ { \\ell } ^ + : = [ \\ell q _ n + m _ n + 1 , \\ell q _ n + b _ n ] R _ { \\ell } ^ - : = [ \\ell q _ n - b _ n , \\ell q _ n + m _ n - 1 ] , \\end{align*}"}
+{"id": "6438.png", "formula": "\\begin{align*} \\begin{aligned} & T ( k _ 1 , \\ldots , k _ n ; ( \\alpha , \\beta ) ) - T ( k _ n , k _ 1 , \\ldots , k _ { n - 1 } ; ( \\alpha , \\beta ) ) \\\\ = & Z _ { I I } ( k _ n , k _ 1 , \\ldots , k _ { n - 2 } , k _ { n - 1 } + 1 ; ( \\alpha , \\beta ) ) \\\\ & - \\sum _ { j = 0 } ^ { k _ n - 2 } Z _ { I } ( j + 1 , k _ 1 , \\ldots , k _ { n - 1 } , k _ n - j ; ( \\alpha , \\beta ) ) , \\end{aligned} \\end{align*}"}
+{"id": "7136.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathcal ( \\delta f ) ^ { - 1 } \\left ( j ^ 2 _ { f ( x ) } \\phi \\right ) & = j ^ 2 _ x ( \\phi \\circ f ) & \\\\ \\mathcal ( \\delta f ) ^ { - 1 } \\left ( j ^ 2 _ { f ( x ) } \\phi \\right ) & = j ^ { 2 ( n _ i + 1 ) } _ x ( \\phi \\circ f ) & \\end{aligned} \\right . \\end{align*}"}
+{"id": "5072.png", "formula": "\\begin{align*} \\begin{aligned} f _ { 1 } ^ { + } & = \\inf \\left \\{ \\int _ { \\Omega } | B | ^ { 2 } \\dd x \\ \\middle | \\ B \\in L ^ { 2 } _ { \\sigma } ( \\Omega ) , \\ \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B \\cdot B \\dd x = 1 \\right \\} , \\\\ - f _ { 1 } ^ { - } & = \\inf \\left \\{ \\int _ { \\Omega } | B | ^ { 2 } \\dd x \\ \\middle | \\ B \\in L ^ { 2 } _ { \\sigma } ( \\Omega ) , \\ \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B \\cdot B \\dd x = - 1 \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "6206.png", "formula": "\\begin{align*} E ' _ 0 = \\kappa \\left ( \\frac { B _ 3 } { 2 \\sqrt { \\kappa B _ 4 } } + L + 5 \\right ) ^ 2 - \\left ( \\frac { Q } { 2 ( L + 2 ) } - \\sqrt { \\kappa B _ 4 } \\right ) ^ 2 , \\end{align*}"}
+{"id": "5620.png", "formula": "\\begin{align*} \\hat { \\mu } ( < a _ n . . . a _ 1 | b _ 1 . . . b _ m > ) = \\mu ( [ a _ n . . . . a _ 1 b _ 1 . . . b _ m ] ) \\ , \\end{align*}"}
+{"id": "9149.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { 1 } y _ \\delta ( x _ a , \\tau ) \\ , d \\tau & = \\dfrac { a _ { 0 , \\delta } ( x _ a ) } { 2 } \\\\ \\displaystyle \\int _ { 0 } ^ { 1 } \\left ( y _ \\delta ( { x _ a } , \\tau ) - \\bar { y } _ a \\right ) u ( \\tau ) \\ , d \\tau & = \\dfrac { b _ { 1 , \\delta } ( x _ a ) } { 2 } \\end{align*}"}
+{"id": "317.png", "formula": "\\begin{align*} g ( f ) ( x ) = \\Big ( \\int _ { 0 } ^ { \\infty } \\Big | \\frac { d } { d t } e ^ { - t L } ( f ) ( x ) \\Big | ^ 2 t d t \\Big ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "5658.png", "formula": "\\begin{align*} E ^ { 1 } _ { p , q } ( n ) & = H _ { q } ( O _ { n , n } , C _ { p } ( n ) ) \\Rightarrow H _ { p + q } ( O _ { n , n } , C _ { * } ( n ) ) . \\end{align*}"}
+{"id": "575.png", "formula": "\\begin{align*} & - \\alpha \\leq \\theta _ 1 \\leq \\theta \\leq \\theta _ 2 \\leq \\alpha , \\\\ & \\omega ( s ) = - s e ^ { - i \\left ( \\theta _ 1 + \\theta _ 2 \\right ) / 2 } , \\\\ & s \\geq 0 . \\end{align*}"}
+{"id": "605.png", "formula": "\\begin{align*} E _ q : = \\{ n \\in N \\ , : \\ , \\pi _ N ( q ^ { - 1 } n ) \\in E \\} \\end{align*}"}
+{"id": "3939.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta w _ { n } & = f ( \\cdot , y ) - \\mathcal { N } _ { n } ( \\cdot , y _ { n } ) , \\ ; \\ ; \\Omega , \\\\ w _ n & = 0 , \\ , \\ ; \\ ; \\qquad \\qquad \\qquad \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5515.png", "formula": "\\begin{align*} \\phi _ p ( x ) = ( 1 + x ) ^ { - p / 2 } \\end{align*}"}
+{"id": "8370.png", "formula": "\\begin{align*} N _ + ( x ; k ) = I + \\mathcal { P } ^ + \\left ( N _ - J \\right ) ( z ) , z \\in \\mathbb { R } . \\end{align*}"}
+{"id": "8554.png", "formula": "\\begin{align*} ( I ^ \\alpha _ { 0 + } \\ , f ) ( t ) : = ( h _ \\alpha \\ , * \\ , f ) ( t ) \\ , = \\ , \\frac { 1 } { \\Gamma ( \\alpha ) } \\ , \\int _ 0 ^ t ( t - \\tau ) ^ { \\alpha - 1 } f ( \\tau ) \\ , d \\tau , \\ t > 0 \\end{align*}"}
+{"id": "7747.png", "formula": "\\begin{align*} F _ k ( x ) = \\max _ { i = 1 , \\ldots , 2 ^ k } x _ i . \\end{align*}"}
+{"id": "8732.png", "formula": "\\begin{align*} \\nu = \\int _ Y \\nu _ y \\ , \\mathrm { d } \\mu . \\end{align*}"}
+{"id": "6345.png", "formula": "\\begin{align*} F ( x ) = & \\frac { 1 } { \\operatorname { B } ( a , b ) } \\sum ^ { \\infty } _ { n = 0 } \\frac { ( - 1 ) ^ n \\Gamma ( b ) } { \\Gamma ( b - n ) n ! } \\int ^ { \\infty } _ { \\frac { \\theta } { x ^ 2 } } \\exp \\{ - ( a + n ) u \\} \\mathrm { d } u \\end{align*}"}
+{"id": "948.png", "formula": "\\begin{align*} w ( t , \\xi _ 0 ) = \\beta ( \\xi _ 0 ) + O ( t ^ { - 1 / 4 + C _ 4 \\varepsilon _ 1 ^ 2 } ) . \\end{align*}"}
+{"id": "4644.png", "formula": "\\begin{align*} d _ { \\widetilde K } = \\# \\{ 1 \\leq r \\leq j : C _ r \\in O _ { \\widetilde K } , \\ \\lambda _ r = 1 \\} . \\end{align*}"}
+{"id": "7282.png", "formula": "\\begin{align*} [ F ( \\gamma ) : F ] = m , \\end{align*}"}
+{"id": "4336.png", "formula": "\\begin{align*} \\int _ { \\{ - t ' _ 1 \\le \\Psi < - t ' _ 2 \\} } | \\tilde { F } | ^ 2 _ h \\le \\frac { G ( T _ 1 ) } { \\int _ { T _ 1 } ^ { + \\infty } c ( s ) e ^ { - s } d s } \\int _ { t ' _ 2 } ^ { t ' _ 1 } e ^ { - s } d s . \\end{align*}"}
+{"id": "932.png", "formula": "\\begin{align*} T ^ { \\ast } : = \\sup \\left \\{ T > 0 : \\sup _ { 0 \\le t < T } \\langle t \\rangle ^ { - \\delta } ( \\| u _ 2 ( t ) \\| _ { H ^ 1 } + \\| J u _ 2 ( t ) \\| _ { L ^ 2 } ) < 4 \\varepsilon _ 2 \\right \\} . \\end{align*}"}
+{"id": "426.png", "formula": "\\begin{align*} ( c _ { i j k } ) _ { i k } = \\theta ^ { j - 1 } \\end{align*}"}
+{"id": "6747.png", "formula": "\\begin{align*} \\mathrm { E } ( X ^ n ) = \\int _ { - \\infty } ^ \\infty x ^ n g ( x ) \\mathrm { d } x = \\frac { 1 } { \\sigma \\mathrm { B } ( \\alpha , \\beta ) } \\sum _ { j = 0 } ^ { \\infty } ( - 1 ) ^ j \\binom { \\beta - 1 } { j } \\ , \\int _ { - \\infty } ^ \\infty x ^ n \\ , h _ j ^ { ( s ) } \\mathrm { d } x . \\end{align*}"}
+{"id": "5093.png", "formula": "\\begin{align*} & \\int _ { \\Omega } ( | u _ n | ^ { 2 } + | B _ n | ^ { 2 } ) \\dd x \\leq \\int _ { \\Omega } ( | u _ { 0 , n } | ^ { 2 } + | B _ { 0 , n } | ^ { 2 } ) \\dd x , \\\\ & \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B _ { n } \\cdot B _ { n } \\dd x = \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B _ { 0 , n } \\cdot B _ { 0 , n } \\dd x . \\end{align*}"}
+{"id": "2875.png", "formula": "\\begin{align*} | \\bar K _ 1 ^ { ( n ) } ( x , y ) | \\le \\frac { C } { 1 + ( x - y ) ^ 2 } , y = 0 , \\ldots , n , \\ , x \\in ( \\delta n , ( 1 - \\delta ) n ) \\end{align*}"}
+{"id": "3510.png", "formula": "\\begin{align*} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\sum _ { i = 1 } ^ { n - 1 } \\left ( i ^ { p } \\ - \\ 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } i ^ { p - 2 j - 1 } \\right ) \\cdot F _ { n + 1 - i } , \\end{align*}"}
+{"id": "7447.png", "formula": "\\begin{align*} \\lambda ( m , k + l ) & \\geq \\max _ { x \\in [ - 1 , 1 ] } \\left ( \\frac { \\delta _ { m , k } ( 1 + 2 x + x ^ 2 ) } { 4 } + \\frac { \\delta _ { m , l } ( 1 - 2 x + x ^ 2 ) } { 4 } + \\frac { \\sqrt { m } } { 2 } ( 1 - x ^ 2 ) \\right ) \\\\ & = \\frac { 1 } { 4 } \\max _ { x \\in [ - 1 , 1 ] } \\left ( a x ^ 2 + 2 b x + c \\right ) , \\end{align*}"}
+{"id": "4802.png", "formula": "\\begin{align*} H _ 0 ^ 2 ( D ) : = \\left \\{ v \\in H ^ 2 ( D ) : v = \\frac { \\partial v } { \\partial \\nu } = 0 \\partial D \\right \\} , \\end{align*}"}
+{"id": "6052.png", "formula": "\\begin{align*} 2 H = \\alpha \\frac { \\nu _ 3 } { z } + \\varpi , \\end{align*}"}
+{"id": "5686.png", "formula": "\\begin{align*} x ^ { t + 1 } = ( I + \\gamma _ t B ) ^ { - 1 } \\left ( x ^ t + \\alpha _ t ( x ^ t - x ^ { t - 1 } ) - \\gamma _ t ( F ( x ^ t ) + \\beta _ t ( F ( x ^ t ) - F ( x ^ { t - 1 } ) ) ) \\right ) , \\end{align*}"}
+{"id": "968.png", "formula": "\\begin{align*} \\gamma _ { s p } ( B ( G _ 1 , G _ 2 , G _ 3 ) ) & = \\gamma _ { s p } ( B ( H , G _ 3 ) ) \\\\ & \\geq \\gamma _ { s p } ( H ) + \\gamma _ { s p } ( G _ 3 ) - 1 \\\\ & = \\gamma _ { s p } ( B ( G _ 1 , G _ 2 ) ) + \\gamma _ { s p } ( G _ 3 ) - 1 \\\\ & \\geq \\gamma _ { s p } ( G _ 1 ) + \\gamma _ { s p } ( G _ 2 ) + \\gamma _ { s p } ( G _ 3 ) - 2 . \\end{align*}"}
+{"id": "5991.png", "formula": "\\begin{align*} \\phi \\rightarrow \\phi e ^ { i \\chi } , \\ \\ A _ { \\mu } = A _ { \\mu } - \\partial _ { \\mu } \\chi , \\end{align*}"}
+{"id": "8215.png", "formula": "\\begin{align*} x _ { i , 1 , k } & = 2 m - 3 i + k + 1 , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 1 , k } & = 4 m + 3 i - k , \\\\ x _ { i , 2 , k } & = 4 m + 3 i - k - 1 , \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 2 , k } & = 2 m - 3 i + k + 2 , \\\\ x _ { i , 3 , k } & = 4 m - 3 i + 2 k + 1 , \\mbox { a n d } \\\\ \\tfrac { 1 } { 2 } S - x _ { i , 3 , k } & = 2 m + 3 i - 2 k . \\end{align*}"}
+{"id": "1224.png", "formula": "\\begin{align*} \\sup _ { y \\in \\Z ^ d } \\sum _ { \\Delta \\ni y : \\ c _ \\Delta > 0 } \\sum _ { z \\neq y } \\sum _ { i = 1 } ^ q \\norm { \\nabla ^ i _ z \\gamma _ \\Delta ( \\cdot | \\cdot ) } _ \\infty < \\infty , \\end{align*}"}
+{"id": "1766.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\int _ 0 ^ T \\int _ { \\Omega } u _ { \\epsilon } ( t , x ) \\phi \\left ( t , x , \\frac { x } { \\epsilon } \\right ) d x d t = \\int _ 0 ^ T \\int _ { \\Omega } \\int _ Y u _ 0 ( t , x , y ) \\phi ( t , x , y ) d y d x d t . \\end{align*}"}
+{"id": "8888.png", "formula": "\\begin{align*} \\epsilon _ { f \\otimes \\chi _ D } = \\begin{cases} - \\lambda _ f ( p ) p ^ { 1 / 2 } & ( p , D ) = 1 D \\equiv \\square \\mod 4 p ; \\\\ 1 & p | D . \\end{cases} \\end{align*}"}
+{"id": "4722.png", "formula": "\\begin{align*} & ( i ) \\ \\ \\ \\nabla ( J + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k f _ k ) ( x ^ * ) = 0 \\\\ & ( i i ) \\ \\ \\gamma _ k f _ k ( x ^ * ) = 0 \\ \\ k \\in \\{ 1 , . . . , m \\} \\\\ & ( i i i ) \\ A _ J + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k A _ k \\succeq 0 \\end{align*}"}
+{"id": "4538.png", "formula": "\\begin{align*} \\mathcal { I } ( x , t ) = \\bigcup _ { i = p + 1 } ^ { n } [ t _ { i , e n } ( x , t ) , t _ { i , e x } ( x , t ) ] . \\end{align*}"}
+{"id": "3856.png", "formula": "\\begin{align*} \\gamma ( s , \\tau ) = \\left ( r _ * \\cos \\left ( \\frac { - s - a _ 1 \\tau } { \\sqrt { k ^ 2 + r _ * ^ 2 } } \\right ) , r _ * \\sin \\left ( \\frac { - s - a _ 1 \\tau } { \\sqrt { k ^ 2 + r _ * ^ 2 } } \\right ) , \\frac { k s - b _ 1 \\tau } { \\sqrt { k ^ 2 + r _ * ^ 2 } } \\right ) ^ t , \\end{align*}"}
+{"id": "5770.png", "formula": "\\begin{align*} A \\circ \\widetilde \\Phi \\circ \\widetilde { \\mathcal { P } } \\circ \\widetilde \\Phi ^ { - 1 } \\circ A ^ { - 1 } = \\mathcal { P } ' \\end{align*}"}
+{"id": "7075.png", "formula": "\\begin{gather*} \\mathcal { K } _ { \\psi } ( \\Omega ) : = \\{ w \\in W ^ { 1 , p } ( \\Omega ) : w \\geq \\psi \\ \\ \\Omega \\} \\end{gather*}"}
+{"id": "796.png", "formula": "\\begin{align*} \\mathrm { D E } ( \\Sigma , c ) ( V , w ) & = \\frac { \\mathrm { d } } { \\mathrm { d } t } \\mathrm { E } \\big ( \\Gamma _ t , c _ t \\big ) _ { \\ , | t = 0 } = \\frac { \\mathrm { d } } { \\mathrm { d } t } \\int _ { \\Gamma _ t } G ( c _ t ) \\ , \\mathrm { d } \\mathcal { H } ^ d _ { \\ , | t = 0 } \\\\ & = \\int _ { \\Sigma } \\partial ^ \\square \\big ( G ( c _ t ) \\big ) _ { \\ , | t = 0 } - G ( c ) H V \\ , \\mathrm { d } \\mathcal { H } ^ d = \\int _ \\Sigma G ' ( c ) w - G ( c ) H V \\ , \\mathrm { d } \\mathcal { H } ^ d \\end{align*}"}
+{"id": "7243.png", "formula": "\\begin{align*} \\sigma ( \\alpha + \\beta ) = \\sigma ( \\alpha ) + \\sigma ( \\beta ) \\end{align*}"}
+{"id": "3193.png", "formula": "\\begin{align*} g ( \\xi , \\hat s ) = ( g \\xi , \\hat s + \\beta ( g ^ { - 1 } o , o , \\xi ) . \\end{align*}"}
+{"id": "1450.png", "formula": "\\begin{align*} y _ { k } \\left [ l \\right ] & = \\mathbf { h } _ { k } ^ { H } \\mathbf { x } \\left [ l \\right ] + z _ { k } \\left [ l \\right ] \\\\ & = \\mathbf { h } _ { k } ^ { H } \\mathbf { p } _ { \\mathrm { c } } s _ { \\mathrm { c } } \\left [ l \\right ] + \\mathbf { h } _ { k } ^ { H } \\sum _ { i \\in \\mathcal { K } } \\mathbf { p } _ { \\mathrm { p } , i } s _ { \\mathrm { p } , i } \\left [ l \\right ] + z _ { k } \\left [ l \\right ] , \\end{align*}"}
+{"id": "7644.png", "formula": "\\begin{align*} \\alpha ^ { * , \\xi } _ t : = - R ^ { - 1 } B ( P _ t x ^ { * , \\xi } _ t + \\varphi _ t ^ { * , \\xi } ) - h ( \\mu _ t ^ { * , \\xi } ) . \\end{align*}"}
+{"id": "5949.png", "formula": "\\begin{align*} \\begin{cases} d u = \\nabla \\cdot [ a ( u ) \\nabla u + b ( u ) ] d t , ( 0 , T ] \\times \\mathbb { T } ^ d , \\\\ u ( 0 ) = u _ 0 , \\end{cases} \\end{align*}"}
+{"id": "1368.png", "formula": "\\begin{align*} ( Z _ d ( n ) ) _ { n \\geq 0 } \\stackrel { d } { = } ( L ( d , n ) ) _ { n \\geq 0 } . \\end{align*}"}
+{"id": "8115.png", "formula": "\\begin{align*} C ( G , x ) : = 1 + \\sum _ { k = 1 } ^ { \\omega ( G ) } c _ { k } ( G ) , \\end{align*}"}
+{"id": "2545.png", "formula": "\\begin{align*} N _ { i t } = - \\log \\left ( \\max \\left ( \\left \\| \\mathbf { r } _ p ( \\mathbf { q } _ w ) \\right \\| , \\left \\| \\mathbf { r } _ d ( \\mathbf { q } _ w ) \\right \\| , \\mu ( \\mathbf { q } _ w ) \\right ) / \\epsilon _ u \\right ) / \\log \\left ( 1 - \\delta / \\sqrt { 2 k + 1 } \\right ) . \\end{align*}"}
+{"id": "2285.png", "formula": "\\begin{align*} \\langle T f , g \\rangle _ \\lambda = \\langle T ^ { ( \\lambda ) } _ a f , g \\rangle _ \\lambda \\end{align*}"}
+{"id": "8611.png", "formula": "\\begin{align*} K \\oplus _ 2 L \\supset K \\widetilde { + } _ 2 L : = \\cup _ { u \\in S ^ { n - 1 } } \\left [ 0 , \\sqrt { \\rho _ K ^ 2 ( u ) + \\rho _ L ^ 2 ( u ) } u \\right ] . \\end{align*}"}
+{"id": "379.png", "formula": "\\begin{align*} \\frac { \\lambda _ { 1 } + \\lambda _ { 2 } } { \\lambda _ { 1 } \\lambda _ { 2 } } = \\frac { - 1 } { 2 } \\frac { ( 2 n - 1 ) } { ( n + 4 ) } + \\frac { n - 1 } { 2 } ( c ^ 2 + ( n - 1 ) a ^ 2 + ( n - 1 ) b ^ 2 ) . \\end{align*}"}
+{"id": "7478.png", "formula": "\\begin{align*} \\left \\| \\div \\big ( | \\nabla u | \\nabla u \\big ) \\right \\| _ { L ^ 1 ( B _ { 3 / 4 } ) } \\leq C \\| \\nabla u \\| _ { L ^ 2 ( B _ { 1 } ) } ^ 2 . \\end{align*}"}
+{"id": "173.png", "formula": "\\begin{align*} R S _ { H ' } ( C ^ \\bullet ) : = R \\Gamma ( H ' , C ^ { \\bullet } ) ^ { R \\Gamma _ { H ' } - l a } . \\end{align*}"}
+{"id": "54.png", "formula": "\\begin{align*} \\begin{cases} R _ 1 | | | \\nabla _ { H } u | | | _ { R _ 1 , 2 R _ 1 } \\leq C ( 1 + K ^ { 1 / 2 } ) | | u \\psi ^ { 1 / 2 } | | _ { 4 R _ 1 } , \\\\ R _ 2 | | | \\nabla _ { H } u | | | _ { R _ 2 , 2 R _ 2 } \\leq C ( 1 + K ^ { 1 / 2 } ) | | u \\psi ^ { 1 / 2 } | | _ { R _ 0 } . \\end{cases} \\end{align*}"}
+{"id": "5178.png", "formula": "\\begin{align*} H [ b _ n ] = 4 \\pi \\int _ { D } ( \\phi _ n - \\phi _ \\infty ) _ + G _ n \\frac { 1 } { r } \\dd z \\dd r + 4 \\pi \\int _ { \\mathbb { R } ^ { 2 } _ { + } \\backslash D } ( \\phi _ n - \\phi _ \\infty ) _ + G _ n \\frac { 1 } { r } \\dd z \\dd r . \\end{align*}"}
+{"id": "7360.png", "formula": "\\begin{gather*} P ( c ) = \\inf _ { Q \\in K } Q ( c ) \\quad \\quad P \\in K \\end{gather*}"}
+{"id": "6717.png", "formula": "\\begin{align*} { \\rm { t r } ( \\pi _ { \\lambda , c _ 1 } ' ( w ) ) } = \\pm { \\rm { t r } } ( \\pi ( p ( w ) ) ) . \\end{align*}"}
+{"id": "7569.png", "formula": "\\begin{align*} y = \\sum _ { m = 0 } ^ \\infty \\frac { \\psi ( \\eta _ m ( y ) ) } { \\gamma ^ { m + 1 } } . \\end{align*}"}
+{"id": "5590.png", "formula": "\\begin{align*} A ( y _ 1 x ) - A ( y _ 1 x ' ) = A ( 1 ^ { k + 1 } 0 . . . ) - A ( 1 0 ^ \\infty ) = c _ { k + 1 } - d \\ . \\end{align*}"}
+{"id": "3820.png", "formula": "\\begin{align*} \\phi ( U | \\bar U ; x , t ) & : = ( \\nabla \\bar A ) ^ { - 1 } ( A - \\bar A ) - ( U - \\bar U ) \\ ; , \\\\ G _ 1 ( U | \\bar U ; x , t ) & : = \\nabla G - \\nabla \\bar G - \\nabla ^ 2 \\bar G \\cdot ( \\nabla \\bar A ) ^ { - 1 } ( A - \\bar A ) \\ ; , \\\\ G _ 2 ( U | \\bar U ; x , t ) & : = G _ { x _ \\alpha } - \\bar G _ { x _ \\alpha } - \\nabla \\bar G _ { x _ \\alpha } \\cdot ( \\nabla \\bar A ) ^ { - 1 } ( A - \\bar A ) \\ ; , \\end{align*}"}
+{"id": "8555.png", "formula": "\\begin{align*} ( D ^ \\alpha _ { 0 + } \\ , f ) ( t ) : = \\frac { d } { d t } \\ , ( I ^ { 1 - \\alpha } _ { 0 + } \\ , f ) ( t ) , \\ t > 0 . \\end{align*}"}
+{"id": "920.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathcal { L } u _ 1 = 3 | u _ 1 | ^ 2 u _ 1 - ( 2 u _ 1 | u _ 2 | ^ 2 + \\overline { u _ 1 } u _ 2 ^ 2 ) , \\\\ & \\mathcal { L } u _ 2 = ( 2 | u _ 1 | ^ 2 u _ 2 + u _ 1 ^ 2 \\overline { u _ 2 } ) - 3 | u _ 2 | ^ 2 u _ 2 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6962.png", "formula": "\\begin{align*} a = \\frac { u _ 1 - u _ 0 \\beta } { \\alpha - \\beta } , b = \\frac { u _ 0 \\alpha - u _ 1 } { \\alpha - \\beta } \\end{align*}"}
+{"id": "2401.png", "formula": "\\begin{align*} l = \\max \\{ \\textnormal { d i s t } ( \\rho , v ) : v \\in T \\} . \\end{align*}"}
+{"id": "1333.png", "formula": "\\begin{align*} y = \\sum _ { j = 1 } ^ m \\alpha _ j ' \\ , m _ { a _ j , b _ j } + \\lambda _ { n + 1 } \\ , m _ { p _ { n + 1 } , q _ { n + 1 } } . \\end{align*}"}
+{"id": "6739.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } \\alpha } \\log \\mathrm { B } ( \\alpha , \\beta ) & = \\psi ( \\alpha ) - \\psi ( \\alpha + \\beta ) . \\end{align*}"}
+{"id": "8909.png", "formula": "\\begin{align*} \\lambda _ { 1 2 } + \\lambda _ { 1 2 3 ' } & \\geq \\log \\lceil 2 ^ { \\lambda _ { 1 2 3 ' } } \\rceil , \\\\ \\lambda _ { 1 2 3 ' } & = \\log m , m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "772.png", "formula": "\\begin{align*} | g ( x , t ) | & \\leq \\int _ { 5 I _ Q } \\frac { | \\partial _ u ^ M \\varphi ( x , u ) - \\partial _ u ^ M \\varphi ( x , t ) | } { | u - t | ^ { \\frac 1 { 2 s } - M + 1 } } d u + \\| \\partial _ t ^ M \\varphi \\| _ \\infty \\int _ { ( 5 I _ Q ) ^ c } \\frac { d u } { | u - s | ^ { \\frac 1 { 2 s } - M + 1 } } = T _ 1 + T _ 2 . \\end{align*}"}
+{"id": "5211.png", "formula": "\\begin{align*} | | \\mathbb { P } \\xi | | _ { L ^ { \\infty } ( \\mathbb { R } ^ { 3 } ) } \\lesssim | | \\mathbb { P } \\xi | | _ { W ^ { 1 , 4 } ( \\mathbb { R } ^ { 3 } ) } \\lesssim | | \\xi | | _ { W ^ { 1 , 4 } ( \\mathbb { R } ^ { 3 } ) } = | | \\xi | | _ { W ^ { 1 , 4 } ( B ( 0 , R ) ) } . \\end{align*}"}
+{"id": "7533.png", "formula": "\\begin{align*} \\deg _ H ( S ) \\leq \\binom { p - \\ell } { k - \\ell } \\leq d ^ { \\frac { k - \\ell } { k - 1 } } . \\end{align*}"}
+{"id": "8590.png", "formula": "\\begin{align*} | P _ { E } \\left ( T \\sum _ { i = 1 } ^ n [ 0 , e _ i ] \\right ) | _ { n - m } = & \\sum _ { | I | = n - m } | \\det ( e _ 1 , \\dots , e _ m , \\{ P _ E T e _ i \\} _ { i \\in I } | _ { n } \\\\ = & \\sum _ { | I | = n - m } | \\det ( e _ 1 , \\dots , e _ m , \\{ T e _ i \\} _ { i \\in I } | _ { n } \\\\ = & | \\det ( T ) | \\sum _ { | I | = n - m } | \\det ( T ^ { - 1 } e _ 1 , \\dots , T ^ { - 1 } e _ m , \\{ e _ i \\} _ { i \\in I } | _ { n } \\\\ = & | \\det ( T ) | \\sum _ { | J | = m } | \\det ( \\{ w _ i ^ J \\} _ { i = 1 } ^ m ) | , \\end{align*}"}
+{"id": "740.png", "formula": "\\begin{align*} \\partial ^ 2 _ { x x } p ^ { \\varepsilon } ( x , t ) = \\exp \\left [ - \\frac { Y _ t h ( x ) } { \\varepsilon } \\right ] \\partial ^ 2 _ { x x } q ^ { \\varepsilon } ( x , t ) & - \\frac { 2 Y _ t h ' ( x ) } { \\varepsilon } p ^ { \\varepsilon } ( x , t ) \\\\ & - p ^ { \\varepsilon } ( x , t ) \\left [ \\frac { \\left ( Y _ t h ' ( x ) \\right ) ^ 2 } { \\varepsilon } + \\frac { Y _ t h '' ( x ) } { \\varepsilon } . \\right ] \\end{align*}"}
+{"id": "6892.png", "formula": "\\begin{align*} \\log \\int _ { \\R ^ n } e ^ f \\ , d \\gamma _ t \\le \\sup _ { y \\in \\R ^ n } \\left ( \\int _ { \\R ^ n } f \\ , d \\gamma _ { y , t } - H ( \\gamma _ { y , t } \\ , | \\ , \\gamma _ t ) \\right ) + \\frac { t ^ 2 } { 2 } \\sum _ { i , j = 1 } ^ n \\int _ { \\R ^ n } | \\partial _ { i j } f | ^ 2 \\ , d \\gamma _ { y ^ * , t } . \\end{align*}"}
+{"id": "4888.png", "formula": "\\begin{align*} \\frac { \\partial f ( z ) } { \\partial z } = & \\frac { 1 } { \\mathrm { B } ( \\alpha , \\beta ) } \\phi ( z ) [ \\Phi ( z ) ] ^ { \\alpha - 2 } [ 1 - \\Phi ( z ) ] ^ { \\beta - 2 } \\times \\\\ & \\Big \\{ ( \\alpha - 1 ) \\phi ( z ) [ 1 - \\Phi ( z ) ] - ( \\beta - 1 ) \\phi ( z ) \\Phi ( z ) - z \\Phi ( z ) [ 1 - \\Phi ( z ) ] \\Big \\} . \\end{align*}"}
+{"id": "2851.png", "formula": "\\begin{align*} \\frak f _ x & : = \\frac 1 { 4 \\gamma } \\left ( q _ { x + 1 } - q _ x \\right ) \\left ( p _ x + p _ { x + 1 } \\right ) + \\frac 1 4 \\left ( q _ { x + 1 } - q _ x \\right ) ^ 2 , x = 0 , \\dots , n - 1 , \\\\ \\mathfrak F _ x & = p _ x ^ 2 + \\left ( q _ { x + 1 } - q _ x \\right ) \\left ( q _ { x } - q _ { x - 1 } \\right ) - \\omega _ 0 ^ 2 q _ x ^ 2 , x = 0 , \\dots , n , \\end{align*}"}
+{"id": "2855.png", "formula": "\\begin{align*} S ( t ) = \\left [ \\begin{array} { c c } { S ^ { ( q ) } ( t ) } & S ^ { ( q , p ) } ( t ) \\\\ S ^ { ( p , q ) } ( t ) & S ^ { ( p ) } ( t ) \\end{array} \\right ] , \\end{align*}"}
+{"id": "2995.png", "formula": "\\begin{align*} \\overline { A s y _ { n } ^ { \\vec { v } } ( X , T ) \\cap \\left ( \\operatorname { s u p p } \\mu _ { x } \\right ) ^ { ( n ) } } = \\left ( \\operatorname { s u p p } \\mu _ { x } \\right ) ^ { ( n ) } \\end{align*}"}
+{"id": "5430.png", "formula": "\\begin{align*} y _ n ^ 2 = \\left ( \\bar { y } _ n + \\sum _ { \\beta = 1 } ^ { n - 1 } \\rho _ \\beta ( x ' _ 0 ) y _ \\beta \\right ) ^ 2 \\leq 2 \\bar { y } _ n ^ 2 + C \\epsilon ^ 2 \\delta ^ 2 b _ \\alpha | y ' | ^ 2 . \\end{align*}"}
+{"id": "4822.png", "formula": "\\begin{align*} \\kappa ( A ) : = \\lVert A \\rVert \\ , \\lVert A ^ { - 1 } \\rVert = \\frac { s _ { \\max } ( A ) } { s _ { \\min } ( A ) } . \\end{align*}"}
+{"id": "4054.png", "formula": "\\begin{align*} e _ 2 ( C ) = 2 - I ( C ) - s ( C ) ; \\end{align*}"}
+{"id": "5655.png", "formula": "\\begin{align*} 1 = - \\sum _ { \\emptyset \\neq I \\subset \\{ 1 , \\dots , m \\} } ( - 1 ) ^ { | I | } , \\end{align*}"}
+{"id": "5180.png", "formula": "\\begin{align*} | H [ b ] - h | = \\lim _ { n \\to \\infty } \\left | H [ b ] - H [ b _ n ] \\right | \\leq \\left ( \\sup _ { n } | | b _ n | | ^ { 5 / 3 } _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } + | | b | | ^ { 5 / 3 } _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } \\right ) \\varepsilon ^ { 1 / 2 } . \\end{align*}"}
+{"id": "3911.png", "formula": "\\begin{align*} w _ \\varepsilon ( x _ 1 , x _ 2 , 0 , t ) = \\frac { 1 } { \\varepsilon ^ 2 } \\left ( u _ \\varepsilon ( \\bar { R } _ { - \\alpha | \\ln \\varepsilon | t } ( x _ 1 , x _ 2 ) ) - \\left ( \\frac { \\alpha | ( x _ 1 , x _ 2 ) | ^ 2 } { 2 } + \\beta \\right ) \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ p _ + . \\end{align*}"}
+{"id": "549.png", "formula": "\\begin{align*} | \\left \\{ z \\in \\{ 1 , 2 , \\ldots , n \\} \\mid \\d ( x , z ) = d \\right \\} | = \\left \\{ \\begin{array} { l l } 1 & \\mbox { i f $ d = \\frac { n } { 2 } $ } \\\\ 2 & \\mbox { i f $ d < \\frac { n } { 2 } $ , } \\end{array} \\right . \\end{align*}"}
+{"id": "8191.png", "formula": "\\begin{align*} x _ { m , 2 j - 1 } & + x _ { m , 2 j } + x _ { 1 , ( n + 1 ) - ( 2 j - 1 ) } + x _ { 1 , ( n + 1 ) - 2 j } \\\\ & = x _ { m , 2 j - 1 } + x _ { m , 2 j } + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { 1 , 2 j - 1 } \\bigr ) + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { 1 , 2 j } \\bigr ) \\\\ & = \\bigl ( n _ 1 m + 1 \\bigr ) + S - \\bigl ( n _ 1 m + 1 \\bigr ) = S . \\end{align*}"}
+{"id": "3200.png", "formula": "\\begin{align*} \\rho ( g , h ) = \\sup _ { x \\in X } d ( g ( x ) , h ( x ) ) + d ( g ^ { - 1 } ( x ) , h ^ { - 1 } ( x ) ) . \\end{align*}"}
+{"id": "1901.png", "formula": "\\begin{align*} P _ { i j } : = \\textsf { P } \\bigl \\{ { Z ( { n + 1 } ) = j \\bigm | { Z ( n ) = i } } \\bigr \\} = \\sum _ { k _ 1 + \\ , \\cdots \\ , + k _ i = j } { p _ { k _ 1 } p _ { k _ 2 } \\ , \\ldots \\ , p _ { k _ i } } \\end{align*}"}
+{"id": "328.png", "formula": "\\begin{align*} L _ { 1 1 } & \\leq C \\Big ( \\int _ { 0 } ^ { 1 } \\Big ( 1 + \\frac { \\sqrt { t } } { \\rho ( x ) } \\Big ) ^ { - 2 ( N - 1 ) } \\Big ( \\frac { \\sqrt { t } } { \\rho ( x ) } \\Big ) ^ { 2 ( M - n - 1 ) } \\Big ) ^ { 1 / 2 } M _ { V , \\eta } f ( x ) \\\\ & \\leq C M _ { V , \\eta } f ( x ) . \\end{align*}"}
+{"id": "7436.png", "formula": "\\begin{align*} a _ k ( x ) f _ k '' ( x ) + b _ k ( x ) f _ k ' ( x ) + p _ k c _ k ( x ) f _ k ( x ) = 0 , \\end{align*}"}
+{"id": "6860.png", "formula": "\\begin{align*} \\sinh ^ 2 \\left ( \\mathrm { d i s t } ( x _ 0 , \\Pi _ { e _ { m + i } } ) \\right ) = - \\frac { ( e _ { m + i } , u _ 0 ) ^ 2 } { ( e _ { m + i } , e _ { m + i } ) ( u _ 0 , u _ 0 ) } , \\end{align*}"}
+{"id": "9019.png", "formula": "\\begin{align*} i _ { 1 3 5 } = - x _ 1 x _ 3 x _ 5 . \\end{align*}"}
+{"id": "1962.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty Q ^ { ( m ) } _ \\nu ( k ) q ^ { - k n } = \\zeta _ \\nu ( n ) \\zeta _ \\nu ( n - 1 ) \\cdots \\zeta _ \\nu ( n - m + 1 ) . \\end{align*}"}
+{"id": "9259.png", "formula": "\\begin{align*} \\begin{aligned} ( \\Delta u ) ^ { m } \\wedge \\beta _ n ^ { n - m } = m ! ( n - m ) ! \\sum _ { 1 \\leq i _ 1 \\leq \\dots \\leq i _ m \\leq n } \\lambda _ { i _ 1 } \\dots \\lambda _ { i _ m } \\Omega _ { 2 n } . \\end{aligned} \\end{align*}"}
+{"id": "8239.png", "formula": "\\begin{align*} ( \\mathcal { R } _ \\lambda ( \\sigma s _ i ) v _ T , v _ T ) = \\pm ( \\mathcal { R } _ \\lambda ( \\sigma ) v _ T , v _ T ) . \\end{align*}"}
+{"id": "82.png", "formula": "\\begin{align*} \\beta _ i ' = s _ { \\beta _ \\ell } ( \\beta _ i ) , i = 1 , \\dotsc , \\ell ' . \\end{align*}"}
+{"id": "5892.png", "formula": "\\begin{align*} & \\lim _ { \\delta \\rightarrow 0 + } \\sup _ { n \\in \\mathbb { N } } \\mathbb { E } \\int _ 0 ^ { T - \\delta } \\Vert Y _ n ^ M ( t + \\delta ) - Y _ n ^ M ( t ) \\Vert _ { H } ^ { \\alpha } d s \\\\ \\leq & C \\bigg ( \\lim _ { \\delta \\rightarrow 0 + } \\sup _ { n \\in \\mathbb { N } } \\mathbb { E } \\int _ 0 ^ { T - \\delta } \\Vert Y _ n ^ M ( t + \\delta ) - Y _ n ^ M ( t ) \\Vert _ { H } ^ { 2 } d s \\bigg ) ^ { \\frac { \\alpha } { 2 } } = 0 . \\end{align*}"}
+{"id": "9026.png", "formula": "\\begin{align*} ( H _ { \\lambda v , \\alpha , x } \\psi ) _ { n } = \\psi _ { n + 1 } + \\psi _ { n - 1 } + \\lambda v ( x + n \\alpha ) \\psi _ { n } , \\end{align*}"}
+{"id": "7468.png", "formula": "\\begin{align*} \\sum _ { X _ k } \\sum _ { X _ k ' } \\int \\widehat { h _ { X _ k } } \\overline { \\widehat { h _ { X _ k ' } } * \\widehat { w _ U } } & = \\sum _ { X _ k } \\sum _ { X _ k ' \\sim X _ k } \\int h _ { X _ k } \\overline { h _ { X _ k ' } } w _ U \\\\ & \\le \\sum _ { X _ k } \\sum _ { X _ k ' \\sim X _ k } \\int ( | h _ { X _ k } | ^ 2 + | h _ { X _ k ' } | ^ 2 ) w _ U \\\\ & \\le \\sum _ { X _ k } \\sum _ { X _ k ' \\sim X _ k } \\int ( | h _ { X _ k } | ^ 2 + | h _ { X _ k ' } | ^ 2 ) w _ U \\le 2 L \\sum _ { X _ k } \\int | h _ { X _ k } | ^ 2 w _ U . \\end{align*}"}
+{"id": "8923.png", "formula": "\\begin{align*} f _ { N } ( x , y | \\rho ) = \\frac { 1 } { 2 \\pi ( 1 - \\rho ^ { 2 } ) } \\exp \\left ( - \\frac { x ^ { 2 } - 2 \\rho x y + y ^ { 2 } } { 2 ( 1 - \\rho ^ { 2 } ) } \\right ) , \\end{align*}"}
+{"id": "6646.png", "formula": "\\begin{align*} & \\xi _ { k } = \\inf \\Big \\{ t \\geq s _ { k } ; | X ^ { \\epsilon } ( t ) | \\geq K \\Big \\} . \\end{align*}"}
+{"id": "90.png", "formula": "\\begin{align*} \\sigma = t _ { \\mu _ \\sigma } \\circ \\sigma _ 1 \\circ \\sigma _ 2 , \\end{align*}"}
+{"id": "7701.png", "formula": "\\begin{align*} & g \\big ( \\lfloor ( 1 - ( 1 + \\tilde \\varepsilon ) ^ { - j } ) r \\rfloor \\big ) \\geq 1 6 \\ , g \\big ( \\lfloor ( 1 - ( 1 + \\tilde \\varepsilon ) ^ { - j - 1 } ) r \\rfloor \\big ) , j = 1 , 2 , \\dots , i - 1 ; \\\\ & g \\big ( \\lfloor ( 1 - ( 1 + \\tilde \\varepsilon ) ^ { - i } ) r \\rfloor \\big ) \\leq \\sqrt { x } . \\end{align*}"}
+{"id": "3973.png", "formula": "\\begin{align*} G = \\begin{pmatrix} 1 & 0 & \\cdots & 0 & a _ 1 & 0 & \\cdots & 0 & 1 \\\\ 0 & 1 & \\cdots & 0 & a _ 2 & 0 & \\cdots & 1 & 0 \\\\ \\vdots & \\vdots & & \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 0 & 0 & \\cdots & 1 & a _ k & 1 & \\cdots & 0 & 0 \\\\ \\end{pmatrix} _ { k \\times ( 2 k + 1 ) } , \\end{align*}"}
+{"id": "5860.png", "formula": "\\begin{align*} { \\left ( { { { { D \\mathord { \\left / { \\vphantom { D D } } \\right . \\kern - \\nulldelimiterspace } D } } _ 0 } } \\right ) ^ 2 } = 1 - K t , \\end{align*}"}
+{"id": "8660.png", "formula": "\\begin{align*} { { \\bf { \\bar x } } _ { \\rm t } } \\left [ m , n \\right ] { \\rm = } \\frac { 1 } { \\sqrt K } \\sum \\limits _ { k = 0 } ^ { K - 1 } { { { \\bf { u } } _ k } s \\left [ m , k \\right ] { e ^ { j \\frac { { 2 \\pi } } { K } k n } } } , \\ n { \\rm = } - { { N } _ { \\rm C P } } . \\cdots , K - 1 . \\end{align*}"}
+{"id": "6266.png", "formula": "\\begin{align*} \\phi _ { i s } ( \\partial _ s ) = \\frac { \\Gamma _ { s i } ^ i } { \\varphi _ s } , \\quad \\forall s \\neq i , \\end{align*}"}
+{"id": "4096.png", "formula": "\\begin{align*} & \\hphantom { { } = { } } ( ( u Y ) ^ { r - 1 } ) ^ i { } _ { \\alpha _ 1 , \\ldots , \\alpha _ { r - 1 } } \\\\ * & = u ^ i { } _ { \\alpha _ 1 , \\ldots , \\alpha _ { r - 1 } , l } Y ^ l \\\\ * & \\hphantom { { } = { } } + ( \\textit { t e r m s w h i c h d o n o t i n v o l v e $ u ^ i { } _ { j _ 1 , \\ldots , j _ r } $ o r $ Y ^ i { } _ { j _ 1 , \\ldots , j _ { r - 1 } } $ } ) \\\\ * & \\hphantom { { } = { } } + u ^ i { } _ l Y ^ l { } _ { \\alpha _ 1 , \\ldots , \\alpha _ { r - 1 } } , \\end{align*}"}
+{"id": "8799.png", "formula": "\\begin{align*} \\begin{cases} d Y _ t = L _ { z } ^ \\perp Y _ t d t + \\Pi _ { \\mathrm { k e r } A ^ \\perp } \\sigma d W _ t \\\\ Y _ t | _ { t = 0 } = Y _ 0 \\in \\mathrm { k e r } A ^ \\perp . \\end{cases} \\end{align*}"}
+{"id": "2424.png", "formula": "\\begin{align*} L _ { q } ^ { - 1 } [ L _ { q } [ f ( t ) ] ] = \\int _ { 0 } ^ { \\infty } d t ^ { \\prime } f ( t ^ { \\prime } ) \\delta ( t - t ^ { \\prime } ) \\equiv f ( t ) . \\end{align*}"}
+{"id": "4838.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c c c } 0 & y & \\mu _ 1 & \\mu _ 2 \\\\ - y & 0 & \\mu _ 3 & \\mu _ 4 \\\\ - \\mu _ 1 & - \\mu _ 3 & 0 & z \\\\ - \\mu _ 2 & - \\mu _ 4 & - z & 0 \\end{array} \\right ) , \\mu _ 1 , \\dots , \\mu _ 4 \\in { \\frak m } _ { x _ 1 , \\dots , x _ { s - 2 } } . \\end{align*}"}
+{"id": "3164.png", "formula": "\\begin{align*} \\langle H \\rangle = \\frac { 1 } { 2 } \\times \\mathop { \\sum \\sum } _ { \\chi _ 4 ( x - y ) \\in \\{ 1 , \\chi _ 4 ( g ) \\} } 1 , \\end{align*}"}
+{"id": "7594.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } T ^ { c ( n ) } ( x _ { 0 0 } , x _ { 0 1 } ) = ( x _ { 1 0 } , x _ { 1 1 } ) \\end{align*}"}
+{"id": "5216.png", "formula": "\\begin{align*} \\int _ 0 ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi ( \\partial _ t \\varphi + ( v + u _ { \\infty } ) \\cdot \\nabla \\varphi ) \\dd x \\dd s = - \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ 0 \\varphi _ 0 \\dd x , \\end{align*}"}
+{"id": "1284.png", "formula": "\\begin{align*} G _ n : = \\prod _ { k = n } ^ { \\infty } \\frac { ( 2 ^ { k + 2 } - 2 ) ^ d } { ( 2 ^ { k + 2 } - 1 ) ^ d } . \\end{align*}"}
+{"id": "6145.png", "formula": "\\begin{align*} & \\max \\Big \\{ \\ ! \\left \\lVert \\mu ( s , x ) - \\mu ( t , x ) \\right \\rVert ^ 2 , \\left \\lVert \\sigma ( s , x ) - \\sigma ( t , x ) \\right \\rVert ^ 2 \\Big \\} \\leq c \\Big [ \\lvert t - s \\rvert + \\ ! \\ ! \\sup _ { \\substack { u , v \\in [ 0 , t ] \\colon \\\\ \\lvert u - v \\rvert \\leq \\lvert t - s \\rvert } } \\left \\lVert x ( u ) - x ( v ) \\right \\rVert ^ 2 \\Big ] , \\end{align*}"}
+{"id": "6291.png", "formula": "\\begin{align*} F ^ * = \\min _ { x } \\ \\{ F ( x ) \\coloneqq f ( x ) + P ( x ) \\} , \\end{align*}"}
+{"id": "808.png", "formula": "\\begin{align*} R ( t ) = F ^ { - 1 } ( t ) \\rightarrow F ^ { - 1 } ( T ) = F ^ { - 1 } \\big ( F ( 0 ) \\big ) = 0 \\end{align*}"}
+{"id": "8541.png", "formula": "\\begin{align*} & \\left \\| r _ { 1 , 2 } ( t ; z ) \\right \\| _ { L ^ { 2 , 1 } } = \\left \\| r _ { 1 , 2 } ( z ) \\right \\| _ { L ^ { 2 , 1 } } , \\\\ & \\left \\| z ^ { - 2 } r _ { 1 , 2 } ( t ; z ) \\right \\| _ { L ^ { 2 , 1 } } = \\left \\| z ^ { - 2 } r _ { 1 , 2 } ( z ) \\right \\| _ { L ^ { 2 , 1 } } . \\end{align*}"}
+{"id": "9203.png", "formula": "\\begin{align*} \\begin{aligned} | \\epsilon _ 2 ( \\eta _ a , t ) | \\le \\int _ N ^ t \\left | h ( x _ a + \\delta u ( \\tau ) ) - h ( \\bar { x } _ 1 ) \\right | \\ , d \\tau \\le L _ r 2 \\delta . \\end{aligned} \\end{align*}"}
+{"id": "7443.png", "formula": "\\begin{align*} \\lambda ( Y ) : = \\sup \\lbrace \\lambda ( Y , X ) : Y \\subset X \\rbrace . \\end{align*}"}
+{"id": "4562.png", "formula": "\\begin{align*} \\frac { f ( v _ n ) } { q ^ { n / 2 } } = \\frac { h ( v _ n ) } { q ^ { n / 2 } } - \\frac { a _ n \\delta ( v _ 0 ) } { q ^ { n / 2 } ( q + 1 ) } - \\sum _ { k = 1 } ^ { n - 1 } \\frac { a _ { n - k } } { q ^ { { ( n - k ) } / 2 } } \\frac { \\delta ( v _ { k } ) } { q ^ { k / 2 } } . \\end{align*}"}
+{"id": "5680.png", "formula": "\\begin{align*} p _ { 1 } \\circ ( f ^ { - 1 } _ { 1 1 } \\circ p _ { 1 } , p _ { 2 } ) \\circ f & = p _ { 1 } \\circ ( f ^ { - 1 } _ { 1 1 } \\circ p _ { 1 } , p _ { 2 } ) \\circ ( f _ { 1 } , f _ { 2 } ) = p _ { 1 } \\circ [ f ^ { - 1 } _ { 1 1 } \\circ p _ { 1 } ( f _ { 1 } , f _ { 2 } ) , f _ { 2 } ] \\\\ & = p _ { 1 } \\circ ( f ^ { - 1 } _ { 1 1 } \\circ f _ { 1 } , f _ { 2 } ) = f ^ { - 1 } _ { 1 1 } \\circ f _ { 1 } = f ^ { - 1 } _ { 1 1 } \\circ f _ { 1 1 } = p _ { 1 } . \\\\ & ~ ~ ~ ~ ~ ~ ( \\because f _ { 1 } = f _ { 1 } \\circ l _ { 1 } = f _ { 1 } \\circ \\iota _ { 1 } ) . \\end{align*}"}
+{"id": "2364.png", "formula": "\\begin{align*} A ( T _ m ) - A ( T _ { m + 1 } ) \\leq 2 \\pi ( m - 1 ) ( T _ m - T _ { m + 1 } ) + \\sum \\limits _ { i = 1 } ^ m A _ i ( T _ m ) - A _ i ( T _ { m + 1 } ) . \\end{align*}"}
+{"id": "8419.png", "formula": "\\begin{align*} \\lim _ { | z | \\rightarrow \\infty } z ( \\Psi ^ - _ { 1 1 } ( x ; z ) - e ^ { - i c _ - ( x ) } ) = \\widehat { \\Psi } ^ - _ { 1 1 } ( x ) . \\end{align*}"}
+{"id": "2429.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } s L _ { q } [ f ( t ) ] = \\lim _ { t \\rightarrow 0 } \\frac { f ( t ) } { 1 + ( 1 - q ) } . \\end{align*}"}
+{"id": "2831.png", "formula": "\\begin{align*} \\begin{aligned} \\| f \\| _ { B M O } & \\leq C \\| f \\| _ { \\dot B ^ 0 _ { \\infty , 2 } } \\\\ & \\leq C \\big ( 1 + \\| f \\| _ { \\dot B ^ 0 _ { \\infty , \\infty } } \\ln ^ { \\frac { 1 } { 2 } } ( 1 + \\| f \\| _ { H ^ s } ) \\big ) , \\end{aligned} \\end{align*}"}
+{"id": "8038.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } b _ \\eta ( \\xi ( s ) ) = - \\infty , \\lim _ { s \\to - \\infty } b _ \\eta ( \\xi ( s ) ) = \\infty , \\end{align*}"}
+{"id": "2730.png", "formula": "\\begin{align*} \\liminf _ { r \\to \\infty } \\frac { \\log \\log M ( r , f ) } { \\log \\log r } = \\infty \\end{align*}"}
+{"id": "2008.png", "formula": "\\begin{align*} \\mathcal { I } & = \\{ ( k , l ) \\in \\Z ^ 2 : k \\leq i \\mbox { o r } l \\leq j \\} \\subseteq \\Z ^ 2 \\\\ m _ { k , l } & = n _ { k , l } \\mbox { i f } k \\leq i \\\\ m _ { k , l } & = 0 \\mbox { i f } l \\leq j \\end{align*}"}
+{"id": "9034.png", "formula": "\\begin{align*} \\| \\tilde { F } \\| _ { h } \\leq \\| F \\| _ { h } \\| P \\| ^ { 2 } \\leq { \\rm e } ^ { 2 | \\log \\varepsilon | ^ { \\frac { 2 } { 2 + \\eta } } } \\varepsilon = : \\tilde { \\varepsilon } . \\end{align*}"}
+{"id": "8582.png", "formula": "\\begin{align*} \\tilde { \\kappa } ( p ) \\cdot \\tilde { k _ 1 } ( p ) \\cdot \\tilde { k _ 2 } ( p ) \\ , = \\ , \\frac { 1 } { p } , \\ \\Re ( p ) > c \\end{align*}"}
+{"id": "2652.png", "formula": "\\begin{align*} & \\sum \\limits _ { n = - \\infty } ^ { \\infty } ( - 1 ) ^ { n } q ^ { n ( 3 n + 1 ) / 2 } \\cdot \\sum \\limits _ { n = 0 } ^ { \\infty } d _ { e } ( n ) q ^ { n } \\\\ & = \\sum \\limits _ { n = - \\infty } ^ { \\infty } ( - 1 ) ^ { n } q ^ { 2 n ( 3 n + 1 ) } \\sum \\limits _ { n = 1 } ^ { \\infty } \\sigma _ { 0 } ( n ) q ^ { 4 n } . \\end{align*}"}
+{"id": "7061.png", "formula": "\\begin{align*} A _ { 1 , 1 } \\ , : = \\ , \\mp \\tfrac { 1 } { 4 } \\ , F _ { 7 , 0 , 0 } \\ , T _ 1 - T _ 2 \\mp \\tfrac { 1 } { 4 } \\ , F _ { 6 , 0 , 1 } \\ , T _ 3 . \\end{align*}"}
+{"id": "6849.png", "formula": "\\begin{align*} J _ { 1 , n } ( x ) = J _ { 1 , n - 1 } ( x ) - e ^ { - \\frac { x } { 2 } } e ( \\frac { i } { 2 } , x ) a _ { n - 1 } ( x ) - \\int _ { x } ^ { \\infty } \\left ( e ( \\frac { i } { 2 } , t ) e ^ { - \\frac { t } { 2 } } \\right ) ^ { \\prime } a _ { n - 1 } ( t ) d t , \\end{align*}"}
+{"id": "7020.png", "formula": "\\begin{align*} \\aligned A _ { 1 , 2 } & \\ , = \\ , 0 , \\\\ D & \\ , = \\ , 0 + 2 \\ , A _ { 1 , 1 } . \\endaligned \\end{align*}"}
+{"id": "1210.png", "formula": "\\begin{align*} \\sum _ { y \\neq x } \\mu ( y ) L _ { y x } = \\sum _ { y \\neq x } \\mu ( x ) L _ { x y } = \\mu ( x ) L _ { x x } . \\end{align*}"}
+{"id": "6263.png", "formula": "\\begin{align*} P = P _ w = _ \\gamma + \\gamma I - B _ w : T L \\rightarrow T L , \\end{align*}"}
+{"id": "8834.png", "formula": "\\begin{align*} X _ { t , 1 } = X _ { 0 , 1 } + t X _ { 0 , 2 } X _ { 0 , n } + X _ { 0 , 2 } \\int _ 0 ^ t E _ s d s \\end{align*}"}
+{"id": "3001.png", "formula": "\\begin{align*} \\varphi ( \\vartheta , z , w ( \\cdot ) ) = \\sigma ( z , w ( \\cdot ) ) , ( z , w ( \\cdot ) ) \\in \\mathbb R ^ n \\times \\mathrm { P L i p } . \\end{align*}"}
+{"id": "4582.png", "formula": "\\begin{align*} \\langle B ^ { \\circ } x ^ { \\circ } , u \\rangle = \\langle x ^ { \\circ } , B u \\rangle = \\lim _ { n \\to \\infty } \\langle x ^ { \\circ } , B u _ { n } \\rangle , \\end{align*}"}
+{"id": "6232.png", "formula": "\\begin{align*} D _ n ( x ) = \\prod _ { k = 0 } ^ { n } m _ \\tau ( ( R ^ { t } ) ^ { k } x ) . \\end{align*}"}
+{"id": "2946.png", "formula": "\\begin{align*} v ( x , y ) : = ( x , - y ) . \\end{align*}"}
+{"id": "4153.png", "formula": "\\begin{align*} \\Phi _ t ( x ' ) = t ^ { 1 - n } ( \\partial _ n K ^ L ) ( x ' / t , 1 ) = t \\ , ( \\partial _ n K ^ L ) ( x ' , t ) = t \\partial _ t K ^ L ( x ' , t ) = t \\partial _ t P _ t ^ L ( x ' ) , \\end{align*}"}
+{"id": "8491.png", "formula": "\\begin{align*} \\| r ( z ) \\| _ { L ^ 2 } ^ 2 = 2 \\pi \\| \\widehat { r } ( \\xi ) \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "6380.png", "formula": "\\begin{align*} f _ { i : n } ( x ) = \\frac { f ( x ) } { \\operatorname { B } ( i , n - i + 1 ) } F ( x ) ^ { i - 1 } \\{ 1 - F ( x ) \\} ^ { n - i } , \\end{align*}"}
+{"id": "192.png", "formula": "\\begin{align*} Y _ { j , ( 1 ) } = \\begin{cases} \\log ( \\frac { [ T _ { j , ( 1 ) } ^ { \\flat } ] } { T _ { j , ( 1 ) } } ) & \\mbox { i f } 1 \\leq j \\leq e \\\\ \\log ( \\frac { [ S _ { j , ( 1 ) } ^ { \\flat } ] } { S _ { j , ( 1 ) } } ) & \\mbox { i f } e + 1 \\leq j \\leq d \\end{cases} \\end{align*}"}
+{"id": "5959.png", "formula": "\\begin{align*} \\nabla \\cdot ( \\nabla n \\otimes \\nabla n ) = \\frac { 1 } { 2 } \\nabla ( | \\nabla n | ^ 2 ) + \\nabla n \\cdot \\Delta n . \\end{align*}"}
+{"id": "6599.png", "formula": "\\begin{align*} X + \\xi \\in E = E ( \\mathfrak { g } ) \\coloneqq \\mathfrak { g } \\oplus \\mathfrak { g } ^ * \\end{align*}"}
+{"id": "3.png", "formula": "\\begin{align*} \\beta _ 1 \\oplus \\beta _ 2 : = \\{ ( \\gamma _ 1 , \\gamma _ 2 ) \\mid ( \\gamma _ 1 , \\gamma _ 2 ) \\in \\mathcal { P } ( \\beta _ 1 + \\beta _ 2 ) \\mid ( \\beta _ 1 , \\beta _ 2 ) < ( \\gamma _ 1 , \\gamma _ 2 ) \\} . \\end{align*}"}
+{"id": "4985.png", "formula": "\\begin{align*} M _ { f _ { 1 , \\infty } } ( n , \\alpha , \\epsilon , Z , \\psi ) : = \\inf \\bigg \\{ \\sum _ { i } ^ { - \\alpha n _ { i } + S _ { 1 , n _ { i } } \\psi ( x _ { i } , \\epsilon ) } : Z \\subseteq \\bigcup _ { i } B _ { n _ { i } } ( x _ { i } , \\epsilon ) \\bigg \\} , \\end{align*}"}
+{"id": "8973.png", "formula": "\\begin{align*} \\hat { k } _ 2 ( { \\Delta } ) & = \\sum _ { j _ 1 , j _ 2 , j _ 3 = 1 , 2 } \\dfrac { ( x _ { 1 , 1 } x _ { 1 , 2 } x _ { 2 , 1 } x _ { 2 , 2 } x _ { 3 , 1 } x _ { 3 , 2 } ) ^ 4 } { x _ { 1 , j _ 1 } x _ { 2 , j _ 2 } x _ { 3 , j _ 3 } } \\\\ & = ( x _ { 1 , 1 } x _ { 1 , 2 } x _ { 2 , 1 } x _ { 2 , 2 } x _ { 3 , 1 } x _ { 3 , 2 } ) ^ 3 ( x _ { 1 , 1 } + x _ { 1 , 2 } ) ( x _ { 2 , 1 } + x _ { 2 , 2 } ) ( x _ { 3 , 1 } + x _ { 3 , 2 } ) . \\end{align*}"}
+{"id": "2388.png", "formula": "\\begin{align*} \\mathcal { O } _ { s } ( k ) = { { 2 s - k - 1 } \\choose { k } } ( C _ { s - k } ) ^ 2 . \\end{align*}"}
+{"id": "8855.png", "formula": "\\begin{align*} \\frac { d } { d t } \\bar { y } _ t = - Z _ { t , 3 } \\bar { y } _ t - \\bar { z } _ { t , 3 } y _ t , \\end{align*}"}
+{"id": "6623.png", "formula": "\\begin{align*} \\left ( \\lambda + \\rho \\right ) ^ 2 = \\left ( \\mu + \\nu \\right ) ^ 2 . \\end{align*}"}
+{"id": "5019.png", "formula": "\\begin{align*} S t _ { D N D C } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( z ^ { 2 } , z ) - z ^ { 2 } | | _ { \\infty } = 0 . \\\\ \\end{align*}"}
+{"id": "3250.png", "formula": "\\begin{align*} \\mathcal P ( m ) = & \\left \\{ p \\subset \\lbrace 1 , . . . , m \\rbrace ^ 2 : \\# p = \\frac m 2 ( x , y ) , ( x ' , y ' ) \\in p , \\right . \\\\ & \\qquad \\left . x < y , x ' < y ' \\right \\} . \\end{align*}"}
+{"id": "7424.png", "formula": "\\begin{align*} | \\xi | _ p \\le \\psi ( p ) = \\psi _ { p _ 0 } [ \\eta ] ( p ) , \\ p \\in U . \\end{align*}"}
+{"id": "8736.png", "formula": "\\begin{align*} d _ \\textrm { H S } ( A , B ) = \\| A - B \\| _ \\textrm { H S } \\forall A , B \\in \\mathrm { U } ( n ) . \\end{align*}"}
+{"id": "6720.png", "formula": "\\begin{align*} F ( x ) = I _ { G ( x ) } ( \\alpha , \\beta ) = \\frac { 1 } { \\mathrm { B } ( \\alpha , \\beta ) } \\int _ 0 ^ { G ( x ) } t ^ { \\alpha - 1 } ( 1 - t ) ^ { \\beta - 1 } \\mathrm { d } t . \\end{align*}"}
+{"id": "6400.png", "formula": "\\begin{align*} ( \\C ^ n , \\omega _ { \\mathrm { s t } } ) , \\omega _ { \\mathrm { s t } } = \\i \\sum d z ^ i \\wedge d \\bar { z } ^ i \\end{align*}"}
+{"id": "428.png", "formula": "\\begin{align*} \\theta ^ { r - 1 } a _ { i } = \\sum _ { j = 1 } ^ { n } q _ { r i j } a _ { j } = \\sum _ { j = 1 } ^ { n } q _ { r i j } \\sum _ { k = 1 } ^ { n } k _ { j k } \\theta ^ { k - 1 } = \\sum _ { k = 1 } ^ { n } \\big ( \\sum _ { j = 1 } ^ { n } q _ { r i j } k _ { j k } \\Big ) \\theta ^ { k - 1 } . \\end{align*}"}
+{"id": "7522.png", "formula": "\\begin{align*} \\sigma ( B ) \\leq \\sum _ { j = 1 } ^ N \\sigma ( B ^ j ) \\mbox { w h e n e v e r } N \\in \\mathbb { Z } ^ + , \\{ B ^ j \\} _ { j = 1 } ^ N \\subset \\mathcal { B } , B \\subset \\bigcup _ { j = 1 } ^ N B ^ j . \\end{align*}"}
+{"id": "3140.png", "formula": "\\begin{align*} \\mathcal { T } \\circ \\beta & = \\alpha \\circ \\mathcal { T } , \\\\ [ T ( a ) , \\mathcal { T } ( b ) ] + [ \\mathcal { T } , ( a ) , T ( b ) ] & = T ( \\varrho ( \\mathcal { T } ( a ) ) ( b ) - \\varrho ( \\mathcal { T } ( b ) ) ( a ) ) + \\mathcal { T } ( \\varrho ( T ( a ) ) ( b ) - \\varrho ( T ( b ) ) ( a ) ) , \\\\ [ \\mathcal { T } ( a ) , \\mathcal { T } ( b ) ] & = \\mathcal { T } ( \\varrho ( \\mathcal { T } ( a ) ) ( b ) - \\varrho ( \\mathcal { T } ( b ) ) ( a ) ) . \\end{align*}"}
+{"id": "1542.png", "formula": "\\begin{align*} \\frac { d L } { d t _ { k } } = \\left [ ( L ^ { k / \\alpha } ) _ { D } , L \\right ] _ \\epsilon = - \\left [ ( L ^ { k } ) _ { S } , L \\right ] \\ ; , k \\geq 1 \\ ; , \\end{align*}"}
+{"id": "8964.png", "formula": "\\begin{align*} v _ 1 ^ { - p ' } ( a ( x ) ) a ' ( x ) + v _ 1 ^ { - p ' } ( b ( x ) ) b ' ( x ) = 2 v _ 1 ^ { - p ' } ( x ) , \\end{align*}"}
+{"id": "6887.png", "formula": "\\begin{align*} P ( d x ) = Z ^ { - 1 } e ^ { f ( x ) } \\mu ^ { \\otimes n } ( d x ) , \\end{align*}"}
+{"id": "5501.png", "formula": "\\begin{align*} F ( p , s ) & \\leq F ( p , 1 . 7 ) ^ { \\frac { 1 0 s - 1 3 } { 4 } } F ( p , 1 . 3 ) ^ { \\frac { 1 7 - 1 0 s } { 4 } } \\\\ & \\leq \\Big ( G ( p , 1 . 7 ) \\Big ) ^ { \\frac { 1 0 s - 1 3 } { 4 } } \\Big ( e ^ { 2 p / 1 7 } G ( p , 1 . 7 ) \\Big ) ^ { \\frac { 1 7 - 1 0 s } { 4 } } \\\\ & = e ^ { \\frac { p } { 2 } \\frac { 1 7 - 1 0 s } { 1 7 } } 1 . 7 ^ { - p / 2 } 2 ^ { 3 p / 2 - 1 } \\Gamma ( p / 2 ) . \\end{align*}"}
+{"id": "4135.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n - 1 } } \\sum _ { 1 \\le \\beta \\le N } ( P ^ L _ { \\alpha \\beta } ) _ t ( x ' - y ' ) C _ { \\beta } \\ , d y ' = C _ { \\alpha } , \\quad \\forall ( x ' , t ) \\in \\R ^ n _ + . \\end{align*}"}
+{"id": "253.png", "formula": "\\begin{align*} ( \\sum _ { j = 1 } ^ s a _ j ( f _ j - 1 ) ) ( h \\otimes 1 ) & = \\sum _ { j = 1 } ^ s a _ j ( f _ j ( h \\otimes 1 ) - 1 ( h \\otimes 1 ) ) \\\\ & = \\sum _ { j = 1 } ^ s a _ j ( h \\otimes ( f _ j ) 1 - h \\otimes 1 ) \\\\ & = \\sum _ { j = 1 } ^ s a _ j ( h \\otimes 1 - h \\otimes 1 ) \\\\ & = 0 , \\end{align*}"}
+{"id": "1439.png", "formula": "\\begin{align*} \\P _ x ( \\tau > n ) \\sim K _ { \\lambda } n ^ { - d ( d - 1 ) / 2 - d / 2 } e ^ { - \\gamma n } e ^ { \\sum _ { i = 1 } ^ d ( \\lambda _ i - \\bar { \\lambda } ) x _ i } h ^ { ( \\bar { \\lambda } , \\ldots , \\bar { \\lambda } ) } ( x ) , n \\rightarrow \\infty \\end{align*}"}
+{"id": "1588.png", "formula": "\\begin{align*} h _ { x , t } : = d x _ 1 ^ 2 + \\phi _ 2 ( t _ 1 , t _ 2 ) ^ 2 d x _ 2 ^ 2 + \\phi _ 3 ( t _ 1 , t _ 2 ) ^ 2 d x _ 3 ^ 2 + t _ 2 ^ 2 d t _ 1 ^ 2 + t _ 1 ^ 2 d t _ 2 ^ 2 . \\end{align*}"}
+{"id": "6197.png", "formula": "\\begin{align*} V ( r ) = \\frac { L ( L + 1 ) } { r ^ 2 } - \\frac { Q } { r } f + \\kappa B _ 1 \\frac { r } { f } + \\kappa B _ 2 \\frac { 1 } { f ^ 2 } + \\kappa B _ 3 \\frac { r } { f ^ 3 } + \\kappa B _ 4 \\frac { 1 } { f ^ 4 } , B _ 4 > 0 , \\end{align*}"}
+{"id": "9220.png", "formula": "\\begin{align*} f _ a ( x ) : = \\dfrac { 1 } { T } \\int _ { 0 } ^ { T } f ( x , t ) \\ , d t . \\end{align*}"}
+{"id": "2507.png", "formula": "\\begin{align*} \\mathbf { X } - \\mathbf { T } _ x = \\mathbf { U } _ { x } = \\left ( \\mathbf { U } _ { x ^ { ( i ) } } : i = 1 , \\cdots , k \\right ) , \\end{align*}"}
+{"id": "159.png", "formula": "\\begin{align*} & \\frac { 1 } { n } \\sum _ { d = 0 } ^ n ( d + 1 ) \\big ( N ( n , r , k - 1 , d ) - N ( n , r , k - 1 , d + 1 ) \\big ) \\\\ = \\ : & \\frac { 1 } { n } \\sum _ { d = 0 } ^ n N ( n , r , k - 1 , d ) \\ , . \\end{align*}"}
+{"id": "2319.png", "formula": "\\begin{align*} s ^ \\star - s ^ g = - 2 \\delta ^ g \\theta - \\| \\theta \\| ^ 2 + 2 \\| N \\| ^ 2 . \\end{align*}"}
+{"id": "5193.png", "formula": "\\begin{align*} \\nabla \\times ( a _ { t } + ( B \\times u ) + \\mu \\nabla \\times b ) = 0 \\textrm { o n } \\ ( L ^ { 2 } _ { \\sigma } \\cap H ^ { 1 } ) ( \\mathbb { R } ^ { 3 } ) ^ { * } . \\end{align*}"}
+{"id": "189.png", "formula": "\\begin{align*} B ^ { i } _ { G ' , n } : = ( B _ { G ' , n } \\widehat { \\otimes } _ { A _ { n } } A _ { n } ^ i ) ^ { \\wedge - u } \\end{align*}"}
+{"id": "4381.png", "formula": "\\begin{align*} & \\lim _ { m ' \\to + \\infty } \\int _ { X _ j } v '' _ { \\epsilon } ( \\Psi ) | f F | ^ 2 _ { h _ { m ' } } e ^ { - u ( - v _ { \\epsilon } ( \\Psi ) ) } \\\\ = & \\int _ { X _ j } v '' _ { \\epsilon } ( \\Psi ) | f F | ^ 2 _ h e ^ { - u ( - v _ { \\epsilon } ( \\Psi ) ) } < + \\infty . \\end{align*}"}
+{"id": "7539.png", "formula": "\\begin{align*} s ( n , k , \\ell ) = \\max \\{ \\vartheta ( S ) : \\ : S \\} \\end{align*}"}
+{"id": "3126.png", "formula": "\\begin{align*} H P M ( x , y , z , t ) & = H P L ( [ x , y ] , \\alpha ( z ) , \\alpha ( t ) ) - H P L ( [ y , x ] , \\alpha ( z ) , \\alpha ( t ) ) \\\\ & \\quad + H P L ( \\alpha ( x ) , \\alpha ( y ) , [ z , t ] ) + H P L ( \\alpha ( y ) , \\alpha ( z ) , [ x , t ] ) \\\\ & \\quad - [ \\alpha ^ { 2 } ( z ) , H P L ( x , y , t ) ] + [ \\alpha ^ { 2 } ( y ) , H P L ( x , z , t ) ] . \\end{align*}"}
+{"id": "4674.png", "formula": "\\begin{align*} \\kappa \\stackrel { d e f } { = } \\lim _ { n \\to \\infty } \\kappa _ n . \\end{align*}"}
+{"id": "2993.png", "formula": "\\begin{align*} \\overline { W ^ { s } ( ( x , t ) , T _ { \\vec { v } } ) \\cap \\operatorname { s u p p } \\widetilde { \\mu } _ { x } } = \\operatorname { s u p p } \\widetilde { \\mu } _ { x } . \\end{align*}"}
+{"id": "2872.png", "formula": "\\begin{align*} & | K _ j ^ { ( n ) } ( x , y ) | \\le C \\left ( \\frac { 1 } { \\chi ^ 2 _ n ( ( x - y ) / 2 ) } + \\frac { 1 } { \\chi ^ 2 _ n ( ( x + y ) / 2 ) } \\right ) , x , y = 0 , \\ldots , n , \\ , n = 1 , 2 , \\ldots \\end{align*}"}
+{"id": "9155.png", "formula": "\\begin{align*} | h ( x + \\delta u ) - h ( x ) | = | R ( x , \\delta u ) | \\le L _ r \\delta . \\end{align*}"}
+{"id": "8495.png", "formula": "\\begin{align*} & \\left ( M _ { - } R \\right ) _ { 1 1 } = e ^ { i c _ + ( x ) \\sigma _ 3 } \\left ( M _ { - } R \\right ) _ { 1 1 } = r _ 2 ( z ) e ^ { 2 i z x } M _ { + , 2 1 } , \\\\ & \\left ( M _ - R \\right ) _ { 2 1 } = e ^ { i c _ + ( x ) \\sigma _ 3 } \\left ( M _ { - } R \\right ) _ { 2 1 } = \\bar { r } _ 1 ( z ) e ^ { - 2 i z x } M _ { - , 1 1 } . \\end{align*}"}
+{"id": "7035.png", "formula": "\\begin{align*} G _ { 5 , 1 } \\ , = \\ , 4 , \\ \\ \\ \\ \\ G _ { 2 , 4 } \\ , = \\ , 4 ! \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0 \\ , = \\ , G _ { 0 , 6 } \\ , = \\ , G _ { 1 , 5 } \\ , = \\ , G _ { 3 , 3 } \\ , = \\ , G _ { 4 , 2 } , \\end{align*}"}
+{"id": "6774.png", "formula": "\\begin{align*} b _ { 0 } ( x ) = g ( \\frac { i } { 2 } , x ) e ^ { - \\frac { x } { 2 } } - 1 . \\end{align*}"}
+{"id": "2026.png", "formula": "\\begin{align*} & \\widehat \\zeta = ( \\widehat y , \\widehat p , \\widehat z , \\widehat q ) = ( y - y ' , p - p ' , z - z ' , q - q ' ) , \\\\ & \\widehat { \\overline \\zeta } = ( \\widehat { \\overline y } , \\widehat { \\overline p } , \\widehat { \\overline z } , \\widehat { \\overline q } ) = ( \\overline y - \\overline y ' , \\overline p - \\overline p ' , \\overline z - \\overline z ' , \\overline q - \\overline q ' ) . \\end{align*}"}
+{"id": "6432.png", "formula": "\\begin{align*} \\Big ( \\frac p q \\Big ) \\Big ( \\frac q p \\Big ) = ( - 1 ) ^ { \\mu + \\nu } . \\end{align*}"}
+{"id": "5386.png", "formula": "\\begin{align*} \\nabla _ { i j } u : = \\nabla _ { e _ i } \\nabla _ { e _ j } u = e _ i ^ k e _ j ^ l \\partial _ { k } \\partial _ l u = e _ i ^ k e _ j ^ l u _ { k l } . \\end{align*}"}
+{"id": "2925.png", "formula": "\\begin{align*} \\left \\Vert \\varphi \\right \\Vert _ { p , r } = \\sup _ { x \\in \\mathbb { R } } \\left \\vert x ^ { p } \\varphi ^ { ( r ) } \\left ( x \\right ) \\right \\vert , \\ ; p , r \\in \\mathbb { N } _ { 0 } \\end{align*}"}
+{"id": "712.png", "formula": "\\begin{align*} \\aligned & \\lim \\limits _ { n \\to \\infty } ( \\| u _ n \\| ^ 2 _ { E _ a } - \\| u _ n - \\tau _ { k _ { 1 , n } } v _ 1 \\| ^ 2 _ { E _ a } ) \\\\ = & \\lim \\limits _ { n \\to \\infty } ( \\| \\tau _ { - k _ { 1 , n } } u _ n \\| ^ 2 _ { E _ a } - \\| \\tau _ { - k _ { 1 , n } } u _ n - v _ 1 \\| ^ 2 _ { E _ a } ) \\\\ = & \\lim \\limits _ { n \\to \\infty } 2 ( \\tau _ { - k _ { 1 , n } } u _ n , v _ 1 ) _ { E _ a } - \\| v _ 1 \\| ^ 2 _ { E _ a } = \\| v _ 1 \\| ^ 2 _ { E _ a } . \\endaligned \\end{align*}"}
+{"id": "5588.png", "formula": "\\begin{align*} A ( y _ { l + 1 } . . . y _ 1 x ) - A ( y _ { l + 1 } . . . y _ 1 x ' ) = A ( 0 1 ^ \\infty ) - A ( 0 1 ^ l 0 ^ \\infty ) = b - b _ l \\ . \\end{align*}"}
+{"id": "6214.png", "formula": "\\begin{align*} W _ + ( r ) & = ( 2 L + 3 ) \\left ( - \\frac { f } { r } + \\Q + \\kappa \\frac { r } { f } \\right ) , \\\\ W _ - ( r ) & = - \\frac { f } { r } - \\Q + \\kappa \\frac { r } { f } , \\end{align*}"}
+{"id": "5724.png", "formula": "\\begin{align*} P _ b ( x ) = x ^ { q + 1 } + x + b , \\end{align*}"}
+{"id": "3999.png", "formula": "\\begin{align*} u = \\arg \\min _ { w \\in L _ 2 ( 0 , T ; \\mathcal { V } ) } \\bigg ( | | \\partial _ t w + \\mathcal { L } w - F | | ^ 2 _ { L _ 2 ( 0 , T ; \\mathcal { H } ) } + | | \\alpha w + \\beta \\mathbf { n } \\cdot \\nabla w - g | | ^ 2 _ { L _ 2 ( 0 , T ; \\partial \\mathcal { H } ) } + | | w ( 0 , \\cdot ) - u _ 0 | | ^ 2 _ { \\mathcal { H } } \\bigg ) , \\end{align*}"}
+{"id": "651.png", "formula": "\\begin{align*} \\max \\{ | X ( t ) - x ( t / n ) n | , | L _ { 1 } ( t ) - \\ell _ { 1 } ( t / n ) n | , | L _ { 2 } ( t ) - \\ell _ { 2 } ( t / n ) n | \\} = o ( n ) . \\end{align*}"}
+{"id": "2255.png", "formula": "\\begin{align*} Z = U D ( x ) V = U D ( \\overline { t } ) D ( x ) D ( t ) V , \\end{align*}"}
+{"id": "8647.png", "formula": "\\begin{align*} y \\left [ n \\right ] = \\left ( { \\sqrt { P \\sum \\nolimits _ { l = 1 } ^ L { { { \\left \\| { { { \\bf { h } } _ l } } \\right \\| } ^ 2 } } } } \\right ) s \\left [ { n - { n _ { \\max } } } \\right ] + z \\left [ n \\right ] . \\end{align*}"}
+{"id": "2700.png", "formula": "\\begin{align*} \\begin{aligned} \\bar \\Delta & = C _ 1 \\min _ { 0 \\le k \\le T - 1 } \\| \\nabla \\phi ( X _ k ) \\| - C _ 2 \\epsilon _ g , \\\\ \\bar \\Delta ' & = \\min _ l \\{ \\gamma ^ l \\delta _ 0 : ~ \\gamma ^ l \\delta _ 0 > \\gamma \\bar \\Delta l \\in \\mathbb { Z } \\} , \\end{aligned} \\end{align*}"}
+{"id": "8035.png", "formula": "\\begin{align*} - \\sum _ { i \\geq 3 } \\sum _ { d \\geq 2 } a _ { i , d } ( \\underline { b } ) = - a ( \\underline { b } ) + \\sum _ { i \\geq 1 } \\left ( a _ { i , 0 } ( \\underline { b } ) + a _ { i , 1 } ( \\underline { b } ) \\right ) + a _ { 2 , 2 } ( \\underline { b } ) . \\end{align*}"}
+{"id": "994.png", "formula": "\\begin{align*} \\forall x \\in \\Omega _ 4 : \\zeta _ x : = \\max \\{ | \\tau \\cap W | : \\tau \\in \\mathcal { U } _ x , \\dim ( \\tau \\cap U ) = 1 \\} . \\end{align*}"}
+{"id": "5441.png", "formula": "\\begin{align*} C u r \\mathcal { G } = \\mathbb { C } [ \\partial ] \\otimes \\mathcal { G } , [ a _ \\lambda b ] = [ a , b ] , \\end{align*}"}
+{"id": "4331.png", "formula": "\\begin{align*} \\int _ { \\{ - t _ 1 \\le \\Psi < - t _ 2 \\} } | \\tilde { F } | ^ 2 _ h c ( - \\Psi ) = \\frac { G ( T _ 1 ) } { \\int _ { T _ 1 } ^ { + \\infty } c ( s ) e ^ { - s } d s } \\int _ { t _ 2 } ^ { t _ 1 } c ( s ) e ^ { - s } d s \\end{align*}"}
+{"id": "9162.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { 0 } ^ { 1 / 2 } \\left [ h ( x + \\delta \\sin ( 2 \\pi t ) ) - h ( x ) \\right ] \\sin ( 2 \\pi t ) \\ , d t \\\\ \\int _ { 0 } ^ { 1 / 2 } \\left [ h ( x ) - h ( x - \\delta \\sin ( 2 \\pi t ) ) \\right ] \\sin ( 2 \\pi t ) \\ , d t . \\end{aligned} \\end{align*}"}
+{"id": "5922.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Vert X _ n ( t , \\omega ) - X ( t , \\omega ) \\Vert _ H = 0 , \\ ( t , \\omega ) . \\end{align*}"}
+{"id": "3767.png", "formula": "\\begin{align*} \\Omega _ w = \\{ ( F _ \\bullet , V _ \\bullet ) ; \\dim V _ i \\cap F _ j \\geq r _ { i , j } ( w ) \\} . \\end{align*}"}
+{"id": "5450.png", "formula": "\\begin{align*} V _ { a , b } = \\mathbb { C } [ \\partial ] v , L _ \\lambda v = ( \\partial + a \\lambda + b ) v , { H _ i } _ \\lambda v = 0 , \\end{align*}"}
+{"id": "4045.png", "formula": "\\begin{align*} s ( G ^ { \\sigma , \\tau } ) & = \\sum _ { \\gamma \\in \\Gamma } s ( G ^ { ( \\gamma ) } ) + \\sum _ { v \\in V } \\sigma ( v ) - 2 ( \\abs { \\Gamma } - 1 ) . \\end{align*}"}
+{"id": "4323.png", "formula": "\\begin{align*} \\frac { G ( t _ 1 ) - G ( t _ 0 ) } { \\int ^ { t _ 0 } _ { t _ 1 } c ( t ) e ^ { - t } d t } \\leq \\liminf \\limits _ { B \\to 0 + 0 } \\frac { G ( t _ 0 ) - G ( t _ 0 + B ) } { \\int _ { t _ 0 } ^ { t _ 0 + B } c ( t ) e ^ { - t } d t } , \\end{align*}"}
+{"id": "3700.png", "formula": "\\begin{align*} h = 0 , \\nabla h = 0 , v = 0 , \\nu ( v ) = 0 \\mbox { o n } \\hat \\Sigma \\end{align*}"}
+{"id": "954.png", "formula": "\\begin{align*} \\tilde { { \\mathcal N } } _ { 1 } ( u _ { 1 } , u _ { 2 } ) & = 3 \\lambda _ 1 | u _ 1 | ^ 2 u _ 1 , \\\\ \\tilde { { \\mathcal N } } _ { 2 } ( u _ { 1 } , u _ { 2 } ) & = \\lambda _ 6 ( 2 | u _ 1 | ^ 2 u _ 2 + u ^ 2 _ 1 \\overline { u } _ 2 ) . \\end{align*}"}
+{"id": "4146.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n - 1 } } \\theta ^ j _ { \\alpha \\beta } ( x ' , t ; y ' ) d y ' = \\int _ { \\R ^ { n - 1 } } t \\ , \\partial _ j K ^ L _ { \\alpha \\beta } ( x ' - y ' , t ) d y ' = t \\ , \\partial _ j \\int _ { \\R ^ { n - 1 } } K ^ L _ { \\alpha \\beta } ( y ' , t ) d y ' = 0 . \\end{align*}"}
+{"id": "3990.png", "formula": "\\begin{align*} \\mathbf { a } + \\mathbf { b } = ( \\alpha _ 1 + \\beta _ 1 , \\alpha _ 2 + \\beta _ 2 , \\alpha _ 3 + b _ 1 , a _ 1 + b _ 2 , a _ 2 + b _ 3 , \\cdots , a _ { 2 k - 2 } + b _ { 2 k - 1 } , a _ { 2 k - 1 } + \\beta _ 3 , \\alpha _ 4 + \\beta _ 4 ) , \\end{align*}"}
+{"id": "769.png", "formula": "\\begin{align*} I _ 2 & = | c | | \\langle f , g \\rangle | = \\left | c \\int ( f - m _ Q f ) \\ ; g \\ ; d m \\right | \\\\ & \\lesssim \\int _ { 2 Q } | f - m _ Q f | \\ ; | g | \\ ; d m + \\int _ { \\R ^ { N + 1 } \\setminus 2 Q } ( f - m _ Q f ) \\ ; g \\ ; d m \\\\ & = I _ { 2 1 } + I _ { 2 2 } . \\end{align*}"}
+{"id": "1986.png", "formula": "\\begin{align*} - x - T ( x ) \\in [ - a - T ( a ) , a - T ( - a ) ] = [ - a - T ( a ) , a - T ( a ) ] \\end{align*}"}
+{"id": "3240.png", "formula": "\\begin{align*} Y _ t ( x ) = & Y _ 0 ( x + t ) + \\int _ 0 ^ t \\alpha _ s ( x + t - s ) d s \\int _ 0 ^ t \\langle \\sigma _ s , \\delta _ { x + t - s } \\rangle d W _ s . \\end{align*}"}
+{"id": "8008.png", "formula": "\\begin{align*} R _ 2 ( g , \\rho ) ( f ) = \\frac { 1 } { 8 \\pi _ N ( x ) ^ 2 L } \\sum _ { ( p , q ) \\atop { p \\neq q \\leq x \\atop { ( p , N ) = ( q , N ) = 1 } } } T _ 1 ( p ) T _ 2 ( q ) T _ 3 ( p , q ) . \\end{align*}"}
+{"id": "8603.png", "formula": "\\begin{align*} | [ - u _ 1 , u _ 1 ] \\oplus _ 2 \\cdots \\oplus _ 2 [ - u _ m , u _ m ] | ^ 2 = | U B _ 2 ^ m | ^ 2 _ n = \\det ( U U ^ * ) | B _ 2 ^ n | ^ 2 = | B _ 2 ^ n | ^ 2 \\sum _ { | I | = n } ( \\det ( u _ i ) _ { i \\in I } ) ^ 2 . \\end{align*}"}
+{"id": "7838.png", "formula": "\\begin{align*} h ( a , s ) : = \\hat \\Phi ( ( a , 0 ) ^ \\top , s ) = \\hat \\Phi ^ \\dagger ( a , s ) : = \\langle z _ 1 ^ * , \\Phi ^ \\dagger ( a u _ \\ell , p ( s ) ) \\rangle . \\end{align*}"}
+{"id": "4701.png", "formula": "\\begin{align*} \\frac { \\sum _ { i = 1 } ^ p \\log R _ i - m _ n } { s _ n } \\overset { d } { \\longrightarrow } S _ \\alpha , n \\to \\infty \\ , , \\end{align*}"}
+{"id": "8212.png", "formula": "\\begin{align*} x _ { 2 i - 1 , n + 1 - j } = \\tfrac { 1 } { 2 } S - x _ { 2 i - 1 , j } . \\end{align*}"}
+{"id": "7949.png", "formula": "\\begin{align*} \\sum _ { Y < p \\leq X } \\frac { 1 - ( p ^ { 2 i D t } ) } { p } & = \\frac { 1 } { 2 \\pi } \\left ( \\int _ 0 ^ { 2 \\pi } ( 1 - \\cos \\alpha ) d \\alpha \\right ) \\cdot \\log \\left ( \\frac { \\log ( 2 D | t | \\log X ) } { \\max \\{ 1 , D | t | \\log N _ { X , \\eta } \\} } \\right ) - O _ K ( 1 ) \\\\ & \\geq \\left ( \\frac { 1 } { 3 } - \\eta \\right ) \\min \\{ \\log \\log X , \\log ( 1 + | t | \\log X ) \\} - O _ K ( 1 ) . \\end{align*}"}
+{"id": "2567.png", "formula": "\\begin{align*} p ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { 1 } } \\frac { N _ { n _ { s } } } { N _ { n _ { 1 } } } + p ^ { n _ { 2 } ^ { \\prime } - n _ { s } ^ { \\prime } } q ^ { n _ { s } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { s } } = q ^ { n _ { 2 } ^ { \\prime } - n _ { 1 } ^ { \\prime } } a _ { n _ { 2 } } \\frac { N _ { n _ { s } } } { N _ { n _ { 2 } } } . \\end{align*}"}
+{"id": "7306.png", "formula": "\\begin{align*} P _ T = P _ { \\rm P B S } \\left ( \\left | { \\mathcal { K } } \\right | - 1 \\right ) + P _ { \\rm M B S } + P _ { \\rm V L C } \\left ( \\left | { \\mathcal { V } } \\right | \\right ) + \\sum \\limits _ { \\forall j } P _ j , \\end{align*}"}
+{"id": "5448.png", "formula": "\\begin{align*} V _ { \\alpha , \\beta , \\gamma } = \\mathbb { C } [ \\partial ] v , L _ \\lambda v = ( \\partial + \\alpha \\lambda + \\beta ) v , H _ \\lambda v = \\gamma v . \\end{align*}"}
+{"id": "7221.png", "formula": "\\begin{align*} S ( u ) = \\frac { 1 } { 2 } \\sum _ { n = 0 } ^ { + \\infty } \\xi _ { n } \\cdot \\nabla \\left ( \\xi _ { n } \\cdot \\nabla u \\right ) . \\end{align*}"}
+{"id": "5251.png", "formula": "\\begin{align*} D ( A ' ) \\coloneqq \\{ x ' \\in ( X , \\upsilon ) ' \\ ; | \\ ; \\exists \\ ; y ' \\in ( X , \\upsilon ) ' \\ ; \\forall \\ ; x \\in D ( A ) : \\ ; \\langle A x , x ' \\rangle = \\langle x , y ' \\rangle \\} \\end{align*}"}
+{"id": "2470.png", "formula": "\\begin{align*} f ( t , q ^ { \\prime } ) = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( - \\frac { 1 } { 1 - q ^ { \\prime } } \\right ) _ { 2 n } \\left ( ( 1 - q ^ { \\prime } ) \\ ; \\alpha t \\right ) ^ { 2 n } } { ( 2 n ) ! } = \\cosh _ { q ^ { \\prime } } \\left ( \\alpha t \\right ) . \\end{align*}"}
+{"id": "1612.png", "formula": "\\begin{align*} \\begin{aligned} z _ m ( & \\phi _ { \\mathrm { t } , m } , \\phi _ { \\mathrm { r } , m } ) = \\sum _ { i \\in \\{ \\mathrm { t , r } \\} } \\Bigg ( [ \\mathbf { V } _ i ] _ { m , m } | \\phi _ { i , m } | ^ 2 \\\\ & + 2 \\Re \\Big \\{ \\underbrace { \\Big ( \\sum _ { n \\ne m } [ \\mathbf { V } _ i ] _ { m , n } \\phi _ { i , n } - [ \\tilde { \\mathbf { v } } _ i ] _ m \\Big ) } _ { = \\chi _ { i , m } } \\phi _ { i , m } ^ * \\Big \\} \\Bigg ) . \\end{aligned} \\end{align*}"}
+{"id": "6176.png", "formula": "\\begin{align*} W ( r ) = \\frac { \\xi } { r } f + \\eta + \\zeta \\frac { r } { f } , \\xi \\le 0 , \\zeta > 0 , \\end{align*}"}
+{"id": "616.png", "formula": "\\begin{align*} \\omega _ T = \\frac { d \\chi _ 1 } { \\chi _ 1 } \\wedge \\ldots \\wedge \\frac { d { \\chi _ r } } { \\chi _ r } . \\end{align*}"}
+{"id": "1205.png", "formula": "\\begin{align*} \\underset { j = 1 } { \\overset { k } { \\sum } } \\lambda _ { j } \\leq \\underset { j = 1 } { \\overset { k } { \\sum } } a _ { j } , \\end{align*}"}
+{"id": "6798.png", "formula": "\\begin{align*} s _ { m } ( x ) : = ( - 1 ) ^ { m + 1 } \\left ( \\sum _ { k = 1 } ^ { N } \\alpha _ { k } ^ { - } e ^ { 2 \\tau _ { k } x } \\frac { \\left ( \\frac { 1 } { 2 } - \\tau _ { k } \\right ) ^ { m } } { \\left ( \\frac { 1 } { 2 } + \\tau _ { k } \\right ) ^ { m + 1 } } + \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } s ^ { - } \\left ( \\rho \\right ) e ^ { - 2 i \\rho x } \\frac { \\left ( \\frac { 1 } { 2 } + i \\rho \\right ) ^ { m } } { \\left ( \\frac { 1 } { 2 } - i \\rho \\right ) ^ { m + 1 } } d \\rho \\right ) . \\end{align*}"}
+{"id": "971.png", "formula": "\\begin{align*} a _ { 2 c - i } & = \\sum _ { j = 0 } ^ { \\lfloor i / 2 \\rfloor } \\binom { c } { j } \\binom { c - j } { i - 2 j } 2 ^ j , \\end{align*}"}
+{"id": "8542.png", "formula": "\\begin{align*} & \\| \\partial _ { z } r _ { 1 , 2 } ( t ; z ) \\| _ { L ^ 2 } = \\big \\| \\partial _ { z } r _ { 1 , 2 } ( z ) e ^ { - 2 i \\eta ^ 2 t } - 2 i \\alpha t ( 1 - \\frac { \\beta ^ 2 } { 4 } z ^ { - 2 } ) r _ { 1 , 2 } ( t ; z ) e ^ { - 2 i \\eta ^ 2 t } \\big \\| _ { L ^ 2 } \\\\ & \\leq 2 \\alpha T \\left \\| r _ { 1 , 2 } ( z ) \\right \\| _ { L ^ 2 } + \\left \\| \\partial _ { z } r _ { 1 , 2 } ( z ) \\right \\| _ { L ^ 2 } + \\frac { 1 } { 2 } \\alpha \\beta ^ 2 T \\| z ^ { - 2 } r _ { 1 , 2 } ( z ) \\| _ { L ^ 2 } . \\end{align*}"}
+{"id": "1744.png", "formula": "\\begin{align*} \\Gamma ( t , x ) : = \\partial B _ { R _ 0 ( t , x ) } ( x ) . \\end{align*}"}
+{"id": "8019.png", "formula": "\\begin{align*} & \\sum _ { l _ 1 , l _ 2 , \\dots , l _ r \\atop { 0 \\leq l _ i \\leq L } } U ( l _ 1 ) U ( l _ 2 ) \\dots U ( l _ r ) \\sum _ { l _ 1 ' , l _ 2 ' , \\dots , l _ r ' \\atop { 0 \\leq l _ i ' \\leq L } } U ( l _ 1 ' ) U ( l _ 2 ' ) \\dots U ( l _ r ' ) \\\\ & \\sum _ { ( s _ 1 , s _ 2 , \\dots , s _ { t } ) } \\sum _ { ( b _ 1 , b _ 2 , \\dots , b _ { t } ) \\atop { b _ u \\in \\mathcal J _ { \\mathcal I ( s _ u ) } } } \\frac { D ( b _ 1 ) D ( b _ 2 ) \\dots D ( b _ { t } ) } { s _ 1 ^ { b _ 1 } s _ 2 ^ { b _ 2 } \\dots s _ { t } ^ { b _ { t } } } . \\end{align*}"}
+{"id": "395.png", "formula": "\\begin{align*} d _ { \\Omega } ( \\omega , \\omega ' ) : = \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { 2 ^ m } \\frac { \\| \\omega - \\omega ' \\| _ { m } } { 1 + \\| \\omega - \\omega ' \\| _ { m } } , \\| \\omega - \\omega ' \\| _ { m } : = \\sup _ { - m \\leq t \\leq m } | \\omega ( t ) - \\omega ' ( t ) | , \\end{align*}"}
+{"id": "1552.png", "formula": "\\begin{align*} \\phi ' _ { x _ 1 , \\dots , x _ k } ( x ) = ( - 1 ) ^ { i } \\end{align*}"}
+{"id": "7197.png", "formula": "\\begin{align*} a _ { k } ( x ) = \\frac { i } { ( 2 \\pi ) ^ { n } } \\int _ { \\mathbb { R } ^ { n - 1 } } \\int _ { \\mathcal { C } } e ^ { - t \\tau } \\operatorname { T r } \\phi _ { - 1 - k } \\ , d \\tau \\ , d \\xi , 0 \\leqslant k \\leqslant n - 1 . \\end{align*}"}
+{"id": "5130.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\nabla \\phi \\cdot \\nabla \\tilde { \\phi } \\frac { 1 } { r } \\dd z \\dd r = \\mu ^ { 2 } \\int _ { \\mathbb { R } ^ { 2 } _ { + } } ( \\phi - \\phi _ { \\infty } ) _ { + } \\tilde { \\phi } \\frac { 1 } { r } \\dd z \\dd r , \\end{align*}"}
+{"id": "3570.png", "formula": "\\begin{align*} \\varphi ( - t ) ^ 2 H ( - t ) + \\varphi ( t ) ^ 2 H ( t ) = 2 \\varphi ( t ^ 4 ) ^ 2 G ( t ^ 4 ) . \\end{align*}"}
+{"id": "4879.png", "formula": "\\begin{align*} \\begin{cases} \\dim H ^ 1 ( S , T _ S ) = 4 0 ; \\\\ \\dim H ^ 2 ( S , T _ S ) = 0 , \\end{cases} \\end{align*}"}
+{"id": "8097.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta _ { p } u & = v ^ { m } | \\nabla u | ^ { \\alpha } & & \\quad \\mbox { i n } \\Omega , \\\\ \\Delta _ { p } v & = v ^ { \\beta } | \\nabla u | ^ q & & \\quad \\mbox { i n } \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7864.png", "formula": "\\begin{align*} a \\approx a ^ \\mathrm { a p p } ( s ) : = \\sqrt { \\frac { - \\gamma _ 2 s } { \\gamma _ 1 } } . \\end{align*}"}
+{"id": "4886.png", "formula": "\\begin{align*} \\left ( P \\circ Q \\right ) = - \\left ( Q \\circ P \\right ) \\ , , \\end{align*}"}
+{"id": "4977.png", "formula": "\\begin{align*} \\begin{aligned} W ( \\xi _ 0 , k + 1 , x , \\mathsf { L } ^ { k + 1 } ) = & E _ { \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { k + 1 } Z _ i ^ { \\mathsf { L } ^ { k + 1 } } \\biggr ) \\biggr ] \\\\ = & E _ { 1 - \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { k + 1 } Z _ i ^ { \\mathsf { R } ^ { k + 1 } } \\biggr ) \\biggr ] \\\\ = & W ( 1 - \\xi _ 0 , k + 1 , x , \\mathsf { R } ^ { k + 1 } ) . \\end{aligned} \\end{align*}"}
+{"id": "5683.png", "formula": "\\begin{align*} p ' _ { j } \\circ p _ { 1 \\ldots ( n - 1 ) } \\circ f \\circ \\iota _ { 1 \\ldots ( n - 1 ) } \\circ u ' _ { j } = p _ { j } \\circ f \\circ u _ { j } \\circ \\iota _ { 1 \\ldots ( n - 1 ) } = p _ { j } \\circ f \\circ \\iota _ { 1 \\ldots ( n - 1 ) } \\\\ = p ' _ { j } \\circ p _ { 1 \\ldots ( n - 1 ) } \\circ f \\circ \\iota _ { 1 \\ldots ( n - 1 ) } ~ ~ ~ ~ \\forall j = 1 , \\ldots , n - 1 . \\end{align*}"}
+{"id": "6013.png", "formula": "\\begin{align*} \\mathop { \\inf } \\limits _ { \\| ( u , v ) \\| _ { E } = r } I ( u , v ) > 0 . \\end{align*}"}
+{"id": "2343.png", "formula": "\\begin{align*} g = d r ^ 2 + e ^ { 2 \\phi ( r ) } d \\theta ^ 2 \\end{align*}"}
+{"id": "1525.png", "formula": "\\begin{align*} \\Delta _ n = \\left \\{ ( x _ 0 , . . . , x _ n ) \\in \\R _ + ^ { n + 1 } | x _ 0 + . . . + x _ n = 1 \\right \\} \\end{align*}"}
+{"id": "5323.png", "formula": "\\begin{align*} \\widehat { f } ( a , z ) = R _ k ^ { n - 1 } ( - \\lambda , f ) e _ { k , \\lambda } ^ { n - 1 } ( z , 0 ) \\end{align*}"}
+{"id": "3649.png", "formula": "\\begin{align*} \\tau _ y = ( \\sigma _ y ^ { - 1 } + 1 / 2 ) ^ { - 1 } , \\end{align*}"}
+{"id": "4961.png", "formula": "\\begin{align*} h ^ \\theta ( x , u ) \\le \\sup _ { \\rho \\in \\Theta _ { n - 1 } } E _ { \\xi _ 1 ^ \\theta ( u ) } \\biggl [ \\varphi \\biggl ( x + u + \\sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \\rho } \\biggr ) \\biggr ] = V \\left ( \\xi _ 1 ^ \\theta ( u ) , n - 1 , x + u \\right ) . \\end{align*}"}
+{"id": "6507.png", "formula": "\\begin{align*} z _ { n - 1 } ' & = \\lambda ( d ( z ) a ^ { - 1 } b ^ { - 1 } , d ( z ) a ^ { - 1 } b ^ { - 1 } a ) \\\\ z _ n ' & = \\lambda ( d ( z ) a ^ { - 1 } b ^ { - 1 } a , d ( z ) a ^ { - 1 } b ^ { - 1 } a b ) \\\\ z _ { n + 1 } ' & = \\lambda ( d ( z ) a ^ { - 1 } b ^ { - 1 } a b , d ( z ) a ^ { - 1 } b ^ { - 1 } a b ^ 2 ) \\\\ z ' & = z _ 1 . . . z _ { n - 2 } z _ { n - 1 } ' z _ n ' z _ { n + 1 } ' \\end{align*}"}
+{"id": "4603.png", "formula": "\\begin{align*} \\frac { ( n - 1 ) x _ 1 } { g } ( x _ 1 - x _ k ) = \\frac { ( n - 1 ) { x _ 1 } ^ 2 } { g } = \\frac { x _ 1 } { 1 } , \\end{align*}"}
+{"id": "7543.png", "formula": "\\begin{align*} U A _ j U ^ * = \\begin{bmatrix} A _ { j , 1 } & 0 & \\ldots & 0 \\\\ 0 & A _ { j , 2 } & \\ddots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\ldots & 0 & A _ { j , \\ell } \\end{bmatrix} . \\end{align*}"}
+{"id": "7315.png", "formula": "\\begin{align*} \\sum _ { r } T _ { j k r } T _ { i \\ell r } = \\sum _ { r } T _ { i j r } T _ { k \\ell r } \\end{align*}"}
+{"id": "2510.png", "formula": "\\begin{align*} \\mathbf { P } _ { x } = \\left ( \\begin{pmatrix} & 0 & 0 \\\\ & 0 & \\mathbf { P } _ { x ^ { ( i ) } } \\end{pmatrix} : i = 1 , \\cdots , k \\right ) . \\end{align*}"}
+{"id": "2463.png", "formula": "\\begin{align*} F _ { q } ^ { ( k ) } ( s ; q ^ { \\prime } ) = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( - \\frac { 1 } { 1 - q ^ { \\prime } } \\right ) _ { n } } { Q _ { 2 n + 1 } ( 2 - q ) } \\frac { ( ( 1 - q ^ { \\prime } ) \\alpha ) ^ { n } } { n ! } ( - 1 ) ^ { k } \\frac { \\Gamma ( k + 2 n + 1 ) } { s ^ { 2 n + k + 1 } } . \\end{align*}"}
+{"id": "5659.png", "formula": "\\begin{align*} E ^ { 1 } _ { p , q } = H _ { q } ( G , C _ { p } ) \\Rightarrow H _ { p + q } ( G , C _ { * } ) \\end{align*}"}
+{"id": "2191.png", "formula": "\\begin{align*} \\mathbf { \\Psi } = ( \\mathbf { u } _ { \\lambda _ { 1 1 } } ) ^ { \\dag } \\mathbf { u } _ { \\lambda _ { 1 2 } } = ( { \\mathbf { u } ^ { H } _ { \\lambda _ { 1 1 } } } \\mathbf { u } _ { \\lambda _ { 1 1 } } ) ^ { - 1 } { \\mathbf { u } ^ { H } _ { \\lambda _ { 1 1 } } } \\mathbf { u } _ { \\lambda _ { 1 2 } } , \\end{align*}"}
+{"id": "5607.png", "formula": "\\begin{align*} \\begin{cases} a _ { \\alpha + n + 1 } - a _ { n + 1 } = d _ { \\alpha + n } - d _ { \\alpha + n + 1 } \\\\ b _ n = a _ { \\alpha + 1 } + ( d _ { \\alpha + 1 } - d _ \\alpha ) = b \\\\ d _ { \\alpha + n } = d _ \\alpha + ( b _ 1 - a _ { \\alpha + 1 } ) + \\sum _ { j = 2 } ^ n ( a _ j - a _ { \\alpha + j } ) \\\\ d = d _ \\alpha + b - a _ { \\alpha + 1 } + \\sum _ { j = 2 } ^ \\infty ( a _ j - a _ { \\alpha + j } ) \\end{cases} \\ \\end{align*}"}
+{"id": "6027.png", "formula": "\\begin{align*} I ( \\gamma _ { u , v } ( t ) ) = & \\frac { t ^ { 2 \\alpha } } { 2 } \\big ( \\| \\nabla u \\| _ { 2 } ^ { 2 } + \\| \\nabla v \\| _ { 2 } ^ { 2 } \\big ) + \\frac { t ^ { 2 ( \\alpha - 1 ) } } { 2 } \\big ( \\| u \\| _ { 2 } ^ { 2 } + \\omega \\| v \\| _ { 2 } ^ { 2 } \\big ) \\\\ & + \\frac { t ^ { 6 \\alpha - 4 } } { 2 } \\Big ( B ( u ) + B ( v ) \\Big ) - \\frac { t ^ { 2 p \\alpha - 2 } } { 2 p } F ( u , v ) . \\end{align*}"}
+{"id": "6961.png", "formula": "\\begin{align*} u _ n = a \\alpha ^ n + b \\beta ^ n , \\end{align*}"}
+{"id": "6360.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty y ^ { - r / 2 } \\exp \\left \\{ - ( a + n ) y \\right \\} \\mathrm { d } y = \\frac { \\Gamma \\left ( 1 - \\frac { r } { 2 } \\right ) } { ( a + n ) ^ { 1 - r / 2 } } , r < 2 . \\end{align*}"}
+{"id": "5022.png", "formula": "\\begin{align*} h ( G ) & = h ( G ' ) + f ( y ) + \\sum _ { j = 1 } ^ y f ( x _ j ) - \\sum _ { j = 1 } ^ y f ( x _ j - 1 ) \\\\ & \\le h ( K _ { 1 , m - y } ) + f ( y ) + y \\left ( f \\left ( \\frac { m } { y } \\right ) - f \\left ( \\frac { m } { y } - 1 \\right ) \\right ) . \\end{align*}"}
+{"id": "694.png", "formula": "\\begin{align*} \\phi _ { \\gamma ^ { - 1 } \\gamma _ 1 , \\gamma _ 2 } = \\sum _ { h \\in \\Gamma } \\beta ^ { \\gamma } _ { \\gamma _ 1 , h } \\phi _ { h , \\gamma _ 2 } = \\sum _ { h \\in \\Gamma } \\phi _ { \\gamma _ 1 , h } \\beta ^ { \\gamma } _ { h , \\gamma _ 2 } = \\phi _ { \\gamma _ 1 , \\gamma \\gamma _ 2 } \\ , , \\ \\ \\ \\end{align*}"}
+{"id": "3253.png", "formula": "\\begin{align*} a _ N ( z ) : = \\sup _ { n \\in \\mathbb N } \\mathbb E \\left [ \\int _ 0 ^ T \\| p _ N \\alpha _ s ^ { \\mathcal S _ n } \\| _ { \\mathcal H } ^ z d s \\right ] , b _ N ( z ) : = \\sup _ { n \\in \\mathbb N } \\mathbb E \\left [ \\int _ 0 ^ T \\| p _ N \\sigma _ s ^ { \\mathcal S _ n } \\| _ { L _ { } ( U , H ) } ^ z d s \\right ] , \\end{align*}"}
+{"id": "2951.png", "formula": "\\begin{align*} \\begin{aligned} E ^ \\Phi _ A ( x ) & = \\Phi ( T _ 1 , x ) , \\\\ E ^ \\Psi _ B \\left ( h ( x ) \\right ) & = \\Psi \\left ( T _ 2 , h ( x ) \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "7150.png", "formula": "\\begin{align*} a _ 0 ( x ) = \\frac { \\Gamma ( n - 1 ) \\operatorname { v o l } ( \\mathbb { S } ^ { n - 2 } ) } { ( 2 \\pi ) ^ { n - 1 } } \\biggl [ \\biggl ( \\frac { \\lambda + 3 \\mu } { 2 \\mu ( \\lambda + \\mu ) } \\biggr ) ^ { n - 1 } + \\frac { 1 } { ( 2 \\mu ) ^ { n - 1 } } + \\frac { n - 2 } { \\mu ^ { n - 1 } } + \\frac { 1 } { \\alpha ^ { n - 1 } } \\biggr ] , \\end{align*}"}
+{"id": "4372.png", "formula": "\\begin{align*} \\sup \\limits _ i \\int _ K | \\tilde { F } _ i - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f F ^ { 1 + \\delta } | ^ 2 _ { h _ { j _ K , 1 } } < + \\infty . \\end{align*}"}
+{"id": "6611.png", "formula": "\\begin{align*} [ v _ a , v _ b ] = \\varepsilon _ c L v _ c , \\quad \\forall \\quad \\mbox { c y c l i c } ( a , b , c ) \\in \\mathfrak { S } _ 3 . \\end{align*}"}
+{"id": "4762.png", "formula": "\\begin{align*} w = 1 + 2 \\gamma _ 1 a _ 1 + 2 \\gamma _ 2 a _ 2 > 0 \\end{align*}"}
+{"id": "8257.png", "formula": "\\begin{align*} \\psi _ i ^ n ( t ) = \\psi _ { 0 i } + \\int _ 0 ^ t \\biggl ( \\psi _ { i - 1 } ^ n ( s ) \\sum _ { j = 1 } ^ { i - 1 } j V _ { i - 1 , j } \\psi _ j ^ n ( s ) - \\psi _ i ^ n ( s ) \\sum _ { j = 1 } ^ { i } j V _ { i , j } \\psi _ j ^ n ( s ) - \\psi _ i ^ n ( s ) \\sum _ { j = i } ^ { n - 1 } V _ { i , j } \\psi _ j ^ n ( s ) \\biggr ) d s . \\end{align*}"}
+{"id": "3943.png", "formula": "\\begin{align*} D _ { \\uparrow } V ( u ) ( h ) = \\liminf _ { \\substack { t \\downarrow 0 \\\\ h ' \\to h } } \\frac { V ( u + t h ' ) - V ( u ) } { t } . \\end{align*}"}
+{"id": "3392.png", "formula": "\\begin{align*} p _ { k , 0 } = { k + 3 \\choose 3 } - { k + 2 \\choose 2 } - { k + 2 \\choose 3 } = 0 . \\end{align*}"}
+{"id": "4542.png", "formula": "\\begin{align*} \\partial _ t U + \\sum _ { j = 1 } ^ d A ^ j ( U ) \\frac { \\partial U } { \\partial x _ j } + B ( x ) V = 0 . \\end{align*}"}
+{"id": "5760.png", "formula": "\\begin{align*} d F ( X ) = \\langle \\langle X \\cdot \\varphi , \\varphi \\rangle \\rangle , \\end{align*}"}
+{"id": "5768.png", "formula": "\\begin{align*} \\nabla _ Y \\mathcal { P } ( X ) = - \\mathcal { P } ( B ^ * ( Y , X ) ) - B ( Y , s ( X ) ) + B ^ * ( Y , t ( X ) ) . \\end{align*}"}
+{"id": "1706.png", "formula": "\\begin{align*} c ( t , \\xi ) = \\sum _ { j = 1 } ^ d ( \\partial _ j W ( t , \\xi ) ) ^ 2 + \\Delta W ( t , \\xi ) - i ( \\mu ( \\xi ) + \\tilde { \\mu } ( \\xi ) ) . \\end{align*}"}
+{"id": "6729.png", "formula": "\\begin{align*} v _ k ( \\alpha ) = \\sum _ { m = k } ^ \\infty ( - 1 ) ^ { k + m } \\ , \\binom { \\alpha } { m } \\ , \\binom { m } { k } \\end{align*}"}
+{"id": "2468.png", "formula": "\\begin{align*} f ( t ) = ( \\alpha t ) ^ { \\delta } { } _ { 0 } F _ { 1 } \\left ( \\ ; ; \\frac { 1 } { 2 } + \\delta ; \\frac { 1 } { 4 } \\alpha ^ { 2 } t ^ { 2 } \\right ) = \\left \\{ \\begin{array} { l l } \\cosh \\alpha t & \\mbox { f o r } \\ ; \\delta = 0 \\\\ \\sinh \\alpha t & \\mbox { f o r } \\ ; \\delta = 1 \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "1502.png", "formula": "\\begin{align*} I _ { a + } ^ { \\nu _ { 1 } } I _ { a + } ^ { \\nu _ { 2 } } u ( t ) = I _ { a + } ^ { \\nu _ { 1 } + \\nu _ { 2 } } u ( t ) = I _ { a + } ^ { \\nu _ { 2 } } I _ { a + } ^ { \\nu _ { 1 } } u ( t ) , \\nu _ { 1 } , \\nu _ { 2 } > 0 ; \\end{align*}"}
+{"id": "8134.png", "formula": "\\begin{align*} t _ 1 = z _ 1 \\cdots z _ { n _ 1 } , t _ 2 = z _ { n _ 1 + 1 } \\cdots z _ { n _ 2 } , \\dots , t _ s = z _ { n _ { s - 1 } + 1 } \\cdots z _ r . \\end{align*}"}
+{"id": "3549.png", "formula": "\\begin{align*} w ( q ) & = \\phi ( - q ^ 3 ) \\frac { \\eta _ 2 ^ 2 \\eta _ 3 } { \\eta _ 1 \\eta _ 6 } \\\\ & = \\phi ( - q ^ 3 ) \\left ( \\phi ( - q ^ 9 ) + \\frac { \\eta _ 3 \\eta _ { 1 8 } ^ 2 } { \\eta _ 6 \\eta _ 9 } \\right ) \\\\ & = \\phi ( - q ^ 3 ) \\phi ( - q ^ 9 ) + \\frac { \\varphi _ 3 ^ 2 } { \\varphi _ 6 } q \\frac { \\varphi _ 3 \\varphi _ { 1 8 } ^ 2 } { \\varphi _ 6 \\varphi _ 9 } \\\\ & = \\phi ( - q ^ 3 ) \\phi ( - q ^ 9 ) + q x ( q ^ 3 ) . \\end{align*}"}
+{"id": "8291.png", "formula": "\\begin{align*} X _ t ^ i = X _ 0 ^ i + \\int _ 0 ^ t \\bigg ( b ( X _ s ^ i ) + \\frac 1 { N - 1 } \\sum _ { j \\neq i } K ( X _ s ^ i - X _ s ^ j ) \\bigg ) \\d s + \\sigma W _ t ^ i , ~ ~ ~ ~ i = 1 , \\cdots , N , \\end{align*}"}
+{"id": "169.png", "formula": "\\begin{align*} Y \\cdot ( f v ) = ( Y \\cdot f ) v + f ( Y \\cdot f ) . \\end{align*}"}
+{"id": "4394.png", "formula": "\\begin{align*} \\sup \\limits _ { X } e ^ { - u ( - v ( \\Psi ) ) } \\le \\sup \\limits _ { t \\in [ T , t _ 0 + B ] } e ^ { - u ( t ) } = \\frac { 1 } { \\delta } c ( T ) e ^ { - T } + \\int ^ { t _ 0 + B } _ T c ( t _ 1 ) e ^ { - t _ 1 } d t _ 1 . \\end{align*}"}
+{"id": "7896.png", "formula": "\\begin{align*} \\gamma _ 1 ^ \\mathfrak { t } & = \\frac { 1 } { 2 } \\left ( c _ 5 ( q , q , p ^ \\mathfrak { t } ( 0 ) ) - \\frac { c _ 2 ( q , q , p ^ \\mathfrak { t } ( 0 ) ) c _ 3 ( q , 2 q , q , p ^ \\mathfrak { t } ( 0 ) ) } { c _ 1 ( q , 2 q , p ^ \\mathfrak { t } ( 0 ) ) } \\right ) \\\\ & = \\gamma _ 1 ^ 0 - \\frac { 1 } { 4 } \\mathfrak { t } \\hat W _ r ( q ) + \\frac { c _ 3 ( q , 2 q , q , p ^ 0 ( 0 ) ) } { 4 c _ 1 ( q , 2 q , p ^ 0 ( 0 ) ) } \\mathfrak { t } ( - \\hat W _ r ( 0 ) + \\hat W _ r ( 2 q ) ) \\\\ & = \\gamma _ 1 ^ 0 + \\mathfrak { t } X ( q , r ) \\end{align*}"}
+{"id": "6309.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { \\lambda _ t } } - 1 \\geq \\frac { 1 } { 2 } \\alpha _ 0 \\sum _ { i = 1 } ^ t \\sqrt { \\gamma _ i / \\gamma _ 0 } \\quad \\Rightarrow \\lambda _ t \\leq 4 \\left ( 2 + \\alpha _ 0 \\sum _ { i = 1 } ^ t \\sqrt { \\gamma _ i / \\gamma _ 0 } \\right ) ^ { - 2 } . \\end{align*}"}
+{"id": "3307.png", "formula": "\\begin{align*} I ( G ) = \\{ \\sigma \\subset V ( G ) \\ | \\ \\{ u , v \\} \\notin E ( G ) \\} . \\end{align*}"}
+{"id": "3218.png", "formula": "\\begin{align*} ( x , . . . , x ) \\in X ^ N = p _ X ^ { \\otimes N } [ \\nu _ 1 \\cdot \\gamma _ 1 , . . . , \\nu _ N \\cdot \\gamma _ N ] \\end{align*}"}
+{"id": "5370.png", "formula": "\\begin{align*} \\begin{aligned} 0 < 2 \\sigma _ 2 ( \\lambda ( D ^ 2 u ) ) = \\ , & 2 \\sum _ { 1 \\leq i < j \\leq n } ( u _ { i i } u _ { j j } - u _ { i j } ^ 2 ) \\\\ = \\ , & ( \\Delta u ) ^ 2 - \\sum _ i u _ { i i } ^ 2 - \\sum _ { i \\neq j } u _ { i j } ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "7263.png", "formula": "\\begin{align*} P ( \\alpha , \\alpha ^ q ) = \\alpha ^ { q ( q + 1 ) } + a \\alpha ^ { 2 q } + b \\alpha ^ { q + 1 } = \\alpha ^ q ( \\alpha ^ { q ^ 2 } + a \\alpha ^ { q } + b \\alpha ) = 0 . \\end{align*}"}
+{"id": "2134.png", "formula": "\\begin{align*} Q _ { 1 1 } = \\sum _ { k = 2 } ^ { 4 } \\mathcal { Q } _ k [ \\varrho ] \\biggl | _ M , \\end{align*}"}
+{"id": "8577.png", "formula": "\\begin{align*} \\kappa ( t ) = h _ \\alpha ( t ) , \\ k _ 1 ( t ) = h _ \\gamma ( t ) , \\ k _ 2 ( t ) = h _ { 1 - \\alpha - \\gamma } ( t ) , \\ t > 0 \\end{align*}"}
+{"id": "4777.png", "formula": "\\begin{align*} F _ n ( \\eta ) : = T _ n - \\frac { 1 } { \\eta } I _ n , \\eta \\in \\Omega . \\end{align*}"}
+{"id": "1739.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d ( r _ n ( x ) , r _ n ( y ) ) } { d ( x , y ) } & \\leq \\frac { d ( x , y ) + 2 r } { d ( x , y ) } \\leq 1 + \\frac { 2 r } { \\rho r } = 1 + \\frac { 2 } { \\rho } . \\end{aligned} \\end{align*}"}
+{"id": "4839.png", "formula": "\\begin{align*} M ( x , y , z ) = \\left ( \\begin{array} { c c c c } 0 & z + \\l _ 0 & \\l _ 1 & \\l _ 2 \\\\ - z - \\l _ 0 & 0 & \\l _ 3 & \\l _ 4 \\\\ - \\l _ 1 & - \\l _ 3 & 0 & z - \\l _ 0 \\\\ - \\l _ 2 & - \\l _ 4 & - z + \\l _ 0 & 0 \\end{array} \\right ) , \\l _ 0 , \\dots , \\l _ 4 \\in \\frak m _ { x , y } . \\end{align*}"}
+{"id": "4464.png", "formula": "\\begin{align*} Q ^ { * } = a _ { 1 } ^ { * } x _ { 1 } ^ { 2 } + a _ { 2 } ^ { * } ( x _ { 2 } + n _ { 2 1 } x _ { 1 } ) ^ { 2 } + a _ { 3 } ^ { * } ( x _ { 3 } + n _ { 3 1 } x _ { 1 } + n _ { 3 2 } x _ { 2 } ) ^ { 2 } + a _ { 4 } ^ { * } ( x _ { 4 } + n _ { 4 1 } x _ { 1 } + n _ { 4 2 } x _ { 2 } + n _ { 4 3 } x _ { 3 } ) ^ { 2 } , \\end{align*}"}
+{"id": "1754.png", "formula": "\\begin{align*} u _ 1 ( t , x , y ) = \\sum _ { i = 1 } ^ n \\partial _ { x _ i } u _ 0 ( t , x ) w _ i ( t , x , y ) , \\end{align*}"}
+{"id": "6814.png", "formula": "\\begin{align*} \\widehat { \\mathbf { H } } ( \\varphi ) = \\left \\{ h _ { k + m } \\right \\} _ { k , m = 0 } ^ { \\infty } , h _ { p } = \\frac { \\left ( - 1 \\right ) ^ { p + 1 } } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\frac { \\varphi \\left ( \\tau \\right ) z ^ { p + 1 } \\left ( \\tau \\right ) d \\tau } { \\frac { 1 } { 4 } + \\tau ^ { 2 } } . \\end{align*}"}
+{"id": "8613.png", "formula": "\\begin{align*} \\frac { | A \\oplus _ 2 B | ^ 2 } { | P _ u ( A \\oplus _ 2 B ) | ^ 2 } & = \\frac { | B _ 2 ^ n | ^ 2 } { | B _ 2 ^ { n - 1 } | ^ 2 } \\rho _ { A \\oplus _ 2 B } ( u ) ^ 2 \\\\ & \\ge \\frac { | B _ 2 ^ n | ^ 2 } { | B _ 2 ^ { n - 1 } | ^ 2 } ( \\rho _ { A } ( u ) ^ 2 + \\rho _ { B } ( u ) ^ 2 ) = \\frac { | A | ^ 2 } { | P _ u A | ^ 2 } + \\frac { | B | ^ 2 } { | P _ u B | ^ 2 } . \\end{align*}"}
+{"id": "4458.png", "formula": "\\begin{align*} \\psi ' ( x ) = 7 2 x K _ { 1 } ( 4 \\pi x ) \\left ( \\frac { K _ { 0 } ( 4 \\pi x ) } { K _ { 1 } ( 4 \\pi x ) } - \\frac { 4 } { 3 } \\pi x \\right ) . \\end{align*}"}
+{"id": "137.png", "formula": "\\begin{align*} f _ 1 & = \\mathcal { F } ( \\gamma ( N _ 1 ^ { ( 5 - 2 \\alpha ) + \\varepsilon } t - n ) \\mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ { N _ 1 } v ) , \\\\ f _ 2 & = \\mathcal { F } ( \\gamma ( N _ 1 ^ { ( 5 - 2 \\alpha ) + \\varepsilon } t - n ) \\mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ { N _ 2 } u _ 1 ) , \\\\ f _ 3 & = \\mathcal { F } ( \\gamma ( N _ 1 ^ { ( 5 - 2 \\alpha ) + \\varepsilon } t - n ) \\mathbf { 1 } _ { [ 0 , T ] } ( t ) P _ N v ) . \\end{align*}"}
+{"id": "5355.png", "formula": "\\begin{align*} E ^ + - E ^ { - } = f ^ * ( f _ * D ) - D = f ^ * ( f _ * N ) - N + f ^ * ( f _ * P ) - P = E _ N ^ + - E _ N ^ { - } + E _ P . \\end{align*}"}
+{"id": "8785.png", "formula": "\\begin{align*} B ( x , x ) = \\begin{pmatrix} x _ 2 x _ 3 \\\\ x _ 1 x _ 3 \\\\ - 2 x _ 1 x _ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "1793.png", "formula": "\\begin{align*} L ( f , 1 ) = \\frac { 1 } { 3 } \\int _ { 0 } ^ { 1 } b ^ { 2 } ( q ) c ( q ^ { 3 } ) \\frac { d q } { q } = \\frac { 1 } { 9 } \\int _ { 0 } ^ { 1 } b ^ { 2 } ( q ) ( a ( q ) - b ( q ) ) \\frac { d q } { q } . \\end{align*}"}
+{"id": "2684.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ d q _ i = \\left ( \\prod _ { i = 1 } ^ d r _ i \\right ) n . \\end{align*}"}
+{"id": "3814.png", "formula": "\\begin{align*} | | \\mathcal { B } _ q ( \\psi _ m ) | | _ { Q M L _ q ( \\mathbb { H } ) } = 1 = | | \\psi _ m | | _ { L ^ 2 ( \\mathbb { R , , \\mathbb { H } } ) } . \\end{align*}"}
+{"id": "166.png", "formula": "\\begin{align*} C ( G _ { ( h ) } , F ) : = C ( G _ { ( h ) } , K ) \\widehat { \\otimes } _ K F . \\end{align*}"}
+{"id": "7930.png", "formula": "\\begin{align*} f ( x ) = \\int _ \\S W _ r ( x - y ) \\prod _ { i = 1 } ^ n ( \\eta _ i ( y ) - \\eta _ i ( x ) ) \\sin ^ { [ n ] } ( \\tilde \\Psi ( y ) - \\tilde \\Psi ( x ) ) \\ \\d y \\end{align*}"}
+{"id": "7667.png", "formula": "\\begin{align*} \\boldsymbol { \\alpha } _ t ^ * : = ( \\alpha _ t ^ { * , i } ) _ { 1 \\leq i \\leq N } = \\left ( - R ^ { - 1 } B y _ t ^ { * , i } - h ( \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } ^ * _ t } ) \\right ) _ { 1 \\leq i \\leq N } \\end{align*}"}
+{"id": "6989.png", "formula": "\\begin{align*} \\prod _ { { \\substack { p < T \\\\ p \\nmid 2 \\alpha \\beta } } } | t _ l | ^ { - s } _ p = A B \\end{align*}"}
+{"id": "1553.png", "formula": "\\begin{align*} \\langle G \\rangle \\supset \\{ \\mathrm { i d } , - \\mathrm { i d } \\} \\circ \\langle F _ 2 \\rangle \\circ ( F _ 0 \\cup F _ 1 ) \\supset \\{ \\mathrm { i d } , - \\mathrm { i d } \\} \\circ \\bigcup _ { k = 0 } ^ \\infty F _ k \\supset \\langle G \\rangle , \\end{align*}"}
+{"id": "9087.png", "formula": "\\begin{align*} \\chi _ l \\leqslant w _ l \\chi + k _ l , ( l = 2 , \\dots , n ) \\end{align*}"}
+{"id": "2342.png", "formula": "\\begin{align*} \\left < \\partial _ t \\gamma , \\nu \\right > _ g = \\kappa \\textnormal { o n } \\mathbb { R } \\times ( 0 , T ) , \\end{align*}"}
+{"id": "3797.png", "formula": "\\begin{align*} B _ q ( \\phi ) ( z ) & = \\int _ { \\mathbb { C } } f ( w ) | w | ^ { \\frac { 2 } { q } - 2 } e ^ { - \\frac { | w | ^ 2 } { q } } \\overline { K _ q ( w , z ) } d A ( w ) \\end{align*}"}
+{"id": "6845.png", "formula": "\\begin{align*} q ( x ) = \\left \\{ \\begin{array} [ c ] { c c } e ^ { x } \\cos 4 x , & x < 0 , \\\\ e ^ { - x } J _ { 0 } ( 2 x ) , & x > 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "2360.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { L } ( u ) = f & \\ \\Omega _ { T } \\\\ u = u _ 0 & \\ \\overline { \\Omega } \\times \\{ 0 \\} \\\\ u = \\psi & \\ \\{ a , b \\} \\times [ 0 , T ] \\end{cases} \\end{align*}"}
+{"id": "1952.png", "formula": "\\begin{align*} = \\frac { \\sqrt { \\pi } \\left ( \\frac { 1 } { 2 } | z | \\right ) ^ { \\nu + \\frac { 1 } { 2 } } } { \\Gamma ( \\nu + 1 ) } \\frac { \\Gamma ( 2 \\nu + 1 , x ) } { x ^ { 2 \\nu + 1 } } , \\end{align*}"}
+{"id": "8110.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & Y ( t _ { j } ) \\leq Y _ 2 , \\\\ & Z ( t _ j ) \\leq Z _ 2 , \\\\ & W ( t _ j ) \\leq W _ 2 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7082.png", "formula": "\\begin{gather*} v _ { 1 } ( x ) = \\eta ^ { 2 } ( x ) [ ( u - \\psi ) ( x + h ) - ( u - \\psi ) ( x ) ] \\end{gather*}"}
+{"id": "2956.png", "formula": "\\begin{align*} T : = T ^ e _ A ( x ) . \\end{align*}"}
+{"id": "4191.png", "formula": "\\begin{align*} \\mathcal { N } _ \\varepsilon = & \\{ u \\in \\mathcal { H } ( \\Omega ) \\setminus \\{ 0 \\} \\mid \\langle I ' _ \\varepsilon ( u ) , u \\rangle = 0 \\} \\\\ = & \\left \\{ u \\in \\mathcal { H } ( \\Omega ) \\setminus \\{ 0 \\} \\mid \\int _ { \\Omega } ( K _ H ( x ) \\nabla u | \\nabla u ) d x = \\frac { 1 } { \\varepsilon ^ 2 } \\int _ { \\Omega } \\left ( u - q \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ { p } _ + u d x \\right \\} . \\end{align*}"}
+{"id": "6275.png", "formula": "\\begin{align*} \\Omega _ { i j } ( \\partial _ j , \\partial _ r ) = \\frac { - \\Gamma _ { j i } ^ i \\Gamma _ { r j } ^ j + \\Gamma _ { r i } ^ i \\Gamma _ { j r } ^ r - \\Gamma _ { r i } ^ i \\Gamma _ { j r } ^ r } { \\varphi _ j } . \\end{align*}"}
+{"id": "5729.png", "formula": "\\begin{align*} r = \\frac { h } { h + h ^ q } . \\end{align*}"}
+{"id": "7408.png", "formula": "\\begin{align*} M ^ { ( 1 ) } = N , M ^ { ( 1 ) } = M ^ { ( 1 ) } _ 1 + M ^ { ( 1 ) } _ 2 , M ^ { ( 1 ) } _ 1 = N _ 1 ~ , M ^ { ( 1 ) } _ 2 = N _ 2 + \\dots + N _ L . \\end{align*}"}
+{"id": "415.png", "formula": "\\begin{align*} X ^ { \\mathfrak h } _ 1 + X ^ { \\mathfrak n } _ 1 , \\ldots , X ^ { \\mathfrak h } _ m + X ^ { \\mathfrak n } _ m , X ^ { \\mathfrak n } _ { m + 1 } , \\ldots , X ^ { \\mathfrak n } _ k \\mbox { i s a b a s i s f o r } \\Delta . \\end{align*}"}
+{"id": "7870.png", "formula": "\\begin{align*} 0 & = ( I - Q ) F _ { \\Psi \\Psi } ( \\Psi ^ q , p _ 0 ) [ \\psi _ p ( 0 , p _ 0 ) p ' ( 0 ) , \\psi _ p ( 0 , p _ 0 ) p ' ( 0 ) ] \\\\ & + ( I - Q ) F _ { \\Psi p } ( \\Psi ^ q , p _ 0 ) [ \\psi _ p ( 0 , p _ 0 ) p ' ( 0 ) , p ' ( 0 ) ] \\\\ & + ( I - Q ) F _ \\Psi ( \\Psi ^ q , p _ 0 ) \\psi _ { p p } ( 0 , p _ 0 ) [ p ' ( 0 ) , p ' ( 0 ) ] \\\\ & + ( I - Q ) F _ { \\Psi p } ( \\Psi ^ q , p _ 0 ) [ \\psi _ p ( 0 , p _ 0 ) p ' ( 0 ) , p ' ( 0 ) ] \\\\ & + ( I - Q ) F _ { p p } ( \\Psi ^ q , p _ 0 ) [ p ' ( 0 ) , p ' ( 0 ) ] . \\end{align*}"}
+{"id": "5064.png", "formula": "\\begin{align*} \\begin{aligned} < u \\times n , \\xi > _ { \\partial \\Omega } & = \\int _ { \\Omega } \\nabla \\times u \\cdot \\xi \\dd x - \\int _ { \\Omega } u \\cdot \\nabla \\times \\xi \\dd x , \\\\ < u \\cdot n , \\varphi > _ { \\partial \\Omega } & = \\int _ { \\Omega } u \\cdot \\nabla \\varphi \\dd x + \\int _ { \\Omega } \\nabla \\cdot u \\varphi \\dd x , \\end{aligned} \\end{align*}"}
+{"id": "3668.png", "formula": "\\begin{align*} \\int _ { M \\setminus \\Omega } \\bigg \\langle S ' ( h , v ) , \\ , \\kappa _ 0 ( \\bar { g } , \\bar { u } , X ) \\bigg \\rangle _ { \\bar g } \\ , d \\mathrm { v o l } _ { \\bar g } & = 0 \\end{align*}"}
+{"id": "700.png", "formula": "\\begin{align*} \\int ( I _ \\alpha \\ast f ) g d x = \\int \\left ( I _ { \\frac { \\alpha } { 2 } } \\ast f \\right ) \\left ( I _ { \\frac { \\alpha } { 2 } } \\ast g \\right ) d x . \\end{align*}"}
+{"id": "3522.png", "formula": "\\begin{align*} - n ^ p - 2 \\sum _ { K = 1 } ^ { p } \\sum _ { j = 0 } ^ { K / 2 } \\binom { p } { p - K } \\binom { K } { 2 j + 1 } B _ { K - 2 j - 1 } \\ , n ^ { p - K } . \\end{align*}"}
+{"id": "609.png", "formula": "\\begin{align*} d _ \\phi ( n _ 1 , n _ 2 ) : = \\frac 1 2 \\left ( d ( 1 , \\pi _ N ( q _ 1 ^ { - 1 } q _ 2 ) ) + d ( 1 , \\pi _ N ( q _ 2 ^ { - 1 } q _ 1 ) ) \\right ) , \\mbox { f o r a l l } n _ 1 , n _ 2 \\in E , \\end{align*}"}
+{"id": "5298.png", "formula": "\\begin{align*} l ( y _ p ) \\leq \\bigvee _ { 1 \\leq i \\leq n } l ( \\Psi ( v _ i ) ) = \\bigvee _ { 1 \\leq i \\leq n } \\Phi ( l ( v _ i ) ) = \\bigvee _ { 1 \\leq i \\leq n } \\Phi ( p _ i ) = \\Phi \\left ( \\bigvee _ { 1 \\leq i \\leq n } p _ i \\right ) = \\Phi ( p ) . \\end{align*}"}
+{"id": "6736.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } s } \\log \\phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) & = \\frac { 1 } { s } + \\frac { \\psi ( 1 / s ) } { s ^ 2 } - \\left | \\frac { x - \\mu } { \\sigma } \\right | ^ s \\log \\left ( \\left | \\frac { x - \\mu } { \\sigma } \\right | \\right ) . \\end{align*}"}
+{"id": "4763.png", "formula": "\\begin{align*} x ^ { * } = - \\frac { 1 } { w } ( \\gamma _ 1 b _ 1 + \\gamma _ 2 b _ 2 ) , \\ \\ ( x ^ { * } ) ^ { T } x ^ { * } = \\frac { 1 } { w ^ { 2 } } ( \\gamma _ 1 b _ 1 + \\gamma _ 2 b _ 2 ) ^ { T } ( \\gamma _ 1 b _ 1 + \\gamma _ 2 b _ 2 ) \\end{align*}"}
+{"id": "5456.png", "formula": "\\begin{align*} q _ X ( \\alpha ) : = \\frac { n } { 2 } \\int _ X ( \\sigma \\overline \\sigma ) ^ { n - 1 } \\alpha ^ 2 + ( 1 - n ) \\int _ X \\sigma ^ n \\overline \\sigma ^ { n - 1 } \\alpha \\int _ X \\sigma ^ { n - 1 } \\overline \\sigma ^ n \\alpha \\end{align*}"}
+{"id": "1701.png", "formula": "\\begin{align*} \\begin{cases} 1 < \\alpha < \\infty , & 1 \\le d \\le 4 , \\\\ 1 < \\alpha < 1 + \\frac { 2 } { d - 2 } , \\ 2 \\le \\alpha < 1 + \\frac { 4 } { d - 4 } , & 5 \\le d \\le 7 , \\\\ 1 < \\alpha < 1 + \\frac { 2 } { d - 2 } , & d \\ge 8 . \\end{cases} \\end{align*}"}
+{"id": "7787.png", "formula": "\\begin{align*} \\dot \\phi _ i = \\omega _ i + \\frac 1 M \\sum _ { j = 1 } ^ M a _ { i j } \\sin ( \\phi _ j - \\phi _ i ) , \\end{align*}"}
+{"id": "1447.png", "formula": "\\begin{align*} \\mathbf { y } _ { \\mathrm { r } } \\left [ l \\right ] & = \\mathbf { H } _ { \\mathrm { r } } \\mathbf { x } \\left [ l \\right ] + \\mathbf { m } \\left [ l \\right ] \\\\ & = \\alpha e ^ { j 2 \\pi \\mathcal { F } _ { D } l T } { \\mathbf { b } } \\left ( \\theta \\right ) { \\mathbf { a } } ^ { H } \\left ( \\theta \\right ) \\mathbf { x } \\left [ l \\right ] + \\mathbf { m } \\left [ l \\right ] , \\end{align*}"}
+{"id": "7519.png", "formula": "\\begin{align*} \\int _ { B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x & \\leq \\theta ^ k \\int _ { B _ { 8 ^ k \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x \\leq \\frac { 1 } { \\theta } \\rho ^ { 2 \\alpha } \\int _ { B _ 1 } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x , \\end{align*}"}
+{"id": "5175.png", "formula": "\\begin{align*} | H [ b _ n ] | = 4 \\pi \\left | \\int _ { \\mathbb { R } ^ { 2 } _ { + } } ( \\phi _ n - \\phi _ \\infty ) _ { + } G _ n \\frac { 1 } { r } \\dd z \\dd r \\right | & \\leq 4 \\pi | | ( \\phi _ n - \\phi _ \\infty ) _ { + } | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } _ { + } ; r ^ { - 1 } ) } | | G _ n | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } _ { + } ; r ^ { - 1 } ) } \\\\ & \\leq 4 \\pi | | ( \\phi _ n - r ^ { 2 } ) _ { + } | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } _ { + } ; r ^ { - 1 } ) } \\left ( \\sup _ { n } | | b _ n | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } \\right ) . \\end{align*}"}
+{"id": "8074.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\mathbf { s } ^ { \\left ( \\right ) ^ H } \\mathbf { s } ^ { \\left ( \\right ) } \\right ] = & \\mathbb { E } \\left [ s _ c ^ * s _ c \\right ] + \\mathbb { E } \\left [ s _ 1 ^ * s _ 1 \\right ] \\cdots + \\mathbb { E } \\left [ s _ M ^ * s _ M \\right ] , = M + 1 . \\end{align*}"}
+{"id": "8184.png", "formula": "\\begin{align*} \\frac { d } { d t } ( \\mathcal { Q } ^ { r } ( t ) ) = \\theta _ { 1 } \\varrho _ { 0 } ^ { 1 - \\sigma } ( r - 1 ) ( \\mathcal { M } ^ { r } ( t ) ) ^ { \\sigma + 1 } \\ge \\theta _ { 1 } \\varrho _ { 0 } ^ { 1 - \\sigma } ( r - 1 ) ( \\mathcal { Q } ^ { r } ( t ) ) ^ { \\sigma + 1 } , t \\in [ \\delta , t _ { 0 } ) . \\end{align*}"}
+{"id": "5987.png", "formula": "\\begin{align*} \\begin{cases} x _ 1 \\mapsto \\frac { 4 a ^ 2 } { s ^ 2 } ( x _ 1 - \\tilde { \\nu _ 1 } ' ( l s ) x _ 2 - \\tilde { \\nu _ 1 } ( l s ) x _ 3 + \\tilde { \\nu _ 1 } ' ( l s ) l s x _ 3 ) \\\\ x _ 2 \\mapsto \\frac { 2 a } { s } ( x _ 2 - l s x _ 3 ) \\\\ x _ 3 \\mapsto x _ 3 \\end{cases} \\end{align*}"}
+{"id": "5698.png", "formula": "\\begin{align*} M ( X ) = \\sum _ { 0 \\le i \\le n - 1 } C _ i X ^ { 2 ^ i } + D . \\end{align*}"}
+{"id": "4118.png", "formula": "\\begin{align*} \\forall ( x , y ) \\in E \\times E \\ , \\ \\ b _ { \\sigma } ( x , y ) = \\sigma ( x ) ( y ) . \\end{align*}"}
+{"id": "3050.png", "formula": "\\begin{align*} \\deg \\varphi = [ S : \\varphi ^ { * } ( S ) ] \\end{align*}"}
+{"id": "2110.png", "formula": "\\begin{align*} R L ( b , c ) - L L ( b , c ) & = \\sum _ { i = 1 } ^ b \\left [ \\Delta _ c \\left ( \\binom { b } { 2 } + i \\right ) - \\Delta _ c ( i ) \\right ] - b \\left ( \\Delta _ c ( b ) - \\Delta _ c ( 1 ) \\right ) \\\\ & > b \\left [ \\Delta _ c \\left ( \\binom { b } { 2 } + b \\right ) - \\Delta _ c ( b ) \\right ] - b \\left ( \\Delta _ c ( b ) - \\Delta _ c ( 1 ) \\right ) \\\\ & = b \\left [ \\Delta _ c \\left ( \\binom { b + 1 } { 2 } \\right ) + \\Delta _ c ( 1 ) - 2 \\Delta _ c ( b ) \\right ] \\end{align*}"}
+{"id": "162.png", "formula": "\\begin{align*} F ( x ) = \\frac { 2 x e ^ { - x } } { 1 - e ^ { - 2 x } } + 2 x \\sum _ { i = 1 } ^ { \\infty } 2 ^ { - i } \\exp ( - 2 ^ { - i } x ) - 4 x ^ 2 \\sum _ { i = 1 } ^ { \\infty } 2 ^ { - 2 i } \\exp ( - 2 ^ { 1 - i } x ) . \\end{align*}"}
+{"id": "7154.png", "formula": "\\begin{align*} \\sum _ { J } \\frac { ( - i ) ^ { | J | } } { J ! } \\partial _ { \\xi } ^ { J } q \\ , \\partial _ { x ^ \\prime } ^ { J } q - \\sum _ { J } \\frac { ( - i ) ^ { | J | } } { J ! } \\partial _ { \\xi } ^ { J } b \\ , \\partial _ { x ^ \\prime } ^ { J } q - \\frac { \\partial q } { \\partial x _ n } + c = 0 , \\end{align*}"}
+{"id": "4001.png", "formula": "\\begin{align*} | | \\nabla ( u - \\tilde { U } ) | | _ { L _ 2 ( \\Omega ) } ^ 2 = | | \\Delta \\tilde { U } + F | | ^ 2 _ { H ^ { - 1 } ( \\Omega ) } \\to \\min \\end{align*}"}
+{"id": "1659.png", "formula": "\\begin{align*} ( f \\star g ) ( z ) = k _ 1 k _ 2 ^ { - 1 } ( c z + d ) k _ 2 ( \\det ( g ) ) f ( g z ) , \\end{align*}"}
+{"id": "3667.png", "formula": "\\begin{align*} \\int _ { M \\setminus \\Omega } \\bigg \\langle S ( g , u ) , \\ , \\kappa _ 0 ( g , u , X ) \\bigg \\rangle _ g \\ , d \\mathrm { v o l } _ g & = 0 , \\end{align*}"}
+{"id": "5171.png", "formula": "\\begin{align*} \\begin{aligned} \\left | H [ b _ 0 ] - H [ b _ 1 ] - H [ b _ 2 ] \\right | & \\lesssim \\left ( \\max _ { 0 \\leq i \\leq 2 } | | b _ i | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } ^ { 5 / 3 } \\right ) | | b _ 0 - b _ 1 - b _ 2 | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } , \\end{aligned} \\end{align*}"}
+{"id": "3868.png", "formula": "\\begin{align*} \\mathcal { L } _ H \\varPhi = W = f _ \\varepsilon \\left ( \\varPhi - \\frac { \\alpha } { 2 } | x ' | ^ 2 | \\ln \\varepsilon | \\right ) \\ \\ \\ B _ { R ^ * } ( 0 ) , \\end{align*}"}
+{"id": "519.png", "formula": "\\begin{align*} S _ { R _ n } ^ { \\beta + \\delta } - \\frac { C _ { \\beta , \\delta } } { C _ { \\beta , \\delta + 1 } } S _ { R _ n } ^ { \\beta + \\delta + 1 } & = C _ { \\beta , \\delta } R _ n ^ { - ( \\beta + \\delta ) } \\int ^ { R _ n } _ 0 ( R _ n - t ) ^ { \\beta - 1 } t ^ \\delta \\big ( S _ t ^ { \\delta } - S _ t ^ { \\delta + 1 } \\big ) d t \\\\ & \\ \\ \\ + C _ { \\beta , \\delta } R _ n ^ { - ( \\beta + \\delta ) } \\int ^ { R _ n } _ 0 ( R _ n - t ) ^ { \\beta - 1 } t ^ \\delta ( 1 - R _ n ^ { - 1 } t ) S _ t ^ { \\delta + 1 } d t \\\\ & = : I _ { R _ n } + I I _ { R _ n } . \\end{align*}"}
+{"id": "4651.png", "formula": "\\begin{align*} \\Vert M _ { \\widetilde { \\phi } } ( a _ 1 , \\ldots , a _ n ) \\Vert _ { S _ p ^ N } \\leq \\Vert T _ { \\phi } ^ { ( N ) } \\Vert \\Vert a _ 1 \\Vert _ { S _ { p _ 1 } ^ N } \\ldots \\Vert a _ n \\Vert _ { S _ { p _ n } ^ N } . \\end{align*}"}
+{"id": "4947.png", "formula": "\\begin{align*} \\begin{aligned} & E _ { P } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\biggr ] \\\\ = & \\xi _ 0 E _ P \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\vert H _ 1 \\biggr ] + ( 1 - \\xi _ 0 ) E _ P \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\vert H _ 2 \\biggr ] , \\end{aligned} \\end{align*}"}
+{"id": "5576.png", "formula": "\\begin{align*} W ( [ 0 ^ l 1 ] | [ 1 ^ k 0 ] ) = b _ k - a + \\sum _ { n = 2 } ^ l ( a _ n - a ) + d _ l - d \\ . \\end{align*}"}
+{"id": "2062.png", "formula": "\\begin{align*} M ^ { k + 1 } ( g h ) & = M M ^ { k } ( g h ) = \\sum ^ { 2 k } _ { j = 0 } M A _ { j , 2 k - j } ( g , h ) . \\end{align*}"}
+{"id": "5038.png", "formula": "\\begin{align*} | G _ { \\gamma , \\omega } ( s ) - 1 | = \\dfrac { | \\alpha \\omega - \\omega | } { | b | | c \\omega + d | } \\leq \\dfrac { 3 ^ { - 2 } } { | b | } < \\varepsilon , & & \\mathrm { i f } \\ | b | > \\frac { 3 ^ { - 2 } } { \\varepsilon } . \\end{align*}"}
+{"id": "2253.png", "formula": "\\begin{align*} h \\cdot a = a , \\end{align*}"}
+{"id": "481.png", "formula": "\\begin{align*} f ( x ) = e ^ { - \\Lambda _ u } f _ r ( x ) + e ^ { - \\Lambda _ u } \\sum _ { n = 1 } ^ \\infty ( \\Lambda _ u ^ n / n ! ) g _ u ^ { \\ast n } \\ast f _ r ( x ) , \\end{align*}"}
+{"id": "6585.png", "formula": "\\begin{align*} w = \\max \\{ f ^ { \\frac { p + 1 } { 2 } } - s ^ { \\frac { p + 1 } { 2 } } , 0 \\} . \\end{align*}"}
+{"id": "5451.png", "formula": "\\begin{align*} \\phi _ { \\lambda } ( \\partial a ) = ( \\partial + \\lambda ) \\phi _ { \\lambda } ( a ) , \\forall \\ a \\in \\mathcal { A } . \\end{align*}"}
+{"id": "1169.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } H _ t ^ k \\leq & \\frac { k } { ( n - 1 ) ^ 2 } \\mathbb { E } \\left [ \\left | \\sum _ { j = 2 } ^ k \\left ( K ( X _ t ^ 1 - X _ t ^ j ) - \\langle \\mu _ t , K ( X _ t ^ 1 - \\cdot ) \\right ) \\right | ^ 2 \\right ] \\\\ & + \\frac { k ( n - k ) ^ 2 } { ( n - 1 ) ^ 2 } \\mathbb { E } \\left [ \\left | \\mathbb { E } [ K ( X _ t ^ 1 - X _ t ^ n ) \\mid X _ t ^ 1 , \\cdots , X _ t ^ k ] - \\langle \\mu _ t , K ( X _ t ^ 1 - \\cdot ) \\right | \\right ] \\end{aligned} . \\end{align*}"}
+{"id": "2358.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\zeta ( t ) = - \\phi ' ( \\zeta ( t ) ) , \\end{align*}"}
+{"id": "3501.png", "formula": "\\begin{align*} I _ 1 : = [ - 2 s q _ n + [ s q _ n / 2 ] + 1 , - s q _ n + [ s q _ n / 2 ] ] , I _ 2 : = [ y - [ s q _ n / 2 ] - s q _ n + 1 , y - [ s q _ n / 2 ] ] . \\end{align*}"}
+{"id": "986.png", "formula": "\\begin{align*} | T \\cap M ' | \\le \\begin{cases} \\theta _ { d - 1 } q ^ { d ^ 2 - d + 1 } & P \\in \\tau _ 1 \\cup \\tau _ 2 , \\\\ ( q ^ d + \\theta _ { d } ) q ^ { d ^ 2 - d - 1 } & \\end{cases} \\end{align*}"}
+{"id": "5637.png", "formula": "\\begin{align*} \\langle u , v \\rangle : = \\ , \\ , ^ { t } u \\psi _ { 2 n } v \\end{align*}"}
+{"id": "4791.png", "formula": "\\begin{align*} a ( u _ n , v _ n ) = \\lambda _ n ( u _ n , v _ n ) v _ n \\in X _ n , \\end{align*}"}
+{"id": "7475.png", "formula": "\\begin{align*} \\int _ { B _ { \\rho } } r ^ { 2 - n } u _ r ^ 2 \\d x & \\leq C \\int _ { B _ { 2 \\rho } \\setminus B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x + C \\varepsilon \\int _ { B _ { 4 \\rho } } r ^ { 3 - n } | \\nabla u | ^ 2 \\d x \\end{align*}"}
+{"id": "3450.png", "formula": "\\begin{align*} 1 \\leq | \\phi ( 0 ) | \\leq & \\frac { | \\tilde { P } _ { m _ n - 1 } ( \\theta _ 1 ) | } { | \\tilde { P } _ { q _ n - 1 } ( \\theta _ { m _ n - q _ n + 1 } ) | } \\prod _ { j = m _ n - q _ n + 1 } ^ 0 | \\cos ( \\pi \\theta _ j ) | \\cdot | \\phi ( m _ n - q _ n ) | \\\\ & + \\frac { | \\tilde { P } _ { - m _ n + q _ n - 1 } ( \\theta _ { m _ n - q _ n + 1 } ) | } { | \\tilde { P } _ { q _ n - 1 } ( \\theta _ { m _ n - q _ n + 1 } ) | } \\prod _ { j = 0 } ^ { m _ n - 1 } | \\cos ( \\pi \\theta _ j ) | \\cdot | \\phi ( m _ n ) | \\end{align*}"}
+{"id": "6428.png", "formula": "\\begin{align*} \\vert \\mathcal { D } _ { n , h } ^ k \\vert = [ t ^ n ] C _ k ( t ) ^ { h + 1 } = \\frac { h + 1 } { k n + h + 1 } \\binom { k n + h + 1 } { n } . \\end{align*}"}
+{"id": "840.png", "formula": "\\begin{align*} \\big [ \\theta _ { \\rho ^ \\varepsilon } ( 0 , z _ l ) \\big ] _ 1 = - \\varepsilon \\rho _ 0 < \\varepsilon \\rho _ 0 = \\big [ \\theta _ { \\rho ^ \\varepsilon } ( 0 , z _ r ) \\big ] _ 1 \\end{align*}"}
+{"id": "8813.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\int _ 0 ^ { \\tau / 2 } | \\Pi _ { \\tilde { V } } e ^ { | z | J ^ \\perp t } \\tilde { Y } _ 0 | d t = \\frac { 1 } { \\tau } \\int _ 0 ^ { \\tau / 2 } | \\Pi _ { \\tilde { V } } \\tilde { Y } _ 0 + t | z | | \\Pi _ V \\tilde { Y } _ 0 | d t \\gtrsim R | z | \\tau ^ { 3 / 2 } . \\end{align*}"}
+{"id": "2115.png", "formula": "\\begin{align*} \\begin{aligned} d ( \\phi ( y ) , \\psi ( y ) ) & \\leq d ( \\phi ( y ) , \\psi ( y ) ) + d ( \\psi ( y ) , \\eta ( y ) ) \\\\ & \\leq \\tilde L \\min \\{ d ( \\eta ( \\hat y ) , \\eta ( y ) ) , d ( \\psi ( \\hat y ) , \\phi ( y ) ) \\} + M _ \\phi + M _ \\eta , \\\\ & = \\tilde L \\min \\{ d ( \\eta ( \\hat y ) , \\eta ( y ) ) , d ( \\eta ( \\hat y ) , \\phi ( y ) ) \\} + M _ \\phi + M _ \\eta , \\\\ \\end{aligned} \\end{align*}"}
+{"id": "3916.png", "formula": "\\begin{align*} | h _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) | + | \\nabla h _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) | \\le C , \\ \\ 1 \\leq i , j \\leq m , \\end{align*}"}
+{"id": "1945.png", "formula": "\\begin{align*} K _ { \\nu } ( z ) = \\sqrt { \\pi } ~ e ^ { z } ~ G _ { { 1 } , { 2 } } ^ { { 2 } , { 0 } } \\left ( 2 z \\bigg { | } \\begin{array} { l l l } \\frac { 1 } { 2 } \\\\ \\nu ~ , ~ - \\nu \\end{array} \\right ) , \\end{align*}"}
+{"id": "2393.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\subset \\mu _ \\ell } c _ { \\ell } \\frac { r _ \\ell } { v ( \\mu _ \\ell ) } + c _ { \\kappa } \\frac { 1 } { v ( \\kappa ) } = 0 , \\sigma \\subset \\kappa , & \\sum _ { \\sigma \\subset \\mu _ \\ell } c _ { \\ell } \\frac { r _ \\ell } { v ( \\mu _ \\ell ) } = 0 , \\sigma \\not \\subset \\kappa , \\end{align*}"}
+{"id": "2221.png", "formula": "\\begin{align*} \\mathbf { v } ^ { 0 } = [ \\cdots , 1 , 0 , \\cdots , 1 , \\cdots , 1 , \\cdots ] ^ { T } , m , n , k = 1 , \\cdots , N . \\end{align*}"}
+{"id": "8545.png", "formula": "\\begin{align*} \\kappa ( t ) = h _ \\alpha ( t ) , \\ \\ k ( t ) = h _ { 1 - \\alpha } ( t ) , \\ 0 < \\alpha < 1 , \\end{align*}"}
+{"id": "972.png", "formula": "\\begin{align*} a _ { 2 c - i } = \\sum _ { j = 0 } ^ { \\lfloor i / 2 \\rfloor } \\frac { c ! \\cdot 2 ^ j } { ( c + j - i ) ! ( i - 2 j ) ! j ! } \\le c ^ i \\sum _ { j = 0 } ^ { \\lfloor i / 2 \\rfloor } \\underbrace { \\frac { 2 ^ j } { c ^ j ( i - 2 j ) ! j ! } } _ { = : b _ { i j } } . \\end{align*}"}
+{"id": "4493.png", "formula": "\\begin{align*} \\Bigg ( \\sum _ { v \\in \\mathbb { N } ^ n } \\frac { c ( v , q ) } { \\mathit { g l } ( v , q ) } X ^ v \\Bigg ) \\ ! \\circ \\mathrm { E x p } \\Bigg ( \\sum _ { v \\in \\mathbb { N } ^ n \\backslash \\{ 0 \\} } \\ ! \\frac { s ( v , q ) } { 1 - q } X ^ v \\Bigg ) = 1 , \\\\ \\end{align*}"}
+{"id": "7166.png", "formula": "\\begin{align*} ( S \\textbf { \\textit { u } } ) ^ i _ j = \\nabla ^ i u _ j + \\nabla _ j u ^ i , \\end{align*}"}
+{"id": "1726.png", "formula": "\\begin{align*} \\int _ 0 ^ t U ( t , s ) ( \\partial _ t G ) ( y ( s ) ) d s = & ( \\alpha - 1 ) \\int _ 0 ^ t U ( t , s ) e ^ { ( \\alpha - 1 ) W ( s ) } g ( y ( s ) ) d W ( s ) \\\\ & + ( \\alpha - 1 ) ^ 2 \\int _ 0 ^ t U ( t , s ) \\mu e ^ { ( \\alpha - 1 ) W ( s ) } g ( y ( s ) ) d s , \\\\ | \\nabla G ( y ) | \\le & | e ^ { ( \\alpha - 1 ) W } | [ ( \\alpha - 1 ) | \\nabla W | | g ( y ) | + | \\nabla g ( y ) | ] . \\end{align*}"}
+{"id": "8089.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ M \\sum \\limits _ { \\substack { j = 1 \\\\ j \\neq l } } ^ M \\sum _ { l = 1 } ^ M a _ l ^ 2 \\phi ^ { \\left ( i , l \\right ) ^ * } \\phi ^ { \\left ( j , l \\right ) } = 2 \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { q = i + 1 } ^ { M } \\sum _ { r = 1 } ^ M a _ r ^ 2 \\Re \\left [ \\phi ^ { \\left ( i , r \\right ) ^ * } \\phi ^ { \\left ( q , r \\right ) } \\right ] . \\end{align*}"}
+{"id": "6886.png", "formula": "\\begin{align*} \\int _ X f \\ , d \\mu = \\int _ X f \\circ \\varphi _ \\alpha \\ , d \\mu , \\end{align*}"}
+{"id": "1956.png", "formula": "\\begin{align*} r ' ( X ~ \\cup ~ \\{ z , \\gamma \\} ) = \\begin{cases} r ( X ) + 1 , ~ e \\in c l ( X ) ; \\\\ r ( X ) + 2 , ~ e \\notin c l ( X ) ; ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{cases} \\end{align*}"}
+{"id": "4893.png", "formula": "\\begin{align*} \\frac { \\partial z } { \\partial \\alpha } & = \\frac { \\phi ( z ) [ 1 - \\Phi ( z ) ] } { - \\frac { \\partial s ( z ) } { \\partial z } } \\mathrm { s i g n } \\left ( \\frac { \\partial z } { \\partial \\alpha } \\right ) = - \\mathrm { s i g n } \\left ( \\frac { \\partial s ( z ) } { \\partial z } \\right ) , \\end{align*}"}
+{"id": "8484.png", "formula": "\\begin{align*} & \\lim _ { \\epsilon \\downarrow 0 } \\frac { 1 } { 2 \\pi i } \\int _ { \\mathbb { R } } \\frac { \\mathrm { e } ^ { i s ( \\xi - 2 x ) } } { s - ( z + i \\epsilon ) } d s = \\lim _ { \\epsilon \\downarrow 0 } \\begin{cases} \\mathrm { e } ^ { i ( z + i \\epsilon ) ( \\xi - 2 x ) } , & \\xi - 2 x > 0 \\\\ 0 , & \\xi - 2 x < 0 \\end{cases} \\\\ [ 6 p t ] & = \\chi ( \\xi - 2 x ) e ^ { i z ( \\xi - 2 x ) } , \\end{align*}"}
+{"id": "1291.png", "formula": "\\begin{align*} \\nu ( \\eta _ { \\Lambda _ n } ) = \\nu ( \\xi _ { \\Delta } \\eta _ { \\Lambda _ n \\setminus \\Delta } ) = 0 . \\end{align*}"}
+{"id": "300.png", "formula": "\\begin{align*} \\operatorname { W S } = \\frac { 1 } { 2 } \\sqrt { \\operatorname { t r } ( \\Sigma _ i ^ { - 1 } ( \\Sigma _ i + \\eta ) - I ) + ( n + 1 ) ^ 2 \\eta ^ 2 \\sum \\Sigma _ i ^ { - 1 } - \\log \\operatorname { d e t } ( ( \\Sigma _ i + \\eta ) \\Sigma _ i ^ { - 1 } ) } \\end{align*}"}
+{"id": "824.png", "formula": "\\begin{align*} \\Gamma _ \\rho ( t ) = \\Bigg \\{ \\begin{bmatrix} x \\\\ w ( t , x ) \\cos \\varphi \\\\ w ( t , x ) \\sin \\varphi \\end{bmatrix} \\ , \\Bigg | \\ , x \\in [ \\tilde { x _ 0 } ( t ) , \\tilde { x _ 5 } ( t ) ] , \\varphi \\in [ 0 , 2 \\pi ] \\Bigg \\} \\end{align*}"}
+{"id": "8749.png", "formula": "\\begin{align*} \\int _ \\Omega \\left ( { \\mathcal D } _ { \\psi } f ( w ) + ( \\frac { \\partial p _ 0 } { \\partial x } + \\frac { \\partial q _ 0 } { \\partial y } ) f + ( \\frac { \\partial p _ 1 } { \\partial x } + \\frac { \\partial q _ 1 } { \\partial y } ) { i } f \\right ) d x d y = \\int _ { \\partial \\Omega } f d \\sigma _ { \\psi } ( w ) , \\end{align*}"}
+{"id": "6189.png", "formula": "\\begin{align*} B _ 1 = Q . \\end{align*}"}
+{"id": "8496.png", "formula": "\\begin{align*} & F _ 1 ( x ; z ) = ( F ( x ; z ) V _ 2 ) _ 1 = \\left ( 0 , \\mathcal { P } ^ { + } ( \\bar { r } _ 1 ( z ) \\mathrm { e } ^ { - 2 i z x } ) \\right ) , \\\\ & F _ 2 ( x ; z ) = ( F ( x ; z ) V _ 1 ) _ 2 = \\left ( \\mathcal { P } ^ { - } \\left ( r _ 2 ( z ) \\mathrm { e } ^ { 2 i z x } \\right ) , 0 \\right ) . \\end{align*}"}
+{"id": "244.png", "formula": "\\begin{align*} l _ S ( g ) = | \\sigma ^ - | + ( \\sigma ^ + - \\sigma ^ - ) + | \\sigma ^ + | + \\sum _ { i = \\sigma ^ - } ^ { \\sigma ^ + } l _ { S _ 0 } \\ ! \\big ( w _ i ( a _ 1 , \\ldots , a _ d ) \\big ) . \\end{align*}"}
+{"id": "3922.png", "formula": "\\begin{align*} \\phi ( e _ 0 u _ 0 ) = \\tau ( u _ 0 , e _ 0 ) \\phi ( e _ 0 ) \\phi ( u _ 0 e _ 1 ) = - \\tau ( u _ 0 , e _ 1 ) \\phi ( e _ 1 ) . \\end{align*}"}
+{"id": "363.png", "formula": "\\begin{align*} l _ k = - \\frac { ( 1 + \\cos ( \\frac { 2 \\pi k } { n - 1 } ) } { \\cos ^ { 2 } ( \\frac { \\pi k } { n - 1 } ) } v _ k - 8 \\cos ^ { 2 } ( \\frac { \\pi k } { n - 1 } ) v _ k . \\end{align*}"}
+{"id": "969.png", "formula": "\\begin{align*} \\chi ( q \\Gamma _ { 2 d + 1 , \\{ d , d + 1 \\} } ) = \\frac { q ^ { d + 2 } - 1 } { q - 1 } - q \\end{align*}"}
+{"id": "8984.png", "formula": "\\begin{align*} \\dfrac { \\hat { k } _ 2 { ( \\Delta ) } } { \\hat { k } _ 2 ( \\Delta \\setminus \\sigma ) } = \\dfrac { D _ { 1 , 2 } D _ { 2 , 2 } D _ { 3 , 3 } } { D _ { 1 , 1 } D _ { 2 , 1 } D _ { 3 , 2 } } = \\dfrac { ( x _ { 1 , 1 } + x _ { 1 , 2 } ) ( x _ { 2 , 1 } + x _ { 2 , 2 } ) ( x _ { 3 , 1 } + x _ { 3 , 2 } + x _ { 3 , 3 } ) } { x _ { 1 , 1 } x _ { 2 , 1 } ( x _ { 3 , 1 } + x _ { 3 , 2 } ) } . \\end{align*}"}
+{"id": "2508.png", "formula": "\\begin{align*} \\mathbf { U } _ { v } = \\begin{pmatrix} & 0 & 0 \\\\ & 0 & ( \\mathbf { v } _ { 1 } - \\beta _ { v } ) \\mathbf { P } _ { v } \\end{pmatrix} \\end{align*}"}
+{"id": "4891.png", "formula": "\\begin{align*} \\frac { \\partial s ( z ) } { \\partial z } = - \\Phi ( z ) \\ , [ 1 - \\Phi ( z ) ] - z \\ , \\phi ( z ) \\ , \\left [ \\alpha - ( \\alpha + \\beta ) \\ , \\Phi ( z ) \\right ] + ( 2 - \\alpha - \\beta ) \\ , \\phi ^ 2 ( z ) . \\end{align*}"}
+{"id": "5584.png", "formula": "\\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) = A ( 1 ^ n 0 ^ k 1 . . . ) - A ( 1 ^ n 0 ^ \\infty ) = c _ n - c _ n = 0 \\ , \\end{align*}"}
+{"id": "4486.png", "formula": "\\begin{align*} X ^ v \\circ X ^ w = q ^ { - \\langle v , w \\rangle } X ^ { v + w } \\ , v , w \\in \\mathbb { N } ^ n . \\end{align*}"}
+{"id": "6163.png", "formula": "\\begin{align*} V _ 1 ( r ) = V ( r ) - E _ 0 , \\end{align*}"}
+{"id": "8530.png", "formula": "\\begin{align*} e ^ { i c _ + ( x ) } \\partial _ { x } \\left ( \\bar { u } _ x ( x ) e ^ { - i c _ - ( x ) } \\right ) = - \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } r _ { \\delta , 2 } ( z ) \\mathrm { e } ^ { 2 i z x } M _ { \\delta , - , 2 2 } ( x ; z ) d z . \\end{align*}"}
+{"id": "3505.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n i ^ 3 F _ i \\ = \\ ( n ^ 3 + 6 n - 1 2 ) F _ { n + 2 } + ( - 3 n ^ 2 + 9 n - 1 9 ) F _ { n + 3 } + 5 0 , \\end{align*}"}
+{"id": "7621.png", "formula": "\\begin{align*} g _ n \\big ( t , q , x \\big ) : = \\sup _ { z \\in \\mathbb { R } ^ d } \\Big ( q \\cdot z - f _ n ( t , z , x ) \\Big ) . \\end{align*}"}
+{"id": "2425.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } s L _ { q } [ f ( t ) ] = \\lim _ { t \\rightarrow 0 } \\frac { f ( t ) } { 1 + ( 1 - q ) } , \\end{align*}"}
+{"id": "1272.png", "formula": "\\begin{align*} \\abs { \\mathbf { I I } } \\leq \\sup _ { \\Delta \\ni 0 : \\xi _ { \\Delta } } \\norm { c _ { \\Delta } ( \\cdot , \\xi _ { \\Delta } ) } _ { \\infty } q ^ { \\Delta } \\log \\left ( \\frac { 1 } { \\delta ( \\mu ) } \\right ) \\abs { \\Lambda _ n \\setminus \\hat { \\Lambda } _ n } = o ( \\abs { \\Lambda _ n } ) , \\end{align*}"}
+{"id": "5667.png", "formula": "\\begin{align*} E ^ { 1 } _ { p , q } ( n ) & = H _ { q } ( O _ { n , n } , C _ { p } ( n ) ) \\Rightarrow H _ { p + q } ( O _ { n , n } , C _ { * } ( n ) ) \\end{align*}"}
+{"id": "7630.png", "formula": "\\begin{align*} \\mathcal { J } ^ i ( \\alpha ^ i , \\boldsymbol { \\alpha } ^ { - i } ) : = \\frac { 1 } { 2 } \\mathbb { E } \\Big \\{ \\int _ 0 ^ T \\Big [ Q \\left ( x ^ i _ t + l ( \\nu ^ { N , i } _ { \\boldsymbol { x } _ t } ) \\right ) ^ 2 + R \\left ( \\alpha ^ i _ t + h ( \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } _ t } ) \\right ) ^ 2 \\Big ] d t + G \\left ( x ^ i _ T + g ( \\nu ^ { N , i } _ { \\boldsymbol { x } _ T } ) \\right ) ^ 2 \\Big \\} , \\end{align*}"}
+{"id": "7572.png", "formula": "\\begin{align*} x = \\sum _ { n = 0 } ^ \\infty \\frac { \\varepsilon _ n } { \\beta ^ { n + 1 } } = \\sum _ { n = 0 } ^ { s - 1 } \\frac { \\varepsilon _ n } { \\beta ^ { n + 1 } } + \\left ( \\sum _ { n = s } ^ { s + t - 1 } \\frac { \\varepsilon _ n } { \\beta ^ { n + 1 } } \\right ) \\frac { \\beta ^ t } { \\beta ^ t - 1 } . \\end{align*}"}
+{"id": "7310.png", "formula": "\\begin{align*} \\begin{array} { l } { \\rm S I N R } _ { k , j } ^ m = 2 y _ { k , j } ^ m \\sqrt { { p _ { k , j } ^ m \\left | G _ { k , j } ^ m \\right | ^ 2 } } - y _ { k , j } ^ m \\left ( { { \\sum \\limits _ { j ' \\ne j } \\sum \\limits _ { k ' \\ne { { k } } } { p _ { k ' , j ' } ^ { { m } } { { { \\left | G _ { k ' , j } ^ m \\right | ^ 2 } } } } } + N _ { \\rm R F } B _ { \\rm R F } } \\right ) , \\end{array} \\end{align*}"}
+{"id": "8707.png", "formula": "\\begin{align*} \\widetilde { \\varphi } ( g ) = \\begin{cases} \\varphi ( g ) & g \\in H \\\\ 0 & g \\notin H . \\end{cases} \\end{align*}"}
+{"id": "6911.png", "formula": "\\begin{align*} H ( Q \\ , | \\ , P ) \\le \\inf \\Big \\{ H ( R \\ , | \\ , P ) : R \\in \\P ( \\R ^ n ) , \\ H ( R ) < \\infty \\Big \\} = 0 , \\end{align*}"}
+{"id": "3591.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\Gamma } = \\left \\lbrace p R \\ , \\vert \\ , p _ { i } = \\dfrac { 1 } { Z ( \\theta ) } \\prod \\limits _ { F \\in \\mathrm { f a c e t } ( \\Gamma ) } \\theta ^ { F } _ { i _ { F } } \\right \\rbrace \\end{align*}"}
+{"id": "6269.png", "formula": "\\begin{align*} \\partial _ r ( d _ { i s } ) + ( d _ { i s } - d _ { i r } ) \\Gamma _ { r s } ^ s = \\sum _ j \\phi _ { i j } ( \\partial _ r ) d _ { j s } \\varphi _ j = \\phi _ { i s } ( \\partial _ r ) ( d _ { s s } - 1 ) \\varphi _ s + \\sum _ j \\phi _ { i j } ( \\partial _ r ) d _ j \\varphi _ j = \\phi _ { i s } ( \\partial _ r ) , \\end{align*}"}
+{"id": "9143.png", "formula": "\\begin{align*} \\underline { \\delta } ^ \\star ( s ) : = 2 \\int _ { 0 } ^ { 1 / 2 } \\alpha ( s \\sin ( 2 \\pi t ) ) \\ , d t \\end{align*}"}
+{"id": "112.png", "formula": "\\begin{align*} \\| \\phi _ { \\lambda } \\| _ { \\dot { H } ^ { s _ 1 , s _ 2 } ( \\R ^ 2 ) } = \\lambda ^ { - \\frac { 3 \\alpha } { 4 } + 1 - s _ 1 - ( \\frac { \\alpha } { 2 } + 1 ) s _ 2 } \\| \\phi \\| _ { \\dot { H } ^ { s _ 1 , s _ 2 } ( \\R ^ 2 ) } . \\end{align*}"}
+{"id": "1814.png", "formula": "\\begin{align*} \\mathcal { Y M } _ e ^ 0 ( \\nabla ) = \\int _ M \\big ( \\exp ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 ) - 1 \\big ) \\ , d v \\ , , \\end{align*}"}
+{"id": "151.png", "formula": "\\begin{align*} A ( 1 + B ) ^ k = \\sum _ { j \\in \\mathbb Z _ 0 } \\lambda _ j , \\ \\ \\ \\ B ( 1 + A ) ^ k = \\sum _ { j \\in \\mathbb Z _ 0 } \\lambda _ j . \\end{align*}"}
+{"id": "4581.png", "formula": "\\begin{align*} D ( A ^ { \\circ } ) & = \\{ x ^ { \\circ } \\in X ^ { \\circ } \\ | \\ \\exists y ^ { \\circ } \\in X ^ { \\circ } \\ \\forall x \\in D ( A ) : \\langle - A x , x ^ { \\circ } \\rangle = \\langle x , y ^ { \\circ } \\rangle \\} , \\\\ - A ^ { \\circ } x ^ { \\circ } & = y ^ { \\circ } \\quad ( x ^ { \\circ } \\in D ( A ^ { \\circ } ) ) , \\end{align*}"}
+{"id": "8347.png", "formula": "\\begin{align*} & S ( k ) = \\sigma _ 2 \\overline { S ( \\bar { k } ) } \\sigma _ 2 , \\ \\ \\ S ( k ) = \\sigma _ 3 S ( - k ) \\sigma _ 3 \\end{align*}"}
+{"id": "6820.png", "formula": "\\begin{align*} J _ { l } \\left ( \\rho \\right ) : = \\int _ { \\mathbb { R } } \\frac { \\varphi _ { 0 } \\left ( \\tau \\right ) \\left ( \\frac { 1 } { 2 } - i \\tau \\right ) ^ { l - 1 } } { \\left ( \\tau - \\rho \\right ) ^ { l + 1 } } d \\tau = \\sum _ { j = 0 } ^ { l - 1 } C _ { l - 1 } ^ { j } \\left ( \\frac { 1 } { 2 } \\right ) ^ { j } \\left ( - i \\right ) ^ { l - 1 - j } \\int _ { \\mathbb { R } } \\frac { \\varphi _ { 0 } \\left ( \\tau \\right ) \\tau ^ { l - 1 - j } } { \\left ( \\tau - \\rho \\right ) ^ { l + 1 } } d \\tau . \\end{align*}"}
+{"id": "7220.png", "formula": "\\begin{align*} d u = \\Delta _ p u \\ , d t + S ( u ) \\ , d t + \\sum _ { n = 0 } ^ { + \\infty } \\xi _ { n } \\cdot \\nabla u \\ , d B _ t ^ { n } , \\end{align*}"}
+{"id": "5002.png", "formula": "\\begin{align*} \\frac 1 p + 2 = \\frac { 1 } { p _ 1 } + \\frac { 1 } { p _ 2 } + \\frac { 1 } { p _ 3 } . \\end{align*}"}
+{"id": "4432.png", "formula": "\\begin{gather*} A ^ * : = \\begin{pmatrix} 0 & 1 & 0 & \\cdots & 0 & - 1 \\\\ 1 & 0 & 1 & \\cdots & 0 & 0 \\\\ 0 & 1 & 0 & \\cdots & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & 0 & 1 \\\\ - 1 & 0 & 0 & \\cdots & 1 & 0 \\\\ \\end{pmatrix} . \\end{gather*}"}
+{"id": "1833.png", "formula": "\\begin{align*} \\mathcal { Y M } _ e ( \\nabla ^ t ) = \\int _ M \\exp \\ , ( \\frac 1 2 | | R ^ \\nabla | | ^ 2 + t \\langle R ^ \\nabla , d ^ \\nabla A \\rangle + \\varepsilon ( t ^ 2 ) ) \\ , d v \\end{align*}"}
+{"id": "1263.png", "formula": "\\begin{align*} \\sum _ { \\Theta \\Subset \\Z ^ d } h _ { \\Theta } ( \\omega ) = 0 . \\end{align*}"}
+{"id": "7688.png", "formula": "\\begin{align*} \\mathbb { E } \\Big [ \\underset { t _ 0 \\leq t \\leq T } { \\sup } | x _ t ^ i - x _ t ^ { * , i } | ^ 2 \\Big ] \\leq \\frac { C } { N ^ 2 } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } \\mathbb { E } [ | \\xi ^ i - \\xi ^ j | ^ 2 ] + \\frac { 1 } { N ^ 2 } \\underset { i , j = 1 } { \\overset { N } { \\sum } } \\mathbb { E } [ | \\xi ^ { i } - \\xi ^ { j } | ^ 2 ] \\Big ) , \\end{align*}"}
+{"id": "7794.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Theta ( t , x ) = \\int _ \\S W _ r ( x - y ) \\sin ( \\Theta ( t , y ) - \\Theta ( t , x ) ) \\ \\d y . \\end{align*}"}
+{"id": "4656.png", "formula": "\\begin{align*} \\Vert M _ { \\widetilde { H } \\vert _ { \\mathbb { Z } _ N \\times \\mathbb { Z } _ N } } : S _ { p _ 1 } ^ { 2 N + 1 } \\times S _ { p _ 2 } ^ { 2 N + 1 } \\rightarrow S _ { 1 } ^ { 2 N + 1 } \\Vert \\leq A _ { p _ 1 , p _ 2 , 2 N + 1 } . \\end{align*}"}
+{"id": "4728.png", "formula": "\\begin{align*} \\Omega _ 0 \\cap \\ ( - \\mathbb { R } _ + ^ { m + 1 } ) = \\emptyset . \\end{align*}"}
+{"id": "3698.png", "formula": "\\begin{align*} h = 0 , \\nabla h = 0 \\mbox { o n } \\hat \\Sigma \\end{align*}"}
+{"id": "4804.png", "formula": "\\begin{align*} a ( u , v ) = ( f , v ) v \\in H _ 0 ^ 2 ( D ) . \\end{align*}"}
+{"id": "45.png", "formula": "\\begin{align*} \\Delta _ H w = | \\nabla _ { H } \\rho | ^ 2 \\left \\{ f '' ( \\rho ) + \\frac { Q - 1 } { \\rho } f ' ( \\rho ) \\right \\} , \\ \\ \\ \\ \\ \\ \\ ; \\mathbb { G } \\setminus \\{ e \\} . \\end{align*}"}
+{"id": "1825.png", "formula": "\\begin{align*} \\int _ M \\langle d ^ \\nabla \\sigma , \\rho \\rangle d v _ g = \\int _ M \\langle \\sigma , \\delta ^ \\nabla \\rho \\rangle d v _ g \\ , , \\end{align*}"}
+{"id": "540.png", "formula": "\\begin{align*} u _ i u _ j & = ( 3 c _ i + 4 s _ { i } + 4 z _ { i } ) ( 3 c _ j + 4 s _ { j } + 4 z _ { j } ) \\\\ & = 9 c _ i c _ j + 1 2 \\left ( c _ i ( s _ { j } + z _ { j } ) + c _ j ( s _ { i } + z _ { i } ) \\right ) \\\\ & \\phantom { { } = { } } + 1 6 ( s _ { i } s _ { j } + s _ i z _ { j } + s _ j z _ { i } + z _ { i } z _ { j } ) \\\\ & = 9 c _ { i , j } - 6 t _ { i , j } + 9 c _ i c _ j + 1 6 ( s _ i z _ { j } + s _ j z _ { i } + z _ { i } z _ { j } ) \\end{align*}"}
+{"id": "1422.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 2 } ( y _ 2 - y _ 1 ) f _ n ( y _ 1 - x _ 1 ) f _ n ( y _ 2 - x _ 2 ) d y _ 1 d y _ 2 = 2 \\P _ { ( x _ 1 , x _ 2 ) } ( S _ 2 ( n ) > S _ 1 ( n ) ) - 1 . \\end{align*}"}
+{"id": "484.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { g _ u ^ { \\ast 2 } \\ast f _ r ( x ) } { g _ u ( x ) } = 2 . \\end{align*}"}
+{"id": "5075.png", "formula": "\\begin{align*} f _ { 1 } ^ { + } = 2 { \\mathcal { I } } _ 1 , - f _ { - } ^ { + } = 2 { \\mathcal { I } } _ { - 1 } . \\end{align*}"}
+{"id": "4576.png", "formula": "\\begin{align*} C _ { \\mathrm { o b s } } = \\frac { C _ 1 } { T ^ { 1 / r } } \\exp \\left ( \\frac { C _ 2 } { T ^ { \\frac { \\gamma _ 1 \\gamma _ 3 } { \\gamma _ 2 - \\gamma _ 1 } } } + C _ 3 T \\right ) , \\end{align*}"}
+{"id": "7281.png", "formula": "\\begin{align*} \\gamma ^ { q ^ n } = \\gamma ^ { q ^ { n - i } } . \\end{align*}"}
+{"id": "3716.png", "formula": "\\begin{align*} ( \\hat h , \\hat v ) : = \\big ( h + L _ V \\bar g , v + V ( \\bar u ) \\big ) \\end{align*}"}
+{"id": "6209.png", "formula": "\\begin{align*} & E _ 0 = 9 \\kappa \\left ( \\frac { ( L + 1 ) \\Q ^ 2 + \\kappa L } { \\Q ^ 2 + 3 \\kappa } \\right ) ^ 2 - \\Q ^ 2 \\left ( \\frac { ( L + 2 ) \\Q ^ 2 + \\kappa ( L + 3 ) } { \\Q ^ 2 + 3 \\kappa } \\right ) ^ 2 , \\\\ & E _ 1 = 9 \\kappa \\left ( \\frac { ( L + 2 ) \\Q ^ 2 + \\kappa ( L + 3 ) } { \\Q ^ 2 + 3 \\kappa } \\right ) ^ 2 - \\Q ^ 2 \\left ( \\frac { ( L + 1 ) \\Q ^ 2 + \\kappa L } { \\Q ^ 2 + 3 \\kappa } \\right ) ^ 2 , \\end{align*}"}
+{"id": "1951.png", "formula": "\\begin{align*} F _ { 1 , p , \\nu } ( b _ { 1 } , b _ { 2 } , b _ { 3 } ; c _ { 1 } ; x , y ) = \\sum _ { n , m = 0 } ^ { \\infty } \\frac { ( b _ { 2 } ) _ { m } ( b _ { 3 } ) _ { n } ~ B _ { p , \\nu } ( b _ { 1 } + m + n , c _ { 1 } - b _ { 1 } ) } { ~ B ( b _ { 1 } , c _ { 1 } - b _ { 1 } ) } \\frac { x ^ { m } } { m ! } \\frac { y ^ { n } } { n ! } , \\end{align*}"}
+{"id": "3691.png", "formula": "\\begin{align*} \\Gamma ( Y ) & = - \\Gamma ( Z ^ { ( i ) } ) \\mbox { i n } M \\setminus \\Omega \\\\ Y & = - Z ^ { ( i ) } \\mbox { o n } \\Sigma . \\end{align*}"}
+{"id": "6317.png", "formula": "\\begin{align*} u _ 0 ^ { p ^ { r } } y _ 0 + u _ 1 ^ { p ^ { r } } y _ 1 + \\cdots + u _ n ^ { p ^ { r } } y _ n = 0 . \\end{align*}"}
+{"id": "8701.png", "formula": "\\begin{align*} P ( X _ a = c X _ b ) = \\int \\int & P ( X _ a = c t | \\sum _ { i = 1 } ^ { a - 1 } X _ i = s X _ b = t ) \\\\ & \\quad \\times ( \\sum _ { i = 1 } ^ { a - 1 } X _ i X _ b s t ) d s d t . \\end{align*}"}
+{"id": "4039.png", "formula": "\\begin{align*} i _ { \\chi \\theta } ( f ) ( F _ \\bullet ) = \\sum _ { F ' _ \\bullet \\in \\partial _ { \\chi \\theta } ^ { - 1 } ( F _ \\bullet ) } f ( F ' _ \\bullet ) . \\end{align*}"}
+{"id": "5807.png", "formula": "\\begin{align*} d v ' = \\left \\{ ( \\log \\sqrt { \\tau _ 0 } ) _ z \\ d z + \\frac { i } { \\sqrt { \\tau _ 0 } } \\left ( Q _ 0 d z + \\frac { \\tau _ 0 } { 4 } d \\overline { z } \\right ) J \\right \\} v ' . \\end{align*}"}
+{"id": "7824.png", "formula": "\\begin{align*} \\Phi ( B _ \\phi v , p ) & = Q F ^ q ( B _ \\phi v + \\psi ( B _ \\phi v , p ) , p ) \\\\ & = Q F ^ q ( B _ \\phi v + B _ \\phi \\psi ( v , p ) , p ) \\\\ & = Q F ^ q ( B _ \\phi [ v + \\psi ( v , p ) ] , p ) \\\\ & = Q B _ \\phi F ^ q ( v + \\psi ( v , p ) , p ) \\\\ & = B _ \\phi Q F ^ q ( v + \\psi ( v , p ) , p ) \\\\ & = B _ \\phi \\Phi ( v , p ) , \\end{align*}"}
+{"id": "4170.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\mathbf { v } + ( \\mathbf { v } \\cdot \\nabla ) \\mathbf { v } = - \\nabla P , \\ \\ & D \\times ( 0 , T ) , \\\\ \\nabla \\cdot \\mathbf { v } = 0 , \\ \\ & D \\times ( 0 , T ) , \\\\ \\mathbf { v } \\cdot \\mathbf { n } = v _ n , \\ \\ & \\partial D \\times ( 0 , T ) , \\\\ \\mathbf { v } ( x , 0 ) = \\mathbf { v } _ 0 ( x ) , \\ \\ & D , \\end{cases} \\end{align*}"}
+{"id": "5872.png", "formula": "\\begin{align*} ( u , v ) = \\langle u , v \\rangle , \\ \\ \\ \\forall \\ , u \\in H , \\ \\ \\forall \\ , v \\in V . \\end{align*}"}
+{"id": "5003.png", "formula": "\\begin{align*} A = \\frac { N ^ { \\frac 1 d } } { ( \\log N ) ^ { \\frac 1 { 1 6 d } } } , R = 1 , T = \\frac 1 { N ( \\log N ) ^ { \\frac 1 { 1 6 } } } . \\end{align*}"}
+{"id": "406.png", "formula": "\\begin{align*} \\phi ( g H ) : = \\pi _ N ( g ) \\psi ( \\pi _ N ( g ) ) , \\forall g \\in G \\end{align*}"}
+{"id": "9311.png", "formula": "\\begin{align*} \\mathbb { Q } ( S , 2 ) & = \\left \\{ b \\in \\mathbb { Q } ^ * / ( \\mathbb { Q } ^ * ) ^ 2 : _ l ( b ) \\equiv 0 ~ ( \\bmod { 2 } ) ~ ~ l \\neq 2 , p , q \\right \\} \\\\ & = \\left \\{ \\pm 1 , \\ ; \\pm 2 , \\ ; \\pm p , \\ ; \\pm q , \\ ; \\pm 2 p , \\ ; \\pm 2 q , \\ ; \\pm p q , \\ ; \\pm 2 p q \\right \\} . \\end{align*}"}
+{"id": "3890.png", "formula": "\\begin{align*} F _ { \\delta , Z } = \\{ u \\in L ^ p ( \\Omega ) \\mid \\int _ { \\Omega } \\frac { \\partial V _ { \\delta , Z , j } } { \\partial z _ { j , h } } u = 0 , \\ \\ \\forall j = 1 , \\cdots , m , \\ h = 1 , 2 \\} , \\end{align*}"}
+{"id": "3411.png", "formula": "\\begin{align*} \\phi ( y ) = G _ { [ x _ 1 , x _ 2 ] } ( x _ 1 , y ) \\phi ( x _ 1 - 1 ) + G _ { [ x _ 1 , x _ 2 ] } ( x _ 2 , y ) \\phi ( x _ 2 + 1 ) . \\end{align*}"}
+{"id": "2836.png", "formula": "\\begin{align*} \\| u \\| _ { 6 ( \\alpha - 2 ) } \\le C \\| u \\| _ 6 ^ { 1 - \\kappa } \\| \\Lambda ^ { s + 1 } u \\| ^ \\kappa = C \\| u \\| _ 6 ^ { \\frac { 2 ( \\alpha - 2 ) s - \\alpha + 3 } { 2 ( \\alpha - 2 ) s } } \\| \\Lambda ^ { s + 1 } u \\| ^ \\frac { \\alpha - 3 } { 2 ( \\alpha - 2 ) s } . \\end{align*}"}
+{"id": "121.png", "formula": "\\begin{align*} \\int _ { \\R ^ 4 } g _ 1 ( \\xi _ 1 , \\eta _ 1 ) & g _ 2 ( \\xi _ 2 , \\eta _ 2 ) g ( \\Omega ( ( \\xi _ 1 , \\eta _ 1 ) , ( \\xi _ 2 , \\eta _ 2 ) ) , \\xi _ 1 + \\xi _ 2 , \\eta _ 1 + \\eta _ 2 ) d \\xi _ 1 d \\xi _ 2 d \\eta _ 1 d \\eta _ 2 \\\\ & \\lesssim L _ { \\max } ^ { \\frac { 1 } { 4 } } N _ 2 ^ { - \\frac { \\alpha } { 2 } } N _ 1 ^ { \\frac { 1 } { 4 } } \\| g _ 1 \\| _ { L ^ 2 } \\| g _ 2 \\| _ { L ^ 2 } \\| g \\| _ { L ^ 2 } , \\end{align*}"}
+{"id": "1940.png", "formula": "\\begin{align*} F ( m ; z ) \\ | \\ T ( l ) = \\sum _ { n = 0 } ^ \\infty \\left ( b _ 3 \\left ( \\frac { m l n - 1 } { 1 2 } \\right ) + \\left ( \\frac { 3 } { l } \\right ) l ^ { 3 m - 4 } b _ 3 \\left ( \\frac { m n - l } { 1 2 l } \\right ) \\right ) q ^ n \\equiv 0 \\ ( \\mathrm { m o d } \\ m ) . \\end{align*}"}
+{"id": "8447.png", "formula": "\\begin{align*} M _ + ( x ; z ) = M _ - ( x ; z ) ( I + R ( x ; z ) ) , z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "314.png", "formula": "\\begin{align*} \\Phi ( t , y ) = H ( t / 4 , x , y ) . \\end{align*}"}
+{"id": "2013.png", "formula": "\\begin{align*} s _ \\alpha \\bullet _ 2 \\lambda & = s _ \\alpha ( \\lambda + 2 \\rho ) - 2 \\rho \\\\ & = s _ \\alpha ( \\lambda ) + 2 \\rho - 2 \\alpha - 2 \\rho \\\\ & = s _ \\alpha ( \\lambda ) - 2 \\alpha \\end{align*}"}
+{"id": "6129.png", "formula": "\\begin{align*} \\widetilde { \\mu } ( n ) \\ r _ n ( m ) = H _ { m } \\ \\widetilde { r } _ { n } ( m + 1 ) + I _ m \\ \\widetilde { r } _ { n } ( m ) + J _ m \\ \\widetilde { r } _ { n } ( m - 1 ) \\ , , \\end{align*}"}
+{"id": "5767.png", "formula": "\\begin{align*} \\nabla _ Y \\mathcal { P } ( X ) = \\mathcal { P } ( B ( Y , X ) ) - B ( Y , f ( X ) ) + B ^ * ( Y , h ( X ) ) . \\end{align*}"}
+{"id": "6048.png", "formula": "\\begin{align*} I ' ( u , v ) = \\mu J ' ( u , v ) . \\end{align*}"}
+{"id": "9043.png", "formula": "\\begin{align*} \\mathcal { I } & = \\bigg \\{ E \\in \\mathbb { C } : \\log | \\lambda | > \\log \\Big | \\frac { E } { 2 } + \\frac { \\sqrt { E ^ { 2 } - 4 } } { 2 } \\Big | \\bigg \\} , \\\\ \\mathcal { O } & = \\bigg \\{ E \\in \\mathbb { C } : \\log | \\lambda | < \\log \\Big | \\frac { E } { 2 } + \\frac { \\sqrt { E ^ { 2 } - 4 } } { 2 } \\Big | \\bigg \\} . \\end{align*}"}
+{"id": "8907.png", "formula": "\\begin{align*} h _ { 1 2 } = h _ { 1 2 3 } . \\end{align*}"}
+{"id": "5447.png", "formula": "\\begin{align*} V _ { \\alpha , \\beta } = \\mathbb { C } [ \\partial ] v , L _ \\lambda v = ( \\partial + \\alpha \\lambda + \\beta ) v . \\end{align*}"}
+{"id": "1497.png", "formula": "\\begin{align*} | | \\mathbf { v } ( T , \\mathbf { v } ^ 0 ) | | _ { \\mathcal { E } } \\leq C | | \\mathbf { v } ^ 0 | | _ { \\mathcal { E } } \\leq C \\gamma = \\gamma _ 1 . \\end{align*}"}
+{"id": "6270.png", "formula": "\\begin{align*} \\partial _ i \\varphi _ i = - 2 \\sum _ { j \\neq i } \\Gamma _ { i j } ^ j \\varphi _ j . \\end{align*}"}
+{"id": "9246.png", "formula": "\\begin{align*} M ( A _ 1 , \\dots , A _ m ) = \\binom { n } { m } \\det ( A _ 1 , \\dots , A _ m , I , \\dots , I ) . \\end{align*}"}
+{"id": "2461.png", "formula": "\\begin{align*} L _ { q } \\left [ \\exp \\left ( - \\alpha t ^ { 2 } \\right ) \\right ] = \\displaystyle \\int _ { 0 } ^ { \\infty } d t \\ ; \\exp _ { q } \\left ( - s t \\right ) \\exp \\left ( - \\alpha t ^ { 2 } \\right ) . \\end{align*}"}
+{"id": "5045.png", "formula": "\\begin{align*} { \\mathcal { I } } _ h = \\inf \\left \\{ \\frac { 1 } { 2 } \\int _ { \\Omega } | B | ^ { 2 } \\dd x \\ \\middle | \\ B \\in L ^ { 2 } _ { \\sigma } ( \\Omega ) , \\ \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B \\cdot B \\dd x = h \\right \\} , \\end{align*}"}
+{"id": "1547.png", "formula": "\\begin{align*} u _ t & { } = ( 1 - t ) u + t h ( u , v ) , & v _ s & { } = ( 1 - s ) v + s k ( u , v ) . \\end{align*}"}
+{"id": "7606.png", "formula": "\\begin{align*} T ( x , y , w , z ) = ( x + \\alpha , y + x , w + \\alpha , z + w ) \\end{align*}"}
+{"id": "3687.png", "formula": "\\begin{align*} N = n + \\frac { n ( n - 1 ) } { 2 } , \\end{align*}"}
+{"id": "87.png", "formula": "\\begin{align*} w ' \\varepsilon ^ { \\mu ' } = w \\varepsilon ^ \\mu s _ { v \\alpha } \\varepsilon ^ { \\Phi ^ + ( - v \\alpha ) v \\alpha ^ \\vee } = w s _ { v \\alpha } \\varepsilon ^ { s _ { v \\alpha } ( \\mu ) + \\Phi ^ + ( - v \\alpha ) v \\alpha ^ \\vee } . \\end{align*}"}
+{"id": "7614.png", "formula": "\\begin{align*} 0 & = e _ y ( u , y ) = f ^ 2 ( u , y ) = f ( u , y ) f ( y , y ) + \\sum _ { u < v < y } f ( u , v ) f ( v , y ) \\\\ & = \\pm f ( u , y ) + \\sum _ { u < v < y } f ( u , v ) f ( v , y ) , \\end{align*}"}
+{"id": "2809.png", "formula": "\\begin{align*} \\mathbf { x } _ { l } = \\sum _ { i = 1 } ^ { K } \\mathbf { D } _ { i l } \\mathbf { w } _ { i l } q _ { i } , \\end{align*}"}
+{"id": "6616.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c } \\varepsilon _ 1 \\mu & \\varepsilon _ 1 \\nu \\\\ \\varepsilon _ 3 \\mu & \\varepsilon _ { 3 } \\nu \\end{array} \\right ) \\end{align*}"}
+{"id": "9175.png", "formula": "\\begin{align*} \\epsilon ( x _ a , t ) : = \\int _ 0 ^ t h ( x _ a + \\delta u ( \\tau ) ) u ( \\tau ) - \\dfrac { b _ { 1 , \\delta } ( x _ a ) } { 2 } \\ , d \\tau \\end{align*}"}
+{"id": "5087.png", "formula": "\\begin{align*} A _ t + B \\times u + \\nabla Q = - \\mu \\nabla \\times B . \\end{align*}"}
+{"id": "2612.png", "formula": "\\begin{align*} a s _ { A } ( x , y , z ) : = \\alpha ( x ) \\cdot ( y \\cdot z ) - ( x \\cdot y ) \\cdot \\alpha ( z ) = 0 . & & \\mbox { ( H o m - a s s o c i a t i v i t y ) } \\end{align*}"}
+{"id": "2279.png", "formula": "\\begin{align*} a \\circ \\widehat { \\varphi } = f \\circ \\rho \\end{align*}"}
+{"id": "5039.png", "formula": "\\begin{align*} \\begin{aligned} u _ t + u \\cdot \\nabla u + \\nabla p & = B \\cdot \\nabla B + \\nu \\Delta u , \\\\ B _ t + u \\cdot \\nabla B & = B \\cdot \\nabla u + \\mu \\Delta B , \\\\ \\nabla \\cdot u = \\nabla \\cdot B & = 0 , \\end{aligned} \\end{align*}"}
+{"id": "1080.png", "formula": "\\begin{align*} w ( \\alpha + w ^ { - 1 } \\gamma ) = \\gamma + w \\alpha \\in \\Phi . \\end{align*}"}
+{"id": "143.png", "formula": "\\begin{align*} Q _ b ( \\omega _ b ) = e ^ { - \\big ( \\Phi _ b ( \\omega _ b ) + | \\partial x | ^ { - 1 } \\Phi _ { \\{ x \\} } ( \\omega ( x ) ) + | \\partial y | ^ { - 1 } \\Phi _ { \\{ y \\} } ( \\omega ( y ) ) \\big ) } , \\end{align*}"}
+{"id": "2332.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 4 g ( N ( X , D ^ g _ { J e _ i } \\theta ^ \\sharp ) , J e _ i ) & = - \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ 4 g ( D ^ g _ { J e _ i } \\theta ^ \\sharp , e _ j ) g ( N ( e _ j , J e _ i ) , X ) , \\\\ & = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ 4 g ( D ^ g _ { e _ i } \\theta ^ \\sharp , e _ j ) g ( N ( e _ i , e _ j ) , X ) , \\\\ & = \\frac { 1 } { 2 } g ( d \\theta , N _ X ) . \\end{align*}"}
+{"id": "1423.png", "formula": "\\begin{align*} \\P _ { ( x _ 1 , x _ 2 ) } ( \\rho \\leq n ) & = 2 \\P _ { ( x _ 1 , x _ 2 ) } ( S _ 2 ( n ) < S _ 1 ( n ) ) . \\end{align*}"}
+{"id": "7052.png", "formula": "\\begin{align*} F _ { 5 , 0 } \\ , : = \\ , \\frac { 2 0 } { 9 } . \\end{align*}"}
+{"id": "2235.png", "formula": "\\begin{align*} M _ \\beta : = \\max _ { \\eta > 0 } \\bigg | \\displaystyle \\int _ 0 ^ \\eta e ^ { - i u } u ^ { - \\beta } d u \\bigg | ^ 2 ; \\end{align*}"}
+{"id": "109.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } \\partial _ t u - D ^ { \\alpha } _ x \\partial _ { x } u - \\partial _ { x } ^ { - 1 } \\partial _ y ^ 2 u & = u \\partial _ x u , ( t , x , y ) \\in \\R \\times \\R \\times \\R , \\\\ u ( 0 ) & = u _ 0 \\in H ^ { s _ 1 , s _ 2 } ( \\R ^ 2 ) , \\end{array} \\right . \\end{align*}"}
+{"id": "3331.png", "formula": "\\begin{align*} \\chi _ i \\cap \\psi _ i & = \\left \\{ \\begin{aligned} & \\psi _ i & & ( i \\neq j ) \\\\ & \\sigma \\cap \\psi _ j & & ( i = j ) \\end{aligned} \\right . \\\\ & = \\left \\{ \\begin{aligned} & \\psi _ i & & ( i \\neq j ) \\\\ & \\psi _ j \\setminus \\{ x \\} & & ( i = j ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "8368.png", "formula": "\\begin{align*} \\varrho - \\mathcal { P } ^ { - } ( \\varrho J ) = I , \\end{align*}"}
+{"id": "729.png", "formula": "\\begin{align*} \\tilde u ( x , t ) : = [ G _ k ( \\cdot , t ) * u _ 0 ( | \\cdot | ) ] ( x ) + \\int _ 0 ^ t [ G _ k ( \\cdot , t - s ) * F ( | \\cdot | , s ) ] ( x ) \\ , { \\rm d } s \\end{align*}"}
+{"id": "1820.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d s } \\mathcal { Y M } _ { e , g _ s } ^ 0 ( { \\nabla } ) _ { \\big { | } _ { s = 0 } } & = \\frac 1 2 \\int _ M \\langle S _ { e , \\mathcal { Y M } ^ 0 } , \\delta g \\rangle d v _ g \\\\ \\end{aligned} \\end{align*}"}
+{"id": "758.png", "formula": "\\begin{align*} f ( a , s ) = \\int _ 0 ^ L \\frac { s } { ( a - x ) ^ 2 + ( s ) ^ 2 } d x = \\arctan ( \\frac { L - a } { s } ) - \\arctan ( \\frac { - a } { s } ) \\le \\pi , \\end{align*}"}
+{"id": "4906.png", "formula": "\\begin{align*} w _ i ( \\alpha , \\beta ) = \\frac { ( - 1 ) ^ i \\ , \\binom { \\beta - 1 } { i } } { \\mathrm { B } ( \\alpha , \\beta ) } w _ { i , j , r } ( \\alpha , \\beta ) = \\frac { ( - 1 ) ^ { i + j + r } \\ , \\binom { \\alpha + i - 1 } { j } \\ , \\binom { \\beta - 1 } { i } \\ , \\binom { j } { r } } { \\mathrm { B } ( \\alpha , \\beta ) } . \\end{align*}"}
+{"id": "1163.png", "formula": "\\begin{align*} H \\left ( P ^ { ( n , k ) } [ t ] \\mid \\mu ^ { \\otimes k } [ t ] \\right ) \\leq \\frac { k } { n } M , k = 1 , \\cdots , n , t \\in [ 0 , T _ M ] . \\end{align*}"}
+{"id": "6947.png", "formula": "\\begin{align*} \\big | T _ { W _ \\ell , \\phi } ( \\mu ) - T _ { W , \\phi } ( \\mu ) \\big | \\le \\sum _ { i = 1 } ^ L | c _ i | d _ \\square ( W _ \\ell , W ) . \\end{align*}"}
+{"id": "6139.png", "formula": "\\begin{align*} d ^ a _ 1 ( x , y ) & = x - \\eta ^ a y \\\\ d ^ a _ 0 ( x , y ) & = \\prod _ { i \\neq a } ( x - \\eta ^ i y ) \\end{align*}"}
+{"id": "3183.png", "formula": "\\begin{align*} & \\lim \\limits _ { q \\to \\infty } \\frac { 2 ^ { m - 1 } ( q - m + 1 ) \\pm 2 ^ { 2 m } ( 1 + 2 ^ m \\cdot 3 m \\sqrt { q } ) ( 3 ^ { m - 1 } - 1 ) } { m \\times 4 ^ { m - 1 } q } \\\\ & = \\frac { 1 } { m \\times 4 ^ { m - 1 } } 2 ^ { m - 1 } \\\\ & = \\frac { 1 } { m \\times 2 ^ { m - 1 } } , \\end{align*}"}
+{"id": "6219.png", "formula": "\\begin{align*} W ( r ) & = \\frac { 1 } { 2 } \\biggl \\{ - ( 2 L + 2 ) \\frac { f } { r } + \\Q [ ( 2 L + 3 ) a _ 0 + 1 ] + \\kappa [ ( 2 L + 3 ) a _ 1 - m ] \\frac { r } { f } \\\\ & \\quad { } + \\kappa ( 2 L + 3 ) \\sum _ { k = 1 } ^ { [ ( m - 1 ) / 2 ] } a _ { 2 k + 1 } \\frac { r } { f ^ { 2 k + 1 } } + ( 2 L + 3 ) \\sum _ { k = 1 } ^ { [ m / 2 ] } a _ { 2 k } \\frac { 1 } { f ^ { 2 k } } \\biggr \\} \\end{align*}"}
+{"id": "5163.png", "formula": "\\begin{align*} \\begin{aligned} \\limsup _ { k \\to \\infty } \\Bigg \\{ & \\left | | | b _ { n _ k } | | _ { L ^ { 2 } ( B ( z _ k e _ z , R _ 0 ) ) } ^ { 2 } - \\alpha \\right | + \\left | | | b _ { n _ k } | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } \\backslash B ( z _ k e _ z , R _ k ) ) } ^ { 2 } - ( l - \\alpha ) \\right | \\\\ & + | | b _ { n _ k } | | _ { L ^ { 2 } ( B ( z _ k e _ z , R _ k ) \\backslash B ( z _ k e _ z , R _ 0 ) ) } ^ { 2 } \\Bigg \\} \\leq \\varepsilon . \\end{aligned} \\end{align*}"}
+{"id": "3035.png", "formula": "\\begin{align*} \\omega = \\frac { 1 } { 2 } \\left ( C \\pi _ { 1 } ^ { \\ast } \\omega ^ { 1 } + \\frac { 1 } { C } \\ , \\pi _ { 2 } ^ { \\ast } \\omega ^ { 2 } \\right ) \\end{align*}"}
+{"id": "1632.png", "formula": "\\begin{align*} \\left < K _ \\ell , x _ { i j } \\right > = q ^ { - \\delta _ { \\ell , i } } \\delta _ { i , j } \\left < E _ \\ell , x _ { i j } \\right > = \\delta _ { \\ell , i } \\delta _ { j , i + 1 } \\left < F _ \\ell , x _ { i j } \\right > = \\delta _ { \\ell , j } \\delta _ { i , j + 1 } \\end{align*}"}
+{"id": "4693.png", "formula": "\\begin{align*} { \\rm V o l } ( { \\tilde \\Sigma } ^ { 3 r } ) \\leq C ( n ) \\sum _ { i = 1 } ^ s { \\rm V o l } ( B _ { p _ i } ( r ) ) \\leq C ( n ) \\sum _ { i = 1 } ^ s { \\rm V o l } ( A \\cap B _ { p _ i } ( r ) ) , \\end{align*}"}
+{"id": "2764.png", "formula": "\\begin{align*} \\begin{aligned} | H ( p ( s , k ) ) | & = \\exp ( k \\eta ( k ) - 2 | s _ 1 | - 2 | s _ 2 | ) \\\\ & \\geq \\exp \\ ! \\left ( \\frac 1 3 k \\eta ( k ) \\right ) \\geq \\exp \\ ! \\left ( \\frac 1 5 | p ( s , k ) | \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "3705.png", "formula": "\\begin{align*} h _ { i j } ( x ) & = \\tfrac { 2 } { n - 2 } c \\delta _ { i j } | x | ^ { 2 - n } + O ^ { 2 , \\alpha } ( | x | ^ { \\max \\{ - 2 q , 1 - n \\} } ) \\\\ v ( x ) & = - c | x | ^ { 2 - n } + O ^ { 2 , \\alpha } ( | x | ^ { \\max \\{ - 2 q , 1 - n \\} } ) \\end{align*}"}
+{"id": "6603.png", "formula": "\\begin{align*} ( \\partial \\alpha ) ( u , v , w ) = \\sum _ { \\mathfrak { S } } \\langle \\alpha _ u v , w \\rangle u , v , w \\in E , \\end{align*}"}
+{"id": "3487.png", "formula": "\\begin{align*} \\tilde { P } _ { 2 s q _ { n - n _ 0 } - 1 } ( \\theta ) = \\sum _ { x _ 1 \\in I _ 1 \\cup I _ 2 } \\tilde { P } _ { 2 s q _ { n - n _ 0 } - 1 } ( \\theta _ { x _ 1 } ) \\prod _ { \\substack { j \\in I _ 1 \\cup I _ 2 \\\\ j \\neq x _ 1 } } \\frac { \\sin \\pi ( \\theta - \\theta _ j ) } { \\sin \\pi ( \\theta _ { x _ 1 } - \\theta _ j ) } . \\end{align*}"}
+{"id": "7074.png", "formula": "\\begin{align*} { } \\hat { T } _ N : = \\arg \\min _ { T \\in \\mathcal { T } } M _ N ( T ) , M _ N ( T ) : = \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ N d ^ 2 _ { \\mathcal { W } } ( T \\# \\mu _ i , \\nu _ i ) . \\end{align*}"}
+{"id": "2773.png", "formula": "\\begin{align*} B ( 0 , R ) \\subset \\overline { B } ( 0 , R _ 1 ) = f ^ k \\ ! \\left ( Y _ 1 \\right ) \\subset f ^ k \\ ! \\left ( B ( x _ 1 , 2 r ) \\right ) . \\end{align*}"}
+{"id": "3633.png", "formula": "\\begin{align*} \\sum _ i F ^ { i i } ( \\nu ^ { n + 1 } ) _ { i i } = & \\ 2 \\sum _ i F ^ { i i } \\frac { u _ i } { u } ( \\nu ^ { n + 1 } ) _ i + ( n - 1 ) \\sigma ( 1 + ( \\nu ^ { n + 1 } ) ^ 2 ) \\\\ & - \\nu ^ { n + 1 } \\left ( \\sum _ i F ^ { i i } + \\sum _ i F ^ { i i } \\kappa _ i ^ 2 \\right ) . \\end{align*}"}
+{"id": "1308.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x _ j } F ( u ( x ) ) & = g ' ( | u ( x ) | ) \\frac { \\partial | u ( x ) | } { \\partial x _ j } \\\\ & = \\frac { g ' ( | u ( x ) | ) } { 2 | u ( x ) | } \\left ( u ( x ) \\frac { \\partial \\overline { u } ( x ) } { \\partial x _ j } + \\frac { \\partial u ( x ) } { \\partial x _ j } \\overline { u } ( x ) \\right ) \\\\ & = { \\rm R e } \\left ( f ( u ( x ) ) \\frac { \\partial \\overline { u } ( x ) } { \\partial x _ j } \\right ) . \\end{align*}"}
+{"id": "3827.png", "formula": "\\begin{align*} \\begin{aligned} | \\langle \\boldsymbol { \\nu } , G ( \\lambda | \\bar U , x , \\tau ) \\rangle | & \\le c _ 3 \\langle \\boldsymbol { \\nu } , \\eta ( \\lambda | \\bar U ; x , t ) \\rangle = c _ 3 \\mathcal { H } ( \\boldsymbol { \\nu } , U , \\bar { U } ; x , t ) \\ ; , \\end{aligned} \\end{align*}"}
+{"id": "9154.png", "formula": "\\begin{align*} h ( x + \\delta u ) = h ( x ) + R ( x , \\delta u ) \\end{align*}"}
+{"id": "5887.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\sup _ { n \\in \\mathbb { N } } \\mathbb { E } | \\langle Y _ n ^ M ( \\zeta _ n + \\delta ) - Y _ n ^ M ( \\zeta _ n ) , e \\rangle | ^ { \\alpha } = 0 . \\end{align*}"}
+{"id": "1405.png", "formula": "\\begin{align*} \\P _ x ( S ( 1 ) \\in d y , \\tau > 1 ) = \\prod _ { i = 1 } ^ d \\lambda _ i e ^ { - \\sum _ { i = 1 } ^ d \\lambda _ i ( y _ i - x _ i ) } \\det ( q _ { 1 + i - j } ( y _ j - x _ i ) ) _ { i , j = 1 } ^ d \\end{align*}"}
+{"id": "7545.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ d A _ { j } ^ * A _ { j } \\ ( x _ 1 \\otimes \\cdots \\otimes x _ d ) & = \\sum \\limits _ { j = 1 } ^ d \\| X _ { j } \\| ^ 2 ( x _ 1 \\otimes \\cdots \\otimes x _ d ) \\\\ & = \\left \\| \\sum \\limits _ { j = 1 } ^ d A _ { j } ^ * A _ { j } \\right \\| ( x _ 1 \\otimes \\cdots \\otimes x _ d ) . \\end{align*}"}
+{"id": "3870.png", "formula": "\\begin{align*} \\begin{cases} - \\varepsilon ^ 2 ( K ( x ) \\nabla u ) = ( u - q | \\ln \\varepsilon | ) ^ { p } _ + , \\ \\ & x \\in \\Omega , \\\\ u = 0 , \\ \\ & x \\in \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "2976.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } \\frac { 1 } { \\# ( \\Lambda _ N ^ { \\vec { v } } ( b ) ) } \\sum _ { ( m , n ) \\in \\Lambda _ N ^ { \\vec { v } } ( b ) } \\max _ { 1 \\leq i < j \\leq k } d ( T ^ { ( m , n ) } x _ i , T ^ { ( m , n ) } x _ j ) = 0 \\end{align*}"}
+{"id": "5847.png", "formula": "\\begin{align*} { f _ { 1 2 } } = { f _ { 1 1 } } - f _ { 1 1 } ^ { \\left ( { e q } \\right ) } + f _ { 1 2 } ^ { \\left ( { e q } \\right ) } - \\delta x - \\delta z , \\end{align*}"}
+{"id": "7289.png", "formula": "\\begin{align*} G ( \\mu ) : & = \\{ ( x , y ) : \\ \\ x \\in ( - \\mu b , \\mu b ) ^ { d } , \\ \\ g ( x ) - \\mu L b < y \\leq g ( x ) \\} . \\end{align*}"}
+{"id": "3454.png", "formula": "\\begin{align*} F ( \\theta _ { ( \\ell + 1 ) q _ n + m _ n } ) = \\left ( \\begin{matrix} \\cos ( \\pi \\theta _ { ( \\ell + 1 ) q _ n + m _ n } ) E - \\lambda \\sin ( \\pi \\theta _ { ( \\ell + 1 ) q _ n + m _ n } ) \\ \\ & - \\cos ( \\pi \\theta _ { ( \\ell + 1 ) q _ n + m _ n } ) \\\\ \\cos ( \\pi \\theta _ { ( \\ell + 1 ) q _ n + m _ n } ) & 0 \\end{matrix} \\right ) \\end{align*}"}
+{"id": "123.png", "formula": "\\begin{align*} & N \\Big ( \\frac { N _ 1 } { N _ { + } } \\Big ) ^ { ( 5 - 2 \\alpha ) + \\varepsilon } \\sum _ { L \\geqslant N _ + ^ { ( 5 - 2 \\alpha ) + \\varepsilon } } L ^ { - \\frac { 1 } { 2 } } \\| \\mathbf { 1 } _ { D _ { N , L } } ( f _ { N _ 1 , L _ 1 } \\ast g _ { N _ 2 , L _ 2 } ) \\| _ { L ^ 2 } \\\\ & \\lesssim N _ 1 ^ { - c _ 1 ( \\varepsilon ) } [ \\min ( N , 1 ) ] ^ { c _ 2 ( \\varepsilon ) } L _ 1 ^ { \\frac { 1 } { 2 } } \\| f _ { N _ 1 , L _ 1 } \\| _ { L ^ 2 } ~ L _ 2 ^ { \\frac { 1 } { 2 } } \\| g _ { N _ 2 , L _ 2 } \\| _ { L ^ 2 } . \\end{align*}"}
+{"id": "2511.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbf { T } _ x \\mathbf { X } ^ { - 1 } = \\mathbf { X } ^ { - 1 } \\mathbf { T } _ x = \\left ( \\mathbf { I } - \\mathbf { U } _ { x ^ { ( i ) } } / \\mathbf { x } ^ { ( i ) } _ 1 : i = 1 , \\cdots , k \\right ) , \\\\ & \\mathbf { T } _ x \\mathbf { X } ^ { - 1 } \\mathbf { e } = \\mathbf { e } ; \\end{aligned} \\end{align*}"}
+{"id": "3473.png", "formula": "\\begin{align*} r _ { \\ell } \\leq & e ^ { 5 0 \\varepsilon q _ n } \\frac { e ^ { - q _ n L } } { \\max ( | \\ell | , 1 ) } \\max ( r _ { \\ell - 1 } , r _ { \\ell } , r _ { \\ell + 1 } ) \\times \\begin{cases} \\max ( | \\ell | , e ^ { \\delta _ n q _ n } ) , \\beta _ n \\geq \\delta _ n + 2 0 0 \\varepsilon , \\\\ e ^ { \\beta _ n q _ n } , \\beta _ n < \\delta _ n + 2 0 0 \\varepsilon \\end{cases} . \\end{align*}"}
+{"id": "4197.png", "formula": "\\begin{align*} c _ \\varepsilon \\leq \\inf \\limits _ { w \\in \\mathcal { H } ( \\Omega ) , w _ + \\neq 0 } \\max \\limits _ { t \\geq 0 } I _ \\varepsilon ( t w ) = \\inf \\limits _ { w \\in \\mathcal { N } _ \\varepsilon } I _ \\varepsilon ( w ) . \\end{align*}"}
+{"id": "3657.png", "formula": "\\begin{align*} D _ { x , 1 } = \\frac { 3 } { 2 } + \\frac { L _ { x y } } { 2 \\mu _ { x } } ( 1 + 2 \\theta ) + \\frac { L _ { l y } } { 2 L _ { x y } } + \\frac { L _ { l x } } { 2 \\mu _ { x } } , \\end{align*}"}
+{"id": "5374.png", "formula": "\\begin{align*} S _ k ^ { i j } H _ { i j } = - \\sum _ i S _ k ^ { p q , r s } u _ { p q i } u _ { r s i } + B ( n - k + 1 ) S _ { k - 1 } + \\Delta f . \\end{align*}"}
+{"id": "7425.png", "formula": "\\begin{align*} | | \\zeta | | G \\kappa \\stackrel { d e f } { = } \\sup _ { p \\in U } \\left \\{ \\ \\frac { | \\zeta | _ p } { \\kappa ( p ) } \\ \\right \\} . \\end{align*}"}
+{"id": "4870.png", "formula": "\\begin{align*} 0 = \\partial ^ 2 _ { x _ i x _ j } \\phi - \\partial ^ 2 _ { x _ i x _ j } \\psi - \\partial ^ 2 _ { x _ i y } \\psi \\partial _ { x _ j } T - \\partial ^ 2 _ { x _ j y } \\psi \\partial _ { x _ i } T - \\partial ^ 2 _ y \\psi \\partial _ { x _ j } T \\partial _ { x _ i } T - \\partial _ y \\psi \\partial ^ 2 _ { x _ j x _ i } T \\\\ + { ( \\partial _ y T ) ^ { - 2 } } ( \\partial _ y T \\partial ^ 3 _ { x _ j x _ i y } T - \\partial ^ 2 _ { x _ i y } T \\partial ^ 2 _ { x _ j y } T ) \\end{align*}"}
+{"id": "8423.png", "formula": "\\begin{align*} & \\Psi ^ \\pm _ { 1 1 } ( x ; k ) = \\psi ^ \\pm _ { 1 1 } ( x ; k ) , \\\\ & \\Psi ^ \\pm _ { 2 1 } ( x ; k ) = - \\bar { u } _ x \\psi ^ \\pm _ { 1 1 } ( x ; k ) + 2 i k \\psi ^ \\pm _ { 2 1 } ( x ; k ) . \\end{align*}"}
+{"id": "3224.png", "formula": "\\begin{align*} d f _ t = \\left ( \\partial _ x f _ t + \\alpha ( \\sigma _ s ) \\right ) d s + \\sigma _ s d W _ s , \\end{align*}"}
+{"id": "4212.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { \\ln \\frac { ( A _ \\varepsilon , \\partial \\Omega ) } { ( A _ \\varepsilon ) } } { \\ln \\frac { 1 } { \\varepsilon } } = 1 . \\end{align*}"}
+{"id": "6674.png", "formula": "\\begin{align*} \\begin{aligned} | \\gamma ^ { \\epsilon } ( s ) | ^ { 2 } & \\leq M ^ 2 \\left ( \\rho ^ { 2 } + | Y ^ { \\epsilon } ( s - ) | ^ { 2 } \\right ) \\\\ & \\leq \\frac { M ^ 2 } { \\big ( 1 - \\epsilon \\lambda ( \\epsilon ) M \\big ) ^ { 2 } } \\cdot \\left ( \\rho ^ { 2 } + | \\tilde { Y } ^ { \\epsilon } ( s ) | ^ { 2 } \\right ) \\\\ & \\leq 2 M ^ { 2 } \\left ( \\rho ^ { 2 } + | \\tilde { Y } ^ { \\epsilon } ( s ) | ^ { 2 } \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "2001.png", "formula": "\\begin{align*} T ( x ) = \\begin{cases} 1 - | x | / 2 , & 1 \\leq | x | \\leq 2 , \\\\ 1 / 2 , & | x | < 1 . \\end{cases} \\end{align*}"}
+{"id": "6510.png", "formula": "\\begin{align*} E _ { d ^ * ( x ) } = E _ { d ^ * ( y ) } \\cup E _ { d ^ * ( y ' f ' ) } \\cup E _ { [ w b ^ { - 1 } , w a ] } . \\end{align*}"}
+{"id": "912.png", "formula": "\\begin{align*} & \\Sigma _ 4 = \\sum \\limits _ { 1 \\leq | m | \\leq M } \\frac { \\Theta _ m } { 2 \\pi i m } \\ , , \\\\ & \\Theta _ m = \\sum \\limits _ { D \\leq d < 2 D } e \\left ( \\frac { \\sqrt { X } m } { d } \\right ) \\sum \\limits _ { n \\in \\mathcal { N } ( d ) } e \\left ( - \\frac { n m } { d } \\right ) \\ , , \\\\ & \\Sigma _ 5 = \\sum \\limits _ { D \\leq d < 2 D } \\sum \\limits _ { n \\in \\mathcal { N } ( d ) } f _ { M } \\left ( \\frac { \\sqrt { X } - n } { d } \\right ) \\ , . \\end{align*}"}
+{"id": "3185.png", "formula": "\\begin{align*} \\lim \\limits _ { q \\to \\infty } \\dfrac { k _ 4 ( P ^ \\ast ( q ) ) } { q ^ 4 } = \\frac { 1 } { 1 5 3 6 } . \\end{align*}"}
+{"id": "99.png", "formula": "\\begin{align*} v ^ { - 1 } \\frac 1 N \\sum _ { k = 1 } ^ N ( \\sigma \\circ w ) ^ k ( \\mu ) \\in X _ \\ast ( T ) _ { \\Gamma _ 0 } \\otimes \\mathbb Q \\end{align*}"}
+{"id": "2266.png", "formula": "\\begin{align*} a ( z ) = a ( | z | ) = a ( | t _ 1 z t _ 2 | ) = a ( t _ 1 z t _ 2 ) \\end{align*}"}
+{"id": "7476.png", "formula": "\\begin{align*} A ( 0 ) \\nabla ( L u ) \\cdot \\nabla u = - f ' ( u ) | \\nabla u | _ { A ( 0 ) } ^ 2 . \\end{align*}"}
+{"id": "3259.png", "formula": "\\begin{align*} U _ n ( X ) _ t = \\sup _ { s \\in [ 0 , t ] } \\| \\Delta _ n ^ { 1 - \\frac m 2 } \\sum _ { i = 1 } ^ { \\lfloor s / \\Delta _ n \\rfloor - k + 1 } \\bigotimes _ { j = 1 } ^ k \\tilde { \\Delta } _ { i + j - 1 } ^ n X ^ { \\otimes m _ j } - \\int _ 0 ^ t \\rho _ { \\Sigma _ s } ^ { \\otimes k } ( m _ 1 , . . . , m _ k ) d s \\| _ { \\mathcal H ^ m } . \\end{align*}"}
+{"id": "5228.png", "formula": "\\begin{align*} 0 & = \\lim _ { k \\to \\infty } \\left \\{ | | v _ { n _ k } | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } + | | b _ { n _ k } ( \\cdot + z _ { k } e _ z ) - b | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } \\right \\} \\\\ & \\geq \\liminf _ { k \\to \\infty } \\left ( | | v _ { n _ k } | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } + \\inf \\left \\{ \\ | | b _ { n _ k } - \\tilde { b } | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } \\ \\middle | \\ \\tilde { b } \\in S _ { h , \\gamma } \\right \\} \\right ) \\geq \\frac { \\varepsilon _ 0 } { 2 } > 0 . \\end{align*}"}
+{"id": "1222.png", "formula": "\\begin{align*} \\eta ^ { z , i } _ x : = \\begin{cases} \\eta _ x , & x \\neq z , \\\\ \\ i , & x = z , \\end{cases} \\end{align*}"}
+{"id": "5236.png", "formula": "\\begin{align*} h _ C = \\left ( \\frac { W } { \\lambda } \\right ) ^ { 2 } \\frac { 1 2 \\pi c _ { 3 / 2 } } { J _ { 5 / 2 } ^ { 2 } ( c _ { 3 / 2 } ) } \\int _ { 0 } ^ { c _ { 3 / 2 } } \\rho J _ { 3 / 2 } ^ { 2 } ( \\rho ) \\dd \\rho . \\end{align*}"}
+{"id": "8303.png", "formula": "\\begin{align*} \\dot y = A y '' + M y , y ( 0 , x ) = y _ 0 ( x ) , \\end{align*}"}
+{"id": "6726.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\Phi _ s ( z ) = \\begin{cases} 0 , & z < - 1 , \\\\ \\frac { 1 } { 2 } ( z + 1 ) , & | z | < 1 , \\\\ 1 , & z > 1 . \\end{cases} \\end{align*}"}
+{"id": "364.png", "formula": "\\begin{align*} \\langle { u _ { i _ 1 } } , { u _ { i _ 2 } } \\rangle = \\frac { 1 } { 6 4 \\cos ^ { 2 } ( \\frac { i _ 1 \\pi } { n - 1 } ) \\cos ^ { 2 } ( \\frac { i _ 2 \\pi } { n - 1 } ) } \\langle S v _ { i _ 1 } , S v _ { i _ 2 } \\rangle . \\end{align*}"}
+{"id": "5864.png", "formula": "\\begin{align*} \\log \\frac { 1 } { x } - \\gamma + \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n - 1 } } { n \\cdot n ! } x ^ { n } \\ \\ ( \\gamma : \\rm { E u l e r ' s \\ c o n s t a n t } ) ; \\end{align*}"}
+{"id": "357.png", "formula": "\\begin{align*} \\nu _ k = ( \\frac { - 1 } { 8 \\cos ^ { 2 } ( \\frac { \\pi k } { n - 1 } ) } ) \\ 1 ' _ { n - 1 } S v _ { k } + 2 \\ 1 _ { n - 1 } ' v _ k . \\end{align*}"}
+{"id": "7760.png", "formula": "\\begin{align*} \\frac { q _ { h + 1 } ( x ) } { p _ { h + 1 } ( x ) } = \\frac { 1 } { 2 \\sqrt x } \\Big ( \\frac { q _ h ( \\sqrt x ) } { p _ h ( \\sqrt { - x } ) } - \\frac { q _ h ( \\sqrt { - x } ) } { p _ h ( \\sqrt x ) } \\Big ) , ~ h = 0 , 1 , \\dots \\end{align*}"}
+{"id": "6384.png", "formula": "\\begin{align*} { P } _ k ( \\mathcal { T } ) : = \\left \\{ q _ h \\in H ^ 1 ( \\Omega ) : q _ h | _ T \\in P _ k ( T ) T \\in \\mathcal { T } \\right \\} , \\end{align*}"}
+{"id": "2746.png", "formula": "\\begin{align*} B ( s , | s | ^ \\beta ) \\cap B ( s ' , | s ' | ^ \\beta ) = \\emptyset \\quad \\ s , s ' \\in S . \\end{align*}"}
+{"id": "2584.png", "formula": "\\begin{align*} ( X \\wedge _ g ^ c Y ) Z = ( X \\wedge _ g Y ) Z + ( J X \\wedge _ g J Y ) Z - 2 g ( J X , Y ) J Z , \\end{align*}"}
+{"id": "5903.png", "formula": "\\begin{align*} \\Lambda : = \\{ n \\in \\mathbb { N } : g _ n ( t , \\omega ) > 0 \\} . \\end{align*}"}
+{"id": "2450.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } d _ { q } s \\ , F _ { q } ( s ) = \\int _ { 0 } ^ { \\infty } \\frac { 1 } { t } \\exp _ { q } ( - s t ) f ( t ) d t = L _ { q } \\left [ \\frac { f ( t ) } { t } \\right ] . \\end{align*}"}
+{"id": "4978.png", "formula": "\\begin{align*} \\mathsf { U } ^ n & = \\{ 1 , 0 , g ( \\xi _ 2 ^ { \\mathsf { U } ^ n } ) , \\cdots , g ( \\xi _ { n - 1 } ^ { \\mathsf { U } ^ n } ) \\} , \\\\ \\mathsf { V } ^ n & = \\{ 0 , 1 , g ( \\xi _ 2 ^ { \\mathsf { V } ^ n } ) , \\cdots , g ( \\xi _ { n - 1 } ^ { \\mathsf { V } ^ n } ) \\} , \\end{align*}"}
+{"id": "8222.png", "formula": "\\begin{align*} \\sigma ( a _ 1 , a _ 2 , \\ldots , a _ { n _ 0 - 1 } , a _ { n _ 0 } ) & = ( S - a _ 1 , S - a _ 2 , \\ldots , S - a _ { n _ 0 - 1 } , S - a _ { n _ 0 } ) . \\end{align*}"}
+{"id": "71.png", "formula": "\\begin{align*} d _ { s _ i } ( [ 1 ] _ \\sigma ) = \\frac 1 2 \\left ( 1 + 1 - 0 - 0 \\right ) = 1 = \\dim X _ { s _ i } ( 1 ) , i = 1 , 2 . \\end{align*}"}
+{"id": "3933.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta y + f ( \\cdot , y ) & = u , \\ ; \\ ; \\Omega , \\\\ y & = 0 , \\ ; \\ ; \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "5564.png", "formula": "\\begin{align*} W ( 0 ^ \\infty | [ 0 ^ k 1 ] ) = \\sum _ { n \\in \\N } ( a _ { n + k } - a ) \\ . \\end{align*}"}
+{"id": "6198.png", "formula": "\\begin{align*} W ( r ) = \\frac { \\xi } { r } f + \\eta + \\zeta \\frac { r } { f } + \\frac { \\sigma } { f ^ 2 } , \\xi \\le 0 , \\sigma > 0 , \\end{align*}"}
+{"id": "4263.png", "formula": "\\begin{align*} & \\alpha _ i ^ { v _ l } = 2 ( ( \\alpha _ i ^ { v } / 2 ) ( \\alpha ^ v _ { i + N _ { v } / 2 } ) ) , \\ 1 \\leq i \\leq N _ v / 2 , \\\\ & \\alpha _ i ^ { v _ r } = \\alpha ^ v _ { i + N _ v / 2 } + ( 1 - 2 \\beta ^ { v _ l } _ i ) \\alpha ^ v _ i , \\ 1 \\leq i \\leq N _ v / 2 . \\end{align*}"}
+{"id": "8820.png", "formula": "\\begin{align*} \\max _ { j = 1 , 2 } \\sup _ { x \\in V _ j \\cap \\S ^ { n - 1 } } ( \\| P _ { x , j } \\| + \\| P _ { x , j } ^ { - 1 } \\| ) \\le C , \\end{align*}"}
+{"id": "9053.png", "formula": "\\begin{align*} r _ { b , k } ( 1 + \\alpha _ { b , k } ) R _ b ( j _ b ) = 0 . \\end{align*}"}
+{"id": "4403.png", "formula": "\\begin{align*} \\sum _ { n < x ^ 2 } \\frac { \\Lambda _ x ( n ) ^ 2 } { n ^ { 2 \\sigma } } & < \\sum _ { n = 1 } ^ \\infty \\frac { \\Lambda ( n ) \\log n } { n ^ { 2 \\sigma } } = \\frac { d } { d s } \\left [ \\frac { \\zeta ' } { \\zeta } ( s ) \\right ] _ { s = 2 \\sigma } . \\end{align*}"}
+{"id": "2942.png", "formula": "\\begin{align*} \\Psi ( t , h ( x ) ) = h \\left ( \\Phi ( t , x ) \\right ) \\end{align*}"}
+{"id": "3681.png", "formula": "\\begin{align*} ( - R _ { k \\ell j i } + R _ { \\ell i k j } ) V _ j V _ \\ell = ( R _ { k \\ell i j } + R _ { \\ell i k j } ) V _ j V _ \\ell = - R _ { i k \\ell j } V _ j V _ \\ell = 0 \\end{align*}"}
+{"id": "3491.png", "formula": "\\begin{align*} \\phi ( y ) = \\sum _ { z \\in \\partial I ( y ) } G _ { I ( y ) } ( z , y ) \\phi ( z ' ) . \\end{align*}"}
+{"id": "4008.png", "formula": "\\begin{align*} l r ( e p o c h s ) = \\begin{cases} 1 0 ^ { - 2 } , \\textit { 1 $ \\leq $ e p o c h s $ < $ 1 0 , 0 0 0 } \\\\ 1 0 ^ { - 3 } , \\textit { 1 0 , 0 0 0 $ \\leq $ e p o c h s $ < $ 4 0 , 0 0 0 } \\\\ 1 0 ^ { - 4 } , \\textit { e p o c h s $ \\geq $ 4 0 , 0 0 0 } . \\end{cases} \\end{align*}"}
+{"id": "5839.png", "formula": "\\begin{align*} { g _ i } \\left ( { { { \\bf { x } } _ b } , t } \\right ) = g _ i ^ { \\left ( { e q } \\right ) } \\left ( { { { \\bf { x } } _ b } , t } \\right ) + g _ i ^ { \\left ( { n e q } \\right ) } \\left ( { { { \\bf { x } } _ b } , t } \\right ) , \\end{align*}"}
+{"id": "5336.png", "formula": "\\begin{align*} P _ { n , k } ( z ) = \\sum _ { i = 0 } ^ { k + d _ n } \\alpha _ { n , k , i } \\binom { z } { i } \\end{align*}"}
+{"id": "6644.png", "formula": "\\begin{align*} \\lim _ { K \\to \\infty } \\limsup _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\sup _ { t \\in [ 0 , 1 ] } | X ^ { \\epsilon } ( t ) | > K \\Big ) = - \\infty . \\end{align*}"}
+{"id": "9279.png", "formula": "\\begin{align*} \\begin{aligned} C _ m ( K ) = \\inf \\left \\{ \\int _ { \\Omega } ( \\Delta u ) ^ m \\wedge \\beta _ n ^ { n - m } : u \\in \\mathcal { U } ^ * ( K , \\Omega ) \\right \\} \\end{aligned} \\end{align*}"}
+{"id": "1131.png", "formula": "\\begin{align*} { \\cal A } = [ A ^ { ( 1 ) } , A ^ { ( 2 ) } , \\dots , A ^ { ( L ) } ] \\in \\R ^ { N \\times N L } . \\end{align*}"}
+{"id": "2722.png", "formula": "\\begin{align*} \\mathbb { E } \\exp ( \\lambda | X | ) = \\mathbb { E } \\sum _ { p = 0 } ^ \\infty \\frac { 1 } { p ! } ( \\lambda | X | ) ^ p = 1 + \\sum _ { p = 1 } ^ \\infty \\frac { 1 } { p ! } \\lambda ^ p \\mathbb { E } | X | ^ p \\end{align*}"}
+{"id": "7070.png", "formula": "\\begin{align*} F _ { 7 , 0 , 0 , 1 } \\ , = \\ , 0 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ F _ { 8 , 0 , 0 , 1 } \\ , = \\ , 0 . \\end{align*}"}
+{"id": "1855.png", "formula": "\\begin{align*} \\delta : = ( n - 1 ) B _ 0 - 2 d _ e \\| R ^ \\nabla \\| ^ 2 _ \\infty \\sqrt { A } \\coth \\sqrt { A } \\geq 0 \\ , , \\end{align*}"}
+{"id": "8132.png", "formula": "\\begin{align*} \\mathcal C '^ \\kappa = \\varprojlim _ n ( \\mathcal C ^ n ) ^ \\kappa = \\varprojlim _ n \\varinjlim _ i ( \\mathcal C _ i ^ n ) ^ \\kappa = \\varinjlim _ i \\varprojlim _ n ( \\mathcal C _ i ^ n ) ^ \\kappa = \\varinjlim _ i \\mathcal C ^ \\kappa = \\mathcal C ^ \\kappa , \\end{align*}"}
+{"id": "5265.png", "formula": "\\begin{align*} \\langle f , g \\rangle = f _ 1 g _ 4 - \\frac { 1 } { 3 } f _ 2 g _ 3 + \\frac { 1 } { 3 } f _ 3 g _ 2 - f _ 4 g _ 1 . \\end{align*}"}
+{"id": "5113.png", "formula": "\\begin{align*} r b = \\nabla \\phi \\times r \\nabla \\theta + G r \\nabla \\theta . \\end{align*}"}
+{"id": "6278.png", "formula": "\\begin{align*} \\tilde { \\nabla } \\xi = d \\xi + \\omega \\xi . \\end{align*}"}
+{"id": "8480.png", "formula": "\\begin{align*} M ( x ; z ) = I + \\mathcal { C } \\left ( M _ { - } ( x ; \\cdot ) R ( x ; \\cdot ) \\right ) , z \\in \\mathbb { C } ^ { \\pm } . \\end{align*}"}
+{"id": "8477.png", "formula": "\\begin{align*} & M _ { \\pm , 1 } ( x ; z ) - e _ 1 = \\mathcal { P } ^ { \\pm } \\left ( M _ - ( x ; \\cdot ) R ( x ; \\cdot ) \\right ) _ { 1 1 } , z \\in \\mathbb { R } , \\\\ & ( 2 i k ) ^ { - 1 } ( M _ { \\pm , 1 } ( x ; z ) - e _ 1 ) = \\mathcal { P } ^ { \\pm } \\left ( ( 2 i k ^ { - 1 } ) M _ - ( x ; \\cdot ) R ( x ; \\cdot ) \\right ) _ { 1 1 } , z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "4226.png", "formula": "\\begin{align*} \\zeta ' & \\ = \\sum _ { z \\in \\mathbb { F } ^ * _ { q ^ n } } \\overline { U _ { q ^ n } } \\Lambda ( z ) { \\bf 1 } _ { - } - ( q ^ n - 1 ) \\overline { U _ { q ^ n } } { \\bf 1 } _ { - } \\\\ & \\ = \\sum _ { x , y \\in \\mathbb { F } ^ * _ { q ^ n } , x y \\ne 1 } \\varepsilon ( x ) \\Lambda ( y ) { \\bf 1 } _ { - } - ( q ^ n - 1 ) \\overline { U _ { q ^ n } } { \\bf 1 } _ { - } \\end{align*}"}
+{"id": "1938.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty b _ 5 \\left ( \\frac { m n - 1 } { 6 } \\right ) q ^ n \\in S _ { 2 m - 2 } ( \\Gamma _ 0 ( 1 8 0 ) , \\chi _ { 5 } ) _ m . \\end{align*}"}
+{"id": "9112.png", "formula": "\\begin{align*} 0 < \\frac { \\sum \\limits _ { i = 1 } ^ j \\chi _ i - r ( j - 1 ) } { \\chi } < \\frac { \\sum \\limits _ { i = 1 } ^ j \\chi _ j - r j } { \\chi } < 1 . \\end{align*}"}
+{"id": "4294.png", "formula": "\\begin{align*} - \\theta \\Delta t \\mathcal { L } ( \\tilde { u } ^ { [ n ] } ) + & \\tilde { u } ^ { [ n ] } ( x ) = \\\\ & \\tilde { u } ^ { [ n - 1 ] } ( x ) + \\theta \\Delta t f ( t _ n , x ) + ( 1 - \\theta ) \\Delta t \\left [ \\mathcal { L } ( \\tilde { u } ^ { [ n - 1 ] } ) + f ( t _ { n - 1 } , x ) \\right ] . \\end{align*}"}
+{"id": "1577.png", "formula": "\\begin{align*} \\phi ( \\tau _ a , p ) : = \\gamma ( \\tau _ a , p ) \\gamma ^ { - 1 } = ( \\tau _ { \\lambda \\iota a } , \\gamma _ N p \\gamma _ N ^ { - 1 } ) . \\end{align*}"}
+{"id": "6709.png", "formula": "\\begin{align*} h ( g \\xi , g \\eta , g \\omega ) - h ( \\xi , \\eta , \\omega ) = 2 \\bar \\beta ( g , \\xi ) . \\end{align*}"}
+{"id": "8676.png", "formula": "\\begin{align*} { \\bf { \\bar U } } ^ { \\star } = { \\left [ { { \\bf { \\tilde \\Sigma } } { { \\bf { \\tilde B } } ^ H } } \\right ] _ { 1 : { M _ t } , : } } , \\end{align*}"}
+{"id": "5997.png", "formula": "\\begin{align*} \\| ( u , v ) \\| _ { 2 p } ^ { 2 p } = \\| u \\| _ { 2 p } ^ { 2 p } + \\| u \\| _ { 2 p } ^ { 2 p } . \\end{align*}"}
+{"id": "963.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 1 } ^ r d _ i p ^ { k _ i n _ i } \\colon n _ i \\in \\N _ 0 \\right \\} , \\end{align*}"}
+{"id": "3614.png", "formula": "\\begin{align*} u = X \\cdot \\mathbf { e } , \\nu ^ { n + 1 } = \\mathbf { e } \\cdot \\nu , \\end{align*}"}
+{"id": "1666.png", "formula": "\\begin{align*} H _ { / f } ^ 1 ( G _ L , \\sigma ) : = H ^ 1 ( G _ L , \\sigma ) / H _ { f } ^ 1 ( G _ L , \\sigma ) . \\end{align*}"}
+{"id": "458.png", "formula": "\\begin{align*} f _ 2 ( x ) = ( e ^ { \\lambda \\Lambda _ 2 } - 1 ) ^ { - 1 } \\sum _ { n = 1 } ^ \\infty \\big ( ( \\lambda \\Lambda _ 2 ) ^ n / n ! \\big ) \\ , g _ 2 ^ { \\ast n } ( x ) , \\end{align*}"}
+{"id": "260.png", "formula": "\\begin{align*} V _ { E _ k } ^ \\# ( D ( p - 1 ) ) \\subseteq V ( p _ k ) \\cup \\bigcup _ { i = 1 } ^ k V ( x _ i ) . \\end{align*}"}
+{"id": "3625.png", "formula": "\\begin{align*} B ( V _ 1 , V _ 2 ) = g ( V _ 1 , V _ 2 ) - g ( V _ 1 , e _ 1 ) g ( V _ 2 , e _ 1 ) , \\end{align*}"}
+{"id": "7137.png", "formula": "\\begin{align*} \\mathcal A _ { \\mathcal X } ^ { \\mathcal D } = \\left \\{ ( s , \\alpha ) | s \\in S , \\alpha \\in A _ { X _ s } ^ { D _ s } \\right \\} \\end{align*}"}
+{"id": "8394.png", "formula": "\\begin{align*} a ( z ) = 1 + k \\int _ { \\mathbb { R } } u _ y \\psi ^ - _ { 2 1 } ( x ; k ) \\mathrm { d } x . \\end{align*}"}
+{"id": "3757.png", "formula": "\\begin{align*} R ^ i _ { n _ 0 ; l _ 1 , \\ldots , l _ { n _ 0 } } = \\underset { n _ 0 \\geq j \\geq i } { \\bigsqcup } Q ^ j _ { n _ 0 ; l _ 1 , \\ldots , l _ { n _ 0 } } . \\end{align*}"}
+{"id": "767.png", "formula": "\\begin{align*} \\gamma _ { \\Theta } ^ s ( E ) = \\sup \\{ | \\langle \\nu , 1 \\rangle | \\} , \\end{align*}"}
+{"id": "4077.png", "formula": "\\begin{align*} A ' _ n ( j ) = A _ n ( j _ V ) \\end{align*}"}
+{"id": "9115.png", "formula": "\\begin{align*} \\frac { \\sum \\limits _ { i = 1 } ^ { j + 1 } \\chi _ i - r ( j + 1 ) } { \\chi } > \\sum \\limits _ { i = 1 } ^ { j + 1 } w _ i > \\frac { \\sum \\limits _ { i = 1 } ^ { j + 1 } \\chi _ i - r j } { \\chi } . \\end{align*}"}
+{"id": "3972.png", "formula": "\\begin{align*} G = \\begin{pmatrix} w ^ { 4 0 } & w ^ { 2 0 } & w ^ { 2 2 } & w ^ 3 & w ^ { 4 9 } & w ^ { 5 5 } & w ^ { 5 4 } & w ^ { 6 5 } \\\\ w ^ { 8 6 } & w ^ { 2 7 } & w ^ { 8 9 } & w ^ { 6 4 } & w ^ { 7 3 } & w ^ { 2 3 } & w ^ { 4 4 } & w ^ { 7 9 } \\\\ w ^ { 8 8 } & w ^ { 1 0 3 } & w ^ { 1 1 0 } & w ^ { 9 7 } & w ^ { 2 1 } & w ^ { 5 1 } & w ^ { 4 7 } & w ^ { 7 0 } \\end{pmatrix} . \\end{align*}"}
+{"id": "4036.png", "formula": "\\begin{align*} a _ I : = \\sum _ { w \\in W _ I } w \\textrm { a n d } b _ I : = \\sum _ { w \\in W _ I } \\epsilon ( w ) \\ , w \\ , , \\end{align*}"}
+{"id": "3322.png", "formula": "\\begin{align*} \\dim K = | \\sigma ^ 1 | - 1 > | \\tau ^ 1 | - 1 = \\dim L , \\end{align*}"}
+{"id": "4152.png", "formula": "\\begin{align*} F ( x ' , t ) : = \\Phi _ t * f ( x ' ) \\quad H ( x ' , t ) : = \\widetilde { \\Phi } _ t * h ( x ' ) , ( x ' , t ) \\in \\R ^ n _ + . \\end{align*}"}
+{"id": "1517.png", "formula": "\\begin{align*} \\frac { F ( x ) G ( y ) - F ( y ) G ( x ) } { x - y } = \\sum _ { i , j = 1 } ^ { m } b _ { i j } x ^ { i - 1 } y ^ { j - 1 } , \\end{align*}"}
+{"id": "115.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } \\partial _ t u - D ^ { 4 } _ x \\partial _ { x } u - \\partial _ { x } ^ { - 1 } \\partial _ y ^ 2 u & = u \\partial _ x u , ( t , x , y ) \\in \\R \\times \\R \\times \\R , \\\\ u ( 0 ) & = u _ 0 \\in H ^ { s _ 1 , s _ 2 } ( \\R ^ 2 ) \\end{array} \\right . \\end{align*}"}
+{"id": "4098.png", "formula": "\\begin{align*} P ^ \\infty & = \\varprojlim P ^ r = \\left \\{ u = ( u ^ r ) _ { r \\in \\mathbb { N } } \\in \\prod _ { r \\in \\mathbb { N } } P ^ r \\ ; \\middle | \\ ; \\pi ^ r _ s u ^ r = u ^ s \\ \\right \\} , \\\\ * \\widetilde { P } ^ \\infty & = \\varprojlim \\widetilde { P } ^ r , \\\\ * G ^ \\infty & = P ^ \\infty ( \\mathbb { R } ^ n , o ) , \\\\ * \\widetilde { G } ^ \\infty & = \\widetilde { P } ^ \\infty ( \\mathbb { R } ^ n , o ) , \\end{align*}"}
+{"id": "5473.png", "formula": "\\begin{align*} \\min \\{ c _ { d , 2 } ( q ) , c _ { d , \\infty } ( q ) \\} = \\begin{cases} c _ { d , 2 } ( q ) , & - ( d - 1 ) < q \\leq q _ d ^ * , \\\\ c _ { d , \\infty } ( q ) , & q _ d ^ * \\leq q \\leq 2 , \\end{cases} \\end{align*}"}
+{"id": "7508.png", "formula": "\\begin{align*} \\| N v \\| _ { L ^ 1 ( \\partial B _ { \\rho } ) } & = \\| ( N v ) ^ { - } \\| _ { L ^ 1 ( \\partial B _ { \\rho } ) } + \\| ( N v ) ^ { + } \\| _ { L ^ 1 ( \\partial B _ { \\rho } ) } \\leq 2 \\| ( N v ) ^ { - } \\| _ { L ^ 1 ( \\partial B _ { \\rho } ) } + \\varepsilon \\| \\nabla u \\| _ { L ^ { 1 } ( B _ { \\rho } ) } \\\\ & \\leq 2 \\| N u \\| _ { L ^ 1 ( \\partial B _ { \\rho } ) } + \\varepsilon \\| \\nabla u \\| _ { L ^ { 1 } ( B _ { \\rho } ) } . \\end{align*}"}
+{"id": "983.png", "formula": "\\begin{align*} | W ' | q ^ { d ^ 2 } \\theta _ { d - 1 } \\le | W ' | q ^ { d ^ 2 - 1 } \\theta _ d = | M ' | \\le d _ 0 + d _ 1 + d _ 2 + d _ 3 , \\end{align*}"}
+{"id": "2275.png", "formula": "\\begin{align*} a ( Z ) = a \\big ( U ( Z ^ * Z ) ^ \\frac { 1 } { 2 } \\big ) = a \\big ( ( Z ^ * Z ) ^ \\frac { 1 } { 2 } \\big ) , \\end{align*}"}
+{"id": "3842.png", "formula": "\\begin{align*} X _ { n + 1 } = X _ n + F _ n ( X _ n , Y _ { n + 1 } ) + G ^ 0 _ n ( X _ n , Y _ { n + 1 } ) \\xi _ { n + 1 } . \\end{align*}"}
+{"id": "4363.png", "formula": "\\begin{align*} \\int _ K | | \\tau _ 0 | | _ { \\omega _ { \\epsilon _ 1 } } \\le & \\liminf _ { \\epsilon \\to 0 } \\int _ K | | \\tau _ { \\epsilon } | | _ { \\omega _ { \\epsilon _ 1 } } \\\\ \\le & \\liminf _ { \\epsilon \\to 0 } \\int _ K | | \\tau _ { \\epsilon } | | _ { \\omega _ { \\epsilon } } \\\\ \\le & \\liminf _ { \\epsilon \\to 0 } \\int _ M | | \\tau _ { \\epsilon } | | _ { \\omega _ { \\epsilon } } . \\end{align*}"}
+{"id": "2854.png", "formula": "\\begin{align*} { \\bf X } ' _ \\ell ( t ) = e ^ { - A ( t + \\ell \\theta ) } { \\bf X } ' + \\int _ { - \\ell \\theta } ^ t e ^ { - A ( t - s ) } \\Sigma \\Big ( { \\bf p } _ \\ell ( s - ) \\Big ) \\dd M ( s ) , t \\ge - \\ell \\theta . \\end{align*}"}
+{"id": "6378.png", "formula": "\\begin{align*} \\int ^ \\infty _ 0 x ^ { - 3 \\alpha } \\exp \\left \\{ - ( a \\alpha + n ) \\frac { \\theta } { x ^ 2 } \\right \\} \\mathrm { d } x = & \\int ^ \\infty _ 0 u ^ { \\frac { 3 ( \\alpha - 1 ) } { 2 } } \\exp \\left \\{ - ( a \\alpha + j ) u \\right \\} \\mathrm { d } u \\\\ = & \\left ( \\frac { 1 } { a \\alpha + n } \\right ) ^ { \\frac { 3 \\alpha - 1 } { 2 } } \\Gamma \\left ( \\frac { 3 \\alpha - 1 } { 2 } \\right ) . \\end{align*}"}
+{"id": "5086.png", "formula": "\\begin{align*} & < u _ t , \\xi > + \\int _ { \\Omega } ( u \\cdot \\nabla u - B \\cdot \\nabla B ) \\cdot \\xi \\dd x + \\nu \\int _ { \\Omega } \\nabla u : \\nabla \\xi \\dd x = 0 , \\\\ & < B _ t , \\zeta > + \\int _ { \\Omega } ( B \\times u ) \\cdot \\nabla \\times \\zeta \\dd x + \\mu \\int _ { \\Omega } \\nabla \\times B \\cdot \\nabla \\times \\zeta \\dd x = 0 , \\end{align*}"}
+{"id": "9067.png", "formula": "\\begin{align*} ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + r ( j - 1 ) < \\chi _ j < ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + r j , \\end{align*}"}
+{"id": "8309.png", "formula": "\\begin{align*} R _ c = \\pi \\sqrt { 1 6 . 6 7 / 0 . 6 5 } \\approx 1 5 . 9 \\ , . \\end{align*}"}
+{"id": "2220.png", "formula": "\\begin{align*} \\mathbf { a } ( \\theta _ { 0 } ) ^ { T } \\mathbf { v } ^ { 0 } = e ^ { - j m \\phi } + e ^ { - j n \\phi } = e ^ { - j m \\phi } ( 1 + e ^ { - j ( n - m ) \\phi } ) \\end{align*}"}
+{"id": "6794.png", "formula": "\\begin{align*} F ^ { \\pm } ( x ) = \\sum _ { k = 1 } ^ { N } \\alpha _ { k } ^ { \\pm } e ^ { \\mp \\tau _ { k } x } + \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } s ^ { \\pm } \\left ( \\rho \\right ) e ^ { \\pm i \\rho x } d \\rho . \\end{align*}"}
+{"id": "6718.png", "formula": "\\begin{align*} \\ell _ \\beta ( \\langle \\gamma \\rangle ) = \\beta ( \\gamma , \\gamma ^ + ) \\end{align*}"}
+{"id": "8437.png", "formula": "\\begin{align*} & k ^ { - 1 } b ( k ) = - \\int _ { - \\infty } ^ { \\infty } \\bar { u } _ y ( y ) \\psi ^ - _ { 1 1 } ( y ; k ) e ^ { - 2 i z y } \\mathrm { d } y , \\\\ & \\psi ^ - _ { 1 1 } ( x ; k ) = 1 - k \\int _ { - \\infty } ^ { x } u _ y ( y ) \\psi ^ - _ { 2 1 } ( y ; k ) \\mathrm { d } y , \\\\ & \\psi ^ - _ { 2 1 } ( x ; k ) = k \\int _ { - \\infty } ^ { x } \\bar { u } _ y ( y ) \\psi ^ - _ { 1 1 } ( y ; k ) e ^ { 2 i z ( x - y ) } \\mathrm { d } y . \\end{align*}"}
+{"id": "7190.png", "formula": "\\begin{align*} d _ 1 = \\begin{pmatrix} 0 & i \\mu ( g ^ { \\alpha \\beta } \\xi _ \\beta ) & 0 \\\\ i \\lambda ( \\xi _ \\beta ) & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , d _ 0 = \\begin{pmatrix} 0 & 0 & 0 \\\\ \\lambda ( \\Gamma ^ \\alpha _ { \\alpha \\beta } ) & \\lambda \\Gamma ^ \\alpha _ { \\alpha n } & - \\beta \\\\ 0 & 0 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "4469.png", "formula": "\\begin{align*} - 4 a \\Delta ^ { 2 } P ( - \\xi , - \\eta ) = 4 a ( 4 a c - b ^ { 2 } ) ( ( b ^ { 2 } - 4 a c ) f + a e ^ { 2 } - b d e + c d ^ { 2 } ) . \\end{align*}"}
+{"id": "19.png", "formula": "\\begin{align*} \\overline { \\mathbf { p } } ^ { \\oplus } = ( \\overline { T _ { \\mathbf { p } } } ^ { \\oplus } ) ^ { \\tau } T _ { \\mathbf { p } } = \\bigcup _ { \\alpha ^ { \\tau } \\in \\mathbf { p } } \\{ \\gamma \\in \\alpha ^ { \\tau } \\} \\subset R ^ - _ { { \\tt A } _ { 2 n - 1 } } . \\end{align*}"}
+{"id": "8683.png", "formula": "\\begin{align*} \\rho ( t ) = \\begin{cases} 0 , & t = 0 , \\\\ \\exp ( - 1 / t ) , & 0 < t < T , \\end{cases} \\end{align*}"}
+{"id": "414.png", "formula": "\\begin{align*} \\pi _ N ( m \\ell n ) = m C _ \\ell ( n ) . \\end{align*}"}
+{"id": "8293.png", "formula": "\\begin{align*} \\gamma ^ i ( x ) = \\frac 1 N \\sum _ { j = 1 } ^ N K ( x ^ i - x ^ j ) \\end{align*}"}
+{"id": "5375.png", "formula": "\\begin{align*} 0 = H _ i = ( \\Delta u ) _ i + B x _ i \\end{align*}"}
+{"id": "7144.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { L } _ g \\textbf { \\textit { U } } = 0 & \\ \\Omega , \\\\ \\textbf { \\textit { U } } = \\textbf { \\textit { V } } & \\ \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "6038.png", "formula": "\\begin{align*} I ( \\widetilde { u } _ { k } , \\widetilde { v } _ { k } ) = I ( \\overline { u } _ { k } , \\overline { v } _ { k } ) + o _ { k } ( 1 ) \\geq \\widetilde { E } _ { 1 } + \\widetilde { E } _ { \\omega } + o _ { k } ( 1 ) . \\end{align*}"}
+{"id": "1969.png", "formula": "\\begin{align*} & V _ n V _ { n ( d - 1 ) } \\int _ 0 ^ { a R } ( R ^ 2 - r ^ 2 ) ^ { n ( d - 1 ) / 2 } r ^ { n - 1 } d r \\\\ & = V _ n V _ { n ( d - 1 ) } R ^ { n d } \\int _ 0 ^ { a R } ( 1 - \\frac { r ^ 2 } { R ^ 2 } ) ^ { n ( d - 1 ) / 2 } \\frac { r ^ { n - 1 } } { R ^ { n } } d r \\\\ & \\leq \\frac { V _ n V _ { n ( d - 1 ) } } { n } a ^ n R ^ { n d } \\\\ & \\leq \\frac { V _ n V _ { n ( d - 1 ) } } { n V _ { n d } } a ^ n \\alpha ^ n ( B ) . \\end{align*}"}
+{"id": "3760.png", "formula": "\\begin{align*} \\# R ^ i _ { n _ 0 ; l _ 1 , \\cdots , l _ { n _ 0 } } = q ^ { n _ 0 - 1 } \\# R ^ i _ { n _ 0 ; l _ 1 - 1 , \\cdots , l _ { n _ 0 } } . \\end{align*}"}
+{"id": "1268.png", "formula": "\\begin{align*} 0 = \\int _ \\Omega \\varphi _ n ( \\eta ) \\mu ( d \\eta ) = \\sum _ { m = 0 } ^ \\infty \\int _ \\Omega ( \\varphi _ n ( r _ m \\eta ) - \\varphi _ n ( r _ { m + 1 } \\eta ) ) \\mu ( d \\eta ) + \\varphi _ n ( \\mathbf { 1 } ) , \\end{align*}"}
+{"id": "8863.png", "formula": "\\begin{align*} \\Sigma ( k ) = \\theta \\big ( \\chi ( 2 k + 1 ) + 1 + \\lceil \\ln ( 2 L ( k + 1 ) ) \\rceil \\big ) + 1 . \\end{align*}"}
+{"id": "8086.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\mathbf { y ' } ^ H \\mathbf { s } ^ { \\left ( \\right ) } \\right ] = a _ c \\sum _ { j = 1 } ^ { M } \\left ( \\mathbf { h } _ { l , * } \\mathbf { p } _ c \\right ) ^ { * } + \\sum _ { l = 1 } ^ M a _ l \\left ( \\mathbf { h } _ { l , * } \\mathbf { p } _ l \\right ) ^ { * } . \\end{align*}"}
+{"id": "5135.png", "formula": "\\begin{align*} j ( \\tau , s ( \\tau ) ) = h . \\end{align*}"}
+{"id": "6006.png", "formula": "\\begin{align*} - \\Delta u + \\omega u + \\big ( \\int _ { | x | } ^ { \\infty } \\frac { h ( s ) } { s } u ^ { 2 } ( s ) d s + \\frac { h ^ { 2 } ( | x | ) } { | x | ^ { 2 } } \\big ) u = | u | ^ { 2 p - 2 } u , \\ x \\in \\R ^ { 2 } , \\end{align*}"}
+{"id": "1303.png", "formula": "\\begin{align*} \\phi = \\alpha + \\beta - \\gamma \\end{align*}"}
+{"id": "3040.png", "formula": "\\begin{align*} d g = \\mathcal { L } _ { g } = _ { } \\mathcal { I } \\circ L _ { g } \\circ \\mathcal { I } ^ { - 1 } : S O \\left ( M _ { \\kappa } \\right ) \\rightarrow S O \\left ( M _ { \\kappa } \\right ) \\end{align*}"}
+{"id": "5896.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\int _ 0 ^ T \\Vert X _ n ( t ) - X ( t ) \\Vert _ H ^ { \\kappa } d t = 0 , \\forall \\ \\kappa \\in [ 1 , \\alpha ) . \\end{align*}"}
+{"id": "2488.png", "formula": "\\begin{align*} \\lambda _ { \\min } ( ( \\mathbf { v } ) ) = \\mathbf { v } _ 1 - \\| \\mathbf { v } _ { 2 : n } \\| , \\lambda _ { \\max } ( ( \\mathbf { v } ) ) = \\mathbf { v } _ 1 + \\| \\mathbf { v } _ { 2 : n } \\| . \\end{align*}"}
+{"id": "6916.png", "formula": "\\begin{align*} G _ 1 ( Q _ 1 , \\dots , Q _ n ) \\coloneqq \\int _ { \\R ^ n } f ( x _ 1 , \\dots , x _ n ) \\prod _ { i = 1 } ^ { n } Q _ i ( d x _ i ) , G _ 2 ( Q _ 1 , \\dots , Q _ n ) \\coloneqq H ( Q _ 1 \\times \\cdots \\times Q _ n ) . \\end{align*}"}
+{"id": "5147.png", "formula": "\\begin{align*} I _ h = \\inf \\left \\{ E [ b ] \\ \\middle | \\ b \\in \\tilde { T } _ h \\right \\} . \\end{align*}"}
+{"id": "153.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } A = f ( B ) ; \\\\ B = f ( A ) . \\end{array} \\right . \\end{align*}"}
+{"id": "8358.png", "formula": "\\begin{align*} \\mathcal { C } ( h ) ( z ) : = \\frac { 1 } { 2 \\pi i } \\int _ { \\mathbb { R } } \\frac { h ( s ) } { s - z } d s , z \\in \\mathbb { C } \\backslash \\mathbb { R } . \\end{align*}"}
+{"id": "9008.png", "formula": "\\begin{align*} s = \\max \\{ q : j _ { r } = \\ell _ { r } r \\in [ q ] \\} . \\end{align*}"}
+{"id": "5009.png", "formula": "\\begin{align*} S t _ { D N P } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( 1 , z ) - 1 | | _ { \\infty } = 0 , \\end{align*}"}
+{"id": "3796.png", "formula": "\\begin{align*} B _ q \\circ T ( f ) = f . \\end{align*}"}
+{"id": "6733.png", "formula": "\\begin{align*} T ( 3 , a , x ) = & G _ { 2 } ^ 3 \\ , { } _ 3 ^ 0 \\left ( x \\left | \\begin{matrix} 0 , 0 \\\\ - 1 , a - 1 , - 1 \\end{matrix} \\right . \\right ) \\\\ = & \\frac { x ^ { a - 1 } { } _ 2 \\mathrm { F } _ 2 \\left ( a , a ; 1 + a , 1 + a ; - x \\right ) } { a ^ 2 } - \\frac { \\Gamma ( a ) \\log ( x ) } { x } + \\frac { \\Gamma ( a ) \\psi ( a ) } { x } , \\end{align*}"}
+{"id": "3350.png", "formula": "\\begin{align*} V ( G [ H ; U ] ) & = V ( G ) \\times V ( H ) , \\\\ E ( G [ H ; U ] ) & = \\left \\{ ( u _ 1 , v _ 1 ) ( u _ 2 , v _ 2 ) \\ \\middle | \\ \\left . \\begin{aligned} & u _ 1 = u _ 2 v _ 1 v _ 2 \\in E ( H ) , \\\\ & \\\\ & u _ 1 u _ 2 \\in E ( G ) v _ 1 , v _ 2 \\in U \\end{aligned} \\right . \\right \\} . \\end{align*}"}
+{"id": "5868.png", "formula": "\\begin{align*} \\Gamma \\left ( X , Y , Z \\right ) \\equiv \\left ( \\nabla _ { Z } G \\right ) \\left ( X , Y \\right ) = Z G \\left ( X , Y \\right ) - G \\left ( \\nabla _ { Z } X , Y \\right ) - G \\left ( X , \\nabla _ { Z } Y \\right ) . \\end{align*}"}
+{"id": "6332.png", "formula": "\\begin{align*} z - z ^ { q ^ 4 } + ( y - y ^ { q ^ 4 } ) x ^ { q ^ 3 } + ( x - x ^ { q ^ 4 } ) x ^ { 2 q ^ 3 } = 0 . \\end{align*}"}
+{"id": "4771.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| p _ n x \\| _ { X _ n } = \\| x \\| _ X , \\ , x \\in X \\lim _ { n \\to \\infty } \\| q _ n y \\| _ { Y _ n } = \\| y \\| _ Y , \\ , y \\in Y . \\end{align*}"}
+{"id": "3863.png", "formula": "\\begin{align*} \\begin{pmatrix} v _ 1 \\\\ v _ 2 \\end{pmatrix} = - \\frac { 1 } { k ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 } \\begin{pmatrix} x _ 1 x _ 2 & - k ^ 2 - x _ 1 ^ 2 \\\\ k ^ 2 + x _ 2 ^ 2 & - x _ 1 x _ 2 \\end{pmatrix} \\begin{pmatrix} \\partial _ { x _ 1 } \\varphi \\\\ \\partial _ { x _ 2 } \\varphi \\end{pmatrix} . \\end{align*}"}
+{"id": "7672.png", "formula": "\\begin{align*} \\mu ^ { N , i } _ { \\boldsymbol { \\alpha } ^ * _ t } = \\rho _ i ^ N ( \\Delta _ { * , t } ^ { N , 1 } , \\ldots , \\Delta _ { * , t } ^ { N , N } ) , \\end{align*}"}
+{"id": "5319.png", "formula": "\\begin{align*} \\varphi _ { k , \\lambda } ^ { n - 1 } ( z ) = ( n - 1 ) ! 2 ^ { n - 1 } \\frac { J _ { n - 1 } ( \\sqrt { ( 2 k + n ) | \\lambda | } \\ , | z | ) } { ( \\sqrt { ( 2 k + n ) | \\lambda | } \\ , | z | ) ^ { n - 1 } } + m _ k ( \\sqrt { | \\lambda | } | z | ) \\end{align*}"}
+{"id": "2130.png", "formula": "\\begin{align*} \\varrho ( z , w ) : = - 1 + | z | ^ 2 + | w | ^ 2 + 2 \\Re ( f ( z , w ) ) = 0 , \\end{align*}"}
+{"id": "388.png", "formula": "\\begin{align*} S ^ { ' } u _ k = ( 1 - 1 + 3 - 3 \\cdots + 3 - 3 ) \\ 1 _ { n - 1 } = \\ 0 . \\end{align*}"}
+{"id": "2665.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) = \\mathcal { P } ^ * ( u _ t ( t ) ) + \\mathcal { A } ( u ( t ) ) , \\end{align*}"}
+{"id": "4137.png", "formula": "\\begin{align*} L K ^ L _ { \\cdot \\beta } = 0 \\ , \\ , \\ , \\ , \\R ^ n _ { + } \\beta \\in \\{ 1 , \\dots , N \\} , \\end{align*}"}
+{"id": "3011.png", "formula": "\\begin{align*} \\varphi ( \\tau , w ( - 0 ) , w ( \\cdot ) ) = \\hat { \\varphi } ( \\tau , w ( \\cdot ) ) , ( \\tau , w ( \\cdot ) ) \\in [ 0 , \\vartheta ] \\times \\mathrm { L i p } . \\end{align*}"}
+{"id": "5904.png", "formula": "\\begin{align*} \\lim _ { \\substack { n \\rightarrow \\infty \\\\ n \\in \\Lambda } } X _ n ( t , \\omega ) = X ( t , \\omega ) , \\end{align*}"}
+{"id": "3448.png", "formula": "\\begin{align*} \\tilde { m } _ n = \\begin{cases} m _ n - q _ { n - 1 } m _ n + q _ n - q _ { n - 1 } , \\ell _ n = 2 \\\\ m _ n + q _ { n - 1 } m _ n - q _ n + q _ { n - 1 } , \\ell _ n = - 2 \\end{cases} . \\end{align*}"}
+{"id": "3616.png", "formula": "\\begin{align*} R _ { i j k l } = - ( \\delta _ { i k } \\delta _ { j l } - \\delta _ { i l } \\delta _ { j k } ) + h _ { i k } h _ { j l } - h _ { i l } h _ { j k } , \\end{align*}"}
+{"id": "2631.png", "formula": "\\begin{align*} & \\circlearrowleft _ { x , y , z } \\varepsilon ( z , x ) \\Big ( 2 \\varepsilon ( h , x + y ) [ \\alpha ( x ) , \\alpha ( y ) ] \\cdot [ \\alpha ( h ) , \\alpha ( z ) ] \\Big ) \\\\ & = \\circlearrowleft _ { x , y , z } \\varepsilon ( z , x ) \\Big ( [ \\alpha ( h ) \\cdot [ x , y ] , \\alpha ^ 2 ( z ) ] + \\varepsilon ( x + y , z ) [ \\alpha ^ 2 ( h ) , \\alpha ( z ) \\cdot [ x , y ] . ] \\Big ) . \\end{align*}"}
+{"id": "7206.png", "formula": "\\begin{align*} \\| u ( t ) \\| _ { L ^ 2 } ^ 2 + 2 \\int _ 0 ^ t \\| \\nabla u ( s ) \\| _ { L ^ p } ^ p \\ , d s = \\| u _ 0 \\| _ { L ^ 2 } ^ 2 ; \\end{align*}"}
+{"id": "7541.png", "formula": "\\begin{align*} f _ k ( d ) = \\lim _ { n \\rightarrow \\infty } \\frac { \\vartheta ( n , d , k ) } { \\log n } , \\end{align*}"}
+{"id": "5715.png", "formula": "\\begin{align*} \\phi _ { f ' } ( x ) = \\alpha ( v x + w ) ^ { q + 1 } ( \\phi _ f \\circ \\mu ( x ) ) , \\end{align*}"}
+{"id": "8259.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\sum _ { j = 1 } ^ { \\infty } V _ { i , j } \\psi _ j ( t ) d t < \\infty , \\end{align*}"}
+{"id": "8705.png", "formula": "\\begin{align*} P _ 1 ( T ) = T ^ 2 + 2 T - 1 , P _ 2 ( T ) = T ^ 6 - 5 T ^ 5 - 3 T ^ 4 + 4 3 T ^ 3 - 3 3 T ^ 2 - 5 9 T + 4 3 ) . \\end{align*}"}
+{"id": "5841.png", "formula": "\\begin{align*} { \\psi _ { x , y , - 1 } } = { \\psi _ { x , y , 1 } } + \\sqrt { { { \\left ( { { \\psi _ { x + 1 , y , 0 } } - { \\psi _ { x - 1 , y , 0 } } } \\right ) } ^ 2 } + { { \\left ( { { \\psi _ { x , y + 1 , 0 } } - { \\psi _ { x , y - 1 , 0 } } } \\right ) } ^ 2 } } \\tan \\left ( { \\frac { \\pi } { 2 } - \\theta } \\right ) , \\end{align*}"}
+{"id": "2599.png", "formula": "\\begin{align*} K ( p , \\tilde { \\pi } ' ) = & K ( p , \\pi ) + \\varepsilon Q ( g , R ) ( \\pi , \\bar { \\pi } ) + \\varepsilon Q ( g , R ) ( \\pi , J \\bar { \\pi } ) + O ( \\varepsilon ^ 2 ) \\\\ = & K ( p , \\pi ) + \\varepsilon Q ^ c ( g , R ) ( \\pi ; \\bar { \\pi } ) + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
+{"id": "1348.png", "formula": "\\begin{align*} s = \\frac { d ( a _ 1 , b _ 1 ) - 2 R } { 2 ( K - 2 ) } . \\end{align*}"}
+{"id": "3033.png", "formula": "\\begin{align*} \\phi : \\mathbb { R } ^ { 3 } \\cong \\operatorname { I m } \\mathbb { H } \\rightarrow \\mathbb { R } ^ { 3 } \\rtimes S ^ { 3 } \\phi \\left ( r u \\right ) = \\left ( \\ell \\left ( r \\right ) u , \\exp \\left ( r u / 2 \\right ) \\right ) \\end{align*}"}
+{"id": "7428.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k } \\dbinom { k } { j } ^ 2 a ^ { p _ k ( k - j ) / k } b ^ { p _ k j / k } \\leq ( a + b ) ^ { p _ k } , \\end{align*}"}
+{"id": "2030.png", "formula": "\\begin{align*} | C \\widehat p _ t + D \\widehat q _ t | ^ 2 = 0 , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "7547.png", "formula": "\\begin{align*} \\boldsymbol { \\lambda } _ k = \\left ( \\langle y _ k | A _ { 1 , k } y _ k \\rangle , \\dots , \\langle y _ k | A _ { d , k } y _ k \\rangle \\right ) , \\end{align*}"}
+{"id": "7718.png", "formula": "\\begin{align*} \\hat M ^ { ( k ) } = G _ k G _ { k - 1 } \\cdots G _ 1 ( M + \\tilde E ^ { ( k ) } ) , \\end{align*}"}
+{"id": "4436.png", "formula": "\\begin{align*} \\Delta [ M _ v ] ( t ) = ( \\Delta M _ v ( t ) ) ^ 2 = e ^ { - 2 \\lambda t } ( v A ^ * _ k ) ^ 2 = e ^ { - 2 \\lambda t } \\lambda ^ 2 v _ { k } ^ 2 , \\end{align*}"}
+{"id": "9056.png", "formula": "\\begin{align*} \\lambda _ s = \\sum _ { a = m - s + 1 } ^ { m } i _ a + \\frac { L } { 2 } \\medspace ( 1 \\leq s \\leq m ) . \\end{align*}"}
+{"id": "7321.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n { ( 1 \\otimes a _ i \\otimes a _ i + a _ i \\otimes 1 \\otimes a _ i + a _ i \\otimes a _ i \\otimes 1 ) } + 1 \\otimes 1 \\otimes y + 1 \\otimes y \\otimes 1 + y \\otimes 1 \\otimes 1 . \\end{align*}"}
+{"id": "7916.png", "formula": "\\begin{align*} \\iota ( \\upsilon ) = \\frac { - 1 } { 2 \\pi } \\sin ( 2 \\pi \\upsilon ) + \\frac { - \\sin ( 2 \\pi \\upsilon ) + 4 \\pi \\upsilon - \\sin ( 4 \\pi \\upsilon ) + \\frac { 1 } { 3 } \\sin ( 6 \\pi \\upsilon ) } { 8 \\left ( - \\sin ( 2 \\pi \\upsilon ) + \\frac 1 3 \\sin ( 6 \\pi \\upsilon ) - ( 2 \\pi \\upsilon + \\frac 1 2 \\sin ( 4 \\pi \\upsilon ) - 2 \\sin ( 2 \\pi \\upsilon ) ) \\right ) } \\left ( - 4 \\upsilon + \\frac { 1 } { \\pi } \\sin ( 4 \\pi \\upsilon ) \\right ) \\end{align*}"}
+{"id": "819.png", "formula": "\\begin{align*} \\partial _ t h _ { i j } = g \\Delta h _ { i j } + \\sum _ { k , l } g ^ { k l } \\nabla _ i g \\nabla _ j h _ { k l } + \\end{align*}"}
+{"id": "6743.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left \\{ 1 - F _ \\Gamma \\left [ \\left ( - \\frac { x - \\mu } { \\sigma } \\right ) ^ s \\right ] \\right \\} = u . \\end{align*}"}
+{"id": "2805.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { 1 6 ^ { m } } \\mathbb { E } [ \\tilde { X } _ { j } 1 _ { \\{ X _ { j } > \\varepsilon \\} } | \\Phi ^ { S _ { n } , \\tilde { S } _ { n , m } } ] \\to 0 , m \\to \\infty \\end{align*}"}
+{"id": "1710.png", "formula": "\\begin{align*} | | u | | ^ 2 _ { \\widetilde { X } _ { [ 0 , T ] } } & = | | \\langle \\xi \\rangle ^ { - 1 } u | | ^ 2 _ { L ^ 2 ( [ 0 , T ] \\times \\mathbb { R } ^ d ) } + \\sum _ { k = - \\infty } ^ { \\infty } 2 ^ k | | S _ k u | | ^ 2 _ { X _ k ( T ) } , \\ d \\not = 2 , \\\\ | | u | | ^ 2 _ { \\widetilde { X } _ { [ 0 , T ] } } & = | | \\langle \\xi \\rangle ^ { - 1 } ( \\log ( 2 + | \\xi | ) ) ^ { - 1 } u | | ^ 2 _ { L ^ 2 ( [ 0 , T ] \\times \\mathbb { R } ^ d ) } + \\sum _ { k = - \\infty } ^ { \\infty } 2 ^ k | | S _ k u | | ^ 2 _ { X _ k ( T ) } , \\ d = 2 . \\end{align*}"}
+{"id": "750.png", "formula": "\\begin{align*} I _ { 2 , Y } & : = \\int _ { r \\in Y , | t | / 2 \\leq | r | \\leq 2 | t | } \\frac { | F _ s ( r ) - F _ s ( t ) | } { | r - t | ^ { 2 - \\frac 1 { 2 s } } } d r \\\\ & \\lesssim \\frac 1 { ( 1 + | t | ^ { \\frac 1 s } ) ^ { \\frac { N + 2 s } 2 } } \\int _ { | r | \\leq 2 | t | } \\frac { d r } { | r - t | ^ { 1 - \\frac 1 { 2 s } } } \\\\ & \\lesssim \\frac { | t | ^ { \\frac 1 { 2 s } } } { ( 1 + | t | ^ { \\frac 1 s } ) ^ { \\frac { N + 2 s } 2 } } \\lesssim \\min \\bigg ( 1 , \\frac 1 { | t | } \\bigg ) . \\end{align*}"}
+{"id": "8792.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } \\tilde { X } _ t = B ( \\tilde { X } _ t , \\tilde { X } _ t ) \\\\ \\tilde { X } _ t | _ { t = 0 } = \\Pi _ { \\mathrm { k e r } A } x _ 0 . \\end{cases} \\end{align*}"}
+{"id": "2004.png", "formula": "\\begin{align*} 0 & = F ^ { ( n + 1 ) } ( x ) + F ^ { ( n + 1 ) } ( x + F ( x ) ) ( 1 + F ^ { \\prime } ( x ) ) ^ { n + 1 } \\\\ & \\qquad + n F ^ { ( n ) } ( x + F ( x ) ) ( 1 + F ^ { \\prime } ( x ) ) ^ { n - 1 } F ^ { \\prime \\prime } ( x ) \\\\ & \\qquad + \\sum _ { j = 1 } ^ { n - 1 } F ^ { ( j + 1 ) } ( x + F ( x ) ) ( 1 + F ^ { \\prime } ( x ) ) G _ { j } ^ { n } ( x ) + \\sum _ { j = 1 } ^ { n - 1 } F ^ { ( j ) } ( x + F ( x ) ) ( G _ { j } ^ { n } ) ^ { \\prime } ( x ) \\end{align*}"}
+{"id": "5238.png", "formula": "\\begin{align*} S _ { h _ C , W , 0 } = \\left \\{ U _ { C } ( \\cdot + z e _ z ) - B _ { \\infty } \\ \\big | \\ z \\in \\mathbb { R } \\ \\right \\} . \\end{align*}"}
+{"id": "8967.png", "formula": "\\begin{align*} \\partial _ { i } [ \\tau ] = \\sum _ { \\hat { \\imath } = 1 } ^ { i + 1 } ( - 1 ) ^ { \\hat { \\imath } - 1 } [ \\tau \\setminus v _ { j _ { \\hat { \\imath } } } ] . \\end{align*}"}
+{"id": "6980.png", "formula": "\\begin{align*} \\rho ^ { k _ 1 } = ( a / b ) ^ { x k _ 1 } ( \\alpha / \\beta ) ^ { y k _ 1 } = ( \\alpha / \\beta ) ^ { x l _ 1 } \\zeta ^ x ( \\alpha / \\beta ) ^ { y k _ 1 } = ( \\alpha / \\beta ) \\zeta ^ x . \\end{align*}"}
+{"id": "7886.png", "formula": "\\begin{align*} \\dot \\theta _ k = \\frac 1 M \\sum _ { j = 1 } ^ M b _ { k - j } \\sin ( \\theta _ j - \\theta _ k ) - \\frac 1 M \\sum _ { j = 1 } ^ M b _ { 1 - j } \\sin ( \\theta _ j ) , \\end{align*}"}
+{"id": "5517.png", "formula": "\\begin{align*} x = a _ 2 ^ 2 + \\dots + a _ n ^ 2 + a _ { n + 1 } ^ 2 \\end{align*}"}
+{"id": "1780.png", "formula": "\\begin{align*} 0 \\equiv \\hat { \\phi } ( \\hat { h } ) = \\hat { h } ( u ^ 2 , \\hat { g } ( u ^ 2 ) ) , \\end{align*}"}
+{"id": "454.png", "formula": "\\begin{align*} ( 1 - e ^ { - 2 } ) f ( x ) = ( 1 - e ^ { - 1 } ) e ^ { - 1 } ( \\underline { f } ( x ) + \\overline { f } ( x ) ) + ( 1 - e ^ { - 1 } ) ^ 2 \\underline { f } \\ast \\overline { f } ( x ) \\end{align*}"}
+{"id": "30.png", "formula": "\\begin{align*} - \\Delta _ { H } u = V u \\ ; \\ ; , \\end{align*}"}
+{"id": "7422.png", "formula": "\\begin{align*} \\psi ( p ) = \\psi _ { p _ 0 } [ \\eta ] ( p ) \\stackrel { d e f } { = } \\inf _ { r \\in R ( p ) } g ( p , r , | \\eta | _ r ) , \\ p > p _ 0 ; \\end{align*}"}
+{"id": "4496.png", "formula": "\\begin{align*} Z _ { V , [ \\alpha ] } ( t ) = \\frac { 1 } { 2 } \\int _ { V } ( t ^ { 2 } \\omega ^ { 2 } - \\alpha ^ { 2 } ) + \\sqrt { - 1 } t \\int _ { V } \\alpha \\wedge \\omega . \\end{align*}"}
+{"id": "318.png", "formula": "\\begin{align*} \\mathcal { T } ^ * ( f ) ( x ) = \\sup _ { t > 0 } | e ^ { - t L } f ( x ) | = \\sup _ { t > 0 } \\Big | \\int _ { \\mathbb { R } ^ n } k _ t ( x , y ) f ( y ) d y \\Big | , \\end{align*}"}
+{"id": "7376.png", "formula": "\\begin{gather*} P ( A ) = P _ 0 ( A ) = 0 \\quad A \\in \\mathcal { R } P ^ * ( A ) = 0 . \\end{gather*}"}
+{"id": "138.png", "formula": "\\begin{align*} \\| v \\| _ { F ^ { s , 0 } ( T ) } ^ 2 & \\lesssim \\| v ( 0 ) \\| _ { H ^ { s , 0 } } ^ 2 + \\| v \\| _ { F ^ { s , 0 } ( T ) } ^ 3 + \\| v \\| _ { F ^ { 0 , 0 } ( T ) } \\| v \\| _ { F ^ { s , 0 } ( T ) } \\| u _ 2 \\| _ { F ^ { 2 s , 0 } ( T ) } . \\end{align*}"}
+{"id": "3836.png", "formula": "\\begin{align*} u v = T _ u v + T _ v u + R ( u , v ) , \\end{align*}"}
+{"id": "3475.png", "formula": "\\begin{align*} \\prod _ { j = x _ 1 } ^ k | \\cos ( \\pi \\theta _ j ) | \\leq C ( \\varepsilon ) e ^ { - ( \\ln 2 - \\varepsilon ) | x _ 1 - k | } , \\prod _ { j = k } ^ { x _ 2 } | \\cos ( \\pi \\theta _ j ) | \\leq C ( \\varepsilon ) e ^ { - ( \\ln 2 - \\varepsilon ) | x _ 2 - k | } c _ { n , \\ell } c _ { n , \\ell + 1 } . \\end{align*}"}
+{"id": "804.png", "formula": "\\begin{align*} V ( t , p ) = \\partial _ t \\theta ( t , z ) \\cdot \\nu ( t , p ) = \\frac { R ' ( t ) } { R _ 0 } z \\cdot \\frac { p } { R ( t ) } = \\frac { R ' ( t ) } { R _ 0 } z \\cdot \\frac { z } { R _ 0 } = R ' ( t ) . \\end{align*}"}
+{"id": "7378.png", "formula": "\\begin{gather*} P ( f ) = P ( \\tilde { f } ) = \\int Q ( \\tilde { f } ) \\ , P _ \\mathcal { U } ( d Q ) = \\int Q ( f ) \\ , P _ \\mathcal { U } ( d Q ) = 0 . \\end{gather*}"}
+{"id": "3629.png", "formula": "\\begin{align*} 0 = \\frac { ( \\tilde { \\kappa } _ 1 ) _ i } { \\tilde { \\kappa } _ 1 } - N \\frac { ( \\nu ^ { n + 1 } ) _ i } { \\nu ^ { n + 1 } } , \\end{align*}"}
+{"id": "3266.png", "formula": "\\begin{align*} \\| p _ N \\mathcal S ( r ) \\sigma _ { s } \\| _ { L _ { } ( U , H ) } ^ 2 = \\| \\sigma _ { s } ^ * \\mathcal S ( r ) ^ * p _ N \\| _ { L _ { } ( H , U ) } ^ 2 = \\sum _ { k = N } ^ { \\infty } \\| \\sigma _ { s } ^ * \\mathcal S ( r ) ^ * e _ k \\| ^ 2 \\to 0 , N \\to \\infty , \\end{align*}"}
+{"id": "8183.png", "formula": "\\begin{align*} \\mathcal { Q } ^ { r } ( t ) : = \\mathcal { M } ^ { r } ( \\delta ) + ( r - 1 ) \\theta _ { 1 } \\varrho _ { 0 } ^ { 1 - \\sigma } \\int _ { \\delta } ^ { t } ( \\mathcal { M } ^ { r } ( s ) ) ^ { \\sigma + 1 } \\ d s . \\end{align*}"}
+{"id": "6018.png", "formula": "\\begin{align*} g ' ( t ) & = t \\| ( u , v ) \\| _ { E } ^ { 2 } + 3 t ^ { 5 } \\Big ( B ( u ) + B ( v ) \\Big ) - t ^ { 2 p - 1 } F ( u , v ) \\\\ & = t ^ { 2 p - 1 } \\Big ( \\frac { 1 } { t ^ { 2 p - 2 } } \\| ( u , v ) \\| _ { E } ^ { 2 } + \\frac { 3 } { t ^ { 2 p - 6 } } \\big ( B ( u ) + B ( v ) \\big ) - F ( u , v ) \\Big ) . \\end{align*}"}
+{"id": "6853.png", "formula": "\\begin{align*} J _ { 1 , n } ^ { \\prime } ( x ) = J _ { 1 , n - 1 } ^ { \\prime } ( x ) - e ^ { - \\frac { x } { 2 } } e ( \\frac { i } { 2 } , x ) a _ { n - 1 } ^ { \\prime } ( x ) , \\end{align*}"}
+{"id": "582.png", "formula": "\\begin{align*} A ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) = \\begin{pmatrix} M _ 1 ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) \\\\ M _ 2 ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) \\\\ \\vdots \\\\ M _ { 2 d } ( ( v _ { i , j } ) _ { 1 \\leq i < j \\leq 2 d } ) \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "3985.png", "formula": "\\begin{align*} \\mathbf { b } + \\mathbf { c } = ( \\beta _ 1 + \\gamma _ 1 , b _ 1 + \\gamma _ 2 , b _ 2 + c _ 1 , b _ 3 + c _ 2 , \\cdots , b _ { 2 k - 1 } + c _ { 2 k - 2 } , \\beta _ 2 + c _ { 2 k - 1 } , \\beta _ 3 + \\gamma _ 3 , \\beta _ 4 + \\gamma _ 4 ) \\end{align*}"}
+{"id": "6844.png", "formula": "\\begin{align*} q ( x ) = - \\frac { c } { 2 } \\operatorname * { s e c h } { } ^ { 2 } \\left ( \\frac { \\sqrt { c } x } { 2 } \\right ) . \\end{align*}"}
+{"id": "1984.png", "formula": "\\begin{align*} & - x - T ( x ) = - g ^ { - 1 } ( z ) - g ^ { - 1 } ( z ) + z = z - 2 g ^ { - 1 } ( z ) , \\end{align*}"}
+{"id": "8740.png", "formula": "\\begin{align*} L \\cong M / N \\cong \\bigoplus _ { i = 1 } ^ l M _ i / N _ i \\end{align*}"}
+{"id": "6122.png", "formula": "\\begin{align*} \\psi _ { n , p } & = \\Big [ { \\frac { n ( 2 j ^ { ( 1 ) } + 1 - n ) ( 2 j ^ { ( 2 ) } + 1 - n ) ( 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } + 2 - n ) ( N - n - p + 2 j ^ { ( 3 ) } + 2 ) ( N - n - p + 1 ) } { ( 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } + 2 - 2 n ) ^ 2 ( 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } + 1 - 2 n ) ( 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } + 3 - 2 n ) } } \\\\ & \\qquad \\times { ( p - n - N + 2 j ^ { ( 1 ) } + 2 j ^ { ( 2 ) } + 1 ) ( n - p - N + 2 j ^ { ( 0 ) } + 2 j ^ { ( 4 ) } ) } \\Big ] ^ { \\frac 1 2 } \\ , , \\end{align*}"}
+{"id": "1378.png", "formula": "\\begin{align*} Z ^ k _ j ( n ) = \\inf \\{ t \\geq 0 : M _ { d - j + 1 } ^ k ( t ) \\geq n \\} , n \\geq 0 . \\end{align*}"}
+{"id": "5649.png", "formula": "\\begin{align*} S t ( e _ { 1 } ) = \\left \\{ \\begin{pmatrix} 1 & c & ^ { t } u \\\\ 0 & 1 & 0 \\\\ 0 & x & B \\end{pmatrix} \\Bigg | \\ , \\ , u + \\ , ^ { t } B \\psi x = 0 ; \\ , \\ , 2 c + \\langle x , x \\rangle = 0 ; \\ , \\ , B \\in O _ { n - 1 , n - 1 } . \\right \\} . \\end{align*}"}
+{"id": "2374.png", "formula": "\\begin{align*} [ \\mathbf { h } ' _ 0 , \\mathbf { h } _ 0 ] = 1 . \\end{align*}"}
+{"id": "293.png", "formula": "\\begin{align*} u ^ K _ { n , h } ( p ) + H ( p , u ^ K _ { n , h } ( p ) , \\nabla _ H u ^ K _ { n , h } ( p ) ) = u ^ K _ { n - 1 } ( p ) \\end{align*}"}
+{"id": "4009.png", "formula": "\\begin{align*} l r ( e p o c h s ) = \\begin{cases} 1 0 ^ { - 3 } , \\textit { 1 $ \\leq $ e p o c h s $ < $ 2 , 0 0 0 } \\\\ 1 0 ^ { - 4 } , \\textit { e p o c h s $ \\geq $ 2 , 0 0 0 } . \\end{cases} \\end{align*}"}
+{"id": "1166.png", "formula": "\\begin{align*} H \\left ( P ^ { ( n , k ) } [ t ] \\mid \\mu ^ { \\otimes k } [ t ] \\right ) \\leq \\frac { k } { n } M , k = 1 , \\cdots , n , t \\in [ 0 , T _ M ] . \\end{align*}"}
+{"id": "7717.png", "formula": "\\begin{align*} \\hat M ^ { ( k ) } _ { i , j } & = \\begin{cases} 0 , & \\mbox { i f } i \\in [ k + 1 , n ] \\mbox { a n d } j = k , \\\\ { \\rm f l } \\Big ( \\hat M ^ { ( k - 1 ) } _ { i , j } - { \\rm f l } \\Big ( { \\rm f l } \\Big ( \\frac { \\hat M ^ { ( k - 1 ) } _ { i , k } } { \\hat M ^ { ( k - 1 ) } _ { k , k } } \\Big ) \\ , \\hat M ^ { ( k - 1 ) } _ { k , j } \\Big ) \\Big ) , & \\mbox { i f } i , j \\in [ k + 1 , n ] , \\\\ \\hat M ^ { ( k - 1 ) } _ { i , j } , & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
+{"id": "8048.png", "formula": "\\begin{align*} \\mathbf { y } ' = \\begin{bmatrix} y _ c \\\\ y _ 1 \\\\ \\vdots \\\\ y _ M \\end{bmatrix} = \\begin{bmatrix} \\sum _ { i = 1 } ^ { M } y _ i \\\\ y _ 1 \\\\ \\vdots \\\\ y _ M \\end{bmatrix} , \\end{align*}"}
+{"id": "4334.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\sum _ { i = 0 } ^ { n - 1 } \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in I _ { i , n } \\backslash U _ k \\} } | \\tilde F | ^ 2 _ h \\\\ \\le & \\frac { G ( T _ 1 ) } { \\int _ { T _ 1 } ^ { + \\infty } c ( s ) e ^ { - s } d s } \\limsup _ { n \\to + \\infty } \\sum _ { i = 0 } ^ { n - 1 } \\frac { 1 } { \\inf _ { I _ { i , n } \\backslash U _ k } c ( t ) } \\int _ { I _ { i , n } \\backslash U _ k } c ( s ) e ^ { - s } d s . \\end{align*}"}
+{"id": "8439.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { - \\infty } ^ { \\infty } \\bar { u } _ y e ^ { - 2 i z y } \\mathrm { d } y = \\bar { u } ( y ) e ^ { - 2 i z y } | _ { - \\infty } ^ { + \\infty } + 2 i z \\int _ { - \\infty } ^ { \\infty } \\bar { u } e ^ { - 2 i z y } \\mathrm { d } y = i z \\widehat { \\bar { u } } ( z ) . \\end{aligned} \\end{align*}"}
+{"id": "7656.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { * , t _ 0 , \\xi } = & ~ [ A x _ t ^ { * , t _ 0 , \\xi } - B ^ 2 R ^ { - 1 } U ( t , x _ t ^ { * , t _ 0 , \\xi } , \\nu _ t ^ { * , t _ 0 , \\xi } ) - B h ( \\mu _ t ^ { * , t _ 0 , \\xi } ) + f ( \\nu _ t ^ { * , t _ 0 , \\xi } ) + b ( \\mu _ t ^ { * , t _ 0 , \\xi } ) ] d t \\\\ & + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ { t _ 0 } ^ { * , t _ 0 , \\xi } = & ~ \\xi , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8128.png", "formula": "\\begin{align*} M ( S ) = \\varinjlim _ i M ( S _ i ) . \\end{align*}"}
+{"id": "4450.png", "formula": "\\begin{align*} a _ { E } ( n ) = \\prod _ { p \\leq \\infty } \\beta _ { p } ( n ) \\end{align*}"}
+{"id": "5780.png", "formula": "\\begin{align*} \\nabla _ X \\Psi _ 1 = - \\frac { 1 } { 2 } \\left ( X - \\langle X , T _ 1 \\rangle ( T _ 1 - f _ 1 ) - \\langle X , T _ 2 \\rangle ( T _ 2 - f _ 2 ) \\right ) \\cdot \\Psi _ 2 - \\frac { 1 } { 2 } S ( X ) \\cdot \\Psi _ 1 \\end{align*}"}
+{"id": "5341.png", "formula": "\\begin{align*} A _ n ( \\ell ) a _ { \\ell } = Q _ { n , 1 } ( \\ell ) \\beta _ { n , 1 } ^ { \\ell } + \\cdots + Q _ { n , s ( n ) } ( \\ell ) \\beta _ { n , s ( n ) } ^ { \\ell } . \\end{align*}"}
+{"id": "6413.png", "formula": "\\begin{align*} \\log \\det ( u _ { , i j } ) = - v _ j x ^ j + u _ { , i } \\xi ^ i + c . \\end{align*}"}
+{"id": "5679.png", "formula": "\\begin{align*} & ( p _ { 1 } , \\ldots , p _ { k } , f _ { k + 1 } , \\ldots , f _ { n } ) \\circ ( p _ { 1 } , \\ldots , p _ { k - 1 } , \\widetilde { f } _ { k } , p _ { k + 1 } , \\ldots , p _ { n } ) = \\\\ & \\Big [ p _ { 1 } , \\ldots , \\widetilde { f } _ { k } , f _ { k + 1 } \\circ ( p _ { 1 } , \\ldots , p _ { k - 1 } , \\widetilde { f } _ { k } , p _ { k + 1 } , \\ldots , p _ { n } ) , \\ldots , f _ { n } \\circ ( p _ { 1 } , \\ldots , p _ { k - 1 } , \\widetilde { f } _ { k } , p _ { k + 1 } , \\ldots , p _ { n } ) \\Big ] \\\\ \\end{align*}"}
+{"id": "2149.png", "formula": "\\begin{align*} 4 s ^ 3 + 8 ( 1 + a ) s ^ 2 + ( 4 + 6 a + 5 a ^ 2 ) s - 2 a = 0 . \\end{align*}"}
+{"id": "9105.png", "formula": "\\begin{align*} \\mathcal { Q } _ { | _ { x \\times C } } \\cong \\bigoplus _ { i = 1 } ^ { n } j _ { i * } ( E _ i ) \\mathcal { R } _ { | _ { x \\times C } } \\cong \\bigoplus _ { i = 1 } ^ { n - 1 } j _ { p _ i * } ( j _ { p _ i } ^ * ( j _ { i + 1 * } ( E _ { i + 1 } ) ) ) . \\end{align*}"}
+{"id": "5126.png", "formula": "\\begin{align*} \\left | \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi ) _ { + } G \\frac { 1 } { r ^ { 2 } } \\dd x \\right | = 2 \\pi \\left | \\int _ { \\mathbb { R } ^ { 2 } _ { + } } ( \\phi - \\phi ) _ { + } G \\frac { 1 } { r } \\dd z \\dd r \\right | & \\leq 2 \\pi | | ( \\phi - \\phi ) _ { + } | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } _ { + } ; r ^ { - 1 } ) } | | G | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 2 } _ { + } ; r ^ { - 1 } ) } \\\\ & \\lesssim | | b | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } ^ { 8 / 3 } . \\end{align*}"}
+{"id": "4229.png", "formula": "\\begin{align*} & \\ \\dot { s } \\varepsilon ( z ) \\dot { s } ( e - \\varepsilon ( 1 ) ) \\lambda _ i = \\sum _ { j , \\mu } g ^ i _ { j , \\mu } \\dot { s } ( \\varepsilon ( z ) - \\varepsilon ( x ^ i _ { j , \\mu } + z ) ) \\lambda _ j \\\\ & \\ = \\sum _ { j , \\mu } g ^ i _ { j , \\mu } \\sum _ { k , \\nu } g ^ j _ { k , \\nu } ( \\varepsilon ( z ^ { - 1 } x ^ j _ { k , \\nu } ) - \\varepsilon ( ( x ^ i _ { j , \\mu } + z ) ^ { - 1 } x ^ j _ { k , \\nu } ) ) \\lambda _ k \\end{align*}"}
+{"id": "66.png", "formula": "\\begin{align*} \\ell ( x , \\alpha ) : = \\langle \\mu , \\alpha \\rangle + \\Phi ^ + ( \\alpha ) - \\Phi ^ + ( w \\alpha ) . \\end{align*}"}
+{"id": "4346.png", "formula": "\\begin{align*} & \\int _ { \\{ - t ' _ 3 \\le \\Psi < - t ' _ 4 \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h \\\\ \\ge & \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\} } | \\tilde F | ^ 2 _ h + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\backslash N \\} } | F _ { t _ 0 , \\tilde { c } } - \\tilde F | ^ 2 _ h \\\\ & + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\cap N \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h . \\end{align*}"}
+{"id": "177.png", "formula": "\\begin{align*} \\gamma _ j ( C ) - C = ( V _ { j , 1 } - 1 ) C V _ { j , 2 } + V _ { j , 1 } C ( V _ { j , 2 } - 1 ) - ( V _ { j , 1 } - 1 ) C ( V _ { j , 2 } - 1 ) . \\end{align*}"}
+{"id": "4540.png", "formula": "\\begin{align*} \\sup _ { x \\in \\mathbb { R } , t > 0 } | \\mathcal { I } ( x , t ) | \\leq \\bar { \\tau } = \\max \\left ( \\sum _ { i = 1 } ^ p \\frac { 2 R } { \\lambda _ i } , \\sum _ { i = p + 1 } ^ n \\frac { 2 R } { \\lambda _ i } \\right ) . \\end{align*}"}
+{"id": "3410.png", "formula": "\\begin{align*} \\| q _ { k - 1 } \\alpha \\| = a _ { k + 1 } \\| q _ k \\alpha \\| + \\| q _ { k + 1 } \\alpha \\| . \\end{align*}"}
+{"id": "4607.png", "formula": "\\begin{align*} \\begin{array} { l } g _ n ( x _ n , x _ { n + 1 } ) = x _ n h _ n ( x _ n , x _ { n + 1 } ) , \\\\ g _ { k - 1 } ( x _ { k - 1 } , x _ k ) = x _ { k - 1 } h _ { k - 1 } ( x _ { k - 1 } , x _ k ) + g _ { k - 1 } ( 0 , x _ k ) . \\ ; \\ ; ( \\ ; 2 \\leq k \\leq n . ) \\end{array} \\end{align*}"}
+{"id": "392.png", "formula": "\\begin{align*} { \\langle { q _ \\delta } , } { q _ \\delta \\rangle } = \\langle u _ \\delta , u _ \\delta \\rangle + \\langle v _ \\delta , v _ \\delta \\rangle = n - 1 . \\end{align*}"}
+{"id": "1774.png", "formula": "\\begin{align*} \\widetilde { v } _ 1 = \\phi _ 1 ( v _ 1 , v _ 1 v _ 2 ) , \\widetilde { v } _ 2 = \\frac { \\phi _ 2 ( v _ 1 , v _ 1 v _ 2 ) } { \\phi _ 1 ( v _ 1 , v _ 1 v _ 2 ) } , \\end{align*}"}
+{"id": "3553.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( \\sum _ { x , y \\in \\Z } q ^ { 2 x ^ 2 - 2 x y + 2 y ^ 2 + x } + \\sum _ { x , y \\in \\Z } q ^ { 2 x ^ 2 - 2 x y + 2 y ^ 2 + 2 x + y + 1 } \\right ) & = \\frac { 1 } { 2 } \\left ( \\sum _ { x , y \\in \\Z } q ^ { f ( x , y ) } + \\sum _ { x , y \\in \\Z } q ^ { f ( x , y ) } \\right ) \\\\ & = \\sum _ { x , y \\in \\Z } q ^ { f ( x , y ) } \\end{align*}"}
+{"id": "7894.png", "formula": "\\begin{align*} p ^ \\mathfrak { t } ( s ) = ( r ^ \\mathfrak { t } ( s ) , \\lambda ^ \\mathfrak { t } ( s ) , \\mu ^ \\mathfrak { t } ( s ) ) ^ \\top = ( r _ 0 , 4 s - 2 \\mathfrak { t } + H ( q , r _ 0 ) / 4 , 2 s + 4 \\mathfrak { t } ) ^ \\top , \\end{align*}"}
+{"id": "1768.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\epsilon \\int _ 0 ^ T \\int _ { \\Gamma _ { \\epsilon } } u _ { \\epsilon } ( t , x ) \\phi \\left ( t , x , \\frac { x } { \\epsilon } \\right ) d \\sigma d t = \\int _ 0 ^ T \\int _ { \\Omega } \\int _ { \\Gamma } u _ 0 ( t , x , y ) \\phi ( t , x , y ) d \\sigma _ y d x d t . \\end{align*}"}
+{"id": "1631.png", "formula": "\\begin{align*} \\left < a _ 1 b _ 1 c _ 1 , a _ 2 b _ 2 c _ 2 \\right > = \\left < b _ 1 , b _ 2 \\right > \\left < S ( c _ 1 ) , a _ 2 \\right > \\left < c _ 2 , S ( a _ 1 ) \\right > \\end{align*}"}
+{"id": "3828.png", "formula": "\\begin{align*} \\begin{aligned} | ( \\langle \\boldsymbol { \\nu } , G \\rangle - \\bar G ) \\cdot ( \\langle \\boldsymbol { \\nu } , R \\rangle - \\bar R ) | & \\le c _ 3 \\langle \\boldsymbol { \\nu } , \\eta ( \\lambda | \\bar U ; x , t ) \\rangle = c _ 3 \\mathcal { H } ( \\boldsymbol { \\nu } , U , \\bar { U } ; x , t ) \\ ; . \\end{aligned} \\end{align*}"}
+{"id": "2841.png", "formula": "\\begin{align*} \\mathcal G _ t \\mathcal E _ x = j _ { x - 1 , x } - j _ { x , x + 1 } , \\end{align*}"}
+{"id": "2099.png", "formula": "\\begin{align*} \\tilde { B } ( \\theta _ 0 , \\varepsilon ) = \\bigcup _ { \\sigma \\in S _ k } \\big \\{ \\theta \\in \\Theta : d _ \\Theta ( \\theta , \\theta _ 0 [ \\sigma ] ) < \\varepsilon \\big \\} . \\end{align*}"}
+{"id": "8486.png", "formula": "\\begin{align*} & \\mathcal { P } ^ - ( f ( z ) e ^ { 2 i z x } ) = \\frac { 1 } { 2 \\pi i } \\lim _ { \\varepsilon \\rightarrow 0 } \\int _ { R } \\frac { f ( s ) e ^ { 2 i s x } } { s - ( z - i \\varepsilon ) } \\mathrm { d } s . \\end{align*}"}
+{"id": "1555.png", "formula": "\\begin{align*} { N } ( \\mathbf { v } ) = \\mathbf { 0 } . \\end{align*}"}
+{"id": "459.png", "formula": "\\begin{align*} ( \\mu _ { 2 } ) _ \\gamma ( d x ) = \\frac { e ^ { \\gamma x } } { \\int _ { [ 0 , \\infty ) } e ^ { \\gamma x } \\mu _ { 2 } ( d x ) } \\mu _ { 2 } ( d x ) . \\end{align*}"}
+{"id": "6246.png", "formula": "\\begin{align*} \\{ ( \\xi , \\rho ) \\} = \\mathcal { S } ( \\beta ) \\cap \\mathcal { S } ( \\beta ) ^ \\perp . \\end{align*}"}
+{"id": "2479.png", "formula": "\\begin{align*} g ( E ) = \\frac { 1 } { \\left ( \\hbar \\omega \\right ) ^ { D N } \\left ( D N - 1 \\right ) ! } E ^ { D N - 1 } , \\end{align*}"}
+{"id": "5809.png", "formula": "\\begin{align*} G = i \\ \\overline { v } \\ \\widehat { v } . \\end{align*}"}
+{"id": "6108.png", "formula": "\\begin{align*} \\begin{aligned} \\psi ^ { + 0 } _ { n , p } & = ( n - p + a _ { 1 2 } - a _ { 1 2 3 } + j ^ { ( 3 ) } ) \\ , \\prod _ { j = - \\infty } ^ p u _ { n , p - j } \\ , \\\\ \\rho ^ { 0 + } _ { n , p } & = ( n - p + a _ { 1 2 } - a _ { 1 2 3 } - j ^ { ( 3 ) } ) \\ , \\prod _ { k = - \\infty } ^ n u _ { n - k , p } \\ , \\end{aligned} \\end{align*}"}
+{"id": "3946.png", "formula": "\\begin{align*} p + \\alpha \\overline { u } + \\mu = 0 , \\\\ u _ { a } \\le \\overline { u } \\le u _ { b } , \\ ; \\ ; \\mu _ { a } \\ge 0 , \\ ; \\ ; \\mu _ { b } \\ge 0 , \\\\ \\mu _ { a } ( u _ { a } - \\overline { u } ) = \\mu _ { b } ( \\overline { u } - u _ { b } ) = 0 , \\end{align*}"}
+{"id": "8280.png", "formula": "\\begin{align*} \\tilde X _ { n + 1 } ^ i = f ( \\tilde X _ n ^ i , \\tau ) + \\gamma ^ i ( \\tilde X _ n ) \\tau + \\sigma W _ \\tau ^ i . \\end{align*}"}
+{"id": "2462.png", "formula": "\\begin{align*} L _ { q } \\left [ \\exp _ { q ^ { \\prime } } \\left ( - \\alpha t ^ { 2 } \\right ) \\right ] = \\displaystyle \\int _ { 0 } ^ { \\infty } d t \\ ; \\exp _ { q } \\left ( - s t \\right ) \\exp _ { q ^ { \\prime } } \\left ( - \\alpha t ^ { 2 } \\right ) , \\end{align*}"}
+{"id": "8535.png", "formula": "\\begin{align*} & \\mathcal { P } ^ + \\left ( f ( z ) e ^ { 2 i z x } \\right ) = - \\int _ { - \\infty } ^ { 2 x } \\widehat { f } ( \\xi ) e ^ { - i z ( \\xi - 2 x ) } \\mathrm { d } \\xi , \\\\ & \\mathcal { P } ^ - \\left ( f ( z ) e ^ { - 2 i z x } \\right ) = \\int _ { - \\infty } ^ { 2 x } \\widehat { f } ( \\xi ) e ^ { i z ( \\xi - 2 x ) } \\mathrm { d } \\xi . \\end{align*}"}
+{"id": "9286.png", "formula": "\\begin{align*} \\begin{aligned} ( \\Delta \\varphi _ { p , N } ^ { + } ) ^ m \\wedge \\beta _ n ^ { n - m } = 2 ^ { - m } ( \\Delta u _ p + \\Delta u _ { p + N } ) ^ m \\wedge \\beta _ n ^ m = 2 ^ { - m } \\sum \\limits _ { k = 0 } ^ m \\binom m k ( \\Delta u _ p ) ^ k \\wedge ( \\Delta u _ { p + N } ) ^ { m - k } \\wedge \\beta _ n ^ m . \\end{aligned} \\end{align*}"}
+{"id": "6454.png", "formula": "\\begin{align*} \\vec { h } = \\mathcal { H } ( \\vec { r } , h _ 1 ) . \\end{align*}"}
+{"id": "7395.png", "formula": "\\begin{align*} [ M ] & = \\frac { [ \\mathbb C _ x ] } { D } \\\\ [ \\mathbb C _ x ] & = \\prod _ { i = 0 } ^ N e ^ { x _ i \\alpha _ i } \\\\ D & = \\prod _ { n = 1 } ^ \\infty ( 1 - e ^ { - n \\delta } ) ^ { N } \\prod _ { n = 1 } ^ \\infty \\prod _ { 1 \\leq i < j \\leq r } ( 1 - e ^ { \\alpha _ j - \\alpha _ i } e ^ { ( 1 - n ) \\delta } ) ( 1 - e ^ { \\alpha _ i - \\alpha _ j } e ^ { - n \\delta } ) . \\end{align*}"}
+{"id": "871.png", "formula": "\\begin{align*} \\left | \\begin{array} { c c } x + y = y ' \\ , \\\\ x ' + y ' = x \\end{array} \\right . . \\end{align*}"}
+{"id": "3439.png", "formula": "\\begin{align*} \\delta ( \\alpha , \\theta ) = \\limsup _ { n \\to \\infty } \\max ( 0 , \\delta _ n ) = \\limsup _ { n \\to \\infty } \\max ( 0 , \\frac { \\ln q _ { n + 1 } + \\ln | c _ { n , 0 } | } { q _ n } ) . \\end{align*}"}
+{"id": "2472.png", "formula": "\\begin{align*} H = \\displaystyle \\sum _ { i = 1 } ^ { D N } \\frac { p _ { i } ^ { 2 } } { 2 m } , \\end{align*}"}
+{"id": "126.png", "formula": "\\begin{align*} f _ i = \\sum _ { L _ i \\geqslant N _ 3 ^ { ( 5 - 2 \\alpha ) + \\varepsilon } } f _ { i , L _ i } . \\end{align*}"}
+{"id": "2143.png", "formula": "\\begin{align*} w = 0 , \\ | z | ^ 2 + \\Re ( a z ^ 2 ) = 1 , \\end{align*}"}
+{"id": "3659.png", "formula": "\\begin{align*} \\Sigma ^ 2 = \\left ( \\frac { 1 } { 2 L _ { x } } + \\frac { \\omega } { L _ { x y } } \\right ) \\frac { \\sigma _ { F , r } ^ 2 } { n } + \\left ( \\frac { 1 } { 2 L _ { y } } + \\frac { 1 } { L _ { x y } \\omega } \\right ) \\frac { \\sigma _ { G , r } ^ 2 } { n } . \\end{align*}"}
+{"id": "8152.png", "formula": "\\begin{align*} \\gamma \\times T ^ n = \\gamma ( n \\bmod \\Z ) \\ , T ^ n \\end{align*}"}
+{"id": "2292.png", "formula": "\\begin{align*} \\Phi = \\{ e _ j - e _ k \\mid j , k = 1 , \\dots , n , \\ ; j \\not = k \\} . \\end{align*}"}
+{"id": "3044.png", "formula": "\\begin{align*} 2 \\pi r \\left \\Vert d \\phi _ { r i } \\left ( z \\right ) \\right \\Vert ^ { 1 / 2 } = \\sqrt { 2 } ~ \\pi \\left ( \\ell \\left ( r \\right ) \\sin r \\right ) ^ { 1 / 2 } \\end{align*}"}
+{"id": "7064.png", "formula": "\\begin{align*} u \\ , = \\ , F ( x , y , z , w ) \\ \\ \\ \\ \\ \\ \\ \\ { \\scriptstyle { ( F ( 0 , 0 , 0 , 0 ) \\ , = \\ , 0 ) } } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ v \\ , = \\ , G ( r , s , t , p ) \\ \\ \\ \\ \\ \\ \\ \\ { \\scriptstyle { ( 0 \\ , = \\ , G ( 0 , 0 , 0 , 0 ) ) } } , \\end{align*}"}
+{"id": "7627.png", "formula": "\\begin{align*} \\widehat \\Phi ^ { \\phi ^ \\varepsilon } ( t ) \\overset { d } { = } \\Phi ^ { q _ \\varepsilon } ( t ) \\end{align*}"}
+{"id": "5766.png", "formula": "\\begin{align*} \\partial _ Y \\left ( \\Phi ( \\mathcal { P } ( X ) ) \\right ) = \\Phi ( \\nabla _ Y \\mathcal { P } ( X ) ) - \\sum _ { i = 1 , 2 } \\sqrt { c _ i } \\ \\langle Y _ i , \\mathcal { P } ( X ) \\rangle N _ i \\\\ + \\Phi ( B ( Y , \\mathcal { P } ( X ) _ T ) - B ^ * ( Y , \\mathcal { P } ( X ) _ N ) ) \\end{align*}"}
+{"id": "154.png", "formula": "\\begin{align*} x _ 0 = \\frac { 1 - x _ 0 } { 1 + \\sum _ { l \\in \\mathbb Z _ 0 } { \\lambda _ l \\widetilde { z } _ l } } + x _ 0 \\cdot c _ { 0 0 } , \\ \\ x _ j = \\frac { ( 1 - x _ 0 ) \\lambda _ { j } \\widetilde { z } _ { j } } { 1 + \\sum _ { l \\in \\mathbb Z _ 0 } { \\lambda _ l \\widetilde { z } _ l } } + x _ 0 \\cdot c _ { 0 j } , \\ \\ j \\in \\mathbb Z _ 0 . \\end{align*}"}
+{"id": "8187.png", "formula": "\\begin{align*} V \\left ( \\mathcal { K } _ { m , n } \\right ) = \\left \\{ \\left ( i , j \\right ) : 1 \\leq i \\leq m , 1 \\leq j \\leq n \\right \\} , \\end{align*}"}
+{"id": "548.png", "formula": "\\begin{align*} w _ 1 ^ 2 & = ( a _ { - 1 } - a _ 1 ) ^ 2 \\\\ & = a _ { - 1 } + a _ 1 - ( a _ { - 1 } + a _ 1 ) - 2 s _ 2 \\\\ & = - 2 s _ 2 = - \\tfrac { 1 } { 8 } ( u _ 2 - 3 v _ 2 ) . \\end{align*}"}
+{"id": "3791.png", "formula": "\\begin{align*} B _ q ( \\psi _ m ) ( z ) = \\sum _ { j = 0 } ^ { \\infty } \\frac { z ^ j } { \\sqrt { \\Gamma ( q j + 1 ) } } ( \\delta _ { j , m } ) = \\frac { z ^ m } { \\sqrt { \\Gamma ( q j + 1 ) } } : = e _ { m , q } ( z ) . \\end{align*}"}
+{"id": "119.png", "formula": "\\begin{align*} & \\sum _ { K _ 1 \\sim K _ 2 } \\Big | \\int ( f _ { N _ 1 , K _ 1 , L _ 1 } * g _ { N _ 2 , K _ 2 , L _ 2 } ) h _ { N _ 3 , \\lesssim K _ 1 , L _ 3 } \\Big | \\\\ & \\lesssim \\sum _ { K _ 1 \\sim K _ 2 \\gtrsim N _ 1 } C ( N _ 1 , N _ 2 , N _ 3 ) \\| f _ { N _ 1 , K _ 1 , L _ 1 } \\| _ { L ^ 2 } \\| g _ { N _ 2 , K _ 2 , L _ 2 } \\| _ { L ^ 2 } \\| h _ { N _ 3 , \\lesssim K _ 1 , L _ 3 } \\| _ { L ^ 2 } \\end{align*}"}
+{"id": "4774.png", "formula": "\\begin{align*} F ( \\eta ) : = T - \\frac { 1 } { \\eta } I , \\eta \\in \\Omega , \\end{align*}"}
+{"id": "3882.png", "formula": "\\begin{align*} \\mathcal { M } = \\{ Z = ( z _ 1 , z _ 2 , \\cdots , z _ m ) \\in \\mathbb { R } ^ { ( 2 m ) } \\mid z _ i \\in B _ { \\bar { \\rho } } ( x _ { 0 , i } ) , \\ i = 1 , \\cdots , m \\} . \\end{align*}"}
+{"id": "571.png", "formula": "\\begin{align*} 2 \\pi i f ( z ) = - \\int _ { e ^ { - i \\alpha } [ 0 , + \\infty ) } f ( \\zeta ) \\int _ { p - i e ^ { i \\alpha } [ 0 , + \\infty ) } e ^ { \\omega ( \\zeta - z ) } d \\omega d \\zeta + \\\\ + \\int _ { e ^ { i \\alpha } [ 0 , + \\infty ) } f ( \\zeta ) \\int _ { p + i e ^ { - i \\alpha } [ 0 , + \\infty ) } e ^ { \\omega ( \\zeta - z ) } d \\omega d \\zeta , z \\in \\Delta . \\end{align*}"}
+{"id": "2011.png", "formula": "\\begin{align*} ( \\epsilon _ N \\circ T _ \\alpha G _ \\alpha f ) ( s \\otimes g ) & = \\epsilon _ N ( s \\otimes ( G _ \\alpha f ) ( g ) ) \\\\ & = \\epsilon _ N ( s \\otimes ( f \\circ g ) ) \\\\ & = f ( g ( s ) ) \\\\ & = f ( \\epsilon _ N ( s \\otimes g ) ) \\\\ & = ( f \\circ \\epsilon _ N ) ( s \\otimes g ) \\end{align*}"}
+{"id": "2046.png", "formula": "\\begin{align*} \\partial _ t M ^ k f + v \\cdot \\partial _ x M ^ k f + \\mathcal L M ^ k f = \\Gamma ( f , \\ M ^ k f ) + M ^ k \\Gamma ( f , f ) - \\Gamma ( f , \\ M ^ k f ) - \\big [ M ^ k , \\ \\mathcal L \\big ] f - k \\partial _ { v _ 1 } ^ 2 M ^ { k - 1 } f . \\end{align*}"}
+{"id": "3021.png", "formula": "\\begin{align*} p _ 0 + \\langle l , p \\rangle > 0 , l \\in L = F ^ \\eta ( z , w ( - h ) ) + \\partial ^ { c i } _ { \\tau , w } g ( \\tau , w ( \\cdot ) ) . \\end{align*}"}
+{"id": "8.png", "formula": "\\begin{align*} \\alpha \\vee \\beta = \\begin{cases} \\{ \\alpha _ { i _ 1 , j _ 2 } , \\alpha _ { i _ 1 , \\overline { j } _ 1 } \\} & i _ 2 \\geq j _ 2 \\\\ \\{ \\alpha _ { i _ 1 , j _ 2 } \\} & \\end{cases} \\end{align*}"}
+{"id": "31.png", "formula": "\\begin{align*} X _ { i } ( g ) = d L _ { g } ( e _ i ) , \\\\ \\ i = 1 , . . . , m \\end{align*}"}
+{"id": "7546.png", "formula": "\\begin{align*} \\boldsymbol { \\lambda } = \\sum \\limits _ { k = 1 } ^ m s _ k \\boldsymbol { \\lambda } _ k . \\end{align*}"}
+{"id": "8312.png", "formula": "\\begin{align*} R _ c = \\pi / 0 . 0 6 7 \\approx 4 6 . 9 . \\end{align*}"}
+{"id": "8442.png", "formula": "\\begin{align*} & \\int _ { - \\infty } ^ { \\infty } \\bar { u } _ y ( y ) \\left ( \\psi ^ - _ { 1 1 } - 1 \\right ) e ^ { - 2 i z y } \\mathrm { d } y = - z \\int _ { - \\infty } ^ { \\infty } \\bar { u } _ y ( y ) e ^ { - 2 i z y } K \\psi ^ - _ { 1 1 } \\mathrm { d } y \\\\ & = - \\frac { 1 } { 2 } z \\widehat { \\bar { u } _ x K \\psi ^ - _ { 1 1 } } ( z ) . \\end{align*}"}
+{"id": "2313.png", "formula": "\\begin{align*} D ^ W _ X Y = D ^ g _ X Y - \\frac { 1 } { 2 } \\theta ( X ) Y - \\frac { 1 } { 2 } \\theta ( Y ) X + \\frac { 1 } { 2 } g ( X , Y ) \\theta ^ \\sharp , \\end{align*}"}
+{"id": "8604.png", "formula": "\\begin{align*} | P _ { u ^ \\bot } ( [ - u _ 1 , u _ 1 ] \\oplus _ 2 \\cdots \\oplus _ 2 [ - u _ m , u _ m ] ) | ^ 2 = | B _ 2 ^ { n - 1 } | ^ 2 \\sum _ { | I | = n - 1 } ( \\det ( P _ { u ^ \\bot } u _ i ) _ { i \\in I } ) ^ 2 . \\end{align*}"}
+{"id": "4454.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s | c _ { i } | \\leq \\sqrt { s } \\sqrt { \\sum _ { i = 1 } ^ s | c _ { i } | ^ { 2 } } . \\end{align*}"}
+{"id": "3845.png", "formula": "\\begin{align*} X ( t ) = X ( 0 ) + \\int _ 0 ^ t F ( X ( s ) ) d s + \\int _ 0 ^ t G ( X ( s ) ) d W ( s ) , \\end{align*}"}
+{"id": "8715.png", "formula": "\\begin{align*} \\psi _ i = m \\cdot \\int _ { \\kappa ^ { - 1 } ( K _ i ) } \\psi \\ , \\mathrm { d } \\mu _ \\varphi ( \\psi ) \\end{align*}"}
+{"id": "5890.png", "formula": "\\begin{align*} & \\mathbb { E } \\Vert Y _ n ^ M ( t + \\delta ) - Y _ n ^ M ( t ) \\Vert _ { H } ^ 2 \\\\ = & \\mathbb { E } \\int _ { t \\wedge \\tau _ n ^ M } ^ { ( t + \\delta ) \\wedge \\tau _ n ^ M } 2 \\langle A ( r , Y _ n ( r ) ) , Y _ n ( r ) - Y _ n ( t \\wedge \\tau _ n ^ M ) \\rangle d r \\\\ & + \\mathbb { E } \\int _ { t \\wedge \\tau _ n ^ M } ^ { ( t + \\delta ) \\wedge \\tau _ n ^ M } \\Vert P _ n B ( r , Y _ n ( r ) ) Q _ n \\Vert _ { L _ 2 } ^ 2 d r . \\end{align*}"}
+{"id": "2924.png", "formula": "\\begin{align*} & J ( u ) = J ( \\bar u ) + J ' ( \\bar u ) ( u - \\bar u ) + \\frac { 1 } { 2 } J '' ( u _ \\theta ) ( u - \\bar u ) ^ 2 \\\\ & \\ge J ( \\bar u ) + \\frac { 1 } { 2 } [ J ' ( \\bar u ) ( u - \\bar u ) + J '' ( \\bar u ) ( u - \\bar u ) ^ 2 ] + \\frac { 1 } { 2 } [ J '' ( u _ \\theta ) - J '' ( \\bar u ) ] ( u - \\bar u ) ^ 2 \\\\ & \\ge J ( \\bar u ) + \\frac { \\gamma } { 2 } \\| z _ { \\bar u , u - \\bar u } \\| ^ 2 _ { L ^ 2 ( \\Omega ) } - \\frac { 1 } { 2 } | [ J '' ( u _ \\theta ) - J '' ( \\bar u ) ] ( u - \\bar u ) ^ 2 | . \\end{align*}"}
+{"id": "2500.png", "formula": "\\begin{align*} \\mathbf { r } _ p ( \\mathbf { x } , \\mathbf { y } , \\mathbf { s } , \\kappa , \\tau ) = \\mathbf { A x } - \\tau \\mathbf { b } , \\mathbf { r } _ d ( \\mathbf { x } , \\mathbf { y } , \\mathbf { s } , \\kappa , \\tau ) = { { \\mathbf { A } } ^ { T } } \\mathbf { y } + \\mathbf { s } - \\tau \\mathbf { c } . \\end{align*}"}
+{"id": "5614.png", "formula": "\\begin{align*} e ^ { c _ { n + 2 } } = 1 - \\left ( 1 - e ^ { a _ { \\alpha + 1 } + d _ { \\alpha + 1 } - d _ \\alpha } \\right ) \\ . \\left ( e ^ { - c _ { n + 1 } } - 1 \\right ) \\ . \\end{align*}"}
+{"id": "1211.png", "formula": "\\begin{align*} \\sum _ { x \\in E } \\sum _ { y \\neq x } \\nu ( x ) \\frac { \\mu ( y ) } { \\mu ( x ) } L _ { y x } = \\sum _ { x \\in E } \\frac { \\nu ( x ) } { \\mu ( x ) } \\sum _ { y \\neq x } \\mu ( y ) L _ { y x } = \\sum _ { x \\in E } \\nu ( x ) L _ { x x } = \\sum _ { x \\in E } \\nu ( x ) \\sum _ { y \\neq x } L _ { x y } , \\end{align*}"}
+{"id": "6005.png", "formula": "\\begin{align*} 1 + \\Big ( p ( 1 + \\tilde { b } ) \\Big ) ^ { \\frac { 1 } { 2 - p } } \\Big ( p - 1 \\Big ) ^ { \\frac { p - 1 } { 2 - p } } ( p - 2 ) = 0 . \\end{align*}"}
+{"id": "4056.png", "formula": "\\begin{align*} \\sum _ { j \\in [ n ] ^ { V ( C ) } } \\beta _ { n } ^ { C } ( j ) = O ( n ^ { e ( C ) } ) = \\bar O ( n ^ { e _ 2 ( C ) } ) . \\end{align*}"}
+{"id": "6521.png", "formula": "\\begin{align*} A ( X _ 1 ) K ( X _ 2 , X _ 5 , X _ 3 , X _ 4 ) + A ( X _ 2 ) K ( X _ 1 , X _ 5 , X _ 3 , X _ 4 ) = 0 \\end{align*}"}
+{"id": "1847.png", "formula": "\\begin{align*} \\int _ { \\partial D } S _ { e , \\mathcal { Y M } ^ 0 } ( X , \\nu ) d s _ g = \\int _ D \\langle S _ { e , \\mathcal { Y M } ^ 0 } , \\nabla X ^ { \\flat } \\rangle d v _ g \\ , , \\end{align*}"}
+{"id": "3444.png", "formula": "\\begin{align*} | m _ n + \\ell _ n q _ n - \\tilde { m } _ n | = q _ { n + 1 } = q _ n + q _ { n - 1 } , \\end{align*}"}
+{"id": "4553.png", "formula": "\\begin{align*} & A _ { w , 1 } f ( v _ { \\ell , \\ell , 0 } ) = ( q ^ 3 + q ^ 2 ) f ( v _ { \\ell , \\ell , 1 } ) + ( q + 1 ) f ( v _ { \\ell + 1 , \\ell , 0 } ) \\\\ & A _ { w , 2 } f ( v _ { \\ell , \\ell , 0 } ) = q ^ 4 f ( v _ { \\ell - 1 , \\ell - 1 , 0 } ) + q ( q + 1 ) ^ 2 f ( v _ { \\ell + 1 , \\ell , 1 } ) + f ( v _ { \\ell + 1 , \\ell + 1 , 0 } ) \\\\ & A _ { w , 3 } f ( v _ { \\ell , \\ell , 0 } ) = ( q ^ 3 + q ^ 2 ) f ( v _ { \\ell , \\ell - 1 , 0 } ) + ( q + 1 ) f ( v _ { \\ell + 1 , \\ell + 1 , 1 } ) . \\end{align*}"}
+{"id": "3478.png", "formula": "\\begin{align*} \\phi ( j q _ n + m _ n + 1 ) + \\phi ( j q _ n + m _ n - 1 ) = ( E - \\lambda \\tan ( \\pi \\theta _ { j q _ n + m _ n } ) ) \\phi ( j q _ n + m _ n ) . \\end{align*}"}
+{"id": "8033.png", "formula": "\\begin{align*} & \\ll \\frac { 1 } { \\pi _ N ( x ) ^ { 2 r } L ^ r } \\sum _ { a = 0 } ^ { 2 r - 1 } \\pi _ N ( x ) ^ a ( \\log \\log x ) ^ { 2 r - a } \\pi _ N ( x ) ^ { r - a } L ^ { 2 r } \\sum _ { a _ 2 = 0 } ^ { [ a / 2 ] } \\left ( \\frac { \\pi _ N ( x ) } { L } \\right ) ^ { a _ 2 } \\\\ & + \\frac { x ^ { E _ 3 ( r ) \\pi _ N ( x ) } 8 ^ { \\nu ( N ) } } { k N } \\\\ & \\ll \\frac { L ^ { 1 / 2 } \\log \\log x } { \\pi _ N ( x ) ^ { 1 / 2 } } + \\frac { x ^ { E _ 3 ( r ) \\pi _ N ( x ) } 8 ^ { \\nu ( N ) } } { k N } . \\end{align*}"}
+{"id": "8003.png", "formula": "\\begin{align*} T ( g , \\rho ) = \\sum _ { l \\geq 1 } ( U ( l ) - U ( l - 1 ) ) ^ 2 \\widehat { g } \\left ( \\frac { l } { \\pi _ N ( x ) } \\right ) , \\end{align*}"}
+{"id": "7411.png", "formula": "\\begin{align*} ^ F V = \\bigoplus _ { p \\in \\frac { 1 } { 2 } \\mathbb Z } F _ p V / F _ { p - \\frac { 1 } { 2 } } V \\end{align*}"}
+{"id": "8830.png", "formula": "\\begin{align*} B _ j ( x , x ) = ( x _ { j + 1 } - x _ { j - 2 } ) x _ { j - 1 } . \\end{align*}"}
+{"id": "6868.png", "formula": "\\begin{align*} \\Psi ( \\hat z _ 1 ) = 0 , \\ ; \\Psi ( \\hat z _ 2 ) = 1 , \\ ; \\Psi ( \\hat z _ 3 ) = 1 + \\i h , \\ ; \\Psi ( \\hat z _ 4 ) = \\i h . \\end{align*}"}
+{"id": "4445.png", "formula": "\\begin{align*} W ( v , x ) \\stackrel { d } { = } e ^ { - \\lambda ^ * t } W ( v , z ) . \\end{align*}"}
+{"id": "6446.png", "formula": "\\begin{align*} \\zeta ^ { \\star } ( 1 , \\{ 2 \\} ^ { s - 1 } ) = 2 \\zeta ( 2 s - 1 ) \\end{align*}"}
+{"id": "2734.png", "formula": "\\begin{align*} Z \\colon Q \\times \\R \\to \\R ^ d , Z ( x _ 1 , \\dots , x _ d ) = e ^ { x _ d } h ( x _ 1 , \\dots , x _ { d - 1 } ) . \\end{align*}"}
+{"id": "8685.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 ( I _ N ^ i ( p ) - J _ N ^ i ( p ) ) = O ( p ^ { \\mu - \\nu } ) \\quad \\mbox { f o r l a r g e } p > 0 , \\end{align*}"}
+{"id": "6660.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\lim _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\chi _ 4 < 1 \\Big ) = - \\infty . \\end{align*}"}
+{"id": "5248.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( | v | ^ { 2 } + | b | ^ { 2 } \\right ) \\dd x + \\int _ { \\delta } ^ { t } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( \\nu | \\nabla v | ^ { 2 } + \\mu | \\nabla b | ^ { 2 } \\right ) \\dd x \\dd s = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( | v | ^ { 2 } + | b | ^ { 2 } \\right ) ( \\delta ) \\dd x . \\end{align*}"}
+{"id": "9210.png", "formula": "\\begin{align*} \\Bigg | \\dfrac { \\partial \\epsilon _ 1 ( \\eta , t ) } { \\partial \\eta } \\Bigg | _ { \\eta = \\eta _ a } \\dot { \\eta } _ a \\Bigg | \\le \\gamma \\delta L _ r ^ 2 3 \\end{align*}"}
+{"id": "1702.png", "formula": "\\begin{align*} X ( t , \\xi ) = e ^ { W ( t , \\xi ) } y ( t , \\xi ) . \\end{align*}"}
+{"id": "2583.png", "formula": "\\begin{align*} L ( p , \\pi , \\bar { \\pi } ) = \\frac { R \\cdot R ( v , w , w , v ; x , y ) } { Q ( g , R ) ( v , w , w , v ; x , y ) } . \\end{align*}"}
+{"id": "766.png", "formula": "\\begin{align*} I _ { 2 2 } & \\lesssim \\sum _ { i = 1 } ^ \\infty \\frac { \\ell ( Q ) ^ { 2 s } } { \\ell ( 2 ^ i Q ) ^ { 1 + 2 s } } \\ , \\Big ( \\ell ( 2 ^ i Q ) ^ { \\frac N q + 2 s } \\ , \\ell ( Q ) ^ { \\frac N { q ' } } + i \\ , \\ell ( Q ) ^ N \\ , \\ell ( 2 ^ i Q ) ^ { 2 s } \\Big ) \\\\ & = \\frac { \\ell ( Q ) ^ { 2 s } \\ell ( Q ) ^ { N + 2 s } } { \\ell ( Q ) ^ { 1 + 2 s } } \\sum _ { i = 1 } ^ \\infty \\ , 2 ^ { i ( \\frac N q - 1 ) } ( 1 + \\frac { i } { 2 ^ i } ) . \\end{align*}"}
+{"id": "1441.png", "formula": "\\begin{align*} \\P _ x ( \\rho > n ) \\sim C _ { \\lambda } n ^ { - d ( d - 1 ) / 2 - d / 2 } e ^ { - \\gamma n } e ^ { \\sum _ { i = 1 } ^ d ( \\lambda _ i - \\bar { \\lambda } ) x _ i } \\Delta ( x ) , n \\rightarrow \\infty \\end{align*}"}
+{"id": "7002.png", "formula": "\\begin{align*} [ J _ + ^ \\alpha , J _ - ^ \\alpha ] = 2 \\omega ^ { - 2 p + 1 } ( \\alpha ) J _ 0 ^ \\alpha . \\end{align*}"}
+{"id": "6996.png", "formula": "\\begin{align*} \\pi _ 0 ( e ) = \\pi _ 0 ( e ^ 2 ) = \\pi _ 0 ( e ) ^ 2 + \\sum _ { \\omega \\in W _ f ^ + } \\pi _ { - \\omega } ( e ) \\pi _ { \\omega } ( e ) . \\end{align*}"}
+{"id": "7383.png", "formula": "\\begin{align*} z = \\prod _ { c = 0 } ^ { \\ell - 1 } X _ c . \\end{align*}"}
+{"id": "509.png", "formula": "\\begin{align*} \\sigma ( \\pi ) = \\sum _ { i \\ge 2 } ( i - 1 ) \\pi _ i . \\end{align*}"}
+{"id": "3431.png", "formula": "\\begin{align*} \\| q _ n ( \\theta - \\frac { 1 } { 2 } ) \\| = e ^ { \\delta _ n q _ n } \\| q _ n \\alpha \\| . \\end{align*}"}
+{"id": "1106.png", "formula": "\\begin{align*} { S ^ { \\rm { N } } } = \\frac { { W I } } { { K L } } { \\widetilde \\varepsilon _ K } \\left ( { { \\gamma ^ { \\rm { N } } } } \\right ) , \\end{align*}"}
+{"id": "7398.png", "formula": "\\begin{align*} [ q ^ \\alpha , q ^ \\beta ] = \\sum _ { \\gamma \\in S } { f ^ { \\alpha \\beta } } _ \\gamma q ^ \\gamma . \\end{align*}"}
+{"id": "146.png", "formula": "\\begin{align*} z _ { i , x } = \\lambda _ i \\prod _ { y \\in S ( x ) } { 1 \\over 1 + \\sum _ { j \\in \\mathbb { Z } _ 0 } z _ { j , y } } , \\ \\ i \\in \\mathbb { Z } _ 0 . \\end{align*}"}
+{"id": "8522.png", "formula": "\\begin{align*} \\left \\| \\partial _ { z } \\mathcal { H } \\log \\left ( 1 + \\bar { r } _ 1 r _ 2 \\right ) \\right \\| _ { L ^ 2 } = \\left \\| \\partial _ { z } \\log \\left ( 1 + \\bar { r } _ 1 r _ 2 \\right ) \\right \\| _ { L ^ 2 } . \\end{align*}"}
+{"id": "579.png", "formula": "\\begin{align*} \\limsup _ { s \\rightarrow + \\infty } \\frac { J _ 3 ( s ) } { s } \\leq - R e \\left ( q e ^ { i \\theta } \\right ) = - \\inf _ { \\omega \\in \\Gamma ^ { \\prime } } R e \\left ( \\omega e ^ { i \\theta } \\right ) . \\end{align*}"}
+{"id": "8127.png", "formula": "\\begin{align*} ( S _ \\bullet \\to S ) = \\varprojlim _ i ( T _ { i , \\bullet } \\to T _ i ) . \\end{align*}"}
+{"id": "6830.png", "formula": "\\begin{align*} \\varphi \\left ( \\rho \\right ) = T _ { + } \\left ( \\rho \\right ) \\varphi _ { 0 } \\left ( \\rho \\right ) \\end{align*}"}
+{"id": "4138.png", "formula": "\\begin{align*} u \\in \\C ^ \\infty ( \\R ^ n _ { + } , \\mathbb { C } ^ N ) , \\quad \\ , L u = 0 \\ , \\ , \\ , \\ , \\R ^ n _ { + } , \\quad \\ , \\ , u \\big | ^ { { } ^ { \\kappa - { \\rm n . t . } } } _ { \\partial \\R ^ n _ { + } } = f \\ , \\R ^ { n - 1 } , \\end{align*}"}
+{"id": "3586.png", "formula": "\\begin{align*} \\mathrm { Q u a d } ^ i : = \\{ z ^ i _ { j k } z ^ i _ { j ' k ' } - z ^ i _ { j k ' } z ^ i _ { j ' k } \\mid j , j ' \\in [ s _ i ] , k , k ' \\in [ t _ i ] \\} . \\end{align*}"}
+{"id": "4222.png", "formula": "\\begin{align*} \\varepsilon ( z ) \\Lambda ( z ^ { - 1 } ) = - \\Lambda ( - z ^ { - 1 } ) , \\dot { s } \\Lambda ( z ) = - \\Lambda ( - z ^ { - 1 } ) , \\quad \\\\ \\dot { s } \\varepsilon ( x ) \\Lambda ( y ) = \\varepsilon ( - x ^ { - 1 } ) \\Lambda ( x ( x y - 1 ) ) + \\Lambda ( x ) - \\Lambda ( y ^ { - 1 } ( x y - 1 ) ) , \\end{align*}"}
+{"id": "744.png", "formula": "\\begin{align*} P _ s ( x , t ) = t ^ { \\frac { - N } { 2 s } } \\phi \\left ( | x | t ^ { - \\frac 1 { 2 s } } \\right ) \\end{align*}"}
+{"id": "8420.png", "formula": "\\begin{align*} \\begin{aligned} & ( I - F ) \\left [ z \\left ( \\Psi ^ { - } _ 1 - e ^ { - i c _ - } e _ 1 \\right ) - \\left ( \\widehat { \\Psi } ^ - _ { 1 1 } e _ 1 + \\widehat { \\Psi } ^ - _ { 2 1 } e _ { 2 } \\right ) \\right ] \\\\ = & z d e _ { 2 } - ( I - F ) \\left ( \\widehat { \\Psi } ^ - _ { 1 1 } e _ 1 + \\widehat { \\Psi } ^ - _ { 2 1 } e _ 2 \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "6453.png", "formula": "\\begin{align*} h _ { n + 1 } = r _ n h _ n + \\frac { r _ n ( r _ n - 1 ) } { 2 } . \\end{align*}"}
+{"id": "8481.png", "formula": "\\begin{align*} \\left \\| R ( x ; z ) V _ { j } ( k ) \\right \\| _ { L _ z ^ { 2 } } \\leq c \\left ( \\left \\| r _ 1 \\right \\| _ { L ^ { 2 } } + \\left \\| r _ 2 \\right \\| _ { L ^ { 2 } } \\right ) , j = 1 , 2 . \\end{align*}"}
+{"id": "7337.png", "formula": "\\begin{align*} | P _ E ( A + B ) + Z + t [ 0 , u ] | _ { k } = | P _ E ( A + B ) + Z | _ k + t | P _ { E \\cap u ^ \\bot } ( A + B + Z ) | _ { k - 1 } \\end{align*}"}
+{"id": "2710.png", "formula": "\\begin{align*} \\bar \\Delta = \\min _ { 0 \\le k \\le T - 1 } \\beta ( X _ k ) . \\end{align*}"}
+{"id": "7964.png", "formula": "\\begin{align*} \\gamma \\varphi ( h ) G ( \\nabla h ) \\sigma _ { k } ( x ) = f ( x ) \\ { \\rm f o r } \\ \\gamma = 1 . \\end{align*}"}
+{"id": "7637.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { P } _ t + 2 A P _ t - B ^ 2 R ^ { - 1 } P ^ 2 _ t + Q = 0 , \\\\ & P _ T = G , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1791.png", "formula": "\\begin{align*} L ( f , n ) = \\frac { ( - 1 ) ^ { n - 1 } } { 3 ( n - 1 ) ! } \\int _ { 0 } ^ { 1 } b ^ { 2 } ( q ) c ( q ^ { 3 } ) ( \\log q ) ^ { n - 1 } \\frac { d q } { q } . \\end{align*}"}
+{"id": "9007.png", "formula": "\\begin{align*} \\langle V , z _ \\tau \\rangle = v _ \\tau + \\sum _ { q = 1 } ^ { d + 1 } ( - 1 ) ^ q v _ { \\tau _ q } . \\end{align*}"}
+{"id": "3060.png", "formula": "\\begin{align*} \\pi _ 1 ^ * ( Z ) = m _ Z E _ 1 + \\tilde { Z } \\end{align*}"}
+{"id": "8843.png", "formula": "\\begin{align*} \\begin{pmatrix} \\bar { x } _ { t , n } \\\\ \\bar { x } _ { t , 1 } \\\\ \\end{pmatrix} = \\int _ 0 ^ t S ( t - s ) \\begin{pmatrix} F ( s ) \\\\ G ( s ) \\\\ \\end{pmatrix} d s - \\int _ 0 ^ t S ( t - s ) \\begin{pmatrix} 0 \\\\ \\sigma _ 1 d W _ s ^ { ( 1 ) } \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "2016.png", "formula": "\\begin{align*} [ D \\widetilde { H } ] ( x _ 0 , Y _ 0 ) ( x , Y ) = \\big < D \\widetilde { H } ( x _ 0 , Y _ 0 ) , ( x , Y ) \\big > _ { \\mathcal H } \\ , . \\end{align*}"}
+{"id": "2007.png", "formula": "\\begin{align*} \\mathcal { I } & = \\{ ( i , j ) : i \\leq 0 \\mbox { o r } j \\leq 0 \\mbox { o r } i \\leq a , j \\leq b \\} \\subseteq \\Z ^ 2 \\\\ m _ { i , j } & = e _ \\alpha ^ { a - i } \\overline { e } _ \\alpha ^ { b - j } \\cdot m \\mbox { i f } i \\leq a , j \\leq b \\\\ m _ { i , j } & = 0 \\mbox { o t h e r w i s e . } \\end{align*}"}
+{"id": "7312.png", "formula": "\\begin{align*} \\begin{array} { l } R _ { \\max } = \\mathop { \\max } \\limits _ { { \\bf p } , { \\bf y } } { R _ T } \\\\ { \\rm { s } } { \\rm { . t } } { \\rm { . } } \\\\ C 2 , \\ , C 3 , \\ , C 8 , \\ , { \\rm a n d } \\ , C 9 . \\end{array} \\end{align*}"}
+{"id": "7639.png", "formula": "\\begin{align*} \\alpha ^ \\xi _ t : = - R ^ { - 1 } B ( P _ t x ^ \\xi _ t + \\varphi _ t ^ \\xi ) - h ( \\mu _ t ) \\end{align*}"}
+{"id": "9124.png", "formula": "\\begin{align*} ( \\epsilon _ 1 , \\epsilon _ 2 ) = ( ( + , + ) , ( + , - ) ) , ( ( - , + ) , ( - , - ) ) , ( ( + , + ) , ( - , + ) ) , ( ( + , - ) , ( - , - ) ) . \\end{align*}"}
+{"id": "1338.png", "formula": "\\begin{align*} \\alpha _ j = \\alpha _ j ' \\dfrac { d ( a _ j , c _ j ) } { d ( a _ j , b _ j ) } , \\alpha _ { m + j } = \\alpha _ j ' \\dfrac { d ( c _ j , d _ j ) } { d ( a _ j , b _ j ) } , \\alpha _ { 2 m + j } = \\alpha _ j ' \\dfrac { d ( d _ j , b _ j ) } { d ( a _ j , b _ j ) } . \\end{align*}"}
+{"id": "3345.png", "formula": "\\begin{align*} K _ i = \\langle \\{ a _ i , b _ i \\} \\rangle , \\ L _ i = \\langle \\{ b _ i \\} \\rangle . \\end{align*}"}
+{"id": "3984.png", "formula": "\\begin{align*} \\gamma _ i + \\bar { \\gamma } _ 4 = 1 \\ , \\ , { \\rm a n d } \\ , \\ , \\gamma _ s + \\bar { \\gamma } _ t = 1 . \\end{align*}"}
+{"id": "526.png", "formula": "\\begin{align*} U ( x , t _ { 1 } + t _ { 2 } ) = \\int _ { \\R ^ { d } } M ( t _ { 1 } , x , y ) u ( y , t _ { 2 } ) d y . \\end{align*}"}
+{"id": "770.png", "formula": "\\begin{align*} I _ { 2 2 } & \\lesssim \\sum _ { i = 1 } ^ \\infty \\frac { \\ell ( Q ) ^ { 2 s } } { \\ell ( 2 ^ i Q ) ^ { 1 + 2 s } } \\ , \\Big ( \\ell ( 2 ^ i Q ) ^ { \\frac N q + 2 s } \\ , \\ell ( Q ) ^ { \\frac N { q ' } } + i \\ , \\ell ( Q ) ^ N \\ , \\ell ( 2 ^ i Q ) ^ { 2 s } \\Big ) \\\\ & = \\frac { \\ell ( Q ) ^ { 2 s } \\ell ( Q ) ^ { N + 2 s } } { \\ell ( Q ) ^ { 1 + 2 s } } \\sum _ { i = 1 } ^ \\infty \\ , \\left ( 2 ^ { i ( \\frac N q - 1 ) } + \\frac i { 2 ^ i } \\right ) . \\end{align*}"}
+{"id": "6920.png", "formula": "\\begin{align*} g _ 1 ( t ) & : = \\int _ { \\R ^ n } f \\big ( ( 1 - t ) x _ 1 + t T _ 1 ( x _ 1 ) , \\ldots , ( 1 - t ) x _ n + t T _ n ( x _ 1 ) \\big ) \\prod _ { i = 1 } ^ n Q _ i ( d x _ i ) , \\\\ g _ 2 ( t ) & : = H \\big ( M _ 1 ( t ) \\times \\cdots \\times M _ n ( t ) \\big ) = \\sum _ { i = 1 } ^ n H ( M _ i ( t ) ) . \\end{align*}"}
+{"id": "8720.png", "formula": "\\begin{align*} \\left [ g , h ^ n \\right ] = \\left [ g , h ^ { n - 1 } \\right ] \\left [ g , h \\right ] ^ { h ^ { n - 1 } } = \\left [ g , h ^ { n - 1 } \\right ] t ^ { h ^ { n - 1 } } = \\left [ g , h ^ { n - 1 } \\right ] t \\end{align*}"}
+{"id": "7032.png", "formula": "\\begin{align*} \\aligned e _ 1 & \\ , : = \\ , ( 1 - y ) \\ , \\partial _ x + x \\ , \\partial _ u , \\\\ e _ 2 & \\ , : = \\ , ( 1 - y ) \\ , \\partial _ y + u \\ , \\partial _ u , \\\\ e _ 3 & \\ , : = \\ , x \\ , \\partial _ x + 2 \\ , u \\ , \\partial _ u , \\\\ e _ 4 & \\ , : = \\ , - \\ , u \\ , \\partial _ x + x \\ , \\partial _ y , \\endaligned \\end{align*}"}
+{"id": "3423.png", "formula": "\\begin{align*} \\frac { \\tilde { P } _ k ( \\theta ) } { ( \\cos { \\pi \\theta } ) ^ k } = : g _ k ( \\tan \\pi \\theta ) . \\end{align*}"}
+{"id": "1761.png", "formula": "\\begin{align*} \\tilde { w } _ i ( t , x , y ) : = w _ i ( t , x , S _ 0 ^ { - 1 } ( t , x , y ) ) \\mbox { f o r } ( t , x , y ) \\in ( 0 , T ) \\times \\Omega \\times Y ^ { \\ast } ( t , x ) , \\end{align*}"}
+{"id": "4691.png", "formula": "\\begin{align*} { \\rm V o l } ( B ) \\geq \\min \\{ { \\rm V o l } ( A ) , { \\rm V o l } ( B ) \\} = \\mathfrak h ^ { - 1 } { \\rm V o l } ( \\Sigma ) . \\end{align*}"}
+{"id": "350.png", "formula": "\\begin{align*} x _ { j } = ( \\alpha _ { j } , \\beta _ { j } \\ 1 _ { n - 1 } ' , \\ 1 _ { n - 1 } ' ) ^ { ' } , \\end{align*}"}
+{"id": "3523.png", "formula": "\\begin{align*} - n ^ p - \\sum _ { K = 1 } ^ { p } \\binom { p } { p - K } \\left ( 2 \\sum _ { j = 0 } ^ { K / 2 } \\binom { K } { 2 j + 1 } B _ { K - 2 j - 1 } \\right ) \\ , n ^ { p - K } . \\end{align*}"}
+{"id": "3314.png", "formula": "\\begin{align*} \\phi ^ { ( k , \\sigma ) } _ i = \\left \\{ \\begin{aligned} & \\phi _ i & & ( i \\neq k ) \\\\ & \\sigma & & ( i = k ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "1523.png", "formula": "\\begin{align*} ( \\varphi _ 1 \\cdot \\varphi _ 2 ) ( Z _ i ) = F _ i \\big ( \\varphi _ 1 ( Z _ 1 ) , \\ldots , \\varphi _ 1 ( Z _ d ) , \\varphi _ 2 ( Z _ 1 ) , \\ldots , \\varphi _ 2 ( Z _ d ) \\big ) , \\end{align*}"}
+{"id": "8490.png", "formula": "\\begin{align*} & \\sup _ { x \\in \\mathbb { R } } \\left \\| \\mathcal { P } ^ + \\left ( z ^ { - i } \\bar { r } _ 1 ( z ) e ^ { - 2 i z x } \\right ) \\right \\| _ { L _ z ^ \\infty } \\leq \\frac { 1 } { \\sqrt { 2 } } \\left \\| z ^ { - i } r _ 1 ( z ) \\right \\| _ { H ^ 1 } , i = 0 , 1 , \\\\ & \\sup _ { x \\in \\mathbb { R } } \\left \\| \\mathcal { P } ^ - \\left ( r _ 2 ( z ) \\mathrm { e } ^ { 2 i z x } \\right ) \\right \\| _ { L _ { z } ^ { \\infty } } \\leq \\frac { 1 } { \\sqrt { 2 } } \\left \\| r _ 2 ( z ) \\right \\| _ { H ^ { 1 } } . \\end{align*}"}
+{"id": "8898.png", "formula": "\\begin{align*} \\mathbf { h } = [ h _ 1 , h _ 2 , h _ 3 , h _ { 1 2 } , h _ { 1 3 } , h _ { 2 3 } , h _ { 1 2 3 } ] ^ \\intercal . \\end{align*}"}
+{"id": "5485.png", "formula": "\\begin{align*} - \\frac { 3 ^ { q + 2 } } { 2 ^ q \\cdot q ( 1 + q ) } h _ q '' ( 0 ) & = \\Gamma ( 2 - q ) - ( 3 / 2 ) ^ { q + 1 } q ( 1 - q ) \\\\ & > 0 . 8 8 - ( 3 / 2 ) ^ 2 \\cdot \\frac { 1 } { 4 } = 0 . 3 1 7 5 . \\end{align*}"}
+{"id": "2260.png", "formula": "\\begin{align*} A D ( x ) + D ( x ) B = D ( v ) . \\end{align*}"}
+{"id": "8141.png", "formula": "\\begin{align*} d _ n ( f ) ( g _ 0 , \\dots , g _ { n + 1 } ) ) = \\sum _ { i = 0 } ^ n ( - 1 ) ^ i f ( g _ 0 , \\dots , \\hat g _ i , \\dots , g _ { n + 1 } ) , . \\end{align*}"}
+{"id": "2732.png", "formula": "\\begin{align*} Q : = \\left \\{ x \\in \\R ^ { d - 1 } \\colon \\| x \\| _ \\infty \\leq 1 \\right \\} = [ - 1 , 1 ] ^ { d - 1 } \\end{align*}"}
+{"id": "2572.png", "formula": "\\begin{align*} p ^ { n } ( l _ { k _ { 1 } } + p ^ { l _ { t } } b _ { t } ) = p ^ { l _ { k } } { b _ { k } } q ^ { n } . \\end{align*}"}
+{"id": "5623.png", "formula": "\\begin{align*} f \\vee g = \\frac { 1 } { 2 } \\big ( f + g + | f - g | \\big ) , \\end{align*}"}
+{"id": "8828.png", "formula": "\\begin{align*} \\lambda _ { - , 1 } = - 2 \\sqrt { \\delta ( 1 - \\delta ) } , E _ { - , 1 } = \\left \\{ \\begin{pmatrix} \\bar { a } _ 1 \\sqrt { \\frac { \\delta } { \\delta - 1 } } \\\\ - \\bar { b } _ 1 \\sqrt { \\frac { \\delta } { \\delta - 1 } } \\\\ 0 \\\\ 1 \\end{pmatrix} , \\begin{pmatrix} - \\bar { b } _ 1 \\sqrt { \\frac { \\delta } { \\delta - 1 } } \\\\ - \\bar { a } _ 1 \\sqrt { \\frac { \\delta } { \\delta - 1 } } \\\\ 1 \\\\ 0 \\end{pmatrix} \\right \\} . \\end{align*}"}
+{"id": "7130.png", "formula": "\\begin{align*} \\begin{aligned} a & = f f '' - 2 ( f ' ) ^ 2 \\\\ b & = - 2 f f ' - t ( f f '' - 2 ( f ' ) ^ 2 ) \\\\ c & = f '' \\\\ d & = - 2 f ' - t f '' \\end{aligned} \\end{align*}"}
+{"id": "6297.png", "formula": "\\begin{align*} \\gamma _ { i - 1 } \\alpha _ i ^ 2 = ( 1 - \\alpha _ i ) \\alpha _ { i - 1 } ^ 2 \\gamma _ i + \\mu \\alpha _ i \\gamma _ i \\gamma _ { i - 1 } \\geq ( 1 - \\alpha _ i ) \\mu \\gamma _ { i - 1 } \\gamma _ i + \\mu \\alpha _ i \\gamma _ i \\gamma _ { i - 1 } = \\mu \\gamma _ i \\gamma _ { i - 1 } , \\end{align*}"}
+{"id": "5266.png", "formula": "\\begin{align*} \\gamma \\cdot f ( x , y ) = \\frac { f ( ( x , y ) \\gamma ) } { | \\det \\gamma | } , \\end{align*}"}
+{"id": "4846.png", "formula": "\\begin{align*} \\mathcal U ( s , i ) = [ 0 , \\bar L ] , ( s , i ) \\in C . \\end{align*}"}
+{"id": "5509.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { ( x + n ) ^ 2 } - 2 \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { ( x + n + \\frac 1 2 ) ^ 2 } > \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { ( \\frac 1 2 + n ) ^ 2 } - 2 \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { ( n + \\frac 1 2 ) ^ 2 } = - \\frac { \\pi ^ 2 } { 2 } + 4 , \\end{align*}"}
+{"id": "1745.png", "formula": "\\begin{align*} \\begin{aligned} \\langle \\partial _ t ( \\bar { J } _ 0 u _ 0 ) , & \\phi \\rangle _ { H ^ 1 ( \\Omega ) ' , H ^ 1 ( \\Omega ) } - \\int _ { \\Omega } q u _ 0 \\phi d x + \\int _ { \\Omega } D _ 0 ^ { \\ast } \\nabla u _ 0 \\cdot \\nabla \\phi d x \\\\ & = \\int _ { \\Omega } \\int _ { Y ^ { \\ast } } J _ 0 f _ 0 d y \\phi d x - \\int _ { \\Omega } \\partial _ t R _ 0 ( u _ 0 - \\rho ) \\phi \\vert \\Gamma ( t , x ) \\vert d x . \\end{aligned} \\end{align*}"}
+{"id": "2060.png", "formula": "\\begin{align*} \\sum _ { \\ell + 2 p = j , \\ \\ell + 2 q = 2 k - j } c ^ { k , j } _ { \\ell , p , q } = { { 2 k } \\choose j } . \\end{align*}"}
+{"id": "7487.png", "formula": "\\begin{align*} \\eta _ { \\delta } = \\min \\{ | x | _ { A ^ { - 1 } ( 0 ) } ^ { - a / 2 } , \\delta ^ { - a / 2 } \\} \\zeta \\end{align*}"}
+{"id": "6082.png", "formula": "\\begin{align*} V ' = \\{ v \\in V \\mid ( v , e _ 1 ) \\in E ( v , f _ 1 ) \\in E \\} . \\end{align*}"}
+{"id": "4695.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ s { \\rm V o l } ( \\Sigma \\cap B _ { p _ i } ( 3 r ) ) \\leq C _ o ( n ) { \\rm V o l } ( \\Sigma ) . \\end{align*}"}
+{"id": "5183.png", "formula": "\\begin{align*} I _ { h , W , \\gamma } = \\left ( \\frac { W } { 2 } \\right ) ^ { - 2 } I _ { \\tilde { h } , 2 , \\tilde { \\gamma } } . \\end{align*}"}
+{"id": "5260.png", "formula": "\\begin{align*} g ^ \\iota = \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} ( g ^ { - 1 } ) ^ t \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "8130.png", "formula": "\\begin{align*} M ( S ) = \\varinjlim _ i M ( S _ i ) . \\end{align*}"}
+{"id": "4618.png", "formula": "\\begin{align*} \\mathcal D _ { E / F } = \\prod _ { \\mathfrak p \\in \\mathcal R _ { E / F } } \\mathfrak p ^ { e _ { \\mathfrak p } - 1 } . \\end{align*}"}
+{"id": "2601.png", "formula": "\\begin{align*} R \\cdot R ( X , J X , J X , X ; U , J U ) = f \\ , \\ , Q ^ c ( g , R ) ( X , J X , J X , X ; U , J U ) \\end{align*}"}
+{"id": "2633.png", "formula": "\\begin{align*} [ \\alpha ( x ) , y \\cdot z ] = \\varepsilon ( x , y ) \\alpha ( y ) \\cdot [ x , z ] + \\varepsilon ( x + y , z ) \\alpha ( z ) \\cdot [ x , y ] . \\end{align*}"}
+{"id": "8723.png", "formula": "\\begin{align*} \\overline { \\pi } ( q _ 1 ) \\overline { \\pi } ( q _ 2 ) = c _ \\chi ( q _ 1 , q _ 2 ) \\overline { \\pi } ( q _ 1 q _ 2 ) \\forall q _ 1 , q _ 2 \\in Q . \\end{align*}"}
+{"id": "7230.png", "formula": "\\begin{align*} \\sigma ( \\alpha + \\beta ) = \\sigma ( \\alpha ) + \\sigma ( \\beta ) \\end{align*}"}
+{"id": "7724.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { l } ( 1 - Z _ i ) \\ge \\exp \\big ( - 2 \\sum _ { i = 1 } ^ { l } Z _ i \\big ) = & \\exp \\bigg ( - 2 \\sum _ { s = 1 } ^ { \\lceil \\log _ 2 ( l / 2 ) \\rceil } 2 ^ s \\log \\big ( \\tau ^ { - 1 / 2 ^ s } l / 2 ^ s \\big ) \\frac { 3 } { l } \\bigg ) . \\end{align*}"}
+{"id": "7715.png", "formula": "\\begin{align*} \\| v \\| _ 2 & = \\sqrt { \\| P _ { F _ r ^ \\perp } v \\| _ 2 ^ 2 + \\| P _ { F _ r } v \\| _ 2 ^ 2 } \\le \\| P _ { F _ r ^ \\perp } v \\| _ 2 \\sqrt { 1 ^ 2 + \\Big ( \\frac { 1 } { 4 } \\frac { n ^ { - \\beta / ( 4 0 p ) } } { 3 \\sqrt { \\beta n } } \\Big ) ^ { - 2 } } \\\\ & \\le O ( \\sqrt { \\beta } ) \\ , n ^ { \\beta / ( 4 0 p ) + \\frac { 1 } { 2 } } \\ , \\| P _ { F _ r ^ \\perp } v \\| _ 2 , \\end{align*}"}
+{"id": "8847.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } X _ { t , 1 } = X _ { t , 1 } X _ { t , 3 } \\\\ \\frac { d } { d t } X _ { t , 3 } = - X _ { t , 1 } ^ 2 \\\\ ( X _ { 0 , 1 } , 0 , X _ { 0 , 3 } ) = \\Pi _ { \\mathrm { k e r } A } x _ 0 . \\end{cases} \\end{align*}"}
+{"id": "7574.png", "formula": "\\begin{align*} B _ 1 + \\cdots + B _ k = \\{ b _ 1 + \\cdots + b _ k : b _ 1 \\in B _ 1 , \\dots , b _ k \\in B _ k \\} \\end{align*}"}
+{"id": "3262.png", "formula": "\\begin{align*} \\mathbb E \\left [ \\sup _ { s \\leq t } \\| \\int _ 0 ^ t \\sigma _ s d W _ s \\| ^ m \\right ] & \\leq C _ m \\mathbb E \\left [ \\left ( \\int _ 0 ^ t ( \\Sigma _ s ) d s \\right ) ^ { \\frac m 2 } \\right ] \\\\ & = C _ m \\mathbb E \\left [ \\left ( \\int _ 0 ^ t \\| \\sigma _ s \\| _ { L _ { } ( U , H ) } ^ 2 d s \\right ) ^ { \\frac m 2 } \\right ] , \\end{align*}"}
+{"id": "6322.png", "formula": "\\begin{align*} z _ 0 ^ { p ^ r } x _ 0 ^ { q ^ { m _ i } } + z _ 1 ^ { p ^ r } x _ 1 ^ { q ^ { m _ i } } + \\cdots + z _ n ^ { p ^ r } x _ n ^ { q ^ { m _ i } } = 0 . \\end{align*}"}
+{"id": "1451.png", "formula": "\\begin{align*} u _ { t } = \\left ( \\frac { u _ { x } } { 1 + u _ { x } ^ { 2 } } \\right ) _ { x } = \\frac { 1 - u _ { x } ^ { 2 } } { ( 1 + u _ { x } ^ { 2 } ) ^ { 2 } } \\ , u _ { x x } , \\end{align*}"}
+{"id": "4827.png", "formula": "\\begin{align*} ( d / d t ) v + A v = 0 , v \\in \\R ^ D , \\end{align*}"}
+{"id": "7236.png", "formula": "\\begin{align*} a ( x ) = \\sum _ { i = 0 } ^ r a _ i x ^ i \\end{align*}"}
+{"id": "2519.png", "formula": "\\begin{align*} \\mathbf { d } _ { h x } = 0 , \\mathbf { d } _ { h s } = 0 . \\end{align*}"}
+{"id": "9275.png", "formula": "\\begin{align*} \\int _ \\Omega \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ m \\wedge \\beta _ n ^ { n - m } = \\int _ \\Omega \\Delta v _ 1 \\wedge \\dots \\wedge \\Delta v _ m \\wedge \\beta _ n ^ { n - m } . \\end{align*}"}
+{"id": "962.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\Phi ( v _ { 1 } , v _ { 2 } ) \\| _ { { { \\bf X } } _ T } & \\le \\tilde { C } ( T ^ { - 2 \\tilde { \\nu } - \\frac 1 2 } + ( \\varepsilon _ { 1 } + \\varepsilon _ { 2 } ) T ^ { - \\tilde { \\nu } - \\frac 1 4 } + \\varepsilon _ { 1 } \\varepsilon _ { 2 } T ^ { - \\frac { \\delta } 2 } \\\\ & \\ \\ \\ \\ \\quad + ( \\varepsilon _ { 1 } + \\varepsilon _ { 2 } ) T ^ { \\tilde { \\nu } - \\frac 1 2 } ( \\log T ) ^ 2 + \\varepsilon _ { 1 } ^ 2 ) \\end{aligned} \\end{align*}"}
+{"id": "1824.png", "formula": "\\begin{align*} ( d ^ \\nabla \\sigma ) ( X _ 1 , X _ 2 ) = ( \\nabla _ { X _ 1 } \\sigma ) ( X _ 2 ) - ( \\nabla _ { X _ 2 } \\sigma ) ( X _ 1 ) \\ , . \\end{align*}"}
+{"id": "3333.png", "formula": "\\begin{align*} \\chi _ i = \\left \\{ \\begin{aligned} & \\psi _ i & & ( i \\neq j ) \\\\ & \\sigma & & ( i = j ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "1349.png", "formula": "\\begin{align*} t _ 1 = \\max \\bigl \\{ t \\colon d ( \\gamma _ { a _ 1 , b _ 1 } ( t ) , a _ 1 ) = R \\bigr \\} \\end{align*}"}
+{"id": "8398.png", "formula": "\\begin{align*} & \\lim _ { | z | \\rightarrow 0 } \\Psi ^ { \\pm } ( x ; z ) = \\begin{pmatrix} 1 & - u _ x ( x ) \\\\ - \\bar { u } _ x ( x ) & 1 \\end{pmatrix} , \\\\ [ 4 p t ] & \\lim _ { | z | \\rightarrow \\infty } \\Psi ^ { \\pm } ( x ; z ) = e ^ { - i c _ \\pm \\sigma _ 3 } . \\end{align*}"}
+{"id": "9059.png", "formula": "\\begin{align*} = [ X _ { \\beta _ { m - k + 1 } } , X _ { - \\beta _ { m - k + 1 } } ] v _ { k - 1 } + \\sum _ { l = 1 } ^ { k - 1 } ( - ) ^ { k - l } X _ { \\beta _ { m - k + 1 } } \\cdots X _ { \\beta _ { m - l } } [ X _ { \\beta _ { m - l + 1 } } , X _ { - \\beta _ { m - k + 1 } } ] v _ { l - 1 } \\end{align*}"}
+{"id": "2837.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\frac { d } { d t } \\| \\Lambda ^ s & \\theta ( t ) \\| ^ 2 + \\mu \\| \\Lambda ^ { s + 1 } \\theta ( t ) \\| ^ 2 \\\\ & = - \\int _ { \\R ^ 3 } \\Lambda ^ s ( ( u \\cdot \\nabla ) \\theta ) \\Lambda ^ s \\theta \\ , d x - \\int _ { \\R ^ 3 } \\Lambda ^ s ( \\mathrm { d i v } \\ , v ) \\Lambda ^ s \\theta \\ , d x . \\end{aligned} \\end{align*}"}
+{"id": "3772.png", "formula": "\\begin{align*} \\mathcal { P } _ s = \\{ V _ 1 \\subset V _ 2 \\subset \\ldots V _ { l - 1 } \\subset V _ { l + 1 } \\subset \\ldots \\subset V _ { n } = \\mathbb { C } ^ n ; \\dim _ { \\mathbb { C } } ( V _ i ) = i \\} . \\end{align*}"}
+{"id": "7222.png", "formula": "\\begin{align*} u ( t ) = u _ 0 + \\int _ 0 ^ t \\Delta _ p u \\ , \\ , d s + \\int _ 0 ^ t S ( u ) \\ , \\ , d s + \\sum _ { n = 0 } ^ { + \\infty } \\int _ 0 ^ t \\xi _ { n } ( x ) \\cdot \\nabla u ( s ) \\ , d B _ s ^ { n } \\end{align*}"}
+{"id": "8052.png", "formula": "\\begin{align*} \\begin{gathered} \\min _ { \\mathbf { a } } \\mathbb { E } \\left [ \\lVert \\mathbf { s } ^ { \\left ( \\right ) } - \\mathbf { y } ' \\rVert ^ 2 \\right ] \\\\ ~ ~ \\left ( \\mathbf { P } ^ { \\left ( \\right ) } \\left ( \\mathbf { a } ^ { \\left ( \\right ) } \\odot \\mathbf { a } ^ { \\left ( \\right ) } \\right ) \\mathbf { P } ^ { \\left ( \\right ) ^ { H } } \\right ) = E _ { t r } , \\end{gathered} \\end{align*}"}
+{"id": "1266.png", "formula": "\\begin{align*} \\varphi _ n ( \\eta ) = \\varphi _ n ( \\mathbf { 1 } ) + \\sum _ { m = 0 } ^ \\infty ( \\varphi _ n ( r _ m \\eta ) - \\varphi _ n ( r _ { m + 1 } \\eta ) ) , \\end{align*}"}
+{"id": "4407.png", "formula": "\\begin{align*} A _ 4 = A _ 2 ^ 2 \\left ( \\frac { 1 } { 2 } + \\frac { 2 A _ 1 ^ 2 c _ 3 } { \\nu ^ 2 } + \\frac { A _ 3 } { 8 } + \\left ( c _ 2 \\nu ^ 3 \\log ^ 3 T _ 0 + \\frac { A _ 1 ^ 2 / 6 0 0 0 } { \\nu ^ 2 \\log ^ 2 T _ 0 } \\right ) \\frac { 1 } { T _ 0 ^ { 1 - 2 \\nu } } + \\frac { c _ 1 ^ 2 e ^ { - 2 \\alpha } / 6 0 0 0 } { \\nu ^ 2 T _ 0 ^ { 5 \\nu } \\log ^ 2 T _ 0 } \\right ) + \\frac { 2 1 5 } { 1 6 T _ 0 } , \\end{align*}"}
+{"id": "5514.png", "formula": "\\begin{align*} \\beta _ { p , d } = \\frac { \\sqrt { \\pi } \\Gamma \\left ( \\frac { d - p } { 2 } \\right ) } { \\Gamma \\left ( \\frac { 1 - p } { 2 } \\right ) \\Gamma \\left ( \\frac { d } { 2 } \\right ) } . \\end{align*}"}
+{"id": "8681.png", "formula": "\\begin{align*} \\mu _ k ^ { \\star } = { \\left [ { \\frac { 1 } { \\nu } - \\frac { 1 } { { { \\left \\| { { { \\bf { \\bar e } } _ k } } \\right \\| } ^ 2 } } } \\right ] ^ + } , \\ \\forall k , \\end{align*}"}
+{"id": "3408.png", "formula": "\\begin{align*} \\| q _ { k - 1 } \\alpha \\| = \\min _ { 1 \\leq n < q _ k } \\| n \\alpha \\| , \\end{align*}"}
+{"id": "6803.png", "formula": "\\begin{align*} y ( \\tau ) = \\left ( z ( \\tau ) + 1 \\right ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } z ^ { n } ( \\tau ) a _ { n } ( x ) = \\frac { 1 } { \\frac { 1 } { 2 } - i \\tau } \\sum _ { n = 0 } ^ { \\infty } \\left ( - 1 \\right ) ^ { n } z ^ { n } ( \\tau ) a _ { n } ( x ) . \\end{align*}"}
+{"id": "6140.png", "formula": "\\begin{align*} f ( x , \\eta ^ { - a } x , z ) - f ( x , \\eta ^ b z , z ) = & d ^ { a + b } _ 1 \\eta ^ { - a } h _ { 0 0 } ( f ( x , y , z ) ) + \\eta ^ { a } h _ { 1 0 } ( d ^ a _ 0 ( x , y ) f ( x , y , z ) \\\\ & + h _ { 0 1 } ( d ^ b _ { 0 } ( y , z ) f ( x , y , z ) ) . \\end{align*}"}
+{"id": "5954.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u ( t ) = - \\Delta ^ 2 u + \\Delta \\varphi ( u ) , \\\\ \\nabla u \\cdot \\nu = \\nabla ( \\Delta u ) \\cdot \\nu = 0 \\ \\partial \\mathcal { O } , \\\\ u ( 0 ) = u _ 0 , \\end{cases} \\end{align*}"}
+{"id": "3784.png", "formula": "\\begin{align*} A _ q ( z , x ) = A _ q ^ z ( x ) : = \\displaystyle \\sum _ { n = 0 } ^ \\infty \\frac { \\overline { z } ^ n } { \\sqrt { \\Gamma ( q n + 1 ) } } \\psi _ n ( x ) , \\end{align*}"}
+{"id": "7392.png", "formula": "\\begin{align*} [ \\mathbb C _ x ] = \\prod _ { i = 0 } ^ r e ^ { x _ i \\omega _ i } \\end{align*}"}
+{"id": "431.png", "formula": "\\begin{align*} a _ { r } = \\sum _ { j = 1 } ^ { n } \\delta _ { r j } a _ { j } , \\end{align*}"}
+{"id": "1440.png", "formula": "\\begin{align*} K _ { \\lambda } = \\chi \\int _ { W ^ d } e ^ { - \\sum _ { j = 1 } ^ d ( \\lambda _ j - \\bar { \\lambda } ) z _ j } \\hat { h } ^ { ( \\bar { \\lambda } , \\ldots , \\bar { \\lambda } ) } ( z ) d z _ 1 \\ldots d z _ d . \\end{align*}"}
+{"id": "3298.png", "formula": "\\begin{align*} \\mathbb E \\left [ \\| \\Delta _ n ^ { - \\frac 1 2 } ( 3 ) _ t \\| _ { \\mathcal H } ^ 2 \\right ] = & 2 \\Delta _ n \\sum _ { i = 1 } ^ n ( i - 1 ) \\int _ { ( i - 1 ) \\Delta _ n } ^ { i \\Delta _ n } \\mathbb E \\left [ \\| \\sigma _ s ^ { \\mathcal S _ n } \\| _ { L _ { } ( U , H ) } ^ 2 \\right ] d s \\leq 2 \\sup _ { r \\in [ 0 , T ] } \\| \\mathcal S ( r ) \\| _ { } ^ 2 . \\end{align*}"}
+{"id": "4502.png", "formula": "\\begin{align*} Z _ { X } ( [ \\alpha ] ) = - \\frac { 1 } { 2 } \\int _ { V } ( \\alpha ^ { 2 } - \\omega ^ { 2 } ) + \\sqrt { - 1 } \\int _ { X } \\alpha \\wedge \\omega = - 1 6 + 1 3 \\sqrt { - 1 } . \\end{align*}"}
+{"id": "5755.png", "formula": "\\begin{align*} \\partial _ X \\langle \\langle \\varphi , \\varphi ' \\rangle \\rangle = \\langle \\langle \\nabla _ X \\varphi , \\varphi ' \\rangle \\rangle + \\langle \\langle \\varphi , \\nabla _ X \\varphi ' \\rangle \\rangle \\end{align*}"}
+{"id": "5784.png", "formula": "\\begin{align*} ( T _ 1 + f _ 1 ) \\cdot ( T _ 2 - f _ 2 ) \\cdot \\Psi _ 2 = i \\Psi _ 2 \\end{align*}"}
+{"id": "8954.png", "formula": "\\begin{align*} \\| { F } ^ { ( \\delta ) } _ { 1 , N } v _ 0 \\| _ p ^ p \\le & \\sum _ { | k | \\le N } \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 0 ^ { p } ( x ) \\biggl ( \\int _ { \\eta _ { k - 1 } } ^ x v _ 1 ^ { - p ' } ( t ) \\bigl | G ^ { ( \\delta ) } _ { 1 , k } ( t ) \\bigr | ^ { p ' - 1 } \\ , d t \\biggr ) ^ p \\ , d x \\\\ \\lesssim & \\sum _ { | k | \\le N } \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { - p ' } \\bigl | G _ { 1 , k } ^ { ( \\delta ) } \\bigr | ^ { p ' } = : \\bigl [ \\mathbf { G } ^ { ( \\delta ) } _ { 1 , N } ( g ) \\bigr ] ^ { p ' } . \\end{align*}"}
+{"id": "3779.png", "formula": "\\begin{align*} \\tau ( K ) : = \\bigcap _ { K \\subset U } \\ker ( r : S H _ { M } ( M ; \\Lambda ) \\rightarrow S H _ { M } ( M - U ; \\Lambda ) ) \\end{align*}"}
+{"id": "2919.png", "formula": "\\begin{align*} & \\mathcal A z + \\frac { \\partial f } { \\partial y } ( x , y _ u ) z = v , \\\\ & \\mathcal A z + \\frac { \\partial f } { \\partial y } ( x , y _ u ) z = - \\frac { \\partial ^ 2 f } { \\partial y ^ 2 } ( x , y _ u ) z _ { u , v _ 1 } z _ { u , v _ 2 } , \\end{align*}"}
+{"id": "1225.png", "formula": "\\begin{align*} c _ \\Delta : = \\sup _ { \\xi _ \\Delta } \\norm { c _ \\Delta ( \\cdot , \\xi _ \\Delta ) } _ \\infty > 0 . \\end{align*}"}
+{"id": "5255.png", "formula": "\\begin{align*} ( \\ell ^ { \\infty } , \\beta _ { 0 } ) ' = ( \\ell ^ { \\infty } , \\mu ( \\ell ^ { \\infty } , \\ell ^ { 1 } ) ) ' = \\ell ^ { 1 } . \\end{align*}"}
+{"id": "4223.png", "formula": "\\begin{align*} \\sum _ { x y \\ne 1 } f _ { x , y } = A + B , \\sum _ { z \\in \\bar { \\mathbb { F } } ^ * _ q } g _ { z } = C , \\sum _ { u \\in \\bar { \\mathbb { F } } _ q } h _ { u } = 0 . \\end{align*}"}
+{"id": "279.png", "formula": "\\begin{align*} u ( p ) + H ( p , u ( p ) , \\nabla _ H u ( p ) ) = f ( p ) , \\end{align*}"}
+{"id": "8836.png", "formula": "\\begin{align*} \\begin{cases} d X _ { t , n } = X _ { t , 1 } X _ { t , n - 1 } d t + \\sigma _ { n } d W _ t ^ { ( n ) } \\\\ d X _ { t , n - 1 } = \\sigma _ { n - 1 } d W _ t ^ { ( n - 1 ) } \\\\ d X _ { t , j } = 0 & j \\not \\in \\{ n , n - 1 \\} \\end{cases} \\end{align*}"}
+{"id": "947.png", "formula": "\\begin{align*} \\begin{aligned} { \\| u _ 2 ( t ) \\| _ { H ^ 1 } } \\le { } & \\| u _ 2 ( 1 ) \\| _ { H ^ 1 } \\\\ & { } + C \\int ^ t _ 1 ( \\| u _ 1 \\| ^ 2 _ { L ^ { \\infty } } \\| u _ 2 \\| _ { H ^ 1 } + \\| u _ 1 \\| _ { L ^ { \\infty } } \\| u _ 2 \\| _ { L ^ { \\infty } } \\| u _ 1 \\| _ { H ^ 1 } ) ( \\tau ) \\ d \\tau . \\end{aligned} \\end{align*}"}
+{"id": "7592.png", "formula": "\\begin{align*} a \\in F \\textup { a n d } \\pi ( b ) = \\pi ( a ) \\implies b \\in F \\end{align*}"}
+{"id": "7966.png", "formula": "\\begin{align*} \\nabla h ( \\Omega , x ) = h _ { i } e _ { i } + h x , F _ { i } ( x ) = ( h _ { i j } + h \\delta _ { i j } ) e _ { j } . \\end{align*}"}
+{"id": "2593.png", "formula": "\\begin{align*} T _ 1 ( u , J u , J u , u , v ) = T _ 2 ( u , J u , J u , u , v ) , \\end{align*}"}
+{"id": "2888.png", "formula": "\\begin{align*} & { \\frak C } \\sum _ { x = 0 } ^ n | \\tilde V _ x ( m ) | ^ 2 \\le \\left [ | \\tilde V _ 0 ( m ) | \\left \\{ \\sum _ { x = 0 } ^ n | M _ { x , 0 } ( m ) | ^ 2 \\right \\} ^ { 1 / 2 } + \\left \\{ \\sum _ { x = 0 } ^ n | \\tilde v _ { x } ( m ) | ^ 2 \\right \\} ^ { 1 / 2 } \\right ] \\left \\{ \\sum _ { x = 0 } ^ n | \\tilde V _ x ( m ) | ^ 2 \\right \\} ^ { 1 / 2 } , \\end{align*}"}
+{"id": "2643.png", "formula": "\\begin{align*} x \\diamond y = x \\cdot D ( y ) , \\forall x , y \\in \\mathcal { H } ( A ) . \\end{align*}"}
+{"id": "7797.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Theta ( t , x ) & = \\int _ \\S W _ r ( x - y ) \\sin ( \\Theta ( t , y ) - \\Theta ( t , x ) ) \\ \\d y \\\\ & + \\lambda \\int _ \\S \\int _ \\S W _ r ( z + y - 2 x ) \\sin ( \\Theta ( t , z ) + \\Theta ( t , y ) - 2 \\Theta ( t , x ) ) \\ \\d y \\d z \\\\ & + \\mu \\int _ \\S \\int _ \\S \\int _ \\S W _ r ( z - y + w - x ) \\sin ( \\Theta ( t , z ) - \\Theta ( t , y ) + \\Theta ( t , w ) - \\Theta ( t , x ) ) \\ \\d w \\d y \\d z , \\end{align*}"}
+{"id": "6185.png", "formula": "\\begin{align*} W ' ( r ) = \\frac { \\xi ' } { r } f + \\eta ' + \\zeta ' \\frac { r } { f } , \\xi ' \\le 0 , \\zeta ' > 0 . \\end{align*}"}
+{"id": "4421.png", "formula": "\\begin{align*} | ( k + 1 ) ^ { 1 - s } - k ^ { 1 - s } | = | 1 - s | \\left \\lvert \\int _ k ^ { k + 1 } y ^ { - s } \\dd y \\right \\rvert \\asymp | 1 - s | k ^ { - \\Re ( s ) } \\end{align*}"}
+{"id": "874.png", "formula": "\\begin{align*} x ' = 2 \\ , . \\end{align*}"}
+{"id": "3070.png", "formula": "\\begin{align*} ( 1 + h ) \\cdot p ( h ) = q \\Big ( \\frac { h } { 1 + h } \\Big ) \\in A _ * \\P ^ n . \\end{align*}"}
+{"id": "157.png", "formula": "\\begin{align*} \\binom { n - c - r d } { r } \\ : = \\ : \\frac { n ^ r } { r ! } \\ , e ^ { - ( 2 d + 1 ) x } \\ , ( 1 + o ( 1 ) ) \\ ; \\ ; \\ ; \\ ; { \\rm a s } \\ ; r \\to \\infty \\ , . \\end{align*}"}
+{"id": "6582.png", "formula": "\\begin{align*} \\begin{aligned} 3 \\int _ { \\Omega } n \\log n & ( S ( x , n , c ) \\cdot \\nabla c ) \\cdot \\nabla \\varphi \\varphi ^ { 2 } \\\\ & \\le \\frac { 1 } { 4 } C _ { 3 } \\| \\nabla c \\| _ { L ^ { 2 } ( \\Omega \\cap B _ { \\delta } ) } \\int _ { \\Omega } \\frac { | \\nabla n | ^ { 2 } } { n } \\varphi ^ { 3 } + \\frac { 1 } { 5 } \\int _ { \\Omega } \\frac { | \\nabla n | ^ { 2 } } { n } \\varphi ^ { 3 } + M . \\end{aligned} \\end{align*}"}
+{"id": "710.png", "formula": "\\begin{align*} \\aligned \\| u _ n ^ + \\| _ { E _ a } ^ 2 = & \\langle \\Phi ' _ { j _ n } ( u _ n ) , u _ n ^ + \\rangle + \\int ( I _ \\alpha \\ast | u _ n | ^ p ) | u _ n | ^ { p - 2 } u _ n u _ n ^ + d x \\\\ \\leq & 1 \\cdot \\| u _ n ^ + \\| _ { E _ a } + \\| u _ n \\| _ { \\mathcal Q ^ { \\alpha , p } } ^ { 2 p - 1 } \\| u _ n ^ + \\| _ { \\mathcal Q ^ { \\alpha , p } } \\\\ \\leq & \\| u _ n ^ + \\| _ { E _ a } + [ C ( 1 + \\| u _ n \\| _ { E _ a } ) ] ^ { 1 - \\frac { 1 } { 2 p } } \\| u _ n ^ + \\| _ { \\mathcal Q ^ { \\alpha , p } } \\endaligned \\end{align*}"}
+{"id": "7382.png", "formula": "\\begin{align*} W = \\bigoplus _ { \\alpha = 1 } ^ r \\Omega ^ { a ( \\alpha ) } \\end{align*}"}
+{"id": "6132.png", "formula": "\\begin{align*} \\widetilde { q } _ { 2 d } ( 2 k ) = e ^ { \\frac { i \\pi ( 2 k ) ^ 2 } { 2 d } } = q _ d ( k ) . \\end{align*}"}
+{"id": "7921.png", "formula": "\\begin{align*} c _ 1 ( q , 1 , p _ 0 ) & = \\frac { 1 } { 4 } ( \\hat W _ { r _ 0 } ( q - 1 ) - 2 \\hat W _ { r _ 0 } ( q ) + \\hat W _ { r _ 0 } ( q + 1 ) ) = 0 \\\\ c _ 1 ( q , 2 , p _ 0 ) & = \\frac { 1 } { 4 } ( \\hat W _ { r _ 0 } ( q - 2 ) - 2 \\hat W _ { r _ 0 } ( q ) + \\hat W _ { r _ 0 } ( q + 2 ) ) < 0 \\\\ c _ 1 ( q + 1 , 1 , p _ 0 ) & = \\frac { 1 } { 4 } ( \\hat W _ { r _ 0 } ( q ) - 2 \\hat W _ { r _ 0 } ( q + 1 ) + \\hat W _ { r _ 0 } ( q + 2 ) ) > 0 \\end{align*}"}
+{"id": "3789.png", "formula": "\\begin{align*} | | B _ q ( \\psi _ m ) | | _ { M L _ q ( \\mathbb { C } ) } = 1 = | | \\psi _ m | | _ { L ^ 2 ( \\mathbb { R } ) } . \\end{align*}"}
+{"id": "2589.png", "formula": "\\begin{align*} T ^ v ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) = T _ 1 ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , v , J v ) . \\end{align*}"}
+{"id": "6966.png", "formula": "\\begin{align*} \\prod ^ e _ { \\substack { i = 1 \\\\ ( i , e ) = 1 } } \\Phi _ { n , e } ^ { ( i ) } ( x , y ) = \\Phi _ { n e } ( x , y ) . \\end{align*}"}
+{"id": "199.png", "formula": "\\begin{align*} f _ { \\alpha } : = D _ { \\theta _ { ( 2 ) } } \\left ( \\frac { \\theta _ { \\alpha , ( 2 ) } ( f ) } { \\alpha ! } \\right ) , \\end{align*}"}
+{"id": "3390.png", "formula": "\\begin{align*} c _ k = \\frac 1 2 \\sum _ { i = k - N } ^ N ( 2 i - k ) ^ 2 = \\frac 1 2 \\sum _ { i = 0 } ^ { 2 N - k } ( 2 i - ( 2 N - k ) ) ^ 2 = { 2 N - k + 2 \\choose 3 } , \\end{align*}"}
+{"id": "7567.png", "formula": "\\begin{align*} x = \\sum _ { m = 0 } ^ \\infty \\frac { \\psi ( \\eta _ m ) } { \\gamma ^ { m + 1 } } . \\end{align*}"}
+{"id": "3327.png", "formula": "\\begin{align*} | \\phi | & = \\sum _ { i \\in \\overline { \\phi } } | \\phi _ i | + \\sum _ { i \\notin \\overline { \\phi } } | \\phi _ i | \\\\ & = | \\overline { \\phi } | ( \\dim K + 1 ) + ( m - | \\overline { \\phi } | ) ( \\dim L + 1 ) \\\\ & = ( \\dim M + 1 ) ( \\dim K + 1 ) + ( m - \\dim M - 1 ) ( \\dim L + 1 ) \\end{align*}"}
+{"id": "6457.png", "formula": "\\begin{align*} \\mathcal { H } _ { i + 1 } ( \\vec { r } + \\vec { e _ i } , h _ 1 ) - \\mathcal { H } _ { i + 1 } ( \\vec { r } , h _ 1 ) = & \\big ( r _ i + 1 \\big ) h _ i + \\frac { ( r _ i + 1 ) r _ i } { 2 } - \\Big ( r _ i h _ i + \\frac { r _ i ( r _ i - 1 ) } { 2 } \\Big ) \\\\ = & h _ i + r _ i \\\\ < & \\Big ( \\frac { 1 } { r _ i } + \\frac { 1 } { h _ i } \\Big ) h _ { i + 1 } \\end{align*}"}
+{"id": "7043.png", "formula": "\\begin{align*} e _ 1 & \\ , : = \\ , \\big ( 1 - y + \\tfrac { 1 } { 3 } \\ , \\theta \\ , u \\big ) \\ , \\partial _ x + \\big ( - \\ , \\tfrac { 1 } { 3 } \\ , \\theta \\ , x - \\tfrac { 1 } { 6 } \\ , u \\big ) \\ , \\partial _ y + x \\ , \\partial _ u , \\\\ e _ 2 & \\ , : = \\ , - \\ , x \\ , \\partial _ x + ( 1 - y ) \\ , \\partial _ y - u \\ , \\partial _ u , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ [ e _ 1 , e _ 2 ] \\ , = \\ , 0 . \\end{align*}"}
+{"id": "1977.png", "formula": "\\begin{align*} \\inf \\{ t \\geq 0 : \\sigma ( x , t ) = 0 \\} & = T ( x ) , \\\\ \\mu ( x , T ( x ) ) & = h ( x ) . \\end{align*}"}
+{"id": "3407.png", "formula": "\\begin{align*} | \\phi ( y ) | \\leq & \\frac { | \\tilde { P } _ { x _ 2 - y } ( \\theta + ( y + 1 ) \\alpha ) | } { | \\tilde { P } _ { 2 q _ n - 1 } ( \\theta + x _ 1 \\alpha ) | } \\prod _ { j = x _ 1 } ^ { y } | \\cos ( \\pi ( \\theta + j \\alpha ) ) | \\cdot | \\phi ( x _ 1 - 1 ) | \\\\ & + \\frac { | \\tilde { P } _ { y - x _ 1 } ( \\theta + x _ 1 \\alpha ) | } { | \\tilde { P } _ { 2 q _ n - 1 } ( \\theta + x _ 1 \\alpha ) | } \\prod _ { j = y } ^ { x _ 2 } | \\cos ( \\pi ( \\theta + j \\alpha ) ) | \\cdot | \\phi ( x _ 2 + 1 ) | , \\end{align*}"}
+{"id": "4165.png", "formula": "\\begin{align*} t ^ { \\alpha } - s ^ { \\alpha } = t ^ { \\alpha - \\varrho } t ^ { \\varrho } - s ^ { \\alpha - \\varrho } s ^ { \\varrho } \\leq s ^ { \\alpha - \\varrho } t ^ { \\varrho } - s ^ { \\alpha - \\varrho } s ^ { \\varrho } \\leq s ^ { \\alpha - \\varrho } ( t - s ) ^ { \\varrho } , \\end{align*}"}
+{"id": "3367.png", "formula": "\\begin{align*} A _ 0 & = \\{ \\omega : \\| u _ 1 - u _ 1 ' \\| > q \\| u _ 0 - u _ 0 ' \\| \\} , \\\\ B _ i & = \\{ \\omega : \\| u _ i - u _ i ' \\| \\leq q \\| u _ { i - 1 } - u _ { i - 1 } ' \\| \\} , \\\\ A _ k & = \\big ( \\cap _ { i = 1 } ^ k B _ i \\big ) \\cap B _ { k + 1 } ^ c , \\forall 1 \\leq k \\leq n - 1 , \\\\ A _ { n } & = \\cap _ { i = 1 } ^ { n } B _ i . \\end{align*}"}
+{"id": "6904.png", "formula": "\\begin{align*} \\partial _ i g ( x ) = \\frac { 1 } { n } D _ m G \\bigg ( \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\delta _ { x _ k } , x _ i \\bigg ) , \\partial _ { i j } g ( x ) = \\frac { 1 } { n ^ 2 } D _ m ^ 2 G \\bigg ( \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\delta _ { x _ k } , x _ i , x _ j \\bigg ) , \\ \\ i \\neq j . \\end{align*}"}
+{"id": "74.png", "formula": "\\begin{align*} 0 = \\ell ( 1 , \\alpha ) = \\ell ( x x ^ { - 1 } , \\alpha ) = \\ell ( x , w ^ { - 1 } \\alpha ) + \\ell ( x ^ { - 1 } , \\alpha ) . \\end{align*}"}
+{"id": "372.png", "formula": "\\begin{align*} C _ { n - 1 - k } ( r , s ) & = \\cos { ( \\frac { 2 \\pi } { n - 1 } ( r - s ) ( n - 1 - k ) ) } \\\\ & = \\cos { ( 2 \\pi ( r - s ) - \\frac { 2 \\pi } { n - 1 } ( r - s ) k ) } \\\\ & = C _ k ( r , s ) , \\end{align*}"}
+{"id": "6389.png", "formula": "\\begin{align*} a _ { j k } : = \\int _ { T } \\mathrm { d i v } ( { \\boldsymbol \\psi } ^ { \\mathrm { R T } _ 2 } _ { j } ) \\chi _ k \\ , d { \\boldsymbol x } = \\frac { 4 \\delta _ { j k } - 3 } { 1 8 0 } j , k = 1 , 2 , 3 , \\end{align*}"}
+{"id": "7090.png", "formula": "\\begin{align*} f ^ z _ { ( n + 1 ) k } & = \\frac { 1 } { \\norm { d } } w _ z f _ { \\ell k } 1 \\leq k \\leq n , \\\\ f ^ z _ { k ( n + 1 ) } & = \\frac { 1 } { \\norm { d } } f ^ z _ { k \\ell } w _ 1 ^ * 1 \\leq k \\leq n , \\\\ f ^ z _ { ( n + 1 ) ( n + 1 ) } & = \\frac { 1 } { \\norm { d } ^ 2 } w _ z w _ 1 ^ * . \\end{align*}"}
+{"id": "5577.png", "formula": "\\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 x ' ) = A ( 1 ^ n 0 ^ k 1 . . . ) - A ( 1 ^ n 0 ^ \\infty ) = \\begin{cases} d _ k - d , \\ \\\\ 0 , \\ \\end{cases} \\ . \\end{align*}"}
+{"id": "7909.png", "formula": "\\begin{align*} u ( \\upsilon ) = \\frac { 4 \\upsilon + \\frac 1 2 f ( 2 \\upsilon ) - 2 f ( \\upsilon ) } { f ( \\upsilon ) } . \\end{align*}"}
+{"id": "8414.png", "formula": "\\begin{align*} I _ 2 = - \\frac { 1 } { 2 i z } ( 1 - e ^ { 2 i z \\delta } ) \\nu ( x ; z ) . \\end{align*}"}
+{"id": "8168.png", "formula": "\\begin{align*} \\int _ { R _ { 0 } } ^ { \\infty } \\nu \\xi ( \\nu , s ) \\ d \\nu = 0 , s \\in ( 0 , t _ { 0 } ) . \\end{align*}"}
+{"id": "1794.png", "formula": "\\begin{align*} ( 1 - x ) ^ { - a } - 1 = a x { } _ { 2 } F _ { 1 } \\left [ \\left . \\begin{matrix} 1 , a + 1 \\\\ 2 \\end{matrix} \\right | x \\right ] , \\end{align*}"}
+{"id": "1227.png", "formula": "\\begin{align*} \\forall x \\in \\Z ^ d \\ \\forall \\Delta \\Subset \\Z ^ d \\ \\forall \\eta \\in \\Omega : c _ { \\Delta + x } ( \\tau _ x \\eta , \\cdot ) = c _ \\Delta ( \\eta , \\cdot ) , \\end{align*}"}
+{"id": "2432.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 } s L _ { q } [ f ( t ) ] = \\lim _ { t \\rightarrow \\infty } \\frac { f ( t ) } { 1 + ( 1 - q ) } . \\end{align*}"}
+{"id": "5116.png", "formula": "\\begin{align*} \\left \\| \\frac { 1 } { s ^ { \\alpha } } \\right \\| _ { L ^ { q } ( \\mathbb { R } ) } = C \\frac { ( r r ' ) ^ { \\alpha } } { | r - r ' | ^ { 2 \\alpha - 1 / q } } , \\alpha > \\frac { 1 } { 2 q } , \\end{align*}"}
+{"id": "3867.png", "formula": "\\begin{align*} \\begin{cases} \\nabla W \\cdot \\nabla ^ \\perp \\left ( \\varPhi - \\frac { \\alpha } { 2 } | x ' | ^ 2 | \\ln \\varepsilon | \\right ) = 0 , \\\\ W = \\mathcal { L } _ H \\varPhi , \\\\ \\varPhi | _ { \\partial B _ { R ^ * } ( 0 ) } = 0 . \\end{cases} \\end{align*}"}
+{"id": "7148.png", "formula": "\\begin{align*} N ( \\eta ) : = \\frac { \\operatorname { v o l } ( B ^ { n - 1 } ) \\operatorname { v o l } ( \\partial \\Omega ) } { ( 2 \\pi ) ^ { n - 1 } } \\eta ^ { n - 1 } + o ( \\eta ^ { n - 1 } ) \\ \\eta \\to + \\infty , \\end{align*}"}
+{"id": "3125.png", "formula": "\\begin{align*} \\alpha ( x \\cdot y ) = \\alpha ( x ) \\cdot \\alpha ( y ) , \\forall x , y \\in A , \\end{align*}"}
+{"id": "3391.png", "formula": "\\begin{align*} P ( x , 0 ) = S _ 0 ^ 4 ( x ) - S _ 0 ^ 3 ( x ) - S _ 0 ( x ) S _ 2 ( x ) + S _ 1 ^ 2 ( x ) . \\end{align*}"}
+{"id": "5764.png", "formula": "\\begin{align*} \\mathcal { P } ' \\left ( \\partial _ Y \\Phi ( X ) \\right ) = \\partial _ Y \\left ( \\Phi \\left ( \\mathcal { P } ( X ) \\right ) \\right ) . \\end{align*}"}
+{"id": "3899.png", "formula": "\\begin{align*} 0 = \\int _ { B _ 1 ( 0 ) } \\phi ^ { p - 1 } _ + \\frac { \\partial \\phi } { \\partial x _ h } \\hat { u } _ i . \\end{align*}"}
+{"id": "1014.png", "formula": "\\begin{align*} \\ell ( x _ 0 ) = \\ell ( x _ 1 ) = \\ell ( x _ L x _ 0 x _ R ) = \\ell ( x _ 0 x _ R ) - \\ell ( x _ L ) . \\end{align*}"}
+{"id": "7405.png", "formula": "\\begin{align*} [ V _ \\lambda ] = q ^ { - \\frac { c } { 2 4 } } q ^ { \\frac { ( \\lambda | \\lambda + 2 \\rho ) } { 2 ( k + h ^ \\vee ) } - \\lambda ( x + y ) } \\ \\prod _ { n = 1 } ^ \\infty \\prod _ { \\alpha \\in \\Delta _ { } ^ { 0 } } ( 1 - q ^ { n } ) ^ { - 1 } \\prod _ { \\alpha \\in \\Delta _ { } ^ { + , \\sharp } } ( 1 - e ^ { \\alpha ( h ) } q ^ { n } ) ^ { - 1 } ( 1 - e ^ { - \\alpha ( h ) } q ^ { n - 1 } ) ^ { - 1 } , \\end{align*}"}
+{"id": "781.png", "formula": "\\begin{align*} \\int _ { \\Gamma ( t ) } c ( t ) \\ , \\mathrm { d } \\mathcal { A } = m \\end{align*}"}
+{"id": "3535.png", "formula": "\\begin{align*} A _ k = \\sum _ { j = 0 } ^ { k - 1 } \\binom { k } { j } B _ j . \\end{align*}"}
+{"id": "5722.png", "formula": "\\begin{align*} t / v = \\mu ( \\infty ) \\textrm { i s a z e r o o f $ f $ } \\iff \\infty \\textrm { i s a z e r o o f } f ' \\end{align*}"}
+{"id": "1729.png", "formula": "\\begin{align*} \\tau _ 1 = \\inf \\{ t \\in [ 0 , T ] : 2 ^ { \\alpha - 1 } C _ t ^ { \\alpha } ( C _ T + ( | | x | | _ { H ^ 2 } + 1 ) & + ( | | x | | _ { H ^ 2 } + 1 ) ^ { \\alpha } ) ^ { \\alpha - 1 } ( t ^ { \\theta } + t ^ { \\alpha } ) \\\\ & + C _ t t ^ { \\alpha } ( | | x | | _ { H ^ 2 } + 1 ) ^ { \\alpha - 1 } > \\frac { 1 } { 2 } \\} \\wedge T , \\end{align*}"}
+{"id": "7.png", "formula": "\\begin{align*} \\beta = \\alpha - \\alpha _ j j \\geq s . \\end{align*}"}
+{"id": "8722.png", "formula": "\\begin{align*} \\beta _ \\eta ( q _ 1 , q _ 2 ) = \\eta ( q _ 1 ) \\eta ( q _ 2 ) \\eta ( q _ 1 q _ 2 ) ^ { - 1 } \\end{align*}"}
+{"id": "3052.png", "formula": "\\begin{align*} d \\log \\varphi _ { \\lambda , u } = \\sum _ { j = 1 } ^ { l } u _ j \\frac { d f _ j } { f _ j } + \\sum _ { i = 0 } ^ { n } \\lambda _ i \\frac { d x _ i } { x _ i } \\end{align*}"}
+{"id": "6042.png", "formula": "\\begin{align*} F ( u , v ) & = \\frac { p } { p \\alpha - 1 } \\Big ( \\alpha a ( u , v ) + ( \\alpha - 1 ) b ( u , v ) + ( 3 \\alpha - 2 ) c ( u , v ) \\Big ) \\\\ & \\geq \\frac { p } { p \\alpha - 1 } ( \\alpha - 1 ) \\| ( u , v ) \\| _ { E } ^ { 2 } . \\end{align*}"}
+{"id": "2096.png", "formula": "\\begin{align*} P _ { w , v } = \\sum _ { i = 1 } ^ k w _ i F _ { v _ i } \\end{align*}"}
+{"id": "8657.png", "formula": "\\begin{align*} y \\left [ i \\right ] = \\sum \\limits _ { l ' = 1 } ^ { L ' } { \\sum \\limits _ { l \\in { { \\bar { \\cal L } } _ { l ' } } } { { \\bf { h } } _ l ^ H { \\bf { \\bar H } } _ { l ' } ^ \\bot { \\bf { \\bar X } } _ { l ' } { \\bf { d } } \\left [ { i - { n _ l } - { \\kappa _ { l ' } } } \\right ] } } + z \\left [ i \\right ] , \\end{align*}"}
+{"id": "1897.png", "formula": "\\begin{align*} x = ( r , 0 , 0 , \\dots , 0 ) , y = ( \\rho , h , 0 , \\dots , 0 ) . \\end{align*}"}
+{"id": "3859.png", "formula": "\\begin{align*} \\mathbf { w } = \\frac { w } { k } \\overrightarrow { \\zeta } , \\end{align*}"}
+{"id": "7510.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } \\int _ { B _ { r _ { j } } \\setminus B _ { r _ { j + 3 } } } r ^ { 2 - n } u _ r ^ 2 \\leq 3 \\int _ { B _ { 1 } } r ^ { 2 - n } u _ r ^ 2 , \\end{align*}"}
+{"id": "164.png", "formula": "\\begin{align*} F \\widehat { \\otimes } _ K F ' : = \\varinjlim _ { n , m } F _ n \\widehat { \\otimes } _ K F _ m ' . \\end{align*}"}
+{"id": "35.png", "formula": "\\begin{align*} \\delta _ { \\lambda } ( g ) = e x p \\circ \\Delta _ { \\lambda } \\circ e x p ^ { - 1 } g , \\end{align*}"}
+{"id": "6994.png", "formula": "\\begin{align*} \\Psi _ k = & \\int \\cdots \\int _ { s < t _ 1 < \\cdots < t _ k < t } \\sum _ { j _ 0 , \\dots , j _ k } \\int _ { H ^ k } q _ { j _ { k - 1 } , j _ { k } } ( x _ k ) \\widetilde { P ^ { ( j _ k ) } } ( t _ k , x _ k ; t , B ) \\\\ & \\times q _ { j _ { k - 2 } , j _ { k - 1 } } ( x _ { k - 1 } ) \\widetilde { P ^ { ( j _ { k - 1 } ) } } ( t _ { k - 1 } , x _ { k - 1 } ; t _ { k } , d x _ { k } ) \\cdots q _ { i , j _ { 1 } } ( x _ { 1 } ) \\widetilde { P ^ { ( j _ 1 ) } } ( t _ { 1 } , x _ { 1 } ; t _ { 2 } , d x _ { 2 } ) \\\\ & \\times \\widetilde { P ^ { ( i ) } } ( s , x ; t _ 1 , d x _ 1 ) d t _ 1 \\cdots d t _ k \\end{align*}"}
+{"id": "6858.png", "formula": "\\begin{align*} \\prod _ { m _ { 1 } , \\ldots , m _ { k } \\geq 0 } ( 1 - x _ { 1 } ^ { m _ { 1 } } \\cdots x _ { k } ^ { m _ { k } } ) ^ { M ( m _ { 1 } , \\ldots , m _ { k } ) } = 1 - x _ { 1 } - \\cdots - x _ { k } . \\end{align*}"}
+{"id": "4175.png", "formula": "\\begin{align*} \\mathbf { w } = \\frac { w } { k } \\overrightarrow { \\zeta } , \\end{align*}"}
+{"id": "3199.png", "formula": "\\begin{align*} r & = f ( n ) + 3 + ( F + 1 ) ( f ( n ) + 2 ) \\\\ & = 2 f ( n ) + I \\tfrac { f ( n ) + 2 } { f ( n ) } + O ( 1 ) \\\\ & = I + \\tfrac { 2 I } { f ( n ) } + 2 f ( n ) + O ( 1 ) . \\end{align*}"}
+{"id": "7359.png", "formula": "\\begin{gather*} 0 = P _ \\alpha ( f ) = P _ \\alpha \\bigl \\{ f \\ , 1 ( \\abs { f } \\le k ) \\bigr \\} + P _ \\alpha \\bigl \\{ f \\ , 1 ( \\abs { f } > k ) \\bigr \\} \\\\ \\le P _ \\alpha \\bigl \\{ f \\ , 1 ( \\abs { f } \\le k ) \\bigr \\} + \\sup _ { Q \\in M } Q \\bigl \\{ \\abs { f } \\ , 1 ( \\abs { f } > k ) \\bigr \\} \\quad \\quad k . \\end{gather*}"}
+{"id": "4688.png", "formula": "\\begin{align*} ( W _ 1 ) = \\int _ { V _ 0 } \\int _ { \\pi _ 1 ( \\pi _ 0 ^ { - 1 } ( x ) \\cap W _ 1 ) } d v ( y ) \\ , d v ( x ) \\geq \\frac 1 2 \\ , ( \\hat A ) \\ , ( \\hat B ) , \\end{align*}"}
+{"id": "1170.png", "formula": "\\begin{align*} H \\left ( \\rho _ n ^ { ( k , \\epsilon ) } ( t , \\cdot ) \\mid \\bar { \\rho } ^ { \\epsilon \\otimes k } ( \\cdot ) \\right ) = \\sup _ { g \\in L ^ \\infty ( \\mathbb { T } ^ { k d } ) } \\mathbb { E } _ { \\rho _ n ^ { ( k , \\epsilon ) } } ( g ) - \\mathbb { E } _ { \\bar { \\rho } ^ { \\epsilon \\otimes k } } ( e ^ g ) + 1 . \\end{align*}"}
+{"id": "8980.png", "formula": "\\begin{align*} D _ { q , j } = \\sum _ { i = 1 } ^ { j } x _ { q , i } . \\end{align*}"}
+{"id": "4528.png", "formula": "\\begin{align*} \\begin{cases} v _ i ( x , t ) = S _ { d , i } ( t , t _ { i , e x } ( x , t ) ) S _ { c , i } ( t _ { i , e x } ( x , t ) , t _ { i , e n } ( x , t ) ) S _ { d , i } ( t _ { i , e n } ( x , t ) , 0 ) v _ { i , 0 } ( x ) & \\forall \\ : i \\in [ 1 , p ] , \\\\ v _ i ( x , t ) = S _ { d , i } ( t , 0 ) v _ { i , 0 } ( x ) & \\forall \\ : i \\in [ p + 1 , n ] ; \\end{cases} \\end{align*}"}
+{"id": "446.png", "formula": "\\begin{align*} \\sup _ { | y | \\le \\alpha ( x ) } | \\beta ( x + y ) - \\beta ( x ) | = o ( \\beta ( x ) ) \\ x \\to \\infty , \\ | y | \\le \\alpha ( x ) . \\end{align*}"}
+{"id": "1861.png", "formula": "\\begin{align*} \\begin{aligned} f ( ) & \\ge \\ , ( f ) \\ , , \\\\ \\big ( \\operatorname { r e s p . } f ( ) & \\le \\ , ( f ) \\ , , \\\\ f ( ) & = \\ , ( f ) \\ , \\big ) . \\end{aligned} \\end{align*}"}
+{"id": "1667.png", "formula": "\\begin{align*} \\dim H _ f ^ 1 ( G _ { F _ v } , \\mathrm { a d } ^ 0 \\rho ) = \\dim H ^ 0 ( G _ { F _ v } , \\mathrm { a d } ^ 0 \\rho ) + \\dim D _ { d R } ( \\mathrm { a d } ^ 0 \\rho ) / D _ { d R } ^ + ( \\mathrm { a d } ^ 0 \\rho ) . \\end{align*}"}
+{"id": "5176.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\left ( \\phi _ n - r ^ { 2 } \\right ) _ { + } ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r = \\int _ { \\{ r < R \\} } \\left ( \\phi _ n - r ^ { 2 } \\right ) _ { + } ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r + \\int _ { \\{ r \\geq R \\} } \\left ( \\phi _ n - r ^ { 2 } \\right ) _ { + } ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r . \\end{align*}"}
+{"id": "8126.png", "formula": "\\begin{align*} \\overline { \\mathcal F } ( X ) = \\varinjlim _ { Y \\in ( \\mathcal C _ 0 ) _ { X / } } F ( Y ) , \\end{align*}"}
+{"id": "1640.png", "formula": "\\begin{align*} x ^ { ( 4 ) } = f ( x ) : = f _ \\uparrow ( x ) + f _ \\downarrow ( x ) , t \\in ( 0 , 1 ) , \\end{align*}"}
+{"id": "2596.png", "formula": "\\begin{align*} Q ^ c ( g , R ) ( \\pi ^ h ; \\bar { \\pi } ) = 2 Q ( g , R ) ( \\pi ^ h ; \\bar { \\pi } ) , \\end{align*}"}
+{"id": "6326.png", "formula": "\\begin{align*} z _ 0 x _ 0 ^ { p ^ { h _ { j } - r } } + z _ 1 x _ 1 ^ { p ^ { h _ { j } - r } } + \\cdots + z _ n x _ n ^ { p ^ { h _ { j } - r } } = 0 . \\end{align*}"}
+{"id": "8133.png", "formula": "\\begin{align*} u z _ 1 \\cdots z _ r = t _ 1 \\cdots t _ s = z _ 1 ^ { l _ { 1 1 } + \\dots + l _ { s 1 } } \\cdots z _ r ^ { l _ { 1 r } + \\dots + l _ { s r } } . \\end{align*}"}
+{"id": "4148.png", "formula": "\\begin{align*} \\widetilde { \\Phi } ( x ' ) : = \\Phi ^ { \\top } ( - x ' ) , \\widetilde { \\Psi } _ { \\varepsilon , \\varepsilon ^ { - 1 } } ( x ' ) : = \\Psi ^ { \\top } _ { \\varepsilon , \\varepsilon ^ { - 1 } } ( - x ' ) , \\widetilde { P } ^ L ( x ' ) : = ( P ^ L ) ^ { \\top } ( - x ' ) , x ' \\in \\R ^ { n - 1 } , \\end{align*}"}
+{"id": "7528.png", "formula": "\\begin{align*} f ( n , \\alpha , t ) \\leq \\left ( \\sum _ { i = 1 } ^ \\alpha \\binom { n } { i } \\right ) ^ t \\leq \\left ( 2 \\binom { n } { \\alpha } \\right ) ^ { t } . \\end{align*}"}
+{"id": "3019.png", "formula": "\\begin{align*} \\overline { t } = \\overline { \\xi } , \\overline { x } = \\overline { y } . \\end{align*}"}
+{"id": "9003.png", "formula": "\\begin{align*} \\sum _ { \\ell _ { M - 1 } < j \\leq \\ell _ M } ( - 1 ) ^ { p ( \\eta , j ) - 1 } i _ { \\eta \\cup j } & = U \\left ( \\prod _ { r = m + 1 } ^ { M - 1 } { D _ { j _ { r } - 1 } } \\right ) \\left ( D _ { \\ell _ M - 1 } - \\sum _ { \\ell _ { M - 1 } < j < \\ell _ M } { x _ { j } } \\right ) \\\\ & = U \\left ( \\prod _ { r = m + 1 } ^ { M - 1 } { D _ { j _ { r } - 1 } } \\right ) D _ { \\ell _ { M - 1 } } \\\\ & = U \\cdot B ( M - 1 ) . \\end{align*}"}
+{"id": "3790.png", "formula": "\\begin{align*} B _ q ( \\psi _ m ) ( z ) & = \\int \\limits _ { \\mathbb R } \\overline { A _ q ( z , x ) } \\psi _ m ( x ) d x \\\\ & = \\int \\limits _ { \\mathbb R } \\left ( \\sum _ { j = 0 } ^ { \\infty } \\frac { z ^ j } { \\sqrt { \\Gamma ( q j + 1 ) } } \\psi _ j ( x ) \\right ) \\psi _ m ( x ) d x , \\\\ & = \\sum _ { j = 0 } ^ { \\infty } \\frac { z ^ j } { \\sqrt { \\Gamma ( q j + 1 ) } } \\left ( \\int \\limits _ { \\mathbb R } \\psi _ j ( x ) \\psi _ m ( x ) d x \\right ) . \\\\ \\end{align*}"}
+{"id": "4596.png", "formula": "\\begin{align*} F \\subseteq \\bigcup _ { n = 0 } ^ \\infty [ x _ n , y _ n ] \\end{align*}"}
+{"id": "1410.png", "formula": "\\begin{align*} \\{ \\widehat S _ i ( n + d - \\pi ( i ) ) \\in d z _ { \\pi ( i ) } , i = 1 , \\ldots , d , \\widehat \\tau < \\infty \\} \\end{align*}"}
+{"id": "4524.png", "formula": "\\begin{align*} v _ 1 ( x , t _ { 1 , e n } ( x , t ) ) = S _ { d , 1 } ( t _ { 1 , e n } ( x , t ) , 0 ) ) v _ 1 ( x , 0 ) . \\end{align*}"}
+{"id": "3317.png", "formula": "\\begin{align*} | S | | \\sigma | + ( m - | S | ) | \\tau ^ 0 | = | S | | \\sigma ' | + ( m - | S | ) | \\tau ^ 0 | . \\end{align*}"}
+{"id": "703.png", "formula": "\\begin{align*} \\kappa ( \\lambda ) : = \\Phi _ \\lambda ( u ) | _ { E ^ + _ \\lambda \\cap \\partial B _ \\rho ( 0 ) } \\geq \\left ( \\frac { 1 } { 8 } - \\frac { 1 } { 2 ^ { 2 p + 1 } } \\right ) \\left ( \\frac { \\beta _ \\lambda ^ p } { 2 C ( N , \\alpha , p ) } \\right ) ^ { \\frac { 1 } { p - 1 } } > 0 . \\end{align*}"}
+{"id": "3543.png", "formula": "\\begin{align*} \\omega _ 1 & = ( 2 , 3 , 4 , 2 ) , & \\omega _ 2 & = ( 3 , 6 , 8 , 4 ) , & \\omega _ 3 & = ( 2 , 4 , 6 , 3 ) , & \\omega _ 4 = ( 1 , 2 , 3 , 2 ) . \\end{align*}"}
+{"id": "5646.png", "formula": "\\begin{align*} 0 = d z _ { q } & = \\sum _ { i } d ( u _ { i } , f _ { i } ) = \\sum _ { i } \\left [ ( d u _ { i } , f _ { i } ) + ( - 1 ) ^ { p + 1 } ( u _ { i } , d f _ { i } ) \\right ] . \\end{align*}"}
+{"id": "2443.png", "formula": "\\begin{align*} L _ { q } [ f ( t ) \\ast g ( t ) ] = F _ { q } ( s ) \\ast G _ { q } ( s ) , \\end{align*}"}
+{"id": "4828.png", "formula": "\\begin{align*} ( d / d t ) v _ k + i \\lambda _ k v _ k = 0 , \\ ; \\ ; 1 \\leqslant k \\leqslant n . \\end{align*}"}
+{"id": "6757.png", "formula": "\\begin{align*} \\Phi _ s ( z ) ^ \\alpha = \\sum _ { r = 0 } ^ \\infty \\sum _ { m = r } ^ \\infty ( - 1 ) ^ { m + r } \\binom { \\alpha } { m } \\binom { m } { r } \\Phi _ s ( z ) ^ r . \\end{align*}"}
+{"id": "474.png", "formula": "\\begin{align*} \\frac { f ( x + y ) } { f ( x ) } & = \\frac { \\inf _ { t \\in [ x _ 0 , x + y ] } f ( t ) } { \\sup _ { s \\ge x } f ( s ) } \\frac { f ( x + y ) } { \\inf _ { t \\in [ x _ 0 , x + y ] } f ( t ) } \\frac { \\sup _ { s \\ge x } f ( s ) } { f ( x ) } \\\\ & \\le \\frac { F ( x + y - c + \\Delta ) } { F ( x + \\Delta ) } \\frac { f ( x + y ) } { \\inf _ { t \\in [ x _ 0 , x + y ] } f ( t ) } \\frac { \\sup _ { s \\ge x } f ( s ) } { f ( x ) } . \\end{align*}"}
+{"id": "8296.png", "formula": "\\begin{align*} \\tilde X _ { n + 1 } ^ i = \\tilde X _ n ^ i + \\bigg ( b ( \\tilde X _ n ^ i ) + \\frac 1 { N - 1 } \\sum _ { j \\neq i } K ( \\tilde X _ n ^ i - \\tilde X _ n ^ j ) \\bigg ) \\tau + \\sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , \\end{align*}"}
+{"id": "1316.png", "formula": "\\begin{align*} { \\rm R e } \\langle u _ { t t } , u \\rangle & = { \\rm R e } \\int _ { \\mathbb { H } ^ n } f ( u ) \\overline { u } d x - \\int _ { \\mathbb { H } ^ n } | \\nabla _ { H } u | ^ 2 d x - m \\int _ { \\mathbb { H } ^ n } | u | ^ 2 d x - b { \\rm R e } \\int _ { \\mathbb { H } ^ n } \\overline { u } u _ t d x \\\\ & = - I ( u ) - b { \\rm R e } \\int _ { \\mathbb { H } ^ n } \\overline { u } u _ t d x . \\end{align*}"}
+{"id": "8745.png", "formula": "\\begin{align*} ( D _ { a ^ + } ^ { \\alpha } { \\bf I } _ { a ^ + } ^ { \\alpha } f ) ( x ) = f ( x ) \\ \\textrm { a n d } \\ ( D _ { b ^ - } ^ { \\alpha } { \\bf I } _ { b ^ - } ^ { \\alpha } f ) ( x ) = f ( x ) . \\end{align*}"}
+{"id": "7884.png", "formula": "\\begin{align*} 2 \\gamma _ 2 \\ddot s ( 0 ) = - \\ddot { \\nu } ( 0 ) , \\end{align*}"}
+{"id": "6988.png", "formula": "\\begin{align*} \\prod ^ { m ( p ) - n } _ { r = 1 } | t _ r | _ 2 = | t _ l | ^ { s _ 1 } _ 2 | \\frac { t _ { 2 l } } { t _ l } | ^ { s _ 2 } _ 2 | s _ 2 ! | _ 2 , \\end{align*}"}
+{"id": "1712.png", "formula": "\\begin{align*} D _ t \\Delta u = & ( - \\Delta + D _ j \\widetilde { b } ^ j + \\widetilde { b } ^ j D _ j + \\widetilde { c } ) \\Delta u + 2 ( D _ j \\nabla \\widetilde { b } ^ j + \\nabla \\widetilde { b } ^ j D _ j + \\nabla \\widetilde { c } ) \\nabla u \\\\ & + ( D _ j \\Delta \\widetilde { b } ^ j + \\Delta \\widetilde { b } ^ j D _ j + \\Delta \\widetilde { c } ) u - i \\Delta f . \\end{align*}"}
+{"id": "4720.png", "formula": "\\begin{align*} & { M i n i m i z e \\ , } _ { x \\in \\mathbb { R } ^ n } \\ \\ J ( x ) : = x ^ T A _ J x + 2 b _ J ^ T x + c _ J \\\\ & s . t . \\ \\ f _ k ( x ) : = x ^ T A _ k x + 2 b _ k ^ T x + c _ k \\le 0 , \\ \\ k = 1 , . . . , m \\end{align*}"}
+{"id": "7725.png", "formula": "\\begin{align*} v _ { [ k ] } = - ( T _ { I , [ k ] } ) ^ { - 1 } T _ { I , \\{ k + 1 \\} } , \\end{align*}"}
+{"id": "7693.png", "formula": "\\begin{align*} | \\phi ^ { N , i } ( 0 , \\boldsymbol { \\xi } ) - \\Psi ^ { N , i } ( 0 , \\boldsymbol { \\xi } ) | \\leq \\frac { C } { N } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 + \\frac { 1 } { N ^ 2 } \\underset { i , j = 1 } { \\overset { N } { \\sum } } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "908.png", "formula": "\\begin{align*} k = \\# \\mathcal { N } ( d ) = \\# \\mathcal { N } ' ( d ) \\ , , \\end{align*}"}
+{"id": "7767.png", "formula": "\\begin{align*} - \\frac { f _ k ' ( 0 ) } { f _ k ( 0 ) } = \\sum _ { j = 1 } ^ { w } z _ j ^ { - k } ~ { \\rm a n d } ~ - \\frac { \\widehat f _ k ' ( 0 ) } { \\widehat f _ k ( 0 ) } = \\sum _ { j = 1 } ^ { w } z _ j ^ { k } ~ { \\rm f o r } ~ k = 0 , 1 , \\dots . \\end{align*}"}
+{"id": "7682.png", "formula": "\\begin{align*} \\mu ^ i + h ( \\mu ^ i ) = - R ^ { - 1 } B \\Big [ P _ t \\nu ^ { N , i } _ { \\boldsymbol { x } } + \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } ) \\Big ] . \\end{align*}"}
+{"id": "8190.png", "formula": "\\begin{align*} x _ { i , j } + x _ { i + 1 , j } + x _ { i + 1 , j } + x _ { i + 1 , j + 1 } = S , \\mbox { \\ \\ f o r a l l } 1 \\leq i \\leq m \\mbox { a n d } 1 \\leq j \\leq n _ 1 . \\end{align*}"}
+{"id": "1086.png", "formula": "\\begin{align*} w ^ { - 1 } s _ \\beta ( \\gamma ) = w ^ { - 1 } \\gamma - \\langle \\beta ^ \\vee , \\gamma \\rangle w ^ { - 1 } \\beta > w ^ { - 1 } \\gamma . \\end{align*}"}
+{"id": "1261.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( \\sum _ { \\Theta \\Subset \\Z ^ d } h _ { \\Theta } ( \\omega ) \\right ) ^ 2 \\mu ( d \\omega ) = 0 . \\end{align*}"}
+{"id": "240.png", "formula": "\\begin{align*} F _ n \\colon R _ { q } ^ \\flat ( n ) & \\to \\mathcal { P } \\ ! \\left ( B _ S ( n + D ' ) \\right ) , \\\\ g & \\mapsto \\big \\{ \\dot g ( J ) \\mid J \\subseteq [ \\sigma _ g ^ - , \\sigma _ g ^ + ] _ \\mathbb { Z } | J | = 2 \\big \\} \\end{align*}"}
+{"id": "4659.png", "formula": "\\begin{align*} m ( s , t ) = \\frac { s } { s - t } , s \\in \\mathbb { R } _ { > 0 } , t \\in \\mathbb { R } _ { < 0 } . \\end{align*}"}
+{"id": "676.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 d x P _ l ( x ) P _ { l ' } ( x ) = \\frac { 2 } { 2 l + 1 } \\delta _ { l , l ' } \\ , , \\end{align*}"}
+{"id": "8235.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\deg _ p ( N _ k ' ) = \\lim _ { k \\to \\infty } \\deg _ p ( N _ k '' ) . \\end{align*}"}
+{"id": "5472.png", "formula": "\\begin{align*} \\left \\| \\sum _ { k = 1 } ^ n a _ k \\xi _ k \\right \\| _ q \\geq c _ d ( q ) \\left \\| \\sum _ { k = 1 } ^ n a _ k \\xi _ k \\right \\| _ 2 . \\end{align*}"}
+{"id": "9127.png", "formula": "\\begin{align*} \\log _ { \\hat { E } } ( d _ { f , n } ) = \\varepsilon _ { f , n } + \\sum _ { j = 0 } ^ { [ ( n + 1 ) / 2 ] } ( - 1 ) ^ j \\frac { \\pi _ { f , n - 2 j } } { p ^ j } , \\end{align*}"}
+{"id": "8219.png", "formula": "\\begin{align*} \\rho ( a _ 1 , a _ 2 , \\ldots , a _ { n _ 0 - 1 } , a _ { n _ 0 } ) & = ( a _ { n _ 0 } , a _ { n _ 0 - 1 } , \\ldots , a _ 2 , a _ 1 ) \\mbox { \\ \\ \\ \\ a n d } \\\\ \\sigma ( a _ 1 , a _ 2 , \\ldots , a _ { n _ 0 - 1 } , a _ { n _ 0 } ) & = ( S - a _ 1 , S - a _ 2 , \\ldots , S - a _ { n _ 0 - 1 } , S - a _ { n _ 0 } ) . \\end{align*}"}
+{"id": "7102.png", "formula": "\\begin{align*} t _ x : = \\sup \\{ t > 0 \\ , | \\ , X _ s ^ x \\in \\Omega \\} \\end{align*}"}
+{"id": "9161.png", "formula": "\\begin{align*} \\dfrac { b _ { 1 , \\delta } ( x ) } { 2 } = \\int _ 0 ^ 1 [ h ( x + \\delta \\sin ( 2 \\pi t ) ) - h ( x ) ] \\sin ( 2 \\pi t ) \\ , d t . \\end{align*}"}
+{"id": "8015.png", "formula": "\\begin{align*} & T _ { 3 , r } ( \\underline { p , q } ) = 2 ^ r \\sum _ { n _ 1 , n _ 2 , \\dots , n _ r \\atop { 1 \\leq n _ i \\leq \\pi _ N ( x ) } } G ( n _ 1 ) G ( n _ 2 ) \\dots G ( n _ r ) \\\\ & \\sum _ { l _ 1 , l _ 2 , \\dots , l _ r \\atop { 0 \\leq l _ i \\leq L } } U ( l _ 1 ) U ( l _ 2 ) \\dots U ( l _ r ) \\sum _ { l _ 1 ' , l _ 2 ' , \\dots , l _ r ' \\atop { 0 \\leq l _ i ' \\leq L } } U ( l _ 1 ' ) U ( l _ 2 ' ) \\dots U ( l _ r ' ) \\prod _ { i = 1 } ^ r I ( p _ i , q _ i , n _ i , l _ i , l _ i ' ) , \\end{align*}"}
+{"id": "586.png", "formula": "\\begin{align*} \\lambda _ { i , j } = \\begin{cases} 0 ~ ~ { \\rm i f } ~ ~ i < a , \\\\ \\\\ \\frac { \\lambda _ { i } \\lambda _ { j } } { \\lambda _ a } ~ ~ { \\rm i f } ~ ~ i \\geq a . \\\\ \\end{cases} \\end{align*}"}
+{"id": "621.png", "formula": "\\begin{align*} | \\prod _ { \\alpha \\in \\Phi ^ + } \\alpha ( X ) | = | D ( X ) | ^ { 1 / 2 } . \\end{align*}"}
+{"id": "3378.png", "formula": "\\begin{align*} p _ \\delta = p + \\delta _ 1 q _ 1 + \\ldots + \\delta _ k q _ k \\end{align*}"}
+{"id": "565.png", "formula": "\\begin{align*} f ( z ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma } g ( t ) e ^ { - z t } d t , \\end{align*}"}
+{"id": "6609.png", "formula": "\\begin{align*} [ u , v ] = L ( u \\times v ) , \\forall u , v \\in \\mathfrak { g } , \\end{align*}"}
+{"id": "791.png", "formula": "\\begin{align*} \\int _ { \\Gamma _ t } c _ t \\ , \\mathrm { d } \\mathcal { H } ^ d & = \\int _ { \\Gamma _ t } ( c + t w ) \\circ \\theta _ t ^ { - 1 } + m _ t \\ , \\mathrm { d } \\mathcal { H } ^ d \\\\ & = \\int _ { \\Gamma _ t } ( c + t w ) \\circ \\theta _ t ^ { - 1 } \\ , \\mathrm { d } \\mathcal { H } ^ d + \\left ( m - \\int _ { \\Gamma _ t } ( c + t w ) \\circ \\theta _ t ^ { - 1 } \\mathrm { d } \\mathcal { H } ^ d \\right ) = m \\end{align*}"}
+{"id": "2092.png", "formula": "\\begin{align*} c _ 1 ( d ) \\| g \\| _ { 1 } - C _ d | \\sigma + \\| g \\| _ { 2 } | \\sqrt { \\frac { \\log ( 2 / \\delta ) } { m } } \\leq \\sum _ { i = 1 } ^ { 3 } \\bar { f } _ { ( i ) } \\leq C _ 1 ( d ) \\| g \\| _ { 1 } + C _ d | \\sigma + \\| g \\| _ { 2 } | \\sqrt { \\frac { \\log ( 2 / \\delta ) } { m } } , \\end{align*}"}
+{"id": "2524.png", "formula": "\\begin{align*} { { \\left ( \\mathbf { x } + \\Delta \\mathbf { x } \\right ) } ^ { T } } \\left ( \\mathbf { s } + \\Delta \\mathbf { s } \\right ) + \\Delta { { \\mathbf { y } } ^ { T } } \\mathbf { A x } - \\Delta { { \\mathbf { x } } ^ { T } } \\left ( { { \\mathbf { A } } ^ { T } } \\mathbf { y } + \\mathbf { C } \\mathbf { x } + \\mathbf { s } \\right ) = \\nu { \\mathbf { x } } ^ { T } \\mathbf { s } + \\Delta \\mathbf { x } ^ T \\Delta \\mathbf { s } . \\end{align*}"}
+{"id": "8227.png", "formula": "\\begin{align*} H ( \\xi ) : = \\begin{cases} 1 / b ' ( \\xi ) & \\ \\xi \\notin ( b ) \\\\ 0 & \\ \\xi \\in ( b ) \\end{cases} \\end{align*}"}
+{"id": "5826.png", "formula": "\\begin{align*} \\rho c _ v \\left [ g _ i + \\Delta t D _ i g _ i + \\frac { \\Delta t ^ 2 } { 2 } D _ i ^ 2 g _ i \\right ] - g _ i + ( 1 - \\rho c _ v ) \\left [ g _ i + \\Delta t d _ i g _ i + \\frac { \\Delta t ^ 2 } { 2 } d _ i ^ 2 g _ i \\right ] = - { \\left ( { { { \\bf { M } } ^ { - 1 } } \\Lambda { \\bf { M } } } \\right ) _ { i j } } \\left [ { g _ i - g _ i ^ { ( e q ) } } \\right ] + \\Delta t { \\bar F _ i } + \\vartheta \\Delta t { S _ i } , \\end{align*}"}
+{"id": "1183.png", "formula": "\\begin{align*} \\sup _ { n } \\mathbb { E } _ { \\mathbb { P } } \\left [ \\exp \\left \\{ \\kappa \\sum _ { i = 1 } ^ { n } \\int _ { T _ { 0 } } ^ { T _ { 0 } + \\delta } \\triangle K _ { t } ^ { i , n } \\cdot d W _ { t } ^ { i } - \\frac { \\kappa } { 2 } \\sum _ { i = 1 } ^ n \\int _ { T _ { 0 } } ^ { T _ { 0 } + \\delta } \\left | \\triangle K _ { t } ^ { i , n } \\right | ^ { 2 } d t \\right \\} \\right ] \\end{align*}"}
+{"id": "4244.png", "formula": "\\begin{align*} \\sup _ { z \\in \\mathbb { B } _ n } \\beta ( \\varphi _ { k j } ( z ) , \\rho ( z ) ) & \\leq \\sup _ { z \\in \\mathbb { B } _ n } k _ { V ^ { - 1 } \\mathbb { B } _ n } ( V ^ { - 1 } \\varphi _ { k j } ( z ) , V ^ { - 1 } \\rho ( z ) ) \\\\ & = \\sup _ { z \\in V ^ { - 1 } \\mathbb { B } _ n } k _ { V ^ { - 1 } \\mathbb { B } _ n } ( V ^ { - 1 } \\varphi _ { k j } V ( z ) , V ^ { - 1 } \\rho V ( z ) ) . \\end{align*}"}
+{"id": "1142.png", "formula": "\\begin{align*} y _ { N ( \\hat { k } - 1 ) + \\hat { \\imath } } \\ , x _ { N ( \\hat { \\ell } - 1 ) + \\hat { \\jmath } } = \\min _ { \\substack { 1 \\leq i , j \\leq N \\\\ [ 1 m m ] 1 \\leq k , \\ell \\leq L \\\\ [ 0 . 5 m m ] w _ { i j } ^ { ( k , \\ell ) } > 0 } } y _ { N ( k - 1 ) + i } \\ , x _ { N ( \\ell - 1 ) + j } . \\end{align*}"}
+{"id": "7363.png", "formula": "\\begin{gather*} \\inf _ { Q \\in M } Q ( c ) = \\sup _ { f _ 1 , \\ldots , f _ n } \\ , \\sum _ { i = 1 } ^ n \\mu _ i ( f _ i ) \\end{gather*}"}
+{"id": "2482.png", "formula": "\\begin{align*} \\begin{aligned} & \\min _ { \\mathbf { x } } \\mathbf { c } ^ T \\mathbf { x } , \\\\ & \\\\ & \\mathbf { A } \\mathbf { x } = \\mathbf { b } , \\\\ & \\mathbf { x } \\in \\mathcal { K } , \\\\ & \\mathcal { K } = \\mathcal { K } _ { L } ^ { l } \\times \\mathcal { K } _ { S } ^ { { { n } _ { l + 1 } } } \\times \\mathcal { K } _ { S } ^ { { { n } _ { l + 2 } } } \\cdots \\times \\mathcal { K } _ { S } ^ { { { n } _ { l + m } } } , \\end{aligned} \\end{align*}"}
+{"id": "3638.png", "formula": "\\begin{align*} - 2 N F ^ { i i } \\frac { u _ i } { u } \\frac { ( \\nu ^ { n + 1 } ) _ i } { \\nu ^ { n + 1 } } = - 2 N F ^ { i i } \\frac { u _ i ^ 2 } { u ^ 2 } \\frac { \\nu ^ { n + 1 } - \\kappa _ i } { \\nu ^ { n + 1 } } . \\end{align*}"}
+{"id": "2994.png", "formula": "\\begin{align*} \\operatorname { s u p p } \\widetilde \\mu _ { ( x , t ) } = \\operatorname { s u p p } \\mu _ { x } \\times \\{ t \\} \\end{align*}"}
+{"id": "6862.png", "formula": "\\begin{align*} F ( r , \\varphi ) = \\int ^ { \\varphi } _ 0 \\frac { d \\theta } { \\sqrt { 1 - r ^ 2 \\sin ^ 2 \\theta } } = \\int ^ z _ 0 \\frac { d t } { \\sqrt { ( 1 - t ^ 2 ) ( 1 - r ^ 2 t ^ 2 ) } } ( 0 \\le r < 1 , \\ z = \\sin \\varphi ) . \\end{align*}"}
+{"id": "7758.png", "formula": "\\begin{align*} p _ { \\rm r e v } ( x ) : = x ^ d p \\Big ( \\frac { 1 } { x } \\Big ) = \\sum _ { i = 0 } ^ d p _ i x ^ { d - i } , ~ p _ { \\rm r e v } ( x ) = p _ 0 \\prod _ { j = 1 } ^ d \\Big ( x - \\frac { 1 } { x _ j } \\Big ) ~ { \\rm i f } ~ p _ 0 \\neq 0 , \\end{align*}"}
+{"id": "8519.png", "formula": "\\begin{align*} \\delta _ { + } ( z ) \\delta _ { - } ^ { - 1 } ( z ) = \\mathrm { e } ^ { \\log \\left ( 1 + \\bar { r } _ 1 ( z ) r _ 2 ( z ) \\right ) } = 1 + \\bar { r } _ 1 ( z ) r _ 2 ( z ) , z \\in \\mathbb { R } . \\end{align*}"}
+{"id": "1275.png", "formula": "\\begin{align*} F _ 0 ( u ) : = \\begin{cases} u - u \\log ( u ) - 1 , & , \\\\ \\ - 1 , & , \\end{cases} \\end{align*}"}
+{"id": "2996.png", "formula": "\\begin{align*} A _ { n + 1 , k } ( x ; q ) \\ = \\ q ^ { n + 1 - k } \\left ( \\ , \\sum _ { i = 1 } ^ { k - 1 } A _ { n , i } ( x ; q ) \\ , + \\ , x \\sum _ { i = k } ^ n A _ { n , i } ( x ; q ) \\right ) \\end{align*}"}
+{"id": "3552.png", "formula": "\\begin{align*} \\sum _ { i , j \\in \\Z } q ^ { g ( i , j ) } & = \\sum _ { i , j \\in \\Z } q ^ { 2 i ^ 2 + 2 i j + 2 j ^ 2 + 2 i + 3 j + 1 } \\\\ & = \\sum _ { k , j \\in \\Z } q ^ { 2 k ^ 2 - 2 k j + 2 j ^ 2 - 2 k + 3 j + 1 } \\\\ & = \\sum _ { l , j \\in \\Z } q ^ { 2 l ^ 2 - 2 l j + 2 j ^ 2 + l + 1 } \\end{align*}"}
+{"id": "1177.png", "formula": "\\begin{align*} \\sup _ { n } \\mathbb { E } _ { \\mathbb { P } } \\left [ \\exp \\left \\{ \\kappa \\sum _ { i = 1 } ^ { n } \\int _ { T _ { 0 } } ^ { T _ { 0 } + \\delta } \\Delta K _ { t } ^ { i , n } \\cdot d W _ { t } ^ { i } - \\frac { \\kappa } { 2 } \\sum _ { i = 1 } ^ n \\int _ { T _ { 0 } } ^ { T _ { 0 } + \\delta } \\left | \\Delta K _ { t } ^ { i , n } \\right | ^ { 2 } d t \\right \\} \\right ] \\end{align*}"}
+{"id": "1757.png", "formula": "\\begin{align*} Y ^ { \\ast } ( t , x ) : = Y \\setminus \\overline { B _ { R _ 0 ( t , x ) } ( x ) } , \\end{align*}"}
+{"id": "7939.png", "formula": "\\begin{align*} \\norm { u _ j } _ { L ^ 2 } ^ 2 & = \\int _ \\S \\left ( \\int _ { x - r } ^ { x + r } \\prod _ { \\stackrel { i = 1 } { i \\neq j } } ^ n ( \\eta _ i ( y ) - \\eta _ i ( x ) ) ( - \\partial \\eta _ j ( x ) ) \\sin ^ { [ n ] } ( \\tilde \\Psi ( y ) - \\tilde \\Psi ( x ) ) \\ \\d y \\right ) ^ 2 \\ \\d x \\\\ & \\le \\int _ \\S \\int _ \\S \\prod _ { \\stackrel { i = 1 } { i \\neq j } } ^ n \\norm { \\partial \\eta _ i } _ { L ^ 2 } ^ 2 ( \\partial \\eta _ j ( x ) ) ^ 2 \\ \\d y \\ \\d x \\\\ & = \\prod _ { i = 1 } ^ n \\norm { \\partial \\eta _ i } _ { L ^ 2 } ^ 2 \\end{align*}"}
+{"id": "4373.png", "formula": "\\begin{align*} \\int _ K | ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f F ^ { 1 + \\delta } | ^ 2 _ { h _ { j _ K , 1 } } \\le C _ K ( \\sup _ K | F ^ { 1 + \\delta } | ^ 2 ) \\int _ { K \\cap \\{ \\Psi < - t _ 0 \\} } | f | ^ 2 _ { \\hat { h } } < + \\infty , \\end{align*}"}
+{"id": "8789.png", "formula": "\\begin{align*} & \\int _ { A _ { n , j } \\cap A _ n } \\int _ 0 ^ { \\tau _ { n + 1 } ( \\omega ) - \\tau _ n ( \\omega ) } D ^ p ( x _ { \\tau _ n + t } ( \\omega ) ) d t d \\P \\\\ & = \\int _ { A _ { n , j } \\cap A _ n } \\int _ 0 ^ { \\eta _ j ( K ) } D ^ p ( x _ { \\tau _ n + t } ( \\omega ) ) d t d \\P + \\int _ { A _ { n , j } \\cap A _ n } \\int _ { \\eta _ j ( K ) } ^ { \\tau _ { n + 1 } ( \\omega ) - \\tau _ n ( \\omega ) } D ^ p ( x _ { \\tau _ n + t } ( \\omega ) ) d t d \\P . \\end{align*}"}
+{"id": "1461.png", "formula": "\\begin{align*} w ( x ) : = \\begin{cases} M z _ { 0 } & x \\in ( 0 , z _ { 0 } ) , \\\\ S _ { H , V } ( x - z _ { 0 } ) + M z _ { 0 } & x \\geq z _ { 0 } , \\end{cases} \\end{align*}"}
+{"id": "6138.png", "formula": "\\begin{align*} \\lambda _ \\mathbb { I } - \\rho _ { \\mathbb { I } } = d ^ { \\mathbb { I } } h + h d ^ { \\mathbb { I I } } , \\end{align*}"}
+{"id": "3792.png", "formula": "\\begin{align*} \\left \\{ e _ { m , q } ( z ) = \\frac { z ^ m } { \\sqrt { \\Gamma ( m q + 1 ) } } \\ ; ; \\ ; m \\geq 0 \\right \\} \\end{align*}"}
+{"id": "4172.png", "formula": "\\begin{align*} h ( H _ { \\rho } ( x ) ) = h ( x ) \\end{align*}"}
+{"id": "4509.png", "formula": "\\begin{align*} \\begin{aligned} \\abs { c _ 3 ( S ^ 3 \\setminus S _ i ) } & \\le \\abs { c _ 2 ( S ^ 2 \\setminus S _ i ) } + \\abs { c _ 3 ( P ^ 3 \\setminus S _ i ) } \\\\ & \\le ( 2 k - t ' - \\abs { S \\cap S _ i } - q _ i - s _ 1 ) + ( s _ 1 - t _ i ) = 2 k - t ' - \\abs { S \\cap S _ i } - p _ i . \\end{aligned} \\end{align*}"}
+{"id": "7912.png", "formula": "\\begin{align*} c _ 3 ( q , 2 q , q , p ^ 0 ( 0 ) ) & = \\frac 1 8 \\left ( \\frac { - 1 } { q } f ( q r ) + 8 r - \\frac 1 q f ( 2 q r ) + \\frac { 1 } { 3 q } f ( 3 q r ) \\right ) \\\\ & = \\frac { 1 } { q } h ( q r ) , \\end{align*}"}
+{"id": "678.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | e ^ { i k \\phi } \\cos ( n \\theta ) \\left | l _ 2 , m _ 2 \\right \\rangle = ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
+{"id": "2907.png", "formula": "\\begin{align*} | \\tilde H ^ { ( n ) } ( y ) | \\le \\frac { C } { \\chi ^ 2 ( y ) } , y = 0 , \\ldots , n , \\ , n \\ge 1 . \\end{align*}"}
+{"id": "2823.png", "formula": "\\begin{align*} \\mathbf { x } ^ { ( k ) } = \\underset { \\mathbf { x } } { \\arg \\min } \\frac { 1 } { 2 } \\left \\| \\mathbf { x } - \\mathbf { r } ^ { ( k ) } \\right \\| ^ { 2 } + \\lambda \\| \\mathcal { F } ( \\mathbf { x } ) \\| _ { 1 } , \\end{align*}"}
+{"id": "1076.png", "formula": "\\begin{align*} s _ { - w ^ { - 1 } \\gamma } ( \\beta ) = \\beta + \\langle - w ^ { - 1 } \\gamma ^ \\vee , \\beta \\rangle w ^ { - 1 } \\gamma . \\end{align*}"}
+{"id": "6031.png", "formula": "\\begin{align*} S _ { \\omega } : = \\{ u \\in H _ { r } ^ { 1 } ( \\R ^ { 2 } ) \\setminus \\{ 0 \\} : J ' _ { \\omega } ( u ) = 0 , J _ { \\omega } ( u ) = E _ { \\omega } \\} . \\end{align*}"}
+{"id": "1281.png", "formula": "\\begin{align*} z \\in \\Lambda _ { n , m } : = [ - 2 ^ n + n + m + 1 , 2 ^ n - n - m - 1 ] ^ d , \\end{align*}"}
+{"id": "2799.png", "formula": "\\begin{align*} \\eta ^ { D } ( \\mathrm { d } x \\mathrm { d } h ) = Z _ { \\lambda } ^ { D } ( \\mathrm { d } x ) \\otimes e ^ { - \\pi \\lambda h } \\mathrm { d } h \\end{align*}"}
+{"id": "2018.png", "formula": "\\begin{align*} D \\widetilde H ( x , Y , Z , U ) = \\Big ( \\partial _ x H ( x , \\mu ) , \\partial _ { \\mu _ y } H ( x , \\mu ) , \\partial _ { \\mu _ z } H ( x , \\mu ) , \\partial _ { \\mu _ u } H ( x , \\mu ) \\Big ) ( Y , Z , U ) . \\end{align*}"}
+{"id": "5247.png", "formula": "\\begin{align*} \\begin{aligned} | | S ( t ) \\varphi _ 0 | | _ { T } & \\leq C | | \\varphi _ 0 | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } , \\\\ \\left \\| \\int _ { 0 } ^ { t } S ( t - s ) \\mathbb { P } \\nabla \\cdot F _ 0 ( s ) \\dd s \\right \\| _ T & \\leq C ' | | F _ 0 | | _ { L ^ { 2 } ( 0 , T ; L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) ) } . \\end{aligned} \\end{align*}"}
+{"id": "830.png", "formula": "\\begin{align*} w ( t , x ) = w _ 0 ( x ) + \\int _ 0 ^ t \\partial _ t w ( s , x ) \\mathrm { d } s \\begin{cases} \\leq w _ 0 ( x ) + \\left ( - g \\big ( c _ 0 ( x ) \\big ) \\frac { 1 } { w _ 0 ( x ) } + \\varepsilon \\right ) \\cdot t , \\\\ \\geq w _ 0 ( x ) + \\left ( - g \\big ( c _ 0 ( x ) \\big ) \\frac { 1 } { w _ 0 ( x ) } - \\varepsilon \\right ) \\cdot t \\end{cases} \\end{align*}"}
+{"id": "4709.png", "formula": "\\begin{align*} D \\sigma _ T ( x ) = \\frac { 1 } { \\pi } \\Big ( \\log \\Big ( \\frac { \\sin ( \\pi x _ 1 ) } { \\sin ( \\pi ( x _ 1 + x _ 2 ) ) } , \\log \\Big ( \\frac { \\sin ( \\pi x _ 2 ) } { \\sin ( \\pi ( x _ 1 + x _ 2 ) ) } \\Big ) \\Big ) . \\end{align*}"}
+{"id": "4847.png", "formula": "\\begin{align*} K ^ { ( 1 ) } ( S _ t , I _ t ) & = \\frac { w } { 2 \\theta ( \\partial _ i V ( S _ t , I _ t ) - \\partial _ s V ( S _ t , I _ t ) ) } , \\\\ K ^ { ( 2 ) } ( S _ t , I _ t ) & = \\frac { w } { 2 \\theta ( 1 - \\theta \\bar L ) ( \\partial _ i V ( S _ t , I _ t ) - \\partial _ s V ( S _ t , I _ t ) ) } . \\end{align*}"}
+{"id": "3254.png", "formula": "\\begin{align*} & \\sup _ { n \\in \\mathbb N } \\mathbb P \\left [ \\sup _ { t \\leq T } \\Delta _ n ^ { 1 - \\frac m 2 } \\| P _ N ^ m S A M P V _ t ^ n ( m _ 1 , . . . , m _ k ) \\| _ { \\mathcal H ^ m } > \\epsilon \\right ] \\to 0 N \\to \\infty . \\end{align*}"}
+{"id": "6564.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } | n _ { - } ( \\cdot , t ) | ^ { 2 } & \\le - 2 \\int _ { 0 } ^ { t } \\int _ { \\Omega } | \\nabla n _ { - } | ^ { 2 } + 2 \\int _ { 0 } ^ { t } \\int _ { \\Omega } | n _ { - } | | S _ { 0 } ( c ) | | \\nabla n _ { - } | | \\nabla c | \\\\ & \\le \\frac { 1 } { 2 } \\| S _ { 0 } \\| _ { \\mathcal { C } ( [ 0 , \\gamma ] ) } ^ { 2 } \\| \\nabla c \\| _ { L ^ { \\infty } ( 0 , T ; L ^ { \\infty } ( \\Omega ) ) } ^ { 2 } \\int _ { 0 } ^ { t } \\int _ { \\Omega } | n _ { - } | ^ { 2 } \\quad \\mbox { f o r } t \\le T \\end{aligned} \\end{align*}"}
+{"id": "6416.png", "formula": "\\begin{align*} \\Gamma _ { i j } ^ k = \\frac { 1 } { 2 } g ^ { l k } u _ { , i j l } \\end{align*}"}
+{"id": "5702.png", "formula": "\\begin{align*} R ( x , y ) & = ( ( a _ 0 x ^ { q + 1 } + b _ 0 x ^ q y + c _ 0 x y ^ q + d _ 0 y ^ { q + 1 } ) , ( a _ 1 x ^ { q + 1 } + b _ 1 x ^ q y + c _ 1 x y ^ q + d _ 1 y ^ { q + 1 } ) ) \\\\ & = ( f ( x , y ) , g ( x , y ) ) \\end{align*}"}
+{"id": "646.png", "formula": "\\begin{align*} a ( b ( x ) ) & \\in g ( b ( x ) ) \\ast g ' ( b ( x ) ) \\subseteq [ g ( f ( x ) \\ast f ' ( x ) ) ] \\ast [ g ' ( f ( x ) \\ast f ' ( x ) ) ] \\\\ & \\subseteq [ g ( f ( x ) ) \\ast g ( f ' ( x ) ) ] \\ast [ g ' ( f ( x ) ) \\ast g ' ( f ' ( x ) ) ] \\\\ & = ( g \\circ f ) ( x ) \\ast ( g \\circ f ' ) ( x ) \\ast ( g ' \\circ f ) ( x ) \\ast ( g ' \\circ f ' ) ( x ) . \\end{align*}"}
+{"id": "5919.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { E } \\Big [ \\sup _ { t \\leq T } \\Vert X ( t , x _ n ) - X ( t , x ) \\Vert _ H ^ p \\Big ] = 0 , \\end{align*}"}
+{"id": "8298.png", "formula": "\\begin{align*} y _ { b e n } ( R ) = y _ { n b } ( R ) , a y ' _ { b e n } ( R ) = b y ' _ { n b } ( R ) . \\end{align*}"}
+{"id": "3096.png", "formula": "\\begin{align*} \\langle y ^ * - \\tilde { y } ^ k , \\tilde { \\lambda } ^ k - \\beta ( \\tilde { y } ^ k - y ^ k ) - \\beta ( y ^ k - \\tilde { y } ^ k ) \\rangle _ { \\mathcal { L } ^ 2 } = \\langle y ^ * - \\tilde { y } ^ k , \\tilde { \\lambda } ^ k \\rangle _ { \\mathcal { L } ^ 2 } \\geq 0 . \\end{align*}"}
+{"id": "1279.png", "formula": "\\begin{align*} \\sum _ { \\eta _ { \\Lambda _ n } } \\gamma _ { \\Lambda _ n } ( \\eta _ { \\Lambda _ n } | r _ n \\eta _ { \\Lambda _ n ^ c } ) \\varphi _ n ( \\eta _ { \\Lambda _ n } ) = 0 . \\end{align*}"}
+{"id": "2555.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { i = 1 } ^ { n } a _ i ^ 2 \\displaystyle \\sum _ { i = 1 } ^ { n } b _ i ^ 2 - { \\Big ( \\displaystyle \\sum _ { i = 1 } ^ { n } a _ i b _ i \\Big ) } ^ 2 \\leq \\frac { n ^ 2 } { 4 } { ( M _ 1 M _ 2 - m _ 1 m _ 2 ) } ^ 2 . \\end{align*}"}
+{"id": "2078.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow { 0 } } \\frac { G ( r + k ) - G ( r ) } { k } = g ( r ) \\end{align*}"}
+{"id": "6795.png", "formula": "\\begin{align*} A _ { m n } ( x ) : = ( - 1 ) ^ { m + n } \\left ( \\sum _ { k = 1 } ^ { N } \\alpha _ { k } ^ { + } e ^ { - 2 \\tau _ { k } x } \\frac { \\left ( \\frac { 1 } { 2 } - \\tau _ { k } \\right ) ^ { m + n } } { \\left ( \\frac { 1 } { 2 } + \\tau _ { k } \\right ) ^ { m + n + 2 } } + \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } s ^ { + } \\left ( \\rho \\right ) e ^ { 2 i \\rho x } \\frac { \\left ( \\frac { 1 } { 2 } + i \\rho \\right ) ^ { m + n } } { \\left ( \\frac { 1 } { 2 } - i \\rho \\right ) ^ { m + n + 2 } } d \\rho \\right ) , \\end{align*}"}
+{"id": "7619.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t \\chi + \\gamma \\nabla \\cdot ( \\chi \\nabla \\Phi ) , & \\Omega \\times ( 0 , T ] , \\\\ & \\partial _ t \\Phi + \\frac { \\gamma } { 2 } \\lvert \\nabla \\Phi \\rvert ^ 2 + \\gamma \\chi = 0 , & \\Omega \\times ( 0 , T ] , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1487.png", "formula": "\\begin{align*} \\partial v = N \\partial _ { z , \\alpha } N ^ { - 1 } v & = \\gamma _ { \\alpha } \\partial _ z ( \\gamma _ { - \\alpha } v ) = \\gamma _ { \\alpha } ( \\gamma _ { - \\alpha } ' v + \\gamma _ { - \\alpha } \\partial _ z v ) \\\\ & = \\gamma _ { \\alpha } ( - \\alpha \\gamma _ { - \\alpha } v + \\gamma _ { - \\alpha } \\partial _ z v ) \\\\ & = ( \\partial _ z - \\alpha ) v . \\end{align*}"}
+{"id": "2546.png", "formula": "\\begin{align*} \\begin{aligned} & \\mu ( { { \\mathbf { q } } _ w } ) \\le { { c } _ { \\mu } } \\mu ( { { \\mathbf { q } } _ c } ) , \\\\ & \\left ( 1 - \\omega \\right ) \\left ( \\mathbf { e } ^ T ( { { \\mathbf { x } } _ { o } } + { { \\mathbf { s } } _ { o } } ) / k + 1 \\right ) = { { c } _ { x s } } , \\\\ & { { c } _ { \\mu } } + { { c } _ { x s } } \\le 1 . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "7777.png", "formula": "\\begin{align*} s _ h : = \\sum _ { j = 1 } ^ w x _ j ^ { h } = \\frac { 1 } { 2 \\pi \\sqrt { - 1 } } \\int _ { \\mathcal C } \\frac { p ' ( x ) } { p ( x ) } ~ x ^ h ~ d x . \\end{align*}"}
+{"id": "2242.png", "formula": "\\begin{align*} \\frac { 1 } { t } \\int _ 0 ^ t \\vert \\langle \\psi _ { 1 / 2 , f _ \\delta } , e ^ { - i s H _ N } \\psi _ { 1 / 2 , f _ \\delta } \\rangle \\vert ^ 2 \\ , d s = O ( \\log ( t ) / t ) ) \\end{align*}"}
+{"id": "6934.png", "formula": "\\begin{align*} z ^ \\top \\nabla ^ 2 f ( x ) z & = \\sum _ { i , j = 1 } ^ n z _ i z _ j \\partial _ { i j } f ( x ) = \\sum _ { i = 1 } ^ n z _ i ^ 2 V '' ( x _ i ) + \\sum _ { i , j = 1 } ^ n \\big ( z _ i ^ 2 - z _ i z _ j \\big ) J _ { i j } K '' ( x _ i - x _ j ) . \\end{align*}"}
+{"id": "8331.png", "formula": "\\begin{align*} c _ \\pm ( x ) = \\frac { 1 } { 2 } \\int _ { \\pm \\infty } ^ x | u _ y ( y , t ) | ^ 2 d y . \\end{align*}"}
+{"id": "7452.png", "formula": "\\begin{align*} Q ( x ) = Q ( x _ 1 , \\dotsc , x _ { 2 s } ) & : = \\sum _ { r = 1 } ^ s x _ { 2 r - 1 } x _ { 2 r } , \\\\ B ( x , y ) = B ( ( x _ 1 , \\dotsc , x _ { 2 s } ) , ( y _ 1 , \\dotsc , y _ { 2 s } ) ) & : = \\sum _ { r = 1 } ^ s ( x _ { 2 r - 1 } y _ { 2 r } + x _ { 2 r } y _ { 2 r - 1 } ) . \\end{align*}"}
+{"id": "5371.png", "formula": "\\begin{align*} S ^ { p q , r s } _ k = \\frac { \\partial ^ 2 S _ k [ D ^ 2 u ] } { \\partial r _ { p q } \\partial r _ { r s } } . \\end{align*}"}
+{"id": "7880.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Psi ( t , x ) = F ( \\Psi , p ) ( x ) . \\end{align*}"}
+{"id": "2113.png", "formula": "\\begin{align*} J _ 1 ( b , c ) = & \\frac { 2 b - 1 } { 2 } \\left ( \\log \\left ( \\binom { b + 1 } { 2 } + c \\right ) - \\log \\left ( \\binom { b } { 2 } + c \\right ) - 2 \\log \\left ( b + c \\right ) + 2 \\log \\left ( b + c - 1 \\right ) \\right ) , \\\\ J _ 2 ( b , c ) = & c \\left ( \\log ( c + 1 ) - \\log ( c ) - \\log ( b + c ) + \\log ( b + c - 1 ) \\right ) , \\\\ J _ 3 ( b , c ) = & \\log \\left ( \\binom { b + 1 } { 2 } + c \\right ) + \\log ( c + 1 ) - 2 \\log ( b + c ) . \\end{align*}"}
+{"id": "5786.png", "formula": "\\begin{align*} \\nabla _ X \\varphi = - \\frac { 1 } { 2 } \\sqrt { | c _ 1 | } \\ X _ 1 \\cdot \\nu _ 1 \\cdot \\varphi - \\frac { 1 } { 2 } B ( X ) \\cdot \\varphi \\end{align*}"}
+{"id": "3483.png", "formula": "\\begin{align*} \\prod _ { \\ell \\in I } | \\cos ( \\pi \\theta _ { \\ell } ) | = \\frac { \\prod _ { \\ell \\in \\tilde { I } } | \\cos ( \\pi \\theta _ { \\ell } ) | } { \\prod _ { \\ell \\in I ^ c } | \\cos ( \\pi \\theta _ { \\ell } ) | } \\geq e ^ { - \\varepsilon ( 2 q _ n - | I | ) } e ^ { ( - \\ln 2 ) | I | } . \\end{align*}"}
+{"id": "6651.png", "formula": "\\begin{align*} \\liminf _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\big ( X ^ { \\epsilon } \\in B ( \\varphi , \\gamma ) \\big ) & \\geq \\lim _ { \\delta \\to 0 } \\liminf _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\hat { X } ^ { \\epsilon } \\in B ( \\varphi , \\delta ) \\Big ) \\\\ & \\geq - \\mathcal { I } ^ { \\varphi } ( \\varphi ) = - \\mathcal { I } ( \\varphi ) , \\end{align*}"}
+{"id": "2551.png", "formula": "\\begin{align*} \\left \\| \\tilde { \\mathbf { U } } \\right \\| \\le ( 1 - \\omega ) \\rho , \\ \\rho = \\max _ { i } \\left ( 1 - \\frac { 2 \\omega ( x _ o ^ { ( i ) } ) _ 1 + 1 - \\omega } { \\beta _ { x _ w ^ { ( i ) } } + \\omega \\beta _ { x _ o ^ { ( i ) } } } \\right ) . \\end{align*}"}
+{"id": "5387.png", "formula": "\\begin{align*} G ^ { \\alpha \\beta } _ 0 = \\frac { \\partial G } { \\partial r _ { \\alpha \\beta } } ( \\nabla _ { \\alpha \\beta } u ( x _ 0 ) ) , 1 \\leq \\alpha , \\beta \\leq n - 1 . \\end{align*}"}
+{"id": "5399.png", "formula": "\\begin{align*} x _ n - y _ n \\geq ( 2 - \\epsilon ) b _ { n - 1 } \\mbox { o n } \\partial \\omega \\cap \\{ x _ n = 2 b _ { n - 1 } \\} . \\end{align*}"}
+{"id": "8501.png", "formula": "\\begin{align*} e ^ { i c _ + ( x ) } \\partial _ { x } \\left ( \\bar { u } _ x ( x ) e ^ { i c _ + ( x ) } \\right ) & = - \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } r _ 2 ( z ) e ^ { 2 i z x } \\left [ M _ { - , 2 2 } ( x ; z ) + \\bar { r } _ 1 ( z ) e ^ { - 2 i z x } M _ { - , 2 1 } ( x ; z ) \\right ] \\mathrm { d } z \\\\ & = - \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } r _ 2 ( z ) e ^ { 2 i s x } M _ { + , 2 2 } ( x ; z ) \\mathrm { d } z , \\end{align*}"}
+{"id": "8391.png", "formula": "\\begin{align*} & w ( x ; z ) = \\sum _ { n = 0 } ^ \\infty w _ n ( x ; z ) = \\sum _ { n = 0 } ^ \\infty F ^ n e _ 1 . \\end{align*}"}
+{"id": "5662.png", "formula": "\\begin{align*} ( f _ { 1 } , \\varphi _ { 1 } ) _ { * } = ( f _ { 2 } , \\varphi _ { 2 } ) _ { * } : H _ { * } ( G , M ) \\rightarrow H _ { * } ( K , N ) . \\end{align*}"}
+{"id": "8320.png", "formula": "\\begin{align*} \\begin{aligned} & y '' _ { b e n } + N _ { b e n } ( E ) y _ { b e n } = 0 \\\\ & y '' _ { n b } + N _ { n b } ( E ) y _ { n b } = 0 . \\end{aligned} \\end{align*}"}
+{"id": "5749.png", "formula": "\\begin{align*} \\gamma ( u ^ q + u + c ) & = ( \\gamma + 1 ) ( v ^ q + v + c ) , \\\\ \\gamma ( u ^ { q + 1 } + d ) & = ( \\gamma + 1 ) ( v ^ { q + 1 } + d ) . \\end{align*}"}
+{"id": "5924.png", "formula": "\\begin{align*} \\tau _ { u } ^ M : = & T \\wedge \\inf \\Big \\{ t \\geq 0 : \\Vert u ( t ) \\Vert _ H ^ 2 > M \\Big \\} \\\\ & \\wedge \\inf \\Big \\{ t \\geq 0 : \\int _ 0 ^ t \\Vert u ( s ) \\Vert _ { V } ^ { \\alpha } d s > M \\Big \\} . \\end{align*}"}
+{"id": "6215.png", "formula": "\\begin{align*} W _ + ( r ) & = ( 2 L + 3 ) \\left ( - \\frac { f } { r } + \\Q \\frac { \\Q ^ 2 + \\kappa } { \\Q ^ 2 + 3 \\kappa } + \\frac { 2 \\kappa \\Q ^ 2 } { \\Q ^ 2 + 3 \\kappa } \\frac { r } { f } + \\frac { 2 \\kappa \\Q } { \\Q ^ 2 + 3 \\kappa } \\frac { 1 } { f ^ 2 } \\right ) , \\\\ W _ - ( r ) & = - \\frac { f } { r } - \\Q + 2 \\kappa \\frac { r } { f } , \\end{align*}"}
+{"id": "6049.png", "formula": "\\begin{align*} \\begin{cases} \\alpha a + ( \\alpha - 1 ) b + ( 3 \\alpha - 2 ) c - \\frac { p \\alpha - 1 } { p } d = 0 ; \\\\ ( 1 - 2 \\alpha \\mu ) a + ( 1 - 2 \\mu ( \\alpha - 1 ) ) b + 3 \\big ( 1 - \\mu ( 6 \\alpha - 4 ) \\big ) c - \\big ( 1 - \\mu ( 2 p \\alpha - 2 ) \\big ) d = 0 ; \\\\ \\big ( 2 \\mu ( \\alpha - 1 ) - 1 \\big ) b + 2 \\big ( \\mu ( 6 \\alpha - 4 ) - 1 \\big ) c - \\frac { \\mu ( 2 p \\alpha - 2 ) - 1 } { p } d = 0 . \\end{cases} \\end{align*}"}
+{"id": "1920.png", "formula": "\\begin{align*} \\| x _ { k + 1 } - x ^ \\star \\| ^ 2 \\leq \\tfrac { 1 } { 2 } \\| x _ k - x ^ \\star \\| ^ 2 , \\textnormal { f o r a l l } k = 0 , 1 , 2 , \\ldots , T . \\end{align*}"}
+{"id": "2628.png", "formula": "\\begin{align*} & \\varepsilon ( h , x + y ) [ [ \\alpha ( x ) , \\alpha ( y ) ] , \\alpha ( h \\cdot z ) ] + \\varepsilon ( x + y , z ) [ [ h \\cdot z , \\alpha ( x ) ] , \\alpha ^ 2 ( y ) ] \\\\ & + \\varepsilon ( x , y + z ) \\big ( 2 [ \\alpha ( h ) \\cdot [ y , z ] , \\alpha ^ 2 ( x ) ] - \\varepsilon ( x , y + z ) [ [ h \\cdot y , \\alpha ( z ) ] , \\alpha ^ 2 ( x ) ] \\big ) = 0 . \\end{align*}"}
+{"id": "7734.png", "formula": "\\begin{align*} \\dot { v } ( s ) = \\begin{cases} \\frac { 1 } { | G / G _ 0 | } \\sum _ { t G _ 0 \\in G / G _ 0 } u ( t ^ { - 1 } s t ) & s \\in G _ 0 \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "835.png", "formula": "\\begin{align*} \\theta _ { \\rho ^ \\varepsilon } ( 0 , z _ 1 ) = F ( z _ 1 ) + \\rho _ 0 ^ \\varepsilon \\nu _ \\Sigma ( z _ 1 ) \\neq F ( z _ 2 ) + \\rho _ 0 ^ \\varepsilon \\nu _ \\Sigma ( z _ 2 ) = \\theta _ { \\rho ^ \\varepsilon } ( 0 , z _ 2 ) \\end{align*}"}
+{"id": "3313.png", "formula": "\\begin{align*} \\phi _ i & = \\phi \\cap V ( K _ i ) \\ ( i \\in [ m ] ) , \\\\ \\overline { \\phi } & = \\{ i \\in [ m ] \\ | \\ \\phi _ i \\notin L _ i \\} . \\end{align*}"}
+{"id": "4681.png", "formula": "\\begin{align*} \\psi _ { K , H } ( t ) = ( n - 1 ) \\sqrt K \\cdot \\frac { ( n - 1 ) \\sqrt K \\sinh \\sqrt K t + H \\cosh \\sqrt K t } { ( n - 1 ) \\sqrt K \\cosh \\sqrt K t + H \\sinh \\sqrt K t } . \\end{align*}"}
+{"id": "7800.png", "formula": "\\begin{align*} \\norm { f } _ { H ^ 1 _ 0 } = \\sqrt { f \\cdot f } = \\sqrt { \\norm { f } _ { L ^ 2 } ^ 2 + \\norm { D f } _ { L ^ 2 } ^ 2 } . \\end{align*}"}
+{"id": "3414.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} \\phi ( k ) \\\\ \\phi ( k - 1 ) \\end{matrix} \\right ) = A _ k ( \\theta , E ) \\left ( \\begin{matrix} \\phi ( 0 ) \\\\ \\phi ( - 1 ) \\end{matrix} \\right ) , \\end{align*}"}
+{"id": "4830.png", "formula": "\\begin{align*} \\langle f \\rangle ( a ) = \\lim _ { { T ' } \\to \\infty } \\frac { 1 } { { T ' } } \\int _ 0 ^ { T ' } f ( \\Phi _ { - t \\Lambda } a ) d t , a \\in \\C ^ n . \\end{align*}"}
+{"id": "217.png", "formula": "\\begin{align*} S = \\{ s _ 1 , \\ldots , s _ 6 \\} s _ 1 = a _ 4 t ^ { - 3 } , \\ , s _ 2 = t ^ { - 2 } , \\ , s _ 3 = 1 , \\ , s _ 4 = a _ 0 , \\ , s _ 5 = t ^ 2 , \\ , s _ 6 = a _ 1 t ^ 3 . \\end{align*}"}
+{"id": "7117.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\mu \\circ ( ( R \\circ \\eta ) \\otimes \\eta ) & = & \\mu \\circ ( R \\otimes i d _ M ) \\circ ( \\eta \\otimes \\eta ) \\\\ & = & \\mu \\circ ( i d _ M \\otimes L ) \\circ ( \\eta \\otimes \\eta ) \\\\ & = & \\mu \\circ ( \\eta \\otimes i d _ M ) \\circ ( i d _ I \\otimes ( L \\circ \\eta ) ) \\\\ & = & \\lambda _ M \\circ ( i d _ I \\otimes ( L \\circ \\eta ) ) \\\\ & = & L \\circ \\eta \\circ \\lambda _ I \\end{array} \\end{align*}"}
+{"id": "3664.png", "formula": "\\begin{align*} m _ { \\mathrm A D M } ( g ) = \\frac { 1 } { 2 ( n - 1 ) \\omega _ { n - 1 } } \\lim _ { r \\to \\infty } \\int _ { | x | = r } \\sum _ { i , j = 1 } ^ n \\left ( \\frac { \\partial g _ { i j } } { \\partial x _ i } - \\frac { \\partial g _ { i j } } { \\partial x _ j } \\right ) \\frac { x _ j } { | x | } \\ , d \\sigma . \\end{align*}"}
+{"id": "4805.png", "formula": "\\begin{align*} a ( u , v ) = \\lambda ( u , v ) v \\in H _ 0 ^ 2 ( D ) . \\end{align*}"}
+{"id": "4636.png", "formula": "\\begin{align*} \\phi ( \\sigma , \\tau ) = \\epsilon ^ { \\sum _ { r = 1 } ^ j \\lambda _ r ( d _ r - 1 ) } . \\end{align*}"}
+{"id": "4006.png", "formula": "\\begin{align*} u ^ k = \\arg \\min _ { w \\in H ^ 1 ( \\Omega ) } \\bigg ( \\frac { | \\Omega | } { 2 \\mathrm { N } _ I } \\sum _ { n = 1 } ^ { \\mathrm { N } _ I } ( w - u ^ { k - 1 } ) ^ 2 ( x _ n ^ I ) + \\tau _ k \\tilde { N } ( w ; w ) \\bigg ) . \\end{align*}"}
+{"id": "5599.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } A ( \\tau _ { y , N + 1 } ( x ) ) + \\sum _ { n = 1 } ^ N \\left [ \\hat { A } \\circ \\hat { \\sigma } ^ { - n } ( y | x ' ) - \\hat { A } \\circ \\hat { \\sigma } ^ { - n } ( \\sigma ( y ) | x ' ) \\right ] \\ \\end{align*}"}
+{"id": "3754.png", "formula": "\\begin{align*} & \\deg ( f _ i m ' _ { i , i } ) \\\\ = & \\deg ( f _ i ) + \\deg ( d _ i ) \\\\ \\overset { ( \\ref { i d e n t i t y 0 } ) } { = } & \\deg ( f _ { l _ 0 } ) + \\deg ( m '' _ { l _ 0 , l _ 0 } ) - \\deg ( m ' _ { i , l _ 0 } ) + \\deg ( d _ i ) \\\\ \\overset { ( \\ref { i d e n t i t y 2 } ) } { > } & k - 1 - \\deg ( d _ 1 ) + \\deg ( m '' _ { l _ 0 , l _ 0 } ) - \\deg ( m ' _ { i , l _ 0 } ) + \\deg ( d _ i ) \\\\ \\geq & k - 1 + \\deg ( m '' _ { l _ 0 , l _ 0 } ) - \\deg ( m ' _ { i , l _ 0 } ) \\\\ = & k - 1 + \\deg ( m ' _ { l _ 0 , l _ 0 } ) - d e g ( m ' _ { i , l _ 0 } ) \\\\ \\overset { ( \\ref { i d e n t i t y 1 } ) } { \\geq } & k \\end{align*}"}
+{"id": "6536.png", "formula": "\\begin{align*} q _ { \\pi ( a ) } = p _ { \\pi ( a ) } . \\end{align*}"}
+{"id": "7617.png", "formula": "\\begin{align*} C _ j ( S _ j ^ { \\top } R _ j ) ^ { \\top } ( S _ j ^ { \\top } R _ j ) C _ j ^ { \\top } v = \\lambda C _ j C _ j ^ { \\top } v , \\end{align*}"}
+{"id": "4207.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\frac { 1 } { \\varepsilon ^ 2 \\ln \\frac { 1 } { \\varepsilon } } \\int _ \\Omega \\left ( v ^ \\tau _ \\varepsilon - q \\ln \\frac { 1 } { \\varepsilon } \\right ) ^ p _ + v ^ \\tau _ \\varepsilon d x = q ^ { p + 1 } ( \\bar { x } ) \\int _ { \\hat { B } _ \\tau ( 0 ) } ( \\hat { U } ( y ) + \\ln \\tau ) ^ { p } _ + d y . \\end{align*}"}
+{"id": "1912.png", "formula": "\\begin{align*} \\pi ( s ) = { \\frac { w ( s ) } { f _ { q } ( s ) } } \\pi \\bigl ( { f _ { q } ( s ) } \\bigr ) . \\end{align*}"}
+{"id": "1919.png", "formula": "\\begin{align*} T _ { \\textnormal { i n n e r } } = \\left \\lceil \\left ( \\tfrac { 2 ^ { p + 2 } ( 5 p + 1 ) } { p ! } \\tfrac { L D ^ { p - 1 } } { \\mu _ { \\textnormal { M i n t y } } } \\right ) ^ { \\frac { 2 } { p } } + \\left ( \\tfrac { 2 ^ { p + 5 } } { p ! } \\tfrac { L D ^ { p - 1 } } { \\mu _ { \\textnormal { M i n t y } } } \\right ) ^ { \\frac { 2 } { p + 1 } } \\right \\rceil , \\end{align*}"}
+{"id": "5798.png", "formula": "\\begin{align*} [ X _ 1 ] = ( \\frac { \\mu } { 2 } d \\overline { z } - \\frac { 2 } { \\mu } h _ z ^ 2 d z + \\frac { \\tau _ 0 } { 2 \\mu } d \\overline { z } ) J - \\frac { 1 } { \\mu } d h \\sqrt { \\tau _ 0 } I . \\end{align*}"}
+{"id": "8242.png", "formula": "\\begin{align*} \\lim \\limits _ { i \\to \\infty } \\left | \\ , ^ { k _ i } \\ ! \\lambda / \\left ( \\ , ^ { k _ i } \\ ! \\lambda _ 1 \\right ) \\right | = \\infty . \\end{align*}"}
+{"id": "4859.png", "formula": "\\begin{align*} e ^ { - \\phi ( x ) } = \\int _ { \\mathbb R } e ^ { - { \\psi } ( x , y ) } d y . \\end{align*}"}
+{"id": "4586.png", "formula": "\\begin{align*} \\norm { \\Theta _ { \\norm { \\cdot } } ( \\nu ) } _ { ( C ( [ 0 , T ] ; ( U , \\norm { \\cdot } _ { U } ) ) , \\norm { \\cdot } _ \\infty ) ' } = \\norm { \\nu } _ { v a r } ( \\nu \\in M ( [ 0 , T ] ; U ' ) ) \\end{align*}"}
+{"id": "2694.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ d \\mu ( q _ i ) = \\prod _ { i = 1 } ^ d ( - 1 ) ^ { | P _ i | } = ( - 1 ) ^ k . \\end{align*}"}
+{"id": "1589.png", "formula": "\\begin{align*} \\phi _ 2 ( t _ 1 , t _ 2 ) : = \\left ( \\frac { \\alpha } { \\sigma ( \\alpha ) } \\right ) ^ { \\ln ( t _ 1 / \\alpha ) } \\left ( \\frac { \\sigma ( \\alpha ) } { \\sigma ^ 2 ( \\alpha ) } \\right ) ^ { \\ln ( t _ 2 / \\sigma ( \\alpha ) ) } \\\\ \\phi _ 3 ( t _ 1 , t _ 2 ) : = \\left ( \\frac { \\alpha } { \\sigma ^ 2 ( \\alpha ) } \\right ) ^ { \\ln ( t _ 1 / \\alpha ) } \\left ( \\frac { \\sigma ( \\alpha ) } { \\alpha } \\right ) ^ { \\ln ( t _ 2 / \\sigma ( \\alpha ) ) } . \\end{align*}"}
+{"id": "1503.png", "formula": "\\begin{align*} D _ { a + } ^ { \\nu _ { 1 } } I _ { a + } ^ { \\nu _ { 2 } } u ( t ) = I _ { a + } ^ { \\nu _ { 2 } - \\nu _ { 1 } } u ( t ) , \\mbox { i f } 0 \\le \\nu _ { 1 } \\le \\nu _ { 2 } ; \\end{align*}"}
+{"id": "8550.png", "formula": "\\begin{align*} f ( t ) = ( \\kappa \\ , * \\ , \\phi ) ( t ) \\ , = \\ , \\int _ 0 ^ t \\kappa ( t - \\tau ) \\phi ( \\tau ) \\ , d \\tau , \\ t > 0 \\end{align*}"}
+{"id": "5988.png", "formula": "\\begin{align*} \\int _ M \\sum _ i \\lambda _ i ^ + ( x ) \\ , d x = h _ \\mu ( f ) , \\end{align*}"}
+{"id": "393.png", "formula": "\\begin{align*} U ^ { \\dagger } = A + H + \\sum _ { k \\in \\nabla } \\Delta _ k . \\end{align*}"}
+{"id": "32.png", "formula": "\\begin{align*} \\Delta _ { H } u = \\sum _ { i = 1 } ^ m X _ { i } ^ 2 u . \\end{align*}"}
+{"id": "2660.png", "formula": "\\begin{align*} \\small [ P ( u _ t ( t ) ) ] _ t + A u ( t ) + B ( t , x , u _ t ( t ) ) = 0 , t \\in J = ( 0 , \\infty ) , \\end{align*}"}
+{"id": "3979.png", "formula": "\\begin{align*} ( \\mathbf { x } , - \\mathbf { y } ) \\underbrace { \\begin{pmatrix} G _ { 1 } & G _ { 2 } & \\cdots & G _ { 2 k - 1 } \\\\ G _ { 2 } & G _ { 3 } & \\cdots & G _ { 2 k } \\end{pmatrix} } _ M = 0 . \\end{align*}"}
+{"id": "9285.png", "formula": "\\begin{align*} C _ m \\left ( \\bigcup _ { N = 1 } ^ \\infty U _ { p , N } ( \\sigma ) \\right ) = C _ m ( U _ p ( \\sigma ) ) = \\lim \\limits _ { N \\to \\infty } C _ m ( U _ { p , N } ( \\sigma ) ) \\end{align*}"}
+{"id": "8389.png", "formula": "\\begin{align*} \\begin{aligned} \\| F f \\| _ { L ^ \\infty } \\leq ( 2 \\| u _ { x } \\| ^ 2 _ { L ^ 2 } + \\| u _ x \\| ^ 3 _ { L ^ 3 } + 2 \\| u _ { x x } \\| _ { L ^ 1 } + \\| u _ { x } \\| _ { L ^ 1 } ) \\| _ { L ^ { \\infty } } . \\end{aligned} \\end{align*}"}
+{"id": "3252.png", "formula": "\\begin{align*} \\langle \\rho _ { \\Sigma } ^ { \\otimes k } ( m _ 1 , . . . , m _ k ) , \\bigotimes _ { l = 1 } ^ { m _ 1 } h _ { 1 , l } \\otimes . . . \\otimes \\bigotimes _ { l = 1 } ^ { m _ k } h _ { k , l } \\rangle _ { \\mathcal H ^ m } = & \\prod _ { j = 1 } ^ { k } \\sum _ { p \\in \\mathcal P ( m _ l ) } \\prod _ { ( x , y ) \\in p } \\langle \\Sigma _ s h _ { x , j } , h _ { y , j } \\rangle . \\end{align*}"}
+{"id": "1433.png", "formula": "\\begin{align*} \\mathfrak { X } & = \\frac { 1 } { \\pi ^ { d / 2 - 1 / 2 } \\prod _ { j = 1 } ^ { d - 1 } j ! } D ( d / 2 - 1 / 2 , - 1 / 2 ) \\\\ & = \\frac { 1 } { \\pi ^ { d / 2 - 1 / 2 } \\prod _ { j = 1 } ^ { d - 1 } j ! } I ( d / 2 - 1 / 2 , 1 / 2 ) . \\end{align*}"}
+{"id": "134.png", "formula": "\\begin{align*} P _ N ( v u _ 1 ) & = P _ N ( P _ { \\ll N } v \\cdot u _ 1 ) + P _ N ( P _ { \\gtrsim N } v \\cdot u _ 1 ) \\\\ & \\sim P _ N ( P _ { \\ll N } v \\cdot P _ N u _ 1 ) + P _ N ( P _ { \\gtrsim N } v \\cdot P _ { \\gtrsim N } u _ 1 ) . \\end{align*}"}
+{"id": "3393.png", "formula": "\\begin{align*} p _ { k , 0 } & = { k + 3 \\choose 3 } - 4 { k - N + 2 \\choose 3 } - { k + 2 \\choose 2 } + 3 { k - N + 1 \\choose 2 } - { 2 N - k + 2 \\choose 3 } \\\\ & = \\frac { 1 } { 6 } ( k - N ) \\left ( - 2 k ^ 2 + 4 k N - 3 k + 4 N ^ 2 + 1 5 N + 5 \\right ) , \\end{align*}"}
+{"id": "8193.png", "formula": "\\begin{align*} x _ { m , 2 j } & + x _ { m , 2 j + 1 } + x _ { 1 , ( n + 1 ) - 2 j } + x _ { 1 , ( n + 1 ) - ( 2 j + 1 ) } \\\\ & = x _ { m , 2 j } + x _ { m , 2 j + 1 } + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { 1 , 2 j } \\bigr ) + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { 1 , 2 j + 1 } \\bigr ) \\\\ & = \\bigl ( ( n _ 1 + 1 ) m + 1 \\bigr ) + S - \\bigl ( ( n _ 1 + 1 ) m + 1 \\bigr ) = S . \\end{align*}"}
+{"id": "5056.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B _ j \\cdot B _ j \\dd x + 2 \\mu _ j \\int _ { 0 } ^ { t } \\int _ { \\Omega } \\nabla \\times B _ j \\cdot B _ j \\dd x \\dd s = \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B _ { 0 } \\cdot B _ 0 \\dd x , \\end{align*}"}
+{"id": "9333.png", "formula": "\\begin{align*} \\mathbb P \\left [ \\frac 1 n \\sum _ { i = 1 } ^ n ( X _ i - \\mathbb E [ X _ i ] ) < - \\epsilon \\right ] \\le \\exp \\left ( - \\frac { n \\epsilon ^ 2 } { 2 \\sigma ^ 2 } \\right ) \\end{align*}"}
+{"id": "1004.png", "formula": "\\begin{align*} \\langle \\omega _ a , \\beta \\rangle = \\begin{cases} 1 , & a = ( \\beta , 0 ) , \\\\ 0 , & a \\neq ( \\beta , 0 ) . \\end{cases} \\end{align*}"}
+{"id": "2346.png", "formula": "\\begin{align*} x _ t = \\frac { x _ { y y } } { e ^ { 2 \\phi } + x _ y ^ 2 } - \\phi ' ( x ) \\left ( 1 + \\frac { x _ y ^ 2 } { e ^ { 2 \\phi } + x _ y ^ 2 } \\right ) = \\frac { \\partial } { \\partial y } \\left ( e ^ { - \\phi } \\tan ^ { - 1 } ( x _ { y } e ^ { - \\phi } ) \\right ) + \\phi ' ( x ) \\left ( x _ y e ^ { - \\phi } \\tan ^ { - 1 } ( x _ y ) e ^ { - \\phi } - 1 \\right ) , \\end{align*}"}
+{"id": "4799.png", "formula": "\\begin{align*} F _ n ( \\eta ) : = T _ n - \\frac { 1 } { \\eta } I _ n , \\end{align*}"}
+{"id": "4714.png", "formula": "\\begin{align*} v ( x ) = ( 1 + 2 d ^ 2 ) x _ 2 \\log ( x _ 2 / d ) + x _ 2 ( ( x _ 1 - r \\gamma ) ^ 2 - d ^ 2 - C _ 0 ) \\end{align*}"}
+{"id": "3551.png", "formula": "\\begin{align*} \\sum _ { \\substack { i , j \\in \\Z \\\\ i \\\\ j } } q ^ { i ^ 2 + i j + j ^ 2 + i + j } & = \\sum _ { m , n \\in \\Z } q ^ { 2 m ^ 2 + 2 m n + 2 n ^ 2 + 2 m + 3 n + 1 } \\\\ & = \\sum _ { m , n \\in \\Z } q ^ { g ( m , n ) } \\end{align*}"}
+{"id": "223.png", "formula": "\\begin{align*} R _ q ^ f ( n ) = R _ { \\mathcal { W } , q } ^ f ( n ) = \\big \\{ g \\in R _ q ( n ) \\mid g _ { | i } \\in B _ { H , S _ 0 } ( f ( n ) ) i \\in \\mathbb { Z } \\big \\} . \\end{align*}"}
+{"id": "8281.png", "formula": "\\begin{align*} \\bar \\pi \\bar p _ T = \\bar \\pi . \\end{align*}"}
+{"id": "6295.png", "formula": "\\begin{align*} \\begin{aligned} \\min & \\ \\frac { 1 } { 2 } x ^ T A x + b ^ T x \\\\ \\mbox { s . t . } & \\ \\frac { 1 } { 2 } x ^ T B _ i x + c _ i ^ T x + d _ i \\leq 0 , i = 1 , \\dots , m , \\\\ & \\ - 1 \\leq x _ i \\leq 1 , i = 1 , \\dots , n , \\end{aligned} \\end{align*}"}
+{"id": "6164.png", "formula": "\\begin{align*} \\hat { H } _ { 1 , 2 } = \\hat { \\pi } _ r ^ 2 + V _ { 1 , 2 } ( r ) + E _ 0 , V _ { 1 , 2 } ( r ) = W ^ 2 ( r ) \\mp f ( r ) \\frac { d W } { d r } , \\end{align*}"}
+{"id": "7743.png", "formula": "\\begin{align*} D ^ a _ F ( 0 ) = \\{ M _ 1 ^ T D _ 1 M _ 2 ^ T D _ 2 \\ldots M _ { L - 1 } ^ T D _ { L - 1 } M _ L ^ T \\} \\end{align*}"}
+{"id": "2981.png", "formula": "\\begin{align*} \\limsup _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\min _ { 1 \\leq i < j \\leq k } { d } ( T ^ { ( n , [ n \\beta + t ] ) } x _ i , T ^ { ( n , [ n \\beta + t ] ) } x _ j ) \\geq \\eta > 0 . \\end{align*}"}
+{"id": "1789.png", "formula": "\\begin{align*} { } _ { A + 1 } F _ { A } \\left [ \\left . \\begin{matrix} a _ { 1 } , a _ { 2 } , \\dots , a _ { A + 1 } \\\\ a _ { 1 } ^ { \\prime } , \\dots , a _ { A } ^ { \\prime } \\end{matrix} \\right | x \\right ] : = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( a _ { 1 } ) _ { n } ( a _ { 2 } ) _ { n } \\cdots ( a _ { A + 1 } ) _ { n } } { ( a _ { 1 } ^ { \\prime } ) _ { n } \\cdots ( a _ { A } ^ { \\prime } ) _ { n } } \\frac { x ^ { n } } { ( 1 ) _ { n } } , \\end{align*}"}
+{"id": "6823.png", "formula": "\\begin{align*} I _ { l } ^ { \\left ( 1 \\right ) } ( s , \\rho ) = \\frac { 1 } { 2 i } \\int _ { \\mathbb { R } } \\frac { e ^ { i S ( \\tau , x - s ) } \\tau ^ { l - 2 } } { \\left ( \\tau - \\rho \\right ) ^ { l + 1 } } d \\tau \\end{align*}"}
+{"id": "6786.png", "formula": "\\begin{align*} u _ { t } - 6 u u _ { x } + u _ { x x x } = 0 . \\end{align*}"}
+{"id": "3804.png", "formula": "\\begin{align*} z ^ q D _ * ^ q ( f ) ( z ) = \\sum _ { n = 1 } ^ { \\infty } a _ n \\frac { \\Gamma ( q n + 1 ) } { \\Gamma ( q ( n - 1 ) + 1 } z ^ { q m } . \\end{align*}"}
+{"id": "9178.png", "formula": "\\begin{align*} \\begin{aligned} \\dfrac { | b _ { 1 , \\delta } ( x _ a ) | } { 2 } = & \\ , \\left | \\int _ 0 ^ 1 h ( x _ a + \\delta u ( t ) ) u ( t ) \\ , d t \\right | \\\\ = & \\ , \\left | \\int _ 0 ^ 1 ( h ( x _ a + \\delta u ( t ) ) - h ( x _ a ) ) u ( t ) \\ , d t \\right | \\\\ \\le & \\ , \\int _ 0 ^ 1 \\left | h ( x _ a + \\delta u ( t ) ) - h ( x _ a ) \\right | \\ , d t \\le L _ r \\delta . \\end{aligned} \\end{align*}"}
+{"id": "2373.png", "formula": "\\begin{align*} C _ 0 ( \\mathcal { H } _ { \\ast } ) = \\mathrm { I m } \\ , \\partial _ 0 \\oplus s _ { _ 0 } ( \\mathrm { I m } \\ , \\varphi _ 0 ) = \\mathrm { I m } \\ , \\partial _ 0 . \\end{align*}"}
+{"id": "2122.png", "formula": "\\begin{align*} z & = \\sqrt { \\frac { a } { a + b } } \\left ( \\frac { \\sqrt { 1 - b } \\ , \\cos ( t ) } { \\sqrt { 1 + a } } + \\frac { i \\sqrt { 1 + b } \\ , \\sin ( t ) } { \\sqrt { 1 - a } } \\right ) , \\\\ w & = \\pm \\sqrt { \\frac { b } { a + b } } \\left ( \\frac { \\sqrt { 1 - a } \\ , \\sin ( t ) } { \\sqrt { 1 + b } } - \\frac { i \\sqrt { 1 + a } \\ , \\cos ( t ) } { \\sqrt { 1 - b } } \\right ) , t \\in [ 0 , 2 \\pi ) , \\end{align*}"}
+{"id": "5005.png", "formula": "\\begin{align*} \\tau = \\sum _ { j = 1 } ^ \\infty \\frac { c _ j } { 2 ^ j } \\end{align*}"}
+{"id": "5100.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } | b | ^ { 2 } \\dd x = 2 \\pi \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\left ( | \\nabla \\phi | ^ { 2 } + | G | ^ { 2 } \\right ) \\frac { 1 } { r } \\dd z \\dd r . \\end{align*}"}
+{"id": "8258.png", "formula": "\\begin{align*} \\sum _ { i = 2 m + 1 } ^ { n - 1 } \\sum _ { j = 1 } ^ { i } j V _ { i , j } \\psi ^ { n } _ i ( s ) \\psi ^ { n } _ j ( s ) & = \\sum _ { i = 2 m + 1 } ^ { n - 1 } \\sum _ { j = 1 } ^ { 2 m } j V _ { i , j } \\psi ^ { n } _ i ( s ) \\psi ^ { n } _ j ( s ) + \\sum _ { i = 2 m + 1 } ^ { n - 1 } \\sum _ { j = i } ^ { n - 1 } i V _ { j , i } \\psi ^ { n } _ j ( s ) \\psi ^ { n } _ i ( s ) . \\end{align*}"}
+{"id": "3913.png", "formula": "\\begin{align*} h ( x , y ) \\ge \\frac { 1 } { 2 \\pi } \\ln \\frac { 1 } { | x - y | + 2 \\max \\{ d i s t ( x , \\partial \\Omega ) , d i s t ( y , \\partial \\Omega ) \\} } . \\end{align*}"}
+{"id": "80.png", "formula": "\\begin{align*} \\langle \\mu , v \\gamma _ 1 \\rangle \\geq - 1 , \\langle \\mu , v \\gamma _ 2 \\rangle \\geq - 1 , \\langle \\mu , v \\gamma _ 1 + v \\gamma _ 2 \\rangle = - 1 . \\end{align*}"}
+{"id": "6403.png", "formula": "\\begin{align*} \\omega = \\sum _ { i = 1 } ^ n d x ^ i \\wedge d \\theta _ i . \\end{align*}"}
+{"id": "3996.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t u + \\mathcal { L } u = & \\ F , ( 0 , T ] \\times \\Omega \\\\ u ( 0 , \\cdot ) = & \\ u _ 0 , \\ , \\Omega \\\\ u = & \\ g _ D ^ { } , ( 0 , T ] \\times \\Gamma _ D ^ { } , \\\\ \\mathbf { n } \\cdot \\nabla u = & \\ g _ N ^ { } , ( 0 , T ] \\times \\Gamma _ N ^ { } , \\end{aligned} \\end{align*}"}
+{"id": "1068.png", "formula": "\\begin{align*} \\langle \\alpha ^ \\vee , 2 \\rho \\rangle = \\sum _ { \\beta \\in I } \\langle \\alpha ^ \\vee , \\beta \\rangle . \\end{align*}"}
+{"id": "3539.png", "formula": "\\begin{align*} n ! F _ { n + 1 } \\ & = \\ \\sum _ { k = 1 } ^ { n } \\begin{bmatrix} n \\\\ k \\end{bmatrix} A _ k , \\\\ n ! F _ { n + 2 } \\ & = \\ \\sum _ { k = 1 } ^ { n } \\begin{bmatrix} n \\\\ k \\end{bmatrix} B _ k . \\\\ \\end{align*}"}
+{"id": "806.png", "formula": "\\begin{align*} \\partial ^ \\square c = - \\frac { m d } { \\alpha _ d } R ^ { - d - 1 } R ' = 0 + \\frac { m } { \\alpha _ d } R ^ { - d } \\cdot \\frac { - d } { R } \\cdot R ' = \\Delta _ \\Gamma \\big ( G ' ( c ) \\big ) + c H V \\end{align*}"}
+{"id": "8595.png", "formula": "\\begin{align*} \\sum _ { | I | = n } | \\det ( \\{ u _ i \\} _ { i \\in I } ) | & \\sum _ { | I | = n - 2 } | \\det ( u , v , ( u _ i ) _ { i \\in I } ) | \\\\ & \\le \\sum _ { | I | = n - 1 } | \\det ( u , ( u _ i ) _ { i \\in I } ) | \\sum _ { | I | = n - 1 } | \\det ( v , ( u _ i ) _ { i \\in I } ) | . \\end{align*}"}
+{"id": "2527.png", "formula": "\\begin{align*} \\Delta { { \\mathbf { x } } ^ { T } } \\Delta \\mathbf { s } = 0 . \\end{align*}"}
+{"id": "4572.png", "formula": "\\begin{align*} & B _ { i , j , 0 } + B _ { i , k , 0 } + B _ { i , l , 0 } = A _ { i , 0 , 0 } \\\\ & z _ j B _ { i , j , 0 } + z _ k B _ { i , k , 0 } + s _ l B _ { i , l , 0 } = \\sqrt { q } A _ { i , 1 , 0 } \\\\ & z _ j ^ 2 B _ { i , j , 0 } + z _ k ^ 2 B _ { i , k , 0 } + s _ l ^ 2 B _ { i , l , 0 } = q A _ { i , 2 , 0 } , \\end{align*}"}
+{"id": "5912.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } ( \\sigma _ { n } ^ M < T ) = 0 . \\end{align*}"}
+{"id": "5821.png", "formula": "\\begin{align*} { \\bf { \\hat \\Lambda } } = d i a g \\left [ { 1 , 1 , 1 , 1 , { { \\hat s } _ e } , { { \\hat s } _ \\upsilon } , { { \\hat s } _ \\upsilon } , { { \\hat s } _ \\upsilon } , { { \\hat s } _ \\upsilon } , { { \\hat s } _ \\upsilon } , { { \\hat s } _ q } , { { \\hat s } _ q } , { { \\hat s } _ q } , { { \\hat s } _ q } , { { \\hat s } _ q } , { { \\hat s } _ q } , { { \\hat s } _ \\pi } , { { \\hat s } _ \\pi } , { { \\hat s } _ \\pi } } \\right ] , \\end{align*}"}
+{"id": "6373.png", "formula": "\\begin{align*} J ( z ) = \\int _ 0 ^ z x f ( x ) \\mathrm { d } x = \\frac { \\sqrt \\theta } { B ( a , b ) } \\int _ { \\theta / z ^ 2 } ^ \\infty y ^ { - 1 / 2 } \\exp ( - a y ) \\left [ 1 - \\exp ( - y ) \\right ] ^ { b - 1 } \\mathrm { d } y . \\end{align*}"}
+{"id": "9167.png", "formula": "\\begin{align*} \\begin{aligned} \\dfrac { b _ { 1 , \\delta } ( x ) } { 2 } & \\geq - A + \\underline { \\delta } ^ \\star ( \\delta ) & \\forall \\ , x \\geq x ^ \\star + \\delta \\\\ \\dfrac { b _ { 1 , \\delta } ( x ) } { 2 } & \\leq A - \\underline { \\delta } ^ \\star ( \\delta ) & \\forall \\ , x \\leq x ^ \\star - \\delta \\ , , \\end{aligned} \\end{align*}"}
+{"id": "7272.png", "formula": "\\begin{align*} \\gamma ^ { q ^ n } \\beta ^ { q ^ n } + \\gamma ^ { q ^ { n - k } } a _ { n - k } \\beta ^ { q ^ { n - k } } + \\cdots + \\gamma a _ 0 \\beta = 0 . \\end{align*}"}
+{"id": "9114.png", "formula": "\\begin{align*} ( \\sum \\limits _ { i = 1 } ^ { j + 1 } w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j } \\chi _ i + r j < \\chi _ { j + 1 } < ( \\sum \\limits _ { i = 1 } ^ { j + 1 } w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j } \\chi _ i + r ( j + 1 ) . \\end{align*}"}
+{"id": "4375.png", "formula": "\\begin{align*} v _ \\epsilon { '' } ( t ) = \\frac { 1 } { B - 4 \\epsilon } \\mathbb { I } _ { ( - t _ 0 - B + 2 \\epsilon , - t _ 0 - 2 \\epsilon ) } * \\rho _ { \\frac { 1 } { 4 } \\epsilon } ( t ) , \\end{align*}"}
+{"id": "7535.png", "formula": "\\begin{align*} \\alpha ( H ) \\leq \\alpha \\left ( \\bigcup _ { i = 1 } ^ k H _ i \\right ) \\leq ( k - 1 ) \\frac { n } { p } \\end{align*}"}
+{"id": "6464.png", "formula": "\\begin{align*} \\Big ( \\frac { 1 } { r _ { j _ { p _ i } } } + \\frac { 1 } { \\mathcal { H } _ { j _ { p _ i } } ( \\vec { r } , h _ 1 ) } \\Big ) & \\leq \\frac { 2 } { r _ { j _ { p _ i } } } \\\\ & = \\frac { 2 } { \\rho _ { p _ i } } < 2 \\epsilon _ { p _ i } . \\end{align*}"}
+{"id": "8931.png", "formula": "\\begin{align*} E ( X - Y ) ^ { n } = \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ \\frac { n ! } { ( n / 2 ) ! } ( 1 - \\rho ) ^ { n / 2 } & & n \\end{array} \\right . . \\end{align*}"}
+{"id": "6180.png", "formula": "\\begin{align*} B _ 2 = \\frac { ( L + 1 ) B _ 1 } { Q } \\left ( \\frac { ( L + 1 ) B _ 1 } { Q } - 1 \\right ) , \\end{align*}"}
+{"id": "3358.png", "formula": "\\begin{align*} S u p p ( F ) = \\bigcup \\limits _ { i = 0 } ^ l S u p p ( s _ { \\lambda ^ i } ) . \\end{align*}"}
+{"id": "5549.png", "formula": "\\begin{align*} \\langle x \\rangle \\vee \\langle y \\rangle = \\langle x \\wedge y \\rangle . \\end{align*}"}
+{"id": "1792.png", "formula": "\\begin{align*} c ( q ^ { 3 } ) = \\frac { a ( q ) - b ( q ) } { 3 } , \\end{align*}"}
+{"id": "9156.png", "formula": "\\begin{align*} \\dot { x } = - \\gamma h ( x + \\delta u ) u = - \\gamma ( h ( x ) + R ( x , \\delta u ) ) u \\end{align*}"}
+{"id": "3348.png", "formula": "\\begin{align*} K _ i = \\langle \\{ a _ i , b _ i \\} , \\{ b _ i , c _ i \\} , \\{ d _ i \\} \\rangle , \\ L _ i = \\langle \\{ c _ i \\} , \\{ d _ i \\} \\rangle . \\end{align*}"}
+{"id": "3012.png", "formula": "\\begin{align*} \\tilde { \\varphi } ( t , x ) = \\varphi ( t , x , \\kappa _ t ( \\cdot ) ) , \\kappa ( \\cdot ) \\in \\Lambda _ 0 ( \\tau , z , w ( \\cdot ) ) , \\end{align*}"}
+{"id": "7703.png", "formula": "\\begin{align*} \\sum _ { q = r + 1 - \\ell _ { i + 1 } } ^ { r - \\ell _ { i + 1 } ' } a _ q ^ 2 \\leq \\frac { \\beta ^ 2 } { \\tau ^ 2 } \\leq \\frac { h ^ 2 } { 1 6 ^ 2 } , \\end{align*}"}
+{"id": "1216.png", "formula": "\\begin{align*} \\gamma _ { \\Lambda } ( \\gamma _ { \\Delta } ( \\eta _ { \\Delta } | \\cdot ) | \\eta _ { \\Lambda ^ c } ) = \\gamma _ \\Lambda ( \\eta _ \\Delta | \\eta _ { \\Lambda ^ c } ) . \\end{align*}"}
+{"id": "5021.png", "formula": "\\begin{align*} P _ i ^ j = P _ 0 ^ 0 + j ( q ^ 2 + 1 ) + i . \\end{align*}"}
+{"id": "8599.png", "formula": "\\begin{align*} P ( 0 ) = | Z | , P ' ( 0 ) = | P _ { u ^ \\perp } Z | + | P _ { v ^ \\perp } Z | \\hbox { a n d } P '' ( 0 ) = 2 | P _ { [ u , v ] ^ \\bot } Z | . \\end{align*}"}
+{"id": "25.png", "formula": "\\begin{align*} \\pi _ A : \\mathbb { R } ^ { | R ^ + | } \\longrightarrow \\mathbb { R } ^ { | A | } , \\ ; e _ \\alpha \\mapsto \\begin{cases} e _ \\alpha & - \\alpha \\in A , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "1955.png", "formula": "\\begin{align*} ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ r ' ( X ~ \\cup ~ \\gamma ) = \\begin{cases} r ( X ) , ~ ~ ~ a , e \\notin X , ~ e \\in c l ( X ) ; ~ \\\\ r ( X ) + 2 , ~ e \\notin c l ( X ) ~ X \\\\ r ( X ) + 1 , \\end{cases} \\end{align*}"}
+{"id": "1635.png", "formula": "\\begin{align*} \\P & = \\left \\{ x \\in \\mathcal { B } : x \\mbox { i s n o n n e g a t i v e , n o n d e c r e a s i n g , a n d } \\right . \\\\ & \\phantom { = = } \\left . ( y - T _ 1 ) ^ k x ( w ) \\le ( w - T _ 1 ) ^ k x ( y ) \\mbox { f o r a l l } y , w \\in [ T _ 1 , T _ 2 ] \\mbox { w i t h } y \\le w \\right \\} . \\end{align*}"}
+{"id": "4922.png", "formula": "\\begin{align*} J ( t , j ) = \\int _ { - \\infty } ^ { \\infty } \\ , x ^ { j } \\ , \\exp \\left ( - t x - \\frac { x ^ 2 } { 2 } \\right ) \\mathrm { d } x = ( - 1 ) ^ j \\ , \\sqrt { 2 \\pi } \\ , \\ , \\ , \\frac { \\partial ^ j } { \\partial t ^ j } \\left \\{ \\exp \\left ( \\frac { t ^ 2 } { 2 } \\right ) \\right \\} \\end{align*}"}
+{"id": "130.png", "formula": "\\begin{align*} & ~ \\sum _ { n \\in A } \\Big | \\int _ { \\R \\times \\R ^ 2 } ( f _ 1 \\ast \\tilde { f } _ 2 ) \\cdot f _ 3 ~ d \\xi d \\tau d \\eta \\Big | \\\\ & \\lesssim N N _ 1 ^ { 5 - 2 \\alpha + \\varepsilon } \\sup _ { n \\in A } \\sum _ { N _ 1 ^ { 5 - 2 \\alpha + \\varepsilon } \\leqslant L _ i \\leqslant N _ 1 ^ \\alpha N } \\Big | \\int _ { \\R \\times \\R ^ 2 } ( f _ { 1 , L _ 1 } \\ast f _ { 2 , L _ 2 } ) \\cdot f _ { 3 , L _ 3 } d \\xi d \\eta d \\tau \\Big | , \\end{align*}"}
+{"id": "7738.png", "formula": "\\begin{align*} \\frac { d } { d t } F ( \\gamma ( t ) ) = \\left \\langle v , \\dot { \\gamma } ( t ) \\right \\rangle \\forall v \\in D _ F ( \\gamma ( t ) ) , t \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "1168.png", "formula": "\\begin{align*} \\left ( \\int \\varphi d | \\mu - \\nu | \\right ) ^ 2 \\leq 4 C H \\left ( \\mu \\mid \\nu \\right ) , C = \\frac { 1 } { 6 } \\int \\varphi ^ 2 d \\mu + \\frac { 1 } { 3 } \\int \\varphi ^ 2 d \\nu , \\end{align*}"}
+{"id": "7097.png", "formula": "\\begin{align*} \\mbox { $ N _ { P _ 2 ^ * } ( c _ 6 ) = \\emptyset $ , a n d $ T $ i s n o t a j u m p a c r o s s $ c _ 1 c _ 7 $ . } \\end{align*}"}
+{"id": "7556.png", "formula": "\\begin{align*} \\prescript { } { l } { b } ( \\lambda ) = \\prescript { } { l } { \\beta } ( \\lambda ) \\prescript { } { l } { \\tilde { p } } ( \\lambda ) \\end{align*}"}
+{"id": "2849.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n \\sum _ x \\varphi \\left ( \\frac x n \\right ) \\int _ 0 ^ \\theta \\left ( \\bar { p ^ 2 _ x } ( t ) - T \\left ( \\frac x n \\right ) \\right ) ^ 2 d t = 0 . \\end{align*}"}
+{"id": "7945.png", "formula": "\\begin{align*} | \\{ p \\leq X : B ( p ) = B \\} | = c ( B ) \\int _ 2 ^ X \\frac { d t } { \\log t } + O _ K \\left ( X e ^ { - c _ K \\sqrt { \\log X } } \\right ) , \\end{align*}"}
+{"id": "5010.png", "formula": "\\begin{align*} S t _ { D N P } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( z , z ) - z | | _ { \\infty } = 0 , \\end{align*}"}
+{"id": "5943.png", "formula": "\\begin{align*} \\lim _ { | z | \\rightarrow \\infty } \\dfrac { \\sup _ { | u | \\leq M } \\sum _ { i = 1 } ^ d a _ i ( t , x , u , z ) z _ i } { | z | + | z | ^ { \\alpha - 1 } } = \\infty . \\end{align*}"}
+{"id": "8433.png", "formula": "\\begin{align*} 2 i k b ( k ) = \\Psi ^ + _ { 1 1 } ( 0 ; z ) \\Psi ^ - _ { 2 1 } ( 0 ; z ) - \\Psi ^ + _ { 2 1 } ( 0 ; z ) \\Psi ^ - _ { 1 1 } ( 0 ; z ) . \\end{align*}"}
+{"id": "1590.png", "formula": "\\begin{align*} \\mathbf { H } _ { \\mathrm { a l l } , k } = \\mathbf { H } _ { \\mathrm { d } , k } + \\overline { \\mathbf { H } } _ { k } \\mathbf { \\Phi } \\overline { \\mathbf { G } } , \\forall k \\in \\mathcal { K } , \\end{align*}"}
+{"id": "158.png", "formula": "\\begin{align*} { \\rm e x } ( n , H _ k ^ r ) \\ : \\geq \\ : \\frac { 1 } { n } \\sum _ { d = 0 } ^ { \\left \\lfloor \\ ! \\frac { ( k - 1 ) n } { r } \\ ! \\right \\rfloor } N ( n , r , k - 1 , d ) \\ , , \\end{align*}"}
+{"id": "5078.png", "formula": "\\begin{align*} \\frac { \\partial j } { \\partial s } ( 0 , 0 ) = 2 \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } U \\cdot U _ 0 \\dd x \\neq 0 . \\end{align*}"}
+{"id": "2514.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\| \\mathbf { R } _ { x s } - \\mathbf { W _ { x s } } \\right \\| \\le 2 \\gamma \\dot { \\nu } , \\\\ & \\left \\| \\mathbf { W } _ { x s } - \\dot { \\nu } \\mathbf { I } \\right \\| \\le \\gamma \\dot { \\nu } , \\\\ & \\left \\| \\mathbf { R } _ { x s } - \\dot { \\nu } \\mathbf { I } \\right \\| \\le 3 \\gamma \\dot { \\nu } , \\end{aligned} \\end{align*}"}
+{"id": "2723.png", "formula": "\\begin{align*} P f ( x ) = \\sum \\limits _ { j = 1 } ^ { n + 1 } f \\left ( x ^ { ( j ) } \\right ) \\lambda _ j ( x ) . \\end{align*}"}
+{"id": "6406.png", "formula": "\\begin{align*} | \\sigma | ^ 2 _ g = \\frac { \\i ^ { n ^ 2 } \\sigma \\wedge \\bar { \\sigma } } { d V _ g } = c _ n \\frac { ( d y _ 1 \\wedge \\dots \\wedge d y _ n ) \\wedge ( d \\theta _ 1 \\wedge \\dots \\wedge d \\theta _ n ) } { ( d x ^ 1 \\wedge \\dots \\wedge d x ^ n ) \\wedge ( d \\theta _ 1 \\wedge \\dots \\wedge d \\theta _ n ) } = c _ n \\frac { d y _ 1 \\wedge \\dots \\wedge d y _ n } { d x ^ 1 \\wedge \\dots \\wedge d x ^ n } , \\end{align*}"}
+{"id": "3355.png", "formula": "\\begin{align*} S u p p ( F ) = \\bigcup \\limits _ { \\mu } S u p p ( s _ \\mu ) . \\end{align*}"}
+{"id": "3000.png", "formula": "\\begin{align*} \\partial ^ { c i } _ { \\tau , w } g ( \\tau , w ( \\cdot ) ) = \\partial ^ { c i } _ { \\tau , w } g _ * ( \\tau , z , w ( \\cdot ) ) . \\end{align*}"}
+{"id": "7663.png", "formula": "\\begin{align*} \\bar \\mu _ t ^ { * , t _ 0 , \\xi } : = \\rho \\big ( - R ^ { - 1 } B \\mathbb E \\big [ \\bar U ( t , \\bar x _ t ^ { * , t _ 0 , \\xi } , \\nu _ t ^ { * , t _ 0 , \\xi } ) | \\mathcal { F } _ t ^ { W ^ 0 } \\big ] \\big ) \\quad \\bar \\nu _ t ^ { * , t _ 0 , \\xi } : = \\mathbb E \\big [ \\bar x _ t ^ { * , t _ 0 , \\xi } | \\mathcal { F } _ t ^ { W ^ 0 } \\big ] . \\end{align*}"}
+{"id": "2439.png", "formula": "\\begin{align*} F _ { q } ( s - s _ { 0 } ) = \\int _ { 0 } ^ { \\infty } \\exp _ { q } ( - s t ) \\exp _ { q } \\bigg ( \\frac { s _ { 0 } t } { 1 - ( 1 - q ) s t } \\bigg ) f ( t ) d t . \\end{align*}"}
+{"id": "7156.png", "formula": "\\begin{align*} \\operatorname { d i v } \\textbf { \\textit { u } } = \\nabla _ i u ^ i = \\frac { \\partial u ^ i } { \\partial x _ i } + \\Gamma ^ j _ { i j } u ^ i , \\textbf { \\textit { u } } = u ^ i \\frac { \\partial } { \\partial x _ i } \\in \\mathfrak { X } ( \\Omega ) , \\end{align*}"}
+{"id": "7825.png", "formula": "\\begin{align*} \\Phi ( a u _ \\ell + b w _ \\ell , p ) = c u _ \\ell + d w _ \\ell \\end{align*}"}
+{"id": "1603.png", "formula": "\\begin{align*} \\min _ { \\mathbf { \\Phi } _ { g } } ~ & \\tilde { f } _ g ( \\mathbf { \\Phi } _ { g } ) = \\mathsf { T r } ( \\mathbf { \\Phi } _ { g } \\mathbf { Y } _ { g , g } \\mathbf { \\Phi } _ { g } ^ H \\mathbf { Z } _ { g } - 2 \\Re \\{ \\mathbf { \\Phi } _ { g } \\tilde { \\mathbf { X } } _ { g } \\} ) \\\\ \\mathrm { s . t . } ~ & \\mathbf { \\Phi } _ { g } ^ H \\mathbf { \\Phi } _ { g } = \\mathbf { I } _ { \\bar { M } } , \\forall g \\in \\mathcal { G } . \\end{align*}"}
+{"id": "8031.png", "formula": "\\begin{align*} C _ 3 ( t ) = \\sum _ { u : \\ , i _ u \\geq 3 \\atop { 2 \\leq d _ u \\leq i _ u } } d _ u = \\sum _ { i \\geq 3 } \\sum _ { d = 2 } ^ { i } d x _ { i , d } , \\\\ \\end{align*}"}
+{"id": "124.png", "formula": "\\begin{align*} N \\Big ( \\frac { N _ 1 } { N _ + } \\Big ) ^ { ( 5 - 2 \\alpha ) + \\varepsilon } \\Big | \\int ( f _ { N _ 1 , L _ 1 } \\ast g _ { N _ 2 , L _ 2 } ) \\cdot h _ { N , L } \\Big | & \\lesssim N _ 1 ^ { - c _ 1 ( \\varepsilon ) } [ \\min ( N , 1 ) ] ^ { c _ 2 ( \\varepsilon ) } ~ ( L _ 1 L _ 2 L ^ { 1 - } ) ^ { \\frac { 1 } { 2 } } \\\\ & \\| f _ { N _ 1 , L _ 1 } \\| _ { L ^ 2 } \\| g _ { N _ 2 , L _ 2 } \\| _ { L ^ 2 } \\| h _ { N , L } \\| _ { L ^ 2 } , \\end{align*}"}
+{"id": "5555.png", "formula": "\\begin{align*} \\int e ^ { W ( y | x ) } d \\nu _ { A } ( y ) = \\varphi _ A ( x ) . \\end{align*}"}
+{"id": "5156.png", "formula": "\\begin{align*} I _ { h / \\tau } \\leq E [ \\tilde { b } ] = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } | \\nabla \\phi | ^ { 2 } \\frac { 1 } { r ^ { 2 } } \\dd x + \\frac { 1 } { 2 \\tau ^ { 2 } } \\int _ { \\mathbb { R } ^ { 3 } } | G | ^ { 2 } \\frac { 1 } { r ^ { 2 } } \\dd x = E [ b ] - \\frac { 1 } { 2 } \\left ( 1 - \\frac { 1 } { \\tau ^ { 2 } } \\right ) \\int _ { \\mathbb { R } ^ { 3 } } | G | ^ { 2 } \\frac { 1 } { r ^ { 2 } } \\dd x < I _ h . \\end{align*}"}
+{"id": "3525.png", "formula": "\\begin{align*} - \\sum _ { K = 0 } ^ { p } \\binom { p } { K } \\ , B _ K \\ , n ^ { p - K } \\end{align*}"}
+{"id": "2883.png", "formula": "\\begin{align*} \\tilde V _ x ( m ) = \\sum _ { x ' = 1 } ^ n M _ { x , x ' } ( m ) \\tilde V _ { x ' } ( m ) + \\tilde v _ x ( m ) . \\end{align*}"}
+{"id": "2827.png", "formula": "\\begin{align*} \\mathcal { L } _ { } ( \\Theta ) = \\mathcal { L } _ { } + \\gamma \\cdot \\mathcal { L } _ { } , \\end{align*}"}
+{"id": "7084.png", "formula": "\\begin{align*} \\varphi _ 1 ( x ) = u ( x ) + t v _ 1 ( x ) , \\end{align*}"}
+{"id": "7265.png", "formula": "\\begin{align*} P ( x _ 1 , \\ldots , x _ n ) = \\lambda _ 0 x _ 1 ^ s + \\lambda _ 1 x _ 1 ^ { s - 1 } x _ 2 + \\cdots + \\lambda _ s x _ 2 ^ s , \\end{align*}"}
+{"id": "4267.png", "formula": "\\begin{align*} 9 1 & = 3 ^ { 3 } + 4 ^ { 3 } = 6 ^ { 3 } + \\left ( - 5 \\right ) ^ { 3 } \\\\ 1 5 2 & = 3 ^ { 3 } + 5 ^ { 3 } = 6 ^ { 3 } + \\left ( - 4 \\right ) ^ { 3 } \\\\ 1 8 9 & = 4 ^ { 3 } + 5 ^ { 3 } = 6 ^ { 3 } + \\left ( - 3 \\right ) ^ { 3 } \\end{align*}"}
+{"id": "2766.png", "formula": "\\begin{align*} | F ( p ( s , k ) ) - a | = | ( 2 s _ 1 , 2 s _ 2 , 0 ) | \\leq \\frac { q } { 1 0 } \\end{align*}"}
+{"id": "8568.png", "formula": "\\begin{align*} k ( t ) = h _ { 1 - \\alpha , \\rho } ( t ) \\ , + \\ , \\rho \\ , \\int _ 0 ^ t h _ { 1 - \\alpha , \\rho } ( \\tau ) \\ , d \\tau , \\end{align*}"}
+{"id": "2641.png", "formula": "\\begin{align*} x \\diamond y = x \\cdot D ( y ) . \\end{align*}"}
+{"id": "148.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb Z _ 0 } z _ j = \\frac { \\sum _ { j \\in \\mathbb Z _ 0 } \\lambda _ j } { ( 1 + A ) ^ k } , \\end{align*}"}
+{"id": "4465.png", "formula": "\\begin{align*} Q ^ { * } = a _ { 1 } ^ { * } x _ { 1 } ^ { 2 } + a _ { 2 } ^ { * } ( x _ { 2 } + n _ { 2 1 } x _ { 1 } ) ^ { 2 } + a _ { 3 } ^ { * } ( x _ { 3 } + n _ { 3 1 } x _ { 1 } + n _ { 3 2 } x _ { 2 } ) ^ { 2 } + a _ { 4 } ^ { * } ( x _ { 4 } + n _ { 4 1 } x _ { 1 } + n _ { 4 2 } x _ { 2 } + n _ { 4 3 } x _ { 3 } ) ^ { 2 } , \\end{align*}"}
+{"id": "5104.png", "formula": "\\begin{align*} \\begin{aligned} { \\mathcal { G } } ( z , r , z ' , r ' ) & = \\frac { \\sqrt { r r ' } } { 2 \\pi } F ( s ) , s = \\frac { | z - z ' | ^ { 2 } + | r - r ' | ^ { 2 } } { r r ' } , \\\\ F ( s ) & = \\int _ { 0 } ^ { \\pi } \\frac { \\cos \\theta } { \\sqrt { 2 ( 1 - \\cos \\theta ) + s } } \\dd \\theta . \\end{aligned} \\end{align*}"}
+{"id": "8626.png", "formula": "\\begin{align*} \\{ w ' \\in \\Omega : O ( \\chi _ { n , m } ' ( w ' , M , \\cdot ) ) = O ( \\chi _ { n , m } ' ( w , M , \\cdot ) ) \\} \\end{align*}"}
+{"id": "2369.png", "formula": "\\begin{align*} C _ 0 ( C _ { \\ast } ) = \\ell _ 0 ( H _ 0 ( C _ { \\ast } ) ) . \\end{align*}"}
+{"id": "3660.png", "formula": "\\begin{align*} N _ { c o m p } ^ i & = N ( r _ { i , f } + r _ { i , g } ) \\\\ & = 2 N + \\mathcal { O } \\left ( \\frac { \\max \\{ \\omega , \\omega ^ { - 1 } \\} } { n L _ { x y } ( 1 - \\theta ) ^ 2 \\epsilon } \\left ( \\left ( \\frac { 1 } { L _ { x } } + \\frac { \\omega } { L _ { x y } } \\right ) \\sigma _ { f , i } ^ 2 + \\left ( \\frac { 1 } { L _ { y } } + \\frac { 1 } { L _ { x y } \\omega } \\right ) \\sigma _ { g , i } ^ 2 \\right ) \\log \\left ( \\frac { \\Psi ^ 0 \\nu } { \\epsilon } \\right ) \\right ) . \\end{align*}"}
+{"id": "16.png", "formula": "\\begin{align*} \\alpha _ { i _ k , j _ k } \\oplus \\alpha _ { i _ { k + 1 } , j _ { k + 1 } } = \\{ \\alpha _ { i _ k , j _ { k + 1 } } , \\alpha _ { j _ k , i _ { k + 1 } } \\} . \\end{align*}"}
+{"id": "8878.png", "formula": "\\begin{align*} h _ { k , S , ( n , g ) } ( \\tau , z ; \\tau _ 0 , z _ 0 ) : = \\det ( \\tau - \\bar { \\tau } _ 0 ) ^ { - k } e \\left ( - ( \\tau - \\bar { \\tau } _ 0 ) ^ { - 1 } S [ z - \\overline { z } _ 0 ] \\right ) . \\end{align*}"}
+{"id": "3894.png", "formula": "\\begin{align*} Q _ { \\delta _ N } L _ { \\delta _ N } u _ N = L _ { \\delta _ N } u _ N - \\sum _ { j = 1 } ^ m \\sum _ { h = 1 } ^ 2 C _ { j , h , N } \\frac { \\partial } { \\partial z _ { j , h } } ( - \\delta _ N ^ 2 ( K ( z _ { N , j } ) \\nabla V _ { \\delta _ N , Z _ N , j } ) ) . \\end{align*}"}
+{"id": "8092.png", "formula": "\\begin{align*} | \\nu _ { p ^ { k + 1 } } | & = 1 + \\sum \\limits _ { j = 1 } ^ { p ^ { k + 1 } - 1 } \\phi ( j ) + | \\nu _ { p ^ k } | \\\\ & = 1 + \\sum \\limits _ { j = 1 } ^ { p ^ { k + 1 } - 1 } \\phi ( j ) + k + \\sum \\limits _ { i = 1 } ^ k \\sum \\limits _ { j = 1 } ^ { p ^ i - 1 } \\phi ( j ) \\\\ & = ( k + 1 ) + \\sum \\limits _ { i = 1 } ^ { k + 1 } \\sum \\limits _ { j = 1 } ^ { p ^ i - 1 } \\phi ( j ) . \\end{align*}"}
+{"id": "5735.png", "formula": "\\begin{align*} \\frac { ( x y ) ^ { q + 1 } + x y } { y ^ { q + 1 } + y } = \\frac { x ^ { q + 1 } y ^ q + x } { y ^ { q } + 1 } = d . \\end{align*}"}
+{"id": "2617.png", "formula": "\\begin{align*} \\begin{array} { l } ( x + a ) \\cdot ( y + b ) = \\big ( x \\cdot _ A y + l _ { B } ( a ) y + r _ { B } ( b ) x \\big ) + \\big ( a \\cdot _ B b + l _ { A } ( x ) b + r _ { A } ( y ) a \\big ) , \\\\ ( \\alpha \\oplus \\beta ) ( x + a ) = \\alpha ( x ) + \\beta ( a ) . \\end{array} \\end{align*}"}
+{"id": "7333.png", "formula": "\\begin{align*} | A | \\ , | A + B + C | = & \\sum _ { k + j + m = n } { n \\choose k , j , m } | A | V ( A [ k ] , B [ j ] , C [ m ] ) \\\\ = & \\sum _ { 0 \\le j + m \\le n } { n \\choose j , m , n - j - m } | A | V ( A [ n - j - m ] , B [ j ] , C [ m ] ) , \\end{align*}"}
+{"id": "5425.png", "formula": "\\begin{align*} l ( y ' ) : = u ( x _ 0 ) + \\sum _ { \\beta = 1 } ^ { n - 1 } \\nabla _ \\beta u ( x _ 0 ) y _ \\beta . \\end{align*}"}
+{"id": "6517.png", "formula": "\\begin{align*} x _ { \\lambda } x _ { \\mu } x _ { \\nu } = x _ { \\lambda } ( x _ { \\mu } x _ { \\nu } ) \\lambda ( \\mu \\nu ) \\end{align*}"}
+{"id": "8206.png", "formula": "\\begin{align*} x _ { m , j } + x _ { m , j + 1 } + x _ { 1 , n + 1 - j } + x _ { 1 , n - j } = S . \\end{align*}"}
+{"id": "5185.png", "formula": "\\begin{align*} \\begin{aligned} A _ { t } + B \\times u + \\nabla Q & = - \\mu \\nabla \\times B . \\end{aligned} \\end{align*}"}
+{"id": "4732.png", "formula": "\\begin{align*} & ( i ) \\ \\ \\ \\nabla ( \\gamma _ 0 J + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k f _ k ) ( x ^ * ) = 0 \\\\ & ( i i ) \\ \\ \\gamma _ k f _ k ( x ^ * ) = 0 \\ \\ k \\in \\{ 1 , . . . , m \\} \\\\ & ( i i i ) \\ \\gamma _ { 0 } A _ J + \\sum \\limits _ { k = 1 } ^ m \\gamma _ k A _ k \\succeq 0 \\end{align*}"}
+{"id": "6441.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { ( - 1 ) ^ r } { r ! } \\frac { \\mathrm { \\partial } ^ r } { \\mathrm { \\partial } \\alpha ^ r } Z ^ { \\star } _ { I } ( \\{ k _ i \\} _ { i = 1 } ^ { n } ; ( \\alpha , \\beta ) ) = \\sum _ { \\begin{subarray} { c } r _ 1 + \\cdots + r _ { n - 1 } = r \\\\ r _ i \\in \\mathbb { Z } _ { \\ge 0 } \\end{subarray} } Z ^ { \\star } _ { I } ( \\{ k _ i , \\{ 1 \\} ^ { r _ i } \\} _ { i = 1 } ^ { n - 1 } , k _ n ; ( \\alpha , \\beta ) ) \\end{aligned} \\end{align*}"}
+{"id": "9052.png", "formula": "\\begin{align*} \\Lambda = \\Lambda ( i _ 1 , \\cdots , i _ m ) = \\prod _ { a = 1 } ^ { m } \\Lambda _ a ^ { i _ a } , \\medspace R = R ( j _ 1 , \\cdots , j _ n ) = \\prod _ { b = 1 } ^ { n } R _ b ( j _ b ) \\cdot C . \\end{align*}"}
+{"id": "8538.png", "formula": "\\begin{align*} & \\psi _ 1 ( x , t ; k ) = \\psi _ 1 ( x ; k ) e ^ { - i \\eta ^ 2 t } , \\\\ & \\psi _ 2 ( x , t ; k ) = \\psi _ 2 ( x ; k ) e ^ { i \\eta ^ 2 t } , \\end{align*}"}
+{"id": "4259.png", "formula": "\\begin{align*} g ( \\Gamma ) = h ^ 1 ( \\Gamma ) + \\sum _ { v \\in V } g ( v ) , h ^ 1 ( \\Gamma ) = | E | - | V | + 1 . \\end{align*}"}
+{"id": "4055.png", "formula": "\\begin{align*} e _ 2 ( C ) = e ( C ) , \\end{align*}"}
+{"id": "7143.png", "formula": "\\begin{align*} \\mathcal { L } _ g : = \\begin{pmatrix} \\mu \\Delta _ { B } + ( \\lambda + \\mu ) \\operatorname { g r a d } \\operatorname { d i v } + \\mu \\operatorname { R i c } + \\rho \\omega ^ 2 & - \\beta \\operatorname { g r a d } \\\\ i \\omega \\theta _ { 0 } \\beta \\operatorname { d i v } & \\alpha \\Delta _ { g } + i \\omega \\gamma \\end{pmatrix} , \\end{align*}"}
+{"id": "3773.png", "formula": "\\begin{align*} f ^ { - 1 } ( \\overline { V } _ \\bullet ) = \\{ V _ \\bullet ; V _ j = \\overline { V } _ j j \\in [ n ] \\setminus \\{ l \\} , V _ { l - 1 } + X V _ i \\subset V _ l \\subset V _ { l + 1 } \\} \\end{align*}"}
+{"id": "6002.png", "formula": "\\begin{align*} & \\frac { R } { 2 } \\int _ { \\partial B _ { R } } | \\nabla v | ^ { 2 } d S _ { x } + \\int _ { B _ { R } } \\omega v ^ { 2 } d x - \\pi g ^ { 2 } ( R ) v ^ { 2 } ( R ) - \\pi \\bigg ( \\int _ { R } ^ { + \\infty } \\frac { g ( s ) } { s } v ^ { 2 } ( s ) d s \\bigg ) v ^ { 2 } ( R ) R ^ { 2 } \\\\ & \\ \\ \\ \\ \\ + 2 \\int _ { \\R ^ { 2 } } \\frac { g ^ { 2 } ( | x | ) } { | x | ^ { 2 } } v ^ { 2 } d x = \\frac { 1 } { p } \\int _ { B _ { R } } v ^ { 2 p } d x - b \\int _ { B _ { R } } | u | ^ { p } | v | ^ { p - 2 } v x \\cdot \\nabla v d x , \\end{align*}"}
+{"id": "8290.png", "formula": "\\begin{align*} \\bar \\pi \\bar p _ t = \\bar \\pi , ~ ~ ~ ~ \\forall t > 0 . \\end{align*}"}
+{"id": "7483.png", "formula": "\\begin{align*} x _ { k } u _ { k } \\ , a _ { i j } ( x ) u _ { i j } = \\partial _ i \\big ( x _ { k } u _ { k } \\ , a _ { i j } ( x ) u _ { j } \\big ) - x _ { k } u _ { k } \\partial _ i a _ { i j } ( x ) u _ { j } - a _ { i j } ( x ) u _ { i } u _ { j } - x _ { k } a _ { i j } ( x ) u _ { i k } u _ { j } . \\end{align*}"}
+{"id": "1290.png", "formula": "\\begin{align*} \\frac { 1 } { \\nu ( \\xi _ { \\Delta } \\eta _ { \\Lambda _ n \\setminus \\Delta } ) } \\int \\mathbf { 1 } _ { \\eta _ { \\Lambda _ n } } ( \\omega ) \\frac { \\gamma _ { \\Delta } ( \\xi _ { \\Delta } | \\omega _ { \\Delta ^ c } ) } { \\gamma _ { \\Delta } ( \\eta _ { \\Delta } | \\omega _ { \\Delta ^ c } ) } \\nu ( d \\omega ) = 1 , \\end{align*}"}
+{"id": "8474.png", "formula": "\\begin{align*} & Q _ { 1 , \\pm } ( x ; k ) = M _ { \\pm } ( x ; z ) V _ 1 ( k ) - V _ 1 ( k ) = \\left ( \\left ( M _ { \\pm , 1 } ( x ; z ) - e _ 1 \\right ) , ( 2 i k ) M _ { \\pm , 2 } ( x ; z ) - e _ 2 \\right ) , \\\\ & Q _ { 2 , \\pm } ( x ; k ) = M _ { \\pm } ( x ; z ) V _ 2 ( k ) - V _ 2 ( k ) = \\left ( ( 2 i k ) ^ { - 1 } \\left ( M _ { \\pm , 1 } ( x ; z ) - e _ 1 \\right ) , M _ { \\pm , 2 } ( x ; z ) - e _ 2 \\right ) , \\end{align*}"}
+{"id": "6792.png", "formula": "\\begin{align*} F ^ { + } ( x + y ) + A ( x , y ) + \\int _ { x } ^ { \\infty } A ( x , t ) F ^ { + } ( t + y ) d t = 0 , y > x \\end{align*}"}
+{"id": "5703.png", "formula": "\\begin{align*} f ( x , y ) = a _ 0 x ^ { q + 1 } + b _ 0 x ^ q y + c _ 0 x y ^ q + d _ 0 y ^ { q + 1 } . \\end{align*}"}
+{"id": "8838.png", "formula": "\\begin{align*} \\begin{aligned} | X _ { t , n } - x _ { t , n } | & \\le 2 K \\int _ 0 ^ t | X _ { s , n - 1 } - x _ { s , n - 1 } | d s + \\int _ 0 ^ t | x _ { s , n - 1 } | | x _ { 0 , 1 } - x _ { s , 1 } | d s \\\\ & + | a _ n | \\int _ 0 ^ t | x _ { s , n } | d s + \\int _ 0 ^ t | x _ { s , n - 2 } | | x _ { s , n - 1 } | d s . \\end{aligned} \\end{align*}"}
+{"id": "9101.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ j \\chi _ i \\leq \\chi ( \\sum \\limits _ { i = 1 } ^ j w _ i ) + j r , \\end{align*}"}
+{"id": "7536.png", "formula": "\\begin{align*} \\left ( 2 \\binom { n } { ( k - 1 ) n / p } \\right ) ^ t \\leq ( c p ) ^ { ( k - 1 ) n t / p } \\end{align*}"}
+{"id": "617.png", "formula": "\\begin{align*} \\begin{aligned} & \\omega _ T = \\frac { d ( x + \\sqrt { \\epsilon } y ) } { x + \\sqrt { \\epsilon } y } \\wedge \\frac { d ( x - \\sqrt { \\epsilon } y ) } { x - \\sqrt { \\epsilon } y } \\\\ & = \\frac { ( d x + \\sqrt { \\epsilon } d y ) \\wedge ( d x - \\sqrt { \\epsilon } d y ) } { x ^ 2 - \\epsilon y ^ 2 } = \\frac { - 2 \\sqrt { \\epsilon } } { N _ { E / F } ( x + \\sqrt { \\epsilon } y ) } d x \\wedge d y , \\end{aligned} \\end{align*}"}
+{"id": "5535.png", "formula": "\\begin{align*} \\int _ { \\mathfrak X } | \\omega | = { \\mathfrak h _ 0 } _ ! \\int _ { \\mathfrak Y } | \\omega | , \\end{align*}"}
+{"id": "2985.png", "formula": "\\begin{align*} T _ { \\vec { v } } ^ { - n } \\phi _ u \\widetilde { B } = & \\phi _ u ( T ^ { ( 0 , 1 ) } \\times i d ) T _ { \\vec { v } } ^ { - n } ( \\widetilde { B } \\cap ( X \\times A _ 1 ^ n ) ) \\\\ & \\cup \\phi _ u T _ { \\vec { v } } ^ { - n } ( \\widetilde { B } \\cap ( X \\times A _ 2 ^ n ) ) \\cup \\phi _ u ( T ^ { - ( 0 , 1 ) } \\times i d ) T _ { \\vec { v } } ^ { - n } ( \\widetilde { B } \\cap ( X \\times A _ 3 ^ n ) ) . \\end{align*}"}
+{"id": "3401.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } a _ n = ( 1 - \\frac 1 n ) a _ { n - 1 } + \\frac 1 n b _ { n - 1 } \\\\ b _ n = ( 1 - \\frac 1 n ) a _ { n - 1 } + \\frac { e ^ t } { n } b _ { n - 1 } \\end{array} \\right . a _ 0 = b _ 0 = 1 . \\end{align*}"}
+{"id": "8407.png", "formula": "\\begin{align*} \\left \\| ( I - F ) ^ { - 1 } \\right \\| _ { L _ x ^ { \\infty } L _ z ^ 2 \\rightarrow L _ x ^ { \\infty } L _ z ^ 2 } \\leq \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { n ! } \\left \\| \\widetilde { Q } ( u ) \\right \\| _ { L ^ 1 } ^ n = \\mathrm { e } ^ { \\left \\| \\widetilde { Q } ( u ) \\right \\| _ { L ^ 1 } } . \\end{align*}"}
+{"id": "1988.png", "formula": "\\begin{align*} - 2 g ^ { - 1 } ( z ) & = 2 g ^ { - 1 } ( z - 2 g ^ { - 1 } ( z ) ) . \\end{align*}"}
+{"id": "6007.png", "formula": "\\begin{align*} J _ { \\omega } = \\frac { 1 } { 2 } \\int _ { \\R ^ { 2 } } ( | \\nabla u | ^ { 2 } + \\omega u ^ { 2 } ) d x + \\frac { 1 } { 2 } B ( u ) - \\frac { 1 } { 2 p } \\int _ { \\R ^ { 2 } } u ^ { 2 p } d x . \\end{align*}"}
+{"id": "2930.png", "formula": "\\begin{align*} \\nu _ { \\alpha } ^ { \\prime } \\left ( d x \\right ) = \\boldsymbol { 1 } _ { \\left \\vert x \\right \\vert > 1 } \\nu _ { \\alpha } \\left ( d x \\right ) \\end{align*}"}
+{"id": "797.png", "formula": "\\begin{align*} \\big \\langle \\mathrm { E } ( \\Sigma , c ) , ( V , w ) \\big \\rangle = \\big \\langle ( V _ g , w _ g ) , ( V , w ) \\big \\rangle = \\int _ \\Sigma \\big ( V _ g - u _ g c H \\big ) V \\ , \\mathrm { d } \\mathcal { H } ^ d + \\int _ \\Sigma u _ g w \\ , \\mathrm { d } \\mathcal { H } ^ d \\end{align*}"}
+{"id": "3753.png", "formula": "\\begin{align*} d e g ( f _ i m ' _ { i , l _ 0 } ) = d e g ( f _ { l _ 0 } m '' _ { l _ 0 , l _ 0 } ) . \\end{align*}"}
+{"id": "8147.png", "formula": "\\begin{align*} g ^ * f _ ! = f ' _ ! g '^ * \\end{align*}"}
+{"id": "1656.png", "formula": "\\begin{align*} U _ p : = \\prod _ { v \\mid p } U ^ { e _ v } _ v , \\ u _ p = \\prod _ { v \\mid p } u ^ { e _ v } _ v , \\ S _ p = \\prod S ^ { e _ v } _ v . \\end{align*}"}
+{"id": "934.png", "formula": "\\begin{align*} J u _ 2 ( t ) = U ( t - t _ 0 ) J u _ 2 ( t _ 0 ) - i \\lambda _ 6 \\int ^ t _ { t _ 0 } U ( t - \\tau ) J \\mathcal { N } _ 2 ( u _ 1 , u _ 2 ) ( \\tau ) \\ d \\tau . \\end{align*}"}
+{"id": "7980.png", "formula": "\\begin{align*} \\partial _ { t } \\frac { w ^ { 1 1 } } { h } \\leq - \\frac { ( w ^ { 1 1 } ) ^ { 2 } } { h } \\sigma _ { k } \\Theta \\left ( c _ { 0 } - c _ { 1 } w _ { 1 1 } - c _ { 2 } w ^ { 2 } _ { 1 1 } + \\frac { 2 - \\vartheta - k } { h } w _ { 1 1 } + \\frac { \\nabla G \\cdot e _ { i } } { G } w _ { 1 1 i } \\right ) + \\frac { 2 w ^ { 1 1 } } { h } . \\end{align*}"}
+{"id": "1971.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { w _ F } \\sum _ { i = 1 } ^ h \\phi ( a _ i x ) . \\end{align*}"}
+{"id": "5668.png", "formula": "\\begin{align*} { } _ { m } E ^ { 1 } _ { p , q } ( n ) = s _ { m } ^ { - 1 } E ^ { 1 } _ { p , q } ( n ) \\Rightarrow s _ { m } ^ { - 1 } H _ { p + q } ( O _ { n , n } , C _ { * } ( n ) ) \\end{align*}"}
+{"id": "7833.png", "formula": "\\begin{align*} ( I - Q ^ \\dagger ) F ^ { q , \\dagger } ( v + \\psi ^ \\dagger ( v , p ) , p ) = 0 \\end{align*}"}
+{"id": "2782.png", "formula": "\\begin{align*} h ^ { V } \\overset { \\mathrm { l a w } } { = } h ^ { U } + \\varphi ^ { V , U } , \\end{align*}"}
+{"id": "4712.png", "formula": "\\begin{align*} v _ { \\alpha } ( x ) = \\begin{cases} x _ n ^ { \\alpha } ( | x ' | ^ 2 - C _ \\alpha ) \\quad C _ { \\alpha } = \\frac { 1 + 2 D ^ 2 } { \\alpha ( 1 - \\alpha ) } & \\alpha \\in [ \\frac { 2 } { n } , 1 ) , \\\\ ( 1 + 2 D ^ 2 ) x _ n \\log ( x _ n / D ) + x _ n ( | x ' | ^ 2 - D ^ 2 ) & \\alpha = 1 . \\end{cases} \\end{align*}"}
+{"id": "2367.png", "formula": "\\begin{align*} C _ { 0 } ( K ; \\mathrm { A d } _ { \\rho } ) = C _ { 0 } ( \\widetilde { K } ; \\mathbb { Z } ) \\otimes \\mathfrak { g } / \\sim \\ ; \\cong \\mathfrak { g } . \\end{align*}"}
+{"id": "3901.png", "formula": "\\begin{align*} \\delta ^ 2 \\int _ { \\Omega } \\left ( K ( x ) \\nabla u | \\nabla \\varphi \\right ) = \\int _ { \\Omega } \\hat { h } \\varphi , \\ \\ \\ \\ \\forall \\varphi \\in H ^ 1 _ 0 ( \\Omega ) . \\end{align*}"}
+{"id": "9305.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\lambda ^ { N _ { i - 1 } } + \\sum _ { \\ell = 1 } ^ { r } c _ \\ell u _ { i - 1 } ^ { \\gamma _ \\ell } \\lambda ^ { ( i - 1 ) \\gamma _ \\ell + N _ { i - 1 } - \\ell } = 0 \\\\ u _ { i - 1 } = \\lambda u _ { i } \\end{array} \\right . \\end{align*}"}
+{"id": "2291.png", "formula": "\\begin{align*} T \\circ \\pi _ 1 ( \\exp ( X ) ) = \\pi _ 2 ( \\exp ( X ) ) \\circ T , \\end{align*}"}
+{"id": "2544.png", "formula": "\\begin{align*} \\begin{aligned} & { { \\mathbf { x } } _ { w } } = \\omega { { \\mathbf { x } } _ { o } } + \\left ( 1 - \\omega \\right ) \\mathbf { e } \\\\ & { { \\mathbf { y } } _ { w } } = \\omega { { \\mathbf { y } } _ { o } } \\\\ & { { \\mathbf { s } } _ { w } } = \\omega { { \\mathbf { s } } _ { o } } + \\left ( 1 - \\omega \\right ) \\mathbf { e } \\\\ & { { \\kappa } _ { w } } = { { \\left ( { { \\mathbf { x } } _ { w } } \\right ) } ^ { T } } { { \\mathbf { s } } _ { w } } / k \\\\ & { { \\tau } _ { w } } = 1 , \\end{aligned} \\end{align*}"}
+{"id": "811.png", "formula": "\\begin{align*} T = \\frac { 1 } { d } \\int _ 0 ^ { R _ 0 } \\frac { z } { g \\left ( \\frac { m } { \\alpha _ d } z ^ { - d } \\right ) } \\ , \\mathrm { d } z = C ( m , d , s ) \\int _ 0 ^ { R _ 0 } z ^ { s d + 1 } \\ , \\mathrm { d } z = \\frac { C ( m , d , s ) } { s d + 2 } \\ , z ^ { s d + 2 } \\Big | _ 0 ^ { R _ 0 } \\end{align*}"}
+{"id": "9214.png", "formula": "\\begin{align*} \\begin{aligned} | \\bar { y } ( t ) & - z _ 2 ( t ) | \\le | \\bar { y } ( 0 ) - z _ 2 ( 0 ) | + t \\gamma ^ 2 \\bar { k } _ 2 ( L _ r , \\delta ) \\\\ & + \\gamma \\int _ 0 ^ t L _ r | x - z _ 1 | + | \\bar { y } ( \\tau ) - z _ 2 ( \\tau ) | \\ , d \\tau \\end{aligned} \\end{align*}"}
+{"id": "6846.png", "formula": "\\begin{align*} \\eta ( x ) : = e ( \\frac { i } { 2 } , x ) \\int _ { 0 } ^ { x } \\frac { d t } { e ^ { 2 } ( \\frac { i } { 2 } , t ) } \\end{align*}"}
+{"id": "5437.png", "formula": "\\begin{align*} v = \\frac { 1 } { \\sqrt { b _ \\alpha } } \\sum _ { \\beta = \\alpha + 1 } ^ { n - 1 } b _ \\beta y _ \\beta ^ 2 + \\sqrt { b _ \\alpha } \\bar { y } _ n . \\end{align*}"}
+{"id": "470.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { f ( x ) } { g ( x ) } = \\lim _ { x \\to \\infty } \\frac { f ( x ) } { f _ 1 ( x ) } \\frac { f _ 1 ( x ) } { g _ 1 ( x ) } \\frac { g _ 1 ( x ) } { g ( x ) } = \\lambda / ( 1 - e ^ { - \\lambda } ) . \\end{align*}"}
+{"id": "4903.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { h _ { } ( x ) } { x } & = \\lim _ { x \\to \\infty } \\frac { \\phi ( \\frac { x - \\mu } { \\sigma } ) \\frac { 1 } { x } } { \\sigma [ 1 - \\Phi ( \\frac { x - \\mu } { \\sigma } ) ] } . \\end{align*}"}
+{"id": "1006.png", "formula": "\\begin{align*} \\ell ( x \\varepsilon ^ \\lambda ) = \\ell ( x ) + \\ell ( \\varepsilon ^ \\lambda ) , \\ell ( x ' \\varepsilon ^ \\lambda ) = \\ell ( x ' ) + \\ell ( \\varepsilon ^ \\lambda ) . \\end{align*}"}
+{"id": "1066.png", "formula": "\\begin{align*} I : = \\{ \\beta \\in \\Phi ^ + \\mid s _ \\alpha ( \\beta ) \\in \\Phi ^ - \\} . \\end{align*}"}
+{"id": "3961.png", "formula": "\\begin{align*} \\mathbf { c } ^ { ( 2 ) } = ( \\star , \\underbrace { 1 , 1 , \\cdots , 1 } _ { 2 k } , \\star , \\star , \\cdots , \\star ) . \\end{align*}"}
+{"id": "7598.png", "formula": "\\begin{align*} F ^ 1 x & = ( x _ { 1 0 0 } , x _ { 1 0 1 } , x _ { 1 1 0 } , x _ { 1 1 1 } ) \\\\ F ^ 2 x & = ( x _ { 0 1 0 } , x _ { 0 1 1 } , x _ { 1 1 0 } , x _ { 1 1 1 } ) \\\\ F ^ 3 x & = ( x _ { 0 0 1 } , x _ { 0 1 1 } , x _ { 1 0 1 } , x _ { 1 1 1 } ) \\end{align*}"}
+{"id": "8178.png", "formula": "\\begin{align*} \\Gamma _ { \\lambda } ( t ) : = \\int _ { 0 } ^ { \\lambda } \\mu \\xi ( \\mu , t ) \\ d \\mu \\mbox { a n d } \\Theta _ { \\lambda } ( t ) : = \\int _ { \\lambda } ^ { \\infty } \\mu \\xi ( \\mu , t ) \\ d \\mu , \\end{align*}"}
+{"id": "4481.png", "formula": "\\begin{align*} \\sum _ { r _ { Q } ( n ) = 0 } n \\ll p ^ { 1 + \\epsilon } m p ^ { - 1 / 4 + \\epsilon } \\ll p ^ { 5 / 2 + \\epsilon } . \\end{align*}"}
+{"id": "5849.png", "formula": "\\begin{align*} { f _ { 1 6 } } = { f _ { 1 5 } } - f _ { 1 5 } ^ { \\left ( { e q } \\right ) } + f _ { 1 6 } ^ { \\left ( { e q } \\right ) } - \\delta y - \\delta z , \\end{align*}"}
+{"id": "4556.png", "formula": "\\begin{align*} A _ { w , 1 } f ( v _ { \\ell , m , m } ) = & q ^ 3 f ( v _ { \\ell - 1 , m - 1 , m - 1 } ) + ( q ^ 2 + q ) f ( v _ { \\ell , m + 1 , m } ) + f ( v _ { \\ell + 1 , m , m } ) \\\\ A _ { w , 2 } f ( v _ { \\ell , m , m } ) = & ( q ^ 4 + q ^ 3 ) f ( v _ { \\ell - 1 , m , m - 1 } ) + q ^ 2 f ( v _ { \\ell , m - 1 , m - 1 } ) \\\\ & + q ^ 2 f ( v _ { \\ell , m + 1 , m + 1 } ) + ( q + 1 ) f ( v _ { \\ell + 1 , m + 1 , m } ) \\\\ A _ { w , 3 } f ( v _ { \\ell , m , m } ) = & q ^ 3 f ( v _ { \\ell - 1 , m , m } ) + ( q ^ 2 + q ) f ( v _ { \\ell , m , m - 1 } ) + f ( v _ { \\ell + 1 , m + 1 , m + 1 } ) \\end{align*}"}
+{"id": "6763.png", "formula": "\\begin{align*} q = \\frac { a _ { 0 } ^ { \\prime \\prime } - a _ { 0 } ^ { \\prime } } { a _ { 0 } + 1 } \\end{align*}"}
+{"id": "5432.png", "formula": "\\begin{align*} \\delta ^ 2 ( 1 - \\epsilon ^ 2 ) b _ { \\alpha } u _ n ( x ) \\leq u ( y ) - u ( x ) \\leq C b _ { \\alpha } ^ 2 , \\mbox { w i t h } y = ( x ' , \\delta ^ 2 b _ \\alpha ) \\in \\partial _ 2 \\omega \\end{align*}"}
+{"id": "4642.png", "formula": "\\begin{align*} S _ 2 \\wr S _ n = ( \\mathbb Z / 2 \\mathbb Z ) \\wr S _ n = ( \\mathbb Z / 2 \\mathbb Z ) ^ n \\rtimes S _ n = B _ n . \\end{align*}"}
+{"id": "8172.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { R _ { 0 } } \\mu \\xi ( \\mu , t _ { 0 } ) \\ d \\mu = \\int _ { 0 } ^ { R _ { 0 } } \\mu \\xi ^ { \\mathrm { i n } } ( \\mu ) \\ d \\mu - R _ { 0 } \\int _ { 0 } ^ { t _ { 0 } } \\int _ { 0 } ^ { R _ { 0 } } \\nu \\xi ( R _ { 0 } , s ) \\xi ( \\nu , s ) \\Lambda ( R _ { 0 } , \\nu ) \\ d \\nu d s . \\end{align*}"}
+{"id": "5065.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\tilde { A } \\cdot B \\dd x = \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B \\cdot B \\dd x , \\end{align*}"}
+{"id": "5342.png", "formula": "\\begin{align*} a _ { \\ell } = R _ 1 ( \\ell ) \\beta _ 1 ^ { \\ell } + \\cdots + R _ s ( \\ell ) \\beta _ s ^ { \\ell } \\end{align*}"}
+{"id": "4413.png", "formula": "\\begin{align*} \\int _ { x } ^ { 2 x } \\left | \\theta ( y + \\delta y ) - \\theta ( y ) - \\delta y \\right | ^ 2 d y & \\geq B \\log ^ 4 x \\int _ I d y = B g x \\log ^ 4 x . \\end{align*}"}
+{"id": "116.png", "formula": "\\begin{align*} ( U _ { \\alpha } ( t ) \\phi ) ^ { \\wedge } ( \\xi , \\eta ) = e ^ { i t \\omega _ \\alpha ( \\xi , \\eta ) } \\hat { \\phi } ( \\xi , \\eta ) = e ^ { i t ( | \\xi | ^ { \\alpha } \\xi + \\frac { \\eta ^ 2 } { \\xi } ) } \\hat { \\phi } ( \\xi , \\eta ) . \\end{align*}"}
+{"id": "4419.png", "formula": "\\begin{align*} \\Lambda ^ { ( s ) } ( h _ k ) = G _ k ( s ) = - \\frac { \\zeta ( s ) } { s } ( k ^ { - s } - k ^ { - 1 } ) \\end{align*}"}
+{"id": "3192.png", "formula": "\\begin{align*} v o l ( \\mathcal { H } ( A ) ) = \\tfrac { 1 } { 2 } \\mathcal { A } . \\end{align*}"}
+{"id": "7276.png", "formula": "\\begin{align*} \\gamma ^ { q ^ n } = \\gamma ^ { q ^ { n - i } } . \\end{align*}"}
+{"id": "6531.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\pi ( a ) } ( x - p _ i ) = \\prod _ { i = 1 } ^ { \\pi ( a ) } p _ i ^ { \\alpha _ i } . \\end{align*}"}
+{"id": "6687.png", "formula": "\\begin{align*} c ( \\gamma , \\gamma ^ + ) + c ( \\gamma , \\gamma ^ - ) = C ( \\gamma \\gamma ^ + , \\gamma \\gamma ^ { - } ) - C ( \\gamma ^ + , \\gamma ^ - ) = 0 \\end{align*}"}
+{"id": "6979.png", "formula": "\\begin{align*} \\rho ^ { l _ 1 } = ( a / b ) ^ { x l _ 1 } ( \\alpha / \\beta ) ^ { y l _ 1 } = ( a / b ) ^ { x l _ 1 } ( a / b ) ^ { y k _ 1 } \\zeta ^ { - y } = ( a / b ) \\zeta ^ { - y } \\end{align*}"}
+{"id": "563.png", "formula": "\\begin{align*} f ( z ) = \\int _ { \\mathbb { C } } e ^ { - \\omega z } d \\mu ( \\omega ) , \\end{align*}"}
+{"id": "6637.png", "formula": "\\begin{align*} & I _ { m , t _ { 1 } , \\cdots , t _ { k } } ( x _ { 1 } , \\cdots , x _ { k } ) \\\\ & = \\inf \\left \\{ J ( \\varphi ) ; ~ \\varphi ( t ) = \\int ^ { t } _ { 0 } \\phi ( r ) d r \\in \\mathcal { A C } , \\int ^ { 1 } _ { 0 } K _ { m } ( t _ { i } , r ) \\phi ( r ) d r = x _ { i } , 1 \\leq i \\leq k , t \\in [ 0 , 1 ] \\right \\} , \\end{align*}"}
+{"id": "4249.png", "formula": "\\begin{align*} \\varepsilon & \\leq \\| \\varphi _ { k m _ { j _ 0 } } - \\rho \\| _ \\infty \\\\ & = \\| \\varphi _ { k j } \\circ \\varphi _ { k ( m _ { j _ 0 } - j ) } - \\rho \\circ \\varphi _ { k ( m _ { j _ 0 } - j ) } \\| _ \\infty \\\\ & = \\sup _ { z \\in \\mathbb { B } _ n } | ( \\varphi _ { k j } - \\rho ) ( \\varphi _ { k ( m _ { j _ 0 } - j ) } ( z ) ) | \\\\ & \\leq \\| \\varphi _ { k j } - \\rho \\| _ \\infty . \\end{align*}"}
+{"id": "4235.png", "formula": "\\begin{align*} \\dot { s } \\varepsilon ( z ) \\dot { s } \\varepsilon ( 1 ) \\xi _ 2 = \\varepsilon ( - z ^ { - 1 } ) \\dot { s } h ( - z ) \\varepsilon ( - z ^ { - 1 } ) \\varepsilon ( 1 ) \\xi _ 2 = \\varepsilon ( - z ^ { - 1 } ) \\dot { s } \\varepsilon ( z ^ 2 - z ) \\xi _ 2 . \\end{align*}"}
+{"id": "1132.png", "formula": "\\begin{align*} \\mathbf { 1 } _ { N L } ^ T \\mathbf { y } \\mathbf { x } ^ T \\mathbf { 1 } _ { N L } = \\mathbf { 1 } _ { N } ^ T Y X ^ T \\mathbf { 1 } _ { N } = \\mathbf { c } _ Y ^ T \\mathbf { c } _ X . \\end{align*}"}
+{"id": "8315.png", "formula": "\\begin{align*} y _ { n b } = A \\exp \\{ \\sqrt { \\frac { \\mu + E } { b } } x \\} + B \\exp \\{ - \\sqrt { \\frac { \\mu + E } { b } } x \\} , x \\in ( R , R + r ) . \\end{align*}"}
+{"id": "1743.png", "formula": "\\begin{align*} q ( t , x ) & : = \\int _ { Y ^ { \\ast } } \\nabla _ y \\cdot V _ 0 ( t , x , y ) d y , \\\\ \\bar { J } _ 0 ( t , x ) & : = \\int _ { Y ^ { \\ast } } J _ 0 ( t , x , y ) d y . \\end{align*}"}
+{"id": "826.png", "formula": "\\begin{align*} \\frac { \\partial _ { x x } w ( t , x ) } { 1 + | \\partial _ x w ( t , x ) | ^ 2 } - \\frac { 1 } { w ( t , x ) } \\begin{cases} \\leq \\frac { R T ^ { 1 / 4 } } { 1 + 0 } - \\frac { 1 } { w _ 0 ( x ) + R T ^ { 1 / 4 } } & \\leq - \\frac { 1 } { w _ 0 ( x ) } + \\tilde { \\varepsilon } , \\\\ \\geq \\frac { - R T ^ { 1 / 4 } } { 1 + R ^ 2 T ^ { 1 / 2 } } - \\frac { 1 } { w _ 0 ( x ) - R T ^ { 1 / 4 } } & \\geq - \\frac { 1 } { w _ 0 ( x ) } - \\tilde { \\varepsilon } \\end{cases} \\end{align*}"}
+{"id": "2702.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { T - 1 } \\Theta _ k ( 1 - \\Lambda _ k ' ) - ( 1 - \\Theta _ k ) ( 1 - \\Lambda _ k ) + ( 1 - \\Theta _ k ) \\Lambda _ k - \\Theta _ k \\Lambda _ k ' \\le \\left | \\log _ \\gamma \\frac { \\bar \\Delta ' } { \\delta _ 0 } \\right | < \\left | \\log _ \\gamma \\frac { \\bar \\Delta } { \\delta _ 0 } \\right | + 1 . \\end{align*}"}
+{"id": "5930.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\liminf _ { n \\rightarrow \\infty } ( I + I I + I I I ) = 0 . \\end{align*}"}
+{"id": "1415.png", "formula": "\\begin{align*} \\mathrm { e } ^ { \\lambda \\sum _ { i = 1 } ^ d ( y _ i - x _ i ) - d n \\ln ( 1 + \\lambda ) } \\le \\mathrm { e } ^ { \\lambda d ( y _ 1 - x _ 1 ) + d ^ 2 - n d \\lambda + n d \\frac { \\lambda ^ 2 } { 2 } } \\le C _ d \\mathrm { e } ^ { d \\frac { ( y _ 1 - x _ 1 - n ) } { \\sqrt n } } . \\end{align*}"}
+{"id": "5232.png", "formula": "\\begin{align*} - \\left ( \\partial _ { \\rho } ^ { 2 } + \\frac { 4 } { \\rho } \\partial _ { \\rho } \\right ) \\tilde { \\varphi } = \\mu ^ { 2 } \\tilde { \\varphi } _ { + } , \\rho > 0 . \\end{align*}"}
+{"id": "196.png", "formula": "\\begin{align*} \\theta _ { j , ( 2 ) } ( z _ { k } ) = \\delta _ { j , k } . \\end{align*}"}
+{"id": "2296.png", "formula": "\\begin{align*} \\mu = m _ 1 e _ 1 + \\dots + m _ n e _ n , \\end{align*}"}
+{"id": "1700.png", "formula": "\\begin{align*} \\frac { 1 } { h ^ + } = \\begin{cases} \\frac { 1 } { h } , & h > 0 , \\\\ \\infty , & h \\le 0 . \\end{cases} \\end{align*}"}
+{"id": "4079.png", "formula": "\\begin{align*} Q = ] a , b [ \\times B ( x _ 0 , r ) , \\mbox { w i t h } 0 < a < b < + \\infty , \\ ; \\ ; x _ 0 \\in \\R \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; 0 < r < + \\infty , \\end{align*}"}
+{"id": "442.png", "formula": "\\begin{align*} F \\ast G ( x ) = \\int _ { - \\infty } ^ \\infty F ( x - y ) G ( d y ) \\end{align*}"}
+{"id": "4661.png", "formula": "\\begin{align*} \\Vert T _ { m \\vert _ { ( K _ N ^ { - 1 } \\mathbb { Z } ) ^ 2 } } ^ { ( N + 1 ) } : L _ { p _ 1 } ( \\mathbb { T } , S _ { p _ 1 } ^ { N + 1 } ) \\times L _ { p _ 2 } ( \\mathbb { T } , S _ { p _ 2 } ^ { N + 1 } ) \\rightarrow L _ { 1 } ( \\mathbb { T } , S _ { 1 } ^ { N + 1 } ) \\Vert \\leq B _ { p _ 1 , p _ 2 , N } . \\end{align*}"}
+{"id": "8574.png", "formula": "\\begin{align*} ( I _ { 0 + } ^ \\alpha \\ , D _ { 0 + } ^ \\alpha \\ , f ) ( t ) \\ , = \\ , f ( t ) - ( I _ { 0 + } ^ { 1 - \\alpha } \\ , f ) ( 0 ) h _ \\alpha ( t ) , \\ t > 0 . \\end{align*}"}
+{"id": "4163.png", "formula": "\\begin{align*} \\Upsilon ( \\cdot ) = \\Phi ( \\cdot ) + \\Psi ( \\cdot ) \\end{align*}"}
+{"id": "4687.png", "formula": "\\begin{align*} & W _ 0 = \\{ ( p , q ) \\in \\hat A \\times \\hat B \\mid d ( q , w ) \\geq d ( p , w ) \\} ; \\\\ & W _ 1 = \\{ ( p , q ) \\in \\hat A \\times \\hat B \\mid d ( p , \\tilde { w } ) \\geq d ( q , \\tilde { w } ) \\} . \\end{align*}"}
+{"id": "1617.png", "formula": "\\begin{align*} \\d : C ^ n _ c ( G , V ) \\to C ^ { n + 1 } _ c ( G , V ) , ( \\d c ) ( x _ 0 , \\ldots , x _ { n + 1 } ) : = \\sum _ { j = 0 } ^ { n + 1 } \\ , ( - 1 ) ^ j \\ , c ( x _ 0 , \\ldots , \\widehat { x } _ j , \\ldots , x _ { n + 1 } ) , \\end{align*}"}
+{"id": "731.png", "formula": "\\begin{align*} \\begin{aligned} & ( i ) & \\left | w ^ \\varepsilon ( x , t ) \\right | & \\leq K , \\medskip \\\\ & ( i i ) & \\left | \\partial _ x w ^ \\varepsilon ( x , t ) \\right | & \\leq K , \\medskip \\\\ & ( i i i ) & \\left | w ^ \\varepsilon ( x , t ) - w ^ \\varepsilon ( x , s ) \\right | & \\leq K \\left ( \\left | t - s \\right | ^ { 1 / 2 } + | t - s | \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "8698.png", "formula": "\\begin{align*} q ( A | E _ i ) = p ( A | E _ i ) , i q ( E _ i ) > 0 A \\in { \\bf A } , \\end{align*}"}
+{"id": "3935.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta y + \\tilde { f } ( y ) & = u _ { 1 } - u _ { 2 } , \\ ; \\ ; \\Omega , \\\\ y & = 0 , \\ ; \\quad \\quad \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4067.png", "formula": "\\begin{align*} n \\sum _ { j \\in [ n ] } T ^ { k _ 1 } _ { n , j } \\ , T ^ { k _ 2 } _ { n , j } = O _ M ( n ^ { 2 + ( - 1 - \\beta ) - H } ) = O _ M ( n ^ { 1 - \\beta - H } ) \\end{align*}"}
+{"id": "3082.png", "formula": "\\begin{align*} f _ n = \\sum _ { j = 0 } ^ n x _ 0 \\dotsb \\hat { x } _ j \\dotsb x _ n \\in \\C [ x _ 0 , \\dotsc , x _ n ] \\end{align*}"}
+{"id": "8044.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathbf { H } \\mathbf { P } ^ { \\left ( \\right ) } \\left ( \\mathbf { a } ^ { \\left ( \\right ) } \\right ) \\mathbf { s } ^ { \\left ( \\right ) } + \\mathbf { n } , \\end{align*}"}
+{"id": "2261.png", "formula": "\\begin{align*} D ( x ) A + B D ( x ) = - D ( v ) , \\end{align*}"}
+{"id": "7079.png", "formula": "\\begin{align*} [ v ] _ { B ^ { 1 + \\gamma } _ { p , \\infty } ( B _ \\rho ) } = \\displaystyle \\sup _ { h \\in \\mathbb { R } ^ { n } } \\biggl ( \\displaystyle \\int _ { B _ \\rho } \\dfrac { | \\tau ^ 2 _ h v ( x ) | ^ { p } } { | h | ^ { ( 1 + \\gamma ) p } } d x \\biggr ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
+{"id": "2706.png", "formula": "\\begin{align*} p _ 0 = 1 - 2 \\exp \\left ( a ( \\epsilon _ f - r / 2 ) \\right ) , \\end{align*}"}
+{"id": "1.png", "formula": "\\begin{align*} R < \\min \\left \\{ \\begin{aligned} & 3 C , C + 2 C _ 0 + F _ 2 , \\\\ & 2 C + 4 C _ 0 + F _ 3 , F _ 5 , 3 C + F _ 7 . \\end{aligned} \\right \\} \\end{align*}"}
+{"id": "1963.png", "formula": "\\begin{align*} Q ^ { ( m ) } _ \\nu ( k ) = \\prod _ { i = 1 } ^ { m - 1 } \\frac { q ^ { k + i } - 1 } { q ^ i - 1 } . \\end{align*}"}
+{"id": "5723.png", "formula": "\\begin{align*} f '' ( x ) = a + b ' x + c ' x ^ q , \\end{align*}"}
+{"id": "7641.png", "formula": "\\begin{align*} \\alpha _ t ( x ) : = - R ^ { - 1 } B ( P _ t x + \\varphi _ t ^ \\xi ) - h ( \\mu _ t ) . \\end{align*}"}
+{"id": "5547.png", "formula": "\\begin{align*} I = ( I \\cap X ) \\vee ( I \\cap Y ) = ( I \\cap X ) \\times ( I \\cap Y ) . \\end{align*}"}
+{"id": "2834.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\Lambda ^ { s - 1 } \\nabla v \\| _ 3 & \\leq C \\| \\nabla v \\| ^ { 1 - \\kappa } \\| \\Lambda ^ { s + 1 } v \\| ^ \\kappa \\end{aligned} \\end{align*}"}
+{"id": "913.png", "formula": "\\begin{align*} \\mathcal { F } ( d ) = \\{ ( u , v ) \\ ; : \\ ; u ^ 2 + u v + v ^ 2 = d , \\ ; \\ ; ( u , v ) = 1 , \\ ; \\ ; u \\in \\mathbb { N } , \\ ; \\ ; v \\in \\mathbb { Z } \\setminus \\{ 0 \\} \\} . \\end{align*}"}
+{"id": "2098.png", "formula": "\\begin{align*} d _ \\Theta ( \\theta , \\theta ' ) = \\left \\{ \\begin{array} { l l } \\min \\{ \\| \\theta - \\theta ' \\| , 1 \\} & k ( \\theta ) = k ( \\theta ' ) , \\\\ 1 & \\end{array} \\right . \\end{align*}"}
+{"id": "8232.png", "formula": "\\begin{align*} \\frac { 1 + b ( z ) } { 1 - b ( z ) } = \\int _ { \\partial \\mathbb { D } } \\frac { t + z } { t - z } d \\mu ( t ) \\ , z \\in \\mathbb { D } \\end{align*}"}
+{"id": "8039.png", "formula": "\\begin{align*} \\theta ( x ) = \\begin{cases} 1 & x \\in q ( x ) ( [ 0 , \\infty ) ) \\\\ - 1 & \\end{cases} \\end{align*}"}
+{"id": "7038.png", "formula": "\\begin{align*} F _ { 0 , 7 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 1 , 6 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 2 , 5 } \\ , : = \\ , 1 2 0 , \\ \\ \\ \\ \\ F _ { 3 , 4 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 4 , 3 } \\ , : = \\ , 0 , \\ \\ \\ \\ \\ F _ { 5 , 2 } \\ , : = \\ , 2 0 , \\ \\ \\ \\ \\ F _ { 6 , 1 } \\ , : = \\ , 0 , \\end{align*}"}
+{"id": "9301.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma ( a , r ) = \\int _ { B ( a , r ) } \\Delta u \\wedge \\beta _ n ^ { n - 1 } . \\end{aligned} \\end{align*}"}
+{"id": "6575.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } \\nabla c \\cdot \\nu c \\psi ^ { 2 } = 0 . \\end{align*}"}
+{"id": "6766.png", "formula": "\\begin{align*} e ( \\rho , x ) = e ^ { i \\rho x } \\left ( 1 + o ( 1 ) \\right ) , e ^ { \\prime } ( \\rho , x ) = i \\rho e ^ { i \\rho x } \\left ( 1 + o ( 1 ) \\right ) , x \\rightarrow \\infty , \\end{align*}"}
+{"id": "43.png", "formula": "\\begin{align*} Z \\psi = 0 . \\end{align*}"}
+{"id": "3081.png", "formula": "\\begin{align*} T _ { \\varphi _ A ^ * ( f ) } = A ^ t \\circ T _ f \\circ \\varphi _ A \\end{align*}"}
+{"id": "9215.png", "formula": "\\begin{align*} \\begin{aligned} f ( t ) \\le & \\ , f ( 0 ) + t \\gamma ^ 2 \\left ( \\bar { k } _ 3 ( L _ r , \\delta ) + \\bar { k } _ 2 ( L _ r , \\delta ) \\right ) \\\\ & + \\gamma A \\int _ 0 ^ t f ( \\tau ) \\ , d \\tau , \\end{aligned} \\end{align*}"}
+{"id": "8114.png", "formula": "\\begin{align*} C ( G , x ) : = 1 + \\sum _ { \\emptyset \\neq U \\subseteq V ( G ) : G [ U ] ~ i s ~ a ~ c l i q u e } x ^ { \\vert U \\vert } , \\end{align*}"}
+{"id": "5557.png", "formula": "\\begin{align*} W ( y | x ) = \\sum _ { n \\in \\N } \\left [ \\hat { A } \\circ \\hat { \\sigma } ^ { - n } ( y | x ) - \\hat { A } \\circ \\hat { \\sigma } ^ { - n } ( y | x ' ) \\right ] \\end{align*}"}
+{"id": "2398.png", "formula": "\\begin{align*} \\widetilde { \\mu } ( k - 1 ) : = \\frac { k \\mu ( k ) } { \\sum _ { k ' = 1 } ^ \\infty k ' \\mu ( k ' ) } , k = 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "9315.png", "formula": "\\begin{align*} b _ { 1 } b _ { 2 } u _ { 3 } ^ { 2 } - b _ { 2 } u _ { 3 } ^ { 2 } = p q ^ { 2 k + 1 } , \\end{align*}"}
+{"id": "2257.png", "formula": "\\begin{align*} U \\cdot ( A , B , x ) = ( U A , B , x ) , ( A , B , x ) \\cdot V = ( A , B V , x ) . \\end{align*}"}
+{"id": "2935.png", "formula": "\\begin{align*} \\Phi \\left ( \\lambda \\right ) = \\int _ { \\left ( 0 , \\infty \\right ) } \\left ( 1 - e ^ { - \\lambda x } \\right ) \\nu _ { \\alpha } ^ { \\prime } \\left ( d x \\right ) \\end{align*}"}
+{"id": "2733.png", "formula": "\\begin{align*} U : = \\left \\{ x \\in \\R ^ { d } \\colon \\| x \\| _ 2 = 1 , x _ d \\geq 0 \\right \\} . \\end{align*}"}
+{"id": "4456.png", "formula": "\\begin{align*} \\xi \\left ( ( a ) \\right ) = \\left ( \\frac { a } { | a | } \\right ) ^ { k - 1 } \\end{align*}"}
+{"id": "6584.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } f ^ { p + 1 } \\le C \\frac { ( p + 1 ) ^ { 2 } } { 2 \\log s } \\int _ { \\Omega } ( f \\log f + e ^ { - 1 } ) \\int _ { \\Omega } f ^ { p - 2 } | \\nabla f | ^ { 2 } + 2 C \\| w \\| _ { L ^ { 1 } ( \\Omega ) } ^ { 2 } + 3 s ^ { p + 1 } | \\Omega | , \\end{aligned} \\end{align*}"}
+{"id": "7169.png", "formula": "\\begin{align*} \\Delta _ g & = \\frac { \\partial ^ 2 } { \\partial x _ n ^ 2 } + \\Gamma ^ \\alpha _ { n \\alpha } \\frac { \\partial } { \\partial x _ n } + g ^ { \\alpha \\beta } \\frac { \\partial ^ 2 } { \\partial x _ \\alpha \\partial x _ \\beta } + \\biggl ( g ^ { \\alpha \\beta } \\Gamma ^ \\gamma _ { \\gamma \\alpha } + \\frac { \\partial g ^ { \\alpha \\beta } } { \\partial x _ \\alpha } \\biggr ) \\frac { \\partial } { \\partial x _ \\beta } . \\end{align*}"}
+{"id": "8188.png", "formula": "\\begin{align*} m n S = & \\sum _ { i = 1 } ^ { m - 1 } \\sum _ { j = 1 } ^ { n } ( x _ { i , j } + x _ { i + 1 , j } + x _ { i , j + 1 } + x _ { i + 1 , j + 1 } ) \\\\ & + \\ ; \\sum _ { i = 1 } ^ { m } ( x _ { m , j } + x _ { m , j + 1 } + x _ { 1 , n - j } + x _ { 1 , n - j + 1 } ) \\ ; \\\\ = \\ ; & 4 \\biggl ( \\sum _ { k = 1 } ^ { m n } k \\biggr ) = ( 2 m n ) ( m n + 1 ) , \\end{align*}"}
+{"id": "5739.png", "formula": "\\begin{align*} \\frac { d _ 1 } { u ^ { q + 1 } } = \\frac { h ^ q } { h } \\end{align*}"}
+{"id": "4767.png", "formula": "\\begin{align*} A _ n : = B _ n + C _ n \\longrightarrow B + C = : A . \\end{align*}"}
+{"id": "2784.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mu _ { n } ^ { D , \\beta } ( A ) = \\mu _ { \\infty } ^ { D , \\beta } ( A ) , \\quad \\mathrm { a . s . } \\end{align*}"}
+{"id": "325.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\rightarrow 0 } \\| g _ { b , \\gamma } ( f ) - g _ b ( f ) \\| _ { L ^ p ( w ) \\rightarrow L ^ p ( w ) } = 0 . \\end{align*}"}
+{"id": "5312.png", "formula": "\\begin{align*} \\| W _ { \\lambda } ( g ) \\| ^ 2 _ { H S } | \\lambda | ^ n = ( 2 \\pi ) ^ n \\| g \\| _ 2 ^ 2 , ~ g \\in L ^ 2 ( \\mathbb { C } ^ n ) . \\end{align*}"}
+{"id": "4344.png", "formula": "\\begin{align*} & \\int _ { \\{ - t ' _ 3 \\le \\Psi < - t ' _ 4 \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h \\\\ \\le & \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\backslash U _ k \\} } | \\tilde F | ^ 2 _ h + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\backslash U _ k \\} } | F _ { t _ 0 , \\tilde { c } } - \\tilde F | ^ 2 _ h \\\\ & + \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in ( t ' _ 4 , t ' _ 3 ] \\cap U _ k \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h . \\end{align*}"}
+{"id": "5870.png", "formula": "\\begin{align*} \\Gamma ^ { \\dag } \\left ( X , Y , Z \\right ) \\equiv \\left ( \\nabla _ { Z } ^ { \\dag } G \\right ) \\left ( X , Y \\right ) = - \\Gamma \\left ( X , Y , Z \\right ) . \\end{align*}"}
+{"id": "8493.png", "formula": "\\begin{align*} M _ { - , 1 } ( x ; z ) - e _ 1 = \\mathcal { P } ^ - \\left ( r _ 2 ( z ) e ^ { 2 i z x } M _ { + , 2 } ( x ; z ) \\right ) ( z ) , z \\in \\mathbb { R } . \\end{align*}"}
+{"id": "1638.png", "formula": "\\begin{align*} \\theta \\left ( S ( r + s ) \\right ) & = \\int ^ { T _ 2 } _ { T _ 1 } G \\left ( \\frac { 3 T _ 1 + T _ 2 } { 4 } , \\tau \\right ) f _ \\downarrow ( r ( \\tau ) + s ( \\tau ) ) \\ ; d \\tau \\\\ & \\le f _ \\downarrow ( 0 ) \\int ^ { T _ 2 } _ { T _ 1 } G \\left ( \\frac { 3 T _ 1 + T _ 2 } { 4 } , \\tau \\right ) d \\tau \\\\ & < c . \\end{align*}"}
+{"id": "6302.png", "formula": "\\begin{align*} x ^ { t + 1 } - y ^ t = \\ & ( 1 - \\alpha _ t ) x ^ t + \\alpha _ t z ^ { t + 1 } - y ^ t = ( 1 - \\alpha _ t ) ( x ^ t - y ^ t ) + \\alpha _ t z ^ { t + 1 } - \\alpha _ t y ^ t \\\\ \\overset { \\eqref { l 2 - e 2 } } { = } \\ & \\alpha _ t ( 1 - \\beta _ t ) ( y ^ t - z ^ t ) + \\alpha _ t z ^ { t + 1 } - \\alpha _ t y ^ t = \\alpha _ t \\left ( z ^ { t + 1 } - \\beta _ t y ^ t - ( 1 - \\beta _ t ) z ^ t \\right ) . \\end{align*}"}
+{"id": "4012.png", "formula": "\\begin{align*} \\sigma ^ L ( G ^ L ( X ) ) = G ( \\{ \\varphi ^ L _ { \\sigma , s } ( E ^ L _ X ( v ) ) \\mid ( s , v ) \\in D \\} ) . \\end{align*}"}
+{"id": "7346.png", "formula": "\\begin{align*} \\epsilon = \\sum _ { i \\in I } \\bar { \\nu } _ i ^ ! & \\ ; & ( \\delta ^ n _ k ) ^ ! = \\sum _ { J = \\{ i _ 0 < \\hdots < \\widehat { i _ k } < \\hdots < i _ n \\} \\subset K = \\{ i _ 0 < \\hdots < i _ n \\} } ( \\nu _ { K } ^ J ) ^ ! \\end{align*}"}
+{"id": "8643.png", "formula": "\\begin{align*} { \\mathbb { E } } \\left [ { { { \\left \\| { { \\bf { x } } \\left [ n \\right ] } \\right \\| } ^ 2 } } \\right ] = \\sum \\limits _ { l = 1 } ^ L { { \\mathbb { E } } \\left [ { { { \\left \\| { { { \\bf { f } } _ l } s \\left [ { n - { \\kappa _ l } } \\right ] } \\right \\| } ^ 2 } } \\right ] } { \\rm = } \\sum \\limits _ { l = 1 } ^ L { { { \\left \\| { { { \\bf { f } } _ l } } \\right \\| } ^ 2 } } \\le P , \\end{align*}"}
+{"id": "0.png", "formula": "\\begin{align*} R < \\min \\left \\{ \\begin{aligned} & 3 C - F _ 1 , C + 2 C _ 0 + F _ 2 , \\\\ & 2 C + 4 C _ 0 + F _ 3 - F _ 4 , \\\\ & F _ 5 , \\frac { 1 } { 2 } \\left [ C + 2 C _ 0 + F _ 2 + F _ 6 \\right ] , \\\\ & C + 2 C _ 0 + \\frac { 1 } { 2 } \\left [ F _ 3 + F _ 6 - F _ 4 \\right ] , \\\\ & C + 2 C _ 0 + \\frac { 1 } { 3 } \\left [ 2 F _ 6 - F _ 1 \\right ] . \\end{aligned} \\right \\} \\end{align*}"}
+{"id": "4560.png", "formula": "\\begin{align*} a _ { \\ell + 2 , 0 , 0 } - { q } ^ { 1 / 2 } \\sigma _ 1 ( \\textbf { z } ) a _ { \\ell + 1 , 0 , 0 } + q \\sigma _ 2 ( \\textbf { z } ) a _ { \\ell , 0 , 0 } - q ^ { 3 / 2 } \\sigma _ 3 ( \\textbf { z } ) a _ { \\ell - 1 , 0 , 0 } + q ^ 2 \\sigma _ 4 ( \\textbf { z } ) a _ { \\ell - 2 , 0 , 0 } = 0 . \\end{align*}"}
+{"id": "7571.png", "formula": "\\begin{align*} x = \\sum _ { n = 0 } ^ \\infty \\frac { \\varepsilon _ n } { \\gamma ^ { n + 1 } } . \\end{align*}"}
+{"id": "2121.png", "formula": "\\begin{align*} \\varrho : = | z | ^ 2 + | w | ^ 2 + \\Re ( a z ^ 2 + b w ^ 2 ) - 1 . \\end{align*}"}
+{"id": "6570.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta ( c ( t ) - c ( s ) ) + n ( t ) ( c ( t ) - c ( s ) ) = - c ( s ) ( n ( t ) - n ( s ) ) , & x \\in \\Omega , \\\\ \\nabla ( c ( t ) - c ( s ) ) \\cdot \\nu + ( c ( t ) - c ( s ) ) = 0 , & x \\in \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "2168.png", "formula": "\\begin{align*} \\min _ { \\substack { \\omega _ { n , k } , \\ , { { \\bf { u } } } _ k } } & \\mathcal { L } ^ * \\\\ & \\omega _ { n , k } \\in \\{ 0 , 1 \\} , \\forall k , \\forall n , \\end{align*}"}
+{"id": "7233.png", "formula": "\\begin{align*} L = \\frac { 1 } { \\delta } \\det A . \\end{align*}"}
+{"id": "4933.png", "formula": "\\begin{align*} \\langle T ( f , g ) , h \\rangle = \\langle T ^ { * 1 } ( h , g ) , f \\rangle = \\langle T ^ { * 2 } ( f , h ) , g \\rangle . \\end{align*}"}
+{"id": "6897.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { X _ i } \\to \\int _ 0 ^ 1 \\mu ^ * _ u \\ , d u , . \\end{align*}"}
+{"id": "7670.png", "formula": "\\begin{align*} \\begin{array} { l } \\mu ^ { N , 1 } _ { \\boldsymbol { \\alpha } ^ * _ t } + h _ 1 ^ N ( \\mu ^ { N , 1 } _ { \\boldsymbol { \\alpha } ^ * _ t } , \\ldots , \\mu ^ { N , N } _ { \\boldsymbol { \\alpha } ^ * _ t } ) = \\Delta ^ { N , 1 } _ { * , t } , \\\\ \\quad \\vdots \\qquad \\qquad ~ ~ \\vdots \\qquad ~ ~ ~ ~ \\vdots \\\\ \\mu ^ { N , N } _ { \\boldsymbol { \\alpha } ^ * _ t } + h _ N ^ N ( \\mu ^ { N , 1 } _ { \\boldsymbol { \\alpha } ^ * _ t } , \\ldots , \\mu ^ { N , N } _ { \\boldsymbol { \\alpha } ^ * _ t } ) = \\Delta ^ { N , N } _ { * , t } , \\end{array} \\end{align*}"}
+{"id": "2501.png", "formula": "\\begin{align*} r _ g ( \\mathbf { x } , \\mathbf { y } , \\mathbf { s } , \\kappa , \\tau ) = { { \\mathbf { b } } ^ { T } } \\mathbf { y } - { { \\mathbf { c } } ^ { T } } \\mathbf { x } - \\kappa . \\end{align*}"}
+{"id": "1344.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { 3 m } \\alpha _ j + \\sum _ { k = 1 } ^ l \\beta _ k < \\sum _ { i = 1 } ^ { n + 1 } \\lambda _ i + \\varepsilon . \\end{align*}"}
+{"id": "7324.png", "formula": "\\begin{align*} \\rho ^ { \\circ } ( A ) = \\inf \\{ t > 0 : A \\mbox { i s c o v e r e d b y f i n i t e l y m a n y s u b s e t s o f s i z e } \\leqslant t \\} , \\end{align*}"}
+{"id": "2499.png", "formula": "\\begin{align*} \\bar { \\mathbf { X } } = ( \\bar { \\mathbf { x } } ) , \\bar { \\mathbf { S } } = ( \\bar { \\mathbf { s } } ) , \\bar { \\mathbf { x } } = \\mathbf { D } ^ { - T } \\mathbf { x } , \\bar { \\mathbf { s } } = \\mathbf { D } \\mathbf { s } , \\mathbf { D } \\in \\mathcal { G } . \\end{align*}"}
+{"id": "2157.png", "formula": "\\begin{align*} \\mathcal { P } [ \\varrho ] = \\frac { 1 } { 2 } a \\left ( \\alpha ^ 2 + \\gamma ^ 2 \\right ) ^ 3 P ( a , \\tau ) \\end{align*}"}
+{"id": "1664.png", "formula": "\\begin{align*} H ^ * ( Y _ { K } ^ 1 , \\mathcal { D } _ { \\Omega } ) = H ^ * ( Y ^ 1 _ { K } , \\mathcal { D } _ { \\Omega } ) _ { \\leq h } \\oplus H ^ * ( Y ^ 1 _ { K } , \\mathcal { D } _ { \\Omega } ) _ { > h } \\end{align*}"}
+{"id": "1045.png", "formula": "\\begin{align*} d ( u \\Rightarrow w v ) = d ( u \\Rightarrow w \\rho _ x ( u ) ) + d ( w \\rho _ x ( u ) \\Rightarrow w v ) . \\end{align*}"}
+{"id": "8018.png", "formula": "\\begin{align*} \\{ l _ 1 , l _ 2 , \\dots , l _ r , l _ 1 ' , l _ 2 ' , \\dots , l _ r ' \\} = \\cup _ { u = 1 } ^ t \\mathcal I ( s _ u ) , \\end{align*}"}
+{"id": "4967.png", "formula": "\\begin{align*} W ( \\xi _ 0 , n , x , \\widetilde \\theta ) = & E _ { \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ n Z ^ { \\widetilde \\theta } \\biggr ) \\biggr ] \\\\ = & E _ { \\xi _ 0 } \\biggl [ V \\biggl ( \\frac { \\xi _ 0 f _ 2 ( Y _ 1 ) } { \\xi _ 0 f _ 2 ( Y _ 1 ) + ( 1 - \\xi _ 0 ) f _ 1 ( Y _ 1 ) } , n - 1 , x + Y _ 1 \\biggr ) \\biggr ] . \\end{align*}"}
+{"id": "3234.png", "formula": "\\begin{align*} \\mathbb E \\left [ [ ( \\mathcal S ( \\Delta _ n ) - I ) Y _ { ( i - 1 ) } ] ^ { \\otimes 2 } \\Big | \\sigma \\right ] = \\int _ 0 ^ { ( i - 1 ) \\Delta _ n } ( \\mathcal S ( \\Delta _ n ) - I ) \\Sigma _ s ^ { \\mathcal S _ n } ( \\mathcal S ( \\Delta _ n ) - I ) ^ * d s . \\end{align*}"}
+{"id": "9186.png", "formula": "\\begin{align*} \\begin{aligned} | x ( t ) - z & ( t ) | \\le | x ( 0 ) - z ( 0 ) | + \\int _ 0 ^ t \\left | \\dot { x } ( \\tau ) - \\dot { z } ( \\tau ) \\right | \\ , d \\tau \\\\ \\le & \\ , | x ( 0 ) - z ( 0 ) | + \\gamma L _ r \\int _ 0 ^ t | x ( \\tau ) - z ( \\tau ) | \\ , d \\tau \\\\ & + \\gamma ^ 2 \\bar k ( L _ r , M _ r , \\delta ) t \\end{aligned} \\end{align*}"}
+{"id": "5985.png", "formula": "\\begin{align*} \\tilde { \\nu _ 1 } ( 0 ) = 0 \\qquad \\frac { d } { d \\tilde { \\nu _ 2 } } \\tilde { \\nu _ 1 } ( 0 ) = 0 . \\end{align*}"}
+{"id": "8919.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } ( - 1 ) ^ { j } \\binom { n } { j } \\frac { \\left ( \\beta \\right ) ^ { \\left ( j + m \\right ) } } { \\left ( \\beta \\right ) ^ { \\left ( j \\right ) } } = ( - 1 ) ^ { n } n ! \\binom { m } { n } \\frac { \\left ( \\beta \\right ) ^ { \\left ( m \\right ) } } { \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } } . \\end{align*}"}
+{"id": "936.png", "formula": "\\begin{align*} \\begin{aligned} \\| u _ 2 ( t ) \\| _ { H ^ 1 } + \\| J u _ 2 ( t ) \\| _ { L ^ 2 } & \\le ( 1 + C \\varepsilon ^ 2 _ 1 \\langle T _ 0 \\rangle ^ { 1 + 2 C _ 1 \\varepsilon _ 1 ^ 2 } e ^ { C \\varepsilon ^ 2 _ 1 \\langle T _ 0 \\rangle ^ { 1 + 2 C _ 1 \\varepsilon _ 1 ^ 2 } } ) \\varepsilon _ 2 \\\\ & < 2 \\varepsilon _ 2 \\end{aligned} \\end{align*}"}
+{"id": "384.png", "formula": "\\begin{align*} H : = \\frac { 1 } { n - 1 } \\begin{bmatrix} 0 & \\ 0 ' & \\ 0 ' \\\\ \\ 0 & T & O \\\\ \\ 0 & O & O \\end{bmatrix} , \\end{align*}"}
+{"id": "6621.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c } - \\varepsilon _ 1 \\mu & \\varepsilon _ 1 \\nu \\\\ \\varepsilon _ 3 \\mu & - \\varepsilon _ { 3 } \\nu \\end{array} \\right ) \\end{align*}"}
+{"id": "892.png", "formula": "\\begin{align*} r = 2 r _ 0 \\ , , q = 2 q _ 0 \\ , . \\end{align*}"}
+{"id": "2847.png", "formula": "\\begin{align*} I _ n ^ { a , a - 1 / 2 } = { \\frak I } ^ { a , a - 1 / 2 } { n ^ { 2 a } } + o ( n ^ { 2 a } ) , \\end{align*}"}
+{"id": "7426.png", "formula": "\\begin{align*} \\exists \\ C = C [ \\psi ] \\ \\in ( 0 , \\infty ) \\ \\Rightarrow \\ | | \\zeta | | G \\psi \\le C ( \\psi ) . \\end{align*}"}
+{"id": "390.png", "formula": "\\begin{align*} A = \\frac { 9 ( n - 1 ) } { ( n + 4 ) ^ 2 } y y ' = { \\lambda } \\frac { y y ' } { { \\| y \\| } ^ 2 } . \\end{align*}"}
+{"id": "1993.png", "formula": "\\begin{align*} \\mu ( x , T ( x ) ) & = f ( x - T ( x ) ) - f ( - x - T ( x ) ) = 2 f ( x - T ( x ) ) = 2 f ( z ) = h ( g ^ { - 1 } ( z ) ) \\\\ & = h ( x ) . \\end{align*}"}
+{"id": "1673.png", "formula": "\\begin{align*} V _ { p , \\alpha _ z } = \\bigoplus _ { v \\mid p } V _ { v , \\alpha _ { z , v } } , \\end{align*}"}
+{"id": "5050.png", "formula": "\\begin{align*} { \\mathcal { E } } = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( | v | ^ { 2 } + | b | ^ { 2 } \\right ) \\dd x , { \\mathcal { H } } = 2 \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi _ { \\infty } ) \\frac { G } { r ^ { 2 } } \\dd x . \\end{align*}"}
+{"id": "4809.png", "formula": "\\begin{align*} a ( u _ n , v _ n ) = \\lambda _ n ( u _ n , v _ n ) v _ n \\in X _ n . \\end{align*}"}
+{"id": "5965.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } \\left ( \\int _ 0 ^ T \\Vert X _ n ( t ) \\Vert _ V ^ p d t > M \\right ) = 0 , \\end{align*}"}
+{"id": "5955.png", "formula": "\\begin{align*} A ( u ) = - \\Delta ^ 2 u + \\Delta \\varphi ( u ) . \\end{align*}"}
+{"id": "1763.png", "formula": "\\begin{align*} 0 \\le \\chi ( z ) & \\le 1 z \\in \\R , \\\\ \\chi ( z ) & = 1 \\mbox { i f } | z | \\le \\frac { \\delta _ 0 } { 2 } , \\\\ \\chi ( z ) & = 0 \\mbox { i f } | z | \\geq \\delta _ 0 , \\\\ z \\chi ' ( z ) & < 0 \\mbox { i f } | z | \\in \\left ( \\frac { \\delta _ 0 } { 2 } , \\delta _ 0 \\right ) . \\end{align*}"}
+{"id": "5434.png", "formula": "\\begin{align*} m : = b _ { n - 1 } \\cdots b _ { \\alpha + 1 } b _ \\alpha ^ { k + \\alpha - n } . \\end{align*}"}
+{"id": "6963.png", "formula": "\\begin{align*} t _ n = \\frac { \\alpha ^ n - \\beta ^ n } { \\alpha - \\beta } \\end{align*}"}
+{"id": "4402.png", "formula": "\\begin{align*} | Z _ 3 | & = \\left | \\frac { 1 } { \\log x } \\sum _ { \\rho } \\frac { x ^ { \\rho - s } - x ^ { 2 ( \\rho - s ) } } { ( s - \\rho ) ^ 2 } \\right | \\\\ & \\leq \\frac { x ^ { \\frac { 1 } { 2 } - \\sigma } + x ^ { 1 - 2 \\sigma } } { \\log x } \\sum _ { \\gamma } \\frac { 1 } { ( \\sigma - \\beta ) ^ 2 + ( t - \\gamma ) ^ 2 } . \\end{align*}"}
+{"id": "573.png", "formula": "\\begin{align*} \\left | e ^ { - \\omega ^ { \\prime } \\zeta } \\right | = 1 . \\end{align*}"}
+{"id": "5891.png", "formula": "\\begin{align*} I _ 1 = & \\mathbb { E } \\int _ 0 ^ { T \\wedge \\tau _ n ^ M } \\big [ 2 \\langle A ( r , Y _ n ( r ) ) , Y _ n ( r ) \\rangle + \\Vert P _ n B ( r , Y _ n ( r ) ) Q _ n \\Vert _ { L _ 2 } ^ 2 \\big ] d r \\int _ { 0 \\vee ( r - \\delta ) } ^ { r } \\mathbf { 1 } _ { \\{ \\tau _ n ^ M > t \\} } d t \\\\ \\leq & \\delta \\ , \\mathbb { E } \\int _ 0 ^ { T \\wedge \\tau _ n ^ M } f ( s ) ( 1 + \\Vert Y _ n ( s ) \\Vert _ H ^ 2 ) d s \\\\ \\leq & C _ M \\delta . \\end{align*}"}
+{"id": "7895.png", "formula": "\\begin{align*} c _ 5 ( q , q , p ^ \\mathfrak { t } ( 0 ) ) & = c _ 5 ( q , q , p ^ 0 ( 0 ) ) - \\frac 1 2 \\mathfrak { t } \\hat W _ r ( q ) \\\\ c _ 2 ( q , q , p ^ \\mathfrak { t } ( 0 ) ) & = c _ 2 ( q , q , p ^ 0 ( 0 ) ) - \\frac 1 2 \\mathfrak { t } ( - \\hat W _ r ( 0 ) + \\hat W _ r ( 2 q ) ) \\\\ c _ 3 ( q , 2 q , q , p ^ \\mathfrak { t } ( 0 ) ) & = c _ 3 ( q , 2 q , q , p ^ 0 ( 0 ) ) \\\\ c _ 1 ( q , 2 q , p ^ \\mathfrak { t } ( 0 ) ) & = c _ 1 ( q , 2 q , p ^ 0 ( 0 ) ) \\end{align*}"}
+{"id": "2940.png", "formula": "\\begin{align*} y ( t + 0 ) = - r y ( t - 0 ) \\end{align*}"}
+{"id": "27.png", "formula": "\\begin{align*} P _ A ( \\lambda ) + P _ A ( \\mu ) = P _ A ( \\lambda + \\mu ) S _ A ( \\lambda ) + S _ A ( \\mu ) = S _ A ( \\lambda + \\mu ) . \\end{align*}"}
+{"id": "9146.png", "formula": "\\begin{align*} \\dot { { x } } _ a = \\ , - \\gamma c _ 1 \\delta \\left . \\dfrac { \\partial h } { \\partial x } \\right | _ { { x } _ a } \\ , . \\end{align*}"}
+{"id": "5561.png", "formula": "\\begin{align*} W ( y | x ) = \\ , \\sum _ { n = 1 } ^ \\infty \\left [ A \\circ \\hat { \\sigma } ^ { - n } ( y | x ) - A \\circ \\hat { \\sigma } ^ { - n } ( y | x ' ) \\right ] , \\end{align*}"}
+{"id": "2354.png", "formula": "\\begin{align*} w _ t = \\nu ( w ) w _ { y y } + \\left ( 2 \\phi ' ( x ) ^ 2 e ^ { 2 \\phi } \\nu ( w ) - \\phi '' ( x ) ( 1 + \\nu ( w ) w ^ 2 ) \\right ) w - 2 \\nu ( w ) ^ 2 ( w _ { y } + \\phi ' ( x ) e ^ { 2 \\phi } ) ^ 2 w . \\end{align*}"}
+{"id": "5151.png", "formula": "\\begin{align*} r ^ { 2 } < \\phi ( z , r ) \\leq \\phi ( z ' , r ) - \\phi ( z ' , 0 ) = \\int _ { 0 } ^ { r } \\partial _ { r ' } \\phi ( z ' , r ' ) \\dd r . \\end{align*}"}
+{"id": "7844.png", "formula": "\\begin{align*} F ^ { q , \\dagger } _ { v v } ( 0 , p ) [ u _ m , u _ k ] & = c _ 3 ( q , m , k , p ) u _ { m - k } + c _ 4 ( q , m , k , p ) u _ { m + k } , \\end{align*}"}
+{"id": "5861.png", "formula": "\\begin{align*} \\rho \\left ( { \\bf { x } } \\right ) = \\frac { { { \\rho _ l } + { \\rho _ v } } } { 2 } - \\frac { { { \\rho _ l } - { \\rho _ v } } } { 2 } \\tanh \\frac { { 2 \\left ( { \\left | { { \\bf { x } } - { { \\bf { x } } _ 0 } } \\right | - { R _ 0 } } \\right ) } } { W } , \\end{align*}"}
+{"id": "1152.png", "formula": "\\begin{align*} d X _ { t } = \\left ( \\langle \\mu _ t , b ( t , X _ t , \\cdot ) \\rangle \\right ) d t + d W _ t , \\mu _ t = \\operatorname { L a w } ( X _ t ) , \\end{align*}"}
+{"id": "3186.png", "formula": "\\begin{align*} \\lim \\limits _ { q \\to \\infty } \\dfrac { k _ 4 ( P ^ \\ast ( q ) ) } { q ^ 4 } = \\frac { 1 } { 1 5 3 6 } + 3 \\times \\lim \\limits _ { q \\to \\infty } { _ { 3 } } F _ { 2 } \\left ( \\begin{array} { c c c } \\chi _ 4 , & \\chi _ 4 , & \\chi _ 4 ^ 3 \\\\ & \\varepsilon , & \\varepsilon \\end{array} | 1 \\right ) . \\end{align*}"}
+{"id": "3099.png", "formula": "\\begin{align*} \\tilde { \\theta } ^ { k } = \\Pi _ { \\mathcal { K } , G } \\big \\{ \\theta ^ { k } - G ^ { - 1 } [ T ( \\tilde { \\theta } ^ { k } ) - \\beta \\Lambda ( y ^ { k } - \\tilde { y } ^ { k } ) + \\zeta ^ k ] \\big \\} . \\end{align*}"}
+{"id": "4081.png", "formula": "\\begin{align*} \\nu _ 0 = 1 - \\frac { q _ 0 - 5 } { 5 q _ 0 } , \\end{align*}"}
+{"id": "2173.png", "formula": "\\begin{align*} \\Lambda = ( \\lambda ^ { ( 0 ) } _ 1 ( t _ 0 ) , \\ldots , \\lambda ^ { ( 0 ) } _ 1 ( t _ N ) , \\lambda ^ { ( 0 ) } _ 2 ( t _ 0 ) , \\ldots , \\lambda ^ { ( 0 ) } _ K ( t _ N ) , \\lambda ^ { ( 1 ) } _ 1 ( t _ 0 ) , \\ldots , \\lambda ^ { ( 1 ) } _ K ( t _ N ) ) . \\end{align*}"}
+{"id": "5353.png", "formula": "\\begin{align*} D + E ^ + - E ^ - = f ^ * ( f _ * D ) . \\end{align*}"}
+{"id": "697.png", "formula": "\\begin{align*} I _ \\alpha \\ast I _ \\beta = I _ { \\alpha + \\beta } , \\forall \\alpha , \\beta > 0 ~ ~ \\alpha + \\beta < N . \\end{align*}"}
+{"id": "6669.png", "formula": "\\begin{align*} \\exp \\bigg [ \\epsilon ^ { - 2 } ( t _ { k } - t _ { k - 1 } ) ( \\Lambda ( \\bar \\alpha _ { k , k } ) - \\delta ) \\bigg ] \\le \\mathbb E \\left ( e ^ { \\Delta _ k } \\Big | \\mathcal { F } _ { t _ { k - 1 } } \\right ) \\le \\exp \\bigg [ \\epsilon ^ { - 2 } ( t _ { k } - t _ { k - 1 } ) ( \\Lambda ( \\bar \\alpha _ { k , k } ) + \\delta ) \\bigg ] . \\end{align*}"}
+{"id": "5190.png", "formula": "\\begin{align*} \\begin{aligned} B \\times u \\in L ^ { 4 / 3 } ( 0 , T ; L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) ) , \\\\ \\nabla \\times B \\in L ^ { 2 } ( 0 , T ; L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) ) . \\end{aligned} \\end{align*}"}
+{"id": "6357.png", "formula": "\\begin{align*} \\operatorname { E } ( X ^ r ) = & \\frac { 2 \\theta } { \\operatorname { B } ( a , b ) } \\int _ 0 ^ \\infty x ^ { r - 3 } \\exp \\left ( - \\frac { a \\theta } { x ^ 2 } \\right ) \\left [ 1 - \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) \\right ] ^ { b - 1 } \\mathrm { d } x . \\end{align*}"}
+{"id": "9211.png", "formula": "\\begin{align*} \\begin{aligned} | x & ( t ) - z _ 1 ( t ) | \\le \\ , | x ( 0 ) - z _ 1 ( 0 ) | + t \\gamma ^ 2 \\bar { k } _ 3 ( L _ r , \\delta ) \\\\ & + \\gamma \\int _ 0 ^ t L _ r | x ( \\tau ) - z _ 1 ( \\tau ) | + | \\bar { y } ( \\tau ) - z _ 2 ( \\tau ) | d \\tau \\end{aligned} \\end{align*}"}
+{"id": "2243.png", "formula": "\\begin{align*} \\widehat { \\Phi _ t } ( \\xi ) = \\frac { 1 } { t } e ^ { - \\frac { \\pi | \\xi | ^ 2 } { t ^ 2 } } , \\xi \\in \\mathbb { R } . \\end{align*}"}
+{"id": "3896.png", "formula": "\\begin{align*} \\int _ { \\Omega } - \\delta _ N ^ 2 \\left ( \\frac { \\partial K ( z _ { N , i } ) } { \\partial z _ { i , h } } \\nabla V _ { \\delta _ N , Z _ N , i } \\right ) u _ N = O \\left ( \\frac { \\varepsilon _ N ^ 2 } { | \\ln \\varepsilon _ N | ^ { p - 1 } } \\right ) . \\end{align*}"}
+{"id": "786.png", "formula": "\\begin{align*} G ( c ) - G ' ( c _ 0 ) c \\geq G ( c _ 0 ) - G ' ( c _ 0 ) c _ 0 = g ( c _ 0 ) > 0 \\phantom { x } \\Leftrightarrow \\phantom { x } \\frac { G ( c ) } { c } \\geq G ' ( c _ 0 ) \\end{align*}"}
+{"id": "8986.png", "formula": "\\begin{align*} \\dfrac { \\hat { k } _ 1 { ( G ) } } { \\hat { k } _ 1 ( G \\setminus \\sigma ) } = \\dfrac { D _ { 1 , \\ell _ 1 } D _ { 2 , \\ell _ 2 } } { D _ { 1 , \\ell _ 1 - 1 } D _ { 2 , \\ell _ 2 - 1 } } \\end{align*}"}
+{"id": "4514.png", "formula": "\\begin{align*} \\lambda _ i \\neq 0 , i = 1 , \\dots , n . \\end{align*}"}
+{"id": "8981.png", "formula": "\\begin{align*} i _ { 2 2 1 } & = - D _ { 1 , 1 } D _ { 2 , 1 } x _ { 3 , 1 } , & i _ { 2 1 2 } & = - D _ { 1 , 1 } x _ { 2 , 1 } D _ { 3 , 1 } , & i _ { 1 2 2 } & = - x _ { 1 , 1 } D _ { 2 , 1 } D _ { 3 , 1 } , \\\\ i _ { 2 1 1 } & = D _ { 1 , 1 } x _ { 2 , 1 } x _ { 3 , 1 } , & i _ { 1 2 1 } & = x _ { 1 , 1 } D _ { 2 , 1 } x _ { 3 , 1 } , & i _ { 1 1 2 } & = x _ { 1 , 1 } x _ { 2 , 1 } D _ { 3 , 1 } , \\\\ i _ { 1 1 1 } & = - x _ { 1 , 1 } x _ { 2 , 1 } x _ { 3 , 1 } , \\end{align*}"}
+{"id": "6347.png", "formula": "\\begin{align*} c _ n ( a , b ) = \\frac { ( - 1 ) ^ n \\Gamma ( a + b ) } { ( a + n ) \\Gamma ( a ) \\Gamma ( b - n ) n ! } , \\end{align*}"}
+{"id": "1865.png", "formula": "\\begin{align*} \\root n \\of { a _ 1 \\cdot a _ n } = \\overset { ` ` f ( a v e r a g e ) \" \\swarrow } { \\exp ( n ^ { - 1 } \\sum _ { j = 1 } ^ n A _ j ) } \\overset { ( \\ , ) } \\le \\overset { ` ` a v e r a g e ( f ) \" \\swarrow } { n ^ { - 1 } \\sum _ { j = 1 } ^ n \\exp A _ j } = n ^ { - 1 } \\sum _ { j = 1 } ^ n a _ j \\ , . \\end{align*}"}
+{"id": "3957.png", "formula": "\\begin{align*} ( f _ { I _ 1 } ( \\mathbf { x } ) , f _ { I _ 2 } ( \\mathbf { x } ) , \\cdots , f _ { I _ s } ( \\mathbf { x } ) ) = ( f _ { J _ 1 } ( \\mathbf { y } ) , f _ { J _ 2 } ( \\mathbf { y } ) , \\cdots , f _ { J _ s } ( \\mathbf { y } ) ) . \\end{align*}"}
+{"id": "720.png", "formula": "\\begin{align*} \\gamma _ { \\textrm { p } , k } = \\frac { { \\lvert \\mathbf { h } _ { k } ^ { H } \\mathbf { p } _ { \\textrm { p } , k } \\rvert } ^ { 2 } } { \\sum _ { i = 1 , i \\neq k } ^ { K } { \\lvert \\mathbf { h } _ { k } ^ { H } \\mathbf { p } _ { \\textrm { p } , i } \\rvert } ^ { 2 } + \\sigma _ { z } ^ { 2 } } . \\end{align*}"}
+{"id": "4164.png", "formula": "\\begin{align*} \\textup { S t o C } : = \\min \\Big \\{ \\alpha + \\gamma - \\frac { 1 } { 2 } - \\frac { \\alpha } { 2 \\beta } ( \\kappa - r ) , 1 - \\varepsilon \\Big \\} = \\min \\Big \\{ \\frac { \\alpha r } { 2 \\beta } + ( \\gamma - \\frac { 1 } { 2 } ) ^ { + } , 1 - \\varepsilon \\Big \\} . \\end{align*}"}
+{"id": "5506.png", "formula": "\\begin{align*} D ( p ) = \\frac { \\Gamma ( 2 x ) } { \\Gamma ( x + \\frac 1 2 ) ^ 2 \\Gamma ( x + \\frac 3 2 ) } = \\frac { 2 ^ { 2 x - 1 } \\Gamma ( x ) } { \\sqrt { \\pi } \\Gamma ( x + \\frac { 1 } { 2 } ) \\Gamma ( x + \\frac { 3 } { 2 } ) } . \\end{align*}"}
+{"id": "8201.png", "formula": "\\begin{align*} x _ { 2 i , n - j } + x _ { 2 i , n - j + 1 } = a _ j , \\mbox { a n d } \\end{align*}"}
+{"id": "5114.png", "formula": "\\begin{align*} 0 = \\lim _ { m \\to \\infty } \\int _ { \\mathbb { R } ^ { 3 } } | b - b _ m | ^ { 2 } \\dd x = \\lim _ { m \\to \\infty } \\int _ { \\mathbb { R } ^ { 2 } _ { + } } \\left ( | \\nabla ( \\phi - \\phi _ m ) | ^ { 2 } + | G - G _ m | ^ { 2 } \\right ) \\frac { 2 \\pi } { r } \\dd z \\dd r . \\end{align*}"}
+{"id": "47.png", "formula": "\\begin{align*} \\int \\rho ^ { - l } \\psi Z ( v ^ 2 ) = - ( Q - l ) \\int \\rho ^ { - l } \\psi v ^ 2 . \\end{align*}"}
+{"id": "4317.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t \\} } | F _ t | ^ 2 _ h c ( - \\Psi ) + \\int _ { \\{ \\Psi < - t \\} } | \\hat { F } - F _ t | ^ 2 _ h c ( - \\Psi ) \\\\ = & \\int _ { \\{ \\Psi < - t \\} } | \\hat { F } | ^ 2 _ h c ( - \\Psi ) . \\end{align*}"}
+{"id": "2830.png", "formula": "\\begin{align*} \\frac { 1 } { q } = \\left ( \\frac { 1 - \\kappa } { p _ 1 } + \\frac { \\kappa } { p _ 2 } \\right ) - \\frac { \\kappa s - r } { n } , \\ , \\ , \\mbox { a n d } \\ , \\ , r \\le \\kappa s . \\end{align*}"}
+{"id": "7962.png", "formula": "\\begin{align*} g _ d ( \\ell ) = \\begin{cases} \\ell , & d = 1 , \\\\ \\log \\ell , & d = 2 , \\\\ 1 , & d \\geq 3 . \\end{cases} \\end{align*}"}
+{"id": "4379.png", "formula": "\\begin{align*} & \\lim _ { m \\to + \\infty } \\int _ { X _ j } v '' _ { \\epsilon } ( \\Psi _ m ) | f F ^ { 1 + \\delta } | ^ 2 _ { h _ { m ' } } e ^ { - \\phi - \\delta M _ { \\eta _ m } } \\\\ = & \\int _ { X _ j } v '' _ { \\epsilon } ( \\Psi ) | f F ^ { 1 + \\delta } | ^ 2 _ { h _ { m ' } } e ^ { - u ( - v _ { \\epsilon } ( \\Psi ) ) - \\delta \\max { \\{ \\psi + T , 2 \\log | F | \\} } } . \\end{align*}"}
+{"id": "6700.png", "formula": "\\begin{align*} s _ \\ast \\beta '' ( \\gamma , \\xi ) = \\beta '' ( s \\gamma s ^ { - 1 } , s \\xi ) ( \\gamma \\in \\Gamma ' ) . \\end{align*}"}
+{"id": "7161.png", "formula": "\\begin{align*} \\Delta _ { g } f = g ^ { i j } \\Bigl ( \\frac { \\partial ^ 2 f } { \\partial x _ i \\partial x _ j } - \\Gamma ^ k _ { i j } \\frac { \\partial f } { \\partial x _ k } \\Bigr ) , f \\in C ^ { \\infty } ( \\Omega ) . \\end{align*}"}
+{"id": "5063.png", "formula": "\\begin{align*} \\begin{aligned} L ^ { p } ( \\Omega ) & = L ^ { p } _ { \\sigma } ( \\Omega ) \\oplus G ^ { p } ( \\Omega ) , 1 < p < \\infty , \\\\ L ^ { p } _ { \\sigma } ( \\Omega ) & = \\{ u \\in L ^ { p } ( \\Omega ) \\ | \\ \\nabla \\cdot u = 0 \\ \\textrm { i n } \\ \\Omega , \\ u \\cdot n = 0 \\ \\textrm { o n } \\ \\partial \\Omega \\} , \\\\ G ^ { p } ( \\Omega ) & = \\{ \\nabla q \\in L ^ { p } ( \\Omega ) \\ | \\ q \\in L ^ { 1 } _ { \\textrm { l o c } } ( \\Omega ) \\} , \\end{aligned} \\end{align*}"}
+{"id": "5088.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B \\cdot B \\dd x = \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B _ 0 \\cdot B _ 0 \\dd x , \\end{align*}"}
+{"id": "2383.png", "formula": "\\begin{align*} C _ { { 3 i + 2 } } ( \\mathcal { H } _ { \\ast } ) = \\mathrm { I m } \\ , \\partial ' _ i \\oplus s _ { _ { 3 i + 2 } } ( \\mathrm { I m } \\ , \\partial _ i ) = s _ { _ { 3 i + 2 } } ( \\mathrm { I m } \\ , \\partial _ i ) . \\end{align*}"}
+{"id": "5106.png", "formula": "\\begin{align*} \\phi ( z , r ) \\longmapsto \\varphi ( y ) = \\frac { \\phi ( z , r ) } { r ^ { 2 } } , y = { } ^ { t } ( y _ 1 , y ' ) , \\ y _ 1 = z , \\ | y ' | = r , \\end{align*}"}
+{"id": "978.png", "formula": "\\begin{align*} g _ i : = \\left | G _ i \\cap \\bigcup _ { j = 1 } ^ { i - 1 } G _ j \\right | . \\end{align*}"}
+{"id": "4340.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi < - t \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h c ( - \\Psi ) = & \\int _ { \\{ \\Psi < - t \\} } | \\tilde { F } | ^ 2 _ h c ( - \\Psi ) \\\\ + & \\int _ { \\{ \\Psi < - t \\} } | F _ { t _ 0 , \\tilde { c } } - \\tilde { F } | ^ 2 _ h c ( - \\Psi ) \\end{align*}"}
+{"id": "5131.png", "formula": "\\begin{align*} h = 2 \\mu \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi - \\phi _ \\infty ) _ { + } ^ { 2 } \\frac { 1 } { r ^ { 2 } } \\dd x . \\end{align*}"}
+{"id": "4861.png", "formula": "\\begin{align*} d \\phi = \\frac { \\int _ { \\mathbb R } e ^ { - \\psi } ( d _ x { \\psi } ) d y } { \\int _ { \\mathbb R } e ^ { - \\psi } d y } \\end{align*}"}
+{"id": "3853.png", "formula": "\\begin{align*} \\partial _ t \\gamma = \\frac { c } { 4 \\pi } | \\ln \\varepsilon | ( \\partial _ s \\gamma \\times \\partial _ { s s } \\gamma ) = \\frac { c \\bar { K } } { 4 \\pi } | \\ln \\varepsilon | \\mathbf { b } _ { \\gamma ( t ) } , \\end{align*}"}
+{"id": "3960.png", "formula": "\\begin{align*} \\mathbf { c } ^ { ( 1 ) } = ( \\underbrace { 1 , 1 , \\cdots , 1 } _ { 2 k } , \\ast , \\ast , \\cdots , \\ast ) . \\end{align*}"}
+{"id": "2076.png", "formula": "\\begin{align*} \\mathcal { I } - d ( t _ 0 , H \\setminus D _ { t _ 0 } ) = 0 , \\ \\mbox { s i n c e } \\ \\mathcal { I } - d ( t _ 0 , D _ { t _ 0 } ) = 1 \\end{align*}"}
+{"id": "6193.png", "formula": "\\begin{align*} E _ 0 = 4 \\kappa ( L + 1 ) ^ 2 - \\frac { Q ^ 2 } { 4 ( L + 1 ) ^ 2 } , E _ 1 = E ' _ 0 = 4 \\kappa ( L + 2 ) ^ 2 - \\frac { Q ^ 2 } { 4 ( L + 2 ) ^ 2 } . \\end{align*}"}
+{"id": "6549.png", "formula": "\\begin{align*} d ( x , y ) = d ( y , x ) = d ( x , x ) = d ( y , y ) \\implies x = y . \\end{align*}"}
+{"id": "7511.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } r _ j \\int _ { B _ { r _ j } } r ^ { 3 - n } | \\nabla u | ^ 2 \\leq \\bigg ( \\sum _ { j = 0 } ^ { \\infty } r _ j \\bigg ) \\int _ { B _ { 1 } } r ^ { 3 - n } | \\nabla u | ^ 2 = 2 \\int _ { B _ { 1 } } r ^ { 3 - n } | \\nabla u | ^ 2 . \\end{align*}"}
+{"id": "8107.png", "formula": "\\begin{align*} u \\ \\mbox { i s b o u n d e d i n } \\ B _ { R } \\ \\mbox { a n d } \\ \\lim \\limits _ { | x | \\nearrow R } v ( x ) = \\infty \\end{align*}"}
+{"id": "3138.png", "formula": "\\begin{align*} T \\circ \\beta & = \\alpha \\circ T , \\\\ T ( a ) \\cdot T ( b ) & = T \\big ( \\ell ( T ( a ) ) b + r ( T ( b ) ) a \\big ) , \\quad \\forall \\ a , b \\in V . \\end{align*}"}
+{"id": "1998.png", "formula": "\\begin{align*} F ( x ) = - \\frac { ( 2 x + 2 ) ^ 2 - e ^ { - 4 } } { 4 x + 4 } , x \\ne - 1 . \\end{align*}"}
+{"id": "7224.png", "formula": "\\begin{align*} K _ Y = ( K _ X + A _ X ) \\vert _ Y = ( 2 - n ) A _ X \\vert _ Y \\end{align*}"}
+{"id": "3354.png", "formula": "\\begin{align*} \\alpha = \\lambda ^ 0 < \\dots < \\lambda ^ l = \\beta , \\end{align*}"}
+{"id": "2083.png", "formula": "\\begin{align*} I _ k \\cap ( H \\setminus P ) = \\bigcup _ { n = 0 } ^ { \\infty } ( I _ k \\cap ( H _ n \\setminus K ) \\cap ( \\bigcap _ { m = 1 } ^ { \\infty } ( H _ n \\setminus P _ m ) ) ) \\end{align*}"}
+{"id": "5806.png", "formula": "\\begin{align*} \\nabla _ X \\psi = - \\frac { 1 } { 2 } S ( X ) \\cdot \\nu _ 1 \\cdot \\psi \\end{align*}"}
+{"id": "7779.png", "formula": "\\begin{align*} | s _ { h , q } - s _ h | \\le \\frac { d \\theta ^ { h } } { \\theta ^ { q } - 1 } ~ { \\rm f o r } ~ h = 0 , 1 , \\dots , q - 1 , \\end{align*}"}
+{"id": "8702.png", "formula": "\\begin{align*} a _ 1 ( F ) > \\sum _ { i = 1 } ^ { \\lfloor ( d - 1 ) / 2 \\rfloor } a _ i ( F ' ) + \\frac { 1 } { 2 } a _ { d / 2 } ( F ' ) \\end{align*}"}
+{"id": "1921.png", "formula": "\\begin{align*} \\small \\nabla ^ { ( m , n ) } \\eta ( z , y ) [ h _ 1 ] ^ m [ h _ 2 ] ^ n = \\tfrac { ( p - 1 ) ! } { ( p - m ) ! } \\cdot \\begin{cases} \\sum _ { i = 1 } ^ d ( z ^ { ( i ) } ) ^ { p - m } ( h _ 1 ^ { ( i ) } ) ^ m h _ 2 ^ { ( i ) } , & \\textnormal { i f } n = 1 , \\\\ \\sum _ { i = 1 } ^ d ( z ^ { ( i ) } ) ^ { p - m } y ^ { ( i ) } ( h _ 1 ^ { ( i ) } ) ^ m , & \\textnormal { i f } n = 0 , \\\\ 0 , & \\textnormal { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "1151.png", "formula": "\\begin{align*} d X _ { t } ^ { i } = \\left ( \\frac { 1 } { n - 1 } \\sum _ { j \\neq i } b \\left ( t , X _ { t } ^ { i } , X _ { t } ^ { j } \\right ) \\right ) d t + d W _ { t } ^ { i } , i = 1 \\cdots n , \\end{align*}"}
+{"id": "8502.png", "formula": "\\begin{align*} e ^ { 2 i ( c _ - ( x ) + c ) } u ( x ) = \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } z ^ { - 1 } \\bar { r } _ 1 ( z ) e ^ { - 2 i z x } M _ { + , 1 1 } ( x ; z ) \\mathrm { d } z . \\end{align*}"}
+{"id": "648.png", "formula": "\\begin{align*} & \\int _ { B _ { R / 2 } } \\vert \\nabla u \\vert ^ 2 d x \\lesssim \\left ( \\int _ { \\mathcal { C } ( R / 2 , R ) } \\vert \\nabla u \\vert ^ { \\frac { p } { 2 } } d x + \\int _ { \\mathcal { C } ( R / 2 , R ) } \\vert u \\vert ^ { p } d x \\right ) ^ { \\frac { 2 } { p } } R ^ { 2 - \\frac { 9 } { p } } \\left ( \\int _ { \\mathcal { C } ( R / 2 , R ) } \\vert u \\vert ^ { p } d x \\right ) ^ { \\frac { 1 } { p } } \\end{align*}"}
+{"id": "1628.png", "formula": "\\begin{align*} Z ^ { i , j } _ \\infty = \\cap _ r \\ , Z _ r ^ { i , j } , \\end{align*}"}
+{"id": "7413.png", "formula": "\\begin{align*} \\beta ^ t _ { a , b } ( z ) \\gamma ^ t _ { a , b } ( w ) = ( z - w ) ^ { - 1 } \\end{align*}"}
+{"id": "3875.png", "formula": "\\begin{align*} \\delta ^ { \\frac { 2 } { p - 1 } } s _ \\delta ^ { - \\frac { 2 } { p - 1 } } \\phi ' ( 1 ) = a / \\ln \\frac { s _ \\delta } { R } . \\end{align*}"}
+{"id": "7710.png", "formula": "\\begin{align*} f _ i ( r ) : = \\Big \\lfloor \\frac { r } { 1 + ( 1 + \\tilde \\varepsilon ) ^ { - i } } \\Big \\rfloor , r \\in \\N . \\end{align*}"}
+{"id": "4702.png", "formula": "\\begin{align*} ( U _ { 1 1 } , \\ldots , U _ { 1 n } ) \\overset { d } { = } \\frac { 1 } { \\sqrt { N _ { 1 1 } ^ 2 + \\cdots + N _ { 1 n } ^ 2 } } ( N _ { 1 1 } , \\ldots , N _ { 1 n } ) \\ , , \\end{align*}"}
+{"id": "5833.png", "formula": "\\begin{align*} \\rho c _ v \\partial _ { t _ 1 } T = { \\bar F } ^ { ( 1 ) } , \\end{align*}"}
+{"id": "1413.png", "formula": "\\begin{align*} \\widetilde G _ { n } ( x , y ) & = \\mathrm { d e t } \\left ( f _ n ^ { ( 1 ) } ( y _ j - x _ i ) \\right ) _ { i , j = 1 } ^ d \\\\ & = \\mathrm { d e t } \\left ( \\mathrm { e } ^ { \\lambda ( y _ j - x _ i ) } ( 1 + \\lambda ) ^ { - n } f _ n ^ { ( 1 + \\lambda ) } ( y _ j - x _ i ) \\right ) _ { i , j = 1 } ^ d \\\\ & = \\mathrm { e } ^ { \\lambda \\sum _ { i = 1 } ^ d ( y _ i - x _ i ) - d n \\ln ( 1 + \\lambda ) } \\widetilde G _ { n } ^ { ( 1 + \\lambda ) } ( x , y ) . \\end{align*}"}
+{"id": "2761.png", "formula": "\\begin{align*} N _ m ( x ) \\leq | h _ m ( x ) | ^ { 3 - 2 \\beta } = | h _ m ( x ) | ^ \\rho . \\end{align*}"}
+{"id": "209.png", "formula": "\\begin{align*} \\lambda _ S ( G ) = \\lim _ { n \\to \\infty } \\frac { | B _ S ( n + 1 ) | } { | B _ S ( n ) | } . \\end{align*}"}
+{"id": "1187.png", "formula": "\\begin{align*} \\int _ { T _ { i - 1 } } ^ { T _ i } \\frac { C _ T } { t ^ { \\frac { d } { p } } } \\| f _ t \\| ^ 2 _ { L ^ p ( \\mathbb { R } ^ d ) } = 0 . 1 1 \\leq i \\leq m - 1 , \\int _ { T _ { m - 1 } } ^ { T m } \\frac { C _ T } { t ^ { \\frac { d } { p } } } \\| f _ t \\| _ { L ^ p ( \\mathbb { R } ^ d ) } ^ 2 \\leq 0 . 1 . \\end{align*}"}
+{"id": "9299.png", "formula": "\\begin{align*} d _ 0 | q | ^ 2 \\wedge d _ 1 | q | ^ 2 = 4 \\sum _ { l = 0 } ^ { n - 1 } | q _ l | ^ 2 \\omega ^ l \\wedge \\omega ^ { n + l } + \\sum _ { | j - k | \\not = n } a _ { j k } \\omega ^ j \\wedge \\omega ^ k \\end{align*}"}
+{"id": "439.png", "formula": "\\begin{align*} \\omega \\kappa \\begin{pmatrix} 1 \\\\ \\vdots \\\\ \\theta ^ { n - 1 } \\end{pmatrix} = v , \\end{align*}"}
+{"id": "9331.png", "formula": "\\begin{align*} m = | M _ 1 | + | M _ 2 | + | M _ 3 | + | M _ 4 | \\geq C k \\log ^ 3 n . \\end{align*}"}
+{"id": "576.png", "formula": "\\begin{align*} & R e \\left ( \\omega ( s ) e ^ { i \\theta } \\right ) \\overset { \\eqref { c o n c a t e n a t e d _ o m e g a } , \\eqref { c o n c a t e n a t e d _ s } } { = } - s R e \\left ( e ^ { i ( \\theta - \\theta _ 2 / 2 - \\theta _ 1 / 2 ) } \\right ) = \\\\ & - s \\cos ( \\theta - \\theta _ 2 / 2 - \\theta _ 1 / 2 ) \\overset { \\eqref { c o n c a t e n a t e d _ s } , \\eqref { c o n c a t e n a t e d _ t h e t a } , \\eqref { a l p h a } } { \\leq } - s \\cos \\left ( \\alpha \\right ) . \\end{align*}"}
+{"id": "7581.png", "formula": "\\begin{align*} \\mu _ N ( E ) = \\dfrac { | A \\cap \\Phi _ N | } { | \\Phi _ N | } \\end{align*}"}
+{"id": "4064.png", "formula": "\\begin{align*} Z _ n = \\delta ( u _ n ) + r _ n N _ n , \\end{align*}"}
+{"id": "2843.png", "formula": "\\begin{align*} j _ { x , x + 1 } = \\mathcal G _ t \\frak f _ x - \\frac 1 { 4 \\gamma } \\nabla \\mathfrak F _ x , \\end{align*}"}
+{"id": "4304.png", "formula": "\\begin{align*} & \\int _ { M } | \\tilde { F } - ( 1 - b _ { t _ 0 , B } ( \\Psi _ 1 ) ) f F ^ { 1 + \\delta } | ^ 2 _ { \\tilde { h } } e ^ { v _ { t _ 0 , B } ( \\Psi _ 1 ) - \\delta \\tilde { M } } c ( - v _ { t _ 0 , B } ( \\Psi _ 1 ) ) \\\\ \\le & \\left ( \\frac { 1 } { \\delta } c ( T _ 1 ) e ^ { - T _ 1 } + \\int _ { T _ 1 } ^ { t _ 0 + B } c ( t ) e ^ { - t } d t \\right ) \\int _ M \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi _ 1 < - t _ 0 \\} } | f F | ^ 2 _ { \\tilde { h } } . \\end{align*}"}
+{"id": "7820.png", "formula": "\\begin{align*} ( I - Q ) F ^ q ( v + \\psi ( v , p ) , p ) = 0 \\end{align*}"}
+{"id": "5459.png", "formula": "\\begin{align*} \\begin{pmatrix} \\theta ^ 2 & \\theta \\cdot \\sigma \\\\ \\theta \\cdot \\sigma & \\sigma ^ 2 \\end{pmatrix} = \\begin{pmatrix} 2 & 0 \\\\ 0 & - 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "5220.png", "formula": "\\begin{align*} T \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } ^ { 2 } \\dd x = \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ + ^ { 2 } \\dd x \\dd t \\leq \\liminf _ { j \\to \\infty } \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { j , + } ^ { 2 } \\dd x \\dd t \\leq T \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ { 0 , + } ^ { 2 } \\dd x . \\end{align*}"}
+{"id": "4796.png", "formula": "\\begin{align*} a _ n ( u _ n , v _ n ) : = \\sum _ { K \\in { \\mathcal T } _ n } \\int _ K \\nabla u _ n \\cdot \\nabla v _ n d K , u _ n , v _ n \\in X _ n . \\end{align*}"}
+{"id": "899.png", "formula": "\\begin{align*} \\Sigma \\big ( X , q \\big ) = X \\frac { \\lambda ( q ) } { q } + \\mathcal { O } \\big ( \\lambda ( q ) \\big ) \\ , . \\end{align*}"}
+{"id": "9255.png", "formula": "\\begin{align*} \\Delta u _ 1 \\wedge \\dots \\wedge \\Delta u _ n = n ! \\det ( A _ 1 , A _ 2 , \\dots , A _ n ) \\Omega _ { 2 n } . \\end{align*}"}
+{"id": "2605.png", "formula": "\\begin{align*} R \\cdot R ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ; u , J u ) = \\frac { 1 } { 2 } \\ , L ( p , \\bar { \\pi } ^ h ) \\ , Q ^ c ( g , R ) ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ; u , J u ) , \\end{align*}"}
+{"id": "9212.png", "formula": "\\begin{align*} \\begin{aligned} | h ( x + & \\delta u ( \\tau ) ) - h ( x _ a + \\delta u ( \\tau ) ) | \\le L _ r | x - x _ a | \\\\ & \\le L _ r ( | x - z _ 1 | + \\gamma | \\epsilon _ 1 ( \\eta _ a , \\tau ) | ) \\\\ & \\le L _ r ( | x - z _ 1 | + \\gamma \\delta L _ r ( 3 + L _ r ) ) . \\end{aligned} \\end{align*}"}
+{"id": "8913.png", "formula": "\\begin{align*} \\lambda _ { 3 } & = h _ { 1 2 3 } - h _ { 1 2 } = \\log \\left ( \\frac { 2 1 6 } { \\zeta } \\right ) , \\\\ \\lambda _ { 1 2 3 ' } & = h _ 3 - \\lambda _ { 3 } = \\log \\left ( \\frac { \\zeta } { 3 6 } \\right ) , \\\\ \\lambda _ { 1 } = \\lambda _ 2 & = h _ { 1 } - \\lambda _ { 1 2 } - \\lambda _ { 1 2 3 ' } = \\log 4 . \\end{align*}"}
+{"id": "6900.png", "formula": "\\begin{align*} d X ^ i _ t = \\alpha _ i ( t , X _ t ) d t + d B ^ i _ t , X ^ i _ 0 = 0 , \\ \\ i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "3956.png", "formula": "\\begin{align*} ( \\mathbf { x } , - \\mathbf { y } ) M _ { I J } = { \\bf 0 } , \\end{align*}"}
+{"id": "7471.png", "formula": "\\begin{align*} - L u = f ( u ) \\Omega , \\end{align*}"}
+{"id": "4299.png", "formula": "\\begin{align*} \\inf \\Bigg \\{ \\int _ { \\{ \\Psi < - t \\} } | \\tilde { f } | ^ 2 _ h c ( - \\Psi ) : \\tilde { f } \\in H ^ 0 ( \\{ \\Psi < - t \\} , \\mathcal { O } ( K _ M \\otimes E ) ) \\\\ \\& \\ , ( \\tilde { f } - f ) _ { z _ 0 } \\in \\mathcal { O } ( K _ M ) _ { z _ 0 } \\otimes J _ { z _ 0 } , z _ 0 \\in Z _ 0 \\Bigg \\} \\end{align*}"}
+{"id": "2803.png", "formula": "\\begin{align*} \\mathbb { E } [ \\langle Z _ { \\lambda } ^ { S _ { n } } , f \\rangle | \\sigma ( \\Phi ^ { S _ { n } , \\tilde { S } _ { n , m } } ) ] = \\langle Y _ { m } ^ { S _ { n } } , f \\rangle . \\end{align*}"}
+{"id": "3149.png", "formula": "\\begin{align*} { A \\choose B } : = \\frac { B ( - 1 ) } { q } J ( A , \\overline { B } ) , \\end{align*}"}
+{"id": "1013.png", "formula": "\\begin{align*} \\ell ( x _ 0 ) \\geq \\ell ( x _ 1 ) = \\ell ( x _ L x _ 0 x _ R ) \\geq \\ell ( x _ 0 x _ R ) - \\ell ( x _ L ) = \\ell ( x _ 0 ) + \\ell ( x _ R ) - \\ell ( x _ L ) . \\end{align*}"}
+{"id": "6885.png", "formula": "\\begin{align*} u ( \\infty ) = \\Re \\ , \\psi ( z _ 0 ) \\end{align*}"}
+{"id": "3951.png", "formula": "\\begin{align*} \\zeta ( x ) : = \\begin{cases} \\frac { \\xi ( x ) } { p ( x ) } , & \\{ p ( x ) \\ne 0 \\} , \\\\ f _ { - } ' ( \\overline { y } ( x ) ) , & \\{ p ( x ) = 0 \\} , \\end{cases} \\end{align*}"}
+{"id": "4415.png", "formula": "\\begin{align*} H ^ 2 ( \\mathbb { D } ) = \\{ f = \\sum _ { n = 0 } ^ { \\infty } a _ n z ^ n \\mid ( a _ n ) _ { n \\in \\mathbb { N } } \\in \\ell ^ 2 \\} . \\end{align*}"}
+{"id": "3214.png", "formula": "\\begin{align*} & : , ~ \\forall n \\in \\N , ~ \\omega _ n = \\alpha _ 1 \\beta _ 1 \\dots \\alpha _ { n - 1 } \\beta _ { n - 1 } \\alpha _ n , \\\\ & : , ~ \\forall n \\in \\N , ~ \\omega _ n = \\beta _ 1 \\alpha _ 1 \\dots \\beta _ { n } \\alpha _ { n } . \\end{align*}"}
+{"id": "553.png", "formula": "\\begin{align*} h ( h e _ n h ) ( h g e _ n h g ) \\cdots ( h g ^ { n - 1 } e _ n h g ^ { n - 1 } ) = ( h e _ n h ) ( h g e _ n h g ) \\cdots ( h g ^ { n - 1 } e _ n h g ^ { n - 1 } ) , \\end{align*}"}
+{"id": "5895.png", "formula": "\\begin{align*} \\widetilde { X } = \\widehat { X } = \\overline { X } , \\widetilde { \\mathbb { P } } \\otimes d t . \\end{align*}"}
+{"id": "8355.png", "formula": "\\begin{align*} N ( x ; k ) : = \\begin{cases} e ^ { i c _ + ( x ) \\sigma _ 3 } \\left ( \\frac { \\psi _ 1 ^ - ( x ; k ) } { a ( k ) } , \\psi _ 2 ^ + ( x ; k ) \\right ) , \\ ; k \\in D ^ + , \\\\ [ 5 p t ] e ^ { i c _ + ( x ) \\sigma _ 3 } \\left ( \\psi _ 1 ^ + ( x ; k ) , \\frac { \\psi _ 2 ^ - ( x ; k ) } { \\overline { a ( \\bar { k } ) } } \\right ) , \\ ; k \\in D ^ - , \\end{cases} \\end{align*}"}
+{"id": "3658.png", "formula": "\\begin{align*} M _ x ^ 2 & = \\frac { \\omega } { 3 L _ { x y } } \\left ( \\Psi ^ 0 + \\frac { 4 } { ( 1 - \\theta ) ^ 2 } ( \\delta _ x + \\delta _ y ) + \\frac { \\Sigma ^ 2 } { 2 ( 1 - \\theta ) } \\right ) , \\\\ M _ y ^ 2 & = \\frac { 1 } { 4 L _ { x y } \\omega } \\left ( \\Psi ^ 0 + \\frac { 4 } { ( 1 - \\theta ) ^ 2 } ( \\delta _ x + \\delta _ y ) + \\frac { \\Sigma ^ 2 } { 2 ( 1 - \\theta ) } \\right ) , \\end{align*}"}
+{"id": "6313.png", "formula": "\\begin{align*} \\frac { 1 } { \\rho _ K } \\| x ^ { K + 1 } - x ^ K \\| \\leq \\frac { r _ 0 + \\theta } { \\rho _ 0 \\zeta ^ K } \\overset { \\eqref { K - s p } } { \\leq } \\frac { \\varepsilon } { 2 } , \\eta _ K = \\eta _ 0 \\sigma ^ K \\overset { \\eqref { K - s p } } { \\leq } \\frac { \\varepsilon } { 2 } , \\end{align*}"}
+{"id": "6143.png", "formula": "\\begin{align*} \\deg ( ( g _ { m , ( m ' ; l ' ) } ^ \\epsilon ) _ { i j } ) = ( - 1 ) ^ j i l + i j ( d - 2 ) + \\frac { ( - 1 ) ^ i } { 2 } \\left ( 1 - \\left ( - 1 \\right ) ^ j \\epsilon \\right ) \\end{align*}"}
+{"id": "679.png", "formula": "\\begin{align*} \\left \\langle l _ 1 , m _ 1 \\right | e ^ { i k \\phi } \\sin ( n \\theta ) \\left | l _ 2 , m _ 2 \\right \\rangle = ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
+{"id": "3776.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\geq p } C F ^ { k } ( H _ { t } ) : = \\lbrace x \\in C F ^ { k } ( H _ { t } ) \\mid \\mathcal { A } ( x ) \\geq p \\rbrace \\end{align*}"}
+{"id": "8192.png", "formula": "\\begin{align*} x _ { 1 , 2 j - 1 } & + x _ { 1 , 2 j } + x _ { m , ( n + 1 ) - ( 2 j - 1 ) } + x _ { m , ( n + 1 ) - 2 j } \\\\ & = x _ { 1 , 2 j - 1 } + x _ { 1 , 2 j } + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { m , 2 j - 1 } \\bigr ) + \\bigl ( \\tfrac { 1 } { 2 } S - x _ { m , 2 j } \\bigr ) \\\\ & = \\bigl ( n _ 1 m + 1 \\bigr ) + S - \\bigl ( n _ 1 m + 1 \\bigr ) = S . \\end{align*}"}
+{"id": "1698.png", "formula": "\\begin{align*} l ' _ { k } - l ' _ { k - 2 } = \\frac { 2 ( 2 k - 1 ) } { h } l _ { k - 1 } , \\forall k \\geq 2 , \\end{align*}"}
+{"id": "5047.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla \\times U = f U , \\nabla \\cdot U & = 0 \\textrm { i n } \\ \\mathbb { R } ^ { 3 } , \\\\ U & \\to B _ { \\infty } \\textrm { a s } \\ | x | \\to \\infty . \\end{aligned} \\end{align*}"}
+{"id": "1024.png", "formula": "\\begin{align*} r _ a ( b ) = \\left ( s _ \\alpha ( \\beta ) , - \\chi _ K ( \\beta ) + \\langle \\alpha ^ \\vee , \\beta \\rangle \\chi _ K ( \\alpha ) \\right ) , \\end{align*}"}
+{"id": "3725.png", "formula": "\\begin{align*} ( T _ h ) ^ i _ { j k } = \\tfrac { 1 } { 2 } ( h ^ i _ { j ; k } + h ^ i _ { k ; j } - h _ { j k ; } ^ { \\ ; \\ ; \\ ; \\ ; \\ ; i } ) \\end{align*}"}
+{"id": "5268.png", "formula": "\\begin{align*} k _ \\theta \\cdot f _ - & = \\frac { 1 } { \\sqrt { 2 } } \\left ( s ( \\theta ) , c ( \\theta ) , s ( \\theta ) , c ( \\theta ) \\right ) \\\\ k _ \\theta \\cdot f _ + & = \\frac { 1 } { ( 1 0 8 ) ^ { \\frac { 1 } { 4 } } } \\left ( s ( 3 \\theta ) , 3 c ( 3 \\theta ) , - 3 s ( 3 \\theta ) , - c ( 3 \\theta ) \\right ) . \\end{align*}"}
+{"id": "250.png", "formula": "\\begin{align*} \\gamma \\cdot \\alpha & = ( \\alpha _ 1 , \\ldots , \\gamma \\alpha _ i , \\ldots , \\alpha _ k ) , \\\\ \\sigma \\cdot \\alpha & = ( \\alpha _ { \\sigma ^ { - 1 } ( 1 ) } , \\alpha _ { \\sigma ^ { - 1 } ( 2 ) } , \\ldots , \\alpha _ { \\sigma ^ { - 1 } ( k ) } ) . \\end{align*}"}
+{"id": "5070.png", "formula": "\\begin{align*} \\int _ { \\Omega } A \\cdot B \\dd x = \\sum _ { j } \\frac { 1 } { \\lambda _ j } c _ j ^ { 2 } = \\sum _ { j } \\frac { 1 } { f _ j ^ { + } } c _ j ^ { 2 } + \\sum _ { j } \\frac { 1 } { f _ j ^ { - } } c _ j ^ { 2 } . \\end{align*}"}
+{"id": "8231.png", "formula": "\\begin{align*} H ^ * ( \\mu ) ( \\xi ) = & \\lim _ { z \\to \\xi } \\left ( \\frac { 1 } { 1 - b ( z ) } - \\frac { 1 } { b ' ( \\xi ) ( 1 - \\bar { \\xi } z ) } \\right ) + C \\\\ = & \\lim _ { z \\to \\xi } \\frac { 1 } { ( 1 - \\bar { \\xi } z ) } \\left ( \\frac { 1 - \\bar { \\xi } z } { 1 - b ( z ) } - \\frac { 1 } { b ' ( z ) } \\right ) + C \\\\ = & - \\frac { b '' ( \\xi ) } { b ' ( \\xi ) ^ 2 } + C \\ , \\end{align*}"}
+{"id": "1857.png", "formula": "\\begin{align*} \\begin{aligned} & \\ln \\big ( \\int _ { B _ { \\rho _ 2 } ( x _ 0 ) - B _ 1 ( x _ 0 ) } \\big ( \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) - 1 \\big ) \\ , d v \\big ) \\\\ & \\ \\ - \\ln \\big ( \\int _ { B _ { \\rho _ 1 } ( x _ 0 ) - B _ 1 ( x _ 0 ) } \\big ( \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) - 1 \\big ) \\ , d v \\big ) \\\\ & \\geq ( 1 + \\delta ) \\ln \\rho _ 2 - ( 1 + \\delta ) \\ln \\rho _ 1 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "2072.png", "formula": "\\begin{align*} m ^ { \\star } ( C _ \\mu \\cap ( \\bigcup _ { k \\in \\mathbb { N } } I _ k ) ) & = m ^ { \\star } ( \\bigcup _ { k \\in \\mathbb { N } } ( C _ \\mu \\cap I _ k ) \\leq \\sum _ { k \\in \\mathbb { N } } m ^ { \\star } ( C _ \\mu \\cap I _ k ) \\leq \\sum _ { k \\in \\mathbb { N } } m ( H \\cap I _ k ) \\\\ & \\leq ( 1 - \\mu ) \\sum _ { k \\in \\mathbb { N } } m ( I _ k ) < ( 1 - \\mu ) m ( G ) < m ^ { \\star } ( C _ { \\mu } ) \\end{align*}"}
+{"id": "4314.png", "formula": "\\begin{align*} \\int _ { U _ 0 \\cap \\{ \\Psi < - T _ 1 - 2 \\} } | \\tilde { F } _ 0 - f _ 0 F ^ 2 | ^ 2 _ h e ^ { - \\varphi _ 1 - \\Psi _ 1 } < + \\infty . \\end{align*}"}
+{"id": "5018.png", "formula": "\\begin{align*} S t _ { D N D C } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( z , z ) - z | | _ { \\infty } = 0 , \\end{align*}"}
+{"id": "8046.png", "formula": "\\begin{align*} \\mathbf { y } _ k = \\mathbf { H } _ k \\sum _ { i \\in \\mathcal { M } _ k } a _ i s _ i \\mathbf { p } _ i + \\mathbf { H } _ k \\sum \\limits _ { \\substack { l = 1 \\\\ l \\neq k } } ^ { K } \\sum \\limits _ { j \\in \\mathcal { M } _ l } a _ j s _ j \\mathbf { p } _ j + \\mathbf { n } _ k \\end{align*}"}
+{"id": "7590.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } T ^ { c _ 1 ( n ) } ( x _ 0 ) = x _ 1 . \\end{align*}"}
+{"id": "1917.png", "formula": "\\begin{align*} T _ { \\textnormal { i n n e r } } = \\left \\lceil \\left ( \\tfrac { 2 ^ { p + 1 } ( 5 p - 2 ) } { p ! } \\tfrac { L D ^ { p - 1 } } { \\mu } \\right ) ^ { \\frac { 2 } { p + 1 } } \\right \\rceil , \\end{align*}"}
+{"id": "5496.png", "formula": "\\begin{align*} h ( p ) = \\sum _ k \\lambda _ k a _ k ^ p \\end{align*}"}
+{"id": "3249.png", "formula": "\\begin{align*} \\langle \\rho _ { \\Sigma _ s } ( m ) , h _ 1 \\otimes . . . \\otimes h _ m \\rangle _ { \\mathcal H ^ m } = \\sum _ { p \\in \\mathcal P ( m ) } \\prod _ { ( x , y ) \\in p } \\langle \\Sigma h _ x , h _ y \\rangle , \\end{align*}"}
+{"id": "3319.png", "formula": "\\begin{align*} | S | | \\sigma ^ 0 | + ( m - | S | ) | \\tau | = | S | | \\sigma ^ 0 | + ( m - | S | ) | \\tau ' | . \\end{align*}"}
+{"id": "6388.png", "formula": "\\begin{align*} \\int _ { T } \\mathrm { d i v } ( { \\boldsymbol \\psi } ^ { \\mathrm { R T } _ 1 } _ { j } ) \\varphi _ k \\ , d \\boldsymbol x = \\frac { 3 \\delta _ { j k } - 1 } { 2 4 } j , k = 1 , 2 , 3 . \\end{align*}"}
+{"id": "3640.png", "formula": "\\begin{align*} Q _ { i - 1 } ( x , w x ) = \\frac { Q _ i ( x , w x ) - Q _ i ( x , s _ { \\alpha _ { i - 1 } } w x ) } { \\alpha _ i ( w x ) } . \\end{align*}"}
+{"id": "5814.png", "formula": "\\begin{align*} f _ i ^ { \\left ( { e q } \\right ) } \\left ( { { \\bf { x } } , t } \\right ) = { { \\hat w } _ i } \\rho \\left [ { 1 + \\frac { { { { \\bf { u } } ^ { e q } } \\cdot { { \\bf { c } } _ i } } } { { \\hat c _ s ^ 2 } } + \\frac { { { { \\left ( { { { \\bf { u } } ^ { e q } } \\cdot { { \\bf { c } } _ i } } \\right ) } ^ 2 } } } { { 2 \\hat c _ s ^ 4 } } - \\frac { { \\left | { { { \\bf { u } } ^ { e q } } } \\right | } } { { 2 \\hat c _ s ^ 2 } } } \\right ] , \\end{align*}"}
+{"id": "6891.png", "formula": "\\begin{align*} y ^ * = t \\int _ { \\R ^ n } \\nabla f \\ , d \\gamma _ { y ^ * , t } , \\end{align*}"}
+{"id": "8533.png", "formula": "\\begin{align*} T _ { \\delta } ( x ; z ) : = \\left ( M _ { \\delta , + , 1 } ( x ; z ) - e _ 1 , M _ { \\delta , - , 2 } ( x ; z ) - e _ 2 \\right ) \\begin{pmatrix} 1 & 0 \\\\ - r _ { \\delta , 2 } ( z ) e ^ { 2 i z x } & 1 \\end{pmatrix} \\end{align*}"}
+{"id": "6050.png", "formula": "\\begin{align*} 0 = & \\mu \\Big ( \\big ( ( 2 \\alpha ^ { 2 } - ( 2 p \\alpha - 2 ) \\alpha \\big ) a ( u , v ) + \\big ( 2 ( \\alpha - 1 ) ^ { 2 } - 2 ( p \\alpha - 1 ) ( \\alpha - 1 ) \\big ) b ( u , v ) \\\\ & + ( 2 ( 3 \\alpha - 2 ) ^ { 2 } - 2 ( p \\alpha - 1 ) ( 3 \\alpha - 2 ) c ( u , v ) \\Big ) . \\end{align*}"}
+{"id": "4939.png", "formula": "\\begin{align*} h ( s ) = h _ { a , \\Delta } ( s ) = \\left ( \\dfrac { a } { a ^ 2 + s ^ 2 } \\right ) \\left ( \\dfrac { e ^ { 2 \\pi a \\Delta } + e ^ { - 2 \\pi a \\Delta } - 2 \\cos ( 2 \\pi \\Delta s ) } { \\left ( e ^ { \\pi a \\Delta } - e ^ { - \\pi a \\Delta } \\right ) ^ 2 } \\right ) \\end{align*}"}
+{"id": "7863.png", "formula": "\\begin{align*} \\hat \\Phi ^ \\dagger ( a , s ) & = \\left [ \\gamma _ 1 a ^ 3 + \\mathcal O ( a ^ 4 ) \\right ] + s \\left [ \\gamma _ 2 a + \\mathcal O ( a ^ 2 ) \\right ] + \\mathcal O ( s ^ 2 ) \\\\ & = a ( \\gamma _ 1 a ^ 2 + \\gamma _ 2 s ) + \\mathcal O ( a ^ 4 + \\abs { s } a ^ 2 + s ^ 2 ) \\end{align*}"}
+{"id": "2927.png", "formula": "\\begin{align*} \\mu \\left ( x \\right ) = D ^ { r } \\left ( e ^ { a \\left \\vert x \\right \\vert } f \\left ( x \\right ) \\right ) \\end{align*}"}
+{"id": "8823.png", "formula": "\\begin{align*} | \\Pi _ { \\mathrm { k e r } A ^ \\perp } X _ t | & = \\left | \\Pi _ { \\mathrm { k e r } A ^ \\perp } X _ 0 + t \\Pi _ { \\mathrm { k e r } A ^ \\perp } B ( X _ 0 , X _ 0 ) + \\Pi _ { \\mathrm { k e r } A ^ \\perp } \\int _ 0 ^ t ( t - s ) \\frac { d } { d s } B ( X _ s , X _ s ) d s \\right | \\\\ & \\ge c t | \\Pi _ { V _ 1 } X _ 0 | | \\Pi _ { V _ 2 } X _ 0 | - C \\sqrt { \\delta } K ^ { r } - C K \\tilde { \\eta } \\int _ 0 ^ t | B ( X _ s , X _ s ) | d s , \\end{align*}"}
+{"id": "3280.png", "formula": "\\begin{align*} \\tilde Z _ n ^ N ( i ) : = & \\Delta _ n ^ { - \\frac 1 2 } \\left ( ( p _ N \\tilde { \\Delta } _ i ^ n Y ) ^ { \\otimes 2 } - \\langle \\langle p _ N \\tilde { \\Delta } _ i ^ n Y \\rangle \\rangle \\right ) \\\\ = & \\Delta _ n ^ { - \\frac 1 2 } \\left ( ( p _ N \\tilde { \\Delta } _ i ^ n Y ) ^ { \\otimes 2 } - \\int _ { t _ { i - 1 } } ^ { t _ i } p _ N \\mathcal S ( t _ i - s ) \\Sigma _ s \\mathcal S ( t _ i - s ) ^ * p _ N d s \\right ) . \\end{align*}"}
+{"id": "8959.png", "formula": "\\begin{align*} \\sum _ { | k | \\le N } \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } \\frac { v _ 1 ^ { - p ' } ( x ) } { \\bigl [ V _ 1 ^ - ( x ) \\bigr ] ^ { p } } \\biggl ( \\int _ { \\eta _ { k - 1 } } ^ x v _ 1 ^ { - p ' } ( t ) \\bigl | G ^ { ( \\delta ) } _ { 1 , k } ( t ) \\bigr | ^ { p ' - 1 } \\ , d t \\biggr ) ^ p \\ , d x \\lesssim \\int _ { \\eta _ { k - 1 } } ^ { \\eta _ k } v _ 1 ^ { - p ' } \\bigl | G _ { 1 , k } ^ { ( \\delta ) } \\bigr | ^ { p ' } = \\bigl [ \\mathbf { G } ^ { ( \\delta ) } _ { 1 , N } ( g ) \\bigr ] ^ { p ' - 1 } , \\end{align*}"}
+{"id": "795.png", "formula": "\\begin{align*} \\partial ^ \\square c _ { t \\ , | t = 0 } = \\partial ^ \\square \\big ( ( c + t w ) \\circ \\theta _ t ^ { - 1 } \\big ) _ { | t = 0 } + \\partial ^ \\square m _ { t \\ , | t = 0 } = w . \\end{align*}"}
+{"id": "4307.png", "formula": "\\begin{align*} & \\int _ { \\{ \\Psi < - t _ 1 \\} } | \\tilde { F } - ( 1 - b _ { t _ 0 , B } ( \\Psi ) ) f | ^ 2 _ h e ^ { v _ { t _ 0 , B } ( \\Psi ) - \\Psi } c ( - v _ { t _ 0 , B } ( \\Psi ) ) \\\\ \\le & \\left ( \\int _ { t _ 1 } ^ { t _ 0 + B } c ( s ) e ^ { - s } d s \\right ) \\int _ { M } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f | ^ 2 _ h e ^ { - \\Psi } . \\end{align*}"}
+{"id": "5139.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } \\left ( \\nabla \\phi \\cdot \\nabla \\tilde { \\phi } + G \\tilde { G } \\right ) \\frac { 1 } { r ^ { 2 } } \\dd x = \\mu \\int _ { \\mathbb { R } ^ { 3 } } \\left ( \\tilde { \\phi } 1 _ { ( 0 , \\infty ) } ( \\phi - \\phi _ { \\infty } ) G + ( \\phi - \\phi _ { \\infty } ) _ { + } \\tilde { G } \\right ) \\frac { 1 } { r ^ { 2 } } \\dd x . \\end{align*}"}
+{"id": "1309.png", "formula": "\\begin{align*} E ( t ) + b \\int _ { 0 } ^ t | | u _ s ( s ) | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } d s = E ( 0 ) , \\end{align*}"}
+{"id": "389.png", "formula": "\\begin{align*} y y ' = \\left [ \\begin{array} { c c c c } 1 & \\frac { ( n - 2 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } ' & \\frac { - ( n + 1 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } ' \\\\ \\frac { ( n - 2 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } & \\frac { ( n - 2 ) ^ 2 } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } & \\frac { - ( n - 2 ) ( n + 1 ) } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } \\\\ \\frac { - ( n + 1 ) } { 3 ( n - 1 ) } \\ 1 _ { n - 1 } & \\frac { - ( n - 2 ) ( n + 1 ) } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } & \\frac { ( n + 1 ) ^ 2 } { 9 ( n - 1 ) ^ 2 } J _ { n - 1 } \\end{array} \\right ] . \\end{align*}"}
+{"id": "2142.png", "formula": "\\begin{align*} \\varrho : = - 1 + | z | ^ 2 + | w | ^ 2 + \\Re ( a z ^ 2 ) = 0 . \\end{align*}"}
+{"id": "5500.png", "formula": "\\begin{align*} F ( p , s ) & \\leq F ( p , 2 ) ^ { \\frac { 8 - 3 s } { 2 } } F ( p , 8 / 3 ) ^ { \\frac { 3 s - 6 } { 2 } } \\\\ & \\leq \\Big ( G ( p , 2 ) \\Big ) ^ { \\frac { 8 - 3 s } { 2 } } \\Big ( e ^ { - p / 6 } G ( p , 2 ) \\Big ) ^ { \\frac { 3 s - 6 } { 2 } } \\\\ & = e ^ { - p \\frac { s - 2 } { 4 } } 2 ^ { p - 1 } \\Gamma ( p / 2 ) . \\end{align*}"}
+{"id": "8016.png", "formula": "\\begin{align*} & I ( p _ i , q _ i , n _ i , l _ i , l _ i ' ) \\\\ & = a _ f ( p _ i ^ { 2 l _ i } ) a _ f ( q _ i ^ { 2 l _ i ' } ) ( a _ f ( p _ i ^ { 2 n _ i } ) - a _ f ( p _ i ^ { 2 n _ i - 2 } ) ) ( a _ f ( q _ i ^ { 2 n _ i } ) - a _ f ( q _ i ^ { 2 n _ i - 2 } ) ) \\\\ \\end{align*}"}
+{"id": "4418.png", "formula": "\\begin{align*} H ^ 2 ( \\mathbb { D } ) & = \\{ \\sum _ { n = 0 } ^ { \\infty } a _ n z ^ n \\mid ( a _ n ) \\in \\ell ^ 2 \\} \\\\ Y & = \\{ \\sum _ { n = 0 } ^ { \\infty } a _ n f _ n \\mid ( a _ n ) \\in \\ell ^ 2 \\} \\end{align*}"}
+{"id": "4949.png", "formula": "\\begin{align*} E _ { \\xi _ 0 } \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\biggr ] = \\xi _ 0 E _ 1 \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\biggr ] + ( 1 - \\xi _ 0 ) E _ 2 \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\biggr ] . \\end{align*}"}
+{"id": "6893.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\int _ { \\R ^ n } e ^ { f ( x ) } d x = \\sup _ { Q \\in \\P ( \\R ) } \\left ( \\int _ { \\R } V \\ , d Q + \\frac 1 2 \\int _ { \\R } \\int _ { \\R } K ( x - y ) Q ( d x ) Q ( d y ) - H ( Q ) \\right ) . \\end{align*}"}
+{"id": "8970.png", "formula": "\\begin{align*} k _ { i } ( \\Delta ) = \\sum _ { \\Upsilon \\in \\mathcal { T } _ { i } ( \\Delta ) } { | \\mathbf { T } ( \\tilde { H } _ { i - 1 } ( \\Upsilon ) ) | ^ 2 } = \\sum _ { \\Upsilon \\in \\mathcal { T } _ { i } ( \\Delta ) } { \\mathbf { t } _ { i - 1 } ( \\Upsilon ) ^ 2 } . \\end{align*}"}
+{"id": "246.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { | R _ q ( n ) | } { | B _ S ( n ) | } = 0 . \\end{align*}"}
+{"id": "903.png", "formula": "\\begin{align*} \\sigma = \\sum \\limits _ { d = 1 } ^ \\infty \\frac { \\mu ( d ) \\lambda ( d ^ 2 ) } { d ^ 2 } = \\prod _ p \\left ( 1 - \\frac { \\lambda ( p ^ 2 ) } { p ^ 2 } \\right ) \\ , . \\end{align*}"}
+{"id": "2833.png", "formula": "\\begin{align*} \\begin{aligned} \\| ( u , v , \\theta ) ( t ) \\| ^ 2 + 2 \\int _ 0 ^ t \\Big ( \\nu \\| \\nabla u \\| ^ 2 + & \\eta \\| \\nabla v \\| ^ 2 + \\mu \\| \\nabla \\theta \\| ^ 2 \\\\ & + \\sigma _ 1 \\| u \\| _ { \\alpha + 1 } ^ { \\alpha + 1 } + \\sigma _ 2 \\| v \\| _ { \\beta + 1 } ^ { \\beta + 1 } \\Big ) \\ , d \\ell = \\| ( u _ 0 , v _ 0 , \\theta _ 0 ) \\| ^ 2 , \\end{aligned} \\end{align*}"}
+{"id": "2998.png", "formula": "\\begin{align*} P ( \\alpha ) = \\big \\{ ( z , w ( \\cdot ) ) \\in \\mathbb R ^ n \\times \\mathrm { P L i p } \\colon \\| z \\| \\leq \\alpha , \\ , \\| w ( \\cdot ) \\| _ \\infty \\leq \\alpha \\big \\} . \\end{align*}"}
+{"id": "6194.png", "formula": "\\begin{align*} \\hat { A } ^ + = - f \\frac { d } { d r } - \\frac { L + 1 } { r } f + \\frac { Q } { 2 ( L + 1 ) } + \\kappa \\left ( L + \\frac { 3 } { 2 } \\right ) \\frac { r } { f } . \\end{align*}"}
+{"id": "341.png", "formula": "\\begin{align*} c _ { k + 1 } < 2 c _ k , \\ \\displaystyle \\lim _ { k \\to \\infty } c _ k = \\infty , \\ \\log c _ k = o ( \\alpha ( 2 ^ { k - 1 } ) ) , \\ \\ \\log c _ k = o ( k ) , \\end{align*}"}
+{"id": "1742.png", "formula": "\\begin{align*} J _ 0 ( t , x , y ) & = \\det ( \\nabla _ y S _ 0 ( t , x , y ) ) , \\\\ V _ 0 ( t , x , y ) & = J _ 0 ( t , x , y ) \\nabla _ y S _ 0 ( t , x , y ) ^ { - 1 } \\partial _ t S _ 0 ( t , x , y ) , \\end{align*}"}
+{"id": "8796.png", "formula": "\\begin{align*} \\left | \\Pi _ { \\mathrm { k e r } A ^ \\perp } \\frac { d ^ { \\ell - 1 } } { d t ^ { \\ell - 1 } } X _ t | _ { t = t _ 0 } \\right | \\ge \\frac { \\gamma ^ 3 } { C _ J } K ^ \\ell . \\end{align*}"}
+{"id": "3517.png", "formula": "\\begin{align*} F _ n - n ^ p \\ \\ = \\ \\ \\mathcal { C } _ n ^ { ( p ) } \\ - \\ 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } \\mathcal { C } _ n ^ { ( p - 2 j - 1 ) } \\ + 2 \\binom { p } { 2 \\lfloor p / 2 \\rfloor + 1 } F _ n \\mbox { f o r $ p $ o d d } . \\end{align*}"}
+{"id": "5699.png", "formula": "\\begin{align*} Q ( X ) = \\sum _ { 0 \\le i \\ne j \\le n - 1 } B _ { i j } X ^ { 2 ^ i + 2 ^ j } + \\sum _ { 0 \\le i \\le n - 1 } C _ i X ^ { 2 ^ i } + D . \\end{align*}"}
+{"id": "7322.png", "formula": "\\begin{align*} & \\phi _ i ( v w ) = v w + ( v w | e _ i ) y = v w + ( v | w e _ i ) y \\\\ & = v w + ( v | w ) y = ( v + ( v | e _ i ) y ) * ( w + ( w | e _ i ) y ) = \\phi _ i ( v ) * \\phi _ i ( w ) . \\end{align*}"}
+{"id": "7507.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } \\div ( A ( x ) \\nabla v ) + \\dd ( x ) \\cdot \\nabla u = 0 & B _ { \\rho } \\\\ v = u & \\partial B _ { \\rho } . \\end{array} \\right . \\end{align*}"}
+{"id": "1999.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} 1 + x , & x \\leq 0 , \\\\ 1 - x , & x > 0 . \\end{cases} \\end{align*}"}
+{"id": "7138.png", "formula": "\\begin{align*} \\mathcal B _ { \\mathcal X } ^ { \\mathfrak a } = \\left \\{ ( s , b ) | s \\in S , b \\in B _ { X _ s } ^ { \\alpha _ s } \\right \\} \\end{align*}"}
+{"id": "64.png", "formula": "\\begin{align*} & \\abs { \\varphi ( v \\beta ' ) - \\langle \\alpha ^ \\vee , \\beta \\rangle \\varphi ( v \\alpha ) - \\varphi ( - v \\beta ) } \\\\ \\leq & \\sum _ { k = 1 } ^ { \\langle \\alpha ^ \\vee , \\beta \\rangle } \\abs { \\varphi ( v ( k \\alpha - \\beta ) ) - \\varphi ( v \\alpha ) - \\varphi ( v ( k - 1 ) \\alpha - \\beta ) } \\\\ \\leq & \\langle \\alpha ^ \\vee , \\beta \\rangle . \\end{align*}"}
+{"id": "3569.png", "formula": "\\begin{align*} \\varphi ( - t ) ^ 2 G ( - t ) - \\varphi ( t ) ^ 2 G ( t ) = 2 t \\varphi ( t ^ 4 ) ^ 2 H ( t ^ 4 ) \\end{align*}"}
+{"id": "684.png", "formula": "\\begin{align*} \\Phi ( X ) = \\left [ \\begin{array} { c c } X & 0 \\\\ 0 & X \\end{array} \\right ] \\quad { \\rm s o \\ t h a t } \\Phi ^ \\dagger \\left ( \\left [ \\begin{array} { c c } A & B \\\\ C & D \\end{array} \\right ] \\right ) = A + D \\ . \\end{align*}"}
+{"id": "3711.png", "formula": "\\begin{align*} h = 0 , \\nabla h = 0 , \\nabla ^ { k + 2 } h = 0 \\mbox { o n } \\hat \\Sigma . \\end{align*}"}
+{"id": "7853.png", "formula": "\\begin{align*} \\hat \\Phi ^ \\dagger ( 0 , 0 ) & = 0 \\end{align*}"}
+{"id": "3097.png", "formula": "\\begin{align*} \\langle \\lambda ^ * - \\tilde { \\lambda } ^ k , \\frac { 1 } { \\beta } ( \\tilde { \\lambda } ^ k - \\lambda ^ k ) \\rangle _ { \\mathcal { L } ^ 2 } = \\langle \\lambda ^ * - \\tilde { \\lambda } ^ k , \\tilde { y } ^ k - \\tilde { x } ^ k \\rangle _ { \\mathcal { L } ^ 2 } = \\langle \\lambda ^ * - \\tilde { \\lambda } ^ k , \\tilde { y } ^ k - y ^ * - ( \\tilde { x } ^ k - x ^ * ) \\rangle _ { \\mathcal { L } ^ 2 } . \\end{align*}"}
+{"id": "3647.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\xi _ x ^ k } \\langle \\nabla f _ { \\delta } ( x _ g ^ k , \\xi _ x ^ k ) - \\nabla f ( x ^ * ) , x ^ { k + 1 } - x ^ * \\rangle & = \\mathbb { E } _ { \\xi _ x ^ k } \\langle \\nabla f _ { \\delta } ( x _ g ^ k , \\xi _ x ^ k ) - \\nabla f _ { \\delta } ( x _ g ^ k ) , x ^ { k + 1 } - x ^ * \\rangle \\\\ & + \\langle \\nabla f _ { \\delta } ( x _ g ^ k ) - \\nabla f ( x ^ * ) , x ^ { k + 1 } - x ^ * \\rangle . \\end{align*}"}
+{"id": "2276.png", "formula": "\\begin{align*} \\widehat { a } = \\chi _ E a . \\end{align*}"}
+{"id": "6606.png", "formula": "\\begin{align*} \\mathcal { B } _ { A B C } \\coloneqq \\mathcal { B } ( e _ A , e _ B , e _ C ) = \\langle [ e _ A , e _ B ] _ H , e _ C \\rangle . \\end{align*}"}
+{"id": "6650.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\lim _ { m \\to \\infty } \\limsup _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\Big ( \\sup _ { t \\in [ 0 , 1 ] } | \\hat { X } ^ { \\epsilon } ( t ) - \\varphi ( t ) | < \\delta , \\chi < 1 \\Big ) = - \\infty . \\end{align*}"}
+{"id": "832.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } F ( z _ l ) & = ( 0 , 0 ) , \\phantom { x x x } & F ( z _ r ) & = ( 0 , 0 ) , \\\\ \\nu _ \\Sigma ( z _ l ) & = ( - 1 , 0 ) , \\phantom { x x x } & \\nu _ \\Sigma ( z _ r ) & = ( + 1 , 0 ) . \\end{alignedat} \\end{align*}"}
+{"id": "2199.png", "formula": "\\begin{align*} \\mathbf { S ( R ) } = \\sum \\limits _ { i = 1 } ^ { N } \\sum \\limits _ { j = 1 } ^ { N } \\mathbf { R } _ { i j } \\end{align*}"}
+{"id": "7440.png", "formula": "\\begin{align*} a = \\Big ( \\frac { 1 - q } { 2 } \\Big ) ^ { k / p _ k } , b = \\Big ( \\frac { q } { 2 } \\Big ) ^ { k / p _ k } . \\end{align*}"}
+{"id": "427.png", "formula": "\\begin{align*} \\theta ^ { r - 1 } a _ { i } = \\sum _ { j = 1 } ^ { n } k _ { i j } \\theta ^ { r - 1 } \\theta ^ { j - 1 } = \\sum _ { j = 1 } ^ { n } k _ { i j } \\sum _ { k = 1 } ^ { n } c _ { r j k } \\theta ^ { k - 1 } = \\sum _ { k = 1 } ^ { n } \\Big ( \\sum _ { j = 1 } ^ { n } k _ { i j } c _ { r j k } \\Big ) \\theta ^ { k - 1 } . \\end{align*}"}
+{"id": "9309.png", "formula": "\\begin{align*} E _ { \\Delta _ { \\tau , n } } : \\ ; \\ ; y ^ { 2 } = x ( x - n \\tau ) ( x + n \\tau ^ { - 1 } ) , \\end{align*}"}
+{"id": "6762.png", "formula": "\\begin{align*} b _ { 0 } ( x ) = g ( \\frac { i } { 2 } , x ) e ^ { - \\frac { x } { 2 } } - 1 . \\end{align*}"}
+{"id": "5467.png", "formula": "\\begin{align*} x _ { i j } : = x _ { i i - 1 } x _ { i - 1 i - 2 } \\cdots x _ { j + 1 j } \\end{align*}"}
+{"id": "8373.png", "formula": "\\begin{align*} s _ 1 ( z ) : = \\mathcal { C } ( s J ) ( z ) s _ 2 ( z ) : = \\mathcal { C } ( s J ) ^ { H } ( z ) , \\end{align*}"}
+{"id": "4318.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi < - t \\} } | \\frac { f _ 1 + f _ 2 } { 2 } | ^ 2 _ h c ( - \\Psi ) + \\int _ { \\{ \\Psi < - t \\} } | \\frac { f _ 1 - f _ 2 } { 2 } | ^ 2 _ h c ( - \\Psi ) \\\\ = \\frac { 1 } { 2 } ( \\int _ { \\{ \\Psi < - t \\} } | f _ 1 | ^ 2 _ h c ( - \\Psi ) + \\int _ { \\{ \\Psi < - t \\} } | f _ 1 | ^ 2 _ h c ( - \\Psi ) ) = G ( t ) , \\end{align*}"}
+{"id": "4725.png", "formula": "\\begin{align*} & ( \\nexists \\ , x \\in \\mathbb { R } ^ n ) \\ \\ s . t . \\\\ & f ( x ) < 0 \\\\ & g ( x ) \\le 0 \\end{align*}"}
+{"id": "5635.png", "formula": "\\begin{align*} O _ { n , n } ( R ) \\subseteq O _ { n + 1 , n + 1 } ( R ) : A \\mapsto \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & A \\end{pmatrix} . \\end{align*}"}
+{"id": "2711.png", "formula": "\\begin{align*} p _ 0 = 1 - 2 \\exp \\left ( \\frac { a } { 2 } ( 2 \\epsilon _ f + \\epsilon _ g ^ { 3 / 2 } - r ) \\right ) . \\end{align*}"}
+{"id": "1693.png", "formula": "\\begin{align*} V _ n ( \\varphi ) = V ( \\varphi ) - \\tilde { V } _ n ( \\varphi ) \\leq - \\eta _ 0 + \\norm { \\tilde { V } _ n ( \\varphi ) } , \\end{align*}"}
+{"id": "4354.png", "formula": "\\begin{align*} \\frac { 1 } { r _ 2 ^ 2 } \\int _ { \\{ 2 a _ 0 ^ f ( \\Psi ; h ) \\Psi \\le 2 \\log r _ 2 \\} } | f | ^ 2 _ h & = \\lim _ { p \\rightarrow 2 a _ 0 ^ f ( \\Psi ; h ) + 0 } \\frac { 1 } { r _ 2 ^ 2 } \\int _ { \\{ p \\Psi < 2 \\log r _ 2 \\} } | f | ^ 2 _ h \\\\ & \\ge G ( 0 ; c \\equiv 1 , \\Psi , h , I _ + ( h , 2 a _ { z _ 0 } ^ f ( \\Psi ; h ) \\Psi ) _ { z _ 0 } , f ) . \\end{align*}"}
+{"id": "8384.png", "formula": "\\begin{align*} ( I - F ) \\Psi ^ - _ 1 = 0 \\Longleftrightarrow F \\Psi ^ - _ 1 = \\Psi ^ - _ 1 . \\end{align*}"}
+{"id": "3860.png", "formula": "\\begin{align*} \\partial _ t w + ( \\mathbf { v } \\cdot \\nabla ) w = 0 . \\end{align*}"}
+{"id": "3274.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sup _ { n \\in \\mathbb N } \\mathbb P \\left [ \\sum _ { k \\geq N } \\langle X _ n e _ k , e _ k \\rangle > \\delta \\right ] = 0 . \\end{align*}"}
+{"id": "1777.png", "formula": "\\begin{align*} D = m \\ , q _ 0 - \\sum _ { j = 1 } ^ s m _ j q _ j \\end{align*}"}
+{"id": "5694.png", "formula": "\\begin{align*} \\| z ^ { s + 1 } - z ^ s \\| \\leq \\| z ^ s - z ^ s _ * \\| + \\| z ^ { s + 1 } - z _ * ^ s \\| \\overset { \\eqref { c 3 } , \\eqref { i n e q } } { \\leq } \\| z ^ s - x ^ * \\| + \\rho _ s \\tau _ s \\overset { \\eqref { p p a e 2 } } { \\leq } \\| z ^ 0 - x ^ * \\| + \\sum _ { k = 0 } ^ { s } \\rho _ k \\tau _ k . \\end{align*}"}
+{"id": "3353.png", "formula": "\\begin{align*} I ( C _ 5 ) = \\langle \\{ a , c \\} , \\{ b , d \\} , \\{ c , e \\} , \\{ d , a \\} , \\{ e , b \\} \\rangle \\end{align*}"}
+{"id": "6165.png", "formula": "\\begin{align*} W ( r ) = - f \\frac { d } { d r } \\log \\psi _ 0 ( r ) - \\frac { 1 } { 2 } \\frac { d f } { d r } . \\end{align*}"}
+{"id": "4084.png", "formula": "\\begin{align*} \\varpi \\partial _ i ( \\partial _ j ( u _ k w _ \\ell ) ) = \\partial _ i ( \\partial _ j ( \\varpi u _ k w _ \\ell ) ) - \\partial _ i ( ( \\partial _ j \\varpi ) u _ k w _ \\ell ) - \\partial _ j ( ( \\partial _ i \\varpi ) u _ k w _ \\ell ) - ( \\partial _ i \\partial _ j \\varpi ) u _ k w _ \\ell . \\end{align*}"}
+{"id": "85.png", "formula": "\\begin{align*} \\langle \\mu _ 1 , \\alpha \\rangle : = \\Phi ^ + ( - y ^ { - 1 } \\alpha ) , \\langle \\mu _ 2 , \\alpha \\rangle : = \\Phi ^ + ( x \\alpha ) . \\end{align*}"}
+{"id": "9051.png", "formula": "\\begin{align*} F ^ L = \\C [ z _ { s , k } | 1 \\leq s \\leq m , 1 \\leq k \\leq L ] \\otimes C ^ { \\C } ( r _ { t , k } , c _ k | 1 \\leq t \\leq n , 1 \\leq k \\leq L ) . \\end{align*}"}
+{"id": "6298.png", "formula": "\\begin{align*} \\gamma _ { t - 1 } \\alpha _ t ^ 2 = ( 1 - \\alpha _ t ) \\alpha _ { t - 1 } ^ 2 \\gamma _ t + \\mu \\alpha _ t \\gamma _ t \\gamma _ { t - 1 } \\leq ( 1 - \\alpha _ t ) \\alpha _ { t - 1 } ^ 2 \\gamma _ t + \\alpha _ t \\gamma _ t \\alpha _ { t - 1 } ^ 2 = \\gamma _ t \\alpha _ { t - 1 } ^ 2 , \\end{align*}"}
+{"id": "3835.png", "formula": "\\begin{align*} & \\Delta _ j \\Delta _ k f = 0 , ~ ~ ~ | j - k | \\geq 2 , \\\\ & \\Delta _ j ( S _ { k - 1 } f \\Delta _ k f ) = 0 , ~ ~ ~ | j - k | \\geq 5 . \\end{align*}"}
+{"id": "4689.png", "formula": "\\begin{align*} \\tilde \\Sigma = \\{ x \\in M \\mid { \\rm V o l } ( A \\cap B _ x ( r ) ) = { \\rm V o l } ( B \\cap B _ x ( r ) ) \\} , \\end{align*}"}
+{"id": "6476.png", "formula": "\\begin{align*} \\exists \\ r = r ( \\theta , p ) \\in [ 1 , p ) , \\ \\exists \\ v = \\ v ( \\theta , p , r ) < \\infty . \\end{align*}"}
+{"id": "2264.png", "formula": "\\begin{align*} a ( U ^ { - 1 } Z V ) = a ( Z ) \\end{align*}"}
+{"id": "4867.png", "formula": "\\begin{align*} \\frac { 1 } { e ^ { - \\phi ( x ) } } e ^ { - \\psi ( x , T ( x , y ) ) } \\partial _ y T ( x , y ) = \\frac { 1 } { e ^ { - \\phi ( x _ 0 ) } } e ^ { - \\psi ( x _ 0 , y ) } . \\end{align*}"}
+{"id": "2972.png", "formula": "\\begin{align*} \\mathcal { T } : = \\bigcup _ { n \\in S ' } \\{ n \\} \\times [ t ( n ) , t \\left ( \\mathrm { s u c c } \\ , ( n ) \\right ) ) \\subset S \\times \\mathbb { R } _ { \\geq 0 } , \\end{align*}"}
+{"id": "7388.png", "formula": "\\begin{align*} \\Delta _ + = \\Delta _ + ^ 0 \\cup \\{ \\alpha + n \\delta , n \\delta | n \\in \\mathbb Z _ { > 0 } , \\alpha \\in \\Delta ^ 0 \\} \\end{align*}"}
+{"id": "5585.png", "formula": "\\begin{align*} W ( [ 1 ^ l 0 ] | [ 0 ^ k 1 ] ) = d _ k - d \\ . \\end{align*}"}
+{"id": "574.png", "formula": "\\begin{align*} 2 \\pi i f ( z ) = - \\int _ { p - i e ^ { i \\alpha } [ 0 , + \\infty ) } \\left ( \\int _ { e ^ { - i \\alpha } [ 0 , + \\infty ) } f ( \\zeta ) e ^ { \\omega \\zeta } d \\zeta \\right ) e ^ { - \\omega z } d \\omega + \\\\ + \\int _ { p + i e ^ { - i \\alpha } [ 0 , + \\infty ) } \\left ( \\int _ { e ^ { i \\alpha } [ 0 , + \\infty ) } f ( \\zeta ) e ^ { \\omega \\zeta } d \\zeta \\right ) e ^ { - \\omega z } d \\omega . \\end{align*}"}
+{"id": "1507.png", "formula": "\\begin{align*} G ( s ^ * , s ^ * ) = \\dfrac { 1 } { \\Gamma ( \\alpha ) } \\left ( \\frac { ( b - a ) ( \\alpha - 1 ) } { 2 \\alpha - 2 - \\beta } \\right ) ^ { \\alpha - 1 } \\left ( \\frac { \\alpha - 1 - \\beta } { 2 \\alpha - 2 - \\beta } \\right ) ^ { \\alpha - 1 - \\beta } . \\end{align*}"}
+{"id": "5310.png", "formula": "\\begin{align*} \\widetilde { f } ( \\lambda , b ) = \\pi _ \\lambda ( f ) Y _ 0 ( b ) , \\ , \\ , \\ , \\pi _ \\lambda ( f ) = \\int _ G f ( g ) \\pi _ \\lambda ( g ) d g . \\end{align*}"}
+{"id": "182.png", "formula": "\\begin{align*} ( \\varinjlim _ { n } X _ { H ' , n } \\widehat { \\otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V ) ) ^ { R G - l a } = 0 . \\end{align*}"}
+{"id": "1037.png", "formula": "\\begin{align*} y _ 1 = w _ 2 ^ { - 1 } \\varepsilon ^ { - w _ 2 \\mu _ 2 } , y _ 2 = w _ 1 ^ { - 1 } \\varepsilon ^ { - w _ 1 \\mu _ 1 } \\end{align*}"}
+{"id": "5294.png", "formula": "\\begin{align*} q ^ \\perp = 1 \\oplus 0 \\oplus 1 \\oplus \\begin{pmatrix} b ^ 2 & - a b \\\\ - a b & a ^ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "9254.png", "formula": "\\begin{align*} \\int _ \\Omega h d _ \\alpha T = - \\int _ \\Omega d _ \\alpha h \\wedge T + \\int _ { \\partial \\Omega } \\sum _ { A = 0 } ^ { 2 n - 1 } h \\ , T _ A \\ , \\tau ( \\mathbf { { n } } ) _ { A \\alpha } \\ , d S , \\alpha = 0 , 1 , \\end{align*}"}
+{"id": "7553.png", "formula": "\\begin{align*} q _ { n , m } \\cdot q _ { l , n } = r ^ { l } _ { n , m } \\cdot q _ { l , m } \\end{align*}"}
+{"id": "674.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 d x T _ n ( x ) = \\int _ 0 ^ \\pi d \\theta \\sin \\theta \\cos ( n \\theta ) = \\end{align*}"}
+{"id": "420.png", "formula": "\\begin{align*} \\pi _ { N } ( ( n \\psi ( n ) ) ^ { - 1 } m \\psi ( m ) ) ) = \\pi _ { N } ( ( n \\psi ( n ) ) ^ { - 1 } m \\psi ( n ) ) ) = ( n \\psi ( n ) ) ^ { - 1 } m \\psi ( n ) , \\end{align*}"}
+{"id": "5532.png", "formula": "\\begin{align*} \\int _ { \\mathfrak U } | \\omega | = \\pi ^ * _ s \\int _ { \\mathfrak X } | \\pi _ { \\eta } ^ * \\omega | \\end{align*}"}
+{"id": "4536.png", "formula": "\\begin{align*} \\int _ { a _ i } ^ { a _ { i + 1 } } | V ( x , t ) | ^ 2 = \\left | \\begin{pmatrix} S _ { d , 1 } ( t , t _ { 1 , e x } ( a _ i ) ) S _ c ( t _ { 1 , e x } ( a _ i ) , t _ { 1 , e n } ( a _ { i + 1 } ) ) ) S _ d ( t _ { 1 , e n } ( a _ { i + 1 } ) , 0 ) v _ { 1 , 0 } ( x ) + R _ { 1 , i } \\\\ S _ { d , 2 } ( t , t _ { 2 , e x } ( a _ i ) ) S _ c ( t _ { 2 , e x } ( a _ i ) , t _ { 2 , e n } ( a _ { i + 1 } ) ) ) S _ d ( t _ { 2 , e n } ( a _ { i + 1 } ) , 0 ) v _ { 1 , 0 } ( x ) + R _ { 2 , i } \\end{pmatrix} \\right | ^ 2 \\end{align*}"}
+{"id": "4806.png", "formula": "\\begin{align*} a ( T f , v ) = ( f , v ) v \\in H _ 0 ^ 2 ( D ) . \\end{align*}"}
+{"id": "9253.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u _ 1 \\wedge \\Delta u _ 2 \\wedge \\dots \\wedge \\Delta u _ n & = d _ 0 ( d _ 1 u _ 1 \\wedge \\Delta u _ 2 \\wedge \\dots \\wedge \\Delta u _ n ) = - d _ 1 ( d _ 0 u _ 1 \\wedge \\Delta u _ 2 \\wedge \\dots \\wedge \\Delta u _ n ) \\\\ & = d _ 0 d _ 1 ( u _ 1 \\Delta u _ 2 \\wedge \\dots \\wedge \\Delta u _ n ) = \\Delta ( u _ 1 \\Delta u _ 2 \\wedge \\dots \\wedge \\Delta u _ n ) . \\end{aligned} \\end{align*}"}
+{"id": "139.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\phi _ n = \\phi . \\end{align*}"}
+{"id": "1892.png", "formula": "\\begin{align*} \\begin{aligned} 1 - \\cos S _ c ( x , y ) = \\frac { 2 | x - y | } { | x + y | + | x - y | } \\ , . \\end{aligned} \\end{align*}"}
+{"id": "9319.png", "formula": "\\begin{align*} 4 z _ { 3 } ^ { 2 } - 2 z _ { 2 } ^ { 2 } = 5 p . \\end{align*}"}
+{"id": "1856.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac 1 { \\rho _ 1 ^ { 1 + \\delta } } \\int _ { B _ { \\rho _ 1 } ( x _ 0 ) - B _ 1 ( x _ 0 ) } \\big ( \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) - 1 \\big ) \\ , d v \\\\ & \\leq \\frac 1 { \\rho _ 2 ^ { 1 + \\delta } } \\int _ { B _ { \\rho _ 2 } ( x _ 0 ) - B _ 1 ( x _ 0 ) } \\big ( \\exp ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) - 1 \\big ) \\ , d v \\ , , \\end{aligned} \\end{align*}"}
+{"id": "8138.png", "formula": "\\begin{align*} \\hat A = A [ [ T ] ] / ( T - x ) \\end{align*}"}
+{"id": "3114.png", "formula": "\\begin{align*} a s _ { \\alpha } ( x , x , y ) = a s _ { \\alpha } ( y , x , x ) = 0 , \\end{align*}"}
+{"id": "9054.png", "formula": "\\begin{align*} \\mu _ t = j _ t - \\frac { L } { 2 } \\medspace ( 1 \\leq t \\leq n ) . \\end{align*}"}
+{"id": "1033.png", "formula": "\\begin{align*} \\prescript L { } { } \\ell { } ^ R ( x , w ^ { - 1 } \\alpha ) = \\prescript L { } { } \\ell { } ^ R ( x , v ( w v ) ^ { - 1 } \\alpha ) \\leq 0 \\end{align*}"}
+{"id": "7678.png", "formula": "\\begin{align*} \\phi ^ { N , i } ( t , \\boldsymbol { x } ) = \\Phi ( t , \\nu ^ { N , i } _ { \\boldsymbol { x } } ) ~ ~ ~ ~ \\nu ^ { N , i } _ { \\boldsymbol { x } } = \\frac { 1 } { N - 1 } \\underset { j \\neq i } { \\sum } x ^ j , \\end{align*}"}
+{"id": "8851.png", "formula": "\\begin{align*} \\begin{cases} d X _ { t , 1 } = X _ { t , 1 } X _ { t , 3 } d t \\\\ d X _ { t , 2 } = - X _ { t , 2 } X _ { t , 3 } d t + \\sum _ { j = 1 } ^ 3 \\sigma _ { 2 j } d W _ t ^ { ( j ) } \\\\ d X _ { t , 3 } = - X _ { t , 1 } ^ 2 d t \\end{cases} \\end{align*}"}
+{"id": "3808.png", "formula": "\\begin{align*} K _ q ( p , s ) = E _ q ^ * ( p \\overline { s } ) : = \\displaystyle \\sum _ { n = 0 } ^ \\infty \\frac { p ^ n \\overline { s } ^ n } { \\Gamma ( q n + 1 ) } , \\end{align*}"}
+{"id": "9141.png", "formula": "\\begin{align*} \\begin{aligned} a _ { k , \\delta } ( x _ a ) & : = 2 \\int _ { 0 } ^ { 1 } y _ \\delta ( x _ a , t ) \\cos ( k 2 \\pi t ) \\ , d t \\\\ b _ { k , \\delta } ( x _ a ) & : = 2 \\int _ { 0 } ^ { 1 } y _ \\delta ( x _ a , t ) \\sin ( k 2 \\pi t ) \\ , d t \\ , . \\end{aligned} \\end{align*}"}
+{"id": "8169.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { R _ { 0 } } \\mu \\xi ( \\mu , t _ { 0 } ) \\ d \\mu = & \\int _ { 0 } ^ { R _ { 0 } } \\mu \\xi ^ { \\mathrm { i n } } ( \\mu ) \\ d \\mu - \\int _ { 0 } ^ { t _ { 0 } } \\int _ { R _ { 0 } } ^ { \\infty } \\int _ { 0 } ^ { \\mu } \\nu \\Lambda ( \\mu , \\nu ) \\xi ( \\mu , s ) \\xi ( \\nu , s ) \\ d \\nu d \\mu d s . \\end{align*}"}
+{"id": "2184.png", "formula": "\\begin{align*} \\overline { d ^ 2 } = \\sum \\limits _ { m = 1 } ^ { N } { d _ m ^ 2 } \\end{align*}"}
+{"id": "3144.png", "formula": "\\begin{align*} N \\circ \\alpha = \\alpha \\circ N , \\end{align*}"}
+{"id": "3092.png", "formula": "\\begin{align*} \\norm { \\theta } _ { \\mathcal { L } ^ 2 , { G } } ^ 2 = \\beta r \\norm { x } _ { \\mathcal { L } ^ 2 } ^ 2 + \\beta \\norm { y } _ { \\mathcal { L } ^ 2 } ^ 2 + \\frac { 1 } { \\beta } \\norm { \\lambda } _ { \\mathcal { L } ^ 2 } ^ 2 , \\end{align*}"}
+{"id": "7297.png", "formula": "\\begin{align*} G ( \\beta , s ) ( 1 ) = [ H ( \\beta , 0 ) \\circ ( g _ { \\beta } ) _ { s } ] ( 1 ) = ( g _ { \\beta } ) _ { s } ( 1 ) = \\beta ( 1 ) . \\end{align*}"}
+{"id": "7240.png", "formula": "\\begin{align*} L = a _ 0 x + a _ 1 x ^ q + a _ 2 x ^ { q ^ 2 } + \\cdots + a _ n x ^ { q ^ n } \\in F [ x ] . \\end{align*}"}
+{"id": "1400.png", "formula": "\\begin{align*} G _ n ( x , z ) d z : = \\P _ x ( S ( n ) \\in d z , \\tau > n ) , x , z \\in W ^ d . \\end{align*}"}
+{"id": "7104.png", "formula": "\\begin{align*} \\Psi ( t , \\cdot ) : = - \\Delta ^ { - 1 } \\omega ( t , \\cdot ) \\end{align*}"}
+{"id": "4597.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ d F _ j = F _ 1 + \\cdots + F _ d = \\left \\{ \\sum _ { j = 1 } ^ d a _ j : a _ j \\in F _ j \\ \\forall 1 \\le j \\le d \\right \\} . \\end{align*}"}
+{"id": "5020.png", "formula": "\\begin{align*} \\mathcal { Y } _ { n } ( f , y ) = [ 1 + F _ { Y _ { m } } ( y ) ] \\mathcal { M } _ { n } ( f ) , \\ ; \\ ; \\ ; ( f \\in \\mathcal { C } [ 0 , 1 ] ) , \\\\ \\end{align*}"}
+{"id": "7789.png", "formula": "\\begin{align*} \\phi ^ q _ i = 2 \\pi q i / M + \\alpha , i = 1 , \\dots , M , \\end{align*}"}
+{"id": "7338.png", "formula": "\\begin{align*} | B - A | | A - C | = \\int _ { \\R ^ n } 1 _ { B - A } * 1 _ { A - C } ( x ) d x \\ge \\int _ { B - C } 1 _ { B - A } * 1 _ { A - C } ( x ) d x , \\end{align*}"}
+{"id": "4662.png", "formula": "\\begin{align*} \\Vert M _ { \\widetilde { m } \\vert _ { \\Lambda _ N \\times \\Lambda _ N \\times \\Lambda _ N } ( \\ : \\cdot \\ : , 0 \\ : \\cdot \\ : ) } : S _ { 1 } ^ { N + 1 } \\rightarrow S _ { 1 } ^ { N + 1 } \\Vert \\leq \\Vert M _ { \\widetilde { m } \\vert _ { \\Lambda _ N \\times \\Lambda _ N \\times \\Lambda _ N } } : S _ { p _ 1 } ^ { N + 1 } \\times S _ { p _ 2 } ^ { N + 1 } \\rightarrow S _ { p } ^ { N + 1 } \\Vert . \\end{align*}"}
+{"id": "2933.png", "formula": "\\begin{align*} \\overline { \\beta } _ { L } = \\sup \\left \\{ \\eta : \\lim \\sup _ { r \\rightarrow \\infty } r ^ { \\eta } \\overline { h } \\left ( r \\right ) = 0 \\right \\} = 0 \\end{align*}"}
+{"id": "3557.png", "formula": "\\begin{align*} c ^ { ( m ) } = 1 - \\frac { 6 } { ( m + 2 ) ( m + 3 ) } \\end{align*}"}
+{"id": "2116.png", "formula": "\\begin{align*} d ( p , g ) \\leq L d ( p , \\pi ^ { - 1 } ( \\pi ( g ) ) ) + M = L d ( p , \\pi ^ { - 1 } ( \\pi ( q ) ) ) + M \\leq L d ( p , q ) + M < r + M , \\end{align*}"}
+{"id": "6410.png", "formula": "\\begin{align*} 0 \\geq \\eta ^ { - 1 } ( - m _ \\delta c _ 0 ( \\delta ) + \\eta '' ) \\Phi ( p ) - 2 \\eta ^ { - 2 } ( \\eta ' ) ^ 2 \\Phi ( p ) + \\eta ^ { - 1 } \\Phi ( p ) ^ 2 \\end{align*}"}
+{"id": "5468.png", "formula": "\\begin{align*} ( \\sigma , \\sigma ( 1 ) , \\ldots , \\sigma ( r - 1 ) ) = [ \\sigma , \\sigma ( r - 1 ) ] = \\Delta \\end{align*}"}
+{"id": "860.png", "formula": "\\begin{align*} \\left | \\begin{array} { c c } x ^ 2 + x y + y ^ 2 = n \\ ; \\ ; \\ , \\\\ x '^ 2 + x ' y ' + y '^ 2 = n \\end{array} \\right . \\ , , \\end{align*}"}
+{"id": "1532.png", "formula": "\\begin{align*} P _ { \\xi } \\circ Q _ { \\xi } = \\sum _ { k \\geq 0 } \\frac { 1 } { k ! } \\ , \\frac { \\partial ^ { k } P _ { \\xi } } { \\partial \\xi ^ { k } } \\ , \\dd ^ { k } Q _ { \\xi } \\ ; , \\end{align*}"}
+{"id": "5105.png", "formula": "\\begin{align*} \\begin{aligned} & F ( s ) \\lesssim \\frac { 1 } { s ^ { \\tau } } , 0 < \\tau \\leq \\frac { 3 } { 2 } , \\\\ & F ^ { ( k ) } ( s ) \\lesssim \\frac { 1 } { s ^ { k + \\tau } } , 0 \\leq \\tau \\leq \\frac { 3 } { 2 } , \\ k \\in \\mathbb { N } . \\end{aligned} \\end{align*}"}
+{"id": "6571.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } | \\nabla c | ^ { 2 } & \\le \\int _ { \\Omega } | \\nabla c | ^ { 2 } + \\int _ { \\Omega } n c ^ { 2 } + \\frac { 1 } { 2 } \\int _ { \\partial \\Omega } c ^ { 2 } \\\\ & = \\int _ { \\partial \\Omega } \\gamma c - \\frac { 1 } { 2 } \\int _ { \\partial \\Omega } c ^ { 2 } \\\\ & \\le \\frac { 1 } { 2 } \\gamma ^ { 2 } | \\partial \\Omega | . \\end{aligned} \\end{align*}"}
+{"id": "5165.png", "formula": "\\begin{align*} \\phi _ { 1 , n } = \\phi _ { n } \\chi _ { R _ 0 } , G _ { 1 , n } = G _ { n } \\chi _ { R _ 0 } , \\end{align*}"}
+{"id": "7591.png", "formula": "\\begin{align*} F ^ * ( x _ { 0 0 } , x _ { 0 1 } , x _ { 1 0 } , x _ { 1 1 } ) = ( x _ { 0 1 } , x _ { 1 0 } , x _ { 1 1 } ) \\end{align*}"}
+{"id": "7609.png", "formula": "\\begin{align*} \\overline { \\{ T ^ n \\pi ( g ) : n \\in \\Z \\} } = g Y . \\end{align*}"}
+{"id": "999.png", "formula": "\\begin{align*} ( w \\varepsilon ^ \\mu ) ( \\alpha , k ) : = ( w \\alpha , k - \\langle \\mu , \\alpha \\rangle ) . \\end{align*}"}
+{"id": "7485.png", "formula": "\\begin{align*} r = | x | u _ r = \\dfrac { x } { | x | } \\cdot \\nabla u . \\end{align*}"}
+{"id": "6078.png", "formula": "\\begin{align*} X _ S = \\boxtimes _ { i \\in S } X _ i . \\end{align*}"}
+{"id": "1816.png", "formula": "\\begin{align*} | | R ^ { \\nabla } | | ^ 2 _ \\infty = \\sup _ { x \\in M } | | R ^ { \\nabla } | | ^ 2 _ x \\ , . \\end{align*}"}
+{"id": "8376.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } s V s ^ { H } d z = 0 \\Longrightarrow \\int _ { \\mathbb { R } } { \\rm R e } ( s V s ^ { H } ) \\mathrm { d } z = 0 , \\end{align*}"}
+{"id": "6417.png", "formula": "\\begin{align*} s = \\frac { 1 } { 4 } ( | u _ { , i j k } | _ g ^ 2 - | g ^ { m n } u _ { , i m n } | _ g ^ 2 ) \\end{align*}"}
+{"id": "2427.png", "formula": "\\begin{align*} L _ { q } [ f ( t ) ] = \\sum _ { n = 0 } ^ { \\infty } a _ { n } \\int _ { 0 } ^ { \\infty } t ^ { n } [ 1 - ( 1 - q ) s t ] ^ { \\frac { 1 } { 1 - q } } d t . \\end{align*}"}
+{"id": "1911.png", "formula": "\\begin{align*} \\pi ( s ) = \\lim _ { n \\to \\infty } w _ n ^ { ( i ) } ( s ) = - s \\mathcal { A } ' _ { \\delta } ( q s ) . \\end{align*}"}
+{"id": "7538.png", "formula": "\\begin{align*} \\tilde { I _ j } = \\{ i \\in [ n / \\ell ] : \\ : V _ i \\subseteq I _ j \\} . \\end{align*}"}
+{"id": "4742.png", "formula": "\\begin{align*} & b _ k ^ T y = 0 \\ \\ k \\in \\{ 0 , . . . , m \\} \\\\ & \\begin{bmatrix} b ^ 0 _ 1 & \\cdots & b ^ 0 _ n \\\\ \\vdots & \\ddots & \\vdots \\\\ b ^ m _ 1 & \\cdots & b ^ m _ n \\end{bmatrix} \\begin{bmatrix} y _ 1 \\\\ \\vdots \\\\ y _ n \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix} \\end{align*}"}
+{"id": "6299.png", "formula": "\\begin{align*} 0 \\leq \\beta _ t = \\mu \\gamma _ t \\alpha _ t ^ { - 1 } \\leq \\sqrt { \\mu \\gamma _ t } \\leq 1 . \\end{align*}"}
+{"id": "3840.png", "formula": "\\begin{align*} & L _ 4 \\colon \\mathbf { Z } = 2 \\eta \\mathbf { X } - \\iota \\mathbf { W } = 0 , & L ' _ 4 \\colon \\mathbf { W } - \\rho \\mathbf { Z } = 2 \\eta ' \\mathbf { X } - \\iota ' \\mathbf { W } = 0 , \\end{align*}"}
+{"id": "4824.png", "formula": "\\begin{align*} \\frac { \\lVert y - x \\rVert _ 2 } { \\lVert x \\rVert _ 2 } = O \\big ( \\delta \\ , \\kappa ( A ) \\big ) \\end{align*}"}
+{"id": "7926.png", "formula": "\\begin{align*} \\abs { \\eta ( y ) - \\eta ( x ) } ^ 2 & = \\abs { \\int _ x ^ y \\partial \\eta ( z ) \\ \\d z } ^ 2 \\le \\int _ \\S \\partial \\eta ( z ) ^ 2 \\ \\d z = \\norm { \\partial \\eta } _ { L ^ 2 } ^ 2 , \\end{align*}"}
+{"id": "3593.png", "formula": "\\begin{align*} R _ B & = \\C [ q _ { u , s ' } ^ { t ( i , \\bullet ) } \\mid u \\in [ \\beta ] , s ' \\in \\{ 0 , \\dots , x _ u - 1 \\} ] , \\\\ R _ C & = \\C [ r _ { v , s '' } ^ { t ( \\bullet , v ) } \\mid v \\in [ \\gamma ] , s '' \\in \\{ 0 , \\dots , y _ v - 1 \\} ] . \\end{align*}"}
+{"id": "3292.png", "formula": "\\begin{align*} A = & \\left \\lbrace \\left | \\langle B _ 1 , B _ 2 \\rangle \\sum _ { i = 1 } ^ n \\int _ { ( i - 1 ) \\Delta _ n } ^ { i \\Delta _ n } \\Delta _ n ^ { - \\frac 1 2 } \\langle ( \\mathcal S ( i \\Delta _ n - s ) - I ) B _ 1 , B _ 2 \\rangle d s \\right | > c \\right \\rbrace . \\end{align*}"}
+{"id": "1409.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\mathcal S _ d } ( \\pi ) \\P _ x ( \\widehat S _ i ( n + d - \\pi ( i ) ) \\in d z _ { \\pi ( i ) } , i = 1 , \\ldots , d , \\widehat \\tau < \\infty ) = 0 . \\end{align*}"}
+{"id": "3023.png", "formula": "\\begin{align*} \\left \\langle \\left ( x , C _ { \\xi } \\right ) , \\left ( y , C _ { \\eta } \\right ) \\right \\rangle = \\tfrac { 1 } { 4 } \\left ( \\left \\langle x , \\eta \\right \\rangle + \\left \\langle y , \\xi \\right \\rangle \\right ) \\end{align*}"}
+{"id": "7727.png", "formula": "\\begin{align*} \\forall i \\notin I _ r A _ { i , [ n ] } ^ { ( r ) } = & A _ { i , [ n ] } - A _ { i , [ r ] } ( A _ { I _ r , [ r ] } ) ^ { - 1 } A _ { I _ r , [ n ] } , \\\\ \\forall s \\in [ r ] , \\ , A _ { i _ s , [ n ] } ^ { ( r ) } = & A _ { i _ s , [ n ] } ^ { ( s ) } = A _ { i _ s , [ n ] } - A _ { i _ s , [ r - 1 ] } ( A _ { I _ { s - 1 } , [ s - 1 ] } ) ^ { - 1 } A _ { I _ { s - 1 } , [ n ] } . \\end{align*}"}
+{"id": "793.png", "formula": "\\begin{align*} \\partial ^ \\square \\big ( ( c + t w ) \\circ \\theta _ t ^ { - 1 } \\big ) _ { | t = 0 } = \\frac { \\mathrm { d } } { \\mathrm { d } t } ( c + t w ) _ { | t = 0 } - \\partial _ t \\theta _ t \\cdot \\nabla _ { \\Gamma _ t } \\big ( ( c + t w ) \\circ \\theta _ t ^ { - 1 } \\big ) _ { | t = 0 } = w - V \\nu \\cdot \\nabla _ \\Sigma c = w \\end{align*}"}
+{"id": "5379.png", "formula": "\\begin{align*} \\Psi = \\rho ( x ' ) - x _ n - \\theta | x ' | ^ 2 + K x _ n ^ 2 . \\end{align*}"}
+{"id": "2663.png", "formula": "\\begin{align*} & [ P ( u _ t ( t ) ) ] _ t + A u ( t ) + B ( t , x , u _ t ( t ) ) = 0 , t > 0 , \\ ; \\ ; x \\in \\mathbb { R } ^ n , \\end{align*}"}
+{"id": "4992.png", "formula": "\\begin{align*} \\mathcal { M } _ { f _ { 1 , \\infty } } ( N , \\alpha + \\delta , 6 \\epsilon , \\bigcup _ { n = N } ^ { \\infty } Z _ { n , n ^ { - 2 } t } , \\psi ) \\leq \\sum _ { n = N } ^ { \\infty } \\mathcal { M } _ { f _ { 1 , \\infty } } ( N , \\alpha + \\delta , 6 \\epsilon , Z _ { n , n ^ { - 2 } t } , \\psi ) . \\end{align*}"}
+{"id": "538.png", "formula": "\\begin{align*} c _ { i , j } & : = - 2 ( c _ i + c _ j ) + c _ { | i - j | } + c _ { i + j } \\\\ t _ { i , j } & : = - 2 ( s _ { i } + s _ { j } ) + s _ { | i - j | } + s _ { i + j } \\\\ u _ { i , j } & : = - 2 ( u _ i + u _ j ) + u _ { | i - j | } + u _ { i + j } \\\\ v _ { i , j } & : = - 2 ( v _ i + v _ j ) + v _ { | i - j | } + v _ { i + j } \\\\ z _ { i , j } & : = - 2 ( z _ { i } + z _ { j } ) + z _ { | i - j | } + z _ { i + j } . \\end{align*}"}
+{"id": "2867.png", "formula": "\\begin{align*} & \\Upsilon _ { j , j ' } ( 1 ) : = \\frac 1 2 \\sum _ { x = 0 } ^ n \\big [ \\psi _ j ( x ) \\psi _ { j ' } ( x - 1 ) + \\psi _ { j ' } ( x ) \\psi _ { j } ( x - 1 ) \\big ] , \\\\ & \\Upsilon _ { j , j ' } ( 2 ) : = \\frac 1 2 \\sum _ { x = 0 } ^ n \\big [ \\psi _ j ( x - 1 ) \\psi _ { j ' } ( x + 1 ) + \\psi _ { j ' } ( x + 1 ) \\psi _ { j } ( x - 1 ) \\big ] . \\end{align*}"}
+{"id": "4399.png", "formula": "\\begin{align*} F ( s , x ) = \\left | \\frac { \\zeta ' } { \\zeta } ( s ) + \\sum _ { n < x ^ 2 } \\frac { \\Lambda _ x ( n ) } { n ^ s } \\right | \\leq | Z _ 1 | + | Z _ 2 | + | Z _ 3 | \\end{align*}"}
+{"id": "8963.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\eta _ { - N - 1 } } v _ 1 ^ { - p ' } ( x ) \\biggl ( \\int _ x ^ { a ^ { - 1 } ( x ) } | \\chi _ { [ \\eta _ { - N } , \\eta _ N ] } g | \\biggr ) ^ { p ' } d x = 0 = \\int _ { \\eta _ { N } } ^ \\infty v _ 1 ^ { - p ' } ( x ) \\biggl ( \\int _ x ^ { a ^ { - 1 } ( x ) } | \\chi _ { [ \\eta _ { - N } , \\eta _ N ] } g | \\biggr ) ^ { p ' } d x . \\end{align*}"}
+{"id": "7586.png", "formula": "\\begin{align*} \\begin{gathered} ( x _ { 1 0 } , x _ { 1 1 } ) \\in \\omega ( ( x _ { 0 0 } , x _ { 0 1 } ) , T \\times T ) \\quad \\bigl ( x _ { 0 1 } , x _ { 1 1 } \\bigr ) \\in \\omega \\bigl ( ( x _ { 0 0 } , x _ { 1 0 } ) , T \\times T \\bigr ) . \\end{gathered} \\end{align*}"}
+{"id": "6958.png", "formula": "\\begin{align*} T _ { W , \\widetilde { K } } ( \\mu ^ t ) & = \\frac 1 2 \\int _ { [ 0 , 1 ] \\times \\R } \\int _ { [ 0 , 1 ] \\times \\R } W ( u , v ) K ( x - y ) \\mu ^ t ( d u , d x ) \\mu ^ t ( d v , d y ) \\\\ & = \\frac 1 2 \\int _ { [ 0 , 1 ] \\times \\R } \\int _ { [ 0 , 1 ] \\times \\R } W ( u , v ) K \\big ( ( 1 - t ) ( x - y ) + t ( T _ u ( x ) - T _ u ( y ) ) \\big ) \\mu ^ 0 ( d u , d x ) \\mu ^ 0 ( d v , d y ) . \\end{align*}"}
+{"id": "4663.png", "formula": "\\begin{align*} \\xi [ X ] \\stackrel { d e f } { = } \\int _ X \\xi ( x ) \\ \\mu ( d x ) \\ - \\end{align*}"}
+{"id": "9341.png", "formula": "\\begin{align*} \\limsup _ { r \\to 0 ^ + } r ^ { 1 - \\frac { 2 } { p } - \\frac { 3 } { q } } \\| v _ 3 \\| _ { L ^ p _ t L ^ q _ x ( Q _ r ( z _ 0 ) ) } = 0 1 \\leq p , q \\leq \\infty . \\end{align*}"}
+{"id": "320.png", "formula": "\\begin{align*} \\mathcal { T } _ b ^ * ( f ) ( x ) = \\sup _ { t > 0 } \\Big | \\int _ { \\mathbb { R } ^ n } k _ t ( x , y ) ( b ( x ) - b ( y ) ) f ( y ) d y \\Big | . \\end{align*}"}
+{"id": "6010.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\omega } : = \\Bigg \\{ u \\in H _ { r } ^ { 1 } ( \\R ^ { 2 } ) \\setminus \\{ 0 \\} : \\int _ { \\R ^ { 2 } } ( | \\nabla u | ^ { 2 } + \\omega u ^ { 2 } ) d x + 3 B ( u ) = \\int _ { \\R ^ { 2 } } u ^ { 2 p } d x \\Bigg \\} . \\end{align*}"}
+{"id": "7516.png", "formula": "\\begin{align*} \\int _ { B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x & \\leq C \\int _ { B _ { 8 \\rho } \\setminus B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x + C \\varepsilon \\int _ { B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x \\\\ & \\rho \\leq 1 / 8 \\varepsilon \\leq \\varepsilon _ 0 . \\end{align*}"}
+{"id": "7631.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { \\xi , \\alpha } = & ~ [ A x _ t ^ { \\xi , \\alpha } + B \\alpha _ t + f ( \\nu _ t ) + b ( \\mu _ t ) ] d t + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ 0 ^ { \\xi , \\alpha } = & ~ \\xi , \\end{aligned} \\right . \\end{align*}"}
+{"id": "37.png", "formula": "\\begin{align*} \\frac { d } { d r } u ( \\delta _ { r } g ) = \\frac { 1 } { r } Z u ( \\delta _ { r } g ) . \\end{align*}"}
+{"id": "1695.png", "formula": "\\begin{align*} \\bar { \\Gamma } _ { j k } = ( - 1 ) ^ { j + k } \\bar { \\Gamma } ^ \\flat _ { j k } , \\forall ( j , k ) \\in \\{ 0 , \\dots , n - 1 \\} ^ 2 . \\end{align*}"}
+{"id": "2306.png", "formula": "\\begin{align*} d _ n ^ { \\mathrm { a v g } } ( \\{ \\xi _ 1 , \\ldots , \\xi _ N \\} , L _ p ( \\Omega ) ) _ p = N ^ { - 1 / p } d _ n ^ { \\mathrm { a v g } } ( ( \\xi _ 1 , \\ldots , \\xi _ N ) , \\ell _ p ^ N ) _ p . \\end{align*}"}
+{"id": "9095.png", "formula": "\\begin{align*} \\mu _ k ( G ) : = \\frac { \\deg ( G ) + k } { ( G ) } . \\end{align*}"}
+{"id": "8774.png", "formula": "\\begin{gather*} \\phi ^ + ( x ) : = \\lim _ { n \\rightarrow \\infty } \\frac 1 n \\sum ^ { n - 1 } _ { k = 0 } \\phi ( f ^ k ( x ) ) , \\\\ \\phi ^ - ( x ) : = \\lim _ { n \\rightarrow \\infty } \\frac 1 n \\sum ^ { n - 1 } _ { k = 0 } \\phi ( f ^ { - k } ( x ) ) , \\\\ \\bar \\phi ( x ) : = \\lim _ { n \\rightarrow \\infty } \\frac 1 { 2 n - 1 } \\sum ^ { n - 1 } _ { k = - ( n - 1 ) } \\phi ( f ^ k ( x ) ) . \\end{gather*}"}
+{"id": "8145.png", "formula": "\\begin{align*} I = \\varinjlim _ { H \\subset G } I ^ H . \\end{align*}"}
+{"id": "6456.png", "formula": "\\begin{align*} \\vec { b } = & \\sum _ { i = j } ^ { n - 1 } b _ i \\vec { e } _ i \\end{align*}"}
+{"id": "1949.png", "formula": "\\begin{align*} = \\frac { z ^ { - \\mu } 4 ^ { \\mu - 1 } } { \\pi } G _ { { 0 } , { 4 } } ^ { { 4 } , { 0 } } \\left ( \\frac { z ^ { 4 } } { 2 5 6 } \\bigg { | } \\begin{array} { l l l } \\\\ \\frac { \\mu + \\nu } { 4 } ~ , \\frac { 2 + \\mu + \\nu } { 4 } , ~ \\frac { \\mu - \\nu } { 4 } , \\frac { 2 + \\mu - \\nu } { 4 } \\end{array} \\right ) , \\end{align*}"}
+{"id": "386.png", "formula": "\\begin{align*} E q _ k = \\left [ \\begin{array} { c c c } \\ 1 _ { n - 1 } ^ { ' } u _ k \\\\ 2 ( J _ { n - 1 } - I _ { n - 1 } ) u _ k \\\\ S ^ { ' } u _ k \\end{array} \\right ] . \\end{align*}"}
+{"id": "904.png", "formula": "\\begin{align*} \\Gamma _ 1 ( X ) = \\sigma X + \\mathcal { O } \\big ( z X ^ \\varepsilon \\big ) \\ , , \\end{align*}"}
+{"id": "9080.png", "formula": "\\begin{align*} ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + r ( j - 1 ) < \\chi _ j < ( \\sum \\limits _ { i = 1 } ^ j w _ i ) \\chi - \\sum \\limits _ { i = 1 } ^ { j - 1 } \\chi _ i + r j , \\end{align*}"}
+{"id": "5744.png", "formula": "\\begin{align*} \\pi ( x ) = \\frac { f ( x , 1 ) } { g ( x , 1 ) } = \\frac { s } { r } \\end{align*}"}
+{"id": "4159.png", "formula": "\\begin{align*} \\beta _ 1 ( u ) = 0 & \\Longrightarrow \\beta _ 1 ( u _ { \\varepsilon } ) = 0 , \\\\ \\beta _ 1 ( u ) = \\beta ' _ 3 ( u ; Q ) = 0 & \\Longrightarrow \\beta ' _ 3 ( u _ { \\varepsilon } ; Q ) = 0 . \\end{align*}"}
+{"id": "3604.png", "formula": "\\begin{align*} m _ { n } ( x ) = \\frac { \\nu _ n ( A _ { h , j } ) } { \\mu _ n ( A _ { h , j } ) } \\end{align*}"}
+{"id": "1935.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty b _ 3 \\left ( \\frac { m n - 1 } { 1 2 } \\right ) q ^ n \\in S _ { 3 m - 3 } ( \\Gamma _ 0 ( 4 3 2 ) , \\chi _ { 1 2 } ) _ m . \\end{align*}"}
+{"id": "6051.png", "formula": "\\begin{align*} p \\Delta \\left ( H - \\frac { \\mu } { 2 } \\right ) ^ { p - 1 } + 2 p \\left ( H - \\frac { \\mu } { 2 } \\right ) ^ { p - 1 } \\left ( 2 H ^ 2 - K \\right ) - 4 H \\left ( \\left ( H - \\frac { \\mu } { 2 } \\right ) ^ { p } + \\varsigma \\right ) = 0 , \\end{align*}"}
+{"id": "313.png", "formula": "\\begin{align*} H ( t , x , y ) = \\mathbb { P } ( t , x , y ) . \\end{align*}"}
+{"id": "7808.png", "formula": "\\begin{align*} X = N \\oplus X _ 0 \\qquad Z = R \\oplus Z _ 0 , \\end{align*}"}
+{"id": "3879.png", "formula": "\\begin{align*} P V _ { \\delta , \\hat { x } , \\hat { q } , z } ( x ) = V _ { \\delta , \\hat { x } , \\hat { q } , z } ( x ) - \\frac { \\hat { q } } { \\ln \\frac { R } { s _ \\delta } } g _ { \\hat { x } } ( T _ { \\hat { x } } x , T _ { \\hat { x } } z ) , \\ \\ \\forall x \\in \\Omega , \\end{align*}"}
+{"id": "3874.png", "formula": "\\begin{align*} \\begin{cases} - \\delta ^ 2 \\Delta w = ( w - a ) ^ { p } _ + , \\ \\ & \\ B _ R ( 0 ) , \\\\ w = 0 , \\ \\ & \\ \\partial B _ R ( 0 ) , \\end{cases} \\end{align*}"}
+{"id": "7675.png", "formula": "\\begin{align*} k _ i ^ N ( t , \\nu ^ { N , 1 } _ { \\boldsymbol { x } } , \\ldots , \\nu ^ { N , N } _ { \\boldsymbol { x } } , \\lambda ^ { N , 1 } _ { \\boldsymbol { \\varphi } } , \\ldots , \\lambda ^ { N , N } _ { \\boldsymbol { \\varphi } } ) : = \\rho _ i ^ N \\big ( - R ^ { - 1 } B [ P _ t \\nu ^ { N , 1 } _ { \\boldsymbol { x } } + \\lambda ^ { N , 1 } _ { \\boldsymbol { \\varphi } } ] , \\ldots , - R ^ { - 1 } B [ P _ t \\nu ^ { N , N } _ { \\boldsymbol { x } } + \\lambda ^ { N , N } _ { \\boldsymbol { \\varphi } } ] \\big ) . \\end{align*}"}
+{"id": "5271.png", "formula": "\\begin{align*} ( D ^ t ) ^ N ( \\psi ) = \\sum _ { \\substack { \\alpha = ( \\alpha _ 0 , . . . , \\alpha _ k ) \\\\ | \\alpha | = N } } C _ { N , \\alpha } \\frac { \\partial ^ { \\alpha _ 0 } \\psi \\partial ^ { \\alpha _ 1 + 1 } \\phi \\partial ^ { \\alpha _ 2 + 1 } \\phi . . . \\partial ^ { \\alpha _ k + 1 } \\phi } { \\lambda ^ N ( \\phi ' ) ^ { N + k } } . \\end{align*}"}
+{"id": "2690.png", "formula": "\\begin{align*} a _ d ( n ) = { } & d a _ d ( n - 1 ) + d a _ d ( n - 2 ) - \\binom { d } { 2 } a _ d ( n - 3 ) + d a _ d ( n - 4 ) - d ^ 2 a _ d ( n - 5 ) \\\\ & + d a _ d ( n - 6 ) - \\sum _ { k \\geq 8 } ^ { n + 1 } \\mu _ d ( k ) a _ d ( n + 1 - k ) \\\\ \\leq { } & ( d + 1 ) a _ d ( n - 1 ) + \\left ( - d ^ 3 \\binom { d } { 2 } + d ( d + 1 ) ^ 2 - d ^ 3 + d + 2 d ^ 2 \\right ) a _ d ( n - 6 ) \\\\ \\leq { } & ( d + 1 ) a _ d ( n - 1 ) , \\end{align*}"}
+{"id": "7279.png", "formula": "\\begin{align*} a _ { n - i } ( \\gamma ^ { ( q ^ { n - i } - q ^ n ) } - 1 ) = 0 \\end{align*}"}
+{"id": "2611.png", "formula": "\\begin{align*} L ( p , \\pi ^ h ) = L ( p , \\pi _ 1 ^ h ) = L ( p , \\pi _ 2 ^ h ) = L ( p , \\bar { \\pi } ^ h ) . \\end{align*}"}
+{"id": "4048.png", "formula": "\\begin{align*} 2 \\bar \\pi = \\sum _ { v \\in V _ 1 } \\pi _ v . \\end{align*}"}
+{"id": "6495.png", "formula": "\\begin{align*} \\mu | _ { 2 , [ m - e _ j , m + e _ 1 ] } ^ * = \\phi ^ j = \\lambda _ x | _ { 2 , [ m - e _ j , m + e _ 1 ] } ^ * \\end{align*}"}
+{"id": "7920.png", "formula": "\\begin{align*} \\hat W _ r ( q + k ) - \\hat W _ r ( 2 q ) & \\le \\frac { 2 } { \\pi } \\left ( \\frac { 1 } { q + k } + \\frac { 1 } { 2 q } \\right ) \\\\ & \\le \\frac { 4 } { \\pi q } \\\\ & \\le 4 r - \\frac { 2 } { \\pi } \\sin ( 2 \\pi r ) \\\\ & = \\hat W _ r ( 0 ) - \\hat W _ r ( 1 ) . \\end{align*}"}
+{"id": "6286.png", "formula": "\\begin{align*} ( \\nabla _ X \\beta ) ( Y , Z ) = ( \\nabla _ Y \\beta ) ( X , Z ) . \\end{align*}"}
+{"id": "8515.png", "formula": "\\begin{align*} \\begin{cases} \\delta _ { + } ( z ) = ( 1 + \\bar { r } _ 1 ( z ) r _ 2 ( z ) ) \\delta _ { - } ( z ) , z \\in \\mathbb { R } \\\\ \\delta _ { \\pm } ( z ) \\rightarrow 1 \\quad | z | \\rightarrow \\infty . \\end{cases} \\end{align*}"}
+{"id": "2932.png", "formula": "\\begin{align*} \\overline { h } \\left ( r \\right ) > \\int _ { \\left \\vert x \\right \\vert > r } \\nu _ { \\alpha } ^ { \\prime } \\left ( d x \\right ) = \\frac { 1 } { \\alpha \\log ^ { \\alpha } \\left ( 1 + r \\right ) } \\end{align*}"}
+{"id": "8136.png", "formula": "\\begin{align*} M / ( x _ 1 ^ n , \\dots , x _ k ^ n ) = ( ( M / x _ 1 ^ n ) / x _ 2 ^ n ) / \\dots / x _ k ^ n \\end{align*}"}
+{"id": "222.png", "formula": "\\begin{align*} S _ 0 = \\{ s _ { | i } \\mid s \\in S , i \\in \\mathbb { Z } \\} = \\left \\{ ( s _ j ) _ { | i } \\mid 1 \\le j \\le d i \\in \\mathbb { Z } \\right \\} \\subseteq H = H _ 0 , \\end{align*}"}
+{"id": "2206.png", "formula": "\\begin{align*} \\mathbf { v } ^ { 0 } = [ 1 , 1 , 1 , \\cdots , 1 ] ^ { T } . \\end{align*}"}
+{"id": "2324.png", "formula": "\\begin{align*} \\frac { s ^ H } { 4 } g & = ( R i c ^ g ) ^ { J , + } + \\frac { 1 } { 4 } \\| N \\| ^ 2 g + ( D ^ g \\theta ) ^ { s y m , J , + } + \\frac { 1 } { 2 } ( \\theta \\otimes \\theta ) ^ { 1 , 1 } . \\end{align*}"}
+{"id": "86.png", "formula": "\\begin{align*} x ' : = w ' \\varepsilon ^ { \\mu ' } : = x r _ { v \\alpha , \\Phi ^ + ( - v \\alpha ) } \\in \\widetilde W \\end{align*}"}
+{"id": "9318.png", "formula": "\\begin{align*} 2 u _ { 1 } ^ { 2 } - 2 p u _ { 2 } ^ { 2 } = 4 ^ { m } p ^ { 2 k + 1 } \\implies u _ { 1 } \\equiv 0 \\pmod p , \\end{align*}"}
+{"id": "4937.png", "formula": "\\begin{align*} \\sum _ { \\gamma _ { \\chi } } \\frac { \\frac { 1 } { 2 } } { \\frac { 1 } { 4 } + \\gamma _ { \\chi } ^ 2 } & \\leq \\left ( \\dfrac { 1 } { 2 } + \\dfrac { 2 } { \\log q - 2 } \\right ) \\log { \\dfrac { q } { \\pi } } + 2 \\left ( 1 - \\dfrac { 2 } { \\log q } \\right ) ^ { - 2 } \\log \\log q \\leq \\dfrac { 1 } { 2 } \\log { \\dfrac { q } { \\pi } } + 2 . 6 \\log \\log q . \\end{align*}"}
+{"id": "2433.png", "formula": "\\begin{align*} L _ { q } [ f ( a t ) ] = \\frac { 1 } { a } F _ { q } ( s / a ) . \\end{align*}"}
+{"id": "3303.png", "formula": "\\begin{align*} \\frac { \\varphi ( N ) } { N ^ { 2 / 3 } W ( N ) } = \\prod _ { i = 1 } ^ k \\frac { q _ i - 1 } { 2 q _ i ^ { 2 / 3 } } , \\end{align*}"}
+{"id": "4705.png", "formula": "\\begin{align*} \\varphi ( t ) : = \\mathrm { K L } \\left ( \\rho _ { t } \\mid \\pi \\right ) \\end{align*}"}
+{"id": "9082.png", "formula": "\\begin{align*} \\chi _ 2 = \\chi - \\chi _ 1 - \\sum \\limits _ { j \\geq 3 } \\chi _ j + ( n - 1 ) r \\end{align*}"}
+{"id": "6339.png", "formula": "\\begin{align*} F ( x ) = \\frac { 2 } { \\pi } \\arcsin \\left \\{ \\exp \\left ( - \\frac { \\theta } { 2 x ^ 2 } \\right ) \\right \\} . \\end{align*}"}
+{"id": "6752.png", "formula": "\\begin{align*} J _ { i , k } ^ { ( s ) } ( l _ 1 , l _ 2 ) & = \\int _ { l _ 1 } ^ { l _ 2 } z ^ i \\phi _ s ( z ) \\Phi _ s ( z ) ^ k \\mathrm { d } z , \\end{align*}"}
+{"id": "699.png", "formula": "\\begin{align*} J ( u ) : = \\int _ { \\R ^ N } \\left ( I _ \\alpha \\ast | u | ^ { p } \\right ) | u | ^ { p } d x \\end{align*}"}
+{"id": "7829.png", "formula": "\\begin{align*} X ^ \\dagger = N ^ \\dagger \\oplus X _ 0 ^ \\dagger Z ^ \\dagger = R ^ \\dagger \\oplus Z _ 0 ^ \\dagger , \\end{align*}"}
+{"id": "7735.png", "formula": "\\begin{align*} A R ( G ) = \\begin{cases} A \\rtimes \\{ I \\} & d \\\\ A \\rtimes \\langle - I \\rangle & d \\end{cases} \\end{align*}"}
+{"id": "2604.png", "formula": "\\begin{align*} R \\cdot R ( x _ n , J x _ n , J x _ n , x _ n ; u _ n , J u _ n ) = \\frac { 1 } { 2 } \\ , L ( p ) \\ , Q ^ c ( g , R ) ( x _ n , J x _ n , J x _ n , x _ n ; u _ n , J u _ n ) , \\end{align*}"}
+{"id": "663.png", "formula": "\\begin{align*} \\mu _ { \\rm e f f } ^ 2 ( r , \\theta ) = \\mu _ r ^ 2 ( r ) \\sin ^ { 2 q } \\theta + \\mu _ 0 ^ 2 ( r ) \\end{align*}"}
+{"id": "8140.png", "formula": "\\begin{align*} A = K \\langle T _ 1 ^ { 1 / p ^ \\infty } , \\dots , T _ n ^ { 1 / p ^ \\infty } \\rangle , A ^ + = K ^ + + A ^ { \\circ \\circ } . \\end{align*}"}
+{"id": "5610.png", "formula": "\\begin{align*} \\hat { \\chi } ^ * _ { [ a _ 0 a _ 1 ) } ( y | x ) = \\chi _ { [ a _ 0 a _ 1 ) } ( y _ 2 , y _ 1 x ) = \\chi _ { ( a _ 1 a _ 0 ] } ( y ) \\ , \\end{align*}"}
+{"id": "7231.png", "formula": "\\begin{align*} \\sigma ( \\lambda \\alpha ) = \\sigma ( \\lambda ) \\sigma ( \\alpha ) = \\lambda \\sigma ( \\alpha ) . \\end{align*}"}
+{"id": "5852.png", "formula": "\\begin{align*} \\delta y = - \\frac { 1 } { 2 } \\left ( { { f _ 3 } - { f _ 4 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) + \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) - \\frac { 1 } { 4 } { F _ y } , \\end{align*}"}
+{"id": "8941.png", "formula": "\\begin{gather*} \\left \\{ \\begin{array} { c c c } 0 & & n \\\\ \\frac { n ! } { ( n / 2 ) ! } \\left ( \\beta \\right ) ^ { \\left ( n / 2 \\right ) } ( 1 - \\rho ) ^ { n / 2 } & & n \\end{array} \\right . = E ( - X + Y ) ^ { n } \\allowbreak \\\\ = \\sum _ { m = 0 } ^ { n } ( - 1 ) ^ { m } \\binom { n } { m } E X ^ { m } Y ^ { n - m } . \\allowbreak \\end{gather*}"}
+{"id": "1075.png", "formula": "\\begin{align*} \\ell ( w s _ { - w ^ { - 1 } \\gamma } ) = \\ell ( w ) - \\ell ( s _ { - w ^ { - 1 } \\gamma } ) . \\end{align*}"}
+{"id": "5801.png", "formula": "\\begin{align*} d G = \\sqrt { \\tau _ 0 } / 4 \\left \\{ \\left ( \\left ( 1 + 4 Q _ 0 / \\tau _ 0 \\right ) d z + \\left ( 1 + 4 \\overline { Q _ 0 } / \\tau _ 0 \\right ) d \\overline { z } \\right ) u _ 1 \\right . \\\\ \\left . - i \\left ( \\left ( 1 - 4 Q _ 0 / \\tau _ 0 \\right ) d z - \\left ( 1 - 4 \\overline { Q _ 0 } / \\tau _ 0 \\right ) d \\overline { z } \\right ) u _ 2 \\right \\} \\end{align*}"}
+{"id": "2928.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\nu _ { \\alpha } \\left ( d x \\right ) = \\frac { 1 } { \\left ( 1 + \\left \\vert x \\right \\vert \\right ) \\log ^ { 1 + \\alpha } \\left ( 1 + \\left \\vert x \\right \\vert \\right ) } d x & & 0 < \\alpha < 2 \\end{array} \\end{align*}"}
+{"id": "6359.png", "formula": "\\begin{align*} S _ r ( a , b ) & = \\int _ 0 ^ \\infty y ^ { - r / 2 } \\exp ( - a y ) \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n \\frac { \\Gamma ( b ) } { \\Gamma ( b - n ) n ! } \\exp ( - n y ) \\mathrm { d } y \\\\ & = \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n \\frac { \\Gamma ( b ) } { \\Gamma ( b - n ) n ! } \\int _ 0 ^ \\infty y ^ { - r / 2 } \\exp \\left \\{ - ( a + n ) y \\right \\} \\mathrm { d } y . \\end{align*}"}
+{"id": "798.png", "formula": "\\begin{align*} \\mathrm { D E } ( \\Sigma , c ) ( V , w ) = \\big \\langle \\mathrm { E } ( \\Sigma , c ) , ( V , w ) \\big \\rangle \\end{align*}"}
+{"id": "8171.png", "formula": "\\begin{align*} \\frac { d } { d s } \\int _ { 0 } ^ { R _ { 0 } } \\mu \\xi ( \\mu , s ) \\ d \\mu = - R _ { 0 } \\int _ { 0 } ^ { R _ { 0 } } \\nu \\xi ( R _ { 0 } , s ) \\xi ( \\nu , s ) \\Lambda ( R _ { 0 } , \\nu ) \\ d \\nu . \\end{align*}"}
+{"id": "2622.png", "formula": "\\begin{align*} ( x \\cdot y ) \\diamond \\alpha ( z ) & = \\varepsilon ( y , z ) ( x \\diamond z ) \\cdot \\alpha ( y ) , \\\\ ( x \\diamond y ) \\cdot \\alpha ( z ) - \\alpha ( x ) \\diamond ( y \\cdot z ) & = \\varepsilon ( x , y ) \\big ( ( y \\diamond x ) \\cdot \\alpha ( z ) - \\alpha ( y ) \\diamond ( x \\cdot z ) \\big ) . \\end{align*}"}
+{"id": "1899.png", "formula": "\\begin{align*} ( \\sin S _ c ^ l - \\sin S _ c ) \\prod _ { m = 1 } ^ { 2 l - 1 } \\partial _ { x } ^ { a _ m } S _ c , ( \\cos S _ c ^ l - \\cos S _ c ) \\prod _ { m = 1 } ^ { 2 l } \\partial _ { x } ^ { b _ m } S _ c , l \\ge 0 , \\end{align*}"}
+{"id": "3897.png", "formula": "\\begin{align*} \\int _ { \\Omega } - \\delta _ N ^ 2 \\left ( ( K ( x ) - K ( z _ { N , i } ) ) \\nabla \\frac { \\partial V _ { \\delta _ N , Z _ N , i } } { \\partial z _ { i , h } } \\right ) u _ N = O \\left ( \\frac { \\varepsilon _ N ^ 2 } { | \\ln \\varepsilon _ N | ^ { p - 1 } } \\right ) . \\end{align*}"}
+{"id": "5781.png", "formula": "\\begin{align*} \\nabla _ X \\Psi _ 2 = - \\frac { 1 } { 2 } \\left ( X - \\langle X , T _ 1 \\rangle ( T _ 1 + f _ 1 ) - \\langle X , T _ 2 \\rangle ( T _ 2 + f _ 2 ) \\right ) \\cdot \\Psi _ 1 + \\frac { 1 } { 2 } S ( X ) \\cdot \\Psi _ 2 , \\end{align*}"}
+{"id": "3360.png", "formula": "\\begin{align*} N e w t o n ( F ) = C o n v ( \\bigcup \\limits _ { i = 0 } ^ l N e w t o n ( s _ { \\lambda ^ i } ) ) . \\end{align*}"}
+{"id": "7123.png", "formula": "\\begin{align*} \\begin{array} { l l l } f ^ \\sharp _ R ( L , R ) ( u ) \\cdot u ' & = & f _ R ( R ( s ) ) ( t ) \\cdot u ' \\\\ & = & t \\cdot f _ L ( R ( s ) ) ( u ' ) \\\\ & = & t \\cdot f _ L ( R ( s ) ) ( f _ L ( s ' ) ( t ' ) ) \\\\ & = & t \\cdot f _ L ( R ( s ) * s ' ) ( t ' ) \\\\ & = & t \\cdot f _ L ( s * L ( s ' ) ) ( t ' ) \\\\ & = & t \\cdot f _ L ( s ) ( f _ L ( L ( s ' ) ( t ' ) ) ) \\\\ & = & t \\cdot f _ L ( s ) ( f ^ \\sharp _ L ( L , R ) ( u ' ) ) \\\\ & = & f _ R ( s ) ( t ) \\cdot f ^ \\sharp _ L ( L , R ) ( u ' ) \\\\ & = & u \\cdot f ^ \\sharp _ L ( L , R ) ( u ' ) . \\end{array} \\end{align*}"}
+{"id": "1452.png", "formula": "\\begin{align*} \\phi ( p ) = \\arctan ( p ^ { 2 } ) \\qquad \\mbox { o r } \\phi ( p ) = ( 1 + p ^ { 2 } ) ^ { \\alpha } \\quad \\alpha \\in ( 0 , 1 / 2 ) , \\end{align*}"}
+{"id": "1248.png", "formula": "\\begin{align*} G _ { \\xi _ { \\Delta } } : \\Omega \\to [ \\xi _ { \\Delta } ] : = \\{ \\omega \\in \\Omega \\colon \\omega _ \\Delta = \\xi _ \\Delta \\} , \\eta \\mapsto \\xi _ { \\Delta } \\eta _ { \\Delta ^ c } . \\end{align*}"}
+{"id": "4366.png", "formula": "\\begin{align*} \\int _ M | | ( \\eta + g ^ { - 1 } ) ^ { \\frac { 1 } { 2 } } u _ 0 | | _ { \\omega } + \\int _ M | | \\tau _ 0 | | _ { \\omega } \\le \\int _ M \\langle ( B + \\lambda I ) ^ { - 1 } v , v \\rangle _ { \\omega , h } d V _ { \\omega } . \\end{align*}"}
+{"id": "1173.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\mathbb { P } } \\left [ \\left | \\sum _ { j = 2 } ^ { n } \\left ( b ( t , X ^ { 1 , \\infty } , X ^ { j , \\infty } ) - \\left \\langle b \\left ( t , X ^ { 1 , \\infty } , \\cdot \\right ) , \\mu \\right \\rangle \\right ) \\right | ^ { 2 p } \\right ] \\leq p ! \\left ( ( n - 1 ) \\beta \\right ) ^ { p } . \\end{align*}"}
+{"id": "1758.png", "formula": "\\begin{align*} \\big ( D _ 0 ( t , x ) ^ { \\ast } \\big ) _ { i j } : = \\int _ { Y ^ { \\ast } } D _ 0 ( t , x , y ) \\big [ \\nabla _ y w _ i ( t , x , y ) + e _ i \\big ] \\cdot \\big [ \\nabla _ y w _ j ( t , x , y ) + e _ j \\big ] d y , \\end{align*}"}
+{"id": "6462.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { ( \\rho _ p + 1 ) ^ { n - j _ p } h _ { j _ p } } { h _ { n + 1 } } & = \\lim _ { n \\to \\infty } \\frac { ( \\rho _ p + 1 ) ^ { n - j _ p } h _ { j _ p } } { \\rho _ p ^ { n - j _ p + 1 } h _ { j _ p } + \\frac { \\rho _ p ( \\rho _ p ^ { n - j _ p + 1 } - 1 ) } { 2 } } \\\\ & \\geq \\lim _ { n \\to \\infty } \\frac { ( \\rho _ p + 1 ) ^ { n - j _ p } h _ { j _ p } } { \\rho _ p ^ { n - j _ p + 1 } h _ { j _ p } + \\rho _ p ^ { n - j + 2 } } \\\\ & = \\infty . \\end{align*}"}
+{"id": "8488.png", "formula": "\\begin{align*} & \\mathcal { P } ^ - ( f ( z ) e ^ { 2 i z x } ) = - \\int _ { 2 x } ^ \\infty \\widehat { f } ( \\xi ) e ^ { - i z ( \\xi - 2 x ) } \\mathrm { d } \\xi , \\end{align*}"}
+{"id": "4868.png", "formula": "\\begin{align*} \\lim _ { y \\to \\pm \\infty } e ^ { - \\psi ( x _ 0 , y ) } d _ x T ( x _ 0 , y ) = 0 . \\end{align*}"}
+{"id": "7542.png", "formula": "\\begin{align*} \\| A _ j x \\| ^ 2 - \\big | \\langle x | A _ j x \\rangle \\big | ^ 2 & = \\| ( A _ j - z _ j I ) x \\| ^ 2 - \\big | \\langle x | ( A _ j - z _ j I ) x \\rangle \\big | ^ 2 \\\\ & \\leq \\| ( A _ j - z _ j I ) x \\| ^ 2 . \\end{align*}"}
+{"id": "7016.png", "formula": "\\begin{align*} u \\ , = \\ , \\tfrac { x ^ 2 } { 2 } + { \\rm O } _ { x , y } ( 3 ) , \\end{align*}"}
+{"id": "7370.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ n \\mu _ i ( f _ i ) \\ge \\inf _ { Q \\in \\mathcal { U } } Q \\Bigl ( \\oplus _ { i = 1 } ^ n f _ i \\Bigr ) \\quad \\quad f _ i \\in B ( \\Omega _ i , \\mathcal { A } _ i ) i \\in I . \\end{gather*}"}
+{"id": "8082.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ M \\sum \\limits _ { \\substack { j = 1 \\\\ j \\neq l } } ^ M \\sum _ { q = 1 } ^ M a _ q ^ 2 \\hat { \\phi } ^ { \\left ( l , q \\right ) ^ * } \\hat { \\phi } ^ { \\left ( j , q \\right ) } = 2 \\sum _ { l = 1 } ^ { M - 1 } \\sum _ { j = i + 1 } ^ { M } \\sum _ { q = 1 } ^ M a _ q ^ 2 \\Re \\left [ \\hat { \\phi } ^ { \\left ( l , q \\right ) ^ * } \\hat { \\phi } ^ { \\left ( j , q \\right ) } \\right ] . \\end{align*}"}
+{"id": "4831.png", "formula": "\\begin{align*} A _ { k j } = \\begin{cases} \\sum _ { l = 1 } ^ { n _ 1 } \\Psi _ { k l } \\bar \\Psi _ { j l } , & \\lambda _ k = \\lambda _ j , \\\\ 0 , & . \\end{cases} \\end{align*}"}
+{"id": "1592.png", "formula": "\\begin{align*} \\mathbf { H } _ { \\mathrm { a l l } , k } = \\left \\{ \\begin{array} { c c } \\mathbf { H } _ { \\mathrm { d } , k } + \\mathbf { H } _ { k } \\mathbf { \\Phi } _ { \\mathrm { r } } \\mathbf { G } , & k \\in \\mathcal { K } _ { \\mathrm { r } } , \\\\ \\mathbf { H } _ { \\mathrm { d } , k } + \\mathbf { H } _ { k } \\mathbf { \\Phi } _ { \\mathrm { t } } \\mathbf { G } , & k \\in \\mathcal { K } _ { \\mathrm { t } } . \\end{array} \\right . \\end{align*}"}
+{"id": "8282.png", "formula": "\\begin{align*} \\big ( \\bar \\pi \\bar p _ t \\big ) \\bar p _ T = \\bar \\pi \\bar p _ t , \\end{align*}"}
+{"id": "834.png", "formula": "\\begin{align*} \\theta _ { \\rho ^ \\varepsilon } ( 0 , \\cdot ) \\circ F ^ { - 1 } : F ( U _ i ) \\rightarrow \\R ^ 2 , \\phantom { x x } p \\mapsto p + \\rho _ 0 ^ \\varepsilon \\nu _ \\Sigma \\big ( F ^ { - 1 } ( p ) \\big ) = p + \\rho _ 0 ^ \\varepsilon \\nu _ { F ( U _ i ) } ( p ) \\end{align*}"}
+{"id": "1389.png", "formula": "\\begin{align*} h _ 1 ( x ) = e ^ { \\sum _ { j = 1 } ^ d x _ j } \\mathrm { d e t } ( ( - 1 ) ^ { j - 1 } \\phi ^ { ( 1 - j ) } _ i ( x _ j ) ) _ { i , j = 1 } ^ d , x \\in W ^ d . \\end{align*}"}
+{"id": "4395.png", "formula": "\\begin{align*} & \\int _ { X } | \\tilde { F } - ( 1 - b ( \\Psi ) ) f F ^ { 1 + \\delta } | _ { h } ^ 2 e ^ { v ( \\Psi ) - \\delta \\max \\{ \\psi + T , 2 \\log | F | \\} } c ( - v ( \\Psi ) ) \\\\ \\le & \\left ( \\frac { 1 } { \\delta } c ( T ) e ^ { - T } + \\int ^ { t _ 0 + B } _ T c ( t _ 1 ) e ^ { - t _ 1 } d t _ 1 \\right ) \\int _ { X } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f F | ^ 2 _ h < + \\infty . \\end{align*}"}
+{"id": "9028.png", "formula": "\\begin{align*} v ( t ) = \\sum _ { \\mathbf { k } = 1 } ^ { \\infty } \\widehat { v } _ { \\mathbf { k } } { \\rm e } ^ { { \\rm i } \\mathbf { k } t } \\sum _ { \\mathbf { k } = 1 } ^ { \\infty } | \\widehat { v } _ { \\mathbf { k } } | < \\infty , \\end{align*}"}
+{"id": "4673.png", "formula": "\\begin{align*} \\forall \\theta > 0 \\ \\Rightarrow \\sup _ { p \\ge 1 } \\left [ \\ \\frac { L ( p ^ { \\theta } ) } { L ( p ) } \\ \\right ] = C ( \\theta ) < \\infty . \\end{align*}"}
+{"id": "7910.png", "formula": "\\begin{align*} c _ 1 ( q , 2 q , p ^ 0 ( 0 ) ) & = \\frac 1 4 \\left ( \\frac { 1 } { - q } f ( - q r ) + \\frac { 1 } { 3 q } f ( 3 q r ) - \\frac { 2 } { q } f ( q r ) - H ( q , r ) \\frac 1 q f ( q r ) \\right ) \\\\ & = \\frac 1 q g ( q r ) , \\end{align*}"}
+{"id": "7296.png", "formula": "\\begin{align*} G ( \\beta , 1 ) = H ( \\beta , 1 ) \\circ [ g _ { \\beta } ] _ { 1 } = \\beta \\circ \\widetilde { \\beta ( 1 ) } \\simeq i d _ { X _ { U ' } ^ { I } } \\end{align*}"}
+{"id": "6429.png", "formula": "\\begin{align*} [ r ^ \\beta ] C _ k ( q , r , t ) = \\frac { C _ k ( t ) ^ { \\beta + 1 } - q ^ { \\beta + 1 } } { 1 - q + q ^ k t } = \\left ( C _ k ( t ) ^ { \\beta + 1 } - q ^ { \\beta + 1 } \\right ) \\left ( 1 + ( q - q ^ k t ) + ( q - q ^ k t ) ^ 2 + \\hdots \\right ) . \\end{align*}"}
+{"id": "1234.png", "formula": "\\begin{align*} \\sum _ { \\xi _ { \\Delta } \\in \\Omega _ { \\Delta } } \\int _ { \\Omega } c _ { \\Delta } ( \\omega , \\xi _ { \\Delta } ) f ( \\omega ) g ( \\xi _ { \\Delta } \\omega _ { \\Delta ^ c } ) \\mu ( d \\omega ) = \\sum _ { \\xi _ { \\Delta } \\in \\Omega _ { \\Delta } } \\int _ { \\Omega } \\hat { c } _ { \\Delta } ( \\omega , \\xi _ { \\Delta } ) f ( \\xi _ { \\Delta } \\omega _ { \\Delta ^ c } ) g ( \\omega ) \\mu ( d \\omega ) , \\end{align*}"}
+{"id": "5061.png", "formula": "\\begin{align*} \\Phi _ { t } + u \\cdot \\nabla \\Phi = 0 . \\end{align*}"}
+{"id": "4535.png", "formula": "\\begin{align*} \\int _ { a _ i } ^ { a _ i + 1 } \\int _ { t _ { 1 , e n } ( x , t ) } ^ { t _ { 1 , e x } ( x , t ) } v _ 1 ( s , x - \\lambda t + \\lambda s ) \\dd s \\dd x & = \\int _ { a _ i } ^ { a _ i + 1 } \\int _ { t _ { 1 , e n } ( x , t ) } ^ { t _ { 1 , e x } ( x , t ) } v _ 1 ^ + - v _ 1 ^ - \\dd s \\dd x . \\end{align*}"}
+{"id": "6288.png", "formula": "\\begin{align*} 0 = ( \\nabla _ T C _ S ( X _ i ) - C _ S C _ T ( X _ i ) - C _ { \\nabla _ T S } ( X _ i ) ) - ( \\nabla _ S C _ T ( X _ i ) - C _ T C _ S ( X _ i ) - C _ { \\nabla _ S T } ( X _ i ) ) = d \\lambda _ i ( T , S ) X _ i . \\end{align*}"}
+{"id": "6461.png", "formula": "\\begin{align*} h _ { n } = \\rho _ p ^ { n - j _ p } h _ { j _ p } + \\frac { \\rho _ p ( \\rho _ p ^ { n - j _ p } - 1 ) } { 2 } . \\end{align*}"}
+{"id": "6957.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } M _ n ^ { \\psi } / n \\ge \\lim _ { n \\to \\infty } M _ n ^ { \\psi } ( Q ^ n ) / n = T _ { W , \\psi } ( \\mu ) - I ( \\mu ) . \\end{align*}"}
+{"id": "2486.png", "formula": "\\begin{align*} \\mathbf { x } \\circ \\mathbf { s } = \\left ( \\mathbf { x } ^ { ( 1 ) } \\circ \\mathbf { s } ^ { ( 1 ) } , \\cdots , \\mathbf { x } ^ { ( k ) } \\circ \\mathbf { s } ^ { ( k ) } \\right ) , \\forall \\mathbf { x } , \\mathbf { s } \\in \\mathcal { R } ^ { n } \\end{align*}"}
+{"id": "1595.png", "formula": "\\begin{align*} | \\phi _ { \\mathrm { r } , m } | ^ { 2 } + | \\phi _ { \\mathrm { t } , m } | ^ { 2 } = 1 , \\forall m \\in \\mathcal { M } . \\end{align*}"}
+{"id": "843.png", "formula": "\\begin{align*} \\int _ { M } c ( t , p ) \\ , \\mathrm { d } \\mathcal { H } ^ d ( p ) = m \\end{align*}"}
+{"id": "5654.png", "formula": "\\begin{align*} s _ { m } = - \\sum _ { \\emptyset \\neq I \\subset \\{ 1 , \\dots , m \\} } ( - 1 ) ^ { | I | } \\langle a _ { I } \\rangle \\in \\mathbb { Z } [ R ^ { * } ] , \\end{align*}"}
+{"id": "2980.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\max _ { 1 \\leq i < j \\leq k } { d } ( T ^ { ( n , [ n \\beta + t ] ) } x _ i , T ^ { ( n , [ n \\beta + t ] ) } x _ j ) = 0 \\end{align*}"}
+{"id": "9172.png", "formula": "\\begin{align*} \\begin{aligned} b _ { 1 , \\delta } ( x ^ \\star ) = & \\ , 2 \\int _ { 0 } ^ { 1 } h ( x ^ \\star + \\delta \\sin ( 2 \\pi t ) ) \\sin ( 2 \\pi t ) \\ , d t \\\\ = & \\ , 2 \\int _ { 0 } ^ { 1 / 2 } h ( x ^ \\star + \\delta \\sin ( 2 \\pi t ) ) \\sin ( 2 \\pi t ) \\ , d t \\\\ & - 2 \\int _ { 0 } ^ { 1 / 2 } h ( x ^ \\star - \\delta \\sin ( 2 \\pi t ) ) \\sin ( 2 \\pi t ) \\ , d t \\end{aligned} \\end{align*}"}
+{"id": "9287.png", "formula": "\\begin{align*} A _ { p , N } : = \\sum _ { A = 0 } ^ { 2 n - 1 } \\int _ S \\left [ v _ p ( d _ 1 v _ { p + N } \\wedge \\alpha ) _ A \\tau ( \\mathbf { { n } } ) _ { A 0 } + v _ { p + N } ( d _ 0 v _ p \\wedge \\alpha ) _ A \\tau ( \\mathbf { { n } } ) _ { A 1 } \\right ] d S , \\end{align*}"}
+{"id": "1775.png", "formula": "\\begin{align*} \\phi _ 1 ( v _ 1 , v _ 1 v _ 2 ) = v _ 1 \\left ( a _ { 1 , 1 } + a _ { 1 , 2 } v _ 2 + v _ 1 \\Phi _ 1 \\right ) , \\phi _ 2 ( v _ 1 , v _ 1 v _ 2 ) = v _ 1 \\left ( a _ { 2 , 1 } + a _ { 2 , 2 } v _ 2 + v _ 1 \\Phi _ 2 \\right ) \\end{align*}"}
+{"id": "1269.png", "formula": "\\begin{align*} \\int _ \\Omega f ( \\eta ) \\mu ( d \\eta ) \\cdot \\left ( \\sum _ { m = 0 } ^ \\infty \\int _ \\Omega ( \\varphi _ n ( r _ m \\eta ) - \\varphi _ n ( r _ { m + 1 } \\eta ) ) \\mu ( d \\eta ) + \\varphi _ n ( \\mathbf { 1 } ) \\right ) = 0 . \\end{align*}"}
+{"id": "8363.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } ( V + V ^ H ) = \\left ( \\begin{array} { c c } 1 + | r ( k ) | ^ 2 & \\overline { r ( k ) } e ^ { - 2 i k ^ 2 x } \\\\ r ( k ) e ^ { 2 i k ^ 2 x } & 1 \\end{array} \\right ) , \\ \\ \\ k \\in \\mathbb { R } , \\\\ & \\frac { 1 } { 2 } \\left ( V + V ^ H \\right ) = \\left ( \\begin{array} { c c } 1 - | r ( k ) | ^ 2 & 0 \\\\ 0 & 1 \\end{array} \\right ) , \\ \\ \\ k \\in i \\mathbb { R } , \\end{align*}"}
+{"id": "6727.png", "formula": "\\begin{align*} \\mathrm { E } ( X ^ n ) = \\frac { 1 } { \\mathrm { B } ( \\alpha , \\beta ) } \\frac { 1 } { \\sigma } \\int _ { - \\infty } ^ \\infty x ^ n \\left [ \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] ^ { \\alpha - 1 } \\left [ 1 - \\Phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right ] ^ { \\beta - 1 } \\phi _ s \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\mathrm { d } x . \\end{align*}"}
+{"id": "868.png", "formula": "\\begin{align*} \\left | \\begin{array} { c c } x + y = x ' \\ , \\\\ x ' + y ' = y \\end{array} \\right . . \\end{align*}"}
+{"id": "2139.png", "formula": "\\begin{align*} \\Re \\varrho _ { Z Z } ( L , L ) & = - a ( b + 1 ) ^ 2 u ^ 2 + a ( b - 1 ) ^ 2 v ^ 2 - b ( 1 + a ) ^ 2 x ^ 2 + b ( a - 1 ) ^ 2 y ^ 2 , \\\\ \\Im \\varrho _ { Z Z } ( L , L ) & = - 2 a \\left ( b ^ 2 - 1 \\right ) u v - 2 b \\left ( a ^ 2 - 1 \\right ) x y . \\end{align*}"}
+{"id": "2317.png", "formula": "\\begin{align*} s ^ W = s ^ g - 3 \\ , \\delta ^ g \\theta - \\frac { 3 } { 2 } \\| \\theta \\| ^ 2 , \\end{align*}"}
+{"id": "1747.png", "formula": "\\begin{align*} J _ 0 ( t , x , y ) & : = \\det \\big ( \\nabla _ y S _ 0 ( t , x , y ) \\big ) , \\\\ D _ 0 ( t , x , y ) & : = J _ 0 ( t , x , y ) \\nabla _ y S _ 0 ( t , x , y ) ^ { - 1 } D ( x ) \\nabla _ y S _ 0 ( t , x , y ) ^ { - T } . \\end{align*}"}
+{"id": "4740.png", "formula": "\\begin{align*} & f _ k ( \\Tilde { x } ) \\le \\lambda f _ k ( x _ v ) + ( 1 - \\lambda ) f _ k ( x _ w ) < \\lambda v _ k + ( 1 - \\lambda ) w _ k . \\\\ & a _ k \\| \\Tilde { x } \\| ^ 2 + 2 b _ k ^ T \\Tilde { x } + c _ k \\le a _ k ( \\lambda \\| x _ v \\| ^ 2 + ( 1 - \\lambda ) \\| x _ w \\| ^ 2 ) + 2 b _ k ^ T ( \\lambda x _ v + ( 1 - \\lambda ) x _ w ) + c _ k . \\end{align*}"}
+{"id": "4401.png", "formula": "\\begin{align*} | Z _ 2 | & = \\left | \\frac { 1 } { \\log x } \\sum _ { q = 1 } ^ \\infty \\frac { x ^ { - ( 2 q + s ) } - x ^ { - 2 ( 2 q + s ) } } { ( 2 q + s ) ^ 2 } \\right | \\\\ & \\leq \\frac { 1 } { \\log x } \\left ( \\frac { x ^ { - ( 2 + \\sigma ) } + x ^ { - 2 ( 2 + \\sigma ) } } { t ^ 2 + 4 } + \\frac { 1 } { t ^ 2 + 1 6 } \\sum _ { q = 2 } ^ \\infty \\left ( x ^ { - ( 2 q + \\sigma ) } + x ^ { - 2 ( 2 q + \\sigma ) } \\right ) \\right ) \\\\ & \\leq \\frac { c _ 1 } { ( t ^ 2 + 4 ) x ^ { 2 + \\sigma } \\log x } , \\end{align*}"}
+{"id": "7731.png", "formula": "\\begin{align*} ( W \\times T \\times L ) / D = ( V \\times U \\times T \\times L ) / D \\cong V \\times K \\end{align*}"}
+{"id": "2986.png", "formula": "\\begin{align*} \\overline { W ^ { s } ( ( x , t ) , T _ { \\vec { v } } ) \\cap \\operatorname { s u p p } \\widetilde \\mu _ { ( x , t ) } } = \\operatorname { s u p p } \\widetilde \\mu _ { ( x , t ) } . \\end{align*}"}
+{"id": "2790.png", "formula": "\\begin{align*} \\mathbb { E } | \\Gamma _ { N } ^ { D } ( b ) \\backslash \\hat { \\Gamma } _ { N } ^ { D , M } ( b ) | \\leq \\ & \\mathbb { E } | \\Gamma _ { N } ^ { D } ( b ) \\backslash D _ { N } ^ { \\varepsilon } | + \\mathbb { E } | \\Gamma _ { N } ^ { D } ( b ' ) | \\\\ & + \\sum _ { x \\in D _ { N } ^ { \\varepsilon } } \\sum _ { k = k _ { N } } ^ { n ( x ) } \\mathbb { P } ( h ^ { D _ { N } } ( x ) - a _ { N } \\in [ b , b ' ) , T _ { N , M } ^ { k } ( x ) ) , \\end{align*}"}
+{"id": "8065.png", "formula": "\\begin{align*} \\left \\lvert 1 - 4 \\mu \\left ( \\sum _ { l = 1 } ^ { M } \\lvert \\phi ^ { \\left ( l , j \\right ) } \\rvert ^ 2 + \\sum _ { q = 1 } ^ { M - 1 } \\sum _ { r = q + 1 } ^ { M } f _ { q , r } ^ { \\left ( j \\right ) } \\right ) \\right \\rvert < 1 \\end{align*}"}
+{"id": "2715.png", "formula": "\\begin{align*} \\begin{aligned} & \\min _ { y _ 1 , y _ 2 } & & y _ 1 \\ ; \\ ; & & \\\\ & & & \\eta _ 1 y _ 2 - L _ 1 y _ 1 \\le ( 2 \\epsilon _ f + r ) / \\delta _ k - L _ 1 \\delta _ k / 2 \\ ; \\ ; & & \\\\ & ~ & & | y _ 1 | \\le \\| x _ k \\| y _ 2 \\ge \\min \\{ 1 0 ^ { - 6 } , 1 0 ^ { - 2 } \\| L _ 1 x _ k \\| \\} , & & \\end{aligned} \\end{align*}"}
+{"id": "1321.png", "formula": "\\begin{align*} M '' ( t ) = 2 | | u _ t | | ^ 2 _ { L ^ 2 ( \\mathbb { H } ^ n ) } - 2 I ( u ) , \\end{align*}"}
+{"id": "3722.png", "formula": "\\begin{align*} 0 & = \\int _ \\Sigma \\Big \\langle Q ( h , v ) , \\big ( k ^ \\intercal , H ' ( k ) \\big ) \\Big \\rangle d \\sigma \\\\ & = \\int _ \\Sigma \\Big \\langle \\big ( v A + \\bar u A ' ( h ) - \\nu ( v ) g ^ \\intercal , 2 v \\big ) , \\big ( - 2 \\zeta A ' ( h ) , \\zeta A \\cdot A ' ( h ) \\big ) \\Big \\rangle \\ , d \\sigma \\\\ & = - \\int _ \\Sigma 2 \\bar u \\zeta | A ' ( h ) | ^ 2 \\ , d \\sigma \\end{align*}"}
+{"id": "6411.png", "formula": "\\begin{align*} \\Phi ( p _ 0 ) = R { s ( p _ 0 ) } \\leq \\Phi ( p ) \\leq C ( \\delta , m _ \\delta ) \\end{align*}"}
+{"id": "7203.png", "formula": "\\begin{align*} \\frac { i } { ( 2 \\pi ) ^ { n } } \\int _ { \\mathbb { R } ^ { n - 1 } } \\int _ { \\mathcal { C } } e ^ { - t \\tau } \\operatorname { T r } \\phi _ { - 2 } \\ , d \\tau \\ , d \\xi = a _ 1 ( x ) t ^ { 2 - n } , \\end{align*}"}
+{"id": "8513.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { x \\in \\left ( x _ 0 , \\infty \\right ) } \\left | \\langle x \\rangle I _ 4 ^ { \\prime } ( x ) \\right | \\leq c \\left \\| r _ 2 \\right \\| _ { H ^ { 1 } \\cap L ^ { 2 , 1 } } \\left \\| r _ 1 \\right \\| _ { H ^ { 1 } \\cap L ^ { 2 , 1 } } \\left ( \\left \\| z ^ { - 1 } r _ 1 \\right \\| _ { H ^ { 1 } \\cap L ^ { 2 , 1 } } + \\left \\| r _ 2 \\right \\| _ { H ^ { 1 } \\cap L ^ { 2 , 1 } } \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "2444.png", "formula": "\\begin{align*} L _ { q } \\left [ \\int _ { 0 } ^ { \\infty } f ( x ) d x \\right ] = \\frac { 2 - q } { s } L _ { \\frac { 1 } { 2 - q } } \\left [ f ( t ) \\right ] \\left ( s ( 2 - q ) \\right ) . \\end{align*}"}
+{"id": "974.png", "formula": "\\begin{align*} e _ 1 : = \\alpha q ^ { d ^ 2 + d - 2 } , \\end{align*}"}
+{"id": "1641.png", "formula": "\\begin{align*} x ( 0 ) = x '' ( 0 ) = x ( 1 ) = x '' ( 1 ) = 0 . \\end{align*}"}
+{"id": "34.png", "formula": "\\begin{align*} | \\nabla _ { H } u | ^ 2 = \\sum _ { i = 1 } ^ m ( X _ i u ) ^ 2 . \\end{align*}"}
+{"id": "3644.png", "formula": "\\begin{align*} \\tau _ x = ( \\sigma _ x ^ { - 1 } + 1 / 2 ) ^ { - 1 } , \\end{align*}"}
+{"id": "394.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\varsigma } = \\{ \\mathcal { A } _ { \\varsigma } ( \\tau , \\omega ) : \\tau \\in \\R , \\omega \\in \\Omega \\} , \\end{align*}"}
+{"id": "3433.png", "formula": "\\begin{align*} | j _ 1 ( \\alpha - \\frac { p _ n } { q _ n } ) | = \\frac { j _ 1 } { q _ n } \\| q _ n \\alpha \\| \\leq \\frac { 1 } { 2 } \\| q _ n \\alpha \\| . \\end{align*}"}
+{"id": "2719.png", "formula": "\\begin{align*} g ( x ) = \\sum _ { i = 1 } ^ n \\frac { f ( x + \\sigma u _ i ) - f ( x ) } { \\sigma } u _ i . \\end{align*}"}
+{"id": "5633.png", "formula": "\\begin{align*} O _ { n , n } ( R ) : = \\{ A \\in G L _ { 2 n } ( R ) | \\ , \\ , ^ { t } A \\psi _ { 2 n } A = \\psi _ { 2 n } \\} . \\end{align*}"}
+{"id": "7700.png", "formula": "\\begin{align*} s _ { \\lfloor ( 1 - ( 1 + \\tilde \\varepsilon ) ^ { - j } ) r \\rfloor } ( F ) \\geq g \\big ( \\lfloor ( 1 - ( 1 + \\tilde \\varepsilon ) ^ { - j } ) r \\rfloor \\big ) , j = 1 , 2 , \\dots , i , \\end{align*}"}
+{"id": "8703.png", "formula": "\\begin{align*} d = 5 , a _ 1 ( F ) = 5 , a _ 1 ( F ' ) = 0 , a _ 2 ( F ' ) = 5 , a _ 3 ( F ' ) = 0 . \\end{align*}"}
+{"id": "7092.png", "formula": "\\begin{align*} N _ { P ^ * } ( c _ 4 ) = \\emptyset . \\end{align*}"}
+{"id": "3346.png", "formula": "\\begin{align*} | \\phi ^ p _ j \\cap \\phi ^ s _ j | = | \\phi ^ s _ j | - 1 . \\end{align*}"}
+{"id": "5382.png", "formula": "\\begin{align*} \\begin{aligned} 0 = \\ , & ( A \\Psi \\pm T _ \\alpha ( u - \\varphi ) ) _ n \\\\ = \\ , & - A + 2 A K x _ n \\\\ & \\pm \\left [ ( u - \\varphi ) _ { \\alpha n } + \\sum _ { \\beta < n } B _ { \\alpha \\beta } ( x _ { \\beta } ( u - \\varphi ) _ { n n } - x _ n ( u - \\varphi ) _ { \\beta n } ) - \\sum _ { \\beta < n } B _ { \\alpha \\beta } ( u - \\varphi ) _ { \\beta } \\right ] \\\\ < \\ , & 0 , \\end{aligned} \\end{align*}"}
+{"id": "3585.png", "formula": "\\begin{align*} \\C [ x ] & = \\C [ x ^ i _ j \\mid i \\in [ r ] , j \\in [ s _ i ] ] , \\\\ \\C [ y ] & = \\C [ y ^ i _ k \\mid i \\in [ r ] , k \\in [ t _ i ] ] , \\\\ \\C [ z ] & = \\C [ z ^ i _ { j k } \\mid i \\in [ r ] , j \\in [ s _ i ] , k \\in [ t _ j ] ] . \\end{align*}"}
+{"id": "8505.png", "formula": "\\begin{align*} I _ 2 ( x ) = - \\int _ { \\mathbb { R } } r _ 2 ( z ) M _ { + , 1 2 } ( x ; z ) \\mathrm { e } ^ { 2 i s x } \\mathcal { P } ^ + \\left ( z ^ { - 1 } \\bar { r } _ 1 ( z ) \\mathrm { e } ^ { - 2 i z x } \\right ) \\mathrm { d } z . \\end{align*}"}
+{"id": "7379.png", "formula": "\\begin{align*} N = m _ 1 s _ 1 + \\cdots + m _ L s _ L \\end{align*}"}
+{"id": "8754.png", "formula": "\\begin{align*} \\mathcal D _ { \\psi } f ( t z ) = \\psi _ 0 \\frac { \\partial f } { \\partial x } ( t z ) + \\psi _ 1 \\frac { \\partial f } { \\partial y } ( t z ) = 0 \\end{align*}"}
+{"id": "3434.png", "formula": "\\begin{align*} m _ n ^ { ( 1 ) } + \\ell _ n ^ { ( 1 ) } q _ n = k _ n . \\end{align*}"}
+{"id": "5856.png", "formula": "\\begin{align*} { f _ { 1 6 } } = { f _ { 1 5 } } + \\frac { 1 } { 2 } \\left ( { { f _ 3 } - { f _ 4 } } \\right ) + \\frac { 1 } { 2 } \\left ( { { f _ 7 } - { f _ 8 } } \\right ) - \\frac { 1 } { 2 } \\left ( { { f _ 9 } - { f _ { 1 0 } } } \\right ) + \\frac { 1 } { 8 } \\left ( { 2 { F _ y } + { F _ z } } \\right ) - \\frac { 1 } { 6 } \\rho { u _ z } , \\end{align*}"}
+{"id": "8112.png", "formula": "\\begin{align*} d ^ { o } _ { j i k } = \\begin{cases} e ^ { ( n ) } _ { 1 , i i k } + \\nu ^ { o } _ { i k } e ^ { * } _ { 3 , i k } & \\ , \\ , \\ , j = i , \\\\ \\frac { e ^ { ( n ) } _ { 1 , j i k } } { 1 + \\nu ^ { o } _ { i k } \\gamma _ { i k } } & \\end{cases} \\end{align*}"}
+{"id": "5508.png", "formula": "\\begin{align*} f '' ( x ) & = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { ( x + n ) ^ 2 } - \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { ( x + n + \\frac 1 2 ) ^ 2 } - \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { ( x + n + 3 / 2 ) ^ 2 } \\\\ & = \\frac { 1 } { x ^ 2 } - \\frac { 1 } { ( x + \\frac 1 2 ) ^ 2 } + \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { ( x + n ) ^ 2 } - 2 \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { ( x + n + \\frac 1 2 ) ^ 2 } . \\end{align*}"}
+{"id": "2947.png", "formula": "\\begin{align*} R _ { \\bar D } ( \\theta ) = \\begin{cases} & ( 0 \\leq \\theta \\leq \\frac { \\pi } { 4 } ) \\\\ \\theta & ( \\frac { \\pi } { 4 } < \\theta < \\frac { 3 } { 4 } \\pi ) \\\\ & ( \\frac { 3 } { 4 } \\pi < \\theta < \\frac { 5 } { 4 } \\pi ) \\\\ \\theta & ( \\frac { 5 } { 4 } \\pi < \\theta < \\frac { 7 } { 4 } \\pi ) \\\\ & ( \\frac { 7 } { 4 } \\pi \\leq \\theta < 2 \\pi ) \\end{cases} \\end{align*}"}
+{"id": "9157.png", "formula": "\\begin{align*} \\dot { x } _ 1 = - \\gamma h ( x _ 1 ) u x ( 0 ) = x _ { 1 0 } \\end{align*}"}
+{"id": "650.png", "formula": "\\begin{align*} P _ u : = R ^ { 2 - \\frac { 9 } { r } } \\left ( R ^ { \\frac { 3 } { r } } \\left ( \\frac { 1 } { R ^ 3 } \\int _ { \\mathcal { C } ( R / 2 , R ) } \\vert u \\vert ^ { p } d x \\right ) ^ { \\frac { 1 } { p } } \\right ) . \\end{align*}"}
+{"id": "2735.png", "formula": "\\begin{align*} h ( x _ 1 , x _ 2 ) = ( x _ 1 , x _ 2 , 1 - \\max \\{ | x _ 1 | , | x _ 2 | \\} ) . \\end{align*}"}
+{"id": "7438.png", "formula": "\\begin{align*} & \\bigg ( \\sum _ { j = 0 } ^ { k } \\mathbb { P } ( Y _ 1 + \\cdots + Y _ k = j , \\ , Z _ 1 + \\cdots + Z _ k = j ) \\bigg ) ^ { 1 / p _ k } \\\\ & \\leq \\mathbb { P } ( Y _ 1 = \\cdots = Y _ k = 0 , \\ , Z _ 1 = \\cdots = Z _ k = 1 ) ^ { 1 / p _ k } \\\\ & + \\mathbb { P } ( Y _ 1 = \\cdots = Y _ k = 1 , \\ , Z _ 1 = \\cdots = Z _ k = 0 ) ^ { 1 / p _ k } . \\end{align*}"}
+{"id": "1360.png", "formula": "\\begin{align*} \\phi ( Y ) = f ^ { - 1 } ( z _ 0 ) . \\end{align*}"}
+{"id": "6552.png", "formula": "\\begin{align*} \\begin{aligned} & \\chi ( x , r , s ) = \\chi ( y , r , s ) \\quad \\mbox { i f } | x | = | y | \\quad \\mbox { a n d } \\\\ & \\chi ( x , r , s ) + | \\partial _ { r } \\chi ( x , r , s ) | \\le \\chi _ { 0 } ( s ) \\quad \\mbox { w i t h s o m e } \\ , \\ , \\chi _ { 0 } \\in \\mathcal { C } ( \\R _ { + } ) . \\end{aligned} \\end{align*}"}
+{"id": "4218.png", "formula": "\\begin{align*} r ^ \\vee ( \\kappa + h ^ \\vee ) ( \\check { \\kappa } + \\check { h } ^ \\vee ) = 1 , r ^ \\vee ( \\check { \\kappa } + n + \\check { h } ^ \\vee ) ( \\varkappa + h ^ \\vee ) = 1 . \\end{align*}"}
+{"id": "8693.png", "formula": "\\begin{align*} \\abs { \\mathcal { T } } \\le 2 \\sum _ { i = 1 } ^ { C _ { B C } } \\abs { \\mathcal { B } ^ { ( i ) } } . \\end{align*}"}
+{"id": "8899.png", "formula": "\\begin{align*} \\Gamma _ 3 = \\mathrm { c o n e } ( \\mathbf { e } _ j , j \\in E ) & \\triangleq \\left \\{ \\sum _ { j \\in E } \\lambda _ j \\mathbf { e } _ j : \\lambda _ j \\geq 0 , j \\in E \\right \\} \\end{align*}"}
+{"id": "7273.png", "formula": "\\begin{align*} a _ { n - k } ( \\gamma ^ { ( q ^ { n - k } - q ^ n ) } - 1 ) \\beta ^ { q ^ { n - k } } + \\cdots + a _ 0 ( \\gamma ^ { 1 - q ^ n } - 1 ) \\beta = 0 . \\end{align*}"}
+{"id": "4275.png", "formula": "\\begin{align*} N = n ^ { 3 } + \\left ( n + 1 \\right ) ^ { 3 } = \\left ( n + a \\right ) ^ { 3 } - \\left ( n + a - \\beta \\right ) ^ { 3 } \\end{align*}"}
+{"id": "3671.png", "formula": "\\begin{align*} ( \\nabla _ V ( \\nabla X ) ) ^ i _ j = V _ k \\ , g ( \\nabla _ { e _ k } \\nabla _ { e _ j } X - \\nabla _ { \\nabla _ { e _ k } e _ j } X , e _ i ) : = V _ k X _ { i ; j k } . \\end{align*}"}
+{"id": "2918.png", "formula": "\\begin{align*} \\int _ { \\Omega } w \\varphi \\ , d x = \\langle \\mathcal A y + a y , \\varphi \\rangle = \\langle \\mathcal A ^ * \\varphi + a \\varphi , y \\rangle = \\langle h , y \\rangle \\ge 0 . \\end{align*}"}
+{"id": "5914.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Vert X ( t , x _ n ) - X ( t , x ) \\Vert _ H = 0 , \\ \\mathbb { P } . \\end{align*}"}
+{"id": "1120.png", "formula": "\\begin{align*} { R ^ * } = \\mathop { \\arg \\max } \\limits _ { { W _ s } \\in \\left [ { { W ^ { l o w } } , { W ^ { u p } } } \\right ] } \\ ; \\left ( { W - { W _ s } } \\right ) { \\log _ 2 } \\left ( { 1 + \\frac { { \\left [ { P - \\overline { { p _ K } \\left ( { \\overline S , { W _ s } } \\right ) } } \\right ] { { \\left | { { h _ b } } \\right | } ^ 2 } } } { { \\left ( { W - { W _ s } } \\right ) { N _ 0 } } } } \\right ) , \\end{align*}"}
+{"id": "7492.png", "formula": "\\begin{align*} & \\frac { ( n - 2 ) ( 1 0 - n ) } { 4 } \\int _ { B _ { \\rho } } r ^ { 2 - n } u _ r ^ 2 \\d x \\\\ & \\leq C \\int _ { B _ { 2 \\rho } \\setminus B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x + C \\varepsilon \\int _ { B _ { 2 \\rho } } r ^ { 4 - n } | D ^ 2 u | | \\nabla u | \\d x \\\\ & + C \\varepsilon \\int _ { B _ { 2 \\rho } } \\big ( r ^ { 3 - n } + \\varepsilon r ^ { 4 - n } \\big ) | \\nabla u | ^ 2 \\d x . \\end{align*}"}
+{"id": "1486.png", "formula": "\\begin{align*} \\det \\big ( M ( \\xi ) - \\lambda I \\big ) & = \\det \\left ( - ( \\xi _ 1 ^ 2 + \\xi _ 2 ^ 2 + \\cdots + \\xi _ d ^ 2 ) \\left ( \\begin{smallmatrix} 1 & 0 \\\\ 0 & \\epsilon \\end{smallmatrix} \\right ) + i \\xi _ 1 c I + \\left ( \\begin{smallmatrix} 0 & e ^ { - \\kappa } \\\\ 0 & - \\kappa e ^ { - \\kappa } \\end{smallmatrix} \\right ) \\right ) \\\\ & = 0 . \\end{align*}"}
+{"id": "6264.png", "formula": "\\begin{align*} \\phi _ { i s } ( \\partial _ r ) = 0 \\quad \\forall r \\neq i \\neq s \\neq r , \\end{align*}"}
+{"id": "2254.png", "formula": "\\begin{align*} Z = U D ( x ) V . \\end{align*}"}
+{"id": "3465.png", "formula": "\\begin{align*} | \\phi ( k ) | \\leq e ^ { 2 4 \\varepsilon q _ n } \\frac { e ^ { \\beta _ n q _ n } } { \\max ( | \\ell | , 1 ) } \\max \\begin{cases} c _ { n , \\ell - 2 } c _ { n , \\ell - 1 } e ^ { - 2 q _ n L } r _ { \\ell - 2 } \\\\ e ^ { - q _ n L } c _ { n , \\ell - 1 } r _ { \\ell - 1 } \\\\ \\max ( ( c _ { n , \\ell - 1 } ) ^ 2 , ( c _ { n , \\ell } ) ^ 2 ) e ^ { - q _ n L } r _ { \\ell } \\\\ c _ { n , \\ell } e ^ { - q _ n L } r _ { \\ell + 1 } \\\\ c _ { n , \\ell } c _ { n , \\ell + 1 } e ^ { - 2 q _ n L } r _ { \\ell + 2 } \\end{cases} \\end{align*}"}
+{"id": "4719.png", "formula": "\\begin{align*} \\frac { \\partial \\tilde y } { \\partial y } = g ^ { \\frac { 1 } { 4 } } + y \\cdot \\biggl ( \\frac { 1 } { 4 } g ^ { - \\frac { 3 } { 4 } } \\cdot \\frac { 1 } { 6 } \\int \\limits _ { 0 } ^ 1 s f _ { y ^ { 5 } } ^ { ( 5 ) } ( x , s y ) ( 1 - s ) ^ 3 \\ , d s \\biggr ) . \\end{align*}"}
+{"id": "6463.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { r _ n ^ 2 } { h _ n } = 0 . \\end{align*}"}
+{"id": "7116.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\mu \\circ ( \\eta \\otimes ( f \\circ \\eta ) ) & = & \\mu \\circ ( \\eta \\otimes i d _ M ) \\circ ( i d _ I \\otimes ( f \\circ \\eta ) ) \\\\ & = & \\lambda _ M \\circ ( i d _ I \\otimes ( f \\circ \\eta ) ) \\\\ & = & ( f \\circ \\eta ) \\circ \\lambda _ I . \\\\ & & \\mbox { ( b y n a t u r a l i t y o f $ \\lambda $ ) } \\end{array} \\end{align*}"}
+{"id": "8549.png", "formula": "\\begin{align*} \\phi ( t ) = \\frac { d } { d t } \\ , \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\ , \\int _ 0 ^ t ( t - \\tau ) ^ { - \\alpha } f ( \\tau ) \\ , d \\tau , \\ t > 0 . \\end{align*}"}
+{"id": "6343.png", "formula": "\\begin{align*} F ( x ) = \\frac { 1 } { \\operatorname { B } ( a , b ) } \\int ^ { \\infty } _ { \\frac { \\theta } { x ^ 2 } } \\exp ( - a u ) \\left \\{ 1 - \\exp ( - u ) \\right \\} ^ { b - 1 } \\mathrm { d } u . \\end{align*}"}
+{"id": "3824.png", "formula": "\\begin{align*} \\langle { \\boldsymbol { \\nu } } _ { x , t } , g ( A ( \\lambda , x , t ) , x , t ) \\rangle = \\langle { \\boldsymbol { \\nu } } _ { x , t } , f ( \\lambda , x , t ) \\rangle = \\langle { \\boldsymbol { N } } _ { x , t } , g ( \\rho , x , t ) \\rangle \\end{align*}"}
+{"id": "8865.png", "formula": "\\begin{align*} \\pi ( [ X ] ) \\wedge \\pi ( [ Y ] ) = [ f _ \\alpha '' X \\cap ( ( \\lambda \\setminus A _ \\alpha ) \\cup f _ \\alpha '' ( Y \\cap A _ \\alpha ) ] = [ f _ \\alpha '' X \\cap f '' _ \\alpha ( Y \\cap A _ \\alpha ) ] = [ f _ \\alpha '' ( X \\cap Y ) ) ] . \\end{align*}"}
+{"id": "5594.png", "formula": "\\begin{align*} W ( 1 0 ^ l | 0 ^ k 1 ) & = \\sum _ { j = 1 } ^ l \\left [ A ( 0 ^ { j + k } 1 . . . ) - A ( 0 ^ { j + \\alpha } 1 . . . ) \\right ] + A ( 1 0 ^ { l + k } 1 . . . ) - A ( 1 0 ^ { j + \\alpha } 1 . . . ) \\\\ & = \\sum _ { j = 1 } ^ l ( a _ { j + k } - a _ { j + \\alpha } ) + d _ { l + k } - d _ { l + \\alpha } \\ . \\end{align*}"}
+{"id": "4790.png", "formula": "\\begin{align*} F _ n ( \\eta ) : = T _ n - \\frac { 1 } { \\eta } I _ n . \\end{align*}"}
+{"id": "5404.png", "formula": "\\begin{align*} - \\epsilon _ 1 b _ { n - 1 } u _ n ( x ) \\leq u ( 0 ) - u ( x ) + \\sum _ { \\alpha = 1 } ^ { n - 1 } x _ \\alpha u _ \\alpha ( x ) \\leq C b _ { n - 1 } ^ 2 . \\end{align*}"}
+{"id": "6970.png", "formula": "\\begin{align*} \\Phi _ { n , e } ^ { ( i ) } ( x , y ) \\Psi _ { n , e } ^ { ( i ) } ( x , y ) = x ^ n - \\zeta ^ i _ { e } y ^ n . \\end{align*}"}
+{"id": "7099.png", "formula": "\\begin{align*} u \\cdot n = 0 \\end{align*}"}
+{"id": "6761.png", "formula": "\\begin{align*} a _ { 0 } ( x ) = e ( \\frac { i } { 2 } , x ) e ^ { \\frac { x } { 2 } } - 1 \\end{align*}"}
+{"id": "3016.png", "formula": "\\begin{align*} \\gamma _ k ( t , x , \\xi , y ) = \\varphi ( t , x , \\kappa _ t ( \\cdot ) ) + k \\| x - y \\| ^ 2 + k ( t - \\xi ) ^ 2 - \\epsilon _ \\circ ( \\xi - \\tau ) , \\end{align*}"}
+{"id": "8109.png", "formula": "\\begin{align*} \\xi _ { t } = h ( \\xi ) \\end{align*}"}
+{"id": "7849.png", "formula": "\\begin{align*} ( I - Q ^ \\dagger ) F ^ { q , \\dagger } _ { v v } ( 0 , p _ 0 ) [ v _ 1 , v _ 2 ] + F ^ { q , \\dagger } _ v ( 0 , p _ 0 ) \\psi ^ \\dagger _ { v v } ( 0 , p _ 0 ) [ v _ 1 , v _ 2 ] = 0 \\end{align*}"}
+{"id": "3316.png", "formula": "\\begin{align*} | \\Phi _ S ( \\sigma , \\tau ) | & = \\sum _ { i \\in [ m ] } | \\Phi _ S ( \\sigma , \\tau ) _ i | = | S | | \\sigma | + ( m - | S | ) | \\tau | . \\end{align*}"}
+{"id": "9283.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\int _ { \\Omega } d _ 0 u \\wedge d _ 1 v \\wedge \\alpha \\right | ^ 2 \\leq \\int _ { \\Omega } d _ 0 u \\wedge d _ 1 u \\wedge \\alpha \\cdot \\int _ { \\Omega } d _ 0 v \\wedge d _ 1 v \\wedge \\alpha . \\end{aligned} \\end{align*}"}
+{"id": "8028.png", "formula": "\\begin{align*} \\sum _ { i \\geq 1 \\atop { d \\geq 0 } } a _ { i , d } ( \\underline { b } ) = a ( \\underline { b } ) . \\end{align*}"}
+{"id": "7500.png", "formula": "\\begin{align*} m _ k : = \\left ( \\frac { 2 ^ { \\star } } { 2 } \\right ) ^ k m _ 0 . \\end{align*}"}
+{"id": "5172.png", "formula": "\\begin{align*} & \\left | \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi _ 1 + \\phi _ 2 - \\phi _ \\infty ) _ { + } G _ 1 \\frac { 1 } { r ^ { 2 } } \\dd x - \\int _ { \\mathbb { R } ^ { 3 } } ( \\phi _ 1 - \\phi _ \\infty ) _ { + } G _ 1 \\frac { 1 } { r ^ { 2 } } \\dd x \\right | \\\\ & \\leq \\int _ { \\mathbb { R } ^ { 3 } } | \\phi _ 2 | 1 _ { ( 0 , \\infty ) } ( | \\phi _ 1 + \\phi _ 2 | + | \\phi _ 1 | - \\phi _ { \\infty } ) | G _ 1 | \\frac { 1 } { r ^ { 2 } } \\dd x = 0 . \\end{align*}"}
+{"id": "1501.png", "formula": "\\begin{align*} I _ { a ^ { + } } ^ { 2 - \\alpha } u ( a ^ { + } ) = I _ { a ^ { + } } ^ { 2 - \\alpha } u ( b ) = 0 , \\end{align*}"}
+{"id": "8200.png", "formula": "\\begin{align*} x _ { 2 i - 1 , n + 1 - j } = \\tfrac { 1 } { 2 } S - x _ { 2 i - 1 , j } ; \\end{align*}"}
+{"id": "5337.png", "formula": "\\begin{align*} H _ { n , k , m } : & = H _ { 0 , m } \\left ( \\sum P _ { n , k } ( \\ell ) a _ { \\ell } z ^ { \\ell } \\right ) \\\\ & = \\begin{pmatrix} P _ { n , k } ( 0 ) a _ 0 & P _ { n , k } ( 1 ) a _ { 1 } & \\ldots & P _ { n , k } ( m ) a _ { m } \\\\ P _ { n , k } ( 1 ) a _ { 1 } & P _ { n , k } ( 2 ) a _ { 2 } & \\ldots & P _ { n , k } ( m + 1 ) a _ { m + 1 } \\\\ \\ldots \\\\ P _ { n , k } ( m ) a _ { m } & P _ { n , k } ( m + 1 ) a _ { m + 1 } & \\ldots & P _ { n , k } ( 2 m ) a _ { 2 m } \\end{pmatrix} , \\end{align*}"}
+{"id": "7209.png", "formula": "\\begin{align*} \\sigma _ { k , i } ( x ) = a _ { k , i } e _ { k } ( x ) , \\end{align*}"}
+{"id": "1809.png", "formula": "\\begin{align*} \\begin{aligned} P \\times V \\times G & \\to P \\times V \\\\ ( p , v ) \\cdot g & = ( p \\cdot g , g ^ { - 1 } v ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "2423.png", "formula": "\\begin{align*} L _ { q } ^ { - 1 } [ L _ { q } [ f ( t ) ] ] \\equiv L _ { q } ^ { - 1 } [ F _ { q } ( s ) ] = L _ { q } ^ { - 1 } \\left [ \\int _ { 0 } ^ { \\infty } d t \\exp _ { q } ( - s t ) f ( t ) \\right ] . \\end{align*}"}
+{"id": "5929.png", "formula": "\\begin{align*} I : = & \\mathbb { E } \\int _ 0 ^ { T \\wedge \\tau _ { X } ^ M } f ( s ) \\Vert X _ n ( s ) - X ( s ) \\Vert _ H ^ 2 d s , \\\\ I I : = & \\mathbb { E } \\int _ 0 ^ { T \\wedge \\tau _ { X } ^ M } \\rho ( X _ n ( s ) ) \\Vert X _ n ( s ) - X ( s ) \\Vert _ H ^ 2 d s , \\\\ I I I : = & \\mathbb { E } \\int _ 0 ^ { T \\wedge \\tau _ { X } ^ M } \\eta ( X ( s ) ) \\Vert X _ n ( s ) - X ( s ) \\Vert _ H ^ 2 d s . \\end{align*}"}
+{"id": "8816.png", "formula": "\\begin{align*} d z _ t = \\Pi _ { \\mathrm { k e r } A } ( B ( y _ t , y _ t ) - A y _ t ) d t + \\Pi _ { \\mathrm { k e r } A } \\sigma d W _ t . \\end{align*}"}
+{"id": "5707.png", "formula": "\\begin{align*} F : ( x , y ) \\mapsto F ( x , y ) = ( ( a _ 0 , b _ 0 , c _ 0 , d _ 0 ) _ q , ( a _ 1 , b _ 1 , c _ 1 , d _ 1 ) _ { q ' } ) , \\end{align*}"}
+{"id": "6117.png", "formula": "\\begin{align*} \\sigma _ \\mathfrak { s } ( \\boldsymbol { J } ; n , p ) = \\sigma _ \\mathfrak { s } ( \\boldsymbol { J } ; q , p ) \\quad \\ | n - q | \\leq 1 \\quad \\sigma _ \\mathfrak { s } ( \\boldsymbol { J } ; n , p ) = \\sigma _ \\mathfrak { s } ( \\boldsymbol { J } ; n , m ) \\quad \\ | p - m | \\leq 1 \\ , . \\end{align*}"}
+{"id": "4279.png", "formula": "\\begin{align*} \\left ( 2 n - \\beta \\left ( 3 \\beta ^ { 2 } + 4 \\right ) + 1 \\right ) \\left ( n ^ { 2 } + \\left ( 2 \\beta + 1 \\right ) n + \\beta ^ { 2 } + \\beta + 1 \\right ) = 0 \\end{align*}"}
+{"id": "8336.png", "formula": "\\begin{align*} m ( x , t ) = \\lim _ { k \\rightarrow \\infty } ( k \\psi ( x , t ; k ) ) _ { 1 2 } . \\end{align*}"}
+{"id": "3017.png", "formula": "\\begin{align*} \\gamma _ k ( t _ k , x _ k , \\xi _ k , y _ k ) = \\min \\limits _ { ( t , x , \\xi , y ) \\in \\Omega _ \\delta \\times \\Omega _ \\delta ^ * } \\gamma _ k ( t , x , \\xi , y ) . \\end{align*}"}
+{"id": "170.png", "formula": "\\begin{gather*} f _ i T _ j ^ { \\frac { 1 } { p ^ n } } = \\zeta _ { p ^ n } ^ { \\delta _ { i , j } } T _ j ^ { \\frac { 1 } { p ^ n } } \\\\ f _ i S _ j ^ { \\frac { 1 } { p ^ n } } = \\zeta _ { p ^ n } ^ { \\delta _ { i , j } } S _ j ^ { \\frac { 1 } { p ^ n } } \\end{gather*}"}
+{"id": "147.png", "formula": "\\begin{align*} z _ i = \\lambda _ i \\cdot \\left ( { 1 \\over 1 + \\sum _ { j \\in \\mathbb Z _ 0 } z _ j } \\right ) ^ k , \\ \\ i \\in \\mathbb Z _ 0 . \\end{align*}"}
+{"id": "8671.png", "formula": "\\begin{align*} { { \\bf { W } } ^ { \\star } } = \\left [ { { { \\bf { W } } ^ { \\star } } , { { \\bf { 0 } } _ { \\left ( { \\sum { { { \\bar r } _ { l ' } } } } \\right ) \\times \\left ( { { M _ t } - K } \\right ) } } } \\right ] \\left [ \\begin{array} { l } { { \\bf { I } } _ K } \\\\ { { \\bf { 0 } } _ { \\left ( { { M _ t } - K } \\right ) \\times K } } \\end{array} \\right ] . \\end{align*}"}
+{"id": "4938.png", "formula": "\\begin{align*} f _ { a } ( x ) = \\dfrac { a } { a ^ 2 + x ^ 2 } . \\end{align*}"}
+{"id": "1624.png", "formula": "\\begin{align*} C ( H , K , M ^ \\vee ) : = \\bigoplus _ { n \\geq 0 } \\ , C _ n ( H , K , M ^ \\vee ) \\end{align*}"}
+{"id": "2959.png", "formula": "\\begin{align*} \\Phi ( t , x , y ) = ( x e ^ { t } , y e ^ { t } ) \\end{align*}"}
+{"id": "2822.png", "formula": "\\begin{align*} \\mathbf { r } ^ { ( k ) } = \\mathbf { x } ^ { ( k - 1 ) } - \\rho \\Phi ^ { \\top } \\left ( \\Phi \\mathbf { x } ^ { ( k - 1 ) } - \\mathbf { y } \\right ) , \\end{align*}"}
+{"id": "7034.png", "formula": "\\begin{align*} A _ { 1 , 1 } \\ , : = \\ , - \\ , \\tfrac { 1 } { 3 } \\ , F _ { 6 , 0 } \\ , T _ 1 - T _ 2 . \\end{align*}"}
+{"id": "1731.png", "formula": "\\begin{align*} \\Delta ( y _ m - y ) = & U ( t , 0 ) \\Delta ( x _ m - x ) + \\int _ 0 ^ t U ( t , s ) \\{ 2 i ( D _ j \\nabla \\tilde { b } ^ j ( s ) + \\nabla \\tilde { b } ^ j ( s ) D _ j + \\nabla \\tilde { c } ( s ) ) \\\\ & \\times \\nabla ( y _ m - y ) + i ( D _ j \\Delta \\tilde { b } ^ j ( s ) + \\Delta \\tilde { b } ^ j ( s ) D _ j + \\Delta \\tilde { c } ( s ) ) ( y _ m - y ) \\\\ & - \\lambda i \\Delta [ e ^ { ( \\alpha - 1 ) W } ( g ( y _ m ( s ) ) - g ( y ( s ) ) ) ] \\} d s , \\end{align*}"}
+{"id": "6061.png", "formula": "\\begin{align*} \\kappa ( s ) = \\alpha \\ , \\frac { \\nu _ { n + 1 } ( s ) } { x _ { n + 1 } ( s , t ) } + \\varpi . \\end{align*}"}
+{"id": "11.png", "formula": "\\begin{align*} \\alpha \\wedge \\beta : = \\{ \\gamma \\in R ^ + \\mid \\gamma = \\alpha + \\beta - \\delta \\delta \\in \\alpha \\vee \\beta \\} \\end{align*}"}
+{"id": "7605.png", "formula": "\\begin{align*} x ( i ) = \\begin{cases} 1 _ { 2 \\Z } ( i ) & i \\in [ a ( n ) , a ( n ) + n ] n \\\\ 1 _ { 3 \\Z } ( i ) & i \\in [ b ( n ) , b ( n ) + n ] n \\\\ 1 _ { 5 \\Z } ( i ) & i \\in [ a ( n ) + b ( n ) , a ( n ) + b ( n ) + n ] n \\\\ 0 & \\\\ \\end{cases} \\end{align*}"}
+{"id": "8220.png", "formula": "\\begin{align*} M N A P P ( m , n ) = & \\biggl \\{ A \\in N A P P ( m , n ) : A \\mbox { i s t h e m i n i m a l e l e m e n t i n t h e s e t } \\\\ & \\{ A , \\rho ( A ) , \\sigma ( A ) , ( \\sigma \\rho ) ( A ) \\} \\mbox { u n d e r t h e l e x i c o g r a p h i c o r d e r } \\biggr \\} \\end{align*}"}
+{"id": "2165.png", "formula": "\\begin{align*} p ( g _ { n , k } ) = ( f _ { n , k , } ) ^ { w _ { n , k } } ( f _ { n , k , } ) ^ { ( 1 - w _ { n , k } ) } , \\end{align*}"}
+{"id": "1226.png", "formula": "\\begin{align*} \\sup _ { y \\in \\Z ^ d } \\sum _ { z \\neq y } \\sum _ { \\Delta \\ni y } \\sum _ { \\xi _ \\Delta } \\sum _ { i = 1 } ^ q \\norm { \\nabla ^ i _ z c _ \\Delta ( \\cdot , \\xi _ \\Delta ) } _ \\infty < \\infty . \\end{align*}"}
+{"id": "6629.png", "formula": "\\begin{align*} \\begin{aligned} X ^ { \\epsilon } ( t ) & = x _ { 0 } + \\int ^ { t } _ { 0 } b ( X ^ { \\epsilon } ( s ) ) d s + \\lambda ( \\epsilon ) \\int ^ { t } _ { 0 } \\sigma ( X ^ { \\epsilon } ( s ) ) d \\Theta _ { \\epsilon } ( s ) \\end{aligned} \\end{align*}"}
+{"id": "4382.png", "formula": "\\begin{align*} \\int _ { X _ j } | u _ { \\epsilon , j } | ^ 2 _ { h } e ^ { v _ \\epsilon ( \\Psi ) - \\delta \\max \\{ \\psi + T , 2 \\log | F | \\} } c ( - v _ \\epsilon ( \\Psi ) ) \\le \\int _ { X _ j } v '' _ { \\epsilon } ( \\Psi ) | f F | ^ 2 _ h e ^ { - u ( - v _ { \\epsilon } ( \\Psi ) ) } < + \\infty . \\end{align*}"}
+{"id": "2673.png", "formula": "\\begin{align*} \\Bigl ( \\frac { 1 } { \\lambda ( t ) } \\mathcal { G } ( t ) \\Bigr ) ^ { \\prime } \\leq - \\frac { k _ 3 } { k _ 2 } \\frac { \\alpha ( t ) } { \\lambda ( t ) } \\Bigl [ \\frac { 1 } { \\lambda ( t ) } \\mathcal { G } ( t ) \\Bigr ] ^ { \\frac { m } { \\ell } } . \\end{align*}"}
+{"id": "9187.png", "formula": "\\begin{align*} \\begin{aligned} | x ( t ) - z ( t ) | \\le & | x ( 0 ) - z ( 0 ) | e ^ { \\gamma L _ r t } \\\\ & + \\left ( e ^ { \\gamma L _ r t } - 1 \\right ) \\dfrac { \\gamma \\bar k ( L _ r , M _ r , \\delta ) } { L _ r } . \\end{aligned} \\end{align*}"}
+{"id": "160.png", "formula": "\\begin{align*} { \\rm e x } ( n , H _ 3 ^ r ) & \\ : \\geq \\ : \\frac { 1 } { n } \\binom { n } { r } \\ : + \\ : \\frac { 1 } { r } \\sum _ { d = 0 } ^ { \\lfloor n / r \\rfloor - 1 } \\binom { n - 1 - r d } { r - 1 } \\ , . \\end{align*}"}
+{"id": "366.png", "formula": "\\begin{align*} P y = y . \\end{align*}"}
+{"id": "531.png", "formula": "\\begin{align*} \\partial _ { t } k _ { t } ( x , y ) = & \\ - [ ( \\sinh 2 t ) ^ { - \\frac { 3 } { 2 } } \\cosh 2 t e ^ { - \\varphi ( t , x , y ) } + ( \\sinh 2 t ) ^ { - \\frac { 1 } { 2 } } e ^ { - \\varphi ( t , x , y ) } \\partial _ { t } \\varphi ( t , x , y ) ] \\\\ = : & \\ - [ A + B ] , \\end{align*}"}
+{"id": "2903.png", "formula": "\\begin{align*} \\mu _ \\infty ( \\dd { \\bf q } ' , \\dd { \\bf p } ' ) = \\mu _ \\infty { \\cal P } _ { 0 , \\theta } ( \\dd { \\bf q } ' , \\dd { \\bf p } ' ) \\ge c _ * \\mu _ \\infty { \\cal Q } _ { 0 , \\theta } ( { \\bf q } ' , { \\bf p } ' ) \\dd { \\bf q } ' \\dd { \\bf p } ' . \\end{align*}"}
+{"id": "3555.png", "formula": "\\begin{align*} h _ { s , t } ^ { m , n } = \\frac { ( m t - n s ) ^ 2 - ( s - t ) ^ 2 } { 4 s t } . \\end{align*}"}
+{"id": "4000.png", "formula": "\\begin{align*} - \\Delta u = F \\quad \\Omega , u = g \\quad \\partial \\Omega \\end{align*}"}
+{"id": "2816.png", "formula": "\\begin{align*} \\alpha _ { k } ^ { \\mathrm { o p t } } = 1 + \\frac { \\sum _ { l \\in \\mathcal { M } _ { k } } \\overline { \\mathbf { h } } _ { k l } ^ { \\mathbf { H } } \\mathbf { z } _ { k l } \\mathbf { z } _ { k l } ^ { \\mathbf { H } } \\overline { \\mathbf { h } } _ { k l } } { \\sum _ { i \\in \\mathcal { K } _ { l } \\setminus \\{ k \\} } \\sum _ { l \\in \\mathcal { M } _ { i } } \\overline { \\mathbf { h } } _ { k l } ^ { \\mathbf { H } } \\mathbf { z } _ { i l } \\mathbf { z } _ { i l } ^ { \\mathbf { H } } \\overline { \\mathbf { h } } _ { k l } + \\delta _ { } ^ 2 } . \\end{align*}"}
+{"id": "93.png", "formula": "\\begin{align*} G ( \\breve F ) = \\bigsqcup _ { x \\in \\widetilde W } I x I , \\end{align*}"}
+{"id": "8078.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ y _ i ^ * y _ i | \\hat { \\mathbf { H } } \\right ] = & a _ c ^ 2 \\left ( \\lvert \\hat { \\phi } ^ { \\left ( i , c \\right ) } \\rvert ^ 2 + \\sigma _ { e , i } ^ 2 \\lVert \\mathbf { p } _ c \\rVert ^ 2 \\right ) + \\sum _ { j = 1 } ^ { K } a _ j ^ 2 \\left ( \\lvert \\hat { \\phi } ^ { \\left ( i , j \\right ) } \\rvert ^ 2 + \\sigma _ { e , i } ^ 2 \\lVert \\mathbf { p } _ j \\rVert ^ 2 \\right ) + \\sigma _ n ^ 2 . \\end{align*}"}
+{"id": "7256.png", "formula": "\\begin{align*} P ^ g ( v _ 1 , \\ldots , v _ n ) = P ( w _ 1 , \\ldots , w _ n ) . \\end{align*}"}
+{"id": "8853.png", "formula": "\\begin{align*} X _ { t , 2 } = e ^ { - \\int _ 0 ^ t X _ { s , 3 } d s } X _ { 0 , 2 } + \\int _ 0 ^ t e ^ { - \\int _ s ^ t X _ { s ' , 3 } d s ' } d W _ s . \\end{align*}"}
+{"id": "5154.png", "formula": "\\begin{align*} I _ h \\leq E [ b ] \\leq \\liminf _ { n \\to \\infty } E [ b _ n ] = I _ h , \\end{align*}"}
+{"id": "559.png", "formula": "\\begin{align*} g _ { \\theta } ( \\omega ) : = \\frac { 1 } { 2 \\pi i } \\int _ { e ^ { i \\theta } [ 0 , + \\infty ) } f ( \\zeta ) e ^ { \\omega \\zeta } d \\zeta , \\omega \\in \\Omega _ { \\theta } , \\end{align*}"}
+{"id": "3268.png", "formula": "\\begin{align*} \\Delta _ n ^ { 1 - \\frac m 2 } \\langle S A M P V _ t ^ n ( m _ 1 , . . . , m _ k ) , & \\bigotimes _ { i = 1 } ^ k \\bigotimes _ { j = 1 } ^ { m _ j } e _ { p _ { i , j } } \\rangle _ { \\mathcal H ^ m } \\\\ & \\stackrel { u . c . p . } { \\longrightarrow } \\langle \\int _ 0 ^ t \\rho _ { \\Sigma _ s } ^ { \\otimes k } ( m _ 1 , . . . , m _ k ) d s , \\bigotimes _ { i = 1 } ^ k \\bigotimes _ { j = 1 } ^ { m _ j } e _ { p _ { i , j } } \\rangle _ { \\mathcal H ^ m } , \\end{align*}"}
+{"id": "938.png", "formula": "\\begin{align*} w ( t , \\xi ) : = e ^ { 3 i \\lambda _ 1 ( \\Phi _ 1 ( t ) - \\theta _ 1 ) } v _ 2 ( t ) . \\end{align*}"}
+{"id": "6352.png", "formula": "\\begin{align*} h ( x ) = \\frac { 2 \\theta } { \\operatorname { B } ( a , b ) \\ , x ^ 3 } \\frac { \\left [ 1 - \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) \\right ] ^ { b - 1 } \\exp \\left ( - \\frac { a \\theta } { x ^ 2 } \\right ) } { I _ { 1 - \\exp \\left ( - \\frac { \\theta } { x ^ 2 } \\right ) } ( b , a ) } . \\end{align*}"}
+{"id": "5237.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 5 } } \\left ( \\varphi _ 1 ( w ) - \\frac { W } { 2 } \\right ) _ { + } ^ { 2 } \\dd w = \\frac { 8 \\pi ^ { 2 } } { 3 } C _ 1 ^ { 2 } \\int _ { 0 } ^ { c _ { 3 / 2 } } \\rho J _ { 3 / 2 } ^ { 2 } ( \\rho ) \\dd \\rho , \\end{align*}"}
+{"id": "2767.png", "formula": "\\begin{align*} F ( P ) = B \\ ! \\left ( a , \\frac { q } { 2 } \\right ) \\subset B \\ ! \\left ( F ( p ( s , k ) ) , \\frac { 3 q } { 5 } \\right ) \\subset B \\ ! \\left ( F ( p ( s , k ) ) , \\frac 2 3 | F ( p ( s , k ) ) | \\right ) . \\end{align*}"}
+{"id": "2712.png", "formula": "\\begin{align*} \\begin{aligned} \\langle x _ k , \\alpha _ 1 x _ k + \\alpha _ 2 v \\rangle & = y _ 1 y _ 2 \\\\ \\| \\alpha _ 1 x _ k + \\alpha _ 2 v \\| & = y _ 2 , \\end{aligned} \\end{align*}"}
+{"id": "5825.png", "formula": "\\begin{align*} S _ i = { { w } _ i } \\rho c _ v \\frac { \\Delta t } { 2 } \\partial _ t ^ 2 T , \\end{align*}"}
+{"id": "61.png", "formula": "\\begin{align*} \\# \\{ \\beta \\in \\Phi ^ - \\setminus ( - I ) \\mid \\varphi ( v ' \\beta ) > 0 \\} = & \\# \\{ \\beta \\in \\Phi ^ - \\setminus ( - I \\cup \\{ - \\alpha \\} ) \\mid \\varphi ( v ' \\beta ) > 0 \\} \\\\ = & \\# \\{ \\beta \\in \\Phi ^ - \\setminus ( - I \\cup \\{ - \\alpha \\} ) \\mid \\varphi ( v \\beta ) > 0 \\} \\\\ \\leq & \\# \\{ \\beta \\in \\Phi ^ - \\setminus ( - I ) \\mid \\varphi ( v \\beta ) > 0 \\} . \\end{align*}"}
+{"id": "2183.png", "formula": "\\begin{align*} C R B = \\frac { \\lambda ^ 2 } { 8 { \\pi } ^ 2 K S N R { c o s } ^ 2 \\theta \\overline { d ^ 2 } } \\end{align*}"}
+{"id": "7493.png", "formula": "\\begin{align*} \\int _ { B _ { 1 / 2 } } r ^ { 4 - n } | D ^ 2 u | | \\nabla u | \\d x & = \\sum _ { j = 0 } ^ { \\infty } \\int _ { B _ { r _ { j + 1 } } \\setminus B _ { r _ { j + 2 } } } r ^ { 4 - n } | D ^ 2 u | | \\nabla u | \\d x \\\\ & \\leq C \\sum _ { j = 0 } ^ { \\infty } r _ j ^ { 4 - n } \\int _ { B _ { r _ { j + 1 } } \\setminus B _ { r _ { j + 2 } } } | D ^ 2 u | | \\nabla u | \\d x , \\end{align*}"}
+{"id": "2262.png", "formula": "\\begin{align*} a _ { j k } x _ k + x _ j b _ { j k } = \\delta _ { j k } v _ j , x _ j a _ { j k } + b _ { j k } x _ k = - \\delta _ { j k } v _ j , \\end{align*}"}
+{"id": "6903.png", "formula": "\\begin{align*} \\alpha _ i ( t , x _ 1 , \\ldots , x _ n ) = \\widehat \\alpha _ i ( t , x _ i ) , \\end{align*}"}
+{"id": "4338.png", "formula": "\\begin{align*} \\int _ { \\{ - t ' _ 1 \\le \\Psi < - t ' _ 2 \\} } | \\tilde { F } | ^ 2 _ h = \\frac { G ( T _ 1 ) } { \\int _ { T _ 1 } ^ { + \\infty } c ( s ) e ^ { - s } d s } \\int _ { t ' _ 2 } ^ { t ' _ 1 } e ^ { - s } d s . \\end{align*}"}
+{"id": "784.png", "formula": "\\begin{align*} g ' ( c ) = \\big ( G ( c ) - G ' ( c ) c \\big ) ' = G ' ( c ) - G '' ( c ) c - G ' ( c ) = - G '' ( c ) c < 0 \\end{align*}"}
+{"id": "319.png", "formula": "\\begin{align*} g _ b ( f ) ( x ) = \\Big ( \\int _ { 0 } ^ { \\infty } \\Big | \\frac { d } { d t } e ^ { - t L } ( b ( x ) - b ( \\cdot ) ) ( f ) ( x ) \\Big | ^ 2 t d t \\Big ) ^ { 1 / 2 } ; \\end{align*}"}
+{"id": "4627.png", "formula": "\\begin{align*} \\phi ( \\sigma , \\tau ) = [ \\widetilde \\sigma , \\widetilde \\tau ] = [ \\widetilde \\tau , \\widetilde \\sigma ] ^ { - 1 } = [ \\widetilde \\tau , \\widetilde \\sigma ] = \\phi ( \\tau , \\sigma ) . \\end{align*}"}
+{"id": "373.png", "formula": "\\begin{align*} \\widetilde { C } _ { n - 1 - k } ( r , s ) & = \\cos { ( \\frac { 2 \\pi } { n - 1 } ( r - s - 1 ) ( n - 1 - k ) ) } \\\\ & = \\cos { ( 2 \\pi ( r - s - 1 ) - \\frac { 2 \\pi } { n - 1 } ( r - s - 1 ) k ) } \\\\ & = \\widetilde { C } _ k ( r , s ) . \\end{align*}"}
+{"id": "4140.png", "formula": "\\begin{align*} \\bigg ( \\frac { 1 } { | Q | } \\iint _ { T _ Q } | \\Theta f ( x ' , t ) | ^ 2 \\frac { d x ' d t } { t } \\bigg ) ^ { \\frac 1 2 } = \\bigg ( \\frac { 1 } { | Q | } \\iint _ { T _ Q } | \\Theta ( f - f _ Q ) ( x ' , t ) | ^ 2 \\frac { d x ' d t } { t } \\bigg ) ^ { \\frac 1 2 } \\le { \\rm I } + { \\rm I I } , \\end{align*}"}
+{"id": "7730.png", "formula": "\\begin{align*} r s [ R , G ] = s r [ R , G ] r s r ' s ' [ R , G ] = r r ' s s ' [ R , G ] . \\end{align*}"}
+{"id": "4850.png", "formula": "\\begin{align*} & G ( c _ 1 , c _ 2 , \\dots , c _ n , m ) \\\\ & = G ( H _ 1 ( c _ 1 , \\dots , c _ n , m ) , \\dots , H _ n ( c _ 1 \\dots , c _ n , m ) ) \\\\ & = ( p _ 1 ( c _ 1 , c _ 2 , \\dots , c _ n ) , \\dots , p _ n ( c _ 1 , . . . , c _ n ) ) \\\\ & = ( c _ 1 , c _ 2 , \\dots , c _ n ) . \\end{align*}"}
+{"id": "4927.png", "formula": "\\begin{align*} \\ln \\left \\{ 1 - \\Phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) \\right \\} = \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ n } { n } \\ , \\Phi \\left ( \\frac { x - \\mu } { \\sigma } \\right ) ^ n . \\end{align*}"}
+{"id": "2615.png", "formula": "\\begin{align*} s ( x \\cdot y ) \\beta ( v ) = s ( \\alpha ( x ) ) s ( y ) v . \\end{align*}"}
+{"id": "3986.png", "formula": "\\begin{align*} \\bar { \\mathbf { b } } + \\bar { \\mathbf { c } } = ( \\bar { \\beta } _ 1 + \\bar { \\gamma } _ 1 , \\bar { \\beta } _ 2 + \\bar { \\gamma } _ 2 , b _ 1 + \\bar { \\gamma } _ 3 , b _ 2 + c _ 1 , b _ 3 + c _ 2 , \\cdots , b _ { 2 k - 1 } + c _ { 2 k - 2 } , \\bar { \\beta } _ 3 + c _ { 2 k - 1 } , \\bar { \\beta } _ 4 + \\bar { \\gamma } _ 4 ) . \\end{align*}"}
+{"id": "5033.png", "formula": "\\begin{align*} \\gamma \\alpha \\omega - \\gamma \\omega = \\dfrac { ( \\alpha \\omega - \\omega ) \\det ( \\gamma ) } { ( c \\omega + d ) ( c \\alpha \\omega + d ) } . \\end{align*}"}
+{"id": "6554.png", "formula": "\\begin{align*} \\mbox { e i t h e r } T _ { \\rm m a x } = \\infty \\mbox { o r } \\limsup _ { t \\nearrow T _ { \\rm m a x } } \\| n ( \\cdot , t ) \\| _ { L ^ { \\infty } ( \\Omega ) } = \\infty . \\end{align*}"}
+{"id": "5040.png", "formula": "\\begin{align*} u = 0 , ( \\nabla \\times B ) \\times n = 0 , B \\cdot n = 0 , \\end{align*}"}
+{"id": "5267.png", "formula": "\\begin{align*} f _ + = \\frac { 1 } { ( 1 0 8 ) ^ { \\frac { 1 } { 4 } } } \\left ( 0 , 3 , 0 , - 1 \\right ) , f _ - = \\frac { 1 } { \\sqrt { 2 } } \\left ( 0 , 1 , 0 , 1 \\right ) . \\end{align*}"}
+{"id": "7446.png", "formula": "\\begin{align*} { U _ \\theta } ^ \\top U _ \\theta = \\begin{bmatrix} \\cos ^ 2 \\theta \\dfrac { m } { k } V ^ \\top V & \\vline & \\cos \\theta \\sin \\theta \\dfrac { m } { \\sqrt { k l } } V ^ \\top W \\\\ \\hline \\cos \\theta \\sin \\theta \\dfrac { m } { \\sqrt { k l } } W ^ \\top V & \\vline & \\sin ^ 2 \\theta \\dfrac { m } { l } W ^ \\top W \\end{bmatrix} . \\end{align*}"}
+{"id": "4276.png", "formula": "\\begin{align*} a = \\left ( \\beta - 1 \\right ) n + \\beta ^ { 2 } + \\beta + 1 \\end{align*}"}
+{"id": "1175.png", "formula": "\\begin{align*} H \\left ( P ^ { ( n , n ) } [ t ] \\mid \\mu ^ { \\otimes n } [ t ] \\right ) = \\mathbb { E } _ \\mathbb { P } [ Z _ t ^ n \\log Z _ t ^ n ] . \\end{align*}"}
+{"id": "3226.png", "formula": "\\begin{align*} \\mathcal S ( t ) h ( x ) : = \\begin{cases} h ( x + t ) , & x + t \\leq 1 , \\\\ h ( 1 ) , & x + t > 1 . \\end{cases} \\end{align*}"}
+{"id": "8366.png", "formula": "\\begin{align*} \\mathcal { C } _ { w } f = \\mathcal { P } ^ + \\left ( f w _ - \\right ) + \\mathcal { P } ^ { - } \\left ( f w _ + \\right ) = \\mathcal { P } ^ - ( f J ) . \\end{align*}"}
+{"id": "5636.png", "formula": "\\begin{align*} \\psi _ { 2 n } = \\begin{pmatrix} \\psi _ { 2 } \\\\ & \\psi _ { 2 } \\\\ & & \\ddots \\\\ & & & \\psi _ { 2 } \\end{pmatrix} = \\bigoplus _ { 1 } ^ { n } \\psi _ { 2 } , \\psi _ { 2 } = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "105.png", "formula": "\\begin{align*} & \\left \\langle \\frac 1 N v ^ { - 1 } \\sum _ { k = 1 } ^ N ( \\sigma \\circ w ) ^ k \\mu , \\alpha \\right \\rangle \\\\ = & \\frac 1 N \\sum _ { k = 1 } ^ N \\langle \\mu , ( \\sigma \\circ w ) ^ k v \\alpha \\rangle \\\\ = & \\frac 1 N \\sum _ { k = 1 } ^ N \\langle \\mu , ( \\sigma \\circ w ) ^ k v \\alpha \\rangle + \\Phi ^ + ( ( \\sigma \\circ w ) ^ k v \\alpha ) - \\Phi ^ + ( ( \\sigma \\circ w ) ^ { k + 1 } v \\alpha ) \\\\ = & \\frac 1 N \\sum _ { k = 1 } ^ N \\ell ( x , ( \\sigma \\circ w ) ^ k v \\alpha ) . \\end{align*}"}
+{"id": "1875.png", "formula": "\\begin{align*} E ( u ) = \\int _ M \\frac 1 p \\bigg ( \\sum _ { i = 1 } ^ m h \\big ( d u ( e _ i ) , d u ( e _ i ) \\big ) \\bigg ) ^ { \\frac p 2 } \\ , d v = \\int _ M \\ , \\frac { 1 } { p } \\big ( { \\sigma _ 1 } ( u ^ { \\ast } ) \\big ) ^ { \\frac p 2 } \\ , d v . \\end{align*}"}
+{"id": "723.png", "formula": "\\begin{align*} R _ { k , i } = \\log _ { 2 } \\bigg ( 1 + \\frac { P _ { k , i } { \\lvert \\mathbf { d } _ { k , i } ^ { H } \\mathbf { h } _ { k } \\rvert } ^ { 2 } } { \\sum _ { \\pi _ { k ' , i ' } > \\pi _ { k , i } } P _ { k ' , i ' } { \\lvert \\mathbf { d } _ { k ' , i ' } ^ { H } \\mathbf { h } _ { k ' } \\rvert } ^ { 2 } + \\sigma _ { \\textrm { u l } } ^ { 2 } } \\bigg ) , \\end{align*}"}
+{"id": "9310.png", "formula": "\\begin{align*} b ( x , y ) = \\begin{cases} ( x , x - 4 ^ { m } p ) & x \\neq 0 , 4 ^ { m } p , \\\\ ( - 1 , - p ) & x = 0 , \\\\ ( p , q ) & x = 4 ^ { m } p , \\\\ ( 1 , 1 ) & P = \\mathcal { O } , \\end{cases} \\end{align*}"}
+{"id": "5524.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial y } ( x , y ) = \\frac { 1 } { y + 1 / 2 } - \\frac { 1 } { x + y } - \\frac { 1 } { 2 } \\frac { 1 } { ( x + y ) ^ 2 } > \\frac { 1 } { y + 1 / 2 } - \\frac { 1 } { 1 + y } - \\frac { 1 } { 2 } \\frac { 1 } { ( 1 + y ) ^ 2 } \\end{align*}"}
+{"id": "7068.png", "formula": "\\begin{align*} F _ { 5 , 1 , 0 , 0 } \\ , = \\ , 0 \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ F _ { 6 , 1 , 0 , 0 } \\ , = \\ , 0 . \\end{align*}"}
+{"id": "1286.png", "formula": "\\begin{align*} \\hat { c } _ \\Delta ^ { B _ { n - 1 } ( x ) } ( \\xi _ \\Delta \\eta _ { \\Delta ^ c } , \\eta _ \\Delta ) = 0 \\end{align*}"}
+{"id": "4685.png", "formula": "\\begin{align*} \\int _ W | \\Delta \\rho | \\leq \\int _ W \\left ( | C - \\Delta \\rho | + C \\right ) = \\int _ W ( 2 C - \\Delta \\rho ) = 2 C \\ , _ n ( W ) + _ { n - 1 } ( \\partial W ) . \\end{align*}"}
+{"id": "167.png", "formula": "\\begin{align*} ( g _ 1 , g _ 2 , g _ 3 ) \\cdot \\phi ( h ) = g _ 3 \\cdot \\phi ( g _ 1 ^ { - 1 } h g _ 2 ) . \\end{align*}"}
+{"id": "7515.png", "formula": "\\begin{align*} \\int _ { B _ { \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x & \\leq C \\int _ { B _ { 4 \\rho } \\setminus B _ { 2 \\rho } } r ^ { 2 - n } | \\nabla u | ^ 2 \\d x + C \\varepsilon ( 1 + \\varepsilon \\rho ) \\int _ { B _ { 8 \\rho } } r ^ { 3 - n } | \\nabla u | ^ 2 \\d x \\\\ & \\rho \\leq 1 / 8 \\varepsilon \\leq \\varepsilon _ 0 . \\end{align*}"}
+{"id": "1554.png", "formula": "\\begin{align*} \\tilde N ( \\mathbf { v } ) = \\mathbf { 0 } . \\end{align*}"}
+{"id": "6281.png", "formula": "\\begin{align*} \\mathcal { U } _ \\mu = \\{ u \\in \\mathcal { U } : u \\mu \\} \\subseteq \\mathcal { U } , \\end{align*}"}
+{"id": "1819.png", "formula": "\\begin{align*} ( i _ X R ^ { \\nabla } ) ( Y _ 1 ) = R ^ { \\nabla } ( X , Y _ 1 ) \\ , , \\end{align*}"}
+{"id": "629.png", "formula": "\\begin{align*} \\mu ^ { T a m a } : = \\rho _ T ^ { - 1 } | \\omega _ \\infty | \\prod _ p L _ p ( 1 , \\sigma _ T ) | \\omega | _ p , \\end{align*}"}
+{"id": "7714.png", "formula": "\\begin{align*} \\big \\| ( A _ { I _ r , [ r + 2 k _ 0 ] } ) ^ \\top \\ , v \\big \\| _ 2 & = \\Bigg \\| \\Bigg ( \\begin{matrix} ( A _ { I _ r , [ r ] } ) ^ \\top v \\\\ ( A _ { I _ r , [ r + 1 , r + 2 k _ 0 ] } ) ^ \\top v \\end{matrix} \\Bigg ) \\Bigg \\| _ 2 \\\\ & \\ge \\max \\Big \\{ n ^ { - \\beta / ( 5 0 p ) } \\ , \\big \\| P _ { F _ r ^ \\perp } \\ , v \\big \\| _ 2 , \\ , n ^ { - \\beta / ( 4 0 p ) } \\ , \\big \\| P _ { F _ r } \\ , v \\big \\| _ 2 - 3 \\sqrt { \\beta n } \\ , \\big \\| P _ { F _ r ^ \\perp } \\ , v \\big \\| _ 2 \\Big \\} , \\end{align*}"}
+{"id": "292.png", "formula": "\\begin{align*} u ^ K _ { n , h } ( p ) + H ( p , u ^ K _ { n , h } ( p ) , \\nabla _ H u ^ K _ { n , h } ( p ) ) = u ^ K _ { n - 1 , h } ( p ) - \\omega _ f ( | h | _ G ) \\end{align*}"}
+{"id": "6662.png", "formula": "\\begin{align*} & \\mathcal { \\tilde { R } } _ { 1 , \\epsilon } ( t ) = \\int ^ { t } _ { 0 } \\big ( b ( \\varphi ( s ) ) - b ( X ^ { \\epsilon } ( s ) ) ) \\big ) d s , \\\\ & \\mathcal { \\tilde { R } } _ { 2 , \\epsilon } ( t ) = \\lambda ( \\epsilon ) \\int ^ { t } _ { 0 } \\big ( \\sigma ( \\varphi _ { m } ( s ) ) - \\sigma ( X ^ { \\epsilon } ( s ) ) \\big ) \\theta _ { \\epsilon } ( s ) d s . \\end{align*}"}
+{"id": "2576.png", "formula": "\\begin{align*} \\rho ^ { - m _ { 1 } } b _ { j _ { 1 } } p ^ { j _ { 2 } + j _ { 3 } } N _ { j _ { 2 } } N _ { j _ { 3 } } - \\rho ^ { - m _ { 2 } } b _ { j _ { 2 } } p ^ { j _ { 1 } + j _ { 3 } } N _ { j _ { 1 } } N _ { j _ { 3 } } = \\rho ^ { - m _ { 3 } } b _ { j _ { 3 } } p ^ { j _ { 1 } + j _ { 2 } } N _ { j _ { 1 } } N _ { j _ { 2 } } . \\end{align*}"}
+{"id": "2208.png", "formula": "\\begin{align*} \\phi = 2 \\pi \\frac { d } { \\lambda } { s i n } \\theta _ 0 = \\frac { 2 \\pi Z } { N } \\end{align*}"}
+{"id": "6745.png", "formula": "\\begin{align*} x = \\mu + \\sigma \\left [ F ^ { - 1 } _ \\Gamma ( 2 u - 1 ) \\right ] ^ { 1 / s } , 1 / 2 < u \\leq 1 . \\end{align*}"}
+{"id": "8406.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { x \\in \\left ( - \\infty , x _ 0 \\right ) } \\| \\langle x \\rangle m ( x ; z ) \\| _ { L _ z ^ 2 ( \\mathbb { R } ) } & \\leq \\sqrt { \\pi } \\left ( \\left \\| u _ { x x } \\right \\| _ { L ^ { 2 , 1 } } + \\frac { 1 } { 2 } \\left \\| u _ x ^ 3 \\right \\| _ { L ^ { 2 , 1 } } \\right ) \\\\ & \\leq c \\left ( \\| u \\| _ { H ^ { 2 , 1 } } + \\| u \\| _ { H ^ { 2 , 1 } } ^ 3 \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "5042.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla \\times U = f U , \\nabla \\cdot U & = 0 \\textrm { i n } \\ \\Omega , \\\\ U \\cdot n & = 0 \\textrm { o n } \\ \\partial \\Omega . \\end{aligned} \\end{align*}"}
+{"id": "6694.png", "formula": "\\begin{align*} | \\delta ( \\zeta \\theta ^ { - 1 } , \\theta \\gamma ^ + ) | = | f ( \\zeta \\gamma ^ + , \\theta \\gamma ^ - ) - f ( \\theta \\gamma ^ + , \\theta \\gamma ^ - ) | \\leq 2 M , \\end{align*}"}
+{"id": "785.png", "formula": "\\begin{align*} G '' ( c ) = - \\frac { g ' ( c ) } { c } \\end{align*}"}
+{"id": "4359.png", "formula": "\\begin{align*} [ \\sqrt { - 1 } \\theta \\wedge \\bar { \\theta } , \\Lambda _ \\omega ] \\alpha = \\bar { \\theta } \\wedge ( \\alpha \\llcorner ( \\bar { \\theta } ) ^ \\sharp ) , \\end{align*}"}
+{"id": "8877.png", "formula": "\\begin{align*} ( \\phi | _ { k , S } \\gamma ) ( \\tau , z ) : = J _ { k , S } ( \\gamma , ( \\tau , z ) ) \\phi ( \\gamma ( \\tau , z ) ) , \\end{align*}"}
+{"id": "3447.png", "formula": "\\begin{align*} \\tilde { m } _ n = \\begin{cases} m _ n + q _ n - q _ { n - 1 } , \\ell _ n = 2 \\\\ m _ n - q _ n + q _ { n - 1 } , \\ell _ n = - 2 \\end{cases} . \\end{align*}"}
+{"id": "9293.png", "formula": "\\begin{align*} \\begin{aligned} \\int u _ k ^ s \\Delta u _ 1 ^ t \\wedge \\dots \\wedge \\Delta u _ { k - 1 } ^ t \\wedge \\beta _ n ^ { n - m } \\wedge \\Delta \\omega = \\int \\Delta u _ 1 ^ t \\wedge \\dots \\wedge \\Delta u _ { k - 1 } ^ t \\wedge \\Delta u _ k ^ s \\wedge \\beta _ n ^ { n - m } \\wedge \\omega \\geq 0 , \\end{aligned} \\end{align*}"}
+{"id": "6875.png", "formula": "\\begin{align*} \\Psi = \\Psi _ 1 ^ { - 1 } \\circ \\Psi _ 2 ^ { - 1 } . \\end{align*}"}
+{"id": "5732.png", "formula": "\\begin{align*} M = \\frac { I _ 1 } { I _ 3 ^ { [ 3 ] } } \\cup \\frac { I _ 3 ^ { [ 3 ] } } { I _ 3 ^ { [ 3 ] } } , \\end{align*}"}
+{"id": "3566.png", "formula": "\\begin{align*} q ^ { - 7 / 6 0 } & \\varphi _ 1 ^ { - 6 } ( \\varphi _ 2 ^ 2 G _ 2 - q ^ { - 7 / 6 0 } \\varphi _ 1 ^ { - 6 } \\varphi _ \\frac { 1 } { 2 } ^ 2 H _ \\frac { 1 } { 2 } \\\\ & = q ^ { - 7 / 6 0 } \\varphi _ 1 ^ { - 6 } ( \\varphi _ 2 ^ 2 G _ 2 - \\varphi _ \\frac { 1 } { 2 } ^ 2 H _ \\frac { 1 } { 2 } ) \\end{align*}"}
+{"id": "8841.png", "formula": "\\begin{align*} \\begin{cases} d \\bar { x } _ { t , n } = x _ { 0 , n - 1 } \\bar { x } _ { t , 1 } d t + x _ { t , 1 } ( x _ { 0 , n - 1 } - x _ { t , n - 1 } ) d t + a _ n x _ { t , n } + x _ { t , n - 2 } x _ { t , n - 1 } d t \\\\ d \\bar { x } _ { t , 1 } = x _ { 0 , 2 } \\bar { x } _ { t , n } d t + x _ { t , n } ( x _ { 0 , 2 } - x _ { t , 2 } ) d t + x _ { t , n - 1 } x _ { t , n } d t - \\sigma _ 1 d W _ t ^ { ( 1 ) } . \\end{cases} \\end{align*}"}
+{"id": "3873.png", "formula": "\\begin{align*} ( T ( x ) ^ { - 1 } ) ( T ( x ) ^ { - 1 } ) ^ t = K ( x ) . \\end{align*}"}
+{"id": "3756.png", "formula": "\\begin{align*} Q ^ { n _ 0 } _ { n _ 0 ; l _ 1 , \\ldots , l _ { n _ 0 } } = P _ { n _ 0 ; l _ 1 , l _ 2 , \\ldots , l _ { n _ 0 } - 1 } = R ^ { n _ 0 } _ { n _ 0 ; l _ 1 , \\ldots , l _ { n _ 0 } } \\end{align*}"}
+{"id": "7608.png", "formula": "\\begin{align*} S _ \\beta ( x , y , z ) = ( x + \\alpha , y + x , z + x + \\beta ) \\end{align*}"}
+{"id": "3749.png", "formula": "\\begin{align*} \\psi \\left ( d _ - ^ { \\ell + 1 } L ( 1 v , 0 w ) \\right ) & = \\psi \\left ( t ^ { - \\ell } d _ - ^ { \\ell + 1 } d _ = L ( v 1 , w \\bullet ) \\right ) \\end{align*}"}
+{"id": "1025.png", "formula": "\\begin{align*} - \\chi _ K ( \\beta ) + \\langle \\alpha ^ \\vee , \\beta \\rangle \\chi _ K ( \\alpha ) \\leq - \\Phi ^ + ( s _ \\alpha ( \\beta ) ) = - \\chi _ K ( s _ \\alpha ( \\beta ) ) . \\end{align*}"}
+{"id": "4026.png", "formula": "\\begin{align*} \\lambda = ( \\lambda _ 1 , \\ldots , \\lambda _ k ) , \\ \\lambda _ 1 \\geq \\ldots \\geq \\lambda _ k > 0 , \\ \\sum \\lambda _ i = m . \\end{align*}"}
+{"id": "462.png", "formula": "\\begin{align*} e ^ { \\gamma x } f _ 1 \\ast f _ 2 ( x ) & = \\int _ { - \\infty } ^ \\infty e ^ { \\gamma ( x - y ) } f _ 2 ( x - y ) e ^ { \\gamma y } f _ 1 ( y ) d y \\\\ & \\le o ( 1 ) \\int _ { - \\infty } ^ 0 e ^ { \\gamma y } f _ 1 ( y ) d y . \\end{align*}"}
+{"id": "688.png", "formula": "\\begin{align*} \\phi _ { g ^ { - 1 } g _ 2 , g _ 1 } = \\phi _ { g _ 2 , g g _ 1 } , \\qquad \\end{align*}"}
+{"id": "8029.png", "formula": "\\begin{align*} C _ 1 ( t ) = \\sum _ { u : \\ , i _ u \\geq 3 \\atop { 0 \\leq d _ u < i _ u } } i _ u = \\sum _ { i \\geq 3 } \\sum _ { d = 0 } ^ { i - 1 } i x _ { i , d } , \\end{align*}"}
+{"id": "2777.png", "formula": "\\begin{align*} a ^ { D } ( x , y ) : = \\tilde { G } ^ { D } ( x , y ) - \\Gamma ( x , y ) ( x , y \\in D , x \\neq y ) , \\end{align*}"}
+{"id": "2394.png", "formula": "\\begin{align*} \\sum _ { n } \\sum _ { \\sigma \\subset \\mu _ \\ell } c _ { \\ell } x _ { \\ell , n } \\pi ^ { n - \\omega ( \\mu _ \\ell ) } + c _ { \\kappa } \\pi ^ { - \\omega ( \\kappa ) } & = 0 \\sigma \\subset \\kappa \\\\ \\sum _ { n } \\sum _ { \\sigma \\subset \\mu _ \\ell } c _ { \\ell } x _ { \\ell , n } \\pi ^ { n - \\omega ( \\mu _ \\ell ) } & = 0 \\sigma \\not \\subset \\kappa . \\end{align*}"}
+{"id": "3615.png", "formula": "\\begin{align*} g _ { i j } = \\left \\langle e _ i , e _ j \\right \\rangle , h _ { i j } = \\left \\langle \\overline { D } _ { e _ i } e _ j , \\mathbf { n } \\right \\rangle , \\end{align*}"}
+{"id": "2381.png", "formula": "\\begin{align*} C _ { { 3 i + 1 } } ( \\mathcal { H } _ { \\ast } ) = \\mathrm { I m } \\ , \\partial _ i \\oplus s _ { _ { 3 i + 1 } } ( \\mathrm { I m } \\ , \\partial '' _ i ) = \\mathrm { I m } \\ , \\partial _ i . \\end{align*}"}
+{"id": "8042.png", "formula": "\\begin{align*} \\xi ( t ) = \\sigma _ { \\xi ( u ) \\xi ( v ) } ( ( t - u ) / ( v - u ) ) \\in \\Sigma _ { \\Gamma ( u ) \\Gamma ( v ) } ( ( t - u ) / ( v - u ) ) . \\end{align*}"}
+{"id": "1959.png", "formula": "\\begin{align*} g _ \\sigma = \\varphi _ { + } ^ { - 1 } \\circ T _ \\sigma \\circ \\varphi _ { - } \\end{align*}"}
+{"id": "8434.png", "formula": "\\begin{align*} \\begin{aligned} 2 i k b ( k ) & = ( \\Psi ^ + _ { 1 1 } ( 0 ; z ) - e ^ { i c _ + ( 0 ) } ) \\Psi ^ - _ { 2 1 } ( 0 ; z ) + e ^ { i c _ + ( 0 ) } \\Psi ^ - _ { 2 1 } ( 0 ; z ) \\\\ & - \\Psi ^ + _ { 2 1 } ( 0 ; z ) ( \\Psi ^ - _ { 1 1 } ( 0 ; z ) - e ^ { i c _ - ( 0 ) } ) - e ^ { i c _ - ( 0 ) } \\Psi ^ + _ { 2 1 } ( 0 ; z ) . \\end{aligned} \\end{align*}"}
+{"id": "7690.png", "formula": "\\begin{align*} | \\phi ^ { N , i } ( t _ 0 , \\boldsymbol { \\xi } ) - \\Psi ^ { N , i } ( t _ 0 , \\boldsymbol { \\xi } ) | \\leq \\frac { C } { N } \\Big ( 1 + \\frac { 1 } { N } \\underset { j = 1 } { \\overset { N } \\sum } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 + \\frac { 1 } { N ^ 2 } \\underset { i , j = 1 } { \\overset { N } { \\sum } } | \\xi ^ { i } - \\xi ^ { j } | ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "2520.png", "formula": "\\begin{align*} { { \\mathbf { A } } ^ { T } } \\mathbf { d } _ { h y } = 0 . \\end{align*}"}
+{"id": "3546.png", "formula": "\\begin{align*} \\xi _ 1 & = ( 2 , 3 ) , & \\xi _ 2 & = ( 1 , 2 ) . \\end{align*}"}
+{"id": "918.png", "formula": "\\begin{align*} i A _ \\ell ' = t ^ { - 1 } N _ \\ell ( A _ 1 , A _ 2 , \\dots , A _ L ) ( \\ell = 1 , 2 , \\dots , L ) . \\end{align*}"}
+{"id": "8503.png", "formula": "\\begin{align*} & e ^ { 2 i ( c _ - ( x ) + c ) } u ( x ) = \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } z ^ { - 1 } \\bar { r } _ 1 ( z ) e ^ { - 2 i z x } \\mathrm { d } z \\\\ & + \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } z ^ { - 1 } \\bar { r } _ 1 ( z ) e ^ { - 2 i z x } \\left ( M _ { - , 1 1 } ( x ; z ) - 1 \\right ) \\mathrm { d } z : = I _ 1 ( x ) + I _ 2 ( x ) . \\end{align*}"}
+{"id": "2881.png", "formula": "\\begin{align*} \\bar { p _ x ^ 2 } ( t ) = T _ - M _ { x , 0 } + \\sum _ { x ' = 1 } ^ n \\int _ 0 ^ \\theta { \\frak g } _ { x , x ' } ( s ) \\bar { p _ { x ' } ^ 2 } ( t - s ) \\dd s + \\bar { p _ x } ^ 2 ( t ) , \\end{align*}"}
+{"id": "1345.png", "formula": "\\begin{align*} \\| y \\| \\geq g \\bigl ( \\sum _ { j = 1 } ^ m \\alpha _ j y _ j \\bigr ) = \\sum _ { j = 1 } ^ m \\alpha _ j g ( y _ j ) = \\sum _ { j = 1 } ^ m \\alpha _ j > 1 - \\frac { \\varepsilon } { 5 } \\end{align*}"}
+{"id": "1575.png", "formula": "\\begin{align*} e ^ { 2 \\varphi ( y ) } & = \\int _ { T _ y N } e ^ { 2 w } \\circ \\exp _ y ( v ) \\chi ( \\Vert v \\Vert ) \\dd V _ y \\\\ & = \\int _ { T _ x N } ( p ^ * e ^ { 2 w } ) \\circ \\exp _ x ( v ) \\chi ( \\Vert v \\Vert ) p ^ * ( \\dd V _ y ) \\\\ & = \\int _ { T _ x N } \\lambda ^ 2 e ^ { 2 w } \\circ \\exp _ x ( v ) \\chi ( \\Vert v \\Vert ) \\dd V _ x \\\\ & = \\lambda ^ 2 e ^ { 2 \\varphi } . \\end{align*}"}
+{"id": "1395.png", "formula": "\\begin{align*} h ^ { ( \\lambda _ 1 , \\ldots , \\lambda _ d ) } ( x ) = \\prod _ { i = 1 } ^ d \\lambda _ i ^ { i - d } e ^ { \\sum _ { i = 1 } ^ d \\lambda _ i x _ i } \\det \\big ( ( - 1 ) ^ { d - j } \\big ( \\frac { d } { d x } \\big ) ^ { d - j } e ^ { - \\lambda _ i x _ j } \\big ) _ { i , j = 1 } ^ d . \\end{align*}"}
+{"id": "8162.png", "formula": "\\begin{align*} T _ { \\mathrm { g e l } } : = \\inf \\left \\{ t \\geq 0 \\ \\ \\mbox { s u c h t h a t } \\ \\ \\mathcal { M } ^ { 1 } ( t ) < \\mathcal { M } ^ { 1 } ( 0 ) = \\varrho _ { 0 } \\right \\} . \\end{align*}"}
+{"id": "6160.png", "formula": "\\begin{align*} & f ( r ) = \\sqrt { 1 - \\kappa r ^ 2 } , \\\\ & V ( r ) = V ( r ; l , Q ) = \\frac { L ( L + 1 ) } { r ^ 2 } - \\frac { Q } { r } f ( r ) , L = l + \\frac { d - 3 } { 2 } , \\\\ & E = { \\cal E } + \\frac { 1 } { 4 } \\kappa ( d - 1 ) ^ 2 , \\\\ & \\psi ( r ) = r ^ { ( d - 1 ) / 2 } f ^ { - 1 / 2 } ( r ) R ( r ) , . \\end{align*}"}
+{"id": "803.png", "formula": "\\begin{align*} \\nu ( t , p ) = \\frac { p } { R ( t ) } \\phantom { x x } \\phantom { x x } H ( t , p ) = - \\frac { d } { R ( t ) } \\end{align*}"}
+{"id": "585.png", "formula": "\\begin{align*} \\lambda _ { i , j } = \\begin{cases} \\lambda _ j ~ ~ { \\rm i f } ~ ~ i = 1 , \\\\ \\\\ \\frac { \\lambda _ { i } \\lambda _ { j } } { \\Lambda } ~ ~ { \\rm i f } ~ ~ i > 1 . \\\\ \\end{cases} \\end{align*}"}
+{"id": "8525.png", "formula": "\\begin{align*} r _ { \\delta , j } ( z ) : = \\bar { \\delta } _ + ( z ) \\bar { \\delta } _ - ( z ) r _ { j } ( z ) , \\ \\ j = 1 , 2 . \\end{align*}"}
+{"id": "4881.png", "formula": "\\begin{align*} S ^ { \\alpha , \\beta } = \\begin{cases} x _ 1 ^ 6 + y _ 1 ^ 6 + x _ 2 ^ 6 + y _ 2 ^ 6 + h _ + ^ 3 + h _ + h _ - ^ 2 = 0 ; \\\\ x _ 1 y _ 1 = \\alpha h _ + + \\beta h _ - ; \\\\ x _ 2 y _ 2 = \\alpha h _ + - \\beta h _ - . \\end{cases} \\end{align*}"}
+{"id": "6668.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb E \\exp \\left ( \\sum _ { i = 1 } ^ k \\alpha _ i \\int _ 0 ^ { \\epsilon ^ { - 2 } t _ i } \\xi ( s ) d s \\right ) & = \\mathbb E \\exp \\left ( \\sum ^ { k } _ { \\ell = 1 } \\Delta _ \\ell \\right ) \\\\ & = \\mathbb E \\left [ \\exp \\left ( \\sum ^ { k - 1 } _ { \\ell = 1 } \\Delta _ \\ell \\right ) \\mathbb E \\left ( e ^ { \\Delta _ k } \\Big | \\mathcal { F } _ { \\epsilon ^ { - 2 } t _ { k - 1 } } \\right ) \\right ] , \\end{aligned} \\end{align*}"}
+{"id": "245.png", "formula": "\\begin{align*} R _ q ( n ) = \\{ g \\in A \\cap B _ S ( n ) \\mid - q ( n ) \\le \\sigma _ g ^ - \\le \\sigma _ g ^ + \\le q ( n ) \\} . \\end{align*}"}
+{"id": "9298.png", "formula": "\\begin{align*} ( \\Delta K _ { m , \\epsilon } ) ^ p \\wedge \\beta _ n ^ { n - p } = ( p A \\wedge B ^ { p - 1 } + B ^ p ) \\wedge \\beta _ n ^ { n - p } , p = 1 , \\ldots , m , \\end{align*}"}
+{"id": "5014.png", "formula": "\\begin{align*} S t _ { D N M } \\lim _ { n \\rightarrow \\infty } | | \\mathcal { Y } _ { n } ( z , z ) - z | | _ { \\infty } = 0 , \\end{align*}"}
+{"id": "8512.png", "formula": "\\begin{align*} \\begin{aligned} I _ 4 ^ { \\prime } ( x ) & = - 4 \\pi ^ { - 1 } \\int _ { - \\infty } ^ { \\infty } r _ 2 ( z ) e ^ { - 2 i z x } \\left ( M _ { + , 2 2 } ( x ; z ) - 1 \\right ) d z \\\\ & - 2 i \\pi ^ { - 1 } \\int _ { - \\infty } ^ { \\infty } s ^ { - 1 } r _ 2 ( z ) e ^ { - 2 i z x } \\partial _ { x } M _ { + , 2 2 } ( x ; z ) d z . \\end{aligned} \\end{align*}"}
+{"id": "5883.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } \\sup _ { n \\in \\mathbb { N } } \\mathbb { P } ( \\tau _ { n } ^ M < T ) = 0 . \\end{align*}"}
+{"id": "9345.png", "formula": "\\begin{align*} \\pi _ { r _ 0 , x _ 0 } & : = \\pi ( x , t ) - c _ { r _ 0 , x _ 0 } ( t ) = \\frac { 1 } { 3 } | v ( x , t ) | ^ 2 + p _ 1 ( x , t ) + p _ 2 ( x , t ) , \\\\ p _ 1 ( x , t ) & : = \\textnormal { p . v . } \\int _ { B _ { 3 r _ 0 } ( x _ 0 ) } K ( x - y ) : ( v \\otimes v ) ( y , t ) \\ , d y , \\\\ p _ 2 ( x , t ) & : = \\int _ { \\mathbb { R } ^ 3 \\setminus B _ { 3 r _ 0 } ( x _ 0 ) } ( K ( x - y ) - K ( x _ 0 - y ) : ( v \\otimes v ) ( y , t ) \\ , d y , \\\\ \\mathcal { E } _ \\gamma ( s ) & : = \\sup _ { r \\in [ \\gamma , r _ 0 ] } E _ r ( s ) . \\end{align*}"}
+{"id": "7742.png", "formula": "\\begin{align*} F \\colon ( x , B , M ) \\mapsto \\frac { 1 } { p } \\sum _ { i = 1 } ^ p | [ U B ^ T M x ] _ i | . \\end{align*}"}
+{"id": "2505.png", "formula": "\\begin{align*} \\hat { \\mathbf { r } } _ p = \\mathbf { A } \\mathbf { x } , \\hat { \\mathbf { r } } _ d = \\mathbf { A } ^ { T } \\mathbf { y } + \\mathbf { C } \\mathbf { x } + \\mathbf { s } . \\end{align*}"}
+{"id": "3998.png", "formula": "\\begin{align*} \\mathcal { W } ^ 2 ( \\mu _ 1 , \\mu _ 2 ) = \\inf \\mathbb { E } [ | X - Y | ^ 2 ] . \\end{align*}"}
+{"id": "9342.png", "formula": "\\begin{align*} \\limsup _ { r \\to 0 ^ + } \\ , ( - \\log ( r ) ) ^ a r ^ { 1 - \\frac { 2 } { p } - \\frac { 3 } { q } } \\| v _ 3 \\| _ { L ^ p _ t L ^ q _ x ( Q _ { \\rho , r } ( z _ 0 ) ) } = 0 , \\end{align*}"}
+{"id": "8499.png", "formula": "\\begin{align*} \\frac { \\psi ^ \\pm _ 2 ( x ; k ) } { 2 i k } = \\Psi ^ \\pm _ 2 ( x ; z ) , \\end{align*}"}
+{"id": "5418.png", "formula": "\\begin{align*} E _ n = \\frac { - \\varepsilon _ 1 } { \\sqrt { \\varepsilon _ 1 ^ 2 + | \\hat { x } | ^ 2 } } e _ 1 + \\frac { | \\hat { x } | } { \\sqrt { \\varepsilon _ 1 ^ 2 + | \\hat { x } | ^ 2 } } e _ 2 . \\end{align*}"}
+{"id": "3762.png", "formula": "\\begin{align*} T _ s ^ 2 = & 1 & & \\\\ T _ s T _ { s ' } = & T _ { s ' } T _ s & & \\\\ T _ s T _ { s ' } T _ s = & T _ { s ' } T _ s T _ { s ' } & & \\end{align*}"}
+{"id": "951.png", "formula": "\\begin{align*} w = \\begin{pmatrix} 1 & 0 \\end{pmatrix} P Q ( t ) P ^ { - 1 } \\begin{pmatrix} W _ 2 \\\\ \\overline { W _ 2 } \\end{pmatrix} + O ( t ^ { - 1 / 4 + C _ 1 \\varepsilon ^ 2 _ 1 + 2 \\mu C _ 2 \\varepsilon ^ 2 _ 1 + C _ 4 \\varepsilon _ 1 ^ 2 } ) \\end{align*}"}
+{"id": "1015.png", "formula": "\\begin{align*} x _ 1 = x _ L x _ 0 x _ R = \\underbrace { ( x _ L r _ a ) } _ { < x _ L } x _ 0 \\underbrace { ( r _ { x _ 0 ^ { - 1 } ( a ) } x _ R ) } _ { < x _ R } , \\end{align*}"}
+{"id": "8719.png", "formula": "\\begin{align*} y _ n = \\left [ g , x ^ { n } \\right ] = \\left [ g , x ^ { n - 1 } \\right ] \\left [ g , x \\right ] ^ { x ^ { n - 1 } } = y _ { n - 1 } y \\left [ y , x ^ { n - 1 } \\right ] \\end{align*}"}
+{"id": "1831.png", "formula": "\\begin{align*} \\begin{aligned} R ^ { \\nabla ^ t } _ { X , Y } & = ( \\nabla _ X + t A _ X ) ( \\nabla _ Y + t A _ Y ) - ( \\nabla _ Y + t A _ Y ) ( \\nabla _ X + t A _ X ) \\\\ & = R ^ \\nabla _ { X , Y } + t [ \\nabla _ X , A _ Y ] - t [ \\nabla _ Y , A _ X ] + t ^ 2 [ A _ X , A _ Y ] \\\\ & = R ^ \\nabla _ { X , Y } + t \\nabla _ X ( A _ Y ) - t \\nabla _ Y ( A _ X ) + t ^ 2 [ A , A ] _ { X , Y } \\\\ & = R ^ \\nabla _ { X , Y } + t ( d ^ \\nabla A ) _ { X , Y } + t ^ 2 [ A , A ] _ { X , Y } \\ , . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "5731.png", "formula": "\\begin{align*} M = \\frac { I _ 1 \\cup I _ 3 ^ { [ 3 ] } } { I _ 3 ^ { [ 3 ] } } . \\end{align*}"}
+{"id": "8784.png", "formula": "\\begin{align*} d u _ m = ( u _ { m + 1 } - u _ { m - 2 } ) u _ { m - 1 } d t - ( A u ) _ m d t + q _ m d W _ t ^ { ( m ) } . \\end{align*}"}
+{"id": "7391.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - y } = \\sum _ { n = 0 } ^ \\infty y ^ n . \\end{align*}"}
+{"id": "5340.png", "formula": "\\begin{align*} r _ { n , k } = \\deg ( C _ { n , k } ) \\leq n . \\end{align*}"}
+{"id": "9321.png", "formula": "\\begin{align*} z _ { 1 } ^ { 2 } - 5 z _ { 2 } ^ { 2 } = 4 p \\equiv 2 8 \\equiv 4 \\pmod 8 , \\\\ z _ { 1 } ^ { 2 } - 5 z _ { 3 } ^ { 2 } = - p \\equiv 1 \\pmod 8 . \\end{align*}"}
+{"id": "2762.png", "formula": "\\begin{align*} Q ( t ) = \\left \\{ x \\in K ( t ) \\colon \\frac { t } { 6 } \\leq x _ 1 \\leq \\frac { t } { 4 } , \\ ; \\frac { t } { 6 } \\leq x _ 2 \\leq \\frac { t } { 4 } , \\ ; | x _ 3 - t | \\leq \\frac { t } { 4 } \\right \\} . \\end{align*}"}
+{"id": "4269.png", "formula": "\\begin{align*} n = \\frac { - 3 \\left ( \\alpha ^ { 2 } + 8 \\right ) \\pm \\sqrt { - 3 \\left ( \\alpha ^ { 4 } + 8 \\left ( \\alpha ^ { 3 } - 6 \\alpha ^ { 2 } + 1 3 \\alpha + 2 \\right ) \\right ) } } { 6 \\left ( \\alpha + 2 \\right ) } \\end{align*}"}
+{"id": "1536.png", "formula": "\\begin{align*} \\widehat { \\mathcal D } _ { \\hat { A } } ^ { \\times } = \\{ P \\in \\widehat { \\mathcal D } _ { \\mathcal { O } _ { \\hat { A } } } \\ ; | \\ ; \\pi ( P ) = 1 \\} \\ ; . \\end{align*}"}
+{"id": "6040.png", "formula": "\\begin{align*} J _ { 1 } ( u ) \\leq \\liminf _ { k \\rightarrow \\infty } J _ { 1 } ( \\overline { u } _ { k } ) = B _ { 1 } \\end{align*}"}
+{"id": "5827.png", "formula": "\\begin{align*} { \\partial _ t } = \\sum \\limits _ { n = 1 } ^ { + \\infty } { { \\varepsilon ^ n } \\partial { t _ n } } , \\ ; \\ ; \\ ; \\nabla = \\varepsilon { \\nabla _ 1 } , \\ ; \\ ; \\ ; { g _ i } = \\sum \\limits _ { n = 1 } ^ { + \\infty } { { \\varepsilon ^ n } g _ i ^ { \\left ( n \\right ) } } , \\ ; \\ ; \\ ; \\bar F = \\varepsilon { \\bar F ^ { \\left ( 1 \\right ) } } , \\end{align*}"}
+{"id": "6784.png", "formula": "\\begin{align*} s ^ { \\pm } ( \\rho ) : = \\mp \\frac { b ( \\mp \\rho ) } { a ( \\rho ) } , \\quad \\rho \\in \\mathbb { R } . \\end{align*}"}
+{"id": "8070.png", "formula": "\\begin{align*} \\lambda _ c = 2 \\sum _ { j = 1 } ^ { M } \\lvert \\phi ^ { \\left ( j , c \\right ) } \\rvert ^ 2 - f ^ { \\left ( c \\right ) } \\end{align*}"}
+{"id": "2081.png", "formula": "\\begin{align*} g ( r ) & = \\lim _ { k \\rightarrow { 0 } } \\frac { G ( r + k ) - G ( r ) } { k } = \\lim _ { n \\rightarrow { \\infty } } \\frac { G ( r + \\frac { 1 } { n } ) - G ( r ) } { \\frac { 1 } { n } } = \\lim _ { n \\rightarrow { \\infty } } n \\{ G ( r + \\frac { 1 } { n } ) - G ( r ) \\} \\end{align*}"}
+{"id": "3171.png", "formula": "\\begin{align*} k _ 3 ( \\langle H \\rangle , a ) = k _ 3 ( \\langle H \\rangle , b ) . \\end{align*}"}
+{"id": "8936.png", "formula": "\\begin{gather*} m _ { n } ( t ) = \\allowbreak \\sqrt { \\frac { n ! } { \\left ( \\beta \\right ) ^ { \\left ( n \\right ) } } } \\frac { 1 } { ( 1 - t ) ^ { \\beta } } \\frac { 1 } { \\Gamma ( \\beta ) } \\times \\\\ \\int _ { 0 } ^ { \\infty } \\left ( \\sum _ { j = 0 } ^ { n } \\left ( \\frac { - t y } { 1 - t } \\right ) ^ { j } L _ { n - j } ( x | ( \\beta + j ) ) / j ! \\right ) y ^ { \\beta - 1 } \\exp ( - y ) d y . \\end{gather*}"}
+{"id": "7750.png", "formula": "\\begin{align*} p ( x ) : = \\sum _ { i = 0 } ^ d p _ i x ^ i = p _ d \\prod _ { j = 1 } ^ d ( x - x _ j ) , ~ p _ d \\neq 0 , \\end{align*}"}
+{"id": "2763.png", "formula": "\\begin{align*} p ( s , k ) = ( 2 s _ 1 , 2 s _ 2 , k \\eta ( k ) ) . \\end{align*}"}
+{"id": "2249.png", "formula": "\\begin{align*} K ( Z , W ) = \\det ( I _ n - Z W ^ * ) ^ { - 2 n } . \\end{align*}"}
+{"id": "8775.png", "formula": "\\begin{align*} \\begin{cases} d x _ t = B ( x _ t , x _ t ) d t - A x _ t d t + \\sigma d W _ t \\\\ x _ t | _ { t = 0 } = x _ 0 \\in \\R ^ n . \\end{cases} \\end{align*}"}
+{"id": "9316.png", "formula": "\\begin{align*} b _ { 1 } u _ { 1 } ^ { 2 } - b _ { 1 } b _ { 2 } u _ { 3 } ^ { 2 } = - p q ^ { 2 k } \\implies b _ { 1 } \\equiv 0 \\pmod { q ^ 2 } , \\end{align*}"}
+{"id": "728.png", "formula": "\\begin{align*} B ( x , t , \\lambda ) = - \\lambda + y ( t ) h ' ( 0 ) . \\end{align*}"}
+{"id": "7014.png", "formula": "\\begin{align*} \\boxed { \\scriptstyle \\sf H r a n k ~ 1 } \\ , : = \\ , \\boxed { \\aligned 0 & \\ , \\neq \\ , F _ { x x } \\\\ 0 & \\ , \\equiv \\ , F _ { x x } \\ , F _ { y y } - F _ { x y } ^ 2 \\endaligned } , \\end{align*}"}
+{"id": "1414.png", "formula": "\\begin{align*} \\lambda \\sum _ { i = 1 } ^ d ( y _ i - x _ i ) = \\lambda d ( y _ 1 - x _ 2 ) + \\lambda \\sum _ { i = 2 } ^ d ( ( y _ i - y _ { 1 } ) - ( x _ i - x _ 1 ) ) \\le \\lambda d ( y _ 1 - x _ 1 ) + | \\lambda | d ^ 2 \\sqrt n . \\end{align*}"}
+{"id": "4522.png", "formula": "\\begin{align*} v _ 1 ( x , t ) & = S _ { d , 1 } ( t , t _ { 1 , e x } ( x , t ) ) v _ 1 ( x , t _ { 1 , e x } ( x , t ) ) . \\end{align*}"}
+{"id": "1329.png", "formula": "\\begin{align*} \\| f \\| = \\sup \\bigl \\{ \\frac { | f ( p ) - f ( q ) | } { d ( p , q ) } \\colon p , q \\in M , p \\neq q \\bigr \\} , \\end{align*}"}
+{"id": "754.png", "formula": "\\begin{align*} \\int _ { \\lambda I _ Q } \\| R _ j \\partial _ j g ( \\cdot , t ) \\| _ { L ^ 1 ( 2 \\lambda Q _ 1 ) } d t & \\lesssim \\int _ { \\lambda I _ Q } \\ell ( Q ) ^ { N / p } \\| R _ j \\partial _ j g ( \\cdot , t ) \\| _ { L ^ q ( 2 \\lambda Q _ 1 ) } d t \\\\ & \\lesssim \\int _ { \\lambda I _ Q } \\ell ( Q ) ^ { N / p } \\| \\partial _ j g ( \\cdot , t ) \\| _ { L ^ q ( 2 \\lambda Q _ 1 ) } d t \\lesssim \\ell ( Q ) ^ { N / p } \\ell ( Q ) ^ { N / q } = \\ell ( Q ) ^ N . \\end{align*}"}
+{"id": "254.png", "formula": "\\begin{align*} \\alpha = \\sum _ { i = 1 } ^ s a _ i w _ i + \\sum _ { i = 1 } ^ t b _ i e _ { j _ i } = \\sum _ { i = 1 } ^ s a _ i \\left ( \\sum _ { j = 1 } ^ k w _ { i j } e _ j \\right ) + \\sum _ { i = 1 } ^ t b _ i e _ { j _ i } \\end{align*}"}
+{"id": "3247.png", "formula": "\\begin{align*} \\sqrt n \\langle \\Pi _ m Q \\Pi _ m h - Q h , h \\rangle \\leq & \\| ( I - \\Pi _ m ) Q h \\| + \\| ( I - \\Pi _ m ) Q h _ m \\| \\\\ = & \\sqrt n \\sqrt { \\Delta _ m } \\left ( \\| ( Q h ) ' \\| + \\| Q \\| _ { L _ { } ( L ^ 2 ( 0 , 1 ) , \\dot H ^ 1 ) } \\| h _ m \\| \\right ) . \\end{align*}"}
+{"id": "3376.png", "formula": "\\begin{align*} V _ i ( n ) = \\left \\{ \\begin{array} { l l } V ( n ) & n \\in [ 0 , n _ i ] \\\\ V ( n _ i ) + V ' ( n _ i ) ( n - n _ i ) & n \\in [ n _ i , L ] \\end{array} \\right . . \\end{align*}"}
+{"id": "1646.png", "formula": "\\begin{align*} \\Pi ( \\widetilde { \\pi } ) : = \\left \\{ \\bigotimes _ v ^ \\prime \\pi _ v \\mid \\pi _ v \\in \\Pi ( \\widetilde { \\pi } _ v ) , \\pi _ v = \\pi ^ 0 _ v \\ , \\ , v \\right \\} . \\end{align*}"}
+{"id": "4011.png", "formula": "\\begin{align*} E ^ L _ Y ( u ) = \\varphi ^ L _ { \\sigma , s } ( E ^ L _ X ( v ) ) \\end{align*}"}
+{"id": "4398.png", "formula": "\\begin{align*} - \\frac { \\zeta ' } { \\zeta } ( s ) & = \\sum _ { n < x ^ 2 } \\frac { \\Lambda _ x ( n ) } { n ^ s } + \\frac { x ^ { 1 - s } - x ^ { 2 ( 1 - s ) } } { ( 1 - s ) ^ 2 \\log x } - \\frac { 1 } { \\log x } \\sum _ { q = 1 } ^ \\infty \\frac { x ^ { - 2 q - s } - x ^ { - 2 ( 2 q + s ) } } { ( 2 q + s ) ^ 2 } \\\\ & - \\frac { 1 } { \\log x } \\sum _ { \\rho } \\frac { x ^ { \\rho - s } - x ^ { 2 ( \\rho - s ) } } { ( s - \\rho ) ^ 2 } . \\end{align*}"}
+{"id": "5119.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\frac { ( r r ' ) ^ { 2 \\tau + 1 } } { | r - r ' | ^ { 4 \\tau + 2 - 2 / q } r ' } \\dd r ' = C r ^ { 2 / q } . \\end{align*}"}
+{"id": "5940.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) = \\nabla \\cdot a \\big ( t , x , u ( t , x ) , \\nabla u ( t , x ) \\big ) - a _ 0 \\big ( t , x , , u ( t , x ) , \\nabla u ( t , x ) \\big ) , \\end{align*}"}
+{"id": "3036.png", "formula": "\\begin{align*} R _ { x } \\left ( y \\right ) = \\cos \\left ( \\left \\vert x \\right \\vert \\right ) y + \\sin \\left ( \\left \\vert x \\right \\vert \\right ) \\left ( x / { \\left \\vert x \\right \\vert } \\right ) \\times y \\end{align*}"}
+{"id": "1965.png", "formula": "\\begin{align*} & Q ^ { ( m - 1 ) } _ \\nu ( a ) - Q ^ { ( m - 1 ) } _ \\nu ( a - 1 ) \\\\ & = \\frac { q ^ { a + 1 } - 1 } { q - 1 } \\cdots \\frac { q ^ { a + m - 2 } - 1 } { q ^ { m - 2 } - 1 } - \\frac { q ^ { a } - 1 } { q - 1 } \\cdots \\frac { q ^ { a + m - 3 } - 1 } { q ^ { m - 2 } - 1 } \\\\ & = q ^ a Q ^ { ( m - 2 ) } _ \\nu ( a ) = \\frac { q ^ a ( q ^ { m - 2 } - 1 ) } { q ^ a - 1 } Q ^ { ( m - 1 ) } _ \\nu ( a - 1 ) . \\end{align*}"}
+{"id": "4301.png", "formula": "\\begin{align*} \\int _ { \\{ - t _ 1 \\le \\Psi < - t _ 2 \\} } | \\tilde { F } | ^ 2 _ h a ( - \\Psi ) = \\frac { G ( T _ 1 ; c ) } { \\int _ { T _ 1 } ^ { + \\infty } c ( t ) e ^ { - t } d t } \\int _ { t _ 2 } ^ { t _ 1 } a ( t ) e ^ { - t } d t \\end{align*}"}
+{"id": "3506.png", "formula": "\\begin{align*} F _ n - n ^ 3 \\ = \\ \\sum _ { i = 1 } ^ { n } \\left ( i ^ 3 - 2 \\left ( \\binom { 3 } { 2 } i ^ 2 + \\binom { 3 } { 0 } i ^ 0 \\right ) \\right ) \\cdot F _ { n - i } . \\end{align*}"}
+{"id": "8224.png", "formula": "\\begin{align*} N K B L ( m , n ) = \\sum _ { { \\scriptscriptstyle A \\in M N A P P ( m , n ) } } N ( A ) \\cdot ( m - 1 ) ! + \\sum _ { { \\scriptscriptstyle A \\in M P A P P ( m , n ) } } N ( A ) \\cdot 2 ^ { m - 1 } ( m - 1 ) ! \\\\ \\ge \\sum _ { i = 1 } ^ { 3 } N ( A _ i ) \\cdot ( m - 1 ) ! + \\sum _ { i = 7 } ^ { 1 0 } N ( A _ i ) \\cdot 2 ^ { m - 1 } ( m - 1 ) ! \\ge ( 5 \\cdot 2 ^ m ) \\cdot ( m - 1 ) ! . \\end{align*}"}
+{"id": "2740.png", "formula": "\\begin{align*} | H ( x ) | \\leq \\max _ { y \\in [ - 1 , 1 ] ^ 2 } | h ( y ) | = \\sqrt { 2 } \\quad \\ x \\in \\R ^ 3 \\setminus \\Omega . \\end{align*}"}
+{"id": "2937.png", "formula": "\\begin{align*} I _ { \\alpha } \\left ( f \\right ) = \\int _ { 1 } ^ { \\infty } \\overline { \\nu _ { \\alpha } ^ { \\prime } } \\left ( f \\left ( x \\right ) \\right ) d x \\end{align*}"}
+{"id": "7361.png", "formula": "\\begin{gather*} K _ 2 = \\bigl \\{ P \\in M : ( \\pi _ 1 , \\ldots , \\pi _ n ) P \\bigr \\} \\quad \\quad \\\\ K _ 3 = \\bigl \\{ P \\in M : P \\ll P ^ * ( \\pi _ 1 , \\ldots , \\pi _ n ) P \\bigr \\} . \\end{gather*}"}
+{"id": "3895.png", "formula": "\\begin{align*} \\hat { u } _ i = c _ 1 \\frac { \\partial \\phi } { \\partial x _ 1 } + c _ 2 \\frac { \\partial \\phi } { \\partial x _ 2 } . \\end{align*}"}
+{"id": "425.png", "formula": "\\begin{align*} \\theta ^ { i - 1 } \\theta ^ { j - 1 } = \\sum _ { k = 1 } ^ { n } c _ { i j k } \\theta ^ { k - 1 } , \\end{align*}"}
+{"id": "5291.png", "formula": "\\begin{align*} H = ( p \\wedge q ^ { \\perp } ) H \\oplus ( p ^ { \\perp } \\wedge q ) H \\oplus ( p \\wedge q ) H \\oplus ( p ^ { \\perp } \\wedge q ^ { \\perp } ) H \\oplus ( e _ 1 + e _ 2 ) H . \\end{align*}"}
+{"id": "7176.png", "formula": "\\begin{align*} q _ 1 ^ 2 - b _ 1 q _ 1 + c _ 2 = 0 . \\end{align*}"}
+{"id": "7396.png", "formula": "\\begin{align*} \\{ | v _ 1 | > | v _ 2 | > \\dots > | v _ N | \\} \\cap \\{ | z v _ i | < | v _ { i + 1 } | | i = 1 , \\dots N - 1 \\} \\subset \\mathbb C ^ N \\times \\mathbb H \\end{align*}"}
+{"id": "3902.png", "formula": "\\begin{align*} u = ( Q _ { \\delta } ( - \\delta ^ 2 ( K ( x ) \\nabla ) ) ) ^ { - 1 } p Q _ \\delta ( V _ { \\delta , Z } - q ) ^ { p - 1 } _ + u + ( Q _ { \\delta } ( - \\delta ^ 2 ( K ( x ) \\nabla ) ) ) ^ { - 1 } h , \\ \\ u \\in E _ { \\delta , Z } . \\end{align*}"}
+{"id": "8388.png", "formula": "\\begin{align*} \\begin{aligned} \\| h _ { 2 } \\| _ { L ^ \\infty } \\leq \\frac { 1 } { 2 } ( 2 \\| u _ { x x } \\| _ { L ^ 1 } + \\| u _ x \\| ^ 3 _ { L ^ 3 } + \\| u _ x \\| ^ 2 _ { L ^ 2 } ) \\| f ( \\cdot ; z ) \\| _ { L ^ { \\infty } } . \\end{aligned} \\end{align*}"}
+{"id": "8764.png", "formula": "\\begin{align*} & F ( W ) c _ { ( \\vartheta , \\varphi ) } = \\int _ { \\gamma } F ( Z ) E _ { ( \\vartheta , \\varphi ) } ( Z , W ) d \\sigma _ { ( \\vartheta , \\varphi ) } ( Z ) - \\int _ { \\Gamma } E _ { ( \\vartheta , \\varphi ) } ( Z , W ) \\frac { \\partial F } { \\partial Z _ { ( \\vartheta , \\varphi ) } } d Z \\wedge d Z ^ { * } , \\end{align*}"}
+{"id": "1160.png", "formula": "\\begin{align*} H \\left ( P ^ { ( n , k ) } [ t ] \\mid \\mu ^ { \\otimes k } [ t ] \\right ) \\leq \\frac { k } { n } M , k = 1 , \\cdots , n , t \\in [ 0 , T _ M ] . \\end{align*}"}
+{"id": "6204.png", "formula": "\\begin{align*} W ' ( r ) = \\frac { \\xi ' } { r } f + \\eta ' + \\zeta ' \\frac { r } { f } + \\frac { \\sigma ' } { f ^ 2 } , \\xi ' \\le 0 , \\sigma ' > 0 , \\end{align*}"}
+{"id": "6311.png", "formula": "\\begin{align*} F _ 0 ( x ^ 0 _ * ) = F ( x ^ 0 _ * ) + \\frac { 1 } { 2 \\rho _ 0 } \\| x ^ 0 _ * - x ^ 0 \\| ^ 2 \\geq F ( x ^ * ) , F _ 0 ( x ^ 0 ) = F ( x ^ 0 ) . \\end{align*}"}
+{"id": "345.png", "formula": "\\begin{align*} D ^ { - 1 } = - \\frac { 1 } { 2 } S + \\frac { 1 } { 2 ( n - 1 ) } \\tau \\tau ' , \\end{align*}"}
+{"id": "7761.png", "formula": "\\begin{align*} \\zeta : = \\zeta _ q : = \\exp \\Big ( \\frac { 2 \\pi \\sqrt { - 1 } } { q } \\Big ) . \\end{align*}"}
+{"id": "9049.png", "formula": "\\begin{align*} \\Delta _ { \\ 1 } ^ + = \\{ e _ i \\pm f _ j | 1 \\leq i \\leq m , 1 \\leq j \\leq n \\} \\cup \\{ e _ i | 1 \\leq i \\leq m \\} . \\end{align*}"}
+{"id": "9190.png", "formula": "\\begin{align*} x _ n ( t ) : = x _ a ( t - n { \\bar { t } } / { \\gamma } , x ( n { \\bar { t } } / { \\gamma } ) ) \\forall \\ , t \\in I _ n . \\end{align*}"}
+{"id": "5704.png", "formula": "\\begin{align*} R : ( x , y ) \\mapsto R ( x , y ) = ( f ( x , y ) , g ( x , y ) ) \\end{align*}"}
+{"id": "9234.png", "formula": "\\begin{align*} \\begin{aligned} \\| x ( t ) \\| _ { \\mathcal { A } } & \\le \\beta ( \\| x ( \\bar { t } / \\gamma ) \\| _ { \\mathcal { A } } , 0 ) + \\tilde { d } \\\\ & \\le \\beta ( \\underline { \\rho } , 0 ) + \\tilde { d } \\\\ & \\le d & & \\forall \\ , t \\in I _ 1 \\end{aligned} \\end{align*}"}
+{"id": "5281.png", "formula": "\\begin{align*} ( n _ { u _ 2 } a _ { t _ 2 } ) ^ { - 1 } k _ { \\theta _ 1 } n _ { u _ 2 } a _ { t _ 2 } = n _ { u _ 1 ' } a _ { t _ 1 ' } k _ { \\theta _ 1 ' } , \\end{align*}"}
+{"id": "3239.png", "formula": "\\begin{align*} \\Pi _ n Y _ { i \\Delta _ n } = \\sum _ { j = 1 } ^ n \\alpha _ { j , i } k ( j \\Delta _ n , \\cdot ) , \\end{align*}"}
+{"id": "4671.png", "formula": "\\begin{align*} \\sup _ { p \\ge 1 } \\left \\{ \\ \\frac { | | \\tau | | _ { p , \\Omega } } { \\psi _ { m , L } ( p ) } \\ \\right \\} = C ( m , L ) < \\infty \\end{align*}"}
+{"id": "2251.png", "formula": "\\begin{align*} ( ( A , B ) \\cdot a ) ( Z ) = a ( A ^ { - 1 } Z B ) \\end{align*}"}
+{"id": "4878.png", "formula": "\\begin{align*} u = \\int ^ x \\frac { \\dd x } { \\sqrt { f _ 5 ( x ) } } + \\int ^ { x ' } \\frac { \\dd x ' } { \\sqrt { f _ 5 ( x ' ) } } , v = \\int ^ x \\frac { x \\dd x } { \\sqrt { f _ 5 ( x ) } } + \\int ^ { x ' } \\frac { x ' \\dd x ' } { \\sqrt { f _ 5 ( x ' ) } } , \\end{align*}"}
+{"id": "9176.png", "formula": "\\begin{align*} \\epsilon ( x _ a , t ) = \\int _ N ^ t h ( x _ a + \\delta u ( \\tau ) ) u ( \\tau ) - \\dfrac { b _ { 1 , \\delta } ( x _ a ) } { 2 } \\ , d \\tau . \\end{align*}"}
+{"id": "1373.png", "formula": "\\begin{align*} ( Z _ 1 ( n ) , Z _ { 2 } ( n ) , \\ldots , Z _ d ( n ) ) _ { n \\geq 0 } = ( Y _ 1 ( n + d - 1 ) , Y _ 2 ( n + d - 2 ) , \\ldots , Y _ d ( n ) ) _ { n \\geq 0 } . \\end{align*}"}
+{"id": "8301.png", "formula": "\\begin{align*} M _ 2 = \\left ( \\begin{aligned} & - m _ 1 b _ 2 \\\\ & \\ , \\ , \\ , b _ 1 - m _ 2 \\end{aligned} \\right ) , M _ 3 = \\left ( \\begin{aligned} & - m _ 1 \\ , \\ , \\ , 0 b _ 3 \\\\ & b _ 1 - m _ 2 \\ , \\ , 0 \\\\ & 0 \\ , \\ , b _ 2 - m _ 3 \\end{aligned} \\right ) . \\end{align*}"}
+{"id": "6489.png", "formula": "\\begin{align*} \\lambda | _ { E _ { k , [ p , q ] } } ^ * ( a ) = \\lambda ( p + a ) \\end{align*}"}
+{"id": "4043.png", "formula": "\\begin{align*} P [ \\Delta _ { M _ n } < s _ n ] = O ( r _ n ^ { 1 + \\kappa } ) \\end{align*}"}
+{"id": "889.png", "formula": "\\begin{align*} x = r \\ , , y = q \\ , . \\end{align*}"}
+{"id": "598.png", "formula": "\\begin{align*} \\tau _ { k } : = \\inf \\bigg \\{ t \\in [ t _ { 0 } , \\infty ) \\ , \\bigg | \\ , | Y _ { t } | _ { V _ { t } } \\geq k \\int _ { t _ { 0 } } ^ { t } | Z _ { s } | + | \\hat { \\Sigma } _ { s } | _ { V _ { s } } ^ { 2 } \\ , d s \\geq k \\bigg \\} \\wedge \\tau \\end{align*}"}
+{"id": "455.png", "formula": "\\begin{align*} \\liminf _ { k \\to \\infty } \\frac { f ( b ^ { n _ k } x _ 0 ) } { \\overline { f } ( b ^ { n _ k } x _ 0 ) } \\ge \\liminf _ { k \\to \\infty } c n _ k ( e - 1 ) ^ { - 1 } n _ k ^ { - 1 / 2 } = \\infty , \\end{align*}"}
+{"id": "3471.png", "formula": "\\begin{align*} r _ { \\ell } ^ - \\leq e ^ { 5 0 \\varepsilon q _ n } \\frac { e ^ { - q _ n L } } { \\max ( | \\ell | , 1 ) } \\max ( r _ { \\ell - 1 } ^ - , r _ { \\ell - 1 } ^ + , r _ { \\ell } ^ + , r _ { \\ell + 1 } ^ - , c _ { n , \\ell + 1 } r _ { \\ell + 1 } ^ + ) \\times \\begin{cases} \\max ( | \\ell | , e ^ { \\delta _ n q _ n } , 1 ) , & \\beta _ n \\geq \\delta _ n + 2 0 0 \\varepsilon \\\\ e ^ { \\beta _ n q _ n } , & \\beta _ n < \\delta _ n + 2 0 0 \\varepsilon \\end{cases} , \\end{align*}"}
+{"id": "6589.png", "formula": "\\begin{align*} Q _ { r } \\ge 0 , c _ { r } = r ^ { 1 - d } \\int _ { 0 } ^ { r } \\rho ^ { d - 1 } n c \\ , d \\rho \\ge 0 , r < R , t < T _ { \\rm m a x } . \\end{align*}"}
+{"id": "4058.png", "formula": "\\begin{align*} \\delta _ H ( k ) = 2 \\phi _ H ( k + 1 ) - ( \\phi _ H ( 2 k ) + \\phi _ H ( 2 ) ) \\end{align*}"}
+{"id": "1362.png", "formula": "\\begin{align*} \\begin{aligned} d ( x _ 1 , x _ 2 ) & \\leq d ( x _ 1 , \\bar x _ 2 ) + d ( \\bar x _ 2 , x _ 2 ) \\\\ & \\leq d ( x _ 1 , \\bar x _ 2 ) + \\lambda d ( f ( \\bar x _ 2 ) , f ( x _ 2 ) ) \\\\ & = d ( x _ 1 , \\bar x _ 2 ) + \\lambda d ( f ( \\bar x _ 2 ) , f ( x _ 1 ) ) \\\\ & \\leq d ( x _ 1 , \\pi ^ { - 1 } ( y _ 2 ) ) + \\lambda ^ 2 d ( x _ 1 , \\pi ^ { - 1 } ( y _ 2 ) ) ^ { \\beta } , \\end{aligned} \\end{align*}"}
+{"id": "120.png", "formula": "\\begin{align*} \\| u v \\| _ { L ^ 2 } & = \\Big \\| \\int _ { \\R \\times \\R ^ 2 } \\hat { u } ( \\tau _ 1 , \\xi _ 1 , \\eta _ 1 ) \\hat { v } ( \\tau - \\tau _ 1 , \\xi - \\xi _ 1 , \\eta - \\eta _ 1 ) d \\tau _ 1 d \\xi _ 1 d \\eta _ 1 \\Big \\| _ { L ^ 2 _ { \\tau , \\xi , \\eta } } \\\\ & \\lesssim \\underline { L } ^ { \\frac { 1 } { 2 } } | E ( \\xi , \\eta ) | ^ { \\frac { 1 } { 2 } } \\| u \\| _ { L ^ 2 } \\| v \\| _ { L ^ 2 } , \\end{align*}"}
+{"id": "7957.png", "formula": "\\begin{align*} \\frac { \\nu _ { N , t } ( \\eta ^ { x - e _ i , x } ) } { \\nu _ { N , t } ( \\eta ) } = \\frac { \\theta _ { x } } { \\theta _ { x - e _ i } } \\frac { \\eta _ { x - e _ i } ( K - \\eta _ x ) } { ( K - \\eta _ { x - e _ i } + 1 ) ( \\eta _ x + 1 ) } , \\end{align*}"}
+{"id": "3452.png", "formula": "\\begin{align*} & | \\tilde { P } _ { q _ n - 1 } ( \\theta _ { m _ n + \\ell q _ n + 1 } ) - \\tilde { P } _ { q _ n - 1 } ( \\theta _ { m _ n - q _ n + 1 } ) | \\\\ \\leq & \\| F _ { q _ n - 1 } ( \\theta _ { m _ n + \\ell q _ n + 1 } ) - F _ { q _ n - 1 } ( \\theta _ { m _ n - q _ n + 1 } ) \\| \\\\ \\leq & \\sum _ { j = 0 } ^ { q _ n - 2 } \\| F _ { q _ n - j - 2 } ( \\theta _ { m _ n + \\ell q _ n + 2 + j } ) \\| \\cdot \\| F ( \\theta _ { m _ n - q _ n + 1 + j } ) - F ( \\theta _ { m _ n + \\ell q _ n + 1 + j } ) \\| \\cdot \\| F _ { j } ( \\theta _ { m _ n - q _ n + 1 } ) \\| , \\end{align*}"}
+{"id": "3101.png", "formula": "\\begin{align*} ( \\alpha - \\alpha ^ 2 ) ( 1 - K _ { 2 } ) ^ 2 \\sum \\limits _ { k = 0 } ^ { \\infty } \\norm { \\theta ^ k - \\tilde { \\theta } ^ k } _ { \\mathcal { L } ^ 2 , G } ^ 2 \\leq \\norm { \\theta ^ 0 - \\theta ^ * } _ { \\mathcal { L } ^ 2 , G } ^ 2 . \\end{align*}"}
+{"id": "3530.png", "formula": "\\begin{align*} - \\sum _ { j = 0 } ^ { p } B _ j n ^ { p - j } \\sum _ { k = j } ^ p ( - 1 ) ^ k \\binom { p } { k } \\binom { k } { j } . \\end{align*}"}
+{"id": "7409.png", "formula": "\\begin{align*} M ^ { ( 2 ) } = M ^ { ( 1 ) } _ 2 , M ^ { ( 2 ) } = M ^ { ( 2 ) } _ 1 + M ^ { ( 2 ) } _ 2 , M ^ { ( 2 ) } = M ^ { ( 1 ) } _ 2 , M ^ { ( 2 ) } _ 1 = N _ 2 ~ , M ^ { ( 2 ) } _ 2 = N _ 3 + \\dots + N _ L . \\end{align*}"}
+{"id": "8179.png", "formula": "\\begin{align*} \\mathcal { U } _ { \\lambda } ( \\tau ) = : \\Theta _ { \\lambda } ( t ) + \\frac { 1 } { 2 } \\varrho _ { 0 } \\lambda ^ { \\beta - 1 } \\int _ { \\tau } ^ { t } \\Theta _ { \\lambda } ( s ) \\ d s \\leq \\Theta _ { \\lambda } ( \\tau ) , 0 \\leq t < \\tau \\leq t _ { 0 } < T _ { \\mathrm { g e l } } . \\end{align*}"}
+{"id": "10.png", "formula": "\\begin{align*} \\alpha \\vee \\beta = \\begin{cases} \\{ \\alpha _ { i _ 1 , j _ 2 } , \\alpha _ { i _ 1 , \\overline { j } _ 1 } \\} & i _ 2 \\geq j _ 2 \\\\ \\{ \\alpha _ { i _ 1 , j _ 2 } \\} & \\end{cases} \\end{align*}"}
+{"id": "6408.png", "formula": "\\begin{align*} \\eta ( t ) = \\begin{cases} R , t \\leq \\delta \\\\ R ( 1 - ( \\frac { t - \\delta } { R - \\delta } ) ^ 2 ) ^ 2 , t > \\delta \\end{cases} \\end{align*}"}
+{"id": "8829.png", "formula": "\\begin{align*} d x _ { t , j } = - a _ j x _ { t , j } d t + B _ j ( x _ t , x _ t ) d t + \\sigma _ j d W _ t ^ { ( j ) } , \\end{align*}"}
+{"id": "4050.png", "formula": "\\begin{align*} \\max _ { v \\in \\Lambda } e ( \\alpha ^ { ( v ) } , G ^ { ( v ) } ) = e ( 0 , G ) + 2 H - 3 / 2 \\end{align*}"}
+{"id": "191.png", "formula": "\\begin{align*} \\partial _ { j , ( 1 ) } = \\sum _ { k = 1 } ^ d a _ { j , k } \\partial _ { k , ( 2 ) } \\end{align*}"}
+{"id": "4324.png", "formula": "\\begin{align*} \\lim \\limits _ { j \\to + \\infty } \\frac { G ( t _ 0 ) - G ( t _ 0 + B _ j ) } { \\int _ { t _ 0 } ^ { t _ 0 + B _ j } c ( t ) e ^ { - t } d t } = \\liminf \\limits _ { B \\to 0 + 0 } \\frac { G ( t _ 0 ) - G ( t _ 0 + B ) } { \\int _ { t _ 0 } ^ { t _ 0 + B } c ( t ) e ^ { - t } d t } \\end{align*}"}
+{"id": "2216.png", "formula": "\\begin{align*} A = \\frac { 2 \\pi } { N } / \\sqrt { N } \\end{align*}"}
+{"id": "5378.png", "formula": "\\begin{align*} S _ k ^ { i j } ( T _ \\alpha u ) _ { i j } = T _ \\alpha f . \\end{align*}"}
+{"id": "2558.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { i = 1 } ^ { n } d _ i ^ 2 \\leq \\frac { m ^ 2 \\Big ( \\triangle ( G ) + \\delta ( G ) \\Big ) ^ 2 } { n \\triangle ( G ) \\delta ( G ) } . \\end{align*}"}
+{"id": "7129.png", "formula": "\\begin{align*} \\mathcal S ( f _ 2 \\circ f _ 1 ) = \\left ( \\mathcal S ( f _ 2 ) \\circ f _ 1 \\right ) ( f _ 1 ' ) ^ 2 + \\mathcal S ( f _ 1 ) \\end{align*}"}
+{"id": "8084.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\varepsilon \\right ] = & { \\mathbb { E } \\left [ \\mathbf { s } ^ { \\left ( \\right ) ^ H } \\mathbf { s } ^ { \\left ( \\right ) } \\right ] } - { \\mathbb { E } \\left [ \\mathbf { s } ^ { \\left ( \\right ) ^ H } \\mathbf { y } ' \\right ] } - { \\mathbb { E } \\left [ \\mathbf { y ' } ^ H \\mathbf { s } ^ { \\left ( \\right ) } \\right ] } + { \\mathbb { E } \\left [ \\mathbf { y ' } ^ H \\mathbf { y } ' \\right ] } . \\end{align*}"}
+{"id": "8840.png", "formula": "\\begin{align*} d ( X _ { t , 3 } - x _ { t , 3 } ) = ( x _ { t , 1 } - x _ { 0 , 1 } ) x _ { t , 2 } d t + x _ { 0 , 1 } ( x _ { t , 2 } - x _ { 0 , 2 } ) d t + a _ 3 x _ { t , 3 } d t - x _ { t , 4 } x _ { t , 2 } d t - \\sigma _ 3 d W _ t ^ { ( 3 ) } . \\end{align*}"}
+{"id": "4071.png", "formula": "\\begin{align*} \\max _ { \\lambda \\in \\Lambda ^ { N ( 2 , 0 ) } _ { 1 } } e ( \\alpha ^ \\lambda , G ^ \\lambda ) = e ( 0 , G ^ { N ( 1 , 0 ) } _ { 1 } ) + 2 H - 3 / 2 = 2 H - 3 / 2 . \\end{align*}"}
+{"id": "8462.png", "formula": "\\begin{align*} V _ 1 ^ { - 1 } ( k ) R ( x ; z ) V _ 1 ( k ) = V _ 2 ^ { - 1 } ( k ) R ( x ; z ) V _ 2 ( k ) = J ( x ; k ) , z \\in \\mathbb { R } , k \\in \\mathbb { R } \\cup i \\mathbb { R } . \\end{align*}"}
+{"id": "5913.png", "formula": "\\begin{align*} \\varphi _ n ( t ) : = \\exp \\left ( - \\int _ 0 ^ t \\big [ f ( r ) + \\rho ( X ( r , x _ n ) ) + \\eta ( X ( r , x ) ) \\big ] d r \\right ) . \\end{align*}"}
+{"id": "3013.png", "formula": "\\begin{align*} \\nu = \\min \\{ 1 , \\varepsilon _ * , \\varepsilon _ * / \\lambda _ 0 , \\varepsilon _ * / \\lambda ^ * , \\nu _ * \\} . \\end{align*}"}
+{"id": "858.png", "formula": "\\begin{align*} \\sigma = \\prod _ p \\left ( 1 - \\frac { \\lambda ( p ^ 2 ) } { p ^ 2 } \\right ) \\ , . \\end{align*}"}
+{"id": "5701.png", "formula": "\\begin{align*} R ( X ) = A X ^ { q + 1 } + B X ^ { 2 ^ l ( q + 1 ) } + C X ^ { q + 2 ^ l } + D X ^ { 2 ^ l q + 1 } , \\end{align*}"}
+{"id": "6514.png", "formula": "\\begin{align*} \\lambda _ y ( w b ^ { - n } , w b ^ { - n + 1 } ) = \\lambda _ x ( w b ^ { - n } , w b ^ { - n + 1 } ) = \\mu ( w b ^ { - n } , w b ^ { - n + 1 } ) \\end{align*}"}
+{"id": "5339.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { \\infty } P _ { n , k } ( \\ell ) a _ { \\ell } z ^ { \\ell } - \\frac { B _ { n , k } ( z ) } { C _ { n , k } ( z ) } = O ( z ^ { 1 0 n + 1 } ) . \\end{align*}"}
+{"id": "5906.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } g _ { n _ i } ( t , \\omega ) = 0 , ( t , \\omega ) . \\end{align*}"}
+{"id": "4350.png", "formula": "\\begin{align*} \\int _ { \\{ \\Psi < - t _ 0 \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h \\mathbb { I } _ E ( - \\Psi ) \\ge & \\lim _ { l \\to + \\infty } \\int _ { \\{ \\Psi < - t _ 0 \\} } | F _ { t _ 0 , \\tilde { c } } | ^ 2 _ h \\mathbb { I } _ { V _ j } ( - \\Psi ) \\\\ \\ge & \\lim _ { l \\to + \\infty } \\int _ { \\{ \\Psi < - t _ 0 \\} } | \\tilde F | ^ 2 _ h \\mathbb { I } _ { V _ j } ( - \\Psi ) \\\\ = & \\int _ { \\{ \\Psi < - t _ 0 \\} } | \\tilde F | ^ 2 _ h \\mathbb { I } _ { V _ j } ( - \\Psi ) . \\end{align*}"}
+{"id": "8054.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\lvert y _ k \\rvert ^ 2 \\right ] = a _ c ^ 2 \\lvert \\mathbf { h } _ k ^ { \\textrm { T } } \\mathbf { p } _ c \\rvert ^ 2 + \\sum _ { i = 1 } ^ K a _ i ^ 2 \\lvert \\mathbf { h } _ k ^ { \\textrm { T } } \\mathbf { p } _ i \\rvert ^ 2 + \\sigma _ w ^ 2 . \\end{align*}"}
+{"id": "4631.png", "formula": "\\begin{align*} \\phi ( \\sigma _ 1 \\sigma _ 2 , \\tau ) = \\phi ( \\tau , \\sigma _ 1 \\sigma _ 2 ) = \\phi ( \\tau , \\sigma _ 1 ) \\phi ( \\tau , \\sigma _ 2 ) = \\phi ( \\sigma _ 1 , \\tau ) \\phi ( \\sigma _ 2 , \\tau ) . \\end{align*}"}
+{"id": "5812.png", "formula": "\\begin{align*} \\underline { u } = \\frac { 1 } { 4 } \\left ( \\sum _ { j = 1 } ^ p e _ j \\cdot u ( e _ j ) + \\sum _ { j = p + 1 } ^ n e _ j \\cdot ( - u ^ * ( e _ j ) ) \\right ) . \\end{align*}"}
+{"id": "5410.png", "formula": "\\begin{align*} \\omega : = \\{ x \\in \\Omega : | x _ \\beta | < \\delta \\frac { b _ \\alpha } { \\sqrt { b _ \\beta } } , \\beta = \\alpha + 1 , \\ldots , n - 1 , x _ n < \\delta ^ 2 b _ \\alpha \\} , \\end{align*}"}
+{"id": "6146.png", "formula": "\\begin{align*} X _ { r } = \\xi _ r \\quad { X } _ t = \\xi _ 0 + \\int _ { 0 } ^ { t } \\mu ( s , X ) \\ , d s + \\int _ { 0 } ^ { t } \\sigma ( s , X ) \\ , d W _ s , \\end{align*}"}
+{"id": "1618.png", "formula": "\\begin{align*} \\phi : C ^ n _ c ( G , V ) ^ G \\to C ^ { n - 1 } _ c ( G , V ) , \\phi ( c ) ( x _ 1 , \\ldots , x _ n ) : = c ( 1 , x _ 1 , x _ 1 x _ 2 , \\ldots , x _ 1 \\cdots x _ n ) , n \\geq 1 \\end{align*}"}
+{"id": "7252.png", "formula": "\\begin{align*} L ( x ) = x ^ { 4 0 9 6 } + ( t ^ { 2 4 } + t ) x ^ { 2 0 4 8 } + t ^ { 1 2 8 } x ^ { 1 0 2 4 } + ( t ^ { 8 8 } + t ^ { 6 5 } ) x ^ { 5 1 2 } + t ^ { 1 6 } x ^ { 3 2 } + \\end{align*}"}
+{"id": "578.png", "formula": "\\begin{align*} g _ { \\theta _ 1 } ( \\omega ) = g _ { \\theta _ 2 } ( \\omega ) , \\omega \\in \\Omega _ { \\theta _ 1 } \\cap \\Omega _ { \\theta _ 2 } . \\end{align*}"}
+{"id": "8765.png", "formula": "\\begin{align*} & \\int _ { J _ U ^ V } \\left ( A _ { \\vartheta \\varphi } { } ^ { ( \\vartheta , \\varphi ) } \\mathcal I _ { a } ^ { \\vec { \\alpha } } [ F ] ( W , Z ) + B _ { \\vartheta \\varphi } { \\bf i } { } ^ { ( \\vartheta , \\varphi ) } \\mathcal I _ { a } ^ { \\vec { \\alpha } } [ F ] ( W , Z ) \\right ) d Z \\wedge d Z ^ { * } \\\\ = & \\int _ { \\partial J _ U ^ V } { } ^ { ( \\vartheta , \\varphi ) } \\mathcal I _ { a } ^ { \\vec { \\alpha } } [ F ] ( W , Z ) d \\sigma _ { ( \\vartheta , \\varphi ) } ( Z ) , \\end{align*}"}
+{"id": "1533.png", "formula": "\\begin{align*} [ P , Q ] = P \\ , Q - Q \\ , P \\ ; . \\end{align*}"}
+{"id": "6427.png", "formula": "\\begin{align*} \\log \\det u _ { , i j } = - v _ p x ^ p + u _ { , q } \\xi ^ q + c . \\end{align*}"}
+{"id": "2659.png", "formula": "\\begin{align*} c _ 2 ^ 2 ( \\mu ) : = \\int _ J \\int _ J \\int _ J \\frac { d \\mu ( z _ 1 ) s \\mu ( z _ 2 ) d \\mu ( z _ 3 ) } { R ^ 2 ( z _ 1 , z _ 2 , z _ 3 ) } < \\infty , \\end{align*}"}
+{"id": "7982.png", "formula": "\\begin{align*} 0 = \\nabla _ { i } E & = - w ^ { 1 1 } \\nabla _ { i } w _ { 1 1 } - d \\frac { h _ { i } } { h } + l \\rho \\rho _ { i } \\\\ & = - w ^ { 1 1 } ( h _ { i 1 1 } + h _ { 1 } \\delta _ { 1 i } ) - d \\frac { h _ { i } } { h } + l \\rho \\rho _ { i } , \\end{align*}"}
+{"id": "3500.png", "formula": "\\begin{align*} \\phi ( k ) = \\sum _ { s ; z _ { i + 1 } \\in I ( z _ i ' ) } G _ { I ( k ) } ( k , z _ 1 ) G _ { I ( z _ 1 ' ) } ( z _ 1 ' , z _ 2 ) \\cdots G _ { I ( z _ t ' ) } ( z _ t ' , z _ { t + 1 } ) \\phi ( { z _ { t + 1 } ' } ) , \\end{align*}"}
+{"id": "3278.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sup _ { n \\in \\mathbb N } \\mathbb E \\left [ \\sup _ { t \\in [ 0 , T ] } \\left \\| P _ N ^ 2 ( I ) _ t ^ n \\right \\| _ { \\mathcal H } \\right ] = 0 . \\end{align*}"}
+{"id": "7134.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathcal H _ X ^ D ( \\alpha ) _ x & = \\mathcal H _ { X , x } & \\\\ \\mathcal H _ X ^ D ( \\alpha ) _ x & = \\alpha _ i \\cap R ^ 0 _ { y _ i , n _ i } & \\end{aligned} \\right . \\end{align*}"}
+{"id": "717.png", "formula": "\\begin{align*} \\frac { N - 2 } { 2 } \\int | \\nabla u | ^ 2 d x - \\frac { N } { 2 } \\lambda \\int | u | ^ 2 d x - \\frac { N + \\alpha } { 2 p } \\int ( I _ \\alpha \\ast | u | ^ { p } ) | u | ^ p d x = 0 . \\end{align*}"}
+{"id": "2760.png", "formula": "\\begin{align*} r _ m ( x ) \\geq \\frac { C _ 1 ^ { - m } } { 4 \\prod _ { j = 1 } ^ { m - 1 } | h _ j ( x ) | } . \\end{align*}"}
+{"id": "6434.png", "formula": "\\begin{align*} \\sum _ { 0 \\le m _ 1 < \\cdots < m _ { n } < \\infty } \\frac { ( \\alpha ) _ { m _ 1 } } { { m _ 1 } ! } \\frac { { m _ n } ! } { ( \\alpha ) _ { m _ n } } \\left \\{ \\prod _ { i = 1 } ^ { n } \\frac { 1 } { ( m _ i + \\alpha ) ^ { a _ i } ( m _ i + \\beta ) ^ { b _ i } } \\right \\} , \\end{align*}"}
+{"id": "7840.png", "formula": "\\begin{align*} \\hat \\Phi ^ \\dagger ( a , s ) : = \\langle z _ 1 ^ * , \\Phi ^ \\dagger ( a u _ \\ell , p ( s ) ) \\rangle , \\end{align*}"}
+{"id": "268.png", "formula": "\\begin{align*} \\mathcal { C } : = \\{ q \\in \\Omega : d _ H ( q , [ 0 , \\xi ] ) < R , 0 < \\langle q , \\xi \\rangle < | \\xi | ^ 2 \\} \\end{align*}"}
+{"id": "4311.png", "formula": "\\begin{align*} & \\int _ { U _ 0 } | \\tilde { F } _ j - ( 1 - b _ { 1 } ( \\Psi _ 1 ) ) f _ j F ^ { 2 } | ^ 2 _ h e ^ { - \\varphi _ 1 + v _ { 1 } ( \\Psi _ 1 ) - \\Psi _ 1 } c ( - v _ { 1 } ( \\Psi _ 1 ) ) \\\\ \\le & \\left ( c ( T _ 1 ) e ^ { - T _ 1 } + \\int _ { T _ 1 } ^ { T _ 1 + 2 } c ( s ) e ^ { - s } d s \\right ) \\int _ { U _ 0 } \\mathbb { I } _ { \\{ - T _ 1 - 2 < \\Psi _ 1 < - T _ 1 - 1 \\} } | f _ j | ^ 2 _ h e ^ { - \\Psi _ 1 } , \\end{align*}"}
+{"id": "5192.png", "formula": "\\begin{align*} b _ t + \\nabla \\times ( B \\times u ) + \\mu \\nabla \\times ( \\nabla \\times b ) = 0 \\textrm { o n } \\ ( L ^ { 2 } _ { \\sigma } \\cap H ^ { 1 } ) ( \\mathbb { R } ^ { 3 } ) ^ { * } , \\end{align*}"}
+{"id": "4284.png", "formula": "\\begin{align*} n ^ { 3 } + \\left ( n + 1 \\right ) ^ { 3 } - \\left ( \\frac { \\beta } { 3 } \\left ( n + 2 \\right ) \\right ) ^ { 3 } + \\left ( \\frac { \\beta } { 3 } \\left ( n - 1 \\right ) \\right ) ^ { 3 } = 0 \\end{align*}"}
+{"id": "5574.png", "formula": "\\begin{align*} A ( y _ n . . . y _ 1 x ) - A ( y _ n . . . y _ 1 ) = A ( 0 ^ n 1 ^ k 0 . . . ) - A ( 0 ^ \\infty ) = a _ n - a \\ . \\end{align*}"}
+{"id": "6057.png", "formula": "\\begin{align*} H = \\frac { \\alpha } { n } \\frac { \\nu _ { n + 1 } } { x _ { n + 1 } } + \\frac { \\varpi } { n } , \\end{align*}"}
+{"id": "783.png", "formula": "\\begin{align*} V = g ( c ) H \\end{align*}"}
+{"id": "1032.png", "formula": "\\begin{align*} \\prescript L { } { } \\ell { } ^ R ( x , w ^ { - 1 } \\alpha ) = - \\Phi ^ + _ R ( w ^ { - 1 } \\alpha ) + \\Phi _ L ^ + ( \\alpha ) = 1 - \\Phi ^ + _ R ( w ^ { - 1 } \\alpha ) . \\end{align*}"}
+{"id": "3782.png", "formula": "\\begin{align*} \\partial _ z ^ * = M _ z , \\end{align*}"}
+{"id": "5477.png", "formula": "\\begin{align*} ( x ) _ k = \\frac { \\Gamma ( x + k ) } { \\Gamma ( x ) } = x ( x + 1 ) \\dots ( x + k - 1 ) \\end{align*}"}
+{"id": "3965.png", "formula": "\\begin{align*} \\mathbf { c } ^ { ( 1 ) } = \\mathbf { c } ^ { ( 2 ) } = \\cdots = \\mathbf { c } ^ { ( n - 2 k + 1 ) } = ( 1 , 1 , \\cdots , 1 ) . \\end{align*}"}
+{"id": "4829.png", "formula": "\\begin{align*} W ( t ) = \\sum _ { l = 1 } ^ { n _ 1 } \\Psi _ { k l } \\beta ^ c _ l ( t ) , k = 1 , \\dots , n . \\end{align*}"}
+{"id": "4715.png", "formula": "\\begin{align*} \\sigma \\geq v + \\sum _ { k = 1 } ^ l a _ k \\hat C _ { q _ k , N } . \\end{align*}"}
+{"id": "5964.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 + } \\sup _ { f \\in \\Upsilon } \\int _ 0 ^ { T - \\delta } \\Vert f ( t + \\delta ) - f ( t ) \\Vert _ Y ^ p d t = 0 , \\end{align*}"}
+{"id": "2550.png", "formula": "\\begin{align*} \\mathbf { U } _ { x _ w ^ { ( i ) } } - \\omega \\mathbf { U } _ { x _ o ^ { ( i ) } } = \\begin{pmatrix} 0 & 0 \\\\ 0 & \\tilde { \\rho } _ i \\mathbf { P } _ { x _ o ^ { ( i ) } } \\end{pmatrix} , \\end{align*}"}
+{"id": "7314.png", "formula": "\\begin{align*} ( \\mu _ T ( y , z ) | \\mu _ T ( x , w ) ) = ( \\mu _ T ( x , y ) | \\mu _ T ( z , w ) ) . \\end{align*}"}
+{"id": "2911.png", "formula": "\\begin{align*} M _ { x , y } ( m ) = \\sum _ { j , j ' = 0 } ^ n \\Theta _ m ( \\mu _ j , \\mu _ { j ' } ) \\psi _ j ( x ) \\psi _ { j ' } ( x ) \\psi _ j ( y ) \\psi _ { j ' } ( y ) \\end{align*}"}
+{"id": "7006.png", "formula": "\\begin{align*} u \\ , = \\ , \\frac { 1 } { 3 \\ , z ^ 2 } \\Big \\{ \\big ( 1 - 2 \\ , y + y ^ 2 - 2 \\ , x z \\big ) ^ { 3 / 2 } - ( 1 - y ) \\ , \\big ( 1 - 2 \\ , y + y ^ 2 - 3 \\ , x z \\big ) \\Big \\} , \\end{align*}"}
+{"id": "1750.png", "formula": "\\begin{align*} - \\nabla \\cdot \\big ( \\nabla \\chi _ i + e _ i \\big ) & = 0 & \\mbox { i n } & Y ^ { \\ast } , \\\\ - \\big ( \\nabla \\chi _ i + e _ i \\big ) \\cdot \\nu & = 0 & \\mbox { o n } & \\Gamma , \\\\ \\chi _ i \\mbox { i s } Y \\mbox { - p e r i o d i c , } \\int _ { Y ^ { \\ast } } \\chi _ i d y = 0 . \\end{align*}"}
+{"id": "4781.png", "formula": "\\begin{align*} a ( u , v ) = ( f , v ) ~ v \\in H ^ 1 _ 0 ( D ) , \\end{align*}"}
+{"id": "5302.png", "formula": "\\begin{align*} y _ 2 ^ { - 1 } y _ 1 \\psi _ 1 ( x ) ( y _ 2 ^ { - 1 } y _ 1 ) ^ { - 1 } = \\psi _ 2 ( x ) = \\psi _ 2 \\circ \\psi _ 1 ^ { - 1 } ( \\psi _ 1 ( x ) ) . \\end{align*}"}
+{"id": "5494.png", "formula": "\\begin{align*} f ( p , s ) = s \\log a + \\log A _ k - \\left ( \\frac { p } { 2 } + 2 \\right ) \\log s + \\log \\Gamma \\left ( \\frac { p } { 2 } + 2 \\right ) . \\end{align*}"}
+{"id": "2052.png", "formula": "\\begin{align*} \\rho _ \\sigma : = \\Big ( ( t - t _ 0 ) \\eta _ 1 ^ 2 + ( t - t _ 0 ) ^ 2 m _ 1 \\eta _ 1 + \\frac { ( t - t _ 0 ) ^ 3 } { 3 } m _ 1 ^ 2 \\Big ) ^ \\sigma , \\sigma \\geq 0 , \\ t \\in [ t _ 0 , 1 ] . \\end{align*}"}
+{"id": "4158.png", "formula": "\\begin{align*} \\beta _ 2 ( u ) = 0 \\Longrightarrow \\beta _ 2 ( u _ { \\varepsilon } ) = 0 . \\end{align*}"}
+{"id": "6318.png", "formula": "\\begin{align*} D _ t ^ { ( 1 ) } u _ k = b _ { k 1 } + 2 b _ { k 2 } t + \\cdots , \\end{align*}"}
+{"id": "790.png", "formula": "\\begin{align*} T _ { ( \\Sigma , c ) } M ^ m = \\left \\{ \\big ( V , w \\big ) \\ , \\big | \\ , V , w : \\Sigma \\rightarrow \\R \\int _ \\Sigma w - c H V \\ , \\mathrm { d } \\mathcal { H } ^ d = 0 \\right \\} . \\end{align*}"}
+{"id": "8589.png", "formula": "\\begin{align*} | P _ { [ u , v ] ^ \\bot } A | _ { n - 2 } \\sqrt { 1 - \\langle u , v \\rangle ^ 2 } = n ( n - 1 ) V ( A [ n - 2 ] , [ 0 , u ] , [ 0 , v ] ) . \\end{align*}"}
+{"id": "5085.png", "formula": "\\begin{align*} { \\mathcal { I } } _ h \\leq \\int _ { \\Omega } | B | ^ { 2 } \\dd x \\leq \\liminf _ { n \\to \\infty } \\int _ { \\Omega } | B _ n | ^ { 2 } \\dd x = { \\mathcal { I } } _ h , \\end{align*}"}
+{"id": "7499.png", "formula": "\\begin{align*} | \\nabla ( \\varphi ^ { + } ) ^ { \\frac { m } { 2 } } | ^ 2 _ { A ( x ) } = \\frac { m ^ 2 } { 4 } ( \\varphi ^ { + } ) ^ { m - 2 } | \\nabla ( \\varphi ^ { + } ) | ^ 2 _ { A ( x ) } , \\end{align*}"}
+{"id": "9035.png", "formula": "\\begin{align*} F _ { + } ( x ) = G ( x ) + \\frac { 1 } { 2 } \\left [ - L , G ( x ) \\right ] + \\frac { 1 } { 1 2 } \\left [ - L , \\left [ - L , G ( x ) \\right ] \\right ] + \\cdots . \\end{align*}"}
+{"id": "5665.png", "formula": "\\begin{align*} \\underline { a } : = \\begin{pmatrix} a \\\\ & a \\\\ & & \\ddots \\\\ & & & a \\\\ & & & & a \\end{pmatrix} , \\end{align*}"}
+{"id": "929.png", "formula": "\\begin{align*} \\Phi _ 1 ( t ) & = \\int ^ t _ 1 \\tau ^ { - 1 } | v _ 1 ( \\tau ) | ^ 2 \\ d \\tau \\\\ & = | \\alpha | ^ 2 \\log t + \\int ^ t _ 1 \\tau ^ { - 1 } ( | e ^ { 3 i \\lambda _ 1 \\Phi _ 1 } v _ 1 ( \\tau ) | ^ 2 - | \\alpha | ^ 2 ) \\ d \\tau . \\end{align*}"}
+{"id": "3302.png", "formula": "\\begin{align*} N ( d ) = \\frac { p + 1 } { 2 d } + \\xi ( d ) , \\end{align*}"}
+{"id": "332.png", "formula": "\\begin{align*} C ( a _ k ) = \\max ( \\{ C ( b _ i ) \\mid b _ i < _ P a _ k \\} \\cup \\{ C ( a _ i ) \\mid 1 \\le i < k \\} ) + 1 . \\end{align*}"}
+{"id": "1996.png", "formula": "\\begin{align*} a & = \\sqrt { 4 - 1 / e ^ 4 } , \\\\ T ( x ) & = 2 - \\sqrt { x ^ 2 + 1 / e ^ 4 } , \\\\ h ( x ) & = 2 \\log \\left ( \\sqrt { x ^ 2 + 1 / e ^ 4 } + x \\right ) + 4 . \\end{align*}"}
+{"id": "8553.png", "formula": "\\begin{align*} \\kappa ( t ) = ( \\sqrt { t } ) ^ { \\alpha - 1 } J _ { \\alpha - 1 } ( 2 \\sqrt { t } ) , \\ k ( t ) = ( \\sqrt { t } ) ^ { - \\alpha } I _ { - \\alpha } ( 2 \\sqrt { t } ) , \\ 0 < \\alpha < 1 , \\end{align*}"}
+{"id": "4434.png", "formula": "\\begin{align*} a \\cos ( x ) + b \\sin ( x ) = r \\cos ( x + \\theta ) . \\end{align*}"}
+{"id": "3508.png", "formula": "\\begin{align*} ( n - 1 ) ^ { p } + n ^ { p } - ( n + 1 ) ^ { p } \\ = \\ n ^ { p } - 2 \\sum _ { j = 0 } ^ { p / 2 } \\binom { p } { 2 j + 1 } n ^ { p - 2 j - 1 } . \\end{align*}"}
+{"id": "3436.png", "formula": "\\begin{align*} \\| \\theta - \\frac { 1 } { 2 } + ( m _ n ^ { ( 1 ) } + \\ell q _ n ) \\alpha \\| \\leq & \\left ( \\left [ \\frac { a _ { n + 1 } + 3 } { 2 } \\right ] + \\left [ \\frac { a _ { n + 1 } } { 6 } \\right ] + \\frac { 1 } { 2 } + \\frac { 1 } { 2 q _ n } \\right ) \\| q _ n \\alpha \\| \\\\ \\leq & \\begin{cases} ( \\frac { 2 } { 3 } a _ { n + 1 } + 2 + \\frac { 1 } { 2 q _ n } ) \\| q _ n \\alpha \\| , a _ { n + 1 } \\geq 7 \\\\ ( a _ { n + 1 } - \\frac { 1 } { 2 } + \\frac { 1 } { 2 q _ n } ) \\| q _ n \\alpha \\| , 4 \\leq a _ { n + 1 } \\leq 6 . \\end{cases} \\end{align*}"}
+{"id": "3386.png", "formula": "\\begin{align*} c _ k = \\left \\{ \\begin{array} { l l } { k + 2 \\choose 3 } & k \\in [ 0 , N ] \\\\ { 2 N - k + 2 \\choose 3 } & k \\in [ N + 1 , 2 N ] \\end{array} \\right . . \\end{align*}"}
+{"id": "8006.png", "formula": "\\begin{align*} & G _ x ( \\theta ) : = \\sum _ { n \\in \\Z } g \\left ( \\pi _ N ( x ) ( \\theta + n ) \\right ) \\\\ & = \\sum _ { | n | \\leq \\pi _ N ( x ) } \\widehat { G _ { x } } ( n ) e ( n \\theta ) \\\\ & = \\widehat { G _ { x } } ( 0 ) + \\sum _ { 1 \\leq n \\leq \\pi _ N ( x ) } \\widehat { G _ { x } } ( n ) 2 \\cos ( 2 \\pi n \\theta ) , \\end{align*}"}
+{"id": "1109.png", "formula": "\\begin{align*} P _ { \\min } ^ { { \\rm { N } } } \\left ( { \\overline S , \\overline \\varepsilon , \\overline R } \\right ) = \\bigcap { \\left \\{ { P \\in { \\mathbb { R } ^ + } : \\left ( { \\overline S , \\overline R } \\right ) \\in { { \\mathcal { R } } } _ { { \\rm { S v B } } } ^ { { \\rm { N } } } \\left ( { W , P , K , \\overline \\varepsilon } \\right ) } \\right \\} } . \\end{align*}"}
+{"id": "7167.png", "formula": "\\begin{align*} ( \\nu S \\textbf { \\textit { u } } ) ^ i = \\nu ^ j ( S \\textbf { \\textit { u } } ) ^ i _ j & = \\nu ^ j ( \\nabla ^ i u _ j + \\nabla _ j u ^ i ) \\end{align*}"}
+{"id": "5122.png", "formula": "\\begin{align*} A \\cdot B = 2 \\Phi \\frac { G } { r ^ { 2 } } + \\nabla \\cdot ( \\Phi \\nabla \\theta \\times A ) . \\end{align*}"}
+{"id": "6939.png", "formula": "\\begin{align*} H \\big ( \\mathrm { L a w } ( t _ { n + 1 } X _ { n + 1 } + ( 1 - t _ { n + 1 } ) \\widetilde { X } ) \\big ) & = H \\big ( \\widetilde { Q } \\circ ( t _ { n + 1 } T + ( 1 - t _ { n + 1 } ) \\mathrm { I d } ) ^ { - 1 } \\big ) \\\\ & \\le t _ { n + 1 } H ( Q _ { n + 1 } ) + ( 1 - t _ { n + 1 } ) H ( \\widetilde { Q } ) . \\end{align*}"}
+{"id": "3540.png", "formula": "\\begin{align*} \\lambda _ 1 & = ( 2 , 3 , 4 , 5 , 6 , 3 , 4 , 2 ) , & \\lambda _ 2 & = ( 3 , 6 , 8 , 1 0 , 1 2 , 6 , 8 , 4 ) , \\\\ \\lambda _ 3 & = ( 4 , 8 , 1 2 , 1 5 , 1 8 , 9 , 1 2 , 6 ) , & \\lambda _ 4 & = ( 5 , 1 0 , 1 5 , 2 0 , 2 4 , 1 2 , 1 6 , 8 ) , \\\\ \\lambda _ 5 & = ( 6 , 1 2 , 1 8 , 2 4 , 3 0 , 1 5 , 2 0 , 1 0 ) , & \\lambda _ 6 & = ( 3 , 6 , 9 , 1 2 , 1 5 , 8 , 1 0 , 5 ) , \\\\ \\lambda _ 7 & = ( 4 , 8 , 1 2 , 1 6 , 2 0 , 1 0 , 1 4 , 7 ) , & \\lambda _ 8 & = ( 2 , 4 , 6 , 8 , 1 0 , 5 , 7 , 4 ) . \\end{align*}"}
+{"id": "4690.png", "formula": "\\begin{align*} { \\rm V o l } ( \\Sigma ) \\geq C ( n ) \\sum _ { i = 1 } ^ { k } { \\rm V o l } ( \\Sigma \\cap B _ { p _ i } ( 3 r ) ) \\geq \\sum _ { i = 1 } ^ { k } \\frac { C ( n ) } { r } { \\rm V o l } ( B \\cap B _ { p _ i } ( r ) ) \\geq \\frac { C ( n ) } { r } { \\rm V o l } ( B ) . \\end{align*}"}
+{"id": "7288.png", "formula": "\\begin{align*} L \\ge L _ G : = 4 \\sqrt { d } \\max _ { x \\in [ - 2 b , 2 b ] ^ d } \\| \\nabla g ( x ) \\| + 1 , \\end{align*}"}
+{"id": "2036.png", "formula": "\\begin{align*} \\theta _ 1 & = \\frac { 1 } { 2 } ( \\arg \\alpha _ 1 - \\arg \\alpha _ 2 + \\arg \\beta _ 1 - \\arg \\beta _ 2 ) , & \\delta _ 1 & = u ( \\theta _ 1 ) \\\\ \\theta _ 2 & = \\frac { 1 } { 2 } ( \\arg \\alpha _ 1 - \\arg \\alpha _ 2 - \\arg \\beta _ 1 + \\arg \\beta _ 2 ) , & \\delta _ 2 & = u ( \\theta _ 2 ) \\end{align*}"}
+{"id": "8638.png", "formula": "\\begin{align*} \\forall \\varepsilon > 0 , \\lim _ { n \\to \\infty } \\sum _ { m = 1 } ^ n \\mathbb { E } _ { n , m } [ \\| \\chi _ { n , m } \\| _ { L ^ 2 ( \\mu ) } ^ 2 : \\| \\chi _ { n , m } \\| _ { L ^ 2 ( \\mu ) } > \\varepsilon ] = 0 \\end{align*}"}
+{"id": "1427.png", "formula": "\\begin{align*} \\frac { \\Gamma ( n + 1 / 2 - k ) } { \\Gamma ( n + 1 ) } = \\frac { 1 } { \\Gamma ( 1 / 2 + k ) } \\int _ 0 ^ { \\infty } e ^ { - ( n + 1 / 2 - k ) v } ( 1 - e ^ { - v } ) ^ { k - 1 / 2 } d v . \\end{align*}"}
+{"id": "4660.png", "formula": "\\begin{align*} & \\Vert T _ { m _ r } ^ { ( N + 1 ) } : L _ { p _ 1 } ( \\mathbb { R } , S _ { p _ 1 } ^ { N + 1 } ) \\times L _ { p _ 2 } ( \\mathbb { R } , S _ { p _ 2 } ^ { N + 1 } ) \\rightarrow L _ { 1 } ( \\mathbb { R } , S _ { 1 } ^ { N + 1 } ) \\Vert \\\\ & \\leq \\Vert T _ { m } ^ { ( N + 1 ) } : L _ { p _ 1 } ( \\mathbb { R } , S _ { p _ 1 } ^ { N + 1 } ) \\times L _ { p _ 2 } ( \\mathbb { R } , S _ { p _ 2 } ^ { N + 1 } ) \\rightarrow L _ { 1 } ( \\mathbb { R } , S _ { 1 } ^ { N + 1 } ) \\Vert = B _ { p _ 1 , p _ 2 , N } . \\end{align*}"}
+{"id": "5205.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ + \\frac { G } { r ^ { 2 } } \\dd x = - \\mu \\int _ { \\mathbb { R } ^ { 3 } } \\nabla \\times B \\cdot B 1 _ { ( 0 , \\infty ) } ( \\Phi ) \\dd x , \\end{align*}"}
+{"id": "127.png", "formula": "\\begin{align*} ( f _ { i , L } ) \\subseteq \\begin{cases} & \\tilde { D } _ { N _ i , \\leqslant L } , \\ ; L = \\max \\{ \\tilde { L } \\in 2 ^ { \\N _ 0 } : \\tilde { L } \\leqslant N _ 3 ^ { 5 - 2 \\alpha + \\varepsilon } \\} , \\\\ & \\tilde { D } _ { N _ i , L } , \\ ; . \\end{cases} \\end{align*}"}
+{"id": "5612.png", "formula": "\\begin{align*} b _ { k + q } - b _ \\alpha = W ( . . . 1 0 ^ { k + q } | 0 ^ { s + p } 1 . . . ) + W ( . . . 1 0 ^ k | 1 ^ s 0 . . . ) . \\end{align*}"}
+{"id": "952.png", "formula": "\\begin{align*} u _ 2 ( t ) & = M ( t ) D ( t ) \\mathcal { F } M ( t ) \\mathcal { F } ^ { - 1 } v _ 2 ( t ) \\\\ & = M ( t ) D ( t ) v _ 2 ( t ) + O ( t ^ { - 3 / 4 + C _ 4 \\varepsilon _ 1 ^ 2 } ) \\\\ & = M ( t ) D ( t ) e ^ { - 3 i \\lambda _ 1 ( \\Phi _ 1 - \\theta _ 1 ) } w + O ( t ^ { - 3 / 4 + C _ 4 \\varepsilon _ 1 ^ 2 } ) . \\end{align*}"}
+{"id": "2984.png", "formula": "\\begin{align*} \\widetilde { \\mu } ( \\phi _ u ( A ) ) = \\widetilde { \\mu } ( A ) . \\end{align*}"}
+{"id": "9061.png", "formula": "\\begin{align*} ( \\lambda + e _ { m - k + 2 } + \\cdots + e _ m - ( k - 1 ) f _ 1 ) ( H _ { \\beta _ { m - k + 1 } } ) v _ { k - 1 } = ( \\lambda _ { m - k + 1 } + \\mu _ 1 - k + 1 ) v _ { k - 1 } . \\end{align*}"}
+{"id": "8761.png", "formula": "\\begin{align*} F ( W ) = \\frac { 1 } { 2 \\pi { \\bf i } } \\int _ { \\gamma } \\frac { F ( Z ) } { Z - W } d Z + \\frac { 1 } { 2 \\pi { \\bf i } } \\int _ { \\Gamma } \\frac { \\dfrac { \\partial F } { \\partial Z ^ * } } { Z - W } d Z \\wedge d Z ^ * , \\end{align*}"}
+{"id": "976.png", "formula": "\\begin{align*} \\left | \\bigcup _ { i \\in I } F _ i \\right | \\le | I | e _ 0 - \\underbrace { | J | | I \\setminus J | } _ { \\mathclap { = c _ 1 ( | I | - c _ 1 ) } } \\theta _ d q ^ { d ^ 2 - d - 2 } \\end{align*}"}
+{"id": "5471.png", "formula": "\\begin{align*} \\widehat { C } _ { \\psi } = C _ { \\widehat { \\psi } } , \\end{align*}"}
+{"id": "2857.png", "formula": "\\begin{align*} \\Sigma _ 2 ( { \\frak y } ) = \\left [ \\begin{array} { c c } 0 _ { n + 1 } & 0 _ { n + 1 } \\\\ 0 _ { n + 1 } & D _ 2 ( { \\frak y } ) \\end{array} \\right ] . \\end{align*}"}
+{"id": "5808.png", "formula": "\\begin{align*} d v v ^ { - 1 } = \\frac { 1 } { 2 } a I + \\frac { 1 } { \\sqrt { \\tau _ 0 } } \\left ( \\frac { \\tau _ 0 } { 4 } d \\overline { z } + Q _ 0 d z \\right ) i J . \\end{align*}"}
+{"id": "4843.png", "formula": "\\begin{align*} H _ { \\mathrm { C V } } ( s , i , p , q , l ) & \\coloneqq ( s + i ) l w + \\beta ( 1 - \\theta l ) ^ 2 s i \\left ( q - p \\right ) + i \\phi ( i ) \\left ( \\frac { w } { r } + \\chi \\right ) - ( \\gamma + \\phi ( i ) ) i q \\\\ & = \\beta \\theta ^ 2 s i ( q - p ) l ^ 2 + \\left [ ( s + i ) w - 2 \\beta \\theta s i ( q - p ) \\right ] l \\\\ & + \\beta s i ( q - p ) + i \\phi ( i ) \\left ( \\frac { w } { r } + \\chi \\right ) - ( \\gamma + \\phi ( i ) ) i q \\ , , \\end{align*}"}
+{"id": "8735.png", "formula": "\\begin{align*} \\| A \\| _ \\textrm { H S } = \\sqrt { \\frac { 1 } { n } \\mathrm { t r } ( A ^ * A ) } \\forall A \\in \\mathrm { U } ( n ) . \\end{align*}"}
+{"id": "614.png", "formula": "\\begin{align*} \\pi \\circ \\phi = \\mbox { i d } _ Y , \\end{align*}"}
+{"id": "3406.png", "formula": "\\begin{align*} \\tilde { \\delta } ( \\alpha , \\theta ) = \\limsup _ { n \\to \\infty } \\frac { \\ln q _ { n + 1 } + \\sum _ { \\theta ' : c _ { \\lambda } ( \\theta ' ) = 0 } \\ln \\| q _ n ( \\theta - \\theta ' ) \\| } { q _ n } , \\end{align*}"}
+{"id": "8850.png", "formula": "\\begin{align*} \\mathcal { L } = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ 3 \\Lambda _ { i j } \\partial _ { x _ i x _ j } + x _ 1 x _ 3 \\partial _ { x _ 1 } - x _ 2 x _ 3 \\partial _ { x _ 2 } + ( x _ 2 ^ 2 - x _ 1 ^ 2 ) \\partial _ { x _ 3 } - x _ 2 \\sum _ { j = 1 } ^ 3 a _ j \\partial _ { x _ j } . \\end{align*}"}
+{"id": "4185.png", "formula": "\\begin{align*} \\begin{cases} \\nabla w \\cdot \\nabla ^ \\perp \\varphi = 0 , \\\\ w = \\mathcal { L } _ H \\varphi , \\\\ \\nabla ^ \\perp \\varphi \\cdot \\nu | _ { \\partial \\Omega } = v _ n | _ { \\partial \\Omega } \\ln \\frac { 1 } { \\varepsilon } . \\end{cases} \\end{align*}"}
+{"id": "3102.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\norm { \\theta ^ k - \\tilde { \\theta } ^ k } _ { \\mathcal { L } ^ 2 , G } ^ 2 = 0 . \\end{align*}"}
+{"id": "9042.png", "formula": "\\begin{align*} L ( E ) = \\lim _ { \\epsilon \\rightarrow + \\infty } L ( E , \\epsilon ) = \\log \\Big | \\frac { E } { 2 } + \\frac { \\sqrt { E ^ { 2 } - 4 } } { 2 } \\Big | . \\end{align*}"}
+{"id": "8393.png", "formula": "\\begin{align*} \\| w _ n \\| _ { L ^ { \\infty } } = \\left \\| F ^ n e _ 1 \\right \\| _ { L ^ { \\infty } } \\leq \\frac { 1 } { n ! } \\left \\| \\widetilde { Q } ( u ) \\right \\| _ { L ^ 1 } ^ n . \\end{align*}"}
+{"id": "5101.png", "formula": "\\begin{align*} a ( x ) = \\nabla \\times ( \\eta ( z , r ) \\nabla \\theta ) + \\phi ( z , r ) \\nabla \\theta , \\end{align*}"}
+{"id": "5537.png", "formula": "\\begin{align*} \\int _ { \\mathfrak X ( n ) } | \\omega ( n ) | = ( \\mathfrak h ( n ) _ 0 ) _ ! \\int _ { \\mathfrak Y ( n ) } | \\mathfrak h ( n ) _ { \\eta } ^ * \\omega ( n ) | . \\end{align*}"}
+{"id": "194.png", "formula": "\\begin{align*} \\theta _ { j , ( 1 ) } = \\sum _ { k = 1 } ^ d - a _ { j , k } \\theta _ { k , ( 2 ) } , \\end{align*}"}
+{"id": "3718.png", "formula": "\\begin{align*} ( p - L _ X h ) ^ \\intercal & = - 2 \\zeta A ' ( h ) \\\\ H ' ( p - L _ X h ) & = \\zeta A \\cdot A ' ( h ) . \\end{align*}"}
+{"id": "3945.png", "formula": "\\begin{align*} - \\Delta p + \\zeta p & = \\overline { y } - g , \\\\ \\langle p + \\alpha \\overline { u } , u - \\overline { u } \\rangle & \\ge 0 u \\in \\mathcal { C } _ { a d } . \\end{align*}"}
+{"id": "6929.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\mathcal { W } _ 2 ^ 2 ( P _ i , Q _ i ) \\le \\mathcal { W } _ 2 ^ 2 ( P , Q ) . \\end{align*}"}
+{"id": "8427.png", "formula": "\\begin{align*} m ( x ; z ) - \\widetilde { m } ( x ; z ) = \\int _ { - \\infty } ^ x e ^ { 2 i z ( x - y ) } [ w ( y ) - \\widetilde { w } ( y ) ] d y , \\end{align*}"}
+{"id": "4865.png", "formula": "\\begin{align*} d _ x T ( x , y ) = - \\frac { d _ x \\Phi ( x , y , T ( x , y ) ) } { \\partial _ z \\Phi ( x , y , T ( x , y ) ) } . \\end{align*}"}
+{"id": "5608.png", "formula": "\\begin{align*} \\begin{cases} c _ { \\alpha + n + 1 } - c _ { n + 1 } = b _ { \\alpha + n } - b _ { \\alpha + n + 1 } \\\\ d _ n = c _ { \\alpha + 1 } + ( b _ { \\alpha + 1 } - b _ \\alpha ) = d \\\\ b _ { \\alpha + n } = b _ \\alpha + ( d _ 1 - c _ { \\alpha + 1 } ) + \\sum _ { j = 2 } ^ n ( c _ j - c _ { \\alpha + j } ) \\\\ b = b _ \\alpha + d - c _ { \\alpha + 1 } + \\sum _ { j = 2 } ^ \\infty ( c _ j - c _ { \\alpha + j } ) \\end{cases} \\end{align*}"}
+{"id": "1124.png", "formula": "\\begin{align*} { R ^ * } = \\mathop { \\arg \\max } \\limits _ { { W _ m } \\in \\left [ { 0 , W } \\right ] } \\ ; R \\left ( { p _ { b , m } ^ * , p _ { b , o } ^ * | { W _ m } , \\overline S } \\right ) . \\end{align*}"}
+{"id": "2580.png", "formula": "\\begin{align*} R \\cdot R ( X _ 1 , X _ 2 , X _ 3 , X _ 4 ; X , Y ) = ( R ( X , Y ) \\cdot R ) ( X _ 1 , X _ 2 , X _ 3 , X _ 4 ) \\\\ = - R ( R ( X , Y ) X _ 1 , X _ 2 , X _ 3 , X _ 4 ) - R ( X _ 1 , R ( X , Y ) X _ 2 , X _ 3 , X _ 4 ) \\\\ - R ( X _ 1 , X _ 2 , R ( X , Y ) X _ 3 , X _ 4 ) - R ( X _ 1 , X _ 2 , X _ 3 , R ( X , Y ) X _ 4 ) , \\end{align*}"}
+{"id": "1470.png", "formula": "\\begin{align*} M s - A = B - M s \\end{align*}"}
+{"id": "8323.png", "formula": "\\begin{align*} & \\phi _ x + i k ^ 2 \\sigma _ 3 \\phi = k P _ x \\phi , \\\\ & \\phi _ t + i \\eta ^ 2 \\sigma _ 3 \\phi = H \\phi , \\end{align*}"}
+{"id": "1114.png", "formula": "\\begin{align*} R _ { b , m } ^ { \\rm { S - N } } = \\min \\left \\{ { R _ { b , m \\to s } ^ { \\rm { S - N } } , R _ { b , m \\to b } ^ { \\rm { S - N } } } \\right \\} \\triangleq { W _ m } { \\log _ 2 } \\left ( { 1 + \\frac { { { p _ { b , m } } { { \\left | { { { \\widetilde h } _ b } } \\right | } ^ 2 } } } { { { p _ s } { { \\left | { { { \\widetilde h } _ b } } \\right | } ^ 2 } + { W _ m } { N _ 0 } } } } \\right ) . \\end{align*}"}
+{"id": "8859.png", "formula": "\\begin{align*} d | \\tilde { x } _ t | ^ 2 = 2 \\lambda _ A | \\tilde { x _ t } | ^ 2 d t - 2 A \\tilde { x } _ t \\cdot \\tilde { x } _ t d t + 2 e ^ { \\lambda _ A t } \\tilde { x } _ t \\cdot \\sigma d W _ t + C _ \\sigma e ^ { 2 \\lambda _ A t } d t . \\end{align*}"}
+{"id": "6652.png", "formula": "\\begin{align*} \\check { Y } ^ { \\epsilon } ( t ) & = \\big ( 3 \\delta ^ { 1 / 2 } + \\sup _ { t \\in [ 0 , 1 ] } | \\varphi ( t ) - \\varphi _ m ( t ) | ^ { 1 / 2 } \\big ) + \\int ^ { t } _ { 0 } \\big ( b ( \\hat { X } ^ { \\epsilon } ( s ) ) - b ( X ^ { \\epsilon } ( s ) ) \\big ) d s \\\\ & \\quad - \\lambda ( \\epsilon ) \\epsilon \\int ^ { t } _ { 0 } \\big ( \\sigma ( \\hat { X } ^ { \\epsilon } ( s - ) ) - \\sigma ( X ^ { \\epsilon } ( s - ) ) \\big ) \\epsilon \\theta ^ { \\epsilon } ( s - ) \\tilde { N } ^ { \\epsilon ^ { - 2 } } ( d s ) , \\end{align*}"}
+{"id": "3281.png", "formula": "\\begin{align*} \\zeta _ t = \\left ( \\psi _ t \\right ) ^ { \\otimes 2 } - \\langle \\langle \\psi \\rangle \\rangle _ t , \\end{align*}"}
+{"id": "6984.png", "formula": "\\begin{align*} T = 2 n \\exp ( \\log 2 n / 1 0 3 . 9 5 \\log \\log 2 n ) , \\end{align*}"}
+{"id": "3202.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } | \\mu ( h ^ { - k } ( A ) \\cap B ) - \\mu ( A ) \\mu ( B ) | = 0 ; \\end{align*}"}
+{"id": "4332.png", "formula": "\\begin{align*} \\int _ { \\{ z \\in M : - \\Psi ( z ) \\in N \\} } | \\tilde { F } | ^ 2 _ h = 0 \\end{align*}"}
+{"id": "4940.png", "formula": "\\begin{align*} \\Re \\bigg \\{ \\left ( \\frac { L ' } { L } \\right ) ' ( s , \\chi ) \\bigg \\} = \\sum _ { \\gamma _ \\chi } g _ { a } ( t - \\gamma _ \\chi ) + \\sum _ { \\gamma } g _ { a } ( \\gamma ) + O ( 1 ) , \\end{align*}"}
+{"id": "7065.png", "formula": "\\begin{align*} u \\ , = \\ , \\tfrac { x ^ 2 } { 2 } + { \\rm O } _ { x , y , z , w } ( 3 ) , \\end{align*}"}
+{"id": "227.png", "formula": "\\begin{align*} \\dot g ( j ) = h _ { j , 1 } \\ , t ^ { - 3 q ( n ) - 4 C } \\ , h _ { j , 2 } \\in B _ S ( n + m ( n ) ) , \\end{align*}"}
+{"id": "3296.png", "formula": "\\begin{align*} & \\left | \\sum _ { i = 1 } ^ n \\int _ { ( i - 1 ) \\Delta _ n } ^ { i \\Delta _ n } \\frac { \\rho } { \\Delta _ n ^ { ( \\mathfrak H + \\epsilon ) } } \\int _ { 2 - ( i \\Delta _ n - s ) } ^ 2 x ^ { \\mathfrak H + \\epsilon } d x d s \\right | \\\\ \\leq & \\rho \\Delta _ n ^ { 1 - ( \\mathfrak H + \\epsilon ) } 2 ^ { \\mathfrak H + \\epsilon } . \\end{align*}"}
+{"id": "580.png", "formula": "\\begin{align*} \\limsup _ { s \\rightarrow + \\infty } \\frac { J _ 2 ( s ) } { s } \\leq - R e \\left ( r e ^ { i \\theta } \\right ) = - \\inf _ { \\omega \\in \\Gamma ^ { \\prime } } R e \\left ( \\omega e ^ { i \\theta } \\right ) . \\end{align*}"}
+{"id": "2509.png", "formula": "\\begin{align*} \\mathbf { P } _ { v } = \\mathbf { I } _ { n - 1 } - \\mathbf { v } _ { 2 : n } ( \\mathbf { v } _ { 2 : n } ) ^ T / \\left \\| \\mathbf { v } _ { 2 : n } \\right \\| ^ 2 . \\end{align*}"}
+{"id": "416.png", "formula": "\\begin{align*} X ^ { \\mathfrak h } _ 1 , \\ldots , X ^ { \\mathfrak h } _ m , X ^ { \\mathfrak n } _ { m + 1 } , \\ldots , X ^ { \\mathfrak n } _ k \\mbox { i s a b a s i s f o r } \\Delta . \\end{align*}"}
+{"id": "1377.png", "formula": "\\begin{align*} D ( j , k ) = \\max ( D ( j , k - 1 ) , D ( j - 1 , k ) ) + e _ { k j } , k \\geq 0 , j = 1 , \\ldots , d . \\end{align*}"}
+{"id": "1444.png", "formula": "\\begin{align*} & \\P _ x ( Z _ d ( n _ 1 ) \\geq \\xi _ 1 , \\ldots , Z _ d ( n _ m ) \\geq \\xi _ m ) \\\\ & = \\frac { 1 } { h ( x ^ 0 ) } \\int _ { I _ { \\xi _ 1 } } \\ldots \\int _ { I _ { \\xi _ m } } \\det ( f _ { n _ 1 - 1 + i } ( x _ j ^ 1 - x _ i ^ 0 ) ) \\det ( f _ { n _ 2 - n _ 1 } ( x _ j ^ 2 - x _ i ^ 1 ) ) \\\\ & \\ldots \\det ( f _ { n _ m - n _ { m - 1 } } ( x _ j ^ m - x _ i ^ { m - 1 } ) ) \\Delta ( x ^ m ) \\prod _ { k = 1 } ^ m \\prod _ { j = 1 } ^ d d x _ j ^ k . \\end{align*}"}
+{"id": "8835.png", "formula": "\\begin{align*} X _ { t , 1 } = X _ { 0 , 1 } + X _ { 0 , 2 } X _ { 0 , n } t + X _ { 0 , 2 } \\int _ 0 ^ t W _ s d s + X _ { 0 , 2 } \\int _ 0 ^ t E _ s d s . \\end{align*}"}
+{"id": "3603.png", "formula": "\\begin{align*} t ( v ' ) t ( w '' ) = t ( w ' ) t ( v '' ) \\end{align*}"}
+{"id": "4911.png", "formula": "\\begin{align*} M = \\frac { Q ( 7 / 8 ) - Q ( 5 / 8 ) + Q ( 3 / 8 ) - Q ( 1 / 8 ) } { Q ( 6 / 8 ) - Q ( 2 / 8 ) } . \\end{align*}"}
+{"id": "7903.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Theta ( t , x ) = \\int _ \\S \\int _ \\S W _ r ( z - x ) W _ r ( y - x ) W _ r ( z - y ) \\sin ( \\Theta ( t , z ) + \\Theta ( t , y ) - 2 \\Theta ( t , x ) ) \\ \\d y \\d z . \\end{align*}"}
+{"id": "2323.png", "formula": "\\begin{align*} R ^ \\nabla _ { X , Y } Z = \\left ( R ^ W _ { X , Y } \\right ) ^ { J , + } Z & - \\frac { 1 } { 2 } ( d J \\theta ) ( X , Y ) J Z - \\frac { 1 } { 2 } ( d \\theta ) ( X , Y ) Z \\\\ & + \\frac { 1 } { 4 } \\left ( \\left ( D ^ W _ X J \\right ) \\left ( D ^ W _ Y J \\right ) - \\left ( D ^ W _ Y J \\right ) \\left ( D ^ W _ X J \\right ) \\right ) Z , \\end{align*}"}
+{"id": "8125.png", "formula": "\\begin{align*} \\bigcup _ { k = 1 } ^ { m } B \\left ( y _ { k } , 2 r \\right ) = \\bigcup _ { k = 1 } ^ { m } E \\left ( \\mathbf { y } \\right ) _ { k } \\end{align*}"}
+{"id": "2080.png", "formula": "\\begin{align*} \\bigg | \\frac { 1 } { k } \\int _ { r } ^ { r + k } g ( t ) d t - g ( r ) \\bigg | & \\leq \\frac { 1 } { k } \\cdot \\frac { \\epsilon } { 2 } \\cdot m ( [ r , r + k ] ) + \\frac { 1 } { k } \\cdot 2 M \\cdot m ( [ r , r + k ] \\setminus B _ r ) \\\\ & < \\frac { 1 } { k } \\cdot \\frac { \\epsilon } { 2 } \\cdot k + \\frac { 1 } { k } \\cdot 2 M \\cdot \\frac { \\epsilon k } { 4 M } \\\\ & = \\epsilon \\end{align*}"}
+{"id": "7837.png", "formula": "\\begin{align*} \\hat \\Phi _ 2 ( ( a , 0 ) ^ \\top , s ) & = \\langle z _ 2 ^ * , \\Phi ( a u _ \\ell , p ( s ) ) \\rangle \\\\ & = \\langle z _ 2 ^ * , Q F ^ q ( a u _ \\ell + \\psi ( a u _ \\ell , p ( s ) ) , p ( s ) ) \\rangle \\\\ & = 0 , \\end{align*}"}
+{"id": "321.png", "formula": "\\begin{align*} \\lim _ { A \\rightarrow + \\infty } A ^ { - n p + n } \\int _ { | x | > 1 } \\frac { w ( A x ) } { { | x | } ^ { n p } } d x = 0 , \\end{align*}"}
+{"id": "4083.png", "formula": "\\begin{align*} & \\psi \\equiv 1 \\ ; \\ ; o n Q _ 0 a n d s u p p ( \\psi ) \\subset Q , \\\\ [ 2 m m ] & \\phi \\equiv 1 \\ ; \\ ; o n \\ ; \\ ; Q _ 1 a n d s u p p ( \\phi ) \\subset Q _ 0 \\subset Q . \\end{align*}"}
+{"id": "2713.png", "formula": "\\begin{align*} \\rho _ k = \\frac { \\phi ( x _ k ) - \\phi ( x _ k + s _ k ) + 2 \\epsilon _ f + r } { m ( x _ k ) - m ( x _ k + s _ k ) } = \\frac { L _ 1 \\| x _ k \\| ^ 2 / 2 - L _ 1 \\| x _ k - \\delta _ k g _ k / \\| g _ k \\| \\| ^ 2 / 2 + 2 \\epsilon _ f + r } { \\| g _ k \\| \\delta _ k } . \\end{align*}"}
+{"id": "2055.png", "formula": "\\begin{align*} M ^ k ( g h ) = ( \\Lambda \\ , \\cdot \\ , \\Lambda ) ^ k ( g , h ) = \\sum ^ { 2 k } _ { j = 0 } A _ { j , 2 k - j } ( g , h ) , \\end{align*}"}
+{"id": "4258.png", "formula": "\\begin{align*} \\mu ( f ^ n ( B ) ) & = \\sum _ { k \\in \\Z } \\mu ( f ^ n ( B _ k ) ) \\\\ & \\geq \\dfrac { 1 } { H } \\sum _ { k \\in \\Z } \\frac { \\mu ( f ^ { k + n } ( W ) ) } { \\mu ( f ^ { k } ( W ) ) } { \\mu ( B _ k ) } \\\\ & \\geq \\dfrac { 1 } { H } { \\mu ( B ) } \\inf _ { k \\in { \\Z } } \\frac { \\mu ( f ^ { k + n } ( W ) ) } { \\mu ( f ^ { k } ( W ) ) } . \\\\ \\end{align*}"}
+{"id": "3032.png", "formula": "\\begin{align*} \\left \\Vert d \\varphi _ { r i } \\left ( i \\right ) \\right \\Vert = \\left \\langle l ^ { \\prime } \\left ( r \\right ) i , \\theta ^ { \\prime } \\left ( r \\right ) i \\right \\rangle = l ^ { \\prime } \\left ( r \\right ) \\theta ^ { \\prime } \\left ( r \\right ) \\end{align*}"}
+{"id": "2465.png", "formula": "\\begin{align*} f ( t ) = \\displaystyle \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( - \\ ; \\alpha ^ { 2 } t ^ { 2 } \\right ) ^ { n } } { ( 2 n ) ! } = \\cos \\alpha t . \\end{align*}"}
+{"id": "7365.png", "formula": "\\begin{gather*} f \\in L , \\ , \\ , f \\le c \\quad Q ( f ) = \\sup _ { g \\in L , \\ , g \\le c } Q ( g ) \\quad Q \\in M ; \\end{gather*}"}
+{"id": "5818.png", "formula": "\\begin{align*} { { \\bf { F } } _ { { \\mathop { \\rm i n t } } } } = - G \\psi \\left ( { \\bf { x } } \\right ) \\sum \\limits _ { i = 1 } ^ { 1 8 } { \\varpi \\left ( { { { \\left | { { { \\bf { c } } _ i } } \\right | } ^ 2 } } \\right ) } \\psi \\left ( { { \\bf { x } } + { { \\bf { c } } _ i } \\Delta t } \\right ) { { \\bf { c } } _ i } , \\end{align*}"}
+{"id": "5504.png", "formula": "\\begin{align*} D ( p ) = \\frac { \\Gamma ( 3 - p ) } { \\left [ \\Gamma \\left ( 2 - \\frac { p } { 2 } \\right ) \\right ] ^ 2 \\Gamma \\left ( 3 - \\frac { p } { 2 } \\right ) } , \\end{align*}"}
+{"id": "7658.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} d x _ t ^ { * , t _ 0 , x _ 0 , \\xi } = & ~ [ A x _ t ^ { * , t _ 0 , x _ 0 , \\xi } - B ^ 2 R ^ { - 1 } U ( t , x _ t ^ { * , t _ 0 , x _ 0 , \\xi } , \\nu _ t ^ { * , t _ 0 , \\xi } ) - B h ( \\mu _ t ^ { * , t _ 0 , \\xi } ) + f ( \\nu _ t ^ { * , t _ 0 , \\xi } ) + b ( \\mu _ t ^ { * , t _ 0 , \\xi } ) ] d t \\\\ & + \\sigma d W _ t + \\sigma _ 0 d W ^ 0 _ t , \\\\ x _ { t _ 0 } ^ { * , t _ 0 , x _ 0 , \\xi } = & ~ x _ 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1293.png", "formula": "\\begin{align*} \\forall \\eta _ { \\Lambda _ n } \\in \\Omega _ { \\Lambda _ n } : \\nu ( \\eta _ { \\Lambda _ n } ) = 0 , \\end{align*}"}
+{"id": "5349.png", "formula": "\\begin{align*} S _ { \\ell } = \\left \\{ \\ k \\in ( p ^ { \\ell - 1 } , p ^ { \\ell } ] : \\ k \\equiv 0 \\bmod p ^ { \\ell - \\lfloor C _ 5 \\log \\ell \\rfloor } \\right \\} \\end{align*}"}
+{"id": "5148.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 } _ { + } } ( \\phi - \\phi _ { \\infty } ) _ { + } ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r = \\int _ { \\mathbb { R } ^ { 2 } _ { + } } ( \\phi ^ { * } - \\phi _ { \\infty } ) _ { + } ^ { 2 } \\frac { 1 } { r } \\dd z \\dd r . \\end{align*}"}
+{"id": "2618.png", "formula": "\\begin{align*} & [ x , y ] = - \\varepsilon ( x , y ) [ y , x ] , & & , \\\\ & \\circlearrowleft _ { x , y , z } \\varepsilon ( z , x ) [ \\alpha ( x ) , [ y , z ] ] = 0 , & & \\end{align*}"}
+{"id": "760.png", "formula": "\\begin{align*} | P _ s * \\mu ( \\bar x ) - P _ s * \\mu ( \\hat x ) | & \\leq \\int _ { | \\bar y - \\bar x _ 0 | _ p \\geq 2 | \\bar x - \\hat x | _ p } | P _ s ( x - y , t - s ) - P _ s ( x - y , u - s ) | \\ , d \\mu ( \\bar y ) \\\\ & + \\int _ { | \\bar y - \\bar x _ 0 | _ p < 2 | \\bar x - \\hat x | _ p } \\ ! | P _ s ( x - y , t - s ) - P _ s ( x - y , u - s ) | \\ , d \\mu ( \\bar y ) \\\\ & = : I _ 1 + I _ 2 . \\end{align*}"}
+{"id": "3104.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\norm { \\tilde { \\theta } ^ k - \\bar { \\theta } ^ k } _ { \\mathcal { L } ^ 2 , G } = \\lim \\limits _ { k \\rightarrow \\infty } \\norm { e _ { G } ( \\tilde { \\theta } ^ k , T , \\mathcal { K } ) } _ { \\mathcal { L } ^ 2 , G } \\leq \\lim \\limits _ { k \\rightarrow \\infty } K \\norm { \\theta ^ k - \\tilde { \\theta } ^ k } _ { \\mathcal { L } ^ 2 , G } = 0 . \\end{align*}"}
+{"id": "91.png", "formula": "\\begin{align*} \\nu _ x = \\nu ^ G \\Bigl ( \\max _ { [ b ] \\in B ( G ) _ x } [ b ] \\Bigr ) = \\nu ^ G \\Bigl ( \\max _ { [ b ] \\in B ( \\tilde G ) _ { x \\gamma } } [ b \\gamma ^ { - 1 } ] \\Bigr ) . \\end{align*}"}
+{"id": "5728.png", "formula": "\\begin{align*} h ^ { q - 1 } = \\frac { r + 1 } { r } = 1 + \\frac { 1 } { r } . \\end{align*}"}
+{"id": "3158.png", "formula": "\\begin{align*} & \\sum _ { y \\neq 0 , 1 } \\chi _ 4 ( y ) \\chi _ 4 ( 1 - y ) \\sum _ { x \\neq 0 , 1 , y } \\overline { \\chi _ 4 } ( x ) \\overline { \\chi _ 4 } ( x - y ) \\\\ & = \\sum _ { y \\neq 0 , 1 } \\chi _ 4 ( y ) \\chi _ 4 ( 1 - y ) \\left [ \\varphi ( y ) \\rho - \\overline { \\chi _ 4 } ( y - 1 ) \\right ] \\\\ & = \\rho \\sum _ y \\overline { \\chi _ 4 } ( y ) \\chi _ 4 ( 1 - y ) - \\sum _ y \\chi _ 4 ( y ) \\\\ & = - \\rho + 1 . \\end{align*}"}
+{"id": "135.png", "formula": "\\begin{align*} P _ N v \\cdot \\partial _ x P _ N ( P _ { \\ll N } v \\cdot P _ N u _ 1 ) = P _ N v \\cdot \\partial _ x P _ { \\ll N } v \\cdot P _ N u _ 1 + P _ N v \\cdot P _ { \\ll N } v \\cdot \\partial _ x P _ N u _ 1 \\end{align*}"}
+{"id": "7889.png", "formula": "\\begin{align*} \\dot \\theta _ k = \\frac { - 1 } M \\sum _ { j = 1 } ^ M b _ { k - j } \\sin ( \\theta _ j - \\theta _ k ) + \\frac { 1 } M \\sum _ { j = 1 } ^ M b _ { 1 - j } \\sin ( \\theta _ j ) . \\end{align*}"}
+{"id": "977.png", "formula": "\\begin{align*} c _ 3 ( e _ 0 - e _ 1 ) + c _ 1 ( | I | - c _ 1 ) \\theta _ d q ^ { d ^ 2 - d - 2 } \\le A : = \\left ( q ^ 2 + \\frac 9 2 q + 1 0 \\right ) q ^ { d ^ 2 + 2 d - 3 } . \\end{align*}"}
+{"id": "4106.png", "formula": "\\begin{align*} \\forall x \\in E , \\forall \\lambda \\in \\mathbf { C } , \\ \\overline { x \\otimes _ \\gamma \\lambda } : = x \\otimes _ { _ { \\overline { \\gamma } } } \\overline { \\lambda } . \\end{align*}"}
+{"id": "5791.png", "formula": "\\begin{align*} d ( h _ z ) = \\frac { 2 } { \\mu } \\mu _ z h _ z d z + \\frac { \\sqrt { \\tau _ 0 } } { 2 } \\left ( \\frac { 1 } { 2 } d \\overline { z } + ( \\alpha - I \\beta ) d z \\right ) \\end{align*}"}
+{"id": "4233.png", "formula": "\\begin{align*} \\dot { s } \\varepsilon ( z ) \\dot { s } \\varepsilon ( 1 ) \\eta = \\dot { s } ( \\varepsilon ( z - 1 ) - \\varepsilon ( z ) ) \\eta + \\dot { s } \\varepsilon ( z ) \\varpi . \\end{align*}"}
+{"id": "1581.png", "formula": "\\begin{align*} h : = b \\vert _ { E _ 1 } + \\sum \\limits _ { k = 2 } ^ \\delta e ^ { 2 f _ k } b \\vert _ { E _ k } + \\sum \\limits _ { k = 2 } ^ \\delta \\sum \\limits _ { k ' = 2 } ^ \\delta e ^ { 2 f _ { k , k ' } } b _ { k , k ' } + g . \\end{align*}"}
+{"id": "5947.png", "formula": "\\begin{align*} \\partial _ { t } u = \\nabla \\cdot ( | \\nabla u | ^ { p - 2 } \\nabla u ) - c | u | ^ { p - 2 } u , \\end{align*}"}
+{"id": "2685.png", "formula": "\\begin{align*} a _ d ( n ) = [ z ^ n ] \\frac { z } { M _ d ( z ) } . \\end{align*}"}
+{"id": "3139.png", "formula": "\\begin{align*} & \\ell ^ \\alpha = \\ell \\circ ( \\alpha ^ 2 \\otimes I d ) r ^ \\alpha = r \\circ ( \\alpha ^ 2 \\otimes I d ) . \\end{align*}"}
+{"id": "21.png", "formula": "\\begin{align*} A _ w : = w ^ { - 1 } ( R ^ + ) \\cap R ^ - . \\end{align*}"}
+{"id": "2640.png", "formula": "\\begin{align*} x \\diamond y = x \\cdot D ( y ) , f o r ~ a l l ~ x , y \\in \\mathcal { H } ( A ) . \\end{align*}"}
+{"id": "5206.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\| \\frac { \\dd } { \\dd t } \\int _ { \\mathbb { R } ^ { 3 } } \\Phi _ + \\frac { G } { r ^ { 2 } } \\dd x \\right \\| _ { L ^ { 2 } ( 0 , T ) } \\leq C \\mu ^ { 1 / 2 } \\left ( | | v _ 0 | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } ^ { 2 } + | | b _ 0 | | _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } ^ { 2 } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "9065.png", "formula": "\\begin{align*} v _ 0 = c X _ { - \\beta _ m } X _ { - \\beta _ { m - 1 } } \\cdots X _ { - \\beta _ { m - k + 1 } } v _ k . \\end{align*}"}
+{"id": "4964.png", "formula": "\\begin{align*} E _ { \\xi ^ { \\bar { \\theta } } _ 1 ( u ) } \\biggl [ \\varphi \\biggl ( x + u + \\sum _ { i = 2 } ^ { n } Z ^ { { \\bar { \\theta } } [ u ] } _ i \\biggr ) \\biggr ] = & E _ { \\xi ^ { \\bar { \\theta } } _ 1 ( u ) } \\biggl [ \\varphi \\biggl ( x + u + \\sum _ { i = 1 } ^ { n - 1 } Z ^ { \\theta ^ { [ n - 1 ] } } _ i \\biggr ) \\biggr ] \\\\ = & V \\bigg ( \\frac { \\xi _ 0 f _ 1 ( u ) } { \\xi _ 0 f _ 1 ( u ) + ( 1 - \\xi _ 0 ) f _ 2 ( u ) } , n - 1 , x + u \\bigg ) . \\end{align*}"}
+{"id": "5629.png", "formula": "\\begin{align*} f _ \\xi ( z ) = \\begin{cases} \\frac { f ( z ) } { \\xi - z } & , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "6661.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { \\epsilon \\to 0 } \\lambda ^ 2 ( \\epsilon ) \\log \\mathbb P \\big ( X ^ { \\epsilon } \\in B ( \\varphi , \\delta ) \\big ) \\leq - \\mathcal { I } ( \\varphi ) . \\end{align*}"}
+{"id": "3426.png", "formula": "\\begin{align*} \\tilde { I } _ 0 & = [ - [ s q _ { n - n _ 0 } / 2 ] - s q _ { n - n _ 0 } , \\ , - [ s q _ { n - n _ 0 } / 2 ] - 1 ] \\cap \\Z , \\\\ \\tilde { I } _ y & = [ y - [ s q _ { n - n _ 0 } / 2 ] - s q _ { n - n _ 0 } , \\ , y - [ s q _ { n - n _ 0 } / 2 ] - 1 ] \\cap \\Z . \\end{align*}"}
+{"id": "98.png", "formula": "\\begin{align*} w \\mu = ( w s _ \\alpha ) ( \\mu - \\langle \\mu , \\alpha \\rangle \\alpha ^ \\vee ) = ( w s _ \\alpha ) \\mu + \\langle \\mu , \\alpha \\rangle w \\alpha ^ \\vee \\geq ( w s _ \\alpha ) \\mu \\underset { } \\geq \\mu . \\end{align*}"}
+{"id": "4162.png", "formula": "\\begin{align*} \\begin{cases} \\min \\{ \\frac { 1 } { 2 } + \\frac { \\alpha } { 2 \\beta } ( r + \\lambda - \\kappa ) , 1 \\} , & \\mbox { i f } \\alpha + \\gamma = 1 , \\\\ \\min \\{ \\frac { \\alpha } { 2 \\beta } \\min \\{ \\kappa , r + \\lambda \\} + ( \\gamma - \\frac { 1 } { 2 } ) ^ { + } , 1 - \\varepsilon \\} , & \\mbox { i f } \\alpha + \\gamma \\neq 1 \\end{cases} \\end{align*}"}
+{"id": "7650.png", "formula": "\\begin{align*} \\eta _ t : = \\exp \\int _ t ^ T ( A - B ^ 2 R ^ { - 1 } P _ s ) d s . \\end{align*}"}
+{"id": "4010.png", "formula": "\\begin{align*} \\sigma \\cdot X : = \\{ S ^ k \\sigma ( x ) \\mid x \\in X , 0 \\leq k < | \\sigma ( x _ 0 ) | \\} . \\end{align*}"}
+{"id": "1468.png", "formula": "\\begin{align*} L _ { 0 } : = \\left ( \\frac { 6 4 \\alpha ^ { 2 } } { \\beta ^ { 2 } M ^ { 3 } } \\right ) ^ { 1 / 5 } . \\end{align*}"}
+{"id": "8806.png", "formula": "\\begin{align*} d z _ t = ( \\Pi _ { \\mathrm { k e r } A } ( B ( y _ t , z _ t ) + B ( z _ t , y _ t ) + B ( y _ t , y _ t ) - A y _ t ) d t + \\Pi _ { \\mathrm { k e r } A } \\sigma d W _ t . \\end{align*}"}
+{"id": "8426.png", "formula": "\\begin{align*} \\begin{aligned} & \\left ( \\Psi ^ - - e ^ { - i c _ - ( x ) \\sigma _ 3 } \\right ) - \\left ( \\widetilde { \\Psi } ^ - - e ^ { - i c _ - ( x ) \\sigma _ 3 } ( \\tilde { u } ) \\right ) \\\\ & = ( I - F ) ^ { - 1 } ( T - \\widetilde { T } ) + ( I - F ) ^ { - 1 } ( F - \\tilde { F } ) ( I - \\tilde { F } ) ^ { - 1 } \\widetilde { T } , \\end{aligned} \\end{align*}"}
+{"id": "1782.png", "formula": "\\begin{align*} \\begin{aligned} d _ { C } ( ( z , t ) , ( z ' , t ' ) ) & = \\big \\lvert \\lvert z - z ' \\rvert ^ { 4 } + \\lvert t - t ' + 2 \\Im ( z \\overline { z ' } ) \\rvert ^ { 2 } \\big \\rvert ^ { 1 / 4 } \\\\ & = \\lvert 2 \\langle \\psi ( z , t , 0 ) , \\psi ( z ' , t ' , 0 ) \\rangle \\rvert ^ { 1 / 2 } , \\end{aligned} \\end{align*}"}
+{"id": "361.png", "formula": "\\begin{align*} w _ k = \\theta _ k u _ k . \\end{align*}"}
+{"id": "1807.png", "formula": "\\begin{align*} \\int _ { B _ \\rho ( x _ 0 ) } F ( \\frac { | | R ^ \\nabla | | ^ 2 } 2 ) \\ , d v = o ( \\rho ^ \\lambda ) \\rho \\rightarrow \\infty , \\end{align*}"}
+{"id": "1703.png", "formula": "\\begin{align*} \\tilde { \\mu } ( \\xi ) = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ N \\mu ^ 2 _ j e _ j ^ 2 ( \\xi ) . \\end{align*}"}
+{"id": "3520.png", "formula": "\\begin{align*} - n ^ p - 2 \\sum _ { j = 0 } ^ { p / 2 } \\sum _ { k = 0 } ^ { p - 2 j - 1 } \\binom { p } { 2 j + 1 } \\binom { { p - 2 j - 1 } } { k } B _ k \\ , n ^ { p - 2 j - 1 - k } . \\end{align*}"}
+{"id": "5433.png", "formula": "\\begin{align*} | F ^ { i j } ( \\nabla _ \\beta \\varphi ) _ { i j } | \\leq C \\sqrt { b _ \\alpha } \\left ( \\sum _ { i = 1 } ^ { \\alpha } F ^ { i i } + \\sum _ { i = \\alpha + 1 } ^ { n - 1 } \\frac { b _ i } { b _ \\alpha } F ^ { i i } \\right ) \\mbox { i n } \\omega _ \\epsilon . \\end{align*}"}
+{"id": "6930.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\mathcal { W } _ 2 ^ 2 ( P _ { S _ i } , Q _ { S _ i } ) \\le \\mathcal { W } _ 2 ^ 2 ( P _ { S _ 1 \\cup \\cdots \\cup S _ m } , Q _ { S _ 1 \\cup \\cdots \\cup S _ m } ) \\le \\mathcal { W } _ 2 ^ 2 ( P , Q ) . \\end{align*}"}
+{"id": "2894.png", "formula": "\\begin{align*} { C ^ { ( p ) } _ { x , x } ( t ) } \\ge T _ - , x = 0 , \\ldots , n . \\end{align*}"}
+{"id": "1135.png", "formula": "\\begin{align*} \\| \\mathbf { c } _ X \\| _ 2 \\leq \\| \\mathbf { c } _ X \\| _ 1 = \\| \\mathbf { x } \\| _ 1 \\leq \\sqrt { N L } \\| \\mathbf { x } \\| _ 2 , \\| \\mathbf { c } _ Y \\| _ 2 \\leq \\| \\mathbf { c } _ Y \\| _ 1 = \\| \\mathbf { y } \\| _ 1 \\leq \\sqrt { N L } \\| \\mathbf { y } \\| _ 2 , \\end{align*}"}
+{"id": "1677.png", "formula": "\\begin{align*} Y _ { i j } = \\bigcap _ { ( a , b ) \\neq ( i , j ) } Z _ { a b } \\end{align*}"}
+{"id": "518.png", "formula": "\\begin{align*} | G ^ { \\delta } _ { R _ n } ( f ) | & = \\Big | \\frac { 1 } { R _ n } \\int ^ { R _ n } _ 0 [ S _ { r } ^ { \\delta + 1 } ( f ) - S _ { r } ^ { \\delta } ( f ) ] d r \\Big | \\le \\Big ( \\int ^ { R _ n } _ 0 \\big | S _ r ^ { \\delta + 1 } ( f ) - S _ r ^ { \\delta } ( f ) \\big | ^ 2 \\frac { d r } { R _ n } \\Big ) ^ { 1 / 2 } \\\\ & \\leq \\Big ( \\int ^ { \\infty } _ 0 \\big | S _ r ^ { \\delta + 1 } ( f ) - S _ r ^ { \\delta } ( f ) \\big | ^ 2 \\frac { d r } { r } \\Big ) ^ { 1 / 2 } = : G ^ { \\delta } ( f ) . \\end{align*}"}
+{"id": "829.png", "formula": "\\begin{align*} \\partial _ t w = g ( c ) \\left ( \\frac { \\partial _ { x x } w } { 1 + | \\partial _ x w | ^ 2 } - \\frac { 1 } { w } \\right ) . \\end{align*}"}
+{"id": "6244.png", "formula": "\\begin{align*} \\partial _ i \\Gamma ^ j _ { k j } + \\Gamma ^ j _ { k j } \\Gamma ^ j _ { i j } - \\Gamma ^ j _ { k j } \\Gamma ^ k _ { i k } - \\Gamma ^ j _ { i j } \\Gamma ^ i _ { k i } + g _ { i k } = 0 . \\end{align*}"}
+{"id": "1966.png", "formula": "\\begin{align*} Q ^ { ( m ) } _ \\nu ( k ) & = Q ^ { ( m - 1 ) } _ \\nu ( k ) \\frac { q ^ { k + 1 } - 1 } { q - 1 } - \\frac { q ^ { m - 2 } - 1 } { q - 1 } \\sum _ { a = 1 } ^ k q ^ a Q ^ { ( m - 1 ) } _ \\nu ( a - 1 ) \\\\ & = Q ^ { ( m - 1 ) } _ \\nu ( k ) \\frac { q ^ { k + 1 } - 1 } { q - 1 } - Q ^ { ( m ) } _ \\nu ( k - 1 ) \\frac { q ^ { m - 1 } - q } { q - 1 } \\\\ & = Q ^ { ( m - 1 ) } _ \\nu ( k ) \\frac { q ^ { k + 1 } - 1 } { q - 1 } - \\left ( Q ^ { ( m ) } _ \\nu ( k ) - q ^ { k } Q ^ { ( m - 1 ) } _ \\nu ( k ) \\right ) \\frac { q ^ { m - 1 } - q } { q - 1 } . \\end{align*}"}
+{"id": "1953.png", "formula": "\\begin{align*} r ' ( X ) = \\begin{cases} r ( X ) + 1 , ~ X ~ ~ n p - ~ ~ ~ \\\\ r ( X ) , \\end{cases} \\end{align*}"}
+{"id": "4973.png", "formula": "\\begin{align*} W ( \\xi _ 0 , k , x , \\mathsf { M } ^ k ) = W ( 1 - \\xi _ 0 , k , x , \\mathsf { M } ^ k ) , ~ \\forall \\xi _ 0 \\in [ 0 , 1 ] , \\forall x \\in \\mathbb { R } . \\end{align*}"}
+{"id": "185.png", "formula": "\\begin{align*} ( C ^ \\bullet ) ^ { \\Pi ' } : = ( S _ { H ' } ( C ^ \\bullet ) ) ^ { \\Pi ' / H ' } . \\end{align*}"}
+{"id": "6135.png", "formula": "\\begin{align*} \\mathrm { c o e v } _ S : = \\mathrm { c o e v } _ { P _ { S } } \\colon & I \\to P _ { S } \\otimes P _ { - S } \\\\ \\mathrm { e v } _ { S } : = \\mathrm { e v } _ { P _ S } \\colon & P _ { - S } \\otimes P _ { S } \\to I \\end{align*}"}
+{"id": "495.png", "formula": "\\begin{align*} e ^ { \\gamma x } f _ r ( x ) = e ^ { \\gamma x } e ^ { - c _ { 1 3 } } f _ 0 ( x ) & + ( 1 - e ^ { - c _ { 1 3 } } ) \\int _ { - \\infty } ^ { x / 2 } e ^ { \\gamma ( x - y ) } f _ 0 ( x - y ) e ^ { \\gamma y } f _ { 1 3 } ( y ) d y \\\\ & + ( 1 - e ^ { - c _ { 1 3 } } ) \\int _ { - \\infty } ^ { x / 2 } e ^ { \\gamma ( x - y ) } f _ { 1 3 } ( x - y ) e ^ { \\gamma y } f _ 0 ( y ) d y . \\end{align*}"}
+{"id": "4778.png", "formula": "\\begin{align*} \\max _ { | \\eta - \\lambda | \\le \\delta } \\max _ { x \\in G ( \\lambda ) } \\| F _ n ( \\eta ) p _ n x - p _ n F ( \\eta ) x \\| = \\max _ { x \\in G ( \\lambda ) } \\left \\| ( T _ n p _ n - p _ n T ) x \\right \\| \\le \\left \\| T _ n p _ n - p _ n T \\right \\| . \\end{align*}"}
+{"id": "1866.png", "formula": "\\begin{align*} \\begin{aligned} \\root n \\of { a _ 1 \\cdot a _ n } & \\le \\frac { a _ 1 + \\cdots + a _ n } { n } \\ , , a _ 1 , \\cdots a _ n > 0 \\ , , \\\\ \\ , ` ` & = \" \\ , \\ , \\ , \\ , \\ , \\ , a _ 1 = \\cdots = a _ n \\ , . \\end{aligned} \\end{align*}"}
+{"id": "2806.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\mathbb { P } \\left ( \\sup _ { x \\in \\tilde { S } _ { n , m } ^ { \\delta } } \\Phi ^ { S _ { n } , \\tilde { S } _ { n , m } } ( x ) > 2 \\sqrt { 2 \\gamma } \\log ( 2 ^ { m } ) + c \\sqrt { \\log ( 2 ^ { m } ) } \\right ) = 0 . \\end{align*}"}
+{"id": "2804.png", "formula": "\\begin{align*} \\mathbb { E } e ^ { - \\langle \\tilde { Z } _ { \\lambda } ^ { S _ { n } , M , m } , f _ { m , \\delta } \\rangle } \\leq \\mathbb { E } \\left [ \\exp \\left ( - e ^ { - \\varepsilon } \\sum _ { j = 1 } ^ { 1 6 ^ { m } } \\mathbb { E } [ \\tilde { X } _ { j } 1 _ { \\{ X _ { j } \\leq \\varepsilon \\} } | \\Phi ^ { S _ { n } , \\tilde { S } _ { n , m } } ] \\right ) \\right ] . \\end{align*}"}
+{"id": "805.png", "formula": "\\begin{align*} \\partial ^ \\square c ( t , p ) = \\partial ^ \\circ c ( t , p ) = \\frac { \\mathrm { d } } { \\mathrm { d } t } c ( t ) = - \\frac { m d } { \\alpha _ d } R ( t ) ^ { - d - 1 } R ' ( t ) \\end{align*}"}
+{"id": "5670.png", "formula": "\\begin{align*} d ^ 1 _ { p , q } = \\begin{cases} 0 , & p \\ , \\ , \\\\ i _ { * } , & p \\ , \\ , , \\end{cases} \\end{align*}"}
+{"id": "7839.png", "formula": "\\begin{align*} T ^ { - 1 } ( \\hat \\Phi ( T ( \\rho , \\chi ) , s ) ) = \\begin{pmatrix} \\hat h ( \\rho , s ) \\\\ 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "5950.png", "formula": "\\begin{align*} \\langle A ( u ) , v \\rangle = - \\int _ { \\mathbb { T } ^ d } \\langle a ( u ( x ) ) \\nabla u ( x ) + b ( u ( x ) ) , \\nabla v ( x ) \\rangle d x . \\end{align*}"}
+{"id": "881.png", "formula": "\\begin{align*} x x ' + x ' y + y y ' = 2 n \\ , . \\end{align*}"}
+{"id": "7073.png", "formula": "\\begin{align*} { } \\nu _ { i } = T _ { \\epsilon _ i } \\# ( T _ 0 \\# \\mu _ i ) , \\{ \\mu _ i , \\nu _ i \\} _ { i = 1 } ^ N , \\end{align*}"}
+{"id": "4749.png", "formula": "\\begin{align*} & M i n i m i z e _ { x \\in \\mathbb { R } ^ n } \\ , T r ( A _ J , X ) + 2 b _ J ^ T x \\\\ & s . t . \\ \\ T r ( A _ k , X ) + 2 b _ k ^ T x + c _ k \\le 0 \\ , k = 1 , . . . , m \\\\ & X = x x ^ T \\end{align*}"}
+{"id": "3993.png", "formula": "\\begin{align*} \\mathbf { a } + \\mathbf { c } = ( \\alpha _ 1 + \\gamma _ 1 , \\alpha _ 2 + \\gamma _ 2 , \\alpha _ 3 + c _ 1 , a _ 1 + c _ 2 , a _ 2 + c _ 3 , \\cdots , a _ { 2 k - 3 } + c _ { 2 k - 2 } , a _ { 2 k - 2 } + \\gamma _ 3 , a _ { 2 k - 1 } + c _ { 2 k - 1 } , \\alpha _ 4 + \\gamma _ 4 ) , \\end{align*}"}
+{"id": "4365.png", "formula": "\\begin{align*} \\int _ K | | ( \\eta + g ^ { - 1 } ) ^ { \\frac { 1 } { 2 } } u _ 0 | | _ { \\omega } + \\int _ K | | \\tau _ 0 | | _ { \\omega } \\le \\int _ M \\langle ( B + \\lambda I ) ^ { - 1 } v , v \\rangle _ { \\omega , h } d V _ { \\omega } . \\end{align*}"}
+{"id": "4946.png", "formula": "\\begin{align*} V ( \\xi _ 0 , n , x ) : = \\sup _ { \\theta \\in \\Theta _ n } W ( \\xi _ 0 , n , x , \\theta ) = \\sup _ { \\theta \\in \\Theta _ n } E _ P \\biggl [ \\varphi \\biggl ( x + \\sum _ { i = 1 } ^ { n } Z _ i ^ \\theta \\biggr ) \\biggr ] . \\end{align*}"}
+{"id": "3396.png", "formula": "\\begin{align*} h ( x ) = x \\cdot \\frac { f ' ( x ) } { f ( x ) } - x \\cdot \\frac { f ' ( x ) } { f ( x ) } \\cdot \\log \\left ( \\frac { f ' ( x ) } { f ( x ) } \\right ) - \\log { f ( x ) } \\end{align*}"}
+{"id": "5992.png", "formula": "\\begin{align*} - \\Delta u + ( \\omega + \\xi ) u + \\Big ( \\int _ { | x | } ^ { \\infty } \\frac { h ( s ) } { s } u ^ { 2 } ( s ) d s + \\frac { h ^ { 2 } ( | x | ) } { | x | ^ { 2 } } \\Big ) u = f ( u ) , \\ \\ \\ \\ x \\in \\mathbb { R } ^ { 2 } , \\end{align*}"}
+{"id": "7918.png", "formula": "\\begin{align*} f ( r ) : = k \\hat W _ { \\frac { r } { k } } ( k ) \\end{align*}"}
+{"id": "5073.png", "formula": "\\begin{align*} { \\mathcal { I } } _ h = \\inf \\left \\{ \\frac { 1 } { 2 } \\int _ { \\Omega } | B | ^ { 2 } \\dd x \\ \\middle | \\ B \\in L ^ { 2 } _ { \\sigma } ( \\Omega ) , \\ \\int _ { \\Omega } \\textrm { c u r l } ^ { - 1 } B \\cdot B \\dd x = h \\right \\} . \\end{align*}"}
+{"id": "3558.png", "formula": "\\begin{align*} h _ { r , s } ^ { ( m ) } = \\frac { [ r ( m + 3 ) - s ( m + 2 ) ] ^ 2 - 1 } { 4 ( m + 2 ) ( m + 3 ) } . \\end{align*}"}
+{"id": "1661.png", "formula": "\\begin{align*} ( z f ) ( \\sigma ) = z _ p \\star f ( \\sigma z ) . \\end{align*}"}
+{"id": "1652.png", "formula": "\\begin{align*} Y \\times H \\to ( Y \\times H ) / H _ r = Y \\times ^ { H _ r } H \\to X ^ { r e p } , \\end{align*}"}
+{"id": "8077.png", "formula": "\\begin{align*} \\mathbf { y ' } ^ H \\mathbf { y } ' = & y _ c ^ * y _ c + y _ 1 ^ * y _ 1 + \\cdots + y _ M ^ * y _ M = \\left ( \\sum _ { l = 1 } ^ M y _ l ^ * \\right ) \\left ( \\sum _ { j = 1 } ^ M y _ j \\right ) + \\sum _ { i = 1 } ^ M y _ i ^ * y _ i . \\end{align*}"}
+{"id": "8529.png", "formula": "\\begin{align*} \\begin{aligned} u ( x ) e ^ { - i ( 2 c _ - ( x ) + c ) } & = \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } z ^ { - 1 } \\bar { r } _ { \\delta , 1 } ( z ) e ^ { - 2 i z x } M _ { \\delta , + , 1 1 } ( x ; z ) d z , \\end{aligned} \\end{align*}"}
+{"id": "2585.png", "formula": "\\begin{align*} R ( X , Y ) Z = \\dfrac { \\tilde { c } } { 4 } ( X \\wedge _ g ^ c Y ) Z , \\end{align*}"}
+{"id": "1508.png", "formula": "\\begin{align*} \\frac { b - s } { b - a } - \\frac { t - s } { t - a } & = \\frac { ( b - s ) ( t - a ) - ( t - s ) ( b - a ) } { ( b - a ) ( t - a ) } \\\\ & = \\frac { ( b - t ) ( s - a ) } { ( b - a ) ( t - a ) } \\ge 0 , \\end{align*}"}
+{"id": "6635.png", "formula": "\\begin{align*} \\begin{aligned} G _ { m , \\epsilon } ( t _ 1 , \\cdots , t _ k ) & = \\left ( \\int ^ { 1 } _ { 0 } K _ { m } ( t _ 1 , r ) \\theta _ { \\epsilon } ( r ) d r , \\cdots , \\int ^ { 1 } _ { 0 } K _ { m } ( t _ k , r ) \\theta _ { \\epsilon } ( r ) d r \\right ) ^ { \\tau } \\\\ & = \\Big ( G _ { m , \\epsilon } ( t _ 1 ) , \\cdots , G _ { m , \\epsilon } ( t _ k ) \\Big ) . \\end{aligned} \\end{align*}"}
+{"id": "5257.png", "formula": "\\begin{align*} D ( M _ { q } ' ) = \\{ f \\in \\ell ^ { 1 } \\ ; | \\ ; q f \\in \\ell ^ { 1 } \\} \\end{align*}"}
+{"id": "280.png", "formula": "\\begin{align*} u _ \\eta ( p ) = u ( \\eta \\cdot p ) , f _ \\eta ( p ) = f ( \\eta \\cdot p ) , p \\in \\Omega _ \\eta . \\end{align*}"}
+{"id": "1559.png", "formula": "\\begin{align*} F ( \\mathbf { v } ) = \\begin{bmatrix} D & 0 & \\ell \\sin \\Psi & - \\cos \\Psi \\\\ 0 & D & - \\ell \\cos \\Psi & - \\sin \\Psi \\\\ 0 & - \\kappa \\ell & D & - \\kappa U \\\\ D ^ 0 _ { - 1 } & 0 & 0 & 0 \\\\ D ^ 0 _ { 1 } & 0 & 0 & 0 \\\\ 0 & 0 & D ^ 0 _ { - 1 } & 0 \\\\ 0 & 0 & D ^ 0 _ { 1 } & 0 \\end{bmatrix} \\mathbf { v } \\end{align*}"}
+{"id": "1691.png", "formula": "\\begin{align*} \\kappa _ 1 = \\underset { [ 0 , h ] } { \\mathrm { m a x } } \\norm { U ( \\tau ) A _ d } , \\kappa _ 2 = \\underset { [ - h , h ] } { \\mathrm { m a x } } \\norm { A _ d ^ \\top U ( \\tau ) A _ d } . \\end{align*}"}
+{"id": "8897.png", "formula": "\\begin{align*} p _ \\alpha = 1 / m _ \\alpha \\end{align*}"}
+{"id": "3002.png", "formula": "\\begin{align*} \\upsilon ( \\tau , z , w ( \\cdot ) ) = \\| z \\| + \\| w ( \\cdot ) \\| _ 1 + \\| w ( - h ) \\| + \\| w ( i h - \\tau ) \\| \\end{align*}"}
+{"id": "2798.png", "formula": "\\begin{align*} m : = [ \\log \\| x - y \\| ] \\end{align*}"}
+{"id": "2174.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta v _ 0 = f & \\Omega , \\\\ v _ 0 = \\gamma ^ - _ { \\partial \\Omega } ( v ) & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "6612.png", "formula": "\\begin{align*} R _ { i a } = \\sum _ { j , b } \\mathcal { B } _ { b i } ^ j \\mathcal { B } _ { a j } ^ b = \\sum _ { j , b } \\mathcal { B } _ { b i j } \\mathcal { B } _ { a j b } ( - \\varepsilon _ { j ' } ) \\varepsilon _ b = \\sum _ { j , b } \\mathcal { B } _ { b i j } \\mathcal { B } _ { j a b } \\varepsilon _ { j ' } \\varepsilon _ b \\end{align*}"}
+{"id": "1909.png", "formula": "\\begin{align*} w _ { n + 1 } ^ { ( i ) } ( s ) = { \\frac { w ( s ) } { f _ { q } ( s ) } } w _ n ^ { ( i ) } \\bigl ( { f _ { q } ( s ) } \\bigr ) , \\end{align*}"}
+{"id": "3175.png", "formula": "\\begin{align*} T _ j & : = \\sum _ { x \\neq 0 , 1 } \\sum _ y \\chi _ 4 ^ { i _ 1 } ( y ) \\chi _ 4 ^ { i _ 2 } ( 1 - y ) \\chi _ 4 ^ { i _ 3 } ( x - y ) , \\\\ U _ { i j } & : = \\sum _ { x \\neq 0 , 1 } \\chi _ 4 ^ l ( m ) \\sum _ y \\chi _ 4 ^ { i _ 1 } ( y ) \\chi _ 4 ^ { i _ 2 } ( 1 - y ) \\chi _ 4 ^ { i _ 3 } ( x - y ) , \\\\ V _ { i j } & : = \\sum _ x \\chi _ 4 ^ { l _ 1 } ( x ) \\chi _ 4 ^ { l _ 2 } ( 1 - x ) \\sum _ y \\chi _ 4 ^ { i _ 1 } ( y ) \\chi _ 4 ^ { i _ 2 } ( 1 - y ) \\chi _ 4 ^ { i _ 3 } ( x - y ) , \\end{align*}"}
+{"id": "1287.png", "formula": "\\begin{align*} \\forall \\Delta \\Subset \\Z ^ d \\ \\forall \\xi _ { \\Delta } : c ^ { B _ { N - 1 } ( x ( \\Delta ) ) } _ { \\Delta } ( \\eta _ { \\Lambda _ N } , \\xi _ { \\Delta } ) > 0 \\forall n \\in \\N : \\ c ^ { B _ { n - 1 } ( x ( \\Delta ) ) } _ { \\Delta } ( \\eta _ { \\Lambda _ n } , \\xi _ { \\Delta } ) = 0 . \\end{align*}"}
+{"id": "5822.png", "formula": "\\begin{align*} \\rho { c _ v } \\frac { { \\partial T } } { { \\partial t } } + \\rho { c _ v } { \\bf { u } } \\cdot \\nabla T = \\nabla \\cdot \\left ( { \\lambda \\nabla T } \\right ) - T { \\left ( { \\frac { { \\partial { p _ { E O S } } } } { { \\partial T } } } \\right ) _ \\rho } \\nabla \\cdot { \\bf { u } } . \\end{align*}"}
+{"id": "3484.png", "formula": "\\begin{align*} \\| A _ { | I | } ( \\theta _ { \\ell _ 1 } ) \\| = \\frac { 1 } { \\prod _ { \\ell \\in I } | \\cos ( \\pi \\theta _ { \\ell } ) | } \\| F _ { | I | } ( \\theta _ { \\ell _ 1 } ) \\| \\leq C ( \\varepsilon ) e ^ { \\varepsilon ( 2 q _ n - | I | ) } e ^ { | I | \\ln 2 } e ^ { | I | ( \\tilde { L } + \\varepsilon ) } \\leq e ^ { 3 \\varepsilon q _ n } e ^ { L | I | } . \\end{align*}"}
+{"id": "7435.png", "formula": "\\begin{align*} j = j _ k = \\Big \\lfloor \\frac { \\omega _ k ( k + 1 ) } { 1 + \\omega _ k } \\Big \\rfloor . \\end{align*}"}
+{"id": "4808.png", "formula": "\\begin{align*} a ( u _ n , v _ n ) = ( p _ n f , v _ n ) v _ n \\in X _ n . \\end{align*}"}
+{"id": "2887.png", "formula": "\\begin{align*} { \\frak M } : = \\sup _ m \\left \\{ \\sum _ { x = 0 } ^ n | M _ { x , 0 } ( m ) | ^ 2 \\right \\} ^ { 1 / 2 } < + \\infty . \\end{align*}"}
+{"id": "8528.png", "formula": "\\begin{align*} M _ \\delta ( x ; z ) = I + \\mathcal { C } \\left ( M _ { \\delta , - } ( x ; \\cdot ) \\hat { R } _ \\delta ( x ; \\cdot ) \\right ) ( z ) , z \\in \\mathbb { C } ^ { \\pm } . \\end{align*}"}
+{"id": "4388.png", "formula": "\\begin{align*} & \\int _ { X _ j } | F _ { j } - ( 1 - b ( \\Psi ) ) f F ^ { 1 + \\delta } | _ { h } ^ 2 e ^ { v ( \\Psi ) - \\delta \\max \\{ \\psi + T , 2 \\log | F | \\} } c ( - v ( \\Psi ) ) \\\\ \\le & \\bigg ( \\sup _ { X _ j } e ^ { - u ( - v ( \\Psi ) ) } \\bigg ) \\int _ { X _ j } \\frac { 1 } { B } \\mathbb { I } _ { \\{ - t _ 0 - B < \\Psi < - t _ 0 \\} } | f F | ^ 2 _ h . \\end{align*}"}
+{"id": "5613.png", "formula": "\\begin{align*} \\exp \\left ( a _ { \\alpha + 1 } + d _ { \\alpha + 1 } - d _ \\alpha \\right ) = 1 - \\exp \\left ( c _ { n + 1 } + b _ { n + 1 } - b _ n \\right ) \\ . \\end{align*}"}
+{"id": "4227.png", "formula": "\\begin{align*} & \\ \\eta = \\sum _ { x \\in \\bar { \\mathbb { F } } _ q } a _ x \\dot { s } \\varepsilon ( u + x ) { \\bf 1 } _ { \\theta } + \\sum _ { y , z \\in \\bar { \\mathbb { F } } _ q } b ' _ { y , z } \\varepsilon ( - ( u + y ) ^ { - 1 } ) \\dot { s } \\varepsilon ( ( u + y ) ^ 2 z - ( u + y ) ) { { \\bf 1 } } _ { \\theta } , \\end{align*}"}
+{"id": "7625.png", "formula": "\\begin{align*} \\partial _ t \\mu ^ \\phi = - \\partial _ x \\Big \\{ \\Big [ b ( t , \\cdot , \\mu ^ \\phi ) + \\sigma \\phi ( t , \\cdot , \\mu ^ \\phi ) \\Big ] \\mu ^ \\phi \\Big \\} . \\end{align*}"}
+{"id": "6946.png", "formula": "\\begin{align*} T _ { W _ \\ell , \\phi } ( \\mu ) = \\sum _ { i = 1 } ^ L c _ i \\int _ { [ 0 , 1 ] ^ 2 } \\bar a _ i ( u ) \\bar b _ i ( v ) W _ \\ell ( u , v ) d u d v , \\end{align*}"}
+{"id": "6721.png", "formula": "\\begin{align*} G ( x ) = \\begin{cases} \\frac { \\Gamma ( 1 / s , ( \\frac { \\mu - x } { \\sigma } ) ^ s ) } { 2 \\Gamma ( 1 / s ) } , & x \\leq \\mu , \\\\ 1 - \\frac { \\Gamma ( 1 / s , ( \\frac { x - \\mu } { \\sigma } ) ^ s ) } { 2 \\Gamma ( 1 / s ) } , & x > \\mu , \\end{cases} \\end{align*}"}
+{"id": "8724.png", "formula": "\\begin{align*} \\overline { \\pi } _ 1 ( q ) = \\eta ( q ) \\overline { \\pi } ( q ) \\end{align*}"}
+{"id": "4959.png", "formula": "\\begin{align*} V ( \\xi _ 0 , n , x ) = \\sup _ { \\theta \\in \\Theta _ n } E _ { \\xi _ 0 } \\left [ h ^ \\theta ( x , Z _ 1 ^ \\theta ) \\right ] , \\end{align*}"}
+{"id": "4461.png", "formula": "\\begin{align*} ( 1 . 4 4 ) ^ { 2 } \\cdot \\frac { 3 2 \\pi ^ { 8 } } { 9 p ^ { 3 } } \\sum _ { n = a p } ^ { b p } n \\log ( n + 2 ) \\sum _ { d = 1 } ^ { \\infty } \\psi \\left ( d \\sqrt { \\frac { n } { p } } \\right ) \\leq \\frac { 6 9 9 5 8 } { p ^ { 3 } } \\cdot \\left [ b p \\log ( b p + 2 ) \\right ] ( b - a ) p \\sum _ { d = 1 } ^ { \\infty } \\psi ( d \\sqrt { a } ) . \\end{align*}"}